LMFDB - Embedded newform 25.8.b.c.24.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,8,Mod(24,25)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	25.b (of order $2$, degree $1$, not 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:	$7.80962563710$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.3
Root			$-2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	25.24
Dual form			25.8.b.c.24.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.28220i q^{2} -79.7424i q^{3} +126.356 q^{4} +102.246 q^{6} -538.197i q^{7} +326.136i q^{8} -4171.85 q^{9} +O(q^{10})$	\(q+1.28220i q^{2} -79.7424i q^{3} +126.356 q^{4} +102.246 q^{6} -538.197i q^{7} +326.136i q^{8} -4171.85 q^{9} -1215.12 q^{11} -10075.9i q^{12} -7070.42i q^{13} +690.077 q^{14} +15755.4 q^{16} -3348.13i q^{17} -5349.15i q^{18} -22169.7 q^{19} -42917.1 q^{21} -1558.03i q^{22} +58513.1i q^{23} +26006.8 q^{24} +9065.71 q^{26} +158276. i q^{27} -68004.4i q^{28} +206301. q^{29} +177822. q^{31} +61947.0i q^{32} +96896.5i q^{33} +4292.98 q^{34} -527138. q^{36} -284128. i q^{37} -28426.0i q^{38} -563812. q^{39} +627353. q^{41} -55028.4i q^{42} +164889. i q^{43} -153538. q^{44} -75025.7 q^{46} -449355. i q^{47} -1.25637e6i q^{48} +533887. q^{49} -266988. q^{51} -893390. i q^{52} +730190. i q^{53} -202942. q^{54} +175525. q^{56} +1.76786e6i q^{57} +264520. i q^{58} -1.42202e6 q^{59} -266326. q^{61} +228004. i q^{62} +2.24527e6i q^{63} +1.93726e6 q^{64} -124241. q^{66} +2.95028e6i q^{67} -423056. i q^{68} +4.66598e6 q^{69} +921138. q^{71} -1.36059e6i q^{72} -4.25657e6i q^{73} +364309. q^{74} -2.80127e6 q^{76} +653973. i q^{77} -722921. i q^{78} -6.28551e6 q^{79} +3.49751e6 q^{81} +804393. i q^{82} +9.17165e6i q^{83} -5.42283e6 q^{84} -211421. q^{86} -1.64510e7i q^{87} -396294. i q^{88} -242643. q^{89} -3.80528e6 q^{91} +7.39348e6i q^{92} -1.41799e7i q^{93} +576164. q^{94} +4.93980e6 q^{96} -2.59198e6i q^{97} +684551. i q^{98} +5.06929e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9}+O(q^{10}) $ 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9	$ 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9} + 9088 q^{11} - 15024 q^{14} + 40704 q^{16} - 77520 q^{19} - 138192 q^{21} + 263040 q^{24} - 114032 q^{26} + 144520 q^{29} + 613648 q^{31} + 906736 q^{34} - 439616 q^{36} - 1549456 q^{39} + 528728 q^{41} - 2868224 q^{44} - 2606832 q^{46} + 2330828 q^{49} + 2332688 q^{51} - 2205920 q^{54} + 1898880 q^{56} + 2240240 q^{59} + 4514088 q^{61} + 10293248 q^{64} - 13128704 q^{66} + 1490544 q^{69} + 1243568 q^{71} - 5302544 q^{74} + 1775360 q^{76} - 8666080 q^{79} - 4795756 q^{81} + 796416 q^{84} + 21596368 q^{86} - 12051240 q^{89} - 10704592 q^{91} - 11721584 q^{94} + 26610688 q^{96} - 5781376 q^{99}+O(q^{100}) $ 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9 + 9088 * q^11 - 15024 * q^14 + 40704 * q^16 - 77520 * q^19 - 138192 * q^21 + 263040 * q^24 - 114032 * q^26 + 144520 * q^29 + 613648 * q^31 + 906736 * q^34 - 439616 * q^36 - 1549456 * q^39 + 528728 * q^41 - 2868224 * q^44 - 2606832 * q^46 + 2330828 * q^49 + 2332688 * q^51 - 2205920 * q^54 + 1898880 * q^56 + 2240240 * q^59 + 4514088 * q^61 + 10293248 * q^64 - 13128704 * q^66 + 1490544 * q^69 + 1243568 * q^71 - 5302544 * q^74 + 1775360 * q^76 - 8666080 * q^79 - 4795756 * q^81 + 796416 * q^84 + 21596368 * q^86 - 12051240 * q^89 - 10704592 * q^91 - 11721584 * q^94 + 26610688 * q^96 - 5781376 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	1.28220i	0.113332i	0.998393	+	0.0566659i	$0.0180470\pi$
$2$	1.28220i	0.113332i	−0.998393	+	0.0566659i	$0.981953\pi$
$3$	− 79.7424i	− 1.70516i	−0.522598	−	0.852579i	$-0.675037\pi$
$3$	− 79.7424i	− 1.70516i	0.522598	−	0.852579i	$-0.324963\pi$
$4$	126.356	0.987156
$5$	0	0
$5$	0	0
$6$	102.246	0.193249
$7$	− 538.197i	− 0.593059i	−0.955024	−	0.296529i	$-0.904171\pi$
$7$	− 538.197i	− 0.593059i	0.955024	−	0.296529i	$-0.0958293\pi$
$8$	326.136i	0.225208i
$9$	−4171.85	−1.90757
$10$	0	0
$11$	−1215.12	−0.275261	−0.137630	−	0.990484i	$-0.543949\pi$
$11$	−1215.12	−0.275261	−0.137630	+	0.990484i	$0.543949\pi$
$12$	− 10075.9i	− 1.68326i
$13$	− 7070.42i	− 0.892573i	−0.894890	−	0.446286i	$-0.852746\pi$
$13$	− 7070.42i	− 0.892573i	0.894890	−	0.446286i	$-0.147254\pi$
$14$	690.077	0.0672124
$15$	0	0
$16$	15755.4	0.961633
$17$	− 3348.13i	− 0.165284i	−0.996579	−	0.0826420i	$-0.973664\pi$
$17$	− 3348.13i	− 0.165284i	0.996579	−	0.0826420i	$-0.0263358\pi$
$18$	− 5349.15i	− 0.216188i
$19$	−22169.7	−0.741519	−0.370759	−	0.928729i	$-0.620903\pi$
$19$	−22169.7	−0.741519	−0.370759	+	0.928729i	$0.620903\pi$
$20$	0	0
$21$	−42917.1	−1.01126
$22$	− 1558.03i	− 0.0311958i
$23$	58513.1i	1.00278i	0.865221	+	0.501390i	$0.167178\pi$
$23$	58513.1i	1.00278i	−0.865221	+	0.501390i	$0.832822\pi$
$24$	26006.8	0.384015
$25$	0	0
$26$	9065.71	0.101157
$27$	158276.i	1.54754i
$28$	− 68004.4i	− 0.585442i
$29$	206301.	1.57076	0.785379	−	0.619015i	$-0.212468\pi$
$29$	206301.	1.57076	0.785379	+	0.619015i	$0.212468\pi$
$30$	0	0
$31$	177822.	1.07206	0.536030	−	0.844199i	$-0.319923\pi$
$31$	177822.	1.07206	0.536030	+	0.844199i	$0.319923\pi$
$32$	61947.0i	0.334191i
$33$	96896.5i	0.469363i
$34$	4292.98	0.0187319
$35$	0	0
$36$	−527138.	−1.88307
$37$	− 284128.i	− 0.922163i	−0.887358	−	0.461081i	$-0.847462\pi$
$37$	− 284128.i	− 0.922163i	0.887358	−	0.461081i	$-0.152538\pi$
$38$	− 28426.0i	− 0.0840376i
$39$	−563812.	−1.52198
$40$	0	0
$41$	627353.	1.42157	0.710785	−	0.703409i	$-0.248340\pi$
$41$	627353.	1.42157	0.710785	+	0.703409i	$0.248340\pi$
$42$	− 55028.4i	− 0.114608i
$43$	164889.i	0.316266i	0.987418	+	0.158133i	$0.0505475\pi$
$43$	164889.i	0.316266i	−0.987418	+	0.158133i	$0.949453\pi$
$44$	−153538.	−0.271725
$45$	0	0
$46$	−75025.7	−0.113647
$47$	− 449355.i	− 0.631316i	−0.948873	−	0.315658i	$-0.897775\pi$
$47$	− 449355.i	− 0.631316i	0.948873	−	0.315658i	$-0.102225\pi$
$48$	− 1.25637e6i	− 1.63974i
$49$	533887.	0.648281
$50$	0	0
$51$	−266988.	−0.281835
$52$	− 893390.i	− 0.881108i
$53$	730190.i	0.673706i	0.941557	+	0.336853i	$0.109362\pi$
$53$	730190.i	0.673706i	−0.941557	+	0.336853i	$0.890638\pi$
