LMFDB - Embedded newform 75.22.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 13701836 x^{6} + 201589162 x^{5} + 55078074399586 x^{4} + \cdots + 23\!\cdots\!25 $ x^8 - 2x^7 - 13701836x^6 + 201589162x^5 + 55078074399586x^4 - 1921148793416750x^3 - 63775892303255475300x^2 - 5448867057840511928250*x + 2374849659307636957978125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{10}\cdot 5^{14}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-249.679$ 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)

$f(q)$	$=$	$q-166.679 q^{2} -59049.0 q^{3} -2.06937e6 q^{4} +9.84223e6 q^{6} -1.02476e9 q^{7} +6.94472e8 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-166.679 q^{2} -59049.0 q^{3} -2.06937e6 q^{4} +9.84223e6 q^{6} -1.02476e9 q^{7} +6.94472e8 q^{8} +3.48678e9 q^{9} +8.41547e10 q^{11} +1.22194e11 q^{12} -6.78961e11 q^{13} +1.70806e11 q^{14} +4.22403e12 q^{16} +5.16667e12 q^{17} -5.81174e11 q^{18} +5.03610e12 q^{19} +6.05111e13 q^{21} -1.40268e13 q^{22} -3.20250e14 q^{23} -4.10079e13 q^{24} +1.13169e14 q^{26} -2.05891e14 q^{27} +2.12061e15 q^{28} +3.64763e15 q^{29} -4.68906e15 q^{31} -2.16047e15 q^{32} -4.96925e15 q^{33} -8.61176e14 q^{34} -7.21545e15 q^{36} -5.20866e16 q^{37} -8.39412e14 q^{38} +4.00920e16 q^{39} -7.94636e16 q^{41} -1.00859e16 q^{42} -1.06423e17 q^{43} -1.74147e17 q^{44} +5.33790e16 q^{46} +1.15217e17 q^{47} -2.49425e17 q^{48} +4.91590e17 q^{49} -3.05087e17 q^{51} +1.40502e18 q^{52} +2.45853e17 q^{53} +3.43177e16 q^{54} -7.11668e17 q^{56} -2.97376e17 q^{57} -6.07983e17 q^{58} -7.28797e18 q^{59} +1.31633e17 q^{61} +7.81568e17 q^{62} -3.57312e18 q^{63} -8.49833e18 q^{64} +8.28270e17 q^{66} +1.83759e19 q^{67} -1.06918e19 q^{68} +1.89104e19 q^{69} -4.72795e19 q^{71} +2.42147e18 q^{72} -3.44105e19 q^{73} +8.68175e18 q^{74} -1.04215e19 q^{76} -8.62385e19 q^{77} -6.68249e18 q^{78} -2.81422e19 q^{79} +1.21577e19 q^{81} +1.32449e19 q^{82} -2.06872e20 q^{83} -1.25220e20 q^{84} +1.77385e19 q^{86} -2.15389e20 q^{87} +5.84431e19 q^{88} -8.54310e19 q^{89} +6.95773e20 q^{91} +6.62716e20 q^{92} +2.76884e20 q^{93} -1.92042e19 q^{94} +1.27574e20 q^{96} -7.33743e20 q^{97} -8.19378e19 q^{98} +2.93429e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) $ 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9	$ 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) $ 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9 - 14410165968 * q^11 - 630755749296 * q^12 - 328226481176 * q^13 + 4068036669186 * q^14 + 18553054308920 * q^16 - 5718214953936 * q^17 + 2322198411066 * q^18 + 75919698170296 * q^19 - 7914601537128 * q^21 - 426897542691372 * q^22 + 49712781537936 * q^23 - 207390009885372 * q^24 - 126056679642054 * q^26 - 1647129056757192 * q^27 + 4235167126399504 * q^28 + 2111332005818352 * q^29 + 5677451410844968 * q^31 + 17528957992420632 * q^32 + 850905890244432 * q^33 + 9944754702742068 * q^34 + 37245496240179504 * q^36 - 26538401323409648 * q^37 + 85729618982552526 * q^38 + 19381445486961624 * q^39 - 9719778050982432 * q^41 - 240213497278764114 * q^42 - 20888744714565080 * q^43 - 388904820117856128 * q^44 - 156886987299808452 * q^46 + 405141072730526160 * q^47 - 1095539303887417080 * q^48 + 1172899395436368720 * q^49 + 337654874814966864 * q^51 - 2264281219115894096 * q^52 + 1890950875516930176 * q^53 - 137123493975036234 * q^54 + 19720793170524660300 * q^56 - 4482982257257808504 * q^57 + 12636224332772065092 * q^58 - 3496737905134765776 * q^59 + 17508733151770015960 * q^61 - 30404101355896200606 * q^62 + 467349306165871272 * q^63 + 27809823573720908096 * q^64 + 25207872998382825228 * q^66 - 29911607071855004216 * q^67 - 29571773973515759808 * q^68 - 2935490037033582864 * q^69 + 52204030316256846816 * q^71 + 12246172693721331228 * q^72 - 39697032757192143344 * q^73 - 32301811215943606764 * q^74 + 383913379529874070384 * q^76 - 202348742072764877136 * q^77 + 7443520876183646646 * q^78 - 274125739348498990400 * q^79 + 97261323672455430408 * q^81 - 231746572148446490256 * q^82 - 276883976086286712912 * q^83 - 250082383646764311696 * q^84 + 486708932755841131242 * q^86 - 124672043611567867248 * q^87 - 2078122304879724561672 * q^88 - 319129684977517046784 * q^89 + 45068531423660000648 * q^91 + 147706203809955886272 * q^92 - 335247828358984515432 * q^93 - 1499781052151613424236 * q^94 - 1035067440494445898968 * q^96 + 451807368184588489864 * q^97 - 2604193962467412900204 * q^98 - 50245141913043465168 * 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 $

$2$	−166.679	−0.115098	−0.0575488	−	0.998343i	$-0.518328\pi$
$2$	−166.679	−0.115098	−0.0575488	+	0.998343i	$0.518328\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−2.06937e6	−0.986753
$5$	0	0
$5$	0	0
$6$	9.84223e6	0.0664516
$7$	−1.02476e9	−1.37118	−0.685588	−	0.727990i	$-0.740455\pi$
$7$	−1.02476e9	−1.37118	−0.685588	+	0.727990i	$0.740455\pi$
$8$	6.94472e8	0.228670
$9$	3.48678e9	0.333333
$10$	0	0
$11$	8.41547e10	0.978262	0.489131	−	0.872210i	$-0.337314\pi$
$11$	8.41547e10	0.978262	0.489131	+	0.872210i	$0.337314\pi$
$12$	1.22194e11	0.569702
$13$	−6.78961e11	−1.36596	−0.682982	−	0.730435i	$-0.739317\pi$
$13$	−6.78961e11	−1.36596	−0.682982	+	0.730435i	$0.739317\pi$
$14$	1.70806e11	0.157819
$15$	0	0
$16$	4.22403e12	0.960433
$17$	5.16667e12	0.621580	0.310790	−	0.950479i	$-0.399406\pi$
$17$	5.16667e12	0.621580	0.310790	+	0.950479i	$0.399406\pi$
$18$	−5.81174e11	−0.0383659
$19$	5.03610e12	0.188444	0.0942218	−	0.995551i	$-0.469964\pi$
$19$	5.03610e12	0.188444	0.0942218	+	0.995551i	$0.469964\pi$
$20$	0	0
$21$	6.05111e13	0.791649
$22$	−1.40268e13	−0.112596
$23$	−3.20250e14	−1.61193	−0.805966	−	0.591961i	$-0.798354\pi$
$23$	−3.20250e14	−1.61193	−0.805966	+	0.591961i	$0.798354\pi$
$24$	−4.10079e13	−0.132023
$25$	0	0
$26$	1.13169e14	0.157219
$27$	−2.05891e14	−0.192450
$28$	2.12061e15	1.35301
$29$	3.64763e15	1.61002	0.805010	−	0.593261i	$-0.202160\pi$
$29$	3.64763e15	1.61002	0.805010	+	0.593261i	$0.202160\pi$
$30$	0	0
$31$	−4.68906e15	−1.02751	−0.513757	−	0.857936i	$-0.671747\pi$
$31$	−4.68906e15	−1.02751	−0.513757	+	0.857936i	$0.671747\pi$
$32$	−2.16047e15	−0.339214
$33$	−4.96925e15	−0.564800
$34$	−8.61176e14	−0.0715423
