LMFDB - Embedded newform 3332.2.a.k.1.1

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:	$0$
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			1.1
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)

$f(q)$	$=$	$q-1.30278 q^{3} -2.30278 q^{5} -1.30278 q^{9} +O(q^{10})$	$q-1.30278 q^{3} -2.30278 q^{5} -1.30278 q^{9} +4.00000 q^{11} -2.60555 q^{13} +3.00000 q^{15} +1.00000 q^{17} -1.39445 q^{19} +4.00000 q^{23} +0.302776 q^{25} +5.60555 q^{27} -5.21110 q^{29} -3.69722 q^{31} -5.21110 q^{33} -11.8167 q^{37} +3.39445 q^{39} -6.51388 q^{41} -0.697224 q^{43} +3.00000 q^{45} -4.60555 q^{47} -1.30278 q^{51} +4.30278 q^{53} -9.21110 q^{55} +1.81665 q^{57} +8.00000 q^{59} +2.51388 q^{61} +6.00000 q^{65} +6.30278 q^{67} -5.21110 q^{69} -10.6056 q^{71} +10.5139 q^{73} -0.394449 q^{75} +4.60555 q^{79} -3.39445 q^{81} +11.2111 q^{83} -2.30278 q^{85} +6.78890 q^{87} +13.8167 q^{89} +4.81665 q^{93} +3.21110 q^{95} +9.69722 q^{97} -5.21110 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} + q^{9}+O(q^{10}) $ 2 * q + q^3 - q^5 + q^9	$ 2 q + q^{3} - q^{5} + q^{9} + 8 q^{11} + 2 q^{13} + 6 q^{15} + 2 q^{17} - 10 q^{19} + 8 q^{23} - 3 q^{25} + 4 q^{27} + 4 q^{29} - 11 q^{31} + 4 q^{33} - 2 q^{37} + 14 q^{39} + 5 q^{41} - 5 q^{43} + 6 q^{45} - 2 q^{47} + q^{51} + 5 q^{53} - 4 q^{55} - 18 q^{57} + 16 q^{59} - 13 q^{61} + 12 q^{65} + 9 q^{67} + 4 q^{69} - 14 q^{71} + 3 q^{73} - 8 q^{75} + 2 q^{79} - 14 q^{81} + 8 q^{83} - q^{85} + 28 q^{87} + 6 q^{89} - 12 q^{93} - 8 q^{95} + 23 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + q^3 - q^5 + q^9 + 8 * q^11 + 2 * q^13 + 6 * q^15 + 2 * q^17 - 10 * q^19 + 8 * q^23 - 3 * q^25 + 4 * q^27 + 4 * q^29 - 11 * q^31 + 4 * q^33 - 2 * q^37 + 14 * q^39 + 5 * q^41 - 5 * q^43 + 6 * q^45 - 2 * q^47 + q^51 + 5 * q^53 - 4 * q^55 - 18 * q^57 + 16 * q^59 - 13 * q^61 + 12 * q^65 + 9 * q^67 + 4 * q^69 - 14 * q^71 + 3 * q^73 - 8 * q^75 + 2 * q^79 - 14 * q^81 + 8 * q^83 - q^85 + 28 * q^87 + 6 * q^89 - 12 * q^93 - 8 * q^95 + 23 * q^97 + 4 * q^99

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
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	0	0
$5$	−2.30278	−1.02983	−0.514916	−	0.857240i	$-0.672177\pi$
$5$	−2.30278	−1.02983	−0.514916	+	0.857240i	$0.672177\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.60555	−0.722650	−0.361325	−	0.932440i	$-0.617675\pi$
$13$	−2.60555	−0.722650	−0.361325	+	0.932440i	$0.617675\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.39445	−0.319908	−0.159954	−	0.987124i	$-0.551135\pi$
$19$	−1.39445	−0.319908	−0.159954	+	0.987124i	$0.551135\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	5.60555	1.07879
$28$	0	0
$29$	−5.21110	−0.967677	−0.483839	−	0.875157i	$-0.660758\pi$
$29$	−5.21110	−0.967677	−0.483839	+	0.875157i	$0.660758\pi$
$30$	0	0
$31$	−3.69722	−0.664041	−0.332021	−	0.943272i	$-0.607730\pi$
$31$	−3.69722	−0.664041	−0.332021	+	0.943272i	$0.607730\pi$
$32$	0	0
$33$	−5.21110	−0.907137
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.8167	−1.94265	−0.971323	−	0.237764i	$-0.923586\pi$
$37$	−11.8167	−1.94265	−0.971323	+	0.237764i	$0.923586\pi$
$38$	0	0
$39$	3.39445	0.543547
$40$	0	0
$41$	−6.51388	−1.01730	−0.508648	−	0.860974i	$-0.669855\pi$
$41$	−6.51388	−1.01730	−0.508648	+	0.860974i	$0.669855\pi$
$42$	0	0
$43$	−0.697224	−0.106326	−0.0531629	−	0.998586i	$-0.516930\pi$
$43$	−0.697224	−0.106326	−0.0531629	+	0.998586i	$0.516930\pi$
$44$	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
$53$	4.30278	0.591032	0.295516	−	0.955338i	$-0.404508\pi$
$53$	4.30278	0.591032	0.295516	+	0.955338i	$0.404508\pi$
$54$	0	0
$55$	−9.21110	−1.24202
$56$	0	0
$57$	1.81665	0.240622
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	2.51388	0.321869	0.160935	−	0.986965i	$-0.448549\pi$
$61$	2.51388	0.321869	0.160935	+	0.986965i	$0.448549\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	6.30278	0.770007	0.385003	−	0.922915i	$-0.374200\pi$
$67$	6.30278	0.770007	0.385003	+	0.922915i	$0.374200\pi$
$68$	0	0
$69$	−5.21110	−0.627343
$70$	0	0
$71$	−10.6056	−1.25865	−0.629324	−	0.777143i	$-0.716668\pi$
$71$	−10.6056	−1.25865	−0.629324	+	0.777143i	$0.716668\pi$
$72$	0	0
$73$	10.5139	1.23056	0.615278	−	0.788310i	$-0.289044\pi$
$73$	10.5139	1.23056	0.615278	+	0.788310i	$0.289044\pi$
$74$	0	0
$75$	−0.394449	−0.0455470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.60555	0.518165	0.259083	−	0.965855i	$-0.416580\pi$
