LMFDB - Embedded newform 75.12.a.c.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1801}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 450 $ x^2 - x - 450
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-20.7191$ 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)

$f(q)$	$=$	\(q+27.7191 q^{2} +243.000 q^{3} -1279.65 q^{4} +6735.74 q^{6} +72157.2 q^{7} -92239.5 q^{8} +59049.0 q^{9} -509499. q^{11} -310955. q^{12} -1.85516e6 q^{13} +2.00013e6 q^{14} +63931.4 q^{16} -5.94676e6 q^{17} +1.63678e6 q^{18} +6.02189e6 q^{19} +1.75342e7 q^{21} -1.41228e7 q^{22} +4.82627e7 q^{23} -2.24142e7 q^{24} -5.14234e7 q^{26} +1.43489e7 q^{27} -9.23361e7 q^{28} -1.13550e7 q^{29} -1.72184e8 q^{31} +1.90679e8 q^{32} -1.23808e8 q^{33} -1.64839e8 q^{34} -7.55622e7 q^{36} -6.25191e8 q^{37} +1.66921e8 q^{38} -4.50805e8 q^{39} -5.53971e8 q^{41} +4.86032e8 q^{42} -1.52106e9 q^{43} +6.51981e8 q^{44} +1.33780e9 q^{46} -1.19456e9 q^{47} +1.55353e7 q^{48} +3.22934e9 q^{49} -1.44506e9 q^{51} +2.37396e9 q^{52} -1.22320e9 q^{53} +3.97739e8 q^{54} -6.65575e9 q^{56} +1.46332e9 q^{57} -3.14750e8 q^{58} -5.83637e9 q^{59} -6.61097e9 q^{61} -4.77277e9 q^{62} +4.26081e9 q^{63} +5.15451e9 q^{64} -3.43185e9 q^{66} -1.66261e10 q^{67} +7.60978e9 q^{68} +1.17278e10 q^{69} +7.36200e9 q^{71} -5.44665e9 q^{72} +6.35726e9 q^{73} -1.73297e10 q^{74} -7.70593e9 q^{76} -3.67640e10 q^{77} -1.24959e10 q^{78} +2.47565e10 q^{79} +3.48678e9 q^{81} -1.53556e10 q^{82} -3.59416e10 q^{83} -2.24377e10 q^{84} -4.21625e10 q^{86} -2.75926e9 q^{87} +4.69959e10 q^{88} +7.47690e10 q^{89} -1.33863e11 q^{91} -6.17594e10 q^{92} -4.18406e10 q^{93} -3.31121e10 q^{94} +4.63349e10 q^{96} +1.66300e10 q^{97} +8.95144e10 q^{98} -3.00854e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 13 q^{2} + 486 q^{3} - 3111 q^{4} + 3159 q^{6} - 7784 q^{7} - 35139 q^{8} + 118098 q^{9} + 295568 q^{11} - 755973 q^{12} - 657492 q^{13} + 3176796 q^{14} + 2974065 q^{16} - 8579948 q^{17} + 767637 q^{18}+ \cdots + 17452994832 q^{99}+O(q^{100}) $ 2 * q + 13 * q^2 + 486 * q^3 - 3111 * q^4 + 3159 * q^6 - 7784 * q^7 - 35139 * q^8 + 118098 * q^9 + 295568 * q^11 - 755973 * q^12 - 657492 * q^13 + 3176796 * q^14 + 2974065 * q^16 - 8579948 * q^17 + 767637 * q^18 + 17627976 * q^19 - 1891512 * q^21 - 25972696 * q^22 + 29841072 * q^23 - 8538777 * q^24 - 69052066 * q^26 + 28697814 * q^27 + 54064108 * q^28 - 201881948 * q^29 - 71057008 * q^31 + 30902261 * q^32 + 71823024 * q^33 - 126080702 * q^34 - 183701439 * q^36 - 705858484 * q^37 - 3909548 * q^38 - 159770556 * q^39 - 327655148 * q^41 + 771961428 * q^42 - 3192552120 * q^43 - 822376568 * q^44 + 1608947832 * q^46 - 2053064720 * q^47 + 722697795 * q^48 + 7642614770 * q^49 - 2084927364 * q^51 + 180610022 * q^52 + 2304299452 * q^53 + 186535791 * q^54 - 11220431460 * q^56 + 4283598168 * q^57 + 2489635114 * q^58 - 1478770576 * q^59 - 8264891460 * q^61 - 6261263448 * q^62 - 459637416 * q^63 + 1546316681 * q^64 - 6311365128 * q^66 - 24212177528 * q^67 + 12432065594 * q^68 + 7251380496 * q^69 - 20218888256 * q^71 - 2074922811 * q^72 - 25879583268 * q^73 - 16142374034 * q^74 - 28960704764 * q^76 - 101122045248 * q^77 - 16779652038 * q^78 + 22324995440 * q^79 + 6973568802 * q^81 - 18686740478 * q^82 - 48014508984 * q^83 + 13137578244 * q^84 - 17559698092 * q^86 - 49057313364 * q^87 + 92965609896 * q^88 + 79209683076 * q^89 - 229606712048 * q^91 - 28023036264 * q^92 - 17266852944 * q^93 - 20475612776 * q^94 + 7509249423 * q^96 + 37075227452 * q^97 + 24554976677 * q^98 + 17452994832 * q^99

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 $

$2$	27.7191	0.612511	0.306256	−	0.951949i	$-0.400924\pi$
$2$	27.7191	0.612511	0.306256	+	0.951949i	$0.400924\pi$
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	−1279.65	−0.624830
$5$	0	0
$5$	0	0
$6$	6735.74	0.353634
$7$	72157.2	1.62271	0.811355	−	0.584554i	$-0.198731\pi$
$7$	72157.2	1.62271	0.811355	+	0.584554i	$0.198731\pi$
$8$	−92239.5	−0.995227
$9$	59049.0	0.333333
$10$	0	0
$11$	−509499.	−0.953857	−0.476929	−	0.878942i	$-0.658250\pi$
$11$	−509499.	−0.953857	−0.476929	+	0.878942i	$0.658250\pi$
$12$	−310955.	−0.360746
$13$	−1.85516e6	−1.38578	−0.692889	−	0.721044i	$-0.743662\pi$
$13$	−1.85516e6	−1.38578	−0.692889	+	0.721044i	$0.743662\pi$
$14$	2.00013e6	0.993928
$15$	0	0
$16$	63931.4	0.0152424
$17$	−5.94676e6	−1.01581	−0.507904	−	0.861414i	$-0.669579\pi$
$17$	−5.94676e6	−1.01581	−0.507904	+	0.861414i	$0.669579\pi$
$18$	1.63678e6	0.204170
$19$	6.02189e6	0.557941	0.278971	−	0.960300i	$-0.410007\pi$
$19$	6.02189e6	0.557941	0.278971	+	0.960300i	$0.410007\pi$
$20$	0	0
$21$	1.75342e7	0.936872
$22$	−1.41228e7	−0.584248
$23$	4.82627e7	1.56354	0.781769	−	0.623568i	$-0.214318\pi$
$23$	4.82627e7	1.56354	0.781769	+	0.623568i	$0.214318\pi$
$24$	−2.24142e7	−0.574594
$25$	0	0
$26$	−5.14234e7	−0.848804
$27$	1.43489e7	0.192450
$28$	−9.23361e7	−1.01392
$29$	−1.13550e7	−0.102801	−0.0514005	−	0.998678i	$-0.516368\pi$
$29$	−1.13550e7	−0.102801	−0.0514005	+	0.998678i	$0.516368\pi$
$30$	0	0
$31$	−1.72184e8	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$31$	−1.72184e8	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$32$	1.90679e8	1.00456
$33$	−1.23808e8	−0.550710
$34$	−1.64839e8	−0.622194
$35$	0	0
$36$	−7.55622e7	−0.208277
$37$	−6.25191e8	−1.48219	−0.741094	−	0.671401i	$-0.765693\pi$
$37$	−6.25191e8	−1.48219	−0.741094	+	0.671401i	$0.765693\pi$
$38$	1.66921e8	0.341745
$39$	−4.50805e8	−0.800079
$40$	0	0
$41$	−5.53971e8	−0.746751	−0.373376	−	0.927680i	$-0.621800\pi$
$41$	−5.53971e8	−0.746751	−0.373376	+	0.927680i	$0.621800\pi$
$42$	4.86032e8	0.573844
$43$	−1.52106e9	−1.57787	−0.788934	−	0.614478i	$-0.789367\pi$
$43$	−1.52106e9	−1.57787	−0.788934	+	0.614478i	$0.789367\pi$
$44$	6.51981e8	0.595998
$45$	0	0
$46$	1.33780e9	0.957685
$47$	−1.19456e9	−0.759747	−0.379873	−	0.925038i	$-0.624032\pi$
$47$	−1.19456e9	−0.759747	−0.379873	+	0.925038i	$0.624032\pi$
$48$	1.55353e7	0.00880023
$49$	3.22934e9	1.63318
$50$	0	0
