LMFDB - Embedded newform 6292.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575 q^{3} -3.64575 q^{5} +1.64575 q^{7} +4.00000 q^{9} +O(q^{10})$	$q-2.64575 q^{3} -3.64575 q^{5} +1.64575 q^{7} +4.00000 q^{9} -1.00000 q^{13} +9.64575 q^{15} -3.00000 q^{17} +1.64575 q^{19} -4.35425 q^{21} -6.64575 q^{23} +8.29150 q^{25} -2.64575 q^{27} -3.00000 q^{29} -5.29150 q^{31} -6.00000 q^{35} +6.93725 q^{37} +2.64575 q^{39} +3.64575 q^{41} +11.9373 q^{43} -14.5830 q^{45} -2.35425 q^{47} -4.29150 q^{49} +7.93725 q^{51} -3.00000 q^{53} -4.35425 q^{57} +7.29150 q^{59} +14.2915 q^{61} +6.58301 q^{63} +3.64575 q^{65} -10.0000 q^{67} +17.5830 q^{69} -8.58301 q^{71} +4.00000 q^{73} -21.9373 q^{75} -8.64575 q^{79} -5.00000 q^{81} +4.70850 q^{83} +10.9373 q^{85} +7.93725 q^{87} +6.00000 q^{89} -1.64575 q^{91} +14.0000 q^{93} -6.00000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} + 8 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^9	$ 2 q - 2 q^{5} - 2 q^{7} + 8 q^{9} - 2 q^{13} + 14 q^{15} - 6 q^{17} - 2 q^{19} - 14 q^{21} - 8 q^{23} + 6 q^{25} - 6 q^{29} - 12 q^{35} - 2 q^{37} + 2 q^{41} + 8 q^{43} - 8 q^{45} - 10 q^{47} + 2 q^{49} - 6 q^{53} - 14 q^{57} + 4 q^{59} + 18 q^{61} - 8 q^{63} + 2 q^{65} - 20 q^{67} + 14 q^{69} + 4 q^{71} + 8 q^{73} - 28 q^{75} - 12 q^{79} - 10 q^{81} + 20 q^{83} + 6 q^{85} + 12 q^{89} + 2 q^{91} + 28 q^{93} - 12 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^9 - 2 * q^13 + 14 * q^15 - 6 * q^17 - 2 * q^19 - 14 * q^21 - 8 * q^23 + 6 * q^25 - 6 * q^29 - 12 * q^35 - 2 * q^37 + 2 * q^41 + 8 * q^43 - 8 * q^45 - 10 * q^47 + 2 * q^49 - 6 * q^53 - 14 * q^57 + 4 * q^59 + 18 * q^61 - 8 * q^63 + 2 * q^65 - 20 * q^67 + 14 * q^69 + 4 * q^71 + 8 * q^73 - 28 * q^75 - 12 * q^79 - 10 * q^81 + 20 * q^83 + 6 * q^85 + 12 * q^89 + 2 * q^91 + 28 * q^93 - 12 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.64575	−1.52753	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	−2.64575	−1.52753	−0.763763	+	0.645497i	$0.776650\pi$
$4$	0	0
$5$	−3.64575	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$6$	0	0
$7$	1.64575	0.622036	0.311018	−	0.950404i	$-0.399330\pi$
$7$	1.64575	0.622036	0.311018	+	0.950404i	$0.399330\pi$
$8$	0	0
$9$	4.00000	1.33333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	9.64575	2.49052
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.64575	0.377561	0.188781	−	0.982019i	$-0.439546\pi$
$19$	1.64575	0.377561	0.188781	+	0.982019i	$0.439546\pi$
$20$	0	0
$21$	−4.35425	−0.950175
$22$	0	0
$23$	−6.64575	−1.38573	−0.692867	−	0.721065i	$-0.743653\pi$
$23$	−6.64575	−1.38573	−0.692867	+	0.721065i	$0.743653\pi$
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	−2.64575	−0.509175
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−5.29150	−0.950382	−0.475191	−	0.879883i	$-0.657621\pi$
$31$	−5.29150	−0.950382	−0.475191	+	0.879883i	$0.657621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.00000	−1.01419
$36$	0	0
$37$	6.93725	1.14048	0.570239	−	0.821479i	$-0.306851\pi$
$37$	6.93725	1.14048	0.570239	+	0.821479i	$0.306851\pi$
$38$	0	0
$39$	2.64575	0.423659
$40$	0	0
$41$	3.64575	0.569371	0.284685	−	0.958621i	$-0.408111\pi$
$41$	3.64575	0.569371	0.284685	+	0.958621i	$0.408111\pi$
$42$	0	0
$43$	11.9373	1.82041	0.910207	−	0.414153i	$-0.135922\pi$
$43$	11.9373	1.82041	0.910207	+	0.414153i	$0.135922\pi$
$44$	0	0
$45$	−14.5830	−2.17391
$46$	0	0
$47$	−2.35425	−0.343402	−0.171701	−	0.985149i	$-0.554926\pi$
$47$	−2.35425	−0.343402	−0.171701	+	0.985149i	$0.554926\pi$
$48$	0	0
$49$	−4.29150	−0.613072
$50$	0	0
$51$	7.93725	1.11144
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.35425	−0.576734
$58$	0	0
$59$	7.29150	0.949273	0.474636	−	0.880182i	$-0.342580\pi$
$59$	7.29150	0.949273	0.474636	+	0.880182i	$0.342580\pi$
$60$	0	0
$61$	14.2915	1.82984	0.914920	−	0.403636i	$-0.132254\pi$
$61$	14.2915	1.82984	0.914920	+	0.403636i	$0.132254\pi$
$62$	0	0
$63$	6.58301	0.829381
$64$	0	0
$65$	3.64575	0.452200
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	17.5830	2.11675
$70$	0	0
$71$	−8.58301	−1.01862	−0.509308	−	0.860585i	$-0.670098\pi$
$71$	−8.58301	−1.01862	−0.509308	+	0.860585i	$0.670098\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	−21.9373	−2.53310
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.64575	−0.972723	−0.486362	−	0.873758i	$-0.661676\pi$
