LMFDB - Embedded newform 1216.3.d.f.191.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(191,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.191"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20] 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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 124 x^{18} + 6294 x^{16} + 169580 x^{14} + 2633777 x^{12} + 23965840 x^{10} + 123396288 x^{8} + \cdots + 6553600 $ x^20 + 124x^18 + 6294x^16 + 169580x^14 + 2633777x^12 + 23965840x^10 + 123396288x^8 + 323597312x^6 + 340403200x^4 + 121339904*x^2 + 6553600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.10
Root			$0.255387i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.f.191.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.255387i q^{3} +0.657472 q^{5} +5.46816i q^{7} +8.93478 q^{9} -7.62075i q^{11} +23.5457 q^{13} -0.167910i q^{15} +1.93859 q^{17} -4.35890i q^{19} +1.39649 q^{21} -15.5276i q^{23} -24.5677 q^{25} -4.58031i q^{27} -21.8489 q^{29} +23.0254i q^{31} -1.94624 q^{33} +3.59516i q^{35} +34.6145 q^{37} -6.01327i q^{39} -33.7873 q^{41} -51.8448i q^{43} +5.87437 q^{45} +90.4336i q^{47} +19.0993 q^{49} -0.495090i q^{51} +88.3584 q^{53} -5.01043i q^{55} -1.11321 q^{57} +19.6684i q^{59} +83.4867 q^{61} +48.8567i q^{63} +15.4807 q^{65} -80.6552i q^{67} -3.96556 q^{69} +65.6448i q^{71} +70.9481 q^{73} +6.27428i q^{75} +41.6715 q^{77} -145.044i q^{79} +79.2432 q^{81} +78.0145i q^{83} +1.27457 q^{85} +5.57992i q^{87} -46.1237 q^{89} +128.752i q^{91} +5.88039 q^{93} -2.86585i q^{95} +24.0304 q^{97} -68.0897i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 68 q^{9} + 16 q^{13} + 8 q^{17} - 64 q^{21} + 196 q^{25} + 88 q^{29} - 184 q^{33} - 16 q^{37} - 16 q^{41} - 16 q^{45} + 52 q^{49} - 88 q^{53} + 208 q^{61} - 192 q^{65} - 248 q^{69} - 152 q^{73} - 312 q^{77}+ \cdots - 72 q^{97}+O(q^{100}) $ 20 * q - 68 * q^9 + 16 * q^13 + 8 * q^17 - 64 * q^21 + 196 * q^25 + 88 * q^29 - 184 * q^33 - 16 * q^37 - 16 * q^41 - 16 * q^45 + 52 * q^49 - 88 * q^53 + 208 * q^61 - 192 * q^65 - 248 * q^69 - 152 * q^73 - 312 * q^77 + 500 * q^81 + 488 * q^85 + 304 * q^89 + 240 * q^93 - 72 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 0.255387i	− 0.0851290i	−0.999094	−	0.0425645i	$-0.986447\pi$
$3$	− 0.255387i	− 0.0851290i	0.999094	−	0.0425645i	$-0.0135528\pi$
$4$	0	0
$5$	0.657472	0.131494	0.0657472	−	0.997836i	$-0.479057\pi$
$5$	0.657472	0.131494	0.0657472	+	0.997836i	$0.479057\pi$
$6$	0	0
$7$	5.46816i	0.781165i	0.920568	+	0.390583i	$0.127726\pi$
$7$	5.46816i	0.781165i	−0.920568	+	0.390583i	$0.872274\pi$
$8$	0	0
$9$	8.93478	0.992753
$10$	0	0
$11$	− 7.62075i	− 0.692796i	−0.938088	−	0.346398i	$-0.887405\pi$
$11$	− 7.62075i	− 0.692796i	0.938088	−	0.346398i	$-0.112595\pi$
$12$	0	0
$13$	23.5457	1.81121	0.905605	−	0.424122i	$-0.139417\pi$
$13$	23.5457	1.81121	0.905605	+	0.424122i	$0.139417\pi$
$14$	0	0
$15$	− 0.167910i	− 0.0111940i
$16$	0	0
$17$	1.93859	0.114035	0.0570173	−	0.998373i	$-0.481841\pi$
$17$	1.93859	0.114035	0.0570173	+	0.998373i	$0.481841\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	1.39649	0.0664998
$22$	0	0
$23$	− 15.5276i	− 0.675115i	−0.941305	−	0.337558i	$-0.890399\pi$
$23$	− 15.5276i	− 0.675115i	0.941305	−	0.337558i	$-0.109601\pi$
$24$	0	0
$25$	−24.5677	−0.982709
$26$	0	0
$27$	− 4.58031i	− 0.169641i
$28$	0	0
$29$	−21.8489	−0.753410	−0.376705	−	0.926333i	$-0.622943\pi$
$29$	−21.8489	−0.753410	−0.376705	+	0.926333i	$0.622943\pi$
$30$	0	0
$31$	23.0254i	0.742756i	0.928482	+	0.371378i	$0.121115\pi$
$31$	23.0254i	0.742756i	−0.928482	+	0.371378i	$0.878885\pi$
$32$	0	0
$33$	−1.94624	−0.0589770
$34$	0	0
$35$	3.59516i	0.102719i
$36$	0	0
$37$	34.6145	0.935527	0.467763	−	0.883854i	$-0.345060\pi$
$37$	34.6145	0.935527	0.467763	+	0.883854i	$0.345060\pi$
$38$	0	0
$39$	− 6.01327i	− 0.154186i
$40$	0	0
$41$	−33.7873	−0.824082	−0.412041	−	0.911165i	$-0.635184\pi$
$41$	−33.7873	−0.824082	−0.412041	+	0.911165i	$0.635184\pi$
$42$	0	0
$43$	− 51.8448i	− 1.20569i	−0.797857	−	0.602846i	$-0.794033\pi$
$43$	− 51.8448i	− 1.20569i	0.797857	−	0.602846i	$-0.205967\pi$
$44$	0	0
$45$	5.87437	0.130541
$46$	0	0
$47$	90.4336i	1.92412i	0.272841	+	0.962059i	$0.412037\pi$
$47$	90.4336i	1.92412i	−0.272841	+	0.962059i	$0.587963\pi$
$48$	0	0
$49$	19.0993	0.389781
$50$	0	0
$51$	− 0.495090i	− 0.00970764i
$52$	0	0
$53$	88.3584	1.66714	0.833570	−	0.552414i	$-0.186293\pi$
$53$	88.3584	1.66714	0.833570	+	0.552414i	$0.186293\pi$
$54$	0	0
$55$	− 5.01043i	− 0.0910988i
$56$	0	0
$57$	−1.11321	−0.0195299
$58$	0	0
$59$	19.6684i	0.333362i	0.986011	+	0.166681i	$0.0533050\pi$
$59$	19.6684i	0.333362i	−0.986011	+	0.166681i	$0.946695\pi$
$60$	0	0
$61$	83.4867	1.36863	0.684317	−	0.729184i	$-0.260100\pi$
$61$	83.4867	1.36863	0.684317	+	0.729184i	$0.260100\pi$
$62$	0	0
