LMFDB - Embedded newform 75.10.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,10,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	75.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:	$38.6276877123$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{79}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.88819$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-35.7764 q^{2} +81.0000 q^{3} +767.950 q^{4} -2897.89 q^{6} +3898.82 q^{7} -9156.97 q^{8} +6561.00 q^{9} +O(q^{10})$	\(q-35.7764 q^{2} +81.0000 q^{3} +767.950 q^{4} -2897.89 q^{6} +3898.82 q^{7} -9156.97 q^{8} +6561.00 q^{9} -23047.7 q^{11} +62203.9 q^{12} -113302. q^{13} -139486. q^{14} -65587.2 q^{16} +369116. q^{17} -234729. q^{18} -1.10031e6 q^{19} +315805. q^{21} +824563. q^{22} +1.32065e6 q^{23} -741714. q^{24} +4.05353e6 q^{26} +531441. q^{27} +2.99410e6 q^{28} +66845.9 q^{29} +6.78077e6 q^{31} +7.03484e6 q^{32} -1.86686e6 q^{33} -1.32056e7 q^{34} +5.03852e6 q^{36} +9.94470e6 q^{37} +3.93651e7 q^{38} -9.17744e6 q^{39} -3.50042e7 q^{41} -1.12984e7 q^{42} -2.55307e7 q^{43} -1.76995e7 q^{44} -4.72481e7 q^{46} -1.54853e7 q^{47} -5.31256e6 q^{48} -2.51528e7 q^{49} +2.98984e7 q^{51} -8.70100e7 q^{52} +1.54895e7 q^{53} -1.90130e7 q^{54} -3.57014e7 q^{56} -8.91251e7 q^{57} -2.39150e6 q^{58} +6.45979e7 q^{59} -1.07375e8 q^{61} -2.42591e8 q^{62} +2.55802e7 q^{63} -2.18101e8 q^{64} +6.67896e7 q^{66} -1.29631e8 q^{67} +2.83463e8 q^{68} +1.06973e8 q^{69} +2.25343e8 q^{71} -6.00789e7 q^{72} -1.72446e8 q^{73} -3.55785e8 q^{74} -8.44983e8 q^{76} -8.98589e7 q^{77} +3.28336e8 q^{78} -3.14962e8 q^{79} +4.30467e7 q^{81} +1.25232e9 q^{82} -4.03289e8 q^{83} +2.42522e8 q^{84} +9.13397e8 q^{86} +5.41452e6 q^{87} +2.11047e8 q^{88} -5.72362e8 q^{89} -4.41743e8 q^{91} +1.01419e9 q^{92} +5.49242e8 q^{93} +5.54007e8 q^{94} +5.69822e8 q^{96} +5.62737e8 q^{97} +8.99875e8 q^{98} -1.51216e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 36 q^{2} + 162 q^{3} + 256 q^{4} - 2916 q^{6} + 3318 q^{7} - 8928 q^{8} + 13122 q^{9}+O(q^{10}) $ 2 * q - 36 * q^2 + 162 * q^3 + 256 * q^4 - 2916 * q^6 + 3318 * q^7 - 8928 * q^8 + 13122 * q^9	\( 2 q - 36 q^{2} + 162 q^{3} + 256 q^{4} - 2916 q^{6} + 3318 q^{7} - 8928 q^{8} + 13122 q^{9} + 24228 q^{11} + 20736 q^{12} - 90934 q^{13} - 139356 q^{14} + 196480 q^{16} + 124236 q^{17} - 236196 q^{18} - 1348846 q^{19} + 268758 q^{21} + 813992 q^{22} + 330444 q^{23} - 723168 q^{24} + 4048524 q^{26} + 1062882 q^{27} + 3291456 q^{28} - 4283172 q^{29} + 1176582 q^{31} + 6859008 q^{32} + 1962468 q^{33} - 13150888 q^{34} + 1679616 q^{36} + 5207668 q^{37} + 39420684 q^{38} - 7365654 q^{39} - 28888488 q^{41} - 11287836 q^{42} + 13370630 q^{43} - 41902272 q^{44} - 47026728 q^{46} - 66459060 q^{47} + 15914880 q^{48} - 65169020 q^{49} + 10063116 q^{51} - 98461184 q^{52} + 3809664 q^{53} - 19131876 q^{54} - 35834400 q^{56} - 109256526 q^{57} - 1418792 q^{58} - 115150284 q^{59} - 231494410 q^{61} - 241338204 q^{62} + 21769398 q^{63} - 352239616 q^{64} + 65933352 q^{66} - 319773234 q^{67} + 408829248 q^{68} + 26765964 q^{69} + 111792024 q^{71} - 58576608 q^{72} - 69141676 q^{73} - 354726216 q^{74} - 717744704 q^{76} - 117317844 q^{77} + 327930444 q^{78} - 334446000 q^{79} + 86093442 q^{81} + 1250955136 q^{82} + 149433012 q^{83} + 266607936 q^{84} + 904698468 q^{86} - 346936932 q^{87} + 221871552 q^{88} - 246671136 q^{89} - 454735218 q^{91} + 1521131328 q^{92} + 95303142 q^{93} + 565405736 q^{94} + 555579648 q^{96} + 696298546 q^{97} + 908823384 q^{98} + 158959908 q^{99}+O(q^{100}) \) 2 * q - 36 * q^2 + 162 * q^3 + 256 * q^4 - 2916 * q^6 + 3318 * q^7 - 8928 * q^8 + 13122 * q^9 + 24228 * q^11 + 20736 * q^12 - 90934 * q^13 - 139356 * q^14 + 196480 * q^16 + 124236 * q^17 - 236196 * q^18 - 1348846 * q^19 + 268758 * q^21 + 813992 * q^22 + 330444 * q^23 - 723168 * q^24 + 4048524 * q^26 + 1062882 * q^27 + 3291456 * q^28 - 4283172 * q^29 + 1176582 * q^31 + 6859008 * q^32 + 1962468 * q^33 - 13150888 * q^34 + 1679616 * q^36 + 5207668 * q^37 + 39420684 * q^38 - 7365654 * q^39 - 28888488 * q^41 - 11287836 * q^42 + 13370630 * q^43 - 41902272 * q^44 - 47026728 * q^46 - 66459060 * q^47 + 15914880 * q^48 - 65169020 * q^49 + 10063116 * q^51 - 98461184 * q^52 + 3809664 * q^53 - 19131876 * q^54 - 35834400 * q^56 - 109256526 * q^57 - 1418792 * q^58 - 115150284 * q^59 - 231494410 * q^61 - 241338204 * q^62 + 21769398 * q^63 - 352239616 * q^64 + 65933352 * q^66 - 319773234 * q^67 + 408829248 * q^68 + 26765964 * q^69 + 111792024 * q^71 - 58576608 * q^72 - 69141676 * q^73 - 354726216 * q^74 - 717744704 * q^76 - 117317844 * q^77 + 327930444 * q^78 - 334446000 * q^79 + 86093442 * q^81 + 1250955136 * q^82 + 149433012 * q^83 + 266607936 * q^84 + 904698468 * q^86 - 346936932 * q^87 + 221871552 * q^88 - 246671136 * q^89 - 454735218 * q^91 + 1521131328 * q^92 + 95303142 * q^93 + 565405736 * q^94 + 555579648 * q^96 + 696298546 * q^97 + 908823384 * q^98 + 158959908 * 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 $

$2$	−35.7764	−1.58111	−0.790554	−	0.612392i	$-0.790207\pi$
$2$	−35.7764	−1.58111	−0.790554	+	0.612392i	$0.790207\pi$
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	767.950	1.49990
$5$	0	0
$5$	0	0
$6$	−2897.89	−0.912853
$7$	3898.82	0.613751	0.306876	−	0.951750i	$-0.400716\pi$
$7$	3898.82	0.613751	0.306876	+	0.951750i	$0.400716\pi$
$8$	−9156.97	−0.790400
$9$	6561.00	0.333333
$10$	0	0
$11$	−23047.7	−0.474636	−0.237318	−	0.971432i	$-0.576268\pi$
$11$	−23047.7	−0.474636	−0.237318	+	0.971432i	$0.576268\pi$
$12$	62203.9	0.865969
$13$	−113302.	−1.10025	−0.550125	−	0.835082i	$-0.685420\pi$
$13$	−113302.	−1.10025	−0.550125	+	0.835082i	$0.685420\pi$
$14$	−139486.	−0.970407
$15$	0	0
$16$	−65587.2	−0.250195
$17$	369116.	1.07187	0.535936	−	0.844259i	$-0.319959\pi$
$17$	369116.	1.07187	0.535936	+	0.844259i	$0.319959\pi$
$18$	−234729.	−0.527036
$19$	−1.10031e6	−1.93697	−0.968487	−	0.249064i	$-0.919877\pi$
$19$	−1.10031e6	−1.93697	−0.968487	+	0.249064i	$0.919877\pi$
$20$	0	0
$21$	315805.	0.354350
$22$	824563.	0.750450
$23$	1.32065e6	0.984041	0.492020	−	0.870584i	$-0.336259\pi$
$23$	1.32065e6	0.984041	0.492020	+	0.870584i	$0.336259\pi$
$24$	−741714.	−0.456337
$25$	0	0
$26$	4.05353e6	1.73961
$27$	531441.	0.192450
$28$	2.99410e6	0.920567
