LMFDB - Embedded newform 1386.2.g.b.881.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(881,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1386.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 $ x^16 - 12x^14 + 72x^12 + 1296x^10 + 8245x^8 + 18780x^6 + 20808x^4 - 3672*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.9
Root			$-0.808328 - 1.95148i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.b.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -3.90295 q^{5} +(1.49520 + 2.18274i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -3.90295 q^{5} +(1.49520 + 2.18274i) q^{7} -1.00000i q^{8} -3.90295i q^{10} +1.00000i q^{11} +6.17373i q^{13} +(-2.18274 + 1.49520i) q^{14} +1.00000 q^{16} -0.462536 q^{17} +1.61666i q^{19} +3.90295 q^{20} -1.00000 q^{22} +10.2330 q^{25} -6.17373 q^{26} +(-1.49520 - 2.18274i) q^{28} -7.80526i q^{29} -5.98215i q^{31} +1.00000i q^{32} -0.462536i q^{34} +(-5.83570 - 8.51915i) q^{35} -9.05749 q^{37} -1.61666 q^{38} +3.90295i q^{40} -0.191589 q^{41} -10.0671 q^{43} -1.00000i q^{44} -5.05707 q^{47} +(-2.52875 + 6.52728i) q^{49} +10.2330i q^{50} -6.17373i q^{52} -6.04790i q^{53} -3.90295i q^{55} +(2.18274 - 1.49520i) q^{56} +7.80526 q^{58} -4.54156 q^{59} -2.55724i q^{61} +5.98215 q^{62} -1.00000 q^{64} -24.0958i q^{65} -5.80526 q^{67} +0.462536 q^{68} +(8.51915 - 5.83570i) q^{70} +8.48528i q^{71} -10.8476i q^{73} -9.05749i q^{74} -1.61666i q^{76} +(-2.18274 + 1.49520i) q^{77} +16.2234 q^{79} -3.90295 q^{80} -0.191589i q^{82} -9.86958 q^{83} +1.80526 q^{85} -10.0671i q^{86} +1.00000 q^{88} -10.1560 q^{89} +(-13.4757 + 9.23097i) q^{91} -5.05707i q^{94} -6.30973i q^{95} -11.0392i q^{97} +(-6.52728 - 2.52875i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^22 + 48 * q^25 - 16 * q^37 - 80 * q^43 + 24 * q^49 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 24 * q^70 + 96 * q^79 - 80 * q^85 + 16 * q^88 - 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−3.90295			−1.74545			−0.872727	−	0.488209i	$-0.837650\pi$
$5$	−3.90295			−1.74545			−0.872727	+	0.488209i	$0.837650\pi$
$6$		0			0
$7$	1.49520	+	2.18274i	0.565133	+	0.825000i
$7$	1.49520	+	2.18274i	0.565133	+	0.825000i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	3.90295i		−	1.23422i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			6.17373i			1.71229i	0.516739	+	0.856143i	$0.327146\pi$
$13$			6.17373i			1.71229i	−0.516739	+	0.856143i	$0.672854\pi$
$14$	−2.18274	+	1.49520i	−0.583363	+	0.399609i
$15$		0			0
$16$	1.00000			0.250000
$17$	−0.462536			−0.112181			−0.0560907	−	0.998426i	$-0.517864\pi$
$17$	−0.462536			−0.112181			−0.0560907	+	0.998426i	$0.517864\pi$
$18$		0			0
$19$			1.61666i			0.370886i	0.982655	+	0.185443i	$0.0593721\pi$
$19$			1.61666i			0.370886i	−0.982655	+	0.185443i	$0.940628\pi$
$20$	3.90295			0.872727
$21$		0			0
$22$	−1.00000			−0.213201
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	10.2330			2.04661
$26$	−6.17373			−1.21077
$27$		0			0
$28$	−1.49520	−	2.18274i	−0.282566	−	0.412500i
$29$		−	7.80526i		−	1.44940i	−0.689065	−	0.724700i	$-0.741978\pi$
$29$		−	7.80526i		−	1.44940i	0.689065	−	0.724700i	$-0.258022\pi$
$30$		0			0
$31$		−	5.98215i		−	1.07442i	−0.843447	−	0.537212i	$-0.819477\pi$
$31$		−	5.98215i		−	1.07442i	0.843447	−	0.537212i	$-0.180523\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	0.462536i		−	0.0793243i
$35$	−5.83570	−	8.51915i	−0.986413	−	1.44000i
$36$		0			0
$37$	−9.05749			−1.48904			−0.744521	−	0.667599i	$-0.767322\pi$
$37$	−9.05749			−1.48904			−0.744521	+	0.667599i	$0.767322\pi$
$38$	−1.61666			−0.262256
$39$		0			0
$40$			3.90295i			0.617111i
$41$	−0.191589			−0.0299211			−0.0149606	−	0.999888i	$-0.504762\pi$
$41$	−0.191589			−0.0299211			−0.0149606	+	0.999888i	$0.504762\pi$
$42$		0			0
$43$	−10.0671			−1.53522			−0.767609	−	0.640919i	$-0.778553\pi$
$43$	−10.0671			−1.53522			−0.767609	+	0.640919i	$0.778553\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$		0			0
$47$	−5.05707			−0.737650			−0.368825	−	0.929499i	$-0.620240\pi$
$47$	−5.05707			−0.737650			−0.368825	+	0.929499i	$0.620240\pi$
$48$		0			0
$49$	−2.52875	+	6.52728i	−0.361250	+	0.932469i
$50$			10.2330i			1.44717i
$51$		0			0
$52$		−	6.17373i		−	0.856143i
$53$		−	6.04790i		−	0.830743i	−0.909652	−	0.415371i	$-0.863652\pi$
$53$		−	6.04790i		−	0.830743i	0.909652	−	0.415371i	$-0.136348\pi$
$54$		0			0
$55$		−	3.90295i		−	0.526274i
$56$	2.18274	−	1.49520i	0.291682	−	0.199805i
$57$		0			0
$58$	7.80526			1.02488
$59$	−4.54156			−0.591261			−0.295630	−	0.955302i	$-0.595530\pi$
$59$	−4.54156			−0.591261			−0.295630	+	0.955302i	$0.595530\pi$
$60$		0			0
$61$		−	2.55724i		−	0.327422i	−0.986508	−	0.163711i	$-0.947654\pi$
$61$		−	2.55724i		−	0.327422i	0.986508	−	0.163711i	$-0.0523464\pi$
$62$	5.98215			0.759733
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	24.0958i		−	2.98872i
$66$		0			0
$67$	−5.80526			−0.709225			−0.354612	−	0.935013i	$-0.615387\pi$
$67$	−5.80526			−0.709225			−0.354612	+	0.935013i	$0.615387\pi$
$68$	0.462536			0.0560907
$69$		0			0
$70$	8.51915	−	5.83570i	1.01823	−	0.697500i
$71$			8.48528i			1.00702i	0.863990	+	0.503509i	$0.167958\pi$
$71$			8.48528i			1.00702i	−0.863990	+	0.503509i	$0.832042\pi$
$72$		0			0
$73$		−	10.8476i		−	1.26962i	−0.772669	−	0.634810i	$-0.781079\pi$
$73$		−	10.8476i		−	1.26962i	0.772669	−	0.634810i	$-0.218921\pi$
$74$		−	9.05749i		−	1.05291i
$75$		0			0
$76$		−	1.61666i		−	0.185443i
$77$	−2.18274	+	1.49520i	−0.248747	+	0.170394i
$78$		0			0
$79$	16.2234			1.82528			0.912640	−	0.408764i	$-0.134040\pi$
$79$	16.2234			1.82528			0.912640	+	0.408764i	$0.134040\pi$
$80$	−3.90295			−0.436363
$81$		0			0