$54$	−202942.	−0.175386
$55$	0	0
$56$	175525.	0.133562
$57$	1.76786e6i	1.26441i
$58$	264520.i	0.178017i
$59$	−1.42202e6	−0.901412	−0.450706	−	0.892673i	$-0.648828\pi$
$59$	−1.42202e6	−0.901412	−0.450706	+	0.892673i	$0.648828\pi$
$60$	0	0
$61$	−266326.	−0.150231	−0.0751153	−	0.997175i	$-0.523932\pi$
$61$	−266326.	−0.150231	−0.0751153	+	0.997175i	$0.523932\pi$
$62$	228004.i	0.121498i
$63$	2.24527e6i	1.13130i
$64$	1.93726e6	0.923758
$65$	0	0
$66$	−124241.	−0.0531938
$67$	2.95028e6i	1.19840i	0.800599	+	0.599200i	$0.204515\pi$
$67$	2.95028e6i	1.19840i	−0.800599	+	0.599200i	$0.795485\pi$
$68$	− 423056.i	− 0.163161i
$69$	4.66598e6	1.70990
$70$	0	0
$71$	921138.	0.305436	0.152718	−	0.988270i	$-0.451197\pi$
$71$	921138.	0.305436	0.152718	+	0.988270i	$0.451197\pi$
$72$	− 1.36059e6i	− 0.429599i
$73$	− 4.25657e6i	− 1.28065i	−0.768105	−	0.640323i	$-0.778800\pi$
$73$	− 4.25657e6i	− 1.28065i	0.768105	−	0.640323i	$-0.221200\pi$
$74$	364309.	0.104510
$75$	0	0
$76$	−2.80127e6	−0.731995
$77$	653973.i	0.163246i
$78$	− 722921.i	− 0.172488i
$79$	−6.28551e6	−1.43432	−0.717159	−	0.696910i	$-0.754558\pi$
$79$	−6.28551e6	−1.43432	−0.717159	+	0.696910i	$0.754558\pi$
$80$	0	0
$81$	3.49751e6	0.731243
$82$	804393.i	0.161109i
$83$	9.17165e6i	1.76065i	0.474367	+	0.880327i	$0.342677\pi$
$83$	9.17165e6i	1.76065i	−0.474367	+	0.880327i	$0.657323\pi$
$84$	−5.42283e6	−0.998271
$85$	0	0
$86$	−211421.	−0.0358430
$87$	− 1.64510e7i	− 2.67839i
$88$	− 396294.i	− 0.0619909i
$89$	−242643.	−0.0364840	−0.0182420	−	0.999834i	$-0.505807\pi$
$89$	−242643.	−0.0364840	−0.0182420	+	0.999834i	$0.505807\pi$
$90$	0	0
$91$	−3.80528e6	−0.529348
$92$	7.39348e6i	0.989901i
$93$	− 1.41799e7i	− 1.82803i
$94$	576164.	0.0715482
$95$	0	0
$96$	4.93980e6	0.569849
$97$	− 2.59198e6i	− 0.288357i	−0.989552	−	0.144179i	$-0.953946\pi$
$97$	− 2.59198e6i	− 0.288357i	0.989552	−	0.144179i	$-0.0460540\pi$
$98$	684551.i	0.0734708i
$99$	5.06929e6	0.525078
$100$	0	0
$101$	3.69169e6	0.356534	0.178267	−	0.983982i	$-0.442951\pi$
$101$	3.69169e6	0.356534	0.178267	+	0.983982i	$0.442951\pi$
$102$	− 342332.i	− 0.0319409i
$103$	− 8.68203e6i	− 0.782873i	−0.920205	−	0.391436i	$-0.871978\pi$
$103$	− 8.68203e6i	− 0.782873i	0.920205	−	0.391436i	$-0.128022\pi$
$104$	2.30592e6	0.201014
$105$	0	0
$106$	−936251.	−0.0763522
$107$	1.52118e7i	1.20043i	0.799838	+	0.600216i	$0.204919\pi$
$107$	1.52118e7i	1.20043i	−0.799838	+	0.600216i	$0.795081\pi$
$108$	1.99992e7i	1.52767i
$109$	1.60843e6	0.118963	0.0594813	−	0.998229i	$-0.481055\pi$
$109$	1.60843e6	0.118963	0.0594813	+	0.998229i	$0.481055\pi$
$110$	0	0
$111$	−2.26570e7	−1.57243
$112$	− 8.47950e6i	− 0.570305i
$113$	2.62766e7i	1.71315i	0.516024	+	0.856574i	$0.327411\pi$
$113$	2.62766e7i	1.71315i	−0.516024	+	0.856574i	$0.672589\pi$
$114$	−2.26676e6	−0.143297
$115$	0	0
$116$	2.60674e7	1.55058
$117$	2.94967e7i	1.70264i
$118$	− 1.82332e6i	− 0.102159i
$119$	−1.80195e6	−0.0980231
$120$	0	0
$121$	−1.80107e7	−0.924231
$122$	− 341483.i	− 0.0170259i
$123$	− 5.00266e7i	− 2.42400i
$124$	2.24688e7	1.05829
$125$	0	0
$126$	−2.87890e6	−0.128212
$127$	− 1.91864e7i	− 0.831152i	−0.909559	−	0.415576i	$-0.863580\pi$
$127$	− 1.91864e7i	− 0.831152i	0.909559	−	0.415576i	$-0.136420\pi$
$128$	1.04132e7i	0.438882i
$129$	1.31487e7	0.539284
$130$	0	0
$131$	3.07018e7	1.19320	0.596602	−	0.802537i	$-0.296517\pi$
$131$	3.07018e7	1.19320	0.596602	+	0.802537i	$0.296517\pi$
$132$	1.22434e7i	0.463335i
$133$	1.19317e7i	0.439764i
$134$	−3.78286e6	−0.135817
$135$	0	0
$136$	1.09194e6	0.0372232
$137$	− 3.16847e7i	− 1.05276i	−0.850251	−	0.526378i	$-0.823550\pi$
$137$	− 3.16847e7i	− 1.05276i	0.850251	−	0.526378i	$-0.176450\pi$
$138$	5.98273e6i	0.193786i
$139$	−2.07581e7	−0.655596	−0.327798	−	0.944748i	$-0.606307\pi$
$139$	−2.07581e7	−0.655596	−0.327798	+	0.944748i	$0.606307\pi$
$140$	0	0
$141$	−3.58326e7	−1.07649
$142$	1.18108e6i	0.0346156i
$143$	8.59140e6i	0.245690i
$144$	−6.57291e7	−1.83438
$145$	0	0
$146$	5.45778e6	0.145138
$147$	− 4.25734e7i	− 1.10542i
$148$	− 3.59012e7i	− 0.910318i
$149$	−2.94607e7	−0.729611	−0.364806	−	0.931084i	$-0.618865\pi$
$149$	−2.94607e7	−0.729611	−0.364806	+	0.931084i	$0.618865\pi$
$150$	0	0
$151$	4.60002e7	1.08728	0.543638	−	0.839320i	$-0.317046\pi$
$151$	4.60002e7	1.08728	0.543638	+	0.839320i	$0.317046\pi$
$152$	− 7.23033e6i	− 0.166996i
$153$	1.39679e7i	0.315290i
$154$	−838526.	−0.0185009
$155$	0	0
$156$	−7.12410e7	−1.50243
$157$	1.92998e7i	0.398019i	0.979998	+	0.199010i	$0.0637725\pi$
$157$	1.92998e7i	0.398019i	−0.979998	+	0.199010i	$0.936227\pi$
$158$	− 8.05929e6i	− 0.162554i
$159$	5.82271e7	1.14878
$160$	0	0
$161$	3.14916e7	0.594708
$162$	4.48452e6i	0.0828730i
$163$	− 2.35624e7i	− 0.426151i	−0.977036	−	0.213076i	$-0.931652\pi$
$163$	− 2.35624e7i	− 0.426151i	0.977036	−	0.213076i	$-0.0683481\pi$
$164$	7.92698e7	1.40331
$165$	0	0
$166$	−1.17599e7	−0.199538
$167$	2.18824e7i	0.363570i	0.983338	+	0.181785i	$0.0581875\pi$
$167$	2.18824e7i	0.363570i	−0.983338	+	0.181785i	$0.941813\pi$
$168$	− 1.39968e7i	− 0.227744i
$169$	1.27577e7	0.203314
$170$	0	0
$171$	9.24886e7	1.41450
$172$	2.08347e7i	0.312204i
$173$	9.62312e7i	1.41304i	0.707693	+	0.706520i	$0.249736\pi$
$173$	9.62312e7i	1.41304i	−0.707693	+	0.706520i	$0.750264\pi$
$174$	2.10935e7	0.303547
$175$	0	0
$176$	−1.91447e7	−0.264700
$177$	1.13395e8i	1.53705i
$178$	− 311117.i	− 0.00413479i
$179$	−8.46776e7	−1.10353	−0.551763	−	0.834001i	$-0.686045\pi$
$179$	−8.46776e7	−1.10353	−0.551763	+	0.834001i	$0.686045\pi$
$180$	0	0
$181$	−9.65249e7	−1.20994	−0.604971	−	0.796248i	$-0.706815\pi$
$181$	−9.65249e7	−1.20994	−0.604971	+	0.796248i	$0.706815\pi$
$182$	− 4.87913e6i	− 0.0599919i
$183$	2.12374e7i	0.256167i
$184$	−1.90832e7	−0.225834
$185$	0	0
$186$	1.81815e7	0.207174
$187$	4.06837e6i	0.0454962i
$188$	− 5.67787e7i	− 0.623208i
$189$	8.51839e7	0.917785
$190$	0	0
$191$	−1.46467e8	−1.52098	−0.760491	−	0.649348i	$-0.775042\pi$
$191$	−1.46467e8	−1.52098	−0.760491	+	0.649348i	$0.775042\pi$
$192$	− 1.54482e8i	− 1.57515i
$193$	8.53275e7i	0.854355i	0.904168	+	0.427177i	$0.140492\pi$
