LMFDB - Embedded newform 3312.2.a.x.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1656)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+0.732051 q^{5} -0.732051 q^{7} +O(q^{10})$	$q+0.732051 q^{5} -0.732051 q^{7} +1.46410 q^{11} -5.46410 q^{13} -0.732051 q^{17} +4.73205 q^{19} -1.00000 q^{23} -4.46410 q^{25} -5.46410 q^{29} +2.53590 q^{31} -0.535898 q^{35} -0.535898 q^{37} +6.92820 q^{41} -0.732051 q^{43} -7.46410 q^{47} -6.46410 q^{49} -14.1962 q^{53} +1.07180 q^{55} +0.535898 q^{59} +6.39230 q^{61} -4.00000 q^{65} +6.19615 q^{67} -10.9282 q^{71} -6.00000 q^{73} -1.07180 q^{77} +9.12436 q^{79} -16.3923 q^{83} -0.535898 q^{85} +6.19615 q^{89} +4.00000 q^{91} +3.46410 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 4 q^{13} + 2 q^{17} + 6 q^{19} - 2 q^{23} - 2 q^{25} - 4 q^{29} + 12 q^{31} - 8 q^{35} - 8 q^{37} + 2 q^{43} - 8 q^{47} - 6 q^{49} - 18 q^{53} + 16 q^{55} + 8 q^{59} - 8 q^{61} - 8 q^{65} + 2 q^{67} - 8 q^{71} - 12 q^{73} - 16 q^{77} - 6 q^{79} - 12 q^{83} - 8 q^{85} + 2 q^{89} + 8 q^{91} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 4 * q^13 + 2 * q^17 + 6 * q^19 - 2 * q^23 - 2 * q^25 - 4 * q^29 + 12 * q^31 - 8 * q^35 - 8 * q^37 + 2 * q^43 - 8 * q^47 - 6 * q^49 - 18 * q^53 + 16 * q^55 + 8 * q^59 - 8 * q^61 - 8 * q^65 + 2 * q^67 - 8 * q^71 - 12 * q^73 - 16 * q^77 - 6 * q^79 - 12 * q^83 - 8 * q^85 + 2 * q^89 + 8 * q^91 - 4 * 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
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.732051	−0.177548	−0.0887742	−	0.996052i	$-0.528295\pi$
$17$	−0.732051	−0.177548	−0.0887742	+	0.996052i	$0.528295\pi$
$18$	0	0
$19$	4.73205	1.08561	0.542803	−	0.839860i	$-0.317363\pi$
$19$	4.73205	1.08561	0.542803	+	0.839860i	$0.317363\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.46410	−1.01466	−0.507329	−	0.861752i	$-0.669367\pi$
$29$	−5.46410	−1.01466	−0.507329	+	0.861752i	$0.669367\pi$
$30$	0	0
$31$	2.53590	0.455461	0.227730	−	0.973724i	$-0.426870\pi$
$31$	2.53590	0.455461	0.227730	+	0.973724i	$0.426870\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.535898	−0.0905834
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	−0.732051	−0.111637	−0.0558184	−	0.998441i	$-0.517777\pi$
$43$	−0.732051	−0.111637	−0.0558184	+	0.998441i	$0.517777\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.46410	−1.08875	−0.544376	−	0.838842i	$-0.683233\pi$
$47$	−7.46410	−1.08875	−0.544376	+	0.838842i	$0.683233\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−14.1962	−1.94999	−0.974996	−	0.222224i	$-0.928669\pi$
$53$	−14.1962	−1.94999	−0.974996	+	0.222224i	$0.928669\pi$
$54$	0	0
$55$	1.07180	0.144521
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.535898	0.0697680	0.0348840	−	0.999391i	$-0.488894\pi$
$59$	0.535898	0.0697680	0.0348840	+	0.999391i	$0.488894\pi$
$60$	0	0
$61$	6.39230	0.818451	0.409225	−	0.912433i	$-0.365799\pi$
$61$	6.39230	0.818451	0.409225	+	0.912433i	$0.365799\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	6.19615	0.756980	0.378490	−	0.925605i	$-0.376443\pi$
$67$	6.19615	0.756980	0.378490	+	0.925605i	$0.376443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.9282	−1.29694	−0.648470	−	0.761241i	$-0.724591\pi$
$71$	−10.9282	−1.29694	−0.648470	+	0.761241i	$0.724591\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.07180	−0.122143
$78$	0	0
$79$	9.12436	1.02657	0.513285	−	0.858218i	$-0.328428\pi$
$79$	9.12436	1.02657	0.513285	+	0.858218i	$0.328428\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.3923	−1.79929	−0.899645	−	0.436623i	$-0.856174\pi$
$83$	−16.3923	−1.79929	−0.899645	+	0.436623i	$0.856174\pi$
$84$	0	0
$85$	−0.535898	−0.0581263
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.19615	0.656791	0.328395	−	0.944540i	$-0.393492\pi$
$89$	6.19615	0.656791	0.328395	+	0.944540i	$0.393492\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9282	−1.08740	−0.543698	−	0.839281i	$-0.682976\pi$
$101$	−10.9282	−1.08740	−0.543698	+	0.839281i	$0.682976\pi$
$102$	0	0
$103$	3.26795	0.322001	0.161000	−	0.986954i	$-0.448528\pi$