$63$	48.8567i	0.775504i
$64$	0	0
$65$	15.4807	0.238164
$66$	0	0
$67$	− 80.6552i	− 1.20381i	−0.798568	−	0.601904i	$-0.794409\pi$
$67$	− 80.6552i	− 1.20381i	0.798568	−	0.601904i	$-0.205591\pi$
$68$	0	0
$69$	−3.96556	−0.0574718
$70$	0	0
$71$	65.6448i	0.924575i	0.886730	+	0.462287i	$0.152971\pi$
$71$	65.6448i	0.924575i	−0.886730	+	0.462287i	$0.847029\pi$
$72$	0	0
$73$	70.9481	0.971891	0.485946	−	0.873989i	$-0.338475\pi$
$73$	70.9481	0.971891	0.485946	+	0.873989i	$0.338475\pi$
$74$	0	0
$75$	6.27428i	0.0836570i
$76$	0	0
$77$	41.6715	0.541188
$78$	0	0
$79$	− 145.044i	− 1.83600i	−0.396580	−	0.918000i	$-0.629803\pi$
$79$	− 145.044i	− 1.83600i	0.396580	−	0.918000i	$-0.370197\pi$
$80$	0	0
$81$	79.2432	0.978312
$82$	0	0
$83$	78.0145i	0.939934i	0.882684	+	0.469967i	$0.155734\pi$
$83$	78.0145i	0.939934i	−0.882684	+	0.469967i	$0.844266\pi$
$84$	0	0
$85$	1.27457	0.0149949
$86$	0	0
$87$	5.57992i	0.0641370i
$88$	0	0
$89$	−46.1237	−0.518243	−0.259122	−	0.965845i	$-0.583433\pi$
$89$	−46.1237	−0.518243	−0.259122	+	0.965845i	$0.583433\pi$
$90$	0	0
$91$	128.752i	1.41485i
$92$	0	0
$93$	5.88039	0.0632300
$94$	0	0
$95$	− 2.86585i	− 0.0301669i
$96$	0	0
$97$	24.0304	0.247737	0.123868	−	0.992299i	$-0.460470\pi$
$97$	24.0304	0.247737	0.123868	+	0.992299i	$0.460470\pi$
$98$	0	0
$99$	− 68.0897i	− 0.687775i
$100$	0	0
$101$	37.3854	0.370153	0.185076	−	0.982724i	$-0.440747\pi$
$101$	37.3854	0.370153	0.185076	+	0.982724i	$0.440747\pi$
$102$	0	0
$103$	108.182i	1.05031i	0.851006	+	0.525156i	$0.175993\pi$
$103$	108.182i	1.05031i	−0.851006	+	0.525156i	$0.824007\pi$
$104$	0	0
$105$	0.918156	0.00874435
$106$	0	0
$107$	− 190.862i	− 1.78376i	−0.452274	−	0.891879i	$-0.649387\pi$
$107$	− 190.862i	− 1.78376i	0.452274	−	0.891879i	$-0.350613\pi$
$108$	0	0
$109$	15.4290	0.141550	0.0707752	−	0.997492i	$-0.477453\pi$
$109$	15.4290	0.141550	0.0707752	+	0.997492i	$0.477453\pi$
$110$	0	0
$111$	− 8.84009i	− 0.0796404i
$112$	0	0
$113$	−20.7286	−0.183439	−0.0917194	−	0.995785i	$-0.529236\pi$
$113$	−20.7286	−0.183439	−0.0917194	+	0.995785i	$0.529236\pi$
$114$	0	0
$115$	− 10.2090i	− 0.0887739i
$116$	0	0
$117$	210.376	1.79808
$118$	0	0
$119$	10.6005i	0.0890798i
$120$	0	0
$121$	62.9241	0.520034
$122$	0	0
$123$	8.62884i	0.0701532i
$124$	0	0
$125$	−32.5894	−0.260715
$126$	0	0
$127$	− 155.513i	− 1.22451i	−0.790660	−	0.612255i	$-0.790263\pi$
$127$	− 155.513i	− 1.22451i	0.790660	−	0.612255i	$-0.209737\pi$
$128$	0	0
$129$	−13.2405	−0.102639
$130$	0	0
$131$	− 51.0526i	− 0.389715i	−0.980832	−	0.194857i	$-0.937576\pi$
$131$	− 51.0526i	− 0.389715i	0.980832	−	0.194857i	$-0.0624244\pi$
$132$	0	0
$133$	23.8351	0.179212
$134$	0	0
$135$	− 3.01142i	− 0.0223068i
$136$	0	0
$137$	136.974	0.999808	0.499904	−	0.866081i	$-0.333369\pi$
$137$	136.974	0.999808	0.499904	+	0.866081i	$0.333369\pi$
$138$	0	0
$139$	8.53883i	0.0614304i	0.999528	+	0.0307152i	$0.00977849\pi$
$139$	8.53883i	0.0614304i	−0.999528	+	0.0307152i	$0.990222\pi$
$140$	0	0
$141$	23.0955	0.163798
$142$	0	0
$143$	− 179.436i	− 1.25480i
$144$	0	0
$145$	−14.3650	−0.0990692
$146$	0	0
$147$	− 4.87771i	− 0.0331817i
$148$	0	0
$149$	−208.907	−1.40206	−0.701029	−	0.713132i	$-0.747276\pi$
$149$	−208.907	−1.40206	−0.701029	+	0.713132i	$0.747276\pi$
$150$	0	0
$151$	− 51.7900i	− 0.342980i	−0.985186	−	0.171490i	$-0.945142\pi$
$151$	− 51.7900i	− 0.342980i	0.985186	−	0.171490i	$-0.0548581\pi$
$152$	0	0
$153$	17.3208	0.113208
$154$	0	0
$155$	15.1386i	0.0976683i
$156$	0	0
$157$	−112.143	−0.714284	−0.357142	−	0.934050i	$-0.616249\pi$
$157$	−112.143	−0.714284	−0.357142	+	0.934050i	$0.616249\pi$
$158$	0	0
$159$	− 22.5656i	− 0.141922i
$160$	0	0
$161$	84.9076	0.527376
$162$	0	0
$163$	257.817i	1.58170i	0.612011	+	0.790849i	$0.290361\pi$
$163$	257.817i	1.58170i	−0.612011	+	0.790849i	$0.709639\pi$
$164$	0	0
$165$	−1.27960	−0.00775514
$166$	0	0
$167$	273.031i	1.63491i	0.575990	+	0.817457i	$0.304617\pi$
$167$	273.031i	1.63491i	−0.575990	+	0.817457i	$0.695383\pi$
$168$	0	0
$169$	385.401	2.28048
$170$	0	0
$171$	− 38.9458i	− 0.227753i
$172$	0	0
$173$	86.9380	0.502532	0.251266	−	0.967918i	$-0.419153\pi$
$173$	86.9380	0.502532	0.251266	+	0.967918i	$0.419153\pi$
$174$	0	0
$175$	− 134.340i	− 0.767658i
$176$	0	0
$177$	5.02304	0.0283788
$178$	0	0
$179$	281.670i	1.57357i	0.617225	+	0.786786i	$0.288257\pi$
$179$	281.670i	1.57357i	−0.617225	+	0.786786i	$0.711743\pi$
$180$	0	0
$181$	−84.3024	−0.465759	−0.232880	−	0.972506i	$-0.574815\pi$
$181$	−84.3024	−0.465759	−0.232880	+	0.972506i	$0.574815\pi$
$182$	0	0
$183$	− 21.3214i	− 0.116510i
$184$	0	0
$185$	22.7581	0.123017
$186$	0	0
$187$	− 14.7735i	− 0.0790026i
$188$	0	0
$189$	25.0458	0.132518
$190$	0	0
$191$	48.9025i	0.256034i	0.991772	+	0.128017i	$0.0408612\pi$
$191$	48.9025i	0.256034i	−0.991772	+	0.128017i	$0.959139\pi$
$192$	0	0
$193$	168.833	0.874781	0.437390	−	0.899272i	$-0.355903\pi$
