LMFDB - Embedded newform 784.4.f.j.783.13

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			783.13
Character	$\chi$	$=$	784.783
Dual form			784.4.f.j.783.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84933 q^{3} -15.9847i q^{5} -23.5800 q^{9} +O(q^{10})$	$q+1.84933 q^{3} -15.9847i q^{5} -23.5800 q^{9} +65.4943i q^{11} -34.4872i q^{13} -29.5611i q^{15} +50.6556i q^{17} +36.7132 q^{19} +196.221i q^{23} -130.512 q^{25} -93.5392 q^{27} +129.098 q^{29} +248.448 q^{31} +121.121i q^{33} +105.266 q^{37} -63.7782i q^{39} +186.044i q^{41} -27.7878i q^{43} +376.920i q^{45} +15.7540 q^{47} +93.6790i q^{51} +420.718 q^{53} +1046.91 q^{55} +67.8949 q^{57} -149.562 q^{59} -794.929i q^{61} -551.269 q^{65} -711.557i q^{67} +362.877i q^{69} -798.886i q^{71} -679.633i q^{73} -241.360 q^{75} +288.078i q^{79} +463.674 q^{81} +1363.56 q^{83} +809.716 q^{85} +238.745 q^{87} +913.742i q^{89} +459.462 q^{93} -586.851i q^{95} +1777.65i q^{97} -1544.35i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.84933	0.355904	0.177952	−	0.984039i	$-0.443053\pi$
$3$	1.84933	0.355904	0.177952	+	0.984039i	$0.443053\pi$
$4$	0	0
$5$	− 15.9847i	− 1.42972i	−0.699268	−	0.714860i	$-0.746491\pi$
$5$	− 15.9847i	− 1.42972i	0.699268	−	0.714860i	$-0.253509\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−23.5800	−0.873332
$10$	0	0
$11$	65.4943i	1.79521i	0.440803	+	0.897604i	$0.354694\pi$
$11$	65.4943i	1.79521i	−0.440803	+	0.897604i	$0.645306\pi$
$12$	0	0
$13$	− 34.4872i	− 0.735771i	−0.929871	−	0.367885i	$-0.880082\pi$
$13$	− 34.4872i	− 0.735771i	0.929871	−	0.367885i	$-0.119918\pi$
$14$	0	0
$15$	− 29.5611i	− 0.508843i
$16$	0	0
$17$	50.6556i	0.722693i	0.932432	+	0.361346i	$0.117683\pi$
$17$	50.6556i	0.722693i	−0.932432	+	0.361346i	$0.882317\pi$
$18$	0	0
$19$	36.7132	0.443294	0.221647	−	0.975127i	$-0.428857\pi$
$19$	36.7132	0.443294	0.221647	+	0.975127i	$0.428857\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	196.221i	1.77891i	0.457028	+	0.889453i	$0.348914\pi$
$23$	196.221i	1.77891i	−0.457028	+	0.889453i	$0.651086\pi$
$24$	0	0
$25$	−130.512	−1.04410
$26$	0	0
$27$	−93.5392	−0.666727
$28$	0	0
$29$	129.098	0.826651	0.413325	−	0.910583i	$-0.364367\pi$
$29$	129.098	0.826651	0.413325	+	0.910583i	$0.364367\pi$
$30$	0	0
$31$	248.448	1.43944	0.719718	−	0.694267i	$-0.244271\pi$
$31$	248.448	1.43944	0.719718	+	0.694267i	$0.244271\pi$
$32$	0	0
$33$	121.121i	0.638922i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	105.266	0.467718	0.233859	−	0.972271i	$-0.424865\pi$
$37$	105.266	0.467718	0.233859	+	0.972271i	$0.424865\pi$
$38$	0	0
$39$	− 63.7782i	− 0.261864i
$40$	0	0
$41$	186.044i	0.708663i	0.935120	+	0.354332i	$0.115292\pi$
$41$	186.044i	0.708663i	−0.935120	+	0.354332i	$0.884708\pi$
$42$	0	0
$43$	− 27.7878i	− 0.0985488i	−0.998785	−	0.0492744i	$-0.984309\pi$
$43$	− 27.7878i	− 0.0985488i	0.998785	−	0.0492744i	$-0.0156909\pi$
$44$	0	0
$45$	376.920i	1.24862i
$46$	0	0
$47$	15.7540	0.0488928	0.0244464	−	0.999701i	$-0.492218\pi$
$47$	15.7540	0.0488928	0.0244464	+	0.999701i	$0.492218\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	93.6790i	0.257209i
$52$	0	0
$53$	420.718	1.09038	0.545190	−	0.838313i	$-0.316458\pi$
$53$	420.718	1.09038	0.545190	+	0.838313i	$0.316458\pi$
$54$	0	0
$55$	1046.91	2.56664
$56$	0	0
$57$	67.8949	0.157770
$58$	0	0
$59$	−149.562	−0.330022	−0.165011	−	0.986292i	$-0.552766\pi$
$59$	−149.562	−0.330022	−0.165011	+	0.986292i	$0.552766\pi$
$60$	0	0
$61$	− 794.929i	− 1.66853i	−0.551365	−	0.834264i	$-0.685893\pi$
$61$	− 794.929i	− 1.66853i	0.551365	−	0.834264i	$-0.314107\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−551.269	−1.05195
$66$	0	0
$67$	− 711.557i	− 1.29747i	−0.761014	−	0.648735i	$-0.775298\pi$
$67$	− 711.557i	− 1.29747i	0.761014	−	0.648735i	$-0.224702\pi$
$68$	0	0
$69$	362.877i	0.633120i
$70$	0	0
$71$	− 798.886i	− 1.33536i	−0.744450	−	0.667679i	$-0.767288\pi$
$71$	− 798.886i	− 1.33536i	0.744450	−	0.667679i	$-0.232712\pi$
$72$	0	0
$73$	− 679.633i	− 1.08966i	−0.838547	−	0.544829i	$-0.816595\pi$
$73$	− 679.633i	− 1.08966i	0.838547	−	0.544829i	$-0.183405\pi$
$74$	0	0
$75$	−241.360	−0.371598
$76$	0	0
$77$	0	0
$78$	0	0
$79$	288.078i	0.410270i	0.978734	+	0.205135i	$0.0657634\pi$
$79$	288.078i	0.410270i	−0.978734	+	0.205135i	$0.934237\pi$
$80$	0	0
$81$	463.674	0.636041
$82$	0	0
$83$	1363.56	1.80326	0.901629	−	0.432510i	$-0.142372\pi$
$83$	1363.56	1.80326	0.901629	+	0.432510i	$0.142372\pi$
$84$	0	0
$85$	809.716	1.03325
$86$	0	0
$87$	238.745	0.294208
$88$	0	0
$89$	913.742i	1.08827i	0.838996	+	0.544137i	$0.183143\pi$
