LMFDB - Embedded newform 5220.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +4.15633 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +4.15633 q^{7} -4.15633 q^{11} -2.96239 q^{13} +0.156325 q^{17} +0.806063 q^{19} +0.806063 q^{23} +1.00000 q^{25} -1.00000 q^{29} -0.806063 q^{31} -4.15633 q^{35} +8.15633 q^{37} -3.73813 q^{41} -10.7308 q^{43} -1.58181 q^{47} +10.2750 q^{49} -12.0508 q^{53} +4.15633 q^{55} +1.73813 q^{59} -14.4993 q^{61} +2.96239 q^{65} -0.806063 q^{67} +11.6629 q^{71} +0.806063 q^{73} -17.2750 q^{77} +0.468976 q^{79} -6.08110 q^{83} -0.156325 q^{85} +13.0132 q^{89} -12.3127 q^{91} -0.806063 q^{95} -4.28233 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7	$ 3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61} - 2 q^{65} - 2 q^{67} + 4 q^{71} + 2 q^{73} - 20 q^{77} - 30 q^{79} + 14 q^{83} + 10 q^{85} - 2 q^{89} - 16 q^{91} - 2 q^{95} + 6 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.15633	1.57094	0.785472	−	0.618898i	$-0.212420\pi$
$7$	4.15633	1.57094	0.785472	+	0.618898i	$0.212420\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.15633	−1.25318	−0.626590	−	0.779349i	$-0.715550\pi$
$11$	−4.15633	−1.25318	−0.626590	+	0.779349i	$0.715550\pi$
$12$	0	0
$13$	−2.96239	−0.821619	−0.410809	−	0.911721i	$-0.634754\pi$
$13$	−2.96239	−0.821619	−0.410809	+	0.911721i	$0.634754\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.156325	0.0379144	0.0189572	−	0.999820i	$-0.493965\pi$
$17$	0.156325	0.0379144	0.0189572	+	0.999820i	$0.493965\pi$
$18$	0	0
$19$	0.806063	0.184924	0.0924618	−	0.995716i	$-0.470526\pi$
$19$	0.806063	0.184924	0.0924618	+	0.995716i	$0.470526\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.806063	0.168076	0.0840379	−	0.996463i	$-0.473218\pi$
$23$	0.806063	0.168076	0.0840379	+	0.996463i	$0.473218\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.806063	−0.144773	−0.0723866	−	0.997377i	$-0.523062\pi$
$31$	−0.806063	−0.144773	−0.0723866	+	0.997377i	$0.523062\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.15633	−0.702547
$36$	0	0
$37$	8.15633	1.34089	0.670446	−	0.741959i	$-0.266103\pi$
$37$	8.15633	1.34089	0.670446	+	0.741959i	$0.266103\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.73813	−0.583799	−0.291899	−	0.956449i	$-0.594287\pi$
$41$	−3.73813	−0.583799	−0.291899	+	0.956449i	$0.594287\pi$
$42$	0	0
$43$	−10.7308	−1.63644	−0.818219	−	0.574907i	$-0.805038\pi$
$43$	−10.7308	−1.63644	−0.818219	+	0.574907i	$0.805038\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.58181	−0.230731	−0.115365	−	0.993323i	$-0.536804\pi$
$47$	−1.58181	−0.230731	−0.115365	+	0.993323i	$0.536804\pi$
$48$	0	0
$49$	10.2750	1.46786
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.0508	−1.65530	−0.827651	−	0.561243i	$-0.810323\pi$
$53$	−12.0508	−1.65530	−0.827651	+	0.561243i	$0.810323\pi$
$54$	0	0
$55$	4.15633	0.560439
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.73813	0.226286	0.113143	−	0.993579i	$-0.463908\pi$
$59$	1.73813	0.226286	0.113143	+	0.993579i	$0.463908\pi$
$60$	0	0
$61$	−14.4993	−1.85644	−0.928222	−	0.372027i	$-0.878663\pi$
$61$	−14.4993	−1.85644	−0.928222	+	0.372027i	$0.878663\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.96239	0.367439
$66$	0	0
$67$	−0.806063	−0.0984763	−0.0492382	−	0.998787i	$-0.515679\pi$
$67$	−0.806063	−0.0984763	−0.0492382	+	0.998787i	$0.515679\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.6629	1.38413	0.692067	−	0.721834i	$-0.256700\pi$
$71$	11.6629	1.38413	0.692067	+	0.721834i	$0.256700\pi$
$72$	0	0
$73$	0.806063	0.0943426	0.0471713	−	0.998887i	$-0.484979\pi$
$73$	0.806063	0.0943426	0.0471713	+	0.998887i	$0.484979\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−17.2750	−1.96867
$78$	0	0
$79$	0.468976	0.0527639	0.0263819	−	0.999652i	$-0.491601\pi$
$79$	0.468976	0.0527639	0.0263819	+	0.999652i	$0.491601\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.08110	−0.667488	−0.333744	−	0.942664i	$-0.608312\pi$
$83$	−6.08110	−0.667488	−0.333744	+	0.942664i	$0.608312\pi$
$84$	0	0
$85$	−0.156325	−0.0169558
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.0132	1.37939	0.689697	−	0.724098i	$-0.257744\pi$
$89$	13.0132	1.37939	0.689697	+	0.724098i	$0.257744\pi$
$90$	0	0
$91$	−12.3127	−1.29072
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.806063	−0.0827004
$96$	0	0
$97$	−4.28233	−0.434805	−0.217403	−	0.976082i	$-0.569758\pi$
