LMFDB - Embedded newform 3312.2.a.z.1.1

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			1.1
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)

$f(q)$	$=$	$q-1.64575 q^{5} -2.00000 q^{7} +O(q^{10})$	$q-1.64575 q^{5} -2.00000 q^{7} -1.64575 q^{11} +5.29150 q^{13} +3.29150 q^{17} -0.354249 q^{19} +1.00000 q^{23} -2.29150 q^{25} -9.29150 q^{29} +1.29150 q^{31} +3.29150 q^{35} +6.93725 q^{37} -6.00000 q^{41} -0.354249 q^{43} +6.00000 q^{47} -3.00000 q^{49} +1.64575 q^{53} +2.70850 q^{55} +0.354249 q^{61} -8.70850 q^{65} +14.9373 q^{67} -6.00000 q^{71} -7.29150 q^{73} +3.29150 q^{77} -8.58301 q^{79} -13.6458 q^{83} -5.41699 q^{85} -10.5830 q^{91} +0.583005 q^{95} -1.29150 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 4 * q^7	$ 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}+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

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
$5$	−1.64575	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$5$	−1.64575	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$6$	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
$11$	−1.64575	−0.496213	−0.248106	−	0.968733i	$-0.579808\pi$
$11$	−1.64575	−0.496213	−0.248106	+	0.968733i	$0.579808\pi$
$12$	0	0
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.29150	0.798307	0.399153	−	0.916884i	$-0.369304\pi$
$17$	3.29150	0.798307	0.399153	+	0.916884i	$0.369304\pi$
$18$	0	0
$19$	−0.354249	−0.0812702	−0.0406351	−	0.999174i	$-0.512938\pi$
$19$	−0.354249	−0.0812702	−0.0406351	+	0.999174i	$0.512938\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.29150	−0.458301
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.29150	−1.72539	−0.862694	−	0.505726i	$-0.831225\pi$
$29$	−9.29150	−1.72539	−0.862694	+	0.505726i	$0.831225\pi$
$30$	0	0
$31$	1.29150	0.231961	0.115980	−	0.993252i	$-0.462999\pi$
$31$	1.29150	0.231961	0.115980	+	0.993252i	$0.462999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.29150	0.556365
$36$	0	0
$37$	6.93725	1.14048	0.570239	−	0.821479i	$-0.306851\pi$
$37$	6.93725	1.14048	0.570239	+	0.821479i	$0.306851\pi$
$38$	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
$43$	−0.354249	−0.0540224	−0.0270112	−	0.999635i	$-0.508599\pi$
$43$	−0.354249	−0.0540224	−0.0270112	+	0.999635i	$0.508599\pi$
$44$	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
$53$	1.64575	0.226061	0.113031	−	0.993592i	$-0.463944\pi$
$53$	1.64575	0.226061	0.113031	+	0.993592i	$0.463944\pi$
$54$	0	0
$55$	2.70850	0.365214
$56$	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
$61$	0.354249	0.0453569	0.0226784	−	0.999743i	$-0.492781\pi$
$61$	0.354249	0.0453569	0.0226784	+	0.999743i	$0.492781\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.70850	−1.08016
$66$	0	0
$67$	14.9373	1.82488	0.912438	−	0.409215i	$-0.134197\pi$
$67$	14.9373	1.82488	0.912438	+	0.409215i	$0.134197\pi$
$68$	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
$73$	−7.29150	−0.853406	−0.426703	−	0.904392i	$-0.640325\pi$
$73$	−7.29150	−0.853406	−0.426703	+	0.904392i	$0.640325\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.29150	0.375102
$78$	0	0
$79$	−8.58301	−0.965664	−0.482832	−	0.875713i	$-0.660392\pi$
$79$	−8.58301	−0.965664	−0.482832	+	0.875713i	$0.660392\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.6458	−1.49782	−0.748908	−	0.662674i	$-0.769421\pi$
$83$	−13.6458	−1.49782	−0.748908	+	0.662674i	$0.769421\pi$
$84$	0	0
$85$	−5.41699	−0.587556
$86$	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
$95$	0.583005	0.0598151
$96$	0	0
$97$	−1.29150	−0.131132	−0.0655661	−	0.997848i	$-0.520885\pi$
$97$	−1.29150	−0.131132	−0.0655661	+	0.997848i	$0.520885\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.29150	−0.924539	−0.462270	−	0.886739i	$-0.652965\pi$
$101$	−9.29150	−0.924539	−0.462270	+	0.886739i	$0.652965\pi$
$102$	0	0
$103$	6.70850	0.661008	0.330504	−	0.943805i	$-0.392781\pi$
