LMFDB - Embedded newform 784.4.f.h.783.6

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:	$6$
Coefficient field:	6.0.12258833328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 $ x^6 - x^5 + 29x^4 - 20x^3 + 808x^2 - 672x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.6
Root			$-2.61524 + 4.52973i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.h.783.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.29424 q^{3} +19.8510i q^{5} +59.3829 q^{9} +O(q^{10})$	$q+9.29424 q^{3} +19.8510i q^{5} +59.3829 q^{9} +11.8356i q^{11} +19.9862i q^{13} +184.500i q^{15} +1.81702i q^{17} -7.32284 q^{19} +106.472i q^{23} -269.062 q^{25} +300.975 q^{27} -191.679 q^{29} +125.150 q^{31} +110.003i q^{33} +316.828 q^{37} +185.756i q^{39} -321.806i q^{41} -74.3089i q^{43} +1178.81i q^{45} +154.562 q^{47} +16.8879i q^{51} -319.421 q^{53} -234.948 q^{55} -68.0602 q^{57} -61.1169 q^{59} +309.324i q^{61} -396.745 q^{65} +594.102i q^{67} +989.576i q^{69} -48.6774i q^{71} -770.269i q^{73} -2500.73 q^{75} +960.207i q^{79} +1193.99 q^{81} -1064.55 q^{83} -36.0697 q^{85} -1781.51 q^{87} +655.567i q^{89} +1163.18 q^{93} -145.366i q^{95} -704.225i q^{97} +702.830i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{3} + 156 q^{9}+O(q^{10}) $ 6 * q + 14 * q^3 + 156 * q^9	$ 6 q + 14 q^{3} + 156 q^{9} - 286 q^{19} - 612 q^{25} + 362 q^{27} - 348 q^{29} - 410 q^{31} + 498 q^{37} - 150 q^{47} + 1290 q^{53} + 918 q^{55} - 6 q^{57} - 642 q^{59} + 1224 q^{65} - 8276 q^{75} - 450 q^{81} + 24 q^{83} + 3786 q^{85} - 7284 q^{87} + 5982 q^{93}+O(q^{100}) $ 6 * q + 14 * q^3 + 156 * q^9 - 286 * q^19 - 612 * q^25 + 362 * q^27 - 348 * q^29 - 410 * q^31 + 498 * q^37 - 150 * q^47 + 1290 * q^53 + 918 * q^55 - 6 * q^57 - 642 * q^59 + 1224 * q^65 - 8276 * q^75 - 450 * q^81 + 24 * q^83 + 3786 * q^85 - 7284 * q^87 + 5982 * 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
$3$	9.29424	1.78868	0.894339	−	0.447390i	$-0.147647\pi$
$3$	9.29424	1.78868	0.894339	+	0.447390i	$0.147647\pi$
$4$	0	0
$5$	19.8510i	1.77553i	0.460300	+	0.887763i	$0.347742\pi$
$5$	19.8510i	1.77553i	−0.460300	+	0.887763i	$0.652258\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	59.3829	2.19937
$10$	0	0
$11$	11.8356i	0.324414i	0.986757	+	0.162207i	$0.0518613\pi$
$11$	11.8356i	0.324414i	−0.986757	+	0.162207i	$0.948139\pi$
$12$	0	0
$13$	19.9862i	0.426398i	0.977009	+	0.213199i	$0.0683882\pi$
$13$	19.9862i	0.426398i	−0.977009	+	0.213199i	$0.931612\pi$
$14$	0	0
$15$	184.500i	3.17584i
$16$	0	0
$17$	1.81702i	0.0259231i	0.999916	+	0.0129616i	$0.00412591\pi$
$17$	1.81702i	0.0259231i	−0.999916	+	0.0129616i	$0.995874\pi$
$18$	0	0
$19$	−7.32284	−0.0884197	−0.0442098	−	0.999022i	$-0.514077\pi$
$19$	−7.32284	−0.0884197	−0.0442098	+	0.999022i	$0.514077\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	106.472i	0.965258i	0.875825	+	0.482629i	$0.160318\pi$
$23$	106.472i	0.965258i	−0.875825	+	0.482629i	$0.839682\pi$
$24$	0	0
$25$	−269.062	−2.15249
$26$	0	0
$27$	300.975	2.14528
$28$	0	0
$29$	−191.679	−1.22737	−0.613687	−	0.789549i	$-0.710315\pi$
$29$	−191.679	−1.22737	−0.613687	+	0.789549i	$0.710315\pi$
$30$	0	0
$31$	125.150	0.725087	0.362543	−	0.931967i	$-0.381908\pi$
$31$	125.150	0.725087	0.362543	+	0.931967i	$0.381908\pi$
$32$	0	0
$33$	110.003i	0.580273i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	316.828	1.40773	0.703867	−	0.710332i	$-0.251455\pi$
$37$	316.828	1.40773	0.703867	+	0.710332i	$0.251455\pi$
$38$	0	0
$39$	185.756i	0.762688i
$40$	0	0
$41$	− 321.806i	− 1.22580i	−0.790162	−	0.612898i	$-0.790003\pi$
$41$	− 321.806i	− 1.22580i	0.790162	−	0.612898i	$-0.209997\pi$
$42$	0	0
$43$	− 74.3089i	− 0.263535i	−0.991281	−	0.131767i	$-0.957935\pi$
$43$	− 74.3089i	− 0.263535i	0.991281	−	0.131767i	$-0.0420652\pi$
$44$	0	0
$45$	1178.81i	3.90504i
$46$	0	0
$47$	154.562	0.479685	0.239842	−	0.970812i	$-0.422904\pi$
$47$	154.562	0.479685	0.239842	+	0.970812i	$0.422904\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.8879i	0.0463681i
$52$	0	0
$53$	−319.421	−0.827847	−0.413923	−	0.910312i	$-0.635842\pi$
$53$	−319.421	−0.827847	−0.413923	+	0.910312i	$0.635842\pi$
$54$	0	0
$55$	−234.948	−0.576006
$56$	0	0
$57$	−68.0602	−0.158154
$58$	0	0
$59$	−61.1169	−0.134860	−0.0674300	−	0.997724i	$-0.521480\pi$
$59$	−61.1169	−0.134860	−0.0674300	+	0.997724i	$0.521480\pi$
$60$	0	0
$61$	309.324i	0.649260i	0.945841	+	0.324630i	$0.105240\pi$
$61$	309.324i	0.649260i	−0.945841	+	0.324630i	$0.894760\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−396.745	−0.757080
$66$	0	0
$67$	594.102i	1.08330i	0.840604	+	0.541650i	$0.182200\pi$
$67$	594.102i	1.08330i	−0.840604	+	0.541650i	$0.817800\pi$
$68$	0	0
$69$	989.576i	1.72654i
$70$	0	0
$71$	− 48.6774i	− 0.0813654i	−0.999172	−	0.0406827i	$-0.987047\pi$
$71$	− 48.6774i	− 0.0813654i	0.999172	−	0.0406827i	$-0.0129533\pi$
