LMFDB - Embedded newform 768.5.g.k.511.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,5,Mod(511,768)] 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("768.511"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-432,0,0,0,0,0,0,0,480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 858x^{12} + 7028x^{10} + 25803x^{8} + 34572x^{6} + 14794x^{4} + 708x^{2} + 9 $ x^16 + 48x^14 + 858x^12 + 7028x^10 + 25803x^8 + 34572x^6 + 14794x^4 + 708*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.11
Root			$-3.55404i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.5.g.k.511.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{3} -13.7025 q^{5} -88.3075i q^{7} -27.0000 q^{9} -170.223i q^{11} +8.23933 q^{13} -71.2001i q^{15} +179.037 q^{17} +87.9274i q^{19} +458.859 q^{21} -697.220i q^{23} -437.243 q^{25} -140.296i q^{27} -84.8258 q^{29} -1129.87i q^{31} +884.505 q^{33} +1210.03i q^{35} +757.702 q^{37} +42.8128i q^{39} +136.341 q^{41} +2920.39i q^{43} +369.966 q^{45} +1670.71i q^{47} -5397.21 q^{49} +930.302i q^{51} -3638.10 q^{53} +2332.47i q^{55} -456.884 q^{57} +1583.53i q^{59} +6578.34 q^{61} +2384.30i q^{63} -112.899 q^{65} -4472.71i q^{67} +3622.86 q^{69} +5473.20i q^{71} +1280.60 q^{73} -2271.98i q^{75} -15032.0 q^{77} -774.773i q^{79} +729.000 q^{81} +2817.15i q^{83} -2453.24 q^{85} -440.768i q^{87} -13256.9 q^{89} -727.595i q^{91} +5870.98 q^{93} -1204.82i q^{95} -3699.17 q^{97} +4596.02i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 432 q^{9} + 480 q^{17} + 1328 q^{25} - 1440 q^{41} - 2480 q^{49} - 7488 q^{57} - 2688 q^{65} + 33760 q^{73} + 11664 q^{81} - 31200 q^{89} - 24352 q^{97}+O(q^{100}) $ 16 * q - 432 * q^9 + 480 * q^17 + 1328 * q^25 - 1440 * q^41 - 2480 * q^49 - 7488 * q^57 - 2688 * q^65 + 33760 * q^73 + 11664 * q^81 - 31200 * q^89 - 24352 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	5.19615i	0.577350i
$3$	5.19615i	0.577350i
$4$	0	0
$5$	−13.7025	−0.548098	−0.274049	−	0.961716i	$-0.588363\pi$
$5$	−13.7025	−0.548098	−0.274049	+	0.961716i	$0.588363\pi$
$6$	0	0
$7$	− 88.3075i	− 1.80219i	−0.433618	−	0.901097i	$-0.642763\pi$
$7$	− 88.3075i	− 1.80219i	0.433618	−	0.901097i	$-0.357237\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	− 170.223i	− 1.40680i	−0.710793	−	0.703401i	$-0.751664\pi$
$11$	− 170.223i	− 1.40680i	0.710793	−	0.703401i	$-0.248336\pi$
$12$	0	0
$13$	8.23933	0.0487534	0.0243767	−	0.999703i	$-0.492240\pi$
$13$	8.23933	0.0487534	0.0243767	+	0.999703i	$0.492240\pi$
$14$	0	0
$15$	− 71.2001i	− 0.316445i
$16$	0	0
$17$	179.037	0.619504	0.309752	−	0.950817i	$-0.399754\pi$
$17$	179.037	0.619504	0.309752	+	0.950817i	$0.399754\pi$
$18$	0	0
$19$	87.9274i	0.243566i	0.992557	+	0.121783i	$0.0388612\pi$
$19$	87.9274i	0.243566i	−0.992557	+	0.121783i	$0.961139\pi$
$20$	0	0
$21$	458.859	1.04050
$22$	0	0
$23$	− 697.220i	− 1.31800i	−0.752145	−	0.658998i	$-0.770981\pi$
$23$	− 697.220i	− 1.31800i	0.752145	−	0.658998i	$-0.229019\pi$
$24$	0	0
$25$	−437.243	−0.699588
$26$	0	0
$27$	− 140.296i	− 0.192450i
$28$	0	0
$29$	−84.8258	−0.100863	−0.0504315	−	0.998728i	$-0.516060\pi$
$29$	−84.8258	−0.100863	−0.0504315	+	0.998728i	$0.516060\pi$
$30$	0	0
$31$	− 1129.87i	− 1.17572i	−0.808961	−	0.587862i	$-0.799970\pi$
$31$	− 1129.87i	− 1.17572i	0.808961	−	0.587862i	$-0.200030\pi$
$32$	0	0
$33$	884.505	0.812217
$34$	0	0
$35$	1210.03i	0.987779i
$36$	0	0
$37$	757.702	0.553471	0.276736	−	0.960946i	$-0.410747\pi$
$37$	757.702	0.553471	0.276736	+	0.960946i	$0.410747\pi$
$38$	0	0
$39$	42.8128i	0.0281478i
$40$	0	0
$41$	136.341	0.0811071	0.0405536	−	0.999177i	$-0.487088\pi$
$41$	136.341	0.0811071	0.0405536	+	0.999177i	$0.487088\pi$
$42$	0	0
$43$	2920.39i	1.57944i	0.613467	+	0.789721i	$0.289774\pi$
$43$	2920.39i	1.57944i	−0.613467	+	0.789721i	$0.710226\pi$
$44$	0	0
$45$	369.966	0.182699
$46$	0	0
$47$	1670.71i	0.756318i	0.925741	+	0.378159i	$0.123443\pi$
$47$	1670.71i	0.756318i	−0.925741	+	0.378159i	$0.876557\pi$
$48$	0	0
$49$	−5397.21	−2.24790
$50$	0	0
$51$	930.302i	0.357671i
$52$	0	0
$53$	−3638.10	−1.29516	−0.647579	−	0.761999i	$-0.724218\pi$
$53$	−3638.10	−1.29516	−0.647579	+	0.761999i	$0.724218\pi$
$54$	0	0
$55$	2332.47i	0.771066i
$56$	0	0
$57$	−456.884	−0.140623
$58$	0	0
$59$	1583.53i	0.454908i	0.973789	+	0.227454i	$0.0730401\pi$
$59$	1583.53i	0.454908i	−0.973789	+	0.227454i	$0.926960\pi$
$60$	0	0
$61$	6578.34	1.76790	0.883948	−	0.467585i	$-0.154876\pi$
$61$	6578.34	1.76790	0.883948	+	0.467585i	$0.154876\pi$
$62$	0	0
$63$	2384.30i	0.600731i
$64$	0	0
$65$	−112.899	−0.0267217
$66$	0	0
$67$	− 4472.71i	− 0.996371i	−0.867070	−	0.498186i	$-0.834000\pi$
$67$	− 4472.71i	− 0.996371i	0.867070	−	0.498186i	$-0.166000\pi$
$68$	0	0
$69$	3622.86	0.760945
$70$	0	0
$71$	5473.20i	1.08574i	0.839818	+	0.542869i	$0.182662\pi$
$71$	5473.20i	1.08574i	−0.839818	+	0.542869i	$0.817338\pi$
