LMFDB - Embedded newform 784.4.f.j.783.17

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			783.17
Character	$\chi$	$=$	784.783
Dual form			784.4.f.j.783.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.30860 q^{3} +7.29492i q^{5} -8.43597 q^{9} +O(q^{10})$	$q+4.30860 q^{3} +7.29492i q^{5} -8.43597 q^{9} +21.8273i q^{11} +30.5508i q^{13} +31.4309i q^{15} -55.0978i q^{17} +58.7072 q^{19} +79.6503i q^{23} +71.7842 q^{25} -152.679 q^{27} -116.193 q^{29} -110.020 q^{31} +94.0451i q^{33} -376.127 q^{37} +131.631i q^{39} +232.841i q^{41} +260.884i q^{43} -61.5397i q^{45} +36.5870 q^{47} -237.394i q^{51} +59.1734 q^{53} -159.228 q^{55} +252.946 q^{57} -707.304 q^{59} +323.781i q^{61} -222.865 q^{65} -239.434i q^{67} +343.181i q^{69} +887.119i q^{71} +550.887i q^{73} +309.289 q^{75} -1248.76i q^{79} -430.063 q^{81} -1393.89 q^{83} +401.934 q^{85} -500.630 q^{87} -982.787i q^{89} -474.033 q^{93} +428.264i q^{95} +441.547i q^{97} -184.135i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.30860	0.829190	0.414595	−	0.910006i	$-0.363923\pi$
$3$	4.30860	0.829190	0.414595	+	0.910006i	$0.363923\pi$
$4$	0	0
$5$	7.29492i	0.652477i	0.945287	+	0.326239i	$0.105781\pi$
$5$	7.29492i	0.652477i	−0.945287	+	0.326239i	$0.894219\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−8.43597	−0.312443
$10$	0	0
$11$	21.8273i	0.598289i	0.954208	+	0.299145i	$0.0967013\pi$
$11$	21.8273i	0.598289i	−0.954208	+	0.299145i	$0.903299\pi$
$12$	0	0
$13$	30.5508i	0.651789i	0.945406	+	0.325895i	$0.105666\pi$
$13$	30.5508i	0.651789i	−0.945406	+	0.325895i	$0.894334\pi$
$14$	0	0
$15$	31.4309i	0.541028i
$16$	0	0
$17$	− 55.0978i	− 0.786070i	−0.919524	−	0.393035i	$-0.871425\pi$
$17$	− 55.0978i	− 0.786070i	0.919524	−	0.393035i	$-0.128575\pi$
$18$	0	0
$19$	58.7072	0.708861	0.354431	−	0.935082i	$-0.384675\pi$
$19$	58.7072	0.708861	0.354431	+	0.935082i	$0.384675\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	79.6503i	0.722097i	0.932547	+	0.361049i	$0.117581\pi$
$23$	79.6503i	0.722097i	−0.932547	+	0.361049i	$0.882419\pi$
$24$	0	0
$25$	71.7842	0.574274
$26$	0	0
$27$	−152.679	−1.08827
$28$	0	0
$29$	−116.193	−0.744018	−0.372009	−	0.928229i	$-0.621331\pi$
$29$	−116.193	−0.744018	−0.372009	+	0.928229i	$0.621331\pi$
$30$	0	0
$31$	−110.020	−0.637426	−0.318713	−	0.947851i	$-0.603251\pi$
$31$	−110.020	−0.637426	−0.318713	+	0.947851i	$0.603251\pi$
$32$	0	0
$33$	94.0451i	0.496096i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−376.127	−1.67121	−0.835607	−	0.549328i	$-0.814884\pi$
$37$	−376.127	−1.67121	−0.835607	+	0.549328i	$0.814884\pi$
$38$	0	0
$39$	131.631i	0.540457i
$40$	0	0
$41$	232.841i	0.886918i	0.896295	+	0.443459i	$0.146249\pi$
$41$	232.841i	0.886918i	−0.896295	+	0.443459i	$0.853751\pi$
$42$	0	0
$43$	260.884i	0.925219i	0.886562	+	0.462610i	$0.153087\pi$
$43$	260.884i	0.925219i	−0.886562	+	0.462610i	$0.846913\pi$
$44$	0	0
$45$	− 61.5397i	− 0.203862i
$46$	0	0
$47$	36.5870	0.113548	0.0567741	−	0.998387i	$-0.481919\pi$
$47$	36.5870	0.113548	0.0567741	+	0.998387i	$0.481919\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 237.394i	− 0.651801i
$52$	0	0
$53$	59.1734	0.153360	0.0766801	−	0.997056i	$-0.475568\pi$
$53$	59.1734	0.153360	0.0766801	+	0.997056i	$0.475568\pi$
$54$	0	0
$55$	−159.228	−0.390370
$56$	0	0
$57$	252.946	0.587781
$58$	0	0
$59$	−707.304	−1.56073	−0.780366	−	0.625323i	$-0.784967\pi$
$59$	−707.304	−1.56073	−0.780366	+	0.625323i	$0.784967\pi$
$60$	0	0
$61$	323.781i	0.679605i	0.940497	+	0.339802i	$0.110360\pi$
$61$	323.781i	0.679605i	−0.940497	+	0.339802i	$0.889640\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−222.865	−0.425278
$66$	0	0
$67$	− 239.434i	− 0.436589i	−0.975883	−	0.218295i	$-0.929951\pi$
$67$	− 239.434i	− 0.436589i	0.975883	−	0.218295i	$-0.0700494\pi$
$68$	0	0
$69$	343.181i	0.598756i
$70$	0	0
$71$	887.119i	1.48284i	0.671042	+	0.741420i	$0.265847\pi$
$71$	887.119i	1.48284i	−0.671042	+	0.741420i	$0.734153\pi$
$72$	0	0
$73$	550.887i	0.883238i	0.897203	+	0.441619i	$0.145596\pi$
$73$	550.887i	0.883238i	−0.897203	+	0.441619i	$0.854404\pi$
$74$	0	0
$75$	309.289	0.476182
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1248.76i	− 1.77843i	−0.457489	−	0.889215i	$-0.651251\pi$
$79$	− 1248.76i	− 1.77843i	0.457489	−	0.889215i	$-0.348749\pi$
$80$	0	0
$81$	−430.063	−0.589936
$82$	0	0
$83$	−1393.89	−1.84336	−0.921681	−	0.387949i	$-0.873184\pi$
$83$	−1393.89	−1.84336	−0.921681	+	0.387949i	$0.873184\pi$
$84$	0	0
$85$	401.934	0.512893
$86$	0	0
$87$	−500.630	−0.616933
$88$	0	0
$89$	− 982.787i	− 1.17051i	−0.810850	−	0.585254i	$-0.800995\pi$
