LMFDB - Embedded newform 25.8.b.b.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:	$4$
Coefficient field:	$\Q(i, \sqrt{649})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 325x^{2} + 26244 $ x^4 + 325*x^2 + 26244
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.2
Root			$-12.2377i$ of defining polynomial
Character	$\chi$	$=$	25.24
Dual form			25.8.b.b.24.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.23774i q^{2} +5.47548i q^{3} +100.566 q^{4} +28.6791 q^{6} +769.970i q^{7} -1197.17i q^{8} +2157.02 q^{9} +O(q^{10})$	\(q-5.23774i q^{2} +5.47548i q^{3} +100.566 q^{4} +28.6791 q^{6} +769.970i q^{7} -1197.17i q^{8} +2157.02 q^{9} +5356.43 q^{11} +550.647i q^{12} -14037.0i q^{13} +4032.90 q^{14} +6602.00 q^{16} +12402.4i q^{17} -11297.9i q^{18} +5038.89 q^{19} -4215.95 q^{21} -28055.6i q^{22} +75190.5i q^{23} +6555.08 q^{24} -73522.1 q^{26} +23785.6i q^{27} +77432.9i q^{28} -195529. q^{29} -93568.2 q^{31} -187817. i q^{32} +29329.0i q^{33} +64960.4 q^{34} +216923. q^{36} +161554. i q^{37} -26392.4i q^{38} +76859.3 q^{39} -28767.5 q^{41} +22082.1i q^{42} -739076. i q^{43} +538676. q^{44} +393828. q^{46} +1.06799e6i q^{47} +36149.1i q^{48} +230689. q^{49} -67908.9 q^{51} -1.41165e6i q^{52} -626442. i q^{53} +124583. q^{54} +921785. q^{56} +27590.3i q^{57} +1.02413e6i q^{58} -2.14861e6 q^{59} -2.57497e6 q^{61} +490086. i q^{62} +1.66084e6i q^{63} -138682. q^{64} +153618. q^{66} -807676. i q^{67} +1.24726e6i q^{68} -411704. q^{69} -1.72722e6 q^{71} -2.58232e6i q^{72} +1.74519e6i q^{73} +846178. q^{74} +506742. q^{76} +4.12429e6i q^{77} -402569. i q^{78} +2.46887e6 q^{79} +4.58716e6 q^{81} +150677. i q^{82} -6.90850e6i q^{83} -423982. q^{84} -3.87109e6 q^{86} -1.07061e6i q^{87} -6.41256e6i q^{88} -2.63567e6 q^{89} +1.08081e7 q^{91} +7.56161e6i q^{92} -512331. i q^{93} +5.59385e6 q^{94} +1.02839e6 q^{96} -1.01234e7i q^{97} -1.20829e6i q^{98} +1.15539e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 362 q^{4} + 1898 q^{6} + 4552 q^{9}+O(q^{10}) $ 4 * q - 362 * q^4 + 1898 * q^6 + 4552 * q^9	$ 4 q - 362 q^{4} + 1898 q^{6} + 4552 q^{9} + 8688 q^{11} + 63516 q^{14} + 66914 q^{16} - 36400 q^{19} - 133032 q^{21} - 269550 q^{24} + 408 q^{26} - 111600 q^{29} - 603552 q^{31} + 353846 q^{34} + 366844 q^{36} - 177616 q^{39} - 216972 q^{41} + 1647486 q^{44} + 4685868 q^{46} - 1645172 q^{49} - 638992 q^{51} + 4493650 q^{54} - 6671700 q^{56} - 4135200 q^{59} + 1164088 q^{61} + 700958 q^{64} - 1556294 q^{66} - 9582936 q^{69} - 9456432 q^{71} - 16453044 q^{74} + 14100050 q^{76} + 14372400 q^{79} + 157124 q^{81} + 34235196 q^{84} + 1458888 q^{86} + 11981700 q^{89} + 11634528 q^{91} - 17950264 q^{94} + 15306718 q^{96} + 22866944 q^{99}+O(q^{100}) $ 4 * q - 362 * q^4 + 1898 * q^6 + 4552 * q^9 + 8688 * q^11 + 63516 * q^14 + 66914 * q^16 - 36400 * q^19 - 133032 * q^21 - 269550 * q^24 + 408 * q^26 - 111600 * q^29 - 603552 * q^31 + 353846 * q^34 + 366844 * q^36 - 177616 * q^39 - 216972 * q^41 + 1647486 * q^44 + 4685868 * q^46 - 1645172 * q^49 - 638992 * q^51 + 4493650 * q^54 - 6671700 * q^56 - 4135200 * q^59 + 1164088 * q^61 + 700958 * q^64 - 1556294 * q^66 - 9582936 * q^69 - 9456432 * q^71 - 16453044 * q^74 + 14100050 * q^76 + 14372400 * q^79 + 157124 * q^81 + 34235196 * q^84 + 1458888 * q^86 + 11981700 * q^89 + 11634528 * q^91 - 17950264 * q^94 + 15306718 * q^96 + 22866944 * 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 $

$2$	− 5.23774i	− 0.462955i	−0.972840	−	0.231478i	$-0.925644\pi$
$2$	− 5.23774i	− 0.462955i	0.972840	−	0.231478i	$-0.0743560\pi$
$3$	5.47548i	0.117084i	0.998285	+	0.0585420i	$0.0186452\pi$
$3$	5.47548i	0.117084i	−0.998285	+	0.0585420i	$0.981355\pi$
$4$	100.566	0.785673
$5$	0	0
$5$	0	0
$6$	28.6791	0.0542047
$7$	769.970i	0.848459i	0.905555	+	0.424229i	$0.139455\pi$
$7$	769.970i	0.848459i	−0.905555	+	0.424229i	$0.860545\pi$
$8$	− 1197.17i	− 0.826686i
$9$	2157.02	0.986291
$10$	0	0
$11$	5356.43	1.21339	0.606696	−	0.794934i	$-0.292494\pi$
$11$	5356.43	1.21339	0.606696	+	0.794934i	$0.292494\pi$
$12$	550.647i	0.0919897i
$13$	− 14037.0i	− 1.77204i	−0.463651	−	0.886018i	$-0.653461\pi$
$13$	− 14037.0i	− 1.77204i	0.463651	−	0.886018i	$-0.346539\pi$
$14$	4032.90	0.392798
$15$	0	0
$16$	6602.00	0.402954
$17$	12402.4i	0.612256i	0.951990	+	0.306128i	$0.0990336\pi$
$17$	12402.4i	0.612256i	−0.951990	+	0.306128i	$0.900966\pi$
$18$	− 11297.9i	− 0.456609i
$19$	5038.89	0.168538	0.0842689	−	0.996443i	$-0.473145\pi$
$19$	5038.89	0.168538	0.0842689	+	0.996443i	$0.473145\pi$
$20$	0	0
$21$	−4215.95	−0.0993410
$22$	− 28055.6i	− 0.561746i
$23$	75190.5i	1.28859i	0.764776	+	0.644296i	$0.222849\pi$
$23$	75190.5i	1.28859i	−0.764776	+	0.644296i	$0.777151\pi$
$24$	6555.08	0.0967918
$25$	0	0
$26$	−73522.1	−0.820373
$27$	23785.6i	0.232563i
$28$	77432.9i	0.666611i
$29$	−195529.	−1.48874	−0.744368	−	0.667770i	$-0.767249\pi$
$29$	−195529.	−1.48874	−0.744368	+	0.667770i	$0.767249\pi$
$30$	0	0
$31$	−93568.2	−0.564108	−0.282054	−	0.959399i	$-0.591016\pi$
$31$	−93568.2	−0.564108	−0.282054	+	0.959399i	$0.591016\pi$
$32$	− 187817.i	− 1.01324i
$33$	29329.0i	0.142069i
$34$	64960.4	0.283447
$35$	0	0
$36$	216923.	0.774902
$37$	161554.i	0.524338i	0.965022	+	0.262169i	$0.0844379\pi$
$37$	161554.i	0.524338i	−0.965022	+	0.262169i	$0.915562\pi$
$38$	− 26392.4i	− 0.0780254i
$39$	76859.3	0.207477
$40$	0	0
$41$	−28767.5	−0.0651867	−0.0325933	−	0.999469i	$-0.510377\pi$
$41$	−28767.5	−0.0651867	−0.0325933	+	0.999469i	$0.510377\pi$
$42$	22082.1i	0.0459904i
$43$	− 739076.i	− 1.41759i	−0.705417	−	0.708793i	$-0.749240\pi$
$43$	− 739076.i	− 1.41759i	0.705417	−	0.708793i	$-0.250760\pi$
$44$	538676.	0.953329
$45$	0	0
$46$	393828.	0.596560
$47$	1.06799e6i	1.50046i	0.661178	+	0.750229i	$0.270057\pi$
$47$	1.06799e6i	1.50046i	−0.661178	+	0.750229i	$0.729943\pi$
$48$	36149.1i	0.0471795i
$49$	230689.	0.280118
$50$	0	0
$51$	−67908.9	−0.0716855
$52$	− 1.41165e6i	− 1.39224i
$53$	− 626442.i	− 0.577983i	−0.957332	−	0.288992i	$-0.906680\pi$
$53$	− 626442.i	− 0.577983i	0.957332	−	0.288992i	$-0.0933200\pi$