$51$	−1.44506e9	−0.586477
$52$	2.37396e9	0.865875
$53$	−1.22320e9	−0.401774	−0.200887	−	0.979614i	$-0.564382\pi$
$53$	−1.22320e9	−0.401774	−0.200887	+	0.979614i	$0.564382\pi$
$54$	3.97739e8	0.117878
$55$	0	0
$56$	−6.65575e9	−1.61496
$57$	1.46332e9	0.322127
$58$	−3.14750e8	−0.0629667
$59$	−5.83637e9	−1.06281	−0.531407	−	0.847117i	$-0.678336\pi$
$59$	−5.83637e9	−1.06281	−0.531407	+	0.847117i	$0.678336\pi$
$60$	0	0
$61$	−6.61097e9	−1.00219	−0.501096	−	0.865392i	$-0.667070\pi$
$61$	−6.61097e9	−1.00219	−0.501096	+	0.865392i	$0.667070\pi$
$62$	−4.77277e9	−0.661632
$63$	4.26081e9	0.540903
$64$	5.15451e9	0.600064
$65$	0	0
$66$	−3.43185e9	−0.337316
$67$	−1.66261e10	−1.50445	−0.752226	−	0.658905i	$-0.771020\pi$
$67$	−1.66261e10	−1.50445	−0.752226	+	0.658905i	$0.771020\pi$
$68$	7.60978e9	0.634707
$69$	1.17278e10	0.902709
$70$	0	0
$71$	7.36200e9	0.484256	0.242128	−	0.970244i	$-0.422155\pi$
$71$	7.36200e9	0.484256	0.242128	+	0.970244i	$0.422155\pi$
$72$	−5.44665e9	−0.331742
$73$	6.35726e9	0.358917	0.179458	−	0.983766i	$-0.442565\pi$
$73$	6.35726e9	0.358917	0.179458	+	0.983766i	$0.442565\pi$
$74$	−1.73297e10	−0.907857
$75$	0	0
$76$	−7.70593e9	−0.348618
$77$	−3.67640e10	−1.54783
$78$	−1.24959e10	−0.490058
$79$	2.47565e10	0.905191	0.452595	−	0.891716i	$-0.350498\pi$
$79$	2.47565e10	0.905191	0.452595	+	0.891716i	$0.350498\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	−1.53556e10	−0.457393
$83$	−3.59416e10	−1.00154	−0.500769	−	0.865581i	$-0.666950\pi$
$83$	−3.59416e10	−1.00154	−0.500769	+	0.865581i	$0.666950\pi$
$84$	−2.24377e10	−0.585385
$85$	0	0
$86$	−4.21625e10	−0.966462
$87$	−2.75926e9	−0.0593522
$88$	4.69959e10	0.949304
$89$	7.47690e10	1.41931	0.709653	−	0.704551i	$-0.248852\pi$
$89$	7.47690e10	1.41931	0.709653	+	0.704551i	$0.248852\pi$
$90$	0	0
$91$	−1.33863e11	−2.24871
$92$	−6.17594e10	−0.976945
$93$	−4.18406e10	−0.623651
$94$	−3.31121e10	−0.465354
$95$	0	0
$96$	4.63349e10	0.579985
$97$	1.66300e10	0.196629	0.0983144	−	0.995155i	$-0.468655\pi$
$97$	1.66300e10	0.196629	0.0983144	+	0.995155i	$0.468655\pi$
$98$	8.95144e10	1.00034
$99$	−3.00854e10	−0.317952
$100$	0	0
$101$	−5.30247e10	−0.502008	−0.251004	−	0.967986i	$-0.580761\pi$
$101$	−5.30247e10	−0.502008	−0.251004	+	0.967986i	$0.580761\pi$
$102$	−4.00558e10	−0.359224
$103$	−1.00437e9	−0.00853666	−0.00426833	−	0.999991i	$-0.501359\pi$
$103$	−1.00437e9	−0.00853666	−0.00426833	+	0.999991i	$0.501359\pi$
$104$	1.71119e11	1.37916
$105$	0	0
$106$	−3.39061e10	−0.246091
$107$	1.65238e11	1.13893	0.569466	−	0.822015i	$-0.307150\pi$
$107$	1.65238e11	1.13893	0.569466	+	0.822015i	$0.307150\pi$
$108$	−1.83616e10	−0.120249
$109$	7.04515e10	0.438576	0.219288	−	0.975660i	$-0.429627\pi$
$109$	7.04515e10	0.438576	0.219288	+	0.975660i	$0.429627\pi$
$110$	0	0
$111$	−1.51921e11	−0.855741
$112$	4.61312e9	0.0247341
$113$	−2.07466e11	−1.05929	−0.529646	−	0.848219i	$-0.677675\pi$
$113$	−2.07466e11	−1.05929	−0.529646	+	0.848219i	$0.677675\pi$
$114$	4.05619e10	0.197307
$115$	0	0
$116$	1.45304e10	0.0642331
$117$	−1.09546e11	−0.461926
$118$	−1.61779e11	−0.650985
$119$	−4.29102e11	−1.64836
$120$	0	0
$121$	−2.57228e10	−0.0901567
$122$	−1.83250e11	−0.613854
$123$	−1.34615e11	−0.431137
$124$	2.20335e11	0.674938
$125$	0	0
$126$	1.18106e11	0.331309
$127$	−1.17811e11	−0.316422	−0.158211	−	0.987405i	$-0.550573\pi$
$127$	−1.17811e11	−0.316422	−0.158211	+	0.987405i	$0.550573\pi$
$128$	−2.47632e11	−0.637017
$129$	−3.69618e11	−0.910983
$130$	0	0
$131$	4.60288e11	1.04241	0.521204	−	0.853432i	$-0.325483\pi$
$131$	4.60288e11	1.04241	0.521204	+	0.853432i	$0.325483\pi$
$132$	1.58431e11	0.344100
$133$	4.34523e11	0.905376
$134$	−4.60860e11	−0.921494
$135$	0	0
$136$	5.48526e11	1.01096
$137$	5.80435e11	1.02752	0.513760	−	0.857934i	$-0.328252\pi$
$137$	5.80435e11	1.02752	0.513760	+	0.857934i	$0.328252\pi$
$138$	3.25085e11	0.552920
$139$	6.94847e11	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$139$	6.94847e11	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$140$	0	0
$141$	−2.90278e11	−0.438640
$142$	2.04068e11	0.296612
$143$	9.45203e11	1.32183
$144$	3.77509e9	0.00508081
$145$	0	0
$146$	1.76217e11	0.219841
$147$	7.84730e11	0.942920
$148$	8.00027e11	0.926115
$149$	−1.12749e11	−0.125774	−0.0628869	−	0.998021i	$-0.520031\pi$
$149$	−1.12749e11	−0.125774	−0.0628869	+	0.998021i	$0.520031\pi$
$150$	0	0
$151$	2.45742e11	0.254746	0.127373	−	0.991855i	$-0.459346\pi$
$151$	2.45742e11	0.254746	0.127373	+	0.991855i	$0.459346\pi$
$152$	−5.55457e11	−0.555278
$153$	−3.51150e11	−0.338602
$154$	−1.01907e12	−0.948065
$155$	0	0
$156$	5.76873e11	0.499913
$157$	5.83317e11	0.488042	0.244021	−	0.969770i	$-0.421533\pi$
$157$	5.83317e11	0.488042	0.244021	+	0.969770i	$0.421533\pi$
$158$	6.86227e11	0.554439
$159$	−2.97239e11	−0.231964
$160$	0	0
$161$	3.48250e12	2.53717
$162$	9.66505e10	0.0680568
$163$	−2.65801e12	−1.80936	−0.904681	−	0.426089i	$-0.859891\pi$
$163$	−2.65801e12	−1.80936	−0.904681	+	0.426089i	$0.859891\pi$
$164$	7.08890e11	0.466592
$165$	0	0
$166$	−9.96269e11	−0.613454
$167$	1.68070e12	1.00126	0.500632	−	0.865660i	$-0.333101\pi$
$167$	1.68070e12	1.00126	0.500632	+	0.865660i	$0.333101\pi$
$168$	−1.61735e12	−0.932400
$169$	1.64947e12	0.920380
$170$	0	0
$171$	3.55587e11	0.185980
$172$	1.94643e12	0.985900
$173$	2.11009e12	1.03525	0.517627	−	0.855606i	$-0.326815\pi$
$173$	2.11009e12	1.03525	0.517627	+	0.855606i	$0.326815\pi$
$174$	−7.64841e10	−0.0363539
$175$	0	0
$176$	−3.25730e10	−0.0145391
$177$	−1.41824e12	−0.613616
$178$	2.07253e12	0.869341
$179$	8.73323e10	0.0355208	0.0177604	−	0.999842i	$-0.494346\pi$
$179$	8.73323e10	0.0355208	0.0177604	+	0.999842i	$0.494346\pi$
$180$	0	0
$181$	2.60066e12	0.995066	0.497533	−	0.867445i	$-0.334239\pi$
$181$	2.60066e12	0.995066	0.497533	+	0.867445i	$0.334239\pi$
$182$	−3.71057e12	−1.37736
$183$	−1.60646e12	−0.578616
$184$	−4.45173e12	−1.55607
$185$	0	0
$186$	−1.15978e12	−0.381993
$187$	3.02987e12	0.968935
$188$	1.52862e12	0.474713
$189$	1.03538e12	0.312291
$190$	0	0
