LMFDB - Embedded newform 2312.2.b.j.577.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(577,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2312.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (of order $2$, degree $1$, not 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:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.3
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.j.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367i q^{3} -1.84776i q^{5} -0.317025i q^{7} +2.41421 q^{9} +O(q^{10})$	$q+0.765367i q^{3} -1.84776i q^{5} -0.317025i q^{7} +2.41421 q^{9} -0.765367i q^{11} +1.41421 q^{15} -2.24264 q^{19} +0.242641 q^{21} -6.62567i q^{23} +1.58579 q^{25} +4.14386i q^{27} -5.54328i q^{29} +2.29610i q^{31} +0.585786 q^{33} -0.585786 q^{35} -0.765367i q^{37} -2.48181i q^{41} +10.2426 q^{43} -4.46088i q^{45} -12.8284 q^{47} +6.89949 q^{49} +3.07107 q^{53} -1.41421 q^{55} -1.71644i q^{57} -7.89949 q^{59} +9.87285i q^{61} -0.765367i q^{63} -9.65685 q^{67} +5.07107 q^{69} -9.68714i q^{71} -7.97069i q^{73} +1.21371i q^{75} -0.242641 q^{77} -14.2793i q^{79} +4.07107 q^{81} +10.2426 q^{83} +4.24264 q^{87} +1.65685 q^{89} -1.75736 q^{93} +4.14386i q^{95} -15.9958i q^{97} -1.84776i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} + 8 q^{19} - 16 q^{21} + 12 q^{25} + 8 q^{33} - 8 q^{35} + 24 q^{43} - 40 q^{47} - 12 q^{49} - 16 q^{53} + 8 q^{59} - 16 q^{67} - 8 q^{69} + 16 q^{77} - 12 q^{81} + 24 q^{83} - 16 q^{89} - 24 q^{93}+O(q^{100}) $ 4 * q + 4 * q^9 + 8 * q^19 - 16 * q^21 + 12 * q^25 + 8 * q^33 - 8 * q^35 + 24 * q^43 - 40 * q^47 - 12 * q^49 - 16 * q^53 + 8 * q^59 - 16 * q^67 - 8 * q^69 + 16 * q^77 - 12 * q^81 + 24 * q^83 - 16 * q^89 - 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0.765367i	0.441885i	0.975287	+	0.220942i	$0.0709133\pi$
$3$	0.765367i	0.441885i	−0.975287	+	0.220942i	$0.929087\pi$
$4$	0	0
$5$	− 1.84776i	− 0.826343i	−0.910653	−	0.413171i	$-0.864421\pi$
$5$	− 1.84776i	− 0.826343i	0.910653	−	0.413171i	$-0.135579\pi$
$6$	0	0
$7$	− 0.317025i	− 0.119824i	−0.998204	−	0.0599122i	$-0.980918\pi$
$7$	− 0.317025i	− 0.119824i	0.998204	−	0.0599122i	$-0.0190821\pi$
$8$	0	0
$9$	2.41421	0.804738
$10$	0	0
$11$	− 0.765367i	− 0.230767i	−0.993321	−	0.115383i	$-0.963190\pi$
$11$	− 0.765367i	− 0.230767i	0.993321	−	0.115383i	$-0.0368097\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−2.24264	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$20$	0	0
$21$	0.242641	0.0529485
$22$	0	0
$23$	− 6.62567i	− 1.38155i	−0.723071	−	0.690774i	$-0.757270\pi$
$23$	− 6.62567i	− 1.38155i	0.723071	−	0.690774i	$-0.242730\pi$
$24$	0	0
$25$	1.58579	0.317157
$26$	0	0
$27$	4.14386i	0.797486i
$28$	0	0
$29$	− 5.54328i	− 1.02936i	−0.857382	−	0.514680i	$-0.827911\pi$
$29$	− 5.54328i	− 1.02936i	0.857382	−	0.514680i	$-0.172089\pi$
$30$	0	0
$31$	2.29610i	0.412392i	0.978511	+	0.206196i	$0.0661084\pi$
$31$	2.29610i	0.412392i	−0.978511	+	0.206196i	$0.933892\pi$
$32$	0	0
$33$	0.585786	0.101972
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	− 0.765367i	− 0.125826i	−0.998019	−	0.0629128i	$-0.979961\pi$
$37$	− 0.765367i	− 0.125826i	0.998019	−	0.0629128i	$-0.0200390\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.48181i	− 0.387594i	−0.981042	−	0.193797i	$-0.937920\pi$
$41$	− 2.48181i	− 0.387594i	0.981042	−	0.193797i	$-0.0620802\pi$
$42$	0	0
$43$	10.2426	1.56199	0.780994	−	0.624538i	$-0.214713\pi$
$43$	10.2426	1.56199	0.780994	+	0.624538i	$0.214713\pi$
$44$	0	0
$45$	− 4.46088i	− 0.664989i
$46$	0	0
$47$	−12.8284	−1.87122	−0.935609	−	0.353037i	$-0.885149\pi$
$47$	−12.8284	−1.87122	−0.935609	+	0.353037i	$0.885149\pi$
$48$	0	0
$49$	6.89949	0.985642
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.07107	0.421844	0.210922	−	0.977503i	$-0.432353\pi$
$53$	3.07107	0.421844	0.210922	+	0.977503i	$0.432353\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	− 1.71644i	− 0.227348i
$58$	0	0
$59$	−7.89949	−1.02843	−0.514213	−	0.857662i	$-0.671916\pi$
$59$	−7.89949	−1.02843	−0.514213	+	0.857662i	$0.671916\pi$
$60$	0	0
$61$	9.87285i	1.26409i	0.774932	+	0.632044i	$0.217784\pi$
$61$	9.87285i	1.26409i	−0.774932	+	0.632044i	$0.782216\pi$
$62$	0	0
$63$	− 0.765367i	− 0.0964272i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	5.07107	0.610485
$70$	0	0
$71$	− 9.68714i	− 1.14965i	−0.818276	−	0.574826i	$-0.805070\pi$
$71$	− 9.68714i	− 1.14965i	0.818276	−	0.574826i	$-0.194930\pi$
$72$	0	0
$73$	− 7.97069i	− 0.932899i	−0.884548	−	0.466450i	$-0.845533\pi$
$73$	− 7.97069i	− 0.932899i	0.884548	−	0.466450i	$-0.154467\pi$
$74$	0	0
$75$	1.21371i	0.140147i
$76$	0	0