$82$		−	0.191589i		−	0.0211574i
$83$	−9.86958			−1.08333			−0.541664	−	0.840595i	$-0.682205\pi$
$83$	−9.86958			−1.08333			−0.541664	+	0.840595i	$0.682205\pi$
$84$		0			0
$85$	1.80526			0.195808
$86$		−	10.0671i		−	1.08556i
$87$		0			0
$88$	1.00000			0.106600
$89$	−10.1560			−1.07654			−0.538269	−	0.842773i	$-0.680922\pi$
$89$	−10.1560			−1.07654			−0.538269	+	0.842773i	$0.680922\pi$
$90$		0			0
$91$	−13.4757	+	9.23097i	−1.41264	+	0.967669i
$92$		0			0
$93$		0			0
$94$		−	5.05707i		−	0.521597i
$95$		−	6.30973i		−	0.647365i
$96$		0			0
$97$		−	11.0392i		−	1.12086i	−0.828201	−	0.560431i	$-0.810635\pi$
$97$		−	11.0392i		−	1.12086i	0.828201	−	0.560431i	$-0.189365\pi$
$98$	−6.52728	−	2.52875i	−0.659355	−	0.255442i
$99$		0			0
$100$	−10.2330			−1.02330
$101$	17.2130			1.71275			0.856376	−	0.516352i	$-0.172710\pi$
$101$	17.2130			1.71275			0.856376	+	0.516352i	$0.172710\pi$
$102$		0			0
$103$		−	12.1339i		−	1.19559i	−0.801648	−	0.597796i	$-0.796043\pi$
$103$		−	12.1339i		−	1.19559i	0.801648	−	0.597796i	$-0.203957\pi$
$104$	6.17373			0.605384
$105$		0			0
$106$	6.04790			0.587424
$107$			17.0383i			1.64715i	0.567204	+	0.823577i	$0.308025\pi$
$107$			17.0383i			1.64715i	−0.567204	+	0.823577i	$0.691975\pi$
$108$		0			0
$109$	14.6608			1.40425			0.702126	−	0.712052i	$-0.252234\pi$
$109$	14.6608			1.40425			0.702126	+	0.712052i	$0.252234\pi$
$110$	3.90295			0.372132
$111$		0			0
$112$	1.49520	+	2.18274i	0.141283	+	0.206250i
$113$			13.4757i			1.26769i	0.773462	+	0.633843i	$0.218523\pi$
$113$			13.4757i			1.26769i	−0.773462	+	0.633843i	$0.781477\pi$
$114$		0			0
$115$		0			0
$116$			7.80526i			0.724700i
$117$		0			0
$118$		−	4.54156i		−	0.418085i
$119$	−0.691584	−	1.00960i	−0.0633974	−	0.0925497i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	2.55724			0.231522
$123$		0			0
$124$			5.98215i			0.537212i
$125$	−20.4243			−1.82681
$126$		0			0
$127$	−3.73816			−0.331708			−0.165854	−	0.986150i	$-0.553038\pi$
$127$	−3.73816			−0.331708			−0.165854	+	0.986150i	$0.553038\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	24.0958			2.11334
$131$	−9.57669			−0.836719			−0.418360	−	0.908281i	$-0.637395\pi$
$131$	−9.57669			−0.836719			−0.418360	+	0.908281i	$0.637395\pi$
$132$		0			0
$133$	−3.52875	+	2.41723i	−0.305981	+	0.209600i
$134$		−	5.80526i		−	0.501498i
$135$		0			0
$136$			0.462536i			0.0396621i
$137$		−	0.418189i		−	0.0357283i	−0.999840	−	0.0178642i	$-0.994313\pi$
$137$		−	0.418189i		−	0.0357283i	0.999840	−	0.0178642i	$-0.00568664\pi$
$138$		0			0
$139$		−	15.5744i		−	1.32100i	−0.750826	−	0.660500i	$-0.770344\pi$
$139$		−	15.5744i		−	1.32100i	0.750826	−	0.660500i	$-0.229656\pi$
$140$	5.83570	+	8.51915i	0.493207	+	0.720000i
$141$		0			0
$142$	−8.48528			−0.712069
$143$	−6.17373			−0.516274
$144$		0			0
$145$			30.4635i			2.52986i
$146$	10.8476			0.897756
$147$		0			0
$148$	9.05749			0.744521
$149$			7.05749i			0.578172i	0.957303	+	0.289086i	$0.0933514\pi$
$149$			7.05749i			0.578172i	−0.957303	+	0.289086i	$0.906649\pi$
$150$		0			0
$151$	21.6776			1.76410			0.882049	−	0.471157i	$-0.156164\pi$
$151$	21.6776			1.76410			0.882049	+	0.471157i	$0.156164\pi$
$152$	1.61666			0.131128
$153$		0			0
$154$	−1.49520	−	2.18274i	−0.120487	−	0.175891i
$155$			23.3480i			1.87536i
$156$		0			0
$157$			5.98215i			0.477427i	0.971090	+	0.238714i	$0.0767257\pi$
$157$			5.98215i			0.477427i	−0.971090	+	0.238714i	$0.923274\pi$
$158$			16.2234i			1.29067i
$159$		0			0
$160$		−	3.90295i		−	0.308556i
$161$		0			0
$162$		0			0
$163$	−9.40859			−0.736938			−0.368469	−	0.929640i	$-0.620118\pi$
$163$	−9.40859			−0.736938			−0.368469	+	0.929640i	$0.620118\pi$
$164$	0.191589			0.0149606
$165$		0			0
$166$		−	9.86958i		−	0.766028i
$167$	−9.05490			−0.700689			−0.350345	−	0.936621i	$-0.613936\pi$
$167$	−9.05490			−0.700689			−0.350345	+	0.936621i	$0.613936\pi$
$168$		0			0
$169$	−25.1150			−1.93192
$170$			1.80526i			0.138457i
$171$		0			0
$172$	10.0671			0.767609
$173$	−5.24866			−0.399048			−0.199524	−	0.979893i	$-0.563940\pi$
$173$	−5.24866			−0.399048			−0.199524	+	0.979893i	$0.563940\pi$
$174$		0			0
$175$	15.3005	+	22.3361i	1.15661	+	1.68845i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$		−	10.1560i		−	0.761228i
$179$		−	4.66083i		−	0.348367i	−0.984713	−	0.174183i	$-0.944271\pi$
$179$		−	4.66083i		−	0.348367i	0.984713	−	0.174183i	$-0.0557286\pi$
$180$		0			0
$181$			8.53939i			0.634728i	0.948304	+	0.317364i	$0.102798\pi$
$181$			8.53939i			0.634728i	−0.948304	+	0.317364i	$0.897202\pi$
$182$	−9.23097	−	13.4757i	−0.684245	−	0.998884i
$183$		0			0
$184$		0			0
$185$	35.3510			2.59906
$186$		0			0
$187$		−	0.462536i		−	0.0338240i
$188$	5.05707			0.368825
$189$		0			0
$190$	6.30973			0.457756
$191$			8.48528i			0.613973i	0.951714	+	0.306987i	$0.0993207\pi$
$191$			8.48528i			0.613973i	−0.951714	+	0.306987i	$0.900679\pi$
$192$		0			0
$193$	−17.6105			−1.26763			−0.633816	−	0.773484i	$-0.718512\pi$
$193$	−17.6105			−1.26763			−0.633816	+	0.773484i	$0.718512\pi$
$194$	11.0392			0.792570
$195$		0			0
$196$	2.52875	−	6.52728i	0.180625	−	0.466235i
$197$		−	9.30078i		−	0.662653i	−0.943516	−	0.331327i	$-0.892504\pi$
$197$		−	9.30078i		−	0.662653i	0.943516	−	0.331327i	$-0.107496\pi$
$198$		0			0
$199$			11.4798i			0.813782i	0.913477	+	0.406891i	$0.133387\pi$
$199$			11.4798i			0.813782i	−0.913477	+	0.406891i	$0.866613\pi$
$200$		−	10.2330i		−	0.723585i
$201$		0			0
$202$			17.2130i			1.21110i
$203$	17.0369	−	11.6704i	1.19575	−	0.819103i
$204$		0			0
$205$	0.747762			0.0522259
$206$	12.1339			0.845411
$207$		0			0
$208$			6.17373i			0.428071i
$209$	−1.61666			−0.111826
$210$		0			0
$211$	−11.5626			−0.796003			−0.398002	−	0.917385i	$-0.630296\pi$
