LMFDB - Embedded newform 416.2.b.c.209.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(209,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.209"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.3
Root			$-0.671462 - 1.24464i$ of defining polynomial
Character	$\chi$	$=$	416.209
Dual form			416.2.b.c.209.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.146365i q^{3} +1.00000i q^{5} +0.146365 q^{7} +2.97858 q^{9} +2.68585i q^{11} -1.00000i q^{13} +0.146365 q^{15} +1.00000 q^{17} +4.00000i q^{19} -0.0214229i q^{21} +6.68585 q^{23} +4.00000 q^{25} -0.875057i q^{27} +4.39312i q^{29} -1.31415 q^{31} +0.393115 q^{33} +0.146365i q^{35} -3.97858i q^{37} -0.146365 q^{39} -6.39312 q^{41} -6.83221i q^{43} +2.97858i q^{45} +7.12494 q^{47} -6.97858 q^{49} -0.146365i q^{51} +8.97858i q^{53} -2.68585 q^{55} +0.585462 q^{57} -12.3503i q^{59} -8.35027i q^{61} +0.435961 q^{63} +1.00000 q^{65} +8.29273i q^{67} -0.978577i q^{69} -5.51806 q^{71} -6.97858 q^{73} -0.585462i q^{75} +0.393115i q^{77} -15.0361 q^{79} +8.80765 q^{81} +4.29273i q^{83} +1.00000i q^{85} +0.643000 q^{87} -5.37169 q^{89} -0.146365i q^{91} +0.192347i q^{93} -4.00000 q^{95} -10.3503 q^{97} +8.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{7} - 12 q^{9} - 2 q^{15} + 6 q^{17} + 16 q^{23} + 24 q^{25} - 32 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{57} + 44 q^{63} + 6 q^{65} + 18 q^{71} - 12 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) $ 6 * q - 2 * q^7 - 12 * q^9 - 2 * q^15 + 6 * q^17 + 16 * q^23 + 24 * q^25 - 32 * q^31 - 16 * q^33 + 2 * q^39 - 20 * q^41 + 10 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^57 + 44 * q^63 + 6 * q^65 + 18 * q^71 - 12 * q^73 + 12 * q^79 + 46 * q^81 - 80 * q^87 + 16 * q^89 - 24 * q^95 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 0.146365i	− 0.0845042i	−0.999107	−	0.0422521i	$-0.986547\pi$
$3$	− 0.146365i	− 0.0845042i	0.999107	−	0.0422521i	$-0.0134533\pi$
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	0.146365	0.0553210	0.0276605	−	0.999617i	$-0.491194\pi$
$7$	0.146365	0.0553210	0.0276605	+	0.999617i	$0.491194\pi$
$8$	0	0
$9$	2.97858	0.992859
$10$	0	0
$11$	2.68585i	0.809813i	0.914358	+	0.404907i	$0.132696\pi$
$11$	2.68585i	0.809813i	−0.914358	+	0.404907i	$0.867304\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0.146365	0.0377914
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 0.0214229i	− 0.00467485i
$22$	0	0
$23$	6.68585	1.39410	0.697048	−	0.717025i	$-0.254497\pi$
$23$	6.68585	1.39410	0.697048	+	0.717025i	$0.254497\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 0.875057i	− 0.168405i
$28$	0	0
$29$	4.39312i	0.815781i	0.913031	+	0.407891i	$0.133735\pi$
$29$	4.39312i	0.815781i	−0.913031	+	0.407891i	$0.866265\pi$
$30$	0	0
$31$	−1.31415	−0.236029	−0.118014	−	0.993012i	$-0.537653\pi$
$31$	−1.31415	−0.236029	−0.118014	+	0.993012i	$0.537653\pi$
$32$	0	0
$33$	0.393115	0.0684326
$34$	0	0
$35$	0.146365i	0.0247403i
$36$	0	0
$37$	− 3.97858i	− 0.654074i	−0.945012	−	0.327037i	$-0.893950\pi$
$37$	− 3.97858i	− 0.654074i	0.945012	−	0.327037i	$-0.106050\pi$
$38$	0	0
$39$	−0.146365	−0.0234372
$40$	0	0
$41$	−6.39312	−0.998437	−0.499218	−	0.866476i	$-0.666379\pi$
$41$	−6.39312	−0.998437	−0.499218	+	0.866476i	$0.666379\pi$
$42$	0	0
$43$	− 6.83221i	− 1.04190i	−0.853586	−	0.520951i	$-0.825577\pi$
$43$	− 6.83221i	− 1.04190i	0.853586	−	0.520951i	$-0.174423\pi$
$44$	0	0
$45$	2.97858i	0.444020i
$46$	0	0
$47$	7.12494	1.03928	0.519640	−	0.854385i	$-0.326066\pi$
$47$	7.12494	1.03928	0.519640	+	0.854385i	$0.326066\pi$
$48$	0	0
$49$	−6.97858	−0.996940
$50$	0	0
$51$	− 0.146365i	− 0.0204953i
$52$	0	0
$53$	8.97858i	1.23330i	0.787236	+	0.616651i	$0.211511\pi$
$53$	8.97858i	1.23330i	−0.787236	+	0.616651i	$0.788489\pi$
$54$	0	0
$55$	−2.68585	−0.362159
$56$	0	0
$57$	0.585462	0.0775463
$58$	0	0
$59$	− 12.3503i	− 1.60787i	−0.594718	−	0.803934i	$-0.702736\pi$
$59$	− 12.3503i	− 1.60787i	0.594718	−	0.803934i	$-0.297264\pi$
$60$	0	0
$61$	− 8.35027i	− 1.06914i	−0.845123	−	0.534571i	$-0.820473\pi$
$61$	− 8.35027i	− 1.06914i	0.845123	−	0.534571i	$-0.179527\pi$
$62$	0	0
$63$	0.435961	0.0549259
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	8.29273i	1.01312i	0.862205	+	0.506559i	$0.169083\pi$
$67$	8.29273i	1.01312i	−0.862205	+	0.506559i	$0.830917\pi$
$68$	0	0
$69$	− 0.978577i	− 0.117807i
$70$	0	0
$71$	−5.51806	−0.654873	−0.327436	−	0.944873i	$-0.606185\pi$
$71$	−5.51806	−0.654873	−0.327436	+	0.944873i	$0.606185\pi$
$72$	0	0
$73$	−6.97858	−0.816781	−0.408390	−	0.912807i	$-0.633910\pi$
$73$	−6.97858	−0.816781	−0.408390	+	0.912807i	$0.633910\pi$
$74$	0	0
$75$	− 0.585462i	− 0.0676033i
$76$	0	0
$77$	0.393115i	0.0447996i
$78$	0	0
$79$	−15.0361	−1.69170	−0.845848	−	0.533425i	$-0.820905\pi$
$79$	−15.0361	−1.69170	−0.845848	+	0.533425i	$0.820905\pi$
