LMFDB - Embedded newform 75.10.a.f.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.26209$ 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)

$f(q)$	$=$	$q-38.7863 q^{2} +81.0000 q^{3} +992.374 q^{4} -3141.69 q^{6} -12272.1 q^{7} -18631.9 q^{8} +6561.00 q^{9} +O(q^{10})$	\(q-38.7863 q^{2} +81.0000 q^{3} +992.374 q^{4} -3141.69 q^{6} -12272.1 q^{7} -18631.9 q^{8} +6561.00 q^{9} -65897.9 q^{11} +80382.3 q^{12} +112300. q^{13} +475990. q^{14} +214567. q^{16} -96634.7 q^{17} -254477. q^{18} -181332. q^{19} -994042. q^{21} +2.55593e6 q^{22} -244145. q^{23} -1.50919e6 q^{24} -4.35569e6 q^{26} +531441. q^{27} -1.21785e7 q^{28} -5.26461e6 q^{29} +1.83457e6 q^{31} +1.21730e6 q^{32} -5.33773e6 q^{33} +3.74810e6 q^{34} +6.51097e6 q^{36} -6.36194e6 q^{37} +7.03318e6 q^{38} +9.09628e6 q^{39} +1.57111e6 q^{41} +3.85552e7 q^{42} +1.99504e7 q^{43} -6.53953e7 q^{44} +9.46946e6 q^{46} -3.00961e7 q^{47} +1.73799e7 q^{48} +1.10251e8 q^{49} -7.82741e6 q^{51} +1.11443e8 q^{52} +2.57306e6 q^{53} -2.06126e7 q^{54} +2.28653e8 q^{56} -1.46879e7 q^{57} +2.04195e8 q^{58} -1.19004e8 q^{59} +1.92875e8 q^{61} -7.11559e7 q^{62} -8.05174e7 q^{63} -1.57073e8 q^{64} +2.07030e8 q^{66} +1.20193e8 q^{67} -9.58978e7 q^{68} -1.97757e7 q^{69} -699549. q^{71} -1.22244e8 q^{72} +8.91287e7 q^{73} +2.46756e8 q^{74} -1.79949e8 q^{76} +8.08707e8 q^{77} -3.52811e8 q^{78} +4.31205e8 q^{79} +4.30467e7 q^{81} -6.09375e7 q^{82} +1.69761e7 q^{83} -9.86461e8 q^{84} -7.73800e8 q^{86} -4.26433e8 q^{87} +1.22780e9 q^{88} -3.09863e6 q^{89} -1.37816e9 q^{91} -2.42283e8 q^{92} +1.48600e8 q^{93} +1.16732e9 q^{94} +9.86009e7 q^{96} -5.72609e8 q^{97} -4.27624e9 q^{98} -4.32356e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 31 q^{2} + 162 q^{3} + 541 q^{4} - 2511 q^{6} - 14112 q^{7} - 26133 q^{8} + 13122 q^{9}+O(q^{10}) $ 2 * q - 31 * q^2 + 162 * q^3 + 541 * q^4 - 2511 * q^6 - 14112 * q^7 - 26133 * q^8 + 13122 * q^9	\( 2 q - 31 q^{2} + 162 q^{3} + 541 q^{4} - 2511 q^{6} - 14112 q^{7} - 26133 q^{8} + 13122 q^{9} - 21512 q^{11} + 43821 q^{12} - 24284 q^{13} + 461664 q^{14} + 387265 q^{16} + 156956 q^{17} - 203391 q^{18} - 95896 q^{19} - 1143072 q^{21} + 2901532 q^{22} + 735264 q^{23} - 2116773 q^{24} - 5419166 q^{26} + 1062882 q^{27} - 11348064 q^{28} - 2678212 q^{29} + 10782432 q^{31} + 6402523 q^{32} - 1742472 q^{33} + 5722622 q^{34} + 3549501 q^{36} - 21968332 q^{37} + 7698404 q^{38} - 1967004 q^{39} + 26060372 q^{41} + 37394784 q^{42} + 7191160 q^{43} - 85429972 q^{44} + 17095392 q^{46} + 31580240 q^{47} + 31368465 q^{48} + 73282930 q^{49} + 12713436 q^{51} + 173093786 q^{52} - 3131116 q^{53} - 16474671 q^{54} + 242454240 q^{56} - 7767576 q^{57} + 224332958 q^{58} - 35494664 q^{59} + 341497340 q^{61} - 1485504 q^{62} - 92588832 q^{63} - 205120471 q^{64} + 235024092 q^{66} + 288195816 q^{67} - 210362042 q^{68} + 59556384 q^{69} + 210286064 q^{71} - 171458613 q^{72} + 232663084 q^{73} + 125240234 q^{74} - 218512364 q^{76} + 727042176 q^{77} - 438952446 q^{78} - 24755040 q^{79} + 86093442 q^{81} + 129742346 q^{82} + 372082152 q^{83} - 919193184 q^{84} - 873146372 q^{86} - 216935172 q^{87} + 894861492 q^{88} - 427639116 q^{89} - 1126859328 q^{91} - 684362592 q^{92} + 873376992 q^{93} + 1647543896 q^{94} + 518604363 q^{96} - 1771658884 q^{97} - 4564085351 q^{98} - 141140232 q^{99}+O(q^{100}) \) 2 * q - 31 * q^2 + 162 * q^3 + 541 * q^4 - 2511 * q^6 - 14112 * q^7 - 26133 * q^8 + 13122 * q^9 - 21512 * q^11 + 43821 * q^12 - 24284 * q^13 + 461664 * q^14 + 387265 * q^16 + 156956 * q^17 - 203391 * q^18 - 95896 * q^19 - 1143072 * q^21 + 2901532 * q^22 + 735264 * q^23 - 2116773 * q^24 - 5419166 * q^26 + 1062882 * q^27 - 11348064 * q^28 - 2678212 * q^29 + 10782432 * q^31 + 6402523 * q^32 - 1742472 * q^33 + 5722622 * q^34 + 3549501 * q^36 - 21968332 * q^37 + 7698404 * q^38 - 1967004 * q^39 + 26060372 * q^41 + 37394784 * q^42 + 7191160 * q^43 - 85429972 * q^44 + 17095392 * q^46 + 31580240 * q^47 + 31368465 * q^48 + 73282930 * q^49 + 12713436 * q^51 + 173093786 * q^52 - 3131116 * q^53 - 16474671 * q^54 + 242454240 * q^56 - 7767576 * q^57 + 224332958 * q^58 - 35494664 * q^59 + 341497340 * q^61 - 1485504 * q^62 - 92588832 * q^63 - 205120471 * q^64 + 235024092 * q^66 + 288195816 * q^67 - 210362042 * q^68 + 59556384 * q^69 + 210286064 * q^71 - 171458613 * q^72 + 232663084 * q^73 + 125240234 * q^74 - 218512364 * q^76 + 727042176 * q^77 - 438952446 * q^78 - 24755040 * q^79 + 86093442 * q^81 + 129742346 * q^82 + 372082152 * q^83 - 919193184 * q^84 - 873146372 * q^86 - 216935172 * q^87 + 894861492 * q^88 - 427639116 * q^89 - 1126859328 * q^91 - 684362592 * q^92 + 873376992 * q^93 + 1647543896 * q^94 + 518604363 * q^96 - 1771658884 * q^97 - 4564085351 * q^98 - 141140232 * 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 $

$2$	−38.7863	−1.71413	−0.857063	−	0.515211i	$-0.827714\pi$
$2$	−38.7863	−1.71413	−0.857063	+	0.515211i	$0.827714\pi$
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	992.374	1.93823
$5$	0	0
$5$	0	0
$6$	−3141.69	−0.989652
$7$	−12272.1	−1.93187	−0.965936	−	0.258780i	$-0.916680\pi$
$7$	−12272.1	−1.93187	−0.965936	+	0.258780i	$0.916680\pi$
$8$	−18631.9	−1.60825
$9$	6561.00	0.333333
$10$	0	0
$11$	−65897.9	−1.35708	−0.678538	−	0.734565i	$-0.737386\pi$
$11$	−65897.9	−1.35708	−0.678538	+	0.734565i	$0.737386\pi$
$12$	80382.3	1.11904
$13$	112300.	1.09052	0.545260	−	0.838267i	$-0.316431\pi$
$13$	112300.	1.09052	0.545260	+	0.838267i	$0.316431\pi$
$14$	475990.	3.31147
$15$	0	0
$16$	214567.	0.818508
$17$	−96634.7	−0.280616	−0.140308	−	0.990108i	$-0.544809\pi$
$17$	−96634.7	−0.280616	−0.140308	+	0.990108i	$0.544809\pi$
$18$	−254477.	−0.571376
$19$	−181332.	−0.319214	−0.159607	−	0.987181i	$-0.551023\pi$
$19$	−181332.	−0.319214	−0.159607	+	0.987181i	$0.551023\pi$
$20$	0	0
$21$	−994042.	−1.11537
$22$	2.55593e6	2.32620
$23$	−244145.	−0.181916	−0.0909582	−	0.995855i	$-0.528993\pi$
$23$	−244145.	−0.181916	−0.0909582	+	0.995855i	$0.528993\pi$
$24$	−1.50919e6	−0.928521
$25$	0	0
$26$	−4.35569e6	−1.86929
$27$	531441.	0.192450
$28$	−1.21785e7	−3.74442
$29$	−5.26461e6	−1.38221	−0.691107	−	0.722752i	$-0.742877\pi$
$29$	−5.26461e6	−1.38221	−0.691107	+	0.722752i	$0.742877\pi$
$30$	0	0
$31$	1.83457e6	0.356784	0.178392	−	0.983959i	$-0.442910\pi$