$72$	0	0
$73$	1280.60	0.240307	0.120154	−	0.992755i	$-0.461661\pi$
$73$	1280.60	0.240307	0.120154	+	0.992755i	$0.461661\pi$
$74$	0	0
$75$	− 2271.98i	− 0.403907i
$76$	0	0
$77$	−15032.0	−2.53533
$78$	0	0
$79$	− 774.773i	− 0.124142i	−0.998072	−	0.0620712i	$-0.980229\pi$
$79$	− 774.773i	− 0.124142i	0.998072	−	0.0620712i	$-0.0197706\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	2817.15i	0.408935i	0.978873	+	0.204468i	$0.0655463\pi$
$83$	2817.15i	0.408935i	−0.978873	+	0.204468i	$0.934454\pi$
$84$	0	0
$85$	−2453.24	−0.339549
$86$	0	0
$87$	− 440.768i	− 0.0582333i
$88$	0	0
$89$	−13256.9	−1.67364	−0.836819	−	0.547480i	$-0.815587\pi$
$89$	−13256.9	−1.67364	−0.836819	+	0.547480i	$0.815587\pi$
$90$	0	0
$91$	− 727.595i	− 0.0878631i
$92$	0	0
$93$	5870.98	0.678805
$94$	0	0
$95$	− 1204.82i	− 0.133498i
$96$	0	0
$97$	−3699.17	−0.393153	−0.196576	−	0.980489i	$-0.562982\pi$
$97$	−3699.17	−0.393153	−0.196576	+	0.980489i	$0.562982\pi$
$98$	0	0
$99$	4596.02i	0.468934i
$100$	0	0
$101$	−1647.79	−0.161532	−0.0807659	−	0.996733i	$-0.525737\pi$
$101$	−1647.79	−0.161532	−0.0807659	+	0.996733i	$0.525737\pi$
$102$	0	0
$103$	− 12740.1i	− 1.20087i	−0.799673	−	0.600436i	$-0.794994\pi$
$103$	− 12740.1i	− 1.20087i	0.799673	−	0.600436i	$-0.205006\pi$
$104$	0	0
$105$	−6287.50	−0.570295
$106$	0	0
$107$	− 3176.78i	− 0.277472i	−0.990329	−	0.138736i	$-0.955696\pi$
$107$	− 3176.78i	− 0.277472i	0.990329	−	0.138736i	$-0.0443039\pi$
$108$	0	0
$109$	−7152.06	−0.601974	−0.300987	−	0.953628i	$-0.597316\pi$
$109$	−7152.06	−0.601974	−0.300987	+	0.953628i	$0.597316\pi$
$110$	0	0
$111$	3937.14i	0.319547i
$112$	0	0
$113$	−10153.1	−0.795137	−0.397568	−	0.917573i	$-0.630146\pi$
$113$	−10153.1	−0.795137	−0.397568	+	0.917573i	$0.630146\pi$
$114$	0	0
$115$	9553.62i	0.722391i
$116$	0	0
$117$	−222.462	−0.0162511
$118$	0	0
$119$	− 15810.3i	− 1.11647i
$120$	0	0
$121$	−14334.9	−0.979091
$122$	0	0
$123$	708.449i	0.0468272i
$124$	0	0
$125$	14555.3	0.931541
$126$	0	0
$127$	− 17055.6i	− 1.05745i	−0.848793	−	0.528725i	$-0.822670\pi$
$127$	− 17055.6i	− 1.05745i	0.848793	−	0.528725i	$-0.177330\pi$
$128$	0	0
$129$	−15174.8	−0.911891
$130$	0	0
$131$	439.351i	0.0256017i	0.999918	+	0.0128008i	$0.00407475\pi$
$131$	439.351i	0.0256017i	−0.999918	+	0.0128008i	$0.995925\pi$
$132$	0	0
$133$	7764.65	0.438954
$134$	0	0
$135$	1922.40i	0.105482i
$136$	0	0
$137$	−9355.78	−0.498470	−0.249235	−	0.968443i	$-0.580179\pi$
$137$	−9355.78	−0.498470	−0.249235	+	0.968443i	$0.580179\pi$
$138$	0	0
$139$	− 12221.4i	− 0.632544i	−0.948669	−	0.316272i	$-0.897569\pi$
$139$	− 12221.4i	− 0.632544i	0.948669	−	0.316272i	$-0.102431\pi$
$140$	0	0
$141$	−8681.25	−0.436661
$142$	0	0
$143$	− 1402.52i	− 0.0685864i
$144$	0	0
$145$	1162.32	0.0552828
$146$	0	0
$147$	− 28044.7i	− 1.29783i
$148$	0	0
$149$	−1012.72	−0.0456158	−0.0228079	−	0.999740i	$-0.507261\pi$
$149$	−1012.72	−0.0456158	−0.0228079	+	0.999740i	$0.507261\pi$
$150$	0	0
$151$	3538.79i	0.155203i	0.996984	+	0.0776017i	$0.0247263\pi$
$151$	3538.79i	0.155203i	−0.996984	+	0.0776017i	$0.975274\pi$
$152$	0	0
$153$	−4833.99	−0.206501
$154$	0	0
$155$	15482.0i	0.644412i
$156$	0	0
$157$	−7215.92	−0.292747	−0.146374	−	0.989229i	$-0.546760\pi$
$157$	−7215.92	−0.292747	−0.146374	+	0.989229i	$0.546760\pi$
$158$	0	0
$159$	− 18904.1i	− 0.747759i
$160$	0	0
$161$	−61569.7	−2.37528
$162$	0	0
$163$	− 28722.0i	− 1.08103i	−0.841333	−	0.540517i	$-0.818229\pi$
$163$	− 28722.0i	− 1.08103i	0.841333	−	0.540517i	$-0.181771\pi$
$164$	0	0
$165$	−12119.9	−0.445175
$166$	0	0
$167$	39422.8i	1.41356i	0.707432	+	0.706781i	$0.249853\pi$
$167$	39422.8i	1.41356i	−0.707432	+	0.706781i	$0.750147\pi$
$168$	0	0
$169$	−28493.1	−0.997623
$170$	0	0
$171$	− 2374.04i	− 0.0811888i
$172$	0	0
$173$	44699.5	1.49352	0.746759	−	0.665094i	$-0.231609\pi$
$173$	44699.5	1.49352	0.746759	+	0.665094i	$0.231609\pi$
$174$	0	0
$175$	38611.8i	1.26079i
$176$	0	0
$177$	−8228.28	−0.262641
$178$	0	0
$179$	− 20791.1i	− 0.648890i	−0.945905	−	0.324445i	$-0.894822\pi$
$179$	− 20791.1i	− 0.648890i	0.945905	−	0.324445i	$-0.105178\pi$
$180$	0	0
$181$	9446.49	0.288346	0.144173	−	0.989553i	$-0.453948\pi$
$181$	9446.49	0.288346	0.144173	+	0.989553i	$0.453948\pi$
$182$	0	0
$183$	34182.1i	1.02070i
$184$	0	0
$185$	−10382.4	−0.303357
$186$	0	0
$187$	− 30476.2i	− 0.871520i
$188$	0	0
$189$	−12389.2	−0.346832
$190$	0	0
$191$	27727.4i	0.760050i	0.924976	+	0.380025i	$0.124085\pi$
$191$	27727.4i	0.760050i	−0.924976	+	0.380025i	$0.875915\pi$
$192$	0	0
$193$	−35343.3	−0.948840	−0.474420	−	0.880299i	$-0.657342\pi$
$193$	−35343.3	−0.948840	−0.474420	+	0.880299i	$0.657342\pi$
$194$	0	0
$195$	− 586.641i	− 0.0154278i
$196$	0	0
$197$	33108.7	0.853120	0.426560	−	0.904459i	$-0.359725\pi$
$197$	33108.7	0.853120	0.426560	+	0.904459i	$0.359725\pi$
$198$	0	0