$211$	−11.5626			−0.796003			−0.398002	+	0.917385i	$0.630296\pi$
$212$			6.04790i			0.415371i
$213$		0			0
$214$	−17.0383			−1.16471
$215$	39.2914			2.67965
$216$		0			0
$217$	13.0575	−	8.94451i	0.886400	−	0.607193i
$218$			14.6608i			0.992957i
$219$		0			0
$220$			3.90295i			0.263137i
$221$		−	2.85557i		−	0.192087i
$222$		0			0
$223$		−	21.5629i		−	1.44396i	−0.691914	−	0.721980i	$-0.743232\pi$
$223$		−	21.5629i		−	1.44396i	0.691914	−	0.721980i	$-0.256768\pi$
$224$	−2.18274	+	1.49520i	−0.145841	+	0.0999023i
$225$		0			0
$226$	−13.4757			−0.896389
$227$	11.8630			0.787374			0.393687	−	0.919245i	$-0.371199\pi$
$227$	11.8630			0.787374			0.393687	+	0.919245i	$0.371199\pi$
$228$		0			0
$229$			6.28413i			0.415267i	0.978207	+	0.207633i	$0.0665761\pi$
$229$			6.28413i			0.415267i	−0.978207	+	0.207633i	$0.933424\pi$
$230$		0			0
$231$		0			0
$232$	−7.80526			−0.512440
$233$		−	22.4469i		−	1.47054i	−0.677772	−	0.735272i	$-0.737054\pi$
$233$		−	22.4469i		−	1.47054i	0.677772	−	0.735272i	$-0.262946\pi$
$234$		0			0
$235$	19.7375			1.28753
$236$	4.54156			0.295630
$237$		0			0
$238$	1.00960	−	0.691584i	0.0654425	−	0.0448287i
$239$			5.40859i			0.349853i	0.984582	+	0.174926i	$0.0559687\pi$
$239$			5.40859i			0.349853i	−0.984582	+	0.174926i	$0.944031\pi$
$240$		0			0
$241$			13.2681i			0.854673i	0.904093	+	0.427337i	$0.140548\pi$
$241$			13.2681i			0.854673i	−0.904093	+	0.427337i	$0.859452\pi$
$242$		−	1.00000i		−	0.0642824i
$243$		0			0
$244$			2.55724i			0.163711i
$245$	9.86958	−	25.4757i	0.630544	−	1.62758i
$246$		0			0
$247$	−9.98080			−0.635063
$248$	−5.98215			−0.379867
$249$		0			0
$250$		−	20.4243i		−	1.29175i
$251$	6.80592			0.429586			0.214793	−	0.976660i	$-0.431092\pi$
$251$	6.80592			0.429586			0.214793	+	0.976660i	$0.431092\pi$
$252$		0			0
$253$		0			0
$254$		−	3.73816i		−	0.234553i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−11.1560			−0.695894			−0.347947	−	0.937514i	$-0.613121\pi$
$257$	−11.1560			−0.695894			−0.347947	+	0.937514i	$0.613121\pi$
$258$		0			0
$259$	−13.5428	−	19.7702i	−0.841507	−	1.22846i
$260$			24.0958i			1.49436i
$261$		0			0
$262$		−	9.57669i		−	0.591650i
$263$			8.17555i			0.504126i	0.967711	+	0.252063i	$0.0811090\pi$
$263$			8.17555i			0.504126i	−0.967711	+	0.252063i	$0.918891\pi$
$264$		0			0
$265$			23.6047i			1.45002i
$266$	−2.41723	−	3.52875i	−0.148210	−	0.216361i
$267$		0			0
$268$	5.80526			0.354612
$269$	−13.2881			−0.810187			−0.405094	−	0.914275i	$-0.632761\pi$
$269$	−13.2881			−0.810187			−0.405094	+	0.914275i	$0.632761\pi$
$270$		0			0
$271$			24.1667i			1.46802i	0.679137	+	0.734012i	$0.262354\pi$
$271$			24.1667i			1.46802i	−0.679137	+	0.734012i	$0.737646\pi$
$272$	−0.462536			−0.0280454
$273$		0			0
$274$	0.418189			0.0252637
$275$			10.2330i			0.617076i
$276$		0			0
$277$	−13.9202			−0.836386			−0.418193	−	0.908358i	$-0.637337\pi$
$277$	−13.9202			−0.836386			−0.418193	+	0.908358i	$0.637337\pi$
$278$	15.5744			0.934088
$279$		0			0
$280$	−8.51915	+	5.83570i	−0.509117	+	0.348750i
$281$			10.1342i			0.604555i	0.953220	+	0.302277i	$0.0977469\pi$
$281$			10.1342i			0.604555i	−0.953220	+	0.302277i	$0.902253\pi$
$282$		0			0
$283$		−	18.2658i		−	1.08579i	−0.839801	−	0.542894i	$-0.817329\pi$
$283$		−	18.2658i		−	1.08579i	0.839801	−	0.542894i	$-0.182671\pi$
$284$		−	8.48528i		−	0.503509i
$285$		0			0
$286$		−	6.17373i		−	0.365061i
$287$	−0.286464	−	0.418189i	−0.0169094	−	0.0246849i
$288$		0			0
$289$	−16.7861			−0.987415
$290$	−30.4635			−1.78888
$291$		0			0
$292$			10.8476i			0.634810i
$293$	−29.5915			−1.72875			−0.864376	−	0.502847i	$-0.832286\pi$
$293$	−29.5915			−1.72875			−0.864376	+	0.502847i	$0.832286\pi$
$294$		0			0
$295$	17.7255			1.03202
$296$			9.05749i			0.526456i
$297$		0			0
$298$	−7.05749			−0.408830
$299$		0			0
$300$		0			0
$301$	−15.0523	−	21.9739i	−0.867602	−	1.26655i
$302$			21.6776i			1.24741i
$303$		0			0
$304$			1.61666i			0.0927216i
$305$			9.98080i			0.571499i
$306$		0			0
$307$		−	4.49782i		−	0.256704i	−0.991729	−	0.128352i	$-0.959031\pi$
$307$		−	4.49782i		−	0.256704i	0.991729	−	0.128352i	$-0.0409688\pi$
$308$	2.18274	−	1.49520i	0.124373	−	0.0851970i
$309$		0			0
$310$	−23.3480			−1.32608
$311$	13.4422			0.762236			0.381118	−	0.924526i	$-0.375539\pi$
$311$	13.4422			0.762236			0.381118	+	0.924526i	$0.375539\pi$
$312$		0			0
$313$		−	17.2440i		−	0.974688i	−0.873210	−	0.487344i	$-0.837966\pi$
$313$		−	17.2440i		−	0.974688i	0.873210	−	0.487344i	$-0.162034\pi$
$314$	−5.98215			−0.337592
$315$		0			0
$316$	−16.2234			−0.912640
$317$			21.6584i			1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$317$			21.6584i			1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$318$		0			0
$319$	7.80526			0.437010
$320$	3.90295			0.218182
$321$		0			0
$322$		0			0
$323$		−	0.747762i		−	0.0416066i
$324$		0			0
$325$			63.1761i			3.50438i
$326$		−	9.40859i		−	0.521094i
$327$		0			0
$328$			0.191589i			0.0105787i
$329$	−7.56134	−	11.0383i	−0.416870	−	0.608561i
$330$		0			0
$331$	−1.31997			−0.0725524			−0.0362762	−	0.999342i	$-0.511550\pi$
$331$	−1.31997			−0.0725524			−0.0362762	+	0.999342i	$0.511550\pi$
$332$	9.86958			0.541664
$333$		0			0
$334$		−	9.05490i		−	0.495462i
$335$	22.6576			1.23792
$336$		0			0
$337$	−26.0958			−1.42153			−0.710764	−	0.703430i	$-0.751651\pi$
$337$	−26.0958			−1.42153			−0.710764	+	0.703430i	$0.751651\pi$
$338$		−	25.1150i		−	1.36608i
$339$		0			0
$340$	−1.80526			−0.0979038
$341$	5.98215			0.323951
$342$		0			0
$343$	−18.0284	+	4.23999i	−0.973441	+	0.228938i
$344$			10.0671i			0.542781i
$345$		0			0
$346$		−	5.24866i		−	0.282170i
$347$		−	2.92331i		−	0.156932i	−0.996917	−	0.0784658i	$-0.974998\pi$
