LMFDB - Embedded newform 3344.2.o.b.1519.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1519,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3344.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.o (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.8
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.b.1519.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58114 q^{3} +4.01759 q^{5} -1.30410i q^{7} +3.66230 q^{9} +O(q^{10})$	$q-2.58114 q^{3} +4.01759 q^{5} -1.30410i q^{7} +3.66230 q^{9} -1.00000i q^{11} +4.08460i q^{13} -10.3700 q^{15} -5.19545 q^{17} +(1.73204 - 4.00000i) q^{19} +3.36606i q^{21} +0.289241i q^{23} +11.1410 q^{25} -1.70949 q^{27} -5.00930i q^{29} -5.32805 q^{31} +2.58114i q^{33} -5.23932i q^{35} -8.08920i q^{37} -10.5429i q^{39} -1.06055i q^{41} -6.01199i q^{43} +14.7136 q^{45} -1.68058i q^{47} +5.29933 q^{49} +13.4102 q^{51} +3.85333i q^{53} -4.01759i q^{55} +(-4.47065 + 10.3246i) q^{57} +2.05637 q^{59} -3.99949 q^{61} -4.77599i q^{63} +16.4102i q^{65} -11.7345 q^{67} -0.746572i q^{69} +10.1887 q^{71} +11.7059 q^{73} -28.7566 q^{75} -1.30410 q^{77} -9.25395 q^{79} -6.57445 q^{81} +12.1991i q^{83} -20.8732 q^{85} +12.9297i q^{87} -16.7730i q^{89} +5.32670 q^{91} +13.7525 q^{93} +(6.95865 - 16.0704i) q^{95} -12.7735i q^{97} -3.66230i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 56 q^{9}+O(q^{10}) $ 64 * q + 56 * q^9	$ 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) $ 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times$.

$n$	$705$	$837$	$2433$	$2927$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−2.58114			−1.49022			−0.745112	−	0.666939i	$-0.767604\pi$
$3$	−2.58114			−1.49022			−0.745112	+	0.666939i	$0.767604\pi$
$4$		0			0
$5$	4.01759			1.79672			0.898361	−	0.439258i	$-0.144759\pi$
$5$	4.01759			1.79672			0.898361	+	0.439258i	$0.144759\pi$
$6$		0			0
$7$		−	1.30410i		−	0.492902i	−0.969155	−	0.246451i	$-0.920736\pi$
$7$		−	1.30410i		−	0.492902i	0.969155	−	0.246451i	$-0.0792644\pi$
$8$		0			0
$9$	3.66230			1.22077
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$			4.08460i			1.13286i	0.824109	+	0.566431i	$0.191676\pi$
$13$			4.08460i			1.13286i	−0.824109	+	0.566431i	$0.808324\pi$
$14$		0			0
$15$	−10.3700			−2.67752
$16$		0			0
$17$	−5.19545			−1.26008			−0.630041	−	0.776562i	$-0.716962\pi$
$17$	−5.19545			−1.26008			−0.630041	+	0.776562i	$0.716962\pi$
$18$		0			0
$19$	1.73204	−	4.00000i	0.397358	−	0.917664i
$19$	1.73204	−	4.00000i	0.397358	−	0.917664i
$20$		0			0
$21$			3.36606i			0.734534i
$22$		0			0
$23$			0.289241i			0.0603109i	0.999545	+	0.0301554i	$0.00960023\pi$
$23$			0.289241i			0.0603109i	−0.999545	+	0.0301554i	$0.990400\pi$
$24$		0			0
$25$	11.1410			2.22821
$26$		0			0
$27$	−1.70949			−0.328992
$28$		0			0
$29$		−	5.00930i		−	0.930203i	−0.885257	−	0.465102i	$-0.846018\pi$
$29$		−	5.00930i		−	0.930203i	0.885257	−	0.465102i	$-0.153982\pi$
$30$		0			0
$31$	−5.32805			−0.956946			−0.478473	−	0.878102i	$-0.658810\pi$
$31$	−5.32805			−0.956946			−0.478473	+	0.878102i	$0.658810\pi$
$32$		0			0
$33$			2.58114i			0.449319i
$34$		0			0
$35$		−	5.23932i		−	0.885607i
$36$		0			0
$37$		−	8.08920i		−	1.32986i	−0.746907	−	0.664928i	$-0.768462\pi$
$37$		−	8.08920i		−	1.32986i	0.746907	−	0.664928i	$-0.231538\pi$
$38$		0			0
$39$		−	10.5429i		−	1.68822i
$40$		0			0
$41$		−	1.06055i		−	0.165630i	−0.996565	−	0.0828148i	$-0.973609\pi$
$41$		−	1.06055i		−	0.165630i	0.996565	−	0.0828148i	$-0.0263910\pi$
$42$		0			0
$43$		−	6.01199i		−	0.916820i	−0.888741	−	0.458410i	$-0.848419\pi$
$43$		−	6.01199i		−	0.916820i	0.888741	−	0.458410i	$-0.151581\pi$
$44$		0			0
$45$	14.7136			2.19338
$46$		0			0
$47$		−	1.68058i		−	0.245138i	−0.992460	−	0.122569i	$-0.960887\pi$
$47$		−	1.68058i		−	0.245138i	0.992460	−	0.122569i	$-0.0391132\pi$
$48$		0			0
$49$	5.29933			0.757048
$50$		0			0
$51$	13.4102			1.87780
$52$		0			0
$53$			3.85333i			0.529296i	0.964345	+	0.264648i	$0.0852557\pi$
$53$			3.85333i			0.529296i	−0.964345	+	0.264648i	$0.914744\pi$
$54$		0			0
$55$		−	4.01759i		−	0.541732i
$56$		0			0
$57$	−4.47065	+	10.3246i	−0.592153	+	1.36752i
$58$		0			0
$59$	2.05637			0.267716			0.133858	−	0.991000i	$-0.457263\pi$
$59$	2.05637			0.267716			0.133858	+	0.991000i	$0.457263\pi$
$60$		0			0
$61$	−3.99949			−0.512082			−0.256041	−	0.966666i	$-0.582418\pi$
$61$	−3.99949			−0.512082			−0.256041	+	0.966666i	$0.582418\pi$
$62$		0			0
$63$		−	4.77599i		−	0.601718i
$64$		0			0
$65$			16.4102i			2.03544i
$66$		0			0
$67$	−11.7345			−1.43360			−0.716798	−	0.697281i	$-0.754393\pi$
$67$	−11.7345			−1.43360			−0.716798	+	0.697281i	$0.754393\pi$
$68$		0			0
$69$		−	0.746572i		−	0.0898767i
$70$		0			0
$71$	10.1887			1.20917			0.604586	−	0.796540i	$-0.293339\pi$
$71$	10.1887			1.20917			0.604586	+	0.796540i	$0.293339\pi$
$72$		0			0
$73$	11.7059			1.37007			0.685036	−	0.728510i	$-0.259787\pi$
$73$	11.7059			1.37007			0.685036	+	0.728510i	$0.259787\pi$
$74$		0			0
$75$	−28.7566			−3.32053
$76$		0			0
$77$	−1.30410			−0.148615
$78$		0			0
$79$	−9.25395			−1.04115			−0.520575	−	0.853816i	$-0.674283\pi$
$79$	−9.25395			−1.04115			−0.520575	+	0.853816i	$0.674283\pi$
