LMFDB - Embedded newform 784.4.f.j.783.3

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			783.3
Character	$\chi$	$=$	784.783
Dual form			784.4.f.j.783.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.34240 q^{3} -1.20378i q^{5} +13.2260 q^{9} +O(q^{10})$	$q-6.34240 q^{3} -1.20378i q^{5} +13.2260 q^{9} -32.0476i q^{11} +59.4009i q^{13} +7.63482i q^{15} -0.0501221i q^{17} +11.3825 q^{19} -75.2486i q^{23} +123.551 q^{25} +87.3601 q^{27} -263.457 q^{29} +207.558 q^{31} +203.259i q^{33} +199.310 q^{37} -376.744i q^{39} +437.976i q^{41} +174.444i q^{43} -15.9211i q^{45} -283.621 q^{47} +0.317894i q^{51} +7.41360 q^{53} -38.5781 q^{55} -72.1924 q^{57} +38.6495 q^{59} +276.420i q^{61} +71.5053 q^{65} -965.988i q^{67} +477.256i q^{69} -580.469i q^{71} -607.215i q^{73} -783.609 q^{75} -827.267i q^{79} -911.175 q^{81} -1203.55 q^{83} -0.0603357 q^{85} +1670.95 q^{87} +312.903i q^{89} -1316.42 q^{93} -13.7020i q^{95} -1247.34i q^{97} -423.862i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−6.34240	−1.22059	−0.610297	−	0.792172i	$-0.708950\pi$
$3$	−6.34240	−1.22059	−0.610297	+	0.792172i	$0.708950\pi$
$4$	0	0
$5$	− 1.20378i	− 0.107669i	−0.998550	−	0.0538345i	$-0.982856\pi$
$5$	− 1.20378i	− 0.107669i	0.998550	−	0.0538345i	$-0.0171443\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	13.2260	0.489852
$10$	0	0
$11$	− 32.0476i	− 0.878429i	−0.898382	−	0.439214i	$-0.855257\pi$
$11$	− 32.0476i	− 0.878429i	0.898382	−	0.439214i	$-0.144743\pi$
$12$	0	0
$13$	59.4009i	1.26730i	0.773622	+	0.633648i	$0.218443\pi$
$13$	59.4009i	1.26730i	−0.773622	+	0.633648i	$0.781557\pi$
$14$	0	0
$15$	7.63482i	0.131420i
$16$	0	0
$17$	− 0.0501221i	0 0.000715082i	−1.00000		0.000357541i	$-0.999886\pi$
$17$	− 0.0501221i	0 0.000715082i	1.00000		0.000357541i	$-0.000113809\pi$
$18$	0	0
$19$	11.3825	0.137438	0.0687191	−	0.997636i	$-0.478109\pi$
$19$	11.3825	0.137438	0.0687191	+	0.997636i	$0.478109\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 75.2486i	− 0.682192i	−0.940029	−	0.341096i	$-0.889202\pi$
$23$	− 75.2486i	− 0.682192i	0.940029	−	0.341096i	$-0.110798\pi$
$24$	0	0
$25$	123.551	0.988407
$26$	0	0
$27$	87.3601	0.622684
$28$	0	0
$29$	−263.457	−1.68699	−0.843497	−	0.537134i	$-0.819507\pi$
$29$	−263.457	−1.68699	−0.843497	+	0.537134i	$0.819507\pi$
$30$	0	0
$31$	207.558	1.20253	0.601267	−	0.799048i	$-0.294663\pi$
$31$	207.558	1.20253	0.601267	+	0.799048i	$0.294663\pi$
$32$	0	0
$33$	203.259i	1.07221i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	199.310	0.885579	0.442789	−	0.896626i	$-0.353989\pi$
$37$	199.310	0.885579	0.442789	+	0.896626i	$0.353989\pi$
$38$	0	0
$39$	− 376.744i	− 1.54686i
$40$	0	0
$41$	437.976i	1.66830i	0.551537	+	0.834150i	$0.314041\pi$
$41$	437.976i	1.66830i	−0.551537	+	0.834150i	$0.685959\pi$
$42$	0	0
$43$	174.444i	0.618661i	0.950955	+	0.309331i	$0.100105\pi$
$43$	174.444i	0.618661i	−0.950955	+	0.309331i	$0.899895\pi$
$44$	0	0
$45$	− 15.9211i	− 0.0527418i
$46$	0	0
$47$	−283.621	−0.880219	−0.440110	−	0.897944i	$-0.645060\pi$
$47$	−283.621	−0.880219	−0.440110	+	0.897944i	$0.645060\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.317894i	0 0.000872825i
$52$	0	0
$53$	7.41360	0.0192139	0.00960694	−	0.999954i	$-0.496942\pi$
$53$	7.41360	0.0192139	0.00960694	+	0.999954i	$0.496942\pi$
$54$	0	0
$55$	−38.5781	−0.0945795
$56$	0	0
$57$	−72.1924	−0.167756
$58$	0	0
$59$	38.6495	0.0852837	0.0426419	−	0.999090i	$-0.486423\pi$
$59$	38.6495	0.0852837	0.0426419	+	0.999090i	$0.486423\pi$
$60$	0	0
$61$	276.420i	0.580196i	0.956997	+	0.290098i	$0.0936879\pi$
$61$	276.420i	0.580196i	−0.956997	+	0.290098i	$0.906312\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	71.5053	0.136448
$66$	0	0
$67$	− 965.988i	− 1.76141i	−0.473669	−	0.880703i	$-0.657071\pi$
$67$	− 965.988i	− 1.76141i	0.473669	−	0.880703i	$-0.342929\pi$
$68$	0	0
$69$	477.256i	0.832680i
$70$	0	0
$71$	− 580.469i	− 0.970268i	−0.874440	−	0.485134i	$-0.838771\pi$
$71$	− 580.469i	− 0.970268i	0.874440	−	0.485134i	$-0.161229\pi$
$72$	0	0
$73$	− 607.215i	− 0.973550i	−0.873527	−	0.486775i	$-0.838173\pi$
$73$	− 607.215i	− 0.973550i	0.873527	−	0.486775i	$-0.161827\pi$
$74$	0	0
$75$	−783.609	−1.20645
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 827.267i	− 1.17816i	−0.808074	−	0.589081i	$-0.799490\pi$
$79$	− 827.267i	− 1.17816i	0.808074	−	0.589081i	$-0.200510\pi$
$80$	0	0
$81$	−911.175	−1.24990
$82$	0	0
$83$	−1203.55	−1.59165	−0.795826	−	0.605525i	$-0.792963\pi$
$83$	−1203.55	−1.59165	−0.795826	+	0.605525i	$0.792963\pi$
$84$	0	0
$85$	−0.0603357	−7.69921e−5 0
$86$	0	0
$87$	1670.95	2.05914