$103$	3.26795	0.322001	0.161000	+	0.986954i	$0.448528\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.39230	0.424620	0.212310	−	0.977202i	$-0.431901\pi$
$107$	4.39230	0.424620	0.212310	+	0.977202i	$0.431901\pi$
$108$	0	0
$109$	−8.53590	−0.817591	−0.408795	−	0.912626i	$-0.634051\pi$
$109$	−8.53590	−0.817591	−0.408795	+	0.912626i	$0.634051\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.73205	−0.445154	−0.222577	−	0.974915i	$-0.571447\pi$
$113$	−4.73205	−0.445154	−0.222577	+	0.974915i	$0.571447\pi$
$114$	0	0
$115$	−0.732051	−0.0682641
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.535898	0.0491257
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	1.46410	0.129918	0.0649590	−	0.997888i	$-0.479308\pi$
$127$	1.46410	0.129918	0.0649590	+	0.997888i	$0.479308\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−3.46410	−0.300376
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.12436	−0.437803	−0.218902	−	0.975747i	$-0.570247\pi$
$137$	−5.12436	−0.437803	−0.218902	+	0.975747i	$0.570247\pi$
$138$	0	0
$139$	17.8564	1.51456	0.757280	−	0.653090i	$-0.226528\pi$
$139$	17.8564	1.51456	0.757280	+	0.653090i	$0.226528\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.12436	0.747496	0.373748	−	0.927530i	$-0.378072\pi$
$149$	9.12436	0.747496	0.373748	+	0.927530i	$0.378072\pi$
$150$	0	0
$151$	−14.9282	−1.21484	−0.607420	−	0.794381i	$-0.707795\pi$
$151$	−14.9282	−1.21484	−0.607420	+	0.794381i	$0.707795\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.85641	0.149110
$156$	0	0
$157$	1.60770	0.128308	0.0641540	−	0.997940i	$-0.479565\pi$
$157$	1.60770	0.128308	0.0641540	+	0.997940i	$0.479565\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.732051	0.0576937
$162$	0	0
$163$	−8.39230	−0.657336	−0.328668	−	0.944446i	$-0.606600\pi$
$163$	−8.39230	−0.657336	−0.328668	+	0.944446i	$0.606600\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.5359	−0.801030	−0.400515	−	0.916290i	$-0.631169\pi$
$173$	−10.5359	−0.801030	−0.400515	+	0.916290i	$0.631169\pi$
$174$	0	0
$175$	3.26795	0.247034
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.4641	−1.45482	−0.727408	−	0.686206i	$-0.759275\pi$
$179$	−19.4641	−1.45482	−0.727408	+	0.686206i	$0.759275\pi$
$180$	0	0
$181$	−1.60770	−0.119499	−0.0597495	−	0.998213i	$-0.519030\pi$
$181$	−1.60770	−0.119499	−0.0597495	+	0.998213i	$0.519030\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.392305	−0.0288428
$186$	0	0
$187$	−1.07180	−0.0783775
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	15.3205	1.10279	0.551397	−	0.834243i	$-0.314095\pi$
$193$	15.3205	1.10279	0.551397	+	0.834243i	$0.314095\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.5359	−1.03564	−0.517820	−	0.855490i	$-0.673256\pi$
$197$	−14.5359	−1.03564	−0.517820	+	0.855490i	$0.673256\pi$
$198$	0	0
$199$	19.6603	1.39368	0.696839	−	0.717227i	$-0.254589\pi$
$199$	19.6603	1.39368	0.696839	+	0.717227i	$0.254589\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	5.07180	0.354230
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.92820	0.479234
$210$	0	0
$211$	−26.2487	−1.80704	−0.903518	−	0.428550i	$-0.859024\pi$
$211$	−26.2487	−1.80704	−0.903518	+	0.428550i	$0.859024\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.535898	−0.0365480
$216$	0	0
$217$	−1.85641	−0.126021
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−1.07180	−0.0717728	−0.0358864	−	0.999356i	$-0.511425\pi$
$223$	−1.07180	−0.0717728	−0.0358864	+	0.999356i	$0.511425\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.3923	−1.35348	−0.676742	−	0.736220i	$-0.736609\pi$
$227$	−20.3923	−1.35348	−0.676742	+	0.736220i	$0.736609\pi$
$228$	0	0
$229$	2.39230	0.158088	0.0790440	−	0.996871i	$-0.474813\pi$
$229$	2.39230	0.158088	0.0790440	+	0.996871i	$0.474813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.07180	−0.332264	−0.166132	−	0.986103i	$-0.553128\pi$
$233$	−5.07180	−0.332264	−0.166132	+	0.986103i	$0.553128\pi$
$234$	0	0
$235$	−5.46410	−0.356439
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.7846	−1.08571	−0.542853	−	0.839828i	$-0.682656\pi$