$89$	913.742i	1.08827i	−0.838996	+	0.544137i	$0.816857\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	459.462	0.512301
$94$	0	0
$95$	− 586.851i	− 0.633786i
$96$	0	0
$97$	1777.65i	1.86076i	0.366601	+	0.930378i	$0.380522\pi$
$97$	1777.65i	1.86076i	−0.366601	+	0.930378i	$0.619478\pi$
$98$	0	0
$99$	− 1544.35i	− 1.56781i
$100$	0	0
$101$	1568.27i	1.54504i	0.634990	+	0.772521i	$0.281004\pi$
$101$	1568.27i	1.54504i	−0.634990	+	0.772521i	$0.718996\pi$
$102$	0	0
$103$	−968.317	−0.926322	−0.463161	−	0.886274i	$-0.653285\pi$
$103$	−968.317	−0.926322	−0.463161	+	0.886274i	$0.653285\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1672.04i	1.51067i	0.655337	+	0.755337i	$0.272527\pi$
$107$	1672.04i	1.51067i	−0.655337	+	0.755337i	$0.727473\pi$
$108$	0	0
$109$	−765.812	−0.672949	−0.336475	−	0.941692i	$-0.609235\pi$
$109$	−765.812	−0.672949	−0.336475	+	0.941692i	$0.609235\pi$
$110$	0	0
$111$	194.671	0.166463
$112$	0	0
$113$	72.8398	0.0606389	0.0303194	−	0.999540i	$-0.490348\pi$
$113$	72.8398	0.0606389	0.0303194	+	0.999540i	$0.490348\pi$
$114$	0	0
$115$	3136.54	2.54333
$116$	0	0
$117$	813.206i	0.642572i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2958.51	−2.22277
$122$	0	0
$123$	344.057i	0.252216i
$124$	0	0
$125$	88.1093i	0.0630459i
$126$	0	0
$127$	834.678i	0.583195i	0.956541	+	0.291597i	$0.0941867\pi$
$127$	834.678i	0.583195i	−0.956541	+	0.291597i	$0.905813\pi$
$128$	0	0
$129$	− 51.3888i	− 0.0350739i
$130$	0	0
$131$	1486.94	0.991712	0.495856	−	0.868405i	$-0.334854\pi$
$131$	1486.94	0.991712	0.495856	+	0.868405i	$0.334854\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1495.20i	0.953232i
$136$	0	0
$137$	40.4652	0.0252349	0.0126174	−	0.999920i	$-0.495984\pi$
$137$	40.4652	0.0252349	0.0126174	+	0.999920i	$0.495984\pi$
$138$	0	0
$139$	1944.82	1.18675	0.593373	−	0.804927i	$-0.297796\pi$
$139$	1944.82	1.18675	0.593373	+	0.804927i	$0.297796\pi$
$140$	0	0
$141$	29.1344	0.0174011
$142$	0	0
$143$	2258.71	1.32086
$144$	0	0
$145$	− 2063.60i	− 1.18188i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−985.482	−0.541838	−0.270919	−	0.962602i	$-0.587328\pi$
$149$	−985.482	−0.541838	−0.270919	+	0.962602i	$0.587328\pi$
$150$	0	0
$151$	2722.07i	1.46701i	0.679681	+	0.733507i	$0.262118\pi$
$151$	2722.07i	1.46701i	−0.679681	+	0.733507i	$0.737882\pi$
$152$	0	0
$153$	− 1194.46i	− 0.631151i
$154$	0	0
$155$	− 3971.37i	− 2.05799i
$156$	0	0
$157$	2404.53i	1.22231i	0.791512	+	0.611154i	$0.209294\pi$
$157$	2404.53i	1.22231i	−0.791512	+	0.611154i	$0.790706\pi$
$158$	0	0
$159$	778.048	0.388071
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1696.94i	0.815429i	0.913109	+	0.407715i	$0.133674\pi$
$163$	1696.94i	0.815429i	−0.913109	+	0.407715i	$0.866326\pi$
$164$	0	0
$165$	1936.08	0.913479
$166$	0	0
$167$	981.444	0.454769	0.227384	−	0.973805i	$-0.426983\pi$
$167$	981.444	0.454769	0.227384	+	0.973805i	$0.426983\pi$
$168$	0	0
$169$	1007.64	0.458641
$170$	0	0
$171$	−865.696	−0.387143
$172$	0	0
$173$	1294.42i	0.568859i	0.958697	+	0.284430i	$0.0918042\pi$
$173$	1294.42i	0.568859i	−0.958697	+	0.284430i	$0.908196\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−276.589	−0.117456
$178$	0	0
$179$	− 1965.61i	− 0.820761i	−0.911914	−	0.410381i	$-0.865396\pi$
$179$	− 1965.61i	− 0.820761i	0.911914	−	0.410381i	$-0.134604\pi$
$180$	0	0
$181$	2200.30i	0.903576i	0.892125	+	0.451788i	$0.149214\pi$
$181$	2200.30i	0.903576i	−0.892125	+	0.451788i	$0.850786\pi$
$182$	0	0
$183$	− 1470.09i	− 0.593836i
$184$	0	0
$185$	− 1682.64i	− 0.668705i
$186$	0	0
$187$	−3317.65	−1.29738
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 559.699i	− 0.212034i	−0.994364	−	0.106017i	$-0.966190\pi$
$191$	− 559.699i	− 0.212034i	0.994364	−	0.106017i	$-0.0338097\pi$
$192$	0	0
$193$	−3056.00	−1.13977	−0.569886	−	0.821724i	$-0.693013\pi$
$193$	−3056.00	−1.13977	−0.569886	+	0.821724i	$0.693013\pi$
$194$	0	0
$195$	−1019.48	−0.374392
$196$	0	0
$197$	1567.28	0.566825	0.283412	−	0.958998i	$-0.408534\pi$
$197$	1567.28	0.566825	0.283412	+	0.958998i	$0.408534\pi$
$198$	0	0
$199$	−1785.64	−0.636085	−0.318043	−	0.948076i	$-0.603026\pi$
$199$	−1785.64	−0.636085	−0.318043	+	0.948076i	$0.603026\pi$
$200$	0	0
$201$	− 1315.91i	− 0.461775i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2973.87	1.01319
$206$	0	0
$207$	− 4626.88i	− 1.55358i
$208$	0	0
$209$	2404.51i	0.795804i
$210$	0	0
$211$	5093.15i	1.66174i	0.556468	+	0.830869i	$0.312156\pi$
$211$	5093.15i	1.66174i	−0.556468	+	0.830869i	$0.687844\pi$
$212$	0	0
$213$	− 1477.41i	− 0.475259i
$214$	0	0
$215$	−444.181	−0.140897
