LMFDB - Embedded newform 3744.2.j.a.2159.33

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3744,2,Mod(2159,3744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3744.2159"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2159.33
Character	$\chi$	$=$	3744.2159
Dual form			3744.2.j.a.2159.34

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.802827 q^{5} +1.93839i q^{7} -1.56258i q^{11} -1.00000i q^{13} +3.54880i q^{17} -4.68717 q^{19} -8.86375 q^{23} -4.35547 q^{25} +8.14367 q^{29} -10.3913i q^{31} +1.55619i q^{35} -7.05279i q^{37} -4.31330i q^{41} +1.21213 q^{43} -10.6556 q^{47} +3.24266 q^{49} -1.19815 q^{53} -1.25448i q^{55} -8.98709i q^{59} -2.66095i q^{61} -0.802827i q^{65} +9.46686 q^{67} -2.27916 q^{71} -12.1500 q^{73} +3.02888 q^{77} -0.0205095i q^{79} -0.956451i q^{83} +2.84908i q^{85} +4.61332i q^{89} +1.93839 q^{91} -3.76299 q^{95} +15.1020 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.802827	0.359035	0.179518	−	0.983755i	$-0.442546\pi$
$5$	0.802827	0.359035	0.179518	+	0.983755i	$0.442546\pi$
$6$	0	0
$7$	1.93839i	0.732641i	0.930489	+	0.366321i	$0.119383\pi$
$7$	1.93839i	0.732641i	−0.930489	+	0.366321i	$0.880617\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.56258i	− 0.471135i	−0.971858	−	0.235567i	$-0.924305\pi$
$11$	− 1.56258i	− 0.471135i	0.971858	−	0.235567i	$-0.0756948\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.54880i	0.860711i	0.902659	+	0.430355i	$0.141612\pi$
$17$	3.54880i	0.860711i	−0.902659	+	0.430355i	$0.858388\pi$
$18$	0	0
$19$	−4.68717	−1.07531	−0.537655	−	0.843165i	$-0.680690\pi$
$19$	−4.68717	−1.07531	−0.537655	+	0.843165i	$0.680690\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.86375	−1.84822	−0.924110	−	0.382126i	$-0.875192\pi$
$23$	−8.86375	−1.84822	−0.924110	+	0.382126i	$0.875192\pi$
$24$	0	0
$25$	−4.35547	−0.871094
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.14367	1.51224	0.756121	−	0.654432i	$-0.227092\pi$
$29$	8.14367	1.51224	0.756121	+	0.654432i	$0.227092\pi$
$30$	0	0
$31$	− 10.3913i	− 1.86634i	−0.359437	−	0.933169i	$-0.617031\pi$
$31$	− 10.3913i	− 1.86634i	0.359437	−	0.933169i	$-0.382969\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.55619i	0.263044i
$36$	0	0
$37$	− 7.05279i	− 1.15947i	−0.814805	−	0.579736i	$-0.803156\pi$
$37$	− 7.05279i	− 1.15947i	0.814805	−	0.579736i	$-0.196844\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.31330i	− 0.673624i	−0.941572	−	0.336812i	$-0.890651\pi$
$41$	− 4.31330i	− 0.673624i	0.941572	−	0.336812i	$-0.109349\pi$
$42$	0	0
$43$	1.21213	0.184849	0.0924244	−	0.995720i	$-0.470538\pi$
$43$	1.21213	0.184849	0.0924244	+	0.995720i	$0.470538\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.6556	−1.55428	−0.777140	−	0.629328i	$-0.783330\pi$
$47$	−10.6556	−1.55428	−0.777140	+	0.629328i	$0.783330\pi$
$48$	0	0
$49$	3.24266	0.463237
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.19815	−0.164579	−0.0822894	−	0.996608i	$-0.526223\pi$
$53$	−1.19815	−0.164579	−0.0822894	+	0.996608i	$0.526223\pi$
$54$	0	0
$55$	− 1.25448i	− 0.169154i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 8.98709i	− 1.17002i	−0.811026	−	0.585010i	$-0.801091\pi$
$59$	− 8.98709i	− 1.17002i	0.811026	−	0.585010i	$-0.198909\pi$
$60$	0	0
$61$	− 2.66095i	− 0.340700i	−0.985384	−	0.170350i	$-0.945510\pi$
$61$	− 2.66095i	− 0.340700i	0.985384	−	0.170350i	$-0.0544898\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.802827i	− 0.0995785i
$66$	0	0
$67$	9.46686	1.15656	0.578280	−	0.815838i	$-0.303724\pi$
$67$	9.46686	1.15656	0.578280	+	0.815838i	$0.303724\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.27916	−0.270486	−0.135243	−	0.990812i	$-0.543182\pi$
$71$	−2.27916	−0.270486	−0.135243	+	0.990812i	$0.543182\pi$
$72$	0	0
$73$	−12.1500	−1.42205	−0.711027	−	0.703164i	$-0.751770\pi$
$73$	−12.1500	−1.42205	−0.711027	+	0.703164i	$0.751770\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.02888	0.345173
$78$	0	0
$79$	− 0.0205095i	− 0.00230750i	−0.999999	−	0.00115375i	$-0.999633\pi$
$79$	− 0.0205095i	− 0.00230750i	0.999999	−	0.00115375i	$-0.000367250\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 0.956451i	− 0.104984i	−0.998621	−	0.0524921i	$-0.983284\pi$
$83$	− 0.956451i	− 0.104984i	0.998621	−	0.0524921i	$-0.0167164\pi$
$84$	0	0
$85$	2.84908i	0.309026i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.61332i	0.489011i	0.969648	+	0.244505i	$0.0786256\pi$
$89$	4.61332i	0.489011i	−0.969648	+	0.244505i	$0.921374\pi$
$90$	0	0
$91$	1.93839	0.203198