$347$		−	2.92331i		−	0.156932i	0.996917	−	0.0784658i	$-0.0250021\pi$
$348$		0			0
$349$			23.1687i			1.24019i	0.784526	+	0.620096i	$0.212907\pi$
$349$			23.1687i			1.24019i	−0.784526	+	0.620096i	$0.787093\pi$
$350$	−22.3361	+	15.3005i	−1.19392	+	0.817844i
$351$		0			0
$352$	−1.00000			−0.0533002
$353$	−21.1514			−1.12577			−0.562887	−	0.826534i	$-0.690309\pi$
$353$	−21.1514			−1.12577			−0.562887	+	0.826534i	$0.690309\pi$
$354$		0			0
$355$		−	33.1177i		−	1.75770i
$356$	10.1560			0.538269
$357$		0			0
$358$	4.66083			0.246333
$359$		−	20.8628i		−	1.10109i	−0.834804	−	0.550547i	$-0.814419\pi$
$359$		−	20.8628i		−	1.10109i	0.834804	−	0.550547i	$-0.185581\pi$
$360$		0			0
$361$	16.3864			0.862443
$362$	−8.53939			−0.448820
$363$		0			0
$364$	13.4757	−	9.23097i	0.706318	−	0.483834i
$365$			42.3378i			2.21606i
$366$		0			0
$367$			28.6708i			1.49660i	0.663358	+	0.748302i	$0.269131\pi$
$367$			28.6708i			1.49660i	−0.663358	+	0.748302i	$0.730869\pi$
$368$		0			0
$369$		0			0
$370$			35.3510i			1.83781i
$371$	13.2010	−	9.04282i	0.685362	−	0.469480i
$372$		0			0
$373$	11.7597			0.608894			0.304447	−	0.952529i	$-0.401528\pi$
$373$	11.7597			0.608894			0.304447	+	0.952529i	$0.401528\pi$
$374$	0.462536			0.0239172
$375$		0			0
$376$			5.05707i			0.260799i
$377$	48.1876			2.48179
$378$		0			0
$379$	−13.4086			−0.688753			−0.344377	−	0.938832i	$-0.611910\pi$
$379$	−13.4086			−0.688753			−0.344377	+	0.938832i	$0.611910\pi$
$380$			6.30973i			0.323682i
$381$		0			0
$382$	−8.48528			−0.434145
$383$	32.9041			1.68132			0.840661	−	0.541562i	$-0.182167\pi$
$383$	32.9041			1.68132			0.840661	+	0.541562i	$0.182167\pi$
$384$		0			0
$385$	8.51915	−	5.83570i	0.434176	−	0.297415i
$386$		−	17.6105i		−	0.896351i
$387$		0			0
$388$			11.0392i			0.560431i
$389$		−	12.5140i		−	0.634484i	−0.948345	−	0.317242i	$-0.897243\pi$
$389$		−	12.5140i		−	0.634484i	0.948345	−	0.317242i	$-0.102757\pi$
$390$		0			0
$391$		0			0
$392$	6.52728	+	2.52875i	0.329678	+	0.127721i
$393$		0			0
$394$	9.30078			0.468566
$395$	−63.3193			−3.18594
$396$		0			0
$397$			25.1575i			1.26262i	0.775532	+	0.631309i	$0.217482\pi$
$397$			25.1575i			1.26262i	−0.775532	+	0.631309i	$0.782518\pi$
$398$	−11.4798			−0.575431
$399$		0			0
$400$	10.2330			0.511652
$401$			26.5332i			1.32500i	0.749060	+	0.662502i	$0.230505\pi$
$401$			26.5332i			1.32500i	−0.749060	+	0.662502i	$0.769495\pi$
$402$		0			0
$403$	36.9322			1.83972
$404$	−17.2130			−0.856376
$405$		0			0
$406$	11.6704	+	17.0369i	0.579194	+	0.845526i
$407$		−	9.05749i		−	0.448963i
$408$		0			0
$409$		−	3.26878i		−	0.161631i	−0.996729	−	0.0808155i	$-0.974248\pi$
$409$		−	3.26878i		−	0.161631i	0.996729	−	0.0808155i	$-0.0257525\pi$
$410$			0.747762i			0.0369293i
$411$		0			0
$412$			12.1339i			0.597796i
$413$	−6.79055	−	9.91307i	−0.334141	−	0.487790i
$414$		0			0
$415$	38.5205			1.89090
$416$	−6.17373			−0.302692
$417$		0			0
$418$		−	1.61666i		−	0.0790732i
$419$	12.8894			0.629687			0.314843	−	0.949144i	$-0.398048\pi$
$419$	12.8894			0.629687			0.314843	+	0.949144i	$0.398048\pi$
$420$		0			0
$421$	−2.94251			−0.143409			−0.0717045	−	0.997426i	$-0.522844\pi$
$421$	−2.94251			−0.143409			−0.0717045	+	0.997426i	$0.522844\pi$
$422$		−	11.5626i		−	0.562859i
$423$		0			0
$424$	−6.04790			−0.293712
$425$	−4.73315			−0.229592
$426$		0			0
$427$	5.58181	−	3.82359i	0.270123	−	0.185037i
$428$		−	17.0383i		−	0.823577i
$429$		0			0
$430$			39.2914i			1.89480i
$431$		−	25.1461i		−	1.21125i	−0.795752	−	0.605623i	$-0.792924\pi$
$431$		−	25.1461i		−	1.21125i	0.795752	−	0.605623i	$-0.207076\pi$
$432$		0			0
$433$			18.1691i			0.873149i	0.899668	+	0.436575i	$0.143808\pi$
$433$			18.1691i			0.873149i	−0.899668	+	0.436575i	$0.856192\pi$
$434$	8.94451	+	13.0575i	0.429350	+	0.626780i
$435$		0			0
$436$	−14.6608			−0.702126
$437$		0			0
$438$		0			0
$439$		−	6.70476i		−	0.320001i	−0.987117	−	0.160000i	$-0.948850\pi$
$439$		−	6.70476i		−	0.320001i	0.987117	−	0.160000i	$-0.0511496\pi$
$440$	−3.90295			−0.186066
$441$		0			0
$442$	2.85557			0.135826
$443$			22.2905i			1.05906i	0.848293	+	0.529528i	$0.177631\pi$
$443$			22.2905i			1.05906i	−0.848293	+	0.529528i	$0.822369\pi$
$444$		0			0
$445$	39.6386			1.87905
$446$	21.5629			1.02103
$447$		0			0
$448$	−1.49520	−	2.18274i	−0.0706416	−	0.103125i
$449$			13.5715i			0.640478i	0.947337	+	0.320239i	$0.103763\pi$
$449$			13.5715i			0.640478i	−0.947337	+	0.320239i	$0.896237\pi$
$450$		0			0
$451$		−	0.191589i		−	0.00902156i
$452$		−	13.4757i		−	0.633843i
$453$		0			0
$454$			11.8630i			0.556758i
$455$	52.5950	−	36.0281i	2.46569	−	1.68902i
$456$		0			0
$457$	−7.62971			−0.356903			−0.178451	−	0.983949i	$-0.557109\pi$
$457$	−7.62971			−0.356903			−0.178451	+	0.983949i	$0.557109\pi$
$458$	−6.28413			−0.293638
$459$		0			0
$460$		0			0
$461$	3.97145			0.184969			0.0924843	−	0.995714i	$-0.470519\pi$
$461$	3.97145			0.184969			0.0924843	+	0.995714i	$0.470519\pi$
$462$		0			0
$463$	−20.9706			−0.974585			−0.487292	−	0.873239i	$-0.662015\pi$
$463$	−20.9706			−0.974585			−0.487292	+	0.873239i	$0.662015\pi$
$464$		−	7.80526i		−	0.362350i
$465$		0			0
$466$	22.4469			1.03983
$467$	12.7617			0.590540			0.295270	−	0.955414i	$-0.404590\pi$
$467$	12.7617			0.590540			0.295270	+	0.955414i	$0.404590\pi$
$468$		0			0
$469$	−8.68003	−	12.6714i	−0.400806	−	0.585110i
$470$			19.7375i			0.910424i
$471$		0			0
$472$			4.54156i			0.209042i
$473$		−	10.0671i		−	0.462885i
$474$		0			0
$475$			16.5433i			0.759059i
$476$	0.691584	+	1.00960i	0.0316987	+	0.0462748i
$477$		0			0
$478$	−5.40859			−0.247383
$479$	−21.2437			−0.970647			−0.485324	−	0.874335i	$-0.661298\pi$
