LMFDB - Embedded newform 25.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,8,Mod(1,25)]

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([0]))

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

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	25.a (trivial)

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:	yes
Analytic conductor:	$7.80962563710$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.35890$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.7178 q^{2} +59.7424 q^{3} +222.356 q^{4} -1118.25 q^{6} -438.197 q^{7} -1766.14 q^{8} +1382.15 q^{9} +O(q^{10})$	\(q-18.7178 q^{2} +59.7424 q^{3} +222.356 q^{4} -1118.25 q^{6} -438.197 q^{7} -1766.14 q^{8} +1382.15 q^{9} +5759.12 q^{11} +13284.1 q^{12} +3530.42 q^{13} +8202.08 q^{14} +4596.61 q^{16} +23991.9 q^{17} -25870.8 q^{18} +16590.3 q^{19} -26178.9 q^{21} -107798. q^{22} +65626.9 q^{23} -105513. q^{24} -66081.7 q^{26} -48083.5 q^{27} -97435.6 q^{28} +134041. q^{29} +129002. q^{31} +140027. q^{32} +344064. q^{33} -449075. q^{34} +307330. q^{36} -161108. q^{37} -310534. q^{38} +210916. q^{39} -362989. q^{41} +490012. q^{42} -588189. q^{43} +1.28057e6 q^{44} -1.22839e6 q^{46} -343895. q^{47} +274612. q^{48} -631527. q^{49} +1.43333e6 q^{51} +785010. q^{52} +1.66139e6 q^{53} +900018. q^{54} +773915. q^{56} +991144. q^{57} -2.50896e6 q^{58} -2.54214e6 q^{59} +2.52337e6 q^{61} -2.41464e6 q^{62} -605655. q^{63} -3.20936e6 q^{64} -6.44011e6 q^{66} -1.56618e6 q^{67} +5.33474e6 q^{68} +3.92071e6 q^{69} -299354. q^{71} -2.44107e6 q^{72} -312494. q^{73} +3.01558e6 q^{74} +3.68895e6 q^{76} -2.52363e6 q^{77} -3.94788e6 q^{78} -1.95247e6 q^{79} -5.89539e6 q^{81} +6.79435e6 q^{82} +621372. q^{83} -5.82104e6 q^{84} +1.10096e7 q^{86} +8.00795e6 q^{87} -1.01714e7 q^{88} +5.78298e6 q^{89} -1.54702e6 q^{91} +1.45925e7 q^{92} +7.70690e6 q^{93} +6.43696e6 q^{94} +8.36554e6 q^{96} -7.20152e6 q^{97} +1.18208e7 q^{98} +7.95998e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{2} - 20 q^{3} + 96 q^{4} - 1016 q^{6} + 100 q^{7} - 1440 q^{8} + 5554 q^{9}+O(q^{10}) $ 2 * q - 20 * q^2 - 20 * q^3 + 96 * q^4 - 1016 * q^6 + 100 * q^7 - 1440 * q^8 + 5554 * q^9	\( 2 q - 20 q^{2} - 20 q^{3} + 96 q^{4} - 1016 q^{6} + 100 q^{7} - 1440 q^{8} + 5554 q^{9} + 4544 q^{11} + 23360 q^{12} - 3540 q^{13} + 7512 q^{14} + 20352 q^{16} + 27340 q^{17} - 31220 q^{18} + 38760 q^{19} - 69096 q^{21} - 106240 q^{22} + 124140 q^{23} - 131520 q^{24} - 57016 q^{26} - 206360 q^{27} - 165440 q^{28} - 72260 q^{29} + 306824 q^{31} + 78080 q^{32} + 440960 q^{33} - 453368 q^{34} - 219808 q^{36} + 123020 q^{37} - 338960 q^{38} + 774728 q^{39} + 264364 q^{41} + 545040 q^{42} - 423300 q^{43} + 1434112 q^{44} - 1303416 q^{46} + 105460 q^{47} - 981760 q^{48} - 1165414 q^{49} + 1166344 q^{51} + 1678400 q^{52} + 2391580 q^{53} + 1102960 q^{54} + 949440 q^{56} - 776720 q^{57} - 2244440 q^{58} - 1120120 q^{59} + 2257044 q^{61} - 2642640 q^{62} + 1639620 q^{63} - 5146624 q^{64} - 6564352 q^{66} - 4516460 q^{67} + 4911680 q^{68} - 745272 q^{69} + 621784 q^{71} - 1080480 q^{72} - 4569060 q^{73} + 2651272 q^{74} + 887680 q^{76} - 3177600 q^{77} - 4670800 q^{78} + 4333040 q^{79} - 2397878 q^{81} + 5989960 q^{82} + 9793020 q^{83} - 398208 q^{84} + 10798184 q^{86} + 24458920 q^{87} - 10567680 q^{88} + 6025620 q^{89} - 5352296 q^{91} + 7199040 q^{92} - 6473040 q^{93} + 5860792 q^{94} + 13305344 q^{96} - 4609540 q^{97} + 12505340 q^{98} + 2890688 q^{99}+O(q^{100}) \) 2 * q - 20 * q^2 - 20 * q^3 + 96 * q^4 - 1016 * q^6 + 100 * q^7 - 1440 * q^8 + 5554 * q^9 + 4544 * q^11 + 23360 * q^12 - 3540 * q^13 + 7512 * q^14 + 20352 * q^16 + 27340 * q^17 - 31220 * q^18 + 38760 * q^19 - 69096 * q^21 - 106240 * q^22 + 124140 * q^23 - 131520 * q^24 - 57016 * q^26 - 206360 * q^27 - 165440 * q^28 - 72260 * q^29 + 306824 * q^31 + 78080 * q^32 + 440960 * q^33 - 453368 * q^34 - 219808 * q^36 + 123020 * q^37 - 338960 * q^38 + 774728 * q^39 + 264364 * q^41 + 545040 * q^42 - 423300 * q^43 + 1434112 * q^44 - 1303416 * q^46 + 105460 * q^47 - 981760 * q^48 - 1165414 * q^49 + 1166344 * q^51 + 1678400 * q^52 + 2391580 * q^53 + 1102960 * q^54 + 949440 * q^56 - 776720 * q^57 - 2244440 * q^58 - 1120120 * q^59 + 2257044 * q^61 - 2642640 * q^62 + 1639620 * q^63 - 5146624 * q^64 - 6564352 * q^66 - 4516460 * q^67 + 4911680 * q^68 - 745272 * q^69 + 621784 * q^71 - 1080480 * q^72 - 4569060 * q^73 + 2651272 * q^74 + 887680 * q^76 - 3177600 * q^77 - 4670800 * q^78 + 4333040 * q^79 - 2397878 * q^81 + 5989960 * q^82 + 9793020 * q^83 - 398208 * q^84 + 10798184 * q^86 + 24458920 * q^87 - 10567680 * q^88 + 6025620 * q^89 - 5352296 * q^91 + 7199040 * q^92 - 6473040 * q^93 + 5860792 * q^94 + 13305344 * q^96 - 4609540 * q^97 + 12505340 * q^98 + 2890688 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−18.7178	−1.65444	−0.827218	−	0.561882i	$-0.810078\pi$
$2$	−18.7178	−1.65444	−0.827218	+	0.561882i	$0.810078\pi$
$3$	59.7424	1.27749	0.638746	−	0.769418i	$-0.279453\pi$
$3$	59.7424	1.27749	0.638746	+	0.769418i	$0.279453\pi$
$4$	222.356	1.73716
$5$	0	0
$5$	0	0
$6$	−1118.25	−2.11353
$7$	−438.197	−0.482865	−0.241433	−	0.970418i	$-0.577617\pi$
$7$	−438.197	−0.482865	−0.241433	+	0.970418i	$0.577617\pi$
$8$	−1766.14	−1.21958
$9$	1382.15	0.631986
$10$	0	0
$11$	5759.12	1.30461	0.652306	−	0.757955i	$-0.273802\pi$
$11$	5759.12	1.30461	0.652306	+	0.757955i	$0.273802\pi$
$12$	13284.1	2.21920
$13$	3530.42	0.445682	0.222841	−	0.974855i	$-0.428467\pi$
$13$	3530.42	0.445682	0.222841	+	0.974855i	$0.428467\pi$
$14$	8202.08	0.798869
$15$	0	0
$16$	4596.61	0.280555
$17$	23991.9	1.18439	0.592193	−	0.805797i	$-0.298262\pi$
$17$	23991.9	1.18439	0.592193	+	0.805797i	$0.298262\pi$
$18$	−25870.8	−1.04558
$19$	16590.3	0.554903	0.277451	−	0.960740i	$-0.410510\pi$
$19$	16590.3	0.554903	0.277451	+	0.960740i	$0.410510\pi$
$20$	0	0
$21$	−26178.9	−0.616856
$22$	−107798.	−2.15840
$23$	65626.9	1.12469	0.562347	−	0.826902i	$-0.309899\pi$
$23$	65626.9	1.12469	0.562347	+	0.826902i	$0.309899\pi$
$24$	−105513.	−1.55800
$25$	0	0
$26$	−66081.7	−0.737351
$27$	−48083.5	−0.470136
$28$	−97435.6	−0.838812
$29$	134041.	1.02058	0.510289	−	0.860003i	$-0.329539\pi$
$29$	134041.	1.02058	0.510289	+	0.860003i	$0.329539\pi$