$193$	8.53275e7i	0.854355i	−0.904168	+	0.427177i	$0.859508\pi$
$194$	3.32345e6	0.0326800
$195$	0	0
$196$	6.74598e7	0.639954
$197$	1.44802e8i	1.34941i	0.738087	+	0.674705i	$0.235729\pi$
$197$	1.44802e8i	1.34941i	−0.738087	+	0.674705i	$0.764271\pi$
$198$	6.49986e6i	0.0595080i
$199$	−1.26870e7	−0.114123	−0.0570617	−	0.998371i	$-0.518173\pi$
$199$	−1.26870e7	−0.114123	−0.0570617	+	0.998371i	$0.518173\pi$
$200$	0	0
$201$	2.35263e8	2.04346
$202$	4.73349e6i	0.0404066i
$203$	− 1.11031e8i	− 0.931552i
$204$	−3.37355e7	−0.278215
$205$	0	0
$206$	1.11321e7	0.0887243
$207$	− 2.44108e8i	− 1.91287i
$208$	− 1.11397e8i	− 0.858327i
$209$	2.69388e7	0.204111
$210$	0	0
$211$	−9.47331e7	−0.694246	−0.347123	−	0.937820i	$-0.612841\pi$
$211$	−9.47331e7	−0.694246	−0.347123	+	0.937820i	$0.612841\pi$
$212$	9.22638e7i	0.665053i
$213$	− 7.34537e7i	− 0.520817i
$214$	−1.95046e7	−0.136047
$215$	0	0
$216$	−5.16196e7	−0.348519
$217$	− 9.57031e7i	− 0.635795i
$218$	2.06234e6i	0.0134822i
$219$	−3.39429e8	−2.18371
$220$	0	0
$221$	−2.36727e7	−0.147528
$222$	− 2.90509e7i	− 0.178207i
$223$	2.28554e8i	1.38014i	0.723744	+	0.690069i	$0.242420\pi$
$223$	2.28554e8i	1.38014i	−0.723744	+	0.690069i	$0.757580\pi$
$224$	3.33397e7	0.198195
$225$	0	0
$226$	−3.36919e7	−0.194154
$227$	− 3.00587e8i	− 1.70561i	−0.522229	−	0.852806i	$-0.674899\pi$
$227$	− 3.00587e8i	− 1.70561i	0.522229	−	0.852806i	$-0.325101\pi$
$228$	2.23380e8i	1.24817i
$229$	1.05343e8	0.579669	0.289835	−	0.957077i	$-0.406400\pi$
$229$	1.05343e8	0.579669	0.289835	+	0.957077i	$0.406400\pi$
$230$	0	0
$231$	5.21494e7	0.278360
$232$	6.72823e7i	0.353747i
$233$	− 3.27196e8i	− 1.69458i	−0.531130	−	0.847291i	$-0.678232\pi$
$233$	− 3.27196e8i	− 1.69458i	0.531130	−	0.847291i	$-0.321768\pi$
$234$	−3.78208e7	−0.192963
$235$	0	0
$236$	−1.79681e8	−0.889834
$237$	5.01221e8i	2.44574i
$238$	− 2.31047e6i	− 0.0111091i
$239$	5.72888e7	0.271442	0.135721	−	0.990747i	$-0.456665\pi$
$239$	5.72888e7	0.271442	0.135721	+	0.990747i	$0.456665\pi$
$240$	0	0
$241$	2.84287e8	1.30827	0.654134	−	0.756379i	$-0.273033\pi$
$241$	2.84287e8	1.30827	0.654134	+	0.756379i	$0.273033\pi$
$242$	− 2.30933e7i	− 0.104745i
$243$	6.72506e7i	0.300659i
$244$	−3.36518e7	−0.148301
$245$	0	0
$246$	6.41442e7	0.274717
$247$	1.56749e8i	0.661859i
$248$	5.79941e7i	0.241436i
$249$	7.31369e8	3.00219
$250$	0	0
$251$	−2.69178e7	−0.107444	−0.0537220	−	0.998556i	$-0.517108\pi$
$251$	−2.69178e7	−0.107444	−0.0537220	+	0.998556i	$0.517108\pi$
$252$	2.83704e8i	1.11677i
$253$	− 7.11004e7i	− 0.276026i
$254$	2.46008e7	0.0941959
$255$	0	0
$256$	2.34618e8	0.874019
$257$	− 3.28749e8i	− 1.20809i	−0.796951	−	0.604044i	$-0.793555\pi$
$257$	− 3.28749e8i	− 1.20809i	0.796951	−	0.604044i	$-0.206445\pi$
$258$	1.68592e7i	0.0611179i
$259$	−1.52917e8	−0.546897
$260$	0	0
$261$	−8.60658e8	−2.99632
$262$	3.93660e7i	0.135228i
$263$	− 1.85590e8i	− 0.629084i	−0.949244	−	0.314542i	$-0.898149\pi$
$263$	− 1.85590e8i	− 0.629084i	0.949244	−	0.314542i	$-0.101851\pi$
$264$	−3.16014e7	−0.105704
$265$	0	0
$266$	−1.52988e7	−0.0498393
$267$	1.93489e7i	0.0622109i
$268$	3.72786e8i	1.18301i
$269$	−2.70876e8	−0.848471	−0.424236	−	0.905552i	$-0.639457\pi$
$269$	−2.70876e8	−0.848471	−0.424236	+	0.905552i	$0.639457\pi$
$270$	0	0
$271$	2.30743e8	0.704263	0.352132	−	0.935950i	$-0.385457\pi$
$271$	2.30743e8	0.704263	0.352132	+	0.935950i	$0.385457\pi$
$272$	− 5.27511e7i	− 0.158942i
$273$	3.03442e8i	0.902623i
$274$	4.06262e7	0.119311
$275$	0	0
$276$	5.89574e8	1.68794
$277$	1.42059e8i	0.401595i	0.979633	+	0.200798i	$0.0643534\pi$
$277$	1.42059e8i	0.401595i	−0.979633	+	0.200798i	$0.935647\pi$
$278$	− 2.66161e7i	− 0.0742998i
$279$	−7.41846e8	−2.04503
$280$	0	0
$281$	−1.15362e8	−0.310164	−0.155082	−	0.987902i	$-0.549564\pi$
$281$	−1.15362e8	−0.310164	−0.155082	+	0.987902i	$0.549564\pi$
$282$	− 4.59447e7i	− 0.122001i
$283$	− 2.18080e8i	− 0.571958i	−0.958236	−	0.285979i	$-0.907681\pi$
$283$	− 2.18080e8i	− 0.571958i	0.958236	−	0.285979i	$-0.0923188\pi$
$284$	1.16391e8	0.301513
$285$	0	0
$286$	−1.10159e7	−0.0278445
$287$	− 3.37639e8i	− 0.843075i
$288$	− 2.58433e8i	− 0.637492i
$289$	3.99129e8	0.972681
$290$	0	0
$291$	−2.06691e8	−0.491695
$292$	− 5.37842e8i	− 1.26420i
$293$	− 4.12384e8i	− 0.957778i	−0.877875	−	0.478889i	$-0.841040\pi$
$293$	− 4.12384e8i	− 0.957778i	0.877875	−	0.478889i	$-0.158960\pi$
$294$	5.45878e7	0.125279
$295$	0	0
$296$	9.26642e7	0.207678
$297$	− 1.92325e8i	− 0.425979i
$298$	− 3.77746e7i	− 0.0826881i
$299$	4.13713e8	0.895055
$300$	0	0
$301$	8.87428e7	0.187564
$302$	5.89815e7i	0.123223i
$303$	− 2.94384e8i	− 0.607946i
$304$	−3.49292e8	−0.713069
$305$	0	0
$306$	−1.79096e7	−0.0357324
$307$	1.57983e8i	0.311621i	0.987787	+	0.155810i	$0.0497989\pi$
$307$	1.57983e8i	0.311621i	−0.987787	+	0.155810i	$0.950201\pi$
$308$	8.26334e7i	0.161149i
$309$	−6.92326e8	−1.33492
$310$	0	0
$311$	−1.69169e8	−0.318904	−0.159452	−	0.987206i	$-0.550973\pi$
$311$	−1.69169e8	−0.318904	−0.159452	+	0.987206i	$0.550973\pi$
$312$	− 1.83879e8i	− 0.342761i
$313$	5.22117e8i	0.962415i	0.876607	+	0.481208i	$0.159802\pi$
$313$	5.22117e8i	0.962415i	−0.876607	+	0.481208i	$0.840198\pi$
$314$	−2.47463e7	−0.0451082
$315$	0	0
$316$	−7.94211e8	−1.41590
$317$	4.73280e8i	0.834470i	0.908799	+	0.417235i	$0.137001\pi$
$317$	4.73280e8i	0.834470i	−0.908799	+	0.417235i	$0.862999\pi$
$318$	7.46589e7i	0.130193i
$319$	−2.50681e8	−0.432368
$320$	0	0
$321$	1.21303e9	2.04693
$322$	4.03786e7i	0.0673993i
$323$	7.42270e7i	0.122561i
$324$	4.41932e8	0.721851
$325$	0	0
$326$	3.02118e7	0.0482965
$327$	− 1.28260e8i	− 0.202850i
$328$	2.04602e8i	0.320149i
$329$	−2.41841e8	−0.374408
$330$	0	0
$331$	−8.75892e7	−0.132756	−0.0663778	−	0.997795i	$-0.521144\pi$
$331$	−8.75892e7	−0.132756	−0.0663778	+	0.997795i	$0.521144\pi$
$332$	1.15889e9i	1.73804i
$333$	1.18534e9i	1.75909i
$334$	−2.80577e7	−0.0412040
$335$	0	0
$336$	−6.76175e8	−0.972460
$337$	− 3.04119e8i	− 0.432851i	−0.976299	−	0.216426i	$-0.930560\pi$
$337$	− 3.04119e8i	− 0.432851i	0.976299	−	0.216426i	$-0.0694399\pi$
$338$	1.63579e7i	0.0230419i
$339$	2.09536e9	2.92119
$340$	0	0
$341$	−2.16075e8	−0.295096
$342$	1.18589e8i	0.160307i