$103$	6.70850	0.661008	0.330504	+	0.943805i	$0.392781\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.6458	−1.31918	−0.659592	−	0.751624i	$-0.729271\pi$
$107$	−13.6458	−1.31918	−0.659592	+	0.751624i	$0.729271\pi$
$108$	0	0
$109$	−18.2288	−1.74600	−0.872999	−	0.487722i	$-0.837828\pi$
$109$	−18.2288	−1.74600	−0.872999	+	0.487722i	$0.837828\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.58301	−0.619277	−0.309639	−	0.950854i	$-0.600208\pi$
$113$	−6.58301	−0.619277	−0.309639	+	0.950854i	$0.600208\pi$
$114$	0	0
$115$	−1.64575	−0.153467
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.58301	−0.603463
$120$	0	0
$121$	−8.29150	−0.753773
$122$	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	$-0.344418\pi$
$127$	10.5830	0.939090	0.469545	+	0.882909i	$0.344418\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21.8745	−1.91118	−0.955592	−	0.294692i	$-0.904783\pi$
$131$	−21.8745	−1.91118	−0.955592	+	0.294692i	$0.904783\pi$
$132$	0	0
$133$	0.708497	0.0614345
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.8745	−1.86887	−0.934433	−	0.356140i	$-0.884093\pi$
$137$	−21.8745	−1.86887	−0.934433	+	0.356140i	$0.884093\pi$
$138$	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
$143$	−8.70850	−0.728241
$144$	0	0
$145$	15.2915	1.26989
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.3542	−0.848253	−0.424127	−	0.905603i	$-0.639419\pi$
$149$	−10.3542	−0.848253	−0.424127	+	0.905603i	$0.639419\pi$
$150$	0	0
$151$	−17.2915	−1.40716	−0.703581	−	0.710615i	$-0.748417\pi$
$151$	−17.2915	−1.40716	−0.703581	+	0.710615i	$0.748417\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.12549	−0.170724
$156$	0	0
$157$	3.64575	0.290963	0.145481	−	0.989361i	$-0.453527\pi$
$157$	3.64575	0.290963	0.145481	+	0.989361i	$0.453527\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	0.708497	0.0554938	0.0277469	−	0.999615i	$-0.491167\pi$
$163$	0.708497	0.0554938	0.0277469	+	0.999615i	$0.491167\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.1660	−1.48311	−0.741555	−	0.670892i	$-0.765911\pi$
$167$	−19.1660	−1.48311	−0.741555	+	0.670892i	$0.765911\pi$
$168$	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	$-0.293789\pi$
$173$	15.8745	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$174$	0	0
$175$	4.58301	0.346443
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.2915	1.14294	0.571470	−	0.820623i	$-0.306373\pi$
$179$	15.2915	1.14294	0.571470	+	0.820623i	$0.306373\pi$
$180$	0	0
$181$	15.6458	1.16294	0.581470	−	0.813568i	$-0.302478\pi$
$181$	15.6458	1.16294	0.581470	+	0.813568i	$0.302478\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.4170	−0.839394
$186$	0	0
$187$	−5.41699	−0.396130
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.29150	0.238165	0.119082	−	0.992884i	$-0.462005\pi$
$191$	3.29150	0.238165	0.119082	+	0.992884i	$0.462005\pi$
$192$	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
$203$	18.5830	1.30427
$204$	0	0
$205$	9.87451	0.689666
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.583005	0.0403273
$210$	0	0
$211$	−11.2915	−0.777339	−0.388670	−	0.921377i	$-0.627065\pi$
$211$	−11.2915	−0.777339	−0.388670	+	0.921377i	$0.627065\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.583005	0.0397606
$216$	0	0
$217$	−2.58301	−0.175346
$218$	0	0
$219$	0	0
$220$	0	0
$221$	17.4170	1.17159
$222$	0	0
$223$	19.8745	1.33090	0.665448	−	0.746444i	$-0.268241\pi$
$223$	19.8745	1.33090	0.665448	+	0.746444i	$0.268241\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.06275	−0.468771	−0.234385	−	0.972144i	$-0.575308\pi$
$227$	−7.06275	−0.468771	−0.234385	+	0.972144i	$0.575308\pi$
$228$	0	0
$229$	9.06275	0.598883	0.299442	−	0.954115i	$-0.403200\pi$
$229$	9.06275	0.598883	0.299442	+	0.954115i	$0.403200\pi$
$230$	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
$235$	−9.87451	−0.644142
$236$	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	$-0.445485\pi$