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.76299	−0.386074
$96$	0	0
$97$	15.1020	1.53337	0.766687	−	0.642020i	$-0.221904\pi$
$97$	15.1020	1.53337	0.766687	+	0.642020i	$0.221904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.0727	−1.30079	−0.650393	−	0.759598i	$-0.725396\pi$
$101$	−13.0727	−1.30079	−0.650393	+	0.759598i	$0.725396\pi$
$102$	0	0
$103$	1.32300i	0.130359i	0.997874	+	0.0651793i	$0.0207619\pi$
$103$	1.32300i	0.130359i	−0.997874	+	0.0651793i	$0.979238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.8445i	1.62842i	0.580572	+	0.814209i	$0.302829\pi$
$107$	16.8445i	1.62842i	−0.580572	+	0.814209i	$0.697171\pi$
$108$	0	0
$109$	− 13.9930i	− 1.34029i	−0.742232	−	0.670143i	$-0.766233\pi$
$109$	− 13.9930i	− 1.34029i	0.742232	−	0.670143i	$-0.233767\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.37554i	0.599760i	0.953977	+	0.299880i	$0.0969467\pi$
$113$	6.37554i	0.599760i	−0.953977	+	0.299880i	$0.903053\pi$
$114$	0	0
$115$	−7.11606	−0.663576
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.87895	−0.630592
$120$	0	0
$121$	8.55835	0.778032
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.51083	−0.671789
$126$	0	0
$127$	− 6.01020i	− 0.533319i	−0.963791	−	0.266660i	$-0.914080\pi$
$127$	− 6.01020i	− 0.533319i	0.963791	−	0.266660i	$-0.0859200\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 10.0777i	− 0.880496i	−0.897876	−	0.440248i	$-0.854891\pi$
$131$	− 10.0777i	− 0.880496i	0.897876	−	0.440248i	$-0.145109\pi$
$132$	0	0
$133$	− 9.08554i	− 0.787816i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.71748i	− 0.403042i	−0.979484	−	0.201521i	$-0.935412\pi$
$137$	− 4.71748i	− 0.403042i	0.979484	−	0.201521i	$-0.0645884\pi$
$138$	0	0
$139$	−3.38023	−0.286707	−0.143354	−	0.989672i	$-0.545789\pi$
$139$	−3.38023	−0.286707	−0.143354	+	0.989672i	$0.545789\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.56258	−0.130669
$144$	0	0
$145$	6.53796	0.542948
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.39241	0.523687	0.261843	−	0.965110i	$-0.415670\pi$
$149$	6.39241	0.523687	0.261843	+	0.965110i	$0.415670\pi$
$150$	0	0
$151$	2.93379i	0.238748i	0.992849	+	0.119374i	$0.0380888\pi$
$151$	2.93379i	0.238748i	−0.992849	+	0.119374i	$0.961911\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.34245i	− 0.670081i
$156$	0	0
$157$	− 15.3371i	− 1.22403i	−0.790846	−	0.612016i	$-0.790359\pi$
$157$	− 15.3371i	− 1.22403i	0.790846	−	0.612016i	$-0.209641\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 17.1814i	− 1.35408i
$162$	0	0
$163$	12.2306	0.957974	0.478987	−	0.877822i	$-0.341004\pi$
$163$	12.2306	0.957974	0.478987	+	0.877822i	$0.341004\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.9067	−0.843987	−0.421993	−	0.906599i	$-0.638669\pi$
$167$	−10.9067	−0.843987	−0.421993	+	0.906599i	$0.638669\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.77362	−0.667046	−0.333523	−	0.942742i	$-0.608237\pi$
$173$	−8.77362	−0.667046	−0.333523	+	0.942742i	$0.608237\pi$
$174$	0	0
$175$	− 8.44258i	− 0.638199i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 13.0905i	− 0.978430i	−0.872163	−	0.489215i	$-0.837283\pi$
$179$	− 13.0905i	− 0.978430i	0.872163	−	0.489215i	$-0.162717\pi$
$180$	0	0
$181$	− 21.1127i	− 1.56929i	−0.619945	−	0.784646i	$-0.712845\pi$
$181$	− 21.1127i	− 1.56929i	0.619945	−	0.784646i	$-0.287155\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.66217i	− 0.416291i
$186$	0	0
$187$	5.54528	0.405511
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.5536	0.908349	0.454174	−	0.890913i	$-0.349934\pi$
$191$	12.5536	0.908349	0.454174	+	0.890913i	$0.349934\pi$
$192$	0	0
$193$	−10.9885	−0.790968	−0.395484	−	0.918473i	$-0.629423\pi$
$193$	−10.9885	−0.790968	−0.395484	+	0.918473i	$0.629423\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.4678	−0.745799	−0.372900	−	0.927872i	$-0.621636\pi$
$197$	−10.4678	−0.745799	−0.372900	+	0.927872i	$0.621636\pi$
$198$	0	0
$199$	− 13.1408i	− 0.931525i	−0.884910	−	0.465762i	$-0.845780\pi$
$199$	− 13.1408i	− 0.931525i	0.884910	−	0.465762i	$-0.154220\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.7856i	1.10793i
$204$	0	0
$205$	− 3.46284i	− 0.241855i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.32406i	0.506616i
$210$	0	0
$211$	−13.4229	−0.924071	−0.462036	−	0.886861i	$-0.652881\pi$
$211$	−13.4229	−0.924071	−0.462036	+	0.886861i	$0.652881\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.973135	0.0663672
$216$	0	0
$217$	20.1424	1.36736
