LMFDB - Embedded newform 784.4.f.j.783.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.2
Character	$\chi$	$=$	784.783
Dual form			784.4.f.j.783.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.86009 q^{3} -7.47612i q^{5} +51.5012 q^{9} +O(q^{10})$	$q-8.86009 q^{3} -7.47612i q^{5} +51.5012 q^{9} -61.2337i q^{11} -12.0928i q^{13} +66.2391i q^{15} -100.394i q^{17} -73.6555 q^{19} +96.8114i q^{23} +69.1076 q^{25} -217.083 q^{27} +284.357 q^{29} +256.587 q^{31} +542.536i q^{33} +408.040 q^{37} +107.143i q^{39} +120.608i q^{41} -308.367i q^{43} -385.029i q^{45} +26.0454 q^{47} +889.504i q^{51} -311.059 q^{53} -457.791 q^{55} +652.595 q^{57} -444.945 q^{59} -488.804i q^{61} -90.4072 q^{65} -585.937i q^{67} -857.758i q^{69} +416.209i q^{71} -819.457i q^{73} -612.300 q^{75} +318.412i q^{79} +532.843 q^{81} -218.850 q^{83} -750.561 q^{85} -2519.43 q^{87} -19.4544i q^{89} -2273.39 q^{93} +550.658i q^{95} -306.810i q^{97} -3153.61i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9}+O(q^{10}) $ 24 * q + 88 * q^9	$ 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 856 * q^25 + 896 * q^29 - 2496 * q^53 + 4416 * q^57 - 4416 * q^65 - 4904 * q^81 - 2432 * q^85 - 11968 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−8.86009	−1.70513	−0.852563	−	0.522625i	$-0.824953\pi$
$3$	−8.86009	−1.70513	−0.852563	+	0.522625i	$0.824953\pi$
$4$	0	0
$5$	− 7.47612i	− 0.668685i	−0.942452	−	0.334342i	$-0.891486\pi$
$5$	− 7.47612i	− 0.668685i	0.942452	−	0.334342i	$-0.108514\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	51.5012	1.90745
$10$	0	0
$11$	− 61.2337i	− 1.67842i	−0.543805	−	0.839212i	$-0.683017\pi$
$11$	− 61.2337i	− 1.67842i	0.543805	−	0.839212i	$-0.316983\pi$
$12$	0	0
$13$	− 12.0928i	− 0.257995i	−0.991645	−	0.128998i	$-0.958824\pi$
$13$	− 12.0928i	− 0.257995i	0.991645	−	0.128998i	$-0.0411759\pi$
$14$	0	0
$15$	66.2391i	1.14019i
$16$	0	0
$17$	− 100.394i	− 1.43231i	−0.697943	−	0.716154i	$-0.745901\pi$
$17$	− 100.394i	− 1.43231i	0.697943	−	0.716154i	$-0.254099\pi$
$18$	0	0
$19$	−73.6555	−0.889355	−0.444677	−	0.895691i	$-0.646682\pi$
$19$	−73.6555	−0.889355	−0.444677	+	0.895691i	$0.646682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	96.8114i	0.877677i	0.898566	+	0.438839i	$0.144610\pi$
$23$	96.8114i	0.877677i	−0.898566	+	0.438839i	$0.855390\pi$
$24$	0	0
$25$	69.1076	0.552861
$26$	0	0
$27$	−217.083	−1.54732
$28$	0	0
$29$	284.357	1.82082	0.910411	−	0.413706i	$-0.135766\pi$
$29$	284.357	1.82082	0.910411	+	0.413706i	$0.135766\pi$
$30$	0	0
$31$	256.587	1.48660	0.743298	−	0.668961i	$-0.233260\pi$
$31$	256.587	1.48660	0.743298	+	0.668961i	$0.233260\pi$
$32$	0	0
$33$	542.536i	2.86192i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	408.040	1.81301	0.906505	−	0.422194i	$-0.138740\pi$
$37$	408.040	1.81301	0.906505	+	0.422194i	$0.138740\pi$
$38$	0	0
$39$	107.143i	0.439914i
$40$	0	0
$41$	120.608i	0.459409i	0.973260	+	0.229705i	$0.0737760\pi$
$41$	120.608i	0.459409i	−0.973260	+	0.229705i	$0.926224\pi$
$42$	0	0
$43$	− 308.367i	− 1.09362i	−0.837257	−	0.546809i	$-0.815842\pi$
$43$	− 308.367i	− 1.09362i	0.837257	−	0.546809i	$-0.184158\pi$
$44$	0	0
$45$	− 385.029i	− 1.27548i
$46$	0	0
$47$	26.0454	0.0808322	0.0404161	−	0.999183i	$-0.487132\pi$
$47$	26.0454	0.0808322	0.0404161	+	0.999183i	$0.487132\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	889.504i	2.44226i
$52$	0	0
$53$	−311.059	−0.806175	−0.403087	−	0.915161i	$-0.632063\pi$
$53$	−311.059	−0.806175	−0.403087	+	0.915161i	$0.632063\pi$
$54$	0	0
$55$	−457.791	−1.12234
$56$	0	0
$57$	652.595	1.51646
$58$	0	0
$59$	−444.945	−0.981812	−0.490906	−	0.871213i	$-0.663334\pi$
$59$	−444.945	−0.981812	−0.490906	+	0.871213i	$0.663334\pi$
$60$	0	0
$61$	− 488.804i	− 1.02598i	−0.858394	−	0.512991i	$-0.828537\pi$
$61$	− 488.804i	− 1.02598i	0.858394	−	0.512991i	$-0.171463\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−90.4072	−0.172517
$66$	0	0
$67$	− 585.937i	− 1.06841i	−0.845354	−	0.534206i	$-0.820611\pi$
$67$	− 585.937i	− 1.06841i	0.845354	−	0.534206i	$-0.179389\pi$
$68$	0	0
$69$	− 857.758i	− 1.49655i
$70$	0	0
$71$	416.209i	0.695704i	0.937550	+	0.347852i	$0.113089\pi$
$71$	416.209i	0.695704i	−0.937550	+	0.347852i	$0.886911\pi$
$72$	0	0
$73$	− 819.457i	− 1.31384i	−0.753961	−	0.656919i	$-0.771859\pi$
$73$	− 819.457i	− 1.31384i	0.753961	−	0.656919i	$-0.228141\pi$
$74$	0	0
$75$	−612.300	−0.942697
$76$	0	0
$77$	0	0
$78$	0	0
$79$	318.412i	0.453471i	0.973956	+	0.226735i	$0.0728053\pi$
$79$	318.412i	0.453471i	−0.973956	+	0.226735i	$0.927195\pi$
$80$	0	0
$81$	532.843	0.730923
$82$	0	0