$72$	0	0
$73$	− 770.269i	− 1.23497i	−0.786581	−	0.617487i	$-0.788151\pi$
$73$	− 770.269i	− 1.23497i	0.786581	−	0.617487i	$-0.211849\pi$
$74$	0	0
$75$	−2500.73	−3.85012
$76$	0	0
$77$	0	0
$78$	0	0
$79$	960.207i	1.36749i	0.729721	+	0.683745i	$0.239650\pi$
$79$	960.207i	1.36749i	−0.729721	+	0.683745i	$0.760350\pi$
$80$	0	0
$81$	1193.99	1.63785
$82$	0	0
$83$	−1064.55	−1.40782	−0.703912	−	0.710288i	$-0.748565\pi$
$83$	−1064.55	−1.40782	−0.703912	+	0.710288i	$0.748565\pi$
$84$	0	0
$85$	−36.0697	−0.0460272
$86$	0	0
$87$	−1781.51	−2.19538
$88$	0	0
$89$	655.567i	0.780786i	0.920648	+	0.390393i	$0.127661\pi$
$89$	655.567i	0.780786i	−0.920648	+	0.390393i	$0.872339\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1163.18	1.29695
$94$	0	0
$95$	− 145.366i	− 0.156991i
$96$	0	0
$97$	− 704.225i	− 0.737147i	−0.929599	−	0.368573i	$-0.879846\pi$
$97$	− 704.225i	− 0.737147i	0.929599	−	0.368573i	$-0.120154\pi$
$98$	0	0
$99$	702.830i	0.713506i
$100$	0	0
$101$	− 690.058i	− 0.679835i	−0.940455	−	0.339918i	$-0.889601\pi$
$101$	− 690.058i	− 0.679835i	0.940455	−	0.339918i	$-0.110399\pi$
$102$	0	0
$103$	1398.27	1.33763	0.668814	−	0.743429i	$-0.266802\pi$
$103$	1398.27	1.33763	0.668814	+	0.743429i	$0.266802\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 31.1751i	− 0.0281665i	−0.999901	−	0.0140832i	$-0.995517\pi$
$107$	− 31.1751i	− 0.0281665i	0.999901	−	0.0140832i	$-0.00448298\pi$
$108$	0	0
$109$	478.157	0.420176	0.210088	−	0.977683i	$-0.432625\pi$
$109$	478.157	0.420176	0.210088	+	0.977683i	$0.432625\pi$
$110$	0	0
$111$	2944.67	2.51798
$112$	0	0
$113$	1861.07	1.54933	0.774666	−	0.632371i	$-0.217918\pi$
$113$	1861.07	1.54933	0.774666	+	0.632371i	$0.217918\pi$
$114$	0	0
$115$	−2113.57	−1.71384
$116$	0	0
$117$	1186.84i	0.937805i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1190.92	0.894755
$122$	0	0
$123$	− 2990.94i	− 2.19256i
$124$	0	0
$125$	− 2859.77i	− 2.04628i
$126$	0	0
$127$	− 1583.04i	− 1.10608i	−0.833155	−	0.553040i	$-0.813468\pi$
$127$	− 1583.04i	− 1.10608i	0.833155	−	0.553040i	$-0.186532\pi$
$128$	0	0
$129$	− 690.645i	− 0.471379i
$130$	0	0
$131$	−2197.20	−1.46542	−0.732712	−	0.680539i	$-0.761746\pi$
$131$	−2197.20	−1.46542	−0.732712	+	0.680539i	$0.761746\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5974.65i	3.80900i
$136$	0	0
$137$	2499.40	1.55867	0.779336	−	0.626607i	$-0.215557\pi$
$137$	2499.40	1.55867	0.779336	+	0.626607i	$0.215557\pi$
$138$	0	0
$139$	2709.21	1.65318	0.826591	−	0.562804i	$-0.190277\pi$
$139$	2709.21	1.65318	0.826591	+	0.562804i	$0.190277\pi$
$140$	0	0
$141$	1436.54	0.858001
$142$	0	0
$143$	−236.548	−0.138329
$144$	0	0
$145$	− 3805.02i	− 2.17924i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1687.58	−0.927867	−0.463933	−	0.885870i	$-0.653562\pi$
$149$	−1687.58	−0.927867	−0.463933	+	0.885870i	$0.653562\pi$
$150$	0	0
$151$	2554.77i	1.37685i	0.725309	+	0.688424i	$0.241697\pi$
$151$	2554.77i	1.37685i	−0.725309	+	0.688424i	$0.758303\pi$
$152$	0	0
$153$	107.900i	0.0570145i
$154$	0	0
$155$	2484.36i	1.28741i
$156$	0	0
$157$	− 288.483i	− 0.146646i	−0.997308	−	0.0733230i	$-0.976640\pi$
$157$	− 288.483i	− 0.146646i	0.997308	−	0.0733230i	$-0.0233604\pi$
$158$	0	0
$159$	−2968.78	−1.48075
$160$	0	0
$161$	0	0
$162$	0	0
$163$	731.683i	0.351594i	0.984426	+	0.175797i	$0.0562502\pi$
$163$	731.683i	0.351594i	−0.984426	+	0.175797i	$0.943750\pi$
$164$	0	0
$165$	−2183.66	−1.03029
$166$	0	0
$167$	−2121.63	−0.983092	−0.491546	−	0.870852i	$-0.663568\pi$
$167$	−2121.63	−0.983092	−0.491546	+	0.870852i	$0.663568\pi$
$168$	0	0
$169$	1797.55	0.818185
$170$	0	0
$171$	−434.851	−0.194467
$172$	0	0
$173$	3179.74i	1.39741i	0.715412	+	0.698703i	$0.246239\pi$
$173$	3179.74i	1.39741i	−0.715412	+	0.698703i	$0.753761\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−568.035	−0.241221
$178$	0	0
$179$	− 2198.12i	− 0.917849i	−0.888475	−	0.458924i	$-0.848235\pi$
$179$	− 2198.12i	− 0.917849i	0.888475	−	0.458924i	$-0.151765\pi$
$180$	0	0
$181$	1266.98i	0.520299i	0.965568	+	0.260150i	$0.0837719\pi$
$181$	1266.98i	0.520299i	−0.965568	+	0.260150i	$0.916228\pi$
$182$	0	0
$183$	2874.93i	1.16132i
$184$	0	0
$185$	6289.34i	2.49947i
$186$	0	0
$187$	−21.5055	−0.00840983
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1741.49i	− 0.659737i	−0.944027	−	0.329869i	$-0.892996\pi$
$191$	− 1741.49i	− 0.659737i	0.944027	−	0.329869i	$-0.107004\pi$
$192$	0	0
$193$	−1278.61	−0.476874	−0.238437	−	0.971158i	$-0.576635\pi$
$193$	−1278.61	−0.476874	−0.238437	+	0.971158i	$0.576635\pi$
$194$	0	0
$195$	−3687.45	−1.35417
$196$	0	0
$197$	760.622	0.275087	0.137543	−	0.990496i	$-0.456079\pi$
$197$	760.622	0.275087	0.137543	+	0.990496i	$0.456079\pi$
$198$	0	0
$199$	4438.89	1.58123	0.790615	−	0.612313i	$-0.209761\pi$
$199$	4438.89	1.58123	0.790615	+	0.612313i	$0.209761\pi$
$200$	0	0