$343$	− 7.30565e8i	− 0.977528i
$344$	−5.37762e7	−0.0712256
$345$	0	0
$346$	−1.23388e8	−0.160142
$347$	3.15347e8i	0.405169i	0.979265	+	0.202584i	$0.0649340\pi$
$347$	3.15347e8i	0.405169i	−0.979265	+	0.202584i	$0.935066\pi$
$348$	− 2.07868e9i	− 2.64399i
$349$	5.23418e8	0.659112	0.329556	−	0.944136i	$-0.393101\pi$
$349$	5.23418e8	0.659112	0.329556	+	0.944136i	$0.393101\pi$
$350$	0	0
$351$	1.11908e9	1.38130
$352$	− 7.52730e7i	− 0.0919898i
$353$	4.13964e8i	0.500900i	0.968130	+	0.250450i	$0.0805786\pi$
$353$	4.13964e8i	0.500900i	−0.968130	+	0.250450i	$0.919421\pi$
$354$	−1.45395e8	−0.174197
$355$	0	0
$356$	−3.06593e7	−0.0360154
$357$	1.43692e8i	0.167145i
$358$	− 1.08574e8i	− 0.125065i
$359$	2.48768e8	0.283768	0.141884	−	0.989883i	$-0.454684\pi$
$359$	2.48768e8	0.283768	0.141884	+	0.989883i	$0.454684\pi$
$360$	0	0
$361$	−4.02376e8	−0.450150
$362$	− 1.23764e8i	− 0.137125i
$363$	1.43621e9i	1.57596i
$364$	−4.80819e8	−0.522549
$365$	0	0
$366$	−2.72307e7	−0.0290319
$367$	1.45884e9i	1.54056i	0.637708	+	0.770278i	$0.279883\pi$
$367$	1.45884e9i	1.54056i	−0.637708	+	0.770278i	$0.720117\pi$
$368$	9.21897e8i	0.964307i
$369$	−2.61722e9	−2.71174
$370$	0	0
$371$	3.92986e8	0.399547
$372$	− 1.79172e9i	− 1.80455i
$373$	− 6.29125e8i	− 0.627706i	−0.949472	−	0.313853i	$-0.898380\pi$
$373$	− 6.29125e8i	− 0.627706i	0.949472	−	0.313853i	$-0.101620\pi$
$374$	−5.21648e6	−0.00515616
$375$	0	0
$376$	1.46551e8	0.142177
$377$	− 1.45864e9i	− 1.40202i
$378$	1.09223e8i	0.104014i
$379$	−8.54658e7	−0.0806409	−0.0403204	−	0.999187i	$-0.512838\pi$
$379$	−8.54658e7	−0.0806409	−0.0403204	+	0.999187i	$0.512838\pi$
$380$	0	0
$381$	−1.52997e9	−1.41725
$382$	− 1.87801e8i	− 0.172376i
$383$	7.56340e8i	0.687893i	0.938989	+	0.343947i	$0.111764\pi$
$383$	7.56340e8i	0.687893i	−0.938989	+	0.343947i	$0.888236\pi$
$384$	8.30371e8	0.748364
$385$	0	0
$386$	−1.09407e8	−0.0968255
$387$	− 6.87892e8i	− 0.603298i
$388$	− 3.27512e8i	− 0.284654i
$389$	−1.42837e9	−1.23032	−0.615160	−	0.788402i	$-0.710908\pi$
$389$	−1.42837e9	−1.23032	−0.615160	+	0.788402i	$0.710908\pi$
$390$	0	0
$391$	1.95909e8	0.165744
$392$	1.74120e8i	0.145998i
$393$	− 2.44824e9i	− 2.03460i
$394$	−1.85666e8	−0.152931
$395$	0	0
$396$	6.40535e8	0.518334
$397$	− 1.67152e8i	− 0.134074i	−0.997750	−	0.0670372i	$-0.978645\pi$
$397$	− 1.67152e8i	− 0.134074i	0.997750	−	0.0670372i	$-0.0213546\pi$
$398$	− 1.62673e7i	− 0.0129338i
$399$	9.51459e8	0.749868
$400$	0	0
$401$	2.32051e9	1.79712	0.898561	−	0.438848i	$-0.144613\pi$
$401$	2.32051e9	1.79712	0.898561	+	0.438848i	$0.144613\pi$
$402$	3.01654e8i	0.231589i
$403$	− 1.25728e9i	− 0.956892i
$404$	4.66467e8	0.351954
$405$	0	0
$406$	1.42364e8	0.105574
$407$	3.45249e8i	0.253835i
$408$	− 8.70742e7i	− 0.0634715i
$409$	8.24735e8	0.596050	0.298025	−	0.954558i	$-0.403672\pi$
$409$	8.24735e8	0.596050	0.298025	+	0.954558i	$0.403672\pi$
$410$	0	0
$411$	−2.52661e9	−1.79512
$412$	− 1.09703e9i	− 0.772817i
$413$	7.65326e8i	0.534590i
$414$	3.12996e8	0.216789
$415$	0	0
$416$	4.37991e8	0.298290
$417$	1.65530e9i	1.11790i
$418$	3.45410e7i	0.0231323i
$419$	−8.23968e8	−0.547219	−0.273610	−	0.961841i	$-0.588218\pi$
$419$	−8.23968e8	−0.547219	−0.273610	+	0.961841i	$0.588218\pi$
$420$	0	0
$421$	−2.12046e8	−0.138498	−0.0692488	−	0.997599i	$-0.522060\pi$
$421$	−2.12046e8	−0.138498	−0.0692488	+	0.997599i	$0.522060\pi$
$422$	− 1.21467e8i	− 0.0786801i
$423$	1.87464e9i	1.20428i
$424$	−2.38141e8	−0.151724
$425$	0	0
$426$	9.41825e7	0.0590251
$427$	1.43336e8i	0.0890956i
$428$	1.92210e9i	1.18501i
$429$	6.85099e8	0.418941
$430$	0	0
$431$	7.44023e8	0.447627	0.223813	−	0.974632i	$-0.428149\pi$
$431$	7.44023e8	0.447627	0.223813	+	0.974632i	$0.428149\pi$
$432$	2.49371e9i	1.48817i
$433$	1.93573e9i	1.14588i	0.819598	+	0.572938i	$0.194197\pi$
$433$	1.93573e9i	1.14588i	−0.819598	+	0.572938i	$0.805803\pi$
$434$	1.22711e8	0.0720557
$435$	0	0
$436$	2.03235e8	0.117435
$437$	− 1.29722e9i	− 0.743581i
$438$	− 4.35216e8i	− 0.247483i
$439$	1.35485e9	0.764300	0.382150	−	0.924100i	$-0.375184\pi$
$439$	1.35485e9	0.764300	0.382150	+	0.924100i	$0.375184\pi$
$440$	0	0
$441$	−2.22730e9	−1.23664
$442$	− 3.03531e7i	− 0.0167196i
$443$	− 1.05985e9i	− 0.579203i	−0.957147	−	0.289601i	$-0.906477\pi$
$443$	− 1.05985e9i	− 0.579203i	0.957147	−	0.289601i	$-0.0935227\pi$
$444$	−2.86285e9	−1.55224
$445$	0	0
$446$	−2.93053e8	−0.156413
$447$	2.34927e9i	1.24410i
$448$	− 1.04263e9i	− 0.547843i
$449$	1.70272e9	0.887730	0.443865	−	0.896094i	$-0.353607\pi$
$449$	1.70272e9	0.887730	0.443865	+	0.896094i	$0.353607\pi$
$450$	0	0
$451$	−7.62309e8	−0.391303
$452$	3.32021e9i	1.69114i
$453$	− 3.66816e9i	− 1.85398i
$454$	3.85414e8	0.193300
$455$	0	0
$456$	−5.76564e8	−0.284754
$457$	− 3.76431e9i	− 1.84493i	−0.386085	−	0.922463i	$-0.626173\pi$
$457$	− 3.76431e9i	− 1.84493i	0.386085	−	0.922463i	$-0.373827\pi$
$458$	1.35071e8i	0.0656949i
$459$	5.29930e8	0.255784
$460$	0	0
$461$	3.59084e9	1.70704	0.853519	−	0.521062i	$-0.174464\pi$
$461$	3.59084e9	1.70704	0.853519	+	0.521062i	$0.174464\pi$
$462$	6.68660e7i	0.0315470i
$463$	2.45649e8i	0.115022i	0.998345	+	0.0575111i	$0.0183164\pi$
$463$	2.45649e8i	0.115022i	−0.998345	+	0.0575111i	$0.981684\pi$
$464$	3.25036e9	1.51049
$465$	0	0
$466$	4.19532e8	0.192050
$467$	− 4.04985e8i	− 0.184005i	−0.995759	−	0.0920025i	$-0.970673\pi$
$467$	− 4.04985e8i	− 0.184005i	0.995759	−	0.0920025i	$-0.0293268\pi$
$468$	3.72709e9i	1.68077i
$469$	1.58783e9	0.710722
$470$	0	0
$471$	1.53901e9	0.678686
$472$	− 4.63771e8i	− 0.203005i
$473$	− 2.00360e8i	− 0.0870556i
$474$	−6.42667e8	−0.277180
$475$	0	0
$476$	−2.27687e8	−0.0967641
$477$	− 3.04624e9i	− 1.28514i
$478$	7.34559e7i	0.0307630i
$479$	−4.18334e9	−1.73920	−0.869598	−	0.493760i	$-0.835622\pi$
$479$	−4.18334e9	−1.73920	−0.869598	+	0.493760i	$0.835622\pi$
$480$	0	0
$481$	−2.00890e9	−0.823097
$482$	3.64513e8i	0.148268i
$483$	− 2.51121e9i	− 1.01407i
$484$	−2.27575e9	−0.912361
$485$	0	0
$486$	−8.62289e7	−0.0340742
$487$	3.25089e9i	1.27541i	0.770280	+	0.637706i	$0.220117\pi$
$487$	3.25089e9i	1.27541i	−0.770280	+	0.637706i	$0.779883\pi$
$488$	− 8.68583e7i	− 0.0338331i
