LMFDB - Embedded newform 784.4.f.g.783.1

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			783.1
Root			$-2.61524 + 4.52973i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.g.783.2

$q$-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	$-0.852353\pi$
$3$	−9.29424	−1.78868	−0.894339	+	0.447390i	$0.852353\pi$
$4$	0	0
$5$	− 19.8510i	− 1.77553i	−0.460300	−	0.887763i	$-0.652258\pi$
$5$	− 19.8510i	− 1.77553i	0.460300	−	0.887763i	$-0.347742\pi$
$6$	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	$-0.931612\pi$
$13$	− 19.9862i	− 0.426398i	0.977009	−	0.213199i	$-0.0683882\pi$
$14$	0	0
$15$	184.500i	3.17584i
$16$	0	0
$17$	− 1.81702i	− 0.0259231i	−0.999916	−	0.0129616i	$-0.995874\pi$
$17$	− 1.81702i	− 0.0259231i	0.999916	−	0.0129616i	$-0.00412591\pi$
$18$	0	0
$19$	7.32284	0.0884197	0.0442098	−	0.999022i	$-0.485923\pi$
$19$	7.32284	0.0884197	0.0442098	+	0.999022i	$0.485923\pi$
$20$	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	$-0.618092\pi$
$31$	−125.150	−0.725087	−0.362543	+	0.931967i	$0.618092\pi$
$32$	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	$0.209997\pi$
$41$	321.806i	1.22580i	−0.790162	+	0.612898i	$0.790003\pi$
$42$	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	$-0.577096\pi$
$47$	−154.562	−0.479685	−0.239842	+	0.970812i	$0.577096\pi$
$48$	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	$-0.478520\pi$
$59$	61.1169	0.134860	0.0674300	+	0.997724i	$0.478520\pi$
$60$	0	0
$61$	− 309.324i	− 0.649260i	−0.945841	−	0.324630i	$-0.894760\pi$
$61$	− 309.324i	− 0.649260i	0.945841	−	0.324630i	$-0.105240\pi$
$62$	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	$0.211849\pi$
$73$	770.269i	1.23497i	−0.786581	+	0.617487i	$0.788151\pi$
$74$	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	$-0.251435\pi$
$83$	1064.55	1.40782	0.703912	+	0.710288i	$0.251435\pi$
$84$	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	$-0.872339\pi$
$89$	− 655.567i	− 0.780786i	0.920648	−	0.390393i	$-0.127661\pi$
$90$	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	$0.120154\pi$
$97$	704.225i	0.737147i	−0.929599	+	0.368573i	$0.879846\pi$
$98$	0	0
$99$	702.830i	0.713506i
$100$	0	0
$101$	690.058i	0.679835i	0.940455	+	0.339918i	$0.110399\pi$
$101$	690.058i	0.679835i	−0.940455	+	0.339918i	$0.889601\pi$
$102$	0	0
$103$	−1398.27	−1.33763	−0.668814	−	0.743429i	$-0.733198\pi$
$103$	−1398.27	−1.33763	−0.668814	+	0.743429i	$0.733198\pi$
$104$	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	$-0.238254\pi$
$131$	2197.20	1.46542	0.732712	+	0.680539i	$0.238254\pi$
$132$	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	$-0.809723\pi$
$139$	−2709.21	−1.65318	−0.826591	+	0.562804i	$0.809723\pi$
$140$	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	$0.0233604\pi$
$157$	288.483i	0.146646i	−0.997308	+	0.0733230i	$0.976640\pi$
$158$	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	$-0.336432\pi$
$167$	2121.63	0.983092	0.491546	+	0.870852i	$0.336432\pi$
$168$	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	$-0.753761\pi$
$173$	− 3179.74i	− 1.39741i	0.715412	−	0.698703i	$-0.246239\pi$
$174$	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	$-0.916228\pi$
$181$	− 1266.98i	− 0.520299i	0.965568	−	0.260150i	$-0.0837719\pi$
$182$	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	$-0.790239\pi$
