LMFDB - Embedded newform 1638.2.bj.g.1135.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(127,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("1638.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.bj (of order $6$, degree $2$, 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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 $ x^12 - 6x^11 + 39x^10 - 140x^9 + 460x^8 - 1066x^7 + 2127x^6 - 3172x^5 + 3842x^4 - 3394x^3 + 2141x^2 - 832*x + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1135.4
Root			$0.500000 - 2.47866i$ of defining polynomial
Character	$\chi$	$=$	1638.1135
Dual form			1638.2.bj.g.127.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.56356i q^{5} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.56356i q^{5} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.781779 - 1.35408i) q^{10} +(2.48215 + 1.43307i) q^{11} +(2.99598 + 2.00602i) q^{13} -1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.11481 + 1.93090i) q^{17} +(-6.26657 + 3.61801i) q^{19} +(1.35408 - 0.781779i) q^{20} +(1.43307 + 2.48215i) q^{22} +(0.833676 - 1.44397i) q^{23} +2.55529 q^{25} +(1.59158 + 3.23525i) q^{26} +(-0.866025 - 0.500000i) q^{28} +(2.41379 - 4.18080i) q^{29} -0.597963i q^{31} +(-0.866025 + 0.500000i) q^{32} +2.22961i q^{34} +(0.781779 + 1.35408i) q^{35} +(0.0333971 + 0.0192818i) q^{37} -7.23602 q^{38} +1.56356 q^{40} +(6.88896 + 3.97734i) q^{41} +(5.04571 + 8.73942i) q^{43} +2.86614i q^{44} +(1.44397 - 0.833676i) q^{46} -7.02636i q^{47} +(0.500000 - 0.866025i) q^{49} +(2.21294 + 1.27764i) q^{50} +(-0.239275 + 3.59760i) q^{52} +5.98404 q^{53} +(2.24069 - 3.88098i) q^{55} +(-0.500000 - 0.866025i) q^{56} +(4.18080 - 2.41379i) q^{58} +(-0.776138 + 0.448103i) q^{59} +(7.12846 + 12.3469i) q^{61} +(0.298982 - 0.517851i) q^{62} -1.00000 q^{64} +(3.13653 - 4.68438i) q^{65} +(-1.42103 - 0.820432i) q^{67} +(-1.11481 + 1.93090i) q^{68} +1.56356i q^{70} +(1.98724 - 1.14733i) q^{71} +11.2277i q^{73} +(0.0192818 + 0.0333971i) q^{74} +(-6.26657 - 3.61801i) q^{76} -2.86614 q^{77} +4.26098 q^{79} +(1.35408 + 0.781779i) q^{80} +(3.97734 + 6.88896i) q^{82} +4.94829i q^{83} +(3.01907 - 1.74306i) q^{85} +10.0914i q^{86} +(-1.43307 + 2.48215i) q^{88} +(-2.09682 - 1.21060i) q^{89} +(-3.59760 - 0.239275i) q^{91} +1.66735 q^{92} +(3.51318 - 6.08501i) q^{94} +(5.65696 + 9.79815i) q^{95} +(-4.23338 + 2.44414i) q^{97} +(0.866025 - 0.500000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{4}+O(q^{10}) $ 12 * q + 6 * q^4	$ 12 q + 6 q^{4} - 2 q^{10} + 18 q^{11} - 8 q^{13} - 12 q^{14} - 6 q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} + 6 q^{23} - 24 q^{25} + 14 q^{26} + 10 q^{29} - 2 q^{35} - 6 q^{37} - 8 q^{38} - 4 q^{40} + 24 q^{41} + 26 q^{43} - 6 q^{46} + 6 q^{49} + 12 q^{50} - 4 q^{52} - 36 q^{53} - 6 q^{55} - 6 q^{56} + 24 q^{58} - 6 q^{59} - 28 q^{61} + 2 q^{62} - 12 q^{64} + 34 q^{65} - 42 q^{67} + 4 q^{68} - 48 q^{71} + 12 q^{76} + 4 q^{77} + 44 q^{79} + 6 q^{82} + 54 q^{85} + 2 q^{88} - 12 q^{89} - 16 q^{91} + 12 q^{92} + 8 q^{94} - 32 q^{95} + 60 q^{97}+O(q^{100}) $ 12 * q + 6 * q^4 - 2 * q^10 + 18 * q^11 - 8 * q^13 - 12 * q^14 - 6 * q^16 - 4 * q^17 + 12 * q^19 - 2 * q^22 + 6 * q^23 - 24 * q^25 + 14 * q^26 + 10 * q^29 - 2 * q^35 - 6 * q^37 - 8 * q^38 - 4 * q^40 + 24 * q^41 + 26 * q^43 - 6 * q^46 + 6 * q^49 + 12 * q^50 - 4 * q^52 - 36 * q^53 - 6 * q^55 - 6 * q^56 + 24 * q^58 - 6 * q^59 - 28 * q^61 + 2 * q^62 - 12 * q^64 + 34 * q^65 - 42 * q^67 + 4 * q^68 - 48 * q^71 + 12 * q^76 + 4 * q^77 + 44 * q^79 + 6 * q^82 + 54 * q^85 + 2 * q^88 - 12 * q^89 - 16 * q^91 + 12 * q^92 + 8 * q^94 - 32 * q^95 + 60 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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 $

$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$		−	1.56356i		−	0.699244i	−0.936891	−	0.349622i	$-0.886310\pi$
$5$		−	1.56356i		−	0.699244i	0.936891	−	0.349622i	$-0.113690\pi$
$6$		0			0
$7$	−0.866025	+	0.500000i	−0.327327	+	0.188982i
$7$	−0.866025	+	0.500000i	−0.327327	+	0.188982i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	0.781779	−	1.35408i	0.247220	−	0.428198i
$11$	2.48215	+	1.43307i	0.748396	+	0.432087i	0.825114	−	0.564966i	$-0.191111\pi$
$11$	2.48215	+	1.43307i	0.748396	+	0.432087i	−0.0767180	+	0.997053i	$0.524444\pi$
$12$		0			0
$13$	2.99598	+	2.00602i	0.830935	+	0.556370i
$13$	2.99598	+	2.00602i	0.830935	+	0.556370i
$14$	−1.00000			−0.267261
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	1.11481	+	1.93090i	0.270380	+	0.468312i	0.968959	−	0.247221i	$-0.0795173\pi$
$17$	1.11481	+	1.93090i	0.270380	+	0.468312i	−0.698579	+	0.715533i	$0.746184\pi$
$18$		0			0
$19$	−6.26657	+	3.61801i	−1.43765	+	0.830028i	−0.997686	−	0.0679872i	$-0.978342\pi$
$19$	−6.26657	+	3.61801i	−1.43765	+	0.830028i	−0.439964	+	0.898015i	$0.645009\pi$
$20$	1.35408	−	0.781779i	0.302782	−	0.174811i
$21$		0			0
$22$	1.43307	+	2.48215i	0.305531	+	0.529196i
$23$	0.833676	−	1.44397i	0.173833	−	0.301088i	−0.765924	−	0.642932i	$-0.777718\pi$
$23$	0.833676	−	1.44397i	0.173833	−	0.301088i	0.939757	+	0.341843i	$0.111051\pi$
$24$		0			0
$25$	2.55529			0.511058
$26$	1.59158	+	3.23525i	0.312135	+	0.634485i
$27$		0			0
$28$	−0.866025	−	0.500000i	−0.163663	−	0.0944911i
$29$	2.41379	−	4.18080i	0.448229	−	0.776356i	−0.550042	−	0.835137i	$-0.685388\pi$
$29$	2.41379	−	4.18080i	0.448229	−	0.776356i	0.998271	+	0.0587816i	$0.0187216\pi$
$30$		0			0
$31$		−	0.597963i		−	0.107397i	−0.998557	−	0.0536987i	$-0.982899\pi$
$31$		−	0.597963i		−	0.107397i	0.998557	−	0.0536987i	$-0.0171010\pi$
$32$	−0.866025	+	0.500000i	−0.153093	+	0.0883883i
$33$		0			0
$34$			2.22961i			0.382375i
$35$	0.781779	+	1.35408i	0.132145	+	0.228881i
$36$		0			0
$37$	0.0333971	+	0.0192818i	0.00549045	+	0.00316991i	0.502743	−	0.864436i	$-0.332324\pi$
$37$	0.0333971	+	0.0192818i	0.00549045	+	0.00316991i	−0.497252	+	0.867606i	$0.665658\pi$
$38$	−7.23602			−1.17384
$39$		0			0
$40$	1.56356			0.247220
