LMFDB - Embedded newform 3312.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.a (trivial)

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:	yes
Analytic conductor:	$26.4464531494$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228 q^{5} -5.16228 q^{7} +O(q^{10})$	$q+3.16228 q^{5} -5.16228 q^{7} +4.00000 q^{13} -7.16228 q^{17} +1.16228 q^{19} -1.00000 q^{23} +5.00000 q^{25} -8.32456 q^{29} +6.32456 q^{31} -16.3246 q^{35} -8.32456 q^{37} +2.00000 q^{41} -1.16228 q^{43} +0.324555 q^{47} +19.6491 q^{49} -5.48683 q^{53} -8.32456 q^{59} +0.324555 q^{61} +12.6491 q^{65} -1.16228 q^{67} -14.6491 q^{73} -13.1623 q^{79} -4.00000 q^{83} -22.6491 q^{85} +9.48683 q^{89} -20.6491 q^{91} +3.67544 q^{95} -3.67544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^7	$ 2 q - 4 q^{7} + 8 q^{13} - 8 q^{17} - 4 q^{19} - 2 q^{23} + 10 q^{25} - 4 q^{29} - 20 q^{35} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 12 q^{47} + 14 q^{49} + 8 q^{53} - 4 q^{59} - 12 q^{61} + 4 q^{67} - 4 q^{73} - 20 q^{79} - 8 q^{83} - 20 q^{85} - 16 q^{91} + 20 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 + 8 * q^13 - 8 * q^17 - 4 * q^19 - 2 * q^23 + 10 * q^25 - 4 * q^29 - 20 * q^35 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 12 * q^47 + 14 * q^49 + 8 * q^53 - 4 * q^59 - 12 * q^61 + 4 * q^67 - 4 * q^73 - 20 * q^79 - 8 * q^83 - 20 * q^85 - 16 * q^91 + 20 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

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
$5$	3.16228	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	3.16228	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	0	0
$7$	−5.16228	−1.95116	−0.975579	−	0.219650i	$-0.929509\pi$
$7$	−5.16228	−1.95116	−0.975579	+	0.219650i	$0.929509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.16228	−1.73711	−0.868554	−	0.495595i	$-0.834950\pi$
$17$	−7.16228	−1.73711	−0.868554	+	0.495595i	$0.834950\pi$
$18$	0	0
$19$	1.16228	0.266645	0.133322	−	0.991073i	$-0.457435\pi$
$19$	1.16228	0.266645	0.133322	+	0.991073i	$0.457435\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.32456	−1.54583	−0.772916	−	0.634509i	$-0.781202\pi$
$29$	−8.32456	−1.54583	−0.772916	+	0.634509i	$0.781202\pi$
$30$	0	0
$31$	6.32456	1.13592	0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456	1.13592	0.567962	+	0.823055i	$0.307732\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16.3246	−2.75935
$36$	0	0
$37$	−8.32456	−1.36855	−0.684274	−	0.729225i	$-0.739881\pi$
$37$	−8.32456	−1.36855	−0.684274	+	0.729225i	$0.739881\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−1.16228	−0.177246	−0.0886228	−	0.996065i	$-0.528247\pi$
$43$	−1.16228	−0.177246	−0.0886228	+	0.996065i	$0.528247\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.324555	0.0473413	0.0236706	−	0.999720i	$-0.492465\pi$
$47$	0.324555	0.0473413	0.0236706	+	0.999720i	$0.492465\pi$
$48$	0	0
$49$	19.6491	2.80702
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.48683	−0.753674	−0.376837	−	0.926279i	$-0.622988\pi$
$53$	−5.48683	−0.753674	−0.376837	+	0.926279i	$0.622988\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.32456	−1.08376	−0.541882	−	0.840454i	$-0.682288\pi$
$59$	−8.32456	−1.08376	−0.541882	+	0.840454i	$0.682288\pi$
$60$	0	0
$61$	0.324555	0.0415551	0.0207775	−	0.999784i	$-0.493386\pi$
$61$	0.324555	0.0415551	0.0207775	+	0.999784i	$0.493386\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.6491	1.56893
$66$	0	0
$67$	−1.16228	−0.141995	−0.0709974	−	0.997477i	$-0.522618\pi$
$67$	−1.16228	−0.141995	−0.0709974	+	0.997477i	$0.522618\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.6491	−1.71455	−0.857274	−	0.514860i	$-0.827844\pi$
$73$	−14.6491	−1.71455	−0.857274	+	0.514860i	$0.827844\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.1623	−1.48087	−0.740436	−	0.672127i	$-0.765381\pi$
$79$	−13.1623	−1.48087	−0.740436	+	0.672127i	$0.765381\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−22.6491	−2.45664