$35$	0	0
$36$	−7.21545e15	−0.328918
$37$	−5.20866e16	−1.78077	−0.890385	−	0.455208i	$-0.849565\pi$
$37$	−5.20866e16	−1.78077	−0.890385	+	0.455208i	$0.849565\pi$
$38$	−8.39412e14	−0.0216894
$39$	4.00920e16	0.788640
$40$	0	0
$41$	−7.94636e16	−0.924565	−0.462283	−	0.886733i	$-0.652969\pi$
$41$	−7.94636e16	−0.924565	−0.462283	+	0.886733i	$0.652969\pi$
$42$	−1.00859e16	−0.0911169
$43$	−1.06423e17	−0.750960	−0.375480	−	0.926830i	$-0.622522\pi$
$43$	−1.06423e17	−0.750960	−0.375480	+	0.926830i	$0.622522\pi$
$44$	−1.74147e17	−0.965303
$45$	0	0
$46$	5.33790e16	0.185530
$47$	1.15217e17	0.319512	0.159756	−	0.987157i	$-0.448929\pi$
$47$	1.15217e17	0.319512	0.159756	+	0.987157i	$0.448929\pi$
$48$	−2.49425e17	−0.554506
$49$	4.91590e17	0.880125
$50$	0	0
$51$	−3.05087e17	−0.358869
$52$	1.40502e18	1.34787
$53$	2.45853e17	0.193098	0.0965491	−	0.995328i	$-0.469220\pi$
$53$	2.45853e17	0.193098	0.0965491	+	0.995328i	$0.469220\pi$
$54$	3.43177e16	0.0221505
$55$	0	0
$56$	−7.11668e17	−0.313547
$57$	−2.97376e17	−0.108798
$58$	−6.07983e17	−0.185309
$59$	−7.28797e18	−1.85635	−0.928177	−	0.372140i	$-0.878624\pi$
$59$	−7.28797e18	−1.85635	−0.928177	+	0.372140i	$0.878624\pi$
$60$	0	0
$61$	1.31633e17	0.0236266	0.0118133	−	0.999930i	$-0.496240\pi$
$61$	1.31633e17	0.0236266	0.0118133	+	0.999930i	$0.496240\pi$
$62$	7.81568e17	0.118264
$63$	−3.57312e18	−0.457059
$64$	−8.49833e18	−0.921390
$65$	0	0
$66$	8.28270e17	0.0650071
$67$	1.83759e19	1.23158	0.615791	−	0.787910i	$-0.288837\pi$
$67$	1.83759e19	1.23158	0.615791	+	0.787910i	$0.288837\pi$
$68$	−1.06918e19	−0.613346
$69$	1.89104e19	0.930650
$70$	0	0
$71$	−4.72795e19	−1.72369	−0.861847	−	0.507168i	$-0.830692\pi$
$71$	−4.72795e19	−1.72369	−0.861847	+	0.507168i	$0.830692\pi$
$72$	2.42147e18	0.0762235
$73$	−3.44105e19	−0.937133	−0.468566	−	0.883428i	$-0.655229\pi$
$73$	−3.44105e19	−0.937133	−0.468566	+	0.883428i	$0.655229\pi$
$74$	8.68175e18	0.204962
$75$	0	0
$76$	−1.04215e19	−0.185947
$77$	−8.62385e19	−1.34137
$78$	−6.68249e18	−0.0907706
$79$	−2.81422e19	−0.334406	−0.167203	−	0.985923i	$-0.553473\pi$
$79$	−2.81422e19	−0.334406	−0.167203	+	0.985923i	$0.553473\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	1.32449e19	0.106415
$83$	−2.06872e20	−1.46346	−0.731732	−	0.681592i	$-0.761288\pi$
$83$	−2.06872e20	−1.46346	−0.731732	+	0.681592i	$0.761288\pi$
$84$	−1.25220e20	−0.781162
$85$	0	0
$86$	1.77385e19	0.0864337
$87$	−2.15389e20	−0.929546
$88$	5.84431e19	0.223700
$89$	−8.54310e19	−0.290416	−0.145208	−	0.989401i	$-0.546385\pi$
$89$	−8.54310e19	−0.290416	−0.145208	+	0.989401i	$0.546385\pi$
$90$	0	0
$91$	6.95773e20	1.87298
$92$	6.62716e20	1.59058
$93$	2.76884e20	0.593236
$94$	−1.92042e19	−0.0367751
$95$	0	0
$96$	1.27574e20	0.195845
$97$	−7.33743e20	−1.01028	−0.505139	−	0.863038i	$-0.668559\pi$
$97$	−7.33743e20	−1.01028	−0.505139	+	0.863038i	$0.668559\pi$
$98$	−8.19378e19	−0.101300
$99$	2.93429e20	0.326087
$100$	0	0
$101$	4.64652e20	0.418556	0.209278	−	0.977856i	$-0.432889\pi$
$101$	4.64652e20	0.418556	0.209278	+	0.977856i	$0.432889\pi$
$102$	5.08516e19	0.0413050
$103$	1.84494e20	0.135267	0.0676336	−	0.997710i	$-0.478455\pi$
$103$	1.84494e20	0.135267	0.0676336	+	0.997710i	$0.478455\pi$
$104$	−4.71519e20	−0.312356
$105$	0	0
$106$	−4.09785e19	−0.0222251
$107$	−4.91756e19	−0.0241669	−0.0120834	−	0.999927i	$-0.503846\pi$
$107$	−4.91756e19	−0.0241669	−0.0120834	+	0.999927i	$0.503846\pi$
$108$	4.26065e20	0.189901
$109$	2.04172e21	0.826071	0.413035	−	0.910715i	$-0.364469\pi$
$109$	2.04172e21	0.826071	0.413035	+	0.910715i	$0.364469\pi$
$110$	0	0
$111$	3.07566e21	1.02813
$112$	−4.32862e21	−1.31692
$113$	−4.00807e21	−1.11074	−0.555369	−	0.831604i	$-0.687423\pi$
$113$	−4.00807e21	−1.11074	−0.555369	+	0.831604i	$0.687423\pi$
$114$	4.95664e19	0.0125224
$115$	0	0
$116$	−7.54829e21	−1.58869
$117$	−2.36739e21	−0.455322
$118$	1.21475e21	0.213662
$119$	−5.29460e21	−0.852296
$120$	0	0
$121$	−3.18234e20	−0.0430031
$122$	−2.19405e19	−0.00271937
$123$	4.69224e21	0.533798
$124$	9.70340e21	1.01390
$125$	0	0
$126$	5.95565e20	0.0526064
$127$	1.49802e22	1.21781	0.608903	−	0.793244i	$-0.291610\pi$
$127$	1.49802e22	1.21781	0.608903	+	0.793244i	$0.291610\pi$
$128$	5.94733e21	0.445264
$129$	6.28417e21	0.433567
$130$	0	0
$131$	−1.97195e22	−1.15757	−0.578784	−	0.815481i	$-0.696473\pi$
$131$	−1.97195e22	−1.15757	−0.578784	+	0.815481i	$0.696473\pi$
$132$	1.02832e22	0.557318
$133$	−5.16080e21	−0.258389
$134$	−3.06288e21	−0.141752
$135$	0	0
$136$	3.58811e21	0.142137
$137$	1.83713e22	0.673867	0.336934	−	0.941528i	$-0.390610\pi$
$137$	1.83713e22	0.673867	0.336934	+	0.941528i	$0.390610\pi$
$138$	−3.15198e21	−0.107116
$139$	−2.43375e22	−0.766690	−0.383345	−	0.923605i	$-0.625228\pi$
$139$	−2.43375e22	−0.766690	−0.383345	+	0.923605i	$0.625228\pi$
$140$	0	0
$141$	−6.80343e21	−0.184471
$142$	7.88051e21	0.198393
$143$	−5.71378e22	−1.33627
$144$	1.47283e22	0.320144
$145$	0	0
$146$	5.73551e21	0.107862
$147$	−2.90279e22	−0.508140
$148$	1.07786e23	1.75718
$149$	3.26853e22	0.496474	0.248237	−	0.968699i	$-0.420149\pi$
$149$	3.26853e22	0.496474	0.248237	+	0.968699i	$0.420149\pi$
$150$	0	0
$151$	−5.37656e22	−0.709981	−0.354990	−	0.934870i	$-0.615516\pi$
$151$	−5.37656e22	−0.709981	−0.354990	+	0.934870i	$0.615516\pi$
$152$	3.49743e21	0.0430915
$153$	1.80151e22	0.207193
$154$	1.43742e22	0.154388
$155$	0	0
$156$	−8.29651e22	−0.778193
$157$	−7.27689e22	−0.638262	−0.319131	−	0.947711i	$-0.603391\pi$
$157$	−7.27689e22	−0.638262	−0.319131	+	0.947711i	$0.603391\pi$
$158$	4.69071e21	0.0384893
$159$	−1.45174e22	−0.111485
$160$	0	0
$161$	3.28180e23	2.21024
$162$	−2.02643e21	−0.0127886
$163$	−2.15971e23	−1.27769	−0.638845	−	0.769336i	$-0.720587\pi$
$163$	−2.15971e23	−1.27769	−0.638845	+	0.769336i	$0.720587\pi$
$164$	1.64440e23	0.912317
$165$	0	0
$166$	3.44813e22	0.168441
$167$	−3.49848e23	−1.60456	−0.802282	−	0.596945i	$-0.796381\pi$
$167$	−3.49848e23	−1.60456	−0.802282	+	0.596945i	$0.796381\pi$
$168$	4.20233e22	0.181027
$169$	2.13923e23	0.865860
$170$	0	0
$171$	1.75598e22	0.0628145
$172$	2.20229e23	0.741012
$173$	−1.37460e23	−0.435205	−0.217602	−	0.976037i	$-0.569824\pi$
$173$	−1.37460e23	−0.435205	−0.217602	+	0.976037i	$0.569824\pi$
