LMFDB - Embedded newform 75.10.a.e.1.2

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			1.2
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)

$f(q)$	$=$	$q-0.223611 q^{2} +81.0000 q^{3} -511.950 q^{4} -18.1125 q^{6} -580.825 q^{7} +228.967 q^{8} +6561.00 q^{9} +O(q^{10})$	\(q-0.223611 q^{2} +81.0000 q^{3} -511.950 q^{4} -18.1125 q^{6} -580.825 q^{7} +228.967 q^{8} +6561.00 q^{9} +47275.7 q^{11} -41467.9 q^{12} +22367.7 q^{13} +129.879 q^{14} +262067. q^{16} -244880. q^{17} -1467.11 q^{18} -248536. q^{19} -47046.8 q^{21} -10571.4 q^{22} -990208. q^{23} +18546.3 q^{24} -5001.67 q^{26} +531441. q^{27} +297353. q^{28} -4.35002e6 q^{29} -5.60419e6 q^{31} -175832. q^{32} +3.82933e6 q^{33} +54758.0 q^{34} -3.35890e6 q^{36} -4.73703e6 q^{37} +55575.5 q^{38} +1.81178e6 q^{39} +6.11568e6 q^{41} +10520.2 q^{42} +3.89014e7 q^{43} -2.42028e7 q^{44} +221422. q^{46} -5.09738e7 q^{47} +2.12274e7 q^{48} -4.00162e7 q^{49} -1.98353e7 q^{51} -1.14511e7 q^{52} -1.16798e7 q^{53} -118836. q^{54} -132990. q^{56} -2.01314e7 q^{57} +972713. q^{58} -1.79748e8 q^{59} -1.24120e8 q^{61} +1.25316e6 q^{62} -3.81079e6 q^{63} -1.34139e8 q^{64} -856281. q^{66} -1.90142e8 q^{67} +1.25366e8 q^{68} -8.02068e7 q^{69} -1.13551e8 q^{71} +1.50225e6 q^{72} +1.03304e8 q^{73} +1.05925e6 q^{74} +1.27238e8 q^{76} -2.74589e7 q^{77} -405135. q^{78} -1.94841e7 q^{79} +4.30467e7 q^{81} -1.36753e6 q^{82} +5.52722e8 q^{83} +2.40856e7 q^{84} -8.69878e6 q^{86} -3.52351e8 q^{87} +1.08246e7 q^{88} +3.25690e8 q^{89} -1.29917e7 q^{91} +5.06937e8 q^{92} -4.53939e8 q^{93} +1.13983e7 q^{94} -1.42424e7 q^{96} +1.33561e8 q^{97} +8.94808e6 q^{98} +3.10176e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.223611	−0.00988231	−0.00494116	−	0.999988i	$-0.501573\pi$
$2$	−0.223611	−0.00988231	−0.00494116	+	0.999988i	$0.501573\pi$
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	−511.950	−0.999902
$5$	0	0
$5$	0	0
$6$	−18.1125	−0.00570555
$7$	−580.825	−0.0914332	−0.0457166	−	0.998954i	$-0.514557\pi$
$7$	−580.825	−0.0914332	−0.0457166	+	0.998954i	$0.514557\pi$
$8$	228.967	0.0197637
$9$	6561.00	0.333333
$10$	0	0
$11$	47275.7	0.973578	0.486789	−	0.873520i	$-0.338168\pi$
$11$	47275.7	0.973578	0.486789	+	0.873520i	$0.338168\pi$
$12$	−41467.9	−0.577294
$13$	22367.7	0.217208	0.108604	−	0.994085i	$-0.465362\pi$
$13$	22367.7	0.217208	0.108604	+	0.994085i	$0.465362\pi$
$14$	129.879	0.000903572 0
$15$	0	0
$16$	262067.	0.999707
$17$	−244880.	−0.711105	−0.355552	−	0.934656i	$-0.615707\pi$
$17$	−244880.	−0.711105	−0.355552	+	0.934656i	$0.615707\pi$
$18$	−1467.11	−0.00329410
$19$	−248536.	−0.437521	−0.218760	−	0.975779i	$-0.570201\pi$
$19$	−248536.	−0.437521	−0.218760	+	0.975779i	$0.570201\pi$
$20$	0	0
$21$	−47046.8	−0.0527890
$22$	−10571.4	−0.00962120
$23$	−990208.	−0.737821	−0.368911	−	0.929465i	$-0.620269\pi$
$23$	−990208.	−0.737821	−0.368911	+	0.929465i	$0.620269\pi$
$24$	18546.3	0.0114106
$25$	0	0
$26$	−5001.67	−0.00214652
$27$	531441.	0.192450
$28$	297353.	0.0914243
$29$	−4.35002e6	−1.14209	−0.571045	−	0.820919i	$-0.693462\pi$
$29$	−4.35002e6	−1.14209	−0.571045	+	0.820919i	$0.693462\pi$
$30$	0	0
$31$	−5.60419e6	−1.08990	−0.544948	−	0.838470i	$-0.683451\pi$
$31$	−5.60419e6	−1.08990	−0.544948	+	0.838470i	$0.683451\pi$
$32$	−175832.	−0.0296431
$33$	3.82933e6	0.562096
$34$	54758.0	0.00702736
$35$	0	0
$36$	−3.35890e6	−0.333301
$37$	−4.73703e6	−0.415526	−0.207763	−	0.978179i	$-0.566618\pi$
$37$	−4.73703e6	−0.415526	−0.207763	+	0.978179i	$0.566618\pi$
$38$	55575.5	0.00432372
$39$	1.81178e6	0.125405
$40$	0	0
$41$	6.11568e6	0.338000	0.169000	−	0.985616i	$-0.445946\pi$
$41$	6.11568e6	0.338000	0.169000	+	0.985616i	$0.445946\pi$
$42$	10520.2	0.000521677 0
$43$	3.89014e7	1.73523	0.867614	−	0.497238i	$-0.165652\pi$
$43$	3.89014e7	1.73523	0.867614	+	0.497238i	$0.165652\pi$
$44$	−2.42028e7	−0.973483
$45$	0	0
$46$	221422.	0.00729138
$47$	−5.09738e7	−1.52372	−0.761862	−	0.647739i	$-0.775715\pi$
$47$	−5.09738e7	−1.52372	−0.761862	+	0.647739i	$0.775715\pi$
$48$	2.12274e7	0.577181
$49$	−4.00162e7	−0.991640
$50$	0	0
$51$	−1.98353e7	−0.410557
$52$	−1.14511e7	−0.217187
$53$	−1.16798e7	−0.203327	−0.101664	−	0.994819i	$-0.532417\pi$
$53$	−1.16798e7	−0.203327	−0.101664	+	0.994819i	$0.532417\pi$
$54$	−118836.	−0.00190185
$55$	0	0
$56$	−132990.	−0.00180706
$57$	−2.01314e7	−0.252603
$58$	972713.	0.0112865
$59$	−1.79748e8	−1.93121	−0.965607	−	0.260006i	$-0.916275\pi$
$59$	−1.79748e8	−1.93121	−0.965607	+	0.260006i	$0.916275\pi$
$60$	0	0
$61$	−1.24120e8	−1.14777	−0.573887	−	0.818935i	$-0.694565\pi$
$61$	−1.24120e8	−1.14777	−0.573887	+	0.818935i	$0.694565\pi$
$62$	1.25316e6	0.0107707
$63$	−3.81079e6	−0.0304777
$64$	−1.34139e8	−0.999414
$65$	0	0
$66$	−856281.	−0.00555480
$67$	−1.90142e8	−1.15277	−0.576383	−	0.817180i	$-0.695536\pi$
$67$	−1.90142e8	−1.15277	−0.576383	+	0.817180i	$0.695536\pi$
$68$	1.25366e8	0.711035
$69$	−8.02068e7	−0.425981
$70$	0	0
$71$	−1.13551e8	−0.530309	−0.265155	−	0.964206i	$-0.585423\pi$
$71$	−1.13551e8	−0.530309	−0.265155	+	0.964206i	$0.585423\pi$
$72$	1.50225e6	0.00658789
$73$	1.03304e8	0.425761	0.212881	−	0.977078i	$-0.431715\pi$
$73$	1.03304e8	0.425761	0.212881	+	0.977078i	$0.431715\pi$
$74$	1.05925e6	0.00410636
$75$	0	0
$76$	1.27238e8	0.437478
$77$	−2.74589e7	−0.0890174
$78$	−405135.	−0.00123929
$79$	−1.94841e7	−0.0562807	−0.0281403	−	0.999604i	$-0.508959\pi$
$79$	−1.94841e7	−0.0562807	−0.0281403	+	0.999604i	$0.508959\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	−1.36753e6	−0.00334023
$83$	5.52722e8	1.27837	0.639183	−	0.769055i	$-0.279273\pi$