$199$	−4438.89	−1.58123	−0.790615	+	0.612313i	$0.790239\pi$
$200$	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	$-0.519103\pi$
$223$	−399.456	−0.119953	−0.0599766	+	0.998200i	$0.519103\pi$
$224$	0	0
$225$	−15977.7	−4.73413
$226$	0	0
$227$	2932.62	0.857466	0.428733	−	0.903431i	$-0.358960\pi$
$227$	2932.62	0.857466	0.428733	+	0.903431i	$0.358960\pi$
$228$	0	0
$229$	1505.24i	0.434363i	0.976131	+	0.217182i	$0.0696864\pi$
$229$	1505.24i	0.434363i	−0.976131	+	0.217182i	$0.930314\pi$
$230$	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	$-0.944665\pi$
$241$	− 1294.25i	− 0.345933i	0.984928	−	0.172966i	$-0.0553352\pi$
$242$	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	$-0.507142\pi$
$251$	−178.426	−0.0448691	−0.0224345	+	0.999748i	$0.507142\pi$
$252$	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	$0.254022\pi$
$257$	5899.74i	1.43197i	−0.698117	+	0.715984i	$0.745978\pi$
$258$	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	$0.0672267\pi$
$269$	1849.77i	0.419265i	−0.977780	+	0.209632i	$0.932773\pi$
$270$	0	0
$271$	−7198.86	−1.61365	−0.806826	−	0.590790i	$-0.798816\pi$
$271$	−7198.86	−1.61365	−0.806826	+	0.590790i	$0.798816\pi$
$272$	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	$-0.701114\pi$
$283$	−5623.58	−1.18123	−0.590614	+	0.806954i	$0.701114\pi$
$284$	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	$-0.983901\pi$
$293$	− 507.111i	− 0.101112i	0.998721	−	0.0505559i	$-0.0160993\pi$
$294$	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	$-0.258434\pi$
$307$	7402.96	1.37625	0.688126	+	0.725591i	$0.258434\pi$
$308$	0	0
$309$	12995.9	2.39259
$310$	0	0
$311$	−4284.78	−0.781245	−0.390623	−	0.920551i	$-0.627740\pi$
$311$	−4284.78	−0.781245	−0.390623	+	0.920551i	$0.627740\pi$
$312$	0	0
$313$	− 1502.00i	− 0.271239i	−0.990761	−	0.135620i	$-0.956698\pi$
$313$	− 1502.00i	− 0.271239i	0.990761	−	0.135620i	$-0.0433025\pi$
$314$	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	$0.163200\pi$
$349$	6396.47i	0.981075i	−0.871420	+	0.490538i	$0.836800\pi$
$350$	0	0
$351$	6015.33i	0.914743i
$352$	0	0
$353$	11472.5i	1.72981i	0.501940	+	0.864903i	$0.332620\pi$
$353$	11472.5i	1.72981i	−0.501940	+	0.864903i	$0.667380\pi$
$354$	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	$-0.240746\pi$
$367$	10227.7	1.45472	0.727362	+	0.686254i	$0.240746\pi$
$368$	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	$-0.446125\pi$
$383$	2525.17	0.336894	0.168447	+	0.985711i	$0.446125\pi$
$384$	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	$0.167983\pi$
$397$	7966.78i	1.00716i	−0.863950	+	0.503578i	$0.832017\pi$
$398$	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	$0.178411\pi$
$409$	8794.35i	1.06321i	−0.846993	+	0.531605i	$0.821589\pi$
$410$	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	$-0.493813\pi$
$419$	333.409	0.0388737	0.0194368	+	0.999811i	$0.493813\pi$
$420$	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	$0.289728\pi$
$433$	14229.4i	1.57926i	−0.613582	+	0.789631i	$0.710272\pi$
$434$	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	$-0.357655\pi$
$439$	7955.09	0.864865	0.432432	+	0.901666i	$0.357655\pi$
$440$	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	$-0.924594\pi$
$461$	− 4645.89i	− 0.469372i	0.972071	−	0.234686i	$-0.0754062\pi$