$83$	−218.850	−0.289421	−0.144710	−	0.989474i	$-0.546225\pi$
$83$	−218.850	−0.289421	−0.144710	+	0.989474i	$0.546225\pi$
$84$	0	0
$85$	−750.561	−0.957762
$86$	0	0
$87$	−2519.43	−3.10473
$88$	0	0
$89$	− 19.4544i	− 0.0231703i	−0.999933	−	0.0115852i	$-0.996312\pi$
$89$	− 19.4544i	− 0.0231703i	0.999933	−	0.0115852i	$-0.00368775\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2273.39	−2.53483
$94$	0	0
$95$	550.658i	0.594698i
$96$	0	0
$97$	− 306.810i	− 0.321153i	−0.987023	−	0.160577i	$-0.948665\pi$
$97$	− 306.810i	− 0.321153i	0.987023	−	0.160577i	$-0.0513354\pi$
$98$	0	0
$99$	− 3153.61i	− 3.20151i
$100$	0	0
$101$	852.974i	0.840337i	0.907446	+	0.420169i	$0.138029\pi$
$101$	852.974i	0.840337i	−0.907446	+	0.420169i	$0.861971\pi$
$102$	0	0
$103$	1780.77	1.70354	0.851770	−	0.523917i	$-0.175530\pi$
$103$	1780.77	1.70354	0.851770	+	0.523917i	$0.175530\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 551.111i	− 0.497924i	−0.968513	−	0.248962i	$-0.919911\pi$
$107$	− 551.111i	− 0.497924i	0.968513	−	0.248962i	$-0.0800895\pi$
$108$	0	0
$109$	1050.78	0.923364	0.461682	−	0.887045i	$-0.347246\pi$
$109$	1050.78	0.923364	0.461682	+	0.887045i	$0.347246\pi$
$110$	0	0
$111$	−3615.27	−3.09141
$112$	0	0
$113$	83.2124	0.0692741	0.0346370	−	0.999400i	$-0.488972\pi$
$113$	83.2124	0.0692741	0.0346370	+	0.999400i	$0.488972\pi$
$114$	0	0
$115$	723.774	0.586889
$116$	0	0
$117$	− 622.793i	− 0.492113i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2418.57	−1.81711
$122$	0	0
$123$	− 1068.60i	− 0.783350i
$124$	0	0
$125$	− 1451.17i	− 1.03837i
$126$	0	0
$127$	− 1318.65i	− 0.921346i	−0.887570	−	0.460673i	$-0.847608\pi$
$127$	− 1318.65i	− 0.921346i	0.887570	−	0.460673i	$-0.152392\pi$
$128$	0	0
$129$	2732.16i	1.86476i
$130$	0	0
$131$	524.951	0.350116	0.175058	−	0.984558i	$-0.443989\pi$
$131$	524.951	0.350116	0.175058	+	0.984558i	$0.443989\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1622.94i	1.03467i
$136$	0	0
$137$	−2010.48	−1.25377	−0.626886	−	0.779111i	$-0.715671\pi$
$137$	−2010.48	−1.25377	−0.626886	+	0.779111i	$0.715671\pi$
$138$	0	0
$139$	1105.17	0.674386	0.337193	−	0.941436i	$-0.390523\pi$
$139$	1105.17	0.674386	0.337193	+	0.941436i	$0.390523\pi$
$140$	0	0
$141$	−230.765	−0.137829
$142$	0	0
$143$	−740.486	−0.433025
$144$	0	0
$145$	− 2125.89i	− 1.21756i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1446.19	−0.795144	−0.397572	−	0.917571i	$-0.630147\pi$
$149$	−1446.19	−0.795144	−0.397572	+	0.917571i	$0.630147\pi$
$150$	0	0
$151$	− 410.819i	− 0.221404i	−0.993854	−	0.110702i	$-0.964690\pi$
$151$	− 410.819i	− 0.221404i	0.993854	−	0.110702i	$-0.0353099\pi$
$152$	0	0
$153$	− 5170.44i	− 2.73206i
$154$	0	0
$155$	− 1918.28i	− 0.994064i
$156$	0	0
$157$	− 2542.49i	− 1.29244i	−0.763151	−	0.646220i	$-0.776349\pi$
$157$	− 2542.49i	− 1.29244i	0.763151	−	0.646220i	$-0.223651\pi$
$158$	0	0
$159$	2756.01	1.37463
$160$	0	0
$161$	0	0
$162$	0	0
$163$	59.6937i	0.0286845i	0.999897	+	0.0143423i	$0.00456544\pi$
$163$	59.6937i	0.0286845i	−0.999897	+	0.0143423i	$0.995435\pi$
$164$	0	0
$165$	4056.07	1.91372
$166$	0	0
$167$	−3811.33	−1.76605	−0.883023	−	0.469329i	$-0.844496\pi$
$167$	−3811.33	−1.76605	−0.883023	+	0.469329i	$0.844496\pi$
$168$	0	0
$169$	2050.76	0.933439
$170$	0	0
$171$	−3793.35	−1.69640
$172$	0	0
$173$	328.670i	0.144441i	0.997389	+	0.0722205i	$0.0230085\pi$
$173$	328.670i	0.144441i	−0.997389	+	0.0722205i	$0.976991\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3942.25	1.67411
$178$	0	0
$179$	− 3648.07i	− 1.52329i	−0.647992	−	0.761647i	$-0.724391\pi$
$179$	− 3648.07i	− 1.52329i	0.647992	−	0.761647i	$-0.275609\pi$
$180$	0	0
$181$	4446.07i	1.82582i	0.408160	+	0.912911i	$0.366171\pi$
$181$	4446.07i	1.82582i	−0.408160	+	0.912911i	$0.633829\pi$
$182$	0	0
$183$	4330.85i	1.74943i
$184$	0	0
$185$	− 3050.56i	− 1.21233i
$186$	0	0
$187$	−6147.52	−2.40402
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 388.447i	− 0.147157i	−0.997289	−	0.0735786i	$-0.976558\pi$
$191$	− 388.447i	− 0.147157i	0.997289	−	0.0735786i	$-0.0234420\pi$
$192$	0	0
$193$	−2363.57	−0.881522	−0.440761	−	0.897624i	$-0.645291\pi$
$193$	−2363.57	−0.881522	−0.440761	+	0.897624i	$0.645291\pi$
$194$	0	0
$195$	801.016	0.294164
$196$	0	0
$197$	−1273.45	−0.460556	−0.230278	−	0.973125i	$-0.573963\pi$
$197$	−1273.45	−0.460556	−0.230278	+	0.973125i	$0.573963\pi$
$198$	0	0
$199$	−902.022	−0.321320	−0.160660	−	0.987010i	$-0.551362\pi$
$199$	−902.022	−0.321320	−0.160660	+	0.987010i	$0.551362\pi$
$200$	0	0
$201$	5191.46i	1.82178i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	901.679	0.307200
$206$	0	0
$207$	4985.91i	1.67413i
$208$	0	0
