LMFDB - Embedded newform 350.8.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,8,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("350.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	350.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:	$109.334758919$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2473x^{2} - 31160x + 389808 $ x^4 - x^3 - 2473x^2 - 31160x + 389808
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-26.1112$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} -36.1742 q^{3} +64.0000 q^{4} +289.394 q^{6} +343.000 q^{7} -512.000 q^{8} -878.424 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -36.1742 q^{3} +64.0000 q^{4} +289.394 q^{6} +343.000 q^{7} -512.000 q^{8} -878.424 q^{9} +6314.71 q^{11} -2315.15 q^{12} +745.251 q^{13} -2744.00 q^{14} +4096.00 q^{16} +34226.8 q^{17} +7027.39 q^{18} -45918.6 q^{19} -12407.8 q^{21} -50517.7 q^{22} +58260.3 q^{23} +18521.2 q^{24} -5962.00 q^{26} +110889. q^{27} +21952.0 q^{28} +98592.5 q^{29} +3824.79 q^{31} -32768.0 q^{32} -228430. q^{33} -273814. q^{34} -56219.1 q^{36} -506799. q^{37} +367349. q^{38} -26958.9 q^{39} +205918. q^{41} +99262.1 q^{42} +872545. q^{43} +404142. q^{44} -466082. q^{46} +938310. q^{47} -148170. q^{48} +117649. q^{49} -1.23813e6 q^{51} +47696.0 q^{52} -483847. q^{53} -887115. q^{54} -175616. q^{56} +1.66107e6 q^{57} -788740. q^{58} -2.72708e6 q^{59} -3.30011e6 q^{61} -30598.3 q^{62} -301299. q^{63} +262144. q^{64} +1.82744e6 q^{66} -4.13387e6 q^{67} +2.19051e6 q^{68} -2.10752e6 q^{69} +4.30629e6 q^{71} +449753. q^{72} +908815. q^{73} +4.05439e6 q^{74} -2.93879e6 q^{76} +2.16595e6 q^{77} +215671. q^{78} -5.74990e6 q^{79} -2.09023e6 q^{81} -1.64734e6 q^{82} -2.66624e6 q^{83} -794097. q^{84} -6.98036e6 q^{86} -3.56651e6 q^{87} -3.23313e6 q^{88} +1.38767e6 q^{89} +255621. q^{91} +3.72866e6 q^{92} -138359. q^{93} -7.50648e6 q^{94} +1.18536e6 q^{96} +1.51141e7 q^{97} -941192. q^{98} -5.54699e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9} + 1324 q^{11} - 2688 q^{12} + 17458 q^{13} - 10976 q^{14} + 16384 q^{16} + 18158 q^{17} - 17680 q^{18} + 11984 q^{19} - 14406 q^{21} - 10592 q^{22} + 145300 q^{23} + 21504 q^{24} - 139664 q^{26} - 62244 q^{27} + 87808 q^{28} - 55578 q^{29} + 17206 q^{31} - 131072 q^{32} + 684320 q^{33} - 145264 q^{34} + 141440 q^{36} + 507898 q^{37} - 95872 q^{38} - 132344 q^{39} - 268660 q^{41} + 115248 q^{42} - 362460 q^{43} + 84736 q^{44} - 1162400 q^{46} + 855988 q^{47} - 172032 q^{48} + 470596 q^{49} - 1347956 q^{51} + 1117312 q^{52} - 1245360 q^{53} + 497952 q^{54} - 702464 q^{56} + 3348414 q^{57} + 444624 q^{58} - 1415834 q^{59} - 4333910 q^{61} - 137648 q^{62} + 758030 q^{63} + 1048576 q^{64} - 5474560 q^{66} + 2271660 q^{67} + 1162112 q^{68} - 10774750 q^{69} - 3370816 q^{71} - 1131520 q^{72} + 4604488 q^{73} - 4063184 q^{74} + 766976 q^{76} + 454132 q^{77} + 1058752 q^{78} - 9036996 q^{79} - 3280024 q^{81} + 2149280 q^{82} - 11603802 q^{83} - 921984 q^{84} + 2899680 q^{86} + 15868398 q^{87} - 677888 q^{88} + 307328 q^{89} + 5988094 q^{91} + 9299200 q^{92} - 30711032 q^{93} - 6847904 q^{94} + 1376256 q^{96} + 21766234 q^{97} - 3764768 q^{98} - 23711228 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9 + 1324 * q^11 - 2688 * q^12 + 17458 * q^13 - 10976 * q^14 + 16384 * q^16 + 18158 * q^17 - 17680 * q^18 + 11984 * q^19 - 14406 * q^21 - 10592 * q^22 + 145300 * q^23 + 21504 * q^24 - 139664 * q^26 - 62244 * q^27 + 87808 * q^28 - 55578 * q^29 + 17206 * q^31 - 131072 * q^32 + 684320 * q^33 - 145264 * q^34 + 141440 * q^36 + 507898 * q^37 - 95872 * q^38 - 132344 * q^39 - 268660 * q^41 + 115248 * q^42 - 362460 * q^43 + 84736 * q^44 - 1162400 * q^46 + 855988 * q^47 - 172032 * q^48 + 470596 * q^49 - 1347956 * q^51 + 1117312 * q^52 - 1245360 * q^53 + 497952 * q^54 - 702464 * q^56 + 3348414 * q^57 + 444624 * q^58 - 1415834 * q^59 - 4333910 * q^61 - 137648 * q^62 + 758030 * q^63 + 1048576 * q^64 - 5474560 * q^66 + 2271660 * q^67 + 1162112 * q^68 - 10774750 * q^69 - 3370816 * q^71 - 1131520 * q^72 + 4604488 * q^73 - 4063184 * q^74 + 766976 * q^76 + 454132 * q^77 + 1058752 * q^78 - 9036996 * q^79 - 3280024 * q^81 + 2149280 * q^82 - 11603802 * q^83 - 921984 * q^84 + 2899680 * q^86 + 15868398 * q^87 - 677888 * q^88 + 307328 * q^89 + 5988094 * q^91 + 9299200 * q^92 - 30711032 * q^93 - 6847904 * q^94 + 1376256 * q^96 + 21766234 * q^97 - 3764768 * q^98 - 23711228 * 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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−36.1742	−0.773526	−0.386763	−	0.922179i	$-0.626407\pi$
$3$	−36.1742	−0.773526	−0.386763	+	0.922179i	$0.626407\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	289.394	0.546966
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	−878.424	−0.401657
$10$	0	0
$11$	6314.71	1.43047	0.715236	−	0.698883i	$-0.246319\pi$
$11$	6314.71	1.43047	0.715236	+	0.698883i	$0.246319\pi$
$12$	−2315.15	−0.386763
$13$	745.251	0.0940807	0.0470404	−	0.998893i	$-0.485021\pi$
$13$	745.251	0.0940807	0.0470404	+	0.998893i	$0.485021\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	34226.8	1.68964	0.844821	−	0.535049i	$-0.179707\pi$
$17$	34226.8	1.68964	0.844821	+	0.535049i	$0.179707\pi$
$18$	7027.39	0.284014
$19$	−45918.6	−1.53586	−0.767929	−	0.640535i	$-0.778713\pi$
$19$	−45918.6	−1.53586	−0.767929	+	0.640535i	$0.778713\pi$
$20$	0	0
$21$	−12407.8	−0.292365
$22$	−50517.7	−1.01150
$23$	58260.3	0.998447	0.499224	−	0.866473i	$-0.333619\pi$
$23$	58260.3	0.998447	0.499224	+	0.866473i	$0.333619\pi$
$24$	18521.2	0.273483
$25$	0	0
$26$	−5962.00	−0.0665251
$27$	110889.	1.08422
$28$	21952.0	0.188982
$29$	98592.5	0.750673	0.375336	−	0.926889i	$-0.377527\pi$
$29$	98592.5	0.750673	0.375336	+	0.926889i	$0.377527\pi$
$30$	0	0
$31$	3824.79	0.0230590	0.0115295	−	0.999934i	$-0.496330\pi$
$31$	3824.79	0.0230590	0.0115295	+	0.999934i	$0.496330\pi$
$32$	−32768.0	−0.176777
$33$	−228430.	−1.10651
$34$	−273814.	−1.19476
$35$	0	0
$36$	−56219.1	−0.200829
$37$	−506799.	−1.64486	−0.822432	−	0.568864i	$-0.807383\pi$