$191$	−2.28556e12	−0.650593	−0.325296	−	0.945612i	$-0.605464\pi$
$191$	−2.28556e12	−0.650593	−0.325296	+	0.945612i	$0.605464\pi$
$192$	1.25255e12	0.346447
$193$	−3.11630e12	−0.837671	−0.418836	−	0.908062i	$-0.637562\pi$
$193$	−3.11630e12	−0.837671	−0.418836	+	0.908062i	$0.637562\pi$
$194$	4.60968e11	0.120437
$195$	0	0
$196$	−4.13243e12	−1.02046
$197$	−3.68099e12	−0.883894	−0.441947	−	0.897041i	$-0.645712\pi$
$197$	−3.68099e12	−0.883894	−0.441947	+	0.897041i	$0.645712\pi$
$198$	−8.33940e11	−0.194749
$199$	−2.20305e12	−0.500417	−0.250208	−	0.968192i	$-0.580499\pi$
$199$	−2.20305e12	−0.500417	−0.250208	+	0.968192i	$0.580499\pi$
$200$	0	0
$201$	−4.04014e12	−0.868596
$202$	−1.46980e12	−0.307485
$203$	−8.19343e11	−0.166816
$204$	1.84918e12	0.366448
$205$	0	0
$206$	−2.78402e10	−0.00522880
$207$	2.84986e12	0.521179
$208$	−1.18603e11	−0.0211226
$209$	−3.06815e12	−0.532196
$210$	0	0
$211$	−1.08697e13	−1.78921	−0.894607	−	0.446853i	$-0.852545\pi$
$211$	−1.08697e13	−1.78921	−0.894607	+	0.446853i	$0.852545\pi$
$212$	1.56528e12	0.251040
$213$	1.78897e12	0.279585
$214$	4.58024e12	0.697609
$215$	0	0
$216$	−1.32354e12	−0.191531
$217$	−1.24243e13	−1.75284
$218$	1.95285e12	0.268633
$219$	1.54481e12	0.207221
$220$	0	0
$221$	1.10322e13	1.40768
$222$	−4.21112e12	−0.524151
$223$	−9.89456e12	−1.20149	−0.600744	−	0.799441i	$-0.705129\pi$
$223$	−9.89456e12	−1.20149	−0.600744	+	0.799441i	$0.705129\pi$
$224$	1.37588e13	1.63011
$225$	0	0
$226$	−5.75078e12	−0.648829
$227$	−5.42930e12	−0.597862	−0.298931	−	0.954275i	$-0.596630\pi$
$227$	−5.42930e12	−0.597862	−0.298931	+	0.954275i	$0.596630\pi$
$228$	−1.87254e12	−0.201275
$229$	2.17366e12	0.228085	0.114042	−	0.993476i	$-0.463620\pi$
$229$	2.17366e12	0.228085	0.114042	+	0.993476i	$0.463620\pi$
$230$	0	0
$231$	−8.93366e12	−0.893642
$232$	1.04738e12	0.102310
$233$	1.98130e13	1.89013	0.945067	−	0.326876i	$-0.105996\pi$
$233$	1.98130e13	1.89013	0.945067	+	0.326876i	$0.105996\pi$
$234$	−3.03650e12	−0.282935
$235$	0	0
$236$	7.46853e12	0.664078
$237$	6.01583e12	0.522612
$238$	−1.18943e13	−1.00964
$239$	1.03457e12	0.0858169	0.0429085	−	0.999079i	$-0.486338\pi$
$239$	1.03457e12	0.0858169	0.0429085	+	0.999079i	$0.486338\pi$
$240$	0	0
$241$	1.36081e13	1.07821	0.539106	−	0.842238i	$-0.318762\pi$
$241$	1.36081e13	1.07821	0.539106	+	0.842238i	$0.318762\pi$
$242$	−7.13011e11	−0.0552220
$243$	8.47289e11	0.0641500
$244$	8.45973e12	0.626199
$245$	0	0
$246$	−3.73140e12	−0.264076
$247$	−1.11716e13	−0.773182
$248$	1.58821e13	1.07504
$249$	−8.73381e12	−0.578239
$250$	0	0
$251$	−2.28023e13	−1.44469	−0.722344	−	0.691534i	$-0.756935\pi$
$251$	−2.28023e13	−1.44469	−0.722344	+	0.691534i	$0.756935\pi$
$252$	−5.45236e12	−0.337972
$253$	−2.45898e13	−1.49139
$254$	−3.26562e12	−0.193812
$255$	0	0
$256$	−1.74206e13	−0.990244
$257$	−6.13077e12	−0.341101	−0.170550	−	0.985349i	$-0.554555\pi$
$257$	−6.13077e12	−0.341101	−0.170550	+	0.985349i	$0.554555\pi$
$258$	−1.02455e13	−0.557987
$259$	−4.51121e13	−2.40516
$260$	0	0
$261$	−6.70500e11	−0.0342670
$262$	1.27588e13	0.638486
$263$	3.25384e13	1.59456	0.797278	−	0.603613i	$-0.206273\pi$
$263$	3.25384e13	1.59456	0.797278	+	0.603613i	$0.206273\pi$
$264$	1.14200e13	0.548081
$265$	0	0
$266$	1.20446e13	0.554553
$267$	1.81689e13	0.819437
$268$	2.12756e13	0.940027
$269$	3.66970e13	1.58852	0.794261	−	0.607577i	$-0.207858\pi$
$269$	3.66970e13	1.58852	0.794261	+	0.607577i	$0.207858\pi$
$270$	0	0
$271$	1.87361e13	0.778659	0.389329	−	0.921099i	$-0.372707\pi$
$271$	1.87361e13	0.778659	0.389329	+	0.921099i	$0.372707\pi$
$272$	−3.80185e11	−0.0154834
$273$	−3.25288e13	−1.29830
$274$	1.60891e13	0.629368
$275$	0	0
$276$	−1.50075e13	−0.564040
$277$	3.03044e13	1.11652	0.558260	−	0.829666i	$-0.311469\pi$
$277$	3.03044e13	1.11652	0.558260	+	0.829666i	$0.311469\pi$
$278$	1.92605e13	0.695700
$279$	−1.01673e13	−0.360065
$280$	0	0
$281$	4.93501e13	1.68037	0.840183	−	0.542304i	$-0.182448\pi$
$281$	4.93501e13	1.68037	0.840183	+	0.542304i	$0.182448\pi$
$282$	−8.04623e12	−0.268672
$283$	5.45122e12	0.178512	0.0892562	−	0.996009i	$-0.471551\pi$
$283$	5.45122e12	0.178512	0.0892562	+	0.996009i	$0.471551\pi$
$284$	−9.42079e12	−0.302578
$285$	0	0
$286$	2.62002e13	0.809638
$287$	−3.99730e13	−1.21176
$288$	1.12594e13	0.334854
$289$	1.09207e12	0.0318649
$290$	0	0
$291$	4.04108e12	0.113524
$292$	−8.13507e12	−0.224262
$293$	−4.85807e13	−1.31429	−0.657146	−	0.753763i	$-0.728237\pi$
$293$	−4.85807e13	−1.31429	−0.657146	+	0.753763i	$0.728237\pi$
$294$	2.17520e13	0.577549
$295$	0	0
$296$	5.76673e13	1.47511
$297$	−7.31075e12	−0.183570
$298$	−3.12531e12	−0.0770378
$299$	−8.95352e13	−2.16672
$300$	0	0
$301$	−1.09756e14	−2.56042
$302$	6.81176e12	0.156035
$303$	−1.28850e13	−0.289834
$304$	3.84988e11	0.00850439
$305$	0	0
$306$	−9.73357e12	−0.207398
$307$	6.68455e13	1.39898	0.699489	−	0.714643i	$-0.253411\pi$
$307$	6.68455e13	1.39898	0.699489	+	0.714643i	$0.253411\pi$
$308$	4.70451e13	0.967132
$309$	−2.44061e11	−0.00492864
$310$	0	0
$311$	7.26443e13	1.41586	0.707928	−	0.706284i	$-0.249630\pi$
$311$	7.26443e13	1.41586	0.707928	+	0.706284i	$0.249630\pi$
$312$	4.15820e13	0.796260
$313$	−6.27938e13	−1.18147	−0.590735	−	0.806866i	$-0.701162\pi$
$313$	−6.27938e13	−1.18147	−0.590735	+	0.806866i	$0.701162\pi$
$314$	1.61690e13	0.298931
$315$	0	0
$316$	−3.16797e13	−0.565590
$317$	4.20135e13	0.737161	0.368581	−	0.929596i	$-0.379844\pi$
$317$	4.20135e13	0.737161	0.368581	+	0.929596i	$0.379844\pi$
$318$	−8.23919e12	−0.142081
$319$	5.78534e12	0.0980574
$320$	0	0
$321$	4.01527e13	0.657563
$322$	9.65318e13	1.55404
$323$	−3.58108e13	−0.566761
$324$	−4.46187e12	−0.0694256
$325$	0	0
$326$	−7.36778e13	−1.10825
$327$	1.71197e13	0.253212
$328$	5.10980e13	0.743187
$329$	−8.61960e13	−1.23285
$330$	0	0
$331$	1.36334e14	1.88604	0.943019	−	0.332740i	$-0.107973\pi$
$331$	1.36334e14	1.88604	0.943019	+	0.332740i	$0.107973\pi$
$332$	4.59927e13	0.625792
$333$	−3.69169e13	−0.494063
$334$	4.65874e13	0.613286
$335$	0	0
$336$	1.12099e12	0.0142802
$337$	−9.94099e13	−1.24585	−0.622924	−	0.782282i	$-0.714056\pi$
