LMFDB - Embedded newform 25.8.a.e.1.2

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{649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 162 $ x^2 - x - 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.2377$ 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)

$f(q)$	$=$	$q+20.2377 q^{2} +45.4755 q^{3} +281.566 q^{4} +920.321 q^{6} -1369.97 q^{7} +3107.83 q^{8} -118.981 q^{9} +O(q^{10})$	\(q+20.2377 q^{2} +45.4755 q^{3} +281.566 q^{4} +920.321 q^{6} -1369.97 q^{7} +3107.83 q^{8} -118.981 q^{9} -1012.43 q^{11} +12804.4 q^{12} +3643.00 q^{13} -27725.1 q^{14} +26855.0 q^{16} -5532.37 q^{17} -2407.90 q^{18} +23238.9 q^{19} -62300.0 q^{21} -20489.4 q^{22} +96310.5 q^{23} +141330. q^{24} +73726.1 q^{26} -104866. q^{27} -385737. q^{28} -139729. q^{29} -208208. q^{31} +145682. q^{32} -46041.0 q^{33} -111963. q^{34} -33501.0 q^{36} +448306. q^{37} +470303. q^{38} +165667. q^{39} -79718.5 q^{41} -1.26081e6 q^{42} +227324. q^{43} -285067. q^{44} +1.94911e6 q^{46} +719892. q^{47} +1.22124e6 q^{48} +1.05328e6 q^{49} -251587. q^{51} +1.02575e6 q^{52} -495702. q^{53} -2.12224e6 q^{54} -4.25763e6 q^{56} +1.05680e6 q^{57} -2.82779e6 q^{58} -81006.9 q^{59} +3.15701e6 q^{61} -4.21366e6 q^{62} +163000. q^{63} -489161. q^{64} -931765. q^{66} +1.06340e6 q^{67} -1.55773e6 q^{68} +4.37976e6 q^{69} -3.00099e6 q^{71} -369772. q^{72} +405758. q^{73} +9.07270e6 q^{74} +6.54328e6 q^{76} +1.38701e6 q^{77} +3.35273e6 q^{78} -4.71733e6 q^{79} -4.50860e6 q^{81} -1.61332e6 q^{82} +5.14106e6 q^{83} -1.75416e7 q^{84} +4.60053e6 q^{86} -6.35423e6 q^{87} -3.14648e6 q^{88} -8.62652e6 q^{89} -4.99080e6 q^{91} +2.71178e7 q^{92} -9.46835e6 q^{93} +1.45690e7 q^{94} +6.62497e6 q^{96} -6.99663e6 q^{97} +2.13159e7 q^{98} +120460. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 15 q^{2} + 40 q^{3} + 181 q^{4} + 949 q^{6} - 600 q^{7} + 4305 q^{8} - 2276 q^{9}+O(q^{10}) $ 2 * q + 15 * q^2 + 40 * q^3 + 181 * q^4 + 949 * q^6 - 600 * q^7 + 4305 * q^8 - 2276 * q^9	\( 2 q + 15 q^{2} + 40 q^{3} + 181 q^{4} + 949 q^{6} - 600 q^{7} + 4305 q^{8} - 2276 q^{9} + 4344 q^{11} + 13355 q^{12} + 17680 q^{13} - 31758 q^{14} + 33457 q^{16} + 6870 q^{17} + 8890 q^{18} + 18200 q^{19} - 66516 q^{21} - 48545 q^{22} + 21120 q^{23} + 134775 q^{24} + 204 q^{26} - 81080 q^{27} - 463170 q^{28} + 55800 q^{29} - 301776 q^{31} - 42135 q^{32} - 75370 q^{33} - 176923 q^{34} + 183422 q^{36} + 609860 q^{37} + 496695 q^{38} + 88808 q^{39} - 108486 q^{41} - 1238730 q^{42} + 966400 q^{43} - 823743 q^{44} + 2342934 q^{46} + 1787880 q^{47} + 1185095 q^{48} + 822586 q^{49} - 319496 q^{51} - 385900 q^{52} + 130740 q^{53} - 2246825 q^{54} - 3335850 q^{56} + 1084390 q^{57} - 3851920 q^{58} + 2067600 q^{59} + 582044 q^{61} - 3723570 q^{62} - 1497840 q^{63} - 350479 q^{64} - 778147 q^{66} + 255720 q^{67} - 2804985 q^{68} + 4791468 q^{69} - 4728216 q^{71} - 2952090 q^{72} - 1339430 q^{73} + 8226522 q^{74} + 7050025 q^{76} + 5511300 q^{77} + 3755300 q^{78} - 7186200 q^{79} + 78562 q^{81} - 1462645 q^{82} + 12049560 q^{83} - 17117598 q^{84} + 729444 q^{86} - 7424840 q^{87} + 3266085 q^{88} - 5990850 q^{89} + 5817264 q^{91} + 34679370 q^{92} - 8956020 q^{93} + 8975132 q^{94} + 7653359 q^{96} - 17120020 q^{97} + 22524195 q^{98} - 11433472 q^{99}+O(q^{100}) \) 2 * q + 15 * q^2 + 40 * q^3 + 181 * q^4 + 949 * q^6 - 600 * q^7 + 4305 * q^8 - 2276 * q^9 + 4344 * q^11 + 13355 * q^12 + 17680 * q^13 - 31758 * q^14 + 33457 * q^16 + 6870 * q^17 + 8890 * q^18 + 18200 * q^19 - 66516 * q^21 - 48545 * q^22 + 21120 * q^23 + 134775 * q^24 + 204 * q^26 - 81080 * q^27 - 463170 * q^28 + 55800 * q^29 - 301776 * q^31 - 42135 * q^32 - 75370 * q^33 - 176923 * q^34 + 183422 * q^36 + 609860 * q^37 + 496695 * q^38 + 88808 * q^39 - 108486 * q^41 - 1238730 * q^42 + 966400 * q^43 - 823743 * q^44 + 2342934 * q^46 + 1787880 * q^47 + 1185095 * q^48 + 822586 * q^49 - 319496 * q^51 - 385900 * q^52 + 130740 * q^53 - 2246825 * q^54 - 3335850 * q^56 + 1084390 * q^57 - 3851920 * q^58 + 2067600 * q^59 + 582044 * q^61 - 3723570 * q^62 - 1497840 * q^63 - 350479 * q^64 - 778147 * q^66 + 255720 * q^67 - 2804985 * q^68 + 4791468 * q^69 - 4728216 * q^71 - 2952090 * q^72 - 1339430 * q^73 + 8226522 * q^74 + 7050025 * q^76 + 5511300 * q^77 + 3755300 * q^78 - 7186200 * q^79 + 78562 * q^81 - 1462645 * q^82 + 12049560 * q^83 - 17117598 * q^84 + 729444 * q^86 - 7424840 * q^87 + 3266085 * q^88 - 5990850 * q^89 + 5817264 * q^91 + 34679370 * q^92 - 8956020 * q^93 + 8975132 * q^94 + 7653359 * q^96 - 17120020 * q^97 + 22524195 * q^98 - 11433472 * 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 $

$2$	20.2377	1.78878	0.894390	−	0.447288i	$-0.147610\pi$
$2$	20.2377	1.78878	0.894390	+	0.447288i	$0.147610\pi$
$3$	45.4755	0.972418	0.486209	−	0.873843i	$-0.338379\pi$
$3$	45.4755	0.972418	0.486209	+	0.873843i	$0.338379\pi$
$4$	281.566	2.19974
$5$	0	0
$5$	0	0
$6$	920.321	1.73944
$7$	−1369.97	−1.50962	−0.754811	−	0.655943i	$-0.772271\pi$
$7$	−1369.97	−1.50962	−0.754811	+	0.655943i	$0.772271\pi$
$8$	3107.83	2.14606
$9$	−118.981	−0.0544037
$10$	0	0
$11$	−1012.43	−0.229347	−0.114673	−	0.993403i	$-0.536582\pi$
$11$	−1012.43	−0.229347	−0.114673	+	0.993403i	$0.536582\pi$
$12$	12804.4	2.13906
$13$	3643.00	0.459894	0.229947	−	0.973203i	$-0.426145\pi$
$13$	3643.00	0.459894	0.229947	+	0.973203i	$0.426145\pi$
$14$	−27725.1	−2.70038
$15$	0	0
$16$	26855.0	1.63910
$17$	−5532.37	−0.273111	−0.136556	−	0.990632i	$-0.543603\pi$
$17$	−5532.37	−0.273111	−0.136556	+	0.990632i	$0.543603\pi$
$18$	−2407.90	−0.0973162
$19$	23238.9	0.777281	0.388640	−	0.921390i	$-0.372945\pi$
$19$	23238.9	0.777281	0.388640	+	0.921390i	$0.372945\pi$
$20$	0	0
$21$	−62300.0	−1.46798
$22$	−20489.4	−0.410251
$23$	96310.5	1.65054	0.825270	−	0.564738i	$-0.191023\pi$
$23$	96310.5	1.65054	0.825270	+	0.564738i	$0.191023\pi$
$24$	141330.	2.08687
$25$	0	0
$26$	73726.1	0.822649
$27$	−104866.	−1.02532
$28$	−385737.	−3.32077
$29$	−139729.	−1.06388	−0.531940	−	0.846782i	$-0.678537\pi$
$29$	−139729.	−1.06388	−0.531940	+	0.846782i	$0.678537\pi$
$30$	0	0
$31$	−208208.	−1.25525	−0.627626	−	0.778515i	$-0.715973\pi$
