LMFDB - Embedded newform 324.3.c.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(161,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, 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:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.3.c.a.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.79555i q^{5} -6.19615 q^{7} +O(q^{10})$	$q-5.79555i q^{5} -6.19615 q^{7} -1.13681i q^{11} -11.3923 q^{13} +31.2514i q^{17} -32.9808 q^{19} -33.6365i q^{23} -8.58846 q^{25} +25.5673i q^{29} -23.1769 q^{31} +35.9101i q^{35} +8.80385 q^{37} -70.6835i q^{41} -59.7654 q^{43} -60.2292i q^{47} -10.6077 q^{49} -19.7718i q^{53} -6.58846 q^{55} +83.4114i q^{59} +30.7654 q^{61} +66.0247i q^{65} -10.5885 q^{67} +3.63342i q^{71} -17.2346 q^{73} +7.04386i q^{77} +113.373 q^{79} -11.8141i q^{83} +181.119 q^{85} -111.252i q^{89} +70.5885 q^{91} +191.142i q^{95} +74.3154 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} - 4 q^{13} - 28 q^{19} + 28 q^{25} + 32 q^{31} + 56 q^{37} - 52 q^{43} - 84 q^{49} + 36 q^{55} - 64 q^{61} + 20 q^{67} - 256 q^{73} + 308 q^{79} + 288 q^{85} + 220 q^{91} - 160 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^13 - 28 * q^19 + 28 * q^25 + 32 * q^31 + 56 * q^37 - 52 * q^43 - 84 * q^49 + 36 * q^55 - 64 * q^61 + 20 * q^67 - 256 * q^73 + 308 * q^79 + 288 * q^85 + 220 * q^91 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$-1$

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
$5$	− 5.79555i	− 1.15911i	−0.814933	−	0.579555i	$-0.803226\pi$
$5$	− 5.79555i	− 1.15911i	0.814933	−	0.579555i	$-0.196774\pi$
$6$	0	0
$7$	−6.19615	−0.885165	−0.442582	−	0.896728i	$-0.645938\pi$
$7$	−6.19615	−0.885165	−0.442582	+	0.896728i	$0.645938\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.13681i	− 0.103347i	−0.998664	−	0.0516733i	$-0.983545\pi$
$11$	− 1.13681i	− 0.103347i	0.998664	−	0.0516733i	$-0.0164554\pi$
$12$	0	0
$13$	−11.3923	−0.876331	−0.438166	−	0.898894i	$-0.644372\pi$
$13$	−11.3923	−0.876331	−0.438166	+	0.898894i	$0.644372\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	31.2514i	1.83832i	0.393887	+	0.919159i	$0.371130\pi$
$17$	31.2514i	1.83832i	−0.393887	+	0.919159i	$0.628870\pi$
$18$	0	0
$19$	−32.9808	−1.73583	−0.867915	−	0.496713i	$-0.834540\pi$
$19$	−32.9808	−1.73583	−0.867915	+	0.496713i	$0.834540\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 33.6365i	− 1.46246i	−0.682132	−	0.731229i	$-0.738947\pi$
$23$	− 33.6365i	− 1.46246i	0.682132	−	0.731229i	$-0.261053\pi$
$24$	0	0
$25$	−8.58846	−0.343538
$26$	0	0
$27$	0	0
$28$	0	0
$29$	25.5673i	0.881632i	0.897597	+	0.440816i	$0.145311\pi$
$29$	25.5673i	0.881632i	−0.897597	+	0.440816i	$0.854689\pi$
$30$	0	0
$31$	−23.1769	−0.747642	−0.373821	−	0.927501i	$-0.621953\pi$
$31$	−23.1769	−0.747642	−0.373821	+	0.927501i	$0.621953\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	35.9101i	1.02600i
$36$	0	0
$37$	8.80385	0.237942	0.118971	−	0.992898i	$-0.462040\pi$
$37$	8.80385	0.237942	0.118971	+	0.992898i	$0.462040\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 70.6835i	− 1.72399i	−0.506919	−	0.861994i	$-0.669216\pi$
$41$	− 70.6835i	− 1.72399i	0.506919	−	0.861994i	$-0.330784\pi$
$42$	0	0
$43$	−59.7654	−1.38989	−0.694946	−	0.719062i	$-0.744572\pi$
$43$	−59.7654	−1.38989	−0.694946	+	0.719062i	$0.744572\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 60.2292i	− 1.28147i	−0.767761	−	0.640736i	$-0.778629\pi$
$47$	− 60.2292i	− 1.28147i	0.767761	−	0.640736i	$-0.221371\pi$
$48$	0	0
$49$	−10.6077	−0.216484
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 19.7718i	− 0.373053i	−0.982450	−	0.186526i	$-0.940277\pi$
$53$	− 19.7718i	− 0.373053i	0.982450	−	0.186526i	$-0.0597229\pi$
$54$	0	0
$55$	−6.58846	−0.119790
$56$	0	0
$57$	0	0
$58$	0	0
$59$	83.4114i	1.41375i	0.707337	+	0.706876i	$0.249896\pi$
$59$	83.4114i	1.41375i	−0.707337	+	0.706876i	$0.750104\pi$
$60$	0	0
$61$	30.7654	0.504350	0.252175	−	0.967682i	$-0.418854\pi$
$61$	30.7654	0.504350	0.252175	+	0.967682i	$0.418854\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	66.0247i	1.01577i
$66$	0	0
$67$	−10.5885	−0.158037	−0.0790183	−	0.996873i	$-0.525179\pi$
$67$	−10.5885	−0.158037	−0.0790183	+	0.996873i	$0.525179\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.63342i	0.0511750i	0.999673	+	0.0255875i	$0.00814564\pi$
$71$	3.63342i	0.0511750i	−0.999673	+	0.0255875i	$0.991854\pi$
$72$	0	0
$73$	−17.2346	−0.236091	−0.118045	−	0.993008i	$-0.537663\pi$
$73$	−17.2346	−0.236091	−0.118045	+	0.993008i	$0.537663\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.04386i	0.0914787i
$78$	0	0
$79$	113.373	1.43510	0.717551	−	0.696506i	$-0.245263\pi$