$77$	−0.242641	−0.0276515
$78$	0	0
$79$	− 14.2793i	− 1.60655i	−0.595608	−	0.803276i	$-0.703089\pi$
$79$	− 14.2793i	− 1.60655i	0.595608	−	0.803276i	$-0.296911\pi$
$80$	0	0
$81$	4.07107	0.452341
$82$	0	0
$83$	10.2426	1.12428	0.562138	−	0.827043i	$-0.309979\pi$
$83$	10.2426	1.12428	0.562138	+	0.827043i	$0.309979\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.24264	0.454859
$88$	0	0
$89$	1.65685	0.175626	0.0878131	−	0.996137i	$-0.472012\pi$
$89$	1.65685	0.175626	0.0878131	+	0.996137i	$0.472012\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.75736	−0.182230
$94$	0	0
$95$	4.14386i	0.425151i
$96$	0	0
$97$	− 15.9958i	− 1.62413i	−0.583570	−	0.812063i	$-0.698345\pi$
$97$	− 15.9958i	− 1.62413i	0.583570	−	0.812063i	$-0.301655\pi$
$98$	0	0
$99$	− 1.84776i	− 0.185707i
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	− 0.448342i	− 0.0437537i
$106$	0	0
$107$	14.4650i	1.39839i	0.714932	+	0.699194i	$0.246458\pi$
$107$	14.4650i	1.39839i	−0.714932	+	0.699194i	$0.753542\pi$
$108$	0	0
$109$	− 11.6662i	− 1.11742i	−0.829363	−	0.558710i	$-0.811296\pi$
$109$	− 11.6662i	− 1.11742i	0.829363	−	0.558710i	$-0.188704\pi$
$110$	0	0
$111$	0.585786	0.0556004
$112$	0	0
$113$	− 8.15640i	− 0.767290i	−0.923481	−	0.383645i	$-0.874669\pi$
$113$	− 8.15640i	− 0.767290i	0.923481	−	0.383645i	$-0.125331\pi$
$114$	0	0
$115$	−12.2426	−1.14163
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.4142	0.946747
$122$	0	0
$123$	1.89949	0.171272
$124$	0	0
$125$	− 12.1689i	− 1.08842i
$126$	0	0
$127$	1.75736	0.155940	0.0779702	−	0.996956i	$-0.475156\pi$
$127$	1.75736	0.155940	0.0779702	+	0.996956i	$0.475156\pi$
$128$	0	0
$129$	7.83938i	0.690219i
$130$	0	0
$131$	− 14.4650i	− 1.26382i	−0.775043	−	0.631909i	$-0.782272\pi$
$131$	− 14.4650i	− 1.26382i	0.775043	−	0.631909i	$-0.217728\pi$
$132$	0	0
$133$	0.710974i	0.0616493i
$134$	0	0
$135$	7.65685	0.658997
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	− 0.502734i	− 0.0426414i	−0.999773	−	0.0213207i	$-0.993213\pi$
$139$	− 0.502734i	− 0.0426414i	0.999773	−	0.0213207i	$-0.00678710\pi$
$140$	0	0
$141$	− 9.81845i	− 0.826863i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.2426	−0.850605
$146$	0	0
$147$	5.28064i	0.435540i
$148$	0	0
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	0	0
$151$	19.8995	1.61940	0.809699	−	0.586845i	$-0.199630\pi$
$151$	19.8995	1.61940	0.809699	+	0.586845i	$0.199630\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.24264	0.340777
$156$	0	0
$157$	15.3137	1.22217	0.611083	−	0.791566i	$-0.290734\pi$
$157$	15.3137	1.22217	0.611083	+	0.791566i	$0.290734\pi$
$158$	0	0
$159$	2.35049i	0.186406i
$160$	0	0
$161$	−2.10051	−0.165543
$162$	0	0
$163$	4.64659i	0.363949i	0.983303	+	0.181975i	$0.0582489\pi$
$163$	4.64659i	0.363949i	−0.983303	+	0.181975i	$0.941751\pi$
$164$	0	0
$165$	− 1.08239i	− 0.0842641i
$166$	0	0
$167$	− 0.765367i	− 0.0592259i	−0.999561	−	0.0296129i	$-0.990573\pi$
$167$	− 0.765367i	− 0.0592259i	0.999561	−	0.0296129i	$-0.00942747\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−5.41421	−0.414035
$172$	0	0
$173$	18.8715i	1.43478i	0.696674	+	0.717388i	$0.254663\pi$
$173$	18.8715i	1.43478i	−0.696674	+	0.717388i	$0.745337\pi$
$174$	0	0
$175$	− 0.502734i	− 0.0380032i
$176$	0	0
$177$	− 6.04601i	− 0.454446i
$178$	0	0
$179$	18.2426	1.36352	0.681759	−	0.731576i	$-0.261215\pi$
$179$	18.2426	1.36352	0.681759	+	0.731576i	$0.261215\pi$
$180$	0	0
$181$	− 8.15640i	− 0.606261i	−0.952949	−	0.303130i	$-0.901968\pi$
$181$	− 8.15640i	− 0.606261i	0.952949	−	0.303130i	$-0.0980318\pi$
$182$	0	0
$183$	−7.55635	−0.558581
$184$	0	0
$185$	−1.41421	−0.103975
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.31371	0.0955582
$190$	0	0
$191$	−8.82843	−0.638803	−0.319401	−	0.947620i	$-0.603482\pi$
$191$	−8.82843	−0.638803	−0.319401	+	0.947620i	$0.603482\pi$
$192$	0	0
$193$	0.765367i	0.0550923i	0.999621	+	0.0275462i	$0.00876932\pi$
$193$	0.765367i	0.0550923i	−0.999621	+	0.0275462i	$0.991231\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.3003i	− 0.876357i	−0.898888	−	0.438179i	$-0.855624\pi$
$197$	− 12.3003i	− 0.876357i	0.898888	−	0.438179i	$-0.144376\pi$
$198$	0	0
$199$	22.4901i	1.59428i	0.603792	+	0.797142i	$0.293656\pi$
$199$	22.4901i	1.59428i	−0.603792	+	0.797142i	$0.706344\pi$
$200$	0	0
$201$	− 7.39104i	− 0.521324i
$202$	0	0
$203$	−1.75736	−0.123342
$204$	0	0
$205$	−4.58579	−0.320285
$206$	0	0
$207$	− 15.9958i	− 1.11178i
$208$	0	0
$209$	1.71644i	0.118729i
$210$	0	0
$211$	26.1857i	1.80269i	0.433097	+	0.901347i	$0.357421\pi$
$211$	26.1857i	1.80269i	−0.433097	+	0.901347i	$0.642579\pi$
$212$	0	0
$213$	7.41421	0.508014
