LMFDB - Embedded newform 75.22.a.g.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 125326x + 2416960 $ x^3 - x^2 - 125326*x + 2416960
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$19.3401$ 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)

$f(q)$	$=$	$q+2395.52 q^{2} -59049.0 q^{3} +3.64136e6 q^{4} -1.41453e8 q^{6} -8.95100e8 q^{7} +3.69919e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+2395.52 q^{2} -59049.0 q^{3} +3.64136e6 q^{4} -1.41453e8 q^{6} -8.95100e8 q^{7} +3.69919e9 q^{8} +3.48678e9 q^{9} -9.77889e10 q^{11} -2.15019e11 q^{12} +8.68503e11 q^{13} -2.14423e12 q^{14} +1.22499e12 q^{16} -5.17763e11 q^{17} +8.35266e12 q^{18} +4.21542e13 q^{19} +5.28548e13 q^{21} -2.34255e14 q^{22} +3.07317e14 q^{23} -2.18433e14 q^{24} +2.08052e15 q^{26} -2.05891e14 q^{27} -3.25938e15 q^{28} -1.24222e15 q^{29} -9.49909e14 q^{31} -4.82328e15 q^{32} +5.77433e15 q^{33} -1.24031e15 q^{34} +1.26966e16 q^{36} -4.84638e15 q^{37} +1.00981e17 q^{38} -5.12842e16 q^{39} -1.03529e17 q^{41} +1.26615e17 q^{42} +6.02217e16 q^{43} -3.56085e17 q^{44} +7.36184e17 q^{46} +2.18229e16 q^{47} -7.23344e16 q^{48} +2.42658e17 q^{49} +3.05734e16 q^{51} +3.16254e18 q^{52} -1.86695e18 q^{53} -4.93216e17 q^{54} -3.31114e18 q^{56} -2.48916e18 q^{57} -2.97576e18 q^{58} -7.16799e18 q^{59} -3.96239e18 q^{61} -2.27553e18 q^{62} -3.12102e18 q^{63} -1.41232e19 q^{64} +1.38325e19 q^{66} -1.14184e19 q^{67} -1.88536e18 q^{68} -1.81468e19 q^{69} -4.31629e19 q^{71} +1.28983e19 q^{72} -2.81764e19 q^{73} -1.16096e19 q^{74} +1.53499e20 q^{76} +8.75308e19 q^{77} -1.22852e20 q^{78} +1.32143e20 q^{79} +1.21577e19 q^{81} -2.48005e20 q^{82} +2.04549e19 q^{83} +1.92463e20 q^{84} +1.44262e20 q^{86} +7.33517e19 q^{87} -3.61740e20 q^{88} -1.36221e20 q^{89} -7.77397e20 q^{91} +1.11905e21 q^{92} +5.60912e19 q^{93} +5.22773e19 q^{94} +2.84810e20 q^{96} +4.85103e20 q^{97} +5.81293e20 q^{98} -3.40969e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9}+O(q^{10}) $ 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9	$ 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9} - 167336332556 q^{11} - 74661555600 q^{12} + 545571033878 q^{13} - 1568858902656 q^{14} + 255267954944 q^{16} + 8104424487194 q^{17} + 8019604122300 q^{18} + 3937700740828 q^{19} + 27497163124728 q^{21} - 114198109969712 q^{22} + 156235274730744 q^{23} - 226740870046656 q^{24} + 29\!\cdots\!56 q^{26}+ \cdots - 58\!\cdots\!56 q^{99}+O(q^{100}) $ 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9 - 167336332556 * q^11 - 74661555600 * q^12 + 545571033878 * q^13 - 1568858902656 * q^14 + 255267954944 * q^16 + 8104424487194 * q^17 + 8019604122300 * q^18 + 3937700740828 * q^19 + 27497163124728 * q^21 - 114198109969712 * q^22 + 156235274730744 * q^23 - 226740870046656 * q^24 + 2972936042645656 * q^26 - 617673396283947 * q^27 - 3826676848454144 * q^28 - 930273612785494 * q^29 - 6257709152718928 * q^31 - 4297283007276032 * q^32 + 9881043101099244 * q^33 - 20587982883981400 * q^34 + 4408690196624400 * q^36 + 22246337613227118 * q^37 + 82202958592557968 * q^38 - 32215423979462022 * q^39 - 186265908060974338 * q^41 + 92639549342934144 * q^42 + 268609288174096316 * q^43 - 284579440966881088 * q^44 + 375824751838355712 * q^46 + 900034127817222032 * q^47 - 15073317471488256 * q^48 - 586354193149814837 * q^49 - 478558161544318506 * q^51 + 3462564631549962144 * q^52 + 1269623243180583374 * q^53 - 473549603817692700 * q^54 - 5236064873787955200 * q^56 - 232517291045152572 * q^57 + 1091617398834773576 * q^58 - 8551487099411338268 * q^59 - 7181148471323735222 * q^61 + 3050004306382316256 * q^62 - 1623679985352063672 * q^63 - 557099647389855744 * q^64 + 6743284195601523888 * q^66 - 2946635148405656396 * q^67 - 10323921738190858400 * q^68 - 9225536737575702456 * q^69 - 37849731561832987624 * q^71 + 13388821635384990144 * q^72 + 7149058835824826594 * q^73 + 50872537645745842424 * q^74 + 200886445168773700032 * q^76 + 110970090560628236256 * q^77 - 175548900382183341144 * q^78 + 309682369826312504000 * q^79 + 36472996377170786403 * q^81 - 265773545783653420264 * q^82 - 199780825758268631580 * q^83 + 225961441224368749056 * q^84 + 267144820042737104944 * q^86 + 54931726561370635206 * q^87 - 749649166937994283776 * q^88 + 162380735751340085838 * q^89 - 558457704043459369328 * q^91 + 1333721831719821170688 * q^92 + 369511467758899979472 * q^93 - 149583353667227862112 * q^94 + 253750264296642413568 * q^96 + 1208943423358924299354 * q^97 + 876777377902285855132 * q^98 - 583465714076809258956 * 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 $

$2$	2395.52	1.65419	0.827094	−	0.562064i	$-0.189992\pi$
$2$	2395.52	1.65419	0.827094	+	0.562064i	$0.189992\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	3.64136e6	1.73634
$5$	0	0
$5$	0	0
$6$	−1.41453e8	−0.955046
$7$	−8.95100e8	−1.19768	−0.598842	−	0.800867i	$-0.704372\pi$
$7$	−8.95100e8	−1.19768	−0.598842	+	0.800867i	$0.704372\pi$
$8$	3.69919e9	1.21804
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−9.77889e10	−1.13675	−0.568377	−	0.822769i	$-0.692428\pi$
$11$	−9.77889e10	−1.13675	−0.568377	+	0.822769i	$0.692428\pi$
$12$	−2.15019e11	−1.00247
$13$	8.68503e11	1.74730	0.873648	−	0.486559i	$-0.161748\pi$
$13$	8.68503e11	1.74730	0.873648	+	0.486559i	$0.161748\pi$
$14$	−2.14423e12	−1.98119
$15$	0	0
$16$	1.22499e12	0.278530
$17$	−5.17763e11	−0.0622898	−0.0311449	−	0.999515i	$-0.509915\pi$
$17$	−5.17763e11	−0.0622898	−0.0311449	+	0.999515i	$0.509915\pi$
$18$	8.35266e12	0.551396
$19$	4.21542e13	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$19$	4.21542e13	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$20$	0	0
$21$	5.28548e13	0.691483
$22$	−2.34255e14	−1.88040
$23$	3.07317e14	1.54684	0.773418	−	0.633896i	$-0.218545\pi$
$23$	3.07317e14	1.54684	0.773418	+	0.633896i	$0.218545\pi$
$24$	−2.18433e14	−0.703236
$25$	0	0
$26$	2.08052e15	2.89035
$27$	−2.05891e14	−0.192450
$28$	−3.25938e15	−2.07958
$29$	−1.24222e15	−0.548300	−0.274150	−	0.961687i	$-0.588397\pi$
$29$	−1.24222e15	−0.548300	−0.274150	+	0.961687i	$0.588397\pi$
$30$	0	0
$31$	−9.49909e14	−0.208154	−0.104077	−	0.994569i	$-0.533189\pi$
$31$	−9.49909e14	−0.208154	−0.104077	+	0.994569i	$0.533189\pi$
