LMFDB - Embedded newform 912.3.m.c.799.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(799,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("912.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.m (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:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 61x^{10} + 1243x^{8} + 9566x^{6} + 25219x^{4} + 13245x^{2} + 841 $ x^12 + 61x^10 + 1243x^8 + 9566x^6 + 25219x^4 + 13245*x^2 + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.10
Root			$4.76380i$ of defining polynomial
Character	$\chi$	$=$	912.799
Dual form			912.3.m.c.799.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -0.0633322 q^{5} -0.702827i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -0.0633322 q^{5} -0.702827i q^{7} -3.00000 q^{9} -4.55087i q^{11} +16.1544 q^{13} -0.109695i q^{15} -0.248456 q^{17} -4.35890i q^{19} +1.21733 q^{21} -29.3347i q^{23} -24.9960 q^{25} -5.19615i q^{27} +7.91476 q^{29} +33.1279i q^{31} +7.88234 q^{33} +0.0445116i q^{35} +46.8507 q^{37} +27.9803i q^{39} +80.1670 q^{41} -61.1094i q^{43} +0.189997 q^{45} -58.8436i q^{47} +48.5060 q^{49} -0.430339i q^{51} +1.09881 q^{53} +0.288217i q^{55} +7.54983 q^{57} +46.8624i q^{59} -33.2896 q^{61} +2.10848i q^{63} -1.02310 q^{65} +94.5098i q^{67} +50.8093 q^{69} +81.6058i q^{71} +78.0320 q^{73} -43.2943i q^{75} -3.19847 q^{77} -25.0820i q^{79} +9.00000 q^{81} -32.8638i q^{83} +0.0157353 q^{85} +13.7088i q^{87} -3.26448 q^{89} -11.3538i q^{91} -57.3791 q^{93} +0.276059i q^{95} +109.394 q^{97} +13.6526i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} - 36 q^{9}+O(q^{10}) $ 12 * q - 4 * q^5 - 36 * q^9	$ 12 q - 4 q^{5} - 36 q^{9} + 20 q^{17} + 88 q^{25} - 24 q^{29} - 72 q^{33} - 40 q^{37} - 32 q^{41} + 12 q^{45} + 128 q^{49} + 184 q^{53} - 276 q^{61} - 232 q^{65} + 120 q^{69} - 92 q^{73} + 308 q^{77} + 108 q^{81} - 244 q^{85} + 72 q^{89} + 144 q^{93} - 280 q^{97}+O(q^{100}) $ 12 * q - 4 * q^5 - 36 * q^9 + 20 * q^17 + 88 * q^25 - 24 * q^29 - 72 * q^33 - 40 * q^37 - 32 * q^41 + 12 * q^45 + 128 * q^49 + 184 * q^53 - 276 * q^61 - 232 * q^65 + 120 * q^69 - 92 * q^73 + 308 * q^77 + 108 * q^81 - 244 * q^85 + 72 * q^89 + 144 * q^93 - 280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−0.0633322	−0.0126664	−0.00633322	−	0.999980i	$-0.502016\pi$
$5$	−0.0633322	−0.0126664	−0.00633322	+	0.999980i	$0.502016\pi$
$6$	0	0
$7$	− 0.702827i	− 0.100404i	−0.998739	−	0.0502019i	$-0.984014\pi$
$7$	− 0.702827i	− 0.100404i	0.998739	−	0.0502019i	$-0.0159865\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 4.55087i	− 0.413715i	−0.978371	−	0.206858i	$-0.933676\pi$
$11$	− 4.55087i	− 0.413715i	0.978371	−	0.206858i	$-0.0663237\pi$
$12$	0	0
$13$	16.1544	1.24265	0.621324	−	0.783554i	$-0.286595\pi$
$13$	16.1544	1.24265	0.621324	+	0.783554i	$0.286595\pi$
$14$	0	0
$15$	− 0.109695i	− 0.00731298i
$16$	0	0
$17$	−0.248456	−0.0146151	−0.00730753	−	0.999973i	$-0.502326\pi$
$17$	−0.248456	−0.0146151	−0.00730753	+	0.999973i	$0.502326\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	1.21733	0.0579682
$22$	0	0
$23$	− 29.3347i	− 1.27542i	−0.770275	−	0.637712i	$-0.779881\pi$
$23$	− 29.3347i	− 1.27542i	0.770275	−	0.637712i	$-0.220119\pi$
$24$	0	0
$25$	−24.9960	−0.999840
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	7.91476	0.272923	0.136461	−	0.990645i	$-0.456427\pi$
$29$	7.91476	0.272923	0.136461	+	0.990645i	$0.456427\pi$
$30$	0	0
$31$	33.1279i	1.06864i	0.845282	+	0.534320i	$0.179432\pi$
$31$	33.1279i	1.06864i	−0.845282	+	0.534320i	$0.820568\pi$
$32$	0	0
$33$	7.88234	0.238859
$34$	0	0
$35$	0.0445116i	0.00127176i
$36$	0	0
$37$	46.8507	1.26623	0.633117	−	0.774056i	$-0.281775\pi$
$37$	46.8507	1.26623	0.633117	+	0.774056i	$0.281775\pi$
$38$	0	0
$39$	27.9803i	0.717443i
$40$	0	0
$41$	80.1670	1.95529	0.977647	−	0.210254i	$-0.0674293\pi$
$41$	80.1670	1.95529	0.977647	+	0.210254i	$0.0674293\pi$
$42$	0	0
$43$	− 61.1094i	− 1.42115i	−0.703622	−	0.710575i	$-0.748435\pi$
$43$	− 61.1094i	− 1.42115i	0.703622	−	0.710575i	$-0.251565\pi$
$44$	0	0
$45$	0.189997	0.00422215
$46$	0	0
$47$	− 58.8436i	− 1.25199i	−0.779827	−	0.625996i	$-0.784693\pi$
$47$	− 58.8436i	− 1.25199i	0.779827	−	0.625996i	$-0.215307\pi$
$48$	0	0
$49$	48.5060	0.989919
$50$	0	0
$51$	− 0.430339i	− 0.00843801i
$52$	0	0
$53$	1.09881	0.0207322	0.0103661	−	0.999946i	$-0.496700\pi$
$53$	1.09881	0.0207322	0.0103661	+	0.999946i	$0.496700\pi$
$54$	0	0
$55$	0.288217i	0.00524030i
$56$	0	0
$57$	7.54983	0.132453
$58$	0	0
$59$	46.8624i	0.794277i	0.917759	+	0.397139i	$0.129997\pi$
$59$	46.8624i	0.794277i	−0.917759	+	0.397139i	$0.870003\pi$
$60$	0	0
$61$	−33.2896	−0.545730	−0.272865	−	0.962052i	$-0.587971\pi$
$61$	−33.2896	−0.545730	−0.272865	+	0.962052i	$0.587971\pi$
$62$	0	0
$63$	2.10848i	0.0334679i
$64$	0	0
$65$	−1.02310	−0.0157399
$66$	0	0
$67$	94.5098i	1.41059i	0.708912	+	0.705297i	$0.249186\pi$
