LMFDB - Embedded newform 25.8.b.a.24.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	25.24
Dual form			25.8.b.a.24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.0000i q^{2} -48.0000i q^{3} -68.0000 q^{4} +672.000 q^{6} +1644.00i q^{7} +840.000i q^{8} -117.000 q^{9} +O(q^{10})$	\(q+14.0000i q^{2} -48.0000i q^{3} -68.0000 q^{4} +672.000 q^{6} +1644.00i q^{7} +840.000i q^{8} -117.000 q^{9} +172.000 q^{11} +3264.00i q^{12} +3862.00i q^{13} -23016.0 q^{14} -20464.0 q^{16} +12254.0i q^{17} -1638.00i q^{18} +25940.0 q^{19} +78912.0 q^{21} +2408.00i q^{22} +12972.0i q^{23} +40320.0 q^{24} -54068.0 q^{26} -99360.0i q^{27} -111792. i q^{28} +81610.0 q^{29} -156888. q^{31} -178976. i q^{32} -8256.00i q^{33} -171556. q^{34} +7956.00 q^{36} -110126. i q^{37} +363160. i q^{38} +185376. q^{39} +467882. q^{41} +1.10477e6i q^{42} -499208. i q^{43} -11696.0 q^{44} -181608. q^{46} +396884. i q^{47} +982272. i q^{48} -1.87919e6 q^{49} +588192. q^{51} -262616. i q^{52} -1.28050e6i q^{53} +1.39104e6 q^{54} -1.38096e6 q^{56} -1.24512e6i q^{57} +1.14254e6i q^{58} +1.33742e6 q^{59} -923978. q^{61} -2.19643e6i q^{62} -192348. i q^{63} -113728. q^{64} +115584. q^{66} +797304. i q^{67} -833272. i q^{68} +622656. q^{69} +5.10339e6 q^{71} -98280.0i q^{72} -4.26748e6i q^{73} +1.54176e6 q^{74} -1.76392e6 q^{76} +282768. i q^{77} +2.59526e6i q^{78} +960.000 q^{79} -5.02516e6 q^{81} +6.55035e6i q^{82} +6.14083e6i q^{83} -5.36602e6 q^{84} +6.98891e6 q^{86} -3.91728e6i q^{87} +144480. i q^{88} -2.01057e6 q^{89} -6.34913e6 q^{91} -882096. i q^{92} +7.53062e6i q^{93} -5.55638e6 q^{94} -8.59085e6 q^{96} +4.88193e6i q^{97} -2.63087e7i q^{98} -20124.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 136 q^{4} + 1344 q^{6} - 234 q^{9}+O(q^{10}) $ 2 * q - 136 * q^4 + 1344 * q^6 - 234 * q^9	$ 2 q - 136 q^{4} + 1344 q^{6} - 234 q^{9} + 344 q^{11} - 46032 q^{14} - 40928 q^{16} + 51880 q^{19} + 157824 q^{21} + 80640 q^{24} - 108136 q^{26} + 163220 q^{29} - 313776 q^{31} - 343112 q^{34} + 15912 q^{36} + 370752 q^{39} + 935764 q^{41} - 23392 q^{44} - 363216 q^{46} - 3758386 q^{49} + 1176384 q^{51} + 2782080 q^{54} - 2761920 q^{56} + 2674840 q^{59} - 1847956 q^{61} - 227456 q^{64} + 231168 q^{66} + 1245312 q^{69} + 10206784 q^{71} + 3083528 q^{74} - 3527840 q^{76} + 1920 q^{79} - 10050318 q^{81} - 10732032 q^{84} + 13977824 q^{86} - 4021140 q^{89} - 12698256 q^{91} - 11112752 q^{94} - 17181696 q^{96} - 40248 q^{99}+O(q^{100}) $ 2 * q - 136 * q^4 + 1344 * q^6 - 234 * q^9 + 344 * q^11 - 46032 * q^14 - 40928 * q^16 + 51880 * q^19 + 157824 * q^21 + 80640 * q^24 - 108136 * q^26 + 163220 * q^29 - 313776 * q^31 - 343112 * q^34 + 15912 * q^36 + 370752 * q^39 + 935764 * q^41 - 23392 * q^44 - 363216 * q^46 - 3758386 * q^49 + 1176384 * q^51 + 2782080 * q^54 - 2761920 * q^56 + 2674840 * q^59 - 1847956 * q^61 - 227456 * q^64 + 231168 * q^66 + 1245312 * q^69 + 10206784 * q^71 + 3083528 * q^74 - 3527840 * q^76 + 1920 * q^79 - 10050318 * q^81 - 10732032 * q^84 + 13977824 * q^86 - 4021140 * q^89 - 12698256 * q^91 - 11112752 * q^94 - 17181696 * q^96 - 40248 * 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 $

$2$	14.0000i	1.23744i	0.785613	+	0.618718i	$0.212348\pi$
$2$	14.0000i	1.23744i	−0.785613	+	0.618718i	$0.787652\pi$
$3$	− 48.0000i	− 1.02640i	−0.858269	−	0.513200i	$-0.828460\pi$
$3$	− 48.0000i	− 1.02640i	0.858269	−	0.513200i	$-0.171540\pi$
$4$	−68.0000	−0.531250
$5$	0	0
$5$	0	0
$6$	672.000	1.27011
$7$	1644.00i	1.81158i	0.423722	+	0.905792i	$0.360723\pi$
$7$	1644.00i	1.81158i	−0.423722	+	0.905792i	$0.639277\pi$
$8$	840.000i	0.580049i
$9$	−117.000	−0.0534979
$10$	0	0
$11$	172.000	0.0389631	0.0194816	−	0.999810i	$-0.493798\pi$
$11$	172.000	0.0389631	0.0194816	+	0.999810i	$0.493798\pi$
$12$	3264.00i	0.545275i
$13$	3862.00i	0.487540i	0.969833	+	0.243770i	$0.0783843\pi$
$13$	3862.00i	0.487540i	−0.969833	+	0.243770i	$0.921616\pi$
$14$	−23016.0	−2.24172
$15$	0	0
$16$	−20464.0	−1.24902
$17$	12254.0i	0.604932i	0.953160	+	0.302466i	$0.0978099\pi$
$17$	12254.0i	0.604932i	−0.953160	+	0.302466i	$0.902190\pi$
$18$	− 1638.00i	− 0.0662003i
$19$	25940.0	0.867626	0.433813	−	0.901003i	$-0.357168\pi$
$19$	25940.0	0.867626	0.433813	+	0.901003i	$0.357168\pi$
$20$	0	0
$21$	78912.0	1.85941
$22$	2408.00i	0.0482144i
$23$	12972.0i	0.222310i	0.993803	+	0.111155i	$0.0354550\pi$
$23$	12972.0i	0.222310i	−0.993803	+	0.111155i	$0.964545\pi$
$24$	40320.0	0.595362
$25$	0	0
$26$	−54068.0	−0.603300
$27$	− 99360.0i	− 0.971490i
$28$	− 111792.i	− 0.962404i
$29$	81610.0	0.621370	0.310685	−	0.950513i	$-0.399442\pi$
$29$	81610.0	0.621370	0.310685	+	0.950513i	$0.399442\pi$
$30$	0	0
$31$	−156888.	−0.945853	−0.472927	−	0.881102i	$-0.656802\pi$
$31$	−156888.	−0.945853	−0.472927	+	0.881102i	$0.656802\pi$
$32$	− 178976.i	− 0.965539i
$33$	− 8256.00i	− 0.0399918i
$34$	−171556.	−0.748565
$35$	0	0
$36$	7956.00	0.0284208
$37$	− 110126.i	− 0.357424i	−0.983901	−	0.178712i	$-0.942807\pi$
$37$	− 110126.i	− 0.357424i	0.983901	−	0.178712i	$-0.0571931\pi$
$38$	363160.i	1.07363i
$39$	185376.	0.500412
$40$	0	0
$41$	467882.	1.06021	0.530106	−	0.847931i	$-0.322152\pi$
$41$	467882.	1.06021	0.530106	+	0.847931i	$0.322152\pi$
$42$	1.10477e6i	2.30090i
$43$	− 499208.i	− 0.957507i	−0.877949	−	0.478753i	$-0.841089\pi$
$43$	− 499208.i	− 0.957507i	0.877949	−	0.478753i	$-0.158911\pi$
$44$	−11696.0	−0.0206992
$45$	0	0
$46$	−181608.	−0.275095
$47$	396884.i	0.557598i	0.960349	+	0.278799i	$0.0899363\pi$
$47$	396884.i	0.557598i	−0.960349	+	0.278799i	$0.910064\pi$
$48$	982272.i	1.28200i
$49$	−1.87919e6	−2.28184
$50$	0	0
$51$	588192.	0.620903
$52$	− 262616.i	− 0.259006i
$53$	− 1.28050e6i	− 1.18144i	−0.806875	−	0.590722i	$-0.798843\pi$
$53$	− 1.28050e6i	− 1.18144i	0.806875	−	0.590722i	$-0.201157\pi$
$54$	1.39104e6	1.20216
$55$	0	0
$56$	−1.38096e6	−1.05081
$57$	− 1.24512e6i	− 0.890531i
$58$	1.14254e6i	0.768906i
$59$	1.33742e6	0.847785	0.423893	−	0.905712i	$-0.360663\pi$
$59$	1.33742e6	0.847785	0.423893	+	0.905712i	$0.360663\pi$
$60$	0	0
$61$	−923978.	−0.521203	−0.260602	−	0.965446i	$-0.583921\pi$
