LMFDB - Embedded newform 1368.2.g.c.685.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(685,1368)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1368.685");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{16} - 8 x^{13} - 6 x^{12} + 12 x^{11} + 20 x^{10} - 16 x^{9} + 40 x^{8} + 48 x^{7} + \cdots + 512 $ x^18 - x^16 - 8x^13 - 6x^12 + 12x^11 + 20x^10 - 16x^9 + 40x^8 + 48x^7 - 48x^6 - 128x^5 - 128x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.11
Root			$1.35992 + 0.388110i$ of defining polynomial
Character	$\chi$	$=$	1368.685
Dual form			1368.2.g.c.685.12

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.388110 - 1.35992i) q^{2} +(-1.69874 - 1.05559i) q^{4} -2.94870i q^{5} -1.28777 q^{7} +(-2.09482 + 1.90046i) q^{8} +O(q^{10})$	\(q+(0.388110 - 1.35992i) q^{2} +(-1.69874 - 1.05559i) q^{4} -2.94870i q^{5} -1.28777 q^{7} +(-2.09482 + 1.90046i) q^{8} +(-4.00998 - 1.14442i) q^{10} -5.49088i q^{11} -3.49605i q^{13} +(-0.499797 + 1.75126i) q^{14} +(1.77144 + 3.58636i) q^{16} +6.29935 q^{17} -1.00000i q^{19} +(-3.11263 + 5.00907i) q^{20} +(-7.46713 - 2.13107i) q^{22} -3.34695 q^{23} -3.69481 q^{25} +(-4.75434 - 1.35685i) q^{26} +(2.18759 + 1.35936i) q^{28} +4.62063i q^{29} -4.44572 q^{31} +(5.56467 - 1.01711i) q^{32} +(2.44484 - 8.56658i) q^{34} +3.79724i q^{35} +1.66388i q^{37} +(-1.35992 - 0.388110i) q^{38} +(5.60387 + 6.17698i) q^{40} -2.50527 q^{41} +7.92235i q^{43} +(-5.79614 + 9.32758i) q^{44} +(-1.29899 + 4.55158i) q^{46} -3.42278 q^{47} -5.34165 q^{49} +(-1.43399 + 5.02463i) q^{50} +(-3.69041 + 5.93889i) q^{52} +6.00788i q^{53} -16.1909 q^{55} +(2.69764 - 2.44735i) q^{56} +(6.28366 + 1.79331i) q^{58} -12.8843i q^{59} +3.93452i q^{61} +(-1.72543 + 6.04580i) q^{62} +(0.776526 - 7.96222i) q^{64} -10.3088 q^{65} +2.20215i q^{67} +(-10.7010 - 6.64956i) q^{68} +(5.16393 + 1.47375i) q^{70} +2.16488 q^{71} +7.34165 q^{73} +(2.26274 + 0.645769i) q^{74} +(-1.05559 + 1.69874i) q^{76} +7.07098i q^{77} +2.68779 q^{79} +(10.5751 - 5.22344i) q^{80} +(-0.972320 + 3.40695i) q^{82} -15.7301i q^{83} -18.5749i q^{85} +(10.7737 + 3.07475i) q^{86} +(10.4352 + 11.5024i) q^{88} +5.60138 q^{89} +4.50211i q^{91} +(5.68561 + 3.53303i) q^{92} +(-1.32842 + 4.65469i) q^{94} -2.94870 q^{95} +10.6487 q^{97} +(-2.07315 + 7.26419i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{2} - 2 q^{4} - 12 q^{7} - 2 q^{8}+O(q^{10}) $ 18 * q - 2 * q^2 - 2 * q^4 - 12 * q^7 - 2 * q^8	$ 18 q - 2 q^{2} - 2 q^{4} - 12 q^{7} - 2 q^{8} + 4 q^{10} + 10 q^{14} + 2 q^{16} - 4 q^{17} - 6 q^{20} - 26 q^{22} + 36 q^{23} - 22 q^{25} - 34 q^{26} + 24 q^{28} + 32 q^{31} + 18 q^{32} + 18 q^{34} - 50 q^{40} + 4 q^{41} - 32 q^{44} + 2 q^{46} - 52 q^{47} + 18 q^{49} - 48 q^{50} + 2 q^{52} - 40 q^{55} + 36 q^{56} + 32 q^{58} + 46 q^{62} - 38 q^{64} + 16 q^{65} - 60 q^{68} - 46 q^{70} + 64 q^{71} + 28 q^{73} + 2 q^{74} + 4 q^{76} + 32 q^{79} + 34 q^{80} + 48 q^{82} + 58 q^{86} - 40 q^{88} - 4 q^{89} - 22 q^{92} - 32 q^{94} - 4 q^{97} - 68 q^{98}+O(q^{100}) $ 18 * q - 2 * q^2 - 2 * q^4 - 12 * q^7 - 2 * q^8 + 4 * q^10 + 10 * q^14 + 2 * q^16 - 4 * q^17 - 6 * q^20 - 26 * q^22 + 36 * q^23 - 22 * q^25 - 34 * q^26 + 24 * q^28 + 32 * q^31 + 18 * q^32 + 18 * q^34 - 50 * q^40 + 4 * q^41 - 32 * q^44 + 2 * q^46 - 52 * q^47 + 18 * q^49 - 48 * q^50 + 2 * q^52 - 40 * q^55 + 36 * q^56 + 32 * q^58 + 46 * q^62 - 38 * q^64 + 16 * q^65 - 60 * q^68 - 46 * q^70 + 64 * q^71 + 28 * q^73 + 2 * q^74 + 4 * q^76 + 32 * q^79 + 34 * q^80 + 48 * q^82 + 58 * q^86 - 40 * q^88 - 4 * q^89 - 22 * q^92 - 32 * q^94 - 4 * q^97 - 68 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.388110	−	1.35992i	0.274435	−	0.961606i
$2$	0.388110	−	1.35992i	0.274435	−	0.961606i
$3$		0			0
$3$		0			0
$4$	−1.69874	−	1.05559i	−0.849370	−	0.527797i
$5$		−	2.94870i		−	1.31870i	−0.751837	−	0.659349i	$-0.770832\pi$
$5$		−	2.94870i		−	1.31870i	0.751837	−	0.659349i	$-0.229168\pi$
$6$		0			0
$7$	−1.28777			−0.486731			−0.243366	−	0.969935i	$-0.578251\pi$
$7$	−1.28777			−0.486731			−0.243366	+	0.969935i	$0.578251\pi$
$8$	−2.09482	+	1.90046i	−0.740630	+	0.671913i
$9$		0			0
$10$	−4.00998	−	1.14442i	−1.26807	−	0.361897i
$11$		−	5.49088i		−	1.65556i	−0.561051	−	0.827781i	$-0.689603\pi$
$11$		−	5.49088i		−	1.65556i	0.561051	−	0.827781i	$-0.310397\pi$
$12$		0			0
$13$		−	3.49605i		−	0.969630i	−0.874617	−	0.484815i	$-0.838887\pi$
$13$		−	3.49605i		−	0.969630i	0.874617	−	0.484815i	$-0.161113\pi$
$14$	−0.499797	+	1.75126i	−0.133576	+	0.468043i
$15$		0			0
$16$	1.77144	+	3.58636i	0.442860	+	0.896591i
$17$	6.29935			1.52782			0.763908	−	0.645325i	$-0.223278\pi$
$17$	6.29935			1.52782			0.763908	+	0.645325i	$0.223278\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	−3.11263	+	5.00907i	−0.696005	+	1.12006i
$21$		0			0
$22$	−7.46713	−	2.13107i	−1.59200	−	0.454345i
$23$	−3.34695			−0.697888			−0.348944	−	0.937144i	$-0.613460\pi$
$23$	−3.34695			−0.697888			−0.348944	+	0.937144i	$0.613460\pi$
$24$		0			0
$25$	−3.69481			−0.738962
$26$	−4.75434	−	1.35685i	−0.932402	−	0.266101i
$27$		0			0
$28$	2.18759	+	1.35936i	0.413415	+	0.256895i
$29$			4.62063i			0.858029i	0.903298	+	0.429015i	$0.141139\pi$
$29$			4.62063i			0.858029i	−0.903298	+	0.429015i	$0.858861\pi$
$30$		0			0
$31$	−4.44572			−0.798475			−0.399237	−	0.916848i	$-0.630725\pi$
$31$	−4.44572			−0.798475			−0.399237	+	0.916848i	$0.630725\pi$
$32$	5.56467	−	1.01711i	0.983703	−	0.179801i
$33$		0			0
$34$	2.44484	−	8.56658i	0.419287	−	1.46916i
$35$			3.79724i			0.641851i
$36$		0			0
$37$			1.66388i			0.273540i	0.990603	+	0.136770i	$0.0436722\pi$
$37$			1.66388i			0.273540i	−0.990603	+	0.136770i	$0.956328\pi$
$38$	−1.35992	−	0.388110i	−0.220607	−	0.0629598i
$39$		0			0
$40$	5.60387	+	6.17698i	0.886050	+	0.976667i
$41$	−2.50527			−0.391257			−0.195629	−	0.980678i	$-0.562675\pi$
$41$	−2.50527			−0.391257			−0.195629	+	0.980678i	$0.562675\pi$
$42$		0			0
$43$			7.92235i			1.20815i	0.796929	+	0.604074i	$0.206457\pi$
$43$			7.92235i			1.20815i	−0.796929	+	0.604074i	$0.793543\pi$
$44$	−5.79614	+	9.32758i	−0.873801	+	1.40619i
$45$		0			0
$46$	−1.29899	+	4.55158i	−0.191525	+	0.671093i
$47$	−3.42278			−0.499264			−0.249632	−	0.968341i	$-0.580310\pi$
$47$	−3.42278			−0.499264			−0.249632	+	0.968341i	$0.580310\pi$
$48$		0			0
$49$	−5.34165			−0.763093
$50$	−1.43399	+	5.02463i	−0.202797	+	0.710590i
$51$		0			0
$52$	−3.69041	+	5.93889i	−0.511768	+	0.823575i
$53$			6.00788i			0.825246i	0.910902	+	0.412623i	$0.135387\pi$
$53$			6.00788i			0.825246i	−0.910902	+	0.412623i	$0.864613\pi$
$54$		0			0
$55$	−16.1909			−2.18318
$56$	2.69764	−	2.44735i	0.360488	−	0.327041i
$57$		0			0
$58$	6.28366	+	1.79331i	0.825086	+	0.235474i
$59$		−	12.8843i		−	1.67740i	−0.544596	−	0.838699i	$-0.683317\pi$