$337$	−9.94099e13	−1.24585	−0.622924	+	0.782282i	$0.714056\pi$
$338$	4.57218e13	0.563743
$339$	−5.04143e13	−0.611583
$340$	0	0
$341$	8.77273e13	1.03035
$342$	9.85655e12	0.113915
$343$	9.03418e13	1.02747
$344$	1.40302e14	1.57034
$345$	0	0
$346$	5.84897e13	0.634105
$347$	−7.96482e13	−0.849892	−0.424946	−	0.905219i	$-0.639707\pi$
$347$	−7.96482e13	−0.849892	−0.424946	+	0.905219i	$0.639707\pi$
$348$	3.53089e12	0.0370850
$349$	−2.33239e13	−0.241135	−0.120568	−	0.992705i	$-0.538471\pi$
$349$	−2.33239e13	−0.241135	−0.120568	+	0.992705i	$0.538471\pi$
$350$	0	0
$351$	−2.66196e13	−0.266693
$352$	−9.71505e13	−0.958209
$353$	7.20065e13	0.699215	0.349608	−	0.936896i	$-0.386315\pi$
$353$	7.20065e13	0.699215	0.349608	+	0.936896i	$0.386315\pi$
$354$	−3.93123e13	−0.375846
$355$	0	0
$356$	−9.56782e13	−0.886825
$357$	−1.04272e14	−0.951681
$358$	2.42077e12	0.0217569
$359$	1.02696e13	0.0908941	0.0454470	−	0.998967i	$-0.485529\pi$
$359$	1.02696e13	0.0908941	0.0454470	+	0.998967i	$0.485529\pi$
$360$	0	0
$361$	−8.02270e13	−0.688702
$362$	7.20880e13	0.609489
$363$	−6.25063e12	−0.0520520
$364$	1.71299e14	1.40506
$365$	0	0
$366$	−4.45298e13	−0.354409
$367$	−1.03116e14	−0.808470	−0.404235	−	0.914655i	$-0.632462\pi$
$367$	−1.03116e14	−0.808470	−0.404235	+	0.914655i	$0.632462\pi$
$368$	3.08550e12	0.0238321
$369$	−3.27114e13	−0.248917
$370$	0	0
$371$	−8.82631e13	−0.651962
$372$	5.35414e13	0.389676
$373$	−1.30172e14	−0.933509	−0.466755	−	0.884387i	$-0.654577\pi$
$373$	−1.30172e14	−0.933509	−0.466755	+	0.884387i	$0.654577\pi$
$374$	8.39852e13	0.593484
$375$	0	0
$376$	1.10185e14	0.756120
$377$	2.10653e13	0.142459
$378$	2.86997e13	0.191281
$379$	−6.26815e13	−0.411740	−0.205870	−	0.978579i	$-0.566002\pi$
$379$	−6.26815e13	−0.411740	−0.205870	+	0.978579i	$0.566002\pi$
$380$	0	0
$381$	−2.86282e13	−0.182686
$382$	−6.33537e13	−0.398495
$383$	−1.06310e14	−0.659145	−0.329572	−	0.944130i	$-0.606905\pi$
$383$	−1.06310e14	−0.659145	−0.329572	+	0.944130i	$0.606905\pi$
$384$	−6.01745e13	−0.367782
$385$	0	0
$386$	−8.63809e13	−0.513083
$387$	−8.98173e13	−0.525956
$388$	−2.12806e13	−0.122860
$389$	−7.42336e13	−0.422549	−0.211275	−	0.977427i	$-0.567761\pi$
$389$	−7.42336e13	−0.422549	−0.211275	+	0.977427i	$0.567761\pi$
$390$	0	0
$391$	−2.87007e14	−1.58825
$392$	−2.97873e14	−1.62539
$393$	1.11850e14	0.601834
$394$	−1.02034e14	−0.541395
$395$	0	0
$396$	3.84988e13	0.198666
$397$	1.48552e14	0.756018	0.378009	−	0.925802i	$-0.376609\pi$
$397$	1.48552e14	0.756018	0.378009	+	0.925802i	$0.376609\pi$
$398$	−6.10665e13	−0.306511
$399$	1.05589e14	0.522719
$400$	0	0
$401$	−9.20974e13	−0.443561	−0.221781	−	0.975097i	$-0.571187\pi$
$401$	−9.20974e13	−0.443561	−0.221781	+	0.975097i	$0.571187\pi$
$402$	−1.11989e14	−0.532025
$403$	3.19429e14	1.49691
$404$	6.78531e13	0.313669
$405$	0	0
$406$	−2.27115e13	−0.102177
$407$	3.18534e14	1.41380
$408$	1.33292e14	0.583677
$409$	−3.23531e13	−0.139778	−0.0698889	−	0.997555i	$-0.522264\pi$
$409$	−3.23531e13	−0.139778	−0.0698889	+	0.997555i	$0.522264\pi$
$410$	0	0
$411$	1.41046e14	0.593239
$412$	1.28524e12	0.00533396
$413$	−4.21137e14	−1.72464
$414$	7.89956e13	0.319228
$415$	0	0
$416$	−3.53740e14	−1.39210
$417$	1.68848e14	0.655764
$418$	−8.50463e13	−0.325976
$419$	−1.61755e14	−0.611900	−0.305950	−	0.952048i	$-0.598974\pi$
$419$	−1.61755e14	−0.611900	−0.305950	+	0.952048i	$0.598974\pi$
$420$	0	0
$421$	−1.14462e14	−0.421803	−0.210902	−	0.977507i	$-0.567640\pi$
$421$	−1.14462e14	−0.421803	−0.210902	+	0.977507i	$0.567640\pi$
$422$	−3.01297e14	−1.09591
$423$	−7.05374e13	−0.253249
$424$	1.12828e14	0.399856
$425$	0	0
$426$	4.95885e13	0.171249
$427$	−4.77029e14	−1.62627
$428$	−2.11447e14	−0.711639
$429$	2.29684e14	0.763161
$430$	0	0
$431$	1.01052e14	0.327279	0.163639	−	0.986520i	$-0.447677\pi$
$431$	1.01052e14	0.327279	0.163639	+	0.986520i	$0.447677\pi$
$432$	9.17346e11	0.00293341
$433$	−1.21838e14	−0.384680	−0.192340	−	0.981328i	$-0.561608\pi$
$433$	−1.21838e14	−0.384680	−0.192340	+	0.981328i	$0.561608\pi$
$434$	−3.44390e14	−1.07364
$435$	0	0
$436$	−9.01534e13	−0.274035
$437$	2.90633e14	0.872362
$438$	4.28208e13	0.126925
$439$	2.85736e14	0.836393	0.418197	−	0.908357i	$-0.362662\pi$
$439$	2.85736e14	0.836393	0.418197	+	0.908357i	$0.362662\pi$
$440$	0	0
$441$	1.90689e14	0.544395
$442$	3.05803e14	0.862222
$443$	−4.82498e14	−1.34362	−0.671808	−	0.740725i	$-0.734482\pi$
$443$	−4.82498e14	−1.34362	−0.671808	+	0.740725i	$0.734482\pi$
$444$	1.94406e14	0.534693
$445$	0	0
$446$	−2.74268e14	−0.735925
$447$	−2.73981e13	−0.0726155
$448$	3.71935e14	0.973729
$449$	6.86974e14	1.77658	0.888291	−	0.459281i	$-0.151893\pi$
$449$	6.86974e14	1.77658	0.888291	+	0.459281i	$0.151893\pi$
$450$	0	0
$451$	2.82248e14	0.712294
$452$	2.65485e14	0.661878
$453$	5.97154e13	0.147078
$454$	−1.50495e14	−0.366197
$455$	0	0
$456$	−1.34976e14	−0.320590
$457$	2.14744e14	0.503944	0.251972	−	0.967734i	$-0.418921\pi$
$457$	2.14744e14	0.503944	0.251972	+	0.967734i	$0.418921\pi$
$458$	6.02519e13	0.139705
$459$	−8.53295e13	−0.195492
$460$	0	0
$461$	−3.22127e14	−0.720563	−0.360281	−	0.932844i	$-0.617319\pi$
$461$	−3.22127e14	−0.720563	−0.360281	+	0.932844i	$0.617319\pi$
$462$	−2.47633e14	−0.547366
$463$	−8.81227e13	−0.192483	−0.0962415	−	0.995358i	$-0.530682\pi$
$463$	−8.81227e13	−0.192483	−0.0962415	+	0.995358i	$0.530682\pi$
$464$	−7.25940e11	−0.00156694
$465$	0	0
$466$	5.49199e14	1.15773
$467$	−3.86746e14	−0.805718	−0.402859	−	0.915262i	$-0.631984\pi$
$467$	−3.86746e14	−0.805718	−0.402859	+	0.915262i	$0.631984\pi$
$468$	1.40180e14	0.288625
$469$	−1.19969e15	−2.44129
$470$	0	0
$471$	1.41746e14	0.281771
$472$	5.38344e14	1.05774
$473$	7.74980e14	1.50506
$474$	1.66753e14	0.320106
$475$	0	0
$476$	5.49101e14	1.02994
$477$	−7.22290e13	−0.133925
$478$	2.86774e13	0.0525638
$479$	−5.88123e14	−1.06567	−0.532835	−	0.846219i	$-0.678873\pi$
$479$	−5.88123e14	−1.06567	−0.532835	+	0.846219i	$0.678873\pi$
$480$	0	0
$481$	1.15983e15	2.05398
$482$	3.77205e14	0.660417
