LMFDB - Embedded newform 192.5.b.a.31.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,5,Mod(31,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("192.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	192.b (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:	$19.8470329121$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.31
Dual form			192.5.b.a.31.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -24.2487i q^{5} -58.0000i q^{7} +27.0000 q^{9} -13.8564 q^{11} +20.7846i q^{13} -126.000i q^{15} -306.000 q^{17} -602.754 q^{19} -301.377i q^{21} +468.000i q^{23} +37.0000 q^{25} +140.296 q^{27} -1465.31i q^{29} -110.000i q^{31} -72.0000 q^{33} -1406.43 q^{35} -1039.23i q^{37} +108.000i q^{39} +2970.00 q^{41} -2889.06 q^{43} -654.715i q^{45} -396.000i q^{47} -963.000 q^{49} -1590.02 q^{51} -1125.83i q^{53} +336.000i q^{55} -3132.00 q^{57} -2681.21 q^{59} -5985.97i q^{61} -1566.00i q^{63} +504.000 q^{65} +4801.24 q^{67} +2431.80i q^{69} +6588.00i q^{71} -5894.00 q^{73} +192.258 q^{75} +803.672i q^{77} -8486.00i q^{79} +729.000 q^{81} +13.8564 q^{83} +7420.11i q^{85} -7614.00i q^{87} +8766.00 q^{89} +1205.51 q^{91} -571.577i q^{93} +14616.0i q^{95} +5918.00 q^{97} -374.123 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{9} - 1224 q^{17} + 148 q^{25} - 288 q^{33} + 11880 q^{41} - 3852 q^{49} - 12528 q^{57} + 2016 q^{65} - 23576 q^{73} + 2916 q^{81} + 35064 q^{89} + 23672 q^{97}+O(q^{100}) $ 4 * q + 108 * q^9 - 1224 * q^17 + 148 * q^25 - 288 * q^33 + 11880 * q^41 - 3852 * q^49 - 12528 * q^57 + 2016 * q^65 - 23576 * q^73 + 2916 * q^81 + 35064 * q^89 + 23672 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	− 24.2487i	− 0.969948i	−0.874528	−	0.484974i	$-0.838829\pi$
$5$	− 24.2487i	− 0.969948i	0.874528	−	0.484974i	$-0.161171\pi$
$6$	0	0
$7$	− 58.0000i	− 1.18367i	−0.806058	−	0.591837i	$-0.798403\pi$
$7$	− 58.0000i	− 1.18367i	0.806058	−	0.591837i	$-0.201597\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	−13.8564	−0.114516	−0.0572579	−	0.998359i	$-0.518236\pi$
$11$	−13.8564	−0.114516	−0.0572579	+	0.998359i	$0.518236\pi$
$12$	0	0
$13$	20.7846i	0.122986i	0.998108	+	0.0614929i	$0.0195862\pi$
$13$	20.7846i	0.122986i	−0.998108	+	0.0614929i	$0.980414\pi$
$14$	0	0
$15$	− 126.000i	− 0.560000i
$16$	0	0
$17$	−306.000	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$17$	−306.000	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$18$	0	0
$19$	−602.754	−1.66968	−0.834839	−	0.550494i	$-0.814439\pi$
$19$	−602.754	−1.66968	−0.834839	+	0.550494i	$0.814439\pi$
$20$	0	0
$21$	− 301.377i	− 0.683394i
$22$	0	0
$23$	468.000i	0.884688i	0.896845	+	0.442344i	$0.145853\pi$
$23$	468.000i	0.884688i	−0.896845	+	0.442344i	$0.854147\pi$
$24$	0	0
$25$	37.0000	0.0592000
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	− 1465.31i	− 1.74235i	−0.490974	−	0.871174i	$-0.663359\pi$
$29$	− 1465.31i	− 1.74235i	0.490974	−	0.871174i	$-0.336641\pi$
$30$	0	0
$31$	− 110.000i	− 0.114464i	−0.998361	−	0.0572320i	$-0.981773\pi$
$31$	− 110.000i	− 0.114464i	0.998361	−	0.0572320i	$-0.0182275\pi$
$32$	0	0
$33$	−72.0000	−0.0661157
$34$	0	0
$35$	−1406.43	−1.14810
$36$	0	0
$37$	− 1039.23i	− 0.759116i	−0.925168	−	0.379558i	$-0.876076\pi$
$37$	− 1039.23i	− 0.759116i	0.925168	−	0.379558i	$-0.123924\pi$
$38$	0	0
$39$	108.000i	0.0710059i
$40$	0	0
$41$	2970.00	1.76681	0.883403	−	0.468615i	$-0.155247\pi$
$41$	2970.00	1.76681	0.883403	+	0.468615i	$0.155247\pi$
$42$	0	0
$43$	−2889.06	−1.56250	−0.781250	−	0.624219i	$-0.785417\pi$
$43$	−2889.06	−1.56250	−0.781250	+	0.624219i	$0.785417\pi$
$44$	0	0
$45$	− 654.715i	− 0.323316i
$46$	0	0
$47$	− 396.000i	− 0.179267i	−0.995975	−	0.0896333i	$-0.971430\pi$
$47$	− 396.000i	− 0.179267i	0.995975	−	0.0896333i	$-0.0285695\pi$
$48$	0	0
$49$	−963.000	−0.401083
$50$	0	0
$51$	−1590.02	−0.611312
$52$	0	0
$53$	− 1125.83i	− 0.400795i	−0.979715	−	0.200397i	$-0.935777\pi$
$53$	− 1125.83i	− 0.400795i	0.979715	−	0.200397i	$-0.0642234\pi$
$54$	0	0
$55$	336.000i	0.111074i
$56$	0	0
$57$	−3132.00	−0.963989
$58$	0	0
$59$	−2681.21	−0.770243	−0.385121	−	0.922866i	$-0.625840\pi$
$59$	−2681.21	−0.770243	−0.385121	+	0.922866i	$0.625840\pi$
$60$	0	0
$61$	− 5985.97i	− 1.60870i	−0.594157	−	0.804349i	$-0.702514\pi$
$61$	− 5985.97i	− 1.60870i	0.594157	−	0.804349i	$-0.297486\pi$
$62$	0	0
$63$	− 1566.00i	− 0.394558i
$64$	0	0
$65$	504.000	0.119290
$66$	0	0
$67$	4801.24	1.06956	0.534779	−	0.844992i	$-0.320395\pi$
$67$	4801.24	1.06956	0.534779	+	0.844992i	$0.320395\pi$
$68$	0	0
$69$	2431.80i	0.510775i
$70$	0	0
$71$	6588.00i	1.30688i	0.756977	+	0.653442i	$0.226676\pi$
$71$	6588.00i	1.30688i	−0.756977	+	0.653442i	$0.773324\pi$
$72$	0	0
$73$	−5894.00	−1.10602	−0.553012	−	0.833173i	$-0.686522\pi$
$73$	−5894.00	−1.10602	−0.553012	+	0.833173i	$0.686522\pi$
$74$	0	0