$67$	94.5098i	1.41059i	−0.708912	+	0.705297i	$0.750814\pi$
$68$	0	0
$69$	50.8093	0.736366
$70$	0	0
$71$	81.6058i	1.14938i	0.818372	+	0.574689i	$0.194877\pi$
$71$	81.6058i	1.14938i	−0.818372	+	0.574689i	$0.805123\pi$
$72$	0	0
$73$	78.0320	1.06893	0.534466	−	0.845190i	$-0.320513\pi$
$73$	78.0320	1.06893	0.534466	+	0.845190i	$0.320513\pi$
$74$	0	0
$75$	− 43.2943i	− 0.577258i
$76$	0	0
$77$	−3.19847	−0.0415386
$78$	0	0
$79$	− 25.0820i	− 0.317494i	−0.987319	−	0.158747i	$-0.949255\pi$
$79$	− 25.0820i	− 0.317494i	0.987319	−	0.158747i	$-0.0507454\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 32.8638i	− 0.395950i	−0.980207	−	0.197975i	$-0.936564\pi$
$83$	− 32.8638i	− 0.395950i	0.980207	−	0.197975i	$-0.0634364\pi$
$84$	0	0
$85$	0.0157353	0.000185121 0
$86$	0	0
$87$	13.7088i	0.157572i
$88$	0	0
$89$	−3.26448	−0.0366796	−0.0183398	−	0.999832i	$-0.505838\pi$
$89$	−3.26448	−0.0366796	−0.0183398	+	0.999832i	$0.505838\pi$
$90$	0	0
$91$	− 11.3538i	− 0.124767i
$92$	0	0
$93$	−57.3791	−0.616980
$94$	0	0
$95$	0.276059i	0.00290588i
$96$	0	0
$97$	109.394	1.12777	0.563887	−	0.825852i	$-0.309305\pi$
$97$	109.394	1.12777	0.563887	+	0.825852i	$0.309305\pi$
$98$	0	0
$99$	13.6526i	0.137905i
$100$	0	0
$101$	191.817	1.89918	0.949589	−	0.313496i	$-0.101500\pi$
$101$	191.817	1.89918	0.949589	+	0.313496i	$0.101500\pi$
$102$	0	0
$103$	60.7251i	0.589564i	0.955564	+	0.294782i	$0.0952471\pi$
$103$	60.7251i	0.589564i	−0.955564	+	0.294782i	$0.904753\pi$
$104$	0	0
$105$	−0.0770963	−0.000734251 0
$106$	0	0
$107$	− 159.580i	− 1.49140i	−0.666283	−	0.745699i	$-0.732116\pi$
$107$	− 159.580i	− 1.49140i	0.666283	−	0.745699i	$-0.267884\pi$
$108$	0	0
$109$	−59.3114	−0.544141	−0.272070	−	0.962277i	$-0.587708\pi$
$109$	−59.3114	−0.544141	−0.272070	+	0.962277i	$0.587708\pi$
$110$	0	0
$111$	81.1478i	0.731061i
$112$	0	0
$113$	49.1732	0.435161	0.217581	−	0.976042i	$-0.430184\pi$
$113$	49.1732	0.435161	0.217581	+	0.976042i	$0.430184\pi$
$114$	0	0
$115$	1.85783i	0.0161551i
$116$	0	0
$117$	−48.4633	−0.414216
$118$	0	0
$119$	0.174622i	0.00146741i
$120$	0	0
$121$	100.290	0.828840
$122$	0	0
$123$	138.853i	1.12889i
$124$	0	0
$125$	3.16636	0.0253309
$126$	0	0
$127$	215.855i	1.69964i	0.527071	+	0.849821i	$0.323290\pi$
$127$	215.855i	1.69964i	−0.527071	+	0.849821i	$0.676710\pi$
$128$	0	0
$129$	105.845	0.820501
$130$	0	0
$131$	− 195.310i	− 1.49092i	−0.666553	−	0.745458i	$-0.732231\pi$
$131$	− 195.310i	− 1.49092i	0.666553	−	0.745458i	$-0.267769\pi$
$132$	0	0
$133$	−3.06355	−0.0230342
$134$	0	0
$135$	0.329084i	0.00243766i
$136$	0	0
$137$	121.510	0.886935	0.443468	−	0.896290i	$-0.353748\pi$
$137$	121.510	0.886935	0.443468	+	0.896290i	$0.353748\pi$
$138$	0	0
$139$	− 12.9451i	− 0.0931300i	−0.998915	−	0.0465650i	$-0.985173\pi$
$139$	− 12.9451i	− 0.0931300i	0.998915	−	0.0465650i	$-0.0148275\pi$
$140$	0	0
$141$	101.920	0.722837
$142$	0	0
$143$	− 73.5167i	− 0.514103i
$144$	0	0
$145$	−0.501260	−0.00345696
$146$	0	0
$147$	84.0149i	0.571530i
$148$	0	0
$149$	−161.851	−1.08625	−0.543123	−	0.839653i	$-0.682758\pi$
$149$	−161.851	−1.08625	−0.543123	+	0.839653i	$0.682758\pi$
$150$	0	0
$151$	− 56.7031i	− 0.375517i	−0.982215	−	0.187759i	$-0.939878\pi$
$151$	− 56.7031i	− 0.375517i	0.982215	−	0.187759i	$-0.0601223\pi$
$152$	0	0
$153$	0.745369	0.00487169
$154$	0	0
$155$	− 2.09806i	− 0.0135359i
$156$	0	0
$157$	−118.680	−0.755926	−0.377963	−	0.925821i	$-0.623375\pi$
$157$	−118.680	−0.755926	−0.377963	+	0.925821i	$0.623375\pi$
$158$	0	0
$159$	1.90319i	0.0119697i
$160$	0	0
$161$	−20.6172	−0.128057
$162$	0	0
$163$	173.693i	1.06560i	0.846241	+	0.532801i	$0.178860\pi$
$163$	173.693i	1.06560i	−0.846241	+	0.532801i	$0.821140\pi$
$164$	0	0
$165$	−0.499206	−0.00302549
$166$	0	0
$167$	− 195.028i	− 1.16783i	−0.811813	−	0.583917i	$-0.801519\pi$
$167$	− 195.028i	− 1.16783i	0.811813	−	0.583917i	$-0.198481\pi$
$168$	0	0
$169$	91.9656	0.544175
$170$	0	0
$171$	13.0767i	0.0764719i
$172$	0	0
$173$	−231.734	−1.33950	−0.669752	−	0.742584i	$-0.733600\pi$
$173$	−231.734	−1.33950	−0.669752	+	0.742584i	$0.733600\pi$
$174$	0	0
$175$	17.5679i	0.100388i
$176$	0	0
$177$	−81.1680	−0.458576
$178$	0	0
$179$	− 179.746i	− 1.00417i	−0.864819	−	0.502083i	$-0.832567\pi$
$179$	− 179.746i	− 1.00417i	0.864819	−	0.502083i	$-0.167433\pi$
$180$	0	0
$181$	78.8630	0.435707	0.217854	−	0.975981i	$-0.430094\pi$
$181$	78.8630	0.435707	0.217854	+	0.975981i	$0.430094\pi$
$182$	0	0
$183$	− 57.6592i	− 0.315078i
$184$	0	0
$185$	−2.96716	−0.0160387
$186$	0	0
$187$	1.13069i	0.00604648i
$188$	0	0
$189$	−3.65200	−0.0193227
$190$	0	0
$191$	− 59.1804i	− 0.309845i	−0.987927	−	0.154922i	$-0.950487\pi$
$191$	− 59.1804i	− 0.309845i	0.987927	−	0.154922i	$-0.0495128\pi$
$192$	0	0
$193$	−145.753	−0.755199	−0.377600	−	0.925969i	$-0.623250\pi$
$193$	−145.753	−0.755199	−0.377600	+	0.925969i	$0.623250\pi$
