LMFDB - Embedded newform 324.5.k.a.89.7

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			89.7
Character	$\chi$	$=$	324.89
Dual form			324.5.k.a.233.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0201223 - 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +O(q^{10})$	$q+(-0.0201223 - 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +(-122.535 + 146.032i) q^{11} +(8.21590 + 46.5947i) q^{13} +(383.840 - 221.610i) q^{17} +(299.637 - 518.987i) q^{19} +(-158.524 + 435.540i) q^{23} +(108.530 - 615.504i) q^{25} +(396.668 + 69.9433i) q^{29} +(-1472.40 - 535.909i) q^{31} +(1.74238 + 1.00596i) q^{35} +(-532.781 - 922.804i) q^{37} +(2327.41 - 410.385i) q^{41} +(1343.05 + 1126.95i) q^{43} +(-864.494 - 2375.18i) q^{47} +(1324.91 - 1111.73i) q^{49} -2410.75i q^{53} +5.96765 q^{55} +(-165.882 - 197.690i) q^{59} +(-3689.65 + 1342.92i) q^{61} +(0.952054 - 1.13461i) q^{65} +(-702.984 - 3986.82i) q^{67} +(2144.37 - 1238.05i) q^{71} +(761.304 - 1318.62i) q^{73} +(4190.34 - 11512.9i) q^{77} +(1351.07 - 7662.27i) q^{79} +(12237.2 + 2157.76i) q^{83} +(-13.0381 - 4.74548i) q^{85} +(-4114.03 - 2375.24i) q^{89} +(-1520.40 - 2633.41i) q^{91} +(-18.4751 + 3.25765i) q^{95} +(-4423.75 - 3711.97i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

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

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
$5$	−0.0201223	−	0.0239808i	−0.000804890	−	0.000959231i	0.765642	−	0.643267i	$-0.222422\pi$
$5$	−0.0201223	−	0.0239808i	−0.000804890	−	0.000959231i	−0.766447	+	0.642308i	$0.777977\pi$
$6$		0			0
$7$	−60.3934	+	21.9814i	−1.23252	+	0.448600i	−0.874459	−	0.485099i	$-0.838783\pi$
$7$	−60.3934	+	21.9814i	−1.23252	+	0.448600i	−0.358059	+	0.933699i	$0.616561\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−122.535	+	146.032i	−1.01269	+	1.20688i	−0.0344472	+	0.999407i	$0.510967\pi$
$11$	−122.535	+	146.032i	−1.01269	+	1.20688i	−0.978242	+	0.207469i	$0.933477\pi$
$12$		0			0
$13$	8.21590	+	46.5947i	0.0486148	+	0.275708i	0.999419	−	0.0340844i	$-0.0108515\pi$
$13$	8.21590	+	46.5947i	0.0486148	+	0.275708i	−0.950804	+	0.309793i	$0.899740\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	383.840	−	221.610i	1.32817	−	0.766817i	0.343150	−	0.939281i	$-0.388506\pi$
$17$	383.840	−	221.610i	1.32817	−	0.766817i	0.985016	+	0.172464i	$0.0551728\pi$
$18$		0			0
$19$	299.637	−	518.987i	0.830020	−	1.43764i	−0.0680020	−	0.997685i	$-0.521662\pi$
$19$	299.637	−	518.987i	0.830020	−	1.43764i	0.898022	−	0.439951i	$-0.145004\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−158.524	+	435.540i	−0.299666	+	0.823327i	0.694889	+	0.719117i	$0.255454\pi$
$23$	−158.524	+	435.540i	−0.299666	+	0.823327i	−0.994555	+	0.104210i	$0.966769\pi$
$24$		0			0
$25$	108.530	−	615.504i	0.173648	−	0.984806i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	396.668	+	69.9433i	0.471663	+	0.0831668i	0.404427	−	0.914570i	$-0.367471\pi$
$29$	396.668	+	69.9433i	0.471663	+	0.0831668i	0.0672356	+	0.997737i	$0.478582\pi$
$30$		0			0
$31$	−1472.40	−	535.909i	−1.53215	−	0.557658i	−0.568005	−	0.823025i	$-0.692285\pi$
$31$	−1472.40	−	535.909i	−1.53215	−	0.557658i	−0.964148	+	0.265367i	$0.914507\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.74238	+	1.00596i	0.00142235	+	0.000821196i
$36$		0			0
$37$	−532.781	−	922.804i	−0.389175	−	0.674071i	0.603163	−	0.797618i	$-0.293907\pi$
$37$	−532.781	−	922.804i	−0.389175	−	0.674071i	−0.992339	+	0.123546i	$0.960573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2327.41	−	410.385i	1.38454	−	0.244132i	0.568764	−	0.822501i	$-0.307422\pi$
$41$	2327.41	−	410.385i	1.38454	−	0.244132i	0.815775	+	0.578369i	$0.196311\pi$
$42$		0			0
$43$	1343.05	+	1126.95i	0.726364	+	0.609491i	0.929138	−	0.369734i	$-0.120551\pi$
$43$	1343.05	+	1126.95i	0.726364	+	0.609491i	−0.202774	+	0.979226i	$0.564996\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−864.494	−	2375.18i	−0.391351	−	1.07523i	−0.966385	−	0.257099i	$-0.917234\pi$
$47$	−864.494	−	2375.18i	−0.391351	−	1.07523i	0.575034	−	0.818129i	$-0.304989\pi$
$48$		0			0
$49$	1324.91	−	1111.73i	0.551814	−	0.463027i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2410.75i		−	0.858225i	−0.903251	−	0.429112i	$-0.858826\pi$
$53$		−	2410.75i		−	0.858225i	0.903251	−	0.429112i	$-0.141174\pi$
$54$		0			0
$55$	5.96765			0.00197278
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−165.882	−	197.690i	−0.0476535	−	0.0567912i	0.741691	−	0.670741i	$-0.234024\pi$
$59$	−165.882	−	197.690i	−0.0476535	−	0.0567912i	−0.789345	+	0.613950i	$0.789580\pi$
$60$		0			0
$61$	−3689.65	+	1342.92i	−0.991574	+	0.360903i	−0.786329	−	0.617807i	$-0.788021\pi$
$61$	−3689.65	+	1342.92i	−0.991574	+	0.360903i	−0.205244	+	0.978711i	$0.565799\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.952054	−	1.13461i	0.000225338	−	0.000268548i
$66$		0			0
$67$	−702.984	−	3986.82i	−0.156601	−	0.888131i	−0.957307	−	0.289073i	$-0.906653\pi$
$67$	−702.984	−	3986.82i	−0.156601	−	0.888131i	0.800706	−	0.599058i	$-0.204458\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2144.37	−	1238.05i	0.425385	−	0.245596i	−0.271994	−	0.962299i	$-0.587683\pi$
$71$	2144.37	−	1238.05i	0.425385	−	0.245596i	0.697379	+	0.716703i	$0.254350\pi$
$72$		0			0
$73$	761.304	−	1318.62i	0.142861	−	0.247442i	−0.785712	−	0.618592i	$-0.787703\pi$
$73$	761.304	−	1318.62i	0.142861	−	0.247442i	0.928573	+	0.371150i	$0.121037\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4190.34	−	11512.9i	0.706753	−	1.94179i
$78$		0			0
$79$	1351.07	−	7662.27i	0.216482	−	1.22773i	−0.661834	−	0.749651i	$-0.730221\pi$
$79$	1351.07	−	7662.27i	0.216482	−	1.22773i	0.878316	−	0.478081i	$-0.158667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12237.2	+	2157.76i	1.77634	+	0.313218i	0.963188	−	0.268828i	$-0.0866362\pi$
$83$	12237.2	+	2157.76i	1.77634	+	0.313218i	0.813156	+	0.582045i	$0.197747\pi$
$84$		0			0
$85$	−13.0381	−	4.74548i	−0.00180458	−	0.000656814i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4114.03	−	2375.24i	−0.519383	−	0.299866i	0.217299	−	0.976105i	$-0.430275\pi$
