LMFDB - Embedded newform 3312.2.a.z.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{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ 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)

$f(q)$	$=$	$q+3.64575 q^{5} -2.00000 q^{7} +3.64575 q^{11} -5.29150 q^{13} -7.29150 q^{17} -5.64575 q^{19} +1.00000 q^{23} +8.29150 q^{25} +1.29150 q^{29} -9.29150 q^{31} -7.29150 q^{35} -8.93725 q^{37} -6.00000 q^{41} -5.64575 q^{43} +6.00000 q^{47} -3.00000 q^{49} -3.64575 q^{53} +13.2915 q^{55} +5.64575 q^{61} -19.2915 q^{65} -0.937254 q^{67} -6.00000 q^{71} +3.29150 q^{73} -7.29150 q^{77} +12.5830 q^{79} -8.35425 q^{83} -26.5830 q^{85} +10.5830 q^{91} -20.5830 q^{95} +9.29150 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7} + 2 q^{11} - 4 q^{17} - 6 q^{19} + 2 q^{23} + 6 q^{25} - 8 q^{29} - 8 q^{31} - 4 q^{35} - 2 q^{37} - 12 q^{41} - 6 q^{43} + 12 q^{47} - 6 q^{49} - 2 q^{53} + 16 q^{55} + 6 q^{61}+ \cdots + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 + 2 * q^11 - 4 * q^17 - 6 * q^19 + 2 * q^23 + 6 * q^25 - 8 * q^29 - 8 * q^31 - 4 * q^35 - 2 * q^37 - 12 * q^41 - 6 * q^43 + 12 * q^47 - 6 * q^49 - 2 * q^53 + 16 * q^55 + 6 * q^61 - 28 * q^65 + 14 * q^67 - 12 * q^71 - 4 * q^73 - 4 * q^77 + 4 * q^79 - 22 * q^83 - 32 * q^85 - 20 * q^95 + 8 * q^97

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
$5$	3.64575	1.63043	0.815215	−	0.579159i	$-0.196619\pi$
$5$	3.64575	1.63043	0.815215	+	0.579159i	$0.196619\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.64575	1.09924	0.549618	−	0.835416i	$-0.314773\pi$
$11$	3.64575	1.09924	0.549618	+	0.835416i	$0.314773\pi$
$12$	0	0
$13$	−5.29150	−1.46760	−0.733799	−	0.679366i	$-0.762255\pi$
$13$	−5.29150	−1.46760	−0.733799	+	0.679366i	$0.762255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.29150	−1.76845	−0.884225	−	0.467062i	$-0.845312\pi$
$17$	−7.29150	−1.76845	−0.884225	+	0.467062i	$0.845312\pi$
$18$	0	0
$19$	−5.64575	−1.29522	−0.647612	−	0.761970i	$-0.724232\pi$
$19$	−5.64575	−1.29522	−0.647612	+	0.761970i	$0.724232\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.29150	0.239826	0.119913	−	0.992784i	$-0.461738\pi$
$29$	1.29150	0.239826	0.119913	+	0.992784i	$0.461738\pi$
$30$	0	0
$31$	−9.29150	−1.66880	−0.834402	−	0.551157i	$-0.814187\pi$
$31$	−9.29150	−1.66880	−0.834402	+	0.551157i	$0.814187\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.29150	−1.23249
$36$	0	0
$37$	−8.93725	−1.46928	−0.734638	−	0.678460i	$-0.762648\pi$
$37$	−8.93725	−1.46928	−0.734638	+	0.678460i	$0.762648\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−5.64575	−0.860969	−0.430485	−	0.902598i	$-0.641657\pi$
$43$	−5.64575	−0.860969	−0.430485	+	0.902598i	$0.641657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.64575	−0.500782	−0.250391	−	0.968145i	$-0.580559\pi$
$53$	−3.64575	−0.500782	−0.250391	+	0.968145i	$0.580559\pi$
$54$	0	0
$55$	13.2915	1.79223
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	5.64575	0.722864	0.361432	−	0.932398i	$-0.382288\pi$
$61$	5.64575	0.722864	0.361432	+	0.932398i	$0.382288\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−19.2915	−2.39282
$66$	0	0
$67$	−0.937254	−0.114504	−0.0572519	−	0.998360i	$-0.518234\pi$
$67$	−0.937254	−0.114504	−0.0572519	+	0.998360i	$0.518234\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	3.29150	0.385241	0.192621	−	0.981273i	$-0.438301\pi$
$73$	3.29150	0.385241	0.192621	+	0.981273i	$0.438301\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.29150	−0.830944
$78$	0	0
$79$	12.5830	1.41570	0.707849	−	0.706363i	$-0.249666\pi$
$79$	12.5830	1.41570	0.707849	+	0.706363i	$0.249666\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.35425	−0.916998	−0.458499	−	0.888695i	$-0.651613\pi$
$83$	−8.35425	−0.916998	−0.458499	+	0.888695i	$0.651613\pi$
$84$	0	0
$85$	−26.5830	−2.88333
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	10.5830	1.10940
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−20.5830	−2.11177
$96$	0	0