$31$	−208208.	−1.25525	−0.627626	+	0.778515i	$0.715973\pi$
$32$	145682.	0.785926
$33$	−46041.0	−0.223021
$34$	−111963.	−0.488536
$35$	0	0
$36$	−33501.0	−0.119674
$37$	448306.	1.45502	0.727509	−	0.686098i	$-0.240678\pi$
$37$	448306.	1.45502	0.727509	+	0.686098i	$0.240678\pi$
$38$	470303.	1.39038
$39$	165667.	0.447209
$40$	0	0
$41$	−79718.5	−0.180641	−0.0903203	−	0.995913i	$-0.528789\pi$
$41$	−79718.5	−0.180641	−0.0903203	+	0.995913i	$0.528789\pi$
$42$	−1.26081e6	−2.62590
$43$	227324.	0.436020	0.218010	−	0.975947i	$-0.430043\pi$
$43$	227324.	0.436020	0.218010	+	0.975947i	$0.430043\pi$
$44$	−285067.	−0.504502
$45$	0	0
$46$	1.94911e6	2.95245
$47$	719892.	1.01140	0.505702	−	0.862708i	$-0.331234\pi$
$47$	719892.	1.01140	0.505702	+	0.862708i	$0.331234\pi$
$48$	1.22124e6	1.59389
$49$	1.05328e6	1.27896
$50$	0	0
$51$	−251587.	−0.265578
$52$	1.02575e6	1.01164
$53$	−495702.	−0.457357	−0.228678	−	0.973502i	$-0.573440\pi$
$53$	−495702.	−0.457357	−0.228678	+	0.973502i	$0.573440\pi$
$54$	−2.12224e6	−1.83407
$55$	0	0
$56$	−4.25763e6	−3.23974
$57$	1.05680e6	0.755841
$58$	−2.82779e6	−1.90305
$59$	−81006.9	−0.0513500	−0.0256750	−	0.999670i	$-0.508173\pi$
$59$	−81006.9	−0.0513500	−0.0256750	+	0.999670i	$0.508173\pi$
$60$	0	0
$61$	3.15701e6	1.78083	0.890414	−	0.455151i	$-0.150415\pi$
$61$	3.15701e6	1.78083	0.890414	+	0.455151i	$0.150415\pi$
$62$	−4.21366e6	−2.24537
$63$	163000.	0.0821290
$64$	−489161.	−0.233250
$65$	0	0
$66$	−931765.	−0.398935
$67$	1.06340e6	0.431950	0.215975	−	0.976399i	$-0.430707\pi$
$67$	1.06340e6	0.431950	0.215975	+	0.976399i	$0.430707\pi$
$68$	−1.55773e6	−0.600773
$69$	4.37976e6	1.60501
$70$	0	0
$71$	−3.00099e6	−0.995087	−0.497543	−	0.867439i	$-0.665765\pi$
$71$	−3.00099e6	−0.995087	−0.497543	+	0.867439i	$0.665765\pi$
$72$	−369772.	−0.116754
$73$	405758.	0.122078	0.0610389	−	0.998135i	$-0.480559\pi$
$73$	405758.	0.122078	0.0610389	+	0.998135i	$0.480559\pi$
$74$	9.07270e6	2.60271
$75$	0	0
$76$	6.54328e6	1.70981
$77$	1.38701e6	0.346227
$78$	3.35273e6	0.799959
$79$	−4.71733e6	−1.07647	−0.538235	−	0.842795i	$-0.680909\pi$
$79$	−4.71733e6	−1.07647	−0.538235	+	0.842795i	$0.680909\pi$
$80$	0	0
$81$	−4.50860e6	−0.942637
$82$	−1.61332e6	−0.323126
$83$	5.14106e6	0.986914	0.493457	−	0.869770i	$-0.335733\pi$
$83$	5.14106e6	0.986914	0.493457	+	0.869770i	$0.335733\pi$
$84$	−1.75416e7	−3.22917
$85$	0	0
$86$	4.60053e6	0.779944
$87$	−6.35423e6	−1.03454
$88$	−3.14648e6	−0.492193
$89$	−8.62652e6	−1.29709	−0.648546	−	0.761175i	$-0.724623\pi$
$89$	−8.62652e6	−1.29709	−0.648546	+	0.761175i	$0.724623\pi$
$90$	0	0
$91$	−4.99080e6	−0.694266
$92$	2.71178e7	3.63075
$93$	−9.46835e6	−1.22063
$94$	1.45690e7	1.80918
$95$	0	0
$96$	6.62497e6	0.764248
$97$	−6.99663e6	−0.778373	−0.389187	−	0.921159i	$-0.627244\pi$
$97$	−6.99663e6	−0.778373	−0.389187	+	0.921159i	$0.627244\pi$
$98$	2.13159e7	2.28777
$99$	120460.	0.0124773
$100$	0	0
$101$	5.34837e6	0.516531	0.258266	−	0.966074i	$-0.416849\pi$
$101$	5.34837e6	0.516531	0.258266	+	0.966074i	$0.416849\pi$
$102$	−5.09155e6	−0.475061
$103$	7.09826e6	0.640061	0.320031	−	0.947407i	$-0.396307\pi$
$103$	7.09826e6	0.640061	0.320031	+	0.947407i	$0.396307\pi$
$104$	1.13218e7	0.986961
$105$	0	0
$106$	−1.00319e7	−0.818111
$107$	−1.01014e7	−0.797145	−0.398573	−	0.917137i	$-0.630494\pi$
$107$	−1.01014e7	−0.797145	−0.398573	+	0.917137i	$0.630494\pi$
$108$	−2.95266e7	−2.25543
$109$	−1.20315e6	−0.0889874	−0.0444937	−	0.999010i	$-0.514167\pi$
$109$	−1.20315e6	−0.0889874	−0.0444937	+	0.999010i	$0.514167\pi$
$110$	0	0
$111$	2.03869e7	1.41489
$112$	−3.67906e7	−2.47442
$113$	−529610.	−0.0345288	−0.0172644	−	0.999851i	$-0.505496\pi$
$113$	−529610.	−0.0345288	−0.0172644	+	0.999851i	$0.505496\pi$
$114$	2.13872e7	1.35203
$115$	0	0
$116$	−3.93428e7	−2.34025
$117$	−433448.	−0.0250199
$118$	−1.63940e6	−0.0918538
$119$	7.57918e6	0.412295
$120$	0	0
$121$	−1.84621e7	−0.947400
$122$	6.38908e7	3.18551
$123$	−3.62524e6	−0.175658
$124$	−5.86243e7	−2.76122
$125$	0	0
$126$	3.29876e6	0.146911
$127$	1.90420e7	0.824897	0.412448	−	0.910981i	$-0.364674\pi$
$127$	1.90420e7	0.824897	0.412448	+	0.910981i	$0.364674\pi$
$128$	−2.85468e7	−1.20316
$129$	1.03377e7	0.423993
$130$	0	0
$131$	4.51565e7	1.75498	0.877488	−	0.479599i	$-0.159218\pi$
$131$	4.51565e7	1.75498	0.877488	+	0.479599i	$0.159218\pi$
$132$	−1.29636e7	−0.490587
$133$	−3.18366e7	−1.17340
$134$	2.15207e7	0.772663
$135$	0	0
$136$	−1.71937e7	−0.586114
$137$	−3.51625e7	−1.16831	−0.584155	−	0.811642i	$-0.698574\pi$
$137$	−3.51625e7	−1.16831	−0.584155	+	0.811642i	$0.698574\pi$
$138$	8.86365e7	2.87102
$139$	−4.10740e7	−1.29722	−0.648612	−	0.761119i	$-0.724650\pi$
$139$	−4.10740e7	−1.29722	−0.648612	+	0.761119i	$0.724650\pi$
$140$	0	0
$141$	3.27374e7	0.983507
$142$	−6.07334e7	−1.77999
$143$	−3.68830e6	−0.105475
$144$	−3.19523e6	−0.0891730
$145$	0	0
$146$	8.21162e6	0.218370
$147$	4.78982e7	1.24368
$148$	1.26228e8	3.20066
$149$	5.46024e7	1.35226	0.676129	−	0.736783i	$-0.263656\pi$
$149$	5.46024e7	1.35226	0.676129	+	0.736783i	$0.263656\pi$
$150$	0	0
$151$	−7.68654e6	−0.181682	−0.0908409	−	0.995865i	$-0.528955\pi$
$151$	−7.68654e6	−0.181682	−0.0908409	+	0.995865i	$0.528955\pi$
$152$	7.22225e7	1.66809
$153$	658246.	0.0148583
$154$	2.80699e7	0.619324
$155$	0	0
$156$	4.66463e7	0.983741
$157$	−4.63213e6	−0.0955283	−0.0477641	−	0.998859i	$-0.515210\pi$
$157$	−4.63213e6	−0.0955283	−0.0477641	+	0.998859i	$0.515210\pi$
$158$	−9.54682e7	−1.92557
$159$	−2.25423e7	−0.444742
$160$	0	0
$161$	−1.31942e8	−2.49169
$162$	−9.12439e7	−1.68617
$163$	2.15892e7	0.390462	0.195231	−	0.980757i	$-0.437454\pi$
$163$	2.15892e7	0.390462	0.195231	+	0.980757i	$0.437454\pi$
$164$	−2.24460e7	−0.397362
$165$	0	0
$166$	1.04043e8	1.76537
$167$	3.71330e7	0.616953	0.308476	−	0.951232i	$-0.400181\pi$
$167$	3.71330e7	0.616953	0.308476	+	0.951232i	$0.400181\pi$
$168$	−1.93618e8	−3.15038
$169$	−4.94771e7	−0.788498
$170$	0	0
$171$	−2.76498e6	−0.0422869
$172$	6.40068e7	0.959128
$173$	6.07796e7	0.892476	0.446238	−	0.894914i	$-0.352763\pi$
$173$	6.07796e7	0.892476	0.446238	+	0.894914i	$0.352763\pi$
$174$	−1.28595e8	−1.85056
$175$	0	0
$176$	−2.71889e7	−0.375922
$177$	−3.68383e6	−0.0499336
$178$	−1.74581e8	−2.32021