$30$	0	0
$31$	129002.	0.777734	0.388867	−	0.921294i	$-0.372867\pi$
$31$	129002.	0.777734	0.388867	+	0.921294i	$0.372867\pi$
$32$	140027.	0.755417
$33$	344064.	1.66663
$34$	−449075.	−1.95949
$35$	0	0
$36$	307330.	1.09786
$37$	−161108.	−0.522890	−0.261445	−	0.965218i	$-0.584199\pi$
$37$	−161108.	−0.522890	−0.261445	+	0.965218i	$0.584199\pi$
$38$	−310534.	−0.918050
$39$	210916.	0.569355
$40$	0	0
$41$	−362989.	−0.822526	−0.411263	−	0.911517i	$-0.634912\pi$
$41$	−362989.	−0.822526	−0.411263	+	0.911517i	$0.634912\pi$
$42$	490012.	1.02055
$43$	−588189.	−1.12818	−0.564089	−	0.825714i	$-0.690772\pi$
$43$	−588189.	−1.12818	−0.564089	+	0.825714i	$0.690772\pi$
$44$	1.28057e6	2.26632
$45$	0	0
$46$	−1.22839e6	−1.86073
$47$	−343895.	−0.483151	−0.241576	−	0.970382i	$-0.577664\pi$
$47$	−343895.	−0.483151	−0.241576	+	0.970382i	$0.577664\pi$
$48$	274612.	0.358406
$49$	−631527.	−0.766841
$50$	0	0
$51$	1.43333e6	1.51304
$52$	785010.	0.774219
$53$	1.66139e6	1.53287	0.766436	−	0.642320i	$-0.222028\pi$
$53$	1.66139e6	1.53287	0.766436	+	0.642320i	$0.222028\pi$
$54$	900018.	0.777809
$55$	0	0
$56$	773915.	0.588891
$57$	991144.	0.708884
$58$	−2.50896e6	−1.68848
$59$	−2.54214e6	−1.61145	−0.805726	−	0.592289i	$-0.798224\pi$
$59$	−2.54214e6	−1.61145	−0.805726	+	0.592289i	$0.798224\pi$
$60$	0	0
$61$	2.52337e6	1.42340	0.711699	−	0.702484i	$-0.247926\pi$
$61$	2.52337e6	1.42340	0.711699	+	0.702484i	$0.247926\pi$
$62$	−2.41464e6	−1.28671
$63$	−605655.	−0.305164
$64$	−3.20936e6	−1.53034
$65$	0	0
$66$	−6.44011e6	−2.75734
$67$	−1.56618e6	−0.636178	−0.318089	−	0.948061i	$-0.603041\pi$
$67$	−1.56618e6	−0.636178	−0.318089	+	0.948061i	$0.603041\pi$
$68$	5.33474e6	2.05746
$69$	3.92071e6	1.43679
$70$	0	0
$71$	−299354.	−0.0992615	−0.0496307	−	0.998768i	$-0.515804\pi$
$71$	−299354.	−0.0992615	−0.0496307	+	0.998768i	$0.515804\pi$
$72$	−2.44107e6	−0.770755
$73$	−312494.	−0.0940183	−0.0470091	−	0.998894i	$-0.514969\pi$
$73$	−312494.	−0.0940183	−0.0470091	+	0.998894i	$0.514969\pi$
$74$	3.01558e6	0.865087
$75$	0	0
$76$	3.68895e6	0.963952
$77$	−2.52363e6	−0.629952
$78$	−3.94788e6	−0.941961
$79$	−1.95247e6	−0.445542	−0.222771	−	0.974871i	$-0.571510\pi$
$79$	−1.95247e6	−0.445542	−0.222771	+	0.974871i	$0.571510\pi$
$80$	0	0
$81$	−5.89539e6	−1.23258
$82$	6.79435e6	1.36082
$83$	621372.	0.119283	0.0596414	−	0.998220i	$-0.481004\pi$
$83$	621372.	0.119283	0.0596414	+	0.998220i	$0.481004\pi$
$84$	−5.82104e6	−1.07158
$85$	0	0
$86$	1.10096e7	1.86650
$87$	8.00795e6	1.30378
$88$	−1.01714e7	−1.59108
$89$	5.78298e6	0.869534	0.434767	−	0.900543i	$-0.356831\pi$
$89$	5.78298e6	0.869534	0.434767	+	0.900543i	$0.356831\pi$
$90$	0	0
$91$	−1.54702e6	−0.215204
$92$	1.45925e7	1.95377
$93$	7.70690e6	0.993549
$94$	6.43696e6	0.799343
$95$	0	0
$96$	8.36554e6	0.965039
$97$	−7.20152e6	−0.801167	−0.400584	−	0.916260i	$-0.631193\pi$
$97$	−7.20152e6	−0.801167	−0.400584	+	0.916260i	$0.631193\pi$
$98$	1.18208e7	1.26869
$99$	7.95998e6	0.824496
$100$	0	0
$101$	−2.91989e6	−0.281995	−0.140997	−	0.990010i	$-0.545031\pi$
$101$	−2.91989e6	−0.281995	−0.140997	+	0.990010i	$0.545031\pi$
$102$	−2.68288e7	−2.50323
$103$	−3.94639e6	−0.355852	−0.177926	−	0.984044i	$-0.556939\pi$
$103$	−3.94639e6	−0.355852	−0.177926	+	0.984044i	$0.556939\pi$
$104$	−6.23520e6	−0.543543
$105$	0	0
$106$	−3.10976e7	−2.53604
$107$	3.81991e6	0.301446	0.150723	−	0.988576i	$-0.451840\pi$
$107$	3.81991e6	0.301446	0.150723	+	0.988576i	$0.451840\pi$
$108$	−1.06917e7	−0.816699
$109$	−8.82259e6	−0.652534	−0.326267	−	0.945278i	$-0.605791\pi$
$109$	−8.82259e6	−0.652534	−0.326267	+	0.945278i	$0.605791\pi$
$110$	0	0
$111$	−9.62496e6	−0.667988
$112$	−2.01422e6	−0.135470
$113$	−2.12074e7	−1.38265	−0.691324	−	0.722545i	$-0.742972\pi$
$113$	−2.12074e7	−1.38265	−0.691324	+	0.722545i	$0.742972\pi$
$114$	−1.85520e7	−1.17280
$115$	0	0
$116$	2.98049e7	1.77290
$117$	4.87958e6	0.281664
$118$	4.75832e7	2.66604
$119$	−1.05132e7	−0.571898
$120$	0	0
$121$	1.36803e7	0.702015
$122$	−4.72319e7	−2.35492
$123$	−2.16858e7	−1.05077
$124$	2.86844e7	1.35105
$125$	0	0
$126$	1.13365e7	0.504874
$127$	2.55822e7	1.10822	0.554108	−	0.832445i	$-0.313059\pi$
$127$	2.55822e7	1.10822	0.554108	+	0.832445i	$0.313059\pi$
$128$	4.21487e7	1.77644
$129$	−3.51398e7	−1.44124
$130$	0	0
$131$	1.30640e7	0.507722	0.253861	−	0.967241i	$-0.418299\pi$
$131$	1.30640e7	0.507722	0.253861	+	0.967241i	$0.418299\pi$
$132$	7.65046e7	2.89520
$133$	−7.26982e6	−0.267943
$134$	2.93154e7	1.05252
$135$	0	0
$136$	−4.23729e7	−1.44445
$137$	−2.14021e7	−0.711106	−0.355553	−	0.934656i	$-0.615707\pi$
$137$	−2.14021e7	−0.711106	−0.355553	+	0.934656i	$0.615707\pi$
$138$	−7.33870e7	−2.37707
$139$	4.00656e7	1.26538	0.632688	−	0.774406i	$-0.281951\pi$
$139$	4.00656e7	1.26538	0.632688	+	0.774406i	$0.281951\pi$
$140$	0	0
$141$	−2.05451e7	−0.617222
$142$	5.60324e6	0.164222
$143$	2.03321e7	0.581442
$144$	6.35321e6	0.177307
$145$	0	0
$146$	5.84921e6	0.155547
$147$	−3.77289e7	−0.979633
$148$	−3.58233e7	−0.908341
$149$	−5.96142e7	−1.47638	−0.738190	−	0.674593i	$-0.764319\pi$
$149$	−5.96142e7	−1.47638	−0.738190	+	0.674593i	$0.764319\pi$
$150$	0	0
$151$	−5.21166e6	−0.123185	−0.0615924	−	0.998101i	$-0.519618\pi$
$151$	−5.21166e6	−0.123185	−0.0615924	+	0.998101i	$0.519618\pi$
$152$	−2.93007e7	−0.676746
$153$	3.31604e7	0.748514
$154$	4.72367e7	1.04222
$155$	0	0
$156$	4.68984e7	0.989058
$157$	1.10197e7	0.227259	0.113630	−	0.993523i	$-0.463752\pi$
$157$	1.10197e7	0.227259	0.113630	+	0.993523i	$0.463752\pi$
$158$	3.65458e7	0.737120
$159$	9.92554e7	1.95823
$160$	0	0
$161$	−2.87575e7	−0.543075
$162$	1.10349e8	2.03922
$163$	−2.32415e7	−0.420346	−0.210173	−	0.977664i	$-0.567403\pi$
$163$	−2.32415e7	−0.420346	−0.210173	+	0.977664i	$0.567403\pi$
$164$	−8.07127e7	−1.42886
$165$	0	0
$166$	−1.16307e7	−0.197346
$167$	5.84152e7	0.970550	0.485275	−	0.874361i	$-0.338719\pi$
$167$	5.84152e7	0.970550	0.485275	+	0.874361i	$0.338719\pi$
$168$	4.62355e7	0.752304
$169$	−5.02846e7	−0.801368
$170$	0	0
$171$	2.29303e7	0.350690
$172$	−1.30787e8	−1.95982
$173$	−1.18828e6	−0.0174485	−0.00872427	−	0.999962i	$-0.502777\pi$
$173$	−1.18828e6	−0.0174485	−0.00872427	+	0.999962i	$0.502777\pi$
$174$	−1.49891e8	−2.15702
$175$	0	0