$88$	0	0
$89$	312.903i	0.372671i	0.982486	+	0.186335i	$0.0596611\pi$
$89$	312.903i	0.372671i	−0.982486	+	0.186335i	$0.940339\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1316.42	−1.46781
$94$	0	0
$95$	− 13.7020i	− 0.0147978i
$96$	0	0
$97$	− 1247.34i	− 1.30565i	−0.757508	−	0.652826i	$-0.773583\pi$
$97$	− 1247.34i	− 1.30565i	0.757508	−	0.652826i	$-0.226417\pi$
$98$	0	0
$99$	− 423.862i	− 0.430300i
$100$	0	0
$101$	− 335.838i	− 0.330862i	−0.986221	−	0.165431i	$-0.947098\pi$
$101$	− 335.838i	− 0.330862i	0.986221	−	0.165431i	$-0.0529016\pi$
$102$	0	0
$103$	−1098.78	−1.05113	−0.525565	−	0.850754i	$-0.676146\pi$
$103$	−1098.78	−1.05113	−0.525565	+	0.850754i	$0.676146\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1311.95i	1.18534i	0.805446	+	0.592669i	$0.201926\pi$
$107$	1311.95i	1.18534i	−0.805446	+	0.592669i	$0.798074\pi$
$108$	0	0
$109$	−2163.44	−1.90110	−0.950548	−	0.310577i	$-0.899478\pi$
$109$	−2163.44	−1.90110	−0.950548	+	0.310577i	$0.899478\pi$
$110$	0	0
$111$	−1264.11	−1.08093
$112$	0	0
$113$	282.482	0.235165	0.117583	−	0.993063i	$-0.462485\pi$
$113$	282.482	0.235165	0.117583	+	0.993063i	$0.462485\pi$
$114$	0	0
$115$	−90.5823	−0.0734508
$116$	0	0
$117$	785.637i	0.620788i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	303.951	0.228363
$122$	0	0
$123$	− 2777.82i	− 2.03632i
$124$	0	0
$125$	− 299.199i	− 0.214090i
$126$	0	0
$127$	− 1431.61i	− 1.00027i	−0.865947	−	0.500135i	$-0.833284\pi$
$127$	− 1431.61i	− 1.00027i	0.865947	−	0.500135i	$-0.166716\pi$
$128$	0	0
$129$	− 1106.39i	− 0.755135i
$130$	0	0
$131$	−2640.46	−1.76106	−0.880528	−	0.473994i	$-0.842812\pi$
$131$	−2640.46	−1.76106	−0.880528	+	0.473994i	$0.842812\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 105.162i	− 0.0670437i
$136$	0	0
$137$	−2066.78	−1.28888	−0.644442	−	0.764653i	$-0.722910\pi$
$137$	−2066.78	−1.28888	−0.644442	+	0.764653i	$0.722910\pi$
$138$	0	0
$139$	−1938.51	−1.18289	−0.591447	−	0.806344i	$-0.701443\pi$
$139$	−1938.51	−1.18289	−0.591447	+	0.806344i	$0.701443\pi$
$140$	0	0
$141$	1798.83	1.07439
$142$	0	0
$143$	1903.66	1.11323
$144$	0	0
$145$	317.144i	0.181637i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−805.415	−0.442833	−0.221417	−	0.975179i	$-0.571068\pi$
$149$	−805.415	−0.442833	−0.221417	+	0.975179i	$0.571068\pi$
$150$	0	0
$151$	448.454i	0.241687i	0.992672	+	0.120843i	$0.0385599\pi$
$151$	448.454i	0.241687i	−0.992672	+	0.120843i	$0.961440\pi$
$152$	0	0
$153$	− 0.662915i	0 0.000350284i
$154$	0	0
$155$	− 249.853i	− 0.129475i
$156$	0	0
$157$	− 1306.65i	− 0.664217i	−0.943241	−	0.332108i	$-0.892240\pi$
$157$	− 1306.65i	− 0.664217i	0.943241	−	0.332108i	$-0.107760\pi$
$158$	0	0
$159$	−47.0200	−0.0234524
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 153.293i	− 0.0736617i	−0.999322	−	0.0368308i	$-0.988274\pi$
$163$	− 153.293i	− 0.0736617i	0.999322	−	0.0368308i	$-0.0117263\pi$
$164$	0	0
$165$	244.678	0.115443
$166$	0	0
$167$	800.313	0.370839	0.185419	−	0.982659i	$-0.440636\pi$
$167$	800.313	0.370839	0.185419	+	0.982659i	$0.440636\pi$
$168$	0	0
$169$	−1331.47	−0.606039
$170$	0	0
$171$	150.545	0.0673244
$172$	0	0
$173$	− 1535.87i	− 0.674971i	−0.941331	−	0.337486i	$-0.890424\pi$
$173$	− 1535.87i	− 0.674971i	0.941331	−	0.337486i	$-0.109576\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−245.131	−0.104097
$178$	0	0
$179$	− 2468.37i	− 1.03070i	−0.856981	−	0.515348i	$-0.827662\pi$
$179$	− 2468.37i	− 1.03070i	0.856981	−	0.515348i	$-0.172338\pi$
$180$	0	0
$181$	3090.01i	1.26894i	0.772947	+	0.634471i	$0.218782\pi$
$181$	3090.01i	1.26894i	−0.772947	+	0.634471i	$0.781218\pi$
$182$	0	0
$183$	− 1753.16i	− 0.708184i
$184$	0	0
$185$	− 239.925i	− 0.0953493i
$186$	0	0
$187$	−1.60629	−0.000628149 0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 4408.01i	− 1.66991i	−0.550320	−	0.834954i	$-0.685494\pi$
$191$	− 4408.01i	− 1.66991i	0.550320	−	0.834954i	$-0.314506\pi$
$192$	0	0
$193$	2099.73	0.783119	0.391559	−	0.920153i	$-0.371936\pi$
$193$	2099.73	0.783119	0.391559	+	0.920153i	$0.371936\pi$
$194$	0	0
$195$	−453.515	−0.166548
$196$	0	0
$197$	−4466.65	−1.61541	−0.807706	−	0.589586i	$-0.799291\pi$
$197$	−4466.65	−1.61541	−0.807706	+	0.589586i	$0.799291\pi$
$198$	0	0
$199$	2908.11	1.03593	0.517967	−	0.855401i	$-0.326689\pi$
$199$	2908.11	1.03593	0.517967	+	0.855401i	$0.326689\pi$
$200$	0	0
$201$	6126.68i	2.14996i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	527.224	0.179624
$206$	0	0
$207$	− 995.238i	− 0.334173i
$208$	0	0
$209$	− 364.782i	− 0.120730i
$210$	0	0
$211$	4943.88i	1.61304i	0.591210	+	0.806518i	$0.298651\pi$