$97$	−4.28233	−0.434805	−0.217403	+	0.976082i	$0.569758\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.31265	0.628132	0.314066	−	0.949401i	$-0.398309\pi$
$101$	6.31265	0.628132	0.314066	+	0.949401i	$0.398309\pi$
$102$	0	0
$103$	−12.2823	−1.21021	−0.605107	−	0.796144i	$-0.706870\pi$
$103$	−12.2823	−1.21021	−0.605107	+	0.796144i	$0.706870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4690	−1.59212	−0.796058	−	0.605220i	$-0.793085\pi$
$107$	−16.4690	−1.59212	−0.796058	+	0.605220i	$0.793085\pi$
$108$	0	0
$109$	−16.0508	−1.53739	−0.768693	−	0.639618i	$-0.779093\pi$
$109$	−16.0508	−1.53739	−0.768693	+	0.639618i	$0.779093\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.89446	0.554504	0.277252	−	0.960797i	$-0.410576\pi$
$113$	5.89446	0.554504	0.277252	+	0.960797i	$0.410576\pi$
$114$	0	0
$115$	−0.806063	−0.0751658
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.649738	0.0595614
$120$	0	0
$121$	6.27504	0.570458
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.03032	−0.357633	−0.178816	−	0.983882i	$-0.557227\pi$
$127$	−4.03032	−0.357633	−0.178816	+	0.983882i	$0.557227\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.89446	−0.165520	−0.0827599	−	0.996570i	$-0.526373\pi$
$131$	−1.89446	−0.165520	−0.0827599	+	0.996570i	$0.526373\pi$
$132$	0	0
$133$	3.35026	0.290505
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.156325	0.0133558	0.00667788	−	0.999978i	$-0.497874\pi$
$137$	0.156325	0.0133558	0.00667788	+	0.999978i	$0.497874\pi$
$138$	0	0
$139$	−4.83638	−0.410216	−0.205108	−	0.978739i	$-0.565755\pi$
$139$	−4.83638	−0.410216	−0.205108	+	0.978739i	$0.565755\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.3127	1.02964
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−8.96239	−0.729349	−0.364674	−	0.931135i	$-0.618820\pi$
$151$	−8.96239	−0.729349	−0.364674	+	0.931135i	$0.618820\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.806063	0.0647446
$156$	0	0
$157$	15.0435	1.20060	0.600301	−	0.799774i	$-0.295048\pi$
$157$	15.0435	1.20060	0.600301	+	0.799774i	$0.295048\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.35026	0.264038
$162$	0	0
$163$	−13.4314	−1.05203	−0.526013	−	0.850477i	$-0.676314\pi$
$163$	−13.4314	−1.05203	−0.526013	+	0.850477i	$0.676314\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.89446	0.146598	0.0732989	−	0.997310i	$-0.476647\pi$
$167$	1.89446	0.146598	0.0732989	+	0.997310i	$0.476647\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.66291	0.430543	0.215272	−	0.976554i	$-0.430936\pi$
$173$	5.66291	0.430543	0.215272	+	0.976554i	$0.430936\pi$
$174$	0	0
$175$	4.15633	0.314189
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.7743	1.62749	0.813745	−	0.581222i	$-0.197425\pi$
$179$	21.7743	1.62749	0.813745	+	0.581222i	$0.197425\pi$
$180$	0	0
$181$	−11.7988	−0.876996	−0.438498	−	0.898732i	$-0.644489\pi$
$181$	−11.7988	−0.876996	−0.438498	+	0.898732i	$0.644489\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.15633	−0.599665
$186$	0	0
$187$	−0.649738	−0.0475136
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−23.1695	−1.67649	−0.838243	−	0.545297i	$-0.816417\pi$
$191$	−23.1695	−1.67649	−0.838243	+	0.545297i	$0.816417\pi$
$192$	0	0
$193$	18.0811	1.30151	0.650753	−	0.759289i	$-0.274453\pi$
$193$	18.0811	1.30151	0.650753	+	0.759289i	$0.274453\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.1622	1.00902	0.504508	−	0.863407i	$-0.331674\pi$
$197$	14.1622	1.00902	0.504508	+	0.863407i	$0.331674\pi$
$198$	0	0
$199$	−26.3634	−1.86885	−0.934427	−	0.356154i	$-0.884088\pi$
$199$	−26.3634	−1.86885	−0.934427	+	0.356154i	$0.884088\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.15633	−0.291717
$204$	0	0
$205$	3.73813	0.261083
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.35026	−0.231742
$210$	0	0
$211$	25.6180	1.76362	0.881808	−	0.471608i	$-0.156326\pi$
$211$	25.6180	1.76362	0.881808	+	0.471608i	$0.156326\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.7308	0.731837
$216$	0	0
$217$	−3.35026	−0.227431
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.463096	−0.0311512
$222$	0	0
$223$	−20.1319	−1.34813	−0.674065	−	0.738672i	$-0.735453\pi$
$223$	−20.1319	−1.34813	−0.674065	+	0.738672i	$0.735453\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.43136	0.0950030	0.0475015	−	0.998871i	$-0.484874\pi$
$227$	1.43136	0.0950030	0.0475015	+	0.998871i	$0.484874\pi$
