LMFDB - Embedded newform 75.22.a.k.1.2

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			1.2
Root			$1827.63$ 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)

$f(q)$	$=$	$q-1910.63 q^{2} +59049.0 q^{3} +1.55335e6 q^{4} -1.12821e8 q^{6} -1.48601e8 q^{7} +1.03900e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-1910.63 q^{2} +59049.0 q^{3} +1.55335e6 q^{4} -1.12821e8 q^{6} -1.48601e8 q^{7} +1.03900e9 q^{8} +3.48678e9 q^{9} +1.36386e11 q^{11} +9.17240e10 q^{12} -5.60884e11 q^{13} +2.83921e11 q^{14} -5.24276e12 q^{16} +1.60332e13 q^{17} -6.66195e12 q^{18} +4.50564e13 q^{19} -8.77474e12 q^{21} -2.60584e14 q^{22} +1.52158e14 q^{23} +6.13517e13 q^{24} +1.07164e15 q^{26} +2.05891e14 q^{27} -2.30830e14 q^{28} +2.68948e15 q^{29} -2.10244e15 q^{31} +7.83804e15 q^{32} +8.05347e15 q^{33} -3.06335e16 q^{34} +5.41621e15 q^{36} -1.10802e16 q^{37} -8.60860e16 q^{38} -3.31196e16 q^{39} +1.74051e16 q^{41} +1.67653e16 q^{42} +1.16017e17 q^{43} +2.11856e17 q^{44} -2.90717e17 q^{46} +3.56887e17 q^{47} -3.09580e17 q^{48} -5.36464e17 q^{49} +9.46743e17 q^{51} -8.71251e17 q^{52} +1.00113e18 q^{53} -3.93382e17 q^{54} -1.54396e17 q^{56} +2.66053e18 q^{57} -5.13861e18 q^{58} +3.90185e18 q^{59} +9.66054e18 q^{61} +4.01698e18 q^{62} -5.18140e17 q^{63} -3.98073e18 q^{64} -1.53872e19 q^{66} -2.77474e18 q^{67} +2.49052e19 q^{68} +8.98477e18 q^{69} +5.04155e19 q^{71} +3.62275e18 q^{72} -1.34316e19 q^{73} +2.11702e19 q^{74} +6.99885e19 q^{76} -2.02671e19 q^{77} +6.32793e19 q^{78} -7.34355e19 q^{79} +1.21577e19 q^{81} -3.32547e19 q^{82} -1.78349e20 q^{83} -1.36303e19 q^{84} -2.21665e20 q^{86} +1.58811e20 q^{87} +1.41705e20 q^{88} -3.57581e20 q^{89} +8.33479e19 q^{91} +2.36355e20 q^{92} -1.24147e20 q^{93} -6.81879e20 q^{94} +4.62828e20 q^{96} -7.34881e20 q^{97} +1.02498e21 q^{98} +4.75549e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1910.63	−1.31935	−0.659677	−	0.751549i	$-0.729307\pi$
$2$	−1910.63	−1.31935	−0.659677	+	0.751549i	$0.729307\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	1.55335e6	0.740697
$5$	0	0
$5$	0	0
$6$	−1.12821e8	−0.761730
$7$	−1.48601e8	−0.198835	−0.0994174	−	0.995046i	$-0.531698\pi$
$7$	−1.48601e8	−0.198835	−0.0994174	+	0.995046i	$0.531698\pi$
$8$	1.03900e9	0.342112
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.36386e11	1.58543	0.792715	−	0.609592i	$-0.208667\pi$
$11$	1.36386e11	1.58543	0.792715	+	0.609592i	$0.208667\pi$
$12$	9.17240e10	0.427642
$13$	−5.60884e11	−1.12841	−0.564206	−	0.825634i	$-0.690818\pi$
$13$	−5.60884e11	−1.12841	−0.564206	+	0.825634i	$0.690818\pi$
$14$	2.83921e11	0.262334
$15$	0	0
$16$	−5.24276e12	−1.19206
$17$	1.60332e13	1.92888	0.964441	−	0.264298i	$-0.0851403\pi$
$17$	1.60332e13	1.92888	0.964441	+	0.264298i	$0.0851403\pi$
$18$	−6.66195e12	−0.439785
$19$	4.50564e13	1.68594	0.842972	−	0.537957i	$-0.180804\pi$
$19$	4.50564e13	1.68594	0.842972	+	0.537957i	$0.180804\pi$
$20$	0	0
$21$	−8.77474e12	−0.114797
$22$	−2.60584e14	−2.09175
$23$	1.52158e14	0.765865	0.382932	−	0.923776i	$-0.374914\pi$
$23$	1.52158e14	0.765865	0.382932	+	0.923776i	$0.374914\pi$
$24$	6.13517e13	0.197519
$25$	0	0
$26$	1.07164e15	1.48878
$27$	2.05891e14	0.192450
$28$	−2.30830e14	−0.147276
$29$	2.68948e15	1.18711	0.593553	−	0.804795i	$-0.297725\pi$
$29$	2.68948e15	1.18711	0.593553	+	0.804795i	$0.297725\pi$
$30$	0	0
$31$	−2.10244e15	−0.460708	−0.230354	−	0.973107i	$-0.573988\pi$
$31$	−2.10244e15	−0.460708	−0.230354	+	0.973107i	$0.573988\pi$
$32$	7.83804e15	1.23064
$33$	8.05347e15	0.915349
$34$	−3.06335e16	−2.54488
$35$	0	0
$36$	5.41621e15	0.246899
$37$	−1.10802e16	−0.378818	−0.189409	−	0.981898i	$-0.560657\pi$
$37$	−1.10802e16	−0.378818	−0.189409	+	0.981898i	$0.560657\pi$
$38$	−8.60860e16	−2.22436
$39$	−3.31196e16	−0.651489
$40$	0	0
$41$	1.74051e16	0.202510	0.101255	−	0.994861i	$-0.467714\pi$
$41$	1.74051e16	0.202510	0.101255	+	0.994861i	$0.467714\pi$
$42$	1.67653e16	0.151458
$43$	1.16017e17	0.818656	0.409328	−	0.912387i	$-0.365763\pi$
$43$	1.16017e17	0.818656	0.409328	+	0.912387i	$0.365763\pi$
$44$	2.11856e17	1.17432
$45$	0	0
$46$	−2.90717e17	−1.01045
$47$	3.56887e17	0.989700	0.494850	−	0.868979i	$-0.335223\pi$
$47$	3.56887e17	0.989700	0.494850	+	0.868979i	$0.335223\pi$
$48$	−3.09580e17	−0.688239
$49$	−5.36464e17	−0.960465
$50$	0	0
$51$	9.46743e17	1.11364
$52$	−8.71251e17	−0.835811
$53$	1.00113e18	0.786312	0.393156	−	0.919472i	$-0.371383\pi$
$53$	1.00113e18	0.786312	0.393156	+	0.919472i	$0.371383\pi$
$54$	−3.93382e17	−0.253910
$55$	0	0
$56$	−1.54396e17	−0.0680238
$57$	2.66053e18	0.973381
$58$	−5.13861e18	−1.56621
$59$	3.90185e18	0.993857	0.496929	−	0.867791i	$-0.334461\pi$
$59$	3.90185e18	0.993857	0.496929	+	0.867791i	$0.334461\pi$
$60$	0	0
$61$	9.66054e18	1.73396	0.866978	−	0.498347i	$-0.166059\pi$
$61$	9.66054e18	1.73396	0.866978	+	0.498347i	$0.166059\pi$
$62$	4.01698e18	0.607837
$63$	−5.18140e17	−0.0662782
$64$	−3.98073e18	−0.431591
$65$	0	0
$66$	−1.53872e19	−1.20767
$67$	−2.77474e18	−0.185968	−0.0929839	−	0.995668i	$-0.529641\pi$
$67$	−2.77474e18	−0.185968	−0.0929839	+	0.995668i	$0.529641\pi$
$68$	2.49052e19	1.42872
$69$	8.98477e18	0.442172
$70$	0	0
$71$	5.04155e19	1.83802	0.919012	−	0.394230i	$-0.128989\pi$
$71$	5.04155e19	1.83802	0.919012	+	0.394230i	$0.128989\pi$
$72$	3.62275e18	0.114037
$73$	−1.34316e19	−0.365794	−0.182897	−	0.983132i	$-0.558547\pi$
$73$	−1.34316e19	−0.365794	−0.182897	+	0.983132i	$0.558547\pi$
$74$	2.11702e19	0.499796
$75$	0	0
$76$	6.99885e19	1.24877
$77$	−2.02671e19	−0.315239
$78$	6.32793e19	0.859545
$79$	−7.34355e19	−0.872613	−0.436306	−	0.899798i	$-0.643714\pi$