$79$	113.373	1.43510	0.717551	+	0.696506i	$0.245263\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 11.8141i	− 0.142339i	−0.997464	−	0.0711693i	$-0.977327\pi$
$83$	− 11.8141i	− 0.142339i	0.997464	−	0.0711693i	$-0.0226730\pi$
$84$	0	0
$85$	181.119	2.13081
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 111.252i	− 1.25003i	−0.780614	−	0.625013i	$-0.785094\pi$
$89$	− 111.252i	− 1.25003i	0.780614	−	0.625013i	$-0.214906\pi$
$90$	0	0
$91$	70.5885	0.775697
$92$	0	0
$93$	0	0
$94$	0	0
$95$	191.142i	2.01202i
$96$	0	0
$97$	74.3154	0.766138	0.383069	−	0.923720i	$-0.374867\pi$
$97$	74.3154	0.766138	0.383069	+	0.923720i	$0.374867\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.2752i	0.171041i	0.996336	+	0.0855206i	$0.0272554\pi$
$101$	17.2752i	0.171041i	−0.996336	+	0.0855206i	$0.972745\pi$
$102$	0	0
$103$	−112.000	−1.08738	−0.543689	−	0.839287i	$-0.682973\pi$
$103$	−112.000	−1.08738	−0.543689	+	0.839287i	$0.682973\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 141.144i	− 1.31910i	−0.751660	−	0.659551i	$-0.770746\pi$
$107$	− 141.144i	− 1.31910i	0.751660	−	0.659551i	$-0.229254\pi$
$108$	0	0
$109$	171.708	1.57530	0.787650	−	0.616123i	$-0.211298\pi$
$109$	171.708	1.57530	0.787650	+	0.616123i	$0.211298\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.57257i	− 0.0493147i	−0.999696	−	0.0246574i	$-0.992151\pi$
$113$	− 5.57257i	− 0.0493147i	0.999696	−	0.0246574i	$-0.00784948\pi$
$114$	0	0
$115$	−194.942	−1.69515
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 193.638i	− 1.62721i
$120$	0	0
$121$	119.708	0.989319
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 95.1140i	− 0.760912i
$126$	0	0
$127$	−68.2346	−0.537281	−0.268640	−	0.963241i	$-0.586574\pi$
$127$	−68.2346	−0.537281	−0.268640	+	0.963241i	$0.586574\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	154.095i	1.17630i	0.808753	+	0.588148i	$0.200143\pi$
$131$	154.095i	1.17630i	−0.808753	+	0.588148i	$0.799857\pi$
$132$	0	0
$133$	204.354	1.53649
$134$	0	0
$135$	0	0
$136$	0	0
$137$	76.4790i	0.558241i	0.960256	+	0.279121i	$0.0900429\pi$
$137$	76.4790i	0.558241i	−0.960256	+	0.279121i	$0.909957\pi$
$138$	0	0
$139$	94.0385	0.676536	0.338268	−	0.941050i	$-0.390159\pi$
$139$	94.0385	0.676536	0.338268	+	0.941050i	$0.390159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.9509i	0.0905658i
$144$	0	0
$145$	148.177	1.02191
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 67.1615i	− 0.450749i	−0.974272	−	0.225374i	$-0.927640\pi$
$149$	− 67.1615i	− 0.450749i	0.974272	−	0.225374i	$-0.0723605\pi$
$150$	0	0
$151$	46.3538	0.306979	0.153490	−	0.988150i	$-0.450949\pi$
$151$	46.3538	0.306979	0.153490	+	0.988150i	$0.450949\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	134.323i	0.866601i
$156$	0	0
$157$	−169.588	−1.08018	−0.540091	−	0.841607i	$-0.681610\pi$
$157$	−169.588	−1.08018	−0.540091	+	0.841607i	$0.681610\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	208.417i	1.29452i
$162$	0	0
$163$	−243.023	−1.49094	−0.745469	−	0.666540i	$-0.767775\pi$
$163$	−243.023	−1.49094	−0.745469	+	0.666540i	$0.767775\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 112.724i	− 0.674992i	−0.941327	−	0.337496i	$-0.890420\pi$
$167$	− 112.724i	− 0.674992i	0.941327	−	0.337496i	$-0.109580\pi$
$168$	0	0
$169$	−39.2154	−0.232044
$170$	0	0
$171$	0	0
$172$	0	0
$173$	187.843i	1.08580i	0.839798	+	0.542898i	$0.182673\pi$
$173$	187.843i	1.08580i	−0.839798	+	0.542898i	$0.817327\pi$
$174$	0	0
$175$	53.2154	0.304088
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 16.3613i	− 0.0914042i	−0.998955	−	0.0457021i	$-0.985448\pi$
$179$	− 16.3613i	− 0.0914042i	0.998955	−	0.0457021i	$-0.0145525\pi$
$180$	0	0
$181$	−227.608	−1.25750	−0.628751	−	0.777607i	$-0.716433\pi$
$181$	−227.608	−1.25750	−0.628751	+	0.777607i	$0.716433\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 51.0232i	− 0.275801i
$186$	0	0
$187$	35.5270	0.189984
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 364.562i	− 1.90870i	−0.298684	−	0.954352i	$-0.596548\pi$
$191$	− 364.562i	− 1.90870i	0.298684	−	0.954352i	$-0.403452\pi$
$192$	0	0
$193$	−24.8846	−0.128936	−0.0644678	−	0.997920i	$-0.520535\pi$
$193$	−24.8846	−0.128936	−0.0644678	+	0.997920i	$0.520535\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	171.259i	0.869333i	0.900592	+	0.434666i	$0.143134\pi$
$197$	171.259i	0.869333i	−0.900592	+	0.434666i	$0.856866\pi$
$198$	0	0
$199$	122.431	0.615230	0.307615	−	0.951511i	$-0.400469\pi$
$199$	122.431	0.615230	0.307615	+	0.951511i	$0.400469\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 158.419i	− 0.780390i
$204$	0	0
