LMFDB - Embedded newform 3744.2.j.a.2159.41

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			2159.41
Character	$\chi$	$=$	3744.2159
Dual form			3744.2.j.a.2159.42

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23777 q^{5} -4.33596i q^{7} +1.18715i q^{11} -1.00000i q^{13} -0.0944593i q^{17} -2.27921 q^{19} -7.64873 q^{23} +5.48318 q^{25} -4.80155 q^{29} -7.57575i q^{31} -14.0389i q^{35} -11.6373i q^{37} +3.47950i q^{41} +9.77740 q^{43} +1.62643 q^{47} -11.8006 q^{49} +10.5857 q^{53} +3.84372i q^{55} -5.55495i q^{59} -8.82142i q^{61} -3.23777i q^{65} -0.890006 q^{67} +5.75108 q^{71} -13.0699 q^{73} +5.14743 q^{77} +15.1770i q^{79} -6.08355i q^{83} -0.305838i q^{85} -2.52869i q^{89} -4.33596 q^{91} -7.37955 q^{95} -17.4724 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.23777	1.44798	0.723988	−	0.689812i	$-0.242307\pi$
$5$	3.23777	1.44798	0.723988	+	0.689812i	$0.242307\pi$
$6$	0	0
$7$	− 4.33596i	− 1.63884i	−0.573194	−	0.819420i	$-0.694296\pi$
$7$	− 4.33596i	− 1.63884i	0.573194	−	0.819420i	$-0.305704\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.18715i	0.357939i	0.983855	+	0.178969i	$0.0572763\pi$
$11$	1.18715i	0.357939i	−0.983855	+	0.178969i	$0.942724\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.0944593i	− 0.0229097i	−0.999934	−	0.0114549i	$-0.996354\pi$
$17$	− 0.0944593i	− 0.0229097i	0.999934	−	0.0114549i	$-0.00364628\pi$
$18$	0	0
$19$	−2.27921	−0.522886	−0.261443	−	0.965219i	$-0.584198\pi$
$19$	−2.27921	−0.522886	−0.261443	+	0.965219i	$0.584198\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.64873	−1.59487	−0.797436	−	0.603404i	$-0.793811\pi$
$23$	−7.64873	−1.59487	−0.797436	+	0.603404i	$0.793811\pi$
$24$	0	0
$25$	5.48318	1.09664
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.80155	−0.891625	−0.445812	−	0.895126i	$-0.647085\pi$
$29$	−4.80155	−0.891625	−0.445812	+	0.895126i	$0.647085\pi$
$30$	0	0
$31$	− 7.57575i	− 1.36065i	−0.732913	−	0.680323i	$-0.761840\pi$
$31$	− 7.57575i	− 1.36065i	0.732913	−	0.680323i	$-0.238160\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 14.0389i	− 2.37300i
$36$	0	0
$37$	− 11.6373i	− 1.91316i	−0.291478	−	0.956578i	$-0.594147\pi$
$37$	− 11.6373i	− 1.91316i	0.291478	−	0.956578i	$-0.405853\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.47950i	0.543407i	0.962381	+	0.271703i	$0.0875870\pi$
$41$	3.47950i	0.543407i	−0.962381	+	0.271703i	$0.912413\pi$
$42$	0	0
$43$	9.77740	1.49104	0.745520	−	0.666484i	$-0.232201\pi$
$43$	9.77740	1.49104	0.745520	+	0.666484i	$0.232201\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.62643	0.237239	0.118620	−	0.992940i	$-0.462153\pi$
$47$	1.62643	0.237239	0.118620	+	0.992940i	$0.462153\pi$
$48$	0	0
$49$	−11.8006	−1.68580
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.5857	1.45406	0.727032	−	0.686604i	$-0.240899\pi$
$53$	10.5857	1.45406	0.727032	+	0.686604i	$0.240899\pi$
$54$	0	0
$55$	3.84372i	0.518287i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.55495i	− 0.723193i	−0.932335	−	0.361596i	$-0.882232\pi$
$59$	− 5.55495i	− 0.723193i	0.932335	−	0.361596i	$-0.117768\pi$
$60$	0	0
$61$	− 8.82142i	− 1.12947i	−0.825273	−	0.564734i	$-0.808979\pi$
$61$	− 8.82142i	− 1.12947i	0.825273	−	0.564734i	$-0.191021\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.23777i	− 0.401596i
$66$	0	0
$67$	−0.890006	−0.108732	−0.0543658	−	0.998521i	$-0.517314\pi$
$67$	−0.890006	−0.108732	−0.0543658	+	0.998521i	$0.517314\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.75108	0.682528	0.341264	−	0.939968i	$-0.389145\pi$
$71$	5.75108	0.682528	0.341264	+	0.939968i	$0.389145\pi$
$72$	0	0
$73$	−13.0699	−1.52972	−0.764859	−	0.644197i	$-0.777191\pi$
$73$	−13.0699	−1.52972	−0.764859	+	0.644197i	$0.777191\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.14743	0.586604
$78$	0	0
$79$	15.1770i	1.70754i	0.520646	+	0.853772i	$0.325691\pi$
$79$	15.1770i	1.70754i	−0.520646	+	0.853772i	$0.674309\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.08355i	− 0.667756i	−0.942616	−	0.333878i	$-0.891643\pi$
$83$	− 6.08355i	− 0.667756i	0.942616	−	0.333878i	$-0.108357\pi$
$84$	0	0
$85$	− 0.305838i	− 0.0331728i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 2.52869i	− 0.268041i	−0.990979	−	0.134020i	$-0.957211\pi$
$89$	− 2.52869i	− 0.268041i	0.990979	−	0.134020i	$-0.0427887\pi$
$90$	0	0
$91$	−4.33596	−0.454532
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.37955	−0.757126
$96$	0	0
$97$	−17.4724	−1.77406	−0.887028	−	0.461716i	$-0.847234\pi$