$241$	5.29150	0.340856	0.170428	+	0.985370i	$0.445485\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.93725	0.315430
$246$	0	0
$247$	−1.87451	−0.119272
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.6458	−1.61875	−0.809373	−	0.587295i	$-0.800193\pi$
$251$	−25.6458	−1.61875	−0.809373	+	0.587295i	$0.800193\pi$
$252$	0	0
$253$	−1.64575	−0.103467
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.5830	−1.53345	−0.766723	−	0.641978i	$-0.778114\pi$
$257$	−24.5830	−1.53345	−0.766723	+	0.641978i	$0.778114\pi$
$258$	0	0
$259$	−13.8745	−0.862120
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.5830	−1.88583	−0.942914	−	0.333035i	$-0.891927\pi$
$263$	−30.5830	−1.88583	−0.942914	+	0.333035i	$0.891927\pi$
$264$	0	0
$265$	−2.70850	−0.166382
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.4170	0.696106	0.348053	−	0.937475i	$-0.386843\pi$
$269$	11.4170	0.696106	0.348053	+	0.937475i	$0.386843\pi$
$270$	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
$275$	3.77124	0.227415
$276$	0	0
$277$	5.29150	0.317936	0.158968	−	0.987284i	$-0.449183\pi$
$277$	5.29150	0.317936	0.158968	+	0.987284i	$0.449183\pi$
$278$	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
$283$	21.5203	1.27925	0.639623	−	0.768689i	$-0.279090\pi$
$283$	21.5203	1.27925	0.639623	+	0.768689i	$0.279090\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−6.16601	−0.362706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	32.2288	1.88282	0.941412	−	0.337259i	$-0.109500\pi$
$293$	32.2288	1.88282	0.941412	+	0.337259i	$0.109500\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.29150	0.306015
$300$	0	0
$301$	0.708497	0.0408371
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.583005	−0.0333828
$306$	0	0
$307$	−29.8745	−1.70503	−0.852514	−	0.522704i	$-0.824923\pi$
$307$	−29.8745	−1.70503	−0.852514	+	0.522704i	$0.824923\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.1660	0.746576	0.373288	−	0.927716i	$-0.378230\pi$
$311$	13.1660	0.746576	0.373288	+	0.927716i	$0.378230\pi$
$312$	0	0
$313$	5.29150	0.299093	0.149547	−	0.988755i	$-0.452219\pi$
$313$	5.29150	0.299093	0.149547	+	0.988755i	$0.452219\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.2915	−1.19585	−0.597925	−	0.801552i	$-0.704008\pi$
$317$	−21.2915	−1.19585	−0.597925	+	0.801552i	$0.704008\pi$
$318$	0	0
$319$	15.2915	0.856160
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.16601	−0.0648786
$324$	0	0
$325$	−12.1255	−0.672601
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	25.8745	1.42219	0.711096	−	0.703095i	$-0.248199\pi$
$331$	25.8745	1.42219	0.711096	+	0.703095i	$0.248199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.5830	−1.34311
$336$	0	0
$337$	15.1660	0.826145	0.413073	−	0.910698i	$-0.364456\pi$
$337$	15.1660	0.826145	0.413073	+	0.910698i	$0.364456\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.12549	−0.115102
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.2915	0.820891	0.410445	−	0.911885i	$-0.365373\pi$
$347$	15.2915	0.820891	0.410445	+	0.911885i	$0.365373\pi$
$348$	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
$353$	6.58301	0.350378	0.175189	−	0.984535i	$-0.443946\pi$
$353$	6.58301	0.350378	0.175189	+	0.984535i	$0.443946\pi$
$354$	0	0
$355$	9.87451	0.524084
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.8745	−1.15449	−0.577246	−	0.816570i	$-0.695873\pi$
$359$	−21.8745	−1.15449	−0.577246	+	0.816570i	$0.695873\pi$
$360$	0	0
$361$	−18.8745	−0.993395
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	11.1660	0.582861	0.291431	−	0.956592i	$-0.405869\pi$
$367$	11.1660	0.582861	0.291431	+	0.956592i	$0.405869\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.29150	−0.170886
$372$	0	0
$373$	−26.9373	−1.39476	−0.697379	−	0.716702i	$-0.745651\pi$
$373$	−26.9373	−1.39476	−0.697379	+	0.716702i	$0.745651\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−49.1660	−2.53218
$378$	0	0