$174$	3.59008e22	0.106988
$175$	0	0
$176$	3.55472e23	0.939555
$177$	4.30347e23	1.07177
$178$	1.42396e22	0.0334262
$179$	−6.53841e23	−1.44716	−0.723578	−	0.690242i	$-0.757504\pi$
$179$	−6.53841e23	−1.44716	−0.723578	+	0.690242i	$0.757504\pi$
$180$	0	0
$181$	−2.73973e22	−0.0539614	−0.0269807	−	0.999636i	$-0.508589\pi$
$181$	−2.73973e22	−0.0539614	−0.0269807	+	0.999636i	$0.508589\pi$
$182$	−1.15971e23	−0.215575
$183$	−7.77280e21	−0.0136408
$184$	−2.22405e23	−0.368601
$185$	0	0
$186$	−4.61508e22	−0.0682800
$187$	4.34800e23	0.608068
$188$	−2.38426e23	−0.315280
$189$	2.10989e23	0.263883
$190$	0	0
$191$	7.68245e23	0.860299	0.430149	−	0.902758i	$-0.358461\pi$
$191$	7.68245e23	0.860299	0.430149	+	0.902758i	$0.358461\pi$
$192$	5.01818e23	0.531965
$193$	3.41926e23	0.343227	0.171613	−	0.985164i	$-0.445102\pi$
$193$	3.41926e23	0.343227	0.171613	+	0.985164i	$0.445102\pi$
$194$	1.22300e23	0.116281
$195$	0	0
$196$	−1.01728e24	−0.868465
$197$	9.27843e23	0.750895	0.375447	−	0.926844i	$-0.377489\pi$
$197$	9.27843e23	0.750895	0.375447	+	0.926844i	$0.377489\pi$
$198$	−4.89085e22	−0.0375319
$199$	2.42106e24	1.76217	0.881086	−	0.472956i	$-0.156813\pi$
$199$	2.42106e24	1.76217	0.881086	+	0.472956i	$0.156813\pi$
$200$	0	0
$201$	−1.08508e24	−0.711054
$202$	−7.74478e22	−0.0481748
$203$	−3.73795e24	−2.20762
$204$	6.31337e23	0.354115
$205$	0	0
$206$	−3.07514e22	−0.0155689
$207$	−1.11664e24	−0.537311
$208$	−2.86795e24	−1.31192
$209$	4.23811e23	0.184347
$210$	0	0
$211$	1.70156e24	0.669704	0.334852	−	0.942271i	$-0.391314\pi$
$211$	1.70156e24	0.669704	0.334852	+	0.942271i	$0.391314\pi$
$212$	−5.08760e23	−0.190540
$213$	2.79181e24	0.995175
$214$	8.19655e21	0.00278155
$215$	0	0
$216$	−1.42986e23	−0.0440076
$217$	4.80517e24	1.40890
$218$	−3.40312e23	−0.0950788
$219$	2.03191e24	0.541054
$220$	0	0
$221$	−3.50797e24	−0.849056
$222$	−5.12649e23	−0.118335
$223$	8.45212e24	1.86108	0.930540	−	0.366191i	$-0.119338\pi$
$223$	8.45212e24	1.86108	0.930540	+	0.366191i	$0.119338\pi$
$224$	2.21397e24	0.465122
$225$	0	0
$226$	6.68061e23	0.127843
$227$	8.35539e24	1.52650	0.763248	−	0.646106i	$-0.223604\pi$
$227$	8.35539e24	1.52650	0.763248	+	0.646106i	$0.223604\pi$
$228$	6.15382e23	0.107357
$229$	8.52372e24	1.42022	0.710112	−	0.704089i	$-0.248644\pi$
$229$	8.52372e24	1.42022	0.710112	+	0.704089i	$0.248644\pi$
$230$	0	0
$231$	5.09230e24	0.774440
$232$	2.53317e24	0.368164
$233$	−1.23141e25	−1.71066	−0.855331	−	0.518081i	$-0.826646\pi$
$233$	−1.23141e25	−1.71066	−0.855331	+	0.518081i	$0.826646\pi$
$234$	3.94594e23	0.0524064
$235$	0	0
$236$	1.50815e25	1.83176
$237$	1.66177e24	0.193069
$238$	8.82500e23	0.0980972
$239$	1.32354e25	1.40786	0.703928	−	0.710271i	$-0.251428\pi$
$239$	1.32354e25	1.40786	0.703928	+	0.710271i	$0.251428\pi$
$240$	0	0
$241$	8.15226e24	0.794509	0.397255	−	0.917708i	$-0.369963\pi$
$241$	8.15226e24	0.794509	0.397255	+	0.917708i	$0.369963\pi$
$242$	5.30429e22	0.00494956
$243$	−7.17898e23	−0.0641500
$244$	−2.72398e23	−0.0233136
$245$	0	0
$246$	−7.82099e23	−0.0614389
$247$	−3.41931e24	−0.257407
$248$	−3.25642e24	−0.234962
$249$	1.22156e25	0.844931
$250$	0	0
$251$	−1.18049e25	−0.750740	−0.375370	−	0.926875i	$-0.622484\pi$
$251$	−1.18049e25	−0.750740	−0.375370	+	0.926875i	$0.622484\pi$
$252$	7.39411e24	0.451004
$253$	−2.69506e25	−1.57689
$254$	−2.49689e24	−0.140167
$255$	0	0
$256$	1.68310e25	0.870142
$257$	−1.84085e24	−0.0913523	−0.0456762	−	0.998956i	$-0.514544\pi$
$257$	−1.84085e24	−0.0913523	−0.0456762	+	0.998956i	$0.514544\pi$
$258$	−1.04744e24	−0.0499025
$259$	5.33763e25	2.44175
$260$	0	0
$261$	1.27185e25	0.536673
$262$	3.28682e24	0.133233
$263$	−4.49232e24	−0.174959	−0.0874794	−	0.996166i	$-0.527881\pi$
$263$	−4.49232e24	−0.174959	−0.0874794	+	0.996166i	$0.527881\pi$
$264$	−3.45101e24	−0.129153
$265$	0	0
$266$	8.60197e23	0.0297400
$267$	5.04462e24	0.167672
$268$	−3.80266e25	−1.21527
$269$	1.48544e25	0.456516	0.228258	−	0.973601i	$-0.426697\pi$
$269$	1.48544e25	0.456516	0.228258	+	0.973601i	$0.426697\pi$
$270$	0	0
$271$	−4.37603e25	−1.24424	−0.622119	−	0.782923i	$-0.713728\pi$
$271$	−4.37603e25	−1.24424	−0.622119	+	0.782923i	$0.713728\pi$
$272$	2.18242e25	0.596986
$273$	−4.10847e25	−1.08136
$274$	−3.06212e24	−0.0775605
$275$	0	0
$276$	−3.91327e25	−0.918321
$277$	3.70091e25	0.836125	0.418063	−	0.908418i	$-0.362709\pi$
$277$	3.70091e25	0.836125	0.418063	+	0.908418i	$0.362709\pi$
$278$	4.05655e24	0.0882442
$279$	−1.63497e25	−0.342505
$280$	0	0
$281$	9.63521e25	1.87260	0.936300	−	0.351202i	$-0.114227\pi$
$281$	9.63521e25	1.87260	0.936300	+	0.351202i	$0.114227\pi$
$282$	1.13399e24	0.0212321
$283$	−6.31416e25	−1.13909	−0.569545	−	0.821960i	$-0.692880\pi$
$283$	−6.31416e25	−1.13909	−0.569545	+	0.821960i	$0.692880\pi$
$284$	9.78388e25	1.70086
$285$	0	0
$286$	9.52367e24	0.153802
$287$	8.14312e25	1.26774
$288$	−7.53310e24	−0.113071
$289$	−4.23975e25	−0.613638
$290$	0	0
$291$	4.33268e25	0.583284
$292$	7.12081e25	0.924718
$293$	−3.04877e25	−0.381957	−0.190978	−	0.981594i	$-0.561166\pi$
$293$	−3.04877e25	−0.381957	−0.190978	+	0.981594i	$0.561166\pi$
$294$	4.83834e24	0.0584857
$295$	0	0
$296$	−3.61727e25	−0.407210
$297$	−1.73267e25	−0.188267
$298$	−5.44795e24	−0.0571429
$299$	2.17437e26	2.20184
$300$	0	0
$301$	1.09058e26	1.02970
$302$	8.96160e24	0.0817171
$303$	−2.74372e25	−0.241653
$304$	2.12726e25	0.180987
$305$	0	0
$306$	−3.00273e24	−0.0238474
$307$	6.23769e25	0.478709	0.239354	−	0.970932i	$-0.423064\pi$
$307$	6.23769e25	0.478709	0.239354	+	0.970932i	$0.423064\pi$
$308$	1.78459e26	1.32360
$309$	−1.08942e25	−0.0780965
$310$	0	0
$311$	−8.51641e25	−0.570522	−0.285261	−	0.958450i	$-0.592080\pi$
$311$	−8.51641e25	−0.570522	−0.285261	+	0.958450i	$0.592080\pi$
$312$	2.78427e25	0.180339
$313$	−4.69068e24	−0.0293778	−0.0146889	−	0.999892i	$-0.504676\pi$
$313$	−4.69068e24	−0.0293778	−0.0146889	+	0.999892i	$0.504676\pi$
$314$	1.21290e25	0.0734625
$315$	0	0
$316$	5.82366e25	0.329976
$317$	−2.31829e26	−1.27071	−0.635353	−	0.772222i	$-0.719145\pi$
$317$	−2.31829e26	−1.27071	−0.635353	+	0.772222i	$0.719145\pi$
$318$	2.41974e24	0.0128317
$319$	3.06965e26	1.57502
$320$	0	0
$321$	2.90377e24	0.0139527
$322$	−5.47007e25	−0.254394