$462$	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	$-0.689474\pi$
$467$	−11317.4	−1.12143	−0.560716	+	0.828008i	$0.689474\pi$
$468$	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	$-0.433877\pi$
$479$	4324.22	0.412482	0.206241	+	0.978501i	$0.433877\pi$
$480$	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	$-0.407946\pi$
$503$	6434.36	0.570366	0.285183	+	0.958473i	$0.407946\pi$
$504$	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	$0.191772\pi$
$509$	13015.0i	1.13336i	−0.823939	+	0.566679i	$0.808228\pi$
$510$	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	$-0.930595\pi$
$521$	− 5144.91i	− 0.432635i	0.976323	−	0.216317i	$-0.0694046\pi$
$522$	0	0
$523$	11197.7	0.936213	0.468106	−	0.883672i	$-0.344936\pi$
$523$	11197.7	0.936213	0.468106	+	0.883672i	$0.344936\pi$
$524$	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	$-0.487120\pi$
$563$	1080.79	0.0809059	0.0404529	+	0.999181i	$0.487120\pi$
$564$	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	$-0.841082\pi$
$577$	− 13271.6i	− 0.957546i	0.877939	−	0.478773i	$-0.158918\pi$
$578$	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	$-0.702407\pi$
$587$	−16892.3	−1.18777	−0.593885	+	0.804550i	$0.702407\pi$
$588$	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	$0.123867\pi$
$593$	10957.3i	0.758787i	−0.925235	+	0.379394i	$0.876133\pi$
$594$	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	$0.203918\pi$
$601$	17612.6i	1.19540i	−0.801721	+	0.597699i	$0.796082\pi$
$602$	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	$-0.315739\pi$
$607$	16363.0	1.09416	0.547080	+	0.837080i	$0.315739\pi$
$608$	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	$-0.694463\pi$
$619$	−17668.2	−1.14725	−0.573624	+	0.819119i	$0.694463\pi$
$620$	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	$-0.394167\pi$
$643$	10643.5	0.652782	0.326391	+	0.945235i	$0.394167\pi$
$644$	0	0
$645$	13710.0	0.836946
$646$	0	0
$647$	18948.4	1.15137	0.575686	−	0.817671i	$-0.304735\pi$
$647$	18948.4	1.15137	0.575686	+	0.817671i	$0.304735\pi$
$648$	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	$-0.844092\pi$
$661$	− 15989.9i	− 0.940898i	0.882427	−	0.470449i	$-0.155908\pi$
$662$	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	$0.279038\pi$
$677$	27077.2i	1.53717i	−0.639750	+	0.768583i	$0.720962\pi$
$678$	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	$-0.356696\pi$
$691$	15808.2	0.870294	0.435147	+	0.900359i	$0.356696\pi$
$692$	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	$-0.319245\pi$
$719$	20738.0	1.07565	0.537827	+	0.843055i	$0.319245\pi$
$720$	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	$-0.522249\pi$
$727$	−2738.00	−0.139679	−0.0698396	+	0.997558i	$0.522249\pi$
$728$	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	$0.142813\pi$
$733$	17216.1i	0.867519i	−0.901029	+	0.433760i	$0.857187\pi$
$734$	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	$-0.747966\pi$
$761$	− 29877.8i	− 1.42322i	0.702575	−	0.711610i	$-0.252034\pi$
$762$	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	$-0.976891\pi$
$769$	− 3093.61i	− 0.145070i	0.997366	−	0.0725348i	$-0.0231089\pi$
$770$	0	0
$771$	− 54833.6i	− 2.56133i
$772$	0	0
$773$	− 24031.7i	− 1.11819i	−0.829104	−	0.559095i	$-0.811149\pi$
