LMFDB - Embedded newform 75.22.a.n.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,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, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
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:	$209.608008215$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 $ x^10 - 5x^9 - 15486550x^8 - 930225020x^7 + 80619202368510x^6 + 14206919536731394x^5 - 159833300365521427920x^4 - 53427455410418539963580x^3 + 93639331506281672292832825x^2 + 44269185428005803484908321595*x + 4729332062567824907768533377518
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2176.21$ 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)

$f(q)$	$=$	$q+2241.21 q^{2} -59049.0 q^{3} +2.92587e6 q^{4} -1.32341e8 q^{6} +1.14262e8 q^{7} +1.85733e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+2241.21 q^{2} -59049.0 q^{3} +2.92587e6 q^{4} -1.32341e8 q^{6} +1.14262e8 q^{7} +1.85733e9 q^{8} +3.48678e9 q^{9} -8.03645e10 q^{11} -1.72770e11 q^{12} -2.07029e11 q^{13} +2.56084e11 q^{14} -1.97333e12 q^{16} -2.89142e12 q^{17} +7.81461e12 q^{18} -1.26461e13 q^{19} -6.74703e12 q^{21} -1.80114e14 q^{22} +1.21633e14 q^{23} -1.09673e14 q^{24} -4.63996e14 q^{26} -2.05891e14 q^{27} +3.34314e14 q^{28} +2.61967e15 q^{29} +6.16148e15 q^{31} -8.31775e15 q^{32} +4.74545e15 q^{33} -6.48028e15 q^{34} +1.02019e16 q^{36} +4.84951e16 q^{37} -2.83425e16 q^{38} +1.22249e16 q^{39} +1.97330e16 q^{41} -1.51215e16 q^{42} -2.05305e17 q^{43} -2.35136e17 q^{44} +2.72605e17 q^{46} -3.51731e16 q^{47} +1.16523e17 q^{48} -5.45490e17 q^{49} +1.70735e17 q^{51} -6.05740e17 q^{52} +5.84445e17 q^{53} -4.61445e17 q^{54} +2.12221e17 q^{56} +7.46738e17 q^{57} +5.87122e18 q^{58} +8.39981e17 q^{59} +3.71348e18 q^{61} +1.38092e19 q^{62} +3.98405e17 q^{63} -1.45034e19 q^{64} +1.06355e19 q^{66} +2.38664e19 q^{67} -8.45991e18 q^{68} -7.18231e18 q^{69} -4.03762e19 q^{71} +6.47609e18 q^{72} +2.77909e19 q^{73} +1.08688e20 q^{74} -3.70008e19 q^{76} -9.18258e18 q^{77} +2.73985e19 q^{78} -2.28507e19 q^{79} +1.21577e19 q^{81} +4.42259e19 q^{82} +2.43889e19 q^{83} -1.97409e19 q^{84} -4.60132e20 q^{86} -1.54689e20 q^{87} -1.49263e20 q^{88} +3.55679e20 q^{89} -2.36555e19 q^{91} +3.55882e20 q^{92} -3.63829e20 q^{93} -7.88302e19 q^{94} +4.91155e20 q^{96} -1.26080e21 q^{97} -1.22256e21 q^{98} -2.80214e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) $ 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9	$ 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} - 593041212045 q^{12} + 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} + 12910340404230 q^{17} + 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} + 461780887241010 q^{22} + 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} - 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) $ 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9 + 5976221790 * q^11 - 593041212045 * q^12 + 842570747430 * q^13 + 1605059874570 * q^14 + 6061861547825 * q^16 + 12910340404230 * q^17 + 2248975938645 * q^18 - 76155422176280 * q^19 - 6811732617210 * q^21 + 461780887241010 * q^22 + 82640920915920 * q^23 - 18549814359015 * q^24 + 286555331159670 * q^26 - 2058911320946490 * q^27 + 1520428218531930 * q^28 - 2553653160731010 * q^29 - 5579671364724940 * q^31 + 29255856594217815 * q^32 - 352889920477710 * q^33 + 36285968426431010 * q^34 + 35018490530045205 * q^36 + 1797908431222890 * q^37 + 10684960744876320 * q^38 - 49752960064994070 * q^39 - 157319729506021440 * q^41 - 94777180533483930 * q^42 - 148924853700743400 * q^43 + 49118820773767290 * q^44 - 238453286038259540 * q^46 - 43171138083158100 * q^47 - 357946862537518425 * q^48 + 453692886452454950 * q^49 - 762342690529377270 * q^51 + 5077473324432504030 * q^52 + 1835165786666252970 * q^53 - 132799780201048605 * q^54 - 5146515624013530450 * q^56 + 4496901524087157720 * q^57 + 3684042125074285440 * q^58 + 4936135121536323630 * q^59 - 4443080871007408300 * q^61 + 39366851097921260280 * q^62 + 402225999313633290 * q^63 - 14877283608188093455 * q^64 - 27267699610694399490 * q^66 + 33664559574358232580 * q^67 + 44791395195934111290 * q^68 - 4879863739164160080 * q^69 - 74662724971386630180 * q^71 + 1095347988085476735 * q^72 + 16062862692682477620 * q^73 - 130249023546610518030 * q^74 - 258736483662380220720 * q^76 + 56216747719240740180 * q^77 - 16920805749647353830 * q^78 - 49130708010799643300 * q^79 + 121576654590569288010 * q^81 + 117185080045656261930 * q^82 + 118408383608453695560 * q^83 - 89779765876091934570 * q^84 - 1007271272794707382260 * q^86 + 150790665488005409490 * q^87 + 41803519849182967710 * q^88 - 572103976713991581780 * q^89 - 1669810835257196575140 * q^91 - 1206545757146297440260 * q^92 + 329474014415642982060 * q^93 - 1671319720124696594920 * q^94 - 1727529076031967757935 * q^96 - 1299275526640742173920 * q^97 - 2683959884079093505455 * q^98 + 20837796914288297790 * 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 $

$2$	2241.21	1.54763	0.773816	−	0.633411i	$-0.218346\pi$
$2$	2241.21	1.54763	0.773816	+	0.633411i	$0.218346\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	2.92587e6	1.39516
$5$	0	0
$5$	0	0
$6$	−1.32341e8	−0.893525
$7$	1.14262e8	0.152887	0.0764435	−	0.997074i	$-0.475644\pi$
$7$	1.14262e8	0.152887	0.0764435	+	0.997074i	$0.475644\pi$
$8$	1.85733e9	0.611566
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−8.03645e10	−0.934203	−0.467102	−	0.884204i	$-0.654702\pi$
$11$	−8.03645e10	−0.934203	−0.467102	+	0.884204i	$0.654702\pi$
$12$	−1.72770e11	−0.805497
$13$	−2.07029e11	−0.416511	−0.208255	−	0.978074i	$-0.566779\pi$
$13$	−2.07029e11	−0.416511	−0.208255	+	0.978074i	$0.566779\pi$
$14$	2.56084e11	0.236613
$15$	0	0
$16$	−1.97333e12	−0.448684
$17$	−2.89142e12	−0.347854	−0.173927	−	0.984759i	$-0.555646\pi$
$17$	−2.89142e12	−0.347854	−0.173927	+	0.984759i	$0.555646\pi$
$18$	7.81461e12	0.515877
$19$	−1.26461e13	−0.473198	−0.236599	−	0.971607i	$-0.576033\pi$
$19$	−1.26461e13	−0.473198	−0.236599	+	0.971607i	$0.576033\pi$
$20$	0	0
$21$	−6.74703e12	−0.0882694
$22$	−1.80114e14	−1.44580
$23$	1.21633e14	0.612222	0.306111	−	0.951996i	$-0.400972\pi$
$23$	1.21633e14	0.612222	0.306111	+	0.951996i	$0.400972\pi$
$24$	−1.09673e14	−0.353088
$25$	0	0