$61$	−923978.	−0.521203	−0.260602	+	0.965446i	$0.583921\pi$
$62$	− 2.19643e6i	− 1.17043i
$63$	− 192348.i	− 0.0969161i
$64$	−113728.	−0.0542297
$65$	0	0
$66$	115584.	0.0494873
$67$	797304.i	0.323864i	0.986802	+	0.161932i	$0.0517725\pi$
$67$	797304.i	0.323864i	−0.986802	+	0.161932i	$0.948228\pi$
$68$	− 833272.i	− 0.321370i
$69$	622656.	0.228179
$70$	0	0
$71$	5.10339e6	1.69221	0.846106	−	0.533015i	$-0.178941\pi$
$71$	5.10339e6	1.69221	0.846106	+	0.533015i	$0.178941\pi$
$72$	− 98280.0i	− 0.0310314i
$73$	− 4.26748e6i	− 1.28393i	−0.766734	−	0.641965i	$-0.778119\pi$
$73$	− 4.26748e6i	− 1.28393i	0.766734	−	0.641965i	$-0.221881\pi$
$74$	1.54176e6	0.442290
$75$	0	0
$76$	−1.76392e6	−0.460926
$77$	282768.i	0.0705850i
$78$	2.59526e6i	0.619228i
$79$	960.000	0.000219067 0	0.000109533	−	1.00000i	$-0.499965\pi$
$79$	960.000	0.000219067 0	0.000109533		1.00000i	$0.499965\pi$
$80$	0	0
$81$	−5.02516e6	−1.05064
$82$	6.55035e6i	1.31195i
$83$	6.14083e6i	1.17884i	0.807828	+	0.589419i	$0.200643\pi$
$83$	6.14083e6i	1.17884i	−0.807828	+	0.589419i	$0.799357\pi$
$84$	−5.36602e6	−0.987812
$85$	0	0
$86$	6.98891e6	1.18485
$87$	− 3.91728e6i	− 0.637775i
$88$	144480.i	0.0226005i
$89$	−2.01057e6	−0.302311	−0.151156	−	0.988510i	$-0.548299\pi$
$89$	−2.01057e6	−0.302311	−0.151156	+	0.988510i	$0.548299\pi$
$90$	0	0
$91$	−6.34913e6	−0.883221
$92$	− 882096.i	− 0.118102i
$93$	7.53062e6i	0.970824i
$94$	−5.55638e6	−0.689992
$95$	0	0
$96$	−8.59085e6	−0.991030
$97$	4.88193e6i	0.543114i	0.962422	+	0.271557i	$0.0875385\pi$
$97$	4.88193e6i	0.543114i	−0.962422	+	0.271557i	$0.912461\pi$
$98$	− 2.63087e7i	− 2.82363i
$99$	−20124.0	−0.00208445
$100$	0	0
$101$	9.72670e6	0.939379	0.469689	−	0.882832i	$-0.344366\pi$
$101$	9.72670e6	0.939379	0.469689	+	0.882832i	$0.344366\pi$
$102$	8.23469e6i	0.768328i
$103$	1.63151e7i	1.47115i	0.677441	+	0.735577i	$0.263089\pi$
$103$	1.63151e7i	1.47115i	−0.677441	+	0.735577i	$0.736911\pi$
$104$	−3.24408e6	−0.282797
$105$	0	0
$106$	1.79270e7	1.46196
$107$	4.08974e6i	0.322740i	0.986894	+	0.161370i	$0.0515913\pi$
$107$	4.08974e6i	0.322740i	−0.986894	+	0.161370i	$0.948409\pi$
$108$	6.75648e6i	0.516104i
$109$	2.68318e7	1.98453	0.992263	−	0.124158i	$-0.0396228\pi$
$109$	2.68318e7	1.98453	0.992263	+	0.124158i	$0.0396228\pi$
$110$	0	0
$111$	−5.28605e6	−0.366860
$112$	− 3.36428e7i	− 2.26271i
$113$	− 1.74810e7i	− 1.13971i	−0.821747	−	0.569853i	$-0.807000\pi$
$113$	− 1.74810e7i	− 1.13971i	0.821747	−	0.569853i	$-0.193000\pi$
$114$	1.74317e7	1.10198
$115$	0	0
$116$	−5.54948e6	−0.330103
$117$	− 451854.i	− 0.0260824i
$118$	1.87239e7i	1.04908i
$119$	−2.01456e7	−1.09589
$120$	0	0
$121$	−1.94576e7	−0.998482
$122$	− 1.29357e7i	− 0.644956i
$123$	− 2.24583e7i	− 1.08820i
$124$	1.06684e7	0.502485
$125$	0	0
$126$	2.69287e6	0.119928
$127$	1.25018e7i	0.541575i	0.962639	+	0.270787i	$0.0872840\pi$
$127$	1.25018e7i	0.541575i	−0.962639	+	0.270787i	$0.912716\pi$
$128$	− 2.45011e7i	− 1.03264i
$129$	−2.39620e7	−0.982786
$130$	0	0
$131$	−7.75619e6	−0.301439	−0.150719	−	0.988577i	$-0.548159\pi$
$131$	−7.75619e6	−0.301439	−0.150719	+	0.988577i	$0.548159\pi$
$132$	561408.i	0.0212456i
$133$	4.26454e7i	1.57178i
$134$	−1.11623e7	−0.400761
$135$	0	0
$136$	−1.02934e7	−0.350890
$137$	− 3.61720e7i	− 1.20185i	−0.799305	−	0.600926i	$-0.794799\pi$
$137$	− 3.61720e7i	− 1.20185i	0.799305	−	0.600926i	$-0.205201\pi$
$138$	8.71718e6i	0.282358i
$139$	−1.09092e7	−0.344542	−0.172271	−	0.985050i	$-0.555110\pi$
$139$	−1.09092e7	−0.344542	−0.172271	+	0.985050i	$0.555110\pi$
$140$	0	0
$141$	1.90504e7	0.572319
$142$	7.14475e7i	2.09401i
$143$	664264.i	0.0189961i
$144$	2.39429e6	0.0668202
$145$	0	0
$146$	5.97447e7	1.58878
$147$	9.02013e7i	2.34208i
$148$	7.48857e6i	0.189882i
$149$	3.64580e7	0.902904	0.451452	−	0.892295i	$-0.350906\pi$
$149$	3.64580e7	0.902904	0.451452	+	0.892295i	$0.350906\pi$
$150$	0	0
$151$	7.18955e6	0.169935	0.0849674	−	0.996384i	$-0.472921\pi$
$151$	7.18955e6	0.169935	0.0849674	+	0.996384i	$0.472921\pi$
$152$	2.17896e7i	0.503265i
$153$	− 1.43372e6i	− 0.0323626i
$154$	−3.95875e6	−0.0873445
$155$	0	0
$156$	−1.26056e7	−0.265844
$157$	− 8.79932e7i	− 1.81468i	−0.420396	−	0.907341i	$-0.638109\pi$
$157$	− 8.79932e7i	− 1.81468i	0.420396	−	0.907341i	$-0.361891\pi$
$158$	13440.0i	0 0.000271081i
$159$	−6.14639e7	−1.21264
$160$	0	0
$161$	−2.13260e7	−0.402734
$162$	− 7.03522e7i	− 1.30010i
$163$	− 5.48875e7i	− 0.992697i	−0.868123	−	0.496349i	$-0.834674\pi$
$163$	− 5.48875e7i	− 0.992697i	0.868123	−	0.496349i	$-0.165326\pi$
$164$	−3.18160e7	−0.563238
$165$	0	0
$166$	−8.59716e7	−1.45874
$167$	− 8.61460e6i	− 0.143129i	−0.997436	−	0.0715644i	$-0.977201\pi$
$167$	− 8.61460e6i	− 0.143129i	0.997436	−	0.0715644i	$-0.0227992\pi$
$168$	6.62861e7i	1.07855i
$169$	4.78335e7	0.762304
$170$	0	0
$171$	−3.03498e6	−0.0464162
$172$	3.39461e7i	0.508676i
$173$	− 5.12524e7i	− 0.752580i	−0.926502	−	0.376290i	$-0.877200\pi$
$173$	− 5.12524e7i	− 0.752580i	0.926502	−	0.376290i	$-0.122800\pi$
$174$	5.48419e7	0.789206
$175$	0	0
$176$	−3.51981e6	−0.0486659
$177$	− 6.41962e7i	− 0.870167i
$178$	− 2.81480e7i	− 0.374091i
$179$	5.01627e7	0.653725	0.326862	−	0.945072i	$-0.394009\pi$
$179$	5.01627e7	0.653725	0.326862	+	0.945072i	$0.394009\pi$
$180$	0	0
$181$	6.90817e7	0.865940	0.432970	−	0.901408i	$-0.357466\pi$
$181$	6.90817e7	0.865940	0.432970	+	0.901408i	$0.357466\pi$
$182$	− 8.88878e7i	− 1.09293i
$183$	4.43509e7i	0.534963i
$184$	−1.08965e7	−0.128951
$185$	0	0
$186$	−1.05429e8	−1.20133
$187$	2.10769e6i	0.0235701i
$188$	− 2.69881e7i	− 0.296224i
$189$	1.63348e8	1.75994
$190$	0	0
$191$	1.54745e8	1.60695	0.803473	−	0.595342i	$-0.202983\pi$
$191$	1.54745e8	1.60695	0.803473	+	0.595342i	$0.202983\pi$
$192$	5.45894e6i	0.0556614i
$193$	− 1.59406e7i	− 0.159607i	−0.996811	−	0.0798037i	$-0.974571\pi$
$193$	− 1.59406e7i	− 0.159607i	0.996811	−	0.0798037i	$-0.0254293\pi$
$194$	−6.83471e7	−0.672069
$195$	0	0
$196$	1.27785e8	1.21223
$197$	1.68188e8i	1.56734i	0.621177	+	0.783670i	$0.286655\pi$
$197$	1.68188e8i	1.56734i	−0.621177	+	0.783670i	$0.713345\pi$
