LMFDB - Embedded newform 75.12.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,12,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, 12, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$57.6257385420$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6154x^{2} - 41770x + 5647125 $ x^4 - 2x^3 - 6154x^2 - 41770*x + 5647125
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$76.7687$ 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)

$f(q)$	$=$	$q+87.7687 q^{2} +243.000 q^{3} +5655.34 q^{4} +21327.8 q^{6} +62441.2 q^{7} +316612. q^{8} +59049.0 q^{9} +O(q^{10})$	\(q+87.7687 q^{2} +243.000 q^{3} +5655.34 q^{4} +21327.8 q^{6} +62441.2 q^{7} +316612. q^{8} +59049.0 q^{9} -382888. q^{11} +1.37425e6 q^{12} +1.05493e6 q^{13} +5.48039e6 q^{14} +1.62065e7 q^{16} -7.93645e6 q^{17} +5.18265e6 q^{18} -1.79788e7 q^{19} +1.51732e7 q^{21} -3.36056e7 q^{22} +1.91419e7 q^{23} +7.69367e7 q^{24} +9.25895e7 q^{26} +1.43489e7 q^{27} +3.53127e8 q^{28} -3.46725e7 q^{29} -1.80961e8 q^{31} +7.74000e8 q^{32} -9.30418e7 q^{33} -6.96572e8 q^{34} +3.33942e8 q^{36} +6.73512e8 q^{37} -1.57798e9 q^{38} +2.56347e8 q^{39} -1.84925e8 q^{41} +1.33173e9 q^{42} +6.10683e8 q^{43} -2.16536e9 q^{44} +1.68006e9 q^{46} +5.06891e8 q^{47} +3.93817e9 q^{48} +1.92158e9 q^{49} -1.92856e9 q^{51} +5.96597e9 q^{52} -1.57990e8 q^{53} +1.25938e9 q^{54} +1.97696e10 q^{56} -4.36885e9 q^{57} -3.04316e9 q^{58} -1.00470e10 q^{59} +6.77049e8 q^{61} -1.58827e10 q^{62} +3.68709e9 q^{63} +3.47421e10 q^{64} -8.16616e9 q^{66} +1.50318e10 q^{67} -4.48834e10 q^{68} +4.65148e9 q^{69} -1.00345e10 q^{71} +1.86956e10 q^{72} +7.06118e9 q^{73} +5.91133e10 q^{74} -1.01676e11 q^{76} -2.39080e10 q^{77} +2.24993e10 q^{78} -4.13923e10 q^{79} +3.48678e9 q^{81} -1.62306e10 q^{82} -4.61318e10 q^{83} +8.58098e10 q^{84} +5.35988e10 q^{86} -8.42542e9 q^{87} -1.21227e11 q^{88} +2.72099e9 q^{89} +6.58709e10 q^{91} +1.08254e11 q^{92} -4.39735e10 q^{93} +4.44891e10 q^{94} +1.88082e11 q^{96} -1.31572e10 q^{97} +1.68655e11 q^{98} -2.26092e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9}+O(q^{10}) $ 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9	\( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9} - 99944 q^{11} + 1129464 q^{12} + 2306276 q^{13} + 3107022 q^{14} + 7622200 q^{16} - 3443816 q^{17} + 2716254 q^{18} - 4214548 q^{19} + 16614396 q^{21} - 34106532 q^{22} + 52691304 q^{23} + 93839796 q^{24} - 8766722 q^{26} + 57395628 q^{27} + 472439576 q^{28} + 217393304 q^{29} - 326317036 q^{31} + 788672072 q^{32} - 24286392 q^{33} - 564836244 q^{34} + 274459752 q^{36} + 273252872 q^{37} - 1762793006 q^{38} + 560425068 q^{39} - 22069456 q^{41} + 755006346 q^{42} - 705091900 q^{43} - 992474176 q^{44} + 2697853524 q^{46} + 187768360 q^{47} + 1852194600 q^{48} + 5213315400 q^{49} - 836847288 q^{51} + 7619620904 q^{52} - 6392224256 q^{53} + 660049722 q^{54} + 20338292700 q^{56} - 1024135164 q^{57} - 16176655332 q^{58} + 36710008 q^{59} + 11538870620 q^{61} - 4825203906 q^{62} + 4037298228 q^{63} + 55398609952 q^{64} - 8287887276 q^{66} + 37721158484 q^{67} - 45682615072 q^{68} + 12803986872 q^{69} + 8211316688 q^{71} + 22803070428 q^{72} + 5713413224 q^{73} + 136155026252 q^{74} - 78094966168 q^{76} + 72854549304 q^{77} - 2130313446 q^{78} + 45026381600 q^{79} + 13947137604 q^{81} + 55157549184 q^{82} - 104211315528 q^{83} + 114802816968 q^{84} + 77092671046 q^{86} + 52826572872 q^{87} - 175877029608 q^{88} + 111829609152 q^{89} + 225968327284 q^{91} + 88286181792 q^{92} - 79295039748 q^{93} + 193370717388 q^{94} + 191647313496 q^{96} + 77104304804 q^{97} - 139912937716 q^{98} - 5901593256 q^{99}+O(q^{100}) \) 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9 - 99944 * q^11 + 1129464 * q^12 + 2306276 * q^13 + 3107022 * q^14 + 7622200 * q^16 - 3443816 * q^17 + 2716254 * q^18 - 4214548 * q^19 + 16614396 * q^21 - 34106532 * q^22 + 52691304 * q^23 + 93839796 * q^24 - 8766722 * q^26 + 57395628 * q^27 + 472439576 * q^28 + 217393304 * q^29 - 326317036 * q^31 + 788672072 * q^32 - 24286392 * q^33 - 564836244 * q^34 + 274459752 * q^36 + 273252872 * q^37 - 1762793006 * q^38 + 560425068 * q^39 - 22069456 * q^41 + 755006346 * q^42 - 705091900 * q^43 - 992474176 * q^44 + 2697853524 * q^46 + 187768360 * q^47 + 1852194600 * q^48 + 5213315400 * q^49 - 836847288 * q^51 + 7619620904 * q^52 - 6392224256 * q^53 + 660049722 * q^54 + 20338292700 * q^56 - 1024135164 * q^57 - 16176655332 * q^58 + 36710008 * q^59 + 11538870620 * q^61 - 4825203906 * q^62 + 4037298228 * q^63 + 55398609952 * q^64 - 8287887276 * q^66 + 37721158484 * q^67 - 45682615072 * q^68 + 12803986872 * q^69 + 8211316688 * q^71 + 22803070428 * q^72 + 5713413224 * q^73 + 136155026252 * q^74 - 78094966168 * q^76 + 72854549304 * q^77 - 2130313446 * q^78 + 45026381600 * q^79 + 13947137604 * q^81 + 55157549184 * q^82 - 104211315528 * q^83 + 114802816968 * q^84 + 77092671046 * q^86 + 52826572872 * q^87 - 175877029608 * q^88 + 111829609152 * q^89 + 225968327284 * q^91 + 88286181792 * q^92 - 79295039748 * q^93 + 193370717388 * q^94 + 191647313496 * q^96 + 77104304804 * q^97 - 139912937716 * q^98 - 5901593256 * 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 $

$2$	87.7687	1.93943	0.969716	−	0.244234i	$-0.0785366\pi$
$2$	87.7687	1.93943	0.969716	+	0.244234i	$0.0785366\pi$
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	5655.34	2.76140
$5$	0	0
$5$	0	0
$6$	21327.8	1.11973
$7$	62441.2	1.40421	0.702105	−	0.712073i	$-0.252244\pi$
$7$	62441.2	1.40421	0.702105	+	0.712073i	$0.252244\pi$
$8$	316612.	3.41611
$9$	59049.0	0.333333
$10$	0	0
$11$	−382888.	−0.716823	−0.358412	−	0.933564i	$-0.616682\pi$
$11$	−382888.	−0.716823	−0.358412	+	0.933564i	$0.616682\pi$
$12$	1.37425e6	1.59429
$13$	1.05493e6	0.788014	0.394007	−	0.919108i	$-0.371089\pi$
$13$	1.05493e6	0.788014	0.394007	+	0.919108i	$0.371089\pi$
$14$	5.48039e6	2.72337
$15$	0	0
$16$	1.62065e7	3.86392
$17$	−7.93645e6	−1.35568	−0.677840	−	0.735209i	$-0.737084\pi$
$17$	−7.93645e6	−1.35568	−0.677840	+	0.735209i	$0.737084\pi$
$18$	5.18265e6	0.646478
$19$	−1.79788e7	−1.66577	−0.832887	−	0.553444i	$-0.813313\pi$
$19$	−1.79788e7	−1.66577	−0.832887	+	0.553444i	$0.813313\pi$
$20$	0	0
$21$	1.51732e7	0.810721
$22$	−3.36056e7	−1.39023
$23$	1.91419e7	0.620129	0.310065	−	0.950715i	$-0.399649\pi$
