LMFDB - Embedded newform 324.4.i.a.73.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(37,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.i (of order $9$, degree $6$, not minimal)

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:	no
Analytic conductor:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			73.9
Character	$\chi$	$=$	324.73
Dual form			324.4.i.a.253.9

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(14.9393 - 12.5356i) q^{5} +(23.5233 - 8.56177i) q^{7} +O(q^{10})$	$q+(14.9393 - 12.5356i) q^{5} +(23.5233 - 8.56177i) q^{7} +(39.5319 + 33.1712i) q^{11} +(-10.2025 - 57.8615i) q^{13} +(3.52013 + 6.09704i) q^{17} +(-42.6669 + 73.9012i) q^{19} +(-32.2883 - 11.7520i) q^{23} +(44.3365 - 251.445i) q^{25} +(-37.0590 + 210.172i) q^{29} +(-133.967 - 48.7600i) q^{31} +(244.095 - 422.784i) q^{35} +(-65.2996 - 113.102i) q^{37} +(8.52460 + 48.3454i) q^{41} +(126.716 + 106.327i) q^{43} +(470.489 - 171.244i) q^{47} +(217.287 - 182.325i) q^{49} +347.277 q^{53} +1006.40 q^{55} +(172.501 - 144.746i) q^{59} +(-577.795 + 210.300i) q^{61} +(-877.746 - 736.517i) q^{65} +(-162.166 - 919.690i) q^{67} +(-46.7209 - 80.9229i) q^{71} +(-133.977 + 232.055i) q^{73} +(1213.92 + 441.832i) q^{77} +(-155.815 + 883.669i) q^{79} +(-6.17283 + 35.0079i) q^{83} +(129.018 + 46.9588i) q^{85} +(-361.721 + 626.519i) q^{89} +(-735.394 - 1273.74i) q^{91} +(288.980 + 1638.89i) q^{95} +(-1214.37 - 1018.98i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 12 q^{5}+O(q^{10}) $ 54 * q - 12 * q^5	$ 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) $ 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{5}{9}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	14.9393	−	12.5356i	1.33621	−	1.12122i	0.353631	−	0.935385i	$-0.384947\pi$
$5$	14.9393	−	12.5356i	1.33621	−	1.12122i	0.982582	−	0.185831i	$-0.0594976\pi$
$6$		0			0
$7$	23.5233	−	8.56177i	1.27014	−	0.462292i	0.382978	−	0.923757i	$-0.374898\pi$
$7$	23.5233	−	8.56177i	1.27014	−	0.462292i	0.887158	+	0.461465i	$0.152676\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	39.5319	+	33.1712i	1.08358	+	0.909227i	0.996213	−	0.0869492i	$-0.0277118\pi$
$11$	39.5319	+	33.1712i	1.08358	+	0.909227i	0.0873623	+	0.996177i	$0.472156\pi$
$12$		0			0
$13$	−10.2025	−	57.8615i	−0.217668	−	1.23445i	−0.876217	−	0.481917i	$-0.839941\pi$
$13$	−10.2025	−	57.8615i	−0.217668	−	1.23445i	0.658550	−	0.752537i	$-0.271170\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.52013	+	6.09704i	0.0502210	+	0.0869853i	0.890043	−	0.455876i	$-0.150674\pi$
$17$	3.52013	+	6.09704i	0.0502210	+	0.0869853i	−0.839822	+	0.542862i	$0.817341\pi$
$18$		0			0
$19$	−42.6669	+	73.9012i	−0.515182	+	0.892321i	0.484663	+	0.874701i	$0.338942\pi$
$19$	−42.6669	+	73.9012i	−0.515182	+	0.892321i	−0.999845	+	0.0176203i	$0.994391\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32.2883	−	11.7520i	−0.292721	−	0.106542i	0.191486	−	0.981495i	$-0.438669\pi$
$23$	−32.2883	−	11.7520i	−0.292721	−	0.106542i	−0.484206	+	0.874954i	$0.660892\pi$
$24$		0			0
$25$	44.3365	−	251.445i	0.354692	−	2.01156i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−37.0590	+	210.172i	−0.237300	+	1.34579i	0.600417	+	0.799687i	$0.295001\pi$
$29$	−37.0590	+	210.172i	−0.237300	+	1.34579i	−0.837717	+	0.546105i	$0.816110\pi$
$30$		0			0
$31$	−133.967	−	48.7600i	−0.776168	−	0.282502i	−0.0765940	−	0.997062i	$-0.524405\pi$
$31$	−133.967	−	48.7600i	−0.776168	−	0.282502i	−0.699574	+	0.714560i	$0.746627\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	244.095	−	422.784i	1.17884	−	2.04182i
$36$		0			0
$37$	−65.2996	−	113.102i	−0.290140	−	0.502538i	0.683702	−	0.729761i	$-0.260369\pi$
$37$	−65.2996	−	113.102i	−0.290140	−	0.502538i	−0.973843	+	0.227223i	$0.927035\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.52460	+	48.3454i	0.0324712	+	0.184153i	0.996729	−	0.0808109i	$-0.0257510\pi$
$41$	8.52460	+	48.3454i	0.0324712	+	0.184153i	−0.964258	+	0.264964i	$0.914640\pi$
$42$		0			0
$43$	126.716	+	106.327i	0.449394	+	0.377086i	0.839211	−	0.543806i	$-0.183017\pi$
$43$	126.716	+	106.327i	0.449394	+	0.377086i	−0.389817	+	0.920892i	$0.627462\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	470.489	−	171.244i	1.46017	−	0.531458i	0.514757	−	0.857336i	$-0.327882\pi$
$47$	470.489	−	171.244i	1.46017	−	0.531458i	0.945412	+	0.325878i	$0.105660\pi$
$48$		0			0
$49$	217.287	−	182.325i	0.633489	−	0.531560i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	347.277			0.900042			0.450021	−	0.893018i	$-0.351416\pi$
$53$	347.277			0.900042			0.450021	+	0.893018i	$0.351416\pi$
$54$		0			0
$55$	1006.40			2.46733
$56$		0			0
$57$		0			0
$58$		0			0
$59$	172.501	−	144.746i	0.380640	−	0.319395i	−0.432314	−	0.901723i	$-0.642303\pi$
$59$	172.501	−	144.746i	0.380640	−	0.319395i	0.812954	+	0.582329i	$0.197858\pi$
$60$		0			0
$61$	−577.795	+	210.300i	−1.21277	+	0.441413i	−0.867664	−	0.497150i	$-0.834380\pi$
$61$	−577.795	+	210.300i	−1.21277	+	0.441413i	−0.345107	+	0.938563i	$0.612158\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−877.746	−	736.517i	−1.67494	−	1.40544i
$66$		0			0
$67$	−162.166	−	919.690i	−0.295698	−	1.67698i	−0.664353	−	0.747419i	$-0.731293\pi$
$67$	−162.166	−	919.690i	−0.295698	−	1.67698i	0.368656	−	0.929566i	$-0.379818\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−46.7209	−	80.9229i	−0.0780950	−	0.135265i	0.824333	−	0.566105i	$-0.191550\pi$
$71$	−46.7209	−	80.9229i	−0.0780950	−	0.135265i	−0.902428	+	0.430841i	$0.858217\pi$
$72$		0			0
$73$	−133.977	+	232.055i	−0.214806	+	0.372055i	−0.953213	−	0.302301i	$-0.902245\pi$
$73$	−133.977	+	232.055i	−0.214806	+	0.372055i	0.738406	+	0.674356i	$0.235579\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1213.92	+	441.832i	1.79662	+	0.653915i
$78$		0			0
$79$	−155.815	+	883.669i	−0.221905	+	1.25849i	0.646608	+	0.762822i	$0.276187\pi$
$79$	−155.815	+	883.669i	−0.221905	+	1.25849i	−0.868514	+	0.495665i	$0.834924\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.17283	+	35.0079i	−0.00816333	+	0.0462966i	−0.988617	−	0.150452i	$-0.951927\pi$