$32$	−4.82328e15	−0.757299
$33$	5.77433e15	0.656305
$34$	−1.24031e15	−0.103039
$35$	0	0
$36$	1.26966e16	0.578779
$37$	−4.84638e15	−0.165691	−0.0828455	−	0.996562i	$-0.526401\pi$
$37$	−4.84638e15	−0.165691	−0.0828455	+	0.996562i	$0.526401\pi$
$38$	1.00981e17	2.60923
$39$	−5.12842e16	−1.00880
$40$	0	0
$41$	−1.03529e17	−1.20456	−0.602282	−	0.798284i	$-0.705742\pi$
$41$	−1.03529e17	−1.20456	−0.602282	+	0.798284i	$0.705742\pi$
$42$	1.26615e17	1.14384
$43$	6.02217e16	0.424946	0.212473	−	0.977167i	$-0.431848\pi$
$43$	6.02217e16	0.424946	0.212473	+	0.977167i	$0.431848\pi$
$44$	−3.56085e17	−1.97379
$45$	0	0
$46$	7.36184e17	2.55876
$47$	2.18229e16	0.0605181	0.0302591	−	0.999542i	$-0.490367\pi$
$47$	2.18229e16	0.0605181	0.0302591	+	0.999542i	$0.490367\pi$
$48$	−7.23344e16	−0.160810
$49$	2.42658e17	0.434446
$50$	0	0
$51$	3.05734e16	0.0359630
$52$	3.16254e18	3.03389
$53$	−1.86695e18	−1.46634	−0.733172	−	0.680043i	$-0.761961\pi$
$53$	−1.86695e18	−1.46634	−0.733172	+	0.680043i	$0.761961\pi$
$54$	−4.93216e17	−0.318349
$55$	0	0
$56$	−3.31114e18	−1.45883
$57$	−2.48916e18	−0.910683
$58$	−2.97576e18	−0.906992
$59$	−7.16799e18	−1.82579	−0.912896	−	0.408192i	$-0.866159\pi$
$59$	−7.16799e18	−1.82579	−0.912896	+	0.408192i	$0.866159\pi$
$60$	0	0
$61$	−3.96239e18	−0.711204	−0.355602	−	0.934638i	$-0.615724\pi$
$61$	−3.96239e18	−0.711204	−0.355602	+	0.934638i	$0.615724\pi$
$62$	−2.27553e18	−0.344325
$63$	−3.12102e18	−0.399228
$64$	−1.41232e19	−1.53125
$65$	0	0
$66$	1.38325e19	1.08565
$67$	−1.14184e19	−0.765280	−0.382640	−	0.923898i	$-0.624985\pi$
$67$	−1.14184e19	−0.765280	−0.382640	+	0.923898i	$0.624985\pi$
$68$	−1.88536e18	−0.108156
$69$	−1.81468e19	−0.893066
$70$	0	0
$71$	−4.31629e19	−1.57361	−0.786806	−	0.617200i	$-0.788267\pi$
$71$	−4.31629e19	−1.57361	−0.786806	+	0.617200i	$0.788267\pi$
$72$	1.28983e19	0.406013
$73$	−2.81764e19	−0.767354	−0.383677	−	0.923467i	$-0.625342\pi$
$73$	−2.81764e19	−0.767354	−0.383677	+	0.923467i	$0.625342\pi$
$74$	−1.16096e19	−0.274084
$75$	0	0
$76$	1.53499e20	2.73881
$77$	8.75308e19	1.36147
$78$	−1.22852e20	−1.66875
$79$	1.32143e20	1.57022	0.785111	−	0.619356i	$-0.212606\pi$
$79$	1.32143e20	1.57022	0.785111	+	0.619356i	$0.212606\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−2.48005e20	−1.99257
$83$	2.04549e19	0.144703	0.0723516	−	0.997379i	$-0.476950\pi$
$83$	2.04549e19	0.144703	0.0723516	+	0.997379i	$0.476950\pi$
$84$	1.92463e20	1.20065
$85$	0	0
$86$	1.44262e20	0.702941
$87$	7.33517e19	0.316561
$88$	−3.61740e20	−1.38461
$89$	−1.36221e20	−0.463072	−0.231536	−	0.972826i	$-0.574375\pi$
$89$	−1.36221e20	−0.463072	−0.231536	+	0.972826i	$0.574375\pi$
$90$	0	0
$91$	−7.77397e20	−2.09271
$92$	1.11905e21	2.68583
$93$	5.60912e19	0.120178
$94$	5.22773e19	0.100108
$95$	0	0
$96$	2.84810e20	0.437227
$97$	4.85103e20	0.667930	0.333965	−	0.942585i	$-0.391613\pi$
$97$	4.85103e20	0.667930	0.333965	+	0.942585i	$0.391613\pi$
$98$	5.81293e20	0.718656
$99$	−3.40969e20	−0.378918
$100$	0	0
$101$	−1.25452e21	−1.13007	−0.565033	−	0.825069i	$-0.691136\pi$
$101$	−1.25452e21	−1.13007	−0.565033	+	0.825069i	$0.691136\pi$
$102$	7.32391e19	0.0594896
$103$	1.27049e20	0.0931498	0.0465749	−	0.998915i	$-0.485169\pi$
$103$	1.27049e20	0.0931498	0.0465749	+	0.998915i	$0.485169\pi$
$104$	3.21276e21	2.12828
$105$	0	0
$106$	−4.47231e21	−2.42561
$107$	3.71056e20	0.182352	0.0911758	−	0.995835i	$-0.470937\pi$
$107$	3.71056e20	0.182352	0.0911758	+	0.995835i	$0.470937\pi$
$108$	−7.49724e20	−0.334158
$109$	1.89361e21	0.766149	0.383075	−	0.923717i	$-0.374865\pi$
$109$	1.89361e21	0.766149	0.383075	+	0.923717i	$0.374865\pi$
$110$	0	0
$111$	2.86174e20	0.0956617
$112$	−1.09649e21	−0.333591
$113$	1.14940e21	0.318528	0.159264	−	0.987236i	$-0.449088\pi$
$113$	1.14940e21	0.318528	0.159264	+	0.987236i	$0.449088\pi$
$114$	−5.96284e21	−1.50644
$115$	0	0
$116$	−4.52336e21	−0.952035
$117$	3.02828e21	0.582432
$118$	−1.71711e22	−3.02020
$119$	4.63449e20	0.0746035
$120$	0	0
$121$	2.16241e21	0.292208
$122$	−9.49199e21	−1.17646
$123$	6.11326e21	0.695455
$124$	−3.45896e21	−0.361425
$125$	0	0
$126$	−7.47647e21	−0.660398
$127$	2.43171e22	1.97684	0.988422	−	0.151730i	$-0.0484845\pi$
$127$	2.43171e22	1.97684	0.988422	+	0.151730i	$0.0484845\pi$
$128$	−2.37174e22	−1.77567
$129$	−3.55603e21	−0.245343
$130$	0	0
$131$	−1.62671e22	−0.954909	−0.477455	−	0.878656i	$-0.658441\pi$
$131$	−1.62671e22	−0.954909	−0.477455	+	0.878656i	$0.658441\pi$
$132$	2.10265e22	1.13957
$133$	−3.77322e22	−1.88917
$134$	−2.73530e22	−1.26592
$135$	0	0
$136$	−1.91530e21	−0.0758715
$137$	−1.96273e22	−0.719938	−0.359969	−	0.932964i	$-0.617213\pi$
$137$	−1.96273e22	−0.719938	−0.359969	+	0.932964i	$0.617213\pi$
$138$	−4.34709e22	−1.47730
$139$	−5.69327e22	−1.79352	−0.896761	−	0.442516i	$-0.854086\pi$
$139$	−5.69327e22	−1.79352	−0.896761	+	0.442516i	$0.854086\pi$
$140$	0	0
$141$	−1.28862e21	−0.0349402
$142$	−1.03398e23	−2.60305
$143$	−8.49299e22	−1.98624
$144$	4.27127e21	0.0928434
$145$	0	0
$146$	−6.74972e22	−1.26935
$147$	−1.43287e22	−0.250828
$148$	−1.76474e22	−0.287696
$149$	−1.92493e22	−0.292388	−0.146194	−	0.989256i	$-0.546702\pi$
$149$	−1.92493e22	−0.292388	−0.146194	+	0.989256i	$0.546702\pi$
$150$	0	0
$151$	3.74189e22	0.494121	0.247060	−	0.969000i	$-0.420535\pi$
$151$	3.74189e22	0.494121	0.247060	+	0.969000i	$0.420535\pi$
$152$	1.55936e23	1.92128
$153$	−1.80533e21	−0.0207633
$154$	2.09682e23	2.25213
$155$	0	0
$156$	−1.86745e23	−1.75162
$157$	1.36574e23	1.19790	0.598950	−	0.800786i	$-0.295585\pi$
$157$	1.36574e23	1.19790	0.598950	+	0.800786i	$0.295585\pi$
$158$	3.16552e23	2.59744
$159$	1.10241e23	0.846594
$160$	0	0
$161$	−2.75080e23	−1.85262
$162$	2.91239e22	0.183799
$163$	−1.51807e23	−0.898096	−0.449048	−	0.893508i	$-0.648237\pi$
$163$	−1.51807e23	−0.898096	−0.449048	+	0.893508i	$0.648237\pi$
$164$	−3.76985e23	−2.09153
$165$	0	0
$166$	4.90002e22	0.239366
$167$	−1.70863e23	−0.783655	−0.391827	−	0.920039i	$-0.628157\pi$
$167$	−1.70863e23	−0.783655	−0.391827	+	0.920039i	$0.628157\pi$
$168$	1.95520e23	0.842254
$169$	5.07233e23	2.05304
$170$	0	0
$171$	1.46983e23	0.525783
$172$	2.19289e23	0.737850
$173$	6.88557e22	0.218000	0.109000	−	0.994042i	$-0.465235\pi$
$173$	6.88557e22	0.218000	0.109000	+	0.994042i	$0.465235\pi$