$201$	5521.73i	1.93767i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6388.17	2.17643
$206$	0	0
$207$	6322.62i	2.12296i
$208$	0	0
$209$	− 86.6699i	− 0.0286846i
$210$	0	0
$211$	− 2084.90i	− 0.680239i	−0.940382	−	0.340119i	$-0.889533\pi$
$211$	− 2084.90i	− 0.680239i	0.940382	−	0.340119i	$-0.110467\pi$
$212$	0	0
$213$	− 452.420i	− 0.145537i
$214$	0	0
$215$	1475.11	0.467913
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 7159.06i	− 2.20897i
$220$	0	0
$221$	−36.3154	−0.0110536
$222$	0	0
$223$	399.456	0.119953	0.0599766	−	0.998200i	$-0.480897\pi$
$223$	399.456	0.119953	0.0599766	+	0.998200i	$0.480897\pi$
$224$	0	0
$225$	−15977.7	−4.73413
$226$	0	0
$227$	−2932.62	−0.857466	−0.428733	−	0.903431i	$-0.641040\pi$
$227$	−2932.62	−0.857466	−0.428733	+	0.903431i	$0.641040\pi$
$228$	0	0
$229$	− 1505.24i	− 0.434363i	−0.976131	−	0.217182i	$-0.930314\pi$
$229$	− 1505.24i	− 0.434363i	0.976131	−	0.217182i	$-0.0696864\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	934.190	0.262664	0.131332	−	0.991338i	$-0.458075\pi$
$233$	934.190	0.262664	0.131332	+	0.991338i	$0.458075\pi$
$234$	0	0
$235$	3068.21i	0.851693i
$236$	0	0
$237$	8924.39i	2.44600i
$238$	0	0
$239$	− 6859.78i	− 1.85658i	−0.371860	−	0.928289i	$-0.621280\pi$
$239$	− 6859.78i	− 1.85658i	0.371860	−	0.928289i	$-0.378720\pi$
$240$	0	0
$241$	1294.25i	0.345933i	0.984928	+	0.172966i	$0.0553352\pi$
$241$	1294.25i	0.345933i	−0.984928	+	0.172966i	$0.944665\pi$
$242$	0	0
$243$	2970.94	0.784304
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 146.356i	− 0.0377019i
$248$	0	0
$249$	−9894.17	−2.51814
$250$	0	0
$251$	178.426	0.0448691	0.0224345	−	0.999748i	$-0.492858\pi$
$251$	178.426	0.0448691	0.0224345	+	0.999748i	$0.492858\pi$
$252$	0	0
$253$	−1260.16	−0.313144
$254$	0	0
$255$	−335.241	−0.0823278
$256$	0	0
$257$	− 5899.74i	− 1.43197i	−0.698117	−	0.715984i	$-0.745978\pi$
$257$	− 5899.74i	− 1.43197i	0.698117	−	0.715984i	$-0.254022\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11382.5	−2.69945
$262$	0	0
$263$	− 2019.42i	− 0.473470i	−0.971574	−	0.236735i	$-0.923923\pi$
$263$	− 2019.42i	− 0.473470i	0.971574	−	0.236735i	$-0.0760774\pi$
$264$	0	0
$265$	− 6340.83i	− 1.46986i
$266$	0	0
$267$	6092.99i	1.39657i
$268$	0	0
$269$	− 1849.77i	− 0.419265i	−0.977780	−	0.209632i	$-0.932773\pi$
$269$	− 1849.77i	− 0.419265i	0.977780	−	0.209632i	$-0.0672267\pi$
$270$	0	0
$271$	7198.86	1.61365	0.806826	−	0.590790i	$-0.201184\pi$
$271$	7198.86	1.61365	0.806826	+	0.590790i	$0.201184\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3184.50i	− 0.698300i
$276$	0	0
$277$	5534.83	1.20056	0.600281	−	0.799789i	$-0.295056\pi$
$277$	5534.83	1.20056	0.600281	+	0.799789i	$0.295056\pi$
$278$	0	0
$279$	7431.80	1.59473
$280$	0	0
$281$	752.919	0.159841	0.0799207	−	0.996801i	$-0.474533\pi$
$281$	752.919	0.159841	0.0799207	+	0.996801i	$0.474533\pi$
$282$	0	0
$283$	5623.58	1.18123	0.590614	−	0.806954i	$-0.298886\pi$
$283$	5623.58	1.18123	0.590614	+	0.806954i	$0.298886\pi$
$284$	0	0
$285$	− 1351.06i	− 0.280807i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4909.70	0.999328
$290$	0	0
$291$	− 6545.24i	− 1.31852i
$292$	0	0
$293$	507.111i	0.101112i	0.998721	+	0.0505559i	$0.0160993\pi$
$293$	507.111i	0.101112i	−0.998721	+	0.0505559i	$0.983901\pi$
$294$	0	0
$295$	− 1213.23i	− 0.239447i
$296$	0	0
$297$	3562.21i	0.695960i
$298$	0	0
$299$	−2127.97	−0.411584
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 6413.57i	− 1.21601i
$304$	0	0
$305$	−6140.38	−1.15278
$306$	0	0
$307$	−7402.96	−1.37625	−0.688126	−	0.725591i	$-0.741566\pi$
$307$	−7402.96	−1.37625	−0.688126	+	0.725591i	$0.741566\pi$
$308$	0	0
$309$	12995.9	2.39259
$310$	0	0
$311$	4284.78	0.781245	0.390623	−	0.920551i	$-0.372260\pi$
$311$	4284.78	0.781245	0.390623	+	0.920551i	$0.372260\pi$
$312$	0	0
$313$	1502.00i	0.271239i	0.990761	+	0.135620i	$0.0433025\pi$
$313$	1502.00i	0.271239i	−0.990761	+	0.135620i	$0.956698\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7097.55	1.25753	0.628767	−	0.777594i	$-0.283560\pi$
$317$	7097.55	1.25753	0.628767	+	0.777594i	$0.283560\pi$
$318$	0	0
$319$	− 2268.63i	− 0.398178i
$320$	0	0
$321$	− 289.749i	− 0.0503808i
$322$	0	0
$323$	− 13.3058i	− 0.00229211i
$324$	0	0
$325$	− 5377.52i	− 0.917818i
$326$	0	0
$327$	4444.11	0.751559
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11497.2i	1.90919i	0.297907	+	0.954595i	$0.403712\pi$
$331$	11497.2i	1.90919i	−0.297907	+	0.954595i	$0.596288\pi$
$332$	0	0
$333$	18814.2	3.09612
$334$	0	0
$335$	−11793.5	−1.92343
$336$	0	0
$337$	6020.26	0.973128	0.486564	−	0.873645i	$-0.338250\pi$
$337$	6020.26	0.973128	0.486564	+	0.873645i	$0.338250\pi$
$338$	0	0
$339$	17297.2	2.77126
$340$	0	0
$341$	1481.23i	0.235229i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−19644.1	−3.06551
$346$	0	0