$239$	−16.7846	−1.08571	−0.542853	+	0.839828i	$0.682656\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.73205	−0.302320
$246$	0	0
$247$	−25.8564	−1.64520
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.60770	0.480193	0.240097	−	0.970749i	$-0.422821\pi$
$251$	7.60770	0.480193	0.240097	+	0.970749i	$0.422821\pi$
$252$	0	0
$253$	−1.46410	−0.0920473
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.7846	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$257$	−20.7846	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$258$	0	0
$259$	0.392305	0.0243766
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.9282	−1.41381	−0.706907	−	0.707307i	$-0.749910\pi$
$263$	−22.9282	−1.41381	−0.706907	+	0.707307i	$0.749910\pi$
$264$	0	0
$265$	−10.3923	−0.638394
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.5359	−0.886269	−0.443135	−	0.896455i	$-0.646134\pi$
$269$	−14.5359	−0.886269	−0.443135	+	0.896455i	$0.646134\pi$
$270$	0	0
$271$	7.32051	0.444689	0.222345	−	0.974968i	$-0.428629\pi$
$271$	7.32051	0.444689	0.222345	+	0.974968i	$0.428629\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.53590	−0.394130
$276$	0	0
$277$	20.9282	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$277$	20.9282	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.58846	0.393034	0.196517	−	0.980500i	$-0.437037\pi$
$281$	6.58846	0.393034	0.196517	+	0.980500i	$0.437037\pi$
$282$	0	0
$283$	−7.66025	−0.455355	−0.227677	−	0.973737i	$-0.573113\pi$
$283$	−7.66025	−0.455355	−0.227677	+	0.973737i	$0.573113\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.07180	−0.299379
$288$	0	0
$289$	−16.4641	−0.968477
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.12436	0.299368	0.149684	−	0.988734i	$-0.452174\pi$
$293$	5.12436	0.299368	0.149684	+	0.988734i	$0.452174\pi$
$294$	0	0
$295$	0.392305	0.0228409
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.46410	0.315997
$300$	0	0
$301$	0.535898	0.0308887
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.67949	0.267947
$306$	0	0
$307$	13.4641	0.768437	0.384218	−	0.923242i	$-0.374471\pi$
$307$	13.4641	0.768437	0.384218	+	0.923242i	$0.374471\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.46410	0.196431	0.0982156	−	0.995165i	$-0.468687\pi$
$311$	3.46410	0.196431	0.0982156	+	0.995165i	$0.468687\pi$
$312$	0	0
$313$	19.0718	1.07800	0.539001	−	0.842305i	$-0.318802\pi$
$313$	19.0718	1.07800	0.539001	+	0.842305i	$0.318802\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.53590	0.142430	0.0712151	−	0.997461i	$-0.477312\pi$
$317$	2.53590	0.142430	0.0712151	+	0.997461i	$0.477312\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.46410	−0.192748
$324$	0	0
$325$	24.3923	1.35304
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.46410	0.301246
$330$	0	0
$331$	−1.46410	−0.0804743	−0.0402372	−	0.999190i	$-0.512811\pi$
$331$	−1.46410	−0.0804743	−0.0402372	+	0.999190i	$0.512811\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.53590	0.247823
$336$	0	0
$337$	27.8564	1.51744	0.758718	−	0.651420i	$-0.225826\pi$
$337$	27.8564	1.51744	0.758718	+	0.651420i	$0.225826\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.71281	0.201060
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.3923	0.987351	0.493675	−	0.869646i	$-0.335653\pi$
$347$	18.3923	0.987351	0.493675	+	0.869646i	$0.335653\pi$
$348$	0	0
$349$	−12.3923	−0.663345	−0.331672	−	0.943395i	$-0.607613\pi$
$349$	−12.3923	−0.663345	−0.331672	+	0.943395i	$0.607613\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.46410	0.503723	0.251862	−	0.967763i	$-0.418957\pi$
$353$	9.46410	0.503723	0.251862	+	0.967763i	$0.418957\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.8564	1.36465	0.682324	−	0.731049i	$-0.260969\pi$
$359$	25.8564	1.36465	0.682324	+	0.731049i	$0.260969\pi$
$360$	0	0
$361$	3.39230	0.178542
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.39230	−0.229904
$366$	0	0
$367$	−37.5167	−1.95835	−0.979177	−	0.203009i	$-0.934928\pi$
$367$	−37.5167	−1.95835	−0.979177	+	0.203009i	$0.934928\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.3923	0.539542
$372$	0	0
