LMFDB - Embedded newform 768.5.g.h.511.2

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 = [8,0,0,0,-96,0,0,0,-216,0,0,0,160,0,0,0,240] 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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 21x^{6} - 2x^{5} + 439x^{4} - 48x^{3} - 2874x^{2} - 3312x + 19044 $ x^8 - 2x^7 - 21x^6 - 2x^5 + 439x^4 - 48x^3 - 2874x^2 - 3312*x + 19044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.2
Root			$-3.01338 + 2.31712i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.5.g.h.511.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -23.5183 q^{5} +40.3412i q^{7} -27.0000 q^{9} +227.579i q^{11} -310.305 q^{13} +122.205i q^{15} -380.831 q^{17} +343.152i q^{19} +209.619 q^{21} -113.116i q^{23} -71.8887 q^{25} +140.296i q^{27} +621.827 q^{29} -1528.61i q^{31} +1182.54 q^{33} -948.758i q^{35} +1385.54 q^{37} +1612.39i q^{39} -2765.92 q^{41} +1018.24i q^{43} +634.995 q^{45} +1335.48i q^{47} +773.586 q^{49} +1978.85i q^{51} +4253.04 q^{53} -5352.28i q^{55} +1783.07 q^{57} +77.6265i q^{59} -1154.74 q^{61} -1089.21i q^{63} +7297.86 q^{65} +3477.85i q^{67} -587.770 q^{69} -2055.93i q^{71} -7351.49 q^{73} +373.545i q^{75} -9180.83 q^{77} -3277.77i q^{79} +729.000 q^{81} -7747.77i q^{83} +8956.49 q^{85} -3231.11i q^{87} +4675.62 q^{89} -12518.1i q^{91} -7942.87 q^{93} -8070.35i q^{95} +408.505 q^{97} -6144.64i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 96 q^{5} - 216 q^{9} + 160 q^{13} + 240 q^{17} - 72 q^{25} + 3360 q^{29} + 1440 q^{33} + 7072 q^{37} + 1776 q^{41} + 2592 q^{45} - 10040 q^{49} + 12960 q^{53} + 3744 q^{57} + 12064 q^{61} + 11520 q^{65}+ \cdots + 2544 q^{97}+O(q^{100}) $ 8 * q - 96 * q^5 - 216 * q^9 + 160 * q^13 + 240 * q^17 - 72 * q^25 + 3360 * q^29 + 1440 * q^33 + 7072 * q^37 + 1776 * q^41 + 2592 * q^45 - 10040 * q^49 + 12960 * q^53 + 3744 * q^57 + 12064 * q^61 + 11520 * q^65 + 1152 * q^69 - 17200 * q^73 + 17664 * q^77 + 5832 * q^81 - 8768 * q^85 + 7824 * q^89 - 8064 * q^93 + 2544 * 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
$5$	−23.5183	−0.940733	−0.470366	−	0.882471i	$-0.655878\pi$
$5$	−23.5183	−0.940733	−0.470366	+	0.882471i	$0.655878\pi$
$6$	0	0
$7$	40.3412i	0.823290i	0.911344	+	0.411645i	$0.135046\pi$
$7$	40.3412i	0.823290i	−0.911344	+	0.411645i	$0.864954\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	227.579i	1.88082i	0.340041	+	0.940411i	$0.389559\pi$
$11$	227.579i	1.88082i	−0.340041	+	0.940411i	$0.610441\pi$
$12$	0	0
$13$	−310.305	−1.83613	−0.918063	−	0.396435i	$-0.870247\pi$
$13$	−310.305	−1.83613	−0.918063	+	0.396435i	$0.870247\pi$
$14$	0	0
$15$	122.205i	0.543132i
$16$	0	0
$17$	−380.831	−1.31775	−0.658876	−	0.752251i	$-0.728968\pi$
$17$	−380.831	−1.31775	−0.658876	+	0.752251i	$0.728968\pi$
$18$	0	0
$19$	343.152i	0.950559i	0.879835	+	0.475279i	$0.157653\pi$
$19$	343.152i	0.950559i	−0.879835	+	0.475279i	$0.842347\pi$
$20$	0	0
$21$	209.619	0.475327
$22$	0	0
$23$	− 113.116i	− 0.213831i	−0.994268	−	0.106915i	$-0.965903\pi$
$23$	− 113.116i	− 0.213831i	0.994268	−	0.106915i	$-0.0340974\pi$
$24$	0	0
$25$	−71.8887	−0.115022
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	621.827	0.739390	0.369695	−	0.929153i	$-0.379462\pi$
$29$	621.827	0.739390	0.369695	+	0.929153i	$0.379462\pi$
$30$	0	0
$31$	− 1528.61i	− 1.59064i	−0.606189	−	0.795321i	$-0.707302\pi$
$31$	− 1528.61i	− 1.59064i	0.606189	−	0.795321i	$-0.292698\pi$
$32$	0	0
$33$	1182.54	1.08589
$34$	0	0
$35$	− 948.758i	− 0.774496i
$36$	0	0
$37$	1385.54	1.01208	0.506042	−	0.862509i	$-0.331108\pi$
$37$	1385.54	1.01208	0.506042	+	0.862509i	$0.331108\pi$
$38$	0	0
$39$	1612.39i	1.06009i
$40$	0	0
$41$	−2765.92	−1.64540	−0.822702	−	0.568474i	$-0.807534\pi$
$41$	−2765.92	−1.64540	−0.822702	+	0.568474i	$0.807534\pi$
$42$	0	0
$43$	1018.24i	0.550698i	0.961344	+	0.275349i	$0.0887934\pi$
$43$	1018.24i	0.550698i	−0.961344	+	0.275349i	$0.911207\pi$
$44$	0	0
$45$	634.995	0.313578
$46$	0	0
$47$	1335.48i	0.604564i	0.953219	+	0.302282i	$0.0977484\pi$
$47$	1335.48i	0.604564i	−0.953219	+	0.302282i	$0.902252\pi$
$48$	0	0
$49$	773.586	0.322193
$50$	0	0
$51$	1978.85i	0.760805i
$52$	0	0
$53$	4253.04	1.51408	0.757038	−	0.653371i	$-0.226646\pi$
$53$	4253.04	1.51408	0.757038	+	0.653371i	$0.226646\pi$
$54$	0	0
$55$	− 5352.28i	− 1.76935i
$56$	0	0
$57$	1783.07	0.548805
$58$	0	0
$59$	77.6265i	0.0223001i	0.999938	+	0.0111500i	$0.00354924\pi$
$59$	77.6265i	0.0223001i	−0.999938	+	0.0111500i	$0.996451\pi$
$60$	0	0
$61$	−1154.74	−0.310332	−0.155166	−	0.987888i	$-0.549591\pi$
$61$	−1154.74	−0.310332	−0.155166	+	0.987888i	$0.549591\pi$
$62$	0	0
$63$	− 1089.21i	− 0.274430i
$64$	0	0
$65$	7297.86	1.72730
$66$	0	0
$67$	3477.85i	0.774748i	0.921923	+	0.387374i	$0.126618\pi$
$67$	3477.85i	0.774748i	−0.921923	+	0.387374i	$0.873382\pi$
$68$	0	0
$69$	−587.770	−0.123455
$70$	0	0
$71$	− 2055.93i	− 0.407841i	−0.978987	−	0.203921i	$-0.934632\pi$