$209$	4510.20i	1.49271i
$210$	0	0
$211$	− 2082.42i	− 0.679429i	−0.940529	−	0.339714i	$-0.889670\pi$
$211$	− 2082.42i	− 0.679429i	0.940529	−	0.339714i	$-0.110330\pi$
$212$	0	0
$213$	− 3687.65i	− 1.18626i
$214$	0	0
$215$	−2305.39	−0.731285
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7260.46i	2.24026i
$220$	0	0
$221$	−1214.05	−0.369528
$222$	0	0
$223$	587.200	0.176331	0.0881655	−	0.996106i	$-0.471900\pi$
$223$	587.200	0.176331	0.0881655	+	0.996106i	$0.471900\pi$
$224$	0	0
$225$	3559.13	1.05456
$226$	0	0
$227$	6394.91	1.86980	0.934901	−	0.354908i	$-0.115488\pi$
$227$	6394.91	1.86980	0.934901	+	0.354908i	$0.115488\pi$
$228$	0	0
$229$	3083.69i	0.889851i	0.895568	+	0.444925i	$0.146770\pi$
$229$	3083.69i	0.889851i	−0.895568	+	0.444925i	$0.853230\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−412.960	−0.116111	−0.0580556	−	0.998313i	$-0.518490\pi$
$233$	−412.960	−0.116111	−0.0580556	+	0.998313i	$0.518490\pi$
$234$	0	0
$235$	− 194.719i	− 0.0540513i
$236$	0	0
$237$	− 2821.16i	− 0.773225i
$238$	0	0
$239$	3721.31i	1.00716i	0.863948	+	0.503581i	$0.167984\pi$
$239$	3721.31i	1.00716i	−0.863948	+	0.503581i	$0.832016\pi$
$240$	0	0
$241$	3230.26i	0.863399i	0.902018	+	0.431699i	$0.142086\pi$
$241$	3230.26i	0.863399i	−0.902018	+	0.431699i	$0.857914\pi$
$242$	0	0
$243$	1140.21	0.301005
$244$	0	0
$245$	0	0
$246$	0	0
$247$	890.701i	0.229449i
$248$	0	0
$249$	1939.03	0.493499
$250$	0	0
$251$	181.422	0.0456226	0.0228113	−	0.999740i	$-0.492738\pi$
$251$	181.422	0.0456226	0.0228113	+	0.999740i	$0.492738\pi$
$252$	0	0
$253$	5928.12	1.47311
$254$	0	0
$255$	6650.04	1.63310
$256$	0	0
$257$	783.196i	0.190095i	0.995473	+	0.0950475i	$0.0303003\pi$
$257$	783.196i	0.190095i	−0.995473	+	0.0950475i	$0.969700\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	14644.7	3.47313
$262$	0	0
$263$	4309.93i	1.01050i	0.862973	+	0.505250i	$0.168600\pi$
$263$	4309.93i	1.01050i	−0.862973	+	0.505250i	$0.831400\pi$
$264$	0	0
$265$	2325.52i	0.539077i
$266$	0	0
$267$	172.367i	0.0395083i
$268$	0	0
$269$	− 7148.05i	− 1.62016i	−0.586316	−	0.810082i	$-0.699422\pi$
$269$	− 7148.05i	− 1.62016i	0.586316	−	0.810082i	$-0.300578\pi$
$270$	0	0
$271$	4098.06	0.918597	0.459298	−	0.888282i	$-0.348101\pi$
$271$	4098.06	0.918597	0.459298	+	0.888282i	$0.348101\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4231.72i	− 0.927935i
$276$	0	0
$277$	−8479.05	−1.83919	−0.919597	−	0.392862i	$-0.871485\pi$
$277$	−8479.05	−1.83919	−0.919597	+	0.392862i	$0.871485\pi$
$278$	0	0
$279$	13214.6	2.83561
$280$	0	0
$281$	1108.37	0.235301	0.117651	−	0.993055i	$-0.462464\pi$
$281$	1108.37	0.235301	0.117651	+	0.993055i	$0.462464\pi$
$282$	0	0
$283$	925.763	0.194455	0.0972277	−	0.995262i	$-0.469002\pi$
$283$	925.763	0.194455	0.0972277	+	0.995262i	$0.469002\pi$
$284$	0	0
$285$	− 4878.88i	− 1.01403i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5166.04	−1.05150
$290$	0	0
$291$	2718.37i	0.547607i
$292$	0	0
$293$	700.439i	0.139659i	0.997559	+	0.0698295i	$0.0222455\pi$
$293$	700.439i	0.139659i	−0.997559	+	0.0698295i	$0.977754\pi$
$294$	0	0
$295$	3326.46i	0.656523i
$296$	0	0
$297$	13292.8i	2.59706i
$298$	0	0
$299$	1170.72	0.226436
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 7557.43i	− 1.43288i
$304$	0	0
$305$	−3654.36	−0.686059
$306$	0	0
$307$	−2959.79	−0.550242	−0.275121	−	0.961410i	$-0.588718\pi$
$307$	−2959.79	−0.550242	−0.275121	+	0.961410i	$0.588718\pi$
$308$	0	0
$309$	−15777.8	−2.90475
$310$	0	0
$311$	−1808.61	−0.329765	−0.164882	−	0.986313i	$-0.552724\pi$
$311$	−1808.61	−0.329765	−0.164882	+	0.986313i	$0.552724\pi$
$312$	0	0
$313$	4552.38i	0.822095i	0.911614	+	0.411047i	$0.134837\pi$
$313$	4552.38i	0.822095i	−0.911614	+	0.411047i	$0.865163\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	730.924	0.129504	0.0647520	−	0.997901i	$-0.479374\pi$
$317$	730.924	0.129504	0.0647520	+	0.997901i	$0.479374\pi$
$318$	0	0
$319$	− 17412.3i	− 3.05611i
$320$	0	0
$321$	4882.89i	0.849023i
$322$	0	0
$323$	7394.61i	1.27383i
$324$	0	0
$325$	− 835.704i	− 0.142635i
$326$	0	0
$327$	−9310.03	−1.57445
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 8937.09i	− 1.48407i	−0.670362	−	0.742035i	$-0.733861\pi$
$331$	− 8937.09i	− 1.48407i	0.670362	−	0.742035i	$-0.266139\pi$
$332$	0	0
$333$	21014.6	3.45823
$334$	0	0
$335$	−4380.54	−0.714431
$336$	0	0
$337$	−6813.13	−1.10129	−0.550645	−	0.834740i	$-0.685618\pi$
$337$	−6813.13	−1.10129	−0.550645	+	0.834740i	$0.685618\pi$
$338$	0	0
$339$	−737.270	−0.118121
$340$	0	0
$341$	− 15711.8i	− 2.49514i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6412.70	−1.00072
$346$	0	0
