LMFDB - Embedded newform 2420.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2420,2,Mod(1,2420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2420, 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("2420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2420.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:	$19.3237972891$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.70636$ of defining polynomial
Character	$\chi$	$=$	2420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70636 q^{3} -1.00000 q^{5} +3.76095 q^{7} +4.32440 q^{9} +3.26981 q^{13} -2.70636 q^{15} -3.90869 q^{17} +4.32440 q^{19} +10.1785 q^{21} -6.41755 q^{23} +1.00000 q^{25} +3.58430 q^{27} +6.50105 q^{29} +5.48817 q^{31} -3.76095 q^{35} -10.0179 q^{37} +8.84928 q^{39} +8.01787 q^{41} +0.906850 q^{43} -4.32440 q^{45} +5.52673 q^{47} +7.14476 q^{49} -10.5783 q^{51} +1.09131 q^{53} +11.7034 q^{57} -6.63406 q^{59} -9.02480 q^{61} +16.2638 q^{63} -3.26981 q^{65} +13.9684 q^{67} -17.3682 q^{69} -10.3730 q^{71} -3.68253 q^{73} +2.70636 q^{75} +7.27463 q^{79} -3.27279 q^{81} -6.11400 q^{83} +3.90869 q^{85} +17.5942 q^{87} -12.1209 q^{89} +12.2976 q^{91} +14.8530 q^{93} -4.32440 q^{95} -12.3393 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{13} - 2 q^{15} + q^{17} + 4 q^{19} + 14 q^{21} - 11 q^{23} + 4 q^{25} + 11 q^{27} + 4 q^{29} - q^{31} - 3 q^{35} - 17 q^{37} + 19 q^{39} + 9 q^{41} + 5 q^{43}+ \cdots + 8 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 + 3 * q^7 + 4 * q^9 + 3 * q^13 - 2 * q^15 + q^17 + 4 * q^19 + 14 * q^21 - 11 * q^23 + 4 * q^25 + 11 * q^27 + 4 * q^29 - q^31 - 3 * q^35 - 17 * q^37 + 19 * q^39 + 9 * q^41 + 5 * q^43 - 4 * q^45 + q^47 + 3 * q^49 + 3 * q^51 + 21 * q^53 + 17 * q^57 - 17 * q^59 + 28 * q^61 + 13 * q^63 - 3 * q^65 + 13 * q^67 - 8 * q^69 + 17 * q^71 + 13 * q^73 + 2 * q^75 + 22 * q^79 - 24 * q^81 - 21 * q^83 - q^85 + 47 * q^87 - 4 * q^89 - 4 * q^91 + 7 * q^93 - 4 * q^95 + 8 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.70636	1.56252	0.781259	−	0.624206i	$-0.214578\pi$
$3$	2.70636	1.56252	0.781259	+	0.624206i	$0.214578\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.76095	1.42151	0.710753	−	0.703442i	$-0.248354\pi$
$7$	3.76095	1.42151	0.710753	+	0.703442i	$0.248354\pi$
$8$	0	0
$9$	4.32440	1.44147
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.26981	0.906881	0.453441	−	0.891287i	$-0.350196\pi$
$13$	3.26981	0.906881	0.453441	+	0.891287i	$0.350196\pi$
$14$	0	0
$15$	−2.70636	−0.698780
$16$	0	0
$17$	−3.90869	−0.947997	−0.473999	−	0.880526i	$-0.657190\pi$
$17$	−3.90869	−0.947997	−0.473999	+	0.880526i	$0.657190\pi$
$18$	0	0
$19$	4.32440	0.992085	0.496042	−	0.868298i	$-0.334786\pi$
$19$	4.32440	0.992085	0.496042	+	0.868298i	$0.334786\pi$
$20$	0	0
$21$	10.1785	2.22113
$22$	0	0
$23$	−6.41755	−1.33815	−0.669075	−	0.743194i	$-0.733310\pi$
$23$	−6.41755	−1.33815	−0.669075	+	0.743194i	$0.733310\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.58430	0.689798
$28$	0	0
$29$	6.50105	1.20722	0.603608	−	0.797282i	$-0.293729\pi$
$29$	6.50105	1.20722	0.603608	+	0.797282i	$0.293729\pi$
$30$	0	0
$31$	5.48817	0.985704	0.492852	−	0.870113i	$-0.335954\pi$
$31$	5.48817	0.985704	0.492852	+	0.870113i	$0.335954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.76095	−0.635717
$36$	0	0
$37$	−10.0179	−1.64693	−0.823464	−	0.567369i	$-0.807961\pi$
$37$	−10.0179	−1.64693	−0.823464	+	0.567369i	$0.807961\pi$
$38$	0	0
$39$	8.84928	1.41702
$40$	0	0
$41$	8.01787	1.25218	0.626091	−	0.779750i	$-0.284654\pi$
$41$	8.01787	1.25218	0.626091	+	0.779750i	$0.284654\pi$
$42$	0	0
$43$	0.906850	0.138293	0.0691467	−	0.997607i	$-0.477972\pi$
$43$	0.906850	0.138293	0.0691467	+	0.997607i	$0.477972\pi$
$44$	0	0
$45$	−4.32440	−0.644643
$46$	0	0
$47$	5.52673	0.806156	0.403078	−	0.915166i	$-0.367940\pi$
$47$	5.52673	0.806156	0.403078	+	0.915166i	$0.367940\pi$
$48$	0	0
$49$	7.14476	1.02068
$50$	0	0
$51$	−10.5783	−1.48126
$52$	0	0
$53$	1.09131	0.149903	0.0749514	−	0.997187i	$-0.476120\pi$
$53$	1.09131	0.149903	0.0749514	+	0.997187i	$0.476120\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.7034	1.55015
$58$	0	0
$59$	−6.63406	−0.863682	−0.431841	−	0.901950i	$-0.642136\pi$
$59$	−6.63406	−0.863682	−0.431841	+	0.901950i	$0.642136\pi$
$60$	0	0
$61$	−9.02480	−1.15551	−0.577754	−	0.816211i	$-0.696071\pi$
$61$	−9.02480	−1.15551	−0.577754	+	0.816211i	$0.696071\pi$
$62$	0	0
$63$	16.2638	2.04905
$64$	0	0
$65$	−3.26981	−0.405570
$66$	0	0
$67$	13.9684	1.70651	0.853254	−	0.521496i	$-0.174626\pi$
$67$	13.9684	1.70651	0.853254	+	0.521496i	$0.174626\pi$
$68$	0	0
$69$	−17.3682	−2.09089
$70$	0	0
$71$	−10.3730	−1.23105	−0.615526	−	0.788117i	$-0.711056\pi$