$41$	6.88896	+	3.97734i	1.07588	+	0.621157i	0.929781	−	0.368114i	$-0.119996\pi$
$41$	6.88896	+	3.97734i	1.07588	+	0.621157i	0.146095	+	0.989271i	$0.453330\pi$
$42$		0			0
$43$	5.04571	+	8.73942i	0.769463	+	1.33275i	0.937854	+	0.347029i	$0.112809\pi$
$43$	5.04571	+	8.73942i	0.769463	+	1.33275i	−0.168391	+	0.985720i	$0.553857\pi$
$44$			2.86614i			0.432087i
$45$		0			0
$46$	1.44397	−	0.833676i	0.212902	−	0.122919i
$47$		−	7.02636i		−	1.02490i	−0.858717	−	0.512450i	$-0.828738\pi$
$47$		−	7.02636i		−	1.02490i	0.858717	−	0.512450i	$-0.171262\pi$
$48$		0			0
$49$	0.500000	−	0.866025i	0.0714286	−	0.123718i
$50$	2.21294	+	1.27764i	0.312958	+	0.180686i
$51$		0			0
$52$	−0.239275	+	3.59760i	−0.0331814	+	0.498898i
$53$	5.98404			0.821971			0.410985	−	0.911642i	$-0.365185\pi$
$53$	5.98404			0.821971			0.410985	+	0.911642i	$0.365185\pi$
$54$		0			0
$55$	2.24069	−	3.88098i	0.302134	−	0.523312i
$56$	−0.500000	−	0.866025i	−0.0668153	−	0.115728i
$57$		0			0
$58$	4.18080	−	2.41379i	0.548966	−	0.316946i
$59$	−0.776138	+	0.448103i	−0.101044	+	0.0583381i	−0.549671	−	0.835381i	$-0.685247\pi$
$59$	−0.776138	+	0.448103i	−0.101044	+	0.0583381i	0.448626	+	0.893719i	$0.351913\pi$
$60$		0			0
$61$	7.12846	+	12.3469i	0.912706	+	1.58085i	0.810225	+	0.586119i	$0.199345\pi$
$61$	7.12846	+	12.3469i	0.912706	+	1.58085i	0.102481	+	0.994735i	$0.467322\pi$
$62$	0.298982	−	0.517851i	0.0379707	−	0.0657672i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	3.13653	−	4.68438i	0.389038	−	0.581026i
$66$		0			0
$67$	−1.42103	−	0.820432i	−0.173606	−	0.100232i	0.410679	−	0.911780i	$-0.365292\pi$
$67$	−1.42103	−	0.820432i	−0.173606	−	0.100232i	−0.584285	+	0.811548i	$0.698625\pi$
$68$	−1.11481	+	1.93090i	−0.135190	+	0.234156i
$69$		0			0
$70$			1.56356i			0.186881i
$71$	1.98724	−	1.14733i	0.235841	−	0.136163i	−0.377422	−	0.926041i	$-0.623190\pi$
$71$	1.98724	−	1.14733i	0.235841	−	0.136163i	0.613264	+	0.789878i	$0.289856\pi$
$72$		0			0
$73$			11.2277i			1.31411i	0.753844	+	0.657054i	$0.228198\pi$
$73$			11.2277i			1.31411i	−0.753844	+	0.657054i	$0.771802\pi$
$74$	0.0192818	+	0.0333971i	0.00224147	+	0.00388233i
$75$		0			0
$76$	−6.26657	−	3.61801i	−0.718825	−	0.415014i
$77$	−2.86614			−0.326627
$78$		0			0
$79$	4.26098			0.479397			0.239699	−	0.970847i	$-0.422951\pi$
$79$	4.26098			0.479397			0.239699	+	0.970847i	$0.422951\pi$
$80$	1.35408	+	0.781779i	0.151391	+	0.0874055i
$81$		0			0
$82$	3.97734	+	6.88896i	0.439224	+	0.760759i
$83$			4.94829i			0.543145i	0.962418	+	0.271572i	$0.0875437\pi$
$83$			4.94829i			0.543145i	−0.962418	+	0.271572i	$0.912456\pi$
$84$		0			0
$85$	3.01907	−	1.74306i	0.327465	−	0.189062i
$86$			10.0914i			1.08818i
$87$		0			0
$88$	−1.43307	+	2.48215i	−0.152766	+	0.264598i
$89$	−2.09682	−	1.21060i	−0.222263	−	0.128323i	0.384735	−	0.923027i	$-0.374293\pi$
$89$	−2.09682	−	1.21060i	−0.222263	−	0.128323i	−0.606997	+	0.794704i	$0.707626\pi$
$90$		0			0
$91$	−3.59760	−	0.239275i	−0.377131	−	0.0250828i
$92$	1.66735			0.173833
$93$		0			0
$94$	3.51318	−	6.08501i	0.362357	−	0.627621i
$95$	5.65696	+	9.79815i	0.580392	+	1.00527i
$96$		0			0
$97$	−4.23338	+	2.44414i	−0.429835	+	0.248165i	−0.699276	−	0.714852i	$-0.746494\pi$
$97$	−4.23338	+	2.44414i	−0.429835	+	0.248165i	0.269442	+	0.963017i	$0.413161\pi$
$98$	0.866025	−	0.500000i	0.0874818	−	0.0505076i
$99$		0			0
$100$	1.27764	+	2.21294i	0.127764	+	0.221294i
$101$	−3.68373	+	6.38042i	−0.366545	+	0.634875i	−0.989023	−	0.147763i	$-0.952793\pi$
$101$	−3.68373	+	6.38042i	−0.366545	+	0.634875i	0.622478	+	0.782638i	$0.286126\pi$
$102$		0			0
$103$	−5.78525			−0.570038			−0.285019	−	0.958522i	$-0.592000\pi$
$103$	−5.78525			−0.570038			−0.285019	+	0.958522i	$0.592000\pi$
$104$	−2.00602	+	2.99598i	−0.196706	+	0.293780i
$105$		0			0
$106$	5.18233	+	2.99202i	0.503352	+	0.290611i
$107$	−0.514478	+	0.891102i	−0.0497365	+	0.0861461i	−0.889822	−	0.456308i	$-0.849171\pi$
$107$	−0.514478	+	0.891102i	−0.0497365	+	0.0861461i	0.840085	+	0.542454i	$0.182505\pi$
$108$		0			0
$109$		−	14.1535i		−	1.35566i	−0.735220	−	0.677829i	$-0.762921\pi$
$109$		−	14.1535i		−	1.35566i	0.735220	−	0.677829i	$-0.237079\pi$
$110$	3.88098	−	2.24069i	0.370037	−	0.213641i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	−6.77051	−	11.7269i	−0.636916	−	1.10317i	−0.986106	−	0.166119i	$-0.946876\pi$
$113$	−6.77051	−	11.7269i	−0.636916	−	1.10317i	0.349189	−	0.937052i	$-0.386457\pi$
$114$		0			0
$115$	−2.25773	−	1.30350i	−0.210534	−	0.121552i
$116$	4.82757			0.448229
$117$		0			0
$118$	−0.896206			−0.0825025
$119$	−1.93090	−	1.11481i	−0.177005	−	0.102194i
$120$		0			0
$121$	−1.39263	−	2.41210i	−0.126602	−	0.219282i
$122$			14.2569i			1.29076i
$123$		0			0
$124$	0.517851	−	0.298982i	0.0465044	−	0.0268493i
$125$		−	11.8131i		−	1.05660i
$126$		0			0
$127$	4.92583	−	8.53178i	0.437096	−	0.757073i	−0.560368	−	0.828244i	$-0.689340\pi$
$127$	4.92583	−	8.53178i	0.437096	−	0.757073i	0.997464	+	0.0711707i	$0.0226735\pi$
$128$	−0.866025	−	0.500000i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	5.05850	−	2.48853i	0.443660	−	0.218259i
$131$	−14.7923			−1.29241			−0.646204	−	0.763165i	$-0.723645\pi$
$131$	−14.7923			−1.29241			−0.646204	+	0.763165i	$0.723645\pi$
$132$		0			0
$133$	3.61801	−	6.26657i	0.313721	−	0.543381i
$134$	−0.820432	−	1.42103i	−0.0708746	−	0.122758i
$135$		0			0
$136$	−1.93090	+	1.11481i	−0.165573	+	0.0955938i
$137$	−0.397503	+	0.229499i	−0.0339610	+	0.0196074i	−0.516884	−	0.856055i	$-0.672908\pi$
$137$	−0.397503	+	0.229499i	−0.0339610	+	0.0196074i	0.482923	+	0.875663i	$0.339575\pi$
$138$		0			0
$139$	−7.65731	−	13.2628i	−0.649485	−	1.12494i	−0.983246	−	0.182283i	$-0.941651\pi$
$139$	−7.65731	−	13.2628i	−0.649485	−	1.12494i	0.333762	−	0.942658i	$-0.391682\pi$
$140$	−0.781779	+	1.35408i	−0.0660724	+	0.114441i
$141$		0			0
$142$	2.29466			0.192564
$143$	4.56170	+	9.27268i	0.381468	+	0.775421i
$144$		0			0
$145$	−6.53693	−	3.77410i	−0.542862	−	0.313422i
$146$	−5.61387	+	9.72351i	−0.464607	+	0.804723i
