LMFDB - Embedded newform 75.22.a.l.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:	$0$
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			1.3
Root			$-1237.86$ 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)

$f(q)$	$=$	$q-1154.86 q^{2} -59049.0 q^{3} -763454. q^{4} +6.81932e7 q^{6} -5.42271e7 q^{7} +3.30359e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-1154.86 q^{2} -59049.0 q^{3} -763454. q^{4} +6.81932e7 q^{6} -5.42271e7 q^{7} +3.30359e9 q^{8} +3.48678e9 q^{9} -9.88393e10 q^{11} +4.50812e10 q^{12} -3.35334e10 q^{13} +6.26246e10 q^{14} -2.21411e12 q^{16} -1.60047e13 q^{17} -4.02674e12 q^{18} -1.23391e13 q^{19} +3.20206e12 q^{21} +1.14145e14 q^{22} +3.49309e14 q^{23} -1.95074e14 q^{24} +3.87263e13 q^{26} -2.05891e14 q^{27} +4.13999e13 q^{28} +9.02810e14 q^{29} -3.33235e15 q^{31} -4.37116e15 q^{32} +5.83636e15 q^{33} +1.84831e16 q^{34} -2.66200e15 q^{36} +4.76742e16 q^{37} +1.42499e16 q^{38} +1.98011e15 q^{39} +7.62667e16 q^{41} -3.69792e15 q^{42} -1.86047e17 q^{43} +7.54593e16 q^{44} -4.03403e17 q^{46} -1.66048e17 q^{47} +1.30741e17 q^{48} -5.55605e17 q^{49} +9.45060e17 q^{51} +2.56012e16 q^{52} -1.04319e18 q^{53} +2.37775e17 q^{54} -1.79144e17 q^{56} +7.28610e17 q^{57} -1.04262e18 q^{58} -1.49458e18 q^{59} -1.01457e18 q^{61} +3.84839e18 q^{62} -1.89078e17 q^{63} +9.69139e18 q^{64} -6.74017e18 q^{66} -1.39620e19 q^{67} +1.22188e19 q^{68} -2.06264e19 q^{69} -1.39714e19 q^{71} +1.15189e19 q^{72} -3.21010e19 q^{73} -5.50570e19 q^{74} +9.42032e18 q^{76} +5.35977e18 q^{77} -2.28675e18 q^{78} +5.59225e19 q^{79} +1.21577e19 q^{81} -8.80772e19 q^{82} -1.95969e20 q^{83} -2.44462e18 q^{84} +2.14858e20 q^{86} -5.33100e19 q^{87} -3.26525e20 q^{88} +4.40698e18 q^{89} +1.81842e18 q^{91} -2.66682e20 q^{92} +1.96772e20 q^{93} +1.91762e20 q^{94} +2.58113e20 q^{96} +6.20950e20 q^{97} +6.41645e20 q^{98} -3.44631e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1154.86	−0.797469	−0.398734	−	0.917066i	$-0.630550\pi$
$2$	−1154.86	−0.797469	−0.398734	+	0.917066i	$0.630550\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−763454.	−0.364043
$5$	0	0
$5$	0	0
$6$	6.81932e7	0.460419
$7$	−5.42271e7	−0.0725583	−0.0362791	−	0.999342i	$-0.511551\pi$
$7$	−5.42271e7	−0.0725583	−0.0362791	+	0.999342i	$0.511551\pi$
$8$	3.30359e9	1.08778
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−9.88393e10	−1.14896	−0.574482	−	0.818517i	$-0.694797\pi$
$11$	−9.88393e10	−1.14896	−0.574482	+	0.818517i	$0.694797\pi$
$12$	4.50812e10	0.210180
$13$	−3.35334e10	−0.0674640	−0.0337320	−	0.999431i	$-0.510739\pi$
$13$	−3.35334e10	−0.0674640	−0.0337320	+	0.999431i	$0.510739\pi$
$14$	6.26246e10	0.0578630
$15$	0	0
$16$	−2.21411e12	−0.503429
$17$	−1.60047e13	−1.92545	−0.962727	−	0.270476i	$-0.912819\pi$
$17$	−1.60047e13	−1.92545	−0.962727	+	0.270476i	$0.912819\pi$
$18$	−4.02674e12	−0.265823
$19$	−1.23391e13	−0.461711	−0.230855	−	0.972988i	$-0.574152\pi$
$19$	−1.23391e13	−0.461711	−0.230855	+	0.972988i	$0.574152\pi$
$20$	0	0
$21$	3.20206e12	0.0418916
$22$	1.14145e14	0.916263
$23$	3.49309e14	1.75820	0.879100	−	0.476638i	$-0.158145\pi$
$23$	3.49309e14	1.75820	0.879100	+	0.476638i	$0.158145\pi$
$24$	−1.95074e14	−0.628031
$25$	0	0
$26$	3.87263e13	0.0538005
$27$	−2.05891e14	−0.192450
$28$	4.13999e13	0.0264144
$29$	9.02810e14	0.398490	0.199245	−	0.979950i	$-0.436151\pi$
$29$	9.02810e14	0.398490	0.199245	+	0.979950i	$0.436151\pi$
$30$	0	0
$31$	−3.33235e15	−0.730218	−0.365109	−	0.930965i	$-0.618968\pi$
$31$	−3.33235e15	−0.730218	−0.365109	+	0.930965i	$0.618968\pi$
$32$	−4.37116e15	−0.686313
$33$	5.83636e15	0.663355
$34$	1.84831e16	1.53549
$35$	0	0
$36$	−2.66200e15	−0.121348
$37$	4.76742e16	1.62992	0.814958	−	0.579520i	$-0.196760\pi$
$37$	4.76742e16	1.62992	0.814958	+	0.579520i	$0.196760\pi$
$38$	1.42499e16	0.368200
$39$	1.98011e15	0.0389504
$40$	0	0
$41$	7.62667e16	0.887369	0.443684	−	0.896183i	$-0.353671\pi$
$41$	7.62667e16	0.887369	0.443684	+	0.896183i	$0.353671\pi$
$42$	−3.69792e15	−0.0334072
$43$	−1.86047e17	−1.31282	−0.656408	−	0.754406i	$-0.727925\pi$
$43$	−1.86047e17	−1.31282	−0.656408	+	0.754406i	$0.727925\pi$
$44$	7.54593e16	0.418273
$45$	0	0
$46$	−4.03403e17	−1.40211
$47$	−1.66048e17	−0.460474	−0.230237	−	0.973135i	$-0.573950\pi$
$47$	−1.66048e17	−0.460474	−0.230237	+	0.973135i	$0.573950\pi$
$48$	1.30741e17	0.290655
$49$	−5.55605e17	−0.994735
$50$	0	0
$51$	9.45060e17	1.11166
$52$	2.56012e16	0.0245598
$53$	−1.04319e18	−0.819345	−0.409673	−	0.912233i	$-0.634357\pi$
$53$	−1.04319e18	−0.819345	−0.409673	+	0.912233i	$0.634357\pi$
$54$	2.37775e17	0.153473
$55$	0	0
$56$	−1.79144e17	−0.0789276
$57$	7.28610e17	0.266569
$58$	−1.04262e18	−0.317783
$59$	−1.49458e18	−0.380691	−0.190345	−	0.981717i	$-0.560961\pi$
$59$	−1.49458e18	−0.380691	−0.190345	+	0.981717i	$0.560961\pi$
$60$	0	0
$61$	−1.01457e18	−0.182105	−0.0910523	−	0.995846i	$-0.529023\pi$
$61$	−1.01457e18	−0.182105	−0.0910523	+	0.995846i	$0.529023\pi$
$62$	3.84839e18	0.582326
$63$	−1.89078e17	−0.0241861
$64$	9.69139e18	1.05074
$65$	0	0
$66$	−6.74017e18	−0.529005
$67$	−1.39620e19	−0.935754	−0.467877	−	0.883794i	$-0.654981\pi$
$67$	−1.39620e19	−0.935754	−0.467877	+	0.883794i	$0.654981\pi$
$68$	1.22188e19	0.700948
$69$	−2.06264e19	−1.01510
$70$	0	0
$71$	−1.39714e19	−0.509362	−0.254681	−	0.967025i	$-0.581970\pi$
$71$	−1.39714e19	−0.509362	−0.254681	+	0.967025i	$0.581970\pi$
$72$	1.15189e19	0.362594
$73$	−3.21010e19	−0.874235	−0.437118	−	0.899404i	$-0.644001\pi$
$73$	−3.21010e19	−0.874235	−0.437118	+	0.899404i	$0.644001\pi$
$74$	−5.50570e19	−1.29981
$75$	0	0
$76$	9.42032e18	0.168083