$23$	1.91419e7	0.620129	0.310065	+	0.950715i	$0.399649\pi$
$24$	7.69367e7	1.97229
$25$	0	0
$26$	9.25895e7	1.52830
$27$	1.43489e7	0.192450
$28$	3.53127e8	3.87759
$29$	−3.46725e7	−0.313904	−0.156952	−	0.987606i	$-0.550167\pi$
$29$	−3.46725e7	−0.313904	−0.156952	+	0.987606i	$0.550167\pi$
$30$	0	0
$31$	−1.80961e8	−1.13526	−0.567631	−	0.823283i	$-0.692140\pi$
$31$	−1.80961e8	−1.13526	−0.567631	+	0.823283i	$0.692140\pi$
$32$	7.74000e8	4.07771
$33$	−9.30418e7	−0.413858
$34$	−6.96572e8	−2.62925
$35$	0	0
$36$	3.33942e8	0.920466
$37$	6.73512e8	1.59675	0.798373	−	0.602163i	$-0.205694\pi$
$37$	6.73512e8	1.59675	0.798373	+	0.602163i	$0.205694\pi$
$38$	−1.57798e9	−3.23065
$39$	2.56347e8	0.454960
$40$	0	0
$41$	−1.84925e8	−0.249278	−0.124639	−	0.992202i	$-0.539777\pi$
$41$	−1.84925e8	−0.249278	−0.124639	+	0.992202i	$0.539777\pi$
$42$	1.33173e9	1.57234
$43$	6.10683e8	0.633489	0.316745	−	0.948511i	$-0.397410\pi$
$43$	6.10683e8	0.633489	0.316745	+	0.948511i	$0.397410\pi$
$44$	−2.16536e9	−1.97943
$45$	0	0
$46$	1.68006e9	1.20270
$47$	5.06891e8	0.322386	0.161193	−	0.986923i	$-0.448466\pi$
$47$	5.06891e8	0.322386	0.161193	+	0.986923i	$0.448466\pi$
$48$	3.93817e9	2.23084
$49$	1.92158e9	0.971808
$50$	0	0
$51$	−1.92856e9	−0.782703
$52$	5.96597e9	2.17602
$53$	−1.57990e8	−0.0518933	−0.0259467	−	0.999663i	$-0.508260\pi$
$53$	−1.57990e8	−0.0518933	−0.0259467	+	0.999663i	$0.508260\pi$
$54$	1.25938e9	0.373244
$55$	0	0
$56$	1.97696e10	4.79694
$57$	−4.36885e9	−0.961735
$58$	−3.04316e9	−0.608795
$59$	−1.00470e10	−1.82958	−0.914792	−	0.403926i	$-0.867645\pi$
$59$	−1.00470e10	−1.82958	−0.914792	+	0.403926i	$0.867645\pi$
$60$	0	0
$61$	6.77049e8	0.102637	0.0513187	−	0.998682i	$-0.483658\pi$
$61$	6.77049e8	0.102637	0.0513187	+	0.998682i	$0.483658\pi$
$62$	−1.58827e10	−2.20176
$63$	3.68709e9	0.468070
$64$	3.47421e10	4.04451
$65$	0	0
$66$	−8.16616e9	−0.802650
$67$	1.50318e10	1.36019	0.680093	−	0.733126i	$-0.261939\pi$
$67$	1.50318e10	1.36019	0.680093	+	0.733126i	$0.261939\pi$
$68$	−4.48834e10	−3.74357
$69$	4.65148e9	0.358032
$70$	0	0
$71$	−1.00345e10	−0.660049	−0.330025	−	0.943972i	$-0.607057\pi$
$71$	−1.00345e10	−0.660049	−0.330025	+	0.943972i	$0.607057\pi$
$72$	1.86956e10	1.13870
$73$	7.06118e9	0.398659	0.199329	−	0.979933i	$-0.436124\pi$
$73$	7.06118e9	0.398659	0.199329	+	0.979933i	$0.436124\pi$
$74$	5.91133e10	3.09678
$75$	0	0
$76$	−1.01676e11	−4.59986
$77$	−2.39080e10	−1.00657
$78$	2.24993e10	0.882364
$79$	−4.13923e10	−1.51346	−0.756729	−	0.653729i	$-0.773204\pi$
$79$	−4.13923e10	−1.51346	−0.756729	+	0.653729i	$0.773204\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	−1.62306e10	−0.483458
$83$	−4.61318e10	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$83$	−4.61318e10	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$84$	8.58098e10	2.23873
$85$	0	0
$86$	5.35988e10	1.22861
$87$	−8.42542e9	−0.181232
$88$	−1.21227e11	−2.44875
$89$	2.72099e9	0.0516514	0.0258257	−	0.999666i	$-0.491779\pi$
$89$	2.72099e9	0.0516514	0.0258257	+	0.999666i	$0.491779\pi$
$90$	0	0
$91$	6.58709e10	1.10654
$92$	1.08254e11	1.71242
$93$	−4.39735e10	−0.655443
$94$	4.44891e10	0.625246
$95$	0	0
$96$	1.88082e11	2.35426
$97$	−1.31572e10	−0.155568	−0.0777839	−	0.996970i	$-0.524784\pi$
$97$	−1.31572e10	−0.155568	−0.0777839	+	0.996970i	$0.524784\pi$
$98$	1.68655e11	1.88476
$99$	−2.26092e10	−0.238941
$100$	0	0
$101$	8.76099e10	0.829441	0.414721	−	0.909949i	$-0.363879\pi$
$101$	8.76099e10	0.829441	0.414721	+	0.909949i	$0.363879\pi$
$102$	−1.69267e11	−1.51800
$103$	3.28747e10	0.279420	0.139710	−	0.990192i	$-0.455383\pi$
$103$	3.28747e10	0.279420	0.139710	+	0.990192i	$0.455383\pi$
$104$	3.34002e11	2.69194
$105$	0	0
$106$	−1.38666e10	−0.100644
$107$	−4.70344e10	−0.324194	−0.162097	−	0.986775i	$-0.551826\pi$
$107$	−4.70344e10	−0.324194	−0.162097	+	0.986775i	$0.551826\pi$
$108$	8.11480e10	0.531431
$109$	−1.88215e9	−0.0117168	−0.00585838	−	0.999983i	$-0.501865\pi$
$109$	−1.88215e9	−0.0117168	−0.00585838	+	0.999983i	$0.501865\pi$
$110$	0	0
$111$	1.63663e11	0.921882
$112$	1.01195e12	5.42576
$113$	2.43261e11	1.24205	0.621027	−	0.783789i	$-0.286716\pi$
$113$	2.43261e11	1.24205	0.621027	+	0.783789i	$0.286716\pi$
$114$	−3.83448e11	−1.86522
$115$	0	0
$116$	−1.96085e11	−0.866813
$117$	6.22924e10	0.262671
$118$	−8.81816e11	−3.54835
$119$	−4.95562e11	−1.90366
$120$	0	0
$121$	−1.38708e11	−0.486165
$122$	5.94237e10	0.199059
$123$	−4.49368e10	−0.143921
$124$	−1.02340e12	−3.13491
$125$	0	0
$126$	3.23611e11	0.907791
$127$	−2.25739e10	−0.0606298	−0.0303149	−	0.999540i	$-0.509651\pi$
$127$	−2.25739e10	−0.0606298	−0.0303149	+	0.999540i	$0.509651\pi$
$128$	1.46412e12	3.76635
$129$	1.48396e11	0.365745
$130$	0	0
$131$	2.51032e11	0.568509	0.284255	−	0.958749i	$-0.408254\pi$
$131$	2.51032e11	0.568509	0.284255	+	0.958749i	$0.408254\pi$
$132$	−5.26183e11	−1.14283
$133$	−1.12262e12	−2.33910
$134$	1.31932e12	2.63799
$135$	0	0
$136$	−2.51278e12	−4.63116
$137$	1.50653e11	0.266694	0.133347	−	0.991069i	$-0.457428\pi$
$137$	1.50653e11	0.266694	0.133347	+	0.991069i	$0.457428\pi$
$138$	4.08255e11	0.694379
$139$	1.92197e11	0.314170	0.157085	−	0.987585i	$-0.449790\pi$
$139$	1.92197e11	0.314170	0.157085	+	0.987585i	$0.449790\pi$
$140$	0	0
$141$	1.23174e11	0.186130
$142$	−8.80718e11	−1.28012
$143$	−4.03919e11	−0.564866
$144$	9.56976e11	1.28797
$145$	0	0
$146$	6.19751e11	0.773172
$147$	4.66944e11	0.561073
$148$	3.80894e12	4.40925
$149$	−3.50463e11	−0.390947	−0.195473	−	0.980709i	$-0.562624\pi$
$149$	−3.50463e11	−0.390947	−0.195473	+	0.980709i	$0.562624\pi$
$150$	0	0
$151$	7.55249e11	0.782919	0.391460	−	0.920195i	$-0.371970\pi$
$151$	7.55249e11	0.782919	0.391460	+	0.920195i	$0.371970\pi$
$152$	−5.69230e12	−5.69047
$153$	−4.68640e11	−0.451894
$154$	−2.09837e12	−1.95218
$155$	0	0
$156$	1.44973e12	1.25633
$157$	−1.05014e12	−0.878619	−0.439309	−	0.898336i	$-0.644777\pi$
$157$	−1.05014e12	−0.878619	−0.439309	+	0.898336i	$0.644777\pi$
$158$	−3.63295e12	−2.93525
$159$	−3.83915e10	−0.0299606
$160$	0	0
$161$	1.19524e12	0.870792
$162$	3.06031e11	0.215493
$163$	−1.64615e12	−1.12057	−0.560283	−	0.828301i	$-0.689308\pi$
$163$	−1.64615e12	−1.12057	−0.560283	+	0.828301i	$0.689308\pi$
$164$	−1.04581e12	−0.688357
$165$	0	0
$166$	−4.04893e12	−2.49314
$167$	−1.04362e12	−0.621731	−0.310866	−	0.950454i	$-0.600619\pi$
$167$	−1.04362e12	−0.621731	−0.310866	+	0.950454i	$0.600619\pi$