$37$	−506799.	−1.64486	−0.822432	+	0.568864i	$0.807383\pi$
$38$	367349.	1.08602
$39$	−26958.9	−0.0727739
$40$	0	0
$41$	205918.	0.466607	0.233303	−	0.972404i	$-0.425046\pi$
$41$	205918.	0.466607	0.233303	+	0.972404i	$0.425046\pi$
$42$	99262.1	0.206734
$43$	872545.	1.67359	0.836793	−	0.547519i	$-0.184427\pi$
$43$	872545.	1.67359	0.836793	+	0.547519i	$0.184427\pi$
$44$	404142.	0.715236
$45$	0	0
$46$	−466082.	−0.706009
$47$	938310.	1.31827	0.659134	−	0.752025i	$-0.270923\pi$
$47$	938310.	1.31827	0.659134	+	0.752025i	$0.270923\pi$
$48$	−148170.	−0.193382
$49$	117649.	0.142857
$50$	0	0
$51$	−1.23813e6	−1.30698
$52$	47696.0	0.0470404
$53$	−483847.	−0.446419	−0.223209	−	0.974771i	$-0.571653\pi$
$53$	−483847.	−0.446419	−0.223209	+	0.974771i	$0.571653\pi$
$54$	−887115.	−0.766658
$55$	0	0
$56$	−175616.	−0.133631
$57$	1.66107e6	1.18803
$58$	−788740.	−0.530806
$59$	−2.72708e6	−1.72868	−0.864341	−	0.502906i	$-0.832264\pi$
$59$	−2.72708e6	−1.72868	−0.864341	+	0.502906i	$0.832264\pi$
$60$	0	0
$61$	−3.30011e6	−1.86154	−0.930772	−	0.365599i	$-0.880864\pi$
$61$	−3.30011e6	−1.86154	−0.930772	+	0.365599i	$0.880864\pi$
$62$	−30598.3	−0.0163052
$63$	−301299.	−0.151812
$64$	262144.	0.125000
$65$	0	0
$66$	1.82744e6	0.782419
$67$	−4.13387e6	−1.67917	−0.839587	−	0.543226i	$-0.817203\pi$
$67$	−4.13387e6	−1.67917	−0.839587	+	0.543226i	$0.817203\pi$
$68$	2.19051e6	0.844821
$69$	−2.10752e6	−0.772325
$70$	0	0
$71$	4.30629e6	1.42790	0.713951	−	0.700195i	$-0.246904\pi$
$71$	4.30629e6	1.42790	0.713951	+	0.700195i	$0.246904\pi$
$72$	449753.	0.142007
$73$	908815.	0.273430	0.136715	−	0.990610i	$-0.456346\pi$
$73$	908815.	0.273430	0.136715	+	0.990610i	$0.456346\pi$
$74$	4.05439e6	1.16309
$75$	0	0
$76$	−2.93879e6	−0.767929
$77$	2.16595e6	0.540667
$78$	215671.	0.0514589
$79$	−5.74990e6	−1.31209	−0.656047	−	0.754720i	$-0.727773\pi$
$79$	−5.74990e6	−1.31209	−0.656047	+	0.754720i	$0.727773\pi$
$80$	0	0
$81$	−2.09023e6	−0.437014
$82$	−1.64734e6	−0.329941
$83$	−2.66624e6	−0.511830	−0.255915	−	0.966699i	$-0.582377\pi$
$83$	−2.66624e6	−0.511830	−0.255915	+	0.966699i	$0.582377\pi$
$84$	−794097.	−0.146183
$85$	0	0
$86$	−6.98036e6	−1.18340
$87$	−3.56651e6	−0.580665
$88$	−3.23313e6	−0.505748
$89$	1.38767e6	0.208651	0.104326	−	0.994543i	$-0.466732\pi$
$89$	1.38767e6	0.208651	0.104326	+	0.994543i	$0.466732\pi$
$90$	0	0
$91$	255621.	0.0355592
$92$	3.72866e6	0.499224
$93$	−138359.	−0.0178368
$94$	−7.50648e6	−0.932156
$95$	0	0
$96$	1.18536e6	0.136741
$97$	1.51141e7	1.68144	0.840719	−	0.541472i	$-0.182133\pi$
$97$	1.51141e7	1.68144	0.840719	+	0.541472i	$0.182133\pi$
$98$	−941192.	−0.101015
$99$	−5.54699e6	−0.574559
$100$	0	0
$101$	−4.26168e6	−0.411581	−0.205791	−	0.978596i	$-0.565977\pi$
$101$	−4.26168e6	−0.411581	−0.205791	+	0.978596i	$0.565977\pi$
$102$	9.90502e6	0.924176
$103$	−1.11866e7	−1.00871	−0.504356	−	0.863496i	$-0.668270\pi$
$103$	−1.11866e7	−1.00871	−0.504356	+	0.863496i	$0.668270\pi$
$104$	−381568.	−0.0332626
$105$	0	0
$106$	3.87078e6	0.315666
$107$	9.03051e6	0.712638	0.356319	−	0.934364i	$-0.384032\pi$
$107$	9.03051e6	0.712638	0.356319	+	0.934364i	$0.384032\pi$
$108$	7.09692e6	0.542109
$109$	1.24806e6	0.0923089	0.0461544	−	0.998934i	$-0.485303\pi$
$109$	1.24806e6	0.0923089	0.0461544	+	0.998934i	$0.485303\pi$
$110$	0	0
$111$	1.83331e7	1.27235
$112$	1.40493e6	0.0944911
$113$	−9.04894e6	−0.589961	−0.294980	−	0.955503i	$-0.595313\pi$
$113$	−9.04894e6	−0.589961	−0.294980	+	0.955503i	$0.595313\pi$
$114$	−1.32886e7	−0.840062
$115$	0	0
$116$	6.30992e6	0.375336
$117$	−654646.	−0.0377882
$118$	2.18166e7	1.22236
$119$	1.17398e7	0.638625
$120$	0	0
$121$	2.03884e7	1.04625
$122$	2.64009e7	1.31631
$123$	−7.44893e6	−0.360933
$124$	244786.	0.0115295
$125$	0	0
$126$	2.41040e6	0.107347
$127$	3.75402e7	1.62623	0.813117	−	0.582100i	$-0.197769\pi$
$127$	3.75402e7	1.62623	0.813117	+	0.582100i	$0.197769\pi$
$128$	−2.09715e6	−0.0883883
$129$	−3.15637e7	−1.29456
$130$	0	0
$131$	−9.77133e6	−0.379756	−0.189878	−	0.981808i	$-0.560809\pi$
$131$	−9.77133e6	−0.379756	−0.189878	+	0.981808i	$0.560809\pi$
$132$	−1.46195e7	−0.553254
$133$	−1.57501e7	−0.580500
$134$	3.30710e7	1.18735
$135$	0	0
$136$	−1.75241e7	−0.597379
$137$	5.06626e7	1.68331	0.841657	−	0.540012i	$-0.181580\pi$
$137$	5.06626e7	1.68331	0.841657	+	0.540012i	$0.181580\pi$
$138$	1.68602e7	0.546117
$139$	2.68319e7	0.847422	0.423711	−	0.905798i	$-0.360727\pi$
$139$	2.68319e7	0.847422	0.423711	+	0.905798i	$0.360727\pi$
$140$	0	0
$141$	−3.39427e7	−1.01972
$142$	−3.44503e7	−1.00968
$143$	4.70604e6	0.134580
$144$	−3.59802e6	−0.100414
$145$	0	0
$146$	−7.27052e6	−0.193344
$147$	−4.25586e6	−0.110504
$148$	−3.24351e7	−0.822432
$149$	−3.77739e7	−0.935492	−0.467746	−	0.883863i	$-0.654934\pi$
$149$	−3.77739e7	−0.935492	−0.467746	+	0.883863i	$0.654934\pi$
$150$	0	0
$151$	8.18626e7	1.93493	0.967467	−	0.252998i	$-0.0814166\pi$
$151$	8.18626e7	1.93493	0.967467	+	0.252998i	$0.0814166\pi$
$152$	2.35103e7	0.543008
$153$	−3.00656e7	−0.678657
$154$	−1.73276e7	−0.382309
$155$	0	0
$156$	−1.72537e6	−0.0363870
$157$	−1.31138e7	−0.270446	−0.135223	−	0.990815i	$-0.543175\pi$
$157$	−1.31138e7	−0.270446	−0.135223	+	0.990815i	$0.543175\pi$
$158$	4.59992e7	0.927791
$159$	1.75028e7	0.345317
$160$	0	0
$161$	1.99833e7	0.377378
$162$	1.67218e7	0.309016
$163$	5.34024e7	0.965838	0.482919	−	0.875665i	$-0.339577\pi$
$163$	5.34024e7	0.965838	0.482919	+	0.875665i	$0.339577\pi$
$164$	1.31788e7	0.233303
$165$	0	0
$166$	2.13299e7	0.361919
$167$	−3.75431e7	−0.623767	−0.311884	−	0.950120i	$-0.600960\pi$
$167$	−3.75431e7	−0.623767	−0.311884	+	0.950120i	$0.600960\pi$
$168$	6.35278e6	0.103367
$169$	−6.21931e7	−0.991149
$170$	0	0
$171$	4.03360e7	0.616888
$172$	5.58429e7	0.836793
$173$	1.03107e8	1.51400	0.756999	−	0.653416i	$-0.226665\pi$
$173$	1.03107e8	1.51400	0.756999	+	0.653416i	$0.226665\pi$
$174$	2.85321e7	0.410592
$175$	0	0
$176$	2.58651e7	0.357618
$177$	9.86499e7	1.33718
$178$	−1.11014e7	−0.147539
$179$	−5.94262e6	−0.0774448	−0.0387224	−	0.999250i	$-0.512329\pi$