$97$	−17.4724	−1.77406	−0.887028	+	0.461716i	$0.847234\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.24871	−0.721273	−0.360637	−	0.932706i	$-0.617441\pi$
$101$	−7.24871	−0.721273	−0.360637	+	0.932706i	$0.617441\pi$
$102$	0	0
$103$	0.154510i	0.0152243i	0.999971	+	0.00761215i	$0.00242305\pi$
$103$	0.154510i	0.0152243i	−0.999971	+	0.00761215i	$0.997577\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.14804i	0.207659i	0.994595	+	0.103829i	$0.0331096\pi$
$107$	2.14804i	0.207659i	−0.994595	+	0.103829i	$0.966890\pi$
$108$	0	0
$109$	12.8734i	1.23305i	0.787337	+	0.616523i	$0.211459\pi$
$109$	12.8734i	1.23305i	−0.787337	+	0.616523i	$0.788541\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.38438i	0.600592i	0.953846	+	0.300296i	$0.0970854\pi$
$113$	6.38438i	0.600592i	−0.953846	+	0.300296i	$0.902915\pi$
$114$	0	0
$115$	−24.7649	−2.30934
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.409572	−0.0375454
$120$	0	0
$121$	9.59068	0.871880
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.56441	0.139925
$126$	0	0
$127$	− 8.10854i	− 0.719517i	−0.933045	−	0.359758i	$-0.882859\pi$
$127$	− 8.10854i	− 0.719517i	0.933045	−	0.359758i	$-0.117141\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 14.9127i	− 1.30293i	−0.758679	−	0.651465i	$-0.774155\pi$
$131$	− 14.9127i	− 1.30293i	0.758679	−	0.651465i	$-0.225845\pi$
$132$	0	0
$133$	9.88255i	0.856926i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.27430i	− 0.621486i	−0.950494	−	0.310743i	$-0.899422\pi$
$137$	− 7.27430i	− 0.621486i	0.950494	−	0.310743i	$-0.100578\pi$
$138$	0	0
$139$	−0.428209	−0.0363202	−0.0181601	−	0.999835i	$-0.505781\pi$
$139$	−0.428209	−0.0363202	−0.0181601	+	0.999835i	$0.505781\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.18715	0.0992743
$144$	0	0
$145$	−15.5463	−1.29105
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.5923	0.949682	0.474841	−	0.880072i	$-0.342506\pi$
$149$	11.5923	0.949682	0.474841	+	0.880072i	$0.342506\pi$
$150$	0	0
$151$	− 7.87771i	− 0.641079i	−0.947235	−	0.320540i	$-0.896136\pi$
$151$	− 7.87771i	− 0.641079i	0.947235	−	0.320540i	$-0.103864\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 24.5286i	− 1.97018i
$156$	0	0
$157$	− 3.66776i	− 0.292719i	−0.989231	−	0.146360i	$-0.953244\pi$
$157$	− 3.66776i	− 0.292719i	0.989231	−	0.146360i	$-0.0467557\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	33.1646i	2.61374i
$162$	0	0
$163$	1.90387	0.149123	0.0745614	−	0.997216i	$-0.476244\pi$
$163$	1.90387	0.149123	0.0745614	+	0.997216i	$0.476244\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.6807	1.52294	0.761470	−	0.648200i	$-0.224478\pi$
$167$	19.6807	1.52294	0.761470	+	0.648200i	$0.224478\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.5528	1.03040	0.515202	−	0.857069i	$-0.327717\pi$
$173$	13.5528	1.03040	0.515202	+	0.857069i	$0.327717\pi$
$174$	0	0
$175$	− 23.7748i	− 1.79721i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.1894i	1.73326i	0.498954	+	0.866629i	$0.333718\pi$
$179$	23.1894i	1.73326i	−0.498954	+	0.866629i	$0.666282\pi$
$180$	0	0
$181$	8.05640i	0.598827i	0.954123	+	0.299414i	$0.0967911\pi$
$181$	8.05640i	0.598827i	−0.954123	+	0.299414i	$0.903209\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 37.6788i	− 2.77020i
$186$	0	0
$187$	0.112137	0.00820028
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.0358	0.798521	0.399261	−	0.916838i	$-0.369267\pi$
$191$	11.0358	0.798521	0.399261	+	0.916838i	$0.369267\pi$
$192$	0	0
$193$	−7.60726	−0.547583	−0.273791	−	0.961789i	$-0.588278\pi$
$193$	−7.60726	−0.547583	−0.273791	+	0.961789i	$0.588278\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.65416	−0.260348	−0.130174	−	0.991491i	$-0.541554\pi$
$197$	−3.65416	−0.260348	−0.130174	+	0.991491i	$0.541554\pi$
$198$	0	0
$199$	5.50005i	0.389888i	0.980814	+	0.194944i	$0.0624525\pi$
$199$	5.50005i	0.389888i	−0.980814	+	0.194944i	$0.937547\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.8193i	1.46123i
$204$	0	0
$205$	11.2658i	0.786840i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.70575i	− 0.187161i
$210$	0	0
$211$	−5.48462	−0.377577	−0.188788	−	0.982018i	$-0.560456\pi$
$211$	−5.48462	−0.377577	−0.188788	+	0.982018i	$0.560456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	31.6570	2.15899
$216$	0	0
$217$	−32.8482	−2.22988
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.0944593	−0.00635402
$222$	0	0
$223$	− 23.2910i	− 1.55968i	−0.625979	−	0.779840i	$-0.715301\pi$