$71$	− 2055.93i	− 0.407841i	0.978987	−	0.203921i	$-0.0653684\pi$
$72$	0	0
$73$	−7351.49	−1.37953	−0.689763	−	0.724035i	$-0.742285\pi$
$73$	−7351.49	−1.37953	−0.689763	+	0.724035i	$0.742285\pi$
$74$	0	0
$75$	373.545i	0.0664079i
$76$	0	0
$77$	−9180.83	−1.54846
$78$	0	0
$79$	− 3277.77i	− 0.525199i	−0.964905	−	0.262600i	$-0.915420\pi$
$79$	− 3277.77i	− 0.525199i	0.964905	−	0.262600i	$-0.0845798\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	− 7747.77i	− 1.12466i	−0.826913	−	0.562329i	$-0.809905\pi$
$83$	− 7747.77i	− 1.12466i	0.826913	−	0.562329i	$-0.190095\pi$
$84$	0	0
$85$	8956.49	1.23965
$86$	0	0
$87$	− 3231.11i	− 0.426887i
$88$	0	0
$89$	4675.62	0.590281	0.295141	−	0.955454i	$-0.404633\pi$
$89$	4675.62	0.590281	0.295141	+	0.955454i	$0.404633\pi$
$90$	0	0
$91$	− 12518.1i	− 1.51166i
$92$	0	0
$93$	−7942.87	−0.918357
$94$	0	0
$95$	− 8070.35i	− 0.894222i
$96$	0	0
$97$	408.505	0.0434164	0.0217082	−	0.999764i	$-0.493090\pi$
$97$	408.505	0.0434164	0.0217082	+	0.999764i	$0.493090\pi$
$98$	0	0
$99$	− 6144.64i	− 0.626940i
$100$	0	0
$101$	6780.48	0.664688	0.332344	−	0.943158i	$-0.392161\pi$
$101$	6780.48	0.664688	0.332344	+	0.943158i	$0.392161\pi$
$102$	0	0
$103$	− 1694.72i	− 0.159744i	−0.996805	−	0.0798719i	$-0.974549\pi$
$103$	− 1694.72i	− 0.159744i	0.996805	−	0.0798719i	$-0.0254511\pi$
$104$	0	0
$105$	−4929.89	−0.447156
$106$	0	0
$107$	− 11254.2i	− 0.982983i	−0.870882	−	0.491492i	$-0.836452\pi$
$107$	− 11254.2i	− 0.982983i	0.870882	−	0.491492i	$-0.163548\pi$
$108$	0	0
$109$	−3445.49	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$109$	−3445.49	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$110$	0	0
$111$	− 7199.49i	− 0.584327i
$112$	0	0
$113$	8660.33	0.678231	0.339116	−	0.940745i	$-0.389872\pi$
$113$	8660.33	0.678231	0.339116	+	0.940745i	$0.389872\pi$
$114$	0	0
$115$	2660.31i	0.201158i
$116$	0	0
$117$	8378.24	0.612042
$118$	0	0
$119$	− 15363.2i	− 1.08489i
$120$	0	0
$121$	−37151.4	−2.53749
$122$	0	0
$123$	14372.2i	0.949974i
$124$	0	0
$125$	16389.7	1.04894
$126$	0	0
$127$	16133.1i	1.00026i	0.865952	+	0.500128i	$0.166714\pi$
$127$	16133.1i	1.00026i	−0.865952	+	0.500128i	$0.833286\pi$
$128$	0	0
$129$	5290.93	0.317946
$130$	0	0
$131$	2698.97i	0.157274i	0.996903	+	0.0786368i	$0.0250567\pi$
$131$	2698.97i	0.157274i	−0.996903	+	0.0786368i	$0.974943\pi$
$132$	0	0
$133$	−13843.2	−0.782586
$134$	0	0
$135$	− 3299.53i	− 0.181044i
$136$	0	0
$137$	5527.88	0.294522	0.147261	−	0.989098i	$-0.452954\pi$
$137$	5527.88	0.294522	0.147261	+	0.989098i	$0.452954\pi$
$138$	0	0
$139$	23618.9i	1.22245i	0.791458	+	0.611224i	$0.209322\pi$
$139$	23618.9i	1.22245i	−0.791458	+	0.611224i	$0.790678\pi$
$140$	0	0
$141$	6939.36	0.349045
$142$	0	0
$143$	− 70619.1i	− 3.45342i
$144$	0	0
$145$	−14624.3	−0.695568
$146$	0	0
$147$	− 4019.67i	− 0.186018i
$148$	0	0
$149$	−3606.61	−0.162453	−0.0812264	−	0.996696i	$-0.525884\pi$
$149$	−3606.61	−0.162453	−0.0812264	+	0.996696i	$0.525884\pi$
$150$	0	0
$151$	− 43619.5i	− 1.91305i	−0.291645	−	0.956527i	$-0.594203\pi$
$151$	− 43619.5i	− 1.91305i	0.291645	−	0.956527i	$-0.405797\pi$
$152$	0	0
$153$	10282.4	0.439251
$154$	0	0
$155$	35950.3i	1.49637i
$156$	0	0
$157$	−39000.2	−1.58222	−0.791111	−	0.611673i	$-0.790497\pi$
$157$	−39000.2	−1.58222	−0.791111	+	0.611673i	$0.790497\pi$
$158$	0	0
$159$	− 22099.4i	− 0.874153i
$160$	0	0
$161$	4563.26	0.176045
$162$	0	0
$163$	13933.2i	0.524415i	0.965012	+	0.262208i	$0.0844505\pi$
$163$	13933.2i	0.524415i	−0.965012	+	0.262208i	$0.915549\pi$
$164$	0	0
$165$	−27811.3	−1.02153
$166$	0	0
$167$	9886.95i	0.354511i	0.984165	+	0.177255i	$0.0567218\pi$
$167$	9886.95i	0.354511i	−0.984165	+	0.177255i	$0.943278\pi$
$168$	0	0
$169$	67728.4	2.37136
$170$	0	0
$171$	− 9265.10i	− 0.316853i
$172$	0	0
$173$	−26467.6	−0.884346	−0.442173	−	0.896930i	$-0.645792\pi$
$173$	−26467.6	−0.884346	−0.442173	+	0.896930i	$0.645792\pi$
$174$	0	0
$175$	− 2900.08i	− 0.0946964i
$176$	0	0
$177$	403.359	0.0128750
$178$	0	0
$179$	33711.5i	1.05213i	0.850443	+	0.526067i	$0.176334\pi$
$179$	33711.5i	1.05213i	−0.850443	+	0.526067i	$0.823666\pi$
$180$	0	0
$181$	−36239.3	−1.10617	−0.553087	−	0.833124i	$-0.686550\pi$
$181$	−36239.3	−1.10617	−0.553087	+	0.833124i	$0.686550\pi$
$182$	0	0
$183$	6000.23i	0.179170i
$184$	0	0
$185$	−32585.6	−0.952100
$186$	0	0
$187$	− 86669.2i	− 2.47846i
$188$	0	0
$189$	−5659.72	−0.158442
$190$	0	0
$191$	20931.9i	0.573774i	0.957964	+	0.286887i	$0.0926205\pi$
$191$	20931.9i	0.573774i	−0.957964	+	0.286887i	$0.907379\pi$
$192$	0	0
$193$	126.993	0.00340929	0.00170465	−	0.999999i	$-0.499457\pi$
$193$	126.993	0.00340929	0.00170465	+	0.999999i	$0.499457\pi$
$194$	0	0
$195$	− 37920.8i	− 0.997259i
$196$	0	0
$197$	8881.31	0.228847	0.114423	−	0.993432i	$-0.463498\pi$
$197$	8881.31	0.228847	0.114423	+	0.993432i	$0.463498\pi$
$198$	0	0
$199$	− 76018.6i	− 1.91961i	−0.280664	−	0.959806i	$-0.590555\pi$