$214$	0	0
$215$	− 18.9259i	− 1.29074i
$216$	0	0
$217$	0.727922	0.0494146
$218$	0	0
$219$	6.10051	0.412234
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−17.5563	−1.17566	−0.587830	−	0.808984i	$-0.700018\pi$
$223$	−17.5563	−1.17566	−0.587830	+	0.808984i	$0.700018\pi$
$224$	0	0
$225$	3.82843	0.255228
$226$	0	0
$227$	− 3.56420i	− 0.236564i	−0.992980	−	0.118282i	$-0.962261\pi$
$227$	− 3.56420i	− 0.236564i	0.992980	−	0.118282i	$-0.0377387\pi$
$228$	0	0
$229$	9.89949	0.654177	0.327089	−	0.944994i	$-0.393932\pi$
$229$	9.89949	0.654177	0.327089	+	0.944994i	$0.393932\pi$
$230$	0	0
$231$	− 0.185709i	− 0.0122188i
$232$	0	0
$233$	0.579658i	0.0379746i	0.999820	+	0.0189873i	$0.00604421\pi$
$233$	0.579658i	0.0379746i	−0.999820	+	0.0189873i	$0.993956\pi$
$234$	0	0
$235$	23.7038i	1.54627i
$236$	0	0
$237$	10.9289	0.709910
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	14.0167i	0.902895i	0.892298	+	0.451448i	$0.149092\pi$
$241$	14.0167i	0.902895i	−0.892298	+	0.451448i	$0.850908\pi$
$242$	0	0
$243$	15.5474i	0.997369i
$244$	0	0
$245$	− 12.7486i	− 0.814478i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7.83938i	0.496800i
$250$	0	0
$251$	−0.828427	−0.0522899	−0.0261449	−	0.999658i	$-0.508323\pi$
$251$	−0.828427	−0.0522899	−0.0261449	+	0.999658i	$0.508323\pi$
$252$	0	0
$253$	−5.07107	−0.318815
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.24264	−0.514162	−0.257081	−	0.966390i	$-0.582761\pi$
$257$	−8.24264	−0.514162	−0.257081	+	0.966390i	$0.582761\pi$
$258$	0	0
$259$	−0.242641	−0.0150770
$260$	0	0
$261$	− 13.3827i	− 0.828366i
$262$	0	0
$263$	−3.89949	−0.240453	−0.120227	−	0.992746i	$-0.538362\pi$
$263$	−3.89949	−0.240453	−0.120227	+	0.992746i	$0.538362\pi$
$264$	0	0
$265$	− 5.67459i	− 0.348588i
$266$	0	0
$267$	1.26810i	0.0776065i
$268$	0	0
$269$	− 14.0167i	− 0.854614i	−0.904107	−	0.427307i	$-0.859462\pi$
$269$	− 14.0167i	− 0.854614i	0.904107	−	0.427307i	$-0.140538\pi$
$270$	0	0
$271$	11.3137	0.687259	0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.21371i	− 0.0731894i
$276$	0	0
$277$	− 9.23880i	− 0.555105i	−0.960710	−	0.277553i	$-0.910477\pi$
$277$	− 9.23880i	− 0.555105i	0.960710	−	0.277553i	$-0.0895233\pi$
$278$	0	0
$279$	5.54328i	0.331867i
$280$	0	0
$281$	−3.27208	−0.195196	−0.0975979	−	0.995226i	$-0.531116\pi$
$281$	−3.27208	−0.195196	−0.0975979	+	0.995226i	$0.531116\pi$
$282$	0	0
$283$	26.2626i	1.56115i	0.625063	+	0.780574i	$0.285073\pi$
$283$	26.2626i	1.56115i	−0.625063	+	0.780574i	$0.714927\pi$
$284$	0	0
$285$	−3.17157	−0.187868
$286$	0	0
$287$	−0.786797	−0.0464431
$288$	0	0
$289$	0	0
$290$	0	0
$291$	12.2426	0.717676
$292$	0	0
$293$	−7.31371	−0.427271	−0.213636	−	0.976913i	$-0.568531\pi$
$293$	−7.31371	−0.427271	−0.213636	+	0.976913i	$0.568531\pi$
$294$	0	0
$295$	14.5964i	0.849833i
$296$	0	0
$297$	3.17157	0.184033
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 3.24718i	− 0.187164i
$302$	0	0
$303$	− 7.65367i	− 0.439692i
$304$	0	0
$305$	18.2426	1.04457
$306$	0	0
$307$	−26.6274	−1.51971	−0.759853	−	0.650094i	$-0.774729\pi$
$307$	−26.6274	−1.51971	−0.759853	+	0.650094i	$0.774729\pi$
$308$	0	0
$309$	− 1.79337i	− 0.102021i
$310$	0	0
$311$	4.01254i	0.227530i	0.993508	+	0.113765i	$0.0362912\pi$
$311$	4.01254i	0.227530i	−0.993508	+	0.113765i	$0.963709\pi$
$312$	0	0
$313$	− 4.27518i	− 0.241647i	−0.992674	−	0.120824i	$-0.961446\pi$
$313$	− 4.27518i	− 0.241647i	0.992674	−	0.120824i	$-0.0385535\pi$
$314$	0	0
$315$	−1.41421	−0.0796819
$316$	0	0
$317$	15.8101i	0.887982i	0.896031	+	0.443991i	$0.146438\pi$
$317$	15.8101i	0.887982i	−0.896031	+	0.443991i	$0.853562\pi$
$318$	0	0
$319$	−4.24264	−0.237542
$320$	0	0
$321$	−11.0711	−0.617927
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.92893	0.493771
$328$	0	0
$329$	4.06694i	0.224217i
$330$	0	0
$331$	13.7574	0.756173	0.378086	−	0.925770i	$-0.376582\pi$
$331$	13.7574	0.756173	0.378086	+	0.925770i	$0.376582\pi$
$332$	0	0
$333$	− 1.84776i	− 0.101257i
$334$	0	0
$335$	17.8435i	0.974897i
$336$	0	0
$337$	15.5474i	0.846923i	0.905914	+	0.423461i	$0.139185\pi$
$337$	15.5474i	0.846923i	−0.905914	+	0.423461i	$0.860815\pi$
$338$	0	0
$339$	6.24264	0.339054
$340$	0	0
$341$	1.75736	0.0951663
$342$	0	0
$343$	− 4.40649i	− 0.237928i
$344$	0	0
$345$	− 9.37011i	− 0.504470i
$346$	0	0
$347$	5.80591i	0.311678i	0.987783	+	0.155839i	$0.0498080\pi$
$347$	5.80591i	0.311678i	−0.987783	+	0.155839i	$0.950192\pi$
$348$	0	0
$349$	−21.8995	−1.17225	−0.586127	−	0.810220i	$-0.699348\pi$
$349$	−21.8995	−1.17225	−0.586127	+	0.810220i	$0.699348\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.9706	1.32905	0.664524	−	0.747266i	$-0.268634\pi$