$489$	−1.87893e9	−0.726656
$490$	0	0
$491$	−3.37360e9	−1.28620	−0.643100	−	0.765782i	$-0.722352\pi$
$491$	−3.37360e9	−1.28620	−0.643100	+	0.765782i	$0.722352\pi$
$492$	− 6.32116e9i	− 2.39287i
$493$	− 6.90723e8i	− 0.259621i
$494$	−2.00984e8	−0.0750097
$495$	0	0
$496$	2.80165e9	1.03093
$497$	− 4.95753e8i	− 0.181142i
$498$	9.37763e8i	0.340244i
$499$	−3.74951e9	−1.35090	−0.675449	−	0.737406i	$-0.736050\pi$
$499$	−3.74951e9	−1.35090	−0.675449	+	0.737406i	$0.736050\pi$
$500$	0	0
$501$	1.74496e9	0.619945
$502$	− 3.45141e7i	− 0.0121768i
$503$	3.09301e9i	1.08366i	0.840488	+	0.541830i	$0.182268\pi$
$503$	3.09301e9i	1.08366i	−0.840488	+	0.541830i	$0.817732\pi$
$504$	−7.32264e8	−0.254777
$505$	0	0
$506$	9.11651e7	0.0312825
$507$	− 1.01733e9i	− 0.346683i
$508$	− 2.42432e9i	− 0.820476i
$509$	−1.36862e9	−0.460015	−0.230007	−	0.973189i	$-0.573875\pi$
$509$	−1.36862e9	−0.460015	−0.230007	+	0.973189i	$0.573875\pi$
$510$	0	0
$511$	−2.29087e9	−0.759499
$512$	1.63371e9i	0.537937i
$513$	− 3.50894e9i	− 1.14753i
$514$	4.21523e8	0.136915
$515$	0	0
$516$	1.66141e9	0.532357
$517$	5.46020e8i	0.173777i
$518$	− 1.96070e8i	− 0.0619808i
$519$	7.67371e9	2.40946
$520$	0	0
$521$	−4.47414e9	−1.38604	−0.693022	−	0.720916i	$-0.743721\pi$
$521$	−4.47414e9	−1.38604	−0.693022	+	0.720916i	$0.743721\pi$
$522$	− 1.10354e9i	− 0.339579i
$523$	3.01993e9i	0.923085i	0.887118	+	0.461542i	$0.152704\pi$
$523$	3.01993e9i	0.923085i	−0.887118	+	0.461542i	$0.847296\pi$
$524$	3.87936e9	1.17788
$525$	0	0
$526$	2.37963e8	0.0712952
$527$	− 5.95370e8i	− 0.177194i
$528$	1.52664e9i	0.451355i
$529$	−1.89619e7	−0.00556913
$530$	0	0
$531$	5.93244e9	1.71950
$532$	1.50764e9i	0.434116i
$533$	− 4.43565e9i	− 1.26885i
$534$	−2.48092e7	−0.00705047
$535$	0	0
$536$	−9.62193e8	−0.269889
$537$	6.75239e9i	1.88169i
$538$	− 3.47318e8i	− 0.0961587i
$539$	−6.48737e8	−0.178446
$540$	0	0
$541$	3.23946e9	0.879593	0.439796	−	0.898097i	$-0.355051\pi$
$541$	3.23946e9	0.879593	0.439796	+	0.898097i	$0.355051\pi$
$542$	2.95859e8i	0.0798154i
$543$	7.69713e9i	2.06314i
$544$	2.07406e8	0.0552365
$545$	0	0
$546$	−3.89074e8	−0.102296
$547$	4.25742e9i	1.11222i	0.831108	+	0.556111i	$0.187707\pi$
$547$	4.25742e9i	1.11222i	−0.831108	+	0.556111i	$0.812293\pi$
$548$	− 4.00355e9i	− 1.03923i
$549$	1.11107e9	0.286575
$550$	0	0
$551$	−4.57364e9	−1.16475
$552$	1.52174e9i	0.385083i
$553$	3.38284e9i	0.850635i
$554$	−1.82148e8	−0.0455135
$555$	0	0
$556$	−2.62291e9	−0.647175
$557$	− 5.65828e9i	− 1.38737i	−0.720280	−	0.693684i	$-0.755987\pi$
$557$	− 5.65828e9i	− 1.38737i	0.720280	−	0.693684i	$-0.244013\pi$
$558$	− 9.51196e8i	− 0.231766i
$559$	1.16584e9	0.282290
$560$	0	0
$561$	3.24422e8	0.0775783
$562$	− 1.47918e8i	− 0.0351515i
$563$	− 4.37511e9i	− 1.03326i	−0.856209	−	0.516630i	$-0.827186\pi$
$563$	− 4.37511e9i	− 1.03326i	0.856209	−	0.516630i	$-0.172814\pi$
$564$	−4.52767e9	−1.06267
$565$	0	0
$566$	2.79623e8	0.0648210
$567$	− 1.88235e9i	− 0.433670i
$568$	3.00416e8i	0.0687866i
$569$	−1.67226e9	−0.380550	−0.190275	−	0.981731i	$-0.560938\pi$
$569$	−1.67226e9	−0.380550	−0.190275	+	0.981731i	$0.560938\pi$
$570$	0	0
$571$	6.39802e8	0.143820	0.0719100	−	0.997411i	$-0.477091\pi$
$571$	6.39802e8	0.143820	0.0719100	+	0.997411i	$0.477091\pi$
$572$	1.08558e9i	0.242535i
$573$	1.16797e10i	2.59352i
$574$	4.32922e8	0.0955472
$575$	0	0
$576$	−8.08196e9	−1.76213
$577$	4.87101e9i	1.05561i	0.849366	+	0.527805i	$0.176985\pi$
$577$	4.87101e9i	1.05561i	−0.849366	+	0.527805i	$0.823015\pi$
$578$	5.11764e8i	0.110236i
$579$	6.80422e9	1.45681
$580$	0	0
$581$	4.93615e9	1.04417
$582$	− 2.65019e8i	− 0.0557246i
$583$	− 8.87268e8i	− 0.185445i
$584$	1.38822e9	0.288412
$585$	0	0
$586$	5.28760e8	0.108547
$587$	− 2.32406e9i	− 0.474258i	−0.971478	−	0.237129i	$-0.923794\pi$
$587$	− 2.32406e9i	− 0.474258i	0.971478	−	0.237129i	$-0.0762064\pi$
$588$	− 5.37941e9i	− 1.09122i
$589$	−3.94226e9	−0.794953
$590$	0	0
$591$	1.15469e10	2.30096
$592$	− 4.47654e9i	− 0.886782i
$593$	− 4.27080e9i	− 0.841043i	−0.907283	−	0.420522i	$-0.861847\pi$
$593$	− 4.27080e9i	− 0.841043i	0.907283	−	0.420522i	$-0.138153\pi$
$594$	2.46599e8	0.0482769
$595$	0	0
$596$	−3.72254e9	−0.720240
$597$	1.01169e9i	0.194598i
$598$	5.30463e8i	0.101438i
$599$	−3.50335e9	−0.666023	−0.333012	−	0.942923i	$-0.608065\pi$
$599$	−3.50335e9	−0.666023	−0.333012	+	0.942923i	$0.608065\pi$
$600$	0	0
$601$	2.98866e9	0.561585	0.280792	−	0.959769i	$-0.409403\pi$
$601$	2.98866e9	0.561585	0.280792	+	0.959769i	$0.409403\pi$
$602$	1.13786e8i	0.0212570i
$603$	− 1.23081e10i	− 2.28603i
$604$	5.81240e9	1.07331
$605$	0	0
$606$	3.77460e8	0.0688996
$607$	− 7.31901e9i	− 1.32829i	−0.747605	−	0.664143i	$-0.768797\pi$
$607$	− 7.31901e9i	− 1.32829i	0.747605	−	0.664143i	$-0.231203\pi$
$608$	− 1.37335e9i	− 0.247809i
$609$	−8.85386e9	−1.58844
$610$	0	0
$611$	−3.17713e9	−0.563496
$612$	1.76492e9i	0.311241i
$613$	8.32799e9i	1.46026i	0.683311	+	0.730128i	$0.260539\pi$
$613$	8.32799e9i	1.46026i	−0.683311	+	0.730128i	$0.739461\pi$
$614$	−2.02566e8	−0.0353165
$615$	0	0
$616$	−2.13284e8	−0.0367643
$617$	− 2.88456e9i	− 0.494403i	−0.968964	−	0.247201i	$-0.920489\pi$
$617$	− 2.88456e9i	− 0.494403i	0.968964	−	0.247201i	$-0.0795109\pi$
$618$	− 8.87702e8i	− 0.151289i
$619$	2.27213e9	0.385049	0.192525	−	0.981292i	$-0.438332\pi$
$619$	2.27213e9	0.385049	0.192525	+	0.981292i	$0.438332\pi$
$620$	0	0
$621$	−9.26125e9	−1.55185
$622$	− 2.16909e8i	− 0.0361420i
$623$	1.30589e8i	0.0216371i
$624$	−8.88308e9	−1.46358
$625$	0	0
$626$	−6.69459e8	−0.109072
$627$	− 2.14817e9i	− 0.348042i
$628$	2.43865e9i	0.392907i
$629$	−9.51296e8	−0.152419
$630$	0	0
$631$	−2.90856e7	−0.00460867	−0.00230434	−	0.999997i	$-0.500733\pi$
$631$	−2.90856e7	−0.00460867	−0.00230434	+	0.999997i	$0.500733\pi$
$632$	− 2.04993e9i	− 0.323020i
$633$	7.55424e9i	1.18380i
$634$	−6.06841e8	−0.0945719
$635$	0	0
$636$	7.35734e9	1.13402
$637$	− 3.77481e9i	− 0.578638i
$638$	− 3.21423e8i	− 0.0490010i
$639$	−3.84285e9	−0.582640
$640$	0	0
$641$	−3.12634e9	−0.468849	−0.234424	−	0.972134i	$-0.575320\pi$
$641$	−3.12634e9	−0.468849	−0.234424	+	0.972134i	$0.575320\pi$