$29$	66845.9	0.0175503	0.00877513	−	0.999961i	$-0.497207\pi$
$29$	66845.9	0.0175503	0.00877513	+	0.999961i	$0.497207\pi$
$30$	0	0
$31$	6.78077e6	1.31872	0.659358	−	0.751829i	$-0.270828\pi$
$31$	6.78077e6	1.31872	0.659358	+	0.751829i	$0.270828\pi$
$32$	7.03484e6	1.18599
$33$	−1.86686e6	−0.274031
$34$	−1.32056e7	−1.69475
$35$	0	0
$36$	5.03852e6	0.499967
$37$	9.94470e6	0.872336	0.436168	−	0.899865i	$-0.356335\pi$
$37$	9.94470e6	0.872336	0.436168	+	0.899865i	$0.356335\pi$
$38$	3.93651e7	3.06256
$39$	−9.17744e6	−0.635230
$40$	0	0
$41$	−3.50042e7	−1.93461	−0.967303	−	0.253625i	$-0.918377\pi$
$41$	−3.50042e7	−1.93461	−0.967303	+	0.253625i	$0.918377\pi$
$42$	−1.12984e7	−0.560265
$43$	−2.55307e7	−1.13882	−0.569410	−	0.822054i	$-0.692828\pi$
$43$	−2.55307e7	−1.13882	−0.569410	+	0.822054i	$0.692828\pi$
$44$	−1.76995e7	−0.711907
$45$	0	0
$46$	−4.72481e7	−1.55587
$47$	−1.54853e7	−0.462891	−0.231445	−	0.972848i	$-0.574345\pi$
$47$	−1.54853e7	−0.462891	−0.231445	+	0.972848i	$0.574345\pi$
$48$	−5.31256e6	−0.144450
$49$	−2.51528e7	−0.623309
$50$	0	0
$51$	2.98984e7	0.618846
$52$	−8.70100e7	−1.65027
$53$	1.54895e7	0.269647	0.134824	−	0.990870i	$-0.456953\pi$
$53$	1.54895e7	0.269647	0.134824	+	0.990870i	$0.456953\pi$
$54$	−1.90130e7	−0.304284
$55$	0	0
$56$	−3.57014e7	−0.485109
$57$	−8.91251e7	−1.11831
$58$	−2.39150e6	−0.0277489
$59$	6.45979e7	0.694039	0.347020	−	0.937858i	$-0.387194\pi$
$59$	6.45979e7	0.694039	0.347020	+	0.937858i	$0.387194\pi$
$60$	0	0
$61$	−1.07375e8	−0.992928	−0.496464	−	0.868057i	$-0.665369\pi$
$61$	−1.07375e8	−0.992928	−0.496464	+	0.868057i	$0.665369\pi$
$62$	−2.42591e8	−2.08503
$63$	2.55802e7	0.204584
$64$	−2.18101e8	−1.62498
$65$	0	0
$66$	6.67896e7	0.433273
$67$	−1.29631e8	−0.785911	−0.392956	−	0.919557i	$-0.628547\pi$
$67$	−1.29631e8	−0.785911	−0.392956	+	0.919557i	$0.628547\pi$
$68$	2.83463e8	1.60770
$69$	1.06973e8	0.568136
$70$	0	0
$71$	2.25343e8	1.05240	0.526201	−	0.850360i	$-0.323616\pi$
$71$	2.25343e8	1.05240	0.526201	+	0.850360i	$0.323616\pi$
$72$	−6.00789e7	−0.263467
$73$	−1.72446e8	−0.710723	−0.355362	−	0.934729i	$-0.615642\pi$
$73$	−1.72446e8	−0.710723	−0.355362	+	0.934729i	$0.615642\pi$
$74$	−3.55785e8	−1.37926
$75$	0	0
$76$	−8.44983e8	−2.90527
$77$	−8.98589e7	−0.291308
$78$	3.28336e8	1.00437
$79$	−3.14962e8	−0.909779	−0.454890	−	0.890548i	$-0.650321\pi$
$79$	−3.14962e8	−0.909779	−0.454890	+	0.890548i	$0.650321\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	1.25232e9	3.05882
$83$	−4.03289e8	−0.932748	−0.466374	−	0.884588i	$-0.654440\pi$
$83$	−4.03289e8	−0.932748	−0.466374	+	0.884588i	$0.654440\pi$
$84$	2.42522e8	0.531490
$85$	0	0
$86$	9.13397e8	1.80060
$87$	5.41452e6	0.0101327
$88$	2.11047e8	0.375152
$89$	−5.72362e8	−0.966975	−0.483488	−	0.875351i	$-0.660630\pi$
$89$	−5.72362e8	−0.966975	−0.483488	+	0.875351i	$0.660630\pi$
$90$	0	0
$91$	−4.41743e8	−0.675280
$92$	1.01419e9	1.47596
$93$	5.49242e8	0.761361
$94$	5.54007e8	0.731881
$95$	0	0
$96$	5.69822e8	0.684729
$97$	5.62737e8	0.645406	0.322703	−	0.946500i	$-0.395409\pi$
$97$	5.62737e8	0.645406	0.322703	+	0.946500i	$0.395409\pi$
$98$	8.99875e8	0.985519
$99$	−1.51216e8	−0.158212
$100$	0	0
$101$	1.92891e9	1.84445	0.922224	−	0.386655i	$-0.126370\pi$
$101$	1.92891e9	1.84445	0.922224	+	0.386655i	$0.126370\pi$
$102$	−1.06966e9	−0.978462
$103$	−1.90431e9	−1.66713	−0.833566	−	0.552420i	$-0.813704\pi$
$103$	−1.90431e9	−1.66713	−0.833566	+	0.552420i	$0.813704\pi$
$104$	1.03750e9	0.869637
$105$	0	0
$106$	−5.54159e8	−0.426342
$107$	−1.05533e9	−0.778323	−0.389162	−	0.921170i	$-0.627235\pi$
$107$	−1.05533e9	−0.778323	−0.389162	+	0.921170i	$0.627235\pi$
$108$	4.08120e8	0.288656
$109$	−5.12411e8	−0.347696	−0.173848	−	0.984773i	$-0.555620\pi$
$109$	−5.12411e8	−0.347696	−0.173848	+	0.984773i	$0.555620\pi$
$110$	0	0
$111$	8.05521e8	0.503644
$112$	−2.55713e8	−0.153558
$113$	−2.18469e9	−1.26048	−0.630242	−	0.776399i	$-0.717044\pi$
$113$	−2.18469e9	−1.26048	−0.630242	+	0.776399i	$0.717044\pi$
$114$	3.18857e9	1.76817
$115$	0	0
$116$	5.13343e7	0.0263237
$117$	−7.43372e8	−0.366750
$118$	−2.31108e9	−1.09735
$119$	1.43912e9	0.657863
$120$	0	0
$121$	−1.82675e9	−0.774721
$122$	3.84148e9	1.56993
$123$	−2.83534e9	−1.11694
$124$	5.20729e9	1.97794
$125$	0	0
$126$	−9.15167e8	−0.323469
$127$	−1.18646e9	−0.404702	−0.202351	−	0.979313i	$-0.564858\pi$
$127$	−1.18646e9	−0.404702	−0.202351	+	0.979313i	$0.564858\pi$
$128$	4.20101e9	1.38328
$129$	−2.06799e9	−0.657498
$130$	0	0
$131$	−3.71076e9	−1.10089	−0.550443	−	0.834873i	$-0.685541\pi$
$131$	−3.71076e9	−1.10089	−0.550443	+	0.834873i	$0.685541\pi$
$132$	−1.43366e9	−0.411020
$133$	−4.28992e9	−1.18882
$134$	4.63774e9	1.24261
$135$	0	0
$136$	−3.37999e9	−0.847207
$137$	3.20066e9	0.776241	0.388120	−	0.921609i	$-0.373124\pi$
$137$	3.20066e9	0.776241	0.388120	+	0.921609i	$0.373124\pi$
$138$	−3.82710e9	−0.898285
$139$	−2.18950e9	−0.497483	−0.248742	−	0.968570i	$-0.580017\pi$
$139$	−2.18950e9	−0.497483	−0.248742	+	0.968570i	$0.580017\pi$
$140$	0	0
$141$	−1.25431e9	−0.267250
$142$	−8.06197e9	−1.66396
$143$	2.61134e9	0.522218
$144$	−4.30318e8	−0.0833984
$145$	0	0
$146$	6.16950e9	1.12373
$147$	−2.03737e9	−0.359868
$148$	7.63703e9	1.30842
$149$	−6.64377e9	−1.10427	−0.552137	−	0.833754i	$-0.686187\pi$
$149$	−6.64377e9	−1.10427	−0.552137	+	0.833754i	$0.686187\pi$
$150$	0	0
$151$	1.18641e10	1.85711	0.928556	−	0.371193i	$-0.121051\pi$
$151$	1.18641e10	1.85711	0.928556	+	0.371193i	$0.121051\pi$
$152$	1.00755e10	1.53098
$153$	2.42177e9	0.357291
$154$	3.21483e9	0.460590
$155$	0	0
$156$	−7.04781e9	−0.952782
$157$	3.30825e9	0.434560	0.217280	−	0.976109i	$-0.430282\pi$
$157$	3.30825e9	0.434560	0.217280	+	0.976109i	$0.430282\pi$
$158$	1.12682e10	1.43846
$159$	1.25465e9	0.155681
$160$	0	0
$161$	5.14899e9	0.603956
$162$	−1.54006e9	−0.175679
$163$	3.84635e9	0.426781	0.213390	−	0.976967i	$-0.431549\pi$
$163$	3.84635e9	0.426781	0.213390	+	0.976967i	$0.431549\pi$
$164$	−2.68814e10	−2.90172
$165$	0	0
$166$	1.44282e10	1.47478
$167$	1.21105e10	1.20486	0.602431	−	0.798171i	$-0.294199\pi$
$167$	1.21105e10	1.20486	0.602431	+	0.798171i	$0.294199\pi$
$168$	−2.89181e9	−0.280078
$169$	2.23278e9	0.210550
$170$	0	0
$171$	−7.21913e9	−0.645658
$172$	−1.96063e10	−1.70812
$173$	−1.36229e10	−1.15628	−0.578138	−	0.815939i	$-0.696220\pi$