$483$	8.46248e14	1.46483
$484$	3.29162e13	0.0563326
$485$	0	0
$486$	2.34861e13	0.0392926
$487$	−3.77139e14	−0.623867	−0.311933	−	0.950104i	$-0.600977\pi$
$487$	−3.77139e14	−0.623867	−0.311933	+	0.950104i	$0.600977\pi$
$488$	6.09792e14	0.997408
$489$	−6.45898e14	−1.04464
$490$	0	0
$491$	−3.71374e13	−0.0587305	−0.0293652	−	0.999569i	$-0.509349\pi$
$491$	−3.71374e13	−0.0587305	−0.0293652	+	0.999569i	$0.509349\pi$
$492$	1.72260e14	0.269387
$493$	6.75253e13	0.104426
$494$	−3.09667e14	−0.473583
$495$	0	0
$496$	−1.10079e13	−0.0164648
$497$	5.31222e14	0.785806
$498$	−2.42093e14	−0.354178
$499$	2.72085e14	0.393688	0.196844	−	0.980435i	$-0.436931\pi$
$499$	2.72085e14	0.393688	0.196844	+	0.980435i	$0.436931\pi$
$500$	0	0
$501$	4.08410e14	0.578081
$502$	−6.32060e14	−0.884888
$503$	−1.30904e15	−1.81272	−0.906358	−	0.422511i	$-0.861149\pi$
$503$	−1.30904e15	−1.81272	−0.906358	+	0.422511i	$0.861149\pi$
$504$	−3.93015e14	−0.538321
$505$	0	0
$506$	−6.81606e14	−0.913494
$507$	4.00821e14	0.531382
$508$	1.50757e14	0.197710
$509$	1.41934e15	1.84135	0.920677	−	0.390326i	$-0.127638\pi$
$509$	1.41934e15	1.84135	0.920677	+	0.390326i	$0.127638\pi$
$510$	0	0
$511$	4.58722e14	0.582418
$512$	2.42674e13	0.0304817
$513$	8.64076e13	0.107376
$514$	−1.69939e14	−0.208928
$515$	0	0
$516$	4.72983e14	0.569209
$517$	6.08626e14	0.724690
$518$	−1.25047e15	−1.47319
$519$	5.12751e14	0.597704
$520$	0	0
$521$	−5.15525e14	−0.588359	−0.294180	−	0.955750i	$-0.595046\pi$
$521$	−5.15525e14	−0.588359	−0.294180	+	0.955750i	$0.595046\pi$
$522$	−1.85856e13	−0.0209889
$523$	−1.15496e15	−1.29064	−0.645322	−	0.763911i	$-0.723277\pi$
$523$	−1.15496e15	−1.29064	−0.645322	+	0.763911i	$0.723277\pi$
$524$	−5.89008e14	−0.651328
$525$	0	0
$526$	9.01935e14	0.976683
$527$	1.02393e15	1.09727
$528$	−7.91524e12	−0.00839416
$529$	1.37648e15	1.44465
$530$	0	0
$531$	−3.44632e14	−0.354271
$532$	−5.56038e14	−0.565706
$533$	1.02771e15	1.03483
$534$	5.03624e14	0.501914
$535$	0	0
$536$	1.53358e15	1.49727
$537$	2.12217e13	0.0205080
$538$	1.01721e15	0.972988
$539$	−1.64534e15	−1.55782
$540$	0	0
$541$	−1.08911e15	−1.01038	−0.505190	−	0.863008i	$-0.668578\pi$
$541$	−1.08911e15	−1.01038	−0.505190	+	0.863008i	$0.668578\pi$
$542$	5.19347e14	0.476937
$543$	6.31961e14	0.574502
$544$	−1.13392e15	−1.02044
$545$	0	0
$546$	−9.01669e14	−0.795221
$547$	−9.36391e14	−0.817574	−0.408787	−	0.912630i	$-0.634048\pi$
$547$	−9.36391e14	−0.817574	−0.408787	+	0.912630i	$0.634048\pi$
$548$	−7.42754e14	−0.642025
$549$	−3.90371e14	−0.334064
$550$	0	0
$551$	−6.83784e13	−0.0573569
$552$	−1.08177e15	−0.898400
$553$	1.78636e15	1.46886
$554$	8.40010e14	0.683881
$555$	0	0
$556$	−8.89163e14	−0.709692
$557$	3.48191e14	0.275178	0.137589	−	0.990489i	$-0.456065\pi$
$557$	3.48191e14	0.275178	0.137589	+	0.990489i	$0.456065\pi$
$558$	−2.81827e14	−0.220544
$559$	2.82182e15	2.18658
$560$	0	0
$561$	7.36258e14	0.559415
$562$	1.36794e15	1.02924
$563$	−8.78985e14	−0.654916	−0.327458	−	0.944866i	$-0.606192\pi$
$563$	−8.78985e14	−0.654916	−0.327458	+	0.944866i	$0.606192\pi$
$564$	3.71454e14	0.274075
$565$	0	0
$566$	1.51103e14	0.109341
$567$	2.51597e14	0.180301
$568$	−6.79067e14	−0.481944
$569$	−1.45202e14	−0.102060	−0.0510299	−	0.998697i	$-0.516250\pi$
$569$	−1.45202e14	−0.102060	−0.0510299	+	0.998697i	$0.516250\pi$
$570$	0	0
$571$	−1.99915e15	−1.37831	−0.689156	−	0.724613i	$-0.742019\pi$
$571$	−1.99915e15	−1.37831	−0.689156	+	0.724613i	$0.742019\pi$
$572$	−1.20953e15	−0.825921
$573$	−5.55391e14	−0.375620
$574$	−1.10802e15	−0.742216
$575$	0	0
$576$	3.04368e14	0.200021
$577$	7.69862e13	0.0501125	0.0250562	−	0.999686i	$-0.492024\pi$
$577$	7.69862e13	0.0501125	0.0250562	+	0.999686i	$0.492024\pi$
$578$	3.02712e13	0.0195176
$579$	−7.57260e14	−0.483630
$580$	0	0
$581$	−2.59345e15	−1.62521
$582$	1.12015e14	0.0695345
$583$	6.23221e14	0.383235
$584$	−5.86390e14	−0.357204
$585$	0	0
$586$	−1.34661e15	−0.805019
$587$	−8.86073e14	−0.524759	−0.262379	−	0.964965i	$-0.584507\pi$
$587$	−8.86073e14	−0.524759	−0.262379	+	0.964965i	$0.584507\pi$
$588$	−1.00418e15	−0.589164
$589$	−1.03687e15	−0.602685
$590$	0	0
$591$	−8.94480e14	−0.510316
$592$	−3.99694e13	−0.0225922
$593$	−9.32426e14	−0.522172	−0.261086	−	0.965316i	$-0.584081\pi$
$593$	−9.32426e14	−0.522172	−0.261086	+	0.965316i	$0.584081\pi$
$594$	−2.02647e14	−0.112439
$595$	0	0
$596$	1.44280e14	0.0785872
$597$	−5.35341e14	−0.288916
$598$	−2.48183e15	−1.32714
$599$	1.16299e15	0.616208	0.308104	−	0.951353i	$-0.400305\pi$
$599$	1.16299e15	0.616208	0.308104	+	0.951353i	$0.400305\pi$
$600$	0	0
$601$	−2.63081e15	−1.36861	−0.684306	−	0.729195i	$-0.739895\pi$
$601$	−2.63081e15	−1.36861	−0.684306	+	0.729195i	$0.739895\pi$
$602$	−3.04233e15	−1.56829
$603$	−9.81753e14	−0.501484
$604$	−3.14465e14	−0.159173
$605$	0	0
$606$	−3.57160e14	−0.177527
$607$	−1.56499e14	−0.0770857	−0.0385429	−	0.999257i	$-0.512272\pi$
$607$	−1.56499e14	−0.0770857	−0.0385429	+	0.999257i	$0.512272\pi$
$608$	1.14825e15	0.560487
$609$	−1.99100e14	−0.0963113
$610$	0	0
$611$	2.21610e15	1.05284
$612$	4.49350e14	0.211569
$613$	1.63056e15	0.760859	0.380429	−	0.924810i	$-0.375776\pi$
$613$	1.63056e15	0.760859	0.380429	+	0.924810i	$0.375776\pi$
$614$	1.85290e15	0.856890
$615$	0	0
$616$	3.39109e15	1.54044
$617$	−3.55060e15	−1.59858	−0.799289	−	0.600947i	$-0.794790\pi$
$617$	−3.55060e15	−1.59858	−0.799289	+	0.600947i	$0.794790\pi$
$618$	−6.76516e12	−0.00301885
$619$	−7.99938e14	−0.353800	−0.176900	−	0.984229i	$-0.556607\pi$
$619$	−7.99938e14	−0.353800	−0.176900	+	0.984229i	$0.556607\pi$
$620$	0	0
$621$	6.92517e14	0.300903
$622$	2.01363e15	0.867228
$623$	5.39512e15	2.30312
$624$	−2.88206e13	−0.0121952
$625$	0	0
$626$	−1.74059e15	−0.723664
$627$	−7.45560e14	−0.307264
$628$	−7.46443e14	−0.304943
$629$	3.71786e15	1.50562
$630$	0	0
$631$	3.38323e14	0.134639	0.0673194	−	0.997731i	$-0.478555\pi$
$631$	3.38323e14	0.134639	0.0673194	+	0.997731i	$0.478555\pi$
$632$	−2.28353e15	−0.900870
$633$	−2.64133e15	−1.03300
$634$	1.16457e15	0.451520
$635$	0	0