$71$	−10.3730	−1.23105	−0.615526	+	0.788117i	$0.711056\pi$
$72$	0	0
$73$	−3.68253	−0.431008	−0.215504	−	0.976503i	$-0.569139\pi$
$73$	−3.68253	−0.431008	−0.215504	+	0.976503i	$0.569139\pi$
$74$	0	0
$75$	2.70636	0.312504
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.27463	0.818460	0.409230	−	0.912431i	$-0.365797\pi$
$79$	7.27463	0.818460	0.409230	+	0.912431i	$0.365797\pi$
$80$	0	0
$81$	−3.27279	−0.363643
$82$	0	0
$83$	−6.11400	−0.671099	−0.335549	−	0.942023i	$-0.608922\pi$
$83$	−6.11400	−0.671099	−0.335549	+	0.942023i	$0.608922\pi$
$84$	0	0
$85$	3.90869	0.423957
$86$	0	0
$87$	17.5942	1.88630
$88$	0	0
$89$	−12.1209	−1.28482	−0.642408	−	0.766363i	$-0.722064\pi$
$89$	−12.1209	−1.28482	−0.642408	+	0.766363i	$0.722064\pi$
$90$	0	0
$91$	12.2976	1.28914
$92$	0	0
$93$	14.8530	1.54018
$94$	0	0
$95$	−4.32440	−0.443674
$96$	0	0
$97$	−12.3393	−1.25286	−0.626432	−	0.779476i	$-0.715486\pi$
$97$	−12.3393	−1.25286	−0.626432	+	0.779476i	$0.715486\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.0832431	0.00828300	0.00414150	−	0.999991i	$-0.498682\pi$
$101$	0.0832431	0.00828300	0.00414150	+	0.999991i	$0.498682\pi$
$102$	0	0
$103$	11.1032	1.09403	0.547016	−	0.837122i	$-0.315764\pi$
$103$	11.1032	1.09403	0.547016	+	0.837122i	$0.315764\pi$
$104$	0	0
$105$	−10.1785	−0.993320
$106$	0	0
$107$	−12.8690	−1.24409	−0.622046	−	0.782980i	$-0.713698\pi$
$107$	−12.8690	−1.24409	−0.622046	+	0.782980i	$0.713698\pi$
$108$	0	0
$109$	7.51183	0.719503	0.359752	−	0.933048i	$-0.382861\pi$
$109$	7.51183	0.719503	0.359752	+	0.933048i	$0.382861\pi$
$110$	0	0
$111$	−27.1120	−2.57336
$112$	0	0
$113$	0.467314	0.0439612	0.0219806	−	0.999758i	$-0.493003\pi$
$113$	0.467314	0.0439612	0.0219806	+	0.999758i	$0.493003\pi$
$114$	0	0
$115$	6.41755	0.598439
$116$	0	0
$117$	14.1399	1.30724
$118$	0	0
$119$	−14.7004	−1.34758
$120$	0	0
$121$	0	0
$122$	0	0
$123$	21.6993	1.95656
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.0516	1.60182	0.800911	−	0.598784i	$-0.204349\pi$
$127$	18.0516	1.60182	0.800911	+	0.598784i	$0.204349\pi$
$128$	0	0
$129$	2.45426	0.216086
$130$	0	0
$131$	6.48010	0.566169	0.283085	−	0.959095i	$-0.408642\pi$
$131$	6.48010	0.566169	0.283085	+	0.959095i	$0.408642\pi$
$132$	0	0
$133$	16.2638	1.41025
$134$	0	0
$135$	−3.58430	−0.308487
$136$	0	0
$137$	1.93761	0.165541	0.0827705	−	0.996569i	$-0.473623\pi$
$137$	1.93761	0.165541	0.0827705	+	0.996569i	$0.473623\pi$
$138$	0	0
$139$	13.7758	1.16845	0.584226	−	0.811591i	$-0.301398\pi$
$139$	13.7758	1.16845	0.584226	+	0.811591i	$0.301398\pi$
$140$	0	0
$141$	14.9573	1.25963
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.50105	−0.539883
$146$	0	0
$147$	19.3363	1.59483
$148$	0	0
$149$	9.68865	0.793725	0.396863	−	0.917878i	$-0.370099\pi$
$149$	9.68865	0.793725	0.396863	+	0.917878i	$0.370099\pi$
$150$	0	0
$151$	−16.6447	−1.35452	−0.677262	−	0.735742i	$-0.736834\pi$
$151$	−16.6447	−1.35452	−0.677262	+	0.735742i	$0.736834\pi$
$152$	0	0
$153$	−16.9027	−1.36650
$154$	0	0
$155$	−5.48817	−0.440820
$156$	0	0
$157$	0.958197	0.0764724	0.0382362	−	0.999269i	$-0.487826\pi$
$157$	0.958197	0.0764724	0.0382362	+	0.999269i	$0.487826\pi$
$158$	0	0
$159$	2.95348	0.234226
$160$	0	0
$161$	−24.1361	−1.90219
$162$	0	0
$163$	−4.44532	−0.348185	−0.174092	−	0.984729i	$-0.555699\pi$
$163$	−4.44532	−0.348185	−0.174092	+	0.984729i	$0.555699\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.9494	1.07943	0.539717	−	0.841847i	$-0.318531\pi$
$167$	13.9494	1.07943	0.539717	+	0.841847i	$0.318531\pi$
$168$	0	0
$169$	−2.30837	−0.177567
$170$	0	0
$171$	18.7004	1.43006
$172$	0	0
$173$	−16.1640	−1.22893	−0.614464	−	0.788945i	$-0.710628\pi$
$173$	−16.1640	−1.22893	−0.614464	+	0.788945i	$0.710628\pi$
$174$	0	0
$175$	3.76095	0.284301
$176$	0	0
$177$	−17.9542	−1.34952
$178$	0	0
$179$	18.3810	1.37386	0.686930	−	0.726724i	$-0.258958\pi$
$179$	18.3810	1.37386	0.686930	+	0.726724i	$0.258958\pi$
$180$	0	0
$181$	11.0775	0.823388	0.411694	−	0.911322i	$-0.364937\pi$
$181$	11.0775	0.823388	0.411694	+	0.911322i	$0.364937\pi$
$182$	0	0
$183$	−24.4244	−1.80550
$184$	0	0
$185$	10.0179	0.736529
$186$	0	0
$187$	0	0
$188$	0	0
$189$	13.4804	0.980552
$190$	0	0
$191$	9.61874	0.695987	0.347994	−	0.937497i	$-0.386863\pi$
$191$	9.61874	0.695987	0.347994	+	0.937497i	$0.386863\pi$
$192$	0	0
$193$	−19.6557	−1.41485	−0.707425	−	0.706789i	$-0.750143\pi$
$193$	−19.6557	−1.41485	−0.707425	+	0.706789i	$0.750143\pi$
$194$	0	0
$195$	−8.84928	−0.633710
$196$	0	0
$197$	−12.1768	−0.867562	−0.433781	−	0.901018i	$-0.642821\pi$
$197$	−12.1768	−0.867562	−0.433781	+	0.901018i	$0.642821\pi$
$198$	0	0
$199$	8.44164	0.598412	0.299206	−	0.954189i	$-0.403278\pi$
$199$	8.44164	0.598412	0.299206	+	0.954189i	$0.403278\pi$
$200$	0	0
$201$	37.8035	2.66645