$54$	124583.	0.107666
$55$	0	0
$56$	921785.	0.701409
$57$	27590.3i	0.0197331i
$58$	1.02413e6i	0.689217i
$59$	−2.14861e6	−1.36199	−0.680997	−	0.732287i	$-0.738453\pi$
$59$	−2.14861e6	−1.36199	−0.680997	+	0.732287i	$0.738453\pi$
$60$	0	0
$61$	−2.57497e6	−1.45251	−0.726253	−	0.687428i	$-0.758740\pi$
$61$	−2.57497e6	−1.45251	−0.726253	+	0.687428i	$0.758740\pi$
$62$	490086.i	0.261157i
$63$	1.66084e6i	0.836827i
$64$	−138682.	−0.0661288
$65$	0	0
$66$	153618.	0.0657715
$67$	− 807676.i	− 0.328077i	−0.986454	−	0.164038i	$-0.947548\pi$
$67$	− 807676.i	− 0.328077i	0.986454	−	0.164038i	$-0.0524521\pi$
$68$	1.24726e6i	0.481033i
$69$	−411704.	−0.150874
$70$	0	0
$71$	−1.72722e6	−0.572722	−0.286361	−	0.958122i	$-0.592446\pi$
$71$	−1.72722e6	−0.572722	−0.286361	+	0.958122i	$0.592446\pi$
$72$	− 2.58232e6i	− 0.815353i
$73$	1.74519e6i	0.525064i	0.964923	+	0.262532i	$0.0845575\pi$
$73$	1.74519e6i	0.525064i	−0.964923	+	0.262532i	$0.915442\pi$
$74$	846178.	0.242745
$75$	0	0
$76$	506742.	0.132416
$77$	4.12429e6i	1.02951i
$78$	− 402569.i	− 0.0960526i
$79$	2.46887e6	0.563382	0.281691	−	0.959505i	$-0.409105\pi$
$79$	2.46887e6	0.563382	0.281691	+	0.959505i	$0.409105\pi$
$80$	0	0
$81$	4.58716e6	0.959062
$82$	150677.i	0.0301785i
$83$	− 6.90850e6i	− 1.32620i	−0.748529	−	0.663102i	$-0.769240\pi$
$83$	− 6.90850e6i	− 1.32620i	0.748529	−	0.663102i	$-0.230760\pi$
$84$	−423982.	−0.0780495
$85$	0	0
$86$	−3.87109e6	−0.656279
$87$	− 1.07061e6i	− 0.174307i
$88$	− 6.41256e6i	− 1.00310i
$89$	−2.63567e6	−0.396302	−0.198151	−	0.980171i	$-0.563494\pi$
$89$	−2.63567e6	−0.396302	−0.198151	+	0.980171i	$0.563494\pi$
$90$	0	0
$91$	1.08081e7	1.50350
$92$	7.56161e6i	1.01241i
$93$	− 512331.i	− 0.0660480i
$94$	5.59385e6	0.694645
$95$	0	0
$96$	1.02839e6	0.118634
$97$	− 1.01234e7i	− 1.12622i	−0.826381	−	0.563112i	$-0.809604\pi$
$97$	− 1.01234e7i	− 1.12622i	0.826381	−	0.563112i	$-0.190396\pi$
$98$	− 1.20829e6i	− 0.129682i
$99$	1.15539e7	1.19676
$100$	0	0
$101$	−8.33196e6	−0.804679	−0.402339	−	0.915491i	$-0.631803\pi$
$101$	−8.33196e6	−0.804679	−0.402339	+	0.915491i	$0.631803\pi$
$102$	355689.i	0.0331872i
$103$	7.75782e6i	0.699535i	0.936837	+	0.349767i	$0.113739\pi$
$103$	7.75782e6i	0.699535i	−0.936837	+	0.349767i	$0.886261\pi$
$104$	−1.68047e7	−1.46492
$105$	0	0
$106$	−3.28114e6	−0.267580
$107$	8.86598e6i	0.699654i	0.936814	+	0.349827i	$0.113760\pi$
$107$	8.86598e6i	0.699654i	−0.936814	+	0.349827i	$0.886240\pi$
$108$	2.39202e6i	0.182718i
$109$	−1.55125e6	−0.114734	−0.0573668	−	0.998353i	$-0.518270\pi$
$109$	−1.55125e6	−0.114734	−0.0573668	+	0.998353i	$0.518270\pi$
$110$	0	0
$111$	−884585.	−0.0613917
$112$	5.08334e6i	0.341890i
$113$	1.80311e7i	1.17557i	0.809019	+	0.587783i	$0.199999\pi$
$113$	1.80311e7i	1.17557i	−0.809019	+	0.587783i	$0.800001\pi$
$114$	144511.	0.00913553
$115$	0	0
$116$	−1.96636e7	−1.16966
$117$	− 3.02781e7i	− 1.74774i
$118$	1.12538e7i	0.630542i
$119$	−9.54945e6	−0.519474
$120$	0	0
$121$	9.20422e6	0.472322
$122$	1.34870e7i	0.672445i
$123$	− 157516.i	− 0.00763232i
$124$	−9.40979e6	−0.443204
$125$	0	0
$126$	8.69905e6	0.387414
$127$	− 1.98814e7i	− 0.861258i	−0.902529	−	0.430629i	$-0.858292\pi$
$127$	− 1.98814e7i	− 0.861258i	0.902529	−	0.430629i	$-0.141708\pi$
$128$	− 2.33142e7i	− 0.982621i
$129$	4.04679e6	0.165977
$130$	0	0
$131$	−5.81990e6	−0.226186	−0.113093	−	0.993584i	$-0.536076\pi$
$131$	−5.81990e6	−0.226186	−0.113093	+	0.993584i	$0.536076\pi$
$132$	2.94951e6i	0.111620i
$133$	3.87979e6i	0.142997i
$134$	−4.23040e6	−0.151885
$135$	0	0
$136$	1.48477e7	0.506144
$137$	3.83328e7i	1.27365i	0.771010	+	0.636823i	$0.219752\pi$
$137$	3.83328e7i	1.27365i	−0.771010	+	0.636823i	$0.780248\pi$
$138$	2.15640e6i	0.0698477i
$139$	3.60826e7	1.13958	0.569791	−	0.821789i	$-0.307024\pi$
$139$	3.60826e7	1.13958	0.569791	+	0.821789i	$0.307024\pi$
$140$	0	0
$141$	−5.84775e6	−0.175680
$142$	9.04673e6i	0.265144i
$143$	− 7.51883e7i	− 2.15018i
$144$	1.42406e7	0.397430
$145$	0	0
$146$	9.14084e6	0.243081
$147$	1.26313e6i	0.0327973i
$148$	1.62469e7i	0.411958i
$149$	4.91341e6	0.121683	0.0608416	−	0.998147i	$-0.480622\pi$
$149$	4.91341e6	0.121683	0.0608416	+	0.998147i	$0.480622\pi$
$150$	0	0
$151$	4.95186e7	1.17044	0.585220	−	0.810874i	$-0.301008\pi$
$151$	4.95186e7	1.17044	0.585220	+	0.810874i	$0.301008\pi$
$152$	− 6.03241e6i	− 0.139328i
$153$	2.67521e7i	0.603863i
$154$	2.16020e7	0.476619
$155$	0	0
$156$	7.72944e6	0.163009
$157$	− 5.84517e6i	− 0.120545i	−0.998182	−	0.0602724i	$-0.980803\pi$
$157$	− 5.84517e6i	− 0.120545i	0.998182	−	0.0602724i	$-0.0191969\pi$
$158$	− 1.29313e7i	− 0.260821i
$159$	3.43007e6	0.0676726
$160$	0	0
$161$	−5.78944e7	−1.09332
$162$	− 2.40264e7i	− 0.444003i
$163$	− 1.79731e7i	− 0.325062i	−0.986703	−	0.162531i	$-0.948034\pi$
$163$	− 1.79731e7i	− 0.325062i	0.986703	−	0.162531i	$-0.0519658\pi$
$164$	−2.89304e6	−0.0512154
$165$	0	0
$166$	−3.61849e7	−0.613973
$167$	5.72460e7i	0.951124i	0.879682	+	0.475562i	$0.157755\pi$
$167$	5.72460e7i	0.951124i	−0.879682	+	0.475562i	$0.842245\pi$
$168$	5.04721e6i	0.0821238i
$169$	−1.34289e8	−2.14011
$170$	0	0
$171$	1.08690e7	0.166227
$172$	− 7.43260e7i	− 1.11376i
$173$	1.09265e8i	1.60442i	0.597039	+	0.802212i	$0.296344\pi$
$173$	1.09265e8i	1.60442i	−0.597039	+	0.802212i	$0.703656\pi$
$174$	−5.60759e6	−0.0806964
$175$	0	0
$176$	3.53632e7	0.488941
$177$	− 1.17647e7i	− 0.159468i
$178$	1.38050e7i	0.183470i
$179$	−1.98072e7	−0.258130	−0.129065	−	0.991636i	$-0.541198\pi$
$179$	−1.98072e7	−0.258130	−0.129065	+	0.991636i	$0.541198\pi$
$180$	0	0
$181$	−1.00547e8	−1.26036	−0.630178	−	0.776451i	$-0.717018\pi$
$181$	−1.00547e8	−1.26036	−0.630178	+	0.776451i	$0.717018\pi$
$182$	− 5.66098e7i	− 0.696053i
$183$	− 1.40992e7i	− 0.170065i
$184$	9.00157e7	1.06526
$185$	0	0
$186$	−2.68345e6	−0.0305773
$187$	6.64325e7i	0.742908i
$188$	1.07403e8i	1.17887i
$189$	−1.83142e7	−0.197320
$190$	0	0
$191$	2.28106e7	0.236875	0.118437	−	0.992962i	$-0.462211\pi$
$191$	2.28106e7	0.236875	0.118437	+	0.992962i	$0.462211\pi$
$192$	− 759351.i	− 0.00774262i
$193$	1.92250e6i	0.0192493i	0.999954	+	0.00962465i	$0.00306367\pi$