$89$	−4114.03	−	2375.24i	−0.519383	−	0.299866i	−0.736682	+	0.676239i	$0.763609\pi$
$90$		0			0
$91$	−1520.40	−	2633.41i	−0.183601	−	0.318007i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−18.4751	+	3.25765i	−0.00204710	+	0.000360959i
$96$		0			0
$97$	−4423.75	−	3711.97i	−0.470162	−	0.394512i	0.376692	−	0.926339i	$-0.377062\pi$
$97$	−4423.75	−	3711.97i	−0.470162	−	0.394512i	−0.846854	+	0.531826i	$0.821506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−191.499	−	526.138i	−0.0187725	−	0.0515771i	0.929952	−	0.367680i	$-0.119848\pi$
$101$	−191.499	−	526.138i	−0.0187725	−	0.0515771i	−0.948725	+	0.316103i	$0.897626\pi$
$102$		0			0
$103$	−5058.09	+	4244.24i	−0.476773	+	0.400060i	−0.849258	−	0.527978i	$-0.822950\pi$
$103$	−5058.09	+	4244.24i	−0.476773	+	0.400060i	0.372485	+	0.928038i	$0.378506\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4467.37i		−	0.390198i	−0.980784	−	0.195099i	$-0.937497\pi$
$107$		−	4467.37i		−	0.390198i	0.980784	−	0.195099i	$-0.0625027\pi$
$108$		0			0
$109$	9782.65			0.823386			0.411693	−	0.911323i	$-0.364938\pi$
$109$	9782.65			0.823386			0.411693	+	0.911323i	$0.364938\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8140.66	+	9701.66i	0.637533	+	0.759782i	0.983978	−	0.178288i	$-0.0570559\pi$
$113$	8140.66	+	9701.66i	0.637533	+	0.759782i	−0.346446	+	0.938070i	$0.612611\pi$
$114$		0			0
$115$	13.6344	−	4.96253i	0.00103096	−	0.000375238i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−18310.1	+	21821.1i	−1.29299	+	1.54093i
$120$		0			0
$121$	−3768.03	−	21369.6i	−0.257362	−	1.45957i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−33.8883	+	19.5654i	−0.00216885	+	0.00125219i
$126$		0			0
$127$	11131.3	−	19280.0i	0.690142	−	1.19536i	−0.281649	−	0.959518i	$-0.590881\pi$
$127$	11131.3	−	19280.0i	0.690142	−	1.19536i	0.971791	−	0.235844i	$-0.0757853\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2326.48	+	6391.94i	−0.135568	+	0.372469i	−0.988837	−	0.149001i	$-0.952394\pi$
$131$	−2326.48	+	6391.94i	−0.135568	+	0.372469i	0.853269	+	0.521470i	$0.174616\pi$
$132$		0			0
$133$	−6688.05	+	37929.8i	−0.378091	+	2.14426i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6309.25	−	1112.49i	−0.336153	−	0.0592728i	0.00302422	−	0.999995i	$-0.499037\pi$
$137$	−6309.25	−	1112.49i	−0.336153	−	0.0592728i	−0.339177	+	0.940723i	$0.610148\pi$
$138$		0			0
$139$	−21971.8	−	7997.08i	−1.13720	−	0.413906i	−0.296297	−	0.955096i	$-0.595752\pi$
$139$	−21971.8	−	7997.08i	−1.13720	−	0.413906i	−0.840901	+	0.541190i	$0.817974\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7811.05	−	4509.71i	−0.381977	−	0.220535i
$144$		0			0
$145$	−6.30457	−	10.9198i	−0.000299860	−	0.000519373i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	33711.6	−	5944.26i	1.51847	−	0.267747i	0.648639	−	0.761096i	$-0.275338\pi$
$149$	33711.6	−	5944.26i	1.51847	−	0.267747i	0.869832	+	0.493348i	$0.164227\pi$
$150$		0			0
$151$	22011.9	+	18470.2i	0.965390	+	0.810059i	0.981822	−	0.189806i	$-0.0607860\pi$
$151$	22011.9	+	18470.2i	0.965390	+	0.810059i	−0.0164311	+	0.999865i	$0.505230\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.7765	+	46.0930i	0.000698292	+	0.00191854i
$156$		0			0
$157$	−5881.23	+	4934.94i	−0.238599	+	0.200208i	−0.754244	−	0.656594i	$-0.771997\pi$
$157$	−5881.23	+	4934.94i	−0.238599	+	0.200208i	0.515645	+	0.856802i	$0.327552\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	29788.3i		−	1.14920i
$162$		0			0
$163$	9213.77			0.346786			0.173393	−	0.984853i	$-0.444527\pi$
$163$	9213.77			0.346786			0.173393	+	0.984853i	$0.444527\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9340.18	+	11131.2i	0.334905	+	0.399125i	0.907047	−	0.421030i	$-0.138331\pi$
$167$	9340.18	+	11131.2i	0.334905	+	0.399125i	−0.572141	+	0.820155i	$0.693887\pi$
$168$		0			0
$169$	24735.0	−	9002.80i	0.866041	−	0.315213i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14480.8	+	17257.6i	−0.483839	+	0.576617i	−0.951639	−	0.307217i	$-0.900602\pi$
$173$	−14480.8	+	17257.6i	−0.483839	+	0.576617i	0.467800	+	0.883834i	$0.345047\pi$
$174$		0			0
$175$	6975.14	+	39558.0i	0.227760	+	1.29169i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−35962.9	+	20763.2i	−1.12240	+	0.648020i	−0.942013	−	0.335576i	$-0.891069\pi$
$179$	−35962.9	+	20763.2i	−1.12240	+	0.648020i	−0.180390	+	0.983595i	$0.557736\pi$
$180$		0			0
$181$	896.024	−	1551.96i	0.0273503	−	0.0473721i	−0.852026	−	0.523499i	$-0.824626\pi$
$181$	896.024	−	1551.96i	0.0273503	−	0.0473721i	0.879377	+	0.476127i	$0.157960\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.4088	+	31.3454i	−0.000333347	+	0.000915863i
$186$		0			0
$187$	−14671.8	+	83208.0i	−0.419566	+	2.37948i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−55200.3	−	9733.31i	−1.51313	−	0.266805i	−0.645399	−	0.763846i	$-0.723309\pi$
$191$	−55200.3	−	9733.31i	−1.51313	−	0.266805i	−0.867727	+	0.497041i	$0.834420\pi$
$192$		0			0
$193$	−39783.2	−	14479.9i	−1.06803	−	0.388733i	−0.252593	−	0.967573i	$-0.581283\pi$
$193$	−39783.2	−	14479.9i	−1.06803	−	0.388733i	−0.815441	+	0.578840i	$0.803506\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−51989.7	−	30016.3i	−1.33963	−	0.773436i	−0.352878	−	0.935669i	$-0.614797\pi$
$197$	−51989.7	−	30016.3i	−1.33963	−	0.773436i	−0.986752	+	0.162233i	$0.948130\pi$
$198$		0			0
$199$	29053.9	+	50322.8i	0.733665	+	1.27074i	0.955307	+	0.295616i	$0.0955250\pi$
$199$	29053.9	+	50322.8i	0.733665	+	1.27074i	−0.221642	+	0.975128i	$0.571142\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−25493.6	+	4495.21i	−0.618641	+	0.109083i
$204$		0			0
$205$	−56.6741	−	47.5552i	−0.00134858	−	0.00113159i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	39072.5	+	107351.i	0.894496	+	2.45761i
$210$		0			0
$211$	−7858.30	+	6593.90i	−0.176508	+	0.148108i	−0.726761	−	0.686890i	$-0.758975\pi$
