LMFDB - Embedded newform 3332.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278 q^{3} +2.30278 q^{5} -1.30278 q^{9} -6.60555 q^{13} +3.00000 q^{15} -1.00000 q^{17} -6.60555 q^{19} +0.302776 q^{25} -5.60555 q^{27} -4.30278 q^{31} -2.60555 q^{37} -8.60555 q^{39} -3.90833 q^{41} +7.30278 q^{43} -3.00000 q^{45} -4.60555 q^{47} -1.30278 q^{51} -3.69722 q^{53} -8.60555 q^{57} -9.21110 q^{59} +7.90833 q^{61} -15.2111 q^{65} -1.69722 q^{67} -7.81665 q^{71} +7.90833 q^{73} +0.394449 q^{75} +12.6056 q^{79} -3.39445 q^{81} -6.00000 q^{83} -2.30278 q^{85} +16.6056 q^{89} -5.60555 q^{93} -15.2111 q^{95} +6.30278 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + q^{5} + q^{9} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{19} - 3 q^{25} - 4 q^{27} - 5 q^{31} + 2 q^{37} - 10 q^{39} + 3 q^{41} + 11 q^{43} - 6 q^{45} - 2 q^{47} + q^{51} - 11 q^{53} - 10 q^{57}+ \cdots + 9 q^{97}+O(q^{100}) $ 2 * q - q^3 + q^5 + q^9 - 6 * q^13 + 6 * q^15 - 2 * q^17 - 6 * q^19 - 3 * q^25 - 4 * q^27 - 5 * q^31 + 2 * q^37 - 10 * q^39 + 3 * q^41 + 11 * q^43 - 6 * q^45 - 2 * q^47 + q^51 - 11 * q^53 - 10 * q^57 - 4 * q^59 + 5 * q^61 - 16 * q^65 - 7 * q^67 + 6 * q^71 + 5 * q^73 + 8 * q^75 + 18 * q^79 - 14 * q^81 - 12 * q^83 - q^85 + 26 * q^89 - 4 * q^93 - 16 * q^95 + 9 * 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
$3$	1.30278	0.752158	0.376079	−	0.926588i	$-0.377272\pi$
$3$	1.30278	0.752158	0.376079	+	0.926588i	$0.377272\pi$
$4$	0	0
$5$	2.30278	1.02983	0.514916	−	0.857240i	$-0.327823\pi$
$5$	2.30278	1.02983	0.514916	+	0.857240i	$0.327823\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.60555	−1.83205	−0.916025	−	0.401121i	$-0.868621\pi$
$13$	−6.60555	−1.83205	−0.916025	+	0.401121i	$0.868621\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.60555	−1.51542	−0.757709	−	0.652593i	$-0.773681\pi$
$19$	−6.60555	−1.51542	−0.757709	+	0.652593i	$0.773681\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	−5.60555	−1.07879
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.30278	−0.772801	−0.386401	−	0.922331i	$-0.626282\pi$
$31$	−4.30278	−0.772801	−0.386401	+	0.922331i	$0.626282\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.60555	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$37$	−2.60555	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$38$	0	0
$39$	−8.60555	−1.37799
$40$	0	0
$41$	−3.90833	−0.610378	−0.305189	−	0.952292i	$-0.598720\pi$
$41$	−3.90833	−0.610378	−0.305189	+	0.952292i	$0.598720\pi$
$42$	0	0
$43$	7.30278	1.11366	0.556831	−	0.830626i	$-0.312017\pi$
$43$	7.30278	1.11366	0.556831	+	0.830626i	$0.312017\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−4.60555	−0.671789	−0.335894	−	0.941900i	$-0.609039\pi$
$47$	−4.60555	−0.671789	−0.335894	+	0.941900i	$0.609039\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.30278	−0.182425
$52$	0	0
$53$	−3.69722	−0.507853	−0.253926	−	0.967224i	$-0.581722\pi$
$53$	−3.69722	−0.507853	−0.253926	+	0.967224i	$0.581722\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.60555	−1.13983
$58$	0	0
$59$	−9.21110	−1.19918	−0.599592	−	0.800306i	$-0.704670\pi$
$59$	−9.21110	−1.19918	−0.599592	+	0.800306i	$0.704670\pi$
$60$	0	0
$61$	7.90833	1.01256	0.506279	−	0.862370i	$-0.331021\pi$
$61$	7.90833	1.01256	0.506279	+	0.862370i	$0.331021\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.2111	−1.88671
$66$	0	0
$67$	−1.69722	−0.207349	−0.103674	−	0.994611i	$-0.533060\pi$
$67$	−1.69722	−0.207349	−0.103674	+	0.994611i	$0.533060\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.81665	−0.927666	−0.463833	−	0.885923i	$-0.653526\pi$
$71$	−7.81665	−0.927666	−0.463833	+	0.885923i	$0.653526\pi$
$72$	0	0
$73$	7.90833	0.925600	0.462800	−	0.886463i	$-0.346845\pi$
$73$	7.90833	0.925600	0.462800	+	0.886463i	$0.346845\pi$
$74$	0	0
$75$	0.394449	0.0455470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.6056	1.41824	0.709118	−	0.705090i	$-0.249093\pi$
$79$	12.6056	1.41824	0.709118	+	0.705090i	$0.249093\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−2.30278	−0.249771
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.6056	1.76018	0.880092	−	0.474802i	$-0.157480\pi$
