LMFDB - Embedded newform 4900.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(1,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, 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("4900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.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:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 700)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.364448 q^{3} -2.86718 q^{9} +O(q^{10})$	$q+0.364448 q^{3} -2.86718 q^{9} +1.36445 q^{11} +2.63555 q^{13} -2.23163 q^{17} +6.23163 q^{19} +6.59607 q^{23} -2.13828 q^{27} +5.50273 q^{29} -4.50273 q^{31} +0.497270 q^{33} -2.09334 q^{37} +0.960522 q^{39} -9.32497 q^{41} -1.86718 q^{43} +6.86718 q^{47} -0.813312 q^{51} -10.1383 q^{53} +2.27110 q^{57} +1.63555 q^{59} -0.0394782 q^{61} -6.72890 q^{67} +2.40393 q^{69} -6.27110 q^{71} +4.00000 q^{73} +5.32497 q^{79} +7.82224 q^{81} +14.7738 q^{83} +2.00546 q^{87} -0.867178 q^{89} -1.64101 q^{93} +16.0988 q^{97} -3.91211 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 + 8 * q^9	$ 3 q + q^{3} + 8 q^{9} + 4 q^{11} + 8 q^{13} + 10 q^{17} + 2 q^{19} + 3 q^{23} + 10 q^{27} + 3 q^{31} + 18 q^{33} - 6 q^{37} - 14 q^{39} - 11 q^{41} + 11 q^{43} + 4 q^{47} - 3 q^{51} - 14 q^{53} + 7 q^{57} + 5 q^{59} - 17 q^{61} - 20 q^{67} + 24 q^{69} - 19 q^{71} + 12 q^{73} - q^{79} + 23 q^{81} + 28 q^{83} - 27 q^{87} + 14 q^{89} + 28 q^{93} + 15 q^{97} + 21 q^{99}+O(q^{100}) $ 3 * q + q^3 + 8 * q^9 + 4 * q^11 + 8 * q^13 + 10 * q^17 + 2 * q^19 + 3 * q^23 + 10 * q^27 + 3 * q^31 + 18 * q^33 - 6 * q^37 - 14 * q^39 - 11 * q^41 + 11 * q^43 + 4 * q^47 - 3 * q^51 - 14 * q^53 + 7 * q^57 + 5 * q^59 - 17 * q^61 - 20 * q^67 + 24 * q^69 - 19 * q^71 + 12 * q^73 - q^79 + 23 * q^81 + 28 * q^83 - 27 * q^87 + 14 * q^89 + 28 * q^93 + 15 * q^97 + 21 * 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
$3$	0.364448	0.210414	0.105207	−	0.994450i	$-0.466449\pi$
$3$	0.364448	0.210414	0.105207	+	0.994450i	$0.466449\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.86718	−0.955726
$10$	0	0
$11$	1.36445	0.411397	0.205698	−	0.978615i	$-0.434053\pi$
$11$	1.36445	0.411397	0.205698	+	0.978615i	$0.434053\pi$
$12$	0	0
$13$	2.63555	0.730971	0.365485	−	0.930817i	$-0.380903\pi$
$13$	2.63555	0.730971	0.365485	+	0.930817i	$0.380903\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.23163	−0.541249	−0.270624	−	0.962685i	$-0.587230\pi$
$17$	−2.23163	−0.541249	−0.270624	+	0.962685i	$0.587230\pi$
$18$	0	0
$19$	6.23163	1.42963	0.714816	−	0.699312i	$-0.246510\pi$
$19$	6.23163	1.42963	0.714816	+	0.699312i	$0.246510\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.59607	1.37538	0.687688	−	0.726006i	$-0.258626\pi$
$23$	6.59607	1.37538	0.687688	+	0.726006i	$0.258626\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.13828	−0.411512
$28$	0	0
$29$	5.50273	1.02183	0.510916	−	0.859631i	$-0.329306\pi$
$29$	5.50273	1.02183	0.510916	+	0.859631i	$0.329306\pi$
$30$	0	0
$31$	−4.50273	−0.808714	−0.404357	−	0.914601i	$-0.632505\pi$
$31$	−4.50273	−0.808714	−0.404357	+	0.914601i	$0.632505\pi$
$32$	0	0
$33$	0.497270	0.0865637
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.09334	−0.344144	−0.172072	−	0.985084i	$-0.555046\pi$
$37$	−2.09334	−0.344144	−0.172072	+	0.985084i	$0.555046\pi$
$38$	0	0
$39$	0.960522	0.153807
$40$	0	0
$41$	−9.32497	−1.45632	−0.728158	−	0.685410i	$-0.759623\pi$
$41$	−9.32497	−1.45632	−0.728158	+	0.685410i	$0.759623\pi$
$42$	0	0
$43$	−1.86718	−0.284742	−0.142371	−	0.989813i	$-0.545473\pi$
$43$	−1.86718	−0.284742	−0.142371	+	0.989813i	$0.545473\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.86718	1.00168	0.500840	−	0.865540i	$-0.333024\pi$
$47$	6.86718	1.00168	0.500840	+	0.865540i	$0.333024\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.813312	−0.113886
$52$	0	0
$53$	−10.1383	−1.39260	−0.696300	−	0.717751i	$-0.745172\pi$
$53$	−10.1383	−1.39260	−0.696300	+	0.717751i	$0.745172\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.27110	0.300815
$58$	0	0
$59$	1.63555	0.212931	0.106465	−	0.994316i	$-0.466047\pi$
$59$	1.63555	0.212931	0.106465	+	0.994316i	$0.466047\pi$
$60$	0	0
$61$	−0.0394782	−0.00505467	−0.00252733	−	0.999997i	$-0.500804\pi$
$61$	−0.0394782	−0.00505467	−0.00252733	+	0.999997i	$0.500804\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.72890	−0.822065	−0.411033	−	0.911621i	$-0.634832\pi$
$67$	−6.72890	−0.822065	−0.411033	+	0.911621i	$0.634832\pi$
$68$	0	0
$69$	2.40393	0.289399
$70$	0	0
$71$	−6.27110	−0.744243	−0.372122	−	0.928184i	$-0.621370\pi$