$80$		0			0
$81$	−6.57445			−0.730495
$82$		0			0
$83$			12.1991i			1.33902i	0.742801	+	0.669512i	$0.233497\pi$
$83$			12.1991i			1.33902i	−0.742801	+	0.669512i	$0.766503\pi$
$84$		0			0
$85$	−20.8732			−2.26402
$86$		0			0
$87$			12.9297i			1.38621i
$88$		0			0
$89$		−	16.7730i		−	1.77794i	−0.457966	−	0.888970i	$-0.651422\pi$
$89$		−	16.7730i		−	1.77794i	0.457966	−	0.888970i	$-0.348578\pi$
$90$		0			0
$91$	5.32670			0.558390
$92$		0			0
$93$	13.7525			1.42606
$94$		0			0
$95$	6.95865	−	16.0704i	0.713942	−	1.64879i
$96$		0			0
$97$		−	12.7735i		−	1.29695i	−0.761234	−	0.648477i	$-0.775406\pi$
$97$		−	12.7735i		−	1.29695i	0.761234	−	0.648477i	$-0.224594\pi$
$98$		0			0
$99$		−	3.66230i		−	0.368075i
$100$		0			0
$101$	−7.11244			−0.707714			−0.353857	−	0.935299i	$-0.615130\pi$
$101$	−7.11244			−0.707714			−0.353857	+	0.935299i	$0.615130\pi$
$102$		0			0
$103$	14.0461			1.38401			0.692003	−	0.721894i	$-0.256728\pi$
$103$	14.0461			1.38401			0.692003	+	0.721894i	$0.256728\pi$
$104$		0			0
$105$			13.5234i			1.31975i
$106$		0			0
$107$	−12.0738			−1.16722			−0.583609	−	0.812035i	$-0.698360\pi$
$107$	−12.0738			−1.16722			−0.583609	+	0.812035i	$0.698360\pi$
$108$		0			0
$109$		−	4.28709i		−	0.410629i	−0.978696	−	0.205315i	$-0.934178\pi$
$109$		−	4.28709i		−	0.410629i	0.978696	−	0.205315i	$-0.0658217\pi$
$110$		0			0
$111$			20.8794i			1.98178i
$112$		0			0
$113$			10.5513i			0.992579i	0.868157	+	0.496289i	$0.165305\pi$
$113$			10.5513i			0.992579i	−0.868157	+	0.496289i	$0.834695\pi$
$114$		0			0
$115$			1.16205i			0.108362i
$116$		0			0
$117$			14.9590i			1.38296i
$118$		0			0
$119$			6.77536i			0.621096i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			2.73742i			0.246825i
$124$		0			0
$125$	24.6722			2.20675
$126$		0			0
$127$	6.95579			0.617226			0.308613	−	0.951188i	$-0.400135\pi$
$127$	6.95579			0.617226			0.308613	+	0.951188i	$0.400135\pi$
$128$		0			0
$129$			15.5178i			1.36627i
$130$		0			0
$131$		−	14.6046i		−	1.27601i	−0.770031	−	0.638006i	$-0.779759\pi$
$131$		−	14.6046i		−	1.27601i	0.770031	−	0.638006i	$-0.220241\pi$
$132$		0			0
$133$	−5.21639	−	2.25875i	−0.452318	−	0.195859i
$134$		0			0
$135$	−6.86805			−0.591107
$136$		0			0
$137$	−13.0182			−1.11222			−0.556111	−	0.831108i	$-0.687707\pi$
$137$	−13.0182			−1.11222			−0.556111	+	0.831108i	$0.687707\pi$
$138$		0			0
$139$			15.9310i			1.35125i	0.737247	+	0.675623i	$0.236125\pi$
$139$			15.9310i			1.35125i	−0.737247	+	0.675623i	$0.763875\pi$
$140$		0			0
$141$			4.33781i			0.365310i
$142$		0			0
$143$	4.08460			0.341571
$144$		0			0
$145$		−	20.1253i		−	1.67132i
$146$		0			0
$147$	−13.6783			−1.12817
$148$		0			0
$149$	20.2366			1.65784			0.828922	−	0.559365i	$-0.188955\pi$
$149$	20.2366			1.65784			0.828922	+	0.559365i	$0.188955\pi$
$150$		0			0
$151$	−0.0187964			−0.00152963			−0.000764814	−	1.00000i	$-0.500243\pi$
$151$	−0.0187964			−0.00152963			−0.000764814		1.00000i	$0.500243\pi$
$152$		0			0
$153$	−19.0273			−1.53827
$154$		0			0
$155$	−21.4059			−1.71937
$156$		0			0
$157$	3.27480			0.261358			0.130679	−	0.991425i	$-0.458284\pi$
$157$	3.27480			0.261358			0.130679	+	0.991425i	$0.458284\pi$
$158$		0			0
$159$		−	9.94600i		−	0.788769i
$160$		0			0
$161$	0.377197			0.0297273
$162$		0			0
$163$			13.8417i			1.08417i	0.840324	+	0.542084i	$0.182364\pi$
$163$			13.8417i			1.08417i	−0.840324	+	0.542084i	$0.817636\pi$
$164$		0			0
$165$			10.3700i			0.807302i
$166$		0			0
$167$	14.0660			1.08846			0.544228	−	0.838937i	$-0.316822\pi$
$167$	14.0660			1.08846			0.544228	+	0.838937i	$0.316822\pi$
$168$		0			0
$169$	−3.68392			−0.283379
$170$		0			0
$171$	6.34327	−	14.6492i	0.485082	−	1.12025i
$172$		0			0
$173$		−	19.4440i		−	1.47830i	−0.673539	−	0.739152i	$-0.735227\pi$
$173$		−	19.4440i		−	1.47830i	0.673539	−	0.739152i	$-0.264773\pi$
$174$		0			0
$175$		−	14.5290i		−	1.09829i
$176$		0			0
$177$	−5.30778			−0.398957
$178$		0			0
$179$	22.5568			1.68597			0.842986	−	0.537936i	$-0.180796\pi$
$179$	22.5568			1.68597			0.842986	+	0.537936i	$0.180796\pi$
$180$		0			0
$181$		−	6.25563i		−	0.464977i	−0.972599	−	0.232489i	$-0.925313\pi$
$181$		−	6.25563i		−	0.464977i	0.972599	−	0.232489i	$-0.0746869\pi$
$182$		0			0
$183$	10.3232			0.763116
$184$		0			0
$185$		−	32.4991i		−	2.38938i
$186$		0			0
$187$			5.19545i			0.379929i
$188$		0			0
$189$			2.22934i			0.162161i
$190$		0			0
$191$		−	21.9881i		−	1.59101i	−0.605950	−	0.795503i	$-0.707207\pi$
$191$		−	21.9881i		−	1.59101i	0.605950	−	0.795503i	$-0.292793\pi$
$192$		0			0
$193$		−	24.2562i		−	1.74600i	−0.487718	−	0.873001i	$-0.662170\pi$
$193$		−	24.2562i		−	1.74600i	0.487718	−	0.873001i	$-0.337830\pi$
$194$		0			0
$195$		−	42.3572i		−	3.03326i
$196$		0			0
$197$	−23.7046			−1.68889			−0.844443	−	0.535646i	$-0.820068\pi$
$197$	−23.7046			−1.68889			−0.844443	+	0.535646i	$0.820068\pi$
$198$		0			0
$199$		−	11.4854i		−	0.814177i	−0.913389	−	0.407089i	$-0.866544\pi$
$199$		−	11.4854i		−	0.814177i	0.913389	−	0.407089i	$-0.133456\pi$
$200$		0			0
$201$	30.2884			2.13638
$202$		0			0
$203$	−6.53260			−0.458499
$204$		0			0
$205$		−	4.26084i		−	0.297590i
$206$		0			0
$207$			1.05929i			0.0736255i
$208$		0			0