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.54880	0.238718
$222$	0	0
$223$	− 2.19718i	− 0.147134i	−0.997290	−	0.0735671i	$-0.976562\pi$
$223$	− 2.19718i	− 0.147134i	0.997290	−	0.0735671i	$-0.0234383\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.70258i	0.643983i	0.946743	+	0.321991i	$0.104352\pi$
$227$	9.70258i	0.643983i	−0.946743	+	0.321991i	$0.895648\pi$
$228$	0	0
$229$	− 4.04664i	− 0.267410i	−0.991021	−	0.133705i	$-0.957313\pi$
$229$	− 4.04664i	− 0.267410i	0.991021	−	0.133705i	$-0.0426874\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.5827i	1.34842i	0.738540	+	0.674210i	$0.235516\pi$
$233$	20.5827i	1.34842i	−0.738540	+	0.674210i	$0.764484\pi$
$234$	0	0
$235$	−8.55461	−0.558041
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.87118	−0.185721	−0.0928605	−	0.995679i	$-0.529601\pi$
$239$	−2.87118	−0.185721	−0.0928605	+	0.995679i	$0.529601\pi$
$240$	0	0
$241$	−16.0104	−1.03132	−0.515661	−	0.856793i	$-0.672454\pi$
$241$	−16.0104	−1.03132	−0.515661	+	0.856793i	$0.672454\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.60329	0.166318
$246$	0	0
$247$	4.68717i	0.298237i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.0749i	0.888399i	0.895928	+	0.444199i	$0.146512\pi$
$251$	14.0749i	0.888399i	−0.895928	+	0.444199i	$0.853488\pi$
$252$	0	0
$253$	13.8503i	0.870761i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.8601i	1.48835i	0.667984	+	0.744176i	$0.267157\pi$
$257$	23.8601i	1.48835i	−0.667984	+	0.744176i	$0.732843\pi$
$258$	0	0
$259$	13.6710	0.849476
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.8432	−1.40857	−0.704286	−	0.709916i	$-0.748733\pi$
$263$	−22.8432	−1.40857	−0.704286	+	0.709916i	$0.748733\pi$
$264$	0	0
$265$	−0.961909	−0.0590896
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.90531	−0.421024	−0.210512	−	0.977591i	$-0.567513\pi$
$269$	−6.90531	−0.421024	−0.210512	+	0.977591i	$0.567513\pi$
$270$	0	0
$271$	11.6435i	0.707293i	0.935379	+	0.353646i	$0.115058\pi$
$271$	11.6435i	0.707293i	−0.935379	+	0.353646i	$0.884942\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.80575i	0.410402i
$276$	0	0
$277$	− 27.1432i	− 1.63088i	−0.578844	−	0.815438i	$-0.696496\pi$
$277$	− 27.1432i	− 1.63088i	0.578844	−	0.815438i	$-0.303504\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 20.3941i	− 1.21661i	−0.793704	−	0.608304i	$-0.791850\pi$
$281$	− 20.3941i	− 1.21661i	0.793704	−	0.608304i	$-0.208150\pi$
$282$	0	0
$283$	−25.1415	−1.49451	−0.747254	−	0.664539i	$-0.768628\pi$
$283$	−25.1415	−1.49451	−0.747254	+	0.664539i	$0.768628\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.36084	0.493525
$288$	0	0
$289$	4.40600	0.259177
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.76876	0.337015	0.168507	−	0.985700i	$-0.446105\pi$
$293$	5.76876	0.337015	0.168507	+	0.985700i	$0.446105\pi$
$294$	0	0
$295$	− 7.21508i	− 0.420078i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.86375i	0.512604i
$300$	0	0
$301$	2.34959i	0.135428i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.13628i	− 0.122323i
$306$	0	0
$307$	16.5948	0.947115	0.473557	−	0.880763i	$-0.342970\pi$
$307$	16.5948	0.947115	0.473557	+	0.880763i	$0.342970\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.27642	0.412608	0.206304	−	0.978488i	$-0.433856\pi$
$311$	7.27642	0.412608	0.206304	+	0.978488i	$0.433856\pi$
$312$	0	0
$313$	4.18644	0.236631	0.118316	−	0.992976i	$-0.462251\pi$
$313$	4.18644	0.236631	0.118316	+	0.992976i	$0.462251\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.8584	−1.84551	−0.922756	−	0.385386i	$-0.874068\pi$
$317$	−32.8584	−1.84551	−0.922756	+	0.385386i	$0.874068\pi$
$318$	0	0
$319$	− 12.7251i	− 0.712470i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16.6338i	− 0.925531i
$324$	0	0
$325$	4.35547i	0.241598i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 20.6547i	− 1.13873i
$330$	0	0
$331$	−11.1690	−0.613903	−0.306951	−	0.951725i	$-0.599309\pi$
$331$	−11.1690	−0.613903	−0.306951	+	0.951725i	$0.599309\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.60025	0.415246
$336$	0	0
$337$	−33.0082	−1.79807	−0.899035	−	0.437876i	$-0.855731\pi$
$337$	−33.0082	−1.79807	−0.899035	+	0.437876i	$0.855731\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.2373	−0.879297
$342$	0	0
$343$	19.8542i	1.07203i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.6685i	− 1.75374i	−0.480731	−	0.876868i	$-0.659629\pi$
$347$	− 32.6685i	− 1.75374i	0.480731	−	0.876868i	$-0.340371\pi$
$348$	0	0
$349$	26.9704i	1.44369i	0.692055	+	0.721845i	$0.256706\pi$