$179$	3.05526e7	0.398164	0.199082	−	0.979983i	$-0.436204\pi$
$179$	3.05526e7	0.398164	0.199082	+	0.979983i	$0.436204\pi$
$180$	0	0
$181$	−1.18125e8	−1.48070	−0.740349	−	0.672223i	$-0.765340\pi$
$181$	−1.18125e8	−1.48070	−0.740349	+	0.672223i	$0.765340\pi$
$182$	−1.01003e8	−1.24189
$183$	1.43567e8	1.73171
$184$	2.99317e8	3.54216
$185$	0	0
$186$	−1.91618e8	−2.18344
$187$	5.60116e6	0.0626372
$188$	2.02697e8	2.22482
$189$	1.43663e8	1.54785
$190$	0	0
$191$	8.20283e7	0.851818	0.425909	−	0.904766i	$-0.359954\pi$
$191$	8.20283e7	0.851818	0.425909	+	0.904766i	$0.359954\pi$
$192$	−2.22448e7	−0.226817
$193$	−1.98988e8	−1.99240	−0.996198	−	0.0871152i	$-0.972235\pi$
$193$	−1.98988e8	−1.99240	−0.996198	+	0.0871152i	$0.972235\pi$
$194$	−1.41596e8	−1.39234
$195$	0	0
$196$	2.96567e8	2.81336
$197$	−8.48880e7	−0.791069	−0.395535	−	0.918451i	$-0.629441\pi$
$197$	−8.48880e7	−0.791069	−0.395535	+	0.918451i	$0.629441\pi$
$198$	2.43785e6	0.0223192
$199$	−4.12974e7	−0.371481	−0.185741	−	0.982599i	$-0.559468\pi$
$199$	−4.12974e7	−0.371481	−0.185741	+	0.982599i	$0.559468\pi$
$200$	0	0
$201$	4.83584e7	0.420036
$202$	1.08239e8	0.923961
$203$	1.91424e8	1.60605
$204$	−7.08384e7	−0.584202
$205$	0	0
$206$	1.43653e8	1.14493
$207$	−1.14591e7	−0.0897955
$208$	9.78328e7	0.753812
$209$	−2.35279e7	−0.178267
$210$	0	0
$211$	−1.55077e7	−0.113647	−0.0568236	−	0.998384i	$-0.518097\pi$
$211$	−1.55077e7	−0.113647	−0.0568236	+	0.998384i	$0.518097\pi$
$212$	−1.39573e8	−1.00606
$213$	−1.36472e8	−0.967640
$214$	−2.04429e8	−1.42592
$215$	0	0
$216$	−3.25904e8	−2.20040
$217$	2.85238e8	1.89496
$218$	−2.43491e7	−0.159179
$219$	1.84520e7	0.118711
$220$	0	0
$221$	−2.01544e7	−0.125602
$222$	4.12585e8	2.53092
$223$	4.16316e6	0.0251395	0.0125697	−	0.999921i	$-0.495999\pi$
$223$	4.16316e6	0.0251395	0.0125697	+	0.999921i	$0.495999\pi$
$224$	−1.99580e8	−1.18645
$225$	0	0
$226$	−1.07181e7	−0.0617645
$227$	6.41270e7	0.363873	0.181937	−	0.983310i	$-0.441763\pi$
$227$	6.41270e7	0.363873	0.181937	+	0.983310i	$0.441763\pi$
$228$	2.97559e8	1.66265
$229$	5.41678e7	0.298069	0.149035	−	0.988832i	$-0.452383\pi$
$229$	5.41678e7	0.298069	0.149035	+	0.988832i	$0.452383\pi$
$230$	0	0
$231$	6.30747e7	0.336677
$232$	−4.34253e8	−2.28315
$233$	−2.74855e8	−1.42350	−0.711752	−	0.702431i	$-0.752098\pi$
$233$	−2.74855e8	−1.42350	−0.711752	+	0.702431i	$0.752098\pi$
$234$	−8.77200e6	−0.0447552
$235$	0	0
$236$	−2.28088e7	−0.112956
$237$	−2.14523e8	−1.04678
$238$	1.53385e8	0.737505
$239$	−1.86000e8	−0.881291	−0.440646	−	0.897681i	$-0.645250\pi$
$239$	−1.86000e8	−0.881291	−0.440646	+	0.897681i	$0.645250\pi$
$240$	0	0
$241$	2.37791e8	1.09430	0.547150	−	0.837034i	$-0.315713\pi$
$241$	2.37791e8	1.09430	0.547150	+	0.837034i	$0.315713\pi$
$242$	−3.73632e8	−1.69469
$243$	2.43102e7	0.108684
$244$	8.88908e8	3.91735
$245$	0	0
$246$	−7.33666e7	−0.314214
$247$	8.46593e7	0.357467
$248$	−6.47075e8	−2.69385
$249$	2.33792e8	0.959693
$250$	0	0
$251$	3.28304e8	1.31044	0.655221	−	0.755438i	$-0.272576\pi$
$251$	3.28304e8	1.31044	0.655221	+	0.755438i	$0.272576\pi$
$252$	4.58953e7	0.180662
$253$	−9.75081e7	−0.378546
$254$	3.85367e8	1.47556
$255$	0	0
$256$	−5.15111e8	−1.91894
$257$	−1.01567e8	−0.373238	−0.186619	−	0.982432i	$-0.559753\pi$
$257$	−1.01567e8	−0.373238	−0.186619	+	0.982432i	$0.559753\pi$
$258$	2.09211e8	0.758431
$259$	−6.14166e8	−2.19653
$260$	0	0
$261$	1.66250e7	0.0578790
$262$	9.13866e8	3.13927
$263$	1.98884e8	0.674147	0.337073	−	0.941478i	$-0.390563\pi$
$263$	1.98884e8	0.674147	0.337073	+	0.941478i	$0.390563\pi$
$264$	−1.43087e8	−0.478617
$265$	0	0
$266$	−6.44301e8	−2.09895
$267$	−3.92295e8	−1.26132
$268$	2.99416e8	0.950175
$269$	2.44943e8	0.767242	0.383621	−	0.923491i	$-0.374677\pi$
$269$	2.44943e8	0.767242	0.383621	+	0.923491i	$0.374677\pi$
$270$	0	0
$271$	−3.91023e8	−1.19347	−0.596733	−	0.802440i	$-0.703535\pi$
$271$	−3.91023e8	−1.19347	−0.596733	+	0.802440i	$0.703535\pi$
$272$	−1.48572e8	−0.447657
$273$	−2.26959e8	−0.675116
$274$	−7.11610e8	−2.08985
$275$	0	0
$276$	1.23319e9	3.53061
$277$	−5.08902e8	−1.43865	−0.719325	−	0.694674i	$-0.755549\pi$
$277$	−5.08902e8	−1.43865	−0.719325	+	0.694674i	$0.755549\pi$
$278$	−8.31245e8	−2.32045
$279$	2.47727e7	0.0682904
$280$	0	0
$281$	−1.55563e8	−0.418248	−0.209124	−	0.977889i	$-0.567061\pi$
$281$	−1.55563e8	−0.418248	−0.209124	+	0.977889i	$0.567061\pi$
$282$	6.62531e8	1.75928
$283$	1.14570e8	0.300481	0.150241	−	0.988649i	$-0.451995\pi$
$283$	1.14570e8	0.300481	0.150241	+	0.988649i	$0.451995\pi$
$284$	−8.44978e8	−2.18893
$285$	0	0
$286$	−7.46429e7	−0.188672
$287$	1.09212e8	0.272699
$288$	−1.73334e7	−0.0427573
$289$	−3.79732e8	−0.925410
$290$	0	0
$291$	−3.18175e8	−0.756904
$292$	1.14248e8	0.268539
$293$	8.42345e8	1.95638	0.978190	−	0.207713i	$-0.0666020\pi$
$293$	8.42345e8	1.95638	0.978190	+	0.207713i	$0.0666020\pi$
$294$	9.69351e8	2.22467
$295$	0	0
$296$	1.39326e9	3.12256
$297$	1.06170e8	0.235154
$298$	1.10503e9	2.41889
$299$	3.50859e8	0.759073
$300$	0	0
$301$	−3.11427e8	−0.658225
$302$	−1.55558e8	−0.324989
$303$	2.43220e8	0.502284
$304$	6.24080e8	1.27404
$305$	0	0
$306$	1.33214e7	0.0265782
$307$	2.38090e7	0.0469631	0.0234816	−	0.999724i	$-0.492525\pi$
$307$	2.38090e7	0.0469631	0.0234816	+	0.999724i	$0.492525\pi$
$308$	3.90534e8	0.761607
$309$	3.22797e8	0.622407
$310$	0	0
$311$	7.31704e8	1.37935	0.689674	−	0.724120i	$-0.257754\pi$
$311$	7.31704e8	1.37935	0.689674	+	0.724120i	$0.257754\pi$
$312$	5.14866e8	0.959739
$313$	−9.21577e8	−1.69874	−0.849369	−	0.527799i	$-0.823017\pi$
$313$	−9.21577e8	−1.69874	−0.849369	+	0.527799i	$0.823017\pi$
$314$	−9.37438e7	−0.170879
$315$	0	0
$316$	−1.32824e9	−2.36795
$317$	−7.85353e7	−0.138471	−0.0692353	−	0.997600i	$-0.522056\pi$
$317$	−7.85353e7	−0.138471	−0.0692353	+	0.997600i	$0.522056\pi$
$318$	−4.56205e8	−0.795545
$319$	1.41466e8	0.243997
$320$	0	0
$321$	−4.59365e8	−0.775158
$322$	−2.67022e9	−4.45709
$323$	−1.28566e8	−0.212284
$324$	−1.26947e9	−2.07355
$325$	0	0
$326$	4.36916e8	0.698451
$327$	−5.47140e7	−0.0865329
$328$	−2.47752e8	−0.387666
$329$	−9.86230e8	−1.52684
$330$	0	0
$331$	3.78878e8	0.574250	0.287125	−	0.957893i	$-0.407300\pi$