$228$	0	0
$229$	26.4749	1.74951	0.874754	−	0.484568i	$-0.161023\pi$
$229$	26.4749	1.74951	0.874754	+	0.484568i	$0.161023\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−29.0738	−1.90469	−0.952344	−	0.305025i	$-0.901335\pi$
$233$	−29.0738	−1.90469	−0.952344	+	0.305025i	$0.901335\pi$
$234$	0	0
$235$	1.58181	0.103186
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.1866	0.788288	0.394144	−	0.919049i	$-0.371041\pi$
$239$	12.1866	0.788288	0.394144	+	0.919049i	$0.371041\pi$
$240$	0	0
$241$	−0.850969	−0.0548157	−0.0274079	−	0.999624i	$-0.508725\pi$
$241$	−0.850969	−0.0548157	−0.0274079	+	0.999624i	$0.508725\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.2750	−0.656448
$246$	0	0
$247$	−2.38787	−0.151937
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.2315	−0.898287	−0.449144	−	0.893460i	$-0.648271\pi$
$251$	−14.2315	−0.898287	−0.449144	+	0.893460i	$0.648271\pi$
$252$	0	0
$253$	−3.35026	−0.210629
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.41090	−0.337522	−0.168761	−	0.985657i	$-0.553977\pi$
$257$	−5.41090	−0.337522	−0.168761	+	0.985657i	$0.553977\pi$
$258$	0	0
$259$	33.9003	2.10646
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.41819	0.395763	0.197881	−	0.980226i	$-0.436594\pi$
$263$	6.41819	0.395763	0.197881	+	0.980226i	$0.436594\pi$
$264$	0	0
$265$	12.0508	0.740274
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−24.1114	−1.47010	−0.735050	−	0.678013i	$-0.762841\pi$
$269$	−24.1114	−1.47010	−0.735050	+	0.678013i	$0.762841\pi$
$270$	0	0
$271$	2.85685	0.173541	0.0867706	−	0.996228i	$-0.472345\pi$
$271$	2.85685	0.173541	0.0867706	+	0.996228i	$0.472345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.15633	−0.250636
$276$	0	0
$277$	3.79877	0.228246	0.114123	−	0.993467i	$-0.463594\pi$
$277$	3.79877	0.228246	0.114123	+	0.993467i	$0.463594\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.5745	−1.22737	−0.613686	−	0.789550i	$-0.710314\pi$
$281$	−20.5745	−1.22737	−0.613686	+	0.789550i	$0.710314\pi$
$282$	0	0
$283$	29.2809	1.74057	0.870285	−	0.492549i	$-0.163935\pi$
$283$	29.2809	1.74057	0.870285	+	0.492549i	$0.163935\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.5369	−0.917114
$288$	0	0
$289$	−16.9756	−0.998562
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.0713	−1.87362	−0.936811	−	0.349835i	$-0.886238\pi$
$293$	−32.0713	−1.87362	−0.936811	+	0.349835i	$0.886238\pi$
$294$	0	0
$295$	−1.73813	−0.101198
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.38787	−0.138094
$300$	0	0
$301$	−44.6009	−2.57075
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.4993	0.830227
$306$	0	0
$307$	1.79289	0.102326	0.0511628	−	0.998690i	$-0.483707\pi$
$307$	1.79289	0.102326	0.0511628	+	0.998690i	$0.483707\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.7586	0.893588	0.446794	−	0.894637i	$-0.352566\pi$
$311$	15.7586	0.893588	0.446794	+	0.894637i	$0.352566\pi$
$312$	0	0
$313$	16.3634	0.924916	0.462458	−	0.886641i	$-0.346968\pi$
$313$	16.3634	0.924916	0.462458	+	0.886641i	$0.346968\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.5296	−0.928395	−0.464198	−	0.885732i	$-0.653657\pi$
$317$	−16.5296	−0.928395	−0.464198	+	0.885732i	$0.653657\pi$
$318$	0	0
$319$	4.15633	0.232710
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.126008	0.00701127
$324$	0	0
$325$	−2.96239	−0.164324
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.57452	−0.362465
$330$	0	0
$331$	−15.8799	−0.872837	−0.436418	−	0.899744i	$-0.643753\pi$
$331$	−15.8799	−0.872837	−0.436418	+	0.899744i	$0.643753\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.806063	0.0440399
$336$	0	0
$337$	9.37073	0.510456	0.255228	−	0.966881i	$-0.417849\pi$
$337$	9.37073	0.510456	0.255228	+	0.966881i	$0.417849\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.35026	0.181427
$342$	0	0
$343$	13.6121	0.734986
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.6458	−1.43042	−0.715210	−	0.698910i	$-0.753669\pi$
$347$	−26.6458	−1.43042	−0.715210	+	0.698910i	$0.753669\pi$
$348$	0	0
$349$	−24.0263	−1.28610	−0.643050	−	0.765824i	$-0.722331\pi$
$349$	−24.0263	−1.28610	−0.643050	+	0.765824i	$0.722331\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.5745	−1.09507	−0.547535	−	0.836782i	$-0.684434\pi$
$353$	−20.5745	−1.09507	−0.547535	+	0.836782i	$0.684434\pi$
$354$	0	0
$355$	−11.6629	−0.619003