$349$	26.9704i	1.44369i	−0.692055	+	0.721845i	$0.743294\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 13.1552i	− 0.700180i	−0.936716	−	0.350090i	$-0.886151\pi$
$353$	− 13.1552i	− 0.700180i	0.936716	−	0.350090i	$-0.113849\pi$
$354$	0	0
$355$	−1.82977	−0.0971141
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.4676	0.974682	0.487341	−	0.873212i	$-0.337967\pi$
$359$	18.4676	0.974682	0.487341	+	0.873212i	$0.337967\pi$
$360$	0	0
$361$	2.96953	0.156291
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.75439	−0.510568
$366$	0	0
$367$	− 21.3061i	− 1.11217i	−0.831126	−	0.556084i	$-0.812303\pi$
$367$	− 21.3061i	− 1.11217i	0.831126	−	0.556084i	$-0.187697\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.32248i	− 0.120577i
$372$	0	0
$373$	33.9820i	1.75952i	0.475419	+	0.879760i	$0.342297\pi$
$373$	33.9820i	1.75952i	−0.475419	+	0.879760i	$0.657703\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.14367i	− 0.419420i
$378$	0	0
$379$	19.1623	0.984303	0.492151	−	0.870510i	$-0.336211\pi$
$379$	19.1623	0.984303	0.492151	+	0.870510i	$0.336211\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.5985	0.541560	0.270780	−	0.962641i	$-0.412718\pi$
$383$	10.5985	0.541560	0.270780	+	0.962641i	$0.412718\pi$
$384$	0	0
$385$	2.43167	0.123929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.4488	−1.69592	−0.847960	−	0.530060i	$-0.822169\pi$
$389$	−33.4488	−1.69592	−0.847960	+	0.530060i	$0.822169\pi$
$390$	0	0
$391$	− 31.4557i	− 1.59078i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 0.0164656i	0 0.000828474i
$396$	0	0
$397$	− 27.8937i	− 1.39995i	−0.714170	−	0.699973i	$-0.753195\pi$
$397$	− 27.8937i	− 1.39995i	0.714170	−	0.699973i	$-0.246805\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.6472i	1.23082i	0.788205	+	0.615412i	$0.211011\pi$
$401$	24.6472i	1.23082i	−0.788205	+	0.615412i	$0.788989\pi$
$402$	0	0
$403$	−10.3913	−0.517629
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.0205	−0.546267
$408$	0	0
$409$	−10.2882	−0.508721	−0.254360	−	0.967109i	$-0.581865\pi$
$409$	−10.2882	−0.508721	−0.254360	+	0.967109i	$0.581865\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	17.4205	0.857204
$414$	0	0
$415$	− 0.767865i	− 0.0376930i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15.4860i	0.756541i	0.925695	+	0.378270i	$0.123481\pi$
$419$	15.4860i	0.756541i	−0.925695	+	0.378270i	$0.876519\pi$
$420$	0	0
$421$	21.3013i	1.03816i	0.854725	+	0.519080i	$0.173725\pi$
$421$	21.3013i	1.03816i	−0.854725	+	0.519080i	$0.826275\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 15.4567i	− 0.749760i
$426$	0	0
$427$	5.15795	0.249611
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.2911	0.495704	0.247852	−	0.968798i	$-0.420275\pi$
$431$	10.2911	0.495704	0.247852	+	0.968798i	$0.420275\pi$
$432$	0	0
$433$	15.9131	0.764734	0.382367	−	0.924011i	$-0.375109\pi$
$433$	15.9131	0.764734	0.382367	+	0.924011i	$0.375109\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.5459	1.98741
$438$	0	0
$439$	− 22.1289i	− 1.05615i	−0.849197	−	0.528076i	$-0.822914\pi$
$439$	− 22.1289i	− 1.05615i	0.849197	−	0.528076i	$-0.177086\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 15.9949i	− 0.759941i	−0.924999	−	0.379970i	$-0.875934\pi$
$443$	− 15.9949i	− 0.759941i	0.924999	−	0.379970i	$-0.124066\pi$
$444$	0	0
$445$	3.70370i	0.175572i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.99142i	0.377138i	0.982060	+	0.188569i	$0.0603850\pi$
$449$	7.99142i	0.377138i	−0.982060	+	0.188569i	$0.939615\pi$
$450$	0	0
$451$	−6.73986	−0.317368
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.55619	0.0729553
$456$	0	0
$457$	−5.45320	−0.255090	−0.127545	−	0.991833i	$-0.540710\pi$
$457$	−5.45320	−0.255090	−0.127545	+	0.991833i	$0.540710\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.1627	−0.939073	−0.469536	−	0.882913i	$-0.655579\pi$
$461$	−20.1627	−0.939073	−0.469536	+	0.882913i	$0.655579\pi$
$462$	0	0
$463$	− 22.6223i	− 1.05135i	−0.850687	−	0.525673i	$-0.823814\pi$
$463$	− 22.6223i	− 1.05135i	0.850687	−	0.525673i	$-0.176186\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.1778i	1.02626i	0.858310	+	0.513132i	$0.171515\pi$
$467$	22.1778i	1.02626i	−0.858310	+	0.513132i	$0.828485\pi$
$468$	0	0
$469$	18.3504i	0.847344i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.89405i	− 0.0870887i
$474$	0	0
$475$	20.4148	0.936695
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.7032	1.40286	0.701432	−	0.712736i	$-0.252544\pi$
$479$	30.7032	1.40286	0.701432	+	0.712736i	$0.252544\pi$