$198$	− 281736.i	− 0.00257937i
$199$	−1.77773e8	−1.59911	−0.799556	−	0.600591i	$-0.794932\pi$
$199$	−1.77773e8	−1.59911	−0.799556	+	0.600591i	$0.794932\pi$
$200$	0	0
$201$	3.82706e7	0.332414
$202$	1.36174e8i	1.16242i
$203$	1.34167e8i	1.12566i
$204$	−3.99971e7	−0.329855
$205$	0	0
$206$	−2.28411e8	−1.82046
$207$	− 1.51772e6i	− 0.0118931i
$208$	− 7.90320e7i	− 0.608949i
$209$	4.46168e6	0.0338054
$210$	0	0
$211$	−1.61996e8	−1.18718	−0.593590	−	0.804767i	$-0.702290\pi$
$211$	−1.61996e8	−1.18718	−0.593590	+	0.804767i	$0.702290\pi$
$212$	8.70739e7i	0.627642i
$213$	− 2.44963e8i	− 1.73689i
$214$	−5.72564e7	−0.399370
$215$	0	0
$216$	8.34624e7	0.563511
$217$	− 2.57924e8i	− 1.71349i
$218$	3.75645e8i	2.45572i
$219$	−2.04839e8	−1.31783
$220$	0	0
$221$	−4.73249e7	−0.294929
$222$	− 7.40047e7i	− 0.453966i
$223$	1.75932e7i	0.106237i	0.998588	+	0.0531187i	$0.0169162\pi$
$223$	1.75932e7i	0.106237i	−0.998588	+	0.0531187i	$0.983084\pi$
$224$	2.94237e8	1.74916
$225$	0	0
$226$	2.44735e8	1.41031
$227$	2.03036e8i	1.15208i	0.817422	+	0.576039i	$0.195403\pi$
$227$	2.03036e8i	1.15208i	−0.817422	+	0.576039i	$0.804597\pi$
$228$	8.46682e7i	0.473095i
$229$	−1.59559e8	−0.878005	−0.439003	−	0.898486i	$-0.644668\pi$
$229$	−1.59559e8	−0.878005	−0.439003	+	0.898486i	$0.644668\pi$
$230$	0	0
$231$	1.35729e7	0.0724485
$232$	6.85524e7i	0.360425i
$233$	− 1.94985e7i	− 0.100985i	−0.998724	−	0.0504924i	$-0.983921\pi$
$233$	− 1.94985e7i	− 0.100985i	0.998724	−	0.0504924i	$-0.0160791\pi$
$234$	6.32596e6	0.0322753
$235$	0	0
$236$	−9.09446e7	−0.450386
$237$	− 46080.0i	0 0.000224850i
$238$	− 2.82038e8i	− 1.35609i
$239$	1.60220e8	0.759146	0.379573	−	0.925162i	$-0.376071\pi$
$239$	1.60220e8	0.759146	0.379573	+	0.925162i	$0.376071\pi$
$240$	0	0
$241$	−3.78779e8	−1.74311	−0.871557	−	0.490294i	$-0.836890\pi$
$241$	−3.78779e8	−1.74311	−0.871557	+	0.490294i	$0.836890\pi$
$242$	− 2.72406e8i	− 1.23556i
$243$	2.39073e7i	0.106883i
$244$	6.28305e7	0.276889
$245$	0	0
$246$	3.14417e8	1.34658
$247$	1.00180e8i	0.423002i
$248$	− 1.31786e8i	− 0.548641i
$249$	2.94760e8	1.20996
$250$	0	0
$251$	1.61304e8	0.643855	0.321927	−	0.946764i	$-0.395669\pi$
$251$	1.61304e8	0.643855	0.321927	+	0.946764i	$0.395669\pi$
$252$	1.30797e7i	0.0514867i
$253$	2.23118e6i	0.00866191i
$254$	−1.75025e8	−0.670164
$255$	0	0
$256$	3.28458e8	1.22360
$257$	− 2.27387e8i	− 0.835603i	−0.908538	−	0.417801i	$-0.862801\pi$
$257$	− 2.27387e8i	− 0.835603i	0.908538	−	0.417801i	$-0.137199\pi$
$258$	− 3.35468e8i	− 1.21614i
$259$	1.81047e8	0.647504
$260$	0	0
$261$	−9.54837e6	−0.0332420
$262$	− 1.08587e8i	− 0.373011i
$263$	4.57728e8i	1.55154i	0.631018	+	0.775768i	$0.282637\pi$
$263$	4.57728e8i	1.55154i	−0.631018	+	0.775768i	$0.717363\pi$
$264$	6.93504e6	0.0231972
$265$	0	0
$266$	−5.97035e8	−1.94498
$267$	9.65074e7i	0.310292i
$268$	− 5.42167e7i	− 0.172053i
$269$	−4.67286e8	−1.46369	−0.731847	−	0.681469i	$-0.761341\pi$
$269$	−4.67286e8	−1.46369	−0.731847	+	0.681469i	$0.761341\pi$
$270$	0	0
$271$	−4.45932e7	−0.136106	−0.0680528	−	0.997682i	$-0.521679\pi$
$271$	−4.45932e7	−0.136106	−0.0680528	+	0.997682i	$0.521679\pi$
$272$	− 2.50766e8i	− 0.755574i
$273$	3.04758e8i	0.906538i
$274$	5.06408e8	1.48722
$275$	0	0
$276$	−4.23406e7	−0.121220
$277$	− 3.16657e8i	− 0.895179i	−0.894239	−	0.447590i	$-0.852283\pi$
$277$	− 3.16657e8i	− 0.895179i	0.894239	−	0.447590i	$-0.147717\pi$
$278$	− 1.52729e8i	− 0.426349i
$279$	1.83559e7	0.0506012
$280$	0	0
$281$	−2.25818e8	−0.607136	−0.303568	−	0.952810i	$-0.598178\pi$
$281$	−2.25818e8	−0.607136	−0.303568	+	0.952810i	$0.598178\pi$
$282$	2.66706e8i	0.708208i
$283$	2.08210e7i	0.0546072i	0.999627	+	0.0273036i	$0.00869208\pi$
$283$	2.08210e7i	0.0546072i	−0.999627	+	0.0273036i	$0.991308\pi$
$284$	−3.47031e8	−0.898987
$285$	0	0
$286$	−9.29970e6	−0.0235065
$287$	7.69198e8i	1.92066i
$288$	2.09402e7i	0.0516544i
$289$	2.60178e8	0.634057
$290$	0	0
$291$	2.34333e8	0.557452
$292$	2.90189e8i	0.682088i
$293$	− 1.78825e8i	− 0.415329i	−0.978200	−	0.207665i	$-0.933414\pi$
$293$	− 1.78825e8i	− 0.415329i	0.978200	−	0.207665i	$-0.0665863\pi$
$294$	−1.26282e9	−2.89818
$295$	0	0
$296$	9.25058e7	0.207323
$297$	− 1.70899e7i	− 0.0378523i
$298$	5.10413e8i	1.11729i
$299$	−5.00979e7	−0.108385
$300$	0	0
$301$	8.20698e8	1.73461
$302$	1.00654e8i	0.210284i
$303$	− 4.66882e8i	− 0.964179i
$304$	−5.30836e8	−1.08368
$305$	0	0
$306$	2.00721e7	0.0400467
$307$	8.55159e7i	0.168680i	0.996437	+	0.0843398i	$0.0268781\pi$
$307$	8.55159e7i	0.168680i	−0.996437	+	0.0843398i	$0.973122\pi$
$308$	− 1.92282e7i	− 0.0374983i
$309$	7.83122e8	1.50999
$310$	0	0
$311$	−4.84706e8	−0.913728	−0.456864	−	0.889537i	$-0.651027\pi$
$311$	−4.84706e8	−0.913728	−0.456864	+	0.889537i	$0.651027\pi$
$312$	1.55716e8i	0.290263i
$313$	5.70821e8i	1.05219i	0.850425	+	0.526096i	$0.176345\pi$
$313$	5.70821e8i	1.05219i	−0.850425	+	0.526096i	$0.823655\pi$
$314$	1.23190e9	2.24555
$315$	0	0
$316$	−65280.0	−0.000116379 0
$317$	5.50191e8i	0.970076i	0.874493	+	0.485038i	$0.161194\pi$
$317$	5.50191e8i	0.970076i	−0.874493	+	0.485038i	$0.838806\pi$
$318$	− 8.60495e8i	− 1.50056i
$319$	1.40369e7	0.0242105
$320$	0	0
$321$	1.96308e8	0.331261
$322$	− 2.98564e8i	− 0.498358i
$323$	3.17869e8i	0.524855i
$324$	3.41711e8	0.558150
$325$	0	0
$326$	7.68425e8	1.22840
$327$	− 1.28792e9i	− 2.03692i
$328$	3.93021e8i	0.614975i
$329$	−6.52477e8	−1.01014
$330$	0	0
$331$	−9.39839e8	−1.42448	−0.712238	−	0.701938i	$-0.752319\pi$
$331$	−9.39839e8	−1.42448	−0.712238	+	0.701938i	$0.752319\pi$
$332$	− 4.17577e8i	− 0.626257i
$333$	1.28847e7i	0.0191215i
$334$	1.20604e8	0.177113
$335$	0	0
$336$	−1.61486e9	−2.32245
$337$	− 5.33632e8i	− 0.759516i	−0.925086	−	0.379758i	$-0.876007\pi$
$337$	− 5.33632e8i	− 0.759516i	0.925086	−	0.379758i	$-0.123993\pi$
$338$	6.69669e8i	0.943304i
$339$	−8.39090e8	−1.16979
$340$	0	0
$341$	−2.69847e7	−0.0368534
$342$	− 4.24897e7i	− 0.0574371i
$343$	− 1.73549e9i	− 2.32216i
$344$	4.19335e8	0.555401
$345$	0	0
$346$	7.17533e8	0.931270
$347$	− 1.07934e9i	− 1.38677i	−0.720567	−	0.693385i	$-0.756118\pi$