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.48683	1.00560	0.502801	−	0.864402i	$-0.332303\pi$
$89$	9.48683	1.00560	0.502801	+	0.864402i	$0.332303\pi$
$90$	0	0
$91$	−20.6491	−2.16461
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.67544	0.377093
$96$	0	0
$97$	−3.67544	−0.373185	−0.186592	−	0.982437i	$-0.559744\pi$
$97$	−3.67544	−0.373185	−0.186592	+	0.982437i	$0.559744\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	7.48683	0.737700	0.368850	−	0.929489i	$-0.379752\pi$
$103$	7.48683	0.737700	0.368850	+	0.929489i	$0.379752\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.1623	−1.05006	−0.525029	−	0.851084i	$-0.675946\pi$
$113$	−11.1623	−1.05006	−0.525029	+	0.851084i	$0.675946\pi$
$114$	0	0
$115$	−3.16228	−0.294884
$116$	0	0
$117$	0	0
$118$	0	0
$119$	36.9737	3.38937
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.32456	−0.206271	−0.103135	−	0.994667i	$-0.532888\pi$
$127$	−2.32456	−0.206271	−0.103135	+	0.994667i	$0.532888\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.64911	0.755676	0.377838	−	0.925872i	$-0.376668\pi$
$131$	8.64911	0.755676	0.377838	+	0.925872i	$0.376668\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.83772	−0.413315	−0.206657	−	0.978413i	$-0.566259\pi$
$137$	−4.83772	−0.413315	−0.206657	+	0.978413i	$0.566259\pi$
$138$	0	0
$139$	16.6491	1.41216	0.706080	−	0.708133i	$-0.250462\pi$
$139$	16.6491	1.41216	0.706080	+	0.708133i	$0.250462\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−26.3246	−2.18614
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.1623	−1.56983	−0.784917	−	0.619600i	$-0.787295\pi$
$149$	−19.1623	−1.56983	−0.784917	+	0.619600i	$0.787295\pi$
$150$	0	0
$151$	4.64911	0.378339	0.189170	−	0.981944i	$-0.439420\pi$
$151$	4.64911	0.378339	0.189170	+	0.981944i	$0.439420\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	12.3246	0.983607	0.491803	−	0.870706i	$-0.336338\pi$
$157$	12.3246	0.983607	0.491803	+	0.870706i	$0.336338\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.16228	0.406844
$162$	0	0
$163$	2.32456	0.182073	0.0910366	−	0.995848i	$-0.470982\pi$
$163$	2.32456	0.182073	0.0910366	+	0.995848i	$0.470982\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.6491	−1.59788	−0.798938	−	0.601413i	$-0.794605\pi$
$167$	−20.6491	−1.59788	−0.798938	+	0.601413i	$0.794605\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.3246	1.54525	0.772624	−	0.634864i	$-0.218944\pi$
$173$	20.3246	1.54525	0.772624	+	0.634864i	$0.218944\pi$
$174$	0	0
$175$	−25.8114	−1.95116
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.67544	−0.573690	−0.286845	−	0.957977i	$-0.592606\pi$
$179$	−7.67544	−0.573690	−0.286845	+	0.957977i	$0.592606\pi$
$180$	0	0
$181$	6.64911	0.494225	0.247112	−	0.968987i	$-0.420518\pi$
$181$	6.64911	0.494225	0.247112	+	0.968987i	$0.420518\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−26.3246	−1.93542
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.9737	−1.37289	−0.686443	−	0.727183i	$-0.740829\pi$
$191$	−18.9737	−1.37289	−0.686443	+	0.727183i	$0.740829\pi$
$192$	0	0
$193$	0.649111	0.0467240	0.0233620	−	0.999727i	$-0.492563\pi$
$193$	0.649111	0.0467240	0.0233620	+	0.999727i	$0.492563\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.67544	0.261865	0.130932	−	0.991391i	$-0.458203\pi$
$197$	3.67544	0.261865	0.130932	+	0.991391i	$0.458203\pi$
$198$	0	0
$199$	9.81139	0.695511	0.347755	−	0.937585i	$-0.386944\pi$
$199$	9.81139	0.695511	0.347755	+	0.937585i	$0.386944\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	42.9737	3.01616
$204$	0	0
$205$	6.32456	0.441726
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.3246	−0.710772	−0.355386	−	0.934720i	$-0.615651\pi$
$211$	−10.3246	−0.710772	−0.355386	+	0.934720i	$0.615651\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.67544	−0.250663
$216$	0	0
$217$	−32.6491	−2.21637
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−28.6491	−1.92715
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.3246	−1.21624	−0.608122	−	0.793844i	$-0.708077\pi$