$83$	5.52722e8	1.27837	0.639183	+	0.769055i	$0.279273\pi$
$84$	2.40856e7	0.0527839
$85$	0	0
$86$	−8.69878e6	−0.0171481
$87$	−3.52351e8	−0.659386
$88$	1.08246e7	0.0192415
$89$	3.25690e8	0.550237	0.275119	−	0.961410i	$-0.411283\pi$
$89$	3.25690e8	0.550237	0.275119	+	0.961410i	$0.411283\pi$
$90$	0	0
$91$	−1.29917e7	−0.0198601
$92$	5.06937e8	0.737749
$93$	−4.53939e8	−0.629251
$94$	1.13983e7	0.0150579
$95$	0	0
$96$	−1.42424e7	−0.0171144
$97$	1.33561e8	0.153182	0.0765910	−	0.997063i	$-0.475596\pi$
$97$	1.33561e8	0.153182	0.0765910	+	0.997063i	$0.475596\pi$
$98$	8.94808e6	0.00979969
$99$	3.10176e8	0.324526
$100$	0	0
$101$	1.72460e8	0.164908	0.0824542	−	0.996595i	$-0.473724\pi$
$101$	1.72460e8	0.164908	0.0824542	+	0.996595i	$0.473724\pi$
$102$	4.43539e6	0.00405725
$103$	−1.72881e9	−1.51349	−0.756746	−	0.653710i	$-0.773212\pi$
$103$	−1.72881e9	−1.51349	−0.756746	+	0.653710i	$0.773212\pi$
$104$	5.12146e6	0.00429283
$105$	0	0
$106$	2.61174e6	0.00200934
$107$	2.12656e9	1.56838	0.784190	−	0.620521i	$-0.213079\pi$
$107$	2.12656e9	1.56838	0.784190	+	0.620521i	$0.213079\pi$
$108$	−2.72071e8	−0.192431
$109$	1.65766e9	1.12480	0.562402	−	0.826864i	$-0.309877\pi$
$109$	1.65766e9	1.12480	0.562402	+	0.826864i	$0.309877\pi$
$110$	0	0
$111$	−3.83700e8	−0.239904
$112$	−1.52215e8	−0.0914065
$113$	−1.83857e9	−1.06079	−0.530394	−	0.847752i	$-0.677956\pi$
$113$	−1.83857e9	−1.06079	−0.530394	+	0.847752i	$0.677956\pi$
$114$	4.50161e6	0.00249630
$115$	0	0
$116$	2.22699e9	1.14198
$117$	1.46754e8	0.0724027
$118$	4.01937e7	0.0190849
$119$	1.42233e8	0.0650186
$120$	0	0
$121$	−1.22956e8	−0.0521454
$122$	2.77545e7	0.0113427
$123$	4.95370e8	0.195145
$124$	2.86906e9	1.08979
$125$	0	0
$126$	852136.	0.000301191 0
$127$	−7.74572e7	−0.0264207	−0.0132104	−	0.999913i	$-0.504205\pi$
$127$	−7.74572e7	−0.0264207	−0.0132104	+	0.999913i	$0.504205\pi$
$128$	1.20021e8	0.0395196
$129$	3.15101e9	1.00183
$130$	0	0
$131$	4.50478e9	1.33645	0.668225	−	0.743960i	$-0.267055\pi$
$131$	4.50478e9	1.33645	0.668225	+	0.743960i	$0.267055\pi$
$132$	−1.96043e9	−0.562041
$133$	1.44356e8	0.0400039
$134$	4.25178e7	0.0113920
$135$	0	0
$136$	−5.60694e7	−0.0140540
$137$	6.80091e9	1.64939	0.824697	−	0.565575i	$-0.191346\pi$
$137$	6.80091e9	1.64939	0.824697	+	0.565575i	$0.191346\pi$
$138$	1.79351e7	0.00420968
$139$	1.95054e9	0.443187	0.221594	−	0.975139i	$-0.428874\pi$
$139$	1.95054e9	0.443187	0.221594	+	0.975139i	$0.428874\pi$
$140$	0	0
$141$	−4.12888e9	−0.879723
$142$	2.53913e7	0.00524068
$143$	1.05745e9	0.211469
$144$	1.71942e9	0.333236
$145$	0	0
$146$	−2.31000e7	−0.00420751
$147$	−3.24132e9	−0.572524
$148$	2.42512e9	0.415486
$149$	−1.06040e10	−1.76251	−0.881257	−	0.472637i	$-0.843302\pi$
$149$	−1.06040e10	−1.76251	−0.881257	+	0.472637i	$0.843302\pi$
$150$	0	0
$151$	−6.54596e9	−1.02465	−0.512327	−	0.858791i	$-0.671216\pi$
$151$	−6.54596e9	−1.02465	−0.512327	+	0.858791i	$0.671216\pi$
$152$	−5.69065e7	−0.00864701
$153$	−1.60666e9	−0.237035
$154$	6.14012e6	0.000879698 0
$155$	0	0
$156$	−9.27543e8	−0.125393
$157$	−4.83783e9	−0.635480	−0.317740	−	0.948178i	$-0.602924\pi$
$157$	−4.83783e9	−0.635480	−0.317740	+	0.948178i	$0.602924\pi$
$158$	4.35687e6	0.000556183 0
$159$	−9.46068e8	−0.117391
$160$	0	0
$161$	5.75137e8	0.0674614
$162$	−9.62573e6	−0.00109803
$163$	−5.42724e9	−0.602191	−0.301096	−	0.953594i	$-0.597352\pi$
$163$	−5.42724e9	−0.602191	−0.301096	+	0.953594i	$0.597352\pi$
$164$	−3.13092e9	−0.337967
$165$	0	0
$166$	−1.23595e8	−0.0126332
$167$	1.36699e10	1.36000	0.680002	−	0.733210i	$-0.261979\pi$
$167$	1.36699e10	1.36000	0.680002	+	0.733210i	$0.261979\pi$
$168$	−1.07722e7	−0.00104330
$169$	−1.01042e10	−0.952821
$170$	0	0
$171$	−1.63065e9	−0.145840
$172$	−1.99156e10	−1.73506
$173$	1.84099e10	1.56259	0.781294	−	0.624163i	$-0.214560\pi$
$173$	1.84099e10	1.56259	0.781294	+	0.624163i	$0.214560\pi$
$174$	7.87897e7	0.00651625
$175$	0	0
$176$	1.23894e10	0.973293
$177$	−1.45596e10	−1.11499
$178$	−7.28280e7	−0.00543762
$179$	−2.21847e10	−1.61516	−0.807580	−	0.589758i	$-0.799223\pi$
$179$	−2.21847e10	−1.61516	−0.807580	+	0.589758i	$0.799223\pi$
$180$	0	0
$181$	−1.26287e10	−0.874593	−0.437297	−	0.899317i	$-0.644064\pi$
$181$	−1.26287e10	−0.874593	−0.437297	+	0.899317i	$0.644064\pi$
$182$	2.90509e6	0.000196263 0
$183$	−1.00537e10	−0.662668
$184$	−2.26725e8	−0.0145820
$185$	0	0
$186$	1.01506e8	0.00621846
$187$	−1.15769e10	−0.692316
$188$	2.60960e10	1.52358
$189$	−3.08674e8	−0.0175963
$190$	0	0
$191$	−3.00606e9	−0.163436	−0.0817179	−	0.996655i	$-0.526041\pi$
$191$	−3.00606e9	−0.163436	−0.0817179	+	0.996655i	$0.526041\pi$
$192$	−1.08653e10	−0.577012
$193$	3.17884e9	0.164915	0.0824576	−	0.996595i	$-0.473723\pi$
$193$	3.17884e9	0.164915	0.0824576	+	0.996595i	$0.473723\pi$
$194$	−2.98658e7	−0.00151379
$195$	0	0
$196$	2.04863e10	0.991543
$197$	2.67639e10	1.26605	0.633026	−	0.774131i	$-0.281813\pi$
$197$	2.67639e10	1.26605	0.633026	+	0.774131i	$0.281813\pi$
$198$	−6.93588e7	−0.00320707
$199$	3.55476e8	0.0160684	0.00803419	−	0.999968i	$-0.497443\pi$
$199$	3.55476e8	0.0160684	0.00803419	+	0.999968i	$0.497443\pi$
$200$	0	0
$201$	−1.54015e10	−0.665550
$202$	−3.85640e7	−0.00162968
$203$	2.52660e9	0.104425
$204$	1.01547e10	0.410516
$205$	0	0
$206$	3.86581e8	0.0149568
$207$	−6.49675e9	−0.245940
$208$	5.86184e9	0.217145
$209$	−1.17497e10	−0.425961
$210$	0	0
$211$	−3.90979e10	−1.35795	−0.678973	−	0.734164i	$-0.737574\pi$
$211$	−3.90979e10	−1.35795	−0.678973	+	0.734164i	$0.737574\pi$
$212$	5.97950e9	0.203307
$213$	−9.19765e9	−0.306174
$214$	−4.75523e8	−0.0154992
$215$	0	0
$216$	1.21682e8	0.00380352
$217$	3.25505e9	0.0996527
$218$	−3.70672e8	−0.0111157