$79$	4.60555	0.518165	0.259083	+	0.965855i	$0.416580\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	11.2111	1.23058	0.615289	−	0.788301i	$-0.289039\pi$
$83$	11.2111	1.23058	0.615289	+	0.788301i	$0.289039\pi$
$84$	0	0
$85$	−2.30278	−0.249771
$86$	0	0
$87$	6.78890	0.727846
$88$	0	0
$89$	13.8167	1.46456	0.732281	−	0.681002i	$-0.238456\pi$
$89$	13.8167	1.46456	0.732281	+	0.681002i	$0.238456\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.81665	0.499464
$94$	0	0
$95$	3.21110	0.329452
$96$	0	0
$97$	9.69722	0.984604	0.492302	−	0.870424i	$-0.336156\pi$
$97$	9.69722	0.984604	0.492302	+	0.870424i	$0.336156\pi$
$98$	0	0
$99$	−5.21110	−0.523736
$100$	0	0
$101$	5.39445	0.536768	0.268384	−	0.963312i	$-0.413510\pi$
$101$	5.39445	0.536768	0.268384	+	0.963312i	$0.413510\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.60555	0.251888	0.125944	−	0.992037i	$-0.459804\pi$
$107$	2.60555	0.251888	0.125944	+	0.992037i	$0.459804\pi$
$108$	0	0
$109$	19.2111	1.84009	0.920045	−	0.391813i	$-0.128152\pi$
$109$	19.2111	1.84009	0.920045	+	0.391813i	$0.128152\pi$
$110$	0	0
$111$	15.3944	1.46118
$112$	0	0
$113$	−16.6056	−1.56212	−0.781059	−	0.624457i	$-0.785320\pi$
$113$	−16.6056	−1.56212	−0.781059	+	0.624457i	$0.785320\pi$
$114$	0	0
$115$	−9.21110	−0.858940
$116$	0	0
$117$	3.39445	0.313817
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	8.48612	0.765168
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	−3.51388	−0.311806	−0.155903	−	0.987772i	$-0.549829\pi$
$127$	−3.51388	−0.311806	−0.155903	+	0.987772i	$0.549829\pi$
$128$	0	0
$129$	0.908327	0.0799737
$130$	0	0
$131$	−5.21110	−0.455296	−0.227648	−	0.973743i	$-0.573104\pi$
$131$	−5.21110	−0.455296	−0.227648	+	0.973743i	$0.573104\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.9083	−1.11097
$136$	0	0
$137$	−7.11943	−0.608254	−0.304127	−	0.952632i	$-0.598365\pi$
$137$	−7.11943	−0.608254	−0.304127	+	0.952632i	$0.598365\pi$
$138$	0	0
$139$	9.51388	0.806957	0.403478	−	0.914989i	$-0.367801\pi$
$139$	9.51388	0.806957	0.403478	+	0.914989i	$0.367801\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	−10.4222	−0.871549
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.697224	−0.0571188	−0.0285594	−	0.999592i	$-0.509092\pi$
$149$	−0.697224	−0.0571188	−0.0285594	+	0.999592i	$0.509092\pi$
$150$	0	0
$151$	15.3028	1.24532	0.622661	−	0.782492i	$-0.286052\pi$
$151$	15.3028	1.24532	0.622661	+	0.782492i	$0.286052\pi$
$152$	0	0
$153$	−1.30278	−0.105323
$154$	0	0
$155$	8.51388	0.683851
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−5.60555	−0.444549
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.60555	0.674039	0.337019	−	0.941498i	$-0.390581\pi$
$163$	8.60555	0.674039	0.337019	+	0.941498i	$0.390581\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	13.1194	1.01521	0.507606	−	0.861589i	$-0.330531\pi$
$167$	13.1194	1.01521	0.507606	+	0.861589i	$0.330531\pi$
$168$	0	0
$169$	−6.21110	−0.477777
$170$	0	0
$171$	1.81665	0.138923
$172$	0	0
$173$	11.7250	0.891434	0.445717	−	0.895174i	$-0.352949\pi$
$173$	11.7250	0.891434	0.445717	+	0.895174i	$0.352949\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.4222	−0.783381
$178$	0	0
$179$	−8.51388	−0.636357	−0.318179	−	0.948031i	$-0.603071\pi$
$179$	−8.51388	−0.636357	−0.318179	+	0.948031i	$0.603071\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−3.27502	−0.242096
$184$	0	0
$185$	27.2111	2.00060
$186$	0	0
$187$	4.00000	0.292509
$188$	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
$193$	21.0278	1.51361	0.756806	−	0.653640i	$-0.226759\pi$
$193$	21.0278	1.51361	0.756806	+	0.653640i	$0.226759\pi$
$194$	0	0
$195$	−7.81665	−0.559762
$196$	0	0
$197$	0.605551	0.0431437	0.0215719	−	0.999767i	$-0.493133\pi$
$197$	0.605551	0.0431437	0.0215719	+	0.999767i	$0.493133\pi$
$198$	0	0
$199$	−13.3028	−0.943009	−0.471504	−	0.881864i	$-0.656289\pi$
$199$	−13.3028	−0.943009	−0.471504	+	0.881864i	$0.656289\pi$
$200$	0	0
$201$	−8.21110	−0.579167
$202$	0	0
$203$	0	0
$204$	0	0
$205$	15.0000	1.04765
$206$	0	0
$207$	−5.21110	−0.362197
$208$	0	0
$209$	−5.57779	−0.385824
$210$	0	0
$211$	27.8167	1.91498	0.957489	−	0.288471i	$-0.0931468\pi$
$211$	27.8167	1.91498	0.957489	+	0.288471i	$0.0931468\pi$
$212$	0	0
$213$	13.8167	0.946702
$214$	0	0
$215$	1.60555	0.109498
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.6972	−0.925573