$347$	− 1.07934e9i	− 1.38677i	0.720567	−	0.693385i	$-0.243882\pi$
$348$	2.66375e8i	0.338818i
$349$	4.27217e8	0.537972	0.268986	−	0.963144i	$-0.413311\pi$
$349$	4.27217e8	0.537972	0.268986	+	0.963144i	$0.413311\pi$
$350$	0	0
$351$	3.83728e8	0.473641
$352$	− 3.07839e7i	− 0.0376204i
$353$	− 1.48966e9i	− 1.80250i	−0.433299	−	0.901250i	$-0.642650\pi$
$353$	− 1.48966e9i	− 1.80250i	0.433299	−	0.901250i	$-0.357350\pi$
$354$	8.98746e8	1.07678
$355$	0	0
$356$	1.36719e8	0.160603
$357$	9.66988e8i	1.12482i
$358$	7.02277e8i	0.808943i
$359$	−8.41275e8	−0.959638	−0.479819	−	0.877367i	$-0.659298\pi$
$359$	−8.41275e8	−0.959638	−0.479819	+	0.877367i	$0.659298\pi$
$360$	0	0
$361$	−2.20988e8	−0.247226
$362$	9.67143e8i	1.07155i
$363$	9.33964e8i	1.02484i
$364$	4.31741e8	0.469211
$365$	0	0
$366$	−6.20913e8	−0.661983
$367$	7.50462e8i	0.792496i	0.918143	+	0.396248i	$0.129688\pi$
$367$	7.50462e8i	0.792496i	−0.918143	+	0.396248i	$0.870312\pi$
$368$	− 2.65459e8i	− 0.277671i
$369$	−5.47422e7	−0.0567192
$370$	0	0
$371$	2.10514e9	2.14029
$372$	− 5.12082e8i	− 0.515750i
$373$	1.71074e8i	0.170688i	0.996352	+	0.0853439i	$0.0271989\pi$
$373$	1.71074e8i	0.170688i	−0.996352	+	0.0853439i	$0.972801\pi$
$374$	−2.95076e7	−0.0291665
$375$	0	0
$376$	−3.33383e8	−0.323434
$377$	3.15178e8i	0.302943i
$378$	2.28687e9i	2.17781i
$379$	−4.66239e7	−0.0439918	−0.0219959	−	0.999758i	$-0.507002\pi$
$379$	−4.66239e7	−0.0439918	−0.0219959	+	0.999758i	$0.507002\pi$
$380$	0	0
$381$	6.00085e8	0.555872
$382$	2.16644e9i	1.98849i
$383$	− 4.42266e8i	− 0.402242i	−0.979566	−	0.201121i	$-0.935542\pi$
$383$	− 4.42266e8i	− 0.402242i	0.979566	−	0.201121i	$-0.0644585\pi$
$384$	−1.17605e9	−1.05991
$385$	0	0
$386$	2.23168e8	0.197504
$387$	5.84073e7i	0.0512247i
$388$	− 3.31972e8i	− 0.288529i
$389$	4.64033e8	0.399691	0.199846	−	0.979827i	$-0.435956\pi$
$389$	4.64033e8	0.399691	0.199846	+	0.979827i	$0.435956\pi$
$390$	0	0
$391$	−1.58959e8	−0.134483
$392$	− 1.57852e9i	− 1.32358i
$393$	3.72297e8i	0.309397i
$394$	−2.35463e9	−1.93948
$395$	0	0
$396$	1.36843e6	0.00110736
$397$	3.17792e8i	0.254904i	0.991845	+	0.127452i	$0.0406799\pi$
$397$	3.17792e8i	0.254904i	−0.991845	+	0.127452i	$0.959320\pi$
$398$	− 2.48882e9i	− 1.97880i
$399$	2.04698e9	1.61327
$400$	0	0
$401$	−1.19563e9	−0.925958	−0.462979	−	0.886369i	$-0.653219\pi$
$401$	−1.19563e9	−0.925958	−0.462979	+	0.886369i	$0.653219\pi$
$402$	5.35788e8i	0.411341i
$403$	− 6.05901e8i	− 0.461142i
$404$	−6.61416e8	−0.499045
$405$	0	0
$406$	−1.87834e9	−1.39294
$407$	− 1.89417e7i	− 0.0139264i
$408$	4.94081e8i	0.360154i
$409$	−2.21305e9	−1.59941	−0.799704	−	0.600395i	$-0.795010\pi$
$409$	−2.21305e9	−1.59941	−0.799704	+	0.600395i	$0.795010\pi$
$410$	0	0
$411$	−1.73626e9	−1.23358
$412$	− 1.10942e9i	− 0.781551i
$413$	2.19872e9i	1.53583i
$414$	2.12481e7	0.0147170
$415$	0	0
$416$	6.91205e8	0.470739
$417$	5.23643e8i	0.353638i
$418$	6.24635e7i	0.0418321i
$419$	8.02299e8	0.532828	0.266414	−	0.963859i	$-0.414161\pi$
$419$	8.02299e8	0.532828	0.266414	+	0.963859i	$0.414161\pi$
$420$	0	0
$421$	3.44713e7	0.0225149	0.0112575	−	0.999937i	$-0.496417\pi$
$421$	3.44713e7	0.0225149	0.0112575	+	0.999937i	$0.496417\pi$
$422$	− 2.26795e9i	− 1.46906i
$423$	− 4.64354e7i	− 0.0298303i
$424$	1.07562e9	0.685295
$425$	0	0
$426$	3.42948e9	2.14929
$427$	− 1.51902e9i	− 0.944204i
$428$	− 2.78103e8i	− 0.171456i
$429$	3.18847e7	0.0194976
$430$	0	0
$431$	1.72692e9	1.03897	0.519485	−	0.854480i	$-0.326124\pi$
$431$	1.72692e9	1.03897	0.519485	+	0.854480i	$0.326124\pi$
$432$	2.03330e9i	1.21341i
$433$	4.88308e8i	0.289059i	0.989501	+	0.144529i	$0.0461668\pi$
$433$	4.88308e8i	0.289059i	−0.989501	+	0.144529i	$0.953833\pi$
$434$	3.61093e9	2.12034
$435$	0	0
$436$	−1.82456e9	−1.05428
$437$	3.36494e8i	0.192882i
$438$	− 2.86775e9i	− 1.63073i
$439$	2.88640e9	1.62828	0.814142	−	0.580665i	$-0.197207\pi$
$439$	2.88640e9	1.62828	0.814142	+	0.580665i	$0.197207\pi$
$440$	0	0
$441$	2.19866e8	0.122074
$442$	− 6.62549e8i	− 0.364956i
$443$	9.26583e8i	0.506374i	0.967417	+	0.253187i	$0.0814788\pi$
$443$	9.26583e8i	0.506374i	−0.967417	+	0.253187i	$0.918521\pi$
$444$	3.59451e8	0.194895
$445$	0	0
$446$	−2.46304e8	−0.131462
$447$	− 1.74999e9i	− 0.926741i
$448$	− 1.86969e8i	− 0.0982418i
$449$	−1.35535e9	−0.706627	−0.353313	−	0.935505i	$-0.614945\pi$
$449$	−1.35535e9	−0.706627	−0.353313	+	0.935505i	$0.614945\pi$
$450$	0	0
$451$	8.04757e7	0.0413092
$452$	1.18871e9i	0.605469i
$453$	− 3.45098e8i	− 0.174421i
$454$	−2.84250e9	−1.42563
$455$	0	0
$456$	1.04590e9	0.516551
$457$	− 4.63429e7i	− 0.0227131i	−0.999936	−	0.0113566i	$-0.996385\pi$
$457$	− 4.63429e7i	− 0.0227131i	0.999936	−	0.0113566i	$-0.00361498\pi$
$458$	− 2.23383e9i	− 1.08648i
$459$	1.21756e9	0.587686
$460$	0	0
$461$	−1.52117e8	−0.0723144	−0.0361572	−	0.999346i	$-0.511512\pi$
$461$	−1.52117e8	−0.0723144	−0.0361572	+	0.999346i	$0.511512\pi$
$462$	1.90020e8i	0.0896505i
$463$	1.63450e9i	0.765337i	0.923886	+	0.382668i	$0.124995\pi$
$463$	1.63450e9i	0.765337i	−0.923886	+	0.382668i	$0.875005\pi$
$464$	−1.67007e9	−0.776106
$465$	0	0
$466$	2.72979e8	0.124962
$467$	1.11380e9i	0.506057i	0.967459	+	0.253029i	$0.0814267\pi$
$467$	1.11380e9i	0.506057i	−0.967459	+	0.253029i	$0.918573\pi$
$468$	3.07261e7i	0.0138563i
$469$	−1.31077e9	−0.586706
$470$	0	0
$471$	−4.22367e9	−1.86259
$472$	1.12343e9i	0.491756i
$473$	− 8.58638e7i	− 0.0373075i
$474$	645120.	0.000278238 0
$475$	0	0
$476$	1.36990e9	0.582189
$477$	1.49818e8i	0.0632049i
$478$	2.24309e9i	0.939395i
$479$	−1.27745e9	−0.531091	−0.265546	−	0.964098i	$-0.585552\pi$
$479$	−1.27745e9	−0.531091	−0.265546	+	0.964098i	$0.585552\pi$
$480$	0	0
$481$	4.25307e8	0.174259
$482$	− 5.30290e9i	− 2.15699i
$483$	1.02365e9i	0.413366i
$484$	1.32312e9	0.530443
$485$	0	0
$486$	−3.34702e8	−0.132261
$487$	− 9.79673e8i	− 0.384352i	−0.981360	−	0.192176i	$-0.938445\pi$
$487$	− 9.79673e8i	− 0.384352i	0.981360	−	0.192176i	$-0.0615545\pi$
$488$	− 7.76142e8i	− 0.302323i
$489$	−2.63460e9	−1.01890
$490$	0	0
$491$	−4.92125e9	−1.87625	−0.938124	−	0.346298i	$-0.887438\pi$
$491$	−4.92125e9	−1.87625	−0.938124	+	0.346298i	$0.887438\pi$