$193$	168.833	0.874781	0.437390	+	0.899272i	$0.355903\pi$
$194$	0	0
$195$	− 3.95356i	− 0.0202747i
$196$	0	0
$197$	77.0588	0.391161	0.195581	−	0.980688i	$-0.437341\pi$
$197$	77.0588	0.391161	0.195581	+	0.980688i	$0.437341\pi$
$198$	0	0
$199$	1.40504i	0.00706052i	0.999994	+	0.00353026i	$0.00112372\pi$
$199$	1.40504i	0.00706052i	−0.999994	+	0.00353026i	$0.998876\pi$
$200$	0	0
$201$	−20.5983	−0.102479
$202$	0	0
$203$	− 119.473i	− 0.588537i
$204$	0	0
$205$	−22.2142	−0.108362
$206$	0	0
$207$	− 138.736i	− 0.670223i
$208$	0	0
$209$	−33.2181	−0.158938
$210$	0	0
$211$	131.181i	0.621710i	0.950457	+	0.310855i	$0.100615\pi$
$211$	131.181i	0.621710i	−0.950457	+	0.310855i	$0.899385\pi$
$212$	0	0
$213$	16.7648	0.0787081
$214$	0	0
$215$	− 34.0865i	− 0.158542i
$216$	0	0
$217$	−125.907	−0.580215
$218$	0	0
$219$	− 18.1192i	− 0.0827361i
$220$	0	0
$221$	45.6455	0.206541
$222$	0	0
$223$	23.9134i	0.107235i	0.998562	+	0.0536176i	$0.0170752\pi$
$223$	23.9134i	0.107235i	−0.998562	+	0.0536176i	$0.982925\pi$
$224$	0	0
$225$	−219.507	−0.975588
$226$	0	0
$227$	311.549i	1.37246i	0.727384	+	0.686231i	$0.240736\pi$
$227$	311.549i	1.37246i	−0.727384	+	0.686231i	$0.759264\pi$
$228$	0	0
$229$	−155.791	−0.680311	−0.340155	−	0.940369i	$-0.610480\pi$
$229$	−155.791	−0.680311	−0.340155	+	0.940369i	$0.610480\pi$
$230$	0	0
$231$	− 10.6423i	− 0.0460707i
$232$	0	0
$233$	−201.438	−0.864542	−0.432271	−	0.901744i	$-0.642288\pi$
$233$	−201.438	−0.864542	−0.432271	+	0.901744i	$0.642288\pi$
$234$	0	0
$235$	59.4575i	0.253011i
$236$	0	0
$237$	−37.0423	−0.156297
$238$	0	0
$239$	92.8528i	0.388505i	0.980952	+	0.194253i	$0.0622282\pi$
$239$	92.8528i	0.388505i	−0.980952	+	0.194253i	$0.937772\pi$
$240$	0	0
$241$	294.032	1.22005	0.610025	−	0.792383i	$-0.291160\pi$
$241$	294.032	1.22005	0.610025	+	0.792383i	$0.291160\pi$
$242$	0	0
$243$	− 61.4604i	− 0.252924i
$244$	0	0
$245$	12.5572	0.0512541
$246$	0	0
$247$	− 102.633i	− 0.415520i
$248$	0	0
$249$	19.9239	0.0800156
$250$	0	0
$251$	− 106.291i	− 0.423471i	−0.977327	−	0.211736i	$-0.932088\pi$
$251$	− 106.291i	− 0.423471i	0.977327	−	0.211736i	$-0.0679115\pi$
$252$	0	0
$253$	−118.332	−0.467717
$254$	0	0
$255$	− 0.325508i	− 0.00127650i
$256$	0	0
$257$	−145.054	−0.564414	−0.282207	−	0.959354i	$-0.591066\pi$
$257$	−145.054	−0.564414	−0.282207	+	0.959354i	$0.591066\pi$
$258$	0	0
$259$	189.277i	0.730801i
$260$	0	0
$261$	−195.215	−0.747950
$262$	0	0
$263$	− 321.321i	− 1.22175i	−0.791726	−	0.610876i	$-0.790817\pi$
$263$	− 321.321i	− 1.22175i	0.791726	−	0.610876i	$-0.209183\pi$
$264$	0	0
$265$	58.0932	0.219220
$266$	0	0
$267$	11.7794i	0.0441175i
$268$	0	0
$269$	−190.119	−0.706761	−0.353381	−	0.935480i	$-0.614968\pi$
$269$	−190.119	−0.706761	−0.353381	+	0.935480i	$0.614968\pi$
$270$	0	0
$271$	− 94.7015i	− 0.349452i	−0.984617	−	0.174726i	$-0.944096\pi$
$271$	− 94.7015i	− 0.349452i	0.984617	−	0.174726i	$-0.0559040\pi$
$272$	0	0
$273$	32.8815	0.120445
$274$	0	0
$275$	187.225i	0.680817i
$276$	0	0
$277$	−166.331	−0.600471	−0.300236	−	0.953865i	$-0.597065\pi$
$277$	−166.331	−0.600471	−0.300236	+	0.953865i	$0.597065\pi$
$278$	0	0
$279$	205.727i	0.737373i
$280$	0	0
$281$	473.516	1.68511	0.842555	−	0.538611i	$-0.181051\pi$
$281$	473.516	1.68511	0.842555	+	0.538611i	$0.181051\pi$
$282$	0	0
$283$	− 322.508i	− 1.13960i	−0.821782	−	0.569802i	$-0.807020\pi$
$283$	− 322.508i	− 1.13960i	0.821782	−	0.569802i	$-0.192980\pi$
$284$	0	0
$285$	−0.731902	−0.00256808
$286$	0	0
$287$	− 184.754i	− 0.643744i
$288$	0	0
$289$	−285.242	−0.986996
$290$	0	0
$291$	− 6.13706i	− 0.0210896i
$292$	0	0
$293$	186.461	0.636387	0.318194	−	0.948026i	$-0.396924\pi$
$293$	186.461	0.636387	0.318194	+	0.948026i	$0.396924\pi$
$294$	0	0
$295$	12.9314i	0.0438352i
$296$	0	0
$297$	−34.9054	−0.117527
$298$	0	0
$299$	− 365.610i	− 1.22278i
$300$	0	0
$301$	283.495	0.941845
$302$	0	0
$303$	− 9.54775i	− 0.0315107i
$304$	0	0
$305$	54.8902	0.179968
$306$	0	0
$307$	− 365.908i	− 1.19188i	−0.803028	−	0.595941i	$-0.796779\pi$
$307$	− 365.908i	− 1.19188i	0.803028	−	0.595941i	$-0.203221\pi$
$308$	0	0
$309$	27.6283	0.0894120
$310$	0	0
$311$	251.100i	0.807395i	0.914893	+	0.403697i	$0.132275\pi$
$311$	251.100i	0.807395i	−0.914893	+	0.403697i	$0.867725\pi$
$312$	0	0
$313$	−436.567	−1.39478	−0.697391	−	0.716691i	$-0.745656\pi$
$313$	−436.567	−1.39478	−0.697391	+	0.716691i	$0.745656\pi$
$314$	0	0
$315$	32.1219i	0.101974i
$316$	0	0
$317$	−292.634	−0.923134	−0.461567	−	0.887105i	$-0.652713\pi$
$317$	−292.634	−0.923134	−0.461567	+	0.887105i	$0.652713\pi$
$318$	0	0
$319$	166.505i	0.521959i
$320$	0	0
$321$	−48.7437	−0.151849
$322$	0	0
$323$	− 8.45011i	− 0.0261613i
$324$	0	0
$325$	−578.465	−1.77989
$326$	0	0
$327$	− 3.94036i	− 0.0120500i
$328$	0	0
$329$	−494.505	−1.50305
$330$	0	0
$331$	154.294i	0.466146i	0.972459	+	0.233073i	$0.0748781\pi$
