LMFDB - Embedded newform 75.22.a.l.1.7

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			1.7
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	$-0.270693\pi$
$2$	1910.63	1.31935	0.659677	+	0.751549i	$0.270693\pi$
$3$	−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	$-0.468302\pi$
$7$	1.48601e8	0.198835	0.0994174	+	0.995046i	$0.468302\pi$
$8$	−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	$-0.309182\pi$
$13$	5.60884e11	1.12841	0.564206	+	0.825634i	$0.309182\pi$
$14$	2.83921e11	0.262334
$15$	0	0
$16$	−5.24276e12	−1.19206
$17$	−1.60332e13	−1.92888	−0.964441	−	0.264298i	$-0.914860\pi$
$17$	−1.60332e13	−1.92888	−0.964441	+	0.264298i	$0.914860\pi$
$18$	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	$-0.625086\pi$
$23$	−1.52158e14	−0.765865	−0.382932	+	0.923776i	$0.625086\pi$
$24$	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	$-0.439343\pi$
$37$	1.10802e16	0.378818	0.189409	+	0.981898i	$0.439343\pi$
$38$	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	$-0.634237\pi$
$43$	−1.16017e17	−0.818656	−0.409328	+	0.912387i	$0.634237\pi$
$44$	2.11856e17	1.17432
$45$	0	0
$46$	−2.90717e17	−1.01045
$47$	−3.56887e17	−0.989700	−0.494850	−	0.868979i	$-0.664777\pi$
$47$	−3.56887e17	−0.989700	−0.494850	+	0.868979i	$0.664777\pi$
$48$	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	$-0.628617\pi$
$53$	−1.00113e18	−0.786312	−0.393156	+	0.919472i	$0.628617\pi$
$54$	−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	$-0.470359\pi$
$67$	2.77474e18	0.185968	0.0929839	+	0.995668i	$0.470359\pi$
$68$	−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	$-0.441453\pi$
$73$	1.34316e19	0.365794	0.182897	+	0.983132i	$0.441453\pi$
$74$	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	$-0.282709\pi$
$83$	1.78349e20	1.26169	0.630844	+	0.775910i	$0.282709\pi$
$84$	−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	$-0.331152\pi$
$97$	7.34881e20	1.01185	0.505923	+	0.862579i	$0.331152\pi$
$98$	−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	$-0.619141\pi$
$103$	−9.97339e20	−0.731227	−0.365613	+	0.930767i	$0.619141\pi$
$104$	−5.82756e20	−0.386044
$105$	0	0
$106$	−1.91279e21	−1.03742
$107$	2.44609e21	1.20211	0.601053	−	0.799209i	$-0.294748\pi$
$107$	2.44609e21	1.20211	0.601053	+	0.799209i	$0.294748\pi$
$108$	−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	$-0.342923\pi$
$113$	3.41858e21	0.947376	0.473688	+	0.880693i	$0.342923\pi$
$114$	−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	$-0.399797\pi$
$127$	7.61735e21	0.619249	0.309624	+	0.950859i	$0.399797\pi$
$128$	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	$-0.437163\pi$
$137$	1.06940e22	0.392260	0.196130	+	0.980578i	$0.437163\pi$
$138$	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	$-0.351948\pi$
$157$	1.02274e23	0.897058	0.448529	+	0.893768i	$0.351948\pi$
$158$	−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	$-0.559239\pi$
$163$	−6.25531e22	−0.370066	−0.185033	+	0.982732i	$0.559239\pi$
$164$	2.70363e22	0.149999
$165$	0	0
$166$	3.40760e23	1.66461
$167$	4.02537e23	1.84622	0.923108	−	0.384541i	$-0.125640\pi$
$167$	4.02537e23	1.84622	0.923108	+	0.384541i	$0.125640\pi$
$168$	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	$-0.345541\pi$
$173$	2.94644e23	0.932852	0.466426	+	0.884560i	$0.345541\pi$
$174$	−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	$-0.474873\pi$
$193$	1.57118e23	0.157715	0.0788577	+	0.996886i	$0.474873\pi$
$194$	1.40409e24	1.33498
$195$	0	0
$196$	−8.33318e23	−0.711414
$197$	−1.38119e24	−1.11778	−0.558892	−	0.829240i	$-0.688773\pi$
$197$	−1.38119e24	−1.11778	−0.558892	+	0.829240i	$0.688773\pi$
$198$	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	$-0.429374\pi$
$223$	1.99883e24	0.440124	0.220062	+	0.975486i	$0.429374\pi$
$224$	−1.16474e24	−0.244695
$225$	0	0
$226$	6.53164e24	1.24992
$227$	4.82296e24	0.881134	0.440567	−	0.897720i	$-0.354778\pi$
$227$	4.82296e24	0.881134	0.440567	+	0.897720i	$0.354778\pi$
$228$	−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	$-0.574805\pi$
$233$	−3.35229e24	−0.465698	−0.232849	+	0.972513i	$0.574805\pi$
$234$	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	$-0.530451\pi$
$257$	−3.84959e24	−0.191037	−0.0955183	+	0.995428i	$0.530451\pi$
$258$	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	$-0.647889\pi$
$263$	−2.30098e25	−0.896142	−0.448071	+	0.893998i	$0.647889\pi$
$264$	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	$-0.467007\pi$