$223$	− 23.2910i	− 1.55968i	0.625979	−	0.779840i	$-0.284699\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.24360i	− 0.613519i	−0.951787	−	0.306760i	$-0.900755\pi$
$227$	− 9.24360i	− 0.613519i	0.951787	−	0.306760i	$-0.0992448\pi$
$228$	0	0
$229$	− 4.11450i	− 0.271894i	−0.990716	−	0.135947i	$-0.956592\pi$
$229$	− 4.11450i	− 0.271894i	0.990716	−	0.135947i	$-0.0434077\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 0.938538i	− 0.0614857i	−0.999527	−	0.0307428i	$-0.990213\pi$
$233$	− 0.938538i	− 0.0614857i	0.999527	−	0.0307428i	$-0.00978729\pi$
$234$	0	0
$235$	5.26601	0.343517
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.8576	−1.54322	−0.771609	−	0.636098i	$-0.780548\pi$
$239$	−23.8576	−1.54322	−0.771609	+	0.636098i	$0.780548\pi$
$240$	0	0
$241$	−6.67466	−0.429953	−0.214976	−	0.976619i	$-0.568967\pi$
$241$	−6.67466	−0.429953	−0.214976	+	0.976619i	$0.568967\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−38.2076	−2.44099
$246$	0	0
$247$	2.27921i	0.145022i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.56246i	0.414219i	0.978318	+	0.207109i	$0.0664056\pi$
$251$	6.56246i	0.414219i	−0.978318	+	0.207109i	$0.933594\pi$
$252$	0	0
$253$	− 9.08018i	− 0.570866i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2751i	1.20235i	0.799117	+	0.601175i	$0.205301\pi$
$257$	19.2751i	1.20235i	−0.799117	+	0.601175i	$0.794699\pi$
$258$	0	0
$259$	−50.4588	−3.13535
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.6440	1.70460	0.852302	−	0.523050i	$-0.175206\pi$
$263$	27.6440	1.70460	0.852302	+	0.523050i	$0.175206\pi$
$264$	0	0
$265$	34.2742	2.10545
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.3377	1.48390	0.741948	−	0.670457i	$-0.233902\pi$
$269$	24.3377	1.48390	0.741948	+	0.670457i	$0.233902\pi$
$270$	0	0
$271$	0.648615i	0.0394006i	0.999806	+	0.0197003i	$0.00627120\pi$
$271$	0.648615i	0.0394006i	−0.999806	+	0.0197003i	$0.993729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.50934i	0.392528i
$276$	0	0
$277$	2.48540i	0.149333i	0.997209	+	0.0746665i	$0.0237892\pi$
$277$	2.48540i	0.149333i	−0.997209	+	0.0746665i	$0.976211\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 26.7449i	− 1.59547i	−0.603011	−	0.797733i	$-0.706032\pi$
$281$	− 26.7449i	− 1.59547i	0.603011	−	0.797733i	$-0.293968\pi$
$282$	0	0
$283$	23.1975	1.37895	0.689474	−	0.724310i	$-0.257842\pi$
$283$	23.1975	1.37895	0.689474	+	0.724310i	$0.257842\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.0870	0.890556
$288$	0	0
$289$	16.9911	0.999475
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.7476	1.62103	0.810516	−	0.585716i	$-0.199187\pi$
$293$	27.7476	1.62103	0.810516	+	0.585716i	$0.199187\pi$
$294$	0	0
$295$	− 17.9857i	− 1.04717i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.64873i	0.442338i
$300$	0	0
$301$	− 42.3944i	− 2.44357i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 28.5618i	− 1.63544i
$306$	0	0
$307$	4.23451	0.241676	0.120838	−	0.992672i	$-0.461442\pi$
$307$	4.23451	0.241676	0.120838	+	0.992672i	$0.461442\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−29.1156	−1.65099	−0.825496	−	0.564408i	$-0.809104\pi$
$311$	−29.1156	−1.65099	−0.825496	+	0.564408i	$0.809104\pi$
$312$	0	0
$313$	23.2500	1.31417	0.657083	−	0.753818i	$-0.271790\pi$
$313$	23.2500	1.31417	0.657083	+	0.753818i	$0.271790\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.81779	−0.382925	−0.191463	−	0.981500i	$-0.561323\pi$
$317$	−6.81779	−0.382925	−0.191463	+	0.981500i	$0.561323\pi$
$318$	0	0
$319$	− 5.70015i	− 0.319147i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.215292i	0.0119792i
$324$	0	0
$325$	− 5.48318i	− 0.304152i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.05214i	− 0.388797i
$330$	0	0
$331$	9.11133	0.500804	0.250402	−	0.968142i	$-0.419437\pi$
$331$	9.11133	0.500804	0.250402	+	0.968142i	$0.419437\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.88164	−0.157441
$336$	0	0
$337$	22.2228	1.21055	0.605276	−	0.796016i	$-0.293063\pi$
$337$	22.2228	1.21055	0.605276	+	0.796016i	$0.293063\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.99354	0.487028
$342$	0	0
$343$	20.8151i	1.12391i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.21378i	0.0651593i	0.999469	+	0.0325796i	$0.0103723\pi$
$347$	1.21378i	0.0651593i	−0.999469	+	0.0325796i	$0.989628\pi$
$348$	0	0
$349$	3.97729i	0.212899i	0.994318	+	0.106450i	$0.0339483\pi$
$349$	3.97729i	0.212899i	−0.994318	+	0.106450i	$0.966052\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.75251i	− 0.412625i	−0.978486	−	0.206312i	$-0.933854\pi$