$227$	−18.3246	−1.21624	−0.608122	+	0.793844i	$0.708077\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.64911	0.435598	0.217799	−	0.975994i	$-0.430112\pi$
$233$	6.64911	0.435598	0.217799	+	0.975994i	$0.430112\pi$
$234$	0	0
$235$	1.02633	0.0669507
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.35089	0.216751	0.108376	−	0.994110i	$-0.465435\pi$
$239$	3.35089	0.216751	0.108376	+	0.994110i	$0.465435\pi$
$240$	0	0
$241$	−20.3246	−1.30922	−0.654610	−	0.755967i	$-0.727167\pi$
$241$	−20.3246	−1.30922	−0.654610	+	0.755967i	$0.727167\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	62.1359	3.96972
$246$	0	0
$247$	4.64911	0.295816
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.3246	−0.904158	−0.452079	−	0.891978i	$-0.649317\pi$
$251$	−14.3246	−0.904158	−0.452079	+	0.891978i	$0.649317\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	42.9737	2.67025
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.3246	1.86989	0.934946	−	0.354790i	$-0.115448\pi$
$263$	30.3246	1.86989	0.934946	+	0.354790i	$0.115448\pi$
$264$	0	0
$265$	−17.3509	−1.06586
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.67544	0.224096	0.112048	−	0.993703i	$-0.464259\pi$
$269$	3.67544	0.224096	0.112048	+	0.993703i	$0.464259\pi$
$270$	0	0
$271$	5.67544	0.344759	0.172379	−	0.985031i	$-0.444855\pi$
$271$	5.67544	0.344759	0.172379	+	0.985031i	$0.444855\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.2982	−1.15952	−0.579759	−	0.814788i	$-0.696853\pi$
$277$	−19.2982	−1.15952	−0.579759	+	0.814788i	$0.696853\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.16228	0.188646	0.0943228	−	0.995542i	$-0.469931\pi$
$281$	3.16228	0.188646	0.0943228	+	0.995542i	$0.469931\pi$
$282$	0	0
$283$	3.48683	0.207271	0.103635	−	0.994615i	$-0.466953\pi$
$283$	3.48683	0.207271	0.103635	+	0.994615i	$0.466953\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.3246	−0.609439
$288$	0	0
$289$	34.2982	2.01754
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.51317	−0.146821	−0.0734104	−	0.997302i	$-0.523388\pi$
$293$	−2.51317	−0.146821	−0.0734104	+	0.997302i	$0.523388\pi$
$294$	0	0
$295$	−26.3246	−1.53267
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.02633	0.0587677
$306$	0	0
$307$	−6.32456	−0.360961	−0.180481	−	0.983579i	$-0.557765\pi$
$307$	−6.32456	−0.360961	−0.180481	+	0.983579i	$0.557765\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.9737	1.64295	0.821473	−	0.570248i	$-0.193153\pi$
$311$	28.9737	1.64295	0.821473	+	0.570248i	$0.193153\pi$
$312$	0	0
$313$	−6.64911	−0.375830	−0.187915	−	0.982185i	$-0.560173\pi$
$313$	−6.64911	−0.375830	−0.187915	+	0.982185i	$0.560173\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.6754	0.655758	0.327879	−	0.944720i	$-0.393666\pi$
$317$	11.6754	0.655758	0.327879	+	0.944720i	$0.393666\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.32456	−0.463191
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.67544	−0.0923703
$330$	0	0
$331$	−6.32456	−0.347629	−0.173814	−	0.984778i	$-0.555609\pi$
$331$	−6.32456	−0.347629	−0.173814	+	0.984778i	$0.555609\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.67544	−0.200811
$336$	0	0
$337$	4.32456	0.235574	0.117787	−	0.993039i	$-0.462420\pi$
$337$	4.32456	0.235574	0.117787	+	0.993039i	$0.462420\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−65.2982	−3.52577
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.3246	−1.52054	−0.760271	−	0.649606i	$-0.774934\pi$
$347$	−28.3246	−1.52054	−0.760271	+	0.649606i	$0.774934\pi$
$348$	0	0
$349$	−8.64911	−0.462976	−0.231488	−	0.972838i	$-0.574359\pi$
$349$	−8.64911	−0.462976	−0.231488	+	0.972838i	$0.574359\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.9737	−0.903417	−0.451709	−	0.892166i	$-0.649185\pi$
$353$	−16.9737	−0.903417	−0.451709	+	0.892166i	$0.649185\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.9737	−1.00139	−0.500696	−	0.865623i	$-0.666923\pi$
$359$	−18.9737	−1.00139	−0.500696	+	0.865623i	$0.666923\pi$