$379$	33.5203	1.72182	0.860910	−	0.508757i	$-0.169895\pi$
$379$	33.5203	1.72182	0.860910	+	0.508757i	$0.169895\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.7490	1.62230	0.811149	−	0.584839i	$-0.198842\pi$
$383$	31.7490	1.62230	0.811149	+	0.584839i	$0.198842\pi$
$384$	0	0
$385$	−5.41699	−0.276076
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.06275	0.358095	0.179048	−	0.983840i	$-0.442698\pi$
$389$	7.06275	0.358095	0.179048	+	0.983840i	$0.442698\pi$
$390$	0	0
$391$	3.29150	0.166458
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.1255	0.710731
$396$	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
$401$	−9.87451	−0.493109	−0.246555	−	0.969129i	$-0.579299\pi$
$401$	−9.87451	−0.493109	−0.246555	+	0.969129i	$0.579299\pi$
$402$	0	0
$403$	6.83399	0.340425
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.4170	−0.565919
$408$	0	0
$409$	15.1660	0.749911	0.374955	−	0.927043i	$-0.377658\pi$
$409$	15.1660	0.749911	0.374955	+	0.927043i	$0.377658\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	22.4575	1.10240
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.9373	0.827439	0.413720	−	0.910404i	$-0.364229\pi$
$419$	16.9373	0.827439	0.413720	+	0.910404i	$0.364229\pi$
$420$	0	0
$421$	22.2288	1.08336	0.541682	−	0.840584i	$-0.317788\pi$
$421$	22.2288	1.08336	0.541682	+	0.840584i	$0.317788\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.54249	−0.365864
$426$	0	0
$427$	−0.708497	−0.0342866
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.8745	−1.63168	−0.815839	−	0.578279i	$-0.803724\pi$
$431$	−33.8745	−1.63168	−0.815839	+	0.578279i	$0.803724\pi$
$432$	0	0
$433$	−18.7085	−0.899073	−0.449537	−	0.893262i	$-0.648411\pi$
$433$	−18.7085	−0.899073	−0.449537	+	0.893262i	$0.648411\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.354249	−0.0169460
$438$	0	0
$439$	−27.7490	−1.32439	−0.662194	−	0.749332i	$-0.730375\pi$
$439$	−27.7490	−1.32439	−0.662194	+	0.749332i	$0.730375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.1660	1.19567	0.597837	−	0.801618i	$-0.296027\pi$
$443$	25.1660	1.19567	0.597837	+	0.801618i	$0.296027\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−11.4170	−0.538801	−0.269401	−	0.963028i	$-0.586826\pi$
$449$	−11.4170	−0.538801	−0.269401	+	0.963028i	$0.586826\pi$
$450$	0	0
$451$	9.87451	0.464972
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17.4170	0.816521
$456$	0	0
$457$	−0.125492	−0.00587027	−0.00293514	−	0.999996i	$-0.500934\pi$
$457$	−0.125492	−0.00587027	−0.00293514	+	0.999996i	$0.500934\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.7490	−1.19925	−0.599626	−	0.800281i	$-0.704684\pi$
$461$	−25.7490	−1.19925	−0.599626	+	0.800281i	$0.704684\pi$
$462$	0	0
$463$	−21.1660	−0.983668	−0.491834	−	0.870689i	$-0.663673\pi$
$463$	−21.1660	−0.983668	−0.491834	+	0.870689i	$0.663673\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.6458	0.631450	0.315725	−	0.948851i	$-0.397752\pi$
$467$	13.6458	0.631450	0.315725	+	0.948851i	$0.397752\pi$
$468$	0	0
$469$	−29.8745	−1.37948
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.583005	0.0268066
$474$	0	0
$475$	0.811762	0.0372462
$476$	0	0
$477$	0	0
$478$	0	0
$479$	42.5830	1.94567	0.972834	−	0.231506i	$-0.0743651\pi$
$479$	42.5830	1.94567	0.972834	+	0.231506i	$0.0743651\pi$
$480$	0	0
$481$	36.7085	1.67376
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.12549	0.0965136
$486$	0	0
$487$	−21.1660	−0.959123	−0.479562	−	0.877508i	$-0.659204\pi$
$487$	−21.1660	−0.959123	−0.479562	+	0.877508i	$0.659204\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.8745	−0.987183	−0.493591	−	0.869694i	$-0.664316\pi$
$491$	−21.8745	−0.987183	−0.493591	+	0.869694i	$0.664316\pi$
$492$	0	0
$493$	−30.5830	−1.37739
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−31.0405	−1.38956	−0.694782	−	0.719220i	$-0.744499\pi$