$179$	−5.94262e6	−0.0774448	−0.0387224	+	0.999250i	$0.512329\pi$
$180$	0	0
$181$	1.24194e8	1.55677	0.778386	−	0.627786i	$-0.216039\pi$
$181$	1.24194e8	1.55677	0.778386	+	0.627786i	$0.216039\pi$
$182$	−2.04497e6	−0.0251441
$183$	1.19379e8	1.43995
$184$	−2.98293e7	−0.353004
$185$	0	0
$186$	1.10687e6	0.0126125
$187$	2.16132e8	2.41698
$188$	6.00518e7	0.659134
$189$	3.80351e7	0.409796
$190$	0	0
$191$	−7.29283e7	−0.757320	−0.378660	−	0.925536i	$-0.623615\pi$
$191$	−7.29283e7	−0.757320	−0.378660	+	0.925536i	$0.623615\pi$
$192$	−9.48286e6	−0.0966908
$193$	3.70753e6	0.0371222	0.0185611	−	0.999828i	$-0.494091\pi$
$193$	3.70753e6	0.0371222	0.0185611	+	0.999828i	$0.494091\pi$
$194$	−1.20913e8	−1.18896
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	−9.26871e7	−0.863749	−0.431875	−	0.901934i	$-0.642148\pi$
$197$	−9.26871e7	−0.863749	−0.431875	+	0.901934i	$0.642148\pi$
$198$	4.43760e7	0.406274
$199$	3.24106e7	0.291542	0.145771	−	0.989318i	$-0.453434\pi$
$199$	3.24106e7	0.291542	0.145771	+	0.989318i	$0.453434\pi$
$200$	0	0
$201$	1.49540e8	1.29888
$202$	3.40934e7	0.291032
$203$	3.38172e7	0.283728
$204$	−7.92402e7	−0.653491
$205$	0	0
$206$	8.94926e7	0.713267
$207$	−5.11772e7	−0.401034
$208$	3.05255e6	0.0235202
$209$	−2.89963e8	−2.19700
$210$	0	0
$211$	1.74124e8	1.27605	0.638027	−	0.770014i	$-0.279751\pi$
$211$	1.74124e8	1.27605	0.638027	+	0.770014i	$0.279751\pi$
$212$	−3.09662e7	−0.223209
$213$	−1.55777e8	−1.10452
$214$	−7.22441e7	−0.503911
$215$	0	0
$216$	−5.67754e7	−0.383329
$217$	1.31190e6	0.00871550
$218$	−9.98450e6	−0.0652722
$219$	−3.28757e7	−0.211505
$220$	0	0
$221$	2.55075e7	0.158963
$222$	−1.46665e8	−0.899684
$223$	4.95838e7	0.299415	0.149707	−	0.988730i	$-0.452167\pi$
$223$	4.95838e7	0.299415	0.149707	+	0.988730i	$0.452167\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	7.23915e7	0.417165
$227$	2.59098e8	1.47019	0.735096	−	0.677963i	$-0.237137\pi$
$227$	2.59098e8	1.47019	0.735096	+	0.677963i	$0.237137\pi$
$228$	1.06309e8	0.594013
$229$	1.83649e8	1.01056	0.505282	−	0.862954i	$-0.331388\pi$
$229$	1.83649e8	1.01056	0.505282	+	0.862954i	$0.331388\pi$
$230$	0	0
$231$	−7.83515e7	−0.418220
$232$	−5.04793e7	−0.265403
$233$	2.06871e6	0.0107140	0.00535702	−	0.999986i	$-0.498295\pi$
$233$	2.06871e6	0.0107140	0.00535702	+	0.999986i	$0.498295\pi$
$234$	5.23717e6	0.0267203
$235$	0	0
$236$	−1.74533e8	−0.864341
$237$	2.07998e8	1.01494
$238$	−9.39183e7	−0.451576
$239$	1.13487e8	0.537716	0.268858	−	0.963180i	$-0.413354\pi$
$239$	1.13487e8	0.537716	0.268858	+	0.963180i	$0.413354\pi$
$240$	0	0
$241$	9.28189e7	0.427147	0.213573	−	0.976927i	$-0.431490\pi$
$241$	9.28189e7	0.427147	0.213573	+	0.976927i	$0.431490\pi$
$242$	−1.63107e8	−0.739809
$243$	−1.66903e8	−0.746176
$244$	−2.11207e8	−0.930772
$245$	0	0
$246$	5.95915e7	0.255218
$247$	−3.42209e7	−0.144495
$248$	−1.95829e6	−0.00815260
$249$	9.64492e7	0.395914
$250$	0	0
$251$	2.07413e8	0.827898	0.413949	−	0.910300i	$-0.364149\pi$
$251$	2.07413e8	0.827898	0.413949	+	0.910300i	$0.364149\pi$
$252$	−1.92832e7	−0.0759061
$253$	3.67897e8	1.42825
$254$	−3.00321e8	−1.14992
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	7.15234e7	0.262834	0.131417	−	0.991327i	$-0.458047\pi$
$257$	7.15234e7	0.262834	0.131417	+	0.991327i	$0.458047\pi$
$258$	2.52509e8	0.915395
$259$	−1.73832e8	−0.621700
$260$	0	0
$261$	−8.66060e7	−0.301513
$262$	7.81706e7	0.268528
$263$	1.88336e8	0.638392	0.319196	−	0.947689i	$-0.396587\pi$
$263$	1.88336e8	0.638392	0.319196	+	0.947689i	$0.396587\pi$
$264$	1.16956e8	0.391209
$265$	0	0
$266$	1.26001e8	0.410475
$267$	−5.01979e7	−0.161397
$268$	−2.64568e8	−0.839587
$269$	1.36086e8	0.426266	0.213133	−	0.977023i	$-0.431633\pi$
$269$	1.36086e8	0.426266	0.213133	+	0.977023i	$0.431633\pi$
$270$	0	0
$271$	−3.36474e8	−1.02697	−0.513486	−	0.858098i	$-0.671646\pi$
$271$	−3.36474e8	−1.02697	−0.513486	+	0.858098i	$0.671646\pi$
$272$	1.40193e8	0.422410
$273$	−9.24690e6	−0.0275060
$274$	−4.05301e8	−1.19028
$275$	0	0
$276$	−1.34881e8	−0.386163
$277$	8.44013e7	0.238600	0.119300	−	0.992858i	$-0.461935\pi$
$277$	8.44013e7	0.238600	0.119300	+	0.992858i	$0.461935\pi$
$278$	−2.14655e8	−0.599218
$279$	−3.35978e6	−0.00926183
$280$	0	0
$281$	6.77804e7	0.182235	0.0911175	−	0.995840i	$-0.470956\pi$
$281$	6.77804e7	0.182235	0.0911175	+	0.995840i	$0.470956\pi$
$282$	2.71541e8	0.721047
$283$	−5.09414e8	−1.33604	−0.668019	−	0.744144i	$-0.732857\pi$
$283$	−5.09414e8	−1.33604	−0.668019	+	0.744144i	$0.732857\pi$
$284$	2.75602e8	0.713951
$285$	0	0
$286$	−3.76483e7	−0.0951623
$287$	7.06299e7	0.176361
$288$	2.87842e7	0.0710036
$289$	7.61133e8	1.85489
$290$	0	0
$291$	−5.46740e8	−1.30064
$292$	5.81642e7	0.136715
$293$	−4.65709e8	−1.08163	−0.540814	−	0.841142i	$-0.681884\pi$
$293$	−4.65709e8	−1.08163	−0.540814	+	0.841142i	$0.681884\pi$
$294$	3.40469e7	0.0781380
$295$	0	0
$296$	2.59481e8	0.581547
$297$	7.00235e8	1.55094
$298$	3.02191e8	0.661493
$299$	4.34185e7	0.0939347
$300$	0	0
$301$	2.99283e8	0.632556
$302$	−6.54901e8	−1.36820
$303$	1.54163e8	0.318369
$304$	−1.88083e8	−0.383964
$305$	0	0
$306$	2.40525e8	0.479883
$307$	−3.67025e8	−0.723954	−0.361977	−	0.932187i	$-0.617898\pi$
$307$	−3.67025e8	−0.723954	−0.361977	+	0.932187i	$0.617898\pi$
$308$	1.38621e8	0.270334
$309$	4.04666e8	0.780265
$310$	0	0
$311$	6.43458e7	0.121299	0.0606497	−	0.998159i	$-0.480683\pi$
$311$	6.43458e7	0.121299	0.0606497	+	0.998159i	$0.480683\pi$
$312$	1.38029e7	0.0257295
$313$	3.89958e8	0.718808	0.359404	−	0.933182i	$-0.382980\pi$
$313$	3.89958e8	0.718808	0.359404	+	0.933182i	$0.382980\pi$
$314$	1.04910e8	0.191234
$315$	0	0
$316$	−3.67993e8	−0.656047
$317$	−2.79985e8	−0.493660	−0.246830	−	0.969059i	$-0.579389\pi$
$317$	−2.79985e8	−0.493660	−0.246830	+	0.969059i	$0.579389\pi$
$318$	−1.40022e8	−0.244176
$319$	6.22583e8	1.07382
$320$	0	0
$321$	−3.26672e8	−0.551244
$322$	−1.59866e8	−0.266846
$323$	−1.57165e9	−2.59505
$324$	−1.33775e8	−0.218507
$325$	0	0
$326$	−4.27219e8	−0.682951
$327$	−4.51477e7	−0.0714033
$328$	−1.05430e8	−0.164970
$329$	3.21840e8	0.498259
$330$	0	0