$79$	−7.34355e19	−0.872613	−0.436306	+	0.899798i	$0.643714\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−3.32547e19	−0.267182
$83$	−1.78349e20	−1.26169	−0.630844	−	0.775910i	$-0.717291\pi$
$83$	−1.78349e20	−1.26169	−0.630844	+	0.775910i	$0.717291\pi$
$84$	−1.36303e19	−0.0850300
$85$	0	0
$86$	−2.21665e20	−1.08010
$87$	1.58811e20	0.685376
$88$	1.41705e20	0.542396
$89$	−3.57581e20	−1.21557	−0.607784	−	0.794103i	$-0.707941\pi$
$89$	−3.57581e20	−1.21557	−0.607784	+	0.794103i	$0.707941\pi$
$90$	0	0
$91$	8.33479e19	0.224367
$92$	2.36355e20	0.567274
$93$	−1.24147e20	−0.265990
$94$	−6.81879e20	−1.30576
$95$	0	0
$96$	4.62828e20	0.710513
$97$	−7.34881e20	−1.01185	−0.505923	−	0.862579i	$-0.668848\pi$
$97$	−7.34881e20	−1.01185	−0.505923	+	0.862579i	$0.668848\pi$
$98$	1.02498e21	1.26719
$99$	4.75549e20	0.528477
$100$	0	0
$101$	1.42597e21	1.28450	0.642251	−	0.766495i	$-0.278001\pi$
$101$	1.42597e21	1.28450	0.642251	+	0.766495i	$0.278001\pi$
$102$	−1.80887e21	−1.46929
$103$	9.97339e20	0.731227	0.365613	−	0.930767i	$-0.380859\pi$
$103$	9.97339e20	0.731227	0.365613	+	0.930767i	$0.380859\pi$
$104$	−5.82756e20	−0.386044
$105$	0	0
$106$	−1.91279e21	−1.03742
$107$	−2.44609e21	−1.20211	−0.601053	−	0.799209i	$-0.705252\pi$
$107$	−2.44609e21	−1.20211	−0.601053	+	0.799209i	$0.705252\pi$
$108$	3.19822e20	0.142547
$109$	4.34723e20	0.175887	0.0879436	−	0.996125i	$-0.471970\pi$
$109$	4.34723e20	0.175887	0.0879436	+	0.996125i	$0.471970\pi$
$110$	0	0
$111$	−6.54277e20	−0.218711
$112$	7.79079e20	0.237024
$113$	−3.41858e21	−0.947376	−0.473688	−	0.880693i	$-0.657077\pi$
$113$	−3.41858e21	−0.947376	−0.473688	+	0.880693i	$0.657077\pi$
$114$	−5.08329e21	−1.28423
$115$	0	0
$116$	4.17772e21	0.879286
$117$	−1.95568e21	−0.376137
$118$	−7.45499e21	−1.31125
$119$	−2.38254e21	−0.383529
$120$	0	0
$121$	1.12009e22	1.51359
$122$	−1.84577e22	−2.28770
$123$	1.02775e21	0.116919
$124$	−3.26583e21	−0.341245
$125$	0	0
$126$	9.89973e20	0.0874445
$127$	−7.61735e21	−0.619249	−0.309624	−	0.950859i	$-0.600203\pi$
$127$	−7.61735e21	−0.619249	−0.309624	+	0.950859i	$0.600203\pi$
$128$	−8.83186e21	−0.661222
$129$	6.85067e21	0.472652
$130$	0	0
$131$	4.93627e21	0.289768	0.144884	−	0.989449i	$-0.453719\pi$
$131$	4.93627e21	0.289768	0.144884	+	0.989449i	$0.453719\pi$
$132$	1.25099e22	0.677996
$133$	−6.69542e21	−0.335224
$134$	5.30151e21	0.245357
$135$	0	0
$136$	1.66584e22	0.659895
$137$	−1.06940e22	−0.392260	−0.196130	−	0.980578i	$-0.562837\pi$
$137$	−1.06940e22	−0.392260	−0.196130	+	0.980578i	$0.562837\pi$
$138$	−1.71666e22	−0.583382
$139$	−4.68793e22	−1.47681	−0.738407	−	0.674355i	$-0.764422\pi$
$139$	−4.68793e22	−1.47681	−0.738407	+	0.674355i	$0.764422\pi$
$140$	0	0
$141$	2.10738e22	0.571403
$142$	−9.63253e22	−2.42501
$143$	−7.64968e22	−1.78902
$144$	−1.82804e22	−0.397355
$145$	0	0
$146$	2.56627e22	0.482612
$147$	−3.16776e22	−0.554525
$148$	−1.72115e22	−0.280590
$149$	−3.57766e22	−0.543429	−0.271714	−	0.962378i	$-0.587591\pi$
$149$	−3.57766e22	−0.543429	−0.271714	+	0.962378i	$0.587591\pi$
$150$	0	0
$151$	8.95656e21	0.118272	0.0591362	−	0.998250i	$-0.481165\pi$
$151$	8.95656e21	0.118272	0.0591362	+	0.998250i	$0.481165\pi$
$152$	4.68134e22	0.576783
$153$	5.59042e22	0.642961
$154$	3.87230e22	0.415912
$155$	0	0
$156$	−5.14465e22	−0.482556
$157$	−1.02274e23	−0.897058	−0.448529	−	0.893768i	$-0.648052\pi$
$157$	−1.02274e23	−0.897058	−0.448529	+	0.893768i	$0.648052\pi$
$158$	1.40308e23	1.15129
$159$	5.91159e22	0.453977
$160$	0	0
$161$	−2.26108e22	−0.152281
$162$	−2.32288e22	−0.146595
$163$	6.25531e22	0.370066	0.185033	−	0.982732i	$-0.440761\pi$
$163$	6.25531e22	0.370066	0.185033	+	0.982732i	$0.440761\pi$
$164$	2.70363e22	0.149999
$165$	0	0
$166$	3.40760e23	1.66461
$167$	−4.02537e23	−1.84622	−0.923108	−	0.384541i	$-0.874360\pi$
$167$	−4.02537e23	−1.84622	−0.923108	+	0.384541i	$0.874360\pi$
$168$	−9.11692e21	−0.0392736
$169$	6.75259e22	0.273313
$170$	0	0
$171$	1.57102e23	0.561981
$172$	1.80215e23	0.606376
$173$	−2.94644e23	−0.932852	−0.466426	−	0.884560i	$-0.654459\pi$
$173$	−2.94644e23	−0.932852	−0.466426	+	0.884560i	$0.654459\pi$
$174$	−3.03429e23	−0.904255
$175$	0	0
$176$	−7.15040e23	−1.88994
$177$	2.30400e23	0.573804
$178$	6.83204e23	1.60376
$179$	1.92162e23	0.425314	0.212657	−	0.977127i	$-0.431788\pi$
$179$	1.92162e23	0.425314	0.212657	+	0.977127i	$0.431788\pi$
$180$	0	0
$181$	−6.24822e23	−1.23064	−0.615321	−	0.788277i	$-0.710973\pi$
$181$	−6.24822e23	−1.23064	−0.615321	+	0.788277i	$0.710973\pi$
$182$	−1.59247e23	−0.296020
$183$	5.70445e23	1.00110
$184$	1.58091e23	0.262012
$185$	0	0
$186$	2.37199e23	0.350935
$187$	2.18670e24	3.05811
$188$	5.54372e23	0.733068
$189$	−3.05956e22	−0.0382658
$190$	0	0
$191$	1.68256e24	1.88417	0.942085	−	0.335375i	$-0.108863\pi$
$191$	1.68256e24	1.88417	0.942085	+	0.335375i	$0.108863\pi$
$192$	−2.35058e23	−0.249179
$193$	−1.57118e23	−0.157715	−0.0788577	−	0.996886i	$-0.525127\pi$
$193$	−1.57118e23	−0.157715	−0.0788577	+	0.996886i	$0.525127\pi$
$194$	1.40409e24	1.33498
$195$	0	0
$196$	−8.33318e23	−0.711414
$197$	1.38119e24	1.11778	0.558892	−	0.829240i	$-0.311227\pi$
$197$	1.38119e24	1.11778	0.558892	+	0.829240i	$0.311227\pi$
$198$	−9.08599e23	−0.697248
$199$	5.55589e23	0.404386	0.202193	−	0.979346i	$-0.435193\pi$
$199$	5.55589e23	0.404386	0.202193	+	0.979346i	$0.435193\pi$
$200$	0	0
$201$	−1.63846e23	−0.107369
$202$	−2.72449e24	−1.69471
$203$	−3.99660e23	−0.236038
$204$	1.47063e24	0.824870
$205$	0	0
$206$	−1.90555e24	−0.964747
$207$	5.30542e23	0.255288
$208$	2.94058e24	1.34514
$209$	6.14506e24	2.67295