$323$	2.60198e25	0.117133
$324$	−2.51587e25	−0.109639
$325$	0	0
$326$	3.59978e25	0.147059
$327$	−1.20561e26	−0.476932
$328$	−5.51852e25	−0.211421
$329$	−1.18070e26	−0.438108
$330$	0	0
$331$	3.64099e25	0.126773	0.0633864	−	0.997989i	$-0.479810\pi$
$331$	3.64099e25	0.126773	0.0633864	+	0.997989i	$0.479810\pi$
$332$	4.28095e26	1.44408
$333$	−1.81615e26	−0.593590
$334$	5.83124e25	0.184681
$335$	0	0
$336$	2.55601e26	0.760326
$337$	−1.45166e26	−0.418554	−0.209277	−	0.977856i	$-0.567111\pi$
$337$	−1.45166e26	−0.418554	−0.209277	+	0.977856i	$0.567111\pi$
$338$	−3.56565e25	−0.0996584
$339$	2.36672e26	0.641285
$340$	0	0
$341$	−3.94607e26	−1.00518
$342$	−2.92685e24	−0.00722980
$343$	6.86138e25	0.164370
$344$	−7.39078e25	−0.171722
$345$	0	0
$346$	2.29118e25	0.0500910
$347$	4.90722e26	1.04082	0.520411	−	0.853916i	$-0.325779\pi$
$347$	4.90722e26	1.04082	0.520411	+	0.853916i	$0.325779\pi$
$348$	4.45719e26	0.917232
$349$	−3.21582e26	−0.642133	−0.321067	−	0.947057i	$-0.604041\pi$
$349$	−3.21582e26	−0.642133	−0.321067	+	0.947057i	$0.604041\pi$
$350$	0	0
$351$	1.39792e26	0.262880
$352$	−1.81814e26	−0.331840
$353$	2.77493e26	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$353$	2.77493e26	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$354$	−7.17299e25	−0.123358
$355$	0	0
$356$	1.76788e26	0.286569
$357$	3.12641e26	0.492073
$358$	1.08982e26	0.166564
$359$	−9.81194e26	−1.45634	−0.728171	−	0.685396i	$-0.759629\pi$
$359$	−9.81194e26	−1.45634	−0.728171	+	0.685396i	$0.759629\pi$
$360$	0	0
$361$	−6.88847e26	−0.964489
$362$	4.56656e24	0.00621082
$363$	1.87914e25	0.0248279
$364$	−1.43981e27	−1.84817
$365$	0	0
$366$	1.29556e24	0.00157003
$367$	−7.12491e25	−0.0839046	−0.0419523	−	0.999120i	$-0.513358\pi$
$367$	−7.12491e25	−0.0839046	−0.0419523	+	0.999120i	$0.513358\pi$
$368$	−1.35275e27	−1.54815
$369$	−2.77072e26	−0.308188
$370$	0	0
$371$	−2.51940e26	−0.264772
$372$	−5.72976e26	−0.585377
$373$	−4.54132e26	−0.451066	−0.225533	−	0.974236i	$-0.572412\pi$
$373$	−4.54132e26	−0.451066	−0.225533	+	0.974236i	$0.572412\pi$
$374$	−7.24720e25	−0.0699872
$375$	0	0
$376$	8.00147e25	0.0730630
$377$	−2.47660e27	−2.19923
$378$	−3.51675e25	−0.0303723
$379$	7.47685e26	0.628068	0.314034	−	0.949412i	$-0.398319\pi$
$379$	7.47685e26	0.628068	0.314034	+	0.949412i	$0.398319\pi$
$380$	0	0
$381$	−8.84566e26	−0.703101
$382$	−1.28050e26	−0.0990183
$383$	−4.08058e25	−0.0306998	−0.0153499	−	0.999882i	$-0.504886\pi$
$383$	−4.08058e25	−0.0306998	−0.0153499	+	0.999882i	$0.504886\pi$
$384$	−3.51184e26	−0.257073
$385$	0	0
$386$	−5.69920e25	−0.0395046
$387$	−3.71074e26	−0.250320
$388$	1.51839e27	0.996894
$389$	−3.70541e26	−0.236791	−0.118396	−	0.992967i	$-0.537775\pi$
$389$	−3.70541e26	−0.236791	−0.118396	+	0.992967i	$0.537775\pi$
$390$	0	0
$391$	−1.65463e27	−1.00195
$392$	3.41396e26	0.201258
$393$	1.16441e27	0.668322
$394$	−1.54652e26	−0.0864262
$395$	0	0
$396$	−6.07214e26	−0.321768
$397$	−4.03203e26	−0.208077	−0.104038	−	0.994573i	$-0.533176\pi$
$397$	−4.03203e26	−0.208077	−0.104038	+	0.994573i	$0.533176\pi$
$398$	−4.03540e26	−0.202822
$399$	3.04740e26	0.149181
$400$	0	0
$401$	3.04218e27	1.41309	0.706544	−	0.707669i	$-0.250253\pi$
$401$	3.04218e27	1.41309	0.706544	+	0.707669i	$0.250253\pi$
$402$	1.80860e26	0.0818406
$403$	3.18369e27	1.40355
$404$	−9.61537e26	−0.413011
$405$	0	0
$406$	6.23037e26	0.254092
$407$	−4.38333e27	−1.74206
$408$	−2.11874e26	−0.0820628
$409$	−4.71550e27	−1.78005	−0.890027	−	0.455907i	$-0.849315\pi$
$409$	−4.71550e27	−1.78005	−0.890027	+	0.455907i	$0.849315\pi$
$410$	0	0
$411$	−1.08481e27	−0.389058
$412$	−3.81787e26	−0.133475
$413$	7.46843e27	2.54539
$414$	1.86121e26	0.0618432
$415$	0	0
$416$	1.46688e27	0.463354
$417$	1.43710e27	0.442649
$418$	−7.06405e25	−0.0212179
$419$	−6.10830e27	−1.78926	−0.894630	−	0.446809i	$-0.852561\pi$
$419$	−6.10830e27	−1.78926	−0.894630	+	0.446809i	$0.852561\pi$
$420$	0	0
$421$	−1.76673e27	−0.492275	−0.246138	−	0.969235i	$-0.579162\pi$
$421$	−1.76673e27	−0.492275	−0.246138	+	0.969235i	$0.579162\pi$
$422$	−2.83615e26	−0.0770813
$423$	4.01736e26	0.106504
$424$	1.70738e26	0.0441558
$425$	0	0
$426$	−4.65336e26	−0.114542
$427$	−1.34893e26	−0.0323963
$428$	1.01763e26	0.0238467
$429$	3.37393e27	0.771497
$430$	0	0
$431$	−6.90869e26	−0.150447	−0.0752236	−	0.997167i	$-0.523967\pi$
$431$	−6.90869e26	−0.150447	−0.0752236	+	0.997167i	$0.523967\pi$
$432$	−8.69690e26	−0.184835
$433$	−3.77275e27	−0.782591	−0.391296	−	0.920265i	$-0.627973\pi$
$433$	−3.77275e27	−0.782591	−0.391296	+	0.920265i	$0.627973\pi$
$434$	−8.00921e26	−0.162161
$435$	0	0
$436$	−4.22507e27	−0.815128
$437$	−1.61281e27	−0.303758
$438$	−3.38676e26	−0.0622740
$439$	6.54147e27	1.17435	0.587175	−	0.809460i	$-0.300240\pi$
$439$	6.54147e27	1.17435	0.587175	+	0.809460i	$0.300240\pi$
$440$	0	0
$441$	1.71407e27	0.293375
$442$	5.84705e26	0.0977243
$443$	−2.70935e27	−0.442208	−0.221104	−	0.975250i	$-0.570966\pi$
$443$	−2.70935e27	−0.442208	−0.221104	+	0.975250i	$0.570966\pi$
$444$	−6.36468e27	−1.01451
$445$	0	0
$446$	−1.40879e27	−0.214206
$447$	−1.93003e27	−0.286639
$448$	8.70876e27	1.26339
$449$	−4.24406e27	−0.601443	−0.300722	−	0.953712i	$-0.597228\pi$
$449$	−4.24406e27	−0.601443	−0.300722	+	0.953712i	$0.597228\pi$
$450$	0	0
$451$	−6.68723e27	−0.904467
$452$	8.29418e27	1.09602
$453$	3.17481e27	0.409908
$454$	−1.39267e27	−0.175696
$455$	0	0
$456$	−2.06520e26	−0.0248789
$457$	−7.12772e27	−0.839132	−0.419566	−	0.907725i	$-0.637818\pi$
$457$	−7.12772e27	−0.839132	−0.419566	+	0.907725i	$0.637818\pi$
$458$	−1.42073e27	−0.163464
$459$	−1.06377e27	−0.119623
$460$	0	0
$461$	−1.68767e28	−1.81312	−0.906560	−	0.422076i	$-0.861302\pi$
$461$	−1.68767e28	−1.81312	−0.906560	+	0.422076i	$0.861302\pi$
$462$	−8.48779e26	−0.0891362
$463$	−1.26513e28	−1.29878	−0.649391	−	0.760455i	$-0.724976\pi$
$463$	−1.26513e28	−1.29878	−0.649391	+	0.760455i	$0.724976\pi$
$464$	1.54077e28	1.54632
$465$	0	0
$466$	2.05250e27	0.196893
$467$	4.32488e27	0.405645	0.202823	−	0.979215i	$-0.434988\pi$
$467$	4.32488e27	0.405645	0.202823	+	0.979215i	$0.434988\pi$
$468$	4.89901e27	0.449290
$469$	−1.88309e28	−1.68872
$470$	0	0
$471$	4.29693e27	0.368501
$472$	−5.06129e27	−0.424493