$202$	0	0
$203$	24.4501	1.71606
$204$	0	0
$205$	−8.01787	−0.559992
$206$	0	0
$207$	−27.7520	−1.92890
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−21.8532	−1.50444	−0.752219	−	0.658913i	$-0.771016\pi$
$211$	−21.8532	−1.50444	−0.752219	+	0.658913i	$0.771016\pi$
$212$	0	0
$213$	−28.0732	−1.92354
$214$	0	0
$215$	−0.906850	−0.0618467
$216$	0	0
$217$	20.6407	1.40118
$218$	0	0
$219$	−9.96626	−0.673458
$220$	0	0
$221$	−12.7807	−0.859721
$222$	0	0
$223$	−26.4245	−1.76951	−0.884757	−	0.466053i	$-0.845676\pi$
$223$	−26.4245	−1.76951	−0.884757	+	0.466053i	$0.845676\pi$
$224$	0	0
$225$	4.32440	0.288293
$226$	0	0
$227$	9.91876	0.658331	0.329166	−	0.944272i	$-0.393233\pi$
$227$	9.91876	0.658331	0.329166	+	0.944272i	$0.393233\pi$
$228$	0	0
$229$	−12.9812	−0.857819	−0.428909	−	0.903348i	$-0.641102\pi$
$229$	−12.9812	−0.857819	−0.428909	+	0.903348i	$0.641102\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.65984	−0.436300	−0.218150	−	0.975915i	$-0.570002\pi$
$233$	−6.65984	−0.436300	−0.218150	+	0.975915i	$0.570002\pi$
$234$	0	0
$235$	−5.52673	−0.360524
$236$	0	0
$237$	19.6878	1.27886
$238$	0	0
$239$	−21.8918	−1.41606	−0.708031	−	0.706181i	$-0.750416\pi$
$239$	−21.8918	−1.41606	−0.708031	+	0.706181i	$0.750416\pi$
$240$	0	0
$241$	10.6517	0.686134	0.343067	−	0.939311i	$-0.388534\pi$
$241$	10.6517	0.686134	0.343067	+	0.939311i	$0.388534\pi$
$242$	0	0
$243$	−19.6102	−1.25800
$244$	0	0
$245$	−7.14476	−0.456462
$246$	0	0
$247$	14.1399	0.899703
$248$	0	0
$249$	−16.5467	−1.04860
$250$	0	0
$251$	−12.1667	−0.767958	−0.383979	−	0.923342i	$-0.625447\pi$
$251$	−12.1667	−0.767958	−0.383979	+	0.923342i	$0.625447\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	10.5783	0.662441
$256$	0	0
$257$	6.11400	0.381381	0.190690	−	0.981650i	$-0.438927\pi$
$257$	6.11400	0.381381	0.190690	+	0.981650i	$0.438927\pi$
$258$	0	0
$259$	−37.6767	−2.34112
$260$	0	0
$261$	28.1131	1.74016
$262$	0	0
$263$	0.446463	0.0275301	0.0137650	−	0.999905i	$-0.495618\pi$
$263$	0.446463	0.0275301	0.0137650	+	0.999905i	$0.495618\pi$
$264$	0	0
$265$	−1.09131	−0.0670385
$266$	0	0
$267$	−32.8036	−2.00755
$268$	0	0
$269$	0.00796455	0.000485607 0	0.000242803	−	1.00000i	$-0.499923\pi$
$269$	0.00796455	0.000485607 0	0.000242803		1.00000i	$0.499923\pi$
$270$	0	0
$271$	18.5880	1.12914	0.564570	−	0.825385i	$-0.309042\pi$
$271$	18.5880	1.12914	0.564570	+	0.825385i	$0.309042\pi$
$272$	0	0
$273$	33.2817	2.01430
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.95960	0.418162	0.209081	−	0.977898i	$-0.432953\pi$
$277$	6.95960	0.418162	0.209081	+	0.977898i	$0.432953\pi$
$278$	0	0
$279$	23.7330	1.42086
$280$	0	0
$281$	−22.3622	−1.33402	−0.667010	−	0.745049i	$-0.732426\pi$
$281$	−22.3622	−1.33402	−0.667010	+	0.745049i	$0.732426\pi$
$282$	0	0
$283$	−23.3482	−1.38791	−0.693954	−	0.720019i	$-0.744133\pi$
$283$	−23.3482	−1.38791	−0.693954	+	0.720019i	$0.744133\pi$
$284$	0	0
$285$	−11.7034	−0.693249
$286$	0	0
$287$	30.1548	1.77998
$288$	0	0
$289$	−1.72213	−0.101302
$290$	0	0
$291$	−33.3946	−1.95762
$292$	0	0
$293$	3.50122	0.204543	0.102272	−	0.994757i	$-0.467389\pi$
$293$	3.50122	0.204543	0.102272	+	0.994757i	$0.467389\pi$
$294$	0	0
$295$	6.63406	0.386250
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.9841	−1.21354
$300$	0	0
$301$	3.41062	0.196585
$302$	0	0
$303$	0.225286	0.0129423
$304$	0	0
$305$	9.02480	0.516758
$306$	0	0
$307$	−8.26358	−0.471628	−0.235814	−	0.971798i	$-0.575776\pi$
$307$	−8.26358	−0.471628	−0.235814	+	0.971798i	$0.575776\pi$
$308$	0	0
$309$	30.0493	1.70945
$310$	0	0
$311$	11.5621	0.655629	0.327814	−	0.944742i	$-0.393688\pi$
$311$	11.5621	0.655629	0.327814	+	0.944742i	$0.393688\pi$
$312$	0	0
$313$	−19.3148	−1.09173	−0.545867	−	0.837872i	$-0.683800\pi$
$313$	−19.3148	−1.09173	−0.545867	+	0.837872i	$0.683800\pi$
$314$	0	0
$315$	−16.2638	−0.916364
$316$	0	0
$317$	22.3736	1.25662	0.628312	−	0.777961i	$-0.283746\pi$
$317$	22.3736	1.25662	0.628312	+	0.777961i	$0.283746\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−34.8282	−1.94392
$322$	0	0
$323$	−16.9027	−0.940493
$324$	0	0
$325$	3.26981	0.181376
$326$	0	0
$327$	20.3297	1.12424
$328$	0	0
$329$	20.7858	1.14596
$330$	0	0
$331$	−5.17876	−0.284650	−0.142325	−	0.989820i	$-0.545458\pi$
$331$	−5.17876	−0.284650	−0.142325	+	0.989820i	$0.545458\pi$
$332$	0	0
$333$	−43.3212	−2.37399
$334$	0	0
$335$	−13.9684	−0.763173
$336$	0	0
$337$	5.52656	0.301051	0.150526	−	0.988606i	$-0.451903\pi$
$337$	5.52656	0.301051	0.150526	+	0.988606i	$0.451903\pi$
$338$	0	0
$339$	1.26472	0.0686902
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.544434	0.0293967
$344$	0	0
$345$	17.3682	0.935073
$346$	0	0
$347$	8.58798	0.461027	0.230513	−	0.973069i	$-0.425959\pi$