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 1256.87i	− 0.387814i
$220$	0	0
$221$	1746.97	0.531736
$222$	0	0
$223$	5458.84	1.63924	0.819621	−	0.572906i	$-0.194184\pi$
$223$	5458.84	1.63924	0.819621	+	0.572906i	$0.194184\pi$
$224$	0	0
$225$	3077.47	0.911843
$226$	0	0
$227$	−3882.56	−1.13522	−0.567610	−	0.823298i	$-0.692132\pi$
$227$	−3882.56	−1.13522	−0.567610	+	0.823298i	$0.692132\pi$
$228$	0	0
$229$	2964.77i	0.855536i	0.903888	+	0.427768i	$0.140700\pi$
$229$	2964.77i	0.855536i	−0.903888	+	0.427768i	$0.859300\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4994.49	1.40429	0.702146	−	0.712033i	$-0.252226\pi$
$233$	4994.49	1.40429	0.702146	+	0.712033i	$0.252226\pi$
$234$	0	0
$235$	− 251.824i	− 0.0699029i
$236$	0	0
$237$	532.752i	0.146017i
$238$	0	0
$239$	879.532i	0.238043i	0.992892	+	0.119021i	$0.0379757\pi$
$239$	879.532i	0.238043i	−0.992892	+	0.119021i	$0.962024\pi$
$240$	0	0
$241$	1590.41i	0.425093i	0.977151	+	0.212546i	$0.0681757\pi$
$241$	1590.41i	0.425093i	−0.977151	+	0.212546i	$0.931824\pi$
$242$	0	0
$243$	3383.05	0.893097
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1266.13i	− 0.326163i
$248$	0	0
$249$	2521.68	0.641787
$250$	0	0
$251$	−2944.10	−0.740358	−0.370179	−	0.928960i	$-0.620704\pi$
$251$	−2944.10	−0.740358	−0.370179	+	0.928960i	$0.620704\pi$
$252$	0	0
$253$	−12851.3	−3.19350
$254$	0	0
$255$	1497.43	0.367737
$256$	0	0
$257$	− 5412.36i	− 1.31367i	−0.754033	−	0.656837i	$-0.771894\pi$
$257$	− 5412.36i	− 1.31367i	0.754033	−	0.656837i	$-0.228106\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3044.12	−0.721941
$262$	0	0
$263$	464.615i	0.108933i	0.998516	+	0.0544665i	$0.0173458\pi$
$263$	464.615i	0.108933i	−0.998516	+	0.0544665i	$0.982654\pi$
$264$	0	0
$265$	− 6725.08i	− 1.55894i
$266$	0	0
$267$	1689.81i	0.387321i
$268$	0	0
$269$	632.708i	0.143409i	0.997426	+	0.0717043i	$0.0228438\pi$
$269$	632.708i	0.143409i	−0.997426	+	0.0717043i	$0.977156\pi$
$270$	0	0
$271$	−1375.45	−0.308313	−0.154156	−	0.988046i	$-0.549266\pi$
$271$	−1375.45	−0.308313	−0.154156	+	0.988046i	$0.549266\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8547.80i	− 1.87437i
$276$	0	0
$277$	5088.73	1.10380	0.551900	−	0.833911i	$-0.313903\pi$
$277$	5088.73	1.10380	0.551900	+	0.833911i	$0.313903\pi$
$278$	0	0
$279$	−5858.38	−1.25711
$280$	0	0
$281$	8628.24	1.83174	0.915868	−	0.401479i	$-0.131504\pi$
$281$	8628.24	1.83174	0.915868	+	0.401479i	$0.131504\pi$
$282$	0	0
$283$	−2305.10	−0.484184	−0.242092	−	0.970253i	$-0.577834\pi$
$283$	−2305.10	−0.484184	−0.242092	+	0.970253i	$0.577834\pi$
$284$	0	0
$285$	− 1085.28i	− 0.225567i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2347.01	0.477715
$290$	0	0
$291$	3287.47i	0.662251i
$292$	0	0
$293$	− 7851.48i	− 1.56549i	−0.622343	−	0.782745i	$-0.713819\pi$
$293$	− 7851.48i	− 1.56549i	0.622343	−	0.782745i	$-0.286181\pi$
$294$	0	0
$295$	2390.71i	0.471838i
$296$	0	0
$297$	− 6126.29i	− 1.19691i
$298$	0	0
$299$	6767.09	1.30887
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2900.26i	0.549887i
$304$	0	0
$305$	−12706.7	−2.38553
$306$	0	0
$307$	−8550.68	−1.58962	−0.794810	−	0.606859i	$-0.792429\pi$
$307$	−8550.68	−1.58962	−0.794810	+	0.606859i	$0.792429\pi$
$308$	0	0
$309$	−1790.74	−0.329682
$310$	0	0
$311$	−3084.99	−0.562488	−0.281244	−	0.959636i	$-0.590747\pi$
$311$	−3084.99	−0.562488	−0.281244	+	0.959636i	$0.590747\pi$
$312$	0	0
$313$	− 88.0581i	− 0.0159020i	−0.999968	−	0.00795102i	$-0.997469\pi$
$313$	− 88.0581i	− 0.0159020i	0.999968	−	0.00795102i	$-0.00253091\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4314.89	−0.764506	−0.382253	−	0.924058i	$-0.624852\pi$
$317$	−4314.89	−0.764506	−0.382253	+	0.924058i	$0.624852\pi$
$318$	0	0
$319$	8455.18i	1.48401i
$320$	0	0
$321$	3092.16i	0.537655i
$322$	0	0
$323$	1859.73i	0.320365i
$324$	0	0
$325$	4500.99i	0.768216i
$326$	0	0
$327$	−1416.24	−0.239506
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2334.92i	0.387730i	0.981028	+	0.193865i	$0.0621024\pi$
$331$	2334.92i	0.387730i	−0.981028	+	0.193865i	$0.937898\pi$
$332$	0	0
$333$	−2482.16	−0.408473
$334$	0	0
$335$	−11374.1	−1.85502
$336$	0	0
$337$	−3300.64	−0.533523	−0.266762	−	0.963763i	$-0.585954\pi$
$337$	−3300.64	−0.533523	−0.266762	+	0.963763i	$0.585954\pi$
$338$	0	0
$339$	134.705	0.0215816
$340$	0	0
$341$	16271.9i	2.58408i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5800.50	0.905184
$346$	0	0
$347$	8102.77i	1.25354i	0.779203	+	0.626771i	$0.215624\pi$
$347$	8102.77i	1.25354i	−0.779203	+	0.626771i	$0.784376\pi$
$348$	0	0
$349$	− 1453.41i	− 0.222920i	−0.993769	−	0.111460i	$-0.964447\pi$