$174$	1.75715e23	0.523652
$175$	0	0
$176$	−1.19790e23	−0.316620
$177$	4.23263e23	1.05412
$178$	−3.26319e23	−0.766007
$179$	−6.71886e23	−1.48710	−0.743548	−	0.668683i	$-0.766858\pi$
$179$	−6.71886e23	−1.48710	−0.743548	+	0.668683i	$0.766858\pi$
$180$	0	0
$181$	1.98744e22	0.0391444	0.0195722	−	0.999808i	$-0.493770\pi$
$181$	1.98744e22	0.0391444	0.0195722	+	0.999808i	$0.493770\pi$
$182$	−1.86227e24	−3.46173
$183$	2.33975e23	0.410614
$184$	1.13682e24	1.88411
$185$	0	0
$186$	1.34368e23	0.198796
$187$	5.06314e22	0.0708082
$188$	7.94652e22	0.105080
$189$	1.84293e23	0.230494
$190$	0	0
$191$	1.61858e24	1.81252	0.906262	−	0.422716i	$-0.138923\pi$
$191$	1.61858e24	1.81252	0.906262	+	0.422716i	$0.138923\pi$
$192$	8.33963e23	0.884065
$193$	6.78885e23	0.681467	0.340733	−	0.940160i	$-0.389325\pi$
$193$	6.78885e23	0.681467	0.340733	+	0.940160i	$0.389325\pi$
$194$	1.16207e24	1.10488
$195$	0	0
$196$	8.83607e23	0.754346
$197$	−1.26752e22	−0.0102579	−0.00512896	−	0.999987i	$-0.501633\pi$
$197$	−1.26752e22	−0.0102579	−0.00512896	+	0.999987i	$0.501633\pi$
$198$	−8.16797e23	−0.626801
$199$	−2.45217e24	−1.78482	−0.892408	−	0.451230i	$-0.850985\pi$
$199$	−2.45217e24	−1.78482	−0.892408	+	0.451230i	$0.850985\pi$
$200$	0	0
$201$	6.74246e23	0.441834
$202$	−3.00523e24	−1.86934
$203$	1.11191e24	0.656691
$204$	1.11329e23	0.0624440
$205$	0	0
$206$	3.04350e23	0.154087
$207$	1.07155e24	0.515612
$208$	1.06391e24	0.486675
$209$	−4.12221e24	−1.79306
$210$	0	0
$211$	−3.42308e24	−1.34726	−0.673630	−	0.739069i	$-0.735266\pi$
$211$	−3.42308e24	−1.34726	−0.673630	+	0.739069i	$0.735266\pi$
$212$	−6.79824e24	−2.54607
$213$	2.54872e24	0.908525
$214$	8.88872e23	0.301644
$215$	0	0
$216$	−7.61630e23	−0.234412
$217$	8.50264e23	0.249302
$218$	4.53619e24	1.26735
$219$	1.66379e24	0.443032
$220$	0	0
$221$	−4.49679e23	−0.108839
$222$	6.85535e23	0.158243
$223$	2.88612e24	0.635498	0.317749	−	0.948175i	$-0.397073\pi$
$223$	2.88612e24	0.635498	0.317749	+	0.948175i	$0.397073\pi$
$224$	4.31732e24	0.907005
$225$	0	0
$226$	2.75341e24	0.526906
$227$	−8.19575e24	−1.49733	−0.748664	−	0.662949i	$-0.769304\pi$
$227$	−8.19575e24	−1.49733	−0.748664	+	0.662949i	$0.769304\pi$
$228$	−9.06395e24	−1.58125
$229$	8.75225e24	1.45830	0.729150	−	0.684353i	$-0.239915\pi$
$229$	8.75225e24	1.45830	0.729150	+	0.684353i	$0.239915\pi$
$230$	0	0
$231$	−5.16861e24	−0.786046
$232$	−4.59520e24	−0.667852
$233$	8.99695e24	1.24985	0.624925	−	0.780685i	$-0.285129\pi$
$233$	8.99695e24	1.24985	0.624925	+	0.780685i	$0.285129\pi$
$234$	7.25431e24	0.963451
$235$	0	0
$236$	−2.61013e25	−3.17019
$237$	−7.80293e24	−0.906568
$238$	1.11020e24	0.123408
$239$	−1.19197e25	−1.26790	−0.633952	−	0.773372i	$-0.718568\pi$
$239$	−1.19197e25	−1.26790	−0.633952	+	0.773372i	$0.718568\pi$
$240$	0	0
$241$	−1.54868e25	−1.50932	−0.754662	−	0.656114i	$-0.772199\pi$
$241$	−1.54868e25	−1.50932	−0.754662	+	0.656114i	$0.772199\pi$
$242$	5.18010e24	0.483367
$243$	−7.17898e23	−0.0641500
$244$	−1.44285e25	−1.23489
$245$	0	0
$246$	1.46444e25	1.15041
$247$	3.66111e25	2.75610
$248$	−3.51389e24	−0.253540
$249$	−1.20784e24	−0.0835444
$250$	0	0
$251$	6.78905e24	0.431752	0.215876	−	0.976421i	$-0.430739\pi$
$251$	6.78905e24	0.431752	0.215876	+	0.976421i	$0.430739\pi$
$252$	−1.13648e25	−0.693194
$253$	−3.00522e25	−1.75837
$254$	5.82521e25	3.27007
$255$	0	0
$256$	−2.71968e25	−1.40604
$257$	−3.14934e25	−1.56287	−0.781433	−	0.623989i	$-0.785511\pi$
$257$	−3.14934e25	−1.56287	−0.781433	+	0.623989i	$0.785511\pi$
$258$	−8.51854e24	−0.405843
$259$	4.33799e24	0.198445
$260$	0	0
$261$	−4.33134e24	−0.182767
$262$	−3.89682e25	−1.57960
$263$	−3.56950e25	−1.39018	−0.695092	−	0.718921i	$-0.744636\pi$
$263$	−3.56950e25	−1.39018	−0.695092	+	0.718921i	$0.744636\pi$
$264$	2.13604e25	0.799406
$265$	0	0
$266$	−9.03883e25	−3.12504
$267$	8.04370e24	0.267355
$268$	−4.15786e25	−1.32878
$269$	2.89075e25	0.888407	0.444204	−	0.895926i	$-0.353487\pi$
$269$	2.89075e25	0.888407	0.444204	+	0.895926i	$0.353487\pi$
$270$	0	0
$271$	1.82449e25	0.518756	0.259378	−	0.965776i	$-0.416482\pi$
$271$	1.82449e25	0.518756	0.259378	+	0.965776i	$0.416482\pi$
$272$	−6.34254e23	−0.0173496
$273$	4.59045e25	1.20822
$274$	−4.70176e25	−1.19091
$275$	0	0
$276$	−6.60790e25	−1.55066
$277$	−5.01498e25	−1.13301	−0.566503	−	0.824060i	$-0.691704\pi$
$277$	−5.01498e25	−1.13301	−0.566503	+	0.824060i	$0.691704\pi$
$278$	−1.36383e26	−2.96682
$279$	−3.31213e24	−0.0693846
$280$	0	0
$281$	5.52534e25	1.07385	0.536924	−	0.843631i	$-0.319586\pi$
$281$	5.52534e25	1.07385	0.536924	+	0.843631i	$0.319586\pi$
$282$	−3.08692e24	−0.0577976
$283$	8.47986e24	0.152979	0.0764894	−	0.997070i	$-0.475629\pi$
$283$	8.47986e24	0.152979	0.0764894	+	0.997070i	$0.475629\pi$
$284$	−1.57172e26	−2.73232
$285$	0	0
$286$	−2.03451e26	−3.28562
$287$	9.26684e25	1.44269
$288$	−1.68177e25	−0.252433
$289$	−6.88239e25	−0.996120
$290$	0	0
$291$	−2.86449e25	−0.385630
$292$	−1.02601e26	−1.33239
$293$	−1.32127e25	−0.165532	−0.0827662	−	0.996569i	$-0.526375\pi$
$293$	−1.32127e25	−0.165532	−0.0827662	+	0.996569i	$0.526375\pi$
$294$	−3.43248e25	−0.414916
$295$	0	0
$296$	−1.79277e25	−0.201818
$297$	2.01339e25	0.218768
$298$	−4.61121e25	−0.483665
$299$	2.66906e26	2.70278
$300$	0	0
$301$	−5.39044e25	−0.508951
$302$	8.96377e25	0.817369
$303$	7.40782e25	0.652444
$304$	5.16384e25	0.439340
$305$	0	0
$306$	−4.32470e24	−0.0343464
$307$	4.16736e25	0.319822	0.159911	−	0.987131i	$-0.448879\pi$
$307$	4.16736e25	0.319822	0.159911	+	0.987131i	$0.448879\pi$
$308$	3.18732e26	2.36397
$309$	−7.50215e24	−0.0537801
$310$	0	0
$311$	2.60761e25	0.174686	0.0873430	−	0.996178i	$-0.472162\pi$
$311$	2.60761e25	0.174686	0.0873430	+	0.996178i	$0.472162\pi$
$312$	−1.89710e26	−1.22876
$313$	1.03276e26	0.646819	0.323409	−	0.946259i	$-0.395171\pi$
$313$	1.03276e26	0.646819	0.323409	+	0.946259i	$0.395171\pi$
$314$	3.27165e26	1.98155
$315$	0	0
$316$	4.81182e26	2.72643
$317$	−1.02803e26	−0.563484	−0.281742	−	0.959490i	$-0.590912\pi$
$317$	−1.02803e26	−0.563484	−0.281742	+	0.959490i	$0.590912\pi$
$318$	2.64086e26	1.40043
$319$	1.21475e26	0.623282
$320$	0	0
$321$	−2.19105e25	−0.105281
$322$	−6.58958e26	−3.06458
$323$	−2.18259e25	−0.0982529
$324$	4.42705e25	0.192926
$325$	0	0
$326$	−3.63657e26	−1.48562