$176$	2.64724e7	0.366015
$177$	−1.51873e8	−2.05862
$178$	−1.08245e8	−1.43859
$179$	1.28635e8	1.67638	0.838191	−	0.545377i	$-0.183613\pi$
$179$	1.28635e8	1.67638	0.838191	+	0.545377i	$0.183613\pi$
$180$	0	0
$181$	1.40320e8	1.75892	0.879458	−	0.475976i	$-0.157905\pi$
$181$	1.40320e8	1.75892	0.879458	+	0.475976i	$0.157905\pi$
$182$	2.89568e7	0.356041
$183$	1.50752e8	1.81838
$184$	−1.15906e8	−1.37165
$185$	0	0
$186$	−1.44256e8	−1.64376
$187$	1.38172e8	1.54516
$188$	−7.64671e7	−0.839309
$189$	2.10700e7	0.227012
$190$	0	0
$191$	−3.81784e7	−0.396461	−0.198231	−	0.980155i	$-0.563519\pi$
$191$	−3.81784e7	−0.396461	−0.198231	+	0.980155i	$0.563519\pi$
$192$	−1.91735e8	−1.95500
$193$	1.35915e8	1.36087	0.680436	−	0.732807i	$-0.261790\pi$
$193$	1.35915e8	1.36087	0.680436	+	0.732807i	$0.261790\pi$
$194$	1.34797e8	1.32548
$195$	0	0
$196$	−1.40424e8	−1.33212
$197$	−6.16154e7	−0.574193	−0.287096	−	0.957902i	$-0.592690\pi$
$197$	−6.16154e7	−0.574193	−0.287096	+	0.957902i	$0.592690\pi$
$198$	−1.48993e8	−1.36408
$199$	−1.84377e8	−1.65852	−0.829261	−	0.558862i	$-0.811238\pi$
$199$	−1.84377e8	−1.65852	−0.829261	+	0.558862i	$0.811238\pi$
$200$	0	0
$201$	−9.35671e7	−0.812713
$202$	5.46539e7	0.466542
$203$	−5.87365e7	−0.492801
$204$	3.18710e8	2.62839
$205$	0	0
$206$	7.38677e7	0.588734
$207$	9.07063e7	0.710790
$208$	1.62280e7	0.125038
$209$	9.55455e7	0.723933
$210$	0	0
$211$	1.71174e8	1.25444	0.627218	−	0.778844i	$-0.284194\pi$
$211$	1.71174e8	1.25444	0.627218	+	0.778844i	$0.284194\pi$
$212$	3.69420e8	2.66284
$213$	−1.78841e7	−0.126806
$214$	−7.15003e7	−0.498723
$215$	0	0
$216$	8.49220e7	0.573366
$217$	−5.65283e7	−0.375541
$218$	1.65139e8	1.07958
$219$	−1.86692e7	−0.120108
$220$	0	0
$221$	8.47014e7	0.527859
$222$	1.80158e8	1.10514
$223$	−2.67014e8	−1.61238	−0.806190	−	0.591657i	$-0.798474\pi$
$223$	−2.67014e8	−1.61238	−0.806190	+	0.591657i	$0.798474\pi$
$224$	−6.13594e7	−0.364765
$225$	0	0
$226$	3.96955e8	2.28750
$227$	−378641.	−0.00214851	−0.00107425	−	0.999999i	$-0.500342\pi$
$227$	−378641.	−0.00214851	−0.00107425	+	0.999999i	$0.500342\pi$
$228$	2.20387e8	1.23144
$229$	−2.24429e8	−1.23497	−0.617483	−	0.786584i	$-0.711848\pi$
$229$	−2.24429e8	−1.23497	−0.617483	+	0.786584i	$0.711848\pi$
$230$	0	0
$231$	−1.50767e8	−0.804759
$232$	−2.36735e8	−1.24467
$233$	−1.55173e8	−0.803656	−0.401828	−	0.915715i	$-0.631625\pi$
$233$	−1.55173e8	−0.803656	−0.401828	+	0.915715i	$0.631625\pi$
$234$	−9.13350e7	−0.465995
$235$	0	0
$236$	−5.65260e8	−2.79934
$237$	−1.16645e8	−0.569176
$238$	1.96783e8	0.946169
$239$	4.07160e7	0.192918	0.0964588	−	0.995337i	$-0.469248\pi$
$239$	4.07160e7	0.192918	0.0964588	+	0.995337i	$0.469248\pi$
$240$	0	0
$241$	−3.06501e8	−1.41050	−0.705249	−	0.708960i	$-0.749165\pi$
$241$	−3.06501e8	−1.41050	−0.705249	+	0.708960i	$0.749165\pi$
$242$	−2.56065e8	−1.16144
$243$	−2.47046e8	−1.10448
$244$	5.61086e8	2.47266
$245$	0	0
$246$	4.05911e8	1.73843
$247$	5.85708e7	0.247310
$248$	−2.27835e8	−0.948506
$249$	3.71222e7	0.152383
$250$	0	0
$251$	−1.30381e8	−0.520421	−0.260211	−	0.965552i	$-0.583792\pi$
$251$	−1.30381e8	−0.520421	−0.260211	+	0.965552i	$0.583792\pi$
$252$	−1.34671e8	−0.530117
$253$	3.77953e8	1.46729
$254$	−4.78842e8	−1.83347
$255$	0	0
$256$	−3.78133e8	−1.40866
$257$	−3.23514e8	−1.18885	−0.594426	−	0.804151i	$-0.702620\pi$
$257$	−3.23514e8	−1.18885	−0.594426	+	0.804151i	$0.702620\pi$
$258$	6.57740e8	2.38443
$259$	7.05969e7	0.252485
$260$	0	0
$261$	1.85266e8	0.644990
$262$	−2.44529e8	−0.839994
$263$	−2.39895e7	−0.0813159	−0.0406579	−	0.999173i	$-0.512945\pi$
$263$	−2.39895e7	−0.0813159	−0.0406579	+	0.999173i	$0.512945\pi$
$264$	−6.07663e8	−2.03259
$265$	0	0
$266$	1.36075e8	0.443295
$267$	3.45489e8	1.11082
$268$	−3.48249e8	−1.10514
$269$	−1.73612e8	−0.543809	−0.271905	−	0.962324i	$-0.587654\pi$
$269$	−1.73612e8	−0.543809	−0.271905	+	0.962324i	$0.587654\pi$
$270$	0	0
$271$	5.08478e8	1.55196	0.775978	−	0.630760i	$-0.217257\pi$
$271$	5.08478e8	1.55196	0.775978	+	0.630760i	$0.217257\pi$
$272$	1.10281e8	0.332285
$273$	−9.24226e7	−0.274922
$274$	4.00600e8	1.17648
$275$	0	0
$276$	8.71792e8	2.49592
$277$	6.01050e8	1.69915	0.849575	−	0.527468i	$-0.176859\pi$
$277$	6.01050e8	1.69915	0.849575	+	0.527468i	$0.176859\pi$
$278$	−7.49940e8	−2.09348
$279$	1.78301e8	0.491517
$280$	0	0
$281$	−6.36212e8	−1.71053	−0.855264	−	0.518193i	$-0.826605\pi$
$281$	−6.36212e8	−1.71053	−0.855264	+	0.518193i	$0.826605\pi$
$282$	3.84559e8	1.02115
$283$	−5.46181e7	−0.143247	−0.0716233	−	0.997432i	$-0.522818\pi$
$283$	−5.46181e7	−0.143247	−0.0716233	+	0.997432i	$0.522818\pi$
$284$	−6.65631e7	−0.172433
$285$	0	0
$286$	−3.80572e8	−0.961958
$287$	1.59061e8	0.397169
$288$	1.93539e8	0.477413
$289$	1.65271e8	0.402768
$290$	0	0
$291$	−4.30236e8	−1.02348
$292$	−6.94850e7	−0.163324
$293$	1.22481e8	0.284467	0.142234	−	0.989833i	$-0.454572\pi$
$293$	1.22481e8	0.284467	0.142234	+	0.989833i	$0.454572\pi$
$294$	7.06202e8	1.62074
$295$	0	0
$296$	2.84538e8	0.637704
$297$	−2.76919e8	−0.613345
$298$	1.11585e9	2.44257
$299$	2.31690e8	0.501255
$300$	0	0
$301$	2.57743e8	0.544758
$302$	9.75509e7	0.203801
$303$	−1.74441e8	−0.360246
$304$	7.62591e7	0.155681
$305$	0	0
$306$	−6.20690e8	−1.23837
$307$	−5.58187e8	−1.10102	−0.550510	−	0.834829i	$-0.685567\pi$
$307$	−5.58187e8	−1.10102	−0.550510	+	0.834829i	$0.685567\pi$
$308$	−5.61143e8	−1.09433
$309$	−2.35767e8	−0.454598
$310$	0	0
$311$	8.94564e7	0.168636	0.0843180	−	0.996439i	$-0.473129\pi$
$311$	8.94564e7	0.168636	0.0843180	+	0.996439i	$0.473129\pi$
$312$	−3.72506e8	−0.694372
$313$	2.75895e8	0.508556	0.254278	−	0.967131i	$-0.418162\pi$
$313$	2.75895e8	0.508556	0.254278	+	0.967131i	$0.418162\pi$
$314$	−2.06265e8	−0.375986
$315$	0	0
$316$	−4.34142e8	−0.773976
$317$	4.40449e8	0.776584	0.388292	−	0.921536i	$-0.373065\pi$
$317$	4.40449e8	0.776584	0.388292	+	0.921536i	$0.373065\pi$
$318$	−1.85784e9	−3.23977
$319$	7.71960e8	1.33146
$320$	0	0
$321$	2.28211e8	0.385095
$322$	5.38277e8	0.898483
$323$	3.98032e8	0.657218
$324$	−1.31088e9	−2.14118
$325$	0	0
$326$	4.35030e8	0.695436
$327$	−5.27082e8	−0.833607
$328$	6.41088e8	1.00313
$329$	1.50694e8	0.233297
$330$	0	0
$331$	1.68079e8	0.254751	0.127376	−	0.991855i	$-0.459345\pi$