$193$	1.92250e6i	0.0192493i	−0.999954	+	0.00962465i	$0.996936\pi$
$194$	−5.30237e7	−0.521391
$195$	0	0
$196$	2.31995e7	0.220081
$197$	− 1.98118e7i	− 0.184626i	−0.995730	−	0.0923129i	$-0.970574\pi$
$197$	− 1.98118e7i	− 0.184626i	0.995730	−	0.0923129i	$-0.0294260\pi$
$198$	− 6.05165e7i	− 0.554046i
$199$	1.14071e8	1.02610	0.513048	−	0.858360i	$-0.328516\pi$
$199$	1.14071e8	1.02610	0.513048	+	0.858360i	$0.328516\pi$
$200$	0	0
$201$	4.42241e6	0.0384125
$202$	4.36407e7i	0.372530i
$203$	− 1.50551e8i	− 1.26313i
$204$	−6.82933e6	−0.0563213
$205$	0	0
$206$	4.06334e7	0.323853
$207$	1.62187e8i	1.27093i
$208$	− 9.26722e7i	− 0.714049i
$209$	2.69905e7	0.204503
$210$	0	0
$211$	9.50285e6	0.0696410	0.0348205	−	0.999394i	$-0.488914\pi$
$211$	9.50285e6	0.0696410	0.0348205	+	0.999394i	$0.488914\pi$
$212$	− 6.29988e7i	− 0.454106i
$213$	− 9.45736e6i	− 0.0670566i
$214$	4.64377e7	0.323909
$215$	0	0
$216$	2.84754e7	0.192257
$217$	− 7.20447e7i	− 0.478622i
$218$	8.12507e6i	0.0531165i
$219$	−9.55574e6	−0.0614766
$220$	0	0
$221$	1.74092e8	1.08494
$222$	4.63323e6i	0.0284216i
$223$	− 1.92299e8i	− 1.16121i	−0.814187	−	0.580603i	$-0.802817\pi$
$223$	− 1.92299e8i	− 1.16121i	0.814187	−	0.580603i	$-0.197183\pi$
$224$	1.44614e8	0.859689
$225$	0	0
$226$	9.44420e7	0.544234
$227$	− 1.60189e8i	− 0.908955i	−0.890758	−	0.454478i	$-0.849826\pi$
$227$	− 1.60189e8i	− 0.908955i	0.890758	−	0.454478i	$-0.150174\pi$
$228$	2.77465e6i	0.0155037i
$229$	−1.97660e8	−1.08766	−0.543832	−	0.839194i	$-0.683027\pi$
$229$	−1.97660e8	−1.08766	−0.543832	+	0.839194i	$0.683027\pi$
$230$	0	0
$231$	−2.25825e7	−0.120540
$232$	2.34081e8i	1.23072i
$233$	1.99180e8i	1.03157i	0.856717	+	0.515786i	$0.172500\pi$
$233$	1.99180e8i	1.03157i	−0.856717	+	0.515786i	$0.827500\pi$
$234$	−1.58589e8	−0.809127
$235$	0	0
$236$	−2.16077e8	−1.07008
$237$	1.35182e7i	0.0659630i
$238$	5.00175e7i	0.240493i
$239$	3.31111e8	1.56885	0.784424	−	0.620225i	$-0.212959\pi$
$239$	3.31111e8	1.56885	0.784424	+	0.620225i	$0.212959\pi$
$240$	0	0
$241$	−3.68270e8	−1.69475	−0.847377	−	0.530991i	$-0.821820\pi$
$241$	−3.68270e8	−1.69475	−0.847377	+	0.530991i	$0.821820\pi$
$242$	− 4.82093e7i	− 0.218664i
$243$	7.71360e7i	0.344854i
$244$	−2.58955e8	−1.14119
$245$	0	0
$246$	−825027.	−0.00353342
$247$	− 7.07309e7i	− 0.298655i
$248$	1.12017e8i	0.466340i
$249$	3.78273e7	0.155277
$250$	0	0
$251$	3.74255e8	1.49386	0.746929	−	0.664903i	$-0.231527\pi$
$251$	3.74255e8	1.49386	0.746929	+	0.664903i	$0.231527\pi$
$252$	1.67024e8i	0.657472i
$253$	4.02753e8i	1.56357i
$254$	−1.04133e8	−0.398724
$255$	0	0
$256$	−1.39865e8	−0.521038
$257$	− 4.57062e8i	− 1.67961i	−0.542885	−	0.839807i	$-0.682668\pi$
$257$	− 4.57062e8i	− 1.67961i	0.542885	−	0.839807i	$-0.317332\pi$
$258$	− 2.11960e7i	− 0.0768397i
$259$	−1.24392e8	−0.444880
$260$	0	0
$261$	−4.21759e8	−1.46833
$262$	3.04831e7i	0.104714i
$263$	− 6.77338e7i	− 0.229594i	−0.993389	−	0.114797i	$-0.963378\pi$
$263$	− 6.77338e7i	− 0.229594i	0.993389	−	0.114797i	$-0.0366217\pi$
$264$	3.51118e7	0.117446
$265$	0	0
$266$	2.03214e7	0.0662014
$267$	− 1.44316e7i	− 0.0464007i
$268$	− 8.12248e7i	− 0.257761i
$269$	−2.21000e7	−0.0692244	−0.0346122	−	0.999401i	$-0.511020\pi$
$269$	−2.21000e7	−0.0692244	−0.0346122	+	0.999401i	$0.511020\pi$
$270$	0	0
$271$	4.22269e8	1.28883	0.644416	−	0.764675i	$-0.277100\pi$
$271$	4.22269e8	1.28883	0.644416	+	0.764675i	$0.277100\pi$
$272$	8.18804e7i	0.246711i
$273$	5.91793e7i	0.176036i
$274$	2.00777e8	0.589641
$275$	0	0
$276$	−4.14034e7	−0.118537
$277$	1.98868e8i	0.562195i	0.959679	+	0.281097i	$0.0906984\pi$
$277$	1.98868e8i	0.562195i	−0.959679	+	0.281097i	$0.909302\pi$
$278$	− 1.88991e8i	− 0.527576i
$279$	−2.01828e8	−0.556375
$280$	0	0
$281$	3.88135e8	1.04354	0.521771	−	0.853085i	$-0.325271\pi$
$281$	3.88135e8	1.04354	0.521771	+	0.853085i	$0.325271\pi$
$282$	3.06290e7i	0.0813318i
$283$	− 2.98951e8i	− 0.784056i	−0.919953	−	0.392028i	$-0.871774\pi$
$283$	− 2.98951e8i	− 0.784056i	0.919953	−	0.392028i	$-0.128226\pi$
$284$	−1.73700e8	−0.449972
$285$	0	0
$286$	−3.93817e8	−0.995435
$287$	− 2.21501e7i	− 0.0553082i
$288$	− 4.05125e8i	− 0.999346i
$289$	2.56520e8	0.625142
$290$	0	0
$291$	5.54304e7	0.131863
$292$	1.75507e8i	0.412528i
$293$	− 1.77029e8i	− 0.411158i	−0.978641	−	0.205579i	$-0.934092\pi$
$293$	− 1.77029e8i	− 0.411158i	0.978641	−	0.205579i	$-0.0659077\pi$
$294$	6.61596e6	0.0151837
$295$	0	0
$296$	1.93408e8	0.433463
$297$	1.27406e8i	0.282190i
$298$	− 2.57351e7i	− 0.0563339i
$299$	1.05545e9	2.28343
$300$	0	0
$301$	5.69066e8	1.20276
$302$	− 2.59366e8i	− 0.541862i
$303$	− 4.56215e7i	− 0.0942150i
$304$	3.32667e7	0.0679130
$305$	0	0
$306$	1.40121e8	0.279562
$307$	2.25726e8i	0.445243i	0.974905	+	0.222622i	$0.0714614\pi$
$307$	2.25726e8i	0.445243i	−0.974905	+	0.222622i	$0.928539\pi$
$308$	4.14764e8i	0.808861i
$309$	−4.24778e7	−0.0819044
$310$	0	0
$311$	−8.14288e8	−1.53503	−0.767515	−	0.641031i	$-0.778507\pi$
$311$	−8.14288e8	−1.53503	−0.767515	+	0.641031i	$0.778507\pi$
$312$	− 9.20136e7i	− 0.171518i
$313$	− 6.60615e7i	− 0.121771i	−0.998145	−	0.0608854i	$-0.980608\pi$
$313$	− 6.60615e7i	− 0.121771i	0.998145	−	0.0608854i	$-0.0193924\pi$
$314$	−3.06155e7	−0.0558068
$315$	0	0
$316$	2.48284e8	0.442634
$317$	6.63770e7i	0.117033i	0.998286	+	0.0585167i	$0.0186371\pi$
$317$	6.63770e7i	0.117033i	−0.998286	+	0.0585167i	$0.981363\pi$
$318$	− 1.79658e7i	− 0.0313294i
$319$	−1.04734e9	−1.80642
$320$	0	0
$321$	−4.85455e7	−0.0819184
$322$	3.03236e8i	0.506157i
$323$	6.24942e7i	0.103188i
$324$	4.61313e8	0.753509
$325$	0	0
$326$	−9.41385e7	−0.150489
$327$	− 8.49386e6i	− 0.0134335i
$328$	3.44396e7i	0.0538889i
$329$	−8.22319e8	−1.27308
$330$	0	0
$331$	−5.59199e8	−0.847557	−0.423778	−	0.905766i	$-0.639296\pi$
$331$	−5.59199e8	−0.847557	−0.423778	+	0.905766i	$0.639296\pi$
$332$	− 6.94761e8i	− 1.04196i
$333$	3.48475e8i	0.517150i
$334$	2.99839e8	0.440328
$335$	0	0
$336$	−2.78337e7	−0.0400298
$337$	− 4.77074e8i	− 0.679018i	−0.940603	−	0.339509i	$-0.889739\pi$
$337$	− 4.77074e8i	− 0.679018i	0.940603	−	0.339509i	$-0.110261\pi$
$338$	7.03370e8i	0.990775i
