LMFDB - Embedded newform 324.6.e.f.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, 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:	$51.9643576194$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 11x^{2} + 10x + 100 $ x^4 - x^3 + 11x^2 + 10x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$-1.35078 + 2.33962i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.6.e.f.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(28.8141 - 49.9074i) q^{5} +(94.4422 + 163.579i) q^{7} +O(q^{10})$	$q+(28.8141 - 49.9074i) q^{5} +(94.4422 + 163.579i) q^{7} +(-352.012 - 609.703i) q^{11} +(-397.769 + 688.956i) q^{13} -180.562 q^{17} -661.769 q^{19} +(-1816.26 + 3145.85i) q^{23} +(-98.0000 - 169.741i) q^{25} +(-4010.93 - 6947.14i) q^{29} +(1501.94 - 2601.43i) q^{31} +10885.0 q^{35} -1520.31 q^{37} +(-1730.78 + 2997.81i) q^{41} +(-5907.88 - 10232.7i) q^{43} +(2989.91 + 5178.68i) q^{47} +(-9435.15 + 16342.2i) q^{49} +13819.2 q^{53} -40571.6 q^{55} +(-11320.1 + 19607.0i) q^{59} +(18702.1 + 32393.0i) q^{61} +(22922.7 + 39703.2i) q^{65} +(-35482.3 + 61457.2i) q^{67} -67767.0 q^{71} -31562.4 q^{73} +(66489.6 - 115163. i) q^{77} +(-31395.8 - 54379.1i) q^{79} +(-46715.7 - 80914.0i) q^{83} +(-5202.73 + 9011.40i) q^{85} -68793.1 q^{89} -150265. q^{91} +(-19068.2 + 33027.2i) q^{95} +(-58887.0 - 101995. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{7}+O(q^{10}) $ 4 * q + 32 * q^7	$ 4 q + 32 q^{7} - 486 q^{11} - 208 q^{13} + 3888 q^{17} - 1264 q^{19} - 6804 q^{23} - 392 q^{25} - 11664 q^{29} - 3328 q^{31} + 39852 q^{35} - 19912 q^{37} - 13608 q^{41} - 4960 q^{43} - 18468 q^{47} - 26676 q^{49} - 23328 q^{53} - 106272 q^{55} - 1944 q^{59} - 8176 q^{61} + 79704 q^{65} - 90064 q^{67} - 89424 q^{71} - 242428 q^{73} + 167184 q^{77} - 28768 q^{79} + 15066 q^{83} - 132840 q^{85} - 357696 q^{89} - 484880 q^{91} - 39852 q^{95} - 88942 q^{97}+O(q^{100}) $ 4 * q + 32 * q^7 - 486 * q^11 - 208 * q^13 + 3888 * q^17 - 1264 * q^19 - 6804 * q^23 - 392 * q^25 - 11664 * q^29 - 3328 * q^31 + 39852 * q^35 - 19912 * q^37 - 13608 * q^41 - 4960 * q^43 - 18468 * q^47 - 26676 * q^49 - 23328 * q^53 - 106272 * q^55 - 1944 * q^59 - 8176 * q^61 + 79704 * q^65 - 90064 * q^67 - 89424 * q^71 - 242428 * q^73 + 167184 * q^77 - 28768 * q^79 + 15066 * q^83 - 132840 * q^85 - 357696 * q^89 - 484880 * q^91 - 39852 * q^95 - 88942 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	28.8141	−	49.9074i	0.515442	−	0.892771i	−0.484398	−	0.874848i	$-0.660961\pi$
$5$	28.8141	−	49.9074i	0.515442	−	0.892771i	0.999839	−	0.0179231i	$-0.00570541\pi$
$6$		0			0
$7$	94.4422	+	163.579i	0.728485	+	1.26177i	0.957523	+	0.288356i	$0.0931087\pi$
$7$	94.4422	+	163.579i	0.728485	+	1.26177i	−0.229038	+	0.973418i	$0.573558\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−352.012	−	609.703i	−0.877155	−	1.51928i	−0.854449	−	0.519535i	$-0.826105\pi$
$11$	−352.012	−	609.703i	−0.877155	−	1.51928i	−0.0227057	−	0.999742i	$-0.507228\pi$
$12$		0			0
$13$	−397.769	+	688.956i	−0.652788	+	1.13066i	0.329655	+	0.944101i	$0.393067\pi$
$13$	−397.769	+	688.956i	−0.652788	+	1.13066i	−0.982443	+	0.186561i	$0.940266\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−180.562			−0.151532			−0.0757661	−	0.997126i	$-0.524140\pi$
$17$	−180.562			−0.151532			−0.0757661	+	0.997126i	$0.524140\pi$
$18$		0			0
$19$	−661.769			−0.420554			−0.210277	−	0.977642i	$-0.567437\pi$
$19$	−661.769			−0.420554			−0.210277	+	0.977642i	$0.567437\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1816.26	+	3145.85i	−0.715909	+	1.23999i	0.246700	+	0.969092i	$0.420654\pi$
$23$	−1816.26	+	3145.85i	−0.715909	+	1.23999i	−0.962608	+	0.270898i	$0.912679\pi$
$24$		0			0
$25$	−98.0000	−	169.741i	−0.0313600	−	0.0543171i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4010.93	−	6947.14i	−0.885626	−	1.53395i	−0.844994	−	0.534776i	$-0.820396\pi$
$29$	−4010.93	−	6947.14i	−0.885626	−	1.53395i	−0.0406324	−	0.999174i	$-0.512937\pi$
$30$		0			0
$31$	1501.94	−	2601.43i	0.280704	−	0.486193i	−0.690855	−	0.722994i	$-0.742766\pi$
$31$	1501.94	−	2601.43i	0.280704	−	0.486193i	0.971558	+	0.236801i	$0.0760989\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10885.0			1.50197
$36$		0			0
$37$	−1520.31			−0.182570			−0.0912848	−	0.995825i	$-0.529097\pi$
$37$	−1520.31			−0.182570			−0.0912848	+	0.995825i	$0.529097\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1730.78	+	2997.81i	−0.160799	+	0.278512i	−0.935155	−	0.354238i	$-0.884740\pi$
$41$	−1730.78	+	2997.81i	−0.160799	+	0.278512i	0.774356	+	0.632750i	$0.218074\pi$
$42$		0			0
$43$	−5907.88	−	10232.7i	−0.487260	−	0.843958i	0.512633	−	0.858608i	$-0.328670\pi$
$43$	−5907.88	−	10232.7i	−0.487260	−	0.843958i	−0.999893	+	0.0146495i	$0.995337\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2989.91	+	5178.68i	0.197430	+	0.341959i	0.947694	−	0.319179i	$-0.103407\pi$
$47$	2989.91	+	5178.68i	0.197430	+	0.341959i	−0.750264	+	0.661138i	$0.770074\pi$
$48$		0			0
$49$	−9435.15	+	16342.2i	−0.561382	+	0.972342i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	13819.2			0.675761			0.337880	−	0.941189i	$-0.390290\pi$
$53$	13819.2			0.675761			0.337880	+	0.941189i	$0.390290\pi$
$54$		0			0
$55$	−40571.6			−1.80849
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11320.1	+	19607.0i	−0.423370	+	0.733298i	−0.996267	−	0.0863297i	$-0.972486\pi$
$59$	−11320.1	+	19607.0i	−0.423370	+	0.733298i	0.572897	+	0.819627i	$0.305820\pi$
$60$		0			0
$61$	18702.1	+	32393.0i	0.643526	+	1.11462i	0.984640	+	0.174598i	$0.0558626\pi$
$61$	18702.1	+	32393.0i	0.643526	+	1.11462i	−0.341113	+	0.940022i	$0.610804\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	22922.7	+	39703.2i	0.672948	+	1.16558i
$66$		0			0
$67$	−35482.3	+	61457.2i	−0.965662	+	1.67258i	−0.257836	+	0.966189i	$0.583009\pi$
$67$	−35482.3	+	61457.2i	−0.965662	+	1.67258i	−0.707826	+	0.706387i	$0.750324\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−67767.0			−1.59541			−0.797705	−	0.603048i	$-0.793953\pi$