$89$	− 982.787i	− 1.17051i	0.810850	−	0.585254i	$-0.199005\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−474.033	−0.528548
$94$	0	0
$95$	428.264i	0.462516i
$96$	0	0
$97$	441.547i	0.462188i	0.972931	+	0.231094i	$0.0742305\pi$
$97$	441.547i	0.462188i	−0.972931	+	0.231094i	$0.925769\pi$
$98$	0	0
$99$	− 184.135i	− 0.186932i
$100$	0	0
$101$	1110.09i	1.09364i	0.837249	+	0.546822i	$0.184163\pi$
$101$	1110.09i	1.09364i	−0.837249	+	0.546822i	$0.815837\pi$
$102$	0	0
$103$	−898.555	−0.859585	−0.429792	−	0.902928i	$-0.641413\pi$
$103$	−898.555	−0.859585	−0.429792	+	0.902928i	$0.641413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1198.80i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$107$	− 1198.80i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$108$	0	0
$109$	1304.01	1.14588	0.572942	−	0.819596i	$-0.305802\pi$
$109$	1304.01	1.14588	0.572942	+	0.819596i	$0.305802\pi$
$110$	0	0
$111$	−1620.58	−1.38575
$112$	0	0
$113$	1948.52	1.62213	0.811067	−	0.584953i	$-0.198887\pi$
$113$	1948.52	1.62213	0.811067	+	0.584953i	$0.198887\pi$
$114$	0	0
$115$	−581.042	−0.471152
$116$	0	0
$117$	− 257.726i	− 0.203647i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	854.569	0.642050
$122$	0	0
$123$	1003.22i	0.735424i
$124$	0	0
$125$	1435.52i	1.02718i
$126$	0	0
$127$	2444.26i	1.70782i	0.520419	+	0.853911i	$0.325776\pi$
$127$	2444.26i	1.70782i	−0.520419	+	0.853911i	$0.674224\pi$
$128$	0	0
$129$	1124.04i	0.767183i
$130$	0	0
$131$	552.936	0.368780	0.184390	−	0.982853i	$-0.440969\pi$
$131$	552.936	0.368780	0.184390	+	0.982853i	$0.440969\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1113.78i	− 0.710068i
$136$	0	0
$137$	76.1775	0.0475057	0.0237529	−	0.999718i	$-0.492439\pi$
$137$	76.1775	0.0475057	0.0237529	+	0.999718i	$0.492439\pi$
$138$	0	0
$139$	−2410.83	−1.47111	−0.735555	−	0.677465i	$-0.763078\pi$
$139$	−2410.83	−1.47111	−0.735555	+	0.677465i	$0.763078\pi$
$140$	0	0
$141$	157.639	0.0941531
$142$	0	0
$143$	−666.841	−0.389959
$144$	0	0
$145$	− 847.619i	− 0.485455i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	745.876	0.410098	0.205049	−	0.978752i	$-0.434265\pi$
$149$	745.876	0.410098	0.205049	+	0.978752i	$0.434265\pi$
$150$	0	0
$151$	2658.43i	1.43271i	0.697735	+	0.716356i	$0.254191\pi$
$151$	2658.43i	1.43271i	−0.697735	+	0.716356i	$0.745809\pi$
$152$	0	0
$153$	464.804i	0.245602i
$154$	0	0
$155$	− 802.588i	− 0.415906i
$156$	0	0
$157$	761.304i	0.386998i	0.981100	+	0.193499i	$0.0619836\pi$
$157$	761.304i	0.386998i	−0.981100	+	0.193499i	$0.938016\pi$
$158$	0	0
$159$	254.955	0.127165
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 1239.24i	− 0.595489i	−0.954646	−	0.297745i	$-0.903766\pi$
$163$	− 1239.24i	− 0.595489i	0.954646	−	0.297745i	$-0.0962344\pi$
$164$	0	0
$165$	−686.051	−0.323691
$166$	0	0
$167$	−1306.07	−0.605192	−0.302596	−	0.953119i	$-0.597853\pi$
$167$	−1306.07	−0.605192	−0.302596	+	0.953119i	$0.597853\pi$
$168$	0	0
$169$	1263.65	0.575170
$170$	0	0
$171$	−495.253	−0.221479
$172$	0	0
$173$	1471.23i	0.646564i	0.946303	+	0.323282i	$0.104786\pi$
$173$	1471.23i	0.646564i	−0.946303	+	0.323282i	$0.895214\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3047.49	−1.29414
$178$	0	0
$179$	− 3129.95i	− 1.30695i	−0.756949	−	0.653474i	$-0.773311\pi$
$179$	− 3129.95i	− 1.30695i	0.756949	−	0.653474i	$-0.226689\pi$
$180$	0	0
$181$	1037.83i	0.426196i	0.977031	+	0.213098i	$0.0683554\pi$
$181$	1037.83i	0.426196i	−0.977031	+	0.213098i	$0.931645\pi$
$182$	0	0
$183$	1395.04i	0.563522i
$184$	0	0
$185$	− 2743.82i	− 1.09043i
$186$	0	0
$187$	1202.64	0.470297
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 229.006i	− 0.0867553i	−0.999059	−	0.0433777i	$-0.986188\pi$
$191$	− 229.006i	− 0.0867553i	0.999059	−	0.0433777i	$-0.0138119\pi$
$192$	0	0
$193$	723.322	0.269771	0.134886	−	0.990861i	$-0.456933\pi$
$193$	723.322	0.269771	0.134886	+	0.990861i	$0.456933\pi$
$194$	0	0
$195$	−960.238	−0.352636
$196$	0	0
$197$	−2377.97	−0.860018	−0.430009	−	0.902825i	$-0.641490\pi$
$197$	−2377.97	−0.860018	−0.430009	+	0.902825i	$0.641490\pi$
$198$	0	0
$199$	−1011.77	−0.360414	−0.180207	−	0.983629i	$-0.557677\pi$
$199$	−1011.77	−0.360414	−0.180207	+	0.983629i	$0.557677\pi$
$200$	0	0
$201$	− 1031.62i	− 0.362016i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1698.56	−0.578694
$206$	0	0
$207$	− 671.928i	− 0.225615i
$208$	0	0
$209$	1281.42i	0.424104i
$210$	0	0
$211$	1861.01i	0.607189i	0.952801	+	0.303595i	$0.0981869\pi$
$211$	1861.01i	0.607189i	−0.952801	+	0.303595i	$0.901813\pi$
$212$	0	0
$213$	3822.24i	1.22956i