$331$	3.78878e8	0.574250	0.287125	+	0.957893i	$0.407300\pi$
$332$	1.44755e9	2.17095
$333$	−5.33398e7	−0.0791584
$334$	7.51487e8	1.10359
$335$	0	0
$336$	−1.67307e9	−2.40617
$337$	1.08031e9	1.53760	0.768800	−	0.639489i	$-0.220854\pi$
$337$	1.08031e9	1.53760	0.768800	+	0.639489i	$0.220854\pi$
$338$	−1.00130e9	−1.41045
$339$	−2.40843e7	−0.0335764
$340$	0	0
$341$	2.10797e8	0.287888
$342$	−5.59570e7	−0.0756420
$343$	−3.14726e8	−0.421117
$344$	7.06485e8	0.935726
$345$	0	0
$346$	1.23004e9	1.59644
$347$	8.77465e8	1.12740	0.563698	−	0.825981i	$-0.309378\pi$
$347$	8.77465e8	1.12740	0.563698	+	0.825981i	$0.309378\pi$
$348$	−1.78913e9	−2.27570
$349$	3.31907e8	0.417953	0.208976	−	0.977921i	$-0.432987\pi$
$349$	3.31907e8	0.417953	0.208976	+	0.977921i	$0.432987\pi$
$350$	0	0
$351$	−3.82026e8	−0.471539
$352$	−1.47494e8	−0.180250
$353$	9.88344e8	1.19590	0.597952	−	0.801532i	$-0.295981\pi$
$353$	9.88344e8	1.19590	0.597952	+	0.801532i	$0.295981\pi$
$354$	−7.45524e7	−0.0893203
$355$	0	0
$356$	−2.42894e9	−2.85326
$357$	3.44667e8	0.400923
$358$	6.18315e8	0.712228
$359$	1.41167e9	1.61028	0.805140	−	0.593084i	$-0.202090\pi$
$359$	1.41167e9	1.61028	0.805140	+	0.593084i	$0.202090\pi$
$360$	0	0
$361$	−3.53826e8	−0.395835
$362$	−2.39058e9	−2.64864
$363$	−8.39575e8	−0.921269
$364$	−1.40524e9	−1.52720
$365$	0	0
$366$	2.90547e9	3.09765
$367$	7.57010e8	0.799412	0.399706	−	0.916643i	$-0.369112\pi$
$367$	7.57010e8	0.799412	0.399706	+	0.916643i	$0.369112\pi$
$368$	2.58642e9	2.70540
$369$	9.48497e6	0.00982752
$370$	0	0
$371$	6.79097e8	0.690435
$372$	−2.66597e9	−2.68506
$373$	2.96259e8	0.295591	0.147796	−	0.989018i	$-0.452782\pi$
$373$	2.96259e8	0.295591	0.147796	+	0.989018i	$0.452782\pi$
$374$	1.13355e8	0.112044
$375$	0	0
$376$	2.23730e9	2.17054
$377$	−5.09032e8	−0.489272
$378$	2.90741e9	2.76876
$379$	1.32923e9	1.25419	0.627093	−	0.778944i	$-0.284244\pi$
$379$	1.32923e9	1.25419	0.627093	+	0.778944i	$0.284244\pi$
$380$	0	0
$381$	8.65944e8	0.802144
$382$	1.66007e9	1.52372
$383$	−8.35565e8	−0.759949	−0.379974	−	0.924997i	$-0.624067\pi$
$383$	−8.35565e8	−0.759949	−0.379974	+	0.924997i	$0.624067\pi$
$384$	−1.29818e9	−1.16997
$385$	0	0
$386$	−4.02706e9	−3.56396
$387$	−2.70472e7	−0.0237211
$388$	−1.97001e9	−1.71221
$389$	1.65397e8	0.142464	0.0712320	−	0.997460i	$-0.477307\pi$
$389$	1.65397e8	0.142464	0.0712320	+	0.997460i	$0.477307\pi$
$390$	0	0
$391$	−5.32825e8	−0.450781
$392$	3.27340e9	2.74472
$393$	2.05351e9	1.70657
$394$	−1.71794e9	−1.41505
$395$	0	0
$396$	3.39176e7	0.0274468
$397$	2.01752e8	0.161827	0.0809134	−	0.996721i	$-0.474216\pi$
$397$	2.01752e8	0.161827	0.0809134	+	0.996721i	$0.474216\pi$
$398$	−8.35765e8	−0.664498
$399$	−1.44778e9	−1.14103
$400$	0	0
$401$	−2.44937e8	−0.189692	−0.0948462	−	0.995492i	$-0.530236\pi$
$401$	−2.44937e8	−0.189692	−0.0948462	+	0.995492i	$0.530236\pi$
$402$	9.78666e8	0.751351
$403$	−7.58502e8	−0.577283
$404$	1.50592e9	1.13623
$405$	0	0
$406$	3.87399e9	2.87288
$407$	−4.53881e8	−0.333704
$408$	−7.81890e8	−0.569948
$409$	1.72583e8	0.124728	0.0623642	−	0.998053i	$-0.480136\pi$
$409$	1.72583e8	0.124728	0.0623642	+	0.998053i	$0.480136\pi$
$410$	0	0
$411$	−1.59903e9	−1.13608
$412$	1.99863e9	1.40797
$413$	1.10977e8	0.0775190
$414$	−2.31906e8	−0.160624
$415$	0	0
$416$	5.30721e8	0.361443
$417$	−1.86786e9	−1.26144
$418$	−4.76151e8	−0.318880
$419$	−2.64174e9	−1.75445	−0.877225	−	0.480079i	$-0.840608\pi$
$419$	−2.64174e9	−1.75445	−0.877225	+	0.480079i	$0.840608\pi$
$420$	0	0
$421$	1.11997e9	0.731509	0.365754	−	0.930711i	$-0.380811\pi$
$421$	1.11997e9	0.731509	0.365754	+	0.930711i	$0.380811\pi$
$422$	−3.13841e8	−0.203290
$423$	−8.56533e7	−0.0550241
$424$	−1.54056e9	−0.981516
$425$	0	0
$426$	−2.76188e9	−1.73090
$427$	−4.32501e9	−2.68838
$428$	−2.84421e9	−1.75351
$429$	−1.67727e8	−0.102566
$430$	0	0
$431$	−2.56747e9	−1.54466	−0.772332	−	0.635219i	$-0.780910\pi$
$431$	−2.56747e9	−1.54466	−0.772332	+	0.635219i	$0.780910\pi$
$432$	−2.81617e9	−1.68060
$433$	9.65650e8	0.571626	0.285813	−	0.958285i	$-0.407736\pi$
$433$	9.65650e8	0.571626	0.285813	+	0.958285i	$0.407736\pi$
$434$	5.77258e9	3.38966
$435$	0	0
$436$	−3.38767e8	−0.195749
$437$	2.23815e9	1.28293
$438$	3.73427e8	0.212347
$439$	2.67168e9	1.50716	0.753580	−	0.657356i	$-0.228325\pi$
$439$	2.67168e9	1.50716	0.753580	+	0.657356i	$0.228325\pi$
$440$	0	0
$441$	−1.25320e8	−0.0695799
$442$	−4.07880e8	−0.224675
$443$	9.25736e8	0.505911	0.252955	−	0.967478i	$-0.418597\pi$
$443$	9.25736e8	0.505911	0.252955	+	0.967478i	$0.418597\pi$
$444$	5.74027e9	3.11237
$445$	0	0
$446$	8.42530e7	0.0449690
$447$	2.48307e9	1.31496
$448$	6.70136e8	0.352119
$449$	−3.43969e9	−1.79332	−0.896658	−	0.442724i	$-0.854012\pi$
$449$	−3.43969e9	−1.79332	−0.896658	+	0.442724i	$0.854012\pi$
$450$	0	0
$451$	8.07098e7	0.0414294
$452$	−1.49120e8	−0.0759542
$453$	−3.49549e8	−0.176671
$454$	1.29778e9	0.650889
$455$	0	0
$456$	3.28435e9	1.62208
$457$	−4.40856e8	−0.216068	−0.108034	−	0.994147i	$-0.534456\pi$
$457$	−4.40856e8	−0.216068	−0.108034	+	0.994147i	$0.534456\pi$
$458$	1.09623e9	0.533180
$459$	5.80155e8	0.280027
$460$	0	0
$461$	2.97064e9	1.41220	0.706100	−	0.708112i	$-0.250453\pi$
$461$	2.97064e9	1.41220	0.706100	+	0.708112i	$0.250453\pi$
$462$	1.27649e9	0.602241
$463$	−1.59975e9	−0.749062	−0.374531	−	0.927214i	$-0.622196\pi$
$463$	−1.59975e9	−0.749062	−0.374531	+	0.927214i	$0.622196\pi$
$464$	−3.75241e9	−1.74380
$465$	0	0
$466$	−5.56245e9	−2.54634
$467$	−1.62694e9	−0.739201	−0.369600	−	0.929191i	$-0.620505\pi$
$467$	−1.62694e9	−0.739201	−0.369600	+	0.929191i	$0.620505\pi$
$468$	−1.22044e8	−0.0550372
$469$	−1.45682e9	−0.652080
$470$	0	0
$471$	−2.10648e8	−0.0928934
$472$	−2.51756e8	−0.110200
$473$	−2.30151e8	−0.0999998
$474$	−4.34146e9	−1.87246
$475$	0	0
$476$	2.13404e9	0.906939
$477$	5.89790e7	0.0248819
$478$	−3.76421e9	−1.57644
$479$	−3.93755e8	−0.163701	−0.0818507	−	0.996645i	$-0.526083\pi$
$479$	−3.93755e8	−0.163701	−0.0818507	+	0.996645i	$0.526083\pi$
$480$	0	0
$481$	1.63318e9	0.669154
$482$	4.81236e9	1.95746
$483$	−6.00015e9	−2.42296
$484$	−5.19831e9	−2.08403
$485$	0	0
$486$	4.91984e8	0.194412
$487$	−2.90720e9	−1.14058	−0.570288	−	0.821445i	$-0.693168\pi$