$331$	154.294i	0.466146i	−0.972459	+	0.233073i	$0.925122\pi$
$332$	0	0
$333$	309.273	0.928747
$334$	0	0
$335$	− 53.0285i	− 0.158294i
$336$	0	0
$337$	136.284	0.404403	0.202202	−	0.979344i	$-0.435190\pi$
$337$	136.284	0.404403	0.202202	+	0.979344i	$0.435190\pi$
$338$	0	0
$339$	5.29381i	0.0156160i
$340$	0	0
$341$	175.471	0.514578
$342$	0	0
$343$	372.377i	1.08565i
$344$	0	0
$345$	−2.60724	−0.00755723
$346$	0	0
$347$	− 218.574i	− 0.629897i	−0.949109	−	0.314949i	$-0.898013\pi$
$347$	− 218.574i	− 0.629897i	0.949109	−	0.314949i	$-0.101987\pi$
$348$	0	0
$349$	69.7726	0.199922	0.0999608	−	0.994991i	$-0.468128\pi$
$349$	69.7726	0.199922	0.0999608	+	0.994991i	$0.468128\pi$
$350$	0	0
$351$	− 107.847i	− 0.307255i
$352$	0	0
$353$	−203.173	−0.575561	−0.287781	−	0.957696i	$-0.592917\pi$
$353$	−203.173	−0.575561	−0.287781	+	0.957696i	$0.592917\pi$
$354$	0	0
$355$	43.1596i	0.121576i
$356$	0	0
$357$	2.70723	0.00758327
$358$	0	0
$359$	173.731i	0.483932i	0.970285	+	0.241966i	$0.0777922\pi$
$359$	173.731i	0.483932i	−0.970285	+	0.241966i	$0.922208\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 16.0700i	− 0.0442700i
$364$	0	0
$365$	46.6464	0.127798
$366$	0	0
$367$	− 363.755i	− 0.991157i	−0.868563	−	0.495579i	$-0.834956\pi$
$367$	− 363.755i	− 0.991157i	0.868563	−	0.495579i	$-0.165044\pi$
$368$	0	0
$369$	−301.882	−0.818109
$370$	0	0
$371$	483.157i	1.30231i
$372$	0	0
$373$	−446.876	−1.19806	−0.599030	−	0.800727i	$-0.704447\pi$
$373$	−446.876	−1.19806	−0.599030	+	0.800727i	$0.704447\pi$
$374$	0	0
$375$	8.32290i	0.0221944i
$376$	0	0
$377$	−514.448	−1.36458
$378$	0	0
$379$	− 215.080i	− 0.567493i	−0.958899	−	0.283746i	$-0.908423\pi$
$379$	− 215.080i	− 0.567493i	0.958899	−	0.283746i	$-0.0915774\pi$
$380$	0	0
$381$	−39.7159	−0.104241
$382$	0	0
$383$	304.799i	0.795821i	0.917424	+	0.397910i	$0.130265\pi$
$383$	304.799i	0.795821i	−0.917424	+	0.397910i	$0.869735\pi$
$384$	0	0
$385$	27.3978	0.0711632
$386$	0	0
$387$	− 463.222i	− 1.19696i
$388$	0	0
$389$	328.735	0.845076	0.422538	−	0.906345i	$-0.361139\pi$
$389$	328.735	0.845076	0.422538	+	0.906345i	$0.361139\pi$
$390$	0	0
$391$	− 30.1017i	− 0.0769864i
$392$	0	0
$393$	−13.0382	−0.0331760
$394$	0	0
$395$	− 95.3624i	− 0.241424i
$396$	0	0
$397$	−375.875	−0.946789	−0.473395	−	0.880850i	$-0.656972\pi$
$397$	−375.875	−0.946789	−0.473395	+	0.880850i	$0.656972\pi$
$398$	0	0
$399$	− 6.08718i	− 0.0152561i
$400$	0	0
$401$	−671.545	−1.67468	−0.837338	−	0.546686i	$-0.815889\pi$
$401$	−671.545	−1.67468	−0.837338	+	0.546686i	$0.815889\pi$
$402$	0	0
$403$	542.151i	1.34529i
$404$	0	0
$405$	52.1002	0.128643
$406$	0	0
$407$	− 263.788i	− 0.648129i
$408$	0	0
$409$	−525.692	−1.28531	−0.642655	−	0.766156i	$-0.722167\pi$
$409$	−525.692	−1.28531	−0.642655	+	0.766156i	$0.722167\pi$
$410$	0	0
$411$	− 34.9813i	− 0.0851126i
$412$	0	0
$413$	−107.550	−0.260411
$414$	0	0
$415$	51.2924i	0.123596i
$416$	0	0
$417$	2.18070	0.00522951
$418$	0	0
$419$	− 419.467i	− 1.00111i	−0.865703	−	0.500557i	$-0.833128\pi$
$419$	− 419.467i	− 1.00111i	0.865703	−	0.500557i	$-0.166872\pi$
$420$	0	0
$421$	202.155	0.480179	0.240090	−	0.970751i	$-0.422823\pi$
$421$	202.155	0.480179	0.240090	+	0.970751i	$0.422823\pi$
$422$	0	0
$423$	808.004i	1.91017i
$424$	0	0
$425$	−47.6267	−0.112063
$426$	0	0
$427$	456.518i	1.06913i
$428$	0	0
$429$	−45.8256	−0.106820
$430$	0	0
$431$	299.444i	0.694766i	0.937723	+	0.347383i	$0.112930\pi$
$431$	299.444i	0.694766i	−0.937723	+	0.347383i	$0.887070\pi$
$432$	0	0
$433$	−193.823	−0.447627	−0.223814	−	0.974632i	$-0.571851\pi$
$433$	−193.823	−0.447627	−0.223814	+	0.974632i	$0.571851\pi$
$434$	0	0
$435$	3.66864i	0.00843365i
$436$	0	0
$437$	−67.6834	−0.154882
$438$	0	0
$439$	661.662i	1.50720i	0.657332	+	0.753601i	$0.271685\pi$
$439$	661.662i	1.50720i	−0.657332	+	0.753601i	$0.728315\pi$
$440$	0	0
$441$	170.648	0.386957
$442$	0	0
$443$	− 548.601i	− 1.23838i	−0.785243	−	0.619188i	$-0.787462\pi$
$443$	− 548.601i	− 1.23838i	0.785243	−	0.619188i	$-0.212538\pi$
$444$	0	0
$445$	−30.3250	−0.0681461
$446$	0	0
$447$	53.3520i	0.119356i
$448$	0	0
$449$	30.2929	0.0674676	0.0337338	−	0.999431i	$-0.489260\pi$
$449$	30.2929	0.0674676	0.0337338	+	0.999431i	$0.489260\pi$
$450$	0	0
$451$	257.485i	0.570920i
$452$	0	0
$453$	−13.2265	−0.0291975
$454$	0	0
$455$	84.6506i	0.186045i
$456$	0	0
$457$	372.917	0.816010	0.408005	−	0.912980i	$-0.366225\pi$
$457$	372.917	0.816010	0.408005	+	0.912980i	$0.366225\pi$
$458$	0	0
$459$	− 8.87932i	− 0.0193449i
$460$	0	0
$461$	−816.298	−1.77071	−0.885355	−	0.464915i	$-0.846085\pi$
$461$	−816.298	−1.77071	−0.885355	+	0.464915i	$0.846085\pi$
$462$	0	0
$463$	− 797.212i	− 1.72184i	−0.508741	−	0.860920i	$-0.669889\pi$
$463$	− 797.212i	− 1.72184i	0.508741	−	0.860920i	$-0.330111\pi$
$464$	0	0
$465$	3.86620	0.00831440
$466$	0	0