$211$	−7858.30	+	6593.90i	−0.176508	+	0.148108i	0.550254	+	0.834998i	$0.314531\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	54.8841i		−	0.00118732i
$216$		0			0
$217$	100703.			2.13857
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13479.4	+	16064.2i	0.275986	+	0.328908i
$222$		0			0
$223$	−74762.5	+	27211.3i	−1.50340	+	0.547192i	−0.956938	−	0.290293i	$-0.906247\pi$
$223$	−74762.5	+	27211.3i	−1.50340	+	0.547192i	−0.546460	+	0.837485i	$0.684025\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14091.0	+	16793.0i	−0.273458	+	0.325895i	−0.885242	−	0.465130i	$-0.846008\pi$
$227$	−14091.0	+	16793.0i	−0.273458	+	0.325895i	0.611784	+	0.791025i	$0.290452\pi$
$228$		0			0
$229$	−4457.80	−	25281.5i	−0.0850061	−	0.482093i	−0.997355	−	0.0726804i	$-0.976845\pi$
$229$	−4457.80	−	25281.5i	−0.0850061	−	0.482093i	0.912349	−	0.409413i	$-0.134266\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	42121.6	−	24318.9i	0.775877	−	0.447953i	−0.0590903	−	0.998253i	$-0.518820\pi$
$233$	42121.6	−	24318.9i	0.775877	−	0.447953i	0.834967	+	0.550300i	$0.185487\pi$
$234$		0			0
$235$	−39.5630	+	68.5252i	−0.000716397	+	0.00124084i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	34021.3	−	93472.8i	0.595601	−	1.63640i	−0.164337	−	0.986404i	$-0.552548\pi$
$239$	34021.3	−	93472.8i	0.595601	−	1.63640i	0.759938	−	0.649996i	$-0.225229\pi$
$240$		0			0
$241$	8476.64	−	48073.4i	0.145945	−	0.827696i	−0.820658	−	0.571419i	$-0.806393\pi$
$241$	8476.64	−	48073.4i	0.145945	−	0.827696i	0.966603	−	0.256277i	$-0.0824958\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−53.3202	−	9.40178i	−0.000888299	−	0.000156631i
$246$		0			0
$247$	26643.8	+	9697.56i	0.436719	+	0.158953i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−35448.9	−	20466.5i	−0.562673	−	0.324859i	0.191545	−	0.981484i	$-0.438650\pi$
$251$	−35448.9	−	20466.5i	−0.562673	−	0.324859i	−0.754217	+	0.656625i	$0.771984\pi$
$252$		0			0
$253$	−44178.0	−	76518.5i	−0.690184	−	1.19543i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	432.210	−	76.2102i	0.00654377	−	0.00115384i	−0.170375	−	0.985379i	$-0.554498\pi$
$257$	432.210	−	76.2102i	0.00654377	−	0.00115384i	0.176919	+	0.984225i	$0.443387\pi$
$258$		0			0
$259$	52461.0	+	44020.0i	0.782054	+	0.656221i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	26508.2	+	72830.8i	0.383239	+	1.05294i	0.969984	+	0.243168i	$0.0781868\pi$
$263$	26508.2	+	72830.8i	0.383239	+	1.05294i	−0.586745	+	0.809772i	$0.699591\pi$
$264$		0			0
$265$	−57.8117	+	48.5098i	−0.000823236	+	0.000690777i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			46665.4i			0.644897i	0.946587	+	0.322449i	$0.104506\pi$
$269$			46665.4i			0.644897i	−0.946587	+	0.322449i	$0.895494\pi$
$270$		0			0
$271$	−64614.8			−0.879819			−0.439910	−	0.898042i	$-0.644989\pi$
$271$	−64614.8			−0.879819			−0.439910	+	0.898042i	$0.644989\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	76584.5	+	91269.8i	1.01269	+	1.20687i
$276$		0			0
$277$	−23345.1	+	8496.92i	−0.304254	+	0.110739i	−0.489635	−	0.871927i	$-0.662870\pi$
$277$	−23345.1	+	8496.92i	−0.304254	+	0.110739i	0.185382	+	0.982667i	$0.440648\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10338.9	−	12321.4i	0.130937	−	0.156044i	−0.696592	−	0.717467i	$-0.745301\pi$
$281$	10338.9	−	12321.4i	0.130937	−	0.156044i	0.827529	+	0.561423i	$0.189746\pi$
$282$		0			0
$283$	−23776.7	−	134845.i	−0.296879	−	1.68368i	−0.659467	−	0.751734i	$-0.729218\pi$
$283$	−23776.7	−	134845.i	−0.296879	−	1.68368i	0.362588	−	0.931950i	$-0.381893\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−131539.	+	75944.3i	−1.59695	+	0.922001i
$288$		0			0
$289$	56461.5	−	97794.3i	0.676016	−	1.17089i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	20530.8	−	56407.8i	0.239150	−	0.657058i	−0.760818	−	0.648966i	$-0.775202\pi$
$293$	20530.8	−	56407.8i	0.239150	−	0.657058i	0.999967	−	0.00809254i	$-0.00257596\pi$
$294$		0			0
$295$	−1.40285	+	7.95594i	−1.61200e−5	+	9.14213e-5i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−21596.3	−	3808.00i	−0.241566	−	0.0425946i
$300$		0			0
$301$	−105883.	−	38538.3i	−1.16867	−	0.425362i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	106.448	+	61.4580i	0.00114430	+	0.000660661i
$306$		0			0
$307$	57552.3	+	99683.5i	0.610641	+	1.05766i	0.991133	+	0.132877i	$0.0424215\pi$
$307$	57552.3	+	99683.5i	0.610641	+	1.05766i	−0.380492	+	0.924784i	$0.624245\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	34145.6	−	6020.79i	0.353032	−	0.0622491i	0.00567992	−	0.999984i	$-0.498192\pi$
$311$	34145.6	−	6020.79i	0.353032	−	0.0622491i	0.347352	+	0.937735i	$0.387081\pi$
$312$		0			0
$313$	89576.8	+	75163.9i	0.914339	+	0.767221i	0.972939	−	0.231060i	$-0.0742195\pi$
$313$	89576.8	+	75163.9i	0.914339	+	0.767221i	−0.0586009	+	0.998281i	$0.518664\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−53640.5	−	147376.i	−0.533795	−	1.46659i	−0.854521	−	0.519418i	$-0.826149\pi$
$317$	−53640.5	−	147376.i	−0.533795	−	1.46659i	0.320726	−	0.947172i	$-0.396073\pi$
$318$		0			0
$319$	−58819.8	+	49355.7i	−0.578019	+	0.485016i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	265610.i		−	2.54589i
$324$		0			0
$325$	29570.9			0.279961
$326$		0			0
$327$		0			0
$328$		0			0
$329$	104419.	+	124442.i	0.964694	+	1.14968i
$330$		0			0
$331$	−2040.65	+	742.736i	−0.0186257	+	0.00677920i	−0.351316	−	0.936257i	$-0.614266\pi$
$331$	−2040.65	+	742.736i	−0.0186257	+	0.00677920i	0.332690	+	0.943036i	$0.392044\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−81.4614	+	97.0819i	−0.000725876	+	0.000865065i
$336$		0			0
$337$	−10362.9	−	58770.8i	−0.0912474	−	0.517490i	−0.995833	−	0.0911974i	$-0.970931\pi$
$337$	−10362.9	−	58770.8i	−0.0912474	−	0.517490i	0.904585	−	0.426292i	$-0.140181\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	258681.	−	149349.i	2.22462	−	1.28438i
$342$		0			0
$343$	21577.1	−	37372.6i	0.183402	−	0.317662i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	31580.8	−	86767.6i	0.262280	−	0.720607i	−0.736733	−	0.676184i	$-0.763633\pi$
