LMFDB - Embedded newform 3332.2.a.k.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:	$0$
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			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+2.30278 q^{3} +1.30278 q^{5} +2.30278 q^{9} +O(q^{10})$	$q+2.30278 q^{3} +1.30278 q^{5} +2.30278 q^{9} +4.00000 q^{11} +4.60555 q^{13} +3.00000 q^{15} +1.00000 q^{17} -8.60555 q^{19} +4.00000 q^{23} -3.30278 q^{25} -1.60555 q^{27} +9.21110 q^{29} -7.30278 q^{31} +9.21110 q^{33} +9.81665 q^{37} +10.6056 q^{39} +11.5139 q^{41} -4.30278 q^{43} +3.00000 q^{45} +2.60555 q^{47} +2.30278 q^{51} +0.697224 q^{53} +5.21110 q^{55} -19.8167 q^{57} +8.00000 q^{59} -15.5139 q^{61} +6.00000 q^{65} +2.69722 q^{67} +9.21110 q^{69} -3.39445 q^{71} -7.51388 q^{73} -7.60555 q^{75} -2.60555 q^{79} -10.6056 q^{81} -3.21110 q^{83} +1.30278 q^{85} +21.2111 q^{87} -7.81665 q^{89} -16.8167 q^{93} -11.2111 q^{95} +13.3028 q^{97} +9.21110 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.30278	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$3$	2.30278	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$4$	0	0
$5$	1.30278	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$5$	1.30278	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.30278	0.767592
$10$	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
$13$	4.60555	1.27735	0.638675	−	0.769477i	$-0.279483\pi$
$13$	4.60555	1.27735	0.638675	+	0.769477i	$0.279483\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−8.60555	−1.97425	−0.987124	−	0.159954i	$-0.948865\pi$
$19$	−8.60555	−1.97425	−0.987124	+	0.159954i	$0.948865\pi$
$20$	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
$25$	−3.30278	−0.660555
$26$	0	0
$27$	−1.60555	−0.308988
$28$	0	0
$29$	9.21110	1.71046	0.855229	−	0.518250i	$-0.173416\pi$
$29$	9.21110	1.71046	0.855229	+	0.518250i	$0.173416\pi$
$30$	0	0
$31$	−7.30278	−1.31162	−0.655809	−	0.754927i	$-0.727672\pi$
$31$	−7.30278	−1.31162	−0.655809	+	0.754927i	$0.727672\pi$
$32$	0	0
$33$	9.21110	1.60345
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.81665	1.61385	0.806924	−	0.590655i	$-0.201131\pi$
$37$	9.81665	1.61385	0.806924	+	0.590655i	$0.201131\pi$
$38$	0	0
$39$	10.6056	1.69825
$40$	0	0
$41$	11.5139	1.79817	0.899083	−	0.437779i	$-0.144235\pi$
$41$	11.5139	1.79817	0.899083	+	0.437779i	$0.144235\pi$
$42$	0	0
$43$	−4.30278	−0.656167	−0.328084	−	0.944649i	$-0.606403\pi$
$43$	−4.30278	−0.656167	−0.328084	+	0.944649i	$0.606403\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	2.60555	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$47$	2.60555	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.30278	0.322453
$52$	0	0
$53$	0.697224	0.0957711	0.0478856	−	0.998853i	$-0.484752\pi$
$53$	0.697224	0.0957711	0.0478856	+	0.998853i	$0.484752\pi$
$54$	0	0
$55$	5.21110	0.702665
$56$	0	0
$57$	−19.8167	−2.62478
$58$	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
$61$	−15.5139	−1.98635	−0.993174	−	0.116640i	$-0.962788\pi$
$61$	−15.5139	−1.98635	−0.993174	+	0.116640i	$0.962788\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	2.69722	0.329518	0.164759	−	0.986334i	$-0.447315\pi$
$67$	2.69722	0.329518	0.164759	+	0.986334i	$0.447315\pi$
$68$	0	0
$69$	9.21110	1.10889
$70$	0	0
$71$	−3.39445	−0.402847	−0.201423	−	0.979504i	$-0.564557\pi$
$71$	−3.39445	−0.402847	−0.201423	+	0.979504i	$0.564557\pi$
$72$	0	0
$73$	−7.51388	−0.879433	−0.439716	−	0.898137i	$-0.644921\pi$
$73$	−7.51388	−0.879433	−0.439716	+	0.898137i	$0.644921\pi$
$74$	0	0
$75$	−7.60555	−0.878213
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.60555	−0.293147	−0.146574	−	0.989200i	$-0.546825\pi$
$79$	−2.60555	−0.293147	−0.146574	+	0.989200i	$0.546825\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−3.21110	−0.352464	−0.176232	−	0.984349i	$-0.556391\pi$
$83$	−3.21110	−0.352464	−0.176232	+	0.984349i	$0.556391\pi$
$84$	0	0
$85$	1.30278	0.141306
$86$	0	0
$87$	21.2111	2.27407
$88$	0	0
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−16.8167	−1.74381
$94$	0	0
$95$	−11.2111	−1.15023
$96$	0	0
$97$	13.3028	1.35069	0.675346	−	0.737501i	$-0.263994\pi$
