LMFDB - Embedded newform 8280.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.20087896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 $ x^5 - x^4 - 21x^3 + 5x^2 + 84*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.43588$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.62057 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.62057 q^{7} +6.55627 q^{11} +7.06828 q^{13} -6.42405 q^{17} -1.80348 q^{19} +1.00000 q^{23} +1.00000 q^{25} +7.17684 q^{29} +4.10856 q^{31} -2.62057 q^{35} +4.62057 q^{37} -4.30508 q^{41} +6.87176 q^{43} -1.80348 q^{47} -0.132589 q^{49} +4.93606 q^{53} -6.55627 q^{55} +7.54623 q^{59} +2.70854 q^{61} -7.06828 q^{65} +7.37337 q^{67} -3.93569 q^{71} -5.04462 q^{73} +17.1812 q^{77} +6.11897 q^{79} +8.42405 q^{83} +6.42405 q^{85} -11.3838 q^{89} +18.5230 q^{91} +1.80348 q^{95} -17.9843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 4 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 - 4 * q^7	$ 5 q - 5 q^{5} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 10 q^{17} - 4 q^{19} + 5 q^{23} + 5 q^{25} - 10 q^{29} + 6 q^{31} + 4 q^{35} + 6 q^{37} - 12 q^{41} - 2 q^{43} - 4 q^{47} + 19 q^{49} - 4 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{65} - 4 q^{67} - 8 q^{71} + 14 q^{73} - 16 q^{77} + 18 q^{79} + 20 q^{83} + 10 q^{85} - 18 q^{89} + 20 q^{91} + 4 q^{95} + 4 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 10 * q^17 - 4 * q^19 + 5 * q^23 + 5 * q^25 - 10 * q^29 + 6 * q^31 + 4 * q^35 + 6 * q^37 - 12 * q^41 - 2 * q^43 - 4 * q^47 + 19 * q^49 - 4 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 14 * q^73 - 16 * q^77 + 18 * q^79 + 20 * q^83 + 10 * q^85 - 18 * q^89 + 20 * q^91 + 4 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.62057	0.990484	0.495242	−	0.868755i	$-0.335079\pi$
$7$	2.62057	0.990484	0.495242	+	0.868755i	$0.335079\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.55627	1.97679	0.988395	−	0.151907i	$-0.0485414\pi$
$11$	6.55627	1.97679	0.988395	+	0.151907i	$0.0485414\pi$
$12$	0	0
$13$	7.06828	1.96039	0.980195	−	0.198037i	$-0.0634565\pi$
$13$	7.06828	1.96039	0.980195	+	0.198037i	$0.0634565\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.42405	−1.55806	−0.779030	−	0.626986i	$-0.784288\pi$
$17$	−6.42405	−1.55806	−0.779030	+	0.626986i	$0.784288\pi$
$18$	0	0
$19$	−1.80348	−0.413746	−0.206873	−	0.978368i	$-0.566329\pi$
$19$	−1.80348	−0.413746	−0.206873	+	0.978368i	$0.566329\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.17684	1.33271	0.666353	−	0.745636i	$-0.267854\pi$
$29$	7.17684	1.33271	0.666353	+	0.745636i	$0.267854\pi$
$30$	0	0
$31$	4.10856	0.737919	0.368960	−	0.929445i	$-0.379714\pi$
$31$	4.10856	0.737919	0.368960	+	0.929445i	$0.379714\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.62057	−0.442958
$36$	0	0
$37$	4.62057	0.759618	0.379809	−	0.925065i	$-0.375990\pi$
$37$	4.62057	0.759618	0.379809	+	0.925065i	$0.375990\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.30508	−0.672341	−0.336171	−	0.941801i	$-0.609132\pi$
$41$	−4.30508	−0.672341	−0.336171	+	0.941801i	$0.609132\pi$
$42$	0	0
$43$	6.87176	1.04793	0.523967	−	0.851739i	$-0.324452\pi$
$43$	6.87176	1.04793	0.523967	+	0.851739i	$0.324452\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.80348	−0.263064	−0.131532	−	0.991312i	$-0.541990\pi$
$47$	−1.80348	−0.263064	−0.131532	+	0.991312i	$0.541990\pi$
$48$	0	0
$49$	−0.132589	−0.0189413
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.93606	0.678021	0.339010	−	0.940783i	$-0.389908\pi$
$53$	4.93606	0.678021	0.339010	+	0.940783i	$0.389908\pi$
$54$	0	0
$55$	−6.55627	−0.884047
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.54623	0.982436	0.491218	−	0.871037i	$-0.336552\pi$
$59$	7.54623	0.982436	0.491218	+	0.871037i	$0.336552\pi$
$60$	0	0
$61$	2.70854	0.346793	0.173396	−	0.984852i	$-0.444526\pi$
$61$	2.70854	0.346793	0.173396	+	0.984852i	$0.444526\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.06828	−0.876713
$66$	0	0
$67$	7.37337	0.900800	0.450400	−	0.892827i	$-0.351281\pi$
$67$	7.37337	0.900800	0.450400	+	0.892827i	$0.351281\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.93569	−0.467081	−0.233541	−	0.972347i	$-0.575031\pi$
$71$	−3.93569	−0.467081	−0.233541	+	0.972347i	$0.575031\pi$
$72$	0	0
$73$	−5.04462	−0.590429	−0.295214	−	0.955431i	$-0.595391\pi$
$73$	−5.04462	−0.590429	−0.295214	+	0.955431i	$0.595391\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.1812	1.95798
$78$	0	0