$209$	−4.00000	−	1.73204i	−0.276686	−	0.119808i
$210$		0			0
$211$	−4.75038			−0.327029			−0.163515	−	0.986541i	$-0.552283\pi$
$211$	−4.75038			−0.327029			−0.163515	+	0.986541i	$0.552283\pi$
$212$		0			0
$213$	−26.2984			−1.80194
$214$		0			0
$215$		−	24.1537i		−	1.64727i
$216$		0			0
$217$			6.94828i			0.471680i
$218$		0			0
$219$	−30.2146			−2.04171
$220$		0			0
$221$		−	21.2213i		−	1.42750i
$222$		0			0
$223$	12.2853			0.822687			0.411343	−	0.911480i	$-0.365060\pi$
$223$	12.2853			0.822687			0.411343	+	0.911480i	$0.365060\pi$
$224$		0			0
$225$	40.8018			2.72012
$226$		0			0
$227$	−10.7612			−0.714248			−0.357124	−	0.934057i	$-0.616243\pi$
$227$	−10.7612			−0.714248			−0.357124	+	0.934057i	$0.616243\pi$
$228$		0			0
$229$	−3.96520			−0.262028			−0.131014	−	0.991381i	$-0.541823\pi$
$229$	−3.96520			−0.262028			−0.131014	+	0.991381i	$0.541823\pi$
$230$		0			0
$231$	3.36606			0.221470
$232$		0			0
$233$	−7.52394			−0.492910			−0.246455	−	0.969154i	$-0.579266\pi$
$233$	−7.52394			−0.492910			−0.246455	+	0.969154i	$0.579266\pi$
$234$		0			0
$235$		−	6.75188i		−	0.440444i
$236$		0			0
$237$	23.8858			1.55155
$238$		0			0
$239$		−	6.57351i		−	0.425205i	−0.977139	−	0.212602i	$-0.931806\pi$
$239$		−	6.57351i		−	0.425205i	0.977139	−	0.212602i	$-0.0681939\pi$
$240$		0			0
$241$		−	13.8628i		−	0.892979i	−0.894789	−	0.446490i	$-0.852674\pi$
$241$		−	13.8628i		−	0.892979i	0.894789	−	0.446490i	$-0.147326\pi$
$242$		0			0
$243$	22.0981			1.41759
$244$		0			0
$245$	21.2906			1.36020
$246$		0			0
$247$	16.3384	+	7.07470i	1.03959	+	0.450152i
$248$		0			0
$249$		−	31.4876i		−	1.99545i
$250$		0			0
$251$		−	11.1157i		−	0.701619i	−0.936447	−	0.350810i	$-0.885906\pi$
$251$		−	11.1157i		−	0.701619i	0.936447	−	0.350810i	$-0.114094\pi$
$252$		0			0
$253$	0.289241			0.0181844
$254$		0			0
$255$	53.8767			3.37389
$256$		0			0
$257$		−	8.87422i		−	0.553559i	−0.960933	−	0.276779i	$-0.910733\pi$
$257$		−	8.87422i		−	0.553559i	0.960933	−	0.276779i	$-0.0892671\pi$
$258$		0			0
$259$	−10.5491			−0.655488
$260$		0			0
$261$		−	18.3456i		−	1.13556i
$262$		0			0
$263$		−	2.28558i		−	0.140935i	−0.997514	−	0.0704675i	$-0.977551\pi$
$263$		−	2.28558i		−	0.140935i	0.997514	−	0.0704675i	$-0.0224491\pi$
$264$		0			0
$265$			15.4811i			0.950997i
$266$		0			0
$267$			43.2936i			2.64953i
$268$		0			0
$269$		−	14.2291i		−	0.867564i	−0.901018	−	0.433782i	$-0.857179\pi$
$269$		−	14.2291i		−	0.867564i	0.901018	−	0.433782i	$-0.142821\pi$
$270$		0			0
$271$			10.5441i			0.640507i	0.947332	+	0.320253i	$0.103768\pi$
$271$			10.5441i			0.640507i	−0.947332	+	0.320253i	$0.896232\pi$
$272$		0			0
$273$	−13.7490			−0.832126
$274$		0			0
$275$		−	11.1410i		−	0.671830i
$276$		0			0
$277$	−5.74383			−0.345114			−0.172557	−	0.985000i	$-0.555203\pi$
$277$	−5.74383			−0.345114			−0.172557	+	0.985000i	$0.555203\pi$
$278$		0			0
$279$	−19.5129			−1.16821
$280$		0			0
$281$		−	12.9126i		−	0.770302i	−0.922854	−	0.385151i	$-0.874149\pi$
$281$		−	12.9126i		−	0.770302i	0.922854	−	0.385151i	$-0.125851\pi$
$282$		0			0
$283$			4.49229i			0.267039i	0.991046	+	0.133520i	$0.0426279\pi$
$283$			4.49229i			0.267039i	−0.991046	+	0.133520i	$0.957372\pi$
$284$		0			0
$285$	−17.9613	+	41.4799i	−1.06393	+	2.45706i
$286$		0			0
$287$	−1.38305			−0.0816391
$288$		0			0
$289$	9.99268			0.587805
$290$		0			0
$291$			32.9703i			1.93275i
$292$		0			0
$293$		−	2.24925i		−	0.131402i	−0.997839	−	0.0657012i	$-0.979072\pi$
$293$		−	2.24925i		−	0.131402i	0.997839	−	0.0657012i	$-0.0209284\pi$
$294$		0			0
$295$	8.26165			0.481012
$296$		0			0
$297$			1.70949i			0.0991949i
$298$		0			0
$299$	−1.18143			−0.0683239
$300$		0			0
$301$	−7.84021			−0.451902
$302$		0			0
$303$	18.3582			1.05465
$304$		0			0
$305$	−16.0683			−0.920068
$306$		0			0
$307$	−5.41994			−0.309332			−0.154666	−	0.987967i	$-0.549430\pi$
$307$	−5.41994			−0.309332			−0.154666	+	0.987967i	$0.549430\pi$
$308$		0			0
$309$	−36.2551			−2.06248
$310$		0			0
$311$		−	16.9863i		−	0.963206i	−0.876390	−	0.481603i	$-0.840055\pi$
$311$		−	16.9863i		−	0.963206i	0.876390	−	0.481603i	$-0.159945\pi$
$312$		0			0
$313$	10.9895			0.621163			0.310581	−	0.950547i	$-0.399476\pi$
$313$	10.9895			0.621163			0.310581	+	0.950547i	$0.399476\pi$
$314$		0			0
$315$		−	19.1880i		−	1.08112i
$316$		0			0
$317$			32.1866i			1.80778i	0.427768	+	0.903889i	$0.359300\pi$
$317$			32.1866i			1.80778i	−0.427768	+	0.903889i	$0.640700\pi$
$318$		0			0
$319$	−5.00930			−0.280467
$320$		0			0
$321$	31.1642			1.73942
$322$		0			0
$323$	−8.99875	+	20.7818i	−0.500704	+	1.15633i
$324$		0			0
$325$			45.5066i			2.52425i
$326$		0			0
$327$			11.0656i			0.611929i
$328$		0			0
$329$	−2.19164			−0.120829
$330$		0			0
$331$	−17.9954			−0.989119			−0.494559	−	0.869144i	$-0.664671\pi$
$331$	−17.9954			−0.989119			−0.494559	+	0.869144i	$0.664671\pi$
$332$		0			0
$333$		−	29.6251i		−	1.62344i
$334$		0			0
$335$	−47.1444			−2.57577
$336$		0			0
$337$			21.5933i			1.17626i	0.808767	+	0.588130i	$0.200136\pi$
$337$			21.5933i			1.17626i	−0.808767	+	0.588130i	$0.799864\pi$
$338$		0			0
$339$		−	27.2343i		−	1.47916i
$340$		0			0
$341$			5.32805i			0.288530i
$342$		0			0