$467$	− 813.132i	− 1.74118i	−0.492008	−	0.870591i	$-0.663737\pi$
$467$	− 813.132i	− 1.74118i	0.492008	−	0.870591i	$-0.336263\pi$
$468$	0	0
$469$	441.035	0.940373
$470$	0	0
$471$	28.6397i	0.0608062i
$472$	0	0
$473$	−395.096	−0.835299
$474$	0	0
$475$	107.088i	0.225449i
$476$	0	0
$477$	789.463	1.65506
$478$	0	0
$479$	− 459.023i	− 0.958295i	−0.877734	−	0.479147i	$-0.840946\pi$
$479$	− 459.023i	− 0.958295i	0.877734	−	0.479147i	$-0.159054\pi$
$480$	0	0
$481$	815.023	1.69444
$482$	0	0
$483$	− 21.6843i	− 0.0448950i
$484$	0	0
$485$	15.7993	0.0325760
$486$	0	0
$487$	134.407i	0.275990i	0.990433	+	0.137995i	$0.0440658\pi$
$487$	134.407i	0.275990i	−0.990433	+	0.137995i	$0.955934\pi$
$488$	0	0
$489$	65.8430	0.134648
$490$	0	0
$491$	− 562.378i	− 1.14537i	−0.819774	−	0.572686i	$-0.805901\pi$
$491$	− 562.378i	− 1.14537i	0.819774	−	0.572686i	$-0.194099\pi$
$492$	0	0
$493$	−42.3560	−0.0859147
$494$	0	0
$495$	− 44.7671i	− 0.0904386i
$496$	0	0
$497$	−358.956	−0.722245
$498$	0	0
$499$	11.3480i	0.0227414i	0.999935	+	0.0113707i	$0.00361949\pi$
$499$	11.3480i	0.0227414i	−0.999935	+	0.0113707i	$0.996381\pi$
$500$	0	0
$501$	69.7284	0.139179
$502$	0	0
$503$	125.309i	0.249124i	0.992212	+	0.124562i	$0.0397526\pi$
$503$	125.309i	0.249124i	−0.992212	+	0.124562i	$0.960247\pi$
$504$	0	0
$505$	24.5799	0.0486730
$506$	0	0
$507$	− 98.4264i	− 0.194135i
$508$	0	0
$509$	−10.6729	−0.0209683	−0.0104841	−	0.999945i	$-0.503337\pi$
$509$	−10.6729	−0.0209683	−0.0104841	+	0.999945i	$0.503337\pi$
$510$	0	0
$511$	387.955i	0.759207i
$512$	0	0
$513$	−19.9651	−0.0389183
$514$	0	0
$515$	71.1267i	0.138110i
$516$	0	0
$517$	689.172	1.33302
$518$	0	0
$519$	− 22.2028i	− 0.0427800i
$520$	0	0
$521$	−775.979	−1.48940	−0.744701	−	0.667398i	$-0.767408\pi$
$521$	−775.979	−1.48940	−0.744701	+	0.667398i	$0.767408\pi$
$522$	0	0
$523$	944.542i	1.80601i	0.429633	+	0.903004i	$0.358643\pi$
$523$	944.542i	1.80601i	−0.429633	+	0.903004i	$0.641357\pi$
$524$	0	0
$525$	−34.3087	−0.0653499
$526$	0	0
$527$	44.6368i	0.0846999i
$528$	0	0
$529$	287.892	0.544220
$530$	0	0
$531$	175.732i	0.330946i
$532$	0	0
$533$	−795.548	−1.49258
$534$	0	0
$535$	− 125.486i	− 0.234554i
$536$	0	0
$537$	71.9347	0.133957
$538$	0	0
$539$	− 145.551i	− 0.270039i
$540$	0	0
$541$	−369.872	−0.683682	−0.341841	−	0.939758i	$-0.611050\pi$
$541$	−369.872	−0.683682	−0.341841	+	0.939758i	$0.611050\pi$
$542$	0	0
$543$	21.5297i	0.0396496i
$544$	0	0
$545$	10.1441	0.0186131
$546$	0	0
$547$	431.881i	0.789545i	0.918779	+	0.394772i	$0.129177\pi$
$547$	431.881i	0.789545i	−0.918779	+	0.394772i	$0.870823\pi$
$548$	0	0
$549$	745.935	1.35872
$550$	0	0
$551$	95.2371i	0.172844i
$552$	0	0
$553$	793.123	1.43422
$554$	0	0
$555$	− 5.81211i	− 0.0104723i
$556$	0	0
$557$	−248.079	−0.445383	−0.222692	−	0.974889i	$-0.571484\pi$
$557$	−248.079	−0.445383	−0.222692	+	0.974889i	$0.571484\pi$
$558$	0	0
$559$	− 1220.72i	− 2.18376i
$560$	0	0
$561$	−3.77296	−0.00672541
$562$	0	0
$563$	− 327.600i	− 0.581882i	−0.956741	−	0.290941i	$-0.906032\pi$
$563$	− 327.600i	− 0.581882i	0.956741	−	0.290941i	$-0.0939684\pi$
$564$	0	0
$565$	−13.6285	−0.0241212
$566$	0	0
$567$	433.314i	0.764223i
$568$	0	0
$569$	−700.289	−1.23074	−0.615368	−	0.788240i	$-0.710993\pi$
$569$	−700.289	−1.23074	−0.615368	+	0.788240i	$0.710993\pi$
$570$	0	0
$571$	− 665.367i	− 1.16527i	−0.812736	−	0.582633i	$-0.802023\pi$
$571$	− 665.367i	− 1.16527i	0.812736	−	0.582633i	$-0.197977\pi$
$572$	0	0
$573$	12.4890	0.0217959
$574$	0	0
$575$	381.479i	0.663442i
$576$	0	0
$577$	−979.815	−1.69812	−0.849060	−	0.528296i	$-0.822831\pi$
$577$	−979.815	−1.69812	−0.849060	+	0.528296i	$0.822831\pi$
$578$	0	0
$579$	− 43.1176i	− 0.0744692i
$580$	0	0
$581$	−426.596	−0.734244
$582$	0	0
$583$	− 673.357i	− 1.15499i
$584$	0	0
$585$	138.316	0.236438
$586$	0	0
$587$	− 178.656i	− 0.304354i	−0.988353	−	0.152177i	$-0.951372\pi$
$587$	− 178.656i	− 0.304354i	0.988353	−	0.152177i	$-0.0486283\pi$
$588$	0	0
$589$	100.366	0.170400
$590$	0	0
$591$	− 19.6798i	− 0.0332992i
$592$	0	0
$593$	−638.505	−1.07674	−0.538368	−	0.842710i	$-0.680959\pi$
$593$	−638.505	−1.07674	−0.538368	+	0.842710i	$0.680959\pi$
$594$	0	0
$595$	6.96953i	0.0117135i
$596$	0	0
$597$	0.358830	0.000601055 0
$598$	0	0
$599$	760.354i	1.26937i	0.772770	+	0.634686i	$0.218871\pi$
$599$	760.354i	1.26937i	−0.772770	+	0.634686i	$0.781129\pi$
$600$	0	0
$601$	904.393	1.50481	0.752406	−	0.658699i	$-0.228893\pi$
$601$	904.393	1.50481	0.752406	+	0.658699i	$0.228893\pi$
$602$	0	0
$603$	− 720.636i	− 1.19508i
$604$	0	0
$605$	41.3709	0.0683816
$606$	0	0
$607$	− 525.160i	− 0.865173i	−0.901592	−	0.432587i	$-0.857601\pi$
$607$	− 525.160i	− 0.865173i	0.901592	−	0.432587i	$-0.142399\pi$
$608$	0	0
$609$	−30.5118	−0.0501016
$610$	0	0
$611$	2129.32i	3.48498i
$612$	0	0
$613$	−883.028	−1.44050	−0.720251	−	0.693713i	$-0.755974\pi$