$71$	−67767.0			−1.59541			−0.797705	+	0.603048i	$0.793953\pi$
$72$		0			0
$73$	−31562.4			−0.693208			−0.346604	−	0.938012i	$-0.612665\pi$
$73$	−31562.4			−0.693208			−0.346604	+	0.938012i	$0.612665\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	66489.6	−	115163.i	1.27799	−	2.21354i
$78$		0			0
$79$	−31395.8	−	54379.1i	−0.565984	−	0.980313i	−0.996957	−	0.0779481i	$-0.975163\pi$
$79$	−31395.8	−	54379.1i	−0.565984	−	0.980313i	0.430974	−	0.902365i	$-0.358170\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−46715.7	−	80914.0i	−0.744334	−	1.28922i	−0.950505	−	0.310709i	$-0.899434\pi$
$83$	−46715.7	−	80914.0i	−0.744334	−	1.28922i	0.206171	−	0.978516i	$-0.433900\pi$
$84$		0			0
$85$	−5202.73	+	9011.40i	−0.0781060	+	0.135284i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−68793.1			−0.920598			−0.460299	−	0.887764i	$-0.652258\pi$
$89$	−68793.1			−0.920598			−0.460299	+	0.887764i	$0.652258\pi$
$90$		0			0
$91$	−150265.			−1.90219
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−19068.2	+	33027.2i	−0.216771	+	0.375459i
$96$		0			0
$97$	−58887.0	−	101995.i	−0.635463	−	1.10065i	−0.986417	−	0.164261i	$-0.947476\pi$
$97$	−58887.0	−	101995.i	−0.635463	−	1.10065i	0.350954	−	0.936393i	$-0.385857\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	80942.8	+	140197.i	0.789542	+	1.36753i	0.926248	+	0.376914i	$0.123015\pi$
$101$	80942.8	+	140197.i	0.789542	+	1.36753i	−0.136707	+	0.990612i	$0.543652\pi$
$102$		0			0
$103$	−29662.3	+	51376.6i	−0.275494	+	0.477169i	−0.970260	−	0.242067i	$-0.922175\pi$
$103$	−29662.3	+	51376.6i	−0.275494	+	0.477169i	0.694766	+	0.719236i	$0.255508\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	21124.1			0.178369			0.0891845	−	0.996015i	$-0.471574\pi$
$107$	21124.1			0.178369			0.0891845	+	0.996015i	$0.471574\pi$
$108$		0			0
$109$	−168898.			−1.36162			−0.680812	−	0.732458i	$-0.738373\pi$
$109$	−168898.			−1.36162			−0.680812	+	0.732458i	$0.738373\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−30637.0	+	53064.8i	−0.225710	+	0.390940i	−0.956532	−	0.291627i	$-0.905803\pi$
$113$	−30637.0	+	53064.8i	−0.225710	+	0.390940i	0.730823	+	0.682567i	$0.239137\pi$
$114$		0			0
$115$	104667.	+	181289.i	0.738018	+	1.27828i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17052.7	−	29536.1i	−0.110389	−	0.191199i
$120$		0			0
$121$	−167300.	+	289772.i	−1.03880	+	1.79926i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	168793.			0.966226
$126$		0			0
$127$	23607.7			0.129881			0.0649404	−	0.997889i	$-0.479314\pi$
$127$	23607.7			0.129881			0.0649404	+	0.997889i	$0.479314\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−103425.	+	179137.i	−0.526557	+	0.912023i	0.472964	+	0.881082i	$0.343184\pi$
$131$	−103425.	+	179137.i	−0.526557	+	0.912023i	−0.999521	+	0.0309416i	$0.990149\pi$
$132$		0			0
$133$	−62498.9	−	108251.i	−0.306368	−	0.530645i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−95747.5	−	165839.i	−0.435839	−	0.754895i	0.561525	−	0.827460i	$-0.310215\pi$
$137$	−95747.5	−	165839.i	−0.435839	−	0.754895i	−0.997364	+	0.0725649i	$0.976882\pi$
$138$		0			0
$139$	−125209.	+	216869.i	−0.549666	+	0.952049i	0.448631	+	0.893717i	$0.351912\pi$
$139$	−125209.	+	216869.i	−0.549666	+	0.952049i	−0.998297	+	0.0583324i	$0.981422\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	560078.			2.29039
$144$		0			0
$145$	−462285.			−1.82595
$146$		0			0
$147$		0			0
$148$		0			0
$149$	100429.	−	173948.i	0.370588	−	0.641878i	−0.619068	−	0.785338i	$-0.712489\pi$
$149$	100429.	−	173948.i	0.370588	−	0.641878i	0.989656	+	0.143460i	$0.0458227\pi$
$150$		0			0
$151$	11631.6	+	20146.5i	0.0415141	+	0.0719046i	0.886036	−	0.463617i	$-0.153449\pi$
$151$	11631.6	+	20146.5i	0.0415141	+	0.0719046i	−0.844522	+	0.535521i	$0.820115\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−86553.9	−	149916.i	−0.289373	−	0.501208i
$156$		0			0
$157$	261993.	−	453785.i	0.848281	−	1.46927i	−0.0344595	−	0.999406i	$-0.510971\pi$
$157$	261993.	−	453785.i	0.848281	−	1.46927i	0.882741	−	0.469860i	$-0.155696\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−686125.			−2.08612
$162$		0			0
$163$	194530.			0.573479			0.286739	−	0.958009i	$-0.407429\pi$
$163$	194530.			0.573479			0.286739	+	0.958009i	$0.407429\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−231379.	+	400759.i	−0.641995	+	1.11197i	0.342991	+	0.939339i	$0.388560\pi$
$167$	−231379.	+	400759.i	−0.641995	+	1.11197i	−0.984987	+	0.172630i	$0.944774\pi$
$168$		0			0
$169$	−130793.	−	226541.i	−0.352265	−	0.610140i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	19945.9	+	34547.3i	0.0506685	+	0.0877604i	0.890247	−	0.455478i	$-0.150531\pi$
$173$	19945.9	+	34547.3i	0.0506685	+	0.0877604i	−0.839579	+	0.543238i	$0.817198\pi$
$174$		0			0
$175$	18510.7	−	32061.4i	0.0456906	−	0.0791385i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	221514.			0.516736			0.258368	−	0.966047i	$-0.416815\pi$
$179$	221514.			0.516736			0.258368	+	0.966047i	$0.416815\pi$
$180$		0			0
$181$	800588.			1.81640			0.908202	−	0.418532i	$-0.137455\pi$
$181$	800588.			1.81640			0.908202	+	0.418532i	$0.137455\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−43806.4	+	75874.9i	−0.0941040	+	0.162993i
$186$		0			0
$187$	63560.2	+	110090.i	0.132917	+	0.230219i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−35275.3	−	61098.6i	−0.0699661	−	0.121185i	0.828920	−	0.559367i	$-0.188956\pi$
$191$	−35275.3	−	61098.6i	−0.0699661	−	0.121185i	−0.898886	+	0.438182i	$0.855622\pi$
$192$		0			0
$193$	229098.	−	396809.i	0.442719	−	0.766811i	−0.555171	−	0.831736i	$-0.687347\pi$
$193$	229098.	−	396809.i	0.442719	−	0.766811i	0.997890	+	0.0649247i	$0.0206807\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−605955.			−1.11244			−0.556218	−	0.831037i	$-0.687748\pi$
$197$	−605955.			−1.11244			−0.556218	+	0.831037i	$0.687748\pi$
$198$		0			0
$199$	−108129.			−0.193557			−0.0967783	−	0.995306i	$-0.530854\pi$
$199$	−108129.			−0.193557			−0.0967783	+	0.995306i	$0.530854\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	757603.	−	1.31221e6i	1.29033	−	2.23492i