$343$		−	16.0395i		−	0.866052i
$344$		0			0
$345$		−	2.99942i		−	0.161483i
$346$		0			0
$347$		−	3.04257i		−	0.163334i	−0.996660	−	0.0816669i	$-0.973976\pi$
$347$		−	3.04257i		−	0.163334i	0.996660	−	0.0816669i	$-0.0260244\pi$
$348$		0			0
$349$	1.69299			0.0906236			0.0453118	−	0.998973i	$-0.485572\pi$
$349$	1.69299			0.0906236			0.0453118	+	0.998973i	$0.485572\pi$
$350$		0			0
$351$		−	6.98259i		−	0.372703i
$352$		0			0
$353$	21.2316			1.13004			0.565021	−	0.825077i	$-0.308868\pi$
$353$	21.2316			1.13004			0.565021	+	0.825077i	$0.308868\pi$
$354$		0			0
$355$	40.9339			2.17255
$356$		0			0
$357$		−	17.4882i		−	0.925572i
$358$		0			0
$359$		−	8.28243i		−	0.437130i	−0.975822	−	0.218565i	$-0.929862\pi$
$359$		−	8.28243i		−	0.437130i	0.975822	−	0.218565i	$-0.0701376\pi$
$360$		0			0
$361$	−13.0000	−	13.8564i	−0.684213	−	0.729282i
$362$		0			0
$363$	2.58114			0.135475
$364$		0			0
$365$	47.0295			2.46164
$366$		0			0
$367$		−	25.5819i		−	1.33537i	−0.744446	−	0.667683i	$-0.767286\pi$
$367$		−	25.5819i		−	1.33537i	0.744446	−	0.667683i	$-0.232714\pi$
$368$		0			0
$369$		−	3.88404i		−	0.202195i
$370$		0			0
$371$	5.02511			0.260891
$372$		0			0
$373$		−	23.1889i		−	1.20068i	−0.799746	−	0.600338i	$-0.795033\pi$
$373$		−	23.1889i		−	1.20068i	0.799746	−	0.600338i	$-0.204967\pi$
$374$		0			0
$375$	−63.6825			−3.28855
$376$		0			0
$377$	20.4610			1.05379
$378$		0			0
$379$	7.00414			0.359778			0.179889	−	0.983687i	$-0.442426\pi$
$379$	7.00414			0.359778			0.179889	+	0.983687i	$0.442426\pi$
$380$		0			0
$381$	−17.9539			−0.919805
$382$		0			0
$383$	−14.3664			−0.734091			−0.367045	−	0.930203i	$-0.619631\pi$
$383$	−14.3664			−0.734091			−0.367045	+	0.930203i	$0.619631\pi$
$384$		0			0
$385$	−5.23932			−0.267021
$386$		0			0
$387$		−	22.0177i		−	1.11922i
$388$		0			0
$389$	27.8434			1.41172			0.705859	−	0.708353i	$-0.250561\pi$
$389$	27.8434			1.41172			0.705859	+	0.708353i	$0.250561\pi$
$390$		0			0
$391$		−	1.50273i		−	0.0759966i
$392$		0			0
$393$			37.6966i			1.90154i
$394$		0			0
$395$	−37.1786			−1.87066
$396$		0			0
$397$	−10.2885			−0.516367			−0.258183	−	0.966096i	$-0.583124\pi$
$397$	−10.2885			−0.516367			−0.258183	+	0.966096i	$0.583124\pi$
$398$		0			0
$399$	13.4642	+	5.83016i	0.674055	+	0.291873i
$400$		0			0
$401$			18.0854i			0.903140i	0.892236	+	0.451570i	$0.149136\pi$
$401$			18.0854i			0.903140i	−0.892236	+	0.451570i	$0.850864\pi$
$402$		0			0
$403$		−	21.7629i		−	1.08409i
$404$		0			0
$405$	−26.4135			−1.31250
$406$		0			0
$407$	−8.08920			−0.400967
$408$		0			0
$409$			38.9165i			1.92430i	0.272526	+	0.962148i	$0.412141\pi$
$409$			38.9165i			1.92430i	−0.272526	+	0.962148i	$0.587859\pi$
$410$		0			0
$411$	33.6019			1.65746
$412$		0			0
$413$		−	2.68170i		−	0.131958i
$414$		0			0
$415$			49.0110i			2.40585i
$416$		0			0
$417$		−	41.1201i		−	2.01366i
$418$		0			0
$419$		−	18.5419i		−	0.905830i	−0.891554	−	0.452915i	$-0.850384\pi$
$419$		−	18.5419i		−	0.905830i	0.891554	−	0.452915i	$-0.149616\pi$
$420$		0			0
$421$			1.47497i			0.0718856i	0.999354	+	0.0359428i	$0.0114434\pi$
$421$			1.47497i			0.0718856i	−0.999354	+	0.0359428i	$0.988557\pi$
$422$		0			0
$423$		−	6.15478i		−	0.299256i
$424$		0			0
$425$	−57.8827			−2.80772
$426$		0			0
$427$			5.21571i			0.252406i
$428$		0			0
$429$	−10.5429			−0.509017
$430$		0			0
$431$	37.7742			1.81952			0.909760	−	0.415134i	$-0.136265\pi$
$431$	37.7742			1.81952			0.909760	+	0.415134i	$0.136265\pi$
$432$		0			0
$433$		−	4.08150i		−	0.196144i	−0.995179	−	0.0980722i	$-0.968732\pi$
$433$		−	4.08150i		−	0.196144i	0.995179	−	0.0980722i	$-0.0312676\pi$
$434$		0			0
$435$			51.9463i			2.49063i
$436$		0			0
$437$	1.15696	+	0.500978i	0.0553451	+	0.0239650i
$438$		0			0
$439$	−18.8066			−0.897590			−0.448795	−	0.893635i	$-0.648147\pi$
$439$	−18.8066			−0.897590			−0.448795	+	0.893635i	$0.648147\pi$
$440$		0			0
$441$	19.4078			0.924179
$442$		0			0
$443$		−	4.18144i		−	0.198666i	−0.995054	−	0.0993330i	$-0.968329\pi$
$443$		−	4.18144i		−	0.198666i	0.995054	−	0.0993330i	$-0.0316709\pi$
$444$		0			0
$445$		−	67.3872i		−	3.19446i
$446$		0			0
$447$	−52.2335			−2.47056
$448$		0			0
$449$			31.8410i			1.50267i	0.659922	+	0.751334i	$0.270589\pi$
$449$			31.8410i			1.50267i	−0.659922	+	0.751334i	$0.729411\pi$
$450$		0			0
$451$	−1.06055			−0.0499392
$452$		0			0
$453$	0.0485162			0.00227949
$454$		0			0
$455$	21.4005			1.00327
$456$		0			0
$457$	11.4130			0.533876			0.266938	−	0.963714i	$-0.413988\pi$
$457$	11.4130			0.533876			0.266938	+	0.963714i	$0.413988\pi$
$458$		0			0
$459$	8.88158			0.414557
$460$		0			0
$461$	−20.2713			−0.944127			−0.472063	−	0.881565i	$-0.656491\pi$
$461$	−20.2713			−0.944127			−0.472063	+	0.881565i	$0.656491\pi$
$462$		0			0
$463$			17.3747i			0.807472i	0.914876	+	0.403736i	$0.132289\pi$
$463$			17.3747i			0.807472i	−0.914876	+	0.403736i	$0.867711\pi$
$464$		0			0
$465$	55.2518			2.56224
$466$		0			0
$467$			35.3741i			1.63692i	0.574566	+	0.818458i	$0.305171\pi$
$467$			35.3741i			1.63692i	−0.574566	+	0.818458i	$0.694829\pi$
$468$		0			0
$469$			15.3029i			0.706622i
$470$		0			0
$471$	−8.45273			−0.389481
$472$		0			0
$473$	−6.01199			−0.276432
$474$		0			0