$83$	−6.17283	+	35.0079i	−0.00816333	+	0.0462966i	0.980454	+	0.196748i	$0.0630382\pi$
$84$		0			0
$85$	129.018	+	46.9588i	0.164635	+	0.0599223i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−361.721	+	626.519i	−0.430813	+	0.746189i	−0.996943	−	0.0781261i	$-0.975106\pi$
$89$	−361.721	+	626.519i	−0.430813	+	0.746189i	0.566131	+	0.824315i	$0.308440\pi$
$90$		0			0
$91$	−735.394	−	1273.74i	−0.847146	−	1.46730i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	288.980	+	1638.89i	0.312092	+	1.76996i
$96$		0			0
$97$	−1214.37	−	1018.98i	−1.27114	−	1.06662i	−0.994401	−	0.105673i	$-0.966300\pi$
$97$	−1214.37	−	1018.98i	−1.27114	−	1.06662i	−0.276744	−	0.960944i	$-0.589255\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1643.15	+	598.056i	−1.61880	+	0.589196i	−0.983153	−	0.182782i	$-0.941490\pi$
$101$	−1643.15	+	598.056i	−1.61880	+	0.589196i	−0.635649	+	0.771978i	$0.719268\pi$
$102$		0			0
$103$	717.798	−	602.304i	0.686667	−	0.576182i	−0.231279	−	0.972888i	$-0.574291\pi$
$103$	717.798	−	602.304i	0.686667	−	0.576182i	0.917946	+	0.396705i	$0.129846\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.1019			0.0145480			0.00727398	−	0.999974i	$-0.497685\pi$
$107$	16.1019			0.0145480			0.00727398	+	0.999974i	$0.497685\pi$
$108$		0			0
$109$	1665.12			1.46321			0.731603	−	0.681731i	$-0.238772\pi$
$109$	1665.12			1.46321			0.731603	+	0.681731i	$0.238772\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	632.247	−	530.518i	0.526343	−	0.441654i	−0.340493	−	0.940247i	$-0.610594\pi$
$113$	632.247	−	530.518i	0.526343	−	0.441654i	0.866836	+	0.498593i	$0.166150\pi$
$114$		0			0
$115$	−629.683	+	229.186i	−0.510593	+	0.185841i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	135.006	+	113.284i	0.104000	+	0.0872664i
$120$		0			0
$121$	231.317	+	1311.87i	0.173792	+	0.985625i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1270.78	−	2201.05i	−0.909295	−	1.57495i
$126$		0			0
$127$	−1130.88	+	1958.74i	−0.790151	+	1.36858i	0.135722	+	0.990747i	$0.456665\pi$
$127$	−1130.88	+	1958.74i	−0.790151	+	1.36858i	−0.925873	+	0.377835i	$0.876669\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1343.42	−	488.966i	−0.895996	−	0.326116i	−0.147349	−	0.989085i	$-0.547074\pi$
$131$	−1343.42	−	488.966i	−0.895996	−	0.326116i	−0.748647	+	0.662969i	$0.769296\pi$
$132$		0			0
$133$	−370.939	+	2103.70i	−0.241839	+	1.37153i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−260.376	+	1476.67i	−0.162375	+	0.920877i	0.789354	+	0.613938i	$0.210416\pi$
$137$	−260.376	+	1476.67i	−0.162375	+	0.920877i	−0.951729	+	0.306938i	$0.900695\pi$
$138$		0			0
$139$	−54.1508	−	19.7093i	−0.0330433	−	0.0120268i	0.325446	−	0.945561i	$-0.394486\pi$
$139$	−54.1508	−	19.7093i	−0.0330433	−	0.0120268i	−0.358489	+	0.933534i	$0.616708\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1516.01	−	2625.81i	0.886540	−	1.53553i
$144$		0			0
$145$	2080.99	+	3604.38i	1.19184	+	2.06433i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	236.688	+	1342.32i	0.130136	+	0.738037i	0.978124	+	0.208022i	$0.0667026\pi$
$149$	236.688	+	1342.32i	0.130136	+	0.738037i	−0.847988	+	0.530015i	$0.822186\pi$
$150$		0			0
$151$	897.782	+	753.329i	0.483844	+	0.405993i	0.851814	−	0.523844i	$-0.175503\pi$
$151$	897.782	+	753.329i	0.483844	+	0.405993i	−0.367970	+	0.929838i	$0.619947\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2612.61	+	950.913i	−1.35387	+	0.492769i
$156$		0			0
$157$	−977.108	+	819.891i	−0.496699	+	0.416780i	−0.856420	−	0.516280i	$-0.827316\pi$
$157$	−977.108	+	819.891i	−0.496699	+	0.416780i	0.359721	+	0.933060i	$0.382872\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−860.144			−0.421049
$162$		0			0
$163$	1616.42			0.776736			0.388368	−	0.921504i	$-0.373039\pi$
$163$	1616.42			0.776736			0.388368	+	0.921504i	$0.373039\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−424.273	+	356.007i	−0.196594	+	0.164962i	−0.735771	−	0.677231i	$-0.763180\pi$
$167$	−424.273	+	356.007i	−0.196594	+	0.164962i	0.539177	+	0.842193i	$0.318736\pi$
$168$		0			0
$169$	−1179.36	+	429.252i	−0.536805	+	0.195381i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−539.141	−	452.393i	−0.236937	−	0.198814i	0.516586	−	0.856235i	$-0.327203\pi$
$173$	−539.141	−	452.393i	−0.236937	−	0.198814i	−0.753523	+	0.657421i	$0.771647\pi$
$174$		0			0
$175$	−1109.87	−	6294.40i	−0.479420	−	2.71892i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	585.812	+	1014.66i	0.244612	+	0.423681i	0.962023	−	0.272970i	$-0.0880059\pi$
$179$	585.812	+	1014.66i	0.244612	+	0.423681i	−0.717410	+	0.696651i	$0.754673\pi$
$180$		0			0
$181$	−386.512	+	669.458i	−0.158725	+	0.274919i	−0.934409	−	0.356202i	$-0.884072\pi$
$181$	−386.512	+	669.458i	−0.158725	+	0.274919i	0.775684	+	0.631121i	$0.217405\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2393.33	−	871.102i	−0.951143	−	0.346188i
$186$		0			0
$187$	−63.0889	+	357.795i	−0.0246712	+	0.139917i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	98.8386	−	560.541i	0.0374435	−	0.212353i	−0.960346	−	0.278812i	$-0.910059\pi$
$191$	98.8386	−	560.541i	0.0374435	−	0.212353i	0.997789	+	0.0664597i	$0.0211704\pi$
$192$		0			0
$193$	−4565.75	−	1661.80i	−1.70285	−	0.619787i	−0.706705	−	0.707509i	$-0.749819\pi$
$193$	−4565.75	−	1661.80i	−1.70285	−	0.619787i	−0.996145	+	0.0877221i	$0.972041\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	946.267	−	1638.98i	0.342227	−	0.592754i	−0.642619	−	0.766186i	$-0.722152\pi$
$197$	946.267	−	1638.98i	0.342227	−	0.592754i	0.984846	+	0.173431i	$0.0554855\pi$
$198$		0			0
$199$	1950.36	+	3378.12i	0.694759	+	1.20336i	0.970262	+	0.242058i	$0.0778225\pi$
$199$	1950.36	+	3378.12i	0.694759	+	1.20336i	−0.275502	+	0.961300i	$0.588844\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	927.696	+	5261.22i	0.320746	+	1.81904i
$204$		0			0