$199$	− 76018.6i	− 1.91961i	0.280664	−	0.959806i	$-0.409445\pi$
$200$	0	0
$201$	18071.4	0.447301
$202$	0	0
$203$	25085.3i	0.608732i
$204$	0	0
$205$	65049.8	1.54788
$206$	0	0
$207$	3054.14i	0.0712769i
$208$	0	0
$209$	−78094.3	−1.78783
$210$	0	0
$211$	54603.0i	1.22645i	0.789907	+	0.613227i	$0.210129\pi$
$211$	54603.0i	1.22645i	−0.789907	+	0.613227i	$0.789871\pi$
$212$	0	0
$213$	−10682.9	−0.235467
$214$	0	0
$215$	− 23947.3i	− 0.518060i
$216$	0	0
$217$	61665.9	1.30956
$218$	0	0
$219$	38199.5i	0.796469i
$220$	0	0
$221$	118174.	2.41956
$222$	0	0
$223$	− 80149.6i	− 1.61173i	−0.592102	−	0.805863i	$-0.701702\pi$
$223$	− 80149.6i	− 1.61173i	0.592102	−	0.805863i	$-0.298298\pi$
$224$	0	0
$225$	1941.00	0.0383406
$226$	0	0
$227$	− 86867.2i	− 1.68579i	−0.538077	−	0.842896i	$-0.680849\pi$
$227$	− 86867.2i	− 1.68579i	0.538077	−	0.842896i	$-0.319151\pi$
$228$	0	0
$229$	36536.4	0.696714	0.348357	−	0.937362i	$-0.386740\pi$
$229$	36536.4	0.696714	0.348357	+	0.937362i	$0.386740\pi$
$230$	0	0
$231$	47705.0i	0.894005i
$232$	0	0
$233$	45963.5	0.846646	0.423323	−	0.905979i	$-0.360864\pi$
$233$	45963.5	0.846646	0.423323	+	0.905979i	$0.360864\pi$
$234$	0	0
$235$	− 31408.3i	− 0.568733i
$236$	0	0
$237$	−17031.8	−0.303224
$238$	0	0
$239$	8705.16i	0.152399i	0.997093	+	0.0761993i	$0.0242785\pi$
$239$	8705.16i	0.152399i	−0.997093	+	0.0761993i	$0.975721\pi$
$240$	0	0
$241$	−4184.78	−0.0720508	−0.0360254	−	0.999351i	$-0.511470\pi$
$241$	−4184.78	−0.0720508	−0.0360254	+	0.999351i	$0.511470\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	−18193.4	−0.303098
$246$	0	0
$247$	− 106482.i	− 1.74535i
$248$	0	0
$249$	−40258.6	−0.649322
$250$	0	0
$251$	23956.2i	0.380251i	0.981760	+	0.190125i	$0.0608895\pi$
$251$	23956.2i	0.380251i	−0.981760	+	0.190125i	$0.939111\pi$
$252$	0	0
$253$	25743.0	0.402177
$254$	0	0
$255$	− 46539.3i	− 0.715714i
$256$	0	0
$257$	−32912.7	−0.498307	−0.249154	−	0.968464i	$-0.580152\pi$
$257$	−32912.7	−0.498307	−0.249154	+	0.968464i	$0.580152\pi$
$258$	0	0
$259$	55894.5i	0.833239i
$260$	0	0
$261$	−16789.3	−0.246463
$262$	0	0
$263$	− 28615.0i	− 0.413696i	−0.978373	−	0.206848i	$-0.933679\pi$
$263$	− 28615.0i	− 0.413696i	0.978373	−	0.206848i	$-0.0663206\pi$
$264$	0	0
$265$	−100024.	−1.42434
$266$	0	0
$267$	− 24295.2i	− 0.340799i
$268$	0	0
$269$	105926.	1.46385	0.731927	−	0.681383i	$-0.238621\pi$
$269$	105926.	1.46385	0.731927	+	0.681383i	$0.238621\pi$
$270$	0	0
$271$	105722.i	1.43955i	0.694206	+	0.719776i	$0.255756\pi$
$271$	105722.i	1.43955i	−0.694206	+	0.719776i	$0.744244\pi$
$272$	0	0
$273$	−65045.9	−0.872760
$274$	0	0
$275$	− 16360.4i	− 0.216336i
$276$	0	0
$277$	−132880.	−1.73181	−0.865903	−	0.500212i	$-0.833256\pi$
$277$	−132880.	−1.73181	−0.865903	+	0.500212i	$0.833256\pi$
$278$	0	0
$279$	41272.4i	0.530214i
$280$	0	0
$281$	75828.4	0.960327	0.480163	−	0.877179i	$-0.340577\pi$
$281$	75828.4	0.960327	0.480163	+	0.877179i	$0.340577\pi$
$282$	0	0
$283$	− 82382.2i	− 1.02863i	−0.857600	−	0.514317i	$-0.828046\pi$
$283$	− 82382.2i	− 1.02863i	0.857600	−	0.514317i	$-0.171954\pi$
$284$	0	0
$285$	−41934.8	−0.516279
$286$	0	0
$287$	− 111581.i	− 1.35464i
$288$	0	0
$289$	61510.9	0.736473
$290$	0	0
$291$	− 2122.65i	− 0.0250665i
$292$	0	0
$293$	50845.8	0.592270	0.296135	−	0.955146i	$-0.404302\pi$
$293$	50845.8	0.592270	0.296135	+	0.955146i	$0.404302\pi$
$294$	0	0
$295$	− 1825.65i	− 0.0209784i
$296$	0	0
$297$	−31928.5	−0.361964
$298$	0	0
$299$	35100.6i	0.392620i
$300$	0	0
$301$	−41077.1	−0.453384
$302$	0	0
$303$	− 35232.4i	− 0.383758i
$304$	0	0
$305$	27157.6	0.291939
$306$	0	0
$307$	− 22007.7i	− 0.233505i	−0.993161	−	0.116753i	$-0.962752\pi$
$307$	− 22007.7i	− 0.233505i	0.993161	−	0.116753i	$-0.0372485\pi$
$308$	0	0
$309$	−8806.03	−0.0922281
$310$	0	0
$311$	161735.i	1.67218i	0.548589	+	0.836092i	$0.315165\pi$
$311$	161735.i	1.67218i	−0.548589	+	0.836092i	$0.684835\pi$
$312$	0	0
$313$	131522.	1.34249	0.671245	−	0.741236i	$-0.265760\pi$
$313$	131522.	1.34249	0.671245	+	0.741236i	$0.265760\pi$
$314$	0	0
$315$	25616.5i	0.258165i
$316$	0	0
$317$	−65811.6	−0.654913	−0.327456	−	0.944866i	$-0.606191\pi$
$317$	−65811.6	−0.654913	−0.327456	+	0.944866i	$0.606191\pi$
$318$	0	0
$319$	141515.i	1.39066i
$320$	0	0
$321$	−58478.4	−0.567526
$322$	0	0
$323$	− 130683.i	− 1.25260i
$324$	0	0
$325$	22307.4	0.211195
$326$	0	0
$327$	17903.3i	0.167432i
$328$	0	0
$329$	−53874.9	−0.497731
$330$	0	0
$331$	6965.35i	0.0635751i	0.999495	+	0.0317875i	$0.0101200\pi$
$331$	6965.35i	0.0635751i	−0.999495	+	0.0317875i	$0.989880\pi$
$332$	0	0
$333$	−37409.7	−0.337361
$334$	0	0
$335$	− 81793.1i	− 0.728831i
$336$	0	0
$337$	−95799.4	−0.843535	−0.421767	−	0.906704i	$-0.638590\pi$
$337$	−95799.4	−0.843535	−0.421767	+	0.906704i	$0.638590\pi$
$338$	0	0
$339$	− 45000.4i	− 0.391577i
$340$	0	0
$341$	347879.	2.99171
$342$	0	0
$343$	128067.i	1.08855i
$344$	0	0
$345$	13823.4	0.116138
