LMFDB - Embedded newform 324.5.k.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(17,324)]

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, 11]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, 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:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			17.1
Character	$\chi$	$=$	324.17
Dual form			324.5.k.a.305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-47.5857 - 8.39064i) q^{5} +(-52.2512 + 43.8440i) q^{7} +O(q^{10})$	$q+(-47.5857 - 8.39064i) q^{5} +(-52.2512 + 43.8440i) q^{7} +(-77.3144 + 13.6326i) q^{11} +(-38.2463 + 13.9205i) q^{13} +(-380.341 - 219.590i) q^{17} +(132.725 + 229.886i) q^{19} +(-384.321 + 458.015i) q^{23} +(1606.68 + 584.785i) q^{25} +(334.742 - 919.697i) q^{29} +(-516.190 - 433.135i) q^{31} +(2854.29 - 1647.92i) q^{35} +(921.369 - 1595.86i) q^{37} +(709.768 + 1950.07i) q^{41} +(281.131 + 1594.37i) q^{43} +(-1008.10 - 1201.41i) q^{47} +(390.965 - 2217.27i) q^{49} -2319.30i q^{53} +3793.44 q^{55} +(1131.36 + 199.489i) q^{59} +(352.780 - 296.017i) q^{61} +(1936.78 - 341.506i) q^{65} +(-7736.59 + 2815.89i) q^{67} +(390.512 + 225.462i) q^{71} +(2556.60 + 4428.15i) q^{73} +(3442.06 - 4102.09i) q^{77} +(-223.536 - 81.3605i) q^{79} +(775.742 - 2131.33i) q^{83} +(16256.3 + 13640.6i) q^{85} +(7644.67 - 4413.65i) q^{89} +(1388.08 - 2404.23i) q^{91} +(-4386.91 - 12052.9i) q^{95} +(-2114.33 - 11991.0i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{5}+O(q^{10}) $ 72 * q - 9 * q^5	$ 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) $ 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{11}{18}\right)$

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
$5$	−47.5857	−	8.39064i	−1.90343	−	0.335625i	−0.907075	−	0.420969i	$-0.861690\pi$
$5$	−47.5857	−	8.39064i	−1.90343	−	0.335625i	−0.996352	+	0.0853435i	$0.972801\pi$
$6$		0			0
$7$	−52.2512	+	43.8440i	−1.06635	+	0.894775i	−0.994717	−	0.102656i	$-0.967266\pi$
$7$	−52.2512	+	43.8440i	−1.06635	+	0.894775i	−0.0716339	+	0.997431i	$0.522821\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−77.3144	+	13.6326i	−0.638962	+	0.112666i	−0.483738	−	0.875213i	$-0.660721\pi$
$11$	−77.3144	+	13.6326i	−0.638962	+	0.112666i	−0.155224	+	0.987879i	$0.549610\pi$
$12$		0			0
$13$	−38.2463	+	13.9205i	−0.226309	+	0.0823698i	−0.452686	−	0.891670i	$-0.649534\pi$
$13$	−38.2463	+	13.9205i	−0.226309	+	0.0823698i	0.226377	+	0.974040i	$0.427312\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−380.341	−	219.590i	−1.31606	−	0.759827i	−0.332967	−	0.942939i	$-0.608050\pi$
$17$	−380.341	−	219.590i	−1.31606	−	0.759827i	−0.983092	+	0.183112i	$0.941383\pi$
$18$		0			0
$19$	132.725	+	229.886i	0.367659	+	0.636803i	0.989199	−	0.146579i	$-0.0468261\pi$
$19$	132.725	+	229.886i	0.367659	+	0.636803i	−0.621540	+	0.783382i	$0.713493\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−384.321	+	458.015i	−0.726504	+	0.865814i	−0.995245	−	0.0973987i	$-0.968948\pi$
$23$	−384.321	+	458.015i	−0.726504	+	0.865814i	0.268742	+	0.963212i	$0.413392\pi$
$24$		0			0
$25$	1606.68	+	584.785i	2.57070	+	0.935657i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	334.742	−	919.697i	0.398029	−	1.09358i	−0.565214	−	0.824944i	$-0.691207\pi$
$29$	334.742	−	919.697i	0.398029	−	1.09358i	0.963243	−	0.268631i	$-0.0865712\pi$
$30$		0			0
$31$	−516.190	−	433.135i	−0.537139	−	0.450713i	0.333419	−	0.942779i	$-0.391798\pi$
$31$	−516.190	−	433.135i	−0.537139	−	0.450713i	−0.870558	+	0.492066i	$0.836242\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2854.29	−	1647.92i	2.33003	−	1.34524i
$36$		0			0
$37$	921.369	−	1595.86i	0.673023	−	1.16571i	−0.304019	−	0.952666i	$-0.598329\pi$
$37$	921.369	−	1595.86i	0.673023	−	1.16571i	0.977042	−	0.213045i	$-0.0683379\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	709.768	+	1950.07i	0.422230	+	1.16007i	0.950428	+	0.310946i	$0.100646\pi$
$41$	709.768	+	1950.07i	0.422230	+	1.16007i	−0.528198	+	0.849121i	$0.677132\pi$
$42$		0			0
$43$	281.131	+	1594.37i	0.152045	+	0.862289i	0.961438	+	0.275022i	$0.0886851\pi$
$43$	281.131	+	1594.37i	0.152045	+	0.862289i	−0.809393	+	0.587267i	$0.800204\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1008.10	−	1201.41i	−0.456361	−	0.543870i	0.487972	−	0.872859i	$-0.337737\pi$
$47$	−1008.10	−	1201.41i	−0.456361	−	0.543870i	−0.944334	+	0.328989i	$0.893292\pi$
$48$		0			0
$49$	390.965	−	2217.27i	0.162834	−	0.923479i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2319.30i		−	0.825668i	−0.910806	−	0.412834i	$-0.864539\pi$
$53$		−	2319.30i		−	0.825668i	0.910806	−	0.412834i	$-0.135461\pi$
$54$		0			0
$55$	3793.44			1.25403
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1131.36	+	199.489i	0.325009	+	0.0573079i	0.333772	−	0.942654i	$-0.391678\pi$
$59$	1131.36	+	199.489i	0.325009	+	0.0573079i	−0.00876307	+	0.999962i	$0.502789\pi$
$60$		0			0
$61$	352.780	−	296.017i	0.0948077	−	0.0795531i	−0.594152	−	0.804353i	$-0.702512\pi$
$61$	352.780	−	296.017i	0.0948077	−	0.0795531i	0.688960	+	0.724800i	$0.258068\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1936.78	−	341.506i	0.458408	−	0.0808298i
$66$		0			0
$67$	−7736.59	+	2815.89i	−1.72346	+	0.627286i	−0.998130	−	0.0611201i	$-0.980533\pi$
$67$	−7736.59	+	2815.89i	−1.72346	+	0.627286i	−0.725325	+	0.688407i	$0.758310\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	390.512	+	225.462i	0.0774671	+	0.0447257i	0.538233	−	0.842796i	$-0.319092\pi$
$71$	390.512	+	225.462i	0.0774671	+	0.0447257i	−0.460766	+	0.887522i	$0.652425\pi$
$72$		0			0
$73$	2556.60	+	4428.15i	0.479751	+	0.830954i	0.999730	−	0.0232253i	$-0.00739350\pi$
$73$	2556.60	+	4428.15i	0.479751	+	0.830954i	−0.519979	+	0.854179i	$0.674060\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3442.06	−	4102.09i	0.580546	−	0.691868i
$78$		0			0
$79$	−223.536	−	81.3605i	−0.0358173	−	0.0130364i	0.324049	−	0.946040i	$-0.394956\pi$