$205$	733.389	+	615.386i	0.249864	+	0.209661i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4138.10	+	1506.15i	−1.36956	+	0.498480i
$210$		0			0
$211$	1142.91	−	959.011i	0.372895	−	0.312896i	−0.437011	−	0.899456i	$-0.643963\pi$
$211$	1142.91	−	959.011i	0.372895	−	0.312896i	0.809906	+	0.586560i	$0.199518\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3225.91			1.02328
$216$		0			0
$217$	−3568.81			−1.11644
$218$		0			0
$219$		0			0
$220$		0			0
$221$	316.870	−	265.885i	0.0964478	−	0.0809293i
$222$		0			0
$223$	−1925.56	+	700.848i	−0.578230	+	0.210459i	−0.614545	−	0.788882i	$-0.710660\pi$
$223$	−1925.56	+	700.848i	−0.578230	+	0.210459i	0.0363147	+	0.999340i	$0.488438\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1243.19	+	1043.16i	0.363496	+	0.305009i	0.806182	−	0.591667i	$-0.201530\pi$
$227$	1243.19	+	1043.16i	0.363496	+	0.305009i	−0.442686	+	0.896677i	$0.645974\pi$
$228$		0			0
$229$	651.857	+	3696.86i	0.188104	+	1.06679i	0.921901	+	0.387425i	$0.126635\pi$
$229$	651.857	+	3696.86i	0.188104	+	1.06679i	−0.733797	+	0.679369i	$0.762254\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	48.1965	+	83.4788i	0.0135513	+	0.0234716i	0.872722	−	0.488218i	$-0.162353\pi$
$233$	48.1965	+	83.4788i	0.0135513	+	0.0234716i	−0.859170	+	0.511690i	$0.829020\pi$
$234$		0			0
$235$	4882.15	−	8456.12i	1.35522	−	2.34731i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4215.25	+	1534.22i	1.14084	+	0.415233i	0.842218	−	0.539137i	$-0.181249\pi$
$239$	4215.25	+	1534.22i	1.14084	+	0.415233i	0.298626	+	0.954370i	$0.403472\pi$
$240$		0			0
$241$	420.948	−	2387.32i	0.112513	−	0.638093i	−0.875438	−	0.483330i	$-0.839427\pi$
$241$	420.948	−	2387.32i	0.112513	−	0.638093i	0.987952	−	0.154764i	$-0.0494616\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	960.563	−	5447.62i	0.250482	−	1.42055i
$246$		0			0
$247$	4711.35	+	1714.79i	1.21367	+	0.441739i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1200.00	−	2078.47i	0.301767	−	0.522677i	−0.674769	−	0.738029i	$-0.735757\pi$
$251$	1200.00	−	2078.47i	0.301767	−	0.522677i	0.976536	+	0.215352i	$0.0690900\pi$
$252$		0			0
$253$	−886.591	−	1535.62i	−0.220314	−	0.381595i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	65.0046	+	368.659i	0.0157777	+	0.0894799i	0.991680	−	0.128729i	$-0.0410897\pi$
$257$	65.0046	+	368.659i	0.0157777	+	0.0894799i	−0.975902	+	0.218209i	$0.929979\pi$
$258$		0			0
$259$	−2504.42	−	2101.45i	−0.600837	−	0.504162i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3775.09	−	1374.02i	0.885102	−	0.322151i	0.140835	−	0.990033i	$-0.455021\pi$
$263$	3775.09	−	1374.02i	0.885102	−	0.322151i	0.744267	+	0.667882i	$0.232799\pi$
$264$		0			0
$265$	5188.09	−	4353.32i	1.20265	−	1.00914i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3102.46			−0.703198			−0.351599	−	0.936151i	$-0.614362\pi$
$269$	−3102.46			−0.703198			−0.351599	+	0.936151i	$0.614362\pi$
$270$		0			0
$271$	−146.797			−0.0329050			−0.0164525	−	0.999865i	$-0.505237\pi$
$271$	−146.797			−0.0329050			−0.0164525	+	0.999865i	$0.505237\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10093.4	−	8469.40i	2.21330	−	1.85718i
$276$		0			0
$277$	−4020.61	+	1463.38i	−0.872111	+	0.317423i	−0.739022	−	0.673681i	$-0.764712\pi$
$277$	−4020.61	+	1463.38i	−0.872111	+	0.317423i	−0.133090	+	0.991104i	$0.542490\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3809.81	+	3196.81i	0.808805	+	0.678668i	0.950322	−	0.311268i	$-0.100754\pi$
$281$	3809.81	+	3196.81i	0.808805	+	0.678668i	−0.141517	+	0.989936i	$0.545198\pi$
$282$		0			0
$283$	−185.085	−	1049.67i	−0.0388770	−	0.220482i	0.959180	−	0.282798i	$-0.0912626\pi$
$283$	−185.085	−	1049.67i	−0.0388770	−	0.220482i	−0.998056	+	0.0623156i	$0.980151\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	614.448	+	1064.26i	0.126375	+	0.218889i
$288$		0			0
$289$	2431.72	−	4211.86i	0.494956	−	0.857288i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6323.47	−	2301.56i	−1.26082	−	0.458902i	−0.376778	−	0.926304i	$-0.622968\pi$
$293$	−6323.47	−	2301.56i	−1.26082	−	0.458902i	−0.884045	+	0.467402i	$0.845190\pi$
$294$		0			0
$295$	762.579	−	4324.80i	0.150505	−	0.853559i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−350.565	+	1988.15i	−0.0678049	+	0.384541i
$300$		0			0
$301$	3891.11	+	1416.25i	0.745116	+	0.271200i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5995.63	+	10384.7i	−1.12560	+	1.94960i
$306$		0			0
$307$	−419.639	−	726.835i	−0.0780132	−	0.135123i	0.824379	−	0.566038i	$-0.191524\pi$
$307$	−419.639	−	726.835i	−0.0780132	−	0.135123i	−0.902393	+	0.430915i	$0.858191\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	782.999	+	4440.61i	0.142765	+	0.809658i	0.969135	+	0.246531i	$0.0792907\pi$
$311$	782.999	+	4440.61i	0.142765	+	0.809658i	−0.826370	+	0.563127i	$0.809598\pi$
$312$		0			0
$313$	3326.13	+	2790.96i	0.600652	+	0.504007i	0.891655	−	0.452715i	$-0.149545\pi$
$313$	3326.13	+	2790.96i	0.600652	+	0.504007i	−0.291003	+	0.956722i	$0.593989\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8805.07	−	3204.78i	1.56007	−	0.567819i	0.589317	−	0.807902i	$-0.299397\pi$
$317$	8805.07	−	3204.78i	1.56007	−	0.567819i	0.970752	+	0.240083i	$0.0771747\pi$
$318$		0			0
$319$	−8436.68	+	7079.22i	−1.48076	+	1.24251i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−600.772			−0.103492
$324$		0			0
$325$	−15001.3			−2.56038
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9601.29	−	8056.44i	1.60893	−	1.35005i
$330$		0			0
$331$	7052.64	−	2566.95i	1.17114	−	0.426261i	0.318077	−	0.948065i	$-0.396963\pi$
$331$	7052.64	−	2566.95i	1.17114	−	0.426261i	0.853066	+	0.521804i	$0.174741\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13951.5	−	11706.7i	−2.27538	−	1.90927i
$336$		0			0
$337$	595.457	+	3377.01i	0.0962511	+	0.545867i	0.994357	+	0.106087i	$0.0338323\pi$
$337$	595.457	+	3377.01i	0.0962511	+	0.545867i	−0.898106	+	0.439780i	$0.855057\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3678.55	−	6371.43i	−0.584178	−	1.01183i