$373$	−6.39230	−0.330981	−0.165490	−	0.986211i	$-0.552921\pi$
$373$	−6.39230	−0.330981	−0.165490	+	0.986211i	$0.552921\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	29.8564	1.53768
$378$	0	0
$379$	−19.6603	−1.00988	−0.504940	−	0.863155i	$-0.668485\pi$
$379$	−19.6603	−1.00988	−0.504940	+	0.863155i	$0.668485\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.07180	−0.259157	−0.129578	−	0.991569i	$-0.541362\pi$
$383$	−5.07180	−0.259157	−0.129578	+	0.991569i	$0.541362\pi$
$384$	0	0
$385$	−0.784610	−0.0399874
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.5885	−1.14528	−0.572640	−	0.819807i	$-0.694081\pi$
$389$	−22.5885	−1.14528	−0.572640	+	0.819807i	$0.694081\pi$
$390$	0	0
$391$	0.732051	0.0370214
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.67949	0.336082
$396$	0	0
$397$	0.143594	0.00720675	0.00360338	−	0.999994i	$-0.498853\pi$
$397$	0.143594	0.00720675	0.00360338	+	0.999994i	$0.498853\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.2679	1.56145	0.780723	−	0.624877i	$-0.214851\pi$
$401$	31.2679	1.56145	0.780723	+	0.624877i	$0.214851\pi$
$402$	0	0
$403$	−13.8564	−0.690237
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.784610	−0.0388917
$408$	0	0
$409$	−1.46410	−0.0723952	−0.0361976	−	0.999345i	$-0.511525\pi$
$409$	−1.46410	−0.0723952	−0.0361976	+	0.999345i	$0.511525\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.392305	−0.0193041
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.3923	1.19164	0.595821	−	0.803117i	$-0.296827\pi$
$419$	24.3923	1.19164	0.595821	+	0.803117i	$0.296827\pi$
$420$	0	0
$421$	5.60770	0.273302	0.136651	−	0.990619i	$-0.456366\pi$
$421$	5.60770	0.273302	0.136651	+	0.990619i	$0.456366\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.26795	0.158519
$426$	0	0
$427$	−4.67949	−0.226456
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.0718	−0.822320	−0.411160	−	0.911563i	$-0.634876\pi$
$431$	−17.0718	−0.822320	−0.411160	+	0.911563i	$0.634876\pi$
$432$	0	0
$433$	4.92820	0.236834	0.118417	−	0.992964i	$-0.462218\pi$
$433$	4.92820	0.236834	0.118417	+	0.992964i	$0.462218\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.73205	−0.226365
$438$	0	0
$439$	−19.7128	−0.940841	−0.470421	−	0.882442i	$-0.655898\pi$
$439$	−19.7128	−0.940841	−0.470421	+	0.882442i	$0.655898\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.7128	−0.936584	−0.468292	−	0.883574i	$-0.655130\pi$
$443$	−19.7128	−0.936584	−0.468292	+	0.883574i	$0.655130\pi$
$444$	0	0
$445$	4.53590	0.215022
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.8564	1.59778	0.798891	−	0.601475i	$-0.205420\pi$
$449$	33.8564	1.59778	0.798891	+	0.601475i	$0.205420\pi$
$450$	0	0
$451$	10.1436	0.477643
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.92820	0.137276
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.3923	−0.949764	−0.474882	−	0.880049i	$-0.657509\pi$
$461$	−20.3923	−0.949764	−0.474882	+	0.880049i	$0.657509\pi$
$462$	0	0
$463$	13.0718	0.607498	0.303749	−	0.952752i	$-0.401762\pi$
$463$	13.0718	0.607498	0.303749	+	0.952752i	$0.401762\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.32051	−0.153655	−0.0768274	−	0.997044i	$-0.524479\pi$
$467$	−3.32051	−0.153655	−0.0768274	+	0.997044i	$0.524479\pi$
$468$	0	0
$469$	−4.53590	−0.209448
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.07180	−0.0492813
$474$	0	0
$475$	−21.1244	−0.969252
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	2.92820	0.133515
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.46410	−0.0664814
$486$	0	0
$487$	16.7846	0.760583	0.380292	−	0.924867i	$-0.375824\pi$
$487$	16.7846	0.760583	0.380292	+	0.924867i	$0.375824\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.39230	−0.107963	−0.0539816	−	0.998542i	$-0.517191\pi$
$491$	−2.39230	−0.107963	−0.0539816	+	0.998542i	$0.517191\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	13.4641	0.602736	0.301368	−	0.953508i	$-0.402557\pi$
$499$	13.4641	0.602736	0.301368	+	0.953508i	$0.402557\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.8564	0.796178	0.398089	−	0.917347i	$-0.369674\pi$