$168$	4.80402e12	2.76952
$169$	−6.79291e11	−0.379034
$170$	0	0
$171$	−1.06163e12	−0.555258
$172$	3.45362e12	1.74932
$173$	−2.55351e12	−1.25281	−0.626403	−	0.779500i	$-0.715473\pi$
$173$	−2.55351e12	−1.25281	−0.626403	+	0.779500i	$0.715473\pi$
$174$	−7.39488e11	−0.351488
$175$	0	0
$176$	−6.20526e12	−2.76975
$177$	−2.44143e12	−1.05631
$178$	2.38818e11	0.100174
$179$	−4.56056e12	−1.85493	−0.927464	−	0.373913i	$-0.878016\pi$
$179$	−4.56056e12	−1.85493	−0.927464	+	0.373913i	$0.878016\pi$
$180$	0	0
$181$	2.60242e12	0.995740	0.497870	−	0.867252i	$-0.334116\pi$
$181$	2.60242e12	0.995740	0.497870	+	0.867252i	$0.334116\pi$
$182$	5.78140e12	2.14605
$183$	1.64523e11	0.0592578
$184$	6.06056e12	2.11843
$185$	0	0
$186$	−3.85950e12	−1.27119
$187$	3.03877e12	0.971783
$188$	2.86664e12	0.890236
$189$	8.95964e11	0.270240
$190$	0	0
$191$	3.68804e12	1.04981	0.524906	−	0.851160i	$-0.324100\pi$
$191$	3.68804e12	1.04981	0.524906	+	0.851160i	$0.324100\pi$
$192$	8.44233e12	2.33510
$193$	−2.81410e12	−0.756440	−0.378220	−	0.925716i	$-0.623464\pi$
$193$	−2.81410e12	−0.756440	−0.378220	+	0.925716i	$0.623464\pi$
$194$	−1.15479e12	−0.301713
$195$	0	0
$196$	1.08672e13	2.68355
$197$	−7.07345e12	−1.69851	−0.849253	−	0.527986i	$-0.822947\pi$
$197$	−7.07345e12	−1.69851	−0.849253	+	0.527986i	$0.822947\pi$
$198$	−1.98438e12	−0.463410
$199$	−3.27038e11	−0.0742858	−0.0371429	−	0.999310i	$-0.511826\pi$
$199$	−3.27038e11	−0.0742858	−0.0371429	+	0.999310i	$0.511826\pi$
$200$	0	0
$201$	3.65272e12	0.785304
$202$	7.68940e12	1.60864
$203$	−2.16500e12	−0.440787
$204$	−1.09067e13	−2.16135
$205$	0	0
$206$	2.88537e12	0.541915
$207$	1.13031e12	0.206710
$208$	1.70966e13	3.04482
$209$	6.88387e12	1.19406
$210$	0	0
$211$	8.57109e12	1.41086	0.705428	−	0.708782i	$-0.250755\pi$
$211$	8.57109e12	1.41086	0.705428	+	0.708782i	$0.250755\pi$
$212$	−8.93486e11	−0.143298
$213$	−2.43839e12	−0.381080
$214$	−4.12815e12	−0.628752
$215$	0	0
$216$	4.54303e12	0.657431
$217$	−1.12994e13	−1.59415
$218$	−1.65194e11	−0.0227239
$219$	1.71587e12	0.230166
$220$	0	0
$221$	−8.37237e12	−1.06829
$222$	1.43645e13	1.78793
$223$	1.06035e13	1.28757	0.643787	−	0.765205i	$-0.277362\pi$
$223$	1.06035e13	1.28757	0.643787	+	0.765205i	$0.277362\pi$
$224$	4.83295e13	5.72596
$225$	0	0
$226$	2.13507e13	2.40888
$227$	1.14613e13	1.26209	0.631046	−	0.775745i	$-0.282626\pi$
$227$	1.14613e13	1.26209	0.631046	+	0.775745i	$0.282626\pi$
$228$	−2.47073e13	−2.65573
$229$	−7.08540e12	−0.743480	−0.371740	−	0.928337i	$-0.621239\pi$
$229$	−7.08540e12	−0.743480	−0.371740	+	0.928337i	$0.621239\pi$
$230$	0	0
$231$	−5.80964e12	−0.581144
$232$	−1.09777e13	−1.07233
$233$	1.35453e13	1.29221	0.646103	−	0.763250i	$-0.276398\pi$
$233$	1.35453e13	1.29221	0.646103	+	0.763250i	$0.276398\pi$
$234$	5.46732e12	0.509433
$235$	0	0
$236$	−5.68195e13	−5.05221
$237$	−1.00583e13	−0.873795
$238$	−4.34948e13	−3.69202
$239$	−8.89346e12	−0.737704	−0.368852	−	0.929488i	$-0.620249\pi$
$239$	−8.89346e12	−0.737704	−0.368852	+	0.929488i	$0.620249\pi$
$240$	0	0
$241$	−1.45080e13	−1.14952	−0.574758	−	0.818323i	$-0.694904\pi$
$241$	−1.45080e13	−1.14952	−0.574758	+	0.818323i	$0.694904\pi$
$242$	−1.21743e13	−0.942883
$243$	8.47289e11	0.0641500
$244$	3.82895e12	0.283423
$245$	0	0
$246$	−3.94404e12	−0.279125
$247$	−1.89663e13	−1.31265
$248$	−5.72944e13	−3.87818
$249$	−1.12100e13	−0.742182
$250$	0	0
$251$	2.43759e13	1.54438	0.772192	−	0.635390i	$-0.219161\pi$
$251$	2.43759e13	1.54438	0.772192	+	0.635390i	$0.219161\pi$
$252$	2.08518e13	1.29253
$253$	−7.32921e12	−0.444523
$254$	−1.98128e12	−0.117587
$255$	0	0
$256$	5.73518e13	3.26007
$257$	1.72520e13	0.959857	0.479928	−	0.877308i	$-0.340663\pi$
$257$	1.72520e13	0.959857	0.479928	+	0.877308i	$0.340663\pi$
$258$	1.30245e13	0.709338
$259$	4.20549e13	2.24217
$260$	0	0
$261$	−2.04738e12	−0.104635
$262$	2.20328e13	1.10259
$263$	7.49613e12	0.367351	0.183675	−	0.982987i	$-0.441201\pi$
$263$	7.49613e12	0.367351	0.183675	+	0.982987i	$0.441201\pi$
$264$	−2.94581e13	−1.41379
$265$	0	0
$266$	−9.85308e13	−4.53652
$267$	6.61201e11	0.0298210
$268$	8.50098e13	3.75602
$269$	1.01471e13	0.439244	0.219622	−	0.975585i	$-0.429518\pi$
$269$	1.01471e13	0.439244	0.219622	+	0.975585i	$0.429518\pi$
$270$	0	0
$271$	−1.91819e12	−0.0797188	−0.0398594	−	0.999205i	$-0.512691\pi$
$271$	−1.91819e12	−0.0797188	−0.0398594	+	0.999205i	$0.512691\pi$
$272$	−1.28622e14	−5.23825
$273$	1.60066e13	0.638860
$274$	1.32226e13	0.517235
$275$	0	0
$276$	2.63057e13	0.988669
$277$	3.51445e13	1.29485	0.647423	−	0.762131i	$-0.275847\pi$
$277$	3.51445e13	1.29485	0.647423	+	0.762131i	$0.275847\pi$
$278$	1.68689e13	0.609312
$279$	−1.06856e13	−0.378420
$280$	0	0
$281$	4.10791e13	1.39874	0.699369	−	0.714760i	$-0.253464\pi$
$281$	4.10791e13	1.39874	0.699369	+	0.714760i	$0.253464\pi$
$282$	1.08109e13	0.360986
$283$	1.59532e13	0.522423	0.261211	−	0.965282i	$-0.415878\pi$
$283$	1.59532e13	0.522423	0.261211	+	0.965282i	$0.415878\pi$
$284$	−5.67487e13	−1.82266
$285$	0	0
$286$	−3.54514e13	−1.09552
$287$	−1.15469e13	−0.350039
$288$	4.57039e13	1.35924
$289$	2.87154e13	0.837870
$290$	0	0
$291$	−3.19720e12	−0.0898171
$292$	3.99334e13	1.10086
$293$	−1.95349e12	−0.0528492	−0.0264246	−	0.999651i	$-0.508412\pi$
$293$	−1.95349e12	−0.0528492	−0.0264246	+	0.999651i	$0.508412\pi$
$294$	4.09831e13	1.08816
$295$	0	0
$296$	2.13242e14	5.45467
$297$	−5.49403e12	−0.137953
$298$	−3.07597e13	−0.758215
$299$	2.01933e13	0.488670
$300$	0	0
$301$	3.81318e13	0.889552
$302$	6.62872e13	1.51842
$303$	2.12892e13	0.478878
$304$	−2.91373e14	−6.43642
$305$	0	0
$306$	−4.11319e13	−0.876417
$307$	−1.43844e13	−0.301044	−0.150522	−	0.988607i	$-0.548096\pi$
$307$	−1.43844e13	−0.301044	−0.150522	+	0.988607i	$0.548096\pi$
$308$	−1.35208e14	−2.77954
$309$	7.98855e12	0.161323
$310$	0	0
$311$	−5.72323e13	−1.11547	−0.557737	−	0.830018i	$-0.688330\pi$
$311$	−5.72323e13	−1.11547	−0.557737	+	0.830018i	$0.688330\pi$
$312$	8.11626e13	1.55419
$313$	7.60555e13	1.43099	0.715495	−	0.698618i	$-0.246201\pi$
$313$	7.60555e13	1.43099	0.715495	+	0.698618i	$0.246201\pi$
$314$	−9.21697e13	−1.70402
$315$	0	0
$316$	−2.34088e14	−4.17926
$317$	7.60401e12	0.133419	0.0667094	−	0.997772i	$-0.478750\pi$
$317$	7.60401e12	0.133419	0.0667094	+	0.997772i	$0.478750\pi$
$318$	−3.36957e12	−0.0581066
$319$	1.32757e13	0.225013
$320$	0	0
$321$	−1.14294e13	−0.187173