$473$	−8.95600e27	−0.734636
$474$	−2.76982e26	−0.0222218
$475$	0	0
$476$	1.09565e28	0.841005
$477$	8.57235e26	0.0643661
$478$	−2.20606e27	−0.162041
$479$	−6.12898e27	−0.440418	−0.220209	−	0.975453i	$-0.570674\pi$
$479$	−6.12898e27	−0.440418	−0.220209	+	0.975453i	$0.570674\pi$
$480$	0	0
$481$	3.53648e28	2.43247
$482$	−1.35881e27	−0.0914461
$483$	−1.93787e28	−1.27609
$484$	6.58544e26	0.0424335
$485$	0	0
$486$	1.19659e26	0.00738351
$487$	1.14277e28	0.690086	0.345043	−	0.938587i	$-0.387864\pi$
$487$	1.14277e28	0.690086	0.345043	+	0.938587i	$0.387864\pi$
$488$	9.14155e25	0.00540271
$489$	1.27529e28	0.737675
$490$	0	0
$491$	−1.32813e27	−0.0736010	−0.0368005	−	0.999323i	$-0.511717\pi$
$491$	−1.32813e27	−0.0736010	−0.0368005	+	0.999323i	$0.511717\pi$
$492$	−9.70999e27	−0.526727
$493$	1.88461e28	1.00076
$494$	5.69928e26	0.0296270
$495$	0	0
$496$	−1.98067e28	−0.986859
$497$	4.84502e28	2.36349
$498$	−2.03608e27	−0.0972495
$499$	3.23815e28	1.51440	0.757200	−	0.653183i	$-0.226567\pi$
$499$	3.23815e28	1.51440	0.757200	+	0.653183i	$0.226567\pi$
$500$	0	0
$501$	2.06582e28	0.926396
$502$	1.96763e27	0.0864083
$503$	2.80905e28	1.20808	0.604041	−	0.796954i	$-0.293556\pi$
$503$	2.80905e28	1.20808	0.604041	+	0.796954i	$0.293556\pi$
$504$	−2.48143e27	−0.104516
$505$	0	0
$506$	4.49209e27	0.181497
$507$	−1.26320e28	−0.499904
$508$	−3.09996e28	−1.20167
$509$	1.04322e28	0.396131	0.198065	−	0.980189i	$-0.436534\pi$
$509$	1.04322e28	0.396131	0.198065	+	0.980189i	$0.436534\pi$
$510$	0	0
$511$	3.52626e28	1.28497
$512$	−1.52778e28	−0.545415
$513$	−1.03689e27	−0.0362660
$514$	3.06830e26	0.0105144
$515$	0	0
$516$	−1.30043e28	−0.427823
$517$	9.69602e27	0.312567
$518$	−8.89672e27	−0.281040
$519$	8.11690e27	0.251266
$520$	0	0
$521$	1.95275e28	0.580565	0.290282	−	0.956941i	$-0.406251\pi$
$521$	1.95275e28	0.580565	0.290282	+	0.956941i	$0.406251\pi$
$522$	−2.11991e27	−0.0617698
$523$	−5.60979e28	−1.60206	−0.801030	−	0.598624i	$-0.795714\pi$
$523$	−5.60979e28	−1.60206	−0.801030	+	0.598624i	$0.795714\pi$
$524$	4.08069e28	1.14223
$525$	0	0
$526$	7.48777e26	0.0201373
$527$	−2.42268e28	−0.638683
$528$	−2.09903e28	−0.542453
$529$	6.30885e28	1.59833
$530$	0	0
$531$	−2.54116e28	−0.618784
$532$	1.06796e28	0.254966
$533$	5.39527e28	1.26292
$534$	−8.40832e26	−0.0192986
$535$	0	0
$536$	1.27616e28	0.281626
$537$	3.86087e28	0.835516
$538$	−2.47592e27	−0.0525439
$539$	4.13696e28	0.860993
$540$	0	0
$541$	−8.18651e28	−1.63881	−0.819403	−	0.573218i	$-0.805695\pi$
$541$	−8.18651e28	−1.63881	−0.819403	+	0.573218i	$0.805695\pi$
$542$	7.29394e27	0.143209
$543$	1.61778e27	0.0311546
$544$	−1.11624e28	−0.210849
$545$	0	0
$546$	6.84796e27	0.124462
$547$	−8.60861e27	−0.153485	−0.0767425	−	0.997051i	$-0.524452\pi$
$547$	−8.60861e27	−0.153485	−0.0767425	+	0.997051i	$0.524452\pi$
$548$	−3.80171e28	−0.664940
$549$	4.58976e26	0.00787555
$550$	0	0
$551$	1.83698e28	0.303398
$552$	1.31328e28	0.212812
$553$	2.88390e28	0.458529
$554$	−6.16865e27	−0.0962360
$555$	0	0
$556$	5.03632e28	0.756534
$557$	1.30988e29	1.93086	0.965430	−	0.260664i	$-0.0839414\pi$
$557$	1.30988e29	1.93086	0.965430	+	0.260664i	$0.0839414\pi$
$558$	2.72516e27	0.0394215
$559$	7.22571e28	1.02579
$560$	0	0
$561$	−2.56745e28	−0.351068
$562$	−1.60599e28	−0.215532
$563$	−7.12976e28	−0.939154	−0.469577	−	0.882891i	$-0.655594\pi$
$563$	−7.12976e28	−0.939154	−0.469577	+	0.882891i	$0.655594\pi$
$564$	1.40788e28	0.182027
$565$	0	0
$566$	1.05244e28	0.131106
$567$	−1.24587e28	−0.152353
$568$	−3.28343e28	−0.394158
$569$	9.25550e27	0.109074	0.0545370	−	0.998512i	$-0.482632\pi$
$569$	9.25550e27	0.109074	0.0545370	+	0.998512i	$0.482632\pi$
$570$	0	0
$571$	−9.74702e28	−1.10712	−0.553558	−	0.832811i	$-0.686730\pi$
$571$	−9.74702e28	−1.10712	−0.553558	+	0.832811i	$0.686730\pi$
$572$	1.18239e29	1.31857
$573$	−4.53641e28	−0.496694
$574$	−1.35729e28	−0.145914
$575$	0	0
$576$	−2.96318e28	−0.307130
$577$	1.53081e29	1.55803	0.779015	−	0.627005i	$-0.215719\pi$
$577$	1.53081e29	1.55803	0.779015	+	0.627005i	$0.215719\pi$
$578$	7.06677e27	0.0706283
$579$	−2.01904e28	−0.198162
$580$	0	0
$581$	2.11995e29	2.00667
$582$	−7.22167e27	−0.0671346
$583$	2.06897e28	0.188901
$584$	−2.38971e28	−0.214295
$585$	0	0
$586$	5.08166e27	0.0439623
$587$	−1.14595e29	−0.973793	−0.486897	−	0.873460i	$-0.661871\pi$
$587$	−1.14595e29	−0.973793	−0.486897	+	0.873460i	$0.661871\pi$
$588$	6.00695e28	0.501409
$589$	−2.36146e28	−0.193629
$590$	0	0
$591$	−5.47882e28	−0.433529
$592$	−2.20015e29	−1.71031
$593$	−9.22603e28	−0.704597	−0.352298	−	0.935888i	$-0.614600\pi$
$593$	−9.22603e28	−0.704597	−0.352298	+	0.935888i	$0.614600\pi$
$594$	2.88800e27	0.0216690
$595$	0	0
$596$	−6.76379e28	−0.489896
$597$	−1.42961e29	−1.01739
$598$	−3.62422e28	−0.253427
$599$	−2.69560e29	−1.85214	−0.926070	−	0.377351i	$-0.876835\pi$
$599$	−2.69560e29	−1.85214	−0.926070	+	0.377351i	$0.876835\pi$
$600$	0	0
$601$	1.71447e29	1.13749	0.568746	−	0.822513i	$-0.307429\pi$
$601$	1.71447e29	1.13749	0.568746	+	0.822513i	$0.307429\pi$
$602$	−1.81777e28	−0.118516
$603$	6.40728e28	0.410527
$604$	1.11261e29	0.700575
$605$	0	0
$606$	4.57322e27	0.0278137
$607$	−1.08981e29	−0.651434	−0.325717	−	0.945467i	$-0.605606\pi$
$607$	−1.08981e29	−0.651434	−0.325717	+	0.945467i	$0.605606\pi$
$608$	−1.08803e28	−0.0639227
$609$	2.20722e29	1.27457
$610$	0	0
$611$	−7.82276e28	−0.436443
$612$	−3.72798e28	−0.204449
$613$	1.05774e29	0.570222	0.285111	−	0.958494i	$-0.407969\pi$
$613$	1.05774e29	0.570222	0.285111	+	0.958494i	$0.407969\pi$
$614$	−1.03969e28	−0.0550982
$615$	0	0
$616$	−5.98902e28	−0.306732
$617$	5.81298e28	0.292688	0.146344	−	0.989234i	$-0.453249\pi$
$617$	5.81298e28	0.292688	0.146344	+	0.989234i	$0.453249\pi$
$618$	1.81584e27	0.00898872
$619$	8.28280e27	0.0403112	0.0201556	−	0.999797i	$-0.493584\pi$
$619$	8.28280e27	0.0403112	0.0201556	+	0.999797i	$0.493584\pi$
$620$	0	0
$621$	6.59366e28	0.310217
$622$	1.41951e28	0.0656657
$623$	8.75464e28	0.398212
$624$	1.69350e29	0.757436
$625$	0	0
$626$	7.81838e26	0.00338132
$627$	−2.50256e28	−0.106433
$628$	1.50586e29	0.629807
$629$	−2.69114e29	−1.10689
$630$	0	0
$631$	4.70876e28	0.187326	0.0936630	−	0.995604i	$-0.470142\pi$
$631$	4.70876e28	0.187326	0.0936630	+	0.995604i	$0.470142\pi$