$147$		0			0
$148$			0.0385636i			0.00316991i
$149$	−8.86563	+	5.11858i	−0.726301	+	0.419330i	−0.817067	−	0.576542i	$-0.804402\pi$
$149$	−8.86563	+	5.11858i	−0.726301	+	0.419330i	0.0907665	+	0.995872i	$0.471068\pi$
$150$		0			0
$151$		−	15.2110i		−	1.23785i	−0.785448	−	0.618927i	$-0.787568\pi$
$151$		−	15.2110i		−	1.23785i	0.785448	−	0.618927i	$-0.212432\pi$
$152$	−3.61801	−	6.26657i	−0.293459	−	0.508286i
$153$		0			0
$154$	−2.48215	−	1.43307i	−0.200017	−	0.115480i
$155$	−0.934950			−0.0750970
$156$		0			0
$157$	−2.27419			−0.181500			−0.0907500	−	0.995874i	$-0.528926\pi$
$157$	−2.27419			−0.181500			−0.0907500	+	0.995874i	$0.528926\pi$
$158$	3.69011	+	2.13049i	0.293570	+	0.169493i
$159$		0			0
$160$	0.781779	+	1.35408i	0.0618050	+	0.107049i
$161$			1.66735i			0.131406i
$162$		0			0
$163$	0.00848066	−	0.00489631i	0.000664256	−	0.000383509i	−0.499668	−	0.866217i	$-0.666545\pi$
$163$	0.00848066	−	0.00489631i	0.000664256	−	0.000383509i	0.500332	+	0.865834i	$0.333211\pi$
$164$			7.95469i			0.621157i
$165$		0			0
$166$	−2.47414	+	4.28534i	−0.192031	+	0.332607i
$167$	−21.6080	−	12.4754i	−1.67208	−	0.965376i	−0.966472	−	0.256773i	$-0.917341\pi$
$167$	−21.6080	−	12.4754i	−1.67208	−	0.965376i	−0.705608	−	0.708603i	$-0.749326\pi$
$168$		0			0
$169$	4.95177	+	12.0200i	0.380906	+	0.924614i
$170$	3.48613			0.267374
$171$		0			0
$172$	−5.04571	+	8.73942i	−0.384731	+	0.666374i
$173$	1.60275	+	2.77604i	0.121855	+	0.211058i	0.920499	−	0.390745i	$-0.127782\pi$
$173$	1.60275	+	2.77604i	0.121855	+	0.211058i	−0.798644	+	0.601803i	$0.794449\pi$
$174$		0			0
$175$	−2.21294	+	1.27764i	−0.167283	+	0.0965808i
$176$	−2.48215	+	1.43307i	−0.187099	+	0.108022i
$177$		0			0
$178$	−1.21060	−	2.09682i	−0.0907383	−	0.157163i
$179$	7.89998	−	13.6832i	0.590472	−	1.02273i	−0.403697	−	0.914893i	$-0.632275\pi$
$179$	7.89998	−	13.6832i	0.590472	−	1.02273i	0.994169	−	0.107835i	$-0.0343917\pi$
$180$		0			0
$181$	9.11907			0.677815			0.338908	−	0.940820i	$-0.389943\pi$
$181$	9.11907			0.677815			0.338908	+	0.940820i	$0.389943\pi$
$182$	−2.99598	−	2.00602i	−0.222077	−	0.148696i
$183$		0			0
$184$	1.44397	+	0.833676i	0.106451	+	0.0614594i
$185$	0.0301482	−	0.0522183i	0.00221654	−	0.00383916i
$186$		0			0
$187$			6.39038i			0.467311i
$188$	6.08501	−	3.51318i	0.443795	−	0.256225i
$189$		0			0
$190$			11.3139i			0.820799i
$191$	−1.63068	−	2.82443i	−0.117992	−	0.204368i	0.800980	−	0.598691i	$-0.204312\pi$
$191$	−1.63068	−	2.82443i	−0.117992	−	0.204368i	−0.918972	+	0.394323i	$0.870979\pi$
$192$		0			0
$193$	19.1158	+	11.0365i	1.37599	+	0.794426i	0.991674	−	0.128777i	$-0.0411051\pi$
$193$	19.1158	+	11.0365i	1.37599	+	0.794426i	0.384313	+	0.923203i	$0.374438\pi$
$194$	−4.88829			−0.350959
$195$		0			0
$196$	1.00000			0.0714286
$197$	4.56660	+	2.63653i	0.325357	+	0.187845i	0.653778	−	0.756687i	$-0.273183\pi$
$197$	4.56660	+	2.63653i	0.325357	+	0.187845i	−0.328421	+	0.944531i	$0.606516\pi$
$198$		0			0
$199$	4.43381	+	7.67958i	0.314304	+	0.544391i	0.979289	−	0.202466i	$-0.0648954\pi$
$199$	4.43381	+	7.67958i	0.314304	+	0.544391i	−0.664985	+	0.746857i	$0.731562\pi$
$200$			2.55529i			0.180686i
$201$		0			0
$202$	−6.38042	+	3.68373i	−0.448924	+	0.259187i
$203$			4.82757i			0.338829i
$204$		0			0
$205$	6.21881	−	10.7713i	0.434340	−	0.752300i
$206$	−5.01017	−	2.89263i	−0.349075	−	0.201539i
$207$		0			0
$208$	−3.23525	+	1.59158i	−0.224324	+	0.110356i
$209$	−20.7394			−1.43458
$210$		0			0
$211$	−3.28453	+	5.68898i	−0.226117	+	0.391646i	−0.956654	−	0.291227i	$-0.905936\pi$
$211$	−3.28453	+	5.68898i	−0.226117	+	0.391646i	0.730537	+	0.682873i	$0.239270\pi$
$212$	2.99202	+	5.18233i	0.205493	+	0.355924i
$213$		0			0
$214$	−0.891102	+	0.514478i	−0.0609145	+	0.0351690i
$215$	13.6646	−	7.88925i	0.931917	−	0.538043i
$216$		0			0
$217$	0.298982	+	0.517851i	0.0202962	+	0.0351540i
$218$	7.07674	−	12.2573i	0.479297	−	0.830167i
$219$		0			0
$220$	4.48137			0.302134
$221$	−0.533490	+	8.02126i	−0.0358864	+	0.539568i
$222$		0			0
$223$	−14.6463	−	8.45606i	−0.980790	−	0.566260i	−0.0782817	−	0.996931i	$-0.524943\pi$
$223$	−14.6463	−	8.45606i	−0.980790	−	0.566260i	−0.902509	+	0.430672i	$0.858277\pi$
$224$	0.500000	−	0.866025i	0.0334077	−	0.0578638i
$225$		0			0
$226$		−	13.5410i		−	0.900736i
$227$	13.9709	−	8.06611i	0.927282	−	0.535367i	0.0413312	−	0.999146i	$-0.486840\pi$
$227$	13.9709	−	8.06611i	0.927282	−	0.535367i	0.885951	+	0.463779i	$0.153507\pi$
$228$		0			0
$229$		−	6.91184i		−	0.456747i	−0.973574	−	0.228374i	$-0.926659\pi$
$229$		−	6.91184i		−	0.456747i	0.973574	−	0.228374i	$-0.0733408\pi$
$230$	−1.30350	−	2.25773i	−0.0859502	−	0.148870i
$231$		0			0
$232$	4.18080	+	2.41379i	0.274483	+	0.158473i
$233$	−7.47405			−0.489641			−0.244821	−	0.969568i	$-0.578729\pi$
$233$	−7.47405			−0.489641			−0.244821	+	0.969568i	$0.578729\pi$
$234$		0			0
$235$	−10.9861			−0.716656
$236$	−0.776138	−	0.448103i	−0.0505222	−	0.0291690i
$237$		0			0
$238$	−1.11481	−	1.93090i	−0.0722621	−	0.125162i
$239$			19.8696i			1.28526i	0.766179	+	0.642628i	$0.222156\pi$
$239$			19.8696i			1.28526i	−0.766179	+	0.642628i	$0.777844\pi$
$240$		0			0
$241$	−9.21842	+	5.32226i	−0.593811	+	0.342837i	−0.766603	−	0.642121i	$-0.778054\pi$
$241$	−9.21842	+	5.32226i	−0.593811	+	0.342837i	0.172792	+	0.984958i	$0.444721\pi$
$242$		−	2.78525i		−	0.179043i
$243$		0			0
$244$	−7.12846	+	12.3469i	−0.456353	+	0.790427i
$245$	−1.35408	−	0.781779i	−0.0865090	−	0.0499460i
$246$		0			0
$247$	−26.0323	−	1.73140i	−1.65640	−	0.110166i
$248$	0.597963			0.0379707
$249$		0			0
$250$	5.90656	−	10.2305i	0.373564	−	0.647032i
$251$	−7.95696	−	13.7819i	−0.502239	−	0.869904i	−0.999997	−	0.00258749i	$-0.999176\pi$
$251$	−7.95696	−	13.7819i	−0.502239	−	0.869904i	0.497757	−	0.867316i	$-0.334157\pi$
$252$		0			0
$253$	4.13861	−	2.38943i	0.260192	−	0.150222i
$254$	8.53178	−	4.92583i	0.535332	−	0.309074i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	15.5509	−	26.9350i	0.970039	−	1.68016i	0.274619	−	0.961553i	$-0.411448\pi$