$199$	3572.48i	0.0902119i	0.998982	+	0.0451059i	$0.0143625\pi$
$199$	3572.48i	0.0902119i	−0.998982	+	0.0451059i	$0.985637\pi$
$200$	0	0
$201$	23240.9	0.575255
$202$	0	0
$203$	7490.75i	0.181775i
$204$	0	0
$205$	−1868.21	−0.0444547
$206$	0	0
$207$	18824.9i	0.439332i
$208$	0	0
$209$	14967.3	0.342649
$210$	0	0
$211$	56070.4i	1.25941i	0.776833	+	0.629707i	$0.216825\pi$
$211$	56070.4i	1.25941i	−0.776833	+	0.629707i	$0.783175\pi$
$212$	0	0
$213$	−28439.6	−0.626851
$214$	0	0
$215$	− 40016.5i	− 0.865689i
$216$	0	0
$217$	−99776.0	−2.11888
$218$	0	0
$219$	6654.17i	0.138741i
$220$	0	0
$221$	1475.14	0.0302030
$222$	0	0
$223$	14086.8i	0.283270i	0.989919	+	0.141635i	$0.0452360\pi$
$223$	14086.8i	0.283270i	−0.989919	+	0.141635i	$0.954764\pi$
$224$	0	0
$225$	11805.6	0.233196
$226$	0	0
$227$	79611.9i	1.54499i	0.635020	+	0.772496i	$0.280992\pi$
$227$	79611.9i	1.54499i	−0.635020	+	0.772496i	$0.719008\pi$
$228$	0	0
$229$	−49793.5	−0.949515	−0.474757	−	0.880117i	$-0.657464\pi$
$229$	−49793.5	−0.949515	−0.474757	+	0.880117i	$0.657464\pi$
$230$	0	0
$231$	− 78108.4i	− 1.46377i
$232$	0	0
$233$	−26278.7	−0.484052	−0.242026	−	0.970270i	$-0.577812\pi$
$233$	−26278.7	−0.484052	−0.242026	+	0.970270i	$0.577812\pi$
$234$	0	0
$235$	− 22892.8i	− 0.414537i
$236$	0	0
$237$	4025.84	0.0716737
$238$	0	0
$239$	− 71671.6i	− 1.25473i	−0.778724	−	0.627366i	$-0.784133\pi$
$239$	− 71671.6i	− 1.25473i	0.778724	−	0.627366i	$-0.215867\pi$
$240$	0	0
$241$	−45225.0	−0.778654	−0.389327	−	0.921100i	$-0.627292\pi$
$241$	−45225.0	−0.778654	−0.389327	+	0.921100i	$0.627292\pi$
$242$	0	0
$243$	3788.00i	0.0641500i
$244$	0	0
$245$	73955.1	1.23207
$246$	0	0
$247$	724.463i	0.0118747i
$248$	0	0
$249$	−14638.4	−0.236099
$250$	0	0
$251$	98930.7i	1.57030i	0.619303	+	0.785152i	$0.287415\pi$
$251$	98930.7i	1.57030i	−0.619303	+	0.785152i	$0.712585\pi$
$252$	0	0
$253$	−118683.	−1.85416
$254$	0	0
$255$	− 12747.4i	− 0.196039i
$256$	0	0
$257$	42850.4	0.648767	0.324383	−	0.945926i	$-0.394843\pi$
$257$	42850.4	0.648767	0.324383	+	0.945926i	$0.394843\pi$
$258$	0	0
$259$	− 66910.8i	− 0.997462i
$260$	0	0
$261$	2290.30	0.0336210
$262$	0	0
$263$	37252.6i	0.538573i	0.963060	+	0.269287i	$0.0867879\pi$
$263$	37252.6i	0.538573i	−0.963060	+	0.269287i	$0.913212\pi$
$264$	0	0
$265$	49850.9	0.709874
$266$	0	0
$267$	− 68884.8i	− 0.966275i
$268$	0	0
$269$	−27373.8	−0.378295	−0.189148	−	0.981949i	$-0.560572\pi$
$269$	−27373.8	−0.378295	−0.189148	+	0.981949i	$0.560572\pi$
$270$	0	0
$271$	− 46285.1i	− 0.630235i	−0.949053	−	0.315117i	$-0.897956\pi$
$271$	− 46285.1i	− 0.630235i	0.949053	−	0.315117i	$-0.102044\pi$
$272$	0	0
$273$	3780.69	0.0507278
$274$	0	0
$275$	74428.8i	0.984182i
$276$	0	0
$277$	−14741.5	−0.192124	−0.0960621	−	0.995375i	$-0.530625\pi$
$277$	−14741.5	−0.192124	−0.0960621	+	0.995375i	$0.530625\pi$
$278$	0	0
$279$	30506.5i	0.391908i
$280$	0	0
$281$	−73086.5	−0.925603	−0.462801	−	0.886462i	$-0.653156\pi$
$281$	−73086.5	−0.925603	−0.462801	+	0.886462i	$0.653156\pi$
$282$	0	0
$283$	64720.8i	0.808111i	0.914734	+	0.404056i	$0.132400\pi$
$283$	64720.8i	0.808111i	−0.914734	+	0.404056i	$0.867600\pi$
$284$	0	0
$285$	6260.44	0.0770753
$286$	0	0
$287$	− 12039.9i	− 0.146171i
$288$	0	0
$289$	−51466.8	−0.616214
$290$	0	0
$291$	− 19221.5i	− 0.226987i
$292$	0	0
$293$	−92603.8	−1.07868	−0.539341	−	0.842087i	$-0.681327\pi$
$293$	−92603.8	−1.07868	−0.539341	+	0.842087i	$0.681327\pi$
$294$	0	0
$295$	− 21698.3i	− 0.249334i
$296$	0	0
$297$	−23881.6	−0.270739
$298$	0	0
$299$	− 5744.62i	− 0.0642568i
$300$	0	0
$301$	257892.	2.84646
$302$	0	0
$303$	− 8562.14i	− 0.0932604i
$304$	0	0
$305$	−90139.4	−0.968981
$306$	0	0
$307$	− 89491.7i	− 0.949524i	−0.880114	−	0.474762i	$-0.842534\pi$
$307$	− 89491.7i	− 0.949524i	0.880114	−	0.474762i	$-0.157466\pi$
$308$	0	0
$309$	66199.3	0.693324
$310$	0	0
$311$	− 155526.i	− 1.60799i	−0.594636	−	0.803995i	$-0.702704\pi$
$311$	− 155526.i	− 1.60799i	0.594636	−	0.803995i	$-0.297296\pi$
$312$	0	0
$313$	131010.	1.33726	0.668630	−	0.743595i	$-0.266881\pi$
$313$	131010.	1.33726	0.668630	+	0.743595i	$0.266881\pi$
$314$	0	0
$315$	− 32670.8i	− 0.329260i
$316$	0	0
$317$	−116023.	−1.15459	−0.577293	−	0.816537i	$-0.695891\pi$
$317$	−116023.	−1.15459	−0.577293	+	0.816537i	$0.695891\pi$
$318$	0	0
$319$	14439.3i	0.141894i
$320$	0	0
$321$	16507.0	0.160199
$322$	0	0
$323$	15742.2i	0.150890i
$324$	0	0
$325$	−3602.59	−0.0341073
$326$	0	0
$327$	− 37163.2i	− 0.347550i
$328$	0	0
$329$	147536.	1.36303
$330$	0	0
$331$	− 1630.76i	− 0.0148845i	−0.999972	−	0.00744223i	$-0.997631\pi$
$331$	− 1630.76i	− 0.0148845i	0.999972	−	0.00744223i	$-0.00236896\pi$
$332$	0	0
$333$	−20458.0	−0.184490
$334$	0	0
$335$	61287.1i	0.546109i
$336$	0	0
$337$	54219.2	0.477412	0.238706	−	0.971092i	$-0.423277\pi$
$337$	54219.2	0.477412	0.238706	+	0.971092i	$0.423277\pi$
$338$	0	0
$339$	− 52757.1i	− 0.459072i