$773$	− 24031.7i	− 1.11819i	0.829104	−	0.559095i	$-0.188851\pi$
$774$	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	$-0.584563\pi$
$787$	−11593.2	−0.525099	−0.262550	+	0.964918i	$0.584563\pi$
$788$	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	$0.0820192\pi$
$797$	11467.4i	0.509658i	−0.966986	+	0.254829i	$0.917981\pi$
$798$	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	$-0.781366\pi$
$811$	−35717.1	−1.54648	−0.773241	+	0.634113i	$0.781366\pi$
$812$	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	$-0.981004\pi$
$829$	− 2847.17i	− 0.119284i	0.998220	−	0.0596419i	$-0.0189959\pi$
$830$	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	$-0.323051\pi$
$839$	25648.9	1.05542	0.527711	+	0.849424i	$0.323051\pi$
$840$	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	$-0.993580\pi$
$853$	− 1004.85i	− 0.0403348i	0.999797	−	0.0201674i	$-0.00641991\pi$
$854$	0	0
$855$	− 8632.23i	− 0.345282i
$856$	0	0
$857$	− 35060.5i	− 1.39748i	−0.715374	−	0.698742i	$-0.753744\pi$
$857$	− 35060.5i	− 1.39748i	0.715374	−	0.698742i	$-0.246256\pi$
$858$	0	0
$859$	−9006.42	−0.357736	−0.178868	−	0.983873i	$-0.557243\pi$
$859$	−9006.42	−0.357736	−0.178868	+	0.983873i	$0.557243\pi$
$860$	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	$-0.710889\pi$
$881$	− 41234.7i	− 1.57688i	0.615111	−	0.788440i	$-0.289111\pi$
$882$	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	$-0.649020\pi$
$887$	−23841.2	−0.902490	−0.451245	+	0.892400i	$0.649020\pi$
$888$	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	$0.181680\pi$
$929$	30596.3i	1.08055i	−0.841488	+	0.540276i	$0.818320\pi$
$930$	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	$0.163110\pi$
$937$	28125.2i	0.980586i	−0.871558	+	0.490293i	$0.836890\pi$
$938$	0	0
$939$	13959.9i	0.485159i
$940$	0	0
$941$	30098.1i	1.04269i	0.853347	+	0.521344i	$0.174569\pi$
$941$	30098.1i	1.04269i	−0.853347	+	0.521344i	$0.825431\pi$
$942$	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	$-0.646113\pi$
$971$	−26812.5	−0.886153	−0.443077	+	0.896484i	$0.646113\pi$
$972$	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	$-0.307580\pi$
$983$	35033.3	1.13671	0.568356	+	0.822783i	$0.307580\pi$
$984$	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	$0.155787\pi$
$997$	29598.9i	0.940226i	−0.882606	+	0.470113i	$0.844213\pi$
$998$	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.g.783.1		6
4.3	odd	2		784.4.f.h.783.5		6
7.2	even	3		112.4.p.g.31.3	yes	6
7.3	odd	6		112.4.p.f.47.1	yes	6
7.6	odd	2		784.4.f.h.783.6		6
28.3	even	6		112.4.p.g.47.3	yes	6
28.23	odd	6		112.4.p.f.31.1	✓	6
28.27	even	2	inner	784.4.f.g.783.2		6
56.3	even	6		448.4.p.f.383.1		6
56.37	even	6		448.4.p.f.255.1		6
56.45	odd	6		448.4.p.g.383.3		6
56.51	odd	6		448.4.p.g.255.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.1	✓	6	28.23	odd	6
112.4.p.f.47.1	yes	6	7.3	odd	6
112.4.p.g.31.3	yes	6	7.2	even	3
112.4.p.g.47.3	yes	6	28.3	even	6
448.4.p.f.255.1		6	56.37	even	6
448.4.p.f.383.1		6	56.3	even	6
448.4.p.g.255.3		6	56.51	odd	6
448.4.p.g.383.3		6	56.45	odd	6
784.4.f.g.783.1		6	1.1	even	1	trivial
784.4.f.g.783.2		6	28.27	even	2	inner
784.4.f.h.783.5		6	4.3	odd	2
784.4.f.h.783.6		6	7.6	odd	2

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

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