$59$		−	12.8843i		−	1.67740i	0.544596	−	0.838699i	$-0.316683\pi$
$60$		0			0
$61$			3.93452i			0.503764i	0.967758	+	0.251882i	$0.0810495\pi$
$61$			3.93452i			0.503764i	−0.967758	+	0.251882i	$0.918950\pi$
$62$	−1.72543	+	6.04580i	−0.219130	+	0.767818i
$63$		0			0
$64$	0.776526	−	7.96222i	0.0970658	−	0.995278i
$65$	−10.3088			−1.27865
$66$		0			0
$67$			2.20215i			0.269035i	0.990911	+	0.134518i	$0.0429485\pi$
$67$			2.20215i			0.269035i	−0.990911	+	0.134518i	$0.957051\pi$
$68$	−10.7010	−	6.64956i	−1.29768	−	0.806377i
$69$		0			0
$70$	5.16393	+	1.47375i	0.617207	+	0.176147i
$71$	2.16488			0.256923			0.128462	−	0.991714i	$-0.458996\pi$
$71$	2.16488			0.256923			0.128462	+	0.991714i	$0.458996\pi$
$72$		0			0
$73$	7.34165			0.859275			0.429638	−	0.903001i	$-0.358641\pi$
$73$	7.34165			0.859275			0.429638	+	0.903001i	$0.358641\pi$
$74$	2.26274	+	0.645769i	0.263038	+	0.0750691i
$75$		0			0
$76$	−1.05559	+	1.69874i	−0.121085	+	0.194859i
$77$			7.07098i			0.805813i
$78$		0			0
$79$	2.68779			0.302400			0.151200	−	0.988503i	$-0.451686\pi$
$79$	2.68779			0.302400			0.151200	+	0.988503i	$0.451686\pi$
$80$	10.5751	−	5.22344i	1.18233	−	0.583998i
$81$		0			0
$82$	−0.972320	+	3.40695i	−0.107375	+	0.376235i
$83$		−	15.7301i		−	1.72660i	−0.504689	−	0.863301i	$-0.668393\pi$
$83$		−	15.7301i		−	1.72660i	0.504689	−	0.863301i	$-0.331607\pi$
$84$		0			0
$85$		−	18.5749i		−	2.01473i
$86$	10.7737	+	3.07475i	1.16176	+	0.331558i
$87$		0			0
$88$	10.4352	+	11.5024i	1.11239	+	1.22616i
$89$	5.60138			0.593745			0.296872	−	0.954917i	$-0.404056\pi$
$89$	5.60138			0.593745			0.296872	+	0.954917i	$0.404056\pi$
$90$		0			0
$91$			4.50211i			0.471949i
$92$	5.68561	+	3.53303i	0.592766	+	0.368344i
$93$		0			0
$94$	−1.32842	+	4.65469i	−0.137016	+	0.480095i
$95$	−2.94870			−0.302530
$96$		0			0
$97$	10.6487			1.08121			0.540604	−	0.841277i	$-0.318196\pi$
$97$	10.6487			1.08121			0.540604	+	0.841277i	$0.318196\pi$
$98$	−2.07315	+	7.26419i	−0.209420	+	0.733794i
$99$		0			0
$100$	6.27653	+	3.90022i	0.627653	+	0.390022i
$101$		−	6.45589i		−	0.642385i	−0.947014	−	0.321193i	$-0.895916\pi$
$101$		−	6.45589i		−	0.642385i	0.947014	−	0.321193i	$-0.104084\pi$
$102$		0			0
$103$	−13.7555			−1.35537			−0.677683	−	0.735354i	$-0.737016\pi$
$103$	−13.7555			−1.35537			−0.677683	+	0.735354i	$0.737016\pi$
$104$	6.64410	+	7.32359i	0.651507	+	0.718137i
$105$		0			0
$106$	8.17021	+	2.33172i	0.793561	+	0.226477i
$107$		−	12.6924i		−	1.22702i	−0.789688	−	0.613508i	$-0.789758\pi$
$107$		−	12.6924i		−	1.22702i	0.789688	−	0.613508i	$-0.210242\pi$
$108$		0			0
$109$			17.2314i			1.65047i	0.564793	+	0.825233i	$0.308956\pi$
$109$			17.2314i			1.65047i	−0.564793	+	0.825233i	$0.691044\pi$
$110$	−6.28387	+	22.0183i	−0.599143	+	2.09936i
$111$		0			0
$112$	−2.28121	−	4.61841i	−0.215554	−	0.436399i
$113$	−5.69865			−0.536084			−0.268042	−	0.963407i	$-0.586377\pi$
$113$	−5.69865			−0.536084			−0.268042	+	0.963407i	$0.586377\pi$
$114$		0			0
$115$			9.86915i			0.920303i
$116$	4.87751	−	7.84925i	0.452865	−	0.728785i
$117$		0			0
$118$	−17.5216	−	5.00054i	−1.61299	−	0.460337i
$119$	−8.11211			−0.743636
$120$		0			0
$121$	−19.1497			−1.74089
$122$	5.35062	+	1.52703i	0.484422	+	0.138251i
$123$		0			0
$124$	7.55212	+	4.69288i	0.678201	+	0.421433i
$125$		−	3.84861i		−	0.344230i
$126$		0			0
$127$	12.9588			1.14991			0.574954	−	0.818186i	$-0.305020\pi$
$127$	12.9588			1.14991			0.574954	+	0.818186i	$0.305020\pi$
$128$	−10.5266	−	4.14623i	−0.930427	−	0.366479i
$129$		0			0
$130$	−4.00095	+	14.0191i	−0.350907	+	1.22956i
$131$			5.93004i			0.518110i	0.965863	+	0.259055i	$0.0834111\pi$
$131$			5.93004i			0.518110i	−0.965863	+	0.259055i	$0.916589\pi$
$132$		0			0
$133$			1.28777i			0.111664i
$134$	2.99474	+	0.854677i	0.258706	+	0.0738328i
$135$		0			0
$136$	−13.1960	+	11.9716i	−1.13155	+	1.02656i
$137$	16.8552			1.44004			0.720020	−	0.693953i	$-0.244132\pi$
$137$	16.8552			1.44004			0.720020	+	0.693953i	$0.244132\pi$
$138$		0			0
$139$			18.4308i			1.56328i	0.623731	+	0.781639i	$0.285616\pi$
$139$			18.4308i			1.56328i	−0.623731	+	0.781639i	$0.714384\pi$
$140$	4.00835	−	6.45053i	0.338767	−	0.545169i
$141$		0			0
$142$	0.840210	−	2.94405i	0.0705089	−	0.247059i
$143$	−19.1964			−1.60528
$144$		0			0
$145$	13.6248			1.13148
$146$	2.84937	−	9.98402i	0.235816	−	0.826284i
$147$		0			0
$148$	1.75638	−	2.82650i	0.144374	−	0.232337i
$149$		−	17.1139i		−	1.40202i	−0.713150	−	0.701012i	$-0.752732\pi$
$149$		−	17.1139i		−	1.40202i	0.713150	−	0.701012i	$-0.247268\pi$
$150$		0			0
$151$	7.67678			0.624728			0.312364	−	0.949963i	$-0.398879\pi$
$151$	7.67678			0.624728			0.312364	+	0.949963i	$0.398879\pi$
$152$	1.90046	+	2.09482i	0.154147	+	0.169912i
$153$		0			0
$154$	9.61594	+	2.74432i	0.774875	+	0.221144i
$155$			13.1091i			1.05295i
$156$		0			0
$157$		−	20.3873i		−	1.62709i	−0.581505	−	0.813543i	$-0.697536\pi$
$157$		−	20.3873i		−	1.62709i	0.581505	−	0.813543i	$-0.302464\pi$
$158$	1.04316	−	3.65516i	0.0829892	−	0.290789i
$159$		0			0
$160$	−2.99913	−	16.4085i	−0.237102	−	1.29721i
$161$	4.31011			0.339684
$162$		0			0
$163$		−	21.0410i		−	1.64806i	−0.566547	−	0.824029i	$-0.691721\pi$
$163$		−	21.0410i		−	1.64806i	0.566547	−	0.824029i	$-0.308279\pi$
$164$	4.25580	+	2.64455i	0.332322	+	0.206504i
$165$		0			0
$166$	−21.3916	−	6.10501i	−1.66031	−	0.473841i
$167$	−17.1563			−1.32759			−0.663795	−	0.747914i	$-0.731055\pi$
$167$	−17.1563			−1.32759			−0.663795	+	0.747914i	$0.731055\pi$
$168$		0			0
$169$	0.777622			0.0598171
$170$	−25.2603	−	7.20910i	−1.93737	−	0.552913i
$171$		0			0
$172$	8.36279	−	13.4580i	0.637657	−	1.02616i
$173$			21.9316i			1.66743i	0.552198	+	0.833713i	$0.313789\pi$
$173$			21.9316i			1.66743i	−0.552198	+	0.833713i	$0.686211\pi$
$174$		0			0
$175$	4.75806			0.359676
$176$	19.6923	−	9.72676i	1.48436	−	0.733182i
$177$		0			0
$178$	2.17395	−	7.61740i	0.162945	−	0.570948i
$179$			3.30598i			0.247100i	0.992338	+	0.123550i	$0.0394280\pi$
$179$			3.30598i			0.247100i	−0.992338	+	0.123550i	$0.960572\pi$
$180$		0			0
$181$		−	16.1379i		−	1.19952i	−0.800181	−	0.599759i	$-0.795263\pi$
$181$		−	16.1379i		−	1.19952i	0.800181	−	0.599759i	$-0.204737\pi$
$182$	6.12249	+	1.74731i	0.453829	+	0.129520i
$183$		0			0
$184$	7.01126	−	6.36074i	0.516877	−	0.468920i
$185$	4.90628			0.360717
$186$		0			0
$187$		−	34.5890i		−	2.52939i
$188$	5.81441	+	3.61307i	0.424060	+	0.263510i
$189$		0			0
$190$	−1.14442	+	4.00998i	−0.0830249	+	0.290914i
$191$	−18.9770			−1.37313			−0.686563	−	0.727071i	$-0.740881\pi$
$191$	−18.9770			−1.37313			−0.686563	+	0.727071i	$0.740881\pi$
$192$		0			0
$193$	7.70837			0.554860			0.277430	−	0.960746i	$-0.410517\pi$
$193$	7.70837			0.554860			0.277430	+	0.960746i	$0.410517\pi$
$194$	4.13285	−	14.4813i	0.296722	−	1.03969i