$632$	−1.95440e28	−0.0764686
$633$	−1.00476e29	−0.386654
$634$	3.86410e28	0.146255
$635$	0	0
$636$	3.00418e28	0.110008
$637$	−3.33770e29	−1.20222
$638$	−5.11646e28	−0.181281
$639$	−1.64853e29	−0.574565
$640$	0	0
$641$	3.35322e29	1.13098	0.565488	−	0.824757i	$-0.308688\pi$
$641$	3.35322e29	1.13098	0.565488	+	0.824757i	$0.308688\pi$
$642$	−4.83998e26	−0.00160593
$643$	2.28214e29	0.744951	0.372476	−	0.928042i	$-0.378509\pi$
$643$	2.28214e29	0.744951	0.372476	+	0.928042i	$0.378509\pi$
$644$	−6.79126e29	−2.18096
$645$	0	0
$646$	−4.33696e27	−0.0134817
$647$	4.05399e29	1.23990	0.619952	−	0.784640i	$-0.287152\pi$
$647$	4.05399e29	1.23990	0.619952	+	0.784640i	$0.287152\pi$
$648$	8.44316e27	0.0254078
$649$	−6.13317e29	−1.81600
$650$	0	0
$651$	−2.83740e29	−0.813431
$652$	4.46924e29	1.26076
$653$	−1.27772e28	−0.0354687	−0.0177344	−	0.999843i	$-0.505645\pi$
$653$	−1.27772e28	−0.0354687	−0.0177344	+	0.999843i	$0.505645\pi$
$654$	2.00951e28	0.0548937
$655$	0	0
$656$	−3.35656e29	−0.887983
$657$	−1.19982e29	−0.312378
$658$	1.96797e28	0.0504251
$659$	6.99206e28	0.176323	0.0881614	−	0.996106i	$-0.471901\pi$
$659$	6.99206e28	0.176323	0.0881614	+	0.996106i	$0.471901\pi$
$660$	0	0
$661$	−4.85023e29	−1.18481	−0.592403	−	0.805642i	$-0.701821\pi$
$661$	−4.85023e29	−1.18481	−0.592403	+	0.805642i	$0.701821\pi$
$662$	−6.06877e27	−0.0145912
$663$	2.07142e29	0.490203
$664$	−1.43667e29	−0.334651
$665$	0	0
$666$	3.02714e28	0.0683208
$667$	−1.16815e30	−2.59524
$668$	7.23966e29	1.58331
$669$	−4.99090e29	−1.07449
$670$	0	0
$671$	1.10775e28	0.0231130
$672$	−1.30733e29	−0.268538
$673$	−1.03296e29	−0.208893	−0.104447	−	0.994530i	$-0.533307\pi$
$673$	−1.03296e29	−0.208893	−0.104447	+	0.994530i	$0.533307\pi$
$674$	2.41962e28	0.0481745
$675$	0	0
$676$	−4.42686e29	−0.854389
$677$	7.81995e29	1.48601	0.743007	−	0.669284i	$-0.233399\pi$
$677$	7.81995e29	1.48601	0.743007	+	0.669284i	$0.233399\pi$
$678$	−3.94483e28	−0.0738103
$679$	7.51911e29	1.38527
$680$	0	0
$681$	−4.93378e29	−0.881322
$682$	6.57727e28	0.115694
$683$	8.70064e29	1.50707	0.753535	−	0.657407i	$-0.228347\pi$
$683$	8.70064e29	1.50707	0.753535	+	0.657407i	$0.228347\pi$
$684$	−3.63377e28	−0.0619824
$685$	0	0
$686$	−1.14365e28	−0.0189186
$687$	−5.03317e29	−0.819966
$688$	−4.49534e29	−0.721247
$689$	−1.66924e29	−0.263765
$690$	0	0
$691$	−6.20998e29	−0.951855	−0.475928	−	0.879484i	$-0.657888\pi$
$691$	−6.20998e29	−0.951855	−0.475928	+	0.879484i	$0.657888\pi$
$692$	2.84457e29	0.429439
$693$	−3.00695e29	−0.447123
$694$	−8.17931e28	−0.119796
$695$	0	0
$696$	−1.49581e29	−0.212560
$697$	−4.10562e29	−0.574691
$698$	5.36011e28	0.0739080
$699$	7.27133e29	0.987652
$700$	0	0
$701$	8.69349e29	1.14592	0.572961	−	0.819583i	$-0.305795\pi$
$701$	8.69349e29	1.14592	0.572961	+	0.819583i	$0.305795\pi$
$702$	−2.33004e28	−0.0302569
$703$	−2.62313e29	−0.335575
$704$	−7.15174e29	−0.901361
$705$	0	0
$706$	−4.62522e28	−0.0565827
$707$	−4.76158e29	−0.573914
$708$	−8.90548e29	−1.05757
$709$	−8.90842e29	−1.04235	−0.521177	−	0.853449i	$-0.674507\pi$
$709$	−8.90842e29	−1.04235	−0.521177	+	0.853449i	$0.674507\pi$
$710$	0	0
$711$	−9.81258e28	−0.111469
$712$	−5.93295e28	−0.0664095
$713$	1.50167e30	1.65628
$714$	−5.21107e28	−0.0566364
$715$	0	0
$716$	1.35304e30	1.42799
$717$	−7.81536e29	−0.812827
$718$	1.63544e29	0.167621
$719$	−1.39858e29	−0.141265	−0.0706323	−	0.997502i	$-0.522502\pi$
$719$	−1.39858e29	−0.141265	−0.0706323	+	0.997502i	$0.522502\pi$
$720$	0	0
$721$	−1.89063e29	−0.185475
$722$	1.14816e29	0.111010
$723$	−4.81383e29	−0.458710
$724$	5.66952e28	0.0532465
$725$	0	0
$726$	−3.13213e27	−0.00285763
$727$	−6.76901e29	−0.608714	−0.304357	−	0.952558i	$-0.598442\pi$
$727$	−6.76901e29	−0.608714	−0.304357	+	0.952558i	$0.598442\pi$
$728$	4.83195e29	0.428295
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−5.49853e29	−0.466782
$732$	1.60848e28	0.0134601
$733$	6.62441e28	0.0546456	0.0273228	−	0.999627i	$-0.491302\pi$
$733$	6.62441e28	0.0546456	0.0273228	+	0.999627i	$0.491302\pi$
$734$	1.18757e28	0.00965722
$735$	0	0
$736$	6.91891e29	0.546790
$737$	1.54642e30	1.20481
$738$	4.61822e28	0.0354717
$739$	1.30334e29	0.0986945	0.0493472	−	0.998782i	$-0.484286\pi$
$739$	1.30334e29	0.0986945	0.0493472	+	0.998782i	$0.484286\pi$
$740$	0	0
$741$	2.01907e29	0.148614
$742$	4.19932e28	0.0304746
$743$	−6.32010e28	−0.0452211	−0.0226105	−	0.999744i	$-0.507198\pi$
$743$	−6.32010e28	−0.0452211	−0.0226105	+	0.999744i	$0.507198\pi$
$744$	1.92288e29	0.135655
$745$	0	0
$746$	7.56944e28	0.0519166
$747$	−7.21319e29	−0.487821
$748$	−8.99761e29	−0.600013
$749$	5.03933e28	0.0331370
$750$	0	0
$751$	2.68105e30	1.71429	0.857147	−	0.515072i	$-0.172235\pi$
$751$	2.68105e30	1.71429	0.857147	+	0.515072i	$0.172235\pi$
$752$	4.86678e29	0.306870
$753$	6.97069e29	0.433440
$754$	4.12797e29	0.253126
$755$	0	0
$756$	−4.36615e29	−0.260387
$757$	1.28440e28	0.00755429	0.00377714	−	0.999993i	$-0.498798\pi$
$757$	1.28440e28	0.00755429	0.00377714	+	0.999993i	$0.498798\pi$
$758$	−1.24623e29	−0.0722891
$759$	1.59140e30	0.910420
$760$	0	0
$761$	9.48703e29	0.527948	0.263974	−	0.964530i	$-0.414967\pi$
$761$	9.48703e29	0.527948	0.263974	+	0.964530i	$0.414967\pi$
$762$	1.47439e29	0.0809252
$763$	−2.09227e30	−1.13269
$764$	−1.58978e30	−0.848902
$765$	0	0
$766$	6.80148e27	0.00353347
$767$	4.94825e30	2.53571
$768$	−9.93853e29	−0.502377
$769$	3.31983e30	1.65535	0.827673	−	0.561210i	$-0.189664\pi$
$769$	3.31983e30	1.65535	0.827673	+	0.561210i	$0.189664\pi$
$770$	0	0
$771$	1.08700e29	0.0527423
$772$	−7.07572e29	−0.338680
$773$	−9.83508e29	−0.464401	−0.232200	−	0.972668i	$-0.574593\pi$
$773$	−9.83508e29	−0.464401	−0.232200	+	0.972668i	$0.574593\pi$
$774$	6.18503e28	0.0288112
$775$	0	0
$776$	−5.09564e29	−0.231021
$777$	−3.15182e30	−1.40975
$778$	6.17615e28	0.0272541
$779$	−4.00186e29	−0.174228
$780$	0	0
$781$	−3.97879e30	−1.68623
$782$	2.75792e29	0.115321
$783$	−7.51014e29	−0.309849
$784$	2.07649e30	0.845301
$785$	0	0
$786$	−1.94084e29	−0.0769223
$787$	−2.08470e30	−0.815283	−0.407642	−	0.913142i	$-0.633649\pi$
$787$	−2.08470e30	−0.815283	−0.407642	+	0.913142i	$0.633649\pi$
$788$	−1.92005e30	−0.740948
$789$	2.65267e29	0.101013