$347$	8890.39i	1.37539i	0.725999	+	0.687696i	$0.241378\pi$
$347$	8890.39i	1.37539i	−0.725999	+	0.687696i	$0.758622\pi$
$348$	0	0
$349$	− 6396.47i	− 0.981075i	−0.871420	−	0.490538i	$-0.836800\pi$
$349$	− 6396.47i	− 0.981075i	0.871420	−	0.490538i	$-0.163200\pi$
$350$	0	0
$351$	6015.33i	0.914743i
$352$	0	0
$353$	− 11472.5i	− 1.72981i	−0.501940	−	0.864903i	$-0.667380\pi$
$353$	− 11472.5i	− 1.72981i	0.501940	−	0.864903i	$-0.332620\pi$
$354$	0	0
$355$	966.295	0.144466
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 195.405i	− 0.0287272i	−0.999897	−	0.0143636i	$-0.995428\pi$
$359$	− 195.405i	− 0.0287272i	0.999897	−	0.0143636i	$-0.00457224\pi$
$360$	0	0
$361$	−6805.38	−0.992182
$362$	0	0
$363$	11068.7	1.60043
$364$	0	0
$365$	15290.6	2.19273
$366$	0	0
$367$	−10227.7	−1.45472	−0.727362	−	0.686254i	$-0.759254\pi$
$367$	−10227.7	−1.45472	−0.727362	+	0.686254i	$0.759254\pi$
$368$	0	0
$369$	− 19109.8i	− 2.69598i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7734.15	−1.07362	−0.536808	−	0.843704i	$-0.680370\pi$
$373$	−7734.15	−1.07362	−0.536808	+	0.843704i	$0.680370\pi$
$374$	0	0
$375$	− 26579.4i	− 3.66014i
$376$	0	0
$377$	− 3830.93i	− 0.523350i
$378$	0	0
$379$	− 9739.99i	− 1.32008i	−0.751231	−	0.660039i	$-0.770540\pi$
$379$	− 9739.99i	− 1.32008i	0.751231	−	0.660039i	$-0.229460\pi$
$380$	0	0
$381$	− 14713.2i	− 1.97842i
$382$	0	0
$383$	−2525.17	−0.336894	−0.168447	−	0.985711i	$-0.553875\pi$
$383$	−2525.17	−0.336894	−0.168447	+	0.985711i	$0.553875\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4412.68i	− 0.579610i
$388$	0	0
$389$	−2665.26	−0.347388	−0.173694	−	0.984800i	$-0.555570\pi$
$389$	−2665.26	−0.347388	−0.173694	+	0.984800i	$0.555570\pi$
$390$	0	0
$391$	−193.462	−0.0250225
$392$	0	0
$393$	−20421.3	−2.62117
$394$	0	0
$395$	−19061.1	−2.42801
$396$	0	0
$397$	− 7966.78i	− 1.00716i	−0.863950	−	0.503578i	$-0.832017\pi$
$397$	− 7966.78i	− 1.00716i	0.863950	−	0.503578i	$-0.167983\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2607.89	0.324768	0.162384	−	0.986728i	$-0.448082\pi$
$401$	2607.89	0.324768	0.162384	+	0.986728i	$0.448082\pi$
$402$	0	0
$403$	2501.28i	0.309175i
$404$	0	0
$405$	23701.9i	2.90805i
$406$	0	0
$407$	3749.83i	0.456689i
$408$	0	0
$409$	− 8794.35i	− 1.06321i	−0.846993	−	0.531605i	$-0.821589\pi$
$409$	− 8794.35i	− 1.06321i	0.846993	−	0.531605i	$-0.178411\pi$
$410$	0	0
$411$	23230.0	2.78796
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 21132.3i	− 2.49963i
$416$	0	0
$417$	25180.0	2.95701
$418$	0	0
$419$	−333.409	−0.0388737	−0.0194368	−	0.999811i	$-0.506187\pi$
$419$	−333.409	−0.0388737	−0.0194368	+	0.999811i	$0.506187\pi$
$420$	0	0
$421$	−1186.07	−0.137305	−0.0686526	−	0.997641i	$-0.521870\pi$
$421$	−1186.07	−0.137305	−0.0686526	+	0.997641i	$0.521870\pi$
$422$	0	0
$423$	9178.34	1.05500
$424$	0	0
$425$	− 488.892i	− 0.0557994i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2198.53	−0.247427
$430$	0	0
$431$	12163.1i	1.35934i	0.733519	+	0.679669i	$0.237876\pi$
$431$	12163.1i	1.35934i	−0.733519	+	0.679669i	$0.762124\pi$
$432$	0	0
$433$	− 14229.4i	− 1.57926i	−0.613582	−	0.789631i	$-0.710272\pi$
$433$	− 14229.4i	− 1.57926i	0.613582	−	0.789631i	$-0.289728\pi$
$434$	0	0
$435$	− 35364.7i	− 3.89795i
$436$	0	0
$437$	− 779.677i	− 0.0853478i
$438$	0	0
$439$	−7955.09	−0.864865	−0.432432	−	0.901666i	$-0.642345\pi$
$439$	−7955.09	−0.864865	−0.432432	+	0.901666i	$0.642345\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6411.91i	− 0.687673i	−0.939030	−	0.343837i	$-0.888273\pi$
$443$	− 6411.91i	− 0.687673i	0.939030	−	0.343837i	$-0.111727\pi$
$444$	0	0
$445$	−13013.6	−1.38631
$446$	0	0
$447$	−15684.8	−1.65965
$448$	0	0
$449$	−5918.99	−0.622125	−0.311063	−	0.950389i	$-0.600685\pi$
$449$	−5918.99	−0.622125	−0.311063	+	0.950389i	$0.600685\pi$
$450$	0	0
$451$	3808.76	0.397666
$452$	0	0
$453$	23744.6i	2.46274i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	868.613	0.0889103	0.0444551	−	0.999011i	$-0.485845\pi$
$457$	868.613	0.0889103	0.0444551	+	0.999011i	$0.485845\pi$
$458$	0	0
$459$	546.878i	0.0556124i
$460$	0	0
$461$	4645.89i	0.469372i	0.972071	+	0.234686i	$0.0754062\pi$
$461$	4645.89i	0.469372i	−0.972071	+	0.234686i	$0.924594\pi$
$462$	0	0
$463$	− 288.831i	− 0.0289916i	−0.999895	−	0.0144958i	$-0.995386\pi$
$463$	− 288.831i	− 0.0289916i	0.999895	−	0.0144958i	$-0.00461433\pi$
$464$	0	0
$465$	23090.2i	2.30276i
$466$	0	0
$467$	11317.4	1.12143	0.560716	−	0.828008i	$-0.310526\pi$
$467$	11317.4	1.12143	0.560716	+	0.828008i	$0.310526\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 2681.23i	− 0.262302i
$472$	0	0
$473$	879.488	0.0854945
$474$	0	0
$475$	1970.30	0.190323
$476$	0	0
$477$	−18968.2	−1.82074
$478$	0	0
$479$	−4324.22	−0.412482	−0.206241	−	0.978501i	$-0.566123\pi$
$479$	−4324.22	−0.412482	−0.206241	+	0.978501i	$0.566123\pi$
$480$	0	0