$195$		0			0
$196$	9.07408	+	5.63862i	0.648149	+	0.402758i
$197$		−	3.69451i		−	0.263223i	−0.991301	−	0.131611i	$-0.957985\pi$
$197$		−	3.69451i		−	0.263223i	0.991301	−	0.131611i	$-0.0420151\pi$
$198$		0			0
$199$	−17.5783			−1.24609			−0.623045	−	0.782186i	$-0.714105\pi$
$199$	−17.5783			−1.24609			−0.623045	+	0.782186i	$0.714105\pi$
$200$	7.73996	−	7.02183i	0.547298	−	0.496518i
$201$		0			0
$202$	−8.77947	−	2.50560i	−0.617721	−	0.176293i
$203$		−	5.95030i		−	0.417629i
$204$		0			0
$205$			7.38727i			0.515950i
$206$	−5.33864	+	18.7063i	−0.371960	+	1.30333i
$207$		0			0
$208$	12.5381	−	6.19305i	0.869362	−	0.429411i
$209$	−5.49088			−0.379812
$210$		0			0
$211$			5.61155i			0.386315i	0.981168	+	0.193157i	$0.0618728\pi$
$211$			5.61155i			0.386315i	−0.981168	+	0.193157i	$0.938127\pi$
$212$	6.34189	−	10.2058i	0.435563	−	0.700939i
$213$		0			0
$214$	−17.2605	−	4.92603i	−1.17991	−	0.336737i
$215$	23.3606			1.59318
$216$		0			0
$217$	5.72506			0.388642
$218$	23.4332	+	6.68767i	1.58710	+	0.452946i
$219$		0			0
$220$	27.5042	+	17.0911i	1.85433	+	1.15228i
$221$		−	22.0228i		−	1.48142i
$222$		0			0
$223$	8.99602			0.602418			0.301209	−	0.953558i	$-0.402610\pi$
$223$	8.99602			0.602418			0.301209	+	0.953558i	$0.402610\pi$
$224$	−7.16600	+	1.30980i	−0.478799	+	0.0875145i
$225$		0			0
$226$	−2.21170	+	7.74968i	−0.147120	+	0.515501i
$227$			27.0232i			1.79359i	0.442446	+	0.896795i	$0.354111\pi$
$227$			27.0232i			1.79359i	−0.442446	+	0.896795i	$0.645889\pi$
$228$		0			0
$229$		−	26.7152i		−	1.76539i	−0.469943	−	0.882697i	$-0.655726\pi$
$229$		−	26.7152i		−	1.76539i	0.469943	−	0.882697i	$-0.344274\pi$
$230$	13.4212	+	3.83032i	0.884969	+	0.252564i
$231$		0			0
$232$	−8.78131	−	9.67938i	−0.576521	−	0.635482i
$233$	−25.7560			−1.68733			−0.843665	−	0.536870i	$-0.819607\pi$
$233$	−25.7560			−1.68733			−0.843665	+	0.536870i	$0.819607\pi$
$234$		0			0
$235$			10.0927i			0.658377i
$236$	−13.6006	+	21.8871i	−0.885326	+	1.42473i
$237$		0			0
$238$	−3.14839	+	11.0318i	−0.204080	+	0.715084i
$239$	27.9612			1.80866			0.904331	−	0.426832i	$-0.140370\pi$
$239$	27.9612			1.80866			0.904331	+	0.426832i	$0.140370\pi$
$240$		0			0
$241$	7.22664			0.465509			0.232754	−	0.972536i	$-0.425226\pi$
$241$	7.22664			0.465509			0.232754	+	0.972536i	$0.425226\pi$
$242$	−7.43221	+	26.0420i	−0.477761	+	1.67404i
$243$		0			0
$244$	4.15326	−	6.68374i	0.265885	−	0.427882i
$245$			15.7509i			1.00629i
$246$		0			0
$247$	−3.49605			−0.222448
$248$	9.31297	−	8.44890i	0.591374	−	0.536505i
$249$		0			0
$250$	−5.23378	−	1.49368i	−0.331013	−	0.0944689i
$251$			6.91493i			0.436466i	0.975897	+	0.218233i	$0.0700293\pi$
$251$			6.91493i			0.436466i	−0.975897	+	0.218233i	$0.929971\pi$
$252$		0			0
$253$			18.3777i			1.15540i
$254$	5.02945	−	17.6229i	0.315576	−	1.10576i
$255$		0			0
$256$	−9.72400	+	12.7061i	−0.607750	+	0.794129i
$257$	29.8393			1.86132			0.930662	−	0.365879i	$-0.119232\pi$
$257$	29.8393			1.86132			0.930662	+	0.365879i	$0.119232\pi$
$258$		0			0
$259$		−	2.14269i		−	0.133141i
$260$	17.5120	+	10.8819i	1.08605	+	0.674867i
$261$		0			0
$262$	8.06436	+	2.30151i	0.498218	+	0.142188i
$263$	−2.59906			−0.160265			−0.0801323	−	0.996784i	$-0.525534\pi$
$263$	−2.59906			−0.160265			−0.0801323	+	0.996784i	$0.525534\pi$
$264$		0			0
$265$	17.7154			1.08825
$266$	1.75126	+	0.499797i	0.107376	+	0.0306445i
$267$		0			0
$268$	2.32458	−	3.74088i	0.141996	−	0.228511i
$269$		−	30.5818i		−	1.86461i	−0.361679	−	0.932303i	$-0.617796\pi$
$269$		−	30.5818i		−	1.86461i	0.361679	−	0.932303i	$-0.382204\pi$
$270$		0			0
$271$	−6.85549			−0.416441			−0.208221	−	0.978082i	$-0.566767\pi$
$271$	−6.85549			−0.416441			−0.208221	+	0.978082i	$0.566767\pi$
$272$	11.1589	+	22.5917i	0.676609	+	1.36983i
$273$		0			0
$274$	6.54169	−	22.9217i	0.395198	−	1.38475i
$275$			20.2878i			1.22340i
$276$		0			0
$277$		−	14.0275i		−	0.842831i	−0.906868	−	0.421416i	$-0.861533\pi$
$277$		−	14.0275i		−	0.842831i	0.906868	−	0.421416i	$-0.138467\pi$
$278$	25.0643	+	7.15318i	1.50326	+	0.429019i
$279$		0			0
$280$	−7.21649	−	7.95453i	−0.431268	−	0.475374i
$281$	−20.0099			−1.19369			−0.596845	−	0.802356i	$-0.703579\pi$
$281$	−20.0099			−1.19369			−0.596845	+	0.802356i	$0.703579\pi$
$282$		0			0
$283$		−	0.712160i		−	0.0423335i	−0.999776	−	0.0211668i	$-0.993262\pi$
$283$		−	0.712160i		−	0.0423335i	0.999776	−	0.0211668i	$-0.00673809\pi$
$284$	−3.67756	−	2.28523i	−0.218223	−	0.135603i
$285$		0			0
$286$	−7.45032	+	26.1055i	−0.440547	+	1.54365i
$287$	3.22621			0.190437
$288$		0			0
$289$	22.6818			1.33422
$290$	5.28794	−	18.5286i	0.310518	−	1.08804i
$291$		0			0
$292$	−12.4716	−	7.74980i	−0.729843	−	0.453523i
$293$		−	22.2392i		−	1.29923i	−0.760265	−	0.649613i	$-0.774931\pi$
$293$		−	22.2392i		−	1.29923i	0.760265	−	0.649613i	$-0.225069\pi$
$294$		0			0
$295$	−37.9920			−2.21198
$296$	−3.16213	−	3.48553i	−0.183795	−	0.202592i
$297$		0			0
$298$	−23.2734	−	6.64207i	−1.34819	−	0.384765i
$299$			11.7011i			0.676694i
$300$		0			0
$301$		−	10.2022i		−	0.588043i
$302$	2.97944	−	10.4398i	0.171447	−	0.600742i
$303$		0			0
$304$	3.58636	−	1.77144i	0.205692	−	0.101599i
$305$	11.6017			0.664312
$306$		0			0
$307$			0.0279474i			0.00159504i	1.00000		0.000797522i	$0.000253859\pi$
$307$			0.0279474i			0.00159504i	−1.00000		0.000797522i	$0.999746\pi$
$308$	7.46409	−	12.0118i	0.425306	−	0.684434i
$309$		0			0
$310$	17.8272	+	5.08777i	1.01252	+	0.288966i
$311$	27.9125			1.58277			0.791387	−	0.611315i	$-0.209359\pi$
$311$	27.9125			1.58277			0.791387	+	0.611315i	$0.209359\pi$
$312$		0			0
$313$	4.82420			0.272680			0.136340	−	0.990662i	$-0.456466\pi$
$313$	4.82420			0.272680			0.136340	+	0.990662i	$0.456466\pi$
$314$	−27.7250	−	7.91253i	−1.56461	−	0.446530i
$315$		0			0
$316$	−4.56585	−	2.83721i	−0.256849	−	0.159606i
$317$		−	21.3190i		−	1.19739i	−0.800976	−	0.598696i	$-0.795686\pi$
$317$		−	21.3190i		−	1.19739i	0.800976	−	0.598696i	$-0.204314\pi$
$318$		0			0
$319$	25.3713			1.42052
$320$	−23.4782	−	2.28974i	−1.31247	−	0.128000i
$321$		0			0
$322$	1.67280	−	5.86138i	0.0932213	−	0.326642i
$323$		−	6.29935i		−	0.350505i
$324$		0			0
$325$			12.9173i			0.716520i
$326$	−28.6140	−	8.16623i	−1.58478	−	0.452286i
$327$		0			0
$328$	5.24808	−	4.76115i	0.289777	−	0.262891i
$329$	4.40775			0.243007
$330$		0			0
$331$		−	11.8507i		−	0.651373i	−0.945478	−	0.325686i	$-0.894405\pi$
$331$		−	11.8507i		−	0.651373i	0.945478	−	0.325686i	$-0.105595\pi$
$332$	−16.6046	+	26.7214i	−0.911296	+	1.46653i
$333$		0			0
$334$	−6.65852	+	23.3311i	−0.364338	+	1.27662i
$335$	6.49347			0.354776
$336$		0			0
$337$	5.98399			0.325969			0.162984	−	0.986629i	$-0.447888\pi$
$337$	5.98399			0.325969			0.162984	+	0.986629i	$0.447888\pi$
$338$	0.301803	−	1.05750i	0.0164159	−	0.0575204i