$613$	−883.028	−1.44050	−0.720251	+	0.693713i	$0.755974\pi$
$614$	0	0
$615$	5.67322i	0.00922475i
$616$	0	0
$617$	−536.131	−0.868932	−0.434466	−	0.900688i	$-0.643063\pi$
$617$	−536.131	−0.868932	−0.434466	+	0.900688i	$0.643063\pi$
$618$	0	0
$619$	− 699.787i	− 1.13051i	−0.824916	−	0.565256i	$-0.808777\pi$
$619$	− 699.787i	− 1.13051i	0.824916	−	0.565256i	$-0.191223\pi$
$620$	0	0
$621$	−71.1214	−0.114527
$622$	0	0
$623$	− 252.211i	− 0.404834i
$624$	0	0
$625$	592.767	0.948427
$626$	0	0
$627$	8.48346i	0.0135302i
$628$	0	0
$629$	67.1032	0.106682
$630$	0	0
$631$	− 607.888i	− 0.963373i	−0.876344	−	0.481686i	$-0.840024\pi$
$631$	− 607.888i	− 0.963373i	0.876344	−	0.481686i	$-0.159976\pi$
$632$	0	0
$633$	33.5019	0.0529255
$634$	0	0
$635$	− 102.245i	− 0.161016i
$636$	0	0
$637$	449.706	0.705976
$638$	0	0
$639$	586.522i	0.917874i
$640$	0	0
$641$	51.2150	0.0798985	0.0399493	−	0.999202i	$-0.487280\pi$
$641$	51.2150	0.0798985	0.0399493	+	0.999202i	$0.487280\pi$
$642$	0	0
$643$	534.899i	0.831880i	0.909392	+	0.415940i	$0.136547\pi$
$643$	534.899i	0.831880i	−0.909392	+	0.415940i	$0.863453\pi$
$644$	0	0
$645$	−8.70524	−0.0134965
$646$	0	0
$647$	32.3797i	0.0500459i	0.999687	+	0.0250229i	$0.00796588\pi$
$647$	32.3797i	0.0500459i	−0.999687	+	0.0250229i	$0.992034\pi$
$648$	0	0
$649$	149.888	0.230952
$650$	0	0
$651$	32.1549i	0.0493931i
$652$	0	0
$653$	−327.004	−0.500772	−0.250386	−	0.968146i	$-0.580558\pi$
$653$	−327.004	−0.500772	−0.250386	+	0.968146i	$0.580558\pi$
$654$	0	0
$655$	− 33.5657i	− 0.0512453i
$656$	0	0
$657$	633.905	0.964848
$658$	0	0
$659$	268.988i	0.408176i	0.978952	+	0.204088i	$0.0654229\pi$
$659$	268.988i	0.408176i	−0.978952	+	0.204088i	$0.934577\pi$
$660$	0	0
$661$	−358.874	−0.542926	−0.271463	−	0.962449i	$-0.587507\pi$
$661$	−358.874	−0.542926	−0.271463	+	0.962449i	$0.587507\pi$
$662$	0	0
$663$	− 11.6572i	− 0.0175826i
$664$	0	0
$665$	15.6709	0.0235653
$666$	0	0
$667$	339.262i	0.508638i
$668$	0	0
$669$	6.10718	0.00912882
$670$	0	0
$671$	− 636.231i	− 0.948184i
$672$	0	0
$673$	754.931	1.12174	0.560870	−	0.827904i	$-0.310467\pi$
$673$	754.931	1.12174	0.560870	+	0.827904i	$0.310467\pi$
$674$	0	0
$675$	112.528i	0.166708i
$676$	0	0
$677$	−1029.69	−1.52095	−0.760477	−	0.649365i	$-0.775035\pi$
$677$	−1029.69	−1.52095	−0.760477	+	0.649365i	$0.775035\pi$
$678$	0	0
$679$	131.402i	0.193523i
$680$	0	0
$681$	79.5655	0.116836
$682$	0	0
$683$	− 284.033i	− 0.415861i	−0.978144	−	0.207930i	$-0.933327\pi$
$683$	− 284.033i	− 0.415861i	0.978144	−	0.207930i	$-0.0666728\pi$
$684$	0	0
$685$	90.0564	0.131469
$686$	0	0
$687$	39.7870i	0.0579141i
$688$	0	0
$689$	2080.46	3.01954
$690$	0	0
$691$	1073.04i	1.55288i	0.630193	+	0.776438i	$0.282976\pi$
$691$	1073.04i	1.55288i	−0.630193	+	0.776438i	$0.717024\pi$
$692$	0	0
$693$	372.325	0.537266
$694$	0	0
$695$	5.61404i	0.00807776i
$696$	0	0
$697$	−65.4997	−0.0939738
$698$	0	0
$699$	51.4447i	0.0735976i
$700$	0	0
$701$	−957.671	−1.36615	−0.683075	−	0.730348i	$-0.739358\pi$
$701$	−957.671	−1.36615	−0.683075	+	0.730348i	$0.739358\pi$
$702$	0	0
$703$	− 150.881i	− 0.214625i
$704$	0	0
$705$	15.1847	0.0215385
$706$	0	0
$707$	204.429i	0.289150i
$708$	0	0
$709$	555.681	0.783753	0.391877	−	0.920018i	$-0.371826\pi$
$709$	555.681	0.783753	0.391877	+	0.920018i	$0.371826\pi$
$710$	0	0
$711$	− 1295.94i	− 1.82269i
$712$	0	0
$713$	357.531	0.501446
$714$	0	0
$715$	− 117.974i	− 0.164999i
$716$	0	0
$717$	23.7134	0.0330731
$718$	0	0
$719$	− 884.813i	− 1.23062i	−0.788287	−	0.615308i	$-0.789032\pi$
$719$	− 884.813i	− 1.23062i	0.788287	−	0.615308i	$-0.210968\pi$
$720$	0	0
$721$	−591.557	−0.820467
$722$	0	0
$723$	− 75.0919i	− 0.103862i
$724$	0	0
$725$	536.777	0.740383
$726$	0	0
$727$	1228.02i	1.68916i	0.535431	+	0.844579i	$0.320149\pi$
$727$	1228.02i	1.68916i	−0.535431	+	0.844579i	$0.679851\pi$
$728$	0	0
$729$	697.493	0.956781
$730$	0	0
$731$	− 100.506i	− 0.137491i
$732$	0	0
$733$	694.837	0.947936	0.473968	−	0.880542i	$-0.342821\pi$
$733$	694.837	0.947936	0.473968	+	0.880542i	$0.342821\pi$
$734$	0	0
$735$	− 3.20696i	− 0.00436320i
$736$	0	0
$737$	−614.653	−0.833994
$738$	0	0
$739$	− 896.588i	− 1.21324i	−0.794990	−	0.606622i	$-0.792524\pi$
$739$	− 896.588i	− 1.21324i	0.794990	−	0.606622i	$-0.207476\pi$
$740$	0	0
$741$	−26.2112	−0.0353728
$742$	0	0
$743$	367.143i	0.494136i	0.968998	+	0.247068i	$0.0794672\pi$
$743$	367.143i	0.494136i	−0.968998	+	0.247068i	$0.920533\pi$
$744$	0	0
$745$	−137.350	−0.184363
$746$	0	0
$747$	697.042i	0.933122i
$748$	0	0
$749$	1043.66	1.39341
$750$	0	0
$751$	142.286i	0.189462i	0.995503	+	0.0947312i	$0.0301992\pi$
$751$	142.286i	0.189462i	−0.995503	+	0.0947312i	$0.969801\pi$
$752$	0	0
$753$	−27.1454	−0.0360496
$754$	0	0
$755$	− 34.0505i	− 0.0451000i
$756$	0	0
$757$	257.791	0.340543	0.170272	−	0.985397i	$-0.445535\pi$