$360$	0	0
$361$	−17.6491	−0.928901
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−46.3246	−2.42474
$366$	0	0
$367$	−29.1623	−1.52226	−0.761129	−	0.648600i	$-0.775355\pi$
$367$	−29.1623	−1.52226	−0.761129	+	0.648600i	$0.775355\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.3246	1.47054
$372$	0	0
$373$	−16.3246	−0.845253	−0.422627	−	0.906304i	$-0.638892\pi$
$373$	−16.3246	−0.845253	−0.422627	+	0.906304i	$0.638892\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−33.2982	−1.71495
$378$	0	0
$379$	−25.1623	−1.29250	−0.646250	−	0.763126i	$-0.723664\pi$
$379$	−25.1623	−1.29250	−0.646250	+	0.763126i	$0.723664\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.9737	1.17390	0.586950	−	0.809623i	$-0.300329\pi$
$383$	22.9737	1.17390	0.586950	+	0.809623i	$0.300329\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.16228	−0.363142	−0.181571	−	0.983378i	$-0.558118\pi$
$389$	−7.16228	−0.363142	−0.181571	+	0.983378i	$0.558118\pi$
$390$	0	0
$391$	7.16228	0.362212
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−41.6228	−2.09427
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.48683	0.273999	0.137000	−	0.990571i	$-0.456254\pi$
$401$	5.48683	0.273999	0.137000	+	0.990571i	$0.456254\pi$
$402$	0	0
$403$	25.2982	1.26019
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	42.9737	2.11460
$414$	0	0
$415$	−12.6491	−0.620920
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.64911	0.422537	0.211268	−	0.977428i	$-0.432241\pi$
$419$	8.64911	0.422537	0.211268	+	0.977428i	$0.432241\pi$
$420$	0	0
$421$	14.6491	0.713954	0.356977	−	0.934113i	$-0.383807\pi$
$421$	14.6491	0.713954	0.356977	+	0.934113i	$0.383807\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−35.8114	−1.73711
$426$	0	0
$427$	−1.67544	−0.0810805
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−18.6491	−0.896219	−0.448110	−	0.893979i	$-0.647903\pi$
$433$	−18.6491	−0.896219	−0.448110	+	0.893979i	$0.647903\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.16228	−0.0555993
$438$	0	0
$439$	−12.6491	−0.603709	−0.301855	−	0.953354i	$-0.597606\pi$
$439$	−12.6491	−0.603709	−0.301855	+	0.953354i	$0.597606\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.6491	−0.791023	−0.395512	−	0.918461i	$-0.629433\pi$
$443$	−16.6491	−0.791023	−0.395512	+	0.918461i	$0.629433\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.6491	1.25765	0.628825	−	0.777547i	$-0.283536\pi$
$449$	26.6491	1.25765	0.628825	+	0.777547i	$0.283536\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−65.2982	−3.06123
$456$	0	0
$457$	16.3246	0.763630	0.381815	−	0.924239i	$-0.375299\pi$
$457$	16.3246	0.763630	0.381815	+	0.924239i	$0.375299\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.973666	0.0453481	0.0226741	−	0.999743i	$-0.492782\pi$
$461$	0.973666	0.0453481	0.0226741	+	0.999743i	$0.492782\pi$
$462$	0	0
$463$	8.64911	0.401958	0.200979	−	0.979596i	$-0.435588\pi$
$463$	8.64911	0.401958	0.200979	+	0.979596i	$0.435588\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.9737	0.877997	0.438998	−	0.898488i	$-0.355333\pi$
$467$	18.9737	0.877997	0.438998	+	0.898488i	$0.355333\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.81139	0.266645
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−33.2982	−1.51827
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.6228	−0.527763
$486$	0	0
$487$	−0.649111	−0.0294140	−0.0147070	−	0.999892i	$-0.504682\pi$
$487$	−0.649111	−0.0294140	−0.0147070	+	0.999892i	$0.504682\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.9737	−0.946528	−0.473264	−	0.880921i	$-0.656924\pi$
$491$	−20.9737	−0.946528	−0.473264	+	0.880921i	$0.656924\pi$
$492$	0	0
$493$	59.6228	2.68527
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.02633	0.0459450	0.0229725	−	0.999736i	$-0.492687\pi$
$499$	1.02633	0.0459450	0.0229725	+	0.999736i	$0.492687\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.2982	0.949641	0.474820	−	0.880083i	$-0.342513\pi$
$503$	21.2982	0.949641	0.474820	+	0.880083i	$0.342513\pi$