$339$		0			0
$340$	−19.6075	+	31.5539i	−1.06337	+	1.71125i
$341$			24.4109i			1.32192i
$342$		0			0
$343$	15.8932			0.858152
$344$	−15.0561	−	16.5959i	−0.811770	−	0.894790i
$345$		0			0
$346$	29.8251	+	8.51187i	1.60341	+	0.457601i
$347$			2.58320i			0.138673i	0.997593	+	0.0693367i	$0.0220883\pi$
$347$			2.58320i			0.138673i	−0.997593	+	0.0693367i	$0.977912\pi$
$348$		0			0
$349$		−	1.50956i		−	0.0808050i	−0.999183	−	0.0404025i	$-0.987136\pi$
$349$		−	1.50956i		−	0.0808050i	0.999183	−	0.0404025i	$-0.0128640\pi$
$350$	1.84665	−	6.47057i	0.0987078	−	0.345866i
$351$		0			0
$352$	−5.58480	−	30.5549i	−0.297671	−	1.62858i
$353$	−35.0753			−1.86687			−0.933435	−	0.358745i	$-0.883205\pi$
$353$	−35.0753			−1.86687			−0.933435	+	0.358745i	$0.883205\pi$
$354$		0			0
$355$		−	6.38356i		−	0.338804i
$356$	−9.51529	−	5.91278i	−0.504309	−	0.313377i
$357$		0			0
$358$	4.49585	+	1.28308i	0.237613	+	0.0678131i
$359$	−18.6870			−0.986263			−0.493132	−	0.869955i	$-0.664148\pi$
$359$	−18.6870			−0.986263			−0.493132	+	0.869955i	$0.664148\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−21.9461	−	6.26327i	−1.15346	−	0.329190i
$363$		0			0
$364$	4.75240	−	7.64791i	0.249093	−	0.400860i
$365$		−	21.6483i		−	1.13312i
$366$		0			0
$367$	6.93930			0.362228			0.181114	−	0.983462i	$-0.442030\pi$
$367$	6.93930			0.362228			0.181114	+	0.983462i	$0.442030\pi$
$368$	−5.92893	−	12.0034i	−0.309067	−	0.625720i
$369$		0			0
$370$	1.90418	−	6.67212i	0.0989935	−	0.346867i
$371$		−	7.73676i		−	0.401673i
$372$		0			0
$373$			5.45591i			0.282497i	0.989974	+	0.141248i	$0.0451116\pi$
$373$			5.45591i			0.282497i	−0.989974	+	0.141248i	$0.954888\pi$
$374$	−47.0381	−	13.4243i	−2.43228	−	0.694156i
$375$		0			0
$376$	7.17010	−	6.50484i	0.369770	−	0.335462i
$377$	16.1540			0.831971
$378$		0			0
$379$		−	2.13324i		−	0.109577i	−0.998498	−	0.0547887i	$-0.982551\pi$
$379$		−	2.13324i		−	0.109577i	0.998498	−	0.0547887i	$-0.0174485\pi$
$380$	5.00907	+	3.11263i	0.256960	+	0.159674i
$381$		0			0
$382$	−7.36516	+	25.8071i	−0.376834	+	1.32040i
$383$	−28.2607			−1.44405			−0.722026	−	0.691866i	$-0.756789\pi$
$383$	−28.2607			−1.44405			−0.722026	+	0.691866i	$0.756789\pi$
$384$		0			0
$385$	20.8502			1.06262
$386$	2.99170	−	10.4827i	0.152273	−	0.533557i
$387$		0			0
$388$	−18.0893	−	11.2407i	−0.918345	−	0.570658i
$389$		−	6.50781i		−	0.329959i	−0.986297	−	0.164979i	$-0.947244\pi$
$389$		−	6.50781i		−	0.329959i	0.986297	−	0.164979i	$-0.0527558\pi$
$390$		0			0
$391$	−21.0836			−1.06625
$392$	11.1898	−	10.1516i	0.565170	−	0.512732i
$393$		0			0
$394$	−5.02422	−	1.43388i	−0.253117	−	0.0722377i
$395$		−	7.92547i		−	0.398774i
$396$		0			0
$397$		−	2.30001i		−	0.115434i	−0.998333	−	0.0577172i	$-0.981618\pi$
$397$		−	2.30001i		−	0.115434i	0.998333	−	0.0577172i	$-0.0183822\pi$
$398$	−6.82230	+	23.9049i	−0.341971	+	1.19825i
$399$		0			0
$400$	−6.54514	−	13.2509i	−0.327257	−	0.662547i
$401$	9.32326			0.465581			0.232791	−	0.972527i	$-0.425214\pi$
$401$	9.32326			0.465581			0.232791	+	0.972527i	$0.425214\pi$
$402$		0			0
$403$			15.5425i			0.774225i
$404$	−6.81480	+	10.9669i	−0.339049	+	0.545623i
$405$		0			0
$406$	−8.09191	−	2.30937i	−0.401595	−	0.114612i
$407$	9.13616			0.452863
$408$		0			0
$409$	−4.55252			−0.225107			−0.112554	−	0.993646i	$-0.535903\pi$
$409$	−4.55252			−0.225107			−0.112554	+	0.993646i	$0.535903\pi$
$410$	10.0461	+	2.86708i	0.496140	+	0.141595i
$411$		0			0
$412$	23.3670	+	14.5202i	1.15121	+	0.715358i
$413$			16.5920i			0.816441i
$414$		0			0
$415$	−46.3833			−2.27687
$416$	−3.55585	−	19.4544i	−0.174340	−	0.953828i
$417$		0			0
$418$	−2.13107	+	7.46713i	−0.104234	+	0.365229i
$419$		−	24.9682i		−	1.21978i	−0.792487	−	0.609889i	$-0.791214\pi$
$419$		−	24.9682i		−	1.21978i	0.792487	−	0.609889i	$-0.208786\pi$
$420$		0			0
$421$			33.8871i			1.65156i	0.563993	+	0.825779i	$0.309264\pi$
$421$			33.8871i			1.65156i	−0.563993	+	0.825779i	$0.690736\pi$
$422$	7.63123	+	2.17790i	0.371482	+	0.106018i
$423$		0			0
$424$	−11.4177	−	12.5854i	−0.554493	−	0.611202i
$425$	−23.2749			−1.12900
$426$		0			0
$427$		−	5.06676i		−	0.245198i
$428$	−13.3980	+	21.5610i	−0.647616	+	1.04219i
$429$		0			0
$430$	9.06649	−	31.7685i	0.437225	−	1.53201i
$431$	−6.05315			−0.291570			−0.145785	−	0.989316i	$-0.546571\pi$
$431$	−6.05315			−0.291570			−0.145785	+	0.989316i	$0.546571\pi$
$432$		0			0
$433$	−11.4663			−0.551034			−0.275517	−	0.961296i	$-0.588849\pi$
$433$	−11.4663			−0.551034			−0.275517	+	0.961296i	$0.588849\pi$
$434$	2.22195	−	7.78560i	0.106657	−	0.373721i
$435$		0			0
$436$	18.1893	−	29.2716i	0.871111	−	1.40186i
$437$			3.34695i			0.160107i
$438$		0			0
$439$	−4.81095			−0.229614			−0.114807	−	0.993388i	$-0.536625\pi$
$439$	−4.81095			−0.229614			−0.114807	+	0.993388i	$0.536625\pi$
$440$	33.9171	−	30.7702i	1.61693	−	1.46691i
$441$		0			0
$442$	−29.9492	−	8.54729i	−1.42454	−	0.406553i
$443$		−	25.3750i		−	1.20560i	−0.797891	−	0.602802i	$-0.794051\pi$
$443$		−	25.3750i		−	1.20560i	0.797891	−	0.602802i	$-0.205949\pi$
$444$		0			0
$445$		−	16.5168i		−	0.782970i
$446$	3.49145	−	12.2338i	0.165325	−	0.579289i
$447$		0			0
$448$	−0.999987	+	10.2535i	−0.0472449	+	0.484433i
$449$	14.0176			0.661531			0.330766	−	0.943713i	$-0.392693\pi$
$449$	14.0176			0.661531			0.330766	+	0.943713i	$0.392693\pi$
$450$		0			0
$451$			13.7561i			0.647750i
$452$	9.68052	+	6.01546i	0.455334	+	0.282943i
$453$		0			0
$454$	36.7492	+	10.4880i	1.72473	+	0.492225i
$455$	13.2754			0.622358
$456$		0			0
$457$	18.0507			0.844376			0.422188	−	0.906508i	$-0.361262\pi$
$457$	18.0507			0.844376			0.422188	+	0.906508i	$0.361262\pi$
$458$	−36.3305	−	10.3685i	−1.69761	−	0.484486i
$459$		0			0
$460$	10.4178	−	16.7651i	0.485734	−	0.781678i
$461$			21.8983i			1.01990i	0.860203	+	0.509952i	$0.170337\pi$
$461$			21.8983i			1.01990i	−0.860203	+	0.509952i	$0.829663\pi$
$462$		0			0
$463$	0.0714519			0.00332065			0.00166033	−	0.999999i	$-0.499472\pi$
$463$	0.0714519			0.00332065			0.00166033	+	0.999999i	$0.499472\pi$
$464$	−16.5713	+	8.18517i	−0.769301	+	0.379987i
$465$		0			0
$466$	−9.99616	+	35.0260i	−0.463063	+	1.62255i
$467$		−	2.23322i		−	0.103341i	−0.998664	−	0.0516705i	$-0.983545\pi$
$467$		−	2.23322i		−	0.103341i	0.998664	−	0.0516705i	$-0.0164545\pi$
$468$		0			0
$469$		−	2.83586i		−	0.130948i
$470$	13.7253	+	3.91709i	0.633099	+	0.180682i
$471$		0			0
$472$	24.4861	+	26.9903i	1.12707	+	1.24233i
$473$	43.5007			2.00016
$474$		0			0
$475$			3.69481i			0.169530i
$476$	13.7804	+	8.56310i	0.631622	+	0.392489i
$477$		0			0
$478$	10.8520	−	38.0249i	0.496361	−	1.73922i
$479$	4.30216			0.196571			0.0982854	−	0.995158i	$-0.468664\pi$
$479$	4.30216			0.196571			0.0982854	+	0.995158i	$0.468664\pi$
$480$		0			0
$481$	5.81701			0.265233