$97$	13.3028	1.35069	0.675346	+	0.737501i	$0.263994\pi$
$98$	0	0
$99$	9.21110	0.925751
$100$	0	0
$101$	12.6056	1.25430	0.627150	−	0.778899i	$-0.284221\pi$
$101$	12.6056	1.25430	0.627150	+	0.778899i	$0.284221\pi$
$102$	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
$107$	−4.60555	−0.445235	−0.222618	−	0.974906i	$-0.571460\pi$
$107$	−4.60555	−0.445235	−0.222618	+	0.974906i	$0.571460\pi$
$108$	0	0
$109$	4.78890	0.458693	0.229347	−	0.973345i	$-0.426341\pi$
$109$	4.78890	0.458693	0.229347	+	0.973345i	$0.426341\pi$
$110$	0	0
$111$	22.6056	2.14562
$112$	0	0
$113$	−9.39445	−0.883755	−0.441878	−	0.897075i	$-0.645687\pi$
$113$	−9.39445	−0.883755	−0.441878	+	0.897075i	$0.645687\pi$
$114$	0	0
$115$	5.21110	0.485938
$116$	0	0
$117$	10.6056	0.980484
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	26.5139	2.39068
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	14.5139	1.28790	0.643949	−	0.765068i	$-0.277295\pi$
$127$	14.5139	1.28790	0.643949	+	0.765068i	$0.277295\pi$
$128$	0	0
$129$	−9.90833	−0.872380
$130$	0	0
$131$	9.21110	0.804778	0.402389	−	0.915469i	$-0.368180\pi$
$131$	9.21110	0.804778	0.402389	+	0.915469i	$0.368180\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.09167	−0.180023
$136$	0	0
$137$	18.1194	1.54805	0.774024	−	0.633157i	$-0.218241\pi$
$137$	18.1194	1.54805	0.774024	+	0.633157i	$0.218241\pi$
$138$	0	0
$139$	−8.51388	−0.722138	−0.361069	−	0.932539i	$-0.617588\pi$
$139$	−8.51388	−0.722138	−0.361069	+	0.932539i	$0.617588\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	18.4222	1.54054
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.30278	−0.352497	−0.176249	−	0.984346i	$-0.556396\pi$
$149$	−4.30278	−0.352497	−0.176249	+	0.984346i	$0.556396\pi$
$150$	0	0
$151$	11.6972	0.951907	0.475953	−	0.879471i	$-0.342103\pi$
$151$	11.6972	0.951907	0.475953	+	0.879471i	$0.342103\pi$
$152$	0	0
$153$	2.30278	0.186168
$154$	0	0
$155$	−9.51388	−0.764173
$156$	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
$159$	1.60555	0.127328
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.39445	0.109222	0.0546108	−	0.998508i	$-0.482608\pi$
$163$	1.39445	0.109222	0.0546108	+	0.998508i	$0.482608\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	−12.1194	−0.937830	−0.468915	−	0.883243i	$-0.655355\pi$
$167$	−12.1194	−0.937830	−0.468915	+	0.883243i	$0.655355\pi$
$168$	0	0
$169$	8.21110	0.631623
$170$	0	0
$171$	−19.8167	−1.51542
$172$	0	0
$173$	−20.7250	−1.57569	−0.787846	−	0.615873i	$-0.788803\pi$
$173$	−20.7250	−1.57569	−0.787846	+	0.615873i	$0.788803\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.4222	1.38470
$178$	0	0
$179$	9.51388	0.711101	0.355550	−	0.934657i	$-0.384293\pi$
$179$	9.51388	0.711101	0.355550	+	0.934657i	$0.384293\pi$
$180$	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
$183$	−35.7250	−2.64087
$184$	0	0
$185$	12.7889	0.940258
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.7250	−0.848390	−0.424195	−	0.905571i	$-0.639443\pi$
$191$	−11.7250	−0.848390	−0.424195	+	0.905571i	$0.639443\pi$
$192$	0	0
$193$	−15.0278	−1.08172	−0.540861	−	0.841112i	$-0.681901\pi$
$193$	−15.0278	−1.08172	−0.540861	+	0.841112i	$0.681901\pi$
$194$	0	0
$195$	13.8167	0.989431
$196$	0	0
$197$	−6.60555	−0.470626	−0.235313	−	0.971920i	$-0.575612\pi$
$197$	−6.60555	−0.470626	−0.235313	+	0.971920i	$0.575612\pi$
$198$	0	0
$199$	−9.69722	−0.687418	−0.343709	−	0.939076i	$-0.611683\pi$
$199$	−9.69722	−0.687418	−0.343709	+	0.939076i	$0.611683\pi$
$200$	0	0
$201$	6.21110	0.438097
$202$	0	0
$203$	0	0
$204$	0	0
$205$	15.0000	1.04765
$206$	0	0
$207$	9.21110	0.640216
$208$	0	0
$209$	−34.4222	−2.38103
$210$	0	0
$211$	6.18335	0.425679	0.212840	−	0.977087i	$-0.431729\pi$
$211$	6.18335	0.425679	0.212840	+	0.977087i	$0.431729\pi$
$212$	0	0
$213$	−7.81665	−0.535588
$214$	0	0
$215$	−5.60555	−0.382295
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−17.3028	−1.16921
$220$	0	0
$221$	4.60555	0.309803
$222$	0	0
$223$	−13.8167	−0.925232	−0.462616	−	0.886559i	$-0.653089\pi$
$223$	−13.8167	−0.925232	−0.462616	+	0.886559i	$0.653089\pi$
$224$	0	0
$225$	−7.60555	−0.507037
$226$	0	0