$31$	1.83457e6	0.356784	0.178392	+	0.983959i	$0.442910\pi$
$32$	1.21730e6	0.205221
$33$	−5.33773e6	−0.783508
$34$	3.74810e6	0.481012
$35$	0	0
$36$	6.51097e6	0.646077
$37$	−6.36194e6	−0.558061	−0.279030	−	0.960282i	$-0.590013\pi$
$37$	−6.36194e6	−0.558061	−0.279030	+	0.960282i	$0.590013\pi$
$38$	7.03318e6	0.547174
$39$	9.09628e6	0.629612
$40$	0	0
$41$	1.57111e6	0.0868319	0.0434159	−	0.999057i	$-0.486176\pi$
$41$	1.57111e6	0.0868319	0.0434159	+	0.999057i	$0.486176\pi$
$42$	3.85552e7	1.91188
$43$	1.99504e7	0.889903	0.444952	−	0.895555i	$-0.353221\pi$
$43$	1.99504e7	0.889903	0.444952	+	0.895555i	$0.353221\pi$
$44$	−6.53953e7	−2.63033
$45$	0	0
$46$	9.46946e6	0.311828
$47$	−3.00961e7	−0.899643	−0.449821	−	0.893119i	$-0.648512\pi$
$47$	−3.00961e7	−0.899643	−0.449821	+	0.893119i	$0.648512\pi$
$48$	1.73799e7	0.472566
$49$	1.10251e8	2.73213
$50$	0	0
$51$	−7.82741e6	−0.162014
$52$	1.11443e8	2.11368
$53$	2.57306e6	0.0447929	0.0223964	−	0.999749i	$-0.492870\pi$
$53$	2.57306e6	0.0447929	0.0223964	+	0.999749i	$0.492870\pi$
$54$	−2.06126e7	−0.329884
$55$	0	0
$56$	2.28653e8	3.10693
$57$	−1.46879e7	−0.184299
$58$	2.04195e8	2.36929
$59$	−1.19004e8	−1.27858	−0.639290	−	0.768965i	$-0.720772\pi$
$59$	−1.19004e8	−1.27858	−0.639290	+	0.768965i	$0.720772\pi$
$60$	0	0
$61$	1.92875e8	1.78358	0.891790	−	0.452449i	$-0.149450\pi$
$61$	1.92875e8	1.78358	0.891790	+	0.452449i	$0.149450\pi$
$62$	−7.11559e7	−0.611573
$63$	−8.05174e7	−0.643958
$64$	−1.57073e8	−1.17028
$65$	0	0
$66$	2.07030e8	1.34303
$67$	1.20193e8	0.728691	0.364345	−	0.931264i	$-0.381293\pi$
$67$	1.20193e8	0.728691	0.364345	+	0.931264i	$0.381293\pi$
$68$	−9.58978e7	−0.543899
$69$	−1.97757e7	−0.105030
$70$	0	0
$71$	−699549.	−0.00326705	−0.00163352	−	0.999999i	$-0.500520\pi$
$71$	−699549.	−0.00326705	−0.00163352	+	0.999999i	$0.500520\pi$
$72$	−1.22244e8	−0.536082
$73$	8.91287e7	0.367337	0.183669	−	0.982988i	$-0.441203\pi$
$73$	8.91287e7	0.367337	0.183669	+	0.982988i	$0.441203\pi$
$74$	2.46756e8	0.956587
$75$	0	0
$76$	−1.79949e8	−0.618711
$77$	8.08707e8	2.62170
$78$	−3.52811e8	−1.07924
$79$	4.31205e8	1.24555	0.622776	−	0.782400i	$-0.286005\pi$
$79$	4.31205e8	1.24555	0.622776	+	0.782400i	$0.286005\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	−6.09375e7	−0.148841
$83$	1.69761e7	0.0392633	0.0196316	−	0.999807i	$-0.493751\pi$
$83$	1.69761e7	0.0392633	0.0196316	+	0.999807i	$0.493751\pi$
$84$	−9.86461e8	−2.16184
$85$	0	0
$86$	−7.73800e8	−1.52541
$87$	−4.26433e8	−0.798022
$88$	1.22780e9	2.18251
$89$	−3.09863e6	−0.00523498	−0.00261749	−	0.999997i	$-0.500833\pi$
$89$	−3.09863e6	−0.00523498	−0.00261749	+	0.999997i	$0.500833\pi$
$90$	0	0
$91$	−1.37816e9	−2.10675
$92$	−2.42283e8	−0.352596
$93$	1.48600e8	0.205989
$94$	1.16732e9	1.54210
$95$	0	0
$96$	9.86009e7	0.118484
$97$	−5.72609e8	−0.656727	−0.328364	−	0.944551i	$-0.606497\pi$
$97$	−5.72609e8	−0.656727	−0.328364	+	0.944551i	$0.606497\pi$
$98$	−4.27624e9	−4.68322
$99$	−4.32356e8	−0.452359
$100$	0	0
$101$	1.78445e9	1.70631	0.853156	−	0.521655i	$-0.174685\pi$
$101$	1.78445e9	1.70631	0.853156	+	0.521655i	$0.174685\pi$
$102$	3.03596e8	0.277712
$103$	−1.16632e9	−1.02106	−0.510528	−	0.859861i	$-0.670550\pi$
$103$	−1.16632e9	−1.02106	−0.510528	+	0.859861i	$0.670550\pi$
$104$	−2.09236e9	−1.75383
$105$	0	0
$106$	−9.97995e7	−0.0767807
$107$	1.72229e9	1.27022	0.635109	−	0.772422i	$-0.280955\pi$
$107$	1.72229e9	1.27022	0.635109	+	0.772422i	$0.280955\pi$
$108$	5.27388e8	0.373013
$109$	2.08468e9	1.41456	0.707279	−	0.706934i	$-0.249922\pi$
$109$	2.08468e9	1.41456	0.707279	+	0.706934i	$0.249922\pi$
$110$	0	0
$111$	−5.15317e8	−0.322197
$112$	−2.63319e9	−1.58125
$113$	1.81995e9	1.05004	0.525020	−	0.851090i	$-0.324058\pi$
$113$	1.81995e9	1.05004	0.525020	+	0.851090i	$0.324058\pi$
$114$	5.69688e8	0.315911
$115$	0	0
$116$	−5.22446e9	−2.67905
$117$	7.36799e8	0.363507
$118$	4.61573e9	2.19165
$119$	1.18591e9	0.542115
$120$	0	0
$121$	1.98458e9	0.841656
$122$	−7.48092e9	−3.05728
$123$	1.27260e8	0.0501324
$124$	1.82058e9	0.691530
$125$	0	0
$126$	3.12297e9	1.10382
$127$	3.74913e9	1.27883	0.639417	−	0.768860i	$-0.279176\pi$
$127$	3.74913e9	1.27883	0.639417	+	0.768860i	$0.279176\pi$
$128$	5.46900e9	1.80079
$129$	1.61598e9	0.513786
$130$	0	0
$131$	−1.96598e9	−0.583256	−0.291628	−	0.956532i	$-0.594197\pi$
$131$	−1.96598e9	−0.583256	−0.291628	+	0.956532i	$0.594197\pi$
$132$	−5.29702e9	−1.51862
$133$	2.22532e9	0.616682
$134$	−4.66184e9	−1.24907
$135$	0	0
$136$	1.80049e9	0.451300
$137$	3.63223e8	0.0880909	0.0440455	−	0.999030i	$-0.485975\pi$
$137$	3.63223e8	0.0880909	0.0440455	+	0.999030i	$0.485975\pi$
$138$	7.67026e8	0.180034
$139$	4.55580e9	1.03514	0.517569	−	0.855642i	$-0.326837\pi$
$139$	4.55580e9	1.03514	0.517569	+	0.855642i	$0.326837\pi$
$140$	0	0
$141$	−2.43779e9	−0.519409
$142$	2.71329e7	0.00560014
$143$	−7.40032e9	−1.47992
$144$	1.40777e9	0.272836
$145$	0	0
$146$	−3.45697e9	−0.629662
$147$	8.93036e9	1.57740
$148$	−6.31342e9	−1.08165
$149$	8.79830e9	1.46238	0.731191	−	0.682173i	$-0.238965\pi$
$149$	8.79830e9	1.46238	0.731191	+	0.682173i	$0.238965\pi$
$150$	0	0
$151$	−2.70702e8	−0.0423736	−0.0211868	−	0.999776i	$-0.506744\pi$
$151$	−2.70702e8	−0.0423736	−0.0211868	+	0.999776i	$0.506744\pi$
$152$	3.37856e9	0.513376
$153$	−6.34020e8	−0.0935388
$154$	−3.13667e10	−4.49392
$155$	0	0
$156$	9.02692e9	1.22033
$157$	−9.73240e9	−1.27841	−0.639207	−	0.769035i	$-0.720737\pi$
$157$	−9.73240e9	−1.27841	−0.639207	+	0.769035i	$0.720737\pi$
$158$	−1.67248e10	−2.13504
$159$	2.08418e8	0.0258612
$160$	0	0
$161$	2.99617e9	0.351439
$162$	−1.66962e9	−0.190459
$163$	−7.76823e9	−0.861941	−0.430971	−	0.902366i	$-0.641829\pi$
$163$	−7.76823e9	−0.861941	−0.430971	+	0.902366i	$0.641829\pi$
$164$	1.55913e9	0.168300
$165$	0	0
$166$	−6.58440e8	−0.0673023
$167$	8.41748e9	0.837448	0.418724	−	0.908114i	$-0.362478\pi$
$167$	8.41748e9	0.837448	0.418724	+	0.908114i	$0.362478\pi$
$168$	1.85209e10	1.79379
$169$	2.00674e9	0.189235
$170$	0	0
$171$	−1.18972e9	−0.106405
$172$	1.97982e10	1.72484
$173$	−3.59838e8	−0.0305422	−0.0152711	−	0.999883i	$-0.504861\pi$
$173$	−3.59838e8	−0.0305422	−0.0152711	+	0.999883i	$0.504861\pi$
$174$	1.65398e10	1.36791
$175$	0	0
$176$	−1.41395e10	−1.11078
$177$	−9.63934e9	−0.738189
$178$	1.20184e8	0.00897342