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.43136	0.497768	0.248884	−	0.968533i	$-0.419936\pi$
$359$	9.43136	0.497768	0.248884	+	0.968533i	$0.419936\pi$
$360$	0	0
$361$	−18.3503	−0.965803
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.806063	−0.0421913
$366$	0	0
$367$	3.50659	0.183042	0.0915212	−	0.995803i	$-0.470827\pi$
$367$	3.50659	0.183042	0.0915212	+	0.995803i	$0.470827\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−50.0870	−2.60039
$372$	0	0
$373$	−16.2374	−0.840742	−0.420371	−	0.907352i	$-0.638100\pi$
$373$	−16.2374	−0.840742	−0.420371	+	0.907352i	$0.638100\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.96239	0.152571
$378$	0	0
$379$	30.8568	1.58501	0.792505	−	0.609866i	$-0.208777\pi$
$379$	30.8568	1.58501	0.792505	+	0.609866i	$0.208777\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.9321	1.06958	0.534789	−	0.844985i	$-0.320391\pi$
$383$	20.9321	1.06958	0.534789	+	0.844985i	$0.320391\pi$
$384$	0	0
$385$	17.2750	0.880418
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.3357	−0.980358	−0.490179	−	0.871622i	$-0.663069\pi$
$389$	−19.3357	−0.980358	−0.490179	+	0.871622i	$0.663069\pi$
$390$	0	0
$391$	0.126008	0.00637250
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.468976	−0.0235967
$396$	0	0
$397$	−3.86414	−0.193936	−0.0969679	−	0.995288i	$-0.530914\pi$
$397$	−3.86414	−0.193936	−0.0969679	+	0.995288i	$0.530914\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.42407	0.420678	0.210339	−	0.977629i	$-0.432543\pi$
$401$	8.42407	0.420678	0.210339	+	0.977629i	$0.432543\pi$
$402$	0	0
$403$	2.38787	0.118948
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−33.9003	−1.68038
$408$	0	0
$409$	24.0508	1.18923	0.594617	−	0.804009i	$-0.297304\pi$
$409$	24.0508	1.18923	0.594617	+	0.804009i	$0.297304\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.22425	0.355482
$414$	0	0
$415$	6.08110	0.298510
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.6009	−1.78807	−0.894035	−	0.447998i	$-0.852137\pi$
$419$	−36.6009	−1.78807	−0.894035	+	0.447998i	$0.852137\pi$
$420$	0	0
$421$	−22.6497	−1.10388	−0.551940	−	0.833884i	$-0.686112\pi$
$421$	−22.6497	−1.10388	−0.551940	+	0.833884i	$0.686112\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.156325	0.00758288
$426$	0	0
$427$	−60.2638	−2.91637
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.4485	0.695960	0.347980	−	0.937502i	$-0.386868\pi$
$431$	14.4485	0.695960	0.347980	+	0.937502i	$0.386868\pi$
$432$	0	0
$433$	−1.55737	−0.0748425	−0.0374213	−	0.999300i	$-0.511914\pi$
$433$	−1.55737	−0.0748425	−0.0374213	+	0.999300i	$0.511914\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.649738	0.0310812
$438$	0	0
$439$	−32.2276	−1.53814	−0.769069	−	0.639166i	$-0.779280\pi$
$439$	−32.2276	−1.53814	−0.769069	+	0.639166i	$0.779280\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.66879	0.364355	0.182178	−	0.983266i	$-0.441685\pi$
$443$	7.66879	0.364355	0.182178	+	0.983266i	$0.441685\pi$
$444$	0	0
$445$	−13.0132	−0.616884
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.6009	−1.63292	−0.816458	−	0.577405i	$-0.804065\pi$
$449$	−34.6009	−1.63292	−0.816458	+	0.577405i	$0.804065\pi$
$450$	0	0
$451$	15.5369	0.731604
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.3127	0.577226
$456$	0	0
$457$	−7.94921	−0.371849	−0.185924	−	0.982564i	$-0.559528\pi$
$457$	−7.94921	−0.371849	−0.185924	+	0.982564i	$0.559528\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.6859	1.42919	0.714593	−	0.699540i	$-0.246612\pi$
$461$	30.6859	1.42919	0.714593	+	0.699540i	$0.246612\pi$
$462$	0	0
$463$	4.74543	0.220539	0.110269	−	0.993902i	$-0.464829\pi$
$463$	4.74543	0.220539	0.110269	+	0.993902i	$0.464829\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.7915	−0.684468	−0.342234	−	0.939615i	$-0.611184\pi$
$467$	−14.7915	−0.684468	−0.342234	+	0.939615i	$0.611184\pi$
$468$	0	0
$469$	−3.35026	−0.154701
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.6009	2.05075
$474$	0	0
$475$	0.806063	0.0369847
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.9452	1.45962	0.729808	−	0.683652i	$-0.239610\pi$
$479$	31.9452	1.45962	0.729808	+	0.683652i	$0.239610\pi$
$480$	0	0
$481$	−24.1622	−1.10170
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.28233	0.194451
$486$	0	0
$487$	−14.9175	−0.675976	−0.337988	−	0.941150i	$-0.609746\pi$