$353$	24.9706	1.32905	0.664524	+	0.747266i	$0.268634\pi$
$354$	0	0
$355$	−17.8995	−0.950007
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.2426	−1.17392	−0.586961	−	0.809615i	$-0.699676\pi$
$359$	−22.2426	−1.17392	−0.586961	+	0.809615i	$0.699676\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	0	0
$363$	7.97069i	0.418353i
$364$	0	0
$365$	−14.7279	−0.770895
$366$	0	0
$367$	2.11039i	0.110162i	0.998482	+	0.0550808i	$0.0175416\pi$
$367$	2.11039i	0.110162i	−0.998482	+	0.0550808i	$0.982458\pi$
$368$	0	0
$369$	− 5.99162i	− 0.311911i
$370$	0	0
$371$	− 0.973606i	− 0.0505471i
$372$	0	0
$373$	−35.9411	−1.86096	−0.930480	−	0.366342i	$-0.880610\pi$
$373$	−35.9411	−1.86096	−0.930480	+	0.366342i	$0.880610\pi$
$374$	0	0
$375$	9.31371	0.480958
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 15.0991i	− 0.775589i	−0.921746	−	0.387794i	$-0.873237\pi$
$379$	− 15.0991i	− 0.775589i	0.921746	−	0.387794i	$-0.126763\pi$
$380$	0	0
$381$	1.34502i	0.0689077i
$382$	0	0
$383$	−10.7279	−0.548171	−0.274086	−	0.961705i	$-0.588375\pi$
$383$	−10.7279	−0.548171	−0.274086	+	0.961705i	$0.588375\pi$
$384$	0	0
$385$	0.448342i	0.0228496i
$386$	0	0
$387$	24.7279	1.25699
$388$	0	0
$389$	−28.7279	−1.45656	−0.728282	−	0.685278i	$-0.759681\pi$
$389$	−28.7279	−1.45656	−0.728282	+	0.685278i	$0.759681\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	11.0711	0.558461
$394$	0	0
$395$	−26.3848	−1.32756
$396$	0	0
$397$	29.0614i	1.45855i	0.684221	+	0.729275i	$0.260143\pi$
$397$	29.0614i	1.45855i	−0.684221	+	0.729275i	$0.739857\pi$
$398$	0	0
$399$	−0.544156	−0.0272419
$400$	0	0
$401$	− 27.0823i	− 1.35243i	−0.736706	−	0.676214i	$-0.763620\pi$
$401$	− 27.0823i	− 1.35243i	0.736706	−	0.676214i	$-0.236380\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 7.52235i	− 0.373789i
$406$	0	0
$407$	−0.585786	−0.0290364
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	10.7151i	0.528539i
$412$	0	0
$413$	2.50434i	0.123231i
$414$	0	0
$415$	− 18.9259i	− 0.929037i
$416$	0	0
$417$	0.384776	0.0188426
$418$	0	0
$419$	6.62567i	0.323685i	0.986817	+	0.161843i	$0.0517437\pi$
$419$	6.62567i	0.323685i	−0.986817	+	0.161843i	$0.948256\pi$
$420$	0	0
$421$	11.3137	0.551396	0.275698	−	0.961244i	$-0.411091\pi$
$421$	11.3137	0.551396	0.275698	+	0.961244i	$0.411091\pi$
$422$	0	0
$423$	−30.9706	−1.50584
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.12994	0.151469
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 2.29610i	− 0.110599i	−0.998470	−	0.0552996i	$-0.982389\pi$
$431$	− 2.29610i	− 0.110599i	0.998470	−	0.0552996i	$-0.0176114\pi$
$432$	0	0
$433$	−5.41421	−0.260190	−0.130095	−	0.991502i	$-0.541528\pi$
$433$	−5.41421	−0.260190	−0.130095	+	0.991502i	$0.541528\pi$
$434$	0	0
$435$	− 7.83938i	− 0.375869i
$436$	0	0
$437$	14.8590i	0.710802i
$438$	0	0
$439$	− 9.42450i	− 0.449807i	−0.974381	−	0.224904i	$-0.927793\pi$
$439$	− 9.42450i	− 0.449807i	0.974381	−	0.224904i	$-0.0722067\pi$
$440$	0	0
$441$	16.6569	0.793184
$442$	0	0
$443$	−40.2843	−1.91396	−0.956982	−	0.290148i	$-0.906295\pi$
$443$	−40.2843	−1.91396	−0.956982	+	0.290148i	$0.906295\pi$
$444$	0	0
$445$	− 3.06147i	− 0.145127i
$446$	0	0
$447$	4.85483i	0.229626i
$448$	0	0
$449$	17.8979i	0.844656i	0.906443	+	0.422328i	$0.138787\pi$
$449$	17.8979i	0.844656i	−0.906443	+	0.422328i	$0.861213\pi$
$450$	0	0
$451$	−1.89949	−0.0894437
$452$	0	0
$453$	15.2304i	0.715587i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.92893	0.230566	0.115283	−	0.993333i	$-0.463223\pi$
$457$	4.92893	0.230566	0.115283	+	0.993333i	$0.463223\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.55635	−0.165636	−0.0828178	−	0.996565i	$-0.526392\pi$
$461$	−3.55635	−0.165636	−0.0828178	+	0.996565i	$0.526392\pi$
$462$	0	0
$463$	24.1421	1.12198	0.560990	−	0.827823i	$-0.310421\pi$
$463$	24.1421	1.12198	0.560990	+	0.827823i	$0.310421\pi$
$464$	0	0
$465$	3.24718i	0.150584i
$466$	0	0
$467$	−13.5563	−0.627313	−0.313657	−	0.949537i	$-0.601554\pi$
$467$	−13.5563	−0.627313	−0.313657	+	0.949537i	$0.601554\pi$
$468$	0	0
$469$	3.06147i	0.141365i
$470$	0	0
$471$	11.7206i	0.540057i
$472$	0	0
$473$	− 7.83938i	− 0.360455i
$474$	0	0
$475$	−3.55635	−0.163176
$476$	0	0
$477$	7.41421	0.339474
$478$	0	0
$479$	− 37.9832i	− 1.73550i	−0.497005	−	0.867748i	$-0.665567\pi$
$479$	− 37.9832i	− 1.73550i	0.497005	−	0.867748i	$-0.334433\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 1.60766i	− 0.0731509i
$484$	0	0
$485$	−29.5563	−1.34208
$486$	0	0
$487$	17.6034i	0.797688i	0.917019	+	0.398844i	$0.130588\pi$
$487$	17.6034i	0.797688i	−0.917019	+	0.398844i	$0.869412\pi$
$488$	0	0