$481$	6332.17i	0.600254i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13979.6	1.30882
$486$	0	0
$487$	− 11480.5i	− 1.06824i	−0.845410	−	0.534119i	$-0.820644\pi$
$487$	− 11480.5i	− 1.06824i	0.845410	−	0.534119i	$-0.179356\pi$
$488$	0	0
$489$	6800.44i	0.628888i
$490$	0	0
$491$	10052.9i	0.923994i	0.886881	+	0.461997i	$0.152867\pi$
$491$	10052.9i	0.923994i	−0.886881	+	0.461997i	$0.847133\pi$
$492$	0	0
$493$	− 348.285i	− 0.0318174i
$494$	0	0
$495$	−13951.9	−1.26685
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 7204.13i	− 0.646295i	−0.946349	−	0.323147i	$-0.895259\pi$
$499$	− 7204.13i	− 0.646295i	0.946349	−	0.323147i	$-0.104741\pi$
$500$	0	0
$501$	−19718.9	−1.75843
$502$	0	0
$503$	−6434.36	−0.570366	−0.285183	−	0.958473i	$-0.592054\pi$
$503$	−6434.36	−0.570366	−0.285183	+	0.958473i	$0.592054\pi$
$504$	0	0
$505$	13698.3	1.20707
$506$	0	0
$507$	16706.9	1.46347
$508$	0	0
$509$	− 13015.0i	− 1.13336i	−0.823939	−	0.566679i	$-0.808228\pi$
$509$	− 13015.0i	− 1.13336i	0.823939	−	0.566679i	$-0.191772\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2203.99	−0.189685
$514$	0	0
$515$	27757.1i	2.37500i
$516$	0	0
$517$	1829.33i	0.155617i
$518$	0	0
$519$	29553.3i	2.49951i
$520$	0	0
$521$	5144.91i	0.432635i	0.976323	+	0.216317i	$0.0694046\pi$
$521$	5144.91i	0.432635i	−0.976323	+	0.216317i	$0.930595\pi$
$522$	0	0
$523$	−11197.7	−0.936213	−0.468106	−	0.883672i	$-0.655064\pi$
$523$	−11197.7	−0.936213	−0.468106	+	0.883672i	$0.655064\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	227.402i	0.0187965i
$528$	0	0
$529$	830.726	0.0682769
$530$	0	0
$531$	−3629.30	−0.296607
$532$	0	0
$533$	6431.67	0.522677
$534$	0	0
$535$	618.857	0.0500104
$536$	0	0
$537$	− 20429.8i	− 1.64174i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21827.4	−1.73463	−0.867313	−	0.497762i	$-0.834155\pi$
$541$	−21827.4	−1.73463	−0.867313	+	0.497762i	$0.834155\pi$
$542$	0	0
$543$	11775.6i	0.930647i
$544$	0	0
$545$	9491.89i	0.746033i
$546$	0	0
$547$	20005.0i	1.56371i	0.623459	+	0.781856i	$0.285727\pi$
$547$	20005.0i	1.56371i	−0.623459	+	0.781856i	$0.714273\pi$
$548$	0	0
$549$	18368.5i	1.42796i
$550$	0	0
$551$	1403.63	0.108524
$552$	0	0
$553$	0	0
$554$	0	0
$555$	58454.7i	4.47074i
$556$	0	0
$557$	−7400.91	−0.562992	−0.281496	−	0.959562i	$-0.590831\pi$
$557$	−7400.91	−0.562992	−0.281496	+	0.959562i	$0.590831\pi$
$558$	0	0
$559$	1485.15	0.112371
$560$	0	0
$561$	−199.877	−0.0150425
$562$	0	0
$563$	−1080.79	−0.0809059	−0.0404529	−	0.999181i	$-0.512880\pi$
$563$	−1080.79	−0.0809059	−0.0404529	+	0.999181i	$0.512880\pi$
$564$	0	0
$565$	36944.0i	2.75088i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4208.13	0.310042	0.155021	−	0.987911i	$-0.450455\pi$
$569$	4208.13	0.310042	0.155021	+	0.987911i	$0.450455\pi$
$570$	0	0
$571$	− 12881.7i	− 0.944105i	−0.881570	−	0.472052i	$-0.843513\pi$
$571$	− 12881.7i	− 0.944105i	0.881570	−	0.472052i	$-0.156487\pi$
$572$	0	0
$573$	− 16185.8i	− 1.18006i
$574$	0	0
$575$	− 28647.5i	− 2.07771i
$576$	0	0
$577$	13271.6i	0.957546i	0.877939	+	0.478773i	$0.158918\pi$
$577$	13271.6i	0.957546i	−0.877939	+	0.478773i	$0.841082\pi$
$578$	0	0
$579$	−11883.7	−0.852974
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 3780.53i	− 0.268565i
$584$	0	0
$585$	−23559.9	−1.66510
$586$	0	0
$587$	16892.3	1.18777	0.593885	−	0.804550i	$-0.297593\pi$
$587$	16892.3	1.18777	0.593885	+	0.804550i	$0.297593\pi$
$588$	0	0
$589$	−916.456	−0.0641119
$590$	0	0
$591$	7069.40	0.492041
$592$	0	0
$593$	− 10957.3i	− 0.758787i	−0.925235	−	0.379394i	$-0.876133\pi$
$593$	− 10957.3i	− 0.758787i	0.925235	−	0.379394i	$-0.123867\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41256.1	2.82831
$598$	0	0
$599$	11575.6i	0.789596i	0.918768	+	0.394798i	$0.129185\pi$
$599$	11575.6i	0.789596i	−0.918768	+	0.394798i	$0.870815\pi$
$600$	0	0
$601$	− 17612.6i	− 1.19540i	−0.801721	−	0.597699i	$-0.796082\pi$
$601$	− 17612.6i	− 1.19540i	0.801721	−	0.597699i	$-0.203918\pi$
$602$	0	0
$603$	35279.5i	2.38257i
$604$	0	0
$605$	23640.9i	1.58866i
$606$	0	0
$607$	−16363.0	−1.09416	−0.547080	−	0.837080i	$-0.684261\pi$
$607$	−16363.0	−1.09416	−0.547080	+	0.837080i	$0.684261\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3089.10i	0.204536i
$612$	0	0
$613$	1007.72	0.0663974	0.0331987	−	0.999449i	$-0.489431\pi$
$613$	1007.72	0.0663974	0.0331987	+	0.999449i	$0.489431\pi$
$614$	0	0
$615$	59373.2	3.89294
$616$	0	0
$617$	−14986.9	−0.977876	−0.488938	−	0.872318i	$-0.662616\pi$
$617$	−14986.9	−0.977876	−0.488938	+	0.872318i	$0.662616\pi$
$618$	0	0
$619$	17668.2	1.14725	0.573624	−	0.819119i	$-0.305537\pi$
$619$	17668.2	1.14725	0.573624	+	0.819119i	$0.305537\pi$
$620$	0	0
$621$	32045.4i	2.07075i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23136.5	1.48074
$626$	0	0
$627$	− 805.531i	− 0.0513075i
$628$	0	0
$629$	575.684i	0.0364929i
$630$	0	0