$347$	31580.8	−	86767.6i	0.262280	−	0.720607i	0.999013	−	0.0444237i	$-0.0141452\pi$
$348$		0			0
$349$	−4641.37	+	26322.5i	−0.0381062	+	0.216111i	−0.997915	−	0.0645444i	$-0.979441\pi$
$349$	−4641.37	+	26322.5i	−0.0381062	+	0.216111i	0.959809	+	0.280655i	$0.0905517\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−51899.9	−	9151.35i	−0.416502	−	0.0734405i	−0.0385297	−	0.999257i	$-0.512267\pi$
$353$	−51899.9	−	9151.35i	−0.416502	−	0.0734405i	−0.377972	+	0.925817i	$0.623379\pi$
$354$		0			0
$355$	−72.8389	−	26.5112i	−0.000577972	−	0.000210364i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31426.2	+	18143.9i	0.243839	+	0.140781i	0.616940	−	0.787010i	$-0.288372\pi$
$359$	31426.2	+	18143.9i	0.243839	+	0.140781i	−0.373101	+	0.927791i	$0.621705\pi$
$360$		0			0
$361$	−114404.	−	198154.i	−0.877865	−	1.52051i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−46.9406	+	8.27690i	−0.000352341	+	6.21272e-5i
$366$		0			0
$367$	−160886.	−	135000.i	−1.19450	−	1.00231i	−0.999770	−	0.0214507i	$-0.993172\pi$
$367$	−160886.	−	135000.i	−1.19450	−	1.00231i	−0.194733	−	0.980856i	$-0.562384\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	52991.7	+	145594.i	0.385000	+	1.05778i
$372$		0			0
$373$	10634.3	−	8923.25i	0.0764350	−	0.0641366i	−0.603770	−	0.797159i	$-0.706335\pi$
$373$	10634.3	−	8923.25i	0.0764350	−	0.0641366i	0.680205	+	0.733022i	$0.261891\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			19057.3i			0.134084i
$378$		0			0
$379$	123341.			0.858677			0.429338	−	0.903144i	$-0.358747\pi$
$379$	123341.			0.858677			0.429338	+	0.903144i	$0.358747\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−158981.	−	189467.i	−1.08380	−	1.29162i	−0.953910	−	0.300093i	$-0.902982\pi$
$383$	−158981.	−	189467.i	−1.08380	−	1.29162i	−0.129889	−	0.991529i	$-0.541462\pi$
$384$		0			0
$385$	−360.406	+	131.177i	−0.00243148	+	0.000884987i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	45832.1	−	54620.6i	0.302880	−	0.360958i	−0.593041	−	0.805173i	$-0.702073\pi$
$389$	45832.1	−	54620.6i	0.302880	−	0.360958i	0.895921	+	0.444214i	$0.146517\pi$
$390$		0			0
$391$	35672.4	+	202308.i	0.233334	+	1.32330i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−210.934	+	121.783i	−0.00135192	+	0.000780533i
$396$		0			0
$397$	35045.3	−	60700.3i	0.222356	−	0.385132i	−0.733167	−	0.680049i	$-0.761959\pi$
$397$	35045.3	−	60700.3i	0.222356	−	0.385132i	0.955523	+	0.294917i	$0.0952919\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	51680.5	−	141991.i	0.321394	−	0.883023i	−0.668815	−	0.743429i	$-0.733198\pi$
$401$	51680.5	−	141991.i	0.321394	−	0.883023i	0.990209	−	0.139594i	$-0.0445797\pi$
$402$		0			0
$403$	12873.4	−	73009.0i	0.0792656	−	0.449538i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	200043.	+	35273.0i	1.20763	+	0.212938i
$408$		0			0
$409$	−35042.7	−	12754.5i	−0.209484	−	0.0762460i	0.235147	−	0.971960i	$-0.424443\pi$
$409$	−35042.7	−	12754.5i	−0.209484	−	0.0762460i	−0.444631	+	0.895714i	$0.646665\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14363.7	+	8292.86i	0.0842102	+	0.0486188i
$414$		0			0
$415$	−194.496	−	336.877i	−0.00112931	−	0.00195603i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−242548.	+	42767.8i	−1.38156	+	0.243607i	−0.814545	−	0.580100i	$-0.803014\pi$
$419$	−242548.	+	42767.8i	−1.38156	+	0.243607i	−0.567017	+	0.823706i	$0.691902\pi$
$420$		0			0
$421$	−13777.8	−	11560.9i	−0.0777348	−	0.0652272i	0.603092	−	0.797671i	$-0.293935\pi$
$421$	−13777.8	−	11560.9i	−0.0777348	−	0.0652272i	−0.680827	+	0.732444i	$0.738379\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−94743.7	−	260306.i	−0.524533	−	1.44114i
$426$		0			0
$427$	193311.	−	162207.i	1.06023	−	0.889640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			314703.i			1.69413i	0.531490	+	0.847064i	$0.321632\pi$
$431$			314703.i			1.69413i	−0.531490	+	0.847064i	$0.678368\pi$
$432$		0			0
$433$	−208556.			−1.11236			−0.556182	−	0.831060i	$-0.687734\pi$
$433$	−208556.			−1.11236			−0.556182	+	0.831060i	$0.687734\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	178540.	+	212776.i	0.934916	+	1.11419i
$438$		0			0
$439$	175311.	−	63807.9i	0.909661	−	0.331090i	0.155544	−	0.987829i	$-0.450287\pi$
$439$	175311.	−	63807.9i	0.909661	−	0.331090i	0.754118	+	0.656739i	$0.228065\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−44298.3	+	52792.7i	−0.225725	+	0.269009i	−0.867006	−	0.498297i	$-0.833959\pi$
$443$	−44298.3	+	52792.7i	−0.225725	+	0.269009i	0.641281	+	0.767306i	$0.278403\pi$
$444$		0			0
$445$	25.8236	+	146.453i	0.000130406	+	0.000739567i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−174409.	+	100695.i	−0.865117	+	0.499476i	−0.865723	−	0.500524i	$-0.833141\pi$
$449$	−174409.	+	100695.i	−0.865117	+	0.499476i	0.000605245		1.00000i	$0.499807\pi$
$450$		0			0
$451$	−225261.	+	390163.i	−1.10747	+	1.91820i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−32.5574	+	89.4506i	−0.000157263	+	0.000432077i
$456$		0			0
$457$	12388.7	−	70259.5i	0.0593187	−	0.336413i	−0.940677	−	0.339303i	$-0.889809\pi$
$457$	12388.7	−	70259.5i	0.0593187	−	0.336413i	0.999996	+	0.00288993i	$0.000919895\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	284665.	+	50194.1i	1.33947	+	0.236184i	0.797044	−	0.603921i	$-0.206396\pi$
$461$	284665.	+	50194.1i	1.33947	+	0.236184i	0.542423	+	0.840105i	$0.317507\pi$
$462$		0			0
$463$	68653.9	+	24988.0i	0.320260	+	0.116565i	0.497148	−	0.867666i	$-0.334381\pi$
$463$	68653.9	+	24988.0i	0.320260	+	0.116565i	−0.176888	+	0.984231i	$0.556603\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	79976.9	+	46174.7i	0.366717	+	0.211724i	0.672023	−	0.740530i	$-0.265426\pi$
$467$	79976.9	+	46174.7i	0.366717	+	0.211724i	−0.305306	+	0.952254i	$0.598759\pi$
$468$		0			0
$469$	130091.	+	225325.i	0.591429	+	1.02439i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−329141.	+	58036.5i	−1.47116	+	0.259405i
$474$		0			0
$475$	−286919.	−	240753.i	−1.27166	−	1.06705i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−94771.9	−	260384.i	−0.413056	−	1.13486i	−0.955557	−	0.294806i	$-0.904745\pi$