$211$	4943.88i	1.61304i	−0.591210	+	0.806518i	$0.701349\pi$
$212$	0	0
$213$	3681.56i	1.18430i
$214$	0	0
$215$	209.991	0.0666106
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3851.20i	1.18831i
$220$	0	0
$221$	2.97730	0.000906220 0
$222$	0	0
$223$	1390.25	0.417481	0.208741	−	0.977971i	$-0.433064\pi$
$223$	1390.25	0.417481	0.208741	+	0.977971i	$0.433064\pi$
$224$	0	0
$225$	1634.09	0.484173
$226$	0	0
$227$	−5056.08	−1.47834	−0.739172	−	0.673517i	$-0.764783\pi$
$227$	−5056.08	−1.47834	−0.739172	+	0.673517i	$0.764783\pi$
$228$	0	0
$229$	− 3722.99i	− 1.07433i	−0.843476	−	0.537166i	$-0.819495\pi$
$229$	− 3722.99i	− 1.07433i	0.843476	−	0.537166i	$-0.180505\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6203.27	1.74416	0.872082	−	0.489361i	$-0.162770\pi$
$233$	6203.27	1.74416	0.872082	+	0.489361i	$0.162770\pi$
$234$	0	0
$235$	341.415i	0.0947722i
$236$	0	0
$237$	5246.86i	1.43806i
$238$	0	0
$239$	2556.16i	0.691818i	0.938268	+	0.345909i	$0.112429\pi$
$239$	2556.16i	0.691818i	−0.938268	+	0.345909i	$0.887571\pi$
$240$	0	0
$241$	− 352.177i	− 0.0941317i	−0.998892	−	0.0470659i	$-0.985013\pi$
$241$	− 352.177i	− 0.0941317i	0.998892	−	0.0470659i	$-0.0149871\pi$
$242$	0	0
$243$	3420.31	0.902934
$244$	0	0
$245$	0	0
$246$	0	0
$247$	676.131i	0.174175i
$248$	0	0
$249$	7633.41	1.94276
$250$	0	0
$251$	5904.72	1.48487	0.742436	−	0.669917i	$-0.233670\pi$
$251$	5904.72	1.48487	0.742436	+	0.669917i	$0.233670\pi$
$252$	0	0
$253$	−2411.54	−0.599257
$254$	0	0
$255$	0.382673	9.39762e−5 0
$256$	0	0
$257$	− 227.216i	− 0.0551492i	−0.999620	−	0.0275746i	$-0.991222\pi$
$257$	− 227.216i	− 0.0551492i	0.999620	−	0.0275746i	$-0.00877838\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3484.49	−0.826377
$262$	0	0
$263$	979.707i	0.229701i	0.993383	+	0.114850i	$0.0366389\pi$
$263$	979.707i	0.229701i	−0.993383	+	0.114850i	$0.963361\pi$
$264$	0	0
$265$	− 8.92430i	− 0.00206874i
$266$	0	0
$267$	− 1984.56i	− 0.454880i
$268$	0	0
$269$	980.037i	0.222134i	0.993813	+	0.111067i	$0.0354267\pi$
$269$	980.037i	0.222134i	−0.993813	+	0.111067i	$0.964573\pi$
$270$	0	0
$271$	−1074.47	−0.240847	−0.120424	−	0.992723i	$-0.538425\pi$
$271$	−1074.47	−0.240847	−0.120424	+	0.992723i	$0.538425\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3959.51i	− 0.868246i
$276$	0	0
$277$	832.519	0.180582	0.0902911	−	0.995915i	$-0.471220\pi$
$277$	832.519	0.180582	0.0902911	+	0.995915i	$0.471220\pi$
$278$	0	0
$279$	2745.16	0.589063
$280$	0	0
$281$	453.251	0.0962230	0.0481115	−	0.998842i	$-0.484680\pi$
$281$	453.251	0.0962230	0.0481115	+	0.998842i	$0.484680\pi$
$282$	0	0
$283$	−441.191	−0.0926716	−0.0463358	−	0.998926i	$-0.514754\pi$
$283$	−441.191	−0.0926716	−0.0463358	+	0.998926i	$0.514754\pi$
$284$	0	0
$285$	86.9034i	0.0180621i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4913.00	0.999999
$290$	0	0
$291$	7911.13i	1.59367i
$292$	0	0
$293$	− 2767.74i	− 0.551853i	−0.961179	−	0.275926i	$-0.911015\pi$
$293$	− 2767.74i	− 0.551853i	0.961179	−	0.275926i	$-0.0889846\pi$
$294$	0	0
$295$	− 46.5253i	− 0.00918241i
$296$	0	0
$297$	− 2799.68i	− 0.546984i
$298$	0	0
$299$	4469.83	0.864539
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2130.02i	0.403849i
$304$	0	0
$305$	332.747	0.0624690
$306$	0	0
$307$	107.837	0.0200476	0.0100238	−	0.999950i	$-0.496809\pi$
$307$	107.837	0.0200476	0.0100238	+	0.999950i	$0.496809\pi$
$308$	0	0
$309$	6968.92	1.28300
$310$	0	0
$311$	−6146.50	−1.12069	−0.560347	−	0.828258i	$-0.689332\pi$
$311$	−6146.50	−1.12069	−0.560347	+	0.828258i	$0.689332\pi$
$312$	0	0
$313$	− 6879.22i	− 1.24229i	−0.783696	−	0.621144i	$-0.786668\pi$
$313$	− 6879.22i	− 1.24229i	0.783696	−	0.621144i	$-0.213332\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7071.92	−1.25299	−0.626497	−	0.779424i	$-0.715512\pi$
$317$	−7071.92	−1.25299	−0.626497	+	0.779424i	$0.715512\pi$
$318$	0	0
$319$	8443.18i	1.48190i
$320$	0	0
$321$	− 8320.92i	− 1.44682i
$322$	0	0
$323$	− 0.570515i	0 9.82796e-5i
$324$	0	0
$325$	7339.04i	1.25260i
$326$	0	0
$327$	13721.4	2.32047
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1142.78i	0.189768i	0.995488	+	0.0948839i	$0.0302480\pi$
$331$	1142.78i	0.189768i	−0.995488	+	0.0948839i	$0.969752\pi$
$332$	0	0
$333$	2636.08	0.433803
$334$	0	0
$335$	−1162.83	−0.189649
$336$	0	0
$337$	−1342.78	−0.217050	−0.108525	−	0.994094i	$-0.534613\pi$
$337$	−1342.78	−0.217050	−0.108525	+	0.994094i	$0.534613\pi$
$338$	0	0
$339$	−1791.61	−0.287042
$340$	0	0
$341$	− 6651.74i	− 1.05634i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	574.509	0.0896537
$346$	0	0
$347$	5757.34i	0.890692i	0.895359	+	0.445346i	$0.146919\pi$