$489$	−3.55635	−0.160824
$490$	0	0
$491$	−20.3848	−0.919952	−0.459976	−	0.887931i	$-0.652142\pi$
$491$	−20.3848	−0.919952	−0.459976	+	0.887931i	$0.652142\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3.41421	−0.153457
$496$	0	0
$497$	−3.07107	−0.137756
$498$	0	0
$499$	16.8155i	0.752767i	0.926464	+	0.376383i	$0.122832\pi$
$499$	16.8155i	0.752767i	−0.926464	+	0.376383i	$0.877168\pi$
$500$	0	0
$501$	0.585786	0.0261710
$502$	0	0
$503$	0.317025i	0.0141355i	0.999975	+	0.00706773i	$0.00224975\pi$
$503$	0.317025i	0.0141355i	−0.999975	+	0.00706773i	$0.997750\pi$
$504$	0	0
$505$	18.4776i	0.822242i
$506$	0	0
$507$	− 9.94977i	− 0.441885i
$508$	0	0
$509$	38.2843	1.69692	0.848460	−	0.529259i	$-0.177530\pi$
$509$	38.2843	1.69692	0.848460	+	0.529259i	$0.177530\pi$
$510$	0	0
$511$	−2.52691	−0.111784
$512$	0	0
$513$	− 9.29319i	− 0.410304i
$514$	0	0
$515$	4.32957i	0.190784i
$516$	0	0
$517$	9.81845i	0.431815i
$518$	0	0
$519$	−14.4437	−0.634006
$520$	0	0
$521$	39.2513i	1.71963i	0.510606	+	0.859815i	$0.329421\pi$
$521$	39.2513i	1.71963i	−0.510606	+	0.859815i	$0.670579\pi$
$522$	0	0
$523$	−4.14214	−0.181123	−0.0905615	−	0.995891i	$-0.528866\pi$
$523$	−4.14214	−0.181123	−0.0905615	+	0.995891i	$0.528866\pi$
$524$	0	0
$525$	0.384776	0.0167930
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−20.8995	−0.908674
$530$	0	0
$531$	−19.0711	−0.827614
$532$	0	0
$533$	0	0
$534$	0	0
$535$	26.7279	1.15555
$536$	0	0
$537$	13.9623i	0.602518i
$538$	0	0
$539$	− 5.28064i	− 0.227453i
$540$	0	0
$541$	25.9999i	1.11782i	0.829227	+	0.558912i	$0.188781\pi$
$541$	25.9999i	1.11782i	−0.829227	+	0.558912i	$0.811219\pi$
$542$	0	0
$543$	6.24264	0.267897
$544$	0	0
$545$	−21.5563	−0.923373
$546$	0	0
$547$	45.1116i	1.92883i	0.264389	+	0.964416i	$0.414830\pi$
$547$	45.1116i	1.92883i	−0.264389	+	0.964416i	$0.585170\pi$
$548$	0	0
$549$	23.8352i	1.01726i
$550$	0	0
$551$	12.4316i	0.529603i
$552$	0	0
$553$	−4.52691	−0.192504
$554$	0	0
$555$	− 1.08239i	− 0.0459450i
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.0416	1.09752	0.548762	−	0.835979i	$-0.315099\pi$
$563$	26.0416	1.09752	0.548762	+	0.835979i	$0.315099\pi$
$564$	0	0
$565$	−15.0711	−0.634045
$566$	0	0
$567$	− 1.29063i	− 0.0542014i
$568$	0	0
$569$	30.1838	1.26537	0.632685	−	0.774410i	$-0.281953\pi$
$569$	30.1838	1.26537	0.632685	+	0.774410i	$0.281953\pi$
$570$	0	0
$571$	− 32.3086i	− 1.35207i	−0.736868	−	0.676036i	$-0.763696\pi$
$571$	− 32.3086i	− 1.35207i	0.736868	−	0.676036i	$-0.236304\pi$
$572$	0	0
$573$	− 6.75699i	− 0.282277i
$574$	0	0
$575$	− 10.5069i	− 0.438168i
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	−0.585786	−0.0243445
$580$	0	0
$581$	− 3.24718i	− 0.134716i
$582$	0	0
$583$	− 2.35049i	− 0.0973475i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.87006	0.366106	0.183053	−	0.983103i	$-0.441402\pi$
$587$	8.87006	0.366106	0.183053	+	0.983103i	$0.441402\pi$
$588$	0	0
$589$	− 5.14933i	− 0.212174i
$590$	0	0
$591$	9.41421	0.387249
$592$	0	0
$593$	−21.6985	−0.891050	−0.445525	−	0.895270i	$-0.646983\pi$
$593$	−21.6985	−0.891050	−0.445525	+	0.895270i	$0.646983\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.2132	−0.704490
$598$	0	0
$599$	44.8284	1.83164	0.915820	−	0.401589i	$-0.131542\pi$
$599$	44.8284	1.83164	0.915820	+	0.401589i	$0.131542\pi$
$600$	0	0
$601$	30.3295i	1.23717i	0.785719	+	0.618583i	$0.212293\pi$
$601$	30.3295i	1.23717i	−0.785719	+	0.618583i	$0.787707\pi$
$602$	0	0
$603$	−23.3137	−0.949408
$604$	0	0
$605$	− 19.2430i	− 0.782337i
$606$	0	0
$607$	39.0656i	1.58562i	0.609467	+	0.792811i	$0.291383\pi$
$607$	39.0656i	1.58562i	−0.609467	+	0.792811i	$0.708617\pi$
$608$	0	0
$609$	− 1.34502i	− 0.0545031i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−29.3137	−1.18397	−0.591985	−	0.805949i	$-0.701655\pi$
$613$	−29.3137	−1.18397	−0.591985	+	0.805949i	$0.701655\pi$
$614$	0	0
$615$	− 3.50981i	− 0.141529i
$616$	0	0
$617$	24.6549i	0.992570i	0.868160	+	0.496285i	$0.165303\pi$
$617$	24.6549i	0.992570i	−0.868160	+	0.496285i	$0.834697\pi$
$618$	0	0
$619$	− 2.11039i	− 0.0848238i	−0.999100	−	0.0424119i	$-0.986496\pi$
$619$	− 2.11039i	− 0.0848238i	0.999100	−	0.0424119i	$-0.0135042\pi$
$620$	0	0
$621$	27.4558	1.10177
$622$	0	0
$623$	− 0.525265i	− 0.0210443i
$624$	0	0
$625$	−14.5563	−0.582254
$626$	0	0
$627$	−1.31371	−0.0524645
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−24.5858	−0.978745	−0.489372	−	0.872075i	$-0.662774\pi$
$631$	−24.5858	−0.978745	−0.489372	+	0.872075i	$0.662774\pi$
$632$	0	0
$633$	−20.0416	−0.796583
$634$	0	0
$635$	− 3.24718i	− 0.128860i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 23.3868i	− 0.925168i