$77$	5.35977e18	0.0833669
$78$	−2.28675e18	−0.0310617
$79$	5.59225e19	0.664511	0.332255	−	0.943189i	$-0.392190\pi$
$79$	5.59225e19	0.664511	0.332255	+	0.943189i	$0.392190\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−8.80772e19	−0.707649
$83$	−1.95969e20	−1.38633	−0.693167	−	0.720777i	$-0.743785\pi$
$83$	−1.95969e20	−1.38633	−0.693167	+	0.720777i	$0.743785\pi$
$84$	−2.44462e18	−0.0152503
$85$	0	0
$86$	2.14858e20	1.04693
$87$	−5.33100e19	−0.230068
$88$	−3.26525e20	−1.24982
$89$	4.40698e18	0.0149812	0.00749059	−	0.999972i	$-0.497616\pi$
$89$	4.40698e18	0.0149812	0.00749059	+	0.999972i	$0.497616\pi$
$90$	0	0
$91$	1.81842e18	0.00489508
$92$	−2.66682e20	−0.640061
$93$	1.96772e20	0.421591
$94$	1.91762e20	0.367214
$95$	0	0
$96$	2.58113e20	0.396243
$97$	6.20950e20	0.854975	0.427487	−	0.904021i	$-0.359399\pi$
$97$	6.20950e20	0.854975	0.427487	+	0.904021i	$0.359399\pi$
$98$	6.41645e20	0.793271
$99$	−3.44631e20	−0.382988
$100$	0	0
$101$	−1.06357e21	−0.958054	−0.479027	−	0.877800i	$-0.659010\pi$
$101$	−1.06357e21	−0.958054	−0.479027	+	0.877800i	$0.659010\pi$
$102$	−1.09141e21	−0.886515
$103$	−8.58566e20	−0.629481	−0.314741	−	0.949178i	$-0.601918\pi$
$103$	−8.58566e20	−0.629481	−0.314741	+	0.949178i	$0.601918\pi$
$104$	−1.10781e20	−0.0733862
$105$	0	0
$106$	1.20474e21	0.653402
$107$	−3.42950e21	−1.68539	−0.842695	−	0.538391i	$-0.819032\pi$
$107$	−3.42950e21	−1.68539	−0.842695	+	0.538391i	$0.819032\pi$
$108$	1.57188e20	0.0700602
$109$	2.39482e21	0.968935	0.484467	−	0.874809i	$-0.339013\pi$
$109$	2.39482e21	0.968935	0.484467	+	0.874809i	$0.339013\pi$
$110$	0	0
$111$	−2.81511e21	−0.941033
$112$	1.20065e20	0.0365280
$113$	−2.79687e21	−0.775084	−0.387542	−	0.921852i	$-0.626676\pi$
$113$	−2.79687e21	−0.775084	−0.387542	+	0.921852i	$0.626676\pi$
$114$	−8.41442e20	−0.212580
$115$	0	0
$116$	−6.89254e20	−0.145068
$117$	−1.16924e20	−0.0224880
$118$	1.72603e21	0.303589
$119$	8.67887e20	0.139708
$120$	0	0
$121$	2.36896e21	0.320119
$122$	1.17169e21	0.145223
$123$	−4.50347e21	−0.512323
$124$	2.54409e21	0.265831
$125$	0	0
$126$	2.18359e20	0.0192877
$127$	−1.65628e22	−1.34646	−0.673232	−	0.739431i	$-0.735095\pi$
$127$	−1.65628e22	−1.34646	−0.673232	+	0.739431i	$0.735095\pi$
$128$	−2.02519e21	−0.151622
$129$	1.09859e22	0.757955
$130$	0	0
$131$	−2.08415e22	−1.22344	−0.611718	−	0.791076i	$-0.709521\pi$
$131$	−2.08415e22	−1.22344	−0.611718	+	0.791076i	$0.709521\pi$
$132$	−4.45580e21	−0.241490
$133$	6.69113e20	0.0335009
$134$	1.61241e22	0.746235
$135$	0	0
$136$	−5.28729e22	−2.09447
$137$	−7.17560e21	−0.263204	−0.131602	−	0.991303i	$-0.542012\pi$
$137$	−7.17560e21	−0.263204	−0.131602	+	0.991303i	$0.542012\pi$
$138$	2.38205e22	0.809508
$139$	2.49666e22	0.786511	0.393255	−	0.919429i	$-0.371349\pi$
$139$	2.49666e22	0.786511	0.393255	+	0.919429i	$0.371349\pi$
$140$	0	0
$141$	9.80495e21	0.265855
$142$	1.61350e22	0.406200
$143$	3.31442e21	0.0775138
$144$	−7.72011e21	−0.167810
$145$	0	0
$146$	3.70721e22	0.697176
$147$	3.28079e22	0.574311
$148$	−3.63971e22	−0.593360
$149$	5.63800e22	0.856386	0.428193	−	0.903687i	$-0.359150\pi$
$149$	5.63800e22	0.856386	0.428193	+	0.903687i	$0.359150\pi$
$150$	0	0
$151$	−4.78403e22	−0.631736	−0.315868	−	0.948803i	$-0.602296\pi$
$151$	−4.78403e22	−0.631736	−0.315868	+	0.948803i	$0.602296\pi$
$152$	−4.07633e22	−0.502241
$153$	−5.58048e22	−0.641818
$154$	−6.18978e21	−0.0664825
$155$	0	0
$156$	−1.51173e21	−0.0141796
$157$	−2.18796e23	−1.91908	−0.959540	−	0.281572i	$-0.909144\pi$
$157$	−2.18796e23	−1.91908	−0.959540	+	0.281572i	$0.909144\pi$
$158$	−6.45825e22	−0.529927
$159$	6.15993e22	0.473049
$160$	0	0
$161$	−1.89420e22	−0.127572
$162$	−1.40404e22	−0.0886077
$163$	1.51641e23	0.897114	0.448557	−	0.893754i	$-0.351938\pi$
$163$	1.51641e23	0.897114	0.448557	+	0.893754i	$0.351938\pi$
$164$	−5.82261e22	−0.323041
$165$	0	0
$166$	2.26317e23	1.10556
$167$	5.39750e22	0.247554	0.123777	−	0.992310i	$-0.460499\pi$
$167$	5.39750e22	0.247554	0.123777	+	0.992310i	$0.460499\pi$
$168$	1.05783e22	0.0455689
$169$	−2.45940e23	−0.995449
$170$	0	0
$171$	−4.30237e22	−0.153904
$172$	1.42038e23	0.477922
$173$	−3.76178e23	−1.19099	−0.595496	−	0.803358i	$-0.703045\pi$
$173$	−3.76178e23	−1.19099	−0.595496	+	0.803358i	$0.703045\pi$
$174$	6.15655e22	0.183472
$175$	0	0
$176$	2.18841e23	0.578422
$177$	8.82533e22	0.219792
$178$	−5.08944e21	−0.0119470
$179$	1.10081e23	0.243643	0.121822	−	0.992552i	$-0.461126\pi$
$179$	1.10081e23	0.243643	0.121822	+	0.992552i	$0.461126\pi$
$180$	0	0
$181$	1.38729e23	0.273238	0.136619	−	0.990624i	$-0.456376\pi$
$181$	1.38729e23	0.273238	0.136619	+	0.990624i	$0.456376\pi$
$182$	−2.10002e21	−0.00390367
$183$	5.99096e22	0.105138
$184$	1.15398e24	1.91254
$185$	0	0
$186$	−2.27243e23	−0.336206
$187$	1.58189e24	2.21228
$188$	1.26770e23	0.167633
$189$	1.11649e22	0.0139639
$190$	0	0
$191$	6.08029e23	0.680885	0.340443	−	0.940265i	$-0.389423\pi$
$191$	6.08029e23	0.680885	0.340443	+	0.940265i	$0.389423\pi$
$192$	−5.72267e23	−0.606646
$193$	−1.31797e24	−1.32299	−0.661493	−	0.749951i	$-0.730077\pi$
$193$	−1.31797e24	−1.32299	−0.661493	+	0.749951i	$0.730077\pi$
$194$	−7.17109e23	−0.681816
$195$	0	0
$196$	4.24179e23	0.362127
$197$	6.55437e23	0.530439	0.265220	−	0.964188i	$-0.414556\pi$
$197$	6.55437e23	0.530439	0.265220	+	0.964188i	$0.414556\pi$
$198$	3.98000e23	0.305421
$199$	−1.28284e24	−0.933718	−0.466859	−	0.884332i	$-0.654615\pi$
$199$	−1.28284e24	−0.933718	−0.466859	+	0.884332i	$0.654615\pi$
$200$	0	0
$201$	8.24441e23	0.540258
$202$	1.22827e24	0.764018
$203$	−4.89568e22	−0.0289138
$204$	−7.21510e23	−0.404693