$79$	−8.64575	−0.972723	−0.486362	+	0.873758i	$0.661676\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	4.70850	0.516825	0.258412	−	0.966035i	$-0.416801\pi$
$83$	4.70850	0.516825	0.258412	+	0.966035i	$0.416801\pi$
$84$	0	0
$85$	10.9373	1.18631
$86$	0	0
$87$	7.93725	0.850963
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.64575	−0.172522
$92$	0	0
$93$	14.0000	1.45173
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.2915	1.02404	0.512021	−	0.858973i	$-0.328897\pi$
$101$	10.2915	1.02404	0.512021	+	0.858973i	$0.328897\pi$
$102$	0	0
$103$	17.2288	1.69760	0.848800	−	0.528714i	$-0.177326\pi$
$103$	17.2288	1.69760	0.848800	+	0.528714i	$0.177326\pi$
$104$	0	0
$105$	15.8745	1.54919
$106$	0	0
$107$	9.22876	0.892178	0.446089	−	0.894989i	$-0.352817\pi$
$107$	9.22876	0.892178	0.446089	+	0.894989i	$0.352817\pi$
$108$	0	0
$109$	13.6458	1.30703	0.653513	−	0.756915i	$-0.273294\pi$
$109$	13.6458	1.30703	0.653513	+	0.756915i	$0.273294\pi$
$110$	0	0
$111$	−18.3542	−1.74211
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	24.2288	2.25934
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−4.93725	−0.452597
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−9.64575	−0.869728
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−13.3542	−1.18500	−0.592499	−	0.805571i	$-0.701859\pi$
$127$	−13.3542	−1.18500	−0.592499	+	0.805571i	$0.701859\pi$
$128$	0	0
$129$	−31.5830	−2.78073
$130$	0	0
$131$	17.3542	1.51625	0.758124	−	0.652111i	$-0.226116\pi$
$131$	17.3542	1.51625	0.758124	+	0.652111i	$0.226116\pi$
$132$	0	0
$133$	2.70850	0.234857
$134$	0	0
$135$	9.64575	0.830174
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	15.3542	1.30233	0.651165	−	0.758936i	$-0.274281\pi$
$139$	15.3542	1.30233	0.651165	+	0.758936i	$0.274281\pi$
$140$	0	0
$141$	6.22876	0.524556
$142$	0	0
$143$	0	0
$144$	0	0
$145$	10.9373	0.908290
$146$	0	0
$147$	11.3542	0.936483
$148$	0	0
$149$	−19.2915	−1.58042	−0.790211	−	0.612835i	$-0.790029\pi$
$149$	−19.2915	−1.58042	−0.790211	+	0.612835i	$0.790029\pi$
$150$	0	0
$151$	23.2915	1.89544	0.947718	−	0.319110i	$-0.103384\pi$
$151$	23.2915	1.89544	0.947718	+	0.319110i	$0.103384\pi$
$152$	0	0
$153$	−12.0000	−0.970143
$154$	0	0
$155$	19.2915	1.54953
$156$	0	0
$157$	0.291503	0.0232644	0.0116322	−	0.999932i	$-0.496297\pi$
$157$	0.291503	0.0232644	0.0116322	+	0.999932i	$0.496297\pi$
$158$	0	0
$159$	7.93725	0.629465
$160$	0	0
$161$	−10.9373	−0.861976
$162$	0	0
$163$	−0.583005	−0.0456645	−0.0228322	−	0.999739i	$-0.507268\pi$
$163$	−0.583005	−0.0456645	−0.0228322	+	0.999739i	$0.507268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.64575	−0.282117	−0.141058	−	0.990001i	$-0.545051\pi$
$167$	−3.64575	−0.282117	−0.141058	+	0.990001i	$0.545051\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.58301	0.503415
$172$	0	0
$173$	11.5830	0.880640	0.440320	−	0.897841i	$-0.354865\pi$
$173$	11.5830	0.880640	0.440320	+	0.897841i	$0.354865\pi$
$174$	0	0
$175$	13.6458	1.03152
$176$	0	0
$177$	−19.2915	−1.45004
$178$	0	0
$179$	−16.5203	−1.23478	−0.617391	−	0.786656i	$-0.711810\pi$
$179$	−16.5203	−1.23478	−0.617391	+	0.786656i	$0.711810\pi$
$180$	0	0
$181$	−24.1660	−1.79625	−0.898123	−	0.439745i	$-0.855069\pi$
$181$	−24.1660	−1.79625	−0.898123	+	0.439745i	$0.855069\pi$
$182$	0	0
$183$	−37.8118	−2.79513
$184$	0	0
$185$	−25.2915	−1.85947
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.35425	−0.316725
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−6.70850	−0.482888	−0.241444	−	0.970415i	$-0.577621\pi$
$193$	−6.70850	−0.482888	−0.241444	+	0.970415i	$0.577621\pi$
$194$	0	0
$195$	−9.64575	−0.690747
$196$	0	0
$197$	−23.1660	−1.65051	−0.825255	−	0.564760i	$-0.808969\pi$
$197$	−23.1660	−1.65051	−0.825255	+	0.564760i	$0.808969\pi$
$198$	0	0
$199$	12.5203	0.887538	0.443769	−	0.896141i	$-0.353641\pi$
$199$	12.5203	0.887538	0.443769	+	0.896141i	$0.353641\pi$
$200$	0	0
$201$	26.4575	1.86617
$202$	0	0
$203$	−4.93725	−0.346527
$204$	0	0
$205$	−13.2915	−0.928319
$206$	0	0
$207$	−26.5830	−1.84765
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.70850	−0.461832	−0.230916	−	0.972974i	$-0.574172\pi$
$211$	−6.70850	−0.461832	−0.230916	+	0.972974i	$0.574172\pi$
$212$	0	0
$213$	22.7085	1.55596
$214$	0	0
$215$	−43.5203	−2.96806
$216$	0	0
$217$	−8.70850	−0.591171
$218$	0	0
$219$	−10.5830	−0.715133