$475$	19.2968	−	44.5642i	0.885397	−	2.04475i
$476$		0			0
$477$			14.1121i			0.646147i
$478$		0			0
$479$		−	7.49116i		−	0.342280i	−0.985247	−	0.171140i	$-0.945255\pi$
$479$		−	7.49116i		−	0.342280i	0.985247	−	0.171140i	$-0.0547451\pi$
$480$		0			0
$481$	33.0411			1.50654
$482$		0			0
$483$	−0.973601			−0.0443004
$484$		0			0
$485$		−	51.3188i		−	2.33027i
$486$		0			0
$487$	14.4026			0.652643			0.326322	−	0.945259i	$-0.394191\pi$
$487$	14.4026			0.652643			0.326322	+	0.945259i	$0.394191\pi$
$488$		0			0
$489$		−	35.7275i		−	1.61565i
$490$		0			0
$491$			14.4256i			0.651020i	0.945539	+	0.325510i	$0.105536\pi$
$491$			14.4256i			0.651020i	−0.945539	+	0.325510i	$0.894464\pi$
$492$		0			0
$493$			26.0255i			1.17213i
$494$		0			0
$495$		−	14.7136i		−	0.661328i
$496$		0			0
$497$		−	13.2870i		−	0.596003i
$498$		0			0
$499$		−	25.9196i		−	1.16032i	−0.814502	−	0.580160i	$-0.802990\pi$
$499$		−	25.9196i		−	1.16032i	0.814502	−	0.580160i	$-0.197010\pi$
$500$		0			0
$501$	−36.3063			−1.62204
$502$		0			0
$503$			16.0988i			0.717811i	0.933374	+	0.358906i	$0.116850\pi$
$503$			16.0988i			0.717811i	−0.933374	+	0.358906i	$0.883150\pi$
$504$		0			0
$505$	−28.5749			−1.27157
$506$		0			0
$507$	9.50873			0.422297
$508$		0			0
$509$			13.3304i			0.590859i	0.955365	+	0.295429i	$0.0954627\pi$
$509$			13.3304i			0.590859i	−0.955365	+	0.295429i	$0.904537\pi$
$510$		0			0
$511$		−	15.2656i		−	0.675311i
$512$		0			0
$513$	−2.96092	+	6.83798i	−0.130728	+	0.301904i
$514$		0			0
$515$	56.4316			2.48667
$516$		0			0
$517$	−1.68058			−0.0739118
$518$		0			0
$519$			50.1879i			2.20300i
$520$		0			0
$521$			38.3148i			1.67860i	0.543667	+	0.839301i	$0.317036\pi$
$521$			38.3148i			1.67860i	−0.543667	+	0.839301i	$0.682964\pi$
$522$		0			0
$523$	−20.8488			−0.911655			−0.455828	−	0.890068i	$-0.650657\pi$
$523$	−20.8488			−0.911655			−0.455828	+	0.890068i	$0.650657\pi$
$524$		0			0
$525$			37.5014i			1.63669i
$526$		0			0
$527$	27.6816			1.20583
$528$		0			0
$529$	22.9163			0.996363
$530$		0			0
$531$	7.53104			0.326819
$532$		0			0
$533$	4.33191			0.187636
$534$		0			0
$535$	−48.5076			−2.09716
$536$		0			0
$537$	−58.2222			−2.51248
$538$		0			0
$539$		−	5.29933i		−	0.228259i
$540$		0			0
$541$	−36.7656			−1.58068			−0.790339	−	0.612669i	$-0.790096\pi$
$541$	−36.7656			−1.58068			−0.790339	+	0.612669i	$0.790096\pi$
$542$		0			0
$543$			16.1467i			0.692920i
$544$		0			0
$545$		−	17.2238i		−	0.737786i
$546$		0			0
$547$	−19.8207			−0.847473			−0.423737	−	0.905785i	$-0.639282\pi$
$547$	−19.8207			−0.847473			−0.423737	+	0.905785i	$0.639282\pi$
$548$		0			0
$549$	−14.6473			−0.625133
$550$		0			0
$551$	−20.0372	−	8.67632i	−0.853613	−	0.369624i
$552$		0			0
$553$			12.0680i			0.513185i
$554$		0			0
$555$			83.8848i			3.56071i
$556$		0			0
$557$	−2.08609			−0.0883906			−0.0441953	−	0.999023i	$-0.514072\pi$
$557$	−2.08609			−0.0883906			−0.0441953	+	0.999023i	$0.514072\pi$
$558$		0			0
$559$	24.5566			1.03863
$560$		0			0
$561$		−	13.4102i		−	0.566179i
$562$		0			0
$563$	9.99476			0.421229			0.210615	−	0.977569i	$-0.432454\pi$
$563$	9.99476			0.421229			0.210615	+	0.977569i	$0.432454\pi$
$564$		0			0
$565$			42.3906i			1.78339i
$566$		0			0
$567$			8.57372i			0.360062i
$568$		0			0
$569$		−	8.69561i		−	0.364539i	−0.983249	−	0.182270i	$-0.941656\pi$
$569$		−	8.69561i		−	0.364539i	0.983249	−	0.182270i	$-0.0583443\pi$
$570$		0			0
$571$			20.0687i			0.839849i	0.907559	+	0.419925i	$0.137943\pi$
$571$			20.0687i			0.839849i	−0.907559	+	0.419925i	$0.862057\pi$
$572$		0			0
$573$			56.7545i			2.37095i
$574$		0			0
$575$			3.22244i			0.134385i
$576$		0			0
$577$	47.7825			1.98921			0.994606	−	0.103727i	$-0.0330768\pi$
$577$	47.7825			1.98921			0.994606	+	0.103727i	$0.0330768\pi$
$578$		0			0
$579$			62.6088i			2.60194i
$580$		0			0
$581$	15.9088			0.660007
$582$		0			0
$583$	3.85333			0.159589
$584$		0			0
$585$			60.0992i			2.48480i
$586$		0			0
$587$		−	23.8347i		−	0.983762i	−0.870662	−	0.491881i	$-0.836309\pi$
$587$		−	23.8347i		−	0.983762i	0.870662	−	0.491881i	$-0.163691\pi$
$588$		0			0
$589$	−9.22842	+	21.3122i	−0.380250	+	0.878154i
$590$		0			0
$591$	61.1851			2.51682
$592$		0			0
$593$	10.6614			0.437810			0.218905	−	0.975746i	$-0.429751\pi$
$593$	10.6614			0.437810			0.218905	+	0.975746i	$0.429751\pi$
$594$		0			0
$595$			27.2206i			1.11594i
$596$		0			0
$597$			29.6454i			1.21331i
$598$		0			0
$599$	−24.0225			−0.981533			−0.490766	−	0.871291i	$-0.663283\pi$
$599$	−24.0225			−0.981533			−0.490766	+	0.871291i	$0.663283\pi$
$600$		0			0
$601$		−	36.6837i		−	1.49636i	−0.663495	−	0.748181i	$-0.730928\pi$
$601$		−	36.6837i		−	1.49636i	0.663495	−	0.748181i	$-0.269072\pi$
$602$		0			0
$603$	−42.9752			−1.75009
$604$		0			0
$605$	−4.01759			−0.163338
$606$		0			0
$607$	16.6721			0.676699			0.338349	−	0.941021i	$-0.390131\pi$
$607$	16.6721			0.676699			0.338349	+	0.941021i	$0.390131\pi$
$608$		0			0
$609$	16.8616			0.683266
$610$		0			0
$611$	6.86448			0.277707
$612$		0			0
$613$	−1.02535			−0.0414133			−0.0207067	−	0.999786i	$-0.506592\pi$
$613$	−1.02535			−0.0414133			−0.0207067	+	0.999786i	$0.506592\pi$
$614$		0			0
$615$			10.9978i			0.443476i