$205$	0	0
$206$	9.91522e23	0.501992
$207$	1.21797e24	0.586066
$208$	7.42465e22	0.0339634
$209$	1.21959e24	0.530489
$210$	0	0
$211$	−6.46432e23	−0.254423	−0.127212	−	0.991876i	$-0.540603\pi$
$211$	−6.46432e23	−0.254423	−0.127212	+	0.991876i	$0.540603\pi$
$212$	7.96428e23	0.298277
$213$	8.24996e23	0.294080
$214$	3.96058e24	1.34405
$215$	0	0
$216$	−6.80181e23	−0.209344
$217$	1.80704e23	0.0529834
$218$	−2.76568e24	−0.772695
$219$	1.89553e24	0.504740
$220$	0	0
$221$	5.36691e23	0.129899
$222$	3.25106e24	0.750444
$223$	2.49752e24	0.549930	0.274965	−	0.961454i	$-0.411334\pi$
$223$	2.49752e24	0.549930	0.274965	+	0.961454i	$0.411334\pi$
$224$	2.37036e23	0.0497977
$225$	0	0
$226$	3.22999e24	0.618106
$227$	9.79134e24	1.78884	0.894418	−	0.447232i	$-0.147590\pi$
$227$	9.79134e24	1.78884	0.894418	+	0.447232i	$0.147590\pi$
$228$	−5.56260e23	−0.0970426
$229$	9.64103e24	1.60639	0.803195	−	0.595716i	$-0.203132\pi$
$229$	9.64103e24	1.60639	0.803195	+	0.595716i	$0.203132\pi$
$230$	0	0
$231$	−3.16489e23	−0.0481319
$232$	2.98252e24	0.433470
$233$	−4.53906e23	−0.0630564	−0.0315282	−	0.999503i	$-0.510037\pi$
$233$	−4.53906e23	−0.0630564	−0.0315282	+	0.999503i	$0.510037\pi$
$234$	1.35030e23	0.0179335
$235$	0	0
$236$	1.14104e24	0.138588
$237$	−3.30217e24	−0.383655
$238$	−1.00229e24	−0.111412
$239$	1.44716e25	1.53936	0.769678	−	0.638432i	$-0.220417\pi$
$239$	1.44716e25	1.53936	0.769678	+	0.638432i	$0.220417\pi$
$240$	0	0
$241$	1.14181e24	0.111280	0.0556399	−	0.998451i	$-0.482280\pi$
$241$	1.14181e24	0.111280	0.0556399	+	0.998451i	$0.482280\pi$
$242$	−2.73582e24	−0.255285
$243$	−7.17898e23	−0.0641500
$244$	7.74581e23	0.0662939
$245$	0	0
$246$	5.20087e24	0.408561
$247$	4.13771e23	0.0311489
$248$	−1.10087e25	−0.794318
$249$	1.15718e25	0.800401
$250$	0	0
$251$	2.09389e25	1.33162	0.665810	−	0.746121i	$-0.268086\pi$
$251$	2.09389e25	1.33162	0.665810	+	0.746121i	$0.268086\pi$
$252$	1.44353e23	0.00880479
$253$	−3.45255e25	−2.02011
$254$	1.91277e25	1.07376
$255$	0	0
$256$	−1.79855e25	−0.929829
$257$	−1.80282e25	−0.894653	−0.447326	−	0.894371i	$-0.647624\pi$
$257$	−1.80282e25	−0.894653	−0.447326	+	0.894371i	$0.647624\pi$
$258$	−1.26871e25	−0.604445
$259$	−2.58523e24	−0.118264
$260$	0	0
$261$	3.14790e24	0.132830
$262$	2.40690e25	0.975652
$263$	−2.96509e25	−1.15479	−0.577395	−	0.816465i	$-0.695931\pi$
$263$	−2.96509e25	−1.15479	−0.577395	+	0.816465i	$0.695931\pi$
$264$	1.92810e25	0.721586
$265$	0	0
$266$	−7.72730e23	−0.0267160
$267$	−2.60228e23	−0.00864939
$268$	1.06593e25	0.340655
$269$	5.52895e25	1.69920	0.849599	−	0.527430i	$-0.176844\pi$
$269$	5.52895e25	1.69920	0.849599	+	0.527430i	$0.176844\pi$
$270$	0	0
$271$	2.65005e25	0.753490	0.376745	−	0.926317i	$-0.377043\pi$
$271$	2.65005e25	0.753490	0.376745	+	0.926317i	$0.377043\pi$
$272$	3.54360e25	0.969329
$273$	−1.07376e23	−0.00282617
$274$	8.28680e24	0.209897
$275$	0	0
$276$	1.57473e25	0.369539
$277$	−1.15811e25	−0.261645	−0.130822	−	0.991406i	$-0.541762\pi$
$277$	−1.15811e25	−0.261645	−0.130822	+	0.991406i	$0.541762\pi$
$278$	−2.88329e25	−0.627218
$279$	−1.16192e25	−0.243406
$280$	0	0
$281$	−7.84037e25	−1.52377	−0.761887	−	0.647710i	$-0.775727\pi$
$281$	−7.84037e25	−1.52377	−0.761887	+	0.647710i	$0.775727\pi$
$282$	−1.13233e25	−0.212011
$283$	−7.91685e25	−1.42822	−0.714109	−	0.700034i	$-0.753168\pi$
$283$	−7.91685e25	−1.42822	−0.714109	+	0.700034i	$0.753168\pi$
$284$	1.06665e25	0.185430
$285$	0	0
$286$	−3.82768e24	−0.0618148
$287$	−4.13572e24	−0.0643860
$288$	−1.52413e25	−0.228771
$289$	1.87058e26	2.70737
$290$	0	0
$291$	−3.66665e25	−0.493620
$292$	2.45076e25	0.318260
$293$	4.94119e25	0.619043	0.309522	−	0.950892i	$-0.399831\pi$
$293$	4.94119e25	0.619043	0.309522	+	0.950892i	$0.399831\pi$
$294$	−3.78885e25	−0.457995
$295$	0	0
$296$	1.57496e26	1.77299
$297$	2.03501e25	0.221118
$298$	−6.51109e25	−0.682941
$299$	−1.17135e25	−0.118615
$300$	0	0
$301$	1.00888e25	0.0952557
$302$	5.52488e25	0.503790
$303$	6.28025e25	0.553132
$304$	2.73200e25	0.232439
$305$	0	0
$306$	6.44467e25	0.511830
$307$	1.81029e26	1.38930	0.694648	−	0.719350i	$-0.255560\pi$
$307$	1.81029e26	1.38930	0.694648	+	0.719350i	$0.255560\pi$
$308$	−4.09194e24	−0.0303492
$309$	5.06975e25	0.363431
$310$	0	0
$311$	1.12773e26	0.755473	0.377737	−	0.925913i	$-0.376702\pi$
$311$	1.12773e26	0.755473	0.377737	+	0.925913i	$0.376702\pi$
$312$	6.54149e24	0.0423695
$313$	−2.07181e26	−1.29758	−0.648790	−	0.760967i	$-0.724725\pi$
$313$	−2.07181e26	−1.29758	−0.648790	+	0.760967i	$0.724725\pi$
$314$	2.52678e26	1.53041
$315$	0	0
$316$	−4.26942e25	−0.241911
$317$	1.12129e26	0.614604	0.307302	−	0.951612i	$-0.400574\pi$
$317$	1.12129e26	0.614604	0.307302	+	0.951612i	$0.400574\pi$
$318$	−7.11385e25	−0.377242
$319$	−8.92331e25	−0.457851
$320$	0	0
$321$	2.02508e26	0.973061
$322$	2.18754e25	0.101735
$323$	1.97483e26	0.889002
$324$	−9.28182e24	−0.0404493
$325$	0	0
$326$	−1.75124e26	−0.715421
$327$	−1.41412e26	−0.559415
$328$	2.51954e26	0.965264
$329$	9.00429e24	0.0334112
$330$	0	0
$331$	−3.52788e26	−1.22834	−0.614172	−	0.789172i	$-0.710510\pi$
$331$	−3.52788e26	−1.22834	−0.614172	+	0.789172i	$0.710510\pi$
$332$	1.49614e26	0.504686
$333$	1.66230e26	0.543306
$334$	−6.23335e25	−0.197417
$335$	0	0
$336$	−7.08969e24	−0.0210894
$337$	3.81011e26	1.09856	0.549279	−	0.835639i	$-0.314902\pi$
$337$	3.81011e26	1.09856	0.549279	+	0.835639i	$0.314902\pi$
$338$	2.84026e26	0.793839
$339$	1.65152e26	0.447495
$340$	0	0
$341$	3.29367e26	0.838994
$342$	4.96863e25	0.122733
$343$	6.04172e25	0.144735
$344$	−6.14624e26	−1.42806
$345$	0	0
$346$	4.34432e26	0.949780