$214$	0	0
$215$	−1903.13	−0.603684
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2373.55i	0.732373i
$220$	0	0
$221$	1683.28	0.512352
$222$	0	0
$223$	6228.99	1.87051	0.935256	−	0.353972i	$-0.115169\pi$
$223$	6228.99	1.87051	0.935256	+	0.353972i	$0.115169\pi$
$224$	0	0
$225$	−605.570	−0.179428
$226$	0	0
$227$	−2708.74	−0.792007	−0.396004	−	0.918249i	$-0.629603\pi$
$227$	−2708.74	−0.792007	−0.396004	+	0.918249i	$0.629603\pi$
$228$	0	0
$229$	2794.23i	0.806321i	0.915129	+	0.403161i	$0.132088\pi$
$229$	2794.23i	0.806321i	−0.915129	+	0.403161i	$0.867912\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2110.75	−0.593476	−0.296738	−	0.954959i	$-0.595899\pi$
$233$	−2110.75	−0.593476	−0.296738	+	0.954959i	$0.595899\pi$
$234$	0	0
$235$	266.899i	0.0740876i
$236$	0	0
$237$	− 5380.39i	− 1.47466i
$238$	0	0
$239$	− 106.409i	− 0.0287993i	−0.999896	−	0.0143997i	$-0.995416\pi$
$239$	− 106.409i	− 0.0287993i	0.999896	−	0.0143997i	$-0.00458372\pi$
$240$	0	0
$241$	− 7211.58i	− 1.92755i	−0.266723	−	0.963773i	$-0.585941\pi$
$241$	− 7211.58i	− 1.92755i	0.266723	−	0.963773i	$-0.414059\pi$
$242$	0	0
$243$	2269.37	0.599097
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1793.55i	0.462028i
$248$	0	0
$249$	−6005.70	−1.52850
$250$	0	0
$251$	925.020	0.232616	0.116308	−	0.993213i	$-0.462894\pi$
$251$	925.020	0.232616	0.116308	+	0.993213i	$0.462894\pi$
$252$	0	0
$253$	−1738.55	−0.432023
$254$	0	0
$255$	1731.77	0.425286
$256$	0	0
$257$	2766.80i	0.671549i	0.941942	+	0.335775i	$0.108998\pi$
$257$	2766.80i	0.671549i	−0.941942	+	0.335775i	$0.891002\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	980.202	0.232464
$262$	0	0
$263$	− 2199.92i	− 0.515791i	−0.966173	−	0.257895i	$-0.916971\pi$
$263$	− 2199.92i	− 0.515791i	0.966173	−	0.257895i	$-0.0830290\pi$
$264$	0	0
$265$	431.665i	0.100064i
$266$	0	0
$267$	− 4234.43i	− 0.970573i
$268$	0	0
$269$	1265.42i	0.286819i	0.989663	+	0.143409i	$0.0458065\pi$
$269$	1265.42i	0.286819i	−0.989663	+	0.143409i	$0.954193\pi$
$270$	0	0
$271$	−486.107	−0.108963	−0.0544814	−	0.998515i	$-0.517351\pi$
$271$	−486.107	−0.108963	−0.0544814	+	0.998515i	$0.517351\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1566.86i	0.343582i
$276$	0	0
$277$	3926.80	0.851764	0.425882	−	0.904779i	$-0.359964\pi$
$277$	3926.80	0.851764	0.425882	+	0.904779i	$0.359964\pi$
$278$	0	0
$279$	928.128	0.199160
$280$	0	0
$281$	−3224.97	−0.684646	−0.342323	−	0.939582i	$-0.611214\pi$
$281$	−3224.97	−0.684646	−0.342323	+	0.939582i	$0.611214\pi$
$282$	0	0
$283$	2706.20	0.568434	0.284217	−	0.958760i	$-0.408266\pi$
$283$	2706.20	0.568434	0.284217	+	0.958760i	$0.408266\pi$
$284$	0	0
$285$	1845.22i	0.383514i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1877.23	0.382094
$290$	0	0
$291$	1902.45i	0.383242i
$292$	0	0
$293$	2199.96i	0.438645i	0.975652	+	0.219323i	$0.0703847\pi$
$293$	2199.96i	0.438645i	−0.975652	+	0.219323i	$0.929615\pi$
$294$	0	0
$295$	− 5159.72i	− 1.01834i
$296$	0	0
$297$	− 3332.58i	− 0.651097i
$298$	0	0
$299$	−2433.38	−0.470655
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4782.93i	0.906839i
$304$	0	0
$305$	−2361.95	−0.443427
$306$	0	0
$307$	8312.27	1.54530	0.772649	−	0.634833i	$-0.218931\pi$
$307$	8312.27	1.54530	0.772649	+	0.634833i	$0.218931\pi$
$308$	0	0
$309$	−3871.51	−0.712759
$310$	0	0
$311$	7991.41	1.45708	0.728539	−	0.685004i	$-0.240200\pi$
$311$	7991.41	1.45708	0.728539	+	0.685004i	$0.240200\pi$
$312$	0	0
$313$	− 6852.64i	− 1.23749i	−0.785592	−	0.618744i	$-0.787642\pi$
$313$	− 6852.64i	− 1.23749i	0.785592	−	0.618744i	$-0.212358\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10508.0	1.86180	0.930898	−	0.365280i	$-0.119027\pi$
$317$	10508.0	1.86180	0.930898	+	0.365280i	$0.119027\pi$
$318$	0	0
$319$	− 2536.18i	− 0.445138i
$320$	0	0
$321$	− 5165.13i	− 0.898099i
$322$	0	0
$323$	− 3234.64i	− 0.557214i
$324$	0	0
$325$	2193.06i	0.374305i
$326$	0	0
$327$	5618.45	0.950157
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4480.16i	0.743963i	0.928240	+	0.371981i	$0.121321\pi$
$331$	4480.16i	0.743963i	−0.928240	+	0.371981i	$0.878679\pi$
$332$	0	0
$333$	3173.00	0.522160
$334$	0	0
$335$	1746.65	0.284865
$336$	0	0
$337$	7920.93	1.28036	0.640179	−	0.768226i	$-0.278860\pi$
$337$	7920.93	1.28036	0.640179	+	0.768226i	$0.278860\pi$
$338$	0	0
$339$	8395.39	1.34506
$340$	0	0
$341$	− 2401.45i	− 0.381365i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2503.48	−0.390675
$346$	0	0
$347$	− 1483.19i	− 0.229457i	−0.993397	−	0.114729i	$-0.963400\pi$
$347$	− 1483.19i	− 0.229457i	0.993397	−	0.114729i	$-0.0365999\pi$