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	−17.2915	−1.15792	−0.578962	−	0.815354i	$-0.696542\pi$
$223$	−17.2915	−1.15792	−0.578962	+	0.815354i	$0.696542\pi$
$224$	0	0
$225$	33.1660	2.21107
$226$	0	0
$227$	15.8745	1.05363	0.526814	−	0.849981i	$-0.323386\pi$
$227$	15.8745	1.05363	0.526814	+	0.849981i	$0.323386\pi$
$228$	0	0
$229$	−24.3542	−1.60937	−0.804687	−	0.593699i	$-0.797667\pi$
$229$	−24.3542	−1.60937	−0.804687	+	0.593699i	$0.797667\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	8.58301	0.559894
$236$	0	0
$237$	22.8745	1.48586
$238$	0	0
$239$	−13.2915	−0.859756	−0.429878	−	0.902887i	$-0.641443\pi$
$239$	−13.2915	−0.859756	−0.429878	+	0.902887i	$0.641443\pi$
$240$	0	0
$241$	20.9373	1.34869	0.674344	−	0.738418i	$-0.264427\pi$
$241$	20.9373	1.34869	0.674344	+	0.738418i	$0.264427\pi$
$242$	0	0
$243$	21.1660	1.35780
$244$	0	0
$245$	15.6458	0.999570
$246$	0	0
$247$	−1.64575	−0.104717
$248$	0	0
$249$	−12.4575	−0.789463
$250$	0	0
$251$	7.93725	0.500995	0.250498	−	0.968117i	$-0.419406\pi$
$251$	7.93725	0.500995	0.250498	+	0.968117i	$0.419406\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−28.9373	−1.81212
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	11.4170	0.709418
$260$	0	0
$261$	−12.0000	−0.742781
$262$	0	0
$263$	−18.6458	−1.14975	−0.574873	−	0.818243i	$-0.694949\pi$
$263$	−18.6458	−1.14975	−0.574873	+	0.818243i	$0.694949\pi$
$264$	0	0
$265$	10.9373	0.671870
$266$	0	0
$267$	−15.8745	−0.971504
$268$	0	0
$269$	11.1660	0.680804	0.340402	−	0.940280i	$-0.389437\pi$
$269$	11.1660	0.680804	0.340402	+	0.940280i	$0.389437\pi$
$270$	0	0
$271$	−23.8745	−1.45027	−0.725137	−	0.688605i	$-0.758223\pi$
$271$	−23.8745	−1.45027	−0.725137	+	0.688605i	$0.758223\pi$
$272$	0	0
$273$	4.35425	0.263531
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.5830	−1.35688	−0.678441	−	0.734655i	$-0.737344\pi$
$277$	−22.5830	−1.35688	−0.678441	+	0.734655i	$0.737344\pi$
$278$	0	0
$279$	−21.1660	−1.26718
$280$	0	0
$281$	−24.4575	−1.45901	−0.729506	−	0.683974i	$-0.760250\pi$
$281$	−24.4575	−1.45901	−0.729506	+	0.683974i	$0.760250\pi$
$282$	0	0
$283$	23.2915	1.38454	0.692268	−	0.721640i	$-0.256612\pi$
$283$	23.2915	1.38454	0.692268	+	0.721640i	$0.256612\pi$
$284$	0	0
$285$	15.8745	0.940325
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−5.29150	−0.310193
$292$	0	0
$293$	−25.2915	−1.47755	−0.738773	−	0.673955i	$-0.764594\pi$
$293$	−25.2915	−1.47755	−0.738773	+	0.673955i	$0.764594\pi$
$294$	0	0
$295$	−26.5830	−1.54772
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.64575	0.384334
$300$	0	0
$301$	19.6458	1.13236
$302$	0	0
$303$	−27.2288	−1.56425
$304$	0	0
$305$	−52.1033	−2.98342
$306$	0	0
$307$	−24.9373	−1.42324	−0.711622	−	0.702562i	$-0.752039\pi$
$307$	−24.9373	−1.42324	−0.711622	+	0.702562i	$0.752039\pi$
$308$	0	0
$309$	−45.5830	−2.59313
$310$	0	0
$311$	−4.52026	−0.256320	−0.128160	−	0.991753i	$-0.540907\pi$
$311$	−4.52026	−0.256320	−0.128160	+	0.991753i	$0.540907\pi$
$312$	0	0
$313$	12.2915	0.694757	0.347378	−	0.937725i	$-0.387072\pi$
$313$	12.2915	0.694757	0.347378	+	0.937725i	$0.387072\pi$
$314$	0	0
$315$	−24.0000	−1.35225
$316$	0	0
$317$	16.9373	0.951291	0.475645	−	0.879637i	$-0.342215\pi$
$317$	16.9373	0.951291	0.475645	+	0.879637i	$0.342215\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−24.4170	−1.36282
$322$	0	0
$323$	−4.93725	−0.274716
$324$	0	0
$325$	−8.29150	−0.459930
$326$	0	0
$327$	−36.1033	−1.99652
$328$	0	0
$329$	−3.87451	−0.213609
$330$	0	0
$331$	−0.354249	−0.0194713	−0.00973563	−	0.999953i	$-0.503099\pi$
$331$	−0.354249	−0.0194713	−0.00973563	+	0.999953i	$0.503099\pi$
$332$	0	0
$333$	27.7490	1.52064
$334$	0	0
$335$	36.4575	1.99189
$336$	0	0
$337$	−0.708497	−0.0385943	−0.0192972	−	0.999814i	$-0.506143\pi$
$337$	−0.708497	−0.0385943	−0.0192972	+	0.999814i	$0.506143\pi$
$338$	0	0
$339$	31.7490	1.72437
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.5830	−1.00339
$344$	0	0
$345$	−64.1033	−3.45120
$346$	0	0
$347$	16.7085	0.896959	0.448480	−	0.893793i	$-0.351966\pi$
$347$	16.7085	0.896959	0.448480	+	0.893793i	$0.351966\pi$
$348$	0	0
$349$	9.16601	0.490645	0.245323	−	0.969441i	$-0.421106\pi$
$349$	9.16601	0.490645	0.245323	+	0.969441i	$0.421106\pi$
$350$	0	0
$351$	2.64575	0.141220
$352$	0	0
$353$	−31.5203	−1.67765	−0.838827	−	0.544398i	$-0.816758\pi$