$173$	−1.36229e10	−1.15628	−0.578138	+	0.815939i	$0.696220\pi$
$174$	−1.93712e8	−0.0160208
$175$	0	0
$176$	1.51163e9	0.118752
$177$	5.23243e9	0.400704
$178$	2.04770e10	1.52889
$179$	−2.50779e10	−1.82580	−0.912898	−	0.408188i	$-0.866161\pi$
$179$	−2.50779e10	−1.82580	−0.912898	+	0.408188i	$0.866161\pi$
$180$	0	0
$181$	−5.53748e9	−0.383494	−0.191747	−	0.981444i	$-0.561415\pi$
$181$	−5.53748e9	−0.383494	−0.191747	+	0.981444i	$0.561415\pi$
$182$	1.58040e10	1.06769
$183$	−8.69735e9	−0.573267
$184$	−1.20932e10	−0.777785
$185$	0	0
$186$	−1.96499e10	−1.20379
$187$	−8.50728e9	−0.508749
$188$	−1.18919e10	−0.694291
$189$	2.07200e9	0.118117
$190$	0	0
$191$	−2.74204e10	−1.49081	−0.745406	−	0.666611i	$-0.767744\pi$
$191$	−2.74204e10	−1.49081	−0.745406	+	0.666611i	$0.767744\pi$
$192$	−1.76661e10	−0.938180
$193$	1.47069e10	0.762978	0.381489	−	0.924373i	$-0.375411\pi$
$193$	1.47069e10	0.762978	0.381489	+	0.924373i	$0.375411\pi$
$194$	−2.01327e10	−1.02046
$195$	0	0
$196$	−1.93161e10	−0.934903
$197$	−6.27835e9	−0.296994	−0.148497	−	0.988913i	$-0.547444\pi$
$197$	−6.27835e9	−0.296994	−0.148497	+	0.988913i	$0.547444\pi$
$198$	5.40996e9	0.250150
$199$	2.30488e10	1.04186	0.520930	−	0.853599i	$-0.325585\pi$
$199$	2.30488e10	1.04186	0.520930	+	0.853599i	$0.325585\pi$
$200$	0	0
$201$	−1.05001e10	−0.453746
$202$	−6.90096e10	−2.91627
$203$	2.60620e8	0.0107715
$204$	2.29605e10	0.928208
$205$	0	0
$206$	6.81293e10	2.63592
$207$	8.66480e9	0.328014
$208$	7.43114e9	0.275277
$209$	2.53596e10	0.919357
$210$	0	0
$211$	−2.61916e10	−0.909684	−0.454842	−	0.890572i	$-0.650304\pi$
$211$	−2.61916e10	−0.909684	−0.454842	+	0.890572i	$0.650304\pi$
$212$	1.18952e10	0.404445
$213$	1.82528e10	0.607605
$214$	3.77558e10	1.23061
$215$	0	0
$216$	−4.86639e9	−0.152112
$217$	2.64370e10	0.809364
$218$	1.83322e10	0.549744
$219$	−1.39681e10	−0.410336
$220$	0	0
$221$	−4.18215e10	−1.17933
$222$	−2.88186e10	−0.796315
$223$	−6.95437e10	−1.88316	−0.941578	−	0.336796i	$-0.890657\pi$
$223$	−6.95437e10	−1.88316	−0.941578	+	0.336796i	$0.890657\pi$
$224$	2.74276e10	0.727900
$225$	0	0
$226$	7.81604e10	1.99296
$227$	−6.83035e9	−0.170737	−0.0853684	−	0.996349i	$-0.527207\pi$
$227$	−6.83035e9	−0.170737	−0.0853684	+	0.996349i	$0.527207\pi$
$228$	−6.84436e10	−1.67736
$229$	2.43923e10	0.586129	0.293065	−	0.956093i	$-0.405325\pi$
$229$	2.43923e10	0.586129	0.293065	+	0.956093i	$0.405325\pi$
$230$	0	0
$231$	−7.27857e9	−0.168187
$232$	−6.12106e8	−0.0138717
$233$	8.36542e10	1.85946	0.929729	−	0.368245i	$-0.120041\pi$
$233$	8.36542e10	1.85946	0.929729	+	0.368245i	$0.120041\pi$
$234$	2.65952e10	0.579871
$235$	0	0
$236$	4.96079e10	1.04099
$237$	−2.55119e10	−0.525261
$238$	−5.14865e10	−1.04015
$239$	−1.66318e10	−0.329723	−0.164862	−	0.986317i	$-0.552718\pi$
$239$	−1.66318e10	−0.329723	−0.164862	+	0.986317i	$0.552718\pi$
$240$	0	0
$241$	−9.26682e10	−1.76951	−0.884757	−	0.466053i	$-0.845676\pi$
$241$	−9.26682e10	−1.76951	−0.884757	+	0.466053i	$0.845676\pi$
$242$	6.53546e10	1.22492
$243$	3.48678e9	0.0641500
$244$	−8.24584e10	−1.48930
$245$	0	0
$246$	1.01438e11	1.76601
$247$	1.24667e11	2.13116
$248$	−6.20913e10	−1.04231
$249$	−3.26664e10	−0.538523
$250$	0	0
$251$	−1.99958e10	−0.317985	−0.158993	−	0.987280i	$-0.550825\pi$
$251$	−1.99958e10	−0.317985	−0.158993	+	0.987280i	$0.550825\pi$
$252$	1.96443e10	0.306856
$253$	−3.04380e10	−0.467061
$254$	4.24472e10	0.639878
$255$	0	0
$256$	−3.86295e10	−0.562134
$257$	1.27910e11	1.82897	0.914483	−	0.404624i	$-0.132598\pi$
$257$	1.27910e11	1.82897	0.914483	+	0.404624i	$0.132598\pi$
$258$	7.39852e10	1.03958
$259$	3.87726e10	0.535398
$260$	0	0
$261$	4.38576e8	0.00585009
$262$	1.32758e11	1.74062
$263$	2.76120e10	0.355875	0.177938	−	0.984042i	$-0.443057\pi$
$263$	2.76120e10	0.355875	0.177938	+	0.984042i	$0.443057\pi$
$264$	1.70948e10	0.216594
$265$	0	0
$266$	1.53478e11	1.87965
$267$	−4.63613e10	−0.558284
$268$	−9.95504e10	−1.17879
$269$	4.03173e10	0.469468	0.234734	−	0.972060i	$-0.424578\pi$
$269$	4.03173e10	0.469468	0.234734	+	0.972060i	$0.424578\pi$
$270$	0	0
$271$	4.36744e10	0.491886	0.245943	−	0.969284i	$-0.420902\pi$
$271$	4.36744e10	0.491886	0.245943	+	0.969284i	$0.420902\pi$
$272$	−2.42093e10	−0.268177
$273$	−3.57812e10	−0.389873
$274$	−1.14508e11	−1.22732
$275$	0	0
$276$	8.21498e10	0.852149
$277$	−4.35371e10	−0.444324	−0.222162	−	0.975010i	$-0.571311\pi$
$277$	−4.35371e10	−0.444324	−0.222162	+	0.975010i	$0.571311\pi$
$278$	7.83325e10	0.786575
$279$	4.44886e10	0.439572
$280$	0	0
$281$	−2.86443e10	−0.274069	−0.137035	−	0.990566i	$-0.543757\pi$
$281$	−2.86443e10	−0.274069	−0.137035	+	0.990566i	$0.543757\pi$
$282$	4.48746e10	0.422551
$283$	2.31306e10	0.214362	0.107181	−	0.994240i	$-0.465818\pi$
$283$	2.31306e10	0.214362	0.107181	+	0.994240i	$0.465818\pi$
$284$	1.73052e11	1.57850
$285$	0	0
$286$	−9.34244e10	−0.825683
$287$	−1.36475e11	−1.18737
$288$	4.61556e10	0.395328
$289$	1.76589e10	0.148910
$290$	0	0
$291$	4.55817e10	0.372625
$292$	−1.32430e11	−1.06602
$293$	−4.76003e10	−0.377316	−0.188658	−	0.982043i	$-0.560414\pi$
$293$	−4.76003e10	−0.377316	−0.188658	+	0.982043i	$0.560414\pi$
$294$	7.28899e10	0.568990
$295$	0	0
$296$	−9.10633e10	−0.689494
$297$	−1.22485e10	−0.0913437
$298$	2.37690e11	1.74598
$299$	−1.49632e11	−1.08269
$300$	0	0
$301$	−9.95398e10	−0.698953
$302$	−4.24454e11	−2.93629
$303$	1.56242e11	1.06489
$304$	7.21662e10	0.484622
$305$	0	0
$306$	−8.66422e10	−0.564915
$307$	2.45646e11	1.57829	0.789145	−	0.614207i	$-0.210524\pi$
$307$	2.45646e11	1.57829	0.789145	+	0.614207i	$0.210524\pi$
$308$	−6.90072e10	−0.436934
$309$	−1.54249e11	−0.962519
$310$	0	0
$311$	−2.62977e10	−0.159403	−0.0797015	−	0.996819i	$-0.525397\pi$
$311$	−2.62977e10	−0.159403	−0.0797015	+	0.996819i	$0.525397\pi$
$312$	8.40375e10	0.502085
$313$	−2.18427e11	−1.28635	−0.643173	−	0.765721i	$-0.722382\pi$
$313$	−2.18427e11	−1.28635	−0.643173	+	0.765721i	$0.722382\pi$
$314$	−1.18357e11	−0.687086
$315$	0	0
$316$	−2.41875e11	−1.36458
$317$	−1.97173e11	−1.09668	−0.548341	−	0.836255i	$-0.684740\pi$
$317$	−1.97173e11	−1.09668	−0.548341	+	0.836255i	$0.684740\pi$
$318$	−4.48869e10	−0.246149
$319$	−1.54064e9	−0.00832998
$320$	0	0
$321$	−8.54815e10	−0.449365
$322$	−1.84212e11	−0.954920
$323$	−4.06142e11	−2.07619
$324$	3.30577e10	0.166656
$325$	0	0
$326$	−1.37609e11	−0.674787
$327$	−4.15053e10	−0.200742