$347$	−3.94952e26	−0.837692	−0.418846	−	0.908057i	$-0.637565\pi$
$347$	−3.94952e26	−0.837692	−0.418846	+	0.908057i	$0.637565\pi$
$348$	4.06998e25	0.0837548
$349$	−9.12481e26	−1.82203	−0.911017	−	0.412368i	$-0.864702\pi$
$349$	−9.12481e26	−1.82203	−0.911017	+	0.412368i	$0.864702\pi$
$350$	0	0
$351$	6.90423e24	0.0129835
$352$	4.32043e26	0.788549
$353$	−3.95907e26	−0.701388	−0.350694	−	0.936490i	$-0.614054\pi$
$353$	−3.95907e26	−0.701388	−0.350694	+	0.936490i	$0.614054\pi$
$354$	−1.01920e26	−0.175277
$355$	0	0
$356$	−3.36453e24	−0.00545380
$357$	−5.12479e25	−0.0806602
$358$	−1.27128e26	−0.194298
$359$	9.83596e26	1.45991	0.729953	−	0.683497i	$-0.239542\pi$
$359$	9.83596e26	1.45991	0.729953	+	0.683497i	$0.239542\pi$
$360$	0	0
$361$	−5.61957e26	−0.786823
$362$	−1.60212e26	−0.217899
$363$	−1.39885e26	−0.184821
$364$	−1.38828e24	−0.00178202
$365$	0	0
$366$	−6.91871e25	−0.0838444
$367$	−9.56476e26	−1.12637	−0.563184	−	0.826331i	$-0.690424\pi$
$367$	−9.56476e26	−1.12637	−0.563184	+	0.826331i	$0.690424\pi$
$368$	−7.73408e26	−0.885129
$369$	2.65925e26	0.295790
$370$	0	0
$371$	5.65692e25	0.0594503
$372$	−1.50226e26	−0.153478
$373$	−2.93918e26	−0.291933	−0.145967	−	0.989290i	$-0.546629\pi$
$373$	−2.93918e26	−0.291933	−0.145967	+	0.989290i	$0.546629\pi$
$374$	−1.82686e27	−1.76422
$375$	0	0
$376$	−5.48555e26	−0.500896
$377$	−3.02743e25	−0.0268837
$378$	−1.28939e25	−0.0111357
$379$	−1.77322e27	−1.48954	−0.744769	−	0.667323i	$-0.767440\pi$
$379$	−1.77322e27	−1.48954	−0.744769	+	0.667323i	$0.767440\pi$
$380$	0	0
$381$	9.78018e26	0.777382
$382$	−7.02187e26	−0.542985
$383$	1.95770e27	1.47285	0.736427	−	0.676517i	$-0.236511\pi$
$383$	1.95770e27	1.47285	0.736427	+	0.676517i	$0.236511\pi$
$384$	1.19585e26	0.0875387
$385$	0	0
$386$	1.52207e27	1.05504
$387$	−6.48706e26	−0.437605
$388$	−4.74067e26	−0.311248
$389$	−4.47898e26	−0.286225	−0.143113	−	0.989706i	$-0.545711\pi$
$389$	−4.47898e26	−0.286225	−0.143113	+	0.989706i	$0.545711\pi$
$390$	0	0
$391$	−5.59058e27	−3.38533
$392$	−1.83549e27	−1.08206
$393$	1.23067e27	0.706351
$394$	−7.56937e26	−0.423009
$395$	0	0
$396$	2.63110e26	0.139424
$397$	−3.07981e26	−0.158936	−0.0794682	−	0.996837i	$-0.525322\pi$
$397$	−3.07981e26	−0.158936	−0.0794682	+	0.996837i	$0.525322\pi$
$398$	1.48150e27	0.744611
$399$	−3.95104e25	−0.0193418
$400$	0	0
$401$	−1.66508e27	−0.773426	−0.386713	−	0.922200i	$-0.626390\pi$
$401$	−1.66508e27	−0.773426	−0.386713	+	0.922200i	$0.626390\pi$
$402$	−9.52113e26	−0.430839
$403$	1.11745e26	0.0492634
$404$	8.11983e26	0.348773
$405$	0	0
$406$	5.65381e25	0.0230578
$407$	−4.71209e27	−1.87272
$408$	3.12209e27	1.20925
$409$	3.42658e27	1.29350	0.646749	−	0.762703i	$-0.276128\pi$
$409$	3.42658e27	1.29350	0.646749	+	0.762703i	$0.276128\pi$
$410$	0	0
$411$	4.23712e26	0.151961
$412$	6.55476e26	0.229158
$413$	8.10466e25	0.0276223
$414$	−1.40658e27	−0.467370
$415$	0	0
$416$	1.46580e26	0.0463015
$417$	−1.47426e27	−0.454092
$418$	−1.40845e27	−0.423049
$419$	−5.73314e27	−1.67937	−0.839683	−	0.543077i	$-0.817259\pi$
$419$	−5.73314e27	−1.67937	−0.839683	+	0.543077i	$0.817259\pi$
$420$	0	0
$421$	−2.33038e26	−0.0649330	−0.0324665	−	0.999473i	$-0.510336\pi$
$421$	−2.33038e26	−0.0649330	−0.0324665	+	0.999473i	$0.510336\pi$
$422$	7.46538e26	0.202895
$423$	−5.78973e26	−0.153491
$424$	−3.44628e27	−0.891269
$425$	0	0
$426$	−9.52753e26	−0.234520
$427$	5.50175e25	0.0132132
$428$	2.61826e27	0.613555
$429$	−1.95713e26	−0.0447526
$430$	0	0
$431$	3.70136e26	0.0806028	0.0403014	−	0.999188i	$-0.487168\pi$
$431$	3.70136e26	0.0806028	0.0403014	+	0.999188i	$0.487168\pi$
$432$	4.55865e26	0.0968850
$433$	3.01062e27	0.624501	0.312251	−	0.950000i	$-0.398917\pi$
$433$	3.01062e27	0.624501	0.312251	+	0.950000i	$0.398917\pi$
$434$	−2.08687e26	−0.0422526
$435$	0	0
$436$	−1.82833e27	−0.352734
$437$	−4.31016e27	−0.811779
$438$	−2.18907e27	−0.402515
$439$	−2.48833e27	−0.446714	−0.223357	−	0.974737i	$-0.571702\pi$
$439$	−2.48833e27	−0.446714	−0.223357	+	0.974737i	$0.571702\pi$
$440$	0	0
$441$	−1.93728e27	−0.331578
$442$	−6.19802e26	−0.103590
$443$	5.05509e27	0.825069	0.412534	−	0.910942i	$-0.364644\pi$
$443$	5.05509e27	0.825069	0.412534	+	0.910942i	$0.364644\pi$
$444$	2.14921e27	0.342577
$445$	0	0
$446$	−2.88428e27	−0.438552
$447$	−3.32918e27	−0.494435
$448$	−5.25536e26	−0.0762401
$449$	4.22145e27	0.598240	0.299120	−	0.954216i	$-0.403307\pi$
$449$	4.22145e27	0.598240	0.299120	+	0.954216i	$0.403307\pi$
$450$	0	0
$451$	−7.53814e27	−1.01956
$452$	2.13528e27	0.282164
$453$	2.82492e27	0.364733
$454$	−1.13076e28	−1.42654
$455$	0	0
$456$	2.40703e27	0.289969
$457$	1.25255e28	1.47460	0.737299	−	0.675567i	$-0.236101\pi$
$457$	1.25255e28	1.47460	0.737299	+	0.675567i	$0.236101\pi$
$458$	−1.11340e28	−1.28105
$459$	3.29522e27	0.370554
$460$	0	0
$461$	1.17590e28	1.26331	0.631655	−	0.775250i	$-0.282376\pi$
$461$	1.17590e28	1.26331	0.631655	+	0.775250i	$0.282376\pi$
$462$	3.65500e26	0.0383837
$463$	3.41343e27	0.350421	0.175210	−	0.984531i	$-0.443939\pi$
$463$	3.41343e27	0.350421	0.175210	+	0.984531i	$0.443939\pi$
$464$	−1.99892e27	−0.200611
$465$	0	0
$466$	5.24198e26	0.0502855
$467$	1.81244e28	1.69995	0.849975	−	0.526823i	$-0.176617\pi$
$467$	1.81244e28	1.69995	0.849975	+	0.526823i	$0.176617\pi$
$468$	8.92659e25	0.00818661
$469$	7.57118e26	0.0678967
$470$	0	0
$471$	1.29197e28	1.10798
$472$	−4.93748e27	−0.414109
$473$	1.83888e28	1.50838
$474$	3.81353e27	0.305953
$475$	0	0
$476$	−6.62592e26	−0.0508596
$477$	−3.63738e27	−0.273115
$478$	−1.67127e28	−1.22759
$479$	−2.72034e27	−0.195479	−0.0977397	−	0.995212i	$-0.531161\pi$