$79$	−223.536	−	81.3605i	−0.0358173	−	0.0130364i	−0.359867	+	0.933004i	$0.617178\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	775.742	−	2131.33i	0.112606	−	0.309382i	−0.870570	−	0.492045i	$-0.836249\pi$
$83$	775.742	−	2131.33i	0.112606	−	0.309382i	0.983176	+	0.182663i	$0.0584717\pi$
$84$		0			0
$85$	16256.3	+	13640.6i	2.25000	+	1.88798i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7644.67	−	4413.65i	0.965114	−	0.557209i	0.0673709	−	0.997728i	$-0.478539\pi$
$89$	7644.67	−	4413.65i	0.965114	−	0.557209i	0.897743	+	0.440519i	$0.145206\pi$
$90$		0			0
$91$	1388.08	−	2404.23i	0.167623	−	0.290331i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4386.91	−	12052.9i	−0.486084	−	1.33550i
$96$		0			0
$97$	−2114.33	−	11991.0i	−0.224714	−	1.27442i	−0.863231	−	0.504809i	$-0.831563\pi$
$97$	−2114.33	−	11991.0i	−0.224714	−	1.27442i	0.638517	−	0.769607i	$-0.279548\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5438.85	+	6481.76i	0.533168	+	0.635405i	0.963642	−	0.267198i	$-0.0860979\pi$
$101$	5438.85	+	6481.76i	0.533168	+	0.635405i	−0.430474	+	0.902603i	$0.641653\pi$
$102$		0			0
$103$	1706.50	−	9678.07i	0.160854	−	0.912251i	−0.792382	−	0.610025i	$-0.791159\pi$
$103$	1706.50	−	9678.07i	0.160854	−	0.912251i	0.953236	−	0.302226i	$-0.0977296\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13666.4i			1.19368i	0.802362	+	0.596838i	$0.203576\pi$
$107$			13666.4i			1.19368i	−0.802362	+	0.596838i	$0.796424\pi$
$108$		0			0
$109$	3077.64			0.259039			0.129519	−	0.991577i	$-0.458657\pi$
$109$	3077.64			0.259039			0.129519	+	0.991577i	$0.458657\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6056.64	−	1067.95i	−0.474323	−	0.0836360i	−0.0686241	−	0.997643i	$-0.521861\pi$
$113$	−6056.64	−	1067.95i	−0.474323	−	0.0836360i	−0.405699	+	0.914007i	$0.632972\pi$
$114$		0			0
$115$	22131.2	−	18570.3i	1.67344	−	1.40418i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	29501.0	−	5201.81i	2.08325	−	0.367334i
$120$		0			0
$121$	−7966.38	+	2899.52i	−0.544114	+	0.198041i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−45394.6	−	26208.6i	−2.90526	−	1.67735i
$126$		0			0
$127$	10767.5	+	18649.9i	0.667587	+	1.15629i	0.978577	+	0.205881i	$0.0660060\pi$
$127$	10767.5	+	18649.9i	0.667587	+	1.15629i	−0.310990	+	0.950413i	$0.600661\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9598.31	+	11438.8i	−0.559310	+	0.666559i	−0.969400	−	0.245486i	$-0.921053\pi$
$131$	−9598.31	+	11438.8i	−0.559310	+	0.666559i	0.410091	+	0.912045i	$0.365497\pi$
$132$		0			0
$133$	−17014.1	−	6192.64i	−0.961849	−	0.350084i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5192.52	+	14266.3i	−0.276654	+	0.760101i	0.721082	+	0.692850i	$0.243645\pi$
$137$	−5192.52	+	14266.3i	−0.276654	+	0.760101i	−0.997736	+	0.0672509i	$0.978577\pi$
$138$		0			0
$139$	5554.88	+	4661.10i	0.287505	+	0.241245i	0.775121	−	0.631813i	$-0.217689\pi$
$139$	5554.88	+	4661.10i	0.287505	+	0.241245i	−0.487616	+	0.873058i	$0.662133\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2767.21	−	1597.65i	0.135323	−	0.0781286i
$144$		0			0
$145$	−23645.8	+	40955.7i	−1.12465	+	1.94795i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6531.41	+	17944.9i	0.294194	+	0.808293i	0.995441	+	0.0953743i	$0.0304048\pi$
$149$	6531.41	+	17944.9i	0.294194	+	0.808293i	−0.701247	+	0.712918i	$0.747373\pi$
$150$		0			0
$151$	−1034.61	−	5867.59i	−0.0453759	−	0.257339i	0.953678	−	0.300829i	$-0.0972634\pi$
$151$	−1034.61	−	5867.59i	−0.0453759	−	0.257339i	−0.999054	+	0.0434900i	$0.986152\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20929.0	+	24942.2i	0.871133	+	1.03818i
$156$		0			0
$157$	4774.11	−	27075.3i	0.193684	−	1.09844i	−0.720597	−	0.693354i	$-0.756132\pi$
$157$	4774.11	−	27075.3i	0.193684	−	1.09844i	0.914281	−	0.405081i	$-0.132757\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	40782.0i		−	1.57332i
$162$		0			0
$163$	9378.23			0.352977			0.176488	−	0.984303i	$-0.443526\pi$
$163$	9378.23			0.352977			0.176488	+	0.984303i	$0.443526\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−36081.3	−	6362.11i	−1.29375	−	0.228123i	−0.515940	−	0.856625i	$-0.672557\pi$
$167$	−36081.3	−	6362.11i	−1.29375	−	0.228123i	−0.777808	+	0.628502i	$0.783668\pi$
$168$		0			0
$169$	−20610.0	+	17293.8i	−0.721613	+	0.605505i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10018.5	+	1766.52i	−0.334741	+	0.0590238i	−0.338492	−	0.940969i	$-0.609917\pi$
$173$	−10018.5	+	1766.52i	−0.334741	+	0.0590238i	0.00375166	+	0.999993i	$0.498806\pi$
$174$		0			0
$175$	−109590.	+	39887.7i	−3.57847	+	1.30245i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	44758.2	+	25841.2i	1.39690	+	0.806503i	0.994067	−	0.108768i	$-0.0346906\pi$
$179$	44758.2	+	25841.2i	1.39690	+	0.806503i	0.402838	+	0.915271i	$0.368024\pi$
$180$		0			0
$181$	−14711.4	−	25480.9i	−0.449052	−	0.777782i	0.549272	−	0.835644i	$-0.314905\pi$
$181$	−14711.4	−	25480.9i	−0.449052	−	0.777782i	−0.998325	+	0.0578619i	$0.981572\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−57234.2	+	68209.1i	−1.67229	+	1.99296i
$186$		0			0
$187$	32399.4	+	11792.4i	0.926518	+	0.337225i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1080.35	−	2968.24i	0.0296141	−	0.0813640i	−0.924005	−	0.382381i	$-0.875104\pi$
$191$	1080.35	−	2968.24i	0.0296141	−	0.0813640i	0.953619	+	0.301017i	$0.0973263\pi$
$192$		0			0
$193$	−42515.0	−	35674.3i	−1.14137	−	0.957725i	−0.141890	−	0.989882i	$-0.545318\pi$
$193$	−42515.0	−	35674.3i	−1.14137	−	0.957725i	−0.999483	+	0.0321572i	$0.989762\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18701.7	+	10797.5i	−0.481892	+	0.278220i	−0.721205	−	0.692722i	$-0.756411\pi$
$197$	−18701.7	+	10797.5i	−0.481892	+	0.278220i	0.239313	+	0.970943i	$0.423078\pi$
$198$		0			0
$199$	26584.7	−	46046.1i	0.671314	−	1.16275i	−0.306217	−	0.951962i	$-0.599063\pi$
$199$	26584.7	−	46046.1i	0.671314	−	1.16275i	0.977532	−	0.210789i	$-0.0676032\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22832.5	+	62731.7i	0.554065	+	1.52228i
$204$		0			0