$210$	0	0
$211$	3.15713e24	1.24259	0.621294	−	0.783577i	$-0.286607\pi$
$211$	3.15713e24	1.24259	0.621294	+	0.783577i	$0.286607\pi$
$212$	1.55511e24	0.582419
$213$	2.97698e24	1.06118
$214$	4.67357e24	1.58600
$215$	0	0
$216$	2.13920e23	0.0658396
$217$	3.12424e23	0.0916047
$218$	−8.30594e23	−0.232058
$219$	−7.93120e23	−0.211191
$220$	0	0
$221$	−8.99274e24	−2.17657
$222$	1.25008e24	0.288557
$223$	−1.99883e24	−0.440124	−0.220062	−	0.975486i	$-0.570626\pi$
$223$	−1.99883e24	−0.440124	−0.220062	+	0.975486i	$0.570626\pi$
$224$	−1.16474e24	−0.244695
$225$	0	0
$226$	6.53164e24	1.24992
$227$	−4.82296e24	−0.881134	−0.440567	−	0.897720i	$-0.645222\pi$
$227$	−4.82296e24	−0.881134	−0.440567	+	0.897720i	$0.645222\pi$
$228$	4.13275e24	0.720980
$229$	−9.84836e21	−0.00164094	−0.000820468	−	1.00000i	$-0.500261\pi$
$229$	−9.84836e21	−0.00164094	−0.000820468		1.00000i	$0.500261\pi$
$230$	0	0
$231$	−1.19675e24	−0.182003
$232$	2.79436e24	0.406124
$233$	3.35229e24	0.465698	0.232849	−	0.972513i	$-0.425195\pi$
$233$	3.35229e24	0.465698	0.232849	+	0.972513i	$0.425195\pi$
$234$	3.73658e24	0.496258
$235$	0	0
$236$	6.06095e24	0.736147
$237$	−4.33629e24	−0.503803
$238$	4.55216e24	0.506011
$239$	7.78507e24	0.828103	0.414052	−	0.910253i	$-0.364113\pi$
$239$	7.78507e24	0.828103	0.414052	+	0.910253i	$0.364113\pi$
$240$	0	0
$241$	−1.33550e25	−1.30156	−0.650779	−	0.759267i	$-0.725558\pi$
$241$	−1.33550e25	−1.30156	−0.650779	+	0.759267i	$0.725558\pi$
$242$	−2.14009e25	−1.99696
$243$	7.17898e23	0.0641500
$244$	1.50062e25	1.28434
$245$	0	0
$246$	−1.96366e24	−0.154258
$247$	−2.52714e25	−1.90244
$248$	−2.18442e24	−0.157614
$249$	−1.05314e25	−0.728436
$250$	0	0
$251$	2.97329e25	1.89088	0.945439	−	0.325801i	$-0.105634\pi$
$251$	2.97329e25	1.89088	0.945439	+	0.325801i	$0.105634\pi$
$252$	−8.04854e23	−0.0490921
$253$	2.07522e25	1.21423
$254$	1.45539e25	0.817009
$255$	0	0
$256$	2.52226e25	1.30398
$257$	3.84959e24	0.191037	0.0955183	−	0.995428i	$-0.469549\pi$
$257$	3.84959e24	0.191037	0.0955183	+	0.995428i	$0.469549\pi$
$258$	−1.30891e25	−0.623595
$259$	1.64653e24	0.0753222
$260$	0	0
$261$	9.37764e24	0.395702
$262$	−9.43139e24	−0.382307
$263$	2.30098e25	0.896142	0.448071	−	0.893998i	$-0.352111\pi$
$263$	2.30098e25	0.896142	0.448071	+	0.893998i	$0.352111\pi$
$264$	8.36752e24	0.313152
$265$	0	0
$266$	1.27925e25	0.442280
$267$	−2.11148e25	−0.701808
$268$	−4.31016e24	−0.137746
$269$	−2.61032e25	−0.802223	−0.401112	−	0.916029i	$-0.631376\pi$
$269$	−2.61032e25	−0.802223	−0.401112	+	0.916029i	$0.631376\pi$
$270$	0	0
$271$	4.04673e25	1.15060	0.575302	−	0.817941i	$-0.304884\pi$
$271$	4.04673e25	1.15060	0.575302	+	0.817941i	$0.304884\pi$
$272$	−8.40580e25	−2.29935
$273$	4.92161e24	0.129539
$274$	2.04323e25	0.517530
$275$	0	0
$276$	1.39565e25	0.327516
$277$	−9.15941e24	−0.206933	−0.103467	−	0.994633i	$-0.532993\pi$
$277$	−9.15941e24	−0.206933	−0.103467	+	0.994633i	$0.532993\pi$
$278$	8.95691e25	1.94844
$279$	−7.33075e24	−0.153569
$280$	0	0
$281$	9.96061e24	0.193584	0.0967920	−	0.995305i	$-0.469142\pi$
$281$	9.96061e24	0.193584	0.0967920	+	0.995305i	$0.469142\pi$
$282$	−4.02643e25	−0.753884
$283$	−5.21338e25	−0.940507	−0.470253	−	0.882531i	$-0.655837\pi$
$283$	−5.21338e25	−0.940507	−0.470253	+	0.882531i	$0.655837\pi$
$284$	7.83131e25	1.36142
$285$	0	0
$286$	1.46157e26	2.36035
$287$	−2.58642e24	−0.0402660
$288$	2.73295e25	0.410215
$289$	1.87971e26	2.72059
$290$	0	0
$291$	−4.33940e25	−0.584189
$292$	−2.08640e25	−0.270942
$293$	−2.24105e25	−0.280765	−0.140382	−	0.990097i	$-0.544833\pi$
$293$	−2.24105e25	−0.280765	−0.140382	+	0.990097i	$0.544833\pi$
$294$	6.05242e25	0.731615
$295$	0	0
$296$	−1.15123e25	−0.129598
$297$	2.80807e25	0.305116
$298$	6.83557e25	0.716975
$299$	−8.53428e25	−0.864211
$300$	0	0
$301$	−1.72402e25	−0.162777
$302$	−1.71127e25	−0.156043
$303$	8.42018e25	0.741607
$304$	−2.36220e26	−2.00976
$305$	0	0
$306$	−1.06812e26	−0.848293
$307$	−4.91767e25	−0.377404	−0.188702	−	0.982034i	$-0.560428\pi$
$307$	−4.91767e25	−0.377404	−0.188702	+	0.982034i	$0.560428\pi$
$308$	−3.14820e25	−0.233496
$309$	5.88919e25	0.422174
$310$	0	0
$311$	−2.74093e25	−0.183618	−0.0918088	−	0.995777i	$-0.529265\pi$
$311$	−2.74093e25	−0.183618	−0.0918088	+	0.995777i	$0.529265\pi$
$312$	−3.44111e25	−0.222882
$313$	−2.38986e26	−1.49678	−0.748388	−	0.663261i	$-0.769172\pi$
$313$	−2.38986e26	−1.49678	−0.748388	+	0.663261i	$0.769172\pi$
$314$	1.95409e26	1.18354
$315$	0	0
$316$	−1.14071e26	−0.646342
$317$	1.92872e26	1.05717	0.528587	−	0.848879i	$-0.322722\pi$
$317$	1.92872e26	1.05717	0.528587	+	0.848879i	$0.322722\pi$
$318$	−1.12949e26	−0.598957
$319$	3.66808e26	1.88207
$320$	0	0
$321$	−1.44439e26	−0.694036
$322$	4.32009e25	0.200912
$323$	7.22396e26	3.25199
$324$	1.88852e25	0.0822997
$325$	0	0
$326$	−1.19516e26	−0.488248
$327$	2.56699e25	0.101549
$328$	1.80838e25	0.0692812
$329$	−5.30338e25	−0.196787
$330$	0	0
$331$	−7.45178e25	−0.259458	−0.129729	−	0.991550i	$-0.541411\pi$
$331$	−7.45178e25	−0.259458	−0.129729	+	0.991550i	$0.541411\pi$
$332$	−2.77040e26	−0.934528
$333$	−3.86344e25	−0.126273
$334$	7.69098e26	2.43581
$335$	0	0
$336$	4.60038e25	0.136846
$337$	5.89369e26	1.69931	0.849656	−	0.527337i	$-0.176809\pi$
$337$	5.89369e26	1.69931	0.849656	+	0.527337i	$0.176809\pi$
$338$	−1.29017e26	−0.360596
$339$	−2.01864e26	−0.546968
$340$	0	0
$341$	−2.86744e26	−0.730420
$342$	−3.00163e26	−0.741453
$343$	1.62719e26	0.389808
$344$	1.20541e26	0.280073
$345$	0	0
$346$	5.62955e26	1.23076
$347$	−5.23771e25	−0.111092	−0.0555459	−	0.998456i	$-0.517690\pi$
$347$	−5.23771e25	−0.111092	−0.0555459	+	0.998456i	$0.517690\pi$