$347$	8017.29i	1.24032i	0.784476	+	0.620159i	$0.212932\pi$
$347$	8017.29i	1.24032i	−0.784476	+	0.620159i	$0.787068\pi$
$348$	0	0
$349$	5566.53i	0.853781i	0.904303	+	0.426890i	$0.140391\pi$
$349$	5566.53i	0.853781i	−0.904303	+	0.426890i	$0.859609\pi$
$350$	0	0
$351$	2625.14i	0.399201i
$352$	0	0
$353$	6508.41i	0.981326i	0.871350	+	0.490663i	$0.163245\pi$
$353$	6508.41i	0.981326i	−0.871350	+	0.490663i	$0.836755\pi$
$354$	0	0
$355$	3111.63	0.465206
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 8739.25i	− 1.28479i	−0.766373	−	0.642396i	$-0.777941\pi$
$359$	− 8739.25i	− 1.28479i	0.766373	−	0.642396i	$-0.222059\pi$
$360$	0	0
$361$	−1433.86	−0.209048
$362$	0	0
$363$	21428.7	3.09839
$364$	0	0
$365$	−6126.36	−0.878543
$366$	0	0
$367$	4591.08	0.653004	0.326502	−	0.945196i	$-0.394130\pi$
$367$	4591.08	0.653004	0.326502	+	0.945196i	$0.394130\pi$
$368$	0	0
$369$	6211.45i	0.876301i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9198.07	−1.27683	−0.638415	−	0.769692i	$-0.720410\pi$
$373$	−9198.07	−1.27683	−0.638415	+	0.769692i	$0.720410\pi$
$374$	0	0
$375$	12857.5i	1.77056i
$376$	0	0
$377$	− 3438.67i	− 0.469763i
$378$	0	0
$379$	6586.09i	0.892625i	0.894877	+	0.446312i	$0.147263\pi$
$379$	6586.09i	0.892625i	−0.894877	+	0.446312i	$0.852737\pi$
$380$	0	0
$381$	11683.3i	1.57101i
$382$	0	0
$383$	−921.517	−0.122943	−0.0614717	−	0.998109i	$-0.519579\pi$
$383$	−921.517	−0.122943	−0.0614717	+	0.998109i	$0.519579\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 15881.3i	− 2.08602i
$388$	0	0
$389$	−8919.80	−1.16260	−0.581301	−	0.813689i	$-0.697456\pi$
$389$	−8919.80	−1.16260	−0.581301	+	0.813689i	$0.697456\pi$
$390$	0	0
$391$	9719.33	1.25710
$392$	0	0
$393$	−4651.11	−0.596991
$394$	0	0
$395$	2380.49	0.303229
$396$	0	0
$397$	− 7287.34i	− 0.921262i	−0.887592	−	0.460631i	$-0.847623\pi$
$397$	− 7287.34i	− 0.921262i	0.887592	−	0.460631i	$-0.152377\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4320.07	−0.537990	−0.268995	−	0.963142i	$-0.586692\pi$
$401$	−4320.07	−0.537990	−0.268995	+	0.963142i	$0.586692\pi$
$402$	0	0
$403$	− 3102.86i	− 0.383534i
$404$	0	0
$405$	− 3983.60i	− 0.488757i
$406$	0	0
$407$	− 24985.8i	− 3.04300i
$408$	0	0
$409$	7792.27i	0.942061i	0.882117	+	0.471030i	$0.156118\pi$
$409$	7792.27i	0.942061i	−0.882117	+	0.471030i	$0.843882\pi$
$410$	0	0
$411$	17813.0	2.13784
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1636.15i	0.193531i
$416$	0	0
$417$	−9791.94	−1.14991
$418$	0	0
$419$	−6318.16	−0.736665	−0.368332	−	0.929694i	$-0.620071\pi$
$419$	−6318.16	−0.736665	−0.368332	+	0.929694i	$0.620071\pi$
$420$	0	0
$421$	122.350	0.0141639	0.00708195	−	0.999975i	$-0.497746\pi$
$421$	122.350	0.0141639	0.00708195	+	0.999975i	$0.497746\pi$
$422$	0	0
$423$	1341.37	0.154184
$424$	0	0
$425$	− 6938.02i	− 0.791867i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6560.78	0.738362
$430$	0	0
$431$	− 4982.96i	− 0.556893i	−0.960452	−	0.278447i	$-0.910180\pi$
$431$	− 4982.96i	− 0.556893i	0.960452	−	0.278447i	$-0.0898195\pi$
$432$	0	0
$433$	− 680.814i	− 0.0755608i	−0.999286	−	0.0377804i	$-0.987971\pi$
$433$	− 680.814i	− 0.0755608i	0.999286	−	0.0377804i	$-0.0120287\pi$
$434$	0	0
$435$	18835.6i	2.07608i
$436$	0	0
$437$	− 7130.70i	− 0.780566i
$438$	0	0
$439$	9117.30	0.991219	0.495610	−	0.868545i	$-0.334945\pi$
$439$	9117.30	0.991219	0.495610	+	0.868545i	$0.334945\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4410.08i	0.472978i	0.971634	+	0.236489i	$0.0759968\pi$
$443$	4410.08i	0.472978i	−0.971634	+	0.236489i	$0.924003\pi$
$444$	0	0
$445$	−145.443	−0.0154936
$446$	0	0
$447$	12813.4	1.35582
$448$	0	0
$449$	14077.1	1.47959	0.739797	−	0.672830i	$-0.234921\pi$
$449$	14077.1	1.47959	0.739797	+	0.672830i	$0.234921\pi$
$450$	0	0
$451$	7385.27	0.771083
$452$	0	0
$453$	3639.90i	0.377522i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7812.25	0.799654	0.399827	−	0.916591i	$-0.369070\pi$
$457$	7812.25	0.799654	0.399827	+	0.916591i	$0.369070\pi$
$458$	0	0
$459$	21793.9i	2.21624i
$460$	0	0
$461$	17552.0i	1.77327i	0.462470	+	0.886635i	$0.346963\pi$
$461$	17552.0i	1.77327i	−0.462470	+	0.886635i	$0.653037\pi$
$462$	0	0
$463$	− 5236.81i	− 0.525648i	−0.964844	−	0.262824i	$-0.915346\pi$
$463$	− 5236.81i	− 0.525648i	0.964844	−	0.262824i	$-0.0846539\pi$
$464$	0	0
$465$	16996.1i	1.69500i
$466$	0	0
$467$	−11941.1	−1.18323	−0.591617	−	0.806219i	$-0.701510\pi$
$467$	−11941.1	−1.18323	−0.591617	+	0.806219i	$0.701510\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	22526.7i	2.20377i
$472$	0	0
$473$	−18882.5	−1.83555
$474$	0	0
$475$	−5090.16	−0.491689
$476$	0	0
$477$	−16019.9	−1.53774