$257$	15.5509	−	26.9350i	0.970039	−	1.68016i	0.695420	−	0.718603i	$-0.255218\pi$
$258$		0			0
$259$	−0.0385636			−0.00239623
$260$	5.62506	+	0.374120i	0.348851	+	0.0232019i
$261$		0			0
$262$	−12.8105	−	7.39614i	−0.791435	−	0.456935i
$263$	−14.1873	+	24.5732i	−0.874829	+	1.51525i	−0.0178837	+	0.999840i	$0.505693\pi$
$263$	−14.1873	+	24.5732i	−0.874829	+	1.51525i	−0.856945	+	0.515408i	$0.827640\pi$
$264$		0			0
$265$		−	9.35639i		−	0.574758i
$266$	6.26657	−	3.61801i	0.384228	−	0.221834i
$267$		0			0
$268$		−	1.64086i		−	0.100232i
$269$	−10.7008	−	18.5344i	−0.652441	−	1.13006i	−0.982529	−	0.186111i	$-0.940412\pi$
$269$	−10.7008	−	18.5344i	−0.652441	−	1.13006i	0.330088	−	0.943950i	$-0.392922\pi$
$270$		0			0
$271$	4.97667	+	2.87328i	0.302311	+	0.174539i	0.643481	−	0.765462i	$-0.277490\pi$
$271$	4.97667	+	2.87328i	0.302311	+	0.174539i	−0.341170	+	0.940002i	$0.610823\pi$
$272$	−2.22961			−0.135190
$273$		0			0
$274$	−0.458997			−0.0277290
$275$	6.34260	+	3.66190i	0.382473	+	0.220821i
$276$		0			0
$277$	13.3010	+	23.0380i	0.799180	+	1.38422i	0.920151	+	0.391564i	$0.128066\pi$
$277$	13.3010	+	23.0380i	0.799180	+	1.38422i	−0.120971	+	0.992656i	$0.538601\pi$
$278$		−	15.3146i		−	0.918510i
$279$		0			0
$280$	−1.35408	+	0.781779i	−0.0809218	+	0.0467202i
$281$			6.69143i			0.399177i	0.979880	+	0.199589i	$0.0639606\pi$
$281$			6.69143i			0.399177i	−0.979880	+	0.199589i	$0.936039\pi$
$282$		0			0
$283$	9.96692	−	17.2632i	0.592472	−	1.02619i	−0.401426	−	0.915891i	$-0.631485\pi$
$283$	9.96692	−	17.2632i	0.592472	−	1.02619i	0.993898	−	0.110300i	$-0.0351812\pi$
$284$	1.98724	+	1.14733i	0.117921	+	0.0680816i
$285$		0			0
$286$	−0.685794	+	10.3112i	−0.0405519	+	0.609716i
$287$	−7.95469			−0.469550
$288$		0			0
$289$	6.01442	−	10.4173i	0.353789	−	0.612781i
$290$	−3.77410	−	6.53693i	−0.221623	−	0.383861i
$291$		0			0
$292$	−9.72351	+	5.61387i	−0.569025	+	0.328527i
$293$	7.67375	−	4.43044i	0.448305	−	0.258829i	−0.258809	−	0.965929i	$-0.583330\pi$
$293$	7.67375	−	4.43044i	0.448305	−	0.258829i	0.707114	+	0.707099i	$0.249997\pi$
$294$		0			0
$295$	0.700635	+	1.21354i	0.0407926	+	0.0706548i
$296$	−0.0192818	+	0.0333971i	−0.00112073	+	0.00194117i
$297$		0			0
$298$	−10.2372			−0.593022
$299$	5.39430	−	2.65373i	0.311961	−	0.153469i
$300$		0			0
$301$	−8.73942	−	5.04571i	−0.503732	−	0.290830i
$302$	7.60551	−	13.1731i	0.437648	−	0.758028i
$303$		0			0
$304$		−	7.23602i		−	0.415014i
$305$	19.3050	−	11.1458i	1.10540	−	0.638205i
$306$		0			0
$307$			8.34636i			0.476352i	0.971222	+	0.238176i	$0.0765495\pi$
$307$			8.34636i			0.476352i	−0.971222	+	0.238176i	$0.923450\pi$
$308$	−1.43307	−	2.48215i	−0.0816567	−	0.141434i
$309$		0			0
$310$	−0.809690	−	0.467475i	−0.0459873	−	0.0265508i
$311$	6.68896			0.379296			0.189648	−	0.981852i	$-0.439265\pi$
$311$	6.68896			0.379296			0.189648	+	0.981852i	$0.439265\pi$
$312$		0			0
$313$	−21.3788			−1.20840			−0.604199	−	0.796833i	$-0.706507\pi$
$313$	−21.3788			−1.20840			−0.604199	+	0.796833i	$0.706507\pi$
$314$	−1.96950	−	1.13709i	−0.111146	−	0.0641699i
$315$		0			0
$316$	2.13049	+	3.69011i	0.119849	+	0.207585i
$317$		−	31.6776i		−	1.77919i	−0.456748	−	0.889596i	$-0.650986\pi$
$317$		−	31.6776i		−	1.77919i	0.456748	−	0.889596i	$-0.349014\pi$
$318$		0			0
$319$	11.9828	−	6.91825i	0.670906	−	0.387348i
$320$			1.56356i			0.0874055i
$321$		0			0
$322$	−0.833676	+	1.44397i	−0.0464589	+	0.0804692i
$323$	−13.9720	−	8.06675i	−0.777424	−	0.448846i
$324$		0			0
$325$	7.65559	+	5.12596i	0.424656	+	0.284337i
$326$	0.00979262			0.000542363
$327$		0			0
$328$	−3.97734	+	6.88896i	−0.219612	+	0.380379i
$329$	3.51318	+	6.08501i	0.193688	+	0.335477i
$330$		0			0
$331$	−21.3644	+	12.3347i	−1.17429	+	0.677979i	−0.954688	−	0.297609i	$-0.903811\pi$
$331$	−21.3644	+	12.3347i	−1.17429	+	0.677979i	−0.219606	+	0.975589i	$0.570477\pi$
$332$	−4.28534	+	2.47414i	−0.235189	+	0.135786i
$333$		0			0
$334$	−12.4754	−	21.6080i	−0.682624	−	1.18234i
$335$	−1.28279	+	2.22186i	−0.0700865	+	0.121393i
$336$		0			0
$337$	28.0871			1.53000			0.765002	−	0.644028i	$-0.222738\pi$
$337$	28.0871			1.53000			0.765002	+	0.644028i	$0.222738\pi$
$338$	−1.72163	+	12.8855i	−0.0936444	+	0.700879i
$339$		0			0
$340$	3.01907	+	1.74306i	0.163732	+	0.0945309i
$341$	0.856923	−	1.48423i	0.0464050	−	0.0803757i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$	−8.73942	+	5.04571i	−0.471198	+	0.272046i
$345$		0			0
$346$			3.20550i			0.172329i
$347$	−5.05398	−	8.75374i	−0.271312	−	0.469926i	0.697886	−	0.716209i	$-0.254124\pi$
$347$	−5.05398	−	8.75374i	−0.271312	−	0.469926i	−0.969198	+	0.246283i	$0.920791\pi$
$348$		0			0
$349$	−15.0596	−	8.69465i	−0.806121	−	0.465414i	0.0394863	−	0.999220i	$-0.487428\pi$
$349$	−15.0596	−	8.69465i	−0.806121	−	0.465414i	−0.845607	+	0.533806i	$0.820761\pi$
$350$	−2.55529			−0.136586
$351$		0			0
$352$	−2.86614			−0.152766
$353$	−2.88091	−	1.66329i	−0.153335	−	0.0885282i	0.421369	−	0.906889i	$-0.361550\pi$
$353$	−2.88091	−	1.66329i	−0.153335	−	0.0885282i	−0.574704	+	0.818361i	$0.694883\pi$
$354$		0			0
$355$	−1.79392	−	3.10716i	−0.0952113	−	0.164911i
$356$		−	2.42120i		−	0.128323i
$357$		0			0
$358$	13.6832	−	7.89998i	0.723177	−	0.417527i
$359$			8.02414i			0.423498i	0.977324	+	0.211749i	$0.0679159\pi$
$359$			8.02414i			0.423498i	−0.977324	+	0.211749i	$0.932084\pi$
$360$		0			0
$361$	16.6800	−	28.8905i	0.877893	−	1.52055i
$362$	7.89735	+	4.55954i	0.415075	+	0.239644i
$363$		0			0
$364$	−1.59158	−	3.23525i	−0.0834216	−	0.169573i
$365$	17.5552			0.918882
$366$		0			0
$367$	−0.519540	+	0.899869i	−0.0271198	+	0.0469728i	−0.879267	−	0.476330i	$-0.841967\pi$
$367$	−0.519540	+	0.899869i	−0.0271198	+	0.0469728i	0.852147	+	0.523302i	$0.175300\pi$
$368$	0.833676	+	1.44397i	0.0434583	+	0.0752721i
$369$		0			0
$370$	0.0522183	−	0.0301482i	0.00271470	−	0.00156733i
$371$	−5.18233	+	2.99202i	−0.269053	+	0.155338i
$372$		0			0
$373$	−13.6562	−	23.6532i	−0.707092	−	1.22472i	−0.965931	−	0.258798i	$-0.916673\pi$