$71$	−6.27110	−0.744243	−0.372122	+	0.928184i	$0.621370\pi$
$72$	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
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.32497	0.599106	0.299553	−	0.954080i	$-0.403162\pi$
$79$	5.32497	0.599106	0.299553	+	0.954080i	$0.403162\pi$
$80$	0	0
$81$	7.82224	0.869138
$82$	0	0
$83$	14.7738	1.62164	0.810819	−	0.585296i	$-0.199022\pi$
$83$	14.7738	1.62164	0.810819	+	0.585296i	$0.199022\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00546	0.215008
$88$	0	0
$89$	−0.867178	−0.0919206	−0.0459603	−	0.998943i	$-0.514635\pi$
$89$	−0.867178	−0.0919206	−0.0459603	+	0.998943i	$0.514635\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.64101	−0.170165
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.0988	1.63459	0.817293	−	0.576222i	$-0.195474\pi$
$97$	16.0988	1.63459	0.817293	+	0.576222i	$0.195474\pi$
$98$	0	0
$99$	−3.91211	−0.393182
$100$	0	0
$101$	15.6949	1.56170	0.780849	−	0.624719i	$-0.214787\pi$
$101$	15.6949	1.56170	0.780849	+	0.624719i	$0.214787\pi$
$102$	0	0
$103$	13.3699	1.31738	0.658688	−	0.752416i	$-0.271112\pi$
$103$	13.3699	1.31738	0.658688	+	0.752416i	$0.271112\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.40393	0.522417	0.261209	−	0.965282i	$-0.415879\pi$
$107$	5.40393	0.522417	0.261209	+	0.965282i	$0.415879\pi$
$108$	0	0
$109$	−15.8277	−1.51602	−0.758009	−	0.652244i	$-0.773828\pi$
$109$	−15.8277	−1.51602	−0.758009	+	0.652244i	$0.773828\pi$
$110$	0	0
$111$	−0.762915	−0.0724127
$112$	0	0
$113$	18.4238	1.73316	0.866581	−	0.499036i	$-0.166312\pi$
$113$	18.4238	1.73316	0.866581	+	0.499036i	$0.166312\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.55660	−0.698608
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.13828	−0.830753
$122$	0	0
$123$	−3.39847	−0.306429
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.32497	0.472515	0.236257	−	0.971691i	$-0.424079\pi$
$127$	5.32497	0.472515	0.236257	+	0.971691i	$0.424079\pi$
$128$	0	0
$129$	−0.680489	−0.0599137
$130$	0	0
$131$	−16.2371	−1.41864	−0.709320	−	0.704886i	$-0.750998\pi$
$131$	−16.2371	−1.41864	−0.709320	+	0.704886i	$0.750998\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.40939	0.376719	0.188360	−	0.982100i	$-0.439683\pi$
$137$	4.40939	0.376719	0.188360	+	0.982100i	$0.439683\pi$
$138$	0	0
$139$	22.3699	1.89739	0.948695	−	0.316192i	$-0.102404\pi$
$139$	22.3699	1.89739	0.948695	+	0.316192i	$0.102404\pi$
$140$	0	0
$141$	2.50273	0.210768
$142$	0	0
$143$	3.59607	0.300719
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.5082	1.84394	0.921971	−	0.387258i	$-0.126578\pi$
$149$	22.5082	1.84394	0.921971	+	0.387258i	$0.126578\pi$
$150$	0	0
$151$	13.5027	1.09884	0.549418	−	0.835547i	$-0.314849\pi$
$151$	13.5027	1.09884	0.549418	+	0.835547i	$0.314849\pi$
$152$	0	0
$153$	6.39847	0.517285
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.5566	1.24155	0.620776	−	0.783988i	$-0.286818\pi$
$157$	15.5566	1.24155	0.620776	+	0.783988i	$0.286818\pi$
$158$	0	0
$159$	−3.69488	−0.293023
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.40393	0.344942	0.172471	−	0.985015i	$-0.444825\pi$
$163$	4.40393	0.344942	0.172471	+	0.985015i	$0.444825\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.09880	0.704087	0.352043	−	0.935984i	$-0.385487\pi$
$167$	9.09880	0.704087	0.352043	+	0.935984i	$0.385487\pi$
$168$	0	0
$169$	−6.05387	−0.465682
$170$	0	0
$171$	−17.8672	−1.36634
$172$	0	0
$173$	19.9605	1.51757	0.758785	−	0.651341i	$-0.225793\pi$
$173$	19.9605	1.51757	0.758785	+	0.651341i	$0.225793\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.596074	0.0448036
$178$	0	0
$179$	−13.1921	−0.986027	−0.493014	−	0.870022i	$-0.664105\pi$
$179$	−13.1921	−0.986027	−0.493014	+	0.870022i	$0.664105\pi$
$180$	0	0
$181$	−10.2316	−0.760511	−0.380255	−	0.924882i	$-0.624164\pi$
$181$	−10.2316	−0.760511	−0.380255	+	0.924882i	$0.624164\pi$
$182$	0	0
$183$	−0.0143878	−0.00106357
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.04494	−0.222668
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.0055	1.01340	0.506700	−	0.862123i	$-0.330865\pi$
$191$	14.0055	1.01340	0.506700	+	0.862123i	$0.330865\pi$
$192$	0	0
$193$	−14.4094	−1.03721	−0.518605	−	0.855014i	$-0.673549\pi$
$193$	−14.4094	−1.03721	−0.518605	+	0.855014i	$0.673549\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.5566	−0.823373	−0.411687	−	0.911325i	$-0.635060\pi$
$197$	−11.5566	−0.823373	−0.411687	+	0.911325i	$0.635060\pi$
$198$	0	0
$199$	5.40939	0.383461	0.191731	−	0.981448i	$-0.438590\pi$
$199$	5.40939	0.383461	0.191731	+	0.981448i	$0.438590\pi$
$200$	0	0