$348$	0	0
$349$	− 8732.67i	− 1.33940i	−0.742634	−	0.669698i	$-0.766424\pi$
$349$	− 8732.67i	− 1.33940i	0.742634	−	0.669698i	$-0.233576\pi$
$350$	0	0
$351$	− 4664.48i	− 0.709320i
$352$	0	0
$353$	− 7881.53i	− 1.18836i	−0.804331	−	0.594181i	$-0.797476\pi$
$353$	− 7881.53i	− 1.18836i	0.804331	−	0.594181i	$-0.202524\pi$
$354$	0	0
$355$	−6471.46	−0.967519
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7699.83i	1.13198i	0.824412	+	0.565991i	$0.191506\pi$
$359$	7699.83i	1.13198i	−0.824412	+	0.565991i	$0.808494\pi$
$360$	0	0
$361$	−3412.46	−0.497516
$362$	0	0
$363$	3681.99	0.532382
$364$	0	0
$365$	−4018.67	−0.576293
$366$	0	0
$367$	1261.06	0.179365	0.0896825	−	0.995970i	$-0.471415\pi$
$367$	1261.06	0.179365	0.0896825	+	0.995970i	$0.471415\pi$
$368$	0	0
$369$	− 1964.24i	− 0.277112i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10012.5	1.38989	0.694944	−	0.719064i	$-0.255429\pi$
$373$	10012.5	1.38989	0.694944	+	0.719064i	$0.255429\pi$
$374$	0	0
$375$	6185.10i	0.851726i
$376$	0	0
$377$	− 3549.79i	− 0.484943i
$378$	0	0
$379$	− 2464.21i	− 0.333979i	−0.985959	−	0.166990i	$-0.946595\pi$
$379$	− 2464.21i	− 0.333979i	0.985959	−	0.166990i	$-0.0534046\pi$
$380$	0	0
$381$	10531.4i	1.41611i
$382$	0	0
$383$	−7470.63	−0.996688	−0.498344	−	0.866979i	$-0.666058\pi$
$383$	−7470.63	−0.996688	−0.498344	+	0.866979i	$0.666058\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2200.81i	− 0.289079i
$388$	0	0
$389$	2044.68	0.266503	0.133251	−	0.991082i	$-0.457458\pi$
$389$	2044.68	0.266503	0.133251	+	0.991082i	$0.457458\pi$
$390$	0	0
$391$	4388.56	0.567619
$392$	0	0
$393$	2382.38	0.305789
$394$	0	0
$395$	9109.57	1.16039
$396$	0	0
$397$	2251.63i	0.284649i	0.989820	+	0.142325i	$0.0454577\pi$
$397$	2251.63i	0.284649i	−0.989820	+	0.142325i	$0.954542\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10049.5	1.25149	0.625745	−	0.780028i	$-0.284795\pi$
$401$	10049.5	1.25149	0.625745	+	0.780028i	$0.284795\pi$
$402$	0	0
$403$	− 3361.20i	− 0.415468i
$404$	0	0
$405$	− 3137.27i	− 0.384919i
$406$	0	0
$407$	− 8209.84i	− 0.999869i
$408$	0	0
$409$	− 7448.43i	− 0.900492i	−0.892905	−	0.450246i	$-0.851336\pi$
$409$	− 7448.43i	− 0.900492i	0.892905	−	0.450246i	$-0.148664\pi$
$410$	0	0
$411$	328.218	0.0393913
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 10168.3i	− 1.20275i
$416$	0	0
$417$	−10387.3	−1.21983
$418$	0	0
$419$	10836.7	1.26350	0.631751	−	0.775171i	$-0.282337\pi$
$419$	10836.7	1.26350	0.631751	+	0.775171i	$0.282337\pi$
$420$	0	0
$421$	−13758.0	−1.59270	−0.796348	−	0.604839i	$-0.793237\pi$
$421$	−13758.0	−1.59270	−0.796348	+	0.604839i	$0.793237\pi$
$422$	0	0
$423$	−308.647	−0.0354774
$424$	0	0
$425$	− 3955.15i	− 0.451419i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2873.15	−0.323350
$430$	0	0
$431$	− 16039.8i	− 1.79260i	−0.443448	−	0.896300i	$-0.646245\pi$
$431$	− 16039.8i	− 1.79260i	0.443448	−	0.896300i	$-0.353755\pi$
$432$	0	0
$433$	− 61.8807i	− 0.00686789i	−0.999994	−	0.00343395i	$-0.998907\pi$
$433$	− 61.8807i	− 0.00686789i	0.999994	−	0.00343395i	$-0.00109306\pi$
$434$	0	0
$435$	− 3652.05i	− 0.402534i
$436$	0	0
$437$	4676.05i	0.511867i
$438$	0	0
$439$	9208.19	1.00110	0.500550	−	0.865708i	$-0.333131\pi$
$439$	9208.19	1.00110	0.500550	+	0.865708i	$0.333131\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11063.1i	1.18651i	0.805014	+	0.593256i	$0.202158\pi$
$443$	11063.1i	1.18651i	−0.805014	+	0.593256i	$0.797842\pi$
$444$	0	0
$445$	7169.35	0.763729
$446$	0	0
$447$	3213.68	0.340049
$448$	0	0
$449$	12834.1	1.34895	0.674473	−	0.738299i	$-0.264371\pi$
$449$	12834.1	1.34895	0.674473	+	0.738299i	$0.264371\pi$
$450$	0	0
$451$	−5082.29	−0.530634
$452$	0	0
$453$	11454.1i	1.18799i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3870.88	−0.396220	−0.198110	−	0.980180i	$-0.563480\pi$
$457$	−3870.88	−0.396220	−0.198110	+	0.980180i	$0.563480\pi$
$458$	0	0
$459$	8412.30i	0.855453i
$460$	0	0
$461$	3071.29i	0.310291i	0.987892	+	0.155146i	$0.0495846\pi$
$461$	3071.29i	0.310291i	−0.987892	+	0.155146i	$0.950415\pi$
$462$	0	0
$463$	− 10063.3i	− 1.01011i	−0.863088	−	0.505054i	$-0.831473\pi$
$463$	− 10063.3i	− 1.01011i	0.863088	−	0.505054i	$-0.168527\pi$
$464$	0	0
$465$	− 3458.03i	− 0.344865i
$466$	0	0
$467$	−7436.68	−0.736892	−0.368446	−	0.929649i	$-0.620110\pi$
$467$	−7436.68	−0.736892	−0.368446	+	0.929649i	$0.620110\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3280.15i	0.320895i
$472$	0	0
$473$	−5694.39	−0.553549
$474$	0	0
$475$	4214.25	0.407080
$476$	0	0
$477$	−499.185	−0.0479164
$478$	0	0
$479$	−11401.1	−1.08753	−0.543767	−	0.839236i	$-0.683003\pi$