$219$	8.36766e9	0.245813
$220$	0	0
$221$	−5.47741e9	−0.154458
$222$	8.57995e7	0.00237081
$223$	−3.00688e10	−0.814225	−0.407112	−	0.913378i	$-0.633464\pi$
$223$	−3.00688e10	−0.814225	−0.407112	+	0.913378i	$0.633464\pi$
$224$	1.02128e8	0.00271036
$225$	0	0
$226$	4.11126e8	0.0104830
$227$	5.17270e10	1.29301	0.646503	−	0.762911i	$-0.276231\pi$
$227$	5.17270e10	1.29301	0.646503	+	0.762911i	$0.276231\pi$
$228$	1.03063e10	0.252578
$229$	4.39043e10	1.05499	0.527494	−	0.849559i	$-0.323132\pi$
$229$	4.39043e10	1.05499	0.527494	+	0.849559i	$0.323132\pi$
$230$	0	0
$231$	−2.22417e9	−0.0513942
$232$	−9.96009e8	−0.0225719
$233$	−5.20811e10	−1.15765	−0.578826	−	0.815451i	$-0.696489\pi$
$233$	−5.20811e10	−1.15765	−0.578826	+	0.815451i	$0.696489\pi$
$234$	−3.28159e7	−0.000715506 0
$235$	0	0
$236$	9.20221e10	1.93103
$237$	−1.57822e9	−0.0324937
$238$	−3.18048e7	−0.000642534 0
$239$	1.25028e9	0.0247866	0.0123933	−	0.999923i	$-0.496055\pi$
$239$	1.25028e9	0.0247866	0.0123933	+	0.999923i	$0.496055\pi$
$240$	0	0
$241$	5.39086e9	0.102939	0.0514696	−	0.998675i	$-0.483609\pi$
$241$	5.39086e9	0.102939	0.0514696	+	0.998675i	$0.483609\pi$
$242$	2.74944e7	0.000515317 0
$243$	3.48678e9	0.0641500
$244$	6.35431e10	1.14766
$245$	0	0
$246$	−1.10770e8	−0.00192848
$247$	−5.55918e9	−0.0950331
$248$	−1.28317e9	−0.0215403
$249$	4.47705e10	0.738065
$250$	0	0
$251$	3.79306e10	0.603196	0.301598	−	0.953435i	$-0.402480\pi$
$251$	3.79306e10	0.603196	0.301598	+	0.953435i	$0.402480\pi$
$252$	1.95094e9	0.0304748
$253$	−4.68128e10	−0.718327
$254$	1.73203e7	0.000261098 0
$255$	0	0
$256$	6.86524e10	0.999024
$257$	1.23297e10	0.176301	0.0881504	−	0.996107i	$-0.471904\pi$
$257$	1.23297e10	0.176301	0.0881504	+	0.996107i	$0.471904\pi$
$258$	−7.04601e8	−0.00990044
$259$	2.75139e9	0.0379929
$260$	0	0
$261$	−2.85405e10	−0.380696
$262$	−1.00732e9	−0.0132072
$263$	−8.86491e10	−1.14255	−0.571273	−	0.820760i	$-0.693550\pi$
$263$	−8.86491e10	−1.14255	−0.571273	+	0.820760i	$0.693550\pi$
$264$	8.76789e8	0.0111091
$265$	0	0
$266$	−3.22796e7	−0.000395331 0
$267$	2.63809e10	0.317680
$268$	9.73431e10	1.15265
$269$	6.53334e10	0.760764	0.380382	−	0.924829i	$-0.375793\pi$
$269$	6.53334e10	0.760764	0.380382	+	0.924829i	$0.375793\pi$
$270$	0	0
$271$	−4.43373e10	−0.499353	−0.249676	−	0.968329i	$-0.580324\pi$
$271$	−4.43373e10	−0.499353	−0.249676	+	0.968329i	$0.580324\pi$
$272$	−6.41751e10	−0.710896
$273$	−1.05233e9	−0.0114662
$274$	−1.52076e9	−0.0162998
$275$	0	0
$276$	4.10619e10	0.425940
$277$	1.89388e11	1.93283	0.966416	−	0.256984i	$-0.0827288\pi$
$277$	1.89388e11	1.93283	0.966416	+	0.256984i	$0.0827288\pi$
$278$	−4.36162e8	−0.00437972
$279$	−3.67691e10	−0.363298
$280$	0	0
$281$	−1.59371e11	−1.52486	−0.762432	−	0.647068i	$-0.775995\pi$
$281$	−1.59371e11	−1.52486	−0.762432	+	0.647068i	$0.775995\pi$
$282$	9.23263e8	0.00869369
$283$	−9.23708e10	−0.856043	−0.428022	−	0.903769i	$-0.640789\pi$
$283$	−9.23708e10	−0.856043	−0.428022	+	0.903769i	$0.640789\pi$
$284$	5.81325e10	0.530257
$285$	0	0
$286$	−2.36457e8	−0.00208980
$287$	−3.55214e9	−0.0309045
$288$	−1.15363e9	−0.00988102
$289$	−5.86215e10	−0.494330
$290$	0	0
$291$	1.08185e10	0.0884396
$292$	−5.28867e10	−0.425720
$293$	1.47354e11	1.16804	0.584018	−	0.811740i	$-0.301480\pi$
$293$	1.47354e11	1.16804	0.584018	+	0.811740i	$0.301480\pi$
$294$	7.24794e8	0.00565786
$295$	0	0
$296$	−1.08462e9	−0.00821232
$297$	2.51242e10	0.187365
$298$	2.37118e9	0.0174177
$299$	−2.21487e10	−0.160261
$300$	0	0
$301$	−2.25949e10	−0.158658
$302$	1.46375e9	0.0101259
$303$	1.39693e10	0.0952099
$304$	−6.51332e10	−0.437392
$305$	0	0
$306$	3.59267e8	0.00234245
$307$	−1.85136e11	−1.18951	−0.594755	−	0.803907i	$-0.702751\pi$
$307$	−1.85136e11	−1.18951	−0.594755	+	0.803907i	$0.702751\pi$
$308$	1.40576e10	0.0890087
$309$	−1.40034e11	−0.873814
$310$	0	0
$311$	−1.63232e11	−0.989429	−0.494715	−	0.869055i	$-0.664727\pi$
$311$	−1.63232e11	−0.989429	−0.494715	+	0.869055i	$0.664727\pi$
$312$	4.14838e8	0.00247847
$313$	1.58814e11	0.935272	0.467636	−	0.883921i	$-0.345106\pi$
$313$	1.58814e11	0.935272	0.467636	+	0.883921i	$0.345106\pi$
$314$	1.08179e9	0.00628001
$315$	0	0
$316$	9.97491e9	0.0562752
$317$	−2.22998e11	−1.24032	−0.620161	−	0.784474i	$-0.712933\pi$
$317$	−2.22998e11	−1.24032	−0.620161	+	0.784474i	$0.712933\pi$
$318$	2.11551e8	0.00116010
$319$	−2.05650e11	−1.11191
$320$	0	0
$321$	1.72252e11	0.905504
$322$	−1.28607e8	−0.000666674 0
$323$	6.08616e10	0.311123
$324$	−2.20378e10	−0.111100
$325$	0	0
$326$	1.21359e9	0.00595104
$327$	1.34271e11	0.649406
$328$	1.40029e9	0.00668012
$329$	2.96068e10	0.139319
$330$	0	0
$331$	2.64535e11	1.21131	0.605657	−	0.795725i	$-0.292910\pi$
$331$	2.64535e11	1.21131	0.605657	+	0.795725i	$0.292910\pi$
$332$	−2.82966e11	−1.27824
$333$	−3.10797e10	−0.138509
$334$	−3.05674e9	−0.0134400
$335$	0	0
$336$	−1.23294e10	−0.0527735
$337$	1.13648e11	0.479986	0.239993	−	0.970775i	$-0.422855\pi$
$337$	1.13648e11	0.479986	0.239993	+	0.970775i	$0.422855\pi$
$338$	2.25941e9	0.00941607
$339$	−1.48925e11	−0.612446
$340$	0	0
$341$	−2.64942e11	−1.06110
$342$	3.64631e8	0.00144124
$343$	4.66808e10	0.182102
$344$	8.90711e9	0.0342945
$345$	0	0
$346$	−4.11667e9	−0.0154420
$347$	−3.78341e11	−1.40088	−0.700441	−	0.713711i	$-0.747013\pi$
$347$	−3.78341e11	−1.40088	−0.700441	+	0.713711i	$0.747013\pi$
$348$	1.80386e11	0.659321
$349$	−4.60370e11	−1.66109	−0.830545	−	0.556952i	$-0.811971\pi$
$349$	−4.60370e11	−1.66109	−0.830545	+	0.556952i	$0.811971\pi$
$350$	0	0
$351$	1.18871e10	0.0418017
$352$	−8.31258e9	−0.0288598
$353$	−1.01736e11	−0.348728	−0.174364	−	0.984681i	$-0.555787\pi$