$75$	192.258	0.0341791
$76$	0	0
$77$	803.672i	0.135549i
$78$	0	0
$79$	− 8486.00i	− 1.35972i	−0.733343	−	0.679859i	$-0.762041\pi$
$79$	− 8486.00i	− 1.35972i	0.733343	−	0.679859i	$-0.237959\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	13.8564	0.00201138	0.00100569	−	0.999999i	$-0.499680\pi$
$83$	13.8564	0.00201138	0.00100569	+	0.999999i	$0.499680\pi$
$84$	0	0
$85$	7420.11i	1.02700i
$86$	0	0
$87$	− 7614.00i	− 1.00595i
$88$	0	0
$89$	8766.00	1.10668	0.553339	−	0.832956i	$-0.313353\pi$
$89$	8766.00	1.10668	0.553339	+	0.832956i	$0.313353\pi$
$90$	0	0
$91$	1205.51	0.145575
$92$	0	0
$93$	− 571.577i	− 0.0660859i
$94$	0	0
$95$	14616.0i	1.61950i
$96$	0	0
$97$	5918.00	0.628972	0.314486	−	0.949262i	$-0.398168\pi$
$97$	5918.00	0.628972	0.314486	+	0.949262i	$0.398168\pi$
$98$	0	0
$99$	−374.123	−0.0381719
$100$	0	0
$101$	16416.4i	1.60929i	0.593756	+	0.804646i	$0.297645\pi$
$101$	16416.4i	1.60929i	−0.593756	+	0.804646i	$0.702355\pi$
$102$	0	0
$103$	− 5342.00i	− 0.503535i	−0.967788	−	0.251767i	$-0.918988\pi$
$103$	− 5342.00i	− 0.503535i	0.967788	−	0.251767i	$-0.0810118\pi$
$104$	0	0
$105$	−7308.00	−0.662857
$106$	0	0
$107$	21733.8	1.89831	0.949156	−	0.314806i	$-0.101940\pi$
$107$	21733.8	1.89831	0.949156	+	0.314806i	$0.101940\pi$
$108$	0	0
$109$	8126.78i	0.684015i	0.939697	+	0.342008i	$0.111107\pi$
$109$	8126.78i	0.684015i	−0.939697	+	0.342008i	$0.888893\pi$
$110$	0	0
$111$	− 5400.00i	− 0.438276i
$112$	0	0
$113$	15354.0	1.20244	0.601222	−	0.799082i	$-0.294681\pi$
$113$	15354.0	1.20244	0.601222	+	0.799082i	$0.294681\pi$
$114$	0	0
$115$	11348.4	0.858102
$116$	0	0
$117$	561.184i	0.0409953i
$118$	0	0
$119$	17748.0i	1.25330i
$120$	0	0
$121$	−14449.0	−0.986886
$122$	0	0
$123$	15432.6	1.02007
$124$	0	0
$125$	− 16052.6i	− 1.02737i
$126$	0	0
$127$	9998.00i	0.619877i	0.950757	+	0.309939i	$0.100309\pi$
$127$	9998.00i	0.619877i	−0.950757	+	0.309939i	$0.899691\pi$
$128$	0	0
$129$	−15012.0	−0.902109
$130$	0	0
$131$	1877.54	0.109408	0.0547038	−	0.998503i	$-0.482579\pi$
$131$	1877.54	0.109408	0.0547038	+	0.998503i	$0.482579\pi$
$132$	0	0
$133$	34959.7i	1.97635i
$134$	0	0
$135$	− 3402.00i	− 0.186667i
$136$	0	0
$137$	27882.0	1.48553	0.742767	−	0.669550i	$-0.233513\pi$
$137$	27882.0	1.48553	0.742767	+	0.669550i	$0.233513\pi$
$138$	0	0
$139$	−11327.6	−0.586285	−0.293142	−	0.956069i	$-0.594701\pi$
$139$	−11327.6	−0.586285	−0.293142	+	0.956069i	$0.594701\pi$
$140$	0	0
$141$	− 2057.68i	− 0.103500i
$142$	0	0
$143$	− 288.000i	− 0.0140838i
$144$	0	0
$145$	−35532.0	−1.68999
$146$	0	0
$147$	−5003.89	−0.231565
$148$	0	0
$149$	33134.1i	1.49246i	0.665688	+	0.746231i	$0.268138\pi$
$149$	33134.1i	1.49246i	−0.665688	+	0.746231i	$0.731862\pi$
$150$	0	0
$151$	− 38794.0i	− 1.70142i	−0.525638	−	0.850708i	$-0.676173\pi$
$151$	− 38794.0i	− 1.70142i	0.525638	−	0.850708i	$-0.323827\pi$
$152$	0	0
$153$	−8262.00	−0.352941
$154$	0	0
$155$	−2667.36	−0.111024
$156$	0	0
$157$	− 35957.4i	− 1.45878i	−0.684100	−	0.729388i	$-0.739805\pi$
$157$	− 35957.4i	− 1.45878i	0.684100	−	0.729388i	$-0.260195\pi$
$158$	0	0
$159$	− 5850.00i	− 0.231399i
$160$	0	0
$161$	27144.0	1.04718
$162$	0	0
$163$	3013.77	0.113432	0.0567159	−	0.998390i	$-0.481937\pi$
$163$	3013.77	0.113432	0.0567159	+	0.998390i	$0.481937\pi$
$164$	0	0
$165$	1745.91i	0.0641288i
$166$	0	0
$167$	11304.0i	0.405321i	0.979249	+	0.202661i	$0.0649588\pi$
$167$	11304.0i	0.405321i	−0.979249	+	0.202661i	$0.935041\pi$
$168$	0	0
$169$	28129.0	0.984874
$170$	0	0
$171$	−16274.3	−0.556559
$172$	0	0
$173$	− 35309.6i	− 1.17978i	−0.807484	−	0.589889i	$-0.799171\pi$
$173$	− 35309.6i	− 1.17978i	0.807484	−	0.589889i	$-0.200829\pi$
$174$	0	0
$175$	− 2146.00i	− 0.0700735i
$176$	0	0
$177$	−13932.0	−0.444700
$178$	0	0
$179$	−6699.57	−0.209094	−0.104547	−	0.994520i	$-0.533339\pi$
$179$	−6699.57	−0.209094	−0.104547	+	0.994520i	$0.533339\pi$
$180$	0	0
$181$	− 14445.3i	− 0.440930i	−0.975395	−	0.220465i	$-0.929243\pi$
$181$	− 14445.3i	− 0.440930i	0.975395	−	0.220465i	$-0.0707575\pi$
$182$	0	0
$183$	− 31104.0i	− 0.928783i
$184$	0	0
$185$	−25200.0	−0.736304
$186$	0	0
$187$	4240.06	0.121252
$188$	0	0
$189$	− 8137.17i	− 0.227798i
$190$	0	0
$191$	35064.0i	0.961158i	0.876951	+	0.480579i	$0.159573\pi$
$191$	35064.0i	0.961158i	−0.876951	+	0.480579i	$0.840427\pi$
$192$	0	0
$193$	−11230.0	−0.301485	−0.150742	−	0.988573i	$-0.548166\pi$
$193$	−11230.0	−0.301485	−0.150742	+	0.988573i	$0.548166\pi$
$194$	0	0
$195$	2618.86	0.0688721
$196$	0	0
$197$	− 28991.1i	− 0.747019i	−0.927626	−	0.373510i	$-0.878154\pi$
$197$	− 28991.1i	− 0.747019i	0.927626	−	0.373510i	$-0.121846\pi$
$198$	0	0
$199$	18226.0i	0.460241i	0.973162	+	0.230120i	$0.0739120\pi$
$199$	18226.0i	0.460241i	−0.973162	+	0.230120i	$0.926088\pi$
$200$	0	0
$201$	24948.0	0.617509
$202$	0	0
$203$	−84988.3	−2.06237
$204$	0	0