$79$	6.11897	0.688437	0.344219	−	0.938889i	$-0.388144\pi$
$79$	6.11897	0.688437	0.344219	+	0.938889i	$0.388144\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.42405	0.924660	0.462330	−	0.886708i	$-0.347014\pi$
$83$	8.42405	0.924660	0.462330	+	0.886708i	$0.347014\pi$
$84$	0	0
$85$	6.42405	0.696786
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.3838	−1.20668	−0.603339	−	0.797485i	$-0.706163\pi$
$89$	−11.3838	−1.20668	−0.603339	+	0.797485i	$0.706163\pi$
$90$	0	0
$91$	18.5230	1.94173
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.80348	0.185033
$96$	0	0
$97$	−17.9843	−1.82603	−0.913014	−	0.407927i	$-0.866252\pi$
$97$	−17.9843	−1.82603	−0.913014	+	0.407927i	$0.866252\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.67126	−0.166296	−0.0831481	−	0.996537i	$-0.526497\pi$
$101$	−1.67126	−0.166296	−0.0831481	+	0.996537i	$0.526497\pi$
$102$	0	0
$103$	6.72913	0.663041	0.331521	−	0.943448i	$-0.392438\pi$
$103$	6.72913	0.663041	0.331521	+	0.943448i	$0.392438\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.9296	−1.73332	−0.866662	−	0.498896i	$-0.833739\pi$
$107$	−17.9296	−1.73332	−0.866662	+	0.498896i	$0.833739\pi$
$108$	0	0
$109$	−12.6692	−1.21349	−0.606744	−	0.794898i	$-0.707525\pi$
$109$	−12.6692	−1.21349	−0.606744	+	0.794898i	$0.707525\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−21.1708	−1.99158	−0.995790	−	0.0916634i	$-0.970782\pi$
$113$	−21.1708	−1.99158	−0.995790	+	0.0916634i	$0.970782\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.8347	−1.54323
$120$	0	0
$121$	31.9847	2.90770
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.04425	−0.358870	−0.179435	−	0.983770i	$-0.557427\pi$
$127$	−4.04425	−0.358870	−0.179435	+	0.983770i	$0.557427\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−4.72614	−0.409808
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.75316	−0.491526	−0.245763	−	0.969330i	$-0.579038\pi$
$137$	−5.75316	−0.491526	−0.245763	+	0.969330i	$0.579038\pi$
$138$	0	0
$139$	4.50161	0.381821	0.190911	−	0.981607i	$-0.438856\pi$
$139$	4.50161	0.381821	0.190911	+	0.981607i	$0.438856\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	46.3416	3.87528
$144$	0	0
$145$	−7.17684	−0.596004
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.1572	1.15980	0.579900	−	0.814688i	$-0.303092\pi$
$149$	14.1572	1.15980	0.579900	+	0.814688i	$0.303092\pi$
$150$	0	0
$151$	−18.9847	−1.54495	−0.772475	−	0.635045i	$-0.780982\pi$
$151$	−18.9847	−1.54495	−0.772475	+	0.635045i	$0.780982\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.10856	−0.330007
$156$	0	0
$157$	8.22753	0.656628	0.328314	−	0.944569i	$-0.393520\pi$
$157$	8.22753	0.656628	0.328314	+	0.944569i	$0.393520\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.62057	0.206530
$162$	0	0
$163$	0.264439	0.0207124	0.0103562	−	0.999946i	$-0.496703\pi$
$163$	0.264439	0.0207124	0.0103562	+	0.999946i	$0.496703\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.45775	−0.731862	−0.365931	−	0.930642i	$-0.619249\pi$
$167$	−9.45775	−0.731862	−0.365931	+	0.930642i	$0.619249\pi$
$168$	0	0
$169$	36.9606	2.84313
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.758851	0.0576944	0.0288472	−	0.999584i	$-0.490816\pi$
$173$	0.758851	0.0576944	0.0288472	+	0.999584i	$0.490816\pi$
$174$	0	0
$175$	2.62057	0.198097
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.9606	−1.64142	−0.820708	−	0.571348i	$-0.806420\pi$
$179$	−21.9606	−1.64142	−0.820708	+	0.571348i	$0.806420\pi$
$180$	0	0
$181$	−14.4727	−1.07574	−0.537872	−	0.843027i	$-0.680772\pi$
$181$	−14.4727	−1.07574	−0.537872	+	0.843027i	$0.680772\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.62057	−0.339711
$186$	0	0
$187$	−42.1178	−3.07996
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.9400	0.863951	0.431976	−	0.901885i	$-0.357817\pi$
$191$	11.9400	0.863951	0.431976	+	0.901885i	$0.357817\pi$
$192$	0	0
$193$	11.6342	0.837448	0.418724	−	0.908114i	$-0.362477\pi$
$193$	11.6342	0.837448	0.418724	+	0.908114i	$0.362477\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.6181	−1.75397	−0.876984	−	0.480519i	$-0.840448\pi$
$197$	−24.6181	−1.75397	−0.876984	+	0.480519i	$0.840448\pi$
$198$	0	0
$199$	18.7291	1.32767	0.663837	−	0.747878i	$-0.268927\pi$
$199$	18.7291	1.32767	0.663837	+	0.747878i	$0.268927\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.8075	1.32002
$204$	0	0
$205$	4.30508	0.300680
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.8241	−0.817888
$210$	0	0