$327$	−1.11816e26	−0.442336
$328$	−3.82972e26	−1.46721
$329$	−1.95337e25	−0.0724816
$330$	0	0
$331$	3.39990e26	1.18378	0.591891	−	0.806018i	$-0.298382\pi$
$331$	3.39990e26	1.18378	0.591891	+	0.806018i	$0.298382\pi$
$332$	7.44838e25	0.251253
$333$	−1.68983e25	−0.0552303
$334$	−4.09305e26	−1.29631
$335$	0	0
$336$	6.47465e25	0.192599
$337$	−3.60222e26	−1.03862	−0.519310	−	0.854586i	$-0.673811\pi$
$337$	−3.60222e26	−1.03862	−0.519310	+	0.854586i	$0.673811\pi$
$338$	1.21509e27	3.39611
$339$	−6.78710e25	−0.183902
$340$	0	0
$341$	9.28905e25	0.236619
$342$	3.52100e26	0.869744
$343$	2.82751e26	0.677354
$344$	2.22771e26	0.517602
$345$	0	0
$346$	1.64945e26	0.360613
$347$	3.16295e26	0.670861	0.335430	−	0.942065i	$-0.391118\pi$
$347$	3.16295e26	0.670861	0.335430	+	0.942065i	$0.391118\pi$
$348$	2.67100e26	0.549657
$349$	−4.86117e26	−0.970675	−0.485338	−	0.874327i	$-0.661303\pi$
$349$	−4.86117e26	−0.970675	−0.485338	+	0.874327i	$0.661303\pi$
$350$	0	0
$351$	−1.78817e26	−0.336267
$352$	4.71663e26	0.860862
$353$	−4.79439e26	−0.849373	−0.424687	−	0.905340i	$-0.639616\pi$
$353$	−4.79439e26	−0.849373	−0.424687	+	0.905340i	$0.639616\pi$
$354$	1.01393e27	1.74372
$355$	0	0
$356$	−4.96029e26	−0.804049
$357$	−2.73662e25	−0.0430724
$358$	−1.60952e27	−2.45993
$359$	2.78861e26	0.413901	0.206951	−	0.978351i	$-0.433646\pi$
$359$	2.78861e26	0.413901	0.206951	+	0.978351i	$0.433646\pi$
$360$	0	0
$361$	1.06277e27	1.48803
$362$	4.76096e25	0.0647522
$363$	−1.27688e26	−0.168706
$364$	−2.83079e27	−3.63365
$365$	0	0
$366$	5.60492e26	0.679232
$367$	1.16376e27	1.37047	0.685233	−	0.728324i	$-0.259700\pi$
$367$	1.16376e27	1.37047	0.685233	+	0.728324i	$0.259700\pi$
$368$	3.76460e26	0.430841
$369$	−3.60982e26	−0.401521
$370$	0	0
$371$	1.67111e27	1.75622
$372$	2.04248e26	0.208669
$373$	−1.59937e27	−1.58857	−0.794287	−	0.607543i	$-0.792155\pi$
$373$	−1.59937e27	−1.58857	−0.794287	+	0.607543i	$0.792155\pi$
$374$	1.21289e26	0.117130
$375$	0	0
$376$	8.07271e25	0.0737135
$377$	−1.07887e27	−0.958043
$378$	4.41478e26	0.381281
$379$	−6.74968e26	−0.566985	−0.283492	−	0.958975i	$-0.591493\pi$
$379$	−6.74968e26	−0.566985	−0.283492	+	0.958975i	$0.591493\pi$
$380$	0	0
$381$	−1.43590e27	−1.14133
$382$	3.87734e27	2.99826
$383$	8.68229e26	0.653201	0.326601	−	0.945162i	$-0.394097\pi$
$383$	8.68229e26	0.653201	0.326601	+	0.945162i	$0.394097\pi$
$384$	1.40049e27	1.02518
$385$	0	0
$386$	1.62628e27	1.12727
$387$	2.09980e26	0.141649
$388$	1.76644e27	1.15975
$389$	7.77026e26	0.496552	0.248276	−	0.968689i	$-0.420136\pi$
$389$	7.77026e26	0.496552	0.248276	+	0.968689i	$0.420136\pi$
$390$	0	0
$391$	−1.59117e26	−0.0963522
$392$	8.97639e26	0.529173
$393$	9.60557e26	0.551317
$394$	−3.03637e25	−0.0169685
$395$	0	0
$396$	−1.24159e27	−0.657929
$397$	−1.66118e27	−0.857266	−0.428633	−	0.903479i	$-0.641005\pi$
$397$	−1.66118e27	−0.857266	−0.428633	+	0.903479i	$0.641005\pi$
$398$	−5.87422e27	−2.95242
$399$	2.22805e27	1.09071
$400$	0	0
$401$	−2.04365e27	−0.949273	−0.474636	−	0.880182i	$-0.657420\pi$
$401$	−2.04365e27	−0.949273	−0.474636	+	0.880182i	$0.657420\pi$
$402$	1.61517e27	0.730877
$403$	−8.24999e26	−0.363706
$404$	−4.56817e27	−1.96217
$405$	0	0
$406$	2.66360e27	1.08629
$407$	4.73922e26	0.188350
$408$	1.13097e26	0.0438044
$409$	6.38571e26	0.241054	0.120527	−	0.992710i	$-0.461542\pi$
$409$	6.38571e26	0.241054	0.120527	+	0.992710i	$0.461542\pi$
$410$	0	0
$411$	1.15897e27	0.415656
$412$	4.62633e26	0.161740
$413$	6.41607e27	2.18672
$414$	2.56692e27	0.852919
$415$	0	0
$416$	−4.18903e27	−1.32322
$417$	3.36182e27	1.03549
$418$	−9.87484e27	−2.96605
$419$	2.49836e27	0.731826	0.365913	−	0.930649i	$-0.380757\pi$
$419$	2.49836e27	0.731826	0.365913	+	0.930649i	$0.380757\pi$
$420$	0	0
$421$	−5.91338e27	−1.64768	−0.823842	−	0.566820i	$-0.808174\pi$
$421$	−5.91338e27	−1.64768	−0.823842	+	0.566820i	$0.808174\pi$
$422$	−8.20006e27	−2.22862
$423$	7.60918e25	0.0201727
$424$	−6.90620e27	−1.78607
$425$	0	0
$426$	6.10552e27	1.50287
$427$	3.54674e27	0.851797
$428$	1.35115e27	0.316624
$429$	5.01503e27	1.14676
$430$	0	0
$431$	7.98822e26	0.173956	0.0869779	−	0.996210i	$-0.472279\pi$
$431$	7.98822e26	0.173956	0.0869779	+	0.996210i	$0.472279\pi$
$432$	−2.52214e26	−0.0536032
$433$	−6.29198e27	−1.30516	−0.652581	−	0.757719i	$-0.726314\pi$
$433$	−6.29198e27	−1.30516	−0.652581	+	0.757719i	$0.726314\pi$
$434$	2.03682e27	0.412393
$435$	0	0
$436$	6.89534e27	1.33029
$437$	1.29547e28	2.43990
$438$	3.98564e27	0.732858
$439$	−8.34934e27	−1.49891	−0.749454	−	0.662056i	$-0.769684\pi$
$439$	−8.34934e27	−1.49891	−0.749454	+	0.662056i	$0.769684\pi$
$440$	0	0
$441$	8.46097e26	0.144815
$442$	−1.07721e27	−0.180040
$443$	1.47979e26	0.0241524	0.0120762	−	0.999927i	$-0.496156\pi$
$443$	1.47979e26	0.0241524	0.0120762	+	0.999927i	$0.496156\pi$
$444$	1.04206e27	0.166101
$445$	0	0
$446$	6.91377e27	1.05123
$447$	1.13665e27	0.168810
$448$	1.26417e28	1.83395
$449$	−4.34849e27	−0.616243	−0.308121	−	0.951347i	$-0.599700\pi$
$449$	−4.34849e27	−0.616243	−0.308121	+	0.951347i	$0.599700\pi$
$450$	0	0
$451$	1.01239e28	1.36929
$452$	4.18539e27	0.553073
$453$	−2.20955e27	−0.285281
$454$	−1.96331e28	−2.47686
$455$	0	0
$456$	−9.20789e27	−1.10925
$457$	1.26213e27	0.148588	0.0742939	−	0.997236i	$-0.476330\pi$
$457$	1.26213e27	0.148588	0.0742939	+	0.997236i	$0.476330\pi$
$458$	2.09662e28	2.41230
$459$	1.06603e26	0.0119877
$460$	0	0
$461$	6.69945e26	0.0719746	0.0359873	−	0.999352i	$-0.488542\pi$
$461$	6.69945e26	0.0719746	0.0359873	+	0.999352i	$0.488542\pi$
$462$	−1.23815e28	−1.30027
$463$	−3.61754e27	−0.371376	−0.185688	−	0.982609i	$-0.559451\pi$
$463$	−3.61754e27	−0.371376	−0.185688	+	0.982609i	$0.559451\pi$
$464$	−1.52170e27	−0.152718
$465$	0	0
$466$	2.15524e28	2.06749
$467$	1.39612e28	1.30947	0.654735	−	0.755859i	$-0.272780\pi$
$467$	1.39612e28	1.30947	0.654735	+	0.755859i	$0.272780\pi$
$468$	1.10271e28	1.01130
$469$	1.02206e28	0.916563
$470$	0	0
$471$	−8.06454e27	−0.691608
$472$	−2.65158e28	−2.22389
$473$	−5.88901e27	−0.483059
$474$	−1.86921e28	−1.49963
$475$	0	0
$476$	1.68759e27	0.129537
$477$	−6.50965e27	−0.488781
$478$	−2.85538e28	−2.09735
$479$	1.23702e28	0.888902	0.444451	−	0.895803i	$-0.353399\pi$
$479$	1.23702e28	0.888902	0.444451	+	0.895803i	$0.353399\pi$