$640$	0	0
$641$	11.7750i	0.465084i	0.972586	+	0.232542i	$0.0747044\pi$
$641$	11.7750i	0.465084i	−0.972586	+	0.232542i	$0.925296\pi$
$642$	0	0
$643$	39.2513i	1.54792i	0.633235	+	0.773960i	$0.281727\pi$
$643$	39.2513i	1.54792i	−0.633235	+	0.773960i	$0.718273\pi$
$644$	0	0
$645$	14.4853	0.570357
$646$	0	0
$647$	45.2548	1.77915	0.889576	−	0.456788i	$-0.151000\pi$
$647$	45.2548	1.77915	0.889576	+	0.456788i	$0.151000\pi$
$648$	0	0
$649$	6.04601i	0.237327i
$650$	0	0
$651$	0.557127i	0.0218355i
$652$	0	0
$653$	25.2890i	0.989634i	0.868997	+	0.494817i	$0.164765\pi$
$653$	25.2890i	0.989634i	−0.868997	+	0.494817i	$0.835235\pi$
$654$	0	0
$655$	−26.7279	−1.04435
$656$	0	0
$657$	− 19.2430i	− 0.750739i
$658$	0	0
$659$	22.7696	0.886976	0.443488	−	0.896280i	$-0.353741\pi$
$659$	22.7696	0.886976	0.443488	+	0.896280i	$0.353741\pi$
$660$	0	0
$661$	26.5858	1.03407	0.517034	−	0.855965i	$-0.327036\pi$
$661$	26.5858	1.03407	0.517034	+	0.855965i	$0.327036\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.31371	0.0509434
$666$	0	0
$667$	−36.7279	−1.42211
$668$	0	0
$669$	− 13.4370i	− 0.519506i
$670$	0	0
$671$	7.55635	0.291710
$672$	0	0
$673$	− 15.3617i	− 0.592151i	−0.955165	−	0.296076i	$-0.904322\pi$
$673$	− 15.3617i	− 0.592151i	0.955165	−	0.296076i	$-0.0956780\pi$
$674$	0	0
$675$	6.57128i	0.252929i
$676$	0	0
$677$	− 10.2124i	− 0.392494i	−0.980554	−	0.196247i	$-0.937124\pi$
$677$	− 10.2124i	− 0.392494i	0.980554	−	0.196247i	$-0.0628755\pi$
$678$	0	0
$679$	−5.07107	−0.194610
$680$	0	0
$681$	2.72792	0.104534
$682$	0	0
$683$	33.9162i	1.29777i	0.760887	+	0.648885i	$0.224764\pi$
$683$	33.9162i	1.29777i	−0.760887	+	0.648885i	$0.775236\pi$
$684$	0	0
$685$	− 25.8686i	− 0.988389i
$686$	0	0
$687$	7.57675i	0.289071i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32.3855i	1.23200i	0.787745	+	0.616001i	$0.211248\pi$
$691$	32.3855i	1.23200i	−0.787745	+	0.616001i	$0.788752\pi$
$692$	0	0
$693$	−0.585786	−0.0222522
$694$	0	0
$695$	−0.928932	−0.0352364
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−0.443651	−0.0167804
$700$	0	0
$701$	28.6863	1.08347	0.541733	−	0.840551i	$-0.317768\pi$
$701$	28.6863	1.08347	0.541733	+	0.840551i	$0.317768\pi$
$702$	0	0
$703$	1.71644i	0.0647369i
$704$	0	0
$705$	−18.1421	−0.683272
$706$	0	0
$707$	3.17025i	0.119230i
$708$	0	0
$709$	37.0096i	1.38992i	0.719047	+	0.694962i	$0.244579\pi$
$709$	37.0096i	1.38992i	−0.719047	+	0.694962i	$0.755421\pi$
$710$	0	0
$711$	− 34.4734i	− 1.29285i
$712$	0	0
$713$	15.2132	0.569739
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.65914i	0.323381i
$718$	0	0
$719$	− 19.4287i	− 0.724567i	−0.932068	−	0.362284i	$-0.881997\pi$
$719$	− 19.4287i	− 0.724567i	0.932068	−	0.362284i	$-0.118003\pi$
$720$	0	0
$721$	0.742837i	0.0276647i
$722$	0	0
$723$	−10.7279	−0.398976
$724$	0	0
$725$	− 8.79045i	− 0.326469i
$726$	0	0
$727$	17.7990	0.660128	0.330064	−	0.943959i	$-0.392930\pi$
$727$	17.7990	0.660128	0.330064	+	0.943959i	$0.392930\pi$
$728$	0	0
$729$	0.313708	0.0116188
$730$	0	0
$731$	0	0
$732$	0	0
$733$	24.7279	0.913347	0.456673	−	0.889634i	$-0.349041\pi$
$733$	24.7279	0.913347	0.456673	+	0.889634i	$0.349041\pi$
$734$	0	0
$735$	9.75736	0.359906
$736$	0	0
$737$	7.39104i	0.272252i
$738$	0	0
$739$	9.07107	0.333685	0.166842	−	0.985984i	$-0.446643\pi$
$739$	9.07107	0.333685	0.166842	+	0.985984i	$0.446643\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 32.3086i	− 1.18529i	−0.805465	−	0.592644i	$-0.798084\pi$
$743$	− 32.3086i	− 1.18529i	0.805465	−	0.592644i	$-0.201916\pi$
$744$	0	0
$745$	− 11.7206i	− 0.429410i
$746$	0	0
$747$	24.7279	0.904747
$748$	0	0
$749$	4.58579	0.167561
$750$	0	0
$751$	24.2066i	0.883311i	0.897185	+	0.441655i	$0.145609\pi$
$751$	24.2066i	0.883311i	−0.897185	+	0.441655i	$0.854391\pi$
$752$	0	0
$753$	− 0.634051i	− 0.0231061i
$754$	0	0
$755$	− 36.7695i	− 1.33818i
$756$	0	0
$757$	14.1005	0.512492	0.256246	−	0.966612i	$-0.417514\pi$
$757$	14.1005	0.512492	0.256246	+	0.966612i	$0.417514\pi$
$758$	0	0
$759$	− 3.88123i	− 0.140880i
$760$	0	0
$761$	3.02944	0.109817	0.0549085	−	0.998491i	$-0.482513\pi$
$761$	3.02944	0.109817	0.0549085	+	0.998491i	$0.482513\pi$
$762$	0	0
$763$	−3.69848	−0.133894
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	43.5980	1.57218	0.786092	−	0.618110i	$-0.212101\pi$
$769$	43.5980	1.57218	0.786092	+	0.618110i	$0.212101\pi$
$770$	0	0
$771$	− 6.30864i	− 0.227200i
$772$	0	0
$773$	32.9289	1.18437	0.592186	−	0.805802i	$-0.298265\pi$
$773$	32.9289	1.18437	0.592186	+	0.805802i	$0.298265\pi$
$774$	0	0
$775$	3.64113i	0.130793i
$776$	0	0
$777$	− 0.185709i	− 0.00666228i
$778$	0	0