$339$	−9.87287e7	−0.137640
$340$	0	0
$341$	−5.01192e8	−0.684485
$342$	− 5.69289e7i	− 0.0769558i
$343$	8.11727e8i	1.08613i
$344$	−8.84799e8	−1.17190
$345$	0	0
$346$	5.72301e8	0.742777
$347$	5.49825e8i	0.706433i	0.935542	+	0.353217i	$0.114912\pi$
$347$	5.49825e8i	0.706433i	−0.935542	+	0.353217i	$0.885088\pi$
$348$	− 1.07667e8i	− 0.136948i
$349$	2.51578e8	0.316799	0.158400	−	0.987375i	$-0.449367\pi$
$349$	2.51578e8	0.316799	0.158400	+	0.987375i	$0.449367\pi$
$350$	0	0
$351$	3.33878e8	0.412110
$352$	− 1.00603e9i	− 1.22945i
$353$	− 8.04432e8i	− 0.973370i	−0.873577	−	0.486685i	$-0.838206\pi$
$353$	− 8.04432e8i	− 0.973370i	0.873577	−	0.486685i	$-0.161794\pi$
$354$	−6.16202e7	−0.0738264
$355$	0	0
$356$	−2.65059e8	−0.311364
$357$	− 5.22878e7i	− 0.0608222i
$358$	1.03745e8i	0.119503i
$359$	−1.13834e8	−0.129850	−0.0649250	−	0.997890i	$-0.520681\pi$
$359$	−1.13834e8	−0.129850	−0.0649250	+	0.997890i	$0.520681\pi$
$360$	0	0
$361$	−8.68481e8	−0.971595
$362$	5.26638e8i	0.583488i
$363$	5.03975e7i	0.0553014i
$364$	1.08693e9	1.18126
$365$	0	0
$366$	−7.38479e7	−0.0787325
$367$	2.25920e8i	0.238574i	0.992860	+	0.119287i	$0.0380609\pi$
$367$	2.25920e8i	0.238574i	−0.992860	+	0.119287i	$0.961939\pi$
$368$	4.96407e8i	0.519243i
$369$	−6.20521e7	−0.0642931
$370$	0	0
$371$	4.82342e8	0.490395
$372$	− 5.15231e7i	− 0.0518921i
$373$	− 8.19128e8i	− 0.817280i	−0.912696	−	0.408640i	$-0.866003\pi$
$373$	− 8.19128e8i	− 0.817280i	0.912696	−	0.408640i	$-0.133997\pi$
$374$	3.47956e8	0.343933
$375$	0	0
$376$	1.27856e9	1.24041
$377$	2.74464e9i	2.63809i
$378$	9.59249e7i	0.0913504i
$379$	1.86356e8	0.175836	0.0879178	−	0.996128i	$-0.471979\pi$
$379$	1.86356e8	0.175836	0.0879178	+	0.996128i	$0.471979\pi$
$380$	0	0
$381$	1.08860e8	0.100840
$382$	− 1.19476e8i	− 0.109662i
$383$	− 1.43839e9i	− 1.30822i	−0.756400	−	0.654109i	$-0.773044\pi$
$383$	− 1.43839e9i	− 1.30822i	0.756400	−	0.654109i	$-0.226956\pi$
$384$	1.27657e8	0.115049
$385$	0	0
$386$	1.00695e7	0.00891156
$387$	− 1.59420e9i	− 1.39815i
$388$	− 1.01807e9i	− 0.884843i
$389$	1.00373e9	0.864558	0.432279	−	0.901740i	$-0.357710\pi$
$389$	1.00373e9	0.864558	0.432279	+	0.901740i	$0.357710\pi$
$390$	0	0
$391$	−9.32540e8	−0.788949
$392$	− 2.76174e8i	− 0.231570i
$393$	− 3.18668e7i	− 0.0264828i
$394$	−1.03769e8	−0.0854735
$395$	0	0
$396$	1.16193e9	0.940261
$397$	− 9.05187e8i	− 0.726058i	−0.931778	−	0.363029i	$-0.881743\pi$
$397$	− 9.05187e8i	− 0.726058i	0.931778	−	0.363029i	$-0.118257\pi$
$398$	− 5.97472e8i	− 0.475036i
$399$	−2.12437e7	−0.0167427
$400$	0	0
$401$	2.05917e9	1.59473	0.797363	−	0.603500i	$-0.206228\pi$
$401$	2.05917e9	1.59473	0.797363	+	0.603500i	$0.206228\pi$
$402$	− 2.31634e7i	− 0.0177833i
$403$	1.31342e9i	0.999619i
$404$	−8.37913e8	−0.632214
$405$	0	0
$406$	−7.88548e8	−0.584773
$407$	8.65354e8i	0.636229i
$408$	8.12985e7i	0.0592614i
$409$	6.09853e8	0.440751	0.220376	−	0.975415i	$-0.429272\pi$
$409$	6.09853e8	0.440751	0.220376	+	0.975415i	$0.429272\pi$
$410$	0	0
$411$	−2.09891e8	−0.149124
$412$	7.80173e8i	0.549605i
$413$	− 1.65436e9i	− 1.15559i
$414$	8.49495e8	0.588382
$415$	0	0
$416$	−2.63639e9	−1.79549
$417$	1.97569e8i	0.133427i
$418$	− 1.41369e8i	− 0.0946755i
$419$	−2.76626e9	−1.83715	−0.918574	−	0.395248i	$-0.870659\pi$
$419$	−2.76626e9	−1.83715	−0.918574	+	0.395248i	$0.870659\pi$
$420$	0	0
$421$	1.11034e9	0.725219	0.362609	−	0.931941i	$-0.381886\pi$
$421$	1.11034e9	0.725219	0.362609	+	0.931941i	$0.381886\pi$
$422$	− 4.97734e7i	− 0.0322407i
$423$	2.30367e9i	1.47989i
$424$	−7.49957e8	−0.477811
$425$	0	0
$426$	−4.95352e7	−0.0310442
$427$	− 1.98265e9i	− 1.23239i
$428$	8.91617e8i	0.549699i
$429$	4.11692e8	0.251751
$430$	0	0
$431$	−1.55134e9	−0.933331	−0.466666	−	0.884434i	$-0.654545\pi$
$431$	−1.55134e9	−0.933331	−0.466666	+	0.884434i	$0.654545\pi$
$432$	1.57032e8i	0.0937122i
$433$	1.19534e9i	0.707593i	0.935322	+	0.353797i	$0.115109\pi$
$433$	1.19534e9i	0.707593i	−0.935322	+	0.353797i	$0.884891\pi$
$434$	−3.77351e8	−0.221581
$435$	0	0
$436$	−1.56004e8	−0.0901430
$437$	3.78876e8i	0.217176i
$438$	5.00505e7i	0.0284609i
$439$	2.81223e8	0.158644	0.0793221	−	0.996849i	$-0.474724\pi$
$439$	2.81223e8	0.158644	0.0793221	+	0.996849i	$0.474724\pi$
$440$	0	0
$441$	4.97601e8	0.276278
$442$	− 9.11849e8i	− 0.502279i
$443$	2.07780e9i	1.13551i	0.823197	+	0.567756i	$0.192188\pi$
$443$	2.07780e9i	1.13551i	−0.823197	+	0.567756i	$0.807812\pi$
$444$	−8.89593e7	−0.0482337
$445$	0	0
$446$	−1.00721e9	−0.537586
$447$	2.69033e7i	0.0142472i
$448$	− 1.06781e8i	− 0.0561075i
$449$	1.73277e9	0.903397	0.451698	−	0.892171i	$-0.350818\pi$
$449$	1.73277e9	0.903397	0.451698	+	0.892171i	$0.350818\pi$
$450$	0	0
$451$	−1.54091e8	−0.0790971
$452$	1.81331e9i	0.923609i
$453$	2.71138e8i	0.137040i
$454$	−8.39029e8	−0.420805
$455$	0	0
$456$	3.30303e7	0.0163131
$457$	− 1.78846e9i	− 0.876541i	−0.898843	−	0.438270i	$-0.855591\pi$
$457$	− 1.78846e9i	− 0.876541i	0.898843	−	0.438270i	$-0.144409\pi$
$458$	1.03529e9i	0.503539i
$459$	−2.94998e8	−0.142388
$460$	0	0
$461$	−3.91667e8	−0.186193	−0.0930965	−	0.995657i	$-0.529677\pi$
$461$	−3.91667e8	−0.186193	−0.0930965	+	0.995657i	$0.529677\pi$
$462$	1.18281e8i	0.0558044i
$463$	1.05509e9i	0.494034i	0.969011	+	0.247017i	$0.0794503\pi$
$463$	1.05509e9i	0.494034i	−0.969011	+	0.247017i	$0.920550\pi$
$464$	−1.29088e9	−0.599892
$465$	0	0
$466$	1.04325e9	0.477572
$467$	− 1.73541e9i	− 0.788485i	−0.919006	−	0.394243i	$-0.871007\pi$
$467$	− 1.73541e9i	− 0.788485i	0.919006	−	0.394243i	$-0.128993\pi$
$468$	− 3.04495e9i	− 1.37315i
$469$	6.21886e8	0.278360
$470$	0	0
$471$	3.20051e7	0.0141139
$472$	2.57225e9i	1.12594i
$473$	− 3.95881e9i	− 1.72009i
$474$	7.08050e7	0.0305379
$475$	0	0
$476$	−9.60351e8	−0.408137
$477$	− 1.35125e9i	− 0.570060i
$478$	− 1.73427e9i	− 0.726306i
$479$	−2.46088e8	−0.102310	−0.0511548	−	0.998691i	$-0.516290\pi$
$479$	−2.46088e8	−0.102310	−0.0511548	+	0.998691i	$0.516290\pi$
$480$	0	0
$481$	2.26773e9	0.929146
$482$	1.92890e9i	0.784595i
$483$	− 3.17000e8i	− 0.128010i
$484$	9.25633e8	0.371091
$485$	0	0
$486$	4.04018e8	0.159652