$492$	1.52717e9i	0.578108i
$493$	1.00005e9i	0.375887i
$494$	−1.40252e9	−0.523439
$495$	0	0
$496$	3.21056e9	1.18139
$497$	8.38998e9i	3.06559i
$498$	4.12664e9i	1.49725i
$499$	3.65786e9	1.31788	0.658940	−	0.752196i	$-0.271005\pi$
$499$	3.65786e9	1.31788	0.658940	+	0.752196i	$0.271005\pi$
$500$	0	0
$501$	−4.13501e8	−0.146908
$502$	2.25826e9i	0.796729i
$503$	− 3.88358e9i	− 1.36064i	−0.732914	−	0.680322i	$-0.761840\pi$
$503$	− 3.88358e9i	− 1.36064i	0.732914	−	0.680322i	$-0.238160\pi$
$504$	1.61572e8	0.0562160
$505$	0	0
$506$	−3.12366e7	−0.0107186
$507$	− 2.29601e9i	− 0.782430i
$508$	− 8.50120e8i	− 0.287711i
$509$	−3.90072e9	−1.31109	−0.655545	−	0.755156i	$-0.727561\pi$
$509$	−3.90072e9	−1.31109	−0.655545	+	0.755156i	$0.727561\pi$
$510$	0	0
$511$	7.01573e9	2.32595
$512$	1.46228e9i	0.481487i
$513$	− 2.57740e9i	− 0.842890i
$514$	3.18342e9	1.03401
$515$	0	0
$516$	1.62941e9	0.522105
$517$	6.82640e7i	0.0217258i
$518$	2.53466e9i	0.801245i
$519$	−2.46011e9	−0.772448
$520$	0	0
$521$	2.88399e9	0.893431	0.446716	−	0.894676i	$-0.352594\pi$
$521$	2.88399e9	0.893431	0.446716	+	0.894676i	$0.352594\pi$
$522$	− 1.33677e8i	− 0.0411349i
$523$	8.77188e8i	0.268125i	0.990973	+	0.134062i	$0.0428023\pi$
$523$	8.77188e8i	0.268125i	−0.990973	+	0.134062i	$0.957198\pi$
$524$	5.27421e8	0.160139
$525$	0	0
$526$	−6.40819e9	−1.91993
$527$	− 1.92251e9i	− 0.572177i
$528$	1.68951e8i	0.0499507i
$529$	3.23655e9	0.950578
$530$	0	0
$531$	−1.56478e8	−0.0453548
$532$	− 2.89988e9i	− 0.835007i
$533$	1.80696e9i	0.516896i
$534$	−1.35110e9	−0.383967
$535$	0	0
$536$	−6.69735e8	−0.187857
$537$	− 2.40781e9i	− 0.670983i
$538$	− 6.54201e9i	− 1.81123i
$539$	−3.23221e8	−0.0889077
$540$	0	0
$541$	−6.53485e8	−0.177437	−0.0887187	−	0.996057i	$-0.528277\pi$
$541$	−6.53485e8	−0.177437	−0.0887187	+	0.996057i	$0.528277\pi$
$542$	− 6.24305e8i	− 0.168422i
$543$	− 3.31592e9i	− 0.888801i
$544$	2.19317e9	0.584086
$545$	0	0
$546$	−4.26661e9	−1.12178
$547$	4.59299e9i	1.19988i	0.800043	+	0.599942i	$0.204810\pi$
$547$	4.59299e9i	1.19988i	−0.800043	+	0.599942i	$0.795190\pi$
$548$	2.45970e9i	0.638484i
$549$	1.08105e8	0.0278833
$550$	0	0
$551$	2.11696e9	0.539117
$552$	5.23031e8i	0.132355i
$553$	1.57824e6i	0 0.000396858i
$554$	4.43320e9	1.10773
$555$	0	0
$556$	7.41827e8	0.183038
$557$	− 6.83164e9i	− 1.67507i	−0.546387	−	0.837533i	$-0.683997\pi$
$557$	− 6.83164e9i	− 1.67507i	0.546387	−	0.837533i	$-0.316003\pi$
$558$	2.56983e8i	0.0626158i
$559$	1.92794e9	0.466823
$560$	0	0
$561$	1.01169e8	0.0241923
$562$	− 3.16145e9i	− 0.751292i
$563$	3.42509e9i	0.808897i	0.914561	+	0.404449i	$0.132537\pi$
$563$	3.42509e9i	0.808897i	−0.914561	+	0.404449i	$0.867463\pi$
$564$	−1.29543e9	−0.304044
$565$	0	0
$566$	−2.91494e8	−0.0675729
$567$	− 8.26136e9i	− 1.90332i
$568$	4.28685e9i	0.981565i
$569$	7.50930e9	1.70886	0.854430	−	0.519566i	$-0.173906\pi$
$569$	7.50930e9	1.70886	0.854430	+	0.519566i	$0.173906\pi$
$570$	0	0
$571$	−1.35841e8	−0.0305355	−0.0152677	−	0.999883i	$-0.504860\pi$
$571$	−1.35841e8	−0.0305355	−0.0152677	+	0.999883i	$0.504860\pi$
$572$	− 4.51700e7i	− 0.0100917i
$573$	− 7.42778e9i	− 1.64937i
$574$	−1.07688e10	−2.37670
$575$	0	0
$576$	1.33062e7	0.00290118
$577$	− 1.63775e9i	− 0.354922i	−0.984128	−	0.177461i	$-0.943212\pi$
$577$	− 1.63775e9i	− 0.354922i	0.984128	−	0.177461i	$-0.0567883\pi$
$578$	3.64249e9i	0.784606i
$579$	−7.65147e8	−0.163821
$580$	0	0
$581$	−1.00955e10	−2.13556
$582$	3.28066e9i	0.689812i
$583$	− 2.20246e8i	− 0.0460328i
$584$	3.58468e9	0.744742
$585$	0	0
$586$	2.50356e9	0.513944
$587$	5.97205e9i	1.21868i	0.792909	+	0.609341i	$0.208566\pi$
$587$	5.97205e9i	1.21868i	−0.792909	+	0.609341i	$0.791434\pi$
$588$	− 6.13369e9i	− 1.24423i
$589$	−4.06967e9	−0.820647
$590$	0	0
$591$	8.07303e9	1.60872
$592$	2.25362e9i	0.446431i
$593$	8.31347e9i	1.63716i	0.574394	+	0.818579i	$0.305238\pi$
$593$	8.31347e9i	1.63716i	−0.574394	+	0.818579i	$0.694762\pi$
$594$	2.39259e8	0.0468399
$595$	0	0
$596$	−2.47915e9	−0.479668
$597$	8.53308e9i	1.64133i
$598$	− 7.01370e8i	− 0.134120i
$599$	−9.78368e9	−1.85998	−0.929990	−	0.367585i	$-0.880185\pi$
$599$	−9.78368e9	−1.85998	−0.929990	+	0.367585i	$0.880185\pi$
$600$	0	0
$601$	5.40159e9	1.01499	0.507494	−	0.861655i	$-0.330572\pi$
$601$	5.40159e9	1.01499	0.507494	+	0.861655i	$0.330572\pi$
$602$	1.14898e10i	2.14646i
$603$	− 9.32846e7i	− 0.0173260i
$604$	−4.88890e8	−0.0902779
$605$	0	0
$606$	6.53634e9	1.19311
$607$	2.84439e9i	0.516214i	0.966116	+	0.258107i	$0.0830986\pi$
$607$	2.84439e9i	0.516214i	−0.966116	+	0.258107i	$0.916901\pi$
$608$	− 4.64264e9i	− 0.837726i
$609$	6.44001e9	1.15538
$610$	0	0
$611$	−1.53277e9	−0.271851
$612$	9.74928e7i	0.0171926i
$613$	− 7.02106e9i	− 1.23109i	−0.788101	−	0.615547i	$-0.788935\pi$
$613$	− 7.02106e9i	− 1.23109i	0.788101	−	0.615547i	$-0.211065\pi$
$614$	−1.19722e9	−0.208730
$615$	0	0
$616$	−2.37525e8	−0.0409428
$617$	− 3.35166e9i	− 0.574462i	−0.957861	−	0.287231i	$-0.907265\pi$
$617$	− 3.35166e9i	− 0.574462i	0.957861	−	0.287231i	$-0.0927347\pi$
$618$	1.09637e10i	1.86852i
$619$	3.92362e9	0.664921	0.332461	−	0.943117i	$-0.392121\pi$
$619$	3.92362e9	0.664921	0.332461	+	0.943117i	$0.392121\pi$
$620$	0	0
$621$	1.28890e9	0.215972
$622$	− 6.78588e9i	− 1.13068i
$623$	− 3.30538e9i	− 0.547662i
$624$	−3.79353e9	−0.625026
$625$	0	0
$626$	−7.99150e9	−1.30202
$627$	− 2.14161e8i	− 0.0346979i
$628$	5.98354e9i	0.964049i
$629$	1.34948e9	0.216217
$630$	0	0
$631$	−6.81545e8	−0.107992	−0.0539960	−	0.998541i	$-0.517196\pi$
$631$	−6.81545e8	−0.107992	−0.0539960	+	0.998541i	$0.517196\pi$
$632$	806400.i	0 0.000127069i
$633$	7.77583e9i	1.21852i
$634$	−7.70267e9	−1.20041
$635$	0	0
$636$	4.17955e9	0.644213
$637$	− 7.25744e9i	− 1.11249i
$638$	1.96517e8i	0.0299590i
$639$	−5.97097e8	−0.0905298
$640$	0	0
$641$	9.65199e9	1.44748	0.723742	−	0.690071i	$-0.242421\pi$
$641$	9.65199e9	1.44748	0.723742	+	0.690071i	$0.242421\pi$
$642$	2.74831e9i	0.409914i
$643$	− 5.07826e9i	− 0.753315i	−0.926353	−	0.376657i	$-0.877073\pi$
$643$	− 5.07826e9i	− 0.753315i	0.926353	−	0.376657i	$-0.122927\pi$
$644$	1.45017e9	0.213952