$97$	9.29150	0.943409	0.471705	−	0.881757i	$-0.343639\pi$
$97$	9.29150	0.943409	0.471705	+	0.881757i	$0.343639\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.29150	0.128509	0.0642547	−	0.997934i	$-0.479533\pi$
$101$	1.29150	0.128509	0.0642547	+	0.997934i	$0.479533\pi$
$102$	0	0
$103$	17.2915	1.70378	0.851891	−	0.523719i	$-0.175456\pi$
$103$	17.2915	1.70378	0.851891	+	0.523719i	$0.175456\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.35425	−0.807636	−0.403818	−	0.914839i	$-0.632317\pi$
$107$	−8.35425	−0.807636	−0.403818	+	0.914839i	$0.632317\pi$
$108$	0	0
$109$	8.22876	0.788172	0.394086	−	0.919074i	$-0.371061\pi$
$109$	8.22876	0.788172	0.394086	+	0.919074i	$0.371061\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.5830	1.37185	0.685927	−	0.727670i	$-0.259397\pi$
$113$	14.5830	1.37185	0.685927	+	0.727670i	$0.259397\pi$
$114$	0	0
$115$	3.64575	0.339968
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.5830	1.33682
$120$	0	0
$121$	2.29150	0.208318
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−10.5830	−0.939090	−0.469545	−	0.882909i	$-0.655582\pi$
$127$	−10.5830	−0.939090	−0.469545	+	0.882909i	$0.655582\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.87451	0.862740	0.431370	−	0.902175i	$-0.358030\pi$
$131$	9.87451	0.862740	0.431370	+	0.902175i	$0.358030\pi$
$132$	0	0
$133$	11.2915	0.979097
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.87451	0.843636	0.421818	−	0.906680i	$-0.361392\pi$
$137$	9.87451	0.843636	0.421818	+	0.906680i	$0.361392\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−19.2915	−1.61324
$144$	0	0
$145$	4.70850	0.391019
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.6458	−1.28175	−0.640875	−	0.767645i	$-0.721428\pi$
$149$	−15.6458	−1.28175	−0.640875	+	0.767645i	$0.721428\pi$
$150$	0	0
$151$	−6.70850	−0.545930	−0.272965	−	0.962024i	$-0.588004\pi$
$151$	−6.70850	−0.545930	−0.272965	+	0.962024i	$0.588004\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−33.8745	−2.72087
$156$	0	0
$157$	−1.64575	−0.131345	−0.0656726	−	0.997841i	$-0.520919\pi$
$157$	−1.64575	−0.131345	−0.0656726	+	0.997841i	$0.520919\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	11.2915	0.884419	0.442209	−	0.896912i	$-0.354195\pi$
$163$	11.2915	0.884419	0.442209	+	0.896912i	$0.354195\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.1660	1.79264	0.896320	−	0.443408i	$-0.146231\pi$
$167$	23.1660	1.79264	0.896320	+	0.443408i	$0.146231\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.8745	−1.20692	−0.603458	−	0.797395i	$-0.706211\pi$
$173$	−15.8745	−1.20692	−0.603458	+	0.797395i	$0.706211\pi$
$174$	0	0
$175$	−16.5830	−1.25356
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.70850	0.351930	0.175965	−	0.984396i	$-0.443696\pi$
$179$	4.70850	0.351930	0.175965	+	0.984396i	$0.443696\pi$
$180$	0	0
$181$	10.3542	0.769625	0.384813	−	0.922995i	$-0.374266\pi$
$181$	10.3542	0.769625	0.384813	+	0.922995i	$0.374266\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−32.5830	−2.39555
$186$	0	0
$187$	−26.5830	−1.94394
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.29150	−0.527595	−0.263797	−	0.964578i	$-0.584975\pi$
$191$	−7.29150	−0.527595	−0.263797	+	0.964578i	$0.584975\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.58301	−0.181291
$204$	0	0
$205$	−21.8745	−1.52778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−20.5830	−1.42376
$210$	0	0
$211$	−0.708497	−0.0487750	−0.0243875	−	0.999703i	$-0.507764\pi$
$211$	−0.708497	−0.0487750	−0.0243875	+	0.999703i	$0.507764\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−20.5830	−1.40375
$216$	0	0
$217$	18.5830	1.26150
$218$	0	0
$219$	0	0
$220$	0	0
$221$	38.5830	2.59537
$222$	0	0
$223$	−11.8745	−0.795176	−0.397588	−	0.917564i	$-0.630153\pi$
$223$	−11.8745	−0.795176	−0.397588	+	0.917564i	$0.630153\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.9373	−1.52240	−0.761200	−	0.648518i	$-0.775389\pi$
$227$	−22.9373	−1.52240	−0.761200	+	0.648518i	$0.775389\pi$