$636$	3.80362e14	0.144938
$637$	−5.99095e15	−2.26323
$638$	1.60364e14	0.0600613
$639$	4.34719e14	0.161419
$640$	0	0
$641$	2.78171e15	1.01529	0.507647	−	0.861565i	$-0.330515\pi$
$641$	2.78171e15	1.01529	0.507647	+	0.861565i	$0.330515\pi$
$642$	1.11300e15	0.402765
$643$	1.37901e15	0.494774	0.247387	−	0.968917i	$-0.420428\pi$
$643$	1.37901e15	0.494774	0.247387	+	0.968917i	$0.420428\pi$
$644$	−4.45639e15	−1.58530
$645$	0	0
$646$	−9.92642e14	−0.347147
$647$	4.51705e15	1.56632	0.783162	−	0.621818i	$-0.213606\pi$
$647$	4.51705e15	1.56632	0.783162	+	0.621818i	$0.213606\pi$
$648$	−3.21619e14	−0.110581
$649$	2.97362e15	1.01377
$650$	0	0
$651$	−3.01910e15	−1.01200
$652$	3.40133e15	1.13054
$653$	4.67633e15	1.54128	0.770642	−	0.637268i	$-0.219936\pi$
$653$	4.67633e15	1.54128	0.770642	+	0.637268i	$0.219936\pi$
$654$	4.74543e14	0.155095
$655$	0	0
$656$	−3.54162e13	−0.0113823
$657$	3.75390e14	0.119639
$658$	−2.38927e15	−0.755133
$659$	−2.44697e15	−0.766935	−0.383468	−	0.923554i	$-0.625270\pi$
$659$	−2.44697e15	−0.766935	−0.383468	+	0.923554i	$0.625270\pi$
$660$	0	0
$661$	−1.00863e15	−0.310902	−0.155451	−	0.987844i	$-0.549683\pi$
$661$	−1.00863e15	−0.310902	−0.155451	+	0.987844i	$0.549683\pi$
$662$	3.77906e15	1.15522
$663$	2.68083e15	0.812726
$664$	3.31523e15	0.996758
$665$	0	0
$666$	−1.02330e15	−0.302619
$667$	−5.48022e14	−0.160733
$668$	−2.15071e15	−0.625620
$669$	−2.40438e15	−0.693680
$670$	0	0
$671$	3.36828e15	0.955948
$672$	3.34340e15	0.941146
$673$	−3.14110e15	−0.877000	−0.438500	−	0.898731i	$-0.644490\pi$
$673$	−3.14110e15	−0.877000	−0.438500	+	0.898731i	$0.644490\pi$
$674$	−2.75555e15	−0.763096
$675$	0	0
$676$	−2.11075e15	−0.575081
$677$	1.77722e15	0.480291	0.240145	−	0.970737i	$-0.422805\pi$
$677$	1.77722e15	0.480291	0.240145	+	0.970737i	$0.422805\pi$
$678$	−1.39744e15	−0.374601
$679$	1.19997e15	0.319071
$680$	0	0
$681$	−1.31932e15	−0.345176
$682$	2.43172e15	0.631102
$683$	4.74129e15	1.22063	0.610313	−	0.792160i	$-0.291044\pi$
$683$	4.74129e15	1.22063	0.610313	+	0.792160i	$0.291044\pi$
$684$	−4.55027e14	−0.116206
$685$	0	0
$686$	2.50419e15	0.629340
$687$	5.28200e14	0.131685
$688$	−9.72438e13	−0.0240506
$689$	2.26924e15	0.556770
$690$	0	0
$691$	2.44203e15	0.589688	0.294844	−	0.955545i	$-0.404732\pi$
$691$	2.44203e15	0.589688	0.294844	+	0.955545i	$0.404732\pi$
$692$	−2.70018e15	−0.646858
$693$	−2.17088e15	−0.515944
$694$	−2.20777e15	−0.520568
$695$	0	0
$696$	2.54513e14	0.0590688
$697$	3.29433e15	0.758555
$698$	−6.46517e14	−0.147698
$699$	4.81456e15	1.09127
$700$	0	0
$701$	3.91146e15	0.872749	0.436375	−	0.899765i	$-0.356262\pi$
$701$	3.91146e15	0.872749	0.436375	+	0.899765i	$0.356262\pi$
$702$	−7.37870e14	−0.163353
$703$	−3.76483e15	−0.826973
$704$	−2.62621e15	−0.572375
$705$	0	0
$706$	1.99596e15	0.428277
$707$	−3.82611e15	−0.814613
$708$	1.81485e15	0.383405
$709$	−7.52555e15	−1.57755	−0.788776	−	0.614681i	$-0.789285\pi$
$709$	−7.52555e15	−1.57755	−0.788776	+	0.614681i	$0.789285\pi$
$710$	0	0
$711$	1.46185e15	0.301730
$712$	−6.89665e15	−1.41253
$713$	−8.31004e15	−1.68893
$714$	−2.89032e15	−0.582915
$715$	0	0
$716$	−1.11755e14	−0.0221945
$717$	2.51401e14	0.0495464
$718$	2.84665e14	0.0556737
$719$	3.82053e15	0.741507	0.370753	−	0.928731i	$-0.379099\pi$
$719$	3.82053e15	0.741507	0.370753	+	0.928731i	$0.379099\pi$
$720$	0	0
$721$	−7.24724e13	−0.0138525
$722$	−2.22382e15	−0.421838
$723$	3.30677e15	0.622506
$724$	−3.32794e15	−0.621747
$725$	0	0
$726$	−1.73262e14	−0.0318824
$727$	7.15305e15	1.30633	0.653163	−	0.757217i	$-0.273442\pi$
$727$	7.15305e15	1.30633	0.653163	+	0.757217i	$0.273442\pi$
$728$	1.23475e16	2.23798
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	9.04540e15	1.60281
$732$	2.05572e15	0.361536
$733$	−3.23209e15	−0.564172	−0.282086	−	0.959389i	$-0.591026\pi$
$733$	−3.23209e15	−0.564172	−0.282086	+	0.959389i	$0.591026\pi$
$734$	−2.85829e15	−0.495197
$735$	0	0
$736$	9.20266e15	1.57067
$737$	8.47096e15	1.43503
$738$	−9.06731e14	−0.152464
$739$	−7.04037e15	−1.17504	−0.587518	−	0.809211i	$-0.699895\pi$
$739$	−7.04037e15	−1.17504	−0.587518	+	0.809211i	$0.699895\pi$
$740$	0	0
$741$	−2.71470e15	−0.446397
$742$	−2.44657e15	−0.399334
$743$	−1.76769e15	−0.286396	−0.143198	−	0.989694i	$-0.545739\pi$
$743$	−1.76769e15	−0.286396	−0.143198	+	0.989694i	$0.545739\pi$
$744$	3.85936e15	0.620674
$745$	0	0
$746$	−3.60825e15	−0.571785
$747$	−2.12232e15	−0.333846
$748$	−3.87717e15	−0.605420
$749$	1.19231e16	1.84816
$750$	0	0
$751$	−6.97015e15	−1.06469	−0.532344	−	0.846528i	$-0.678689\pi$
$751$	−6.97015e15	−1.06469	−0.532344	+	0.846528i	$0.678689\pi$
$752$	−7.63698e13	−0.0115804
$753$	−5.54097e15	−0.834091
$754$	5.83912e14	0.0872579
$755$	0	0
$756$	−1.32492e15	−0.195128
$757$	1.10779e16	1.61968	0.809841	−	0.586650i	$-0.199553\pi$
$757$	1.10779e16	1.61968	0.809841	+	0.586650i	$0.199553\pi$
$758$	−1.73747e15	−0.252196
$759$	−5.97532e15	−0.861055
$760$	0	0
$761$	1.03991e15	0.147700	0.0738500	−	0.997269i	$-0.476471\pi$
$761$	1.03991e15	0.147700	0.0738500	+	0.997269i	$0.476471\pi$
$762$	−7.93546e14	−0.111897
$763$	5.08359e15	0.711681
$764$	2.92472e15	0.406510
$765$	0	0
$766$	−2.94682e15	−0.403734
$767$	1.08274e16	1.47282
$768$	−4.23319e15	−0.571717
$769$	−5.06484e15	−0.679158	−0.339579	−	0.940578i	$-0.610285\pi$
$769$	−5.06484e15	−0.679158	−0.339579	+	0.940578i	$0.610285\pi$
$770$	0	0
$771$	−1.48978e15	−0.196935
$772$	3.98777e15	0.523402
$773$	9.32127e15	1.21475	0.607376	−	0.794414i	$-0.292222\pi$
$773$	9.32127e15	1.21475	0.607376	+	0.794414i	$0.292222\pi$
$774$	−2.48965e15	−0.322154
$775$	0	0
$776$	−1.53394e15	−0.195690
$777$	−1.09622e16	−1.38862
$778$	−2.05769e15	−0.258816
$779$	−3.33596e15	−0.416643
$780$	0	0
$781$	−3.75093e15	−0.461911
$782$	−7.95557e15	−0.972823
$783$	−1.62931e14	−0.0197841
$784$	2.06456e14	0.0248937
$785$	0	0
$786$	3.10038e15	0.368630
$787$	−8.41800e15	−0.993912	−0.496956	−	0.867776i	$-0.665549\pi$
$787$	−8.41800e15	−0.993912	−0.496956	+	0.867776i	$0.665549\pi$
$788$	4.71038e15	0.552283
$789$	7.90683e15	0.920617