$201$	−2.45233	−0.172974
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.9121	−1.31448
$208$	0	0
$209$	8.50273	0.588146
$210$	0	0
$211$	−10.9660	−0.754929	−0.377465	−	0.926024i	$-0.623204\pi$
$211$	−10.9660	−0.754929	−0.377465	+	0.926024i	$0.623204\pi$
$212$	0	0
$213$	−2.28549	−0.156599
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.45779	0.0985085
$220$	0	0
$221$	−5.88157	−0.395637
$222$	0	0
$223$	21.5566	1.44354	0.721768	−	0.692135i	$-0.243330\pi$
$223$	21.5566	1.44354	0.721768	+	0.692135i	$0.243330\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.4238	−0.824595	−0.412297	−	0.911049i	$-0.635274\pi$
$227$	−12.4238	−0.824595	−0.412297	+	0.911049i	$0.635274\pi$
$228$	0	0
$229$	8.36445	0.552738	0.276369	−	0.961052i	$-0.410869\pi$
$229$	8.36445	0.552738	0.276369	+	0.961052i	$0.410869\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0055	0.917528	0.458764	−	0.888558i	$-0.348292\pi$
$233$	14.0055	0.917528	0.458764	+	0.888558i	$0.348292\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.94067	0.126060
$238$	0	0
$239$	16.9605	1.09708	0.548542	−	0.836123i	$-0.315183\pi$
$239$	16.9605	1.09708	0.548542	+	0.836123i	$0.315183\pi$
$240$	0	0
$241$	−10.3304	−0.665441	−0.332721	−	0.943025i	$-0.607967\pi$
$241$	−10.3304	−0.665441	−0.332721	+	0.943025i	$0.607967\pi$
$242$	0	0
$243$	9.26564	0.594391
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.4238	1.04502
$248$	0	0
$249$	5.38429	0.341216
$250$	0	0
$251$	−19.5082	−1.23135	−0.615673	−	0.788002i	$-0.711116\pi$
$251$	−19.5082	−1.23135	−0.615673	+	0.788002i	$0.711116\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.4633	1.77549	0.887744	−	0.460337i	$-0.152271\pi$
$257$	28.4633	1.77549	0.887744	+	0.460337i	$0.152271\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−15.7773	−0.976591
$262$	0	0
$263$	−28.5621	−1.76121	−0.880606	−	0.473849i	$-0.842864\pi$
$263$	−28.5621	−1.76121	−0.880606	+	0.473849i	$0.842864\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.316041	−0.0193414
$268$	0	0
$269$	9.19761	0.560788	0.280394	−	0.959885i	$-0.409535\pi$
$269$	9.19761	0.560788	0.280394	+	0.959885i	$0.409535\pi$
$270$	0	0
$271$	−3.86172	−0.234583	−0.117291	−	0.993098i	$-0.537421\pi$
$271$	−3.86172	−0.234583	−0.117291	+	0.993098i	$0.537421\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.6410	−0.639356	−0.319678	−	0.947526i	$-0.603575\pi$
$277$	−10.6410	−0.639356	−0.319678	+	0.947526i	$0.603575\pi$
$278$	0	0
$279$	12.9101	0.772909
$280$	0	0
$281$	1.73436	0.103463	0.0517315	−	0.998661i	$-0.483526\pi$
$281$	1.73436	0.103463	0.0517315	+	0.998661i	$0.483526\pi$
$282$	0	0
$283$	−5.91211	−0.351439	−0.175719	−	0.984440i	$-0.556225\pi$
$283$	−5.91211	−0.351439	−0.175719	+	0.984440i	$0.556225\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.0198	−0.707050
$290$	0	0
$291$	5.86718	0.343940
$292$	0	0
$293$	24.3250	1.42108	0.710540	−	0.703657i	$-0.248451\pi$
$293$	24.3250	1.42108	0.710540	+	0.703657i	$0.248451\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.91757	−0.169295
$298$	0	0
$299$	17.3843	1.00536
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.71997	0.328604
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.5226	−1.34250	−0.671252	−	0.741229i	$-0.734243\pi$
$307$	−23.5226	−1.34250	−0.671252	+	0.741229i	$0.734243\pi$
$308$	0	0
$309$	4.87264	0.277195
$310$	0	0
$311$	8.81877	0.500067	0.250033	−	0.968237i	$-0.419558\pi$
$311$	8.81877	0.500067	0.250033	+	0.968237i	$0.419558\pi$
$312$	0	0
$313$	−0.0933442	−0.00527612	−0.00263806	−	0.999997i	$-0.500840\pi$
$313$	−0.0933442	−0.00527612	−0.00263806	+	0.999997i	$0.500840\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.54221	−0.198950	−0.0994751	−	0.995040i	$-0.531716\pi$
$317$	−3.54221	−0.198950	−0.0994751	+	0.995040i	$0.531716\pi$
$318$	0	0
$319$	7.50819	0.420378
$320$	0	0
$321$	1.96945	0.109924
$322$	0	0
$323$	−13.9067	−0.773787
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.76837	−0.318992
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.6499	−1.08006	−0.540029	−	0.841646i	$-0.681587\pi$
$331$	−19.6499	−1.08006	−0.540029	+	0.841646i	$0.681587\pi$
$332$	0	0
$333$	6.00199	0.328907
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.0449	1.25534	0.627669	−	0.778480i	$-0.284009\pi$
$337$	23.0449	1.25534	0.627669	+	0.778480i	$0.284009\pi$
$338$	0	0
$339$	6.71451	0.364682
$340$	0	0
$341$	−6.14374	−0.332702
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.5621	1.21119	0.605597	−	0.795771i	$-0.292934\pi$
$347$	22.5621	1.21119	0.605597	+	0.795771i	$0.292934\pi$
$348$	0	0