$346$	0	0
$347$	− 65514.1i	− 0.544096i	−0.962284	−	0.272048i	$-0.912299\pi$
$347$	− 65514.1i	− 0.544096i	0.962284	−	0.272048i	$-0.0877010\pi$
$348$	0	0
$349$	−124174.	−1.01948	−0.509740	−	0.860329i	$-0.670258\pi$
$349$	−124174.	−1.01948	−0.509740	+	0.860329i	$0.670258\pi$
$350$	0	0
$351$	− 43534.6i	− 0.353363i
$352$	0	0
$353$	144384.	1.15870	0.579350	−	0.815079i	$-0.303306\pi$
$353$	144384.	1.15870	0.579350	+	0.815079i	$0.303306\pi$
$354$	0	0
$355$	48351.9i	0.383670i
$356$	0	0
$357$	−79829.4	−0.626363
$358$	0	0
$359$	− 125622.i	− 0.974715i	−0.873203	−	0.487357i	$-0.837961\pi$
$359$	− 125622.i	− 0.974715i	0.873203	−	0.487357i	$-0.162039\pi$
$360$	0	0
$361$	12567.9	0.0964377
$362$	0	0
$363$	193044.i	1.46502i
$364$	0	0
$365$	172895.	1.29776
$366$	0	0
$367$	132797.i	0.985950i	0.870044	+	0.492975i	$0.164091\pi$
$367$	132797.i	0.985950i	−0.870044	+	0.492975i	$0.835909\pi$
$368$	0	0
$369$	74679.9	0.548468
$370$	0	0
$371$	171573.i	1.24652i
$372$	0	0
$373$	−83521.7	−0.600319	−0.300159	−	0.953889i	$-0.597040\pi$
$373$	−83521.7	−0.600319	−0.300159	+	0.953889i	$0.597040\pi$
$374$	0	0
$375$	− 85163.1i	− 0.605604i
$376$	0	0
$377$	−192956.	−1.35761
$378$	0	0
$379$	85671.5i	0.596428i	0.954499	+	0.298214i	$0.0963909\pi$
$379$	85671.5i	0.596428i	−0.954499	+	0.298214i	$0.903609\pi$
$380$	0	0
$381$	83830.2	0.577498
$382$	0	0
$383$	270564.i	1.84448i	0.386621	+	0.922239i	$0.373642\pi$
$383$	270564.i	1.84448i	−0.386621	+	0.922239i	$0.626358\pi$
$384$	0	0
$385$	215918.	1.45669
$386$	0	0
$387$	− 27492.5i	− 0.183566i
$388$	0	0
$389$	28603.7	0.189027	0.0945133	−	0.995524i	$-0.469870\pi$
$389$	28603.7	0.189027	0.0945133	+	0.995524i	$0.469870\pi$
$390$	0	0
$391$	43078.2i	0.281776i
$392$	0	0
$393$	14024.3	0.0908019
$394$	0	0
$395$	77087.6i	0.494072i
$396$	0	0
$397$	−215256.	−1.36576	−0.682880	−	0.730530i	$-0.739273\pi$
$397$	−215256.	−1.36576	−0.682880	+	0.730530i	$0.739273\pi$
$398$	0	0
$399$	71931.2i	0.451826i
$400$	0	0
$401$	79995.2	0.497479	0.248740	−	0.968570i	$-0.419984\pi$
$401$	79995.2	0.497479	0.248740	+	0.968570i	$0.419984\pi$
$402$	0	0
$403$	474335.i	2.92062i
$404$	0	0
$405$	−17144.9	−0.104526
$406$	0	0
$407$	315321.i	1.90355i
$408$	0	0
$409$	−205058.	−1.22583	−0.612915	−	0.790149i	$-0.710003\pi$
$409$	−205058.	−1.22583	−0.612915	+	0.790149i	$0.710003\pi$
$410$	0	0
$411$	− 28723.7i	− 0.170042i
$412$	0	0
$413$	−3131.55	−0.0183594
$414$	0	0
$415$	182215.i	1.05800i
$416$	0	0
$417$	122727.	0.705780
$418$	0	0
$419$	− 56983.1i	− 0.324577i	−0.986743	−	0.162289i	$-0.948112\pi$
$419$	− 56983.1i	− 0.324577i	0.986743	−	0.162289i	$-0.0518876\pi$
$420$	0	0
$421$	243587.	1.37432	0.687162	−	0.726504i	$-0.258856\pi$
$421$	243587.	1.37432	0.687162	+	0.726504i	$0.258856\pi$
$422$	0	0
$423$	− 36058.0i	− 0.201521i
$424$	0	0
$425$	27377.4	0.151570
$426$	0	0
$427$	− 46583.8i	− 0.255493i
$428$	0	0
$429$	−366947.	−1.99384
$430$	0	0
$431$	− 67838.5i	− 0.365192i	−0.983188	−	0.182596i	$-0.941550\pi$
$431$	− 67838.5i	− 0.365192i	0.983188	−	0.182596i	$-0.0584501\pi$
$432$	0	0
$433$	103864.	0.553974	0.276987	−	0.960874i	$-0.410664\pi$
$433$	103864.	0.553974	0.276987	+	0.960874i	$0.410664\pi$
$434$	0	0
$435$	75990.2i	0.401586i
$436$	0	0
$437$	38816.1	0.203259
$438$	0	0
$439$	124230.i	0.644612i	0.946636	+	0.322306i	$0.104458\pi$
$439$	124230.i	0.644612i	−0.946636	+	0.322306i	$0.895542\pi$
$440$	0	0
$441$	−20886.8	−0.107398
$442$	0	0
$443$	− 313124.i	− 1.59554i	−0.602960	−	0.797772i	$-0.706012\pi$
$443$	− 313124.i	− 1.59554i	0.602960	−	0.797772i	$-0.293988\pi$
$444$	0	0
$445$	−109963.	−0.555297
$446$	0	0
$447$	18740.5i	0.0937922i
$448$	0	0
$449$	−220308.	−1.09279	−0.546395	−	0.837528i	$-0.684000\pi$
$449$	−220308.	−1.09279	−0.546395	+	0.837528i	$0.684000\pi$
$450$	0	0
$451$	− 629467.i	− 3.09471i
$452$	0	0
$453$	−226654.	−1.10450
$454$	0	0
$455$	294405.i	1.42207i
$456$	0	0
$457$	−143104.	−0.685202	−0.342601	−	0.939481i	$-0.611308\pi$
$457$	−143104.	−0.685202	−0.342601	+	0.939481i	$0.611308\pi$
$458$	0	0
$459$	− 53429.1i	− 0.253602i
$460$	0	0
$461$	82138.4	0.386496	0.193248	−	0.981150i	$-0.438098\pi$
$461$	82138.4	0.386496	0.193248	+	0.981150i	$0.438098\pi$
$462$	0	0
$463$	102147.i	0.476501i	0.971204	+	0.238251i	$0.0765740\pi$
$463$	102147.i	0.476501i	−0.971204	+	0.238251i	$0.923426\pi$
$464$	0	0
$465$	186803.	0.863929
$466$	0	0
$467$	− 8057.50i	− 0.0369459i	−0.999829	−	0.0184730i	$-0.994120\pi$
$467$	− 8057.50i	− 0.0369459i	0.999829	−	0.0184730i	$-0.00588046\pi$
$468$	0	0
$469$	−140301.	−0.637843
$470$	0	0
$471$	202651.i	0.913496i
$472$	0	0
$473$	−231731.	−1.03576
$474$	0	0
$475$	− 24668.7i	− 0.109335i
$476$	0	0
$477$	−114832.	−0.504692
$478$	0	0
$479$	112208.i	0.489051i	0.969643	+	0.244525i	$0.0786321\pi$
$479$	112208.i	0.489051i	−0.969643	+	0.244525i	$0.921368\pi$
$480$	0	0
$481$	−429941.	−1.85831
$482$	0	0
$483$	− 23711.4i	− 0.101639i
$484$	0	0
$485$	−9607.35	−0.0408432