$616$		0			0
$617$	−9.70762			−0.390814			−0.195407	−	0.980722i	$-0.562603\pi$
$617$	−9.70762			−0.390814			−0.195407	+	0.980722i	$0.562603\pi$
$618$		0			0
$619$		−	47.6309i		−	1.91445i	−0.289346	−	0.957224i	$-0.593438\pi$
$619$		−	47.6309i		−	1.91445i	0.289346	−	0.957224i	$-0.406562\pi$
$620$		0			0
$621$		−	0.494455i		−	0.0198418i
$622$		0			0
$623$	−21.8737			−0.876349
$624$		0			0
$625$	43.4176			1.73670
$626$		0			0
$627$	10.3246	+	4.47065i	0.412324	+	0.178541i
$628$		0			0
$629$			42.0270i			1.67573i
$630$		0			0
$631$			33.3061i			1.32589i	0.748666	+	0.662947i	$0.230695\pi$
$631$			33.3061i			1.32589i	−0.748666	+	0.662947i	$0.769305\pi$
$632$		0			0
$633$	12.2614			0.487347
$634$		0			0
$635$	27.9455			1.10898
$636$		0			0
$637$			21.6456i			0.857631i
$638$		0			0
$639$	37.3140			1.47612
$640$		0			0
$641$		−	32.9918i		−	1.30310i	−0.758607	−	0.651548i	$-0.774120\pi$
$641$		−	32.9918i		−	1.30310i	0.758607	−	0.651548i	$-0.225880\pi$
$642$		0			0
$643$			26.2499i			1.03519i	0.855625	+	0.517597i	$0.173173\pi$
$643$			26.2499i			1.03519i	−0.855625	+	0.517597i	$0.826827\pi$
$644$		0			0
$645$			62.3442i			2.45480i
$646$		0			0
$647$			41.7317i			1.64064i	0.571903	+	0.820321i	$0.306205\pi$
$647$			41.7317i			1.64064i	−0.571903	+	0.820321i	$0.693795\pi$
$648$		0			0
$649$		−	2.05637i		−	0.0807195i
$650$		0			0
$651$		−	17.9345i		−	0.702909i
$652$		0			0
$653$	−48.1086			−1.88264			−0.941318	−	0.337522i	$-0.890411\pi$
$653$	−48.1086			−1.88264			−0.941318	+	0.337522i	$0.890411\pi$
$654$		0			0
$655$		−	58.6754i		−	2.29264i
$656$		0			0
$657$	42.8705			1.67254
$658$		0			0
$659$	−37.2415			−1.45072			−0.725362	−	0.688368i	$-0.758328\pi$
$659$	−37.2415			−1.45072			−0.725362	+	0.688368i	$0.758328\pi$
$660$		0			0
$661$		−	37.5272i		−	1.45964i	−0.683641	−	0.729819i	$-0.739604\pi$
$661$		−	37.5272i		−	1.45964i	0.683641	−	0.729819i	$-0.260396\pi$
$662$		0			0
$663$			54.7752i			2.12729i
$664$		0			0
$665$	−20.9573	−	9.07474i	−0.812689	−	0.351903i
$666$		0			0
$667$	1.44889			0.0561013
$668$		0			0
$669$	−31.7102			−1.22599
$670$		0			0
$671$			3.99949i			0.154398i
$672$		0			0
$673$			22.4855i			0.866754i	0.901213	+	0.433377i	$0.142678\pi$
$673$			22.4855i			0.866754i	−0.901213	+	0.433377i	$0.857322\pi$
$674$		0			0
$675$	−19.0455			−0.733063
$676$		0			0
$677$			45.2921i			1.74072i	0.492417	+	0.870359i	$0.336113\pi$
$677$			45.2921i			1.74072i	−0.492417	+	0.870359i	$0.663887\pi$
$678$		0			0
$679$	−16.6579			−0.639271
$680$		0			0
$681$	27.7763			1.06439
$682$		0			0
$683$	27.4399			1.04996			0.524979	−	0.851115i	$-0.324073\pi$
$683$	27.4399			1.04996			0.524979	+	0.851115i	$0.324073\pi$
$684$		0			0
$685$	−52.3020			−1.99835
$686$		0			0
$687$	10.2348			0.390480
$688$		0			0
$689$	−15.7393			−0.599620
$690$		0			0
$691$		−	15.3482i		−	0.583873i	−0.956438	−	0.291937i	$-0.905700\pi$
$691$		−	15.3482i		−	0.583873i	0.956438	−	0.291937i	$-0.0942996\pi$
$692$		0			0
$693$	−4.77599			−0.181425
$694$		0			0
$695$			64.0040i			2.42781i
$696$		0			0
$697$			5.51002i			0.208707i
$698$		0			0
$699$	19.4204			0.734546
$700$		0			0
$701$	−30.4544			−1.15025			−0.575124	−	0.818066i	$-0.695046\pi$
$701$	−30.4544			−1.15025			−0.575124	+	0.818066i	$0.695046\pi$
$702$		0			0
$703$	−32.3568	−	14.0108i	−1.22036	−	0.528429i
$704$		0			0
$705$			17.4276i			0.656360i
$706$		0			0
$707$			9.27530i			0.348834i
$708$		0			0
$709$	17.0793			0.641426			0.320713	−	0.947176i	$-0.396078\pi$
$709$	17.0793			0.641426			0.320713	+	0.947176i	$0.396078\pi$
$710$		0			0
$711$	−33.8908			−1.27100
$712$		0			0
$713$		−	1.54109i		−	0.0577142i
$714$		0			0
$715$	16.4102			0.613708
$716$		0			0
$717$			16.9672i			0.633650i
$718$		0			0
$719$			25.9329i			0.967136i	0.875307	+	0.483568i	$0.160659\pi$
$719$			25.9329i			0.967136i	−0.875307	+	0.483568i	$0.839341\pi$
$720$		0			0
$721$		−	18.3175i		−	0.682179i
$722$		0			0
$723$			35.7818i			1.33074i
$724$		0			0
$725$		−	55.8088i		−	2.07269i
$726$		0			0
$727$		−	11.5381i		−	0.427924i	−0.976842	−	0.213962i	$-0.931363\pi$
$727$		−	11.5381i		−	0.427924i	0.976842	−	0.213962i	$-0.0686369\pi$
$728$		0			0
$729$	−37.3150			−1.38204
$730$		0			0
$731$			31.2350i			1.15527i
$732$		0			0
$733$	−1.19954			−0.0443060			−0.0221530	−	0.999755i	$-0.507052\pi$
$733$	−1.19954			−0.0443060			−0.0221530	+	0.999755i	$0.507052\pi$
$734$		0			0
$735$	−54.9540			−2.02701
$736$		0			0
$737$			11.7345i			0.432246i
$738$		0			0
$739$		−	51.1176i		−	1.88039i	−0.340636	−	0.940195i	$-0.610642\pi$
$739$		−	51.1176i		−	1.88039i	0.340636	−	0.940195i	$-0.389358\pi$
$740$		0			0
$741$	−42.1717	−	18.2608i	−1.54922	−	0.670828i
$742$		0			0
$743$	15.6075			0.572584			0.286292	−	0.958142i	$-0.407577\pi$
$743$	15.6075			0.572584			0.286292	+	0.958142i	$0.407577\pi$
$744$		0			0
$745$	81.3022			2.97868
$746$		0			0
$747$			44.6767i			1.63464i
$748$		0			0
$749$			15.7454i			0.575324i
$750$		0			0
$751$	21.6930			0.791588			0.395794	−	0.918339i	$-0.370469\pi$
$751$	21.6930			0.791588			0.395794	+	0.918339i	$0.370469\pi$
$752$		0			0
$753$			28.6913i			1.04557i
$754$		0			0
$755$	−0.0755162			−0.00274831
$756$		0			0