$499$	−31.0405	−1.38956	−0.694782	+	0.719220i	$0.744499\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.12549	−0.0947710	−0.0473855	−	0.998877i	$-0.515089\pi$
$503$	−2.12549	−0.0947710	−0.0473855	+	0.998877i	$0.515089\pi$
$504$	0	0
$505$	15.2915	0.680463
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.8745	1.23552	0.617758	−	0.786368i	$-0.288041\pi$
$509$	27.8745	1.23552	0.617758	+	0.786368i	$0.288041\pi$
$510$	0	0
$511$	14.5830	0.645114
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.0405	−0.486503
$516$	0	0
$517$	−9.87451	−0.434280
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.4575	1.24675	0.623373	−	0.781925i	$-0.285762\pi$
$521$	28.4575	1.24675	0.623373	+	0.781925i	$0.285762\pi$
$522$	0	0
$523$	5.06275	0.221378	0.110689	−	0.993855i	$-0.464694\pi$
$523$	5.06275	0.221378	0.110689	+	0.993855i	$0.464694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.25098	0.185176
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−31.7490	−1.37520
$534$	0	0
$535$	22.4575	0.970923
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.93725	0.212663
$540$	0	0
$541$	20.5830	0.884933	0.442466	−	0.896785i	$-0.354104\pi$
$541$	20.5830	0.884933	0.442466	+	0.896785i	$0.354104\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.0000	1.28506
$546$	0	0
$547$	−16.7085	−0.714404	−0.357202	−	0.934027i	$-0.616269\pi$
$547$	−16.7085	−0.714404	−0.357202	+	0.934027i	$0.616269\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.29150	0.140223
$552$	0	0
$553$	17.1660	0.729973
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.1033	1.78397	0.891986	−	0.452062i	$-0.149312\pi$
$557$	42.1033	1.78397	0.891986	+	0.452062i	$0.149312\pi$
$558$	0	0
$559$	−1.87451	−0.0792832
$560$	0	0
$561$	0	0
$562$	0	0
$563$	45.3948	1.91316	0.956581	−	0.291468i	$-0.0941436\pi$
$563$	45.3948	1.91316	0.956581	+	0.291468i	$0.0941436\pi$
$564$	0	0
$565$	10.8340	0.455789
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.70850	0.365079	0.182540	−	0.983199i	$-0.441568\pi$
$569$	8.70850	0.365079	0.182540	+	0.983199i	$0.441568\pi$
$570$	0	0
$571$	−18.9373	−0.792499	−0.396250	−	0.918143i	$-0.629689\pi$
$571$	−18.9373	−0.792499	−0.396250	+	0.918143i	$0.629689\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.29150	−0.0955623
$576$	0	0
$577$	28.7085	1.19515	0.597575	−	0.801813i	$-0.296131\pi$
$577$	28.7085	1.19515	0.597575	+	0.801813i	$0.296131\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	27.2915	1.13224
$582$	0	0
$583$	−2.70850	−0.112174
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.7085	−0.854731	−0.427366	−	0.904079i	$-0.640558\pi$
$587$	−20.7085	−0.854731	−0.427366	+	0.904079i	$0.640558\pi$
$588$	0	0
$589$	−0.457513	−0.0188515
$590$	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
$595$	10.8340	0.444150
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.5830	1.24959	0.624794	−	0.780790i	$-0.285183\pi$
$599$	30.5830	1.24959	0.624794	+	0.780790i	$0.285183\pi$
$600$	0	0
$601$	−32.4575	−1.32397	−0.661985	−	0.749517i	$-0.730286\pi$
$601$	−32.4575	−1.32397	−0.661985	+	0.749517i	$0.730286\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13.6458	0.554779
$606$	0	0
$607$	−35.8745	−1.45610	−0.728051	−	0.685523i	$-0.759573\pi$
$607$	−35.8745	−1.45610	−0.728051	+	0.685523i	$0.759573\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.7490	1.28443
$612$	0	0
$613$	−2.93725	−0.118635	−0.0593173	−	0.998239i	$-0.518892\pi$
$613$	−2.93725	−0.118635	−0.0593173	+	0.998239i	$0.518892\pi$
$614$	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
$619$	−12.3542	−0.496559	−0.248280	−	0.968688i	$-0.579865\pi$
$619$	−12.3542	−0.496559	−0.248280	+	0.968688i	$0.579865\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.29150	−0.331660
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.8340	0.910451
$630$	0	0
$631$	−23.8745	−0.950429	−0.475215	−	0.879870i	$-0.657630\pi$
$631$	−23.8745	−0.950429	−0.475215	+	0.879870i	$0.657630\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17.4170	−0.691172