$486$	0	0
$487$	− 220829.i	− 0.931102i	−0.885021	−	0.465551i	$-0.845856\pi$
$487$	− 220829.i	− 0.931102i	0.885021	−	0.465551i	$-0.154144\pi$
$488$	0	0
$489$	72399.0	0.302771
$490$	0	0
$491$	181363.i	0.752290i	0.926561	+	0.376145i	$0.122751\pi$
$491$	181363.i	0.752290i	−0.926561	+	0.376145i	$0.877249\pi$
$492$	0	0
$493$	−236811.	−0.974333
$494$	0	0
$495$	144512.i	0.589783i
$496$	0	0
$497$	82938.6	0.335772
$498$	0	0
$499$	1211.21i	0.00486429i	0.999997	+	0.00243214i	$0.000774176\pi$
$499$	1211.21i	0.00486429i	−0.999997	+	0.00243214i	$0.999226\pi$
$500$	0	0
$501$	51374.1	0.204677
$502$	0	0
$503$	− 316513.i	− 1.25099i	−0.780227	−	0.625497i	$-0.784896\pi$
$503$	− 316513.i	− 1.25099i	0.780227	−	0.625497i	$-0.215104\pi$
$504$	0	0
$505$	−159466.	−0.625294
$506$	0	0
$507$	− 351927.i	− 1.36910i
$508$	0	0
$509$	60847.8	0.234860	0.117430	−	0.993081i	$-0.462534\pi$
$509$	60847.8	0.234860	0.117430	+	0.993081i	$0.462534\pi$
$510$	0	0
$511$	− 296568.i	− 1.13575i
$512$	0	0
$513$	−48142.9	−0.182935
$514$	0	0
$515$	39857.0i	0.150276i
$516$	0	0
$517$	−303928.	−1.13708
$518$	0	0
$519$	137530.i	0.510577i
$520$	0	0
$521$	224829.	0.828278	0.414139	−	0.910214i	$-0.364083\pi$
$521$	224829.	0.828278	0.414139	+	0.910214i	$0.364083\pi$
$522$	0	0
$523$	− 93536.3i	− 0.341961i	−0.985274	−	0.170981i	$-0.945306\pi$
$523$	− 93536.3i	− 0.341961i	0.985274	−	0.170981i	$-0.0546936\pi$
$524$	0	0
$525$	−15069.3	−0.0546730
$526$	0	0
$527$	582140.i	2.09607i
$528$	0	0
$529$	267046.	0.954276
$530$	0	0
$531$	− 2095.92i	− 0.00743336i
$532$	0	0
$533$	858280.	3.02117
$534$	0	0
$535$	264679.i	0.924724i
$536$	0	0
$537$	175170.	0.607450
$538$	0	0
$539$	176052.i	0.605988i
$540$	0	0
$541$	23161.3	0.0791351	0.0395675	−	0.999217i	$-0.487402\pi$
$541$	23161.3	0.0791351	0.0395675	+	0.999217i	$0.487402\pi$
$542$	0	0
$543$	188305.i	0.638649i
$544$	0	0
$545$	81032.1	0.272813
$546$	0	0
$547$	− 355721.i	− 1.18887i	−0.804143	−	0.594436i	$-0.797375\pi$
$547$	− 355721.i	− 1.18887i	0.804143	−	0.594436i	$-0.202625\pi$
$548$	0	0
$549$	31178.1	0.103444
$550$	0	0
$551$	213381.i	0.702834i
$552$	0	0
$553$	132229.	0.432391
$554$	0	0
$555$	169320.i	0.549695i
$556$	0	0
$557$	350706.	1.13040	0.565200	−	0.824954i	$-0.308799\pi$
$557$	350706.	1.13040	0.565200	+	0.824954i	$0.308799\pi$
$558$	0	0
$559$	− 315966.i	− 1.01115i
$560$	0	0
$561$	−450346.	−1.43094
$562$	0	0
$563$	220322.i	0.695089i	0.937663	+	0.347545i	$0.112984\pi$
$563$	220322.i	0.695089i	−0.937663	+	0.347545i	$0.887016\pi$
$564$	0	0
$565$	−203676.	−0.638034
$566$	0	0
$567$	29408.8i	0.0914767i
$568$	0	0
$569$	540989.	1.67095	0.835476	−	0.549526i	$-0.185192\pi$
$569$	540989.	1.67095	0.835476	+	0.549526i	$0.185192\pi$
$570$	0	0
$571$	479447.i	1.47051i	0.677790	+	0.735255i	$0.262938\pi$
$571$	479447.i	1.47051i	−0.677790	+	0.735255i	$0.737062\pi$
$572$	0	0
$573$	108765.	0.331269
$574$	0	0
$575$	8131.80i	0.0245952i
$576$	0	0
$577$	−299723.	−0.900261	−0.450131	−	0.892963i	$-0.648623\pi$
$577$	−299723.	−0.900261	−0.450131	+	0.892963i	$0.648623\pi$
$578$	0	0
$579$	− 659.874i	− 0.00196836i
$580$	0	0
$581$	312555.	0.925920
$582$	0	0
$583$	967904.i	2.84771i
$584$	0	0
$585$	−197042.	−0.575768
$586$	0	0
$587$	205529.i	0.596480i	0.954491	+	0.298240i	$0.0963996\pi$
$587$	205529.i	0.596480i	−0.954491	+	0.298240i	$0.903600\pi$
$588$	0	0
$589$	524544.	1.51200
$590$	0	0
$591$	− 46148.7i	− 0.132125i
$592$	0	0
$593$	566721.	1.61161	0.805805	−	0.592181i	$-0.201733\pi$
$593$	566721.	1.61161	0.805805	+	0.592181i	$0.201733\pi$
$594$	0	0
$595$	361316.i	1.02059i
$596$	0	0
$597$	−395004.	−1.10829
$598$	0	0
$599$	− 224984.i	− 0.627044i	−0.949581	−	0.313522i	$-0.898491\pi$
$599$	− 224984.i	− 0.627044i	0.949581	−	0.313522i	$-0.101509\pi$
$600$	0	0
$601$	−50655.9	−0.140243	−0.0701215	−	0.997538i	$-0.522339\pi$
$601$	−50655.9	−0.140243	−0.0701215	+	0.997538i	$0.522339\pi$
$602$	0	0
$603$	− 93901.8i	− 0.258249i
$604$	0	0
$605$	873738.	2.38710
$606$	0	0
$607$	− 225688.i	− 0.612534i	−0.951946	−	0.306267i	$-0.900920\pi$
$607$	− 225688.i	− 0.612534i	0.951946	−	0.306267i	$-0.0990801\pi$
$608$	0	0
$609$	130347.	0.351452
$610$	0	0
$611$	− 414407.i	− 1.11005i
$612$	0	0
$613$	−200659.	−0.533994	−0.266997	−	0.963697i	$-0.586031\pi$
$613$	−200659.	−0.533994	−0.266997	+	0.963697i	$0.586031\pi$
$614$	0	0
$615$	− 338009.i	− 0.893672i
$616$	0	0
$617$	−14910.4	−0.0391669	−0.0195835	−	0.999808i	$-0.506234\pi$
$617$	−14910.4	−0.0391669	−0.0195835	+	0.999808i	$0.506234\pi$
$618$	0	0
$619$	− 160175.i	− 0.418037i	−0.977912	−	0.209018i	$-0.932973\pi$
$619$	− 160175.i	− 0.418037i	0.977912	−	0.209018i	$-0.0670268\pi$
$620$	0	0
$621$	15869.8	0.0411517
$622$	0	0
$623$	188620.i	0.485973i
$624$	0	0
$625$	−340527.	−0.871748
$626$	0	0
$627$	405790.i	1.03220i
$628$	0	0
$629$	−527657.	−1.33368
$630$	0	0
$631$	− 238216.i	− 0.598291i	−0.954208	−	0.299145i	$-0.903298\pi$