$478$	0	0
$479$	5774.82	0.550852	0.275426	−	0.961322i	$-0.411181\pi$
$479$	5774.82	0.550852	0.275426	+	0.961322i	$0.411181\pi$
$480$	0	0
$481$	− 4934.34i	− 0.467748i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2293.75	−0.214750
$486$	0	0
$487$	2923.50i	0.272026i	0.990707	+	0.136013i	$0.0434289\pi$
$487$	2923.50i	0.272026i	−0.990707	+	0.136013i	$0.956571\pi$
$488$	0	0
$489$	− 528.892i	− 0.0489107i
$490$	0	0
$491$	14580.3i	1.34012i	0.742308	+	0.670059i	$0.233731\pi$
$491$	14580.3i	1.34012i	−0.742308	+	0.670059i	$0.766269\pi$
$492$	0	0
$493$	− 28547.9i	− 2.60798i
$494$	0	0
$495$	−23576.8	−2.14080
$496$	0	0
$497$	0	0
$498$	0	0
$499$	874.052i	0.0784127i	0.999231	+	0.0392064i	$0.0124830\pi$
$499$	874.052i	0.0784127i	−0.999231	+	0.0392064i	$0.987517\pi$
$500$	0	0
$501$	33768.8	3.01133
$502$	0	0
$503$	5065.24	0.449002	0.224501	−	0.974474i	$-0.427925\pi$
$503$	5065.24	0.449002	0.224501	+	0.974474i	$0.427925\pi$
$504$	0	0
$505$	6376.94	0.561921
$506$	0	0
$507$	−18170.0	−1.59163
$508$	0	0
$509$	15623.5i	1.36051i	0.732978	+	0.680253i	$0.238130\pi$
$509$	15623.5i	1.36051i	−0.732978	+	0.680253i	$0.761870\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	15989.4	1.37612
$514$	0	0
$515$	− 13313.3i	− 1.13913i
$516$	0	0
$517$	− 1594.86i	− 0.135671i
$518$	0	0
$519$	− 2912.04i	− 0.246290i
$520$	0	0
$521$	− 8426.11i	− 0.708550i	−0.935141	−	0.354275i	$-0.884728\pi$
$521$	− 8426.11i	− 0.708550i	0.935141	−	0.354275i	$-0.115272\pi$
$522$	0	0
$523$	−1830.09	−0.153010	−0.0765050	−	0.997069i	$-0.524376\pi$
$523$	−1830.09	−0.153010	−0.0765050	+	0.997069i	$0.524376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 25760.0i	− 2.12926i
$528$	0	0
$529$	2794.55	0.229683
$530$	0	0
$531$	−22915.2	−1.87276
$532$	0	0
$533$	1458.49	0.118525
$534$	0	0
$535$	−4120.17	−0.332954
$536$	0	0
$537$	32322.2i	2.59741i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3788.15	−0.301045	−0.150522	−	0.988607i	$-0.548096\pi$
$541$	−3788.15	−0.301045	−0.150522	+	0.988607i	$0.548096\pi$
$542$	0	0
$543$	− 39392.6i	− 3.11325i
$544$	0	0
$545$	− 7855.78i	− 0.617440i
$546$	0	0
$547$	6463.45i	0.505223i	0.967568	+	0.252612i	$0.0812895\pi$
$547$	6463.45i	0.505223i	−0.967568	+	0.252612i	$0.918710\pi$
$548$	0	0
$549$	− 25174.0i	− 1.95701i
$550$	0	0
$551$	−20944.5	−1.61936
$552$	0	0
$553$	0	0
$554$	0	0
$555$	27028.2i	2.06718i
$556$	0	0
$557$	−1915.55	−0.145717	−0.0728587	−	0.997342i	$-0.523212\pi$
$557$	−1915.55	−0.145717	−0.0728587	+	0.997342i	$0.523212\pi$
$558$	0	0
$559$	−3729.02	−0.282148
$560$	0	0
$561$	54467.6	4.09915
$562$	0	0
$563$	220.298	0.0164911	0.00824553	−	0.999966i	$-0.497375\pi$
$563$	220.298	0.0164911	0.00824553	+	0.999966i	$0.497375\pi$
$564$	0	0
$565$	− 622.106i	− 0.0463225i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18909.3	1.39318	0.696590	−	0.717470i	$-0.254700\pi$
$569$	18909.3	1.39318	0.696590	+	0.717470i	$0.254700\pi$
$570$	0	0
$571$	25001.7i	1.83238i	0.400748	+	0.916188i	$0.368750\pi$
$571$	25001.7i	1.83238i	−0.400748	+	0.916188i	$0.631250\pi$
$572$	0	0
$573$	3441.68i	0.250922i
$574$	0	0
$575$	6690.41i	0.485233i
$576$	0	0
$577$	− 19401.0i	− 1.39978i	−0.714251	−	0.699890i	$-0.753233\pi$
$577$	− 19401.0i	− 1.39978i	0.714251	−	0.699890i	$-0.246767\pi$
$578$	0	0
$579$	20941.5	1.50311
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19047.3i	1.35310i
$584$	0	0
$585$	−4656.08	−0.329069
$586$	0	0
$587$	−16828.3	−1.18327	−0.591633	−	0.806208i	$-0.701516\pi$
$587$	−16828.3	−1.18327	−0.591633	+	0.806208i	$0.701516\pi$
$588$	0	0
$589$	−18899.1	−1.32211
$590$	0	0
$591$	11282.9	0.785305
$592$	0	0
$593$	− 5988.30i	− 0.414688i	−0.978268	−	0.207344i	$-0.933518\pi$
$593$	− 5988.30i	− 0.414688i	0.978268	−	0.207344i	$-0.0664820\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7992.00	0.547890
$598$	0	0
$599$	14163.0i	0.966087i	0.875596	+	0.483043i	$0.160469\pi$
$599$	14163.0i	0.966087i	−0.875596	+	0.483043i	$0.839531\pi$
$600$	0	0
$601$	16419.5i	1.11442i	0.830371	+	0.557210i	$0.188128\pi$
$601$	16419.5i	1.11442i	−0.830371	+	0.557210i	$0.811872\pi$
$602$	0	0
$603$	− 30176.5i	− 2.03795i
$604$	0	0
$605$	18081.5i	1.21507i
$606$	0	0
$607$	18589.4	1.24303	0.621517	−	0.783401i	$-0.286517\pi$
$607$	18589.4	1.24303	0.621517	+	0.783401i	$0.286517\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 314.962i	− 0.0208543i
$612$	0	0
$613$	−5262.43	−0.346734	−0.173367	−	0.984857i	$-0.555465\pi$
$613$	−5262.43	−0.346734	−0.173367	+	0.984857i	$0.555465\pi$
$614$	0	0
$615$	−7988.96	−0.523814
$616$	0	0
$617$	−4798.33	−0.313085	−0.156543	−	0.987671i	$-0.550035\pi$