$322$	1.04905e14	1.68884
$323$	1.42688e14	2.25826
$324$	1.97190e13	0.306822
$325$	0	0
$326$	−1.44480e14	−2.17326
$327$	−4.57362e11	−0.00676468
$328$	−5.85495e13	−0.851563
$329$	3.16509e13	0.452698
$330$	0	0
$331$	1.63382e13	0.226021	0.113011	−	0.993594i	$-0.463951\pi$
$331$	1.63382e13	0.226021	0.113011	+	0.993594i	$0.463951\pi$
$332$	−2.60891e14	−3.54977
$333$	3.97702e13	0.532249
$334$	−9.15973e13	−1.20581
$335$	0	0
$336$	2.45904e14	3.13257
$337$	9.03816e13	1.13270	0.566351	−	0.824164i	$-0.308355\pi$
$337$	9.03816e13	1.13270	0.566351	+	0.824164i	$0.308355\pi$
$338$	−5.96205e13	−0.735112
$339$	5.91124e13	0.717100
$340$	0	0
$341$	6.92878e13	0.813782
$342$	−9.31779e13	−1.07688
$343$	−3.48081e12	−0.0395879
$344$	1.93349e14	2.16407
$345$	0	0
$346$	−2.24118e14	−2.42973
$347$	−1.02560e14	−1.09437	−0.547185	−	0.837011i	$-0.684301\pi$
$347$	−1.02560e14	−1.09437	−0.547185	+	0.837011i	$0.684301\pi$
$348$	−4.76487e13	−0.500455
$349$	6.90914e13	0.714305	0.357153	−	0.934046i	$-0.383748\pi$
$349$	6.90914e13	0.714305	0.357153	+	0.934046i	$0.383748\pi$
$350$	0	0
$351$	1.51370e13	0.151653
$352$	−2.96355e14	−2.92299
$353$	8.61771e13	0.836818	0.418409	−	0.908259i	$-0.362588\pi$
$353$	8.61771e13	0.836818	0.418409	+	0.908259i	$0.362588\pi$
$354$	−2.14281e14	−2.04864
$355$	0	0
$356$	1.53882e13	0.142630
$357$	−1.20422e14	−1.09908
$358$	−4.00275e14	−3.59751
$359$	3.91904e13	0.346865	0.173432	−	0.984846i	$-0.444514\pi$
$359$	3.91904e13	0.346865	0.173432	+	0.984846i	$0.444514\pi$
$360$	0	0
$361$	2.06747e14	1.77480
$362$	2.28411e14	1.93117
$363$	−3.37061e13	−0.280687
$364$	3.72523e14	3.05559
$365$	0	0
$366$	1.44400e13	0.114926
$367$	5.17829e13	0.405997	0.202999	−	0.979179i	$-0.434931\pi$
$367$	5.17829e13	0.405997	0.202999	+	0.979179i	$0.434931\pi$
$368$	3.10223e14	2.39613
$369$	−1.09196e13	−0.0830928
$370$	0	0
$371$	−9.86507e12	−0.0728692
$372$	−2.48685e14	−1.80994
$373$	−7.37267e13	−0.528720	−0.264360	−	0.964424i	$-0.585161\pi$
$373$	−7.37267e13	−0.528720	−0.264360	+	0.964424i	$0.585161\pi$
$374$	2.66709e14	1.88471
$375$	0	0
$376$	1.60488e14	1.10131
$377$	−3.65770e13	−0.247360
$378$	7.86376e13	0.524113
$379$	2.87720e14	1.88997	0.944985	−	0.327114i	$-0.106076\pi$
$379$	2.87720e14	1.88997	0.944985	+	0.327114i	$0.106076\pi$
$380$	0	0
$381$	−5.48546e12	−0.0350046
$382$	3.23694e14	2.03604
$383$	1.74954e14	1.08475	0.542375	−	0.840137i	$-0.317525\pi$
$383$	1.74954e14	1.08475	0.542375	+	0.840137i	$0.317525\pi$
$384$	3.55780e14	2.17450
$385$	0	0
$386$	−2.46990e14	−1.46706
$387$	3.60602e13	0.211163
$388$	−7.44086e13	−0.429584
$389$	−3.48617e14	−1.98438	−0.992191	−	0.124725i	$-0.960195\pi$
$389$	−3.48617e14	−1.98438	−0.992191	+	0.124725i	$0.960195\pi$
$390$	0	0
$391$	−1.51919e14	−0.840697
$392$	6.08396e14	3.31981
$393$	6.10008e13	0.328229
$394$	−6.20828e14	−3.29414
$395$	0	0
$396$	−1.27863e14	−0.659812
$397$	−2.11610e14	−1.07693	−0.538465	−	0.842648i	$-0.680996\pi$
$397$	−2.11610e14	−1.07693	−0.538465	+	0.842648i	$0.680996\pi$
$398$	−2.87037e13	−0.144072
$399$	−2.72796e14	−1.35048
$400$	0	0
$401$	−2.06412e14	−0.994127	−0.497064	−	0.867714i	$-0.665588\pi$
$401$	−2.06412e14	−0.994127	−0.497064	+	0.867714i	$0.665588\pi$
$402$	3.20594e14	1.52304
$403$	−1.90901e14	−0.894601
$404$	4.95464e14	2.29042
$405$	0	0
$406$	−1.90019e14	−0.854877
$407$	−2.57880e14	−1.14459
$408$	−6.10604e14	−2.67380
$409$	−2.96368e13	−0.128042	−0.0640212	−	0.997949i	$-0.520393\pi$
$409$	−2.96368e13	−0.128042	−0.0640212	+	0.997949i	$0.520393\pi$
$410$	0	0
$411$	3.66086e13	0.153976
$412$	1.85918e14	0.771589
$413$	−6.27350e14	−2.56912
$414$	9.92059e13	0.400900
$415$	0	0
$416$	8.16513e14	3.21329
$417$	4.67038e13	0.181386
$418$	6.04188e14	2.31581
$419$	−1.39207e14	−0.526604	−0.263302	−	0.964713i	$-0.584812\pi$
$419$	−1.39207e14	−0.526604	−0.263302	+	0.964713i	$0.584812\pi$
$420$	0	0
$421$	4.16722e14	1.53566	0.767830	−	0.640653i	$-0.221336\pi$
$421$	4.16722e14	1.53566	0.767830	+	0.640653i	$0.221336\pi$
$422$	7.52273e14	2.73626
$423$	2.99314e13	0.107462
$424$	−5.00214e13	−0.177273
$425$	0	0
$426$	−2.14014e14	−0.739078
$427$	4.22758e13	0.144125
$428$	−2.65996e14	−0.895228
$429$	−9.81523e13	−0.326126
$430$	0	0
$431$	5.08053e14	1.64545	0.822724	−	0.568441i	$-0.192453\pi$
$431$	5.08053e14	1.64545	0.822724	+	0.568441i	$0.192453\pi$
$432$	2.32545e14	0.743612
$433$	−1.86523e14	−0.588909	−0.294454	−	0.955666i	$-0.595138\pi$
$433$	−1.86523e14	−0.588909	−0.294454	+	0.955666i	$0.595138\pi$
$434$	−9.91736e14	−3.09174
$435$	0	0
$436$	−1.06442e13	−0.0323547
$437$	−3.44149e14	−1.03299
$438$	1.50599e14	0.446391
$439$	2.69745e14	0.789583	0.394792	−	0.918771i	$-0.370817\pi$
$439$	2.69745e14	0.789583	0.394792	+	0.918771i	$0.370817\pi$
$440$	0	0
$441$	1.13467e14	0.323936
$442$	−7.34832e14	−2.07189
$443$	5.01997e14	1.39791	0.698957	−	0.715163i	$-0.253648\pi$
$443$	5.01997e14	1.39791	0.698957	+	0.715163i	$0.253648\pi$
$444$	9.25573e14	2.54568
$445$	0	0
$446$	9.30655e14	2.49716
$447$	−8.51625e13	−0.225713
$448$	2.16934e15	5.67935
$449$	−4.61993e14	−1.19476	−0.597379	−	0.801959i	$-0.703791\pi$
$449$	−4.61993e14	−1.19476	−0.597379	+	0.801959i	$0.703791\pi$
$450$	0	0
$451$	7.08056e13	0.178688
$452$	1.37572e15	3.42981
$453$	1.83525e14	0.452019
$454$	1.00594e15	2.44774
$455$	0	0
$456$	−1.38323e15	−3.28539
$457$	3.87521e14	0.909402	0.454701	−	0.890644i	$-0.349746\pi$
$457$	3.87521e14	0.909402	0.454701	+	0.890644i	$0.349746\pi$
$458$	−6.21876e14	−1.44193
$459$	−1.13879e14	−0.260901
$460$	0	0
$461$	3.45901e13	0.0773744	0.0386872	−	0.999251i	$-0.487682\pi$
$461$	3.45901e13	0.0773744	0.0386872	+	0.999251i	$0.487682\pi$
$462$	−5.09905e14	−1.12709
$463$	−8.83904e14	−1.93068	−0.965339	−	0.261001i	$-0.915948\pi$
$463$	−8.83904e14	−1.93068	−0.965339	+	0.261001i	$0.915948\pi$
$464$	−5.61919e14	−1.21290
$465$	0	0
$466$	1.18886e15	2.50615
$467$	−4.52395e14	−0.942485	−0.471243	−	0.882004i	$-0.656194\pi$
$467$	−4.52395e14	−0.942485	−0.471243	+	0.882004i	$0.656194\pi$
$468$	3.52285e14	0.725340
$469$	9.38602e14	1.90999
$470$	0	0
$471$	−2.55185e14	−0.507271
$472$	−3.18101e15	−6.25006
$473$	−2.33823e14	−0.454100
$474$	−8.82806e14	−1.69467
$475$	0	0
$476$	−2.80257e15	−5.25677
$477$	−9.32913e12	−0.0172978
$478$	−7.80567e14	−1.43073
$479$	1.86634e14	0.338179	0.169089	−	0.985601i	$-0.445917\pi$
$479$	1.86634e14	0.338179	0.169089	+	0.985601i	$0.445917\pi$