$204$		0			0
$205$	99741.9	+	172758.i	0.165765	+	0.287113i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	232951.	+	403483.i	0.368891	+	0.638939i
$210$		0			0
$211$	−137461.	+	238090.i	−0.212556	+	0.368158i	−0.952514	−	0.304495i	$-0.901512\pi$
$211$	−137461.	+	238090.i	−0.212556	+	0.368158i	0.739958	+	0.672654i	$0.234846\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−680920.			−1.00462
$216$		0			0
$217$	567385.			0.817954
$218$		0			0
$219$		0			0
$220$		0			0
$221$	71822.1	−	124399.i	0.0989184	−	0.171332i
$222$		0			0
$223$	251923.	+	436343.i	0.339239	+	0.587579i	0.984290	−	0.176561i	$-0.0564972\pi$
$223$	251923.	+	436343.i	0.339239	+	0.587579i	−0.645051	+	0.764140i	$0.723164\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	194863.	+	337512.i	0.250994	+	0.434735i	0.963800	−	0.266627i	$-0.0859091\pi$
$227$	194863.	+	337512.i	0.250994	+	0.434735i	−0.712805	+	0.701362i	$0.752576\pi$
$228$		0			0
$229$	465752.	−	806707.i	0.586903	−	1.01655i	−0.407732	−	0.913102i	$-0.633680\pi$
$229$	465752.	−	806707.i	0.586903	−	1.01655i	0.994635	−	0.103444i	$-0.0329864\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	967962.			1.16807			0.584034	−	0.811729i	$-0.301473\pi$
$233$	967962.			1.16807			0.584034	+	0.811729i	$0.301473\pi$
$234$		0			0
$235$	344606.			0.407055
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−167948.	+	290894.i	−0.190187	+	0.329413i	−0.945312	−	0.326168i	$-0.894243\pi$
$239$	−167948.	+	290894.i	−0.190187	+	0.329413i	0.755125	+	0.655580i	$0.227576\pi$
$240$		0			0
$241$	273571.	+	473839.i	0.303408	+	0.525519i	0.976906	−	0.213671i	$-0.0685420\pi$
$241$	273571.	+	473839.i	0.303408	+	0.525519i	−0.673497	+	0.739190i	$0.735209\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	543730.	+	941768.i	0.578719	+	1.00237i
$246$		0			0
$247$	263231.	−	455929.i	0.274533	−	0.475505i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	36479.6			0.0365482			0.0182741	−	0.999833i	$-0.494183\pi$
$251$	36479.6			0.0365482			0.0182741	+	0.999833i	$0.494183\pi$
$252$		0			0
$253$	2.55738e6			2.51185
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−599841.	+	1.03896e6i	−0.566505	+	0.981215i	0.430403	+	0.902637i	$0.358371\pi$
$257$	−599841.	+	1.03896e6i	−0.566505	+	0.981215i	−0.996908	+	0.0785784i	$0.974962\pi$
$258$		0			0
$259$	−143582.	−	248691.i	−0.132999	−	0.230362i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	439254.	+	760810.i	0.391585	+	0.678246i	0.992659	−	0.120948i	$-0.0385935\pi$
$263$	439254.	+	760810.i	0.391585	+	0.678246i	−0.601074	+	0.799194i	$0.705260\pi$
$264$		0			0
$265$	398187.	−	689680.i	0.348315	−	0.603299i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	434159.			0.365820			0.182910	−	0.983130i	$-0.441448\pi$
$269$	434159.			0.365820			0.182910	+	0.983130i	$0.441448\pi$
$270$		0			0
$271$	799583.			0.661364			0.330682	−	0.943742i	$-0.392721\pi$
$271$	799583.			0.661364			0.330682	+	0.943742i	$0.392721\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−68994.4	+	119502.i	−0.0550152	+	0.0952891i
$276$		0			0
$277$	−815212.	−	1.41199e6i	−0.638368	−	1.10569i	−0.985791	−	0.167977i	$-0.946277\pi$
$277$	−815212.	−	1.41199e6i	−0.638368	−	1.10569i	0.347423	−	0.937708i	$-0.387057\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.26849e6	−	2.19709e6i	−0.958344	−	1.65990i	−0.726523	−	0.687142i	$-0.758865\pi$
$281$	−1.26849e6	−	2.19709e6i	−0.958344	−	1.65990i	−0.231821	−	0.972759i	$-0.574468\pi$
$282$		0			0
$283$	1.03662e6	−	1.79548e6i	0.769402	−	1.33264i	−0.168486	−	0.985704i	$-0.553888\pi$
$283$	1.03662e6	−	1.79548e6i	0.769402	−	1.33264i	0.937888	−	0.346939i	$-0.112779\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−653836.			−0.468559
$288$		0			0
$289$	−1.38725e6			−0.977038
$290$		0			0
$291$		0			0
$292$		0			0
$293$	333384.	−	577438.i	0.226869	−	0.392949i	−0.730009	−	0.683437i	$-0.760484\pi$
$293$	333384.	−	577438.i	0.226869	−	0.392949i	0.956879	+	0.290488i	$0.0938176\pi$
$294$		0			0
$295$	652355.	+	1.12991e6i	0.436445	+	0.755944i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.44490e6	−	2.50264e6i	−0.934673	−	1.61890i
$300$		0			0
$301$	1.11591e6	−	1.93281e6i	0.709923	−	1.22962i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.15554e6			1.32680
$306$		0			0
$307$	811069.			0.491148			0.245574	−	0.969378i	$-0.421024\pi$
$307$	811069.			0.491148			0.245574	+	0.969378i	$0.421024\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−492519.	+	853067.i	−0.288750	+	0.500129i	−0.973512	−	0.228638i	$-0.926573\pi$
$311$	−492519.	+	853067.i	−0.288750	+	0.500129i	0.684762	+	0.728767i	$0.259906\pi$
$312$		0			0
$313$	−391592.	−	678257.i	−0.225929	−	0.391321i	0.730668	−	0.682732i	$-0.239209\pi$
$313$	−391592.	−	678257.i	−0.225929	−	0.391321i	−0.956598	+	0.291411i	$0.905875\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.33259e6	−	2.30812e6i	−0.744817	−	1.29006i	−0.950280	−	0.311396i	$-0.899203\pi$
$317$	−1.33259e6	−	2.30812e6i	−0.744817	−	1.29006i	0.205463	−	0.978665i	$-0.434130\pi$
$318$		0			0
$319$	−2.82380e6	+	4.89096e6i	−1.55366	+	2.69102i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	119491.			0.0637275
$324$		0			0
$325$	155925.			0.0818857
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−564748.	+	978171.i	−0.287650	+	0.498225i
$330$		0			0
$331$	1.17076e6	+	2.02782e6i	0.587353	+	1.01733i	0.994578	+	0.103998i	$0.0331635\pi$
$331$	1.17076e6	+	2.02782e6i	0.587353	+	1.01733i	−0.407224	+	0.913328i	$0.633503\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.04478e6	+	3.54166e6i	0.995485	+	1.72423i
$336$		0			0
$337$	355371.	−	615521.i	0.170454	−	0.295235i	−0.768125	−	0.640300i	$-0.778810\pi$
$337$	355371.	−	615521.i	0.170454	−	0.295235i	0.938579	+	0.345065i	$0.112143\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.11480e6			−0.984882
$342$		0			0
$343$	−389725.			−0.178864
$344$		0			0
$345$		0			0
$346$		0			0