$487$	2.60089e9i	1.02040i	0.860055	+	0.510201i	$0.170429\pi$
$487$	2.60089e9i	1.02040i	−0.860055	+	0.510201i	$0.829571\pi$
$488$	3.08267e9i	1.20077i
$489$	9.84114e7	0.0380596
$490$	0	0
$491$	3.18729e9	1.21517	0.607584	−	0.794256i	$-0.292139\pi$
$491$	3.18729e9	1.21517	0.607584	+	0.794256i	$0.292139\pi$
$492$	− 1.58408e7i	− 0.00599651i
$493$	− 2.42502e9i	− 0.911488i
$494$	−3.70470e8	−0.138264
$495$	0	0
$496$	−6.17737e8	−0.227310
$497$	− 1.32991e9i	− 0.485931i
$498$	− 1.98130e8i	− 0.0718864i
$499$	3.08098e9	1.11004	0.555018	−	0.831838i	$-0.312711\pi$
$499$	3.08098e9	1.11004	0.555018	+	0.831838i	$0.312711\pi$
$500$	0	0
$501$	−3.13449e8	−0.111361
$502$	− 1.96025e9i	− 0.691590i
$503$	− 2.17354e9i	− 0.761517i	−0.924674	−	0.380759i	$-0.875663\pi$
$503$	− 2.17354e9i	− 0.761517i	0.924674	−	0.380759i	$-0.124337\pi$
$504$	1.98831e9	0.691794
$505$	0	0
$506$	2.10951e9	0.723862
$507$	− 7.35295e8i	− 0.250573i
$508$	− 1.99939e9i	− 0.676667i
$509$	1.91522e9	0.643733	0.321866	−	0.946785i	$-0.395690\pi$
$509$	1.91522e9	0.643733	0.321866	+	0.946785i	$0.395690\pi$
$510$	0	0
$511$	−1.34374e9	−0.445495
$512$	− 2.25164e9i	− 0.741404i
$513$	1.19853e8i	0.0391957i
$514$	−2.39397e9	−0.777586
$515$	0	0
$516$	4.06970e8	0.130403
$517$	5.72061e9i	1.82065i
$518$	6.51532e8i	0.205959i
$519$	−5.98278e8	−0.187853
$520$	0	0
$521$	−6.78513e8	−0.210197	−0.105098	−	0.994462i	$-0.533516\pi$
$521$	−6.78513e8	−0.210197	−0.105098	+	0.994462i	$0.533516\pi$
$522$	2.20906e9i	0.679769i
$523$	5.85186e9i	1.78870i	0.447366	+	0.894351i	$0.352362\pi$
$523$	5.85186e9i	1.78870i	−0.447366	+	0.894351i	$0.647638\pi$
$524$	−5.85285e8	−0.177708
$525$	0	0
$526$	−3.54772e8	−0.106292
$527$	− 1.16047e9i	− 0.345379i
$528$	1.93630e8i	0.0572472i
$529$	−2.24878e9	−0.660468
$530$	0	0
$531$	−4.63459e9	−1.34332
$532$	3.90176e8i	0.112349i
$533$	4.03810e8i	0.115513i
$534$	−7.55889e7	−0.0214814
$535$	0	0
$536$	−9.66925e8	−0.271216
$537$	− 1.08454e8i	− 0.0302229i
$538$	1.15754e8i	0.0320478i
$539$	1.23567e9	0.339893
$540$	0	0
$541$	−7.71811e8	−0.209566	−0.104783	−	0.994495i	$-0.533415\pi$
$541$	−7.71811e8	−0.209566	−0.104783	+	0.994495i	$0.533415\pi$
$542$	− 2.21173e9i	− 0.596672i
$543$	− 5.50542e8i	− 0.147568i
$544$	2.32938e9	0.620360
$545$	0	0
$546$	3.09966e8	0.0814967
$547$	− 5.26686e9i	− 1.37593i	−0.725744	−	0.687965i	$-0.758504\pi$
$547$	− 5.26686e9i	− 1.37593i	0.725744	−	0.687965i	$-0.241496\pi$
$548$	3.85498e9i	1.00067i
$549$	−5.55426e9	−1.43259
$550$	0	0
$551$	−9.85247e8	−0.250908
$552$	4.92879e8i	0.124725i
$553$	1.90095e9i	0.478006i
$554$	1.04162e9	0.260271
$555$	0	0
$556$	3.62869e9	0.895339
$557$	− 8.78323e7i	− 0.0215358i	−0.999942	−	0.0107679i	$-0.996572\pi$
$557$	− 8.78323e7i	− 0.0215358i	0.999942	−	0.0107679i	$-0.00342760\pi$
$558$	1.05712e9i	0.257577i
$559$	−1.03744e10	−2.51201
$560$	0	0
$561$	−3.63750e8	−0.0869826
$562$	− 2.03295e9i	− 0.483114i
$563$	− 5.32226e8i	− 0.125695i	−0.998023	−	0.0628473i	$-0.979982\pi$
$563$	− 5.32226e8i	− 0.125695i	0.998023	−	0.0628473i	$-0.0200181\pi$
$564$	−5.88085e8	−0.138027
$565$	0	0
$566$	−1.56583e9	−0.362983
$567$	3.53198e9i	0.813724i
$568$	2.06778e9i	0.473461i
$569$	7.55216e9	1.71861	0.859306	−	0.511461i	$-0.170896\pi$
$569$	7.55216e9	1.71861	0.859306	+	0.511461i	$0.170896\pi$
$570$	0	0
$571$	−2.55147e9	−0.573541	−0.286771	−	0.957999i	$-0.592582\pi$
$571$	−2.55147e9	−0.573541	−0.286771	+	0.957999i	$0.592582\pi$
$572$	− 7.56139e9i	− 1.68933i
$573$	1.24899e8i	0.0277343i
$574$	−1.16017e8	−0.0256052
$575$	0	0
$576$	−2.99140e8	−0.0652222
$577$	5.53852e9i	1.20027i	0.799899	+	0.600135i	$0.204886\pi$
$577$	5.53852e9i	1.20027i	−0.799899	+	0.600135i	$0.795114\pi$
$578$	− 1.34358e9i	− 0.289413i
$579$	−1.05266e7	−0.00225379
$580$	0	0
$581$	5.31934e9	1.12523
$582$	− 2.90330e8i	− 0.0610466i
$583$	− 3.35550e9i	− 0.701321i
$584$	2.08929e9	0.434063
$585$	0	0
$586$	−9.27233e8	−0.190348
$587$	4.73642e9i	0.966533i	0.875473	+	0.483266i	$0.160550\pi$
$587$	4.73642e9i	0.966533i	−0.875473	+	0.483266i	$0.839450\pi$
$588$	1.27028e8i	0.0257680i
$589$	−4.71480e8	−0.0950735
$590$	0	0
$591$	1.08479e8	0.0216167
$592$	1.06658e9i	0.211284i
$593$	− 1.44860e9i	− 0.285272i	−0.989775	−	0.142636i	$-0.954442\pi$
$593$	− 1.44860e9i	− 0.285272i	0.989775	−	0.142636i	$-0.0455578\pi$
$594$	6.67319e8	0.130641
$595$	0	0
$596$	4.94122e8	0.0956032
$597$	6.24591e8i	0.120139i
$598$	− 5.52816e9i	− 1.05713i
$599$	−1.04150e9	−0.198001	−0.0990005	−	0.995087i	$-0.531565\pi$
$599$	−1.04150e9	−0.198001	−0.0990005	+	0.995087i	$0.531565\pi$
$600$	0	0
$601$	−7.31995e9	−1.37546	−0.687729	−	0.725967i	$-0.741392\pi$
$601$	−7.31995e9	−1.37546	−0.687729	+	0.725967i	$0.741392\pi$
$602$	− 2.98062e9i	− 0.556825i
$603$	− 1.74217e9i	− 0.323579i
$604$	4.97990e9	0.919583
$605$	0	0
$606$	−2.38953e8	−0.0436173
$607$	6.16291e9i	1.11847i	0.829008	+	0.559237i	$0.188906\pi$
$607$	6.16291e9i	1.11847i	−0.829008	+	0.559237i	$0.811094\pi$
$608$	− 9.46391e8i	− 0.170769i
$609$	8.24340e8	0.147892
$610$	0	0
$611$	1.49914e10	2.65887
$612$	2.69036e9i	0.474439i
$613$	− 3.18382e8i	− 0.0558260i	−0.999610	−	0.0279130i	$-0.991114\pi$
$613$	− 3.18382e8i	− 0.0558260i	0.999610	−	0.0279130i	$-0.00888614\pi$
$614$	1.18229e9	0.206128
$615$	0	0
$616$	4.93748e9	0.851085
$617$	− 7.13555e9i	− 1.22301i	−0.791241	−	0.611505i	$-0.790565\pi$
$617$	− 7.13555e9i	− 1.22301i	0.791241	−	0.611505i	$-0.209435\pi$
$618$	2.22487e8i	0.0379180i
$619$	9.96303e9	1.68840	0.844198	−	0.536031i	$-0.180077\pi$
$619$	9.96303e9	1.68840	0.844198	+	0.536031i	$0.180077\pi$
$620$	0	0
$621$	−1.78845e9	−0.299679
$622$	4.26503e9i	0.710650i
$623$	− 2.02939e9i	− 0.336246i
$624$	5.07425e8	0.0836037
$625$	0	0
$626$	−3.46013e8	−0.0563744
$627$	1.47786e8i	0.0239440i
$628$	− 5.87826e8i	− 0.0947087i
$629$	−2.00365e9	−0.321030
$630$	0	0
$631$	−4.94300e9	−0.783227	−0.391614	−	0.920130i	$-0.628083\pi$
$631$	−4.94300e9	−0.783227	−0.391614	+	0.920130i	$0.628083\pi$
$632$	− 2.95565e9i	− 0.465740i
$633$	5.20326e7i	0.00815385i
$634$	3.47665e8	0.0541813
$635$	0	0
$636$	3.44949e8	0.0531685
$637$	− 3.23818e9i	− 0.496379i
$638$	5.48568e9i	0.836292i