$479$	−21.2437			−0.970647			−0.485324	+	0.874335i	$0.661298\pi$
$480$		0			0
$481$		−	55.9186i		−	2.54967i
$482$	−13.2681			−0.604345
$483$		0			0
$484$	1.00000			0.0454545
$485$			43.0856i			1.95641i
$486$		0			0
$487$	−32.4085			−1.46857			−0.734285	−	0.678842i	$-0.762482\pi$
$487$	−32.4085			−1.46857			−0.734285	+	0.678842i	$0.762482\pi$
$488$	−2.55724			−0.115761
$489$		0			0
$490$	25.4757	+	9.86958i	1.15087	+	0.445862i
$491$			19.8939i			0.897798i	0.893583	+	0.448899i	$0.148184\pi$
$491$			19.8939i			0.897798i	−0.893583	+	0.448899i	$0.851816\pi$
$492$		0			0
$493$			3.61021i			0.162596i
$494$		−	9.98080i		−	0.449058i
$495$		0			0
$496$		−	5.98215i		−	0.268606i
$497$	−18.5212	+	12.6872i	−0.830789	+	0.569099i
$498$		0			0
$499$	−18.1947			−0.814509			−0.407254	−	0.913315i	$-0.633514\pi$
$499$	−18.1947			−0.814509			−0.407254	+	0.913315i	$0.633514\pi$
$500$	20.4243			0.913403
$501$		0			0
$502$			6.80592i			0.303763i
$503$	32.9835			1.47066			0.735330	−	0.677709i	$-0.237027\pi$
$503$	32.9835			1.47066			0.735330	+	0.677709i	$0.237027\pi$
$504$		0			0
$505$	−67.1813			−2.98953
$506$		0			0
$507$		0			0
$508$	3.73816			0.165854
$509$	−41.8331			−1.85422			−0.927110	−	0.374789i	$-0.877715\pi$
$509$	−41.8331			−1.85422			−0.927110	+	0.374789i	$0.877715\pi$
$510$		0			0
$511$	23.6776	−	16.2194i	1.04744	−	0.717503i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	11.1560i		−	0.492072i
$515$			47.3582i			2.08685i
$516$		0			0
$517$		−	5.05707i		−	0.222410i
$518$	19.7702	−	13.5428i	0.868653	−	0.595035i
$519$		0			0
$520$	−24.0958			−1.05667
$521$	16.5788			0.726330			0.363165	−	0.931725i	$-0.381696\pi$
$521$	16.5788			0.726330			0.363165	+	0.931725i	$0.381696\pi$
$522$		0			0
$523$			29.3360i			1.28277i	0.767217	+	0.641387i	$0.221641\pi$
$523$			29.3360i			1.28277i	−0.767217	+	0.641387i	$0.778359\pi$
$524$	9.57669			0.418360
$525$		0			0
$526$	−8.17555			−0.356471
$527$			2.76696i			0.120531i
$528$		0			0
$529$	23.0000			1.00000
$530$	−23.6047			−1.02532
$531$		0			0
$532$	3.52875	−	2.41723i	0.152991	−	0.104800i
$533$		−	1.18282i		−	0.0512335i
$534$		0			0
$535$		−	66.4997i		−	2.87503i
$536$			5.80526i			0.250749i
$537$		0			0
$538$		−	13.2881i		−	0.572889i
$539$	−6.52728	−	2.52875i	−0.281150	−	0.108921i
$540$		0			0
$541$	−12.0293			−0.517182			−0.258591	−	0.965987i	$-0.583258\pi$
$541$	−12.0293			−0.517182			−0.258591	+	0.965987i	$0.583258\pi$
$542$	−24.1667			−1.03805
$543$		0			0
$544$		−	0.462536i		−	0.0198311i
$545$	−57.2205			−2.45106
$546$		0			0
$547$	13.6776			0.584812			0.292406	−	0.956294i	$-0.405544\pi$
$547$	13.6776			0.584812			0.292406	+	0.956294i	$0.405544\pi$
$548$			0.418189i			0.0178642i
$549$		0			0
$550$	−10.2330			−0.436338
$551$	12.6184			0.537563
$552$		0			0
$553$	24.2573	+	35.4116i	1.03153	+	1.50586i
$554$		−	13.9202i		−	0.591415i
$555$		0			0
$556$			15.5744i			0.660500i
$557$			21.8245i			0.924732i	0.886689	+	0.462366i	$0.152999\pi$
$557$			21.8245i			0.924732i	−0.886689	+	0.462366i	$0.847001\pi$
$558$		0			0
$559$		−	62.1515i		−	2.62873i
$560$	−5.83570	−	8.51915i	−0.246603	−	0.360000i
$561$		0			0
$562$	−10.1342			−0.427485
$563$	−8.12518			−0.342435			−0.171218	−	0.985233i	$-0.554770\pi$
$563$	−8.12518			−0.342435			−0.171218	+	0.985233i	$0.554770\pi$
$564$		0			0
$565$		−	52.5950i		−	2.21269i
$566$	18.2658			0.767768
$567$		0			0
$568$	8.48528			0.356034
$569$			22.3511i			0.937007i	0.883462	+	0.468503i	$0.155207\pi$
$569$			22.3511i			0.937007i	−0.883462	+	0.468503i	$0.844793\pi$
$570$		0			0
$571$	−16.5716			−0.693499			−0.346749	−	0.937958i	$-0.612715\pi$
$571$	−16.5716			−0.693499			−0.346749	+	0.937958i	$0.612715\pi$
$572$	6.17373			0.258137
$573$		0			0
$574$	0.418189	−	0.286464i	0.0174549	−	0.0119568i
$575$		0			0
$576$		0			0
$577$		−	0.925072i		−	0.0385112i	−0.999815	−	0.0192556i	$-0.993870\pi$
$577$		−	0.925072i		−	0.0385112i	0.999815	−	0.0192556i	$-0.00612963\pi$
$578$		−	16.7861i		−	0.698208i
$579$		0			0
$580$		−	30.4635i		−	1.26493i
$581$	−14.7570	−	21.5428i	−0.612224	−	0.893745i
$582$		0			0
$583$	6.04790			0.250478
$584$	−10.8476			−0.448878
$585$		0			0
$586$		−	29.5915i		−	1.22241i
$587$	20.1973			0.833630			0.416815	−	0.908991i	$-0.363146\pi$
$587$	20.1973			0.833630			0.416815	+	0.908991i	$0.363146\pi$
$588$		0			0
$589$	9.67107			0.398490
$590$			17.7255i			0.729747i
$591$		0			0
$592$	−9.05749			−0.372261
$593$	−46.5818			−1.91288			−0.956442	−	0.291921i	$-0.905705\pi$
$593$	−46.5818			−1.91288			−0.956442	+	0.291921i	$0.905705\pi$
$594$		0			0
$595$	2.69922	+	3.94041i	0.110657	+	0.161541i
$596$		−	7.05749i		−	0.289086i
$597$		0			0
$598$		0			0
$599$		−	6.15338i		−	0.251420i	−0.992067	−	0.125710i	$-0.959879\pi$
$599$		−	6.15338i		−	0.251420i	0.992067	−	0.125710i	$-0.0401209\pi$
$600$		0			0
$601$		−	2.69586i		−	0.109966i	−0.998487	−	0.0549831i	$-0.982489\pi$
$601$		−	2.69586i		−	0.109966i	0.998487	−	0.0549831i	$-0.0175105\pi$
$602$	21.9739	−	15.0523i	0.895589	−	0.613487i
$603$		0			0
$604$	−21.6776			−0.882049
$605$	3.90295			0.158678
$606$		0			0
$607$			3.05724i			0.124090i	0.998073	+	0.0620448i	$0.0197622\pi$
$607$			3.05724i			0.124090i	−0.998073	+	0.0620448i	$0.980238\pi$
$608$	−1.61666			−0.0655641
$609$		0			0
$610$	−9.98080			−0.404111
$611$		−	31.2210i		−	1.26307i
$612$		0			0
$613$	−26.9706			−1.08933			−0.544665	−	0.838653i	$-0.683343\pi$
$613$	−26.9706			−1.08933			−0.544665	+	0.838653i	$0.683343\pi$
$614$	4.49782			0.181517
$615$		0			0
$616$	1.49520	+	2.18274i	0.0602434	+	0.0879453i
$617$		−	44.6895i		−	1.79913i	−0.436784	−	0.899566i	$-0.643883\pi$
$617$		−	44.6895i		−	1.79913i	0.436784	−	0.899566i	$-0.356117\pi$
$618$		0			0