$779$	5.56581i	0.199416i
$780$	0	0
$781$	−7.41421	−0.265301
$782$	0	0
$783$	22.9706	0.820901
$784$	0	0
$785$	− 28.2960i	− 1.00993i
$786$	0	0
$787$	− 11.2948i	− 0.402616i	−0.979528	−	0.201308i	$-0.935481\pi$
$787$	− 11.2948i	− 0.402616i	0.979528	−	0.201308i	$-0.0645192\pi$
$788$	0	0
$789$	− 2.98454i	− 0.106253i
$790$	0	0
$791$	−2.58579	−0.0919400
$792$	0	0
$793$	0	0
$794$	0	0
$795$	4.34315	0.154036
$796$	0	0
$797$	−13.4142	−0.475156	−0.237578	−	0.971368i	$-0.576354\pi$
$797$	−13.4142	−0.475156	−0.237578	+	0.971368i	$0.576354\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	−6.10051	−0.215282
$804$	0	0
$805$	3.88123i	0.136795i
$806$	0	0
$807$	10.7279	0.377641
$808$	0	0
$809$	15.9958i	0.562382i	0.959652	+	0.281191i	$0.0907294\pi$
$809$	15.9958i	0.562382i	−0.959652	+	0.281191i	$0.909271\pi$
$810$	0	0
$811$	− 42.7611i	− 1.50154i	−0.660561	−	0.750772i	$-0.729681\pi$
$811$	− 42.7611i	− 1.50154i	0.660561	−	0.750772i	$-0.270319\pi$
$812$	0	0
$813$	8.65914i	0.303689i
$814$	0	0
$815$	8.58579	0.300747
$816$	0	0
$817$	−22.9706	−0.803638
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.0961i	1.67857i	0.543694	+	0.839283i	$0.317025\pi$
$821$	48.0961i	1.67857i	−0.543694	+	0.839283i	$0.682975\pi$
$822$	0	0
$823$	− 9.50143i	− 0.331199i	−0.986193	−	0.165599i	$-0.947044\pi$
$823$	− 9.50143i	− 0.331199i	0.986193	−	0.165599i	$-0.0529559\pi$
$824$	0	0
$825$	0.928932	0.0323413
$826$	0	0
$827$	− 8.15640i	− 0.283626i	−0.989893	−	0.141813i	$-0.954707\pi$
$827$	− 8.15640i	− 0.283626i	0.989893	−	0.141813i	$-0.0452932\pi$
$828$	0	0
$829$	−53.9411	−1.87345	−0.936726	−	0.350062i	$-0.886160\pi$
$829$	−53.9411	−1.87345	−0.936726	+	0.350062i	$0.886160\pi$
$830$	0	0
$831$	7.07107	0.245293
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.41421	−0.0489409
$836$	0	0
$837$	−9.51472	−0.328877
$838$	0	0
$839$	− 34.9217i	− 1.20563i	−0.797880	−	0.602816i	$-0.794045\pi$
$839$	− 34.9217i	− 1.20563i	0.797880	−	0.602816i	$-0.205955\pi$
$840$	0	0
$841$	−1.72792	−0.0595835
$842$	0	0
$843$	− 2.50434i	− 0.0862541i
$844$	0	0
$845$	24.0209i	0.826343i
$846$	0	0
$847$	− 3.30157i	− 0.113443i
$848$	0	0
$849$	−20.1005	−0.689848
$850$	0	0
$851$	−5.07107	−0.173834
$852$	0	0
$853$	15.8101i	0.541327i	0.962674	+	0.270663i	$0.0872430\pi$
$853$	15.8101i	0.541327i	−0.962674	+	0.270663i	$0.912757\pi$
$854$	0	0
$855$	10.0042i	0.342135i
$856$	0	0
$857$	7.97069i	0.272274i	0.990690	+	0.136137i	$0.0434687\pi$
$857$	7.97069i	0.272274i	−0.990690	+	0.136137i	$0.956531\pi$
$858$	0	0
$859$	−42.0416	−1.43444	−0.717221	−	0.696846i	$-0.754586\pi$
$859$	−42.0416	−1.43444	−0.717221	+	0.696846i	$0.754586\pi$
$860$	0	0
$861$	− 0.602188i	− 0.0205225i
$862$	0	0
$863$	15.8579	0.539808	0.269904	−	0.962887i	$-0.413008\pi$
$863$	15.8579	0.539808	0.269904	+	0.962887i	$0.413008\pi$
$864$	0	0
$865$	34.8701	1.18562
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.9289	−0.370739
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 38.6172i	− 1.30700i
$874$	0	0
$875$	−3.85786	−0.130420
$876$	0	0
$877$	7.33664i	0.247741i	0.992298	+	0.123870i	$0.0395307\pi$
$877$	7.33664i	0.247741i	−0.992298	+	0.123870i	$0.960469\pi$
$878$	0	0
$879$	− 5.59767i	− 0.188805i
$880$	0	0
$881$	24.2066i	0.815540i	0.913085	+	0.407770i	$0.133694\pi$
$881$	24.2066i	0.815540i	−0.913085	+	0.407770i	$0.866306\pi$
$882$	0	0
$883$	−41.2548	−1.38834	−0.694168	−	0.719813i	$-0.744227\pi$
$883$	−41.2548	−1.38834	−0.694168	+	0.719813i	$0.744227\pi$
$884$	0	0
$885$	−11.1716	−0.375528
$886$	0	0
$887$	− 15.2848i	− 0.513214i	−0.966516	−	0.256607i	$-0.917395\pi$
$887$	− 15.2848i	− 0.513214i	0.966516	−	0.256607i	$-0.0826045\pi$
$888$	0	0
$889$	− 0.557127i	− 0.0186855i
$890$	0	0
$891$	− 3.11586i	− 0.104385i
$892$	0	0
$893$	28.7696	0.962736
$894$	0	0
$895$	− 33.7080i	− 1.12673i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.7279	0.424500
$900$	0	0
$901$	0	0
$902$	0	0
$903$	2.48528	0.0827050
$904$	0	0
$905$	−15.0711	−0.500979
$906$	0	0
$907$	23.4637i	0.779101i	0.921005	+	0.389550i	$0.127370\pi$
$907$	23.4637i	0.779101i	−0.921005	+	0.389550i	$0.872630\pi$
$908$	0	0
$909$	−24.1421	−0.800744
$910$	0	0
$911$	29.1383i	0.965396i	0.875787	+	0.482698i	$0.160343\pi$
$911$	29.1383i	0.965396i	−0.875787	+	0.482698i	$0.839657\pi$
$912$	0	0
$913$	− 7.83938i	− 0.259446i
$914$	0	0
$915$	13.9623i	0.461580i
$916$	0	0
$917$	−4.58579	−0.151436
$918$	0	0
$919$	44.2843	1.46080	0.730402	−	0.683018i	$-0.239333\pi$
$919$	44.2843	1.46080	0.730402	+	0.683018i	$0.239333\pi$
$920$	0	0
$921$	− 20.3797i	− 0.671535i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 1.21371i	− 0.0399065i
$926$	0	0