$220$	0	0
$221$	−2.60555	−0.175268
$222$	0	0
$223$	7.81665	0.523442	0.261721	−	0.965144i	$-0.415710\pi$
$223$	7.81665	0.523442	0.261721	+	0.965144i	$0.415710\pi$
$224$	0	0
$225$	−0.394449	−0.0262966
$226$	0	0
$227$	−12.9083	−0.856756	−0.428378	−	0.903600i	$-0.640915\pi$
$227$	−12.9083	−0.856756	−0.428378	+	0.903600i	$0.640915\pi$
$228$	0	0
$229$	−13.2111	−0.873014	−0.436507	−	0.899701i	$-0.643785\pi$
$229$	−13.2111	−0.873014	−0.436507	+	0.899701i	$0.643785\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.3944	1.27057	0.635286	−	0.772277i	$-0.280882\pi$
$233$	19.3944	1.27057	0.635286	+	0.772277i	$0.280882\pi$
$234$	0	0
$235$	10.6056	0.691830
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	−7.11943	−0.460518	−0.230259	−	0.973129i	$-0.573957\pi$
$239$	−7.11943	−0.460518	−0.230259	+	0.973129i	$0.573957\pi$
$240$	0	0
$241$	8.48612	0.546639	0.273320	−	0.961923i	$-0.411878\pi$
$241$	8.48612	0.546639	0.273320	+	0.961923i	$0.411878\pi$
$242$	0	0
$243$	−12.3944	−0.795104
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.63331	0.231182
$248$	0	0
$249$	−14.6056	−0.925589
$250$	0	0
$251$	23.8167	1.50329	0.751647	−	0.659566i	$-0.229260\pi$
$251$	23.8167	1.50329	0.751647	+	0.659566i	$0.229260\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−24.4222	−1.52342	−0.761708	−	0.647921i	$-0.775639\pi$
$257$	−24.4222	−1.52342	−0.761708	+	0.647921i	$0.775639\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.78890	0.420222
$262$	0	0
$263$	10.4222	0.642661	0.321330	−	0.946967i	$-0.395870\pi$
$263$	10.4222	0.642661	0.321330	+	0.946967i	$0.395870\pi$
$264$	0	0
$265$	−9.90833	−0.608664
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	−27.2111	−1.65909	−0.829545	−	0.558440i	$-0.811400\pi$
$269$	−27.2111	−1.65909	−0.829545	+	0.558440i	$0.811400\pi$
$270$	0	0
$271$	−18.4222	−1.11907	−0.559535	−	0.828807i	$-0.689020\pi$
$271$	−18.4222	−1.11907	−0.559535	+	0.828807i	$0.689020\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.21110	0.0730322
$276$	0	0
$277$	30.4222	1.82789	0.913947	−	0.405835i	$-0.133019\pi$
$277$	30.4222	1.82789	0.913947	+	0.405835i	$0.133019\pi$
$278$	0	0
$279$	4.81665	0.288366
$280$	0	0
$281$	6.11943	0.365055	0.182527	−	0.983201i	$-0.441572\pi$
$281$	6.11943	0.365055	0.182527	+	0.983201i	$0.441572\pi$
$282$	0	0
$283$	19.3305	1.14908	0.574540	−	0.818476i	$-0.305181\pi$
$283$	19.3305	1.14908	0.574540	+	0.818476i	$0.305181\pi$
$284$	0	0
$285$	−4.18335	−0.247800
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.6333	−0.740578
$292$	0	0
$293$	−3.21110	−0.187595	−0.0937973	−	0.995591i	$-0.529901\pi$
$293$	−3.21110	−0.187595	−0.0937973	+	0.995591i	$0.529901\pi$
$294$	0	0
$295$	−18.4222	−1.07258
$296$	0	0
$297$	22.4222	1.30107
$298$	0	0
$299$	−10.4222	−0.602732
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.02776	−0.403734
$304$	0	0
$305$	−5.78890	−0.331471
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	18.2389	1.03757
$310$	0	0
$311$	−14.7250	−0.834977	−0.417489	−	0.908682i	$-0.637090\pi$
$311$	−14.7250	−0.834977	−0.417489	+	0.908682i	$0.637090\pi$
$312$	0	0
$313$	6.90833	0.390482	0.195241	−	0.980755i	$-0.437451\pi$
$313$	6.90833	0.390482	0.195241	+	0.980755i	$0.437451\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.63331	0.541060	0.270530	−	0.962711i	$-0.412801\pi$
$317$	9.63331	0.541060	0.270530	+	0.962711i	$0.412801\pi$
$318$	0	0
$319$	−20.8444	−1.16706
$320$	0	0
$321$	−3.39445	−0.189460
$322$	0	0
$323$	−1.39445	−0.0775892
$324$	0	0
$325$	−0.788897	−0.0437602
$326$	0	0
$327$	−25.0278	−1.38404
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.7250	−1.57887	−0.789434	−	0.613836i	$-0.789626\pi$
$331$	−28.7250	−1.57887	−0.789434	+	0.613836i	$0.789626\pi$
$332$	0	0
$333$	15.3944	0.843611
$334$	0	0
$335$	−14.5139	−0.792978
$336$	0	0
$337$	14.4222	0.785628	0.392814	−	0.919618i	$-0.371502\pi$
$337$	14.4222	0.785628	0.392814	+	0.919618i	$0.371502\pi$
$338$	0	0
$339$	21.6333	1.17496
$340$	0	0
$341$	−14.7889	−0.800864
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−30.8444	−1.65581	−0.827907	−	0.560865i	$-0.810469\pi$
$347$	−30.8444	−1.65581	−0.827907	+	0.560865i	$0.810469\pi$
$348$	0	0
$349$	10.4222	0.557888	0.278944	−	0.960307i	$-0.410016\pi$
$349$	10.4222	0.557888	0.278944	+	0.960307i	$0.410016\pi$
$350$	0	0
$351$	−14.6056	−0.779587
$352$	0	0
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	0	0