$487$	−14.9175	−0.675976	−0.337988	+	0.941150i	$0.609746\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.25457	0.146877	0.0734384	−	0.997300i	$-0.476603\pi$
$491$	3.25457	0.146877	0.0734384	+	0.997300i	$0.476603\pi$
$492$	0	0
$493$	−0.156325	−0.00704053
$494$	0	0
$495$	0	0
$496$	0	0
$497$	48.4749	2.17439
$498$	0	0
$499$	20.3733	0.912033	0.456017	−	0.889971i	$-0.349276\pi$
$499$	20.3733	0.912033	0.456017	+	0.889971i	$0.349276\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.2809	−0.770518	−0.385259	−	0.922808i	$-0.625888\pi$
$503$	−17.2809	−0.770518	−0.385259	+	0.922808i	$0.625888\pi$
$504$	0	0
$505$	−6.31265	−0.280909
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.3488	−0.990595	−0.495298	−	0.868723i	$-0.664941\pi$
$509$	−22.3488	−0.990595	−0.495298	+	0.868723i	$0.664941\pi$
$510$	0	0
$511$	3.35026	0.148207
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.2823	0.541224
$516$	0	0
$517$	6.57452	0.289147
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.5256	−1.24973	−0.624865	−	0.780732i	$-0.714846\pi$
$521$	−28.5256	−1.24973	−0.624865	+	0.780732i	$0.714846\pi$
$522$	0	0
$523$	12.8061	0.559970	0.279985	−	0.960004i	$-0.409670\pi$
$523$	12.8061	0.559970	0.279985	+	0.960004i	$0.409670\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.126008	−0.00548899
$528$	0	0
$529$	−22.3503	−0.971751
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.0738	0.479660
$534$	0	0
$535$	16.4690	0.712016
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−42.7064	−1.83950
$540$	0	0
$541$	20.0870	0.863607	0.431803	−	0.901968i	$-0.357877\pi$
$541$	20.0870	0.863607	0.431803	+	0.901968i	$0.357877\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.0508	0.687540
$546$	0	0
$547$	−32.9321	−1.40807	−0.704037	−	0.710163i	$-0.748621\pi$
$547$	−32.9321	−1.40807	−0.704037	+	0.710163i	$0.748621\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.806063	−0.0343395
$552$	0	0
$553$	1.94921	0.0828890
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.8510	0.713998	0.356999	−	0.934105i	$-0.383800\pi$
$557$	16.8510	0.713998	0.356999	+	0.934105i	$0.383800\pi$
$558$	0	0
$559$	31.7889	1.34453
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.2071	1.27308	0.636539	−	0.771245i	$-0.280365\pi$
$563$	30.2071	1.27308	0.636539	+	0.771245i	$0.280365\pi$
$564$	0	0
$565$	−5.89446	−0.247982
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.97556	0.0828199	0.0414099	−	0.999142i	$-0.486815\pi$
$569$	1.97556	0.0828199	0.0414099	+	0.999142i	$0.486815\pi$
$570$	0	0
$571$	2.32724	0.0973919	0.0486960	−	0.998814i	$-0.484493\pi$
$571$	2.32724	0.0973919	0.0486960	+	0.998814i	$0.484493\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.806063	0.0336152
$576$	0	0
$577$	−21.2447	−0.884429	−0.442215	−	0.896909i	$-0.645807\pi$
$577$	−21.2447	−0.884429	−0.442215	+	0.896909i	$0.645807\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−25.2750	−1.04859
$582$	0	0
$583$	50.0870	2.07439
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.6834	0.564774	0.282387	−	0.959301i	$-0.408874\pi$
$587$	13.6834	0.564774	0.282387	+	0.959301i	$0.408874\pi$
$588$	0	0
$589$	−0.649738	−0.0267720
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.3488	1.73906	0.869529	−	0.493881i	$-0.164422\pi$
$593$	42.3488	1.73906	0.869529	+	0.493881i	$0.164422\pi$
$594$	0	0
$595$	−0.649738	−0.0266367
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.85685	0.116728	0.0583638	−	0.998295i	$-0.481412\pi$
$599$	2.85685	0.116728	0.0583638	+	0.998295i	$0.481412\pi$
$600$	0	0
$601$	12.7367	0.519542	0.259771	−	0.965670i	$-0.416353\pi$
$601$	12.7367	0.519542	0.259771	+	0.965670i	$0.416353\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.27504	−0.255117
$606$	0	0
$607$	−38.6213	−1.56759	−0.783796	−	0.621019i	$-0.786719\pi$
$607$	−38.6213	−1.56759	−0.783796	+	0.621019i	$0.786719\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.68594	0.189573
$612$	0	0
$613$	37.3258	1.50758	0.753788	−	0.657118i	$-0.228225\pi$
$613$	37.3258	1.50758	0.753788	+	0.657118i	$0.228225\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.19394	−0.289617	−0.144808	−	0.989460i	$-0.546257\pi$
$617$	−7.19394	−0.289617	−0.144808	+	0.989460i	$0.546257\pi$
$618$	0	0
$619$	−20.0059	−0.804104	−0.402052	−	0.915617i	$-0.631703\pi$
$619$	−20.0059	−0.804104	−0.402052	+	0.915617i	$0.631703\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	54.0870	2.16695