$479$	−11401.1	−1.08753	−0.543767	+	0.839236i	$0.683003\pi$
$480$	0	0
$481$	− 11491.0i	− 1.08928i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3221.05	−0.301567
$486$	0	0
$487$	− 1925.77i	− 0.179189i	−0.995978	−	0.0895945i	$-0.971443\pi$
$487$	− 1925.77i	− 0.179189i	0.995978	−	0.0895945i	$-0.0285571\pi$
$488$	0	0
$489$	− 5339.39i	− 0.493774i
$490$	0	0
$491$	20769.7i	1.90901i	0.298192	+	0.954506i	$0.403617\pi$
$491$	20769.7i	1.90901i	−0.298192	+	0.954506i	$0.596383\pi$
$492$	0	0
$493$	6401.99i	0.584850i
$494$	0	0
$495$	1343.25	0.121969
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 4017.64i	− 0.360429i	−0.983627	−	0.180215i	$-0.942321\pi$
$499$	− 4017.64i	− 0.360429i	0.983627	−	0.180215i	$-0.0576792\pi$
$500$	0	0
$501$	−5627.35	−0.501819
$502$	0	0
$503$	−12683.3	−1.12429	−0.562147	−	0.827037i	$-0.690025\pi$
$503$	−12683.3	−1.12429	−0.562147	+	0.827037i	$0.690025\pi$
$504$	0	0
$505$	−8098.01	−0.713578
$506$	0	0
$507$	5444.56	0.476926
$508$	0	0
$509$	16422.8i	1.43011i	0.699068	+	0.715055i	$0.253598\pi$
$509$	16422.8i	1.43011i	−0.699068	+	0.715055i	$0.746402\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8963.39	−0.771429
$514$	0	0
$515$	− 6554.88i	− 0.560859i
$516$	0	0
$517$	798.596i	0.0679347i
$518$	0	0
$519$	6338.94i	0.536124i
$520$	0	0
$521$	− 12611.5i	− 1.06050i	−0.847843	−	0.530248i	$-0.822099\pi$
$521$	− 12611.5i	− 1.06050i	0.847843	−	0.530248i	$-0.177901\pi$
$522$	0	0
$523$	8227.18	0.687858	0.343929	−	0.938996i	$-0.388242\pi$
$523$	8227.18	0.687858	0.343929	+	0.938996i	$0.388242\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6061.88i	0.501062i
$528$	0	0
$529$	5822.83	0.478575
$530$	0	0
$531$	5966.80	0.487640
$532$	0	0
$533$	−7113.47	−0.578084
$534$	0	0
$535$	8745.12	0.706700
$536$	0	0
$537$	− 13485.7i	− 1.08371i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−14128.7	−1.12281	−0.561404	−	0.827542i	$-0.689739\pi$
$541$	−14128.7	−1.12281	−0.561404	+	0.827542i	$0.689739\pi$
$542$	0	0
$543$	4471.60i	0.353398i
$544$	0	0
$545$	9512.64i	0.747664i
$546$	0	0
$547$	5978.72i	0.467334i	0.972317	+	0.233667i	$0.0750725\pi$
$547$	5978.72i	0.467334i	−0.972317	+	0.233667i	$0.924927\pi$
$548$	0	0
$549$	− 2731.41i	− 0.212338i
$550$	0	0
$551$	−6821.38	−0.527406
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 11822.0i	− 0.904173i
$556$	0	0
$557$	17085.3	1.29969	0.649844	−	0.760068i	$-0.274834\pi$
$557$	17085.3	1.29969	0.649844	+	0.760068i	$0.274834\pi$
$558$	0	0
$559$	−7970.21	−0.603048
$560$	0	0
$561$	5181.68	0.389966
$562$	0	0
$563$	−18164.9	−1.35979	−0.679893	−	0.733312i	$-0.737974\pi$
$563$	−18164.9	−1.35979	−0.679893	+	0.733312i	$0.737974\pi$
$564$	0	0
$565$	14214.3i	1.05841i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2716.25	−0.200125	−0.100063	−	0.994981i	$-0.531904\pi$
$569$	−2716.25	−0.200125	−0.100063	+	0.994981i	$0.531904\pi$
$570$	0	0
$571$	16040.2i	1.17559i	0.809009	+	0.587796i	$0.200004\pi$
$571$	16040.2i	1.17559i	−0.809009	+	0.587796i	$0.799996\pi$
$572$	0	0
$573$	− 986.694i	− 0.0719367i
$574$	0	0
$575$	5717.63i	0.414681i
$576$	0	0
$577$	− 8635.18i	− 0.623028i	−0.950242	−	0.311514i	$-0.899164\pi$
$577$	− 8635.18i	− 0.623028i	0.950242	−	0.311514i	$-0.100836\pi$
$578$	0	0
$579$	3116.51	0.223692
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1291.60i	0.0917538i
$584$	0	0
$585$	1880.09	0.132875
$586$	0	0
$587$	−27896.0	−1.96148	−0.980742	−	0.195307i	$-0.937430\pi$
$587$	−27896.0	−1.96148	−0.980742	+	0.195307i	$0.937430\pi$
$588$	0	0
$589$	−6458.98	−0.451847
$590$	0	0
$591$	−10245.7	−0.713119
$592$	0	0
$593$	11146.7i	0.771903i	0.922519	+	0.385952i	$0.126127\pi$
$593$	11146.7i	0.771903i	−0.922519	+	0.385952i	$0.873873\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4359.31	−0.298852
$598$	0	0
$599$	2535.71i	0.172966i	0.996253	+	0.0864829i	$0.0275628\pi$
$599$	2535.71i	0.172966i	−0.996253	+	0.0864829i	$0.972437\pi$
$600$	0	0
$601$	24136.0i	1.63815i	0.573689	+	0.819073i	$0.305512\pi$
$601$	24136.0i	1.63815i	−0.573689	+	0.819073i	$0.694488\pi$
$602$	0	0
$603$	2019.86i	0.136409i
$604$	0	0
$605$	6234.01i	0.418923i
$606$	0	0
$607$	26255.6	1.75565	0.877825	−	0.478981i	$-0.158994\pi$
$607$	26255.6	1.75565	0.877825	+	0.478981i	$0.158994\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1117.76i	0.0740095i
$612$	0	0
$613$	3121.21	0.205651	0.102826	−	0.994699i	$-0.467212\pi$
$613$	3121.21	0.205651	0.102826	+	0.994699i	$0.467212\pi$
$614$	0	0
$615$	−7318.39	−0.479847
$616$	0	0
$617$	24316.0	1.58659	0.793295	−	0.608837i	$-0.208364\pi$