$480$	0	0
$481$	−7.05279	−0.321579
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.1243	0.550536
$486$	0	0
$487$	30.0364i	1.36108i	0.732711	+	0.680540i	$0.238255\pi$
$487$	30.0364i	1.36108i	−0.732711	+	0.680540i	$0.761745\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.60047i	0.207616i	0.994597	+	0.103808i	$0.0331028\pi$
$491$	4.60047i	0.207616i	−0.994597	+	0.103808i	$0.966897\pi$
$492$	0	0
$493$	28.9003i	1.30160i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.41789i	− 0.198169i
$498$	0	0
$499$	41.5377	1.85948	0.929742	−	0.368212i	$-0.120030\pi$
$499$	41.5377	1.85948	0.929742	+	0.368212i	$0.120030\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.33119	−0.371469	−0.185735	−	0.982600i	$-0.559467\pi$
$503$	−8.33119	−0.371469	−0.185735	+	0.982600i	$0.559467\pi$
$504$	0	0
$505$	−10.4952	−0.467028
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.1613	1.46985	0.734924	−	0.678149i	$-0.237218\pi$
$509$	33.1613	1.46985	0.734924	+	0.678149i	$0.237218\pi$
$510$	0	0
$511$	− 23.5515i	− 1.04186i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.06214i	0.0468033i
$516$	0	0
$517$	16.6502i	0.732275i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.5036i	1.42401i	0.702176	+	0.712004i	$0.252212\pi$
$521$	32.5036i	1.42401i	−0.702176	+	0.712004i	$0.747788\pi$
$522$	0	0
$523$	−11.2030	−0.489872	−0.244936	−	0.969539i	$-0.578767\pi$
$523$	−11.2030	−0.489872	−0.244936	+	0.969539i	$0.578767\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.8768	1.60638
$528$	0	0
$529$	55.5662	2.41592
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.31330	−0.186830
$534$	0	0
$535$	13.5232i	0.584659i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.06690i	− 0.218247i
$540$	0	0
$541$	− 5.13802i	− 0.220901i	−0.993882	−	0.110450i	$-0.964771\pi$
$541$	− 5.13802i	− 0.220901i	0.993882	−	0.110450i	$-0.0352293\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 11.2340i	− 0.481210i
$546$	0	0
$547$	−22.5750	−0.965239	−0.482620	−	0.875830i	$-0.660315\pi$
$547$	−22.5750	−0.965239	−0.482620	+	0.875830i	$0.660315\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−38.1707	−1.62613
$552$	0	0
$553$	0.0397553	0.00169057
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.3702	−1.32920	−0.664599	−	0.747200i	$-0.731398\pi$
$557$	−31.3702	−1.32920	−0.664599	+	0.747200i	$0.731398\pi$
$558$	0	0
$559$	− 1.21213i	− 0.0512678i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.80199i	0.0759450i	0.999279	+	0.0379725i	$0.0120899\pi$
$563$	1.80199i	0.0759450i	−0.999279	+	0.0379725i	$0.987910\pi$
$564$	0	0
$565$	5.11846i	0.215335i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.95840i	0.249789i	0.992170	+	0.124894i	$0.0398592\pi$
$569$	5.95840i	0.249789i	−0.992170	+	0.124894i	$0.960141\pi$
$570$	0	0
$571$	21.0692	0.881718	0.440859	−	0.897576i	$-0.354674\pi$
$571$	21.0692	0.881718	0.440859	+	0.897576i	$0.354674\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.6058	1.60997
$576$	0	0
$577$	−11.1836	−0.465580	−0.232790	−	0.972527i	$-0.574785\pi$
$577$	−11.1836	−0.465580	−0.232790	+	0.972527i	$0.574785\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.85397	0.0769157
$582$	0	0
$583$	1.87220i	0.0775388i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.31524i	0.0542859i	0.999632	+	0.0271429i	$0.00864092\pi$
$587$	1.31524i	0.0542859i	−0.999632	+	0.0271429i	$0.991359\pi$
$588$	0	0
$589$	48.7059i	2.00689i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 34.1415i	− 1.40202i	−0.713150	−	0.701011i	$-0.752732\pi$
$593$	− 34.1415i	− 1.40202i	0.713150	−	0.701011i	$-0.247268\pi$
$594$	0	0
$595$	−5.52261	−0.226405
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.6319	−0.556984	−0.278492	−	0.960439i	$-0.589835\pi$
$599$	−13.6319	−0.556984	−0.278492	+	0.960439i	$0.589835\pi$
$600$	0	0
$601$	−43.0603	−1.75647	−0.878234	−	0.478232i	$-0.841278\pi$
$601$	−43.0603	−1.75647	−0.878234	+	0.478232i	$0.841278\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.87088	0.279341
$606$	0	0
$607$	27.2168i	1.10470i	0.833613	+	0.552349i	$0.186268\pi$
$607$	27.2168i	1.10470i	−0.833613	+	0.552349i	$0.813732\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.6556i	0.431080i
$612$	0	0
$613$	− 3.98645i	− 0.161011i	−0.996754	−	0.0805055i	$-0.974347\pi$
$613$	− 3.98645i	− 0.161011i	0.996754	−	0.0805055i	$-0.0256535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.9892i	− 1.32809i	−0.747691	−	0.664047i	$-0.768837\pi$
$617$	− 32.9892i	− 1.32809i	0.747691	−	0.664047i	$-0.231163\pi$