$642$	1.55534e9i	0.231982i
$643$	− 2.48120e9i	− 0.368064i	−0.982920	−	0.184032i	$-0.941085\pi$
$643$	− 2.48120e9i	− 0.368064i	0.982920	−	0.184032i	$-0.0589150\pi$
$644$	3.97915e9	0.587070
$645$	0	0
$646$	−9.51740e7	−0.0138901
$647$	3.10080e9i	0.450100i	0.974347	+	0.225050i	$0.0722545\pi$
$647$	3.10080e9i	0.450100i	−0.974347	+	0.225050i	$0.927746\pi$
$648$	1.14066e9i	0.164682i
$649$	1.72792e9	0.248123
$650$	0	0
$651$	−7.63159e9	−1.08413
$652$	− 2.97726e9i	− 0.420678i
$653$	− 9.64911e8i	− 0.135610i	−0.997699	−	0.0678049i	$-0.978400\pi$
$653$	− 9.64911e8i	− 0.135610i	0.997699	−	0.0678049i	$-0.0215995\pi$
$654$	1.64456e8	0.0229894
$655$	0	0
$656$	9.88419e9	1.36703
$657$	1.77577e10i	2.44292i
$658$	− 3.10089e8i	− 0.0424323i
$659$	7.70039e9	1.04813	0.524063	−	0.851679i	$-0.324416\pi$
$659$	7.70039e9	1.04813	0.524063	+	0.851679i	$0.324416\pi$
$660$	0	0
$661$	1.30650e10	1.75956	0.879779	−	0.475382i	$-0.157690\pi$
$661$	1.30650e10	1.75956	0.879779	+	0.475382i	$0.157690\pi$
$662$	− 1.12307e8i	− 0.0150454i
$663$	1.88772e9i	0.251559i
$664$	−2.99120e9	−0.396513
$665$	0	0
$666$	−1.51984e9	−0.199360
$667$	1.20713e10i	1.57513i
$668$	2.76498e9i	0.358900i
$669$	1.82255e10	2.35335
$670$	0	0
$671$	3.23617e8	0.0413526
$672$	− 2.65858e9i	− 0.337954i
$673$	− 1.31771e10i	− 1.66636i	−0.553003	−	0.833179i	$-0.686518\pi$
$673$	− 1.31771e10i	− 1.66636i	0.553003	−	0.833179i	$-0.313482\pi$
$674$	3.89942e8	0.0490558
$675$	0	0
$676$	1.61201e9	0.200703
$677$	− 3.18151e9i	− 0.394069i	−0.980397	−	0.197035i	$-0.936869\pi$
$677$	− 3.18151e9i	− 0.394069i	0.980397	−	0.197035i	$-0.0631311\pi$
$678$	2.68667e9i	0.331063i
$679$	−1.39500e9	−0.171013
$680$	0	0
$681$	−2.39695e10	−2.90834
$682$	− 2.77051e8i	− 0.0334438i
$683$	3.33139e8i	0.0400086i	0.999800	+	0.0200043i	$0.00636799\pi$
$683$	3.33139e8i	0.0400086i	−0.999800	+	0.0200043i	$0.993632\pi$
$684$	1.16865e10	1.39633
$685$	0	0
$686$	9.36731e8	0.110785
$687$	− 8.40028e9i	− 0.988428i
$688$	2.59789e9i	0.304132i
$689$	5.16275e9	0.601331
$690$	0	0
$691$	−4.45248e9	−0.513368	−0.256684	−	0.966495i	$-0.582630\pi$
$691$	−4.45248e9	−0.513368	−0.256684	+	0.966495i	$0.582630\pi$
$692$	1.21594e10i	1.39489i
$693$	− 2.72828e9i	− 0.311402i
$694$	−4.04339e8	−0.0459185
$695$	0	0
$696$	5.36525e9	0.603195
$697$	− 2.10046e9i	− 0.234963i
$698$	6.71127e8i	0.0746983i
$699$	−2.60914e10	−2.88953
$700$	0	0
$701$	−1.91380e9	−0.209838	−0.104919	−	0.994481i	$-0.533458\pi$
$701$	−1.91380e9	−0.209838	−0.104919	+	0.994481i	$0.533458\pi$
$702$	1.43489e9i	0.156545i
$703$	6.29902e9i	0.683801i
$704$	−2.35400e9	−0.254274
$705$	0	0
$706$	−5.30785e8	−0.0567679
$707$	− 1.98686e9i	− 0.211445i
$708$	1.43282e10i	1.51731i
$709$	−1.38704e9	−0.146160	−0.0730799	−	0.997326i	$-0.523283\pi$
$709$	−1.38704e9	−0.146160	−0.0730799	+	0.997326i	$0.523283\pi$
$710$	0	0
$711$	2.62222e10	2.73606
$712$	− 7.91344e7i	− 0.00821647i
$713$	1.04049e10i	1.07504i
$714$	−1.84242e8	−0.0189428
$715$	0	0
$716$	−1.06995e10	−1.08935
$717$	− 4.56835e9i	− 0.462852i
$718$	3.18970e8i	0.0321599i
$719$	−9.42958e9	−0.946109	−0.473055	−	0.881033i	$-0.656849\pi$
$719$	−9.42958e9	−0.946109	−0.473055	+	0.881033i	$0.656849\pi$
$720$	0	0
$721$	−4.67264e9	−0.464290
$722$	− 5.15928e8i	− 0.0510163i
$723$	− 2.26697e10i	− 2.23081i
$724$	−1.21965e10	−1.19440
$725$	0	0
$726$	−1.84151e9	−0.178606
$727$	− 2.99387e9i	− 0.288976i	−0.989507	−	0.144488i	$-0.953846\pi$
$727$	− 2.99387e9i	− 0.288976i	0.989507	−	0.144488i	$-0.0461535\pi$
$728$	− 1.24104e9i	− 0.119213i
$729$	1.30118e10	1.24391
$730$	0	0
$731$	5.52070e8	0.0522737
$732$	2.68348e9i	0.252877i
$733$	− 1.11711e10i	− 1.04768i	−0.851815	−	0.523842i	$-0.824498\pi$
$733$	− 1.11711e10i	− 1.04768i	0.851815	−	0.523842i	$-0.175502\pi$
$734$	−1.87053e9	−0.174594
$735$	0	0
$736$	−3.62471e9	−0.335121
$737$	− 3.58495e9i	− 0.329873i
$738$	− 3.35581e9i	− 0.307326i
$739$	6.96449e9	0.634796	0.317398	−	0.948292i	$-0.397191\pi$
$739$	6.96449e9	0.634796	0.317398	+	0.948292i	$0.397191\pi$
$740$	0	0
$741$	1.24995e10	1.12858
$742$	5.03887e8i	0.0452814i
$743$	− 1.27161e9i	− 0.113734i	−0.998382	−	0.0568672i	$-0.981889\pi$
$743$	− 1.27161e9i	− 0.113734i	0.998382	−	0.0568672i	$-0.0181112\pi$
$744$	4.62458e9	0.411687
$745$	0	0
$746$	8.06666e8	0.0711390
$747$	− 3.82627e10i	− 3.35856i
$748$	5.14063e8i	0.0449119i
$749$	8.18695e9	0.711927
$750$	0	0
$751$	−1.70771e10	−1.47121	−0.735603	−	0.677413i	$-0.763101\pi$
$751$	−1.70771e10	−1.47121	−0.735603	+	0.677413i	$0.763101\pi$
$752$	− 7.07976e9i	− 0.607094i
$753$	2.14649e9i	0.183209i
$754$	1.87027e9	0.158893
$755$	0	0
$756$	1.07635e10	0.905997
$757$	1.37933e10i	1.15567i	0.816155	+	0.577833i	$0.196101\pi$
$757$	1.37933e10i	1.15567i	−0.816155	+	0.577833i	$0.803899\pi$
$758$	− 1.09584e8i	− 0.00913917i
$759$	−5.66972e9	−0.470669
$760$	0	0
$761$	1.71364e10	1.40953	0.704765	−	0.709441i	$-0.251053\pi$
$761$	1.71364e10	1.40953	0.704765	+	0.709441i	$0.251053\pi$
$762$	− 1.96173e9i	− 0.160619i
$763$	− 8.65654e8i	− 0.0705518i
$764$	−1.85070e10	−1.50145
$765$	0	0
$766$	−9.69780e8	−0.0779602
$767$	1.00543e10i	0.804575i
$768$	− 1.87090e10i	− 1.49034i
$769$	4.41529e9	0.350120	0.175060	−	0.984558i	$-0.443988\pi$
$769$	4.41529e9	0.350120	0.175060	+	0.984558i	$0.443988\pi$
$770$	0	0
$771$	−2.62152e10	−2.05998
$772$	1.07816e10i	0.843381i
$773$	5.87300e8i	0.0457332i	0.999739	+	0.0228666i	$0.00727930\pi$
$773$	5.87300e8i	0.0457332i	−0.999739	+	0.0228666i	$0.992721\pi$
$774$	8.82017e8	0.0683728
$775$	0	0
$776$	8.45338e8	0.0649403
$777$	1.21939e10i	0.932546i
$778$	− 1.83146e9i	− 0.139434i
$779$	−1.39082e10	−1.05412
$780$	0	0
$781$	−1.11929e9	−0.0840746
$782$	2.51195e8i	0.0187840i
$783$	3.26527e10i	2.43082i
$784$	8.41160e9	0.623408
$785$	0	0
$786$	3.13914e9	0.230585
$787$	− 2.30069e10i	− 1.68247i	−0.540670	−	0.841235i	$-0.681829\pi$
$787$	− 2.30069e10i	− 1.68247i	0.540670	−	0.841235i	$-0.318171\pi$
$788$	1.82967e10i	1.33208i
$789$	−1.47994e10	−1.07269
$790$	0	0
$791$	1.41420e10	1.01600
$792$	1.65328e9i	0.118252i
$793$	1.88303e9i	0.134092i
$794$	2.14323e8	0.0151949
$795$	0	0
$796$	−1.60308e9	−0.112658
$797$	2.49526e10i	1.74587i	0.487836	+	0.872935i	$0.337787\pi$