$194$	0	0
$195$	− 1.77205i	− 0.00908746i
$196$	0	0
$197$	−6.23462	−0.0316478	−0.0158239	−	0.999875i	$-0.505037\pi$
$197$	−6.23462	−0.0316478	−0.0158239	+	0.999875i	$0.505037\pi$
$198$	0	0
$199$	− 133.843i	− 0.672576i	−0.941759	−	0.336288i	$-0.890829\pi$
$199$	− 133.843i	− 0.672576i	0.941759	−	0.336288i	$-0.109171\pi$
$200$	0	0
$201$	−163.696	−0.814406
$202$	0	0
$203$	− 5.56271i	− 0.0274025i
$204$	0	0
$205$	−5.07716	−0.0247666
$206$	0	0
$207$	88.0042i	0.425141i
$208$	0	0
$209$	−19.8368	−0.0949128
$210$	0	0
$211$	− 71.9633i	− 0.341058i	−0.985353	−	0.170529i	$-0.945452\pi$
$211$	− 71.9633i	− 0.341058i	0.985353	−	0.170529i	$-0.0545477\pi$
$212$	0	0
$213$	−141.345	−0.663593
$214$	0	0
$215$	3.87020i	0.0180009i
$216$	0	0
$217$	23.2832	0.107296
$218$	0	0
$219$	135.155i	0.617148i
$220$	0	0
$221$	−4.01367	−0.0181614
$222$	0	0
$223$	57.6075i	0.258330i	0.991623	+	0.129165i	$0.0412296\pi$
$223$	57.6075i	0.258330i	−0.991623	+	0.129165i	$0.958770\pi$
$224$	0	0
$225$	74.9880	0.333280
$226$	0	0
$227$	33.8787i	0.149246i	0.997212	+	0.0746228i	$0.0237753\pi$
$227$	33.8787i	0.149246i	−0.997212	+	0.0746228i	$0.976225\pi$
$228$	0	0
$229$	−118.065	−0.515568	−0.257784	−	0.966203i	$-0.582992\pi$
$229$	−118.065	−0.515568	−0.257784	+	0.966203i	$0.582992\pi$
$230$	0	0
$231$	− 5.53992i	− 0.0239823i
$232$	0	0
$233$	−108.676	−0.466420	−0.233210	−	0.972426i	$-0.574923\pi$
$233$	−108.676	−0.466420	−0.233210	+	0.972426i	$0.574923\pi$
$234$	0	0
$235$	3.72670i	0.0158583i
$236$	0	0
$237$	43.4433	0.183305
$238$	0	0
$239$	− 56.0439i	− 0.234493i	−0.993103	−	0.117247i	$-0.962593\pi$
$239$	− 56.0439i	− 0.234493i	0.993103	−	0.117247i	$-0.0374068\pi$
$240$	0	0
$241$	48.7383	0.202234	0.101117	−	0.994875i	$-0.467758\pi$
$241$	48.7383	0.202234	0.101117	+	0.994875i	$0.467758\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−3.07200	−0.0125388
$246$	0	0
$247$	− 70.4155i	− 0.285083i
$248$	0	0
$249$	56.9218	0.228602
$250$	0	0
$251$	− 418.698i	− 1.66812i	−0.551676	−	0.834059i	$-0.686011\pi$
$251$	− 418.698i	− 1.66812i	0.551676	−	0.834059i	$-0.313989\pi$
$252$	0	0
$253$	−133.499	−0.527662
$254$	0	0
$255$	0.0272543i	0 0.000106880i
$256$	0	0
$257$	430.138	1.67369	0.836844	−	0.547442i	$-0.184398\pi$
$257$	430.138	1.67369	0.836844	+	0.547442i	$0.184398\pi$
$258$	0	0
$259$	− 32.9279i	− 0.127135i
$260$	0	0
$261$	−23.7443	−0.0909743
$262$	0	0
$263$	− 39.4055i	− 0.149831i	−0.997190	−	0.0749153i	$-0.976131\pi$
$263$	− 39.4055i	− 0.149831i	0.997190	−	0.0749153i	$-0.0238687\pi$
$264$	0	0
$265$	−0.0695899	−0.000262603 0
$266$	0	0
$267$	− 5.65425i	− 0.0211770i
$268$	0	0
$269$	−116.020	−0.431302	−0.215651	−	0.976471i	$-0.569187\pi$
$269$	−116.020	−0.431302	−0.215651	+	0.976471i	$0.569187\pi$
$270$	0	0
$271$	176.392i	0.650894i	0.945560	+	0.325447i	$0.105515\pi$
$271$	176.392i	0.650894i	−0.945560	+	0.325447i	$0.894485\pi$
$272$	0	0
$273$	19.6653	0.0720341
$274$	0	0
$275$	113.753i	0.413649i
$276$	0	0
$277$	−447.543	−1.61568	−0.807840	−	0.589402i	$-0.799364\pi$
$277$	−447.543	−1.61568	−0.807840	+	0.589402i	$0.799364\pi$
$278$	0	0
$279$	− 99.3836i	− 0.356214i
$280$	0	0
$281$	−468.474	−1.66717	−0.833584	−	0.552392i	$-0.813715\pi$
$281$	−468.474	−1.66717	−0.833584	+	0.552392i	$0.813715\pi$
$282$	0	0
$283$	469.390i	1.65862i	0.558787	+	0.829311i	$0.311267\pi$
$283$	469.390i	1.65862i	−0.558787	+	0.829311i	$0.688733\pi$
$284$	0	0
$285$	−0.478148	−0.00167771
$286$	0	0
$287$	− 56.3435i	− 0.196319i
$288$	0	0
$289$	−288.938	−0.999786
$290$	0	0
$291$	189.476i	0.651120i
$292$	0	0
$293$	−309.886	−1.05763	−0.528815	−	0.848737i	$-0.677364\pi$
$293$	−309.886	−1.05763	−0.528815	+	0.848737i	$0.677364\pi$
$294$	0	0
$295$	− 2.96790i	− 0.0100607i
$296$	0	0
$297$	−23.6470	−0.0796196
$298$	0	0
$299$	− 473.886i	− 1.58490i
$300$	0	0
$301$	−42.9493	−0.142689
$302$	0	0
$303$	332.237i	1.09649i
$304$	0	0
$305$	2.10830	0.00691247
$306$	0	0
$307$	145.394i	0.473598i	0.971559	+	0.236799i	$0.0760982\pi$
$307$	145.394i	0.473598i	−0.971559	+	0.236799i	$0.923902\pi$
$308$	0	0
$309$	−105.179	−0.340385
$310$	0	0
$311$	82.2155i	0.264358i	0.991226	+	0.132179i	$0.0421974\pi$
$311$	82.2155i	0.264358i	−0.991226	+	0.132179i	$0.957803\pi$
$312$	0	0
$313$	84.7272	0.270694	0.135347	−	0.990798i	$-0.456785\pi$
$313$	84.7272	0.270694	0.135347	+	0.990798i	$0.456785\pi$
$314$	0	0
$315$	− 0.133535i	0 0.000423920i
$316$	0	0
$317$	−518.411	−1.63537	−0.817683	−	0.575669i	$-0.804742\pi$
$317$	−518.411	−1.63537	−0.817683	+	0.575669i	$0.804742\pi$
$318$	0	0
$319$	− 36.0191i	− 0.112912i
$320$	0	0
$321$	276.400	0.861059
$322$	0	0
$323$	1.08300i	0.00335293i
$324$	0	0
$325$	−403.796	−1.24245
$326$	0	0
$327$	− 102.730i	− 0.314160i
$328$	0	0
$329$	−41.3569	−0.125705
$330$	0	0
$331$	211.916i	0.640229i	0.947379	+	0.320115i	$0.103721\pi$
$331$	211.916i	0.640229i	−0.947379	+	0.320115i	$0.896279\pi$
$332$	0	0
$333$	−140.552	−0.422078