$331$	−9.73110e8	−1.47490	−0.737452	−	0.675400i	$-0.763971\pi$
$331$	−9.73110e8	−1.47490	−0.737452	+	0.675400i	$0.763971\pi$
$332$	−1.70639e8	−0.255915
$333$	4.45185e8	0.660671
$334$	3.00345e8	0.441070
$335$	0	0
$336$	−5.08222e7	−0.0730914
$337$	6.83968e8	0.973489	0.486744	−	0.873544i	$-0.338184\pi$
$337$	6.83968e8	0.973489	0.486744	+	0.873544i	$0.338184\pi$
$338$	4.97545e8	0.700848
$339$	3.27338e8	0.456350
$340$	0	0
$341$	2.41524e7	0.0329853
$342$	−3.22688e8	−0.436206
$343$	4.03536e7	0.0539949
$344$	−4.46743e8	−0.591702
$345$	0	0
$346$	−8.24853e8	−1.07056
$347$	1.22484e9	1.57372	0.786860	−	0.617131i	$-0.211705\pi$
$347$	1.22484e9	1.57372	0.786860	+	0.617131i	$0.211705\pi$
$348$	−2.28257e8	−0.290333
$349$	−4.04463e8	−0.509319	−0.254660	−	0.967031i	$-0.581963\pi$
$349$	−4.04463e8	−0.509319	−0.254660	+	0.967031i	$0.581963\pi$
$350$	0	0
$351$	8.26404e7	0.102004
$352$	−2.06920e8	−0.252874
$353$	−7.05601e8	−0.853783	−0.426891	−	0.904303i	$-0.640391\pi$
$353$	−7.05601e8	−0.853783	−0.426891	+	0.904303i	$0.640391\pi$
$354$	−7.89199e8	−0.945530
$355$	0	0
$356$	8.88108e7	0.104326
$357$	−4.24678e8	−0.493993
$358$	4.75409e7	0.0547617
$359$	3.36353e8	0.383676	0.191838	−	0.981427i	$-0.438555\pi$
$359$	3.36353e8	0.383676	0.191838	+	0.981427i	$0.438555\pi$
$360$	0	0
$361$	1.21465e9	1.35886
$362$	−9.93551e8	−1.10080
$363$	−7.37535e8	−0.809300
$364$	1.63597e7	0.0177796
$365$	0	0
$366$	−9.55031e8	−1.01820
$367$	−8.74364e8	−0.923339	−0.461670	−	0.887052i	$-0.652749\pi$
$367$	−8.74364e8	−0.923339	−0.461670	+	0.887052i	$0.652749\pi$
$368$	2.38634e8	0.249612
$369$	−1.80883e8	−0.187416
$370$	0	0
$371$	−1.65960e8	−0.168730
$372$	−8.85496e6	−0.00891839
$373$	−1.07213e9	−1.06971	−0.534854	−	0.844944i	$-0.679633\pi$
$373$	−1.07213e9	−1.06971	−0.534854	+	0.844944i	$0.679633\pi$
$374$	−1.72906e9	−1.70907
$375$	0	0
$376$	−4.80415e8	−0.466078
$377$	7.34761e7	0.0706238
$378$	−3.04281e8	−0.289770
$379$	1.08023e9	1.01925	0.509623	−	0.860398i	$-0.329785\pi$
$379$	1.08023e9	1.01925	0.509623	+	0.860398i	$0.329785\pi$
$380$	0	0
$381$	−1.35799e9	−1.25793
$382$	5.83426e8	0.535506
$383$	4.80369e8	0.436898	0.218449	−	0.975848i	$-0.429900\pi$
$383$	4.80369e8	0.436898	0.218449	+	0.975848i	$0.429900\pi$
$384$	7.58629e7	0.0683707
$385$	0	0
$386$	−2.96602e7	−0.0262494
$387$	−7.66465e8	−0.672208
$388$	9.67301e8	0.840719
$389$	1.08897e9	0.937981	0.468990	−	0.883203i	$-0.344618\pi$
$389$	1.08897e9	0.937981	0.468990	+	0.883203i	$0.344618\pi$
$390$	0	0
$391$	1.99406e9	1.68702
$392$	−6.02363e7	−0.0505076
$393$	3.53470e8	0.293751
$394$	7.41497e8	0.610763
$395$	0	0
$396$	−3.55008e8	−0.287279
$397$	−4.04505e8	−0.324457	−0.162229	−	0.986753i	$-0.551868\pi$
$397$	−4.04505e8	−0.324457	−0.162229	+	0.986753i	$0.551868\pi$
$398$	−2.59285e8	−0.206152
$399$	5.69747e8	0.449032
$400$	0	0
$401$	−5.30245e7	−0.0410649	−0.0205325	−	0.999789i	$-0.506536\pi$
$401$	−5.30245e7	−0.0410649	−0.0205325	+	0.999789i	$0.506536\pi$
$402$	−1.19632e9	−0.918450
$403$	2.85042e6	0.00216941
$404$	−2.72747e8	−0.205791
$405$	0	0
$406$	−2.70538e8	−0.200626
$407$	−3.20029e9	−2.35293
$408$	6.33921e8	0.462088
$409$	−1.01143e9	−0.730975	−0.365488	−	0.930816i	$-0.619098\pi$
$409$	−1.01143e9	−0.730975	−0.365488	+	0.930816i	$0.619098\pi$
$410$	0	0
$411$	−1.83268e9	−1.30209
$412$	−7.15941e8	−0.504356
$413$	−9.35387e8	−0.653380
$414$	4.09418e8	0.283574
$415$	0	0
$416$	−2.44204e7	−0.0166313
$417$	−9.70623e8	−0.655503
$418$	2.31970e9	1.55351
$419$	1.47653e9	0.980600	0.490300	−	0.871554i	$-0.336887\pi$
$419$	1.47653e9	0.980600	0.490300	+	0.871554i	$0.336887\pi$
$420$	0	0
$421$	1.14680e9	0.749034	0.374517	−	0.927220i	$-0.377809\pi$
$421$	1.14680e9	0.749034	0.374517	+	0.927220i	$0.377809\pi$
$422$	−1.39299e9	−0.902307
$423$	−8.24234e8	−0.529492
$424$	2.47730e8	0.157833
$425$	0	0
$426$	1.24621e9	0.781014
$427$	−1.13194e9	−0.703598
$428$	5.77953e8	0.356319
$429$	−1.70238e8	−0.104101
$430$	0	0
$431$	−1.52732e9	−0.918885	−0.459442	−	0.888208i	$-0.651951\pi$
$431$	−1.52732e9	−0.918885	−0.459442	+	0.888208i	$0.651951\pi$
$432$	4.54203e8	0.271055
$433$	−1.48978e9	−0.881891	−0.440946	−	0.897534i	$-0.645357\pi$
$433$	−1.48978e9	−0.881891	−0.440946	+	0.897534i	$0.645357\pi$
$434$	−1.04952e7	−0.00616279
$435$	0	0
$436$	7.98760e7	0.0461544
$437$	−2.67523e9	−1.53347
$438$	2.63006e8	0.149557
$439$	−6.65614e8	−0.375489	−0.187744	−	0.982218i	$-0.560118\pi$
$439$	−6.65614e8	−0.375489	−0.187744	+	0.982218i	$0.560118\pi$
$440$	0	0
$441$	−1.03346e8	−0.0573796
$442$	−2.04060e8	−0.112404
$443$	3.52783e8	0.192795	0.0963973	−	0.995343i	$-0.469268\pi$
$443$	3.52783e8	0.192795	0.0963973	+	0.995343i	$0.469268\pi$
$444$	1.17332e9	0.636173
$445$	0	0
$446$	−3.96671e8	−0.211718
$447$	1.36644e9	0.723628
$448$	8.99154e7	0.0472456
$449$	1.02175e9	0.532698	0.266349	−	0.963877i	$-0.414183\pi$
$449$	1.02175e9	0.532698	0.266349	+	0.963877i	$0.414183\pi$
$450$	0	0
$451$	1.30031e9	0.667468
$452$	−5.79132e8	−0.294980
$453$	−2.96132e9	−1.49672
$454$	−2.07279e9	−1.03958
$455$	0	0
$456$	−8.50468e8	−0.420031
$457$	1.88738e9	0.925025	0.462513	−	0.886613i	$-0.346948\pi$
$457$	1.88738e9	0.925025	0.462513	+	0.886613i	$0.346948\pi$
$458$	−1.46919e9	−0.714577
$459$	3.79539e9	1.83194
$460$	0	0
$461$	−7.70081e8	−0.366086	−0.183043	−	0.983105i	$-0.558595\pi$
$461$	−7.70081e8	−0.366086	−0.183043	+	0.983105i	$0.558595\pi$
$462$	6.26812e8	0.295726
$463$	−1.68300e9	−0.788043	−0.394021	−	0.919101i	$-0.628916\pi$
$463$	−1.68300e9	−0.788043	−0.394021	+	0.919101i	$0.628916\pi$
$464$	4.03835e8	0.187668
$465$	0	0
$466$	−1.65496e7	−0.00757596
$467$	−2.16210e9	−0.982351	−0.491176	−	0.871061i	$-0.663433\pi$
$467$	−2.16210e9	−0.982351	−0.491176	+	0.871061i	$0.663433\pi$
$468$	−4.18973e7	−0.0188941
$469$	−1.41792e9	−0.634668
$470$	0	0
$471$	4.74382e8	0.209197
$472$	1.39626e9	0.611181
$473$	5.50987e9	2.39402
$474$	−1.66399e9	−0.717671
$475$	0	0
$476$	7.51346e8	0.319312
$477$	4.25023e8	0.179307
$478$	−9.07896e8	−0.380223
$479$	−2.42537e9	−1.00833	−0.504166	−	0.863607i	$-0.668200\pi$
$479$	−2.42537e9	−1.00833	−0.504166	+	0.863607i	$0.668200\pi$