$480$	0	0
$481$	7.10506e14	1.25826
$482$	−1.27335e15	−2.22941
$483$	2.90444e14	0.502752
$484$	−7.84444e14	−1.34249
$485$	0	0
$486$	7.43654e13	0.124415
$487$	8.05753e14	1.33288	0.666442	−	0.745557i	$-0.267816\pi$
$487$	8.05753e14	1.33288	0.666442	+	0.745557i	$0.267816\pi$
$488$	2.14362e14	0.350621
$489$	−4.00014e14	−0.646959
$490$	0	0
$491$	1.76869e13	0.0279707	0.0139854	−	0.999902i	$-0.495548\pi$
$491$	1.76869e13	0.0279707	0.0139854	+	0.999902i	$0.495548\pi$
$492$	−2.54133e14	−0.397423
$493$	2.75177e14	0.425553
$494$	−1.66465e15	−2.54580
$495$	0	0
$496$	−2.93274e15	−4.38656
$497$	−6.26569e14	−0.926848
$498$	−9.83890e14	−1.43941
$499$	−2.58112e14	−0.373470	−0.186735	−	0.982410i	$-0.559791\pi$
$499$	−2.58112e14	−0.373470	−0.186735	+	0.982410i	$0.559791\pi$
$500$	0	0
$501$	−2.53600e14	−0.358957
$502$	2.13944e15	2.99523
$503$	−1.06608e15	−1.47627	−0.738135	−	0.674653i	$-0.764293\pi$
$503$	−1.06608e15	−1.47627	−0.738135	+	0.674653i	$0.764293\pi$
$504$	1.16738e15	1.59898
$505$	0	0
$506$	−6.43275e14	−0.862122
$507$	−1.65068e14	−0.218836
$508$	−1.27663e14	−0.167423
$509$	4.34497e14	0.563689	0.281844	−	0.959460i	$-0.409054\pi$
$509$	4.34497e14	0.563689	0.281844	+	0.959460i	$0.409054\pi$
$510$	0	0
$511$	4.40909e14	0.559801
$512$	2.03518e15	2.55634
$513$	−2.57976e14	−0.320578
$514$	1.51418e15	1.86158
$515$	0	0
$516$	8.39230e14	1.00997
$517$	−1.94082e14	−0.231094
$518$	3.69111e15	4.34854
$519$	−6.20503e14	−0.723308
$520$	0	0
$521$	−1.04036e15	−1.18734	−0.593671	−	0.804708i	$-0.702322\pi$
$521$	−1.04036e15	−1.18734	−0.593671	+	0.804708i	$0.702322\pi$
$522$	−1.79696e14	−0.202932
$523$	7.19418e14	0.803937	0.401969	−	0.915653i	$-0.368326\pi$
$523$	7.19418e14	0.803937	0.401969	+	0.915653i	$0.368326\pi$
$524$	1.41967e15	1.56988
$525$	0	0
$526$	6.57926e14	0.712452
$527$	1.43619e15	1.53905
$528$	−1.50788e15	−1.59912
$529$	−5.86397e14	−0.615440
$530$	0	0
$531$	−5.93268e14	−0.609861
$532$	−6.34879e15	−6.45918
$533$	−1.95082e14	−0.196435
$534$	5.80328e13	0.0578358
$535$	0	0
$536$	4.75924e15	4.64655
$537$	−1.10822e15	−1.07094
$538$	8.90601e14	0.851885
$539$	−7.35751e14	−0.696614
$540$	0	0
$541$	−3.27090e14	−0.303447	−0.151723	−	0.988423i	$-0.548482\pi$
$541$	−3.27090e14	−0.303447	−0.151723	+	0.988423i	$0.548482\pi$
$542$	−1.68357e14	−0.154609
$543$	6.32389e14	0.574891
$544$	−6.14281e15	−5.52807
$545$	0	0
$546$	1.40488e15	1.23902
$547$	1.96798e15	1.71827	0.859133	−	0.511752i	$-0.171003\pi$
$547$	1.96798e15	1.71827	0.859133	+	0.511752i	$0.171003\pi$
$548$	8.51992e14	0.736449
$549$	3.99791e13	0.0342125
$550$	0	0
$551$	6.23370e14	0.522892
$552$	1.47272e15	1.22308
$553$	−2.58459e15	−2.12521
$554$	3.08458e15	2.51127
$555$	0	0
$556$	1.08694e15	0.867549
$557$	1.56986e15	1.24068	0.620338	−	0.784335i	$-0.286995\pi$
$557$	1.56986e15	1.24068	0.620338	+	0.784335i	$0.286995\pi$
$558$	−9.37858e14	−0.733921
$559$	6.44225e14	0.499198
$560$	0	0
$561$	7.38422e14	0.561059
$562$	3.60546e15	2.71276
$563$	2.04917e15	1.52680	0.763400	−	0.645926i	$-0.223528\pi$
$563$	2.04917e15	1.52680	0.763400	+	0.645926i	$0.223528\pi$
$564$	6.96594e14	0.513978
$565$	0	0
$566$	1.40019e15	1.01320
$567$	2.17719e14	0.156023
$568$	−3.17705e15	−2.25480
$569$	−3.03071e14	−0.213023	−0.106511	−	0.994311i	$-0.533968\pi$
$569$	−3.03071e14	−0.213023	−0.106511	+	0.994311i	$0.533968\pi$
$570$	0	0
$571$	−1.32350e15	−0.912482	−0.456241	−	0.889856i	$-0.650804\pi$
$571$	−1.32350e15	−0.912482	−0.456241	+	0.889856i	$0.650804\pi$
$572$	−2.28430e15	−1.55982
$573$	8.96193e14	0.606110
$574$	−1.01346e15	−0.678877
$575$	0	0
$576$	2.05149e15	1.34817
$577$	6.81552e14	0.443641	0.221821	−	0.975088i	$-0.428800\pi$
$577$	6.81552e14	0.443641	0.221821	+	0.975088i	$0.428800\pi$
$578$	2.52031e15	1.62499
$579$	−6.83827e14	−0.436731
$580$	0	0
$581$	−2.88053e15	−1.80511
$582$	−2.80614e14	−0.174194
$583$	6.04924e13	0.0371983
$584$	2.23565e15	1.36186
$585$	0	0
$586$	−1.71455e14	−0.102497
$587$	−1.94324e15	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$587$	−1.94324e15	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$588$	2.64073e15	1.54935
$589$	3.25346e15	1.89109
$590$	0	0
$591$	−1.71885e15	−0.980633
$592$	1.09153e16	6.16971
$593$	−1.93834e14	−0.108550	−0.0542750	−	0.998526i	$-0.517285\pi$
$593$	−1.93834e14	−0.108550	−0.0542750	+	0.998526i	$0.517285\pi$
$594$	−4.82203e14	−0.267550
$595$	0	0
$596$	−1.98199e15	−1.07956
$597$	−7.94702e13	−0.0428889
$598$	1.77234e15	0.947743
$599$	−1.40207e15	−0.742886	−0.371443	−	0.928456i	$-0.621137\pi$
$599$	−1.40207e15	−0.742886	−0.371443	+	0.928456i	$0.621137\pi$
$600$	0	0
$601$	−1.07407e15	−0.558759	−0.279379	−	0.960181i	$-0.590129\pi$
$601$	−1.07407e15	−0.558759	−0.279379	+	0.960181i	$0.590129\pi$
$602$	3.34678e15	1.72523
$603$	8.87611e14	0.453396
$604$	4.27119e15	2.16195
$605$	0	0
$606$	1.86853e15	0.928752
$607$	4.44600e14	0.218994	0.109497	−	0.993987i	$-0.465076\pi$
$607$	4.44600e14	0.218994	0.109497	+	0.993987i	$0.465076\pi$
$608$	−1.39156e16	−6.79253
$609$	−5.26094e14	−0.254488
$610$	0	0
$611$	5.34732e14	0.254045
$612$	−2.65032e15	−1.24786
$613$	−1.50291e15	−0.701295	−0.350648	−	0.936507i	$-0.614038\pi$
$613$	−1.50291e15	−0.701295	−0.350648	+	0.936507i	$0.614038\pi$
$614$	−1.26250e15	−0.583855
$615$	0	0
$616$	−7.56956e15	−3.43856
$617$	9.90889e14	0.446125	0.223063	−	0.974804i	$-0.428395\pi$
$617$	9.90889e14	0.446125	0.223063	+	0.974804i	$0.428395\pi$
$618$	7.01145e14	0.312875
$619$	2.38830e15	1.05631	0.528153	−	0.849149i	$-0.322885\pi$
$619$	2.38830e15	1.05631	0.528153	+	0.849149i	$0.322885\pi$
$620$	0	0
$621$	2.74666e14	0.119344
$622$	−5.02320e15	−2.16338
$623$	1.69902e14	0.0725295
$624$	4.15448e15	1.75793
$625$	0	0
$626$	6.67529e15	2.77531
$627$	1.67278e15	0.689394
$628$	−5.93892e15	−2.42622
$629$	−5.34530e15	−2.16468
$630$	0	0
$631$	−3.39467e15	−1.35094	−0.675469	−	0.737388i	$-0.736059\pi$
$631$	−3.39467e15	−1.35094	−0.675469	+	0.737388i	$0.736059\pi$
$632$	−1.31053e16	−5.17014
$633$	2.08277e15	0.814558
$634$	6.67394e14	0.258757
$635$	0	0
$636$	−2.17117e14	−0.0827332
$637$	2.02713e15	0.765798
$638$	1.16519e15	0.436398
$639$	−5.92529e14	−0.220016
$640$	0	0
$641$	−2.03644e15	−0.743280	−0.371640	−	0.928377i	$-0.621204\pi$
$641$	−2.03644e15	−0.743280	−0.371640	+	0.928377i	$0.621204\pi$
$642$	−1.00314e15	−0.363010
$643$	5.15241e15	1.84863	0.924315	−	0.381630i	$-0.124637\pi$