$348$	2.46690e26	0.507656
$349$	−6.83224e26	−1.36426	−0.682128	−	0.731233i	$-0.738945\pi$
$349$	−6.83224e26	−1.36426	−0.682128	+	0.731233i	$0.738945\pi$
$350$	0	0
$351$	−1.15481e26	−0.217163
$352$	1.06900e27	1.95110
$353$	−2.64431e26	−0.468466	−0.234233	−	0.972180i	$-0.575258\pi$
$353$	−2.64431e26	−0.468466	−0.234233	+	0.972180i	$0.575258\pi$
$354$	−4.40209e26	−0.757051
$355$	0	0
$356$	−5.55450e26	−0.900367
$357$	−1.40687e26	−0.221430
$358$	−3.67150e26	−0.561140
$359$	7.58347e26	1.12558	0.562790	−	0.826600i	$-0.309728\pi$
$359$	7.58347e26	1.12558	0.562790	+	0.826600i	$0.309728\pi$
$360$	0	0
$361$	1.31587e27	1.84241
$362$	1.19380e27	1.62365
$363$	6.61404e26	0.873871
$364$	1.29469e26	0.166188
$365$	0	0
$366$	−1.08991e27	−1.32081
$367$	1.63074e27	1.92039	0.960197	−	0.279324i	$-0.0901102\pi$
$367$	1.63074e27	1.92039	0.960197	+	0.279324i	$0.0901102\pi$
$368$	−7.97727e26	−0.912960
$369$	6.06879e25	0.0675033
$370$	0	0
$371$	−1.48769e26	−0.156346
$372$	−1.92844e26	−0.197018
$373$	−1.70221e27	−1.69072	−0.845358	−	0.534200i	$-0.820613\pi$
$373$	−1.70221e27	−1.69072	−0.845358	+	0.534200i	$0.820613\pi$
$374$	−4.17798e27	−4.03473
$375$	0	0
$376$	3.70804e26	0.338589
$377$	−1.50849e27	−1.33954
$378$	5.84569e25	0.0504861
$379$	4.61712e26	0.387846	0.193923	−	0.981017i	$-0.437879\pi$
$379$	4.61712e26	0.387846	0.193923	+	0.981017i	$0.437879\pi$
$380$	0	0
$381$	−4.49797e26	−0.357523
$382$	−3.21475e27	−2.48589
$383$	4.50472e26	0.338907	0.169453	−	0.985538i	$-0.445800\pi$
$383$	4.50472e26	0.338907	0.169453	+	0.985538i	$0.445800\pi$
$384$	−5.21512e26	−0.381757
$385$	0	0
$386$	3.00194e26	0.208083
$387$	4.04525e26	0.272885
$388$	−1.14153e27	−0.749471
$389$	−1.11428e27	−0.712069	−0.356035	−	0.934473i	$-0.615872\pi$
$389$	−1.11428e27	−0.712069	−0.356035	+	0.934473i	$0.615872\pi$
$390$	0	0
$391$	2.43957e27	1.47726
$392$	−5.57383e26	−0.328587
$393$	2.91482e26	0.167298
$394$	−2.63894e27	−1.47475
$395$	0	0
$396$	7.38697e26	0.391441
$397$	3.08694e27	1.59304	0.796522	−	0.604610i	$-0.206671\pi$
$397$	3.08694e27	1.59304	0.796522	+	0.604610i	$0.206671\pi$
$398$	−1.06153e27	−0.533529
$399$	−3.95358e26	−0.193542
$400$	0	0
$401$	−3.42137e27	−1.58922	−0.794610	−	0.607120i	$-0.792325\pi$
$401$	−3.42137e27	−1.58922	−0.794610	+	0.607120i	$0.792325\pi$
$402$	3.13049e26	0.141657
$403$	1.17922e27	0.519868
$404$	2.21503e27	0.951426
$405$	0	0
$406$	7.63602e26	0.311418
$407$	−1.51119e27	−0.600590
$408$	9.83661e26	0.380990
$409$	−4.79772e27	−1.81109	−0.905545	−	0.424251i	$-0.860537\pi$
$409$	−4.79772e27	−1.81109	−0.905545	+	0.424251i	$0.860537\pi$
$410$	0	0
$411$	−6.31470e26	−0.226471
$412$	1.54922e27	0.541618
$413$	−5.79818e26	−0.197613
$414$	−1.01367e27	−0.336816
$415$	0	0
$416$	−4.39623e27	−1.38867
$417$	−2.76818e27	−0.852640
$418$	−1.17409e28	−3.52657
$419$	−3.60023e27	−1.05459	−0.527295	−	0.849683i	$-0.676794\pi$
$419$	−3.60023e27	−1.05459	−0.527295	+	0.849683i	$0.676794\pi$
$420$	0	0
$421$	5.98328e27	1.66716	0.833580	−	0.552398i	$-0.186287\pi$
$421$	5.98328e27	1.66716	0.833580	+	0.552398i	$0.186287\pi$
$422$	−6.03212e27	−1.63942
$423$	1.24439e27	0.329900
$424$	1.04017e27	0.269007
$425$	0	0
$426$	−5.68791e27	−1.40008
$427$	−1.43557e27	−0.344771
$428$	−3.79964e27	−0.890396
$429$	−4.51706e27	−1.03289
$430$	0	0
$431$	−1.55512e27	−0.338651	−0.169326	−	0.985560i	$-0.554159\pi$
$431$	−1.55512e27	−0.338651	−0.169326	+	0.985560i	$0.554159\pi$
$432$	−1.07944e27	−0.229413
$433$	4.63906e26	0.0962292	0.0481146	−	0.998842i	$-0.484679\pi$
$433$	4.63906e26	0.0962292	0.0481146	+	0.998842i	$0.484679\pi$
$434$	−5.96927e26	−0.120859
$435$	0	0
$436$	6.75279e26	0.130279
$437$	6.85568e27	1.29121
$438$	1.51536e27	0.278636
$439$	4.07981e27	0.732424	0.366212	−	0.930531i	$-0.380654\pi$
$439$	4.07981e27	0.732424	0.366212	+	0.930531i	$0.380654\pi$
$440$	0	0
$441$	−1.87053e27	−0.320155
$442$	1.71818e28	2.87167
$443$	4.21850e26	0.0688523	0.0344262	−	0.999407i	$-0.489040\pi$
$443$	4.21850e26	0.0688523	0.0344262	+	0.999407i	$0.489040\pi$
$444$	−1.01632e27	−0.161998
$445$	0	0
$446$	3.81902e27	0.580679
$447$	−2.11257e27	−0.313749
$448$	5.91540e26	0.0858153
$449$	−1.99365e27	−0.282529	−0.141264	−	0.989972i	$-0.545117\pi$
$449$	−1.99365e27	−0.282529	−0.141264	+	0.989972i	$0.545117\pi$
$450$	0	0
$451$	2.37382e27	0.321065
$452$	−5.31027e27	−0.701718
$453$	5.28876e26	0.0682846
$454$	9.21488e27	1.16253
$455$	0	0
$456$	2.76428e27	0.333006
$457$	8.48597e27	0.999036	0.499518	−	0.866303i	$-0.333510\pi$
$457$	8.48597e27	0.999036	0.499518	+	0.866303i	$0.333510\pi$
$458$	1.88166e25	0.00216498
$459$	3.30109e27	0.371214
$460$	0	0
$461$	−4.64937e27	−0.499499	−0.249749	−	0.968311i	$-0.580348\pi$
$461$	−4.64937e27	−0.499499	−0.249749	+	0.968311i	$0.580348\pi$
$462$	2.28655e27	0.240127
$463$	2.51235e27	0.257917	0.128959	−	0.991650i	$-0.458837\pi$
$463$	2.51235e27	0.257917	0.128959	+	0.991650i	$0.458837\pi$
$464$	−1.41003e28	−1.41511
$465$	0	0
$466$	−6.40498e27	−0.614421
$467$	3.55621e27	0.333550	0.166775	−	0.985995i	$-0.446665\pi$
$467$	3.55621e27	0.333550	0.166775	+	0.985995i	$0.446665\pi$
$468$	−3.03786e27	−0.278604
$469$	4.12330e26	0.0369768
$470$	0	0
$471$	−6.03920e27	−0.517917
$472$	4.05400e27	0.340011
$473$	1.58231e28	1.29792
$474$	8.28505e27	0.664695
$475$	0	0
$476$	−3.70094e27	−0.284079
$477$	3.49073e27	0.262104
$478$	−1.48744e28	−1.09256
$479$	−2.78288e27	−0.199973	−0.0999866	−	0.994989i	$-0.531880\pi$
$479$	−2.78288e27	−0.199973	−0.0999866	+	0.994989i	$0.531880\pi$
$480$	0	0
$481$	6.21472e27	0.427463
$482$	2.55164e28	1.71722
$483$	−1.33515e27	−0.0879192