$342$		0			0
$343$	−742.883	+	1286.71i	−0.116944	+	0.202553i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10147.3	−	3693.32i	−1.56984	−	0.571377i	−0.596880	−	0.802331i	$-0.703593\pi$
$347$	−10147.3	−	3693.32i	−1.56984	−	0.571377i	−0.972965	+	0.230954i	$0.925815\pi$
$348$		0			0
$349$	−221.987	+	1258.95i	−0.0340479	+	0.193095i	−0.997088	−	0.0762641i	$-0.975701\pi$
$349$	−221.987	+	1258.95i	−0.0340479	+	0.193095i	0.963040	+	0.269359i	$0.0868119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1599.06	+	9068.74i	−0.241104	+	1.36737i	0.588267	+	0.808667i	$0.299811\pi$
$353$	−1599.06	+	9068.74i	−0.241104	+	1.36737i	−0.829370	+	0.558700i	$0.811301\pi$
$354$		0			0
$355$	−1712.39	−	623.260i	−0.256012	−	0.0931809i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	331.332	−	573.884i	0.0487104	−	0.0843689i	−0.840642	−	0.541591i	$-0.817822\pi$
$359$	331.332	−	573.884i	0.0487104	−	0.0843689i	0.889353	+	0.457222i	$0.151156\pi$
$360$		0			0
$361$	−211.429	−	366.206i	−0.0308250	−	0.0533905i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	907.419	+	5146.23i	0.130127	+	0.737989i
$366$		0			0
$367$	−4961.59	−	4163.27i	−0.705702	−	0.592155i	0.217687	−	0.976019i	$-0.430149\pi$
$367$	−4961.59	−	4163.27i	−0.705702	−	0.592155i	−0.923390	+	0.383864i	$0.874593\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8169.10	−	2973.31i	1.14318	−	0.416082i
$372$		0			0
$373$	5459.75	−	4581.27i	0.757896	−	0.635950i	−0.179682	−	0.983725i	$-0.557507\pi$
$373$	5459.75	−	4581.27i	0.757896	−	0.635950i	0.937578	+	0.347775i	$0.113063\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12539.0			1.71297
$378$		0			0
$379$	7451.59			1.00993			0.504964	−	0.863140i	$-0.331506\pi$
$379$	7451.59			1.00993			0.504964	+	0.863140i	$0.331506\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10198.1	+	8557.21i	−1.36057	+	1.14165i	−0.384762	+	0.923016i	$0.625717\pi$
$383$	−10198.1	+	8557.21i	−1.36057	+	1.14165i	−0.975806	+	0.218637i	$0.929839\pi$
$384$		0			0
$385$	23673.8	−	8616.56i	3.13384	−	1.14063i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10703.4	+	8981.22i	1.39508	+	1.17061i	0.963236	+	0.268656i	$0.0865796\pi$
$389$	10703.4	+	8981.22i	1.39508	+	1.17061i	0.431839	+	0.901951i	$0.357865\pi$
$390$		0			0
$391$	−42.0066	−	238.232i	−0.00543316	−	0.0308130i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8749.53	+	15154.6i	1.11452	+	1.93041i
$396$		0			0
$397$	−4359.70	+	7551.22i	−0.551151	+	0.954622i	0.447041	+	0.894514i	$0.352478\pi$
$397$	−4359.70	+	7551.22i	−0.551151	+	0.954622i	−0.998192	+	0.0601082i	$0.980855\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12751.4	−	4641.12i	−1.58796	−	0.577971i	−0.611046	−	0.791595i	$-0.709251\pi$
$401$	−12751.4	−	4641.12i	−1.58796	−	0.577971i	−0.976916	+	0.213624i	$0.931473\pi$
$402$		0			0
$403$	−1454.52	+	8249.02i	−0.179789	+	1.01963i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1170.32	−	6637.22i	0.142532	−	0.808341i
$408$		0			0
$409$	−8465.11	−	3081.05i	−1.02341	−	0.372489i	−0.224839	−	0.974396i	$-0.572186\pi$
$409$	−8465.11	−	3081.05i	−1.02341	−	0.372489i	−0.798566	+	0.601907i	$0.794408\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2818.51	−	4881.81i	0.335811	−	0.581642i
$414$		0			0
$415$	346.626	+	600.374i	0.0410005	+	0.0710149i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−769.338	−	4363.13i	−0.0897008	−	0.508718i	−0.996243	−	0.0866031i	$-0.972399\pi$
$419$	−769.338	−	4363.13i	−0.0897008	−	0.508718i	0.906542	−	0.422115i	$-0.138712\pi$
$420$		0			0
$421$	−472.103	−	396.141i	−0.0546529	−	0.0458592i	0.615052	−	0.788487i	$-0.289135\pi$
$421$	−472.103	−	396.141i	−0.0546529	−	0.0458592i	−0.669705	+	0.742627i	$0.733579\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1689.14	−	614.796i	0.192789	−	0.0701694i
$426$		0			0
$427$	−11791.1	+	9893.89i	−1.33632	+	1.12131i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9458.47			−1.05707			−0.528537	−	0.848910i	$-0.677259\pi$
$431$	−9458.47			−1.05707			−0.528537	+	0.848910i	$0.677259\pi$
$432$		0			0
$433$	5691.49			0.631676			0.315838	−	0.948813i	$-0.397714\pi$
$433$	5691.49			0.631676			0.315838	+	0.948813i	$0.397714\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2246.13	−	1884.72i	0.245874	−	0.206313i
$438$		0			0
$439$	−12777.3	+	4650.57i	−1.38913	+	0.505602i	−0.924933	−	0.380129i	$-0.875880\pi$
$439$	−12777.3	+	4650.57i	−1.38913	+	0.505602i	−0.464198	+	0.885732i	$0.653657\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8689.72	+	7291.54i	0.931966	+	0.782013i	0.976169	−	0.217010i	$-0.0696304\pi$
$443$	8689.72	+	7291.54i	0.931966	+	0.782013i	−0.0442031	+	0.999023i	$0.514075\pi$
$444$		0			0
$445$	2449.91	+	13894.1i	0.260982	+	1.48010i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2603.29	−	4509.03i	−0.273623	−	0.473930i	0.696163	−	0.717883i	$-0.254889\pi$
$449$	−2603.29	−	4509.03i	−0.273623	−	0.473930i	−0.969787	+	0.243954i	$0.921556\pi$
$450$		0			0
$451$	−1266.68	+	2193.96i	−0.132252	+	0.229067i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−26953.3	−	9810.21i	−2.77713	−	1.01079i
$456$		0			0
$457$	2794.59	−	15848.9i	0.286051	−	1.62228i	−0.415454	−	0.909614i	$-0.636377\pi$
$457$	2794.59	−	15848.9i	0.286051	−	1.62228i	0.701506	−	0.712664i	$-0.252512\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2417.98	+	13713.0i	−0.244287	+	1.38542i	0.577855	+	0.816139i	$0.303890\pi$
$461$	−2417.98	+	13713.0i	−0.244287	+	1.38542i	−0.822143	+	0.569282i	$0.807221\pi$
$462$		0			0
$463$	−683.331	−	248.712i	−0.0685898	−	0.0249646i	0.307497	−	0.951549i	$-0.400508\pi$
$463$	−683.331	−	248.712i	−0.0685898	−	0.0249646i	−0.376087	+	0.926584i	$0.622731\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6275.96	+	10870.3i	−0.621877	+	1.07712i	0.367259	+	0.930119i	$0.380296\pi$
$467$	−6275.96	+	10870.3i	−0.621877	+	1.07712i	−0.989136	+	0.147004i	$0.953037\pi$
$468$		0			0