$617$	−4798.33	−0.313085	−0.156543	+	0.987671i	$0.550035\pi$
$618$	0	0
$619$	5492.16	0.356622	0.178311	−	0.983974i	$-0.442937\pi$
$619$	5492.16	0.356622	0.178311	+	0.983974i	$0.442937\pi$
$620$	0	0
$621$	− 21016.1i	− 1.35805i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−2210.69	−0.141484
$626$	0	0
$627$	− 39960.8i	− 2.54526i
$628$	0	0
$629$	− 40965.0i	− 2.59679i
$630$	0	0
$631$	4615.33i	0.291178i	0.989345	+	0.145589i	$0.0465077\pi$
$631$	4615.33i	0.291178i	−0.989345	+	0.145589i	$0.953492\pi$
$632$	0	0
$633$	18450.4i	1.15851i
$634$	0	0
$635$	−9858.36	−0.616090
$636$	0	0
$637$	0	0
$638$	0	0
$639$	21435.3i	1.32702i
$640$	0	0
$641$	−2406.03	−0.148257	−0.0741284	−	0.997249i	$-0.523617\pi$
$641$	−2406.03	−0.148257	−0.0741284	+	0.997249i	$0.523617\pi$
$642$	0	0
$643$	−21395.2	−1.31220	−0.656098	−	0.754675i	$-0.727794\pi$
$643$	−21395.2	−1.31220	−0.656098	+	0.754675i	$0.727794\pi$
$644$	0	0
$645$	20426.0	1.24693
$646$	0	0
$647$	4800.83	0.291716	0.145858	−	0.989306i	$-0.453406\pi$
$647$	4800.83	0.291716	0.145858	+	0.989306i	$0.453406\pi$
$648$	0	0
$649$	27245.6i	1.64790i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10334.4	0.619323	0.309661	−	0.950847i	$-0.399784\pi$
$653$	10334.4	0.619323	0.309661	+	0.950847i	$0.399784\pi$
$654$	0	0
$655$	− 3924.60i	− 0.234117i
$656$	0	0
$657$	− 42203.0i	− 2.50608i
$658$	0	0
$659$	− 29036.0i	− 1.71636i	−0.513346	−	0.858182i	$-0.671594\pi$
$659$	− 29036.0i	− 1.71636i	0.513346	−	0.858182i	$-0.328406\pi$
$660$	0	0
$661$	4690.96i	0.276032i	0.990430	+	0.138016i	$0.0440725\pi$
$661$	4690.96i	0.276032i	−0.990430	+	0.138016i	$0.955927\pi$
$662$	0	0
$663$	10756.6	0.630092
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27529.0i	1.59809i
$668$	0	0
$669$	−5202.64	−0.300666
$670$	0	0
$671$	−29931.3	−1.72203
$672$	0	0
$673$	−13736.5	−0.786779	−0.393389	−	0.919372i	$-0.628698\pi$
$673$	−13736.5	−0.786779	−0.393389	+	0.919372i	$0.628698\pi$
$674$	0	0
$675$	−15002.1	−0.855453
$676$	0	0
$677$	− 17305.2i	− 0.982414i	−0.871043	−	0.491207i	$-0.836556\pi$
$677$	− 17305.2i	− 0.982414i	0.871043	−	0.491207i	$-0.163444\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−56659.5	−3.18825
$682$	0	0
$683$	26134.4i	1.46413i	0.681233	+	0.732067i	$0.261444\pi$
$683$	26134.4i	1.46413i	−0.681233	+	0.732067i	$0.738556\pi$
$684$	0	0
$685$	15030.6i	0.838378i
$686$	0	0
$687$	− 27321.8i	− 1.51731i
$688$	0	0
$689$	3761.57i	0.207989i
$690$	0	0
$691$	28243.1	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$691$	28243.1	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 8262.41i	− 0.450951i
$696$	0	0
$697$	12108.4	0.658015
$698$	0	0
$699$	3658.86	0.197984
$700$	0	0
$701$	29732.6	1.60198	0.800988	−	0.598680i	$-0.204308\pi$
$701$	29732.6	1.60198	0.800988	+	0.598680i	$0.204308\pi$
$702$	0	0
$703$	−30054.4	−1.61241
$704$	0	0
$705$	1725.23i	0.0921642i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18124.3	−0.960047	−0.480023	−	0.877256i	$-0.659372\pi$
$709$	−18124.3	−0.960047	−0.480023	+	0.877256i	$0.659372\pi$
$710$	0	0
$711$	16398.6i	0.864974i
$712$	0	0
$713$	24840.6i	1.30475i
$714$	0	0
$715$	5535.97i	0.289557i
$716$	0	0
$717$	− 32971.2i	− 1.71734i
$718$	0	0
$719$	−22481.7	−1.16610	−0.583050	−	0.812436i	$-0.698141\pi$
$719$	−22481.7	−1.16610	−0.583050	+	0.812436i	$0.698141\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 28620.4i	− 1.47220i
$724$	0	0
$725$	19651.3	1.00666
$726$	0	0
$727$	27637.9	1.40995	0.704974	−	0.709233i	$-0.250959\pi$
$727$	27637.9	1.40995	0.704974	+	0.709233i	$0.250959\pi$
$728$	0	0
$729$	−24489.1	−1.24417
$730$	0	0
$731$	−30958.3	−1.56640
$732$	0	0
$733$	− 7075.60i	− 0.356539i	−0.983982	−	0.178270i	$-0.942950\pi$
$733$	− 7075.60i	− 0.356539i	0.983982	−	0.178270i	$-0.0570499\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35879.1	−1.79325
$738$	0	0
$739$	− 29891.3i	− 1.48792i	−0.668226	−	0.743958i	$-0.732946\pi$
$739$	− 29891.3i	− 1.48792i	0.668226	−	0.743958i	$-0.267054\pi$
$740$	0	0
$741$	− 7891.69i	− 0.391240i
$742$	0	0
$743$	− 11676.8i	− 0.576553i	−0.957547	−	0.288277i	$-0.906918\pi$
$743$	− 11676.8i	− 0.576553i	0.957547	−	0.288277i	$-0.0930823\pi$
$744$	0	0
$745$	10811.9i	0.531700i
$746$	0	0
$747$	−11271.0	−0.552056
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 30942.3i	− 1.50346i	−0.659468	−	0.751732i	$-0.729218\pi$
$751$	− 30942.3i	− 1.50346i	0.659468	−	0.751732i	$-0.270782\pi$
$752$	0	0
$753$	−1607.42	−0.0777923
$754$	0	0
$755$	−3071.34	−0.148049
$756$	0	0
$757$	−28884.0	−1.38680	−0.693399	−	0.720554i	$-0.743888\pi$
$757$	−28884.0	−1.38680	−0.693399	+	0.720554i	$0.743888\pi$
$758$	0	0
$759$	−52523.7	−2.51184