$353$	−1.01736e11	−0.348728	−0.174364	+	0.984681i	$0.555787\pi$
$354$	3.25569e9	0.0110186
$355$	0	0
$356$	−1.66737e11	−0.550184
$357$	1.15208e10	0.0375385
$358$	4.96075e9	0.0159615
$359$	6.76144e10	0.214839	0.107420	−	0.994214i	$-0.465741\pi$
$359$	6.76144e10	0.214839	0.107420	+	0.994214i	$0.465741\pi$
$360$	0	0
$361$	−2.60917e11	−0.808576
$362$	2.82392e9	0.00864300
$363$	−9.95945e9	−0.0301062
$364$	6.65111e9	0.0198581
$365$	0	0
$366$	2.24812e9	0.00654869
$367$	5.81071e11	1.67198	0.835991	−	0.548743i	$-0.184893\pi$
$367$	5.81071e11	1.67198	0.835991	+	0.548743i	$0.184893\pi$
$368$	−2.59501e11	−0.737605
$369$	4.01250e10	0.112667
$370$	0	0
$371$	6.78395e9	0.0185909
$372$	2.32394e11	0.629190
$373$	3.30933e11	0.885219	0.442610	−	0.896714i	$-0.354053\pi$
$373$	3.30933e11	0.885219	0.442610	+	0.896714i	$0.354053\pi$
$374$	2.58872e9	0.00684168
$375$	0	0
$376$	−1.16713e10	−0.0301144
$377$	−9.72999e10	−0.248071
$378$	6.90230e7	0.000173892 0
$379$	5.19853e11	1.29421	0.647104	−	0.762401i	$-0.275980\pi$
$379$	5.19853e11	1.29421	0.647104	+	0.762401i	$0.275980\pi$
$380$	0	0
$381$	−6.27403e9	−0.0152540
$382$	6.72188e8	0.00161512
$383$	−4.51303e11	−1.07170	−0.535850	−	0.844313i	$-0.680009\pi$
$383$	−4.51303e11	−1.07170	−0.535850	+	0.844313i	$0.680009\pi$
$384$	9.72170e9	0.0228166
$385$	0	0
$386$	−7.10824e8	−0.00162974
$387$	2.55232e11	0.578410
$388$	−6.83767e10	−0.153167
$389$	4.61900e11	1.02276	0.511381	−	0.859354i	$-0.329134\pi$
$389$	4.61900e11	1.02276	0.511381	+	0.859354i	$0.329134\pi$
$390$	0	0
$391$	2.42482e11	0.524668
$392$	−9.16239e9	−0.0195984
$393$	3.64887e11	0.771599
$394$	−5.98471e9	−0.0125115
$395$	0	0
$396$	−1.58795e11	−0.324494
$397$	−4.81112e11	−0.972052	−0.486026	−	0.873944i	$-0.661554\pi$
$397$	−4.81112e11	−0.972052	−0.486026	+	0.873944i	$0.661554\pi$
$398$	−7.94885e7	−0.000158793 0
$399$	1.16928e10	0.0230963
$400$	0	0
$401$	4.12911e11	0.797457	0.398728	−	0.917069i	$-0.369452\pi$
$401$	4.12911e11	0.797457	0.398728	+	0.917069i	$0.369452\pi$
$402$	3.44395e9	0.00657717
$403$	−1.25353e11	−0.236734
$404$	−8.82910e10	−0.164892
$405$	0	0
$406$	−5.64976e8	−0.00103196
$407$	−2.23947e11	−0.404547
$408$	−4.54162e9	−0.00811410
$409$	6.65886e11	1.17664	0.588322	−	0.808627i	$-0.299789\pi$
$409$	6.65886e11	1.17664	0.588322	+	0.808627i	$0.299789\pi$
$410$	0	0
$411$	5.50874e11	0.952278
$412$	8.85064e11	1.51334
$413$	1.04402e11	0.176577
$414$	1.45275e9	0.00243046
$415$	0	0
$416$	−3.93296e9	−0.00643872
$417$	1.57993e11	0.255874
$418$	2.62737e9	0.00420948
$419$	7.79778e11	1.23597	0.617985	−	0.786190i	$-0.287949\pi$
$419$	7.79778e11	1.23597	0.617985	+	0.786190i	$0.287949\pi$
$420$	0	0
$421$	9.59110e11	1.48799	0.743994	−	0.668187i	$-0.232929\pi$
$421$	9.59110e11	1.48799	0.743994	+	0.668187i	$0.232929\pi$
$422$	8.74272e9	0.0134196
$423$	−3.34439e11	−0.507908
$424$	−2.67430e9	−0.00401849
$425$	0	0
$426$	2.05670e9	0.00302571
$427$	7.20918e10	0.104945
$428$	−1.08869e12	−1.56823
$429$	8.56533e10	0.122092
$430$	0	0
$431$	1.89582e10	0.0264636	0.0132318	−	0.999912i	$-0.495788\pi$
$431$	1.89582e10	0.0264636	0.0132318	+	0.999912i	$0.495788\pi$
$432$	1.39273e11	0.192394
$433$	−8.39898e11	−1.14824	−0.574118	−	0.818773i	$-0.694655\pi$
$433$	−8.39898e11	−1.14824	−0.574118	+	0.818773i	$0.694655\pi$
$434$	−7.27866e8	−0.000984799 0
$435$	0	0
$436$	−8.48641e11	−1.12469
$437$	2.46103e11	0.322812
$438$	−1.87110e9	−0.00242920
$439$	−3.74391e11	−0.481100	−0.240550	−	0.970637i	$-0.577328\pi$
$439$	−3.74391e11	−0.481100	−0.240550	+	0.970637i	$0.577328\pi$
$440$	0	0
$441$	−2.62547e11	−0.330547
$442$	1.22481e9	0.00152640
$443$	7.04057e11	0.868542	0.434271	−	0.900782i	$-0.357006\pi$
$443$	7.04057e11	0.868542	0.434271	+	0.900782i	$0.357006\pi$
$444$	1.96435e11	0.239881
$445$	0	0
$446$	6.72372e9	0.00804642
$447$	−8.58926e11	−1.01759
$448$	7.79113e10	0.0913797
$449$	3.02133e11	0.350824	0.175412	−	0.984495i	$-0.443874\pi$
$449$	3.02133e11	0.350824	0.175412	+	0.984495i	$0.443874\pi$
$450$	0	0
$451$	2.89123e11	0.329070
$452$	9.41258e11	1.06068
$453$	−5.30223e11	−0.591584
$454$	−1.15667e10	−0.0127779
$455$	0	0
$456$	−4.60943e9	−0.00499235
$457$	−2.44866e11	−0.262607	−0.131303	−	0.991342i	$-0.541916\pi$
$457$	−2.44866e11	−0.262607	−0.131303	+	0.991342i	$0.541916\pi$
$458$	−9.81749e9	−0.0104257
$459$	−1.30139e11	−0.136852
$460$	0	0
$461$	−5.68075e11	−0.585803	−0.292902	−	0.956143i	$-0.594621\pi$
$461$	−5.68075e11	−0.585803	−0.292902	+	0.956143i	$0.594621\pi$
$462$	4.97350e8	0.000507894 0
$463$	−3.27020e11	−0.330720	−0.165360	−	0.986233i	$-0.552879\pi$
$463$	−3.27020e11	−0.330720	−0.165360	+	0.986233i	$0.552879\pi$
$464$	−1.14000e12	−1.14175
$465$	0	0
$466$	1.16459e10	0.0114403
$467$	7.11477e11	0.692205	0.346102	−	0.938197i	$-0.387505\pi$
$467$	7.11477e11	0.692205	0.346102	+	0.938197i	$0.387505\pi$
$468$	−7.51310e10	−0.0723957
$469$	1.10439e11	0.105401
$470$	0	0
$471$	−3.91864e11	−0.366894
$472$	−4.11563e10	−0.0381678
$473$	1.83909e12	1.68938
$474$	3.52907e8	0.000321113 0
$475$	0	0
$476$	−7.28160e10	−0.0650123
$477$	−7.66315e10	−0.0677758
$478$	−2.79577e8	−0.000244949 0
$479$	3.36653e11	0.292195	0.146098	−	0.989270i	$-0.453329\pi$
$479$	3.36653e11	0.292195	0.146098	+	0.989270i	$0.453329\pi$
$480$	0	0
$481$	−1.05957e11	−0.0902557
$482$	−1.20546e9	−0.00101728
$483$	4.65861e10	0.0389488
$484$	6.29474e10	0.0521403
$485$	0	0
$486$	−7.79684e8	−0.000633951 0
$487$	−1.66615e12	−1.34225	−0.671127	−	0.741342i	$-0.734190\pi$
$487$	−1.66615e12	−1.34225	−0.671127	+	0.741342i	$0.734190\pi$
$488$	−2.84193e10	−0.0226842
$489$	−4.39606e11	−0.347675
$490$	0	0