$636$	0	0
$637$	−15.8745	−0.628971
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.4575	1.12400	0.562002	−	0.827136i	$-0.310031\pi$
$641$	28.4575	1.12400	0.562002	+	0.827136i	$0.310031\pi$
$642$	0	0
$643$	−24.3542	−0.960438	−0.480219	−	0.877149i	$-0.659443\pi$
$643$	−24.3542	−0.960438	−0.480219	+	0.877149i	$0.659443\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.1660	0.517609	0.258805	−	0.965930i	$-0.416671\pi$
$647$	13.1660	0.517609	0.258805	+	0.965930i	$0.416671\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.8745	0.621217	0.310609	−	0.950538i	$-0.399467\pi$
$653$	15.8745	0.621217	0.310609	+	0.950538i	$0.399467\pi$
$654$	0	0
$655$	36.0000	1.40664
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.06275	0.275126	0.137563	−	0.990493i	$-0.456073\pi$
$659$	7.06275	0.275126	0.137563	+	0.990493i	$0.456073\pi$
$660$	0	0
$661$	27.6458	1.07530	0.537648	−	0.843170i	$-0.319313\pi$
$661$	27.6458	1.07530	0.537648	+	0.843170i	$0.319313\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.16601	−0.0452159
$666$	0	0
$667$	−9.29150	−0.359768
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.583005	−0.0225067
$672$	0	0
$673$	−12.7085	−0.489877	−0.244938	−	0.969539i	$-0.578768\pi$
$673$	−12.7085	−0.489877	−0.244938	+	0.969539i	$0.578768\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.06275	0.271443	0.135722	−	0.990747i	$-0.456665\pi$
$677$	7.06275	0.271443	0.135722	+	0.990747i	$0.456665\pi$
$678$	0	0
$679$	2.58301	0.0991266
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33.8745	−1.29617	−0.648086	−	0.761567i	$-0.724430\pi$
$683$	−33.8745	−1.29617	−0.648086	+	0.761567i	$0.724430\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.70850	0.331767
$690$	0	0
$691$	−17.8745	−0.679978	−0.339989	−	0.940429i	$-0.610423\pi$
$691$	−17.8745	−0.679978	−0.339989	+	0.940429i	$0.610423\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.58301	−0.249708
$696$	0	0
$697$	−19.7490	−0.748047
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.0627	0.719990	0.359995	−	0.932954i	$-0.382778\pi$
$701$	19.0627	0.719990	0.359995	+	0.932954i	$0.382778\pi$
$702$	0	0
$703$	−2.45751	−0.0926869
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.5830	0.698886
$708$	0	0
$709$	−38.9373	−1.46232	−0.731160	−	0.682206i	$-0.761021\pi$
$709$	−38.9373	−1.46232	−0.731160	+	0.682206i	$0.761021\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.29150	0.0483672
$714$	0	0
$715$	14.3320	0.535987
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.7490	−0.736514	−0.368257	−	0.929724i	$-0.620045\pi$
$719$	−19.7490	−0.736514	−0.368257	+	0.929724i	$0.620045\pi$
$720$	0	0
$721$	−13.4170	−0.499675
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21.2915	0.790747
$726$	0	0
$727$	−40.3320	−1.49583	−0.747916	−	0.663793i	$-0.768945\pi$
$727$	−40.3320	−1.49583	−0.747916	+	0.663793i	$0.768945\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.16601	−0.0431265
$732$	0	0
$733$	−31.3948	−1.15959	−0.579796	−	0.814762i	$-0.696868\pi$
$733$	−31.3948	−1.15959	−0.579796	+	0.814762i	$0.696868\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.5830	−0.905527
$738$	0	0
$739$	10.5830	0.389302	0.194651	−	0.980873i	$-0.437643\pi$
$739$	10.5830	0.389302	0.194651	+	0.980873i	$0.437643\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.7490	−0.724521	−0.362261	−	0.932077i	$-0.617995\pi$
$743$	−19.7490	−0.724521	−0.362261	+	0.932077i	$0.617995\pi$
$744$	0	0
$745$	17.0405	0.624316
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27.2915	0.997210
$750$	0	0
$751$	−42.4575	−1.54930	−0.774648	−	0.632392i	$-0.782073\pi$
$751$	−42.4575	−1.54930	−0.774648	+	0.632392i	$0.782073\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28.4575	1.03567
$756$	0	0
$757$	5.77124	0.209759	0.104880	−	0.994485i	$-0.466554\pi$
$757$	5.77124	0.209759	0.104880	+	0.994485i	$0.466554\pi$