$757$	3.19589			0.116156			0.0580782	−	0.998312i	$-0.481503\pi$
$757$	3.19589			0.116156			0.0580782	+	0.998312i	$0.481503\pi$
$758$		0			0
$759$	−0.746572			−0.0270988
$760$		0			0
$761$	12.4414			0.450999			0.225500	−	0.974243i	$-0.427599\pi$
$761$	12.4414			0.450999			0.225500	+	0.974243i	$0.427599\pi$
$762$		0			0
$763$	−5.59078			−0.202400
$764$		0			0
$765$	−76.4439			−2.76383
$766$		0			0
$767$			8.39943i			0.303286i
$768$		0			0
$769$	24.0034			0.865584			0.432792	−	0.901494i	$-0.357528\pi$
$769$	24.0034			0.865584			0.432792	+	0.901494i	$0.357528\pi$
$770$		0			0
$771$			22.9056i			0.824926i
$772$		0			0
$773$		−	12.7936i		−	0.460154i	−0.973172	−	0.230077i	$-0.926102\pi$
$773$		−	12.7936i		−	0.460154i	0.973172	−	0.230077i	$-0.0738979\pi$
$774$		0			0
$775$	−59.3600			−2.13227
$776$		0			0
$777$	27.2287			0.976824
$778$		0			0
$779$	−4.24219	−	1.83691i	−0.151992	−	0.0658143i
$780$		0			0
$781$		−	10.1887i		−	0.364579i
$782$		0			0
$783$			8.56336i			0.306030i
$784$		0			0
$785$	13.1568			0.469587
$786$		0			0
$787$	37.6889			1.34346			0.671731	−	0.740795i	$-0.265551\pi$
$787$	37.6889			1.34346			0.671731	+	0.740795i	$0.265551\pi$
$788$		0			0
$789$			5.89941i			0.210025i
$790$		0			0
$791$	13.7598			0.489244
$792$		0			0
$793$		−	16.3363i		−	0.580118i
$794$		0			0
$795$		−	39.9590i		−	1.41720i
$796$		0			0
$797$			23.9720i			0.849133i	0.905397	+	0.424567i	$0.139574\pi$
$797$			23.9720i			0.849133i	−0.905397	+	0.424567i	$0.860426\pi$
$798$		0			0
$799$			8.73136i			0.308893i
$800$		0			0
$801$		−	61.4279i		−	2.17045i
$802$		0			0
$803$		−	11.7059i		−	0.413092i
$804$		0			0
$805$	1.51543			0.0534117
$806$		0			0
$807$			36.7274i			1.29286i
$808$		0			0
$809$	−1.54279			−0.0542417			−0.0271209	−	0.999632i	$-0.508634\pi$
$809$	−1.54279			−0.0542417			−0.0271209	+	0.999632i	$0.508634\pi$
$810$		0			0
$811$	−32.2082			−1.13098			−0.565491	−	0.824755i	$-0.691313\pi$
$811$	−32.2082			−1.13098			−0.565491	+	0.824755i	$0.691313\pi$
$812$		0			0
$813$		−	27.2158i		−	0.954498i
$814$		0			0
$815$			55.6104i			1.94795i
$816$		0			0
$817$	−24.0480	−	10.4130i	−0.841333	−	0.364306i
$818$		0			0
$819$	19.5080			0.681664
$820$		0			0
$821$	−28.8624			−1.00731			−0.503653	−	0.863906i	$-0.668011\pi$
$821$	−28.8624			−1.00731			−0.503653	+	0.863906i	$0.668011\pi$
$822$		0			0
$823$			7.64173i			0.266374i	0.991091	+	0.133187i	$0.0425211\pi$
$823$			7.64173i			0.266374i	−0.991091	+	0.133187i	$0.957479\pi$
$824$		0			0
$825$			28.7566i			1.00118i
$826$		0			0
$827$	9.62215			0.334595			0.167297	−	0.985906i	$-0.446496\pi$
$827$	9.62215			0.334595			0.167297	+	0.985906i	$0.446496\pi$
$828$		0			0
$829$			11.4396i			0.397313i	0.980069	+	0.198657i	$0.0636578\pi$
$829$			11.4396i			0.397313i	−0.980069	+	0.198657i	$0.936342\pi$
$830$		0			0
$831$	14.8257			0.514296
$832$		0			0
$833$	−27.5324			−0.953942
$834$		0			0
$835$	56.5113			1.95565
$836$		0			0
$837$	9.10827			0.314828
$838$		0			0
$839$	12.8385			0.443234			0.221617	−	0.975134i	$-0.428867\pi$
$839$	12.8385			0.443234			0.221617	+	0.975134i	$0.428867\pi$
$840$		0			0
$841$	3.90695			0.134722
$842$		0			0
$843$			33.3293i			1.14792i
$844$		0			0
$845$	−14.8005			−0.509152
$846$		0			0
$847$			1.30410i			0.0448093i
$848$		0			0
$849$		−	11.5953i		−	0.397948i
$850$		0			0
$851$	2.33973			0.0802047
$852$		0			0
$853$	−21.4597			−0.734767			−0.367383	−	0.930070i	$-0.619746\pi$
$853$	−21.4597			−0.734767			−0.367383	+	0.930070i	$0.619746\pi$
$854$		0			0
$855$	25.4847	−	58.8546i	0.871557	−	2.01278i
$856$		0			0
$857$			25.1636i			0.859571i	0.902931	+	0.429786i	$0.141411\pi$
$857$			25.1636i			0.859571i	−0.902931	+	0.429786i	$0.858589\pi$
$858$		0			0
$859$			47.9012i			1.63437i	0.576378	+	0.817183i	$0.304466\pi$
$859$			47.9012i			1.63437i	−0.576378	+	0.817183i	$0.695534\pi$
$860$		0			0
$861$	3.56986			0.121661
$862$		0			0
$863$	7.72645			0.263011			0.131506	−	0.991315i	$-0.458019\pi$
$863$	7.72645			0.263011			0.131506	+	0.991315i	$0.458019\pi$
$864$		0			0
$865$		−	78.1182i		−	2.65610i
$866$		0			0
$867$	−25.7925			−0.875961
$868$		0			0
$869$			9.25395i			0.313919i
$870$		0			0
$871$		−	47.9306i		−	1.62407i
$872$		0			0
$873$		−	46.7805i		−	1.58328i
$874$		0			0
$875$		−	32.1749i		−	1.08771i
$876$		0			0
$877$		−	36.6502i		−	1.23759i	−0.785553	−	0.618794i	$-0.787622\pi$
$877$		−	36.6502i		−	1.23759i	0.785553	−	0.618794i	$-0.212378\pi$
$878$		0			0
$879$			5.80563i			0.195819i
$880$		0			0
$881$	2.10689			0.0709828			0.0354914	−	0.999370i	$-0.488700\pi$
$881$	2.10689			0.0709828			0.0354914	+	0.999370i	$0.488700\pi$
$882$		0			0
$883$			1.03287i			0.0347590i	0.999849	+	0.0173795i	$0.00553234\pi$
$883$			1.03287i			0.0347590i	−0.999849	+	0.0173795i	$0.994468\pi$
$884$		0			0
$885$	−21.3245			−0.716815
$886$		0			0
$887$	26.5213			0.890497			0.445248	−	0.895407i	$-0.353115\pi$
$887$	26.5213			0.890497			0.445248	+	0.895407i	$0.353115\pi$
$888$		0			0
$889$		−	9.07101i		−	0.304232i
$890$		0			0
$891$			6.57445i			0.220253i
$892$		0			0
$893$	−6.72232	−	2.91084i	−0.224954	−	0.0974074i
$894$		0			0
$895$	90.6239			3.02922
$896$		0			0
$897$	3.04944			0.101818
$898$		0			0