$179$	−8.65338e9	−0.630009	−0.315005	−	0.949090i	$-0.602006\pi$
$179$	−8.65338e9	−0.630009	−0.315005	+	0.949090i	$0.602006\pi$
$180$	0	0
$181$	6.09713e9	0.422252	0.211126	−	0.977459i	$-0.432287\pi$
$181$	6.09713e9	0.422252	0.211126	+	0.977459i	$0.432287\pi$
$182$	5.34535e10	3.61123
$183$	1.56229e10	1.02975
$184$	4.54888e9	0.292566
$185$	0	0
$186$	−5.76363e9	−0.353092
$187$	6.36802e9	0.380818
$188$	−2.98666e10	−1.74371
$189$	−6.52191e9	−0.371789
$190$	0	0
$191$	−2.96238e10	−1.61061	−0.805306	−	0.592859i	$-0.797999\pi$
$191$	−2.96238e10	−1.61061	−0.805306	+	0.592859i	$0.797999\pi$
$192$	−1.27229e10	−0.675662
$193$	4.63438e8	0.0240427	0.0120214	−	0.999928i	$-0.496173\pi$
$193$	4.63438e8	0.0240427	0.0120214	+	0.999928i	$0.496173\pi$
$194$	2.22094e10	1.12571
$195$	0	0
$196$	1.09411e11	5.29550
$197$	−1.23414e10	−0.583803	−0.291902	−	0.956448i	$-0.594288\pi$
$197$	−1.23414e10	−0.583803	−0.291902	+	0.956448i	$0.594288\pi$
$198$	1.67695e10	0.775400
$199$	−1.00354e10	−0.453624	−0.226812	−	0.973939i	$-0.572830\pi$
$199$	−1.00354e10	−0.453624	−0.226812	+	0.973939i	$0.572830\pi$
$200$	0	0
$201$	9.73565e9	0.420710
$202$	−6.92122e10	−2.92484
$203$	6.46080e10	2.67026
$204$	−7.76772e9	−0.314020
$205$	0	0
$206$	4.52371e10	1.75022
$207$	−1.60183e9	−0.0606388
$208$	2.40958e10	0.892599
$209$	1.19494e10	0.433198
$210$	0	0
$211$	−3.56268e10	−1.23739	−0.618693	−	0.785633i	$-0.712338\pi$
$211$	−3.56268e10	−1.23739	−0.618693	+	0.785633i	$0.712338\pi$
$212$	2.55344e9	0.0868189
$213$	−5.66635e7	−0.00188623
$214$	−6.68011e10	−2.17732
$215$	0	0
$216$	−9.90176e9	−0.309507
$217$	−2.25140e10	−0.689262
$218$	−8.08571e10	−2.42473
$219$	7.21943e9	0.212082
$220$	0	0
$221$	−1.08521e10	−0.306018
$222$	1.99872e10	0.552286
$223$	3.41646e10	0.925133	0.462566	−	0.886585i	$-0.346929\pi$
$223$	3.41646e10	0.925133	0.462566	+	0.886585i	$0.346929\pi$
$224$	−1.49388e10	−0.396460
$225$	0	0
$226$	−7.05889e10	−1.79990
$227$	−3.64936e10	−0.912221	−0.456110	−	0.889923i	$-0.650758\pi$
$227$	−3.64936e10	−0.912221	−0.456110	+	0.889923i	$0.650758\pi$
$228$	−1.45759e10	−0.357213
$229$	3.87446e10	0.931005	0.465502	−	0.885047i	$-0.345874\pi$
$229$	3.87446e10	0.931005	0.465502	+	0.885047i	$0.345874\pi$
$230$	0	0
$231$	6.55052e10	1.51364
$232$	9.80898e10	2.22294
$233$	2.68717e10	0.597302	0.298651	−	0.954362i	$-0.403463\pi$
$233$	2.68717e10	0.597302	0.298651	+	0.954362i	$0.403463\pi$
$234$	−2.85777e10	−0.623097
$235$	0	0
$236$	−1.18097e11	−2.47818
$237$	3.49276e10	0.719120
$238$	−4.59971e10	−0.929254
$239$	1.34711e10	0.267063	0.133531	−	0.991045i	$-0.457368\pi$
$239$	1.34711e10	0.267063	0.133531	+	0.991045i	$0.457368\pi$
$240$	0	0
$241$	−1.85860e10	−0.354902	−0.177451	−	0.984130i	$-0.556785\pi$
$241$	−1.85860e10	−0.354902	−0.177451	+	0.984130i	$0.556785\pi$
$242$	−7.69745e10	−1.44271
$243$	3.48678e9	0.0641500
$244$	1.91405e11	3.45699
$245$	0	0
$246$	−4.93593e9	−0.0859333
$247$	−2.03635e10	−0.348110
$248$	−3.41815e10	−0.573797
$249$	1.37506e9	0.0226687
$250$	0	0
$251$	4.30327e10	0.684332	0.342166	−	0.939640i	$-0.388839\pi$
$251$	4.30327e10	0.684332	0.342166	+	0.939640i	$0.388839\pi$
$252$	−7.99034e10	−1.24814
$253$	1.60886e10	0.246875
$254$	−1.45415e11	−2.19208
$255$	0	0
$256$	−1.31701e11	−1.91650
$257$	6.26858e8	0.00896335	0.00448168	−	0.999990i	$-0.498573\pi$
$257$	6.26858e8	0.00896335	0.00448168	+	0.999990i	$0.498573\pi$
$258$	−6.26778e10	−0.880694
$259$	7.80745e10	1.07810
$260$	0	0
$261$	−3.45411e10	−0.460738
$262$	7.62532e10	0.999774
$263$	−6.21446e10	−0.800944	−0.400472	−	0.916309i	$-0.631154\pi$
$263$	−6.21446e10	−0.800944	−0.400472	+	0.916309i	$0.631154\pi$
$264$	9.94521e10	1.26007
$265$	0	0
$266$	−8.63120e10	−1.05707
$267$	−2.50989e8	−0.00302242
$268$	1.19277e11	1.41237
$269$	−5.44876e10	−0.634472	−0.317236	−	0.948347i	$-0.602755\pi$
$269$	−5.44876e10	−0.634472	−0.317236	+	0.948347i	$0.602755\pi$
$270$	0	0
$271$	−9.74930e10	−1.09802	−0.549012	−	0.835815i	$-0.684996\pi$
$271$	−9.74930e10	−1.09802	−0.549012	+	0.835815i	$0.684996\pi$
$272$	−2.07346e10	−0.229687
$273$	−1.11631e11	−1.21633
$274$	−1.40881e10	−0.150999
$275$	0	0
$276$	−1.96249e10	−0.203571
$277$	−1.43195e11	−1.46140	−0.730702	−	0.682697i	$-0.760807\pi$
$277$	−1.43195e11	−1.46140	−0.730702	+	0.682697i	$0.760807\pi$
$278$	−1.76702e11	−1.77436
$279$	1.20366e10	0.118928
$280$	0	0
$281$	6.51280e10	0.623146	0.311573	−	0.950222i	$-0.399144\pi$
$281$	6.51280e10	0.623146	0.311573	+	0.950222i	$0.399144\pi$
$282$	9.45526e10	0.890333
$283$	1.65431e11	1.53312	0.766561	−	0.642171i	$-0.221966\pi$
$283$	1.65431e11	1.53312	0.766561	+	0.642171i	$0.221966\pi$
$284$	−6.94214e8	−0.00633229
$285$	0	0
$286$	2.87031e11	2.53677
$287$	−1.92808e10	−0.167748
$288$	7.98667e9	0.0684069
$289$	−1.09250e11	−0.921254
$290$	0	0
$291$	−4.63813e10	−0.379162
$292$	8.84490e10	0.711984
$293$	−5.64261e10	−0.447277	−0.223638	−	0.974672i	$-0.571793\pi$
$293$	−5.64261e10	−0.447277	−0.223638	+	0.974672i	$0.571793\pi$
$294$	−3.46375e11	−2.70386
$295$	0	0
$296$	1.18535e11	0.897499
$297$	−3.50208e10	−0.261169
$298$	−3.41253e11	−2.50671
$299$	−2.74174e10	−0.198384
$300$	0	0
$301$	−2.44833e11	−1.71918
$302$	1.04995e10	0.0726338
$303$	1.44541e11	0.985140
$304$	−3.89078e10	−0.261279
$305$	0	0
$306$	2.45913e10	0.160337
$307$	−1.79360e10	−0.115240	−0.0576200	−	0.998339i	$-0.518351\pi$
$307$	−1.79360e10	−0.115240	−0.0576200	+	0.998339i	$0.518351\pi$
$308$	8.02540e11	5.08146
$309$	−9.44717e10	−0.589507
$310$	0	0
$311$	3.19953e11	1.93938	0.969692	−	0.244330i	$-0.0785680\pi$
$311$	3.19953e11	1.93938	0.969692	+	0.244330i	$0.0785680\pi$
$312$	−1.69481e11	−1.01257
$313$	3.16987e11	1.86677	0.933387	−	0.358872i	$-0.116838\pi$
$313$	3.16987e11	1.86677	0.933387	+	0.358872i	$0.116838\pi$
$314$	3.77483e11	2.19136
$315$	0	0
$316$	4.27917e11	2.41417
$317$	3.08157e11	1.71398	0.856990	−	0.515334i	$-0.172332\pi$
$317$	3.08157e11	1.71398	0.856990	+	0.515334i	$0.172332\pi$
$318$	−8.08376e9	−0.0443293
$319$	3.46927e11	1.87577
$320$	0	0
$321$	1.39505e11	0.733361
$322$	−1.16210e11	−0.602412
$323$	1.75229e10	0.0895768
$324$	4.27185e10	0.215359
$325$	0	0
$326$	3.01301e11	1.47748
$327$	1.68859e11	0.816696
$328$	−2.92728e10	−0.139647
$329$	3.69343e11	1.73800
$330$	0	0
$331$	−1.79184e11	−0.820490	−0.410245	−	0.911975i	$-0.634557\pi$