$205$	− 72018.7i	− 1.71371i
$206$	0	0
$207$	12636.0i	0.294896i
$208$	0	0
$209$	8352.00	0.191204
$210$	0	0
$211$	−37266.8	−0.837061	−0.418531	−	0.908203i	$-0.637455\pi$
$211$	−37266.8	−0.837061	−0.418531	+	0.908203i	$0.637455\pi$
$212$	0	0
$213$	34232.3i	0.754530i
$214$	0	0
$215$	70056.0i	1.51554i
$216$	0	0
$217$	−6380.00	−0.135488
$218$	0	0
$219$	−30626.1	−0.638563
$220$	0	0
$221$	− 6360.09i	− 0.130220i
$222$	0	0
$223$	− 10162.0i	− 0.204348i	−0.994767	−	0.102174i	$-0.967420\pi$
$223$	− 10162.0i	− 0.204348i	0.994767	−	0.102174i	$-0.0325798\pi$
$224$	0	0
$225$	999.000	0.0197333
$226$	0	0
$227$	−15214.3	−0.295258	−0.147629	−	0.989043i	$-0.547164\pi$
$227$	−15214.3	−0.295258	−0.147629	+	0.989043i	$0.547164\pi$
$228$	0	0
$229$	7711.09i	0.147043i	0.997294	+	0.0735216i	$0.0234238\pi$
$229$	7711.09i	0.147043i	−0.997294	+	0.0735216i	$0.976576\pi$
$230$	0	0
$231$	4176.00i	0.0782594i
$232$	0	0
$233$	−21258.0	−0.391571	−0.195786	−	0.980647i	$-0.562726\pi$
$233$	−21258.0	−0.391571	−0.195786	+	0.980647i	$0.562726\pi$
$234$	0	0
$235$	−9602.49	−0.173879
$236$	0	0
$237$	− 44094.5i	− 0.785034i
$238$	0	0
$239$	− 97056.0i	− 1.69913i	−0.527484	−	0.849565i	$-0.676865\pi$
$239$	− 97056.0i	− 1.69913i	0.527484	−	0.849565i	$-0.323135\pi$
$240$	0	0
$241$	47242.0	0.813381	0.406691	−	0.913566i	$-0.366683\pi$
$241$	47242.0	0.813381	0.406691	+	0.913566i	$0.366683\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	23351.5i	0.389030i
$246$	0	0
$247$	− 12528.0i	− 0.205347i
$248$	0	0
$249$	72.0000	0.00116127
$250$	0	0
$251$	78191.7	1.24112	0.620559	−	0.784160i	$-0.286906\pi$
$251$	78191.7	1.24112	0.620559	+	0.784160i	$0.286906\pi$
$252$	0	0
$253$	− 6484.80i	− 0.101311i
$254$	0	0
$255$	38556.0i	0.592941i
$256$	0	0
$257$	23922.0	0.362186	0.181093	−	0.983466i	$-0.442037\pi$
$257$	23922.0	0.362186	0.181093	+	0.983466i	$0.442037\pi$
$258$	0	0
$259$	−60275.4	−0.898546
$260$	0	0
$261$	− 39563.5i	− 0.580783i
$262$	0	0
$263$	− 84528.0i	− 1.22205i	−0.791611	−	0.611025i	$-0.790757\pi$
$263$	− 84528.0i	− 1.22205i	0.791611	−	0.611025i	$-0.209243\pi$
$264$	0	0
$265$	−27300.0	−0.388750
$266$	0	0
$267$	45549.5	0.638941
$268$	0	0
$269$	− 92751.3i	− 1.28179i	−0.767630	−	0.640893i	$-0.778564\pi$
$269$	− 92751.3i	− 1.28179i	0.767630	−	0.640893i	$-0.221436\pi$
$270$	0	0
$271$	61118.0i	0.832205i	0.909318	+	0.416103i	$0.136604\pi$
$271$	61118.0i	0.832205i	−0.909318	+	0.416103i	$0.863396\pi$
$272$	0	0
$273$	6264.00	0.0840478
$274$	0	0
$275$	−512.687	−0.00677933
$276$	0	0
$277$	69441.4i	0.905021i	0.891759	+	0.452511i	$0.149472\pi$
$277$	69441.4i	0.905021i	−0.891759	+	0.452511i	$0.850528\pi$
$278$	0	0
$279$	− 2970.00i	− 0.0381547i
$280$	0	0
$281$	−60570.0	−0.767088	−0.383544	−	0.923523i	$-0.625296\pi$
$281$	−60570.0	−0.767088	−0.383544	+	0.923523i	$0.625296\pi$
$282$	0	0
$283$	87565.6	1.09335	0.546677	−	0.837344i	$-0.315893\pi$
$283$	87565.6	1.09335	0.546677	+	0.837344i	$0.315893\pi$
$284$	0	0
$285$	75947.0i	0.935020i
$286$	0	0
$287$	− 172260.i	− 2.09132i
$288$	0	0
$289$	10115.0	0.121107
$290$	0	0
$291$	30750.8	0.363137
$292$	0	0
$293$	77162.9i	0.898821i	0.893325	+	0.449410i	$0.148366\pi$
$293$	77162.9i	0.898821i	−0.893325	+	0.449410i	$0.851634\pi$
$294$	0	0
$295$	65016.0i	0.747096i
$296$	0	0
$297$	−1944.00	−0.0220386
$298$	0	0
$299$	−9727.20	−0.108804
$300$	0	0
$301$	167566.i	1.84949i
$302$	0	0
$303$	85302.0i	0.929125i
$304$	0	0
$305$	−145152.	−1.56035
$306$	0	0
$307$	−12657.8	−0.134302	−0.0671510	−	0.997743i	$-0.521391\pi$
$307$	−12657.8	−0.134302	−0.0671510	+	0.997743i	$0.521391\pi$
$308$	0	0
$309$	− 27757.8i	− 0.290716i
$310$	0	0
$311$	137592.i	1.42257i	0.702906	+	0.711283i	$0.251886\pi$
$311$	137592.i	1.42257i	−0.702906	+	0.711283i	$0.748114\pi$
$312$	0	0
$313$	13198.0	0.134716	0.0673580	−	0.997729i	$-0.478543\pi$
$313$	13198.0	0.134716	0.0673580	+	0.997729i	$0.478543\pi$
$314$	0	0
$315$	−37973.5	−0.382701
$316$	0	0
$317$	134106.i	1.33453i	0.744820	+	0.667266i	$0.232536\pi$
$317$	134106.i	1.33453i	−0.744820	+	0.667266i	$0.767464\pi$
$318$	0	0
$319$	20304.0i	0.199526i
$320$	0	0
$321$	112932.	1.09599
$322$	0	0
$323$	184443.	1.76789
$324$	0	0
$325$	769.031i	0.00728076i
$326$	0	0
$327$	42228.0i	0.394916i
$328$	0	0
$329$	−22968.0	−0.212193
$330$	0	0
$331$	52107.0	0.475598	0.237799	−	0.971314i	$-0.423574\pi$
$331$	52107.0	0.475598	0.237799	+	0.971314i	$0.423574\pi$
$332$	0	0
$333$	− 28059.2i	− 0.253039i
$334$	0	0
$335$	− 116424.i	− 1.03742i
$336$	0	0
$337$	84470.0	0.743777	0.371888	−	0.928277i	$-0.378710\pi$
$337$	84470.0	0.743777	0.371888	+	0.928277i	$0.378710\pi$
$338$	0	0
$339$	79781.7	0.694231
$340$	0	0
$341$	1524.20i	0.0131079i
$342$	0	0
$343$	− 83404.0i	− 0.708922i
$344$	0	0
$345$	58968.0	0.495425
$346$	0	0
$347$	57670.4	0.478954	0.239477	−	0.970902i	$-0.423024\pi$
$347$	57670.4	0.478954	0.239477	+	0.970902i	$0.423024\pi$