$80$	0	0
$81$	8.80765	0.978628
$82$	0	0
$83$	4.29273i	0.471188i	0.971852	+	0.235594i	$0.0757036\pi$
$83$	4.29273i	0.471188i	−0.971852	+	0.235594i	$0.924296\pi$
$84$	0	0
$85$	1.00000i	0.108465i
$86$	0	0
$87$	0.643000	0.0689369
$88$	0	0
$89$	−5.37169	−0.569398	−0.284699	−	0.958617i	$-0.591894\pi$
$89$	−5.37169	−0.569398	−0.284699	+	0.958617i	$0.591894\pi$
$90$	0	0
$91$	− 0.146365i	− 0.0153433i
$92$	0	0
$93$	0.192347i	0.0199454i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−10.3503	−1.05091	−0.525455	−	0.850821i	$-0.676105\pi$
$97$	−10.3503	−1.05091	−0.525455	+	0.850821i	$0.676105\pi$
$98$	0	0
$99$	8.00000i	0.804030i
$100$	0	0
$101$	− 17.9572i	− 1.78680i	−0.449258	−	0.893402i	$-0.648312\pi$
$101$	− 17.9572i	− 1.78680i	0.449258	−	0.893402i	$-0.351688\pi$
$102$	0	0
$103$	−10.2499	−1.00995	−0.504976	−	0.863134i	$-0.668499\pi$
$103$	−10.2499	−1.00995	−0.504976	+	0.863134i	$0.668499\pi$
$104$	0	0
$105$	0.0214229	0.00209066
$106$	0	0
$107$	− 14.6858i	− 1.41973i	−0.704336	−	0.709867i	$-0.748755\pi$
$107$	− 14.6858i	− 1.41973i	0.704336	−	0.709867i	$-0.251245\pi$
$108$	0	0
$109$	− 14.7648i	− 1.41421i	−0.707108	−	0.707106i	$-0.750000\pi$
$109$	− 14.7648i	− 1.41421i	0.707108	−	0.707106i	$-0.250000\pi$
$110$	0	0
$111$	−0.582326	−0.0552720
$112$	0	0
$113$	3.95715	0.372258	0.186129	−	0.982525i	$-0.440406\pi$
$113$	3.95715	0.372258	0.186129	+	0.982525i	$0.440406\pi$
$114$	0	0
$115$	6.68585i	0.623458i
$116$	0	0
$117$	− 2.97858i	− 0.275370i
$118$	0	0
$119$	0.146365	0.0134173
$120$	0	0
$121$	3.78623	0.344203
$122$	0	0
$123$	0.935731i	0.0843721i
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	6.10038	0.541322	0.270661	−	0.962675i	$-0.412758\pi$
$127$	6.10038	0.541322	0.270661	+	0.962675i	$0.412758\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	14.8322i	1.29590i	0.761684	+	0.647948i	$0.224373\pi$
$131$	14.8322i	1.29590i	−0.761684	+	0.647948i	$0.775627\pi$
$132$	0	0
$133$	0.585462i	0.0507660i
$134$	0	0
$135$	0.875057	0.0753129
$136$	0	0
$137$	9.76481	0.834264	0.417132	−	0.908846i	$-0.363035\pi$
$137$	9.76481	0.834264	0.417132	+	0.908846i	$0.363035\pi$
$138$	0	0
$139$	− 19.4752i	− 1.65187i	−0.563768	−	0.825933i	$-0.690649\pi$
$139$	− 19.4752i	− 1.65187i	0.563768	−	0.825933i	$-0.309351\pi$
$140$	0	0
$141$	− 1.04285i	− 0.0878235i
$142$	0	0
$143$	2.68585	0.224602
$144$	0	0
$145$	−4.39312	−0.364828
$146$	0	0
$147$	1.02142i	0.0842455i
$148$	0	0
$149$	11.9572i	0.979568i	0.871844	+	0.489784i	$0.162924\pi$
$149$	11.9572i	0.979568i	−0.871844	+	0.489784i	$0.837076\pi$
$150$	0	0
$151$	−12.7894	−1.04078	−0.520392	−	0.853928i	$-0.674214\pi$
$151$	−12.7894	−1.04078	−0.520392	+	0.853928i	$0.674214\pi$
$152$	0	0
$153$	2.97858	0.240804
$154$	0	0
$155$	− 1.31415i	− 0.105555i
$156$	0	0
$157$	21.7220i	1.73360i	0.498655	+	0.866801i	$0.333828\pi$
$157$	21.7220i	1.73360i	−0.498655	+	0.866801i	$0.666172\pi$
$158$	0	0
$159$	1.31415	0.104219
$160$	0	0
$161$	0.978577	0.0771227
$162$	0	0
$163$	− 20.6430i	− 1.61688i	−0.588575	−	0.808442i	$-0.700311\pi$
$163$	− 20.6430i	− 1.61688i	0.588575	−	0.808442i	$-0.299689\pi$
$164$	0	0
$165$	0.393115i	0.0306040i
$166$	0	0
$167$	18.0147	1.39402	0.697009	−	0.717062i	$-0.254514\pi$
$167$	18.0147	1.39402	0.697009	+	0.717062i	$0.254514\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	11.9143i	0.911110i
$172$	0	0
$173$	− 6.58546i	− 0.500683i	−0.968158	−	0.250342i	$-0.919457\pi$
$173$	− 6.58546i	− 0.500683i	0.968158	−	0.250342i	$-0.0805430\pi$
$174$	0	0
$175$	0.585462	0.0442568
$176$	0	0
$177$	−1.80765	−0.135872
$178$	0	0
$179$	7.56090i	0.565128i	0.959248	+	0.282564i	$0.0911850\pi$
$179$	7.56090i	0.565128i	−0.959248	+	0.282564i	$0.908815\pi$
$180$	0	0
$181$	0.628308i	0.0467017i	0.999727	+	0.0233509i	$0.00743349\pi$
$181$	0.628308i	0.0467017i	−0.999727	+	0.0233509i	$0.992567\pi$
$182$	0	0
$183$	−1.22219	−0.0903470
$184$	0	0
$185$	3.97858	0.292511
$186$	0	0
$187$	2.68585i	0.196409i
$188$	0	0
$189$	− 0.128078i	− 0.00931632i
$190$	0	0
$191$	10.1004	0.730838	0.365419	−	0.930843i	$-0.380926\pi$
$191$	10.1004	0.730838	0.365419	+	0.930843i	$0.380926\pi$
$192$	0	0
$193$	−3.37169	−0.242700	−0.121350	−	0.992610i	$-0.538722\pi$
$193$	−3.37169	−0.242700	−0.121350	+	0.992610i	$0.538722\pi$
$194$	0	0
$195$	− 0.146365i	− 0.0104815i
$196$	0	0
$197$	14.7220i	1.04890i	0.851442	+	0.524448i	$0.175728\pi$
$197$	14.7220i	1.04890i	−0.851442	+	0.524448i	$0.824272\pi$
$198$	0	0
$199$	13.6644	0.968645	0.484323	−	0.874889i	$-0.339066\pi$
$199$	13.6644	0.968645	0.484323	+	0.874889i	$0.339066\pi$
$200$	0	0
$201$	1.21377	0.0856127
$202$	0	0
$203$	0.643000i	0.0451298i
$204$	0	0
$205$	− 6.39312i	− 0.446515i
$206$	0	0
$207$	19.9143	1.38414
$208$	0	0
$209$	−10.7434	−0.743135
$210$	0	0
$211$	1.46052i	0.100546i	0.998736	+	0.0502731i	$0.0160092\pi$