$334$	0	0
$335$	− 5.98551i	− 0.0178672i
$336$	0	0
$337$	−44.1806	−0.131100	−0.0655498	−	0.997849i	$-0.520880\pi$
$337$	−44.1806	−0.131100	−0.0655498	+	0.997849i	$0.520880\pi$
$338$	0	0
$339$	85.1705i	0.251240i
$340$	0	0
$341$	150.761	0.442113
$342$	0	0
$343$	− 68.5299i	− 0.199796i
$344$	0	0
$345$	−3.21786	−0.00932714
$346$	0	0
$347$	− 398.702i	− 1.14900i	−0.818506	−	0.574498i	$-0.805197\pi$
$347$	− 398.702i	− 1.14900i	0.818506	−	0.574498i	$-0.194803\pi$
$348$	0	0
$349$	−107.790	−0.308854	−0.154427	−	0.988004i	$-0.549353\pi$
$349$	−107.790	−0.308854	−0.154427	+	0.988004i	$0.549353\pi$
$350$	0	0
$351$	− 83.9409i	− 0.239148i
$352$	0	0
$353$	440.477	1.24781	0.623906	−	0.781500i	$-0.285545\pi$
$353$	440.477	1.24781	0.623906	+	0.781500i	$0.285545\pi$
$354$	0	0
$355$	− 5.16828i	− 0.0145585i
$356$	0	0
$357$	−0.302454	−0.000847209 0
$358$	0	0
$359$	466.457i	1.29932i	0.760223	+	0.649662i	$0.225090\pi$
$359$	466.457i	1.29932i	−0.760223	+	0.649662i	$0.774910\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	173.707i	0.478531i
$364$	0	0
$365$	−4.94194	−0.0135396
$366$	0	0
$367$	− 458.614i	− 1.24963i	−0.780773	−	0.624815i	$-0.785174\pi$
$367$	− 458.614i	− 1.24963i	0.780773	−	0.624815i	$-0.214826\pi$
$368$	0	0
$369$	−240.501	−0.651764
$370$	0	0
$371$	− 0.772271i	− 0.00208159i
$372$	0	0
$373$	506.489	1.35788	0.678940	−	0.734194i	$-0.262440\pi$
$373$	506.489	1.35788	0.678940	+	0.734194i	$0.262440\pi$
$374$	0	0
$375$	5.48429i	0.0146248i
$376$	0	0
$377$	127.859	0.339147
$378$	0	0
$379$	202.109i	0.533269i	0.963798	+	0.266634i	$0.0859116\pi$
$379$	202.109i	0.533269i	−0.963798	+	0.266634i	$0.914088\pi$
$380$	0	0
$381$	−373.871	−0.981289
$382$	0	0
$383$	− 96.6625i	− 0.252383i	−0.992006	−	0.126191i	$-0.959725\pi$
$383$	− 96.6625i	− 0.252383i	0.992006	−	0.126191i	$-0.0402753\pi$
$384$	0	0
$385$	0.202566	0.000526147 0
$386$	0	0
$387$	183.328i	0.473716i
$388$	0	0
$389$	−559.129	−1.43735	−0.718675	−	0.695346i	$-0.755251\pi$
$389$	−559.129	−1.43735	−0.718675	+	0.695346i	$0.755251\pi$
$390$	0	0
$391$	7.28840i	0.0186404i
$392$	0	0
$393$	338.287	0.860780
$394$	0	0
$395$	1.58850i	0.00402152i
$396$	0	0
$397$	−230.679	−0.581054	−0.290527	−	0.956867i	$-0.593831\pi$
$397$	−230.679	−0.581054	−0.290527	+	0.956867i	$0.593831\pi$
$398$	0	0
$399$	− 5.30623i	− 0.0132988i
$400$	0	0
$401$	73.0255	0.182108	0.0910542	−	0.995846i	$-0.470976\pi$
$401$	73.0255	0.182108	0.0910542	+	0.995846i	$0.470976\pi$
$402$	0	0
$403$	535.162i	1.32794i
$404$	0	0
$405$	−0.569990	−0.00140738
$406$	0	0
$407$	− 213.211i	− 0.523861i
$408$	0	0
$409$	−319.156	−0.780333	−0.390167	−	0.920744i	$-0.627583\pi$
$409$	−319.156	−0.780333	−0.390167	+	0.920744i	$0.627583\pi$
$410$	0	0
$411$	210.462i	0.512072i
$412$	0	0
$413$	32.9361	0.0797485
$414$	0	0
$415$	2.08134i	0.00501527i
$416$	0	0
$417$	22.4215	0.0537686
$418$	0	0
$419$	− 82.8978i	− 0.197847i	−0.995095	−	0.0989234i	$-0.968460\pi$
$419$	− 82.8978i	− 0.197847i	0.995095	−	0.0989234i	$-0.0315399\pi$
$420$	0	0
$421$	208.527	0.495313	0.247656	−	0.968848i	$-0.420340\pi$
$421$	208.527	0.495313	0.247656	+	0.968848i	$0.420340\pi$
$422$	0	0
$423$	176.531i	0.417330i
$424$	0	0
$425$	6.21041	0.0146127
$426$	0	0
$427$	23.3968i	0.0547934i
$428$	0	0
$429$	127.335	0.296817
$430$	0	0
$431$	− 549.352i	− 1.27460i	−0.770616	−	0.637300i	$-0.780051\pi$
$431$	− 549.352i	− 1.27460i	0.770616	−	0.637300i	$-0.219949\pi$
$432$	0	0
$433$	738.404	1.70532	0.852661	−	0.522465i	$-0.174987\pi$
$433$	738.404	1.70532	0.852661	+	0.522465i	$0.174987\pi$
$434$	0	0
$435$	− 0.868207i	− 0.00199588i
$436$	0	0
$437$	−127.867	−0.292602
$438$	0	0
$439$	758.730i	1.72832i	0.503221	+	0.864158i	$0.332148\pi$
$439$	758.730i	1.72832i	−0.503221	+	0.864158i	$0.667852\pi$
$440$	0	0
$441$	−145.518	−0.329973
$442$	0	0
$443$	− 741.372i	− 1.67353i	−0.547565	−	0.836763i	$-0.684445\pi$
$443$	− 741.372i	− 1.67353i	0.547565	−	0.836763i	$-0.315555\pi$
$444$	0	0
$445$	0.206747	0.000464600 0
$446$	0	0
$447$	− 280.333i	− 0.627144i
$448$	0	0
$449$	278.924	0.621212	0.310606	−	0.950539i	$-0.399468\pi$
$449$	278.924	0.621212	0.310606	+	0.950539i	$0.399468\pi$
$450$	0	0
$451$	− 364.830i	− 0.808935i
$452$	0	0
$453$	98.2127	0.216805
$454$	0	0
$455$	0.719059i	0.00158035i
$456$	0	0
$457$	−248.933	−0.544712	−0.272356	−	0.962197i	$-0.587803\pi$
$457$	−248.933	−0.544712	−0.272356	+	0.962197i	$0.587803\pi$
$458$	0	0
$459$	1.29102i	0.00281267i
$460$	0	0
$461$	−420.060	−0.911193	−0.455596	−	0.890186i	$-0.650574\pi$
$461$	−420.060	−0.911193	−0.455596	+	0.890186i	$0.650574\pi$
$462$	0	0
$463$	398.717i	0.861160i	0.902552	+	0.430580i	$0.141691\pi$
$463$	398.717i	0.861160i	−0.902552	+	0.430580i	$0.858309\pi$
$464$	0	0
$465$	3.63395	0.00781495
$466$	0	0
$467$	694.518i	1.48719i	0.668630	+	0.743595i	$0.266881\pi$
$467$	694.518i	1.48719i	−0.668630	+	0.743595i	$0.733119\pi$
$468$	0	0