$618$	0	0
$619$	25.2629	1.01540	0.507701	−	0.861534i	$-0.330496\pi$
$619$	25.2629	1.01540	0.507701	+	0.861534i	$0.330496\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.94239	−0.358269
$624$	0	0
$625$	15.7474	0.629898
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.0289	0.997970
$630$	0	0
$631$	18.6763i	0.743490i	0.928335	+	0.371745i	$0.121240\pi$
$631$	18.6763i	0.743490i	−0.928335	+	0.371745i	$0.878760\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 4.82516i	− 0.191480i
$636$	0	0
$637$	− 3.24266i	− 0.128479i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.11079i	0.241362i	0.992691	+	0.120681i	$0.0385078\pi$
$641$	6.11079i	0.241362i	−0.992691	+	0.120681i	$0.961492\pi$
$642$	0	0
$643$	−11.4130	−0.450085	−0.225043	−	0.974349i	$-0.572252\pi$
$643$	−11.4130	−0.450085	−0.225043	+	0.974349i	$0.572252\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.21134	−0.0476225	−0.0238113	−	0.999716i	$-0.507580\pi$
$647$	−1.21134	−0.0476225	−0.0238113	+	0.999716i	$0.507580\pi$
$648$	0	0
$649$	−14.0430	−0.551237
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.5708	1.39199	0.695996	−	0.718046i	$-0.254963\pi$
$653$	35.5708	1.39199	0.695996	+	0.718046i	$0.254963\pi$
$654$	0	0
$655$	− 8.09068i	− 0.316129i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 15.1253i	− 0.589198i	−0.955621	−	0.294599i	$-0.904814\pi$
$659$	− 15.1253i	− 0.589198i	0.955621	−	0.294599i	$-0.0951860\pi$
$660$	0	0
$661$	− 15.7791i	− 0.613736i	−0.951752	−	0.306868i	$-0.900719\pi$
$661$	− 15.7791i	− 0.613736i	0.951752	−	0.306868i	$-0.0992810\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7.29412i	− 0.282854i
$666$	0	0
$667$	−72.1835	−2.79496
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.15794	−0.160515
$672$	0	0
$673$	−5.40534	−0.208361	−0.104180	−	0.994558i	$-0.533222\pi$
$673$	−5.40534	−0.208361	−0.104180	+	0.994558i	$0.533222\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.8031	0.530495	0.265247	−	0.964180i	$-0.414546\pi$
$677$	13.8031	0.530495	0.265247	+	0.964180i	$0.414546\pi$
$678$	0	0
$679$	29.2735i	1.12341i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.83627i	0.146791i	0.997303	+	0.0733955i	$0.0233835\pi$
$683$	3.83627i	0.146791i	−0.997303	+	0.0733955i	$0.976616\pi$
$684$	0	0
$685$	− 3.78732i	− 0.144706i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.19815i	0.0456459i
$690$	0	0
$691$	9.16893	0.348802	0.174401	−	0.984675i	$-0.444201\pi$
$691$	9.16893	0.348802	0.174401	+	0.984675i	$0.444201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.71374	−0.102938
$696$	0	0
$697$	15.3070	0.579796
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.0528	1.36170	0.680849	−	0.732424i	$-0.261611\pi$
$701$	36.0528	1.36170	0.680849	+	0.732424i	$0.261611\pi$
$702$	0	0
$703$	33.0576i	1.24679i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 25.3400i	− 0.953010i
$708$	0	0
$709$	36.9645i	1.38823i	0.719864	+	0.694115i	$0.244204\pi$
$709$	36.9645i	1.38823i	−0.719864	+	0.694115i	$0.755796\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	92.1062i	3.44941i
$714$	0	0
$715$	−1.25448	−0.0469149
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.5007	0.391609	0.195805	−	0.980643i	$-0.437268\pi$
$719$	10.5007	0.391609	0.195805	+	0.980643i	$0.437268\pi$
$720$	0	0
$721$	−2.56448	−0.0955061
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−35.4695	−1.31730
$726$	0	0
$727$	48.8568i	1.81200i	0.423279	+	0.905999i	$0.360879\pi$
$727$	48.8568i	1.81200i	−0.423279	+	0.905999i	$0.639121\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.30163i	0.159101i
$732$	0	0
$733$	− 41.0047i	− 1.51454i	−0.653099	−	0.757272i	$-0.726532\pi$
$733$	− 41.0047i	− 1.51454i	0.653099	−	0.757272i	$-0.273468\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.7927i	− 0.544896i
$738$	0	0
$739$	−45.7034	−1.68123	−0.840614	−	0.541635i	$-0.817805\pi$
$739$	−45.7034	−1.68123	−0.840614	+	0.541635i	$0.817805\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.54515	−0.203432	−0.101716	−	0.994813i	$-0.532433\pi$
$743$	−5.54515	−0.203432	−0.101716	+	0.994813i	$0.532433\pi$
$744$	0	0
$745$	5.13200	0.188022
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−32.6511	−1.19305
$750$	0	0
$751$	14.7605i	0.538619i	0.963054	+	0.269310i	$0.0867955\pi$
$751$	14.7605i	0.538619i	−0.963054	+	0.269310i	$0.913204\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.35532i	0.0857190i
$756$	0	0
$757$	1.55725i	0.0565992i	0.999599	+	0.0282996i	$0.00900925\pi$