$205$	−409.650	−1.99829
$206$	0	0
$207$	0	0
$208$	0	0
$209$	37.4929i	0.179392i
$210$	0	0
$211$	−372.512	−1.76546	−0.882729	−	0.469883i	$-0.844296\pi$
$211$	−372.512	−1.76546	−0.882729	+	0.469883i	$0.844296\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	346.373i	1.61104i
$216$	0	0
$217$	143.608	0.661787
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 356.025i	− 1.61097i
$222$	0	0
$223$	−320.119	−1.43551	−0.717756	−	0.696295i	$-0.754831\pi$
$223$	−320.119	−1.43551	−0.717756	+	0.696295i	$0.754831\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.2245i	0.0670684i	0.999438	+	0.0335342i	$0.0106763\pi$
$227$	15.2245i	0.0670684i	−0.999438	+	0.0335342i	$0.989324\pi$
$228$	0	0
$229$	−179.708	−0.784750	−0.392375	−	0.919805i	$-0.628346\pi$
$229$	−179.708	−0.784750	−0.392375	+	0.919805i	$0.628346\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 150.796i	− 0.647193i	−0.946195	−	0.323596i	$-0.895108\pi$
$233$	− 150.796i	− 0.647193i	0.946195	−	0.323596i	$-0.104892\pi$
$234$	0	0
$235$	−349.061	−1.48537
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 264.545i	− 1.10688i	−0.832888	−	0.553441i	$-0.813314\pi$
$239$	− 264.545i	− 1.10688i	0.832888	−	0.553441i	$-0.186686\pi$
$240$	0	0
$241$	228.450	0.947925	0.473963	−	0.880545i	$-0.342823\pi$
$241$	228.450	0.947925	0.473963	+	0.880545i	$0.342823\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	61.4775i	0.250928i
$246$	0	0
$247$	375.727	1.52116
$248$	0	0
$249$	0	0
$250$	0	0
$251$	47.2783i	0.188360i	0.995555	+	0.0941798i	$0.0300229\pi$
$251$	47.2783i	0.188360i	−0.995555	+	0.0941798i	$0.969977\pi$
$252$	0	0
$253$	−38.2384	−0.151140
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 25.7903i	− 0.100351i	−0.998740	−	0.0501757i	$-0.984022\pi$
$257$	− 25.7903i	− 0.100351i	0.998740	−	0.0501757i	$-0.0159781\pi$
$258$	0	0
$259$	−54.5500	−0.210618
$260$	0	0
$261$	0	0
$262$	0	0
$263$	285.230i	1.08453i	0.840209	+	0.542263i	$0.182432\pi$
$263$	285.230i	1.08453i	−0.840209	+	0.542263i	$0.817568\pi$
$264$	0	0
$265$	−114.588	−0.432409
$266$	0	0
$267$	0	0
$268$	0	0
$269$	205.341i	0.763350i	0.924297	+	0.381675i	$0.124653\pi$
$269$	205.341i	0.763350i	−0.924297	+	0.381675i	$0.875347\pi$
$270$	0	0
$271$	282.004	1.04060	0.520302	−	0.853982i	$-0.325819\pi$
$271$	282.004	1.04060	0.520302	+	0.853982i	$0.325819\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.76346i	0.0355035i
$276$	0	0
$277$	48.7077	0.175840	0.0879200	−	0.996128i	$-0.471978\pi$
$277$	48.7077	0.175840	0.0879200	+	0.996128i	$0.471978\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 246.712i	− 0.877980i	−0.898492	−	0.438990i	$-0.855336\pi$
$281$	− 246.712i	− 0.877980i	0.898492	−	0.438990i	$-0.144664\pi$
$282$	0	0
$283$	−352.946	−1.24716	−0.623580	−	0.781760i	$-0.714322\pi$
$283$	−352.946	−1.24716	−0.623580	+	0.781760i	$0.714322\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	437.966i	1.52601i
$288$	0	0
$289$	−687.650	−2.37941
$290$	0	0
$291$	0	0
$292$	0	0
$293$	279.658i	0.954464i	0.878777	+	0.477232i	$0.158360\pi$
$293$	279.658i	0.954464i	−0.878777	+	0.477232i	$0.841640\pi$
$294$	0	0
$295$	483.415	1.63870
$296$	0	0
$297$	0	0
$298$	0	0
$299$	383.197i	1.28160i
$300$	0	0
$301$	370.315	1.23028
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 178.302i	− 0.584598i
$306$	0	0
$307$	234.708	0.764520	0.382260	−	0.924055i	$-0.375146\pi$
$307$	234.708	0.764520	0.382260	+	0.924055i	$0.375146\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	235.700i	0.757879i	0.925421	+	0.378940i	$0.123711\pi$
$311$	235.700i	0.757879i	−0.925421	+	0.378940i	$0.876289\pi$
$312$	0	0
$313$	−357.669	−1.14271	−0.571357	−	0.820702i	$-0.693583\pi$
$313$	−357.669	−1.14271	−0.571357	+	0.820702i	$0.693583\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 603.986i	− 1.90532i	−0.304040	−	0.952659i	$-0.598336\pi$
$317$	− 603.986i	− 1.90532i	0.304040	−	0.952659i	$-0.401664\pi$
$318$	0	0
$319$	29.0653	0.0911137
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1030.69i	− 3.19101i
$324$	0	0
$325$	97.8423	0.301053
$326$	0	0
$327$	0	0
$328$	0	0
$329$	373.189i	1.13431i
$330$	0	0
$331$	−36.7424	−0.111004	−0.0555021	−	0.998459i	$-0.517676\pi$
$331$	−36.7424	−0.111004	−0.0555021	+	0.998459i	$0.517676\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	61.3660i	0.183182i
$336$	0	0
$337$	168.277	0.499338	0.249669	−	0.968331i	$-0.419678\pi$
$337$	168.277	0.499338	0.249669	+	0.968331i	$0.419678\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.3478i	0.0772663i
$342$	0	0
$343$	369.338	1.07679
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 176.608i	− 0.508957i	−0.967078	−	0.254479i	$-0.918096\pi$