$347$	351093.	−	608112.i	0.156531	−	0.271119i	−0.777085	−	0.629396i	$-0.783302\pi$
$347$	351093.	−	608112.i	0.156531	−	0.271119i	0.933615	+	0.358277i	$0.116636\pi$
$348$		0			0
$349$	69349.7	+	120117.i	0.0304776	+	0.0527888i	0.880862	−	0.473373i	$-0.156964\pi$
$349$	69349.7	+	120117.i	0.0304776	+	0.0527888i	−0.850384	+	0.526162i	$0.823630\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	27547.1	+	47712.9i	0.0117663	+	0.0203798i	0.871849	−	0.489776i	$-0.162921\pi$
$353$	27547.1	+	47712.9i	0.0117663	+	0.0203798i	−0.860082	+	0.510155i	$0.829588\pi$
$354$		0			0
$355$	−1.95264e6	+	3.38207e6i	−0.822340	+	1.42434i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.94311e6			−0.795722			−0.397861	−	0.917446i	$-0.630247\pi$
$359$	−1.94311e6			−0.795722			−0.397861	+	0.917446i	$0.630247\pi$
$360$		0			0
$361$	−2.03816e6			−0.823134
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−909442.	+	1.57520e6i	−0.357308	+	0.618876i
$366$		0			0
$367$	747966.	+	1.29551e6i	0.289879	+	0.502085i	0.973781	−	0.227489i	$-0.0730517\pi$
$367$	747966.	+	1.29551e6i	0.289879	+	0.502085i	−0.683902	+	0.729574i	$0.739718\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.30511e6	+	2.26052e6i	0.492282	+	0.852657i
$372$		0			0
$373$	858481.	−	1.48693e6i	0.319491	−	0.553375i	−0.660891	−	0.750482i	$-0.729821\pi$
$373$	858481.	−	1.48693e6i	0.319491	−	0.553375i	0.980382	+	0.197107i	$0.0631548\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.38170e6			2.31251
$378$		0			0
$379$	−3.54794e6			−1.26876			−0.634378	−	0.773023i	$-0.718744\pi$
$379$	−3.54794e6			−1.26876			−0.634378	+	0.773023i	$0.718744\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.37651e6	−	2.38419e6i	0.479493	−	0.830507i	−0.520230	−	0.854026i	$-0.674154\pi$
$383$	1.37651e6	−	2.38419e6i	0.479493	−	0.830507i	0.999723	+	0.0235192i	$0.00748709\pi$
$384$		0			0
$385$	−3.83167e6	−	6.63665e6i	−1.31746	−	2.28190i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.47686e6	+	2.55799e6i	0.494839	+	0.857087i	0.999982	−	0.00594868i	$-0.00189353\pi$
$389$	1.47686e6	+	2.55799e6i	0.494839	+	0.857087i	−0.505143	+	0.863036i	$0.668560\pi$
$390$		0			0
$391$	327948.	−	568022.i	0.108483	−	0.187898i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.61856e6			−1.16693
$396$		0			0
$397$	−2.21949e6			−0.706767			−0.353383	−	0.935479i	$-0.614969\pi$
$397$	−2.21949e6			−0.706767			−0.353383	+	0.935479i	$0.614969\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.21911e6	−	3.84361e6i	0.689157	−	1.19365i	−0.282954	−	0.959133i	$-0.591315\pi$
$401$	2.21911e6	−	3.84361e6i	0.689157	−	1.19365i	0.972111	−	0.234521i	$-0.0753521\pi$
$402$		0			0
$403$	1.19485e6	+	2.06954e6i	0.366480	+	0.634762i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	535169.	+	926940.i	0.160142	+	0.277374i
$408$		0			0
$409$	165800.	−	287175.i	0.0490092	−	0.0848864i	−0.840480	−	0.541842i	$-0.817727\pi$
$409$	165800.	−	287175.i	0.0490092	−	0.0848864i	0.889489	+	0.456956i	$0.151060\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.27637e6			−1.23367
$414$		0			0
$415$	−5.38428e6			−1.53464
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.24743e6	−	2.16061e6i	0.347121	−	0.601232i	−0.638616	−	0.769526i	$-0.720493\pi$
$419$	1.24743e6	−	2.16061e6i	0.347121	−	0.601232i	0.985737	+	0.168294i	$0.0538259\pi$
$420$		0			0
$421$	−2.27578e6	−	3.94176e6i	−0.625784	−	1.08389i	−0.988389	−	0.151946i	$-0.951446\pi$
$421$	−2.27578e6	−	3.94176e6i	−0.625784	−	1.08389i	0.362605	−	0.931943i	$-0.381887\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	17695.1	+	30648.8i	0.00475205	+	0.00823079i
$426$		0			0
$427$	−3.53254e6	+	6.11854e6i	−0.937599	+	1.62397i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−614314.			−0.159293			−0.0796466	−	0.996823i	$-0.525379\pi$
$431$	−614314.			−0.159293			−0.0796466	+	0.996823i	$0.525379\pi$
$432$		0			0
$433$	2.42759e6			0.622236			0.311118	−	0.950371i	$-0.399297\pi$
$433$	2.42759e6			0.622236			0.311118	+	0.950371i	$0.399297\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.20194e6	−	2.08182e6i	0.301079	−	0.521483i
$438$		0			0
$439$	1.15900e6	+	2.00745e6i	0.287027	+	0.497145i	0.973099	−	0.230389i	$-0.0739998\pi$
$439$	1.15900e6	+	2.00745e6i	0.287027	+	0.497145i	−0.686072	+	0.727534i	$0.740666\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.61824e6	−	4.53493e6i	−0.633870	−	1.09790i	−0.986753	−	0.162228i	$-0.948132\pi$
$443$	−2.61824e6	−	4.53493e6i	−0.633870	−	1.09790i	0.352883	−	0.935667i	$-0.385201\pi$
$444$		0			0
$445$	−1.98221e6	+	3.43329e6i	−0.474515	+	0.821883i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	985716.			0.230747			0.115373	−	0.993322i	$-0.463194\pi$
$449$	985716.			0.230747			0.115373	+	0.993322i	$0.463194\pi$
$450$		0			0
$451$	2.43703e6			0.564183
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.32973e6	+	7.49932e6i	−0.980466	+	1.69822i
$456$		0			0
$457$	2.54748e6	+	4.41237e6i	0.570585	+	0.988283i	0.996506	+	0.0835223i	$0.0266170\pi$
$457$	2.54748e6	+	4.41237e6i	0.570585	+	0.988283i	−0.425920	+	0.904761i	$0.640050\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.44756e6	+	7.70340e6i	0.974697	+	1.68822i	0.680932	+	0.732347i	$0.261575\pi$
$461$	4.44756e6	+	7.70340e6i	0.974697	+	1.68822i	0.293765	+	0.955878i	$0.405092\pi$
$462$		0			0
$463$	−1.63133e6	+	2.82555e6i	−0.353664	+	0.612563i	−0.986888	−	0.161405i	$-0.948398\pi$
$463$	−1.63133e6	+	2.82555e6i	−0.353664	+	0.612563i	0.633225	+	0.773968i	$0.281731\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.63482e6			0.346879			0.173439	−	0.984845i	$-0.444512\pi$
$467$	1.63482e6			0.346879			0.173439	+	0.984845i	$0.444512\pi$
$468$		0			0
$469$	−1.34041e7			−2.81388
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.15929e6	+	7.20411e6i	−0.854804	+	1.48056i
$474$		0			0
$475$	64853.3	+	112329.i	0.0131886	+	0.0228433i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.28557e6	−	7.42282e6i	−0.853433	−	1.47819i	−0.878091	−	0.478493i	$-0.841183\pi$
$479$	−4.28557e6	−	7.42282e6i	−0.853433	−	1.47819i	0.0246586	−	0.999696i	$-0.492150\pi$