$758$	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
$763$	36.4575	1.31985
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	25.0405	0.902984	0.451492	−	0.892275i	$-0.350892\pi$
$769$	25.0405	0.902984	0.451492	+	0.892275i	$0.350892\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.64575	0.0591936	0.0295968	−	0.999562i	$-0.490578\pi$
$773$	1.64575	0.0591936	0.0295968	+	0.999562i	$0.490578\pi$
$774$	0	0
$775$	−2.95948	−0.106308
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.12549	0.0761537
$780$	0	0
$781$	9.87451	0.353338
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	44.3542	1.58106	0.790529	−	0.612424i	$-0.209806\pi$
$787$	44.3542	1.58106	0.790529	+	0.612424i	$0.209806\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.1660	0.468129
$792$	0	0
$793$	1.87451	0.0665657
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.5203	1.25819	0.629096	−	0.777328i	$-0.283425\pi$
$797$	35.5203	1.25819	0.629096	+	0.777328i	$0.283425\pi$
$798$	0	0
$799$	19.7490	0.698670
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	3.29150	0.116010
$806$	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
$811$	−29.8745	−1.04904	−0.524518	−	0.851399i	$-0.675754\pi$
$811$	−29.8745	−1.04904	−0.524518	+	0.851399i	$0.675754\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.16601	−0.0408436
$816$	0	0
$817$	0.125492	0.00439041
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.12549	0.283582	0.141791	−	0.989897i	$-0.454714\pi$
$821$	8.12549	0.283582	0.141791	+	0.989897i	$0.454714\pi$
$822$	0	0
$823$	13.2915	0.463313	0.231656	−	0.972798i	$-0.425586\pi$
$823$	13.2915	0.463313	0.231656	+	0.972798i	$0.425586\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.8118	−0.515055	−0.257528	−	0.966271i	$-0.582908\pi$
$827$	−14.8118	−0.515055	−0.257528	+	0.966271i	$0.582908\pi$
$828$	0	0
$829$	−15.4170	−0.535454	−0.267727	−	0.963495i	$-0.586273\pi$
$829$	−15.4170	−0.535454	−0.267727	+	0.963495i	$0.586273\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.87451	−0.342131
$834$	0	0
$835$	31.5425	1.09157
$836$	0	0
$837$	0	0
$838$	0	0
$839$	43.7490	1.51038	0.755192	−	0.655504i	$-0.227544\pi$
$839$	43.7490	1.51038	0.755192	+	0.655504i	$0.227544\pi$
$840$	0	0
$841$	57.3320	1.97697
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24.6863	−0.849233
$846$	0	0
$847$	16.5830	0.569799
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.93725	0.237806
$852$	0	0
$853$	27.1660	0.930146	0.465073	−	0.885272i	$-0.346028\pi$
$853$	27.1660	0.930146	0.465073	+	0.885272i	$0.346028\pi$
$854$	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
$859$	−13.4170	−0.457782	−0.228891	−	0.973452i	$-0.573510\pi$
$859$	−13.4170	−0.457782	−0.228891	+	0.973452i	$0.573510\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.8340	0.368793	0.184397	−	0.982852i	$-0.440967\pi$
$863$	10.8340	0.368793	0.184397	+	0.982852i	$0.440967\pi$
$864$	0	0
$865$	−26.1255	−0.888293
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.1255	0.479175
$870$	0	0
$871$	79.0405	2.67819
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−13.2915	−0.448822	−0.224411	−	0.974495i	$-0.572046\pi$
$877$	−13.2915	−0.448822	−0.224411	+	0.974495i	$0.572046\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31.7490	−1.06965	−0.534826	−	0.844962i	$-0.679623\pi$
$881$	−31.7490	−1.06965	−0.534826	+	0.844962i	$0.679623\pi$
$882$	0	0
$883$	7.29150	0.245379	0.122689	−	0.992445i	$-0.460848\pi$
$883$	7.29150	0.245379	0.122689	+	0.992445i	$0.460848\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.7490	0.864567	0.432284	−	0.901738i	$-0.357708\pi$
$887$	25.7490	0.864567	0.432284	+	0.901738i	$0.357708\pi$
$888$	0	0
$889$	−21.1660	−0.709885
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.12549	−0.0711269
$894$	0	0
$895$	−25.1660	−0.841207
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	5.41699	0.180466
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25.7490	−0.855926