$347$	− 176.608i	− 0.508957i	0.967078	−	0.254479i	$-0.0819038\pi$
$348$	0	0
$349$	131.215	0.375975	0.187988	−	0.982171i	$-0.439804\pi$
$349$	131.215	0.375975	0.187988	+	0.982171i	$0.439804\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 13.6199i	− 0.0385832i	−0.999814	−	0.0192916i	$-0.993859\pi$
$353$	− 13.6199i	− 0.0385832i	0.999814	−	0.0192916i	$-0.00614109\pi$
$354$	0	0
$355$	21.0577	0.0593175
$356$	0	0
$357$	0	0
$358$	0	0
$359$	96.5853i	0.269040i	0.990911	+	0.134520i	$0.0429492\pi$
$359$	96.5853i	0.269040i	−0.990911	+	0.134520i	$0.957051\pi$
$360$	0	0
$361$	726.731	2.01310
$362$	0	0
$363$	0	0
$364$	0	0
$365$	99.8842i	0.273655i
$366$	0	0
$367$	254.946	0.694676	0.347338	−	0.937740i	$-0.387086\pi$
$367$	254.946	0.694676	0.347338	+	0.937740i	$0.387086\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	122.509i	0.330213i
$372$	0	0
$373$	−91.1000	−0.244236	−0.122118	−	0.992516i	$-0.538969\pi$
$373$	−91.1000	−0.244236	−0.122118	+	0.992516i	$0.538969\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 291.271i	− 0.772602i
$378$	0	0
$379$	193.454	0.510432	0.255216	−	0.966884i	$-0.417853\pi$
$379$	193.454	0.510432	0.255216	+	0.966884i	$0.417853\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	332.955i	0.869334i	0.900591	+	0.434667i	$0.143134\pi$
$383$	332.955i	0.869334i	−0.900591	+	0.434667i	$0.856866\pi$
$384$	0	0
$385$	40.8231	0.106034
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 41.1264i	− 0.105723i	−0.998602	−	0.0528616i	$-0.983166\pi$
$389$	− 41.1264i	− 0.105723i	0.998602	−	0.0528616i	$-0.0168342\pi$
$390$	0	0
$391$	1051.19	2.68846
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 657.060i	− 1.66344i
$396$	0	0
$397$	−176.881	−0.445544	−0.222772	−	0.974871i	$-0.571510\pi$
$397$	−176.881	−0.445544	−0.222772	+	0.974871i	$0.571510\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	437.185i	1.09024i	0.838359	+	0.545119i	$0.183515\pi$
$401$	437.185i	1.09024i	−0.838359	+	0.545119i	$0.816485\pi$
$402$	0	0
$403$	264.038	0.655182
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.0083i	− 0.0245905i
$408$	0	0
$409$	−145.904	−0.356733	−0.178367	−	0.983964i	$-0.557081\pi$
$409$	−145.904	−0.356733	−0.178367	+	0.983964i	$0.557081\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 516.830i	− 1.25140i
$414$	0	0
$415$	−68.4693	−0.164986
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 133.431i	− 0.318451i	−0.987242	−	0.159226i	$-0.949100\pi$
$419$	− 133.431i	− 0.318451i	0.987242	−	0.159226i	$-0.0508998\pi$
$420$	0	0
$421$	288.692	0.685730	0.342865	−	0.939385i	$-0.388603\pi$
$421$	288.692	0.685730	0.342865	+	0.939385i	$0.388603\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 268.401i	− 0.631532i
$426$	0	0
$427$	−190.627	−0.446433
$428$	0	0
$429$	0	0
$430$	0	0
$431$	669.565i	1.55351i	0.629800	+	0.776757i	$0.283137\pi$
$431$	669.565i	1.55351i	−0.629800	+	0.776757i	$0.716863\pi$
$432$	0	0
$433$	−510.654	−1.17934	−0.589669	−	0.807645i	$-0.700742\pi$
$433$	−510.654	−1.17934	−0.589669	+	0.807645i	$0.700742\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1109.36i	2.53858i
$438$	0	0
$439$	77.2999	0.176082	0.0880409	−	0.996117i	$-0.471939\pi$
$439$	77.2999	0.176082	0.0880409	+	0.996117i	$0.471939\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	759.351i	1.71411i	0.515224	+	0.857055i	$0.327709\pi$
$443$	759.351i	1.71411i	−0.515224	+	0.857055i	$0.672291\pi$
$444$	0	0
$445$	−644.769	−1.44892
$446$	0	0
$447$	0	0
$448$	0	0
$449$	541.149i	1.20523i	0.798032	+	0.602616i	$0.205875\pi$
$449$	541.149i	1.20523i	−0.798032	+	0.602616i	$0.794125\pi$
$450$	0	0
$451$	−80.3538	−0.178168
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 409.099i	− 0.899119i
$456$	0	0
$457$	−55.5885	−0.121638	−0.0608189	−	0.998149i	$-0.519371\pi$
$457$	−55.5885	−0.121638	−0.0608189	+	0.998149i	$0.519371\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 250.702i	− 0.543822i	−0.962322	−	0.271911i	$-0.912344\pi$
$461$	− 250.702i	− 0.543822i	0.962322	−	0.271911i	$-0.0876557\pi$
$462$	0	0
$463$	571.023	1.23331	0.616656	−	0.787233i	$-0.288487\pi$
$463$	571.023	1.23331	0.616656	+	0.787233i	$0.288487\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 72.2663i	− 0.154746i	−0.997002	−	0.0773729i	$-0.975347\pi$
$467$	− 72.2663i	− 0.154746i	0.997002	−	0.0773729i	$-0.0246532\pi$
$468$	0	0
$469$	65.6077	0.139888
$470$	0	0
$471$	0	0
$472$	0	0
$473$	67.9420i	0.143641i
$474$	0	0
$475$	283.254	0.596324
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 707.526i	− 1.47709i	−0.674205	−	0.738544i	$-0.735514\pi$
$479$	− 707.526i	− 1.47709i	0.674205	−	0.738544i	$-0.264486\pi$