$619$		−	13.3036i		−	0.534716i	−0.963597	−	0.267358i	$-0.913849\pi$
$619$		−	13.3036i		−	0.534716i	0.963597	−	0.267358i	$-0.0861506\pi$
$620$		−	23.3480i		−	0.937680i
$621$		0			0
$622$			13.4422i			0.538983i
$623$	−15.1853	−	22.1681i	−0.608387	−	0.888144i
$624$		0			0
$625$	28.5500			1.14200
$626$	17.2440			0.689208
$627$		0			0
$628$		−	5.98215i		−	0.238714i
$629$	4.18942			0.167043
$630$		0			0
$631$	46.2108			1.83962			0.919811	−	0.392361i	$-0.128342\pi$
$631$	46.2108			1.83962			0.919811	+	0.392361i	$0.128342\pi$
$632$		−	16.2234i		−	0.645334i
$633$		0			0
$634$	−21.6584			−0.860165
$635$	14.5899			0.578982
$636$		0			0
$637$	−40.2977	−	15.6118i	−1.59665	−	0.618563i
$638$			7.80526i			0.309013i
$639$		0			0
$640$			3.90295i			0.154278i
$641$		−	1.91074i		−	0.0754697i	−0.999288	−	0.0377348i	$-0.987986\pi$
$641$		−	1.91074i		−	0.0754697i	0.999288	−	0.0377348i	$-0.0120142\pi$
$642$		0			0
$643$		−	6.39171i		−	0.252064i	−0.992026	−	0.126032i	$-0.959776\pi$
$643$		−	6.39171i		−	0.252064i	0.992026	−	0.126032i	$-0.0402242\pi$
$644$		0			0
$645$		0			0
$646$	0.747762			0.0294203
$647$	2.01979			0.0794061			0.0397031	−	0.999212i	$-0.487359\pi$
$647$	2.01979			0.0794061			0.0397031	+	0.999212i	$0.487359\pi$
$648$		0			0
$649$		−	4.54156i		−	0.178272i
$650$	−63.1761			−2.47797
$651$		0			0
$652$	9.40859			0.368469
$653$		−	6.70704i		−	0.262467i	−0.991351	−	0.131233i	$-0.958106\pi$
$653$		−	6.70704i		−	0.262467i	0.991351	−	0.131233i	$-0.0418937\pi$
$654$		0			0
$655$	37.3774			1.46045
$656$	−0.191589			−0.00748028
$657$		0			0
$658$	11.0383	−	7.56134i	0.430318	−	0.294772i
$659$			21.3217i			0.830574i	0.909690	+	0.415287i	$0.136319\pi$
$659$			21.3217i			0.830574i	−0.909690	+	0.415287i	$0.863681\pi$
$660$		0			0
$661$		−	0.830346i		−	0.0322967i	−0.999870	−	0.0161484i	$-0.994860\pi$
$661$		−	0.830346i		−	0.0322967i	0.999870	−	0.0161484i	$-0.00514040\pi$
$662$		−	1.31997i		−	0.0513023i
$663$		0			0
$664$			9.86958i			0.383014i
$665$	13.7725	−	9.43432i	0.534076	−	0.365847i
$666$		0			0
$667$		0			0
$668$	9.05490			0.350345
$669$		0			0
$670$			22.6576i			0.875341i
$671$	2.55724			0.0987213
$672$		0			0
$673$	−6.03839			−0.232763			−0.116381	−	0.993205i	$-0.537130\pi$
$673$	−6.03839			−0.232763			−0.116381	+	0.993205i	$0.537130\pi$
$674$		−	26.0958i		−	1.00517i
$675$		0			0
$676$	25.1150			0.965961
$677$	44.2910			1.70224			0.851122	−	0.524969i	$-0.175923\pi$
$677$	44.2910			1.70224			0.851122	+	0.524969i	$0.175923\pi$
$678$		0			0
$679$	24.0958	−	16.5059i	0.924712	−	0.633436i
$680$		−	1.80526i		−	0.0692284i
$681$		0			0
$682$			5.98215i			0.229068i
$683$		−	35.8819i		−	1.37298i	−0.727139	−	0.686490i	$-0.759150\pi$
$683$		−	35.8819i		−	1.37298i	0.727139	−	0.686490i	$-0.240850\pi$
$684$		0			0
$685$			1.63217i			0.0623621i
$686$	−4.23999	−	18.0284i	−0.161884	−	0.688327i
$687$		0			0
$688$	−10.0671			−0.383804
$689$	37.3381			1.42247
$690$		0			0
$691$		−	12.3164i		−	0.468539i	−0.972172	−	0.234270i	$-0.924730\pi$
$691$		−	12.3164i		−	0.468539i	0.972172	−	0.234270i	$-0.0752699\pi$
$692$	5.24866			0.199524
$693$		0			0
$694$	2.92331			0.110967
$695$			60.7860i			2.30574i
$696$		0			0
$697$	0.0886166			0.00335659
$698$	−23.1687			−0.876949
$699$		0			0
$700$	−15.3005	−	22.3361i	−0.578303	−	0.844226i
$701$		−	10.0384i		−	0.379145i	−0.981867	−	0.189572i	$-0.939290\pi$
$701$		−	10.0384i		−	0.379145i	0.981867	−	0.189572i	$-0.0607102\pi$
$702$		0			0
$703$		−	14.6429i		−	0.552266i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$		−	21.1514i		−	0.796043i
$707$	25.7368	+	37.5715i	0.967933	+	1.41302i
$708$		0			0
$709$	46.1916			1.73476			0.867381	−	0.497645i	$-0.165802\pi$
$709$	46.1916			1.73476			0.867381	+	0.497645i	$0.165802\pi$
$710$	33.1177			1.24288
$711$		0			0
$712$			10.1560i			0.380614i
$713$		0			0
$714$		0			0
$715$	24.0958			0.901132
$716$			4.66083i			0.174183i
$717$		0			0
$718$	20.8628			0.778591
$719$	−4.27526			−0.159440			−0.0797202	−	0.996817i	$-0.525403\pi$
$719$	−4.27526			−0.159440			−0.0797202	+	0.996817i	$0.525403\pi$
$720$		0			0
$721$	26.4853	−	18.1427i	0.986363	−	0.675668i
$722$			16.3864i			0.609840i
$723$		0			0
$724$		−	8.53939i		−	0.317364i
$725$		−	79.8715i		−	2.96635i
$726$		0			0
$727$			3.16304i			0.117311i	0.998278	+	0.0586554i	$0.0186813\pi$
$727$			3.16304i			0.117311i	−0.998278	+	0.0586554i	$0.981319\pi$
$728$	9.23097	+	13.4757i	0.342123	+	0.499442i
$729$		0			0
$730$	−42.3378			−1.56699
$731$	4.65639			0.172223
$732$		0			0
$733$			27.7542i			1.02512i	0.858650	+	0.512562i	$0.171303\pi$
$733$			27.7542i			1.02512i	−0.858650	+	0.512562i	$0.828697\pi$
$734$	−28.6708			−1.05826
$735$		0			0
$736$		0			0
$737$		−	5.80526i		−	0.213839i
$738$		0			0
$739$	−3.38875			−0.124657			−0.0623286	−	0.998056i	$-0.519853\pi$
$739$	−3.38875			−0.124657			−0.0623286	+	0.998056i	$0.519853\pi$
$740$	−35.3510			−1.29953
$741$		0			0
$742$	9.04282	+	13.2010i	0.331972	+	0.484624i
$743$		−	19.1533i		−	0.702666i	−0.936251	−	0.351333i	$-0.885729\pi$
$743$		−	19.1533i		−	0.702666i	0.936251	−	0.351333i	$-0.114271\pi$
$744$		0			0
$745$		−	27.5451i		−	1.00917i
$746$			11.7597i			0.430553i
$747$		0			0
$748$			0.462536i			0.0169120i
$749$	−37.1903	+	25.4757i	−1.35890	+	0.930861i
$750$		0			0
$751$	50.1724			1.83082			0.915408	−	0.402527i	$-0.131868\pi$
$751$	50.1724			1.83082			0.915408	+	0.402527i	$0.131868\pi$
$752$	−5.05707			−0.184412
$753$		0			0
$754$			48.1876i			1.75489i
$755$	−84.6067			−3.07915
$756$		0			0
$757$	−50.3408			−1.82967			−0.914833	−	0.403832i	$-0.867678\pi$
$757$	−50.3408			−1.82967			−0.914833	+	0.403832i	$0.867678\pi$
$758$		−	13.4086i		−	0.487022i
$759$		0			0
$760$	−6.30973			−0.228878