$639$	−3.72565e9	−0.564871
$640$	0	0
$641$	−2.18024e9	−0.326965	−0.163482	−	0.986546i	$-0.552273\pi$
$641$	−2.18024e9	−0.326965	−0.163482	+	0.986546i	$0.552273\pi$
$642$	2.54269e8i	0.0379245i
$643$	− 8.70849e9i	− 1.29183i	−0.763411	−	0.645913i	$-0.776477\pi$
$643$	− 8.70849e9i	− 1.29183i	0.763411	−	0.645913i	$-0.223523\pi$
$644$	−5.82221e9	−0.858989
$645$	0	0
$646$	3.27328e8	0.0477716
$647$	4.57610e9i	0.664249i	0.943236	+	0.332124i	$0.107765\pi$
$647$	4.57610e9i	0.664249i	−0.943236	+	0.332124i	$0.892235\pi$
$648$	− 5.49161e9i	− 0.792843i
$649$	−1.15089e10	−1.65263
$650$	0	0
$651$	3.94479e8	0.0560390
$652$	− 1.80749e9i	− 0.255393i
$653$	9.23098e9i	1.29733i	0.761072	+	0.648667i	$0.224673\pi$
$653$	9.23098e9i	1.29733i	−0.761072	+	0.648667i	$0.775327\pi$
$654$	−4.44886e7	−0.00621909
$655$	0	0
$656$	−1.89923e8	−0.0262672
$657$	3.76440e9i	0.517866i
$658$	4.30709e9i	0.589378i
$659$	8.30221e9	1.13004	0.565021	−	0.825077i	$-0.308868\pi$
$659$	8.30221e9	1.13004	0.565021	+	0.825077i	$0.308868\pi$
$660$	0	0
$661$	2.67892e9	0.360791	0.180395	−	0.983594i	$-0.442262\pi$
$661$	2.67892e9	0.360791	0.180395	+	0.983594i	$0.442262\pi$
$662$	2.92894e9i	0.392381i
$663$	9.53237e8i	0.127029i
$664$	−8.27064e9	−1.09635
$665$	0	0
$666$	1.82522e9	0.239417
$667$	− 1.47019e10i	− 1.91837i
$668$	5.75700e9i	0.747272i
$669$	1.05293e9	0.135959
$670$	0	0
$671$	−1.37927e10	−1.76246
$672$	7.91829e8i	0.100656i
$673$	8.63768e9i	1.09231i	0.837685	+	0.546153i	$0.183908\pi$
$673$	8.63768e9i	1.09231i	−0.837685	+	0.546153i	$0.816092\pi$
$674$	−2.49879e9	−0.314355
$675$	0	0
$676$	−1.35049e10	−1.68143
$677$	− 8.17701e9i	− 1.01282i	−0.862291	−	0.506412i	$-0.830971\pi$
$677$	− 8.17701e9i	− 1.01282i	0.862291	−	0.506412i	$-0.169029\pi$
$678$	5.17115e8i	0.0637211i
$679$	7.79471e9	0.955555
$680$	0	0
$681$	8.77112e8	0.106424
$682$	2.62511e9i	0.316886i
$683$	3.81811e9i	0.458539i	0.973363	+	0.229269i	$0.0736337\pi$
$683$	3.81811e9i	0.458539i	−0.973363	+	0.229269i	$0.926366\pi$
$684$	1.09305e9	0.130600
$685$	0	0
$686$	4.25162e9	0.502828
$687$	− 1.08228e9i	− 0.127348i
$688$	− 4.87938e9i	− 0.571222i
$689$	−8.79336e9	−1.02421
$690$	0	0
$691$	1.38590e10	1.59794	0.798969	−	0.601372i	$-0.205379\pi$
$691$	1.38590e10	1.59794	0.798969	+	0.601372i	$0.205379\pi$
$692$	1.09883e10i	1.26055i
$693$	8.89618e9i	1.01540i
$694$	2.87984e9	0.327047
$695$	0	0
$696$	−1.28171e9	−0.144097
$697$	− 3.56785e8i	− 0.0399110i
$698$	− 1.31770e9i	− 0.146664i
$699$	−1.09061e9	−0.120781
$700$	0	0
$701$	−4.38274e9	−0.480543	−0.240271	−	0.970706i	$-0.577236\pi$
$701$	−4.38274e9	−0.480543	−0.240271	+	0.970706i	$0.577236\pi$
$702$	− 1.74877e9i	− 0.190788i
$703$	8.14053e8i	0.0883708i
$704$	−7.42842e8	−0.0802402
$705$	0	0
$706$	−4.21341e9	−0.450627
$707$	− 6.41536e9i	− 0.682737i
$708$	− 1.18312e9i	− 0.125289i
$709$	5.46537e9	0.575914	0.287957	−	0.957643i	$-0.407024\pi$
$709$	5.46537e9	0.575914	0.287957	+	0.957643i	$0.407024\pi$
$710$	0	0
$711$	5.32539e9	0.555659
$712$	3.15535e9i	0.327618i
$713$	− 7.03543e9i	− 0.726905i
$714$	−2.73870e8	−0.0281579
$715$	0	0
$716$	−1.99194e9	−0.202806
$717$	1.81299e9i	0.183687i
$718$	5.96233e8i	0.0601147i
$719$	7.24738e9	0.727160	0.363580	−	0.931563i	$-0.381554\pi$
$719$	7.24738e9	0.727160	0.363580	+	0.931563i	$0.381554\pi$
$720$	0	0
$721$	−5.97329e9	−0.593526
$722$	4.54888e9i	0.449805i
$723$	− 2.01646e9i	− 0.198429i
$724$	−1.01116e10	−0.990227
$725$	0	0
$726$	2.63969e8	0.0256021
$727$	1.61701e10i	1.56078i	0.625291	+	0.780392i	$0.284980\pi$
$727$	1.61701e10i	1.56078i	−0.625291	+	0.780392i	$0.715020\pi$
$728$	− 1.29391e10i	− 1.24292i
$729$	9.60977e9	0.918685
$730$	0	0
$731$	9.16629e9	0.867926
$732$	− 1.41790e9i	− 0.133616i
$733$	− 6.59871e9i	− 0.618864i	−0.950922	−	0.309432i	$-0.899861\pi$
$733$	− 6.59871e9i	− 0.618864i	0.950922	−	0.309432i	$-0.100139\pi$
$734$	1.18331e9	0.110449
$735$	0	0
$736$	1.41221e10	1.30565
$737$	− 4.32626e9i	− 0.398086i
$738$	3.25013e8i	0.0297648i
$739$	4.52765e8	0.0412683	0.0206342	−	0.999787i	$-0.493431\pi$
$739$	4.52765e8	0.0412683	0.0206342	+	0.999787i	$0.493431\pi$
$740$	0	0
$741$	3.87285e8	0.0349677
$742$	− 2.52638e9i	− 0.227031i
$743$	− 2.03152e10i	− 1.81702i	−0.417864	−	0.908509i	$-0.637221\pi$
$743$	− 2.03152e10i	− 1.81702i	0.417864	−	0.908509i	$-0.362779\pi$
$744$	−6.13346e8	−0.0546010
$745$	0	0
$746$	−4.29038e9	−0.378364
$747$	− 1.49018e10i	− 1.30802i
$748$	6.68085e9i	0.583682i
$749$	−6.82654e9	−0.593628
$750$	0	0
$751$	1.40167e10	1.20755	0.603776	−	0.797154i	$-0.293662\pi$
$751$	1.40167e10	1.20755	0.603776	+	0.797154i	$0.293662\pi$
$752$	7.05086e9i	0.604616i
$753$	2.04922e9i	0.174907i
$754$	1.43757e10	1.22132
$755$	0	0
$756$	−1.84179e9	−0.155029
$757$	− 1.74979e10i	− 1.46605i	−0.680200	−	0.733026i	$-0.738107\pi$
$757$	− 1.74979e10i	− 1.46605i	0.680200	−	0.733026i	$-0.261893\pi$
$758$	− 9.76086e8i	− 0.0814040i
$759$	−2.20526e9	−0.183069
$760$	0	0
$761$	7.69298e9	0.632774	0.316387	−	0.948630i	$-0.397530\pi$
$761$	7.69298e9	0.632774	0.316387	+	0.948630i	$0.397530\pi$
$762$	− 5.70180e8i	− 0.0466842i
$763$	− 1.19442e9i	− 0.0973466i
$764$	2.29397e9	0.186106
$765$	0	0
$766$	−7.53390e9	−0.605646
$767$	3.01600e10i	2.41350i
$768$	− 7.65829e8i	− 0.0610053i
$769$	6.85942e9	0.543933	0.271967	−	0.962307i	$-0.412326\pi$
$769$	6.85942e9	0.543933	0.271967	+	0.962307i	$0.412326\pi$
$770$	0	0
$771$	2.50263e9	0.196656
$772$	1.93338e8i	0.0151236i
$773$	8.93170e9i	0.695514i	0.937585	+	0.347757i	$0.113057\pi$
$773$	8.93170e9i	0.695514i	−0.937585	+	0.347757i	$0.886943\pi$
$774$	−8.35001e9	−0.647282
$775$	0	0
$776$	−1.21194e10	−0.931034
$777$	− 6.81104e8i	− 0.0520883i
$778$	− 5.25728e9i	− 0.400251i
$779$	−1.44956e8	−0.0109864
$780$	0	0
$781$	−9.25175e9	−0.694937
$782$	4.88440e9i	0.365248i
$783$	− 4.65076e9i	− 0.346225i
$784$	1.52301e9	0.112875
$785$	0	0
$786$	−1.66910e8	−0.0122603
$787$	− 5.20674e9i	− 0.380762i	−0.981710	−	0.190381i	$-0.939028\pi$
$787$	− 5.20674e9i	− 0.380762i	0.981710	−	0.190381i	$-0.0609724\pi$
$788$	− 1.99240e9i	− 0.145055i
$789$	3.70875e8	0.0268818
$790$	0	0
$791$	−1.38834e10	−0.997419