$373$	−13.6562	−	23.6532i	−0.707092	−	1.22472i	0.258840	−	0.965920i	$-0.416660\pi$
$374$	−3.19519	+	5.53423i	−0.165219	+	0.286168i
$375$		0			0
$376$	7.02636			0.362357
$377$	15.6184	−	7.68349i	0.804390	−	0.395720i
$378$		0			0
$379$	−1.91535	−	1.10583i	−0.0983850	−	0.0568026i	0.450000	−	0.893028i	$-0.351424\pi$
$379$	−1.91535	−	1.10583i	−0.0983850	−	0.0568026i	−0.548385	+	0.836226i	$0.684757\pi$
$380$	−5.65696	+	9.79815i	−0.290196	+	0.502634i
$381$		0			0
$382$		−	3.26137i		−	0.166866i
$383$	−19.3458	+	11.1693i	−0.988522	+	0.570724i	−0.904832	−	0.425768i	$-0.860004\pi$
$383$	−19.3458	+	11.1693i	−0.988522	+	0.570724i	−0.0836900	+	0.996492i	$0.526671\pi$
$384$		0			0
$385$			4.48137i			0.228392i
$386$	11.0365	+	19.1158i	0.561744	+	0.972969i
$387$		0			0
$388$	−4.23338	−	2.44414i	−0.214917	−	0.124083i
$389$	12.2604			0.621629			0.310814	−	0.950471i	$-0.399398\pi$
$389$	12.2604			0.621629			0.310814	+	0.950471i	$0.399398\pi$
$390$		0			0
$391$	3.71755			0.188004
$392$	0.866025	+	0.500000i	0.0437409	+	0.0252538i
$393$		0			0
$394$	2.63653	+	4.56660i	0.132826	+	0.230062i
$395$		−	6.66228i		−	0.335216i
$396$		0			0
$397$	2.54193	−	1.46759i	0.127576	−	0.0736561i	−0.434854	−	0.900501i	$-0.643200\pi$
$397$	2.54193	−	1.46759i	0.127576	−	0.0736561i	0.562430	+	0.826845i	$0.309867\pi$
$398$			8.86762i			0.444493i
$399$		0			0
$400$	−1.27764	+	2.21294i	−0.0638822	+	0.110647i
$401$	−18.9229	−	10.9251i	−0.944963	−	0.545575i	−0.0534502	−	0.998571i	$-0.517022\pi$
$401$	−18.9229	−	10.9251i	−0.944963	−	0.545575i	−0.891513	+	0.452996i	$0.850355\pi$
$402$		0			0
$403$	1.19953	−	1.79148i	0.0597526	−	0.0892402i
$404$	−7.36747			−0.366545
$405$		0			0
$406$	−2.41379	+	4.18080i	−0.119794	+	0.207490i
$407$	0.0552644	+	0.0957207i	0.00273935	+	0.00474470i
$408$		0			0
$409$	−6.39292	+	3.69095i	−0.316109	+	0.182506i	−0.649657	−	0.760227i	$-0.725088\pi$
$409$	−6.39292	+	3.69095i	−0.316109	+	0.182506i	0.333548	+	0.942733i	$0.391754\pi$
$410$	10.7713	−	6.21881i	0.531956	−	0.307125i
$411$		0			0
$412$	−2.89263	−	5.01017i	−0.142509	−	0.246834i
$413$	0.448103	−	0.776138i	0.0220497	−	0.0381912i
$414$		0			0
$415$	7.73693			0.379791
$416$	−3.59760	−	0.239275i	−0.176387	−	0.0117314i
$417$		0			0
$418$	−17.9609	−	10.3697i	−0.878495	−	0.507199i
$419$	−4.29137	+	7.43287i	−0.209647	+	0.363119i	−0.951603	−	0.307329i	$-0.900565\pi$
$419$	−4.29137	+	7.43287i	−0.209647	+	0.363119i	0.741956	+	0.670448i	$0.233898\pi$
$420$		0			0
$421$			7.49525i			0.365296i	0.983178	+	0.182648i	$0.0584669\pi$
$421$			7.49525i			0.365296i	−0.983178	+	0.182648i	$0.941533\pi$
$422$	−5.68898	+	3.28453i	−0.276935	+	0.159889i
$423$		0			0
$424$			5.98404i			0.290611i
$425$	2.84865	+	4.93401i	0.138180	+	0.239334i
$426$		0			0
$427$	−12.3469	−	7.12846i	−0.597507	−	0.344971i
$428$	−1.02896			−0.0497365
$429$		0			0
$430$	15.7785			0.760907
$431$	14.2713	+	8.23956i	0.687426	+	0.396886i	0.802647	−	0.596454i	$-0.203424\pi$
$431$	14.2713	+	8.23956i	0.687426	+	0.396886i	−0.115221	+	0.993340i	$0.536758\pi$
$432$		0			0
$433$	−12.7805	−	22.1365i	−0.614192	−	1.06381i	−0.990526	−	0.137328i	$-0.956149\pi$
$433$	−12.7805	−	22.1365i	−0.614192	−	1.06381i	0.376333	−	0.926484i	$-0.377185\pi$
$434$			0.597963i			0.0287031i
$435$		0			0
$436$	12.2573	−	7.07674i	0.587017	−	0.338914i
$437$			12.0650i			0.577146i
$438$		0			0
$439$	4.60420	−	7.97470i	0.219746	−	0.380612i	−0.734984	−	0.678084i	$-0.762810\pi$
$439$	4.60420	−	7.97470i	0.219746	−	0.380612i	0.954730	+	0.297473i	$0.0961437\pi$
$440$	3.88098	+	2.24069i	0.185019	+	0.106821i
$441$		0			0
$442$	−4.47264	+	6.67987i	−0.212742	+	0.317729i
$443$	6.20759			0.294931			0.147466	−	0.989067i	$-0.452888\pi$
$443$	6.20759			0.294931			0.147466	+	0.989067i	$0.452888\pi$
$444$		0			0
$445$	−1.89284	+	3.27850i	−0.0897294	+	0.155416i
$446$	−8.45606	−	14.6463i	−0.400406	−	0.693523i
$447$		0			0
$448$	0.866025	−	0.500000i	0.0409159	−	0.0236228i
$449$	−4.51968	+	2.60944i	−0.213297	+	0.123147i	−0.602843	−	0.797860i	$-0.705965\pi$
$449$	−4.51968	+	2.60944i	−0.213297	+	0.123147i	0.389546	+	0.921007i	$0.372632\pi$
$450$		0			0
$451$	11.3996	+	19.7447i	0.536787	+	0.929743i
$452$	6.77051	−	11.7269i	0.318458	−	0.551586i
$453$		0			0
$454$	16.1322			0.757123
$455$	−0.374120	+	5.62506i	−0.0175390	+	0.263707i
$456$		0			0
$457$	−29.3870	−	16.9666i	−1.37467	−	0.793664i	−0.383155	−	0.923684i	$-0.625162\pi$
$457$	−29.3870	−	16.9666i	−1.37467	−	0.793664i	−0.991511	+	0.130021i	$0.958496\pi$
$458$	3.45592	−	5.98583i	0.161484	−	0.279699i
$459$		0			0
$460$		−	2.60700i		−	0.121552i
$461$	−0.731583	+	0.422380i	−0.0340732	+	0.0196722i	−0.516940	−	0.856022i	$-0.672929\pi$
$461$	−0.731583	+	0.422380i	−0.0340732	+	0.0196722i	0.482867	+	0.875694i	$0.339596\pi$
$462$		0			0
$463$		−	6.50221i		−	0.302183i	−0.988520	−	0.151092i	$-0.951721\pi$
$463$		−	6.50221i		−	0.302183i	0.988520	−	0.151092i	$-0.0482789\pi$
$464$	2.41379	+	4.18080i	0.112057	+	0.194089i
$465$		0			0
$466$	−6.47272	−	3.73702i	−0.299843	−	0.173114i
$467$	−9.52759			−0.440884			−0.220442	−	0.975400i	$-0.570750\pi$
$467$	−9.52759			−0.440884			−0.220442	+	0.975400i	$0.570750\pi$
$468$		0			0
$469$	1.64086			0.0757681
$470$	−9.51426	−	5.49306i	−0.438860	−	0.253376i
$471$		0			0
$472$	−0.448103	−	0.776138i	−0.0206256	−	0.0357246i
$473$			28.9234i			1.32990i
$474$		0			0
$475$	−16.0129	+	9.24505i	−0.734722	+	0.424192i
$476$		−	2.22961i		−	0.102194i
$477$		0			0
$478$	−9.93478	+	17.2075i	−0.454406	+	0.787055i
$479$	2.46123	+	1.42099i	0.112457	+	0.0649268i	0.555173	−	0.831735i	$-0.312652\pi$
$479$	2.46123	+	1.42099i	0.112457	+	0.0649268i	−0.442717	+	0.896662i	$0.645985\pi$
$480$		0			0
$481$	0.0613773	+	0.124763i	0.00279856	+	0.00568871i
$482$	−10.6445			−0.484844
$483$		0			0
$484$	1.39263	−	2.41210i	0.0633011	−	0.109641i
$485$	3.82156	+	6.61913i	0.173528	+	0.300559i
$486$		0			0
$487$	−4.55853	+	2.63187i	−0.206567	+	0.119261i	−0.599715	−	0.800214i	$-0.704719\pi$