$353$	−31.5203	−1.67765	−0.838827	+	0.544398i	$0.816758\pi$
$354$	0	0
$355$	31.2915	1.66078
$356$	0	0
$357$	13.0627	0.691354
$358$	0	0
$359$	26.8118	1.41507	0.707535	−	0.706678i	$-0.249807\pi$
$359$	26.8118	1.41507	0.707535	+	0.706678i	$0.249807\pi$
$360$	0	0
$361$	−16.2915	−0.857448
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.5830	−0.763309
$366$	0	0
$367$	3.29150	0.171815	0.0859075	−	0.996303i	$-0.472621\pi$
$367$	3.29150	0.171815	0.0859075	+	0.996303i	$0.472621\pi$
$368$	0	0
$369$	14.5830	0.759161
$370$	0	0
$371$	−4.93725	−0.256329
$372$	0	0
$373$	−15.2915	−0.791764	−0.395882	−	0.918301i	$-0.629561\pi$
$373$	−15.2915	−0.791764	−0.395882	+	0.918301i	$0.629561\pi$
$374$	0	0
$375$	31.7490	1.63951
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	−17.5203	−0.899955	−0.449978	−	0.893040i	$-0.648568\pi$
$379$	−17.5203	−0.899955	−0.449978	+	0.893040i	$0.648568\pi$
$380$	0	0
$381$	35.3320	1.81011
$382$	0	0
$383$	−4.93725	−0.252282	−0.126141	−	0.992012i	$-0.540259\pi$
$383$	−4.93725	−0.252282	−0.126141	+	0.992012i	$0.540259\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	47.7490	2.42722
$388$	0	0
$389$	−21.4575	−1.08794	−0.543970	−	0.839105i	$-0.683079\pi$
$389$	−21.4575	−1.08794	−0.543970	+	0.839105i	$0.683079\pi$
$390$	0	0
$391$	19.9373	1.00827
$392$	0	0
$393$	−45.9150	−2.31611
$394$	0	0
$395$	31.5203	1.58596
$396$	0	0
$397$	−24.3542	−1.22230	−0.611152	−	0.791513i	$-0.709294\pi$
$397$	−24.3542	−1.22230	−0.611152	+	0.791513i	$0.709294\pi$
$398$	0	0
$399$	−7.16601	−0.358749
$400$	0	0
$401$	33.8745	1.69161	0.845806	−	0.533490i	$-0.179120\pi$
$401$	33.8745	1.69161	0.845806	+	0.533490i	$0.179120\pi$
$402$	0	0
$403$	5.29150	0.263589
$404$	0	0
$405$	18.2288	0.905794
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	−31.7490	−1.56606
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	−17.1660	−0.842646
$416$	0	0
$417$	−40.6235	−1.98934
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	5.64575	0.275157	0.137579	−	0.990491i	$-0.456068\pi$
$421$	5.64575	0.275157	0.137579	+	0.990491i	$0.456068\pi$
$422$	0	0
$423$	−9.41699	−0.457870
$424$	0	0
$425$	−24.8745	−1.20659
$426$	0	0
$427$	23.5203	1.13823
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.2915	1.50726	0.753629	−	0.657300i	$-0.228301\pi$
$431$	31.2915	1.50726	0.753629	+	0.657300i	$0.228301\pi$
$432$	0	0
$433$	−8.29150	−0.398464	−0.199232	−	0.979952i	$-0.563845\pi$
$433$	−8.29150	−0.398464	−0.199232	+	0.979952i	$0.563845\pi$
$434$	0	0
$435$	−28.9373	−1.38744
$436$	0	0
$437$	−10.9373	−0.523200
$438$	0	0
$439$	−4.77124	−0.227719	−0.113860	−	0.993497i	$-0.536321\pi$
$439$	−4.77124	−0.227719	−0.113860	+	0.993497i	$0.536321\pi$
$440$	0	0
$441$	−17.1660	−0.817429
$442$	0	0
$443$	−23.8118	−1.13133	−0.565665	−	0.824635i	$-0.691381\pi$
$443$	−23.8118	−1.13133	−0.565665	+	0.824635i	$0.691381\pi$
$444$	0	0
$445$	−21.8745	−1.03695
$446$	0	0
$447$	51.0405	2.41413
$448$	0	0
$449$	32.5830	1.53769	0.768844	−	0.639437i	$-0.220832\pi$
$449$	32.5830	1.53769	0.768844	+	0.639437i	$0.220832\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−61.6235	−2.89533
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	−23.6458	−1.10610	−0.553051	−	0.833148i	$-0.686536\pi$
$457$	−23.6458	−1.10610	−0.553051	+	0.833148i	$0.686536\pi$
$458$	0	0
$459$	7.93725	0.370479
$460$	0	0
$461$	−29.1660	−1.35840	−0.679198	−	0.733955i	$-0.737672\pi$
$461$	−29.1660	−1.35840	−0.679198	+	0.733955i	$0.737672\pi$
$462$	0	0
$463$	−12.5830	−0.584782	−0.292391	−	0.956299i	$-0.594451\pi$
$463$	−12.5830	−0.584782	−0.292391	+	0.956299i	$0.594451\pi$
$464$	0	0
$465$	−51.0405	−2.36695
$466$	0	0
$467$	−4.52026	−0.209173	−0.104586	−	0.994516i	$-0.533352\pi$
$467$	−4.52026	−0.209173	−0.104586	+	0.994516i	$0.533352\pi$
$468$	0	0
$469$	−16.4575	−0.759937
$470$	0	0
$471$	−0.771243	−0.0355370
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13.6458	0.626110
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	4.93725	0.225589	0.112794	−	0.993618i	$-0.464020\pi$
$479$	4.93725	0.225589	0.112794	+	0.993618i	$0.464020\pi$
$480$	0	0
$481$	−6.93725	−0.316312
$482$	0	0
$483$	28.9373	1.31669
$484$	0	0
$485$	−7.29150	−0.331090
$486$	0	0
$487$	12.9373	0.586243	0.293121	−	0.956075i	$-0.405306\pi$
$487$	12.9373	0.586243	0.293121	+	0.956075i	$0.405306\pi$
$488$	0	0
$489$	1.54249	0.0697537
$490$	0	0