$205$	−17412.4	−	98750.9i	−0.414335	−	2.34981i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13395.5	−	15964.1i	−0.306666	−	0.365470i
$210$		0			0
$211$	−4276.20	+	24251.5i	−0.0960490	+	0.544721i	0.898372	+	0.439236i	$0.144751\pi$
$211$	−4276.20	+	24251.5i	−0.0960490	+	0.544721i	−0.994421	+	0.105485i	$0.966360\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	78228.1i		−	1.69233i
$216$		0			0
$217$	45961.9			0.976065
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17603.4	+	3103.96i	0.360423	+	0.0635523i
$222$		0			0
$223$	4228.35	−	3548.01i	0.0850279	−	0.0713468i	−0.599283	−	0.800537i	$-0.704548\pi$
$223$	4228.35	−	3548.01i	0.0850279	−	0.0713468i	0.684311	+	0.729190i	$0.260103\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	46836.4	−	8258.53i	0.908933	−	0.160269i	0.300414	−	0.953809i	$-0.402875\pi$
$227$	46836.4	−	8258.53i	0.908933	−	0.160269i	0.608519	+	0.793539i	$0.291764\pi$
$228$		0			0
$229$	−51370.7	+	18697.4i	−0.979589	+	0.356541i	−0.781681	−	0.623679i	$-0.785637\pi$
$229$	−51370.7	+	18697.4i	−0.979589	+	0.356541i	−0.197909	+	0.980220i	$0.563415\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	31489.0	+	18180.2i	0.580025	+	0.334878i	0.761143	−	0.648584i	$-0.224638\pi$
$233$	31489.0	+	18180.2i	0.580025	+	0.334878i	−0.181118	+	0.983461i	$0.557972\pi$
$234$		0			0
$235$	37890.6	+	65628.5i	0.686114	+	1.18838i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2388.77	+	2846.82i	−0.0418195	+	0.0498385i	−0.786549	−	0.617528i	$-0.788134\pi$
$239$	−2388.77	+	2846.82i	−0.0418195	+	0.0498385i	0.744729	+	0.667367i	$0.232579\pi$
$240$		0			0
$241$	−58565.7	−	21316.2i	−1.00835	−	0.367008i	−0.215548	−	0.976493i	$-0.569154\pi$
$241$	−58565.7	−	21316.2i	−1.00835	−	0.367008i	−0.792797	+	0.609486i	$0.791376\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−37208.7	+	102230.i	−0.619886	+	1.70312i
$246$		0			0
$247$	−8276.36	−	6944.69i	−0.135658	−	0.113831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1259.51	−	727.177i	0.0199919	−	0.0115423i	−0.489971	−	0.871739i	$-0.662993\pi$
$251$	1259.51	−	727.177i	0.0199919	−	0.0115423i	0.509963	+	0.860197i	$0.329659\pi$
$252$		0			0
$253$	23469.6	−	40650.5i	0.366660	−	0.635074i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−19302.1	−	53032.2i	−0.292240	−	0.802922i	−0.995738	−	0.0922252i	$-0.970602\pi$
$257$	−19302.1	−	53032.2i	−0.292240	−	0.802922i	0.703498	−	0.710697i	$-0.251620\pi$
$258$		0			0
$259$	21826.1	+	123782.i	0.325369	+	1.84526i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−40224.9	−	47938.1i	−0.581545	−	0.693058i	0.392412	−	0.919789i	$-0.371641\pi$
$263$	−40224.9	−	47938.1i	−0.581545	−	0.693058i	−0.973957	+	0.226731i	$0.927196\pi$
$264$		0			0
$265$	−19460.4	+	110366.i	−0.277115	+	1.57160i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	13805.0i		−	0.190780i	−0.995440	−	0.0953900i	$-0.969590\pi$
$269$		−	13805.0i		−	0.190780i	0.995440	−	0.0953900i	$-0.0304098\pi$
$270$		0			0
$271$	36775.8			0.500753			0.250376	−	0.968149i	$-0.419446\pi$
$271$	36775.8			0.500753			0.250376	+	0.968149i	$0.419446\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−132192.	−	23309.0i	−1.74799	−	0.308218i
$276$		0			0
$277$	86455.4	−	72544.7i	1.12676	−	0.945467i	0.127837	−	0.991795i	$-0.459196\pi$
$277$	86455.4	−	72544.7i	1.12676	−	0.945467i	0.998926	+	0.0463282i	$0.0147520\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	38119.8	−	6721.55i	0.482768	−	0.0851250i	0.0730327	−	0.997330i	$-0.476732\pi$
$281$	38119.8	−	6721.55i	0.482768	−	0.0851250i	0.409735	+	0.912205i	$0.365621\pi$
$282$		0			0
$283$	110746.	−	40308.3i	1.38279	−	0.503293i	0.459765	−	0.888041i	$-0.347934\pi$
$283$	110746.	−	40308.3i	1.38279	−	0.503293i	0.923022	+	0.384747i	$0.125711\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−122585.	−	70774.6i	−1.48824	−	0.859238i
$288$		0			0
$289$	54679.0	+	94706.8i	0.654674	+	1.13393i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−30534.9	+	36390.1i	−0.355682	+	0.423885i	−0.913982	−	0.405754i	$-0.867009\pi$
$293$	−30534.9	+	36390.1i	−0.355682	+	0.423885i	0.558301	+	0.829639i	$0.311453\pi$
$294$		0			0
$295$	−52162.6	−	18985.6i	−0.599397	−	0.218163i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8323.02	−	22867.3i	0.0930976	−	0.255784i
$300$		0			0
$301$	−84593.0	−	70982.0i	−0.933687	−	0.783457i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19271.0	+	11126.1i	−0.207160	+	0.119604i
$306$		0			0
$307$	−84477.1	+	146319.i	−0.896319	+	1.55247i	−0.0641547	+	0.997940i	$0.520435\pi$
$307$	−84477.1	+	146319.i	−0.896319	+	1.55247i	−0.832164	+	0.554530i	$0.812898\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	19992.0	+	54927.7i	0.206698	+	0.567898i	0.999114	−	0.0420829i	$-0.0133993\pi$
$311$	19992.0	+	54927.7i	0.206698	+	0.567898i	−0.792416	+	0.609981i	$0.791177\pi$
$312$		0			0
$313$	−31970.0	−	181311.i	−0.326327	−	1.85069i	−0.500179	−	0.865922i	$-0.666733\pi$
$313$	−31970.0	−	181311.i	−0.326327	−	1.85069i	0.173852	−	0.984772i	$-0.444379\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	61514.6	+	73310.2i	0.612152	+	0.729535i	0.979700	−	0.200470i	$-0.0642469\pi$
$317$	61514.6	+	73310.2i	0.612152	+	0.729535i	−0.367547	+	0.930005i	$0.619802\pi$
$318$		0			0
$319$	−13342.5	+	75669.2i	−0.131116	+	0.743597i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	116580.i		−	1.11743i
$324$		0			0
$325$	−69590.2			−0.658842
$326$		0			0
$327$		0			0
$328$		0			0
$329$	105349.	+	18575.9i	0.973283	+	0.171616i
$330$		0			0
$331$	123147.	−	103333.i	1.12400	−	0.943151i	0.125204	−	0.992131i	$-0.460041\pi$
$331$	123147.	−	103333.i	1.12400	−	0.943151i	0.998800	+	0.0489797i	$0.0155970\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	391778.	−	69081.0i	3.49100	−	0.615558i
$336$		0			0
$337$	207738.	−	75610.3i	1.82918	−	0.665766i	0.836059	−	0.548639i	$-0.184854\pi$
$337$	207738.	−	75610.3i	1.82918	−	0.665766i	0.993117	−	0.117126i	$-0.0373682\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	45813.7	+	26450.5i	0.393991	+	0.227471i