$277$	9.15941e24	0.206933	0.103467	+	0.994633i	$0.467007\pi$
$278$	−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	$-0.344163\pi$
$283$	5.21338e25	0.940507	0.470253	+	0.882531i	$0.344163\pi$
$284$	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	$-0.455167\pi$
$293$	2.24105e25	0.280765	0.140382	+	0.990097i	$0.455167\pi$
$294$	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	$-0.439572\pi$
$307$	4.91767e25	0.377404	0.188702	+	0.982034i	$0.439572\pi$
$308$	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	$-0.230828\pi$
$313$	2.38986e26	1.49678	0.748388	+	0.663261i	$0.230828\pi$
$314$	1.95409e26	1.18354
$315$	0	0
$316$	−1.14071e26	−0.646342
$317$	−1.92872e26	−1.05717	−0.528587	−	0.848879i	$-0.677278\pi$
$317$	−1.92872e26	−1.05717	−0.528587	+	0.848879i	$0.677278\pi$
$318$	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	$-0.823191\pi$
$337$	−5.89369e26	−1.69931	−0.849656	+	0.527337i	$0.823191\pi$
$338$	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	$-0.482310\pi$
$347$	5.23771e25	0.111092	0.0555459	+	0.998456i	$0.482310\pi$
$348$	−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	$-0.424742\pi$
$353$	2.64431e26	0.468466	0.234233	+	0.972180i	$0.424742\pi$
$354$	−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	$-0.909890\pi$
$367$	−1.63074e27	−1.92039	−0.960197	+	0.279324i	$0.909890\pi$
$368$	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	$-0.179387\pi$
$373$	1.70221e27	1.69072	0.845358	+	0.534200i	$0.179387\pi$
$374$	−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	$-0.554200\pi$
$383$	−4.50472e26	−0.338907	−0.169453	+	0.985538i	$0.554200\pi$
$384$	−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	$-0.793329\pi$
$397$	−3.08694e27	−1.59304	−0.796522	+	0.604610i	$0.793329\pi$
$398$	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	$-0.515321\pi$
$433$	−4.63906e26	−0.0962292	−0.0481146	+	0.998842i	$0.515321\pi$
$434$	−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	$-0.510960\pi$
$443$	−4.21850e26	−0.0688523	−0.0344262	+	0.999407i	$0.510960\pi$
$444$	−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	$-0.666490\pi$
$457$	−8.48597e27	−0.999036	−0.499518	+	0.866303i	$0.666490\pi$
$458$	−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	$-0.541163\pi$
$463$	−2.51235e27	−0.257917	−0.128959	+	0.991650i	$0.541163\pi$
$464$	−1.41003e28	−1.41511
$465$	0	0
$466$	−6.40498e27	−0.614421
$467$	−3.55621e27	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$467$	−3.55621e27	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$468$	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	$-0.470134\pi$
$487$	3.10293e27	0.187378	0.0936888	+	0.995602i	$0.470134\pi$
$488$	−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	$-0.720563\pi$
$503$	−2.97064e28	−1.27757	−0.638787	+	0.769384i	$0.720563\pi$
$504$	−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	$-0.345061\pi$
$523$	3.27582e28	0.935518	0.467759	+	0.883856i	$0.345061\pi$
$524$	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	$-0.240267\pi$
$547$	8.17077e28	1.45679	0.728394	+	0.685159i	$0.240267\pi$
$548$	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	$-0.405362\pi$
$557$	3.97472e28	0.585905	0.292953	+	0.956127i	$0.405362\pi$
$558$	−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	$-0.610875\pi$
$563$	−5.18243e28	−0.682645	−0.341323	+	0.939946i	$0.610875\pi$
$564$	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	$-0.217975\pi$
$577$	1.52204e29	1.54911	0.774553	+	0.632509i	$0.217975\pi$
$578$	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	$-0.714427\pi$
$587$	−1.46825e29	−1.24767	−0.623836	+	0.781556i	$0.714427\pi$
$588$	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	$-0.434259\pi$
$593$	5.37030e28	0.410133	0.205066	+	0.978748i	$0.434259\pi$
$594$	−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	$-0.517278\pi$
$607$	−1.81524e28	−0.108505	−0.0542527	+	0.998527i	$0.517278\pi$
$608$	−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	$-0.494470\pi$
$613$	6.44516e27	0.0347455	0.0173727	+	0.999849i	$0.494470\pi$
$614$	9.39585e28	0.497930
$615$	0	0
$616$	−2.10575e28	−0.107847
$617$	4.01342e28	0.202079	0.101039	−	0.994882i	$-0.467783\pi$
$617$	4.01342e28	0.202079	0.101039	+	0.994882i	$0.467783\pi$
$618$	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	$-0.861828\pi$
$643$	−5.55873e29	−1.81451	−0.907257	+	0.420577i	$0.861828\pi$