$491$	7.80759e11	0.606248	0.303124	−	0.952951i	$-0.401970\pi$
$491$	7.80759e11	0.606248	0.303124	+	0.952951i	$0.401970\pi$
$492$	−2.53605e11	−0.195126
$493$	1.06523e12	0.812145
$494$	1.24310e9	0.000939146 0
$495$	0	0
$496$	−1.46867e12	−1.08958
$497$	6.59534e10	0.0484879
$498$	−1.00112e10	−0.00729378
$499$	−5.98100e11	−0.431838	−0.215919	−	0.976411i	$-0.569275\pi$
$499$	−5.98100e11	−0.431838	−0.215919	+	0.976411i	$0.569275\pi$
$500$	0	0
$501$	1.10726e12	0.785199
$502$	−8.48171e9	−0.00596097
$503$	2.15422e12	1.50050	0.750248	−	0.661156i	$-0.229934\pi$
$503$	2.15422e12	1.50050	0.750248	+	0.661156i	$0.229934\pi$
$504$	−8.72544e8	−0.000602352 0
$505$	0	0
$506$	1.04679e10	0.00709873
$507$	−8.18439e11	−0.550111
$508$	3.96542e10	0.0264181
$509$	−2.28931e12	−1.51173	−0.755866	−	0.654726i	$-0.772784\pi$
$509$	−2.28931e12	−1.51173	−0.755866	+	0.654726i	$0.772784\pi$
$510$	0	0
$511$	−6.00018e10	−0.0389287
$512$	−7.68022e10	−0.0493923
$513$	−1.32082e11	−0.0842009
$514$	−2.75706e9	−0.00174226
$515$	0	0
$516$	−1.61316e12	−1.00174
$517$	−2.40982e12	−1.48347
$518$	−6.15241e8	−0.000375458 0
$519$	1.49120e12	0.902161
$520$	0	0
$521$	1.81335e12	1.07823	0.539114	−	0.842233i	$-0.318759\pi$
$521$	1.81335e12	1.07823	0.539114	+	0.842233i	$0.318759\pi$
$522$	6.38197e9	0.00376216
$523$	2.18455e11	0.127675	0.0638373	−	0.997960i	$-0.479666\pi$
$523$	2.18455e11	0.127675	0.0638373	+	0.997960i	$0.479666\pi$
$524$	−2.30622e12	−1.33632
$525$	0	0
$526$	1.98229e10	0.0112910
$527$	1.37235e12	0.775030
$528$	1.00354e12	0.561931
$529$	−8.20641e11	−0.455620
$530$	0	0
$531$	−1.17933e12	−0.643738
$532$	−7.39031e10	−0.0400000
$533$	1.36794e11	0.0734165
$534$	−5.89907e9	−0.00313941
$535$	0	0
$536$	−4.35361e10	−0.0227829
$537$	−1.79696e12	−0.932513
$538$	−1.46093e10	−0.00751811
$539$	−1.89180e12	−0.965439
$540$	0	0
$541$	−1.00780e12	−0.505811	−0.252906	−	0.967491i	$-0.581386\pi$
$541$	−1.00780e12	−0.505811	−0.252906	+	0.967491i	$0.581386\pi$
$542$	9.91431e9	0.00493476
$543$	−1.02293e12	−0.504947
$544$	4.30578e10	0.0210793
$545$	0	0
$546$	2.35313e8	0.000113313 0
$547$	1.06674e12	0.509468	0.254734	−	0.967011i	$-0.418012\pi$
$547$	1.06674e12	0.509468	0.254734	+	0.967011i	$0.418012\pi$
$548$	−3.48173e12	−1.64923
$549$	−8.14349e11	−0.382591
$550$	0	0
$551$	1.08114e12	0.499688
$552$	−1.83647e10	−0.00841895
$553$	1.13169e10	0.00514593
$554$	−4.23493e10	−0.0191008
$555$	0	0
$556$	−9.98577e11	−0.443144
$557$	2.02018e12	0.889286	0.444643	−	0.895708i	$-0.353331\pi$
$557$	2.02018e12	0.889286	0.444643	+	0.895708i	$0.353331\pi$
$558$	8.22197e9	0.00359023
$559$	8.70134e11	0.376906
$560$	0	0
$561$	−9.37728e11	−0.399709
$562$	3.56372e10	0.0150692
$563$	5.92218e11	0.248424	0.124212	−	0.992256i	$-0.460360\pi$
$563$	5.92218e11	0.248424	0.124212	+	0.992256i	$0.460360\pi$
$564$	2.11378e12	0.879637
$565$	0	0
$566$	2.06551e10	0.00845968
$567$	−2.50026e10	−0.0101592
$568$	−2.59994e10	−0.0104808
$569$	−1.47778e12	−0.591025	−0.295512	−	0.955339i	$-0.595490\pi$
$569$	−1.47778e12	−0.591025	−0.295512	+	0.955339i	$0.595490\pi$
$570$	0	0
$571$	1.63251e11	0.0642677	0.0321339	−	0.999484i	$-0.489770\pi$
$571$	1.63251e11	0.0642677	0.0321339	+	0.999484i	$0.489770\pi$
$572$	−5.41361e11	−0.211449
$573$	−2.43491e11	−0.0943597
$574$	7.94298e8	0.000305408 0
$575$	0	0
$576$	−8.80087e11	−0.333138
$577$	−1.06656e12	−0.400584	−0.200292	−	0.979736i	$-0.564189\pi$
$577$	−1.06656e12	−0.400584	−0.200292	+	0.979736i	$0.564189\pi$
$578$	1.31084e10	0.00488512
$579$	2.57486e11	0.0952138
$580$	0	0
$581$	−3.21035e11	−0.116885
$582$	−2.41913e9	−0.000873988 0
$583$	−5.52173e11	−0.197955
$584$	2.36533e10	0.00841460
$585$	0	0
$586$	−3.29499e10	−0.0115429
$587$	2.49441e12	0.867153	0.433577	−	0.901117i	$-0.357251\pi$
$587$	2.49441e12	0.867153	0.433577	+	0.901117i	$0.357251\pi$
$588$	1.65939e12	0.572468
$589$	1.39284e12	0.476852
$590$	0	0
$591$	2.16788e12	0.730955
$592$	−1.24142e12	−0.415405
$593$	1.76944e12	0.587610	0.293805	−	0.955865i	$-0.405078\pi$
$593$	1.76944e12	0.587610	0.293805	+	0.955865i	$0.405078\pi$
$594$	−5.61806e9	−0.00185160
$595$	0	0
$596$	5.42873e12	1.76234
$597$	2.87936e10	0.00927708
$598$	4.95269e9	0.00158375
$599$	5.49741e12	1.74477	0.872383	−	0.488823i	$-0.162574\pi$
$599$	5.49741e12	1.74477	0.872383	+	0.488823i	$0.162574\pi$
$600$	0	0
$601$	4.24474e11	0.132714	0.0663569	−	0.997796i	$-0.478862\pi$
$601$	4.24474e11	0.132714	0.0663569	+	0.997796i	$0.478862\pi$
$602$	5.05247e9	0.00156790
$603$	−1.24752e12	−0.384255
$604$	3.35120e12	1.02455
$605$	0	0
$606$	−3.12369e9	−0.000940894 0
$607$	2.57263e12	0.769181	0.384590	−	0.923087i	$-0.374343\pi$
$607$	2.57263e12	0.769181	0.384590	+	0.923087i	$0.374343\pi$
$608$	4.37006e10	0.0129695
$609$	2.04655e11	0.0602898
$610$	0	0
$611$	−1.14017e12	−0.330965
$612$	8.22529e11	0.237012
$613$	−3.01773e12	−0.863194	−0.431597	−	0.902066i	$-0.642050\pi$
$613$	−3.01773e12	−0.863194	−0.431597	+	0.902066i	$0.642050\pi$
$614$	4.13985e10	0.0117551
$615$	0	0
$616$	−6.28717e9	−0.00175931
$617$	2.87854e12	0.799631	0.399816	−	0.916596i	$-0.369074\pi$
$617$	2.87854e12	0.799631	0.399816	+	0.916596i	$0.369074\pi$
$618$	3.13131e10	0.00863531
$619$	5.36553e12	1.46894	0.734471	−	0.678640i	$-0.237431\pi$
$619$	5.36553e12	1.46894	0.734471	+	0.678640i	$0.237431\pi$
$620$	0	0
$621$	−5.26237e11	−0.141994
$622$	3.65006e10	0.00977785
$623$	−1.89169e11	−0.0503100
$624$	4.74809e11	0.125368
$625$	0	0
$626$	−3.55125e10	−0.00924265
$627$	−9.51728e11	−0.245928
$628$	2.47672e12	0.635418
$629$	1.16001e12	0.295483
$630$	0	0
$631$	3.28798e12	0.825653	0.412826	−	0.910810i	$-0.364542\pi$
$631$	3.28798e12	0.825653	0.412826	+	0.910810i	$0.364542\pi$