$480$	0	0
$481$	−3.77692e8	−0.154750
$482$	−7.42551e8	−0.302038
$483$	−7.22880e8	−0.291912
$484$	1.30486e9	0.523124
$485$	0	0
$486$	1.33522e9	0.527626
$487$	2.13448e8	0.0837414	0.0418707	−	0.999123i	$-0.486668\pi$
$487$	2.13448e8	0.0837414	0.0418707	+	0.999123i	$0.486668\pi$
$488$	1.68965e9	0.658155
$489$	−1.93179e9	−0.747101
$490$	0	0
$491$	3.12035e9	1.18965	0.594823	−	0.803857i	$-0.297222\pi$
$491$	3.12035e9	1.18965	0.594823	+	0.803857i	$0.297222\pi$
$492$	−4.76732e8	−0.180466
$493$	3.37450e9	1.26837
$494$	2.73767e8	0.102173
$495$	0	0
$496$	1.56663e7	0.00576476
$497$	1.47706e9	0.539697
$498$	−7.71594e8	−0.279954
$499$	4.48059e9	1.61430	0.807148	−	0.590350i	$-0.201010\pi$
$499$	4.48059e9	1.61430	0.807148	+	0.590350i	$0.201010\pi$
$500$	0	0
$501$	1.35809e9	0.482501
$502$	−1.65930e9	−0.585412
$503$	4.45243e9	1.55994	0.779972	−	0.625815i	$-0.215233\pi$
$503$	4.45243e9	1.55994	0.779972	+	0.625815i	$0.215233\pi$
$504$	1.54265e8	0.0536737
$505$	0	0
$506$	−2.94318e9	−1.00993
$507$	2.24979e9	0.766680
$508$	2.40257e9	0.813117
$509$	5.18082e9	1.74135	0.870675	−	0.491859i	$-0.163682\pi$
$509$	5.18082e9	1.74135	0.870675	+	0.491859i	$0.163682\pi$
$510$	0	0
$511$	3.11724e8	0.103347
$512$	−1.34218e8	−0.0441942
$513$	−5.09189e9	−1.66521
$514$	−5.72187e8	−0.185852
$515$	0	0
$516$	−2.02007e9	−0.647282
$517$	5.92516e9	1.88574
$518$	1.39066e9	0.439608
$519$	−3.72981e9	−1.17112
$520$	0	0
$521$	4.00499e9	1.24071	0.620353	−	0.784323i	$-0.286989\pi$
$521$	4.00499e9	1.24071	0.620353	+	0.784323i	$0.286989\pi$
$522$	6.92848e8	0.213202
$523$	1.37822e9	0.421271	0.210635	−	0.977565i	$-0.432447\pi$
$523$	1.37822e9	0.421271	0.210635	+	0.977565i	$0.432447\pi$
$524$	−6.25365e8	−0.189878
$525$	0	0
$526$	−1.50669e9	−0.451411
$527$	1.30910e8	0.0389615
$528$	−9.35649e8	−0.276627
$529$	−1.05639e7	−0.00310262
$530$	0	0
$531$	2.39553e9	0.694337
$532$	−1.00801e9	−0.290250
$533$	1.53461e8	0.0438987
$534$	4.01583e8	0.114125
$535$	0	0
$536$	2.11654e9	0.593677
$537$	2.14970e8	0.0599056
$538$	−1.08869e9	−0.301416
$539$	7.42919e8	0.204353
$540$	0	0
$541$	1.45566e9	0.395249	0.197625	−	0.980278i	$-0.436677\pi$
$541$	1.45566e9	0.395249	0.197625	+	0.980278i	$0.436677\pi$
$542$	2.69179e9	0.726179
$543$	−4.49262e9	−1.20420
$544$	−1.12154e9	−0.298689
$545$	0	0
$546$	7.39752e7	0.0194496
$547$	−5.71801e9	−1.49379	−0.746894	−	0.664943i	$-0.768456\pi$
$547$	−5.71801e9	−1.49379	−0.746894	+	0.664943i	$0.768456\pi$
$548$	3.24240e9	0.841657
$549$	2.89889e9	0.747703
$550$	0	0
$551$	−4.52723e9	−1.15293
$552$	1.07905e9	0.273058
$553$	−1.97221e9	−0.495925
$554$	−6.75211e8	−0.168716
$555$	0	0
$556$	1.71724e9	0.423711
$557$	4.85544e9	1.19052	0.595258	−	0.803535i	$-0.297050\pi$
$557$	4.85544e9	1.19052	0.595258	+	0.803535i	$0.297050\pi$
$558$	2.68783e7	0.00654910
$559$	6.50265e8	0.157452
$560$	0	0
$561$	−7.81842e9	−1.86960
$562$	−5.42243e8	−0.128860
$563$	−4.85954e9	−1.14767	−0.573834	−	0.818972i	$-0.694544\pi$
$563$	−4.85954e9	−1.14767	−0.573834	+	0.818972i	$0.694544\pi$
$564$	−2.17233e9	−0.509858
$565$	0	0
$566$	4.07532e9	0.944721
$567$	−7.16948e8	−0.165176
$568$	−2.20482e9	−0.504840
$569$	2.60749e9	0.593376	0.296688	−	0.954974i	$-0.404118\pi$
$569$	2.60749e9	0.593376	0.296688	+	0.954974i	$0.404118\pi$
$570$	0	0
$571$	−5.96951e9	−1.34187	−0.670937	−	0.741514i	$-0.734108\pi$
$571$	−5.96951e9	−1.34187	−0.670937	+	0.741514i	$0.734108\pi$
$572$	3.01187e8	0.0672899
$573$	2.63813e9	0.585807
$574$	−5.65039e8	−0.124706
$575$	0	0
$576$	−2.30274e8	−0.0502071
$577$	4.48735e9	0.972467	0.486234	−	0.873829i	$-0.338370\pi$
$577$	4.48735e9	0.972467	0.486234	+	0.873829i	$0.338370\pi$
$578$	−6.08907e9	−1.31161
$579$	−1.34117e8	−0.0287150
$580$	0	0
$581$	−9.14520e8	−0.193454
$582$	4.37392e9	0.919688
$583$	−3.05535e9	−0.638589
$584$	−4.65313e8	−0.0966720
$585$	0	0
$586$	3.72567e9	0.764826
$587$	4.05343e9	0.827160	0.413580	−	0.910468i	$-0.364278\pi$
$587$	4.05343e9	0.827160	0.413580	+	0.910468i	$0.364278\pi$
$588$	−2.72375e8	−0.0552519
$589$	−1.75629e8	−0.0354154
$590$	0	0
$591$	3.35289e9	0.668133
$592$	−2.07585e9	−0.411216
$593$	8.05283e9	1.58583	0.792916	−	0.609332i	$-0.208562\pi$
$593$	8.05283e9	1.58583	0.792916	+	0.609332i	$0.208562\pi$
$594$	−5.60188e9	−1.09668
$595$	0	0
$596$	−2.41753e9	−0.467746
$597$	−1.17243e9	−0.225516
$598$	−3.47348e8	−0.0664218
$599$	−8.60034e9	−1.63501	−0.817507	−	0.575919i	$-0.804645\pi$
$599$	−8.60034e9	−1.63501	−0.817507	+	0.575919i	$0.804645\pi$
$600$	0	0
$601$	9.34002e9	1.75504	0.877521	−	0.479539i	$-0.159196\pi$
$601$	9.34002e9	1.75504	0.877521	+	0.479539i	$0.159196\pi$
$602$	−2.39426e9	−0.447285
$603$	3.63130e9	0.674452
$604$	5.23920e9	0.967467
$605$	0	0
$606$	−1.23330e9	−0.225121
$607$	2.63602e9	0.478396	0.239198	−	0.970971i	$-0.423115\pi$
$607$	2.63602e9	0.478396	0.239198	+	0.970971i	$0.423115\pi$
$608$	1.50466e9	0.271504
$609$	−1.22331e9	−0.219471
$610$	0	0
$611$	6.99276e8	0.124024
$612$	−1.92420e9	−0.339328
$613$	9.14317e9	1.60319	0.801595	−	0.597867i	$-0.203985\pi$
$613$	9.14317e9	1.60319	0.801595	+	0.597867i	$0.203985\pi$
$614$	2.93620e9	0.511913
$615$	0	0
$616$	−1.10896e9	−0.191155
$617$	3.00021e8	0.0514225	0.0257113	−	0.999669i	$-0.491815\pi$
$617$	3.00021e8	0.0514225	0.0257113	+	0.999669i	$0.491815\pi$
$618$	−3.23733e9	−0.551730
$619$	4.97273e9	0.842708	0.421354	−	0.906896i	$-0.361555\pi$
$619$	4.97273e9	0.842708	0.421354	+	0.906896i	$0.361555\pi$
$620$	0	0
$621$	6.46045e9	1.08254
$622$	−5.14767e8	−0.0857717
$623$	4.75971e8	0.0788627
$624$	−1.10424e8	−0.0181935
$625$	0	0
$626$	−3.11966e9	−0.508274
$627$	1.04892e10	1.69944
$628$	−8.39284e8	−0.135223
$629$	−1.73461e10	−2.77923
$630$	0	0
$631$	−2.62435e9	−0.415834	−0.207917	−	0.978147i	$-0.566668\pi$
$631$	−2.62435e9	−0.415834	−0.207917	+	0.978147i	$0.566668\pi$
$632$	2.94395e9	0.463896
$633$	−6.29879e9	−0.987062
$634$	2.23988e9	0.349070
$635$	0	0
$636$	1.12018e9	0.172658
$637$	8.76780e7	0.0134401
$638$	−4.98066e9	−0.759303
$639$	−3.78275e9	−0.573527
$640$	0	0
$641$	8.56442e9	1.28438	0.642192	−	0.766544i	$-0.278025\pi$