$227$	−2.09167	−0.138829	−0.0694146	−	0.997588i	$-0.522113\pi$
$227$	−2.09167	−0.138829	−0.0694146	+	0.997588i	$0.522113\pi$
$228$	0	0
$229$	1.21110	0.0800319	0.0400160	−	0.999199i	$-0.487259\pi$
$229$	1.21110	0.0800319	0.0400160	+	0.999199i	$0.487259\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.6056	1.74299	0.871494	−	0.490407i	$-0.163152\pi$
$233$	26.6056	1.74299	0.871494	+	0.490407i	$0.163152\pi$
$234$	0	0
$235$	3.39445	0.221429
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	18.1194	1.17205	0.586024	−	0.810294i	$-0.300692\pi$
$239$	18.1194	1.17205	0.586024	+	0.810294i	$0.300692\pi$
$240$	0	0
$241$	26.5139	1.70791	0.853955	−	0.520348i	$-0.174198\pi$
$241$	26.5139	1.70791	0.853955	+	0.520348i	$0.174198\pi$
$242$	0	0
$243$	−19.6056	−1.25770
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−39.6333	−2.52181
$248$	0	0
$249$	−7.39445	−0.468604
$250$	0	0
$251$	2.18335	0.137812	0.0689058	−	0.997623i	$-0.478049\pi$
$251$	2.18335	0.137812	0.0689058	+	0.997623i	$0.478049\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	4.42221	0.275850	0.137925	−	0.990443i	$-0.455957\pi$
$257$	4.42221	0.275850	0.137925	+	0.990443i	$0.455957\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.2111	1.31293
$262$	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
$265$	0.908327	0.0557981
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	−12.7889	−0.779753	−0.389876	−	0.920867i	$-0.627482\pi$
$269$	−12.7889	−0.779753	−0.389876	+	0.920867i	$0.627482\pi$
$270$	0	0
$271$	10.4222	0.633104	0.316552	−	0.948575i	$-0.397475\pi$
$271$	10.4222	0.633104	0.316552	+	0.948575i	$0.397475\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.2111	−0.796659
$276$	0	0
$277$	1.57779	0.0948005	0.0474003	−	0.998876i	$-0.484906\pi$
$277$	1.57779	0.0948005	0.0474003	+	0.998876i	$0.484906\pi$
$278$	0	0
$279$	−16.8167	−1.00679
$280$	0	0
$281$	−19.1194	−1.14057	−0.570285	−	0.821447i	$-0.693167\pi$
$281$	−19.1194	−1.14057	−0.570285	+	0.821447i	$0.693167\pi$
$282$	0	0
$283$	−20.3305	−1.20852	−0.604262	−	0.796785i	$-0.706532\pi$
$283$	−20.3305	−1.20852	−0.604262	+	0.796785i	$0.706532\pi$
$284$	0	0
$285$	−25.8167	−1.52925
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	30.6333	1.79576
$292$	0	0
$293$	11.2111	0.654960	0.327480	−	0.944858i	$-0.393801\pi$
$293$	11.2111	0.654960	0.327480	+	0.944858i	$0.393801\pi$
$294$	0	0
$295$	10.4222	0.606804
$296$	0	0
$297$	−6.42221	−0.372654
$298$	0	0
$299$	18.4222	1.06538
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.0278	1.66760
$304$	0	0
$305$	−20.2111	−1.15728
$306$	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
$309$	−32.2389	−1.83400
$310$	0	0
$311$	17.7250	1.00509	0.502546	−	0.864551i	$-0.332397\pi$
$311$	17.7250	1.00509	0.502546	+	0.864551i	$0.332397\pi$
$312$	0	0
$313$	−3.90833	−0.220912	−0.110456	−	0.993881i	$-0.535231\pi$
$313$	−3.90833	−0.220912	−0.110456	+	0.993881i	$0.535231\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.6333	−1.88903	−0.944517	−	0.328461i	$-0.893470\pi$
$317$	−33.6333	−1.88903	−0.944517	+	0.328461i	$0.893470\pi$
$318$	0	0
$319$	36.8444	2.06289
$320$	0	0
$321$	−10.6056	−0.591944
$322$	0	0
$323$	−8.60555	−0.478826
$324$	0	0
$325$	−15.2111	−0.843760
$326$	0	0
$327$	11.0278	0.609836
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3.72498	0.204743	0.102372	−	0.994746i	$-0.467357\pi$
$331$	3.72498	0.204743	0.102372	+	0.994746i	$0.467357\pi$
$332$	0	0
$333$	22.6056	1.23878
$334$	0	0
$335$	3.51388	0.191984
$336$	0	0
$337$	−14.4222	−0.785628	−0.392814	−	0.919618i	$-0.628498\pi$
$337$	−14.4222	−0.785628	−0.392814	+	0.919618i	$0.628498\pi$
$338$	0	0
$339$	−21.6333	−1.17496
$340$	0	0
$341$	−29.2111	−1.58187
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	26.8444	1.44108	0.720542	−	0.693412i	$-0.243893\pi$
$347$	26.8444	1.44108	0.720542	+	0.693412i	$0.243893\pi$
$348$	0	0
$349$	−18.4222	−0.986118	−0.493059	−	0.869996i	$-0.664121\pi$
$349$	−18.4222	−0.986118	−0.493059	+	0.869996i	$0.664121\pi$
$350$	0	0
$351$	−7.39445	−0.394686
$352$	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
$355$	−4.42221	−0.234706