$482$	2.80473	−	9.82762i	0.127752	−	0.447636i
$483$		0			0
$484$	32.5304	+	20.2144i	1.47866	+	0.918834i
$485$		−	31.3997i		−	1.42578i
$486$		0			0
$487$	−7.32726			−0.332030			−0.166015	−	0.986123i	$-0.553090\pi$
$487$	−7.32726			−0.332030			−0.166015	+	0.986123i	$0.553090\pi$
$488$	−7.47739	−	8.24211i	−0.338486	−	0.373103i
$489$		0			0
$490$	21.4199	+	6.11309i	0.967653	+	0.276161i
$491$			26.4080i			1.19178i	0.803067	+	0.595889i	$0.203200\pi$
$491$			26.4080i			1.19178i	−0.803067	+	0.595889i	$0.796800\pi$
$492$		0			0
$493$			29.1070i			1.31091i
$494$	−1.35685	+	4.75434i	−0.0610477	+	0.213908i
$495$		0			0
$496$	−7.87533	−	15.9440i	−0.353613	−	0.715905i
$497$	−2.78786			−0.125053
$498$		0			0
$499$		−	0.185693i		−	0.00831278i	−0.999991	−	0.00415639i	$-0.998677\pi$
$499$		−	0.185693i		−	0.00831278i	0.999991	−	0.00415639i	$-0.00132302\pi$
$500$	−4.06257	+	6.53778i	−0.181684	+	0.292379i
$501$		0			0
$502$	9.40372	+	2.68375i	0.419708	+	0.119782i
$503$	−19.8460			−0.884891			−0.442446	−	0.896795i	$-0.645889\pi$
$503$	−19.8460			−0.884891			−0.442446	+	0.896795i	$0.645889\pi$
$504$		0			0
$505$	−19.0365			−0.847111
$506$	24.9921	+	7.13258i	1.11104	+	0.317082i
$507$		0			0
$508$	−22.0137	−	13.6793i	−0.976698	−	0.606919i
$509$			18.3142i			0.811763i	0.913926	+	0.405881i	$0.133035\pi$
$509$			18.3142i			0.811763i	−0.913926	+	0.405881i	$0.866965\pi$
$510$		0			0
$511$	−9.45435			−0.418236
$512$	13.5052	+	18.1552i	0.596850	+	0.802353i
$513$		0			0
$514$	11.5809	−	40.5789i	0.510813	−	1.78986i
$515$			40.5607i			1.78732i
$516$		0			0
$517$			18.7941i			0.826562i
$518$	−2.91388	−	0.831602i	−0.128029	−	0.0365385i
$519$		0			0
$520$	21.5951	−	19.5914i	0.947006	−	0.859141i
$521$	3.02014			0.132315			0.0661574	−	0.997809i	$-0.478926\pi$
$521$	3.02014			0.132315			0.0661574	+	0.997809i	$0.478926\pi$
$522$		0			0
$523$		−	10.0602i		−	0.439902i	−0.975511	−	0.219951i	$-0.929410\pi$
$523$		−	10.0602i		−	0.439902i	0.975511	−	0.219951i	$-0.0705897\pi$
$524$	6.25972	−	10.0736i	0.273457	−	0.440067i
$525$		0			0
$526$	−1.00872	+	3.53450i	−0.0439823	+	0.154111i
$527$	−28.0051			−1.21992
$528$		0			0
$529$	−11.7979			−0.512952
$530$	6.87554	−	24.0915i	0.298654	−	1.04647i
$531$		0			0
$532$	1.35936	−	2.18759i	0.0589358	−	0.0948439i
$533$			8.75855i			0.379375i
$534$		0			0
$535$	−37.4259			−1.61806
$536$	−4.18509	−	4.61310i	−0.180768	−	0.199256i
$537$		0			0
$538$	−41.5887	−	11.8691i	−1.79301	−	0.511714i
$539$			29.3303i			1.26335i
$540$		0			0
$541$			40.6502i			1.74769i	0.486207	+	0.873844i	$0.338380\pi$
$541$			40.6502i			1.74769i	−0.486207	+	0.873844i	$0.661620\pi$
$542$	−2.66069	+	9.32288i	−0.114286	+	0.400452i
$543$		0			0
$544$	35.0538	−	6.40710i	1.50292	−	0.274702i
$545$	50.8101			2.17646
$546$		0			0
$547$		−	24.6605i		−	1.05441i	−0.849739	−	0.527204i	$-0.823240\pi$
$547$		−	24.6605i		−	1.05441i	0.849739	−	0.527204i	$-0.176760\pi$
$548$	−28.6327	−	17.7923i	−1.22313	−	0.760049i
$549$		0			0
$550$	27.5896	+	7.87389i	1.17643	+	0.335744i
$551$	4.62063			0.196845
$552$		0			0
$553$	−3.46125			−0.147187
$554$	−19.0762	−	5.44422i	−0.810471	−	0.231303i
$555$		0			0
$556$	19.4554	−	31.3091i	0.825094	−	1.32780i
$557$		−	4.92892i		−	0.208845i	−0.994533	−	0.104423i	$-0.966701\pi$
$557$		−	4.92892i		−	0.208845i	0.994533	−	0.104423i	$-0.0332994\pi$
$558$		0			0
$559$	27.6969			1.17146
$560$	−13.6183	+	6.72659i	−0.575477	+	0.284250i
$561$		0			0
$562$	−7.76605	+	27.2118i	−0.327591	+	1.14786i
$563$		−	22.5955i		−	0.952287i	−0.879368	−	0.476143i	$-0.842034\pi$
$563$		−	22.5955i		−	0.952287i	0.879368	−	0.476143i	$-0.157966\pi$
$564$		0			0
$565$			16.8036i			0.706932i
$566$	−0.968478	−	0.276397i	−0.0407082	−	0.0116178i
$567$		0			0
$568$	−4.53502	+	4.11425i	−0.190285	+	0.172630i
$569$	−10.6678			−0.447219			−0.223609	−	0.974679i	$-0.571784\pi$
$569$	−10.6678			−0.447219			−0.223609	+	0.974679i	$0.571784\pi$
$570$		0			0
$571$			11.4121i			0.477580i	0.971071	+	0.238790i	$0.0767507\pi$
$571$			11.4121i			0.477580i	−0.971071	+	0.238790i	$0.923249\pi$
$572$	32.6097	+	20.2636i	1.36348	+	0.847264i
$573$		0			0
$574$	1.25212	−	4.38737i	0.0522626	−	0.183125i
$575$	12.3664			0.515713
$576$		0			0
$577$	−18.8240			−0.783655			−0.391827	−	0.920039i	$-0.628157\pi$
$577$	−18.8240			−0.783655			−0.391827	+	0.920039i	$0.628157\pi$
$578$	8.80304	−	30.8453i	0.366158	−	1.28300i
$579$		0			0
$580$	−23.1451	−	14.3823i	−0.961046	−	0.597192i
$581$			20.2567i			0.840391i
$582$		0			0
$583$	32.9885			1.36625
$584$	−15.3794	+	13.9525i	−0.636405	+	0.577358i
$585$		0			0
$586$	−30.2434	−	8.63125i	−1.24934	−	0.356554i
$587$			4.16283i			0.171818i	0.996303	+	0.0859091i	$0.0273795\pi$
$587$			4.16283i			0.171818i	−0.996303	+	0.0859091i	$0.972621\pi$
$588$		0			0
$589$			4.44572i			0.183183i
$590$	−14.7451	+	51.6659i	−0.607045	+	2.12705i
$591$		0			0
$592$	−5.96728	+	2.94747i	−0.245254	+	0.121140i
$593$	13.5867			0.557939			0.278970	−	0.960300i	$-0.410007\pi$
$593$	13.5867			0.557939			0.278970	+	0.960300i	$0.410007\pi$
$594$		0			0
$595$			23.9201i			0.980630i
$596$	−18.0653	+	29.0720i	−0.739984	+	1.19084i
$597$		0			0
$598$	15.9125	+	4.54133i	0.650712	+	0.185709i
$599$	3.85199			0.157388			0.0786939	−	0.996899i	$-0.474925\pi$
$599$	3.85199			0.157388			0.0786939	+	0.996899i	$0.474925\pi$
$600$		0			0
$601$	30.9493			1.26245			0.631225	−	0.775600i	$-0.282553\pi$
$601$	30.9493			1.26245			0.631225	+	0.775600i	$0.282553\pi$
$602$	−13.8741	−	3.95956i	−0.565465	−	0.161380i
$603$		0			0
$604$	−13.0409	−	8.10357i	−0.530625	−	0.329730i
$605$			56.4668i			2.29570i
$606$		0			0
$607$	10.9960			0.446315			0.223158	−	0.974782i	$-0.428364\pi$
$607$	10.9960			0.446315			0.223158	+	0.974782i	$0.428364\pi$
$608$	−1.01711	−	5.56467i	−0.0412491	−	0.225677i
$609$		0			0
$610$	4.50275	−	15.7774i	0.182311	−	0.638807i
$611$			11.9662i			0.484101i
$612$		0			0
$613$			25.3152i			1.02247i	0.859440	+	0.511236i	$0.170812\pi$
$613$			25.3152i			1.02247i	−0.859440	+	0.511236i	$0.829188\pi$
$614$	0.0380061	+	0.0108467i	0.00153380	+	0.000437736i
$615$		0			0
$616$	−13.4381	−	14.8124i	−0.541436	−	0.596810i
$617$	−30.1624			−1.21429			−0.607146	−	0.794590i	$-0.707686\pi$
$617$	−30.1624			−1.21429			−0.607146	+	0.794590i	$0.707686\pi$
$618$		0			0
$619$		−	6.84302i		−	0.275044i	−0.990499	−	0.137522i	$-0.956086\pi$
$619$		−	6.84302i		−	0.275044i	0.990499	−	0.137522i	$-0.0439138\pi$
$620$	13.8379	−	22.2689i	0.555742	−	0.894341i
$621$		0			0
$622$	10.8331	−	37.9587i	0.434369	−	1.52200i
$623$	−7.21328			−0.288994
$624$		0			0
$625$	−29.8224			−1.19290
$626$	1.87232	−	6.56051i	0.0748331	−	0.262211i
$627$		0			0
$628$	−21.5207	+	34.6328i	−0.858771	+	1.38200i
$629$			10.4814i			0.417919i
$630$		0			0