$487$	−4.55853	+	2.63187i	−0.206567	+	0.119261i	0.393148	+	0.919475i	$0.371386\pi$
$488$	−12.3469	+	7.12846i	−0.558916	+	0.322690i
$489$		0			0
$490$	−0.781779	−	1.35408i	−0.0353172	−	0.0611711i
$491$	−11.4457	+	19.8245i	−0.516536	+	0.894666i	0.483280	+	0.875466i	$0.339445\pi$
$491$	−11.4457	+	19.8245i	−0.516536	+	0.894666i	−0.999816	+	0.0192004i	$0.993888\pi$
$492$		0			0
$493$	10.7636			0.484769
$494$	−21.6789	−	14.5156i	−0.975382	−	0.653087i
$495$		0			0
$496$	0.517851	+	0.298982i	0.0232522	+	0.0134247i
$497$	−1.14733	+	1.98724i	−0.0514648	+	0.0891397i
$498$		0			0
$499$			6.79877i			0.304355i	0.988353	+	0.152177i	$0.0486285\pi$
$499$			6.79877i			0.304355i	−0.988353	+	0.152177i	$0.951372\pi$
$500$	10.2305	−	5.90656i	0.457520	−	0.264150i
$501$		0			0
$502$		−	15.9139i		−	0.710273i
$503$	5.40300	+	9.35827i	0.240908	+	0.417265i	0.960973	−	0.276642i	$-0.0892215\pi$
$503$	5.40300	+	9.35827i	0.240908	+	0.417265i	−0.720065	+	0.693906i	$0.755888\pi$
$504$		0			0
$505$	9.97615	+	5.75973i	0.443933	+	0.256305i
$506$	4.77886			0.212446
$507$		0			0
$508$	9.85165			0.437096
$509$	23.6593	+	13.6597i	1.04868	+	0.605455i	0.922279	−	0.386524i	$-0.126324\pi$
$509$	23.6593	+	13.6597i	1.04868	+	0.605455i	0.126400	+	0.991979i	$0.459658\pi$
$510$		0			0
$511$	−5.61387	−	9.72351i	−0.248343	−	0.430143i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	26.9350	−	15.5509i	1.18805	−	0.685921i
$515$			9.04557i			0.398595i
$516$		0			0
$517$	10.0693	−	17.4405i	0.442846	−	0.767031i
$518$	−0.0333971	−	0.0192818i	−0.00146738	−	0.000847194i
$519$		0			0
$520$	4.68438	+	3.13653i	0.205424	+	0.137546i
$521$	3.63580			0.159287			0.0796437	−	0.996823i	$-0.474622\pi$
$521$	3.63580			0.159287			0.0796437	+	0.996823i	$0.474622\pi$
$522$		0			0
$523$	3.59223	−	6.22193i	0.157077	−	0.272066i	−0.776736	−	0.629826i	$-0.783126\pi$
$523$	3.59223	−	6.22193i	0.157077	−	0.272066i	0.933813	+	0.357760i	$0.116459\pi$
$524$	−7.39614	−	12.8105i	−0.323102	−	0.559629i
$525$		0			0
$526$	−24.5732	+	14.1873i	−1.07144	+	0.618597i
$527$	1.15461	−	0.666613i	0.0502955	−	0.0290381i
$528$		0			0
$529$	10.1100	+	17.5110i	0.439564	+	0.761347i
$530$	4.67819	−	8.10287i	0.203208	−	0.351966i
$531$		0			0
$532$	7.23602			0.313721
$533$	12.6606	+	25.7354i	0.548389	+	1.11473i
$534$		0			0
$535$	1.39329	+	0.804416i	0.0602371	+	0.0347779i
$536$	0.820432	−	1.42103i	0.0354373	−	0.0613792i
$537$		0			0
$538$		−	21.4017i		−	0.922691i
$539$	2.48215	−	1.43307i	0.106914	−	0.0617267i
$540$		0			0
$541$			0.445063i			0.0191347i	0.999954	+	0.00956737i	$0.00304543\pi$
$541$			0.445063i			0.0191347i	−0.999954	+	0.00956737i	$0.996955\pi$
$542$	2.87328	+	4.97667i	0.123418	+	0.213766i
$543$		0			0
$544$	−1.93090	−	1.11481i	−0.0827867	−	0.0477969i
$545$	−22.1298			−0.947935
$546$		0			0
$547$	−5.67129			−0.242487			−0.121243	−	0.992623i	$-0.538688\pi$
$547$	−5.67129			−0.242487			−0.121243	+	0.992623i	$0.538688\pi$
$548$	−0.397503	−	0.229499i	−0.0169805	−	0.00980370i
$549$		0			0
$550$	3.66190	+	6.34260i	0.156144	+	0.270450i
$551$			34.9324i			1.48817i
$552$		0			0
$553$	−3.69011	+	2.13049i	−0.156920	+	0.0905976i
$554$			26.6020i			1.13021i
$555$		0			0
$556$	7.65731	−	13.2628i	0.324742	−	0.562470i
$557$	−22.9561	−	13.2537i	−0.972683	−	0.561579i	−0.0726298	−	0.997359i	$-0.523139\pi$
$557$	−22.9561	−	13.2537i	−0.972683	−	0.561579i	−0.900053	+	0.435780i	$0.856472\pi$
$558$		0			0
$559$	−2.41462	+	36.3049i	−0.102128	+	1.53553i
$560$	−1.56356			−0.0660724
$561$		0			0
$562$	−3.34571	+	5.79495i	−0.141130	+	0.244445i
$563$	5.76880	+	9.99186i	0.243126	+	0.421107i	0.961603	−	0.274444i	$-0.0884938\pi$
$563$	5.76880	+	9.99186i	0.243126	+	0.421107i	−0.718477	+	0.695551i	$0.755160\pi$
$564$		0			0
$565$	−18.3356	+	10.5861i	−0.771386	+	0.445360i
$566$	17.2632	−	9.96692i	0.725627	−	0.418941i
$567$		0			0
$568$	1.14733	+	1.98724i	0.0481409	+	0.0833826i
$569$	16.0791	−	27.8497i	0.674069	−	1.16752i	−0.302671	−	0.953095i	$-0.597878\pi$
$569$	16.0791	−	27.8497i	0.674069	−	1.16752i	0.976740	−	0.214427i	$-0.0687883\pi$
$570$		0			0
$571$	−11.5540			−0.483521			−0.241761	−	0.970336i	$-0.577725\pi$
$571$	−11.5540			−0.483521			−0.241761	+	0.970336i	$0.577725\pi$
$572$	−5.74953	+	8.58689i	−0.240400	+	0.359036i
$573$		0			0
$574$	−6.88896	−	3.97734i	−0.287540	−	0.166011i
$575$	2.13028	−	3.68976i	0.0888389	−	0.153873i
$576$		0			0
$577$		−	9.14050i		−	0.380524i	−0.981733	−	0.190262i	$-0.939066\pi$
$577$		−	9.14050i		−	0.380524i	0.981733	−	0.190262i	$-0.0609337\pi$
$578$	10.4173	−	6.01442i	0.433301	−	0.250167i
$579$		0			0
$580$		−	7.54819i		−	0.313422i
$581$	−2.47414	−	4.28534i	−0.102645	−	0.177786i
$582$		0			0
$583$	14.8533	+	8.57554i	0.615160	+	0.355163i
$584$	−11.2277			−0.464607
$585$		0			0
$586$	8.86088			0.366040
$587$	−37.0629	−	21.3983i	−1.52975	−	0.883201i	−0.999372	−	0.0354398i	$-0.988717\pi$
$587$	−37.0629	−	21.3983i	−1.52975	−	0.883201i	−0.530378	−	0.847761i	$-0.677950\pi$
$588$		0			0
$589$	2.16344	+	3.74718i	0.0891428	+	0.154400i
$590$			1.40127i			0.0576894i
$591$		0			0
$592$	−0.0333971	+	0.0192818i	−0.00137261	+	0.000792478i
$593$		−	14.2439i		−	0.584927i	−0.956277	−	0.292463i	$-0.905525\pi$
$593$		−	14.2439i		−	0.584927i	0.956277	−	0.292463i	$-0.0944750\pi$
$594$		0			0
$595$	−1.74306	+	3.01907i	−0.0714586	+	0.123770i
$596$	−8.86563	−	5.11858i	−0.363150	−	0.209665i
$597$		0			0
$598$	5.99847	+	0.398955i	0.245296	+	0.0163145i
$599$	−3.74735			−0.153113			−0.0765563	−	0.997065i	$-0.524392\pi$
$599$	−3.74735			−0.153113			−0.0765563	+	0.997065i	$0.524392\pi$
$600$		0			0
$601$	−5.33462	+	9.23984i	−0.217604	+	0.376901i	−0.954075	−	0.299568i	$-0.903157\pi$
$601$	−5.33462	+	9.23984i	−0.217604	+	0.376901i	0.736471	+	0.676469i	$0.236491\pi$
$602$	−5.04571	−	8.73942i	−0.205648	−	0.356192i
$603$		0			0
$604$	13.1731	−	7.60551i	0.536007	−	0.309464i
$605$	−3.77145	+	2.17745i	−0.153331	+	0.0885259i
$606$		0			0
$607$	−4.82628	−	8.35936i	−0.195893	−	0.339296i	0.751300	−	0.659961i	$-0.229427\pi$