$228$	0	0
$229$	24.9373	1.64790	0.823950	−	0.566662i	$-0.191766\pi$
$229$	24.9373	1.64790	0.823950	+	0.566662i	$0.191766\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	21.8745	1.42694
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−5.29150	−0.340856	−0.170428	−	0.985370i	$-0.554515\pi$
$241$	−5.29150	−0.340856	−0.170428	+	0.985370i	$0.554515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.9373	−0.698756
$246$	0	0
$247$	29.8745	1.90087
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.3542	−1.28475	−0.642374	−	0.766391i	$-0.722051\pi$
$251$	−20.3542	−1.28475	−0.642374	+	0.766391i	$0.722051\pi$
$252$	0	0
$253$	3.64575	0.229206
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.41699	−0.213146	−0.106573	−	0.994305i	$-0.533988\pi$
$257$	−3.41699	−0.213146	−0.106573	+	0.994305i	$0.533988\pi$
$258$	0	0
$259$	17.8745	1.11067
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.41699	−0.580677	−0.290338	−	0.956924i	$-0.593768\pi$
$263$	−9.41699	−0.580677	−0.290338	+	0.956924i	$0.593768\pi$
$264$	0	0
$265$	−13.2915	−0.816491
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.5830	1.98662	0.993310	−	0.115474i	$-0.0368389\pi$
$269$	32.5830	1.98662	0.993310	+	0.115474i	$0.0368389\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	30.2288	1.82286
$276$	0	0
$277$	−5.29150	−0.317936	−0.158968	−	0.987284i	$-0.550817\pi$
$277$	−5.29150	−0.317936	−0.158968	+	0.987284i	$0.550817\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−15.5203	−0.922584	−0.461292	−	0.887248i	$-0.652614\pi$
$283$	−15.5203	−0.922584	−0.461292	+	0.887248i	$0.652614\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	36.1660	2.12741
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.77124	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$293$	5.77124	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.29150	−0.306015
$300$	0	0
$301$	11.2915	0.650831
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.5830	1.17858
$306$	0	0
$307$	1.87451	0.106984	0.0534919	−	0.998568i	$-0.482965\pi$
$307$	1.87451	0.106984	0.0534919	+	0.998568i	$0.482965\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−29.1660	−1.65385	−0.826926	−	0.562310i	$-0.809913\pi$
$311$	−29.1660	−1.65385	−0.826926	+	0.562310i	$0.809913\pi$
$312$	0	0
$313$	−5.29150	−0.299093	−0.149547	−	0.988755i	$-0.547781\pi$
$313$	−5.29150	−0.299093	−0.149547	+	0.988755i	$0.547781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.7085	−0.601449	−0.300725	−	0.953711i	$-0.597228\pi$
$317$	−10.7085	−0.601449	−0.300725	+	0.953711i	$0.597228\pi$
$318$	0	0
$319$	4.70850	0.263625
$320$	0	0
$321$	0	0
$322$	0	0
$323$	41.1660	2.29054
$324$	0	0
$325$	−43.8745	−2.43372
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−5.87451	−0.322892	−0.161446	−	0.986882i	$-0.551616\pi$
$331$	−5.87451	−0.322892	−0.161446	+	0.986882i	$0.551616\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.41699	−0.186690
$336$	0	0
$337$	−27.1660	−1.47983	−0.739913	−	0.672702i	$-0.765134\pi$
$337$	−27.1660	−1.47983	−0.739913	+	0.672702i	$0.765134\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−33.8745	−1.83441
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.70850	0.252765	0.126383	−	0.991982i	$-0.459663\pi$
$347$	4.70850	0.252765	0.126383	+	0.991982i	$0.459663\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.5830	−0.776175	−0.388088	−	0.921623i	$-0.626864\pi$
$353$	−14.5830	−0.776175	−0.388088	+	0.921623i	$0.626864\pi$
$354$	0	0
$355$	−21.8745	−1.16098
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.87451	0.521157	0.260578	−	0.965453i	$-0.416087\pi$
$359$	9.87451	0.521157	0.260578	+	0.965453i	$0.416087\pi$
$360$	0	0
$361$	12.8745	0.677606
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	−31.1660	−1.62685	−0.813426	−	0.581668i	$-0.802400\pi$
$367$	−31.1660	−1.62685	−0.813426	+	0.581668i	$0.802400\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.29150	0.378556
$372$	0	0