$89$	16.6056	1.76018	0.880092	+	0.474802i	$0.157480\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.60555	−0.581269
$94$	0	0
$95$	−15.2111	−1.56063
$96$	0	0
$97$	6.30278	0.639950	0.319975	−	0.947426i	$-0.396325\pi$
$97$	6.30278	0.639950	0.319975	+	0.947426i	$0.396325\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.8167	−1.97183	−0.985915	−	0.167245i	$-0.946513\pi$
$101$	−19.8167	−1.97183	−0.985915	+	0.167245i	$0.946513\pi$
$102$	0	0
$103$	7.21110	0.710531	0.355266	−	0.934765i	$-0.384390\pi$
$103$	7.21110	0.710531	0.355266	+	0.934765i	$0.384390\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.39445	−0.134806	−0.0674032	−	0.997726i	$-0.521471\pi$
$107$	−1.39445	−0.134806	−0.0674032	+	0.997726i	$0.521471\pi$
$108$	0	0
$109$	8.42221	0.806701	0.403350	−	0.915046i	$-0.367846\pi$
$109$	8.42221	0.806701	0.403350	+	0.915046i	$0.367846\pi$
$110$	0	0
$111$	−3.39445	−0.322187
$112$	0	0
$113$	4.60555	0.433254	0.216627	−	0.976254i	$-0.430494\pi$
$113$	4.60555	0.433254	0.216627	+	0.976254i	$0.430494\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.60555	0.795583
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−5.09167	−0.459101
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	14.9083	1.32290	0.661450	−	0.749989i	$-0.269941\pi$
$127$	14.9083	1.32290	0.661450	+	0.749989i	$0.269941\pi$
$128$	0	0
$129$	9.51388	0.837650
$130$	0	0
$131$	21.2111	1.85322	0.926611	−	0.376021i	$-0.122708\pi$
$131$	21.2111	1.85322	0.926611	+	0.376021i	$0.122708\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.9083	−1.11097
$136$	0	0
$137$	−12.6972	−1.08480	−0.542399	−	0.840121i	$-0.682484\pi$
$137$	−12.6972	−1.08480	−0.542399	+	0.840121i	$0.682484\pi$
$138$	0	0
$139$	−7.09167	−0.601508	−0.300754	−	0.953702i	$-0.597238\pi$
$139$	−7.09167	−0.601508	−0.300754	+	0.953702i	$0.597238\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.7250	1.45209	0.726044	−	0.687649i	$-0.241357\pi$
$149$	17.7250	1.45209	0.726044	+	0.687649i	$0.241357\pi$
$150$	0	0
$151$	−11.1194	−0.904886	−0.452443	−	0.891793i	$-0.649447\pi$
$151$	−11.1194	−0.904886	−0.452443	+	0.891793i	$0.649447\pi$
$152$	0	0
$153$	1.30278	0.105323
$154$	0	0
$155$	−9.90833	−0.795856
$156$	0	0
$157$	7.21110	0.575509	0.287754	−	0.957704i	$-0.407091\pi$
$157$	7.21110	0.575509	0.287754	+	0.957704i	$0.407091\pi$
$158$	0	0
$159$	−4.81665	−0.381985
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.6056	0.987343	0.493671	−	0.869648i	$-0.335655\pi$
$163$	12.6056	0.987343	0.493671	+	0.869648i	$0.335655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.7250	1.83589	0.917947	−	0.396703i	$-0.129846\pi$
$167$	23.7250	1.83589	0.917947	+	0.396703i	$0.129846\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	0	0
$171$	8.60555	0.658083
$172$	0	0
$173$	−17.3028	−1.31551	−0.657753	−	0.753234i	$-0.728493\pi$
$173$	−17.3028	−1.31551	−0.657753	+	0.753234i	$0.728493\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	9.90833	0.740583	0.370292	−	0.928916i	$-0.379258\pi$
$179$	9.90833	0.740583	0.370292	+	0.928916i	$0.379258\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	10.3028	0.761603
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.7250	1.49961	0.749803	−	0.661661i	$-0.230148\pi$
$191$	20.7250	1.49961	0.749803	+	0.661661i	$0.230148\pi$
$192$	0	0
$193$	−11.8167	−0.850581	−0.425291	−	0.905057i	$-0.639828\pi$
$193$	−11.8167	−0.850581	−0.425291	+	0.905057i	$0.639828\pi$
$194$	0	0
$195$	−19.8167	−1.41910
$196$	0	0
$197$	−4.60555	−0.328132	−0.164066	−	0.986449i	$-0.552461\pi$
$197$	−4.60555	−0.328132	−0.164066	+	0.986449i	$0.552461\pi$
$198$	0	0
$199$	−21.1194	−1.49712	−0.748558	−	0.663069i	$-0.769254\pi$
$199$	−21.1194	−1.49712	−0.748558	+	0.663069i	$0.769254\pi$
$200$	0	0
$201$	−2.21110	−0.155959
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−21.0278	−1.44761	−0.723805	−	0.690004i	$-0.757609\pi$
$211$	−21.0278	−1.44761	−0.723805	+	0.690004i	$0.757609\pi$
$212$	0	0
$213$	−10.1833	−0.697751
$214$	0	0
$215$	16.8167	1.14689
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.3028	0.696197
$220$	0	0
$221$	6.60555	0.444337
$222$	0	0