$484$	1.73990e28	1.12111
$485$	0	0
$486$	−1.37164e27	−0.0846367
$487$	−3.10293e27	−0.187378	−0.0936888	−	0.995602i	$-0.529866\pi$
$487$	−3.10293e27	−0.187378	−0.0936888	+	0.995602i	$0.529866\pi$
$488$	1.00373e28	0.593208
$489$	3.69370e27	0.213658
$490$	0	0
$491$	2.09221e28	1.15944	0.579722	−	0.814814i	$-0.303161\pi$
$491$	2.09221e28	1.15944	0.579722	+	0.814814i	$0.303161\pi$
$492$	1.59647e27	0.0866017
$493$	4.31209e28	2.28979
$494$	4.82842e28	2.50999
$495$	0	0
$496$	1.10226e28	0.549193
$497$	−7.49179e27	−0.365463
$498$	2.01215e28	0.961065
$499$	1.04897e28	0.490579	0.245290	−	0.969450i	$-0.421117\pi$
$499$	1.04897e28	0.490579	0.245290	+	0.969450i	$0.421117\pi$
$500$	0	0
$501$	−2.37694e28	−1.06591
$502$	−5.68086e28	−2.49474
$503$	2.97064e28	1.27757	0.638787	−	0.769384i	$-0.279437\pi$
$503$	2.97064e28	1.27757	0.638787	+	0.769384i	$0.279437\pi$
$504$	−5.38345e26	−0.0226746
$505$	0	0
$506$	−3.96498e28	−1.60199
$507$	3.98733e27	0.157797
$508$	−1.18325e28	−0.458676
$509$	2.64672e27	0.100501	0.0502506	−	0.998737i	$-0.483998\pi$
$509$	2.64672e27	0.100501	0.0502506	+	0.998737i	$0.483998\pi$
$510$	0	0
$511$	1.99594e27	0.0727325
$512$	−2.96693e28	−1.05919
$513$	9.27670e27	0.324460
$514$	−7.35514e27	−0.252045
$515$	0	0
$516$	1.06415e28	0.350092
$517$	4.86745e28	1.56910
$518$	−3.14592e27	−0.0993767
$519$	−1.73984e28	−0.538582
$520$	0	0
$521$	6.28960e28	1.86994	0.934968	−	0.354732i	$-0.115428\pi$
$521$	6.28960e28	1.86994	0.934968	+	0.354732i	$0.115428\pi$
$522$	−1.79172e28	−0.522072
$523$	−3.27582e28	−0.935518	−0.467759	−	0.883856i	$-0.654939\pi$
$523$	−3.27582e28	−0.935518	−0.467759	+	0.883856i	$0.654939\pi$
$524$	7.66778e27	0.214630
$525$	0	0
$526$	−4.39632e28	−1.18233
$527$	−3.37087e28	−0.888651
$528$	−4.22224e28	−1.09116
$529$	−1.63196e28	−0.413451
$530$	0	0
$531$	1.36049e28	0.331286
$532$	−1.04004e28	−0.248300
$533$	−9.76224e27	−0.228515
$534$	4.03425e28	0.925934
$535$	0	0
$536$	−2.88295e27	−0.0636219
$537$	1.13469e28	0.245555
$538$	4.98736e28	1.05842
$539$	−7.31662e28	−1.52275
$540$	0	0
$541$	6.14011e28	1.22915	0.614575	−	0.788858i	$-0.289327\pi$
$541$	6.14011e28	1.22915	0.614575	+	0.788858i	$0.289327\pi$
$542$	−7.73180e28	−1.51806
$543$	−3.68951e28	−0.710511
$544$	1.25669e29	2.37377
$545$	0	0
$546$	−9.40337e27	−0.170907
$547$	−8.17077e28	−1.45679	−0.728394	−	0.685159i	$-0.759733\pi$
$547$	−8.17077e28	−1.45679	−0.728394	+	0.685159i	$0.759733\pi$
$548$	−1.66116e28	−0.290546
$549$	3.36842e28	0.577985
$550$	0	0
$551$	1.21178e29	2.00140
$552$	9.33514e27	0.151273
$553$	1.09126e28	0.173506
$554$	1.75002e28	0.273018
$555$	0	0
$556$	−7.28202e28	−1.09387
$557$	−3.97472e28	−0.585905	−0.292953	−	0.956127i	$-0.594638\pi$
$557$	−3.97472e28	−0.585905	−0.292953	+	0.956127i	$0.594638\pi$
$558$	1.40063e28	0.202612
$559$	−6.50719e28	−0.923781
$560$	0	0
$561$	1.29123e29	1.76560
$562$	−1.90310e28	−0.255406
$563$	5.18243e28	0.682645	0.341323	−	0.939946i	$-0.389125\pi$
$563$	5.18243e28	0.682645	0.341323	+	0.939946i	$0.389125\pi$
$564$	3.27351e28	0.423237
$565$	0	0
$566$	9.96084e28	1.24086
$567$	−1.80664e27	−0.0220927
$568$	5.23815e28	0.628811
$569$	1.24791e29	1.47063	0.735316	−	0.677724i	$-0.237034\pi$
$569$	1.24791e29	1.47063	0.735316	+	0.677724i	$0.237034\pi$
$570$	0	0
$571$	−1.35062e29	−1.53410	−0.767050	−	0.641587i	$-0.778276\pi$
$571$	−1.35062e29	−1.53410	−0.767050	+	0.641587i	$0.778276\pi$
$572$	−1.18827e29	−1.32512
$573$	9.93534e28	1.08783
$574$	4.94168e27	0.0531251
$575$	0	0
$576$	−1.38799e28	−0.143864
$577$	−1.52204e29	−1.54911	−0.774553	−	0.632509i	$-0.782025\pi$
$577$	−1.52204e29	−1.54911	−0.774553	+	0.632509i	$0.782025\pi$
$578$	−3.59142e29	−3.58942
$579$	−9.27766e27	−0.0910571
$580$	0	0
$581$	2.65029e28	0.250867
$582$	8.29099e28	0.770753
$583$	1.36541e29	1.24664
$584$	−1.39553e28	−0.125143
$585$	0	0
$586$	4.28182e28	0.370428
$587$	1.46825e29	1.24767	0.623836	−	0.781556i	$-0.285573\pi$
$587$	1.46825e29	1.24767	0.623836	+	0.781556i	$0.285573\pi$
$588$	−4.92066e28	−0.410735
$589$	−9.47282e28	−0.776728
$590$	0	0
$591$	8.15579e28	0.645353
$592$	5.80910e28	0.451576
$593$	−5.37030e28	−0.410133	−0.205066	−	0.978748i	$-0.565741\pi$
$593$	−5.37030e28	−0.410133	−0.205066	+	0.978748i	$0.565741\pi$
$594$	−5.36518e28	−0.402557
$595$	0	0
$596$	−5.55737e28	−0.402516
$597$	3.28070e28	0.233473
$598$	1.63059e29	1.14020
$599$	−1.28904e29	−0.885698	−0.442849	−	0.896596i	$-0.646032\pi$
$599$	−1.28904e29	−0.885698	−0.442849	+	0.896596i	$0.646032\pi$
$600$	0	0
$601$	−6.99775e28	−0.464277	−0.232138	−	0.972683i	$-0.574572\pi$
$601$	−6.99775e28	−0.464277	−0.232138	+	0.972683i	$0.574572\pi$
$602$	3.29396e28	0.214761
$603$	−9.67494e27	−0.0619893
$604$	1.39127e28	0.0876040
$605$	0	0
$606$	−1.60879e29	−0.978443
$607$	1.81524e28	0.108505	0.0542527	−	0.998527i	$-0.482722\pi$
$607$	1.81524e28	0.108505	0.0542527	+	0.998527i	$0.482722\pi$
$608$	3.53153e29	2.07480
$609$	−2.35995e28	−0.136277
$610$	0	0
$611$	−2.00172e29	−1.11679
$612$	8.68390e28	0.476239
$613$	−6.44516e27	−0.0347455	−0.0173727	−	0.999849i	$-0.505530\pi$
$613$	−6.44516e27	−0.0347455	−0.0173727	+	0.999849i	$0.505530\pi$
$614$	9.39585e28	0.497930
$615$	0	0
$616$	−2.10575e28	−0.107847
$617$	−4.01342e28	−0.202079	−0.101039	−	0.994882i	$-0.532217\pi$
$617$	−4.01342e28	−0.202079	−0.101039	+	0.994882i	$0.532217\pi$
$618$	−1.12521e29	−0.556997
$619$	−1.87217e29	−0.911155	−0.455578	−	0.890196i	$-0.650567\pi$
$619$	−1.87217e29	−0.911155	−0.455578	+	0.890196i	$0.650567\pi$
$620$	0	0
$621$	3.13280e28	0.147391
$622$	5.23691e28	0.242257
$623$	5.31368e28	0.241697
$624$	1.73638e29	0.776617