$503$	17.8564	0.796178	0.398089	+	0.917347i	$0.369674\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.3205	−1.56555	−0.782777	−	0.622302i	$-0.786197\pi$
$509$	−35.3205	−1.56555	−0.782777	+	0.622302i	$0.786197\pi$
$510$	0	0
$511$	4.39230	0.194304
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.39230	0.105418
$516$	0	0
$517$	−10.9282	−0.480622
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19.2679	0.844144	0.422072	−	0.906562i	$-0.361303\pi$
$521$	19.2679	0.844144	0.422072	+	0.906562i	$0.361303\pi$
$522$	0	0
$523$	22.9808	1.00488	0.502439	−	0.864612i	$-0.332436\pi$
$523$	22.9808	1.00488	0.502439	+	0.864612i	$0.332436\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.85641	−0.0808663
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−37.8564	−1.63974
$534$	0	0
$535$	3.21539	0.139013
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.46410	−0.407648
$540$	0	0
$541$	18.2487	0.784573	0.392287	−	0.919843i	$-0.371684\pi$
$541$	18.2487	0.784573	0.392287	+	0.919843i	$0.371684\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.24871	−0.267665
$546$	0	0
$547$	−19.6077	−0.838365	−0.419182	−	0.907902i	$-0.637683\pi$
$547$	−19.6077	−0.838365	−0.419182	+	0.907902i	$0.637683\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−25.8564	−1.10152
$552$	0	0
$553$	−6.67949	−0.284041
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.5167	−1.08118	−0.540588	−	0.841288i	$-0.681798\pi$
$557$	−25.5167	−1.08118	−0.540588	+	0.841288i	$0.681798\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.5359	−0.612615	−0.306308	−	0.951933i	$-0.599094\pi$
$563$	−14.5359	−0.612615	−0.306308	+	0.951933i	$0.599094\pi$
$564$	0	0
$565$	−3.46410	−0.145736
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.7321	1.53989	0.769944	−	0.638112i	$-0.220284\pi$
$569$	36.7321	1.53989	0.769944	+	0.638112i	$0.220284\pi$
$570$	0	0
$571$	28.4449	1.19038	0.595190	−	0.803585i	$-0.297077\pi$
$571$	28.4449	1.19038	0.595190	+	0.803585i	$0.297077\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.46410	0.186166
$576$	0	0
$577$	−14.5359	−0.605137	−0.302569	−	0.953128i	$-0.597844\pi$
$577$	−14.5359	−0.605137	−0.302569	+	0.953128i	$0.597844\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−20.7846	−0.860811
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	0	0
$595$	0.392305	0.0160829
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.7846	1.66641	0.833207	−	0.552961i	$-0.186502\pi$
$599$	40.7846	1.66641	0.833207	+	0.552961i	$0.186502\pi$
$600$	0	0
$601$	7.32051	0.298610	0.149305	−	0.988791i	$-0.452296\pi$
$601$	7.32051	0.298610	0.149305	+	0.988791i	$0.452296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.48334	−0.263585
$606$	0	0
$607$	−1.07180	−0.0435029	−0.0217514	−	0.999763i	$-0.506924\pi$
$607$	−1.07180	−0.0435029	−0.0217514	+	0.999763i	$0.506924\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.7846	1.64997
$612$	0	0
$613$	−25.3205	−1.02269	−0.511343	−	0.859377i	$-0.670852\pi$
$613$	−25.3205	−1.02269	−0.511343	+	0.859377i	$0.670852\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.1962	0.571516	0.285758	−	0.958302i	$-0.407755\pi$
$617$	14.1962	0.571516	0.285758	+	0.958302i	$0.407755\pi$
$618$	0	0
$619$	−23.2679	−0.935218	−0.467609	−	0.883935i	$-0.654884\pi$
$619$	−23.2679	−0.935218	−0.467609	+	0.883935i	$0.654884\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.53590	−0.181727
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.392305	0.0156422
$630$	0	0
$631$	5.80385	0.231048	0.115524	−	0.993305i	$-0.463145\pi$
$631$	5.80385	0.231048	0.115524	+	0.993305i	$0.463145\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.07180	0.0425330
$636$	0	0
$637$	35.3205	1.39945
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.19615	0.0867428	0.0433714	−	0.999059i	$-0.486190\pi$
$641$	2.19615	0.0867428	0.0433714	+	0.999059i	$0.486190\pi$
$642$	0	0
$643$	29.1244	1.14855	0.574276	−	0.818662i	$-0.305284\pi$
$643$	29.1244	1.14855	0.574276	+	0.818662i	$0.305284\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.5359	1.43637	0.718187	−	0.695850i	$-0.244972\pi$