$797$	2.49526e10i	1.74587i	−0.487836	+	0.872935i	$0.662213\pi$
$798$	1.21996e9i	0.0849838i
$799$	−1.50450e9	−0.104346
$800$	0	0
$801$	1.01227e9	0.0695956
$802$	2.97536e9i	0.203671i
$803$	5.17223e9i	0.352512i
$804$	2.97268e10	2.01722
$805$	0	0
$806$	1.61208e9	0.108446
$807$	2.16003e10i	1.44678i
$808$	1.20399e9i	0.0802942i
$809$	1.29758e10	0.861616	0.430808	−	0.902443i	$-0.358229\pi$
$809$	1.29758e10	0.861616	0.430808	+	0.902443i	$0.358229\pi$
$810$	0	0
$811$	−2.04887e10	−1.34878	−0.674391	−	0.738374i	$-0.735594\pi$
$811$	−2.04887e10	−1.34878	−0.674391	+	0.738374i	$0.735594\pi$
$812$	− 1.40294e10i	− 0.919587i
$813$	− 1.84000e10i	− 1.20088i
$814$	−4.42679e8	−0.0287676
$815$	0	0
$816$	−4.20649e9	−0.271022
$817$	− 3.65554e9i	− 0.234517i
$818$	1.05748e9i	0.0675514i
$819$	1.58750e10	1.00977
$820$	0	0
$821$	−7.97556e9	−0.502991	−0.251495	−	0.967858i	$-0.580922\pi$
$821$	−7.97556e9	−0.502991	−0.251495	+	0.967858i	$0.580922\pi$
$822$	− 3.23963e9i	− 0.203444i
$823$	− 5.64462e9i	− 0.352968i	−0.984304	−	0.176484i	$-0.943528\pi$
$823$	− 5.64462e9i	− 0.352968i	0.984304	−	0.176484i	$-0.0564724\pi$
$824$	2.83152e9	0.176309
$825$	0	0
$826$	−9.81302e8	−0.0605860
$827$	1.22946e10i	0.755866i	0.925833	+	0.377933i	$0.123365\pi$
$827$	1.22946e10i	0.755866i	−0.925833	+	0.377933i	$0.876635\pi$
$828$	− 3.08445e10i	− 1.88830i
$829$	1.72844e10	1.05369	0.526845	−	0.849962i	$-0.323375\pi$
$829$	1.72844e10	1.05369	0.526845	+	0.849962i	$0.323375\pi$
$830$	0	0
$831$	1.13281e10	0.684784
$832$	− 1.36973e10i	− 0.824521i
$833$	− 1.78752e9i	− 0.107150i
$834$	−2.12243e9	−0.126693
$835$	0	0
$836$	3.40388e9	0.201489
$837$	2.81450e10i	1.65906i
$838$	− 1.05649e9i	− 0.0620173i
$839$	4.87580e9	0.285023	0.142511	−	0.989793i	$-0.454482\pi$
$839$	4.87580e9	0.285023	0.142511	+	0.989793i	$0.454482\pi$
$840$	0	0
$841$	2.53104e10	1.46728
$842$	− 2.71885e8i	− 0.0156962i
$843$	9.19927e9i	0.528879i
$844$	−1.19701e10	−0.685329
$845$	0	0
$846$	−2.40367e9	−0.136483
$847$	9.69328e9i	0.548124i
$848$	1.15044e10i	0.647857i
$849$	−1.73903e10	−0.975279
$850$	0	0
$851$	1.66252e10	0.924727
$852$	− 9.28132e9i	− 0.514128i
$853$	1.67542e10i	0.924277i	0.886808	+	0.462138i	$0.152918\pi$
$853$	1.67542e10i	0.924277i	−0.886808	+	0.462138i	$0.847082\pi$
$854$	−1.83785e8	−0.0100974
$855$	0	0
$856$	−4.96112e9	−0.270347
$857$	− 1.45198e10i	− 0.788001i	−0.919110	−	0.394001i	$-0.871091\pi$
$857$	− 1.45198e10i	− 0.788001i	0.919110	−	0.394001i	$-0.128909\pi$
$858$	8.78435e8i	0.0474793i
$859$	−2.44108e10	−1.31403	−0.657016	−	0.753877i	$-0.728181\pi$
$859$	−2.44108e10	−1.31403	−0.657016	+	0.753877i	$0.728181\pi$
$860$	0	0
$861$	−2.69242e10	−1.43758
$862$	9.53988e8i	0.0507303i
$863$	2.03914e10i	1.07996i	0.841677	+	0.539981i	$0.181569\pi$
$863$	2.03914e10i	1.07996i	−0.841677	+	0.539981i	$0.818431\pi$
$864$	−9.80475e9	−0.517176
$865$	0	0
$866$	−2.48200e9	−0.129864
$867$	− 3.18275e10i	− 1.65858i
$868$	− 1.20927e10i	− 0.627629i
$869$	7.63764e9	0.394812
$870$	0	0
$871$	2.08597e10	1.06966
$872$	5.24568e8i	0.0267913i
$873$	1.08134e10i	0.550061i
$874$	1.66330e9	0.0842713
$875$	0	0
$876$	−4.28888e10	−2.15566
$877$	− 3.75418e10i	− 1.87939i	−0.342018	−	0.939693i	$-0.611110\pi$
$877$	− 3.75418e10i	− 1.87939i	0.342018	−	0.939693i	$-0.388890\pi$
$878$	1.73719e9i	0.0866194i
$879$	−3.28845e10	−1.63316
$880$	0	0
$881$	1.05852e10	0.521533	0.260767	−	0.965402i	$-0.416025\pi$
$881$	1.05852e10	0.521533	0.260767	+	0.965402i	$0.416025\pi$
$882$	− 2.85584e9i	− 0.140150i
$883$	− 3.45129e10i	− 1.68702i	−0.537116	−	0.843509i	$-0.680486\pi$
$883$	− 3.45129e10i	− 1.68702i	0.537116	−	0.843509i	$-0.319514\pi$
$884$	−2.99118e9	−0.145633
$885$	0	0
$886$	1.35894e9	0.0656420
$887$	1.68685e10i	0.811602i	0.913961	+	0.405801i	$0.133007\pi$
$887$	1.68685e10i	0.811602i	−0.913961	+	0.405801i	$0.866993\pi$
$888$	− 7.38926e9i	− 0.354124i
$889$	−1.03261e10	−0.492922
$890$	0	0
$891$	−4.24990e9	−0.201283
$892$	2.88792e10i	1.36241i
$893$	9.96206e9i	0.468133i
$894$	−3.01224e9	−0.140996
$895$	0	0
$896$	5.60434e9	0.260283
$897$	− 3.29904e10i	− 1.52621i
$898$	2.18323e9i	0.100608i
$899$	3.66849e10	1.68395
$900$	0	0
$901$	2.44477e9	0.111353
$902$	− 9.77434e8i	− 0.0443470i
$903$	− 7.07656e9i	− 0.319827i
$904$	−8.56974e9	−0.385814
$905$	0	0
$906$	4.70333e9	0.210115
$907$	− 1.14713e10i	− 0.510492i	−0.966876	−	0.255246i	$-0.917844\pi$
$907$	− 1.14713e10i	− 0.510492i	0.966876	−	0.255246i	$-0.0821564\pi$
$908$	− 3.79810e10i	− 1.68370i
$909$	−1.54012e10	−0.680112
$910$	0	0
$911$	1.62306e10	0.711248	0.355624	−	0.934629i	$-0.384268\pi$
$911$	1.62306e10	0.711248	0.355624	+	0.934629i	$0.384268\pi$
$912$	2.78534e10i	1.21590i
$913$	− 1.11446e10i	− 0.484639i
$914$	4.82661e9	0.209089
$915$	0	0
$916$	1.33107e10	0.572224
$917$	− 1.65236e10i	− 0.707641i
$918$	6.79477e8i	0.0289885i
$919$	4.36259e10	1.85413	0.927065	−	0.374900i	$-0.122323\pi$
$919$	4.36259e10	1.85413	0.927065	+	0.374900i	$0.122323\pi$
$920$	0	0
$921$	1.25979e10	0.531363
$922$	4.60419e9i	0.193462i
$923$	− 6.51283e9i	− 0.272624i
$924$	6.58938e9	0.274785
$925$	0	0
$926$	−3.14972e8	−0.0130357
$927$	3.62201e10i	1.49338i
$928$	1.27797e10i	0.524934i
$929$	−1.67801e10	−0.686656	−0.343328	−	0.939216i	$-0.611554\pi$
$929$	−1.67801e10	−0.686656	−0.343328	+	0.939216i	$0.611554\pi$
$930$	0	0
$931$	−1.18361e10	−0.480713
$932$	− 4.13432e10i	− 1.67282i
$933$	1.34900e10i	0.543782i
$934$	5.19273e8	0.0208536
$935$	0	0
$936$	−9.61993e9	−0.383448
$937$	1.94934e10i	0.774105i	0.922058	+	0.387053i	$0.126507\pi$
$937$	1.94934e10i	0.774105i	−0.922058	+	0.387053i	$0.873493\pi$
$938$	2.03592e9i	0.0805474i
$939$	4.16348e10	1.64107
$940$	0	0
$941$	−4.91523e10	−1.92300	−0.961502	−	0.274797i	$-0.911389\pi$
$941$	−4.91523e10	−1.92300	−0.961502	+	0.274797i	$0.911389\pi$
$942$	1.97333e9i	0.0769167i
$943$	3.67084e10i	1.42552i
$944$	−2.24045e10	−0.866827
$945$	0	0
$946$	2.56902e8	0.00986617
$947$	3.01570e10i	1.15389i	0.816784	+	0.576944i	$0.195755\pi$
$947$	3.01570e10i	1.15389i	−0.816784	+	0.576944i	$0.804245\pi$
$948$	6.33323e10i	2.41433i
$949$	−3.00957e10	−1.14307
$950$	0	0