$331$	−1.79184e11	−0.820490	−0.410245	+	0.911975i	$0.634557\pi$
$332$	1.68466e10	0.0761013
$333$	−4.17407e10	−0.186020
$334$	−3.26482e11	−1.43549
$335$	0	0
$336$	−2.13288e11	−0.912937
$337$	9.73351e10	0.411088	0.205544	−	0.978648i	$-0.434104\pi$
$337$	9.73351e10	0.411088	0.205544	+	0.978648i	$0.434104\pi$
$338$	−7.78340e10	−0.324373
$339$	1.47416e11	0.606241
$340$	0	0
$341$	−1.20894e11	−0.484183
$342$	4.61447e10	0.182391
$343$	−8.57794e11	−3.34626
$344$	−3.71713e11	−1.43118
$345$	0	0
$346$	1.39568e10	0.0523531
$347$	3.47750e11	1.28761	0.643806	−	0.765189i	$-0.277354\pi$
$347$	3.47750e11	1.28761	0.643806	+	0.765189i	$0.277354\pi$
$348$	−4.23182e11	−1.54675
$349$	−1.10131e11	−0.397372	−0.198686	−	0.980063i	$-0.563667\pi$
$349$	−1.10131e11	−0.397372	−0.198686	+	0.980063i	$0.563667\pi$
$350$	0	0
$351$	5.96807e10	0.209871
$352$	−8.02172e10	−0.278500
$353$	3.40052e11	1.16563	0.582814	−	0.812606i	$-0.301952\pi$
$353$	3.40052e11	1.16563	0.582814	+	0.812606i	$0.301952\pi$
$354$	3.73874e11	1.26535
$355$	0	0
$356$	−3.07500e9	−0.0101466
$357$	9.60589e10	0.312990
$358$	3.35632e11	1.07992
$359$	−2.75063e11	−0.873993	−0.436996	−	0.899463i	$-0.643958\pi$
$359$	−2.75063e11	−0.873993	−0.436996	+	0.899463i	$0.643958\pi$
$360$	0	0
$361$	−2.89807e11	−0.898102
$362$	−2.36485e11	−0.723793
$363$	1.60751e11	0.485930
$364$	−1.36765e12	−4.08336
$365$	0	0
$366$	−6.05954e11	−1.76512
$367$	2.12218e11	0.610639	0.305320	−	0.952250i	$-0.401237\pi$
$367$	2.12218e11	0.610639	0.305320	+	0.952250i	$0.401237\pi$
$368$	−5.23854e10	−0.148900
$369$	1.03081e10	0.0289440
$370$	0	0
$371$	−3.15769e10	−0.0865341
$372$	1.47467e11	0.399255
$373$	−6.08324e11	−1.62721	−0.813607	−	0.581415i	$-0.802499\pi$
$373$	−6.08324e11	−1.62721	−0.813607	+	0.581415i	$0.802499\pi$
$374$	−2.46992e11	−0.652770
$375$	0	0
$376$	5.60748e11	1.44685
$377$	−5.91215e11	−1.50733
$378$	2.52960e11	0.637294
$379$	4.64136e11	1.15550	0.577749	−	0.816214i	$-0.303931\pi$
$379$	4.64136e11	1.15550	0.577749	+	0.816214i	$0.303931\pi$
$380$	0	0
$381$	3.03680e11	0.738335
$382$	1.14900e12	2.76079
$383$	−5.22749e11	−1.24136	−0.620681	−	0.784063i	$-0.713144\pi$
$383$	−5.22749e11	−1.24136	−0.620681	+	0.784063i	$0.713144\pi$
$384$	4.42989e11	1.03969
$385$	0	0
$386$	−1.79750e10	−0.0412123
$387$	1.30894e11	0.296634
$388$	−5.68242e11	−1.27289
$389$	−1.18899e11	−0.263271	−0.131636	−	0.991298i	$-0.542023\pi$
$389$	−1.18899e11	−0.263271	−0.131636	+	0.991298i	$0.542023\pi$
$390$	0	0
$391$	2.35928e10	0.0510487
$392$	−2.05419e12	−4.39394
$393$	−1.59245e11	−0.336743
$394$	4.78677e11	1.00071
$395$	0	0
$396$	−4.29059e11	−0.876776
$397$	2.76954e11	0.559564	0.279782	−	0.960064i	$-0.409738\pi$
$397$	2.76954e11	0.559564	0.279782	+	0.960064i	$0.409738\pi$
$398$	3.89236e11	0.777569
$399$	1.80251e11	0.356041
$400$	0	0
$401$	−1.53482e11	−0.296421	−0.148210	−	0.988956i	$-0.547351\pi$
$401$	−1.53482e11	−0.296421	−0.148210	+	0.988956i	$0.547351\pi$
$402$	−3.77609e11	−0.721150
$403$	2.06021e11	0.389080
$404$	1.77084e12	3.30723
$405$	0	0
$406$	−2.50590e12	−4.57717
$407$	4.19238e11	0.757331
$408$	1.45840e11	0.260558
$409$	−1.52364e11	−0.269232	−0.134616	−	0.990898i	$-0.542980\pi$
$409$	−1.52364e11	−0.269232	−0.134616	+	0.990898i	$0.542980\pi$
$410$	0	0
$411$	2.94211e10	0.0508593
$412$	−1.15742e12	−1.97904
$413$	1.46043e12	2.47006
$414$	6.21291e10	0.103943
$415$	0	0
$416$	1.36702e11	0.223797
$417$	3.69020e11	0.597637
$418$	−4.63472e11	−0.742557
$419$	−9.09824e11	−1.44210	−0.721049	−	0.692885i	$-0.756340\pi$
$419$	−9.09824e11	−1.44210	−0.721049	+	0.692885i	$0.756340\pi$
$420$	0	0
$421$	9.88529e11	1.53363	0.766814	−	0.641869i	$-0.221841\pi$
$421$	9.88529e11	1.53363	0.766814	+	0.641869i	$0.221841\pi$
$422$	1.38183e12	2.12104
$423$	−1.97461e11	−0.299881
$424$	−4.79411e10	−0.0720380
$425$	0	0
$426$	2.19776e9	0.00323324
$427$	−2.36699e12	−3.44565
$428$	1.70915e12	2.46198
$429$	−5.99426e11	−0.854432
$430$	0	0
$431$	2.03278e11	0.283755	0.141878	−	0.989884i	$-0.454686\pi$
$431$	2.03278e11	0.283755	0.141878	+	0.989884i	$0.454686\pi$
$432$	1.14030e11	0.157522
$433$	2.10847e11	0.288251	0.144126	−	0.989559i	$-0.453963\pi$
$433$	2.10847e11	0.288251	0.144126	+	0.989559i	$0.453963\pi$
$434$	8.73234e11	1.18148
$435$	0	0
$436$	2.06879e12	2.74174
$437$	4.42712e10	0.0580704
$438$	−2.80015e11	−0.363536
$439$	2.11413e11	0.271670	0.135835	−	0.990732i	$-0.456628\pi$
$439$	2.11413e11	0.271670	0.135835	+	0.990732i	$0.456628\pi$
$440$	0	0
$441$	7.23359e11	0.910711
$442$	4.20911e11	0.524553
$443$	−1.45517e12	−1.79514	−0.897569	−	0.440874i	$-0.854669\pi$
$443$	−1.45517e12	−1.79514	−0.897569	+	0.440874i	$0.854669\pi$
$444$	−5.11387e11	−0.624491
$445$	0	0
$446$	−1.32512e12	−1.58579
$447$	7.12663e11	0.844306
$448$	1.92761e12	2.26084
$449$	−2.19470e11	−0.254840	−0.127420	−	0.991849i	$-0.540670\pi$
$449$	−2.19470e11	−0.254840	−0.127420	+	0.991849i	$0.540670\pi$
$450$	0	0
$451$	−1.03533e11	−0.117837
$452$	1.80607e12	2.03522
$453$	−2.19269e10	−0.0244644
$454$	1.41545e12	1.56366
$455$	0	0
$456$	2.73663e11	0.296397
$457$	−7.40816e11	−0.794488	−0.397244	−	0.917713i	$-0.630033\pi$
$457$	−7.40816e11	−0.794488	−0.397244	+	0.917713i	$0.630033\pi$
$458$	−1.50276e12	−1.59586
$459$	−5.13556e10	−0.0540046
$460$	0	0
$461$	2.80170e11	0.288913	0.144457	−	0.989511i	$-0.453857\pi$
$461$	2.80170e11	0.288913	0.144457	+	0.989511i	$0.453857\pi$
$462$	−2.54070e12	−2.59457
$463$	5.07092e10	0.0512828	0.0256414	−	0.999671i	$-0.491837\pi$
$463$	5.07092e10	0.0512828	0.0256414	+	0.999671i	$0.491837\pi$
$464$	−1.12961e12	−1.13135
$465$	0	0
$466$	−1.04225e12	−1.02385
$467$	−1.77228e12	−1.72427	−0.862135	−	0.506678i	$-0.830873\pi$
$467$	−1.77228e12	−1.72427	−0.862135	+	0.506678i	$0.830873\pi$
$468$	7.31180e11	0.704560
$469$	−1.47503e12	−1.40774
$470$	0	0
$471$	−7.88324e11	−0.738092
$472$	2.21728e12	2.05627
$473$	−1.31469e12	−1.20767
$474$	−1.35471e12	−1.23266
$475$	0	0
$476$	1.17687e12	1.05074
$477$	1.68819e10	0.0149310
$478$	−5.22495e11	−0.457779
$479$	−7.79555e11	−0.676608	−0.338304	−	0.941037i	$-0.609853\pi$
$479$	−7.79555e11	−0.676608	−0.338304	+	0.941037i	$0.609853\pi$
$480$	0	0
$481$	−7.14444e11	−0.608577
$482$	7.20880e11	0.608347
$483$	2.42690e11	0.202904
$484$	1.96945e12	1.63132
$485$	0	0
$486$	−1.35239e11	−0.109961
$487$	1.33958e12	1.07916	0.539582	−	0.841933i	$-0.318582\pi$