$479$	−94771.9	−	260384.i	−0.413056	−	1.13486i	0.542501	−	0.840055i	$-0.317477\pi$
$480$		0			0
$481$	38620.5	−	32406.4i	0.166927	−	0.140069i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			180.778i			0.000768533i
$486$		0			0
$487$	−231576.			−0.976416			−0.488208	−	0.872727i	$-0.662349\pi$
$487$	−231576.			−0.976416			−0.488208	+	0.872727i	$0.662349\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	271099.	+	323083.i	1.12451	+	1.34014i	0.933509	+	0.358553i	$0.116730\pi$
$491$	271099.	+	323083.i	1.12451	+	1.34014i	0.191003	+	0.981589i	$0.438826\pi$
$492$		0			0
$493$	167757.	−	61058.6i	0.690220	−	0.251219i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−102291.	+	121906.i	−0.414120	+	0.493529i
$498$		0			0
$499$	−12436.4	−	70530.2i	−0.0499451	−	0.283253i	0.949598	−	0.313470i	$-0.101491\pi$
$499$	−12436.4	−	70530.2i	−0.0499451	−	0.283253i	−0.999543	+	0.0302167i	$0.990380\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−259012.	+	149541.i	−1.02373	+	0.591048i	−0.915181	−	0.403043i	$-0.867952\pi$
$503$	−259012.	+	149541.i	−1.02373	+	0.591048i	−0.108545	+	0.994092i	$0.534619\pi$
$504$		0			0
$505$	−8.76382	+	15.1794i	−3.43645e−5	+	5.95211e-5i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	108249.	−	297413.i	0.417820	−	1.14795i	−0.535115	−	0.844779i	$-0.679732\pi$
$509$	108249.	−	297413.i	0.417820	−	1.14795i	0.952936	−	0.303173i	$-0.0980460\pi$
$510$		0			0
$511$	−16992.7	+	96370.3i	−0.0650759	+	0.369064i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	203.560	+	35.8932i	0.000767500	+	0.000135331i
$516$		0			0
$517$	452783.	+	164800.i	1.69398	+	0.616559i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	175042.	+	101061.i	0.644862	+	0.372311i	0.786485	−	0.617609i	$-0.211899\pi$
$521$	175042.	+	101061.i	0.644862	+	0.372311i	−0.141623	+	0.989921i	$0.545232\pi$
$522$		0			0
$523$	−53298.1	−	92314.9i	−0.194853	−	0.337496i	0.751999	−	0.659164i	$-0.229090\pi$
$523$	−53298.1	−	92314.9i	−0.194853	−	0.337496i	−0.946852	+	0.321668i	$0.895756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−683928.	+	120595.i	−2.46257	+	0.434218i
$528$		0			0
$529$	49805.3	+	41791.6i	0.177977	+	0.149341i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	38243.5	+	105073.i	0.134618	+	0.369860i
$534$		0			0
$535$	−107.131	+	89.8936i	−0.000374290	+	0.000314066i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			329704.i			1.13487i
$540$		0			0
$541$	−53821.8			−0.183892			−0.0919462	−	0.995764i	$-0.529309\pi$
$541$	−53821.8			−0.183892			−0.0919462	+	0.995764i	$0.529309\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−196.849	−	234.596i	−0.000662736	−	0.000789818i
$546$		0			0
$547$	−35772.0	+	13019.9i	−0.119555	+	0.0435145i	−0.401105	−	0.916032i	$-0.631374\pi$
$547$	−35772.0	+	13019.9i	−0.119555	+	0.0435145i	0.281550	+	0.959547i	$0.409152\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	155156.	−	184908.i	0.511053	−	0.609049i
$552$		0			0
$553$	86832.0	+	492449.i	0.283942	+	1.61032i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	369490.	−	213325.i	1.19095	−	0.687594i	0.232426	−	0.972614i	$-0.425334\pi$
$557$	369490.	−	213325.i	1.19095	−	0.687594i	0.958522	+	0.285020i	$0.0920002\pi$
$558$		0			0
$559$	−41475.5	+	71837.7i	−0.132730	+	0.229895i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	77677.9	−	213418.i	0.245065	−	0.673310i	−0.754785	−	0.655972i	$-0.772259\pi$
$563$	77677.9	−	213418.i	0.245065	−	0.673310i	0.999850	−	0.0173374i	$-0.00551896\pi$
$564$		0			0
$565$	68.8448	−	390.438i	0.000215662	−	0.00122308i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−508866.	−	89726.9i	−1.57173	−	0.277139i	−0.681215	−	0.732084i	$-0.738548\pi$
$569$	−508866.	−	89726.9i	−1.57173	−	0.277139i	−0.890520	+	0.454945i	$0.849659\pi$
$570$		0			0
$571$	470667.	+	171309.i	1.44358	+	0.525421i	0.940791	−	0.338987i	$-0.110084\pi$
$571$	470667.	+	171309.i	1.44358	+	0.525421i	0.502791	+	0.864408i	$0.332306\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	250872.	+	144841.i	0.758781	+	0.438082i
$576$		0			0
$577$	139405.	+	241457.i	0.418723	+	0.725250i	0.995811	−	0.0914318i	$-0.0291444\pi$
$577$	139405.	+	241457.i	0.418723	+	0.725250i	−0.577088	+	0.816682i	$0.695811\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−786479.	+	138677.i	−2.32989	+	0.410822i
$582$		0			0
$583$	352047.	+	295403.i	1.03577	+	0.869115i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−107700.	−	295904.i	−0.312565	−	0.858764i	−0.992137	−	0.125156i	$-0.960057\pi$
$587$	−107700.	−	295904.i	−0.312565	−	0.858764i	0.679572	−	0.733608i	$-0.262165\pi$
$588$		0			0
$589$	−719315.	+	603577.i	−2.07343	+	1.73981i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	36185.2i		−	0.102902i	−0.998676	−	0.0514508i	$-0.983615\pi$
$593$		−	36185.2i		−	0.102902i	0.998676	−	0.0514508i	$-0.0163845\pi$
$594$		0			0
$595$	891.727			0.00251883
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−65094.9	−	77577.1i	−0.181424	−	0.216212i	0.667666	−	0.744461i	$-0.267293\pi$
$599$	−65094.9	−	77577.1i	−0.181424	−	0.216212i	−0.849090	+	0.528249i	$0.822849\pi$
$600$		0			0
$601$	−267323.	+	97297.5i	−0.740094	+	0.269372i	−0.684431	−	0.729077i	$-0.739949\pi$
$601$	−267323.	+	97297.5i	−0.740094	+	0.269372i	−0.0556627	+	0.998450i	$0.517727\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−436.638	+	520.364i	−0.00119292	+	0.00142166i
$606$		0			0
$607$	71270.1	+	404193.i	0.193433	+	1.09701i	0.914633	+	0.404285i	$0.132480\pi$
$607$	71270.1	+	404193.i	0.193433	+	1.09701i	−0.721201	+	0.692726i	$0.756409\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	103568.	−	59795.1i	0.277424	−	0.160171i
$612$		0			0
$613$	−163074.	+	282453.i	−0.433974	+	0.751666i	−0.997211	−	0.0746297i	$-0.976223\pi$
$613$	−163074.	+	282453.i	−0.433974	+	0.751666i	0.563237	+	0.826295i	$0.309556\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−249596.	+	685760.i	−0.655643	+	1.80137i	−0.0598795	+	0.998206i	$0.519072\pi$