$342$		0			0
$343$	−5099.36	−	8832.36i	−0.0433439	−	0.0750738i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	106819.	−	127302.i	0.887135	−	1.05725i	−0.110852	−	0.993837i	$-0.535358\pi$
$347$	106819.	−	127302.i	0.887135	−	1.05725i	0.997988	−	0.0634098i	$-0.0201975\pi$
$348$		0			0
$349$	154320.	+	56167.9i	1.26699	+	0.461145i	0.886107	−	0.463480i	$-0.153399\pi$
$349$	154320.	+	56167.9i	1.26699	+	0.461145i	0.380879	+	0.924625i	$0.375622\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5400.36	−	14837.4i	0.0433384	−	0.119071i	−0.916136	−	0.400869i	$-0.868708\pi$
$353$	5400.36	−	14837.4i	0.0433384	−	0.119071i	0.959474	+	0.281797i	$0.0909306\pi$
$354$		0			0
$355$	−16691.0	−	14005.4i	−0.132442	−	0.111132i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	61732.4	−	35641.2i	0.478988	−	0.276544i	−0.241007	−	0.970523i	$-0.577478\pi$
$359$	61732.4	−	35641.2i	0.478988	−	0.276544i	0.719995	+	0.693980i	$0.244144\pi$
$360$		0			0
$361$	29928.8	−	51838.1i	0.229654	−	0.397773i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−84502.3	−	232168.i	−0.634282	−	1.74268i
$366$		0			0
$367$	−6066.96	−	34407.4i	−0.0450442	−	0.255458i	0.953967	−	0.299911i	$-0.0969568\pi$
$367$	−6066.96	−	34407.4i	−0.0450442	−	0.255458i	−0.999012	+	0.0444523i	$0.985846\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	101687.	+	121186.i	0.738787	+	0.880452i
$372$		0			0
$373$	−14638.0	+	83016.5i	−0.105212	+	0.596687i	0.885923	+	0.463832i	$0.153526\pi$
$373$	−14638.0	+	83016.5i	−0.105212	+	0.596687i	−0.991135	+	0.132855i	$0.957585\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			39834.7i			0.280272i
$378$		0			0
$379$	55881.5			0.389036			0.194518	−	0.980899i	$-0.437686\pi$
$379$	55881.5			0.389036			0.194518	+	0.980899i	$0.437686\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	87237.6	+	15382.3i	0.594711	+	0.104864i	0.462899	−	0.886411i	$-0.346809\pi$
$383$	87237.6	+	15382.3i	0.594711	+	0.104864i	0.131812	+	0.991275i	$0.457920\pi$
$384$		0			0
$385$	−198212.	+	166319.i	−1.33724	+	1.12207i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21860.4	−	3854.58i	0.144464	−	0.0254729i	−0.100948	−	0.994892i	$-0.532188\pi$
$389$	21860.4	−	3854.58i	0.144464	−	0.0254729i	0.245412	+	0.969419i	$0.421077\pi$
$390$		0			0
$391$	246748.	−	89809.1i	1.61399	−	0.587444i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9954.44	+	5747.20i	0.0638003	+	0.0368351i
$396$		0			0
$397$	2098.88	+	3635.36i	0.0133170	+	0.0230657i	0.872607	−	0.488423i	$-0.162428\pi$
$397$	2098.88	+	3635.36i	0.0133170	+	0.0230657i	−0.859290	+	0.511489i	$0.829094\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13156.4	−	15679.2i	0.0818181	−	0.0975070i	−0.723582	−	0.690238i	$-0.757506\pi$
$401$	13156.4	−	15679.2i	0.0818181	−	0.0975070i	0.805401	+	0.592731i	$0.201950\pi$
$402$		0			0
$403$	25771.8	+	9380.17i	0.158685	+	0.0577565i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−49479.3	+	135943.i	−0.298700	+	0.820671i
$408$		0			0
$409$	−58453.3	−	49048.1i	−0.349432	−	0.293208i	0.451130	−	0.892458i	$-0.351021\pi$
$409$	−58453.3	−	49048.1i	−0.349432	−	0.293208i	−0.800562	+	0.599250i	$0.795465\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−67861.2	+	39179.7i	−0.397852	+	0.229700i
$414$		0			0
$415$	−54797.5	+	94912.0i	−0.318174	+	0.551093i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	74622.6	+	205024.i	0.425052	+	1.16782i	0.948780	+	0.315936i	$0.102319\pi$
$419$	74622.6	+	205024.i	0.425052	+	1.16782i	−0.523728	+	0.851885i	$0.675459\pi$
$420$		0			0
$421$	15845.6	+	89864.6i	0.0894012	+	0.507019i	0.996320	+	0.0857134i	$0.0273170\pi$
$421$	15845.6	+	89864.6i	0.0894012	+	0.507019i	−0.906919	+	0.421306i	$0.861572\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−482675.	−	575230.i	−2.67225	−	3.18466i
$426$		0			0
$427$	−5454.59	+	30934.5i	−0.0299162	+	0.169663i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			97952.6i			0.527304i	0.964618	+	0.263652i	$0.0849271\pi$
$431$			97952.6i			0.527304i	−0.964618	+	0.263652i	$0.915073\pi$
$432$		0			0
$433$	66312.8			0.353689			0.176845	−	0.984239i	$-0.443411\pi$
$433$	66312.8			0.353689			0.176845	+	0.984239i	$0.443411\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−156300.	−	27559.9i	−0.818459	−	0.144316i
$438$		0			0
$439$	44409.2	−	37263.8i	0.230433	−	0.193356i	−0.520259	−	0.854008i	$-0.674165\pi$
$439$	44409.2	−	37263.8i	0.230433	−	0.193356i	0.750692	+	0.660652i	$0.229720\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−273335.	+	48196.3i	−1.39279	+	0.245587i	−0.819179	−	0.573537i	$-0.805571\pi$
$443$	−273335.	+	48196.3i	−1.39279	+	0.245587i	−0.573615	+	0.819125i	$0.694460\pi$
$444$		0			0
$445$	−400810.	+	145883.i	−2.02404	+	0.736689i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18456.0	−	10655.6i	−0.0915474	−	0.0528549i	0.453527	−	0.891242i	$-0.350165\pi$
$449$	−18456.0	−	10655.6i	−0.0915474	−	0.0528549i	−0.545075	+	0.838387i	$0.683499\pi$
$450$		0			0
$451$	−81459.9	−	141093.i	−0.400489	−	0.693667i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−86225.9	+	102760.i	−0.416500	+	0.496365i
$456$		0			0
$457$	95765.7	+	34855.9i	0.458540	+	0.166895i	0.560954	−	0.827847i	$-0.310434\pi$
$457$	95765.7	+	34855.9i	0.458540	+	0.166895i	−0.102414	+	0.994742i	$0.532657\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−75918.6	+	208585.i	−0.357229	+	0.981478i	0.622758	+	0.782415i	$0.286012\pi$
$461$	−75918.6	+	208585.i	−0.357229	+	0.981478i	−0.979987	+	0.199063i	$0.936210\pi$
$462$		0			0
$463$	76646.9	+	64314.4i	0.357547	+	0.300017i	0.803812	−	0.594883i	$-0.202802\pi$
$463$	76646.9	+	64314.4i	0.357547	+	0.300017i	−0.446265	+	0.894901i	$0.647246\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	349382.	−	201716.i	1.60202	−	0.924924i	0.610932	−	0.791683i	$-0.290795\pi$
$467$	349382.	−	201716.i	1.60202	−	0.924924i	0.991084	−	0.133241i	$-0.0425383\pi$
$468$		0			0
$469$	280786.	−	486336.i	1.27653	−	2.21101i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−43470.9	−	119435.i	−0.194302	−	0.533839i