$487$	−2.90720e9	−1.14058	−0.570288	+	0.821445i	$0.693168\pi$
$488$	9.81146e9	3.82177
$489$	9.81777e8	0.379692
$490$	0	0
$491$	1.49340e9	0.569364	0.284682	−	0.958622i	$-0.408112\pi$
$491$	1.49340e9	0.569364	0.284682	+	0.958622i	$0.408112\pi$
$492$	−1.02074e9	−0.386401
$493$	7.73030e8	0.290558
$494$	1.71331e9	0.639429
$495$	0	0
$496$	−5.59142e9	−2.05748
$497$	4.11127e9	1.50220
$498$	4.73143e9	1.71668
$499$	2.32305e9	0.836965	0.418482	−	0.908225i	$-0.362562\pi$
$499$	2.32305e9	0.836965	0.418482	+	0.908225i	$0.362562\pi$
$500$	0	0
$501$	1.68864e9	0.599936
$502$	6.64412e9	2.34409
$503$	4.58290e9	1.60566	0.802828	−	0.596210i	$-0.203328\pi$
$503$	4.58290e9	1.60566	0.802828	+	0.596210i	$0.203328\pi$
$504$	5.06577e8	0.176254
$505$	0	0
$506$	−1.97334e9	−0.677136
$507$	−2.24999e9	−0.766749
$508$	5.36158e9	1.81455
$509$	−3.08836e9	−1.03804	−0.519022	−	0.854761i	$-0.673704\pi$
$509$	−3.08836e9	−1.03804	−0.519022	+	0.854761i	$0.673704\pi$
$510$	0	0
$511$	−5.55876e8	−0.184291
$512$	−6.77068e9	−2.22940
$513$	−2.43696e9	−0.796962
$514$	−2.05548e9	−0.667641
$515$	0	0
$516$	2.91074e9	0.932673
$517$	−7.28843e8	−0.231962
$518$	−1.24293e10	−3.92910
$519$	2.76398e9	0.867860
$520$	0	0
$521$	3.95517e9	1.22527	0.612637	−	0.790364i	$-0.290109\pi$
$521$	3.95517e9	1.22527	0.612637	+	0.790364i	$0.290109\pi$
$522$	3.36453e8	0.103533
$523$	2.97750e9	0.910113	0.455057	−	0.890463i	$-0.349619\pi$
$523$	2.97750e9	0.910113	0.455057	+	0.890463i	$0.349619\pi$
$524$	1.27145e10	3.86048
$525$	0	0
$526$	4.02496e9	1.20590
$527$	1.15188e9	0.342824
$528$	−1.23643e9	−0.365553
$529$	5.87088e9	1.72428
$530$	0	0
$531$	9.63828e6	0.00279363
$532$	−8.96410e9	−2.58117
$533$	−2.90415e8	−0.0830756
$534$	−7.93917e9	−2.25622
$535$	0	0
$536$	3.30485e9	0.926991
$537$	1.38939e9	0.387182
$538$	4.95710e9	1.37243
$539$	−1.06637e9	−0.293324
$540$	0	0
$541$	−4.75653e9	−1.29152	−0.645758	−	0.763542i	$-0.723459\pi$
$541$	−4.75653e9	−1.29152	−0.645758	+	0.763542i	$0.723459\pi$
$542$	−7.91342e9	−2.13485
$543$	−5.37179e9	−1.43986
$544$	−8.05968e8	−0.214645
$545$	0	0
$546$	−4.59314e9	−1.20763
$547$	4.58098e9	1.19675	0.598374	−	0.801217i	$-0.295814\pi$
$547$	4.58098e9	1.19675	0.598374	+	0.801217i	$0.295814\pi$
$548$	−9.90057e9	−2.56997
$549$	−3.75624e8	−0.0968836
$550$	0	0
$551$	−3.24714e9	−0.826933
$552$	1.36116e10	3.44446
$553$	6.46260e9	1.62506
$554$	−1.02990e10	−2.57343
$555$	0	0
$556$	−1.15650e10	−2.85355
$557$	1.98969e9	0.487856	0.243928	−	0.969793i	$-0.421564\pi$
$557$	1.98969e9	0.487856	0.243928	+	0.969793i	$0.421564\pi$
$558$	5.01344e8	0.122156
$559$	8.28143e8	0.200523
$560$	0	0
$561$	2.54716e8	0.0609096
$562$	−3.14824e9	−0.748154
$563$	−1.51071e9	−0.356781	−0.178390	−	0.983960i	$-0.557089\pi$
$563$	−1.51071e9	−0.356781	−0.178390	+	0.983960i	$0.557089\pi$
$564$	9.21774e9	2.16345
$565$	0	0
$566$	2.31863e9	0.537495
$567$	6.17665e9	1.42302
$568$	−9.32658e9	−2.13552
$569$	1.75619e8	0.0399650	0.0199825	−	0.999800i	$-0.493639\pi$
$569$	1.75619e8	0.0399650	0.0199825	+	0.999800i	$0.493639\pi$
$570$	0	0
$571$	−2.21767e8	−0.0498506	−0.0249253	−	0.999689i	$-0.507935\pi$
$571$	−2.21767e8	−0.0498506	−0.0249253	+	0.999689i	$0.507935\pi$
$572$	−1.03850e9	−0.232018
$573$	3.73028e9	0.828323
$574$	2.21020e9	0.487798
$575$	0	0
$576$	5.82008e7	0.0126897
$577$	−7.51519e9	−1.62864	−0.814319	−	0.580418i	$-0.802889\pi$
$577$	−7.51519e9	−1.62864	−0.814319	+	0.580418i	$0.802889\pi$
$578$	−7.68491e9	−1.65536
$579$	−9.04906e9	−1.93744
$580$	0	0
$581$	−7.04310e9	−1.48987
$582$	−6.43915e9	−1.35393
$583$	5.01866e8	0.104893
$584$	1.26103e9	0.261987
$585$	0	0
$586$	1.70472e10	3.49953
$587$	−5.57215e9	−1.13708	−0.568538	−	0.822657i	$-0.692491\pi$
$587$	−5.57215e9	−1.13708	−0.568538	+	0.822657i	$0.692491\pi$
$588$	1.34865e10	2.73576
$589$	−4.83852e9	−0.975683
$590$	0	0
$591$	−3.86032e9	−0.769250
$592$	1.20393e10	2.38492
$593$	2.27962e9	0.448922	0.224461	−	0.974483i	$-0.427938\pi$
$593$	2.27962e9	0.448922	0.224461	+	0.974483i	$0.427938\pi$
$594$	2.14863e9	0.420639
$595$	0	0
$596$	1.53742e10	2.97461
$597$	−1.87802e9	−0.361235
$598$	7.10060e9	1.35782
$599$	3.90822e9	0.742993	0.371497	−	0.928434i	$-0.378845\pi$
$599$	3.90822e9	0.742993	0.371497	+	0.928434i	$0.378845\pi$
$600$	0	0
$601$	−2.15505e9	−0.404946	−0.202473	−	0.979288i	$-0.564898\pi$
$601$	−2.15505e9	−0.404946	−0.202473	+	0.979288i	$0.564898\pi$
$602$	−6.30259e9	−1.17742
$603$	−1.26524e8	−0.0234997
$604$	−2.16427e9	−0.399652
$605$	0	0
$606$	4.92221e9	0.898476
$607$	5.06857e9	0.919867	0.459933	−	0.887953i	$-0.347873\pi$
$607$	5.06857e9	0.919867	0.459933	+	0.887953i	$0.347873\pi$
$608$	3.38549e9	0.610885
$609$	8.70510e9	1.56176
$610$	0	0
$611$	2.62257e9	0.465138
$612$	1.85340e8	0.0326843
$613$	−6.56079e9	−1.15039	−0.575194	−	0.818017i	$-0.695073\pi$
$613$	−6.56079e9	−1.15039	−0.575194	+	0.818017i	$0.695073\pi$
$614$	4.81841e8	0.0840068
$615$	0	0
$616$	4.31058e9	0.743024
$617$	1.11814e8	0.0191645	0.00958223	−	0.999954i	$-0.496950\pi$
$617$	1.11814e8	0.0191645	0.00958223	+	0.999954i	$0.496950\pi$
$618$	6.53268e9	1.11335
$619$	1.48440e9	0.251556	0.125778	−	0.992058i	$-0.459857\pi$
$619$	1.48440e9	0.251556	0.125778	+	0.992058i	$0.459857\pi$
$620$	0	0
$621$	−1.00997e10	−1.69233
$622$	1.48080e10	2.46735
$623$	1.18181e10	1.95812
$624$	4.44900e9	0.733020
$625$	0	0
$626$	−1.86506e10	−3.03867
$627$	−1.06994e9	−0.173350
$628$	−1.30425e9	−0.210137
$629$	−2.48019e9	−0.397382
$630$	0	0
$631$	6.22722e9	0.986714	0.493357	−	0.869827i	$-0.335770\pi$
$631$	6.22722e9	0.986714	0.493357	+	0.869827i	$0.335770\pi$
$632$	−1.46607e10	−2.31017
$633$	−7.05220e8	−0.110513
$634$	−1.58938e9	−0.247693
$635$	0	0
$636$	−6.34714e9	−0.978314
$637$	3.83708e9	0.588184
$638$	2.86295e9	0.436458
$639$	3.57061e8	0.0541364
$640$	0	0
$641$	6.72145e9	1.00800	0.503999	−	0.863704i	$-0.331861\pi$
$641$	6.72145e9	1.00800	0.503999	+	0.863704i	$0.331861\pi$
$642$	−9.29651e9	−1.38659
$643$	−6.13000e9	−0.909332	−0.454666	−	0.890662i	$-0.650241\pi$
$643$	−6.13000e9	−0.909332	−0.454666	+	0.890662i	$0.650241\pi$
$644$	−3.71505e10	−5.48106
$645$	0	0
$646$	−2.60189e9	−0.379730
$647$	1.10514e10	1.60418	0.802089	−	0.597204i	$-0.203722\pi$