$347$	5757.34i	0.890692i	−0.895359	+	0.445346i	$0.853081\pi$
$348$	0	0
$349$	− 1592.10i	− 0.244193i	−0.992518	−	0.122097i	$-0.961038\pi$
$349$	− 1592.10i	− 0.244193i	0.992518	−	0.122097i	$-0.0389618\pi$
$350$	0	0
$351$	5189.27i	0.789125i
$352$	0	0
$353$	1917.78i	0.289159i	0.989493	+	0.144580i	$0.0461830\pi$
$353$	1917.78i	0.289159i	−0.989493	+	0.144580i	$0.953817\pi$
$354$	0	0
$355$	−698.754	−0.104468
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7999.18i	1.17599i	0.808864	+	0.587996i	$0.200083\pi$
$359$	7999.18i	1.17599i	−0.808864	+	0.587996i	$0.799917\pi$
$360$	0	0
$361$	−6729.44	−0.981111
$362$	0	0
$363$	−1927.78	−0.278738
$364$	0	0
$365$	−730.950	−0.104821
$366$	0	0
$367$	−12487.3	−1.77611	−0.888055	−	0.459736i	$-0.847944\pi$
$367$	−12487.3	−1.77611	−0.888055	+	0.459736i	$0.847944\pi$
$368$	0	0
$369$	5792.67i	0.817220i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7697.61	−1.06855	−0.534273	−	0.845312i	$-0.679414\pi$
$373$	−7697.61	−1.06855	−0.534273	+	0.845312i	$0.679414\pi$
$374$	0	0
$375$	1897.64i	0.261317i
$376$	0	0
$377$	− 15649.6i	− 2.13792i
$378$	0	0
$379$	− 436.305i	− 0.0591332i	−0.999563	−	0.0295666i	$-0.990587\pi$
$379$	− 436.305i	− 0.0591332i	0.999563	−	0.0295666i	$-0.00941271\pi$
$380$	0	0
$381$	9079.81i	1.22093i
$382$	0	0
$383$	−6527.29	−0.870833	−0.435417	−	0.900229i	$-0.643399\pi$
$383$	−6527.29	−0.870833	−0.435417	+	0.900229i	$0.643399\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2307.20i	0.303053i
$388$	0	0
$389$	−4911.51	−0.640163	−0.320082	−	0.947390i	$-0.603710\pi$
$389$	−4911.51	−0.640163	−0.320082	+	0.947390i	$0.603710\pi$
$390$	0	0
$391$	−3.77162	−0.000487823 0
$392$	0	0
$393$	16746.9	2.14954
$394$	0	0
$395$	−995.844	−0.126851
$396$	0	0
$397$	− 13192.4i	− 1.66778i	−0.551933	−	0.833889i	$-0.686110\pi$
$397$	− 13192.4i	− 1.66778i	0.551933	−	0.833889i	$-0.313890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11606.3	1.44536	0.722681	−	0.691182i	$-0.242910\pi$
$401$	11606.3	1.44536	0.722681	+	0.691182i	$0.242910\pi$
$402$	0	0
$403$	12329.1i	1.52397i
$404$	0	0
$405$	1096.85i	0.134575i
$406$	0	0
$407$	− 6387.42i	− 0.777918i
$408$	0	0
$409$	− 11332.9i	− 1.37011i	−0.728493	−	0.685053i	$-0.759779\pi$
$409$	− 11332.9i	− 1.37011i	0.728493	−	0.685053i	$-0.240221\pi$
$410$	0	0
$411$	13108.4	1.57321
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1448.81i	0.171371i
$416$	0	0
$417$	12294.8	1.44383
$418$	0	0
$419$	6871.75	0.801210	0.400605	−	0.916251i	$-0.368800\pi$
$419$	6871.75	0.801210	0.400605	+	0.916251i	$0.368800\pi$
$420$	0	0
$421$	4187.77	0.484797	0.242398	−	0.970177i	$-0.422066\pi$
$421$	4187.77	0.484797	0.242398	+	0.970177i	$0.422066\pi$
$422$	0	0
$423$	−3751.17	−0.431177
$424$	0	0
$425$	− 6.19263i	0 0.000706792i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12073.8	−1.35880
$430$	0	0
$431$	10863.3i	1.21408i	0.794673	+	0.607038i	$0.207642\pi$
$431$	10863.3i	1.21408i	−0.794673	+	0.607038i	$0.792358\pi$
$432$	0	0
$433$	− 7048.70i	− 0.782307i	−0.920325	−	0.391154i	$-0.872076\pi$
$433$	− 7048.70i	− 0.782307i	0.920325	−	0.391154i	$-0.127924\pi$
$434$	0	0
$435$	− 2011.45i	− 0.221705i
$436$	0	0
$437$	− 856.517i	− 0.0937592i
$438$	0	0
$439$	7070.05	0.768645	0.384323	−	0.923199i	$-0.374435\pi$
$439$	7070.05	0.768645	0.384323	+	0.923199i	$0.374435\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8168.28i	− 0.876042i	−0.898965	−	0.438021i	$-0.855680\pi$
$443$	− 8168.28i	− 0.876042i	0.898965	−	0.438021i	$-0.144320\pi$
$444$	0	0
$445$	376.665	0.0401250
$446$	0	0
$447$	5108.26	0.540520
$448$	0	0
$449$	−10582.8	−1.11233	−0.556164	−	0.831073i	$-0.687727\pi$
$449$	−10582.8	−1.11233	−0.556164	+	0.831073i	$0.687727\pi$
$450$	0	0
$451$	14036.1	1.46548
$452$	0	0
$453$	− 2844.28i	− 0.295002i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7352.74	0.752619	0.376309	−	0.926494i	$-0.377193\pi$
$457$	7352.74	0.752619	0.376309	+	0.926494i	$0.377193\pi$
$458$	0	0
$459$	− 4.37867i	0 0.000445270i
$460$	0	0
$461$	13541.4i	1.36809i	0.729442	+	0.684043i	$0.239780\pi$
$461$	13541.4i	1.36809i	−0.729442	+	0.684043i	$0.760220\pi$
$462$	0	0
$463$	16798.3i	1.68615i	0.537800	+	0.843073i	$0.319256\pi$
$463$	16798.3i	1.68615i	−0.537800	+	0.843073i	$0.680744\pi$
$464$	0	0
$465$	1584.67i	0.158037i
$466$	0	0
$467$	−16085.1	−1.59385	−0.796926	−	0.604076i	$-0.793542\pi$
$467$	−16085.1	−1.59385	−0.796926	+	0.604076i	$0.793542\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8287.29i	0.810739i
$472$	0	0
$473$	5590.51	0.543450
$474$	0	0
$475$	1406.32	0.135845
$476$	0	0