$480$	0	0
$481$	−4.20909e27	−0.289511
$482$	−3.70989e28	−2.49671
$483$	1.62432e28	1.06961
$484$	7.87412e27	0.507371
$485$	0	0
$486$	−1.71974e27	−0.106116
$487$	−6.65171e27	−0.401679	−0.200839	−	0.979624i	$-0.564367\pi$
$487$	−6.65171e27	−0.401679	−0.200839	+	0.979624i	$0.564367\pi$
$488$	−1.46576e28	−0.866275
$489$	8.96407e27	0.518516
$490$	0	0
$491$	9.32561e27	0.516799	0.258399	−	0.966038i	$-0.416805\pi$
$491$	9.32561e27	0.516799	0.258399	+	0.966038i	$0.416805\pi$
$492$	2.22606e28	1.20754
$493$	6.43174e26	0.0341535
$494$	8.77025e28	4.55910
$495$	0	0
$496$	−1.16363e27	−0.0579771
$497$	3.86351e28	1.88469
$498$	−2.89341e27	−0.138198
$499$	3.90533e28	1.82643	0.913213	−	0.407482i	$-0.133593\pi$
$499$	3.90533e28	1.82643	0.913213	+	0.407482i	$0.133593\pi$
$500$	0	0
$501$	1.00893e28	0.452443
$502$	1.62633e28	0.714200
$503$	8.11507e27	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$503$	8.11507e27	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$504$	−1.15452e28	−0.486276
$505$	0	0
$506$	−7.19906e28	−2.90868
$507$	−2.99516e28	−1.18532
$508$	8.85473e28	3.43247
$509$	4.86176e27	0.184611	0.0923053	−	0.995731i	$-0.470576\pi$
$509$	4.86176e27	0.184611	0.0923053	+	0.995731i	$0.470576\pi$
$510$	0	0
$511$	2.52207e28	0.919048
$512$	−1.54116e28	−0.550192
$513$	−8.67918e27	−0.303561
$514$	−7.54430e28	−2.58527
$515$	0	0
$516$	−1.29488e28	−0.425998
$517$	−2.13404e27	−0.0687942
$518$	1.03917e28	0.328266
$519$	−4.06586e27	−0.125862
$520$	0	0
$521$	1.58212e28	0.470373	0.235187	−	0.971950i	$-0.424430\pi$
$521$	1.58212e28	0.470373	0.235187	+	0.971950i	$0.424430\pi$
$522$	−1.03758e28	−0.302331
$523$	−3.16532e28	−0.903961	−0.451980	−	0.892028i	$-0.649282\pi$
$523$	−3.16532e28	−0.903961	−0.451980	+	0.892028i	$0.649282\pi$
$524$	−5.92345e28	−1.65804
$525$	0	0
$526$	−8.55081e28	−2.29963
$527$	4.91827e26	0.0129659
$528$	7.07350e27	0.182801
$529$	5.49722e28	1.39270
$530$	0	0
$531$	−2.49932e28	−0.608597
$532$	−1.37397e29	−3.28023
$533$	−8.99149e28	−2.10473
$534$	1.92688e28	0.442255
$535$	0	0
$536$	−4.22389e28	−0.932141
$537$	3.96742e28	0.858575
$538$	6.92486e28	1.46959
$539$	−2.37293e28	−0.493858
$540$	0	0
$541$	−3.79937e28	−0.760572	−0.380286	−	0.924869i	$-0.624175\pi$
$541$	−3.79937e28	−0.760572	−0.380286	+	0.924869i	$0.624175\pi$
$542$	4.37060e28	0.858120
$543$	−1.17356e27	−0.0226000
$544$	2.49731e27	0.0471720
$545$	0	0
$546$	1.09965e29	1.99863
$547$	−1.84275e28	−0.328549	−0.164274	−	0.986415i	$-0.552528\pi$
$547$	−1.84275e28	−0.328549	−0.164274	+	0.986415i	$0.552528\pi$
$548$	−7.14702e28	−1.25005
$549$	−1.38160e28	−0.237068
$550$	0	0
$551$	−5.23647e28	−0.864862
$552$	−6.71283e28	−1.08779
$553$	−1.18282e29	−1.88063
$554$	−1.20135e29	−1.87420
$555$	0	0
$556$	−2.07313e29	−3.11416
$557$	9.84927e28	1.45186	0.725930	−	0.687769i	$-0.241410\pi$
$557$	9.84927e28	1.45186	0.725930	+	0.687769i	$0.241410\pi$
$558$	−7.93427e27	−0.114775
$559$	5.23027e28	0.742506
$560$	0	0
$561$	−2.98973e27	−0.0408811
$562$	1.32361e29	1.77635
$563$	−4.45134e28	−0.586344	−0.293172	−	0.956060i	$-0.594711\pi$
$563$	−4.45134e28	−0.586344	−0.293172	+	0.956060i	$0.594711\pi$
$564$	−4.69234e27	−0.0606679
$565$	0	0
$566$	2.03137e28	0.253056
$567$	−1.08823e28	−0.133076
$568$	−1.59668e29	−1.91672
$569$	−8.45042e28	−0.995863	−0.497932	−	0.867216i	$-0.665907\pi$
$569$	−8.45042e28	−0.995863	−0.497932	+	0.867216i	$0.665907\pi$
$570$	0	0
$571$	4.52445e27	0.0513909	0.0256955	−	0.999670i	$-0.491820\pi$
$571$	4.52445e27	0.0513909	0.0256955	+	0.999670i	$0.491820\pi$
$572$	−3.09261e29	−3.44879
$573$	−9.55756e28	−1.04646
$574$	2.21989e29	2.38647
$575$	0	0
$576$	−4.92447e28	−0.510415
$577$	1.25450e29	1.27681	0.638403	−	0.769702i	$-0.279595\pi$
$577$	1.25450e29	1.27681	0.638403	+	0.769702i	$0.279595\pi$
$578$	−1.64869e29	−1.64777
$579$	−4.00875e28	−0.393445
$580$	0	0
$581$	−1.83092e28	−0.173309
$582$	−6.86194e28	−0.637904
$583$	1.82567e29	1.66687
$584$	−1.04230e29	−0.934668
$585$	0	0
$586$	−3.16514e28	−0.273822
$587$	4.77782e27	0.0406004	0.0203002	−	0.999794i	$-0.493538\pi$
$587$	4.77782e27	0.0406004	0.0203002	+	0.999794i	$0.493538\pi$
$588$	−5.21761e28	−0.435522
$589$	−4.00427e28	−0.328331
$590$	0	0
$591$	7.48458e26	0.00592241
$592$	−5.93676e27	−0.0461500
$593$	6.12849e28	0.468036	0.234018	−	0.972232i	$-0.424812\pi$
$593$	6.12849e28	0.468036	0.234018	+	0.972232i	$0.424812\pi$
$594$	4.82311e28	0.361884
$595$	0	0
$596$	−7.00938e28	−0.507684
$597$	1.44798e29	1.03046
$598$	6.39378e29	4.47091
$599$	2.07417e29	1.42516	0.712579	−	0.701592i	$-0.247527\pi$
$599$	2.07417e29	1.42516	0.712579	+	0.701592i	$0.247527\pi$
$600$	0	0
$601$	9.65863e28	0.640816	0.320408	−	0.947280i	$-0.396180\pi$
$601$	9.65863e28	0.640816	0.320408	+	0.947280i	$0.396180\pi$
$602$	−1.29129e29	−0.841901
$603$	−3.98135e28	−0.255093
$604$	1.36256e29	0.857961
$605$	0	0
$606$	1.77456e29	1.07926
$607$	1.65622e29	0.990006	0.495003	−	0.868891i	$-0.335167\pi$
$607$	1.65622e29	0.990006	0.495003	+	0.868891i	$0.335167\pi$
$608$	−2.03321e29	−1.19453
$609$	−6.56571e28	−0.379140
$610$	0	0
$611$	1.89533e28	0.105743
$612$	−6.57385e27	−0.0360521
$613$	3.16231e27	0.0170478	0.00852391	−	0.999964i	$-0.497287\pi$
$613$	3.16231e27	0.0170478	0.00852391	+	0.999964i	$0.497287\pi$
$614$	9.98299e28	0.529045
$615$	0	0
$616$	3.23793e29	1.65833
$617$	−2.68077e29	−1.34979	−0.674894	−	0.737915i	$-0.735811\pi$
$617$	−2.68077e29	−1.34979	−0.674894	+	0.737915i	$0.735811\pi$
$618$	−1.79715e28	−0.0889624
$619$	2.92975e29	1.42587	0.712934	−	0.701231i	$-0.247366\pi$
$619$	2.92975e29	1.42587	0.712934	+	0.701231i	$0.247366\pi$
$620$	0	0
$621$	−6.32739e28	−0.297689
$622$	6.24657e28	0.288963
$623$	1.21931e29	0.554613
$624$	−6.28227e28	−0.280982
$625$	0	0
$626$	2.47399e29	1.06996
$627$	2.43412e29	1.03522
$628$	4.97314e29	2.07996
$629$	2.50927e27	0.0103209
$630$	0	0
$631$	4.71183e29	1.87448	0.937240	−	0.348684i	$-0.113371\pi$
$631$	4.71183e29	1.87448	0.937240	+	0.348684i	$0.113371\pi$
$632$	4.88823e29	1.91259
$633$	2.02129e29	0.777841
$634$	−2.46266e29	−0.932109
$635$	0	0
$636$	4.01429e29	1.46997
$637$	2.10750e29	0.759106
$638$	2.90996e29	1.03103
$639$	−1.50500e29	−0.524537
$640$	0	0
$641$	−2.29916e29	−0.775460	−0.387730	−	0.921773i	$-0.626741\pi$