$625$	0	0
$626$	4.56614e29	1.97478
$627$	3.62860e29	1.54323
$628$	−1.58868e29	−0.664449
$629$	−1.77651e29	−0.730696
$630$	0	0
$631$	−2.81317e29	−1.11915	−0.559574	−	0.828781i	$-0.689035\pi$
$631$	−2.81317e29	−1.11915	−0.559574	+	0.828781i	$0.689035\pi$
$632$	−7.62992e28	−0.298532
$633$	1.86426e29	0.717409
$634$	−3.68507e29	−1.39479
$635$	0	0
$636$	9.18279e28	0.336260
$637$	3.00894e29	1.08380
$638$	−7.00835e29	−2.48312
$639$	1.75788e29	0.612675
$640$	0	0
$641$	−1.86851e29	−0.630212	−0.315106	−	0.949056i	$-0.602040\pi$
$641$	−1.86851e29	−0.630212	−0.315106	+	0.949056i	$0.602040\pi$
$642$	2.75970e29	0.915679
$643$	5.55873e29	1.81451	0.907257	−	0.420577i	$-0.138172\pi$
$643$	5.55873e29	1.81451	0.907257	+	0.420577i	$0.138172\pi$
$644$	−3.51226e28	−0.112794
$645$	0	0
$646$	−1.38023e30	−4.29053
$647$	3.28170e29	1.00370	0.501850	−	0.864955i	$-0.332653\pi$
$647$	3.28170e29	1.00370	0.501850	+	0.864955i	$0.332653\pi$
$648$	1.26318e28	0.0380125
$649$	5.32158e29	1.57569
$650$	0	0
$651$	1.84483e28	0.0528880
$652$	9.71672e28	0.274107
$653$	−1.46885e29	−0.407745	−0.203872	−	0.978997i	$-0.565353\pi$
$653$	−1.46885e29	−0.407745	−0.203872	+	0.978997i	$0.565353\pi$
$654$	−4.90458e28	−0.133979
$655$	0	0
$656$	−9.12508e28	−0.241405
$657$	−4.68329e28	−0.121931
$658$	1.01328e29	0.259631
$659$	1.14775e29	0.289434	0.144717	−	0.989473i	$-0.453773\pi$
$659$	1.14775e29	0.289434	0.144717	+	0.989473i	$0.453773\pi$
$660$	0	0
$661$	1.56152e29	0.381446	0.190723	−	0.981644i	$-0.438917\pi$
$661$	1.56152e29	0.381446	0.190723	+	0.981644i	$0.438917\pi$
$662$	1.42376e29	0.342317
$663$	−5.31012e29	−1.25664
$664$	−1.85304e29	−0.431639
$665$	0	0
$666$	7.38160e28	0.166599
$667$	4.09226e29	0.909163
$668$	−6.25282e29	−1.36749
$669$	−1.18029e29	−0.254106
$670$	0	0
$671$	1.31756e30	2.74907
$672$	−6.87767e28	−0.141275
$673$	−1.17411e29	−0.237438	−0.118719	−	0.992928i	$-0.537879\pi$
$673$	−1.17411e29	−0.237438	−0.118719	+	0.992928i	$0.537879\pi$
$674$	−1.12607e30	−2.24200
$675$	0	0
$676$	1.04892e29	0.202442
$677$	−5.88223e29	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$677$	−5.88223e29	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$678$	3.85687e29	0.721644
$679$	1.09204e29	0.201190
$680$	0	0
$681$	−2.84791e29	−0.508723
$682$	5.47861e29	0.963683
$683$	−7.06012e29	−1.22291	−0.611455	−	0.791279i	$-0.709416\pi$
$683$	−7.06012e29	−1.22291	−0.611455	+	0.791279i	$0.709416\pi$
$684$	2.44035e29	0.416258
$685$	0	0
$686$	−3.10897e29	−0.514296
$687$	−5.81536e26	−0.000947394 0
$688$	−6.08247e29	−0.975892
$689$	−5.61519e29	−0.887283
$690$	0	0
$691$	−1.36189e29	−0.208748	−0.104374	−	0.994538i	$-0.533284\pi$
$691$	−1.36189e29	−0.208748	−0.104374	+	0.994538i	$0.533284\pi$
$692$	−4.57686e29	−0.690961
$693$	−7.06671e28	−0.105080
$694$	1.00073e29	0.146570
$695$	0	0
$696$	1.65004e29	0.234476
$697$	2.79059e29	0.390618
$698$	1.30539e30	1.79994
$699$	1.97949e29	0.268871
$700$	0	0
$701$	−7.43242e29	−0.979695	−0.489847	−	0.871808i	$-0.662948\pi$
$701$	−7.43242e29	−0.979695	−0.489847	+	0.871808i	$0.662948\pi$
$702$	2.20641e29	0.286515
$703$	−4.99235e29	−0.638666
$704$	−5.42916e29	−0.684258
$705$	0	0
$706$	5.05230e29	0.618073
$707$	−2.11900e29	−0.255403
$708$	3.57893e29	0.425015
$709$	−5.48308e29	−0.641562	−0.320781	−	0.947153i	$-0.603945\pi$
$709$	−5.48308e29	−0.641562	−0.320781	+	0.947153i	$0.603945\pi$
$710$	0	0
$711$	−2.56054e29	−0.290871
$712$	−3.71525e29	−0.415861
$713$	−3.19902e29	−0.352840
$714$	2.68801e29	0.292145
$715$	0	0
$716$	2.98495e29	0.315029
$717$	4.59701e29	0.478106
$718$	−1.44892e30	−1.48504
$719$	−4.78069e29	−0.482878	−0.241439	−	0.970416i	$-0.577619\pi$
$719$	−4.78069e29	−0.482878	−0.241439	+	0.970416i	$0.577619\pi$
$720$	0	0
$721$	−1.48206e29	−0.145393
$722$	−2.51413e30	−2.43079
$723$	−7.88597e29	−0.751455
$724$	−9.70571e29	−0.911533
$725$	0	0
$726$	−1.26370e30	−1.15295
$727$	−1.28823e30	−1.15847	−0.579233	−	0.815162i	$-0.696648\pi$
$727$	−1.28823e30	−1.15847	−0.579233	+	0.815162i	$0.696648\pi$
$728$	8.65981e28	0.0767589
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.86012e30	1.57909
$732$	8.86103e29	0.741512
$733$	−3.07979e29	−0.254056	−0.127028	−	0.991899i	$-0.540544\pi$
$733$	−3.07979e29	−0.254056	−0.127028	+	0.991899i	$0.540544\pi$
$734$	−3.11574e30	−2.53368
$735$	0	0
$736$	1.19262e30	0.942507
$737$	−3.78437e29	−0.294839
$738$	−1.15952e29	−0.0890608
$739$	−2.56564e29	−0.194280	−0.0971401	−	0.995271i	$-0.530969\pi$
$739$	−2.56564e29	−0.194280	−0.0971401	+	0.995271i	$0.530969\pi$
$740$	0	0
$741$	−1.49225e30	−1.09837
$742$	2.84243e29	0.206276
$743$	1.41998e30	1.01601	0.508006	−	0.861354i	$-0.330383\pi$
$743$	1.41998e30	1.01601	0.508006	+	0.861354i	$0.330383\pi$
$744$	−1.28988e29	−0.0909984
$745$	0	0
$746$	3.25230e30	2.23065
$747$	−6.21866e29	−0.420562
$748$	3.39672e30	2.26513
$749$	3.63491e29	0.239020
$750$	0	0
$751$	−5.33570e29	−0.341171	−0.170585	−	0.985343i	$-0.554566\pi$
$751$	−5.33570e29	−0.341171	−0.170585	+	0.985343i	$0.554566\pi$
$752$	−1.87107e30	−1.17979
$753$	1.75570e30	1.09170
$754$	2.88216e30	1.76733
$755$	0	0
$756$	−4.75259e28	−0.0283433
$757$	2.22525e30	1.30880	0.654398	−	0.756150i	$-0.272922\pi$
$757$	2.22525e30	1.30880	0.654398	+	0.756150i	$0.272922\pi$
$758$	−8.82160e29	−0.511706
$759$	1.22540e30	0.701033
$760$	0	0
$761$	−1.05556e30	−0.587411	−0.293705	−	0.955896i	$-0.594888\pi$
$761$	−1.05556e30	−0.587411	−0.293705	+	0.955896i	$0.594888\pi$
$762$	8.59396e29	0.471700
$763$	−6.46002e28	−0.0349725
$764$	2.61361e30	1.39560
$765$	0	0
$766$	−8.60685e29	−0.447138
$767$	−2.18848e30	−1.12148
$768$	1.48937e30	0.752852