$644$	−3.51226e28	−0.112794
$645$	0	0
$646$	−1.38023e30	−4.29053
$647$	−3.28170e29	−1.00370	−0.501850	−	0.864955i	$-0.667347\pi$
$647$	−3.28170e29	−1.00370	−0.501850	+	0.864955i	$0.667347\pi$
$648$	−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	$-0.434647\pi$
$653$	1.46885e29	0.407745	0.203872	+	0.978997i	$0.434647\pi$
$654$	−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	$-0.462121\pi$
$673$	1.17411e29	0.237438	0.118719	+	0.992928i	$0.462121\pi$
$674$	−1.12607e30	−2.24200
$675$	0	0
$676$	1.04892e29	0.202442
$677$	5.88223e29	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$677$	5.88223e29	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$678$	−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	$-0.290584\pi$
$683$	7.06012e29	1.22291	0.611455	+	0.791279i	$0.290584\pi$
$684$	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	$-0.303352\pi$
$727$	1.28823e30	1.15847	0.579233	+	0.815162i	$0.303352\pi$
$728$	−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	$-0.459456\pi$
$733$	3.07979e29	0.254056	0.127028	+	0.991899i	$0.459456\pi$
$734$	−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	$-0.669617\pi$
$743$	−1.41998e30	−1.01601	−0.508006	+	0.861354i	$0.669617\pi$
$744$	−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	$-0.727078\pi$
$757$	−2.22525e30	−1.30880	−0.654398	+	0.756150i	$0.727078\pi$
$758$	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	$-0.276131\pi$
$773$	2.73934e30	1.29348	0.646742	+	0.762709i	$0.276131\pi$
$774$	−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	$-0.673739\pi$
$787$	−2.65478e30	−1.03823	−0.519116	+	0.854704i	$0.673739\pi$
$788$	−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	$-0.587743\pi$
$797$	−1.58925e30	−0.544351	−0.272175	+	0.962248i	$0.587743\pi$
$798$	−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	$-0.390743\pi$
$823$	2.75275e30	0.673084	0.336542	+	0.941669i	$0.390743\pi$
$824$	1.03623e30	0.250162
$825$	0	0
$826$	1.10782e30	0.260722
$827$	5.37854e30	1.24985	0.624923	−	0.780687i	$-0.285130\pi$
$827$	5.37854e30	1.24985	0.624923	+	0.780687i	$0.285130\pi$
$828$	−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	$-0.692586\pi$
$853$	−6.77552e30	−1.13757	−0.568785	+	0.822486i	$0.692586\pi$
$854$	2.74283e30	0.454875
$855$	0	0
$856$	−2.54148e30	−0.411255
$857$	5.10163e30	0.815475	0.407737	−	0.913099i	$-0.366318\pi$
$857$	5.10163e30	0.815475	0.407737	+	0.913099i	$0.366318\pi$
$858$	−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	$-0.602369\pi$
$863$	−4.25548e30	−0.632172	−0.316086	+	0.948731i	$0.602369\pi$
$864$	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	$-0.252412\pi$
$877$	1.11865e31	1.40346	0.701729	+	0.712444i	$0.252412\pi$
$878$	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	$-0.277675\pi$
$883$	1.10116e31	1.28607	0.643036	+	0.765836i	$0.277675\pi$
$884$	−1.39689e31	−1.61218
$885$	0	0
$886$	−8.05998e29	−0.0908406
$887$	1.01581e31	1.13139	0.565695	−	0.824614i	$-0.308608\pi$
$887$	1.01581e31	1.13139	0.565695	+	0.824614i	$0.308608\pi$
$888$	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	$-0.877276\pi$
$907$	−2.10279e31	−1.85319	−0.926593	+	0.376067i	$0.877276\pi$
$908$	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	$-0.282306\pi$
$937$	2.01787e31	1.26365	0.631826	+	0.775110i	$0.282306\pi$
$938$	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	$-0.610797\pi$
$947$	−1.21780e31	−0.682184	−0.341092	+	0.940030i	$0.610797\pi$
$948$	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	$-0.234972\pi$
$953$	2.82199e31	1.47938	0.739691	+	0.672946i	$0.234972\pi$
$954$	−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	$-0.693177\pi$
$967$	−2.53583e31	−1.14062	−0.570311	+	0.821429i	$0.693177\pi$
$968$	−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	$-0.573977\pi$
$977$	−1.14091e31	−0.460635	−0.230318	+	0.973116i	$0.573977\pi$
$978$	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	$-0.631550\pi$
$983$	−2.12152e31	−0.803223	−0.401611	+	0.915810i	$0.631550\pi$
$984$	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	$-0.507528\pi$
$997$	−1.44914e30	−0.0472945	−0.0236473	+	0.999720i	$0.507528\pi$
$998$	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.l.1.7	yes	8
5.2	odd	4		75.22.b.j.49.13		16
5.3	odd	4		75.22.b.j.49.4		16
5.4	even	2		75.22.a.k.1.2	✓	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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