$487$	1.33958e12	1.07916	0.539582	+	0.841933i	$0.318582\pi$
$488$	−3.59364e12	−2.86844
$489$	−6.29226e11	−0.497642
$490$	0	0
$491$	−9.76349e11	−0.758121	−0.379061	−	0.925372i	$-0.623753\pi$
$491$	−9.76349e11	−0.758121	−0.379061	+	0.925372i	$0.623753\pi$
$492$	1.26289e11	0.0971682
$493$	5.08744e11	0.387872
$494$	7.89825e11	0.596705
$495$	0	0
$496$	3.93637e11	0.292031
$497$	8.58495e9	0.00631152
$498$	−5.33336e10	−0.0388570
$499$	1.32057e12	0.953472	0.476736	−	0.879046i	$-0.341820\pi$
$499$	1.32057e12	0.953472	0.476736	+	0.879046i	$0.341820\pi$
$500$	0	0
$501$	6.81815e11	0.483501
$502$	−1.66908e12	−1.17303
$503$	1.16421e12	0.810917	0.405459	−	0.914113i	$-0.367112\pi$
$503$	1.16421e12	0.810917	0.405459	+	0.914113i	$0.367112\pi$
$504$	1.50019e12	1.03564
$505$	0	0
$506$	−6.24017e11	−0.423174
$507$	1.62546e11	0.109255
$508$	3.72054e12	2.47868
$509$	8.32285e11	0.549594	0.274797	−	0.961502i	$-0.411389\pi$
$509$	8.32285e11	0.549594	0.274797	+	0.961502i	$0.411389\pi$
$510$	0	0
$511$	−1.09380e12	−0.709649
$512$	2.30806e12	1.48434
$513$	−9.63671e10	−0.0614329
$514$	−2.43135e10	−0.0153643
$515$	0	0
$516$	1.60366e12	0.995835
$517$	1.98327e12	1.22088
$518$	−3.02822e12	−1.84800
$519$	−2.91469e10	−0.0176335
$520$	0	0
$521$	2.43459e12	1.44763	0.723813	−	0.689997i	$-0.242388\pi$
$521$	2.43459e12	1.44763	0.723813	+	0.689997i	$0.242388\pi$
$522$	1.33972e12	0.789763
$523$	1.07631e12	0.629043	0.314522	−	0.949250i	$-0.398156\pi$
$523$	1.07631e12	0.629043	0.314522	+	0.949250i	$0.398156\pi$
$524$	−1.95099e12	−1.13048
$525$	0	0
$526$	2.41036e12	1.37292
$527$	−1.77283e11	−0.100119
$528$	−1.14530e12	−0.641308
$529$	−1.74155e12	−0.966906
$530$	0	0
$531$	−7.80787e11	−0.426194
$532$	2.20835e12	1.19527
$533$	1.76435e11	0.0946919
$534$	9.73494e9	0.00518081
$535$	0	0
$536$	−2.23943e12	−1.17191
$537$	−7.00923e11	−0.363736
$538$	2.11337e12	1.08757
$539$	−7.26533e12	−3.70771
$540$	0	0
$541$	−3.12532e12	−1.56858	−0.784291	−	0.620393i	$-0.786973\pi$
$541$	−3.12532e12	−1.56858	−0.784291	+	0.620393i	$0.786973\pi$
$542$	3.78139e12	1.88215
$543$	4.93867e11	0.243787
$544$	−1.17633e11	−0.0575883
$545$	0	0
$546$	4.32974e12	2.08495
$547$	−1.59061e12	−0.759663	−0.379832	−	0.925056i	$-0.624018\pi$
$547$	−1.59061e12	−0.759663	−0.379832	+	0.925056i	$0.624018\pi$
$548$	3.60453e11	0.170741
$549$	1.26546e12	0.594527
$550$	0	0
$551$	9.54641e11	0.441223
$552$	3.68460e11	0.168913
$553$	−5.29180e12	−2.40625
$554$	5.55402e12	2.50503
$555$	0	0
$556$	4.52106e12	2.00633
$557$	−3.76617e12	−1.65787	−0.828937	−	0.559343i	$-0.811054\pi$
$557$	−3.76617e12	−1.65787	−0.828937	+	0.559343i	$0.811054\pi$
$558$	−4.66854e11	−0.203858
$559$	2.24042e12	0.970458
$560$	0	0
$561$	5.15810e11	0.219865
$562$	−2.52607e12	−1.06815
$563$	2.32486e12	0.975234	0.487617	−	0.873058i	$-0.337866\pi$
$563$	2.32486e12	0.975234	0.487617	+	0.873058i	$0.337866\pi$
$564$	−2.41919e12	−1.00673
$565$	0	0
$566$	−6.41643e12	−2.62797
$567$	−5.28275e11	−0.214653
$568$	1.30339e10	0.00525422
$569$	3.30244e11	0.132078	0.0660388	−	0.997817i	$-0.478964\pi$
$569$	3.30244e11	0.132078	0.0660388	+	0.997817i	$0.478964\pi$
$570$	0	0
$571$	3.83190e12	1.50852	0.754261	−	0.656575i	$-0.227995\pi$
$571$	3.83190e12	1.50852	0.754261	+	0.656575i	$0.227995\pi$
$572$	−7.34388e12	−2.86843
$573$	−2.39953e12	−0.929887
$574$	7.47832e11	0.287542
$575$	0	0
$576$	−1.03055e12	−0.390094
$577$	1.66616e12	0.625786	0.312893	−	0.949788i	$-0.398702\pi$
$577$	1.66616e12	0.625786	0.312893	+	0.949788i	$0.398702\pi$
$578$	4.23738e12	1.57915
$579$	3.75384e10	0.0138811
$580$	0	0
$581$	−2.08333e11	−0.0758517
$582$	1.79896e12	0.649931
$583$	−1.69559e11	−0.0607874
$584$	−1.66064e12	−0.590769
$585$	0	0
$586$	2.18856e12	0.766689
$587$	−2.00482e12	−0.696953	−0.348476	−	0.937318i	$-0.613301\pi$
$587$	−2.00482e12	−0.696953	−0.348476	+	0.937318i	$0.613301\pi$
$588$	8.86226e12	3.05736
$589$	−3.32665e11	−0.113891
$590$	0	0
$591$	−9.99654e11	−0.337059
$592$	−1.36506e12	−0.456777
$593$	−1.46848e12	−0.487666	−0.243833	−	0.969817i	$-0.578405\pi$
$593$	−1.46848e12	−0.487666	−0.243833	+	0.969817i	$0.578405\pi$
$594$	1.35833e12	0.447678
$595$	0	0
$596$	8.73121e12	2.83443
$597$	−8.12868e11	−0.261900
$598$	1.06342e12	0.340055
$599$	5.67302e12	1.80050	0.900251	−	0.435370i	$-0.143383\pi$
$599$	5.67302e12	1.80050	0.900251	+	0.435370i	$0.143383\pi$
$600$	0	0
$601$	2.23747e12	0.699555	0.349778	−	0.936833i	$-0.386257\pi$
$601$	2.23747e12	0.699555	0.349778	+	0.936833i	$0.386257\pi$
$602$	9.49617e12	2.94689
$603$	7.88587e11	0.242897
$604$	−2.68638e11	−0.0821299
$605$	0	0
$606$	−5.60619e12	−1.68865
$607$	−7.87986e11	−0.235597	−0.117798	−	0.993038i	$-0.537584\pi$
$607$	−7.87986e11	−0.235597	−0.117798	+	0.993038i	$0.537584\pi$
$608$	−2.20734e11	−0.0655094
$609$	5.23324e12	1.54168
$610$	0	0
$611$	−3.37979e12	−0.981079
$612$	−6.29185e11	−0.181300
$613$	4.35930e12	1.24694	0.623468	−	0.781848i	$-0.285723\pi$
$613$	4.35930e12	1.24694	0.623468	+	0.781848i	$0.285723\pi$
$614$	6.95671e11	0.197536
$615$	0	0
$616$	−1.50678e13	−4.21634
$617$	2.04791e12	0.568889	0.284444	−	0.958693i	$-0.408191\pi$
$617$	2.04791e12	0.568889	0.284444	+	0.958693i	$0.408191\pi$
$618$	3.66421e12	1.01049
$619$	−6.31079e12	−1.72773	−0.863865	−	0.503723i	$-0.831963\pi$
$619$	−6.31079e12	−1.72773	−0.863865	+	0.503723i	$0.831963\pi$
$620$	0	0
$621$	−1.29748e11	−0.0350098
$622$	−1.24098e13	−3.32435
$623$	3.80268e10	0.0101133
$624$	1.95176e12	0.515342
$625$	0	0
$626$	−1.22947e13	−3.19989
$627$	9.67899e11	0.250107
$628$	−9.65818e12	−2.47786
$629$	6.14784e11	0.156601
$630$	0	0
$631$	4.94106e12	1.24076	0.620380	−	0.784301i	$-0.286978\pi$
$631$	4.94106e12	1.24076	0.620380	+	0.784301i	$0.286978\pi$
$632$	−8.03418e12	−2.00316
$633$	−2.88577e12	−0.714405
$634$	−1.19523e13	−2.93798
$635$	0	0
$636$	2.06829e11	0.0501249
$637$	1.23812e13	2.97945
$638$	−1.34560e13	−3.21531
$639$	−4.58974e9	−0.00108902
$640$	0	0
$641$	−6.55819e11	−0.153434	−0.0767172	−	0.997053i	$-0.524444\pi$
$641$	−6.55819e11	−0.153434	−0.0767172	+	0.997053i	$0.524444\pi$
$642$	−5.41089e12	−1.25707
$643$	−5.76324e11	−0.132959	−0.0664794	−	0.997788i	$-0.521177\pi$
$643$	−5.76324e11	−0.132959	−0.0664794	+	0.997788i	$0.521177\pi$
$644$	2.97332e12	0.681171
$645$	0	0
$646$	−6.79649e11	−0.153546