$757$	1.55725i	0.0565992i	−0.999599	+	0.0282996i	$0.990991\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.7799i	1.55077i	0.631488	+	0.775386i	$0.282445\pi$
$761$	42.7799i	1.55077i	−0.631488	+	0.775386i	$0.717555\pi$
$762$	0	0
$763$	27.1238	0.981949
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.98709	−0.324505
$768$	0	0
$769$	9.58861	0.345774	0.172887	−	0.984942i	$-0.444690\pi$
$769$	9.58861	0.345774	0.172887	+	0.984942i	$0.444690\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	43.9844	1.58201	0.791005	−	0.611809i	$-0.209558\pi$
$773$	43.9844	1.58201	0.791005	+	0.611809i	$0.209558\pi$
$774$	0	0
$775$	45.2591i	1.62576i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.2172i	0.724355i
$780$	0	0
$781$	3.56136i	0.127435i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 12.3130i	− 0.439470i
$786$	0	0
$787$	22.7288	0.810194	0.405097	−	0.914274i	$-0.367238\pi$
$787$	22.7288	0.810194	0.405097	+	0.914274i	$0.367238\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.3583	−0.439409
$792$	0	0
$793$	−2.66095	−0.0944931
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.8732	0.597678	0.298839	−	0.954304i	$-0.403401\pi$
$797$	16.8732	0.597678	0.298839	+	0.954304i	$0.403401\pi$
$798$	0	0
$799$	− 37.8146i	− 1.33779i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18.9854i	0.669979i
$804$	0	0
$805$	− 13.7937i	− 0.486163i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.7940i	0.379496i	0.981833	+	0.189748i	$0.0607670\pi$
$809$	10.7940i	0.379496i	−0.981833	+	0.189748i	$0.939233\pi$
$810$	0	0
$811$	40.2601	1.41372	0.706861	−	0.707352i	$-0.250111\pi$
$811$	40.2601	1.41372	0.706861	+	0.707352i	$0.250111\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.81905	0.343946
$816$	0	0
$817$	−5.68148	−0.198770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.9301	−0.765365	−0.382683	−	0.923880i	$-0.625000\pi$
$821$	−21.9301	−0.765365	−0.382683	+	0.923880i	$0.625000\pi$
$822$	0	0
$823$	41.8759i	1.45970i	0.683605	+	0.729852i	$0.260411\pi$
$823$	41.8759i	1.45970i	−0.683605	+	0.729852i	$0.739589\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.4860i	1.26874i	0.773028	+	0.634372i	$0.218741\pi$
$827$	36.4860i	1.26874i	−0.773028	+	0.634372i	$0.781259\pi$
$828$	0	0
$829$	− 26.5146i	− 0.920890i	−0.887688	−	0.460445i	$-0.847690\pi$
$829$	− 26.5146i	− 0.920890i	0.887688	−	0.460445i	$-0.152310\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.5076i	0.398713i
$834$	0	0
$835$	−8.75620	−0.303021
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.41523	−0.325050	−0.162525	−	0.986704i	$-0.551964\pi$
$839$	−9.41523	−0.325050	−0.162525	+	0.986704i	$0.551964\pi$
$840$	0	0
$841$	37.3194	1.28688
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.802827	−0.0276181
$846$	0	0
$847$	16.5894i	0.570018i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	62.5142i	2.14296i
$852$	0	0
$853$	17.6829i	0.605449i	0.953078	+	0.302725i	$0.0978963\pi$
$853$	17.6829i	0.605449i	−0.953078	+	0.302725i	$0.902104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 0.161548i	− 0.00551836i	−0.999996	−	0.00275918i	$-0.999122\pi$
$857$	− 0.161548i	− 0.00551836i	0.999996	−	0.00275918i	$-0.000878275\pi$
$858$	0	0
$859$	−30.8841	−1.05375	−0.526876	−	0.849942i	$-0.676637\pi$
$859$	−30.8841	−1.05375	−0.526876	+	0.849942i	$0.676637\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.8502	1.15227	0.576137	−	0.817353i	$-0.304560\pi$
$863$	33.8502	1.15227	0.576137	+	0.817353i	$0.304560\pi$
$864$	0	0
$865$	−7.04370	−0.239493
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.0320477	−0.00108714
$870$	0	0
$871$	− 9.46686i	− 0.320772i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 14.5589i	− 0.492180i
$876$	0	0
$877$	− 0.270189i	− 0.00912362i	−0.999990	−	0.00456181i	$-0.998548\pi$
$877$	− 0.270189i	− 0.00912362i	0.999990	−	0.00456181i	$-0.00145207\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 13.3907i	− 0.451144i	−0.974226	−	0.225572i	$-0.927575\pi$
$881$	− 13.3907i	− 0.451144i	0.974226	−	0.225572i	$-0.0724251\pi$
$882$	0	0
$883$	4.00234	0.134689	0.0673447	−	0.997730i	$-0.478547\pi$
$883$	4.00234	0.134689	0.0673447	+	0.997730i	$0.478547\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.3603	0.750786	0.375393	−	0.926866i	$-0.377508\pi$
$887$	22.3603	0.750786	0.375393	+	0.926866i	$0.377508\pi$
$888$	0	0
$889$	11.6501	0.390732
$890$	0	0
$891$	0	0
$892$	0	0
$893$	49.9446	1.67133
$894$	0	0
$895$	− 10.5094i	− 0.351291i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 84.6236i	− 2.82236i