$340$	0	0
$341$	−192330.	−1.65401
$342$	0	0
$343$	264588.i	2.24896i
$344$	0	0
$345$	−49642.1	−0.417073
$346$	0	0
$347$	142151.i	1.18057i	0.807195	+	0.590285i	$0.200985\pi$
$347$	142151.i	1.18057i	−0.807195	+	0.590285i	$0.799015\pi$
$348$	0	0
$349$	52378.2	0.430031	0.215015	−	0.976611i	$-0.431020\pi$
$349$	52378.2	0.430031	0.215015	+	0.976611i	$0.431020\pi$
$350$	0	0
$351$	− 1155.95i	− 0.00938260i
$352$	0	0
$353$	97057.1	0.778893	0.389447	−	0.921049i	$-0.372666\pi$
$353$	97057.1	0.778893	0.389447	+	0.921049i	$0.372666\pi$
$354$	0	0
$355$	− 74996.3i	− 0.595091i
$356$	0	0
$357$	82152.6	0.644592
$358$	0	0
$359$	190534.i	1.47837i	0.673500	+	0.739187i	$0.264790\pi$
$359$	190534.i	1.47837i	−0.673500	+	0.739187i	$0.735210\pi$
$360$	0	0
$361$	122590.	0.940675
$362$	0	0
$363$	− 74486.2i	− 0.565279i
$364$	0	0
$365$	−17547.3	−0.131712
$366$	0	0
$367$	23079.6i	0.171355i	0.996323	+	0.0856774i	$0.0273054\pi$
$367$	23079.6i	0.171355i	−0.996323	+	0.0856774i	$0.972695\pi$
$368$	0	0
$369$	−3681.21	−0.0270357
$370$	0	0
$371$	321271.i	2.33412i
$372$	0	0
$373$	225921.	1.62382	0.811911	−	0.583781i	$-0.198427\pi$
$373$	225921.	1.62382	0.811911	+	0.583781i	$0.198427\pi$
$374$	0	0
$375$	75631.7i	0.537826i
$376$	0	0
$377$	−698.908	−0.00491742
$378$	0	0
$379$	82454.6i	0.574033i	0.957926	+	0.287016i	$0.0926634\pi$
$379$	82454.6i	0.574033i	−0.957926	+	0.287016i	$0.907337\pi$
$380$	0	0
$381$	88623.6	0.610519
$382$	0	0
$383$	201710.i	1.37509i	0.726143	+	0.687544i	$0.241311\pi$
$383$	201710.i	1.37509i	−0.726143	+	0.687544i	$0.758689\pi$
$384$	0	0
$385$	205975.	1.38961
$386$	0	0
$387$	− 78850.4i	− 0.526480i
$388$	0	0
$389$	67700.5	0.447396	0.223698	−	0.974658i	$-0.428187\pi$
$389$	67700.5	0.447396	0.223698	+	0.974658i	$0.428187\pi$
$390$	0	0
$391$	− 124828.i	− 0.816504i
$392$	0	0
$393$	−2282.93	−0.0147811
$394$	0	0
$395$	10616.3i	0.0680423i
$396$	0	0
$397$	−264317.	−1.67704	−0.838521	−	0.544869i	$-0.816579\pi$
$397$	−264317.	−1.67704	−0.838521	+	0.544869i	$0.816579\pi$
$398$	0	0
$399$	40346.3i	0.253430i
$400$	0	0
$401$	215092.	1.33763	0.668815	−	0.743429i	$-0.266802\pi$
$401$	215092.	1.33763	0.668815	+	0.743429i	$0.266802\pi$
$402$	0	0
$403$	− 9309.38i	− 0.0573206i
$404$	0	0
$405$	−9989.09	−0.0608998
$406$	0	0
$407$	− 128978.i	− 0.778624i
$408$	0	0
$409$	77503.9	0.463316	0.231658	−	0.972797i	$-0.425585\pi$
$409$	77503.9	0.463316	0.231658	+	0.972797i	$0.425585\pi$
$410$	0	0
$411$	− 48614.1i	− 0.287792i
$412$	0	0
$413$	139838.	0.819832
$414$	0	0
$415$	− 38601.9i	− 0.224137i
$416$	0	0
$417$	63504.2	0.365200
$418$	0	0
$419$	− 64473.4i	− 0.367242i	−0.982997	−	0.183621i	$-0.941218\pi$
$419$	− 64473.4i	− 0.367242i	0.982997	−	0.183621i	$-0.0587819\pi$
$420$	0	0
$421$	269.777	0.00152209	0.000761045	−	1.00000i	$-0.499758\pi$
$421$	269.777	0.00152209	0.000761045		1.00000i	$0.499758\pi$
$422$	0	0
$423$	− 45109.1i	− 0.252106i
$424$	0	0
$425$	−78282.5	−0.433398
$426$	0	0
$427$	− 580917.i	− 3.18609i
$428$	0	0
$429$	7287.73	0.0395984
$430$	0	0
$431$	− 231809.i	− 1.24789i	−0.781470	−	0.623943i	$-0.785530\pi$
$431$	− 231809.i	− 1.24789i	0.781470	−	0.623943i	$-0.214470\pi$
$432$	0	0
$433$	239278.	1.27623	0.638113	−	0.769943i	$-0.279715\pi$
$433$	239278.	1.27623	0.638113	+	0.769943i	$0.279715\pi$
$434$	0	0
$435$	6039.60i	0.0319176i
$436$	0	0
$437$	61304.7	0.321019
$438$	0	0
$439$	− 188317.i	− 0.977148i	−0.872522	−	0.488574i	$-0.837517\pi$
$439$	− 188317.i	− 0.977148i	0.872522	−	0.488574i	$-0.162483\pi$
$440$	0	0
$441$	145725.	0.749301
$442$	0	0
$443$	147266.i	0.750405i	0.926943	+	0.375202i	$0.122427\pi$
$443$	147266.i	0.750405i	−0.926943	+	0.375202i	$0.877573\pi$
$444$	0	0
$445$	181652.	0.917318
$446$	0	0
$447$	− 5262.22i	− 0.0263363i
$448$	0	0
$449$	−198631.	−0.985267	−0.492633	−	0.870237i	$-0.663966\pi$
$449$	−198631.	−0.985267	−0.492633	+	0.870237i	$0.663966\pi$
$450$	0	0
$451$	− 23208.4i	− 0.114102i
$452$	0	0
$453$	−18388.1	−0.0896067
$454$	0	0
$455$	9969.84i	0.0481576i
$456$	0	0
$457$	−109365.	−0.523657	−0.261829	−	0.965114i	$-0.584325\pi$
$457$	−109365.	−0.523657	−0.261829	+	0.965114i	$0.584325\pi$
$458$	0	0
$459$	− 25118.2i	− 0.119224i
$460$	0	0
$461$	−366442.	−1.72426	−0.862131	−	0.506686i	$-0.830870\pi$
$461$	−366442.	−1.72426	−0.862131	+	0.506686i	$0.830870\pi$
$462$	0	0
$463$	41097.6i	0.191714i	0.995395	+	0.0958571i	$0.0305592\pi$
$463$	41097.6i	0.191714i	−0.995395	+	0.0958571i	$0.969441\pi$
$464$	0	0
$465$	−80446.9	−0.372052
$466$	0	0
$467$	8542.52i	0.0391699i	0.999808	+	0.0195849i	$0.00623448\pi$
$467$	8542.52i	0.0391699i	−0.999808	+	0.0195849i	$0.993766\pi$
$468$	0	0
$469$	−394974.	−1.79565
$470$	0	0
$471$	− 37495.0i	− 0.169018i
$472$	0	0
$473$	497117.	2.22196
$474$	0	0
$475$	− 38445.6i	− 0.170396i
$476$	0	0
$477$	98228.6	0.431719
$478$	0	0
$479$	− 118476.i	− 0.516368i	−0.966096	−	0.258184i	$-0.916876\pi$