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.27504	0.0508391
$630$	0	0
$631$	42.7123	1.70035	0.850175	−	0.526501i	$-0.176496\pi$
$631$	42.7123	1.70035	0.850175	+	0.526501i	$0.176496\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.03032	0.159938
$636$	0	0
$637$	−30.4387	−1.20602
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.15888	0.0457730	0.0228865	−	0.999738i	$-0.492714\pi$
$641$	1.15888	0.0457730	0.0228865	+	0.999738i	$0.492714\pi$
$642$	0	0
$643$	40.3839	1.59259	0.796293	−	0.604911i	$-0.206791\pi$
$643$	40.3839	1.59259	0.796293	+	0.604911i	$0.206791\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.2170	1.26658	0.633290	−	0.773915i	$-0.281704\pi$
$647$	32.2170	1.26658	0.633290	+	0.773915i	$0.281704\pi$
$648$	0	0
$649$	−7.22425	−0.283577
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.7294	−1.71126	−0.855632	−	0.517584i	$-0.826831\pi$
$653$	−43.7294	−1.71126	−0.855632	+	0.517584i	$0.826831\pi$
$654$	0	0
$655$	1.89446	0.0740227
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.6688	1.07782	0.538912	−	0.842362i	$-0.318836\pi$
$659$	27.6688	1.07782	0.538912	+	0.842362i	$0.318836\pi$
$660$	0	0
$661$	−1.22899	−0.0478023	−0.0239011	−	0.999714i	$-0.507609\pi$
$661$	−1.22899	−0.0478023	−0.0239011	+	0.999714i	$0.507609\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.35026	−0.129918
$666$	0	0
$667$	−0.806063	−0.0312109
$668$	0	0
$669$	0	0
$670$	0	0
$671$	60.2638	2.32646
$672$	0	0
$673$	18.9380	0.730004	0.365002	−	0.931007i	$-0.381068\pi$
$673$	18.9380	0.730004	0.365002	+	0.931007i	$0.381068\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.4591	0.863174	0.431587	−	0.902071i	$-0.357954\pi$
$677$	22.4591	0.863174	0.431587	+	0.902071i	$0.357954\pi$
$678$	0	0
$679$	−17.7988	−0.683054
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.6702	0.561340	0.280670	−	0.959804i	$-0.409443\pi$
$683$	14.6702	0.561340	0.280670	+	0.959804i	$0.409443\pi$
$684$	0	0
$685$	−0.156325	−0.00597288
$686$	0	0
$687$	0	0
$688$	0	0
$689$	35.6991	1.36003
$690$	0	0
$691$	39.3014	1.49510	0.747548	−	0.664208i	$-0.231231\pi$
$691$	39.3014	1.49510	0.747548	+	0.664208i	$0.231231\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.83638	0.183454
$696$	0	0
$697$	−0.584365	−0.0221344
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−48.0216	−1.81375	−0.906876	−	0.421399i	$-0.861539\pi$
$701$	−48.0216	−1.81375	−0.906876	+	0.421399i	$0.861539\pi$
$702$	0	0
$703$	6.57452	0.247963
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.2374	0.986760
$708$	0	0
$709$	8.57452	0.322023	0.161011	−	0.986953i	$-0.448524\pi$
$709$	8.57452	0.322023	0.161011	+	0.986953i	$0.448524\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.649738	−0.0243329
$714$	0	0
$715$	−12.3127	−0.460467
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.1114	0.377092	0.188546	−	0.982064i	$-0.439622\pi$
$719$	10.1114	0.377092	0.188546	+	0.982064i	$0.439622\pi$
$720$	0	0
$721$	−51.0494	−1.90118
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	47.2057	1.75076	0.875381	−	0.483433i	$-0.160610\pi$
$727$	47.2057	1.75076	0.875381	+	0.483433i	$0.160610\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.67750	−0.0620446
$732$	0	0
$733$	48.7210	1.79955	0.899775	−	0.436353i	$-0.143730\pi$
$733$	48.7210	1.79955	0.899775	+	0.436353i	$0.143730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.35026	0.123408
$738$	0	0
$739$	0.432779	0.0159200	0.00796001	−	0.999968i	$-0.497466\pi$
$739$	0.432779	0.0159200	0.00796001	+	0.999968i	$0.497466\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.2692	0.780290	0.390145	−	0.920753i	$-0.372425\pi$
$743$	21.2692	0.780290	0.390145	+	0.920753i	$0.372425\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−68.4504	−2.50112
$750$	0	0
$751$	12.4328	0.453679	0.226839	−	0.973932i	$-0.427161\pi$
$751$	12.4328	0.453679	0.226839	+	0.973932i	$0.427161\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.96239	0.326175
$756$	0	0
$757$	28.8061	1.04697	0.523487	−	0.852034i	$-0.324631\pi$
$757$	28.8061	1.04697	0.523487	+	0.852034i	$0.324631\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.17679	0.296409	0.148204	−	0.988957i	$-0.452651\pi$
$761$	8.17679	0.296409	0.148204	+	0.988957i	$0.452651\pi$
$762$	0	0
$763$	−66.7123	−2.41515
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.14903	−0.185921
$768$	0	0
$769$	−29.5369	−1.06513	−0.532564	−	0.846390i	$-0.678771\pi$