$631$	− 3794.54i	− 0.239395i	−0.992810	−	0.119698i	$-0.961807\pi$
$631$	− 3794.54i	− 0.239395i	0.992810	−	0.119698i	$-0.0381925\pi$
$632$	0	0
$633$	− 19377.6i	− 1.21673i
$634$	0	0
$635$	31424.9	1.96387
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 2890.61i	− 0.178952i
$640$	0	0
$641$	14251.5	0.878157	0.439079	−	0.898449i	$-0.355305\pi$
$641$	14251.5	0.878157	0.439079	+	0.898449i	$0.355305\pi$
$642$	0	0
$643$	−10643.5	−0.652782	−0.326391	−	0.945235i	$-0.605833\pi$
$643$	−10643.5	−0.652782	−0.326391	+	0.945235i	$0.605833\pi$
$644$	0	0
$645$	13710.0	0.836946
$646$	0	0
$647$	−18948.4	−1.15137	−0.575686	−	0.817671i	$-0.695265\pi$
$647$	−18948.4	−1.15137	−0.575686	+	0.817671i	$0.695265\pi$
$648$	0	0
$649$	− 723.353i	− 0.0437505i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21346.3	1.27924	0.639620	−	0.768691i	$-0.279092\pi$
$653$	21346.3	1.27924	0.639620	+	0.768691i	$0.279092\pi$
$654$	0	0
$655$	− 43616.7i	− 2.60190i
$656$	0	0
$657$	− 45740.8i	− 2.71616i
$658$	0	0
$659$	3245.68i	0.191857i	0.995388	+	0.0959284i	$0.0305820\pi$
$659$	3245.68i	0.191857i	−0.995388	+	0.0959284i	$0.969418\pi$
$660$	0	0
$661$	15989.9i	0.940898i	0.882427	+	0.470449i	$0.155908\pi$
$661$	15989.9i	0.940898i	−0.882427	+	0.470449i	$0.844092\pi$
$662$	0	0
$663$	−337.524	−0.0197713
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 20408.4i	− 1.18473i
$668$	0	0
$669$	3712.64	0.214558
$670$	0	0
$671$	−3661.02	−0.210629
$672$	0	0
$673$	−3473.65	−0.198959	−0.0994796	−	0.995040i	$-0.531718\pi$
$673$	−3473.65	−0.198959	−0.0994796	+	0.995040i	$0.531718\pi$
$674$	0	0
$675$	−80980.8	−4.61771
$676$	0	0
$677$	− 27077.2i	− 1.53717i	−0.639750	−	0.768583i	$-0.720962\pi$
$677$	− 27077.2i	− 1.53717i	0.639750	−	0.768583i	$-0.279038\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−27256.5	−1.53373
$682$	0	0
$683$	− 6615.17i	− 0.370604i	−0.982682	−	0.185302i	$-0.940674\pi$
$683$	− 6615.17i	− 0.370604i	0.982682	−	0.185302i	$-0.0593263\pi$
$684$	0	0
$685$	49615.5i	2.76746i
$686$	0	0
$687$	− 13990.1i	− 0.776936i
$688$	0	0
$689$	− 6384.01i	− 0.352992i
$690$	0	0
$691$	−15808.2	−0.870294	−0.435147	−	0.900359i	$-0.643304\pi$
$691$	−15808.2	−0.870294	−0.435147	+	0.900359i	$0.643304\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	53780.5i	2.93527i
$696$	0	0
$697$	584.730	0.0317765
$698$	0	0
$699$	8682.59	0.469822
$700$	0	0
$701$	9851.70	0.530804	0.265402	−	0.964138i	$-0.414495\pi$
$701$	9851.70	0.530804	0.265402	+	0.964138i	$0.414495\pi$
$702$	0	0
$703$	−2320.08	−0.124471
$704$	0	0
$705$	28516.7i	1.52340i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10767.2	−0.570342	−0.285171	−	0.958477i	$-0.592050\pi$
$709$	−10767.2	−0.570342	−0.285171	+	0.958477i	$0.592050\pi$
$710$	0	0
$711$	57019.9i	3.00761i
$712$	0	0
$713$	13325.0i	0.699896i
$714$	0	0
$715$	− 4695.71i	− 0.245608i
$716$	0	0
$717$	− 63756.4i	− 3.32082i
$718$	0	0
$719$	−20738.0	−1.07565	−0.537827	−	0.843055i	$-0.680755\pi$
$719$	−20738.0	−1.07565	−0.537827	+	0.843055i	$0.680755\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12029.0i	0.618762i
$724$	0	0
$725$	51573.5	2.64192
$726$	0	0
$727$	2738.00	0.139679	0.0698396	−	0.997558i	$-0.477751\pi$
$727$	2738.00	0.139679	0.0698396	+	0.997558i	$0.477751\pi$
$728$	0	0
$729$	−4625.18	−0.234984
$730$	0	0
$731$	135.021	0.00683165
$732$	0	0
$733$	− 17216.1i	− 0.867519i	−0.901029	−	0.433760i	$-0.857187\pi$
$733$	− 17216.1i	− 0.867519i	0.901029	−	0.433760i	$-0.142813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7031.53	−0.351438
$738$	0	0
$739$	5523.75i	0.274959i	0.990505	+	0.137479i	$0.0439001\pi$
$739$	5523.75i	0.274959i	−0.990505	+	0.137479i	$0.956100\pi$
$740$	0	0
$741$	− 1360.26i	− 0.0674366i
$742$	0	0
$743$	20424.6i	1.00849i	0.863561	+	0.504244i	$0.168229\pi$
$743$	20424.6i	1.00849i	−0.863561	+	0.504244i	$0.831771\pi$
$744$	0	0
$745$	− 33500.2i	− 1.64745i
$746$	0	0
$747$	−63216.0	−3.09632
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10763.8i	0.523003i	0.965203	+	0.261502i	$0.0842177\pi$
$751$	10763.8i	0.523003i	−0.965203	+	0.261502i	$0.915782\pi$
$752$	0	0
$753$	1658.33	0.0802563
$754$	0	0
$755$	−50714.7	−2.44463
$756$	0	0
$757$	8591.14	0.412484	0.206242	−	0.978501i	$-0.433877\pi$
$757$	8591.14	0.412484	0.206242	+	0.978501i	$0.433877\pi$
$758$	0	0
$759$	−11712.2	−0.560113
$760$	0	0
$761$	29877.8i	1.42322i	0.702575	+	0.711610i	$0.252034\pi$
$761$	29877.8i	1.42322i	−0.702575	+	0.711610i	$0.747966\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2141.93	−0.101231
$766$	0	0
$767$	− 1221.49i	− 0.0575040i
$768$	0	0
$769$	3093.61i	0.145070i	0.997366	+	0.0725348i	$0.0231089\pi$
$769$	3093.61i	0.145070i	−0.997366	+	0.0725348i	$0.976891\pi$
$770$	0	0
$771$	− 54833.6i	− 2.56133i
$772$	0	0
$773$	24031.7i	1.11819i	0.829104	+	0.559095i	$0.188851\pi$
$773$	24031.7i	1.11819i	−0.829104	+	0.559095i	$0.811149\pi$