$645$	0	0
$646$	−4.45016e9	−0.649474
$647$	2.08330e9i	0.302404i	0.988503	+	0.151202i	$0.0483144\pi$
$647$	2.08330e9i	0.302404i	−0.988503	+	0.151202i	$0.951686\pi$
$648$	− 4.22113e9i	− 0.609420i
$649$	2.30036e8	0.0330324
$650$	0	0
$651$	−1.23803e10	−1.75873
$652$	3.73235e9i	0.527370i
$653$	6.84881e9i	0.962540i	0.876572	+	0.481270i	$0.159824\pi$
$653$	6.84881e9i	0.962540i	−0.876572	+	0.481270i	$0.840176\pi$
$654$	1.80309e10	2.52056
$655$	0	0
$656$	−9.57474e9	−1.32423
$657$	4.99295e8i	0.0686876i
$658$	− 9.13468e9i	− 1.24998i
$659$	−6.99913e9	−0.952676	−0.476338	−	0.879262i	$-0.658036\pi$
$659$	−6.99913e9	−0.952676	−0.476338	+	0.879262i	$0.658036\pi$
$660$	0	0
$661$	−1.99594e9	−0.268809	−0.134404	−	0.990927i	$-0.542912\pi$
$661$	−1.99594e9	−0.268809	−0.134404	+	0.990927i	$0.542912\pi$
$662$	− 1.31577e10i	− 1.76270i
$663$	2.27160e9i	0.302715i
$664$	−5.15830e9	−0.683783
$665$	0	0
$666$	−1.80386e8	−0.0236616
$667$	1.05864e9i	0.138137i
$668$	5.85793e8i	0.0760372i
$669$	8.44472e8	0.109042
$670$	0	0
$671$	−1.58924e8	−0.0203077
$672$	− 1.41234e10i	− 1.79533i
$673$	− 1.17939e10i	− 1.49144i	−0.666261	−	0.745718i	$-0.732106\pi$
$673$	− 1.17939e10i	− 1.49144i	0.666261	−	0.745718i	$-0.267894\pi$
$674$	7.47085e9	0.939853
$675$	0	0
$676$	−3.25268e9	−0.404974
$677$	− 7.80222e9i	− 0.966402i	−0.875509	−	0.483201i	$-0.839474\pi$
$677$	− 7.80222e9i	− 0.966402i	0.875509	−	0.483201i	$-0.160526\pi$
$678$	− 1.17473e10i	− 1.44755i
$679$	−8.02590e9	−0.983897
$680$	0	0
$681$	9.74572e9	1.18249
$682$	− 3.77786e8i	− 0.0456038i
$683$	4.51153e9i	0.541815i	0.962605	+	0.270908i	$0.0873238\pi$
$683$	4.51153e9i	0.541815i	−0.962605	+	0.270908i	$0.912676\pi$
$684$	2.06379e8	0.0246586
$685$	0	0
$686$	2.42968e10	2.87353
$687$	7.65883e9i	0.901185i
$688$	1.02158e10i	1.19595i
$689$	4.94528e9	0.576002
$690$	0	0
$691$	−1.06331e10	−1.22599	−0.612997	−	0.790086i	$-0.710036\pi$
$691$	−1.06331e10	−1.22599	−0.612997	+	0.790086i	$0.710036\pi$
$692$	3.48516e9i	0.399808i
$693$	− 3.30839e7i	− 0.00377615i
$694$	1.51107e10	1.71604
$695$	0	0
$696$	3.29052e9	0.369940
$697$	5.73343e9i	0.641356i
$698$	5.98104e9i	0.665707i
$699$	−9.35929e8	−0.103651
$700$	0	0
$701$	−4.38514e9	−0.480807	−0.240403	−	0.970673i	$-0.577280\pi$
$701$	−4.38514e9	−0.480807	−0.240403	+	0.970673i	$0.577280\pi$
$702$	5.37220e9i	0.586100i
$703$	− 2.85667e9i	− 0.310110i
$704$	−1.95612e7	−0.00211296
$705$	0	0
$706$	2.08552e10	2.23048
$707$	1.59907e10i	1.70176i
$708$	4.36534e9i	0.462276i
$709$	5.98805e9	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$709$	5.98805e9	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$710$	0	0
$711$	−112320.	−1.17196e−5 0
$712$	− 1.68888e9i	− 0.175355i
$713$	− 2.03515e9i	− 0.210273i
$714$	−1.35378e10	−1.39189
$715$	0	0
$716$	−3.41106e9	−0.347291
$717$	− 7.69058e9i	− 0.779188i
$718$	− 1.17779e10i	− 1.18749i
$719$	1.17768e10	1.18161	0.590807	−	0.806813i	$-0.298809\pi$
$719$	1.17768e10	1.18161	0.590807	+	0.806813i	$0.298809\pi$
$720$	0	0
$721$	−2.68219e10	−2.66512
$722$	− 3.09383e9i	− 0.305926i
$723$	1.81814e10i	1.78913i
$724$	−4.69755e9	−0.460031
$725$	0	0
$726$	−1.30755e10	−1.26818
$727$	− 8.41051e9i	− 0.811805i	−0.913916	−	0.405902i	$-0.866957\pi$
$727$	− 8.41051e9i	− 0.811805i	0.913916	−	0.405902i	$-0.133043\pi$
$728$	− 5.33327e9i	− 0.512311i
$729$	−9.84247e9	−0.940931
$730$	0	0
$731$	6.11729e9	0.579227
$732$	− 3.01586e9i	− 0.284199i
$733$	1.44084e10i	1.35130i	0.737223	+	0.675650i	$0.236137\pi$
$733$	1.44084e10i	1.35130i	−0.737223	+	0.675650i	$0.763863\pi$
$734$	−1.05065e10	−0.980664
$735$	0	0
$736$	2.32168e9	0.214649
$737$	1.37136e8i	0.0126187i
$738$	− 7.66391e8i	− 0.0701864i
$739$	−8.21708e9	−0.748966	−0.374483	−	0.927234i	$-0.622180\pi$
$739$	−8.21708e9	−0.748966	−0.374483	+	0.927234i	$0.622180\pi$
$740$	0	0
$741$	4.80865e9	0.434170
$742$	2.94719e10i	2.64847i
$743$	− 1.72531e10i	− 1.54314i	−0.636144	−	0.771570i	$-0.719472\pi$
$743$	− 1.72531e10i	− 1.54314i	0.636144	−	0.771570i	$-0.280528\pi$
$744$	−6.32572e9	−0.563125
$745$	0	0
$746$	−2.39503e9	−0.211215
$747$	− 7.18477e8i	− 0.0630654i
$748$	− 1.43323e8i	− 0.0125216i
$749$	−6.72354e9	−0.584671
$750$	0	0
$751$	1.58498e10	1.36548	0.682739	−	0.730662i	$-0.260789\pi$
$751$	1.58498e10	1.36548	0.682739	+	0.730662i	$0.260789\pi$
$752$	− 8.12183e9i	− 0.696453i
$753$	− 7.74260e9i	− 0.660853i
$754$	−4.41249e9	−0.374873
$755$	0	0
$756$	−1.11077e10	−0.934966
$757$	− 7.13856e9i	− 0.598102i	−0.954237	−	0.299051i	$-0.903330\pi$
$757$	− 7.13856e9i	− 0.598102i	0.954237	−	0.299051i	$-0.0966701\pi$
$758$	− 6.52735e8i	− 0.0544371i
$759$	1.07097e8	0.00889059
$760$	0	0
$761$	−2.59993e9	−0.213853	−0.106926	−	0.994267i	$-0.534101\pi$
$761$	−2.59993e9	−0.213853	−0.106926	+	0.994267i	$0.534101\pi$
$762$	8.40119e9i	0.687857i
$763$	4.41114e10i	3.59514i
$764$	−1.05227e10	−0.853690
$765$	0	0
$766$	6.19172e9	0.497749
$767$	5.16512e9i	0.413329i
$768$	− 1.57660e10i	− 1.25591i
$769$	4.96477e9	0.393692	0.196846	−	0.980434i	$-0.436930\pi$
$769$	4.96477e9	0.393692	0.196846	+	0.980434i	$0.436930\pi$
$770$	0	0
$771$	−1.09146e10	−0.857663
$772$	1.08396e9i	0.0847914i
$773$	− 1.49681e10i	− 1.16557i	−0.812626	−	0.582786i	$-0.801963\pi$
$773$	− 1.49681e10i	− 1.16557i	0.812626	−	0.582786i	$-0.198037\pi$
$774$	−8.17703e8	−0.0633873
$775$	0	0
$776$	−4.10082e9	−0.315032
$777$	− 8.69026e9i	− 0.664598i
$778$	6.49646e9i	0.494593i
$779$	1.21369e10	0.919867
$780$	0	0
$781$	8.77783e8	0.0659339
$782$	− 2.22542e9i	− 0.166414i
$783$	− 8.10877e9i	− 0.603655i
$784$	3.84558e10	2.85007
$785$	0	0
$786$	−5.21216e9	−0.382859
$787$	9.79990e9i	0.716655i	0.933596	+	0.358328i	$0.116653\pi$
$787$	9.79990e9i	0.716655i	−0.933596	+	0.358328i	$0.883347\pi$
$788$	− 1.14368e10i	− 0.832650i
$789$	2.19709e10	1.59250
$790$	0	0
$791$	2.87388e10	2.06467
$792$	− 1.69042e7i	− 0.00120908i
$793$	− 3.56840e9i	− 0.254108i
$794$	−4.44909e9	−0.315428
$795$	0	0
$796$	1.20885e10	0.849529
$797$	− 3.93169e9i	− 0.275090i	−0.990495	−	0.137545i	$-0.956079\pi$
$797$	− 3.93169e9i	− 0.275090i	0.990495	−	0.137545i	$-0.0439212\pi$