$328$	3.20532e11	1.52911
$329$	−6.03744e10	−0.284100
$330$	0	0
$331$	7.19139e10	0.329296	0.164648	−	0.986352i	$-0.447351\pi$
$331$	7.19139e10	0.329296	0.164648	+	0.986352i	$0.447351\pi$
$332$	−3.09705e11	−1.39903
$333$	6.52472e10	0.290779
$334$	−4.33269e11	−1.90502
$335$	0	0
$336$	−2.07128e10	−0.0886566
$337$	9.49224e10	0.400898	0.200449	−	0.979704i	$-0.435760\pi$
$337$	9.49224e10	0.400898	0.200449	+	0.979704i	$0.435760\pi$
$338$	−7.98807e10	−0.332902
$339$	−1.76960e11	−0.727741
$340$	0	0
$341$	−1.56281e11	−0.625910
$342$	2.58274e11	1.02085
$343$	−2.55398e11	−0.996308
$344$	2.33784e11	0.900123
$345$	0	0
$346$	4.87377e11	1.82820
$347$	2.17424e11	0.805054	0.402527	−	0.915408i	$-0.368132\pi$
$347$	2.17424e11	0.805054	0.402527	+	0.915408i	$0.368132\pi$
$348$	4.15808e9	0.0151980
$349$	2.50076e11	0.902315	0.451157	−	0.892444i	$-0.351011\pi$
$349$	2.50076e11	0.902315	0.451157	+	0.892444i	$0.351011\pi$
$350$	0	0
$351$	−6.02132e10	−0.211743
$352$	−1.62137e11	−0.562911
$353$	4.64191e11	1.59115	0.795573	−	0.605857i	$-0.207170\pi$
$353$	4.64191e11	1.59115	0.795573	+	0.605857i	$0.207170\pi$
$354$	−1.87197e11	−0.633556
$355$	0	0
$356$	−4.39545e11	−1.45037
$357$	1.16569e11	0.379817
$358$	8.97196e11	2.88678
$359$	−2.17323e11	−0.690526	−0.345263	−	0.938506i	$-0.612210\pi$
$359$	−2.17323e11	−0.690526	−0.345263	+	0.938506i	$0.612210\pi$
$360$	0	0
$361$	8.87994e11	2.75187
$362$	1.98111e11	0.606345
$363$	−1.47967e11	−0.447285
$364$	−3.39237e11	−1.01285
$365$	0	0
$366$	3.11160e11	0.906398
$367$	−5.69129e11	−1.63762	−0.818811	−	0.574063i	$-0.805366\pi$
$367$	−5.69129e11	−1.63762	−0.818811	+	0.574063i	$0.805366\pi$
$368$	−8.66178e10	−0.246202
$369$	−2.29662e11	−0.644868
$370$	0	0
$371$	6.03909e10	0.165497
$372$	4.21791e11	1.14197
$373$	2.82198e10	0.0754856	0.0377428	−	0.999287i	$-0.487983\pi$
$373$	2.82198e10	0.0754856	0.0377428	+	0.999287i	$0.487983\pi$
$374$	3.04360e11	0.804387
$375$	0	0
$376$	1.41798e11	0.365869
$377$	−7.57375e9	−0.0193097
$378$	−7.41285e10	−0.186755
$379$	1.30996e10	0.0326124	0.0163062	−	0.999867i	$-0.494809\pi$
$379$	1.30996e10	0.0326124	0.0163062	+	0.999867i	$0.494809\pi$
$380$	0	0
$381$	−9.61031e10	−0.233655
$382$	9.81001e11	2.35714
$383$	2.80400e11	0.665861	0.332931	−	0.942951i	$-0.391962\pi$
$383$	2.80400e11	0.665861	0.332931	+	0.942951i	$0.391962\pi$
$384$	3.40282e11	0.798635
$385$	0	0
$386$	−5.26158e11	−1.20635
$387$	−1.67507e11	−0.379607
$388$	4.32154e11	0.968046
$389$	−3.02952e11	−0.670811	−0.335405	−	0.942074i	$-0.608873\pi$
$389$	−3.02952e11	−0.670811	−0.335405	+	0.942074i	$0.608873\pi$
$390$	0	0
$391$	4.87474e11	1.05477
$392$	2.30323e11	0.492663
$393$	−3.00572e11	−0.635597
$394$	2.24617e11	0.469580
$395$	0	0
$396$	−1.16126e11	−0.237302
$397$	7.31203e11	1.47734	0.738670	−	0.674067i	$-0.235454\pi$
$397$	7.31203e11	1.47734	0.738670	+	0.674067i	$0.235454\pi$
$398$	−8.24603e11	−1.64729
$399$	−3.47483e11	−0.686366
$400$	0	0
$401$	−3.27396e9	−0.00632301	−0.00316150	−	0.999995i	$-0.501006\pi$
$401$	−3.27396e9	−0.00632301	−0.00316150	+	0.999995i	$0.501006\pi$
$402$	3.75657e11	0.717422
$403$	−7.68272e11	−1.45092
$404$	1.48131e12	2.76649
$405$	0	0
$406$	−9.32406e9	−0.0170309
$407$	−2.29202e11	−0.414042
$408$	−2.73779e11	−0.489135
$409$	6.73551e11	1.19019	0.595094	−	0.803656i	$-0.297115\pi$
$409$	6.73551e11	1.19019	0.595094	+	0.803656i	$0.297115\pi$
$410$	0	0
$411$	2.59253e11	0.448163
$412$	−1.46241e12	−2.50053
$413$	2.51856e11	0.425968
$414$	−3.09995e11	−0.518625
$415$	0	0
$416$	−7.97059e11	−1.30488
$417$	−1.77350e11	−0.287222
$418$	−9.07275e11	−1.45360
$419$	9.58213e11	1.51879	0.759397	−	0.650628i	$-0.225494\pi$
$419$	9.58213e11	1.51879	0.759397	+	0.650628i	$0.225494\pi$
$420$	0	0
$421$	−2.79406e11	−0.433477	−0.216739	−	0.976230i	$-0.569542\pi$
$421$	−2.79406e11	−0.433477	−0.216739	+	0.976230i	$0.569542\pi$
$422$	9.37040e11	1.43831
$423$	−1.01599e11	−0.154297
$424$	−1.41837e11	−0.213129
$425$	0	0
$426$	−6.53019e11	−0.960689
$427$	−4.18635e11	−0.609411
$428$	−8.10438e11	−1.16741
$429$	2.11519e11	0.301503
$430$	0	0
$431$	−1.85936e11	−0.259546	−0.129773	−	0.991544i	$-0.541425\pi$
$431$	−1.85936e11	−0.259546	−0.129773	+	0.991544i	$0.541425\pi$
$432$	−3.48557e10	−0.0481501
$433$	2.36668e11	0.323551	0.161776	−	0.986828i	$-0.448278\pi$
$433$	2.36668e11	0.323551	0.161776	+	0.986828i	$0.448278\pi$
$434$	−9.45821e11	−1.27969
$435$	0	0
$436$	−3.93506e11	−0.521509
$437$	−1.45313e12	−1.90606
$438$	4.99730e11	0.648786
$439$	1.10414e12	1.41885	0.709423	−	0.704783i	$-0.248956\pi$
$439$	1.10414e12	1.41885	0.709423	+	0.704783i	$0.248956\pi$
$440$	0	0
$441$	−1.65027e11	−0.207770
$442$	1.49622e12	1.86464
$443$	2.82238e11	0.348175	0.174088	−	0.984730i	$-0.444302\pi$
$443$	2.82238e11	0.348175	0.174088	+	0.984730i	$0.444302\pi$
$444$	6.18600e11	0.755416
$445$	0	0
$446$	2.48802e12	2.97747
$447$	−5.38146e11	−0.637553
$448$	−8.50336e11	−0.997331
$449$	−5.99349e11	−0.695939	−0.347970	−	0.937506i	$-0.613129\pi$
$449$	−5.99349e11	−0.695939	−0.347970	+	0.937506i	$0.613129\pi$
$450$	0	0
$451$	8.06765e11	0.918233
$452$	−1.67773e12	−1.89060
$453$	9.60991e11	1.07220
$454$	2.44365e11	0.269953
$455$	0	0
$456$	8.16115e11	0.883914
$457$	−4.94163e11	−0.529965	−0.264983	−	0.964253i	$-0.585366\pi$
$457$	−4.94163e11	−0.529965	−0.264983	+	0.964253i	$0.585366\pi$
$458$	−8.72669e11	−0.926734
$459$	1.96164e11	0.206282
$460$	0	0
$461$	6.25704e11	0.645230	0.322615	−	0.946530i	$-0.395438\pi$
$461$	6.25704e11	0.645230	0.322615	+	0.946530i	$0.395438\pi$
$462$	2.60401e11	0.265922
$463$	8.05319e11	0.814429	0.407215	−	0.913332i	$-0.366500\pi$
$463$	8.05319e11	0.814429	0.407215	+	0.913332i	$0.366500\pi$
$464$	−4.38423e9	−0.00439099
$465$	0	0
$466$	−2.99285e12	−2.94000
$467$	2.95042e11	0.287050	0.143525	−	0.989647i	$-0.454156\pi$
$467$	2.95042e11	0.287050	0.143525	+	0.989647i	$0.454156\pi$
$468$	−5.70873e11	−0.550089
$469$	−5.05410e11	−0.482354
$470$	0	0
$471$	2.67968e11	0.250893
$472$	−5.91520e11	−0.548568
$473$	5.88424e11	0.540525
$474$	9.12724e11	0.830495
$475$	0	0
$476$	1.10517e12	0.986730
$477$	1.01627e11	0.0898825
$478$	5.95027e11	0.521328
$479$	4.71656e11	0.409369	0.204685	−	0.978828i	$-0.434383\pi$
$479$	4.71656e11	0.409369	0.204685	+	0.978828i	$0.434383\pi$
$480$	0	0
$481$	−1.12675e12	−0.959788
$482$	3.31533e12	2.79779
$483$	4.17068e11	0.348694