$480$		0			0
$481$	604733.	−	1.04743e6i	0.119179	−	0.206425i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.78709e6			−1.31018
$486$		0			0
$487$	−5.12742e6			−0.979662			−0.489831	−	0.871817i	$-0.662942\pi$
$487$	−5.12742e6			−0.979662			−0.489831	+	0.871817i	$0.662942\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.66995e6	−	6.35654e6i	0.687000	−	1.18992i	−0.285804	−	0.958288i	$-0.592261\pi$
$491$	3.66995e6	−	6.35654e6i	0.687000	−	1.18992i	0.972804	−	0.231630i	$-0.0744060\pi$
$492$		0			0
$493$	724224.	+	1.25439e6i	0.134201	+	0.232443i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.40006e6	−	1.10852e7i	−1.16223	−	2.01305i
$498$		0			0
$499$	−3.04443e6	+	5.27310e6i	−0.547336	+	0.948015i	0.451119	+	0.892464i	$0.351025\pi$
$499$	−3.04443e6	+	5.27310e6i	−0.547336	+	0.948015i	−0.998456	+	0.0555509i	$0.982308\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.11508e6			1.60635			0.803175	−	0.595743i	$-0.203142\pi$
$503$	9.11508e6			1.60635			0.803175	+	0.595743i	$0.203142\pi$
$504$		0			0
$505$	9.32917e6			1.62785
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.59082e6	−	2.75538e6i	0.272162	−	0.471398i	−0.697253	−	0.716825i	$-0.745595\pi$
$509$	1.59082e6	−	2.75538e6i	0.272162	−	0.471398i	0.969415	+	0.245427i	$0.0789281\pi$
$510$		0			0
$511$	−2.98082e6	−	5.16294e6i	−0.504992	−	0.874671i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.70938e6	+	2.96074e6i	0.284002	+	0.491905i
$516$		0			0
$517$	2.10497e6	−	3.64592e6i	0.346354	−	0.599902i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27154.5			−0.00438276			−0.00219138	−	0.999998i	$-0.500698\pi$
$521$	−27154.5			−0.00438276			−0.00219138	+	0.999998i	$0.500698\pi$
$522$		0			0
$523$	−3.93041e6			−0.628324			−0.314162	−	0.949369i	$-0.601724\pi$
$523$	−3.93041e6			−0.628324			−0.314162	+	0.949369i	$0.601724\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−271194.	+	469721.i	−0.0425356	+	0.0736739i
$528$		0			0
$529$	−3.37940e6	−	5.85330e6i	−0.525050	−	0.909413i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.37690e6	−	2.38487e6i	−0.209935	−	0.363619i
$534$		0			0
$535$	608672.	−	1.05425e6i	0.0919388	−	0.159243i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.32852e7			1.96968
$540$		0			0
$541$	8.38180e6			1.23124			0.615622	−	0.788042i	$-0.288905\pi$
$541$	8.38180e6			1.23124			0.615622	+	0.788042i	$0.288905\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.86662e6	+	8.42924e6i	−0.701837	+	1.21562i
$546$		0			0
$547$	3.75274e6	+	6.49994e6i	0.536266	+	0.928840i	0.999101	+	0.0423954i	$0.0134989\pi$
$547$	3.75274e6	+	6.49994e6i	0.536266	+	0.928840i	−0.462835	+	0.886444i	$0.653168\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.65431e6	+	4.59740e6i	0.372454	+	0.645109i
$552$		0			0
$553$	5.93018e6	−	1.02714e7i	0.824622	−	1.42829i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.03110e6			0.140820			0.0704099	−	0.997518i	$-0.477569\pi$
$557$	1.03110e6			0.140820			0.0704099	+	0.997518i	$0.477569\pi$
$558$		0			0
$559$	9.39988e6			1.27231
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.59812e6	+	1.14283e7i	−0.877303	+	1.51953i	−0.0230137	+	0.999735i	$0.507326\pi$
$563$	−6.59812e6	+	1.14283e7i	−0.877303	+	1.51953i	−0.854289	+	0.519798i	$0.826007\pi$
$564$		0			0
$565$	1.76555e6	+	3.05802e6i	0.232680	+	0.403014i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.74578e6	+	3.02377e6i	0.226052	+	0.391533i	0.956634	−	0.291291i	$-0.0940848\pi$
$569$	1.74578e6	+	3.02377e6i	0.226052	+	0.391533i	−0.730583	+	0.682824i	$0.760752\pi$
$570$		0			0
$571$	4.74727e6	−	8.22252e6i	0.609332	−	1.05539i	−0.382019	−	0.924155i	$-0.624771\pi$
$571$	4.74727e6	−	8.22252e6i	0.609332	−	1.05539i	0.991351	−	0.131239i	$-0.0418957\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	711972.			0.0898036
$576$		0			0
$577$	2.60369e6			0.325574			0.162787	−	0.986661i	$-0.447952\pi$
$577$	2.60369e6			0.325574			0.162787	+	0.986661i	$0.447952\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.82387e6	−	1.52834e7i	1.08447	−	1.87836i
$582$		0			0
$583$	−4.86453e6	−	8.42561e6i	−0.592747	−	1.02667i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.52049e6	+	4.36562e6i	0.301919	+	0.522939i	0.976571	−	0.215197i	$-0.0690394\pi$
$587$	2.52049e6	+	4.36562e6i	0.301919	+	0.522939i	−0.674652	+	0.738136i	$0.735706\pi$
$588$		0			0
$589$	−993936.	+	1.72155e6i	−0.118051	+	0.204471i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.35192e7			−1.57875			−0.789374	−	0.613912i	$-0.789595\pi$
$593$	−1.35192e7			−1.57875			−0.789374	+	0.613912i	$0.789595\pi$
$594$		0			0
$595$	−1.96543e6			−0.227596
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−407892.	+	706490.i	−0.0464492	+	0.0804524i	−0.888315	−	0.459234i	$-0.848124\pi$
$599$	−407892.	+	706490.i	−0.0464492	+	0.0804524i	0.841866	+	0.539687i	$0.181457\pi$
$600$		0			0
$601$	3.26529e6	+	5.65565e6i	0.368753	+	0.638699i	0.989371	−	0.145414i	$-0.0464514\pi$
$601$	3.26529e6	+	5.65565e6i	0.368753	+	0.638699i	−0.620618	+	0.784113i	$0.713118\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.64119e6	+	1.66990e7i	1.07088	+	1.85482i
$606$		0			0
$607$	550482.	−	953464.i	0.0606418	−	0.105035i	−0.834111	−	0.551597i	$-0.814019\pi$
$607$	550482.	−	953464.i	0.0606418	−	0.105035i	0.894752	+	0.446562i	$0.147352\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.75717e6			−0.515520
$612$		0			0
$613$	6.09992e6			0.655651			0.327825	−	0.944738i	$-0.393684\pi$
$613$	6.09992e6			0.655651			0.327825	+	0.944738i	$0.393684\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	282912.	−	490019.i	0.0299184	−	0.0518203i	−0.850678	−	0.525686i	$-0.823809\pi$
$617$	282912.	−	490019.i	0.0299184	−	0.0518203i	0.880597	+	0.473866i	$0.157142\pi$
$618$		0			0
$619$	6.87546e6	+	1.19086e7i	0.721232	+	1.24921i	0.960506	+	0.278259i	$0.0897573\pi$
$619$	6.87546e6	+	1.19086e7i	0.721232	+	1.24921i	−0.239274	+	0.970952i	$0.576909\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.49697e6	−	1.12531e7i	−0.670642	−	1.16159i