$900$	0	0
$901$	− 4.25200i	− 0.141655i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 16.9498i	− 0.563431i
$906$	0	0
$907$	−39.1830	−1.30105	−0.650525	−	0.759484i	$-0.725451\pi$
$907$	−39.1830	−1.30105	−0.650525	+	0.759484i	$0.725451\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.9229	1.19018	0.595089	−	0.803660i	$-0.297117\pi$
$911$	35.9229	1.19018	0.595089	+	0.803660i	$0.297117\pi$
$912$	0	0
$913$	−1.49453	−0.0494617
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.5345	0.645087
$918$	0	0
$919$	7.29406i	0.240609i	0.992737	+	0.120304i	$0.0383871\pi$
$919$	7.29406i	0.240609i	−0.992737	+	0.120304i	$0.961613\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.27916i	0.0750194i
$924$	0	0
$925$	30.7182i	1.01001i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 7.63172i	− 0.250389i	−0.992132	−	0.125194i	$-0.960045\pi$
$929$	− 7.63172i	− 0.250389i	0.992132	−	0.125194i	$-0.0399554\pi$
$930$	0	0
$931$	−15.1989	−0.498123
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.45190	0.145593
$936$	0	0
$937$	32.9234	1.07556	0.537780	−	0.843085i	$-0.319263\pi$
$937$	32.9234	1.07556	0.537780	+	0.843085i	$0.319263\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.2462	1.05119	0.525597	−	0.850733i	$-0.323842\pi$
$941$	32.2462	1.05119	0.525597	+	0.850733i	$0.323842\pi$
$942$	0	0
$943$	38.2320i	1.24501i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34.5486i	− 1.12268i	−0.827585	−	0.561340i	$-0.810286\pi$
$947$	− 34.5486i	− 1.12268i	0.827585	−	0.561340i	$-0.189714\pi$
$948$	0	0
$949$	12.1500i	0.394407i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 12.5966i	− 0.408045i	−0.978966	−	0.204022i	$-0.934598\pi$
$953$	− 12.5966i	− 0.408045i	0.978966	−	0.204022i	$-0.0654015\pi$
$954$	0	0
$955$	10.0784	0.326129
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.14430	0.295285
$960$	0	0
$961$	−76.9798	−2.48322
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.82185	−0.283985
$966$	0	0
$967$	30.2118i	0.971547i	0.874085	+	0.485774i	$0.161462\pi$
$967$	30.2118i	0.971547i	−0.874085	+	0.485774i	$0.838538\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 38.8754i	− 1.24757i	−0.781596	−	0.623785i	$-0.785594\pi$
$971$	− 38.8754i	− 1.24757i	0.781596	−	0.623785i	$-0.214406\pi$
$972$	0	0
$973$	− 6.55218i	− 0.210053i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 14.2864i	− 0.457064i	−0.973536	−	0.228532i	$-0.926607\pi$
$977$	− 14.2864i	− 0.457064i	0.973536	−	0.228532i	$-0.0733925\pi$
$978$	0	0
$979$	7.20867	0.230390
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.9024	−1.75112	−0.875558	−	0.483113i	$-0.839506\pi$
$983$	−54.9024	−1.75112	−0.875558	+	0.483113i	$0.839506\pi$
$984$	0	0
$985$	−8.40383	−0.267768
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.7441	−0.341641
$990$	0	0
$991$	2.25689i	0.0716924i	0.999357	+	0.0358462i	$0.0114126\pi$
$991$	2.25689i	0.0716924i	−0.999357	+	0.0358462i	$0.988587\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 10.5498i	− 0.334450i
$996$	0	0
$997$	− 20.1398i	− 0.637833i	−0.947783	−	0.318916i	$-0.896681\pi$
$997$	− 20.1398i	− 0.637833i	0.947783	−	0.318916i	$-0.103319\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.j.a.2159.33		48
3.2	odd	2	inner	3744.2.j.a.2159.15		48
4.3	odd	2		936.2.j.a.755.26	yes	48
8.3	odd	2	inner	3744.2.j.a.2159.16		48
8.5	even	2		936.2.j.a.755.24	yes	48
12.11	even	2		936.2.j.a.755.23	✓	48
24.5	odd	2		936.2.j.a.755.25	yes	48
24.11	even	2	inner	3744.2.j.a.2159.34		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.j.a.755.23	✓	48	12.11	even	2
936.2.j.a.755.24	yes	48	8.5	even	2
936.2.j.a.755.25	yes	48	24.5	odd	2
936.2.j.a.755.26	yes	48	4.3	odd	2
3744.2.j.a.2159.15		48	3.2	odd	2	inner
3744.2.j.a.2159.16		48	8.3	odd	2	inner
3744.2.j.a.2159.33		48	1.1	even	1	trivial
3744.2.j.a.2159.34		48	24.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.802827 q^{5} +1.93839i q^{7} -1.56258i q^{11} -1.00000i q^{13} +3.54880i q^{17} -4.68717 q^{19} -8.86375 q^{23} -4.35547 q^{25} +8.14367 q^{29} -10.3913i q^{31} +1.55619i q^{35} -7.05279i q^{37} -4.31330i q^{41} +1.21213 q^{43} -10.6556 q^{47} +3.24266 q^{49} -1.19815 q^{53} -1.25448i q^{55} -8.98709i q^{59} -2.66095i q^{61} -0.802827i q^{65} +9.46686 q^{67} -2.27916 q^{71} -12.1500 q^{73} +3.02888 q^{77} -0.0205095i q^{79} -0.956451i q^{83} +2.84908i q^{85} +4.61332i q^{89} +1.93839 q^{91} -3.76299 q^{95} +15.1020 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

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