$479$	− 118476.i	− 0.516368i	0.966096	−	0.258184i	$-0.0831240\pi$
$480$	0	0
$481$	6242.96	0.0269836
$482$	0	0
$483$	− 319926.i	− 1.37137i
$484$	0	0
$485$	50687.8	0.215486
$486$	0	0
$487$	− 346308.i	− 1.46017i	−0.683355	−	0.730086i	$-0.739480\pi$
$487$	− 346308.i	− 1.46017i	0.683355	−	0.730086i	$-0.260520\pi$
$488$	0	0
$489$	149244.	0.624135
$490$	0	0
$491$	246817.i	1.02379i	0.859048	+	0.511896i	$0.171057\pi$
$491$	246817.i	1.02379i	−0.859048	+	0.511896i	$0.828943\pi$
$492$	0	0
$493$	−15186.9	−0.0624851
$494$	0	0
$495$	− 62976.8i	− 0.257022i
$496$	0	0
$497$	483325.	1.95671
$498$	0	0
$499$	− 266566.i	− 1.07054i	−0.844681	−	0.535270i	$-0.820210\pi$
$499$	− 266566.i	− 1.07054i	0.844681	−	0.535270i	$-0.179790\pi$
$500$	0	0
$501$	−204847.	−0.816121
$502$	0	0
$503$	265145.i	1.04797i	0.851728	+	0.523984i	$0.175555\pi$
$503$	265145.i	1.04797i	−0.851728	+	0.523984i	$0.824445\pi$
$504$	0	0
$505$	22578.7	0.0885353
$506$	0	0
$507$	− 148055.i	− 0.575978i
$508$	0	0
$509$	353828.	1.36570	0.682852	−	0.730557i	$-0.260739\pi$
$509$	353828.	1.36570	0.682852	+	0.730557i	$0.260739\pi$
$510$	0	0
$511$	− 113086.i	− 0.433080i
$512$	0	0
$513$	12335.9	0.0468744
$514$	0	0
$515$	174570.i	0.658196i
$516$	0	0
$517$	284393.	1.06399
$518$	0	0
$519$	232266.i	0.862283i
$520$	0	0
$521$	212416.	0.782551	0.391276	−	0.920274i	$-0.372034\pi$
$521$	212416.	0.782551	0.391276	+	0.920274i	$0.372034\pi$
$522$	0	0
$523$	− 258355.i	− 0.944525i	−0.881458	−	0.472263i	$-0.843437\pi$
$523$	− 258355.i	− 0.944525i	0.881458	−	0.472263i	$-0.156563\pi$
$524$	0	0
$525$	−200633.	−0.727919
$526$	0	0
$527$	− 202288.i	− 0.728366i
$528$	0	0
$529$	−206274.	−0.737113
$530$	0	0
$531$	− 42755.4i	− 0.151636i
$532$	0	0
$533$	1123.36	0.00395425
$534$	0	0
$535$	43529.6i	0.152082i
$536$	0	0
$537$	108034.	0.374637
$538$	0	0
$539$	918730.i	3.16235i
$540$	0	0
$541$	−445618.	−1.52254	−0.761269	−	0.648436i	$-0.775423\pi$
$541$	−445618.	−1.52254	−0.761269	+	0.648436i	$0.775423\pi$
$542$	0	0
$543$	49085.4i	0.166476i
$544$	0	0
$545$	98000.8	0.329941
$546$	0	0
$547$	148402.i	0.495982i	0.968762	+	0.247991i	$0.0797704\pi$
$547$	148402.i	0.495982i	−0.968762	+	0.247991i	$0.920230\pi$
$548$	0	0
$549$	−177615.	−0.589299
$550$	0	0
$551$	− 7458.51i	− 0.0245668i
$552$	0	0
$553$	−68418.3	−0.223729
$554$	0	0
$555$	− 53948.4i	− 0.175143i
$556$	0	0
$557$	348085.	1.12195	0.560976	−	0.827832i	$-0.310426\pi$
$557$	348085.	1.12195	0.560976	+	0.827832i	$0.310426\pi$
$558$	0	0
$559$	24062.0i	0.0770032i
$560$	0	0
$561$	158359.	0.503172
$562$	0	0
$563$	484922.i	1.52987i	0.644106	+	0.764936i	$0.277230\pi$
$563$	484922.i	1.52987i	−0.644106	+	0.764936i	$0.722770\pi$
$564$	0	0
$565$	139122.	0.435813
$566$	0	0
$567$	− 64376.2i	− 0.200244i
$568$	0	0
$569$	81752.4	0.252508	0.126254	−	0.991998i	$-0.459705\pi$
$569$	81752.4	0.252508	0.126254	+	0.991998i	$0.459705\pi$
$570$	0	0
$571$	522522.i	1.60263i	0.598244	+	0.801314i	$0.295865\pi$
$571$	522522.i	1.60263i	−0.598244	+	0.801314i	$0.704135\pi$
$572$	0	0
$573$	−144076.	−0.438815
$574$	0	0
$575$	304854.i	0.922054i
$576$	0	0
$577$	331590.	0.995979	0.497990	−	0.867183i	$-0.334072\pi$
$577$	331590.	0.995979	0.497990	+	0.867183i	$0.334072\pi$
$578$	0	0
$579$	− 183649.i	− 0.547813i
$580$	0	0
$581$	248776.	0.736980
$582$	0	0
$583$	619288.i	1.82203i
$584$	0	0
$585$	3048.28	0.00890723
$586$	0	0
$587$	− 643493.i	− 1.86753i	−0.357885	−	0.933766i	$-0.616502\pi$
$587$	− 643493.i	− 1.86753i	0.357885	−	0.933766i	$-0.383498\pi$
$588$	0	0
$589$	99346.6	0.286367
$590$	0	0
$591$	172038.i	0.492549i
$592$	0	0
$593$	−524010.	−1.49015	−0.745075	−	0.666981i	$-0.767586\pi$
$593$	−524010.	−1.49015	−0.745075	+	0.666981i	$0.767586\pi$
$594$	0	0
$595$	216640.i	0.611934i
$596$	0	0
$597$	−18563.2	−0.0520838
$598$	0	0
$599$	109551.i	0.305326i	0.988278	+	0.152663i	$0.0487849\pi$
$599$	109551.i	0.305326i	−0.988278	+	0.152663i	$0.951215\pi$
$600$	0	0
$601$	87917.6	0.243404	0.121702	−	0.992567i	$-0.461165\pi$
$601$	87917.6	0.243404	0.121702	+	0.992567i	$0.461165\pi$
$602$	0	0
$603$	120763.i	0.332124i
$604$	0	0
$605$	196423.	0.536638
$606$	0	0
$607$	75114.8i	0.203868i	0.994791	+	0.101934i	$0.0325030\pi$
$607$	75114.8i	0.203868i	−0.994791	+	0.101934i	$0.967497\pi$
$608$	0	0
$609$	−38923.1	−0.104948
$610$	0	0
$611$	13765.5i	0.0368731i
$612$	0	0
$613$	−672859.	−1.79062	−0.895309	−	0.445446i	$-0.853045\pi$
$613$	−672859.	−1.79062	−0.895309	+	0.445446i	$0.853045\pi$
$614$	0	0
$615$	− 9707.49i	− 0.0256659i
$616$	0	0
$617$	−383301.	−1.00686	−0.503431	−	0.864035i	$-0.667929\pi$
$617$	−383301.	−1.00686	−0.503431	+	0.864035i	$0.667929\pi$
$618$	0	0
$619$	− 404659.i	− 1.05611i	−0.849211	−	0.528054i	$-0.822922\pi$
$619$	− 404659.i	− 1.05611i	0.849211	−	0.528054i	$-0.177078\pi$
$620$	0	0
$621$	−97817.2	−0.253648
$622$	0	0
$623$	1.17068e6i	3.01622i