$761$	−35.5297			−1.28795			−0.643976	−	0.765046i	$-0.722716\pi$
$761$	−35.5297			−1.28795			−0.643976	+	0.765046i	$0.722716\pi$
$762$		0			0
$763$	21.9209	+	32.0008i	0.793589	+	1.15851i
$764$		−	8.48528i		−	0.306987i
$765$		0			0
$766$			32.9041i			1.18887i
$767$		−	28.0384i		−	1.01241i
$768$		0			0
$769$			1.91405i			0.0690223i	0.999404	+	0.0345112i	$0.0109874\pi$
$769$			1.91405i			0.0690223i	−0.999404	+	0.0345112i	$0.989013\pi$
$770$	5.83570	+	8.51915i	0.210304	+	0.307009i
$771$		0			0
$772$	17.6105			0.633816
$773$	−37.9830			−1.36615			−0.683077	−	0.730347i	$-0.739359\pi$
$773$	−37.9830			−1.36615			−0.683077	+	0.730347i	$0.739359\pi$
$774$		0			0
$775$		−	61.2155i		−	2.19893i
$776$	−11.0392			−0.396285
$777$		0			0
$778$	12.5140			0.448648
$779$		−	0.309733i		−	0.0110973i
$780$		0			0
$781$	−8.48528			−0.303627
$782$		0			0
$783$		0			0
$784$	−2.52875	+	6.52728i	−0.0903124	+	0.233117i
$785$		−	23.3480i		−	0.833327i
$786$		0			0
$787$			24.4614i			0.871956i	0.899957	+	0.435978i	$0.143598\pi$
$787$			24.4614i			0.871956i	−0.899957	+	0.435978i	$0.856402\pi$
$788$			9.30078i			0.331327i
$789$		0			0
$790$		−	63.3193i		−	2.25280i
$791$	−29.4140	+	20.1489i	−1.04584	+	0.716411i
$792$		0			0
$793$	15.7877			0.560639
$794$	−25.1575			−0.892805
$795$		0			0
$796$		−	11.4798i		−	0.406891i
$797$	−12.6712			−0.448838			−0.224419	−	0.974493i	$-0.572048\pi$
$797$	−12.6712			−0.448838			−0.224419	+	0.974493i	$0.572048\pi$
$798$		0			0
$799$	2.33908			0.0827506
$800$			10.2330i			0.361793i
$801$		0			0
$802$	−26.5332			−0.936919
$803$	10.8476			0.382805
$804$		0			0
$805$		0			0
$806$			36.9322i			1.30088i
$807$		0			0
$808$		−	17.2130i		−	0.605550i
$809$			2.38949i			0.0840099i	0.999117	+	0.0420050i	$0.0133745\pi$
$809$			2.38949i			0.0840099i	−0.999117	+	0.0420050i	$0.986625\pi$
$810$		0			0
$811$			28.0805i			0.986041i	0.870018	+	0.493021i	$0.164107\pi$
$811$			28.0805i			0.986041i	−0.870018	+	0.493021i	$0.835893\pi$
$812$	−17.0369	+	11.6704i	−0.597877	+	0.409552i
$813$		0			0
$814$	9.05749			0.317465
$815$	36.7213			1.28629
$816$		0			0
$817$		−	16.2750i		−	0.569391i
$818$	3.26878			0.114290
$819$		0			0
$820$	−0.747762			−0.0261130
$821$			1.86582i			0.0651174i	0.999470	+	0.0325587i	$0.0103656\pi$
$821$			1.86582i			0.0651174i	−0.999470	+	0.0325587i	$0.989634\pi$
$822$		0			0
$823$	24.2300			0.844604			0.422302	−	0.906455i	$-0.361222\pi$
$823$	24.2300			0.844604			0.422302	+	0.906455i	$0.361222\pi$
$824$	−12.1339			−0.422706
$825$		0			0
$826$	9.91307	−	6.79055i	0.344920	−	0.236273i
$827$			40.1622i			1.39658i	0.715816	+	0.698289i	$0.246055\pi$
$827$			40.1622i			1.39658i	−0.715816	+	0.698289i	$0.753945\pi$
$828$		0			0
$829$		−	14.2461i		−	0.494789i	−0.968915	−	0.247395i	$-0.920426\pi$
$829$		−	14.2461i		−	0.494789i	0.968915	−	0.247395i	$-0.0795744\pi$
$830$			38.5205i			1.33707i
$831$		0			0
$832$		−	6.17373i		−	0.214036i
$833$	1.16964	−	3.01910i	0.0405255	−	0.104606i
$834$		0			0
$835$	35.3409			1.22302
$836$	1.61666			0.0559132
$837$		0			0
$838$			12.8894i			0.445256i
$839$	38.0186			1.31255			0.656274	−	0.754523i	$-0.272131\pi$
$839$	38.0186			1.31255			0.656274	+	0.754523i	$0.272131\pi$
$840$		0			0
$841$	−31.9220			−1.10076
$842$		−	2.94251i		−	0.101405i
$843$		0			0
$844$	11.5626			0.398002
$845$	98.0226			3.37208
$846$		0			0
$847$	−1.49520	−	2.18274i	−0.0513757	−	0.0750000i
$848$		−	6.04790i		−	0.207686i
$849$		0			0
$850$		−	4.73315i		−	0.162346i
$851$		0			0
$852$		0			0
$853$		−	8.54404i		−	0.292542i	−0.989245	−	0.146271i	$-0.953273\pi$
$853$		−	8.54404i		−	0.292542i	0.989245	−	0.146271i	$-0.0467272\pi$
$854$	3.82359	+	5.58181i	0.130841	+	0.191006i
$855$		0			0
$856$	17.0383			0.582357
$857$	29.3469			1.00247			0.501235	−	0.865311i	$-0.332879\pi$
$857$	29.3469			1.00247			0.501235	+	0.865311i	$0.332879\pi$
$858$		0			0
$859$			51.4209i			1.75446i	0.480072	+	0.877229i	$0.340610\pi$
$859$			51.4209i			1.75446i	−0.480072	+	0.877229i	$0.659390\pi$
$860$	−39.2914			−1.33983
$861$		0			0
$862$	25.1461			0.856480
$863$			5.84662i			0.199021i	0.995036	+	0.0995106i	$0.0317277\pi$
$863$			5.84662i			0.199021i	−0.995036	+	0.0995106i	$0.968272\pi$
$864$		0			0
$865$	20.4853			0.696520
$866$	−18.1691			−0.617410
$867$		0			0
$868$	−13.0575	+	8.94451i	−0.443200	+	0.303596i
$869$			16.2234i			0.550343i
$870$		0			0
$871$		−	35.8401i		−	1.21440i
$872$		−	14.6608i		−	0.496478i
$873$		0			0
$874$		0			0
$875$	−30.5385	−	44.5811i	−1.03239	−	1.50712i
$876$		0			0
$877$	7.18450			0.242603			0.121302	−	0.992616i	$-0.461293\pi$
$877$	7.18450			0.242603			0.121302	+	0.992616i	$0.461293\pi$
$878$	6.70476			0.226275
$879$		0			0
$880$		−	3.90295i		−	0.131569i
$881$	48.9958			1.65071			0.825356	−	0.564613i	$-0.190975\pi$
$881$	48.9958			1.65071			0.825356	+	0.564613i	$0.190975\pi$
$882$		0			0
$883$	−5.67825			−0.191088			−0.0955441	−	0.995425i	$-0.530459\pi$
$883$	−5.67825			−0.191088			−0.0955441	+	0.995425i	$0.530459\pi$
$884$			2.85557i			0.0960433i
$885$		0			0
$886$	−22.2905			−0.748865
$887$	−12.3036			−0.413114			−0.206557	−	0.978435i	$-0.566226\pi$
$887$	−12.3036			−0.413114			−0.206557	+	0.978435i	$0.566226\pi$
$888$		0			0
$889$	−5.58931	−	8.15946i	−0.187459	−	0.273659i
$890$			39.6386i			1.32869i
$891$		0			0
$892$			21.5629i			0.721980i
$893$		−	8.17555i		−	0.273584i
$894$		0			0
$895$			18.1910i			0.608058i
$896$	2.18274	−	1.49520i	0.0729204	−	0.0499512i
$897$		0			0
$898$	−13.5715			−0.452886
$899$	−46.6922			−1.55727
$900$		0			0
$901$			2.79737i			0.0931939i
$902$	0.191589			0.00637920
$903$		0			0
$904$	13.4757			0.448195
$905$		−	33.3288i		−	1.10789i
$906$		0			0