$211$	1.46052i	0.100546i	−0.998736	+	0.0502731i	$0.983991\pi$
$212$	0	0
$213$	0.807653i	0.0553395i
$214$	0	0
$215$	6.83221	0.465953
$216$	0	0
$217$	−0.192347	−0.0130573
$218$	0	0
$219$	1.02142i	0.0690214i
$220$	0	0
$221$	− 1.00000i	− 0.0672673i
$222$	0	0
$223$	−7.12494	−0.477121	−0.238561	−	0.971128i	$-0.576676\pi$
$223$	−7.12494	−0.477121	−0.238561	+	0.971128i	$0.576676\pi$
$224$	0	0
$225$	11.9143	0.794287
$226$	0	0
$227$	24.9357i	1.65504i	0.561434	+	0.827521i	$0.310250\pi$
$227$	24.9357i	1.65504i	−0.561434	+	0.827521i	$0.689750\pi$
$228$	0	0
$229$	− 2.95715i	− 0.195414i	−0.995215	−	0.0977071i	$-0.968849\pi$
$229$	− 2.95715i	− 0.195414i	0.995215	−	0.0977071i	$-0.0311508\pi$
$230$	0	0
$231$	0.0575385	0.00378576
$232$	0	0
$233$	−5.19235	−0.340162	−0.170081	−	0.985430i	$-0.554403\pi$
$233$	−5.19235	−0.340162	−0.170081	+	0.985430i	$0.554403\pi$
$234$	0	0
$235$	7.12494i	0.464780i
$236$	0	0
$237$	2.20077i	0.142955i
$238$	0	0
$239$	14.3963	0.931216	0.465608	−	0.884991i	$-0.345836\pi$
$239$	14.3963	0.931216	0.465608	+	0.884991i	$0.345836\pi$
$240$	0	0
$241$	23.3288	1.50274	0.751372	−	0.659879i	$-0.229393\pi$
$241$	23.3288	1.50274	0.751372	+	0.659879i	$0.229393\pi$
$242$	0	0
$243$	− 3.91431i	− 0.251103i
$244$	0	0
$245$	− 6.97858i	− 0.445845i
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0.628308	0.0398174
$250$	0	0
$251$	− 15.2713i	− 0.963916i	−0.876194	−	0.481958i	$-0.839926\pi$
$251$	− 15.2713i	− 0.963916i	0.876194	−	0.481958i	$-0.160074\pi$
$252$	0	0
$253$	17.9572i	1.12896i
$254$	0	0
$255$	0.146365	0.00916576
$256$	0	0
$257$	20.9143	1.30460	0.652299	−	0.757961i	$-0.273805\pi$
$257$	20.9143	1.30460	0.652299	+	0.757961i	$0.273805\pi$
$258$	0	0
$259$	− 0.582326i	− 0.0361840i
$260$	0	0
$261$	13.0852i	0.809956i
$262$	0	0
$263$	−13.0214	−0.802935	−0.401468	−	0.915873i	$-0.631500\pi$
$263$	−13.0214	−0.802935	−0.401468	+	0.915873i	$0.631500\pi$
$264$	0	0
$265$	−8.97858	−0.551550
$266$	0	0
$267$	0.786230i	0.0481165i
$268$	0	0
$269$	− 24.3503i	− 1.48466i	−0.670033	−	0.742331i	$-0.733720\pi$
$269$	− 24.3503i	− 1.48466i	0.670033	−	0.742331i	$-0.266280\pi$
$270$	0	0
$271$	−25.5756	−1.55361	−0.776803	−	0.629743i	$-0.783160\pi$
$271$	−25.5756	−1.55361	−0.776803	+	0.629743i	$0.783160\pi$
$272$	0	0
$273$	−0.0214229	−0.00129657
$274$	0	0
$275$	10.7434i	0.647850i
$276$	0	0
$277$	− 4.43596i	− 0.266531i	−0.991080	−	0.133266i	$-0.957454\pi$
$277$	− 4.43596i	− 0.266531i	0.991080	−	0.133266i	$-0.0425463\pi$
$278$	0	0
$279$	−3.91431	−0.234344
$280$	0	0
$281$	11.4145	0.680934	0.340467	−	0.940256i	$-0.389415\pi$
$281$	11.4145	0.680934	0.340467	+	0.940256i	$0.389415\pi$
$282$	0	0
$283$	− 19.4721i	− 1.15749i	−0.815507	−	0.578747i	$-0.803542\pi$
$283$	− 19.4721i	− 1.15749i	0.815507	−	0.578747i	$-0.196458\pi$
$284$	0	0
$285$	0.585462i	0.0346798i
$286$	0	0
$287$	−0.935731	−0.0552345
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	1.51492i	0.0888063i
$292$	0	0
$293$	− 10.8077i	− 0.631390i	−0.948861	−	0.315695i	$-0.897762\pi$
$293$	− 10.8077i	− 0.631390i	0.948861	−	0.315695i	$-0.102238\pi$
$294$	0	0
$295$	12.3503	0.719060
$296$	0	0
$297$	2.35027	0.136376
$298$	0	0
$299$	− 6.68585i	− 0.386652i
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	−2.62831	−0.150992
$304$	0	0
$305$	8.35027	0.478135
$306$	0	0
$307$	4.93573i	0.281697i	0.990031	+	0.140849i	$0.0449831\pi$
$307$	4.93573i	0.281697i	−0.990031	+	0.140849i	$0.955017\pi$
$308$	0	0
$309$	1.50023i	0.0853451i
$310$	0	0
$311$	−0.786230	−0.0445830	−0.0222915	−	0.999752i	$-0.507096\pi$
$311$	−0.786230	−0.0445830	−0.0222915	+	0.999752i	$0.507096\pi$
$312$	0	0
$313$	−13.9357	−0.787694	−0.393847	−	0.919176i	$-0.628856\pi$
$313$	−13.9357	−0.787694	−0.393847	+	0.919176i	$0.628856\pi$
$314$	0	0
$315$	0.435961i	0.0245636i
$316$	0	0
$317$	19.9572i	1.12091i	0.828186	+	0.560453i	$0.189373\pi$
$317$	19.9572i	1.12091i	−0.828186	+	0.560453i	$0.810627\pi$
$318$	0	0
$319$	−11.7992	−0.660630
$320$	0	0
$321$	−2.14950	−0.119973
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	−2.16106	−0.119507
$328$	0	0
$329$	1.04285	0.0574939
$330$	0	0
$331$	11.7073i	0.643490i	0.946826	+	0.321745i	$0.104269\pi$
$331$	11.7073i	0.643490i	−0.946826	+	0.321745i	$0.895731\pi$
$332$	0	0
$333$	− 11.8505i	− 0.649403i
$334$	0	0
$335$	−8.29273	−0.453080
$336$	0	0
$337$	−0.807653	−0.0439957	−0.0219978	−	0.999758i	$-0.507003\pi$
$337$	−0.807653	−0.0439957	−0.0219978	+	0.999758i	$0.507003\pi$
$338$	0	0
$339$	− 0.579191i	− 0.0314573i
$340$	0	0
$341$	− 3.52962i	− 0.191139i
$342$	0	0
$343$	−2.04598	−0.110473
$344$	0	0
$345$	0.978577	0.0526848
$346$	0	0
$347$	2.59702i	0.139415i	0.997567	+	0.0697076i	$0.0222066\pi$
$347$	2.59702i	0.139415i	−0.997567	+	0.0697076i	$0.977793\pi$
$348$	0	0
$349$	16.1709i	0.865610i	0.901488	+	0.432805i	$0.142476\pi$