$348$	0	0
$349$	− 43315.1i	− 0.355622i	−0.984065	−	0.177811i	$-0.943098\pi$
$349$	− 43315.1i	− 0.355622i	0.984065	−	0.177811i	$-0.0569016\pi$
$350$	0	0
$351$	2916.00i	0.0236686i
$352$	0	0
$353$	−188118.	−1.50967	−0.754833	−	0.655917i	$-0.772282\pi$
$353$	−188118.	−1.50967	−0.754833	+	0.655917i	$0.772282\pi$
$354$	0	0
$355$	159751.	1.26761
$356$	0	0
$357$	92221.3i	0.723594i
$358$	0	0
$359$	− 14148.0i	− 0.109776i	−0.998493	−	0.0548878i	$-0.982520\pi$
$359$	− 14148.0i	− 0.109776i	0.998493	−	0.0548878i	$-0.0174801\pi$
$360$	0	0
$361$	232991.	1.78782
$362$	0	0
$363$	−75079.2	−0.569779
$364$	0	0
$365$	142922.i	1.07279i
$366$	0	0
$367$	265810.i	1.97351i	0.162220	+	0.986755i	$0.448135\pi$
$367$	265810.i	1.97351i	−0.162220	+	0.986755i	$0.551865\pi$
$368$	0	0
$369$	80190.0	0.588935
$370$	0	0
$371$	−65298.3	−0.474410
$372$	0	0
$373$	− 131774.i	− 0.947138i	−0.880757	−	0.473569i	$-0.842965\pi$
$373$	− 131774.i	− 0.947138i	0.880757	−	0.473569i	$-0.157035\pi$
$374$	0	0
$375$	− 83412.0i	− 0.593152i
$376$	0	0
$377$	30456.0	0.214284
$378$	0	0
$379$	91930.3	0.640001	0.320000	−	0.947417i	$-0.396317\pi$
$379$	91930.3	0.640001	0.320000	+	0.947417i	$0.396317\pi$
$380$	0	0
$381$	51951.1i	0.357886i
$382$	0	0
$383$	− 189000.i	− 1.28844i	−0.764840	−	0.644220i	$-0.777182\pi$
$383$	− 189000.i	− 1.28844i	0.764840	−	0.644220i	$-0.222818\pi$
$384$	0	0
$385$	19488.0	0.131476
$386$	0	0
$387$	−78004.6	−0.520833
$388$	0	0
$389$	− 180906.i	− 1.19551i	−0.801679	−	0.597755i	$-0.796060\pi$
$389$	− 180906.i	− 1.19551i	0.801679	−	0.597755i	$-0.203940\pi$
$390$	0	0
$391$	− 143208.i	− 0.936729i
$392$	0	0
$393$	9756.00	0.0631665
$394$	0	0
$395$	−205775.	−1.31886
$396$	0	0
$397$	140005.i	0.888307i	0.895951	+	0.444153i	$0.146495\pi$
$397$	140005.i	0.888307i	−0.895951	+	0.444153i	$0.853505\pi$
$398$	0	0
$399$	181656.i	1.14105i
$400$	0	0
$401$	−208674.	−1.29772	−0.648858	−	0.760910i	$-0.724753\pi$
$401$	−208674.	−1.29772	−0.648858	+	0.760910i	$0.724753\pi$
$402$	0	0
$403$	2286.31	0.0140775
$404$	0	0
$405$	− 17677.3i	− 0.107772i
$406$	0	0
$407$	14400.0i	0.0869308i
$408$	0	0
$409$	−194078.	−1.16019	−0.580096	−	0.814548i	$-0.696985\pi$
$409$	−194078.	−1.16019	−0.580096	+	0.814548i	$0.696985\pi$
$410$	0	0
$411$	144879.	0.857674
$412$	0	0
$413$	155510.i	0.911716i
$414$	0	0
$415$	− 336.000i	− 0.00195094i
$416$	0	0
$417$	−58860.0	−0.338492
$418$	0	0
$419$	−72829.3	−0.414837	−0.207419	−	0.978252i	$-0.566506\pi$
$419$	−72829.3	−0.414837	−0.207419	+	0.978252i	$0.566506\pi$
$420$	0	0
$421$	− 107893.i	− 0.608736i	−0.952555	−	0.304368i	$-0.901555\pi$
$421$	− 107893.i	− 0.608736i	0.952555	−	0.304368i	$-0.0984452\pi$
$422$	0	0
$423$	− 10692.0i	− 0.0597555i
$424$	0	0
$425$	−11322.0	−0.0626824
$426$	0	0
$427$	−347186.	−1.90417
$428$	0	0
$429$	− 1496.49i	− 0.00813130i
$430$	0	0
$431$	− 151380.i	− 0.814918i	−0.913223	−	0.407459i	$-0.866415\pi$
$431$	− 151380.i	− 0.814918i	0.913223	−	0.407459i	$-0.133585\pi$
$432$	0	0
$433$	−13922.0	−0.0742550	−0.0371275	−	0.999311i	$-0.511821\pi$
$433$	−13922.0	−0.0742550	−0.0371275	+	0.999311i	$0.511821\pi$
$434$	0	0
$435$	−184630.	−0.975715
$436$	0	0
$437$	− 282089.i	− 1.47714i
$438$	0	0
$439$	171130.i	0.887968i	0.896035	+	0.443984i	$0.146435\pi$
$439$	171130.i	0.887968i	−0.896035	+	0.443984i	$0.853565\pi$
$440$	0	0
$441$	−26001.0	−0.133694
$442$	0	0
$443$	226719.	1.15526	0.577630	−	0.816299i	$-0.303978\pi$
$443$	226719.	1.15526	0.577630	+	0.816299i	$0.303978\pi$
$444$	0	0
$445$	− 212564.i	− 1.07342i
$446$	0	0
$447$	172170.i	0.861673i
$448$	0	0
$449$	160830.	0.797764	0.398882	−	0.917002i	$-0.369398\pi$
$449$	160830.	0.797764	0.398882	+	0.917002i	$0.369398\pi$
$450$	0	0
$451$	−41153.5	−0.202327
$452$	0	0
$453$	− 201580.i	− 0.982313i
$454$	0	0
$455$	− 29232.0i	− 0.141200i
$456$	0	0
$457$	−146030.	−0.699213	−0.349607	−	0.936897i	$-0.613685\pi$
$457$	−146030.	−0.699213	−0.349607	+	0.936897i	$0.613685\pi$
$458$	0	0
$459$	−42930.6	−0.203771
$460$	0	0
$461$	− 99561.7i	− 0.468480i	−0.972179	−	0.234240i	$-0.924740\pi$
$461$	− 99561.7i	− 0.468480i	0.972179	−	0.234240i	$-0.0752601\pi$
$462$	0	0
$463$	− 47194.0i	− 0.220153i	−0.993923	−	0.110077i	$-0.964890\pi$
$463$	− 47194.0i	− 0.220153i	0.993923	−	0.110077i	$-0.0351096\pi$
$464$	0	0
$465$	−13860.0	−0.0640999
$466$	0	0
$467$	−279872.	−1.28329	−0.641646	−	0.767001i	$-0.721748\pi$
$467$	−279872.	−1.28329	−0.641646	+	0.767001i	$0.721748\pi$
$468$	0	0
$469$	− 278472.i	− 1.26601i
$470$	0	0
$471$	− 186840.i	− 0.842225i
$472$	0	0
$473$	40032.0	0.178931
$474$	0	0
$475$	−22301.9	−0.0988449
$476$	0	0
$477$	− 30397.5i	− 0.133598i
$478$	0	0
$479$	− 126828.i	− 0.552770i	−0.961047	−	0.276385i	$-0.910864\pi$
$479$	− 126828.i	− 0.552770i	0.961047	−	0.276385i	$-0.0891364\pi$
$480$	0	0
$481$	21600.0	0.0933606
$482$	0	0