$790$	0	0
$791$	4.10731e30	1.52302
$792$	2.03778e29	0.0745665
$793$	−8.93737e28	−0.0322732
$794$	6.72055e28	0.0239491
$795$	0	0
$796$	−5.01007e30	−1.73883
$797$	5.27961e30	1.80838	0.904189	−	0.427132i	$-0.140476\pi$
$797$	5.27961e30	1.80838	0.904189	+	0.427132i	$0.140476\pi$
$798$	−5.07938e28	−0.0171704
$799$	5.95286e29	0.198602
$800$	0	0
$801$	−2.97880e29	−0.0968053
$802$	−5.07068e29	−0.162643
$803$	−2.89581e30	−0.916762
$804$	2.24543e30	0.701635
$805$	0	0
$806$	−5.30654e29	−0.161545
$807$	−8.77138e29	−0.263570
$808$	3.22688e29	0.0957114
$809$	1.28746e30	0.376942	0.188471	−	0.982079i	$-0.439647\pi$
$809$	1.28746e30	0.376942	0.188471	+	0.982079i	$0.439647\pi$
$810$	0	0
$811$	1.27883e30	0.364834	0.182417	−	0.983221i	$-0.441608\pi$
$811$	1.27883e30	0.364834	0.182417	+	0.983221i	$0.441608\pi$
$812$	7.73519e30	2.17838
$813$	2.58400e30	0.718361
$814$	7.30610e29	0.200507
$815$	0	0
$816$	−1.28870e30	−0.344670
$817$	−5.35957e29	−0.141514
$818$	7.85976e29	0.204880
$819$	2.42601e30	0.624326
$820$	0	0
$821$	5.58684e30	1.40140	0.700702	−	0.713454i	$-0.252870\pi$
$821$	5.58684e30	1.40140	0.700702	+	0.713454i	$0.252870\pi$
$822$	1.80815e29	0.0447796
$823$	−2.60095e30	−0.635965	−0.317983	−	0.948097i	$-0.603005\pi$
$823$	−2.60095e30	−0.635965	−0.317983	+	0.948097i	$0.603005\pi$
$824$	1.28126e29	0.0309316
$825$	0	0
$826$	−1.24483e30	−0.292968
$827$	2.27379e30	0.528374	0.264187	−	0.964471i	$-0.414896\pi$
$827$	2.27379e30	0.528374	0.264187	+	0.964471i	$0.414896\pi$
$828$	2.31075e30	0.530193
$829$	7.80903e30	1.76919	0.884595	−	0.466360i	$-0.154435\pi$
$829$	7.80903e30	1.76919	0.884595	+	0.466360i	$0.154435\pi$
$830$	0	0
$831$	−2.18535e30	−0.482737
$832$	5.77003e30	1.25859
$833$	2.53988e30	0.547068
$834$	−2.39535e29	−0.0509478
$835$	0	0
$836$	−8.77022e29	−0.181905
$837$	9.65436e29	0.197745
$838$	1.01813e30	0.205939
$839$	−9.06205e30	−1.81020	−0.905099	−	0.425201i	$-0.860203\pi$
$839$	−9.06205e30	−1.81020	−0.905099	+	0.425201i	$0.860203\pi$
$840$	0	0
$841$	8.17233e30	1.59217
$842$	2.94477e29	0.0566597
$843$	−5.68950e30	−1.08115
$844$	−3.52117e30	−0.660832
$845$	0	0
$846$	−6.69609e28	−0.0122584
$847$	3.26114e29	0.0589649
$848$	1.03849e30	0.185458
$849$	3.72845e30	0.657654
$850$	0	0
$851$	1.66807e31	2.87048
$852$	−5.77728e30	−0.981992
$853$	4.21419e29	0.0707537	0.0353769	−	0.999374i	$-0.488737\pi$
$853$	4.21419e29	0.0707537	0.0353769	+	0.999374i	$0.488737\pi$
$854$	2.24838e28	0.00372873
$855$	0	0
$856$	−3.41511e28	−0.00552625
$857$	−1.02690e31	−1.64146	−0.820729	−	0.571318i	$-0.806432\pi$
$857$	−1.02690e31	−1.64146	−0.820729	+	0.571318i	$0.806432\pi$
$858$	−5.62363e29	−0.0887974
$859$	1.23241e31	1.92232	0.961161	−	0.275988i	$-0.0890049\pi$
$859$	1.23241e31	1.92232	0.961161	+	0.275988i	$0.0890049\pi$
$860$	0	0
$861$	−4.80843e30	−0.731931
$862$	1.15153e29	0.0173161
$863$	8.10217e30	1.20362	0.601808	−	0.798641i	$-0.294447\pi$
$863$	8.10217e30	1.20362	0.601808	+	0.798641i	$0.294447\pi$
$864$	4.44822e29	0.0652817
$865$	0	0
$866$	6.28838e29	0.0900743
$867$	2.50353e30	0.354284
$868$	−9.94367e30	−1.39024
$869$	−2.36830e30	−0.327136
$870$	0	0
$871$	−1.24765e31	−1.68230
$872$	1.41792e30	0.188898
$873$	−2.55840e30	−0.336759
$874$	2.68822e29	0.0349618
$875$	0	0
$876$	−4.20477e30	−0.533886
$877$	−2.22627e30	−0.279307	−0.139654	−	0.990200i	$-0.544599\pi$
$877$	−2.22627e30	−0.279307	−0.139654	+	0.990200i	$0.544599\pi$
$878$	−1.09033e30	−0.135165
$879$	1.80027e30	0.220523
$880$	0	0
$881$	2.32997e29	0.0278678	0.0139339	−	0.999903i	$-0.495565\pi$
$881$	2.32997e29	0.0278678	0.0139339	+	0.999903i	$0.495565\pi$
$882$	−2.85699e29	−0.0337667
$883$	1.22673e31	1.43272	0.716361	−	0.697730i	$-0.245807\pi$
$883$	1.22673e31	1.43272	0.716361	+	0.697730i	$0.245807\pi$
$884$	7.25928e30	0.837809
$885$	0	0
$886$	4.51593e29	0.0508971
$887$	7.18852e30	0.800647	0.400324	−	0.916374i	$-0.368898\pi$
$887$	7.18852e30	0.800647	0.400324	+	0.916374i	$0.368898\pi$
$888$	2.13596e30	0.235103
$889$	−1.53511e31	−1.66983
$890$	0	0
$891$	1.02312e30	0.108696
$892$	−1.74906e31	−1.83642
$893$	5.80242e29	0.0602100
$894$	3.21696e29	0.0329915
$895$	0	0
$896$	−6.09459e30	−0.610535
$897$	−1.28395e31	−1.27124
$898$	7.07396e29	0.0692247
$899$	−1.71039e31	−1.65432
$900$	0	0
$901$	1.27024e30	0.120026
$902$	1.11462e30	0.104102
$903$	−6.43978e30	−0.594497
$904$	−2.78349e30	−0.253993
$905$	0	0
$906$	−5.29174e29	−0.0471794
$907$	−4.25095e30	−0.374636	−0.187318	−	0.982299i	$-0.559979\pi$
$907$	−4.25095e30	−0.374636	−0.187318	+	0.982299i	$0.559979\pi$
$908$	−1.72904e31	−1.50627
$909$	1.62014e30	0.139519
$910$	0	0
$911$	1.77961e31	1.49755	0.748775	−	0.662825i	$-0.230643\pi$
$911$	1.77961e31	1.49755	0.748775	+	0.662825i	$0.230643\pi$
$912$	−1.25613e30	−0.104493
$913$	−1.74093e31	−1.43165
$914$	1.18804e30	0.0965820
$915$	0	0
$916$	−1.76387e31	−1.40141
$917$	2.02077e31	1.58723
$918$	1.77308e29	0.0137683
$919$	−1.40986e31	−1.08234	−0.541169	−	0.840914i	$-0.682018\pi$
$919$	−1.40986e31	−1.08234	−0.541169	+	0.840914i	$0.682018\pi$
$920$	0	0
$921$	−3.68330e30	−0.276383
$922$	2.81299e30	0.208686
$923$	3.21009e31	2.35451
$924$	−1.05378e31	−0.764181
$925$	0	0
$926$	2.10872e30	0.149487
$927$	6.43292e29	0.0450891
$928$	−7.88059e30	−0.546141
$929$	−3.66823e29	−0.0251358	−0.0125679	−	0.999921i	$-0.504001\pi$
$929$	−3.66823e29	−0.0251358	−0.0125679	+	0.999921i	$0.504001\pi$
$930$	0	0
$931$	2.47569e30	0.165854
$932$	2.54824e31	1.68800
$933$	5.02885e30	0.329391
$934$	−7.20867e29	−0.0466888
$935$	0	0
$936$	−1.64409e30	−0.104119
$937$	−1.83413e31	−1.14859	−0.574295	−	0.818649i	$-0.694724\pi$
$937$	−1.83413e31	−1.14859	−0.574295	+	0.818649i	$0.694724\pi$
$938$	3.13872e30	0.194367
$939$	2.76980e29	0.0169613
$940$	0	0
$941$	−1.86445e31	−1.11650	−0.558250	−	0.829673i	$-0.688527\pi$
$941$	−1.86445e31	−1.11650	−0.558250	+	0.829673i	$0.688527\pi$
$942$	−7.16208e29	−0.0424136
$943$	2.54482e31	1.49034
$944$	−3.07846e31	−1.78290
$945$	0	0
$946$	1.49278e30	0.0845548
$947$	−7.59041e30	−0.425197	−0.212599	−	0.977140i	$-0.568193\pi$
$947$	−7.59041e30	−0.425197	−0.212599	+	0.977140i	$0.568193\pi$
$948$	−3.43881e30	−0.190511
$949$	2.33634e31	1.28009
$950$	0	0
$951$	1.36892e31	0.733642
$952$	−3.67695e30	−0.194895