$647$	36.5359	1.43637	0.718187	+	0.695850i	$0.244972\pi$
$648$	0	0
$649$	0.784610	0.0307986
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.1436	0.396949	0.198475	−	0.980106i	$-0.436401\pi$
$653$	10.1436	0.396949	0.198475	+	0.980106i	$0.436401\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.5359	−0.722056	−0.361028	−	0.932555i	$-0.617574\pi$
$659$	−18.5359	−0.722056	−0.361028	+	0.932555i	$0.617574\pi$
$660$	0	0
$661$	−39.1769	−1.52381	−0.761903	−	0.647692i	$-0.775735\pi$
$661$	−39.1769	−1.52381	−0.761903	+	0.647692i	$0.775735\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.53590	−0.0983379
$666$	0	0
$667$	5.46410	0.211571
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.35898	0.361300
$672$	0	0
$673$	18.5359	0.714506	0.357253	−	0.934008i	$-0.383713\pi$
$673$	18.5359	0.714506	0.357253	+	0.934008i	$0.383713\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.0526	1.38561	0.692806	−	0.721124i	$-0.256374\pi$
$677$	36.0526	1.38561	0.692806	+	0.721124i	$0.256374\pi$
$678$	0	0
$679$	1.46410	0.0561871
$680$	0	0
$681$	0	0
$682$	0	0
$683$	43.7128	1.67262	0.836312	−	0.548254i	$-0.184707\pi$
$683$	43.7128	1.67262	0.836312	+	0.548254i	$0.184707\pi$
$684$	0	0
$685$	−3.75129	−0.143329
$686$	0	0
$687$	0	0
$688$	0	0
$689$	77.5692	2.95515
$690$	0	0
$691$	6.53590	0.248637	0.124319	−	0.992242i	$-0.460325\pi$
$691$	6.53590	0.248637	0.124319	+	0.992242i	$0.460325\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.0718	0.495842
$696$	0	0
$697$	−5.07180	−0.192108
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.58846	0.248843	0.124421	−	0.992229i	$-0.460293\pi$
$701$	6.58846	0.248843	0.124421	+	0.992229i	$0.460293\pi$
$702$	0	0
$703$	−2.53590	−0.0956432
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	21.3205	0.800708	0.400354	−	0.916360i	$-0.368887\pi$
$709$	21.3205	0.800708	0.400354	+	0.916360i	$0.368887\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.53590	−0.0949701
$714$	0	0
$715$	−5.85641	−0.219017
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.4641	−0.725889	−0.362944	−	0.931811i	$-0.618228\pi$
$719$	−19.4641	−0.725889	−0.362944	+	0.931811i	$0.618228\pi$
$720$	0	0
$721$	−2.39230	−0.0890941
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.3923	0.905907
$726$	0	0
$727$	37.5167	1.39142	0.695708	−	0.718325i	$-0.255091\pi$
$727$	37.5167	1.39142	0.695708	+	0.718325i	$0.255091\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.535898	0.0198209
$732$	0	0
$733$	−51.1769	−1.89026	−0.945131	−	0.326691i	$-0.894066\pi$
$733$	−51.1769	−1.89026	−0.945131	+	0.326691i	$0.894066\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.07180	0.334164
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.0718	1.06654	0.533270	−	0.845945i	$-0.320963\pi$
$743$	29.0718	1.06654	0.533270	+	0.845945i	$0.320963\pi$
$744$	0	0
$745$	6.67949	0.244718
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.21539	−0.117488
$750$	0	0
$751$	48.8372	1.78209	0.891047	−	0.453911i	$-0.149972\pi$
$751$	48.8372	1.78209	0.891047	+	0.453911i	$0.149972\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.9282	−0.397718
$756$	0	0
$757$	−43.1769	−1.56929	−0.784646	−	0.619944i	$-0.787155\pi$
$757$	−43.1769	−1.56929	−0.784646	+	0.619944i	$0.787155\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.1769	−0.767663	−0.383831	−	0.923403i	$-0.625396\pi$
$761$	−21.1769	−0.767663	−0.383831	+	0.923403i	$0.625396\pi$
$762$	0	0
$763$	6.24871	0.226219
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.92820	−0.105731
$768$	0	0
$769$	−20.1436	−0.726397	−0.363198	−	0.931712i	$-0.618315\pi$
$769$	−20.1436	−0.726397	−0.363198	+	0.931712i	$0.618315\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−37.5167	−1.34938	−0.674690	−	0.738101i	$-0.735723\pi$
$773$	−37.5167	−1.34938	−0.674690	+	0.738101i	$0.735723\pi$
$774$	0	0
$775$	−11.3205	−0.406645
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.7846	1.17463
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.17691	0.0420059
$786$	0	0
$787$	−44.4449	−1.58429	−0.792144	−	0.610334i	$-0.791035\pi$