$484$	−1.40285e12	−1.16201
$485$	0	0
$486$	−1.24745e11	−0.101428
$487$	2.13496e12	1.71993	0.859964	−	0.510355i	$-0.170486\pi$
$487$	2.13496e12	1.71993	0.859964	+	0.510355i	$0.170486\pi$
$488$	9.83227e11	0.784810
$489$	3.11555e11	0.246402
$490$	0	0
$491$	−1.15317e12	−0.895418	−0.447709	−	0.894179i	$-0.647760\pi$
$491$	−1.15317e12	−0.895418	−0.447709	+	0.894179i	$0.647760\pi$
$492$	−2.17740e12	−1.67531
$493$	2.46739e10	0.0188116
$494$	−4.46013e12	−3.36959
$495$	0	0
$496$	−4.44732e11	−0.329937
$497$	8.78574e11	0.645914
$498$	1.16868e12	0.851462
$499$	−9.44785e10	−0.0682151	−0.0341075	−	0.999418i	$-0.510859\pi$
$499$	−9.44785e10	−0.0682151	−0.0341075	+	0.999418i	$0.510859\pi$
$500$	0	0
$501$	9.80949e11	0.695627
$502$	7.15377e11	0.502769
$503$	1.99443e12	1.38920	0.694598	−	0.719398i	$-0.255582\pi$
$503$	1.99443e12	1.38920	0.694598	+	0.719398i	$0.255582\pi$
$504$	−2.34237e11	−0.161703
$505$	0	0
$506$	1.08896e12	0.738474
$507$	1.80855e11	0.121561
$508$	−9.11141e11	−0.607014
$509$	1.29001e12	0.851847	0.425923	−	0.904759i	$-0.359949\pi$
$509$	1.29001e12	0.851847	0.425923	+	0.904759i	$0.359949\pi$
$510$	0	0
$511$	−6.72337e11	−0.436208
$512$	−7.68892e11	−0.494482
$513$	−5.84750e11	−0.372771
$514$	−4.57616e12	−2.89179
$515$	0	0
$516$	−1.58811e12	−0.986183
$517$	3.56900e11	0.219705
$518$	−1.38715e12	−0.846522
$519$	−1.10345e12	−0.667576
$520$	0	0
$521$	1.19407e12	0.710003	0.355001	−	0.934866i	$-0.384480\pi$
$521$	1.19407e12	0.710003	0.355001	+	0.934866i	$0.384480\pi$
$522$	−1.56907e10	−0.00924962
$523$	8.52995e9	0.00498527	0.00249264	−	0.999997i	$-0.499207\pi$
$523$	8.52995e9	0.00498527	0.00249264	+	0.999997i	$0.499207\pi$
$524$	−2.84968e12	−1.65122
$525$	0	0
$526$	−9.87859e11	−0.562677
$527$	2.50289e12	1.41349
$528$	1.22442e11	0.0685613
$529$	−5.70317e10	−0.0316640
$530$	0	0
$531$	4.23827e11	0.231346
$532$	−3.29444e12	−1.78311
$533$	3.96603e12	2.12855
$534$	1.65864e12	0.882707
$535$	0	0
$536$	1.18703e12	0.621184
$537$	−2.03131e12	−1.05412
$538$	−1.44241e12	−0.742280
$539$	5.79713e11	0.295845
$540$	0	0
$541$	2.03592e11	0.102182	0.0510908	−	0.998694i	$-0.483730\pi$
$541$	2.03592e11	0.102182	0.0510908	+	0.998694i	$0.483730\pi$
$542$	−1.56251e12	−0.777726
$543$	−4.48536e11	−0.221410
$544$	2.59667e12	1.27122
$545$	0	0
$546$	1.28012e12	0.616431
$547$	−2.64470e12	−1.26309	−0.631545	−	0.775339i	$-0.717579\pi$
$547$	−2.64470e12	−1.26309	−0.631545	+	0.775339i	$0.717579\pi$
$548$	2.45794e12	1.16429
$549$	−7.04486e11	−0.330976
$550$	0	0
$551$	−7.35512e10	−0.0339944
$552$	−9.79546e11	−0.449055
$553$	−1.22798e12	−0.558378
$554$	1.55760e12	0.702525
$555$	0	0
$556$	−1.68143e12	−0.746177
$557$	−1.68698e12	−0.742613	−0.371307	−	0.928510i	$-0.621090\pi$
$557$	−1.68698e12	−0.742613	−0.371307	+	0.928510i	$0.621090\pi$
$558$	−1.59164e12	−0.695011
$559$	2.89267e12	1.25299
$560$	0	0
$561$	−6.89090e11	−0.293726
$562$	1.02479e12	0.433333
$563$	5.77046e11	0.242060	0.121030	−	0.992649i	$-0.461380\pi$
$563$	5.77046e11	0.242060	0.121030	+	0.992649i	$0.461380\pi$
$564$	−9.63246e11	−0.400849
$565$	0	0
$566$	−8.27531e11	−0.338930
$567$	1.67832e11	0.0681946
$568$	−2.06346e12	−0.831819
$569$	−5.33359e11	−0.213311	−0.106656	−	0.994296i	$-0.534014\pi$
$569$	−5.33359e11	−0.213311	−0.106656	+	0.994296i	$0.534014\pi$
$570$	0	0
$571$	2.56331e12	1.00911	0.504556	−	0.863379i	$-0.331656\pi$
$571$	2.56331e12	1.00911	0.504556	+	0.863379i	$0.331656\pi$
$572$	2.00538e12	0.783276
$573$	−2.22105e12	−0.860721
$574$	4.88259e12	1.87736
$575$	0	0
$576$	−1.43096e12	−0.541659
$577$	−4.17162e12	−1.56680	−0.783400	−	0.621518i	$-0.786516\pi$
$577$	−4.17162e12	−1.56680	−0.783400	+	0.621518i	$0.786516\pi$
$578$	−6.31772e11	−0.235443
$579$	1.19126e12	0.440505
$580$	0	0
$581$	−1.57235e12	−0.572476
$582$	−1.63075e12	−0.589161
$583$	−3.56998e11	−0.127984
$584$	1.57908e12	0.561755
$585$	0	0
$586$	1.70297e12	0.596577
$587$	1.53800e12	0.534668	0.267334	−	0.963604i	$-0.413857\pi$
$587$	1.53800e12	0.534668	0.267334	+	0.963604i	$0.413857\pi$
$588$	−1.56460e12	−0.539766
$589$	−7.46094e12	−2.55432
$590$	0	0
$591$	−5.08546e11	−0.171470
$592$	−6.52245e11	−0.218254
$593$	−8.68139e11	−0.288299	−0.144150	−	0.989556i	$-0.546045\pi$
$593$	−8.68139e11	−0.288299	−0.144150	+	0.989556i	$0.546045\pi$
$594$	4.38207e11	0.144424
$595$	0	0
$596$	−5.10209e12	−1.65630
$597$	1.86695e12	0.601519
$598$	5.35330e12	1.71185
$599$	4.89825e12	1.55461	0.777303	−	0.629127i	$-0.216587\pi$
$599$	4.89825e12	1.55461	0.777303	+	0.629127i	$0.216587\pi$
$600$	0	0
$601$	1.06839e12	0.334036	0.167018	−	0.985954i	$-0.446586\pi$
$601$	1.06839e12	0.334036	0.167018	+	0.985954i	$0.446586\pi$
$602$	3.56118e12	1.10512
$603$	−8.50511e11	−0.261970
$604$	9.11103e12	2.78549
$605$	0	0
$606$	−5.58977e12	−1.68371
$607$	−3.11476e11	−0.0931268	−0.0465634	−	0.998915i	$-0.514827\pi$
$607$	−3.11476e11	−0.0931268	−0.0465634	+	0.998915i	$0.514827\pi$
$608$	−7.74050e12	−2.29722
$609$	2.11103e10	0.00621893
$610$	0	0
$611$	1.75451e12	0.509296
$612$	1.85980e12	0.535901
$613$	−3.17050e10	−0.00906891	−0.00453445	−	0.999990i	$-0.501443\pi$
$613$	−3.17050e10	−0.00906891	−0.00453445	+	0.999990i	$0.501443\pi$
$614$	−8.78832e12	−2.49545
$615$	0	0
$616$	8.22835e11	0.230250
$617$	4.71027e12	1.30847	0.654233	−	0.756293i	$-0.272992\pi$
$617$	4.71027e12	1.30847	0.654233	+	0.756293i	$0.272992\pi$
$618$	5.51847e12	1.52185
$619$	−4.40729e12	−1.20660	−0.603301	−	0.797513i	$-0.706148\pi$
$619$	−4.40729e12	−1.20660	−0.603301	+	0.797513i	$0.706148\pi$
$620$	0	0
$621$	7.01848e11	0.189379
$622$	9.40837e11	0.252033
$623$	−2.23154e12	−0.593483
$624$	6.01922e11	0.158931
$625$	0	0
$626$	7.81454e12	2.03385
$627$	2.05413e12	0.530791
$628$	2.54057e12	0.651798
$629$	3.67075e12	0.935033
$630$	0	0
$631$	−1.28716e12	−0.323221	−0.161610	−	0.986855i	$-0.551669\pi$
$631$	−1.28716e12	−0.323221	−0.161610	+	0.986855i	$0.551669\pi$
$632$	2.88410e12	0.719089
$633$	−2.12152e12	−0.525206
$634$	7.05414e12	1.73397
$635$	0	0
$636$	9.63509e11	0.233506
$637$	2.84985e12	0.685796
$638$	5.51187e10	0.0131706
$639$	1.47848e12	0.350801
$640$	0	0
$641$	−5.95493e12	−1.39321	−0.696603	−	0.717457i	$-0.745306\pi$
$641$	−5.95493e12	−1.39321	−0.696603	+	0.717457i	$0.745306\pi$
$642$	3.05822e12	0.710495
$643$	1.38447e12	0.319399	0.159700	−	0.987166i	$-0.448947\pi$