$631$	35.9723			1.43204			0.716018	−	0.698082i	$-0.245963\pi$
$631$	35.9723			1.43204			0.716018	+	0.698082i	$0.245963\pi$
$632$	−5.63043	+	5.10802i	−0.223966	+	0.203186i
$633$		0			0
$634$	−28.9920	−	8.27411i	−1.15142	−	0.328607i
$635$		−	38.2116i		−	1.51638i
$636$		0			0
$637$			18.6747i			0.739918i
$638$	9.84687	−	34.5028i	0.389841	−	1.36598i
$639$		0			0
$640$	−12.2260	+	31.0397i	−0.483274	+	1.22695i
$641$	24.6201			0.972434			0.486217	−	0.873838i	$-0.338376\pi$
$641$	24.6201			0.972434			0.486217	+	0.873838i	$0.338376\pi$
$642$		0			0
$643$		−	5.24598i		−	0.206881i	−0.994636	−	0.103441i	$-0.967015\pi$
$643$		−	5.24598i		−	0.206881i	0.994636	−	0.103441i	$-0.0329852\pi$
$644$	−7.32175	−	4.54972i	−0.288517	−	0.179284i
$645$		0			0
$646$	−8.56658	−	2.44484i	−0.337048	−	0.0961910i
$647$	22.3096			0.877080			0.438540	−	0.898712i	$-0.355496\pi$
$647$	22.3096			0.877080			0.438540	+	0.898712i	$0.355496\pi$
$648$		0			0
$649$	−70.7463			−2.77704
$650$	17.5664	+	5.01332i	0.689010	+	0.196639i
$651$		0			0
$652$	−22.2108	+	35.7432i	−0.869841	+	1.39981i
$653$			40.7754i			1.59566i	0.602880	+	0.797832i	$0.294020\pi$
$653$			40.7754i			1.59566i	−0.602880	+	0.797832i	$0.705980\pi$
$654$		0			0
$655$	17.4859			0.683230
$656$	−4.43793	−	8.98480i	−0.173272	−	0.350797i
$657$		0			0
$658$	1.71069	−	5.99417i	0.0666897	−	0.233677i
$659$			3.88737i			0.151430i	0.997129	+	0.0757152i	$0.0241240\pi$
$659$			3.88737i			0.151430i	−0.997129	+	0.0757152i	$0.975876\pi$
$660$		0			0
$661$			32.3636i			1.25880i	0.777082	+	0.629399i	$0.216699\pi$
$661$			32.3636i			1.25880i	−0.777082	+	0.629399i	$0.783301\pi$
$662$	−16.1159	−	4.59937i	−0.626364	−	0.178760i
$663$		0			0
$664$	29.8944	+	32.9517i	1.16013	+	1.27877i
$665$	3.79724			0.147251
$666$		0			0
$667$		−	15.4650i		−	0.598809i
$668$	29.1440	+	18.1100i	1.12762	+	0.700699i
$669$		0			0
$670$	2.52018	−	8.83057i	0.0973632	−	0.341155i
$671$	21.6040			0.834013
$672$		0			0
$673$	35.5763			1.37137			0.685684	−	0.727900i	$-0.259503\pi$
$673$	35.5763			1.37137			0.685684	+	0.727900i	$0.259503\pi$
$674$	2.32245	−	8.13772i	0.0894573	−	0.313453i
$675$		0			0
$676$	−1.32098	−	0.820854i	−0.0508069	−	0.0315713i
$677$		−	31.2202i		−	1.19989i	−0.800041	−	0.599945i	$-0.795189\pi$
$677$		−	31.2202i		−	1.19989i	0.800041	−	0.599945i	$-0.204811\pi$
$678$		0			0
$679$	−13.7130			−0.526257
$680$	35.3007	+	38.9110i	1.35372	+	1.49217i
$681$		0			0
$682$	33.1968	+	9.47412i	1.27117	+	0.362783i
$683$			32.1850i			1.23153i	0.787932	+	0.615763i	$0.211152\pi$
$683$			32.1850i			1.23153i	−0.787932	+	0.615763i	$0.788848\pi$
$684$		0			0
$685$		−	49.7010i		−	1.89898i
$686$	6.16831	−	21.6134i	0.235507	−	0.825204i
$687$		0			0
$688$	−28.4124	+	14.0340i	−1.08321	+	0.535040i
$689$	21.0039			0.800183
$690$		0			0
$691$		−	8.80188i		−	0.334839i	−0.985886	−	0.167420i	$-0.946457\pi$
$691$		−	8.80188i		−	0.334839i	0.985886	−	0.167420i	$-0.0535435\pi$
$692$	23.1508	−	37.2560i	0.880063	−	1.41626i
$693$		0			0
$694$	3.51293	+	1.00257i	0.133349	+	0.0380569i
$695$	54.3468			2.06149
$696$		0			0
$697$	−15.7816			−0.597769
$698$	−2.05288	−	0.585876i	−0.0777025	−	0.0221757i
$699$		0			0
$700$	−8.08272	−	5.02259i	−0.305498	−	0.189836i
$701$			24.5610i			0.927657i	0.885925	+	0.463828i	$0.153525\pi$
$701$			24.5610i			0.927657i	−0.885925	+	0.463828i	$0.846475\pi$
$702$		0			0
$703$	1.66388			0.0627544
$704$	−43.7196	−	4.26381i	−1.64774	−	0.160698i
$705$		0			0
$706$	−13.6131	+	47.6995i	−0.512335	+	1.79519i
$707$			8.31370i			0.312669i
$708$		0			0
$709$		−	24.4013i		−	0.916411i	−0.888846	−	0.458205i	$-0.848492\pi$
$709$		−	24.4013i		−	0.916411i	0.888846	−	0.458205i	$-0.151508\pi$
$710$	−8.68110	−	2.47753i	−0.325796	−	0.0929799i
$711$		0			0
$712$	−11.7339	+	10.6452i	−0.439745	+	0.398945i
$713$	14.8796			0.557246
$714$		0			0
$715$			56.6043i			2.11688i
$716$	3.48977	−	5.61600i	0.130419	−	0.209880i
$717$		0			0
$718$	−7.25262	+	25.4128i	−0.270666	+	0.948396i
$719$	21.0555			0.785237			0.392619	−	0.919701i	$-0.371569\pi$
$719$	21.0555			0.785237			0.392619	+	0.919701i	$0.371569\pi$
$720$		0			0
$721$	17.7139			0.659699
$722$	−0.388110	+	1.35992i	−0.0144440	+	0.0506108i
$723$		0			0
$724$	−17.0350	+	27.4140i	−0.633102	+	1.01884i
$725$		−	17.0723i		−	0.634051i
$726$		0			0
$727$	−37.2079			−1.37996			−0.689981	−	0.723827i	$-0.742381\pi$
$727$	−37.2079			−1.37996			−0.689981	+	0.723827i	$0.742381\pi$
$728$	−8.55606	−	9.43110i	−0.317109	−	0.349540i
$729$		0			0
$730$	−29.4399	−	8.40193i	−1.08962	−	0.310969i
$731$			49.9056i			1.84583i
$732$		0			0
$733$		−	27.9903i		−	1.03384i	−0.856033	−	0.516922i	$-0.827078\pi$
$733$		−	27.9903i		−	1.03384i	0.856033	−	0.516922i	$-0.172922\pi$
$734$	2.69321	−	9.43686i	0.0994083	−	0.348321i
$735$		0			0
$736$	−18.6247	+	3.40421i	−0.686515	+	0.125481i
$737$	12.0917			0.445405
$738$		0			0
$739$			18.5138i			0.681039i	0.940237	+	0.340520i	$0.110603\pi$
$739$			18.5138i			0.681039i	−0.940237	+	0.340520i	$0.889397\pi$
$740$	−8.33450	−	5.17904i	−0.306382	−	0.190385i
$741$		0			0
$742$	−10.5213	−	3.00272i	−0.386251	−	0.110233i
$743$	43.4882			1.59543			0.797714	−	0.603036i	$-0.206043\pi$
$743$	43.4882			1.59543			0.797714	+	0.603036i	$0.206043\pi$
$744$		0			0
$745$	−50.4636			−1.84884
$746$	7.41958	+	2.11750i	0.271650	+	0.0775271i
$747$		0			0
$748$	−36.5119	+	58.7577i	−1.33501	+	2.14839i
$749$			16.3448i			0.597227i
$750$		0			0
$751$	1.20498			0.0439704			0.0219852	−	0.999758i	$-0.493001\pi$
$751$	1.20498			0.0439704			0.0219852	+	0.999758i	$0.493001\pi$
$752$	−6.06325	−	12.2753i	−0.221104	−	0.447635i
$753$		0			0
$754$	6.26952	−	21.9680i	0.228322	−	0.800028i
$755$		−	22.6365i		−	0.823827i
$756$		0			0
$757$		−	26.3897i		−	0.959148i	−0.877501	−	0.479574i	$-0.840791\pi$
$757$		−	26.3897i		−	0.959148i	0.877501	−	0.479574i	$-0.159209\pi$
$758$	−2.90103	−	0.827934i	−0.105370	−	0.0300719i
$759$		0			0
$760$	6.17698	−	5.60387i	0.224063	−	0.203274i
$761$	−25.1822			−0.912853			−0.456426	−	0.889761i	$-0.650871\pi$
$761$	−25.1822			−0.912853			−0.456426	+	0.889761i	$0.650871\pi$
$762$		0			0
$763$		−	22.1900i		−	0.803333i
$764$	32.2369	+	20.0320i	1.16629	+	0.724732i
$765$		0			0
$766$	−10.9683	+	38.4321i	−0.396299	+	1.38861i
$767$	−45.0443			−1.62646
$768$		0			0
$769$	39.4670			1.42322			0.711609	−	0.702576i	$-0.247967\pi$
$769$	39.4670			1.42322			0.711609	+	0.702576i	$0.247967\pi$
$770$	8.09217	−	28.3545i	0.291622	−	1.02182i
$771$		0			0
$772$	−13.0945	−	8.13691i	−0.471282	−	0.292854i
$773$			15.0642i			0.541822i	0.962604	+	0.270911i	$0.0873249\pi$
$773$			15.0642i			0.541822i	−0.962604	+	0.270911i	$0.912675\pi$
$774$		0			0
$775$	16.4261			0.590043
$776$	−22.3070	+	20.2373i	−0.800774	+	0.726477i
$777$		0			0
$778$	−8.85007	−	2.52575i	−0.317290	−	0.0905524i