$769$	−29.5369	−1.06513	−0.532564	+	0.846390i	$0.678771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−30.3331	−1.09101	−0.545503	−	0.838109i	$-0.683661\pi$
$773$	−30.3331	−1.09101	−0.545503	+	0.838109i	$0.683661\pi$
$774$	0	0
$775$	−0.806063	−0.0289547
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.01317	−0.107958
$780$	0	0
$781$	−48.4749	−1.73457
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15.0435	−0.536925
$786$	0	0
$787$	20.1681	0.718915	0.359457	−	0.933162i	$-0.382962\pi$
$787$	20.1681	0.718915	0.359457	+	0.933162i	$0.382962\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.4993	0.871095
$792$	0	0
$793$	42.9525	1.52529
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.2677	0.647077	0.323538	−	0.946215i	$-0.395128\pi$
$797$	18.2677	0.647077	0.323538	+	0.946215i	$0.395128\pi$
$798$	0	0
$799$	−0.247277	−0.00874802
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.35026	−0.118228
$804$	0	0
$805$	−3.35026	−0.118081
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.5355	0.651673	0.325837	−	0.945426i	$-0.394354\pi$
$809$	18.5355	0.651673	0.325837	+	0.945426i	$0.394354\pi$
$810$	0	0
$811$	−41.1608	−1.44535	−0.722675	−	0.691188i	$-0.757088\pi$
$811$	−41.1608	−1.44535	−0.722675	+	0.691188i	$0.757088\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.4314	0.470480
$816$	0	0
$817$	−8.64974	−0.302616
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.5501	−1.55481	−0.777404	−	0.629001i	$-0.783464\pi$
$821$	−44.5501	−1.55481	−0.777404	+	0.629001i	$0.783464\pi$
$822$	0	0
$823$	21.3298	0.743510	0.371755	−	0.928331i	$-0.378756\pi$
$823$	21.3298	0.743510	0.371755	+	0.928331i	$0.378756\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.2692	0.461414	0.230707	−	0.973023i	$-0.425896\pi$
$827$	13.2692	0.461414	0.230707	+	0.973023i	$0.425896\pi$
$828$	0	0
$829$	−5.91493	−0.205434	−0.102717	−	0.994711i	$-0.532754\pi$
$829$	−5.91493	−0.205434	−0.102717	+	0.994711i	$0.532754\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.60625	0.0556532
$834$	0	0
$835$	−1.89446	−0.0655605
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.3317	−0.943595	−0.471798	−	0.881707i	$-0.656395\pi$
$839$	−27.3317	−0.943595	−0.471798	+	0.881707i	$0.656395\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.22425	0.145319
$846$	0	0
$847$	26.0811	0.896157
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.57452	0.225371
$852$	0	0
$853$	32.8423	1.12450	0.562249	−	0.826968i	$-0.309936\pi$
$853$	32.8423	1.12450	0.562249	+	0.826968i	$0.309936\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.9610	1.22840	0.614202	−	0.789149i	$-0.289478\pi$
$857$	35.9610	1.22840	0.614202	+	0.789149i	$0.289478\pi$
$858$	0	0
$859$	34.1417	1.16490	0.582451	−	0.812866i	$-0.302094\pi$
$859$	34.1417	1.16490	0.582451	+	0.812866i	$0.302094\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.6312	1.79159	0.895793	−	0.444471i	$-0.146608\pi$
$863$	52.6312	1.79159	0.895793	+	0.444471i	$0.146608\pi$
$864$	0	0
$865$	−5.66291	−0.192545
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.94921	−0.0661226
$870$	0	0
$871$	2.38787	0.0809100
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.15633	−0.140509
$876$	0	0
$877$	30.9741	1.04592	0.522961	−	0.852356i	$-0.324827\pi$
$877$	30.9741	1.04592	0.522961	+	0.852356i	$0.324827\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.5111	−1.56700	−0.783499	−	0.621394i	$-0.786567\pi$
$881$	−46.5111	−1.56700	−0.783499	+	0.621394i	$0.786567\pi$
$882$	0	0
$883$	37.8291	1.27305	0.636525	−	0.771256i	$-0.280371\pi$
$883$	37.8291	1.27305	0.636525	+	0.771256i	$0.280371\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.5672	−0.657003	−0.328501	−	0.944503i	$-0.606544\pi$
$887$	−19.5672	−0.657003	−0.328501	+	0.944503i	$0.606544\pi$
$888$	0	0
$889$	−16.7513	−0.561821
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.27504	−0.0426676
$894$	0	0
$895$	−21.7743	−0.727836
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.806063	0.0268837
$900$	0	0
$901$	−1.88384	−0.0627598
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.7988	0.392204
$906$	0	0
$907$	44.2335	1.46875	0.734374	−	0.678745i	$-0.237476\pi$
$907$	44.2335	1.46875	0.734374	+	0.678745i	$0.237476\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.95509	−0.0647752	−0.0323876	−	0.999475i	$-0.510311\pi$