$899$			26.6898i			0.890154i
$900$		0			0
$901$		−	20.0198i		−	0.666956i
$902$		0			0
$903$	20.2367			0.673436
$904$		0			0
$905$		−	25.1326i		−	0.835435i
$906$		0			0
$907$	−32.5066			−1.07937			−0.539683	−	0.841869i	$-0.681456\pi$
$907$	−32.5066			−1.07937			−0.539683	+	0.841869i	$0.681456\pi$
$908$		0			0
$909$	−26.0479			−0.863954
$910$		0			0
$911$	22.5115			0.745840			0.372920	−	0.927864i	$-0.378357\pi$
$911$	22.5115			0.745840			0.372920	+	0.927864i	$0.378357\pi$
$912$		0			0
$913$	12.1991			0.403731
$914$		0			0
$915$	41.4746			1.37111
$916$		0			0
$917$	−19.0458			−0.628949
$918$		0			0
$919$			1.07456i			0.0354463i	0.999843	+	0.0177232i	$0.00564175\pi$
$919$			1.07456i			0.0354463i	−0.999843	+	0.0177232i	$0.994358\pi$
$920$		0			0
$921$	13.9896			0.460974
$922$		0			0
$923$			41.6166i			1.36983i
$924$		0			0
$925$		−	90.1221i		−	2.96320i
$926$		0			0
$927$	51.4412			1.68955
$928$		0			0
$929$	−1.62067			−0.0531725			−0.0265862	−	0.999647i	$-0.508464\pi$
$929$	−1.62067			−0.0531725			−0.0265862	+	0.999647i	$0.508464\pi$
$930$		0			0
$931$	9.17868	−	21.1974i	0.300819	−	0.694715i
$932$		0			0
$933$			43.8441i			1.43539i
$934$		0			0
$935$			20.8732i			0.682626i
$936$		0			0
$937$	3.39960			0.111060			0.0555300	−	0.998457i	$-0.482315\pi$
$937$	3.39960			0.111060			0.0555300	+	0.998457i	$0.482315\pi$
$938$		0			0
$939$	−28.3655			−0.925672
$940$		0			0
$941$			52.6979i			1.71790i	0.512057	+	0.858952i	$0.328884\pi$
$941$			52.6979i			1.71790i	−0.512057	+	0.858952i	$0.671116\pi$
$942$		0			0
$943$	0.306753			0.00998926
$944$		0			0
$945$			8.95659i			0.291358i
$946$		0			0
$947$			31.6242i			1.02765i	0.857895	+	0.513825i	$0.171772\pi$
$947$			31.6242i			1.02765i	−0.857895	+	0.513825i	$0.828228\pi$
$948$		0			0
$949$			47.8138i			1.55210i
$950$		0			0
$951$		−	83.0781i		−	2.69399i
$952$		0			0
$953$			52.3361i			1.69533i	0.530529	+	0.847667i	$0.321993\pi$
$953$			52.3361i			1.69533i	−0.530529	+	0.847667i	$0.678007\pi$
$954$		0			0
$955$		−	88.3393i		−	2.85859i
$956$		0			0
$957$	12.9297			0.417958
$958$		0			0
$959$			16.9770i			0.548217i
$960$		0			0
$961$	−2.61189			−0.0842545
$962$		0			0
$963$	−44.2179			−1.42490
$964$		0			0
$965$		−	97.4517i		−	3.13708i
$966$		0			0
$967$			39.8205i			1.28054i	0.768149	+	0.640271i	$0.221178\pi$
$967$			39.8205i			1.28054i	−0.768149	+	0.640271i	$0.778822\pi$
$968$		0			0
$969$	23.2271	−	53.6408i	0.746160	−	1.72319i
$970$		0			0
$971$	36.5458			1.17281			0.586405	−	0.810018i	$-0.300543\pi$
$971$	36.5458			1.17281			0.586405	+	0.810018i	$0.300543\pi$
$972$		0			0
$973$	20.7755			0.666031
$974$		0			0
$975$		−	117.459i		−	3.76170i
$976$		0			0
$977$			14.6279i			0.467987i	0.972238	+	0.233994i	$0.0751795\pi$
$977$			14.6279i			0.467987i	−0.972238	+	0.233994i	$0.924821\pi$
$978$		0			0
$979$	−16.7730			−0.536069
$980$		0			0
$981$		−	15.7006i		−	0.501282i
$982$		0			0
$983$	−42.3229			−1.34989			−0.674945	−	0.737868i	$-0.735833\pi$
$983$	−42.3229			−1.34989			−0.674945	+	0.737868i	$0.735833\pi$
$984$		0			0
$985$	−95.2355			−3.03446
$986$		0			0
$987$	5.65692			0.180062
$988$		0			0
$989$	1.73891			0.0552942
$990$		0			0
$991$	−24.6118			−0.781819			−0.390910	−	0.920429i	$-0.627840\pi$
$991$	−24.6118			−0.781819			−0.390910	+	0.920429i	$0.627840\pi$
$992$		0			0
$993$	46.4488			1.47401
$994$		0			0
$995$		−	46.1436i		−	1.46285i
$996$		0			0
$997$	22.6612			0.717689			0.358844	−	0.933397i	$-0.383171\pi$
$997$	22.6612			0.717689			0.358844	+	0.933397i	$0.383171\pi$
$998$		0			0
$999$			13.8284i			0.437512i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.8	yes	64
4.3	odd	2	inner	3344.2.o.b.1519.57	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.58	yes	64
76.75	even	2	inner	3344.2.o.b.1519.7	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.7	✓	64	76.75	even	2	inner
3344.2.o.b.1519.8	yes	64	1.1	even	1	trivial
3344.2.o.b.1519.57	yes	64	4.3	odd	2	inner
3344.2.o.b.1519.58	yes	64	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.58114 q^{3} +4.01759 q^{5} -1.30410i q^{7} +3.66230 q^{9} +O(q^{10})\)	\(q-2.58114 q^{3} +4.01759 q^{5} -1.30410i q^{7} +3.66230 q^{9} -1.00000i q^{11} +4.08460i q^{13} -10.3700 q^{15} -5.19545 q^{17} +(1.73204 - 4.00000i) q^{19} +3.36606i q^{21} +0.289241i q^{23} +11.1410 q^{25} -1.70949 q^{27} -5.00930i q^{29} -5.32805 q^{31} +2.58114i q^{33} -5.23932i q^{35} -8.08920i q^{37} -10.5429i q^{39} -1.06055i q^{41} -6.01199i q^{43} +14.7136 q^{45} -1.68058i q^{47} +5.29933 q^{49} +13.4102 q^{51} +3.85333i q^{53} -4.01759i q^{55} +(-4.47065 + 10.3246i) q^{57} +2.05637 q^{59} -3.99949 q^{61} -4.77599i q^{63} +16.4102i q^{65} -11.7345 q^{67} -0.746572i q^{69} +10.1887 q^{71} +11.7059 q^{73} -28.7566 q^{75} -1.30410 q^{77} -9.25395 q^{79} -6.57445 q^{81} +12.1991i q^{83} -20.8732 q^{85} +12.9297i q^{87} -16.7730i q^{89} +5.32670 q^{91} +13.7525 q^{93} +(6.95865 - 16.0704i) q^{95} -12.7735i q^{97} -3.66230i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 64 * q + 56 * q^9	\( 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) \) 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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