$607$	−4.82628	−	8.35936i	−0.195893	−	0.339296i	−0.947193	+	0.320665i	$0.896094\pi$
$608$	3.61801	−	6.26657i	0.146730	−	0.254143i
$609$		0			0
$610$	22.2915			0.902558
$611$	14.0950	−	21.0508i	0.570223	−	0.851625i
$612$		0			0
$613$	−3.45968	−	1.99745i	−0.139735	−	0.0806761i	0.428503	−	0.903541i	$-0.359041\pi$
$613$	−3.45968	−	1.99745i	−0.139735	−	0.0806761i	−0.568238	+	0.822864i	$0.692375\pi$
$614$	−4.17318	+	7.22816i	−0.168416	+	0.291705i
$615$		0			0
$616$		−	2.86614i		−	0.115480i
$617$	−2.80199	+	1.61773i	−0.112804	+	0.0651273i	−0.555340	−	0.831623i	$-0.687412\pi$
$617$	−2.80199	+	1.61773i	−0.112804	+	0.0651273i	0.442537	+	0.896750i	$0.354079\pi$
$618$		0			0
$619$			39.2679i			1.57831i	0.614193	+	0.789156i	$0.289482\pi$
$619$			39.2679i			1.57831i	−0.614193	+	0.789156i	$0.710518\pi$
$620$	−0.467475	−	0.809690i	−0.0187742	−	0.0325179i
$621$		0			0
$622$	5.79281	+	3.34448i	0.232270	+	0.134101i
$623$	2.42120			0.0970033
$624$		0			0
$625$	−5.69406			−0.227763
$626$	−18.5145	−	10.6894i	−0.739990	−	0.427233i
$627$		0			0
$628$	−1.13709	−	1.96950i	−0.0453750	−	0.0785918i
$629$			0.0859819i			0.00342832i
$630$		0			0
$631$	25.4983	−	14.7215i	1.01507	−	0.586052i	0.102400	−	0.994743i	$-0.467348\pi$
$631$	25.4983	−	14.7215i	1.01507	−	0.586052i	0.912673	+	0.408691i	$0.134015\pi$
$632$			4.26098i			0.169493i
$633$		0			0
$634$	15.8388	−	27.4336i	0.629040	−	1.08953i
$635$	−13.3399	−	7.70181i	−0.529379	−	0.305637i
$636$		0			0
$637$	3.23525	−	1.59158i	0.128185	−	0.0630608i
$638$	13.8365			0.547792
$639$		0			0
$640$	−0.781779	+	1.35408i	−0.0309025	+	0.0535247i
$641$	−2.04559	−	3.54307i	−0.0807961	−	0.139943i	0.822796	−	0.568337i	$-0.192413\pi$
$641$	−2.04559	−	3.54307i	−0.0807961	−	0.139943i	−0.903592	+	0.428394i	$0.859080\pi$
$642$		0			0
$643$	19.2672	−	11.1239i	0.759825	−	0.438685i	−0.0694080	−	0.997588i	$-0.522111\pi$
$643$	19.2672	−	11.1239i	0.759825	−	0.438685i	0.829233	+	0.558903i	$0.188778\pi$
$644$	−1.44397	+	0.833676i	−0.0569003	+	0.0328514i
$645$		0			0
$646$	−8.06675	−	13.9720i	−0.317382	−	0.549722i
$647$	24.9292	−	43.1786i	0.980066	−	1.69752i	0.317980	−	0.948097i	$-0.396996\pi$
$647$	24.9292	−	43.1786i	0.980066	−	1.69752i	0.662086	−	0.749427i	$-0.269671\pi$
$648$		0			0
$649$	−2.56865			−0.100828
$650$	4.06695	+	8.26700i	0.159519	+	0.324259i
$651$		0			0
$652$	0.00848066	+	0.00489631i	0.000332128	+	0.000191754i
$653$	3.70177	−	6.41165i	0.144861	−	0.250907i	−0.784460	−	0.620180i	$-0.787060\pi$
$653$	3.70177	−	6.41165i	0.144861	−	0.250907i	0.929321	+	0.369272i	$0.120393\pi$
$654$		0			0
$655$			23.1286i			0.903709i
$656$	−6.88896	+	3.97734i	−0.268969	+	0.155289i
$657$		0			0
$658$			7.02636i			0.273916i
$659$	15.0410	+	26.0518i	0.585914	+	1.01483i	0.994761	+	0.102230i	$0.0325977\pi$
$659$	15.0410	+	26.0518i	0.585914	+	1.01483i	−0.408847	+	0.912603i	$0.634069\pi$
$660$		0			0
$661$	23.0639	+	13.3160i	0.897082	+	0.517931i	0.876252	−	0.481852i	$-0.160036\pi$
$661$	23.0639	+	13.3160i	0.897082	+	0.517931i	0.0208300	+	0.999783i	$0.493369\pi$
$662$	−24.6695			−0.958807
$663$		0			0
$664$	−4.94829			−0.192031
$665$	−9.79815	−	5.65696i	−0.379956	−	0.219368i
$666$		0			0
$667$	−4.02463	−	6.97087i	−0.155834	−	0.269913i
$668$		−	24.9508i		−	0.965376i
$669$		0			0
$670$	−2.22186	+	1.28279i	−0.0858380	+	0.0495586i
$671$			40.8623i			1.57747i
$672$		0			0
$673$	−1.84652	+	3.19827i	−0.0711783	+	0.123284i	−0.899418	−	0.437090i	$-0.856009\pi$
$673$	−1.84652	+	3.19827i	−0.0711783	+	0.123284i	0.828240	+	0.560374i	$0.189343\pi$
$674$	24.3242	+	14.0436i	0.936932	+	0.540938i
$675$		0			0
$676$	−7.93372	+	10.2984i	−0.305143	+	0.396090i
$677$	35.6533			1.37027			0.685134	−	0.728417i	$-0.259744\pi$
$677$	35.6533			1.37027			0.685134	+	0.728417i	$0.259744\pi$
$678$		0			0
$679$	2.44414	−	4.23338i	0.0937976	−	0.162462i
$680$	1.74306	+	3.01907i	0.0668434	+	0.115776i
$681$		0			0
$682$	1.48423	−	0.856923i	0.0568342	−	0.0328133i
$683$	26.2105	−	15.1326i	1.00292	−	0.579034i	0.0938062	−	0.995590i	$-0.470097\pi$
$683$	26.2105	−	15.1326i	1.00292	−	0.579034i	0.909110	+	0.416557i	$0.136763\pi$
$684$		0			0
$685$	0.358834	+	0.621519i	0.0137104	+	0.0237470i
$686$	−0.500000	+	0.866025i	−0.0190901	+	0.0330650i
$687$		0			0
$688$	−10.0914			−0.384731
$689$	17.9280	+	12.0041i	0.683004	+	0.457320i
$690$		0			0
$691$	3.03377	+	1.75155i	0.115410	+	0.0666320i	0.556594	−	0.830785i	$-0.312108\pi$
$691$	3.03377	+	1.75155i	0.115410	+	0.0666320i	−0.441184	+	0.897417i	$0.645441\pi$
$692$	−1.60275	+	2.77604i	−0.0609273	+	0.105529i
$693$		0			0
$694$		−	10.1080i		−	0.383693i
$695$	−20.7372	+	11.9726i	−0.786608	+	0.454148i
$696$		0			0
$697$			17.7359i			0.671794i
$698$	−8.69465	−	15.0596i	−0.329097	−	0.570013i
$699$		0			0
$700$	−2.21294	−	1.27764i	−0.0836414	−	0.0482904i
$701$	45.2243			1.70810			0.854048	−	0.520194i	$-0.174140\pi$
$701$	45.2243			1.70810			0.854048	+	0.520194i	$0.174140\pi$
$702$		0			0
$703$	−0.279047			−0.0105245
$704$	−2.48215	−	1.43307i	−0.0935495	−	0.0540108i
$705$		0			0
$706$	−1.66329	−	2.88091i	−0.0625989	−	0.108424i
$707$		−	7.36747i		−	0.277082i
$708$		0			0
$709$	2.59657	−	1.49913i	0.0975161	−	0.0563009i	−0.450449	−	0.892802i	$-0.648736\pi$
$709$	2.59657	−	1.49913i	0.0975161	−	0.0563009i	0.547965	+	0.836501i	$0.315403\pi$
$710$		−	3.58784i		−	0.134649i
$711$		0			0
$712$	1.21060	−	2.09682i	0.0453692	−	0.0785817i
$713$	−0.863440	−	0.498507i	−0.0323361	−	0.0186692i
$714$		0			0
$715$	14.4984	−	7.13248i	0.542208	−	0.266740i
$716$	15.8000			0.590472
$717$		0			0
$718$	−4.01207	+	6.94911i	−0.149729	+	0.259339i
$719$	10.0397	+	17.3892i	0.374417	+	0.648509i	0.990240	−	0.139376i	$-0.0445096\pi$
$719$	10.0397	+	17.3892i	0.374417	+	0.648509i	−0.615823	+	0.787885i	$0.711176\pi$
$720$		0			0
$721$	5.01017	−	2.89263i	0.186589	−	0.107727i
$722$	28.8905	−	16.6800i	1.07519	−	0.620764i
$723$		0			0
$724$	4.55954	+	7.89735i	0.169454	+	0.293503i
$725$	6.16792	−	10.6832i	0.229071	−	0.396762i