$474$		0			0
$475$	78812.9	+	446970.i	0.349309	+	1.98103i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−292457.	−	348537.i	−1.27465	−	1.51907i	−0.736974	−	0.675921i	$-0.763746\pi$
$479$	−292457.	−	348537.i	−1.27465	−	1.51907i	−0.537678	−	0.843150i	$-0.680698\pi$
$480$		0			0
$481$	−13023.8	+	73861.5i	−0.0562920	+	0.319248i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			588339.i			2.50118i
$486$		0			0
$487$	39017.6			0.164514			0.0822569	−	0.996611i	$-0.473787\pi$
$487$	39017.6			0.164514			0.0822569	+	0.996611i	$0.473787\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	396398.	+	69895.6i	1.64425	+	0.289926i	0.917726	−	0.397214i	$-0.130023\pi$
$491$	396398.	+	69895.6i	1.64425	+	0.289926i	0.726525	+	0.687140i	$0.241134\pi$
$492$		0			0
$493$	−329272.	+	276292.i	−1.35476	+	1.13678i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−30289.8	+	5340.92i	−0.122626	+	0.0216224i
$498$		0			0
$499$	229151.	−	83404.1i	0.920281	−	0.334955i	0.161931	−	0.986802i	$-0.448228\pi$
$499$	229151.	−	83404.1i	0.920281	−	0.334955i	0.758350	+	0.651847i	$0.226006\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−71653.8	−	41369.4i	−0.283207	−	0.163509i	0.351668	−	0.936125i	$-0.385615\pi$
$503$	−71653.8	−	41369.4i	−0.283207	−	0.163509i	−0.634874	+	0.772616i	$0.718948\pi$
$504$		0			0
$505$	−204425.	−	354074.i	−0.801588	−	1.38839i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	124541.	−	148422.i	0.480702	−	0.572879i	−0.470125	−	0.882600i	$-0.655791\pi$
$509$	124541.	−	148422.i	0.480702	−	0.572879i	0.950827	+	0.309721i	$0.100236\pi$
$510$		0			0
$511$	−327733.	−	119285.i	−1.25510	−	0.456819i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−162410.	+	446219.i	−0.612349	+	1.68242i
$516$		0			0
$517$	94319.1	+	79143.1i	0.352873	+	0.296096i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	320701.	−	185157.i	1.18148	−	0.682126i	0.225121	−	0.974331i	$-0.427722\pi$
$521$	320701.	−	185157.i	1.18148	−	0.682126i	0.956356	+	0.292205i	$0.0943889\pi$
$522$		0			0
$523$	−74878.3	+	129693.i	−0.273749	+	0.474147i	−0.969819	−	0.243827i	$-0.921597\pi$
$523$	−74878.3	+	129693.i	−0.273749	+	0.474147i	0.696070	+	0.717974i	$0.254930\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	101216.	+	278089.i	0.364442	+	1.00130i
$528$		0			0
$529$	−13481.9	−	76459.9i	−0.0481772	−	0.273226i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−54292.0	−	64702.7i	−0.191109	−	0.227755i
$534$		0			0
$535$	114670.	−	650324.i	0.400628	−	2.27207i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			176757.i			0.608414i
$540$		0			0
$541$	−11020.0			−0.0376518			−0.0188259	−	0.999823i	$-0.505993\pi$
$541$	−11020.0			−0.0376518			−0.0188259	+	0.999823i	$0.505993\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−146452.	−	25823.4i	−0.493061	−	0.0869400i
$546$		0			0
$547$	89576.9	−	75163.9i	0.299379	−	0.251209i	−0.480707	−	0.876881i	$-0.659620\pi$
$547$	89576.9	−	75163.9i	0.299379	−	0.251209i	0.780086	+	0.625673i	$0.215175\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	255854.	−	45114.0i	0.842731	−	0.148596i
$552$		0			0
$553$	15247.2	−	5549.52i	0.0498585	−	0.0181470i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	102639.	+	59258.7i	0.330828	+	0.191004i	0.656209	−	0.754579i	$-0.272159\pi$
$557$	102639.	+	59258.7i	0.330828	+	0.191004i	−0.325381	+	0.945583i	$0.605492\pi$
$558$		0			0
$559$	−32946.7	−	57065.3i	−0.105436	−	0.182620i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	117645.	−	140203.i	0.371155	−	0.442325i	−0.547847	−	0.836579i	$-0.684552\pi$
$563$	117645.	−	140203.i	0.371155	−	0.442325i	0.919002	+	0.394254i	$0.128997\pi$
$564$		0			0
$565$	279248.	+	101638.i	0.874770	+	0.318390i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14205.9	+	39030.3i	−0.0438777	+	0.120553i	−0.959696	−	0.281039i	$-0.909321\pi$
$569$	−14205.9	+	39030.3i	−0.0438777	+	0.120553i	0.915819	+	0.401592i	$0.131543\pi$
$570$		0			0
$571$	−373980.	−	313806.i	−1.14703	−	0.962476i	−0.147388	−	0.989079i	$-0.547086\pi$
$571$	−373980.	−	313806.i	−1.14703	−	0.962476i	−0.999646	+	0.0266033i	$0.991531\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−885323.	+	511141.i	−2.67773	+	1.54599i
$576$		0			0
$577$	−281929.	+	488316.i	−0.846815	+	1.46673i	0.0372198	+	0.999307i	$0.488150\pi$
$577$	−281929.	+	488316.i	−0.846815	+	1.46673i	−0.884035	+	0.467420i	$0.845184\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	52912.7	+	145376.i	0.156750	+	0.430667i
$582$		0			0
$583$	31618.1	+	179315.i	0.0930249	+	0.527570i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−170276.	−	202927.i	−0.494172	−	0.588931i	0.460102	−	0.887866i	$-0.347813\pi$
$587$	−170276.	−	202927.i	−0.494172	−	0.588931i	−0.954273	+	0.298935i	$0.903369\pi$
$588$		0			0
$589$	31060.5	−	176153.i	0.0895318	−	0.507760i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			283761.i			0.806945i	0.914992	+	0.403472i	$0.132197\pi$
$593$			283761.i			0.806945i	−0.914992	+	0.403472i	$0.867803\pi$
$594$		0			0
$595$	−1.44747e6			−4.08861
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−238299.	−	42018.5i	−0.664153	−	0.117108i	−0.168598	−	0.985685i	$-0.553924\pi$
$599$	−238299.	−	42018.5i	−0.664153	−	0.117108i	−0.495554	+	0.868577i	$0.665035\pi$
$600$		0			0
$601$	−352750.	+	295992.i	−0.976603	+	0.819468i	−0.983573	−	0.180508i	$-0.942226\pi$
$601$	−352750.	+	295992.i	−0.976603	+	0.819468i	0.00697012	+	0.999976i	$0.497781\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	403414.	−	71132.8i	1.10215	−	0.194339i
$606$		0			0
$607$	−477941.	+	173956.i	−1.29717	+	0.472132i	−0.896074	−	0.443904i	$-0.853593\pi$
$607$	−477941.	+	173956.i	−1.29717	+	0.472132i	−0.401096	+	0.916036i	$0.631371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	55280.4	+	31916.1i	0.148077	+	0.0854925i
$612$		0			0
$613$	−260763.	−	451654.i	−0.693944	−	1.20195i	−0.970535	−	0.240958i	$-0.922538\pi$