$480$	0	0
$481$	−100.296	−0.208516
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 430.699i	− 0.888039i
$486$	0	0
$487$	360.908	0.741083	0.370542	−	0.928816i	$-0.379172\pi$
$487$	360.908	0.741083	0.370542	+	0.928816i	$0.379172\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 680.019i	− 1.38497i	−0.721433	−	0.692484i	$-0.756516\pi$
$491$	− 680.019i	− 1.38497i	0.721433	−	0.692484i	$-0.243484\pi$
$492$	0	0
$493$	−799.015	−1.62072
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 22.5133i	− 0.0452983i
$498$	0	0
$499$	236.704	0.474356	0.237178	−	0.971466i	$-0.423777\pi$
$499$	236.704	0.474356	0.237178	+	0.971466i	$0.423777\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 95.6715i	− 0.190202i	−0.995468	−	0.0951009i	$-0.969683\pi$
$503$	− 95.6715i	− 0.190202i	0.995468	−	0.0951009i	$-0.0303174\pi$
$504$	0	0
$505$	100.119	0.198256
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 435.224i	− 0.855057i	−0.904002	−	0.427529i	$-0.859384\pi$
$509$	− 435.224i	− 0.855057i	0.904002	−	0.427529i	$-0.140616\pi$
$510$	0	0
$511$	106.788	0.208979
$512$	0	0
$513$	0	0
$514$	0	0
$515$	649.102i	1.26039i
$516$	0	0
$517$	−68.4693	−0.132436
$518$	0	0
$519$	0	0
$520$	0	0
$521$	794.771i	1.52547i	0.646709	+	0.762737i	$0.276145\pi$
$521$	794.771i	1.52547i	−0.646709	+	0.762737i	$0.723855\pi$
$522$	0	0
$523$	−594.481	−1.13667	−0.568337	−	0.822796i	$-0.692413\pi$
$523$	−594.481	−1.13667	−0.568337	+	0.822796i	$0.692413\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 724.311i	− 1.37440i
$528$	0	0
$529$	−602.415	−1.13878
$530$	0	0
$531$	0	0
$532$	0	0
$533$	805.248i	1.51078i
$534$	0	0
$535$	−818.008	−1.52899
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0590i	0.0223728i
$540$	0	0
$541$	219.508	0.405744	0.202872	−	0.979205i	$-0.434972\pi$
$541$	219.508	0.405744	0.202872	+	0.979205i	$0.434972\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 995.141i	− 1.82595i
$546$	0	0
$547$	−584.592	−1.06872	−0.534362	−	0.845256i	$-0.679448\pi$
$547$	−584.592	−1.06872	−0.534362	+	0.845256i	$0.679448\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 843.230i	− 1.53036i
$552$	0	0
$553$	−702.477	−1.27030
$554$	0	0
$555$	0	0
$556$	0	0
$557$	597.187i	1.07215i	0.844171	+	0.536075i	$0.180093\pi$
$557$	597.187i	1.07215i	−0.844171	+	0.536075i	$0.819907\pi$
$558$	0	0
$559$	680.865	1.21801
$560$	0	0
$561$	0	0
$562$	0	0
$563$	360.216i	0.639816i	0.947449	+	0.319908i	$0.103652\pi$
$563$	360.216i	0.639816i	−0.947449	+	0.319908i	$0.896348\pi$
$564$	0	0
$565$	−32.2961	−0.0571613
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 561.723i	− 0.987211i	−0.869686	−	0.493605i	$-0.835679\pi$
$569$	− 561.723i	− 0.987211i	0.869686	−	0.493605i	$-0.164321\pi$
$570$	0	0
$571$	−38.0770	−0.0666847	−0.0333423	−	0.999444i	$-0.510615\pi$
$571$	−38.0770	−0.0666847	−0.0333423	+	0.999444i	$0.510615\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	288.886i	0.502410i
$576$	0	0
$577$	−747.008	−1.29464	−0.647320	−	0.762218i	$-0.724110\pi$
$577$	−747.008	−1.29464	−0.647320	+	0.762218i	$0.724110\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	73.2020i	0.125993i
$582$	0	0
$583$	−22.4768	−0.0385537
$584$	0	0
$585$	0	0
$586$	0	0
$587$	599.773i	1.02176i	0.859652	+	0.510880i	$0.170680\pi$
$587$	599.773i	1.02176i	−0.859652	+	0.510880i	$0.829320\pi$
$588$	0	0
$589$	764.392	1.29778
$590$	0	0
$591$	0	0
$592$	0	0
$593$	212.184i	0.357814i	0.983866	+	0.178907i	$0.0572561\pi$
$593$	212.184i	0.357814i	−0.983866	+	0.178907i	$0.942744\pi$
$594$	0	0
$595$	−1122.24	−1.88612
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 325.220i	− 0.542938i	−0.962447	−	0.271469i	$-0.912491\pi$
$599$	− 325.220i	− 0.542938i	0.962447	−	0.271469i	$-0.0875095\pi$
$600$	0	0
$601$	747.477	1.24372	0.621861	−	0.783128i	$-0.286377\pi$
$601$	747.477	1.24372	0.621861	+	0.783128i	$0.286377\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 693.772i	− 1.14673i
$606$	0	0
$607$	−123.727	−0.203833	−0.101917	−	0.994793i	$-0.532498\pi$
$607$	−123.727	−0.203833	−0.101917	+	0.994793i	$0.532498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	686.149i	1.12299i
$612$	0	0
$613$	−954.008	−1.55629	−0.778146	−	0.628083i	$-0.783840\pi$
$613$	−954.008	−1.55629	−0.778146	+	0.628083i	$0.783840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 250.123i	− 0.405385i	−0.979242	−	0.202693i	$-0.935031\pi$
$617$	− 250.123i	− 0.405385i	0.979242	−	0.202693i	$-0.0649692\pi$
$618$	0	0
$619$	−1017.54	−1.64384	−0.821921	−	0.569601i	$-0.807098\pi$
$619$	−1017.54	−1.64384	−0.821921	+	0.569601i	$0.807098\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	689.337i	1.10648i