$479$	−2.72034e27	−0.195479	−0.0977397	+	0.995212i	$0.531161\pi$
$480$	0	0
$481$	−1.59868e27	−0.109961
$482$	−1.31863e27	−0.0887422
$483$	1.11851e27	0.0736537
$484$	−1.80859e27	−0.116537
$485$	0	0
$486$	8.29071e26	0.0511577
$487$	8.83077e27	0.533267	0.266633	−	0.963798i	$-0.414089\pi$
$487$	8.83077e27	0.533267	0.266633	+	0.963798i	$0.414089\pi$
$488$	−3.35174e27	−0.198090
$489$	−8.95427e27	−0.517949
$490$	0	0
$491$	8.20139e27	0.454498	0.227249	−	0.973837i	$-0.427027\pi$
$491$	8.20139e27	0.454498	0.227249	+	0.973837i	$0.427027\pi$
$492$	3.43819e27	0.186508
$493$	−1.44492e28	−0.767274
$494$	−4.77847e26	−0.0248403
$495$	0	0
$496$	7.37817e27	0.367613
$497$	7.57627e26	0.0369584
$498$	−1.33638e28	−0.638295
$499$	1.79278e26	0.00838437	0.00419218	−	0.999991i	$-0.498666\pi$
$499$	1.79278e26	0.00838437	0.00419218	+	0.999991i	$0.498666\pi$
$500$	0	0
$501$	−3.18717e27	−0.142925
$502$	−2.41815e28	−1.06193
$503$	−2.00700e28	−0.863145	−0.431572	−	0.902078i	$-0.642041\pi$
$503$	−2.00700e28	−0.863145	−0.431572	+	0.902078i	$0.642041\pi$
$504$	−6.24638e26	−0.0263092
$505$	0	0
$506$	3.98721e28	1.61097
$507$	1.45225e28	0.574723
$508$	1.26449e28	0.490172
$509$	1.92770e28	0.731988	0.365994	−	0.930617i	$-0.380729\pi$
$509$	1.92770e28	0.731988	0.365994	+	0.930617i	$0.380729\pi$
$510$	0	0
$511$	1.74074e27	0.0634330
$512$	2.50178e28	0.893131
$513$	2.54051e27	0.0888563
$514$	2.08200e28	0.713458
$515$	0	0
$516$	−8.38722e27	−0.275928
$517$	1.64120e28	0.529069
$518$	2.98558e27	0.0943118
$519$	2.22129e28	0.687620
$520$	0	0
$521$	−2.71956e28	−0.808542	−0.404271	−	0.914639i	$-0.632475\pi$
$521$	−2.71956e28	−0.808542	−0.404271	+	0.914639i	$0.632475\pi$
$522$	−3.63538e27	−0.105928
$523$	1.07341e28	0.306547	0.153273	−	0.988184i	$-0.451018\pi$
$523$	1.07341e28	0.306547	0.153273	+	0.988184i	$0.451018\pi$
$524$	1.59116e28	0.445383
$525$	0	0
$526$	3.42426e28	0.920909
$527$	5.33331e28	1.40600
$528$	−1.29223e28	−0.333952
$529$	8.25455e28	2.09126
$530$	0	0
$531$	−5.21127e27	−0.126897
$532$	−5.10837e26	−0.0121958
$533$	−2.55748e27	−0.0598655
$534$	3.00526e26	0.00689762
$535$	0	0
$536$	−4.61247e28	−1.01790
$537$	−6.50016e27	−0.140668
$538$	−6.38515e28	−1.35506
$539$	5.49157e28	1.14292
$540$	0	0
$541$	1.57220e28	0.314728	0.157364	−	0.987541i	$-0.449700\pi$
$541$	1.57220e28	0.314728	0.157364	+	0.987541i	$0.449700\pi$
$542$	−3.06044e28	−0.600885
$543$	−8.19178e27	−0.157754
$544$	6.99590e28	1.32146
$545$	0	0
$546$	1.24004e26	0.00225379
$547$	8.57716e27	0.152924	0.0764622	−	0.997072i	$-0.475638\pi$
$547$	8.57716e27	0.152924	0.0764622	+	0.997072i	$0.475638\pi$
$548$	5.47824e27	0.0958175
$549$	−3.53760e27	−0.0607015
$550$	0	0
$551$	−1.11398e28	−0.183987
$552$	−6.81412e28	−1.10420
$553$	−3.03251e27	−0.0482158
$554$	1.33745e28	0.208653
$555$	0	0
$556$	−1.90609e28	−0.286324
$557$	1.39112e28	0.205062	0.102531	−	0.994730i	$-0.467306\pi$
$557$	1.39112e28	0.205062	0.102531	+	0.994730i	$0.467306\pi$
$558$	1.34185e28	0.194109
$559$	6.23879e27	0.0885679
$560$	0	0
$561$	−9.34091e28	−1.27726
$562$	9.05452e28	1.21516
$563$	3.60324e28	0.474629	0.237315	−	0.971433i	$-0.423733\pi$
$563$	3.60324e28	0.474629	0.237315	+	0.971433i	$0.423733\pi$
$564$	−7.48563e27	−0.0967827
$565$	0	0
$566$	9.14284e28	1.13896
$567$	−6.59275e26	−0.00806203
$568$	−4.61558e28	−0.554075
$569$	−7.59750e28	−0.895348	−0.447674	−	0.894197i	$-0.647747\pi$
$569$	−7.59750e28	−0.895348	−0.447674	+	0.894197i	$0.647747\pi$
$570$	0	0
$571$	−3.16584e27	−0.0359592	−0.0179796	−	0.999838i	$-0.505723\pi$
$571$	−3.16584e27	−0.0359592	−0.0179796	+	0.999838i	$0.505723\pi$
$572$	−2.53041e27	−0.0282184
$573$	−3.59035e28	−0.393109
$574$	4.77617e27	0.0513458
$575$	0	0
$576$	3.37918e28	0.350248
$577$	−1.53123e29	−1.55845	−0.779227	−	0.626742i	$-0.784388\pi$
$577$	−1.53123e29	−1.55845	−0.779227	+	0.626742i	$0.784388\pi$
$578$	−2.16025e29	−2.15904
$579$	7.78251e28	0.763826
$580$	0	0
$581$	1.06269e28	0.100590
$582$	4.23446e28	0.393647
$583$	1.03108e29	0.941399
$584$	−1.06049e29	−0.950978
$585$	0	0
$586$	−5.70637e28	−0.493668
$587$	1.31923e29	1.12104	0.560519	−	0.828141i	$-0.310602\pi$
$587$	1.31923e29	1.12104	0.560519	+	0.828141i	$0.310602\pi$
$588$	−2.50474e28	−0.209074
$589$	4.11181e28	0.337149
$590$	0	0
$591$	−3.87029e28	−0.306249
$592$	−1.05556e29	−0.820548
$593$	5.00314e28	0.382093	0.191046	−	0.981581i	$-0.438812\pi$
$593$	5.00314e28	0.382093	0.191046	+	0.981581i	$0.438812\pi$
$594$	−2.35015e28	−0.176335
$595$	0	0
$596$	−4.30436e28	−0.311761
$597$	7.57505e28	0.539083
$598$	1.35275e28	0.0945920
$599$	2.24144e29	1.54009	0.770043	−	0.637993i	$-0.220235\pi$
$599$	2.24144e29	1.54009	0.770043	+	0.637993i	$0.220235\pi$
$600$	0	0
$601$	2.91222e29	1.93215	0.966077	−	0.258255i	$-0.0831475\pi$
$601$	2.91222e29	1.93215	0.966077	+	0.258255i	$0.0831475\pi$
$602$	−1.16511e28	−0.0759635
$603$	−4.86824e28	−0.311918
$604$	3.65239e28	0.229979
$605$	0	0
$606$	−7.25280e28	−0.441106
$607$	−2.63905e29	−1.57749	−0.788743	−	0.614723i	$-0.789268\pi$
$607$	−2.63905e29	−1.57749	−0.788743	+	0.614723i	$0.789268\pi$
$608$	5.39361e28	0.316878
$609$	2.89085e27	0.0166934
$610$	0	0
$611$	5.56815e27	0.0310655
$612$	4.26044e28	0.233649
$613$	2.31019e29	1.24541	0.622705	−	0.782457i	$-0.286034\pi$
$613$	2.31019e29	1.24541	0.622705	+	0.782457i	$0.286034\pi$
$614$	−2.09063e29	−1.10792
$615$	0	0
$616$	1.77065e28	0.0906850
$617$	3.69003e29	1.85796	0.928978	−	0.370136i	$-0.120689\pi$
$617$	3.69003e29	1.85796	0.928978	+	0.370136i	$0.120689\pi$
$618$	−5.85484e28	−0.289825
$619$	−1.15989e29	−0.564501	−0.282251	−	0.959341i	$-0.591081\pi$