$613$	−260763.	−	451654.i	−0.693944	−	1.20195i	0.276592	−	0.960988i	$-0.410795\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11636.2	+	13867.5i	−0.0305662	+	0.0364274i	−0.781111	−	0.624392i	$-0.785347\pi$
$617$	−11636.2	+	13867.5i	−0.0305662	+	0.0364274i	0.750545	+	0.660820i	$0.229791\pi$
$618$		0			0
$619$	−535558.	−	194927.i	−1.39774	−	0.508734i	−0.470230	−	0.882544i	$-0.655829\pi$
$619$	−535558.	−	194927.i	−1.39774	−	0.508734i	−0.927505	+	0.373810i	$0.878051\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−205931.	+	565791.i	−0.530574	+	1.45774i
$624$		0			0
$625$	1.12161e6	+	941145.i	2.87133	+	2.40933i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−700869.	+	404647.i	−1.77148	+	1.02276i
$630$		0			0
$631$	−73354.3	+	127053.i	−0.184233	+	0.319100i	−0.943318	−	0.331891i	$-0.892313\pi$
$631$	−73354.3	+	127053.i	−0.184233	+	0.319100i	0.759085	+	0.650992i	$0.225647\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−355895.	−	977813.i	−0.882621	−	2.42498i
$636$		0			0
$637$	15912.6	+	90244.9i	0.0392159	+	0.222405i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−143698.	−	171253.i	−0.349732	−	0.416794i	0.562287	−	0.826942i	$-0.309921\pi$
$641$	−143698.	−	171253.i	−0.349732	−	0.416794i	−0.912019	+	0.410148i	$0.865477\pi$
$642$		0			0
$643$	57503.2	−	326117.i	0.139082	−	0.788772i	−0.832848	−	0.553502i	$-0.813291\pi$
$643$	57503.2	−	326117.i	0.139082	−	0.788772i	0.971930	−	0.235271i	$-0.0755976\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	801131.i		−	1.91379i	−0.290430	−	0.956896i	$-0.593798\pi$
$647$		−	801131.i		−	1.91379i	0.290430	−	0.956896i	$-0.406202\pi$
$648$		0			0
$649$	−90189.7			−0.214125
$650$		0			0
$651$		0			0
$652$		0			0
$653$	798904.	+	140868.i	1.87356	+	0.330360i	0.990347	−	0.138607i	$-0.0442625\pi$
$653$	798904.	+	140868.i	1.87356	+	0.330360i	0.883216	+	0.468967i	$0.155374\pi$
$654$		0			0
$655$	552721.	−	463788.i	1.28832	−	1.08103i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	121439.	−	21412.9i	0.279632	−	0.0493066i	−0.0320735	−	0.999486i	$-0.510211\pi$
$659$	121439.	−	21412.9i	0.279632	−	0.0493066i	0.311705	+	0.950179i	$0.399100\pi$
$660$		0			0
$661$	432224.	−	157317.i	0.989250	−	0.360058i	0.203820	−	0.979008i	$-0.434664\pi$
$661$	432224.	−	157317.i	0.989250	−	0.360058i	0.785430	+	0.618951i	$0.212442\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	757669.	+	437440.i	1.71331	+	0.989181i
$666$		0			0
$667$	292587.	+	506775.i	0.657663	+	1.13911i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−23239.4	+	27695.7i	−0.0516156	+	0.0615130i
$672$		0			0
$673$	−66334.1	−	24143.6i	−0.146456	−	0.0533055i	0.267752	−	0.963488i	$-0.413719\pi$
$673$	−66334.1	−	24143.6i	−0.146456	−	0.0533055i	−0.414208	+	0.910182i	$0.635941\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−145933.	+	400948.i	−0.318403	+	0.874804i	0.672485	+	0.740111i	$0.265227\pi$
$677$	−145933.	+	400948.i	−0.318403	+	0.874804i	−0.990887	+	0.134693i	$0.956995\pi$
$678$		0			0
$679$	636208.	+	533842.i	1.37994	+	1.15791i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−190402.	+	109929.i	−0.408160	+	0.235652i	−0.689999	−	0.723810i	$-0.742389\pi$
$683$	−190402.	+	109929.i	−0.408160	+	0.235652i	0.281839	+	0.959462i	$0.409056\pi$
$684$		0			0
$685$	366793.	−	635304.i	0.781700	−	1.35394i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	32285.9	+	88704.7i	0.0680102	+	0.186856i
$690$		0			0
$691$	34511.7	+	195726.i	0.0722787	+	0.409913i	0.999383	+	0.0351107i	$0.0111784\pi$
$691$	34511.7	+	195726.i	0.0722787	+	0.409913i	−0.927105	+	0.374802i	$0.877711\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−225223.	−	268411.i	−0.466277	−	0.555687i
$696$		0			0
$697$	158262.	−	897550.i	0.325771	−	1.84754i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			339044.i			0.689953i	0.938611	+	0.344976i	$0.112113\pi$
$701$			339044.i			0.689953i	−0.938611	+	0.344976i	$0.887887\pi$
$702$		0			0
$703$	489154.			0.989771
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−568372.	−	100219.i	−1.13709	−	0.200499i
$708$		0			0
$709$	−465553.	+	390646.i	−0.926141	+	0.777125i	−0.975121	−	0.221675i	$-0.928848\pi$
$709$	−465553.	+	390646.i	−0.926141	+	0.777125i	0.0489796	+	0.998800i	$0.484403\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	396765.	−	69960.4i	0.780467	−	0.137617i
$714$		0			0
$715$	−145085.	+	52806.6i	−0.283799	+	0.103294i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4521.85	+	2610.69i	0.00874698	+	0.00505007i	0.504367	−	0.863489i	$-0.331726\pi$
$719$	4521.85	+	2610.69i	0.00874698	+	0.00505007i	−0.495620	+	0.868539i	$0.665059\pi$
$720$		0			0
$721$	335158.	+	580511.i	0.644732	+	1.11671i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.07565e6	−	1.28191e6i	2.04642	−	2.43883i
$726$		0			0
$727$	738284.	+	268713.i	1.39686	+	0.508417i	0.927246	−	0.374452i	$-0.122169\pi$
$727$	738284.	+	268713.i	1.39686	+	0.508417i	0.469619	+	0.882869i	$0.344391\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	243183.	−	668139.i	0.455090	−	1.25035i
$732$		0			0
$733$	72309.6	+	60674.9i	0.134582	+	0.112928i	0.707594	−	0.706619i	$-0.249781\pi$
$733$	72309.6	+	60674.9i	0.134582	+	0.112928i	−0.573012	+	0.819547i	$0.694225\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	559762.	−	323179.i	1.03055	−	0.594987i
$738$		0			0
$739$	61958.1	−	107315.i	0.113451	−	0.196503i	−0.803708	−	0.595023i	$-0.797143\pi$
$739$	61958.1	−	107315.i	0.113451	−	0.196503i	0.917160	+	0.398520i	$0.130476\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	161118.	+	442667.i	0.291854	+	0.801863i	0.995795	+	0.0916045i	$0.0291995\pi$
$743$	161118.	+	442667.i	0.291854	+	0.801863i	−0.703941	+	0.710258i	$0.748578\pi$
$744$		0			0
$745$	−160232.	−	908723.i	−0.288694	−	1.63726i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−599189.	−	714085.i	−1.06807	−	1.27288i
$750$		0			0
$751$	28489.9	−	161574.i	0.0505139	−	0.286479i	−0.949078	−	0.315041i	$-0.897982\pi$