$469$	−11688.8	−	20245.7i	−1.15083	−	1.99330i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1482.31	+	8406.62i	0.144095	+	0.817203i
$474$		0			0
$475$	16690.4	+	14004.9i	1.61222	+	1.35282i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8706.20	+	3168.80i	−0.830472	+	0.302267i	−0.722053	−	0.691838i	$-0.756801\pi$
$479$	−8706.20	+	3168.80i	−0.830472	+	0.302267i	−0.108420	+	0.994105i	$0.534579\pi$
$480$		0			0
$481$	−5878.05	+	4932.27i	−0.557206	+	0.467551i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30915.4			−2.89443
$486$		0			0
$487$	−7806.10			−0.726341			−0.363170	−	0.931723i	$-0.618306\pi$
$487$	−7806.10			−0.726341			−0.363170	+	0.931723i	$0.618306\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4101.97	−	3441.96i	0.377025	−	0.316361i	−0.434508	−	0.900668i	$-0.643078\pi$
$491$	4101.97	−	3441.96i	0.377025	−	0.316361i	0.811533	+	0.584306i	$0.198633\pi$
$492$		0			0
$493$	−1411.88	+	513.882i	−0.128982	+	0.0469454i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1791.87	−	1503.56i	−0.161723	−	0.135702i
$498$		0			0
$499$	256.588	+	1455.18i	0.0230190	+	0.130547i	0.994152	−	0.107993i	$-0.0344423\pi$
$499$	256.588	+	1455.18i	0.0230190	+	0.130547i	−0.971133	+	0.238540i	$0.923331\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6261.59	+	10845.4i	0.555051	+	0.961376i	0.997900	+	0.0647801i	$0.0206346\pi$
$503$	6261.59	+	10845.4i	0.555051	+	0.961376i	−0.442849	+	0.896596i	$0.646032\pi$
$504$		0			0
$505$	−17050.5	+	29532.3i	−1.50245	+	2.60232i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−980.124	−	356.736i	−0.0853502	−	0.0310649i	0.298992	−	0.954256i	$-0.403350\pi$
$509$	−980.124	−	356.736i	−0.0853502	−	0.0310649i	−0.384342	+	0.923191i	$0.625572\pi$
$510$		0			0
$511$	−1164.78	+	6605.78i	−0.100835	+	0.571864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3173.18	−	17996.0i	0.271509	−	1.53980i
$516$		0			0
$517$	24279.7	+	8837.10i	2.06542	+	0.751751i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4911.87	+	8507.61i	−0.413038	+	0.715403i	−0.995220	−	0.0976552i	$-0.968866\pi$
$521$	−4911.87	+	8507.61i	−0.413038	+	0.715403i	0.582182	+	0.813058i	$0.302199\pi$
$522$		0			0
$523$	−904.471	−	1566.59i	−0.0756209	−	0.130979i	0.825735	−	0.564058i	$-0.190761\pi$
$523$	−904.471	−	1566.59i	−0.0756209	−	0.130979i	−0.901356	+	0.433079i	$0.857427\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−174.289	−	988.444i	−0.0144064	−	0.0817027i
$528$		0			0
$529$	−8416.04	−	7061.89i	−0.691710	−	0.580414i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2710.37	−	986.492i	0.220261	−	0.0801683i
$534$		0			0
$535$	240.552	−	201.847i	0.0194392	−	0.0163114i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14637.7			1.16974
$540$		0			0
$541$	5724.91			0.454960			0.227480	−	0.973783i	$-0.426951\pi$
$541$	5724.91			0.454960			0.227480	+	0.973783i	$0.426951\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	24875.7	−	20873.2i	1.95516	−	1.64057i
$546$		0			0
$547$	6752.26	−	2457.62i	0.527798	−	0.192103i	−0.0643567	−	0.997927i	$-0.520500\pi$
$547$	6752.26	−	2457.62i	0.527798	−	0.192103i	0.592155	+	0.805824i	$0.298277\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13950.8	−	11706.1i	−1.07863	−	0.905075i
$552$		0			0
$553$	3900.50	+	22120.8i	0.299939	+	1.70104i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	800.454	+	1386.43i	0.0608910	+	0.105466i	0.894864	−	0.446339i	$-0.147272\pi$
$557$	800.454	+	1386.43i	0.0608910	+	0.105466i	−0.833973	+	0.551805i	$0.813939\pi$
$558$		0			0
$559$	4859.42	−	8416.77i	0.367677	−	0.636836i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12290.5	+	4473.39i	0.920042	+	0.334868i	0.758255	−	0.651958i	$-0.226052\pi$
$563$	12290.5	+	4473.39i	0.920042	+	0.334868i	0.161787	+	0.986826i	$0.448274\pi$
$564$		0			0
$565$	2794.98	−	15851.1i	0.208117	−	1.18029i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3956.96	−	22441.0i	0.291537	−	1.65339i	−0.389419	−	0.921061i	$-0.627324\pi$
$569$	3956.96	−	22441.0i	0.291537	−	1.65339i	0.680955	−	0.732325i	$-0.261565\pi$
$570$		0			0
$571$	10197.6	+	3711.63i	0.747386	+	0.272026i	0.687505	−	0.726179i	$-0.258706\pi$
$571$	10197.6	+	3711.63i	0.747386	+	0.272026i	0.0598805	+	0.998206i	$0.480928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4386.52	+	7597.68i	−0.318140	+	0.551035i
$576$		0			0
$577$	−10891.7	−	18865.0i	−0.785835	−	1.36111i	−0.928499	−	0.371335i	$-0.878900\pi$
$577$	−10891.7	−	18865.0i	−0.785835	−	1.36111i	0.142664	−	0.989771i	$-0.454433\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	154.524	+	876.350i	0.0110340	+	0.0625768i
$582$		0			0
$583$	13728.5	+	11519.6i	0.975263	+	0.818343i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9679.74	−	3523.14i	0.680623	−	0.247726i	0.0215079	−	0.999769i	$-0.493153\pi$
$587$	9679.74	−	3523.14i	0.680623	−	0.247726i	0.659115	+	0.752042i	$0.270931\pi$
$588$		0			0
$589$	9319.39	−	7819.90i	0.651950	−	0.547051i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4017.02			−0.278178			−0.139089	−	0.990280i	$-0.544417\pi$
$593$	−4017.02			−0.278178			−0.139089	+	0.990280i	$0.544417\pi$
$594$		0			0
$595$	3436.98			0.236811
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3884.65	−	3259.61i	0.264979	−	0.222344i	−0.500611	−	0.865672i	$-0.666891\pi$
$599$	3884.65	−	3259.61i	0.264979	−	0.222344i	0.765590	+	0.643328i	$0.222447\pi$
$600$		0			0
$601$	−6769.34	+	2463.84i	−0.459446	+	0.167225i	−0.561366	−	0.827568i	$-0.689724\pi$
$601$	−6769.34	+	2463.84i	−0.459446	+	0.167225i	0.101919	+	0.994793i	$0.467502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19900.7	+	16698.7i	1.33732	+	1.12215i
$606$		0			0
$607$	−829.608	−	4704.94i	−0.0554741	−	0.314609i	0.944426	−	0.328723i	$-0.106618\pi$
$607$	−829.608	−	4704.94i	−0.0554741	−	0.314609i	−0.999900	+	0.0141141i	$0.995507\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14708.6	−	25476.1i	−0.973892	−	1.68683i
$612$		0			0
$613$	9124.06	−	15803.3i	0.601170	−	1.04126i	−0.391474	−	0.920189i	$-0.628035\pi$