$927$	−5.65685	−0.185795
$928$	0	0
$929$	− 27.7933i	− 0.911869i	−0.890013	−	0.455934i	$-0.849305\pi$
$929$	− 27.7933i	− 0.911869i	0.890013	−	0.455934i	$-0.150695\pi$
$930$	0	0
$931$	−15.4731	−0.507110
$932$	0	0
$933$	−3.07107	−0.100542
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.443651	0.0144934	0.00724672	−	0.999974i	$-0.497693\pi$
$937$	0.443651	0.0144934	0.00724672	+	0.999974i	$0.497693\pi$
$938$	0	0
$939$	3.27208	0.106780
$940$	0	0
$941$	38.9886i	1.27099i	0.772103	+	0.635497i	$0.219205\pi$
$941$	38.9886i	1.27099i	−0.772103	+	0.635497i	$0.780795\pi$
$942$	0	0
$943$	−16.4437	−0.535479
$944$	0	0
$945$	− 2.42742i	− 0.0789639i
$946$	0	0
$947$	− 51.4202i	− 1.67093i	−0.549541	−	0.835466i	$-0.685198\pi$
$947$	− 51.4202i	− 1.67093i	0.549541	−	0.835466i	$-0.314802\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−12.1005	−0.392386
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	16.3128i	0.527870i
$956$	0	0
$957$	− 3.24718i	− 0.104966i
$958$	0	0
$959$	− 4.43835i	− 0.143322i
$960$	0	0
$961$	25.7279	0.829933
$962$	0	0
$963$	34.9217i	1.12534i
$964$	0	0
$965$	1.41421	0.0455251
$966$	0	0
$967$	34.5269	1.11031	0.555155	−	0.831747i	$-0.312659\pi$
$967$	34.5269	1.11031	0.555155	+	0.831747i	$0.312659\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	54.7279	1.75630	0.878151	−	0.478383i	$-0.158777\pi$
$971$	54.7279	1.75630	0.878151	+	0.478383i	$0.158777\pi$
$972$	0	0
$973$	−0.159380	−0.00510947
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.3848	−1.10007	−0.550033	−	0.835143i	$-0.685385\pi$
$977$	−34.3848	−1.10007	−0.550033	+	0.835143i	$0.685385\pi$
$978$	0	0
$979$	− 1.26810i	− 0.0405287i
$980$	0	0
$981$	− 28.1647i	− 0.899231i
$982$	0	0
$983$	− 38.9886i	− 1.24354i	−0.783198	−	0.621772i	$-0.786413\pi$
$983$	− 38.9886i	− 1.24354i	0.783198	−	0.621772i	$-0.213587\pi$
$984$	0	0
$985$	−22.7279	−0.724172
$986$	0	0
$987$	−3.11270	−0.0990783
$988$	0	0
$989$	− 67.8644i	− 2.15796i
$990$	0	0
$991$	− 50.7862i	− 1.61328i	−0.591046	−	0.806638i	$-0.701285\pi$
$991$	− 50.7862i	− 1.61328i	0.591046	−	0.806638i	$-0.298715\pi$
$992$	0	0
$993$	10.5294i	0.334141i
$994$	0	0
$995$	41.5563	1.31742
$996$	0	0
$997$	− 8.68167i	− 0.274951i	−0.990505	−	0.137476i	$-0.956101\pi$
$997$	− 8.68167i	− 0.274951i	0.990505	−	0.137476i	$-0.0438989\pi$
$998$	0	0
$999$	3.17157	0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.j.577.3		4
17.3	odd	16		136.2.n.a.25.1	✓	4
17.4	even	4		2312.2.a.s.1.3		4
17.11	odd	16		136.2.n.a.49.1	yes	4
17.13	even	4		2312.2.a.s.1.2		4
17.16	even	2	inner	2312.2.b.j.577.2		4
51.11	even	16		1224.2.bq.a.865.1		4
51.20	even	16		1224.2.bq.a.433.1		4
68.3	even	16		272.2.v.e.161.1		4
68.11	even	16		272.2.v.e.49.1		4
68.47	odd	4		4624.2.a.bm.1.3		4
68.55	odd	4		4624.2.a.bm.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.a.25.1	✓	4	17.3	odd	16
136.2.n.a.49.1	yes	4	17.11	odd	16
272.2.v.e.49.1		4	68.11	even	16
272.2.v.e.161.1		4	68.3	even	16
1224.2.bq.a.433.1		4	51.20	even	16
1224.2.bq.a.865.1		4	51.11	even	16
2312.2.a.s.1.2		4	17.13	even	4
2312.2.a.s.1.3		4	17.4	even	4
2312.2.b.j.577.2		4	17.16	even	2	inner
2312.2.b.j.577.3		4	1.1	even	1	trivial
4624.2.a.bm.1.2		4	68.55	odd	4
4624.2.a.bm.1.3		4	68.47	odd	4

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.765367i q^{3} -1.84776i q^{5} -0.317025i q^{7} +2.41421 q^{9} +O(q^{10})\)	\(q+0.765367i q^{3} -1.84776i q^{5} -0.317025i q^{7} +2.41421 q^{9} -0.765367i q^{11} +1.41421 q^{15} -2.24264 q^{19} +0.242641 q^{21} -6.62567i q^{23} +1.58579 q^{25} +4.14386i q^{27} -5.54328i q^{29} +2.29610i q^{31} +0.585786 q^{33} -0.585786 q^{35} -0.765367i q^{37} -2.48181i q^{41} +10.2426 q^{43} -4.46088i q^{45} -12.8284 q^{47} +6.89949 q^{49} +3.07107 q^{53} -1.41421 q^{55} -1.71644i q^{57} -7.89949 q^{59} +9.87285i q^{61} -0.765367i q^{63} -9.65685 q^{67} +5.07107 q^{69} -9.68714i q^{71} -7.97069i q^{73} +1.21371i q^{75} -0.242641 q^{77} -14.2793i q^{79} +4.07107 q^{81} +10.2426 q^{83} +4.24264 q^{87} +1.65685 q^{89} -1.75736 q^{93} +4.14386i q^{95} -15.9958i q^{97} -1.84776i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} + 8 q^{19} - 16 q^{21} + 12 q^{25} + 8 q^{33} - 8 q^{35} + 24 q^{43} - 40 q^{47} - 12 q^{49} - 16 q^{53} + 8 q^{59} - 16 q^{67} - 8 q^{69} + 16 q^{77} - 12 q^{81} + 24 q^{83} - 16 q^{89} - 24 q^{93}+O(q^{100}) \) 4 * q + 4 * q^9 + 8 * q^19 - 16 * q^21 + 12 * q^25 + 8 * q^33 - 8 * q^35 + 24 * q^43 - 40 * q^47 - 12 * q^49 - 16 * q^53 + 8 * q^59 - 16 * q^67 - 8 * q^69 + 16 * q^77 - 12 * q^81 + 24 * q^83 - 16 * q^89 - 24 * q^93

Introduction

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