$349$	− 1453.41i	− 0.222920i	0.993769	−	0.111460i	$-0.0355528\pi$
$350$	0	0
$351$	3225.90i	0.490558i
$352$	0	0
$353$	− 9186.41i	− 1.38511i	−0.721366	−	0.692554i	$-0.756485\pi$
$353$	− 9186.41i	− 1.38511i	0.721366	−	0.692554i	$-0.243515\pi$
$354$	0	0
$355$	−12770.0	−1.90919
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6069.62i	− 0.892318i	−0.894954	−	0.446159i	$-0.852792\pi$
$359$	− 6069.62i	− 0.892318i	0.894954	−	0.446159i	$-0.147208\pi$
$360$	0	0
$361$	−5511.14	−0.803491
$362$	0	0
$363$	−5471.26	−0.791093
$364$	0	0
$365$	−10863.8	−1.55790
$366$	0	0
$367$	−8481.43	−1.20634	−0.603170	−	0.797613i	$-0.706096\pi$
$367$	−8481.43	−1.20634	−0.603170	+	0.797613i	$0.706096\pi$
$368$	0	0
$369$	− 4386.91i	− 0.618898i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2921.96	0.405612	0.202806	−	0.979219i	$-0.434994\pi$
$373$	2921.96	0.405612	0.202806	+	0.979219i	$0.434994\pi$
$374$	0	0
$375$	162.943i	0.0224383i
$376$	0	0
$377$	− 4452.22i	− 0.608225i
$378$	0	0
$379$	6409.71i	0.868720i	0.900739	+	0.434360i	$0.143025\pi$
$379$	6409.71i	0.868720i	−0.900739	+	0.434360i	$0.856975\pi$
$380$	0	0
$381$	1543.60i	0.207561i
$382$	0	0
$383$	8391.29	1.11952	0.559759	−	0.828656i	$-0.310894\pi$
$383$	8391.29	1.11952	0.559759	+	0.828656i	$0.310894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	655.235i	0.0860658i
$388$	0	0
$389$	3488.48	0.454687	0.227343	−	0.973815i	$-0.426996\pi$
$389$	3488.48	0.454687	0.227343	+	0.973815i	$0.426996\pi$
$390$	0	0
$391$	−9939.66	−1.28560
$392$	0	0
$393$	2749.84	0.352954
$394$	0	0
$395$	4604.86	0.586571
$396$	0	0
$397$	− 3697.26i	− 0.467407i	−0.972308	−	0.233703i	$-0.924916\pi$
$397$	− 3697.26i	− 0.467407i	0.972308	−	0.233703i	$-0.0750844\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−775.525	−0.0965783	−0.0482891	−	0.998833i	$-0.515377\pi$
$401$	−775.525	−0.0965783	−0.0482891	+	0.998833i	$0.515377\pi$
$402$	0	0
$403$	− 8568.25i	− 1.05909i
$404$	0	0
$405$	− 7411.71i	− 0.909360i
$406$	0	0
$407$	6894.30i	0.839651i
$408$	0	0
$409$	− 10288.6i	− 1.24385i	−0.783075	−	0.621927i	$-0.786350\pi$
$409$	− 10288.6i	− 1.24385i	0.783075	−	0.621927i	$-0.213650\pi$
$410$	0	0
$411$	74.8336	0.00898120
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 21796.2i	− 2.57815i
$416$	0	0
$417$	3596.63	0.422368
$418$	0	0
$419$	13768.7	1.60536	0.802680	−	0.596410i	$-0.203407\pi$
$419$	13768.7	1.60536	0.802680	+	0.596410i	$0.203407\pi$
$420$	0	0
$421$	998.377	0.115577	0.0577885	−	0.998329i	$-0.481595\pi$
$421$	998.377	0.115577	0.0577885	+	0.998329i	$0.481595\pi$
$422$	0	0
$423$	−371.479	−0.0426996
$424$	0	0
$425$	− 6611.16i	− 0.754561i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4177.11	0.470100
$430$	0	0
$431$	− 2285.74i	− 0.255453i	−0.991809	−	0.127727i	$-0.959232\pi$
$431$	− 2285.74i	− 0.255453i	0.991809	−	0.127727i	$-0.0407680\pi$
$432$	0	0
$433$	− 2548.84i	− 0.282885i	−0.989946	−	0.141443i	$-0.954826\pi$
$433$	− 2548.84i	− 0.282885i	0.989946	−	0.141443i	$-0.0451741\pi$
$434$	0	0
$435$	− 3816.28i	− 0.420636i
$436$	0	0
$437$	7203.88i	0.788578i
$438$	0	0
$439$	9944.48	1.08115	0.540574	−	0.841296i	$-0.318207\pi$
$439$	9944.48	1.08115	0.540574	+	0.841296i	$0.318207\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 10138.2i	− 1.08732i	−0.839306	−	0.543660i	$-0.817038\pi$
$443$	− 10138.2i	− 1.08732i	0.839306	−	0.543660i	$-0.182962\pi$
$444$	0	0
$445$	14605.9	1.55593
$446$	0	0
$447$	−1822.48	−0.192842
$448$	0	0
$449$	−8414.78	−0.884450	−0.442225	−	0.896904i	$-0.645811\pi$
$449$	−8414.78	−0.884450	−0.442225	+	0.896904i	$0.645811\pi$
$450$	0	0
$451$	−12184.8	−1.27220
$452$	0	0
$453$	5034.02i	0.522117i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16404.4	−1.67914	−0.839570	−	0.543252i	$-0.817193\pi$
$457$	−16404.4	−1.67914	−0.839570	+	0.543252i	$0.817193\pi$
$458$	0	0
$459$	− 4738.28i	− 0.481839i
$460$	0	0
$461$	7609.41i	0.768775i	0.923172	+	0.384388i	$0.125587\pi$
$461$	7609.41i	0.768775i	−0.923172	+	0.384388i	$0.874413\pi$
$462$	0	0
$463$	− 1078.98i	− 0.108303i	−0.998533	−	0.0541515i	$-0.982755\pi$
$463$	− 1078.98i	− 0.108303i	0.998533	−	0.0541515i	$-0.0172454\pi$
$464$	0	0
$465$	− 7344.38i	− 0.732447i
$466$	0	0
$467$	3846.75	0.381170	0.190585	−	0.981671i	$-0.438961\pi$
$467$	3846.75	0.381170	0.190585	+	0.981671i	$0.438961\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4446.77i	0.435024i
$472$	0	0
$473$	1819.94	0.176915
$474$	0	0
$475$	−4791.51	−0.462842
$476$	0	0
$477$	−9920.53	−0.952263
$478$	0	0
$479$	−17003.3	−1.62192	−0.810961	−	0.585100i	$-0.801055\pi$
$479$	−17003.3	−1.62192	−0.810961	+	0.585100i	$0.801055\pi$