$349$	−14.3250	−0.766798	−0.383399	−	0.923583i	$-0.625247\pi$
$349$	−14.3250	−0.766798	−0.383399	+	0.923583i	$0.625247\pi$
$350$	0	0
$351$	−5.63555	−0.300804
$352$	0	0
$353$	32.7003	1.74046	0.870232	−	0.492643i	$-0.163969\pi$
$353$	32.7003	1.74046	0.870232	+	0.492643i	$0.163969\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.1581	−1.38057	−0.690287	−	0.723536i	$-0.742516\pi$
$359$	−26.1581	−1.38057	−0.690287	+	0.723536i	$0.742516\pi$
$360$	0	0
$361$	19.8332	1.04385
$362$	0	0
$363$	−3.33043	−0.174802
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.2316	−1.31708	−0.658540	−	0.752546i	$-0.728826\pi$
$367$	−25.2316	−1.31708	−0.658540	+	0.752546i	$0.728826\pi$
$368$	0	0
$369$	26.7363	1.39184
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.69488	0.294870	0.147435	−	0.989072i	$-0.452898\pi$
$373$	5.69488	0.294870	0.147435	+	0.989072i	$0.452898\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5027	0.746929
$378$	0	0
$379$	14.9660	0.768751	0.384375	−	0.923177i	$-0.374417\pi$
$379$	14.9660	0.768751	0.384375	+	0.923177i	$0.374417\pi$
$380$	0	0
$381$	1.94067	0.0994238
$382$	0	0
$383$	−24.4776	−1.25075	−0.625374	−	0.780325i	$-0.715054\pi$
$383$	−24.4776	−1.25075	−0.625374	+	0.780325i	$0.715054\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.35353	0.272135
$388$	0	0
$389$	−31.2460	−1.58424	−0.792118	−	0.610368i	$-0.791022\pi$
$389$	−31.2460	−1.58424	−0.792118	+	0.610368i	$0.791022\pi$
$390$	0	0
$391$	−14.7200	−0.744421
$392$	0	0
$393$	−5.91757	−0.298502
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.13282	−0.0568547	−0.0284274	−	0.999596i	$-0.509050\pi$
$397$	−1.13282	−0.0568547	−0.0284274	+	0.999596i	$0.509050\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.82224	−0.190874	−0.0954368	−	0.995435i	$-0.530425\pi$
$401$	−3.82224	−0.190874	−0.0954368	+	0.995435i	$0.530425\pi$
$402$	0	0
$403$	−11.8672	−0.591146
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.85626	−0.141580
$408$	0	0
$409$	−33.6609	−1.66442	−0.832211	−	0.554459i	$-0.812925\pi$
$409$	−33.6609	−1.66442	−0.832211	+	0.554459i	$0.812925\pi$
$410$	0	0
$411$	1.60699	0.0792671
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.15267	0.399238
$418$	0	0
$419$	0.497270	0.0242933	0.0121466	−	0.999926i	$-0.496134\pi$
$419$	0.497270	0.0242933	0.0121466	+	0.999926i	$0.496134\pi$
$420$	0	0
$421$	−1.28003	−0.0623850	−0.0311925	−	0.999513i	$-0.509930\pi$
$421$	−1.28003	−0.0623850	−0.0311925	+	0.999513i	$0.509930\pi$
$422$	0	0
$423$	−19.6894	−0.957332
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.31058	0.0632755
$430$	0	0
$431$	−10.2910	−0.495698	−0.247849	−	0.968799i	$-0.579724\pi$
$431$	−10.2910	−0.495698	−0.247849	+	0.968799i	$0.579724\pi$
$432$	0	0
$433$	29.8870	1.43628	0.718139	−	0.695899i	$-0.244994\pi$
$433$	29.8870	1.43628	0.718139	+	0.695899i	$0.244994\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.1043	1.96628
$438$	0	0
$439$	5.04841	0.240947	0.120474	−	0.992717i	$-0.461559\pi$
$439$	5.04841	0.240947	0.120474	+	0.992717i	$0.461559\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.7738	0.559392	0.279696	−	0.960089i	$-0.409766\pi$
$443$	11.7738	0.559392	0.279696	+	0.960089i	$0.409766\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.20307	0.387992
$448$	0	0
$449$	−11.3843	−0.537258	−0.268629	−	0.963244i	$-0.586571\pi$
$449$	−11.3843	−0.537258	−0.268629	+	0.963244i	$0.586571\pi$
$450$	0	0
$451$	−12.7234	−0.599123
$452$	0	0
$453$	4.92104	0.231211
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.2515	−1.74255	−0.871275	−	0.490795i	$-0.836706\pi$
$457$	−37.2515	−1.74255	−0.871275	+	0.490795i	$0.836706\pi$
$458$	0	0
$459$	4.77184	0.222731
$460$	0	0
$461$	15.0449	0.700713	0.350356	−	0.936616i	$-0.386061\pi$
$461$	15.0449	0.700713	0.350356	+	0.936616i	$0.386061\pi$
$462$	0	0
$463$	31.2031	1.45013	0.725065	−	0.688681i	$-0.241810\pi$
$463$	31.2031	1.45013	0.725065	+	0.688681i	$0.241810\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.7684	0.868497	0.434248	−	0.900793i	$-0.357014\pi$
$467$	18.7684	0.868497	0.434248	+	0.900793i	$0.357014\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.66957	0.261240
$472$	0	0
$473$	−2.54767	−0.117142
$474$	0	0
$475$	0	0
$476$	0	0
$477$	29.0683	1.33094
$478$	0	0
$479$	0.714508	0.0326467	0.0163234	−	0.999867i	$-0.494804\pi$
$479$	0.714508	0.0326467	0.0163234	+	0.999867i	$0.494804\pi$
$480$	0	0
$481$	−5.51712	−0.251559
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.6949	−0.756517	−0.378259	−	0.925700i	$-0.623477\pi$
$487$	−16.6949	−0.756517	−0.378259	+	0.925700i	$0.623477\pi$