$643$	1.38447e12	0.319399	0.159700	+	0.987166i	$0.448947\pi$
$644$	3.95417e12	0.905876
$645$	0	0
$646$	1.45303e13	3.28268
$647$	−1.41737e12	−0.317991	−0.158996	−	0.987279i	$-0.550826\pi$
$647$	−1.41737e12	−0.317991	−0.158996	+	0.987279i	$0.550826\pi$
$648$	−3.94177e11	−0.0878222
$649$	−1.48883e12	−0.329416
$650$	0	0
$651$	2.14140e12	0.467286
$652$	2.95381e12	0.640130
$653$	−7.68868e12	−1.65479	−0.827394	−	0.561621i	$-0.810178\pi$
$653$	−7.68868e12	−1.65479	−0.827394	+	0.561621i	$0.810178\pi$
$654$	1.48491e12	0.317395
$655$	0	0
$656$	2.29583e12	0.484029
$657$	−1.13142e12	−0.236908
$658$	2.15998e12	0.449193
$659$	−8.26006e12	−1.70608	−0.853039	−	0.521847i	$-0.825243\pi$
$659$	−8.26006e12	−1.70608	−0.853039	+	0.521847i	$0.825243\pi$
$660$	0	0
$661$	4.87919e12	0.994126	0.497063	−	0.867714i	$-0.334412\pi$
$661$	4.87919e12	0.994126	0.497063	+	0.867714i	$0.334412\pi$
$662$	−2.57282e12	−0.520653
$663$	−3.38754e12	−0.680885
$664$	3.69290e12	0.737244
$665$	0	0
$666$	−2.33431e12	−0.459753
$667$	8.82801e10	0.0172702
$668$	9.30024e12	1.80718
$669$	−5.63304e12	−1.08724
$670$	0	0
$671$	2.47474e12	0.471279
$672$	2.22164e12	0.420253
$673$	3.86510e12	0.726261	0.363130	−	0.931738i	$-0.381708\pi$
$673$	3.86510e12	0.726261	0.363130	+	0.931738i	$0.381708\pi$
$674$	−3.39598e12	−0.633863
$675$	0	0
$676$	1.71466e12	0.315804
$677$	−2.05208e12	−0.375444	−0.187722	−	0.982222i	$-0.560110\pi$
$677$	−2.05208e12	−0.375444	−0.187722	+	0.982222i	$0.560110\pi$
$678$	6.33099e12	1.15064
$679$	2.19401e12	0.396119
$680$	0	0
$681$	−5.53259e11	−0.0985749
$682$	5.59117e12	0.989631
$683$	−4.25955e12	−0.748981	−0.374490	−	0.927231i	$-0.622182\pi$
$683$	−4.25955e12	−0.748981	−0.374490	+	0.927231i	$0.622182\pi$
$684$	−5.54393e12	−0.968424
$685$	0	0
$686$	9.13721e12	1.57527
$687$	1.97578e12	0.338402
$688$	1.67449e12	0.284927
$689$	−1.75499e12	−0.296680
$690$	0	0
$691$	−3.86518e12	−0.644938	−0.322469	−	0.946580i	$-0.604513\pi$
$691$	−3.86518e12	−0.644938	−0.322469	+	0.946580i	$0.604513\pi$
$692$	−1.04617e13	−1.73430
$693$	−5.89564e11	−0.0971028
$694$	−7.77865e12	−1.27288
$695$	0	0
$696$	−4.95806e10	−0.00800884
$697$	−1.29206e13	−2.07365
$698$	−8.94683e12	−1.42666
$699$	6.77599e12	1.07356
$700$	0	0
$701$	−1.03789e13	−1.62338	−0.811692	−	0.584086i	$-0.801453\pi$
$701$	−1.03789e13	−1.62338	−0.811692	+	0.584086i	$0.801453\pi$
$702$	2.15421e12	0.334789
$703$	−1.09423e13	−1.68969
$704$	5.02671e12	0.771272
$705$	0	0
$706$	−1.66071e13	−2.51577
$707$	7.52050e12	1.13203
$708$	4.01824e12	0.601017
$709$	−8.16100e12	−1.21293	−0.606465	−	0.795110i	$-0.707413\pi$
$709$	−8.16100e12	−1.21293	−0.606465	+	0.795110i	$0.707413\pi$
$710$	0	0
$711$	−2.06646e12	−0.303260
$712$	5.24110e12	0.764297
$713$	8.95503e12	1.29767
$714$	−4.17041e12	−0.600532
$715$	0	0
$716$	−1.92586e13	−2.73851
$717$	−1.34718e12	−0.190366
$718$	7.77502e12	1.09180
$719$	9.74567e12	1.35998	0.679989	−	0.733223i	$-0.261985\pi$
$719$	9.74567e12	1.35998	0.679989	+	0.733223i	$0.261985\pi$
$720$	0	0
$721$	−7.42457e12	−1.02320
$722$	−3.17692e13	−4.35100
$723$	−7.50612e12	−1.02163
$724$	−4.25250e12	−0.575203
$725$	0	0
$726$	5.29372e12	0.707206
$727$	9.02097e12	1.19770	0.598850	−	0.800861i	$-0.295624\pi$
$727$	9.02097e12	1.19770	0.598850	+	0.800861i	$0.295624\pi$
$728$	4.04503e12	0.533741
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−9.42381e12	−1.22067
$732$	−6.67913e12	−0.859845
$733$	−9.01918e12	−1.15398	−0.576991	−	0.816750i	$-0.695773\pi$
$733$	−9.01918e12	−1.15398	−0.576991	+	0.816750i	$0.695773\pi$
$734$	2.03614e13	2.58926
$735$	0	0
$736$	9.29057e12	1.16706
$737$	2.98770e12	0.373022
$738$	8.21649e12	1.01961
$739$	−6.50657e12	−0.802513	−0.401257	−	0.915966i	$-0.631426\pi$
$739$	−6.50657e12	−0.802513	−0.401257	+	0.915966i	$0.631426\pi$
$740$	0	0
$741$	1.00980e13	1.23042
$742$	−2.16057e12	−0.261668
$743$	9.86689e12	1.18777	0.593883	−	0.804552i	$-0.297594\pi$
$743$	9.86689e12	1.18777	0.593883	+	0.804552i	$0.297594\pi$
$744$	−5.02939e12	−0.601779
$745$	0	0
$746$	−1.00960e12	−0.119351
$747$	−2.64598e12	−0.310916
$748$	−6.53317e12	−0.763074
$749$	−4.11453e12	−0.477697
$750$	0	0
$751$	−9.02486e12	−1.03529	−0.517644	−	0.855596i	$-0.673191\pi$
$751$	−9.02486e12	−1.03529	−0.517644	+	0.855596i	$0.673191\pi$
$752$	1.01564e12	0.115813
$753$	−1.61966e12	−0.183589
$754$	2.70962e11	0.0305307
$755$	0	0
$756$	1.59119e12	0.177163
$757$	−1.47517e13	−1.63271	−0.816356	−	0.577549i	$-0.804009\pi$
$757$	−1.47517e13	−1.63271	−0.816356	+	0.577549i	$0.804009\pi$
$758$	−4.68657e11	−0.0515637
$759$	−2.46548e12	−0.269658
$760$	0	0
$761$	1.39852e13	1.51160	0.755802	−	0.654800i	$-0.227247\pi$
$761$	1.39852e13	1.51160	0.755802	+	0.654800i	$0.227247\pi$
$762$	3.43822e12	0.369434
$763$	−1.99780e12	−0.213399
$764$	−2.10575e13	−2.23607
$765$	0	0
$766$	−1.00317e13	−1.05280
$767$	−7.31905e12	−0.763617
$768$	−3.12899e12	−0.324548
$769$	−1.52150e13	−1.56893	−0.784464	−	0.620175i	$-0.787062\pi$
$769$	−1.52150e13	−1.56893	−0.784464	+	0.620175i	$0.787062\pi$
$770$	0	0
$771$	1.03607e13	1.05595
$772$	1.12941e13	1.14439
$773$	1.35332e13	1.36330	0.681652	−	0.731677i	$-0.261262\pi$
$773$	1.35332e13	1.36330	0.681652	+	0.731677i	$0.261262\pi$
$774$	5.99280e12	0.600199
$775$	0	0
$776$	−5.15297e12	−0.510129
$777$	3.14058e12	0.309112
$778$	1.08385e13	1.06062
$779$	3.85154e13	3.74728
$780$	0	0
$781$	−5.19364e12	−0.499508
$782$	−1.74401e13	−1.66770
$783$	3.55246e10	0.00337755
$784$	1.64970e12	0.155949
$785$	0	0
$786$	1.07534e13	1.00495
$787$	9.11709e12	0.847169	0.423585	−	0.905857i	$-0.360772\pi$
$787$	9.11709e12	0.847169	0.423585	+	0.905857i	$0.360772\pi$
$788$	−4.82146e12	−0.445462
$789$	2.23658e12	0.205465
$790$	0	0
$791$	−8.51773e12	−0.773624
$792$	1.38468e12	0.125051
$793$	1.21657e13	1.09247
$794$	−2.61598e13	−2.33584
$795$	0	0
$796$	1.77003e13	1.56269
$797$	6.45668e12	0.566822	0.283411	−	0.958999i	$-0.408534\pi$
$797$	6.45668e12	0.566822	0.283411	+	0.958999i	$0.408534\pi$
$798$	1.24317e13	1.08522
$799$	−5.71587e12	−0.496160
$800$	0	0
$801$	−3.75526e12	−0.322325
$802$	1.17130e11	0.00999736
$803$	3.97449e12	0.337335
$804$	−8.06358e12	−0.680575
$805$	0	0
$806$	2.74860e13	2.29406
$807$	3.26570e12	0.271048
$808$	−1.76630e13	−1.45785
$809$	−1.53222e13	−1.25763	−0.628815	−	0.777555i	$-0.716460\pi$
$809$	−1.53222e13	−1.25763	−0.628815	+	0.777555i	$0.716460\pi$