$347$	8.58798	0.461027	0.230513	+	0.973069i	$0.425959\pi$
$348$	0	0
$349$	14.6729	0.785422	0.392711	−	0.919662i	$-0.371537\pi$
$349$	14.6729	0.785422	0.392711	+	0.919662i	$0.371537\pi$
$350$	0	0
$351$	11.7200	0.625565
$352$	0	0
$353$	5.79041	0.308192	0.154096	−	0.988056i	$-0.450753\pi$
$353$	5.79041	0.308192	0.154096	+	0.988056i	$0.450753\pi$
$354$	0	0
$355$	10.3730	0.550543
$356$	0	0
$357$	−39.7846	−2.10562
$358$	0	0
$359$	−11.3630	−0.599714	−0.299857	−	0.953984i	$-0.596939\pi$
$359$	−11.3630	−0.599714	−0.299857	+	0.953984i	$0.596939\pi$
$360$	0	0
$361$	−0.299598	−0.0157683
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.68253	0.192752
$366$	0	0
$367$	−4.58446	−0.239307	−0.119653	−	0.992816i	$-0.538178\pi$
$367$	−4.58446	−0.239307	−0.119653	+	0.992816i	$0.538178\pi$
$368$	0	0
$369$	34.6725	1.80498
$370$	0	0
$371$	4.10436	0.213088
$372$	0	0
$373$	−22.8487	−1.18306	−0.591530	−	0.806283i	$-0.701476\pi$
$373$	−22.8487	−1.18306	−0.591530	+	0.806283i	$0.701476\pi$
$374$	0	0
$375$	−2.70636	−0.139756
$376$	0	0
$377$	21.2572	1.09480
$378$	0	0
$379$	−21.0506	−1.08130	−0.540648	−	0.841249i	$-0.681821\pi$
$379$	−21.0506	−1.08130	−0.540648	+	0.841249i	$0.681821\pi$
$380$	0	0
$381$	48.8542	2.50288
$382$	0	0
$383$	−0.259899	−0.0132802	−0.00664012	−	0.999978i	$-0.502114\pi$
$383$	−0.259899	−0.0132802	−0.00664012	+	0.999978i	$0.502114\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.92158	0.199345
$388$	0	0
$389$	−15.7617	−0.799148	−0.399574	−	0.916701i	$-0.630842\pi$
$389$	−15.7617	−0.799148	−0.399574	+	0.916701i	$0.630842\pi$
$390$	0	0
$391$	25.0842	1.26856
$392$	0	0
$393$	17.5375	0.884650
$394$	0	0
$395$	−7.27463	−0.366026
$396$	0	0
$397$	7.62978	0.382928	0.191464	−	0.981500i	$-0.438677\pi$
$397$	7.62978	0.382928	0.191464	+	0.981500i	$0.438677\pi$
$398$	0	0
$399$	44.0159	2.20355
$400$	0	0
$401$	−30.1505	−1.50565	−0.752823	−	0.658223i	$-0.771309\pi$
$401$	−30.1505	−1.50565	−0.752823	+	0.658223i	$0.771309\pi$
$402$	0	0
$403$	17.9452	0.893916
$404$	0	0
$405$	3.27279	0.162626
$406$	0	0
$407$	0	0
$408$	0	0
$409$	31.1480	1.54017	0.770085	−	0.637942i	$-0.220214\pi$
$409$	31.1480	1.54017	0.770085	+	0.637942i	$0.220214\pi$
$410$	0	0
$411$	5.24387	0.258661
$412$	0	0
$413$	−24.9504	−1.22773
$414$	0	0
$415$	6.11400	0.300125
$416$	0	0
$417$	37.2824	1.82573
$418$	0	0
$419$	13.3733	0.653328	0.326664	−	0.945141i	$-0.394075\pi$
$419$	13.3733	0.653328	0.326664	+	0.945141i	$0.394075\pi$
$420$	0	0
$421$	−10.9730	−0.534793	−0.267396	−	0.963587i	$-0.586163\pi$
$421$	−10.9730	−0.534793	−0.267396	+	0.963587i	$0.586163\pi$
$422$	0	0
$423$	23.8998	1.16205
$424$	0	0
$425$	−3.90869	−0.189599
$426$	0	0
$427$	−33.9418	−1.64256
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.2159	−0.781093	−0.390547	−	0.920583i	$-0.627714\pi$
$431$	−16.2159	−0.781093	−0.390547	+	0.920583i	$0.627714\pi$
$432$	0	0
$433$	−21.4642	−1.03150	−0.515751	−	0.856739i	$-0.672487\pi$
$433$	−21.4642	−1.03150	−0.515751	+	0.856739i	$0.672487\pi$
$434$	0	0
$435$	−17.5942	−0.843577
$436$	0	0
$437$	−27.7520	−1.32756
$438$	0	0
$439$	5.94741	0.283855	0.141927	−	0.989877i	$-0.454670\pi$
$439$	5.94741	0.283855	0.141927	+	0.989877i	$0.454670\pi$
$440$	0	0
$441$	30.8968	1.47127
$442$	0	0
$443$	−23.7410	−1.12797	−0.563984	−	0.825786i	$-0.690732\pi$
$443$	−23.7410	−1.12797	−0.563984	+	0.825786i	$0.690732\pi$
$444$	0	0
$445$	12.1209	0.574587
$446$	0	0
$447$	26.2210	1.24021
$448$	0	0
$449$	34.4170	1.62424	0.812119	−	0.583492i	$-0.198314\pi$
$449$	34.4170	1.62424	0.812119	+	0.583492i	$0.198314\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−45.0465	−2.11647
$454$	0	0
$455$	−12.2976	−0.576520
$456$	0	0
$457$	14.8301	0.693723	0.346862	−	0.937916i	$-0.387247\pi$
$457$	14.8301	0.693723	0.346862	+	0.937916i	$0.387247\pi$
$458$	0	0
$459$	−14.0099	−0.653926
$460$	0	0
$461$	29.8441	1.38998	0.694990	−	0.719020i	$-0.255409\pi$
$461$	29.8441	1.38998	0.694990	+	0.719020i	$0.255409\pi$
$462$	0	0
$463$	−6.87337	−0.319433	−0.159716	−	0.987163i	$-0.551058\pi$
$463$	−6.87337	−0.319433	−0.159716	+	0.987163i	$0.551058\pi$
$464$	0	0
$465$	−14.8530	−0.688790
$466$	0	0
$467$	−1.40932	−0.0652155	−0.0326077	−	0.999468i	$-0.510381\pi$
$467$	−1.40932	−0.0652155	−0.0326077	+	0.999468i	$0.510381\pi$
$468$	0	0
$469$	52.5344	2.42581
$470$	0	0
$471$	2.59323	0.119490
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.32440	0.198417
$476$	0	0
$477$	4.71925	0.216080
$478$	0	0
$479$	−23.2691	−1.06319	−0.531596	−	0.846998i	$-0.678408\pi$
$479$	−23.2691	−1.06319	−0.531596	+	0.846998i	$0.678408\pi$
$480$	0	0
$481$	−32.7565	−1.49357
$482$	0	0
$483$	−65.3210	−2.97221
$484$	0	0
$485$	12.3393	0.560298
$486$	0	0
$487$	18.6543	0.845308	0.422654	−	0.906291i	$-0.361099\pi$