$223$	9.02776	0.604543	0.302272	−	0.953222i	$-0.402255\pi$
$223$	9.02776	0.604543	0.302272	+	0.953222i	$0.402255\pi$
$224$	0	0
$225$	−0.394449	−0.0262966
$226$	0	0
$227$	−5.51388	−0.365969	−0.182984	−	0.983116i	$-0.558576\pi$
$227$	−5.51388	−0.365969	−0.182984	+	0.983116i	$0.558576\pi$
$228$	0	0
$229$	−10.7889	−0.712950	−0.356475	−	0.934305i	$-0.616022\pi$
$229$	−10.7889	−0.712950	−0.356475	+	0.934305i	$0.616022\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.8167	−0.905159	−0.452580	−	0.891724i	$-0.649496\pi$
$233$	−13.8167	−0.905159	−0.452580	+	0.891724i	$0.649496\pi$
$234$	0	0
$235$	−10.6056	−0.691830
$236$	0	0
$237$	16.4222	1.06674
$238$	0	0
$239$	11.3028	0.731116	0.365558	−	0.930789i	$-0.380878\pi$
$239$	11.3028	0.731116	0.365558	+	0.930789i	$0.380878\pi$
$240$	0	0
$241$	−8.48612	−0.546639	−0.273320	−	0.961923i	$-0.588122\pi$
$241$	−8.48612	−0.546639	−0.273320	+	0.961923i	$0.588122\pi$
$242$	0	0
$243$	12.3944	0.795104
$244$	0	0
$245$	0	0
$246$	0	0
$247$	43.6333	2.77632
$248$	0	0
$249$	−7.81665	−0.495360
$250$	0	0
$251$	−4.18335	−0.264050	−0.132025	−	0.991246i	$-0.542148\pi$
$251$	−4.18335	−0.264050	−0.132025	+	0.991246i	$0.542148\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−3.21110	−0.200303	−0.100152	−	0.994972i	$-0.531933\pi$
$257$	−3.21110	−0.200303	−0.100152	+	0.994972i	$0.531933\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.4222	−1.13596	−0.567981	−	0.823042i	$-0.692275\pi$
$263$	−18.4222	−1.13596	−0.567981	+	0.823042i	$0.692275\pi$
$264$	0	0
$265$	−8.51388	−0.523003
$266$	0	0
$267$	21.6333	1.32394
$268$	0	0
$269$	−15.2111	−0.927437	−0.463719	−	0.885983i	$-0.653485\pi$
$269$	−15.2111	−0.927437	−0.463719	+	0.885983i	$0.653485\pi$
$270$	0	0
$271$	1.21110	0.0735692	0.0367846	−	0.999323i	$-0.488288\pi$
$271$	1.21110	0.0735692	0.0367846	+	0.999323i	$0.488288\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.4222	−1.34722	−0.673610	−	0.739087i	$-0.735257\pi$
$277$	−22.4222	−1.34722	−0.673610	+	0.739087i	$0.735257\pi$
$278$	0	0
$279$	5.60555	0.335596
$280$	0	0
$281$	3.69722	0.220558	0.110279	−	0.993901i	$-0.464826\pi$
$281$	3.69722	0.220558	0.110279	+	0.993901i	$0.464826\pi$
$282$	0	0
$283$	9.51388	0.565541	0.282771	−	0.959188i	$-0.408746\pi$
$283$	9.51388	0.565541	0.282771	+	0.959188i	$0.408746\pi$
$284$	0	0
$285$	−19.8167	−1.17384
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.21110	0.481343
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	−21.2111	−1.23496
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−25.8167	−1.48313
$304$	0	0
$305$	18.2111	1.04276
$306$	0	0
$307$	1.21110	0.0691213	0.0345606	−	0.999403i	$-0.488997\pi$
$307$	1.21110	0.0691213	0.0345606	+	0.999403i	$0.488997\pi$
$308$	0	0
$309$	9.39445	0.534432
$310$	0	0
$311$	1.88057	0.106637	0.0533187	−	0.998578i	$-0.483020\pi$
$311$	1.88057	0.106637	0.0533187	+	0.998578i	$0.483020\pi$
$312$	0	0
$313$	−33.3305	−1.88395	−0.941977	−	0.335679i	$-0.891034\pi$
$313$	−33.3305	−1.88395	−0.941977	+	0.335679i	$0.891034\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.81665	−0.101396
$322$	0	0
$323$	6.60555	0.367543
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	10.9722	0.606766
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.69722	0.313148	0.156574	−	0.987666i	$-0.449955\pi$
$331$	5.69722	0.313148	0.156574	+	0.987666i	$0.449955\pi$
$332$	0	0
$333$	3.39445	0.186015
$334$	0	0
$335$	−3.90833	−0.213535
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	0	0
$349$	−29.2111	−1.56363	−0.781817	−	0.623508i	$-0.785707\pi$
$349$	−29.2111	−1.56363	−0.781817	+	0.623508i	$0.785707\pi$
$350$	0	0
$351$	37.0278	1.97640
$352$	0	0
$353$	15.6333	0.832077	0.416039	−	0.909347i	$-0.363418\pi$
$353$	15.6333	0.832077	0.416039	+	0.909347i	$0.363418\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.90833	0.364608	0.182304	−	0.983242i	$-0.441645\pi$
$359$	6.90833	0.364608	0.182304	+	0.983242i	$0.441645\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	0	0
$363$	−14.3305	−0.752158
$364$	0	0
$365$	18.2111	0.953213
$366$	0	0
$367$	−6.33053	−0.330451	−0.165226	−	0.986256i	$-0.552835\pi$
$367$	−6.33053	−0.330451	−0.165226	+	0.986256i	$0.552835\pi$