$779$			2.50527i			0.0897605i
$780$		0			0
$781$		−	11.8871i		−	0.425353i
$782$	−8.18278	+	28.6720i	−0.292615	+	1.02531i
$783$		0			0
$784$	−9.46242	−	19.1571i	−0.337943	−	0.684182i
$785$	−60.1160			−2.14563
$786$		0			0
$787$		−	27.3618i		−	0.975344i	−0.873027	−	0.487672i	$-0.837846\pi$
$787$		−	27.3618i		−	0.975344i	0.873027	−	0.487672i	$-0.162154\pi$
$788$	−3.89990	+	6.27601i	−0.138928	+	0.223574i
$789$		0			0
$790$	−10.7780	−	3.07596i	−0.383463	−	0.109438i
$791$	7.33854			0.260929
$792$		0			0
$793$	13.7553			0.488465
$794$	−3.12783	−	0.892659i	−0.111002	−	0.0316793i
$795$		0			0
$796$	29.8609	+	18.5555i	1.05839	+	0.657683i
$797$			34.6794i			1.22841i	0.789148	+	0.614204i	$0.210523\pi$
$797$			34.6794i			1.22841i	−0.789148	+	0.614204i	$0.789477\pi$
$798$		0			0
$799$	−21.5613			−0.762783
$800$	−20.5604	+	3.75801i	−0.726919	+	0.132866i
$801$		0			0
$802$	3.61845	−	12.6788i	0.127772	−	0.447706i
$803$		−	40.3121i		−	1.42258i
$804$		0			0
$805$		−	12.7092i		−	0.447940i
$806$	21.1364	+	6.03219i	0.744499	+	0.212475i
$807$		0			0
$808$	12.2691	+	13.5239i	0.431627	+	0.475770i
$809$	−34.3845			−1.20889			−0.604447	−	0.796645i	$-0.706606\pi$
$809$	−34.3845			−1.20889			−0.604447	+	0.796645i	$0.706606\pi$
$810$		0			0
$811$		−	16.7760i		−	0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$811$		−	16.7760i		−	0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$812$	−6.28111	+	10.1080i	−0.220424	+	0.354722i
$813$		0			0
$814$	3.54584	−	12.4244i	0.124282	−	0.435475i
$815$	−62.0435			−2.17329
$816$		0			0
$817$	7.92235			0.277168
$818$	−1.76688	+	6.19104i	−0.0617775	+	0.216465i
$819$		0			0
$820$	7.79797	−	12.5491i	0.272317	−	0.438232i
$821$		−	19.3580i		−	0.675600i	−0.941218	−	0.337800i	$-0.890317\pi$
$821$		−	19.3580i		−	0.675600i	0.941218	−	0.337800i	$-0.109683\pi$
$822$		0			0
$823$	52.2622			1.82175			0.910873	−	0.412686i	$-0.135409\pi$
$823$	52.2622			1.82175			0.910873	+	0.412686i	$0.135409\pi$
$824$	28.8152	−	26.1417i	1.00382	−	0.910688i
$825$		0			0
$826$	22.5638	+	6.43955i	0.785095	+	0.224060i
$827$			9.59663i			0.333707i	0.985982	+	0.166854i	$0.0533608\pi$
$827$			9.59663i			0.333707i	−0.985982	+	0.166854i	$0.946639\pi$
$828$		0			0
$829$			47.7719i			1.65919i	0.558368	+	0.829593i	$0.311428\pi$
$829$			47.7719i			1.65919i	−0.558368	+	0.829593i	$0.688572\pi$
$830$	−18.0018	+	63.0773i	−0.624853	+	2.18945i
$831$		0			0
$832$	−27.8363	−	2.71478i	−0.965052	−	0.0941179i
$833$	−33.6489			−1.16587
$834$		0			0
$835$			50.5886i			1.75069i
$836$	9.32758	+	5.79614i	0.322601	+	0.200464i
$837$		0			0
$838$	−33.9547	−	9.69043i	−1.17295	−	0.334750i
$839$	27.1885			0.938652			0.469326	−	0.883025i	$-0.344497\pi$
$839$	27.1885			0.938652			0.469326	+	0.883025i	$0.344497\pi$
$840$		0			0
$841$	7.64979			0.263786
$842$	46.0837	+	13.1520i	1.58815	+	0.453246i
$843$		0			0
$844$	5.92352	−	9.53256i	0.203896	−	0.328124i
$845$		−	2.29297i		−	0.0788806i
$846$		0			0
$847$	24.6604			0.847343
$848$	−21.5464	+	10.6426i	−0.739908	+	0.365469i
$849$		0			0
$850$	−9.03323	+	31.6519i	−0.309837	+	1.08565i
$851$		−	5.56893i		−	0.190901i
$852$		0			0
$853$		−	38.7736i		−	1.32758i	−0.747918	−	0.663791i	$-0.768946\pi$
$853$		−	38.7736i		−	1.32758i	0.747918	−	0.663791i	$-0.231054\pi$
$854$	−6.89037	−	1.96646i	−0.235783	−	0.0672909i
$855$		0			0
$856$	24.1213	+	26.5882i	0.824448	+	0.908765i
$857$	−5.86582			−0.200372			−0.100186	−	0.994969i	$-0.531944\pi$
$857$	−5.86582			−0.200372			−0.100186	+	0.994969i	$0.531944\pi$
$858$		0			0
$859$			6.83567i			0.233230i	0.993177	+	0.116615i	$0.0372043\pi$
$859$			6.83567i			0.233230i	−0.993177	+	0.116615i	$0.962796\pi$
$860$	−39.6836	−	24.6593i	−1.35320	−	0.840876i
$861$		0			0
$862$	−2.34929	+	8.23177i	−0.0800171	+	0.280375i
$863$	−44.0758			−1.50036			−0.750179	−	0.661235i	$-0.770033\pi$
$863$	−44.0758			−1.50036			−0.750179	+	0.661235i	$0.770033\pi$
$864$		0			0
$865$	64.6695			2.19883
$866$	−4.45018	+	15.5932i	−0.151223	+	0.529878i
$867$		0			0
$868$	−9.72539	−	6.04334i	−0.330101	−	0.205124i
$869$		−	14.7583i		−	0.500641i
$870$		0			0
$871$	7.69883			0.260865
$872$	−32.7475	−	36.0966i	−1.10897	−	1.22238i
$873$		0			0
$874$	4.55158	+	1.29899i	0.153959	+	0.0439389i
$875$			4.95612i			0.167547i
$876$		0			0
$877$		−	43.5149i		−	1.46939i	−0.678396	−	0.734697i	$-0.737324\pi$
$877$		−	43.5149i		−	1.46939i	0.678396	−	0.734697i	$-0.262676\pi$
$878$	−1.86718	+	6.54249i	−0.0630143	+	0.220798i
$879$		0			0
$880$	−28.6813	−	58.0666i	−0.966846	−	1.95742i
$881$	−15.7048			−0.529107			−0.264553	−	0.964371i	$-0.585225\pi$
$881$	−15.7048			−0.529107			−0.264553	+	0.964371i	$0.585225\pi$
$882$		0			0
$883$			56.2699i			1.89363i	0.321772	+	0.946817i	$0.395722\pi$
$883$			56.2699i			1.89363i	−0.321772	+	0.946817i	$0.604278\pi$
$884$	−23.2472	+	37.4111i	−0.781888	+	1.25827i
$885$		0			0
$886$	−34.5079	−	9.84831i	−1.15931	−	0.330860i
$887$	55.0760			1.84927			0.924636	−	0.380851i	$-0.124369\pi$
$887$	55.0760			1.84927			0.924636	+	0.380851i	$0.124369\pi$
$888$		0			0
$889$	−16.6880			−0.559696
$890$	−22.4614	−	6.41033i	−0.752908	−	0.214875i
$891$		0			0
$892$	−15.2819	−	9.49615i	−0.511676	−	0.317955i
$893$			3.42278i			0.114539i
$894$		0			0
$895$	9.74833			0.325851
$896$	13.5558	+	5.33939i	0.452867	+	0.178376i
$897$		0			0
$898$	5.44037	−	19.0628i	0.181548	−	0.636132i
$899$		−	20.5420i		−	0.685115i
$900$		0			0
$901$			37.8457i			1.26082i
$902$	18.7072	+	5.33889i	0.622880	+	0.177766i
$903$		0			0
$904$	11.9376	−	10.8300i	0.397040	−	0.360202i
$905$	−47.5857			−1.58180
$906$		0			0
$907$			18.0558i			0.599532i	0.954013	+	0.299766i	$0.0969086\pi$
$907$			18.0558i			0.599532i	−0.954013	+	0.299766i	$0.903091\pi$
$908$	28.5255	−	45.9053i	0.946652	−	1.52342i
$909$		0			0
$910$	5.15230	−	18.0534i	0.170797	−	0.598463i
$911$	9.88512			0.327509			0.163754	−	0.986501i	$-0.447640\pi$
$911$	9.88512			0.327509			0.163754	+	0.986501i	$0.447640\pi$
$912$		0			0
$913$	−86.3720			−2.85850
$914$	7.00566	−	24.5474i	0.231727	−	0.811957i
$915$		0			0
$916$	−28.2005	+	45.3823i	−0.931770	+	1.49947i
$917$		−	7.63653i		−	0.252180i
$918$		0			0
$919$	17.5136			0.577720			0.288860	−	0.957371i	$-0.406724\pi$
$919$	17.5136			0.577720			0.288860	+	0.957371i	$0.406724\pi$
$920$	−18.7559	−	20.6741i	−0.618364	−	0.681604i
$921$		0			0
$922$	29.7798	+	8.49895i	0.980746	+	0.279898i
$923$		−	7.56852i		−	0.249121i
$924$		0			0
$925$		−	6.14772i		−	0.202136i
$926$	0.0277312	−	0.0971685i	0.000911304	−	0.00319316i
$927$		0			0
$928$	4.69967	+	25.7123i	0.154274	+	0.844046i
$929$	26.2472			0.861143			0.430571	−	0.902557i	$-0.358312\pi$
$929$	26.2472			0.861143			0.430571	+	0.902557i	$0.358312\pi$
$930$		0			0
$931$			5.34165i			0.175066i
$932$	43.7527	+	27.1879i	1.43317	+	0.890568i