$480$	0	0
$481$	− 3630.31i	− 0.344133i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28415.3	2.66036
$486$	0	0
$487$	− 15452.5i	− 1.43782i	−0.695102	−	0.718911i	$-0.744641\pi$
$487$	− 15452.5i	− 1.43782i	0.695102	−	0.718911i	$-0.255359\pi$
$488$	0	0
$489$	3138.22i	0.290215i
$490$	0	0
$491$	6984.48i	0.641965i	0.947085	+	0.320983i	$0.104013\pi$
$491$	6984.48i	0.641965i	−0.947085	+	0.320983i	$0.895987\pi$
$492$	0	0
$493$	6539.53i	0.597415i
$494$	0	0
$495$	−24686.1	−2.24153
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10058.4i	0.902355i	0.892434	+	0.451177i	$0.148996\pi$
$499$	10058.4i	0.902355i	−0.892434	+	0.451177i	$0.851004\pi$
$500$	0	0
$501$	1815.02	0.161854
$502$	0	0
$503$	10851.5	0.961920	0.480960	−	0.876743i	$-0.340288\pi$
$503$	10851.5	0.961920	0.480960	+	0.876743i	$0.340288\pi$
$504$	0	0
$505$	25068.5	2.20898
$506$	0	0
$507$	1863.45	0.163232
$508$	0	0
$509$	5347.37i	0.465655i	0.972518	+	0.232827i	$0.0747977\pi$
$509$	5347.37i	0.465655i	−0.972518	+	0.232827i	$0.925202\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3434.12	−0.295556
$514$	0	0
$515$	15478.3i	1.32438i
$516$	0	0
$517$	1031.80i	0.0877727i
$518$	0	0
$519$	2393.81i	0.202459i
$520$	0	0
$521$	− 13051.0i	− 1.09746i	−0.836001	−	0.548728i	$-0.815112\pi$
$521$	− 13051.0i	− 1.09746i	0.836001	−	0.548728i	$-0.184888\pi$
$522$	0	0
$523$	1273.65	0.106487	0.0532437	−	0.998582i	$-0.483044\pi$
$523$	1273.65	0.106487	0.0532437	+	0.998582i	$0.483044\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12585.2i	1.04027i
$528$	0	0
$529$	−26335.5	−2.16450
$530$	0	0
$531$	3526.66	0.288219
$532$	0	0
$533$	6416.13	0.521414
$534$	0	0
$535$	26727.1	2.15984
$536$	0	0
$537$	− 3635.06i	− 0.292112i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7201.83	0.572331	0.286165	−	0.958180i	$-0.407619\pi$
$541$	7201.83	0.572331	0.286165	+	0.958180i	$0.407619\pi$
$542$	0	0
$543$	4069.09i	0.321586i
$544$	0	0
$545$	12241.3i	0.962129i
$546$	0	0
$547$	− 10450.0i	− 0.816840i	−0.912794	−	0.408420i	$-0.866080\pi$
$547$	− 10450.0i	− 0.816840i	0.912794	−	0.408420i	$-0.133920\pi$
$548$	0	0
$549$	18744.4i	1.45718i
$550$	0	0
$551$	4739.59	0.366449
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 3111.77i	− 0.237995i
$556$	0	0
$557$	−13177.3	−1.00241	−0.501203	−	0.865329i	$-0.667109\pi$
$557$	−13177.3	−1.00241	−0.501203	+	0.865329i	$0.667109\pi$
$558$	0	0
$559$	−958.322	−0.0725093
$560$	0	0
$561$	−6135.44	−0.461744
$562$	0	0
$563$	−623.652	−0.0466852	−0.0233426	−	0.999728i	$-0.507431\pi$
$563$	−623.652	−0.0466852	−0.0233426	+	0.999728i	$0.507431\pi$
$564$	0	0
$565$	− 1164.33i	− 0.0866965i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18387.9	−1.35477	−0.677384	−	0.735630i	$-0.736886\pi$
$569$	−18387.9	−1.35477	−0.677384	+	0.735630i	$0.736886\pi$
$570$	0	0
$571$	− 4490.64i	− 0.329119i	−0.986367	−	0.164560i	$-0.947380\pi$
$571$	− 4490.64i	− 0.329119i	0.986367	−	0.164560i	$-0.0526203\pi$
$572$	0	0
$573$	− 1035.07i	− 0.0754636i
$574$	0	0
$575$	− 25609.2i	− 1.85735i
$576$	0	0
$577$	− 11841.5i	− 0.854362i	−0.904166	−	0.427181i	$-0.859507\pi$
$577$	− 11841.5i	− 0.854362i	0.904166	−	0.427181i	$-0.140493\pi$
$578$	0	0
$579$	−5651.57	−0.405649
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27554.7i	1.95746i
$584$	0	0
$585$	12998.9	0.918698
$586$	0	0
$587$	7718.07	0.542690	0.271345	−	0.962482i	$-0.412532\pi$
$587$	7718.07	0.542690	0.271345	+	0.962482i	$0.412532\pi$
$588$	0	0
$589$	9121.30	0.638093
$590$	0	0
$591$	2898.43	0.201735
$592$	0	0
$593$	7796.15i	0.539881i	0.962877	+	0.269940i	$0.0870040\pi$
$593$	7796.15i	0.539881i	−0.962877	+	0.269940i	$0.912996\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3302.25	−0.226386
$598$	0	0
$599$	21487.5i	1.46570i	0.680389	+	0.732851i	$0.261811\pi$
$599$	21487.5i	1.46570i	−0.680389	+	0.732851i	$0.738189\pi$
$600$	0	0
$601$	− 20466.4i	− 1.38909i	−0.719450	−	0.694544i	$-0.755606\pi$
$601$	− 20466.4i	− 1.38909i	0.719450	−	0.694544i	$-0.244394\pi$
$602$	0	0
$603$	16778.5i	1.13312i
$604$	0	0
$605$	47291.0i	3.17794i
$606$	0	0
$607$	20787.7	1.39002	0.695012	−	0.718998i	$-0.255399\pi$
$607$	20787.7	1.39002	0.695012	+	0.718998i	$0.255399\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 543.312i	− 0.0359739i
$612$	0	0
$613$	21713.9	1.43069	0.715346	−	0.698770i	$-0.246269\pi$
$613$	21713.9	1.43069	0.715346	+	0.698770i	$0.246269\pi$
$614$	0	0
$615$	5499.67	0.360598
$616$	0	0
$617$	−8.20003	−0.000535042 0	−0.000267521	−	1.00000i	$-0.500085\pi$
$617$	−8.20003	−0.000535042 0	−0.000267521		1.00000i	$0.500085\pi$
$618$	0	0