$349$	16.1709i	0.865610i	−0.901488	+	0.432805i	$0.857524\pi$
$350$	0	0
$351$	−0.875057	−0.0467071
$352$	0	0
$353$	−5.32885	−0.283626	−0.141813	−	0.989893i	$-0.545293\pi$
$353$	−5.32885	−0.283626	−0.141813	+	0.989893i	$0.545293\pi$
$354$	0	0
$355$	− 5.51806i	− 0.292868i
$356$	0	0
$357$	− 0.0214229i	− 0.00113382i
$358$	0	0
$359$	−34.6002	−1.82613	−0.913063	−	0.407818i	$-0.866290\pi$
$359$	−34.6002	−1.82613	−0.913063	+	0.407818i	$0.866290\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 0.554173i	− 0.0290866i
$364$	0	0
$365$	− 6.97858i	− 0.365275i
$366$	0	0
$367$	14.2499	0.743838	0.371919	−	0.928265i	$-0.378700\pi$
$367$	14.2499	0.743838	0.371919	+	0.928265i	$0.378700\pi$
$368$	0	0
$369$	−19.0424	−0.991307
$370$	0	0
$371$	1.31415i	0.0682275i
$372$	0	0
$373$	− 15.1281i	− 0.783302i	−0.920114	−	0.391651i	$-0.871904\pi$
$373$	− 15.1281i	− 0.783302i	0.920114	−	0.391651i	$-0.128096\pi$
$374$	0	0
$375$	1.31729	0.0680245
$376$	0	0
$377$	4.39312	0.226257
$378$	0	0
$379$	− 5.22846i	− 0.268568i	−0.990943	−	0.134284i	$-0.957127\pi$
$379$	− 5.22846i	− 0.268568i	0.990943	−	0.134284i	$-0.0428735\pi$
$380$	0	0
$381$	− 0.892886i	− 0.0457439i
$382$	0	0
$383$	−27.3257	−1.39628	−0.698139	−	0.715962i	$-0.745988\pi$
$383$	−27.3257	−1.39628	−0.698139	+	0.715962i	$0.745988\pi$
$384$	0	0
$385$	−0.393115	−0.0200350
$386$	0	0
$387$	− 20.3503i	− 1.03446i
$388$	0	0
$389$	− 9.02142i	− 0.457404i	−0.973496	−	0.228702i	$-0.926552\pi$
$389$	− 9.02142i	− 0.457404i	0.973496	−	0.228702i	$-0.0734482\pi$
$390$	0	0
$391$	6.68585	0.338118
$392$	0	0
$393$	2.17092	0.109509
$394$	0	0
$395$	− 15.0361i	− 0.756549i
$396$	0	0
$397$	− 35.0852i	− 1.76088i	−0.474160	−	0.880439i	$-0.657248\pi$
$397$	− 35.0852i	− 1.76088i	0.474160	−	0.880439i	$-0.342752\pi$
$398$	0	0
$399$	0.0856914	0.00428994
$400$	0	0
$401$	−1.06427	−0.0531470	−0.0265735	−	0.999647i	$-0.508460\pi$
$401$	−1.06427	−0.0531470	−0.0265735	+	0.999647i	$0.508460\pi$
$402$	0	0
$403$	1.31415i	0.0654627i
$404$	0	0
$405$	8.80765i	0.437656i
$406$	0	0
$407$	10.6858	0.529678
$408$	0	0
$409$	−31.7220	−1.56855	−0.784275	−	0.620413i	$-0.786965\pi$
$409$	−31.7220	−1.56855	−0.784275	+	0.620413i	$0.786965\pi$
$410$	0	0
$411$	− 1.42923i	− 0.0704988i
$412$	0	0
$413$	− 1.80765i	− 0.0889488i
$414$	0	0
$415$	−4.29273	−0.210722
$416$	0	0
$417$	−2.85050	−0.139590
$418$	0	0
$419$	3.06740i	0.149852i	0.997189	+	0.0749262i	$0.0238721\pi$
$419$	3.06740i	0.149852i	−0.997189	+	0.0749262i	$0.976128\pi$
$420$	0	0
$421$	− 2.21377i	− 0.107893i	−0.998544	−	0.0539463i	$-0.982820\pi$
$421$	− 2.21377i	− 0.107893i	0.998544	−	0.0539463i	$-0.0171800\pi$
$422$	0	0
$423$	21.2222	1.03186
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	− 1.22219i	− 0.0591460i
$428$	0	0
$429$	− 0.393115i	− 0.0189798i
$430$	0	0
$431$	−18.8898	−0.909887	−0.454944	−	0.890520i	$-0.650341\pi$
$431$	−18.8898	−0.909887	−0.454944	+	0.890520i	$0.650341\pi$
$432$	0	0
$433$	−21.8929	−1.05210	−0.526052	−	0.850452i	$-0.676328\pi$
$433$	−21.8929	−1.05210	−0.526052	+	0.850452i	$0.676328\pi$
$434$	0	0
$435$	0.643000i	0.0308295i
$436$	0	0
$437$	26.7434i	1.27931i
$438$	0	0
$439$	−13.8077	−0.659003	−0.329502	−	0.944155i	$-0.606881\pi$
$439$	−13.8077	−0.659003	−0.329502	+	0.944155i	$0.606881\pi$
$440$	0	0
$441$	−20.7862	−0.989820
$442$	0	0
$443$	25.5756i	1.21513i	0.794269	+	0.607567i	$0.207854\pi$
$443$	25.5756i	1.21513i	−0.794269	+	0.607567i	$0.792146\pi$
$444$	0	0
$445$	− 5.37169i	− 0.254643i
$446$	0	0
$447$	1.75011	0.0827776
$448$	0	0
$449$	−24.3074	−1.14714	−0.573569	−	0.819157i	$-0.694442\pi$
$449$	−24.3074	−1.14714	−0.573569	+	0.819157i	$0.694442\pi$
$450$	0	0
$451$	− 17.1709i	− 0.808547i
$452$	0	0
$453$	1.87192i	0.0879506i
$454$	0	0
$455$	0.146365	0.00686172
$456$	0	0
$457$	12.2008	0.570728	0.285364	−	0.958419i	$-0.407886\pi$
$457$	12.2008	0.570728	0.285364	+	0.958419i	$0.407886\pi$
$458$	0	0
$459$	− 0.875057i	− 0.0408442i
$460$	0	0
$461$	− 8.02142i	− 0.373595i	−0.982398	−	0.186797i	$-0.940189\pi$
$461$	− 8.02142i	− 0.373595i	0.982398	−	0.186797i	$-0.0598108\pi$
$462$	0	0
$463$	36.0575	1.67574	0.837868	−	0.545873i	$-0.183802\pi$
$463$	36.0575	1.67574	0.837868	+	0.545873i	$0.183802\pi$
$464$	0	0
$465$	−0.192347	−0.00891987
$466$	0	0
$467$	− 19.1856i	− 0.887804i	−0.896075	−	0.443902i	$-0.853594\pi$
$467$	− 19.1856i	− 0.887804i	0.896075	−	0.443902i	$-0.146406\pi$
$468$	0	0
$469$	1.21377i	0.0560467i
$470$	0	0
$471$	3.17935	0.146497
$472$	0	0
$473$	18.3503	0.843746
$474$	0	0
$475$	16.0000i	0.734130i
$476$	0	0
$477$	26.7434i	1.22450i
$478$	0	0
$479$	11.9112	0.544235	0.272118	−	0.962264i	$-0.412276\pi$
$479$	11.9112	0.544235	0.272118	+	0.962264i	$0.412276\pi$
$480$	0	0
$481$	−3.97858	−0.181408
$482$	0	0
$483$	− 0.143230i	− 0.00651719i
$484$	0	0
$485$	− 10.3503i	− 0.469982i
$486$	0	0
$487$	15.8568	0.718539	0.359269	−	0.933234i	$-0.383026\pi$