$790$	0	0
$791$	−1.49702e16	−1.71892
$792$	2.77506e15	0.316435
$793$	1.22644e16	1.38882
$794$	4.11774e15	0.463069
$795$	0	0
$796$	2.81913e15	0.312676
$797$	−1.32426e16	−1.45866	−0.729328	−	0.684164i	$-0.760167\pi$
$797$	−1.32426e16	−1.45866	−0.729328	+	0.684164i	$0.760167\pi$
$798$	2.92684e15	0.320171
$799$	7.10375e15	0.771757
$800$	0	0
$801$	4.41503e15	0.473102
$802$	−2.55286e15	−0.271686
$803$	−3.23901e15	−0.342356
$804$	5.16997e15	0.542725
$805$	0	0
$806$	8.85427e15	0.916875
$807$	8.91738e15	0.917134
$808$	4.89097e15	0.499611
$809$	−2.72166e15	−0.276132	−0.138066	−	0.990423i	$-0.544089\pi$
$809$	−2.72166e15	−0.276132	−0.138066	+	0.990423i	$0.544089\pi$
$810$	0	0
$811$	1.27431e16	1.27544	0.637722	−	0.770267i	$-0.279877\pi$
$811$	1.27431e16	1.27544	0.637722	+	0.770267i	$0.279877\pi$
$812$	1.04847e15	0.104232
$813$	4.55286e15	0.449559
$814$	8.82947e15	0.865965
$815$	0	0
$816$	−9.23850e13	−0.00893934
$817$	−9.15968e15	−0.880358
$818$	−8.96800e14	−0.0856155
$819$	−7.90450e15	−0.749571
$820$	0	0
$821$	2.99243e15	0.279986	0.139993	−	0.990152i	$-0.455292\pi$
$821$	2.99243e15	0.279986	0.139993	+	0.990152i	$0.455292\pi$
$822$	3.90966e15	0.363366
$823$	−1.31983e16	−1.21848	−0.609239	−	0.792987i	$-0.708525\pi$
$823$	−1.31983e16	−1.21848	−0.609239	+	0.792987i	$0.708525\pi$
$824$	9.26423e13	0.00849591
$825$	0	0
$826$	−1.16735e16	−1.05636
$827$	8.07855e15	0.726195	0.363097	−	0.931751i	$-0.381719\pi$
$827$	8.07855e15	0.726195	0.363097	+	0.931751i	$0.381719\pi$
$828$	−3.64683e15	−0.325648
$829$	−9.24683e15	−0.820244	−0.410122	−	0.912031i	$-0.634514\pi$
$829$	−9.24683e15	−0.820244	−0.410122	+	0.912031i	$0.634514\pi$
$830$	0	0
$831$	7.36397e15	0.644623
$832$	−9.56245e15	−0.831555
$833$	−1.92041e16	−1.65900
$834$	4.68031e15	0.401663
$835$	0	0
$836$	3.92616e15	0.332532
$837$	−2.47065e15	−0.207884
$838$	−4.48370e15	−0.374796
$839$	−7.02444e14	−0.0583339	−0.0291669	−	0.999575i	$-0.509285\pi$
$839$	−7.02444e14	−0.0583339	−0.0291669	+	0.999575i	$0.509285\pi$
$840$	0	0
$841$	−1.20716e16	−0.989432
$842$	−3.17278e15	−0.258359
$843$	1.19921e16	0.970159
$844$	1.39094e16	1.11795
$845$	0	0
$846$	−1.95523e15	−0.155118
$847$	−1.85608e15	−0.146298
$848$	−7.82012e13	−0.00612402
$849$	1.32465e15	0.103064
$850$	0	0
$851$	−3.01734e16	−2.31746
$852$	−2.28925e15	−0.174693
$853$	−4.32915e15	−0.328234	−0.164117	−	0.986441i	$-0.552477\pi$
$853$	−4.32915e15	−0.328234	−0.164117	+	0.986441i	$0.552477\pi$
$854$	−1.32228e16	−0.996106
$855$	0	0
$856$	−1.52414e16	−1.13350
$857$	−2.43435e16	−1.79882	−0.899412	−	0.437103i	$-0.856005\pi$
$857$	−2.43435e16	−1.79882	−0.899412	+	0.437103i	$0.856005\pi$
$858$	6.36664e15	0.467445
$859$	1.46780e16	1.07079	0.535394	−	0.844603i	$-0.320163\pi$
$859$	1.46780e16	1.07079	0.535394	+	0.844603i	$0.320163\pi$
$860$	0	0
$861$	−9.71344e15	−0.699610
$862$	2.80106e15	0.200462
$863$	−2.68085e16	−1.90640	−0.953199	−	0.302342i	$-0.902232\pi$
$863$	−2.68085e16	−1.90640	−0.953199	+	0.302342i	$0.902232\pi$
$864$	2.73603e15	0.193328
$865$	0	0
$866$	−3.37724e15	−0.235621
$867$	2.65373e14	0.0183972
$868$	1.58988e16	1.09523
$869$	−1.26134e16	−0.863422
$870$	0	0
$871$	3.08441e16	2.08484
$872$	−6.49841e15	−0.436482
$873$	9.81983e14	0.0655429
$874$	8.05608e15	0.534332
$875$	0	0
$876$	−1.97682e15	−0.129478
$877$	−1.33110e16	−0.866386	−0.433193	−	0.901301i	$-0.642613\pi$
$877$	−1.33110e16	−0.866386	−0.433193	+	0.901301i	$0.642613\pi$
$878$	7.92035e15	0.512300
$879$	−1.18051e16	−0.758807
$880$	0	0
$881$	−1.89533e16	−1.20314	−0.601570	−	0.798820i	$-0.705458\pi$
$881$	−1.89533e16	−1.20314	−0.601570	+	0.798820i	$0.705458\pi$
$882$	5.28573e15	0.333448
$883$	3.10076e15	0.194395	0.0971975	−	0.995265i	$-0.469012\pi$
$883$	3.10076e15	0.194395	0.0971975	+	0.995265i	$0.469012\pi$
$884$	−1.41174e16	−0.879563
$885$	0	0
$886$	−1.33744e16	−0.822980
$887$	2.17069e16	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$887$	2.17069e16	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$888$	1.40132e16	0.851657
$889$	−8.50094e15	−0.513461
$890$	0	0
$891$	−1.77651e15	−0.105984
$892$	1.26616e16	0.750726
$893$	−7.19350e15	−0.423894
$894$	−7.59451e14	−0.0444778
$895$	0	0
$896$	−1.78684e16	−1.03369
$897$	−2.17570e16	−1.25095
$898$	1.90423e16	1.08818
$899$	1.95514e15	0.111045
$900$	0	0
$901$	7.27411e15	0.408125
$902$	7.82365e15	0.436288
$903$	−2.66706e16	−1.47826
$904$	1.91366e16	1.05424
$905$	0	0
$906$	1.65526e15	0.0900866
$907$	7.15520e15	0.387063	0.193532	−	0.981094i	$-0.438006\pi$
$907$	7.15520e15	0.387063	0.193532	+	0.981094i	$0.438006\pi$
$908$	6.94761e15	0.373562
$909$	−3.13105e15	−0.167336
$910$	0	0
$911$	2.19413e16	1.15854	0.579270	−	0.815135i	$-0.303338\pi$
$911$	2.19413e16	1.15854	0.579270	+	0.815135i	$0.303338\pi$
$912$	9.35522e13	0.00491001
$913$	1.83122e16	0.955325
$914$	5.95252e15	0.308672
$915$	0	0
$916$	−2.78153e15	−0.142514
$917$	3.32131e16	1.69152
$918$	−2.36526e15	−0.119741
$919$	−3.69170e15	−0.185777	−0.0928884	−	0.995677i	$-0.529610\pi$
$919$	−3.69170e15	−0.185777	−0.0928884	+	0.995677i	$0.529610\pi$
$920$	0	0
$921$	1.62435e16	0.807700
$922$	−8.92906e15	−0.441353
$923$	−1.36577e16	−0.671071
$924$	1.14320e16	0.558374
$925$	0	0
$926$	−2.44268e15	−0.117898
$927$	−5.93069e13	−0.00284555
$928$	−2.16515e15	−0.103270
$929$	2.85838e16	1.35530	0.677648	−	0.735387i	$-0.263000\pi$
$929$	2.85838e16	1.35530	0.677648	+	0.735387i	$0.263000\pi$
$930$	0	0
$931$	1.94467e16	0.911221
$932$	−2.53537e16	−1.18101
$933$	1.76526e16	0.817445
$934$	−1.07203e16	−0.493512
$935$	0	0
$936$	1.01044e16	0.459721
$937$	−1.92375e16	−0.870125	−0.435062	−	0.900400i	$-0.643274\pi$
$937$	−1.92375e16	−0.870125	−0.435062	+	0.900400i	$0.643274\pi$
$938$	−3.32544e16	−1.49532
$939$	−1.52589e16	−0.682122
$940$	0	0
$941$	−2.64255e16	−1.16756	−0.583781	−	0.811911i	$-0.698427\pi$
$941$	−2.64255e16	−1.16756	−0.583781	+	0.811911i	$0.698427\pi$
$942$	3.92907e15	0.172588
$943$	−2.67361e16	−1.16757
$944$	−3.73128e14	−0.0161999
$945$	0	0
$946$	2.14817e16	0.921867