$211$	3.49839	0.240839	0.120420	−	0.992723i	$-0.461576\pi$
$211$	3.49839	0.240839	0.120420	+	0.992723i	$0.461576\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.87176	−0.468650
$216$	0	0
$217$	10.7668	0.730897
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−45.4070	−3.05441
$222$	0	0
$223$	−13.8477	−0.927313	−0.463656	−	0.886015i	$-0.653463\pi$
$223$	−13.8477	−0.927313	−0.463656	+	0.886015i	$0.653463\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.5772	−1.29939	−0.649693	−	0.760196i	$-0.725103\pi$
$227$	−19.5772	−1.29939	−0.649693	+	0.760196i	$0.725103\pi$
$228$	0	0
$229$	25.4959	1.68482	0.842410	−	0.538838i	$-0.181136\pi$
$229$	25.4959	1.68482	0.842410	+	0.538838i	$0.181136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.9847	−1.11270	−0.556351	−	0.830947i	$-0.687799\pi$
$233$	−16.9847	−1.11270	−0.556351	+	0.830947i	$0.687799\pi$
$234$	0	0
$235$	1.80348	0.117646
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.5459	−1.07026	−0.535131	−	0.844769i	$-0.679738\pi$
$239$	−16.5459	−1.07026	−0.535131	+	0.844769i	$0.679738\pi$
$240$	0	0
$241$	−10.1572	−0.654280	−0.327140	−	0.944976i	$-0.606085\pi$
$241$	−10.1572	−0.654280	−0.327140	+	0.944976i	$0.606085\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.132589	0.00847081
$246$	0	0
$247$	−12.7475	−0.811103
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.72950	−0.361643	−0.180822	−	0.983516i	$-0.557876\pi$
$251$	−5.72950	−0.361643	−0.180822	+	0.983516i	$0.557876\pi$
$252$	0	0
$253$	6.55627	0.412189
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.5269	1.21806	0.609028	−	0.793149i	$-0.291560\pi$
$257$	19.5269	1.21806	0.609028	+	0.793149i	$0.291560\pi$
$258$	0	0
$259$	12.1086	0.752389
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.0726	1.79270	0.896348	−	0.443352i	$-0.146211\pi$
$263$	29.0726	1.79270	0.896348	+	0.443352i	$0.146211\pi$
$264$	0	0
$265$	−4.93606	−0.303220
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.8078	−0.719936	−0.359968	−	0.932965i	$-0.617212\pi$
$269$	−11.8078	−0.719936	−0.359968	+	0.932965i	$0.617212\pi$
$270$	0	0
$271$	9.13259	0.554765	0.277383	−	0.960760i	$-0.410533\pi$
$271$	9.13259	0.554765	0.277383	+	0.960760i	$0.410533\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.55627	0.395358
$276$	0	0
$277$	27.8564	1.67373	0.836865	−	0.547409i	$-0.184386\pi$
$277$	27.8564	1.67373	0.836865	+	0.547409i	$0.184386\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.9461	0.712645	0.356322	−	0.934363i	$-0.384031\pi$
$281$	11.9461	0.712645	0.356322	+	0.934363i	$0.384031\pi$
$282$	0	0
$283$	−17.0076	−1.01099	−0.505497	−	0.862828i	$-0.668691\pi$
$283$	−17.0076	−1.01099	−0.505497	+	0.862828i	$0.668691\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.2818	−0.665943
$288$	0	0
$289$	24.2684	1.42755
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.9472	1.39901	0.699506	−	0.714626i	$-0.253403\pi$
$293$	23.9472	1.39901	0.699506	+	0.714626i	$0.253403\pi$
$294$	0	0
$295$	−7.54623	−0.439359
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.06828	0.408769
$300$	0	0
$301$	18.0080	1.03796
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.70854	−0.155091
$306$	0	0
$307$	15.9400	0.909746	0.454873	−	0.890556i	$-0.349685\pi$
$307$	15.9400	0.909746	0.454873	+	0.890556i	$0.349685\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.16023	−0.462724	−0.231362	−	0.972868i	$-0.574318\pi$
$311$	−8.16023	−0.462724	−0.231362	+	0.972868i	$0.574318\pi$
$312$	0	0
$313$	2.84375	0.160738	0.0803692	−	0.996765i	$-0.474390\pi$
$313$	2.84375	0.160738	0.0803692	+	0.996765i	$0.474390\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.5269	1.32140	0.660702	−	0.750648i	$-0.270259\pi$
$317$	23.5269	1.32140	0.660702	+	0.750648i	$0.270259\pi$
$318$	0	0
$319$	47.0533	2.63448
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.5856	0.644641
$324$	0	0
$325$	7.06828	0.392078
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.72614	−0.260561
$330$	0	0
$331$	−11.9326	−0.655877	−0.327938	−	0.944699i	$-0.606354\pi$
$331$	−11.9326	−0.655877	−0.327938	+	0.944699i	$0.606354\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.37337	−0.402850
$336$	0	0
$337$	12.6073	0.686765	0.343382	−	0.939196i	$-0.388427\pi$
$337$	12.6073	0.686765	0.343382	+	0.939196i	$0.388427\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.9368	1.45871
$342$	0	0
$343$	−18.6915	−1.00925
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.85132	0.314115	0.157058	−	0.987589i	$-0.449799\pi$