$26$	−4.63996e14	−0.644605
$27$	−2.05891e14	−0.192450
$28$	3.34314e14	0.213302
$29$	2.61967e15	1.15629	0.578146	−	0.815934i	$-0.303777\pi$
$29$	2.61967e15	1.15629	0.578146	+	0.815934i	$0.303777\pi$
$30$	0	0
$31$	6.16148e15	1.35017	0.675083	−	0.737742i	$-0.264108\pi$
$31$	6.16148e15	1.35017	0.675083	+	0.737742i	$0.264108\pi$
$32$	−8.31775e15	−1.30596
$33$	4.74545e15	0.539362
$34$	−6.48028e15	−0.538350
$35$	0	0
$36$	1.02019e16	0.465054
$37$	4.84951e16	1.65798	0.828991	−	0.559262i	$-0.188916\pi$
$37$	4.84951e16	1.65798	0.828991	+	0.559262i	$0.188916\pi$
$38$	−2.83425e16	−0.732336
$39$	1.22249e16	0.240473
$40$	0	0
$41$	1.97330e16	0.229596	0.114798	−	0.993389i	$-0.463378\pi$
$41$	1.97330e16	0.229596	0.114798	+	0.993389i	$0.463378\pi$
$42$	−1.51215e16	−0.136608
$43$	−2.05305e17	−1.44871	−0.724355	−	0.689427i	$-0.757862\pi$
$43$	−2.05305e17	−1.44871	−0.724355	+	0.689427i	$0.757862\pi$
$44$	−2.35136e17	−1.30337
$45$	0	0
$46$	2.72605e17	0.947494
$47$	−3.51731e16	−0.0975400	−0.0487700	−	0.998810i	$-0.515530\pi$
$47$	−3.51731e16	−0.0975400	−0.0487700	+	0.998810i	$0.515530\pi$
$48$	1.16523e17	0.259048
$49$	−5.45490e17	−0.976626
$50$	0	0
$51$	1.70735e17	0.200834
$52$	−6.05740e17	−0.581100
$53$	5.84445e17	0.459036	0.229518	−	0.973304i	$-0.426285\pi$
$53$	5.84445e17	0.459036	0.229518	+	0.973304i	$0.426285\pi$
$54$	−4.61445e17	−0.297842
$55$	0	0
$56$	2.12221e17	0.0935005
$57$	7.46738e17	0.273201
$58$	5.87122e18	1.78951
$59$	8.39981e17	0.213955	0.106978	−	0.994261i	$-0.465883\pi$
$59$	8.39981e17	0.213955	0.106978	+	0.994261i	$0.465883\pi$
$60$	0	0
$61$	3.71348e18	0.666527	0.333264	−	0.942834i	$-0.391850\pi$
$61$	3.71348e18	0.666527	0.333264	+	0.942834i	$0.391850\pi$
$62$	1.38092e19	2.08956
$63$	3.98405e17	0.0509623
$64$	−1.45034e19	−1.57247
$65$	0	0
$66$	1.06355e19	0.834734
$67$	2.38664e19	1.59957	0.799783	−	0.600289i	$-0.204948\pi$
$67$	2.38664e19	1.59957	0.799783	+	0.600289i	$0.204948\pi$
$68$	−8.45991e18	−0.485313
$69$	−7.18231e18	−0.353467
$70$	0	0
$71$	−4.03762e19	−1.47202	−0.736009	−	0.676972i	$-0.763292\pi$
$71$	−4.03762e19	−1.47202	−0.736009	+	0.676972i	$0.763292\pi$
$72$	6.47609e18	0.203855
$73$	2.77909e19	0.756854	0.378427	−	0.925631i	$-0.376465\pi$
$73$	2.77909e19	0.756854	0.378427	+	0.925631i	$0.376465\pi$
$74$	1.08688e20	2.56594
$75$	0	0
$76$	−3.70008e19	−0.660188
$77$	−9.18258e18	−0.142828
$78$	2.73985e19	0.372163
$79$	−2.28507e19	−0.271528	−0.135764	−	0.990741i	$-0.543349\pi$
$79$	−2.28507e19	−0.271528	−0.135764	+	0.990741i	$0.543349\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	4.42259e19	0.355329
$83$	2.43889e19	0.172533	0.0862664	−	0.996272i	$-0.472506\pi$
$83$	2.43889e19	0.172533	0.0862664	+	0.996272i	$0.472506\pi$
$84$	−1.97409e19	−0.123150
$85$	0	0
$86$	−4.60132e20	−2.24207
$87$	−1.54689e20	−0.667585
$88$	−1.49263e20	−0.571327
$89$	3.55679e20	1.20910	0.604551	−	0.796566i	$-0.293352\pi$
$89$	3.55679e20	1.20910	0.604551	+	0.796566i	$0.293352\pi$
$90$	0	0
$91$	−2.36555e19	−0.0636791
$92$	3.55882e20	0.854150
$93$	−3.63829e20	−0.779519
$94$	−7.88302e19	−0.150956
$95$	0	0
$96$	4.91155e20	0.753998
$97$	−1.26080e21	−1.73597	−0.867983	−	0.496593i	$-0.834584\pi$
$97$	−1.26080e21	−1.73597	−0.867983	+	0.496593i	$0.834584\pi$
$98$	−1.22256e21	−1.51146
$99$	−2.80214e20	−0.311401
$100$	0	0
$101$	1.20165e21	1.08244	0.541219	−	0.840882i	$-0.317963\pi$
$101$	1.20165e21	1.08244	0.541219	+	0.840882i	$0.317963\pi$
$102$	3.82654e20	0.310817
$103$	1.96753e21	1.44255	0.721274	−	0.692650i	$-0.243557\pi$
$103$	1.96753e21	1.44255	0.721274	+	0.692650i	$0.243557\pi$
$104$	−3.84521e20	−0.254724
$105$	0	0
$106$	1.30986e21	0.710419
$107$	−3.27093e21	−1.60746	−0.803732	−	0.594992i	$-0.797155\pi$
$107$	−3.27093e21	−1.60746	−0.803732	+	0.594992i	$0.797155\pi$
$108$	−6.02410e20	−0.268499
$109$	4.56726e21	1.84790	0.923948	−	0.382518i	$-0.124943\pi$
$109$	4.56726e21	1.84790	0.923948	+	0.382518i	$0.124943\pi$
$110$	0	0
$111$	−2.86359e21	−0.957236
$112$	−2.25476e20	−0.0685980
$113$	4.36233e21	1.20891	0.604456	−	0.796638i	$-0.293390\pi$
$113$	4.36233e21	1.20891	0.604456	+	0.796638i	$0.293390\pi$
$114$	1.67360e21	0.422815
$115$	0	0
$116$	7.66480e21	1.61321
$117$	−7.21866e20	−0.138837
$118$	1.88257e21	0.331124
$119$	−3.30378e20	−0.0531824
$120$	0	0
$121$	−9.41790e20	−0.127265
$122$	8.32269e21	1.03154
$123$	−1.16522e21	−0.132557
$124$	1.80277e22	1.88370
$125$	0	0
$126$	8.92910e20	0.0788709
$127$	1.10108e22	0.895120	0.447560	−	0.894254i	$-0.352293\pi$
$127$	1.10108e22	0.895120	0.447560	+	0.894254i	$0.352293\pi$
$128$	−1.50617e22	−1.12763
$129$	1.21231e22	0.836413
$130$	0	0
$131$	1.86145e22	1.09271	0.546353	−	0.837555i	$-0.316016\pi$
$131$	1.86145e22	1.09271	0.546353	+	0.837555i	$0.316016\pi$
$132$	1.38845e22	0.752498
$133$	−1.44496e21	−0.0723459
$134$	5.34897e22	2.47554
$135$	0	0
$136$	−5.37031e21	−0.212736
$137$	3.77702e22	1.38543	0.692714	−	0.721213i	$-0.256415\pi$
$137$	3.77702e22	1.38543	0.692714	+	0.721213i	$0.256415\pi$
$138$	−1.60971e22	−0.547036
$139$	4.76118e22	1.49989	0.749944	−	0.661501i	$-0.230080\pi$
$139$	4.76118e22	1.49989	0.749944	+	0.661501i	$0.230080\pi$
$140$	0	0
$141$	2.07693e21	0.0563147
$142$	−9.04916e22	−2.27814
$143$	1.66378e22	0.389106
$144$	−6.88059e21	−0.149561
$145$	0	0
$146$	6.22852e22	1.17133
$147$	3.22106e22	0.563855
$148$	1.41890e23	2.31315
$149$	7.22628e22	1.09764	0.548819	−	0.835941i	$-0.315078\pi$
$149$	7.22628e22	1.09764	0.548819	+	0.835941i	$0.315078\pi$
$150$	0	0
$151$	1.81223e22	0.239308	0.119654	−	0.992816i	$-0.461822\pi$
$151$	1.81223e22	0.239308	0.119654	+	0.992816i	$0.461822\pi$
$152$	−2.34879e22	−0.289392
$153$	−1.00818e22	−0.115951
$154$	−2.05801e22	−0.221044
$155$	0	0
$156$	3.57683e22	0.335498
$157$	1.94846e22	0.170901	0.0854504	−	0.996342i	$-0.472767\pi$
$157$	1.94846e22	0.170901	0.0854504	+	0.996342i	$0.472767\pi$
$158$	−5.12131e22	−0.420225
$159$	−3.45109e22	−0.265025
$160$	0	0
$161$	1.38980e22	0.0936009
$162$	2.72479e22	0.171959
$163$	−1.56023e23	−0.923033	−0.461517	−	0.887132i	$-0.652695\pi$
$163$	−1.56023e23	−0.923033	−0.461517	+	0.887132i	$0.652695\pi$
$164$	5.77363e22	0.320323
$165$	0	0
$166$	5.46606e22	0.267017
$167$	1.05286e23	0.482890	0.241445	−	0.970414i	$-0.422379\pi$
$167$	1.05286e23	0.482890	0.241445	+	0.970414i	$0.422379\pi$