$355$	24.4222	1.29620
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.5139	−1.02990	−0.514952	−	0.857219i	$-0.672190\pi$
$359$	−19.5139	−1.02990	−0.514952	+	0.857219i	$0.672190\pi$
$360$	0	0
$361$	−17.0555	−0.897659
$362$	0	0
$363$	−6.51388	−0.341890
$364$	0	0
$365$	−24.2111	−1.26727
$366$	0	0
$367$	3.90833	0.204013	0.102007	−	0.994784i	$-0.467474\pi$
$367$	3.90833	0.204013	0.102007	+	0.994784i	$0.467474\pi$
$368$	0	0
$369$	8.48612	0.441770
$370$	0	0
$371$	0	0
$372$	0	0
$373$	24.3305	1.25979	0.629894	−	0.776681i	$-0.283098\pi$
$373$	24.3305	1.25979	0.629894	+	0.776681i	$0.283098\pi$
$374$	0	0
$375$	−14.0917	−0.727691
$376$	0	0
$377$	13.5778	0.699292
$378$	0	0
$379$	38.4222	1.97362	0.986808	−	0.161895i	$-0.0517605\pi$
$379$	38.4222	1.97362	0.986808	+	0.161895i	$0.0517605\pi$
$380$	0	0
$381$	4.57779	0.234528
$382$	0	0
$383$	30.4222	1.55450	0.777251	−	0.629191i	$-0.216614\pi$
$383$	30.4222	1.55450	0.777251	+	0.629191i	$0.216614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.908327	0.0461729
$388$	0	0
$389$	2.72498	0.138162	0.0690810	−	0.997611i	$-0.477993\pi$
$389$	2.72498	0.138162	0.0690810	+	0.997611i	$0.477993\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	6.78890	0.342455
$394$	0	0
$395$	−10.6056	−0.533623
$396$	0	0
$397$	22.9083	1.14974	0.574868	−	0.818246i	$-0.305053\pi$
$397$	22.9083	1.14974	0.574868	+	0.818246i	$0.305053\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.60555	−0.229990	−0.114995	−	0.993366i	$-0.536685\pi$
$401$	−4.60555	−0.229990	−0.114995	+	0.993366i	$0.536685\pi$
$402$	0	0
$403$	9.63331	0.479869
$404$	0	0
$405$	7.81665	0.388413
$406$	0	0
$407$	−47.2666	−2.34292
$408$	0	0
$409$	−15.8167	−0.782083	−0.391042	−	0.920373i	$-0.627885\pi$
$409$	−15.8167	−0.782083	−0.391042	+	0.920373i	$0.627885\pi$
$410$	0	0
$411$	9.27502	0.457503
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−25.8167	−1.26729
$416$	0	0
$417$	−12.3944	−0.606959
$418$	0	0
$419$	−9.51388	−0.464783	−0.232392	−	0.972622i	$-0.574655\pi$
$419$	−9.51388	−0.464783	−0.232392	+	0.972622i	$0.574655\pi$
$420$	0	0
$421$	21.9083	1.06775	0.533873	−	0.845565i	$-0.320736\pi$
$421$	21.9083	1.06775	0.533873	+	0.845565i	$0.320736\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	0.302776	0.0146868
$426$	0	0
$427$	0	0
$428$	0	0
$429$	13.5778	0.655542
$430$	0	0
$431$	28.8444	1.38939	0.694693	−	0.719306i	$-0.255540\pi$
$431$	28.8444	1.38939	0.694693	+	0.719306i	$0.255540\pi$
$432$	0	0
$433$	14.6056	0.701898	0.350949	−	0.936395i	$-0.385859\pi$
$433$	14.6056	0.701898	0.350949	+	0.936395i	$0.385859\pi$
$434$	0	0
$435$	−15.6333	−0.749560
$436$	0	0
$437$	−5.57779	−0.266822
$438$	0	0
$439$	−31.9083	−1.52290	−0.761451	−	0.648223i	$-0.775513\pi$
$439$	−31.9083	−1.52290	−0.761451	+	0.648223i	$0.775513\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.7889	−0.892687	−0.446344	−	0.894862i	$-0.647274\pi$
$443$	−18.7889	−0.892687	−0.446344	+	0.894862i	$0.647274\pi$
$444$	0	0
$445$	−31.8167	−1.50825
$446$	0	0
$447$	0.908327	0.0429624
$448$	0	0
$449$	31.0278	1.46429	0.732145	−	0.681149i	$-0.238519\pi$
$449$	31.0278	1.46429	0.732145	+	0.681149i	$0.238519\pi$
$450$	0	0
$451$	−26.0555	−1.22691
$452$	0	0
$453$	−19.9361	−0.936679
$454$	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
$461$	−30.6056	−1.42544	−0.712721	−	0.701447i	$-0.752538\pi$
$461$	−30.6056	−1.42544	−0.712721	+	0.701447i	$0.752538\pi$
$462$	0	0
$463$	25.7250	1.19554	0.597771	−	0.801667i	$-0.296053\pi$
$463$	25.7250	1.19554	0.597771	+	0.801667i	$0.296053\pi$
$464$	0	0
$465$	−11.0917	−0.514364
$466$	0	0
$467$	9.39445	0.434723	0.217362	−	0.976091i	$-0.430255\pi$
$467$	9.39445	0.434723	0.217362	+	0.976091i	$0.430255\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.60555	0.120057
$472$	0	0
$473$	−2.78890	−0.128234
$474$	0	0
$475$	−0.422205	−0.0193721
$476$	0	0
$477$	−5.60555	−0.256661
$478$	0	0
$479$	−27.9083	−1.27516	−0.637582	−	0.770382i	$-0.720065\pi$
$479$	−27.9083	−1.27516	−0.637582	+	0.770382i	$0.720065\pi$
$480$	0	0
$481$	30.7889	1.40385
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.3305	−1.01398
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−11.2111	−0.506984
$490$	0	0
$491$	−4.48612	−0.202456	−0.101228	−	0.994863i	$-0.532277\pi$
$491$	−4.48612	−0.202456	−0.101228	+	0.994863i	$0.532277\pi$
$492$	0	0
$493$	−5.21110	−0.234696