$347$	5.85132	0.314115	0.157058	+	0.987589i	$0.449799\pi$
$348$	0	0
$349$	−29.7034	−1.58999	−0.794993	−	0.606618i	$-0.792526\pi$
$349$	−29.7034	−1.58999	−0.794993	+	0.606618i	$0.792526\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.1160	0.538419	0.269209	−	0.963082i	$-0.413238\pi$
$353$	10.1160	0.538419	0.269209	+	0.963082i	$0.413238\pi$
$354$	0	0
$355$	3.93569	0.208885
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.45775	−0.0769369	−0.0384684	−	0.999260i	$-0.512248\pi$
$359$	−1.45775	−0.0769369	−0.0384684	+	0.999260i	$0.512248\pi$
$360$	0	0
$361$	−15.7475	−0.828815
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.04462	0.264048
$366$	0	0
$367$	1.98959	0.103856	0.0519280	−	0.998651i	$-0.483463\pi$
$367$	1.98959	0.103856	0.0519280	+	0.998651i	$0.483463\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.9353	0.671569
$372$	0	0
$373$	5.12218	0.265217	0.132608	−	0.991169i	$-0.457665\pi$
$373$	5.12218	0.265217	0.132608	+	0.991169i	$0.457665\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	50.7280	2.61262
$378$	0	0
$379$	−9.79084	−0.502922	−0.251461	−	0.967867i	$-0.580911\pi$
$379$	−9.79084	−0.502922	−0.251461	+	0.967867i	$0.580911\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.4176	−0.787804	−0.393902	−	0.919152i	$-0.628875\pi$
$383$	−15.4176	−0.787804	−0.393902	+	0.919152i	$0.628875\pi$
$384$	0	0
$385$	−17.1812	−0.875635
$386$	0	0
$387$	0	0
$388$	0	0
$389$	38.3616	1.94501	0.972506	−	0.232877i	$-0.0748138\pi$
$389$	38.3616	1.94501	0.972506	+	0.232877i	$0.0748138\pi$
$390$	0	0
$391$	−6.42405	−0.324878
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.11897	−0.307879
$396$	0	0
$397$	2.87102	0.144092	0.0720462	−	0.997401i	$-0.477047\pi$
$397$	2.87102	0.144092	0.0720462	+	0.997401i	$0.477047\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.8453	1.49040	0.745201	−	0.666840i	$-0.232354\pi$
$401$	29.8453	1.49040	0.745201	+	0.666840i	$0.232354\pi$
$402$	0	0
$403$	29.0405	1.44661
$404$	0	0
$405$	0	0
$406$	0	0
$407$	30.2937	1.50160
$408$	0	0
$409$	−6.21788	−0.307454	−0.153727	−	0.988113i	$-0.549128\pi$
$409$	−6.21788	−0.307454	−0.153727	+	0.988113i	$0.549128\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.7755	0.973087
$414$	0	0
$415$	−8.42405	−0.413520
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.6695	1.64487	0.822433	−	0.568863i	$-0.192616\pi$
$419$	33.6695	1.64487	0.822433	+	0.568863i	$0.192616\pi$
$420$	0	0
$421$	17.0149	0.829256	0.414628	−	0.909991i	$-0.363912\pi$
$421$	17.0149	0.829256	0.414628	+	0.909991i	$0.363912\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.42405	−0.311612
$426$	0	0
$427$	7.09793	0.343493
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.6422	−0.946129	−0.473065	−	0.881028i	$-0.656852\pi$
$431$	−19.6422	−0.946129	−0.473065	+	0.881028i	$0.656852\pi$
$432$	0	0
$433$	−5.24476	−0.252047	−0.126023	−	0.992027i	$-0.540221\pi$
$433$	−5.24476	−0.252047	−0.126023	+	0.992027i	$0.540221\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.80348	−0.0862719
$438$	0	0
$439$	17.9194	0.855249	0.427624	−	0.903957i	$-0.359351\pi$
$439$	17.9194	0.855249	0.427624	+	0.903957i	$0.359351\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.75616	0.463529	0.231764	−	0.972772i	$-0.425550\pi$
$443$	9.75616	0.463529	0.231764	+	0.972772i	$0.425550\pi$
$444$	0	0
$445$	11.3838	0.539643
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.5703	1.63147	0.815736	−	0.578425i	$-0.196332\pi$
$449$	34.5703	1.63147	0.815736	+	0.578425i	$0.196332\pi$
$450$	0	0
$451$	−28.2253	−1.32908
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.5230	−0.868370
$456$	0	0
$457$	30.0455	1.40547	0.702736	−	0.711451i	$-0.251962\pi$
$457$	30.0455	1.40547	0.702736	+	0.711451i	$0.251962\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.2849	0.711888	0.355944	−	0.934507i	$-0.384159\pi$
$461$	15.2849	0.711888	0.355944	+	0.934507i	$0.384159\pi$
$462$	0	0
$463$	4.89235	0.227367	0.113684	−	0.993517i	$-0.463735\pi$
$463$	4.89235	0.227367	0.113684	+	0.993517i	$0.463735\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.4972	1.64262	0.821308	−	0.570485i	$-0.193245\pi$
$467$	35.4972	1.64262	0.821308	+	0.570485i	$0.193245\pi$
$468$	0	0
$469$	19.3225	0.892228
$470$	0	0
$471$	0	0
$472$	0	0
$473$	45.0531	2.07154
$474$	0	0
$475$	−1.80348	−0.0827491
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.80421	−0.356584	−0.178292	−	0.983978i	$-0.557057\pi$