$619$	−1.15989e29	−0.564501	−0.282251	+	0.959341i	$0.591081\pi$
$620$	0	0
$621$	−7.19197e28	−0.338366
$622$	−1.30236e29	−0.602467
$623$	−2.38978e26	−0.00108701
$624$	−4.38418e27	−0.0196088
$625$	0	0
$626$	2.39265e29	1.03478
$627$	−7.20153e28	−0.306278
$628$	1.67041e29	0.698628
$629$	−7.63010e29	−3.13833
$630$	0	0
$631$	5.06645e27	0.0201556	0.0100778	−	0.999949i	$-0.496792\pi$
$631$	5.06645e27	0.0201556	0.0100778	+	0.999949i	$0.496792\pi$
$632$	1.84745e29	0.722843
$633$	3.81712e28	0.146891
$634$	−1.29493e29	−0.490128
$635$	0	0
$636$	−4.70283e28	−0.172210
$637$	1.86313e28	0.0671089
$638$	1.03052e29	0.365122
$639$	−4.87152e28	−0.169787
$640$	0	0
$641$	9.31381e28	0.314136	0.157068	−	0.987588i	$-0.449796\pi$
$641$	9.31381e28	0.314136	0.157068	+	0.987588i	$0.449796\pi$
$642$	−2.33868e29	−0.775986
$643$	−2.35019e29	−0.767164	−0.383582	−	0.923507i	$-0.625310\pi$
$643$	−2.35019e29	−0.767164	−0.383582	+	0.923507i	$0.625310\pi$
$644$	1.44614e28	0.0464417
$645$	0	0
$646$	−2.28065e29	−0.708952
$647$	−2.34044e29	−0.715817	−0.357909	−	0.933757i	$-0.616510\pi$
$647$	−2.34044e29	−0.715817	−0.357909	+	0.933757i	$0.616510\pi$
$648$	4.01640e28	0.120865
$649$	1.47723e29	0.437400
$650$	0	0
$651$	−1.06704e28	−0.0305900
$652$	−1.15771e29	−0.326588
$653$	4.42897e29	1.22946	0.614730	−	0.788738i	$-0.289265\pi$
$653$	4.42897e29	1.22946	0.614730	+	0.788738i	$0.289265\pi$
$654$	1.63310e29	0.446116
$655$	0	0
$656$	−1.68862e29	−0.446727
$657$	−1.11929e29	−0.291412
$658$	−1.03987e28	−0.0266444
$659$	−2.44989e29	−0.617803	−0.308901	−	0.951094i	$-0.599961\pi$
$659$	−2.44989e29	−0.617803	−0.308901	+	0.951094i	$0.599961\pi$
$660$	0	0
$661$	−4.68385e28	−0.114416	−0.0572082	−	0.998362i	$-0.518220\pi$
$661$	−4.68385e28	−0.114416	−0.0572082	+	0.998362i	$0.518220\pi$
$662$	4.07420e29	0.979567
$663$	−3.16911e28	−0.0749972
$664$	−6.47403e29	−1.50803
$665$	0	0
$666$	−1.91972e29	−0.433269
$667$	3.15360e29	0.700625
$668$	−4.12074e28	−0.0901203
$669$	−1.47476e29	−0.317502
$670$	0	0
$671$	1.00280e29	0.209232
$672$	−1.39967e28	−0.0287507
$673$	6.92309e29	1.40004	0.700022	−	0.714121i	$-0.253174\pi$
$673$	6.92309e29	1.40004	0.700022	+	0.714121i	$0.253174\pi$
$674$	−4.40013e29	−0.876066
$675$	0	0
$676$	1.87764e29	0.362386
$677$	4.53998e28	0.0862726	0.0431363	−	0.999069i	$-0.486265\pi$
$677$	4.53998e28	0.0862726	0.0431363	+	0.999069i	$0.486265\pi$
$678$	−1.90728e29	−0.356863
$679$	−3.36723e28	−0.0620355
$680$	0	0
$681$	−5.78169e29	−1.03279
$682$	−3.80372e29	−0.669072
$683$	7.04765e29	1.22075	0.610375	−	0.792112i	$-0.291019\pi$
$683$	7.04765e29	1.22075	0.610375	+	0.792112i	$0.291019\pi$
$684$	3.28466e28	0.0560276
$685$	0	0
$686$	−6.97733e28	−0.115421
$687$	−5.69293e29	−0.927450
$688$	4.11928e29	0.660910
$689$	3.49817e28	0.0552763
$690$	0	0
$691$	−7.53737e29	−1.15532	−0.577658	−	0.816279i	$-0.696033\pi$
$691$	−7.53737e29	−1.15532	−0.577658	+	0.816279i	$0.696033\pi$
$692$	2.87195e29	0.433573
$693$	1.86884e28	0.0277890
$694$	4.56113e29	0.668033
$695$	0	0
$696$	−1.76115e29	−0.250264
$697$	−1.22062e30	−1.70859
$698$	1.05379e30	1.45302
$699$	2.68027e28	0.0364056
$700$	0	0
$701$	−6.61496e29	−0.871943	−0.435971	−	0.899961i	$-0.643595\pi$
$701$	−6.61496e29	−0.871943	−0.435971	+	0.899961i	$0.643595\pi$
$702$	−7.97341e27	−0.0103539
$703$	−5.88256e29	−0.752550
$704$	−9.57890e29	−1.20727
$705$	0	0
$706$	4.57216e29	0.559335
$707$	5.76741e28	0.0695147
$708$	−6.73774e28	−0.0800138
$709$	1.51806e30	1.77625	0.888125	−	0.459601i	$-0.152008\pi$
$709$	1.51806e30	1.77625	0.888125	+	0.459601i	$0.152008\pi$
$710$	0	0
$711$	1.94990e29	0.221504
$712$	1.45589e28	0.0162963
$713$	−1.16402e30	−1.28387
$714$	5.91840e28	0.0643240
$715$	0	0
$716$	−8.40416e28	−0.0886967
$717$	−8.54535e29	−0.888748
$718$	−1.13591e30	−1.16423
$719$	−7.62329e29	−0.769997	−0.384999	−	0.922917i	$-0.625798\pi$
$719$	−7.62329e29	−0.769997	−0.384999	+	0.922917i	$0.625798\pi$
$720$	0	0
$721$	4.65575e28	0.0456741
$722$	6.48980e29	0.627467
$723$	−6.74230e28	−0.0642474
$724$	−1.05913e29	−0.0994704
$725$	0	0
$726$	1.61547e29	0.147389
$727$	−1.25316e30	−1.12692	−0.563461	−	0.826142i	$-0.690531\pi$
$727$	−1.25316e30	−1.12692	−0.563461	+	0.826142i	$0.690531\pi$
$728$	6.00732e27	0.00532478
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	2.97762e30	2.52777
$732$	−4.57382e28	−0.0382748
$733$	−9.66845e29	−0.797563	−0.398782	−	0.917046i	$-0.630567\pi$
$733$	−9.66845e29	−0.797563	−0.398782	+	0.917046i	$0.630567\pi$
$734$	1.10459e30	0.898244
$735$	0	0
$736$	−1.52689e30	−1.20667
$737$	1.37999e30	1.07515
$738$	−3.07106e29	−0.235883
$739$	−1.13028e30	−0.855892	−0.427946	−	0.903804i	$-0.640763\pi$
$739$	−1.13028e30	−0.855892	−0.427946	+	0.903804i	$0.640763\pi$
$740$	0	0
$741$	−2.44328e28	−0.0179838
$742$	−6.53294e28	−0.0474098
$743$	1.52259e30	1.08943	0.544717	−	0.838620i	$-0.316637\pi$
$743$	1.52259e30	1.08943	0.544717	+	0.838620i	$0.316637\pi$
$744$	6.50054e29	0.458600
$745$	0	0
$746$	3.39434e29	0.232808
$747$	−6.83303e29	−0.462112
$748$	−1.20770e30	−0.805365
$749$	1.85972e29	0.122289
$750$	0	0
$751$	−2.27958e30	−1.45759	−0.728796	−	0.684731i	$-0.759920\pi$
$751$	−2.27958e30	−1.45759	−0.728796	+	0.684731i	$0.759920\pi$
$752$	3.67647e29	0.231816
$753$	−1.23642e30	−0.768811
$754$	3.49625e28	0.0214390
$755$	0	0
$756$	−8.52387e27	−0.00508345
$757$	−2.33908e30	−1.37575	−0.687874	−	0.725830i	$-0.741456\pi$
$757$	−2.33908e30	−1.37575	−0.687874	+	0.725830i	$0.741456\pi$
$758$	2.04782e30	1.18786
$759$	2.03870e30	1.16631
$760$	0	0
$761$	2.09867e30	1.16790	0.583951	−	0.811789i	$-0.301506\pi$
$761$	2.09867e30	1.16790	0.583951	+	0.811789i	$0.301506\pi$