$757$	257.791	0.340543	0.170272	+	0.985397i	$0.445535\pi$
$758$	0	0
$759$	30.2205i	0.0398162i
$760$	0	0
$761$	1242.01	1.63208	0.816040	−	0.577995i	$-0.196165\pi$
$761$	1242.01	1.63208	0.816040	+	0.577995i	$0.196165\pi$
$762$	0	0
$763$	84.3682i	0.110574i
$764$	0	0
$765$	11.3880	0.0148862
$766$	0	0
$767$	463.106i	0.603789i
$768$	0	0
$769$	1208.34	1.57131	0.785654	−	0.618666i	$-0.212326\pi$
$769$	1208.34	1.57131	0.785654	+	0.618666i	$0.212326\pi$
$770$	0	0
$771$	37.0450i	0.0480480i
$772$	0	0
$773$	950.549	1.22969	0.614844	−	0.788649i	$-0.289219\pi$
$773$	950.549	1.22969	0.614844	+	0.788649i	$0.289219\pi$
$774$	0	0
$775$	− 565.683i	− 0.729913i
$776$	0	0
$777$	48.3390	0.0622123
$778$	0	0
$779$	147.276i	0.189057i
$780$	0	0
$781$	500.263	0.640541
$782$	0	0
$783$	100.075i	0.127809i
$784$	0	0
$785$	−73.7306	−0.0939243
$786$	0	0
$787$	− 470.589i	− 0.597953i	−0.954260	−	0.298977i	$-0.903355\pi$
$787$	− 470.589i	− 0.597953i	0.954260	−	0.298977i	$-0.0966452\pi$
$788$	0	0
$789$	−82.0611	−0.104007
$790$	0	0
$791$	− 113.347i	− 0.143296i
$792$	0	0
$793$	1965.76	2.47888
$794$	0	0
$795$	− 14.8362i	− 0.0186619i
$796$	0	0
$797$	486.651	0.610603	0.305302	−	0.952256i	$-0.401243\pi$
$797$	486.651	0.610603	0.305302	+	0.952256i	$0.401243\pi$
$798$	0	0
$799$	175.313i	0.219416i
$800$	0	0
$801$	−412.105	−0.514488
$802$	0	0
$803$	− 540.678i	− 0.673322i
$804$	0	0
$805$	55.8244	0.0693470
$806$	0	0
$807$	48.5539i	0.0601659i
$808$	0	0
$809$	−16.1795	−0.0199994	−0.00999972	−	0.999950i	$-0.503183\pi$
$809$	−16.1795	−0.0199994	−0.00999972	+	0.999950i	$0.503183\pi$
$810$	0	0
$811$	73.2570i	0.0903292i	0.998980	+	0.0451646i	$0.0143812\pi$
$811$	73.2570i	0.0903292i	−0.998980	+	0.0451646i	$0.985619\pi$
$812$	0	0
$813$	−24.1855	−0.0297485
$814$	0	0
$815$	169.507i	0.207984i
$816$	0	0
$817$	−225.986	−0.276605
$818$	0	0
$819$	1150.37i	1.40460i
$820$	0	0
$821$	−97.2100	−0.118404	−0.0592022	−	0.998246i	$-0.518856\pi$
$821$	−97.2100	−0.118404	−0.0592022	+	0.998246i	$0.518856\pi$
$822$	0	0
$823$	660.502i	0.802554i	0.915957	+	0.401277i	$0.131434\pi$
$823$	660.502i	0.802554i	−0.915957	+	0.401277i	$0.868566\pi$
$824$	0	0
$825$	47.8147	0.0579572
$826$	0	0
$827$	1295.04i	1.56594i	0.622057	+	0.782972i	$0.286297\pi$
$827$	1295.04i	1.56594i	−0.622057	+	0.782972i	$0.713703\pi$
$828$	0	0
$829$	711.988	0.858852	0.429426	−	0.903102i	$-0.358716\pi$
$829$	711.988	0.858852	0.429426	+	0.903102i	$0.358716\pi$
$830$	0	0
$831$	42.4786i	0.0511175i
$832$	0	0
$833$	37.0256	0.0444485
$834$	0	0
$835$	179.510i	0.214982i
$836$	0	0
$837$	105.464	0.126002
$838$	0	0
$839$	− 202.080i	− 0.240858i	−0.992722	−	0.120429i	$-0.961573\pi$
$839$	− 202.080i	− 0.240858i	0.992722	−	0.120429i	$-0.0384270\pi$
$840$	0	0
$841$	−363.627	−0.432374
$842$	0	0
$843$	− 120.930i	− 0.143452i
$844$	0	0
$845$	253.391	0.299871
$846$	0	0
$847$	344.079i	0.406232i
$848$	0	0
$849$	−82.3643	−0.0970133
$850$	0	0
$851$	− 537.482i	− 0.631588i
$852$	0	0
$853$	−1556.36	−1.82457	−0.912286	−	0.409555i	$-0.865684\pi$
$853$	−1556.36	−1.82457	−0.912286	+	0.409555i	$0.865684\pi$
$854$	0	0
$855$	− 25.6058i	− 0.0299483i
$856$	0	0
$857$	−1025.20	−1.19627	−0.598134	−	0.801396i	$-0.704091\pi$
$857$	−1025.20	−1.19627	−0.598134	+	0.801396i	$0.704091\pi$
$858$	0	0
$859$	705.285i	0.821053i	0.911849	+	0.410527i	$0.134655\pi$
$859$	705.285i	0.821053i	−0.911849	+	0.410527i	$0.865345\pi$
$860$	0	0
$861$	−47.1839	−0.0548012
$862$	0	0
$863$	− 662.244i	− 0.767375i	−0.923463	−	0.383687i	$-0.874654\pi$
$863$	− 662.244i	− 0.767375i	0.923463	−	0.383687i	$-0.125346\pi$
$864$	0	0
$865$	57.1593	0.0660801
$866$	0	0
$867$	72.8470i	0.0840219i
$868$	0	0
$869$	−1105.34	−1.27197
$870$	0	0
$871$	− 1899.09i	− 2.18035i
$872$	0	0
$873$	214.707	0.245941
$874$	0	0
$875$	− 178.204i	− 0.203662i
$876$	0	0
$877$	−337.589	−0.384936	−0.192468	−	0.981303i	$-0.561649\pi$
$877$	−337.589	−0.384936	−0.192468	+	0.981303i	$0.561649\pi$
$878$	0	0
$879$	− 47.6198i	− 0.0541750i
$880$	0	0
$881$	−88.0746	−0.0999712	−0.0499856	−	0.998750i	$-0.515918\pi$
$881$	−88.0746	−0.0999712	−0.0499856	+	0.998750i	$0.515918\pi$
$882$	0	0
$883$	574.901i	0.651077i	0.945529	+	0.325539i	$0.105546\pi$
$883$	574.901i	0.651077i	−0.945529	+	0.325539i	$0.894454\pi$
$884$	0	0
$885$	3.30251	0.00373165
$886$	0	0
$887$	656.925i	0.740615i	0.928909	+	0.370307i	$0.120748\pi$
$887$	656.925i	0.740615i	−0.928909	+	0.370307i	$0.879252\pi$
$888$	0	0
$889$	850.368	0.956544
$890$	0	0
$891$	− 603.893i	− 0.677770i
$892$	0	0
$893$	394.191	0.441423
$894$	0	0
$895$	185.190i	0.206916i
$896$	0	0
$897$	−93.3719	−0.104094
$898$	0	0
$899$	− 503.080i	− 0.559600i
$900$	0	0
$901$	171.290	0.190112
$902$	0	0
$903$	− 72.4010i	− 0.0801783i
$904$	0	0
$905$	−55.4265	−0.0612448
$906$	0	0
$907$	− 1480.35i	− 1.63214i	−0.577954	−	0.816070i	$-0.696149\pi$