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−14.7889	−0.662042	−0.331021	−	0.943623i	$-0.607393\pi$
$499$	−14.7889	−0.662042	−0.331021	+	0.943623i	$0.607393\pi$
$500$	0	0
$501$	−17.0917	−0.763600
$502$	0	0
$503$	5.88057	0.262202	0.131101	−	0.991369i	$-0.458149\pi$
$503$	5.88057	0.262202	0.131101	+	0.991369i	$0.458149\pi$
$504$	0	0
$505$	−12.4222	−0.552781
$506$	0	0
$507$	8.09167	0.359364
$508$	0	0
$509$	0.605551	0.0268406	0.0134203	−	0.999910i	$-0.495728\pi$
$509$	0.605551	0.0268406	0.0134203	+	0.999910i	$0.495728\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.81665	−0.345114
$514$	0	0
$515$	32.2389	1.42061
$516$	0	0
$517$	−18.4222	−0.810208
$518$	0	0
$519$	−15.2750	−0.670499
$520$	0	0
$521$	18.5139	0.811108	0.405554	−	0.914071i	$-0.367079\pi$
$521$	18.5139	0.811108	0.405554	+	0.914071i	$0.367079\pi$
$522$	0	0
$523$	34.0555	1.48914	0.744572	−	0.667542i	$-0.232654\pi$
$523$	34.0555	1.48914	0.744572	+	0.667542i	$0.232654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.69722	−0.161054
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−10.4222	−0.452285
$532$	0	0
$533$	16.9722	0.735149
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	11.0917	0.478641
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	0	0
$543$	−7.81665	−0.335445
$544$	0	0
$545$	−44.2389	−1.89498
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	0	0
$549$	−3.27502	−0.139774
$550$	0	0
$551$	7.26662	0.309568
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−35.4500	−1.50477
$556$	0	0
$557$	−35.2111	−1.49194	−0.745971	−	0.665978i	$-0.768014\pi$
$557$	−35.2111	−1.49194	−0.745971	+	0.665978i	$0.768014\pi$
$558$	0	0
$559$	1.81665	0.0768363
$560$	0	0
$561$	−5.21110	−0.220013
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	38.2389	1.60872
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.11943	−0.130773	−0.0653866	−	0.997860i	$-0.520828\pi$
$569$	−3.11943	−0.130773	−0.0653866	+	0.997860i	$0.520828\pi$
$570$	0	0
$571$	19.4500	0.813956	0.406978	−	0.913438i	$-0.366583\pi$
$571$	19.4500	0.813956	0.406978	+	0.913438i	$0.366583\pi$
$572$	0	0
$573$	−27.0000	−1.12794
$574$	0	0
$575$	1.21110	0.0505065
$576$	0	0
$577$	29.8167	1.24128	0.620642	−	0.784094i	$-0.286872\pi$
$577$	29.8167	1.24128	0.620642	+	0.784094i	$0.286872\pi$
$578$	0	0
$579$	−27.3944	−1.13847
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.2111	0.712811
$584$	0	0
$585$	−7.81665	−0.323179
$586$	0	0
$587$	18.6056	0.767933	0.383967	−	0.923347i	$-0.374558\pi$
$587$	18.6056	0.767933	0.383967	+	0.923347i	$0.374558\pi$
$588$	0	0
$589$	5.15559	0.212432
$590$	0	0
$591$	−0.788897	−0.0324509
$592$	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
$597$	17.3305	0.709291
$598$	0	0
$599$	−5.72498	−0.233916	−0.116958	−	0.993137i	$-0.537314\pi$
$599$	−5.72498	−0.233916	−0.116958	+	0.993137i	$0.537314\pi$
$600$	0	0
$601$	41.2666	1.68330	0.841650	−	0.540023i	$-0.181585\pi$
$601$	41.2666	1.68330	0.841650	+	0.540023i	$0.181585\pi$
$602$	0	0
$603$	−8.21110	−0.334382
$604$	0	0
$605$	−11.5139	−0.468106
$606$	0	0
$607$	39.3305	1.59638	0.798189	−	0.602408i	$-0.205792\pi$
$607$	39.3305	1.59638	0.798189	+	0.602408i	$0.205792\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	10.7250	0.433178	0.216589	−	0.976263i	$-0.430507\pi$
$613$	10.7250	0.433178	0.216589	+	0.976263i	$0.430507\pi$
$614$	0	0
$615$	−19.5416	−0.787995
$616$	0	0
$617$	−35.6333	−1.43454	−0.717271	−	0.696794i	$-0.754609\pi$
$617$	−35.6333	−1.43454	−0.717271	+	0.696794i	$0.754609\pi$
$618$	0	0
$619$	−22.7889	−0.915963	−0.457982	−	0.888962i	$-0.651427\pi$
$619$	−22.7889	−0.915963	−0.457982	+	0.888962i	$0.651427\pi$
$620$	0	0
$621$	22.4222	0.899772
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	7.26662	0.290201
$628$	0	0
$629$	−11.8167	−0.471161
$630$	0	0
$631$	−16.1194	−0.641704	−0.320852	−	0.947129i	$-0.603969\pi$
$631$	−16.1194	−0.641704	−0.320852	+	0.947129i	$0.603969\pi$
$632$	0	0
$633$	−36.2389	−1.44037
$634$	0	0
$635$	8.09167	0.321108
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.8167	0.546578
$640$	0	0
$641$	0.366692	0.0144835	0.00724174	−	0.999974i	$-0.497695\pi$
$641$	0.366692	0.0144835	0.00724174	+	0.999974i	$0.497695\pi$
$642$	0	0
$643$	−33.1194	−1.30610	−0.653051	−	0.757314i	$-0.726511\pi$
$643$	−33.1194	−1.30610	−0.653051	+	0.757314i	$0.726511\pi$
$644$	0	0
$645$	−2.09167	−0.0823595
$646$	0	0