$373$	−11.0627	−0.572807	−0.286404	−	0.958109i	$-0.592460\pi$
$373$	−11.0627	−0.572807	−0.286404	+	0.958109i	$0.592460\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.83399	−0.351968
$378$	0	0
$379$	−3.52026	−0.180824	−0.0904118	−	0.995904i	$-0.528818\pi$
$379$	−3.52026	−0.180824	−0.0904118	+	0.995904i	$0.528818\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.7490	−1.62230	−0.811149	−	0.584839i	$-0.801158\pi$
$383$	−31.7490	−1.62230	−0.811149	+	0.584839i	$0.801158\pi$
$384$	0	0
$385$	−26.5830	−1.35480
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.9373	1.16296	0.581482	−	0.813559i	$-0.302473\pi$
$389$	22.9373	1.16296	0.581482	+	0.813559i	$0.302473\pi$
$390$	0	0
$391$	−7.29150	−0.368747
$392$	0	0
$393$	0	0
$394$	0	0
$395$	45.8745	2.30820
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.8745	1.09236	0.546180	−	0.837668i	$-0.316081\pi$
$401$	21.8745	1.09236	0.546180	+	0.837668i	$0.316081\pi$
$402$	0	0
$403$	49.1660	2.44913
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.5830	−1.61508
$408$	0	0
$409$	−27.1660	−1.34327	−0.671636	−	0.740881i	$-0.734408\pi$
$409$	−27.1660	−1.34327	−0.671636	+	0.740881i	$0.734408\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−30.4575	−1.49510
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.06275	0.0519185	0.0259593	−	0.999663i	$-0.491736\pi$
$419$	1.06275	0.0519185	0.0259593	+	0.999663i	$0.491736\pi$
$420$	0	0
$421$	−4.22876	−0.206097	−0.103048	−	0.994676i	$-0.532860\pi$
$421$	−4.22876	−0.206097	−0.103048	+	0.994676i	$0.532860\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−60.4575	−2.93262
$426$	0	0
$427$	−11.2915	−0.546434
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.12549	−0.102381	−0.0511907	−	0.998689i	$-0.516302\pi$
$431$	−2.12549	−0.102381	−0.0511907	+	0.998689i	$0.516302\pi$
$432$	0	0
$433$	−29.2915	−1.40766	−0.703830	−	0.710369i	$-0.748528\pi$
$433$	−29.2915	−1.40766	−0.703830	+	0.710369i	$0.748528\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.64575	−0.270073
$438$	0	0
$439$	35.7490	1.70621	0.853104	−	0.521741i	$-0.174717\pi$
$439$	35.7490	1.70621	0.853104	+	0.521741i	$0.174717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.1660	−0.815582	−0.407791	−	0.913075i	$-0.633701\pi$
$443$	−17.1660	−0.815582	−0.407791	+	0.913075i	$0.633701\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.5830	−1.53769	−0.768844	−	0.639437i	$-0.779168\pi$
$449$	−32.5830	−1.53769	−0.768844	+	0.639437i	$0.779168\pi$
$450$	0	0
$451$	−21.8745	−1.03003
$452$	0	0
$453$	0	0
$454$	0	0
$455$	38.5830	1.80880
$456$	0	0
$457$	−31.8745	−1.49103	−0.745513	−	0.666491i	$-0.767796\pi$
$457$	−31.8745	−1.49103	−0.745513	+	0.666491i	$0.767796\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.7490	1.75815	0.879073	−	0.476686i	$-0.158162\pi$
$461$	37.7490	1.75815	0.879073	+	0.476686i	$0.158162\pi$
$462$	0	0
$463$	21.1660	0.983668	0.491834	−	0.870689i	$-0.336327\pi$
$463$	21.1660	0.983668	0.491834	+	0.870689i	$0.336327\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.35425	0.386589	0.193294	−	0.981141i	$-0.438083\pi$
$467$	8.35425	0.386589	0.193294	+	0.981141i	$0.438083\pi$
$468$	0	0
$469$	1.87451	0.0865567
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−20.5830	−0.946408
$474$	0	0
$475$	−46.8118	−2.14787
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.4170	0.978567	0.489284	−	0.872125i	$-0.337258\pi$
$479$	21.4170	0.978567	0.489284	+	0.872125i	$0.337258\pi$
$480$	0	0
$481$	47.2915	2.15631
$482$	0	0
$483$	0	0
$484$	0	0
$485$	33.8745	1.53816
$486$	0	0
$487$	21.1660	0.959123	0.479562	−	0.877508i	$-0.340796\pi$
$487$	21.1660	0.959123	0.479562	+	0.877508i	$0.340796\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.87451	0.445630	0.222815	−	0.974861i	$-0.428475\pi$
$491$	9.87451	0.445630	0.222815	+	0.974861i	$0.428475\pi$
$492$	0	0
$493$	−9.41699	−0.424120
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	43.0405	1.92676	0.963379	−	0.268143i	$-0.0864100\pi$