$491$	−0.645751	−0.0291423	−0.0145712	−	0.999894i	$-0.504638\pi$
$491$	−0.645751	−0.0291423	−0.0145712	+	0.999894i	$0.504638\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.1255	−0.633615
$498$	0	0
$499$	−0.354249	−0.0158583	−0.00792917	−	0.999969i	$-0.502524\pi$
$499$	−0.354249	−0.0158583	−0.00792917	+	0.999969i	$0.502524\pi$
$500$	0	0
$501$	9.64575	0.430940
$502$	0	0
$503$	−4.06275	−0.181149	−0.0905744	−	0.995890i	$-0.528870\pi$
$503$	−4.06275	−0.181149	−0.0905744	+	0.995890i	$0.528870\pi$
$504$	0	0
$505$	−37.5203	−1.66963
$506$	0	0
$507$	−2.64575	−0.117502
$508$	0	0
$509$	13.0627	0.578996	0.289498	−	0.957179i	$-0.406512\pi$
$509$	13.0627	0.578996	0.289498	+	0.957179i	$0.406512\pi$
$510$	0	0
$511$	6.58301	0.291215
$512$	0	0
$513$	−4.35425	−0.192245
$514$	0	0
$515$	−62.8118	−2.76782
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−30.6458	−1.34520
$520$	0	0
$521$	22.2915	0.976608	0.488304	−	0.872673i	$-0.337616\pi$
$521$	22.2915	0.976608	0.488304	+	0.872673i	$0.337616\pi$
$522$	0	0
$523$	6.77124	0.296086	0.148043	−	0.988981i	$-0.452703\pi$
$523$	6.77124	0.296086	0.148043	+	0.988981i	$0.452703\pi$
$524$	0	0
$525$	−36.1033	−1.57568
$526$	0	0
$527$	15.8745	0.691504
$528$	0	0
$529$	21.1660	0.920261
$530$	0	0
$531$	29.1660	1.26570
$532$	0	0
$533$	−3.64575	−0.157915
$534$	0	0
$535$	−33.6458	−1.45463
$536$	0	0
$537$	43.7085	1.88616
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−26.2288	−1.12766	−0.563831	−	0.825890i	$-0.690673\pi$
$541$	−26.2288	−1.12766	−0.563831	+	0.825890i	$0.690673\pi$
$542$	0	0
$543$	63.9373	2.74381
$544$	0	0
$545$	−49.7490	−2.13101
$546$	0	0
$547$	16.6458	0.711721	0.355860	−	0.934539i	$-0.384188\pi$
$547$	16.6458	0.711721	0.355860	+	0.934539i	$0.384188\pi$
$548$	0	0
$549$	57.1660	2.43979
$550$	0	0
$551$	−4.93725	−0.210334
$552$	0	0
$553$	−14.2288	−0.605068
$554$	0	0
$555$	66.9150	2.84038
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−11.9373	−0.504892
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.22876	−0.136076	−0.0680379	−	0.997683i	$-0.521674\pi$
$563$	−3.22876	−0.136076	−0.0680379	+	0.997683i	$0.521674\pi$
$564$	0	0
$565$	43.7490	1.84053
$566$	0	0
$567$	−8.22876	−0.345575
$568$	0	0
$569$	−44.5830	−1.86902	−0.934508	−	0.355941i	$-0.884160\pi$
$569$	−44.5830	−1.86902	−0.934508	+	0.355941i	$0.884160\pi$
$570$	0	0
$571$	37.2288	1.55797	0.778987	−	0.627039i	$-0.215734\pi$
$571$	37.2288	1.55797	0.778987	+	0.627039i	$0.215734\pi$
$572$	0	0
$573$	−31.7490	−1.32633
$574$	0	0
$575$	−55.1033	−2.29796
$576$	0	0
$577$	−27.3948	−1.14046	−0.570230	−	0.821485i	$-0.693146\pi$
$577$	−27.3948	−1.14046	−0.570230	+	0.821485i	$0.693146\pi$
$578$	0	0
$579$	17.7490	0.737624
$580$	0	0
$581$	7.74902	0.321483
$582$	0	0
$583$	0	0
$584$	0	0
$585$	14.5830	0.602933
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−8.70850	−0.358827
$590$	0	0
$591$	61.2915	2.52120
$592$	0	0
$593$	−24.2288	−0.994956	−0.497478	−	0.867477i	$-0.665740\pi$
$593$	−24.2288	−0.994956	−0.497478	+	0.867477i	$0.665740\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	0	0
$597$	−33.1255	−1.35574
$598$	0	0
$599$	−8.12549	−0.331999	−0.165999	−	0.986126i	$-0.553085\pi$
$599$	−8.12549	−0.331999	−0.165999	+	0.986126i	$0.553085\pi$
$600$	0	0
$601$	22.4170	0.914408	0.457204	−	0.889362i	$-0.348851\pi$
$601$	22.4170	0.914408	0.457204	+	0.889362i	$0.348851\pi$
$602$	0	0
$603$	−40.0000	−1.62893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.64575	−0.107388	−0.0536939	−	0.998557i	$-0.517100\pi$
$607$	−2.64575	−0.107388	−0.0536939	+	0.998557i	$0.517100\pi$
$608$	0	0
$609$	13.0627	0.529329
$610$	0	0
$611$	2.35425	0.0952427
$612$	0	0
$613$	−22.5830	−0.912119	−0.456059	−	0.889949i	$-0.650740\pi$
$613$	−22.5830	−0.912119	−0.456059	+	0.889949i	$0.650740\pi$
$614$	0	0
$615$	35.1660	1.41803
$616$	0	0
$617$	25.2915	1.01820	0.509099	−	0.860708i	$-0.329979\pi$
$617$	25.2915	1.01820	0.509099	+	0.860708i	$0.329979\pi$
$618$	0	0
$619$	−17.0627	−0.685810	−0.342905	−	0.939370i	$-0.611411\pi$
$619$	−17.0627	−0.685810	−0.342905	+	0.939370i	$0.611411\pi$
$620$	0	0
$621$	17.5830	0.705582
$622$	0	0
$623$	9.87451	0.395614
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.8118	−0.829819
$630$	0	0
$631$	44.2288	1.76072	0.880359	−	0.474307i	$-0.157301\pi$
$631$	44.2288	1.76072	0.880359	+	0.474307i	$0.157301\pi$