$641$	8.56442e9	1.28438	0.642192	+	0.766544i	$0.278025\pi$
$642$	2.61338e9	0.389789
$643$	−3.18507e9	−0.472477	−0.236238	−	0.971695i	$-0.575915\pi$
$643$	−3.18507e9	−0.472477	−0.236238	+	0.971695i	$0.575915\pi$
$644$	1.27893e9	0.188689
$645$	0	0
$646$	1.25732e10	1.83498
$647$	4.57365e9	0.663893	0.331946	−	0.943298i	$-0.392295\pi$
$647$	4.57365e9	0.663893	0.331946	+	0.943298i	$0.392295\pi$
$648$	1.07020e9	0.154508
$649$	−1.72207e10	−2.47283
$650$	0	0
$651$	−4.74571e7	−0.00674167
$652$	3.41776e9	0.482919
$653$	8.95531e9	1.25859	0.629295	−	0.777166i	$-0.283344\pi$
$653$	8.95531e9	1.25859	0.629295	+	0.777166i	$0.283344\pi$
$654$	3.61182e8	0.0504898
$655$	0	0
$656$	8.43441e8	0.116652
$657$	−7.98325e8	−0.109825
$658$	−2.57472e9	−0.352322
$659$	−9.21115e9	−1.25376	−0.626881	−	0.779115i	$-0.715669\pi$
$659$	−9.21115e9	−1.25376	−0.626881	+	0.779115i	$0.715669\pi$
$660$	0	0
$661$	−4.11253e9	−0.553865	−0.276933	−	0.960889i	$-0.589318\pi$
$661$	−4.11253e9	−0.553865	−0.276933	+	0.960889i	$0.589318\pi$
$662$	7.78488e9	1.04291
$663$	−9.22715e8	−0.122962
$664$	1.36512e9	0.180959
$665$	0	0
$666$	−3.56148e9	−0.467165
$667$	5.74403e9	0.749507
$668$	−2.40276e9	−0.311884
$669$	−1.79366e9	−0.231605
$670$	0	0
$671$	−2.08392e10	−2.66289
$672$	4.06578e8	0.0516834
$673$	6.99579e9	0.884676	0.442338	−	0.896848i	$-0.354149\pi$
$673$	6.99579e9	0.884676	0.442338	+	0.896848i	$0.354149\pi$
$674$	−5.47174e9	−0.688361
$675$	0	0
$676$	−3.98036e9	−0.495574
$677$	−7.67846e9	−0.951072	−0.475536	−	0.879696i	$-0.657746\pi$
$677$	−7.67846e9	−0.951072	−0.475536	+	0.879696i	$0.657746\pi$
$678$	−2.61871e9	−0.322688
$679$	5.18413e9	0.635523
$680$	0	0
$681$	−9.37268e9	−1.13723
$682$	−1.93219e8	−0.0233241
$683$	−2.66479e9	−0.320030	−0.160015	−	0.987115i	$-0.551154\pi$
$683$	−2.66479e9	−0.320030	−0.160015	+	0.987115i	$0.551154\pi$
$684$	2.58150e9	0.308444
$685$	0	0
$686$	−3.22829e8	−0.0381802
$687$	−6.64336e9	−0.781698
$688$	3.57394e9	0.418397
$689$	−3.60587e8	−0.0419994
$690$	0	0
$691$	−1.15699e10	−1.33400	−0.667000	−	0.745058i	$-0.732422\pi$
$691$	−1.15699e10	−1.33400	−0.667000	+	0.745058i	$0.732422\pi$
$692$	6.59883e9	0.756999
$693$	−1.90262e9	−0.217163
$694$	−9.79875e9	−1.11279
$695$	0	0
$696$	1.82605e9	0.205296
$697$	7.04791e9	0.788398
$698$	3.23571e9	0.360143
$699$	−7.48339e7	−0.00828758
$700$	0	0
$701$	1.12934e10	1.23826	0.619128	−	0.785290i	$-0.287486\pi$
$701$	1.12934e10	1.23826	0.619128	+	0.785290i	$0.287486\pi$
$702$	−6.61123e8	−0.0721278
$703$	2.32715e10	2.52628
$704$	1.65536e9	0.178809
$705$	0	0
$706$	5.64481e9	0.603716
$707$	−1.46176e9	−0.155563
$708$	6.31359e9	0.668591
$709$	4.03637e9	0.425333	0.212667	−	0.977125i	$-0.431785\pi$
$709$	4.03637e9	0.425333	0.212667	+	0.977125i	$0.431785\pi$
$710$	0	0
$711$	5.05085e9	0.527012
$712$	−7.10487e8	−0.0737693
$713$	2.22833e8	0.0230232
$714$	3.39742e9	0.349306
$715$	0	0
$716$	−3.80328e8	−0.0387224
$717$	−4.10530e9	−0.415938
$718$	−2.69083e9	−0.271300
$719$	−6.75174e9	−0.677430	−0.338715	−	0.940889i	$-0.609992\pi$
$719$	−6.75174e9	−0.677430	−0.338715	+	0.940889i	$0.609992\pi$
$720$	0	0
$721$	−3.83700e9	−0.381257
$722$	−9.71717e9	−0.960859
$723$	−3.35765e9	−0.330409
$724$	7.94841e9	0.778386
$725$	0	0
$726$	5.90028e9	0.572262
$727$	9.82216e9	0.948062	0.474031	−	0.880508i	$-0.342799\pi$
$727$	9.82216e9	0.948062	0.474031	+	0.880508i	$0.342799\pi$
$728$	−1.30878e8	−0.0125721
$729$	1.06089e10	1.01420
$730$	0	0
$731$	2.98644e10	2.82776
$732$	7.64025e9	0.719977
$733$	−1.00939e9	−0.0946666	−0.0473333	−	0.998879i	$-0.515072\pi$
$733$	−1.00939e9	−0.0946666	−0.0473333	+	0.998879i	$0.515072\pi$
$734$	6.99492e9	0.652899
$735$	0	0
$736$	−1.90907e9	−0.176502
$737$	−2.61042e10	−2.40201
$738$	1.44707e9	0.132523
$739$	1.40892e9	0.128419	0.0642097	−	0.997936i	$-0.479547\pi$
$739$	1.40892e9	0.128419	0.0642097	+	0.997936i	$0.479547\pi$
$740$	0	0
$741$	1.23791e9	0.111770
$742$	1.32768e9	0.119310
$743$	−5.38430e9	−0.481580	−0.240790	−	0.970577i	$-0.577407\pi$
$743$	−5.38430e9	−0.481580	−0.240790	+	0.970577i	$0.577407\pi$
$744$	7.08397e7	0.00630625
$745$	0	0
$746$	8.57701e9	0.756398
$747$	2.34209e9	0.205580
$748$	1.38325e10	1.20849
$749$	3.09747e9	0.269352
$750$	0	0
$751$	−1.16269e10	−1.00167	−0.500836	−	0.865542i	$-0.666974\pi$
$751$	−1.16269e10	−1.00167	−0.500836	+	0.865542i	$0.666974\pi$
$752$	3.84332e9	0.329567
$753$	−7.50299e9	−0.640401
$754$	−5.87809e8	−0.0499386
$755$	0	0
$756$	2.43424e9	0.204898
$757$	4.02535e9	0.337262	0.168631	−	0.985679i	$-0.446065\pi$
$757$	4.02535e9	0.337262	0.168631	+	0.985679i	$0.446065\pi$
$758$	−8.64184e9	−0.720716
$759$	−1.33084e10	−1.10479
$760$	0	0
$761$	−1.18196e10	−0.972198	−0.486099	−	0.873904i	$-0.661581\pi$
$761$	−1.18196e10	−0.972198	−0.486099	+	0.873904i	$0.661581\pi$
$762$	1.08639e10	0.889494
$763$	4.28085e8	0.0348895
$764$	−4.66741e9	−0.378660
$765$	0	0
$766$	−3.84296e9	−0.308933
$767$	−2.03235e9	−0.162636
$768$	−6.06903e8	−0.0483454
$769$	−7.07925e9	−0.561364	−0.280682	−	0.959801i	$-0.590561\pi$
$769$	−7.07925e9	−0.561364	−0.280682	+	0.959801i	$0.590561\pi$
$770$	0	0
$771$	−2.58730e9	−0.203309
$772$	2.37282e8	0.0185611
$773$	3.65977e9	0.284987	0.142494	−	0.989796i	$-0.454488\pi$
$773$	3.65977e9	0.284987	0.142494	+	0.989796i	$0.454488\pi$
$774$	6.13172e9	0.475323
$775$	0	0
$776$	−7.73841e9	−0.594478
$777$	6.28825e9	0.480901
$778$	−8.71179e9	−0.663252
$779$	−9.45547e9	−0.716642
$780$	0	0
$781$	2.71930e10	2.04257
$782$	−1.59525e10	−1.19290
$783$	1.09329e10	0.813894
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	−2.82776e9	−0.207713
$787$	−2.00996e9	−0.146986	−0.0734931	−	0.997296i	$-0.523415\pi$
$787$	−2.00996e9	−0.146986	−0.0734931	+	0.997296i	$0.523415\pi$
$788$	−5.93198e9	−0.431875
$789$	−6.81290e9	−0.493813
$790$	0	0
$791$	−3.10379e9	−0.222984
$792$	2.84006e9	0.203137
$793$	−2.45941e9	−0.175135
$794$	3.23604e9	0.229426
$795$	0	0
$796$	2.07428e9	0.145771
$797$	1.43563e10	1.00447	0.502236	−	0.864731i	$-0.332511\pi$
$797$	1.43563e10	1.00447	0.502236	+	0.864731i	$0.332511\pi$
$798$	−4.55798e9	−0.317513