$331$	1.68079e8	0.254751	0.127376	+	0.991855i	$0.459345\pi$
$332$	1.38166e8	0.207213
$333$	−2.22675e8	−0.330459
$334$	−1.09340e9	−1.60571
$335$	0	0
$336$	−1.20334e8	−0.173062
$337$	−8.38651e8	−1.19365	−0.596824	−	0.802372i	$-0.703571\pi$
$337$	−8.38651e8	−1.19365	−0.596824	+	0.802372i	$0.703571\pi$
$338$	9.41218e8	1.32581
$339$	−1.26698e9	−1.76632
$340$	0	0
$341$	7.42939e8	1.01464
$342$	−4.29205e8	−0.580195
$343$	6.37607e8	0.853146
$344$	1.03882e9	1.37590
$345$	0	0
$346$	2.22421e7	0.0288675
$347$	1.16128e9	1.49205	0.746023	−	0.665921i	$-0.231961\pi$
$347$	1.16128e9	1.49205	0.746023	+	0.665921i	$0.231961\pi$
$348$	1.78062e9	2.26487
$349$	−8.37482e8	−1.05460	−0.527298	−	0.849680i	$-0.676795\pi$
$349$	−8.37482e8	−1.05460	−0.527298	+	0.849680i	$0.676795\pi$
$350$	0	0
$351$	−1.69755e8	−0.209531
$352$	8.06432e8	0.985527
$353$	−7.61561e8	−0.921496	−0.460748	−	0.887531i	$-0.652419\pi$
$353$	−7.61561e8	−0.921496	−0.460748	+	0.887531i	$0.652419\pi$
$354$	2.84274e9	3.40585
$355$	0	0
$356$	1.28588e9	1.51052
$357$	−6.28081e8	−0.730596
$358$	−2.40776e9	−2.77347
$359$	3.19728e8	0.364712	0.182356	−	0.983233i	$-0.441628\pi$
$359$	3.19728e8	0.364712	0.182356	+	0.983233i	$0.441628\pi$
$360$	0	0
$361$	−6.18634e8	−0.692083
$362$	−2.62649e9	−2.91001
$363$	8.17293e8	0.896818
$364$	−3.43989e8	−0.373843
$365$	0	0
$366$	−2.82175e9	−3.00839
$367$	−1.25415e8	−0.132440	−0.0662199	−	0.997805i	$-0.521094\pi$
$367$	−1.25415e8	−0.132440	−0.0662199	+	0.997805i	$0.521094\pi$
$368$	3.01661e8	0.315538
$369$	−5.01706e8	−0.519825
$370$	0	0
$371$	−7.28016e8	−0.740171
$372$	1.71367e9	1.72595
$373$	−1.71505e9	−1.71118	−0.855588	−	0.517657i	$-0.826804\pi$
$373$	−1.71505e9	−1.71118	−0.855588	+	0.517657i	$0.826804\pi$
$374$	−2.58628e9	−2.55637
$375$	0	0
$376$	6.07365e8	0.589240
$377$	4.73223e8	0.454853
$378$	−3.94385e8	−0.375577
$379$	−1.07297e8	−0.101239	−0.0506196	−	0.998718i	$-0.516120\pi$
$379$	−1.07297e8	−0.101239	−0.0506196	+	0.998718i	$0.516120\pi$
$380$	0	0
$381$	1.52834e9	1.41574
$382$	7.14615e8	0.655919
$383$	6.77468e8	0.616160	0.308080	−	0.951361i	$-0.400314\pi$
$383$	6.77468e8	0.616160	0.308080	+	0.951361i	$0.400314\pi$
$384$	2.51807e9	2.26938
$385$	0	0
$386$	−2.54403e9	−2.25148
$387$	−8.12967e8	−0.712992
$388$	−1.60130e9	−1.39175
$389$	1.94836e9	1.67821	0.839104	−	0.543971i	$-0.183080\pi$
$389$	1.94836e9	1.67821	0.839104	+	0.543971i	$0.183080\pi$
$390$	0	0
$391$	1.57451e9	1.33207
$392$	1.11536e9	0.935222
$393$	7.80474e8	0.648611
$394$	1.15331e9	0.949965
$395$	0	0
$396$	1.76995e9	1.43228
$397$	1.11752e9	0.896369	0.448185	−	0.893941i	$-0.352071\pi$
$397$	1.11752e9	0.896369	0.448185	+	0.893941i	$0.352071\pi$
$398$	3.45113e9	2.74392
$399$	−4.34316e8	−0.342295
$400$	0	0
$401$	1.61315e9	1.24931	0.624655	−	0.780901i	$-0.285240\pi$
$401$	1.61315e9	1.24931	0.624655	+	0.780901i	$0.285240\pi$
$402$	1.75137e9	1.34458
$403$	4.55432e8	0.346622
$404$	−6.49254e8	−0.489869
$405$	0	0
$406$	1.09942e9	0.815308
$407$	−9.27838e8	−0.682169
$408$	−2.53146e9	−1.84527
$409$	1.32866e7	0.00960248	0.00480124	−	0.999988i	$-0.498472\pi$
$409$	1.32866e7	0.00960248	0.00480124	+	0.999988i	$0.498472\pi$
$410$	0	0
$411$	−1.27861e9	−0.908432
$412$	−8.77503e8	−0.618170
$413$	1.11396e9	0.778114
$414$	−1.69782e9	−1.17596
$415$	0	0
$416$	4.94354e8	0.336676
$417$	2.39361e9	1.61651
$418$	−1.78840e9	−1.19770
$419$	−5.93249e7	−0.0393992	−0.0196996	−	0.999806i	$-0.506271\pi$
$419$	−5.93249e7	−0.0393992	−0.0196996	+	0.999806i	$0.506271\pi$
$420$	0	0
$421$	−2.93348e9	−1.91600	−0.958000	−	0.286769i	$-0.907419\pi$
$421$	−2.93348e9	−1.91600	−0.958000	+	0.286769i	$0.907419\pi$
$422$	−3.20399e9	−2.07538
$423$	−4.75315e8	−0.305345
$424$	−2.93424e9	−1.86946
$425$	0	0
$426$	3.34751e8	0.209792
$427$	−1.10573e9	−0.687310
$428$	8.49380e8	0.523659
$429$	1.21469e9	0.742788
$430$	0	0
$431$	−1.96307e8	−0.118104	−0.0590520	−	0.998255i	$-0.518808\pi$
$431$	−1.96307e8	−0.118104	−0.0590520	+	0.998255i	$0.518808\pi$
$432$	−2.21021e8	−0.131899
$433$	7.68256e8	0.454777	0.227389	−	0.973804i	$-0.426981\pi$
$433$	7.68256e8	0.454777	0.227389	+	0.973804i	$0.426981\pi$
$434$	1.05809e9	0.621308
$435$	0	0
$436$	−1.96175e9	−1.13355
$437$	1.08877e9	0.624095
$438$	3.49446e8	0.198710
$439$	−1.35112e9	−0.762197	−0.381099	−	0.924534i	$-0.624454\pi$
$439$	−1.35112e9	−0.762197	−0.381099	+	0.924534i	$0.624454\pi$
$440$	0	0
$441$	−8.72866e8	−0.484633
$442$	−1.58542e9	−0.873308
$443$	−2.20956e9	−1.20752	−0.603759	−	0.797167i	$-0.706331\pi$
$443$	−2.20956e9	−1.20752	−0.603759	+	0.797167i	$0.706331\pi$
$444$	−2.14017e9	−1.16040
$445$	0	0
$446$	4.99792e9	2.66758
$447$	−3.56150e9	−1.88606
$448$	1.40633e9	0.738950
$449$	3.16264e9	1.64887	0.824436	−	0.565955i	$-0.191493\pi$
$449$	3.16264e9	1.64887	0.824436	+	0.565955i	$0.191493\pi$
$450$	0	0
$451$	−2.09050e9	−1.07308
$452$	−4.71558e9	−2.40188
$453$	−3.11357e8	−0.157368
$454$	7.08732e6	0.00355457
$455$	0	0
$456$	−1.75050e9	−0.864538
$457$	−5.97325e8	−0.292755	−0.146377	−	0.989229i	$-0.546761\pi$
$457$	−5.97325e8	−0.292755	−0.146377	+	0.989229i	$0.546761\pi$
$458$	4.20082e9	2.04317
$459$	−1.15361e9	−0.556821
$460$	0	0
$461$	−2.01803e9	−0.959346	−0.479673	−	0.877447i	$-0.659245\pi$
$461$	−2.01803e9	−0.959346	−0.479673	+	0.877447i	$0.659245\pi$
$462$	2.82204e9	1.33142
$463$	−1.32264e8	−0.0619310	−0.0309655	−	0.999520i	$-0.509858\pi$
$463$	−1.32264e8	−0.0619310	−0.0309655	+	0.999520i	$0.509858\pi$
$464$	6.16136e8	0.286328
$465$	0	0
$466$	2.90449e9	1.32960
$467$	3.62018e9	1.64483	0.822416	−	0.568887i	$-0.192626\pi$
$467$	3.62018e9	1.64483	0.822416	+	0.568887i	$0.192626\pi$
$468$	1.08500e9	0.489295
$469$	6.86293e8	0.307188
$470$	0	0
$471$	6.58344e8	0.290322
$472$	4.48976e9	1.96529
$473$	−3.38745e9	−1.47183
$474$	2.18334e9	0.941665
$475$	0	0
$476$	−2.33766e9	−0.993477
$477$	2.29629e9	0.968753
$478$	−7.62113e8	−0.319170
$479$	−2.77900e9	−1.15535	−0.577675	−	0.816267i	$-0.696040\pi$
$479$	−2.77900e9	−1.15535	−0.577675	+	0.816267i	$0.696040\pi$
$480$	0	0
$481$	−5.68778e8	−0.233042
$482$	5.73703e9	2.33358
$483$	−1.71804e9	−0.693774
$484$	3.04189e9	1.21951
$485$	0	0
$486$	4.62416e9	1.82728
$487$	−1.86684e9	−0.732414	−0.366207	−	0.930533i	$-0.619344\pi$
$487$	−1.86684e9	−0.732414	−0.366207	+	0.930533i	$0.619344\pi$