$798$	2.86577e10i	1.99632i
$799$	−4.86342e9	−0.337309
$800$	0	0
$801$	2.35237e8	0.0161730
$802$	− 1.67388e10i	− 1.14581i
$803$	− 7.34006e8i	− 0.0500259i
$804$	−2.60240e9	−0.176595
$805$	0	0
$806$	8.48262e9	0.570634
$807$	2.24297e10i	1.50234i
$808$	8.17043e9i	0.544885i
$809$	−1.26324e10	−0.838816	−0.419408	−	0.907798i	$-0.637762\pi$
$809$	−1.26324e10	−0.838816	−0.419408	+	0.907798i	$0.637762\pi$
$810$	0	0
$811$	1.16653e10	0.767934	0.383967	−	0.923347i	$-0.374558\pi$
$811$	1.16653e10	0.767934	0.383967	+	0.923347i	$0.374558\pi$
$812$	− 9.12335e9i	− 0.598009i
$813$	2.14047e9i	0.139699i
$814$	2.65183e8	0.0172330
$815$	0	0
$816$	−1.20368e10	−0.775522
$817$	− 1.29495e10i	− 0.830758i
$818$	− 3.09827e10i	− 1.97917i
$819$	7.42848e8	0.0472505
$820$	0	0
$821$	−8.17500e9	−0.515569	−0.257784	−	0.966202i	$-0.582992\pi$
$821$	−8.17500e9	−0.515569	−0.257784	+	0.966202i	$0.582992\pi$
$822$	− 2.43076e10i	− 1.52648i
$823$	− 1.75211e10i	− 1.09563i	−0.836601	−	0.547813i	$-0.815460\pi$
$823$	− 1.75211e10i	− 1.09563i	0.836601	−	0.547813i	$-0.184540\pi$
$824$	−1.37046e10	−0.853341
$825$	0	0
$826$	−3.07821e10	−1.90050
$827$	− 1.22225e10i	− 0.751437i	−0.926734	−	0.375718i	$-0.877396\pi$
$827$	− 1.22225e10i	− 0.751437i	0.926734	−	0.375718i	$-0.122604\pi$
$828$	1.03205e8i	0.00631823i
$829$	1.06634e10	0.650063	0.325032	−	0.945703i	$-0.394625\pi$
$829$	1.06634e10	0.650063	0.325032	+	0.945703i	$0.394625\pi$
$830$	0	0
$831$	−1.51995e10	−0.918812
$832$	− 4.39218e8i	− 0.0264392i
$833$	− 2.30276e10i	− 1.38036i
$834$	−7.33100e9	−0.437605
$835$	0	0
$836$	−3.03394e8	−0.0179591
$837$	1.55884e10i	0.918887i
$838$	1.12322e10i	0.659341i
$839$	2.31400e9	0.135268	0.0676342	−	0.997710i	$-0.478455\pi$
$839$	2.31400e9	0.135268	0.0676342	+	0.997710i	$0.478455\pi$
$840$	0	0
$841$	−1.05897e10	−0.613899
$842$	4.82599e8i	0.0278608i
$843$	1.08392e10i	0.623164i
$844$	1.10158e10	0.630690
$845$	0	0
$846$	6.50096e8	0.0369132
$847$	− 3.19883e10i	− 1.80883i
$848$	2.62041e10i	1.47565i
$849$	9.99410e8	0.0560488
$850$	0	0
$851$	1.42855e9	0.0794590
$852$	1.66575e10i	0.922721i
$853$	− 4.22377e9i	− 0.233012i	−0.993190	−	0.116506i	$-0.962831\pi$
$853$	− 4.22377e9i	− 0.233012i	0.993190	−	0.116506i	$-0.0371694\pi$
$854$	2.12663e10	1.16839
$855$	0	0
$856$	−3.43538e9	−0.187205
$857$	3.52104e9i	0.191090i	0.995425	+	0.0955450i	$0.0304594\pi$
$857$	3.52104e9i	0.191090i	−0.995425	+	0.0955450i	$0.969541\pi$
$858$	4.46385e8i	0.0241271i
$859$	−2.44930e10	−1.31846	−0.659229	−	0.751943i	$-0.729117\pi$
$859$	−2.44930e10	−1.31846	−0.659229	+	0.751943i	$0.729117\pi$
$860$	0	0
$861$	3.69215e10	1.97137
$862$	2.41769e10i	1.28566i
$863$	− 5.40573e9i	− 0.286297i	−0.989701	−	0.143148i	$-0.954277\pi$
$863$	− 5.40573e9i	− 0.286297i	0.989701	−	0.143148i	$-0.0457226\pi$
$864$	−1.77831e10	−0.938012
$865$	0	0
$866$	−6.83631e9	−0.357692
$867$	− 1.24886e10i	− 0.650797i
$868$	1.75388e10i	0.910293i
$869$	165120.	8.53553e−6 0
$870$	0	0
$871$	−3.07919e9	−0.157897
$872$	2.25387e10i	1.15112i
$873$	− 5.71186e8i	− 0.0290555i
$874$	−4.71091e9	−0.238679
$875$	0	0
$876$	1.39290e10	0.700095
$877$	2.89155e10i	1.44755i	0.690039	+	0.723773i	$0.257594\pi$
$877$	2.89155e10i	1.44755i	−0.690039	+	0.723773i	$0.742406\pi$
$878$	4.04096e10i	2.01490i
$879$	−8.58362e9	−0.426294
$880$	0	0
$881$	7.80643e8	0.0384624	0.0192312	−	0.999815i	$-0.493878\pi$
$881$	7.80643e8	0.0384624	0.0192312	+	0.999815i	$0.493878\pi$
$882$	3.07812e9i	0.151059i
$883$	− 1.36907e10i	− 0.669211i	−0.942358	−	0.334605i	$-0.891397\pi$
$883$	− 1.36907e10i	− 0.669211i	0.942358	−	0.334605i	$-0.108603\pi$
$884$	3.21810e9	0.156681
$885$	0	0
$886$	−1.29722e10	−0.626606
$887$	− 3.58403e9i	− 0.172441i	−0.996276	−	0.0862203i	$-0.972521\pi$
$887$	− 3.58403e9i	− 0.172441i	0.996276	−	0.0862203i	$-0.0274789\pi$
$888$	− 4.44028e9i	− 0.212797i
$889$	−2.05529e10	−0.981108
$890$	0	0
$891$	−8.64327e8	−0.0409361
$892$	− 1.19634e9i	− 0.0564386i
$893$	1.02952e10i	0.483786i
$894$	2.44998e10	1.14678
$895$	0	0
$896$	4.02798e10	1.87072
$897$	2.40470e9i	0.111247i
$898$	− 1.89749e10i	− 0.874406i
$899$	−1.28036e10	−0.587725
$900$	0	0
$901$	1.56912e10	0.714694
$902$	1.12666e9i	0.0511175i
$903$	− 3.93935e10i	− 1.78040i
$904$	1.46841e10	0.661085
$905$	0	0
$906$	4.83138e9	0.215835
$907$	3.01108e10i	1.33998i	0.742372	+	0.669988i	$0.233701\pi$
$907$	3.01108e10i	1.33998i	−0.742372	+	0.669988i	$0.766299\pi$
$908$	− 1.38064e10i	− 0.612042i
$909$	−1.13802e9	−0.0502548
$910$	0	0
$911$	−2.48800e10	−1.09027	−0.545137	−	0.838347i	$-0.683523\pi$
$911$	−2.48800e10	−1.09027	−0.545137	+	0.838347i	$0.683523\pi$
$912$	2.54801e10i	1.11229i
$913$	1.05622e9i	0.0459312i
$914$	6.48801e8	0.0281060
$915$	0	0
$916$	1.08500e10	0.466440
$917$	− 1.27512e10i	− 0.546082i
$918$	1.70458e10i	0.727224i
$919$	−1.08420e10	−0.460794	−0.230397	−	0.973097i	$-0.574003\pi$
$919$	−1.08420e10	−0.460794	−0.230397	+	0.973097i	$0.574003\pi$
$920$	0	0
$921$	4.10477e9	0.173133
$922$	− 2.12964e9i	− 0.0894846i
$923$	1.97093e10i	0.825021i
$924$	−9.22955e8	−0.0384883
$925$	0	0
$926$	−2.28831e10	−0.947056
$927$	− 1.90886e9i	− 0.0787037i
$928$	− 1.46062e10i	− 0.599957i
$929$	−1.07045e10	−0.438038	−0.219019	−	0.975721i	$-0.570286\pi$
$929$	−1.07045e10	−0.438038	−0.219019	+	0.975721i	$0.570286\pi$
$930$	0	0
$931$	−4.87463e10	−1.97978
$932$	1.32590e9i	0.0536482i
$933$	2.32659e10i	0.937851i
$934$	−1.55933e10	−0.626214
$935$	0	0
$936$	3.79557e8	0.0151291
$937$	− 3.42787e10i	− 1.36124i	−0.732635	−	0.680621i	$-0.761710\pi$
$937$	− 3.42787e10i	− 1.36124i	0.732635	−	0.680621i	$-0.238290\pi$
$938$	− 1.83507e10i	− 0.726012i
$939$	2.73994e10	1.07997
$940$	0	0
$941$	−3.73695e9	−0.146202	−0.0731010	−	0.997325i	$-0.523290\pi$
$941$	−3.73695e9	−0.146202	−0.0731010	+	0.997325i	$0.523290\pi$
$942$	− 5.91314e10i	− 2.30484i
$943$	6.06937e9i	0.235696i
$944$	−2.73690e10	−1.05890
$945$	0	0
$946$	1.20209e9	0.0461657
$947$	− 3.32150e10i	− 1.27089i	−0.772145	−	0.635447i	$-0.780816\pi$
$947$	− 3.32150e10i	− 1.27089i	0.772145	−	0.635447i	$-0.219184\pi$
$948$	3.13344e6i	0 0.000119452i
$949$	1.64810e10	0.625968
$950$	0	0