$751$	28489.9	−	161574.i	0.0505139	−	0.286479i	0.999592	+	0.0285622i	$0.00909286\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			287894.i			0.505056i
$756$		0			0
$757$	919126.			1.60392			0.801961	−	0.597376i	$-0.203790\pi$
$757$	919126.			1.60392			0.801961	+	0.597376i	$0.203790\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	834832.	+	147203.i	1.44155	+	0.254184i	0.839102	−	0.543974i	$-0.183081\pi$
$761$	834832.	+	147203.i	1.44155	+	0.254184i	0.602448	+	0.798158i	$0.294192\pi$
$762$		0			0
$763$	−160810.	+	134936.i	−0.276226	+	0.231781i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−46047.2	+	8119.36i	−0.0782731	+	0.0138017i
$768$		0			0
$769$	151062.	−	54982.2i	0.255449	−	0.0929758i	−0.211122	−	0.977460i	$-0.567712\pi$
$769$	151062.	−	54982.2i	0.255449	−	0.0929758i	0.466570	+	0.884484i	$0.345489\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−397507.	−	229501.i	−0.665251	−	0.384083i	0.129024	−	0.991641i	$-0.458816\pi$
$773$	−397507.	−	229501.i	−0.665251	−	0.384083i	−0.794275	+	0.607559i	$0.792149\pi$
$774$		0			0
$775$	−576064.	−	997772.i	−0.959108	−	1.66122i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−354091.	+	421989.i	−0.583498	+	0.695386i
$780$		0			0
$781$	−33265.8	−	12107.8i	−0.0545376	−	0.0198501i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−454359.	+	1.24834e6i	−0.737326	+	2.02579i
$786$		0			0
$787$	−363174.	−	304739.i	−0.586361	−	0.492015i	0.300668	−	0.953729i	$-0.402790\pi$
$787$	−363174.	−	304739.i	−0.586361	−	0.492015i	−0.887029	+	0.461714i	$0.847235\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	363290.	−	209745.i	0.580631	−	0.335227i
$792$		0			0
$793$	−9371.79	+	16232.4i	−0.0149031	+	0.0258129i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−344884.	−	947562.i	−0.542946	−	1.49173i	−0.843055	−	0.537827i	$-0.819245\pi$
$797$	−344884.	−	947562.i	−0.542946	−	1.49173i	0.300109	−	0.953905i	$-0.402977\pi$
$798$		0			0
$799$	119605.	+	678314.i	0.187351	+	1.06252i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−258029.	−	307507.i	−0.400163	−	0.476896i
$804$		0			0
$805$	−342187.	+	1.94064e6i	−0.528046	+	2.99470i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	820719.i		−	1.25400i	−0.779019	−	0.627000i	$-0.784283\pi$
$809$		−	820719.i		−	1.25400i	0.779019	−	0.627000i	$-0.215717\pi$
$810$		0			0
$811$	223347.			0.339577			0.169789	−	0.985481i	$-0.445692\pi$
$811$	223347.			0.339577			0.169789	+	0.985481i	$0.445692\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−446270.	−	78689.4i	−0.671865	−	0.118468i
$816$		0			0
$817$	−329211.	+	276241.i	−0.493208	+	0.413851i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−600428.	+	105872.i	−0.890789	+	0.157070i	−0.600268	−	0.799799i	$-0.704940\pi$
$821$	−600428.	+	105872.i	−0.890789	+	0.157070i	−0.290521	+	0.956869i	$0.593828\pi$
$822$		0			0
$823$	56527.4	−	20574.3i	0.0834564	−	0.0303756i	−0.299955	−	0.953953i	$-0.596972\pi$
$823$	56527.4	−	20574.3i	0.0834564	−	0.0303756i	0.383411	+	0.923578i	$0.374749\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	547746.	+	316241.i	0.800882	+	0.462389i	0.843779	−	0.536690i	$-0.180326\pi$
$827$	547746.	+	316241.i	0.800882	+	0.462389i	−0.0428977	+	0.999079i	$0.513659\pi$
$828$		0			0
$829$	−62874.8	−	108902.i	−0.0914887	−	0.158463i	0.816649	−	0.577135i	$-0.195829\pi$
$829$	−62874.8	−	108902.i	−0.0914887	−	0.158463i	−0.908138	+	0.418672i	$0.862496\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−635591.	+	757468.i	−0.915984	+	1.09163i
$834$		0			0
$835$	1.66357e6	+	605491.i	2.38599	+	0.868430i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	138269.	−	379890.i	0.196426	−	0.539677i	−0.801903	−	0.597454i	$-0.796179\pi$
$839$	138269.	−	379890.i	0.196426	−	0.539677i	0.998330	+	0.0577771i	$0.0184013\pi$
$840$		0			0
$841$	−191981.	−	161091.i	−0.271435	−	0.227761i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.12585e6	−	650008.i	1.57676	−	0.910343i
$846$		0			0
$847$	289126.	−	500781.i	0.403014	−	0.698041i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	376826.	+	1.03532e6i	0.520334	+	1.42961i
$852$		0			0
$853$	29591.0	+	167819.i	0.0406688	+	0.230644i	0.998367	−	0.0571314i	$-0.0181954\pi$
$853$	29591.0	+	167819.i	0.0406688	+	0.230644i	−0.957698	+	0.287776i	$0.907084\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−692880.	−	825742.i	−0.943401	−	1.12430i	−0.992095	−	0.125488i	$-0.959950\pi$
$857$	−692880.	−	825742.i	−0.943401	−	1.12430i	0.0486944	−	0.998814i	$-0.484494\pi$
$858$		0			0
$859$	−119.204	+	676.037i	−0.000161549	+	0.000916187i	−0.984888	−	0.173190i	$-0.944593\pi$
$859$	−119.204	+	676.037i	−0.000161549	+	0.000916187i	0.984727	+	0.174106i	$0.0557036\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	184442.i		−	0.247649i	−0.992304	−	0.123825i	$-0.960484\pi$
$863$		−	184442.i		−	0.247649i	0.992304	−	0.123825i	$-0.0395160\pi$
$864$		0			0
$865$	491557.			0.656964
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18391.7	+	3242.95i	0.0243547	+	0.00429439i
$870$		0			0
$871$	256697.	−	215394.i	0.338364	−	0.283921i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.52101e6	−	620849.i	4.59887	−	0.810905i
$876$		0			0
$877$	−640397.	+	233085.i	−0.832626	+	0.303051i	−0.722936	−	0.690915i	$-0.757208\pi$
$877$	−640397.	+	233085.i	−0.832626	+	0.303051i	−0.109690	+	0.993966i	$0.534986\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	525075.	+	303152.i	0.676502	+	0.390579i	0.798536	−	0.601947i	$-0.205608\pi$
$881$	525075.	+	303152.i	0.676502	+	0.390579i	−0.122034	+	0.992526i	$0.538942\pi$
$882$		0			0
$883$	194188.	+	336344.i	0.249058	+	0.431382i	0.963265	−	0.268553i	$-0.0865455\pi$
$883$	194188.	+	336344.i	0.249058	+	0.431382i	−0.714206	+	0.699935i	$0.753212\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−698209.	+	832093.i	−0.887439	+	1.05761i	0.110528	+	0.993873i	$0.464746\pi$
$887$	−698209.	+	832093.i	−0.887439	+	1.05761i	−0.997967	+	0.0637353i	$0.979699\pi$
$888$		0			0