$624$	0	0
$625$	73832.8	0.189012
$626$	0	0
$627$	77772.2i	0.197829i
$628$	0	0
$629$	135657.	0.342878
$630$	0	0
$631$	− 348605.i	− 0.875538i	−0.899087	−	0.437769i	$-0.855769\pi$
$631$	− 348605.i	− 0.875538i	0.899087	−	0.437769i	$-0.144231\pi$
$632$	0	0
$633$	−291350.	−0.727123
$634$	0	0
$635$	233704.i	0.579587i
$636$	0	0
$637$	−44469.4	−0.109593
$638$	0	0
$639$	− 147776.i	− 0.361913i
$640$	0	0
$641$	356791.	0.868355	0.434178	−	0.900827i	$-0.357039\pi$
$641$	356791.	0.868355	0.434178	+	0.900827i	$0.357039\pi$
$642$	0	0
$643$	− 298557.i	− 0.722113i	−0.932544	−	0.361057i	$-0.882416\pi$
$643$	− 298557.i	− 0.722113i	0.932544	−	0.361057i	$-0.117584\pi$
$644$	0	0
$645$	207932.	0.499806
$646$	0	0
$647$	− 635093.i	− 1.51715i	−0.651585	−	0.758575i	$-0.725896\pi$
$647$	− 635093.i	− 1.51715i	0.651585	−	0.758575i	$-0.274104\pi$
$648$	0	0
$649$	269554.	0.639965
$650$	0	0
$651$	− 518452.i	− 1.22334i
$652$	0	0
$653$	190899.	0.447691	0.223846	−	0.974625i	$-0.428139\pi$
$653$	190899.	0.447691	0.223846	+	0.974625i	$0.428139\pi$
$654$	0	0
$655$	− 6020.18i	− 0.0140322i
$656$	0	0
$657$	−34576.1	−0.0801023
$658$	0	0
$659$	836840.i	1.92696i	0.267787	+	0.963478i	$0.413708\pi$
$659$	836840.i	1.92696i	−0.267787	+	0.963478i	$0.586292\pi$
$660$	0	0
$661$	525360.	1.20241	0.601207	−	0.799093i	$-0.294687\pi$
$661$	525360.	1.20241	0.601207	+	0.799093i	$0.294687\pi$
$662$	0	0
$663$	7665.07i	0.0174377i
$664$	0	0
$665$	−106395.	−0.240590
$666$	0	0
$667$	59142.2i	0.132937i
$668$	0	0
$669$	−73196.9	−0.163546
$670$	0	0
$671$	− 1.11978e6i	− 2.48708i
$672$	0	0
$673$	117394.	0.259189	0.129595	−	0.991567i	$-0.458632\pi$
$673$	117394.	0.259189	0.129595	+	0.991567i	$0.458632\pi$
$674$	0	0
$675$	61343.4i	0.134636i
$676$	0	0
$677$	109743.	0.239441	0.119721	−	0.992808i	$-0.461800\pi$
$677$	109743.	0.239441	0.119721	+	0.992808i	$0.461800\pi$
$678$	0	0
$679$	326665.i	0.708537i
$680$	0	0
$681$	−413675.	−0.892001
$682$	0	0
$683$	214566.i	0.459960i	0.973195	+	0.229980i	$0.0738661\pi$
$683$	214566.i	0.459960i	−0.973195	+	0.229980i	$0.926134\pi$
$684$	0	0
$685$	128197.	0.273211
$686$	0	0
$687$	− 258735.i	− 0.548202i
$688$	0	0
$689$	−29975.5	−0.0631434
$690$	0	0
$691$	− 548085.i	− 1.14787i	−0.818902	−	0.573933i	$-0.805417\pi$
$691$	− 548085.i	− 1.14787i	0.818902	−	0.573933i	$-0.194583\pi$
$692$	0	0
$693$	405863.	0.845110
$694$	0	0
$695$	167463.i	0.346696i
$696$	0	0
$697$	24410.1	0.0502462
$698$	0	0
$699$	− 136548.i	− 0.279468i
$700$	0	0
$701$	−665706.	−1.35471	−0.677355	−	0.735656i	$-0.736874\pi$
$701$	−665706.	−1.35471	−0.677355	+	0.735656i	$0.736874\pi$
$702$	0	0
$703$	66622.8i	0.134807i
$704$	0	0
$705$	118954.	0.239333
$706$	0	0
$707$	145512.i	0.291111i
$708$	0	0
$709$	9799.47	0.0194944	0.00974721	−	0.999952i	$-0.496897\pi$
$709$	9799.47	0.0194944	0.00974721	+	0.999952i	$0.496897\pi$
$710$	0	0
$711$	20918.9i	0.0413808i
$712$	0	0
$713$	−787768.	−1.54960
$714$	0	0
$715$	19218.0i	0.0375921i
$716$	0	0
$717$	372416.	0.724420
$718$	0	0
$719$	− 471450.i	− 0.911964i	−0.889989	−	0.455982i	$-0.849288\pi$
$719$	− 471450.i	− 0.911964i	0.889989	−	0.455982i	$-0.150712\pi$
$720$	0	0
$721$	−1.12504e6	−2.16420
$722$	0	0
$723$	− 234996.i	− 0.449556i
$724$	0	0
$725$	37089.4	0.0705626
$726$	0	0
$727$	− 458566.i	− 0.867628i	−0.901003	−	0.433814i	$-0.857168\pi$
$727$	− 458566.i	− 0.867628i	0.901003	−	0.433814i	$-0.142832\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	522856.i	0.978471i
$732$	0	0
$733$	376476.	0.700695	0.350348	−	0.936620i	$-0.386063\pi$
$733$	376476.	0.700695	0.350348	+	0.936620i	$0.386063\pi$
$734$	0	0
$735$	384282.i	0.711337i
$736$	0	0
$737$	−761358.	−1.40170
$738$	0	0
$739$	− 773462.i	− 1.41628i	−0.706070	−	0.708142i	$-0.749534\pi$
$739$	− 773462.i	− 1.41628i	0.706070	−	0.708142i	$-0.250466\pi$
$740$	0	0
$741$	−3764.42	−0.00685586
$742$	0	0
$743$	− 361051.i	− 0.654019i	−0.945021	−	0.327010i	$-0.893959\pi$
$743$	− 361051.i	− 0.654019i	0.945021	−	0.327010i	$-0.106041\pi$
$744$	0	0
$745$	13876.7	0.0250019
$746$	0	0
$747$	− 76063.2i	− 0.136312i
$748$	0	0
$749$	−280533.	−0.500058
$750$	0	0
$751$	− 848488.i	− 1.50441i	−0.658930	−	0.752204i	$-0.728991\pi$
$751$	− 848488.i	− 1.50441i	0.658930	−	0.752204i	$-0.271009\pi$
$752$	0	0
$753$	−514059.	−0.906615
$754$	0	0
$755$	− 48490.2i	− 0.0850667i
$756$	0	0
$757$	−873979.	−1.52514	−0.762570	−	0.646906i	$-0.776062\pi$
$757$	−873979.	−1.52514	−0.762570	+	0.646906i	$0.776062\pi$
$758$	0	0
$759$	− 616694.i	− 1.07050i
$760$	0	0
$761$	−81423.0	−0.140598	−0.0702988	−	0.997526i	$-0.522395\pi$
$761$	−81423.0	−0.140598	−0.0702988	+	0.997526i	$0.522395\pi$
$762$	0	0
$763$	631580.i	1.08487i
$764$	0	0
$765$	66237.6	0.113183
$766$	0	0
$767$	13047.3i	0.0221783i
$768$	0	0
$769$	−231240.	−0.391030	−0.195515	−	0.980701i	$-0.562638\pi$
$769$	−231240.	−0.391030	−0.195515	+	0.980701i	$0.562638\pi$