$774$	0	0
$775$	−33673.2	−1.56075
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2356.53i	0.108385i
$780$	0	0
$781$	576.125	0.0263961
$782$	0	0
$783$	−57690.5	−2.63307
$784$	0	0
$785$	5726.67	0.260374
$786$	0	0
$787$	11593.2	0.525099	0.262550	−	0.964918i	$-0.415437\pi$
$787$	11593.2	0.525099	0.262550	+	0.964918i	$0.415437\pi$
$788$	0	0
$789$	− 18769.0i	− 0.846886i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6182.20	−0.276843
$794$	0	0
$795$	− 58933.2i	− 2.62911i
$796$	0	0
$797$	− 11467.4i	− 0.509658i	−0.966986	−	0.254829i	$-0.917981\pi$
$797$	− 11467.4i	− 0.509658i	0.966986	−	0.254829i	$-0.0820192\pi$
$798$	0	0
$799$	280.843i	0.0124349i
$800$	0	0
$801$	38929.5i	1.71723i
$802$	0	0
$803$	9116.57	0.400643
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 17192.2i	− 0.749930i
$808$	0	0
$809$	36381.7	1.58110	0.790551	−	0.612396i	$-0.209794\pi$
$809$	36381.7	1.58110	0.790551	+	0.612396i	$0.209794\pi$
$810$	0	0
$811$	35717.1	1.54648	0.773241	−	0.634113i	$-0.218634\pi$
$811$	35717.1	1.54648	0.773241	+	0.634113i	$0.218634\pi$
$812$	0	0
$813$	66907.9	2.88630
$814$	0	0
$815$	−14524.6	−0.624264
$816$	0	0
$817$	544.152i	0.0233017i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31319.1	−1.33136	−0.665678	−	0.746239i	$-0.731858\pi$
$821$	−31319.1	−1.33136	−0.665678	+	0.746239i	$0.731858\pi$
$822$	0	0
$823$	4176.34i	0.176887i	0.996081	+	0.0884435i	$0.0281893\pi$
$823$	4176.34i	0.176887i	−0.996081	+	0.0884435i	$0.971811\pi$
$824$	0	0
$825$	− 29597.5i	− 1.24903i
$826$	0	0
$827$	7930.70i	0.333467i	0.986002	+	0.166734i	$0.0533220\pi$
$827$	7930.70i	0.333467i	−0.986002	+	0.166734i	$0.946678\pi$
$828$	0	0
$829$	2847.17i	0.119284i	0.998220	+	0.0596419i	$0.0189959\pi$
$829$	2847.17i	0.119284i	−0.998220	+	0.0596419i	$0.981004\pi$
$830$	0	0
$831$	51442.0	2.14742
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 42116.4i	− 1.74551i
$836$	0	0
$837$	37667.1	1.55552
$838$	0	0
$839$	−25648.9	−1.05542	−0.527711	−	0.849424i	$-0.676949\pi$
$839$	−25648.9	−1.05542	−0.527711	+	0.849424i	$0.676949\pi$
$840$	0	0
$841$	12351.8	0.506449
$842$	0	0
$843$	6997.82	0.285905
$844$	0	0
$845$	35683.2i	1.45271i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	52266.9	2.11283
$850$	0	0
$851$	33733.3i	1.35883i
$852$	0	0
$853$	1004.85i	0.0403348i	0.999797	+	0.0201674i	$0.00641991\pi$
$853$	1004.85i	0.0403348i	−0.999797	+	0.0201674i	$0.993580\pi$
$854$	0	0
$855$	− 8632.23i	− 0.345282i
$856$	0	0
$857$	35060.5i	1.39748i	0.715374	+	0.698742i	$0.246256\pi$
$857$	35060.5i	1.39748i	−0.715374	+	0.698742i	$0.753744\pi$
$858$	0	0
$859$	9006.42	0.357736	0.178868	−	0.983873i	$-0.442757\pi$
$859$	9006.42	0.357736	0.178868	+	0.983873i	$0.442757\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43533.3i	1.71714i	0.512696	+	0.858570i	$0.328647\pi$
$863$	43533.3i	1.71714i	−0.512696	+	0.858570i	$0.671353\pi$
$864$	0	0
$865$	−63121.0	−2.48113
$866$	0	0
$867$	45631.9	1.78748
$868$	0	0
$869$	−11364.6	−0.443633
$870$	0	0
$871$	−11873.8	−0.461916
$872$	0	0
$873$	− 41818.9i	− 1.62126i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23537.6	−0.906280	−0.453140	−	0.891439i	$-0.649696\pi$
$877$	−23537.6	−0.906280	−0.453140	+	0.891439i	$0.649696\pi$
$878$	0	0
$879$	4713.21i	0.180856i
$880$	0	0
$881$	41234.7i	1.57688i	0.615111	+	0.788440i	$0.289111\pi$
$881$	41234.7i	1.57688i	−0.615111	+	0.788440i	$0.710889\pi$
$882$	0	0
$883$	2981.04i	0.113613i	0.998385	+	0.0568063i	$0.0180917\pi$
$883$	2981.04i	0.113613i	−0.998385	+	0.0568063i	$0.981908\pi$
$884$	0	0
$885$	− 11276.1i	− 0.428294i
$886$	0	0
$887$	23841.2	0.902490	0.451245	−	0.892400i	$-0.350980\pi$
$887$	23841.2	0.902490	0.451245	+	0.892400i	$0.350980\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	14131.6i	0.531342i
$892$	0	0
$893$	−1131.83	−0.0424136
$894$	0	0
$895$	43634.8	1.62967
$896$	0	0
$897$	−19777.8	−0.736191
$898$	0	0
$899$	−23988.7	−0.889953
$900$	0	0
$901$	− 580.396i	− 0.0214604i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25150.9	−0.923805
$906$	0	0
$907$	− 33942.4i	− 1.24260i	−0.783572	−	0.621301i	$-0.786605\pi$
$907$	− 33942.4i	− 1.24260i	0.783572	−	0.621301i	$-0.213395\pi$
$908$	0	0
$909$	− 40977.7i	− 1.49521i
$910$	0	0
$911$	− 18217.3i	− 0.662531i	−0.943538	−	0.331265i	$-0.892524\pi$
$911$	− 18217.3i	− 0.662531i	0.943538	−	0.331265i	$-0.107476\pi$
$912$	0	0
$913$	− 12599.5i	− 0.456718i
$914$	0	0
$915$	−57070.2	−2.06195
$916$	0	0
$917$	0	0
$918$	0	0
$919$	40860.7i	1.46667i	0.679867	+	0.733335i	$0.262037\pi$
$919$	40860.7i	1.46667i	−0.679867	+	0.733335i	$0.737963\pi$
$920$	0	0
$921$	−68804.9	−2.46167
$922$	0	0
$923$	972.875	0.0346940
$924$	0	0
$925$	−85246.2	−3.03014
$926$	0	0
$927$	83033.4	2.94194
$928$	0	0
$929$	− 30596.3i	− 1.08055i	−0.841488	−	0.540276i	$-0.818320\pi$