$168$	−1.25314e22	−0.0539825
$169$	−2.04203e23	−0.826519
$170$	0	0
$171$	−4.40942e22	−0.157733
$172$	−6.00696e23	−2.02119
$173$	−7.86959e22	−0.249154	−0.124577	−	0.992210i	$-0.539757\pi$
$173$	−7.86959e22	−0.249154	−0.124577	+	0.992210i	$0.539757\pi$
$174$	−3.46690e23	−1.03318
$175$	0	0
$176$	1.58586e23	0.419162
$177$	−4.96000e22	−0.123527
$178$	7.97151e23	1.87125
$179$	2.91149e23	0.644404	0.322202	−	0.946671i	$-0.395577\pi$
$179$	2.91149e23	0.644404	0.322202	+	0.946671i	$0.395577\pi$
$180$	0	0
$181$	−8.82086e23	−1.73734	−0.868672	−	0.495388i	$-0.835026\pi$
$181$	−8.82086e23	−1.73734	−0.868672	+	0.495388i	$0.835026\pi$
$182$	−5.30169e22	−0.0985518
$183$	−2.19277e23	−0.384820
$184$	2.25912e23	0.374414
$185$	0	0
$186$	−8.15417e23	−1.20641
$187$	2.32368e23	0.324967
$188$	−1.02912e23	−0.136084
$189$	−2.35254e22	−0.0294231
$190$	0	0
$191$	5.40009e23	0.604715	0.302358	−	0.953195i	$-0.402226\pi$
$191$	5.40009e23	0.604715	0.302358	+	0.953195i	$0.402226\pi$
$192$	8.56413e23	0.907863
$193$	1.01784e24	1.02171	0.510854	−	0.859668i	$-0.329329\pi$
$193$	1.01784e24	1.02171	0.510854	+	0.859668i	$0.329329\pi$
$194$	−2.82571e24	−2.68664
$195$	0	0
$196$	−1.59603e24	−1.36255
$197$	7.12880e23	0.576927	0.288464	−	0.957491i	$-0.406856\pi$
$197$	7.12880e23	0.576927	0.288464	+	0.957491i	$0.406856\pi$
$198$	−6.28018e23	−0.481934
$199$	−2.33381e23	−0.169867	−0.0849335	−	0.996387i	$-0.527068\pi$
$199$	−2.33381e23	−0.169867	−0.0849335	+	0.996387i	$0.527068\pi$
$200$	0	0
$201$	−1.40929e24	−0.923510
$202$	2.69315e24	1.67522
$203$	2.99327e23	0.176782
$204$	4.99549e23	0.280196
$205$	0	0
$206$	4.40964e24	2.23253
$207$	4.24108e23	0.204074
$208$	4.08538e23	0.186882
$209$	1.01630e24	0.442063
$210$	0	0
$211$	−3.08845e24	−1.21556	−0.607778	−	0.794107i	$-0.707939\pi$
$211$	−3.08845e24	−1.21556	−0.607778	+	0.794107i	$0.707939\pi$
$212$	1.71001e24	0.640430
$213$	2.38418e24	0.849870
$214$	−7.33083e24	−2.48776
$215$	0	0
$216$	−3.82407e23	−0.117696
$217$	7.04020e23	0.206423
$218$	1.02362e25	2.85986
$219$	−1.64102e24	−0.436970
$220$	0	0
$221$	5.98608e23	0.144885
$222$	−6.41790e24	−1.48145
$223$	2.01600e24	0.443904	0.221952	−	0.975058i	$-0.428757\pi$
$223$	2.01600e24	0.443904	0.221952	+	0.975058i	$0.428757\pi$
$224$	−9.50399e23	−0.199665
$225$	0	0
$226$	9.77690e24	1.87095
$227$	−5.14661e24	−0.940264	−0.470132	−	0.882596i	$-0.655794\pi$
$227$	−5.14661e24	−0.940264	−0.470132	+	0.882596i	$0.655794\pi$
$228$	2.18486e24	0.381160
$229$	−1.72303e24	−0.287092	−0.143546	−	0.989644i	$-0.545851\pi$
$229$	−1.72303e24	−0.287092	−0.143546	+	0.989644i	$0.545851\pi$
$230$	0	0
$231$	5.42222e23	0.0824615
$232$	4.86558e24	0.707148
$233$	−5.07503e24	−0.705020	−0.352510	−	0.935808i	$-0.614672\pi$
$233$	−5.07503e24	−0.705020	−0.352510	+	0.935808i	$0.614672\pi$
$234$	−1.61785e24	−0.214868
$235$	0	0
$236$	2.45767e24	0.298503
$237$	1.34931e24	0.156767
$238$	−7.40447e23	−0.0823068
$239$	1.28772e25	1.36976	0.684881	−	0.728655i	$-0.259854\pi$
$239$	1.28772e25	1.36976	0.684881	+	0.728655i	$0.259854\pi$
$240$	0	0
$241$	−4.48716e24	−0.437313	−0.218656	−	0.975802i	$-0.570167\pi$
$241$	−4.48716e24	−0.437313	−0.218656	+	0.975802i	$0.570167\pi$
$242$	−2.11075e24	−0.196959
$243$	−7.17898e23	−0.0641500
$244$	1.08652e25	0.929914
$245$	0	0
$246$	−2.61149e24	−0.205149
$247$	2.61811e24	0.197092
$248$	1.14439e25	0.825715
$249$	−1.44014e24	−0.0996119
$250$	0	0
$251$	−1.61181e25	−1.02504	−0.512520	−	0.858675i	$-0.671288\pi$
$251$	−1.61181e25	−1.02504	−0.512520	+	0.858675i	$0.671288\pi$
$252$	1.16568e24	0.0711008
$253$	−9.77498e24	−0.571940
$254$	2.46776e25	1.38532
$255$	0	0
$256$	−3.34041e24	−0.172695
$257$	8.54506e24	0.424050	0.212025	−	0.977264i	$-0.431994\pi$
$257$	8.54506e24	0.424050	0.212025	+	0.977264i	$0.431994\pi$
$258$	2.71703e25	1.29446
$259$	5.54112e24	0.253484
$260$	0	0
$261$	9.13422e24	0.385430
$262$	4.17190e25	1.69110
$263$	2.95578e25	1.15116	0.575582	−	0.817744i	$-0.304776\pi$
$263$	2.95578e25	1.15116	0.575582	+	0.817744i	$0.304776\pi$
$264$	8.81384e24	0.329856
$265$	0	0
$266$	−3.23846e24	−0.111965
$267$	−2.10025e25	−0.698076
$268$	6.98300e25	2.23165
$269$	1.66707e25	0.512337	0.256168	−	0.966632i	$-0.417540\pi$
$269$	1.66707e25	0.512337	0.256168	+	0.966632i	$0.417540\pi$
$270$	0	0
$271$	2.36122e25	0.671364	0.335682	−	0.941975i	$-0.391033\pi$
$271$	2.36122e25	0.671364	0.335682	+	0.941975i	$0.391033\pi$
$272$	5.70574e24	0.156077
$273$	1.39683e24	0.0367652
$274$	8.46510e25	2.14413
$275$	0	0
$276$	−2.10145e25	−0.493144
$277$	6.30266e25	1.42392	0.711961	−	0.702219i	$-0.247807\pi$
$277$	6.30266e25	1.42392	0.711961	+	0.702219i	$0.247807\pi$
$278$	1.06708e26	2.32127
$279$	2.14838e25	0.450055
$280$	0	0
$281$	3.64193e23	0.00707807	0.00353903	−	0.999994i	$-0.498873\pi$
$281$	3.64193e23	0.00707807	0.00353903	+	0.999994i	$0.498873\pi$
$282$	4.65484e24	0.0871544
$283$	1.07819e25	0.194508	0.0972540	−	0.995260i	$-0.468994\pi$
$283$	1.07819e25	0.194508	0.0972540	+	0.995260i	$0.468994\pi$
$284$	−1.18135e26	−2.05370
$285$	0	0
$286$	3.72888e25	0.602192
$287$	2.25473e24	0.0351022
$288$	−2.90022e25	−0.435321
$289$	−6.07316e25	−0.878997
$290$	0	0
$291$	7.44487e25	1.00226
$292$	8.13124e25	1.05593
$293$	−8.81247e25	−1.10405	−0.552024	−	0.833829i	$-0.686144\pi$
$293$	−8.81247e25	−1.10405	−0.552024	+	0.833829i	$0.686144\pi$
$294$	7.21908e25	0.872640
$295$	0	0
$296$	9.00712e25	1.01396
$297$	1.65463e25	0.179787
$298$	1.61956e26	1.69874
$299$	−2.51816e25	−0.254997
$300$	0	0
$301$	−2.34585e25	−0.221489
$302$	4.06160e25	0.370360
$303$	−7.09562e25	−0.624946
$304$	2.49549e25	0.212317
$305$	0	0
$306$	−2.25953e25	−0.179450
$307$	−1.43462e26	−1.10099	−0.550495	−	0.834839i	$-0.685561\pi$
$307$	−1.43462e26	−1.10099	−0.550495	+	0.834839i	$0.685561\pi$
$308$	−2.68670e25	−0.199268
$309$	−1.16181e26	−0.832855
$310$	0	0
$311$	−2.53384e26	−1.69744	−0.848720	−	0.528843i	$-0.822626\pi$
$311$	−2.53384e26	−1.69744	−0.848720	+	0.528843i	$0.822626\pi$
$312$	2.27056e25	0.147065
$313$	2.99850e26	1.87797	0.938986	−	0.343955i	$-0.111767\pi$
$313$	2.99850e26	1.87797	0.938986	+	0.343955i	$0.111767\pi$
$314$	4.36690e25	0.264491
$315$	0	0
$316$	−6.68580e25	−0.378825
$317$	−2.74765e26	−1.50605	−0.753024	−	0.657993i	$-0.771406\pi$
$317$	−2.74765e26	−1.50605	−0.753024	+	0.657993i	$0.771406\pi$