$487$	15.8568	0.718539	0.359269	+	0.933234i	$0.383026\pi$
$488$	0	0
$489$	−3.02142	−0.136633
$490$	0	0
$491$	3.91117i	0.176509i	0.996098	+	0.0882544i	$0.0281288\pi$
$491$	3.91117i	0.176509i	−0.996098	+	0.0882544i	$0.971871\pi$
$492$	0	0
$493$	4.39312i	0.197856i
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	−0.807653	−0.0362282
$498$	0	0
$499$	− 23.1793i	− 1.03765i	−0.854880	−	0.518825i	$-0.826370\pi$
$499$	− 23.1793i	− 1.03765i	0.854880	−	0.518825i	$-0.173630\pi$
$500$	0	0
$501$	− 2.63673i	− 0.117800i
$502$	0	0
$503$	−18.5426	−0.826774	−0.413387	−	0.910555i	$-0.635654\pi$
$503$	−18.5426	−0.826774	−0.413387	+	0.910555i	$0.635654\pi$
$504$	0	0
$505$	17.9572	0.799083
$506$	0	0
$507$	0.146365i	0.00650032i
$508$	0	0
$509$	25.9143i	1.14863i	0.818634	+	0.574316i	$0.194732\pi$
$509$	25.9143i	1.14863i	−0.818634	+	0.574316i	$0.805268\pi$
$510$	0	0
$511$	−1.02142	−0.0451851
$512$	0	0
$513$	3.50023	0.154539
$514$	0	0
$515$	− 10.2499i	− 0.451664i
$516$	0	0
$517$	19.1365i	0.841622i
$518$	0	0
$519$	−0.963884	−0.0423098
$520$	0	0
$521$	−12.7648	−0.559236	−0.279618	−	0.960111i	$-0.590208\pi$
$521$	−12.7648	−0.559236	−0.279618	+	0.960111i	$0.590208\pi$
$522$	0	0
$523$	27.1856i	1.18874i	0.804190	+	0.594372i	$0.202599\pi$
$523$	27.1856i	1.18874i	−0.804190	+	0.594372i	$0.797401\pi$
$524$	0	0
$525$	− 0.0856914i	− 0.00373988i
$526$	0	0
$527$	−1.31415	−0.0572454
$528$	0	0
$529$	21.7005	0.943502
$530$	0	0
$531$	− 36.7862i	− 1.59639i
$532$	0	0
$533$	6.39312i	0.276917i
$534$	0	0
$535$	14.6858	0.634924
$536$	0	0
$537$	1.10666	0.0477557
$538$	0	0
$539$	− 18.7434i	− 0.807335i
$540$	0	0
$541$	6.76481i	0.290842i	0.989370	+	0.145421i	$0.0464536\pi$
$541$	6.76481i	0.290842i	−0.989370	+	0.145421i	$0.953546\pi$
$542$	0	0
$543$	0.0919626	0.00394649
$544$	0	0
$545$	14.7648	0.632455
$546$	0	0
$547$	15.3832i	0.657740i	0.944375	+	0.328870i	$0.106668\pi$
$547$	15.3832i	0.657740i	−0.944375	+	0.328870i	$0.893332\pi$
$548$	0	0
$549$	− 24.8719i	− 1.06151i
$550$	0	0
$551$	−17.5725	−0.748612
$552$	0	0
$553$	−2.20077	−0.0935862
$554$	0	0
$555$	− 0.582326i	− 0.0247184i
$556$	0	0
$557$	− 7.59388i	− 0.321763i	−0.986974	−	0.160882i	$-0.948566\pi$
$557$	− 7.59388i	− 0.321763i	0.986974	−	0.160882i	$-0.0514337\pi$
$558$	0	0
$559$	−6.83221	−0.288972
$560$	0	0
$561$	0.393115	0.0165973
$562$	0	0
$563$	14.9754i	0.631140i	0.948902	+	0.315570i	$0.102196\pi$
$563$	14.9754i	0.631140i	−0.948902	+	0.315570i	$0.897804\pi$
$564$	0	0
$565$	3.95715i	0.166479i
$566$	0	0
$567$	1.28914	0.0541386
$568$	0	0
$569$	−4.25662	−0.178447	−0.0892233	−	0.996012i	$-0.528438\pi$
$569$	−4.25662	−0.178447	−0.0892233	+	0.996012i	$0.528438\pi$
$570$	0	0
$571$	43.5903i	1.82420i	0.409971	+	0.912098i	$0.365539\pi$
$571$	43.5903i	1.82420i	−0.409971	+	0.912098i	$0.634461\pi$
$572$	0	0
$573$	− 1.47835i	− 0.0617589i
$574$	0	0
$575$	26.7434	1.11528
$576$	0	0
$577$	43.5934	1.81482	0.907409	−	0.420249i	$-0.138057\pi$
$577$	43.5934	1.81482	0.907409	+	0.420249i	$0.138057\pi$
$578$	0	0
$579$	0.493499i	0.0205091i
$580$	0	0
$581$	0.628308i	0.0260666i
$582$	0	0
$583$	−24.1151	−0.998744
$584$	0	0
$585$	2.97858	0.123149
$586$	0	0
$587$	1.31415i	0.0542409i	0.999632	+	0.0271205i	$0.00863377\pi$
$587$	1.31415i	0.0542409i	−0.999632	+	0.0271205i	$0.991366\pi$
$588$	0	0
$589$	− 5.25662i	− 0.216595i
$590$	0	0
$591$	2.15479	0.0886361
$592$	0	0
$593$	−0.777809	−0.0319408	−0.0159704	−	0.999872i	$-0.505084\pi$
$593$	−0.777809	−0.0319408	−0.0159704	+	0.999872i	$0.505084\pi$
$594$	0	0
$595$	0.146365i	0.00600040i
$596$	0	0
$597$	− 2.00000i	− 0.0818546i
$598$	0	0
$599$	−3.36327	−0.137420	−0.0687098	−	0.997637i	$-0.521888\pi$
$599$	−3.36327	−0.137420	−0.0687098	+	0.997637i	$0.521888\pi$
$600$	0	0
$601$	4.17092	0.170136	0.0850678	−	0.996375i	$-0.472889\pi$
$601$	4.17092	0.170136	0.0850678	+	0.996375i	$0.472889\pi$
$602$	0	0
$603$	24.7005i	1.00588i
$604$	0	0
$605$	3.78623i	0.153932i
$606$	0	0
$607$	34.8929	1.41626	0.708129	−	0.706083i	$-0.249539\pi$
$607$	34.8929	1.41626	0.708129	+	0.706083i	$0.249539\pi$
$608$	0	0
$609$	0.0941131	0.00381365
$610$	0	0
$611$	− 7.12494i	− 0.288244i
$612$	0	0
$613$	− 17.9143i	− 0.723552i	−0.932265	−	0.361776i	$-0.882171\pi$
$613$	− 17.9143i	− 0.723552i	0.932265	−	0.361776i	$-0.117829\pi$
$614$	0	0
$615$	−0.935731	−0.0377323
$616$	0	0
$617$	23.9143	0.962754	0.481377	−	0.876514i	$-0.340137\pi$
$617$	23.9143	0.962754	0.481377	+	0.876514i	$0.340137\pi$
$618$	0	0
$619$	27.9656i	1.12403i	0.827127	+	0.562016i	$0.189974\pi$
$619$	27.9656i	1.12403i	−0.827127	+	0.562016i	$0.810026\pi$
$620$	0	0
$621$	− 5.85050i	− 0.234772i
$622$	0	0
$623$	−0.786230	−0.0314997
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	1.57246i	0.0627980i
$628$	0	0
$629$	− 3.97858i	− 0.158636i
$630$	0	0
$631$	28.4966	1.13443	0.567217	−	0.823569i	$-0.308020\pi$