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.48612	−0.0784345	−0.0392173	−	0.999231i	$-0.512486\pi$
$359$	−1.48612	−0.0784345	−0.0392173	+	0.999231i	$0.512486\pi$
$360$	0	0
$361$	55.0555	2.89766
$362$	0	0
$363$	11.5139	0.604322
$364$	0	0
$365$	−9.78890	−0.512374
$366$	0	0
$367$	−6.90833	−0.360612	−0.180306	−	0.983611i	$-0.557709\pi$
$367$	−6.90833	−0.360612	−0.180306	+	0.983611i	$0.557709\pi$
$368$	0	0
$369$	26.5139	1.38026
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.3305	−0.793785	−0.396892	−	0.917865i	$-0.629911\pi$
$373$	−15.3305	−0.793785	−0.396892	+	0.917865i	$0.629911\pi$
$374$	0	0
$375$	−24.9083	−1.28626
$376$	0	0
$377$	42.4222	2.18485
$378$	0	0
$379$	9.57779	0.491978	0.245989	−	0.969273i	$-0.420887\pi$
$379$	9.57779	0.491978	0.245989	+	0.969273i	$0.420887\pi$
$380$	0	0
$381$	33.4222	1.71227
$382$	0	0
$383$	1.57779	0.0806216	0.0403108	−	0.999187i	$-0.487165\pi$
$383$	1.57779	0.0806216	0.0403108	+	0.999187i	$0.487165\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.90833	−0.503669
$388$	0	0
$389$	−29.7250	−1.50712	−0.753558	−	0.657381i	$-0.771664\pi$
$389$	−29.7250	−1.50712	−0.753558	+	0.657381i	$0.771664\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	21.2111	1.06996
$394$	0	0
$395$	−3.39445	−0.170793
$396$	0	0
$397$	12.0917	0.606864	0.303432	−	0.952853i	$-0.401868\pi$
$397$	12.0917	0.606864	0.303432	+	0.952853i	$0.401868\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.60555	0.130115	0.0650575	−	0.997882i	$-0.479277\pi$
$401$	2.60555	0.130115	0.0650575	+	0.997882i	$0.479277\pi$
$402$	0	0
$403$	−33.6333	−1.67539
$404$	0	0
$405$	−13.8167	−0.686555
$406$	0	0
$407$	39.2666	1.94637
$408$	0	0
$409$	5.81665	0.287615	0.143808	−	0.989606i	$-0.454065\pi$
$409$	5.81665	0.287615	0.143808	+	0.989606i	$0.454065\pi$
$410$	0	0
$411$	41.7250	2.05814
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.18335	−0.205352
$416$	0	0
$417$	−19.6056	−0.960088
$418$	0	0
$419$	8.51388	0.415930	0.207965	−	0.978136i	$-0.433316\pi$
$419$	8.51388	0.415930	0.207965	+	0.978136i	$0.433316\pi$
$420$	0	0
$421$	11.0917	0.540575	0.270288	−	0.962780i	$-0.412881\pi$
$421$	11.0917	0.540575	0.270288	+	0.962780i	$0.412881\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	−3.30278	−0.160208
$426$	0	0
$427$	0	0
$428$	0	0
$429$	42.4222	2.04816
$430$	0	0
$431$	−28.8444	−1.38939	−0.694693	−	0.719306i	$-0.744460\pi$
$431$	−28.8444	−1.38939	−0.694693	+	0.719306i	$0.744460\pi$
$432$	0	0
$433$	7.39445	0.355355	0.177677	−	0.984089i	$-0.443142\pi$
$433$	7.39445	0.355355	0.177677	+	0.984089i	$0.443142\pi$
$434$	0	0
$435$	27.6333	1.32492
$436$	0	0
$437$	−34.4222	−1.64664
$438$	0	0
$439$	−21.0917	−1.00665	−0.503325	−	0.864097i	$-0.667890\pi$
$439$	−21.0917	−1.00665	−0.503325	+	0.864097i	$0.667890\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.2111	−1.57791	−0.788954	−	0.614453i	$-0.789377\pi$
$443$	−33.2111	−1.57791	−0.788954	+	0.614453i	$0.789377\pi$
$444$	0	0
$445$	−10.1833	−0.482737
$446$	0	0
$447$	−9.90833	−0.468648
$448$	0	0
$449$	−5.02776	−0.237274	−0.118637	−	0.992938i	$-0.537853\pi$
$449$	−5.02776	−0.237274	−0.118637	+	0.992938i	$0.537853\pi$
$450$	0	0
$451$	46.0555	2.16867
$452$	0	0
$453$	26.9361	1.26557
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.9083	−0.650604	−0.325302	−	0.945610i	$-0.605466\pi$
$457$	−13.9083	−0.650604	−0.325302	+	0.945610i	$0.605466\pi$
$458$	0	0
$459$	−1.60555	−0.0749407
$460$	0	0
$461$	−23.3944	−1.08959	−0.544794	−	0.838570i	$-0.683392\pi$
$461$	−23.3944	−1.08959	−0.544794	+	0.838570i	$0.683392\pi$
$462$	0	0
$463$	−6.72498	−0.312536	−0.156268	−	0.987715i	$-0.549946\pi$
$463$	−6.72498	−0.312536	−0.156268	+	0.987715i	$0.549946\pi$
$464$	0	0
$465$	−21.9083	−1.01597
$466$	0	0
$467$	16.6056	0.768413	0.384207	−	0.923247i	$-0.374475\pi$
$467$	16.6056	0.768413	0.384207	+	0.923247i	$0.374475\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.60555	−0.212213
$472$	0	0
$473$	−17.2111	−0.791367
$474$	0	0
$475$	28.4222	1.30410
$476$	0	0
$477$	1.60555	0.0735131
$478$	0	0
$479$	−17.0917	−0.780938	−0.390469	−	0.920616i	$-0.627687\pi$