$929$	− 30596.3i	− 1.08055i	0.841488	−	0.540276i	$-0.181680\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	39823.7	1.39740
$934$	0	0
$935$	− 426.906i	− 0.0149319i
$936$	0	0
$937$	− 28125.2i	− 0.980586i	−0.871558	−	0.490293i	$-0.836890\pi$
$937$	− 28125.2i	− 0.980586i	0.871558	−	0.490293i	$-0.163110\pi$
$938$	0	0
$939$	13959.9i	0.485159i
$940$	0	0
$941$	− 30098.1i	− 1.04269i	−0.853347	−	0.521344i	$-0.825431\pi$
$941$	− 30098.1i	− 1.04269i	0.853347	−	0.521344i	$-0.174569\pi$
$942$	0	0
$943$	34263.3	1.18321
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41414.0i	− 1.42109i	−0.703650	−	0.710547i	$-0.748448\pi$
$947$	− 41414.0i	− 1.42109i	0.703650	−	0.710547i	$-0.251552\pi$
$948$	0	0
$949$	15394.7	0.526590
$950$	0	0
$951$	65966.3	2.24932
$952$	0	0
$953$	−33248.1	−1.13013	−0.565063	−	0.825047i	$-0.691148\pi$
$953$	−33248.1	−1.13013	−0.565063	+	0.825047i	$0.691148\pi$
$954$	0	0
$955$	34570.3	1.17138
$956$	0	0
$957$	− 21085.2i	− 0.712212i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−14128.4	−0.474249
$962$	0	0
$963$	− 1851.27i	− 0.0619485i
$964$	0	0
$965$	− 25381.8i	− 0.846702i
$966$	0	0
$967$	39769.8i	1.32255i	0.750142	+	0.661277i	$0.229985\pi$
$967$	39769.8i	1.32255i	−0.750142	+	0.661277i	$0.770015\pi$
$968$	0	0
$969$	− 123.667i	− 0.00409985i
$970$	0	0
$971$	26812.5	0.886153	0.443077	−	0.896484i	$-0.353887\pi$
$971$	26812.5	0.886153	0.443077	+	0.896484i	$0.353887\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 49979.9i	− 1.64168i
$976$	0	0
$977$	15822.0	0.518106	0.259053	−	0.965863i	$-0.416590\pi$
$977$	15822.0	0.518106	0.259053	+	0.965863i	$0.416590\pi$
$978$	0	0
$979$	−7759.00	−0.253298
$980$	0	0
$981$	28394.4	0.924120
$982$	0	0
$983$	−35033.3	−1.13671	−0.568356	−	0.822783i	$-0.692420\pi$
$983$	−35033.3	−1.13671	−0.568356	+	0.822783i	$0.692420\pi$
$984$	0	0
$985$	15099.1i	0.488424i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7911.82	0.254379
$990$	0	0
$991$	25557.0i	0.819217i	0.912261	+	0.409608i	$0.134335\pi$
$991$	25557.0i	0.819217i	−0.912261	+	0.409608i	$0.865665\pi$
$992$	0	0
$993$	106858.i	3.41493i
$994$	0	0
$995$	88116.4i	2.80752i
$996$	0	0
$997$	− 29598.9i	− 0.940226i	−0.882606	−	0.470113i	$-0.844213\pi$
$997$	− 29598.9i	− 0.940226i	0.882606	−	0.470113i	$-0.155787\pi$
$998$	0	0
$999$	95357.1	3.01999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.h.783.6		6
4.3	odd	2		784.4.f.g.783.2		6
7.4	even	3		112.4.p.f.47.1	yes	6
7.5	odd	6		112.4.p.g.31.3	yes	6
7.6	odd	2		784.4.f.g.783.1		6
28.11	odd	6		112.4.p.g.47.3	yes	6
28.19	even	6		112.4.p.f.31.1	✓	6
28.27	even	2	inner	784.4.f.h.783.5		6
56.5	odd	6		448.4.p.f.255.1		6
56.11	odd	6		448.4.p.f.383.1		6
56.19	even	6		448.4.p.g.255.3		6
56.53	even	6		448.4.p.g.383.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.1	✓	6	28.19	even	6
112.4.p.f.47.1	yes	6	7.4	even	3
112.4.p.g.31.3	yes	6	7.5	odd	6
112.4.p.g.47.3	yes	6	28.11	odd	6
448.4.p.f.255.1		6	56.5	odd	6
448.4.p.f.383.1		6	56.11	odd	6
448.4.p.g.255.3		6	56.19	even	6
448.4.p.g.383.3		6	56.53	even	6
784.4.f.g.783.1		6	7.6	odd	2
784.4.f.g.783.2		6	4.3	odd	2
784.4.f.h.783.5		6	28.27	even	2	inner
784.4.f.h.783.6		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+9.29424 q^{3} +19.8510i q^{5} +59.3829 q^{9} +O(q^{10})\)	\(q+9.29424 q^{3} +19.8510i q^{5} +59.3829 q^{9} +11.8356i q^{11} +19.9862i q^{13} +184.500i q^{15} +1.81702i q^{17} -7.32284 q^{19} +106.472i q^{23} -269.062 q^{25} +300.975 q^{27} -191.679 q^{29} +125.150 q^{31} +110.003i q^{33} +316.828 q^{37} +185.756i q^{39} -321.806i q^{41} -74.3089i q^{43} +1178.81i q^{45} +154.562 q^{47} +16.8879i q^{51} -319.421 q^{53} -234.948 q^{55} -68.0602 q^{57} -61.1169 q^{59} +309.324i q^{61} -396.745 q^{65} +594.102i q^{67} +989.576i q^{69} -48.6774i q^{71} -770.269i q^{73} -2500.73 q^{75} +960.207i q^{79} +1193.99 q^{81} -1064.55 q^{83} -36.0697 q^{85} -1781.51 q^{87} +655.567i q^{89} +1163.18 q^{93} -145.366i q^{95} -704.225i q^{97} +702.830i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{3} + 156 q^{9}+O(q^{10}) \) 6 * q + 14 * q^3 + 156 * q^9	\( 6 q + 14 q^{3} + 156 q^{9} - 286 q^{19} - 612 q^{25} + 362 q^{27} - 348 q^{29} - 410 q^{31} + 498 q^{37} - 150 q^{47} + 1290 q^{53} + 918 q^{55} - 6 q^{57} - 642 q^{59} + 1224 q^{65} - 8276 q^{75} - 450 q^{81} + 24 q^{83} + 3786 q^{85} - 7284 q^{87} + 5982 q^{93}+O(q^{100}) \) 6 * q + 14 * q^3 + 156 * q^9 - 286 * q^19 - 612 * q^25 + 362 * q^27 - 348 * q^29 - 410 * q^31 + 498 * q^37 - 150 * q^47 + 1290 * q^53 + 918 * q^55 - 6 * q^57 - 642 * q^59 + 1224 * q^65 - 8276 * q^75 - 450 * q^81 + 24 * q^83 + 3786 * q^85 - 7284 * q^87 + 5982 * q^93

Introduction

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Modular forms

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Database

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