$617$	24316.0	1.58659	0.793295	+	0.608837i	$0.208364\pi$
$618$	0	0
$619$	−6864.03	−0.445701	−0.222850	−	0.974853i	$-0.571536\pi$
$619$	−6864.03	−0.445701	−0.222850	+	0.974853i	$0.571536\pi$
$620$	0	0
$621$	− 12161.0i	− 0.785834i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1499.00	−0.0959362
$626$	0	0
$627$	5521.13i	0.351663i
$628$	0	0
$629$	20723.8i	1.31369i
$630$	0	0
$631$	5366.60i	0.338575i	0.985567	+	0.169288i	$0.0541467\pi$
$631$	5366.60i	0.338575i	−0.985567	+	0.169288i	$0.945853\pi$
$632$	0	0
$633$	8018.33i	0.503475i
$634$	0	0
$635$	−17830.7	−1.11431
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 7483.71i	− 0.463304i
$640$	0	0
$641$	−30589.1	−1.88486	−0.942431	−	0.334399i	$-0.891467\pi$
$641$	−30589.1	−1.88486	−0.942431	+	0.334399i	$0.891467\pi$
$642$	0	0
$643$	−15703.2	−0.963098	−0.481549	−	0.876419i	$-0.659926\pi$
$643$	−15703.2	−0.963098	−0.481549	+	0.876419i	$0.659926\pi$
$644$	0	0
$645$	−8199.81	−0.500569
$646$	0	0
$647$	620.287	0.0376909	0.0188454	−	0.999822i	$-0.494001\pi$
$647$	620.287	0.0376909	0.0188454	+	0.999822i	$0.494001\pi$
$648$	0	0
$649$	− 15438.5i	− 0.933769i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2820.96	0.169055	0.0845275	−	0.996421i	$-0.473062\pi$
$653$	2820.96	0.169055	0.0845275	+	0.996421i	$0.473062\pi$
$654$	0	0
$655$	4033.62i	0.240621i
$656$	0	0
$657$	− 4647.27i	− 0.275962i
$658$	0	0
$659$	12783.8i	0.755668i	0.925873	+	0.377834i	$0.123331\pi$
$659$	12783.8i	0.755668i	−0.925873	+	0.377834i	$0.876669\pi$
$660$	0	0
$661$	26233.0i	1.54364i	0.635841	+	0.771820i	$0.280653\pi$
$661$	26233.0i	1.54364i	−0.635841	+	0.771820i	$0.719347\pi$
$662$	0	0
$663$	7252.59	0.424837
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9254.82i	− 0.537254i
$668$	0	0
$669$	26838.2	1.55101
$670$	0	0
$671$	−7067.27	−0.406600
$672$	0	0
$673$	11105.6	0.636091	0.318046	−	0.948075i	$-0.396973\pi$
$673$	11105.6	0.636091	0.318046	+	0.948075i	$0.396973\pi$
$674$	0	0
$675$	−10960.0	−0.624962
$676$	0	0
$677$	18882.3i	1.07195i	0.844235	+	0.535973i	$0.180055\pi$
$677$	18882.3i	1.07195i	−0.844235	+	0.535973i	$0.819945\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11670.9	−0.656725
$682$	0	0
$683$	5600.98i	0.313785i	0.987616	+	0.156893i	$0.0501477\pi$
$683$	5600.98i	0.313785i	−0.987616	+	0.156893i	$0.949852\pi$
$684$	0	0
$685$	555.709i	0.0309964i
$686$	0	0
$687$	12039.2i	0.668594i
$688$	0	0
$689$	1807.79i	0.0999586i
$690$	0	0
$691$	7871.27	0.433339	0.216670	−	0.976245i	$-0.430481\pi$
$691$	7871.27	0.433339	0.216670	+	0.976245i	$0.430481\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 17586.8i	− 0.959865i
$696$	0	0
$697$	12829.0	0.697180
$698$	0	0
$699$	−9094.39	−0.492105
$700$	0	0
$701$	366.142	0.0197275	0.00986377	−	0.999951i	$-0.496860\pi$
$701$	366.142	0.0197275	0.00986377	+	0.999951i	$0.496860\pi$
$702$	0	0
$703$	−22081.4	−1.18466
$704$	0	0
$705$	1149.96i	0.0614327i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6287.99	0.333076	0.166538	−	0.986035i	$-0.446741\pi$
$709$	6287.99	0.333076	0.166538	+	0.986035i	$0.446741\pi$
$710$	0	0
$711$	10534.5i	0.555659i
$712$	0	0
$713$	− 8763.15i	− 0.460284i
$714$	0	0
$715$	− 4864.55i	− 0.254439i
$716$	0	0
$717$	− 458.475i	− 0.0238801i
$718$	0	0
$719$	−37302.8	−1.93485	−0.967427	−	0.253149i	$-0.918534\pi$
$719$	−37302.8	−1.93485	−0.967427	+	0.253149i	$0.918534\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 31071.8i	− 1.59830i
$724$	0	0
$725$	−8340.83	−0.427270
$726$	0	0
$727$	−21013.4	−1.07200	−0.536000	−	0.844218i	$-0.680065\pi$
$727$	−21013.4	−1.07200	−0.536000	+	0.844218i	$0.680065\pi$
$728$	0	0
$729$	21389.5	1.08670
$730$	0	0
$731$	14374.1	0.727287
$732$	0	0
$733$	− 28825.5i	− 1.45252i	−0.687423	−	0.726258i	$-0.741258\pi$
$733$	− 28825.5i	− 1.45252i	0.687423	−	0.726258i	$-0.258742\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5226.19	0.261207
$738$	0	0
$739$	− 15856.8i	− 0.789312i	−0.918829	−	0.394656i	$-0.870864\pi$
$739$	− 15856.8i	− 0.789312i	0.918829	−	0.394656i	$-0.129136\pi$
$740$	0	0
$741$	7727.70i	0.383109i
$742$	0	0
$743$	17589.1i	0.868481i	0.900797	+	0.434241i	$0.142983\pi$
$743$	17589.1i	0.868481i	−0.900797	+	0.434241i	$0.857017\pi$
$744$	0	0
$745$	5441.11i	0.267579i
$746$	0	0
$747$	11758.8	0.575946
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 37263.1i	− 1.81059i	−0.424786	−	0.905294i	$-0.639651\pi$
$751$	− 37263.1i	− 1.81059i	0.424786	−	0.905294i	$-0.360349\pi$
$752$	0	0
$753$	3985.54	0.192883
$754$	0	0
$755$	−19393.0	−0.934812
$756$	0	0
$757$	28065.2	1.34748	0.673742	−	0.738967i	$-0.264686\pi$