$617$	−249596.	+	685760.i	−0.655643	+	1.80137i	−0.595764	+	0.803160i	$0.703151\pi$
$618$		0			0
$619$	119433.	−	677338.i	0.311705	−	1.76776i	−0.278427	−	0.960457i	$-0.589813\pi$
$619$	119433.	−	677338.i	0.311705	−	1.76776i	0.590131	−	0.807307i	$-0.299076\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	300671.	+	53016.4i	0.774668	+	0.136595i
$624$		0			0
$625$	−367066.	−	133601.i	−0.939688	−	0.342019i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−409005.	−	236139.i	−1.03378	−	0.596852i
$630$		0			0
$631$	−43866.1	−	75978.2i	−0.110172	−	0.190823i	0.805668	−	0.592368i	$-0.201807\pi$
$631$	−43866.1	−	75978.2i	−0.110172	−	0.190823i	−0.915839	+	0.401545i	$0.868473\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−686.336	+	121.020i	−0.00170212	+	0.000300129i
$636$		0			0
$637$	62685.9	+	52599.7i	0.154487	+	0.129630i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	95751.6	+	263075.i	0.233040	+	0.640272i	0.999999	−	0.00142423i	$-0.000453347\pi$
$641$	95751.6	+	263075.i	0.233040	+	0.640272i	−0.766959	+	0.641696i	$0.778231\pi$
$642$		0			0
$643$	395316.	−	331709.i	0.956142	−	0.802298i	−0.0241793	−	0.999708i	$-0.507697\pi$
$643$	395316.	−	331709.i	0.956142	−	0.802298i	0.980321	+	0.197409i	$0.0632528\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			230598.i			0.550868i	0.961320	+	0.275434i	$0.0888215\pi$
$647$			230598.i			0.550868i	−0.961320	+	0.275434i	$0.911178\pi$
$648$		0			0
$649$	49195.4			0.116798
$650$		0			0
$651$		0			0
$652$		0			0
$653$	344862.	+	410991.i	0.808760	+	0.963842i	0.999843	−	0.0177317i	$-0.00564448\pi$
$653$	344862.	+	410991.i	0.808760	+	0.963842i	−0.191083	+	0.981574i	$0.561200\pi$
$654$		0			0
$655$	200.098	−	72.8296i	0.000466401	−	0.000169756i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	432211.	−	515089.i	0.995234	−	1.18607i	0.0127133	−	0.999919i	$-0.495953\pi$
$659$	432211.	−	515089.i	0.995234	−	1.18607i	0.982521	−	0.186154i	$-0.0596024\pi$
$660$		0			0
$661$	−90978.6	−	515965.i	−0.208227	−	1.18091i	−0.892281	−	0.451481i	$-0.850896\pi$
$661$	−90978.6	−	515965.i	−0.208227	−	1.18091i	0.684054	−	0.729431i	$-0.260215\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1044.16	−	602.849i	0.00236116	−	0.00136322i
$666$		0			0
$667$	−93344.4	+	161677.i	−0.209815	+	0.363410i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	256003.	−	703362.i	0.568590	−	1.56219i
$672$		0			0
$673$	2915.44	−	16534.3i	0.00643685	−	0.0365052i	−0.981420	−	0.191871i	$-0.938544\pi$
$673$	2915.44	−	16534.3i	0.00643685	−	0.0365052i	0.987857	+	0.155366i	$0.0496556\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−886585.	−	156329.i	−1.93438	−	0.341084i	−0.934495	−	0.355975i	$-0.884149\pi$
$677$	−886585.	−	156329.i	−1.93438	−	0.341084i	−0.999889	+	0.0148909i	$0.995260\pi$
$678$		0			0
$679$	348759.	+	126938.i	0.756461	+	0.275329i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	515809.	+	297802.i	1.10573	+	0.638391i	0.937719	−	0.347395i	$-0.112934\pi$
$683$	515809.	+	297802.i	1.10573	+	0.638391i	0.168007	+	0.985786i	$0.446267\pi$
$684$		0			0
$685$	100.278	+	173.686i	0.000213710	+	0.000370156i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	112328.	−	19806.5i	0.236620	−	0.0417224i
$690$		0			0
$691$	−285792.	−	239808.i	−0.598541	−	0.502236i	0.292435	−	0.956285i	$-0.405534\pi$
$691$	−285792.	−	239808.i	−0.598541	−	0.502236i	−0.890976	+	0.454050i	$0.849979\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	250.346	+	687.820i	0.000518288	+	0.00142398i
$696$		0			0
$697$	802407.	−	673300.i	1.65169	−	1.38594i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			182277.i			0.370934i	0.982650	+	0.185467i	$0.0593798\pi$
$701$			182277.i			0.370934i	−0.982650	+	0.185467i	$0.940620\pi$
$702$		0			0
$703$	−638564.			−1.29209
$704$		0			0
$705$		0			0
$706$		0			0
$707$	23130.5	+	27565.9i	0.0462750	+	0.0551484i
$708$		0			0
$709$	219638.	−	79941.7i	0.436933	−	0.159031i	−0.114182	−	0.993460i	$-0.536425\pi$
$709$	219638.	−	79941.7i	0.436933	−	0.159031i	0.551115	+	0.834429i	$0.314202\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	466820.	−	556334.i	0.918270	−	1.09435i
$714$		0			0
$715$	49.0296	+	278.061i	9.59061e−5	+	0.000543910i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−355326.	+	205148.i	−0.687336	+	0.396834i	−0.802613	−	0.596500i	$-0.796558\pi$
$719$	−355326.	+	205148.i	−0.687336	+	0.396834i	0.115277	+	0.993333i	$0.463224\pi$
$720$		0			0
$721$	212181.	−	367508.i	0.408164	−	0.706962i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	86100.8	−	236560.i	0.163806	−	0.450054i
$726$		0			0
$727$	167973.	−	952624.i	0.317813	−	1.80241i	−0.238183	−	0.971220i	$-0.576552\pi$
$727$	167973.	−	952624.i	0.317813	−	1.80241i	0.555996	−	0.831185i	$-0.312337\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	765258.	+	134936.i	1.43210	+	0.252518i
$732$		0			0
$733$	−165250.	−	60146.2i	−0.307563	−	0.111944i	0.183628	−	0.982996i	$-0.441216\pi$
$733$	−165250.	−	60146.2i	−0.307563	−	0.111944i	−0.491191	+	0.871052i	$0.663438\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	668343.	+	385868.i	1.23045	+	0.710402i
$738$		0			0
$739$	−366610.	−	634988.i	−0.671299	−	1.16272i	−0.977536	−	0.210769i	$-0.932403\pi$
$739$	−366610.	−	634988.i	−0.671299	−	1.16272i	0.306237	−	0.951955i	$-0.400930\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−730405.	+	128790.i	−1.32308	+	0.233295i	−0.790175	−	0.612881i	$-0.790011\pi$
$743$	−730405.	+	128790.i	−1.32308	+	0.233295i	−0.532904	+	0.846176i	$0.678899\pi$
$744$		0			0
$745$	−820.901	−	688.818i	−0.00147903	−	0.00124106i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	98199.0	+	269800.i	0.175043	+	0.480925i
$750$		0			0
$751$	−72745.4	+	61040.7i	−0.128981	+	0.108228i	−0.704996	−	0.709211i	$-0.749051\pi$
$751$	−72745.4	+	61040.7i	−0.128981	+	0.108228i	0.576015	+	0.817439i	$0.304607\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	899.523i		−	0.00157804i