$368$	0	0
$369$	5.09167	0.265062
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.9361	−1.60181	−0.800905	−	0.598792i	$-0.795648\pi$
$373$	−30.9361	−1.60181	−0.800905	+	0.598792i	$0.795648\pi$
$374$	0	0
$375$	−14.0917	−0.727691
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.78890	−0.348722	−0.174361	−	0.984682i	$-0.555786\pi$
$379$	−6.78890	−0.348722	−0.174361	+	0.984682i	$0.555786\pi$
$380$	0	0
$381$	19.4222	0.995030
$382$	0	0
$383$	−18.4222	−0.941331	−0.470665	−	0.882312i	$-0.655986\pi$
$383$	−18.4222	−0.941331	−0.470665	+	0.882312i	$0.655986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.51388	−0.483618
$388$	0	0
$389$	−10.1194	−0.513075	−0.256538	−	0.966534i	$-0.582582\pi$
$389$	−10.1194	−0.513075	−0.256538	+	0.966534i	$0.582582\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	27.6333	1.39392
$394$	0	0
$395$	29.0278	1.46054
$396$	0	0
$397$	21.9361	1.10094	0.550470	−	0.834855i	$-0.314448\pi$
$397$	21.9361	1.10094	0.550470	+	0.834855i	$0.314448\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.6056	−0.829242	−0.414621	−	0.909994i	$-0.636086\pi$
$401$	−16.6056	−0.829242	−0.414621	+	0.909994i	$0.636086\pi$
$402$	0	0
$403$	28.4222	1.41581
$404$	0	0
$405$	−7.81665	−0.388413
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.6056	1.31556	0.657780	−	0.753210i	$-0.271496\pi$
$409$	26.6056	1.31556	0.657780	+	0.753210i	$0.271496\pi$
$410$	0	0
$411$	−16.5416	−0.815939
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.8167	−0.678233
$416$	0	0
$417$	−9.23886	−0.452429
$418$	0	0
$419$	17.5139	0.855609	0.427804	−	0.903871i	$-0.359287\pi$
$419$	17.5139	0.855609	0.427804	+	0.903871i	$0.359287\pi$
$420$	0	0
$421$	−30.9361	−1.50773	−0.753866	−	0.657028i	$-0.771813\pi$
$421$	−30.9361	−1.50773	−0.753866	+	0.657028i	$0.771813\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	−0.302776	−0.0146868
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.6333	−0.753030	−0.376515	−	0.926411i	$-0.622878\pi$
$431$	−15.6333	−0.753030	−0.376515	+	0.926411i	$0.622878\pi$
$432$	0	0
$433$	−3.81665	−0.183417	−0.0917083	−	0.995786i	$-0.529233\pi$
$433$	−3.81665	−0.183417	−0.0917083	+	0.995786i	$0.529233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	7.90833	0.377444	0.188722	−	0.982031i	$-0.439566\pi$
$439$	7.90833	0.377444	0.188722	+	0.982031i	$0.439566\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.2111	−1.00777	−0.503885	−	0.863771i	$-0.668096\pi$
$443$	−21.2111	−1.00777	−0.503885	+	0.863771i	$0.668096\pi$
$444$	0	0
$445$	38.2389	1.81270
$446$	0	0
$447$	23.0917	1.09220
$448$	0	0
$449$	−32.2389	−1.52145	−0.760723	−	0.649077i	$-0.775155\pi$
$449$	−32.2389	−1.52145	−0.760723	+	0.649077i	$0.775155\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−14.4861	−0.680617
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.09167	−0.144622	−0.0723112	−	0.997382i	$-0.523037\pi$
$457$	−3.09167	−0.144622	−0.0723112	+	0.997382i	$0.523037\pi$
$458$	0	0
$459$	5.60555	0.261645
$460$	0	0
$461$	31.8167	1.48185	0.740925	−	0.671588i	$-0.234388\pi$
$461$	31.8167	1.48185	0.740925	+	0.671588i	$0.234388\pi$
$462$	0	0
$463$	−16.6972	−0.775986	−0.387993	−	0.921662i	$-0.626832\pi$
$463$	−16.6972	−0.775986	−0.387993	+	0.921662i	$0.626832\pi$
$464$	0	0
$465$	−12.9083	−0.598609
$466$	0	0
$467$	35.4500	1.64043	0.820214	−	0.572056i	$-0.193854\pi$
$467$	35.4500	1.64043	0.820214	+	0.572056i	$0.193854\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.39445	0.432873
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	4.81665	0.220539
$478$	0	0
$479$	−14.5139	−0.663156	−0.331578	−	0.943428i	$-0.607581\pi$
$479$	−14.5139	−0.663156	−0.331578	+	0.943428i	$0.607581\pi$
$480$	0	0
$481$	17.2111	0.784759
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.5139	0.659041
$486$	0	0
$487$	5.21110	0.236138	0.118069	−	0.993005i	$-0.462330\pi$
$487$	5.21110	0.236138	0.118069	+	0.993005i	$0.462330\pi$
$488$	0	0
$489$	16.4222	0.742638
$490$	0	0
$491$	−25.3305	−1.14315	−0.571575	−	0.820550i	$-0.693668\pi$
$491$	−25.3305	−1.14315	−0.571575	+	0.820550i	$0.693668\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.57779	0.0706318	0.0353159	−	0.999376i	$-0.488756\pi$