$757$	28065.2	1.34748	0.673742	+	0.738967i	$0.264686\pi$
$758$	0	0
$759$	−7490.72	−0.358229
$760$	0	0
$761$	− 13162.9i	− 0.627013i	−0.949586	−	0.313506i	$-0.898496\pi$
$761$	− 13162.9i	− 0.627013i	0.949586	−	0.313506i	$-0.101504\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3390.71	−0.160250
$766$	0	0
$767$	− 21608.7i	− 1.01727i
$768$	0	0
$769$	86.1491i	0.00403981i	0.999998	+	0.00201991i	$0.000642957\pi$
$769$	86.1491i	0.00403981i	−0.999998	+	0.00201991i	$0.999357\pi$
$770$	0	0
$771$	11921.0i	0.556842i
$772$	0	0
$773$	21795.5i	1.01414i	0.861905	+	0.507070i	$0.169272\pi$
$773$	21795.5i	1.01414i	−0.861905	+	0.507070i	$0.830728\pi$
$774$	0	0
$775$	−7897.71	−0.366057
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13669.4i	0.628702i
$780$	0	0
$781$	−19363.4	−0.887167
$782$	0	0
$783$	17740.3	0.809689
$784$	0	0
$785$	−5553.65	−0.252507
$786$	0	0
$787$	−15336.7	−0.694655	−0.347328	−	0.937744i	$-0.612911\pi$
$787$	−15336.7	−0.694655	−0.347328	+	0.937744i	$0.612911\pi$
$788$	0	0
$789$	− 9478.58i	− 0.427689i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9891.76	−0.442959
$794$	0	0
$795$	1859.87i	0.0829721i
$796$	0	0
$797$	6023.38i	0.267703i	0.991001	+	0.133851i	$0.0427345\pi$
$797$	6023.38i	0.267703i	−0.991001	+	0.133851i	$0.957266\pi$
$798$	0	0
$799$	− 2015.87i	− 0.0892568i
$800$	0	0
$801$	8290.76i	0.365717i
$802$	0	0
$803$	−12024.4	−0.528432
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5452.20i	0.237827i
$808$	0	0
$809$	−10694.5	−0.464770	−0.232385	−	0.972624i	$-0.574653\pi$
$809$	−10694.5	−0.464770	−0.232385	+	0.972624i	$0.574653\pi$
$810$	0	0
$811$	11140.5	0.482362	0.241181	−	0.970480i	$-0.422465\pi$
$811$	11140.5	0.482362	0.241181	+	0.970480i	$0.422465\pi$
$812$	0	0
$813$	−2094.44	−0.0903508
$814$	0	0
$815$	9040.15	0.388543
$816$	0	0
$817$	15315.8i	0.655852i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20793.3	0.883912	0.441956	−	0.897037i	$-0.354285\pi$
$821$	20793.3	0.883912	0.441956	+	0.897037i	$0.354285\pi$
$822$	0	0
$823$	34204.9i	1.44874i	0.689414	+	0.724368i	$0.257868\pi$
$823$	34204.9i	1.44874i	−0.689414	+	0.724368i	$0.742132\pi$
$824$	0	0
$825$	6750.95i	0.284895i
$826$	0	0
$827$	− 13200.0i	− 0.555028i	−0.960722	−	0.277514i	$-0.910490\pi$
$827$	− 13200.0i	− 0.555028i	0.960722	−	0.277514i	$-0.0895104\pi$
$828$	0	0
$829$	21786.2i	0.912745i	0.889789	+	0.456373i	$0.150852\pi$
$829$	21786.2i	0.912745i	−0.889789	+	0.456373i	$0.849148\pi$
$830$	0	0
$831$	16919.0	0.706275
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 9527.70i	− 0.394874i
$836$	0	0
$837$	16797.8	0.693689
$838$	0	0
$839$	−34045.3	−1.40092	−0.700462	−	0.713690i	$-0.747022\pi$
$839$	−34045.3	−1.40092	−0.700462	+	0.713690i	$0.747022\pi$
$840$	0	0
$841$	−10888.2	−0.446437
$842$	0	0
$843$	−13895.1	−0.567702
$844$	0	0
$845$	9218.22i	0.375286i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11659.9	0.471340
$850$	0	0
$851$	− 29958.7i	− 1.20678i
$852$	0	0
$853$	− 24531.8i	− 0.984703i	−0.870397	−	0.492351i	$-0.836137\pi$
$853$	− 24531.8i	− 0.984703i	0.870397	−	0.492351i	$-0.163863\pi$
$854$	0	0
$855$	− 3612.83i	− 0.144510i
$856$	0	0
$857$	− 704.704i	− 0.0280890i	−0.999901	−	0.0140445i	$-0.995529\pi$
$857$	− 704.704i	− 0.0280890i	0.999901	−	0.0140445i	$-0.00447064\pi$
$858$	0	0
$859$	44971.5	1.78627	0.893137	−	0.449785i	$-0.148499\pi$
$859$	44971.5	1.78627	0.893137	+	0.449785i	$0.148499\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10675.4i	0.421084i	0.977585	+	0.210542i	$0.0675229\pi$
$863$	10675.4i	0.421084i	−0.977585	+	0.210542i	$0.932477\pi$
$864$	0	0
$865$	−10732.5	−0.421868
$866$	0	0
$867$	8088.23	0.316829
$868$	0	0
$869$	27257.0	1.06402
$870$	0	0
$871$	7314.89	0.284564
$872$	0	0
$873$	− 3724.88i	− 0.144408i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21401.4	−0.824030	−0.412015	−	0.911177i	$-0.635175\pi$
$877$	−21401.4	−0.824030	−0.412015	+	0.911177i	$0.635175\pi$
$878$	0	0
$879$	9478.75i	0.363720i
$880$	0	0
$881$	45960.0i	1.75758i	0.477205	+	0.878792i	$0.341650\pi$
$881$	45960.0i	1.75758i	−0.477205	+	0.878792i	$0.658350\pi$
$882$	0	0
$883$	− 22863.2i	− 0.871357i	−0.900102	−	0.435679i	$-0.856508\pi$
$883$	− 22863.2i	− 0.871357i	0.900102	−	0.435679i	$-0.143492\pi$
$884$	0	0
$885$	− 22231.2i	− 0.844399i
$886$	0	0
$887$	−28962.2	−1.09634	−0.548170	−	0.836367i	$-0.684675\pi$
$887$	−28962.2	−1.09634	−0.548170	+	0.836367i	$0.684675\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 9387.12i	− 0.352952i
$892$	0	0
$893$	2147.92	0.0804899
$894$	0	0
$895$	22832.7	0.852754
$896$	0	0
$897$	−10484.5	−0.390263
$898$	0	0