$762$	−1.12947e30	−0.619938
$763$	−1.29864e29	−0.0703043
$764$	−4.64202e29	−0.247872
$765$	0	0
$766$	−2.26087e30	−1.17456
$767$	5.01183e28	0.0256829
$768$	1.06203e30	0.536837
$769$	−1.66515e29	−0.0830286	−0.0415143	−	0.999138i	$-0.513218\pi$
$769$	−1.66515e29	−0.0830286	−0.0415143	+	0.999138i	$0.513218\pi$
$770$	0	0
$771$	1.06455e30	0.516528
$772$	1.00621e30	0.481624
$773$	−3.54312e30	−1.67302	−0.836509	−	0.547953i	$-0.815407\pi$
$773$	−3.54312e30	−1.67302	−0.836509	+	0.547953i	$0.815407\pi$
$774$	7.49163e29	0.348977
$775$	0	0
$776$	2.05137e30	0.930026
$777$	1.52656e29	0.0682797
$778$	5.17259e29	0.228256
$779$	−9.41060e29	−0.409708
$780$	0	0
$781$	1.38092e30	0.585239
$782$	6.45633e30	2.69970
$783$	−1.85881e29	−0.0766894
$784$	1.23017e30	0.500779
$785$	0	0
$786$	−1.42125e30	−0.563293
$787$	4.26632e30	1.66847	0.834236	−	0.551408i	$-0.185909\pi$
$787$	4.26632e30	1.66847	0.834236	+	0.551408i	$0.185909\pi$
$788$	−5.00396e29	−0.193103
$789$	1.75086e30	0.666718
$790$	0	0
$791$	1.51666e29	0.0562388
$792$	−1.13852e30	−0.416608
$793$	3.40221e28	0.0122855
$794$	3.55674e29	0.126747
$795$	0	0
$796$	9.79391e29	0.339914
$797$	2.93870e30	1.00657	0.503284	−	0.864121i	$-0.332125\pi$
$797$	2.93870e30	1.00657	0.503284	+	0.864121i	$0.332125\pi$
$798$	4.56289e28	0.0154245
$799$	2.65754e30	0.886622
$800$	0	0
$801$	1.53662e28	0.00499373
$802$	1.92293e30	0.616784
$803$	3.17284e30	1.00447
$804$	−6.29423e29	−0.196677
$805$	0	0
$806$	−1.29050e29	−0.0392861
$807$	−3.26479e30	−0.981032
$808$	−3.51359e30	−1.04215
$809$	2.08466e30	0.610345	0.305172	−	0.952297i	$-0.401286\pi$
$809$	2.08466e30	0.610345	0.305172	+	0.952297i	$0.401286\pi$
$810$	0	0
$811$	8.65885e29	0.247025	0.123513	−	0.992343i	$-0.460584\pi$
$811$	8.65885e29	0.247025	0.123513	+	0.992343i	$0.460584\pi$
$812$	3.73763e28	0.0105259
$813$	−1.56483e30	−0.435027
$814$	5.44179e30	1.49343
$815$	0	0
$816$	−2.09246e30	−0.559643
$817$	2.29565e30	0.606141
$818$	−3.95721e30	−1.03152
$819$	6.34044e27	0.00163169
$820$	0	0
$821$	7.71132e30	1.93431	0.967154	−	0.254190i	$-0.0818090\pi$
$821$	7.71132e30	1.93431	0.967154	+	0.254190i	$0.0818090\pi$
$822$	−4.89327e29	−0.121184
$823$	−7.28859e30	−1.78215	−0.891077	−	0.453852i	$-0.850049\pi$
$823$	−7.28859e30	−1.78215	−0.891077	+	0.453852i	$0.850049\pi$
$824$	−2.83635e30	−0.684738
$825$	0	0
$826$	−9.35974e28	−0.0220279
$827$	7.57300e30	1.75979	0.879894	−	0.475170i	$-0.157614\pi$
$827$	7.57300e30	1.75979	0.879894	+	0.475170i	$0.157614\pi$
$828$	−9.29862e29	−0.213354
$829$	−1.01498e30	−0.229951	−0.114975	−	0.993368i	$-0.536679\pi$
$829$	−1.01498e30	−0.229951	−0.114975	+	0.993368i	$0.536679\pi$
$830$	0	0
$831$	6.83852e29	0.151061
$832$	−3.24985e29	−0.0708873
$833$	8.89228e30	1.91532
$834$	1.70256e30	0.362125
$835$	0	0
$836$	−9.31098e29	−0.193121
$837$	6.86101e29	0.140530
$838$	6.62097e30	1.33924
$839$	−4.31459e30	−0.861865	−0.430933	−	0.902384i	$-0.641815\pi$
$839$	−4.31459e30	−0.861865	−0.430933	+	0.902384i	$0.641815\pi$
$840$	0	0
$841$	−4.31778e30	−0.841206
$842$	2.69126e29	0.0517821
$843$	4.62966e30	0.879751
$844$	4.93521e29	0.0926212
$845$	0	0
$846$	6.68631e29	0.122405
$847$	−1.28462e29	−0.0232273
$848$	2.30973e30	0.412482
$849$	4.67482e30	0.824582
$850$	0	0
$851$	1.66531e31	2.86572
$852$	−6.29846e29	−0.107058
$853$	−4.19489e30	−0.704297	−0.352149	−	0.935944i	$-0.614549\pi$
$853$	−4.19489e30	−0.704297	−0.352149	+	0.935944i	$0.614549\pi$
$854$	−6.35374e28	−0.0105371
$855$	0	0
$856$	−1.13297e31	−1.83334
$857$	5.58348e30	0.892497	0.446248	−	0.894909i	$-0.352760\pi$
$857$	5.58348e30	0.892497	0.446248	+	0.894909i	$0.352760\pi$
$858$	2.26021e29	0.0356888
$859$	4.37363e30	0.682203	0.341102	−	0.940026i	$-0.389200\pi$
$859$	4.37363e30	0.682203	0.341102	+	0.940026i	$0.389200\pi$
$860$	0	0
$861$	2.44210e29	0.0371733
$862$	−4.27455e29	−0.0642782
$863$	−3.15953e30	−0.469363	−0.234682	−	0.972072i	$-0.575405\pi$
$863$	−3.15953e30	−0.469363	−0.234682	+	0.972072i	$0.575405\pi$
$864$	8.99984e29	0.132081
$865$	0	0
$866$	−3.47684e30	−0.498020
$867$	−1.10456e31	−1.56310
$868$	−1.37959e29	−0.0192882
$869$	−5.52734e30	−0.763499
$870$	0	0
$871$	4.68193e29	0.0631297
$872$	7.91151e30	1.05399
$873$	2.16512e30	0.284992
$874$	4.97762e30	0.647369
$875$	0	0
$876$	−1.44715e30	−0.183747
$877$	−1.26020e31	−1.58104	−0.790519	−	0.612438i	$-0.790189\pi$
$877$	−1.26020e31	−1.58104	−0.790519	+	0.612438i	$0.790189\pi$
$878$	2.87367e30	0.356241
$879$	−2.91772e30	−0.357405
$880$	0	0
$881$	1.94046e30	0.232091	0.116046	−	0.993244i	$-0.462978\pi$
$881$	1.94046e30	0.232091	0.116046	+	0.993244i	$0.462978\pi$
$882$	2.23728e30	0.264424
$883$	−9.71752e30	−1.13493	−0.567464	−	0.823398i	$-0.692075\pi$
$883$	−9.71752e30	−1.13493	−0.567464	+	0.823398i	$0.692075\pi$
$884$	−4.09739e29	−0.0472888
$885$	0	0
$886$	−5.83792e30	−0.657967
$887$	−1.21031e31	−1.34803	−0.674014	−	0.738719i	$-0.735431\pi$
$887$	−1.21031e31	−1.34803	−0.674014	+	0.738719i	$0.735431\pi$
$888$	−9.30000e30	−1.02364
$889$	8.98154e29	0.0976972
$890$	0	0
$891$	−1.20166e30	−0.127663
$892$	−1.90674e30	−0.200198
$893$	2.04888e30	0.212606
$894$	3.84474e30	0.394296
$895$	0	0
$896$	1.09820e29	0.0110014
$897$	6.91672e29	0.0684825
$898$	−4.87518e30	−0.477078
$899$	−3.00848e30	−0.290984
$900$	0	0
$901$	1.66959e31	1.57761
$902$	8.70549e30	0.813064
$903$	−5.95733e29	−0.0549959
$904$	−9.23973e30	−0.843123
$905$	0	0
$906$	−3.26238e30	−0.290863
$907$	−1.16242e31	−1.02444	−0.512221	−	0.858854i	$-0.671177\pi$
$907$	−1.16242e31	−1.02444	−0.512221	+	0.858854i	$0.671177\pi$
$908$	−7.47524e30	−0.651214
$909$	−3.70842e30	−0.319351
$910$	0	0