$488$	−4.45661e9	−1.73594
$489$	−1.38850e9	−0.536989
$490$	0	0
$491$	5.06515e9	1.93111	0.965555	−	0.260198i	$-0.0837879\pi$
$491$	5.06515e9	1.93111	0.965555	+	0.260198i	$0.0837879\pi$
$492$	−4.82197e9	−1.82535
$493$	3.21590e9	1.20876
$494$	−1.09632e9	−0.409158
$495$	0	0
$496$	5.92973e8	0.218197
$497$	1.31176e8	0.0479299
$498$	−6.94846e8	−0.252108
$499$	−5.56606e8	−0.200538	−0.100269	−	0.994960i	$-0.531970\pi$
$499$	−5.56606e8	−0.200538	−0.100269	+	0.994960i	$0.531970\pi$
$500$	0	0
$501$	3.48986e9	1.23987
$502$	2.44044e9	0.861004
$503$	7.79533e8	0.273116	0.136558	−	0.990632i	$-0.456396\pi$
$503$	7.79533e8	0.273116	0.136558	+	0.990632i	$0.456396\pi$
$504$	1.06967e9	0.372171
$505$	0	0
$506$	−7.07445e9	−2.42754
$507$	−3.00412e9	−1.02374
$508$	5.68835e9	1.92515
$509$	−1.12351e9	−0.377629	−0.188814	−	0.982013i	$-0.560464\pi$
$509$	−1.12351e9	−0.377629	−0.188814	+	0.982013i	$0.560464\pi$
$510$	0	0
$511$	1.36934e8	0.0453981
$512$	1.68278e9	0.554094
$513$	−7.97720e8	−0.260879
$514$	6.05547e9	1.96688
$515$	0	0
$516$	−7.81355e9	−2.50365
$517$	−1.98053e9	−0.630326
$518$	−1.32142e9	−0.417721
$519$	−7.09909e7	−0.0222904
$520$	0	0
$521$	−6.43550e8	−0.199366	−0.0996828	−	0.995019i	$-0.531783\pi$
$521$	−6.43550e8	−0.199366	−0.0996828	+	0.995019i	$0.531783\pi$
$522$	−3.46777e9	−1.06709
$523$	−6.30863e9	−1.92832	−0.964159	−	0.265324i	$-0.914521\pi$
$523$	−6.30863e9	−1.92832	−0.964159	+	0.265324i	$0.914521\pi$
$524$	2.90486e9	0.881993
$525$	0	0
$526$	4.49030e8	0.134532
$527$	3.09500e9	0.921137
$528$	1.58153e9	0.467582
$529$	9.02060e8	0.264936
$530$	0	0
$531$	−3.51362e9	−1.01841
$532$	−1.61649e9	−0.465459
$533$	−1.28150e9	−0.366585
$534$	−6.46679e9	−1.83778
$535$	0	0
$536$	2.76608e9	0.775868
$537$	7.68495e9	2.14156
$538$	3.24963e9	0.899697
$539$	−3.63704e9	−1.00043
$540$	0	0
$541$	−3.71277e9	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$541$	−3.71277e9	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$542$	−9.51758e9	−2.56761
$543$	8.38306e9	2.24700
$544$	3.35951e9	0.894705
$545$	0	0
$546$	1.72995e9	0.454840
$547$	−4.19241e9	−1.09524	−0.547619	−	0.836728i	$-0.684466\pi$
$547$	−4.19241e9	−1.09524	−0.547619	+	0.836728i	$0.684466\pi$
$548$	−4.75888e9	−1.23530
$549$	3.48768e9	0.899567
$550$	0	0
$551$	2.22379e9	0.566321
$552$	−6.92450e9	−1.75227
$553$	8.55564e8	0.215137
$554$	−1.12503e10	−2.81113
$555$	0	0
$556$	8.90883e9	2.19816
$557$	−2.52750e9	−0.619723	−0.309861	−	0.950782i	$-0.600283\pi$
$557$	−2.52750e9	−0.619723	−0.309861	+	0.950782i	$0.600283\pi$
$558$	−3.33740e9	−0.813182
$559$	−2.07656e9	−0.502808
$560$	0	0
$561$	8.25473e9	1.97393
$562$	1.19085e10	2.82996
$563$	4.12649e9	0.974543	0.487272	−	0.873250i	$-0.337992\pi$
$563$	4.12649e9	0.974543	0.487272	+	0.873250i	$0.337992\pi$
$564$	−4.56833e9	−1.07221
$565$	0	0
$566$	1.02233e9	0.236992
$567$	2.58334e9	0.595170
$568$	5.28700e8	0.121057
$569$	5.27044e8	0.119937	0.0599686	−	0.998200i	$-0.480900\pi$
$569$	5.27044e8	0.119937	0.0599686	+	0.998200i	$0.480900\pi$
$570$	0	0
$571$	−3.18083e9	−0.715014	−0.357507	−	0.933911i	$-0.616373\pi$
$571$	−3.18083e9	−0.715014	−0.357507	+	0.933911i	$0.616373\pi$
$572$	4.52097e9	1.01006
$573$	−2.28087e9	−0.506476
$574$	−2.97726e9	−0.657091
$575$	0	0
$576$	−4.43583e9	−0.967155
$577$	−4.88332e9	−1.05828	−0.529139	−	0.848535i	$-0.677485\pi$
$577$	−4.88332e9	−1.05828	−0.529139	+	0.848535i	$0.677485\pi$
$578$	−3.09351e9	−0.666354
$579$	8.11990e9	1.73850
$580$	0	0
$581$	−2.72283e8	−0.0575976
$582$	8.05307e9	1.69329
$583$	9.56814e9	1.99981
$584$	5.51907e8	0.114662
$585$	0	0
$586$	−2.29257e9	−0.470632
$587$	−8.66750e8	−0.176873	−0.0884363	−	0.996082i	$-0.528187\pi$
$587$	−8.66750e8	−0.176873	−0.0884363	+	0.996082i	$0.528187\pi$
$588$	−8.38925e9	−1.70178
$589$	2.14019e9	0.431567
$590$	0	0
$591$	−3.68105e9	−0.733527
$592$	−7.40549e8	−0.146699
$593$	1.43710e9	0.283007	0.141503	−	0.989938i	$-0.454806\pi$
$593$	1.43710e9	0.283007	0.141503	+	0.989938i	$0.454806\pi$
$594$	5.18331e9	1.01474
$595$	0	0
$596$	−1.32556e10	−2.56470
$597$	−1.10151e10	−2.11875
$598$	−4.33674e9	−0.829294
$599$	7.50069e9	1.42596	0.712980	−	0.701185i	$-0.247345\pi$
$599$	7.50069e9	1.42596	0.712980	+	0.701185i	$0.247345\pi$
$600$	0	0
$601$	5.05436e9	0.949741	0.474870	−	0.880056i	$-0.342495\pi$
$601$	5.05436e9	0.949741	0.474870	+	0.880056i	$0.342495\pi$
$602$	−4.82437e9	−0.901266
$603$	−2.16469e9	−0.402055
$604$	−1.15884e9	−0.213991
$605$	0	0
$606$	3.26515e9	0.596004
$607$	9.94598e8	0.180504	0.0902521	−	0.995919i	$-0.471233\pi$
$607$	9.94598e8	0.180504	0.0902521	+	0.995919i	$0.471233\pi$
$608$	2.32309e9	0.419183
$609$	−3.50906e9	−0.629550
$610$	0	0
$611$	−1.21409e9	−0.215332
$612$	7.37342e9	1.30029
$613$	−1.58283e9	−0.277539	−0.138769	−	0.990325i	$-0.544315\pi$
$613$	−1.58283e9	−0.277539	−0.138769	+	0.990325i	$0.544315\pi$
$614$	1.04480e10	1.82157
$615$	0	0
$616$	4.45707e9	0.768275
$617$	8.35079e9	1.43130	0.715648	−	0.698461i	$-0.246132\pi$
$617$	8.35079e9	1.43130	0.715648	+	0.698461i	$0.246132\pi$
$618$	4.41303e9	0.752103
$619$	1.12049e9	0.189886	0.0949428	−	0.995483i	$-0.469733\pi$
$619$	1.12049e9	0.189886	0.0949428	+	0.995483i	$0.469733\pi$
$620$	0	0
$621$	−3.15557e9	−0.528758
$622$	−1.67443e9	−0.278997
$623$	−2.53408e9	−0.419868
$624$	9.69497e8	0.159735
$625$	0	0
$626$	−5.16415e9	−0.841372
$627$	5.70812e9	0.924819
$628$	2.45030e9	0.394785
$629$	−3.86528e9	−0.619303
$630$	0	0
$631$	3.72303e9	0.589921	0.294961	−	0.955509i	$-0.404693\pi$
$631$	3.72303e9	0.589921	0.294961	+	0.955509i	$0.404693\pi$
$632$	3.44832e9	0.543372
$633$	1.02263e10	1.60253
$634$	−8.24424e9	−1.28481
$635$	0	0
$636$	2.20700e10	3.40176
$637$	−2.22956e9	−0.341767
$638$	−1.44494e10	−2.20281
$639$	−4.13753e8	−0.0627318
$640$	0	0
$641$	−1.08733e9	−0.163065	−0.0815323	−	0.996671i	$-0.525981\pi$
$641$	−1.08733e9	−0.163065	−0.0815323	+	0.996671i	$0.525981\pi$
$642$	−4.27160e9	−0.637115
$643$	−9.30287e9	−1.38000	−0.689999	−	0.723810i	$-0.742389\pi$
$643$	−9.30287e9	−1.38000	−0.689999	+	0.723810i	$0.742389\pi$
$644$	−6.39440e9	−0.943407
$645$	0	0
$646$	−7.45029e9	−1.08733
$647$	3.75633e9	0.545254	0.272627	−	0.962120i	$-0.412108\pi$
$647$	3.75633e9	0.545254	0.272627	+	0.962120i	$0.412108\pi$
$648$	1.04121e10	1.50323