$479$	−7.80421	−0.356584	−0.178292	+	0.983978i	$0.557057\pi$
$480$	0	0
$481$	32.6595	1.48915
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.9843	0.816625
$486$	0	0
$487$	−21.5498	−0.976517	−0.488258	−	0.872699i	$-0.662368\pi$
$487$	−21.5498	−0.976517	−0.488258	+	0.872699i	$0.662368\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.8315	0.714465	0.357232	−	0.934016i	$-0.383720\pi$
$491$	15.8315	0.714465	0.357232	+	0.934016i	$0.383720\pi$
$492$	0	0
$493$	−46.1044	−2.07644
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.3138	−0.462636
$498$	0	0
$499$	−5.76357	−0.258013	−0.129006	−	0.991644i	$-0.541179\pi$
$499$	−5.76357	−0.258013	−0.129006	+	0.991644i	$0.541179\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.8075	0.838583	0.419291	−	0.907852i	$-0.362279\pi$
$503$	18.8075	0.838583	0.419291	+	0.907852i	$0.362279\pi$
$504$	0	0
$505$	1.67126	0.0743699
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.21675	−0.142580	−0.0712900	−	0.997456i	$-0.522712\pi$
$509$	−3.21675	−0.142580	−0.0712900	+	0.997456i	$0.522712\pi$
$510$	0	0
$511$	−13.2198	−0.584810
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.72913	−0.296521
$516$	0	0
$517$	−11.8241	−0.520022
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.9100	−0.653217	−0.326609	−	0.945160i	$-0.605906\pi$
$521$	−14.9100	−0.653217	−0.326609	+	0.945160i	$0.605906\pi$
$522$	0	0
$523$	12.3982	0.542134	0.271067	−	0.962561i	$-0.412624\pi$
$523$	12.3982	0.542134	0.271067	+	0.962561i	$0.412624\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.3936	−1.14972
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−30.4296	−1.31805
$534$	0	0
$535$	17.9296	0.775166
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.869290	−0.0374430
$540$	0	0
$541$	−30.8954	−1.32830	−0.664149	−	0.747600i	$-0.731206\pi$
$541$	−30.8954	−1.32830	−0.664149	+	0.747600i	$0.731206\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.6692	0.542688
$546$	0	0
$547$	14.2251	0.608220	0.304110	−	0.952637i	$-0.401641\pi$
$547$	14.2251	0.608220	0.304110	+	0.952637i	$0.401641\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.9433	−0.551401
$552$	0	0
$553$	16.0352	0.681886
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.6989	0.622811	0.311406	−	0.950277i	$-0.399200\pi$
$557$	14.6989	0.622811	0.311406	+	0.950277i	$0.399200\pi$
$558$	0	0
$559$	48.5715	2.05436
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.4972	−0.821710	−0.410855	−	0.911701i	$-0.634770\pi$
$563$	−19.4972	−0.821710	−0.410855	+	0.911701i	$0.634770\pi$
$564$	0	0
$565$	21.1708	0.890662
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.8854	0.749795	0.374898	−	0.927066i	$-0.377678\pi$
$569$	17.8854	0.749795	0.374898	+	0.927066i	$0.377678\pi$
$570$	0	0
$571$	−23.3904	−0.978856	−0.489428	−	0.872044i	$-0.662794\pi$
$571$	−23.3904	−0.978856	−0.489428	+	0.872044i	$0.662794\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−45.7515	−1.90466	−0.952329	−	0.305072i	$-0.901320\pi$
$577$	−45.7515	−1.90466	−0.952329	+	0.305072i	$0.901320\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.0759	0.915861
$582$	0	0
$583$	32.3622	1.34030
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.9839	−1.03120	−0.515599	−	0.856830i	$-0.672430\pi$
$587$	−24.9839	−1.03120	−0.515599	+	0.856830i	$0.672430\pi$
$588$	0	0
$589$	−7.40969	−0.305311
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.3178	−1.03968	−0.519838	−	0.854265i	$-0.674008\pi$
$593$	−25.3178	−1.03968	−0.519838	+	0.854265i	$0.674008\pi$
$594$	0	0
$595$	16.8347	0.690155
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.23756	−0.214001	−0.107000	−	0.994259i	$-0.534125\pi$
$599$	−5.23756	−0.214001	−0.107000	+	0.994259i	$0.534125\pi$
$600$	0	0
$601$	−26.1173	−1.06535	−0.532673	−	0.846321i	$-0.678812\pi$
$601$	−26.1173	−1.06535	−0.532673	+	0.846321i	$0.678812\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−31.9847	−1.30036
$606$	0	0
$607$	−24.6631	−1.00105	−0.500523	−	0.865723i	$-0.666859\pi$
$607$	−24.6631	−1.00105	−0.500523	+	0.865723i	$0.666859\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.7475	−0.515708
$612$	0	0
$613$	−4.60052	−0.185813	−0.0929067	−	0.995675i	$-0.529616\pi$
$613$	−4.60052	−0.185813	−0.0929067	+	0.995675i	$0.529616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.9143	−0.761461	−0.380731	−	0.924686i	$-0.624327\pi$
$617$	−18.9143	−0.761461	−0.380731	+	0.924686i	$0.624327\pi$
$618$	0	0
$619$	−47.9686	−1.92802	−0.964010	−	0.265865i	$-0.914343\pi$