$479$	−17.0917	−0.780938	−0.390469	+	0.920616i	$0.627687\pi$
$480$	0	0
$481$	45.2111	2.06145
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.3305	0.786939
$486$	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
$489$	3.21110	0.145211
$490$	0	0
$491$	−22.5139	−1.01604	−0.508019	−	0.861346i	$-0.669622\pi$
$491$	−22.5139	−1.01604	−0.508019	+	0.861346i	$0.669622\pi$
$492$	0	0
$493$	9.21110	0.414847
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−29.2111	−1.30767	−0.653834	−	0.756638i	$-0.726841\pi$
$499$	−29.2111	−1.30767	−0.653834	+	0.756638i	$0.726841\pi$
$500$	0	0
$501$	−27.9083	−1.24685
$502$	0	0
$503$	31.1194	1.38755	0.693773	−	0.720193i	$-0.255947\pi$
$503$	31.1194	1.38755	0.693773	+	0.720193i	$0.255947\pi$
$504$	0	0
$505$	16.4222	0.730779
$506$	0	0
$507$	18.9083	0.839748
$508$	0	0
$509$	−6.60555	−0.292786	−0.146393	−	0.989227i	$-0.546766\pi$
$509$	−6.60555	−0.292786	−0.146393	+	0.989227i	$0.546766\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	13.8167	0.610020
$514$	0	0
$515$	−18.2389	−0.803700
$516$	0	0
$517$	10.4222	0.458368
$518$	0	0
$519$	−47.7250	−2.09489
$520$	0	0
$521$	0.486122	0.0212974	0.0106487	−	0.999943i	$-0.496610\pi$
$521$	0.486122	0.0212974	0.0106487	+	0.999943i	$0.496610\pi$
$522$	0	0
$523$	−38.0555	−1.66405	−0.832026	−	0.554737i	$-0.812819\pi$
$523$	−38.0555	−1.66405	−0.832026	+	0.554737i	$0.812819\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.30278	−0.318114
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	18.4222	0.799456
$532$	0	0
$533$	53.0278	2.29689
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	21.9083	0.945414
$538$	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
$543$	13.8167	0.592929
$544$	0	0
$545$	6.23886	0.267243
$546$	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
$549$	−35.7250	−1.52471
$550$	0	0
$551$	−79.2666	−3.37687
$552$	0	0
$553$	0	0
$554$	0	0
$555$	29.4500	1.25008
$556$	0	0
$557$	−20.7889	−0.880854	−0.440427	−	0.897788i	$-0.645173\pi$
$557$	−20.7889	−0.880854	−0.440427	+	0.897788i	$0.645173\pi$
$558$	0	0
$559$	−19.8167	−0.838155
$560$	0	0
$561$	9.21110	0.388893
$562$	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
$565$	−12.2389	−0.514893
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.1194	0.927295	0.463647	−	0.886020i	$-0.346540\pi$
$569$	22.1194	0.927295	0.463647	+	0.886020i	$0.346540\pi$
$570$	0	0
$571$	−45.4500	−1.90202	−0.951011	−	0.309158i	$-0.899953\pi$
$571$	−45.4500	−1.90202	−0.951011	+	0.309158i	$0.899953\pi$
$572$	0	0
$573$	−27.0000	−1.12794
$574$	0	0
$575$	−13.2111	−0.550941
$576$	0	0
$577$	8.18335	0.340677	0.170339	−	0.985386i	$-0.445514\pi$
$577$	8.18335	0.340677	0.170339	+	0.985386i	$0.445514\pi$
$578$	0	0
$579$	−34.6056	−1.43816
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.78890	0.115504
$584$	0	0
$585$	13.8167	0.571248
$586$	0	0
$587$	11.3944	0.470299	0.235150	−	0.971959i	$-0.424442\pi$
$587$	11.3944	0.470299	0.235150	+	0.971959i	$0.424442\pi$
$588$	0	0
$589$	62.8444	2.58946
$590$	0	0
$591$	−15.2111	−0.625701
$592$	0	0
$593$	−22.1833	−0.910961	−0.455480	−	0.890246i	$-0.650532\pi$
$593$	−22.1833	−0.910961	−0.455480	+	0.890246i	$0.650532\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.3305	−0.913928
$598$	0	0
$599$	26.7250	1.09195	0.545977	−	0.837800i	$-0.316159\pi$
$599$	26.7250	1.09195	0.545977	+	0.837800i	$0.316159\pi$
$600$	0	0
$601$	−45.2666	−1.84646	−0.923232	−	0.384243i	$-0.874462\pi$
$601$	−45.2666	−1.84646	−0.923232	+	0.384243i	$0.874462\pi$
$602$	0	0
$603$	6.21110	0.252936
$604$	0	0
$605$	6.51388	0.264827
$606$	0	0
$607$	−0.330532	−0.0134159	−0.00670794	−	0.999978i	$-0.502135\pi$
$607$	−0.330532	−0.0134159	−0.00670794	+	0.999978i	$0.502135\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−21.7250	−0.877464	−0.438732	−	0.898618i	$-0.644572\pi$
$613$	−21.7250	−0.877464	−0.438732	+	0.898618i	$0.644572\pi$
$614$	0	0
$615$	34.5416	1.39285
$616$	0	0
$617$	7.63331	0.307305	0.153653	−	0.988125i	$-0.450896\pi$
$617$	7.63331	0.307305	0.153653	+	0.988125i	$0.450896\pi$
$618$	0	0
$619$	−37.2111	−1.49564	−0.747820	−	0.663901i	$-0.768900\pi$