$613$	9124.06	−	15803.3i	0.601170	−	1.04126i	0.992644	−	0.121068i	$-0.0386319\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9238.38	−	3362.50i	−0.602793	−	0.219399i	0.0225540	−	0.999746i	$-0.492820\pi$
$617$	−9238.38	−	3362.50i	−0.602793	−	0.219399i	−0.625347	+	0.780347i	$0.715042\pi$
$618$		0			0
$619$	−2028.41	+	11503.7i	−0.131710	+	0.746966i	0.845384	+	0.534159i	$0.179372\pi$
$619$	−2028.41	+	11503.7i	−0.131710	+	0.746966i	−0.977094	+	0.212807i	$0.931739\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3144.74	+	17834.7i	−0.202234	+	1.14692i
$624$		0			0
$625$	−16585.3	−	6036.55i	−1.06146	−	0.386339i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	459.726	−	796.269i	0.0291423	−	0.0504759i
$630$		0			0
$631$	−5452.85	−	9444.62i	−0.344017	−	0.595855i	0.641158	−	0.767409i	$-0.278454\pi$
$631$	−5452.85	−	9444.62i	−0.344017	−	0.595855i	−0.985175	+	0.171554i	$0.945121\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7659.36	+	43438.4i	0.478665	+	2.71465i
$636$		0			0
$637$	−12766.5	−	10712.4i	−0.794077	−	0.666309i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−19546.8	+	7114.47i	−1.20445	+	0.438385i	−0.864776	−	0.502158i	$-0.832540\pi$
$641$	−19546.8	+	7114.47i	−1.20445	+	0.438385i	−0.339676	+	0.940542i	$0.610318\pi$
$642$		0			0
$643$	6541.99	−	5489.38i	0.401230	−	0.336672i	−0.419739	−	0.907645i	$-0.637878\pi$
$643$	6541.99	−	5489.38i	0.401230	−	0.336672i	0.820969	+	0.570973i	$0.193434\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17690.0			1.07491			0.537454	−	0.843293i	$-0.319386\pi$
$647$	17690.0			1.07491			0.537454	+	0.843293i	$0.319386\pi$
$648$		0			0
$649$	11620.7			0.702854
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10827.1	−	9085.01i	0.648847	−	0.544447i	−0.257874	−	0.966179i	$-0.583022\pi$
$653$	10827.1	−	9085.01i	0.648847	−	0.544447i	0.906721	+	0.421732i	$0.138578\pi$
$654$		0			0
$655$	−26199.3	+	9535.77i	−1.56289	+	0.568845i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7705.44	+	6465.63i	0.455480	+	0.382193i	0.841465	−	0.540312i	$-0.181694\pi$
$659$	7705.44	+	6465.63i	0.455480	+	0.382193i	−0.385985	+	0.922505i	$0.626138\pi$
$660$		0			0
$661$	−3544.49	−	20101.8i	−0.208570	−	1.18286i	−0.891723	−	0.452582i	$-0.850503\pi$
$661$	−3544.49	−	20101.8i	−0.208570	−	1.18286i	0.683153	−	0.730275i	$-0.260608\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20829.5	+	36077.8i	1.21464	+	2.10382i
$666$		0			0
$667$	3666.51	−	6350.58i	0.212845	−	0.368659i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−29817.3	−	10852.6i	−1.71547	−	0.624381i
$672$		0			0
$673$	2017.24	−	11440.4i	0.115541	−	0.655265i	−0.870940	−	0.491389i	$-0.836489\pi$
$673$	2017.24	−	11440.4i	0.115541	−	0.655265i	0.986481	−	0.163876i	$-0.0523996\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3453.35	−	19584.9i	0.196046	−	1.11183i	−0.714876	−	0.699252i	$-0.753517\pi$
$677$	3453.35	−	19584.9i	0.196046	−	1.11183i	0.910921	−	0.412580i	$-0.135372\pi$
$678$		0			0
$679$	−37290.3	−	13572.6i	−2.10762	−	0.767109i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5188.26	−	8986.33i	0.290664	−	0.503444i	−0.683303	−	0.730135i	$-0.739457\pi$
$683$	5188.26	−	8986.33i	0.290664	−	0.503444i	0.973967	+	0.226690i	$0.0727906\pi$
$684$		0			0
$685$	14621.0	+	25324.3i	0.815533	+	1.41255i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3543.12	−	20094.0i	−0.195910	−	1.11106i
$690$		0			0
$691$	−23664.3	−	19856.7i	−1.30279	−	1.09317i	−0.989655	−	0.143468i	$-0.954175\pi$
$691$	−23664.3	−	19856.7i	−1.30279	−	1.09317i	−0.313140	−	0.949707i	$-0.601381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1056.04	+	384.368i	−0.0576374	+	0.0209783i
$696$		0			0
$697$	−264.756	+	222.157i	−0.0143879	+	0.0120729i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28142.8			1.51632			0.758158	−	0.652071i	$-0.226099\pi$
$701$	28142.8			1.51632			0.758158	+	0.652071i	$0.226099\pi$
$702$		0			0
$703$	11144.5			0.597900
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−33531.7	+	28136.5i	−1.78372	+	1.49672i
$708$		0			0
$709$	20208.5	−	7355.30i	1.07045	−	0.389611i	0.254103	−	0.967177i	$-0.418220\pi$
$709$	20208.5	−	7355.30i	1.07045	−	0.389611i	0.816344	+	0.577566i	$0.195997\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3752.54	+	3148.76i	0.197102	+	0.165388i
$714$		0			0
$715$	−10267.8	−	58231.9i	−0.537057	−	3.04580i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18578.1	−	32178.3i	−0.963626	−	1.66905i	−0.713261	−	0.700899i	$-0.752783\pi$
$719$	−18578.1	−	32178.3i	−0.963626	−	1.66905i	−0.250366	−	0.968151i	$-0.580551\pi$
$720$		0			0
$721$	11728.2	−	20313.8i	0.605797	−	1.04927i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	51203.6	+	18636.6i	2.62297	+	0.954683i
$726$		0			0
$727$	−2772.11	+	15721.4i	−0.141419	+	0.802029i	0.828753	+	0.559614i	$0.189051\pi$
$727$	−2772.11	+	15721.4i	−0.141419	+	0.802029i	−0.970173	+	0.242415i	$0.922060\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−202.225	+	1146.87i	−0.0102320	+	0.0580283i
$732$		0			0
$733$	31627.8	+	11511.6i	1.59372	+	0.580067i	0.978129	−	0.207998i	$-0.0666948\pi$
$733$	31627.8	+	11511.6i	1.59372	+	0.580067i	0.615592	+	0.788065i	$0.288917\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24096.5	−	41736.4i	1.20435	−	2.08600i
$738$		0			0
$739$	14420.2	+	24976.5i	0.717801	+	1.24327i	0.961869	+	0.273510i	$0.0881845\pi$
$739$	14420.2	+	24976.5i	0.717801	+	1.24327i	−0.244068	+	0.969758i	$0.578482\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1573.69	+	8924.86i	0.0777028	+	0.440675i	0.998694	+	0.0510912i	$0.0162699\pi$
$743$	1573.69	+	8924.86i	0.0777028	+	0.440675i	−0.920991	+	0.389583i	$0.872619\pi$
$744$		0			0
$745$	20362.8	+	17086.4i	1.00139	+	0.840264i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	378.770	−	137.861i	0.0184779	−	0.00672541i
$750$		0			0
$751$	4600.11	−	3859.95i	0.223516	−	0.187552i	−0.524152	−	0.851625i	$-0.675618\pi$