$499$	1.57779	0.0706318	0.0353159	+	0.999376i	$0.488756\pi$
$500$	0	0
$501$	30.9083	1.38088
$502$	0	0
$503$	−21.1472	−0.942906	−0.471453	−	0.881891i	$-0.656270\pi$
$503$	−21.1472	−0.942906	−0.471453	+	0.881891i	$0.656270\pi$
$504$	0	0
$505$	−45.6333	−2.03066
$506$	0	0
$507$	39.9083	1.77239
$508$	0	0
$509$	11.0278	0.488797	0.244398	−	0.969675i	$-0.421410\pi$
$509$	11.0278	0.488797	0.244398	+	0.969675i	$0.421410\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	37.0278	1.63482
$514$	0	0
$515$	16.6056	0.731728
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−22.5416	−0.989468
$520$	0	0
$521$	−39.3583	−1.72432	−0.862159	−	0.506638i	$-0.830888\pi$
$521$	−39.3583	−1.72432	−0.862159	+	0.506638i	$0.830888\pi$
$522$	0	0
$523$	−10.7889	−0.471766	−0.235883	−	0.971782i	$-0.575798\pi$
$523$	−10.7889	−0.471766	−0.235883	+	0.971782i	$0.575798\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.30278	0.187432
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	25.8167	1.11824
$534$	0	0
$535$	−3.21110	−0.138828
$536$	0	0
$537$	12.9083	0.557035
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.0555	1.80811	0.904054	−	0.427419i	$-0.140577\pi$
$541$	42.0555	1.80811	0.904054	+	0.427419i	$0.140577\pi$
$542$	0	0
$543$	−18.2389	−0.782704
$544$	0	0
$545$	19.3944	0.830767
$546$	0	0
$547$	−31.2111	−1.33449	−0.667245	−	0.744838i	$-0.732527\pi$
$547$	−31.2111	−1.33449	−0.667245	+	0.744838i	$0.732527\pi$
$548$	0	0
$549$	−10.3028	−0.439712
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.81665	−0.331798
$556$	0	0
$557$	33.6333	1.42509	0.712544	−	0.701627i	$-0.247543\pi$
$557$	33.6333	1.42509	0.712544	+	0.701627i	$0.247543\pi$
$558$	0	0
$559$	−48.2389	−2.04029
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.6333	1.16461	0.582303	−	0.812972i	$-0.302152\pi$
$563$	27.6333	1.16461	0.582303	+	0.812972i	$0.302152\pi$
$564$	0	0
$565$	10.6056	0.446179
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.88057	0.204604	0.102302	−	0.994753i	$-0.467379\pi$
$569$	4.88057	0.204604	0.102302	+	0.994753i	$0.467379\pi$
$570$	0	0
$571$	−9.02776	−0.377800	−0.188900	−	0.981996i	$-0.560492\pi$
$571$	−9.02776	−0.377800	−0.188900	+	0.981996i	$0.560492\pi$
$572$	0	0
$573$	27.0000	1.12794
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.2389	−1.17560	−0.587800	−	0.809007i	$-0.700006\pi$
$577$	−28.2389	−1.17560	−0.587800	+	0.809007i	$0.700006\pi$
$578$	0	0
$579$	−15.3944	−0.639771
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	19.8167	0.819318
$586$	0	0
$587$	10.6056	0.437738	0.218869	−	0.975754i	$-0.429763\pi$
$587$	10.6056	0.437738	0.218869	+	0.975754i	$0.429763\pi$
$588$	0	0
$589$	28.4222	1.17112
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−43.8167	−1.79933	−0.899667	−	0.436576i	$-0.856191\pi$
$593$	−43.8167	−1.79933	−0.899667	+	0.436576i	$0.856191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−27.5139	−1.12607
$598$	0	0
$599$	43.9638	1.79631	0.898157	−	0.439675i	$-0.144906\pi$
$599$	43.9638	1.79631	0.898157	+	0.439675i	$0.144906\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	2.21110	0.0900431
$604$	0	0
$605$	−25.3305	−1.02983
$606$	0	0
$607$	21.5139	0.873221	0.436611	−	0.899651i	$-0.356179\pi$
$607$	21.5139	0.873221	0.436611	+	0.899651i	$0.356179\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.4222	1.23075
$612$	0	0
$613$	29.1472	1.17724	0.588622	−	0.808408i	$-0.299671\pi$
$613$	29.1472	1.17724	0.588622	+	0.808408i	$0.299671\pi$
$614$	0	0
$615$	−11.7250	−0.472797
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	0	0
$619$	43.6333	1.75377	0.876885	−	0.480700i	$-0.159617\pi$
$619$	43.6333	1.75377	0.876885	+	0.480700i	$0.159617\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.60555	0.103890
$630$	0	0
$631$	−8.11943	−0.323229	−0.161615	−	0.986854i	$-0.551670\pi$
$631$	−8.11943	−0.323229	−0.161615	+	0.986854i	$0.551670\pi$
$632$	0	0
$633$	−27.3944	−1.08883
$634$	0	0
$635$	34.3305	1.36237
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.1833	0.402847
$640$	0	0
$641$	−12.8444	−0.507324	−0.253662	−	0.967293i	$-0.581635\pi$