$624$		0			0
$625$	5.16985e6	−	8.95445e6i	0.529393	−	0.916936i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	274511.			0.0276652
$630$		0			0
$631$	420973.			0.0420902			0.0210451	−	0.999779i	$-0.493301\pi$
$631$	420973.			0.0420902			0.0210451	+	0.999779i	$0.493301\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	680234.	−	1.17820e6i	0.0669459	−	0.115954i
$636$		0			0
$637$	−7.50601e6	−	1.30008e7i	−0.732927	−	1.26947i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−157211.	−	272297.i	−0.0151125	−	0.0261757i	0.858370	−	0.513031i	$-0.171477\pi$
$641$	−157211.	−	272297.i	−0.0151125	−	0.0261757i	−0.873483	+	0.486855i	$0.838144\pi$
$642$		0			0
$643$	−6.90197e6	+	1.19546e7i	−0.658333	+	1.14027i	0.322714	+	0.946496i	$0.395405\pi$
$643$	−6.90197e6	+	1.19546e7i	−0.658333	+	1.14027i	−0.981047	+	0.193770i	$0.937929\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.43225e6			−0.322343			−0.161172	−	0.986926i	$-0.551527\pi$
$647$	−3.43225e6			−0.322343			−0.161172	+	0.986926i	$0.551527\pi$
$648$		0			0
$649$	1.59392e7			1.48544
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8.74805e6	−	1.51521e7i	0.802839	−	1.39056i	−0.114902	−	0.993377i	$-0.536655\pi$
$653$	8.74805e6	−	1.51521e7i	0.802839	−	1.39056i	0.917741	−	0.397181i	$-0.130011\pi$
$654$		0			0
$655$	5.96016e6	+	1.03233e7i	0.542819	+	0.940189i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.03433e6	−	1.04518e7i	−0.541272	−	0.937510i	−0.998831	−	0.0483313i	$-0.984610\pi$
$659$	−6.03433e6	−	1.04518e7i	−0.541272	−	0.937510i	0.457560	−	0.889179i	$-0.348724\pi$
$660$		0			0
$661$	90936.0	−	157506.i	0.00809529	−	0.0140215i	−0.861949	−	0.506994i	$-0.830756\pi$
$661$	90936.0	−	157506.i	0.00809529	−	0.0140215i	0.870045	+	0.492973i	$0.164090\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.20339e6			−0.631659
$666$		0			0
$667$	2.91395e7			2.53611
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.31668e7	−	2.28055e7i	1.12894	−	1.95539i
$672$		0			0
$673$	5.31915e6	+	9.21304e6i	0.452694	+	0.784089i	0.998552	−	0.0537884i	$-0.0171296\pi$
$673$	5.31915e6	+	9.21304e6i	0.452694	+	0.784089i	−0.545858	+	0.837878i	$0.683796\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.33956e6	−	7.51633e6i	−0.363893	−	0.630281i	0.624705	−	0.780861i	$-0.285219\pi$
$677$	−4.33956e6	−	7.51633e6i	−0.363893	−	0.630281i	−0.988598	+	0.150580i	$0.951886\pi$
$678$		0			0
$679$	1.11228e7	−	1.92653e7i	0.925850	−	1.60362i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.67179e7			−1.37129			−0.685647	−	0.727934i	$-0.740481\pi$
$683$	−1.67179e7			−1.37129			−0.685647	+	0.727934i	$0.740481\pi$
$684$		0			0
$685$	−1.10355e7			−0.898598
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.49684e6	+	9.52081e6i	−0.441128	+	0.764057i
$690$		0			0
$691$	1.92775e6	+	3.33897e6i	0.153588	+	0.266022i	0.932544	−	0.361057i	$-0.117584\pi$
$691$	1.92775e6	+	3.33897e6i	0.153588	+	0.266022i	−0.778956	+	0.627078i	$0.784251\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.21556e6	+	1.24977e7i	0.566641	+	0.981452i
$696$		0			0
$697$	312515.	−	541291.i	0.0243662	−	0.0422035i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.86020e6			−0.142976			−0.0714882	−	0.997441i	$-0.522775\pi$
$701$	−1.86020e6			−0.142976			−0.0714882	+	0.997441i	$0.522775\pi$
$702$		0			0
$703$	1.00610e6			0.0767805
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.52888e7	+	2.64810e7i	−1.15034	+	1.99245i
$708$		0			0
$709$	−7.78040e6	−	1.34760e7i	−0.581281	−	1.00681i	−0.995328	−	0.0965531i	$-0.969218\pi$
$709$	−7.78040e6	−	1.34760e7i	−0.581281	−	1.00681i	0.414046	−	0.910256i	$-0.364115\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.45581e6	+	9.44974e6i	0.401916	+	0.696139i
$714$		0			0
$715$	1.61381e7	−	2.79521e7i	1.18056	−	2.04479i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.35632e6			−0.314266			−0.157133	−	0.987577i	$-0.550225\pi$
$719$	−4.35632e6			−0.314266			−0.157133	+	0.987577i	$0.550225\pi$
$720$		0			0
$721$	−1.12055e7			−0.802772
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−786143.	+	1.36164e6i	−0.0555465	+	0.0962093i
$726$		0			0
$727$	−1.48078e6	−	2.56478e6i	−0.103909	−	0.179976i	0.809383	−	0.587281i	$-0.199802\pi$
$727$	−1.48078e6	−	2.56478e6i	−0.103909	−	0.179976i	−0.913292	+	0.407305i	$0.866468\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.06674e6	+	1.84765e6i	0.0738355	+	0.127887i
$732$		0			0
$733$	480426.	−	832122.i	0.0330268	−	0.0572041i	−0.849040	−	0.528329i	$-0.822819\pi$
$733$	480426.	−	832122.i	0.0330268	−	0.0572041i	0.882066	+	0.471125i	$0.156152\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.99609e7			3.38814
$738$		0			0
$739$	1.36123e7			0.916898			0.458449	−	0.888721i	$-0.348405\pi$
$739$	1.36123e7			0.916898			0.458449	+	0.888721i	$0.348405\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.86858e6	+	4.96853e6i	−0.190632	+	0.330184i	−0.945460	−	0.325739i	$-0.894387\pi$
$743$	−2.86858e6	+	4.96853e6i	−0.190632	+	0.330184i	0.754828	+	0.655923i	$0.227720\pi$
$744$		0			0
$745$	−5.78751e6	−	1.00243e7i	−0.382033	−	0.661701i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.99501e6	+	3.45546e6i	0.129939	+	0.225061i
$750$		0			0
$751$	−1.22046e7	+	2.11390e7i	−0.789631	+	1.36768i	0.136562	+	0.990632i	$0.456395\pi$
$751$	−1.22046e7	+	2.11390e7i	−0.789631	+	1.36768i	−0.926193	+	0.377049i	$0.876939\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.34061e6			0.0855924
$756$		0			0
$757$	5.16979e6			0.327894			0.163947	−	0.986469i	$-0.447577\pi$
$757$	5.16979e6			0.327894			0.163947	+	0.986469i	$0.447577\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.42163e6	+	9.39054e6i	−0.339366	+	0.587799i	−0.984314	−	0.176428i	$-0.943546\pi$
$761$	−5.42163e6	+	9.39054e6i	−0.339366	+	0.587799i	0.644948	+	0.764227i	$0.276879\pi$
$762$		0			0
$763$	−1.59510e7	−	2.76280e7i	−0.991923	−	1.71806i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.00555e6	−	1.55981e7i	−0.552741	−	0.957376i
$768$		0			0