$499$	43.0405	1.92676	0.963379	+	0.268143i	$0.0864100\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.8745	−1.51039	−0.755195	−	0.655500i	$-0.772458\pi$
$503$	−33.8745	−1.51039	−0.755195	+	0.655500i	$0.772458\pi$
$504$	0	0
$505$	4.70850	0.209525
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.87451	−0.171735	−0.0858673	−	0.996307i	$-0.527366\pi$
$509$	−3.87451	−0.171735	−0.0858673	+	0.996307i	$0.527366\pi$
$510$	0	0
$511$	−6.58301	−0.291215
$512$	0	0
$513$	0	0
$514$	0	0
$515$	63.0405	2.77790
$516$	0	0
$517$	21.8745	0.962040
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.4575	−1.07150	−0.535752	−	0.844376i	$-0.679972\pi$
$521$	−24.4575	−1.07150	−0.535752	+	0.844376i	$0.679972\pi$
$522$	0	0
$523$	20.9373	0.915522	0.457761	−	0.889075i	$-0.348652\pi$
$523$	20.9373	0.915522	0.457761	+	0.889075i	$0.348652\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	67.7490	2.95119
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.7490	1.37520
$534$	0	0
$535$	−30.4575	−1.31679
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.9373	−0.471101
$540$	0	0
$541$	−0.583005	−0.0250654	−0.0125327	−	0.999921i	$-0.503989\pi$
$541$	−0.583005	−0.0250654	−0.0125327	+	0.999921i	$0.503989\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.0000	1.28506
$546$	0	0
$547$	−27.2915	−1.16690	−0.583450	−	0.812149i	$-0.698298\pi$
$547$	−27.2915	−1.16690	−0.583450	+	0.812149i	$0.698298\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.29150	−0.310628
$552$	0	0
$553$	−25.1660	−1.07017
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.1033	−0.682317	−0.341159	−	0.940006i	$-0.610819\pi$
$557$	−16.1033	−0.682317	−0.341159	+	0.940006i	$0.610819\pi$
$558$	0	0
$559$	29.8745	1.26356
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−23.3948	−0.985972	−0.492986	−	0.870037i	$-0.664095\pi$
$563$	−23.3948	−0.985972	−0.492986	+	0.870037i	$0.664095\pi$
$564$	0	0
$565$	53.1660	2.23671
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.2915	0.808742	0.404371	−	0.914595i	$-0.367490\pi$
$569$	19.2915	0.808742	0.404371	+	0.914595i	$0.367490\pi$
$570$	0	0
$571$	−3.06275	−0.128172	−0.0640860	−	0.997944i	$-0.520413\pi$
$571$	−3.06275	−0.128172	−0.0640860	+	0.997944i	$0.520413\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.29150	0.345780
$576$	0	0
$577$	39.2915	1.63573	0.817863	−	0.575413i	$-0.195158\pi$
$577$	39.2915	1.63573	0.817863	+	0.575413i	$0.195158\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.7085	0.693185
$582$	0	0
$583$	−13.2915	−0.550478
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.2915	−1.29154	−0.645769	−	0.763533i	$-0.723463\pi$
$587$	−31.2915	−1.29154	−0.645769	+	0.763533i	$0.723463\pi$
$588$	0	0
$589$	52.4575	2.16147
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	53.1660	2.17959
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.41699	0.384768	0.192384	−	0.981320i	$-0.438378\pi$
$599$	9.41699	0.384768	0.192384	+	0.981320i	$0.438378\pi$
$600$	0	0
$601$	20.4575	0.834479	0.417240	−	0.908796i	$-0.362998\pi$
$601$	20.4575	0.834479	0.417240	+	0.908796i	$0.362998\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.35425	0.339649
$606$	0	0
$607$	−4.12549	−0.167449	−0.0837243	−	0.996489i	$-0.526682\pi$
$607$	−4.12549	−0.167449	−0.0837243	+	0.996489i	$0.526682\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.7490	−1.28443
$612$	0	0
$613$	12.9373	0.522531	0.261265	−	0.965267i	$-0.415860\pi$
$613$	12.9373	0.522531	0.261265	+	0.965267i	$0.415860\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	−17.6458	−0.709243	−0.354621	−	0.935010i	$-0.615390\pi$
$619$	−17.6458	−0.709243	−0.354621	+	0.935010i	$0.615390\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	0	0
$628$	0	0
$629$	65.1660	2.59834
$630$	0	0
$631$	7.87451	0.313479	0.156740	−	0.987640i	$-0.449902\pi$
$631$	7.87451	0.313479	0.156740	+	0.987640i	$0.449902\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−38.5830	−1.53112
$636$	0	0
$637$	15.8745	0.628971