$483$	141044.	0.604591
$484$	0	0
$485$	− 143504.i	− 0.610071i
$486$	0	0
$487$	− 177010.i	− 0.746345i	−0.927762	−	0.373173i	$-0.878270\pi$
$487$	− 177010.i	− 0.746345i	0.927762	−	0.373173i	$-0.121730\pi$
$488$	0	0
$489$	15660.0	0.0654899
$490$	0	0
$491$	−85667.2	−0.355346	−0.177673	−	0.984090i	$-0.556857\pi$
$491$	−85667.2	−0.355346	−0.177673	+	0.984090i	$0.556857\pi$
$492$	0	0
$493$	448386.i	1.84484i
$494$	0	0
$495$	9072.00i	0.0370248i
$496$	0	0
$497$	382104.	1.54692
$498$	0	0
$499$	−318566.	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$499$	−318566.	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$500$	0	0
$501$	58737.3i	0.234012i
$502$	0	0
$503$	379404.i	1.49957i	0.661683	+	0.749784i	$0.269842\pi$
$503$	379404.i	1.49957i	−0.661683	+	0.749784i	$0.730158\pi$
$504$	0	0
$505$	398076.	1.56093
$506$	0	0
$507$	146163.	0.568618
$508$	0	0
$509$	− 234973.i	− 0.906950i	−0.891269	−	0.453475i	$-0.850184\pi$
$509$	− 234973.i	− 0.906950i	0.891269	−	0.453475i	$-0.149816\pi$
$510$	0	0
$511$	341852.i	1.30917i
$512$	0	0
$513$	−84564.0	−0.321330
$514$	0	0
$515$	−129537.	−0.488403
$516$	0	0
$517$	5487.14i	0.0205289i
$518$	0	0
$519$	− 183474.i	− 0.681145i
$520$	0	0
$521$	−289422.	−1.06624	−0.533121	−	0.846039i	$-0.678981\pi$
$521$	−289422.	−1.06624	−0.533121	+	0.846039i	$0.678981\pi$
$522$	0	0
$523$	−382790.	−1.39945	−0.699725	−	0.714412i	$-0.746694\pi$
$523$	−382790.	−1.39945	−0.699725	+	0.714412i	$0.746694\pi$
$524$	0	0
$525$	− 11150.9i	− 0.0404569i
$526$	0	0
$527$	33660.0i	0.121197i
$528$	0	0
$529$	60817.0	0.217327
$530$	0	0
$531$	−72392.8	−0.256748
$532$	0	0
$533$	61730.3i	0.217292i
$534$	0	0
$535$	− 527016.i	− 1.84126i
$536$	0	0
$537$	−34812.0	−0.120720
$538$	0	0
$539$	13343.7	0.0459303
$540$	0	0
$541$	79043.9i	0.270068i	0.990841	+	0.135034i	$0.0431144\pi$
$541$	79043.9i	0.270068i	−0.990841	+	0.135034i	$0.956886\pi$
$542$	0	0
$543$	− 75060.0i	− 0.254571i
$544$	0	0
$545$	197064.	0.663459
$546$	0	0
$547$	305970.	1.02260	0.511299	−	0.859403i	$-0.329165\pi$
$547$	305970.	1.02260	0.511299	+	0.859403i	$0.329165\pi$
$548$	0	0
$549$	− 161621.i	− 0.536233i
$550$	0	0
$551$	883224.i	2.90916i
$552$	0	0
$553$	−492188.	−1.60946
$554$	0	0
$555$	−130943.	−0.425105
$556$	0	0
$557$	333714.i	1.07563i	0.843062	+	0.537817i	$0.180751\pi$
$557$	333714.i	1.07563i	−0.843062	+	0.537817i	$0.819249\pi$
$558$	0	0
$559$	− 60048.0i	− 0.192165i
$560$	0	0
$561$	22032.0	0.0700049
$562$	0	0
$563$	472088.	1.48938	0.744691	−	0.667410i	$-0.232597\pi$
$563$	472088.	1.48938	0.744691	+	0.667410i	$0.232597\pi$
$564$	0	0
$565$	− 372315.i	− 1.16631i
$566$	0	0
$567$	− 42282.0i	− 0.131519i
$568$	0	0
$569$	21258.0	0.0656595	0.0328298	−	0.999461i	$-0.489548\pi$
$569$	21258.0	0.0656595	0.0328298	+	0.999461i	$0.489548\pi$
$570$	0	0
$571$	490205.	1.50351	0.751754	−	0.659444i	$-0.229208\pi$
$571$	490205.	1.50351	0.751754	+	0.659444i	$0.229208\pi$
$572$	0	0
$573$	182198.i	0.554925i
$574$	0	0
$575$	17316.0i	0.0523735i
$576$	0	0
$577$	−454610.	−1.36549	−0.682743	−	0.730658i	$-0.739213\pi$
$577$	−454610.	−1.36549	−0.682743	+	0.730658i	$0.739213\pi$
$578$	0	0
$579$	−58352.8	−0.174062
$580$	0	0
$581$	− 803.672i	− 0.00238082i
$582$	0	0
$583$	15600.0i	0.0458973i
$584$	0	0
$585$	13608.0	0.0397633
$586$	0	0
$587$	35243.8	0.102284	0.0511418	−	0.998691i	$-0.483714\pi$
$587$	35243.8	0.102284	0.0511418	+	0.998691i	$0.483714\pi$
$588$	0	0
$589$	66302.9i	0.191118i
$590$	0	0
$591$	− 150642.i	− 0.431292i
$592$	0	0
$593$	197658.	0.562089	0.281044	−	0.959695i	$-0.409319\pi$
$593$	197658.	0.562089	0.281044	+	0.959695i	$0.409319\pi$
$594$	0	0
$595$	430366.	1.21564
$596$	0	0
$597$	94705.1i	0.265720i
$598$	0	0
$599$	204156.i	0.568995i	0.958677	+	0.284498i	$0.0918268\pi$
$599$	204156.i	0.568995i	−0.958677	+	0.284498i	$0.908173\pi$
$600$	0	0
$601$	294242.	0.814621	0.407311	−	0.913290i	$-0.366467\pi$
$601$	294242.	0.814621	0.407311	+	0.913290i	$0.366467\pi$
$602$	0	0
$603$	129634.	0.356519
$604$	0	0
$605$	350370.i	0.957229i
$606$	0	0
$607$	− 331762.i	− 0.900429i	−0.892921	−	0.450214i	$-0.851348\pi$
$607$	− 331762.i	− 0.900429i	0.892921	−	0.450214i	$-0.148652\pi$
$608$	0	0
$609$	−441612.	−1.19071
$610$	0	0
$611$	8230.71	0.0220473
$612$	0	0
$613$	448698.i	1.19408i	0.802212	+	0.597040i	$0.203657\pi$
$613$	448698.i	1.19408i	−0.802212	+	0.597040i	$0.796343\pi$
$614$	0	0
$615$	− 374220.i	− 0.989411i
$616$	0	0
$617$	358470.	0.941635	0.470817	−	0.882231i	$-0.343959\pi$
$617$	358470.	0.941635	0.470817	+	0.882231i	$0.343959\pi$
$618$	0	0
$619$	−321808.	−0.839877	−0.419939	−	0.907553i	$-0.637948\pi$
$619$	−321808.	−0.839877	−0.419939	+	0.907553i	$0.637948\pi$
$620$	0	0
$621$	65658.6i	0.170258i
$622$	0	0
$623$	− 508428.i	− 1.30995i
$624$	0	0
$625$	−366131.	−0.937295
$626$	0	0
$627$	43398.3	0.110392
$628$	0	0
$629$	318005.i	0.803770i