$469$	66.4240	0.141629
$470$	0	0
$471$	− 205.561i	− 0.436434i
$472$	0	0
$473$	−278.101	−0.587951
$474$	0	0
$475$	108.955i	0.229379i
$476$	0	0
$477$	−3.29642	−0.00691074
$478$	0	0
$479$	515.537i	1.07628i	0.842856	+	0.538139i	$0.180872\pi$
$479$	515.537i	1.07628i	−0.842856	+	0.538139i	$0.819128\pi$
$480$	0	0
$481$	756.846	1.57348
$482$	0	0
$483$	− 35.7101i	− 0.0739340i
$484$	0	0
$485$	−6.92817	−0.0142849
$486$	0	0
$487$	− 823.031i	− 1.69000i	−0.534764	−	0.845001i	$-0.679600\pi$
$487$	− 823.031i	− 1.69000i	0.534764	−	0.845001i	$-0.320400\pi$
$488$	0	0
$489$	−300.845	−0.615225
$490$	0	0
$491$	425.194i	0.865976i	0.901400	+	0.432988i	$0.142541\pi$
$491$	425.194i	0.865976i	−0.901400	+	0.432988i	$0.857459\pi$
$492$	0	0
$493$	−1.96647	−0.00398879
$494$	0	0
$495$	− 0.864650i	− 0.00174677i
$496$	0	0
$497$	57.3548	0.115402
$498$	0	0
$499$	− 314.189i	− 0.629637i	−0.949152	−	0.314818i	$-0.898056\pi$
$499$	− 314.189i	− 0.629637i	0.949152	−	0.314818i	$-0.101944\pi$
$500$	0	0
$501$	337.799	0.674249
$502$	0	0
$503$	23.9798i	0.0476735i	0.999716	+	0.0238368i	$0.00758820\pi$
$503$	23.9798i	0.0476735i	−0.999716	+	0.0238368i	$0.992412\pi$
$504$	0	0
$505$	−12.1482	−0.0240558
$506$	0	0
$507$	159.289i	0.314180i
$508$	0	0
$509$	431.558	0.847855	0.423927	−	0.905696i	$-0.360651\pi$
$509$	431.558	0.847855	0.423927	+	0.905696i	$0.360651\pi$
$510$	0	0
$511$	− 54.8430i	− 0.107325i
$512$	0	0
$513$	−22.6495	−0.0441511
$514$	0	0
$515$	− 3.84586i	− 0.00746769i
$516$	0	0
$517$	−267.789	−0.517968
$518$	0	0
$519$	− 401.376i	− 0.773363i
$520$	0	0
$521$	86.2058	0.165462	0.0827311	−	0.996572i	$-0.473636\pi$
$521$	86.2058	0.165462	0.0827311	+	0.996572i	$0.473636\pi$
$522$	0	0
$523$	− 758.701i	− 1.45067i	−0.688396	−	0.725335i	$-0.741685\pi$
$523$	− 758.701i	− 1.45067i	0.688396	−	0.725335i	$-0.258315\pi$
$524$	0	0
$525$	−30.4284	−0.0579589
$526$	0	0
$527$	− 8.23082i	− 0.0156183i
$528$	0	0
$529$	−331.527	−0.626705
$530$	0	0
$531$	− 140.587i	− 0.264759i
$532$	0	0
$533$	1295.05	2.42974
$534$	0	0
$535$	10.1065i	0.0188907i
$536$	0	0
$537$	311.329	0.579756
$538$	0	0
$539$	− 220.745i	− 0.409545i
$540$	0	0
$541$	−440.995	−0.815148	−0.407574	−	0.913172i	$-0.633625\pi$
$541$	−440.995	−0.815148	−0.407574	+	0.913172i	$0.633625\pi$
$542$	0	0
$543$	136.595i	0.251556i
$544$	0	0
$545$	3.75632	0.00689233
$546$	0	0
$547$	101.935i	0.186353i	0.995650	+	0.0931765i	$0.0297021\pi$
$547$	101.935i	0.186353i	−0.995650	+	0.0931765i	$0.970298\pi$
$548$	0	0
$549$	99.8687	0.181910
$550$	0	0
$551$	− 34.4997i	− 0.0626128i
$552$	0	0
$553$	−17.6283	−0.0318776
$554$	0	0
$555$	− 5.13927i	− 0.00925994i
$556$	0	0
$557$	149.483	0.268372	0.134186	−	0.990956i	$-0.457158\pi$
$557$	149.483	0.268372	0.134186	+	0.990956i	$0.457158\pi$
$558$	0	0
$559$	− 987.188i	− 1.76599i
$560$	0	0
$561$	−1.95842	−0.00349094
$562$	0	0
$563$	− 486.988i	− 0.864988i	−0.901637	−	0.432494i	$-0.857634\pi$
$563$	− 486.988i	− 0.864988i	0.901637	−	0.432494i	$-0.142366\pi$
$564$	0	0
$565$	−3.11425	−0.00551195
$566$	0	0
$567$	− 6.32544i	− 0.0111560i
$568$	0	0
$569$	826.938	1.45332	0.726659	−	0.686998i	$-0.241072\pi$
$569$	826.938	1.45332	0.726659	+	0.686998i	$0.241072\pi$
$570$	0	0
$571$	11.5733i	0.0202685i	0.999949	+	0.0101343i	$0.00322589\pi$
$571$	11.5733i	0.0202685i	−0.999949	+	0.0101343i	$0.996774\pi$
$572$	0	0
$573$	102.503	0.178889
$574$	0	0
$575$	733.251i	1.27522i
$576$	0	0
$577$	747.156	1.29490	0.647449	−	0.762109i	$-0.275836\pi$
$577$	747.156	1.29490	0.647449	+	0.762109i	$0.275836\pi$
$578$	0	0
$579$	− 252.452i	− 0.436014i
$580$	0	0
$581$	−23.0976	−0.0397548
$582$	0	0
$583$	− 5.00053i	− 0.00857723i
$584$	0	0
$585$	3.06929	0.00524665
$586$	0	0
$587$	960.068i	1.63555i	0.575537	+	0.817775i	$0.304793\pi$
$587$	960.068i	1.63555i	−0.575537	+	0.817775i	$0.695207\pi$
$588$	0	0
$589$	144.401	0.245163
$590$	0	0
$591$	− 10.7987i	− 0.0182719i
$592$	0	0
$593$	−786.436	−1.32620	−0.663100	−	0.748531i	$-0.730760\pi$
$593$	−786.436	−1.32620	−0.663100	+	0.748531i	$0.730760\pi$
$594$	0	0
$595$	− 0.0110592i	0 1.85869e-5i
$596$	0	0
$597$	231.822	0.388312
$598$	0	0
$599$	478.815i	0.799358i	0.916655	+	0.399679i	$0.130878\pi$
$599$	478.815i	0.799358i	−0.916655	+	0.399679i	$0.869122\pi$
$600$	0	0
$601$	441.719	0.734973	0.367486	−	0.930029i	$-0.380218\pi$
$601$	441.719	0.734973	0.367486	+	0.930029i	$0.380218\pi$
$602$	0	0
$603$	− 283.529i	− 0.470198i
$604$	0	0
$605$	−6.35156	−0.0104985
$606$	0	0
$607$	860.514i	1.41765i	0.705384	+	0.708825i	$0.250775\pi$
$607$	860.514i	1.41765i	−0.705384	+	0.708825i	$0.749225\pi$
$608$	0	0
$609$	9.63489	0.0158208
$610$	0	0
$611$	− 950.585i	− 1.55579i
$612$	0	0
$613$	551.707	0.900012	0.450006	−	0.893026i	$-0.351422\pi$
$613$	551.707	0.900012	0.450006	+	0.893026i	$0.351422\pi$
$614$	0	0
$615$	− 8.79389i	− 0.0142990i
$616$	0	0
$617$	299.911	0.486080	0.243040	−	0.970016i	$-0.421855\pi$