$619$	22039.5	1.43109	0.715543	−	0.698568i	$-0.246179\pi$
$619$	22039.5	1.43109	0.715543	+	0.698568i	$0.246179\pi$
$620$	0	0
$621$	− 18354.3i	− 1.18604i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−14905.6	−0.953959
$626$	0	0
$627$	4446.73i	0.283230i
$628$	0	0
$629$	5332.29i	0.338016i
$630$	0	0
$631$	− 22284.8i	− 1.40593i	−0.711222	−	0.702967i	$-0.751858\pi$
$631$	− 22284.8i	− 1.40593i	0.711222	−	0.702967i	$-0.248142\pi$
$632$	0	0
$633$	9418.92i	0.591419i
$634$	0	0
$635$	13342.1	0.833805
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18837.7i	1.16621i
$640$	0	0
$641$	−3206.78	−0.197598	−0.0987988	−	0.995107i	$-0.531500\pi$
$641$	−3206.78	−0.197598	−0.0987988	+	0.995107i	$0.531500\pi$
$642$	0	0
$643$	−12460.1	−0.764197	−0.382098	−	0.924122i	$-0.624798\pi$
$643$	−12460.1	−0.764197	−0.382098	+	0.924122i	$0.624798\pi$
$644$	0	0
$645$	−821.438	−0.0501459
$646$	0	0
$647$	20581.0	1.25058	0.625288	−	0.780394i	$-0.284982\pi$
$647$	20581.0	1.25058	0.625288	+	0.780394i	$0.284982\pi$
$648$	0	0
$649$	− 9795.45i	− 0.592458i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19747.9	−1.18345	−0.591726	−	0.806139i	$-0.701553\pi$
$653$	−19747.9	−1.18345	−0.591726	+	0.806139i	$0.701553\pi$
$654$	0	0
$655$	− 23768.3i	− 1.41787i
$656$	0	0
$657$	16025.7i	0.951633i
$658$	0	0
$659$	− 3970.57i	− 0.234707i	−0.993090	−	0.117353i	$-0.962559\pi$
$659$	− 3970.57i	− 0.234707i	0.993090	−	0.117353i	$-0.0374410\pi$
$660$	0	0
$661$	− 13941.5i	− 0.820365i	−0.912004	−	0.410182i	$-0.865465\pi$
$661$	− 13941.5i	− 0.820365i	0.912004	−	0.410182i	$-0.134535\pi$
$662$	0	0
$663$	3230.72	0.189247
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25331.7i	1.47053i
$668$	0	0
$669$	10095.2	0.583413
$670$	0	0
$671$	52063.3	2.99535
$672$	0	0
$673$	17087.4	0.978707	0.489353	−	0.872086i	$-0.337233\pi$
$673$	17087.4	0.978707	0.489353	+	0.872086i	$0.337233\pi$
$674$	0	0
$675$	12208.0	0.696127
$676$	0	0
$677$	20387.7i	1.15741i	0.815538	+	0.578703i	$0.196441\pi$
$677$	20387.7i	1.15741i	−0.815538	+	0.578703i	$0.803559\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7180.15	−0.404029
$682$	0	0
$683$	1186.60i	0.0664773i	0.999447	+	0.0332386i	$0.0105821\pi$
$683$	1186.60i	0.0664773i	−0.999447	+	0.0332386i	$0.989418\pi$
$684$	0	0
$685$	− 646.826i	− 0.0360788i
$686$	0	0
$687$	5482.85i	0.304489i
$688$	0	0
$689$	− 14509.4i	− 0.802269i
$690$	0	0
$691$	−13680.2	−0.753138	−0.376569	−	0.926389i	$-0.622896\pi$
$691$	−13680.2	−0.753138	−0.376569	+	0.926389i	$0.622896\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 31087.5i	− 1.69671i
$696$	0	0
$697$	−9424.16	−0.512146
$698$	0	0
$699$	9236.47	0.499793
$700$	0	0
$701$	3385.95	0.182433	0.0912165	−	0.995831i	$-0.470924\pi$
$701$	3385.95	0.182433	0.0912165	+	0.995831i	$0.470924\pi$
$702$	0	0
$703$	3864.64	0.207336
$704$	0	0
$705$	− 465.706i	− 0.0248787i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2007.77	0.106352	0.0531758	−	0.998585i	$-0.483066\pi$
$709$	2007.77	0.106352	0.0531758	+	0.998585i	$0.483066\pi$
$710$	0	0
$711$	− 6792.88i	− 0.358302i
$712$	0	0
$713$	48750.5i	2.56062i
$714$	0	0
$715$	− 36105.0i	− 1.88846i
$716$	0	0
$717$	1626.55i	0.0847204i
$718$	0	0
$719$	−18230.2	−0.945582	−0.472791	−	0.881175i	$-0.656753\pi$
$719$	−18230.2	−0.945582	−0.472791	+	0.881175i	$0.656753\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2941.20i	0.151292i
$724$	0	0
$725$	−16848.8	−0.863103
$726$	0	0
$727$	3284.58	0.167563	0.0837814	−	0.996484i	$-0.473300\pi$
$727$	3284.58	0.167563	0.0837814	+	0.996484i	$0.473300\pi$
$728$	0	0
$729$	−6262.82	−0.318184
$730$	0	0
$731$	1407.61	0.0712205
$732$	0	0
$733$	10527.1i	0.530459i	0.964185	+	0.265229i	$0.0854477\pi$
$733$	10527.1i	0.530459i	−0.964185	+	0.265229i	$0.914552\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46602.9	2.32923
$738$	0	0
$739$	− 37002.4i	− 1.84189i	−0.389697	−	0.920943i	$-0.627420\pi$
$739$	− 37002.4i	− 1.84189i	0.389697	−	0.920943i	$-0.372580\pi$
$740$	0	0
$741$	− 2341.50i	− 0.116083i
$742$	0	0
$743$	5683.82i	0.280645i	0.990106	+	0.140322i	$0.0448139\pi$
$743$	5683.82i	0.280645i	−0.990106	+	0.140322i	$0.955186\pi$
$744$	0	0
$745$	15752.7i	0.774676i
$746$	0	0
$747$	−32152.8	−1.57484
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15316.9i	0.744234i	0.928186	+	0.372117i	$0.121368\pi$
$751$	15316.9i	0.744234i	−0.928186	+	0.372117i	$0.878632\pi$
$752$	0	0
$753$	−5444.62	−0.263497
$754$	0	0
$755$	43511.7	2.09742
$756$	0	0
$757$	−16385.8	−0.786728	−0.393364	−	0.919383i	$-0.628689\pi$
$757$	−16385.8	−0.786728	−0.393364	+	0.919383i	$0.628689\pi$
$758$	0	0