$504$	0	0
$505$	−18.9737	−0.844317
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.3246	−1.25546	−0.627732	−	0.778430i	$-0.716017\pi$
$509$	−28.3246	−1.25546	−0.627732	+	0.778430i	$0.716017\pi$
$510$	0	0
$511$	75.6228	3.34535
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.6754	1.04326
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.18861	0.358750	0.179375	−	0.983781i	$-0.442593\pi$
$521$	8.18861	0.358750	0.179375	+	0.983781i	$0.442593\pi$
$522$	0	0
$523$	−13.8114	−0.603930	−0.301965	−	0.953319i	$-0.597643\pi$
$523$	−13.8114	−0.603930	−0.301965	+	0.953319i	$0.597643\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.2982	−1.97322
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−50.5964	−2.18747
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	44.2719	1.89640
$546$	0	0
$547$	−22.9737	−0.982283	−0.491142	−	0.871080i	$-0.663420\pi$
$547$	−22.9737	−0.982283	−0.491142	+	0.871080i	$0.663420\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.67544	−0.412188
$552$	0	0
$553$	67.9473	2.88941
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.7851	1.30440	0.652202	−	0.758045i	$-0.273845\pi$
$557$	30.7851	1.30440	0.652202	+	0.758045i	$0.273845\pi$
$558$	0	0
$559$	−4.64911	−0.196636
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.3246	1.44661	0.723304	−	0.690530i	$-0.242623\pi$
$563$	34.3246	1.44661	0.723304	+	0.690530i	$0.242623\pi$
$564$	0	0
$565$	−35.2982	−1.48501
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.4868	−0.565397	−0.282699	−	0.959209i	$-0.591230\pi$
$569$	−13.4868	−0.565397	−0.282699	+	0.959209i	$0.591230\pi$
$570$	0	0
$571$	42.4605	1.77692	0.888458	−	0.458957i	$-0.151777\pi$
$571$	42.4605	1.77692	0.888458	+	0.458957i	$0.151777\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.00000	−0.208514
$576$	0	0
$577$	15.3509	0.639066	0.319533	−	0.947575i	$-0.396474\pi$
$577$	15.3509	0.639066	0.319533	+	0.947575i	$0.396474\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.6491	0.856669
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.64911	0.356987	0.178494	−	0.983941i	$-0.442878\pi$
$587$	8.64911	0.356987	0.178494	+	0.983941i	$0.442878\pi$
$588$	0	0
$589$	7.35089	0.302888
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.6491	1.42287	0.711434	−	0.702753i	$-0.248046\pi$
$593$	34.6491	1.42287	0.711434	+	0.702753i	$0.248046\pi$
$594$	0	0
$595$	116.921	4.79329
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.35089	−0.136914	−0.0684568	−	0.997654i	$-0.521808\pi$
$599$	−3.35089	−0.136914	−0.0684568	+	0.997654i	$0.521808\pi$
$600$	0	0
$601$	8.64911	0.352805	0.176402	−	0.984318i	$-0.443554\pi$
$601$	8.64911	0.352805	0.176402	+	0.984318i	$0.443554\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−34.7851	−1.41421
$606$	0	0
$607$	44.6491	1.81225	0.906126	−	0.423008i	$-0.139026\pi$
$607$	44.6491	1.81225	0.906126	+	0.423008i	$0.139026\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.29822	0.0525204
$612$	0	0
$613$	37.6228	1.51957	0.759785	−	0.650175i	$-0.225304\pi$
$613$	37.6228	1.51957	0.759785	+	0.650175i	$0.225304\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.1623	−1.41558	−0.707790	−	0.706423i	$-0.750308\pi$
$617$	−35.1623	−1.41558	−0.707790	+	0.706423i	$0.750308\pi$
$618$	0	0
$619$	5.81139	0.233579	0.116790	−	0.993157i	$-0.462740\pi$
$619$	5.81139	0.233579	0.116790	+	0.993157i	$0.462740\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−48.9737	−1.96209
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	59.6228	2.37732
$630$	0	0
$631$	25.8114	1.02754	0.513768	−	0.857929i	$-0.328249\pi$
$631$	25.8114	1.02754	0.513768	+	0.857929i	$0.328249\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.35089	−0.291711
$636$	0	0
$637$	78.5964	3.11410
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−47.1623	−1.86280	−0.931399	−	0.364000i	$-0.881411\pi$
$641$	−47.1623	−1.86280	−0.931399	+	0.364000i	$0.881411\pi$
$642$	0	0
$643$	−9.16228	−0.361325	−0.180662	−	0.983545i	$-0.557824\pi$