$630$	0	0
$631$	− 26002.0i	− 0.0653052i	−0.999467	−	0.0326526i	$-0.989605\pi$
$631$	− 26002.0i	− 0.0653052i	0.999467	−	0.0326526i	$-0.0103955\pi$
$632$	0	0
$633$	−193644.	−0.483278
$634$	0	0
$635$	242439.	0.601249
$636$	0	0
$637$	− 20015.6i	− 0.0493275i
$638$	0	0
$639$	177876.i	0.435628i
$640$	0	0
$641$	322758.	0.785527	0.392763	−	0.919640i	$-0.371519\pi$
$641$	322758.	0.785527	0.392763	+	0.919640i	$0.371519\pi$
$642$	0	0
$643$	596788.	1.44344	0.721720	−	0.692186i	$-0.243352\pi$
$643$	596788.	1.44344	0.721720	+	0.692186i	$0.243352\pi$
$644$	0	0
$645$	364022.i	0.874999i
$646$	0	0
$647$	345348.i	0.824989i	0.910960	+	0.412495i	$0.135342\pi$
$647$	345348.i	0.824989i	−0.910960	+	0.412495i	$0.864658\pi$
$648$	0	0
$649$	37152.0	0.0882049
$650$	0	0
$651$	−33151.5	−0.0782241
$652$	0	0
$653$	76095.9i	0.178458i	0.996011	+	0.0892288i	$0.0284402\pi$
$653$	76095.9i	0.178458i	−0.996011	+	0.0892288i	$0.971560\pi$
$654$	0	0
$655$	− 45528.0i	− 0.106120i
$656$	0	0
$657$	−159138.	−0.368675
$658$	0	0
$659$	−303712.	−0.699344	−0.349672	−	0.936872i	$-0.613707\pi$
$659$	−303712.	−0.699344	−0.349672	+	0.936872i	$0.613707\pi$
$660$	0	0
$661$	595978.i	1.36404i	0.731333	+	0.682020i	$0.238898\pi$
$661$	595978.i	1.36404i	−0.731333	+	0.682020i	$0.761102\pi$
$662$	0	0
$663$	− 33048.0i	− 0.0751827i
$664$	0	0
$665$	847728.	1.91696
$666$	0	0
$667$	685767.	1.54143
$668$	0	0
$669$	− 52803.3i	− 0.117980i
$670$	0	0
$671$	82944.0i	0.184221i
$672$	0	0
$673$	362542.	0.800439	0.400219	−	0.916419i	$-0.368934\pi$
$673$	362542.	0.800439	0.400219	+	0.916419i	$0.368934\pi$
$674$	0	0
$675$	5190.96	0.0113930
$676$	0	0
$677$	109026.i	0.237876i	0.992902	+	0.118938i	$0.0379490\pi$
$677$	109026.i	0.237876i	−0.992902	+	0.118938i	$0.962051\pi$
$678$	0	0
$679$	− 343244.i	− 0.744498i
$680$	0	0
$681$	−79056.0	−0.170467
$682$	0	0
$683$	−210382.	−0.450990	−0.225495	−	0.974244i	$-0.572400\pi$
$683$	−210382.	−0.450990	−0.225495	+	0.974244i	$0.572400\pi$
$684$	0	0
$685$	− 676103.i	− 1.44089i
$686$	0	0
$687$	40068.0i	0.0848954i
$688$	0	0
$689$	23400.0	0.0492921
$690$	0	0
$691$	−521673.	−1.09255	−0.546276	−	0.837605i	$-0.683955\pi$
$691$	−521673.	−1.09255	−0.546276	+	0.837605i	$0.683955\pi$
$692$	0	0
$693$	21699.1i	0.0451831i
$694$	0	0
$695$	274680.i	0.568666i
$696$	0	0
$697$	−908820.	−1.87074
$698$	0	0
$699$	−110460.	−0.226074
$700$	0	0
$701$	264813.i	0.538894i	0.963015	+	0.269447i	$0.0868410\pi$
$701$	264813.i	0.538894i	−0.963015	+	0.269447i	$0.913159\pi$
$702$	0	0
$703$	626400.i	1.26748i
$704$	0	0
$705$	−49896.0	−0.100389
$706$	0	0
$707$	952150.	1.90488
$708$	0	0
$709$	− 143767.i	− 0.286001i	−0.989723	−	0.143000i	$-0.954325\pi$
$709$	− 143767.i	− 0.286001i	0.989723	−	0.143000i	$-0.0456750\pi$
$710$	0	0
$711$	− 229122.i	− 0.453239i
$712$	0	0
$713$	51480.0	0.101265
$714$	0	0
$715$	−6983.63	−0.0136606
$716$	0	0
$717$	− 504318.i	− 0.980993i
$718$	0	0
$719$	283212.i	0.547840i	0.961752	+	0.273920i	$0.0883204\pi$
$719$	283212.i	0.547840i	−0.961752	+	0.273920i	$0.911680\pi$
$720$	0	0
$721$	−309836.	−0.596021
$722$	0	0
$723$	245477.	0.469606
$724$	0	0
$725$	− 54216.7i	− 0.103147i
$726$	0	0
$727$	225086.i	0.425873i	0.977066	+	0.212936i	$0.0683027\pi$
$727$	225086.i	0.425873i	−0.977066	+	0.212936i	$0.931697\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	884053.	1.65441
$732$	0	0
$733$	368158.i	0.685214i	0.939479	+	0.342607i	$0.111310\pi$
$733$	368158.i	0.685214i	−0.939479	+	0.342607i	$0.888690\pi$
$734$	0	0
$735$	121338.i	0.224606i
$736$	0	0
$737$	−66528.0	−0.122481
$738$	0	0
$739$	−1.06623e6	−1.95237	−0.976184	−	0.216942i	$-0.930392\pi$
$739$	−1.06623e6	−1.95237	−0.976184	+	0.216942i	$0.930392\pi$
$740$	0	0
$741$	− 65097.4i	− 0.118557i
$742$	0	0
$743$	903024.i	1.63577i	0.575383	+	0.817884i	$0.304853\pi$
$743$	903024.i	1.63577i	−0.575383	+	0.817884i	$0.695147\pi$
$744$	0	0
$745$	803460.	1.44761
$746$	0	0
$747$	374.123	0.000670460 0
$748$	0	0
$749$	− 1.26056e6i	− 2.24698i
$750$	0	0
$751$	− 765038.i	− 1.35645i	−0.734855	−	0.678224i	$-0.762750\pi$
$751$	− 765038.i	− 1.35645i	0.734855	−	0.678224i	$-0.237250\pi$
$752$	0	0
$753$	406296.	0.716560
$754$	0	0
$755$	−940705.	−1.65029
$756$	0	0
$757$	583944.i	1.01901i	0.860467	+	0.509506i	$0.170172\pi$
$757$	583944.i	1.01901i	−0.860467	+	0.509506i	$0.829828\pi$
$758$	0	0
$759$	− 33696.0i	− 0.0584918i
$760$	0	0
$761$	−20430.0	−0.0352776	−0.0176388	−	0.999844i	$-0.505615\pi$
$761$	−20430.0	−0.0352776	−0.0176388	+	0.999844i	$0.505615\pi$
$762$	0	0
$763$	471353.	0.809650
$764$	0	0
$765$	200343.i	0.342335i
$766$	0	0
$767$	− 55728.0i	− 0.0947290i
$768$	0	0
$769$	−172654.	−0.291960	−0.145980	−	0.989288i	$-0.546634\pi$
$769$	−172654.	−0.291960	−0.145980	+	0.989288i	$0.546634\pi$
$770$	0	0
$771$	124302.	0.209108
$772$	0	0
$773$	402525.i	0.673650i	0.941567	+	0.336825i	$0.109353\pi$