$632$	0	0
$633$	17.7490	0.705460
$634$	0	0
$635$	48.6863	1.93206
$636$	0	0
$637$	4.29150	0.170036
$638$	0	0
$639$	−34.3320	−1.35815
$640$	0	0
$641$	−10.2915	−0.406490	−0.203245	−	0.979128i	$-0.565149\pi$
$641$	−10.2915	−0.406490	−0.203245	+	0.979128i	$0.565149\pi$
$642$	0	0
$643$	−18.8118	−0.741863	−0.370932	−	0.928660i	$-0.620962\pi$
$643$	−18.8118	−0.741863	−0.370932	+	0.928660i	$0.620962\pi$
$644$	0	0
$645$	115.144	4.53378
$646$	0	0
$647$	33.6863	1.32434	0.662172	−	0.749352i	$-0.269635\pi$
$647$	33.6863	1.32434	0.662172	+	0.749352i	$0.269635\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	23.0405	0.903029
$652$	0	0
$653$	−18.4170	−0.720713	−0.360356	−	0.932815i	$-0.617345\pi$
$653$	−18.4170	−0.720713	−0.360356	+	0.932815i	$0.617345\pi$
$654$	0	0
$655$	−63.2693	−2.47214
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	−36.6458	−1.42752	−0.713758	−	0.700393i	$-0.753008\pi$
$659$	−36.6458	−1.42752	−0.713758	+	0.700393i	$0.753008\pi$
$660$	0	0
$661$	21.5203	0.837041	0.418521	−	0.908207i	$-0.362549\pi$
$661$	21.5203	0.837041	0.418521	+	0.908207i	$0.362549\pi$
$662$	0	0
$663$	−7.93725	−0.308257
$664$	0	0
$665$	−9.87451	−0.382917
$666$	0	0
$667$	19.9373	0.771974
$668$	0	0
$669$	45.7490	1.76876
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.5830	−0.407945	−0.203972	−	0.978977i	$-0.565385\pi$
$673$	−10.5830	−0.407945	−0.203972	+	0.978977i	$0.565385\pi$
$674$	0	0
$675$	−21.9373	−0.844365
$676$	0	0
$677$	34.3320	1.31949	0.659743	−	0.751491i	$-0.270665\pi$
$677$	34.3320	1.31949	0.659743	+	0.751491i	$0.270665\pi$
$678$	0	0
$679$	3.29150	0.126316
$680$	0	0
$681$	−42.0000	−1.60944
$682$	0	0
$683$	34.9373	1.33684	0.668418	−	0.743785i	$-0.266972\pi$
$683$	34.9373	1.33684	0.668418	+	0.743785i	$0.266972\pi$
$684$	0	0
$685$	−43.7490	−1.67156
$686$	0	0
$687$	64.4353	2.45836
$688$	0	0
$689$	3.00000	0.114291
$690$	0	0
$691$	3.52026	0.133917	0.0669585	−	0.997756i	$-0.478670\pi$
$691$	3.52026	0.133917	0.0669585	+	0.997756i	$0.478670\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−55.9778	−2.12336
$696$	0	0
$697$	−10.9373	−0.414278
$698$	0	0
$699$	31.7490	1.20086
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	11.4170	0.430600
$704$	0	0
$705$	−22.7085	−0.855251
$706$	0	0
$707$	16.9373	0.636991
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	−34.5830	−1.29696
$712$	0	0
$713$	35.1660	1.31698
$714$	0	0
$715$	0	0
$716$	0	0
$717$	35.1660	1.31330
$718$	0	0
$719$	−27.2288	−1.01546	−0.507731	−	0.861516i	$-0.669515\pi$
$719$	−27.2288	−1.01546	−0.507731	+	0.861516i	$0.669515\pi$
$720$	0	0
$721$	28.3542	1.05597
$722$	0	0
$723$	−55.3948	−2.06015
$724$	0	0
$725$	−24.8745	−0.923816
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	−35.8118	−1.32455
$732$	0	0
$733$	40.6863	1.50278	0.751391	−	0.659857i	$-0.229383\pi$
$733$	40.6863	1.50278	0.751391	+	0.659857i	$0.229383\pi$
$734$	0	0
$735$	−41.3948	−1.52687
$736$	0	0
$737$	0	0
$738$	0	0
$739$	17.0627	0.627663	0.313832	−	0.949479i	$-0.398387\pi$
$739$	17.0627	0.627663	0.313832	+	0.949479i	$0.398387\pi$
$740$	0	0
$741$	4.35425	0.159957
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	70.3320	2.57677
$746$	0	0
$747$	18.8340	0.689100
$748$	0	0
$749$	15.1882	0.554966
$750$	0	0
$751$	18.7085	0.682683	0.341341	−	0.939939i	$-0.389119\pi$
$751$	18.7085	0.682683	0.341341	+	0.939939i	$0.389119\pi$
$752$	0	0
$753$	−21.0000	−0.765283
$754$	0	0
$755$	−84.9150	−3.09037
$756$	0	0
$757$	11.8340	0.430114	0.215057	−	0.976602i	$-0.431006\pi$
$757$	11.8340	0.430114	0.215057	+	0.976602i	$0.431006\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.8745	0.575450	0.287725	−	0.957713i	$-0.407101\pi$
$761$	15.8745	0.575450	0.287725	+	0.957713i	$0.407101\pi$
$762$	0	0
$763$	22.4575	0.813017
$764$	0	0
$765$	43.7490	1.58175
$766$	0	0
$767$	−7.29150	−0.263281
$768$	0	0
$769$	−0.708497	−0.0255491	−0.0127745	−	0.999918i	$-0.504066\pi$
$769$	−0.708497	−0.0255491	−0.0127745	+	0.999918i	$0.504066\pi$
$770$	0	0
$771$	31.7490	1.14341
$772$	0	0
$773$	0.228757	0.00822780	0.00411390	−	0.999992i	$-0.498691\pi$
$773$	0.228757	0.00822780	0.00411390	+	0.999992i	$0.498691\pi$
$774$	0	0
$775$	−43.8745	−1.57602
$776$	0	0
$777$	−30.2065	−1.08365
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	7.93725	0.283654
$784$	0	0
$785$	−1.06275	−0.0379310
$786$	0	0