$617$	299.911	0.486080	0.243040	+	0.970016i	$0.421855\pi$
$618$	0	0
$619$	351.780i	0.568303i	0.958779	+	0.284152i	$0.0917119\pi$
$619$	351.780i	0.568303i	−0.958779	+	0.284152i	$0.908288\pi$
$620$	0	0
$621$	−152.428	−0.245455
$622$	0	0
$623$	2.29437i	0.00368277i
$624$	0	0
$625$	624.699	0.999519
$626$	0	0
$627$	− 34.3583i	− 0.0547979i
$628$	0	0
$629$	−11.6403	−0.0185061
$630$	0	0
$631$	− 915.116i	− 1.45026i	−0.688611	−	0.725131i	$-0.741779\pi$
$631$	− 915.116i	− 1.45026i	0.688611	−	0.725131i	$-0.258221\pi$
$632$	0	0
$633$	124.644	0.196910
$634$	0	0
$635$	− 13.6706i	− 0.0215284i
$636$	0	0
$637$	783.587	1.23012
$638$	0	0
$639$	− 244.817i	− 0.383126i
$640$	0	0
$641$	486.647	0.759199	0.379600	−	0.925151i	$-0.376062\pi$
$641$	486.647	0.759199	0.379600	+	0.925151i	$0.376062\pi$
$642$	0	0
$643$	660.302i	1.02691i	0.858117	+	0.513454i	$0.171634\pi$
$643$	660.302i	1.02691i	−0.858117	+	0.513454i	$0.828366\pi$
$644$	0	0
$645$	−6.70338	−0.0103928
$646$	0	0
$647$	− 407.589i	− 0.629967i	−0.949097	−	0.314983i	$-0.898001\pi$
$647$	− 407.589i	− 0.629967i	0.949097	−	0.314983i	$-0.101999\pi$
$648$	0	0
$649$	213.265	0.328605
$650$	0	0
$651$	40.3276i	0.0619472i
$652$	0	0
$653$	−10.2696	−0.0157268	−0.00786340	−	0.999969i	$-0.502503\pi$
$653$	−10.2696	−0.0157268	−0.00786340	+	0.999969i	$0.502503\pi$
$654$	0	0
$655$	12.3694i	0.0188846i
$656$	0	0
$657$	−234.096	−0.356310
$658$	0	0
$659$	576.160i	0.874294i	0.899390	+	0.437147i	$0.144011\pi$
$659$	576.160i	0.874294i	−0.899390	+	0.437147i	$0.855989\pi$
$660$	0	0
$661$	−838.383	−1.26836	−0.634178	−	0.773187i	$-0.718661\pi$
$661$	−838.383	−1.26836	−0.634178	+	0.773187i	$0.718661\pi$
$662$	0	0
$663$	− 6.95188i	− 0.0104855i
$664$	0	0
$665$	0.194022	0.000291762 0
$666$	0	0
$667$	− 232.178i	− 0.348092i
$668$	0	0
$669$	−99.7791	−0.149147
$670$	0	0
$671$	151.496i	0.225777i
$672$	0	0
$673$	−237.621	−0.353077	−0.176538	−	0.984294i	$-0.556490\pi$
$673$	−237.621	−0.353077	−0.176538	+	0.984294i	$0.556490\pi$
$674$	0	0
$675$	129.883i	0.192419i
$676$	0	0
$677$	−171.963	−0.254007	−0.127004	−	0.991902i	$-0.540536\pi$
$677$	−171.963	−0.254007	−0.127004	+	0.991902i	$0.540536\pi$
$678$	0	0
$679$	− 76.8851i	− 0.113233i
$680$	0	0
$681$	−58.6797	−0.0861669
$682$	0	0
$683$	216.652i	0.317207i	0.987342	+	0.158603i	$0.0506991\pi$
$683$	216.652i	0.317207i	−0.987342	+	0.158603i	$0.949301\pi$
$684$	0	0
$685$	−7.69551	−0.0112343
$686$	0	0
$687$	− 204.495i	− 0.297663i
$688$	0	0
$689$	17.7506	0.0257628
$690$	0	0
$691$	102.023i	0.147645i	0.997271	+	0.0738227i	$0.0235199\pi$
$691$	102.023i	0.147645i	−0.997271	+	0.0738227i	$0.976480\pi$
$692$	0	0
$693$	9.59542	0.0138462
$694$	0	0
$695$	0.819840i	0.00117963i
$696$	0	0
$697$	−19.9180	−0.0285767
$698$	0	0
$699$	− 188.232i	− 0.269288i
$700$	0	0
$701$	9.18056	0.0130964	0.00654819	−	0.999979i	$-0.497916\pi$
$701$	9.18056	0.0130964	0.00654819	+	0.999979i	$0.497916\pi$
$702$	0	0
$703$	− 204.217i	− 0.290494i
$704$	0	0
$705$	−6.45483	−0.00915578
$706$	0	0
$707$	− 134.814i	− 0.190685i
$708$	0	0
$709$	−243.144	−0.342940	−0.171470	−	0.985189i	$-0.554852\pi$
$709$	−243.144	−0.342940	−0.171470	+	0.985189i	$0.554852\pi$
$710$	0	0
$711$	75.2461i	0.105831i
$712$	0	0
$713$	971.797	1.36297
$714$	0	0
$715$	4.65598i	0.00651186i
$716$	0	0
$717$	97.0708	0.135385
$718$	0	0
$719$	738.875i	1.02764i	0.857897	+	0.513821i	$0.171771\pi$
$719$	738.875i	1.02764i	−0.857897	+	0.513821i	$0.828229\pi$
$720$	0	0
$721$	42.6793	0.0591945
$722$	0	0
$723$	84.4173i	0.116760i
$724$	0	0
$725$	−197.837	−0.272879
$726$	0	0
$727$	− 1000.17i	− 1.37575i	−0.725827	−	0.687877i	$-0.758543\pi$
$727$	− 1000.17i	− 1.37575i	0.725827	−	0.687877i	$-0.241457\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	15.1830i	0.0207702i
$732$	0	0
$733$	452.712	0.617615	0.308808	−	0.951125i	$-0.400070\pi$
$733$	452.712	0.617615	0.308808	+	0.951125i	$0.400070\pi$
$734$	0	0
$735$	− 5.32085i	− 0.00723925i
$736$	0	0
$737$	430.102	0.583584
$738$	0	0
$739$	995.848i	1.34756i	0.738931	+	0.673781i	$0.235331\pi$
$739$	995.848i	1.34756i	−0.738931	+	0.673781i	$0.764669\pi$
$740$	0	0
$741$	121.963	0.164593
$742$	0	0
$743$	1382.64i	1.86089i	0.366430	+	0.930445i	$0.380580\pi$
$743$	1382.64i	1.86089i	−0.366430	+	0.930445i	$0.619420\pi$
$744$	0	0
$745$	10.2504	0.0137589
$746$	0	0
$747$	98.5914i	0.131983i
$748$	0	0
$749$	−112.157	−0.149742
$750$	0	0
$751$	984.927i	1.31149i	0.754984	+	0.655744i	$0.227645\pi$
$751$	984.927i	1.31149i	−0.754984	+	0.655744i	$0.772355\pi$
$752$	0	0
$753$	725.205	0.963088
$754$	0	0
$755$	3.59114i	0.00475647i
$756$	0	0
$757$	880.633	1.16332	0.581660	−	0.813432i	$-0.302404\pi$
$757$	880.633	1.16332	0.581660	+	0.813432i	$0.302404\pi$
$758$	0	0
$759$	− 231.226i	− 0.304646i
$760$	0	0
$761$	−504.649	−0.663140	−0.331570	−	0.943431i	$-0.607578\pi$
$761$	−504.649	−0.663140	−0.331570	+	0.943431i	$0.607578\pi$
$762$	0	0