$773$	402525.i	0.673650i	−0.941567	+	0.336825i	$0.890647\pi$
$774$	0	0
$775$	− 4070.00i	− 0.00677627i
$776$	0	0
$777$	−313200.	−0.518776
$778$	0	0
$779$	−1.79018e6	−2.95000
$780$	0	0
$781$	− 91286.0i	− 0.149659i
$782$	0	0
$783$	− 205578.i	− 0.335315i
$784$	0	0
$785$	−871920.	−1.41494
$786$	0	0
$787$	−938986.	−1.51604	−0.758018	−	0.652233i	$-0.773832\pi$
$787$	−938986.	−1.51604	−0.758018	+	0.652233i	$0.773832\pi$
$788$	0	0
$789$	− 439220.i	− 0.705551i
$790$	0	0
$791$	− 890532.i	− 1.42330i
$792$	0	0
$793$	124416.	0.197847
$794$	0	0
$795$	−141855.	−0.224445
$796$	0	0
$797$	− 1.01547e6i	− 1.59864i	−0.600906	−	0.799320i	$-0.705193\pi$
$797$	− 1.01547e6i	− 1.59864i	0.600906	−	0.799320i	$-0.294807\pi$
$798$	0	0
$799$	121176.i	0.189812i
$800$	0	0
$801$	236682.	0.368893
$802$	0	0
$803$	81669.7	0.126657
$804$	0	0
$805$	− 658207.i	− 1.01571i
$806$	0	0
$807$	− 481950.i	− 0.740040i
$808$	0	0
$809$	−572742.	−0.875109	−0.437554	−	0.899192i	$-0.644155\pi$
$809$	−572742.	−0.875109	−0.437554	+	0.899192i	$0.644155\pi$
$810$	0	0
$811$	−503590.	−0.765659	−0.382830	−	0.923819i	$-0.625050\pi$
$811$	−503590.	−0.765659	−0.382830	+	0.923819i	$0.625050\pi$
$812$	0	0
$813$	317578.i	0.480474i
$814$	0	0
$815$	− 73080.0i	− 0.110023i
$816$	0	0
$817$	1.74139e6	2.60887
$818$	0	0
$819$	32548.7	0.0485250
$820$	0	0
$821$	345783.i	0.513000i	0.966544	+	0.256500i	$0.0825694\pi$
$821$	345783.i	0.513000i	−0.966544	+	0.256500i	$0.917431\pi$
$822$	0	0
$823$	− 739546.i	− 1.09186i	−0.837832	−	0.545928i	$-0.816177\pi$
$823$	− 739546.i	− 1.09186i	0.837832	−	0.545928i	$-0.183823\pi$
$824$	0	0
$825$	−2664.00	−0.00391405
$826$	0	0
$827$	551769.	0.806764	0.403382	−	0.915032i	$-0.367835\pi$
$827$	551769.	0.806764	0.403382	+	0.915032i	$0.367835\pi$
$828$	0	0
$829$	− 885944.i	− 1.28913i	−0.764549	−	0.644566i	$-0.777038\pi$
$829$	− 885944.i	− 1.28913i	0.764549	−	0.644566i	$-0.222962\pi$
$830$	0	0
$831$	360828.i	0.522514i
$832$	0	0
$833$	294678.	0.424676
$834$	0	0
$835$	274107.	0.393141
$836$	0	0
$837$	− 15432.6i	− 0.0220286i
$838$	0	0
$839$	− 418140.i	− 0.594016i	−0.954875	−	0.297008i	$-0.904011\pi$
$839$	− 418140.i	− 0.594016i	0.954875	−	0.297008i	$-0.0959887\pi$
$840$	0	0
$841$	−1.43987e6	−2.03578
$842$	0	0
$843$	−314731.	−0.442878
$844$	0	0
$845$	− 682092.i	− 0.955277i
$846$	0	0
$847$	838042.i	1.16815i
$848$	0	0
$849$	455004.	0.631248
$850$	0	0
$851$	486360.	0.671581
$852$	0	0
$853$	422634.i	0.580854i	0.956897	+	0.290427i	$0.0937973\pi$
$853$	422634.i	0.580854i	−0.956897	+	0.290427i	$0.906203\pi$
$854$	0	0
$855$	394632.i	0.539834i
$856$	0	0
$857$	49914.0	0.0679612	0.0339806	−	0.999422i	$-0.489182\pi$
$857$	49914.0	0.0679612	0.0339806	+	0.999422i	$0.489182\pi$
$858$	0	0
$859$	−726443.	−0.984499	−0.492249	−	0.870454i	$-0.663825\pi$
$859$	−726443.	−0.984499	−0.492249	+	0.870454i	$0.663825\pi$
$860$	0	0
$861$	− 895089.i	− 1.20742i
$862$	0	0
$863$	− 375912.i	− 0.504736i	−0.967631	−	0.252368i	$-0.918791\pi$
$863$	− 375912.i	− 0.504736i	0.967631	−	0.252368i	$-0.0812094\pi$
$864$	0	0
$865$	−856212.	−1.14432
$866$	0	0
$867$	52559.1	0.0699213
$868$	0	0
$869$	117585.i	0.155709i
$870$	0	0
$871$	99792.0i	0.131540i
$872$	0	0
$873$	159786.	0.209657
$874$	0	0
$875$	−931054.	−1.21607
$876$	0	0
$877$	− 1.41335e6i	− 1.83760i	−0.394720	−	0.918801i	$-0.629159\pi$
$877$	− 1.41335e6i	− 1.83760i	0.394720	−	0.918801i	$-0.370841\pi$
$878$	0	0
$879$	400950.i	0.518934i
$880$	0	0
$881$	−689742.	−0.888658	−0.444329	−	0.895864i	$-0.646558\pi$
$881$	−689742.	−0.888658	−0.444329	+	0.895864i	$0.646558\pi$
$882$	0	0
$883$	707695.	0.907663	0.453832	−	0.891087i	$-0.350057\pi$
$883$	707695.	0.907663	0.453832	+	0.891087i	$0.350057\pi$
$884$	0	0
$885$	337833.i	0.431336i
$886$	0	0
$887$	− 706032.i	− 0.897382i	−0.893687	−	0.448691i	$-0.851890\pi$
$887$	− 706032.i	− 0.897382i	0.893687	−	0.448691i	$-0.148110\pi$
$888$	0	0
$889$	579884.	0.733732
$890$	0	0
$891$	−10101.3	−0.0127240
$892$	0	0
$893$	238690.i	0.299318i
$894$	0	0
$895$	162456.i	0.202810i
$896$	0	0
$897$	−50544.0	−0.0628181
$898$	0	0
$899$	−161185.	−0.199436
$900$	0	0
$901$	344505.i	0.424371i
$902$	0	0
$903$	870696.i	1.06780i
$904$	0	0
$905$	−350280.	−0.427679
$906$	0	0
$907$	−13364.5	−0.0162457	−0.00812285	−	0.999967i	$-0.502586\pi$
$907$	−13364.5	−0.0162457	−0.00812285	+	0.999967i	$0.502586\pi$
$908$	0	0
$909$	443242.i	0.536430i
$910$	0	0
$911$	− 194544.i	− 0.234413i	−0.993108	−	0.117206i	$-0.962606\pi$
$911$	− 194544.i	− 0.234413i	0.993108	−	0.117206i	$-0.0373939\pi$
$912$	0	0
$913$	−192.000	−0.000230335 0
$914$	0	0
$915$	−754232.	−0.900871
$916$	0	0
$917$	− 108897.i	− 0.129503i
$918$	0	0
$919$	1.17935e6i	1.39641i	0.715900	+	0.698203i	$0.246017\pi$
$919$	1.17935e6i	1.39641i	−0.715900	+	0.698203i	$0.753983\pi$
$920$	0	0
$921$	−65772.0	−0.0775393
$922$	0	0