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.4575	−0.966014	−0.483007	−	0.875617i	$-0.660455\pi$
$641$	−24.4575	−0.966014	−0.483007	+	0.875617i	$0.660455\pi$
$642$	0	0
$643$	−29.6458	−1.16911	−0.584557	−	0.811353i	$-0.698732\pi$
$643$	−29.6458	−1.16911	−0.584557	+	0.811353i	$0.698732\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.1660	−1.14663	−0.573317	−	0.819334i	$-0.694344\pi$
$647$	−29.1660	−1.14663	−0.573317	+	0.819334i	$0.694344\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.8745	−0.621217	−0.310609	−	0.950538i	$-0.600533\pi$
$653$	−15.8745	−0.621217	−0.310609	+	0.950538i	$0.600533\pi$
$654$	0	0
$655$	36.0000	1.40664
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.9373	0.893509	0.446754	−	0.894657i	$-0.352580\pi$
$659$	22.9373	0.893509	0.446754	+	0.894657i	$0.352580\pi$
$660$	0	0
$661$	22.3542	0.869480	0.434740	−	0.900556i	$-0.356840\pi$
$661$	22.3542	0.869480	0.434740	+	0.900556i	$0.356840\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	41.1660	1.59635
$666$	0	0
$667$	1.29150	0.0500072
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20.5830	0.794598
$672$	0	0
$673$	−23.2915	−0.897821	−0.448911	−	0.893577i	$-0.648188\pi$
$673$	−23.2915	−0.897821	−0.448911	+	0.893577i	$0.648188\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.9373	0.881550	0.440775	−	0.897618i	$-0.354704\pi$
$677$	22.9373	0.881550	0.440775	+	0.897618i	$0.354704\pi$
$678$	0	0
$679$	−18.5830	−0.713150
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.12549	−0.0813297	−0.0406648	−	0.999173i	$-0.512948\pi$
$683$	−2.12549	−0.0813297	−0.0406648	+	0.999173i	$0.512948\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.2915	0.734948
$690$	0	0
$691$	13.8745	0.527811	0.263906	−	0.964549i	$-0.414989\pi$
$691$	13.8745	0.527811	0.263906	+	0.964549i	$0.414989\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.5830	0.553165
$696$	0	0
$697$	43.7490	1.65711
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.9373	1.31956	0.659781	−	0.751458i	$-0.270649\pi$
$701$	34.9373	1.31956	0.659781	+	0.751458i	$0.270649\pi$
$702$	0	0
$703$	50.4575	1.90304
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.58301	−0.0971439
$708$	0	0
$709$	−23.0627	−0.866140	−0.433070	−	0.901360i	$-0.642570\pi$
$709$	−23.0627	−0.866140	−0.433070	+	0.901360i	$0.642570\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.29150	−0.347970
$714$	0	0
$715$	−70.3320	−2.63027
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.7490	1.63156	0.815781	−	0.578360i	$-0.196307\pi$
$719$	43.7490	1.63156	0.815781	+	0.578360i	$0.196307\pi$
$720$	0	0
$721$	−34.5830	−1.28794
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.7085	0.397704
$726$	0	0
$727$	44.3320	1.64418	0.822092	−	0.569355i	$-0.192807\pi$
$727$	44.3320	1.64418	0.822092	+	0.569355i	$0.192807\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	41.1660	1.52258
$732$	0	0
$733$	37.3948	1.38121	0.690604	−	0.723233i	$-0.257345\pi$
$733$	37.3948	1.38121	0.690604	+	0.723233i	$0.257345\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.41699	−0.125867
$738$	0	0
$739$	−10.5830	−0.389302	−0.194651	−	0.980873i	$-0.562357\pi$
$739$	−10.5830	−0.389302	−0.194651	+	0.980873i	$0.562357\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	43.7490	1.60500	0.802498	−	0.596655i	$-0.203504\pi$
$743$	43.7490	1.60500	0.802498	+	0.596655i	$0.203504\pi$
$744$	0	0
$745$	−57.0405	−2.08980
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.7085	0.610515
$750$	0	0
$751$	10.4575	0.381600	0.190800	−	0.981629i	$-0.438892\pi$
$751$	10.4575	0.381600	0.190800	+	0.981629i	$0.438892\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−24.4575	−0.890100
$756$	0	0
$757$	32.2288	1.17137	0.585687	−	0.810537i	$-0.300825\pi$
$757$	32.2288	1.17137	0.585687	+	0.810537i	$0.300825\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	−16.4575	−0.595802
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−49.0405	−1.76845	−0.884223	−	0.467065i	$-0.845312\pi$