$643$	−9.16228	−0.361325	−0.180662	+	0.983545i	$0.557824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.2719	1.66188	0.830940	−	0.556363i	$-0.187803\pi$
$647$	42.2719	1.66188	0.830940	+	0.556363i	$0.187803\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.9473	1.40673	0.703364	−	0.710830i	$-0.251680\pi$
$653$	35.9473	1.40673	0.703364	+	0.710830i	$0.251680\pi$
$654$	0	0
$655$	27.3509	1.06869
$656$	0	0
$657$	0	0
$658$	0	0
$659$	47.6228	1.85512	0.927560	−	0.373674i	$-0.121902\pi$
$659$	47.6228	1.85512	0.927560	+	0.373674i	$0.121902\pi$
$660$	0	0
$661$	−19.6754	−0.765286	−0.382643	−	0.923896i	$-0.624986\pi$
$661$	−19.6754	−0.765286	−0.382643	+	0.923896i	$0.624986\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−18.9737	−0.735767
$666$	0	0
$667$	8.32456	0.322328
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20.6491	−0.795965	−0.397982	−	0.917393i	$-0.630289\pi$
$673$	−20.6491	−0.795965	−0.397982	+	0.917393i	$0.630289\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.8377	1.10832	0.554162	−	0.832409i	$-0.313039\pi$
$677$	28.8377	1.10832	0.554162	+	0.832409i	$0.313039\pi$
$678$	0	0
$679$	18.9737	0.728142
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.2982	−0.814954	−0.407477	−	0.913216i	$-0.633591\pi$
$683$	−21.2982	−0.814954	−0.407477	+	0.913216i	$0.633591\pi$
$684$	0	0
$685$	−15.2982	−0.584515
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−21.9473	−0.836127
$690$	0	0
$691$	9.02633	0.343378	0.171689	−	0.985151i	$-0.445078\pi$
$691$	9.02633	0.343378	0.171689	+	0.985151i	$0.445078\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	52.6491	1.99709
$696$	0	0
$697$	−14.3246	−0.542581
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.8377	−1.39134	−0.695671	−	0.718361i	$-0.744893\pi$
$701$	−36.8377	−1.39134	−0.695671	+	0.718361i	$0.744893\pi$
$702$	0	0
$703$	−9.67544	−0.364916
$704$	0	0
$705$	0	0
$706$	0	0
$707$	30.9737	1.16488
$708$	0	0
$709$	35.9473	1.35003	0.675015	−	0.737804i	$-0.264137\pi$
$709$	35.9473	1.35003	0.675015	+	0.737804i	$0.264137\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.32456	−0.236856
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	48.9737	1.82641	0.913205	−	0.407501i	$-0.133600\pi$
$719$	48.9737	1.82641	0.913205	+	0.407501i	$0.133600\pi$
$720$	0	0
$721$	−38.6491	−1.43937
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−41.6228	−1.54583
$726$	0	0
$727$	8.51317	0.315736	0.157868	−	0.987460i	$-0.449538\pi$
$727$	8.51317	0.315736	0.157868	+	0.987460i	$0.449538\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.32456	0.307895
$732$	0	0
$733$	18.0000	0.664845	0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000	0.664845	0.332423	+	0.943131i	$0.392134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.2719	−1.77092	−0.885462	−	0.464712i	$-0.846158\pi$
$743$	−48.2719	−1.77092	−0.885462	+	0.464712i	$0.846158\pi$
$744$	0	0
$745$	−60.5964	−2.22008
$746$	0	0
$747$	0	0
$748$	0	0
$749$	82.5964	3.01801
$750$	0	0
$751$	23.4868	0.857047	0.428523	−	0.903531i	$-0.359034\pi$
$751$	23.4868	0.857047	0.428523	+	0.903531i	$0.359034\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.7018	0.535053
$756$	0	0
$757$	−27.9473	−1.01576	−0.507882	−	0.861427i	$-0.669571\pi$
$757$	−27.9473	−1.01576	−0.507882	+	0.861427i	$0.669571\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.0263	−0.399704	−0.199852	−	0.979826i	$-0.564046\pi$
$761$	−11.0263	−0.399704	−0.199852	+	0.979826i	$0.564046\pi$
$762$	0	0
$763$	−72.2719	−2.61642
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.2982	−1.20233
$768$	0	0
$769$	−30.6491	−1.10524	−0.552618	−	0.833435i	$-0.686371\pi$
$769$	−30.6491	−1.10524	−0.552618	+	0.833435i	$0.686371\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.1359	1.37165	0.685827	−	0.727764i	$-0.259441\pi$
$773$	38.1359	1.37165	0.685827	+	0.727764i	$0.259441\pi$
$774$	0	0
$775$	31.6228	1.13592
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.32456	0.0832858