$647$	−7.63331	−0.300096	−0.150048	−	0.988679i	$-0.547943\pi$
$647$	−7.63331	−0.300096	−0.150048	+	0.988679i	$0.547943\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.5778	0.453074	0.226537	−	0.974003i	$-0.427260\pi$
$653$	11.5778	0.453074	0.226537	+	0.974003i	$0.427260\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−13.6972	−0.534380
$658$	0	0
$659$	−4.11943	−0.160470	−0.0802351	−	0.996776i	$-0.525567\pi$
$659$	−4.11943	−0.160470	−0.0802351	+	0.996776i	$0.525567\pi$
$660$	0	0
$661$	−31.0278	−1.20684	−0.603420	−	0.797424i	$-0.706196\pi$
$661$	−31.0278	−1.20684	−0.603420	+	0.797424i	$0.706196\pi$
$662$	0	0
$663$	3.39445	0.131829
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.8444	−0.807099
$668$	0	0
$669$	−10.1833	−0.393711
$670$	0	0
$671$	10.0555	0.388189
$672$	0	0
$673$	29.0278	1.11894	0.559469	−	0.828851i	$-0.311005\pi$
$673$	29.0278	1.11894	0.559469	+	0.828851i	$0.311005\pi$
$674$	0	0
$675$	1.69722	0.0653262
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	16.8167	0.644416
$682$	0	0
$683$	20.6056	0.788450	0.394225	−	0.919014i	$-0.371013\pi$
$683$	20.6056	0.788450	0.394225	+	0.919014i	$0.371013\pi$
$684$	0	0
$685$	16.3944	0.626400
$686$	0	0
$687$	17.2111	0.656645
$688$	0	0
$689$	−11.2111	−0.427109
$690$	0	0
$691$	−44.9361	−1.70945	−0.854725	−	0.519082i	$-0.826274\pi$
$691$	−44.9361	−1.70945	−0.854725	+	0.519082i	$0.826274\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.9083	−0.831030
$696$	0	0
$697$	−6.51388	−0.246731
$698$	0	0
$699$	−25.2666	−0.955671
$700$	0	0
$701$	−40.4222	−1.52673	−0.763363	−	0.645970i	$-0.776453\pi$
$701$	−40.4222	−1.52673	−0.763363	+	0.645970i	$0.776453\pi$
$702$	0	0
$703$	16.4777	0.621469
$704$	0	0
$705$	−13.8167	−0.520365
$706$	0	0
$707$	0	0
$708$	0	0
$709$	46.2389	1.73654	0.868268	−	0.496095i	$-0.165233\pi$
$709$	46.2389	1.73654	0.868268	+	0.496095i	$0.165233\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	−14.7889	−0.553849
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	9.27502	0.346382
$718$	0	0
$719$	−13.3028	−0.496110	−0.248055	−	0.968746i	$-0.579791\pi$
$719$	−13.3028	−0.496110	−0.248055	+	0.968746i	$0.579791\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−11.0555	−0.411159
$724$	0	0
$725$	−1.57779	−0.0585978
$726$	0	0
$727$	−14.4222	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$727$	−14.4222	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	−0.697224	−0.0257878
$732$	0	0
$733$	34.2389	1.26464	0.632321	−	0.774707i	$-0.282103\pi$
$733$	34.2389	1.26464	0.632321	+	0.774707i	$0.282103\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.2111	0.928663
$738$	0	0
$739$	16.3305	0.600728	0.300364	−	0.953825i	$-0.402892\pi$
$739$	16.3305	0.600728	0.300364	+	0.953825i	$0.402892\pi$
$740$	0	0
$741$	−4.73338	−0.173885
$742$	0	0
$743$	−27.3944	−1.00500	−0.502502	−	0.864576i	$-0.667587\pi$
$743$	−27.3944	−1.00500	−0.502502	+	0.864576i	$0.667587\pi$
$744$	0	0
$745$	1.60555	0.0588228
$746$	0	0
$747$	−14.6056	−0.534389
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.6056	0.387002	0.193501	−	0.981100i	$-0.438016\pi$
$751$	10.6056	0.387002	0.193501	+	0.981100i	$0.438016\pi$
$752$	0	0
$753$	−31.0278	−1.13071
$754$	0	0
$755$	−35.2389	−1.28247
$756$	0	0
$757$	−11.3028	−0.410806	−0.205403	−	0.978677i	$-0.565851\pi$
$757$	−11.3028	−0.410806	−0.205403	+	0.978677i	$0.565851\pi$
$758$	0	0
$759$	−20.8444	−0.756604
$760$	0	0
$761$	−0.238859	−0.00865863	−0.00432931	−	0.999991i	$-0.501378\pi$
$761$	−0.238859	−0.00865863	−0.00432931	+	0.999991i	$0.501378\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	−20.8444	−0.752648
$768$	0	0
$769$	−36.0555	−1.30020	−0.650098	−	0.759851i	$-0.725272\pi$
$769$	−36.0555	−1.30020	−0.650098	+	0.759851i	$0.725272\pi$
$770$	0	0
$771$	31.8167	1.14585
$772$	0	0
$773$	19.0278	0.684381	0.342190	−	0.939631i	$-0.388831\pi$
$773$	19.0278	0.684381	0.342190	+	0.939631i	$0.388831\pi$
$774$	0	0
$775$	−1.11943	−0.0402111
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.08327	0.325442
$780$	0	0
$781$	−42.4222	−1.51799
$782$	0	0
$783$	−29.2111	−1.04392
$784$	0	0
$785$	4.60555	0.164379
$786$	0	0
$787$	−21.5778	−0.769165	−0.384583	−	0.923091i	$-0.625655\pi$
$787$	−21.5778	−0.769165	−0.384583	+	0.923091i	$0.625655\pi$
$788$	0	0
$789$	−13.5778	−0.483382
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.55004	−0.232599
$794$	0	0
$795$	12.9083	0.457811