$649$	−1.46405e10	−2.10232
$650$	0	0
$651$	−3.37714e9	−0.479750
$652$	−5.16788e9	−0.730207
$653$	−6.47262e9	−0.909671	−0.454835	−	0.890576i	$-0.650302\pi$
$653$	−6.47262e9	−0.909671	−0.454835	+	0.890576i	$0.650302\pi$
$654$	9.86582e9	1.37915
$655$	0	0
$656$	−1.66852e9	−0.230764
$657$	−4.31915e8	−0.0594182
$658$	−2.82065e9	−0.385975
$659$	8.31257e9	1.13145	0.565726	−	0.824593i	$-0.308596\pi$
$659$	8.31257e9	1.13145	0.565726	+	0.824593i	$0.308596\pi$
$660$	0	0
$661$	−1.42966e9	−0.192543	−0.0962714	−	0.995355i	$-0.530692\pi$
$661$	−1.42966e9	−0.192543	−0.0962714	+	0.995355i	$0.530692\pi$
$662$	−3.14608e9	−0.421470
$663$	5.06026e9	0.674335
$664$	−1.09743e9	−0.145475
$665$	0	0
$666$	4.16799e9	0.546723
$667$	8.79672e9	1.14784
$668$	1.29890e10	1.68600
$669$	−1.59521e10	−2.05980
$670$	0	0
$671$	1.45324e10	1.85698
$672$	−3.66575e9	−0.465984
$673$	−7.03224e9	−0.889285	−0.444642	−	0.895708i	$-0.646669\pi$
$673$	−7.03224e9	−0.889285	−0.444642	+	0.895708i	$0.646669\pi$
$674$	1.56977e10	1.97481
$675$	0	0
$676$	−1.11811e10	−1.39210
$677$	5.75322e8	0.0712608	0.0356304	−	0.999365i	$-0.488656\pi$
$677$	5.75322e8	0.0712608	0.0356304	+	0.999365i	$0.488656\pi$
$678$	2.37150e10	2.92227
$679$	3.15568e9	0.386856
$680$	0	0
$681$	−2.26209e7	−0.00274470
$682$	−1.39062e10	−1.67866
$683$	−7.67069e9	−0.921217	−0.460609	−	0.887603i	$-0.652369\pi$
$683$	−7.67069e9	−0.921217	−0.460609	+	0.887603i	$0.652369\pi$
$684$	5.09870e9	0.609204
$685$	0	0
$686$	−1.19346e10	−1.41148
$687$	−1.34079e10	−1.57766
$688$	−2.70368e9	−0.316516
$689$	5.86541e9	0.683173
$690$	0	0
$691$	3.62855e9	0.418369	0.209185	−	0.977876i	$-0.432919\pi$
$691$	3.62855e9	0.418369	0.209185	+	0.977876i	$0.432919\pi$
$692$	−2.64222e8	−0.0303108
$693$	−3.48804e9	−0.398121
$694$	−2.17365e10	−2.46849
$695$	0	0
$696$	−1.41431e10	−1.59006
$697$	−8.70878e9	−0.974188
$698$	1.56758e10	1.74476
$699$	−9.27040e9	−1.02666
$700$	0	0
$701$	−1.71375e10	−1.87904	−0.939518	−	0.342499i	$-0.888727\pi$
$701$	−1.71375e10	−1.87904	−0.939518	+	0.342499i	$0.888727\pi$
$702$	3.17744e9	0.346655
$703$	−2.67283e9	−0.290153
$704$	−1.84831e10	−1.99651
$705$	0	0
$706$	1.42547e10	1.52456
$707$	1.27948e9	0.136166
$708$	−3.37700e10	−3.57614
$709$	1.34055e10	1.41260	0.706302	−	0.707911i	$-0.250362\pi$
$709$	1.34055e10	1.41260	0.706302	+	0.707911i	$0.250362\pi$
$710$	0	0
$711$	−2.69860e9	−0.281576
$712$	−1.02135e10	−1.06046
$713$	8.46601e9	0.874712
$714$	1.17563e10	1.20872
$715$	0	0
$716$	2.86027e10	2.91214
$717$	2.43247e9	0.246451
$718$	−5.98460e9	−0.603392
$719$	−5.99982e9	−0.601987	−0.300994	−	0.953626i	$-0.597318\pi$
$719$	−5.99982e9	−0.601987	−0.300994	+	0.953626i	$0.597318\pi$
$720$	0	0
$721$	1.72929e9	0.171829
$722$	1.15795e10	1.14501
$723$	−1.83111e10	−1.80190
$724$	3.12010e10	3.05551
$725$	0	0
$726$	−1.52979e10	−1.48373
$727$	1.84312e10	1.77903	0.889513	−	0.456909i	$-0.151044\pi$
$727$	1.84312e10	1.77903	0.889513	+	0.456909i	$0.151044\pi$
$728$	2.73225e9	0.262458
$729$	−1.86590e9	−0.178378
$730$	0	0
$731$	−1.41118e10	−1.33620
$732$	3.35206e10	3.15881
$733$	7.01069e9	0.657502	0.328751	−	0.944417i	$-0.393372\pi$
$733$	7.01069e9	0.657502	0.328751	+	0.944417i	$0.393372\pi$
$734$	2.34749e9	0.219113
$735$	0	0
$736$	9.18953e9	0.849613
$737$	−9.01980e9	−0.829966
$738$	9.39083e9	0.860016
$739$	3.58081e9	0.326382	0.163191	−	0.986595i	$-0.447821\pi$
$739$	3.58081e9	0.326382	0.163191	+	0.986595i	$0.447821\pi$
$740$	0	0
$741$	3.49916e9	0.315936
$742$	1.36268e10	1.22456
$743$	4.23347e9	0.378648	0.189324	−	0.981915i	$-0.439370\pi$
$743$	4.23347e9	0.378648	0.189324	+	0.981915i	$0.439370\pi$
$744$	−1.36114e10	−1.21171
$745$	0	0
$746$	3.21019e10	2.83103
$747$	8.58830e8	0.0753851
$748$	3.07234e10	2.68419
$749$	−1.67387e9	−0.145558
$750$	0	0
$751$	3.94072e9	0.339497	0.169749	−	0.985487i	$-0.445704\pi$
$751$	3.94072e9	0.339497	0.169749	+	0.985487i	$0.445704\pi$
$752$	−1.58075e9	−0.135550
$753$	−7.78925e9	−0.664834
$754$	−8.85769e9	−0.752524
$755$	0	0
$756$	4.68505e9	0.394355
$757$	8.01225e8	0.0671303	0.0335652	−	0.999437i	$-0.489314\pi$
$757$	8.01225e8	0.0671303	0.0335652	+	0.999437i	$0.489314\pi$
$758$	2.00836e9	0.167494
$759$	2.25798e10	1.87445
$760$	0	0
$761$	−3.84688e9	−0.316419	−0.158209	−	0.987406i	$-0.550572\pi$
$761$	−3.84688e9	−0.316419	−0.158209	+	0.987406i	$0.550572\pi$
$762$	−2.86072e10	−2.34225
$763$	3.86603e9	0.315086
$764$	−8.48919e9	−0.688715
$765$	0	0
$766$	−1.26807e10	−1.01940
$767$	−8.97482e9	−0.718194
$768$	−2.25906e10	−1.79955
$769$	4.40578e9	0.349366	0.174683	−	0.984625i	$-0.444110\pi$
$769$	4.40578e9	0.349366	0.174683	+	0.984625i	$0.444110\pi$
$770$	0	0
$771$	−1.93275e10	−1.51875
$772$	3.02216e10	2.36405
$773$	1.54451e10	1.20272	0.601359	−	0.798979i	$-0.294626\pi$
$773$	1.54451e10	1.20272	0.601359	+	0.798979i	$0.294626\pi$
$774$	1.52170e10	1.17960
$775$	0	0
$776$	1.27189e10	0.977085
$777$	4.21762e9	0.322548
$778$	−3.64690e10	−2.77649
$779$	−6.02210e9	−0.456422
$780$	0	0
$781$	−1.72401e9	−0.129498
$782$	−2.94714e10	−2.20382
$783$	−6.44518e9	−0.479810
$784$	−2.90288e9	−0.215141
$785$	0	0
$786$	−1.46088e10	−1.07309
$787$	2.40331e10	1.75751	0.878755	−	0.477274i	$-0.158375\pi$
$787$	2.40331e10	1.75751	0.878755	+	0.477274i	$0.158375\pi$
$788$	−1.37006e10	−0.997463
$789$	−1.43319e9	−0.103880
$790$	0	0
$791$	9.29299e9	0.667633
$792$	−1.40584e10	−1.00554
$793$	8.90856e9	0.634383
$794$	−2.09174e10	−1.48299
$795$	0	0
$796$	−4.09973e10	−2.88111
$797$	−1.37624e10	−0.962918	−0.481459	−	0.876469i	$-0.659893\pi$
$797$	−1.37624e10	−0.962918	−0.481459	+	0.876469i	$0.659893\pi$
$798$	8.12944e9	0.566305
$799$	−8.25068e9	−0.572237
$800$	0	0
$801$	7.99296e9	0.549533
$802$	−3.01947e10	−2.06690
$803$	−1.79969e9	−0.122657
$804$	−2.08052e10	−1.41181
$805$	0	0
$806$	−8.52468e9	−0.573463
$807$	−1.03720e10	−0.694712
$808$	5.15692e9	0.343914
$809$	5.00163e9	0.332118	0.166059	−	0.986116i	$-0.446896\pi$
$809$	5.00163e9	0.332118	0.166059	+	0.986116i	$0.446896\pi$
$810$	0	0
$811$	8.14966e9	0.536496	0.268248	−	0.963350i	$-0.413555\pi$
$811$	8.14966e9	0.536496	0.268248	+	0.963350i	$0.413555\pi$
$812$	−1.30604e10	−0.856073
$813$	3.03777e10	1.98261
$814$	1.73671e10	1.12860
$815$	0	0
$816$	6.58847e9	0.424491
$817$	−9.75824e9	−0.626029