$624$	0	0
$625$	−765.950	−1.22552
$626$	0	0
$627$	0	0
$628$	0	0
$629$	275.133i	0.437413i
$630$	0	0
$631$	−75.6001	−0.119810	−0.0599050	−	0.998204i	$-0.519080\pi$
$631$	−75.6001	−0.119810	−0.0599050	+	0.998204i	$0.519080\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	395.458i	0.622768i
$636$	0	0
$637$	120.846	0.189711
$638$	0	0
$639$	0	0
$640$	0	0
$641$	242.366i	0.378106i	0.981967	+	0.189053i	$0.0605418\pi$
$641$	242.366i	0.378106i	−0.981967	+	0.189053i	$0.939458\pi$
$642$	0	0
$643$	−733.184	−1.14026	−0.570128	−	0.821556i	$-0.693106\pi$
$643$	−733.184	−1.14026	−0.570128	+	0.821556i	$0.693106\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1177.72i	1.82028i	0.414296	+	0.910142i	$0.364028\pi$
$647$	1177.72i	1.82028i	−0.414296	+	0.910142i	$0.635972\pi$
$648$	0	0
$649$	94.8231	0.146106
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 635.259i	− 0.972832i	−0.873727	−	0.486416i	$-0.838304\pi$
$653$	− 635.259i	− 0.972832i	0.873727	−	0.486416i	$-0.161696\pi$
$654$	0	0
$655$	893.065	1.36346
$656$	0	0
$657$	0	0
$658$	0	0
$659$	916.389i	1.39057i	0.718732	+	0.695287i	$0.244723\pi$
$659$	916.389i	1.39057i	−0.718732	+	0.695287i	$0.755277\pi$
$660$	0	0
$661$	413.358	0.625352	0.312676	−	0.949860i	$-0.398775\pi$
$661$	413.358	0.625352	0.312676	+	0.949860i	$0.398775\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1184.34i	− 1.78097i
$666$	0	0
$667$	859.996	1.28935
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 34.9744i	− 0.0521229i
$672$	0	0
$673$	−732.454	−1.08834	−0.544171	−	0.838975i	$-0.683156\pi$
$673$	−732.454	−1.08834	−0.544171	+	0.838975i	$0.683156\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 585.686i	− 0.865119i	−0.901605	−	0.432559i	$-0.857611\pi$
$677$	− 585.686i	− 0.865119i	0.901605	−	0.432559i	$-0.142389\pi$
$678$	0	0
$679$	−460.469	−0.678158
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1082.97i	− 1.58560i	−0.609480	−	0.792801i	$-0.708622\pi$
$683$	− 1082.97i	− 1.58560i	0.609480	−	0.792801i	$-0.291378\pi$
$684$	0	0
$685$	443.238	0.647063
$686$	0	0
$687$	0	0
$688$	0	0
$689$	225.246i	0.326918i
$690$	0	0
$691$	−178.358	−0.258115	−0.129058	−	0.991637i	$-0.541195\pi$
$691$	−178.358	−0.258115	−0.129058	+	0.991637i	$0.541195\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 545.005i	− 0.784180i
$696$	0	0
$697$	2208.96	3.16924
$698$	0	0
$699$	0	0
$700$	0	0
$701$	638.090i	0.910257i	0.890426	+	0.455129i	$0.150407\pi$
$701$	638.090i	0.910257i	−0.890426	+	0.455129i	$0.849593\pi$
$702$	0	0
$703$	−290.358	−0.413026
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 107.040i	− 0.151400i
$708$	0	0
$709$	824.846	1.16339	0.581697	−	0.813406i	$-0.302389\pi$
$709$	824.846	1.16339	0.581697	+	0.813406i	$0.302389\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	779.591i	1.09340i
$714$	0	0
$715$	75.0577	0.104976
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 748.451i	− 1.04096i	−0.853874	−	0.520480i	$-0.825753\pi$
$719$	− 748.451i	− 1.04096i	0.853874	−	0.520480i	$-0.174247\pi$
$720$	0	0
$721$	693.969	0.962509
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 219.584i	− 0.302874i
$726$	0	0
$727$	26.3576	0.0362553	0.0181276	−	0.999836i	$-0.494229\pi$
$727$	26.3576	0.0362553	0.0181276	+	0.999836i	$0.494229\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1867.75i	− 2.55506i
$732$	0	0
$733$	−1072.95	−1.46377	−0.731887	−	0.681426i	$-0.761360\pi$
$733$	−1072.95	−1.46377	−0.731887	+	0.681426i	$0.761360\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0371i	0.0163325i
$738$	0	0
$739$	158.831	0.214926	0.107463	−	0.994209i	$-0.465727\pi$
$739$	158.831	0.214926	0.107463	+	0.994209i	$0.465727\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	164.059i	0.220807i	0.993887	+	0.110403i	$0.0352143\pi$
$743$	164.059i	0.220807i	−0.993887	+	0.110403i	$0.964786\pi$
$744$	0	0
$745$	−389.238	−0.522468
$746$	0	0
$747$	0	0
$748$	0	0
$749$	874.549i	1.16762i
$750$	0	0
$751$	232.396	0.309449	0.154724	−	0.987958i	$-0.450551\pi$
$751$	232.396	0.309449	0.154724	+	0.987958i	$0.450551\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 268.646i	− 0.355823i
$756$	0	0
$757$	1012.72	1.33781	0.668905	−	0.743348i	$-0.266763\pi$
$757$	1012.72	1.33781	0.668905	+	0.743348i	$0.266763\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	578.998i	0.760838i	0.924814	+	0.380419i	$0.124220\pi$
$761$	578.998i	0.760838i	−0.924814	+	0.380419i	$0.875780\pi$
$762$	0	0
$763$	−1063.93	−1.39440
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 950.248i	− 1.23892i
$768$	0	0
$769$	−267.269	−0.347554	−0.173777	−	0.984785i	$-0.555597\pi$