$763$	41.6856i	0.0546338i
$764$	0	0
$765$	−0.0472059	−6.17070e−5 0
$766$	0	0
$767$	757.035i	0.987008i
$768$	0	0
$769$	−157.844	−0.205259	−0.102630	−	0.994720i	$-0.532726\pi$
$769$	−157.844	−0.205259	−0.102630	+	0.994720i	$0.532726\pi$
$770$	0	0
$771$	745.020i	0.966304i
$772$	0	0
$773$	1322.66	1.71108	0.855539	−	0.517738i	$-0.173226\pi$
$773$	1322.66	1.71108	0.855539	+	0.517738i	$0.173226\pi$
$774$	0	0
$775$	− 828.064i	− 1.06847i
$776$	0	0
$777$	57.0328	0.0734013
$778$	0	0
$779$	− 349.440i	− 0.448575i
$780$	0	0
$781$	371.377	0.475515
$782$	0	0
$783$	− 41.1263i	− 0.0525240i
$784$	0	0
$785$	7.51630	0.00957490
$786$	0	0
$787$	− 1148.86i	− 1.45979i	−0.683558	−	0.729896i	$-0.739569\pi$
$787$	− 1148.86i	− 1.45979i	0.683558	−	0.729896i	$-0.260431\pi$
$788$	0	0
$789$	68.2523	0.0865048
$790$	0	0
$791$	− 34.5603i	− 0.0436918i
$792$	0	0
$793$	−537.774	−0.678151
$794$	0	0
$795$	− 0.120533i	0 0.000151614i
$796$	0	0
$797$	−283.241	−0.355384	−0.177692	−	0.984086i	$-0.556863\pi$
$797$	−283.241	−0.355384	−0.177692	+	0.984086i	$0.556863\pi$
$798$	0	0
$799$	14.6201i	0.0182979i
$800$	0	0
$801$	9.79345	0.0122265
$802$	0	0
$803$	− 355.113i	− 0.442233i
$804$	0	0
$805$	1.30574	0.00162203
$806$	0	0
$807$	− 200.953i	− 0.249012i
$808$	0	0
$809$	−1049.05	−1.29672	−0.648360	−	0.761334i	$-0.724545\pi$
$809$	−1049.05	−1.29672	−0.648360	+	0.761334i	$0.724545\pi$
$810$	0	0
$811$	− 255.833i	− 0.315453i	−0.987483	−	0.157727i	$-0.949584\pi$
$811$	− 255.833i	− 0.315453i	0.987483	−	0.157727i	$-0.0504165\pi$
$812$	0	0
$813$	−305.521	−0.375794
$814$	0	0
$815$	− 11.0004i	− 0.0134974i
$816$	0	0
$817$	−266.370	−0.326034
$818$	0	0
$819$	34.0613i	0.0415889i
$820$	0	0
$821$	880.419	1.07237	0.536187	−	0.844099i	$-0.319864\pi$
$821$	880.419	1.07237	0.536187	+	0.844099i	$0.319864\pi$
$822$	0	0
$823$	611.336i	0.742814i	0.928470	+	0.371407i	$0.121125\pi$
$823$	611.336i	0.742814i	−0.928470	+	0.371407i	$0.878875\pi$
$824$	0	0
$825$	−197.027	−0.238820
$826$	0	0
$827$	263.878i	0.319079i	0.987192	+	0.159539i	$0.0510009\pi$
$827$	263.878i	0.319079i	−0.987192	+	0.159539i	$0.948999\pi$
$828$	0	0
$829$	−695.570	−0.839047	−0.419524	−	0.907744i	$-0.637803\pi$
$829$	−695.570	−0.839047	−0.419524	+	0.907744i	$0.637803\pi$
$830$	0	0
$831$	− 775.168i	− 0.932814i
$832$	0	0
$833$	−12.0516	−0.0144677
$834$	0	0
$835$	12.3516i	0.0147923i
$836$	0	0
$837$	172.137	0.205660
$838$	0	0
$839$	1172.50i	1.39750i	0.715368	+	0.698748i	$0.246259\pi$
$839$	1172.50i	1.39750i	−0.715368	+	0.698748i	$0.753741\pi$
$840$	0	0
$841$	−778.357	−0.925513
$842$	0	0
$843$	− 811.421i	− 0.962540i
$844$	0	0
$845$	−5.82439	−0.00689277
$846$	0	0
$847$	− 70.4862i	− 0.0832187i
$848$	0	0
$849$	−813.008	−0.957606
$850$	0	0
$851$	− 1374.35i	− 1.61499i
$852$	0	0
$853$	1375.94	1.61306	0.806530	−	0.591193i	$-0.201343\pi$
$853$	1375.94	1.61306	0.806530	+	0.591193i	$0.201343\pi$
$854$	0	0
$855$	− 0.828176i	0 0.000968627i
$856$	0	0
$857$	−1036.71	−1.20969	−0.604847	−	0.796342i	$-0.706766\pi$
$857$	−1036.71	−1.20969	−0.604847	+	0.796342i	$0.706766\pi$
$858$	0	0
$859$	− 1318.45i	− 1.53487i	−0.641128	−	0.767434i	$-0.721533\pi$
$859$	− 1318.45i	− 1.53487i	0.641128	−	0.767434i	$-0.278467\pi$
$860$	0	0
$861$	97.5899	0.113345
$862$	0	0
$863$	617.016i	0.714966i	0.933920	+	0.357483i	$0.116365\pi$
$863$	617.016i	0.714966i	−0.933920	+	0.357483i	$0.883635\pi$
$864$	0	0
$865$	14.6763	0.0169668
$866$	0	0
$867$	− 500.456i	− 0.577227i
$868$	0	0
$869$	−114.145	−0.131352
$870$	0	0
$871$	1526.75i	1.75287i
$872$	0	0
$873$	−328.182	−0.375925
$874$	0	0
$875$	− 2.22540i	− 0.00254332i
$876$	0	0
$877$	−721.074	−0.822206	−0.411103	−	0.911589i	$-0.634856\pi$
$877$	−721.074	−0.822206	−0.411103	+	0.911589i	$0.634856\pi$
$878$	0	0
$879$	− 536.738i	− 0.610623i
$880$	0	0
$881$	300.789	0.341418	0.170709	−	0.985322i	$-0.445394\pi$
$881$	300.789	0.341418	0.170709	+	0.985322i	$0.445394\pi$
$882$	0	0
$883$	313.616i	0.355171i	0.984105	+	0.177586i	$0.0568287\pi$
$883$	313.616i	0.355171i	−0.984105	+	0.177586i	$0.943171\pi$
$884$	0	0
$885$	5.14055	0.00580853
$886$	0	0
$887$	1734.73i	1.95573i	0.209236	+	0.977865i	$0.432902\pi$
$887$	1734.73i	1.95573i	−0.209236	+	0.977865i	$0.567098\pi$
$888$	0	0
$889$	151.708	0.170651
$890$	0	0
$891$	− 40.9578i	− 0.0459684i
$892$	0	0
$893$	−256.493	−0.287226
$894$	0	0
$895$	11.3837i	0.0127192i
$896$	0	0
$897$	820.795	0.915044
$898$	0	0
$899$	262.199i	0.291657i
$900$	0	0
$901$	−0.273005	−0.000303003 0
$902$	0	0
$903$	− 74.3904i	− 0.0823814i
$904$	0	0
$905$	−4.99457	−0.00551887
$906$	0	0
$907$	− 69.0770i	− 0.0761599i	−0.999275	−	0.0380800i	$-0.987876\pi$
$907$	− 69.0770i	− 0.0761599i	0.999275	−	0.0380800i	$-0.0121242\pi$
$908$	0	0
$909$	−575.451	−0.633060
$910$	0	0
$911$	1343.29i	1.47452i	0.675607	+	0.737262i	$0.263882\pi$
$911$	1343.29i	1.47452i	−0.675607	+	0.737262i	$0.736118\pi$
$912$	0	0