$911$	−1.25241e31	−1.05391	−0.526956	−	0.849893i	$-0.676667\pi$
$911$	−1.25241e31	−1.05391	−0.526956	+	0.849893i	$0.676667\pi$
$912$	−1.61322e30	−0.134199
$913$	1.93695e31	1.59285
$914$	−1.44651e31	−1.17595
$915$	0	0
$916$	−7.36049e30	−0.584796
$917$	1.13018e30	0.0887704
$918$	−3.80551e30	−0.295505
$919$	−2.43739e30	−0.187116	−0.0935582	−	0.995614i	$-0.529824\pi$
$919$	−2.43739e30	−0.187116	−0.0935582	+	0.995614i	$0.529824\pi$
$920$	0	0
$921$	−1.06896e31	−0.802110
$922$	−1.35800e31	−1.00745
$923$	4.68508e29	0.0343636
$924$	2.41625e29	0.0175221
$925$	0	0
$926$	−3.94202e30	−0.279450
$927$	−2.99363e30	−0.209827
$928$	−3.94633e30	−0.273489
$929$	−1.07248e30	−0.0734891	−0.0367446	−	0.999325i	$-0.511699\pi$
$929$	−1.07248e30	−0.0734891	−0.0367446	+	0.999325i	$0.511699\pi$
$930$	0	0
$931$	6.85566e30	0.459280
$932$	3.46537e29	0.0229553
$933$	−6.65910e30	−0.436173
$934$	−2.09311e31	−1.35566
$935$	0	0
$936$	−3.86269e29	−0.0244621
$937$	−1.98552e31	−1.24339	−0.621697	−	0.783258i	$-0.713556\pi$
$937$	−1.98552e31	−1.24339	−0.621697	+	0.783258i	$0.713556\pi$
$938$	−8.74364e29	−0.0541455
$939$	1.22338e31	0.749158
$940$	0	0
$941$	1.14059e31	0.683026	0.341513	−	0.939877i	$-0.389061\pi$
$941$	1.14059e31	0.683026	0.341513	+	0.939877i	$0.389061\pi$
$942$	−1.49204e31	−0.883581
$943$	2.66407e31	1.56017
$944$	3.30915e30	0.191651
$945$	0	0
$946$	−2.12364e31	−1.20289
$947$	3.85453e30	0.215922	0.107961	−	0.994155i	$-0.465568\pi$
$947$	3.85453e30	0.215922	0.107961	+	0.994155i	$0.465568\pi$
$948$	2.52105e30	0.139667
$949$	1.07646e30	0.0589795
$950$	0	0
$951$	−6.62110e30	−0.354842
$952$	2.86715e30	0.151971
$953$	3.66219e31	1.91984	0.959921	−	0.280271i	$-0.0904242\pi$
$953$	3.66219e31	1.91984	0.959921	+	0.280271i	$0.0904242\pi$
$954$	4.20066e30	0.217801
$955$	0	0
$956$	−1.10484e31	−0.560393
$957$	5.26913e30	0.264340
$958$	3.14161e30	0.155889
$959$	3.89112e29	0.0190976
$960$	0	0
$961$	−9.72097e30	−0.466782
$962$	1.84625e30	0.0876903
$963$	−1.19579e31	−0.561797
$964$	−8.71723e29	−0.0405107
$965$	0	0
$966$	−1.29172e30	−0.0587365
$967$	1.78235e31	0.801706	0.400853	−	0.916142i	$-0.368714\pi$
$967$	1.78235e31	0.801706	0.400853	+	0.916142i	$0.368714\pi$
$968$	7.82609e30	0.348220
$969$	−1.16612e31	−0.513266
$970$	0	0
$971$	−1.90769e31	−0.821684	−0.410842	−	0.911707i	$-0.634765\pi$
$971$	−1.90769e31	−0.821684	−0.410842	+	0.911707i	$0.634765\pi$
$972$	5.48082e29	0.0233534
$973$	−1.35387e30	−0.0570679
$974$	−1.01983e31	−0.425264
$975$	0	0
$976$	2.24638e30	0.0916767
$977$	7.85849e29	0.0317282	0.0158641	−	0.999874i	$-0.494950\pi$
$977$	7.85849e29	0.0317282	0.0158641	+	0.999874i	$0.494950\pi$
$978$	1.03409e31	0.413048
$979$	−4.35583e29	−0.0172128
$980$	0	0
$981$	8.35022e30	0.322978
$982$	−9.47145e30	−0.362448
$983$	1.71763e31	0.650307	0.325154	−	0.945661i	$-0.394584\pi$
$983$	1.71763e31	0.650307	0.325154	+	0.945661i	$0.394584\pi$
$984$	−1.48776e31	−0.557295
$985$	0	0
$986$	1.66868e31	0.611877
$987$	−5.31694e29	−0.0192900
$988$	−3.15895e29	−0.0113395
$989$	−6.49880e31	−2.30819
$990$	0	0
$991$	4.47902e31	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$991$	4.47902e31	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$992$	1.45662e31	0.501158
$993$	2.08318e31	0.709185
$994$	−8.74952e29	−0.0294732
$995$	0	0
$996$	−8.83453e30	−0.291381
$997$	−5.08780e31	−1.66047	−0.830235	−	0.557414i	$-0.811794\pi$
$997$	−5.08780e31	−1.66047	−0.830235	+	0.557414i	$0.811794\pi$
$998$	−2.07040e29	−0.00668627
$999$	−9.81570e30	−0.313678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.l.1.3	yes	8
5.2	odd	4		75.22.b.j.49.6		16
5.3	odd	4		75.22.b.j.49.11		16
5.4	even	2		75.22.a.k.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.k.1.6	✓	8	5.4	even	2
75.22.a.l.1.3	yes	8	1.1	even	1	trivial
75.22.b.j.49.6		16	5.2	odd	4
75.22.b.j.49.11		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)

\(f(q)\)	\(=\)	\(q-1154.86 q^{2} -59049.0 q^{3} -763454. q^{4} +6.81932e7 q^{6} -5.42271e7 q^{7} +3.30359e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-1154.86 q^{2} -59049.0 q^{3} -763454. q^{4} +6.81932e7 q^{6} -5.42271e7 q^{7} +3.30359e9 q^{8} +3.48678e9 q^{9} -9.88393e10 q^{11} +4.50812e10 q^{12} -3.35334e10 q^{13} +6.26246e10 q^{14} -2.21411e12 q^{16} -1.60047e13 q^{17} -4.02674e12 q^{18} -1.23391e13 q^{19} +3.20206e12 q^{21} +1.14145e14 q^{22} +3.49309e14 q^{23} -1.95074e14 q^{24} +3.87263e13 q^{26} -2.05891e14 q^{27} +4.13999e13 q^{28} +9.02810e14 q^{29} -3.33235e15 q^{31} -4.37116e15 q^{32} +5.83636e15 q^{33} +1.84831e16 q^{34} -2.66200e15 q^{36} +4.76742e16 q^{37} +1.42499e16 q^{38} +1.98011e15 q^{39} +7.62667e16 q^{41} -3.69792e15 q^{42} -1.86047e17 q^{43} +7.54593e16 q^{44} -4.03403e17 q^{46} -1.66048e17 q^{47} +1.30741e17 q^{48} -5.55605e17 q^{49} +9.45060e17 q^{51} +2.56012e16 q^{52} -1.04319e18 q^{53} +2.37775e17 q^{54} -1.79144e17 q^{56} +7.28610e17 q^{57} -1.04262e18 q^{58} -1.49458e18 q^{59} -1.01457e18 q^{61} +3.84839e18 q^{62} -1.89078e17 q^{63} +9.69139e18 q^{64} -6.74017e18 q^{66} -1.39620e19 q^{67} +1.22188e19 q^{68} -2.06264e19 q^{69} -1.39714e19 q^{71} +1.15189e19 q^{72} -3.21010e19 q^{73} -5.50570e19 q^{74} +9.42032e18 q^{76} +5.35977e18 q^{77} -2.28675e18 q^{78} +5.59225e19 q^{79} +1.21577e19 q^{81} -8.80772e19 q^{82} -1.95969e20 q^{83} -2.44462e18 q^{84} +2.14858e20 q^{86} -5.33100e19 q^{87} -3.26525e20 q^{88} +4.40698e18 q^{89} +1.81842e18 q^{91} -2.66682e20 q^{92} +1.96772e20 q^{93} +1.91762e20 q^{94} +2.58113e20 q^{96} +6.20950e20 q^{97} +6.41645e20 q^{98} -3.44631e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more