$631$	− 238216.i	− 0.598291i	0.954208	−	0.299145i	$-0.0967016\pi$
$632$	0	0
$633$	283725.	0.708094
$634$	0	0
$635$	− 379424.i	− 0.940973i
$636$	0	0
$637$	−240048.	−0.591587
$638$	0	0
$639$	55510.0i	0.135947i
$640$	0	0
$641$	−100397.	−0.244346	−0.122173	−	0.992509i	$-0.538986\pi$
$641$	−100397.	−0.244346	−0.122173	+	0.992509i	$0.538986\pi$
$642$	0	0
$643$	− 265567.i	− 0.642321i	−0.947025	−	0.321160i	$-0.895927\pi$
$643$	− 265567.i	− 0.642321i	0.947025	−	0.321160i	$-0.104073\pi$
$644$	0	0
$645$	−124434.	−0.299102
$646$	0	0
$647$	− 210025.i	− 0.501721i	−0.968023	−	0.250860i	$-0.919287\pi$
$647$	− 210025.i	− 0.501721i	0.968023	−	0.250860i	$-0.0807135\pi$
$648$	0	0
$649$	−17666.2	−0.0419424
$650$	0	0
$651$	− 320425.i	− 0.756075i
$652$	0	0
$653$	44076.3	0.103366	0.0516831	−	0.998664i	$-0.483541\pi$
$653$	44076.3	0.103366	0.0516831	+	0.998664i	$0.483541\pi$
$654$	0	0
$655$	− 63475.3i	− 0.147952i
$656$	0	0
$657$	198490.	0.459842
$658$	0	0
$659$	706672.i	1.62722i	0.581409	+	0.813611i	$0.302502\pi$
$659$	706672.i	1.62722i	−0.581409	+	0.813611i	$0.697498\pi$
$660$	0	0
$661$	−506522.	−1.15930	−0.579649	−	0.814866i	$-0.696810\pi$
$661$	−506522.	−1.15930	−0.579649	+	0.814866i	$0.696810\pi$
$662$	0	0
$663$	− 614049.i	− 1.39693i
$664$	0	0
$665$	325568.	0.736204
$666$	0	0
$667$	− 70338.8i	− 0.158104i
$668$	0	0
$669$	−416469.	−0.930531
$670$	0	0
$671$	− 262796.i	− 0.583678i
$672$	0	0
$673$	−72766.9	−0.160659	−0.0803293	−	0.996768i	$-0.525597\pi$
$673$	−72766.9	−0.160659	−0.0803293	+	0.996768i	$0.525597\pi$
$674$	0	0
$675$	− 10085.7i	− 0.0221360i
$676$	0	0
$677$	−95915.9	−0.209273	−0.104637	−	0.994511i	$-0.533368\pi$
$677$	−95915.9	−0.209273	−0.104637	+	0.994511i	$0.533368\pi$
$678$	0	0
$679$	16479.6i	0.0357443i
$680$	0	0
$681$	−451375.	−0.973292
$682$	0	0
$683$	83040.2i	0.178011i	0.996031	+	0.0890055i	$0.0283689\pi$
$683$	83040.2i	0.178011i	−0.996031	+	0.0890055i	$0.971631\pi$
$684$	0	0
$685$	−130006.	−0.277066
$686$	0	0
$687$	− 189849.i	− 0.402248i
$688$	0	0
$689$	−1.31974e6	−2.78004
$690$	0	0
$691$	− 337325.i	− 0.706469i	−0.935535	−	0.353234i	$-0.885082\pi$
$691$	− 337325.i	− 0.706469i	0.935535	−	0.353234i	$-0.114918\pi$
$692$	0	0
$693$	247882.	0.516154
$694$	0	0
$695$	− 555477.i	− 1.15000i
$696$	0	0
$697$	1.05335e6	2.16823
$698$	0	0
$699$	− 238834.i	− 0.488811i
$700$	0	0
$701$	442010.	0.899490	0.449745	−	0.893157i	$-0.351515\pi$
$701$	442010.	0.899490	0.449745	+	0.893157i	$0.351515\pi$
$702$	0	0
$703$	475451.i	0.962045i
$704$	0	0
$705$	−163202.	−0.328358
$706$	0	0
$707$	273533.i	0.547231i
$708$	0	0
$709$	380072.	0.756089	0.378045	−	0.925787i	$-0.376597\pi$
$709$	380072.	0.756089	0.378045	+	0.925787i	$0.376597\pi$
$710$	0	0
$711$	88499.7i	0.175066i
$712$	0	0
$713$	−172911.	−0.340128
$714$	0	0
$715$	1.66084e6i	3.24875i
$716$	0	0
$717$	45233.3	0.0879874
$718$	0	0
$719$	− 564351.i	− 1.09167i	−0.837893	−	0.545835i	$-0.816213\pi$
$719$	− 564351.i	− 1.09167i	0.837893	−	0.545835i	$-0.183787\pi$
$720$	0	0
$721$	68367.1	0.131515
$722$	0	0
$723$	21744.8i	0.0415986i
$724$	0	0
$725$	−44702.3	−0.0850460
$726$	0	0
$727$	− 25822.2i	− 0.0488568i	−0.999702	−	0.0244284i	$-0.992223\pi$
$727$	− 25822.2i	− 0.0488568i	0.999702	−	0.0244284i	$-0.00777658\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 387777.i	− 0.725684i
$732$	0	0
$733$	−191223.	−0.355904	−0.177952	−	0.984039i	$-0.556947\pi$
$733$	−191223.	−0.355904	−0.177952	+	0.984039i	$0.556947\pi$
$734$	0	0
$735$	94535.9i	0.174993i
$736$	0	0
$737$	−791486.	−1.45716
$738$	0	0
$739$	− 954938.i	− 1.74858i	−0.485400	−	0.874292i	$-0.661326\pi$
$739$	− 954938.i	− 1.74858i	0.485400	−	0.874292i	$-0.338674\pi$
$740$	0	0
$741$	−553296.	−1.00768
$742$	0	0
$743$	− 417501.i	− 0.756275i	−0.925749	−	0.378138i	$-0.876565\pi$
$743$	− 417501.i	− 0.756275i	0.925749	−	0.378138i	$-0.123435\pi$
$744$	0	0
$745$	84821.5	0.152825
$746$	0	0
$747$	209190.i	0.374886i
$748$	0	0
$749$	454007.	0.809280
$750$	0	0
$751$	− 776902.i	− 1.37748i	−0.725007	−	0.688742i	$-0.758163\pi$
$751$	− 776902.i	− 1.37748i	0.725007	−	0.688742i	$-0.241837\pi$
$752$	0	0
$753$	124480.	0.219538
$754$	0	0
$755$	1.02586e6i	1.79967i
$756$	0	0
$757$	522866.	0.912429	0.456214	−	0.889870i	$-0.349205\pi$
$757$	522866.	0.912429	0.456214	+	0.889870i	$0.349205\pi$
$758$	0	0
$759$	− 133764.i	− 0.232197i
$760$	0	0
$761$	−929896.	−1.60570	−0.802851	−	0.596180i	$-0.796684\pi$
$761$	−929896.	−1.60570	−0.802851	+	0.596180i	$0.796684\pi$
$762$	0	0
$763$	− 138995.i	− 0.238754i
$764$	0	0
$765$	−241825.	−0.413218
$766$	0	0
$767$	− 24087.9i	− 0.0409457i
$768$	0	0
$769$	890030.	1.50505	0.752527	−	0.658562i	$-0.228835\pi$
$769$	890030.	1.50505	0.752527	+	0.658562i	$0.228835\pi$
$770$	0	0
$771$	171019.i	0.287698i
$772$	0	0
$773$	774804.	1.29668	0.648340	−	0.761351i	$-0.275464\pi$
$773$	774804.	1.29668	0.648340	+	0.761351i	$0.275464\pi$
$774$	0	0
$775$	109890.i	0.182959i
$776$	0	0
$777$	290436.	0.481071