$769$	2.30142e30	1.14754	0.573772	−	0.819015i	$-0.305480\pi$
$769$	2.30142e30	1.14754	0.573772	+	0.819015i	$0.305480\pi$
$770$	0	0
$771$	2.27314e29	0.110295
$772$	−2.44060e29	−0.116819
$773$	−2.73934e30	−1.29348	−0.646742	−	0.762709i	$-0.723869\pi$
$773$	−2.73934e30	−1.29348	−0.646742	+	0.762709i	$0.723869\pi$
$774$	−7.72898e29	−0.360033
$775$	0	0
$776$	−7.63539e29	−0.346165
$777$	9.72262e28	0.0434873
$778$	2.12897e30	0.939472
$779$	7.84211e29	0.341420
$780$	0	0
$781$	6.87597e30	2.91406
$782$	−4.66112e30	−1.94903
$783$	5.53741e29	0.228459
$784$	2.81255e30	1.14494
$785$	0	0
$786$	−5.56914e29	−0.220725
$787$	2.65478e30	1.03823	0.519116	−	0.854704i	$-0.326261\pi$
$787$	2.65478e30	1.03823	0.519116	+	0.854704i	$0.326261\pi$
$788$	2.14548e30	0.827940
$789$	1.35870e30	0.517388
$790$	0	0
$791$	5.08004e29	0.188371
$792$	4.94094e29	0.180799
$793$	−5.41844e30	−1.95662
$794$	−5.89800e30	−2.10179
$795$	0	0
$796$	8.63027e29	0.299528
$797$	1.58925e30	0.544351	0.272175	−	0.962248i	$-0.412257\pi$
$797$	1.58925e30	0.544351	0.272175	+	0.962248i	$0.412257\pi$
$798$	7.55382e29	0.255350
$799$	5.72203e30	1.90901
$800$	0	0
$801$	−1.24681e30	−0.405189
$802$	6.53697e30	2.09675
$803$	−1.83188e30	−0.579941
$804$	−2.54511e29	−0.0795276
$805$	0	0
$806$	−2.25306e30	−0.685890
$807$	−1.54137e30	−0.463164
$808$	1.48157e30	0.439444
$809$	2.00336e30	0.586543	0.293271	−	0.956029i	$-0.405256\pi$
$809$	2.00336e30	0.586543	0.293271	+	0.956029i	$0.405256\pi$
$810$	0	0
$811$	−4.22476e30	−1.20527	−0.602633	−	0.798019i	$-0.705882\pi$
$811$	−4.22476e30	−1.20527	−0.602633	+	0.798019i	$0.705882\pi$
$812$	−6.20813e29	−0.174833
$813$	2.38955e30	0.664302
$814$	2.88733e30	0.792391
$815$	0	0
$816$	−4.96354e30	−1.32753
$817$	5.22729e30	1.38021
$818$	9.16666e30	2.38947
$819$	2.90616e29	0.0747891
$820$	0	0
$821$	−3.19976e30	−0.802629	−0.401315	−	0.915940i	$-0.631447\pi$
$821$	−3.19976e30	−0.802629	−0.401315	+	0.915940i	$0.631447\pi$
$822$	1.20650e30	0.298796
$823$	−2.75275e30	−0.673084	−0.336542	−	0.941669i	$-0.609257\pi$
$823$	−2.75275e30	−0.673084	−0.336542	+	0.941669i	$0.609257\pi$
$824$	1.03623e30	0.250162
$825$	0	0
$826$	1.10782e30	0.260722
$827$	−5.37854e30	−1.24985	−0.624923	−	0.780687i	$-0.714870\pi$
$827$	−5.37854e30	−1.24985	−0.624923	+	0.780687i	$0.714870\pi$
$828$	8.24119e29	0.189091
$829$	2.39350e30	0.542265	0.271132	−	0.962542i	$-0.412602\pi$
$829$	2.39350e30	0.542265	0.271132	+	0.962542i	$0.412602\pi$
$830$	0	0
$831$	−5.40854e29	−0.119473
$832$	2.23272e30	0.487013
$833$	−8.60121e30	−1.85262
$834$	5.28896e30	1.12493
$835$	0	0
$836$	9.54546e30	1.97984
$837$	−4.32873e29	−0.0886632
$838$	6.87871e30	1.39138
$839$	8.32505e30	1.66298	0.831488	−	0.555542i	$-0.187489\pi$
$839$	8.32505e30	1.66298	0.831488	+	0.555542i	$0.187489\pi$
$840$	0	0
$841$	2.10047e30	0.409222
$842$	−1.14318e31	−2.19958
$843$	5.88164e29	0.111766
$844$	4.90415e30	0.920382
$845$	0	0
$846$	−2.37757e30	−0.435255
$847$	−1.66447e30	−0.300954
$848$	−5.24869e30	−0.937335
$849$	−3.07845e30	−0.543002
$850$	0	0
$851$	−1.68594e30	−0.290123
$852$	4.62431e30	0.786016
$853$	6.77552e30	1.13757	0.568785	−	0.822486i	$-0.307414\pi$
$853$	6.77552e30	1.13757	0.568785	+	0.822486i	$0.307414\pi$
$854$	2.74283e30	0.454875
$855$	0	0
$856$	−2.54148e30	−0.411255
$857$	−5.10163e30	−0.815475	−0.407737	−	0.913099i	$-0.633682\pi$
$857$	−5.10163e30	−0.815475	−0.407737	+	0.913099i	$0.633682\pi$
$858$	8.63043e30	1.36275
$859$	−1.56926e30	−0.244775	−0.122387	−	0.992482i	$-0.539055\pi$
$859$	−1.56926e30	−0.244775	−0.122387	+	0.992482i	$0.539055\pi$
$860$	0	0
$861$	−1.52725e29	−0.0232476
$862$	2.97126e30	0.446801
$863$	4.25548e30	0.632172	0.316086	−	0.948731i	$-0.397631\pi$
$863$	4.25548e30	0.632172	0.316086	+	0.948731i	$0.397631\pi$
$864$	1.61378e30	0.236838
$865$	0	0
$866$	−8.86352e29	−0.126960
$867$	1.10995e31	1.57073
$868$	4.85306e29	0.0678513
$869$	−1.00156e31	−1.38347
$870$	0	0
$871$	1.55631e30	0.209848
$872$	4.51675e29	0.0601732
$873$	−2.56237e30	−0.337282
$874$	−1.30987e31	−1.70356
$875$	0	0
$876$	−1.23200e30	−0.156429
$877$	−1.11865e31	−1.40346	−0.701729	−	0.712444i	$-0.747588\pi$
$877$	−1.11865e31	−1.40346	−0.701729	+	0.712444i	$0.747588\pi$
$878$	−7.79501e30	−0.966328
$879$	−1.32332e30	−0.162099
$880$	0	0
$881$	−3.16790e29	−0.0378900	−0.0189450	−	0.999821i	$-0.506031\pi$
$881$	−3.16790e29	−0.0378900	−0.0189450	+	0.999821i	$0.506031\pi$
$882$	3.57390e30	0.422398
$883$	−1.10116e31	−1.28607	−0.643036	−	0.765836i	$-0.722325\pi$
$883$	−1.10116e31	−1.28607	−0.643036	+	0.765836i	$0.722325\pi$
$884$	−1.39689e31	−1.61218
$885$	0	0
$886$	−8.05998e29	−0.0908406
$887$	−1.01581e31	−1.13139	−0.565695	−	0.824614i	$-0.691392\pi$
$887$	−1.01581e31	−1.13139	−0.565695	+	0.824614i	$0.691392\pi$
$888$	−6.79791e29	−0.0748237
$889$	1.13195e30	0.123128
$890$	0	0
$891$	1.65814e30	0.176159
$892$	−3.10489e30	−0.325998
$893$	1.60800e31	1.66858
$894$	4.03634e30	0.413946
$895$	0	0
$896$	1.31242e30	0.131474
$897$	−5.03941e30	−0.498952
$898$	3.80913e30	0.372756
$899$	−5.65447e30	−0.546909
$900$	0	0
$901$	1.60513e31	1.51670
$902$	−4.53549e30	−0.423599
$903$	−1.01802e30	−0.0939795
$904$	−3.55189e30	−0.324109
$905$	0	0
$906$	−1.01049e30	−0.0900916
$907$	2.10279e31	1.85319	0.926593	−	0.376067i	$-0.122724\pi$
$907$	2.10279e31	1.85319	0.926593	+	0.376067i	$0.122724\pi$
$908$	−7.49176e30	−0.652653
$909$	4.97203e30	0.428167
$910$	0	0
$911$	2.09268e31	1.76100	0.880499	−	0.474047i	$-0.157207\pi$
$911$	2.09268e31	1.76100	0.880499	+	0.474047i	$0.157207\pi$
$912$	−1.39485e31	−1.16033
$913$	−2.43244e31	−2.00032