$488$	0	0
$489$	1.60500	0.0725807
$490$	0	0
$491$	6.19761	0.279694	0.139847	−	0.990173i	$-0.455339\pi$
$491$	6.19761	0.279694	0.139847	+	0.990173i	$0.455339\pi$
$492$	0	0
$493$	−12.2800	−0.553065
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14.4687	0.647708	0.323854	−	0.946107i	$-0.395021\pi$
$499$	14.4687	0.647708	0.323854	+	0.946107i	$0.395021\pi$
$500$	0	0
$501$	3.31604	0.148150
$502$	0	0
$503$	−30.6949	−1.36862	−0.684308	−	0.729193i	$-0.739896\pi$
$503$	−30.6949	−1.36862	−0.684308	+	0.729193i	$0.739896\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.20632	−0.0979861
$508$	0	0
$509$	−44.4742	−1.97128	−0.985641	−	0.168852i	$-0.945994\pi$
$509$	−44.4742	−1.97128	−0.985641	+	0.168852i	$0.945994\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−13.3250	−0.588312
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.36991	0.412088
$518$	0	0
$519$	7.27457	0.319318
$520$	0	0
$521$	12.6499	0.554204	0.277102	−	0.960841i	$-0.410626\pi$
$521$	12.6499	0.554204	0.277102	+	0.960841i	$0.410626\pi$
$522$	0	0
$523$	10.2711	0.449124	0.224562	−	0.974460i	$-0.427905\pi$
$523$	10.2711	0.449124	0.224562	+	0.974460i	$0.427905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.0484	0.437715
$528$	0	0
$529$	20.5082	0.891660
$530$	0	0
$531$	−4.68942	−0.203503
$532$	0	0
$533$	−24.5764	−1.06452
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.80785	−0.207474
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.13828	−0.0919319	−0.0459660	−	0.998943i	$-0.514637\pi$
$541$	−2.13828	−0.0919319	−0.0459660	+	0.998943i	$0.514637\pi$
$542$	0	0
$543$	−3.72890	−0.160022
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−32.0198	−1.36907	−0.684535	−	0.728980i	$-0.739995\pi$
$547$	−32.0198	−1.36907	−0.684535	+	0.728980i	$0.739995\pi$
$548$	0	0
$549$	0.113191	0.00483088
$550$	0	0
$551$	34.2910	1.46084
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.33043	−0.352972	−0.176486	−	0.984303i	$-0.556473\pi$
$557$	−8.33043	−0.352972	−0.176486	+	0.984303i	$0.556473\pi$
$558$	0	0
$559$	−4.92104	−0.208138
$560$	0	0
$561$	−1.10972	−0.0468525
$562$	0	0
$563$	19.4633	0.820278	0.410139	−	0.912023i	$-0.365480\pi$
$563$	19.4633	0.820278	0.410139	+	0.912023i	$0.365480\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.8925	0.792014	0.396007	−	0.918247i	$-0.370396\pi$
$569$	18.8925	0.792014	0.396007	+	0.918247i	$0.370396\pi$
$570$	0	0
$571$	−18.6301	−0.779645	−0.389823	−	0.920890i	$-0.627464\pi$
$571$	−18.6301	−0.779645	−0.389823	+	0.920890i	$0.627464\pi$
$572$	0	0
$573$	5.10426	0.213234
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.5926	0.774020	0.387010	−	0.922075i	$-0.373508\pi$
$577$	18.5926	0.774020	0.387010	+	0.922075i	$0.373508\pi$
$578$	0	0
$579$	−5.25147	−0.218244
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.8332	−0.572911
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.1383	−0.418452	−0.209226	−	0.977867i	$-0.567094\pi$
$587$	−10.1383	−0.418452	−0.209226	+	0.977867i	$0.567094\pi$
$588$	0	0
$589$	−28.0593	−1.15616
$590$	0	0
$591$	−4.21178	−0.173249
$592$	0	0
$593$	17.0055	0.698331	0.349165	−	0.937061i	$-0.386465\pi$
$593$	17.0055	0.698331	0.349165	+	0.937061i	$0.386465\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.97144	0.0806857
$598$	0	0
$599$	−8.18669	−0.334499	−0.167250	−	0.985915i	$-0.553489\pi$
$599$	−8.18669	−0.334499	−0.167250	+	0.985915i	$0.553489\pi$
$600$	0	0
$601$	35.3359	1.44138	0.720690	−	0.693257i	$-0.243825\pi$
$601$	35.3359	1.44138	0.720690	+	0.693257i	$0.243825\pi$
$602$	0	0
$603$	19.2929	0.785669
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.2316	−1.14589	−0.572943	−	0.819595i	$-0.694198\pi$
$607$	−28.2316	−1.14589	−0.572943	+	0.819595i	$0.694198\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.0988	0.732199
$612$	0	0
$613$	31.1581	1.25846	0.629232	−	0.777217i	$-0.283369\pi$
$613$	31.1581	1.25846	0.629232	+	0.777217i	$0.283369\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.1527	−1.57623	−0.788114	−	0.615530i	$-0.788942\pi$
$617$	−39.1527	−1.57623	−0.788114	+	0.615530i	$0.788942\pi$
$618$	0	0
$619$	39.3843	1.58299	0.791494	−	0.611177i	$-0.209304\pi$
$619$	39.3843	1.58299	0.791494	+	0.611177i	$0.209304\pi$
$620$	0	0
$621$	−14.1043	−0.565985
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.09880	0.123754
$628$	0	0
$629$	4.67156	0.186267
$630$	0	0
$631$	22.7003	0.903686	0.451843	−	0.892097i	$-0.350767\pi$
$631$	22.7003	0.903686	0.451843	+	0.892097i	$0.350767\pi$
$632$	0	0
$633$	−3.99653	−0.158848