$641$	−2.29916e29	−0.775460	−0.387730	+	0.921773i	$0.626741\pi$
$642$	−5.24870e28	−0.174154
$643$	−6.65872e28	−0.217358	−0.108679	−	0.994077i	$-0.534662\pi$
$643$	−6.65872e28	−0.217358	−0.108679	+	0.994077i	$0.534662\pi$
$644$	−1.00166e30	−3.21678
$645$	0	0
$646$	−5.22843e28	−0.162529
$647$	5.21267e28	0.159428	0.0797141	−	0.996818i	$-0.474599\pi$
$647$	5.21267e28	0.159428	0.0797141	+	0.996818i	$0.474599\pi$
$648$	4.49735e28	0.135338
$649$	7.00950e29	2.07548
$650$	0	0
$651$	−5.02072e28	−0.143935
$652$	−5.52785e29	−1.55940
$653$	2.06014e29	0.571885	0.285943	−	0.958247i	$-0.407693\pi$
$653$	2.06014e29	0.571885	0.285943	+	0.958247i	$0.407693\pi$
$654$	−2.67858e29	−0.731707
$655$	0	0
$656$	−1.26821e29	−0.335507
$657$	−9.82451e28	−0.255785
$658$	−4.67934e28	−0.119898
$659$	4.49618e29	1.13383	0.566914	−	0.823777i	$-0.308137\pi$
$659$	4.49618e29	1.13383	0.566914	+	0.823777i	$0.308137\pi$
$660$	0	0
$661$	−7.39634e29	−1.80676	−0.903382	−	0.428837i	$-0.858923\pi$
$661$	−7.39634e29	−1.80676	−0.903382	+	0.428837i	$0.858923\pi$
$662$	8.14452e29	1.95820
$663$	2.65531e28	0.0628381
$664$	7.56666e28	0.176254
$665$	0	0
$666$	−4.04801e28	−0.0913614
$667$	−3.81755e29	−0.848131
$668$	−6.22174e29	−1.36069
$669$	−1.70423e29	−0.366905
$670$	0	0
$671$	3.87478e29	0.808463
$672$	−2.54933e29	−0.523659
$673$	−1.13374e29	−0.229273	−0.114637	−	0.993407i	$-0.536570\pi$
$673$	−1.13374e29	−0.229273	−0.114637	+	0.993407i	$0.536570\pi$
$674$	−8.62920e29	−1.71807
$675$	0	0
$676$	1.84702e30	3.56477
$677$	−1.51989e29	−0.288823	−0.144411	−	0.989518i	$-0.546129\pi$
$677$	−1.51989e29	−0.288823	−0.144411	+	0.989518i	$0.546129\pi$
$678$	−1.62586e29	−0.304209
$679$	−4.34216e29	−0.799969
$680$	0	0
$681$	4.83951e29	0.864483
$682$	2.22521e29	0.391413
$683$	−7.58437e29	−1.31372	−0.656859	−	0.754013i	$-0.728115\pi$
$683$	−7.58437e29	−1.31372	−0.656859	+	0.754013i	$0.728115\pi$
$684$	5.35217e29	0.912937
$685$	0	0
$686$	6.77336e29	1.12047
$687$	−5.16812e29	−0.841950
$688$	7.37709e28	0.118360
$689$	−1.62145e30	−2.56214
$690$	0	0
$691$	2.90381e29	0.445092	0.222546	−	0.974922i	$-0.428563\pi$
$691$	2.90381e29	0.445092	0.222546	+	0.974922i	$0.428563\pi$
$692$	2.50729e29	0.378521
$693$	3.05201e29	0.453824
$694$	7.57690e29	1.10973
$695$	0	0
$696$	2.71342e29	0.385585
$697$	5.36032e28	0.0750320
$698$	−1.16450e30	−1.60568
$699$	−5.31261e29	−0.721602
$700$	0	0
$701$	2.33143e28	0.0307315	0.0153657	−	0.999882i	$-0.495109\pi$
$701$	2.33143e28	0.0307315	0.0153657	+	0.999882i	$0.495109\pi$
$702$	−4.28360e29	−0.556249
$703$	−2.04295e29	−0.261353
$704$	1.38110e30	1.74065
$705$	0	0
$706$	−1.14850e30	−1.40502
$707$	1.12292e30	1.35346
$708$	1.54125e30	1.83031
$709$	−8.71248e29	−1.01943	−0.509714	−	0.860344i	$-0.670249\pi$
$709$	−8.71248e29	−1.01943	−0.509714	+	0.860344i	$0.670249\pi$
$710$	0	0
$711$	4.60755e29	0.523407
$712$	−5.03906e29	−0.564040
$713$	−2.91923e29	−0.321980
$714$	−6.55563e28	−0.0712498
$715$	0	0
$716$	−2.44658e30	−2.58210
$717$	7.03845e29	0.732025
$718$	6.68018e29	0.684670
$719$	1.35598e30	1.36962	0.684809	−	0.728722i	$-0.259885\pi$
$719$	1.35598e30	1.36962	0.684809	+	0.728722i	$0.259885\pi$
$720$	0	0
$721$	−1.13722e29	−0.111564
$722$	2.54588e30	2.46149
$723$	9.14479e29	0.871409
$724$	7.23700e28	0.0679678
$725$	0	0
$726$	−3.05880e29	−0.279072
$727$	−1.22439e30	−1.10106	−0.550528	−	0.834816i	$-0.685574\pi$
$727$	−1.22439e30	−1.10106	−0.550528	+	0.834816i	$0.685574\pi$
$728$	−2.87574e30	−2.54900
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−3.11805e28	−0.0264698
$732$	8.51989e29	0.712964
$733$	8.89865e29	0.734062	0.367031	−	0.930209i	$-0.380374\pi$
$733$	8.89865e29	0.734062	0.367031	+	0.930209i	$0.380374\pi$
$734$	2.78780e30	2.26701
$735$	0	0
$736$	−1.48228e30	−1.17142
$737$	1.11659e30	0.869934
$738$	−8.64739e29	−0.664191
$739$	−1.60693e30	−1.21683	−0.608415	−	0.793619i	$-0.708194\pi$
$739$	−1.60693e30	−1.21683	−0.608415	+	0.793619i	$0.708194\pi$
$740$	0	0
$741$	−2.16185e30	−1.59123
$742$	4.00317e30	2.90511
$743$	−9.94863e29	−0.711837	−0.355919	−	0.934517i	$-0.615832\pi$
$743$	−9.94863e29	−0.711837	−0.355919	+	0.934517i	$0.615832\pi$
$744$	2.07492e29	0.146381
$745$	0	0
$746$	−3.83133e30	−2.62780
$747$	7.13219e28	0.0482344
$748$	1.84367e29	0.122947
$749$	−3.32132e29	−0.218400
$750$	0	0
$751$	1.58627e30	1.01428	0.507140	−	0.861864i	$-0.330703\pi$
$751$	1.58627e30	1.01428	0.507140	+	0.861864i	$0.330703\pi$
$752$	2.67328e28	0.0168561
$753$	−4.00886e29	−0.249272
$754$	−2.58445e30	−1.58478
$755$	0	0
$756$	6.71078e29	0.400216
$757$	−3.32796e30	−1.95736	−0.978682	−	0.205380i	$-0.934157\pi$
$757$	−3.32796e30	−1.95736	−0.978682	+	0.205380i	$0.934157\pi$
$758$	−1.61690e30	−0.937899
$759$	1.77455e30	1.01520
$760$	0	0
$761$	−2.63740e30	−1.46770	−0.733849	−	0.679312i	$-0.762278\pi$
$761$	−2.63740e30	−1.46770	−0.733849	+	0.679312i	$0.762278\pi$
$762$	−3.43973e30	−1.88798
$763$	−1.69497e30	−0.917604
$764$	5.89384e30	3.14715
$765$	0	0
$766$	2.07986e30	1.08052
$767$	−6.22542e30	−3.19020
$768$	1.60595e30	0.811780
$769$	−3.09799e30	−1.54473	−0.772366	−	0.635178i	$-0.780927\pi$
$769$	−3.09799e30	−1.54473	−0.772366	+	0.635178i	$0.780927\pi$
$770$	0	0
$771$	1.85965e30	0.902321
$772$	2.47207e30	1.18326
$773$	1.95261e30	0.921999	0.460999	−	0.887400i	$-0.347491\pi$
$773$	1.95261e30	0.921999	0.460999	+	0.887400i	$0.347491\pi$
$774$	5.03011e29	0.234314
$775$	0	0
$776$	1.79449e30	0.813566
$777$	−2.56154e29	−0.114573
$778$	1.86138e30	0.821391
$779$	−4.36416e30	−1.90002
$780$	0	0
$781$	4.22085e30	1.78881
$782$	−3.81169e29	−0.159385
$783$	2.55762e29	0.105520
$784$	2.97254e29	0.121007
$785$	0	0
$786$	2.30103e30	0.911982
$787$	−1.80423e30	−0.705598	−0.352799	−	0.935699i	$-0.614770\pi$
$787$	−1.80423e30	−0.705598	−0.352799	+	0.935699i	$0.614770\pi$
$788$	−4.61550e28	−0.0178112
$789$	2.10776e30	0.802623
$790$	0	0
$791$	−1.02883e30	−0.381496
$792$	−1.26131e30	−0.461537
$793$	−3.44135e30	−1.24268
$794$	−3.97938e30	−1.41808
$795$	0	0
$796$	−8.92924e30	−3.09904
$797$	−2.29947e29	−0.0787617	−0.0393808	−	0.999224i	$-0.512539\pi$
$797$	−2.29947e29	−0.0787617	−0.0393808	+	0.999224i	$0.512539\pi$
$798$	5.33734e30	1.80424
$799$	−1.12991e28	−0.00376966
$800$	0	0
$801$	−4.74972e29	−0.154357
$802$	−4.89561e30	−1.57028
$803$	2.75534e30	0.872292