$647$	3.52372e12	0.790556	0.395278	−	0.918562i	$-0.370648\pi$
$647$	3.52372e12	0.790556	0.395278	+	0.918562i	$0.370648\pi$
$648$	−8.02043e11	−0.178694
$649$	7.84212e12	1.73513
$650$	0	0
$651$	−1.82364e12	−0.397945
$652$	−7.70899e12	−1.67064
$653$	4.37841e11	0.0942338	0.0471169	−	0.998889i	$-0.484997\pi$
$653$	4.37841e11	0.0942338	0.0471169	+	0.998889i	$0.484997\pi$
$654$	−6.54942e12	−1.39992
$655$	0	0
$656$	3.37108e11	0.0710725
$657$	5.84774e11	0.122446
$658$	−1.43254e13	−2.97914
$659$	1.98477e12	0.409945	0.204972	−	0.978768i	$-0.434290\pi$
$659$	1.98477e12	0.409945	0.204972	+	0.978768i	$0.434290\pi$
$660$	0	0
$661$	2.80945e12	0.572420	0.286210	−	0.958167i	$-0.407604\pi$
$661$	2.80945e12	0.572420	0.286210	+	0.958167i	$0.407604\pi$
$662$	6.94988e12	1.40642
$663$	−8.79016e11	−0.176679
$664$	−3.16297e11	−0.0631450
$665$	0	0
$666$	1.61896e12	0.318862
$667$	1.28533e12	0.251447
$668$	8.35328e12	1.62317
$669$	2.76733e12	0.534126
$670$	0	0
$671$	−1.27101e13	−2.42045
$672$	−1.21004e12	−0.228896
$673$	2.08224e12	0.391258	0.195629	−	0.980678i	$-0.437325\pi$
$673$	2.08224e12	0.391258	0.195629	+	0.980678i	$0.437325\pi$
$674$	−3.77526e12	−0.704657
$675$	0	0
$676$	1.99144e12	0.366781
$677$	6.44805e12	1.17972	0.589861	−	0.807505i	$-0.299183\pi$
$677$	6.44805e12	1.17972	0.589861	+	0.807505i	$0.299183\pi$
$678$	−5.71770e12	−1.03917
$679$	7.02712e12	1.26871
$680$	0	0
$681$	−2.95598e12	−0.526671
$682$	4.68902e12	0.829952
$683$	4.02024e12	0.706902	0.353451	−	0.935453i	$-0.385008\pi$
$683$	4.02024e12	0.706902	0.353451	+	0.935453i	$0.385008\pi$
$684$	−1.18064e12	−0.206237
$685$	0	0
$686$	3.32706e13	5.73591
$687$	3.13832e12	0.537516
$688$	4.28069e12	0.728392
$689$	2.88954e11	0.0488476
$690$	0	0
$691$	−8.43146e12	−1.40686	−0.703431	−	0.710763i	$-0.748350\pi$
$691$	−8.43146e12	−1.40686	−0.703431	+	0.710763i	$0.748350\pi$
$692$	−3.57094e11	−0.0591978
$693$	5.30592e12	0.873900
$694$	−1.34879e13	−2.20713
$695$	0	0
$696$	7.94527e12	1.28342
$697$	−1.51824e11	−0.0243664
$698$	4.27159e12	0.681145
$699$	2.17661e12	0.344853
$700$	0	0
$701$	6.55956e12	1.02599	0.512995	−	0.858391i	$-0.328536\pi$
$701$	6.55956e12	1.02599	0.512995	+	0.858391i	$0.328536\pi$
$702$	−2.31479e12	−0.359745
$703$	1.15362e12	0.178141
$704$	1.03507e13	1.58816
$705$	0	0
$706$	−1.31894e13	−1.99803
$707$	−2.18990e13	−3.29638
$708$	−9.56583e12	−1.43078
$709$	−5.30959e12	−0.789138	−0.394569	−	0.918866i	$-0.629106\pi$
$709$	−5.30959e12	−0.789138	−0.394569	+	0.918866i	$0.629106\pi$
$710$	0	0
$711$	2.82914e12	0.415184
$712$	5.77335e10	0.00841914
$713$	−4.47899e11	−0.0649049
$714$	−3.72577e12	−0.536505
$715$	0	0
$716$	−8.58739e12	−1.22110
$717$	1.09116e12	0.154189
$718$	1.06687e13	1.49813
$719$	1.02053e13	1.42411	0.712056	−	0.702123i	$-0.247764\pi$
$719$	1.02053e13	1.42411	0.712056	+	0.702123i	$0.247764\pi$
$720$	0	0
$721$	1.43132e13	1.97255
$722$	1.12405e13	1.53946
$723$	−1.50546e12	−0.204903
$724$	6.05063e12	0.818422
$725$	0	0
$726$	−6.23493e12	−0.832946
$727$	7.45813e12	0.990206	0.495103	−	0.868834i	$-0.335130\pi$
$727$	7.45813e12	0.990206	0.495103	+	0.868834i	$0.335130\pi$
$728$	2.56777e13	3.38817
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−1.92790e12	−0.249721
$732$	1.55038e13	1.99589
$733$	1.06610e13	1.36405	0.682024	−	0.731329i	$-0.261100\pi$
$733$	1.06610e13	1.36405	0.682024	+	0.731329i	$0.261100\pi$
$734$	−8.23114e12	−1.04671
$735$	0	0
$736$	−2.97196e11	−0.0373330
$737$	−7.92047e12	−0.988889
$738$	−3.99811e11	−0.0496136
$739$	2.33179e12	0.287601	0.143800	−	0.989607i	$-0.454068\pi$
$739$	2.33179e12	0.287601	0.143800	+	0.989607i	$0.454068\pi$
$740$	0	0
$741$	−1.64944e12	−0.200981
$742$	1.22475e12	0.148330
$743$	7.80869e12	0.940001	0.470001	−	0.882666i	$-0.344254\pi$
$743$	7.80869e12	0.940001	0.470001	+	0.882666i	$0.344254\pi$
$744$	−2.76870e12	−0.331282
$745$	0	0
$746$	2.35946e13	2.78925
$747$	1.11380e11	0.0130878
$748$	6.31946e12	0.738113
$749$	−2.11361e13	−2.45390
$750$	0	0
$751$	4.20664e12	0.482565	0.241283	−	0.970455i	$-0.422432\pi$
$751$	4.20664e12	0.482565	0.241283	+	0.970455i	$0.422432\pi$
$752$	−6.45763e12	−0.736364
$753$	3.48565e12	0.395099
$754$	2.29310e13	2.58376
$755$	0	0
$756$	−6.47217e12	−0.720613
$757$	1.54834e13	1.71370	0.856848	−	0.515570i	$-0.172420\pi$
$757$	1.54834e13	1.71370	0.856848	+	0.515570i	$0.172420\pi$
$758$	−1.80021e13	−1.98067
$759$	1.30318e12	0.142533
$760$	0	0
$761$	9.77615e12	1.05666	0.528332	−	0.849038i	$-0.322818\pi$
$761$	9.77615e12	1.05666	0.528332	+	0.849038i	$0.322818\pi$
$762$	−1.17786e13	−1.26560
$763$	−2.55835e13	−2.73275
$764$	−2.93979e13	−3.12174
$765$	0	0
$766$	2.02755e13	2.12785
$767$	−1.33641e13	−1.39432
$768$	−1.06678e13	−1.10649
$769$	3.35411e12	0.345867	0.172933	−	0.984934i	$-0.444675\pi$
$769$	3.35411e12	0.345867	0.172933	+	0.984934i	$0.444675\pi$
$770$	0	0
$771$	5.07755e10	0.00517499
$772$	4.59904e11	0.0466003
$773$	6.04474e12	0.608933	0.304467	−	0.952523i	$-0.401522\pi$
$773$	6.04474e12	0.608933	0.304467	+	0.952523i	$0.401522\pi$
$774$	−5.07690e12	−0.508469
$775$	0	0
$776$	1.06688e13	1.05618
$777$	6.32403e12	0.622443
$778$	4.61163e12	0.451280
$779$	−2.84892e11	−0.0277180
$780$	0	0
$781$	4.60988e10	0.00443364
$782$	−9.15078e11	−0.0875040
$783$	−2.79783e12	−0.266007
$784$	2.36563e13	2.23627
$785$	0	0
$786$	6.17651e12	0.577220
$787$	−4.16503e12	−0.387019	−0.193509	−	0.981098i	$-0.561987\pi$
$787$	−4.16503e12	−0.387019	−0.193509	+	0.981098i	$0.561987\pi$
$788$	−1.22473e13	−1.13155
$789$	−5.03371e12	−0.462425
$790$	0	0
$791$	−2.23346e13	−2.02854
$792$	8.05562e12	0.727504
$793$	2.16599e13	1.94503
$794$	−1.07420e13	−0.959164
$795$	0	0
$796$	−9.95887e12	−0.879228
$797$	−2.14334e13	−1.88160	−0.940802	−	0.338957i	$-0.889926\pi$
$797$	−2.14334e13	−1.88160	−0.940802	+	0.338957i	$0.889926\pi$
$798$	−6.99128e12	−0.610300
$799$	2.90833e12	0.252454
$800$	0	0
$801$	−2.03301e10	−0.00174499
$802$	5.95301e12	0.508103
$803$	−5.87339e12	−0.498504
$804$	9.66140e12	0.815432
$805$	0	0
$806$	−7.99080e12	−0.666933
$807$	−4.41350e12	−0.366313
$808$	−3.32478e13	−2.74417
$809$	−1.82139e11	−0.0149498	−0.00747490	−	0.999972i	$-0.502379\pi$
$809$	−1.82139e11	−0.0149498	−0.00747490	+	0.999972i	$0.502379\pi$
$810$	0	0
$811$	−1.32191e13	−1.07302	−0.536511	−	0.843894i	$-0.680258\pi$