$631$	28.4966	1.13443	0.567217	+	0.823569i	$0.308020\pi$
$632$	0	0
$633$	0.213770	0.00849658
$634$	0	0
$635$	6.10038i	0.242086i
$636$	0	0
$637$	6.97858i	0.276501i
$638$	0	0
$639$	−16.4360	−0.650197
$640$	0	0
$641$	13.9143	0.549582	0.274791	−	0.961504i	$-0.411391\pi$
$641$	13.9143	0.549582	0.274791	+	0.961504i	$0.411391\pi$
$642$	0	0
$643$	− 19.0361i	− 0.750711i	−0.926881	−	0.375356i	$-0.877521\pi$
$643$	− 19.0361i	− 0.750711i	0.926881	−	0.375356i	$-0.122479\pi$
$644$	0	0
$645$	− 1.00000i	− 0.0393750i
$646$	0	0
$647$	17.6069	0.692198	0.346099	−	0.938198i	$-0.387506\pi$
$647$	17.6069	0.692198	0.346099	+	0.938198i	$0.387506\pi$
$648$	0	0
$649$	33.1709	1.30207
$650$	0	0
$651$	0.0281529i	0.00110340i
$652$	0	0
$653$	− 33.8715i	− 1.32549i	−0.748844	−	0.662746i	$-0.769391\pi$
$653$	− 33.8715i	− 1.32549i	0.748844	−	0.662746i	$-0.230609\pi$
$654$	0	0
$655$	−14.8322	−0.579542
$656$	0	0
$657$	−20.7862	−0.810948
$658$	0	0
$659$	27.1856i	1.05900i	0.848310	+	0.529501i	$0.177621\pi$
$659$	27.1856i	1.05900i	−0.848310	+	0.529501i	$0.822379\pi$
$660$	0	0
$661$	48.2730i	1.87760i	0.344460	+	0.938801i	$0.388062\pi$
$661$	48.2730i	1.87760i	−0.344460	+	0.938801i	$0.611938\pi$
$662$	0	0
$663$	−0.146365	−0.00568436
$664$	0	0
$665$	−0.585462	−0.0227032
$666$	0	0
$667$	29.3717i	1.13728i
$668$	0	0
$669$	1.04285i	0.0403187i
$670$	0	0
$671$	22.4275	0.865806
$672$	0	0
$673$	5.78623	0.223043	0.111521	−	0.993762i	$-0.464428\pi$
$673$	5.78623	0.223043	0.111521	+	0.993762i	$0.464428\pi$
$674$	0	0
$675$	− 3.50023i	− 0.134724i
$676$	0	0
$677$	− 37.3288i	− 1.43466i	−0.696731	−	0.717332i	$-0.745363\pi$
$677$	− 37.3288i	− 1.43466i	0.696731	−	0.717332i	$-0.254637\pi$
$678$	0	0
$679$	−1.51492	−0.0581374
$680$	0	0
$681$	3.64973	0.139858
$682$	0	0
$683$	17.6644i	0.675910i	0.941162	+	0.337955i	$0.109735\pi$
$683$	17.6644i	0.675910i	−0.941162	+	0.337955i	$0.890265\pi$
$684$	0	0
$685$	9.76481i	0.373094i
$686$	0	0
$687$	−0.432825	−0.0165133
$688$	0	0
$689$	8.97858	0.342057
$690$	0	0
$691$	− 44.4998i	− 1.69285i	−0.532507	−	0.846426i	$-0.678750\pi$
$691$	− 44.4998i	− 1.69285i	0.532507	−	0.846426i	$-0.321250\pi$
$692$	0	0
$693$	1.17092i	0.0444797i
$694$	0	0
$695$	19.4752	0.738737
$696$	0	0
$697$	−6.39312	−0.242157
$698$	0	0
$699$	0.759980i	0.0287451i
$700$	0	0
$701$	17.0643i	0.644509i	0.946653	+	0.322254i	$0.104441\pi$
$701$	17.0643i	0.644509i	−0.946653	+	0.322254i	$0.895559\pi$
$702$	0	0
$703$	15.9143	0.600220
$704$	0	0
$705$	1.04285	0.0392758
$706$	0	0
$707$	− 2.62831i	− 0.0988477i
$708$	0	0
$709$	0.427539i	0.0160566i	0.999968	+	0.00802829i	$0.00255551\pi$
$709$	0.427539i	0.0160566i	−0.999968	+	0.00802829i	$0.997444\pi$
$710$	0	0
$711$	−44.7862	−1.67961
$712$	0	0
$713$	−8.78623	−0.329047
$714$	0	0
$715$	2.68585i	0.100445i
$716$	0	0
$717$	− 2.10711i	− 0.0786916i
$718$	0	0
$719$	−37.3780	−1.39396	−0.696981	−	0.717089i	$-0.745474\pi$
$719$	−37.3780	−1.39396	−0.696981	+	0.717089i	$0.745474\pi$
$720$	0	0
$721$	−1.50023	−0.0558715
$722$	0	0
$723$	− 3.41454i	− 0.126988i
$724$	0	0
$725$	17.5725i	0.652625i
$726$	0	0
$727$	−14.4851	−0.537222	−0.268611	−	0.963249i	$-0.586565\pi$
$727$	−14.4851	−0.537222	−0.268611	+	0.963249i	$0.586565\pi$
$728$	0	0
$729$	25.8500	0.957409
$730$	0	0
$731$	− 6.83221i	− 0.252698i
$732$	0	0
$733$	0.299461i	0.0110608i	0.999985	+	0.00553042i	$0.00176040\pi$
$733$	0.299461i	0.0110608i	−0.999985	+	0.00553042i	$0.998240\pi$
$734$	0	0
$735$	−1.02142	−0.0376757
$736$	0	0
$737$	−22.2730	−0.820436
$738$	0	0
$739$	− 18.9786i	− 0.698138i	−0.937097	−	0.349069i	$-0.886498\pi$
$739$	− 18.9786i	− 0.698138i	0.937097	−	0.349069i	$-0.113502\pi$
$740$	0	0
$741$	− 0.585462i	− 0.0215075i
$742$	0	0
$743$	26.7465	0.981235	0.490617	−	0.871375i	$-0.336771\pi$
$743$	26.7465	0.981235	0.490617	+	0.871375i	$0.336771\pi$
$744$	0	0
$745$	−11.9572	−0.438076
$746$	0	0
$747$	12.7862i	0.467824i
$748$	0	0
$749$	− 2.14950i	− 0.0785411i
$750$	0	0
$751$	14.1923	0.517886	0.258943	−	0.965893i	$-0.416626\pi$
$751$	14.1923	0.517886	0.258943	+	0.965893i	$0.416626\pi$
$752$	0	0
$753$	−2.23519	−0.0814549
$754$	0	0
$755$	− 12.7894i	− 0.465453i
$756$	0	0
$757$	45.5934i	1.65712i	0.559899	+	0.828561i	$0.310840\pi$
$757$	45.5934i	1.65712i	−0.559899	+	0.828561i	$0.689160\pi$
$758$	0	0
$759$	2.62831	0.0954015
$760$	0	0
$761$	−37.6363	−1.36431	−0.682157	−	0.731206i	$-0.738958\pi$
$761$	−37.6363	−1.36431	−0.682157	+	0.731206i	$0.738958\pi$
$762$	0	0
$763$	− 2.16106i	− 0.0782356i
$764$	0	0
$765$	2.97858i	0.107691i
$766$	0	0
$767$	−12.3503	−0.445942
$768$	0	0
$769$	0.700539	0.0252621	0.0126310	−	0.999920i	$-0.495979\pi$
$769$	0.700539	0.0252621	0.0126310	+	0.999920i	$0.495979\pi$
$770$	0	0
$771$	− 3.06113i	− 0.110244i
$772$	0	0
$773$	5.08569i	0.182920i	0.995809	+	0.0914598i	$0.0291533\pi$