$792$	− 1.38320e10i	− 0.989344i
$793$	3.61448e10i	2.57389i
$794$	−4.74113e9	−0.336132
$795$	0	0
$796$	1.14716e10	0.806176
$797$	1.96111e10i	1.37214i	0.727538	+	0.686068i	$0.240665\pi$
$797$	1.96111e10i	1.37214i	−0.727538	+	0.686068i	$0.759335\pi$
$798$	1.11269e8i	0.00775112i
$799$	−1.32456e10	−0.918666
$800$	0	0
$801$	−5.68520e9	−0.390870
$802$	− 1.07854e10i	− 0.738287i
$803$	9.34798e9i	0.637109i
$804$	4.44745e8	0.0301797
$805$	0	0
$806$	6.87933e9	0.462779
$807$	− 1.21008e8i	− 0.00810508i
$808$	9.97477e9i	0.665217i
$809$	−1.99512e10	−1.32479	−0.662397	−	0.749153i	$-0.730461\pi$
$809$	−1.99512e10	−1.32479	−0.662397	+	0.749153i	$0.730461\pi$
$810$	0	0
$811$	2.18039e9	0.143536	0.0717679	−	0.997421i	$-0.477136\pi$
$811$	2.18039e9	0.143536	0.0717679	+	0.997421i	$0.477136\pi$
$812$	− 1.51403e10i	− 0.992407i
$813$	2.31212e9i	0.150902i
$814$	4.53250e9	0.294545
$815$	0	0
$816$	−4.48334e8	−0.0288859
$817$	− 3.72412e9i	− 0.238917i
$818$	− 3.19425e9i	− 0.204048i
$819$	2.33132e10	1.48289
$820$	0	0
$821$	−2.52847e10	−1.59462	−0.797310	−	0.603570i	$-0.793744\pi$
$821$	−2.52847e10	−1.59462	−0.797310	+	0.603570i	$0.793744\pi$
$822$	1.09935e9i	0.0690375i
$823$	− 1.86851e10i	− 1.16841i	−0.811606	−	0.584205i	$-0.801406\pi$
$823$	− 1.86851e10i	− 1.16841i	0.811606	−	0.584205i	$-0.198594\pi$
$824$	9.28742e9	0.578296
$825$	0	0
$826$	−8.66512e9	−0.534989
$827$	− 7.19901e9i	− 0.442592i	−0.975207	−	0.221296i	$-0.928971\pi$
$827$	− 7.19901e9i	− 0.442592i	0.975207	−	0.221296i	$-0.0710287\pi$
$828$	1.63105e10i	0.998532i
$829$	−1.83330e10	−1.11762	−0.558808	−	0.829297i	$-0.688741\pi$
$829$	−1.83330e10	−1.11762	−0.558808	+	0.829297i	$0.688741\pi$
$830$	0	0
$831$	−1.08890e9	−0.0658240
$832$	1.94668e9i	0.117183i
$833$	2.86109e9i	0.171504i
$834$	1.03482e9	0.0617707
$835$	0	0
$836$	2.71433e9	0.160672
$837$	− 2.22557e9i	− 0.131191i
$838$	1.44890e10i	0.850517i
$839$	−2.42360e10	−1.41675	−0.708376	−	0.705835i	$-0.750572\pi$
$839$	−2.42360e10	−1.41675	−0.708376	+	0.705835i	$0.750572\pi$
$840$	0	0
$841$	2.09816e10	1.21633
$842$	− 5.81568e9i	− 0.335744i
$843$	2.12522e9i	0.122182i
$844$	9.55664e8	0.0547150
$845$	0	0
$846$	1.20660e10	0.685122
$847$	7.08698e9i	0.400746i
$848$	− 4.13577e9i	− 0.232901i
$849$	1.63690e9	0.0918004
$850$	0	0
$851$	−1.21473e10	−0.675658
$852$	− 9.51090e8i	− 0.0526845i
$853$	− 3.21411e10i	− 1.77312i	−0.462612	−	0.886561i	$-0.653088\pi$
$853$	− 3.21411e10i	− 1.77312i	0.462612	−	0.886561i	$-0.346912\pi$
$854$	−1.03846e10	−0.570541
$855$	0	0
$856$	1.06141e10	0.578395
$857$	9.48584e9i	0.514805i	0.966304	+	0.257403i	$0.0828667\pi$
$857$	9.48584e9i	0.514805i	−0.966304	+	0.257403i	$0.917133\pi$
$858$	− 2.15633e9i	− 0.116550i
$859$	1.33721e10	0.719819	0.359910	−	0.932987i	$-0.382808\pi$
$859$	1.33721e10	0.719819	0.359910	+	0.932987i	$0.382808\pi$
$860$	0	0
$861$	1.21283e8	0.00647571
$862$	8.12550e9i	0.432091i
$863$	− 1.13928e10i	− 0.603381i	−0.953406	−	0.301690i	$-0.902449\pi$
$863$	− 1.13928e10i	− 0.603381i	0.953406	−	0.301690i	$-0.0975509\pi$
$864$	4.46734e9	0.235641
$865$	0	0
$866$	6.26088e9	0.327584
$867$	1.40457e9i	0.0731941i
$868$	− 7.24525e9i	− 0.376040i
$869$	1.32243e10	0.683604
$870$	0	0
$871$	−1.13373e10	−0.581364
$872$	1.85711e9i	0.0948486i
$873$	− 2.18363e10i	− 1.11079i
$874$	1.98446e9	0.100543
$875$	0	0
$876$	−9.60983e8	−0.0483005
$877$	1.45509e10i	0.728435i	0.931314	+	0.364218i	$0.118664\pi$
$877$	1.45509e10i	0.728435i	−0.931314	+	0.364218i	$0.881336\pi$
$878$	− 1.47297e9i	− 0.0734452i
$879$	9.69320e8	0.0481400
$880$	0	0
$881$	−5.46095e9	−0.269062	−0.134531	−	0.990909i	$-0.542953\pi$
$881$	−5.46095e9	−0.269062	−0.134531	+	0.990909i	$0.542953\pi$
$882$	− 2.60630e9i	− 0.127904i
$883$	1.95772e10i	0.956948i	0.878102	+	0.478474i	$0.158810\pi$
$883$	1.95772e10i	0.956948i	−0.878102	+	0.478474i	$0.841190\pi$
$884$	1.75078e10	0.852408
$885$	0	0
$886$	1.08830e10	0.525691
$887$	1.35654e10i	0.652678i	0.945253	+	0.326339i	$0.105815\pi$
$887$	1.35654e10i	0.652678i	−0.945253	+	0.326339i	$0.894185\pi$
$888$	1.05900e9i	0.0507516i
$889$	1.53081e10	0.730742
$890$	0	0
$891$	2.45708e10	1.16372
$892$	− 1.93387e10i	− 0.912328i
$893$	5.38148e9i	0.252884i
$894$	1.40912e8	0.00659580
$895$	0	0
$896$	1.79513e10	0.833714
$897$	5.77908e9i	0.267353i
$898$	− 9.07579e9i	− 0.418232i
$899$	1.82953e10	0.839807
$900$	0	0
$901$	7.76936e9	0.353874
$902$	8.07090e8i	0.0366184i
$903$	3.11591e9i	0.140824i
$904$	2.15862e10	0.971824
$905$	0	0
$906$	1.42015e9	0.0634433
$907$	− 7.63541e9i	− 0.339787i	−0.985462	−	0.169894i	$-0.945658\pi$
$907$	− 7.63541e9i	− 0.339787i	0.985462	−	0.169894i	$-0.0543424\pi$
$908$	− 1.61096e10i	− 0.714141i
$909$	−1.79722e10	−0.793648
$910$	0	0
$911$	−2.03069e10	−0.889875	−0.444937	−	0.895562i	$-0.646774\pi$
$911$	−2.03069e10	−0.889875	−0.444937	+	0.895562i	$0.646774\pi$
$912$	1.82151e8i	0.00795152i
$913$	− 3.70049e10i	− 1.60921i
$914$	−9.36748e9	−0.405799
$915$	0	0
$916$	−1.98779e10	−0.854547
$917$	− 4.48115e9i	− 0.191910i
$918$	1.54512e9i	0.0659193i
$919$	−1.93832e10	−0.823797	−0.411899	−	0.911230i	$-0.635134\pi$
$919$	−1.93832e10	−0.823797	−0.411899	+	0.911230i	$0.635134\pi$
$920$	0	0
$921$	−1.23596e9	−0.0521309
$922$	2.05145e9i	0.0861990i
$923$	2.42450e10i	1.01488i
$924$	−2.27103e9	−0.0947047
$925$	0	0
$926$	5.52629e9	0.228715
$927$	1.67338e10i	0.689945i
$928$	3.67237e10i	1.50844i
$929$	−1.17298e10	−0.479993	−0.239997	−	0.970774i	$-0.577146\pi$
$929$	−1.17298e10	−0.479993	−0.239997	+	0.970774i	$0.577146\pi$
$930$	0	0
$931$	1.16242e9	0.0472104
$932$	2.00307e10i	0.810478i
$933$	− 4.45862e9i	− 0.179727i
$934$	−9.08964e9	−0.365033
$935$	0	0
$936$	−3.62480e10	−1.44484
$937$	− 1.41104e10i	− 0.560340i	−0.959950	−	0.280170i	$-0.909609\pi$
$937$	− 1.41104e10i	− 0.560340i	0.959950	−	0.280170i	$-0.0903908\pi$
$938$	− 3.25728e9i	− 0.128868i
$939$	3.61718e8	0.0142574
$940$	0	0
$941$	1.98332e10	0.775942	0.387971	−	0.921672i	$-0.373176\pi$
$941$	1.98332e10	0.775942	0.387971	+	0.921672i	$0.373176\pi$
$942$	− 1.67634e8i	− 0.00653409i
$943$	− 2.16304e9i	− 0.0839990i
$944$	−1.41851e10	−0.548821