$778$	0	0
$779$	− 949131.i	− 1.56405i
$780$	0	0
$781$	467887.	0.767076
$782$	0	0
$783$	87239.9i	0.142296i
$784$	0	0
$785$	917219.	1.48845
$786$	0	0
$787$	797643.i	1.28783i	0.765096	+	0.643916i	$0.222691\pi$
$787$	797643.i	1.28783i	−0.765096	+	0.643916i	$0.777309\pi$
$788$	0	0
$789$	−148688.	−0.238848
$790$	0	0
$791$	349368.i	0.558381i
$792$	0	0
$793$	358323.	0.569808
$794$	0	0
$795$	519742.i	0.822344i
$796$	0	0
$797$	−51158.3	−0.0805378	−0.0402689	−	0.999189i	$-0.512821\pi$
$797$	−51158.3	−0.0805378	−0.0402689	+	0.999189i	$0.512821\pi$
$798$	0	0
$799$	− 508592.i	− 0.796666i
$800$	0	0
$801$	−126242.	−0.196760
$802$	0	0
$803$	− 1.67305e6i	− 2.59464i
$804$	0	0
$805$	−107320.	−0.165611
$806$	0	0
$807$	− 550408.i	− 0.845157i
$808$	0	0
$809$	97436.9	0.148877	0.0744383	−	0.997226i	$-0.476284\pi$
$809$	97436.9	0.148877	0.0744383	+	0.997226i	$0.476284\pi$
$810$	0	0
$811$	− 52742.9i	− 0.0801904i	−0.999196	−	0.0400952i	$-0.987234\pi$
$811$	− 52742.9i	− 0.0801904i	0.999196	−	0.0400952i	$-0.0127661\pi$
$812$	0	0
$813$	549349.	0.831126
$814$	0	0
$815$	− 327685.i	− 0.493334i
$816$	0	0
$817$	−349411.	−0.523471
$818$	0	0
$819$	337989.i	0.503888i
$820$	0	0
$821$	−435598.	−0.646248	−0.323124	−	0.946357i	$-0.604733\pi$
$821$	−435598.	−0.646248	−0.323124	+	0.946357i	$0.604733\pi$
$822$	0	0
$823$	164224.i	0.242459i	0.992625	+	0.121229i	$0.0386837\pi$
$823$	164224.i	0.242459i	−0.992625	+	0.121229i	$0.961316\pi$
$824$	0	0
$825$	−85011.1	−0.124901
$826$	0	0
$827$	− 652007.i	− 0.953326i	−0.879086	−	0.476663i	$-0.841846\pi$
$827$	− 652007.i	− 0.953326i	0.879086	−	0.476663i	$-0.158154\pi$
$828$	0	0
$829$	−112061.	−0.163060	−0.0815299	−	0.996671i	$-0.525981\pi$
$829$	−112061.	−0.163060	−0.0815299	+	0.996671i	$0.525981\pi$
$830$	0	0
$831$	690464.i	0.999859i
$832$	0	0
$833$	−294605.	−0.424571
$834$	0	0
$835$	− 232524.i	− 0.333500i
$836$	0	0
$837$	214458.	0.306119
$838$	0	0
$839$	− 890284.i	− 1.26475i	−0.774663	−	0.632375i	$-0.782080\pi$
$839$	− 890284.i	− 1.26475i	0.774663	−	0.632375i	$-0.217920\pi$
$840$	0	0
$841$	−320612.	−0.453303
$842$	0	0
$843$	− 394016.i	− 0.554445i
$844$	0	0
$845$	−1.59286e6	−2.23081
$846$	0	0
$847$	− 1.49873e6i	− 2.08909i
$848$	0	0
$849$	−428071.	−0.593882
$850$	0	0
$851$	− 156728.i	− 0.216415i
$852$	0	0
$853$	−850265.	−1.16857	−0.584287	−	0.811547i	$-0.698626\pi$
$853$	−850265.	−1.16857	−0.584287	+	0.811547i	$0.698626\pi$
$854$	0	0
$855$	217900.i	0.298074i
$856$	0	0
$857$	−1.00134e6	−1.36339	−0.681694	−	0.731638i	$-0.738756\pi$
$857$	−1.00134e6	−1.36339	−0.681694	+	0.731638i	$0.738756\pi$
$858$	0	0
$859$	− 1.14307e6i	− 1.54913i	−0.632495	−	0.774564i	$-0.717969\pi$
$859$	− 1.14307e6i	− 1.54913i	0.632495	−	0.774564i	$-0.282031\pi$
$860$	0	0
$861$	−579790.	−0.782104
$862$	0	0
$863$	− 1.01524e6i	− 1.36317i	−0.731741	−	0.681583i	$-0.761292\pi$
$863$	− 1.01524e6i	− 1.36317i	0.731741	−	0.681583i	$-0.238708\pi$
$864$	0	0
$865$	622473.	0.831933
$866$	0	0
$867$	− 319620.i	− 0.425203i
$868$	0	0
$869$	745952.	0.987806
$870$	0	0
$871$	− 1.07919e6i	− 1.42254i
$872$	0	0
$873$	−11029.6	−0.0144721
$874$	0	0
$875$	661179.i	0.863580i
$876$	0	0
$877$	−1.03625e6	−1.34730	−0.673652	−	0.739049i	$-0.735275\pi$
$877$	−1.03625e6	−1.34730	−0.673652	+	0.739049i	$0.735275\pi$
$878$	0	0
$879$	− 264202.i	− 0.341947i
$880$	0	0
$881$	1.33421e6	1.71899	0.859494	−	0.511146i	$-0.170779\pi$
$881$	1.33421e6	1.71899	0.859494	+	0.511146i	$0.170779\pi$
$882$	0	0
$883$	− 922293.i	− 1.18290i	−0.806342	−	0.591449i	$-0.798556\pi$
$883$	− 922293.i	− 1.18290i	0.806342	−	0.591449i	$-0.201444\pi$
$884$	0	0
$885$	−9486.33	−0.0121119
$886$	0	0
$887$	− 870179.i	− 1.10602i	−0.833176	−	0.553008i	$-0.813480\pi$
$887$	− 870179.i	− 1.10602i	0.833176	−	0.553008i	$-0.186520\pi$
$888$	0	0
$889$	−650830.	−0.823501
$890$	0	0
$891$	165905.i	0.208980i
$892$	0	0
$893$	−458273.	−0.574673
$894$	0	0
$895$	− 792837.i	− 0.989778i
$896$	0	0
$897$	182388.	0.226679
$898$	0	0
$899$	− 950529.i	− 1.17610i
$900$	0	0
$901$	−1.61969e6	−1.99518
$902$	0	0
$903$	213443.i	0.261762i
$904$	0	0
$905$	852288.	1.04061
$906$	0	0
$907$	1.42374e6i	1.73068i	0.501184	+	0.865341i	$0.332898\pi$
$907$	1.42374e6i	1.73068i	−0.501184	+	0.865341i	$0.667102\pi$
$908$	0	0
$909$	−183073.	−0.221563
$910$	0	0
$911$	784773.i	0.945600i	0.881170	+	0.472800i	$0.156757\pi$
$911$	784773.i	0.945600i	−0.881170	+	0.472800i	$0.843243\pi$
$912$	0	0
$913$	1.76323e6	2.11528
$914$	0	0
$915$	− 141115.i	− 0.168551i
$916$	0	0
$917$	−108880.	−0.129482
$918$	0	0
$919$	− 252716.i	− 0.299227i	−0.988745	−	0.149614i	$-0.952197\pi$
$919$	− 252716.i	− 0.299227i	0.988745	−	0.149614i	$-0.0478030\pi$
$920$	0	0
$921$	−114355.	−0.134814
$922$	0	0
$923$	637965.i	0.748848i
$924$	0	0
$925$	−99604.9	−0.116412
$926$	0	0
$927$	45757.5i	0.0532479i
$928$	0	0
$929$	1.07539e6	1.24605	0.623025	−	0.782202i	$-0.285903\pi$
$929$	1.07539e6	1.24605	0.623025	+	0.782202i	$0.285903\pi$