$619$	−47.9686	−1.92802	−0.964010	+	0.265865i	$0.914343\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−29.8320	−1.19519
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.6828	−1.18353
$630$	0	0
$631$	−29.5445	−1.17615	−0.588074	−	0.808807i	$-0.700114\pi$
$631$	−29.5445	−1.17615	−0.588074	+	0.808807i	$0.700114\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.04425	0.160491
$636$	0	0
$637$	−0.937178	−0.0371323
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.2557	1.70850	0.854248	−	0.519865i	$-0.174018\pi$
$641$	43.2557	1.70850	0.854248	+	0.519865i	$0.174018\pi$
$642$	0	0
$643$	−37.2455	−1.46882	−0.734410	−	0.678707i	$-0.762541\pi$
$643$	−37.2455	−1.46882	−0.734410	+	0.678707i	$0.762541\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.9007	−1.64729	−0.823643	−	0.567109i	$-0.808062\pi$
$647$	−41.9007	−1.64729	−0.823643	+	0.567109i	$0.808062\pi$
$648$	0	0
$649$	49.4751	1.94207
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.4697	−1.42717	−0.713584	−	0.700570i	$-0.752929\pi$
$653$	−36.4697	−1.42717	−0.713584	+	0.700570i	$0.752929\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.8247	0.694350	0.347175	−	0.937800i	$-0.387141\pi$
$659$	17.8247	0.694350	0.347175	+	0.937800i	$0.387141\pi$
$660$	0	0
$661$	−2.70189	−0.105091	−0.0525456	−	0.998619i	$-0.516733\pi$
$661$	−2.70189	−0.105091	−0.0525456	+	0.998619i	$0.516733\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.72614	0.183272
$666$	0	0
$667$	7.17684	0.277889
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17.7579	0.685537
$672$	0	0
$673$	8.19652	0.315953	0.157976	−	0.987443i	$-0.449503\pi$
$673$	8.19652	0.315953	0.157976	+	0.987443i	$0.449503\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.8759	−1.03292	−0.516462	−	0.856310i	$-0.672751\pi$
$677$	−26.8759	−1.03292	−0.516462	+	0.856310i	$0.672751\pi$
$678$	0	0
$679$	−47.1292	−1.80865
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17.1252	0.655277	0.327638	−	0.944803i	$-0.393747\pi$
$683$	17.1252	0.655277	0.327638	+	0.944803i	$0.393747\pi$
$684$	0	0
$685$	5.75316	0.219817
$686$	0	0
$687$	0	0
$688$	0	0
$689$	34.8895	1.32918
$690$	0	0
$691$	15.7042	0.597414	0.298707	−	0.954345i	$-0.403445\pi$
$691$	15.7042	0.597414	0.298707	+	0.954345i	$0.403445\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.50161	−0.170756
$696$	0	0
$697$	27.6561	1.04755
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.81955	0.257571	0.128785	−	0.991672i	$-0.458892\pi$
$701$	6.81955	0.257571	0.128785	+	0.991672i	$0.458892\pi$
$702$	0	0
$703$	−8.33309	−0.314289
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.37965	−0.164714
$708$	0	0
$709$	23.0955	0.867368	0.433684	−	0.901065i	$-0.357213\pi$
$709$	23.0955	0.867368	0.433684	+	0.901065i	$0.357213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.10856	0.153867
$714$	0	0
$715$	−46.3416	−1.73308
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.5152	0.839675	0.419838	−	0.907599i	$-0.362087\pi$
$719$	22.5152	0.839675	0.419838	+	0.907599i	$0.362087\pi$
$720$	0	0
$721$	17.6342	0.656732
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.17684	0.266541
$726$	0	0
$727$	−3.45998	−0.128323	−0.0641617	−	0.997940i	$-0.520437\pi$
$727$	−3.45998	−0.128323	−0.0641617	+	0.997940i	$0.520437\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−44.1445	−1.63274
$732$	0	0
$733$	−16.2835	−0.601446	−0.300723	−	0.953711i	$-0.597228\pi$
$733$	−16.2835	−0.601446	−0.300723	+	0.953711i	$0.597228\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	48.3418	1.78069
$738$	0	0
$739$	24.2459	0.891899	0.445949	−	0.895058i	$-0.352866\pi$
$739$	24.2459	0.891899	0.445949	+	0.895058i	$0.352866\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.7202	−1.12702	−0.563508	−	0.826111i	$-0.690549\pi$
$743$	−30.7202	−1.12702	−0.563508	+	0.826111i	$0.690549\pi$
$744$	0	0
$745$	−14.1572	−0.518678
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46.9860	−1.71683
$750$	0	0
$751$	−37.3493	−1.36290	−0.681448	−	0.731867i	$-0.738649\pi$
$751$	−37.3493	−1.36290	−0.681448	+	0.731867i	$0.738649\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.9847	0.690923
$756$	0	0
$757$	18.8930	0.686677	0.343338	−	0.939212i	$-0.388442\pi$
$757$	18.8930	0.686677	0.343338	+	0.939212i	$0.388442\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	37.5549	1.36137	0.680683	−	0.732578i	$-0.261683\pi$
$761$	37.5549	1.36137	0.680683	+	0.732578i	$0.261683\pi$
$762$	0	0
$763$	−33.2005	−1.20194