$318$	−7.73462e25	−0.410161
$319$	−2.10528e26	−1.08021
$320$	0	0
$321$	1.93145e26	0.928069
$322$	3.11483e25	0.144860
$323$	3.65651e25	0.164604
$324$	3.55717e25	0.155018
$325$	0	0
$326$	−3.49679e26	−1.42852
$327$	−2.69692e26	−1.06688
$328$	3.66507e25	0.140413
$329$	−4.01893e24	−0.0149126
$330$	0	0
$331$	3.10260e26	1.08027	0.540135	−	0.841579i	$-0.318373\pi$
$331$	3.10260e26	1.08027	0.540135	+	0.841579i	$0.318373\pi$
$332$	7.13586e25	0.240711
$333$	1.69092e26	0.552660
$334$	2.35968e26	0.747336
$335$	0	0
$336$	1.33141e25	0.0396051
$337$	3.91809e26	1.12969	0.564846	−	0.825196i	$-0.308936\pi$
$337$	3.91809e26	1.12969	0.564846	+	0.825196i	$0.308936\pi$
$338$	−4.57663e26	−1.27915
$339$	−2.57591e26	−0.697966
$340$	0	0
$341$	−4.95164e26	−1.26133
$342$	−9.88242e25	−0.244112
$343$	−1.26149e26	−0.302200
$344$	−3.81319e26	−0.885981
$345$	0	0
$346$	−1.76374e26	−0.385599
$347$	7.36212e26	1.56150	0.780752	−	0.624841i	$-0.214836\pi$
$347$	7.36212e26	1.56150	0.780752	+	0.624841i	$0.214836\pi$
$348$	−4.52599e26	−0.931390
$349$	−6.78361e26	−1.35455	−0.677273	−	0.735732i	$-0.736838\pi$
$349$	−6.78361e26	−1.35455	−0.677273	+	0.735732i	$0.736838\pi$
$350$	0	0
$351$	4.26255e25	0.0801576
$352$	6.68452e26	1.22004
$353$	3.38643e26	0.599940	0.299970	−	0.953949i	$-0.403023\pi$
$353$	3.38643e26	0.599940	0.299970	+	0.953949i	$0.403023\pi$
$354$	−1.11164e26	−0.191175
$355$	0	0
$356$	1.04067e27	1.68689
$357$	1.95085e25	0.0307049
$358$	6.52526e26	0.997300
$359$	−6.09645e26	−0.904868	−0.452434	−	0.891798i	$-0.649444\pi$
$359$	−6.09645e26	−0.904868	−0.452434	+	0.891798i	$0.649444\pi$
$360$	0	0
$361$	−5.54286e26	−0.776083
$362$	−1.97694e27	−2.68877
$363$	5.56117e25	0.0734762
$364$	−6.92128e25	−0.0888427
$365$	0	0
$366$	−4.91447e26	−0.595559
$367$	1.02782e27	1.21038	0.605190	−	0.796081i	$-0.293097\pi$
$367$	1.02782e27	1.21038	0.605190	+	0.796081i	$0.293097\pi$
$368$	−2.40023e26	−0.274695
$369$	6.88049e25	0.0765319
$370$	0	0
$371$	6.67796e25	0.0701807
$372$	−1.06452e27	−1.08756
$373$	−5.42369e25	−0.0538706	−0.0269353	−	0.999637i	$-0.508575\pi$
$373$	−5.42369e25	−0.0538706	−0.0269353	+	0.999637i	$0.508575\pi$
$374$	5.20785e26	0.502929
$375$	0	0
$376$	−6.53278e25	−0.0596521
$377$	−5.42348e26	−0.481608
$378$	−5.27254e25	−0.0455361
$379$	4.95344e26	0.416098	0.208049	−	0.978118i	$-0.433289\pi$
$379$	4.95344e26	0.416098	0.208049	+	0.978118i	$0.433289\pi$
$380$	0	0
$381$	−6.50179e26	−0.516798
$382$	1.21027e27	0.935876
$383$	−1.74581e27	−1.31344	−0.656718	−	0.754136i	$-0.728056\pi$
$383$	−1.74581e27	−1.31344	−0.656718	+	0.754136i	$0.728056\pi$
$384$	8.89375e26	0.651039
$385$	0	0
$386$	2.28119e27	1.58123
$387$	−7.15855e26	−0.482903
$388$	−3.68892e27	−2.42196
$389$	1.30302e27	0.832682	0.416341	−	0.909209i	$-0.363312\pi$
$389$	1.30302e27	0.832682	0.416341	+	0.909209i	$0.363312\pi$
$390$	0	0
$391$	−3.51692e26	−0.212964
$392$	−1.01315e27	−0.597271
$393$	−1.09917e27	−0.630874
$394$	1.59771e27	0.892871
$395$	0	0
$396$	−8.19869e26	−0.434455
$397$	2.59311e27	1.33820	0.669099	−	0.743173i	$-0.266680\pi$
$397$	2.59311e27	1.33820	0.669099	+	0.743173i	$0.266680\pi$
$398$	−5.23057e26	−0.262892
$399$	8.53235e25	0.0417689
$400$	0	0
$401$	2.16689e27	1.00652	0.503258	−	0.864136i	$-0.332134\pi$
$401$	2.16689e27	1.00652	0.503258	+	0.864136i	$0.332134\pi$
$402$	−3.15851e27	−1.42925
$403$	−1.27561e27	−0.562359
$404$	3.51586e27	1.51018
$405$	0	0
$406$	6.70855e26	0.273593
$407$	−3.89729e27	−1.54889
$408$	3.17111e26	0.122823
$409$	6.68068e24	0.00252189	0.00126094	−	0.999999i	$-0.499599\pi$
$409$	6.68068e24	0.00252189	0.00126094	+	0.999999i	$0.499599\pi$
$410$	0	0
$411$	−2.23029e27	−0.799877
$412$	5.75673e27	2.01259
$413$	9.59775e25	0.0327110
$414$	9.50515e26	0.315831
$415$	0	0
$416$	1.72202e27	0.543948
$417$	−2.81143e27	−0.865961
$418$	2.27773e27	0.684151
$419$	3.80383e27	1.11423	0.557114	−	0.830436i	$-0.311909\pi$
$419$	3.80383e27	1.11423	0.557114	+	0.830436i	$0.311909\pi$
$420$	0	0
$421$	−4.72025e27	−1.31523	−0.657617	−	0.753352i	$-0.728436\pi$
$421$	−4.72025e27	−1.31523	−0.657617	+	0.753352i	$0.728436\pi$
$422$	−6.92187e27	−1.88123
$423$	−1.22641e26	−0.0325133
$424$	1.08550e27	0.280731
$425$	0	0
$426$	5.34344e27	1.31529
$427$	4.24308e26	0.101903
$428$	−9.57030e27	−2.24267
$429$	−9.82446e26	−0.224650
$430$	0	0
$431$	−6.21329e27	−1.35304	−0.676519	−	0.736425i	$-0.736512\pi$
$431$	−6.21329e27	−1.35304	−0.676519	+	0.736425i	$0.736512\pi$
$432$	4.06292e26	0.0863493
$433$	3.48587e27	0.723082	0.361541	−	0.932356i	$-0.382251\pi$
$433$	3.48587e27	0.723082	0.361541	+	0.932356i	$0.382251\pi$
$434$	1.57786e27	0.319467
$435$	0	0
$436$	1.33632e28	2.57811
$437$	−1.53818e27	−0.289703
$438$	−3.67788e27	−0.676268
$439$	−7.74923e26	−0.139117	−0.0695587	−	0.997578i	$-0.522159\pi$
$439$	−7.74923e26	−0.139117	−0.0695587	+	0.997578i	$0.522159\pi$
$440$	0	0
$441$	−1.90201e27	−0.325542
$442$	1.34161e27	0.224229
$443$	−4.57529e26	−0.0746757	−0.0373379	−	0.999303i	$-0.511888\pi$
$443$	−4.57529e26	−0.0746757	−0.0373379	+	0.999303i	$0.511888\pi$
$444$	−8.37847e27	−1.33550
$445$	0	0
$446$	4.51827e27	0.687000
$447$	−4.26705e27	−0.633721
$448$	−1.65718e27	−0.240410
$449$	−6.18212e27	−0.876094	−0.438047	−	0.898952i	$-0.644330\pi$
$449$	−6.18212e27	−0.876094	−0.438047	+	0.898952i	$0.644330\pi$
$450$	0	0
$451$	−1.58584e27	−0.214489
$452$	1.27636e28	1.68663
$453$	−1.07011e27	−0.138164
$454$	−1.15346e28	−1.45518
$455$	0	0
$456$	1.38694e27	0.167080
$457$	8.54345e27	1.00580	0.502902	−	0.864344i	$-0.332266\pi$
$457$	8.54345e27	1.00580	0.502902	+	0.864344i	$0.332266\pi$
$458$	−3.86168e27	−0.444313
$459$	5.95318e26	0.0669446
$460$	0	0
$461$	−3.86582e27	−0.415319	−0.207660	−	0.978201i	$-0.566585\pi$
$461$	−3.86582e27	−0.415319	−0.207660	+	0.978201i	$0.566585\pi$
$462$	1.21523e27	0.127620
$463$	−3.75823e27	−0.385819	−0.192909	−	0.981217i	$-0.561792\pi$
$463$	−3.75823e27	−0.385819	−0.192909	+	0.981217i	$0.561792\pi$
$464$	−5.16948e27	−0.518810
$465$	0	0
$466$	−1.13742e28	−1.09111
$467$	−3.49287e26	−0.0327609	−0.0163804	−	0.999866i	$-0.505214\pi$
$467$	−3.49287e26	−0.0327609	−0.0163804	+	0.999866i	$0.505214\pi$
$468$	−2.11208e27	−0.193700
$469$	2.72702e27	0.244553
$470$	0	0
$471$	−1.15054e27	−0.0986697
$472$	1.56012e27	0.130848
$473$	1.64993e28	1.35339
$474$	3.02408e27	0.242617