$913$	−149.559	−0.163810
$914$	0	0
$915$	3.65169i	0.00399091i
$916$	0	0
$917$	−137.269	−0.149694
$918$	0	0
$919$	− 1033.02i	− 1.12407i	−0.827112	−	0.562037i	$-0.810018\pi$
$919$	− 1033.02i	− 1.12407i	0.827112	−	0.562037i	$-0.189982\pi$
$920$	0	0
$921$	−251.831	−0.273432
$922$	0	0
$923$	1318.30i	1.42827i
$924$	0	0
$925$	−1171.08	−1.26603
$926$	0	0
$927$	− 182.175i	− 0.196521i
$928$	0	0
$929$	−1139.71	−1.22681	−0.613407	−	0.789767i	$-0.710202\pi$
$929$	−1139.71	−1.22681	−0.613407	+	0.789767i	$0.710202\pi$
$930$	0	0
$931$	− 211.433i	− 0.227103i
$932$	0	0
$933$	−142.401	−0.152627
$934$	0	0
$935$	− 0.0716092i	0 7.65874e-5i
$936$	0	0
$937$	−1168.48	−1.24704	−0.623520	−	0.781807i	$-0.714298\pi$
$937$	−1168.48	−1.24704	−0.623520	+	0.781807i	$0.714298\pi$
$938$	0	0
$939$	146.752i	0.156285i
$940$	0	0
$941$	−546.839	−0.581126	−0.290563	−	0.956856i	$-0.593843\pi$
$941$	−546.839	−0.581126	−0.290563	+	0.956856i	$0.593843\pi$
$942$	0	0
$943$	− 2351.68i	− 2.49383i
$944$	0	0
$945$	0.231289	0.000244750 0
$946$	0	0
$947$	139.348i	0.147147i	0.997290	+	0.0735736i	$0.0234404\pi$
$947$	139.348i	0.147147i	−0.997290	+	0.0735736i	$0.976560\pi$
$948$	0	0
$949$	1260.56	1.32831
$950$	0	0
$951$	− 897.914i	− 0.944179i
$952$	0	0
$953$	−208.451	−0.218731	−0.109366	−	0.994002i	$-0.534882\pi$
$953$	−208.451	−0.218731	−0.109366	+	0.994002i	$0.534882\pi$
$954$	0	0
$955$	3.74802i	0.00392463i
$956$	0	0
$957$	62.3868	0.0651900
$958$	0	0
$959$	− 85.4006i	− 0.0890517i
$960$	0	0
$961$	−136.456	−0.141993
$962$	0	0
$963$	478.739i	0.497133i
$964$	0	0
$965$	9.23089	0.00956569
$966$	0	0
$967$	651.467i	0.673699i	0.941559	+	0.336849i	$0.109361\pi$
$967$	651.467i	0.673699i	−0.941559	+	0.336849i	$0.890639\pi$
$968$	0	0
$969$	−1.87580	−0.00193581
$970$	0	0
$971$	1430.30i	1.47302i	0.676425	+	0.736511i	$0.263528\pi$
$971$	1430.30i	1.47302i	−0.676425	+	0.736511i	$0.736472\pi$
$972$	0	0
$973$	−9.09814	−0.00935061
$974$	0	0
$975$	− 699.395i	− 0.717328i
$976$	0	0
$977$	921.541	0.943235	0.471618	−	0.881803i	$-0.343670\pi$
$977$	921.541	0.943235	0.471618	+	0.881803i	$0.343670\pi$
$978$	0	0
$979$	14.8562i	0.0151749i
$980$	0	0
$981$	177.934	0.181380
$982$	0	0
$983$	177.459i	0.180528i	0.995918	+	0.0902638i	$0.0287710\pi$
$983$	177.459i	0.180528i	−0.995918	+	0.0902638i	$0.971229\pi$
$984$	0	0
$985$	0.394853	0.000400866 0
$986$	0	0
$987$	− 71.6322i	− 0.0725757i
$988$	0	0
$989$	−1792.63	−1.81257
$990$	0	0
$991$	− 1181.85i	− 1.19258i	−0.802768	−	0.596292i	$-0.796640\pi$
$991$	− 1181.85i	− 1.19258i	0.802768	−	0.596292i	$-0.203360\pi$
$992$	0	0
$993$	−367.049	−0.369637
$994$	0	0
$995$	8.47655i	0.00851914i
$996$	0	0
$997$	973.593	0.976523	0.488261	−	0.872697i	$-0.337631\pi$
$997$	973.593	0.976523	0.488261	+	0.872697i	$0.337631\pi$
$998$	0	0
$999$	− 243.443i	− 0.243687i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.3.m.c.799.10	yes	12
3.2	odd	2		2736.3.m.d.1711.5		12
4.3	odd	2	inner	912.3.m.c.799.4	✓	12
12.11	even	2		2736.3.m.d.1711.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.c.799.4	✓	12	4.3	odd	2	inner
912.3.m.c.799.10	yes	12	1.1	even	1	trivial
2736.3.m.d.1711.5		12	3.2	odd	2
2736.3.m.d.1711.6		12	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -0.0633322 q^{5} -0.702827i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -0.0633322 q^{5} -0.702827i q^{7} -3.00000 q^{9} -4.55087i q^{11} +16.1544 q^{13} -0.109695i q^{15} -0.248456 q^{17} -4.35890i q^{19} +1.21733 q^{21} -29.3347i q^{23} -24.9960 q^{25} -5.19615i q^{27} +7.91476 q^{29} +33.1279i q^{31} +7.88234 q^{33} +0.0445116i q^{35} +46.8507 q^{37} +27.9803i q^{39} +80.1670 q^{41} -61.1094i q^{43} +0.189997 q^{45} -58.8436i q^{47} +48.5060 q^{49} -0.430339i q^{51} +1.09881 q^{53} +0.288217i q^{55} +7.54983 q^{57} +46.8624i q^{59} -33.2896 q^{61} +2.10848i q^{63} -1.02310 q^{65} +94.5098i q^{67} +50.8093 q^{69} +81.6058i q^{71} +78.0320 q^{73} -43.2943i q^{75} -3.19847 q^{77} -25.0820i q^{79} +9.00000 q^{81} -32.8638i q^{83} +0.0157353 q^{85} +13.7088i q^{87} -3.26448 q^{89} -11.3538i q^{91} -57.3791 q^{93} +0.276059i q^{95} +109.394 q^{97} +13.6526i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} - 36 q^{9}+O(q^{10}) \) 12 * q - 4 * q^5 - 36 * q^9	\( 12 q - 4 q^{5} - 36 q^{9} + 20 q^{17} + 88 q^{25} - 24 q^{29} - 72 q^{33} - 40 q^{37} - 32 q^{41} + 12 q^{45} + 128 q^{49} + 184 q^{53} - 276 q^{61} - 232 q^{65} + 120 q^{69} - 92 q^{73} + 308 q^{77} + 108 q^{81} - 244 q^{85} + 72 q^{89} + 144 q^{93} - 280 q^{97}+O(q^{100}) \) 12 * q - 4 * q^5 - 36 * q^9 + 20 * q^17 + 88 * q^25 - 24 * q^29 - 72 * q^33 - 40 * q^37 - 32 * q^41 + 12 * q^45 + 128 * q^49 + 184 * q^53 - 276 * q^61 - 232 * q^65 + 120 * q^69 - 92 * q^73 + 308 * q^77 + 108 * q^81 - 244 * q^85 + 72 * q^89 + 144 * q^93 - 280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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