$760$	0	0
$761$	− 22478.8i	− 1.07077i	−0.844608	−	0.535385i	$-0.820167\pi$
$761$	− 22478.8i	− 1.07077i	0.844608	−	0.535385i	$-0.179833\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−38654.8	−1.82689
$766$	0	0
$767$	5380.63i	0.253303i
$768$	0	0
$769$	− 30021.9i	− 1.40783i	−0.710287	−	0.703913i	$-0.751435\pi$
$769$	− 30021.9i	− 1.40783i	0.710287	−	0.703913i	$-0.248565\pi$
$770$	0	0
$771$	− 6939.19i	− 0.324136i
$772$	0	0
$773$	− 2446.25i	− 0.113823i	−0.998379	−	0.0569117i	$-0.981875\pi$
$773$	− 2446.25i	− 0.113823i	0.998379	−	0.0569117i	$-0.0181253\pi$
$774$	0	0
$775$	17732.1	0.821881
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8883.44i	− 0.408578i
$780$	0	0
$781$	25486.0	1.16769
$782$	0	0
$783$	−61729.2	−2.81739
$784$	0	0
$785$	−19008.0	−0.864234
$786$	0	0
$787$	2274.33	0.103013	0.0515064	−	0.998673i	$-0.483598\pi$
$787$	2274.33	0.103013	0.0515064	+	0.998673i	$0.483598\pi$
$788$	0	0
$789$	− 38186.4i	− 1.72303i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5911.01	−0.264699
$794$	0	0
$795$	− 20604.3i	− 0.919193i
$796$	0	0
$797$	− 25454.5i	− 1.13130i	−0.824646	−	0.565649i	$-0.808626\pi$
$797$	− 25454.5i	− 1.13130i	0.824646	−	0.565649i	$-0.191374\pi$
$798$	0	0
$799$	− 2614.82i	− 0.115777i
$800$	0	0
$801$	− 1001.92i	− 0.0441963i
$802$	0	0
$803$	−50178.4	−2.20518
$804$	0	0
$805$	0	0
$806$	0	0
$807$	63332.4i	2.76258i
$808$	0	0
$809$	−2042.65	−0.0887709	−0.0443855	−	0.999014i	$-0.514133\pi$
$809$	−2042.65	−0.0887709	−0.0443855	+	0.999014i	$0.514133\pi$
$810$	0	0
$811$	33523.4	1.45150	0.725750	−	0.687958i	$-0.241493\pi$
$811$	33523.4	1.45150	0.725750	+	0.687958i	$0.241493\pi$
$812$	0	0
$813$	−36309.2	−1.56632
$814$	0	0
$815$	446.278	0.0191809
$816$	0	0
$817$	22713.0i	0.972614i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28446.0	1.20922	0.604611	−	0.796521i	$-0.293328\pi$
$821$	28446.0	1.20922	0.604611	+	0.796521i	$0.293328\pi$
$822$	0	0
$823$	3185.80i	0.134933i	0.997722	+	0.0674666i	$0.0214916\pi$
$823$	3185.80i	0.134933i	−0.997722	+	0.0674666i	$0.978508\pi$
$824$	0	0
$825$	37493.4i	1.58225i
$826$	0	0
$827$	− 21966.6i	− 0.923644i	−0.886973	−	0.461822i	$-0.847196\pi$
$827$	− 21966.6i	− 0.923644i	0.886973	−	0.461822i	$-0.152804\pi$
$828$	0	0
$829$	41832.8i	1.75261i	0.481756	+	0.876306i	$0.339999\pi$
$829$	41832.8i	1.75261i	−0.481756	+	0.876306i	$0.660001\pi$
$830$	0	0
$831$	75125.2	3.13606
$832$	0	0
$833$	0	0
$834$	0	0
$835$	28494.0i	1.18093i
$836$	0	0
$837$	−55700.8	−2.30024
$838$	0	0
$839$	−13093.9	−0.538799	−0.269400	−	0.963028i	$-0.586825\pi$
$839$	−13093.9	−0.538799	−0.269400	+	0.963028i	$0.586825\pi$
$840$	0	0
$841$	56470.1	2.31539
$842$	0	0
$843$	−9820.23	−0.401218
$844$	0	0
$845$	− 15331.8i	− 0.624176i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8202.34	−0.331571
$850$	0	0
$851$	39503.0i	1.59124i
$852$	0	0
$853$	− 23013.9i	− 0.923777i	−0.886938	−	0.461889i	$-0.847172\pi$
$853$	− 23013.9i	− 0.923777i	0.886938	−	0.461889i	$-0.152828\pi$
$854$	0	0
$855$	28359.5i	1.13436i
$856$	0	0
$857$	18808.6i	0.749696i	0.927086	+	0.374848i	$0.122305\pi$
$857$	18808.6i	0.749696i	−0.927086	+	0.374848i	$0.877695\pi$
$858$	0	0
$859$	−24257.3	−0.963504	−0.481752	−	0.876308i	$-0.659999\pi$
$859$	−24257.3	−0.963504	−0.481752	+	0.876308i	$0.659999\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 11594.2i	− 0.457326i	−0.973506	−	0.228663i	$-0.926565\pi$
$863$	− 11594.2i	− 0.457326i	0.973506	−	0.228663i	$-0.0734354\pi$
$864$	0	0
$865$	2457.17	0.0965854
$866$	0	0
$867$	45771.6	1.79295
$868$	0	0
$869$	19497.6	0.761116
$870$	0	0
$871$	−7085.61	−0.275645
$872$	0	0
$873$	− 15801.1i	− 0.612585i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6641.93	0.255738	0.127869	−	0.991791i	$-0.459186\pi$
$877$	6641.93	0.255738	0.127869	+	0.991791i	$0.459186\pi$
$878$	0	0
$879$	− 6205.95i	− 0.238136i
$880$	0	0
$881$	− 13406.2i	− 0.512674i	−0.966588	−	0.256337i	$-0.917484\pi$
$881$	− 13406.2i	− 0.512674i	0.966588	−	0.256337i	$-0.0825156\pi$
$882$	0	0
$883$	− 6095.70i	− 0.232318i	−0.993231	−	0.116159i	$-0.962942\pi$
$883$	− 6095.70i	− 0.232318i	0.993231	−	0.116159i	$-0.0370582\pi$
$884$	0	0
$885$	− 29472.8i	− 1.11945i
$886$	0	0
$887$	23852.8	0.902929	0.451464	−	0.892289i	$-0.350902\pi$
$887$	23852.8	0.902929	0.451464	+	0.892289i	$0.350902\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 32628.0i	− 1.22680i
$892$	0	0
$893$	−1918.39	−0.0718885
$894$	0	0
$895$	−27273.4	−1.01860
$896$	0	0
$897$	−10372.7	−0.386102
$898$	0	0
$899$	72962.5	2.70683
$900$	0	0
$901$	31228.6i	1.15469i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33239.3	1.22090