$787$	−44.4449	−1.58429	−0.792144	+	0.610334i	$0.791035\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.46410	0.123169
$792$	0	0
$793$	−34.9282	−1.24034
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.2295	1.74380	0.871899	−	0.489686i	$-0.162889\pi$
$797$	49.2295	1.74380	0.871899	+	0.489686i	$0.162889\pi$
$798$	0	0
$799$	5.46410	0.193306
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.78461	−0.310002
$804$	0	0
$805$	0.535898	0.0188879
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.5359	0.792320	0.396160	−	0.918181i	$-0.370343\pi$
$809$	22.5359	0.792320	0.396160	+	0.918181i	$0.370343\pi$
$810$	0	0
$811$	27.6077	0.969437	0.484719	−	0.874670i	$-0.338922\pi$
$811$	27.6077	0.969437	0.484719	+	0.874670i	$0.338922\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.14359	−0.215201
$816$	0	0
$817$	−3.46410	−0.121194
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.78461	0.306585	0.153292	−	0.988181i	$-0.451012\pi$
$821$	8.78461	0.306585	0.153292	+	0.988181i	$0.451012\pi$
$822$	0	0
$823$	15.6077	0.544050	0.272025	−	0.962290i	$-0.412307\pi$
$823$	15.6077	0.544050	0.272025	+	0.962290i	$0.412307\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.3205	−1.22821	−0.614107	−	0.789223i	$-0.710484\pi$
$827$	−35.3205	−1.22821	−0.614107	+	0.789223i	$0.710484\pi$
$828$	0	0
$829$	−53.1769	−1.84691	−0.923455	−	0.383706i	$-0.874648\pi$
$829$	−53.1769	−1.84691	−0.923455	+	0.383706i	$0.874648\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.73205	0.163956
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.7128	0.542467	0.271233	−	0.962514i	$-0.412569\pi$
$839$	15.7128	0.542467	0.271233	+	0.962514i	$0.412569\pi$
$840$	0	0
$841$	0.856406	0.0295313
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.3397	0.424500
$846$	0	0
$847$	6.48334	0.222770
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.535898	0.0183704
$852$	0	0
$853$	15.8564	0.542913	0.271457	−	0.962451i	$-0.412495\pi$
$853$	15.8564	0.542913	0.271457	+	0.962451i	$0.412495\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.9282	−0.373300	−0.186650	−	0.982426i	$-0.559763\pi$
$857$	−10.9282	−0.373300	−0.186650	+	0.982426i	$0.559763\pi$
$858$	0	0
$859$	48.7846	1.66451	0.832255	−	0.554393i	$-0.187050\pi$
$859$	48.7846	1.66451	0.832255	+	0.554393i	$0.187050\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.67949	−0.227373	−0.113686	−	0.993517i	$-0.536266\pi$
$863$	−6.67949	−0.227373	−0.113686	+	0.993517i	$0.536266\pi$
$864$	0	0
$865$	−7.71281	−0.262244
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.3590	0.453172
$870$	0	0
$871$	−33.8564	−1.14718
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.07180	0.171458
$876$	0	0
$877$	42.7846	1.44473	0.722367	−	0.691510i	$-0.243054\pi$
$877$	42.7846	1.44473	0.722367	+	0.691510i	$0.243054\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40.0526	−1.34940	−0.674702	−	0.738090i	$-0.735728\pi$
$881$	−40.0526	−1.34940	−0.674702	+	0.738090i	$0.735728\pi$
$882$	0	0
$883$	28.3923	0.955477	0.477739	−	0.878502i	$-0.341457\pi$
$883$	28.3923	0.955477	0.477739	+	0.878502i	$0.341457\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.4641	1.32507	0.662537	−	0.749029i	$-0.269480\pi$
$887$	39.4641	1.32507	0.662537	+	0.749029i	$0.269480\pi$
$888$	0	0
$889$	−1.07180	−0.0359469
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−35.3205	−1.18196
$894$	0	0
$895$	−14.2487	−0.476282
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.8564	−0.462137
$900$	0	0
$901$	10.3923	0.346218
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.17691	−0.0391220
$906$	0	0
$907$	−16.4449	−0.546043	−0.273021	−	0.962008i	$-0.588023\pi$
$907$	−16.4449	−0.546043	−0.273021	+	0.962008i	$0.588023\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.0718	−1.36077	−0.680385	−	0.732855i	$-0.738187\pi$
$911$	−41.0718	−1.36077	−0.680385	+	0.732855i	$0.738187\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−13.1244	−0.432933	−0.216466	−	0.976290i	$-0.569453\pi$
$919$	−13.1244	−0.432933	−0.216466	+	0.976290i	$0.569453\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59.7128	1.96547