$759$	−23766.4	−1.13658
$760$	0	0
$761$	1478.50i	0.0704280i	0.999380	+	0.0352140i	$0.0112113\pi$
$761$	1478.50i	0.0704280i	−0.999380	+	0.0352140i	$0.988789\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−19093.1	−0.902369
$766$	0	0
$767$	5157.96i	0.242820i
$768$	0	0
$769$	14145.4i	0.663324i	0.943398	+	0.331662i	$0.107609\pi$
$769$	14145.4i	0.663324i	−0.943398	+	0.331662i	$0.892391\pi$
$770$	0	0
$771$	− 10009.3i	− 0.467542i
$772$	0	0
$773$	− 5731.86i	− 0.266702i	−0.991069	−	0.133351i	$-0.957426\pi$
$773$	− 5731.86i	− 0.266702i	0.991069	−	0.133351i	$-0.0425737\pi$
$774$	0	0
$775$	−32425.4	−1.50291
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6830.27i	0.314146i
$780$	0	0
$781$	52322.5	2.39724
$782$	0	0
$783$	−12075.7	−0.551150
$784$	0	0
$785$	38435.8	1.74756
$786$	0	0
$787$	−24156.5	−1.09414	−0.547070	−	0.837087i	$-0.684257\pi$
$787$	−24156.5	−1.09414	−0.547070	+	0.837087i	$0.684257\pi$
$788$	0	0
$789$	859.227i	0.0387697i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−27414.8	−1.22765
$794$	0	0
$795$	− 12436.9i	− 0.554832i
$796$	0	0
$797$	9638.73i	0.428383i	0.976792	+	0.214192i	$0.0687117\pi$
$797$	9638.73i	0.428383i	−0.976792	+	0.214192i	$0.931288\pi$
$798$	0	0
$799$	798.029i	0.0353345i
$800$	0	0
$801$	− 21546.0i	− 0.950425i
$802$	0	0
$803$	44512.1	1.95616
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1170.09i	0.0510397i
$808$	0	0
$809$	7783.24	0.338250	0.169125	−	0.985595i	$-0.445906\pi$
$809$	7783.24	0.338250	0.169125	+	0.985595i	$0.445906\pi$
$810$	0	0
$811$	25896.1	1.12125	0.560625	−	0.828070i	$-0.310561\pi$
$811$	25896.1	1.12125	0.560625	+	0.828070i	$0.310561\pi$
$812$	0	0
$813$	−2543.67	−0.109730
$814$	0	0
$815$	27125.2	1.16583
$816$	0	0
$817$	− 1020.18i	− 0.0436861i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14182.6	−0.602894	−0.301447	−	0.953483i	$-0.597470\pi$
$821$	−14182.6	−0.602894	−0.301447	+	0.953483i	$0.597470\pi$
$822$	0	0
$823$	− 8428.64i	− 0.356991i	−0.983941	−	0.178496i	$-0.942877\pi$
$823$	− 8428.64i	− 0.356991i	0.983941	−	0.178496i	$-0.0571231\pi$
$824$	0	0
$825$	− 15807.7i	− 0.667096i
$826$	0	0
$827$	− 32452.7i	− 1.36456i	−0.731090	−	0.682281i	$-0.760988\pi$
$827$	− 32452.7i	− 1.36456i	0.731090	−	0.682281i	$-0.239012\pi$
$828$	0	0
$829$	− 9759.86i	− 0.408895i	−0.978878	−	0.204447i	$-0.934460\pi$
$829$	− 9759.86i	− 0.408895i	0.978878	−	0.204447i	$-0.0655397\pi$
$830$	0	0
$831$	9410.76	0.392847
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 15688.1i	− 0.650192i
$836$	0	0
$837$	−23239.6	−0.959710
$838$	0	0
$839$	−8141.90	−0.335029	−0.167515	−	0.985870i	$-0.553574\pi$
$839$	−8141.90	−0.335029	−0.167515	+	0.985870i	$0.553574\pi$
$840$	0	0
$841$	−7722.74	−0.316649
$842$	0	0
$843$	15956.5	0.651923
$844$	0	0
$845$	− 16106.8i	− 0.655728i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4262.90	−0.172323
$850$	0	0
$851$	20655.3i	0.832026i
$852$	0	0
$853$	30918.0i	1.24105i	0.784189	+	0.620523i	$0.213079\pi$
$853$	30918.0i	1.24105i	−0.784189	+	0.620523i	$0.786921\pi$
$854$	0	0
$855$	13837.9i	0.553505i
$856$	0	0
$857$	− 244.236i	− 0.00973507i	−0.999988	−	0.00486753i	$-0.998451\pi$
$857$	− 244.236i	− 0.00973507i	0.999988	−	0.00486753i	$-0.00154939\pi$
$858$	0	0
$859$	12448.5	0.494456	0.247228	−	0.968957i	$-0.420480\pi$
$859$	12448.5	0.494456	0.247228	+	0.968957i	$0.420480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 38578.5i	− 1.52170i	−0.648927	−	0.760850i	$-0.724782\pi$
$863$	− 38578.5i	− 1.52170i	0.648927	−	0.760850i	$-0.275218\pi$
$864$	0	0
$865$	20690.9	0.813309
$866$	0	0
$867$	4340.41	0.170021
$868$	0	0
$869$	−18867.5	−0.736520
$870$	0	0
$871$	−24539.6	−0.954641
$872$	0	0
$873$	− 41917.0i	− 1.62506i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12784.2	−0.492237	−0.246119	−	0.969240i	$-0.579155\pi$
$877$	−12784.2	−0.492237	−0.246119	+	0.969240i	$0.579155\pi$
$878$	0	0
$879$	− 14520.0i	− 0.557164i
$880$	0	0
$881$	− 47480.9i	− 1.81575i	−0.419244	−	0.907874i	$-0.637705\pi$
$881$	− 47480.9i	− 1.81575i	0.419244	−	0.907874i	$-0.362295\pi$
$882$	0	0
$883$	24333.9i	0.927409i	0.885990	+	0.463705i	$0.153480\pi$
$883$	24333.9i	0.927409i	−0.885990	+	0.463705i	$0.846520\pi$
$884$	0	0
$885$	4421.21i	0.167929i
$886$	0	0
$887$	44646.9	1.69007	0.845037	−	0.534709i	$-0.179579\pi$
$887$	44646.9	1.69007	0.845037	+	0.534709i	$0.179579\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	30368.0i	1.14183i
$892$	0	0
$893$	578.380	0.0216739
$894$	0	0
$895$	−31419.7	−1.17346
$896$	0	0
$897$	12514.6	0.465831
$898$	0	0
$899$	32074.0	1.18991
$900$	0	0