$947$	1.92888e16	0.822961	0.411481	−	0.911418i	$-0.365012\pi$
$947$	1.92888e16	0.822961	0.411481	+	0.911418i	$0.365012\pi$
$948$	−7.69816e15	−0.326544
$949$	−1.17937e16	−0.497379
$950$	0	0
$951$	1.02093e16	0.425600
$952$	3.95801e16	1.64049
$953$	−1.47724e16	−0.608750	−0.304375	−	0.952552i	$-0.598448\pi$
$953$	−1.47724e16	−0.608750	−0.304375	+	0.952552i	$0.598448\pi$
$954$	−2.00212e15	−0.0820304
$955$	0	0
$956$	−1.32389e15	−0.0536210
$957$	1.40584e15	0.0566135
$958$	−1.63022e16	−0.652735
$959$	4.18826e16	1.66737
$960$	0	0
$961$	4.23870e15	0.166822
$962$	3.21495e16	1.25809
$963$	9.75711e15	0.379644
$964$	−1.74137e16	−0.673699
$965$	0	0
$966$	2.34572e16	0.897228
$967$	6.09293e15	0.231729	0.115865	−	0.993265i	$-0.463036\pi$
$967$	6.09293e15	0.231729	0.115865	+	0.993265i	$0.463036\pi$
$968$	2.37265e15	0.0897263
$969$	−8.70202e15	−0.327219
$970$	0	0
$971$	2.81975e16	1.04835	0.524174	−	0.851611i	$-0.324374\pi$
$971$	2.81975e16	1.04835	0.524174	+	0.851611i	$0.324374\pi$
$972$	−1.08423e15	−0.0400829
$973$	5.01383e16	1.84310
$974$	−1.04539e16	−0.382125
$975$	0	0
$976$	−4.22649e14	−0.0152759
$977$	1.84959e16	0.664745	0.332372	−	0.943148i	$-0.392151\pi$
$977$	1.84959e16	0.664745	0.332372	+	0.943148i	$0.392151\pi$
$978$	−1.79037e16	−0.639851
$979$	−3.80947e16	−1.35382
$980$	0	0
$981$	4.16009e15	0.146192
$982$	−1.02942e15	−0.0359731
$983$	2.21617e16	0.770121	0.385060	−	0.922891i	$-0.374181\pi$
$983$	2.21617e16	0.770121	0.385060	+	0.922891i	$0.374181\pi$
$984$	1.24168e16	0.429079
$985$	0	0
$986$	1.87174e15	0.0639621
$987$	−2.09456e16	−0.711785
$988$	1.42958e16	0.483108
$989$	−7.34106e16	−2.46706
$990$	0	0
$991$	7.87374e15	0.261683	0.130842	−	0.991403i	$-0.458232\pi$
$991$	7.87374e15	0.261683	0.130842	+	0.991403i	$0.458232\pi$
$992$	−3.28317e16	−1.08512
$993$	3.31292e16	1.08890
$994$	1.47250e16	0.481315
$995$	0	0
$996$	1.11762e16	0.361301
$997$	6.16649e15	0.198251	0.0991254	−	0.995075i	$-0.468396\pi$
$997$	6.16649e15	0.198251	0.0991254	+	0.995075i	$0.468396\pi$
$998$	7.54196e15	0.241138
$999$	−8.97081e15	−0.285247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.a.c.1.2		2
3.2	odd	2		225.12.a.i.1.1		2
5.2	odd	4		75.12.b.d.49.4		4
5.3	odd	4		75.12.b.d.49.1		4
5.4	even	2		15.12.a.c.1.1	✓	2
15.2	even	4		225.12.b.i.199.1		4
15.8	even	4		225.12.b.i.199.4		4
15.14	odd	2		45.12.a.c.1.2		2
20.19	odd	2		240.12.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.1	✓	2	5.4	even	2
45.12.a.c.1.2		2	15.14	odd	2
75.12.a.c.1.2		2	1.1	even	1	trivial
75.12.b.d.49.1		4	5.3	odd	4
75.12.b.d.49.4		4	5.2	odd	4
225.12.a.i.1.1		2	3.2	odd	2
225.12.b.i.199.1		4	15.2	even	4
225.12.b.i.199.4		4	15.8	even	4
240.12.a.m.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+27.7191 q^{2} +243.000 q^{3} -1279.65 q^{4} +6735.74 q^{6} +72157.2 q^{7} -92239.5 q^{8} +59049.0 q^{9} -509499. q^{11} -310955. q^{12} -1.85516e6 q^{13} +2.00013e6 q^{14} +63931.4 q^{16} -5.94676e6 q^{17} +1.63678e6 q^{18} +6.02189e6 q^{19} +1.75342e7 q^{21} -1.41228e7 q^{22} +4.82627e7 q^{23} -2.24142e7 q^{24} -5.14234e7 q^{26} +1.43489e7 q^{27} -9.23361e7 q^{28} -1.13550e7 q^{29} -1.72184e8 q^{31} +1.90679e8 q^{32} -1.23808e8 q^{33} -1.64839e8 q^{34} -7.55622e7 q^{36} -6.25191e8 q^{37} +1.66921e8 q^{38} -4.50805e8 q^{39} -5.53971e8 q^{41} +4.86032e8 q^{42} -1.52106e9 q^{43} +6.51981e8 q^{44} +1.33780e9 q^{46} -1.19456e9 q^{47} +1.55353e7 q^{48} +3.22934e9 q^{49} -1.44506e9 q^{51} +2.37396e9 q^{52} -1.22320e9 q^{53} +3.97739e8 q^{54} -6.65575e9 q^{56} +1.46332e9 q^{57} -3.14750e8 q^{58} -5.83637e9 q^{59} -6.61097e9 q^{61} -4.77277e9 q^{62} +4.26081e9 q^{63} +5.15451e9 q^{64} -3.43185e9 q^{66} -1.66261e10 q^{67} +7.60978e9 q^{68} +1.17278e10 q^{69} +7.36200e9 q^{71} -5.44665e9 q^{72} +6.35726e9 q^{73} -1.73297e10 q^{74} -7.70593e9 q^{76} -3.67640e10 q^{77} -1.24959e10 q^{78} +2.47565e10 q^{79} +3.48678e9 q^{81} -1.53556e10 q^{82} -3.59416e10 q^{83} -2.24377e10 q^{84} -4.21625e10 q^{86} -2.75926e9 q^{87} +4.69959e10 q^{88} +7.47690e10 q^{89} -1.33863e11 q^{91} -6.17594e10 q^{92} -4.18406e10 q^{93} -3.31121e10 q^{94} +4.63349e10 q^{96} +1.66300e10 q^{97} +8.95144e10 q^{98} -3.00854e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 13 q^{2} + 486 q^{3} - 3111 q^{4} + 3159 q^{6} - 7784 q^{7} - 35139 q^{8} + 118098 q^{9} + 295568 q^{11} - 755973 q^{12} - 657492 q^{13} + 3176796 q^{14} + 2974065 q^{16} - 8579948 q^{17} + 767637 q^{18}+ \cdots + 17452994832 q^{99}+O(q^{100}) \) 2 * q + 13 * q^2 + 486 * q^3 - 3111 * q^4 + 3159 * q^6 - 7784 * q^7 - 35139 * q^8 + 118098 * q^9 + 295568 * q^11 - 755973 * q^12 - 657492 * q^13 + 3176796 * q^14 + 2974065 * q^16 - 8579948 * q^17 + 767637 * q^18 + 17627976 * q^19 - 1891512 * q^21 - 25972696 * q^22 + 29841072 * q^23 - 8538777 * q^24 - 69052066 * q^26 + 28697814 * q^27 + 54064108 * q^28 - 201881948 * q^29 - 71057008 * q^31 + 30902261 * q^32 + 71823024 * q^33 - 126080702 * q^34 - 183701439 * q^36 - 705858484 * q^37 - 3909548 * q^38 - 159770556 * q^39 - 327655148 * q^41 + 771961428 * q^42 - 3192552120 * q^43 - 822376568 * q^44 + 1608947832 * q^46 - 2053064720 * q^47 + 722697795 * q^48 + 7642614770 * q^49 - 2084927364 * q^51 + 180610022 * q^52 + 2304299452 * q^53 + 186535791 * q^54 - 11220431460 * q^56 + 4283598168 * q^57 + 2489635114 * q^58 - 1478770576 * q^59 - 8264891460 * q^61 - 6261263448 * q^62 - 459637416 * q^63 + 1546316681 * q^64 - 6311365128 * q^66 - 24212177528 * q^67 + 12432065594 * q^68 + 7251380496 * q^69 - 20218888256 * q^71 - 2074922811 * q^72 - 25879583268 * q^73 - 16142374034 * q^74 - 28960704764 * q^76 - 101122045248 * q^77 - 16779652038 * q^78 + 22324995440 * q^79 + 6973568802 * q^81 - 18686740478 * q^82 - 48014508984 * q^83 + 13137578244 * q^84 - 17559698092 * q^86 - 49057313364 * q^87 + 92965609896 * q^88 + 79209683076 * q^89 - 229606712048 * q^91 - 28023036264 * q^92 - 17266852944 * q^93 - 20475612776 * q^94 + 7509249423 * q^96 + 37075227452 * q^97 + 24554976677 * q^98 + 17452994832 * q^99

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