$475$	0	0
$476$	−9.66643e26	−0.0741981
$477$	2.03783e27	0.153012
$478$	2.88606e28	2.11988
$479$	−2.41521e28	−1.73553	−0.867766	−	0.496973i	$-0.834445\pi$
$479$	−2.41521e28	−1.73553	−0.867766	+	0.496973i	$0.834445\pi$
$480$	0	0
$481$	−1.00399e28	−0.690567
$482$	−1.00567e28	−0.676799
$483$	−8.20662e26	−0.0540405
$484$	−2.75555e27	−0.177555
$485$	0	0
$486$	−1.60896e27	−0.0992806
$487$	5.91212e27	0.357017	0.178509	−	0.983938i	$-0.442873\pi$
$487$	5.91212e27	0.357017	0.178509	+	0.983938i	$0.442873\pi$
$488$	6.89714e27	0.407625
$489$	9.21297e27	0.532914
$490$	0	0
$491$	−2.46927e28	−1.36840	−0.684200	−	0.729295i	$-0.739848\pi$
$491$	−2.46927e28	−1.36840	−0.684200	+	0.729295i	$0.739848\pi$
$492$	−3.40927e27	−0.184939
$493$	−7.57456e27	−0.402221
$494$	5.86773e27	0.305026
$495$	0	0
$496$	−1.21587e28	−0.605798
$497$	−4.61345e27	−0.225052
$498$	−3.22765e27	−0.154162
$499$	−5.83166e27	−0.272732	−0.136366	−	0.990659i	$-0.543542\pi$
$499$	−5.83166e27	−0.272732	−0.136366	+	0.990659i	$0.543542\pi$
$500$	0	0
$501$	−6.21704e27	−0.278797
$502$	−3.61241e28	−1.58638
$503$	−1.91670e28	−0.824309	−0.412154	−	0.911114i	$-0.635224\pi$
$503$	−1.91670e28	−0.824309	−0.412154	+	0.911114i	$0.635224\pi$
$504$	7.39968e26	0.0311668
$505$	0	0
$506$	−2.19078e28	−0.885152
$507$	1.20580e28	0.477191
$508$	3.22162e28	1.24884
$509$	−3.23438e28	−1.22816	−0.614080	−	0.789244i	$-0.710473\pi$
$509$	−3.23438e28	−1.22816	−0.614080	+	0.789244i	$0.710473\pi$
$510$	0	0
$511$	3.17543e27	0.115713
$512$	2.41000e28	0.860365
$513$	2.60372e27	0.0910670
$514$	1.91513e28	0.656274
$515$	0	0
$516$	3.54705e28	1.16693
$517$	2.82667e27	0.0911221
$518$	1.24188e28	0.392300
$519$	4.64691e27	0.143849
$520$	0	0
$521$	2.29721e28	0.682976	0.341488	−	0.939886i	$-0.389069\pi$
$521$	2.29721e28	0.682976	0.341488	+	0.939886i	$0.389069\pi$
$522$	2.04717e28	0.596504
$523$	−2.66045e27	−0.0759780	−0.0379890	−	0.999278i	$-0.512095\pi$
$523$	−2.66045e27	−0.0759780	−0.0379890	+	0.999278i	$0.512095\pi$
$524$	5.44636e28	1.52450
$525$	0	0
$526$	6.62453e28	1.78158
$527$	−1.78154e28	−0.469661
$528$	−9.36435e27	−0.242003
$529$	−2.46770e28	−0.625184
$530$	0	0
$531$	2.92883e27	0.0713185
$532$	−4.22776e27	−0.100934
$533$	−4.08532e27	−0.0956291
$534$	−4.70710e28	−1.08036
$535$	0	0
$536$	4.43277e28	0.978240
$537$	−1.71921e28	−0.372047
$538$	3.73626e28	0.792908
$539$	4.38381e28	0.912367
$540$	0	0
$541$	−2.31491e28	−0.463406	−0.231703	−	0.972787i	$-0.574430\pi$
$541$	−2.31491e28	−0.463406	−0.231703	+	0.972787i	$0.574430\pi$
$542$	5.29198e28	1.03902
$543$	5.20863e28	1.00306
$544$	2.40501e28	0.454285
$545$	0	0
$546$	3.13059e27	0.0568989
$547$	4.35759e28	0.776926	0.388463	−	0.921464i	$-0.373006\pi$
$547$	4.35759e28	0.776926	0.388463	+	0.921464i	$0.373006\pi$
$548$	1.10511e29	1.93290
$549$	1.29481e28	0.222176
$550$	0	0
$551$	−3.31285e28	−0.547155
$552$	−1.33399e28	−0.216168
$553$	−2.61095e27	−0.0415131
$554$	1.41256e29	2.20371
$555$	0	0
$556$	1.39306e29	2.09259
$557$	2.43763e28	0.359326	0.179663	−	0.983728i	$-0.442499\pi$
$557$	2.43763e28	0.359326	0.179663	+	0.983728i	$0.442499\pi$
$558$	4.81496e28	0.696520
$559$	4.25042e28	0.603403
$560$	0	0
$561$	−1.37211e28	−0.187620
$562$	8.16232e26	0.0109542
$563$	1.40547e29	1.85132	0.925662	−	0.378351i	$-0.123509\pi$
$563$	1.40547e29	1.85132	0.925662	+	0.378351i	$0.123509\pi$
$564$	6.07683e27	0.0785682
$565$	0	0
$566$	2.41645e28	0.301027
$567$	1.38915e27	0.0169874
$568$	−7.49918e28	−0.900235
$569$	3.22894e27	0.0380523	0.0190261	−	0.999819i	$-0.493943\pi$
$569$	3.22894e27	0.0380523	0.0190261	+	0.999819i	$0.493943\pi$
$570$	0	0
$571$	−5.77191e28	−0.655602	−0.327801	−	0.944747i	$-0.606308\pi$
$571$	−5.77191e28	−0.655602	−0.327801	+	0.944747i	$0.606308\pi$
$572$	4.86800e28	0.542866
$573$	−3.18870e28	−0.349132
$574$	5.05332e27	0.0543252
$575$	0	0
$576$	−5.05704e28	−0.524155
$577$	1.35599e27	0.0138010	0.00690052	−	0.999976i	$-0.497803\pi$
$577$	1.35599e27	0.0138010	0.00690052	+	0.999976i	$0.497803\pi$
$578$	−1.36112e29	−1.36036
$579$	−6.01022e28	−0.589883
$580$	0	0
$581$	2.78671e27	0.0263780
$582$	1.66855e29	1.55113
$583$	−4.69687e28	−0.428833
$584$	5.16167e28	0.462866
$585$	0	0
$586$	−1.97506e29	−1.70866
$587$	7.75267e28	0.658797	0.329398	−	0.944191i	$-0.393154\pi$
$587$	7.75267e28	0.658797	0.329398	+	0.944191i	$0.393154\pi$
$588$	9.42441e28	0.786669
$589$	−7.79186e28	−0.638896
$590$	0	0
$591$	−4.20948e28	−0.333089
$592$	−9.56970e28	−0.743910
$593$	3.22759e28	0.246493	0.123246	−	0.992376i	$-0.460669\pi$
$593$	3.22759e28	0.246493	0.123246	+	0.992376i	$0.460669\pi$
$594$	3.70838e28	0.278245
$595$	0	0
$596$	2.11431e29	1.53138
$597$	1.37809e28	0.0980728
$598$	−5.64372e28	−0.394642
$599$	2.20890e29	1.51773	0.758864	−	0.651249i	$-0.225755\pi$
$599$	2.20890e29	1.51773	0.758864	+	0.651249i	$0.225755\pi$
$600$	0	0
$601$	−9.50760e28	−0.630796	−0.315398	−	0.948959i	$-0.602138\pi$
$601$	−9.50760e28	−0.630796	−0.315398	+	0.948959i	$0.602138\pi$
$602$	−5.25754e28	−0.342783
$603$	8.32171e28	0.533189
$604$	5.30236e28	0.333873
$605$	0	0
$606$	−1.59028e29	−0.967186
$607$	−2.76276e29	−1.65144	−0.825718	−	0.564083i	$-0.809230\pi$
$607$	−2.76276e29	−1.65144	−0.825718	+	0.564083i	$0.809230\pi$
$608$	1.05187e29	0.617980
$609$	−1.76750e28	−0.102065
$610$	0	0
$611$	7.28185e27	0.0406265
$612$	−2.94979e28	−0.161771
$613$	−3.49899e29	−1.88628	−0.943142	−	0.332390i	$-0.892145\pi$
$613$	−3.49899e29	−1.88628	−0.943142	+	0.332390i	$0.892145\pi$
$614$	−3.21528e29	−1.70393
$615$	0	0
$616$	−1.70550e28	−0.0873484
$617$	−7.63276e28	−0.384315	−0.192158	−	0.981364i	$-0.561548\pi$
$617$	−7.63276e28	−0.384315	−0.192158	+	0.981364i	$0.561548\pi$
$618$	−2.60385e29	−1.28895
$619$	−2.56723e29	−1.24943	−0.624717	−	0.780851i	$-0.714786\pi$
$619$	−2.56723e29	−1.24943	−0.624717	+	0.780851i	$0.714786\pi$
$620$	0	0
$621$	−2.50432e28	−0.117822
$622$	−5.67886e29	−2.62701
$623$	4.06404e28	0.184856
$624$	−2.41237e28	−0.107896
$625$	0	0
$626$	6.72028e29	2.90641
$627$	−6.00113e28	−0.255225
$628$	5.70092e28	0.238434
$629$	−1.40220e29	−0.576736
$630$	0	0
$631$	−4.13211e29	−1.64386	−0.821928	−	0.569591i	$-0.807102\pi$
$631$	−4.13211e29	−1.64386	−0.821928	+	0.569591i	$0.807102\pi$
$632$	−4.24411e28	−0.166057
$633$	1.82370e29	0.701802
$634$	−6.15805e29	−2.33081
$635$	0	0