$811$	−1.32191e13	−1.07302	−0.536511	+	0.843894i	$0.680258\pi$
$812$	6.41153e13	5.17558
$813$	−7.89693e12	−0.633944
$814$	−1.62607e13	−1.29816
$815$	0	0
$816$	−1.67950e12	−0.132610
$817$	−3.61763e12	−0.284070
$818$	5.90963e12	0.461498
$819$	−9.04209e12	−0.702249
$820$	0	0
$821$	−1.15652e13	−0.888398	−0.444199	−	0.895928i	$-0.646512\pi$
$821$	−1.15652e13	−0.888398	−0.444199	+	0.895928i	$0.646512\pi$
$822$	−1.14113e12	−0.0871793
$823$	−9.35914e12	−0.711110	−0.355555	−	0.934655i	$-0.615708\pi$
$823$	−9.35914e12	−0.711110	−0.355555	+	0.934655i	$0.615708\pi$
$824$	2.17307e13	1.64211
$825$	0	0
$826$	−5.66448e13	−4.23399
$827$	2.25652e13	1.67751	0.838753	−	0.544512i	$-0.183285\pi$
$827$	2.25652e13	1.67751	0.838753	+	0.544512i	$0.183285\pi$
$828$	−1.58962e12	−0.117532
$829$	1.22484e13	0.900705	0.450353	−	0.892851i	$-0.351298\pi$
$829$	1.22484e13	0.900705	0.450353	+	0.892851i	$0.351298\pi$
$830$	0	0
$831$	−1.15988e13	−0.843742
$832$	−1.76392e13	−1.27622
$833$	−1.06541e13	−0.766681
$834$	−1.43129e13	−1.02442
$835$	0	0
$836$	1.18582e13	0.839638
$837$	9.74963e11	0.0686631
$838$	3.52887e13	2.47194
$839$	4.67277e12	0.325571	0.162786	−	0.986661i	$-0.447952\pi$
$839$	4.67277e12	0.325571	0.162786	+	0.986661i	$0.447952\pi$
$840$	0	0
$841$	1.32090e13	0.910516
$842$	−3.83414e13	−2.62883
$843$	5.27537e12	0.359773
$844$	−3.53551e13	−2.39834
$845$	0	0
$846$	7.65876e12	0.514034
$847$	−2.43550e13	−1.62597
$848$	5.52094e11	0.0366633
$849$	1.33999e13	0.885149
$850$	0	0
$851$	1.55323e12	0.101520
$852$	−5.62314e10	−0.00365595
$853$	−4.04678e12	−0.261721	−0.130861	−	0.991401i	$-0.541774\pi$
$853$	−4.04678e12	−0.261721	−0.130861	+	0.991401i	$0.541774\pi$
$854$	9.18067e13	5.90628
$855$	0	0
$856$	−3.20895e13	−2.04282
$857$	2.35372e13	1.49053	0.745267	−	0.666766i	$-0.232322\pi$
$857$	2.35372e13	1.49053	0.745267	+	0.666766i	$0.232322\pi$
$858$	2.32495e13	1.46460
$859$	2.19727e13	1.37694	0.688470	−	0.725265i	$-0.258283\pi$
$859$	2.19727e13	1.37694	0.688470	+	0.725265i	$0.258283\pi$
$860$	0	0
$861$	−1.56175e12	−0.0968494
$862$	−7.88441e12	−0.486392
$863$	2.06939e13	1.26997	0.634984	−	0.772525i	$-0.281007\pi$
$863$	2.06939e13	1.26997	0.634984	+	0.772525i	$0.281007\pi$
$864$	6.46921e11	0.0394947
$865$	0	0
$866$	−8.17796e12	−0.494099
$867$	−8.84922e12	−0.531887
$868$	−2.23423e13	−1.33595
$869$	−2.84155e13	−1.69031
$870$	0	0
$871$	1.34977e13	0.794652
$872$	−3.88416e13	−2.27496
$873$	−3.75689e12	−0.218909
$874$	−1.71711e12	−0.0995400
$875$	0	0
$876$	7.16437e12	0.411064
$877$	8.18344e12	0.467130	0.233565	−	0.972341i	$-0.424961\pi$
$877$	8.18344e12	0.467130	0.233565	+	0.972341i	$0.424961\pi$
$878$	−8.19991e12	−0.465676
$879$	−4.57052e12	−0.258235
$880$	0	0
$881$	2.43747e13	1.36316	0.681581	−	0.731743i	$-0.261293\pi$
$881$	2.43747e13	1.36316	0.681581	+	0.731743i	$0.261293\pi$
$882$	−2.80564e13	−1.56107
$883$	1.17064e13	0.648039	0.324019	−	0.946050i	$-0.394966\pi$
$883$	1.17064e13	0.648039	0.324019	+	0.946050i	$0.394966\pi$
$884$	−1.07693e13	−0.593133
$885$	0	0
$886$	5.64407e13	3.07709
$887$	−2.83247e13	−1.53642	−0.768210	−	0.640198i	$-0.778852\pi$
$887$	−2.83247e13	−1.53642	−0.768210	+	0.640198i	$0.778852\pi$
$888$	9.60134e12	0.518171
$889$	−4.60098e13	−2.47055
$890$	0	0
$891$	−2.83669e12	−0.150786
$892$	3.39040e13	1.79312
$893$	5.45738e12	0.287179
$894$	−2.76415e13	−1.44725
$895$	0	0
$896$	−6.71163e13	−3.47890
$897$	−2.22081e12	−0.114537
$898$	8.51244e12	0.436828
$899$	−9.65827e12	−0.493152
$900$	0	0
$901$	−2.48647e11	−0.0125696
$902$	4.01565e12	0.201988
$903$	−1.98315e13	−0.992569
$904$	−3.39091e13	−1.68872
$905$	0	0
$906$	8.50462e11	0.0419351
$907$	−5.36313e12	−0.263139	−0.131570	−	0.991307i	$-0.542002\pi$
$907$	−5.36313e12	−0.263139	−0.131570	+	0.991307i	$0.542002\pi$
$908$	−3.62153e13	−1.76809
$909$	1.17078e13	0.568771
$910$	0	0
$911$	3.43721e13	1.65338	0.826691	−	0.562656i	$-0.190220\pi$
$911$	3.43721e13	1.65338	0.826691	+	0.562656i	$0.190220\pi$
$912$	−3.15153e12	−0.150850
$913$	−1.11869e12	−0.0532833
$914$	2.87335e13	1.36185
$915$	0	0
$916$	3.84492e13	1.80450
$917$	2.41268e13	1.12678
$918$	1.99189e12	0.0925708
$919$	−2.98037e13	−1.37832	−0.689160	−	0.724609i	$-0.742020\pi$
$919$	−2.98037e13	−1.37832	−0.689160	+	0.724609i	$0.742020\pi$
$920$	0	0
$921$	−1.45282e12	−0.0665339
$922$	−1.08668e13	−0.495234
$923$	−7.85592e10	−0.00356278
$924$	6.50057e13	2.93378
$925$	0	0
$926$	−1.96682e12	−0.0879053
$927$	−7.65221e12	−0.340352
$928$	−6.40859e12	−0.283659
$929$	−2.03150e13	−0.894843	−0.447421	−	0.894323i	$-0.647658\pi$
$929$	−2.03150e13	−0.894843	−0.447421	+	0.894323i	$0.647658\pi$
$930$	0	0
$931$	−1.99921e13	−0.872136
$932$	2.66668e13	1.15771
$933$	2.59162e13	1.11970
$934$	6.87399e13	2.95562
$935$	0	0
$936$	−1.37280e13	−0.584609
$937$	−2.73553e13	−1.15935	−0.579674	−	0.814848i	$-0.696820\pi$
$937$	−2.73553e13	−1.15935	−0.579674	+	0.814848i	$0.696820\pi$
$938$	5.72107e13	2.41304
$939$	2.56759e13	1.07778
$940$	0	0
$941$	−4.24123e13	−1.76335	−0.881675	−	0.471857i	$-0.843584\pi$
$941$	−4.24123e13	−1.76335	−0.881675	+	0.471857i	$0.843584\pi$
$942$	3.05761e13	1.26518
$943$	−3.83578e11	−0.0157961
$944$	−2.55344e13	−1.04653
$945$	0	0
$946$	5.09918e13	2.07009
$947$	−9.23741e12	−0.373229	−0.186614	−	0.982433i	$-0.559752\pi$
$947$	−9.23741e12	−0.373229	−0.186614	+	0.982433i	$0.559752\pi$
$948$	3.46613e13	1.39382
$949$	1.00091e13	0.400589
$950$	0	0
$951$	2.49607e13	0.989566
$952$	−2.20958e13	−0.871855
$953$	−1.11343e13	−0.437265	−0.218633	−	0.975807i	$-0.570160\pi$
$953$	−1.11343e13	−0.437265	−0.218633	+	0.975807i	$0.570160\pi$
$954$	−6.54784e11	−0.0255936
$955$	0	0
$956$	1.33684e13	0.517629
$957$	2.81011e13	1.08298
$958$	3.02360e13	1.15979
$959$	−4.45752e12	−0.170180
$960$	0	0
$961$	−2.30740e13	−0.872705
$962$	2.77106e13	1.04318
$963$	1.12999e13	0.423406
$964$	−1.84442e13	−0.687882
$965$	0	0
$966$	−9.41304e12	−0.347803
$967$	5.48630e12	0.201772	0.100886	−	0.994898i	$-0.467832\pi$
$967$	5.48630e12	0.201772	0.100886	+	0.994898i	$0.467832\pi$
$968$	−3.69766e13	−1.35359
$969$	1.41936e12	0.0517172
$970$	0	0
$971$	−2.07887e13	−0.750484	−0.375242	−	0.926927i	$-0.622440\pi$
$971$	−2.07887e13	−0.750484	−0.375242	+	0.926927i	$0.622440\pi$
$972$	3.46019e12	0.124338
$973$	−5.59093e13	−1.99975
$974$	−5.19572e13	−1.84982
$975$	0	0
$976$	4.13847e13	1.45987