$726$		0			0
$727$	−32.5895			−1.20868			−0.604338	−	0.796728i	$-0.706562\pi$
$727$	−32.5895			−1.20868			−0.604338	+	0.796728i	$0.706562\pi$
$728$	0.239275	−	3.59760i	0.00886811	−	0.133336i
$729$		0			0
$730$	15.2033	+	8.77761i	0.562698	+	0.324874i
$731$	−11.2500	+	19.4855i	−0.416095	+	0.720698i
$732$		0			0
$733$		−	18.5190i		−	0.684017i	−0.939697	−	0.342008i	$-0.888893\pi$
$733$		−	18.5190i		−	0.684017i	0.939697	−	0.342008i	$-0.111107\pi$
$734$	−0.899869	+	0.519540i	−0.0332148	+	0.0191766i
$735$		0			0
$736$			1.66735i			0.0614594i
$737$	−2.35147	−	4.07287i	−0.0866176	−	0.150026i
$738$		0			0
$739$	13.7968	+	7.96559i	0.507524	+	0.293019i	0.731815	−	0.681503i	$-0.238673\pi$
$739$	13.7968	+	7.96559i	0.507524	+	0.293019i	−0.224291	+	0.974522i	$0.572007\pi$
$740$	0.0602965			0.00221654
$741$		0			0
$742$	−5.98404			−0.219681
$743$	−10.5962	−	6.11773i	−0.388738	−	0.224438i	0.292875	−	0.956151i	$-0.405388\pi$
$743$	−10.5962	−	6.11773i	−0.388738	−	0.224438i	−0.681613	+	0.731713i	$0.738721\pi$
$744$		0			0
$745$	8.00319	+	13.8619i	0.293214	+	0.507862i
$746$		−	27.3124i		−	0.999979i
$747$		0			0
$748$	−5.53423	+	3.19519i	−0.202351	+	0.116828i
$749$		−	1.02896i		−	0.0375972i
$750$		0			0
$751$	−10.4107	+	18.0318i	−0.379891	+	0.657990i	−0.991046	−	0.133521i	$-0.957372\pi$
$751$	−10.4107	+	18.0318i	−0.379891	+	0.657990i	0.611155	+	0.791511i	$0.290705\pi$
$752$	6.08501	+	3.51318i	0.221897	+	0.128113i
$753$		0			0
$754$	17.3677	+	1.15512i	0.632494	+	0.0420669i
$755$	−23.7833			−0.865563
$756$		0			0
$757$	−13.5575	+	23.4823i	−0.492757	+	0.853480i	−0.999965	−	0.00834344i	$-0.997344\pi$
$757$	−13.5575	+	23.4823i	−0.492757	+	0.853480i	0.507208	+	0.861824i	$0.330678\pi$
$758$	−1.10583	−	1.91535i	−0.0401655	−	0.0695687i
$759$		0			0
$760$	−9.79815	+	5.65696i	−0.355416	+	0.205200i
$761$	−41.8920	+	24.1864i	−1.51858	+	0.876755i	−0.518824	+	0.854881i	$0.673630\pi$
$761$	−41.8920	+	24.1864i	−1.51858	+	0.876755i	−0.999761	+	0.0218739i	$0.993037\pi$
$762$		0			0
$763$	7.07674	+	12.2573i	0.256195	+	0.443743i
$764$	1.63068	−	2.82443i	0.0589961	−	0.102184i
$765$		0			0
$766$	−22.3386			−0.807125
$767$	−3.22419	−	0.214439i	−0.116419	−	0.00774296i
$768$		0			0
$769$	−15.0214	−	8.67264i	−0.541687	−	0.312743i	0.204075	−	0.978955i	$-0.434581\pi$
$769$	−15.0214	−	8.67264i	−0.541687	−	0.312743i	−0.745762	+	0.666212i	$0.767915\pi$
$770$	−2.24069	+	3.88098i	−0.0807487	+	0.139861i
$771$		0			0
$772$			22.0730i			0.794426i
$773$	12.1659	−	7.02398i	0.437576	−	0.252635i	−0.264993	−	0.964250i	$-0.585370\pi$
$773$	12.1659	−	7.02398i	0.437576	−	0.252635i	0.702569	+	0.711616i	$0.252036\pi$
$774$		0			0
$775$		−	1.52797i		−	0.0548862i
$776$	−2.44414	−	4.23338i	−0.0877396	−	0.151970i
$777$		0			0
$778$	10.6178	+	6.13022i	0.380668	+	0.219779i
$779$	−57.5603			−2.06231
$780$		0			0
$781$	6.57682			0.235337
$782$

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.bj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.56356i q^{5} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.56356i q^{5} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.781779 - 1.35408i) q^{10} +(2.48215 + 1.43307i) q^{11} +(2.99598 + 2.00602i) q^{13} -1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.11481 + 1.93090i) q^{17} +(-6.26657 + 3.61801i) q^{19} +(1.35408 - 0.781779i) q^{20} +(1.43307 + 2.48215i) q^{22} +(0.833676 - 1.44397i) q^{23} +2.55529 q^{25} +(1.59158 + 3.23525i) q^{26} +(-0.866025 - 0.500000i) q^{28} +(2.41379 - 4.18080i) q^{29} -0.597963i q^{31} +(-0.866025 + 0.500000i) q^{32} +2.22961i q^{34} +(0.781779 + 1.35408i) q^{35} +(0.0333971 + 0.0192818i) q^{37} -7.23602 q^{38} +1.56356 q^{40} +(6.88896 + 3.97734i) q^{41} +(5.04571 + 8.73942i) q^{43} +2.86614i q^{44} +(1.44397 - 0.833676i) q^{46} -7.02636i q^{47} +(0.500000 - 0.866025i) q^{49} +(2.21294 + 1.27764i) q^{50} +(-0.239275 + 3.59760i) q^{52} +5.98404 q^{53} +(2.24069 - 3.88098i) q^{55} +(-0.500000 - 0.866025i) q^{56} +(4.18080 - 2.41379i) q^{58} +(-0.776138 + 0.448103i) q^{59} +(7.12846 + 12.3469i) q^{61} +(0.298982 - 0.517851i) q^{62} -1.00000 q^{64} +(3.13653 - 4.68438i) q^{65} +(-1.42103 - 0.820432i) q^{67} +(-1.11481 + 1.93090i) q^{68} +1.56356i q^{70} +(1.98724 - 1.14733i) q^{71} +11.2277i q^{73} +(0.0192818 + 0.0333971i) q^{74} +(-6.26657 - 3.61801i) q^{76} -2.86614 q^{77} +4.26098 q^{79} +(1.35408 + 0.781779i) q^{80} +(3.97734 + 6.88896i) q^{82} +4.94829i q^{83} +(3.01907 - 1.74306i) q^{85} +10.0914i q^{86} +(-1.43307 + 2.48215i) q^{88} +(-2.09682 - 1.21060i) q^{89} +(-3.59760 - 0.239275i) q^{91} +1.66735 q^{92} +(3.51318 - 6.08501i) q^{94} +(5.65696 + 9.79815i) q^{95} +(-4.23338 + 2.44414i) q^{97} +(0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{4}+O(q^{10}) \) 12 * q + 6 * q^4	\( 12 q + 6 q^{4} - 2 q^{10} + 18 q^{11} - 8 q^{13} - 12 q^{14} - 6 q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} + 6 q^{23} - 24 q^{25} + 14 q^{26} + 10 q^{29} - 2 q^{35} - 6 q^{37} - 8 q^{38} - 4 q^{40} + 24 q^{41} + 26 q^{43} - 6 q^{46} + 6 q^{49} + 12 q^{50} - 4 q^{52} - 36 q^{53} - 6 q^{55} - 6 q^{56} + 24 q^{58} - 6 q^{59} - 28 q^{61} + 2 q^{62} - 12 q^{64} + 34 q^{65} - 42 q^{67} + 4 q^{68} - 48 q^{71} + 12 q^{76} + 4 q^{77} + 44 q^{79} + 6 q^{82} + 54 q^{85} + 2 q^{88} - 12 q^{89} - 16 q^{91} + 12 q^{92} + 8 q^{94} - 32 q^{95} + 60 q^{97}+O(q^{100}) \) 12 * q + 6 * q^4 - 2 * q^10 + 18 * q^11 - 8 * q^13 - 12 * q^14 - 6 * q^16 - 4 * q^17 + 12 * q^19 - 2 * q^22 + 6 * q^23 - 24 * q^25 + 14 * q^26 + 10 * q^29 - 2 * q^35 - 6 * q^37 - 8 * q^38 - 4 * q^40 + 24 * q^41 + 26 * q^43 - 6 * q^46 + 6 * q^49 + 12 * q^50 - 4 * q^52 - 36 * q^53 - 6 * q^55 - 6 * q^56 + 24 * q^58 - 6 * q^59 - 28 * q^61 + 2 * q^62 - 12 * q^64 + 34 * q^65 - 42 * q^67 + 4 * q^68 - 48 * q^71 + 12 * q^76 + 4 * q^77 + 44 * q^79 + 6 * q^82 + 54 * q^85 + 2 * q^88 - 12 * q^89 - 16 * q^91 + 12 * q^92 + 8 * q^94 - 32 * q^95 + 60 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more