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.9804	0.711292
$640$	0	0
$641$	8.41285	0.332288	0.166144	−	0.986102i	$-0.446868\pi$
$641$	8.41285	0.332288	0.166144	+	0.986102i	$0.446868\pi$
$642$	0	0
$643$	−29.7398	−1.17282	−0.586412	−	0.810013i	$-0.699460\pi$
$643$	−29.7398	−1.17282	−0.586412	+	0.810013i	$0.699460\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.6859	−0.852563	−0.426281	−	0.904591i	$-0.640177\pi$
$647$	−21.6859	−0.852563	−0.426281	+	0.904591i	$0.640177\pi$
$648$	0	0
$649$	2.23163	0.0875990
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.0702	−1.17674	−0.588370	−	0.808592i	$-0.700230\pi$
$653$	−30.0702	−1.17674	−0.588370	+	0.808592i	$0.700230\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−11.4687	−0.447437
$658$	0	0
$659$	2.37537	0.0925311	0.0462656	−	0.998929i	$-0.485268\pi$
$659$	2.37537	0.0925311	0.0462656	+	0.998929i	$0.485268\pi$
$660$	0	0
$661$	14.4633	0.562555	0.281278	−	0.959626i	$-0.409242\pi$
$661$	14.4633	0.562555	0.281278	+	0.959626i	$0.409242\pi$
$662$	0	0
$663$	−2.14353	−0.0832476
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.2964	1.40540
$668$	0	0
$669$	7.85626	0.303741
$670$	0	0
$671$	−0.0538660	−0.00207947
$672$	0	0
$673$	−8.99653	−0.346791	−0.173395	−	0.984852i	$-0.555474\pi$
$673$	−8.99653	−0.346791	−0.173395	+	0.984852i	$0.555474\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.49181	0.287934	0.143967	−	0.989583i	$-0.454014\pi$
$677$	7.49181	0.287934	0.143967	+	0.989583i	$0.454014\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.52782	−0.173506
$682$	0	0
$683$	42.3250	1.61952	0.809760	−	0.586761i	$-0.199597\pi$
$683$	42.3250	1.61952	0.809760	+	0.586761i	$0.199597\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.04841	0.116304
$688$	0	0
$689$	−26.7200	−1.01795
$690$	0	0
$691$	−28.6465	−1.08976	−0.544882	−	0.838513i	$-0.683425\pi$
$691$	−28.6465	−1.08976	−0.544882	+	0.838513i	$0.683425\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.8098	0.788229
$698$	0	0
$699$	5.10426	0.193061
$700$	0	0
$701$	−20.4292	−0.771601	−0.385801	−	0.922582i	$-0.626075\pi$
$701$	−20.4292	−0.771601	−0.385801	+	0.922582i	$0.626075\pi$
$702$	0	0
$703$	−13.0449	−0.491999
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.8277	1.08265	0.541323	−	0.840814i	$-0.317923\pi$
$709$	28.8277	1.08265	0.541323	+	0.840814i	$0.317923\pi$
$710$	0	0
$711$	−15.2676	−0.572581
$712$	0	0
$713$	−29.7003	−1.11229
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.18123	0.230842
$718$	0	0
$719$	−31.3106	−1.16769	−0.583844	−	0.811866i	$-0.698452\pi$
$719$	−31.3106	−1.16769	−0.583844	+	0.811866i	$0.698452\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3.76490	−0.140018
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.17230	−0.303094	−0.151547	−	0.988450i	$-0.548425\pi$
$727$	−8.17230	−0.303094	−0.151547	+	0.988450i	$0.548425\pi$
$728$	0	0
$729$	−20.0899	−0.744069
$730$	0	0
$731$	4.16684	0.154116
$732$	0	0
$733$	−21.4183	−0.791103	−0.395552	−	0.918444i	$-0.629447\pi$
$733$	−21.4183	−0.791103	−0.395552	+	0.918444i	$0.629447\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.18123	−0.338195
$738$	0	0
$739$	4.35006	0.160020	0.0800098	−	0.996794i	$-0.474505\pi$
$739$	4.35006	0.160020	0.0800098	+	0.996794i	$0.474505\pi$
$740$	0	0
$741$	5.98561	0.219887
$742$	0	0
$743$	−6.53328	−0.239683	−0.119841	−	0.992793i	$-0.538239\pi$
$743$	−6.53328	−0.239683	−0.119841	+	0.992793i	$0.538239\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−42.3592	−1.54984
$748$	0	0
$749$	0	0
$750$	0	0
$751$	34.4831	1.25831	0.629153	−	0.777281i	$-0.283402\pi$
$751$	34.4831	1.25831	0.629153	+	0.777281i	$0.283402\pi$
$752$	0	0
$753$	−7.10972	−0.259093
$754$	0	0
$755$	0	0
$756$	0	0
$757$	44.2425	1.60802	0.804011	−	0.594614i	$-0.202695\pi$
$757$	44.2425	1.60802	0.804011	+	0.594614i	$0.202695\pi$
$758$	0	0
$759$	3.28003	0.119058
$760$	0	0
$761$	23.8188	0.863430	0.431715	−	0.902010i	$-0.357909\pi$
$761$	23.8188	0.863430	0.431715	+	0.902010i	$0.357909\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.31058	0.155646
$768$	0	0
$769$	0.158128	0.00570225	0.00285113	−	0.999996i	$-0.499092\pi$
$769$	0.158128	0.00570225	0.00285113	+	0.999996i	$0.499092\pi$
$770$	0	0
$771$	10.3734	0.373588
$772$	0	0
$773$	−17.7991	−0.640191	−0.320095	−	0.947385i	$-0.603715\pi$
$773$	−17.7991	−0.640191	−0.320095	+	0.947385i	$0.603715\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−58.1097	−2.08200
$780$	0	0
$781$	−8.55660	−0.306179
$782$	0	0
$783$	−11.7664	−0.420496
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−41.5226	−1.48012	−0.740060	−	0.672541i	$-0.765203\pi$