$770$	0	0
$771$	222657.i	0.374566i
$772$	0	0
$773$	−1.13648e6	−1.90197	−0.950985	−	0.309236i	$-0.899927\pi$
$773$	−1.13648e6	−1.90197	−0.950985	+	0.309236i	$0.899927\pi$
$774$	0	0
$775$	494028.i	0.822523i
$776$	0	0
$777$	347679.	0.575885
$778$	0	0
$779$	11988.1i	0.0197550i
$780$	0	0
$781$	931665.	1.52742
$782$	0	0
$783$	11900.7i	0.0194111i
$784$	0	0
$785$	98875.9	0.160454
$786$	0	0
$787$	808537.i	1.30542i	0.757608	+	0.652710i	$0.226368\pi$
$787$	808537.i	1.30542i	−0.757608	+	0.652710i	$0.773632\pi$
$788$	0	0
$789$	−193570.	−0.310945
$790$	0	0
$791$	896595.i	1.43299i
$792$	0	0
$793$	54201.1	0.0861910
$794$	0	0
$795$	259033.i	0.409846i
$796$	0	0
$797$	−1.07779e6	−1.69675	−0.848375	−	0.529395i	$-0.822419\pi$
$797$	−1.07779e6	−1.69675	−0.848375	+	0.529395i	$0.822419\pi$
$798$	0	0
$799$	299118.i	0.468542i
$800$	0	0
$801$	357936.	0.557879
$802$	0	0
$803$	− 217987.i	− 0.338064i
$804$	0	0
$805$	843657.	1.30189
$806$	0	0
$807$	− 142239.i	− 0.218409i
$808$	0	0
$809$	53549.4	0.0818197	0.0409098	−	0.999163i	$-0.486974\pi$
$809$	53549.4	0.0818197	0.0409098	+	0.999163i	$0.486974\pi$
$810$	0	0
$811$	− 895246.i	− 1.36113i	−0.732686	−	0.680567i	$-0.761734\pi$
$811$	− 895246.i	− 1.36113i	0.732686	−	0.680567i	$-0.238266\pi$
$812$	0	0
$813$	240504.	0.363866
$814$	0	0
$815$	393562.i	0.592513i
$816$	0	0
$817$	−256782.	−0.384699
$818$	0	0
$819$	19645.1i	0.0292877i
$820$	0	0
$821$	−102742.	−0.152427	−0.0762134	−	0.997092i	$-0.524283\pi$
$821$	−102742.	−0.152427	−0.0762134	+	0.997092i	$0.524283\pi$
$822$	0	0
$823$	445678.i	0.657994i	0.944331	+	0.328997i	$0.106711\pi$
$823$	445678.i	0.657994i	−0.944331	+	0.328997i	$0.893289\pi$
$824$	0	0
$825$	−386743.	−0.568218
$826$	0	0
$827$	357500.i	0.522716i	0.965242	+	0.261358i	$0.0841703\pi$
$827$	357500.i	0.522716i	−0.965242	+	0.261358i	$0.915830\pi$
$828$	0	0
$829$	−366247.	−0.532923	−0.266462	−	0.963846i	$-0.585855\pi$
$829$	−366247.	−0.532923	−0.266462	+	0.963846i	$0.585855\pi$
$830$	0	0
$831$	− 76599.1i	− 0.110923i
$832$	0	0
$833$	−966299.	−1.39258
$834$	0	0
$835$	− 540190.i	− 0.774771i
$836$	0	0
$837$	−158516.	−0.226268
$838$	0	0
$839$	− 703317.i	− 0.999142i	−0.866273	−	0.499571i	$-0.833491\pi$
$839$	− 703317.i	− 0.999142i	0.866273	−	0.499571i	$-0.166509\pi$
$840$	0	0
$841$	−700086.	−0.989827
$842$	0	0
$843$	− 379769.i	− 0.534397i
$844$	0	0
$845$	390426.	0.546796
$846$	0	0
$847$	1.26588e6i	1.76451i
$848$	0	0
$849$	−336299.	−0.466563
$850$	0	0
$851$	− 528285.i	− 0.729473i
$852$	0	0
$853$	838604.	1.15255	0.576274	−	0.817257i	$-0.304506\pi$
$853$	838604.	1.15255	0.576274	+	0.817257i	$0.304506\pi$
$854$	0	0
$855$	32530.2i	0.0444994i
$856$	0	0
$857$	−578897.	−0.788205	−0.394103	−	0.919066i	$-0.628945\pi$
$857$	−578897.	−0.788205	−0.394103	+	0.919066i	$0.628945\pi$
$858$	0	0
$859$	− 70062.4i	− 0.0949508i	−0.998872	−	0.0474754i	$-0.984882\pi$
$859$	− 70062.4i	− 0.0949508i	0.998872	−	0.0474754i	$-0.0151176\pi$
$860$	0	0
$861$	62561.3	0.0843917
$862$	0	0
$863$	− 97607.4i	− 0.131057i	−0.997851	−	0.0655286i	$-0.979127\pi$
$863$	− 97607.4i	− 0.131057i	0.997851	−	0.0655286i	$-0.0208734\pi$
$864$	0	0
$865$	−612493.	−0.818595
$866$	0	0
$867$	− 267430.i	− 0.355772i
$868$	0	0
$869$	−131884.	−0.174644
$870$	0	0
$871$	− 36852.1i	− 0.0485765i
$872$	0	0
$873$	99877.7	0.131051
$874$	0	0
$875$	− 1.28535e6i	− 1.67882i
$876$	0	0
$877$	−559203.	−0.727060	−0.363530	−	0.931582i	$-0.618429\pi$
$877$	−559203.	−0.727060	−0.363530	+	0.931582i	$0.618429\pi$
$878$	0	0
$879$	− 481184.i	− 0.622778i
$880$	0	0
$881$	1.21896e6	1.57050	0.785248	−	0.619181i	$-0.212535\pi$
$881$	1.21896e6	1.57050	0.785248	+	0.619181i	$0.212535\pi$
$882$	0	0
$883$	642583.i	0.824152i	0.911149	+	0.412076i	$0.135196\pi$
$883$	642583.i	0.824152i	−0.911149	+	0.412076i	$0.864804\pi$
$884$	0	0
$885$	112748.	0.143953
$886$	0	0
$887$	384494.i	0.488700i	0.969687	+	0.244350i	$0.0785746\pi$
$887$	384494.i	0.488700i	−0.969687	+	0.244350i	$0.921425\pi$
$888$	0	0
$889$	−1.50614e6	−1.90573
$890$	0	0
$891$	− 124093.i	− 0.156311i
$892$	0	0
$893$	−146901.	−0.184214
$894$	0	0
$895$	284889.i	0.355655i
$896$	0	0
$897$	29849.9	0.0370987
$898$	0	0
$899$	95842.2i	0.118587i
$900$	0	0
$901$	−651353.	−0.802356
$902$	0	0
$903$	1.34005e6i	1.64340i
$904$	0	0
$905$	−129440.	−0.158042
$906$	0	0
$907$	− 517177.i	− 0.628672i	−0.949312	−	0.314336i	$-0.898218\pi$
$907$	− 517177.i	− 0.628672i	0.949312	−	0.314336i	$-0.101782\pi$
$908$	0	0
$909$	44490.2	0.0538439
$910$	0	0
$911$	11448.1i	0.0137943i	0.999976	+	0.00689713i	$0.00219544\pi$
$911$	11448.1i	0.0137943i	−0.999976	+	0.00689713i	$0.997805\pi$
$912$	0	0
$913$	479545.	0.575291
$914$	0	0
$915$	− 468378.i	− 0.559441i
$916$	0	0
$917$	38797.9	0.0461392
$918$	0	0
$919$	− 376317.i	− 0.445578i	−0.974867	−	0.222789i	$-0.928484\pi$
$919$	− 376317.i	− 0.445578i	0.974867	−	0.222789i	$-0.0715160\pi$