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	38.9737	1.39103
$786$	0	0
$787$	−33.1623	−1.18211	−0.591054	−	0.806632i	$-0.701288\pi$
$787$	−33.1623	−1.18211	−0.591054	+	0.806632i	$0.701288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	57.6228	2.04883
$792$	0	0
$793$	1.29822	0.0461012
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.8114	1.26850	0.634252	−	0.773126i	$-0.281308\pi$
$797$	35.8114	1.26850	0.634252	+	0.773126i	$0.281308\pi$
$798$	0	0
$799$	−2.32456	−0.0822369
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	16.3246	0.575365
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.97367	0.315497	0.157749	−	0.987479i	$-0.449576\pi$
$809$	8.97367	0.315497	0.157749	+	0.987479i	$0.449576\pi$
$810$	0	0
$811$	5.67544	0.199292	0.0996459	−	0.995023i	$-0.468229\pi$
$811$	5.67544	0.199292	0.0996459	+	0.995023i	$0.468229\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.35089	0.257490
$816$	0	0
$817$	−1.35089	−0.0472616
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.2982	−1.37152	−0.685759	−	0.727829i	$-0.740529\pi$
$821$	−39.2982	−1.37152	−0.685759	+	0.727829i	$0.740529\pi$
$822$	0	0
$823$	35.6228	1.24173	0.620866	−	0.783917i	$-0.286781\pi$
$823$	35.6228	1.24173	0.620866	+	0.783917i	$0.286781\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.02633	0.174783	0.0873914	−	0.996174i	$-0.472147\pi$
$827$	5.02633	0.174783	0.0873914	+	0.996174i	$0.472147\pi$
$828$	0	0
$829$	33.2982	1.15650	0.578248	−	0.815861i	$-0.303737\pi$
$829$	33.2982	1.15650	0.578248	+	0.815861i	$0.303737\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−140.732	−4.87609
$834$	0	0
$835$	−65.2982	−2.25974
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.6228	−0.815549	−0.407774	−	0.913083i	$-0.633695\pi$
$839$	−23.6228	−0.815549	−0.407774	+	0.913083i	$0.633695\pi$
$840$	0	0
$841$	40.2982	1.38959
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.48683	0.326357
$846$	0	0
$847$	56.7851	1.95116
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.32456	0.285362
$852$	0	0
$853$	27.2982	0.934673	0.467337	−	0.884079i	$-0.345214\pi$
$853$	27.2982	0.934673	0.467337	+	0.884079i	$0.345214\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.35089	−0.319420	−0.159710	−	0.987164i	$-0.551056\pi$
$857$	−9.35089	−0.319420	−0.159710	+	0.987164i	$0.551056\pi$
$858$	0	0
$859$	−41.2982	−1.40908	−0.704539	−	0.709666i	$-0.748846\pi$
$859$	−41.2982	−1.40908	−0.704539	+	0.709666i	$0.748846\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.02633	0.239179	0.119590	−	0.992823i	$-0.461842\pi$
$863$	7.02633	0.239179	0.119590	+	0.992823i	$0.461842\pi$
$864$	0	0
$865$	64.2719	2.18531
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.64911	−0.157529
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.9473	−0.943714	−0.471857	−	0.881675i	$-0.656416\pi$
$877$	−27.9473	−0.943714	−0.471857	+	0.881675i	$0.656416\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.16228	0.106540	0.0532699	−	0.998580i	$-0.483036\pi$
$881$	3.16228	0.106540	0.0532699	+	0.998580i	$0.483036\pi$
$882$	0	0
$883$	−14.3246	−0.482060	−0.241030	−	0.970518i	$-0.577485\pi$
$883$	−14.3246	−0.482060	−0.241030	+	0.970518i	$0.577485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.6754	−0.660637	−0.330318	−	0.943870i	$-0.607156\pi$
$887$	−19.6754	−0.660637	−0.330318	+	0.943870i	$0.607156\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.377223	0.0126233
$894$	0	0
$895$	−24.2719	−0.811320
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−52.6491	−1.75595
$900$	0	0
$901$	39.2982	1.30921
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.0263	0.698939
$906$	0	0
$907$	6.83772	0.227043	0.113521	−	0.993536i	$-0.463787\pi$
$907$	6.83772	0.227043	0.113521	+	0.993536i	$0.463787\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−44.6491	−1.47444
$918$	0	0
$919$	30.4605	1.00480	0.502400	−	0.864636i	$-0.332451\pi$
$919$	30.4605	1.00480	0.502400	+	0.864636i	$0.332451\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−41.6228	−1.36855