$769$	4.63905e6	−	8.03507e6i	0.282887	−	0.489975i	−0.689208	−	0.724564i	$-0.742041\pi$
$769$	4.63905e6	−	8.03507e6i	0.282887	−	0.489975i	0.972095	+	0.234589i	$0.0753745\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.33311e7			0.802450			0.401225	−	0.915980i	$-0.368585\pi$
$773$	1.33311e7			0.802450			0.401225	+	0.915980i	$0.368585\pi$
$774$		0			0
$775$	−588760.			−0.0352115
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.14538e6	−	1.98385e6i	0.0676247	−	0.117129i
$780$		0			0
$781$	2.38548e7	+	4.13178e7i	1.39942	+	2.42387i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.50981e7	−	2.61507e7i	−0.874479	−	1.51464i
$786$		0			0
$787$	−3.17909e6	+	5.50634e6i	−0.182964	+	0.316903i	−0.942889	−	0.333108i	$-0.891903\pi$
$787$	−3.17909e6	+	5.50634e6i	−0.182964	+	0.316903i	0.759925	+	0.650011i	$0.225236\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.15737e7			−0.657704
$792$		0			0
$793$	−2.97565e7			−1.68035
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.29576e6	+	3.97638e6i	−0.128021	+	0.221739i	−0.922910	−	0.385016i	$-0.874196\pi$
$797$	−2.29576e6	+	3.97638e6i	−0.128021	+	0.221739i	0.794889	+	0.606755i	$0.207529\pi$
$798$		0			0
$799$	−539866.	−	935074.i	−0.0299170	−	0.0518178i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.11104e7	+	1.92437e7i	0.608050	+	1.05317i
$804$		0			0
$805$	−1.97700e7	+	3.42427e7i	−1.07527	+	1.86242i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.91492e6			0.478901			0.239451	−	0.970909i	$-0.423033\pi$
$809$	8.91492e6			0.478901			0.239451	+	0.970909i	$0.423033\pi$
$810$		0			0
$811$	−3.29527e6			−0.175930			−0.0879648	−	0.996124i	$-0.528036\pi$
$811$	−3.29527e6			−0.175930			−0.0879648	+	0.996124i	$0.528036\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.60520e6	−	9.70849e6i	0.295595	−	0.511985i
$816$		0			0
$817$	3.90965e6	+	6.77171e6i	0.204919	+	0.354930i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.20912e6	+	3.82631e6i	0.114383	+	0.198117i	0.917533	−	0.397660i	$-0.130178\pi$
$821$	2.20912e6	+	3.82631e6i	0.114383	+	0.198117i	−0.803150	+	0.595777i	$0.796844\pi$
$822$		0			0
$823$	2.79130e6	−	4.83467e6i	0.143650	−	0.248809i	−0.785218	−	0.619219i	$-0.787449\pi$
$823$	2.79130e6	−	4.83467e6i	0.143650	−	0.248809i	0.928869	+	0.370410i	$0.120783\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.18571e7			1.11130			0.555648	−	0.831418i	$-0.312470\pi$
$827$	2.18571e7			1.11130			0.555648	+	0.831418i	$0.312470\pi$
$828$		0			0
$829$	1.89089e7			0.955608			0.477804	−	0.878466i	$-0.341433\pi$
$829$	1.89089e7			0.955608			0.477804	+	0.878466i	$0.341433\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.70363e6	−	2.95078e6i	0.0850675	−	0.147341i
$834$		0			0
$835$	1.33339e7	+	2.30950e7i	0.661822	+	1.14631i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.27488e6	−	9.13636e6i	−0.258707	−	0.448093i	0.707189	−	0.707025i	$-0.249963\pi$
$839$	−5.27488e6	−	9.13636e6i	−0.258707	−	0.448093i	−0.965896	+	0.258931i	$0.916630\pi$
$840$		0			0
$841$	−2.19196e7	+	3.79659e7i	−1.06867	+	1.85099i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.50748e7			−0.726287
$846$		0			0
$847$	−6.32007e7			−3.02701
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.76128e6	−	4.78267e6i	0.130703	−	0.226385i
$852$		0			0
$853$	320767.	+	555585.i	0.0150945	+	0.0261444i	0.873474	−	0.486871i	$-0.161862\pi$
$853$	320767.	+	555585.i	0.0150945	+	0.0261444i	−0.858380	+	0.513015i	$0.828528\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.60855e7	−	2.78610e7i	−0.748141	−	1.29582i	−0.948713	−	0.316139i	$-0.897613\pi$
$857$	−1.60855e7	−	2.78610e7i	−0.748141	−	1.29582i	0.200572	−	0.979679i	$-0.435720\pi$
$858$		0			0
$859$	−723696.	+	1.25348e6i	−0.0334637	+	0.0579608i	−0.882272	−	0.470740i	$-0.843987\pi$
$859$	−723696.	+	1.25348e6i	−0.0334637	+	0.0579608i	0.848809	+	0.528700i	$0.177320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.16338e6			0.418821			0.209411	−	0.977828i	$-0.432845\pi$
$863$	9.16338e6			0.418821			0.209411	+	0.977828i	$0.432845\pi$
$864$		0			0
$865$	2.29889e6			0.104467
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.21034e7	+	3.82843e7i	−0.992911	+	1.71977i
$870$		0			0
$871$	−2.82275e7	−	4.88915e7i	−1.26075	−	2.18367i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.59412e7	+	2.76109e7i	0.703882	+	1.21916i
$876$		0			0
$877$	1.07618e7	−	1.86400e7i	0.472483	−	0.818365i	−0.527021	−	0.849852i	$-0.676691\pi$
$877$	1.07618e7	−	1.86400e7i	0.472483	−	0.818365i	0.999504	+	0.0314872i	$0.0100243\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.58447e7			−1.98998			−0.994991	−	0.0999625i	$-0.968128\pi$
$881$	−4.58447e7			−1.98998			−0.994991	+	0.0999625i	$0.968128\pi$
$882$		0			0
$883$	−3.10719e7			−1.34111			−0.670557	−	0.741858i	$-0.733945\pi$
$883$	−3.10719e7			−1.34111			−0.670557	+	0.741858i	$0.733945\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.41068e6	+	9.37158e6i	−0.230910	+	0.399948i	−0.958076	−	0.286513i	$-0.907504\pi$
$887$	−5.41068e6	+	9.37158e6i	−0.230910	+	0.399948i	0.727166	+	0.686462i	$0.240837\pi$
$888$		0			0
$889$	2.22956e6	+	3.86172e6i	0.0946162	+	0.163880i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.97863e6	−	3.42709e6i	−0.0830302	−	0.143812i
$894$		0			0
$895$	6.38272e6	−	1.10552e7i	0.266347	−	0.461327i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.40967e7			−0.994394
$900$		0			0
$901$	−2.49523e6			−0.102399
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.30682e7	−	3.99553e7i	0.936250	−	1.62163i
$906$		0			0
$907$	−1.85893e7	−	3.21976e7i	−0.750316	−	1.29959i	−0.947670	−	0.319253i	$-0.896568\pi$
$907$	−1.85893e7	−	3.21976e7i	−0.750316	−	1.29959i	0.197354	−	0.980332i	$-0.436765\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.38254e7	+	2.39464e7i	0.551929	+	0.955969i	0.998135	+	0.0610389i	$0.0194414\pi$
$911$	1.38254e7	+	2.39464e7i	0.551929	+	0.955969i	−0.446206	+	0.894930i	$0.647225\pi$
$912$		0			0
$913$	−3.28890e7	+	5.69655e7i	−1.30579	+	2.26170i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.90705e7			−1.53436