$769$	−267.269	−0.347554	−0.173777	+	0.984785i	$0.555597\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1356.98i	− 1.75548i	−0.479140	−	0.877739i	$-0.659051\pi$
$773$	− 1356.98i	− 1.75548i	0.479140	−	0.877739i	$-0.340949\pi$
$774$	0	0
$775$	199.054	0.256844
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2331.19i	2.99255i
$780$	0	0
$781$	4.13052	0.00528876
$782$	0	0
$783$	0	0
$784$	0	0
$785$	982.859i	1.25205i
$786$	0	0
$787$	57.8808	0.0735461	0.0367731	−	0.999324i	$-0.488292\pi$
$787$	57.8808	0.0735461	0.0367731	+	0.999324i	$0.488292\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	34.5285i	0.0436517i
$792$	0	0
$793$	−350.488	−0.441978
$794$	0	0
$795$	0	0
$796$	0	0
$797$	164.884i	0.206880i	0.994636	+	0.103440i	$0.0329850\pi$
$797$	164.884i	0.206880i	−0.994636	+	0.103440i	$0.967015\pi$
$798$	0	0
$799$	1882.25	2.35575
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.5925i	0.0243992i
$804$	0	0
$805$	1207.89	1.50049
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 981.255i	− 1.21292i	−0.795113	−	0.606461i	$-0.792588\pi$
$809$	− 981.255i	− 1.21292i	0.795113	−	0.606461i	$-0.207412\pi$
$810$	0	0
$811$	−289.877	−0.357432	−0.178716	−	0.983901i	$-0.557194\pi$
$811$	−289.877	−0.357432	−0.178716	+	0.983901i	$0.557194\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1408.45i	1.72816i
$816$	0	0
$817$	1971.11	2.41262
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 132.139i	− 0.160949i	−0.996757	−	0.0804745i	$-0.974356\pi$
$821$	− 132.139i	− 0.160949i	0.996757	−	0.0804745i	$-0.0256435\pi$
$822$	0	0
$823$	486.192	0.590756	0.295378	−	0.955380i	$-0.404554\pi$
$823$	486.192	0.590756	0.295378	+	0.955380i	$0.404554\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	258.192i	0.312203i	0.987741	+	0.156101i	$0.0498927\pi$
$827$	258.192i	0.312203i	−0.987741	+	0.156101i	$0.950107\pi$
$828$	0	0
$829$	−376.400	−0.454041	−0.227020	−	0.973890i	$-0.572898\pi$
$829$	−376.400	−0.454041	−0.227020	+	0.973890i	$0.572898\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 331.505i	− 0.397966i
$834$	0	0
$835$	−653.296	−0.782391
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 291.584i	− 0.347537i	−0.984787	−	0.173768i	$-0.944406\pi$
$839$	− 291.584i	− 0.347537i	0.984787	−	0.173768i	$-0.0555944\pi$
$840$	0	0
$841$	187.311	0.222724
$842$	0	0
$843$	0	0
$844$	0	0
$845$	227.275i	0.268964i
$846$	0	0
$847$	−741.727	−0.875711
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 296.131i	− 0.347980i
$852$	0	0
$853$	193.538	0.226891	0.113446	−	0.993544i	$-0.463811\pi$
$853$	193.538	0.226891	0.113446	+	0.993544i	$0.463811\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 191.677i	− 0.223661i	−0.993727	−	0.111830i	$-0.964329\pi$
$857$	− 191.677i	− 0.223661i	0.993727	−	0.111830i	$-0.0356714\pi$
$858$	0	0
$859$	1196.52	1.39292	0.696458	−	0.717597i	$-0.254758\pi$
$859$	1196.52	1.39292	0.696458	+	0.717597i	$0.254758\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 692.323i	− 0.802228i	−0.916028	−	0.401114i	$-0.868623\pi$
$863$	− 692.323i	− 0.802228i	0.916028	−	0.401114i	$-0.131377\pi$
$864$	0	0
$865$	1088.65	1.25856
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 128.884i	− 0.148313i
$870$	0	0
$871$	120.627	0.138492
$872$	0	0
$873$	0	0
$874$	0	0
$875$	589.341i	0.673532i
$876$	0	0
$877$	−1026.61	−1.17059	−0.585297	−	0.810819i	$-0.699022\pi$
$877$	−1026.61	−1.17059	−0.585297	+	0.810819i	$0.699022\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	984.820i	1.11784i	0.829220	+	0.558922i	$0.188785\pi$
$881$	984.820i	1.11784i	−0.829220	+	0.558922i	$0.811215\pi$
$882$	0	0
$883$	1038.19	1.17576	0.587878	−	0.808950i	$-0.299964\pi$
$883$	1038.19	1.17576	0.587878	+	0.808950i	$0.299964\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	169.564i	0.191166i	0.995421	+	0.0955830i	$0.0304715\pi$
$887$	169.564i	0.191166i	−0.995421	+	0.0955830i	$0.969528\pi$
$888$	0	0
$889$	422.792	0.475582
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1986.40i	2.22442i
$894$	0	0
$895$	−94.8231	−0.105948
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 592.572i	− 0.659146i
$900$	0	0
$901$	617.896	0.685789
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1319.11i	1.45758i
$906$	0	0
$907$	731.962	0.807014	0.403507	−	0.914977i	$-0.367791\pi$
$907$	731.962	0.807014	0.403507	+	0.914977i	$0.367791\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 985.712i	− 1.08201i	−0.841019	−	0.541006i	$-0.818044\pi$
$911$	− 985.712i	− 1.08201i	0.841019	−	0.541006i	$-0.181956\pi$
$912$	0	0
$913$	−13.4304	−0.0147102
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 954.795i	− 1.04122i
$918$	0	0