$636$	−1.00974e29	−0.369753
$637$	1.12932e29	0.406775
$638$	−4.71838e29	−1.67177
$639$	−1.40783e29	−0.490673
$640$	0	0
$641$	−5.15627e29	−1.73911	−0.869554	−	0.493839i	$-0.835593\pi$
$641$	−5.15627e29	−1.73911	−0.869554	+	0.493839i	$0.835593\pi$
$642$	4.32878e29	1.43631
$643$	−3.07884e29	−1.00501	−0.502507	−	0.864573i	$-0.667589\pi$
$643$	−3.07884e29	−1.00501	−0.502507	+	0.864573i	$0.667589\pi$
$644$	4.06636e28	0.130588
$645$	0	0
$646$	8.19501e28	0.254747
$647$	2.34348e29	0.716747	0.358374	−	0.933578i	$-0.383331\pi$
$647$	2.34348e29	0.716747	0.358374	+	0.933578i	$0.383331\pi$
$648$	2.25807e28	0.0679517
$649$	−6.75047e28	−0.199878
$650$	0	0
$651$	−4.15717e28	−0.119178
$652$	−4.56501e29	−1.28778
$653$	−2.65412e29	−0.736771	−0.368386	−	0.929673i	$-0.620089\pi$
$653$	−2.65412e29	−0.736771	−0.368386	+	0.929673i	$0.620089\pi$
$654$	−6.04436e29	−1.65114
$655$	0	0
$656$	−3.89399e28	−0.103016
$657$	9.69008e28	0.252285
$658$	−9.00726e27	−0.0230792
$659$	5.54851e29	1.39920	0.699599	−	0.714536i	$-0.253362\pi$
$659$	5.54851e29	1.39920	0.699599	+	0.714536i	$0.253362\pi$
$660$	0	0
$661$	8.03149e29	1.96192	0.980959	−	0.194214i	$-0.0622157\pi$
$661$	8.03149e29	1.96192	0.980959	+	0.194214i	$0.0622157\pi$
$662$	6.95358e29	1.67186
$663$	−3.53472e28	−0.0836495
$664$	4.52981e28	0.105515
$665$	0	0
$666$	3.78970e29	0.855315
$667$	3.18638e29	0.707907
$668$	3.08053e29	0.673710
$669$	−1.19043e29	−0.256288
$670$	0	0
$671$	−2.98432e29	−0.622672
$672$	5.61201e28	0.115277
$673$	5.51247e29	1.11478	0.557388	−	0.830252i	$-0.311803\pi$
$673$	5.51247e29	1.11478	0.557388	+	0.830252i	$0.311803\pi$
$674$	8.78125e29	1.74835
$675$	0	0
$676$	−5.97472e29	−1.15313
$677$	−6.54093e28	−0.124296	−0.0621482	−	0.998067i	$-0.519795\pi$
$677$	−6.54093e28	−0.124296	−0.0621482	+	0.998067i	$0.519795\pi$
$678$	−5.77316e29	−1.08019
$679$	−1.44060e29	−0.265407
$680$	0	0
$681$	3.03902e29	0.542862
$682$	−1.10977e30	−1.95207
$683$	−3.31361e29	−0.573963	−0.286982	−	0.957936i	$-0.592652\pi$
$683$	−3.31361e29	−0.573963	−0.286982	+	0.957936i	$0.592652\pi$
$684$	−1.29014e29	−0.220063
$685$	0	0
$686$	−2.82726e29	−0.467695
$687$	1.01743e29	0.165753
$688$	4.05136e29	0.650013
$689$	−1.20997e29	−0.191194
$690$	0	0
$691$	−9.41077e28	−0.144247	−0.0721233	−	0.997396i	$-0.522978\pi$
$691$	−9.41077e28	−0.144247	−0.0721233	+	0.997396i	$0.522978\pi$
$692$	−2.30254e29	−0.347610
$693$	−3.20177e28	−0.0476092
$694$	1.65000e30	2.41663
$695$	0	0
$696$	−2.87307e29	−0.408272
$697$	−5.70565e28	−0.0798659
$698$	−1.52035e30	−2.09634
$699$	2.99675e29	0.407043
$700$	0	0
$701$	−4.47905e28	−0.0590401	−0.0295200	−	0.999564i	$-0.509398\pi$
$701$	−4.47905e28	−0.0590401	−0.0295200	+	0.999564i	$0.509398\pi$
$702$	9.55326e28	0.124054
$703$	−6.13273e29	−0.784554
$704$	1.16556e30	1.46900
$705$	0	0
$706$	7.58970e29	0.928486
$707$	1.37302e29	0.165491
$708$	−1.45123e29	−0.172341
$709$	−2.43421e29	−0.284822	−0.142411	−	0.989808i	$-0.545485\pi$
$709$	−2.43421e29	−0.284822	−0.142411	+	0.989808i	$0.545485\pi$
$710$	0	0
$711$	−7.96753e28	−0.0905093
$712$	6.60612e29	0.739446
$713$	7.49439e29	0.826602
$714$	4.37226e28	0.0475199
$715$	0	0
$716$	8.51863e29	0.899048
$717$	−7.60388e29	−0.790832
$718$	−1.36634e30	−1.40040
$719$	−2.06022e29	−0.208095	−0.104047	−	0.994572i	$-0.533179\pi$
$719$	−2.06022e29	−0.208095	−0.104047	+	0.994572i	$0.533179\pi$
$720$	0	0
$721$	2.24813e29	0.220547
$722$	−1.24227e30	−1.20109
$723$	2.64962e29	0.252483
$724$	−2.58087e30	−2.42388
$725$	0	0
$726$	1.24638e29	0.113714
$727$	1.28019e30	1.15124	0.575618	−	0.817719i	$-0.304762\pi$
$727$	1.28019e30	1.15124	0.575618	+	0.817719i	$0.304762\pi$
$728$	−4.39359e28	−0.0389440
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	5.93624e29	0.503940
$732$	−6.41577e29	−0.536886
$733$	1.78868e27	0.00147550	0.000737752	−	1.00000i	$-0.499765\pi$
$733$	1.78868e27	0.00147550	0.000737752		1.00000i	$0.499765\pi$
$734$	2.30355e30	1.87322
$735$	0	0
$736$	−1.01171e30	−0.799540
$737$	−1.91802e30	−1.49432
$738$	1.54206e29	0.118443
$739$	2.29344e30	1.73668	0.868342	−	0.495965i	$-0.165186\pi$
$739$	2.29344e30	1.73668	0.868342	+	0.495965i	$0.165186\pi$
$740$	0	0
$741$	−1.54597e29	−0.113791
$742$	1.49667e29	0.108614
$743$	7.02085e29	0.502351	0.251176	−	0.967942i	$-0.419183\pi$
$743$	7.02085e29	0.502351	0.251176	+	0.967942i	$0.419183\pi$
$744$	−6.75749e29	−0.476727
$745$	0	0
$746$	−1.21556e29	−0.0833719
$747$	8.50387e28	0.0575109
$748$	6.79877e29	0.453381
$749$	−3.73741e29	−0.245760
$750$	0	0
$751$	1.73004e30	1.10621	0.553104	−	0.833112i	$-0.313443\pi$
$751$	1.73004e30	1.10621	0.553104	+	0.833112i	$0.313443\pi$
$752$	6.94082e28	0.0437647
$753$	9.51760e29	0.591807
$754$	−1.21551e30	−0.745351
$755$	0	0
$756$	−6.88323e28	−0.0410500
$757$	1.79655e30	1.05665	0.528327	−	0.849041i	$-0.322819\pi$
$757$	1.79655e30	1.05665	0.528327	+	0.849041i	$0.322819\pi$
$758$	1.11017e30	0.643966
$759$	5.77203e29	0.330210
$760$	0	0
$761$	2.45713e30	1.36738	0.683689	−	0.729774i	$-0.260375\pi$
$761$	2.45713e30	1.36738	0.683689	+	0.729774i	$0.260375\pi$
$762$	−1.45719e30	−0.799812
$763$	5.21862e29	0.282519
$764$	1.58000e30	0.843676
$765$	0	0
$766$	−3.91272e30	−2.03271
$767$	−1.73901e29	−0.0891148
$768$	1.97248e29	0.0997055
$769$	−2.70584e30	−1.34920	−0.674599	−	0.738184i	$-0.735683\pi$
$769$	−2.70584e30	−1.34920	−0.674599	+	0.738184i	$0.735683\pi$
$770$	0	0
$771$	−5.04578e29	−0.244826
$772$	2.97806e30	1.42545
$773$	1.85688e30	0.876796	0.438398	−	0.898781i	$-0.355546\pi$
$773$	1.85688e30	0.876796	0.438398	+	0.898781i	$0.355546\pi$
$774$	−1.60438e30	−0.747356
$775$	0	0
$776$	−2.34171e30	−1.06166
$777$	−3.27198e29	−0.146349
$778$	2.92033e30	1.28868
$779$	−2.49546e29	−0.108644
$780$	0	0
$781$	3.24482e30	1.37516
$782$	−7.88216e29	−0.329590
$783$	−5.39366e29	−0.222528
$784$	1.07643e30	0.438197
$785$	0	0
$786$	−2.46347e30	−0.976360
$787$	2.99802e30	1.17246	0.586232	−	0.810143i	$-0.300611\pi$
$787$	2.99802e30	1.17246	0.586232	+	0.810143i	$0.300611\pi$
$788$	2.08579e30	0.804907
$789$	−1.74536e30	−0.664625
$790$	0	0
$791$	4.98447e29	0.184827
$792$	−5.20448e29	−0.190442
$793$	−7.68799e29	−0.277616
$794$	5.81170e30	2.07104
$795$	0	0
$796$	−6.82843e29	−0.236992
$797$	4.97662e30	1.70460	0.852300	−	0.523054i	$-0.175207\pi$
$797$	4.97662e30	1.70460	0.852300	+	0.523054i	$0.175207\pi$