$918$		0			0
$919$	2.24727e6			0.0877743			0.0438872	−	0.999036i	$-0.486026\pi$
$919$	2.24727e6			0.0877743			0.0438872	+	0.999036i	$0.486026\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.69556e7	−	4.66884e7i	1.04146	−	1.80387i
$924$		0			0
$925$	148991.	+	258059.i	0.00572539	+	0.00991666i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−395902.	−	685723.i	−0.0150504	−	0.0260681i	0.858402	−	0.512977i	$-0.171458\pi$
$929$	−395902.	−	685723.i	−0.0150504	−	0.0260681i	−0.873453	+	0.486909i	$0.838124\pi$
$930$		0			0
$931$	6.24389e6	−	1.08147e7i	0.236092	−	0.408923i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.32571e6			0.274044
$936$		0			0
$937$	−9.61458e6			−0.357751			−0.178876	−	0.983872i	$-0.557246\pi$
$937$	−9.61458e6			−0.357751			−0.178876	+	0.983872i	$0.557246\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.51978e6	+	1.30246e7i	−0.276842	+	0.479504i	−0.970598	−	0.240706i	$-0.922621\pi$
$941$	−7.51978e6	+	1.30246e7i	−0.276842	+	0.479504i	0.693757	+	0.720210i	$0.255954\pi$
$942$		0			0
$943$	−6.28710e6	−	1.08896e7i	−0.230235	−	0.398778i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.75767e6	+	1.34367e7i	0.281097	+	0.486874i	0.971655	−	0.236402i	$-0.0759684\pi$
$947$	7.75767e6	+	1.34367e7i	0.281097	+	0.486874i	−0.690558	+	0.723277i	$0.742635\pi$
$948$		0			0
$949$	1.25545e7	−	2.17451e7i	0.452518	−	0.783784i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.16599e7			1.84256			0.921280	−	0.388899i	$-0.127145\pi$
$953$	5.16599e7			1.84256			0.921280	+	0.388899i	$0.127145\pi$
$954$		0			0
$955$	−4.06570e6			−0.144254
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.80852e7	−	3.13245e7i	0.635005	−	1.09986i
$960$		0			0
$961$	9.80294e6	+	1.69792e7i	0.342411	+	0.593073i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.32025e7	−	2.28674e7i	−0.456391	−	0.790493i
$966$		0			0
$967$	−3.09797e6	+	5.36585e6i	−0.106540	+	0.184532i	−0.914366	−	0.404888i	$-0.867310\pi$
$967$	−3.09797e6	+	5.36585e6i	−0.106540	+	0.184532i	0.807827	+	0.589420i	$0.200644\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.67480e7			−1.59116			−0.795582	−	0.605846i	$-0.792835\pi$
$971$	−4.67480e7			−1.59116			−0.795582	+	0.605846i	$0.792835\pi$
$972$		0			0
$973$	−4.73001e7			−1.60169
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.31953e6	+	4.01755e6i	−0.0777435	+	0.134656i	−0.902276	−	0.431159i	$-0.858105\pi$
$977$	−2.31953e6	+	4.01755e6i	−0.0777435	+	0.134656i	0.824533	+	0.565815i	$0.191438\pi$
$978$		0			0
$979$	2.42160e7	+	4.19434e7i	0.807507	+	1.39864i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.37725e7	−	2.38546e7i	−0.454598	−	0.787387i	0.544067	−	0.839042i	$-0.316884\pi$
$983$	−1.37725e7	−	2.38546e7i	−0.454598	−	0.787387i	−0.998665	+	0.0516545i	$0.983551\pi$
$984$		0			0
$985$	−1.74600e7	+	3.02416e7i	−0.573395	+	0.993150i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.29209e7			1.39533
$990$		0			0
$991$	1.83035e7			0.592038			0.296019	−	0.955182i	$-0.404341\pi$
$991$	1.83035e7			0.592038			0.296019	+	0.955182i	$0.404341\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.11563e6	+	5.39642e6i	−0.0997672	+	0.172802i
$996$		0			0
$997$	2.22445e7	+	3.85286e7i	0.708736	+	1.22757i	0.965326	+	0.261047i	$0.0840677\pi$
$997$	2.22445e7	+	3.85286e7i	0.708736	+	1.22757i	−0.256590	+	0.966520i	$0.582599\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.e.f.109.2		4
3.2	odd	2		324.6.e.g.109.1		4
9.2	odd	6		324.6.e.g.217.1		4
9.4	even	3		108.6.a.c.1.1	yes	2
9.5	odd	6		108.6.a.b.1.2	✓	2
9.7	even	3	inner	324.6.e.f.217.2		4
36.23	even	6		432.6.a.r.1.2		2
36.31	odd	6		432.6.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.a.b.1.2	✓	2	9.5	odd	6
108.6.a.c.1.1	yes	2	9.4	even	3
324.6.e.f.109.2		4	1.1	even	1	trivial
324.6.e.f.217.2		4	9.7	even	3	inner
324.6.e.g.109.1		4	3.2	odd	2
324.6.e.g.217.1		4	9.2	odd	6
432.6.a.q.1.1		2	36.31	odd	6
432.6.a.r.1.2		2	36.23	even	6

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(28.8141 - 49.9074i) q^{5} +(94.4422 + 163.579i) q^{7} +O(q^{10})\)	\(q+(28.8141 - 49.9074i) q^{5} +(94.4422 + 163.579i) q^{7} +(-352.012 - 609.703i) q^{11} +(-397.769 + 688.956i) q^{13} -180.562 q^{17} -661.769 q^{19} +(-1816.26 + 3145.85i) q^{23} +(-98.0000 - 169.741i) q^{25} +(-4010.93 - 6947.14i) q^{29} +(1501.94 - 2601.43i) q^{31} +10885.0 q^{35} -1520.31 q^{37} +(-1730.78 + 2997.81i) q^{41} +(-5907.88 - 10232.7i) q^{43} +(2989.91 + 5178.68i) q^{47} +(-9435.15 + 16342.2i) q^{49} +13819.2 q^{53} -40571.6 q^{55} +(-11320.1 + 19607.0i) q^{59} +(18702.1 + 32393.0i) q^{61} +(22922.7 + 39703.2i) q^{65} +(-35482.3 + 61457.2i) q^{67} -67767.0 q^{71} -31562.4 q^{73} +(66489.6 - 115163. i) q^{77} +(-31395.8 - 54379.1i) q^{79} +(-46715.7 - 80914.0i) q^{83} +(-5202.73 + 9011.40i) q^{85} -68793.1 q^{89} -150265. q^{91} +(-19068.2 + 33027.2i) q^{95} +(-58887.0 - 101995. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{7}+O(q^{10}) \) 4 * q + 32 * q^7	\( 4 q + 32 q^{7} - 486 q^{11} - 208 q^{13} + 3888 q^{17} - 1264 q^{19} - 6804 q^{23} - 392 q^{25} - 11664 q^{29} - 3328 q^{31} + 39852 q^{35} - 19912 q^{37} - 13608 q^{41} - 4960 q^{43} - 18468 q^{47} - 26676 q^{49} - 23328 q^{53} - 106272 q^{55} - 1944 q^{59} - 8176 q^{61} + 79704 q^{65} - 90064 q^{67} - 89424 q^{71} - 242428 q^{73} + 167184 q^{77} - 28768 q^{79} + 15066 q^{83} - 132840 q^{85} - 357696 q^{89} - 484880 q^{91} - 39852 q^{95} - 88942 q^{97}+O(q^{100}) \) 4 * q + 32 * q^7 - 486 * q^11 - 208 * q^13 + 3888 * q^17 - 1264 * q^19 - 6804 * q^23 - 392 * q^25 - 11664 * q^29 - 3328 * q^31 + 39852 * q^35 - 19912 * q^37 - 13608 * q^41 - 4960 * q^43 - 18468 * q^47 - 26676 * q^49 - 23328 * q^53 - 106272 * q^55 - 1944 * q^59 - 8176 * q^61 + 79704 * q^65 - 90064 * q^67 - 89424 * q^71 - 242428 * q^73 + 167184 * q^77 - 28768 * q^79 + 15066 * q^83 - 132840 * q^85 - 357696 * q^89 - 484880 * q^91 - 39852 * q^95 - 88942 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more