$923$	−136929.	−0.160728
$924$	0	0
$925$	− 38451.5i	− 0.0449397i
$926$	0	0
$927$	− 144234.i	− 0.167845i
$928$	0	0
$929$	232110.	0.268944	0.134472	−	0.990917i	$-0.457066\pi$
$929$	232110.	0.268944	0.134472	+	0.990917i	$0.457066\pi$
$930$	0	0
$931$	580452.	0.669679
$932$	0	0
$933$	714949.i	0.821319i
$934$	0	0
$935$	− 102816.i	− 0.117608i
$936$	0	0
$937$	1.21008e6	1.37827	0.689135	−	0.724633i	$-0.257991\pi$
$937$	1.21008e6	1.37827	0.689135	+	0.724633i	$0.257991\pi$
$938$	0	0
$939$	68578.8	0.0777784
$940$	0	0
$941$	1.19574e6i	1.35038i	0.737644	+	0.675190i	$0.235938\pi$
$941$	1.19574e6i	1.35038i	−0.737644	+	0.675190i	$0.764062\pi$
$942$	0	0
$943$	1.38996e6i	1.56307i
$944$	0	0
$945$	−197316.	−0.220952
$946$	0	0
$947$	−986985.	−1.10055	−0.550276	−	0.834983i	$-0.685477\pi$
$947$	−986985.	−1.10055	−0.550276	+	0.834983i	$0.685477\pi$
$948$	0	0
$949$	− 122504.i	− 0.136025i
$950$	0	0
$951$	696834.i	0.770492i
$952$	0	0
$953$	−156078.	−0.171853	−0.0859263	−	0.996301i	$-0.527385\pi$
$953$	−156078.	−0.171853	−0.0859263	+	0.996301i	$0.527385\pi$
$954$	0	0
$955$	850257.	0.932274
$956$	0	0
$957$	105503.i	0.115197i
$958$	0	0
$959$	− 1.61716e6i	− 1.75839i
$960$	0	0
$961$	911421.	0.986898
$962$	0	0
$963$	586812.	0.632771
$964$	0	0
$965$	272313.i	0.292425i
$966$	0	0
$967$	− 1.04281e6i	− 1.11520i	−0.830109	−	0.557601i	$-0.811722\pi$
$967$	− 1.04281e6i	− 1.11520i	0.830109	−	0.557601i	$-0.188278\pi$
$968$	0	0
$969$	958392.	1.02069
$970$	0	0
$971$	1.15445e6	1.22443	0.612217	−	0.790690i	$-0.290278\pi$
$971$	1.15445e6	1.22443	0.612217	+	0.790690i	$0.290278\pi$
$972$	0	0
$973$	657002.i	0.693970i
$974$	0	0
$975$	3996.00i	0.00420355i
$976$	0	0
$977$	−1.24965e6	−1.30918	−0.654590	−	0.755984i	$-0.727159\pi$
$977$	−1.24965e6	−1.30918	−0.654590	+	0.755984i	$0.727159\pi$
$978$	0	0
$979$	−121465.	−0.126732
$980$	0	0
$981$	219423.i	0.228005i
$982$	0	0
$983$	− 839088.i	− 0.868361i	−0.900826	−	0.434181i	$-0.857038\pi$
$983$	− 839088.i	− 0.868361i	0.900826	−	0.434181i	$-0.142962\pi$
$984$	0	0
$985$	−702996.	−0.724570
$986$	0	0
$987$	−119345.	−0.122510
$988$	0	0
$989$	− 1.35208e6i	− 1.38232i
$990$	0	0
$991$	− 422558.i	− 0.430268i	−0.976585	−	0.215134i	$-0.930981\pi$
$991$	− 422558.i	− 0.430268i	0.976585	−	0.215134i	$-0.0690188\pi$
$992$	0	0
$993$	270756.	0.274587
$994$	0	0
$995$	441957.	0.446410
$996$	0	0
$997$	945533.i	0.951232i	0.879653	+	0.475616i	$0.157775\pi$
$997$	945533.i	0.951232i	−0.879653	+	0.475616i	$0.842225\pi$
$998$	0	0
$999$	− 145800.i	− 0.146092i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.5.b.a.31.3	yes	4
3.2	odd	2		576.5.b.f.415.3		4
4.3	odd	2	inner	192.5.b.a.31.1	✓	4
8.3	odd	2	inner	192.5.b.a.31.4	yes	4
8.5	even	2	inner	192.5.b.a.31.2	yes	4
12.11	even	2		576.5.b.f.415.4		4
16.3	odd	4		768.5.g.e.511.1		4
16.5	even	4		768.5.g.e.511.2		4
16.11	odd	4		768.5.g.e.511.4		4
16.13	even	4		768.5.g.e.511.3		4
24.5	odd	2		576.5.b.f.415.1		4
24.11	even	2		576.5.b.f.415.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.5.b.a.31.1	✓	4	4.3	odd	2	inner
192.5.b.a.31.2	yes	4	8.5	even	2	inner
192.5.b.a.31.3	yes	4	1.1	even	1	trivial
192.5.b.a.31.4	yes	4	8.3	odd	2	inner
576.5.b.f.415.1		4	24.5	odd	2
576.5.b.f.415.2		4	24.11	even	2
576.5.b.f.415.3		4	3.2	odd	2
576.5.b.f.415.4		4	12.11	even	2
768.5.g.e.511.1		4	16.3	odd	4
768.5.g.e.511.2		4	16.5	even	4
768.5.g.e.511.3		4	16.13	even	4
768.5.g.e.511.4		4	16.11	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	192.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -24.2487i q^{5} -58.0000i q^{7} +27.0000 q^{9} -13.8564 q^{11} +20.7846i q^{13} -126.000i q^{15} -306.000 q^{17} -602.754 q^{19} -301.377i q^{21} +468.000i q^{23} +37.0000 q^{25} +140.296 q^{27} -1465.31i q^{29} -110.000i q^{31} -72.0000 q^{33} -1406.43 q^{35} -1039.23i q^{37} +108.000i q^{39} +2970.00 q^{41} -2889.06 q^{43} -654.715i q^{45} -396.000i q^{47} -963.000 q^{49} -1590.02 q^{51} -1125.83i q^{53} +336.000i q^{55} -3132.00 q^{57} -2681.21 q^{59} -5985.97i q^{61} -1566.00i q^{63} +504.000 q^{65} +4801.24 q^{67} +2431.80i q^{69} +6588.00i q^{71} -5894.00 q^{73} +192.258 q^{75} +803.672i q^{77} -8486.00i q^{79} +729.000 q^{81} +13.8564 q^{83} +7420.11i q^{85} -7614.00i q^{87} +8766.00 q^{89} +1205.51 q^{91} -571.577i q^{93} +14616.0i q^{95} +5918.00 q^{97} -374.123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{9} - 1224 q^{17} + 148 q^{25} - 288 q^{33} + 11880 q^{41} - 3852 q^{49} - 12528 q^{57} + 2016 q^{65} - 23576 q^{73} + 2916 q^{81} + 35064 q^{89} + 23672 q^{97}+O(q^{100}) \) 4 * q + 108 * q^9 - 1224 * q^17 + 148 * q^25 - 288 * q^33 + 11880 * q^41 - 3852 * q^49 - 12528 * q^57 + 2016 * q^65 - 23576 * q^73 + 2916 * q^81 + 35064 * q^89 + 23672 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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