$914$	−1.62135e31	−1.31808
$915$	0	0
$916$	−1.52980e28	−0.00121544
$917$	−7.33535e29	−0.0576159
$918$	−6.30716e30	−0.489762
$919$	6.78455e30	0.520844	0.260422	−	0.965495i	$-0.416138\pi$
$919$	6.78455e30	0.520844	0.260422	+	0.965495i	$0.416138\pi$
$920$	0	0
$921$	−2.90384e30	−0.217894
$922$	8.88322e30	0.659016
$923$	−2.82772e31	−2.07405
$924$	−1.85898e30	−0.134809
$925$	0	0
$926$	−4.80017e30	−0.340284
$927$	3.47751e30	0.243742
$928$	2.10803e31	1.46091
$929$	−1.81521e31	−1.24383	−0.621915	−	0.783085i	$-0.713645\pi$
$929$	−1.81521e31	−1.24383	−0.621915	+	0.783085i	$0.713645\pi$
$930$	0	0
$931$	−2.41711e31	−1.61929
$932$	5.20729e30	0.344941
$933$	−1.61849e30	−0.106012
$934$	−6.79461e30	−0.440071
$935$	0	0
$936$	−2.03194e30	−0.128681
$937$	−2.01787e31	−1.26365	−0.631826	−	0.775110i	$-0.717694\pi$
$937$	−2.01787e31	−1.26365	−0.631826	+	0.775110i	$0.717694\pi$
$938$	−7.87810e29	−0.0487856
$939$	−1.41119e31	−0.864164
$940$	0	0
$941$	3.28574e31	1.96762	0.983812	−	0.179204i	$-0.0573522\pi$
$941$	3.28574e31	1.96762	0.983812	+	0.179204i	$0.0573522\pi$
$942$	1.15387e31	0.683316
$943$	2.64832e30	0.155095
$944$	−2.04564e31	−1.18474
$945$	0	0
$946$	−3.02320e31	−1.71242
$947$	1.21780e31	0.682184	0.341092	−	0.940030i	$-0.389203\pi$
$947$	1.21780e31	0.682184	0.341092	+	0.940030i	$0.389203\pi$
$948$	−6.73580e30	−0.373166
$949$	7.53354e30	0.412766
$950$	0	0
$951$	1.13889e31	0.610360
$952$	−2.47545e30	−0.131210
$953$	−2.82199e31	−1.47938	−0.739691	−	0.672946i	$-0.765028\pi$
$953$	−2.82199e31	−1.47938	−0.739691	+	0.672946i	$0.765028\pi$
$954$	−6.66950e30	−0.345808
$955$	0	0
$956$	1.20930e31	0.613374
$957$	2.16597e31	1.08662
$958$	5.31706e30	0.263836
$959$	1.58914e30	0.0779949
$960$	0	0
$961$	−1.64053e31	−0.787748
$962$	−1.18740e31	−0.563975
$963$	−8.52898e30	−0.400702
$964$	−2.07450e31	−0.964060
$965$	0	0
$966$	2.55097e30	0.115997
$967$	2.53583e31	1.14062	0.570311	−	0.821429i	$-0.306823\pi$
$967$	2.53583e31	1.14062	0.570311	+	0.821429i	$0.306823\pi$
$968$	1.16377e31	0.517818
$969$	4.26568e31	1.87754
$970$	0	0
$971$	2.75874e31	1.18825	0.594127	−	0.804372i	$-0.297498\pi$
$971$	2.75874e31	1.18825	0.594127	+	0.804372i	$0.297498\pi$
$972$	1.11515e30	0.0475157
$973$	6.96632e30	0.293642
$974$	5.92855e30	0.247218
$975$	0	0
$976$	−5.06478e31	−2.06699
$977$	1.14091e31	0.460635	0.230318	−	0.973116i	$-0.426023\pi$
$977$	1.14091e31	0.460635	0.230318	+	0.973116i	$0.426023\pi$
$978$	−7.05729e30	−0.281890
$979$	−4.87691e31	−1.92720
$980$	0	0
$981$	1.51578e30	0.0586291
$982$	−3.99744e31	−1.52972
$983$	2.12152e31	0.803223	0.401611	−	0.915810i	$-0.368450\pi$
$983$	2.12152e31	0.803223	0.401611	+	0.915810i	$0.368450\pi$
$984$	1.06783e30	0.0399995
$985$	0	0
$986$	−8.23881e31	−3.02104
$987$	−3.13159e30	−0.113615
$988$	−3.92554e31	−1.40913
$989$	1.76528e31	0.626980
$990$	0	0
$991$	1.91318e31	0.665247	0.332624	−	0.943060i	$-0.392066\pi$
$991$	1.91318e31	0.665247	0.332624	+	0.943060i	$0.392066\pi$
$992$	−1.64790e31	−0.566967
$993$	−4.40020e30	−0.149798
$994$	1.43140e31	0.482175
$995$	0	0
$996$	−1.63589e31	−0.539550
$997$	1.44914e30	0.0472945	0.0236473	−	0.999720i	$-0.492472\pi$
$997$	1.44914e30	0.0472945	0.0236473	+	0.999720i	$0.492472\pi$
$998$	−2.00420e31	−0.647248
$999$	−2.28132e30	−0.0729036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.k.1.2	✓	8
5.2	odd	4		75.22.b.j.49.4		16
5.3	odd	4		75.22.b.j.49.13		16
5.4	even	2		75.22.a.l.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.k.1.2	✓	8	1.1	even	1	trivial
75.22.a.l.1.7	yes	8	5.4	even	2
75.22.b.j.49.4		16	5.2	odd	4
75.22.b.j.49.13		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)

\(f(q)\)	\(=\)	\(q-1910.63 q^{2} +59049.0 q^{3} +1.55335e6 q^{4} -1.12821e8 q^{6} -1.48601e8 q^{7} +1.03900e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-1910.63 q^{2} +59049.0 q^{3} +1.55335e6 q^{4} -1.12821e8 q^{6} -1.48601e8 q^{7} +1.03900e9 q^{8} +3.48678e9 q^{9} +1.36386e11 q^{11} +9.17240e10 q^{12} -5.60884e11 q^{13} +2.83921e11 q^{14} -5.24276e12 q^{16} +1.60332e13 q^{17} -6.66195e12 q^{18} +4.50564e13 q^{19} -8.77474e12 q^{21} -2.60584e14 q^{22} +1.52158e14 q^{23} +6.13517e13 q^{24} +1.07164e15 q^{26} +2.05891e14 q^{27} -2.30830e14 q^{28} +2.68948e15 q^{29} -2.10244e15 q^{31} +7.83804e15 q^{32} +8.05347e15 q^{33} -3.06335e16 q^{34} +5.41621e15 q^{36} -1.10802e16 q^{37} -8.60860e16 q^{38} -3.31196e16 q^{39} +1.74051e16 q^{41} +1.67653e16 q^{42} +1.16017e17 q^{43} +2.11856e17 q^{44} -2.90717e17 q^{46} +3.56887e17 q^{47} -3.09580e17 q^{48} -5.36464e17 q^{49} +9.46743e17 q^{51} -8.71251e17 q^{52} +1.00113e18 q^{53} -3.93382e17 q^{54} -1.54396e17 q^{56} +2.66053e18 q^{57} -5.13861e18 q^{58} +3.90185e18 q^{59} +9.66054e18 q^{61} +4.01698e18 q^{62} -5.18140e17 q^{63} -3.98073e18 q^{64} -1.53872e19 q^{66} -2.77474e18 q^{67} +2.49052e19 q^{68} +8.98477e18 q^{69} +5.04155e19 q^{71} +3.62275e18 q^{72} -1.34316e19 q^{73} +2.11702e19 q^{74} +6.99885e19 q^{76} -2.02671e19 q^{77} +6.32793e19 q^{78} -7.34355e19 q^{79} +1.21577e19 q^{81} -3.32547e19 q^{82} -1.78349e20 q^{83} -1.36303e19 q^{84} -2.21665e20 q^{86} +1.58811e20 q^{87} +1.41705e20 q^{88} -3.57581e20 q^{89} +8.33479e19 q^{91} +2.36355e20 q^{92} -1.24147e20 q^{93} -6.81879e20 q^{94} +4.62828e20 q^{96} -7.34881e20 q^{97} +1.02498e21 q^{98} +4.75549e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 666 q^{2} + 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} - 134034472 q^{7} - 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) 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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