$804$	2.45517e30	0.767174
$805$	0	0
$806$	−1.97630e30	−0.601638
$807$	−1.70696e30	−0.512922
$808$	−4.64071e30	−1.37647
$809$	4.34067e29	0.127086	0.0635430	−	0.997979i	$-0.479760\pi$
$809$	4.34067e29	0.127086	0.0635430	+	0.997979i	$0.479760\pi$
$810$	0	0
$811$	2.19906e30	0.627361	0.313680	−	0.949529i	$-0.398438\pi$
$811$	2.19906e30	0.627361	0.313680	+	0.949529i	$0.398438\pi$
$812$	4.04886e30	1.14024
$813$	−1.07734e30	−0.299504
$814$	1.13529e30	0.311566
$815$	0	0
$816$	3.74521e28	0.0100168
$817$	2.53860e30	0.670289
$818$	1.52971e30	0.398749
$819$	−2.71062e30	−0.697569
$820$	0	0
$821$	1.63563e30	0.410282	0.205141	−	0.978732i	$-0.434235\pi$
$821$	1.63563e30	0.410282	0.205141	+	0.978732i	$0.434235\pi$
$822$	2.77634e30	0.687573
$823$	−1.02581e30	−0.250825	−0.125412	−	0.992105i	$-0.540025\pi$
$823$	−1.02581e30	−0.250825	−0.125412	+	0.992105i	$0.540025\pi$
$824$	4.69980e29	0.113460
$825$	0	0
$826$	1.53698e31	3.61725
$827$	2.60042e30	0.604276	0.302138	−	0.953264i	$-0.402300\pi$
$827$	2.60042e30	0.604276	0.302138	+	0.953264i	$0.402300\pi$
$828$	3.90190e30	0.895277
$829$	3.77733e30	0.855780	0.427890	−	0.903831i	$-0.359257\pi$
$829$	3.77733e30	0.855780	0.427890	+	0.903831i	$0.359257\pi$
$830$	0	0
$831$	2.96130e30	0.654141
$832$	−1.22661e31	−2.67554
$833$	−1.25639e29	−0.0270616
$834$	8.05331e30	1.71289
$835$	0	0
$836$	−1.50105e31	−3.11335
$837$	1.95578e29	0.0400592
$838$	5.98487e30	1.21058
$839$	−5.77342e29	−0.115327	−0.0576637	−	0.998336i	$-0.518365\pi$
$839$	−5.77342e29	−0.115327	−0.0576637	+	0.998336i	$0.518365\pi$
$840$	0	0
$841$	−3.58974e30	−0.699367
$842$	−1.41656e31	−2.72558
$843$	−3.26266e30	−0.619986
$844$	−1.24647e31	−2.33930
$845$	0	0
$846$	1.82280e29	0.0333694
$847$	−1.93557e30	−0.349973
$848$	−2.28699e30	−0.408421
$849$	−5.00728e29	−0.0883224
$850$	0	0
$851$	−1.48937e30	−0.256297
$852$	9.28083e30	1.57751
$853$	1.06255e31	1.78396	0.891979	−	0.452077i	$-0.149317\pi$
$853$	1.06255e31	1.78396	0.891979	+	0.452077i	$0.149317\pi$
$854$	8.49628e30	1.40903
$855$	0	0
$856$	1.37261e30	0.222112
$857$	−4.53080e30	−0.724229	−0.362115	−	0.932134i	$-0.617945\pi$
$857$	−4.53080e30	−0.724229	−0.362115	+	0.932134i	$0.617945\pi$
$858$	1.20136e31	1.89695
$859$	1.45535e30	0.227007	0.113504	−	0.993538i	$-0.463793\pi$
$859$	1.45535e30	0.227007	0.113504	+	0.993538i	$0.463793\pi$
$860$	0	0
$861$	−5.47198e30	−0.832935
$862$	1.91360e30	0.287756
$863$	2.75426e30	0.409158	0.204579	−	0.978850i	$-0.434417\pi$
$863$	2.75426e30	0.409158	0.204579	+	0.978850i	$0.434417\pi$
$864$	9.93070e29	0.145742
$865$	0	0
$866$	−1.50726e31	−2.15898
$867$	4.06398e30	0.575110
$868$	3.09612e30	0.432873
$869$	−1.29221e31	−1.78495
$870$	0	0
$871$	−9.91692e30	−1.33717
$872$	7.00484e30	0.933201
$873$	1.69145e30	0.222643
$874$	3.10333e31	4.03606
$875$	0	0
$876$	6.05846e30	0.769253
$877$	5.92980e30	0.743951	0.371975	−	0.928243i	$-0.378681\pi$
$877$	5.92980e30	0.743951	0.371975	+	0.928243i	$0.378681\pi$
$878$	−2.00010e31	−2.47948
$879$	7.80199e29	0.0955701
$880$	0	0
$881$	3.23525e30	0.386955	0.193478	−	0.981105i	$-0.438023\pi$
$881$	3.23525e30	0.386955	0.193478	+	0.981105i	$0.438023\pi$
$882$	2.02684e30	0.239552
$883$	−8.45781e30	−0.987804	−0.493902	−	0.869518i	$-0.664430\pi$
$883$	−8.45781e30	−0.987804	−0.493902	+	0.869518i	$0.664430\pi$
$884$	−1.63744e30	−0.188981
$885$	0	0
$886$	3.54486e29	0.0399526
$887$	−4.01399e30	−0.447073	−0.223536	−	0.974696i	$-0.571760\pi$
$887$	−4.01399e30	−0.447073	−0.223536	+	0.974696i	$0.571760\pi$
$888$	1.05861e30	0.116520
$889$	−2.17662e31	−2.36763
$890$	0	0
$891$	−1.18888e30	−0.126306
$892$	1.05094e31	1.10344
$893$	9.19928e29	0.0954583
$894$	2.72288e30	0.279244
$895$	0	0
$896$	2.12294e31	2.12669
$897$	−1.57605e31	−1.56045
$898$	−1.04169e31	−1.01938
$899$	1.17999e30	0.114131
$900$	0	0
$901$	9.66637e29	0.0913383
$902$	2.42521e31	2.26507
$903$	3.18300e30	0.293843
$904$	4.25185e30	0.387980
$905$	0	0
$906$	−5.29302e30	−0.471908
$907$	−4.80307e29	−0.0423294	−0.0211647	−	0.999776i	$-0.506737\pi$
$907$	−4.80307e29	−0.0423294	−0.0211647	+	0.999776i	$0.506737\pi$
$908$	−2.98437e31	−2.59987
$909$	−4.37424e30	−0.376688
$910$	0	0
$911$	−7.50459e30	−0.631516	−0.315758	−	0.948840i	$-0.602259\pi$
$911$	−7.50459e30	−0.631516	−0.315758	+	0.948840i	$0.602259\pi$
$912$	−3.04920e30	−0.253653
$913$	−2.00026e30	−0.164492
$914$	3.02345e30	0.245792
$915$	0	0
$916$	3.18701e31	2.53210
$917$	1.45607e31	1.14368
$918$	2.55369e29	0.0198299
$919$	−1.20781e31	−0.927225	−0.463613	−	0.886038i	$-0.653447\pi$
$919$	−1.20781e31	−0.927225	−0.463613	+	0.886038i	$0.653447\pi$
$920$	0	0
$921$	−2.46078e30	−0.184649
$922$	1.60487e30	0.119060
$923$	−3.74871e31	−2.74956
$924$	−1.88208e31	−1.36484
$925$	0	0
$926$	−8.66590e30	−0.614325
$927$	4.42994e29	0.0310499
$928$	5.99156e30	0.415227
$929$	−8.20870e30	−0.562483	−0.281241	−	0.959637i	$-0.590746\pi$
$929$	−8.20870e30	−0.562483	−0.281241	+	0.959637i	$0.590746\pi$
$930$	0	0
$931$	1.02291e31	0.685274
$932$	3.27612e31	2.17016
$933$	−1.53977e30	−0.100855
$934$	3.34443e31	2.16611
$935$	0	0
$936$	1.12022e31	0.709425
$937$	8.49649e29	0.0532076	0.0266038	−	0.999646i	$-0.491531\pi$
$937$	8.49649e29	0.0532076	0.0266038	+	0.999646i	$0.491531\pi$
$938$	2.44837e31	1.51617
$939$	−6.09833e30	−0.373441
$940$	0	0
$941$	1.00788e31	0.603558	0.301779	−	0.953378i	$-0.402419\pi$
$941$	1.00788e31	0.603558	0.301779	+	0.953378i	$0.402419\pi$
$942$	−1.93188e31	−1.14405
$943$	−3.18161e31	−1.86326
$944$	−8.78071e30	−0.508538
$945$	0	0
$946$	−1.41072e31	−0.799070
$947$	−6.41829e30	−0.359538	−0.179769	−	0.983709i	$-0.557535\pi$
$947$	−6.41829e30	−0.359538	−0.179769	+	0.983709i	$0.557535\pi$
$948$	−2.84133e31	−1.57411
$949$	−2.44713e31	−1.34079
$950$	0	0
$951$	6.07039e30	0.325328
$952$	1.71439e30	0.0908701
$953$	1.05328e31	0.552166	0.276083	−	0.961134i	$-0.410964\pi$
$953$	1.05328e31	0.552166	0.276083	+	0.961134i	$0.410964\pi$
$954$	−1.55940e31	−0.808536
$955$	0	0
$956$	−4.34039e31	−2.20151
$957$	−7.17298e30	−0.359852
$958$	2.96331e31	1.47041
$959$	1.75684e31	0.862258
$960$	0	0
$961$	−1.99232e31	−0.956672
$962$	−1.00830e31	−0.478906
$963$	1.29379e30	0.0607839
$964$	−5.63930e31	−2.62070
$965$	0	0
$966$	3.89108e31	1.76934
$967$	5.89411e30	0.265119	0.132559	−	0.991175i	$-0.457680\pi$