$951$	2.64091e10	0.995686
$952$	− 1.69223e10i	− 0.635667i
$953$	4.69895e10i	1.75864i	0.476235	+	0.879318i	$0.342001\pi$
$953$	4.69895e10i	1.75864i	−0.476235	+	0.879318i	$0.657999\pi$
$954$	−2.09746e9	−0.0782120
$955$	0	0
$956$	−1.08950e10	−0.403296
$957$	− 6.73772e8i	− 0.0248497i
$958$	− 1.78843e10i	− 0.657192i
$959$	5.94668e10	2.17726
$960$	0	0
$961$	−2.89877e9	−0.105361
$962$	5.95429e9i	0.215634i
$963$	− 4.78500e8i	− 0.0172659i
$964$	2.57570e10	0.926030
$965$	0	0
$966$	−1.43311e10	−0.511515
$967$	1.42294e10i	0.506050i	0.967460	+	0.253025i	$0.0814254\pi$
$967$	1.42294e10i	0.506050i	−0.967460	+	0.253025i	$0.918575\pi$
$968$	− 1.63444e10i	− 0.579168i
$969$	1.52577e10	0.538711
$970$	0	0
$971$	5.45474e8	0.0191208	0.00956041	−	0.999954i	$-0.496957\pi$
$971$	5.45474e8	0.0191208	0.00956041	+	0.999954i	$0.496957\pi$
$972$	− 1.62570e9i	− 0.0567816i
$973$	− 1.79348e10i	− 0.624167i
$974$	1.37154e10	0.475612
$975$	0	0
$976$	1.89083e10	0.650995
$977$	1.97916e10i	0.678968i	0.940612	+	0.339484i	$0.110252\pi$
$977$	1.97916e10i	0.678968i	−0.940612	+	0.339484i	$0.889748\pi$
$978$	− 3.68844e10i	− 1.26083i
$979$	−3.45818e8	−0.0117790
$980$	0	0
$981$	−3.13932e9	−0.106168
$982$	− 6.88975e10i	− 2.32174i
$983$	− 4.71503e10i	− 1.58324i	−0.611013	−	0.791620i	$-0.709238\pi$
$983$	− 4.71503e10i	− 1.58324i	0.611013	−	0.791620i	$-0.290762\pi$
$984$	1.88650e10	0.631210
$985$	0	0
$986$	−1.40007e10	−0.465136
$987$	3.13189e10i	1.03680i
$988$	− 6.81226e9i	− 0.224720i
$989$	6.47573e9	0.212864
$990$	0	0
$991$	3.87968e10	1.26631	0.633153	−	0.774027i	$-0.281761\pi$
$991$	3.87968e10	1.26631	0.633153	+	0.774027i	$0.281761\pi$
$992$	2.80792e10i	0.913258i
$993$	4.51123e10i	1.46208i
$994$	−1.17460e11	−3.79347
$995$	0	0
$996$	−2.00437e10	−0.642791
$997$	5.66394e10i	1.81003i	0.425380	+	0.905015i	$0.360140\pi$
$997$	5.66394e10i	1.81003i	−0.425380	+	0.905015i	$0.639860\pi$
$998$	5.12101e10i	1.63079i
$999$	−1.09421e10	−0.347234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.b.a.24.2		2
3.2	odd	2		225.8.b.b.199.1		2
4.3	odd	2		400.8.c.e.49.2		2
5.2	odd	4		5.8.a.a.1.1	✓	1
5.3	odd	4		25.8.a.a.1.1		1
5.4	even	2	inner	25.8.b.a.24.1		2
15.2	even	4		45.8.a.f.1.1		1
15.8	even	4		225.8.a.b.1.1		1
15.14	odd	2		225.8.b.b.199.2		2
20.3	even	4		400.8.a.e.1.1		1
20.7	even	4		80.8.a.d.1.1		1
20.19	odd	2		400.8.c.e.49.1		2
35.27	even	4		245.8.a.a.1.1		1
40.27	even	4		320.8.a.a.1.1		1
40.37	odd	4		320.8.a.h.1.1		1
55.32	even	4		605.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.a.1.1	✓	1	5.2	odd	4
25.8.a.a.1.1		1	5.3	odd	4
25.8.b.a.24.1		2	5.4	even	2	inner
25.8.b.a.24.2		2	1.1	even	1	trivial
45.8.a.f.1.1		1	15.2	even	4
80.8.a.d.1.1		1	20.7	even	4
225.8.a.b.1.1		1	15.8	even	4
225.8.b.b.199.1		2	3.2	odd	2
225.8.b.b.199.2		2	15.14	odd	2
245.8.a.a.1.1		1	35.27	even	4
320.8.a.a.1.1		1	40.27	even	4
320.8.a.h.1.1		1	40.37	odd	4
400.8.a.e.1.1		1	20.3	even	4
400.8.c.e.49.1		2	20.19	odd	2
400.8.c.e.49.2		2	4.3	odd	2
605.8.a.c.1.1		1	55.32	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)

\(f(q)\)	\(=\)	\(q+14.0000i q^{2} -48.0000i q^{3} -68.0000 q^{4} +672.000 q^{6} +1644.00i q^{7} +840.000i q^{8} -117.000 q^{9} +O(q^{10})\)	\(q+14.0000i q^{2} -48.0000i q^{3} -68.0000 q^{4} +672.000 q^{6} +1644.00i q^{7} +840.000i q^{8} -117.000 q^{9} +172.000 q^{11} +3264.00i q^{12} +3862.00i q^{13} -23016.0 q^{14} -20464.0 q^{16} +12254.0i q^{17} -1638.00i q^{18} +25940.0 q^{19} +78912.0 q^{21} +2408.00i q^{22} +12972.0i q^{23} +40320.0 q^{24} -54068.0 q^{26} -99360.0i q^{27} -111792. i q^{28} +81610.0 q^{29} -156888. q^{31} -178976. i q^{32} -8256.00i q^{33} -171556. q^{34} +7956.00 q^{36} -110126. i q^{37} +363160. i q^{38} +185376. q^{39} +467882. q^{41} +1.10477e6i q^{42} -499208. i q^{43} -11696.0 q^{44} -181608. q^{46} +396884. i q^{47} +982272. i q^{48} -1.87919e6 q^{49} +588192. q^{51} -262616. i q^{52} -1.28050e6i q^{53} +1.39104e6 q^{54} -1.38096e6 q^{56} -1.24512e6i q^{57} +1.14254e6i q^{58} +1.33742e6 q^{59} -923978. q^{61} -2.19643e6i q^{62} -192348. i q^{63} -113728. q^{64} +115584. q^{66} +797304. i q^{67} -833272. i q^{68} +622656. q^{69} +5.10339e6 q^{71} -98280.0i q^{72} -4.26748e6i q^{73} +1.54176e6 q^{74} -1.76392e6 q^{76} +282768. i q^{77} +2.59526e6i q^{78} +960.000 q^{79} -5.02516e6 q^{81} +6.55035e6i q^{82} +6.14083e6i q^{83} -5.36602e6 q^{84} +6.98891e6 q^{86} -3.91728e6i q^{87} +144480. i q^{88} -2.01057e6 q^{89} -6.34913e6 q^{91} -882096. i q^{92} +7.53062e6i q^{93} -5.55638e6 q^{94} -8.59085e6 q^{96} +4.88193e6i q^{97} -2.63087e7i q^{98} -20124.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 136 q^{4} + 1344 q^{6} - 234 q^{9}+O(q^{10}) \) 2 * q - 136 * q^4 + 1344 * q^6 - 234 * q^9	\( 2 q - 136 q^{4} + 1344 q^{6} - 234 q^{9} + 344 q^{11} - 46032 q^{14} - 40928 q^{16} + 51880 q^{19} + 157824 q^{21} + 80640 q^{24} - 108136 q^{26} + 163220 q^{29} - 313776 q^{31} - 343112 q^{34} + 15912 q^{36} + 370752 q^{39} + 935764 q^{41} - 23392 q^{44} - 363216 q^{46} - 3758386 q^{49} + 1176384 q^{51} + 2782080 q^{54} - 2761920 q^{56} + 2674840 q^{59} - 1847956 q^{61} - 227456 q^{64} + 231168 q^{66} + 1245312 q^{69} + 10206784 q^{71} + 3083528 q^{74} - 3527840 q^{76} + 1920 q^{79} - 10050318 q^{81} - 10732032 q^{84} + 13977824 q^{86} - 4021140 q^{89} - 12698256 q^{91} - 11112752 q^{94} - 17181696 q^{96} - 40248 q^{99}+O(q^{100}) \) 2 * q - 136 * q^4 + 1344 * q^6 - 234 * q^9 + 344 * q^11 - 46032 * q^14 - 40928 * q^16 + 51880 * q^19 + 157824 * q^21 + 80640 * q^24 - 108136 * q^26 + 163220 * q^29 - 313776 * q^31 - 343112 * q^34 + 15912 * q^36 + 370752 * q^39 + 935764 * q^41 - 23392 * q^44 - 363216 * q^46 - 3758386 * q^49 + 1176384 * q^51 + 2782080 * q^54 - 2761920 * q^56 + 2674840 * q^59 - 1847956 * q^61 - 227456 * q^64 + 231168 * q^66 + 1245312 * q^69 + 10206784 * q^71 + 3083528 * q^74 - 3527840 * q^76 + 1920 * q^79 - 10050318 * q^81 - 10732032 * q^84 + 13977824 * q^86 - 4021140 * q^89 - 12698256 * q^91 - 11112752 * q^94 - 17181696 * q^96 - 40248 * q^99

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