$353$	− 7.75251i	− 0.412625i	0.978486	−	0.206312i	$-0.0661463\pi$
$354$	0	0
$355$	18.6207	0.988284
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.3649	−1.18037	−0.590187	−	0.807267i	$-0.700946\pi$
$359$	−22.3649	−1.18037	−0.590187	+	0.807267i	$0.700946\pi$
$360$	0	0
$361$	−13.8052	−0.726591
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−42.3174	−2.21500
$366$	0	0
$367$	14.6130i	0.762793i	0.924411	+	0.381397i	$0.124557\pi$
$367$	14.6130i	0.762793i	−0.924411	+	0.381397i	$0.875443\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 45.8994i	− 2.38298i
$372$	0	0
$373$	− 35.5065i	− 1.83846i	−0.393722	−	0.919229i	$-0.628813\pi$
$373$	− 35.5065i	− 1.83846i	0.393722	−	0.919229i	$-0.371187\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.80155i	0.247292i
$378$	0	0
$379$	2.22424	0.114251	0.0571257	−	0.998367i	$-0.481806\pi$
$379$	2.22424	0.114251	0.0571257	+	0.998367i	$0.481806\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−37.1436	−1.89795	−0.948976	−	0.315349i	$-0.897879\pi$
$383$	−37.1436	−1.89795	−0.948976	+	0.315349i	$0.897879\pi$
$384$	0	0
$385$	16.6662	0.849389
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.1794	0.617519	0.308760	−	0.951140i	$-0.400086\pi$
$389$	12.1794	0.617519	0.308760	+	0.951140i	$0.400086\pi$
$390$	0	0
$391$	0.722494i	0.0365381i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	49.1397i	2.47248i
$396$	0	0
$397$	20.5977i	1.03377i	0.856054	+	0.516886i	$0.172909\pi$
$397$	20.5977i	1.03377i	−0.856054	+	0.516886i	$0.827091\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.1123i	0.554923i	0.960737	+	0.277462i	$0.0894931\pi$
$401$	11.1123i	0.554923i	−0.960737	+	0.277462i	$0.910507\pi$
$402$	0	0
$403$	−7.57575	−0.377375
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.8152	0.684792
$408$	0	0
$409$	−30.0867	−1.48769	−0.743846	−	0.668351i	$-0.767000\pi$
$409$	−30.0867	−1.48769	−0.743846	+	0.668351i	$0.767000\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−24.0861	−1.18520
$414$	0	0
$415$	− 19.6972i	− 0.966895i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 16.9123i	− 0.826219i	−0.910681	−	0.413109i	$-0.864443\pi$
$419$	− 16.9123i	− 0.826219i	0.910681	−	0.413109i	$-0.135557\pi$
$420$	0	0
$421$	− 22.0720i	− 1.07572i	−0.843033	−	0.537862i	$-0.819232\pi$
$421$	− 22.0720i	− 1.07572i	0.843033	−	0.537862i	$-0.180768\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 0.517937i	− 0.0251236i
$426$	0	0
$427$	−38.2494	−1.85102
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.8351	0.618247	0.309123	−	0.951022i	$-0.399964\pi$
$431$	12.8351	0.618247	0.309123	+	0.951022i	$0.399964\pi$
$432$	0	0
$433$	−22.3461	−1.07389	−0.536943	−	0.843618i	$-0.680421\pi$
$433$	−22.3461	−1.07389	−0.536943	+	0.843618i	$0.680421\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.4330	0.833935
$438$	0	0
$439$	30.4945i	1.45542i	0.685884	+	0.727711i	$0.259416\pi$
$439$	30.4945i	1.45542i	−0.685884	+	0.727711i	$0.740584\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.66810i	0.126765i	0.997989	+	0.0633826i	$0.0201888\pi$
$443$	2.66810i	0.126765i	−0.997989	+	0.0633826i	$0.979811\pi$
$444$	0	0
$445$	− 8.18733i	− 0.388116i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 38.8868i	− 1.83518i	−0.397526	−	0.917591i	$-0.630131\pi$
$449$	− 38.8868i	− 1.83518i	0.397526	−	0.917591i	$-0.369869\pi$
$450$	0	0
$451$	−4.13068	−0.194506
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.0389	−0.658152
$456$	0	0
$457$	31.8203	1.48849	0.744246	−	0.667905i	$-0.232809\pi$
$457$	31.8203	1.48849	0.744246	+	0.667905i	$0.232809\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.9499	1.20861	0.604303	−	0.796755i	$-0.293452\pi$
$461$	25.9499	1.20861	0.604303	+	0.796755i	$0.293452\pi$
$462$	0	0
$463$	2.03696i	0.0946657i	0.998879	+	0.0473329i	$0.0150721\pi$
$463$	2.03696i	0.0946657i	−0.998879	+	0.0473329i	$0.984928\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.91994i	0.181393i	0.995879	+	0.0906967i	$0.0289094\pi$
$467$	3.91994i	0.181393i	−0.995879	+	0.0906967i	$0.971091\pi$
$468$	0	0
$469$	3.85903i	0.178194i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.6072i	0.533700i
$474$	0	0
$475$	−12.4973	−0.573415
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.5822	−1.03181	−0.515904	−	0.856646i	$-0.672544\pi$
$479$	−22.5822	−1.03181	−0.515904	+	0.856646i	$0.672544\pi$
$480$	0	0
$481$	−11.6373	−0.530614
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−56.5717	−2.56879