$799$	3.21153e10	2.22740
$800$	0	0
$801$	−1.21896e9	−0.0838062
$802$	4.24196e8	0.0290373
$803$	5.73891e9	0.391133
$804$	9.57055e9	0.649442
$805$	0	0
$806$	−2.28034e7	−0.00153401
$807$	−4.92281e9	−0.329728
$808$	2.18198e9	0.145516
$809$	1.51420e10	1.00546	0.502728	−	0.864445i	$-0.332330\pi$
$809$	1.51420e10	1.00546	0.502728	+	0.864445i	$0.332330\pi$
$810$	0	0
$811$	8.79310e9	0.578854	0.289427	−	0.957200i	$-0.406535\pi$
$811$	8.79310e9	0.578854	0.289427	+	0.957200i	$0.406535\pi$
$812$	2.16430e9	0.141864
$813$	1.21717e10	0.794390
$814$	2.56023e10	1.66377
$815$	0	0
$816$	−5.07137e9	−0.326746
$817$	−4.00660e10	−2.57039
$818$	8.09141e9	0.516878
$819$	−2.24544e8	−0.0142826
$820$	0	0
$821$	2.73656e10	1.72585	0.862927	−	0.505328i	$-0.168629\pi$
$821$	2.73656e10	1.72585	0.862927	+	0.505328i	$0.168629\pi$
$822$	1.46614e10	0.920715
$823$	−1.91658e10	−1.19847	−0.599235	−	0.800573i	$-0.704528\pi$
$823$	−1.91658e10	−1.19847	−0.599235	+	0.800573i	$0.704528\pi$
$824$	5.72753e9	0.356633
$825$	0	0
$826$	7.48310e9	0.462010
$827$	−2.95985e10	−1.81970	−0.909852	−	0.414934i	$-0.863805\pi$
$827$	−2.95985e10	−1.81970	−0.909852	+	0.414934i	$0.863805\pi$
$828$	−3.27534e9	−0.200517
$829$	1.32270e10	0.806343	0.403171	−	0.915124i	$-0.367908\pi$
$829$	1.32270e10	0.806343	0.403171	+	0.915124i	$0.367908\pi$
$830$	0	0
$831$	−3.05315e9	−0.184563
$832$	1.95363e8	0.0117601
$833$	4.02675e9	0.241377
$834$	7.76499e9	0.463511
$835$	0	0
$836$	−1.85576e10	−1.09850
$837$	4.24128e8	0.0250010
$838$	−1.18122e10	−0.693389
$839$	−3.23344e10	−1.89016	−0.945078	−	0.326844i	$-0.894015\pi$
$839$	−3.23344e10	−1.89016	−0.945078	+	0.326844i	$0.894015\pi$
$840$	0	0
$841$	−7.52940e9	−0.436490
$842$	−9.17442e9	−0.529647
$843$	−2.45190e9	−0.140964
$844$	1.11439e10	0.638027
$845$	0	0
$846$	6.59387e9	0.374407
$847$	6.99322e9	0.395444
$848$	−1.98184e9	−0.111605
$849$	1.84277e10	1.03346
$850$	0	0
$851$	−2.95263e10	−1.64231
$852$	−9.96971e9	−0.552260
$853$	−2.51999e9	−0.139020	−0.0695099	−	0.997581i	$-0.522144\pi$
$853$	−2.51999e9	−0.139020	−0.0695099	+	0.997581i	$0.522144\pi$
$854$	9.05549e9	0.497519
$855$	0	0
$856$	−4.62362e9	−0.251956
$857$	−1.00917e10	−0.547687	−0.273843	−	0.961774i	$-0.588295\pi$
$857$	−1.00917e10	−0.547687	−0.273843	+	0.961774i	$0.588295\pi$
$858$	1.36190e9	0.0736105
$859$	−1.35017e10	−0.726797	−0.363399	−	0.931634i	$-0.618384\pi$
$859$	−1.35017e10	−0.726797	−0.363399	+	0.931634i	$0.618384\pi$
$860$	0	0
$861$	−2.55498e9	−0.136420
$862$	1.22186e10	0.649749
$863$	5.67430e9	0.300521	0.150260	−	0.988646i	$-0.451989\pi$
$863$	5.67430e9	0.300521	0.150260	+	0.988646i	$0.451989\pi$
$864$	−3.63362e9	−0.191665
$865$	0	0
$866$	1.19183e10	0.623591
$867$	−2.75334e10	−1.43481
$868$	8.39617e7	0.00435775
$869$	−3.63089e10	−1.87691
$870$	0	0
$871$	−3.08077e9	−0.157978
$872$	−6.39008e8	−0.0326361
$873$	−1.32766e10	−0.675361
$874$	2.14018e10	1.08433
$875$	0	0
$876$	−2.10404e9	−0.105753
$877$	5.31339e9	0.265995	0.132997	−	0.991116i	$-0.457540\pi$
$877$	5.31339e9	0.265995	0.132997	+	0.991116i	$0.457540\pi$
$878$	5.32491e9	0.265511
$879$	1.68467e10	0.836668
$880$	0	0
$881$	1.94889e9	0.0960223	0.0480111	−	0.998847i	$-0.484712\pi$
$881$	1.94889e9	0.0960223	0.0480111	+	0.998847i	$0.484712\pi$
$882$	8.26766e8	0.0405735
$883$	1.65273e10	0.807865	0.403932	−	0.914789i	$-0.367643\pi$
$883$	1.65273e10	0.807865	0.403932	+	0.914789i	$0.367643\pi$
$884$	1.63248e9	0.0794814
$885$	0	0
$886$	−2.82227e9	−0.136326
$887$	−8.46336e9	−0.407202	−0.203601	−	0.979054i	$-0.565265\pi$
$887$	−8.46336e9	−0.407202	−0.203601	+	0.979054i	$0.565265\pi$
$888$	−9.38654e9	−0.449842
$889$	1.28763e10	0.614659
$890$	0	0
$891$	−1.31992e10	−0.625137
$892$	3.17336e9	0.149707
$893$	−4.30859e10	−2.02467
$894$	−1.09315e10	−0.511682
$895$	0	0
$896$	−7.19323e8	−0.0334077
$897$	−1.57063e9	−0.0726609
$898$	−8.17398e9	−0.376675
$899$	3.77095e8	0.0173098
$900$	0	0
$901$	−1.65605e10	−0.754288
$902$	−1.04025e10	−0.471971
$903$	−1.08263e10	−0.489299
$904$	4.63306e9	0.208583
$905$	0	0
$906$	2.36905e10	1.05834
$907$	−2.64091e10	−1.17524	−0.587622	−	0.809135i	$-0.699936\pi$
$907$	−2.64091e10	−1.17524	−0.587622	+	0.809135i	$0.699936\pi$
$908$	1.65823e10	0.735096
$909$	3.74356e9	0.165315
$910$	0	0
$911$	−1.37372e10	−0.601984	−0.300992	−	0.953627i	$-0.597318\pi$
$911$	−1.37372e10	−0.601984	−0.300992	+	0.953627i	$0.597318\pi$
$912$	6.80375e9	0.297007
$913$	−1.68365e10	−0.732158
$914$	−1.50991e10	−0.654092
$915$	0	0
$916$	1.17535e10	0.505282
$917$	−3.35157e9	−0.143534
$918$	−3.03631e10	−1.29538
$919$	−4.52249e8	−0.0192209	−0.00961045	−	0.999954i	$-0.503059\pi$
$919$	−4.52249e8	−0.0192209	−0.00961045	+	0.999954i	$0.503059\pi$
$920$	0	0
$921$	1.32768e10	0.559997
$922$	6.16065e9	0.258862
$923$	3.20926e9	0.134338
$924$	−5.01449e9	−0.209110
$925$	0	0
$926$	1.34640e10	0.557230
$927$	9.82656e9	0.405156
$928$	−3.23068e9	−0.132701
$929$	−3.74640e10	−1.53306	−0.766531	−	0.642208i	$-0.778019\pi$
$929$	−3.74640e10	−1.53306	−0.766531	+	0.642208i	$0.778019\pi$
$930$	0	0
$931$	−5.40228e9	−0.219408
$932$	1.32397e8	0.00535702
$933$	−2.32766e9	−0.0938283
$934$	1.72968e10	0.694627
$935$	0	0
$936$	3.35179e8	0.0133601
$937$	−1.56051e9	−0.0619695	−0.0309848	−	0.999520i	$-0.509864\pi$
$937$	−1.56051e9	−0.0619695	−0.0309848	+	0.999520i	$0.509864\pi$
$938$	1.13434e10	0.448778
$939$	−1.41064e10	−0.556017
$940$	0	0
$941$	−3.71636e10	−1.45397	−0.726984	−	0.686655i	$-0.759078\pi$
$941$	−3.71636e10	−1.45397	−0.726984	+	0.686655i	$0.759078\pi$
$942$	−3.79506e9	−0.147925
$943$	1.19968e10	0.465882
$944$	−1.11701e10	−0.432171
$945$	0	0
$946$	−4.40790e10	−1.69283
$947$	3.62768e10	1.38805	0.694024	−	0.719952i	$-0.255836\pi$
$947$	3.62768e10	1.38805	0.694024	+	0.719952i	$0.255836\pi$
$948$	1.33119e10	0.507470
$949$	6.77295e8	0.0257245
$950$	0	0
$951$	1.01283e10	0.381859
$952$	−6.01077e9	−0.225788
$953$	3.71732e10	1.39125	0.695624	−	0.718406i	$-0.255128\pi$
$953$	3.71732e10	1.39125	0.695624	+	0.718406i	$0.255128\pi$
$954$	−3.40018e9	−0.126789
$955$	0	0
$956$	7.26316e9	0.268858