$907$	−24.7566			−0.822030			−0.411015	−	0.911629i	$-0.634826\pi$
$907$	−24.7566			−0.822030			−0.411015	+	0.911629i	$0.634826\pi$
$908$	−11.8630			−0.393687
$909$		0			0
$910$	36.0281	+	52.5950i	1.19432	+	1.74351i
$911$			54.8746i			1.81808i	0.416713	+	0.909038i	$0.363182\pi$
$911$			54.8746i			1.81808i	−0.416713	+	0.909038i	$0.636818\pi$
$912$		0			0
$913$		−	9.86958i		−	0.326635i
$914$		−	7.62971i		−	0.252368i
$915$		0			0
$916$		−	6.28413i		−	0.207633i
$917$	−14.3191	−	20.9035i	−0.472858	−	0.690293i
$918$		0			0
$919$	−20.7070			−0.683062			−0.341531	−	0.939870i	$-0.610945\pi$
$919$	−20.7070			−0.683062			−0.341531	+	0.939870i	$0.610945\pi$
$920$		0			0
$921$		0			0
$922$			3.97145i			0.130793i
$923$	−52.3859			−1.72430
$924$		0			0
$925$	−92.6857			−3.04749
$926$		−	20.9706i		−	0.689135i
$927$		0			0
$928$	7.80526			0.256220
$929$	−36.6612			−1.20282			−0.601408	−	0.798942i	$-0.705393\pi$
$929$	−36.6612			−1.20282			−0.601408	+	0.798942i	$0.705393\pi$
$930$		0			0
$931$	−10.5524	−	4.08811i	−0.345840	−	0.133983i
$932$			22.4469i			0.735272i
$933$		0			0
$934$			12.7617i			0.417575i
$935$			1.80526i			0.0590382i
$936$		0			0
$937$			34.5426i			1.12846i	0.825619	+	0.564228i	$0.190826\pi$
$937$			34.5426i			1.12846i	−0.825619	+	0.564228i	$0.809174\pi$
$938$	12.6714	−	8.68003i	0.413736	−	0.283413i
$939$		0			0
$940$	−19.7375			−0.643767
$941$	−1.71597			−0.0559390			−0.0279695	−	0.999609i	$-0.508904\pi$
$941$	−1.71597			−0.0559390			−0.0279695	+	0.999609i	$0.508904\pi$
$942$		0			0
$943$		0			0
$944$	−4.54156			−0.147815
$945$		0			0
$946$	10.0671			0.327309
$947$		−	23.3510i		−	0.758806i	−0.925232	−	0.379403i	$-0.876129\pi$
$947$		−	23.3510i		−	0.758806i	0.925232	−	0.379403i	$-0.123871\pi$
$948$		0			0
$949$	66.9704			2.17395
$950$	−16.5433			−0.536736
$951$		0			0
$952$	−1.00960	+	0.691584i	−0.0327213	+	0.0224144i
$953$		−	49.2210i		−	1.59443i	−0.603699	−	0.797213i	$-0.706307\pi$
$953$		−	49.2210i		−	1.59443i	0.603699	−	0.797213i	$-0.293693\pi$
$954$		0			0
$955$		−	33.1177i		−	1.07166i
$956$		−	5.40859i		−	0.174926i
$957$		0			0
$958$		−	21.2437i		−	0.686351i
$959$	0.912800	−	0.625277i	0.0294758	−	0.0201912i
$960$		0			0
$961$	−4.78606			−0.154389
$962$	55.9186			1.80289
$963$		0			0
$964$		−	13.2681i		−	0.427337i
$965$	68.7330			2.21259
$966$		0			0
$967$	60.0568			1.93130			0.965648	−	0.259855i	$-0.0836749\pi$
$967$	60.0568			1.93130			0.965648	+	0.259855i	$0.0836749\pi$
$968$			1.00000i			0.0321412i
$969$		0			0
$970$	−43.0856			−1.38339
$971$	−56.1083			−1.80060			−0.900301	−	0.435269i	$-0.856653\pi$
$971$	−56.1083			−1.80060			−0.900301	+	0.435269i	$0.856653\pi$
$972$		0			0
$973$	33.9948	−	23.2868i	1.08982	−	0.746540i
$974$		−	32.4085i		−	1.03844i
$975$		0			0
$976$		−	2.55724i		−	0.0818554i
$977$			29.6170i			0.947533i	0.880650	+	0.473767i	$0.157106\pi$
$977$			29.6170i			0.947533i	−0.880650	+	0.473767i	$0.842894\pi$
$978$		0			0
$979$		−	10.1560i		−	0.324589i
$980$	−9.86958	+	25.4757i	−0.315272	+	0.813791i
$981$		0			0
$982$	−19.8939			−0.634839
$983$	−18.5194			−0.590676			−0.295338	−	0.955393i	$-0.595432\pi$
$983$	−18.5194			−0.590676			−0.295338	+	0.955393i	$0.595432\pi$
$984$		0			0
$985$			36.3005i			1.15663i
$986$	−3.61021			−0.114973
$987$		0			0
$988$	9.98080			0.317532
$989$		0			0
$990$		0			0
$991$	−47.5708			−1.51114			−0.755569	−	0.655069i	$-0.772639\pi$
$991$	−47.5708			−1.51114			−0.755569	+	0.655069i	$0.772639\pi$
$992$	5.98215			0.189933
$993$		0			0
$994$	−12.6872	−	18.5212i	−0.402414	−	0.587457i
$995$		−	44.8052i		−	1.42042i
$996$		0			0
$997$			10.1295i			0.320805i	0.987052	+	0.160402i	$0.0512792\pi$
$997$			10.1295i			0.320805i	−0.987052	+	0.160402i	$0.948721\pi$
$998$		−	18.1947i		−	0.575945i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.g.b.881.9	yes	16
3.2	odd	2	inner	1386.2.g.b.881.8	yes	16
7.6	odd	2	inner	1386.2.g.b.881.16	yes	16
21.20	even	2	inner	1386.2.g.b.881.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.g.b.881.1	✓	16	21.20	even	2	inner
1386.2.g.b.881.8	yes	16	3.2	odd	2	inner
1386.2.g.b.881.9	yes	16	1.1	even	1	trivial
1386.2.g.b.881.16	yes	16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -3.90295 q^{5} +(1.49520 + 2.18274i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -3.90295 q^{5} +(1.49520 + 2.18274i) q^{7} -1.00000i q^{8} -3.90295i q^{10} +1.00000i q^{11} +6.17373i q^{13} +(-2.18274 + 1.49520i) q^{14} +1.00000 q^{16} -0.462536 q^{17} +1.61666i q^{19} +3.90295 q^{20} -1.00000 q^{22} +10.2330 q^{25} -6.17373 q^{26} +(-1.49520 - 2.18274i) q^{28} -7.80526i q^{29} -5.98215i q^{31} +1.00000i q^{32} -0.462536i q^{34} +(-5.83570 - 8.51915i) q^{35} -9.05749 q^{37} -1.61666 q^{38} +3.90295i q^{40} -0.191589 q^{41} -10.0671 q^{43} -1.00000i q^{44} -5.05707 q^{47} +(-2.52875 + 6.52728i) q^{49} +10.2330i q^{50} -6.17373i q^{52} -6.04790i q^{53} -3.90295i q^{55} +(2.18274 - 1.49520i) q^{56} +7.80526 q^{58} -4.54156 q^{59} -2.55724i q^{61} +5.98215 q^{62} -1.00000 q^{64} -24.0958i q^{65} -5.80526 q^{67} +0.462536 q^{68} +(8.51915 - 5.83570i) q^{70} +8.48528i q^{71} -10.8476i q^{73} -9.05749i q^{74} -1.61666i q^{76} +(-2.18274 + 1.49520i) q^{77} +16.2234 q^{79} -3.90295 q^{80} -0.191589i q^{82} -9.86958 q^{83} +1.80526 q^{85} -10.0671i q^{86} +1.00000 q^{88} -10.1560 q^{89} +(-13.4757 + 9.23097i) q^{91} -5.05707i q^{94} -6.30973i q^{95} -11.0392i q^{97} +(-6.52728 - 2.52875i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^22 + 48 * q^25 - 16 * q^37 - 80 * q^43 + 24 * q^49 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 24 * q^70 + 96 * q^79 - 80 * q^85 + 16 * q^88 - 32 * q^91

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