$487$	18.6543	0.845308	0.422654	+	0.906291i	$0.361099\pi$
$488$	0	0
$489$	−12.0307	−0.544045
$490$	0	0
$491$	−42.0138	−1.89605	−0.948027	−	0.318190i	$-0.896925\pi$
$491$	−42.0138	−1.89605	−0.948027	+	0.318190i	$0.896925\pi$
$492$	0	0
$493$	−25.4106	−1.14444
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.0125	−1.74995
$498$	0	0
$499$	−5.22231	−0.233783	−0.116891	−	0.993145i	$-0.537293\pi$
$499$	−5.22231	−0.233783	−0.116891	+	0.993145i	$0.537293\pi$
$500$	0	0
$501$	37.7520	1.68664
$502$	0	0
$503$	28.8361	1.28574	0.642868	−	0.765977i	$-0.277744\pi$
$503$	28.8361	1.28574	0.642868	+	0.765977i	$0.277744\pi$
$504$	0	0
$505$	−0.0832431	−0.00370427
$506$	0	0
$507$	−6.24728	−0.277451
$508$	0	0
$509$	5.52488	0.244886	0.122443	−	0.992476i	$-0.460927\pi$
$509$	5.52488	0.244886	0.122443	+	0.992476i	$0.460927\pi$
$510$	0	0
$511$	−13.8498	−0.612680
$512$	0	0
$513$	15.4999	0.684338
$514$	0	0
$515$	−11.1032	−0.489266
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−43.7457	−1.92022
$520$	0	0
$521$	−29.4551	−1.29045	−0.645226	−	0.763991i	$-0.723237\pi$
$521$	−29.4551	−1.29045	−0.645226	+	0.763991i	$0.723237\pi$
$522$	0	0
$523$	−2.21539	−0.0968722	−0.0484361	−	0.998826i	$-0.515424\pi$
$523$	−2.21539	−0.0968722	−0.0484361	+	0.998826i	$0.515424\pi$
$524$	0	0
$525$	10.1785	0.444226
$526$	0	0
$527$	−21.4515	−0.934444
$528$	0	0
$529$	18.1849	0.790648
$530$	0	0
$531$	−28.6883	−1.24497
$532$	0	0
$533$	26.2169	1.13558
$534$	0	0
$535$	12.8690	0.556375
$536$	0	0
$537$	49.7456	2.14668
$538$	0	0
$539$	0	0
$540$	0	0
$541$	33.7116	1.44938	0.724688	−	0.689077i	$-0.241984\pi$
$541$	33.7116	1.44938	0.724688	+	0.689077i	$0.241984\pi$
$542$	0	0
$543$	29.9799	1.28656
$544$	0	0
$545$	−7.51183	−0.321772
$546$	0	0
$547$	−28.0701	−1.20019	−0.600096	−	0.799928i	$-0.704871\pi$
$547$	−28.0701	−1.20019	−0.600096	+	0.799928i	$0.704871\pi$
$548$	0	0
$549$	−39.0268	−1.66562
$550$	0	0
$551$	28.1131	1.19766
$552$	0	0
$553$	27.3595	1.16345
$554$	0	0
$555$	27.1120	1.15084
$556$	0	0
$557$	15.5191	0.657565	0.328782	−	0.944406i	$-0.393362\pi$
$557$	15.5191	0.657565	0.328782	+	0.944406i	$0.393362\pi$
$558$	0	0
$559$	2.96522	0.125416
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.1441	−0.848971	−0.424485	−	0.905435i	$-0.639545\pi$
$563$	−20.1441	−0.848971	−0.424485	+	0.905435i	$0.639545\pi$
$564$	0	0
$565$	−0.467314	−0.0196601
$566$	0	0
$567$	−12.3088	−0.516921
$568$	0	0
$569$	31.4731	1.31942	0.659711	−	0.751520i	$-0.270679\pi$
$569$	31.4731	1.31942	0.659711	+	0.751520i	$0.270679\pi$
$570$	0	0
$571$	−14.3295	−0.599673	−0.299836	−	0.953991i	$-0.596932\pi$
$571$	−14.3295	−0.599673	−0.299836	+	0.953991i	$0.596932\pi$
$572$	0	0
$573$	26.0318	1.08749
$574$	0	0
$575$	−6.41755	−0.267630
$576$	0	0
$577$	−11.4614	−0.477142	−0.238571	−	0.971125i	$-0.576679\pi$
$577$	−11.4614	−0.477142	−0.238571	+	0.971125i	$0.576679\pi$
$578$	0	0
$579$	−53.1955	−2.21073
$580$	0	0
$581$	−22.9945	−0.953971
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−14.1399	−0.584614
$586$	0	0
$587$	9.42183	0.388880	0.194440	−	0.980914i	$-0.437711\pi$
$587$	9.42183	0.388880	0.194440	+	0.980914i	$0.437711\pi$
$588$	0	0
$589$	23.7330	0.977901
$590$	0	0
$591$	−32.9549	−1.35558
$592$	0	0
$593$	−28.2270	−1.15914	−0.579571	−	0.814922i	$-0.696780\pi$
$593$	−28.2270	−1.15914	−0.579571	+	0.814922i	$0.696780\pi$
$594$	0	0
$595$	14.7004	0.602658
$596$	0	0
$597$	22.8461	0.935030
$598$	0	0
$599$	4.03320	0.164792	0.0823960	−	0.996600i	$-0.473743\pi$
$599$	4.03320	0.164792	0.0823960	+	0.996600i	$0.473743\pi$
$600$	0	0
$601$	37.1844	1.51679	0.758393	−	0.651798i	$-0.225985\pi$
$601$	37.1844	1.51679	0.758393	+	0.651798i	$0.225985\pi$
$602$	0	0
$603$	60.4048	2.45987
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.1300	−0.735872	−0.367936	−	0.929851i	$-0.619936\pi$
$607$	−18.1300	−0.735872	−0.367936	+	0.929851i	$0.619936\pi$
$608$	0	0
$609$	66.1710	2.68138
$610$	0	0
$611$	18.0713	0.731087
$612$	0	0
$613$	−8.53512	−0.344730	−0.172365	−	0.985033i	$-0.555141\pi$
$613$	−8.53512	−0.344730	−0.172365	+	0.985033i	$0.555141\pi$
$614$	0	0
$615$	−21.6993	−0.874999
$616$	0	0
$617$	−24.5659	−0.988985	−0.494492	−	0.869182i	$-0.664646\pi$
$617$	−24.5659	−0.988985	−0.494492	+	0.869182i	$0.664646\pi$
$618$	0	0
$619$	−38.2737	−1.53835	−0.769174	−	0.639039i	$-0.779332\pi$
$619$	−38.2737	−1.53835	−0.769174	+	0.639039i	$0.779332\pi$
$620$	0	0
$621$	−23.0024	−0.923054
$622$	0	0
$623$	−45.5862	−1.82637
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	39.1568	1.56128
$630$	0	0
$631$	20.6926	0.823759	0.411880	−	0.911238i	$-0.364872\pi$
$631$	20.6926	0.823759	0.411880	+	0.911238i	$0.364872\pi$
$632$	0	0
$633$	−59.1427	−2.35071
$634$	0	0
$635$	−18.0516	−0.716356
$636$	0	0