$641$	−12.8444	−0.507324	−0.253662	+	0.967293i	$0.581635\pi$
$642$	0	0
$643$	−38.1472	−1.50438	−0.752189	−	0.658947i	$-0.771002\pi$
$643$	−38.1472	−1.50438	−0.752189	+	0.658947i	$0.771002\pi$
$644$	0	0
$645$	21.9083	0.862640
$646$	0	0
$647$	−18.4222	−0.724252	−0.362126	−	0.932129i	$-0.617949\pi$
$647$	−18.4222	−0.724252	−0.362126	+	0.932129i	$0.617949\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.0555	1.09790	0.548949	−	0.835856i	$-0.315028\pi$
$653$	28.0555	1.09790	0.548949	+	0.835856i	$0.315028\pi$
$654$	0	0
$655$	48.8444	1.90851
$656$	0	0
$657$	−10.3028	−0.401950
$658$	0	0
$659$	38.3028	1.49206	0.746032	−	0.665910i	$-0.231957\pi$
$659$	38.3028	1.49206	0.746032	+	0.665910i	$0.231957\pi$
$660$	0	0
$661$	14.1833	0.551668	0.275834	−	0.961205i	$-0.411046\pi$
$661$	14.1833	0.551668	0.275834	+	0.961205i	$0.411046\pi$
$662$	0	0
$663$	8.60555	0.334212
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	11.7611	0.454712
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20.1833	−0.778011	−0.389005	−	0.921235i	$-0.627181\pi$
$673$	−20.1833	−0.778011	−0.389005	+	0.921235i	$0.627181\pi$
$674$	0	0
$675$	−1.69722	−0.0653262
$676$	0	0
$677$	−48.4222	−1.86102	−0.930508	−	0.366271i	$-0.880634\pi$
$677$	−48.4222	−1.86102	−0.930508	+	0.366271i	$0.880634\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.18335	−0.275266
$682$	0	0
$683$	41.4500	1.58604	0.793019	−	0.609196i	$-0.208508\pi$
$683$	41.4500	1.58604	0.793019	+	0.609196i	$0.208508\pi$
$684$	0	0
$685$	−29.2389	−1.11716
$686$	0	0
$687$	−14.0555	−0.536251
$688$	0	0
$689$	24.4222	0.930412
$690$	0	0
$691$	−29.4861	−1.12170	−0.560852	−	0.827916i	$-0.689527\pi$
$691$	−29.4861	−1.12170	−0.560852	+	0.827916i	$0.689527\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.3305	−0.619452
$696$	0	0
$697$	3.90833	0.148038
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−18.8444	−0.711744	−0.355872	−	0.934535i	$-0.615816\pi$
$701$	−18.8444	−0.711744	−0.355872	+	0.934535i	$0.615816\pi$
$702$	0	0
$703$	17.2111	0.649129
$704$	0	0
$705$	−13.8167	−0.520365
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.1833	0.457555	0.228778	−	0.973479i	$-0.426527\pi$
$709$	12.1833	0.457555	0.228778	+	0.973479i	$0.426527\pi$
$710$	0	0
$711$	−16.4222	−0.615881
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.7250	0.549914
$718$	0	0
$719$	29.3028	1.09281	0.546405	−	0.837521i	$-0.315996\pi$
$719$	29.3028	1.09281	0.546405	+	0.837521i	$0.315996\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−11.0555	−0.411159
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22.4222	0.831594	0.415797	−	0.909458i	$-0.363503\pi$
$727$	22.4222	0.831594	0.415797	+	0.909458i	$0.363503\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	−7.30278	−0.270103
$732$	0	0
$733$	30.2389	1.11690	0.558449	−	0.829539i	$-0.311397\pi$
$733$	30.2389	1.11690	0.558449	+	0.829539i	$0.311397\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−25.3583	−0.932820	−0.466410	−	0.884569i	$-0.654453\pi$
$739$	−25.3583	−0.932820	−0.466410	+	0.884569i	$0.654453\pi$
$740$	0	0
$741$	56.8444	2.08823
$742$	0	0
$743$	1.81665	0.0666466	0.0333233	−	0.999445i	$-0.489391\pi$
$743$	1.81665	0.0666466	0.0333233	+	0.999445i	$0.489391\pi$
$744$	0	0
$745$	40.8167	1.49541
$746$	0	0
$747$	7.81665	0.285996
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.6056	0.678926	0.339463	−	0.940619i	$-0.389755\pi$
$751$	18.6056	0.678926	0.339463	+	0.940619i	$0.389755\pi$
$752$	0	0
$753$	−5.44996	−0.198608
$754$	0	0
$755$	−25.6056	−0.931881
$756$	0	0
$757$	−45.7250	−1.66190	−0.830951	−	0.556345i	$-0.812203\pi$
$757$	−45.7250	−1.66190	−0.830951	+	0.556345i	$0.812203\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.8167	−1.37085	−0.685426	−	0.728142i	$-0.740384\pi$
$761$	−37.8167	−1.37085	−0.685426	+	0.728142i	$0.740384\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	60.8444	2.19696
$768$	0	0
$769$	53.2666	1.92084	0.960422	−	0.278550i	$-0.0898539\pi$
$769$	53.2666	1.92084	0.960422	+	0.278550i	$0.0898539\pi$
$770$	0	0
$771$	−4.18335	−0.150660
$772$	0	0
$773$	−25.8167	−0.928560	−0.464280	−	0.885688i	$-0.653687\pi$
$773$	−25.8167	−0.928560	−0.464280	+	0.885688i	$0.653687\pi$