$967$	5.89411e30	0.265119	0.132559	+	0.991175i	$0.457680\pi$
$968$	7.99917e30	0.355921
$969$	1.28880e30	0.0567263
$970$	0	0
$971$	2.48314e31	1.06954	0.534772	−	0.844997i	$-0.320398\pi$
$971$	2.48314e31	1.06954	0.534772	+	0.844997i	$0.320398\pi$
$972$	−2.61413e30	−0.111386
$973$	5.09605e31	2.14807
$974$	−1.59343e31	−0.664452
$975$	0	0
$976$	−4.85389e30	−0.198092
$977$	4.17133e31	1.68416	0.842078	−	0.539356i	$-0.181332\pi$
$977$	4.17133e31	1.68416	0.842078	+	0.539356i	$0.181332\pi$
$978$	2.14736e31	0.857723
$979$	1.33209e31	0.526398
$980$	0	0
$981$	6.60262e30	0.255383
$982$	2.23397e31	0.854882
$983$	−3.40441e31	−1.28893	−0.644466	−	0.764633i	$-0.722920\pi$
$983$	−3.40441e31	−1.28893	−0.644466	+	0.764633i	$0.722920\pi$
$984$	2.26141e31	0.847092
$985$	0	0
$986$	1.54074e30	0.0564964
$987$	1.15345e30	0.0418473
$988$	1.33314e32	4.78551
$989$	1.85071e31	0.657322
$990$	0	0
$991$	−2.06241e31	−0.717137	−0.358569	−	0.933503i	$-0.616735\pi$
$991$	−2.06241e31	−0.717137	−0.358569	+	0.933503i	$0.616735\pi$
$992$	4.58167e30	0.157635
$993$	−2.00760e31	−0.683457
$994$	9.25511e31	3.11763
$995$	0	0
$996$	−4.39819e30	−0.145061
$997$	−2.00871e31	−0.655568	−0.327784	−	0.944753i	$-0.606302\pi$
$997$	−2.00871e31	−0.655568	−0.327784	+	0.944753i	$0.606302\pi$
$998$	9.35530e31	3.02125
$999$	9.97826e29	0.0318872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.g.1.3		3
5.2	odd	4		75.22.b.g.49.6		6
5.3	odd	4		75.22.b.g.49.1		6
5.4	even	2		15.22.a.b.1.1	✓	3
15.14	odd	2		45.22.a.e.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.b.1.1	✓	3	5.4	even	2
45.22.a.e.1.3		3	15.14	odd	2
75.22.a.g.1.3		3	1.1	even	1	trivial
75.22.b.g.49.1		6	5.3	odd	4
75.22.b.g.49.6		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+2395.52 q^{2} -59049.0 q^{3} +3.64136e6 q^{4} -1.41453e8 q^{6} -8.95100e8 q^{7} +3.69919e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+2395.52 q^{2} -59049.0 q^{3} +3.64136e6 q^{4} -1.41453e8 q^{6} -8.95100e8 q^{7} +3.69919e9 q^{8} +3.48678e9 q^{9} -9.77889e10 q^{11} -2.15019e11 q^{12} +8.68503e11 q^{13} -2.14423e12 q^{14} +1.22499e12 q^{16} -5.17763e11 q^{17} +8.35266e12 q^{18} +4.21542e13 q^{19} +5.28548e13 q^{21} -2.34255e14 q^{22} +3.07317e14 q^{23} -2.18433e14 q^{24} +2.08052e15 q^{26} -2.05891e14 q^{27} -3.25938e15 q^{28} -1.24222e15 q^{29} -9.49909e14 q^{31} -4.82328e15 q^{32} +5.77433e15 q^{33} -1.24031e15 q^{34} +1.26966e16 q^{36} -4.84638e15 q^{37} +1.00981e17 q^{38} -5.12842e16 q^{39} -1.03529e17 q^{41} +1.26615e17 q^{42} +6.02217e16 q^{43} -3.56085e17 q^{44} +7.36184e17 q^{46} +2.18229e16 q^{47} -7.23344e16 q^{48} +2.42658e17 q^{49} +3.05734e16 q^{51} +3.16254e18 q^{52} -1.86695e18 q^{53} -4.93216e17 q^{54} -3.31114e18 q^{56} -2.48916e18 q^{57} -2.97576e18 q^{58} -7.16799e18 q^{59} -3.96239e18 q^{61} -2.27553e18 q^{62} -3.12102e18 q^{63} -1.41232e19 q^{64} +1.38325e19 q^{66} -1.14184e19 q^{67} -1.88536e18 q^{68} -1.81468e19 q^{69} -4.31629e19 q^{71} +1.28983e19 q^{72} -2.81764e19 q^{73} -1.16096e19 q^{74} +1.53499e20 q^{76} +8.75308e19 q^{77} -1.22852e20 q^{78} +1.32143e20 q^{79} +1.21577e19 q^{81} -2.48005e20 q^{82} +2.04549e19 q^{83} +1.92463e20 q^{84} +1.44262e20 q^{86} +7.33517e19 q^{87} -3.61740e20 q^{88} -1.36221e20 q^{89} -7.77397e20 q^{91} +1.11905e21 q^{92} +5.60912e19 q^{93} +5.22773e19 q^{94} +2.84810e20 q^{96} +4.85103e20 q^{97} +5.81293e20 q^{98} -3.40969e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9}+O(q^{10}) \) 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9	\( 3 q + 2300 q^{2} - 177147 q^{3} + 1264400 q^{4} - 135812700 q^{6} - 465666872 q^{7} + 3839876544 q^{8} + 10460353203 q^{9} - 167336332556 q^{11} - 74661555600 q^{12} + 545571033878 q^{13} - 1568858902656 q^{14} + 255267954944 q^{16} + 8104424487194 q^{17} + 8019604122300 q^{18} + 3937700740828 q^{19} + 27497163124728 q^{21} - 114198109969712 q^{22} + 156235274730744 q^{23} - 226740870046656 q^{24} + 29\!\cdots\!56 q^{26}+ \cdots - 58\!\cdots\!56 q^{99}+O(q^{100}) \) 3 * q + 2300 * q^2 - 177147 * q^3 + 1264400 * q^4 - 135812700 * q^6 - 465666872 * q^7 + 3839876544 * q^8 + 10460353203 * q^9 - 167336332556 * q^11 - 74661555600 * q^12 + 545571033878 * q^13 - 1568858902656 * q^14 + 255267954944 * q^16 + 8104424487194 * q^17 + 8019604122300 * q^18 + 3937700740828 * q^19 + 27497163124728 * q^21 - 114198109969712 * q^22 + 156235274730744 * q^23 - 226740870046656 * q^24 + 2972936042645656 * q^26 - 617673396283947 * q^27 - 3826676848454144 * q^28 - 930273612785494 * q^29 - 6257709152718928 * q^31 - 4297283007276032 * q^32 + 9881043101099244 * q^33 - 20587982883981400 * q^34 + 4408690196624400 * q^36 + 22246337613227118 * q^37 + 82202958592557968 * q^38 - 32215423979462022 * q^39 - 186265908060974338 * q^41 + 92639549342934144 * q^42 + 268609288174096316 * q^43 - 284579440966881088 * q^44 + 375824751838355712 * q^46 + 900034127817222032 * q^47 - 15073317471488256 * q^48 - 586354193149814837 * q^49 - 478558161544318506 * q^51 + 3462564631549962144 * q^52 + 1269623243180583374 * q^53 - 473549603817692700 * q^54 - 5236064873787955200 * q^56 - 232517291045152572 * q^57 + 1091617398834773576 * q^58 - 8551487099411338268 * q^59 - 7181148471323735222 * q^61 + 3050004306382316256 * q^62 - 1623679985352063672 * q^63 - 557099647389855744 * q^64 + 6743284195601523888 * q^66 - 2946635148405656396 * q^67 - 10323921738190858400 * q^68 - 9225536737575702456 * q^69 - 37849731561832987624 * q^71 + 13388821635384990144 * q^72 + 7149058835824826594 * q^73 + 50872537645745842424 * q^74 + 200886445168773700032 * q^76 + 110970090560628236256 * q^77 - 175548900382183341144 * q^78 + 309682369826312504000 * q^79 + 36472996377170786403 * q^81 - 265773545783653420264 * q^82 - 199780825758268631580 * q^83 + 225961441224368749056 * q^84 + 267144820042737104944 * q^86 + 54931726561370635206 * q^87 - 749649166937994283776 * q^88 + 162380735751340085838 * q^89 - 558457704043459369328 * q^91 + 1333721831719821170688 * q^92 + 369511467758899979472 * q^93 - 149583353667227862112 * q^94 + 253750264296642413568 * q^96 + 1208943423358924299354 * q^97 + 876777377902285855132 * q^98 - 583465714076809258956 * q^99

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