$924$	0	0
$925$	2.39230	0.0786585
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.7128	−1.04046	−0.520232	−	0.854025i	$-0.674154\pi$
$929$	−31.7128	−1.04046	−0.520232	+	0.854025i	$0.674154\pi$
$930$	0	0
$931$	−30.5885	−1.00250
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.784610	−0.0256595
$936$	0	0
$937$	32.6410	1.06634	0.533168	−	0.846010i	$-0.321001\pi$
$937$	32.6410	1.06634	0.533168	+	0.846010i	$0.321001\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29.9090	−0.975004	−0.487502	−	0.873122i	$-0.662092\pi$
$941$	−29.9090	−0.975004	−0.487502	+	0.873122i	$0.662092\pi$
$942$	0	0
$943$	−6.92820	−0.225613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.1051	1.49822	0.749108	−	0.662448i	$-0.230483\pi$
$947$	46.1051	1.49822	0.749108	+	0.662448i	$0.230483\pi$
$948$	0	0
$949$	32.7846	1.06423
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.8038	0.835869	0.417934	−	0.908477i	$-0.362754\pi$
$953$	25.8038	0.835869	0.417934	+	0.908477i	$0.362754\pi$
$954$	0	0
$955$	15.2154	0.492358
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.75129	0.121135
$960$	0	0
$961$	−24.5692	−0.792555
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.2154	0.361036
$966$	0	0
$967$	−51.3205	−1.65036	−0.825178	−	0.564873i	$-0.808925\pi$
$967$	−51.3205	−1.65036	−0.825178	+	0.564873i	$0.808925\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.5359	−0.466479	−0.233240	−	0.972419i	$-0.574933\pi$
$971$	−14.5359	−0.466479	−0.233240	+	0.972419i	$0.574933\pi$
$972$	0	0
$973$	−13.0718	−0.419063
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.4115	0.940959	0.470479	−	0.882411i	$-0.344081\pi$
$977$	29.4115	0.940959	0.470479	+	0.882411i	$0.344081\pi$
$978$	0	0
$979$	9.07180	0.289936
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.6410	−1.36004	−0.680019	−	0.733195i	$-0.738028\pi$
$983$	−42.6410	−1.36004	−0.680019	+	0.733195i	$0.738028\pi$
$984$	0	0
$985$	−10.6410	−0.339051
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.732051	0.0232779
$990$	0	0
$991$	35.3205	1.12199	0.560996	−	0.827818i	$-0.310418\pi$
$991$	35.3205	1.12199	0.560996	+	0.827818i	$0.310418\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.3923	0.456267
$996$	0	0
$997$	−10.2487	−0.324580	−0.162290	−	0.986743i	$-0.551888\pi$
$997$	−10.2487	−0.324580	−0.162290	+	0.986743i	$0.551888\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.x.1.2		2
3.2	odd	2		3312.2.a.bd.1.1		2
4.3	odd	2		1656.2.a.k.1.2	✓	2
12.11	even	2		1656.2.a.m.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.a.k.1.2	✓	2	4.3	odd	2
1656.2.a.m.1.1	yes	2	12.11	even	2
3312.2.a.x.1.2		2	1.1	even	1	trivial
3312.2.a.bd.1.1		2	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+0.732051 q^{5} -0.732051 q^{7} +O(q^{10})\)	\(q+0.732051 q^{5} -0.732051 q^{7} +1.46410 q^{11} -5.46410 q^{13} -0.732051 q^{17} +4.73205 q^{19} -1.00000 q^{23} -4.46410 q^{25} -5.46410 q^{29} +2.53590 q^{31} -0.535898 q^{35} -0.535898 q^{37} +6.92820 q^{41} -0.732051 q^{43} -7.46410 q^{47} -6.46410 q^{49} -14.1962 q^{53} +1.07180 q^{55} +0.535898 q^{59} +6.39230 q^{61} -4.00000 q^{65} +6.19615 q^{67} -10.9282 q^{71} -6.00000 q^{73} -1.07180 q^{77} +9.12436 q^{79} -16.3923 q^{83} -0.535898 q^{85} +6.19615 q^{89} +4.00000 q^{91} +3.46410 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 4 q^{13} + 2 q^{17} + 6 q^{19} - 2 q^{23} - 2 q^{25} - 4 q^{29} + 12 q^{31} - 8 q^{35} - 8 q^{37} + 2 q^{43} - 8 q^{47} - 6 q^{49} - 18 q^{53} + 16 q^{55} + 8 q^{59} - 8 q^{61} - 8 q^{65} + 2 q^{67} - 8 q^{71} - 12 q^{73} - 16 q^{77} - 6 q^{79} - 12 q^{83} - 8 q^{85} + 2 q^{89} + 8 q^{91} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 4 * q^13 + 2 * q^17 + 6 * q^19 - 2 * q^23 - 2 * q^25 - 4 * q^29 + 12 * q^31 - 8 * q^35 - 8 * q^37 + 2 * q^43 - 8 * q^47 - 6 * q^49 - 18 * q^53 + 16 * q^55 + 8 * q^59 - 8 * q^61 - 8 * q^65 + 2 * q^67 - 8 * q^71 - 12 * q^73 - 16 * q^77 - 6 * q^79 - 12 * q^83 - 8 * q^85 + 2 * q^89 + 8 * q^91 - 4 * q^97

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