$632$	−4.46122e9	−0.00111231
$633$	−3.16693e12	−0.784010
$634$	4.98649e10	0.0122573
$635$	0	0
$636$	4.84339e11	0.117380
$637$	−8.95071e11	−0.215392
$638$	4.59857e10	0.0109883
$639$	−7.45009e11	−0.176770
$640$	0	0
$641$	−4.40972e12	−1.03169	−0.515846	−	0.856681i	$-0.672522\pi$
$641$	−4.40972e12	−1.03169	−0.515846	+	0.856681i	$0.672522\pi$
$642$	−3.85174e10	−0.00894848
$643$	−4.91168e12	−1.13313	−0.566566	−	0.824016i	$-0.691728\pi$
$643$	−4.91168e12	−1.13313	−0.566566	+	0.824016i	$0.691728\pi$
$644$	−2.94442e11	−0.0674548
$645$	0	0
$646$	−1.36093e10	−0.00307461
$647$	2.80990e12	0.630408	0.315204	−	0.949024i	$-0.397927\pi$
$647$	2.80990e12	0.630408	0.315204	+	0.949024i	$0.397927\pi$
$648$	9.85626e9	0.00219596
$649$	−8.49772e12	−1.88019
$650$	0	0
$651$	2.63659e11	0.0575345
$652$	2.77847e12	0.602133
$653$	2.56025e12	0.551028	0.275514	−	0.961297i	$-0.411152\pi$
$653$	2.56025e12	0.551028	0.275514	+	0.961297i	$0.411152\pi$
$654$	−3.00244e10	−0.00641763
$655$	0	0
$656$	1.60272e12	0.337901
$657$	6.77781e11	0.141920
$658$	−6.62042e9	−0.00137679
$659$	8.27996e12	1.71019	0.855094	−	0.518472i	$-0.173499\pi$
$659$	8.27996e12	1.71019	0.855094	+	0.518472i	$0.173499\pi$
$660$	0	0
$661$	−3.18671e11	−0.0649286	−0.0324643	−	0.999473i	$-0.510336\pi$
$661$	−3.18671e11	−0.0649286	−0.0324643	+	0.999473i	$0.510336\pi$
$662$	−5.91530e10	−0.0119706
$663$	−4.43670e11	−0.0891762
$664$	1.26555e11	0.0252652
$665$	0	0
$666$	6.94976e9	0.00136879
$667$	4.30742e12	0.842658
$668$	−6.99829e12	−1.35987
$669$	−2.43557e12	−0.470093
$670$	0	0
$671$	−5.86784e12	−1.11745
$672$	8.27234e9	0.00156483
$673$	−6.44097e12	−1.21027	−0.605136	−	0.796122i	$-0.706881\pi$
$673$	−6.44097e12	−1.21027	−0.605136	+	0.796122i	$0.706881\pi$
$674$	−2.54130e10	−0.00474337
$675$	0	0
$676$	5.17284e12	0.952728
$677$	−7.45494e12	−1.36394	−0.681969	−	0.731381i	$-0.738876\pi$
$677$	−7.45494e12	−1.36394	−0.681969	+	0.731381i	$0.738876\pi$
$678$	3.33012e10	0.00605238
$679$	−7.75757e10	−0.0140059
$680$	0	0
$681$	4.18989e12	0.746518
$682$	5.92439e10	0.0104861
$683$	1.88685e11	0.0331776	0.0165888	−	0.999862i	$-0.494719\pi$
$683$	1.88685e11	0.0331776	0.0165888	+	0.999862i	$0.494719\pi$
$684$	8.34809e11	0.145826
$685$	0	0
$686$	−1.04384e10	−0.00179959
$687$	3.55625e12	0.609097
$688$	1.01948e13	1.73472
$689$	−2.61251e11	−0.0441644
$690$	0	0
$691$	−4.12572e12	−0.688412	−0.344206	−	0.938894i	$-0.611852\pi$
$691$	−4.12572e12	−0.688412	−0.344206	+	0.938894i	$0.611852\pi$
$692$	−9.42497e12	−1.56244
$693$	−1.80158e11	−0.0296725
$694$	8.46014e10	0.0138439
$695$	0	0
$696$	−8.06767e10	−0.0130319
$697$	−1.49761e12	−0.240354
$698$	1.02944e11	0.0164154
$699$	−4.21857e12	−0.668371
$700$	0	0
$701$	−6.26038e12	−0.979196	−0.489598	−	0.871948i	$-0.662857\pi$
$701$	−6.26038e12	−0.979196	−0.489598	+	0.871948i	$0.662857\pi$
$702$	−2.65809e9	−0.000413098 0
$703$	1.17732e12	0.181801
$704$	−6.34152e12	−0.973008
$705$	0	0
$706$	2.27492e10	0.00344624
$707$	−1.00169e11	−0.0150781
$708$	7.45379e12	1.11488
$709$	−8.40293e12	−1.24888	−0.624442	−	0.781071i	$-0.714674\pi$
$709$	−8.40293e12	−1.24888	−0.624442	+	0.781071i	$0.714674\pi$
$710$	0	0
$711$	−1.27835e11	−0.0187602
$712$	7.45723e10	0.0108747
$713$	5.54931e12	0.804148
$714$	−2.57619e9	−0.000370967 0
$715$	0	0
$716$	1.13575e13	1.61500
$717$	1.01273e11	0.0143105
$718$	−1.51193e10	−0.00212311
$719$	−3.44185e11	−0.0480300	−0.0240150	−	0.999712i	$-0.507645\pi$
$719$	−3.44185e11	−0.0480300	−0.0240150	+	0.999712i	$0.507645\pi$
$720$	0	0
$721$	1.00414e12	0.138383
$722$	5.83440e10	0.00799060
$723$	4.36659e11	0.0594320
$724$	6.46528e12	0.874508
$725$	0	0
$726$	2.22704e9	0.000297519 0
$727$	−8.67049e12	−1.15117	−0.575584	−	0.817743i	$-0.695225\pi$
$727$	−8.67049e12	−1.15117	−0.575584	+	0.817743i	$0.695225\pi$
$728$	−2.97467e9	−0.000392507 0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−9.52617e12	−1.23393
$732$	5.14699e12	0.662603
$733$	1.10927e13	1.41929	0.709644	−	0.704561i	$-0.248856\pi$
$733$	1.10927e13	1.41929	0.709644	+	0.704561i	$0.248856\pi$
$734$	−1.29934e11	−0.0165230
$735$	0	0
$736$	1.74110e11	0.0218713
$737$	−8.98909e12	−1.12231
$738$	−8.97239e9	−0.00111341
$739$	−6.37409e12	−0.786173	−0.393087	−	0.919501i	$-0.628593\pi$
$739$	−6.37409e12	−0.786173	−0.393087	+	0.919501i	$0.628593\pi$
$740$	0	0
$741$	−4.50294e11	−0.0548674
$742$	−1.51697e9	−0.000183721 0
$743$	−6.36760e12	−0.766525	−0.383263	−	0.923639i	$-0.625200\pi$
$743$	−6.36760e12	−0.766525	−0.383263	+	0.923639i	$0.625200\pi$
$744$	−1.03937e11	−0.0124363
$745$	0	0
$746$	−7.40004e10	−0.00874801
$747$	3.62641e12	0.426122
$748$	5.92679e12	0.692249
$749$	−1.23516e12	−0.143402
$750$	0	0
$751$	−3.40024e12	−0.390058	−0.195029	−	0.980797i	$-0.562480\pi$
$751$	−3.40024e12	−0.390058	−0.195029	+	0.980797i	$0.562480\pi$
$752$	−1.33586e13	−1.52328
$753$	3.07238e12	0.348255
$754$	2.17573e10	0.00245152
$755$	0	0
$756$	1.58026e11	0.0175946
$757$	1.38522e13	1.53316	0.766581	−	0.642148i	$-0.221956\pi$
$757$	1.38522e13	1.53316	0.766581	+	0.642148i	$0.221956\pi$
$758$	−1.16245e11	−0.0127898
$759$	−3.79183e12	−0.414726
$760$	0	0
$761$	−1.25728e13	−1.35894	−0.679471	−	0.733703i	$-0.737790\pi$
$761$	−1.25728e13	−1.35894	−0.679471	+	0.733703i	$0.737790\pi$
$762$	1.40294e9	0.000150745 0
$763$	−9.62812e11	−0.102845
$764$	1.53895e12	0.163420
$765$	0	0
$766$	1.00916e11	0.0105909
$767$	−4.02055e12	−0.419475
$768$	5.56084e12	0.576787
$769$	−1.11440e13	−1.14914	−0.574569	−	0.818456i	$-0.694830\pi$
$769$	−1.11440e13	−1.14914	−0.574569	+	0.818456i	$0.694830\pi$
$770$	0	0
$771$	9.98707e11	0.101787