$810$	0	0
$811$	7.37601e12	0.598725	0.299363	−	0.954139i	$-0.403226\pi$
$811$	7.37601e12	0.598725	0.299363	+	0.954139i	$0.403226\pi$
$812$	2.00143e11	0.0161562
$813$	3.53762e12	0.283991
$814$	8.20004e12	0.654645
$815$	0	0
$816$	−1.96095e12	−0.154832
$817$	2.80917e13	2.20586
$818$	−2.40972e13	−1.88182
$819$	−2.89828e12	−0.225093
$820$	0	0
$821$	−1.51211e13	−1.16156	−0.580779	−	0.814062i	$-0.697252\pi$
$821$	−1.51211e13	−1.16156	−0.580779	+	0.814062i	$0.697252\pi$
$822$	−9.27514e12	−0.708594
$823$	−4.00688e12	−0.304444	−0.152222	−	0.988346i	$-0.548643\pi$
$823$	−4.00688e12	−0.304444	−0.152222	+	0.988346i	$0.548643\pi$
$824$	1.74377e13	1.31770
$825$	0	0
$826$	−9.01049e12	−0.673501
$827$	1.91371e13	1.42266	0.711331	−	0.702858i	$-0.248093\pi$
$827$	1.91371e13	1.42266	0.711331	+	0.702858i	$0.248093\pi$
$828$	6.65413e12	0.491988
$829$	−9.83691e11	−0.0723375	−0.0361687	−	0.999346i	$-0.511515\pi$
$829$	−9.83691e11	−0.0723375	−0.0361687	+	0.999346i	$0.511515\pi$
$830$	0	0
$831$	−3.52650e12	−0.256531
$832$	2.47112e13	1.78788
$833$	−9.28430e12	−0.668108
$834$	6.34493e12	0.454129
$835$	0	0
$836$	1.94749e13	1.37895
$837$	3.60358e12	0.253787
$838$	−3.42814e13	−2.40138
$839$	2.22885e12	0.155293	0.0776464	−	0.996981i	$-0.475259\pi$
$839$	2.22885e12	0.155293	0.0776464	+	0.996981i	$0.475259\pi$
$840$	0	0
$841$	−1.45027e13	−0.999692
$842$	9.99614e12	0.685374
$843$	−2.32019e12	−0.158234
$844$	−2.01138e13	−1.36444
$845$	0	0
$846$	3.63484e12	0.243960
$847$	−7.12218e12	−0.475486
$848$	−1.01591e12	−0.0674645
$849$	1.87358e12	0.123762
$850$	0	0
$851$	1.31335e13	0.858414
$852$	1.40172e13	0.911348
$853$	2.18337e13	1.41207	0.706036	−	0.708176i	$-0.250482\pi$
$853$	2.18337e13	1.41207	0.706036	+	0.708176i	$0.250482\pi$
$854$	1.49773e13	0.963545
$855$	0	0
$856$	9.66359e12	0.615186
$857$	−9.68361e12	−0.613230	−0.306615	−	0.951834i	$-0.599196\pi$
$857$	−9.68361e12	−0.613230	−0.306615	+	0.951834i	$0.599196\pi$
$858$	−7.56738e12	−0.476708
$859$	2.42005e13	1.51654	0.758271	−	0.651940i	$-0.226044\pi$
$859$	2.42005e13	1.51654	0.758271	+	0.651940i	$0.226044\pi$
$860$	0	0
$861$	−1.10545e13	−0.685527
$862$	6.65210e12	0.410371
$863$	8.31922e12	0.510545	0.255272	−	0.966869i	$-0.417835\pi$
$863$	8.31922e12	0.510545	0.255272	+	0.966869i	$0.417835\pi$
$864$	3.73860e12	0.228243
$865$	0	0
$866$	−8.46711e12	−0.511570
$867$	1.43037e12	0.0859732
$868$	2.03023e13	1.21397
$869$	7.25915e12	0.431814
$870$	0	0
$871$	1.46875e13	0.864699
$872$	4.69213e12	0.274818
$873$	3.69212e12	0.215135
$874$	5.19876e13	3.01369
$875$	0	0
$876$	−1.07268e13	−0.615464
$877$	−5.75721e12	−0.328635	−0.164317	−	0.986408i	$-0.552542\pi$
$877$	−5.75721e12	−0.328635	−0.164317	+	0.986408i	$0.552542\pi$
$878$	−3.95023e13	−2.24335
$879$	−3.85562e12	−0.217843
$880$	0	0
$881$	−1.44674e13	−0.809093	−0.404546	−	0.914517i	$-0.632571\pi$
$881$	−1.44674e13	−0.809093	−0.404546	+	0.914517i	$0.632571\pi$
$882$	5.90408e12	0.328506
$883$	2.35593e12	0.130418	0.0652092	−	0.997872i	$-0.479229\pi$
$883$	2.35593e12	0.130418	0.0652092	+	0.997872i	$0.479229\pi$
$884$	−3.21168e13	−1.76888
$885$	0	0
$886$	−1.00974e13	−0.550503
$887$	8.46394e12	0.459110	0.229555	−	0.973296i	$-0.426273\pi$
$887$	8.46394e12	0.459110	0.229555	+	0.973296i	$0.426273\pi$
$888$	−7.37613e12	−0.398080
$889$	−4.62579e12	−0.248387
$890$	0	0
$891$	−9.92128e11	−0.0527373
$892$	−5.34061e13	−2.82455
$893$	1.70386e13	0.896608
$894$	1.92529e13	1.00804
$895$	0	0
$896$	1.63790e13	0.848988
$897$	−1.21202e13	−0.625092
$898$	2.14426e13	1.10036
$899$	4.53266e11	0.0231438
$900$	0	0
$901$	5.71743e12	0.289028
$902$	−2.88632e13	−1.45183
$903$	−8.06273e12	−0.403540
$904$	2.00052e13	0.996286
$905$	0	0
$906$	−3.43808e13	−1.69527
$907$	−1.03461e13	−0.507627	−0.253813	−	0.967253i	$-0.581685\pi$
$907$	−1.03461e13	−0.507627	−0.253813	+	0.967253i	$0.581685\pi$
$908$	−5.24537e12	−0.256088
$909$	1.26556e13	0.614816
$910$	0	0
$911$	1.71086e13	0.822967	0.411484	−	0.911417i	$-0.365011\pi$
$911$	1.71086e13	0.822967	0.411484	+	0.911417i	$0.365011\pi$
$912$	5.84546e12	0.279797
$913$	9.29487e12	0.442716
$914$	1.76794e13	0.837933
$915$	0	0
$916$	1.87321e13	0.879137
$917$	−1.44676e13	−0.675670
$918$	−7.01802e12	−0.326154
$919$	1.22871e13	0.568239	0.284120	−	0.958789i	$-0.408299\pi$
$919$	1.22871e13	0.568239	0.284120	+	0.958789i	$0.408299\pi$
$920$	0	0
$921$	1.98973e13	0.911226
$922$	−2.23854e13	−1.02018
$923$	−2.55318e13	−1.15791
$924$	−5.58958e12	−0.252264
$925$	0	0
$926$	−2.88114e13	−1.28770
$927$	−1.24942e13	−0.555711
$928$	4.70250e11	0.0208144
$929$	−2.64336e11	−0.0116436	−0.00582178	−	0.999983i	$-0.501853\pi$
$929$	−2.64336e11	−0.0116436	−0.00582178	+	0.999983i	$0.501853\pi$
$930$	0	0
$931$	2.76758e13	1.20733
$932$	6.42423e13	2.78900
$933$	−2.13011e12	−0.0920313
$934$	−1.05555e13	−0.453857
$935$	0	0
$936$	6.80704e12	0.289879
$937$	−4.09385e13	−1.73502	−0.867508	−	0.497424i	$-0.834279\pi$
$937$	−4.09385e13	−1.73502	−0.867508	+	0.497424i	$0.834279\pi$
$938$	1.80817e13	0.762654
$939$	−1.76926e13	−0.742672
$940$	0	0
$941$	−3.12966e13	−1.30120	−0.650601	−	0.759420i	$-0.725483\pi$
$941$	−3.12966e13	−1.30120	−0.650601	+	0.759420i	$0.725483\pi$
$942$	−9.58694e12	−0.396690
$943$	−4.62283e13	−1.90373
$944$	−4.23679e12	−0.173645
$945$	0	0
$946$	−2.10517e13	−0.854628
$947$	2.81260e13	1.13641	0.568203	−	0.822889i	$-0.307639\pi$
$947$	2.81260e13	1.13641	0.568203	+	0.822889i	$0.307639\pi$
$948$	−1.95919e13	−0.787841
$949$	1.95384e13	0.781973
$950$	0	0
$951$	−1.59710e13	−0.633170
$952$	−1.31780e13	−0.519975
$953$	4.17759e13	1.64062	0.820309	−	0.571920i	$-0.193801\pi$
$953$	4.17759e13	1.64062	0.820309	+	0.571920i	$0.193801\pi$
$954$	−3.63584e12	−0.142114
$955$	0	0
$956$	−1.27724e13	−0.494553
$957$	−1.24792e11	−0.00480932
$958$	−1.68741e13	−0.647257
$959$	1.24788e13	0.476419
$960$	0	0
$961$	1.95392e13	0.739011
$962$	4.03111e13	1.51753
$963$	−6.92400e12	−0.259441
$964$	−7.11645e13	−2.65410
$965$	0	0
$966$	−1.49212e13	−0.551323
$967$	6.22122e12	0.228800	0.114400	−	0.993435i	$-0.463505\pi$
$967$	6.22122e12	0.228800	0.114400	+	0.993435i	$0.463505\pi$
$968$	1.67275e13	0.612339
$969$	−3.28975e13	−1.19869
$970$	0	0
$971$	4.60161e12	0.166120	0.0830602	−	0.996545i	$-0.473531\pi$
$971$	4.60161e12	0.166120	0.0830602	+	0.996545i	$0.473531\pi$
$972$	2.67768e12	0.0962188