$764$	0	0
$765$	0	0
$766$	0	0
$767$	53.3389	1.92596
$768$	0	0
$769$	−35.8799	−1.29386	−0.646931	−	0.762549i	$-0.723948\pi$
$769$	−35.8799	−1.29386	−0.646931	+	0.762549i	$0.723948\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.3745	1.16443	0.582215	−	0.813035i	$-0.302186\pi$
$773$	32.3745	1.16443	0.582215	+	0.813035i	$0.302186\pi$
$774$	0	0
$775$	4.10856	0.147584
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.76411	0.278178
$780$	0	0
$781$	−25.8035	−0.923321
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.22753	−0.293653
$786$	0	0
$787$	−47.7278	−1.70131	−0.850656	−	0.525723i	$-0.823795\pi$
$787$	−47.7278	−1.70131	−0.850656	+	0.525723i	$0.823795\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−55.4796	−1.97263
$792$	0	0
$793$	19.1447	0.679849
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.8427	0.454910	0.227455	−	0.973789i	$-0.426959\pi$
$797$	12.8427	0.454910	0.227455	+	0.973789i	$0.426959\pi$
$798$	0	0
$799$	11.5856	0.409870
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−33.0739	−1.16715
$804$	0	0
$805$	−2.62057	−0.0923631
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.84864	−0.311102	−0.155551	−	0.987828i	$-0.549715\pi$
$809$	−8.84864	−0.311102	−0.155551	+	0.987828i	$0.549715\pi$
$810$	0	0
$811$	30.4149	1.06801	0.534006	−	0.845480i	$-0.320686\pi$
$811$	30.4149	1.06801	0.534006	+	0.845480i	$0.320686\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.264439	−0.00926289
$816$	0	0
$817$	−12.3930	−0.433578
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31.2463	−1.09050	−0.545251	−	0.838273i	$-0.683566\pi$
$821$	−31.2463	−1.09050	−0.545251	+	0.838273i	$0.683566\pi$
$822$	0	0
$823$	−50.6232	−1.76462	−0.882308	−	0.470673i	$-0.844011\pi$
$823$	−50.6232	−1.76462	−0.882308	+	0.470673i	$0.844011\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.5478	1.13180	0.565898	−	0.824475i	$-0.308529\pi$
$827$	32.5478	1.13180	0.565898	+	0.824475i	$0.308529\pi$
$828$	0	0
$829$	−29.0119	−1.00763	−0.503813	−	0.863813i	$-0.668070\pi$
$829$	−29.0119	−1.00763	−0.503813	+	0.863813i	$0.668070\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.851759	0.0295117
$834$	0	0
$835$	9.45775	0.327299
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.97993	0.0683547	0.0341774	−	0.999416i	$-0.489119\pi$
$839$	1.97993	0.0683547	0.0341774	+	0.999416i	$0.489119\pi$
$840$	0	0
$841$	22.5071	0.776106
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−36.9606	−1.27148
$846$	0	0
$847$	83.8182	2.88003
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.62057	0.158391
$852$	0	0
$853$	−11.1781	−0.382732	−0.191366	−	0.981519i	$-0.561292\pi$
$853$	−11.1781	−0.382732	−0.191366	+	0.981519i	$0.561292\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.2491	−0.999130	−0.499565	−	0.866276i	$-0.666507\pi$
$857$	−29.2491	−0.999130	−0.499565	+	0.866276i	$0.666507\pi$
$858$	0	0
$859$	47.5143	1.62117	0.810584	−	0.585623i	$-0.199150\pi$
$859$	47.5143	1.62117	0.810584	+	0.585623i	$0.199150\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.5628	0.563806	0.281903	−	0.959443i	$-0.409034\pi$
$863$	16.5628	0.563806	0.281903	+	0.959443i	$0.409034\pi$
$864$	0	0
$865$	−0.758851	−0.0258017
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40.1176	1.36090
$870$	0	0
$871$	52.1171	1.76592
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.62057	−0.0885916
$876$	0	0
$877$	−4.80976	−0.162414	−0.0812070	−	0.996697i	$-0.525877\pi$
$877$	−4.80976	−0.162414	−0.0812070	+	0.996697i	$0.525877\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.8046	−1.00414	−0.502072	−	0.864826i	$-0.667429\pi$
$881$	−29.8046	−1.00414	−0.502072	+	0.864826i	$0.667429\pi$
$882$	0	0
$883$	−22.1579	−0.745673	−0.372836	−	0.927897i	$-0.621615\pi$
$883$	−22.1579	−0.745673	−0.372836	+	0.927897i	$0.621615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.3425	−1.05238	−0.526189	−	0.850367i	$-0.676380\pi$
$887$	−31.3425	−1.05238	−0.526189	+	0.850367i	$0.676380\pi$
$888$	0	0
$889$	−10.5983	−0.355455
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.25252	0.108842
$894$	0	0
$895$	21.9606	0.734063
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29.4865	0.983430
$900$	0	0
$901$	−31.7095	−1.05640
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.4727	0.481087
$906$	0	0
$907$	11.3541	0.377006	0.188503	−	0.982073i	$-0.439636\pi$
$907$	11.3541	0.377006	0.188503	+	0.982073i	$0.439636\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.23245	−0.173359	−0.0866794	−	0.996236i	$-0.527626\pi$