$773$	5.08569i	0.182920i	−0.995809	+	0.0914598i	$0.970847\pi$
$774$	0	0
$775$	−5.25662	−0.188823
$776$	0	0
$777$	−0.0852325	−0.00305770
$778$	0	0
$779$	− 25.5725i	− 0.916228i
$780$	0	0
$781$	− 14.8207i	− 0.530325i
$782$	0	0
$783$	3.84423	0.137381
$784$	0	0
$785$	−21.7220	−0.775290
$786$	0	0
$787$	14.3074i	0.510005i	0.966940	+	0.255002i	$0.0820762\pi$
$787$	14.3074i	0.510005i	−0.966940	+	0.255002i	$0.917924\pi$
$788$	0	0
$789$	1.90589i	0.0678514i
$790$	0	0
$791$	0.579191	0.0205937
$792$	0	0
$793$	−8.35027	−0.296527
$794$	0	0
$795$	1.31415i	0.0466082i
$796$	0	0
$797$	2.54262i	0.0900641i	0.998986	+	0.0450320i	$0.0143390\pi$
$797$	2.54262i	0.0900641i	−0.998986	+	0.0450320i	$0.985661\pi$
$798$	0	0
$799$	7.12494	0.252062
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	− 18.7434i	− 0.661440i
$804$	0	0
$805$	0.978577i	0.0344903i
$806$	0	0
$807$	−3.56404	−0.125460
$808$	0	0
$809$	−33.1495	−1.16547	−0.582737	−	0.812661i	$-0.698018\pi$
$809$	−33.1495	−1.16547	−0.582737	+	0.812661i	$0.698018\pi$
$810$	0	0
$811$	− 1.28600i	− 0.0451576i	−0.999745	−	0.0225788i	$-0.992812\pi$
$811$	− 1.28600i	− 0.0451576i	0.999745	−	0.0225788i	$-0.00718767\pi$
$812$	0	0
$813$	3.74338i	0.131286i
$814$	0	0
$815$	20.6430	0.723093
$816$	0	0
$817$	27.3288	0.956115
$818$	0	0
$819$	− 0.435961i	− 0.0152337i
$820$	0	0
$821$	− 11.7005i	− 0.408352i	−0.978934	−	0.204176i	$-0.934549\pi$
$821$	− 11.7005i	− 0.408352i	0.978934	−	0.204176i	$-0.0654514\pi$
$822$	0	0
$823$	38.9786	1.35871	0.679354	−	0.733811i	$-0.262260\pi$
$823$	38.9786	1.35871	0.679354	+	0.733811i	$0.262260\pi$
$824$	0	0
$825$	1.57246	0.0547461
$826$	0	0
$827$	− 43.5296i	− 1.51367i	−0.653604	−	0.756837i	$-0.726744\pi$
$827$	− 43.5296i	− 1.51367i	0.653604	−	0.756837i	$-0.273256\pi$
$828$	0	0
$829$	35.0852i	1.21856i	0.792955	+	0.609280i	$0.208542\pi$
$829$	35.0852i	1.21856i	−0.792955	+	0.609280i	$0.791458\pi$
$830$	0	0
$831$	−0.649272	−0.0225230
$832$	0	0
$833$	−6.97858	−0.241793
$834$	0	0
$835$	18.0147i	0.623424i
$836$	0	0
$837$	1.14996i	0.0397484i
$838$	0	0
$839$	10.4851	0.361985	0.180993	−	0.983484i	$-0.442069\pi$
$839$	10.4851	0.361985	0.180993	+	0.983484i	$0.442069\pi$
$840$	0	0
$841$	9.70054	0.334501
$842$	0	0
$843$	− 1.67069i	− 0.0575418i
$844$	0	0
$845$	− 1.00000i	− 0.0344010i
$846$	0	0
$847$	0.554173	0.0190416
$848$	0	0
$849$	−2.85004	−0.0978131
$850$	0	0
$851$	− 26.6002i	− 0.911842i
$852$	0	0
$853$	− 40.5296i	− 1.38771i	−0.720116	−	0.693854i	$-0.755911\pi$
$853$	− 40.5296i	− 1.38771i	0.720116	−	0.693854i	$-0.244089\pi$
$854$	0	0
$855$	−11.9143	−0.407461
$856$	0	0
$857$	5.61531	0.191815	0.0959076	−	0.995390i	$-0.469425\pi$
$857$	5.61531	0.191815	0.0959076	+	0.995390i	$0.469425\pi$
$858$	0	0
$859$	− 27.4721i	− 0.937335i	−0.883375	−	0.468668i	$-0.844734\pi$
$859$	− 27.4721i	− 0.937335i	0.883375	−	0.468668i	$-0.155266\pi$
$860$	0	0
$861$	0.136959i	0.00466754i
$862$	0	0
$863$	−42.8898	−1.45998	−0.729992	−	0.683456i	$-0.760476\pi$
$863$	−42.8898	−1.45998	−0.729992	+	0.683456i	$0.760476\pi$
$864$	0	0
$865$	6.58546	0.223912
$866$	0	0
$867$	2.34185i	0.0795333i
$868$	0	0
$869$	− 40.3847i	− 1.36996i
$870$	0	0
$871$	8.29273	0.280988
$872$	0	0
$873$	−30.8291	−1.04341
$874$	0	0
$875$	1.31729i	0.0445325i
$876$	0	0
$877$	− 31.7005i	− 1.07045i	−0.844709	−	0.535226i	$-0.820227\pi$
$877$	− 31.7005i	− 1.07045i	0.844709	−	0.535226i	$-0.179773\pi$
$878$	0	0
$879$	−1.58187	−0.0533551
$880$	0	0
$881$	−44.4011	−1.49591	−0.747955	−	0.663749i	$-0.768964\pi$
$881$	−44.4011	−1.49591	−0.747955	+	0.663749i	$0.768964\pi$
$882$	0	0
$883$	1.28287i	0.0431719i	0.999767	+	0.0215859i	$0.00687155\pi$
$883$	1.28287i	0.0431719i	−0.999767	+	0.0215859i	$0.993128\pi$
$884$	0	0
$885$	− 1.80765i	− 0.0607636i
$886$	0	0
$887$	−41.5212	−1.39415	−0.697073	−	0.717001i	$-0.745514\pi$
$887$	−41.5212	−1.39415	−0.697073	+	0.717001i	$0.745514\pi$
$888$	0	0
$889$	0.892886	0.0299464
$890$	0	0
$891$	23.6560i	0.792506i
$892$	0	0
$893$	28.4998i	0.953708i
$894$	0	0
$895$	−7.56090	−0.252733
$896$	0	0
$897$	−0.978577	−0.0326737
$898$	0	0
$899$	− 5.77323i	− 0.192548i
$900$	0	0
$901$	8.97858i	0.299120i
$902$	0	0
$903$	−0.146365	−0.00487074
$904$	0	0
$905$	−0.628308	−0.0208857
$906$	0	0
$907$	41.9834i	1.39404i	0.717054	+	0.697018i	$0.245490\pi$
$907$	41.9834i	1.39404i	−0.717054	+	0.697018i	$0.754510\pi$
$908$	0	0
$909$	− 53.4868i	− 1.77404i
$910$	0	0
$911$	37.5443	1.24390	0.621949	−	0.783058i	$-0.286341\pi$
$911$	37.5443	1.24390	0.621949	+	0.783058i	$0.286341\pi$
$912$	0	0
$913$	−11.5296	−0.381575
$914$	0	0
$915$	− 1.22219i	− 0.0404044i
$916$	0	0
$917$	2.17092i	0.0716902i
$918$	0	0
$919$	−48.7005	−1.60648	−0.803241	−	0.595654i	$-0.796893\pi$
$919$	−48.7005	−1.60648	−0.803241	+	0.595654i	$0.796893\pi$
$920$	0	0
$921$	0.722421	0.0238046
$922$	0	0
$923$	5.51806i	0.181629i
$924$	0	0
$925$	− 15.9143i	− 0.523259i