$772$	−1.62741e12	−0.164899
$773$	−4.40857e12	−0.444110	−0.222055	−	0.975034i	$-0.571276\pi$
$773$	−4.40857e12	−0.444110	−0.222055	+	0.975034i	$0.571276\pi$
$774$	−5.70727e10	−0.00571602
$775$	0	0
$776$	3.05811e10	0.00302744
$777$	2.22862e11	0.0219352
$778$	−1.03286e11	−0.0101073
$779$	−1.51997e12	−0.147882
$780$	0	0
$781$	−5.36821e12	−0.516297
$782$	−5.42217e10	−0.00518493
$783$	−2.31178e12	−0.219795
$784$	−1.04869e13	−0.991349
$785$	0	0
$786$	−8.15928e10	−0.00762518
$787$	1.39083e13	1.29237	0.646186	−	0.763180i	$-0.276363\pi$
$787$	1.39083e13	1.29237	0.646186	+	0.763180i	$0.276363\pi$
$788$	−1.37018e13	−1.26593
$789$	−7.18057e12	−0.659649
$790$	0	0
$791$	1.06789e12	0.0969912
$792$	7.10199e10	0.00641382
$793$	−2.77627e12	−0.249306
$794$	1.07582e11	0.00960612
$795$	0	0
$796$	−1.81986e11	−0.0160668
$797$	−1.11093e13	−0.975266	−0.487633	−	0.873049i	$-0.662139\pi$
$797$	−1.11093e13	−0.975266	−0.487633	+	0.873049i	$0.662139\pi$
$798$	−2.61465e9	−0.000228245 0
$799$	1.24825e13	1.08353
$800$	0	0
$801$	2.13686e12	0.183412
$802$	−9.23316e10	−0.00788072
$803$	4.88379e12	0.414512
$804$	7.88479e12	0.665485
$805$	0	0
$806$	2.80303e10	0.00233948
$807$	5.29201e12	0.439227
$808$	3.94876e10	0.00325919
$809$	8.47032e12	0.695235	0.347617	−	0.937636i	$-0.386991\pi$
$809$	8.47032e12	0.695235	0.347617	+	0.937636i	$0.386991\pi$
$810$	0	0
$811$	1.98968e13	1.61506	0.807532	−	0.589823i	$-0.200803\pi$
$811$	1.98968e13	1.61506	0.807532	+	0.589823i	$0.200803\pi$
$812$	−1.29349e12	−0.104415
$813$	−3.59132e12	−0.288301
$814$	5.00769e10	0.00399786
$815$	0	0
$816$	−5.19818e12	−0.410436
$817$	−9.66840e12	−0.759198
$818$	−1.48900e11	−0.0116280
$819$	−8.52387e10	−0.00662002
$820$	0	0
$821$	−7.94161e11	−0.0610049	−0.0305024	−	0.999535i	$-0.509711\pi$
$821$	−7.94161e11	−0.0610049	−0.0305024	+	0.999535i	$0.509711\pi$
$822$	−1.23182e11	−0.00941071
$823$	−1.36089e13	−1.03401	−0.517004	−	0.855983i	$-0.672953\pi$
$823$	−1.36089e13	−1.03401	−0.517004	+	0.855983i	$0.672953\pi$
$824$	−3.95840e11	−0.0299121
$825$	0	0
$826$	−2.33455e10	−0.00174499
$827$	1.79571e13	1.33494	0.667470	−	0.744637i	$-0.267377\pi$
$827$	1.79571e13	1.33494	0.667470	+	0.744637i	$0.267377\pi$
$828$	3.32601e12	0.245916
$829$	−3.69953e12	−0.272052	−0.136026	−	0.990705i	$-0.543433\pi$
$829$	−3.69953e12	−0.272052	−0.136026	+	0.990705i	$0.543433\pi$
$830$	0	0
$831$	1.53404e13	1.11592
$832$	−3.00038e12	−0.217081
$833$	9.79919e12	0.705160
$834$	−3.53291e10	−0.00252863
$835$	0	0
$836$	6.01527e12	0.425919
$837$	−2.97829e12	−0.209750
$838$	−1.74367e11	−0.0122142
$839$	2.50920e13	1.74826	0.874132	−	0.485688i	$-0.161431\pi$
$839$	2.50920e13	1.74826	0.874132	+	0.485688i	$0.161431\pi$
$840$	0	0
$841$	4.41551e12	0.304368
$842$	−2.14468e11	−0.0147048
$843$	−1.29091e13	−0.880381
$844$	2.00162e13	1.35781
$845$	0	0
$846$	7.47843e10	0.00501931
$847$	7.14160e10	0.00476782
$848$	−3.06091e12	−0.203268
$849$	−7.48203e12	−0.494237
$850$	0	0
$851$	4.69065e12	0.306584
$852$	4.70874e12	0.306144
$853$	6.79534e12	0.439481	0.219741	−	0.975558i	$-0.429479\pi$
$853$	6.79534e12	0.439481	0.219741	+	0.975558i	$0.429479\pi$
$854$	−1.61205e10	−0.00103710
$855$	0	0
$856$	4.86912e11	0.0309969
$857$	1.31867e13	0.835070	0.417535	−	0.908661i	$-0.362894\pi$
$857$	1.31867e13	0.835070	0.417535	+	0.908661i	$0.362894\pi$
$858$	−1.91530e10	−0.00120655
$859$	1.27461e13	0.798747	0.399374	−	0.916788i	$-0.369228\pi$
$859$	1.27461e13	0.798747	0.399374	+	0.916788i	$0.369228\pi$
$860$	0	0
$861$	−2.87723e11	−0.0178427
$862$	−4.23926e9	−0.000261522 0
$863$	8.12433e12	0.498585	0.249292	−	0.968428i	$-0.419802\pi$
$863$	8.12433e12	0.498585	0.249292	+	0.968428i	$0.419802\pi$
$864$	−9.34444e10	−0.00570481
$865$	0	0
$866$	1.87811e11	0.0113472
$867$	−4.74835e12	−0.285402
$868$	−1.66642e12	−0.0996429
$869$	−9.21126e11	−0.0547937
$870$	0	0
$871$	−4.25304e12	−0.250390
$872$	3.79550e11	0.0222303
$873$	8.76295e11	0.0510607
$874$	−5.50313e10	−0.00319013
$875$	0	0
$876$	−4.28382e12	−0.245789
$877$	6.46585e12	0.369086	0.184543	−	0.982824i	$-0.440920\pi$
$877$	6.46585e12	0.369086	0.184543	+	0.982824i	$0.440920\pi$
$878$	8.37180e10	0.00475438
$879$	1.19356e13	0.674366
$880$	0	0
$881$	−1.69271e13	−0.946653	−0.473326	−	0.880887i	$-0.656947\pi$
$881$	−1.69271e13	−0.946653	−0.473326	+	0.880887i	$0.656947\pi$
$882$	5.87084e10	0.00326656
$883$	−3.56100e13	−1.97128	−0.985642	−	0.168846i	$-0.945996\pi$
$883$	−3.56100e13	−1.97128	−0.985642	+	0.168846i	$0.945996\pi$
$884$	2.80416e12	0.154443
$885$	0	0
$886$	−1.57435e11	−0.00858321
$887$	2.18278e13	1.18400	0.592002	−	0.805936i	$-0.298338\pi$
$887$	2.18278e13	1.18400	0.592002	+	0.805936i	$0.298338\pi$
$888$	−8.78544e10	−0.00474139
$889$	4.49891e10	0.00241573
$890$	0	0
$891$	2.03506e12	0.108175
$892$	1.53937e13	0.814145
$893$	1.26688e13	0.666661
$894$	1.92066e11	0.0100561
$895$	0	0
$896$	−6.97112e10	−0.00361340
$897$	−1.79404e12	−0.0925266
$898$	−6.75602e10	−0.00346695
$899$	2.43783e13	1.24476
$900$	0	0
$901$	2.86016e12	0.144587
$902$	−6.46511e10	−0.00325197
$903$	−1.83019e12	−0.0916010
$904$	−4.20972e11	−0.0209650
$905$	0	0
$906$	1.18564e11	0.00584622
$907$	4.79737e12	0.235380	0.117690	−	0.993050i	$-0.462451\pi$
$907$	4.79737e12	0.235380	0.117690	+	0.993050i	$0.462451\pi$
$908$	−2.64816e13	−1.29288
$909$	1.13151e12	0.0549695
$910$	0	0
$911$	1.70045e13	0.817957	0.408979	−	0.912544i	$-0.365885\pi$
$911$	1.70045e13	0.817957	0.408979	+	0.912544i	$0.365885\pi$
$912$	−5.27579e12	−0.252529
$913$	2.61303e13	1.24459
$914$	5.47549e10	0.00259516
$915$	0	0
$916$	−2.24768e13	−1.05488
$917$	−2.61649e12	−0.122196
$918$	2.91006e10	0.00135242