$769$	−49.0405	−1.76845	−0.884223	+	0.467065i	$0.845312\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.64575	−0.131129	−0.0655643	−	0.997848i	$-0.520885\pi$
$773$	−3.64575	−0.131129	−0.0655643	+	0.997848i	$0.520885\pi$
$774$	0	0
$775$	−77.0405	−2.76738
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.8745	1.21368
$780$	0	0
$781$	−21.8745	−0.782731
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	49.6458	1.76968	0.884840	−	0.465895i	$-0.154268\pi$
$787$	49.6458	1.76968	0.884840	+	0.465895i	$0.154268\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29.1660	−1.03702
$792$	0	0
$793$	−29.8745	−1.06087
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.52026	−0.0538503	−0.0269252	−	0.999637i	$-0.508572\pi$
$797$	−1.52026	−0.0538503	−0.0269252	+	0.999637i	$0.508572\pi$
$798$	0	0
$799$	−43.7490	−1.54773
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−7.29150	−0.256992
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	1.87451	0.0658229	0.0329114	−	0.999458i	$-0.489522\pi$
$811$	1.87451	0.0658229	0.0329114	+	0.999458i	$0.489522\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	41.1660	1.44198
$816$	0	0
$817$	31.8745	1.11515
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.8745	1.39163	0.695815	−	0.718221i	$-0.255043\pi$
$821$	39.8745	1.39163	0.695815	+	0.718221i	$0.255043\pi$
$822$	0	0
$823$	2.70850	0.0944123	0.0472061	−	0.998885i	$-0.484968\pi$
$823$	2.70850	0.0944123	0.0472061	+	0.998885i	$0.484968\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.8118	1.14098	0.570488	−	0.821306i	$-0.306754\pi$
$827$	32.8118	1.14098	0.570488	+	0.821306i	$0.306754\pi$
$828$	0	0
$829$	−36.5830	−1.27058	−0.635290	−	0.772274i	$-0.719119\pi$
$829$	−36.5830	−1.27058	−0.635290	+	0.772274i	$0.719119\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	21.8745	0.757907
$834$	0	0
$835$	84.4575	2.92277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.7490	−0.681812	−0.340906	−	0.940097i	$-0.610734\pi$
$839$	−19.7490	−0.681812	−0.340906	+	0.940097i	$0.610734\pi$
$840$	0	0
$841$	−27.3320	−0.942483
$842$	0	0
$843$	0	0
$844$	0	0
$845$	54.6863	1.88126
$846$	0	0
$847$	−4.58301	−0.157474
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.93725	−0.306365
$852$	0	0
$853$	−15.1660	−0.519274	−0.259637	−	0.965706i	$-0.583603\pi$
$853$	−15.1660	−0.519274	−0.259637	+	0.965706i	$0.583603\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	0	0
$859$	−34.5830	−1.17996	−0.589978	−	0.807419i	$-0.700864\pi$
$859$	−34.5830	−1.17996	−0.589978	+	0.807419i	$0.700864\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.1660	1.80979	0.904896	−	0.425633i	$-0.139948\pi$
$863$	53.1660	1.80979	0.904896	+	0.425633i	$0.139948\pi$
$864$	0	0
$865$	−57.8745	−1.96779
$866$	0	0
$867$	0	0
$868$	0	0
$869$	45.8745	1.55619
$870$	0	0
$871$	4.95948	0.168046
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−2.70850	−0.0914595	−0.0457297	−	0.998954i	$-0.514561\pi$
$877$	−2.70850	−0.0914595	−0.0457297	+	0.998954i	$0.514561\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.7490	1.06965	0.534826	−	0.844962i	$-0.320377\pi$
$881$	31.7490	1.06965	0.534826	+	0.844962i	$0.320377\pi$
$882$	0	0
$883$	−3.29150	−0.110768	−0.0553839	−	0.998465i	$-0.517638\pi$
$883$	−3.29150	−0.110768	−0.0553839	+	0.998465i	$0.517638\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.7490	−1.26749	−0.633744	−	0.773543i	$-0.718483\pi$
$887$	−37.7490	−1.26749	−0.633744	+	0.773543i	$0.718483\pi$
$888$	0	0
$889$	21.1660	0.709885
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−33.8745	−1.13357
$894$	0	0
$895$	17.1660	0.573796
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	26.5830	0.885608
$902$	0	0
$903$	0	0
$904$	0	0
$905$	37.7490	1.25482
$906$	0	0
$907$	8.47974	0.281565	0.140783	−	0.990041i	$-0.455038\pi$
$907$	8.47974	0.281565	0.140783	+	0.990041i	$0.455038\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.29150	−0.241578	−0.120789	−	0.992678i	$-0.538542\pi$
$911$	−7.29150	−0.241578	−0.120789	+	0.992678i	$0.538542\pi$