$798$	1.91228e29	0.0646429
$799$	1.01700e29	0.0339297
$800$	0	0
$801$	1.24018e30	0.403034
$802$	4.85645e30	1.55772
$803$	−2.23340e30	−0.707056
$804$	−4.12339e30	−1.28845
$805$	0	0
$806$	−2.85890e30	−0.870324
$807$	−9.84390e29	−0.295798
$808$	2.23185e30	0.661982
$809$	2.09042e30	0.612032	0.306016	−	0.952026i	$-0.401004\pi$
$809$	2.09042e30	0.612032	0.306016	+	0.952026i	$0.401004\pi$
$810$	0	0
$811$	−4.87546e30	−1.39090	−0.695450	−	0.718574i	$-0.744795\pi$
$811$	−4.87546e30	−1.39090	−0.695450	+	0.718574i	$0.744795\pi$
$812$	8.75792e29	0.246640
$813$	−1.39427e30	−0.387612
$814$	−8.73463e30	−2.39711
$815$	0	0
$816$	−3.36918e29	−0.0901110
$817$	2.59631e30	0.685527
$818$	1.49728e28	0.00390296
$819$	−8.24815e28	−0.0212264
$820$	0	0
$821$	6.75479e30	1.69437	0.847186	−	0.531296i	$-0.178295\pi$
$821$	6.75479e30	1.69437	0.847186	+	0.531296i	$0.178295\pi$
$822$	−4.99856e30	−1.23791
$823$	1.79953e30	0.440008	0.220004	−	0.975499i	$-0.429393\pi$
$823$	1.79953e30	0.440008	0.220004	+	0.975499i	$0.429393\pi$
$824$	3.65434e30	0.882213
$825$	0	0
$826$	2.15106e29	0.0506246
$827$	−4.29149e30	−0.997242	−0.498621	−	0.866820i	$-0.666160\pi$
$827$	−4.29149e30	−0.997242	−0.498621	+	0.866820i	$0.666160\pi$
$828$	1.24088e30	0.284717
$829$	−3.65085e30	−0.827125	−0.413563	−	0.910476i	$-0.635716\pi$
$829$	−3.65085e30	−0.827125	−0.413563	+	0.910476i	$0.635716\pi$
$830$	0	0
$831$	−3.72166e30	−0.822102
$832$	3.00263e30	0.654949
$833$	1.57724e30	0.339724
$834$	−6.30100e30	−1.34019
$835$	0	0
$836$	2.97355e30	0.616750
$837$	−1.26859e30	−0.259840
$838$	8.52519e30	1.72441
$839$	−8.23722e28	−0.0164543	−0.00822716	−	0.999966i	$-0.502619\pi$
$839$	−8.23722e28	−0.0164543	−0.00822716	+	0.999966i	$0.502619\pi$
$840$	0	0
$841$	1.72982e30	0.337010
$842$	−1.05791e31	−2.03550
$843$	−2.15052e28	−0.00408652
$844$	−9.03640e30	−1.69590
$845$	0	0
$846$	−2.74864e29	−0.0503186
$847$	−1.07610e29	−0.0194571
$848$	−1.15331e30	−0.205962
$849$	−6.36660e29	−0.112299
$850$	0	0
$851$	5.89860e30	1.01505
$852$	6.97578e30	1.18571
$853$	4.91881e30	0.825838	0.412919	−	0.910768i	$-0.364509\pi$
$853$	4.91881e30	0.825838	0.412919	+	0.910768i	$0.364509\pi$
$854$	9.50964e29	0.157709
$855$	0	0
$856$	−6.07517e30	−0.983069
$857$	−1.29758e29	−0.0207413	−0.0103706	−	0.999946i	$-0.503301\pi$
$857$	−1.29758e29	−0.0207413	−0.0103706	+	0.999946i	$0.503301\pi$
$858$	−2.20187e30	−0.347676
$859$	9.96267e30	1.55399	0.776994	−	0.629509i	$-0.216744\pi$
$859$	9.96267e30	1.55399	0.776994	+	0.629509i	$0.216744\pi$
$860$	0	0
$861$	−1.33139e29	−0.0202663
$862$	−1.39253e31	−2.09400
$863$	−1.99413e30	−0.296238	−0.148119	−	0.988970i	$-0.547322\pi$
$863$	−1.99413e30	−0.296238	−0.148119	+	0.988970i	$0.547322\pi$
$864$	1.71255e30	0.251333
$865$	0	0
$866$	7.81256e30	1.11906
$867$	3.58614e30	0.507489
$868$	2.05987e30	0.287993
$869$	1.83638e30	0.253662
$870$	0	0
$871$	−4.94105e30	−0.666237
$872$	8.48289e30	1.13011
$873$	−4.39612e30	−0.578656
$874$	−3.44739e30	−0.448353
$875$	0	0
$876$	−4.80142e30	−0.609644
$877$	−2.12503e30	−0.266606	−0.133303	−	0.991075i	$-0.542558\pi$
$877$	−2.12503e30	−0.266606	−0.133303	+	0.991075i	$0.542558\pi$
$878$	−1.73676e30	−0.215302
$879$	5.20368e30	0.637422
$880$	0	0
$881$	1.39848e31	1.67267	0.836333	−	0.548221i	$-0.184695\pi$
$881$	1.39848e31	1.67267	0.836333	+	0.548221i	$0.184695\pi$
$882$	−4.26280e30	−0.503819
$883$	2.96651e30	0.346464	0.173232	−	0.984881i	$-0.444579\pi$
$883$	2.96651e30	0.346464	0.173232	+	0.984881i	$0.444579\pi$
$884$	1.75145e30	0.202138
$885$	0	0
$886$	−1.02542e30	−0.115570
$887$	6.70105e30	0.746354	0.373177	−	0.927760i	$-0.378268\pi$
$887$	6.70105e30	0.746354	0.373177	+	0.927760i	$0.378268\pi$
$888$	−5.31861e30	−0.585413
$889$	1.25811e30	0.136852
$890$	0	0
$891$	−9.77045e29	−0.103800
$892$	5.89854e30	0.619318
$893$	4.44801e29	0.0461557
$894$	−9.56335e30	−0.980767
$895$	0	0
$896$	−1.72097e30	−0.172401
$897$	1.48695e30	0.147223
$898$	−1.38554e31	−1.35587
$899$	1.61410e31	1.56119
$900$	0	0
$901$	−1.68988e30	−0.159678
$902$	−3.55419e30	−0.331950
$903$	1.38520e30	0.127877
$904$	8.10227e30	0.739329
$905$	0	0
$906$	−2.39833e30	−0.213827
$907$	1.08445e30	0.0955729	0.0477865	−	0.998858i	$-0.484783\pi$
$907$	1.08445e30	0.0955729	0.0477865	+	0.998858i	$0.484783\pi$
$908$	−1.50583e31	−1.31182
$909$	4.18989e30	0.360813
$910$	0	0
$911$	−2.33979e31	−1.96895	−0.984475	−	0.175523i	$-0.943838\pi$
$911$	−2.33979e31	−1.96895	−0.984475	+	0.175523i	$0.943838\pi$
$912$	−1.47356e30	−0.122581
$913$	−1.96000e30	−0.161181
$914$	1.91477e31	1.55661
$915$	0	0
$916$	−5.04137e30	−0.400540
$917$	2.12692e30	0.167060
$918$	1.33423e30	0.103606
$919$	1.44771e31	1.11139	0.555697	−	0.831385i	$-0.312451\pi$
$919$	1.44771e31	1.11139	0.555697	+	0.831385i	$0.312451\pi$
$920$	0	0
$921$	8.47127e30	0.635657
$922$	−8.66411e30	−0.642761
$923$	8.35906e30	0.613111
$924$	1.58647e30	0.115047
$925$	0	0
$926$	−8.42299e30	−0.597105
$927$	6.86035e30	0.480849
$928$	−2.17897e31	−1.51007
$929$	7.21832e30	0.494619	0.247310	−	0.968936i	$-0.420453\pi$
$929$	7.21832e30	0.494619	0.247310	+	0.968936i	$0.420453\pi$
$930$	0	0
$931$	6.89831e30	0.462138
$932$	−1.48489e31	−0.983617
$933$	1.49620e31	0.980017
$934$	−7.82826e29	−0.0507018
$935$	0	0
$936$	−1.34074e30	−0.0849079
$937$	−7.45714e30	−0.466989	−0.233495	−	0.972358i	$-0.575016\pi$
$937$	−7.45714e30	−0.466989	−0.233495	+	0.972358i	$0.575016\pi$
$938$	6.11181e30	0.378478
$939$	−1.77059e31	−1.08425
$940$	0	0
$941$	−1.70393e31	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$941$	−1.70393e31	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$942$	−2.57861e30	−0.152704
$943$	2.40019e30	0.140564
$944$	−1.65756e30	−0.0959984
$945$	0	0
$946$	3.69783e31	2.09455
$947$	3.40600e31	1.90797	0.953983	−	0.299862i	$-0.0969406\pi$
$947$	3.40600e31	1.90797	0.953983	+	0.299862i	$0.0969406\pi$
$948$	3.94790e30	0.218715
$949$	−5.75352e30	−0.315238
$950$	0	0
$951$	1.62246e31	0.869517
$952$	−6.13620e29	−0.0325246
$953$	−9.56557e30	−0.501459	−0.250729	−	0.968057i	$-0.580670\pi$
$953$	−9.56557e30	−0.501459	−0.250729	+	0.968057i	$0.580670\pi$
$954$	4.56721e30	0.236806
$955$	0	0
$956$	3.76771e31	1.91104
$957$	1.24315e31	0.623660
$958$	−5.41300e31	−2.68596
$959$	4.31568e30	0.211814
$960$	0	0
$961$	1.71383e31	0.822949