$919$	−1.12820e12	−0.0521756	−0.0260878	−	0.999660i	$-0.508305\pi$
$919$	−1.12820e12	−0.0521756	−0.0260878	+	0.999660i	$0.508305\pi$
$920$	0	0
$921$	−1.49960e13	−0.686764
$922$	1.27028e11	0.00578909
$923$	−2.53988e12	−0.115188
$924$	1.13866e12	0.0513892
$925$	0	0
$926$	7.31253e10	0.00326827
$927$	−1.13427e13	−0.504497
$928$	7.64873e11	0.0338550
$929$	−3.97727e13	−1.75192	−0.875960	−	0.482384i	$-0.839771\pi$
$929$	−3.97727e13	−1.75192	−0.875960	+	0.482384i	$0.839771\pi$
$930$	0	0
$931$	9.94549e12	0.433863
$932$	2.66629e13	1.15754
$933$	−1.32218e13	−0.571247
$934$	−1.59094e11	−0.00684058
$935$	0	0
$936$	3.36019e10	0.00143094
$937$	−3.91551e13	−1.65944	−0.829718	−	0.558184i	$-0.811499\pi$
$937$	−3.91551e13	−1.65944	−0.829718	+	0.558184i	$0.811499\pi$
$938$	−2.46954e10	−0.00104161
$939$	1.28639e13	0.539980
$940$	0	0
$941$	2.55120e13	1.06070	0.530348	−	0.847780i	$-0.322061\pi$
$941$	2.55120e13	1.06070	0.530348	+	0.847780i	$0.322061\pi$
$942$	8.76251e10	0.00362577
$943$	−6.05579e12	−0.249384
$944$	−4.71061e13	−1.93065
$945$	0	0
$946$	−4.11241e11	−0.0166950
$947$	−1.92093e12	−0.0776133	−0.0388066	−	0.999247i	$-0.512356\pi$
$947$	−1.92093e12	−0.0776133	−0.0388066	+	0.999247i	$0.512356\pi$
$948$	8.07967e11	0.0324905
$949$	2.31068e12	0.0924789
$950$	0	0
$951$	−1.80629e13	−0.716101
$952$	3.25665e10	0.00128501
$953$	3.70640e13	1.45558	0.727788	−	0.685802i	$-0.240549\pi$
$953$	3.70640e13	1.45558	0.727788	+	0.685802i	$0.240549\pi$
$954$	1.71357e10	0.000669781 0
$955$	0	0
$956$	−6.40081e11	−0.0247842
$957$	−1.66577e13	−0.641963
$958$	−7.52794e10	−0.00288756
$959$	−3.95014e12	−0.150809
$960$	0	0
$961$	4.96727e12	0.187872
$962$	2.36931e10	0.000891935 0
$963$	1.39524e13	0.522793
$964$	−2.75985e12	−0.102929
$965$	0	0
$966$	−1.04172e10	−0.000384905 0
$967$	−2.97114e13	−1.09271	−0.546353	−	0.837555i	$-0.683984\pi$
$967$	−2.97114e13	−1.09271	−0.546353	+	0.837555i	$0.683984\pi$
$968$	−2.81529e10	−0.00103058
$969$	4.92979e12	0.179627
$970$	0	0
$971$	4.42639e13	1.59795	0.798975	−	0.601364i	$-0.205376\pi$
$971$	4.42639e13	1.59795	0.798975	+	0.601364i	$0.205376\pi$
$972$	−1.78506e12	−0.0641438
$973$	−1.13292e12	−0.0405221
$974$	3.72571e11	0.0132646
$975$	0	0
$976$	−3.25277e13	−1.14744
$977$	−3.02165e12	−0.106101	−0.0530503	−	0.998592i	$-0.516894\pi$
$977$	−3.02165e12	−0.106101	−0.0530503	+	0.998592i	$0.516894\pi$
$978$	9.83008e10	0.00343584
$979$	1.53972e13	0.535699
$980$	0	0
$981$	1.08759e13	0.374935
$982$	−1.74586e11	−0.00599113
$983$	−2.56151e13	−0.874994	−0.437497	−	0.899220i	$-0.644135\pi$
$983$	−2.56151e13	−0.874994	−0.437497	+	0.899220i	$0.644135\pi$
$984$	1.13423e11	0.00385677
$985$	0	0
$986$	−2.38198e11	−0.00802587
$987$	2.39815e12	0.0804359
$988$	2.84602e12	0.0950238
$989$	−3.85204e13	−1.28029
$990$	0	0
$991$	4.40641e12	0.145129	0.0725644	−	0.997364i	$-0.476882\pi$
$991$	4.40641e12	0.145129	0.0725644	+	0.997364i	$0.476882\pi$
$992$	9.85396e11	0.0323079
$993$	2.14273e13	0.699353
$994$	−1.47479e10	−0.000479172 0
$995$	0	0
$996$	−2.29202e13	−0.737993
$997$	−4.49122e13	−1.43958	−0.719790	−	0.694192i	$-0.755762\pi$
$997$	−4.49122e13	−1.43958	−0.719790	+	0.694192i	$0.755762\pi$
$998$	1.33742e11	0.00426756
$999$	−2.51745e12	−0.0799681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.e.1.2	✓	2
3.2	odd	2		225.10.a.l.1.1		2
5.2	odd	4		75.10.b.g.49.2		4
5.3	odd	4		75.10.b.g.49.3		4
5.4	even	2		75.10.a.h.1.1	yes	2
15.2	even	4		225.10.b.j.199.3		4
15.8	even	4		225.10.b.j.199.2		4
15.14	odd	2		225.10.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.10.a.e.1.2	✓	2	1.1	even	1	trivial
75.10.a.h.1.1	yes	2	5.4	even	2
75.10.b.g.49.2		4	5.2	odd	4
75.10.b.g.49.3		4	5.3	odd	4
225.10.a.g.1.2		2	15.14	odd	2
225.10.a.l.1.1		2	3.2	odd	2
225.10.b.j.199.2		4	15.8	even	4
225.10.b.j.199.3		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)

\(f(q)\)	\(=\)	\(q-0.223611 q^{2} +81.0000 q^{3} -511.950 q^{4} -18.1125 q^{6} -580.825 q^{7} +228.967 q^{8} +6561.00 q^{9} +O(q^{10})\)	\(q-0.223611 q^{2} +81.0000 q^{3} -511.950 q^{4} -18.1125 q^{6} -580.825 q^{7} +228.967 q^{8} +6561.00 q^{9} +47275.7 q^{11} -41467.9 q^{12} +22367.7 q^{13} +129.879 q^{14} +262067. q^{16} -244880. q^{17} -1467.11 q^{18} -248536. q^{19} -47046.8 q^{21} -10571.4 q^{22} -990208. q^{23} +18546.3 q^{24} -5001.67 q^{26} +531441. q^{27} +297353. q^{28} -4.35002e6 q^{29} -5.60419e6 q^{31} -175832. q^{32} +3.82933e6 q^{33} +54758.0 q^{34} -3.35890e6 q^{36} -4.73703e6 q^{37} +55575.5 q^{38} +1.81178e6 q^{39} +6.11568e6 q^{41} +10520.2 q^{42} +3.89014e7 q^{43} -2.42028e7 q^{44} +221422. q^{46} -5.09738e7 q^{47} +2.12274e7 q^{48} -4.00162e7 q^{49} -1.98353e7 q^{51} -1.14511e7 q^{52} -1.16798e7 q^{53} -118836. q^{54} -132990. q^{56} -2.01314e7 q^{57} +972713. q^{58} -1.79748e8 q^{59} -1.24120e8 q^{61} +1.25316e6 q^{62} -3.81079e6 q^{63} -1.34139e8 q^{64} -856281. q^{66} -1.90142e8 q^{67} +1.25366e8 q^{68} -8.02068e7 q^{69} -1.13551e8 q^{71} +1.50225e6 q^{72} +1.03304e8 q^{73} +1.05925e6 q^{74} +1.27238e8 q^{76} -2.74589e7 q^{77} -405135. q^{78} -1.94841e7 q^{79} +4.30467e7 q^{81} -1.36753e6 q^{82} +5.52722e8 q^{83} +2.40856e7 q^{84} -8.69878e6 q^{86} -3.52351e8 q^{87} +1.08246e7 q^{88} +3.25690e8 q^{89} -1.29917e7 q^{91} +5.06937e8 q^{92} -4.53939e8 q^{93} +1.13983e7 q^{94} -1.42424e7 q^{96} +1.33561e8 q^{97} +8.94808e6 q^{98} +3.10176e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 36 q^{2} + 162 q^{3} + 256 q^{4} - 2916 q^{6} + 3318 q^{7} - 8928 q^{8} + 13122 q^{9}+O(q^{10}) \) 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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