$637$	23.3620	0.925635
$638$	0	0
$639$	−44.8571	−1.77452
$640$	0	0
$641$	−8.35542	−0.330019	−0.165010	−	0.986292i	$-0.552766\pi$
$641$	−8.35542	−0.330019	−0.165010	+	0.986292i	$0.552766\pi$
$642$	0	0
$643$	29.4013	1.15948	0.579738	−	0.814803i	$-0.303155\pi$
$643$	29.4013	1.15948	0.579738	+	0.814803i	$0.303155\pi$
$644$	0	0
$645$	−2.45426	−0.0966366
$646$	0	0
$647$	23.4990	0.923843	0.461921	−	0.886921i	$-0.347160\pi$
$647$	23.4990	0.923843	0.461921	+	0.886921i	$0.347160\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	55.8613	2.18938
$652$	0	0
$653$	15.4179	0.603348	0.301674	−	0.953411i	$-0.402455\pi$
$653$	15.4179	0.603348	0.301674	+	0.953411i	$0.402455\pi$
$654$	0	0
$655$	−6.48010	−0.253198
$656$	0	0
$657$	−15.9247	−0.621283
$658$	0	0
$659$	12.2785	0.478301	0.239151	−	0.970982i	$-0.423131\pi$
$659$	12.2785	0.478301	0.239151	+	0.970982i	$0.423131\pi$
$660$	0	0
$661$	25.8163	1.00414	0.502070	−	0.864827i	$-0.332572\pi$
$661$	25.8163	1.00414	0.502070	+	0.864827i	$0.332572\pi$
$662$	0	0
$663$	−34.5891	−1.34333
$664$	0	0
$665$	−16.2638	−0.630685
$666$	0	0
$667$	−41.7208	−1.61544
$668$	0	0
$669$	−71.5142	−2.76490
$670$	0	0
$671$	0	0
$672$	0	0
$673$	32.3331	1.24635	0.623174	−	0.782083i	$-0.285843\pi$
$673$	32.3331	1.24635	0.623174	+	0.782083i	$0.285843\pi$
$674$	0	0
$675$	3.58430	0.137960
$676$	0	0
$677$	−31.4919	−1.21033	−0.605167	−	0.796099i	$-0.706894\pi$
$677$	−31.4919	−1.21033	−0.605167	+	0.796099i	$0.706894\pi$
$678$	0	0
$679$	−46.4075	−1.78096
$680$	0	0
$681$	26.8438	1.02866
$682$	0	0
$683$	−13.3758	−0.511812	−0.255906	−	0.966702i	$-0.582374\pi$
$683$	−13.3758	−0.511812	−0.255906	+	0.966702i	$0.582374\pi$
$684$	0	0
$685$	−1.93761	−0.0740322
$686$	0	0
$687$	−35.1317	−1.34036
$688$	0	0
$689$	3.56837	0.135944
$690$	0	0
$691$	−27.4503	−1.04426	−0.522129	−	0.852866i	$-0.674862\pi$
$691$	−27.4503	−1.04426	−0.522129	+	0.852866i	$0.674862\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.7758	−0.522548
$696$	0	0
$697$	−31.3394	−1.18706
$698$	0	0
$699$	−18.0239	−0.681728
$700$	0	0
$701$	−0.216417	−0.00817395	−0.00408698	−	0.999992i	$-0.501301\pi$
$701$	−0.216417	−0.00817395	−0.00408698	+	0.999992i	$0.501301\pi$
$702$	0	0
$703$	−43.3212	−1.63389
$704$	0	0
$705$	−14.9573	−0.563325
$706$	0	0
$707$	0.313073	0.0117743
$708$	0	0
$709$	−50.7909	−1.90749	−0.953746	−	0.300613i	$-0.902809\pi$
$709$	−50.7909	−1.90749	−0.953746	+	0.300613i	$0.902809\pi$
$710$	0	0
$711$	31.4584	1.17978
$712$	0	0
$713$	−35.2206	−1.31902
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−59.2471	−2.21262
$718$	0	0
$719$	−34.0411	−1.26952	−0.634759	−	0.772710i	$-0.718901\pi$
$719$	−34.0411	−1.26952	−0.634759	+	0.772710i	$0.718901\pi$
$720$	0	0
$721$	41.7587	1.55517
$722$	0	0
$723$	28.8273	1.07210
$724$	0	0
$725$	6.50105	0.241443
$726$	0	0
$727$	16.2361	0.602162	0.301081	−	0.953599i	$-0.402652\pi$
$727$	16.2361	0.602162	0.301081	+	0.953599i	$0.402652\pi$
$728$	0	0
$729$	−43.2540	−1.60200
$730$	0	0
$731$	−3.54460	−0.131102
$732$	0	0
$733$	−29.7989	−1.10065	−0.550324	−	0.834951i	$-0.685496\pi$
$733$	−29.7989	−1.10065	−0.550324	+	0.834951i	$0.685496\pi$
$734$	0	0
$735$	−19.3363	−0.713230
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−42.2471	−1.55408	−0.777042	−	0.629448i	$-0.783281\pi$
$739$	−42.2471	−1.55408	−0.777042	+	0.629448i	$0.783281\pi$
$740$	0	0
$741$	38.2678	1.40580
$742$	0	0
$743$	20.9125	0.767207	0.383603	−	0.923498i	$-0.374683\pi$
$743$	20.9125	0.767207	0.383603	+	0.923498i	$0.374683\pi$
$744$	0	0
$745$	−9.68865	−0.354965
$746$	0	0
$747$	−26.4394	−0.967366
$748$	0	0
$749$	−48.3997	−1.76849
$750$	0	0
$751$	28.4428	1.03789	0.518946	−	0.854807i	$-0.326325\pi$
$751$	28.4428	1.03789	0.518946	+	0.854807i	$0.326325\pi$
$752$	0	0
$753$	−32.9276	−1.19995
$754$	0	0
$755$	16.6447	0.605762
$756$	0	0
$757$	−39.6707	−1.44186	−0.720928	−	0.693010i	$-0.756284\pi$
$757$	−39.6707	−1.44186	−0.720928	+	0.693010i	$0.756284\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−48.7803	−1.76829	−0.884143	−	0.467217i	$-0.845257\pi$
$761$	−48.7803	−1.76829	−0.884143	+	0.467217i	$0.845257\pi$
$762$	0	0
$763$	28.2516	1.02278
$764$	0	0
$765$	16.9027	0.611120
$766$	0	0
$767$	−21.6921	−0.783256
$768$	0	0
$769$	−10.7632	−0.388132	−0.194066	−	0.980988i	$-0.562168\pi$
$769$	−10.7632	−0.388132	−0.194066	+	0.980988i	$0.562168\pi$
$770$	0	0
$771$	16.5467	0.595915
$772$	0	0
$773$	24.4695	0.880106	0.440053	−	0.897972i	$-0.354960\pi$
$773$	24.4695	0.880106	0.440053	+	0.897972i	$0.354960\pi$
$774$	0	0
$775$	5.48817	0.197141
$776$	0	0
$777$	−101.967	−3.65804
$778$	0	0
$779$	34.6725	1.24227
$780$	0	0
$781$	0	0
$782$	0	0
$783$	23.3017	0.832735
$784$	0	0
$785$	−0.958197	−0.0341995
$786$	0	0