$751$	4600.11	−	3859.95i	0.223516	−	0.187552i	0.747668	+	0.664072i	$0.231173\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22855.7			1.10172
$756$		0			0
$757$	−25006.2			−1.20061			−0.600307	−	0.799770i	$-0.704955\pi$
$757$	−25006.2			−1.20061			−0.600307	+	0.799770i	$0.704955\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16281.6	−	13661.8i	0.775566	−	0.650777i	−0.166562	−	0.986031i	$-0.553267\pi$
$761$	16281.6	−	13661.8i	0.775566	−	0.650777i	0.942128	+	0.335254i	$0.108822\pi$
$762$		0			0
$763$	39169.0	−	14256.4i	1.85847	−	0.676429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10135.2	−	8504.41i	−0.477131	−	0.400360i
$768$		0			0
$769$	598.520	+	3394.38i	0.0280666	+	0.159173i	0.995620	−	0.0934935i	$-0.0298034\pi$
$769$	598.520	+	3394.38i	0.0280666	+	0.159173i	−0.967553	+	0.252667i	$0.918692\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9004.85	−	15596.8i	−0.418993	−	0.725718i	0.576845	−	0.816853i	$-0.304284\pi$
$773$	−9004.85	−	15596.8i	−0.418993	−	0.725718i	−0.995838	+	0.0911359i	$0.970950\pi$
$774$		0			0
$775$	−18200.1	+	31523.5i	−0.843569	+	1.46110i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3936.50	−	1432.77i	−0.181052	−	0.0658977i
$780$		0			0
$781$	837.347	−	4748.83i	0.0383644	−	0.217575i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4319.52	+	24497.2i	−0.196395	+	1.11381i
$786$		0			0
$787$	−16461.5	−	5991.50i	−0.745603	−	0.271377i	−0.0588488	−	0.998267i	$-0.518743\pi$
$787$	−16461.5	−	5991.50i	−0.745603	−	0.271377i	−0.686754	+	0.726890i	$0.740965\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	10330.3	−	17892.7i	0.464354	−	0.804285i
$792$		0			0
$793$	18063.3	+	31286.5i	0.808885	+	1.40103i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1312.20	−	7441.84i	−0.0583192	−	0.330745i	0.941664	−	0.336555i	$-0.109262\pi$
$797$	−1312.20	−	7441.84i	−0.0583192	−	0.330745i	−0.999983	+	0.00581003i	$0.998151\pi$
$798$		0			0
$799$	2700.27	+	2265.79i	0.119560	+	0.100323i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12993.9	+	4729.41i	−0.571041	+	0.207842i
$804$		0			0
$805$	−12850.0	+	10782.4i	−0.562610	+	0.472086i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−38981.7			−1.69410			−0.847048	−	0.531517i	$-0.821622\pi$
$809$	−38981.7			−1.69410			−0.847048	+	0.531517i	$0.821622\pi$
$810$		0			0
$811$	−3470.10			−0.150249			−0.0751244	−	0.997174i	$-0.523935\pi$
$811$	−3470.10			−0.150249			−0.0751244	+	0.997174i	$0.523935\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24148.2	−	20262.8i	1.03788	−	0.870888i
$816$		0			0
$817$	−13264.3	+	4827.80i	−0.568002	+	0.206736i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18661.9	−	15659.2i	−0.793305	−	0.665662i	0.153256	−	0.988187i	$-0.451024\pi$
$821$	−18661.9	−	15659.2i	−0.793305	−	0.665662i	−0.946561	+	0.322524i	$0.895469\pi$
$822$		0			0
$823$	6973.29	+	39547.5i	0.295351	+	1.67502i	0.665774	+	0.746154i	$0.268102\pi$
$823$	6973.29	+	39547.5i	0.295351	+	1.67502i	−0.370423	+	0.928863i	$0.620787\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4564.78	+	7906.43i	0.191938	+	0.332447i	0.945893	−	0.324480i	$-0.105189\pi$
$827$	4564.78	+	7906.43i	0.191938	+	0.332447i	−0.753954	+	0.656927i	$0.771856\pi$
$828$		0			0
$829$	11850.3	−	20525.3i	0.496475	−	0.859920i	−0.503517	−	0.863985i	$-0.667961\pi$
$829$	11850.3	−	20525.3i	0.496475	−	0.859920i	0.999992	+	0.00406556i	$0.00129411\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1876.52	+	682.998i	0.0780523	+	0.0284087i
$834$		0			0
$835$	−1875.59	+	10637.0i	−0.0777336	+	0.440849i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5839.10	+	33115.2i	−0.240272	+	1.36265i	0.590950	+	0.806708i	$0.298753\pi$
$839$	−5839.10	+	33115.2i	−0.240272	+	1.36265i	−0.831222	+	0.555941i	$0.812358\pi$
$840$		0			0
$841$	−19880.8	−	7236.01i	−0.815153	−	0.296692i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12237.9	+	21196.7i	−0.498221	+	0.862945i
$846$		0			0
$847$	16673.2	+	28878.9i	0.676386	+	1.17154i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	779.238	+	4419.28i	0.0313889	+	0.178015i
$852$		0			0
$853$	−20152.9	−	16910.3i	−0.808937	−	0.678778i	0.141417	−	0.989950i	$-0.454834\pi$
$853$	−20152.9	−	16910.3i	−0.808937	−	0.678778i	−0.950354	+	0.311172i	$0.899279\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−151.335	+	55.0813i	−0.00603208	+	0.00219550i	−0.345034	−	0.938590i	$-0.612133\pi$
$857$	−151.335	+	55.0813i	−0.00603208	+	0.00219550i	0.339002	+	0.940786i	$0.389911\pi$
$858$		0			0
$859$	37816.0	−	31731.4i	1.50206	−	1.26038i	0.624382	−	0.781119i	$-0.285351\pi$
$859$	37816.0	−	31731.4i	1.50206	−	1.26038i	0.877675	−	0.479256i	$-0.159094\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−19390.7			−0.764853			−0.382426	−	0.923986i	$-0.624911\pi$
$863$	−19390.7			−0.764853			−0.382426	+	0.923986i	$0.624911\pi$
$864$		0			0
$865$	−13725.4			−0.539512
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−35472.0	+	29764.6i	−1.38470	+	1.16190i
$870$		0			0
$871$	−51560.1	+	18766.4i	−2.00580	+	0.730050i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−48737.7	−	40895.8i	−1.88301	−	1.58004i
$876$		0			0
$877$	−4605.11	−	26116.9i	−0.177313	−	1.00559i	−0.935440	−	0.353486i	$-0.884996\pi$
$877$	−4605.11	−	26116.9i	−0.177313	−	1.00559i	0.758127	−	0.652107i	$-0.226115\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4444.25	−	7697.66i	−0.169955	−	0.294371i	0.768449	−	0.639911i	$-0.221029\pi$
$881$	−4444.25	−	7697.66i	−0.169955	−	0.294371i	−0.938404	+	0.345540i	$0.887696\pi$
$882$		0			0
$883$	9393.58	−	16270.2i	0.358006	−	0.620084i	−0.629622	−	0.776902i	$-0.716790\pi$
$883$	9393.58	−	16270.2i	0.358006	−	0.620084i	0.987628	+	0.156818i	$0.0501235\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16935.5	+	6164.02i	0.641081	+	0.233334i	0.642047	−	0.766665i	$-0.278085\pi$
$887$	16935.5	+	6164.02i	0.641081	+	0.233334i	−0.000966332		1.00000i	$0.500308\pi$
$888$		0			0