$945$	0	0
$946$	−2.07352e10	−0.796324
$947$	− 4.96249e10i	− 1.89878i	−0.314098	−	0.949390i	$-0.601702\pi$
$947$	− 4.96249e10i	− 1.89878i	0.314098	−	0.949390i	$-0.398298\pi$
$948$	1.35948e9i	0.0518253i
$949$	2.44972e10	0.930432
$950$	0	0
$951$	−3.63446e8	−0.0137028
$952$	1.14323e10i	0.429442i
$953$	3.24475e10i	1.21438i	0.794555	+	0.607192i	$0.207704\pi$
$953$	3.24475e10i	1.21438i	−0.794555	+	0.607192i	$0.792296\pi$
$954$	−7.07748e9	−0.263912
$955$	0	0
$956$	3.32985e10	1.23260
$957$	− 5.73467e9i	− 0.211503i
$958$	1.28895e9i	0.0473648i
$959$	−2.95151e10	−1.08064
$960$	0	0
$961$	−1.87576e10	−0.681782
$962$	− 1.18778e10i	− 0.430153i
$963$	1.91241e10i	0.690063i
$964$	−3.70355e10	−1.33152
$965$	0	0
$966$	−1.66036e9	−0.0592629
$967$	6.43301e9i	0.228782i	0.993436	+	0.114391i	$0.0364917\pi$
$967$	6.43301e9i	0.228782i	−0.993436	+	0.114391i	$0.963508\pi$
$968$	− 1.10190e10i	− 0.390462i
$969$	−3.42186e8	−0.0120817
$970$	0	0
$971$	7.99885e9	0.280389	0.140194	−	0.990124i	$-0.455227\pi$
$971$	7.99885e9	0.280389	0.140194	+	0.990124i	$0.455227\pi$
$972$	7.75726e9i	0.270942i
$973$	2.77825e10i	0.966889i
$974$	1.36228e10	0.472400
$975$	0	0
$976$	−1.69999e10	−0.585293
$977$	− 2.43029e10i	− 0.833733i	−0.908968	−	0.416866i	$-0.863128\pi$
$977$	− 2.43029e10i	− 0.833733i	0.908968	−	0.416866i	$-0.136872\pi$
$978$	− 5.15453e8i	− 0.0176199i
$979$	−1.41178e10	−0.480871
$980$	0	0
$981$	−3.34609e9	−0.113161
$982$	− 1.66942e10i	− 0.562568i
$983$	2.05054e10i	0.688544i	0.938870	+	0.344272i	$0.111874\pi$
$983$	2.05054e10i	0.688544i	−0.938870	+	0.344272i	$0.888126\pi$
$984$	−1.88573e8	−0.00630953
$985$	0	0
$986$	−1.27016e10	−0.421978
$987$	− 4.50259e9i	− 0.149057i
$988$	− 7.11313e9i	− 0.234645i
$989$	5.55714e10	1.82669
$990$	0	0
$991$	−3.93587e10	−1.28464	−0.642322	−	0.766435i	$-0.722029\pi$
$991$	−3.93587e10	−1.28464	−0.642322	+	0.766435i	$0.722029\pi$
$992$	1.75737e10i	0.571574i
$993$	− 3.06188e9i	− 0.0992353i
$994$	−6.96571e9	−0.224964
$995$	0	0
$996$	3.80415e9	0.121997
$997$	2.18052e10i	0.696831i	0.937340	+	0.348416i	$0.113280\pi$
$997$	2.18052e10i	0.696831i	−0.937340	+	0.348416i	$0.886720\pi$
$998$	− 1.61374e10i	− 0.513897i
$999$	−3.84266e9	−0.121942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.b.b.24.2		4
3.2	odd	2		225.8.b.l.199.3		4
4.3	odd	2		400.8.c.s.49.2		4
5.2	odd	4		25.8.a.c.1.2	✓	2
5.3	odd	4		25.8.a.e.1.1	yes	2
5.4	even	2	inner	25.8.b.b.24.3		4
15.2	even	4		225.8.a.v.1.1		2
15.8	even	4		225.8.a.k.1.2		2
15.14	odd	2		225.8.b.l.199.2		4
20.3	even	4		400.8.a.v.1.2		2
20.7	even	4		400.8.a.bd.1.1		2
20.19	odd	2		400.8.c.s.49.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.a.c.1.2	✓	2	5.2	odd	4
25.8.a.e.1.1	yes	2	5.3	odd	4
25.8.b.b.24.2		4	1.1	even	1	trivial
25.8.b.b.24.3		4	5.4	even	2	inner
225.8.a.k.1.2		2	15.8	even	4
225.8.a.v.1.1		2	15.2	even	4
225.8.b.l.199.2		4	15.14	odd	2
225.8.b.l.199.3		4	3.2	odd	2
400.8.a.v.1.2		2	20.3	even	4
400.8.a.bd.1.1		2	20.7	even	4
400.8.c.s.49.2		4	4.3	odd	2
400.8.c.s.49.3		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.23774i q^{2} +5.47548i q^{3} +100.566 q^{4} +28.6791 q^{6} +769.970i q^{7} -1197.17i q^{8} +2157.02 q^{9} +O(q^{10})\)	\(q-5.23774i q^{2} +5.47548i q^{3} +100.566 q^{4} +28.6791 q^{6} +769.970i q^{7} -1197.17i q^{8} +2157.02 q^{9} +5356.43 q^{11} +550.647i q^{12} -14037.0i q^{13} +4032.90 q^{14} +6602.00 q^{16} +12402.4i q^{17} -11297.9i q^{18} +5038.89 q^{19} -4215.95 q^{21} -28055.6i q^{22} +75190.5i q^{23} +6555.08 q^{24} -73522.1 q^{26} +23785.6i q^{27} +77432.9i q^{28} -195529. q^{29} -93568.2 q^{31} -187817. i q^{32} +29329.0i q^{33} +64960.4 q^{34} +216923. q^{36} +161554. i q^{37} -26392.4i q^{38} +76859.3 q^{39} -28767.5 q^{41} +22082.1i q^{42} -739076. i q^{43} +538676. q^{44} +393828. q^{46} +1.06799e6i q^{47} +36149.1i q^{48} +230689. q^{49} -67908.9 q^{51} -1.41165e6i q^{52} -626442. i q^{53} +124583. q^{54} +921785. q^{56} +27590.3i q^{57} +1.02413e6i q^{58} -2.14861e6 q^{59} -2.57497e6 q^{61} +490086. i q^{62} +1.66084e6i q^{63} -138682. q^{64} +153618. q^{66} -807676. i q^{67} +1.24726e6i q^{68} -411704. q^{69} -1.72722e6 q^{71} -2.58232e6i q^{72} +1.74519e6i q^{73} +846178. q^{74} +506742. q^{76} +4.12429e6i q^{77} -402569. i q^{78} +2.46887e6 q^{79} +4.58716e6 q^{81} +150677. i q^{82} -6.90850e6i q^{83} -423982. q^{84} -3.87109e6 q^{86} -1.07061e6i q^{87} -6.41256e6i q^{88} -2.63567e6 q^{89} +1.08081e7 q^{91} +7.56161e6i q^{92} -512331. i q^{93} +5.59385e6 q^{94} +1.02839e6 q^{96} -1.01234e7i q^{97} -1.20829e6i q^{98} +1.15539e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 362 q^{4} + 1898 q^{6} + 4552 q^{9}+O(q^{10}) \) 4 * q - 362 * q^4 + 1898 * q^6 + 4552 * q^9	\( 4 q - 362 q^{4} + 1898 q^{6} + 4552 q^{9} + 8688 q^{11} + 63516 q^{14} + 66914 q^{16} - 36400 q^{19} - 133032 q^{21} - 269550 q^{24} + 408 q^{26} - 111600 q^{29} - 603552 q^{31} + 353846 q^{34} + 366844 q^{36} - 177616 q^{39} - 216972 q^{41} + 1647486 q^{44} + 4685868 q^{46} - 1645172 q^{49} - 638992 q^{51} + 4493650 q^{54} - 6671700 q^{56} - 4135200 q^{59} + 1164088 q^{61} + 700958 q^{64} - 1556294 q^{66} - 9582936 q^{69} - 9456432 q^{71} - 16453044 q^{74} + 14100050 q^{76} + 14372400 q^{79} + 157124 q^{81} + 34235196 q^{84} + 1458888 q^{86} + 11981700 q^{89} + 11634528 q^{91} - 17950264 q^{94} + 15306718 q^{96} + 22866944 q^{99}+O(q^{100}) \) 4 * q - 362 * q^4 + 1898 * q^6 + 4552 * q^9 + 8688 * q^11 + 63516 * q^14 + 66914 * q^16 - 36400 * q^19 - 133032 * q^21 - 269550 * q^24 + 408 * q^26 - 111600 * q^29 - 603552 * q^31 + 353846 * q^34 + 366844 * q^36 - 177616 * q^39 - 216972 * q^41 + 1647486 * q^44 + 4685868 * q^46 - 1645172 * q^49 - 638992 * q^51 + 4493650 * q^54 - 6671700 * q^56 - 4135200 * q^59 + 1164088 * q^61 + 700958 * q^64 - 1556294 * q^66 - 9582936 * q^69 - 9456432 * q^71 - 16453044 * q^74 + 14100050 * q^76 + 14372400 * q^79 + 157124 * q^81 + 34235196 * q^84 + 1458888 * q^86 + 11981700 * q^89 + 11634528 * q^91 - 17950264 * q^94 + 15306718 * q^96 + 22866944 * q^99

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