$787$	−4.85410	−0.173030	−0.0865150	−	0.996251i	$-0.527573\pi$
$787$	−4.85410	−0.173030	−0.0865150	+	0.996251i	$0.527573\pi$
$788$	0	0
$789$	1.20829	0.0430163
$790$	0	0
$791$	1.75755	0.0624911
$792$	0	0
$793$	−29.5093	−1.04791
$794$	0	0
$795$	−2.95348	−0.104749
$796$	0	0
$797$	−4.60683	−0.163182	−0.0815911	−	0.996666i	$-0.526000\pi$
$797$	−4.60683	−0.163182	−0.0815911	+	0.996666i	$0.526000\pi$
$798$	0	0
$799$	−21.6023	−0.764233
$800$	0	0
$801$	−52.4157	−1.85202
$802$	0	0
$803$	0	0
$804$	0	0
$805$	24.1361	0.850685
$806$	0	0
$807$	0.0215549	0.000758770 0
$808$	0	0
$809$	31.6970	1.11441	0.557203	−	0.830376i	$-0.311874\pi$
$809$	31.6970	1.11441	0.557203	+	0.830376i	$0.311874\pi$
$810$	0	0
$811$	14.4303	0.506715	0.253358	−	0.967373i	$-0.418465\pi$
$811$	14.4303	0.506715	0.253358	+	0.967373i	$0.418465\pi$
$812$	0	0
$813$	50.3058	1.76430
$814$	0	0
$815$	4.44532	0.155713
$816$	0	0
$817$	3.92158	0.137199
$818$	0	0
$819$	53.1796	1.85825
$820$	0	0
$821$	54.4551	1.90050	0.950249	−	0.311493i	$-0.100829\pi$
$821$	54.4551	1.90050	0.950249	+	0.311493i	$0.100829\pi$
$822$	0	0
$823$	0.0607126	0.00211631	0.00105815	−	0.999999i	$-0.499663\pi$
$823$	0.0607126	0.00211631	0.00105815	+	0.999999i	$0.499663\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.8274	−0.411279	−0.205639	−	0.978628i	$-0.565927\pi$
$827$	−11.8274	−0.411279	−0.205639	+	0.978628i	$0.565927\pi$
$828$	0	0
$829$	16.7497	0.581743	0.290871	−	0.956762i	$-0.406055\pi$
$829$	16.7497	0.581743	0.290871	+	0.956762i	$0.406055\pi$
$830$	0	0
$831$	18.8352	0.653386
$832$	0	0
$833$	−27.9267	−0.967602
$834$	0	0
$835$	−13.9494	−0.482737
$836$	0	0
$837$	19.6712	0.679936
$838$	0	0
$839$	−0.907817	−0.0313413	−0.0156707	−	0.999877i	$-0.504988\pi$
$839$	−0.907817	−0.0313413	−0.0156707	+	0.999877i	$0.504988\pi$
$840$	0	0
$841$	13.2637	0.457368
$842$	0	0
$843$	−60.5203	−2.08443
$844$	0	0
$845$	2.30837	0.0794102
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−63.1888	−2.16863
$850$	0	0
$851$	64.2902	2.20384
$852$	0	0
$853$	53.2766	1.82415	0.912077	−	0.410018i	$-0.134478\pi$
$853$	53.2766	1.82415	0.912077	+	0.410018i	$0.134478\pi$
$854$	0	0
$855$	−18.7004	−0.639540
$856$	0	0
$857$	37.1487	1.26898	0.634488	−	0.772933i	$-0.281211\pi$
$857$	37.1487	1.26898	0.634488	+	0.772933i	$0.281211\pi$
$858$	0	0
$859$	26.8141	0.914887	0.457443	−	0.889239i	$-0.348765\pi$
$859$	26.8141	0.914887	0.457443	+	0.889239i	$0.348765\pi$
$860$	0	0
$861$	81.6099	2.78126
$862$	0	0
$863$	−17.5306	−0.596748	−0.298374	−	0.954449i	$-0.596444\pi$
$863$	−17.5306	−0.596748	−0.298374	+	0.954449i	$0.596444\pi$
$864$	0	0
$865$	16.1640	0.549594
$866$	0	0
$867$	−4.66070	−0.158286
$868$	0	0
$869$	0	0
$870$	0	0
$871$	45.6739	1.54760
$872$	0	0
$873$	−53.3600	−1.80596
$874$	0	0
$875$	−3.76095	−0.127143
$876$	0	0
$877$	20.7757	0.701545	0.350772	−	0.936461i	$-0.385919\pi$
$877$	20.7757	0.701545	0.350772	+	0.936461i	$0.385919\pi$
$878$	0	0
$879$	9.47556	0.319603
$880$	0	0
$881$	36.3078	1.22324	0.611621	−	0.791151i	$-0.290518\pi$
$881$	36.3078	1.22324	0.611621	+	0.791151i	$0.290518\pi$
$882$	0	0
$883$	−37.1002	−1.24852	−0.624261	−	0.781216i	$-0.714600\pi$
$883$	−37.1002	−1.24852	−0.624261	+	0.781216i	$0.714600\pi$
$884$	0	0
$885$	17.9542	0.603523
$886$	0	0
$887$	21.9815	0.738068	0.369034	−	0.929416i	$-0.379689\pi$
$887$	21.9815	0.738068	0.369034	+	0.929416i	$0.379689\pi$
$888$	0	0
$889$	67.8912	2.27700
$890$	0	0
$891$	0	0
$892$	0	0
$893$	23.8998	0.799775
$894$	0	0
$895$	−18.3810	−0.614409
$896$	0	0
$897$	−56.7907	−1.89619
$898$	0	0
$899$	35.6789	1.18996
$900$	0	0
$901$	−4.26559	−0.142107
$902$	0	0
$903$	9.23037	0.307168
$904$	0	0
$905$	−11.0775	−0.368230
$906$	0	0
$907$	20.6799	0.686666	0.343333	−	0.939214i	$-0.388444\pi$
$907$	20.6799	0.686666	0.343333	+	0.939214i	$0.388444\pi$
$908$	0	0
$909$	0.359976	0.0119397
$910$	0	0
$911$	−34.2643	−1.13523	−0.567613	−	0.823296i	$-0.692133\pi$
$911$	−34.2643	−1.13523	−0.567613	+	0.823296i	$0.692133\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	24.4244	0.807445
$916$	0	0
$917$	24.3713	0.804813
$918$	0	0
$919$	16.6707	0.549915	0.274958	−	0.961456i	$-0.411336\pi$
$919$	16.6707	0.549915	0.274958	+	0.961456i	$0.411336\pi$
$920$	0	0
$921$	−22.3642	−0.736927
$922$	0	0
$923$	−33.9178	−1.11642
$924$	0	0
$925$	−10.0179	−0.329386
$926$	0	0
$927$	48.0147	1.57701
$928$	0	0
$929$	37.2041	1.22063	0.610313	−	0.792160i	$-0.291043\pi$
$929$	37.2041	1.22063	0.610313	+	0.792160i	$0.291043\pi$
$930$	0	0
$931$	30.8968	1.01260
$932$	0	0
$933$	31.2913	1.02443
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.6477	0.511187	0.255593	−	0.966784i	$-0.417729\pi$
$937$	15.6477	0.511187	0.255593	+	0.966784i	$0.417729\pi$
$938$	0	0
$939$	−52.2727	−1.70586
$940$	0	0