$477$	98.0522	0.00941196
$478$	0	0
$479$	−15911.2	−1.51775	−0.758874	−	0.651237i	$-0.774250\pi$
$479$	−15911.2	−1.51775	−0.758874	+	0.651237i	$0.774250\pi$
$480$	0	0
$481$	11839.2i	1.12229i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1501.52	−0.140578
$486$	0	0
$487$	− 16419.4i	− 1.52779i	−0.645339	−	0.763897i	$-0.723284\pi$
$487$	− 16419.4i	− 1.52779i	0.645339	−	0.763897i	$-0.276716\pi$
$488$	0	0
$489$	972.247i	0.0899111i
$490$	0	0
$491$	− 17000.0i	− 1.56253i	−0.624202	−	0.781263i	$-0.714576\pi$
$491$	− 17000.0i	− 1.56253i	0.624202	−	0.781263i	$-0.285424\pi$
$492$	0	0
$493$	13.2050i	0.00120634i
$494$	0	0
$495$	−510.234	−0.0463300
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12104.6i	− 1.08592i	−0.839758	−	0.542961i	$-0.817303\pi$
$499$	− 12104.6i	− 1.08592i	0.839758	−	0.542961i	$-0.182697\pi$
$500$	0	0
$501$	−5075.90	−0.452644
$502$	0	0
$503$	−2835.69	−0.251366	−0.125683	−	0.992070i	$-0.540112\pi$
$503$	−2835.69	−0.251366	−0.125683	+	0.992070i	$0.540112\pi$
$504$	0	0
$505$	−404.273	−0.0356236
$506$	0	0
$507$	8444.70	0.739728
$508$	0	0
$509$	− 6737.27i	− 0.586688i	−0.956007	−	0.293344i	$-0.905232\pi$
$509$	− 6737.27i	− 0.586688i	0.956007	−	0.293344i	$-0.0947682\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	994.377	0.0855806
$514$	0	0
$515$	1322.69i	0.113174i
$516$	0	0
$517$	9089.36i	0.773210i
$518$	0	0
$519$	9741.10i	0.823866i
$520$	0	0
$521$	15506.8i	1.30397i	0.758234	+	0.651983i	$0.226063\pi$
$521$	15506.8i	1.30397i	−0.758234	+	0.651983i	$0.773937\pi$
$522$	0	0
$523$	17300.8	1.44648	0.723241	−	0.690595i	$-0.242651\pi$
$523$	17300.8	1.44648	0.723241	+	0.690595i	$0.242651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.4032i	0 0.000859910i
$528$	0	0
$529$	6504.65	0.534614
$530$	0	0
$531$	511.179	0.0417764
$532$	0	0
$533$	−26016.2	−2.11423
$534$	0	0
$535$	1579.30	0.127624
$536$	0	0
$537$	15655.4i	1.25806i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16817.1	1.33646	0.668228	−	0.743957i	$-0.267053\pi$
$541$	16817.1	1.33646	0.668228	+	0.743957i	$0.267053\pi$
$542$	0	0
$543$	− 19598.0i	− 1.54886i
$544$	0	0
$545$	2604.29i	0.204689i
$546$	0	0
$547$	21898.3i	1.71171i	0.517217	+	0.855855i	$0.326968\pi$
$547$	21898.3i	1.71171i	−0.517217	+	0.855855i	$0.673032\pi$
$548$	0	0
$549$	3655.93i	0.284210i
$550$	0	0
$551$	−2998.81	−0.231857
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1521.70i	0.116383i
$556$	0	0
$557$	6338.38	0.482165	0.241082	−	0.970505i	$-0.422498\pi$
$557$	6338.38	0.482165	0.241082	+	0.970505i	$0.422498\pi$
$558$	0	0
$559$	−10362.1	−0.784027
$560$	0	0
$561$	10.1878	0.000766715 0
$562$	0	0
$563$	−6280.96	−0.470180	−0.235090	−	0.971974i	$-0.575538\pi$
$563$	−6280.96	−0.470180	−0.235090	+	0.971974i	$0.575538\pi$
$564$	0	0
$565$	− 340.045i	− 0.0253200i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4614.36	0.339972	0.169986	−	0.985446i	$-0.445628\pi$
$569$	4614.36	0.339972	0.169986	+	0.985446i	$0.445628\pi$
$570$	0	0
$571$	− 10508.4i	− 0.770160i	−0.922883	−	0.385080i	$-0.874174\pi$
$571$	− 10508.4i	− 0.770160i	0.922883	−	0.385080i	$-0.125826\pi$
$572$	0	0
$573$	27957.4i	2.03828i
$574$	0	0
$575$	− 9297.03i	− 0.674283i
$576$	0	0
$577$	− 19662.2i	− 1.41863i	−0.704892	−	0.709314i	$-0.749005\pi$
$577$	− 19662.2i	− 1.41863i	0.704892	−	0.709314i	$-0.250995\pi$
$578$	0	0
$579$	−13317.3	−0.955871
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 237.588i	− 0.0168780i
$584$	0	0
$585$	945.730	0.0668395
$586$	0	0
$587$	14249.8	1.00196	0.500980	−	0.865459i	$-0.332973\pi$
$587$	14249.8	1.00196	0.500980	+	0.865459i	$0.332973\pi$
$588$	0	0
$589$	2362.53	0.165274
$590$	0	0
$591$	28329.3	1.97176
$592$	0	0
$593$	14904.4i	1.03212i	0.856552	+	0.516061i	$0.172602\pi$
$593$	14904.4i	1.03212i	−0.856552	+	0.516061i	$0.827398\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18444.4	−1.26445
$598$	0	0
$599$	− 20362.1i	− 1.38894i	−0.719522	−	0.694469i	$-0.755639\pi$
$599$	− 20362.1i	− 1.38894i	0.719522	−	0.694469i	$-0.244361\pi$
$600$	0	0
$601$	3974.81i	0.269777i	0.990861	+	0.134889i	$0.0430676\pi$
$601$	3974.81i	0.269777i	−0.990861	+	0.134889i	$0.956932\pi$
$602$	0	0
$603$	− 12776.2i	− 0.862829i
$604$	0	0
$605$	− 365.888i	− 0.0245875i
$606$	0	0
$607$	−26181.2	−1.75068	−0.875339	−	0.483510i	$-0.839362\pi$
$607$	−26181.2	−1.75068	−0.875339	+	0.483510i	$0.839362\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 16847.3i	− 1.11550i
$612$	0	0
$613$	−4131.71	−0.272232	−0.136116	−	0.990693i	$-0.543462\pi$
$613$	−4131.71	−0.272232	−0.136116	+	0.990693i	$0.543462\pi$
$614$	0	0
$615$	−3343.87	−0.219248
$616$	0	0