$920$	0	0
$921$	465013.	0.548208
$922$	0	0
$923$	45095.5i	0.0529334i
$924$	0	0
$925$	−331300.	−0.387202
$926$	0	0
$927$	343981.i	0.400291i
$928$	0	0
$929$	−1.17387e6	−1.36016	−0.680078	−	0.733139i	$-0.738054\pi$
$929$	−1.17387e6	−1.36016	−0.680078	+	0.733139i	$0.738054\pi$
$930$	0	0
$931$	− 474563.i	− 0.547513i
$932$	0	0
$933$	808139.	0.928373
$934$	0	0
$935$	417599.i	0.477679i
$936$	0	0
$937$	−1.66559e6	−1.89710	−0.948548	−	0.316635i	$-0.897447\pi$
$937$	−1.66559e6	−1.89710	−0.948548	+	0.316635i	$0.897447\pi$
$938$	0	0
$939$	680748.i	0.772068i
$940$	0	0
$941$	−416169.	−0.469992	−0.234996	−	0.971996i	$-0.575508\pi$
$941$	−416169.	−0.469992	−0.234996	+	0.971996i	$0.575508\pi$
$942$	0	0
$943$	− 95059.7i	− 0.106899i
$944$	0	0
$945$	169762.	0.190098
$946$	0	0
$947$	− 456383.i	− 0.508896i	−0.967086	−	0.254448i	$-0.918106\pi$
$947$	− 456383.i	− 0.508896i	0.967086	−	0.254448i	$-0.0818938\pi$
$948$	0	0
$949$	10551.3	0.0117158
$950$	0	0
$951$	− 602874.i	− 0.666600i
$952$	0	0
$953$	669115.	0.736741	0.368371	−	0.929679i	$-0.379916\pi$
$953$	669115.	0.736741	0.368371	+	0.929679i	$0.379916\pi$
$954$	0	0
$955$	− 379933.i	− 0.416582i
$956$	0	0
$957$	−75028.8	−0.0819227
$958$	0	0
$959$	826186.i	0.898339i
$960$	0	0
$961$	−353087.	−0.382327
$962$	0	0
$963$	85773.0i	0.0924906i
$964$	0	0
$965$	484291.	0.520058
$966$	0	0
$967$	− 309216.i	− 0.330681i	−0.986237	−	0.165340i	$-0.947128\pi$
$967$	− 309216.i	− 0.330681i	0.986237	−	0.165340i	$-0.0528722\pi$
$968$	0	0
$969$	−81799.1	−0.0871166
$970$	0	0
$971$	− 761927.i	− 0.808118i	−0.914733	−	0.404059i	$-0.867599\pi$
$971$	− 761927.i	− 0.808118i	0.914733	−	0.404059i	$-0.132401\pi$
$972$	0	0
$973$	−1.07924e6	−1.13997
$974$	0	0
$975$	− 18719.6i	− 0.0196919i
$976$	0	0
$977$	−689123.	−0.721951	−0.360975	−	0.932575i	$-0.617556\pi$
$977$	−689123.	−0.721951	−0.360975	+	0.932575i	$0.617556\pi$
$978$	0	0
$979$	2.25663e6i	2.35448i
$980$	0	0
$981$	193106.	0.200658
$982$	0	0
$983$	− 68098.1i	− 0.0704739i	−0.999379	−	0.0352369i	$-0.988781\pi$
$983$	− 68098.1i	− 0.0704739i	0.999379	−	0.0352369i	$-0.0112186\pi$
$984$	0	0
$985$	−453671.	−0.467594
$986$	0	0
$987$	766619.i	0.786947i
$988$	0	0
$989$	2.03615e6	2.08170
$990$	0	0
$991$	− 179708.i	− 0.182987i	−0.995806	−	0.0914935i	$-0.970836\pi$
$991$	− 179708.i	− 0.182987i	0.995806	−	0.0914935i	$-0.0291641\pi$
$992$	0	0
$993$	8473.66	0.00859355
$994$	0	0
$995$	− 48951.8i	− 0.0494450i
$996$	0	0
$997$	509791.	0.512863	0.256432	−	0.966562i	$-0.417453\pi$
$997$	509791.	0.512863	0.256432	+	0.966562i	$0.417453\pi$
$998$	0	0
$999$	− 106303.i	− 0.106516i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.g.k.511.11		16
4.3	odd	2	inner	768.5.g.k.511.3		16
8.3	odd	2	inner	768.5.g.k.511.14		16
8.5	even	2	inner	768.5.g.k.511.6		16
16.3	odd	4		96.5.b.a.79.7		8
16.5	even	4		96.5.b.a.79.6		8
16.11	odd	4		24.5.b.a.19.4	yes	8
16.13	even	4		24.5.b.a.19.3	✓	8
48.5	odd	4		288.5.b.d.271.6		8
48.11	even	4		72.5.b.d.19.5		8
48.29	odd	4		72.5.b.d.19.6		8
48.35	even	4		288.5.b.d.271.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.5.b.a.19.3	✓	8	16.13	even	4
24.5.b.a.19.4	yes	8	16.11	odd	4
72.5.b.d.19.5		8	48.11	even	4
72.5.b.d.19.6		8	48.29	odd	4
96.5.b.a.79.6		8	16.5	even	4
96.5.b.a.79.7		8	16.3	odd	4
288.5.b.d.271.3		8	48.35	even	4
288.5.b.d.271.6		8	48.5	odd	4
768.5.g.k.511.3		16	4.3	odd	2	inner
768.5.g.k.511.6		16	8.5	even	2	inner
768.5.g.k.511.11		16	1.1	even	1	trivial
768.5.g.k.511.14		16	8.3	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{3} -13.7025 q^{5} -88.3075i q^{7} -27.0000 q^{9} -170.223i q^{11} +8.23933 q^{13} -71.2001i q^{15} +179.037 q^{17} +87.9274i q^{19} +458.859 q^{21} -697.220i q^{23} -437.243 q^{25} -140.296i q^{27} -84.8258 q^{29} -1129.87i q^{31} +884.505 q^{33} +1210.03i q^{35} +757.702 q^{37} +42.8128i q^{39} +136.341 q^{41} +2920.39i q^{43} +369.966 q^{45} +1670.71i q^{47} -5397.21 q^{49} +930.302i q^{51} -3638.10 q^{53} +2332.47i q^{55} -456.884 q^{57} +1583.53i q^{59} +6578.34 q^{61} +2384.30i q^{63} -112.899 q^{65} -4472.71i q^{67} +3622.86 q^{69} +5473.20i q^{71} +1280.60 q^{73} -2271.98i q^{75} -15032.0 q^{77} -774.773i q^{79} +729.000 q^{81} +2817.15i q^{83} -2453.24 q^{85} -440.768i q^{87} -13256.9 q^{89} -727.595i q^{91} +5870.98 q^{93} -1204.82i q^{95} -3699.17 q^{97} +4596.02i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 432 q^{9} + 480 q^{17} + 1328 q^{25} - 1440 q^{41} - 2480 q^{49} - 7488 q^{57} - 2688 q^{65} + 33760 q^{73} + 11664 q^{81} - 31200 q^{89} - 24352 q^{97}+O(q^{100}) \) 16 * q - 432 * q^9 + 480 * q^17 + 1328 * q^25 - 1440 * q^41 - 2480 * q^49 - 7488 * q^57 - 2688 * q^65 + 33760 * q^73 + 11664 * q^81 - 31200 * q^89 - 24352 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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