$953$	1.85218e31	0.970974	0.485487	−	0.874244i	$-0.338642\pi$
$953$	1.85218e31	0.970974	0.485487	+	0.874244i	$0.338642\pi$
$954$	−1.42883e29	−0.00740838
$955$	0	0
$956$	−2.73889e31	−1.38921
$957$	−1.81260e31	−0.909339
$958$	1.02157e30	0.0506911
$959$	−1.88262e31	−0.923991
$960$	0	0
$961$	1.16178e30	0.0557865
$962$	−5.89457e30	−0.279971
$963$	−1.71465e29	−0.00805562
$964$	−1.68700e31	−0.783984
$965$	0	0
$966$	3.23002e30	0.146874
$967$	−1.68944e31	−0.759915	−0.379957	−	0.925004i	$-0.624061\pi$
$967$	−1.68944e31	−0.759915	−0.379957	+	0.925004i	$0.624061\pi$
$968$	−2.21005e29	−0.00983354
$969$	−1.53645e30	−0.0676266
$970$	0	0
$971$	−4.19989e31	−1.80899	−0.904494	−	0.426487i	$-0.859751\pi$
$971$	−4.19989e31	−1.80899	−0.904494	+	0.426487i	$0.859751\pi$
$972$	1.48560e30	0.0633002
$973$	2.49401e31	1.05127
$974$	−1.90475e30	−0.0794272
$975$	0	0
$976$	5.56022e29	0.0226918
$977$	−2.58281e31	−1.04280	−0.521399	−	0.853313i	$-0.674590\pi$
$977$	−2.58281e31	−1.04280	−0.521399	+	0.853313i	$0.674590\pi$
$978$	−2.12564e30	−0.0849045
$979$	−7.18942e30	−0.284103
$980$	0	0
$981$	7.11903e30	0.275357
$982$	2.21371e29	0.00847130
$983$	2.49436e31	0.944382	0.472191	−	0.881496i	$-0.343463\pi$
$983$	2.49436e31	0.944382	0.472191	+	0.881496i	$0.343463\pi$
$984$	3.25863e30	0.122064
$985$	0	0
$986$	−3.14125e30	−0.115185
$987$	6.97189e30	0.252942
$988$	7.07582e30	0.253997
$989$	3.40820e31	1.21050
$990$	0	0
$991$	3.31091e31	1.15126	0.575631	−	0.817710i	$-0.304757\pi$
$991$	3.31091e31	1.15126	0.575631	+	0.817710i	$0.304757\pi$
$992$	1.01306e31	0.348547
$993$	−2.14997e30	−0.0731923
$994$	−8.07564e30	−0.272032
$995$	0	0
$996$	−2.52786e31	−0.833738
$997$	−4.06660e31	−1.32719	−0.663594	−	0.748093i	$-0.730970\pi$
$997$	−4.06660e31	−1.32719	−0.663594	+	0.748093i	$0.730970\pi$
$998$	−5.39731e30	−0.174304
$999$	1.07242e31	0.342709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.l.1.4	yes	8
5.2	odd	4		75.22.b.j.49.8		16
5.3	odd	4		75.22.b.j.49.9		16
5.4	even	2		75.22.a.k.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.k.1.5	✓	8	5.4	even	2
75.22.a.l.1.4	yes	8	1.1	even	1	trivial
75.22.b.j.49.8		16	5.2	odd	4
75.22.b.j.49.9		16	5.3	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)

\(f(q)\)	\(=\)	\(q-166.679 q^{2} -59049.0 q^{3} -2.06937e6 q^{4} +9.84223e6 q^{6} -1.02476e9 q^{7} +6.94472e8 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-166.679 q^{2} -59049.0 q^{3} -2.06937e6 q^{4} +9.84223e6 q^{6} -1.02476e9 q^{7} +6.94472e8 q^{8} +3.48678e9 q^{9} +8.41547e10 q^{11} +1.22194e11 q^{12} -6.78961e11 q^{13} +1.70806e11 q^{14} +4.22403e12 q^{16} +5.16667e12 q^{17} -5.81174e11 q^{18} +5.03610e12 q^{19} +6.05111e13 q^{21} -1.40268e13 q^{22} -3.20250e14 q^{23} -4.10079e13 q^{24} +1.13169e14 q^{26} -2.05891e14 q^{27} +2.12061e15 q^{28} +3.64763e15 q^{29} -4.68906e15 q^{31} -2.16047e15 q^{32} -4.96925e15 q^{33} -8.61176e14 q^{34} -7.21545e15 q^{36} -5.20866e16 q^{37} -8.39412e14 q^{38} +4.00920e16 q^{39} -7.94636e16 q^{41} -1.00859e16 q^{42} -1.06423e17 q^{43} -1.74147e17 q^{44} +5.33790e16 q^{46} +1.15217e17 q^{47} -2.49425e17 q^{48} +4.91590e17 q^{49} -3.05087e17 q^{51} +1.40502e18 q^{52} +2.45853e17 q^{53} +3.43177e16 q^{54} -7.11668e17 q^{56} -2.97376e17 q^{57} -6.07983e17 q^{58} -7.28797e18 q^{59} +1.31633e17 q^{61} +7.81568e17 q^{62} -3.57312e18 q^{63} -8.49833e18 q^{64} +8.28270e17 q^{66} +1.83759e19 q^{67} -1.06918e19 q^{68} +1.89104e19 q^{69} -4.72795e19 q^{71} +2.42147e18 q^{72} -3.44105e19 q^{73} +8.68175e18 q^{74} -1.04215e19 q^{76} -8.62385e19 q^{77} -6.68249e18 q^{78} -2.81422e19 q^{79} +1.21577e19 q^{81} +1.32449e19 q^{82} -2.06872e20 q^{83} -1.25220e20 q^{84} +1.77385e19 q^{86} -2.15389e20 q^{87} +5.84431e19 q^{88} -8.54310e19 q^{89} +6.95773e20 q^{91} +6.62716e20 q^{92} +2.76884e20 q^{93} -1.92042e19 q^{94} +1.27574e20 q^{96} -7.33743e20 q^{97} -8.19378e19 q^{98} +2.93429e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) \) 8 * q + 666 * q^2 - 472392 * q^3 + 10681904 * q^4 - 39326634 * q^6 + 134034472 * q^7 + 3512168028 * q^8 + 27894275208 * q^9 - 14410165968 * q^11 - 630755749296 * q^12 - 328226481176 * q^13 + 4068036669186 * q^14 + 18553054308920 * q^16 - 5718214953936 * q^17 + 2322198411066 * q^18 + 75919698170296 * q^19 - 7914601537128 * q^21 - 426897542691372 * q^22 + 49712781537936 * q^23 - 207390009885372 * q^24 - 126056679642054 * q^26 - 1647129056757192 * q^27 + 4235167126399504 * q^28 + 2111332005818352 * q^29 + 5677451410844968 * q^31 + 17528957992420632 * q^32 + 850905890244432 * q^33 + 9944754702742068 * q^34 + 37245496240179504 * q^36 - 26538401323409648 * q^37 + 85729618982552526 * q^38 + 19381445486961624 * q^39 - 9719778050982432 * q^41 - 240213497278764114 * q^42 - 20888744714565080 * q^43 - 388904820117856128 * q^44 - 156886987299808452 * q^46 + 405141072730526160 * q^47 - 1095539303887417080 * q^48 + 1172899395436368720 * q^49 + 337654874814966864 * q^51 - 2264281219115894096 * q^52 + 1890950875516930176 * q^53 - 137123493975036234 * q^54 + 19720793170524660300 * q^56 - 4482982257257808504 * q^57 + 12636224332772065092 * q^58 - 3496737905134765776 * q^59 + 17508733151770015960 * q^61 - 30404101355896200606 * q^62 + 467349306165871272 * q^63 + 27809823573720908096 * q^64 + 25207872998382825228 * q^66 - 29911607071855004216 * q^67 - 29571773973515759808 * q^68 - 2935490037033582864 * q^69 + 52204030316256846816 * q^71 + 12246172693721331228 * q^72 - 39697032757192143344 * q^73 - 32301811215943606764 * q^74 + 383913379529874070384 * q^76 - 202348742072764877136 * q^77 + 7443520876183646646 * q^78 - 274125739348498990400 * q^79 + 97261323672455430408 * q^81 - 231746572148446490256 * q^82 - 276883976086286712912 * q^83 - 250082383646764311696 * q^84 + 486708932755841131242 * q^86 - 124672043611567867248 * q^87 - 2078122304879724561672 * q^88 - 319129684977517046784 * q^89 + 45068531423660000648 * q^91 + 147706203809955886272 * q^92 - 335247828358984515432 * q^93 - 1499781052151613424236 * q^94 - 1035067440494445898968 * q^96 + 451807368184588489864 * q^97 - 2604193962467412900204 * q^98 - 50245141913043465168 * q^99

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