$619$	−37.2111	−1.49564	−0.747820	+	0.663901i	$0.768900\pi$
$620$	0	0
$621$	−6.42221	−0.257714
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	−79.2666	−3.16560
$628$	0	0
$629$	9.81665	0.391416
$630$	0	0
$631$	9.11943	0.363039	0.181519	−	0.983387i	$-0.441898\pi$
$631$	9.11943	0.363039	0.181519	+	0.983387i	$0.441898\pi$
$632$	0	0
$633$	14.2389	0.565944
$634$	0	0
$635$	18.9083	0.750354
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.81665	−0.309222
$640$	0	0
$641$	43.6333	1.72341	0.861706	−	0.507408i	$-0.169396\pi$
$641$	43.6333	1.72341	0.861706	+	0.507408i	$0.169396\pi$
$642$	0	0
$643$	−7.88057	−0.310779	−0.155390	−	0.987853i	$-0.549663\pi$
$643$	−7.88057	−0.310779	−0.155390	+	0.987853i	$0.549663\pi$
$644$	0	0
$645$	−12.9083	−0.508265
$646$	0	0
$647$	35.6333	1.40089	0.700445	−	0.713706i	$-0.252985\pi$
$647$	35.6333	1.40089	0.700445	+	0.713706i	$0.252985\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.4222	1.58184	0.790922	−	0.611918i	$-0.209602\pi$
$653$	40.4222	1.58184	0.790922	+	0.611918i	$0.209602\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−17.3028	−0.675046
$658$	0	0
$659$	21.1194	0.822696	0.411348	−	0.911478i	$-0.365058\pi$
$659$	21.1194	0.822696	0.411348	+	0.911478i	$0.365058\pi$
$660$	0	0
$661$	5.02776	0.195557	0.0977785	−	0.995208i	$-0.468826\pi$
$661$	5.02776	0.195557	0.0977785	+	0.995208i	$0.468826\pi$
$662$	0	0
$663$	10.6056	0.411885
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.8444	1.42662
$668$	0	0
$669$	−31.8167	−1.23010
$670$	0	0
$671$	−62.0555	−2.39563
$672$	0	0
$673$	−7.02776	−0.270900	−0.135450	−	0.990784i	$-0.543248\pi$
$673$	−7.02776	−0.270900	−0.135450	+	0.990784i	$0.543248\pi$
$674$	0	0
$675$	5.30278	0.204104
$676$	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
$681$	−4.81665	−0.184575
$682$	0	0
$683$	13.3944	0.512524	0.256262	−	0.966607i	$-0.417509\pi$
$683$	13.3944	0.512524	0.256262	+	0.966607i	$0.417509\pi$
$684$	0	0
$685$	23.6056	0.901922
$686$	0	0
$687$	2.78890	0.106403
$688$	0	0
$689$	3.21110	0.122333
$690$	0	0
$691$	1.93608	0.0736521	0.0368260	−	0.999322i	$-0.488275\pi$
$691$	1.93608	0.0736521	0.0368260	+	0.999322i	$0.488275\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.0917	−0.420731
$696$	0	0
$697$	11.5139	0.436119
$698$	0	0
$699$	61.2666	2.31732
$700$	0	0
$701$	−11.5778	−0.437287	−0.218644	−	0.975805i	$-0.570163\pi$
$701$	−11.5778	−0.437287	−0.218644	+	0.975805i	$0.570163\pi$
$702$	0	0
$703$	−84.4777	−3.18614
$704$	0	0
$705$	7.81665	0.294392
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.23886	−0.159194	−0.0795968	−	0.996827i	$-0.525363\pi$
$709$	−4.23886	−0.159194	−0.0795968	+	0.996827i	$0.525363\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	−29.2111	−1.09396
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	41.7250	1.55825
$718$	0	0
$719$	−9.69722	−0.361645	−0.180823	−	0.983516i	$-0.557876\pi$
$719$	−9.69722	−0.361645	−0.180823	+	0.983516i	$0.557876\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	61.0555	2.27068
$724$	0	0
$725$	−30.4222	−1.12985
$726$	0	0
$727$	14.4222	0.534890	0.267445	−	0.963573i	$-0.413821\pi$
$727$	14.4222	0.534890	0.267445	+	0.963573i	$0.413821\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	−4.30278	−0.159144
$732$	0	0
$733$	−16.2389	−0.599796	−0.299898	−	0.953971i	$-0.596953\pi$
$733$	−16.2389	−0.599796	−0.299898	+	0.953971i	$0.596953\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.7889	0.397414
$738$	0	0
$739$	−23.3305	−0.858227	−0.429114	−	0.903250i	$-0.641174\pi$
$739$	−23.3305	−0.858227	−0.429114	+	0.903250i	$0.641174\pi$
$740$	0	0
$741$	−91.2666	−3.35276
$742$	0	0
$743$	−34.6056	−1.26955	−0.634777	−	0.772695i	$-0.718908\pi$
$743$	−34.6056	−1.26955	−0.634777	+	0.772695i	$0.718908\pi$
$744$	0	0
$745$	−5.60555	−0.205372
$746$	0	0
$747$	−7.39445	−0.270549
$748$	0	0
$749$	0	0
$750$	0	0
$751$	3.39445	0.123865	0.0619326	−	0.998080i	$-0.480274\pi$
$751$	3.39445	0.123865	0.0619326	+	0.998080i	$0.480274\pi$
$752$	0	0
$753$	5.02776	0.183222
$754$	0	0
$755$	15.2389	0.554599
$756$	0	0
$757$	−7.69722	−0.279760	−0.139880	−	0.990168i	$-0.544672\pi$