$787$	18.8118	0.670567	0.335283	−	0.942117i	$-0.391168\pi$
$787$	18.8118	0.670567	0.335283	+	0.942117i	$0.391168\pi$
$788$	0	0
$789$	49.3320	1.75627
$790$	0	0
$791$	−19.7490	−0.702194
$792$	0	0
$793$	−14.2915	−0.507506
$794$	0	0
$795$	−28.9373	−1.02630
$796$	0	0
$797$	−31.2915	−1.10840	−0.554201	−	0.832383i	$-0.686976\pi$
$797$	−31.2915	−1.10840	−0.554201	+	0.832383i	$0.686976\pi$
$798$	0	0
$799$	7.06275	0.249862
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	0	0
$804$	0	0
$805$	39.8745	1.40539
$806$	0	0
$807$	−29.5425	−1.03994
$808$	0	0
$809$	−24.8745	−0.874541	−0.437271	−	0.899330i	$-0.644055\pi$
$809$	−24.8745	−0.874541	−0.437271	+	0.899330i	$0.644055\pi$
$810$	0	0
$811$	−3.06275	−0.107548	−0.0537738	−	0.998553i	$-0.517125\pi$
$811$	−3.06275	−0.107548	−0.0537738	+	0.998553i	$0.517125\pi$
$812$	0	0
$813$	63.1660	2.21533
$814$	0	0
$815$	2.12549	0.0744527
$816$	0	0
$817$	19.6458	0.687318
$818$	0	0
$819$	−6.58301	−0.230029
$820$	0	0
$821$	1.29150	0.0450738	0.0225369	−	0.999746i	$-0.492826\pi$
$821$	1.29150	0.0450738	0.0225369	+	0.999746i	$0.492826\pi$
$822$	0	0
$823$	1.35425	0.0472061	0.0236031	−	0.999721i	$-0.492486\pi$
$823$	1.35425	0.0472061	0.0236031	+	0.999721i	$0.492486\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.2288	−0.633876	−0.316938	−	0.948446i	$-0.602655\pi$
$827$	−18.2288	−0.633876	−0.316938	+	0.948446i	$0.602655\pi$
$828$	0	0
$829$	14.8745	0.516613	0.258307	−	0.966063i	$-0.416836\pi$
$829$	14.8745	0.516613	0.258307	+	0.966063i	$0.416836\pi$
$830$	0	0
$831$	59.7490	2.07267
$832$	0	0
$833$	12.8745	0.446075
$834$	0	0
$835$	13.2915	0.459971
$836$	0	0
$837$	14.0000	0.483911
$838$	0	0
$839$	46.9373	1.62045	0.810227	−	0.586116i	$-0.199344\pi$
$839$	46.9373	1.62045	0.810227	+	0.586116i	$0.199344\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	64.7085	2.22868
$844$	0	0
$845$	−3.64575	−0.125418
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−61.6235	−2.11491
$850$	0	0
$851$	−46.1033	−1.58040
$852$	0	0
$853$	−35.0405	−1.19976	−0.599882	−	0.800088i	$-0.704786\pi$
$853$	−35.0405	−1.19976	−0.599882	+	0.800088i	$0.704786\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	0	0
$857$	−16.2915	−0.556507	−0.278254	−	0.960508i	$-0.589756\pi$
$857$	−16.2915	−0.556507	−0.278254	+	0.960508i	$0.589756\pi$
$858$	0	0
$859$	−53.9373	−1.84032	−0.920158	−	0.391548i	$-0.871940\pi$
$859$	−53.9373	−1.84032	−0.920158	+	0.391548i	$0.871940\pi$
$860$	0	0
$861$	−15.8745	−0.541002
$862$	0	0
$863$	34.9373	1.18928	0.594639	−	0.803993i	$-0.297295\pi$
$863$	34.9373	1.18928	0.594639	+	0.803993i	$0.297295\pi$
$864$	0	0
$865$	−42.2288	−1.43582
$866$	0	0
$867$	21.1660	0.718835
$868$	0	0
$869$	0	0
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−19.7490	−0.667639
$876$	0	0
$877$	−17.4170	−0.588130	−0.294065	−	0.955785i	$-0.595008\pi$
$877$	−17.4170	−0.588130	−0.294065	+	0.955785i	$0.595008\pi$
$878$	0	0
$879$	66.9150	2.25699
$880$	0	0
$881$	53.6235	1.80662	0.903311	−	0.428986i	$-0.141129\pi$
$881$	53.6235	1.80662	0.903311	+	0.428986i	$0.141129\pi$
$882$	0	0
$883$	39.7490	1.33766	0.668830	−	0.743415i	$-0.266795\pi$
$883$	39.7490	1.33766	0.668830	+	0.743415i	$0.266795\pi$
$884$	0	0
$885$	70.3320	2.36419
$886$	0	0
$887$	−25.2915	−0.849206	−0.424603	−	0.905380i	$-0.639586\pi$
$887$	−25.2915	−0.849206	−0.424603	+	0.905380i	$0.639586\pi$
$888$	0	0
$889$	−21.9778	−0.737111
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.87451	−0.129655
$894$	0	0
$895$	60.2288	2.01323
$896$	0	0
$897$	−17.5830	−0.587079
$898$	0	0
$899$	15.8745	0.529444
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	−51.9778	−1.72971
$904$	0	0
$905$	88.1033	2.92865
$906$	0	0
$907$	17.2288	0.572071	0.286036	−	0.958219i	$-0.407662\pi$
$907$	17.2288	0.572071	0.286036	+	0.958219i	$0.407662\pi$
$908$	0	0
$909$	41.1660	1.36539
$910$	0	0
$911$	29.8118	0.987708	0.493854	−	0.869545i	$-0.335588\pi$
$911$	29.8118	0.987708	0.493854	+	0.869545i	$0.335588\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	137.852	4.55726
$916$	0	0
$917$	28.5608	0.943160
$918$	0	0
$919$	47.1033	1.55379	0.776897	−	0.629628i	$-0.216793\pi$
$919$	47.1033	1.55379	0.776897	+	0.629628i	$0.216793\pi$
$920$	0	0
$921$	65.9778	2.17404
$922$	0	0
$923$	8.58301	0.282513
$924$	0	0
$925$	57.5203	1.89125
$926$	0	0
$927$	68.9150	2.26347
$928$	0	0