$756$		0			0
$757$	−499006.			−0.870791			−0.435396	−	0.900239i	$-0.643391\pi$
$757$	−499006.			−0.870791			−0.435396	+	0.900239i	$0.643391\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	314122.	+	374356.i	0.542412	+	0.646421i	0.965727	−	0.259561i	$-0.0835780\pi$
$761$	314122.	+	374356.i	0.542412	+	0.646421i	−0.423315	+	0.905983i	$0.639134\pi$
$762$		0			0
$763$	−590807.	+	215036.i	−1.01484	+	0.369371i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7848.44	−	9353.41i	0.0133411	−	0.0158993i
$768$		0			0
$769$	96572.4	+	547689.i	0.163305	+	0.926150i	0.950795	+	0.309822i	$0.100269\pi$
$769$	96572.4	+	547689.i	0.163305	+	0.926150i	−0.787489	+	0.616328i	$0.788619\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−297119.	+	171542.i	−0.497247	+	0.287086i	−0.727576	−	0.686027i	$-0.759353\pi$
$773$	−297119.	+	171542.i	−0.497247	+	0.287086i	0.230329	+	0.973113i	$0.426020\pi$
$774$		0			0
$775$	−489654.	+	848105.i	−0.815240	+	1.41204i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	484394.	−	1.33086e6i	0.798222	−	2.19310i
$780$		0			0
$781$	−81965.7	+	464851.i	−0.134379	+	0.762099i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	236.687	+	41.7344i	0.000384092	+	6.77258e-5i
$786$		0			0
$787$	−191036.	−	69531.5i	−0.308437	−	0.112262i	0.183164	−	0.983082i	$-0.441366\pi$
$787$	−191036.	−	69531.5i	−0.308437	−	0.112262i	−0.491601	+	0.870820i	$0.663588\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−704898.	−	406973.i	−1.12661	−	0.650448i
$792$		0			0
$793$	−92886.8	−	160885.i	−0.147709	−	0.255840i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−750642.	+	132358.i	−1.18172	+	0.208370i	−0.729783	−	0.683678i	$-0.760379\pi$
$797$	−750642.	+	132358.i	−1.18172	+	0.208370i	−0.451940	+	0.892048i	$0.649268\pi$
$798$		0			0
$799$	−858191.	−	720108.i	−1.34428	−	1.12799i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	99273.6	+	272752.i	0.153958	+	0.422997i
$804$		0			0
$805$	−714.346	+	599.408i	−0.00110234	+	0.000924976i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	524519.i		−	0.801428i	−0.916203	−	0.400714i	$-0.868762\pi$
$809$		−	524519.i		−	0.801428i	0.916203	−	0.400714i	$-0.131238\pi$
$810$		0			0
$811$	1.30005e6			1.97660			0.988298	−	0.152532i	$-0.0487428\pi$
$811$	1.30005e6			1.97660			0.988298	+	0.152532i	$0.0487428\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−185.402	−	220.953i	−0.000279125	−	0.000332648i
$816$		0			0
$817$	987298.	−	359347.i	1.47912	−	0.538357i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−38691.2	+	46110.3i	−0.0574018	+	0.0684088i	−0.793981	−	0.607942i	$-0.791995\pi$
$821$	−38691.2	+	46110.3i	−0.0574018	+	0.0684088i	0.736580	+	0.676351i	$0.236440\pi$
$822$		0			0
$823$	−144022.	−	816790.i	−0.212632	−	1.20590i	−0.884968	−	0.465652i	$-0.845820\pi$
$823$	−144022.	−	816790.i	−0.212632	−	1.20590i	0.672336	−	0.740246i	$-0.265291\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.00315e6	−	579170.i	1.46675	−	0.846827i	0.467440	−	0.884025i	$-0.345176\pi$
$827$	1.00315e6	−	579170.i	1.46675	−	0.846827i	0.999308	+	0.0371972i	$0.0118430\pi$
$828$		0			0
$829$	565538.	−	979540.i	0.822910	−	1.42532i	−0.0805960	−	0.996747i	$-0.525682\pi$
$829$	565538.	−	979540.i	0.822910	−	1.42532i	0.903506	−	0.428575i	$-0.140984\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	262181.	−	720338.i	0.377844	−	1.03812i
$834$		0			0
$835$	78.9891	−	447.969i	0.000113291	−	0.000642503i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	736507.	+	129866.i	1.04629	+	0.184490i	0.670267	−	0.742120i	$-0.266180\pi$
$839$	736507.	+	129866.i	1.04629	+	0.184490i	0.376025	+	0.926609i	$0.377291\pi$
$840$		0			0
$841$	−512173.	−	186416.i	−0.724144	−	0.263567i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−713.618	−	412.008i	−0.000999430	−	0.000577021i
$846$		0			0
$847$	697297.	+	1.20775e6i	0.971966	+	1.68349i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	486376.	−	85761.3i	0.671604	−	0.118422i
$852$		0			0
$853$	20927.0	+	17559.8i	0.0287613	+	0.0241336i	0.657055	−	0.753843i	$-0.271802\pi$
$853$	20927.0	+	17559.8i	0.0287613	+	0.0241336i	−0.628294	+	0.777976i	$0.716246\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−274182.	−	753309.i	−0.373316	−	1.02568i	−0.974071	−	0.226245i	$-0.927355\pi$
$857$	−274182.	−	753309.i	−0.373316	−	1.02568i	0.600754	−	0.799434i	$-0.294867\pi$
$858$		0			0
$859$	68325.2	−	57331.6i	0.0925965	−	0.0776977i	−0.595314	−	0.803493i	$-0.702972\pi$
$859$	68325.2	−	57331.6i	0.0925965	−	0.0776977i	0.687911	+	0.725795i	$0.258528\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	822294.i		−	1.10409i	−0.833813	−	0.552047i	$-0.813847\pi$
$863$		−	822294.i		−	1.10409i	0.833813	−	0.552047i	$-0.186153\pi$
$864$		0			0
$865$	705.236			0.000942546
$866$		0			0
$867$		0			0
$868$		0			0
$869$	953383.	+	1.13620e6i	1.26249	+	1.50458i
$870$		0			0
$871$	179989.	−	65510.6i	0.237252	−	0.0863526i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1616.55	−	1926.53i	0.00211142	−	0.00251629i
$876$		0			0
$877$	−1480.55	−	8396.64i	−0.00192498	−	0.0109171i	0.983830	−	0.179104i	$-0.0573199\pi$
$877$	−1480.55	−	8396.64i	−0.00192498	−	0.0109171i	−0.985755	+	0.168187i	$0.946209\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	442659.	−	255570.i	0.570319	−	0.329274i	−0.186958	−	0.982368i	$-0.559863\pi$
$881$	442659.	−	255570.i	0.570319	−	0.329274i	0.757277	+	0.653094i	$0.226529\pi$
$882$		0			0
$883$	590245.	−	1.02233e6i	0.757026	−	1.31121i	−0.187335	−	0.982296i	$-0.559985\pi$
$883$	590245.	−	1.02233e6i	0.757026	−	1.31121i	0.944361	−	0.328912i	$-0.106682\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16612.2	+	45641.5i	−0.0211144	+	0.0580113i	−0.949802	−	0.312852i	$-0.898716\pi$
$887$	−16612.2	+	45641.5i	−0.0211144	+	0.0580113i	0.928688	+	0.370863i	$0.120938\pi$
$888$		0			0
$889$	−248456.	+	1.40906e6i	−0.314374	+	1.78290i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.49172e6	−	263031.i	−1.87062	−	0.329840i
$894$		0			0