$926$	0	0
$927$	−30.5301	−1.00274
$928$	0	0
$929$	−11.1281	−0.365100	−0.182550	−	0.983197i	$-0.558435\pi$
$929$	−11.1281	−0.365100	−0.182550	+	0.983197i	$0.558435\pi$
$930$	0	0
$931$	− 27.9143i	− 0.914855i
$932$	0	0
$933$	0.115077i	0.00376745i
$934$	0	0
$935$	−2.68585	−0.0878366
$936$	0	0
$937$	17.6153	0.575467	0.287733	−	0.957711i	$-0.407098\pi$
$937$	17.6153	0.575467	0.287733	+	0.957711i	$0.407098\pi$
$938$	0	0
$939$	2.03971i	0.0665634i
$940$	0	0
$941$	35.6577i	1.16241i	0.813758	+	0.581204i	$0.197418\pi$
$941$	35.6577i	1.16241i	−0.813758	+	0.581204i	$0.802582\pi$
$942$	0	0
$943$	−42.7434	−1.39192
$944$	0	0
$945$	0.128078	0.00416638
$946$	0	0
$947$	22.7715i	0.739976i	0.929037	+	0.369988i	$0.120638\pi$
$947$	22.7715i	0.739976i	−0.929037	+	0.369988i	$0.879362\pi$
$948$	0	0
$949$	6.97858i	0.226534i
$950$	0	0
$951$	2.92104	0.0947212
$952$	0	0
$953$	−1.74338	−0.0564738	−0.0282369	−	0.999601i	$-0.508989\pi$
$953$	−1.74338	−0.0564738	−0.0282369	+	0.999601i	$0.508989\pi$
$954$	0	0
$955$	10.1004i	0.326841i
$956$	0	0
$957$	1.72700i	0.0558260i
$958$	0	0
$959$	1.42923	0.0461523
$960$	0	0
$961$	−29.2730	−0.944290
$962$	0	0
$963$	− 43.7429i	− 1.40960i
$964$	0	0
$965$	− 3.37169i	− 0.108539i
$966$	0	0
$967$	33.8402	1.08823	0.544113	−	0.839012i	$-0.316866\pi$
$967$	33.8402	1.08823	0.544113	+	0.839012i	$0.316866\pi$
$968$	0	0
$969$	0.585462	0.0188077
$970$	0	0
$971$	40.0031i	1.28376i	0.766804	+	0.641881i	$0.221846\pi$
$971$	40.0031i	1.28376i	−0.766804	+	0.641881i	$0.778154\pi$
$972$	0	0
$973$	− 2.85050i	− 0.0913828i
$974$	0	0
$975$	−0.585462	−0.0187498
$976$	0	0
$977$	38.7005	1.23814	0.619070	−	0.785336i	$-0.287510\pi$
$977$	38.7005	1.23814	0.619070	+	0.785336i	$0.287510\pi$
$978$	0	0
$979$	− 14.4275i	− 0.461106i
$980$	0	0
$981$	− 43.9781i	− 1.40411i
$982$	0	0
$983$	−23.8536	−0.760813	−0.380406	−	0.924819i	$-0.624216\pi$
$983$	−23.8536	−0.760813	−0.380406	+	0.924819i	$0.624216\pi$
$984$	0	0
$985$	−14.7220	−0.469081
$986$	0	0
$987$	− 0.152637i	− 0.00485848i
$988$	0	0
$989$	− 45.6791i	− 1.45251i
$990$	0	0
$991$	3.01300	0.0957111	0.0478556	−	0.998854i	$-0.484761\pi$
$991$	3.01300	0.0957111	0.0478556	+	0.998854i	$0.484761\pi$
$992$	0	0
$993$	1.71354	0.0543776
$994$	0	0
$995$	13.6644i	0.433191i
$996$	0	0
$997$	− 7.13650i	− 0.226015i	−0.993594	−	0.113008i	$-0.963952\pi$
$997$	− 7.13650i	− 0.226015i	0.993594	−	0.113008i	$-0.0360484\pi$
$998$	0	0
$999$	−3.48148	−0.110149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.b.c.209.3		6
3.2	odd	2		3744.2.g.c.1873.2		6
4.3	odd	2		104.2.b.c.53.6	yes	6
8.3	odd	2		104.2.b.c.53.5	✓	6
8.5	even	2	inner	416.2.b.c.209.4		6
12.11	even	2		936.2.g.c.469.1		6
16.3	odd	4		3328.2.a.bh.1.2		3
16.5	even	4		3328.2.a.bg.1.2		3
16.11	odd	4		3328.2.a.be.1.2		3
16.13	even	4		3328.2.a.bf.1.2		3
24.5	odd	2		3744.2.g.c.1873.5		6
24.11	even	2		936.2.g.c.469.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.c.53.5	✓	6	8.3	odd	2
104.2.b.c.53.6	yes	6	4.3	odd	2
416.2.b.c.209.3		6	1.1	even	1	trivial
416.2.b.c.209.4		6	8.5	even	2	inner
936.2.g.c.469.1		6	12.11	even	2
936.2.g.c.469.2		6	24.11	even	2
3328.2.a.be.1.2		3	16.11	odd	4
3328.2.a.bf.1.2		3	16.13	even	4
3328.2.a.bg.1.2		3	16.5	even	4
3328.2.a.bh.1.2		3	16.3	odd	4
3744.2.g.c.1873.2		6	3.2	odd	2
3744.2.g.c.1873.5		6	24.5	odd	2

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.146365i q^{3} +1.00000i q^{5} +0.146365 q^{7} +2.97858 q^{9} +2.68585i q^{11} -1.00000i q^{13} +0.146365 q^{15} +1.00000 q^{17} +4.00000i q^{19} -0.0214229i q^{21} +6.68585 q^{23} +4.00000 q^{25} -0.875057i q^{27} +4.39312i q^{29} -1.31415 q^{31} +0.393115 q^{33} +0.146365i q^{35} -3.97858i q^{37} -0.146365 q^{39} -6.39312 q^{41} -6.83221i q^{43} +2.97858i q^{45} +7.12494 q^{47} -6.97858 q^{49} -0.146365i q^{51} +8.97858i q^{53} -2.68585 q^{55} +0.585462 q^{57} -12.3503i q^{59} -8.35027i q^{61} +0.435961 q^{63} +1.00000 q^{65} +8.29273i q^{67} -0.978577i q^{69} -5.51806 q^{71} -6.97858 q^{73} -0.585462i q^{75} +0.393115i q^{77} -15.0361 q^{79} +8.80765 q^{81} +4.29273i q^{83} +1.00000i q^{85} +0.643000 q^{87} -5.37169 q^{89} -0.146365i q^{91} +0.192347i q^{93} -4.00000 q^{95} -10.3503 q^{97} +8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{7} - 12 q^{9} - 2 q^{15} + 6 q^{17} + 16 q^{23} + 24 q^{25} - 32 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{57} + 44 q^{63} + 6 q^{65} + 18 q^{71} - 12 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 6 * q - 2 * q^7 - 12 * q^9 - 2 * q^15 + 6 * q^17 + 16 * q^23 + 24 * q^25 - 32 * q^31 - 16 * q^33 + 2 * q^39 - 20 * q^41 + 10 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^57 + 44 * q^63 + 6 * q^65 + 18 * q^71 - 12 * q^73 + 12 * q^79 + 46 * q^81 - 80 * q^87 + 16 * q^89 - 24 * q^95 + 16 * q^97

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