$643$	5.15241e15	1.84863	0.924315	+	0.381630i	$0.124637\pi$
$644$	6.75952e15	2.40460
$645$	0	0
$646$	1.25235e16	4.37974
$647$	−4.93499e15	−1.71125	−0.855623	−	0.517599i	$-0.826826\pi$
$647$	−4.93499e15	−1.71125	−0.855623	+	0.517599i	$0.826826\pi$
$648$	1.10396e15	0.379568
$649$	3.84689e15	1.31149
$650$	0	0
$651$	−2.74576e15	−0.920381
$652$	−9.30954e15	−3.09433
$653$	2.13069e15	0.702259	0.351129	−	0.936327i	$-0.385798\pi$
$653$	2.13069e15	0.702259	0.351129	+	0.936327i	$0.385798\pi$
$654$	−4.01420e13	−0.0131196
$655$	0	0
$656$	−2.99698e15	−0.963192
$657$	4.16956e14	0.132886
$658$	2.77796e15	0.877977
$659$	5.09427e15	1.59666	0.798330	−	0.602220i	$-0.205717\pi$
$659$	5.09427e15	1.59666	0.798330	+	0.602220i	$0.205717\pi$
$660$	0	0
$661$	−1.60078e15	−0.493429	−0.246714	−	0.969088i	$-0.579351\pi$
$661$	−1.60078e15	−0.493429	−0.246714	+	0.969088i	$0.579351\pi$
$662$	1.43398e15	0.438353
$663$	−2.03449e15	−0.616780
$664$	−1.46059e16	−4.39141
$665$	0	0
$666$	3.49058e15	1.03226
$667$	−6.63698e14	−0.194661
$668$	−5.90204e15	−1.71685
$669$	2.57665e15	0.743382
$670$	0	0
$671$	−2.59234e14	−0.0735729
$672$	1.17441e16	3.30588
$673$	3.62623e15	1.01245	0.506223	−	0.862403i	$-0.331041\pi$
$673$	3.62623e15	1.01245	0.506223	+	0.862403i	$0.331041\pi$
$674$	7.93268e15	2.19680
$675$	0	0
$676$	−3.84162e15	−1.04667
$677$	5.31766e15	1.43709	0.718543	−	0.695482i	$-0.244809\pi$
$677$	5.31766e15	1.43709	0.718543	+	0.695482i	$0.244809\pi$
$678$	5.18822e15	1.39077
$679$	−8.21553e14	−0.218450
$680$	0	0
$681$	2.78509e15	0.728670
$682$	6.08130e15	1.57827
$683$	−3.08846e15	−0.795113	−0.397556	−	0.917578i	$-0.630142\pi$
$683$	−3.08846e15	−0.795113	−0.397556	+	0.917578i	$0.630142\pi$
$684$	−6.00388e15	−1.53329
$685$	0	0
$686$	−3.05506e14	−0.0767781
$687$	−1.72175e15	−0.429248
$688$	9.89701e15	2.44775
$689$	−1.66668e14	−0.0408926
$690$	0	0
$691$	−1.12109e15	−0.270714	−0.135357	−	0.990797i	$-0.543218\pi$
$691$	−1.12109e15	−0.270714	−0.135357	+	0.990797i	$0.543218\pi$
$692$	−1.44410e16	−3.45950
$693$	−1.41174e15	−0.335524
$694$	−9.00153e15	−2.12246
$695$	0	0
$696$	−2.66759e15	−0.619110
$697$	1.46765e15	0.337942
$698$	6.06406e15	1.38535
$699$	3.29151e15	0.746055
$700$	0	0
$701$	7.71502e15	1.72142	0.860712	−	0.509092i	$-0.170019\pi$
$701$	7.71502e15	1.72142	0.860712	+	0.509092i	$0.170019\pi$
$702$	1.32856e15	0.294121
$703$	−1.21089e16	−2.65982
$704$	−1.33023e16	−2.89920
$705$	0	0
$706$	7.56365e15	1.62295
$707$	5.47047e15	1.16471
$708$	−1.38071e16	−2.91689
$709$	−1.58078e15	−0.331374	−0.165687	−	0.986178i	$-0.552984\pi$
$709$	−1.58078e15	−0.331374	−0.165687	+	0.986178i	$0.552984\pi$
$710$	0	0
$711$	−2.44417e15	−0.504486
$712$	8.61499e14	0.176447
$713$	−3.46394e15	−0.704009
$714$	−1.05692e16	−2.13159
$715$	0	0
$716$	−2.57916e16	−5.12219
$717$	−2.16111e15	−0.425914
$718$	3.43969e15	0.672720
$719$	−9.88218e15	−1.91798	−0.958989	−	0.283443i	$-0.908523\pi$
$719$	−9.88218e15	−1.91798	−0.958989	+	0.283443i	$0.908523\pi$
$720$	0	0
$721$	2.05274e15	0.392364
$722$	1.81459e16	3.44211
$723$	−3.52546e15	−0.663674
$724$	1.47176e16	2.74963
$725$	0	0
$726$	−2.95834e15	−0.544374
$727$	−2.98469e14	−0.0545080	−0.0272540	−	0.999629i	$-0.508676\pi$
$727$	−2.98469e14	−0.0545080	−0.0272540	+	0.999629i	$0.508676\pi$
$728$	2.08555e16	3.78006
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−4.84666e15	−0.858809
$732$	9.30434e14	0.163634
$733$	2.66302e15	0.464839	0.232420	−	0.972616i	$-0.425336\pi$
$733$	2.66302e15	0.464839	0.232420	+	0.972616i	$0.425336\pi$
$734$	4.54492e15	0.787404
$735$	0	0
$736$	1.48158e16	2.52870
$737$	−5.75548e15	−0.975013
$738$	−9.58402e14	−0.161153
$739$	−5.23337e15	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$739$	−5.23337e15	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$740$	0	0
$741$	−4.60881e15	−0.757860
$742$	−8.65845e14	−0.141325
$743$	−1.41918e15	−0.229932	−0.114966	−	0.993369i	$-0.536676\pi$
$743$	−1.41918e15	−0.229932	−0.114966	+	0.993369i	$0.536676\pi$
$744$	−1.39225e16	−2.23907
$745$	0	0
$746$	−6.47089e15	−1.02542
$747$	−2.72404e15	−0.428499
$748$	1.71853e16	2.68348
$749$	−2.93689e15	−0.455236
$750$	0	0
$751$	9.75692e15	1.49037	0.745183	−	0.666860i	$-0.232362\pi$
$751$	9.75692e15	1.49037	0.745183	+	0.666860i	$0.232362\pi$
$752$	8.21491e15	1.24567
$753$	5.92334e15	0.891650
$754$	−3.21031e15	−0.479739
$755$	0	0
$756$	5.06698e15	0.746242
$757$	1.43310e15	0.209532	0.104766	−	0.994497i	$-0.466591\pi$
$757$	1.43310e15	0.209532	0.104766	+	0.994497i	$0.466591\pi$
$758$	2.52528e16	3.66547
$759$	−1.78100e15	−0.256645
$760$	0	0
$761$	−3.16730e15	−0.449856	−0.224928	−	0.974375i	$-0.572215\pi$
$761$	−3.16730e15	−0.449856	−0.224928	+	0.974375i	$0.572215\pi$
$762$	−4.81452e14	−0.0678892
$763$	−1.17524e14	−0.0164528
$764$	2.08571e16	2.89895
$765$	0	0
$766$	1.53555e16	2.10380
$767$	−1.05989e16	−1.44174
$768$	1.39365e16	1.88220
$769$	−4.04915e15	−0.542961	−0.271481	−	0.962444i	$-0.587513\pi$
$769$	−4.04915e15	−0.542961	−0.271481	+	0.962444i	$0.587513\pi$
$770$	0	0
$771$	4.19223e15	0.554174
$772$	−1.59147e16	−2.08883
$773$	−1.35963e16	−1.77187	−0.885936	−	0.463807i	$-0.846483\pi$
$773$	−1.35963e16	−1.77187	−0.885936	+	0.463807i	$0.846483\pi$
$774$	3.16496e15	0.409537
$775$	0	0
$776$	−4.16573e15	−0.531437
$777$	1.02194e16	1.29452
$778$	−3.05976e16	−3.84858
$779$	3.32473e15	0.415241
$780$	0	0
$781$	3.84210e15	0.473139
$782$	−1.33337e16	−1.63048
$783$	−4.97513e14	−0.0604108
$784$	3.11421e16	3.75499
$785$	0	0
$786$	5.35396e15	0.636578
$787$	−3.82272e15	−0.451348	−0.225674	−	0.974203i	$-0.572458\pi$
$787$	−3.82272e15	−0.451348	−0.225674	+	0.974203i	$0.572458\pi$
$788$	−4.00028e16	−4.69025
$789$	1.82156e15	0.212090
$790$	0	0
$791$	1.51895e16	1.74411
$792$	−7.15833e15	−0.816250
$793$	7.14237e14	0.0808797
$794$	−1.85727e16	−2.08863
$795$	0	0
$796$	−1.84951e15	−0.205133
$797$	8.10418e15	0.892665	0.446333	−	0.894867i	$-0.352730\pi$
$797$	8.10418e15	0.892665	0.446333	+	0.894867i	$0.352730\pi$
$798$	−2.39430e16	−2.61916
$799$	−4.02291e15	−0.437052
$800$	0	0
$801$	1.60672e14	0.0172171
$802$	−1.81166e16	−1.92804
$803$	−2.70364e15	−0.285768
$804$	2.06574e16	2.16854
$805$	0	0
$806$	−1.67551e16	−1.73502
$807$	2.46576e15	0.253598
$808$	2.77383e16	2.83346
$809$	1.04602e16	1.06126	0.530630	−	0.847604i	$-0.321956\pi$
$809$	1.04602e16	1.06126	0.530630	+	0.847604i	$0.321956\pi$
$810$	0	0