$911$	−1.95509	−0.0647752	−0.0323876	+	0.999475i	$0.510311\pi$
$912$	0	0
$913$	25.2750	0.836482
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.87399	−0.260022
$918$	0	0
$919$	46.3634	1.52939	0.764694	−	0.644393i	$-0.222890\pi$
$919$	46.3634	1.52939	0.764694	+	0.644393i	$0.222890\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−34.5501	−1.13723
$924$	0	0
$925$	8.15633	0.268178
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.0460	0.395218	0.197609	−	0.980281i	$-0.436682\pi$
$929$	12.0460	0.395218	0.197609	+	0.980281i	$0.436682\pi$
$930$	0	0
$931$	8.28233	0.271443
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.649738	0.0212487
$936$	0	0
$937$	6.27645	0.205043	0.102521	−	0.994731i	$-0.467309\pi$
$937$	6.27645	0.205043	0.102521	+	0.994731i	$0.467309\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52.1768	1.70091	0.850457	−	0.526044i	$-0.176325\pi$
$941$	52.1768	1.70091	0.850457	+	0.526044i	$0.176325\pi$
$942$	0	0
$943$	−3.01317	−0.0981224
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.3707	−0.694455	−0.347228	−	0.937781i	$-0.612877\pi$
$947$	−21.3707	−0.694455	−0.347228	+	0.937781i	$0.612877\pi$
$948$	0	0
$949$	−2.38787	−0.0775136
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.1016	−0.715940	−0.357970	−	0.933733i	$-0.616531\pi$
$953$	−22.1016	−0.715940	−0.357970	+	0.933733i	$0.616531\pi$
$954$	0	0
$955$	23.1695	0.749747
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.649738	0.0209811
$960$	0	0
$961$	−30.3503	−0.979041
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18.0811	−0.582051
$966$	0	0
$967$	−31.0844	−0.999608	−0.499804	−	0.866139i	$-0.666595\pi$
$967$	−31.0844	−0.999608	−0.499804	+	0.866139i	$0.666595\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.87987	−0.252877	−0.126439	−	0.991974i	$-0.540355\pi$
$971$	−7.87987	−0.252877	−0.126439	+	0.991974i	$0.540355\pi$
$972$	0	0
$973$	−20.1016	−0.644427
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.5877	−1.26652	−0.633261	−	0.773938i	$-0.718284\pi$
$977$	−39.5877	−1.26652	−0.633261	+	0.773938i	$0.718284\pi$
$978$	0	0
$979$	−54.0870	−1.72863
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.3444	0.361830	0.180915	−	0.983499i	$-0.442094\pi$
$983$	11.3444	0.361830	0.180915	+	0.983499i	$0.442094\pi$
$984$	0	0
$985$	−14.1622	−0.451245
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.64974	−0.275046
$990$	0	0
$991$	−17.4664	−0.554839	−0.277420	−	0.960749i	$-0.589479\pi$
$991$	−17.4664	−0.554839	−0.277420	+	0.960749i	$0.589479\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	26.3634	0.835777
$996$	0	0
$997$	−51.7901	−1.64021	−0.820104	−	0.572215i	$-0.806084\pi$
$997$	−51.7901	−1.64021	−0.820104	+	0.572215i	$0.806084\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.w.1.3		3
3.2	odd	2		580.2.a.d.1.2	✓	3
12.11	even	2		2320.2.a.o.1.2		3
15.2	even	4		2900.2.c.g.349.3		6
15.8	even	4		2900.2.c.g.349.4		6
15.14	odd	2		2900.2.a.f.1.2		3
24.5	odd	2		9280.2.a.bh.1.2		3
24.11	even	2		9280.2.a.bt.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.d.1.2	✓	3	3.2	odd	2
2320.2.a.o.1.2		3	12.11	even	2
2900.2.a.f.1.2		3	15.14	odd	2
2900.2.c.g.349.3		6	15.2	even	4
2900.2.c.g.349.4		6	15.8	even	4
5220.2.a.w.1.3		3	1.1	even	1	trivial
9280.2.a.bh.1.2		3	24.5	odd	2
9280.2.a.bt.1.2		3	24.11	even	2

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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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.15633 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +4.15633 q^{7} -4.15633 q^{11} -2.96239 q^{13} +0.156325 q^{17} +0.806063 q^{19} +0.806063 q^{23} +1.00000 q^{25} -1.00000 q^{29} -0.806063 q^{31} -4.15633 q^{35} +8.15633 q^{37} -3.73813 q^{41} -10.7308 q^{43} -1.58181 q^{47} +10.2750 q^{49} -12.0508 q^{53} +4.15633 q^{55} +1.73813 q^{59} -14.4993 q^{61} +2.96239 q^{65} -0.806063 q^{67} +11.6629 q^{71} +0.806063 q^{73} -17.2750 q^{77} +0.468976 q^{79} -6.08110 q^{83} -0.156325 q^{85} +13.0132 q^{89} -12.3127 q^{91} -0.806063 q^{95} -4.28233 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7	\( 3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61} - 2 q^{65} - 2 q^{67} + 4 q^{71} + 2 q^{73} - 20 q^{77} - 30 q^{79} + 14 q^{83} + 10 q^{85} - 2 q^{89} - 16 q^{91} - 2 q^{95} + 6 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * q^97

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