$617$	−5746.66	−0.374962	−0.187481	−	0.982268i	$-0.560032\pi$
$617$	−5746.66	−0.374962	−0.187481	+	0.982268i	$0.560032\pi$
$618$	0	0
$619$	15861.2	1.02991	0.514956	−	0.857216i	$-0.327808\pi$
$619$	15861.2	1.02991	0.514956	+	0.857216i	$0.327808\pi$
$620$	0	0
$621$	− 6573.73i	− 0.424790i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15083.7	0.965357
$626$	0	0
$627$	2313.59i	0.147362i
$628$	0	0
$629$	− 9.98985i	0 0.000633261i
$630$	0	0
$631$	− 22206.1i	− 1.40097i	−0.713669	−	0.700483i	$-0.752968\pi$
$631$	− 22206.1i	− 1.40097i	0.713669	−	0.700483i	$-0.247032\pi$
$632$	0	0
$633$	− 31356.0i	− 1.96886i
$634$	0	0
$635$	−1723.33	−0.107698
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 7677.29i	− 0.475288i
$640$	0	0
$641$	−19154.1	−1.18025	−0.590125	−	0.807312i	$-0.700922\pi$
$641$	−19154.1	−1.18025	−0.590125	+	0.807312i	$0.700922\pi$
$642$	0	0
$643$	−10627.2	−0.651782	−0.325891	−	0.945407i	$-0.605664\pi$
$643$	−10627.2	−0.651782	−0.325891	+	0.945407i	$0.605664\pi$
$644$	0	0
$645$	−1331.85	−0.0813046
$646$	0	0
$647$	−200.922	−0.0122088	−0.00610438	−	0.999981i	$-0.501943\pi$
$647$	−200.922	−0.0122088	−0.00610438	+	0.999981i	$0.501943\pi$
$648$	0	0
$649$	− 1238.63i	− 0.0749157i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20890.6	−1.25193	−0.625965	−	0.779851i	$-0.715295\pi$
$653$	−20890.6	−1.25193	−0.625965	+	0.779851i	$0.715295\pi$
$654$	0	0
$655$	3178.52i	0.189611i
$656$	0	0
$657$	− 8031.03i	− 0.476895i
$658$	0	0
$659$	− 5837.60i	− 0.345069i	−0.985003	−	0.172535i	$-0.944804\pi$
$659$	− 5837.60i	− 0.345069i	0.985003	−	0.172535i	$-0.0551957\pi$
$660$	0	0
$661$	12540.0i	0.737898i	0.929450	+	0.368949i	$0.120282\pi$
$661$	12540.0i	0.737898i	−0.929450	+	0.368949i	$0.879718\pi$
$662$	0	0
$663$	−18.8832	−0.00110613
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19824.8i	1.15085i
$668$	0	0
$669$	−8817.54	−0.509575
$670$	0	0
$671$	8858.60	0.509661
$672$	0	0
$673$	1101.18	0.0630720	0.0315360	−	0.999503i	$-0.489960\pi$
$673$	1101.18	0.0630720	0.0315360	+	0.999503i	$0.489960\pi$
$674$	0	0
$675$	10793.4	0.615465
$676$	0	0
$677$	21392.2i	1.21443i	0.794537	+	0.607216i	$0.207714\pi$
$677$	21392.2i	1.21443i	−0.794537	+	0.607216i	$0.792286\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	32067.7	1.80446
$682$	0	0
$683$	− 27751.0i	− 1.55471i	−0.629065	−	0.777353i	$-0.716562\pi$
$683$	− 27751.0i	− 1.55471i	0.629065	−	0.777353i	$-0.283438\pi$
$684$	0	0
$685$	2487.94i	0.138773i
$686$	0	0
$687$	23612.7i	1.31133i
$688$	0	0
$689$	440.374i	0.0243497i
$690$	0	0
$691$	−14965.7	−0.823912	−0.411956	−	0.911204i	$-0.635154\pi$
$691$	−14965.7	−0.823912	−0.411956	+	0.911204i	$0.635154\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2333.53i	0.127361i
$696$	0	0
$697$	21.9523	0.00119297
$698$	0	0
$699$	−39343.6	−2.12892
$700$	0	0
$701$	18759.8	1.01077	0.505383	−	0.862895i	$-0.331351\pi$
$701$	18759.8	1.01077	0.505383	+	0.862895i	$0.331351\pi$
$702$	0	0
$703$	2268.65	0.121712
$704$	0	0
$705$	− 2165.39i	− 0.115679i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25289.7	−1.33960	−0.669800	−	0.742542i	$-0.733620\pi$
$709$	−25289.7	−1.33960	−0.669800	+	0.742542i	$0.733620\pi$
$710$	0	0
$711$	− 10941.4i	− 0.577125i
$712$	0	0
$713$	− 15618.4i	− 0.820358i
$714$	0	0
$715$	− 2291.58i	− 0.119860i
$716$	0	0
$717$	− 16212.2i	− 0.844429i
$718$	0	0
$719$	−29721.2	−1.54161	−0.770803	−	0.637074i	$-0.780145\pi$
$719$	−29721.2	−1.54161	−0.770803	+	0.637074i	$0.780145\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2233.65i	0.114897i
$724$	0	0
$725$	−32550.4	−1.66744
$726$	0	0
$727$	−32242.6	−1.64486	−0.822429	−	0.568868i	$-0.807381\pi$
$727$	−32242.6	−1.64486	−0.822429	+	0.568868i	$0.807381\pi$
$728$	0	0
$729$	2908.76	0.147780
$730$	0	0
$731$	8.74349	0.000442394 0
$732$	0	0
$733$	5731.71i	0.288821i	0.989518	+	0.144410i	$0.0461286\pi$
$733$	5731.71i	0.288821i	−0.989518	+	0.144410i	$0.953871\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30957.6	−1.54727
$738$	0	0
$739$	− 12713.2i	− 0.632833i	−0.948620	−	0.316416i	$-0.897520\pi$
$739$	− 12713.2i	− 0.632833i	0.948620	−	0.316416i	$-0.102480\pi$
$740$	0	0
$741$	− 4288.29i	− 0.212597i
$742$	0	0
$743$	− 33625.5i	− 1.66029i	−0.557544	−	0.830147i	$-0.688256\pi$
$743$	− 33625.5i	− 1.66029i	0.557544	−	0.830147i	$-0.311744\pi$
$744$	0	0
$745$	969.539i	0.0476794i
$746$	0	0
$747$	−15918.2	−0.779674
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 3188.67i	− 0.154935i	−0.996995	−	0.0774675i	$-0.975317\pi$
$751$	− 3188.67i	− 0.154935i	0.996995	−	0.0774675i	$-0.0246834\pi$
$752$	0	0
$753$	−37450.1	−1.81243