$811$	−1.14632e16	−1.14734	−0.573669	−	0.819087i	$-0.694480\pi$
$811$	−1.14632e16	−1.14734	−0.573669	+	0.819087i	$0.694480\pi$
$812$	−1.22438e16	−1.21719
$813$	−4.66120e14	−0.0460257
$814$	−2.26338e16	−2.21985
$815$	0	0
$816$	−3.12551e16	−3.02430
$817$	−1.09793e16	−1.05525
$818$	−2.60119e15	−0.248329
$819$	3.88961e15	0.368846
$820$	0	0
$821$	1.19070e16	1.11408	0.557040	−	0.830486i	$-0.311937\pi$
$821$	1.19070e16	1.11408	0.557040	+	0.830486i	$0.311937\pi$
$822$	3.21309e15	0.298626
$823$	7.60886e15	0.702458	0.351229	−	0.936290i	$-0.385764\pi$
$823$	7.60886e15	0.702458	0.351229	+	0.936290i	$0.385764\pi$
$824$	1.04085e16	0.954529
$825$	0	0
$826$	−5.50617e16	−4.98264
$827$	6.80604e15	0.611807	0.305903	−	0.952063i	$-0.401042\pi$
$827$	6.80604e15	0.611807	0.305903	+	0.952063i	$0.401042\pi$
$828$	6.39230e15	0.570808
$829$	2.91128e15	0.258247	0.129123	−	0.991629i	$-0.458784\pi$
$829$	2.91128e15	0.258247	0.129123	+	0.991629i	$0.458784\pi$
$830$	0	0
$831$	8.54011e15	0.747580
$832$	3.66503e16	3.18713
$833$	−1.52505e16	−1.31746
$834$	4.09914e15	0.351786
$835$	0	0
$836$	3.89306e16	3.29729
$837$	−2.59659e15	−0.218481
$838$	−1.22180e16	−1.02131
$839$	−2.16788e16	−1.80030	−0.900151	−	0.435579i	$-0.856544\pi$
$839$	−2.16788e16	−1.80030	−0.900151	+	0.435579i	$0.856544\pi$
$840$	0	0
$841$	−1.09983e16	−0.901464
$842$	3.65752e16	2.97831
$843$	9.98223e15	0.807562
$844$	4.84725e16	3.89593
$845$	0	0
$846$	2.62704e15	0.208415
$847$	−8.66112e15	−0.682677
$848$	−2.56046e15	−0.200512
$849$	3.87663e15	0.301621
$850$	0	0
$851$	1.28923e16	0.990189
$852$	−1.37899e16	−1.05231
$853$	1.42605e16	1.08122	0.540611	−	0.841273i	$-0.318193\pi$
$853$	1.42605e16	1.08122	0.540611	+	0.841273i	$0.318193\pi$
$854$	3.71049e15	0.279520
$855$	0	0
$856$	−1.48916e16	−1.10748
$857$	−9.55449e15	−0.706014	−0.353007	−	0.935621i	$-0.614841\pi$
$857$	−9.55449e15	−0.706014	−0.353007	+	0.935621i	$0.614841\pi$
$858$	−8.61470e15	−0.632499
$859$	−1.90758e16	−1.39162	−0.695811	−	0.718225i	$-0.744955\pi$
$859$	−1.90758e16	−1.39162	−0.695811	+	0.718225i	$0.744955\pi$
$860$	0	0
$861$	−2.80591e15	−0.202095
$862$	4.45912e16	3.19124
$863$	−5.99685e14	−0.0426446	−0.0213223	−	0.999773i	$-0.506788\pi$
$863$	−5.99685e14	−0.0426446	−0.0213223	+	0.999773i	$0.506788\pi$
$864$	1.11060e16	0.784755
$865$	0	0
$866$	−1.63708e16	−1.14215
$867$	6.97784e15	0.483744
$868$	−6.39022e16	−4.40207
$869$	1.58486e16	1.08488
$870$	0	0
$871$	1.58574e16	1.07185
$872$	−5.95910e14	−0.0400258
$873$	−7.76920e14	−0.0518559
$874$	−3.02055e16	−2.00342
$875$	0	0
$876$	9.70382e15	0.635580
$877$	−2.15536e16	−1.40289	−0.701444	−	0.712725i	$-0.747461\pi$
$877$	−2.15536e16	−1.40289	−0.701444	+	0.712725i	$0.747461\pi$
$878$	2.36751e16	1.53134
$879$	−4.74697e14	−0.0305125
$880$	0	0
$881$	1.52938e16	0.970840	0.485420	−	0.874281i	$-0.338667\pi$
$881$	1.52938e16	0.970840	0.485420	+	0.874281i	$0.338667\pi$
$882$	9.95889e15	0.628252
$883$	2.81706e16	1.76609	0.883043	−	0.469291i	$-0.155491\pi$
$883$	2.81706e16	1.76609	0.883043	+	0.469291i	$0.155491\pi$
$884$	−4.73487e16	−2.94999
$885$	0	0
$886$	4.40596e16	2.71116
$887$	−2.31839e16	−1.41777	−0.708887	−	0.705322i	$-0.750802\pi$
$887$	−2.31839e16	−1.41777	−0.708887	+	0.705322i	$0.750802\pi$
$888$	5.18178e16	3.14925
$889$	−1.40954e15	−0.0851371
$890$	0	0
$891$	−1.33505e15	−0.0796470
$892$	5.99664e16	3.55551
$893$	−9.11329e15	−0.537022
$894$	−7.47460e15	−0.437756
$895$	0	0
$896$	9.14212e16	5.28875
$897$	4.90697e15	0.282134
$898$	−4.05485e16	−2.31715
$899$	6.27438e15	0.356363
$900$	0	0
$901$	1.25388e15	0.0703508
$902$	6.21451e15	0.346554
$903$	9.26603e15	0.513583
$904$	7.70193e16	4.24300
$905$	0	0
$906$	1.61078e16	0.876660
$907$	−2.02187e16	−1.09374	−0.546870	−	0.837218i	$-0.684181\pi$
$907$	−2.02187e16	−1.09374	−0.546870	+	0.837218i	$0.684181\pi$
$908$	6.48175e16	3.48514
$909$	5.17328e15	0.276480
$910$	0	0
$911$	8.97403e15	0.473845	0.236922	−	0.971529i	$-0.423861\pi$
$911$	8.97403e15	0.473845	0.236922	+	0.971529i	$0.423861\pi$
$912$	−7.08036e16	−3.71607
$913$	1.76633e16	0.921475
$914$	3.40122e16	1.76372
$915$	0	0
$916$	−4.00704e16	−2.05304
$917$	1.56748e16	0.798307
$918$	−9.99505e15	−0.506000
$919$	−2.17613e16	−1.09509	−0.547544	−	0.836777i	$-0.684437\pi$
$919$	−2.17613e16	−1.09509	−0.547544	+	0.836777i	$0.684437\pi$
$920$	0	0
$921$	−3.49541e15	−0.173808
$922$	3.03593e15	0.150062
$923$	−1.05857e16	−0.520128
$924$	−3.28555e16	−1.60477
$925$	0	0
$926$	−7.75791e16	−3.74442
$927$	1.94122e15	0.0931399
$928$	−2.68365e16	−1.28001
$929$	3.23873e16	1.53564	0.767818	−	0.640668i	$-0.221343\pi$
$929$	3.23873e16	1.53564	0.767818	+	0.640668i	$0.221343\pi$
$930$	0	0
$931$	−3.45477e16	−1.61881
$932$	7.66035e16	3.56830
$933$	−1.39074e16	−0.644019
$934$	−3.97061e16	−1.82789
$935$	0	0
$936$	1.97225e16	0.897315
$937$	4.28972e16	1.94027	0.970133	−	0.242574i	$-0.0779916\pi$
$937$	4.28972e16	1.94027	0.970133	+	0.242574i	$0.0779916\pi$
$938$	8.23799e16	3.70429
$939$	1.84815e16	0.826182
$940$	0	0
$941$	1.98759e16	0.878179	0.439090	−	0.898443i	$-0.355301\pi$
$941$	1.98759e16	0.878179	0.439090	+	0.898443i	$0.355301\pi$
$942$	−2.23972e16	−0.983817
$943$	−3.53982e15	−0.154585
$944$	−1.62827e17	−7.06937
$945$	0	0
$946$	−2.05224e16	−0.880696
$947$	2.02074e15	0.0862155	0.0431077	−	0.999070i	$-0.486274\pi$
$947$	2.02074e15	0.0862155	0.0431077	+	0.999070i	$0.486274\pi$
$948$	−5.68833e16	−2.41290
$949$	7.44903e15	0.314149
$950$	0	0
$951$	1.84777e15	0.0770293
$952$	−1.56901e17	−6.50312
$953$	1.91443e15	0.0788911	0.0394456	−	0.999222i	$-0.487441\pi$
$953$	1.91443e15	0.0788911	0.0394456	+	0.999222i	$0.487441\pi$
$954$	−8.18806e14	−0.0335479
$955$	0	0
$956$	−5.02956e16	−2.03710
$957$	3.22599e15	0.129912
$958$	1.63806e16	0.655874
$959$	9.40693e15	0.374495
$960$	0	0
$961$	7.33843e15	0.288818
$962$	6.23602e16	2.44031
$963$	−2.77733e15	−0.108065
$964$	−8.20480e16	−3.17427
$965$	0	0
$966$	2.54919e16	0.975054
$967$	3.24352e16	1.23359	0.616796	−	0.787123i	$-0.288431\pi$
$967$	3.24352e16	1.23359	0.616796	+	0.787123i	$0.288431\pi$
$968$	−4.39167e16	−1.66079
$969$	3.46732e16	1.30380
$970$	0	0
$971$	−3.71058e16	−1.37955	−0.689773	−	0.724025i	$-0.742290\pi$
$971$	−3.71058e16	−1.37955	−0.689773	+	0.724025i	$0.742290\pi$
$972$	4.79171e15	0.177144
$973$	1.20010e16	0.441161
$974$	7.07199e16	2.58504
$975$	0	0
$976$	1.09726e16	0.396583