$919$	−839.650	−0.913656	−0.456828	−	0.889555i	$-0.651015\pi$
$919$	−839.650	−0.913656	−0.456828	+	0.889555i	$0.651015\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 41.3931i	− 0.0448462i
$924$	0	0
$925$	−75.6115	−0.0817421
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 599.706i	− 0.645539i	−0.946478	−	0.322769i	$-0.895386\pi$
$929$	− 599.706i	− 0.645539i	0.946478	−	0.322769i	$-0.104614\pi$
$930$	0	0
$931$	349.850	0.375779
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 205.899i	− 0.220212i
$936$	0	0
$937$	1610.24	1.71850	0.859252	−	0.511553i	$-0.170930\pi$
$937$	1610.24	1.71850	0.859252	+	0.511553i	$0.170930\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 880.100i	− 0.935282i	−0.883919	−	0.467641i	$-0.845104\pi$
$941$	− 880.100i	− 0.935282i	0.883919	−	0.467641i	$-0.154896\pi$
$942$	0	0
$943$	−2377.55	−2.52126
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 985.043i	− 1.04017i	−0.854114	−	0.520086i	$-0.825900\pi$
$947$	− 985.043i	− 1.04017i	0.854114	−	0.520086i	$-0.174100\pi$
$948$	0	0
$949$	196.342	0.206894
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 845.550i	− 0.887251i	−0.896212	−	0.443625i	$-0.853692\pi$
$953$	− 845.550i	− 0.887251i	0.896212	−	0.443625i	$-0.146308\pi$
$954$	0	0
$955$	−2112.84	−2.21240
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 473.876i	− 0.494135i
$960$	0	0
$961$	−423.831	−0.441031
$962$	0	0
$963$	0	0
$964$	0	0
$965$	144.220i	0.149451i
$966$	0	0
$967$	−1026.94	−1.06199	−0.530994	−	0.847376i	$-0.678181\pi$
$967$	−1026.94	−1.06199	−0.530994	+	0.847376i	$0.678181\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	338.171i	0.348271i	0.984722	+	0.174135i	$0.0557130\pi$
$971$	338.171i	0.348271i	−0.984722	+	0.174135i	$0.944287\pi$
$972$	0	0
$973$	−582.677	−0.598846
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 911.618i	− 0.933079i	−0.884500	−	0.466540i	$-0.845501\pi$
$977$	− 911.618i	− 0.933079i	0.884500	−	0.466540i	$-0.154499\pi$
$978$	0	0
$979$	−126.473	−0.129186
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 957.961i	− 0.974528i	−0.873255	−	0.487264i	$-0.837995\pi$
$983$	− 957.961i	− 0.974528i	0.873255	−	0.487264i	$-0.162005\pi$
$984$	0	0
$985$	992.538	1.00765
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2010.30i	2.03266i
$990$	0	0
$991$	1771.18	1.78727	0.893633	−	0.448798i	$-0.148148\pi$
$991$	1771.18	1.78727	0.893633	+	0.448798i	$0.148148\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 709.554i	− 0.713120i
$996$	0	0
$997$	−1088.65	−1.09193	−0.545963	−	0.837809i	$-0.683836\pi$
$997$	−1088.65	−1.09193	−0.545963	+	0.837809i	$0.683836\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.c.a.161.1	✓	4
3.2	odd	2	inner	324.3.c.a.161.4	yes	4
4.3	odd	2		1296.3.e.f.161.1		4
9.2	odd	6		324.3.g.d.53.1		8
9.4	even	3		324.3.g.d.269.1		8
9.5	odd	6		324.3.g.d.269.4		8
9.7	even	3		324.3.g.d.53.4		8
12.11	even	2		1296.3.e.f.161.4		4
36.7	odd	6		1296.3.q.n.1025.4		8
36.11	even	6		1296.3.q.n.1025.1		8
36.23	even	6		1296.3.q.n.593.4		8
36.31	odd	6		1296.3.q.n.593.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.c.a.161.1	✓	4	1.1	even	1	trivial
324.3.c.a.161.4	yes	4	3.2	odd	2	inner
324.3.g.d.53.1		8	9.2	odd	6
324.3.g.d.53.4		8	9.7	even	3
324.3.g.d.269.1		8	9.4	even	3
324.3.g.d.269.4		8	9.5	odd	6
1296.3.e.f.161.1		4	4.3	odd	2
1296.3.e.f.161.4		4	12.11	even	2
1296.3.q.n.593.1		8	36.31	odd	6
1296.3.q.n.593.4		8	36.23	even	6
1296.3.q.n.1025.1		8	36.11	even	6
1296.3.q.n.1025.4		8	36.7	odd	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.79555i q^{5} -6.19615 q^{7} +O(q^{10})\)	\(q-5.79555i q^{5} -6.19615 q^{7} -1.13681i q^{11} -11.3923 q^{13} +31.2514i q^{17} -32.9808 q^{19} -33.6365i q^{23} -8.58846 q^{25} +25.5673i q^{29} -23.1769 q^{31} +35.9101i q^{35} +8.80385 q^{37} -70.6835i q^{41} -59.7654 q^{43} -60.2292i q^{47} -10.6077 q^{49} -19.7718i q^{53} -6.58846 q^{55} +83.4114i q^{59} +30.7654 q^{61} +66.0247i q^{65} -10.5885 q^{67} +3.63342i q^{71} -17.2346 q^{73} +7.04386i q^{77} +113.373 q^{79} -11.8141i q^{83} +181.119 q^{85} -111.252i q^{89} +70.5885 q^{91} +191.142i q^{95} +74.3154 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} - 4 q^{13} - 28 q^{19} + 28 q^{25} + 32 q^{31} + 56 q^{37} - 52 q^{43} - 84 q^{49} + 36 q^{55} - 64 q^{61} + 20 q^{67} - 256 q^{73} + 308 q^{79} + 288 q^{85} + 220 q^{91} - 160 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^13 - 28 * q^19 + 28 * q^25 + 32 * q^31 + 56 * q^37 - 52 * q^43 - 84 * q^49 + 36 * q^55 - 64 * q^61 + 20 * q^67 - 256 * q^73 + 308 * q^79 + 288 * q^85 + 220 * q^91 - 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more