$818$	−2.48697e8	−0.0158867
$819$	−2.13822e9	−0.136006
$820$	0	0
$821$	1.30820e10	0.825035	0.412518	−	0.910950i	$-0.364650\pi$
$821$	1.30820e10	0.825035	0.412518	+	0.910950i	$0.364650\pi$
$822$	2.39328e10	1.50294
$823$	3.12969e10	1.95705	0.978524	−	0.206133i	$-0.0660881\pi$
$823$	3.12969e10	1.95705	0.978524	+	0.206133i	$0.0660881\pi$
$824$	6.96985e9	0.433989
$825$	0	0
$826$	−2.08508e10	−1.28734
$827$	1.98645e10	1.22126	0.610630	−	0.791916i	$-0.290916\pi$
$827$	1.98645e10	1.22126	0.610630	+	0.791916i	$0.290916\pi$
$828$	2.01691e10	1.23475
$829$	1.06183e10	0.647310	0.323655	−	0.946175i	$-0.395088\pi$
$829$	1.06183e10	0.647310	0.323655	+	0.946175i	$0.395088\pi$
$830$	0	0
$831$	3.59082e10	2.17065
$832$	−1.13304e10	−0.682046
$833$	−1.51515e10	−0.908235
$834$	−4.48032e10	−2.67441
$835$	0	0
$836$	2.12451e10	1.25758
$837$	−6.20288e9	−0.365640
$838$	1.11043e9	0.0651834
$839$	4.19640e9	0.245307	0.122654	−	0.992450i	$-0.460860\pi$
$839$	4.19640e9	0.245307	0.122654	+	0.992450i	$0.460860\pi$
$840$	0	0
$841$	7.17225e8	0.0415785
$842$	5.49082e10	3.16990
$843$	−3.80088e10	−2.18518
$844$	3.80615e10	2.17915
$845$	0	0
$846$	8.89685e9	0.505173
$847$	−5.99465e9	−0.338979
$848$	7.63676e9	0.430055
$849$	−3.26302e9	−0.182996
$850$	0	0
$851$	−1.05730e10	−0.588091
$852$	−3.97664e9	−0.220281
$853$	−2.52606e10	−1.39355	−0.696775	−	0.717289i	$-0.745383\pi$
$853$	−2.52606e10	−1.39355	−0.696775	+	0.717289i	$0.745383\pi$
$854$	2.06969e10	1.13711
$855$	0	0
$856$	−6.74648e9	−0.367637
$857$	−1.07248e10	−0.582046	−0.291023	−	0.956716i	$-0.593996\pi$
$857$	−1.07248e10	−0.582046	−0.291023	+	0.956716i	$0.593996\pi$
$858$	−2.27363e10	−1.22889
$859$	−3.02643e10	−1.62913	−0.814563	−	0.580075i	$-0.803023\pi$
$859$	−3.02643e10	−1.62913	−0.814563	+	0.580075i	$0.803023\pi$
$860$	0	0
$861$	9.50266e9	0.507381
$862$	3.67443e9	0.195396
$863$	1.39177e10	0.737106	0.368553	−	0.929607i	$-0.379853\pi$
$863$	1.39177e10	0.737106	0.368553	+	0.929607i	$0.379853\pi$
$864$	−6.73299e9	−0.355148
$865$	0	0
$866$	−1.43801e10	−0.752399
$867$	9.87370e9	0.514533
$868$	−1.25694e10	−0.652373
$869$	−1.12445e10	−0.581260
$870$	0	0
$871$	−5.52926e9	−0.283533
$872$	1.55819e10	0.795815
$873$	−9.95360e9	−0.506326
$874$	−2.03794e10	−1.03253
$875$	0	0
$876$	−4.15120e9	−0.208646
$877$	1.81702e10	0.909622	0.454811	−	0.890588i	$-0.349707\pi$
$877$	1.81702e10	0.909622	0.454811	+	0.890588i	$0.349707\pi$
$878$	2.52900e10	1.26101
$879$	7.31731e9	0.363404
$880$	0	0
$881$	−3.14352e10	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$881$	−3.14352e10	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$882$	1.63381e10	0.801793
$883$	2.11722e10	1.03491	0.517457	−	0.855709i	$-0.326879\pi$
$883$	2.11722e10	1.03491	0.517457	+	0.855709i	$0.326879\pi$
$884$	1.88339e10	0.916973
$885$	0	0
$886$	4.13582e10	1.99776
$887$	−3.72771e10	−1.79353	−0.896766	−	0.442505i	$-0.854090\pi$
$887$	−3.72771e10	−1.79353	−0.896766	+	0.442505i	$0.854090\pi$
$888$	1.69990e10	0.814662
$889$	−1.12100e10	−0.535119
$890$	0	0
$891$	−3.39523e10	−1.60804
$892$	−5.93722e10	−2.80095
$893$	−5.70532e9	−0.268102
$894$	6.66634e10	3.12037
$895$	0	0
$896$	−1.84694e10	−0.857780
$897$	1.38417e10	0.640350
$898$	−5.91976e10	−2.72795
$899$	1.72916e10	0.793738
$900$	0	0
$901$	3.98599e10	1.81551
$902$	3.91295e10	1.77534
$903$	1.53982e10	0.695923
$904$	3.74551e10	1.68625
$905$	0	0
$906$	5.82792e9	0.260354
$907$	−1.00189e10	−0.445854	−0.222927	−	0.974835i	$-0.571561\pi$
$907$	−1.00189e10	−0.445854	−0.222927	+	0.974835i	$0.571561\pi$
$908$	−8.41930e7	−0.00373229
$909$	−4.03573e9	−0.178217
$910$	0	0
$911$	3.16088e10	1.38514	0.692571	−	0.721350i	$-0.256478\pi$
$911$	3.16088e10	1.38514	0.692571	+	0.721350i	$0.256478\pi$
$912$	4.55590e9	0.198881
$913$	3.57855e9	0.155618
$914$	1.11806e10	0.484344
$915$	0	0
$916$	−4.99032e10	−2.14533
$917$	−5.72460e9	−0.245161
$918$	2.15931e10	0.921225
$919$	7.18397e8	0.0305324	0.0152662	−	0.999883i	$-0.495140\pi$
$919$	7.18397e8	0.0305324	0.0152662	+	0.999883i	$0.495140\pi$
$920$	0	0
$921$	−3.33474e10	−1.40654
$922$	3.77732e10	1.58718
$923$	−1.05685e9	−0.0442390
$924$	−3.35240e10	−1.39799
$925$	0	0
$926$	2.47569e9	0.102461
$927$	−5.45451e9	−0.224893
$928$	1.87694e10	0.770962
$929$	−1.63954e10	−0.670915	−0.335458	−	0.942055i	$-0.608891\pi$
$929$	−1.63954e10	−0.670915	−0.335458	+	0.942055i	$0.608891\pi$
$930$	0	0
$931$	−1.04772e10	−0.425522
$932$	−3.45036e10	−1.39608
$933$	5.34434e9	0.215431
$934$	−6.77618e10	−2.72127
$935$	0	0
$936$	−8.61800e9	−0.343511
$937$	9.79890e9	0.389125	0.194562	−	0.980890i	$-0.437671\pi$
$937$	9.79890e9	0.389125	0.194562	+	0.980890i	$0.437671\pi$
$938$	−1.28459e10	−0.508223
$939$	1.64826e10	0.649676
$940$	0	0
$941$	1.68542e10	0.659394	0.329697	−	0.944087i	$-0.393053\pi$
$941$	1.68542e10	0.659394	0.329697	+	0.944087i	$0.393053\pi$
$942$	−1.23227e10	−0.480319
$943$	−2.38218e10	−0.925090
$944$	−1.16852e10	−0.452100
$945$	0	0
$946$	6.34056e10	2.43506
$947$	−2.47225e10	−0.945948	−0.472974	−	0.881076i	$-0.656820\pi$
$947$	−2.47225e10	−0.945948	−0.472974	+	0.881076i	$0.656820\pi$
$948$	−2.59367e10	−0.988748
$949$	−1.10324e9	−0.0419022
$950$	0	0
$951$	2.63135e10	0.992080
$952$	1.85677e10	0.697474
$953$	4.36738e9	0.163454	0.0817271	−	0.996655i	$-0.473956\pi$
$953$	4.36738e9	0.163454	0.0817271	+	0.996655i	$0.473956\pi$
$954$	−4.29816e10	−1.60274
$955$	0	0
$956$	9.05343e9	0.335128
$957$	4.61188e10	1.70093
$958$	5.20167e10	1.91145
$959$	9.37832e9	0.343368
$960$	0	0
$961$	−1.08711e10	−0.395130
$962$	1.06463e10	0.385554
$963$	5.27970e9	0.190510
$964$	−6.81523e10	−2.45025
$965$	0	0
$966$	3.21579e10	1.14780
$967$	1.00818e10	0.358547	0.179274	−	0.983799i	$-0.442625\pi$
$967$	1.00818e10	0.358547	0.179274	+	0.983799i	$0.442625\pi$
$968$	−2.41612e10	−0.856161
$969$	2.37794e10	0.839591
$970$	0	0
$971$	−1.19066e10	−0.417369	−0.208685	−	0.977983i	$-0.566918\pi$
$971$	−1.19066e10	−0.417369	−0.208685	+	0.977983i	$0.566918\pi$
$972$	−5.49322e10	−1.91865
$973$	−1.75566e10	−0.611006
$974$	3.49432e10	1.21173
$975$	0	0
$976$	1.15989e10	0.399341
$977$	−1.99598e10	−0.684741	−0.342370	−	0.939565i	$-0.611230\pi$
$977$	−1.99598e10	−0.684741	−0.342370	+	0.939565i	$0.611230\pi$
$978$	2.59897e10	0.888414
$979$	3.33049e10	1.13441