$929$	−23.3948	−0.767557	−0.383779	−	0.923425i	$-0.625377\pi$
$929$	−23.3948	−0.767557	−0.383779	+	0.923425i	$0.625377\pi$
$930$	0	0
$931$	−7.06275	−0.231472
$932$	0	0
$933$	11.9595	0.391536
$934$	0	0
$935$	0	0
$936$	0	0
$937$	46.8745	1.53132	0.765662	−	0.643243i	$-0.222412\pi$
$937$	46.8745	1.53132	0.765662	+	0.643243i	$0.222412\pi$
$938$	0	0
$939$	−32.5203	−1.06126
$940$	0	0
$941$	−18.8340	−0.613971	−0.306985	−	0.951714i	$-0.599320\pi$
$941$	−18.8340	−0.613971	−0.306985	+	0.951714i	$0.599320\pi$
$942$	0	0
$943$	−24.2288	−0.788997
$944$	0	0
$945$	15.8745	0.516398
$946$	0	0
$947$	4.10326	0.133338	0.0666691	−	0.997775i	$-0.478763\pi$
$947$	4.10326	0.133338	0.0666691	+	0.997775i	$0.478763\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−44.8118	−1.45312
$952$	0	0
$953$	−42.8745	−1.38884	−0.694421	−	0.719569i	$-0.744339\pi$
$953$	−42.8745	−1.38884	−0.694421	+	0.719569i	$0.744339\pi$
$954$	0	0
$955$	−43.7490	−1.41568
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.7490	0.637729
$960$	0	0
$961$	−3.00000	−0.0967742
$962$	0	0
$963$	36.9150	1.18957
$964$	0	0
$965$	24.4575	0.787315
$966$	0	0
$967$	−52.5830	−1.69096	−0.845478	−	0.534011i	$-0.820684\pi$
$967$	−52.5830	−1.69096	−0.845478	+	0.534011i	$0.820684\pi$
$968$	0	0
$969$	13.0627	0.419636
$970$	0	0
$971$	7.47974	0.240036	0.120018	−	0.992772i	$-0.461705\pi$
$971$	7.47974	0.240036	0.120018	+	0.992772i	$0.461705\pi$
$972$	0	0
$973$	25.2693	0.810096
$974$	0	0
$975$	21.9373	0.702554
$976$	0	0
$977$	−61.9778	−1.98284	−0.991422	−	0.130697i	$-0.958278\pi$
$977$	−61.9778	−1.98284	−0.991422	+	0.130697i	$0.958278\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	54.5830	1.74270
$982$	0	0
$983$	43.5203	1.38808	0.694040	−	0.719936i	$-0.255829\pi$
$983$	43.5203	1.38808	0.694040	+	0.719936i	$0.255829\pi$
$984$	0	0
$985$	84.4575	2.69104
$986$	0	0
$987$	10.2510	0.326292
$988$	0	0
$989$	−79.3320	−2.52261
$990$	0	0
$991$	−53.1033	−1.68688	−0.843440	−	0.537223i	$-0.819473\pi$
$991$	−53.1033	−1.68688	−0.843440	+	0.537223i	$0.819473\pi$
$992$	0	0
$993$	0.937254	0.0297429
$994$	0	0
$995$	−45.6458	−1.44707
$996$	0	0
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	0	0
$999$	−18.3542	−0.580703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.l.1.1	✓	2
11.10	odd	2		6292.2.a.m.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.l.1.1	✓	2	1.1	even	1	trivial
6292.2.a.m.1.1	yes	2	11.10	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:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64575 q^{3} -3.64575 q^{5} +1.64575 q^{7} +4.00000 q^{9} +O(q^{10})\)	\(q-2.64575 q^{3} -3.64575 q^{5} +1.64575 q^{7} +4.00000 q^{9} -1.00000 q^{13} +9.64575 q^{15} -3.00000 q^{17} +1.64575 q^{19} -4.35425 q^{21} -6.64575 q^{23} +8.29150 q^{25} -2.64575 q^{27} -3.00000 q^{29} -5.29150 q^{31} -6.00000 q^{35} +6.93725 q^{37} +2.64575 q^{39} +3.64575 q^{41} +11.9373 q^{43} -14.5830 q^{45} -2.35425 q^{47} -4.29150 q^{49} +7.93725 q^{51} -3.00000 q^{53} -4.35425 q^{57} +7.29150 q^{59} +14.2915 q^{61} +6.58301 q^{63} +3.64575 q^{65} -10.0000 q^{67} +17.5830 q^{69} -8.58301 q^{71} +4.00000 q^{73} -21.9373 q^{75} -8.64575 q^{79} -5.00000 q^{81} +4.70850 q^{83} +10.9373 q^{85} +7.93725 q^{87} +6.00000 q^{89} -1.64575 q^{91} +14.0000 q^{93} -6.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} + 8 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^9	\( 2 q - 2 q^{5} - 2 q^{7} + 8 q^{9} - 2 q^{13} + 14 q^{15} - 6 q^{17} - 2 q^{19} - 14 q^{21} - 8 q^{23} + 6 q^{25} - 6 q^{29} - 12 q^{35} - 2 q^{37} + 2 q^{41} + 8 q^{43} - 8 q^{45} - 10 q^{47} + 2 q^{49} - 6 q^{53} - 14 q^{57} + 4 q^{59} + 18 q^{61} - 8 q^{63} + 2 q^{65} - 20 q^{67} + 14 q^{69} + 4 q^{71} + 8 q^{73} - 28 q^{75} - 12 q^{79} - 10 q^{81} + 20 q^{83} + 6 q^{85} + 12 q^{89} + 2 q^{91} + 28 q^{93} - 12 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^9 - 2 * q^13 + 14 * q^15 - 6 * q^17 - 2 * q^19 - 14 * q^21 - 8 * q^23 + 6 * q^25 - 6 * q^29 - 12 * q^35 - 2 * q^37 + 2 * q^41 + 8 * q^43 - 8 * q^45 - 10 * q^47 + 2 * q^49 - 6 * q^53 - 14 * q^57 + 4 * q^59 + 18 * q^61 - 8 * q^63 + 2 * q^65 - 20 * q^67 + 14 * q^69 + 4 * q^71 + 8 * q^73 - 28 * q^75 - 12 * q^79 - 10 * q^81 + 20 * q^83 + 6 * q^85 + 12 * q^89 + 2 * q^91 + 28 * q^93 - 12 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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