$486$	0	0
$487$	10.9764i	0.497386i	0.968582	+	0.248693i	$0.0800011\pi$
$487$	10.9764i	0.497386i	−0.968582	+	0.248693i	$0.919999\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.57966i	0.116419i	0.998304	+	0.0582093i	$0.0185391\pi$
$491$	2.57966i	0.116419i	−0.998304	+	0.0582093i	$0.981461\pi$
$492$	0	0
$493$	0.453551i	0.0204269i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 24.9365i	− 1.11855i
$498$	0	0
$499$	−5.21886	−0.233628	−0.116814	−	0.993154i	$-0.537268\pi$
$499$	−5.21886	−0.233628	−0.116814	+	0.993154i	$0.537268\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.0588	1.25108	0.625539	−	0.780193i	$-0.284879\pi$
$503$	28.0588	1.25108	0.625539	+	0.780193i	$0.284879\pi$
$504$	0	0
$505$	−23.4697	−1.04439
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.51782	−0.421870	−0.210935	−	0.977500i	$-0.567651\pi$
$509$	−9.51782	−0.421870	−0.210935	+	0.977500i	$0.567651\pi$
$510$	0	0
$511$	56.6707i	2.50696i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.500268i	0.0220444i
$516$	0	0
$517$	1.93081i	0.0849171i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.21823i	0.140993i	0.997512	+	0.0704966i	$0.0224584\pi$
$521$	3.21823i	0.140993i	−0.997512	+	0.0704966i	$0.977542\pi$
$522$	0	0
$523$	23.4163	1.02392	0.511962	−	0.859008i	$-0.328919\pi$
$523$	23.4163	1.02392	0.511962	+	0.859008i	$0.328919\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.715600	−0.0311720
$528$	0	0
$529$	35.5031	1.54362
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.47950	0.150714
$534$	0	0
$535$	6.95487i	0.300685i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 14.0090i	− 0.603411i
$540$	0	0
$541$	31.3574i	1.34816i	0.738659	+	0.674079i	$0.235459\pi$
$541$	31.3574i	1.34816i	−0.738659	+	0.674079i	$0.764541\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	41.6811i	1.78542i
$546$	0	0
$547$	20.2955	0.867774	0.433887	−	0.900967i	$-0.357142\pi$
$547$	20.2955	0.867774	0.433887	+	0.900967i	$0.357142\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.9437	0.466218
$552$	0	0
$553$	65.8069	2.79839
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.5330	−1.20898	−0.604492	−	0.796611i	$-0.706624\pi$
$557$	−28.5330	−1.20898	−0.604492	+	0.796611i	$0.706624\pi$
$558$	0	0
$559$	− 9.77740i	− 0.413540i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.5762i	0.951474i	0.879588	+	0.475737i	$0.157819\pi$
$563$	22.5762i	0.951474i	−0.879588	+	0.475737i	$0.842181\pi$
$564$	0	0
$565$	20.6712i	0.869643i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 2.31847i	− 0.0971951i	−0.998818	−	0.0485976i	$-0.984525\pi$
$569$	− 2.31847i	− 0.0971951i	0.998818	−	0.0485976i	$-0.0154752\pi$
$570$	0	0
$571$	8.29865	0.347288	0.173644	−	0.984809i	$-0.444446\pi$
$571$	8.29865	0.347288	0.173644	+	0.984809i	$0.444446\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−41.9394	−1.74899
$576$	0	0
$577$	24.1082	1.00364	0.501818	−	0.864973i	$-0.332665\pi$
$577$	24.1082	1.00364	0.501818	+	0.864973i	$0.332665\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−26.3780	−1.09435
$582$	0	0
$583$	12.5668i	0.520465i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.7910i	1.68362i	0.539771	+	0.841812i	$0.318511\pi$
$587$	40.7910i	1.68362i	−0.539771	+	0.841812i	$0.681489\pi$
$588$	0	0
$589$	17.2667i	0.711462i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.44136i	0.387710i	0.981030	+	0.193855i	$0.0620992\pi$
$593$	9.44136i	0.387710i	−0.981030	+	0.193855i	$0.937901\pi$
$594$	0	0
$595$	−1.32610	−0.0543648
$596$	0	0
$597$	0	0
$598$	0	0
$599$	41.8793	1.71114	0.855572	−	0.517684i	$-0.173206\pi$
$599$	41.8793	1.71114	0.855572	+	0.517684i	$0.173206\pi$
$600$	0	0
$601$	−39.8407	−1.62514	−0.812568	−	0.582866i	$-0.801931\pi$
$601$	−39.8407	−1.62514	−0.812568	+	0.582866i	$0.801931\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	31.0524	1.26246
$606$	0	0
$607$	− 19.3389i	− 0.784944i	−0.919764	−	0.392472i	$-0.871620\pi$
$607$	− 19.3389i	− 0.784944i	0.919764	−	0.392472i	$-0.128380\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.62643i	− 0.0657983i
$612$	0	0
$613$	42.2369i	1.70593i	0.521967	+	0.852966i	$0.325199\pi$
$613$	42.2369i	1.70593i	−0.521967	+	0.852966i	$0.674801\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.7584i	0.714927i	0.933927	+	0.357464i	$0.116358\pi$
$617$	17.7584i	0.714927i	−0.933927	+	0.357464i	$0.883642\pi$
$618$	0	0
$619$	−24.6547	−0.990956	−0.495478	−	0.868620i	$-0.665007\pi$