$901$	21311.7i	0.788009i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	35171.3	1.29186
$906$	0	0
$907$	33634.9i	1.23134i	0.788003	+	0.615671i	$0.211115\pi$
$907$	33634.9i	1.23134i	−0.788003	+	0.615671i	$0.788885\pi$
$908$	0	0
$909$	− 36979.9i	− 1.34933i
$910$	0	0
$911$	13365.2i	0.486069i	0.970018	+	0.243034i	$0.0781428\pi$
$911$	13365.2i	0.486069i	−0.970018	+	0.243034i	$0.921857\pi$
$912$	0	0
$913$	89305.6i	3.23722i
$914$	0	0
$915$	−23499.0	−0.849019
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26476.2i	0.950347i	0.879892	+	0.475174i	$0.157615\pi$
$919$	26476.2i	0.950347i	−0.879892	+	0.475174i	$0.842385\pi$
$920$	0	0
$921$	−15813.1	−0.565752
$922$	0	0
$923$	−27551.3	−0.982517
$924$	0	0
$925$	−13738.4	−0.488343
$926$	0	0
$927$	22832.9	0.808986
$928$	0	0
$929$	38557.8i	1.36172i	0.732412	+	0.680862i	$0.238395\pi$
$929$	38557.8i	1.36172i	−0.732412	+	0.680862i	$0.761605\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5705.17	−0.200192
$934$	0	0
$935$	53031.8i	1.85489i
$936$	0	0
$937$	− 29805.5i	− 1.03917i	−0.854418	−	0.519586i	$-0.826086\pi$
$937$	− 29805.5i	− 1.03917i	0.854418	−	0.519586i	$-0.173914\pi$
$938$	0	0
$939$	− 162.849i	− 0.00565960i
$940$	0	0
$941$	− 37423.5i	− 1.29646i	−0.761444	−	0.648231i	$-0.775509\pi$
$941$	− 37423.5i	− 1.29646i	0.761444	−	0.648231i	$-0.224491\pi$
$942$	0	0
$943$	−36505.7	−1.26064
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5727.25i	− 0.196526i	−0.995160	−	0.0982632i	$-0.968671\pi$
$947$	− 5727.25i	− 0.196526i	0.995160	−	0.0982632i	$-0.0313287\pi$
$948$	0	0
$949$	−23438.6	−0.801738
$950$	0	0
$951$	−7979.67	−0.272091
$952$	0	0
$953$	−20927.1	−0.711326	−0.355663	−	0.934614i	$-0.615745\pi$
$953$	−20927.1	−0.711326	−0.355663	+	0.934614i	$0.615745\pi$
$954$	0	0
$955$	−8946.65	−0.303148
$956$	0	0
$957$	15636.4i	0.528165i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31935.2	1.07197
$962$	0	0
$963$	− 39426.6i	− 1.31932i
$964$	0	0
$965$	48849.4i	1.62955i
$966$	0	0
$967$	6581.50i	0.218870i	0.993994	+	0.109435i	$0.0349041\pi$
$967$	6581.50i	0.218870i	−0.993994	+	0.109435i	$0.965096\pi$
$968$	0	0
$969$	3439.25i	0.114019i
$970$	0	0
$971$	7190.75	0.237654	0.118827	−	0.992915i	$-0.462087\pi$
$971$	7190.75	0.237654	0.118827	+	0.992915i	$0.462087\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8323.83i	0.273411i
$976$	0	0
$977$	−43183.2	−1.41408	−0.707038	−	0.707175i	$-0.749969\pi$
$977$	−43183.2	−1.41408	−0.707038	+	0.707175i	$0.749969\pi$
$978$	0	0
$979$	−59844.9	−1.95368
$980$	0	0
$981$	18057.8	0.587708
$982$	0	0
$983$	−16162.3	−0.524413	−0.262206	−	0.965012i	$-0.584450\pi$
$983$	−16162.3	−0.524413	−0.262206	+	0.965012i	$0.584450\pi$
$984$	0	0
$985$	− 25052.7i	− 0.810400i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5452.53	0.175309
$990$	0	0
$991$	32288.1i	1.03498i	0.855690	+	0.517489i	$0.173133\pi$
$991$	32288.1i	1.03498i	−0.855690	+	0.517489i	$0.826867\pi$
$992$	0	0
$993$	4318.04i	0.137995i
$994$	0	0
$995$	28543.1i	0.909424i
$996$	0	0
$997$	− 29674.9i	− 0.942643i	−0.881961	−	0.471321i	$-0.843777\pi$
$997$	− 29674.9i	− 0.942643i	0.881961	−	0.471321i	$-0.156223\pi$
$998$	0	0
$999$	−9846.46	−0.311840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.j.783.13	yes	24
4.3	odd	2	inner	784.4.f.j.783.11	✓	24
7.6	odd	2	inner	784.4.f.j.783.12	yes	24
28.27	even	2	inner	784.4.f.j.783.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.j.783.11	✓	24	4.3	odd	2	inner
784.4.f.j.783.12	yes	24	7.6	odd	2	inner
784.4.f.j.783.13	yes	24	1.1	even	1	trivial
784.4.f.j.783.14	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)

\(f(q)\)	\(=\)	\(q+1.84933 q^{3} -15.9847i q^{5} -23.5800 q^{9} +O(q^{10})\)	\(q+1.84933 q^{3} -15.9847i q^{5} -23.5800 q^{9} +65.4943i q^{11} -34.4872i q^{13} -29.5611i q^{15} +50.6556i q^{17} +36.7132 q^{19} +196.221i q^{23} -130.512 q^{25} -93.5392 q^{27} +129.098 q^{29} +248.448 q^{31} +121.121i q^{33} +105.266 q^{37} -63.7782i q^{39} +186.044i q^{41} -27.7878i q^{43} +376.920i q^{45} +15.7540 q^{47} +93.6790i q^{51} +420.718 q^{53} +1046.91 q^{55} +67.8949 q^{57} -149.562 q^{59} -794.929i q^{61} -551.269 q^{65} -711.557i q^{67} +362.877i q^{69} -798.886i q^{71} -679.633i q^{73} -241.360 q^{75} +288.078i q^{79} +463.674 q^{81} +1363.56 q^{83} +809.716 q^{85} +238.745 q^{87} +913.742i q^{89} +459.462 q^{93} -586.851i q^{95} +1777.65i q^{97} -1544.35i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 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

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Database

Properties

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