$787$	−41.5226	−1.48012	−0.740060	+	0.672541i	$0.765203\pi$
$788$	0	0
$789$	−10.4094	−0.370584
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.104047	−0.00369481
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.0933	0.888852	0.444426	−	0.895816i	$-0.353408\pi$
$797$	25.0933	0.888852	0.444426	+	0.895816i	$0.353408\pi$
$798$	0	0
$799$	−15.3250	−0.542158
$800$	0	0
$801$	2.48635	0.0878509
$802$	0	0
$803$	5.45779	0.192601
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.35205	0.117998
$808$	0	0
$809$	12.6949	0.446328	0.223164	−	0.974781i	$-0.428361\pi$
$809$	12.6949	0.446328	0.223164	+	0.974781i	$0.428361\pi$
$810$	0	0
$811$	−17.9749	−0.631184	−0.315592	−	0.948895i	$-0.602203\pi$
$811$	−17.9749	−0.631184	−0.315592	+	0.948895i	$0.602203\pi$
$812$	0	0
$813$	−1.40740	−0.0493595
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.6356	−0.407076
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.0683	0.700387	0.350193	−	0.936677i	$-0.386116\pi$
$821$	20.0683	0.700387	0.350193	+	0.936677i	$0.386116\pi$
$822$	0	0
$823$	10.6105	0.369857	0.184929	−	0.982752i	$-0.440795\pi$
$823$	10.6105	0.369857	0.184929	+	0.982752i	$0.440795\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.7793	−1.20939	−0.604697	−	0.796455i	$-0.706706\pi$
$827$	−34.7793	−1.20939	−0.604697	+	0.796455i	$0.706706\pi$
$828$	0	0
$829$	26.2282	0.910942	0.455471	−	0.890251i	$-0.349471\pi$
$829$	26.2282	0.910942	0.455471	+	0.890251i	$0.349471\pi$
$830$	0	0
$831$	−3.87810	−0.134530
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.62810	0.332796
$838$	0	0
$839$	−33.9210	−1.17108	−0.585542	−	0.810642i	$-0.699118\pi$
$839$	−33.9210	−1.17108	−0.585542	+	0.810642i	$0.699118\pi$
$840$	0	0
$841$	1.28003	0.0441391
$842$	0	0
$843$	0.632082	0.0217701
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.15466	−0.0739477
$850$	0	0
$851$	−13.8079	−0.473327
$852$	0	0
$853$	−57.6698	−1.97458	−0.987288	−	0.158942i	$-0.949192\pi$
$853$	−57.6698	−1.97458	−0.987288	+	0.158942i	$0.949192\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.4039	1.20938	0.604688	−	0.796463i	$-0.293298\pi$
$857$	35.4039	1.20938	0.604688	+	0.796463i	$0.293298\pi$
$858$	0	0
$859$	13.4434	0.458683	0.229342	−	0.973346i	$-0.426343\pi$
$859$	13.4434	0.458683	0.229342	+	0.973346i	$0.426343\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.06478	0.240488	0.120244	−	0.992744i	$-0.461632\pi$
$863$	7.06478	0.240488	0.120244	+	0.992744i	$0.461632\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.38061	−0.148773
$868$	0	0
$869$	7.26564	0.246470
$870$	0	0
$871$	−17.7344	−0.600906
$872$	0	0
$873$	−46.1581	−1.56222
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15.0055	−0.506698	−0.253349	−	0.967375i	$-0.581532\pi$
$877$	−15.0055	−0.506698	−0.253349	+	0.967375i	$0.581532\pi$
$878$	0	0
$879$	8.86519	0.299015
$880$	0	0
$881$	39.3699	1.32641	0.663203	−	0.748440i	$-0.269197\pi$
$881$	39.3699	1.32641	0.663203	+	0.748440i	$0.269197\pi$
$882$	0	0
$883$	−58.2229	−1.95936	−0.979679	−	0.200574i	$-0.935719\pi$
$883$	−58.2229	−1.95936	−0.979679	+	0.200574i	$0.935719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.2460	0.847678	0.423839	−	0.905738i	$-0.360682\pi$
$887$	25.2460	0.847678	0.423839	+	0.905738i	$0.360682\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.6730	0.357560
$892$	0	0
$893$	42.7937	1.43204
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.33567	0.211542
$898$	0	0
$899$	−24.7773	−0.826369
$900$	0	0
$901$	22.6248	0.753743
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.1867	−0.902719	−0.451360	−	0.892342i	$-0.649061\pi$
$907$	−27.1867	−0.902719	−0.451360	+	0.892342i	$0.649061\pi$
$908$	0	0
$909$	−45.0000	−1.49256
$910$	0	0
$911$	−40.7882	−1.35137	−0.675687	−	0.737189i	$-0.736153\pi$
$911$	−40.7882	−1.35137	−0.675687	+	0.737189i	$0.736153\pi$
$912$	0	0
$913$	20.1581	0.667137
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−47.0737	−1.55282	−0.776409	−	0.630229i	$-0.782961\pi$
$919$	−47.0737	−1.55282	−0.776409	+	0.630229i	$0.782961\pi$
$920$	0	0
$921$	−8.57276	−0.282482
$922$	0	0
$923$	−16.5278	−0.544020
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−38.3339	−1.25905
$928$	0	0
$929$	−17.9911	−0.590268	−0.295134	−	0.955456i	$-0.595364\pi$
$929$	−17.9911	−0.590268	−0.295134	+	0.955456i	$0.595364\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.21398	0.105221
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.7003	−1.23162	−0.615808	−	0.787896i	$-0.711170\pi$
$937$	−37.7003	−1.23162	−0.615808	+	0.787896i	$0.711170\pi$