$619$	−24.6547	−0.990956	−0.495478	+	0.868620i	$0.665007\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.9643	−0.439276
$624$	0	0
$625$	−22.3507	−0.894027
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.09925	−0.0438299
$630$	0	0
$631$	7.10652i	0.282906i	0.989945	+	0.141453i	$0.0451774\pi$
$631$	7.10652i	0.282906i	−0.989945	+	0.141453i	$0.954823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 26.2536i	− 1.04184i
$636$	0	0
$637$	11.8006i	0.467555i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 26.3422i	− 1.04046i	−0.854028	−	0.520228i	$-0.825847\pi$
$641$	− 26.3422i	− 1.04046i	0.854028	−	0.520228i	$-0.174153\pi$
$642$	0	0
$643$	−15.8987	−0.626983	−0.313491	−	0.949591i	$-0.601499\pi$
$643$	−15.8987	−0.626983	−0.313491	+	0.949591i	$0.601499\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.85006	−0.387246	−0.193623	−	0.981076i	$-0.562024\pi$
$647$	−9.85006	−0.387246	−0.193623	+	0.981076i	$0.562024\pi$
$648$	0	0
$649$	6.59455	0.258859
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.8492	0.581094	0.290547	−	0.956861i	$-0.406163\pi$
$653$	14.8492	0.581094	0.290547	+	0.956861i	$0.406163\pi$
$654$	0	0
$655$	− 48.2840i	− 1.88661i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.2506i	1.64585i	0.568149	+	0.822925i	$0.307660\pi$
$659$	42.2506i	1.64585i	−0.568149	+	0.822925i	$0.692340\pi$
$660$	0	0
$661$	− 20.8921i	− 0.812608i	−0.913738	−	0.406304i	$-0.866817\pi$
$661$	− 20.8921i	− 0.812608i	0.913738	−	0.406304i	$-0.133183\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	31.9974i	1.24081i
$666$	0	0
$667$	36.7258	1.42203
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.4723	0.404280
$672$	0	0
$673$	4.04031	0.155742	0.0778712	−	0.996963i	$-0.475188\pi$
$673$	4.04031	0.155742	0.0778712	+	0.996963i	$0.475188\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.6247	0.984836	0.492418	−	0.870359i	$-0.336113\pi$
$677$	25.6247	0.984836	0.492418	+	0.870359i	$0.336113\pi$
$678$	0	0
$679$	75.7597i	2.90739i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.10338i	− 0.118747i	−0.998236	−	0.0593737i	$-0.981090\pi$
$683$	− 3.10338i	− 0.118747i	0.998236	−	0.0593737i	$-0.0189104\pi$
$684$	0	0
$685$	− 23.5525i	− 0.899896i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 10.5857i	− 0.403285i
$690$	0	0
$691$	−15.5146	−0.590202	−0.295101	−	0.955466i	$-0.595353\pi$
$691$	−15.5146	−0.590202	−0.295101	+	0.955466i	$0.595353\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.38644	−0.0525908
$696$	0	0
$697$	0.328671	0.0124493
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.73072	0.367524	0.183762	−	0.982971i	$-0.441172\pi$
$701$	9.73072	0.367524	0.183762	+	0.982971i	$0.441172\pi$
$702$	0	0
$703$	26.5237i	1.00036i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31.4301i	1.18205i
$708$	0	0
$709$	− 24.9061i	− 0.935367i	−0.883896	−	0.467684i	$-0.845089\pi$
$709$	− 24.9061i	− 0.935367i	0.883896	−	0.467684i	$-0.154911\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	57.9449i	2.17006i
$714$	0	0
$715$	3.84372	0.143747
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.8333	−0.739658	−0.369829	−	0.929100i	$-0.620584\pi$
$719$	−19.8333	−0.739658	−0.369829	+	0.929100i	$0.620584\pi$
$720$	0	0
$721$	0.669949	0.0249502
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−26.3277	−0.977787
$726$	0	0
$727$	0.331359i	0.0122894i	0.999981	+	0.00614472i	$0.00195594\pi$
$727$	0.331359i	0.0122894i	−0.999981	+	0.00614472i	$0.998044\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 0.923566i	− 0.0341593i
$732$	0	0
$733$	− 9.92295i	− 0.366512i	−0.983065	−	0.183256i	$-0.941336\pi$
$733$	− 9.92295i	− 0.366512i	0.983065	−	0.183256i	$-0.0586638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.05657i	− 0.0389192i
$738$	0	0
$739$	11.4372	0.420724	0.210362	−	0.977624i	$-0.432536\pi$
$739$	11.4372	0.420724	0.210362	+	0.977624i	$0.432536\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.1594	0.666204	0.333102	−	0.942891i	$-0.391905\pi$
$743$	18.1594	0.666204	0.333102	+	0.942891i	$0.391905\pi$
$744$	0	0
$745$	37.5334	1.37512
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.31382	0.340320
$750$	0	0
$751$	− 5.84293i	− 0.213211i	−0.994301	−	0.106606i	$-0.966002\pi$
$751$	− 5.84293i	− 0.213211i	0.994301	−	0.106606i	$-0.0339983\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 25.5063i	− 0.928268i
$756$	0	0
$757$	41.0906i	1.49346i	0.665126	+	0.746731i	$0.268378\pi$
$757$	41.0906i	1.49346i	−0.665126	+	0.746731i	$0.731622\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0805i	1.09042i	0.838301	+	0.545208i	$0.183549\pi$