$977$	−2.25372e13	−0.791362	−0.395681	−	0.918388i	$-0.629491\pi$
$977$	−2.25372e13	−0.791362	−0.395681	+	0.918388i	$0.629491\pi$
$978$	2.44053e13	0.853022
$979$	2.04193e11	0.00710427
$980$	0	0
$981$	1.36776e13	0.471519
$982$	3.78689e13	1.29952
$983$	2.20727e13	0.753989	0.376994	−	0.926216i	$-0.376958\pi$
$983$	2.20727e13	0.753989	0.376994	+	0.926216i	$0.376958\pi$
$984$	−2.37110e12	−0.0806253
$985$	0	0
$986$	−1.97323e13	−0.664861
$987$	2.99168e13	1.00343
$988$	−2.02082e13	−0.674717
$989$	−4.87077e12	−0.161888
$990$	0	0
$991$	−5.27788e13	−1.73831	−0.869157	−	0.494536i	$-0.835338\pi$
$991$	−5.27788e13	−1.73831	−0.869157	+	0.494536i	$0.835338\pi$
$992$	2.23321e12	0.0732195
$993$	−1.45139e13	−0.473710
$994$	−3.32978e11	−0.0108188
$995$	0	0
$996$	1.36458e12	0.0439371
$997$	4.19157e13	1.34353	0.671767	−	0.740763i	$-0.265536\pi$
$997$	4.19157e13	1.34353	0.671767	+	0.740763i	$0.265536\pi$
$998$	−5.12199e13	−1.63437
$999$	−3.38099e12	−0.107399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.f.1.1		2
3.2	odd	2		225.10.a.k.1.2		2
5.2	odd	4		75.10.b.f.49.1		4
5.3	odd	4		75.10.b.f.49.4		4
5.4	even	2		15.10.a.d.1.2	✓	2
15.2	even	4		225.10.b.i.199.4		4
15.8	even	4		225.10.b.i.199.1		4
15.14	odd	2		45.10.a.d.1.1		2
20.19	odd	2		240.10.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.d.1.2	✓	2	5.4	even	2
45.10.a.d.1.1		2	15.14	odd	2
75.10.a.f.1.1		2	1.1	even	1	trivial
75.10.b.f.49.1		4	5.2	odd	4
75.10.b.f.49.4		4	5.3	odd	4
225.10.a.k.1.2		2	3.2	odd	2
225.10.b.i.199.1		4	15.8	even	4
225.10.b.i.199.4		4	15.2	even	4
240.10.a.r.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-38.7863 q^{2} +81.0000 q^{3} +992.374 q^{4} -3141.69 q^{6} -12272.1 q^{7} -18631.9 q^{8} +6561.00 q^{9} +O(q^{10})\)	\(q-38.7863 q^{2} +81.0000 q^{3} +992.374 q^{4} -3141.69 q^{6} -12272.1 q^{7} -18631.9 q^{8} +6561.00 q^{9} -65897.9 q^{11} +80382.3 q^{12} +112300. q^{13} +475990. q^{14} +214567. q^{16} -96634.7 q^{17} -254477. q^{18} -181332. q^{19} -994042. q^{21} +2.55593e6 q^{22} -244145. q^{23} -1.50919e6 q^{24} -4.35569e6 q^{26} +531441. q^{27} -1.21785e7 q^{28} -5.26461e6 q^{29} +1.83457e6 q^{31} +1.21730e6 q^{32} -5.33773e6 q^{33} +3.74810e6 q^{34} +6.51097e6 q^{36} -6.36194e6 q^{37} +7.03318e6 q^{38} +9.09628e6 q^{39} +1.57111e6 q^{41} +3.85552e7 q^{42} +1.99504e7 q^{43} -6.53953e7 q^{44} +9.46946e6 q^{46} -3.00961e7 q^{47} +1.73799e7 q^{48} +1.10251e8 q^{49} -7.82741e6 q^{51} +1.11443e8 q^{52} +2.57306e6 q^{53} -2.06126e7 q^{54} +2.28653e8 q^{56} -1.46879e7 q^{57} +2.04195e8 q^{58} -1.19004e8 q^{59} +1.92875e8 q^{61} -7.11559e7 q^{62} -8.05174e7 q^{63} -1.57073e8 q^{64} +2.07030e8 q^{66} +1.20193e8 q^{67} -9.58978e7 q^{68} -1.97757e7 q^{69} -699549. q^{71} -1.22244e8 q^{72} +8.91287e7 q^{73} +2.46756e8 q^{74} -1.79949e8 q^{76} +8.08707e8 q^{77} -3.52811e8 q^{78} +4.31205e8 q^{79} +4.30467e7 q^{81} -6.09375e7 q^{82} +1.69761e7 q^{83} -9.86461e8 q^{84} -7.73800e8 q^{86} -4.26433e8 q^{87} +1.22780e9 q^{88} -3.09863e6 q^{89} -1.37816e9 q^{91} -2.42283e8 q^{92} +1.48600e8 q^{93} +1.16732e9 q^{94} +9.86009e7 q^{96} -5.72609e8 q^{97} -4.27624e9 q^{98} -4.32356e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 31 q^{2} + 162 q^{3} + 541 q^{4} - 2511 q^{6} - 14112 q^{7} - 26133 q^{8} + 13122 q^{9}+O(q^{10}) \) 2 * q - 31 * q^2 + 162 * q^3 + 541 * q^4 - 2511 * q^6 - 14112 * q^7 - 26133 * q^8 + 13122 * q^9	\( 2 q - 31 q^{2} + 162 q^{3} + 541 q^{4} - 2511 q^{6} - 14112 q^{7} - 26133 q^{8} + 13122 q^{9} - 21512 q^{11} + 43821 q^{12} - 24284 q^{13} + 461664 q^{14} + 387265 q^{16} + 156956 q^{17} - 203391 q^{18} - 95896 q^{19} - 1143072 q^{21} + 2901532 q^{22} + 735264 q^{23} - 2116773 q^{24} - 5419166 q^{26} + 1062882 q^{27} - 11348064 q^{28} - 2678212 q^{29} + 10782432 q^{31} + 6402523 q^{32} - 1742472 q^{33} + 5722622 q^{34} + 3549501 q^{36} - 21968332 q^{37} + 7698404 q^{38} - 1967004 q^{39} + 26060372 q^{41} + 37394784 q^{42} + 7191160 q^{43} - 85429972 q^{44} + 17095392 q^{46} + 31580240 q^{47} + 31368465 q^{48} + 73282930 q^{49} + 12713436 q^{51} + 173093786 q^{52} - 3131116 q^{53} - 16474671 q^{54} + 242454240 q^{56} - 7767576 q^{57} + 224332958 q^{58} - 35494664 q^{59} + 341497340 q^{61} - 1485504 q^{62} - 92588832 q^{63} - 205120471 q^{64} + 235024092 q^{66} + 288195816 q^{67} - 210362042 q^{68} + 59556384 q^{69} + 210286064 q^{71} - 171458613 q^{72} + 232663084 q^{73} + 125240234 q^{74} - 218512364 q^{76} + 727042176 q^{77} - 438952446 q^{78} - 24755040 q^{79} + 86093442 q^{81} + 129742346 q^{82} + 372082152 q^{83} - 919193184 q^{84} - 873146372 q^{86} - 216935172 q^{87} + 894861492 q^{88} - 427639116 q^{89} - 1126859328 q^{91} - 684362592 q^{92} + 873376992 q^{93} + 1647543896 q^{94} + 518604363 q^{96} - 1771658884 q^{97} - 4564085351 q^{98} - 141140232 q^{99}+O(q^{100}) \) 2 * q - 31 * q^2 + 162 * q^3 + 541 * q^4 - 2511 * q^6 - 14112 * q^7 - 26133 * q^8 + 13122 * q^9 - 21512 * q^11 + 43821 * q^12 - 24284 * q^13 + 461664 * q^14 + 387265 * q^16 + 156956 * q^17 - 203391 * q^18 - 95896 * q^19 - 1143072 * q^21 + 2901532 * q^22 + 735264 * q^23 - 2116773 * q^24 - 5419166 * q^26 + 1062882 * q^27 - 11348064 * q^28 - 2678212 * q^29 + 10782432 * q^31 + 6402523 * q^32 - 1742472 * q^33 + 5722622 * q^34 + 3549501 * q^36 - 21968332 * q^37 + 7698404 * q^38 - 1967004 * q^39 + 26060372 * q^41 + 37394784 * q^42 + 7191160 * q^43 - 85429972 * q^44 + 17095392 * q^46 + 31580240 * q^47 + 31368465 * q^48 + 73282930 * q^49 + 12713436 * q^51 + 173093786 * q^52 - 3131116 * q^53 - 16474671 * q^54 + 242454240 * q^56 - 7767576 * q^57 + 224332958 * q^58 - 35494664 * q^59 + 341497340 * q^61 - 1485504 * q^62 - 92588832 * q^63 - 205120471 * q^64 + 235024092 * q^66 + 288195816 * q^67 - 210362042 * q^68 + 59556384 * q^69 + 210286064 * q^71 - 171458613 * q^72 + 232663084 * q^73 + 125240234 * q^74 - 218512364 * q^76 + 727042176 * q^77 - 438952446 * q^78 - 24755040 * q^79 + 86093442 * q^81 + 129742346 * q^82 + 372082152 * q^83 - 919193184 * q^84 - 873146372 * q^86 - 216935172 * q^87 + 894861492 * q^88 - 427639116 * q^89 - 1126859328 * q^91 - 684362592 * q^92 + 873376992 * q^93 + 1647543896 * q^94 + 518604363 * q^96 - 1771658884 * q^97 - 4564085351 * q^98 - 141140232 * q^99

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