$907$	− 1480.35i	− 1.63214i	0.577954	−	0.816070i	$-0.303851\pi$
$908$	0	0
$909$	334.031	0.367470
$910$	0	0
$911$	− 1640.44i	− 1.80071i	−0.435161	−	0.900353i	$-0.643309\pi$
$911$	− 1640.44i	− 1.80071i	0.435161	−	0.900353i	$-0.356691\pi$
$912$	0	0
$913$	594.529	0.651182
$914$	0	0
$915$	− 14.0182i	− 0.0153205i
$916$	0	0
$917$	279.164	0.304432
$918$	0	0
$919$	− 754.817i	− 0.821346i	−0.911783	−	0.410673i	$-0.865294\pi$
$919$	− 754.817i	− 0.821346i	0.911783	−	0.410673i	$-0.134706\pi$
$920$	0	0
$921$	−93.4480	−0.101464
$922$	0	0
$923$	1545.65i	1.67460i
$924$	0	0
$925$	−850.399	−0.919351
$926$	0	0
$927$	966.583i	1.04270i
$928$	0	0
$929$	375.308	0.403992	0.201996	−	0.979386i	$-0.435257\pi$
$929$	375.308	0.403992	0.201996	+	0.979386i	$0.435257\pi$
$930$	0	0
$931$	− 83.2518i	− 0.0894219i
$932$	0	0
$933$	64.1276	0.0687327
$934$	0	0
$935$	− 9.71316i	− 0.0103884i
$936$	0	0
$937$	−866.580	−0.924845	−0.462422	−	0.886660i	$-0.653020\pi$
$937$	−866.580	−0.924845	−0.462422	+	0.886660i	$0.653020\pi$
$938$	0	0
$939$	111.493i	0.118736i
$940$	0	0
$941$	438.764	0.466274	0.233137	−	0.972444i	$-0.425101\pi$
$941$	438.764	0.466274	0.233137	+	0.972444i	$0.425101\pi$
$942$	0	0
$943$	524.638i	0.556350i
$944$	0	0
$945$	16.4669	0.0174253
$946$	0	0
$947$	− 1417.70i	− 1.49704i	−0.663111	−	0.748521i	$-0.730764\pi$
$947$	− 1417.70i	− 1.49704i	0.663111	−	0.748521i	$-0.269236\pi$
$948$	0	0
$949$	1670.52	1.76030
$950$	0	0
$951$	74.7348i	0.0785855i
$952$	0	0
$953$	−1140.01	−1.19623	−0.598116	−	0.801410i	$-0.704084\pi$
$953$	−1140.01	−1.19623	−0.598116	+	0.801410i	$0.704084\pi$
$954$	0	0
$955$	32.1520i	0.0336670i
$956$	0	0
$957$	42.5232	0.0444338
$958$	0	0
$959$	748.994i	0.781015i
$960$	0	0
$961$	430.829	0.448313
$962$	0	0
$963$	− 1705.31i	− 1.77083i
$964$	0	0
$965$	111.003	0.115029
$966$	0	0
$967$	1183.86i	1.22426i	0.790758	+	0.612129i	$0.209687\pi$
$967$	1183.86i	1.22426i	−0.790758	+	0.612129i	$0.790313\pi$
$968$	0	0
$969$	−2.15805	−0.00222709
$970$	0	0
$971$	1064.70i	1.09650i	0.836314	+	0.548250i	$0.184706\pi$
$971$	1064.70i	1.09650i	−0.836314	+	0.548250i	$0.815294\pi$
$972$	0	0
$973$	−46.6916	−0.0479873
$974$	0	0
$975$	147.732i	0.151520i
$976$	0	0
$977$	−1948.31	−1.99418	−0.997088	−	0.0762643i	$-0.975701\pi$
$977$	−1948.31	−1.99418	−0.997088	+	0.0762643i	$0.975701\pi$
$978$	0	0
$979$	351.497i	0.359037i
$980$	0	0
$981$	137.855	0.140525
$982$	0	0
$983$	1858.53i	1.89068i	0.326092	+	0.945338i	$0.394268\pi$
$983$	1858.53i	1.89068i	−0.326092	+	0.945338i	$0.605732\pi$
$984$	0	0
$985$	50.6640	0.0514355
$986$	0	0
$987$	126.290i	0.127953i
$988$	0	0
$989$	−805.027	−0.813981
$990$	0	0
$991$	793.544i	0.800751i	0.916351	+	0.400376i	$0.131120\pi$
$991$	793.544i	0.800751i	−0.916351	+	0.400376i	$0.868880\pi$
$992$	0	0
$993$	39.4047	0.0396825
$994$	0	0
$995$	0.923777i	0 0.000928419i
$996$	0	0
$997$	939.137	0.941963	0.470981	−	0.882143i	$-0.343900\pi$
$997$	939.137	0.941963	0.470981	+	0.882143i	$0.343900\pi$
$998$	0	0
$999$	− 158.545i	− 0.158704i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.f.191.10		20
4.3	odd	2	inner	1216.3.d.f.191.11		20
8.3	odd	2		608.3.d.b.191.10	✓	20
8.5	even	2		608.3.d.b.191.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.d.b.191.10	✓	20	8.3	odd	2
608.3.d.b.191.11	yes	20	8.5	even	2
1216.3.d.f.191.10		20	1.1	even	1	trivial
1216.3.d.f.191.11		20	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.255387i q^{3} +0.657472 q^{5} +5.46816i q^{7} +8.93478 q^{9} -7.62075i q^{11} +23.5457 q^{13} -0.167910i q^{15} +1.93859 q^{17} -4.35890i q^{19} +1.39649 q^{21} -15.5276i q^{23} -24.5677 q^{25} -4.58031i q^{27} -21.8489 q^{29} +23.0254i q^{31} -1.94624 q^{33} +3.59516i q^{35} +34.6145 q^{37} -6.01327i q^{39} -33.7873 q^{41} -51.8448i q^{43} +5.87437 q^{45} +90.4336i q^{47} +19.0993 q^{49} -0.495090i q^{51} +88.3584 q^{53} -5.01043i q^{55} -1.11321 q^{57} +19.6684i q^{59} +83.4867 q^{61} +48.8567i q^{63} +15.4807 q^{65} -80.6552i q^{67} -3.96556 q^{69} +65.6448i q^{71} +70.9481 q^{73} +6.27428i q^{75} +41.6715 q^{77} -145.044i q^{79} +79.2432 q^{81} +78.0145i q^{83} +1.27457 q^{85} +5.57992i q^{87} -46.1237 q^{89} +128.752i q^{91} +5.88039 q^{93} -2.86585i q^{95} +24.0304 q^{97} -68.0897i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 68 q^{9} + 16 q^{13} + 8 q^{17} - 64 q^{21} + 196 q^{25} + 88 q^{29} - 184 q^{33} - 16 q^{37} - 16 q^{41} - 16 q^{45} + 52 q^{49} - 88 q^{53} + 208 q^{61} - 192 q^{65} - 248 q^{69} - 152 q^{73} - 312 q^{77}+ \cdots - 72 q^{97}+O(q^{100}) \) 20 * q - 68 * q^9 + 16 * q^13 + 8 * q^17 - 64 * q^21 + 196 * q^25 + 88 * q^29 - 184 * q^33 - 16 * q^37 - 16 * q^41 - 16 * q^45 + 52 * q^49 - 88 * q^53 + 208 * q^61 - 192 * q^65 - 248 * q^69 - 152 * q^73 - 312 * q^77 + 500 * q^81 + 488 * q^85 + 304 * q^89 + 240 * q^93 - 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more