$761$	30.0805i	1.09042i	−0.838301	+	0.545208i	$0.816451\pi$
$762$	0	0
$763$	55.8185	2.02077
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.55495	−0.200578
$768$	0	0
$769$	27.7812	1.00182	0.500908	−	0.865501i	$-0.333000\pi$
$769$	27.7812	1.00182	0.500908	+	0.865501i	$0.333000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	50.7995	1.82713	0.913566	−	0.406691i	$-0.133317\pi$
$773$	50.7995	1.82713	0.913566	+	0.406691i	$0.133317\pi$
$774$	0	0
$775$	− 41.5392i	− 1.49213i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 7.93050i	− 0.284140i
$780$	0	0
$781$	6.82738i	0.244303i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 11.8754i	− 0.423850i
$786$	0	0
$787$	13.8683	0.494352	0.247176	−	0.968971i	$-0.420497\pi$
$787$	13.8683	0.494352	0.247176	+	0.968971i	$0.420497\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	27.6824	0.984274
$792$	0	0
$793$	−8.82142	−0.313258
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.5890	1.57942	0.789712	−	0.613478i	$-0.210230\pi$
$797$	44.5890	1.57942	0.789712	+	0.613478i	$0.210230\pi$
$798$	0	0
$799$	− 0.153631i	− 0.00543509i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 15.5159i	− 0.547545i
$804$	0	0
$805$	107.380i	3.78463i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.1233i	1.27003i	0.772501	+	0.635014i	$0.219006\pi$
$809$	36.1233i	1.27003i	−0.772501	+	0.635014i	$0.780994\pi$
$810$	0	0
$811$	−5.44424	−0.191173	−0.0955865	−	0.995421i	$-0.530473\pi$
$811$	−5.44424	−0.191173	−0.0955865	+	0.995421i	$0.530473\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.16431	0.215926
$816$	0	0
$817$	−22.2847	−0.779643
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.7998	1.45882	0.729412	−	0.684075i	$-0.239794\pi$
$821$	41.7998	1.45882	0.729412	+	0.684075i	$0.239794\pi$
$822$	0	0
$823$	− 40.2621i	− 1.40345i	−0.712448	−	0.701725i	$-0.752414\pi$
$823$	− 40.2621i	− 1.40345i	0.712448	−	0.701725i	$-0.247586\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 0.370314i	− 0.0128771i	−0.999979	−	0.00643854i	$-0.997951\pi$
$827$	− 0.370314i	− 0.0128771i	0.999979	−	0.00643854i	$-0.00204947\pi$
$828$	0	0
$829$	− 24.4071i	− 0.847695i	−0.905733	−	0.423848i	$-0.860679\pi$
$829$	− 24.4071i	− 0.847695i	0.905733	−	0.423848i	$-0.139321\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.11467i	0.0386211i
$834$	0	0
$835$	63.7217	2.20518
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.88667	−0.237754	−0.118877	−	0.992909i	$-0.537929\pi$
$839$	−6.88667	−0.237754	−0.118877	+	0.992909i	$0.537929\pi$
$840$	0	0
$841$	−5.94515	−0.205005
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.23777	−0.111383
$846$	0	0
$847$	− 41.5848i	− 1.42887i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	89.0104i	3.05124i
$852$	0	0
$853$	1.95999i	0.0671087i	0.999437	+	0.0335544i	$0.0106827\pi$
$853$	1.95999i	0.0671087i	−0.999437	+	0.0335544i	$0.989317\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.2994i	− 0.830050i	−0.909810	−	0.415025i	$-0.863773\pi$
$857$	− 24.2994i	− 0.830050i	0.909810	−	0.415025i	$-0.136227\pi$
$858$	0	0
$859$	14.9524	0.510171	0.255085	−	0.966919i	$-0.417896\pi$
$859$	14.9524	0.510171	0.255085	+	0.966919i	$0.417896\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.3312	−0.419760	−0.209880	−	0.977727i	$-0.567307\pi$
$863$	−12.3312	−0.419760	−0.209880	+	0.977727i	$0.567307\pi$
$864$	0	0
$865$	43.8810	1.49200
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18.0173	−0.611196
$870$	0	0
$871$	0.890006i	0.0301567i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 6.78323i	− 0.229315i
$876$	0	0
$877$	18.4786i	0.623980i	0.950085	+	0.311990i	$0.100995\pi$
$877$	18.4786i	0.623980i	−0.950085	+	0.311990i	$0.899005\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.9527i	0.975440i	0.873000	+	0.487720i	$0.162171\pi$
$881$	28.9527i	0.975440i	−0.873000	+	0.487720i	$0.837829\pi$
$882$	0	0
$883$	33.0699	1.11289	0.556445	−	0.830884i	$-0.312165\pi$
$883$	33.0699	1.11289	0.556445	+	0.830884i	$0.312165\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.3010	−0.849523	−0.424762	−	0.905305i	$-0.639642\pi$
$887$	−25.3010	−0.849523	−0.424762	+	0.905305i	$0.639642\pi$
$888$	0	0
$889$	−35.1583	−1.17917
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.70697	−0.124049
$894$	0	0
$895$	75.0820i	2.50971i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	36.3753i	1.21319i
$900$	0	0