$906$	0	0
$907$	− 43014.8i	− 1.57473i	−0.616485	−	0.787367i	$-0.711444\pi$
$907$	− 43014.8i	− 1.57473i	0.616485	−	0.787367i	$-0.288556\pi$
$908$	0	0
$909$	43929.2i	1.60290i
$910$	0	0
$911$	11732.6i	0.426694i	0.976977	+	0.213347i	$0.0684364\pi$
$911$	11732.6i	0.426694i	−0.976977	+	0.213347i	$0.931564\pi$
$912$	0	0
$913$	13401.0i	0.485771i
$914$	0	0
$915$	32378.0	1.16982
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 31129.0i	− 1.11736i	−0.829385	−	0.558678i	$-0.811309\pi$
$919$	− 31129.0i	− 1.11736i	0.829385	−	0.558678i	$-0.188691\pi$
$920$	0	0
$921$	26224.0	0.938232
$922$	0	0
$923$	5033.13	0.179488
$924$	0	0
$925$	28198.7	1.00234
$926$	0	0
$927$	91711.9	3.24942
$928$	0	0
$929$	− 32972.4i	− 1.16447i	−0.813021	−	0.582234i	$-0.802179\pi$
$929$	− 32972.4i	− 1.16447i	0.813021	−	0.582234i	$-0.197821\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	16024.4	0.562290
$934$	0	0
$935$	45959.6i	1.60753i
$936$	0	0
$937$	− 43429.1i	− 1.51416i	−0.653323	−	0.757080i	$-0.726625\pi$
$937$	− 43429.1i	− 1.51416i	0.653323	−	0.757080i	$-0.273375\pi$
$938$	0	0
$939$	− 40334.5i	− 1.40177i
$940$	0	0
$941$	− 20473.6i	− 0.709267i	−0.935005	−	0.354634i	$-0.884606\pi$
$941$	− 20473.6i	− 0.709267i	0.935005	−	0.354634i	$-0.115394\pi$
$942$	0	0
$943$	−11676.2	−0.403213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4525.11i	0.155276i	0.996982	+	0.0776379i	$0.0247378\pi$
$947$	4525.11i	0.155276i	−0.996982	+	0.0776379i	$0.975262\pi$
$948$	0	0
$949$	−9909.52	−0.338964
$950$	0	0
$951$	−6476.05	−0.220821
$952$	0	0
$953$	−11974.9	−0.407037	−0.203518	−	0.979071i	$-0.565238\pi$
$953$	−11974.9	−0.407037	−0.203518	+	0.979071i	$0.565238\pi$
$954$	0	0
$955$	−2904.08	−0.0984018
$956$	0	0
$957$	154274.i	5.21105i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	36046.1	1.20997
$962$	0	0
$963$	− 28382.9i	− 0.949767i
$964$	0	0
$965$	17670.4i	0.589460i
$966$	0	0
$967$	36345.2i	1.20867i	0.796730	+	0.604335i	$0.206561\pi$
$967$	36345.2i	1.20867i	−0.796730	+	0.604335i	$0.793439\pi$
$968$	0	0
$969$	− 65516.9i	− 2.17204i
$970$	0	0
$971$	−45749.3	−1.51201	−0.756006	−	0.654564i	$-0.772852\pi$
$971$	−45749.3	−1.51201	−0.756006	+	0.654564i	$0.772852\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7404.41i	0.243211i
$976$	0	0
$977$	−23883.6	−0.782091	−0.391045	−	0.920371i	$-0.627886\pi$
$977$	−23883.6	−0.782091	−0.391045	+	0.920371i	$0.627886\pi$
$978$	0	0
$979$	−1191.26	−0.0388896
$980$	0	0
$981$	54116.6	1.76127
$982$	0	0
$983$	−55247.4	−1.79259	−0.896296	−	0.443457i	$-0.853752\pi$
$983$	−55247.4	−1.79259	−0.896296	+	0.443457i	$0.853752\pi$
$984$	0	0
$985$	9520.45i	0.307966i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29853.5	0.959843
$990$	0	0
$991$	27352.7i	0.876779i	0.898785	+	0.438389i	$0.144451\pi$
$991$	27352.7i	0.876779i	−0.898785	+	0.438389i	$0.855549\pi$
$992$	0	0
$993$	79183.4i	2.53052i
$994$	0	0
$995$	6743.62i	0.214862i
$996$	0	0
$997$	26683.5i	0.847619i	0.905751	+	0.423809i	$0.139307\pi$
$997$	26683.5i	0.847619i	−0.905751	+	0.423809i	$0.860693\pi$
$998$	0	0
$999$	−88578.6	−2.80531

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.j.783.2	yes	24
4.3	odd	2	inner	784.4.f.j.783.24	yes	24
7.6	odd	2	inner	784.4.f.j.783.23	yes	24
28.27	even	2	inner	784.4.f.j.783.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.j.783.1	✓	24	28.27	even	2	inner
784.4.f.j.783.2	yes	24	1.1	even	1	trivial
784.4.f.j.783.23	yes	24	7.6	odd	2	inner
784.4.f.j.783.24	yes	24	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.86009 q^{3} -7.47612i q^{5} +51.5012 q^{9} +O(q^{10})\)	\(q-8.86009 q^{3} -7.47612i q^{5} +51.5012 q^{9} -61.2337i q^{11} -12.0928i q^{13} +66.2391i q^{15} -100.394i q^{17} -73.6555 q^{19} +96.8114i q^{23} +69.1076 q^{25} -217.083 q^{27} +284.357 q^{29} +256.587 q^{31} +542.536i q^{33} +408.040 q^{37} +107.143i q^{39} +120.608i q^{41} -308.367i q^{43} -385.029i q^{45} +26.0454 q^{47} +889.504i q^{51} -311.059 q^{53} -457.791 q^{55} +652.595 q^{57} -444.945 q^{59} -488.804i q^{61} -90.4072 q^{65} -585.937i q^{67} -857.758i q^{69} +416.209i q^{71} -819.457i q^{73} -612.300 q^{75} +318.412i q^{79} +532.843 q^{81} -218.850 q^{83} -750.561 q^{85} -2519.43 q^{87} -19.4544i q^{89} -2273.39 q^{93} +550.658i q^{95} -306.810i q^{97} -3153.61i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 24 * q + 88 * q^9	\( 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 856 * q^25 + 896 * q^29 - 2496 * q^53 + 4416 * q^57 - 4416 * q^65 - 4904 * q^81 - 2432 * q^85 - 11968 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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