$889$	−1.38030e6	−	502388.i	−1.74650	−	0.635676i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	142387.	−	391205.i	0.178553	−	0.490571i
$894$		0			0
$895$	−1.91303e6	−	1.60522e6i	−2.38822	−	2.00396i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−571144.	+	329750.i	−0.706685	+	0.408005i
$900$		0			0
$901$	−509296.	+	882126.i	−0.627365	+	1.08663i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	486251.	+	1.33596e6i	0.593695	+	1.63116i
$906$		0			0
$907$	204852.	+	1.16177e6i	0.249015	+	1.41223i	0.810981	+	0.585073i	$0.198934\pi$
$907$	204852.	+	1.16177e6i	0.249015	+	1.41223i	−0.561966	+	0.827160i	$0.689955\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−43204.8	−	51489.5i	−0.0520589	−	0.0620414i	0.739387	−	0.673281i	$-0.235116\pi$
$911$	−43204.8	−	51489.5i	−0.0520589	−	0.0620414i	−0.791446	+	0.611240i	$0.790671\pi$
$912$		0			0
$913$	−30920.4	+	175358.i	−0.0370940	+	0.210370i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1.01852e6i		−	1.21124i
$918$		0			0
$919$	−5673.89			−0.00671815			−0.00335907	−	0.999994i	$-0.501069\pi$
$919$	−5673.89			−0.00671815			−0.00335907	+	0.999994i	$0.501069\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18074.2	−	3186.96i	−0.0212156	−	0.00374088i
$924$		0			0
$925$	2.41358e6	−	2.02524e6i	2.82084	−	2.36697i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−423766.	+	74721.5i	−0.491015	+	0.0865793i	−0.413673	−	0.910425i	$-0.635754\pi$
$929$	−423766.	+	74721.5i	−0.491015	+	0.0865793i	−0.0773420	+	0.997005i	$0.524643\pi$
$930$		0			0
$931$	561611.	−	204410.i	0.647942	−	0.235832i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.44280e6	−	833002.i	−1.65038	−	0.952846i
$936$		0			0
$937$	778624.	+	1.34862e6i	0.886846	+	1.53606i	0.843583	+	0.536999i	$0.180442\pi$
$937$	778624.	+	1.34862e6i	0.886846	+	1.53606i	0.0432635	+	0.999064i	$0.486225\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	893353.	−	1.06466e6i	1.00889	−	1.20235i	0.0296671	−	0.999560i	$-0.490555\pi$
$941$	893353.	−	1.06466e6i	1.00889	−	1.20235i	0.979223	−	0.202788i	$-0.0650003\pi$
$942$		0			0
$943$	−1.16594e6	−	424368.i	−1.31115	−	0.477221i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−574435.	+	1.57825e6i	−0.640532	+	1.75985i	0.00951636	+	0.999955i	$0.496971\pi$
$947$	−574435.	+	1.57825e6i	−0.640532	+	1.75985i	−0.650048	+	0.759893i	$0.725251\pi$
$948$		0			0
$949$	−159422.	−	133771.i	−0.177018	−	0.148536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−70784.8	+	40867.6i	−0.0779389	+	0.0449980i	−0.538463	−	0.842649i	$-0.680995\pi$
$953$	−70784.8	+	40867.6i	−0.0779389	+	0.0449980i	0.460524	+	0.887647i	$0.347661\pi$
$954$		0			0
$955$	−76314.6	+	132181.i	−0.0836760	+	0.144931i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−354177.	−	973093.i	−0.385108	−	1.05808i
$960$		0			0
$961$	−81521.3	−	462330.i	−0.0882723	−	0.500617i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.72377e6	+	2.05431e6i	1.85108	+	2.20603i
$966$		0			0
$967$	108012.	−	612565.i	0.115510	−	0.655088i	−0.870987	−	0.491306i	$-0.836520\pi$
$967$	108012.	−	612565.i	0.115510	−	0.655088i	0.986497	−	0.163782i	$-0.0523692\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.42070e6i		−	1.50683i	−0.657546	−	0.753414i	$-0.728406\pi$
$971$		−	1.42070e6i		−	1.50683i	0.657546	−	0.753414i	$-0.271594\pi$
$972$		0			0
$973$	−494610.			−0.522441
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−432227.	−	76213.3i	−0.452817	−	0.0798439i	−0.0574117	−	0.998351i	$-0.518285\pi$
$977$	−432227.	−	76213.3i	−0.452817	−	0.0798439i	−0.395405	+	0.918507i	$0.629396\pi$
$978$		0			0
$979$	−530873.	+	445455.i	−0.553892	+	0.464771i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−98179.9	+	17311.8i	−0.101605	+	0.0179157i	−0.224220	−	0.974539i	$-0.571983\pi$
$983$	−98179.9	+	17311.8i	−0.101605	+	0.0179157i	0.122614	+	0.992454i	$0.460872\pi$
$984$		0			0
$985$	980533.	−	356885.i	1.01062	−	0.367837i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−838292.	−	483988.i	−0.857043	−	0.494814i
$990$		0			0
$991$	−390738.	−	676778.i	−0.397867	−	0.689126i	0.595595	−	0.803285i	$-0.296916\pi$
$991$	−390738.	−	676778.i	−0.397867	−	0.689126i	−0.993463	+	0.114158i	$0.963583\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.65141e6	+	1.96807e6i	−1.66805	+	1.98790i
$996$		0			0
$997$	16274.8	+	5923.53i	0.0163729	+	0.00595923i	0.350194	−	0.936677i	$-0.386116\pi$
$997$	16274.8	+	5923.53i	0.0163729	+	0.00595923i	−0.333821	+	0.942637i	$0.608338\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.k.a.17.1		72
3.2	odd	2		108.5.k.a.77.4	✓	72
27.7	even	9		108.5.k.a.101.4	yes	72
27.20	odd	18	inner	324.5.k.a.305.1		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.77.4	✓	72	3.2	odd	2
108.5.k.a.101.4	yes	72	27.7	even	9
324.5.k.a.17.1		72	1.1	even	1	trivial
324.5.k.a.305.1		72	27.20	odd	18	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-47.5857 - 8.39064i) q^{5} +(-52.2512 + 43.8440i) q^{7} +O(q^{10})\)	\(q+(-47.5857 - 8.39064i) q^{5} +(-52.2512 + 43.8440i) q^{7} +(-77.3144 + 13.6326i) q^{11} +(-38.2463 + 13.9205i) q^{13} +(-380.341 - 219.590i) q^{17} +(132.725 + 229.886i) q^{19} +(-384.321 + 458.015i) q^{23} +(1606.68 + 584.785i) q^{25} +(334.742 - 919.697i) q^{29} +(-516.190 - 433.135i) q^{31} +(2854.29 - 1647.92i) q^{35} +(921.369 - 1595.86i) q^{37} +(709.768 + 1950.07i) q^{41} +(281.131 + 1594.37i) q^{43} +(-1008.10 - 1201.41i) q^{47} +(390.965 - 2217.27i) q^{49} -2319.30i q^{53} +3793.44 q^{55} +(1131.36 + 199.489i) q^{59} +(352.780 - 296.017i) q^{61} +(1936.78 - 341.506i) q^{65} +(-7736.59 + 2815.89i) q^{67} +(390.512 + 225.462i) q^{71} +(2556.60 + 4428.15i) q^{73} +(3442.06 - 4102.09i) q^{77} +(-223.536 - 81.3605i) q^{79} +(775.742 - 2131.33i) q^{83} +(16256.3 + 13640.6i) q^{85} +(7644.67 - 4413.65i) q^{89} +(1388.08 - 2404.23i) q^{91} +(-4386.91 - 12052.9i) q^{95} +(-2114.33 - 11991.0i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 72 * q - 9 * q^5	\( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more