$774$	0	0
$775$	−1.30278	−0.0467971
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.8167	0.924978
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16.6056	0.592678
$786$	0	0
$787$	−2.42221	−0.0863423	−0.0431711	−	0.999068i	$-0.513746\pi$
$787$	−2.42221	−0.0863423	−0.0431711	+	0.999068i	$0.513746\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−52.2389	−1.85506
$794$	0	0
$795$	−11.0917	−0.393381
$796$	0	0
$797$	−0.422205	−0.0149553	−0.00747764	−	0.999972i	$-0.502380\pi$
$797$	−0.422205	−0.0149553	−0.00747764	+	0.999972i	$0.502380\pi$
$798$	0	0
$799$	4.60555	0.162933
$800$	0	0
$801$	−21.6333	−0.764375
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.8167	−0.697579
$808$	0	0
$809$	40.6056	1.42762	0.713808	−	0.700342i	$-0.246969\pi$
$809$	40.6056	1.42762	0.713808	+	0.700342i	$0.246969\pi$
$810$	0	0
$811$	−4.09167	−0.143678	−0.0718390	−	0.997416i	$-0.522887\pi$
$811$	−4.09167	−0.143678	−0.0718390	+	0.997416i	$0.522887\pi$
$812$	0	0
$813$	1.57779	0.0553357
$814$	0	0
$815$	29.0278	1.01680
$816$	0	0
$817$	−48.2389	−1.68766
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.6056	−0.370136	−0.185068	−	0.982726i	$-0.559251\pi$
$821$	−10.6056	−0.370136	−0.185068	+	0.982726i	$0.559251\pi$
$822$	0	0
$823$	43.8722	1.52929	0.764644	−	0.644453i	$-0.222915\pi$
$823$	43.8722	1.52929	0.764644	+	0.644453i	$0.222915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.6333	0.960904	0.480452	−	0.877021i	$-0.340473\pi$
$827$	27.6333	0.960904	0.480452	+	0.877021i	$0.340473\pi$
$828$	0	0
$829$	26.6056	0.924049	0.462024	−	0.886867i	$-0.347123\pi$
$829$	26.6056	0.924049	0.462024	+	0.886867i	$0.347123\pi$
$830$	0	0
$831$	−29.2111	−1.01332
$832$	0	0
$833$	0	0
$834$	0	0
$835$	54.6333	1.89066
$836$	0	0
$837$	24.1194	0.833689
$838$	0	0
$839$	−18.4222	−0.636005	−0.318003	−	0.948090i	$-0.603012\pi$
$839$	−18.4222	−0.636005	−0.318003	+	0.948090i	$0.603012\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	4.81665	0.165894
$844$	0	0
$845$	70.5416	2.42671
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12.3944	0.425376
$850$	0	0
$851$	0	0
$852$	0	0
$853$	16.4222	0.562286	0.281143	−	0.959666i	$-0.409287\pi$
$853$	16.4222	0.562286	0.281143	+	0.959666i	$0.409287\pi$
$854$	0	0
$855$	19.8167	0.677715
$856$	0	0
$857$	34.5416	1.17992	0.589960	−	0.807433i	$-0.299144\pi$
$857$	34.5416	1.17992	0.589960	+	0.807433i	$0.299144\pi$
$858$	0	0
$859$	−37.4500	−1.27778	−0.638888	−	0.769300i	$-0.720605\pi$
$859$	−37.4500	−1.27778	−0.638888	+	0.769300i	$0.720605\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.9083	0.643647	0.321823	−	0.946800i	$-0.395704\pi$
$863$	18.9083	0.643647	0.321823	+	0.946800i	$0.395704\pi$
$864$	0	0
$865$	−39.8444	−1.35475
$866$	0	0
$867$	1.30278	0.0442446
$868$	0	0
$869$	0	0
$870$	0	0
$871$	11.2111	0.379874
$872$	0	0
$873$	−8.21110	−0.277904
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.6056	−1.10101	−0.550506	−	0.834831i	$-0.685565\pi$
$877$	−32.6056	−1.10101	−0.550506	+	0.834831i	$0.685565\pi$
$878$	0	0
$879$	28.1833	0.950601
$880$	0	0
$881$	9.69722	0.326708	0.163354	−	0.986568i	$-0.447769\pi$
$881$	9.69722	0.326708	0.163354	+	0.986568i	$0.447769\pi$
$882$	0	0
$883$	−14.6695	−0.493667	−0.246833	−	0.969058i	$-0.579390\pi$
$883$	−14.6695	−0.493667	−0.246833	+	0.969058i	$0.579390\pi$
$884$	0	0
$885$	−27.6333	−0.928883
$886$	0	0
$887$	48.9083	1.64218	0.821090	−	0.570798i	$-0.193366\pi$
$887$	48.9083	1.64218	0.821090	+	0.570798i	$0.193366\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	30.4222	1.01804
$894$	0	0
$895$	22.8167	0.762677
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	3.69722	0.123172
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−32.2389	−1.07166
$906$	0	0
$907$	−22.8444	−0.758536	−0.379268	−	0.925287i	$-0.623824\pi$
$907$	−22.8444	−0.758536	−0.379268	+	0.925287i	$0.623824\pi$
$908$	0	0
$909$	25.8167	0.856284
$910$	0	0
$911$	37.8167	1.25292	0.626461	−	0.779453i	$-0.284503\pi$
$911$	37.8167	1.25292	0.626461	+	0.779453i	$0.284503\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	23.7250	0.784324
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.3305	0.802590	0.401295	−	0.915949i	$-0.368560\pi$