$889$	−9831.68	+	55758.2i	−0.370915	+	2.10357i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7419.17	+	42076.2i	−0.278021	+	1.57674i
$894$		0			0
$895$	21470.9	+	7814.77i	0.801892	+	0.291865i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	15212.7	−	26349.1i	0.564373	−	0.977523i
$900$		0			0
$901$	1222.46	+	2117.37i	0.0452010	+	0.0782904i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2617.82	+	14846.4i	0.0961538	+	0.545315i
$906$		0			0
$907$	−23612.4	−	19813.2i	−0.864430	−	0.725343i	0.0984877	−	0.995138i	$-0.468599\pi$
$907$	−23612.4	−	19813.2i	−0.864430	−	0.725343i	−0.962918	+	0.269795i	$0.913044\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	43762.2	−	15928.1i	1.59155	−	0.579278i	0.613878	−	0.789401i	$-0.289609\pi$
$911$	43762.2	−	15928.1i	1.59155	−	0.579278i	0.977675	+	0.210123i	$0.0673864\pi$
$912$		0			0
$913$	−1405.28	+	1179.17i	−0.0509397	+	0.0427435i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−35788.1			−1.28880
$918$		0			0
$919$	3405.86			0.122251			0.0611256	−	0.998130i	$-0.480531\pi$
$919$	3405.86			0.122251			0.0611256	+	0.998130i	$0.480531\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4205.65	+	3528.96i	−0.149979	+	0.125847i
$924$		0			0
$925$	−31334.1	+	11404.7i	−1.11379	+	0.405388i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23801.4	+	19971.7i	0.840580	+	0.705330i	0.957694	−	0.287788i	$-0.0929199\pi$
$929$	23801.4	+	19971.7i	0.840580	+	0.705330i	−0.117114	+	0.993118i	$0.537364\pi$
$930$		0			0
$931$	4203.11	+	23837.0i	0.147961	+	0.839126i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3542.66	+	6136.06i	0.123912	+	0.214621i
$936$		0			0
$937$	14196.9	−	24589.8i	0.494976	−	0.857324i	−0.505007	−	0.863115i	$-0.668510\pi$
$937$	14196.9	−	24589.8i	0.494976	−	0.857324i	0.999983	+	0.00579118i	$0.00184340\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8174.57	+	2975.30i	0.283192	+	0.103073i	0.479711	−	0.877427i	$-0.340742\pi$
$941$	8174.57	+	2975.30i	0.283192	+	0.103073i	−0.196519	+	0.980500i	$0.562964\pi$
$942$		0			0
$943$	292.909	−	1661.17i	0.0101150	−	0.0573650i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1821.30	+	10329.1i	−0.0624967	+	0.354436i	0.937483	+	0.348032i	$0.113150\pi$
$947$	−1821.30	+	10329.1i	−0.0624967	+	0.354436i	−0.999979	+	0.00640472i	$0.997961\pi$
$948$		0			0
$949$	14794.0	+	5384.57i	0.506041	+	0.184184i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16369.0	+	28351.9i	−0.556393	+	0.963701i	0.441401	+	0.897310i	$0.354482\pi$
$953$	−16369.0	+	28351.9i	−0.556393	+	0.963701i	−0.997794	+	0.0663907i	$0.978852\pi$
$954$		0			0
$955$	−5550.13	−	9613.10i	−0.188061	−	0.325731i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6517.98	+	36965.3i	0.219475	+	1.24470i
$960$		0			0
$961$	−7251.59	−	6084.80i	−0.243415	−	0.204250i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−89040.8	+	32408.2i	−2.97028	+	1.08109i
$966$		0			0
$967$	−12731.9	+	10683.3i	−0.423403	+	0.355277i	−0.829456	−	0.558572i	$-0.811349\pi$
$967$	−12731.9	+	10683.3i	−0.423403	+	0.355277i	0.406053	+	0.913850i	$0.366905\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30671.4			1.01369			0.506844	−	0.862038i	$-0.330812\pi$
$971$	30671.4			1.01369			0.506844	+	0.862038i	$0.330812\pi$
$972$		0			0
$973$	−1442.55			−0.0475293
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7083.39	+	5943.67i	−0.231953	+	0.194631i	−0.751355	−	0.659899i	$-0.770599\pi$
$977$	−7083.39	+	5943.67i	−0.231953	+	0.194631i	0.519402	+	0.854530i	$0.326155\pi$
$978$		0			0
$979$	−35081.9	+	12768.8i	−1.14527	+	0.416845i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−39119.3	−	32825.0i	−1.26929	−	1.06506i	−0.994627	−	0.103519i	$-0.966990\pi$
$983$	−39119.3	−	32825.0i	−1.26929	−	1.06506i	−0.274662	−	0.961541i	$-0.588566\pi$
$984$		0			0
$985$	−6409.00	−	36347.3i	−0.207318	−	1.17576i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2841.88	−	4922.28i	−0.0913715	−	0.158260i
$990$		0			0
$991$	19142.6	−	33156.0i	0.613609	−	1.06280i	−0.377018	−	0.926206i	$-0.623051\pi$
$991$	19142.6	−	33156.0i	0.613609	−	1.06280i	0.990627	−	0.136595i	$-0.0436160\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	71483.6	+	26017.9i	2.27757	+	0.828968i
$996$		0			0
$997$	1194.72	−	6775.57i	0.0379509	−	0.215230i	−0.959935	−	0.280223i	$-0.909591\pi$
$997$	1194.72	−	6775.57i	0.0379509	−	0.215230i	0.997886	+	0.0649932i	$0.0207026\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.i.a.73.9		54
3.2	odd	2		108.4.i.a.25.3	yes	54
27.13	even	9	inner	324.4.i.a.253.9		54
27.14	odd	18		108.4.i.a.13.3	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.3	✓	54	27.14	odd	18
108.4.i.a.25.3	yes	54	3.2	odd	2
324.4.i.a.73.9		54	1.1	even	1	trivial
324.4.i.a.253.9		54	27.13	even	9	inner

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(14.9393 - 12.5356i) q^{5} +(23.5233 - 8.56177i) q^{7} +O(q^{10})\)	\(q+(14.9393 - 12.5356i) q^{5} +(23.5233 - 8.56177i) q^{7} +(39.5319 + 33.1712i) q^{11} +(-10.2025 - 57.8615i) q^{13} +(3.52013 + 6.09704i) q^{17} +(-42.6669 + 73.9012i) q^{19} +(-32.2883 - 11.7520i) q^{23} +(44.3365 - 251.445i) q^{25} +(-37.0590 + 210.172i) q^{29} +(-133.967 - 48.7600i) q^{31} +(244.095 - 422.784i) q^{35} +(-65.2996 - 113.102i) q^{37} +(8.52460 + 48.3454i) q^{41} +(126.716 + 106.327i) q^{43} +(470.489 - 171.244i) q^{47} +(217.287 - 182.325i) q^{49} +347.277 q^{53} +1006.40 q^{55} +(172.501 - 144.746i) q^{59} +(-577.795 + 210.300i) q^{61} +(-877.746 - 736.517i) q^{65} +(-162.166 - 919.690i) q^{67} +(-46.7209 - 80.9229i) q^{71} +(-133.977 + 232.055i) q^{73} +(1213.92 + 441.832i) q^{77} +(-155.815 + 883.669i) q^{79} +(-6.17283 + 35.0079i) q^{83} +(129.018 + 46.9588i) q^{85} +(-361.721 + 626.519i) q^{89} +(-735.394 - 1273.74i) q^{91} +(288.980 + 1638.89i) q^{95} +(-1214.37 - 1018.98i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 54 * q - 12 * q^5	\( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Introduction

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