$754$	0	0
$755$	539.838	0.0260222
$756$	0	0
$757$	−5260.96	−0.252593	−0.126297	−	0.991993i	$-0.540309\pi$
$757$	−5260.96	−0.252593	−0.126297	+	0.991993i	$0.540309\pi$
$758$	0	0
$759$	15294.9	0.731450
$760$	0	0
$761$	− 948.174i	− 0.0451659i	−0.999745	−	0.0225830i	$-0.992811\pi$
$761$	− 948.174i	− 0.0451659i	0.999745	−	0.0225830i	$-0.00718899\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.798001	−3.77147e−5 0
$766$	0	0
$767$	2295.82i	0.108080i
$768$	0	0
$769$	− 11408.5i	− 0.534983i	−0.963560	−	0.267492i	$-0.913805\pi$
$769$	− 11408.5i	− 0.534983i	0.963560	−	0.267492i	$-0.0861947\pi$
$770$	0	0
$771$	1441.09i	0.0673149i
$772$	0	0
$773$	15877.3i	0.738767i	0.929277	+	0.369384i	$0.120431\pi$
$773$	15877.3i	0.738767i	−0.929277	+	0.369384i	$0.879569\pi$
$774$	0	0
$775$	25644.0	1.18859
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4985.26i	0.229288i
$780$	0	0
$781$	−18602.6	−0.852311
$782$	0	0
$783$	−23015.7	−1.05046
$784$	0	0
$785$	−1572.91	−0.0715155
$786$	0	0
$787$	−4441.58	−0.201176	−0.100588	−	0.994928i	$-0.532072\pi$
$787$	−4441.58	−0.201176	−0.100588	+	0.994928i	$0.532072\pi$
$788$	0	0
$789$	− 6213.69i	− 0.280372i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16419.6	−0.735280
$794$	0	0
$795$	56.6015i	0.00252509i
$796$	0	0
$797$	− 31356.7i	− 1.39361i	−0.717258	−	0.696807i	$-0.754603\pi$
$797$	− 31356.7i	− 1.39361i	0.717258	−	0.696807i	$-0.245397\pi$
$798$	0	0
$799$	14.2157i	0 0.000629429i
$800$	0	0
$801$	4138.46i	0.182553i
$802$	0	0
$803$	−19459.8	−0.855194
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6215.78i	− 0.271135i
$808$	0	0
$809$	17706.0	0.769483	0.384741	−	0.923024i	$-0.374291\pi$
$809$	17706.0	0.769483	0.384741	+	0.923024i	$0.374291\pi$
$810$	0	0
$811$	395.892	0.0171414	0.00857069	−	0.999963i	$-0.497272\pi$
$811$	395.892	0.0171414	0.00857069	+	0.999963i	$0.497272\pi$
$812$	0	0
$813$	6814.74	0.293977
$814$	0	0
$815$	−184.531	−0.00793107
$816$	0	0
$817$	1985.61i	0.0850277i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17612.5	0.748696	0.374348	−	0.927288i	$-0.377867\pi$
$821$	17612.5	0.748696	0.374348	+	0.927288i	$0.377867\pi$
$822$	0	0
$823$	27818.7i	1.17825i	0.808043	+	0.589124i	$0.200527\pi$
$823$	27818.7i	1.17825i	−0.808043	+	0.589124i	$0.799473\pi$
$824$	0	0
$825$	25112.8i	1.05978i
$826$	0	0
$827$	− 20407.9i	− 0.858106i	−0.903279	−	0.429053i	$-0.858847\pi$
$827$	− 20407.9i	− 0.858106i	0.903279	−	0.429053i	$-0.141153\pi$
$828$	0	0
$829$	− 37155.2i	− 1.55664i	−0.627870	−	0.778318i	$-0.716073\pi$
$829$	− 37155.2i	− 1.55664i	0.627870	−	0.778318i	$-0.283927\pi$
$830$	0	0
$831$	−5280.17	−0.220418
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 963.397i	− 0.0399278i
$836$	0	0
$837$	18132.3	0.748798
$838$	0	0
$839$	43415.1	1.78648	0.893239	−	0.449582i	$-0.148427\pi$
$839$	43415.1	1.78648	0.893239	+	0.449582i	$0.148427\pi$
$840$	0	0
$841$	45020.8	1.84595
$842$	0	0
$843$	−2874.70	−0.117449
$844$	0	0
$845$	1602.79i	0.0652516i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2798.21	0.113114
$850$	0	0
$851$	− 14997.8i	− 0.604135i
$852$	0	0
$853$	23345.0i	0.937067i	0.883446	+	0.468533i	$0.155217\pi$
$853$	23345.0i	0.937067i	−0.883446	+	0.468533i	$0.844783\pi$
$854$	0	0
$855$	− 181.222i	− 0.00724874i
$856$	0	0
$857$	− 32275.9i	− 1.28649i	−0.765660	−	0.643245i	$-0.777588\pi$
$857$	− 32275.9i	− 1.28649i	0.765660	−	0.643245i	$-0.222412\pi$
$858$	0	0
$859$	1427.27	0.0566913	0.0283457	−	0.999598i	$-0.490976\pi$
$859$	1427.27	0.0566913	0.0283457	+	0.999598i	$0.490976\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 40868.3i	− 1.61202i	−0.591902	−	0.806010i	$-0.701623\pi$
$863$	− 40868.3i	− 1.61202i	0.591902	−	0.806010i	$-0.298377\pi$
$864$	0	0
$865$	−1848.84	−0.0726734
$866$	0	0
$867$	−31160.2	−1.22059
$868$	0	0
$869$	−26511.9	−1.03493
$870$	0	0
$871$	57380.6	2.23222
$872$	0	0
$873$	− 16497.3i	− 0.639576i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11616.7	−0.447285	−0.223643	−	0.974671i	$-0.571795\pi$
$877$	−11616.7	−0.447285	−0.223643	+	0.974671i	$0.571795\pi$
$878$	0	0
$879$	17554.1i	0.673589i
$880$	0	0
$881$	− 24754.9i	− 0.946668i	−0.880883	−	0.473334i	$-0.843050\pi$
$881$	− 24754.9i	− 0.946668i	0.880883	−	0.473334i	$-0.156950\pi$
$882$	0	0
$883$	26120.2i	0.995485i	0.867325	+	0.497743i	$0.165838\pi$
$883$	26120.2i	0.995485i	−0.867325	+	0.497743i	$0.834162\pi$
$884$	0	0
$885$	295.082i	0.0112080i
$886$	0	0
$887$	899.397	0.0340460	0.0170230	−	0.999855i	$-0.494581\pi$
$887$	899.397	0.0340460	0.0170230	+	0.999855i	$0.494581\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	29201.0i	1.09795i