$919$	24.3305	0.802590	0.401295	+	0.915949i	$0.368560\pi$
$920$	0	0
$921$	1.57779	0.0519901
$922$	0	0
$923$	51.6333	1.69953
$924$	0	0
$925$	−0.788897	−0.0259388
$926$	0	0
$927$	−9.39445	−0.308554
$928$	0	0
$929$	−53.9361	−1.76959	−0.884793	−	0.465985i	$-0.845700\pi$
$929$	−53.9361	−1.76959	−0.884793	+	0.465985i	$0.845700\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.44996	0.0802081
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.78890	0.221784	0.110892	−	0.993832i	$-0.464629\pi$
$937$	6.78890	0.221784	0.110892	+	0.993832i	$0.464629\pi$
$938$	0	0
$939$	−43.4222	−1.41703
$940$	0	0
$941$	0.275019	0.00896537	0.00448269	−	0.999990i	$-0.498573\pi$
$941$	0.275019	0.00896537	0.00448269	+	0.999990i	$0.498573\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.8444	1.58723	0.793615	−	0.608420i	$-0.208196\pi$
$947$	48.8444	1.58723	0.793615	+	0.608420i	$0.208196\pi$
$948$	0	0
$949$	−52.2389	−1.69575
$950$	0	0
$951$	−7.81665	−0.253472
$952$	0	0
$953$	17.5139	0.567330	0.283665	−	0.958923i	$-0.408450\pi$
$953$	17.5139	0.567330	0.283665	+	0.958923i	$0.408450\pi$
$954$	0	0
$955$	47.7250	1.54434
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−12.4861	−0.402778
$962$	0	0
$963$	1.81665	0.0585409
$964$	0	0
$965$	−27.2111	−0.875956
$966$	0	0
$967$	16.5139	0.531051	0.265525	−	0.964104i	$-0.414455\pi$
$967$	16.5139	0.531051	0.265525	+	0.964104i	$0.414455\pi$
$968$	0	0
$969$	8.60555	0.276450
$970$	0	0
$971$	−36.8444	−1.18239	−0.591197	−	0.806527i	$-0.701344\pi$
$971$	−36.8444	−1.18239	−0.591197	+	0.806527i	$0.701344\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.60555	−0.0834444
$976$	0	0
$977$	−33.1472	−1.06047	−0.530236	−	0.847850i	$-0.677897\pi$
$977$	−33.1472	−1.06047	−0.530236	+	0.847850i	$0.677897\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−10.9722	−0.350317
$982$	0	0
$983$	−23.0917	−0.736510	−0.368255	−	0.929725i	$-0.620045\pi$
$983$	−23.0917	−0.736510	−0.368255	+	0.929725i	$0.620045\pi$
$984$	0	0
$985$	−10.6056	−0.337921
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	7.42221	0.235537
$994$	0	0
$995$	−48.6333	−1.54178
$996$	0	0
$997$	−14.1472	−0.448046	−0.224023	−	0.974584i	$-0.571919\pi$
$997$	−14.1472	−0.448046	−0.224023	+	0.974584i	$0.571919\pi$
$998$	0	0
$999$	14.6056	0.462099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.j.1.2		2
7.6	odd	2		476.2.a.d.1.1	✓	2
21.20	even	2		4284.2.a.n.1.2		2
28.27	even	2		1904.2.a.h.1.2		2
56.13	odd	2		7616.2.a.q.1.2		2
56.27	even	2		7616.2.a.v.1.1		2
119.118	odd	2		8092.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.d.1.1	✓	2	7.6	odd	2
1904.2.a.h.1.2		2	28.27	even	2
3332.2.a.j.1.2		2	1.1	even	1	trivial
4284.2.a.n.1.2		2	21.20	even	2
7616.2.a.q.1.2		2	56.13	odd	2
7616.2.a.v.1.1		2	56.27	even	2
8092.2.a.k.1.2		2	119.118	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{3} +2.30278 q^{5} -1.30278 q^{9} -6.60555 q^{13} +3.00000 q^{15} -1.00000 q^{17} -6.60555 q^{19} +0.302776 q^{25} -5.60555 q^{27} -4.30278 q^{31} -2.60555 q^{37} -8.60555 q^{39} -3.90833 q^{41} +7.30278 q^{43} -3.00000 q^{45} -4.60555 q^{47} -1.30278 q^{51} -3.69722 q^{53} -8.60555 q^{57} -9.21110 q^{59} +7.90833 q^{61} -15.2111 q^{65} -1.69722 q^{67} -7.81665 q^{71} +7.90833 q^{73} +0.394449 q^{75} +12.6056 q^{79} -3.39445 q^{81} -6.00000 q^{83} -2.30278 q^{85} +16.6056 q^{89} -5.60555 q^{93} -15.2111 q^{95} +6.30278 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{5} + q^{9} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{19} - 3 q^{25} - 4 q^{27} - 5 q^{31} + 2 q^{37} - 10 q^{39} + 3 q^{41} + 11 q^{43} - 6 q^{45} - 2 q^{47} + q^{51} - 11 q^{53} - 10 q^{57}+ \cdots + 9 q^{97}+O(q^{100}) \) 2 * q - q^3 + q^5 + q^9 - 6 * q^13 + 6 * q^15 - 2 * q^17 - 6 * q^19 - 3 * q^25 - 4 * q^27 - 5 * q^31 + 2 * q^37 - 10 * q^39 + 3 * q^41 + 11 * q^43 - 6 * q^45 - 2 * q^47 + q^51 - 11 * q^53 - 10 * q^57 - 4 * q^59 + 5 * q^61 - 16 * q^65 - 7 * q^67 + 6 * q^71 + 5 * q^73 + 8 * q^75 + 18 * q^79 - 14 * q^81 - 12 * q^83 - q^85 + 26 * q^89 - 4 * q^93 - 16 * q^95 + 9 * q^97

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