$977$	−3.72467e16	−1.33865	−0.669326	−	0.742969i	$-0.733417\pi$
$977$	−3.72467e16	−1.33865	−0.669326	+	0.742969i	$0.733417\pi$
$978$	−3.51087e16	−1.25473
$979$	−1.04184e15	−0.0370249
$980$	0	0
$981$	−1.11139e14	−0.00390559
$982$	1.55236e15	0.0542473
$983$	−1.26252e16	−0.438725	−0.219362	−	0.975643i	$-0.570398\pi$
$983$	−1.26252e16	−0.438725	−0.219362	+	0.975643i	$0.570398\pi$
$984$	−1.42275e16	−0.491650
$985$	0	0
$986$	2.41519e16	0.825332
$987$	7.69117e15	0.261365
$988$	−1.07261e17	−3.62476
$989$	1.16896e16	0.392845
$990$	0	0
$991$	−1.40424e16	−0.466699	−0.233350	−	0.972393i	$-0.574969\pi$
$991$	−1.40424e16	−0.466699	−0.233350	+	0.972393i	$0.574969\pi$
$992$	−1.40064e17	−4.62926
$993$	3.97017e15	0.130493
$994$	−5.49931e16	−1.79756
$995$	0	0
$996$	−6.33966e16	−2.04946
$997$	−2.46726e16	−0.793217	−0.396608	−	0.917988i	$-0.629813\pi$
$997$	−2.46726e16	−0.793217	−0.396608	+	0.917988i	$0.629813\pi$
$998$	−2.26542e16	−0.724319
$999$	9.66417e15	0.307294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.a.i.1.4	yes	4
3.2	odd	2		225.12.a.q.1.1		4
5.2	odd	4		75.12.b.g.49.8		8
5.3	odd	4		75.12.b.g.49.1		8
5.4	even	2		75.12.a.h.1.1	✓	4
15.2	even	4		225.12.b.o.199.1		8
15.8	even	4		225.12.b.o.199.8		8
15.14	odd	2		225.12.a.s.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.12.a.h.1.1	✓	4	5.4	even	2
75.12.a.i.1.4	yes	4	1.1	even	1	trivial
75.12.b.g.49.1		8	5.3	odd	4
75.12.b.g.49.8		8	5.2	odd	4
225.12.a.q.1.1		4	3.2	odd	2
225.12.a.s.1.4		4	15.14	odd	2
225.12.b.o.199.1		8	15.2	even	4
225.12.b.o.199.8		8	15.8	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+87.7687 q^{2} +243.000 q^{3} +5655.34 q^{4} +21327.8 q^{6} +62441.2 q^{7} +316612. q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q+87.7687 q^{2} +243.000 q^{3} +5655.34 q^{4} +21327.8 q^{6} +62441.2 q^{7} +316612. q^{8} +59049.0 q^{9} -382888. q^{11} +1.37425e6 q^{12} +1.05493e6 q^{13} +5.48039e6 q^{14} +1.62065e7 q^{16} -7.93645e6 q^{17} +5.18265e6 q^{18} -1.79788e7 q^{19} +1.51732e7 q^{21} -3.36056e7 q^{22} +1.91419e7 q^{23} +7.69367e7 q^{24} +9.25895e7 q^{26} +1.43489e7 q^{27} +3.53127e8 q^{28} -3.46725e7 q^{29} -1.80961e8 q^{31} +7.74000e8 q^{32} -9.30418e7 q^{33} -6.96572e8 q^{34} +3.33942e8 q^{36} +6.73512e8 q^{37} -1.57798e9 q^{38} +2.56347e8 q^{39} -1.84925e8 q^{41} +1.33173e9 q^{42} +6.10683e8 q^{43} -2.16536e9 q^{44} +1.68006e9 q^{46} +5.06891e8 q^{47} +3.93817e9 q^{48} +1.92158e9 q^{49} -1.92856e9 q^{51} +5.96597e9 q^{52} -1.57990e8 q^{53} +1.25938e9 q^{54} +1.97696e10 q^{56} -4.36885e9 q^{57} -3.04316e9 q^{58} -1.00470e10 q^{59} +6.77049e8 q^{61} -1.58827e10 q^{62} +3.68709e9 q^{63} +3.47421e10 q^{64} -8.16616e9 q^{66} +1.50318e10 q^{67} -4.48834e10 q^{68} +4.65148e9 q^{69} -1.00345e10 q^{71} +1.86956e10 q^{72} +7.06118e9 q^{73} +5.91133e10 q^{74} -1.01676e11 q^{76} -2.39080e10 q^{77} +2.24993e10 q^{78} -4.13923e10 q^{79} +3.48678e9 q^{81} -1.62306e10 q^{82} -4.61318e10 q^{83} +8.58098e10 q^{84} +5.35988e10 q^{86} -8.42542e9 q^{87} -1.21227e11 q^{88} +2.72099e9 q^{89} +6.58709e10 q^{91} +1.08254e11 q^{92} -4.39735e10 q^{93} +4.44891e10 q^{94} +1.88082e11 q^{96} -1.31572e10 q^{97} +1.68655e11 q^{98} -2.26092e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9}+O(q^{10}) \) 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9	\( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9} - 99944 q^{11} + 1129464 q^{12} + 2306276 q^{13} + 3107022 q^{14} + 7622200 q^{16} - 3443816 q^{17} + 2716254 q^{18} - 4214548 q^{19} + 16614396 q^{21} - 34106532 q^{22} + 52691304 q^{23} + 93839796 q^{24} - 8766722 q^{26} + 57395628 q^{27} + 472439576 q^{28} + 217393304 q^{29} - 326317036 q^{31} + 788672072 q^{32} - 24286392 q^{33} - 564836244 q^{34} + 274459752 q^{36} + 273252872 q^{37} - 1762793006 q^{38} + 560425068 q^{39} - 22069456 q^{41} + 755006346 q^{42} - 705091900 q^{43} - 992474176 q^{44} + 2697853524 q^{46} + 187768360 q^{47} + 1852194600 q^{48} + 5213315400 q^{49} - 836847288 q^{51} + 7619620904 q^{52} - 6392224256 q^{53} + 660049722 q^{54} + 20338292700 q^{56} - 1024135164 q^{57} - 16176655332 q^{58} + 36710008 q^{59} + 11538870620 q^{61} - 4825203906 q^{62} + 4037298228 q^{63} + 55398609952 q^{64} - 8287887276 q^{66} + 37721158484 q^{67} - 45682615072 q^{68} + 12803986872 q^{69} + 8211316688 q^{71} + 22803070428 q^{72} + 5713413224 q^{73} + 136155026252 q^{74} - 78094966168 q^{76} + 72854549304 q^{77} - 2130313446 q^{78} + 45026381600 q^{79} + 13947137604 q^{81} + 55157549184 q^{82} - 104211315528 q^{83} + 114802816968 q^{84} + 77092671046 q^{86} + 52826572872 q^{87} - 175877029608 q^{88} + 111829609152 q^{89} + 225968327284 q^{91} + 88286181792 q^{92} - 79295039748 q^{93} + 193370717388 q^{94} + 191647313496 q^{96} + 77104304804 q^{97} - 139912937716 q^{98} - 5901593256 q^{99}+O(q^{100}) \) 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9 - 99944 * q^11 + 1129464 * q^12 + 2306276 * q^13 + 3107022 * q^14 + 7622200 * q^16 - 3443816 * q^17 + 2716254 * q^18 - 4214548 * q^19 + 16614396 * q^21 - 34106532 * q^22 + 52691304 * q^23 + 93839796 * q^24 - 8766722 * q^26 + 57395628 * q^27 + 472439576 * q^28 + 217393304 * q^29 - 326317036 * q^31 + 788672072 * q^32 - 24286392 * q^33 - 564836244 * q^34 + 274459752 * q^36 + 273252872 * q^37 - 1762793006 * q^38 + 560425068 * q^39 - 22069456 * q^41 + 755006346 * q^42 - 705091900 * q^43 - 992474176 * q^44 + 2697853524 * q^46 + 187768360 * q^47 + 1852194600 * q^48 + 5213315400 * q^49 - 836847288 * q^51 + 7619620904 * q^52 - 6392224256 * q^53 + 660049722 * q^54 + 20338292700 * q^56 - 1024135164 * q^57 - 16176655332 * q^58 + 36710008 * q^59 + 11538870620 * q^61 - 4825203906 * q^62 + 4037298228 * q^63 + 55398609952 * q^64 - 8287887276 * q^66 + 37721158484 * q^67 - 45682615072 * q^68 + 12803986872 * q^69 + 8211316688 * q^71 + 22803070428 * q^72 + 5713413224 * q^73 + 136155026252 * q^74 - 78094966168 * q^76 + 72854549304 * q^77 - 2130313446 * q^78 + 45026381600 * q^79 + 13947137604 * q^81 + 55157549184 * q^82 - 104211315528 * q^83 + 114802816968 * q^84 + 77092671046 * q^86 + 52826572872 * q^87 - 175877029608 * q^88 + 111829609152 * q^89 + 225968327284 * q^91 + 88286181792 * q^92 - 79295039748 * q^93 + 193370717388 * q^94 + 191647313496 * q^96 + 77104304804 * q^97 - 139912937716 * q^98 - 5901593256 * q^99

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