$930$	0	0
$931$	265457.i	0.306264i
$932$	0	0
$933$	840402.	0.965436
$934$	0	0
$935$	2.03831e6i	2.33157i
$936$	0	0
$937$	−503408.	−0.573378	−0.286689	−	0.958024i	$-0.592555\pi$
$937$	−503408.	−0.573378	−0.286689	+	0.958024i	$0.592555\pi$
$938$	0	0
$939$	− 683410.i	− 0.775087i
$940$	0	0
$941$	−85259.4	−0.0962860	−0.0481430	−	0.998840i	$-0.515330\pi$
$941$	−85259.4	−0.0962860	−0.0481430	+	0.998840i	$0.515330\pi$
$942$	0	0
$943$	312871.i	0.351838i
$944$	0	0
$945$	133107.	0.149052
$946$	0	0
$947$	874018.i	0.974587i	0.873238	+	0.487293i	$0.162016\pi$
$947$	874018.i	0.974587i	−0.873238	+	0.487293i	$0.837984\pi$
$948$	0	0
$949$	2.28121e6	2.53298
$950$	0	0
$951$	341967.i	0.378114i
$952$	0	0
$953$	−653547.	−0.719600	−0.359800	−	0.933029i	$-0.617155\pi$
$953$	−653547.	−0.719600	−0.359800	+	0.933029i	$0.617155\pi$
$954$	0	0
$955$	− 492282.i	− 0.539768i
$956$	0	0
$957$	735333.	0.802898
$958$	0	0
$959$	223002.i	0.242477i
$960$	0	0
$961$	−1.41312e6	−1.53014
$962$	0	0
$963$	303863.i	0.327661i
$964$	0	0
$965$	−2986.66	−0.00320723
$966$	0	0
$967$	− 1.57343e6i	− 1.68265i	−0.540527	−	0.841326i	$-0.681775\pi$
$967$	− 1.57343e6i	− 1.68265i	0.540527	−	0.841326i	$-0.318225\pi$
$968$	0	0
$969$	−679047.	−0.723190
$970$	0	0
$971$	280331.i	0.297326i	0.988888	+	0.148663i	$0.0474970\pi$
$971$	280331.i	0.297326i	−0.988888	+	0.148663i	$0.952503\pi$
$972$	0	0
$973$	−952816.	−1.00643
$974$	0	0
$975$	− 115913.i	− 0.121933i
$976$	0	0
$977$	−1.19063e6	−1.24735	−0.623673	−	0.781685i	$-0.714360\pi$
$977$	−1.19063e6	−1.24735	−0.623673	+	0.781685i	$0.714360\pi$
$978$	0	0
$979$	1.06407e6i	1.11021i
$980$	0	0
$981$	93028.2	0.0966667
$982$	0	0
$983$	− 1.13386e6i	− 1.17341i	−0.809800	−	0.586706i	$-0.800424\pi$
$983$	− 1.13386e6i	− 1.17341i	0.809800	−	0.586706i	$-0.199576\pi$
$984$	0	0
$985$	−208874.	−0.215284
$986$	0	0
$987$	279942.i	0.287365i
$988$	0	0
$989$	115180.	0.117756
$990$	0	0
$991$	− 535087.i	− 0.544850i	−0.962177	−	0.272425i	$-0.912174\pi$
$991$	− 535087.i	− 0.544850i	0.962177	−	0.272425i	$-0.0878256\pi$
$992$	0	0
$993$	36193.0	0.0367051
$994$	0	0
$995$	1.78783e6i	1.80584i
$996$	0	0
$997$	1.02178e6	1.02794	0.513970	−	0.857808i	$-0.328174\pi$
$997$	1.02178e6	1.02794	0.513970	+	0.857808i	$0.328174\pi$
$998$	0	0
$999$	194386.i	0.194776i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.g.h.511.2		8
4.3	odd	2	inner	768.5.g.h.511.6		8
8.3	odd	2		768.5.g.j.511.3		8
8.5	even	2		768.5.g.j.511.7		8
16.3	odd	4		384.5.b.d.319.7	yes	16
16.5	even	4		384.5.b.d.319.2	✓	16
16.11	odd	4		384.5.b.d.319.10	yes	16
16.13	even	4		384.5.b.d.319.15	yes	16
48.5	odd	4		1152.5.b.m.703.13		16
48.11	even	4		1152.5.b.m.703.14		16
48.29	odd	4		1152.5.b.m.703.3		16
48.35	even	4		1152.5.b.m.703.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.d.319.2	✓	16	16.5	even	4
384.5.b.d.319.7	yes	16	16.3	odd	4
384.5.b.d.319.10	yes	16	16.11	odd	4
384.5.b.d.319.15	yes	16	16.13	even	4
768.5.g.h.511.2		8	1.1	even	1	trivial
768.5.g.h.511.6		8	4.3	odd	2	inner
768.5.g.j.511.3		8	8.3	odd	2
768.5.g.j.511.7		8	8.5	even	2
1152.5.b.m.703.3		16	48.29	odd	4
1152.5.b.m.703.4		16	48.35	even	4
1152.5.b.m.703.13		16	48.5	odd	4
1152.5.b.m.703.14		16	48.11	even	4

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)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -23.5183 q^{5} +40.3412i q^{7} -27.0000 q^{9} +227.579i q^{11} -310.305 q^{13} +122.205i q^{15} -380.831 q^{17} +343.152i q^{19} +209.619 q^{21} -113.116i q^{23} -71.8887 q^{25} +140.296i q^{27} +621.827 q^{29} -1528.61i q^{31} +1182.54 q^{33} -948.758i q^{35} +1385.54 q^{37} +1612.39i q^{39} -2765.92 q^{41} +1018.24i q^{43} +634.995 q^{45} +1335.48i q^{47} +773.586 q^{49} +1978.85i q^{51} +4253.04 q^{53} -5352.28i q^{55} +1783.07 q^{57} +77.6265i q^{59} -1154.74 q^{61} -1089.21i q^{63} +7297.86 q^{65} +3477.85i q^{67} -587.770 q^{69} -2055.93i q^{71} -7351.49 q^{73} +373.545i q^{75} -9180.83 q^{77} -3277.77i q^{79} +729.000 q^{81} -7747.77i q^{83} +8956.49 q^{85} -3231.11i q^{87} +4675.62 q^{89} -12518.1i q^{91} -7942.87 q^{93} -8070.35i q^{95} +408.505 q^{97} -6144.64i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 96 q^{5} - 216 q^{9} + 160 q^{13} + 240 q^{17} - 72 q^{25} + 3360 q^{29} + 1440 q^{33} + 7072 q^{37} + 1776 q^{41} + 2592 q^{45} - 10040 q^{49} + 12960 q^{53} + 3744 q^{57} + 12064 q^{61} + 11520 q^{65}+ \cdots + 2544 q^{97}+O(q^{100}) \) 8 * q - 96 * q^5 - 216 * q^9 + 160 * q^13 + 240 * q^17 - 72 * q^25 + 3360 * q^29 + 1440 * q^33 + 7072 * q^37 + 1776 * q^41 + 2592 * q^45 - 10040 * q^49 + 12960 * q^53 + 3744 * q^57 + 12064 * q^61 + 11520 * q^65 + 1152 * q^69 - 17200 * q^73 + 17664 * q^77 + 5832 * q^81 - 8768 * q^85 + 7824 * q^89 - 8064 * q^93 + 2544 * q^97

Introduction

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Modular forms

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