$957$	−2.25215e10	−0.830625
$958$	1.94029e10	0.712998
$959$	1.73773e10	0.636233
$960$	0	0
$961$	−2.74980e10	−0.999468
$962$	3.02154e9	0.109425
$963$	−7.93262e9	−0.286236
$964$	5.94041e9	0.213573
$965$	0	0
$966$	5.78304e9	0.206413
$967$	8.67923e9	0.308666	0.154333	−	0.988019i	$-0.450677\pi$
$967$	8.67923e9	0.308666	0.154333	+	0.988019i	$0.450677\pi$
$968$	−1.04389e10	−0.369904
$969$	5.68531e10	2.00734
$970$	0	0
$971$	4.98093e9	0.174600	0.0872998	−	0.996182i	$-0.472176\pi$
$971$	4.98093e9	0.174600	0.0872998	+	0.996182i	$0.472176\pi$
$972$	−1.06818e10	−0.373088
$973$	9.20334e9	0.320295
$974$	−1.70758e9	−0.0592141
$975$	0	0
$976$	−1.35172e10	−0.465386
$977$	−1.29609e10	−0.444635	−0.222318	−	0.974974i	$-0.571362\pi$
$977$	−1.29609e10	−0.444635	−0.222318	+	0.974974i	$0.571362\pi$
$978$	1.54543e10	0.528280
$979$	8.76273e9	0.298470
$980$	0	0
$981$	−1.09633e9	−0.0370765
$982$	−2.49628e10	−0.841207
$983$	−2.23078e10	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$983$	−2.23078e10	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$984$	3.81385e9	0.127609
$985$	0	0
$986$	−2.69960e10	−0.896872
$987$	−1.16423e10	−0.385416
$988$	−2.19014e9	−0.0722473
$989$	5.08347e10	1.67099
$990$	0	0
$991$	5.41745e10	1.76822	0.884111	−	0.467276i	$-0.154765\pi$
$991$	5.41745e10	1.76822	0.884111	+	0.467276i	$0.154765\pi$
$992$	−1.25331e8	−0.00407630
$993$	3.52015e10	1.14088
$994$	−1.18164e10	−0.381623
$995$	0	0
$996$	6.17275e9	0.197957
$997$	−5.05453e10	−1.61528	−0.807639	−	0.589677i	$-0.799255\pi$
$997$	−5.05453e10	−1.61528	−0.807639	+	0.589677i	$0.799255\pi$
$998$	−3.58447e10	−1.14148
$999$	−5.61987e10	−1.78339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.v.1.2	✓	4
5.2	odd	4		350.8.c.o.99.3		8
5.3	odd	4		350.8.c.o.99.6		8
5.4	even	2		350.8.a.y.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.8.a.v.1.2	✓	4	1.1	even	1	trivial
350.8.a.y.1.3	yes	4	5.4	even	2
350.8.c.o.99.3		8	5.2	odd	4
350.8.c.o.99.6		8	5.3	odd	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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -36.1742 q^{3} +64.0000 q^{4} +289.394 q^{6} +343.000 q^{7} -512.000 q^{8} -878.424 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -36.1742 q^{3} +64.0000 q^{4} +289.394 q^{6} +343.000 q^{7} -512.000 q^{8} -878.424 q^{9} +6314.71 q^{11} -2315.15 q^{12} +745.251 q^{13} -2744.00 q^{14} +4096.00 q^{16} +34226.8 q^{17} +7027.39 q^{18} -45918.6 q^{19} -12407.8 q^{21} -50517.7 q^{22} +58260.3 q^{23} +18521.2 q^{24} -5962.00 q^{26} +110889. q^{27} +21952.0 q^{28} +98592.5 q^{29} +3824.79 q^{31} -32768.0 q^{32} -228430. q^{33} -273814. q^{34} -56219.1 q^{36} -506799. q^{37} +367349. q^{38} -26958.9 q^{39} +205918. q^{41} +99262.1 q^{42} +872545. q^{43} +404142. q^{44} -466082. q^{46} +938310. q^{47} -148170. q^{48} +117649. q^{49} -1.23813e6 q^{51} +47696.0 q^{52} -483847. q^{53} -887115. q^{54} -175616. q^{56} +1.66107e6 q^{57} -788740. q^{58} -2.72708e6 q^{59} -3.30011e6 q^{61} -30598.3 q^{62} -301299. q^{63} +262144. q^{64} +1.82744e6 q^{66} -4.13387e6 q^{67} +2.19051e6 q^{68} -2.10752e6 q^{69} +4.30629e6 q^{71} +449753. q^{72} +908815. q^{73} +4.05439e6 q^{74} -2.93879e6 q^{76} +2.16595e6 q^{77} +215671. q^{78} -5.74990e6 q^{79} -2.09023e6 q^{81} -1.64734e6 q^{82} -2.66624e6 q^{83} -794097. q^{84} -6.98036e6 q^{86} -3.56651e6 q^{87} -3.23313e6 q^{88} +1.38767e6 q^{89} +255621. q^{91} +3.72866e6 q^{92} -138359. q^{93} -7.50648e6 q^{94} +1.18536e6 q^{96} +1.51141e7 q^{97} -941192. q^{98} -5.54699e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9} + 1324 q^{11} - 2688 q^{12} + 17458 q^{13} - 10976 q^{14} + 16384 q^{16} + 18158 q^{17} - 17680 q^{18} + 11984 q^{19} - 14406 q^{21} - 10592 q^{22} + 145300 q^{23} + 21504 q^{24} - 139664 q^{26} - 62244 q^{27} + 87808 q^{28} - 55578 q^{29} + 17206 q^{31} - 131072 q^{32} + 684320 q^{33} - 145264 q^{34} + 141440 q^{36} + 507898 q^{37} - 95872 q^{38} - 132344 q^{39} - 268660 q^{41} + 115248 q^{42} - 362460 q^{43} + 84736 q^{44} - 1162400 q^{46} + 855988 q^{47} - 172032 q^{48} + 470596 q^{49} - 1347956 q^{51} + 1117312 q^{52} - 1245360 q^{53} + 497952 q^{54} - 702464 q^{56} + 3348414 q^{57} + 444624 q^{58} - 1415834 q^{59} - 4333910 q^{61} - 137648 q^{62} + 758030 q^{63} + 1048576 q^{64} - 5474560 q^{66} + 2271660 q^{67} + 1162112 q^{68} - 10774750 q^{69} - 3370816 q^{71} - 1131520 q^{72} + 4604488 q^{73} - 4063184 q^{74} + 766976 q^{76} + 454132 q^{77} + 1058752 q^{78} - 9036996 q^{79} - 3280024 q^{81} + 2149280 q^{82} - 11603802 q^{83} - 921984 q^{84} + 2899680 q^{86} + 15868398 q^{87} - 677888 q^{88} + 307328 q^{89} + 5988094 q^{91} + 9299200 q^{92} - 30711032 q^{93} - 6847904 q^{94} + 1376256 q^{96} + 21766234 q^{97} - 3764768 q^{98} - 23711228 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9 + 1324 * q^11 - 2688 * q^12 + 17458 * q^13 - 10976 * q^14 + 16384 * q^16 + 18158 * q^17 - 17680 * q^18 + 11984 * q^19 - 14406 * q^21 - 10592 * q^22 + 145300 * q^23 + 21504 * q^24 - 139664 * q^26 - 62244 * q^27 + 87808 * q^28 - 55578 * q^29 + 17206 * q^31 - 131072 * q^32 + 684320 * q^33 - 145264 * q^34 + 141440 * q^36 + 507898 * q^37 - 95872 * q^38 - 132344 * q^39 - 268660 * q^41 + 115248 * q^42 - 362460 * q^43 + 84736 * q^44 - 1162400 * q^46 + 855988 * q^47 - 172032 * q^48 + 470596 * q^49 - 1347956 * q^51 + 1117312 * q^52 - 1245360 * q^53 + 497952 * q^54 - 702464 * q^56 + 3348414 * q^57 + 444624 * q^58 - 1415834 * q^59 - 4333910 * q^61 - 137648 * q^62 + 758030 * q^63 + 1048576 * q^64 - 5474560 * q^66 + 2271660 * q^67 + 1162112 * q^68 - 10774750 * q^69 - 3370816 * q^71 - 1131520 * q^72 + 4604488 * q^73 - 4063184 * q^74 + 766976 * q^76 + 454132 * q^77 + 1058752 * q^78 - 9036996 * q^79 - 3280024 * q^81 + 2149280 * q^82 - 11603802 * q^83 - 921984 * q^84 + 2899680 * q^86 + 15868398 * q^87 - 677888 * q^88 + 307328 * q^89 + 5988094 * q^91 + 9299200 * q^92 - 30711032 * q^93 - 6847904 * q^94 + 1376256 * q^96 + 21766234 * q^97 - 3764768 * q^98 - 23711228 * q^99

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