$647$	1.10514e10	1.60418	0.802089	+	0.597204i	$0.203722\pi$
$648$	−1.40120e10	−2.02296
$649$	8.20142e7	0.0117769
$650$	0	0
$651$	1.29714e10	1.84269
$652$	6.07878e9	0.858914
$653$	−1.90379e9	−0.267561	−0.133781	−	0.991011i	$-0.542712\pi$
$653$	−1.90379e9	−0.267561	−0.133781	+	0.991011i	$0.542712\pi$
$654$	−1.10729e9	−0.154788
$655$	0	0
$656$	−2.14084e9	−0.296088
$657$	−4.82774e7	−0.00664149
$658$	−1.99591e10	−2.73117
$659$	1.16276e9	0.158268	0.0791339	−	0.996864i	$-0.474785\pi$
$659$	1.16276e9	0.158268	0.0791339	+	0.996864i	$0.474785\pi$
$660$	0	0
$661$	−2.20847e9	−0.297431	−0.148716	−	0.988880i	$-0.547514\pi$
$661$	−2.20847e9	−0.297431	−0.148716	+	0.988880i	$0.547514\pi$
$662$	7.66763e9	1.02721
$663$	−9.16532e8	−0.122138
$664$	1.59775e10	2.11798
$665$	0	0
$666$	−1.07948e9	−0.141597
$667$	−1.34573e10	−1.75598
$668$	1.04554e10	1.35713
$669$	1.89322e8	0.0244461
$670$	0	0
$671$	−3.19627e9	−0.408427
$672$	−9.07601e9	−1.15373
$673$	−9.22930e9	−1.16712	−0.583561	−	0.812070i	$-0.698341\pi$
$673$	−9.22930e9	−1.16712	−0.583561	+	0.812070i	$0.698341\pi$
$674$	2.18630e10	2.75043
$675$	0	0
$676$	−1.39311e10	−1.73449
$677$	−5.75119e9	−0.712357	−0.356178	−	0.934418i	$-0.615920\pi$
$677$	−5.75119e9	−0.712357	−0.356178	+	0.934418i	$0.615920\pi$
$678$	−4.87411e8	−0.0600609
$679$	9.58518e9	1.17505
$680$	0	0
$681$	2.91620e9	0.353837
$682$	4.26605e9	0.514969
$683$	2.96482e9	0.356063	0.178031	−	0.984025i	$-0.443027\pi$
$683$	2.96482e9	0.356063	0.178031	+	0.984025i	$0.443027\pi$
$684$	−7.78526e8	−0.0930200
$685$	0	0
$686$	−6.36934e9	−0.753287
$687$	2.46331e9	0.289848
$688$	6.10479e9	0.714680
$689$	−1.80584e9	−0.210336
$690$	0	0
$691$	1.35266e10	1.55961	0.779805	−	0.626023i	$-0.215318\pi$
$691$	1.35266e10	1.55961	0.779805	+	0.626023i	$0.215318\pi$
$692$	1.71135e10	1.96321
$693$	−1.65027e8	−0.0188360
$694$	1.77579e10	2.01666
$695$	0	0
$696$	−1.97479e10	−2.22018
$697$	4.41032e8	0.0493350
$698$	6.71704e9	0.747626
$699$	−1.24992e10	−1.38424
$700$	0	0
$701$	8.89669e9	0.975473	0.487737	−	0.872991i	$-0.337823\pi$
$701$	8.89669e9	0.975473	0.487737	+	0.872991i	$0.337823\pi$
$702$	−7.73133e9	−0.843480
$703$	1.04181e10	1.13096
$704$	4.95244e8	0.0534952
$705$	0	0
$706$	2.00018e10	2.13921
$707$	−7.32710e9	−0.779766
$708$	−1.03724e9	−0.109841
$709$	−4.02759e9	−0.424408	−0.212204	−	0.977225i	$-0.568064\pi$
$709$	−4.02759e9	−0.424408	−0.212204	+	0.977225i	$0.568064\pi$
$710$	0	0
$711$	5.61272e8	0.0585639
$712$	−2.68098e10	−2.78364
$713$	−2.00526e10	−2.07184
$714$	6.97528e9	0.717163
$715$	0	0
$716$	8.60256e9	0.875855
$717$	−8.45842e9	−0.856983
$718$	2.85689e10	2.88044
$719$	−9.22714e9	−0.925798	−0.462899	−	0.886411i	$-0.653191\pi$
$719$	−9.22714e9	−0.925798	−0.462899	+	0.886411i	$0.653191\pi$
$720$	0	0
$721$	−9.72440e9	−0.966250
$722$	−7.16063e9	−0.708062
$723$	1.08137e10	1.06412
$724$	−3.32600e10	−3.25714
$725$	0	0
$726$	−1.69911e10	−1.64795
$727$	9.12103e9	0.880386	0.440193	−	0.897903i	$-0.354910\pi$
$727$	9.12103e9	0.880386	0.440193	+	0.897903i	$0.354910\pi$
$728$	−1.55106e10	−1.48994
$729$	1.09658e10	1.04832
$730$	0	0
$731$	−1.25764e9	−0.119082
$732$	4.04235e10	3.80930
$733$	−1.72029e10	−1.61338	−0.806691	−	0.590974i	$-0.798744\pi$
$733$	−1.72029e10	−1.61338	−0.806691	+	0.590974i	$0.798744\pi$
$734$	1.53202e10	1.42997
$735$	0	0
$736$	1.40307e10	1.29720
$737$	−1.07662e9	−0.0990663
$738$	1.91954e8	0.0175793
$739$	−1.73905e10	−1.58510	−0.792552	−	0.609805i	$-0.791248\pi$
$739$	−1.73905e10	−1.58510	−0.792552	+	0.609805i	$0.791248\pi$
$740$	0	0
$741$	3.84992e9	0.347607
$742$	1.37434e10	1.23504
$743$	−4.62464e9	−0.413635	−0.206817	−	0.978380i	$-0.566311\pi$
$743$	−4.62464e9	−0.413635	−0.206817	+	0.978380i	$0.566311\pi$
$744$	−2.94260e10	−2.61955
$745$	0	0
$746$	5.99562e9	0.528748
$747$	−6.11688e8	−0.0536918
$748$	1.57710e9	0.137785
$749$	1.38386e10	1.20339
$750$	0	0
$751$	−1.36007e10	−1.17171	−0.585856	−	0.810415i	$-0.699242\pi$
$751$	−1.36007e10	−1.17171	−0.585856	+	0.810415i	$0.699242\pi$
$752$	1.93327e10	1.65779
$753$	1.49298e10	1.27430
$754$	−1.03017e10	−0.875200
$755$	0	0
$756$	4.04505e10	3.40485
$757$	1.78432e10	1.49499	0.747493	−	0.664269i	$-0.231257\pi$
$757$	1.78432e10	1.49499	0.747493	+	0.664269i	$0.231257\pi$
$758$	2.69006e10	2.24346
$759$	−4.43423e9	−0.368105
$760$	0	0
$761$	−1.41211e10	−1.16151	−0.580754	−	0.814079i	$-0.697242\pi$
$761$	−1.41211e10	−1.16151	−0.580754	+	0.814079i	$0.697242\pi$
$762$	1.75248e10	1.43486
$763$	1.64829e9	0.134337
$764$	2.30964e10	1.87377
$765$	0	0
$766$	−1.69099e10	−1.35938
$767$	−2.95108e8	−0.0236155
$768$	−2.34249e10	−1.86601
$769$	−6.98040e9	−0.553526	−0.276763	−	0.960938i	$-0.589262\pi$
$769$	−6.98040e9	−0.553526	−0.276763	+	0.960938i	$0.589262\pi$
$770$	0	0
$771$	−4.61880e9	−0.362943
$772$	−5.60282e10	−4.38274
$773$	5.57792e9	0.434354	0.217177	−	0.976132i	$-0.430315\pi$
$773$	5.57792e9	0.434354	0.217177	+	0.976132i	$0.430315\pi$
$774$	−5.47375e8	−0.0424318
$775$	0	0
$776$	−2.17443e10	−1.67044
$777$	−2.79295e10	−2.13594
$778$	3.34727e9	0.254837
$779$	−1.85257e9	−0.140408
$780$	0	0
$781$	3.03831e9	0.228220
$782$	−1.07832e10	−0.806349
$783$	1.46527e10	1.09082
$784$	2.82857e10	2.09634
$785$	0	0
$786$	4.15585e10	3.05268
$787$	−1.26349e10	−0.923978	−0.461989	−	0.886886i	$-0.652864\pi$
$787$	−1.26349e10	−0.923978	−0.461989	+	0.886886i	$0.652864\pi$
$788$	−2.39016e10	−1.74014
$789$	9.04434e9	0.655552
$790$	0	0
$791$	7.25550e8	0.0521254
$792$	3.74370e8	0.0267771
$793$	1.15010e10	0.818992
$794$	4.08300e9	0.289473
$795$	0	0
$796$	−1.16279e10	−0.817160
$797$	−9.24597e9	−0.646916	−0.323458	−	0.946242i	$-0.604846\pi$
$797$	−9.24597e9	−0.646916	−0.323458	+	0.946242i	$0.604846\pi$
$798$	−2.92999e10	−2.04106
$799$	−3.98271e9	−0.276226
$800$	0	0
$801$	1.02639e9	0.0705666
$802$	−4.95698e9	−0.339318
$803$	−4.10803e8	−0.0279982
$804$	1.36161e10	0.923967
$805$	0	0
$806$	−1.53504e10	−1.03263
$807$	1.11389e10	0.746080
$808$	1.66218e10	1.10851
$809$	3.77594e9	0.250729	0.125365	−	0.992111i	$-0.459990\pi$
$809$	3.77594e9	0.250729	0.125365	+	0.992111i	$0.459990\pi$
$810$	0	0
$811$	−8.24089e9	−0.542502	−0.271251	−	0.962509i	$-0.587437\pi$
$811$	−8.24089e9	−0.542502	−0.271251	+	0.962509i	$0.587437\pi$