$962$	−2.25015e31	−1.06874
$963$	−1.14050e31	−0.535821
$964$	−1.31288e31	−0.610122
$965$	0	0
$966$	−1.83927e30	−0.0836347
$967$	−7.75754e30	−0.348936	−0.174468	−	0.984663i	$-0.555821\pi$
$967$	−7.75754e30	−0.348936	−0.174468	+	0.984663i	$0.555821\pi$
$968$	−1.74921e30	−0.0778307
$969$	−2.15913e30	−0.0950342
$970$	0	0
$971$	−2.98641e31	−1.28631	−0.643157	−	0.765734i	$-0.722376\pi$
$971$	−2.98641e31	−1.28631	−0.643157	+	0.765734i	$0.722376\pi$
$972$	−2.10047e30	−0.0894997
$973$	5.44019e30	0.229313
$974$	1.32503e31	0.552531
$975$	0	0
$976$	−7.32794e30	−0.299060
$977$	4.06815e31	1.64250	0.821248	−	0.570571i	$-0.193278\pi$
$977$	4.06815e31	1.64250	0.821248	+	0.570571i	$0.193278\pi$
$978$	2.06482e31	0.824754
$979$	−2.85840e31	−1.12955
$980$	0	0
$981$	1.59250e31	0.615965
$982$	−5.53415e31	−2.11778
$983$	−1.43896e31	−0.544800	−0.272400	−	0.962184i	$-0.587817\pi$
$983$	−1.43896e31	−0.544800	−0.272400	+	0.962184i	$0.587817\pi$
$984$	−2.16419e30	−0.0810674
$985$	0	0
$986$	−1.69762e31	−0.622490
$987$	2.37314e29	0.00860979
$988$	7.66024e30	0.274976
$989$	−2.49719e31	−0.886932
$990$	0	0
$991$	6.65548e30	0.231423	0.115711	−	0.993283i	$-0.463085\pi$
$991$	6.65548e30	0.231423	0.115711	+	0.993283i	$0.463085\pi$
$992$	−5.12496e31	−1.76327
$993$	−1.83205e31	−0.623694
$994$	−1.03397e31	−0.348298
$995$	0	0
$996$	−4.21365e30	−0.138975
$997$	−1.16327e31	−0.379648	−0.189824	−	0.981818i	$-0.560792\pi$
$997$	−1.16327e31	−0.379648	−0.189824	+	0.981818i	$0.560792\pi$
$998$	−1.30700e31	−0.422089
$999$	−9.98471e30	−0.319079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.n.1.9		10
5.2	odd	4		15.22.b.a.4.17	yes	20
5.3	odd	4		15.22.b.a.4.4	✓	20
5.4	even	2		75.22.a.m.1.2		10
15.2	even	4		45.22.b.d.19.4		20
15.8	even	4		45.22.b.d.19.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.b.a.4.4	✓	20	5.3	odd	4
15.22.b.a.4.17	yes	20	5.2	odd	4
45.22.b.d.19.4		20	15.2	even	4
45.22.b.d.19.17		20	15.8	even	4
75.22.a.m.1.2		10	5.4	even	2
75.22.a.n.1.9		10	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+2241.21 q^{2} -59049.0 q^{3} +2.92587e6 q^{4} -1.32341e8 q^{6} +1.14262e8 q^{7} +1.85733e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+2241.21 q^{2} -59049.0 q^{3} +2.92587e6 q^{4} -1.32341e8 q^{6} +1.14262e8 q^{7} +1.85733e9 q^{8} +3.48678e9 q^{9} -8.03645e10 q^{11} -1.72770e11 q^{12} -2.07029e11 q^{13} +2.56084e11 q^{14} -1.97333e12 q^{16} -2.89142e12 q^{17} +7.81461e12 q^{18} -1.26461e13 q^{19} -6.74703e12 q^{21} -1.80114e14 q^{22} +1.21633e14 q^{23} -1.09673e14 q^{24} -4.63996e14 q^{26} -2.05891e14 q^{27} +3.34314e14 q^{28} +2.61967e15 q^{29} +6.16148e15 q^{31} -8.31775e15 q^{32} +4.74545e15 q^{33} -6.48028e15 q^{34} +1.02019e16 q^{36} +4.84951e16 q^{37} -2.83425e16 q^{38} +1.22249e16 q^{39} +1.97330e16 q^{41} -1.51215e16 q^{42} -2.05305e17 q^{43} -2.35136e17 q^{44} +2.72605e17 q^{46} -3.51731e16 q^{47} +1.16523e17 q^{48} -5.45490e17 q^{49} +1.70735e17 q^{51} -6.05740e17 q^{52} +5.84445e17 q^{53} -4.61445e17 q^{54} +2.12221e17 q^{56} +7.46738e17 q^{57} +5.87122e18 q^{58} +8.39981e17 q^{59} +3.71348e18 q^{61} +1.38092e19 q^{62} +3.98405e17 q^{63} -1.45034e19 q^{64} +1.06355e19 q^{66} +2.38664e19 q^{67} -8.45991e18 q^{68} -7.18231e18 q^{69} -4.03762e19 q^{71} +6.47609e18 q^{72} +2.77909e19 q^{73} +1.08688e20 q^{74} -3.70008e19 q^{76} -9.18258e18 q^{77} +2.73985e19 q^{78} -2.28507e19 q^{79} +1.21577e19 q^{81} +4.42259e19 q^{82} +2.43889e19 q^{83} -1.97409e19 q^{84} -4.60132e20 q^{86} -1.54689e20 q^{87} -1.49263e20 q^{88} +3.55679e20 q^{89} -2.36555e19 q^{91} +3.55882e20 q^{92} -3.63829e20 q^{93} -7.88302e19 q^{94} +4.91155e20 q^{96} -1.26080e21 q^{97} -1.22256e21 q^{98} -2.80214e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9	\( 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} - 593041212045 q^{12} + 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} + 12910340404230 q^{17} + 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} + 461780887241010 q^{22} + 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} - 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) \) 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9 + 5976221790 * q^11 - 593041212045 * q^12 + 842570747430 * q^13 + 1605059874570 * q^14 + 6061861547825 * q^16 + 12910340404230 * q^17 + 2248975938645 * q^18 - 76155422176280 * q^19 - 6811732617210 * q^21 + 461780887241010 * q^22 + 82640920915920 * q^23 - 18549814359015 * q^24 + 286555331159670 * q^26 - 2058911320946490 * q^27 + 1520428218531930 * q^28 - 2553653160731010 * q^29 - 5579671364724940 * q^31 + 29255856594217815 * q^32 - 352889920477710 * q^33 + 36285968426431010 * q^34 + 35018490530045205 * q^36 + 1797908431222890 * q^37 + 10684960744876320 * q^38 - 49752960064994070 * q^39 - 157319729506021440 * q^41 - 94777180533483930 * q^42 - 148924853700743400 * q^43 + 49118820773767290 * q^44 - 238453286038259540 * q^46 - 43171138083158100 * q^47 - 357946862537518425 * q^48 + 453692886452454950 * q^49 - 762342690529377270 * q^51 + 5077473324432504030 * q^52 + 1835165786666252970 * q^53 - 132799780201048605 * q^54 - 5146515624013530450 * q^56 + 4496901524087157720 * q^57 + 3684042125074285440 * q^58 + 4936135121536323630 * q^59 - 4443080871007408300 * q^61 + 39366851097921260280 * q^62 + 402225999313633290 * q^63 - 14877283608188093455 * q^64 - 27267699610694399490 * q^66 + 33664559574358232580 * q^67 + 44791395195934111290 * q^68 - 4879863739164160080 * q^69 - 74662724971386630180 * q^71 + 1095347988085476735 * q^72 + 16062862692682477620 * q^73 - 130249023546610518030 * q^74 - 258736483662380220720 * q^76 + 56216747719240740180 * q^77 - 16920805749647353830 * q^78 - 49130708010799643300 * q^79 + 121576654590569288010 * q^81 + 117185080045656261930 * q^82 + 118408383608453695560 * q^83 - 89779765876091934570 * q^84 - 1007271272794707382260 * q^86 + 150790665488005409490 * q^87 + 41803519849182967710 * q^88 - 572103976713991581780 * q^89 - 1669810835257196575140 * q^91 - 1206545757146297440260 * q^92 + 329474014415642982060 * q^93 - 1671319720124696594920 * q^94 - 1727529076031967757935 * q^96 - 1299275526640742173920 * q^97 - 2683959884079093505455 * q^98 + 20837796914288297790 * q^99

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