$938$	0	0
$939$	−0.0340191	−0.00111017
$940$	0	0
$941$	−4.83316	−0.157556	−0.0787782	−	0.996892i	$-0.525102\pi$
$941$	−4.83316	−0.157556	−0.0787782	+	0.996892i	$0.525102\pi$
$942$	0	0
$943$	−61.5082	−2.00298
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.83663	−0.0921780	−0.0460890	−	0.998937i	$-0.514676\pi$
$947$	−2.83663	−0.0921780	−0.0460890	+	0.998937i	$0.514676\pi$
$948$	0	0
$949$	10.5422	0.342215
$950$	0	0
$951$	−1.29095	−0.0418619
$952$	0	0
$953$	17.7991	0.576571	0.288285	−	0.957545i	$-0.406915\pi$
$953$	17.7991	0.576571	0.288285	+	0.957545i	$0.406915\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.73634	0.0884535
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−10.7254	−0.345982
$962$	0	0
$963$	−15.4940	−0.499288
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.1581	−0.487453	−0.243726	−	0.969844i	$-0.578370\pi$
$967$	−15.1581	−0.487453	−0.243726	+	0.969844i	$0.578370\pi$
$968$	0	0
$969$	−5.06825	−0.162816
$970$	0	0
$971$	47.3843	1.52063	0.760317	−	0.649552i	$-0.225044\pi$
$971$	47.3843	1.52063	0.760317	+	0.649552i	$0.225044\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.7200	0.758869	0.379434	−	0.925219i	$-0.376119\pi$
$977$	23.7200	0.758869	0.379434	+	0.925219i	$0.376119\pi$
$978$	0	0
$979$	−1.18322	−0.0378158
$980$	0	0
$981$	45.3808	1.44890
$982$	0	0
$983$	−34.4543	−1.09892	−0.549461	−	0.835519i	$-0.685167\pi$
$983$	−34.4543	−1.09892	−0.549461	+	0.835519i	$0.685167\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.3160	−0.391627
$990$	0	0
$991$	−41.6609	−1.32340	−0.661700	−	0.749768i	$-0.730165\pi$
$991$	−41.6609	−1.32340	−0.661700	+	0.749768i	$0.730165\pi$
$992$	0	0
$993$	−7.16138	−0.227260
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.7882	−0.785051	−0.392525	−	0.919741i	$-0.628399\pi$
$997$	−24.7882	−0.785051	−0.392525	+	0.919741i	$0.628399\pi$
$998$	0	0
$999$	4.47616	0.141619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bd.1.2		3
5.2	odd	4		4900.2.e.s.2549.3		6
5.3	odd	4		4900.2.e.s.2549.4		6
5.4	even	2		4900.2.a.bb.1.2		3
7.3	odd	6		700.2.i.e.401.2	yes	6
7.5	odd	6		700.2.i.e.501.2	yes	6
7.6	odd	2		4900.2.a.ba.1.2		3
35.3	even	12		700.2.r.d.149.3		12
35.12	even	12		700.2.r.d.249.3		12
35.13	even	4		4900.2.e.t.2549.3		6
35.17	even	12		700.2.r.d.149.4		12
35.19	odd	6		700.2.i.d.501.2	yes	6
35.24	odd	6		700.2.i.d.401.2	✓	6
35.27	even	4		4900.2.e.t.2549.4		6
35.33	even	12		700.2.r.d.249.4		12
35.34	odd	2		4900.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.2	✓	6	35.24	odd	6
700.2.i.d.501.2	yes	6	35.19	odd	6
700.2.i.e.401.2	yes	6	7.3	odd	6
700.2.i.e.501.2	yes	6	7.5	odd	6
700.2.r.d.149.3		12	35.3	even	12
700.2.r.d.149.4		12	35.17	even	12
700.2.r.d.249.3		12	35.12	even	12
700.2.r.d.249.4		12	35.33	even	12
4900.2.a.ba.1.2		3	7.6	odd	2
4900.2.a.bb.1.2		3	5.4	even	2
4900.2.a.bc.1.2		3	35.34	odd	2
4900.2.a.bd.1.2		3	1.1	even	1	trivial
4900.2.e.s.2549.3		6	5.2	odd	4
4900.2.e.s.2549.4		6	5.3	odd	4
4900.2.e.t.2549.3		6	35.13	even	4
4900.2.e.t.2549.4		6	35.27	even	4

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q+0.364448 q^{3} -2.86718 q^{9} +O(q^{10})\)	\(q+0.364448 q^{3} -2.86718 q^{9} +1.36445 q^{11} +2.63555 q^{13} -2.23163 q^{17} +6.23163 q^{19} +6.59607 q^{23} -2.13828 q^{27} +5.50273 q^{29} -4.50273 q^{31} +0.497270 q^{33} -2.09334 q^{37} +0.960522 q^{39} -9.32497 q^{41} -1.86718 q^{43} +6.86718 q^{47} -0.813312 q^{51} -10.1383 q^{53} +2.27110 q^{57} +1.63555 q^{59} -0.0394782 q^{61} -6.72890 q^{67} +2.40393 q^{69} -6.27110 q^{71} +4.00000 q^{73} +5.32497 q^{79} +7.82224 q^{81} +14.7738 q^{83} +2.00546 q^{87} -0.867178 q^{89} -1.64101 q^{93} +16.0988 q^{97} -3.91211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 + 8 * q^9	\( 3 q + q^{3} + 8 q^{9} + 4 q^{11} + 8 q^{13} + 10 q^{17} + 2 q^{19} + 3 q^{23} + 10 q^{27} + 3 q^{31} + 18 q^{33} - 6 q^{37} - 14 q^{39} - 11 q^{41} + 11 q^{43} + 4 q^{47} - 3 q^{51} - 14 q^{53} + 7 q^{57} + 5 q^{59} - 17 q^{61} - 20 q^{67} + 24 q^{69} - 19 q^{71} + 12 q^{73} - q^{79} + 23 q^{81} + 28 q^{83} - 27 q^{87} + 14 q^{89} + 28 q^{93} + 15 q^{97} + 21 q^{99}+O(q^{100}) \) 3 * q + q^3 + 8 * q^9 + 4 * q^11 + 8 * q^13 + 10 * q^17 + 2 * q^19 + 3 * q^23 + 10 * q^27 + 3 * q^31 + 18 * q^33 - 6 * q^37 - 14 * q^39 - 11 * q^41 + 11 * q^43 + 4 * q^47 - 3 * q^51 - 14 * q^53 + 7 * q^57 + 5 * q^59 - 17 * q^61 - 20 * q^67 + 24 * q^69 - 19 * q^71 + 12 * q^73 - q^79 + 23 * q^81 + 28 * q^83 - 27 * q^87 + 14 * q^89 + 28 * q^93 + 15 * q^97 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more