$757$	−7.69722	−0.279760	−0.139880	+	0.990168i	$0.544672\pi$
$758$	0	0
$759$	36.8444	1.33737
$760$	0	0
$761$	50.2389	1.82116	0.910579	−	0.413336i	$-0.135636\pi$
$761$	50.2389	1.82116	0.910579	+	0.413336i	$0.135636\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	36.8444	1.33037
$768$	0	0
$769$	36.0555	1.30020	0.650098	−	0.759851i	$-0.274728\pi$
$769$	36.0555	1.30020	0.650098	+	0.759851i	$0.274728\pi$
$770$	0	0
$771$	10.1833	0.366744
$772$	0	0
$773$	−17.0278	−0.612446	−0.306223	−	0.951960i	$-0.599065\pi$
$773$	−17.0278	−0.612446	−0.306223	+	0.951960i	$0.599065\pi$
$774$	0	0
$775$	24.1194	0.866395
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−99.0833	−3.55003
$780$	0	0
$781$	−13.5778	−0.485852
$782$	0	0
$783$	−14.7889	−0.528512
$784$	0	0
$785$	−2.60555	−0.0929961
$786$	0	0
$787$	−50.4222	−1.79736	−0.898679	−	0.438607i	$-0.855472\pi$
$787$	−50.4222	−1.79736	−0.898679	+	0.438607i	$0.855472\pi$
$788$	0	0
$789$	−42.4222	−1.51027
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−71.4500	−2.53726
$794$	0	0
$795$	2.09167	0.0741840
$796$	0	0
$797$	−48.0555	−1.70221	−0.851107	−	0.524993i	$-0.824068\pi$
$797$	−48.0555	−1.70221	−0.851107	+	0.524993i	$0.824068\pi$
$798$	0	0
$799$	2.60555	0.0921778
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	−30.0555	−1.06064
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29.4500	−1.03669
$808$	0	0
$809$	−19.8167	−0.696716	−0.348358	−	0.937361i	$-0.613261\pi$
$809$	−19.8167	−0.696716	−0.348358	+	0.937361i	$0.613261\pi$
$810$	0	0
$811$	4.48612	0.157529	0.0787645	−	0.996893i	$-0.474902\pi$
$811$	4.48612	0.157529	0.0787645	+	0.996893i	$0.474902\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	0	0
$815$	1.81665	0.0636346
$816$	0	0
$817$	37.0278	1.29544
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.23886	−0.287538	−0.143769	−	0.989611i	$-0.545922\pi$
$821$	−8.23886	−0.287538	−0.143769	+	0.989611i	$0.545922\pi$
$822$	0	0
$823$	−38.6056	−1.34570	−0.672852	−	0.739777i	$-0.734931\pi$
$823$	−38.6056	−1.34570	−0.672852	+	0.739777i	$0.734931\pi$
$824$	0	0
$825$	−30.4222	−1.05917
$826$	0	0
$827$	45.2111	1.57214	0.786072	−	0.618135i	$-0.212111\pi$
$827$	45.2111	1.57214	0.786072	+	0.618135i	$0.212111\pi$
$828$	0	0
$829$	16.2389	0.563999	0.281999	−	0.959415i	$-0.409002\pi$
$829$	16.2389	0.563999	0.281999	+	0.959415i	$0.409002\pi$
$830$	0	0
$831$	3.63331	0.126038
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15.7889	−0.546397
$836$	0	0
$837$	11.7250	0.405275
$838$	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
$841$	55.8444	1.92567
$842$	0	0
$843$	−44.0278	−1.51640
$844$	0	0
$845$	10.6972	0.367996
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−46.8167	−1.60674
$850$	0	0
$851$	39.2666	1.34604
$852$	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
$855$	−25.8167	−0.882911
$856$	0	0
$857$	45.9638	1.57009	0.785047	−	0.619436i	$-0.212639\pi$
$857$	45.9638	1.57009	0.785047	+	0.619436i	$0.212639\pi$
$858$	0	0
$859$	54.6056	1.86312	0.931559	−	0.363591i	$-0.118449\pi$
$859$	54.6056	1.86312	0.931559	+	0.363591i	$0.118449\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.3583	0.795125	0.397563	−	0.917575i	$-0.369856\pi$
$863$	23.3583	0.795125	0.397563	+	0.917575i	$0.369856\pi$
$864$	0	0
$865$	−27.0000	−0.918028
$866$	0	0
$867$	2.30278	0.0782064
$868$	0	0
$869$	−10.4222	−0.353549
$870$	0	0
$871$	12.4222	0.420910
$872$	0	0
$873$	30.6333	1.03678
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.39445	−0.0470872	−0.0235436	−	0.999723i	$-0.507495\pi$
$877$	−1.39445	−0.0470872	−0.0235436	+	0.999723i	$0.507495\pi$
$878$	0	0
$879$	25.8167	0.870774
$880$	0	0
$881$	13.1194	0.442005	0.221002	−	0.975273i	$-0.429067\pi$
$881$	13.1194	0.442005	0.221002	+	0.975273i	$0.429067\pi$
$882$	0	0
$883$	12.0917	0.406917	0.203459	−	0.979084i	$-0.434782\pi$
$883$	12.0917	0.406917	0.203459	+	0.979084i	$0.434782\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	−55.7805	−1.87293	−0.936463	−	0.350767i	$-0.885921\pi$
$887$	−55.7805	−1.87293	−0.936463	+	0.350767i	$0.885921\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−42.4222	−1.42120