$892$	0	0
$893$	−3228.31	−0.120976
$894$	0	0
$895$	−2971.36	−0.110974
$896$	0	0
$897$	−28349.5	−1.05525
$898$	0	0
$899$	−54682.7	−2.02867
$900$	0	0
$901$	− 0.371585i	0 1.37395e-5i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3719.67	0.136626
$906$	0	0
$907$	− 31077.6i	− 1.13772i	−0.822433	−	0.568862i	$-0.807384\pi$
$907$	− 31077.6i	− 1.13772i	0.822433	−	0.568862i	$-0.192616\pi$
$908$	0	0
$909$	− 4441.79i	− 0.162074i
$910$	0	0
$911$	− 34740.9i	− 1.26347i	−0.775186	−	0.631734i	$-0.782344\pi$
$911$	− 34740.9i	− 1.26347i	0.775186	−	0.631734i	$-0.217656\pi$
$912$	0	0
$913$	38571.0i	1.39815i
$914$	0	0
$915$	−2110.42	−0.0762494
$916$	0	0
$917$	0	0
$918$	0	0
$919$	39231.5i	1.40819i	0.710104	+	0.704096i	$0.248648\pi$
$919$	39231.5i	1.40819i	−0.710104	+	0.704096i	$0.751352\pi$
$920$	0	0
$921$	−683.947	−0.0244700
$922$	0	0
$923$	34480.4	1.22962
$924$	0	0
$925$	24625.0	0.875313
$926$	0	0
$927$	−14532.5	−0.514898
$928$	0	0
$929$	45468.4i	1.60578i	0.596127	+	0.802890i	$0.296705\pi$
$929$	45468.4i	1.60578i	−0.596127	+	0.802890i	$0.703295\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	38983.5	1.36791
$934$	0	0
$935$	1.93362i	0 6.76321e-5i
$936$	0	0
$937$	48136.2i	1.67827i	0.543922	+	0.839136i	$0.316939\pi$
$937$	48136.2i	1.67827i	−0.543922	+	0.839136i	$0.683061\pi$
$938$	0	0
$939$	43630.7i	1.51633i
$940$	0	0
$941$	54900.0i	1.90190i	0.309342	+	0.950951i	$0.399891\pi$
$941$	54900.0i	1.90190i	−0.309342	+	0.950951i	$0.600109\pi$
$942$	0	0
$943$	32957.0	1.13810
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18393.7i	0.631167i	0.948898	+	0.315584i	$0.102200\pi$
$947$	18393.7i	0.631167i	−0.948898	+	0.315584i	$0.897800\pi$
$948$	0	0
$949$	36069.1	1.23378
$950$	0	0
$951$	44853.0	1.52940
$952$	0	0
$953$	−19200.5	−0.652641	−0.326320	−	0.945259i	$-0.605809\pi$
$953$	−19200.5	−0.652641	−0.326320	+	0.945259i	$0.605809\pi$
$954$	0	0
$955$	−5306.25	−0.179797
$956$	0	0
$957$	− 53550.0i	− 1.80881i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	13289.4	0.446087
$962$	0	0
$963$	17351.9i	0.580640i
$964$	0	0
$965$	− 2527.60i	− 0.0843176i
$966$	0	0
$967$	34750.8i	1.15565i	0.816162	+	0.577823i	$0.196097\pi$
$967$	34750.8i	1.15565i	−0.816162	+	0.577823i	$0.803903\pi$
$968$	0	0
$969$	3.61843i	0 0.000119960i
$970$	0	0
$971$	18750.2	0.619693	0.309846	−	0.950787i	$-0.399722\pi$
$971$	18750.2	0.619693	0.309846	+	0.950787i	$0.399722\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 46547.1i	− 1.52892i
$976$	0	0
$977$	54959.8	1.79971	0.899856	−	0.436187i	$-0.143671\pi$
$977$	54959.8	1.79971	0.899856	+	0.436187i	$0.143671\pi$
$978$	0	0
$979$	10027.8	0.327365
$980$	0	0
$981$	−28613.6	−0.931256
$982$	0	0
$983$	−35869.1	−1.16383	−0.581916	−	0.813249i	$-0.697697\pi$
$983$	−35869.1	−1.16383	−0.581916	+	0.813249i	$0.697697\pi$
$984$	0	0
$985$	5376.85i	0.173930i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13126.7	0.422046
$990$	0	0
$991$	28471.6i	0.912644i	0.889815	+	0.456322i	$0.150833\pi$
$991$	28471.6i	0.912644i	−0.889815	+	0.456322i	$0.849167\pi$
$992$	0	0
$993$	− 7247.99i	− 0.231630i
$994$	0	0
$995$	− 3500.71i	− 0.111538i
$996$	0	0
$997$	− 17267.9i	− 0.548524i	−0.961655	−	0.274262i	$-0.911566\pi$
$997$	− 17267.9i	− 0.548524i	0.961655	−	0.274262i	$-0.0884336\pi$
$998$	0	0
$999$	17411.8	0.551436

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.j.783.3	✓	24
4.3	odd	2	inner	784.4.f.j.783.21	yes	24
7.6	odd	2	inner	784.4.f.j.783.22	yes	24
28.27	even	2	inner	784.4.f.j.783.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.j.783.3	✓	24	1.1	even	1	trivial
784.4.f.j.783.4	yes	24	28.27	even	2	inner
784.4.f.j.783.21	yes	24	4.3	odd	2	inner
784.4.f.j.783.22	yes	24	7.6	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)

\(f(q)\)	\(=\)	\(q-6.34240 q^{3} -1.20378i q^{5} +13.2260 q^{9} +O(q^{10})\)	\(q-6.34240 q^{3} -1.20378i q^{5} +13.2260 q^{9} -32.0476i q^{11} +59.4009i q^{13} +7.63482i q^{15} -0.0501221i q^{17} +11.3825 q^{19} -75.2486i q^{23} +123.551 q^{25} +87.3601 q^{27} -263.457 q^{29} +207.558 q^{31} +203.259i q^{33} +199.310 q^{37} -376.744i q^{39} +437.976i q^{41} +174.444i q^{43} -15.9211i q^{45} -283.621 q^{47} +0.317894i q^{51} +7.41360 q^{53} -38.5781 q^{55} -72.1924 q^{57} +38.6495 q^{59} +276.420i q^{61} +71.5053 q^{65} -965.988i q^{67} +477.256i q^{69} -580.469i q^{71} -607.215i q^{73} -783.609 q^{75} -827.267i q^{79} -911.175 q^{81} -1203.55 q^{83} -0.0603357 q^{85} +1670.95 q^{87} +312.903i q^{89} -1316.42 q^{93} -13.7020i q^{95} -1247.34i q^{97} -423.862i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more