$906$	0	0
$907$	45.5203	1.51148	0.755738	−	0.654874i	$-0.227278\pi$
$907$	45.5203	1.51148	0.755738	+	0.654874i	$0.227278\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.29150	0.109052	0.0545262	−	0.998512i	$-0.482635\pi$
$911$	3.29150	0.109052	0.0545262	+	0.998512i	$0.482635\pi$
$912$	0	0
$913$	22.4575	0.743235
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43.7490	1.44472
$918$	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
$925$	−15.8967	−0.522681
$926$	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
$931$	1.06275	0.0348301
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.91503	0.291553
$936$	0	0
$937$	−28.5830	−0.933766	−0.466883	−	0.884319i	$-0.654623\pi$
$937$	−28.5830	−0.933766	−0.466883	+	0.884319i	$0.654623\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.8118	0.874038	0.437019	−	0.899452i	$-0.356034\pi$
$941$	26.8118	0.874038	0.437019	+	0.899452i	$0.356034\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.12549	0.0690692	0.0345346	−	0.999404i	$-0.489005\pi$
$947$	2.12549	0.0690692	0.0345346	+	0.999404i	$0.489005\pi$
$948$	0	0
$949$	−38.5830	−1.25246
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.0405	0.746356	0.373178	−	0.927760i	$-0.378268\pi$
$953$	23.0405	0.746356	0.373178	+	0.927760i	$0.378268\pi$
$954$	0	0
$955$	−5.41699	−0.175290
$956$	0	0
$957$	0	0
$958$	0	0
$959$	43.7490	1.41273
$960$	0	0
$961$	−29.3320	−0.946194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	36.2065	1.16553
$966$	0	0
$967$	−23.8745	−0.767752	−0.383876	−	0.923385i	$-0.625411\pi$
$967$	−23.8745	−0.767752	−0.383876	+	0.923385i	$0.625411\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.2288	−0.649172	−0.324586	−	0.945856i	$-0.605225\pi$
$971$	−20.2288	−0.649172	−0.324586	+	0.945856i	$0.605225\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.8745	0.699828	0.349914	−	0.936782i	$-0.386211\pi$
$977$	21.8745	0.699828	0.349914	+	0.936782i	$0.386211\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.8745	−0.697688	−0.348844	−	0.937181i	$-0.613426\pi$
$983$	−21.8745	−0.697688	−0.348844	+	0.937181i	$0.613426\pi$
$984$	0	0
$985$	−9.87451	−0.314628
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.354249	−0.0112645
$990$	0	0
$991$	14.4575	0.459258	0.229629	−	0.973278i	$-0.426249\pi$
$991$	14.4575	0.459258	0.229629	+	0.973278i	$0.426249\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	23.0405	0.730434
$996$	0	0
$997$	−40.5830	−1.28528	−0.642638	−	0.766170i	$-0.722160\pi$
$997$	−40.5830	−1.28528	−0.642638	+	0.766170i	$0.722160\pi$
$998$	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.1.1		2
3.2	odd	2		3312.2.a.v.1.2		2
4.3	odd	2		414.2.a.g.1.1	yes	2
12.11	even	2		414.2.a.e.1.2	✓	2
92.91	even	2		9522.2.a.bc.1.2		2
276.275	odd	2		9522.2.a.bb.1.1		2

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

\(f(q)\)	\(=\)	\(q-1.64575 q^{5} -2.00000 q^{7} +O(q^{10})\)	\(q-1.64575 q^{5} -2.00000 q^{7} -1.64575 q^{11} +5.29150 q^{13} +3.29150 q^{17} -0.354249 q^{19} +1.00000 q^{23} -2.29150 q^{25} -9.29150 q^{29} +1.29150 q^{31} +3.29150 q^{35} +6.93725 q^{37} -6.00000 q^{41} -0.354249 q^{43} +6.00000 q^{47} -3.00000 q^{49} +1.64575 q^{53} +2.70850 q^{55} +0.354249 q^{61} -8.70850 q^{65} +14.9373 q^{67} -6.00000 q^{71} -7.29150 q^{73} +3.29150 q^{77} -8.58301 q^{79} -13.6458 q^{83} -5.41699 q^{85} -10.5830 q^{91} +0.583005 q^{95} -1.29150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{5} - 4 q^{7} + 2 q^{11} - 4 q^{17} - 6 q^{19} + 2 q^{23} + 6 q^{25} - 8 q^{29} - 8 q^{31} - 4 q^{35} - 2 q^{37} - 12 q^{41} - 6 q^{43} + 12 q^{47} - 6 q^{49} - 2 q^{53} + 16 q^{55} + 6 q^{61} - 28 q^{65} + 14 q^{67} - 12 q^{71} - 4 q^{73} - 4 q^{77} + 4 q^{79} - 22 q^{83} - 32 q^{85} - 20 q^{95} + 8 q^{97}+O(q^{100}) \) 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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