$895$	1221.57	+	444.616i	0.00152501	+	0.000555059i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−546571.	−	315563.i	−0.676281	−	0.390451i
$900$		0			0
$901$	−534247.	−	925343.i	−0.658101	−	1.13986i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−55.2472	+	9.74157i	−6.74548e−5	+	1.18941e-5i
$906$		0			0
$907$	171541.	+	143940.i	0.208523	+	0.174972i	0.741068	−	0.671430i	$-0.234320\pi$
$907$	171541.	+	143940.i	0.208523	+	0.174972i	−0.532545	+	0.846402i	$0.678764\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	117743.	+	323495.i	0.141872	+	0.389790i	0.990196	−	0.139688i	$-0.0446100\pi$
$911$	117743.	+	323495.i	0.141872	+	0.389790i	−0.848324	+	0.529478i	$0.822388\pi$
$912$		0			0
$913$	−1.81460e6	+	1.52263e6i	−2.17690	+	1.82664i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	437170.i		−	0.519890i
$918$		0			0
$919$	−744201.			−0.881169			−0.440585	−	0.897711i	$-0.645229\pi$
$919$	−744201.			−0.881169			−0.440585	+	0.897711i	$0.645229\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	75304.4	+	89744.3i	0.0883929	+	0.105343i
$924$		0			0
$925$	−625812.	+	227777.i	−0.731409	+	0.266211i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−861768.	+	1.02701e6i	−0.998525	+	1.19000i	−0.0167668	+	0.999859i	$0.505337\pi$
$929$	−861768.	+	1.02701e6i	−0.998525	+	1.19000i	−0.981758	+	0.190136i	$0.939107\pi$
$930$		0			0
$931$	−179981.	−	1.02072e6i	−0.207648	−	1.17763i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2290.62	−	1322.49i	0.00262017	−	0.00151276i
$936$		0			0
$937$	41933.4	−	72630.8i	0.0477618	−	0.0827259i	−0.841156	−	0.540792i	$-0.818124\pi$
$937$	41933.4	−	72630.8i	0.0477618	−	0.0827259i	0.888918	+	0.458066i	$0.151458\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	60997.9	−	167590.i	0.0688867	−	0.189265i	−0.900472	−	0.434914i	$-0.856779\pi$
$941$	60997.9	−	167590.i	0.0688867	−	0.189265i	0.969359	+	0.245649i	$0.0790011\pi$
$942$		0			0
$943$	−190210.	+	1.07874e6i	−0.213900	+	1.21309i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	319363.	+	56312.3i	0.356110	+	0.0627918i	0.348841	−	0.937182i	$-0.386575\pi$
$947$	319363.	+	56312.3i	0.356110	+	0.0627918i	0.00726867	+	0.999974i	$0.497686\pi$
$948$		0			0
$949$	67695.4	+	24639.1i	0.0751669	+	0.0273585i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−111927.	−	64620.9i	−0.123239	−	0.0711520i	0.437113	−	0.899406i	$-0.356001\pi$
$953$	−111927.	−	64620.9i	−0.123239	−	0.0711520i	−0.560352	+	0.828254i	$0.689334\pi$
$954$		0			0
$955$	877.343	+	1519.60i	0.000961973	+	0.00166619i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	405491.	−	71499.0i	0.440904	−	0.0777432i
$960$		0			0
$961$	1.17330e6	+	984517.i	1.27047	+	1.06605i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	453.289	+	1245.40i	0.000486766	+	0.00133738i
$966$		0			0
$967$	866669.	−	727222.i	0.926831	−	0.777703i	−0.0484150	−	0.998827i	$-0.515417\pi$
$967$	866669.	−	727222.i	0.926831	−	0.777703i	0.975246	+	0.221124i	$0.0709725\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.53192e6i			1.62479i	0.583105	+	0.812397i	$0.301838\pi$
$971$			1.53192e6i			1.62479i	−0.583105	+	0.812397i	$0.698162\pi$
$972$		0			0
$973$	1.50274e6			1.58729
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15911.4	−	18962.5i	−0.0166694	−	0.0198658i	0.757645	−	0.652666i	$-0.226350\pi$
$977$	−15911.4	−	18962.5i	−0.0166694	−	0.0198658i	−0.774315	+	0.632801i	$0.781905\pi$
$978$		0			0
$979$	850974.	−	309729.i	0.887873	−	0.323159i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−632658.	+	753973.i	−0.654730	+	0.780277i	−0.986619	−	0.163042i	$-0.947869\pi$
$983$	−632658.	+	753973.i	−0.654730	+	0.780277i	0.331889	+	0.943318i	$0.392314\pi$
$984$		0			0
$985$	326.337	+	1850.75i	0.000336352	+	0.00190755i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−703736.	+	406302.i	−0.719477	+	0.415391i
$990$		0			0
$991$	−351721.	+	609199.i	−0.358138	+	0.620314i	−0.987650	−	0.156677i	$-0.949922\pi$
$991$	−351721.	+	609199.i	−0.358138	+	0.620314i	0.629511	+	0.776991i	$0.283255\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	622.149	−	1709.34i	0.000628418	−	0.00172656i
$996$		0			0
$997$	−139091.	+	788824.i	−0.139929	+	0.793579i	0.831370	+	0.555719i	$0.187557\pi$
$997$	−139091.	+	788824.i	−0.139929	+	0.793579i	−0.971300	+	0.237860i	$0.923554\pi$
$998$		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.89.7		72
3.2	odd	2		108.5.k.a.29.4	✓	72
27.13	even	9		108.5.k.a.41.4	yes	72
27.14	odd	18	inner	324.5.k.a.233.7		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.4	✓	72	3.2	odd	2
108.5.k.a.41.4	yes	72	27.13	even	9
324.5.k.a.89.7		72	1.1	even	1	trivial
324.5.k.a.233.7		72	27.14	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)

\(f(q)\)	\(=\)	\(q+(-0.0201223 - 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +O(q^{10})\)	\(q+(-0.0201223 - 0.0239808i) q^{5} +(-60.3934 + 21.9814i) q^{7} +(-122.535 + 146.032i) q^{11} +(8.21590 + 46.5947i) q^{13} +(383.840 - 221.610i) q^{17} +(299.637 - 518.987i) q^{19} +(-158.524 + 435.540i) q^{23} +(108.530 - 615.504i) q^{25} +(396.668 + 69.9433i) q^{29} +(-1472.40 - 535.909i) q^{31} +(1.74238 + 1.00596i) q^{35} +(-532.781 - 922.804i) q^{37} +(2327.41 - 410.385i) q^{41} +(1343.05 + 1126.95i) q^{43} +(-864.494 - 2375.18i) q^{47} +(1324.91 - 1111.73i) q^{49} -2410.75i q^{53} +5.96765 q^{55} +(-165.882 - 197.690i) q^{59} +(-3689.65 + 1342.92i) q^{61} +(0.952054 - 1.13461i) q^{65} +(-702.984 - 3986.82i) q^{67} +(2144.37 - 1238.05i) q^{71} +(761.304 - 1318.62i) q^{73} +(4190.34 - 11512.9i) q^{77} +(1351.07 - 7662.27i) q^{79} +(12237.2 + 2157.76i) q^{83} +(-13.0381 - 4.74548i) q^{85} +(-4114.03 - 2375.24i) q^{89} +(-1520.40 - 2633.41i) q^{91} +(-18.4751 + 3.25765i) q^{95} +(-4423.75 - 3711.97i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 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

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Database

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