$911$	−5.23245	−0.173359	−0.0866794	+	0.996236i	$0.527626\pi$
$912$	0	0
$913$	55.2303	1.82786
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	58.1349	1.91769	0.958846	−	0.283925i	$-0.0916368\pi$
$919$	58.1349	1.91769	0.958846	+	0.283925i	$0.0916368\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−27.8186	−0.915661
$924$	0	0
$925$	4.62057	0.151924
$926$	0	0
$927$	0	0
$928$	0	0
$929$	31.9681	1.04884	0.524419	−	0.851460i	$-0.324283\pi$
$929$	31.9681	1.04884	0.524419	+	0.851460i	$0.324283\pi$
$930$	0	0
$931$	0.239121	0.00783688
$932$	0	0
$933$	0	0
$934$	0	0
$935$	42.1178	1.37740
$936$	0	0
$937$	44.2567	1.44580	0.722902	−	0.690951i	$-0.242808\pi$
$937$	44.2567	1.44580	0.722902	+	0.690951i	$0.242808\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.4144	−0.600292	−0.300146	−	0.953893i	$-0.597035\pi$
$941$	−18.4144	−0.600292	−0.300146	+	0.953893i	$0.597035\pi$
$942$	0	0
$943$	−4.30508	−0.140193
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.8625	−1.58782	−0.793909	−	0.608037i	$-0.791957\pi$
$947$	−48.8625	−1.58782	−0.793909	+	0.608037i	$0.791957\pi$
$948$	0	0
$949$	−35.6568	−1.15747
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49.0643	1.58935	0.794674	−	0.607037i	$-0.207642\pi$
$953$	49.0643	1.58935	0.794674	+	0.607037i	$0.207642\pi$
$954$	0	0
$955$	−11.9400	−0.386371
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.0766	−0.486849
$960$	0	0
$961$	−14.1197	−0.455475
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11.6342	−0.374518
$966$	0	0
$967$	−20.4921	−0.658981	−0.329491	−	0.944159i	$-0.606877\pi$
$967$	−20.4921	−0.658981	−0.329491	+	0.944159i	$0.606877\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.8581	0.444728	0.222364	−	0.974964i	$-0.428623\pi$
$971$	13.8581	0.444728	0.222364	+	0.974964i	$0.428623\pi$
$972$	0	0
$973$	11.7968	0.378188
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.1668	0.325266	0.162633	−	0.986687i	$-0.448001\pi$
$977$	10.1668	0.325266	0.162633	+	0.986687i	$0.448001\pi$
$978$	0	0
$979$	−74.6351	−2.38535
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.6875	−1.61668	−0.808341	−	0.588715i	$-0.799634\pi$
$983$	−50.6875	−1.61668	−0.808341	+	0.588715i	$0.799634\pi$
$984$	0	0
$985$	24.6181	0.784399
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.87176	0.218509
$990$	0	0
$991$	19.9799	0.634684	0.317342	−	0.948311i	$-0.397210\pi$
$991$	19.9799	0.634684	0.317342	+	0.948311i	$0.397210\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.7291	−0.593753
$996$	0	0
$997$	−7.41454	−0.234821	−0.117410	−	0.993083i	$-0.537459\pi$
$997$	−7.41454	−0.234821	−0.117410	+	0.993083i	$0.537459\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.br.1.4		5
3.2	odd	2		2760.2.a.w.1.4	✓	5
12.11	even	2		5520.2.a.cc.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.w.1.4	✓	5	3.2	odd	2
5520.2.a.cc.1.2		5	12.11	even	2
8280.2.a.br.1.4		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.62057 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.62057 q^{7} +6.55627 q^{11} +7.06828 q^{13} -6.42405 q^{17} -1.80348 q^{19} +1.00000 q^{23} +1.00000 q^{25} +7.17684 q^{29} +4.10856 q^{31} -2.62057 q^{35} +4.62057 q^{37} -4.30508 q^{41} +6.87176 q^{43} -1.80348 q^{47} -0.132589 q^{49} +4.93606 q^{53} -6.55627 q^{55} +7.54623 q^{59} +2.70854 q^{61} -7.06828 q^{65} +7.37337 q^{67} -3.93569 q^{71} -5.04462 q^{73} +17.1812 q^{77} +6.11897 q^{79} +8.42405 q^{83} +6.42405 q^{85} -11.3838 q^{89} +18.5230 q^{91} +1.80348 q^{95} -17.9843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 4 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 - 4 * q^7	\( 5 q - 5 q^{5} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 10 q^{17} - 4 q^{19} + 5 q^{23} + 5 q^{25} - 10 q^{29} + 6 q^{31} + 4 q^{35} + 6 q^{37} - 12 q^{41} - 2 q^{43} - 4 q^{47} + 19 q^{49} - 4 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{65} - 4 q^{67} - 8 q^{71} + 14 q^{73} - 16 q^{77} + 18 q^{79} + 20 q^{83} + 10 q^{85} - 18 q^{89} + 20 q^{91} + 4 q^{95} + 4 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 10 * q^17 - 4 * q^19 + 5 * q^23 + 5 * q^25 - 10 * q^29 + 6 * q^31 + 4 * q^35 + 6 * q^37 - 12 * q^41 - 2 * q^43 - 4 * q^47 + 19 * q^49 - 4 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 14 * q^73 - 16 * q^77 + 18 * q^79 + 20 * q^83 + 10 * q^85 - 18 * q^89 + 20 * q^91 + 4 * q^95 + 4 * q^97

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