$796$	0	0
$797$	24.0555	0.852090	0.426045	−	0.904702i	$-0.359907\pi$
$797$	24.0555	0.852090	0.426045	+	0.904702i	$0.359907\pi$
$798$	0	0
$799$	−4.60555	−0.162933
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	42.0555	1.48411
$804$	0	0
$805$	0	0
$806$	0	0
$807$	35.4500	1.24790
$808$	0	0
$809$	1.81665	0.0638701	0.0319351	−	0.999490i	$-0.489833\pi$
$809$	1.81665	0.0638701	0.0319351	+	0.999490i	$0.489833\pi$
$810$	0	0
$811$	22.5139	0.790569	0.395285	−	0.918559i	$-0.370646\pi$
$811$	22.5139	0.790569	0.395285	+	0.918559i	$0.370646\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	0	0
$815$	−19.8167	−0.694147
$816$	0	0
$817$	0.972244	0.0340145
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.2389	1.47415	0.737073	−	0.675813i	$-0.236207\pi$
$821$	42.2389	1.47415	0.737073	+	0.675813i	$0.236207\pi$
$822$	0	0
$823$	−31.3944	−1.09434	−0.547171	−	0.837021i	$-0.684295\pi$
$823$	−31.3944	−1.09434	−0.547171	+	0.837021i	$0.684295\pi$
$824$	0	0
$825$	−1.57779	−0.0549318
$826$	0	0
$827$	30.7889	1.07063	0.535317	−	0.844651i	$-0.320192\pi$
$827$	30.7889	1.07063	0.535317	+	0.844651i	$0.320192\pi$
$828$	0	0
$829$	−34.2389	−1.18916	−0.594582	−	0.804035i	$-0.702683\pi$
$829$	−34.2389	−1.18916	−0.594582	+	0.804035i	$0.702683\pi$
$830$	0	0
$831$	−39.6333	−1.37486
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−30.2111	−1.04550
$836$	0	0
$837$	−20.7250	−0.716360
$838$	0	0
$839$	10.4222	0.359814	0.179907	−	0.983684i	$-0.442420\pi$
$839$	10.4222	0.359814	0.179907	+	0.983684i	$0.442420\pi$
$840$	0	0
$841$	−1.84441	−0.0636004
$842$	0	0
$843$	−7.97224	−0.274579
$844$	0	0
$845$	14.3028	0.492030
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−25.1833	−0.864290
$850$	0	0
$851$	−47.2666	−1.62028
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	−4.18335	−0.143067
$856$	0	0
$857$	−36.9638	−1.26266	−0.631330	−	0.775514i	$-0.717491\pi$
$857$	−36.9638	−1.26266	−0.631330	+	0.775514i	$0.717491\pi$
$858$	0	0
$859$	47.3944	1.61708	0.808539	−	0.588443i	$-0.200259\pi$
$859$	47.3944	1.61708	0.808539	+	0.588443i	$0.200259\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−52.3583	−1.78230	−0.891148	−	0.453712i	$-0.850100\pi$
$863$	−52.3583	−1.78230	−0.891148	+	0.453712i	$0.850100\pi$
$864$	0	0
$865$	−27.0000	−0.918028
$866$	0	0
$867$	−1.30278	−0.0442446
$868$	0	0
$869$	18.4222	0.624931
$870$	0	0
$871$	−16.4222	−0.556445
$872$	0	0
$873$	−12.6333	−0.427573
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.60555	−0.290589	−0.145294	−	0.989388i	$-0.546413\pi$
$877$	−8.60555	−0.290589	−0.145294	+	0.989388i	$0.546413\pi$
$878$	0	0
$879$	4.18335	0.141101
$880$	0	0
$881$	−12.1194	−0.408314	−0.204157	−	0.978938i	$-0.565445\pi$
$881$	−12.1194	−0.408314	−0.204157	+	0.978938i	$0.565445\pi$
$882$	0	0
$883$	22.9083	0.770927	0.385463	−	0.922723i	$-0.374042\pi$
$883$	22.9083	0.770927	0.385463	+	0.922723i	$0.374042\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	48.7805	1.63789	0.818944	−	0.573873i	$-0.194560\pi$
$887$	48.7805	1.63789	0.818944	+	0.573873i	$0.194560\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−13.5778	−0.454873
$892$	0	0
$893$	6.42221	0.214911
$894$	0	0
$895$	19.6056	0.655341
$896$	0	0
$897$	13.5778	0.453349
$898$	0	0
$899$	19.2666	0.642578
$900$	0	0
$901$	4.30278	0.143346
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.8167	−0.459281
$906$	0	0
$907$	12.7889	0.424648	0.212324	−	0.977199i	$-0.431897\pi$
$907$	12.7889	0.424648	0.212324	+	0.977199i	$0.431897\pi$
$908$	0	0
$909$	−7.02776	−0.233096
$910$	0	0
$911$	5.81665	0.192714	0.0963572	−	0.995347i	$-0.469281\pi$
$911$	5.81665	0.192714	0.0963572	+	0.995347i	$0.469281\pi$
$912$	0	0
$913$	44.8444	1.48413
$914$	0	0
$915$	7.54163	0.249319
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−49.3583	−1.62818	−0.814090	−	0.580739i	$-0.802764\pi$
$919$	−49.3583	−1.62818	−0.814090	+	0.580739i	$0.802764\pi$
$920$	0	0
$921$	5.21110	0.171712
$922$	0	0
$923$	27.6333	0.909561
$924$	0	0
$925$	−3.57779	−0.117637
$926$	0	0
$927$	18.2389	0.599043
$928$	0	0
$929$	−46.9083	−1.53901	−0.769506	−	0.638639i	$-0.779498\pi$
$929$	−46.9083	−1.53901	−0.769506	+	0.638639i	$0.779498\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	19.1833	0.628035
$934$	0	0
$935$	−9.21110	−0.301235
$936$	0	0
$937$	22.7889	0.744481	0.372240	−	0.928136i	$-0.378590\pi$