$933$		0			0
$934$	−3.03699	−	0.866734i	−0.0993732	−	0.0283604i
$935$	−101.992			−3.33551
$936$		0			0
$937$	35.2758			1.15241			0.576205	−	0.817305i	$-0.304533\pi$
$937$	35.2758			1.15241			0.576205	+	0.817305i	$0.304533\pi$
$938$	−3.85653	−	1.10063i	−0.125920	−	0.0359367i
$939$		0			0
$940$	10.6538	−	17.1449i	0.347490	−	0.559206i
$941$		−	1.19310i		−	0.0388938i	−0.999811	−	0.0194469i	$-0.993809\pi$
$941$		−	1.19310i		−	0.0388938i	0.999811	−	0.0194469i	$-0.00619053\pi$
$942$		0			0
$943$	8.38502			0.273054
$944$	46.2079	−	22.8238i	1.50394	−	0.742852i
$945$		0			0
$946$	16.8831	−	59.1572i	0.548915	−	1.92337i
$947$		−	43.9229i		−	1.42730i	−0.700502	−	0.713651i	$-0.747040\pi$
$947$		−	43.9229i		−	1.42730i	0.700502	−	0.713651i	$-0.252960\pi$
$948$		0			0
$949$		−	25.6668i		−	0.833179i
$950$	5.02463	+	1.43399i	0.163021	+	0.0465249i
$951$		0			0
$952$	16.9934	−	15.4167i	0.550759	−	0.499658i
$953$	9.41243			0.304898			0.152449	−	0.988311i	$-0.451284\pi$
$953$	9.41243			0.304898			0.152449	+	0.988311i	$0.451284\pi$
$954$		0			0
$955$			55.9573i			1.81074i
$956$	−47.4989	−	29.5157i	−1.53622	−	0.954607i
$957$		0			0
$958$	1.66971	−	5.85058i	0.0539460	−	0.189024i
$959$	−21.7057			−0.700912
$960$		0			0
$961$	−11.2356			−0.362438
$962$	2.25764	−	7.91065i	0.0727893	−	0.255049i
$963$		0			0
$964$	−12.2762	−	7.62840i	−0.395390	−	0.245694i
$965$		−	22.7296i		−	0.731693i
$966$		0			0
$967$	4.18043			0.134433			0.0672167	−	0.997738i	$-0.478588\pi$
$967$	4.18043			0.134433			0.0672167	+	0.997738i	$0.478588\pi$
$968$	40.1152	−	36.3933i	1.28935	−	1.16972i
$969$		0			0
$970$	−42.7009	−	12.1865i	−1.37104	−	0.391286i
$971$			1.49104i			0.0478497i	0.999714	+	0.0239249i	$0.00761625\pi$
$971$			1.49104i			0.0478497i	−0.999714	+	0.0239249i	$0.992384\pi$
$972$		0			0
$973$		−	23.7346i		−	0.760896i
$974$	−2.84379	+	9.96446i	−0.0911208	+	0.319282i
$975$		0			0
$976$	−14.1106	+	6.96978i	−0.451670	+	0.223097i
$977$	−40.4152			−1.29300			−0.646498	−	0.762915i	$-0.723767\pi$
$977$	−40.4152			−1.29300			−0.646498	+	0.762915i	$0.723767\pi$
$978$		0			0
$979$		−	30.7565i		−	0.982981i
$980$	16.6266	−	26.7567i	0.531116	−	0.854712i
$981$		0			0
$982$	35.9127	+	10.2492i	1.14602	+	0.327066i
$983$	30.4847			0.972312			0.486156	−	0.873872i	$-0.338399\pi$
$983$	30.4847			0.972312			0.486156	+	0.873872i	$0.338399\pi$
$984$		0			0
$985$	−10.8940			−0.347111
$986$	39.5830	+	11.2967i	1.26058	+	0.359760i
$987$		0			0
$988$	5.93889	+	3.69041i	0.188941	+	0.117408i
$989$		−	26.5157i		−	0.843152i
$990$		0			0
$991$	55.5877			1.76580			0.882900	−	0.469561i	$-0.155588\pi$
$991$	55.5877			1.76580			0.882900	+	0.469561i	$0.155588\pi$
$992$	−24.7389	+	4.52176i	−0.785462	+	0.143566i
$993$		0			0
$994$	−1.08200	+	3.79125i	−0.0343189	+	0.120251i
$995$			51.8329i			1.64321i
$996$		0			0
$997$		−	34.9498i		−	1.10687i	−0.832892	−	0.553435i	$-0.813317\pi$
$997$		−	34.9498i		−	1.10687i	0.832892	−	0.553435i	$-0.186683\pi$
$998$	−0.252527	−	0.0720695i	−0.00799362	−	0.00228132i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.g.c.685.11		18
3.2	odd	2		456.2.g.a.229.8	yes	18
4.3	odd	2		5472.2.g.d.2737.3		18
8.3	odd	2		5472.2.g.d.2737.16		18
8.5	even	2	inner	1368.2.g.c.685.12		18
12.11	even	2		1824.2.g.b.913.8		18
24.5	odd	2		456.2.g.a.229.7	✓	18
24.11	even	2		1824.2.g.b.913.11		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.g.a.229.7	✓	18	24.5	odd	2
456.2.g.a.229.8	yes	18	3.2	odd	2
1368.2.g.c.685.11		18	1.1	even	1	trivial
1368.2.g.c.685.12		18	8.5	even	2	inner
1824.2.g.b.913.8		18	12.11	even	2
1824.2.g.b.913.11		18	24.11	even	2
5472.2.g.d.2737.3		18	4.3	odd	2
5472.2.g.d.2737.16		18	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.388110 - 1.35992i) q^{2} +(-1.69874 - 1.05559i) q^{4} -2.94870i q^{5} -1.28777 q^{7} +(-2.09482 + 1.90046i) q^{8} +O(q^{10})\)	\(q+(0.388110 - 1.35992i) q^{2} +(-1.69874 - 1.05559i) q^{4} -2.94870i q^{5} -1.28777 q^{7} +(-2.09482 + 1.90046i) q^{8} +(-4.00998 - 1.14442i) q^{10} -5.49088i q^{11} -3.49605i q^{13} +(-0.499797 + 1.75126i) q^{14} +(1.77144 + 3.58636i) q^{16} +6.29935 q^{17} -1.00000i q^{19} +(-3.11263 + 5.00907i) q^{20} +(-7.46713 - 2.13107i) q^{22} -3.34695 q^{23} -3.69481 q^{25} +(-4.75434 - 1.35685i) q^{26} +(2.18759 + 1.35936i) q^{28} +4.62063i q^{29} -4.44572 q^{31} +(5.56467 - 1.01711i) q^{32} +(2.44484 - 8.56658i) q^{34} +3.79724i q^{35} +1.66388i q^{37} +(-1.35992 - 0.388110i) q^{38} +(5.60387 + 6.17698i) q^{40} -2.50527 q^{41} +7.92235i q^{43} +(-5.79614 + 9.32758i) q^{44} +(-1.29899 + 4.55158i) q^{46} -3.42278 q^{47} -5.34165 q^{49} +(-1.43399 + 5.02463i) q^{50} +(-3.69041 + 5.93889i) q^{52} +6.00788i q^{53} -16.1909 q^{55} +(2.69764 - 2.44735i) q^{56} +(6.28366 + 1.79331i) q^{58} -12.8843i q^{59} +3.93452i q^{61} +(-1.72543 + 6.04580i) q^{62} +(0.776526 - 7.96222i) q^{64} -10.3088 q^{65} +2.20215i q^{67} +(-10.7010 - 6.64956i) q^{68} +(5.16393 + 1.47375i) q^{70} +2.16488 q^{71} +7.34165 q^{73} +(2.26274 + 0.645769i) q^{74} +(-1.05559 + 1.69874i) q^{76} +7.07098i q^{77} +2.68779 q^{79} +(10.5751 - 5.22344i) q^{80} +(-0.972320 + 3.40695i) q^{82} -15.7301i q^{83} -18.5749i q^{85} +(10.7737 + 3.07475i) q^{86} +(10.4352 + 11.5024i) q^{88} +5.60138 q^{89} +4.50211i q^{91} +(5.68561 + 3.53303i) q^{92} +(-1.32842 + 4.65469i) q^{94} -2.94870 q^{95} +10.6487 q^{97} +(-2.07315 + 7.26419i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{2} - 2 q^{4} - 12 q^{7} - 2 q^{8}+O(q^{10}) \) 18 * q - 2 * q^2 - 2 * q^4 - 12 * q^7 - 2 * q^8	\( 18 q - 2 q^{2} - 2 q^{4} - 12 q^{7} - 2 q^{8} + 4 q^{10} + 10 q^{14} + 2 q^{16} - 4 q^{17} - 6 q^{20} - 26 q^{22} + 36 q^{23} - 22 q^{25} - 34 q^{26} + 24 q^{28} + 32 q^{31} + 18 q^{32} + 18 q^{34} - 50 q^{40} + 4 q^{41} - 32 q^{44} + 2 q^{46} - 52 q^{47} + 18 q^{49} - 48 q^{50} + 2 q^{52} - 40 q^{55} + 36 q^{56} + 32 q^{58} + 46 q^{62} - 38 q^{64} + 16 q^{65} - 60 q^{68} - 46 q^{70} + 64 q^{71} + 28 q^{73} + 2 q^{74} + 4 q^{76} + 32 q^{79} + 34 q^{80} + 48 q^{82} + 58 q^{86} - 40 q^{88} - 4 q^{89} - 22 q^{92} - 32 q^{94} - 4 q^{97} - 68 q^{98}+O(q^{100}) \) 18 * q - 2 * q^2 - 2 * q^4 - 12 * q^7 - 2 * q^8 + 4 * q^10 + 10 * q^14 + 2 * q^16 - 4 * q^17 - 6 * q^20 - 26 * q^22 + 36 * q^23 - 22 * q^25 - 34 * q^26 + 24 * q^28 + 32 * q^31 + 18 * q^32 + 18 * q^34 - 50 * q^40 + 4 * q^41 - 32 * q^44 + 2 * q^46 - 52 * q^47 + 18 * q^49 - 48 * q^50 + 2 * q^52 - 40 * q^55 + 36 * q^56 + 32 * q^58 + 46 * q^62 - 38 * q^64 + 16 * q^65 - 60 * q^68 - 46 * q^70 + 64 * q^71 + 28 * q^73 + 2 * q^74 + 4 * q^76 + 32 * q^79 + 34 * q^80 + 48 * q^82 + 58 * q^86 - 40 * q^88 - 4 * q^89 - 22 * q^92 - 32 * q^94 - 4 * q^97 - 68 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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