$980$	0	0
$981$	−1.21942e10	−0.412392
$982$	−9.48085e10	−3.19490
$983$	3.22193e10	1.08188	0.540940	−	0.841061i	$-0.318069\pi$
$983$	3.22193e10	1.08188	0.540940	+	0.841061i	$0.318069\pi$
$984$	3.83001e10	1.28150
$985$	0	0
$986$	−6.01947e10	−1.99981
$987$	9.00280e9	0.298035
$988$	1.30236e10	0.429616
$989$	−3.86010e10	−1.26885
$990$	0	0
$991$	−1.42552e10	−0.465281	−0.232640	−	0.972563i	$-0.574737\pi$
$991$	−1.42552e10	−0.465281	−0.232640	+	0.972563i	$0.574737\pi$
$992$	1.80638e10	0.587513
$993$	1.00415e10	0.325443
$994$	−2.45532e9	−0.0792969
$995$	0	0
$996$	8.25435e9	0.264713
$997$	2.73327e10	0.873475	0.436737	−	0.899589i	$-0.356134\pi$
$997$	2.73327e10	0.873475	0.436737	+	0.899589i	$0.356134\pi$
$998$	1.04184e10	0.331776
$999$	7.74662e9	0.245829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.a.b.1.1		2
3.2	odd	2		225.8.a.w.1.2		2
4.3	odd	2		400.8.a.bb.1.1		2
5.2	odd	4		25.8.b.c.24.1		4
5.3	odd	4		25.8.b.c.24.4		4
5.4	even	2		5.8.a.b.1.2	✓	2
15.2	even	4		225.8.b.m.199.4		4
15.8	even	4		225.8.b.m.199.1		4
15.14	odd	2		45.8.a.h.1.1		2
20.3	even	4		400.8.c.m.49.2		4
20.7	even	4		400.8.c.m.49.3		4
20.19	odd	2		80.8.a.g.1.2		2
35.34	odd	2		245.8.a.c.1.2		2
40.19	odd	2		320.8.a.u.1.1		2
40.29	even	2		320.8.a.l.1.2		2
55.54	odd	2		605.8.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.b.1.2	✓	2	5.4	even	2
25.8.a.b.1.1		2	1.1	even	1	trivial
25.8.b.c.24.1		4	5.2	odd	4
25.8.b.c.24.4		4	5.3	odd	4
45.8.a.h.1.1		2	15.14	odd	2
80.8.a.g.1.2		2	20.19	odd	2
225.8.a.w.1.2		2	3.2	odd	2
225.8.b.m.199.1		4	15.8	even	4
225.8.b.m.199.4		4	15.2	even	4
245.8.a.c.1.2		2	35.34	odd	2
320.8.a.l.1.2		2	40.29	even	2
320.8.a.u.1.1		2	40.19	odd	2
400.8.a.bb.1.1		2	4.3	odd	2
400.8.c.m.49.2		4	20.3	even	4
400.8.c.m.49.3		4	20.7	even	4
605.8.a.d.1.1		2	55.54	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.a (trivial)

\(f(q)\)	\(=\)	\(q-18.7178 q^{2} +59.7424 q^{3} +222.356 q^{4} -1118.25 q^{6} -438.197 q^{7} -1766.14 q^{8} +1382.15 q^{9} +O(q^{10})\)	\(q-18.7178 q^{2} +59.7424 q^{3} +222.356 q^{4} -1118.25 q^{6} -438.197 q^{7} -1766.14 q^{8} +1382.15 q^{9} +5759.12 q^{11} +13284.1 q^{12} +3530.42 q^{13} +8202.08 q^{14} +4596.61 q^{16} +23991.9 q^{17} -25870.8 q^{18} +16590.3 q^{19} -26178.9 q^{21} -107798. q^{22} +65626.9 q^{23} -105513. q^{24} -66081.7 q^{26} -48083.5 q^{27} -97435.6 q^{28} +134041. q^{29} +129002. q^{31} +140027. q^{32} +344064. q^{33} -449075. q^{34} +307330. q^{36} -161108. q^{37} -310534. q^{38} +210916. q^{39} -362989. q^{41} +490012. q^{42} -588189. q^{43} +1.28057e6 q^{44} -1.22839e6 q^{46} -343895. q^{47} +274612. q^{48} -631527. q^{49} +1.43333e6 q^{51} +785010. q^{52} +1.66139e6 q^{53} +900018. q^{54} +773915. q^{56} +991144. q^{57} -2.50896e6 q^{58} -2.54214e6 q^{59} +2.52337e6 q^{61} -2.41464e6 q^{62} -605655. q^{63} -3.20936e6 q^{64} -6.44011e6 q^{66} -1.56618e6 q^{67} +5.33474e6 q^{68} +3.92071e6 q^{69} -299354. q^{71} -2.44107e6 q^{72} -312494. q^{73} +3.01558e6 q^{74} +3.68895e6 q^{76} -2.52363e6 q^{77} -3.94788e6 q^{78} -1.95247e6 q^{79} -5.89539e6 q^{81} +6.79435e6 q^{82} +621372. q^{83} -5.82104e6 q^{84} +1.10096e7 q^{86} +8.00795e6 q^{87} -1.01714e7 q^{88} +5.78298e6 q^{89} -1.54702e6 q^{91} +1.45925e7 q^{92} +7.70690e6 q^{93} +6.43696e6 q^{94} +8.36554e6 q^{96} -7.20152e6 q^{97} +1.18208e7 q^{98} +7.95998e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{2} - 20 q^{3} + 96 q^{4} - 1016 q^{6} + 100 q^{7} - 1440 q^{8} + 5554 q^{9}+O(q^{10}) \) 2 * q - 20 * q^2 - 20 * q^3 + 96 * q^4 - 1016 * q^6 + 100 * q^7 - 1440 * q^8 + 5554 * q^9	\( 2 q - 20 q^{2} - 20 q^{3} + 96 q^{4} - 1016 q^{6} + 100 q^{7} - 1440 q^{8} + 5554 q^{9} + 4544 q^{11} + 23360 q^{12} - 3540 q^{13} + 7512 q^{14} + 20352 q^{16} + 27340 q^{17} - 31220 q^{18} + 38760 q^{19} - 69096 q^{21} - 106240 q^{22} + 124140 q^{23} - 131520 q^{24} - 57016 q^{26} - 206360 q^{27} - 165440 q^{28} - 72260 q^{29} + 306824 q^{31} + 78080 q^{32} + 440960 q^{33} - 453368 q^{34} - 219808 q^{36} + 123020 q^{37} - 338960 q^{38} + 774728 q^{39} + 264364 q^{41} + 545040 q^{42} - 423300 q^{43} + 1434112 q^{44} - 1303416 q^{46} + 105460 q^{47} - 981760 q^{48} - 1165414 q^{49} + 1166344 q^{51} + 1678400 q^{52} + 2391580 q^{53} + 1102960 q^{54} + 949440 q^{56} - 776720 q^{57} - 2244440 q^{58} - 1120120 q^{59} + 2257044 q^{61} - 2642640 q^{62} + 1639620 q^{63} - 5146624 q^{64} - 6564352 q^{66} - 4516460 q^{67} + 4911680 q^{68} - 745272 q^{69} + 621784 q^{71} - 1080480 q^{72} - 4569060 q^{73} + 2651272 q^{74} + 887680 q^{76} - 3177600 q^{77} - 4670800 q^{78} + 4333040 q^{79} - 2397878 q^{81} + 5989960 q^{82} + 9793020 q^{83} - 398208 q^{84} + 10798184 q^{86} + 24458920 q^{87} - 10567680 q^{88} + 6025620 q^{89} - 5352296 q^{91} + 7199040 q^{92} - 6473040 q^{93} + 5860792 q^{94} + 13305344 q^{96} - 4609540 q^{97} + 12505340 q^{98} + 2890688 q^{99}+O(q^{100}) \) 2 * q - 20 * q^2 - 20 * q^3 + 96 * q^4 - 1016 * q^6 + 100 * q^7 - 1440 * q^8 + 5554 * q^9 + 4544 * q^11 + 23360 * q^12 - 3540 * q^13 + 7512 * q^14 + 20352 * q^16 + 27340 * q^17 - 31220 * q^18 + 38760 * q^19 - 69096 * q^21 - 106240 * q^22 + 124140 * q^23 - 131520 * q^24 - 57016 * q^26 - 206360 * q^27 - 165440 * q^28 - 72260 * q^29 + 306824 * q^31 + 78080 * q^32 + 440960 * q^33 - 453368 * q^34 - 219808 * q^36 + 123020 * q^37 - 338960 * q^38 + 774728 * q^39 + 264364 * q^41 + 545040 * q^42 - 423300 * q^43 + 1434112 * q^44 - 1303416 * q^46 + 105460 * q^47 - 981760 * q^48 - 1165414 * q^49 + 1166344 * q^51 + 1678400 * q^52 + 2391580 * q^53 + 1102960 * q^54 + 949440 * q^56 - 776720 * q^57 - 2244440 * q^58 - 1120120 * q^59 + 2257044 * q^61 - 2642640 * q^62 + 1639620 * q^63 - 5146624 * q^64 - 6564352 * q^66 - 4516460 * q^67 + 4911680 * q^68 - 745272 * q^69 + 621784 * q^71 - 1080480 * q^72 - 4569060 * q^73 + 2651272 * q^74 + 887680 * q^76 - 3177600 * q^77 - 4670800 * q^78 + 4333040 * q^79 - 2397878 * q^81 + 5989960 * q^82 + 9793020 * q^83 - 398208 * q^84 + 10798184 * q^86 + 24458920 * q^87 - 10567680 * q^88 + 6025620 * q^89 - 5352296 * q^91 + 7199040 * q^92 - 6473040 * q^93 + 5860792 * q^94 + 13305344 * q^96 - 4609540 * q^97 + 12505340 * q^98 + 2890688 * q^99

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