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22.6491	0.743093	0.371547	−	0.928414i	$-0.378828\pi$
$929$	22.6491	0.743093	0.371547	+	0.928414i	$0.378828\pi$
$930$	0	0
$931$	22.8377	0.748476
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	43.1623	1.40705	0.703525	−	0.710670i	$-0.251608\pi$
$941$	43.1623	1.40705	0.703525	+	0.710670i	$0.251608\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.9737	−0.551570	−0.275785	−	0.961219i	$-0.588938\pi$
$947$	−16.9737	−0.551570	−0.275785	+	0.961219i	$0.588938\pi$
$948$	0	0
$949$	−58.5964	−1.90212
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26.1359	−0.846626	−0.423313	−	0.905983i	$-0.639133\pi$
$953$	−26.1359	−0.846626	−0.423313	+	0.905983i	$0.639133\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.9737	0.806442
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.05267	0.0660777
$966$	0	0
$967$	−10.9737	−0.352889	−0.176445	−	0.984311i	$-0.556460\pi$
$967$	−10.9737	−0.352889	−0.176445	+	0.984311i	$0.556460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.62278	−0.116260	−0.0581302	−	0.998309i	$-0.518514\pi$
$971$	−3.62278	−0.116260	−0.0581302	+	0.998309i	$0.518514\pi$
$972$	0	0
$973$	−85.9473	−2.75534
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.1359	0.452249	0.226124	−	0.974098i	$-0.427394\pi$
$977$	14.1359	0.452249	0.226124	+	0.974098i	$0.427394\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53.2982	−1.69995	−0.849975	−	0.526824i	$-0.823383\pi$
$983$	−53.2982	−1.69995	−0.849975	+	0.526824i	$0.823383\pi$
$984$	0	0
$985$	11.6228	0.370332
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.16228	0.0369583
$990$	0	0
$991$	42.3246	1.34448	0.672242	−	0.740332i	$-0.265331\pi$
$991$	42.3246	1.34448	0.672242	+	0.740332i	$0.265331\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	31.0263	0.983601
$996$	0	0
$997$	−0.649111	−0.0205575	−0.0102788	−	0.999947i	$-0.503272\pi$
$997$	−0.649111	−0.0205575	−0.0102788	+	0.999947i	$0.503272\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.y.1.2		2
3.2	odd	2		1104.2.a.n.1.1		2
4.3	odd	2		828.2.a.f.1.2		2
12.11	even	2		276.2.a.a.1.1	✓	2
24.5	odd	2		4416.2.a.be.1.2		2
24.11	even	2		4416.2.a.bk.1.2		2
60.23	odd	4		6900.2.f.k.6349.1		4
60.47	odd	4		6900.2.f.k.6349.4		4
60.59	even	2		6900.2.a.p.1.1		2
276.275	odd	2		6348.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.a.a.1.1	✓	2	12.11	even	2
828.2.a.f.1.2		2	4.3	odd	2
1104.2.a.n.1.1		2	3.2	odd	2
3312.2.a.y.1.2		2	1.1	even	1	trivial
4416.2.a.be.1.2		2	24.5	odd	2
4416.2.a.bk.1.2		2	24.11	even	2
6348.2.a.e.1.2		2	276.275	odd	2
6900.2.a.p.1.1		2	60.59	even	2
6900.2.f.k.6349.1		4	60.23	odd	4
6900.2.f.k.6349.4		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.16228 q^{5} -5.16228 q^{7} +O(q^{10})\)	\(q+3.16228 q^{5} -5.16228 q^{7} +4.00000 q^{13} -7.16228 q^{17} +1.16228 q^{19} -1.00000 q^{23} +5.00000 q^{25} -8.32456 q^{29} +6.32456 q^{31} -16.3246 q^{35} -8.32456 q^{37} +2.00000 q^{41} -1.16228 q^{43} +0.324555 q^{47} +19.6491 q^{49} -5.48683 q^{53} -8.32456 q^{59} +0.324555 q^{61} +12.6491 q^{65} -1.16228 q^{67} -14.6491 q^{73} -13.1623 q^{79} -4.00000 q^{83} -22.6491 q^{85} +9.48683 q^{89} -20.6491 q^{91} +3.67544 q^{95} -3.67544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^7	\( 2 q - 4 q^{7} + 8 q^{13} - 8 q^{17} - 4 q^{19} - 2 q^{23} + 10 q^{25} - 4 q^{29} - 20 q^{35} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 12 q^{47} + 14 q^{49} + 8 q^{53} - 4 q^{59} - 12 q^{61} + 4 q^{67} - 4 q^{73} - 20 q^{79} - 8 q^{83} - 20 q^{85} - 16 q^{91} + 20 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 + 8 * q^13 - 8 * q^17 - 4 * q^19 - 2 * q^23 + 10 * q^25 - 4 * q^29 - 20 * q^35 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 12 * q^47 + 14 * q^49 + 8 * q^53 - 4 * q^59 - 12 * q^61 + 4 * q^67 - 4 * q^73 - 20 * q^79 - 8 * q^83 - 20 * q^85 - 16 * q^91 + 20 * q^95 - 20 * q^97

Introduction

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