$941$	−14.4769	−0.471932	−0.235966	−	0.971761i	$-0.575825\pi$
$941$	−14.4769	−0.471932	−0.235966	+	0.971761i	$0.575825\pi$
$942$	0	0
$943$	−51.4551	−1.67561
$944$	0	0
$945$	−13.4804	−0.438516
$946$	0	0
$947$	−56.2394	−1.82754	−0.913768	−	0.406238i	$-0.866841\pi$
$947$	−56.2394	−1.82754	−0.913768	+	0.406238i	$0.866841\pi$
$948$	0	0
$949$	−12.0412	−0.390873
$950$	0	0
$951$	60.5510	1.96350
$952$	0	0
$953$	−22.1291	−0.716830	−0.358415	−	0.933562i	$-0.616683\pi$
$953$	−22.1291	−0.716830	−0.358415	+	0.933562i	$0.616683\pi$
$954$	0	0
$955$	−9.61874	−0.311255
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.28725	0.235318
$960$	0	0
$961$	−0.880038	−0.0283883
$962$	0	0
$963$	−55.6506	−1.79332
$964$	0	0
$965$	19.6557	0.632740
$966$	0	0
$967$	32.2808	1.03808	0.519039	−	0.854750i	$-0.326290\pi$
$967$	32.2808	1.03808	0.519039	+	0.854750i	$0.326290\pi$
$968$	0	0
$969$	−45.7449	−1.46954
$970$	0	0
$971$	−27.8700	−0.894390	−0.447195	−	0.894437i	$-0.647577\pi$
$971$	−27.8700	−0.894390	−0.447195	+	0.894437i	$0.647577\pi$
$972$	0	0
$973$	51.8103	1.66096
$974$	0	0
$975$	8.84928	0.283404
$976$	0	0
$977$	10.6628	0.341133	0.170567	−	0.985346i	$-0.445440\pi$
$977$	10.6628	0.341133	0.170567	+	0.985346i	$0.445440\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	32.4841	1.03714
$982$	0	0
$983$	30.6163	0.976508	0.488254	−	0.872702i	$-0.337634\pi$
$983$	30.6163	0.976508	0.488254	+	0.872702i	$0.337634\pi$
$984$	0	0
$985$	12.1768	0.387986
$986$	0	0
$987$	56.2538	1.79058
$988$	0	0
$989$	−5.81975	−0.185057
$990$	0	0
$991$	12.2099	0.387859	0.193929	−	0.981015i	$-0.437877\pi$
$991$	12.2099	0.387859	0.193929	+	0.981015i	$0.437877\pi$
$992$	0	0
$993$	−14.0156	−0.444772
$994$	0	0
$995$	−8.44164	−0.267618
$996$	0	0
$997$	−13.5307	−0.428521	−0.214260	−	0.976777i	$-0.568734\pi$
$997$	−13.5307	−0.428521	−0.214260	+	0.976777i	$0.568734\pi$
$998$	0	0
$999$	−35.9070	−1.13605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.2.a.n.1.4		4
4.3	odd	2		9680.2.a.ck.1.1		4
11.3	even	5		220.2.m.a.141.1	✓	8
11.4	even	5		220.2.m.a.181.1	yes	8
11.10	odd	2		2420.2.a.m.1.4		4
33.14	odd	10		1980.2.z.c.361.1		8
33.26	odd	10		1980.2.z.c.181.1		8
44.3	odd	10		880.2.bo.f.801.2		8
44.15	odd	10		880.2.bo.f.401.2		8
44.43	even	2		9680.2.a.cl.1.1		4
55.3	odd	20		1100.2.cb.c.449.1		16
55.4	even	10		1100.2.n.c.401.2		8
55.14	even	10		1100.2.n.c.801.2		8
55.37	odd	20		1100.2.cb.c.49.1		16
55.47	odd	20		1100.2.cb.c.449.4		16
55.48	odd	20		1100.2.cb.c.49.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.m.a.141.1	✓	8	11.3	even	5
220.2.m.a.181.1	yes	8	11.4	even	5
880.2.bo.f.401.2		8	44.15	odd	10
880.2.bo.f.801.2		8	44.3	odd	10
1100.2.n.c.401.2		8	55.4	even	10
1100.2.n.c.801.2		8	55.14	even	10
1100.2.cb.c.49.1		16	55.37	odd	20
1100.2.cb.c.49.4		16	55.48	odd	20
1100.2.cb.c.449.1		16	55.3	odd	20
1100.2.cb.c.449.4		16	55.47	odd	20
1980.2.z.c.181.1		8	33.26	odd	10
1980.2.z.c.361.1		8	33.14	odd	10
2420.2.a.m.1.4		4	11.10	odd	2
2420.2.a.n.1.4		4	1.1	even	1	trivial
9680.2.a.ck.1.1		4	4.3	odd	2
9680.2.a.cl.1.1		4	44.43	even	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:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2420.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70636 q^{3} -1.00000 q^{5} +3.76095 q^{7} +4.32440 q^{9} +3.26981 q^{13} -2.70636 q^{15} -3.90869 q^{17} +4.32440 q^{19} +10.1785 q^{21} -6.41755 q^{23} +1.00000 q^{25} +3.58430 q^{27} +6.50105 q^{29} +5.48817 q^{31} -3.76095 q^{35} -10.0179 q^{37} +8.84928 q^{39} +8.01787 q^{41} +0.906850 q^{43} -4.32440 q^{45} +5.52673 q^{47} +7.14476 q^{49} -10.5783 q^{51} +1.09131 q^{53} +11.7034 q^{57} -6.63406 q^{59} -9.02480 q^{61} +16.2638 q^{63} -3.26981 q^{65} +13.9684 q^{67} -17.3682 q^{69} -10.3730 q^{71} -3.68253 q^{73} +2.70636 q^{75} +7.27463 q^{79} -3.27279 q^{81} -6.11400 q^{83} +3.90869 q^{85} +17.5942 q^{87} -12.1209 q^{89} +12.2976 q^{91} +14.8530 q^{93} -4.32440 q^{95} -12.3393 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{13} - 2 q^{15} + q^{17} + 4 q^{19} + 14 q^{21} - 11 q^{23} + 4 q^{25} + 11 q^{27} + 4 q^{29} - q^{31} - 3 q^{35} - 17 q^{37} + 19 q^{39} + 9 q^{41} + 5 q^{43}+ \cdots + 8 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 + 3 * q^7 + 4 * q^9 + 3 * q^13 - 2 * q^15 + q^17 + 4 * q^19 + 14 * q^21 - 11 * q^23 + 4 * q^25 + 11 * q^27 + 4 * q^29 - q^31 - 3 * q^35 - 17 * q^37 + 19 * q^39 + 9 * q^41 + 5 * q^43 - 4 * q^45 + q^47 + 3 * q^49 + 3 * q^51 + 21 * q^53 + 17 * q^57 - 17 * q^59 + 28 * q^61 + 13 * q^63 - 3 * q^65 + 13 * q^67 - 8 * q^69 + 17 * q^71 + 13 * q^73 + 2 * q^75 + 22 * q^79 - 24 * q^81 - 21 * q^83 - q^85 + 47 * q^87 - 4 * q^89 - 4 * q^91 + 7 * q^93 - 4 * q^95 + 8 * q^97

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