$899$	12783.6	0.474257
$900$	0	0
$901$	− 3260.33i	− 0.120552i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7570.90	−0.278083
$906$	0	0
$907$	12131.8i	0.444133i	0.975032	+	0.222067i	$0.0712802\pi$
$907$	12131.8i	0.444133i	−0.975032	+	0.222067i	$0.928720\pi$
$908$	0	0
$909$	− 9364.69i	− 0.341702i
$910$	0	0
$911$	37975.1i	1.38109i	0.723291	+	0.690543i	$0.242629\pi$
$911$	37975.1i	1.38109i	−0.723291	+	0.690543i	$0.757371\pi$
$912$	0	0
$913$	− 30424.8i	− 1.10286i
$914$	0	0
$915$	−10176.7	−0.367685
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8404.14i	0.301662i	0.988560	+	0.150831i	$0.0481949\pi$
$919$	8404.14i	0.301662i	−0.988560	+	0.150831i	$0.951805\pi$
$920$	0	0
$921$	35814.3	1.28135
$922$	0	0
$923$	−27102.2	−0.966499
$924$	0	0
$925$	−27000.0	−0.959734
$926$	0	0
$927$	7580.18	0.268572
$928$	0	0
$929$	34512.7i	1.21886i	0.792838	+	0.609432i	$0.208603\pi$
$929$	34512.7i	1.21886i	−0.792838	+	0.609432i	$0.791397\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	34431.8	1.20820
$934$	0	0
$935$	8773.14i	0.306858i
$936$	0	0
$937$	− 41349.8i	− 1.44166i	−0.693110	−	0.720832i	$-0.743760\pi$
$937$	− 41349.8i	− 1.44166i	0.693110	−	0.720832i	$-0.256240\pi$
$938$	0	0
$939$	− 29525.3i	− 1.02611i
$940$	0	0
$941$	− 19799.0i	− 0.685898i	−0.939354	−	0.342949i	$-0.888574\pi$
$941$	− 19799.0i	− 0.685898i	0.939354	−	0.342949i	$-0.111426\pi$
$942$	0	0
$943$	−18545.9	−0.640441
$944$	0	0
$945$	0	0
$946$	0	0
$947$	54100.3i	1.85641i	0.372066	+	0.928206i	$0.378649\pi$
$947$	54100.3i	1.85641i	−0.372066	+	0.928206i	$0.621351\pi$
$948$	0	0
$949$	−16830.0	−0.575685
$950$	0	0
$951$	45274.8	1.54378
$952$	0	0
$953$	−36760.6	−1.24952	−0.624761	−	0.780816i	$-0.714803\pi$
$953$	−36760.6	−1.24952	−0.624761	+	0.780816i	$0.714803\pi$
$954$	0	0
$955$	1670.58	0.0566059
$956$	0	0
$957$	− 10927.4i	− 0.369104i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17686.5	−0.593688
$962$	0	0
$963$	10113.0i	0.338409i
$964$	0	0
$965$	5276.58i	0.176020i
$966$	0	0
$967$	3530.73i	0.117415i	0.998275	+	0.0587077i	$0.0186980\pi$
$967$	3530.73i	0.117415i	−0.998275	+	0.0587077i	$0.981302\pi$
$968$	0	0
$969$	− 13936.8i	− 0.462037i
$970$	0	0
$971$	−20605.2	−0.681000	−0.340500	−	0.940245i	$-0.610596\pi$
$971$	−20605.2	−0.681000	−0.340500	+	0.940245i	$0.610596\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	9449.03i	0.310370i
$976$	0	0
$977$	−6352.62	−0.208023	−0.104011	−	0.994576i	$-0.533168\pi$
$977$	−6352.62	−0.208023	−0.104011	+	0.994576i	$0.533168\pi$
$978$	0	0
$979$	21451.6	0.700302
$980$	0	0
$981$	−11000.6	−0.358024
$982$	0	0
$983$	5353.15	0.173692	0.0868459	−	0.996222i	$-0.472321\pi$
$983$	5353.15	0.173692	0.0868459	+	0.996222i	$0.472321\pi$
$984$	0	0
$985$	− 17347.1i	− 0.561142i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20779.5	−0.668098
$990$	0	0
$991$	6358.28i	0.203812i	0.994794	+	0.101906i	$0.0324941\pi$
$991$	6358.28i	0.203812i	−0.994794	+	0.101906i	$0.967506\pi$
$992$	0	0
$993$	19303.2i	0.616887i
$994$	0	0
$995$	− 7380.77i	− 0.235162i
$996$	0	0
$997$	− 27647.2i	− 0.878232i	−0.898430	−	0.439116i	$-0.855292\pi$
$997$	− 27647.2i	− 0.878232i	0.898430	−	0.439116i	$-0.144708\pi$
$998$	0	0
$999$	57426.9	1.81872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.j.783.17	yes	24
4.3	odd	2	inner	784.4.f.j.783.7	✓	24
7.6	odd	2	inner	784.4.f.j.783.8	yes	24
28.27	even	2	inner	784.4.f.j.783.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.j.783.7	✓	24	4.3	odd	2	inner
784.4.f.j.783.8	yes	24	7.6	odd	2	inner
784.4.f.j.783.17	yes	24	1.1	even	1	trivial
784.4.f.j.783.18	yes	24	28.27	even	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)

\(f(q)\)	\(=\)	\(q+4.30860 q^{3} +7.29492i q^{5} -8.43597 q^{9} +O(q^{10})\)	\(q+4.30860 q^{3} +7.29492i q^{5} -8.43597 q^{9} +21.8273i q^{11} +30.5508i q^{13} +31.4309i q^{15} -55.0978i q^{17} +58.7072 q^{19} +79.6503i q^{23} +71.7842 q^{25} -152.679 q^{27} -116.193 q^{29} -110.020 q^{31} +94.0451i q^{33} -376.127 q^{37} +131.631i q^{39} +232.841i q^{41} +260.884i q^{43} -61.5397i q^{45} +36.5870 q^{47} -237.394i q^{51} +59.1734 q^{53} -159.228 q^{55} +252.946 q^{57} -707.304 q^{59} +323.781i q^{61} -222.865 q^{65} -239.434i q^{67} +343.181i q^{69} +887.119i q^{71} +550.887i q^{73} +309.289 q^{75} -1248.76i q^{79} -430.063 q^{81} -1393.89 q^{83} +401.934 q^{85} -500.630 q^{87} -982.787i q^{89} -474.033 q^{93} +428.264i q^{95} +441.547i q^{97} -184.135i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 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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