$951$	3.77405e10	1.42290
$952$	− 5.87681e8i	− 0.0220756i
$953$	1.86916e10i	0.699554i	0.936833	+	0.349777i	$0.113743\pi$
$953$	1.86916e10i	0.699554i	−0.936833	+	0.349777i	$0.886257\pi$
$954$	3.90590e9	0.145647
$955$	0	0
$956$	7.23879e9	0.267956
$957$	1.99899e10i	0.737256i
$958$	− 5.36388e9i	− 0.197106i
$959$	−1.70526e10	−0.624346
$960$	0	0
$961$	4.10799e9	0.149313
$962$	− 2.57582e9i	− 0.0932830i
$963$	− 6.34614e10i	− 2.28990i
$964$	3.59213e10	1.29146
$965$	0	0
$966$	3.21988e9	0.114926
$967$	− 7.61328e9i	− 0.270757i	−0.990794	−	0.135378i	$-0.956775\pi$
$967$	− 7.61328e9i	− 0.270757i	0.990794	−	0.135378i	$-0.0432250\pi$
$968$	− 5.87392e9i	− 0.208144i
$969$	5.91903e9	0.208986
$970$	0	0
$971$	−4.83242e10	−1.69394	−0.846969	−	0.531643i	$-0.821575\pi$
$971$	−4.83242e10	−1.69394	−0.846969	+	0.531643i	$0.821575\pi$
$972$	8.49752e9i	0.296797i
$973$	1.11720e10i	0.388807i
$974$	−4.16829e9	−0.144545
$975$	0	0
$976$	−4.19606e9	−0.144467
$977$	1.42491e9i	0.0488829i	0.999701	+	0.0244414i	$0.00778072\pi$
$977$	1.42491e9i	0.0488829i	−0.999701	+	0.0244414i	$0.992219\pi$
$978$	− 2.40916e9i	− 0.0823531i
$979$	2.94840e8	0.0100426
$980$	0	0
$981$	−6.71014e9	−0.226929
$982$	− 4.32564e9i	− 0.145767i
$983$	− 4.85675e10i	− 1.63083i	−0.578877	−	0.815415i	$-0.696509\pi$
$983$	− 4.85675e10i	− 1.63083i	0.578877	−	0.815415i	$-0.303491\pi$
$984$	1.63155e10	0.545905
$985$	0	0
$986$	8.85647e8	0.0294233
$987$	1.92850e10i	0.638425i
$988$	1.98062e10i	0.653358i
$989$	−9.64818e9	−0.317145
$990$	0	0
$991$	−1.69341e10	−0.552719	−0.276360	−	0.961054i	$-0.589128\pi$
$991$	−1.69341e10	−0.552719	−0.276360	+	0.961054i	$0.589128\pi$
$992$	1.10155e10i	0.358273i
$993$	6.98457e9i	0.226369i
$994$	6.35656e8	0.0205291
$995$	0	0
$996$	9.24128e10	2.96363
$997$	2.36849e10i	0.756900i	0.925622	+	0.378450i	$0.123543\pi$
$997$	2.36849e10i	0.756900i	−0.925622	+	0.378450i	$0.876457\pi$
$998$	− 4.80763e9i	− 0.153100i
$999$	4.49707e10	1.42709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.b.c.24.3		4
3.2	odd	2		225.8.b.m.199.2		4
4.3	odd	2		400.8.c.m.49.4		4
5.2	odd	4		25.8.a.b.1.2		2
5.3	odd	4		5.8.a.b.1.1	✓	2
5.4	even	2	inner	25.8.b.c.24.2		4
15.2	even	4		225.8.a.w.1.1		2
15.8	even	4		45.8.a.h.1.2		2
15.14	odd	2		225.8.b.m.199.3		4
20.3	even	4		80.8.a.g.1.1		2
20.7	even	4		400.8.a.bb.1.2		2
20.19	odd	2		400.8.c.m.49.1		4
35.13	even	4		245.8.a.c.1.1		2
40.3	even	4		320.8.a.u.1.2		2
40.13	odd	4		320.8.a.l.1.1		2
55.43	even	4		605.8.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.b.1.1	✓	2	5.3	odd	4
25.8.a.b.1.2		2	5.2	odd	4
25.8.b.c.24.2		4	5.4	even	2	inner
25.8.b.c.24.3		4	1.1	even	1	trivial
45.8.a.h.1.2		2	15.8	even	4
80.8.a.g.1.1		2	20.3	even	4
225.8.a.w.1.1		2	15.2	even	4
225.8.b.m.199.2		4	3.2	odd	2
225.8.b.m.199.3		4	15.14	odd	2
245.8.a.c.1.1		2	35.13	even	4
320.8.a.l.1.1		2	40.13	odd	4
320.8.a.u.1.2		2	40.3	even	4
400.8.a.bb.1.2		2	20.7	even	4
400.8.c.m.49.1		4	20.19	odd	2
400.8.c.m.49.4		4	4.3	odd	2
605.8.a.d.1.2		2	55.43	even	4

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:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.28220i q^{2} -79.7424i q^{3} +126.356 q^{4} +102.246 q^{6} -538.197i q^{7} +326.136i q^{8} -4171.85 q^{9} +O(q^{10})\)	\(q+1.28220i q^{2} -79.7424i q^{3} +126.356 q^{4} +102.246 q^{6} -538.197i q^{7} +326.136i q^{8} -4171.85 q^{9} -1215.12 q^{11} -10075.9i q^{12} -7070.42i q^{13} +690.077 q^{14} +15755.4 q^{16} -3348.13i q^{17} -5349.15i q^{18} -22169.7 q^{19} -42917.1 q^{21} -1558.03i q^{22} +58513.1i q^{23} +26006.8 q^{24} +9065.71 q^{26} +158276. i q^{27} -68004.4i q^{28} +206301. q^{29} +177822. q^{31} +61947.0i q^{32} +96896.5i q^{33} +4292.98 q^{34} -527138. q^{36} -284128. i q^{37} -28426.0i q^{38} -563812. q^{39} +627353. q^{41} -55028.4i q^{42} +164889. i q^{43} -153538. q^{44} -75025.7 q^{46} -449355. i q^{47} -1.25637e6i q^{48} +533887. q^{49} -266988. q^{51} -893390. i q^{52} +730190. i q^{53} -202942. q^{54} +175525. q^{56} +1.76786e6i q^{57} +264520. i q^{58} -1.42202e6 q^{59} -266326. q^{61} +228004. i q^{62} +2.24527e6i q^{63} +1.93726e6 q^{64} -124241. q^{66} +2.95028e6i q^{67} -423056. i q^{68} +4.66598e6 q^{69} +921138. q^{71} -1.36059e6i q^{72} -4.25657e6i q^{73} +364309. q^{74} -2.80127e6 q^{76} +653973. i q^{77} -722921. i q^{78} -6.28551e6 q^{79} +3.49751e6 q^{81} +804393. i q^{82} +9.17165e6i q^{83} -5.42283e6 q^{84} -211421. q^{86} -1.64510e7i q^{87} -396294. i q^{88} -242643. q^{89} -3.80528e6 q^{91} +7.39348e6i q^{92} -1.41799e7i q^{93} +576164. q^{94} +4.93980e6 q^{96} -2.59198e6i q^{97} +684551. i q^{98} +5.06929e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9}+O(q^{10}) \) 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9	\( 4 q - 192 q^{4} - 2032 q^{6} - 11108 q^{9} + 9088 q^{11} - 15024 q^{14} + 40704 q^{16} - 77520 q^{19} - 138192 q^{21} + 263040 q^{24} - 114032 q^{26} + 144520 q^{29} + 613648 q^{31} + 906736 q^{34} - 439616 q^{36} - 1549456 q^{39} + 528728 q^{41} - 2868224 q^{44} - 2606832 q^{46} + 2330828 q^{49} + 2332688 q^{51} - 2205920 q^{54} + 1898880 q^{56} + 2240240 q^{59} + 4514088 q^{61} + 10293248 q^{64} - 13128704 q^{66} + 1490544 q^{69} + 1243568 q^{71} - 5302544 q^{74} + 1775360 q^{76} - 8666080 q^{79} - 4795756 q^{81} + 796416 q^{84} + 21596368 q^{86} - 12051240 q^{89} - 10704592 q^{91} - 11721584 q^{94} + 26610688 q^{96} - 5781376 q^{99}+O(q^{100}) \) 4 * q - 192 * q^4 - 2032 * q^6 - 11108 * q^9 + 9088 * q^11 - 15024 * q^14 + 40704 * q^16 - 77520 * q^19 - 138192 * q^21 + 263040 * q^24 - 114032 * q^26 + 144520 * q^29 + 613648 * q^31 + 906736 * q^34 - 439616 * q^36 - 1549456 * q^39 + 528728 * q^41 - 2868224 * q^44 - 2606832 * q^46 + 2330828 * q^49 + 2332688 * q^51 - 2205920 * q^54 + 1898880 * q^56 + 2240240 * q^59 + 4514088 * q^61 + 10293248 * q^64 - 13128704 * q^66 + 1490544 * q^69 + 1243568 * q^71 - 5302544 * q^74 + 1775360 * q^76 - 8666080 * q^79 - 4795756 * q^81 + 796416 * q^84 + 21596368 * q^86 - 12051240 * q^89 - 10704592 * q^91 - 11721584 * q^94 + 26610688 * q^96 - 5781376 * q^99

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