$937$	22.7889	0.744481	0.372240	+	0.928136i	$0.378590\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	0	0
$941$	37.3028	1.21604	0.608018	−	0.793923i	$-0.291965\pi$
$941$	37.3028	1.21604	0.608018	+	0.793923i	$0.291965\pi$
$942$	0	0
$943$	−26.0555	−0.848484
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.8444	−1.06730	−0.533650	−	0.845705i	$-0.679180\pi$
$947$	−32.8444	−1.06730	−0.533650	+	0.845705i	$0.679180\pi$
$948$	0	0
$949$	−27.3944	−0.889261
$950$	0	0
$951$	−12.5500	−0.406963
$952$	0	0
$953$	54.3583	1.76084	0.880419	−	0.474197i	$-0.157262\pi$
$953$	54.3583	1.76084	0.880419	+	0.474197i	$0.157262\pi$
$954$	0	0
$955$	−47.7250	−1.54434
$956$	0	0
$957$	27.1556	0.877816
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.3305	−0.559049
$962$	0	0
$963$	−3.39445	−0.109385
$964$	0	0
$965$	−48.4222	−1.55877
$966$	0	0
$967$	8.51388	0.273788	0.136894	−	0.990586i	$-0.456288\pi$
$967$	8.51388	0.273788	0.136894	+	0.990586i	$0.456288\pi$
$968$	0	0
$969$	1.81665	0.0583593
$970$	0	0
$971$	−17.2111	−0.552331	−0.276165	−	0.961110i	$-0.589064\pi$
$971$	−17.2111	−0.552331	−0.276165	+	0.961110i	$0.589064\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.02776	0.0329145
$976$	0	0
$977$	−12.3028	−0.393601	−0.196800	−	0.980444i	$-0.563055\pi$
$977$	−12.3028	−0.393601	−0.196800	+	0.980444i	$0.563055\pi$
$978$	0	0
$979$	55.2666	1.76633
$980$	0	0
$981$	−25.0278	−0.799075
$982$	0	0
$983$	56.7805	1.81102	0.905508	−	0.424329i	$-0.139490\pi$
$983$	56.7805	1.81102	0.905508	+	0.424329i	$0.139490\pi$
$984$	0	0
$985$	−1.39445	−0.0444308
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.78890	−0.0886818
$990$	0	0
$991$	3.21110	0.102004	0.0510020	−	0.998699i	$-0.483759\pi$
$991$	3.21110	0.102004	0.0510020	+	0.998699i	$0.483759\pi$
$992$	0	0
$993$	37.4222	1.18756
$994$	0	0
$995$	30.6333	0.971141
$996$	0	0
$997$	49.3028	1.56143	0.780717	−	0.624884i	$-0.214854\pi$
$997$	49.3028	1.56143	0.780717	+	0.624884i	$0.214854\pi$
$998$	0	0
$999$	−66.2389	−2.09570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.k.1.1		2
7.6	odd	2		476.2.a.c.1.2	✓	2
21.20	even	2		4284.2.a.l.1.1		2
28.27	even	2		1904.2.a.k.1.1		2
56.13	odd	2		7616.2.a.t.1.1		2
56.27	even	2		7616.2.a.o.1.2		2
119.118	odd	2		8092.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.c.1.2	✓	2	7.6	odd	2
1904.2.a.k.1.1		2	28.27	even	2
3332.2.a.k.1.1		2	1.1	even	1	trivial
4284.2.a.l.1.1		2	21.20	even	2
7616.2.a.o.1.2		2	56.27	even	2
7616.2.a.t.1.1		2	56.13	odd	2
8092.2.a.l.1.1		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)

\(f(q)\)	\(=\)	\(q-1.30278 q^{3} -2.30278 q^{5} -1.30278 q^{9} +O(q^{10})\)	\(q-1.30278 q^{3} -2.30278 q^{5} -1.30278 q^{9} +4.00000 q^{11} -2.60555 q^{13} +3.00000 q^{15} +1.00000 q^{17} -1.39445 q^{19} +4.00000 q^{23} +0.302776 q^{25} +5.60555 q^{27} -5.21110 q^{29} -3.69722 q^{31} -5.21110 q^{33} -11.8167 q^{37} +3.39445 q^{39} -6.51388 q^{41} -0.697224 q^{43} +3.00000 q^{45} -4.60555 q^{47} -1.30278 q^{51} +4.30278 q^{53} -9.21110 q^{55} +1.81665 q^{57} +8.00000 q^{59} +2.51388 q^{61} +6.00000 q^{65} +6.30278 q^{67} -5.21110 q^{69} -10.6056 q^{71} +10.5139 q^{73} -0.394449 q^{75} +4.60555 q^{79} -3.39445 q^{81} +11.2111 q^{83} -2.30278 q^{85} +6.78890 q^{87} +13.8167 q^{89} +4.81665 q^{93} +3.21110 q^{95} +9.69722 q^{97} -5.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} + q^{9}+O(q^{10}) \) 2 * q + q^3 - q^5 + q^9	\( 2 q + q^{3} - q^{5} + q^{9} + 8 q^{11} + 2 q^{13} + 6 q^{15} + 2 q^{17} - 10 q^{19} + 8 q^{23} - 3 q^{25} + 4 q^{27} + 4 q^{29} - 11 q^{31} + 4 q^{33} - 2 q^{37} + 14 q^{39} + 5 q^{41} - 5 q^{43} + 6 q^{45} - 2 q^{47} + q^{51} + 5 q^{53} - 4 q^{55} - 18 q^{57} + 16 q^{59} - 13 q^{61} + 12 q^{65} + 9 q^{67} + 4 q^{69} - 14 q^{71} + 3 q^{73} - 8 q^{75} + 2 q^{79} - 14 q^{81} + 8 q^{83} - q^{85} + 28 q^{87} + 6 q^{89} - 12 q^{93} - 8 q^{95} + 23 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + q^3 - q^5 + q^9 + 8 * q^11 + 2 * q^13 + 6 * q^15 + 2 * q^17 - 10 * q^19 + 8 * q^23 - 3 * q^25 + 4 * q^27 + 4 * q^29 - 11 * q^31 + 4 * q^33 - 2 * q^37 + 14 * q^39 + 5 * q^41 - 5 * q^43 + 6 * q^45 - 2 * q^47 + q^51 + 5 * q^53 - 4 * q^55 - 18 * q^57 + 16 * q^59 - 13 * q^61 + 12 * q^65 + 9 * q^67 + 4 * q^69 - 14 * q^71 + 3 * q^73 - 8 * q^75 + 2 * q^79 - 14 * q^81 + 8 * q^83 - q^85 + 28 * q^87 + 6 * q^89 - 12 * q^93 - 8 * q^95 + 23 * q^97 + 4 * q^99

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