$901$	− 0.999922i	− 0.0333122i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.0848i	0.867088i
$906$	0	0
$907$	−45.0830	−1.49696	−0.748478	−	0.663160i	$-0.769215\pi$
$907$	−45.0830	−1.49696	−0.748478	+	0.663160i	$0.769215\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.9238	0.626973	0.313486	−	0.949593i	$-0.398503\pi$
$911$	18.9238	0.626973	0.313486	+	0.949593i	$0.398503\pi$
$912$	0	0
$913$	7.22207	0.239016
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−64.6609	−2.13529
$918$	0	0
$919$	16.4591i	0.542937i	0.962447	+	0.271468i	$0.0875092\pi$
$919$	16.4591i	0.542937i	−0.962447	+	0.271468i	$0.912491\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 5.75108i	− 0.189299i
$924$	0	0
$925$	− 63.8092i	− 2.09803i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 30.3566i	− 0.995966i	−0.867187	−	0.497983i	$-0.834074\pi$
$929$	− 30.3566i	− 0.995966i	0.867187	−	0.497983i	$-0.165926\pi$
$930$	0	0
$931$	26.8959	0.881478
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.363075	0.0118738
$936$	0	0
$937$	−9.81295	−0.320575	−0.160288	−	0.987070i	$-0.551242\pi$
$937$	−9.81295	−0.320575	−0.160288	+	0.987070i	$0.551242\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.31708	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$941$	9.31708	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$942$	0	0
$943$	− 26.6138i	− 0.866664i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 56.9545i	− 1.85077i	−0.379025	−	0.925386i	$-0.623741\pi$
$947$	− 56.9545i	− 1.85077i	0.379025	−	0.925386i	$-0.376259\pi$
$948$	0	0
$949$	13.0699i	0.424268i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.0413i	0.973133i	0.873643	+	0.486567i	$0.161751\pi$
$953$	30.0413i	0.973133i	−0.873643	+	0.486567i	$0.838249\pi$
$954$	0	0
$955$	35.7314	1.15624
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.5411	−1.01852
$960$	0	0
$961$	−26.3921	−0.851357
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−24.6306	−0.792887
$966$	0	0
$967$	− 25.3713i	− 0.815885i	−0.913008	−	0.407942i	$-0.866246\pi$
$967$	− 25.3713i	− 0.815885i	0.913008	−	0.407942i	$-0.133754\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.4712i	0.464404i	0.972668	+	0.232202i	$0.0745931\pi$
$971$	14.4712i	0.464404i	−0.972668	+	0.232202i	$0.925407\pi$
$972$	0	0
$973$	1.85670i	0.0595230i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.5139i	0.528325i	0.964478	+	0.264163i	$0.0850955\pi$
$977$	16.5139i	0.528325i	−0.964478	+	0.264163i	$0.914904\pi$
$978$	0	0
$979$	3.00193	0.0959421
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.69363	−0.149703	−0.0748517	−	0.997195i	$-0.523848\pi$
$983$	−4.69363	−0.149703	−0.0748517	+	0.997195i	$0.523848\pi$
$984$	0	0
$985$	−11.8313	−0.376977
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−74.7847	−2.37802
$990$	0	0
$991$	− 49.9083i	− 1.58539i	−0.609618	−	0.792695i	$-0.708677\pi$
$991$	− 49.9083i	− 1.58539i	0.609618	−	0.792695i	$-0.291323\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.8079i	0.564548i
$996$	0	0
$997$	− 47.8441i	− 1.51524i	−0.652697	−	0.757619i	$-0.726363\pi$
$997$	− 47.8441i	− 1.51524i	0.652697	−	0.757619i	$-0.273637\pi$
$998$	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.2159.41		48
3.2	odd	2	inner	3744.2.j.a.2159.7		48
4.3	odd	2		936.2.j.a.755.39	yes	48
8.3	odd	2	inner	3744.2.j.a.2159.8		48
8.5	even	2		936.2.j.a.755.9	✓	48
12.11	even	2		936.2.j.a.755.10	yes	48
24.5	odd	2		936.2.j.a.755.40	yes	48
24.11	even	2	inner	3744.2.j.a.2159.42		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.j.a.755.9	✓	48	8.5	even	2
936.2.j.a.755.10	yes	48	12.11	even	2
936.2.j.a.755.39	yes	48	4.3	odd	2
936.2.j.a.755.40	yes	48	24.5	odd	2
3744.2.j.a.2159.7		48	3.2	odd	2	inner
3744.2.j.a.2159.8		48	8.3	odd	2	inner
3744.2.j.a.2159.41		48	1.1	even	1	trivial
3744.2.j.a.2159.42		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)

\(f(q)\)	\(=\)	\(q+3.23777 q^{5} -4.33596i q^{7} +1.18715i q^{11} -1.00000i q^{13} -0.0944593i q^{17} -2.27921 q^{19} -7.64873 q^{23} +5.48318 q^{25} -4.80155 q^{29} -7.57575i q^{31} -14.0389i q^{35} -11.6373i q^{37} +3.47950i q^{41} +9.77740 q^{43} +1.62643 q^{47} -11.8006 q^{49} +10.5857 q^{53} +3.84372i q^{55} -5.55495i q^{59} -8.82142i q^{61} -3.23777i q^{65} -0.890006 q^{67} +5.75108 q^{71} -13.0699 q^{73} +5.14743 q^{77} +15.1770i q^{79} -6.08355i q^{83} -0.305838i q^{85} -2.52869i q^{89} -4.33596 q^{91} -7.37955 q^{95} -17.4724 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

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