$912$	0	0
$913$	−30.4575	−1.00800
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.7490	−0.652170
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31.7490	1.04503
$924$	0	0
$925$	−74.1033	−2.43650
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	16.9373	0.555096
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−96.9150	−3.16946
$936$	0	0
$937$	−7.41699	−0.242303	−0.121151	−	0.992634i	$-0.538659\pi$
$937$	−7.41699	−0.242303	−0.121151	+	0.992634i	$0.538659\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.8118	−0.678444	−0.339222	−	0.940706i	$-0.610164\pi$
$941$	−20.8118	−0.678444	−0.339222	+	0.940706i	$0.610164\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.8745	1.10077	0.550387	−	0.834910i	$-0.314480\pi$
$947$	33.8745	1.10077	0.550387	+	0.834910i	$0.314480\pi$
$948$	0	0
$949$	−17.4170	−0.565380
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−51.0405	−1.65336	−0.826682	−	0.562669i	$-0.809775\pi$
$953$	−51.0405	−1.65336	−0.826682	+	0.562669i	$0.809775\pi$
$954$	0	0
$955$	−26.5830	−0.860206
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.7490	−0.637729
$960$	0	0
$961$	55.3320	1.78490
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−80.2065	−2.58194
$966$	0	0
$967$	7.87451	0.253227	0.126614	−	0.991952i	$-0.459589\pi$
$967$	7.87451	0.253227	0.126614	+	0.991952i	$0.459589\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.22876	0.199890	0.0999452	−	0.994993i	$-0.468133\pi$
$971$	6.22876	0.199890	0.0999452	+	0.994993i	$0.468133\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.87451	−0.315913	−0.157957	−	0.987446i	$-0.550491\pi$
$977$	−9.87451	−0.315913	−0.157957	+	0.987446i	$0.550491\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.87451	0.314948	0.157474	−	0.987523i	$-0.449665\pi$
$983$	9.87451	0.314948	0.157474	+	0.987523i	$0.449665\pi$
$984$	0	0
$985$	21.8745	0.696980
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.64575	−0.179524
$990$	0	0
$991$	−38.4575	−1.22164	−0.610822	−	0.791768i	$-0.709161\pi$
$991$	−38.4575	−1.22164	−0.610822	+	0.791768i	$0.709161\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−51.0405	−1.61809
$996$	0	0
$997$	−19.4170	−0.614942	−0.307471	−	0.951557i	$-0.599483\pi$
$997$	−19.4170	−0.614942	−0.307471	+	0.951557i	$0.599483\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.z.1.2		2
3.2	odd	2		3312.2.a.v.1.1		2
4.3	odd	2		414.2.a.g.1.2	yes	2
12.11	even	2		414.2.a.e.1.1	✓	2
92.91	even	2		9522.2.a.bc.1.1		2
276.275	odd	2		9522.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.a.e.1.1	✓	2	12.11	even	2
414.2.a.g.1.2	yes	2	4.3	odd	2
3312.2.a.v.1.1		2	3.2	odd	2
3312.2.a.z.1.2		2	1.1	even	1	trivial
9522.2.a.bb.1.2		2	276.275	odd	2
9522.2.a.bc.1.1		2	92.91	even	2

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)

\(f(q)\)	\(=\)	\(q+3.64575 q^{5} -2.00000 q^{7} +3.64575 q^{11} -5.29150 q^{13} -7.29150 q^{17} -5.64575 q^{19} +1.00000 q^{23} +8.29150 q^{25} +1.29150 q^{29} -9.29150 q^{31} -7.29150 q^{35} -8.93725 q^{37} -6.00000 q^{41} -5.64575 q^{43} +6.00000 q^{47} -3.00000 q^{49} -3.64575 q^{53} +13.2915 q^{55} +5.64575 q^{61} -19.2915 q^{65} -0.937254 q^{67} -6.00000 q^{71} +3.29150 q^{73} -7.29150 q^{77} +12.5830 q^{79} -8.35425 q^{83} -26.5830 q^{85} +10.5830 q^{91} -20.5830 q^{95} +9.29150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7} + 2 q^{11} - 4 q^{17} - 6 q^{19} + 2 q^{23} + 6 q^{25} - 8 q^{29} - 8 q^{31} - 4 q^{35} - 2 q^{37} - 12 q^{41} - 6 q^{43} + 12 q^{47} - 6 q^{49} - 2 q^{53} + 16 q^{55} + 6 q^{61}+ \cdots + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 + 2 * q^11 - 4 * q^17 - 6 * q^19 + 2 * q^23 + 6 * q^25 - 8 * q^29 - 8 * q^31 - 4 * q^35 - 2 * q^37 - 12 * q^41 - 6 * q^43 + 12 * q^47 - 6 * q^49 - 2 * q^53 + 16 * q^55 + 6 * q^61 - 28 * q^65 + 14 * q^67 - 12 * q^71 - 4 * q^73 - 4 * q^77 + 4 * q^79 - 22 * q^83 - 32 * q^85 - 20 * q^95 + 8 * q^97

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