$812$	5.38985e10	3.53290
$813$	−1.77820e10	−1.16055
$814$	−9.18552e9	−0.596923
$815$	0	0
$816$	−6.75637e9	−0.435309
$817$	5.28276e9	0.338910
$818$	3.49268e9	0.223112
$819$	5.93810e8	0.0377706
$820$	0	0
$821$	8.07649e9	0.509356	0.254678	−	0.967026i	$-0.418031\pi$
$821$	8.07649e9	0.509356	0.254678	+	0.967026i	$0.418031\pi$
$822$	−3.23608e10	−2.03221
$823$	1.76555e10	1.10403	0.552016	−	0.833834i	$-0.313859\pi$
$823$	1.76555e10	1.10403	0.552016	+	0.833834i	$0.313859\pi$
$824$	2.20602e10	1.37361
$825$	0	0
$826$	2.24593e9	0.138664
$827$	−2.60329e9	−0.160049	−0.0800246	−	0.996793i	$-0.525500\pi$
$827$	−2.60329e9	−0.160049	−0.0800246	+	0.996793i	$0.525500\pi$
$828$	−3.22649e9	−0.197526
$829$	9.16053e9	0.558444	0.279222	−	0.960227i	$-0.409923\pi$
$829$	9.16053e9	0.558444	0.279222	+	0.960227i	$0.409923\pi$
$830$	0	0
$831$	−2.31426e10	−1.39897
$832$	−1.78202e9	−0.107270
$833$	−5.82711e9	−0.349297
$834$	−3.78013e10	−2.25645
$835$	0	0
$836$	−6.62465e9	−0.392140
$837$	2.18338e10	1.28704
$838$	−5.34629e10	−3.13833
$839$	2.51395e10	1.46957	0.734784	−	0.678301i	$-0.237283\pi$
$839$	2.51395e10	1.46957	0.734784	+	0.678301i	$0.237283\pi$
$840$	0	0
$841$	2.27422e9	0.131840
$842$	2.26657e10	1.30851
$843$	−7.07430e9	−0.406712
$844$	−4.36644e9	−0.249994
$845$	0	0
$846$	−1.73343e9	−0.0984260
$847$	2.52926e10	1.43022
$848$	−1.33121e10	−0.749653
$849$	5.21012e9	0.292193
$850$	0	0
$851$	4.31766e10	2.40157
$852$	−3.84258e10	−2.12855
$853$	−1.37385e10	−0.757908	−0.378954	−	0.925415i	$-0.623716\pi$
$853$	−1.37385e10	−0.757908	−0.378954	+	0.925415i	$0.623716\pi$
$854$	−8.75285e10	−4.80891
$855$	0	0
$856$	−3.13934e10	−1.71072
$857$	2.41390e10	1.31004	0.655022	−	0.755610i	$-0.272659\pi$
$857$	2.41390e10	1.31004	0.655022	+	0.755610i	$0.272659\pi$
$858$	−3.39442e9	−0.183468
$859$	−2.62094e10	−1.41085	−0.705426	−	0.708783i	$-0.749244\pi$
$859$	−2.62094e10	−1.41085	−0.705426	+	0.708783i	$0.749244\pi$
$860$	0	0
$861$	4.96646e9	0.265177
$862$	−5.19597e10	−2.76307
$863$	1.61957e10	0.857750	0.428875	−	0.903364i	$-0.358910\pi$
$863$	1.61957e10	0.857750	0.428875	+	0.903364i	$0.358910\pi$
$864$	−1.52771e10	−0.805826
$865$	0	0
$866$	1.95426e10	1.02251
$867$	−1.72685e10	−0.899885
$868$	8.03135e10	4.16840
$869$	4.77599e9	0.246885
$870$	0	0
$871$	3.87395e9	0.198651
$872$	−3.73920e9	−0.190973
$873$	8.32465e8	0.0423464
$874$	4.52951e10	2.29488
$875$	0	0
$876$	5.19547e9	0.261132
$877$	1.25717e10	0.629356	0.314678	−	0.949198i	$-0.398103\pi$
$877$	1.25717e10	0.629356	0.314678	+	0.949198i	$0.398103\pi$
$878$	5.40688e10	2.69598
$879$	3.83060e10	1.90242
$880$	0	0
$881$	−3.40786e10	−1.67906	−0.839530	−	0.543313i	$-0.817170\pi$
$881$	−3.40786e10	−1.67906	−0.839530	+	0.543313i	$0.817170\pi$
$882$	−2.53618e9	−0.124463
$883$	−2.78949e10	−1.36352	−0.681762	−	0.731574i	$-0.738786\pi$
$883$	−2.78949e10	−1.36352	−0.681762	+	0.731574i	$0.738786\pi$
$884$	−5.67480e9	−0.276292
$885$	0	0
$886$	1.87348e10	0.904963
$887$	3.29903e10	1.58728	0.793641	−	0.608387i	$-0.208183\pi$
$887$	3.29903e10	1.58728	0.793641	+	0.608387i	$0.208183\pi$
$888$	6.33591e10	3.03643
$889$	−2.60870e10	−1.24528
$890$	0	0
$891$	4.56466e9	0.216191
$892$	1.17221e9	0.0553002
$893$	1.67295e10	0.786144
$894$	5.02517e10	2.35218
$895$	0	0
$896$	3.91083e10	1.81631
$897$	1.59555e10	0.738136
$898$	−6.96115e10	−3.20785
$899$	2.90926e10	1.33544
$900$	0	0
$901$	2.74241e9	0.124909
$902$	1.63338e9	0.0741080
$903$	−1.41623e10	−0.640069
$904$	−1.64594e9	−0.0741010
$905$	0	0
$906$	−7.07408e9	−0.316025
$907$	−1.66008e10	−0.738759	−0.369380	−	0.929279i	$-0.620430\pi$
$907$	−1.66008e10	−0.738759	−0.369380	+	0.929279i	$0.620430\pi$
$908$	1.80560e10	0.800425
$909$	−6.36353e8	−0.0281012
$910$	0	0
$911$	−2.80189e9	−0.122782	−0.0613912	−	0.998114i	$-0.519554\pi$
$911$	−2.80189e9	−0.122782	−0.0613912	+	0.998114i	$0.519554\pi$
$912$	2.83804e10	1.23890
$913$	−5.20499e9	−0.226346
$914$	−8.92193e9	−0.386498
$915$	0	0
$916$	1.52518e10	0.655673
$917$	−6.18631e10	−2.64935
$918$	1.17410e10	0.500907
$919$	−7.97528e9	−0.338955	−0.169477	−	0.985534i	$-0.554208\pi$
$919$	−7.97528e9	−0.338955	−0.169477	+	0.985534i	$0.554208\pi$
$920$	0	0
$921$	1.08273e9	0.0456678
$922$	6.01190e10	2.52612
$923$	−1.09326e10	−0.457635
$924$	1.77597e10	0.740600
$925$	0	0
$926$	−3.23753e10	−1.33991
$927$	−8.44557e8	−0.0348217
$928$	−2.03560e10	−0.836131
$929$	−4.40021e10	−1.80061	−0.900303	−	0.435264i	$-0.856655\pi$
$929$	−4.40021e10	−1.80061	−0.900303	+	0.435264i	$0.856655\pi$
$930$	0	0
$931$	2.44769e10	0.994107
$932$	−7.73900e10	−3.13133
$933$	3.32746e10	1.34130
$934$	−3.29256e10	−1.32227
$935$	0	0
$936$	−1.34708e9	−0.0536943
$937$	1.85516e10	0.736702	0.368351	−	0.929687i	$-0.379922\pi$
$937$	1.85516e10	0.736702	0.368351	+	0.929687i	$0.379922\pi$
$938$	−2.94828e10	−1.16643
$939$	−4.19092e10	−1.65188
$940$	0	0
$941$	−3.98382e8	−0.0155860	−0.00779302	−	0.999970i	$-0.502481\pi$
$941$	−3.98382e8	−0.0155860	−0.00779302	+	0.999970i	$0.502481\pi$
$942$	−4.26305e9	−0.166166
$943$	−7.67772e9	−0.298155
$944$	−2.17544e9	−0.0841677
$945$	0	0
$946$	−4.65774e9	−0.178878
$947$	−1.22967e10	−0.470504	−0.235252	−	0.971934i	$-0.575591\pi$
$947$	−1.22967e10	−0.470504	−0.235252	+	0.971934i	$0.575591\pi$
$948$	−6.04024e10	−2.30263
$949$	1.47818e9	0.0561429
$950$	0	0
$951$	−3.57143e9	−0.134651
$952$	2.35548e10	0.884810
$953$	−2.00329e10	−0.749754	−0.374877	−	0.927075i	$-0.622315\pi$
$953$	−2.00329e10	−0.749754	−0.374877	+	0.927075i	$0.622315\pi$
$954$	1.19360e9	0.0445082
$955$	0	0
$956$	−5.23712e10	−1.93861
$957$	6.43324e9	0.237267
$958$	−7.96872e9	−0.292826
$959$	4.81716e10	1.76370
$960$	0	0
$961$	1.58379e10	0.575659
$962$	3.30519e10	1.19697
$963$	1.20187e9	0.0433676
$964$	6.69540e10	2.40717
$965$	0	0
$966$	−1.21429e11	−4.33415
$967$	2.01344e10	0.716055	0.358027	−	0.933711i	$-0.383449\pi$
$967$	2.01344e10	0.716055	0.358027	+	0.933711i	$0.383449\pi$
$968$	−5.73772e10	−2.03318
$969$	−5.84661e9	−0.206429
$970$	0	0
$971$	−5.02424e10	−1.76118	−0.880589	−	0.473881i	$-0.842853\pi$
$971$	−5.02424e10	−1.76118	−0.880589	+	0.473881i	$0.842853\pi$
$972$	6.84493e9	0.239077
$973$	5.62702e10	1.95832
$974$	−5.88352e10	−2.04024
$975$	0	0