$973$	−8.53648e12	−0.305331
$974$	−7.63813e13	−2.71939
$975$	0	0
$976$	7.04241e12	0.248426
$977$	−1.25975e13	−0.442344	−0.221172	−	0.975235i	$-0.570988\pi$
$977$	−1.25975e13	−0.442344	−0.221172	+	0.975235i	$0.570988\pi$
$978$	−1.11463e13	−0.389588
$979$	1.31916e13	0.458961
$980$	0	0
$981$	−3.36193e12	−0.115899
$982$	4.12562e13	1.41575
$983$	5.00583e12	0.170996	0.0854978	−	0.996338i	$-0.472752\pi$
$983$	5.00583e12	0.170996	0.0854978	+	0.996338i	$0.472752\pi$
$984$	2.59631e13	0.882833
$985$	0	0
$986$	−8.82743e11	−0.0297432
$987$	−4.89033e12	−0.164025
$988$	9.57380e13	3.19652
$989$	−3.37172e13	−1.12065
$990$	0	0
$991$	4.38829e13	1.44532	0.722660	−	0.691204i	$-0.242919\pi$
$991$	4.38829e13	1.44532	0.722660	+	0.691204i	$0.242919\pi$
$992$	4.77016e13	1.56398
$993$	5.82503e12	0.190119
$994$	−3.14322e13	−1.02126
$995$	0	0
$996$	−2.50861e13	−0.807731
$997$	−2.53017e13	−0.811000	−0.405500	−	0.914095i	$-0.632903\pi$
$997$	−2.53017e13	−0.811000	−0.405500	+	0.914095i	$0.632903\pi$
$998$	3.38010e12	0.107855
$999$	5.28502e12	0.167881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.e.1.1	✓	2
3.2	odd	2		225.10.a.l.1.2		2
5.2	odd	4		75.10.b.g.49.1		4
5.3	odd	4		75.10.b.g.49.4		4
5.4	even	2		75.10.a.h.1.2	yes	2
15.2	even	4		225.10.b.j.199.4		4
15.8	even	4		225.10.b.j.199.1		4
15.14	odd	2		225.10.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.10.a.e.1.1	✓	2	1.1	even	1	trivial
75.10.a.h.1.2	yes	2	5.4	even	2
75.10.b.g.49.1		4	5.2	odd	4
75.10.b.g.49.4		4	5.3	odd	4
225.10.a.g.1.1		2	15.14	odd	2
225.10.a.l.1.2		2	3.2	odd	2
225.10.b.j.199.1		4	15.8	even	4
225.10.b.j.199.4		4	15.2	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-35.7764 q^{2} +81.0000 q^{3} +767.950 q^{4} -2897.89 q^{6} +3898.82 q^{7} -9156.97 q^{8} +6561.00 q^{9} +O(q^{10})\)	\(q-35.7764 q^{2} +81.0000 q^{3} +767.950 q^{4} -2897.89 q^{6} +3898.82 q^{7} -9156.97 q^{8} +6561.00 q^{9} -23047.7 q^{11} +62203.9 q^{12} -113302. q^{13} -139486. q^{14} -65587.2 q^{16} +369116. q^{17} -234729. q^{18} -1.10031e6 q^{19} +315805. q^{21} +824563. q^{22} +1.32065e6 q^{23} -741714. q^{24} +4.05353e6 q^{26} +531441. q^{27} +2.99410e6 q^{28} +66845.9 q^{29} +6.78077e6 q^{31} +7.03484e6 q^{32} -1.86686e6 q^{33} -1.32056e7 q^{34} +5.03852e6 q^{36} +9.94470e6 q^{37} +3.93651e7 q^{38} -9.17744e6 q^{39} -3.50042e7 q^{41} -1.12984e7 q^{42} -2.55307e7 q^{43} -1.76995e7 q^{44} -4.72481e7 q^{46} -1.54853e7 q^{47} -5.31256e6 q^{48} -2.51528e7 q^{49} +2.98984e7 q^{51} -8.70100e7 q^{52} +1.54895e7 q^{53} -1.90130e7 q^{54} -3.57014e7 q^{56} -8.91251e7 q^{57} -2.39150e6 q^{58} +6.45979e7 q^{59} -1.07375e8 q^{61} -2.42591e8 q^{62} +2.55802e7 q^{63} -2.18101e8 q^{64} +6.67896e7 q^{66} -1.29631e8 q^{67} +2.83463e8 q^{68} +1.06973e8 q^{69} +2.25343e8 q^{71} -6.00789e7 q^{72} -1.72446e8 q^{73} -3.55785e8 q^{74} -8.44983e8 q^{76} -8.98589e7 q^{77} +3.28336e8 q^{78} -3.14962e8 q^{79} +4.30467e7 q^{81} +1.25232e9 q^{82} -4.03289e8 q^{83} +2.42522e8 q^{84} +9.13397e8 q^{86} +5.41452e6 q^{87} +2.11047e8 q^{88} -5.72362e8 q^{89} -4.41743e8 q^{91} +1.01419e9 q^{92} +5.49242e8 q^{93} +5.54007e8 q^{94} +5.69822e8 q^{96} +5.62737e8 q^{97} +8.99875e8 q^{98} -1.51216e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 36 q^{2} + 162 q^{3} + 256 q^{4} - 2916 q^{6} + 3318 q^{7} - 8928 q^{8} + 13122 q^{9}+O(q^{10}) \) 2 * q - 36 * q^2 + 162 * q^3 + 256 * q^4 - 2916 * q^6 + 3318 * q^7 - 8928 * q^8 + 13122 * q^9	\( 2 q - 36 q^{2} + 162 q^{3} + 256 q^{4} - 2916 q^{6} + 3318 q^{7} - 8928 q^{8} + 13122 q^{9} + 24228 q^{11} + 20736 q^{12} - 90934 q^{13} - 139356 q^{14} + 196480 q^{16} + 124236 q^{17} - 236196 q^{18} - 1348846 q^{19} + 268758 q^{21} + 813992 q^{22} + 330444 q^{23} - 723168 q^{24} + 4048524 q^{26} + 1062882 q^{27} + 3291456 q^{28} - 4283172 q^{29} + 1176582 q^{31} + 6859008 q^{32} + 1962468 q^{33} - 13150888 q^{34} + 1679616 q^{36} + 5207668 q^{37} + 39420684 q^{38} - 7365654 q^{39} - 28888488 q^{41} - 11287836 q^{42} + 13370630 q^{43} - 41902272 q^{44} - 47026728 q^{46} - 66459060 q^{47} + 15914880 q^{48} - 65169020 q^{49} + 10063116 q^{51} - 98461184 q^{52} + 3809664 q^{53} - 19131876 q^{54} - 35834400 q^{56} - 109256526 q^{57} - 1418792 q^{58} - 115150284 q^{59} - 231494410 q^{61} - 241338204 q^{62} + 21769398 q^{63} - 352239616 q^{64} + 65933352 q^{66} - 319773234 q^{67} + 408829248 q^{68} + 26765964 q^{69} + 111792024 q^{71} - 58576608 q^{72} - 69141676 q^{73} - 354726216 q^{74} - 717744704 q^{76} - 117317844 q^{77} + 327930444 q^{78} - 334446000 q^{79} + 86093442 q^{81} + 1250955136 q^{82} + 149433012 q^{83} + 266607936 q^{84} + 904698468 q^{86} - 346936932 q^{87} + 221871552 q^{88} - 246671136 q^{89} - 454735218 q^{91} + 1521131328 q^{92} + 95303142 q^{93} + 565405736 q^{94} + 555579648 q^{96} + 696298546 q^{97} + 908823384 q^{98} + 158959908 q^{99}+O(q^{100}) \) 2 * q - 36 * q^2 + 162 * q^3 + 256 * q^4 - 2916 * q^6 + 3318 * q^7 - 8928 * q^8 + 13122 * q^9 + 24228 * q^11 + 20736 * q^12 - 90934 * q^13 - 139356 * q^14 + 196480 * q^16 + 124236 * q^17 - 236196 * q^18 - 1348846 * q^19 + 268758 * q^21 + 813992 * q^22 + 330444 * q^23 - 723168 * q^24 + 4048524 * q^26 + 1062882 * q^27 + 3291456 * q^28 - 4283172 * q^29 + 1176582 * q^31 + 6859008 * q^32 + 1962468 * q^33 - 13150888 * q^34 + 1679616 * q^36 + 5207668 * q^37 + 39420684 * q^38 - 7365654 * q^39 - 28888488 * q^41 - 11287836 * q^42 + 13370630 * q^43 - 41902272 * q^44 - 47026728 * q^46 - 66459060 * q^47 + 15914880 * q^48 - 65169020 * q^49 + 10063116 * q^51 - 98461184 * q^52 + 3809664 * q^53 - 19131876 * q^54 - 35834400 * q^56 - 109256526 * q^57 - 1418792 * q^58 - 115150284 * q^59 - 231494410 * q^61 - 241338204 * q^62 + 21769398 * q^63 - 352239616 * q^64 + 65933352 * q^66 - 319773234 * q^67 + 408829248 * q^68 + 26765964 * q^69 + 111792024 * q^71 - 58576608 * q^72 - 69141676 * q^73 - 354726216 * q^74 - 717744704 * q^76 - 117317844 * q^77 + 327930444 * q^78 - 334446000 * q^79 + 86093442 * q^81 + 1250955136 * q^82 + 149433012 * q^83 + 266607936 * q^84 + 904698468 * q^86 - 346936932 * q^87 + 221871552 * q^88 - 246671136 * q^89 - 454735218 * q^91 + 1521131328 * q^92 + 95303142 * q^93 + 565405736 * q^94 + 555579648 * q^96 + 696298546 * q^97 + 908823384 * q^98 + 158959908 * q^99

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