$892$	0	0
$893$	−22.4222	−0.750330
$894$	0	0
$895$	12.3944	0.414301
$896$	0	0
$897$	42.4222	1.41644
$898$	0	0
$899$	−67.2666	−2.24347
$900$	0	0
$901$	0.697224	0.0232279
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.81665	0.259834
$906$	0	0
$907$	27.2111	0.903530	0.451765	−	0.892137i	$-0.350795\pi$
$907$	27.2111	0.903530	0.451765	+	0.892137i	$0.350795\pi$
$908$	0	0
$909$	29.0278	0.962790
$910$	0	0
$911$	−15.8167	−0.524029	−0.262015	−	0.965064i	$-0.584387\pi$
$911$	−15.8167	−0.524029	−0.262015	+	0.965064i	$0.584387\pi$
$912$	0	0
$913$	−12.8444	−0.425088
$914$	0	0
$915$	−46.5416	−1.53862
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.3583	0.869480	0.434740	−	0.900556i	$-0.356840\pi$
$919$	26.3583	0.869480	0.434740	+	0.900556i	$0.356840\pi$
$920$	0	0
$921$	−9.21110	−0.303516
$922$	0	0
$923$	−15.6333	−0.514577
$924$	0	0
$925$	−32.4222	−1.06604
$926$	0	0
$927$	−32.2389	−1.05886
$928$	0	0
$929$	−36.0917	−1.18413	−0.592065	−	0.805890i	$-0.701687\pi$
$929$	−36.0917	−1.18413	−0.592065	+	0.805890i	$0.701687\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	40.8167	1.33628
$934$	0	0
$935$	5.21110	0.170421
$936$	0	0
$937$	37.2111	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$937$	37.2111	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	0	0
$941$	33.6972	1.09850	0.549249	−	0.835659i	$-0.314914\pi$
$941$	33.6972	1.09850	0.549249	+	0.835659i	$0.314914\pi$
$942$	0	0
$943$	46.0555	1.49977
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.8444	0.807335	0.403667	−	0.914906i	$-0.367735\pi$
$947$	24.8444	0.807335	0.403667	+	0.914906i	$0.367735\pi$
$948$	0	0
$949$	−34.6056	−1.12334
$950$	0	0
$951$	−77.4500	−2.51149
$952$	0	0
$953$	−21.3583	−0.691863	−0.345931	−	0.938260i	$-0.612437\pi$
$953$	−21.3583	−0.691863	−0.345931	+	0.938260i	$0.612437\pi$
$954$	0	0
$955$	−15.2750	−0.494288
$956$	0	0
$957$	84.8444	2.74263
$958$	0	0
$959$	0	0
$960$	0	0
$961$	22.3305	0.720340
$962$	0	0
$963$	−10.6056	−0.341759
$964$	0	0
$965$	−19.5778	−0.630232
$966$	0	0
$967$	−9.51388	−0.305946	−0.152973	−	0.988230i	$-0.548885\pi$
$967$	−9.51388	−0.305946	−0.152973	+	0.988230i	$0.548885\pi$
$968$	0	0
$969$	−19.8167	−0.636603
$970$	0	0
$971$	−2.78890	−0.0895000	−0.0447500	−	0.998998i	$-0.514249\pi$
$971$	−2.78890	−0.0895000	−0.0447500	+	0.998998i	$0.514249\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−35.0278	−1.12179
$976$	0	0
$977$	−8.69722	−0.278249	−0.139124	−	0.990275i	$-0.544429\pi$
$977$	−8.69722	−0.278249	−0.139124	+	0.990275i	$0.544429\pi$
$978$	0	0
$979$	−31.2666	−0.999285
$980$	0	0
$981$	11.0278	0.352089
$982$	0	0
$983$	−47.7805	−1.52396	−0.761981	−	0.647600i	$-0.775773\pi$
$983$	−47.7805	−1.52396	−0.761981	+	0.647600i	$0.775773\pi$
$984$	0	0
$985$	−8.60555	−0.274196
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17.2111	−0.547281
$990$	0	0
$991$	−11.2111	−0.356132	−0.178066	−	0.984019i	$-0.556984\pi$
$991$	−11.2111	−0.356132	−0.178066	+	0.984019i	$0.556984\pi$
$992$	0	0
$993$	8.57779	0.272208
$994$	0	0
$995$	−12.6333	−0.400503
$996$	0	0
$997$	45.6972	1.44725	0.723623	−	0.690196i	$-0.242476\pi$
$997$	45.6972	1.44725	0.723623	+	0.690196i	$0.242476\pi$
$998$	0	0
$999$	−15.7611	−0.498660

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.30278 q^{3} +1.30278 q^{5} +2.30278 q^{9} +O(q^{10})\)	\(q+2.30278 q^{3} +1.30278 q^{5} +2.30278 q^{9} +4.00000 q^{11} +4.60555 q^{13} +3.00000 q^{15} +1.00000 q^{17} -8.60555 q^{19} +4.00000 q^{23} -3.30278 q^{25} -1.60555 q^{27} +9.21110 q^{29} -7.30278 q^{31} +9.21110 q^{33} +9.81665 q^{37} +10.6056 q^{39} +11.5139 q^{41} -4.30278 q^{43} +3.00000 q^{45} +2.60555 q^{47} +2.30278 q^{51} +0.697224 q^{53} +5.21110 q^{55} -19.8167 q^{57} +8.00000 q^{59} -15.5139 q^{61} +6.00000 q^{65} +2.69722 q^{67} +9.21110 q^{69} -3.39445 q^{71} -7.51388 q^{73} -7.60555 q^{75} -2.60555 q^{79} -10.6056 q^{81} -3.21110 q^{83} +1.30278 q^{85} +21.2111 q^{87} -7.81665 q^{89} -16.8167 q^{93} -11.2111 q^{95} +13.3028 q^{97} +9.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} + q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more