$976$	8.47816e10	2.91895
$977$	−3.03613e10	−1.04157	−0.520786	−	0.853687i	$-0.674361\pi$
$977$	−3.03613e10	−1.04157	−0.520786	+	0.853687i	$0.674361\pi$
$978$	1.98690e10	0.679186
$979$	8.73379e9	0.297484
$980$	0	0
$981$	1.43152e8	0.00484124
$982$	3.02230e10	1.01847
$983$	−3.64309e10	−1.22330	−0.611650	−	0.791129i	$-0.709494\pi$
$983$	−3.64309e10	−1.22330	−0.611650	+	0.791129i	$0.709494\pi$
$984$	−1.12666e10	−0.376973
$985$	0	0
$986$	1.56444e10	0.519744
$987$	−4.48493e10	−1.48472
$988$	2.38372e10	0.786332
$989$	2.18937e10	0.719668
$990$	0	0
$991$	−2.79867e10	−0.913470	−0.456735	−	0.889603i	$-0.650981\pi$
$991$	−2.79867e10	−0.913470	−0.456735	+	0.889603i	$0.650981\pi$
$992$	−3.03322e10	−0.986536
$993$	1.72296e10	0.558411
$994$	8.32029e10	2.68711
$995$	0	0
$996$	6.58280e10	2.11107
$997$	−3.06088e10	−0.978169	−0.489084	−	0.872236i	$-0.662669\pi$
$997$	−3.06088e10	−0.978169	−0.489084	+	0.872236i	$0.662669\pi$
$998$	4.70133e10	1.49715
$999$	−4.70119e10	−1.49186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.a.e.1.2	yes	2
3.2	odd	2		225.8.a.k.1.1		2
4.3	odd	2		400.8.a.v.1.1		2
5.2	odd	4		25.8.b.b.24.4		4
5.3	odd	4		25.8.b.b.24.1		4
5.4	even	2		25.8.a.c.1.1	✓	2
15.2	even	4		225.8.b.l.199.1		4
15.8	even	4		225.8.b.l.199.4		4
15.14	odd	2		225.8.a.v.1.2		2
20.3	even	4		400.8.c.s.49.1		4
20.7	even	4		400.8.c.s.49.4		4
20.19	odd	2		400.8.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.a.c.1.1	✓	2	5.4	even	2
25.8.a.e.1.2	yes	2	1.1	even	1	trivial
25.8.b.b.24.1		4	5.3	odd	4
25.8.b.b.24.4		4	5.2	odd	4
225.8.a.k.1.1		2	3.2	odd	2
225.8.a.v.1.2		2	15.14	odd	2
225.8.b.l.199.1		4	15.2	even	4
225.8.b.l.199.4		4	15.8	even	4
400.8.a.v.1.1		2	4.3	odd	2
400.8.a.bd.1.2		2	20.19	odd	2
400.8.c.s.49.1		4	20.3	even	4
400.8.c.s.49.4		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+20.2377 q^{2} +45.4755 q^{3} +281.566 q^{4} +920.321 q^{6} -1369.97 q^{7} +3107.83 q^{8} -118.981 q^{9} +O(q^{10})\)	\(q+20.2377 q^{2} +45.4755 q^{3} +281.566 q^{4} +920.321 q^{6} -1369.97 q^{7} +3107.83 q^{8} -118.981 q^{9} -1012.43 q^{11} +12804.4 q^{12} +3643.00 q^{13} -27725.1 q^{14} +26855.0 q^{16} -5532.37 q^{17} -2407.90 q^{18} +23238.9 q^{19} -62300.0 q^{21} -20489.4 q^{22} +96310.5 q^{23} +141330. q^{24} +73726.1 q^{26} -104866. q^{27} -385737. q^{28} -139729. q^{29} -208208. q^{31} +145682. q^{32} -46041.0 q^{33} -111963. q^{34} -33501.0 q^{36} +448306. q^{37} +470303. q^{38} +165667. q^{39} -79718.5 q^{41} -1.26081e6 q^{42} +227324. q^{43} -285067. q^{44} +1.94911e6 q^{46} +719892. q^{47} +1.22124e6 q^{48} +1.05328e6 q^{49} -251587. q^{51} +1.02575e6 q^{52} -495702. q^{53} -2.12224e6 q^{54} -4.25763e6 q^{56} +1.05680e6 q^{57} -2.82779e6 q^{58} -81006.9 q^{59} +3.15701e6 q^{61} -4.21366e6 q^{62} +163000. q^{63} -489161. q^{64} -931765. q^{66} +1.06340e6 q^{67} -1.55773e6 q^{68} +4.37976e6 q^{69} -3.00099e6 q^{71} -369772. q^{72} +405758. q^{73} +9.07270e6 q^{74} +6.54328e6 q^{76} +1.38701e6 q^{77} +3.35273e6 q^{78} -4.71733e6 q^{79} -4.50860e6 q^{81} -1.61332e6 q^{82} +5.14106e6 q^{83} -1.75416e7 q^{84} +4.60053e6 q^{86} -6.35423e6 q^{87} -3.14648e6 q^{88} -8.62652e6 q^{89} -4.99080e6 q^{91} +2.71178e7 q^{92} -9.46835e6 q^{93} +1.45690e7 q^{94} +6.62497e6 q^{96} -6.99663e6 q^{97} +2.13159e7 q^{98} +120460. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 15 q^{2} + 40 q^{3} + 181 q^{4} + 949 q^{6} - 600 q^{7} + 4305 q^{8} - 2276 q^{9}+O(q^{10}) \) 2 * q + 15 * q^2 + 40 * q^3 + 181 * q^4 + 949 * q^6 - 600 * q^7 + 4305 * q^8 - 2276 * q^9	\( 2 q + 15 q^{2} + 40 q^{3} + 181 q^{4} + 949 q^{6} - 600 q^{7} + 4305 q^{8} - 2276 q^{9} + 4344 q^{11} + 13355 q^{12} + 17680 q^{13} - 31758 q^{14} + 33457 q^{16} + 6870 q^{17} + 8890 q^{18} + 18200 q^{19} - 66516 q^{21} - 48545 q^{22} + 21120 q^{23} + 134775 q^{24} + 204 q^{26} - 81080 q^{27} - 463170 q^{28} + 55800 q^{29} - 301776 q^{31} - 42135 q^{32} - 75370 q^{33} - 176923 q^{34} + 183422 q^{36} + 609860 q^{37} + 496695 q^{38} + 88808 q^{39} - 108486 q^{41} - 1238730 q^{42} + 966400 q^{43} - 823743 q^{44} + 2342934 q^{46} + 1787880 q^{47} + 1185095 q^{48} + 822586 q^{49} - 319496 q^{51} - 385900 q^{52} + 130740 q^{53} - 2246825 q^{54} - 3335850 q^{56} + 1084390 q^{57} - 3851920 q^{58} + 2067600 q^{59} + 582044 q^{61} - 3723570 q^{62} - 1497840 q^{63} - 350479 q^{64} - 778147 q^{66} + 255720 q^{67} - 2804985 q^{68} + 4791468 q^{69} - 4728216 q^{71} - 2952090 q^{72} - 1339430 q^{73} + 8226522 q^{74} + 7050025 q^{76} + 5511300 q^{77} + 3755300 q^{78} - 7186200 q^{79} + 78562 q^{81} - 1462645 q^{82} + 12049560 q^{83} - 17117598 q^{84} + 729444 q^{86} - 7424840 q^{87} + 3266085 q^{88} - 5990850 q^{89} + 5817264 q^{91} + 34679370 q^{92} - 8956020 q^{93} + 8975132 q^{94} + 7653359 q^{96} - 17120020 q^{97} + 22524195 q^{98} - 11433472 q^{99}+O(q^{100}) \) 2 * q + 15 * q^2 + 40 * q^3 + 181 * q^4 + 949 * q^6 - 600 * q^7 + 4305 * q^8 - 2276 * q^9 + 4344 * q^11 + 13355 * q^12 + 17680 * q^13 - 31758 * q^14 + 33457 * q^16 + 6870 * q^17 + 8890 * q^18 + 18200 * q^19 - 66516 * q^21 - 48545 * q^22 + 21120 * q^23 + 134775 * q^24 + 204 * q^26 - 81080 * q^27 - 463170 * q^28 + 55800 * q^29 - 301776 * q^31 - 42135 * q^32 - 75370 * q^33 - 176923 * q^34 + 183422 * q^36 + 609860 * q^37 + 496695 * q^38 + 88808 * q^39 - 108486 * q^41 - 1238730 * q^42 + 966400 * q^43 - 823743 * q^44 + 2342934 * q^46 + 1787880 * q^47 + 1185095 * q^48 + 822586 * q^49 - 319496 * q^51 - 385900 * q^52 + 130740 * q^53 - 2246825 * q^54 - 3335850 * q^56 + 1084390 * q^57 - 3851920 * q^58 + 2067600 * q^59 + 582044 * q^61 - 3723570 * q^62 - 1497840 * q^63 - 350479 * q^64 - 778147 * q^66 + 255720 * q^67 - 2804985 * q^68 + 4791468 * q^69 - 4728216 * q^71 - 2952090 * q^72 - 1339430 * q^73 + 8226522 * q^74 + 7050025 * q^76 + 5511300 * q^77 + 3755300 * q^78 - 7186200 * q^79 + 78562 * q^81 - 1462645 * q^82 + 12049560 * q^83 - 17117598 * q^84 + 729444 * q^86 - 7424840 * q^87 + 3266085 * q^88 - 5990850 * q^89 + 5817264 * q^91 + 34679370 * q^92 - 8956020 * q^93 + 8975132 * q^94 + 7653359 * q^96 - 17120020 * q^97 + 22524195 * q^98 - 11433472 * q^99

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