LMFDB - Embedded newform 8280.2.a.bu.1.1

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 16x^{4} + 26x^{3} + 52x^{2} - 48x + 9 $ x^6 - 2x^5 - 16x^4 + 26x^3 + 52x^2 - 48*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.485614$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -4.41777 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.41777 q^{7} -0.245598 q^{11} +0.725630 q^{13} -0.737339 q^{17} +1.75440 q^{19} +1.00000 q^{23} +1.00000 q^{25} -10.2258 q^{29} -3.46297 q^{31} -4.41777 q^{35} +0.273677 q^{37} +6.36154 q^{41} +1.02877 q^{43} +3.89849 q^{47} +12.5166 q^{49} -5.12696 q^{53} -0.245598 q^{55} +3.69214 q^{59} -9.32805 q^{61} +0.725630 q^{65} +8.56185 q^{67} -2.72091 q^{71} +13.2828 q^{73} +1.08499 q^{77} -13.0825 q^{79} -16.2960 q^{83} -0.737339 q^{85} +14.1978 q^{89} -3.20566 q^{91} +1.75440 q^{95} +18.6778 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 2 q^{7}+O(q^{10}) $ 6 * q + 6 * q^5 - 2 * q^7	$ 6 q + 6 q^{5} - 2 q^{7} - 4 q^{11} + 4 q^{17} + 8 q^{19} + 6 q^{23} + 6 q^{25} + 18 q^{29} - 8 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 2 q^{59} - 8 q^{61} - 2 q^{67} + 2 q^{71} + 8 q^{73} + 16 q^{77} - 28 q^{79} + 24 q^{83} + 4 q^{85} + 4 q^{89} - 8 q^{91} + 8 q^{95} + 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^5 - 2 * q^7 - 4 * q^11 + 4 * q^17 + 8 * q^19 + 6 * q^23 + 6 * q^25 + 18 * q^29 - 8 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^41 + 8 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 2 * q^59 - 8 * q^61 - 2 * q^67 + 2 * q^71 + 8 * q^73 + 16 * q^77 - 28 * q^79 + 24 * q^83 + 4 * q^85 + 4 * q^89 - 8 * q^91 + 8 * q^95 + 24 * 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
$7$	−4.41777	−1.66976	−0.834879	−	0.550433i	$-0.814463\pi$
$7$	−4.41777	−1.66976	−0.834879	+	0.550433i	$0.814463\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.245598	−0.0740505	−0.0370253	−	0.999314i	$-0.511788\pi$
$11$	−0.245598	−0.0740505	−0.0370253	+	0.999314i	$0.511788\pi$
$12$	0	0
$13$	0.725630	0.201253	0.100627	−	0.994924i	$-0.467915\pi$
$13$	0.725630	0.201253	0.100627	+	0.994924i	$0.467915\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.737339	−0.178831	−0.0894155	−	0.995994i	$-0.528500\pi$
$17$	−0.737339	−0.178831	−0.0894155	+	0.995994i	$0.528500\pi$
$18$	0	0
$19$	1.75440	0.402487	0.201244	−	0.979541i	$-0.435502\pi$
$19$	1.75440	0.402487	0.201244	+	0.979541i	$0.435502\pi$
$20$	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
$29$	−10.2258	−1.89889	−0.949446	−	0.313930i	$-0.898354\pi$
$29$	−10.2258	−1.89889	−0.949446	+	0.313930i	$0.898354\pi$
$30$	0	0
$31$	−3.46297	−0.621968	−0.310984	−	0.950415i	$-0.600658\pi$
$31$	−3.46297	−0.621968	−0.310984	+	0.950415i	$0.600658\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.41777	−0.746739
$36$	0	0
$37$	0.273677	0.0449923	0.0224961	−	0.999747i	$-0.492839\pi$
$37$	0.273677	0.0449923	0.0224961	+	0.999747i	$0.492839\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.36154	0.993506	0.496753	−	0.867892i	$-0.334525\pi$
$41$	6.36154	0.993506	0.496753	+	0.867892i	$0.334525\pi$
$42$	0	0
$43$	1.02877	0.156886	0.0784432	−	0.996919i	$-0.475005\pi$
$43$	1.02877	0.156886	0.0784432	+	0.996919i	$0.475005\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.89849	0.568653	0.284327	−	0.958727i	$-0.408230\pi$
$47$	3.89849	0.568653	0.284327	+	0.958727i	$0.408230\pi$
$48$	0	0
$49$	12.5166	1.78809
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.12696	−0.704243	−0.352121	−	0.935954i	$-0.614540\pi$
$53$	−5.12696	−0.704243	−0.352121	+	0.935954i	$0.614540\pi$
$54$	0	0
$55$	−0.245598	−0.0331164
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.69214	0.480675	0.240338	−	0.970689i	$-0.422742\pi$
$59$	3.69214	0.480675	0.240338	+	0.970689i	$0.422742\pi$
$60$	0	0
$61$	−9.32805	−1.19433	−0.597167	−	0.802117i	$-0.703707\pi$
$61$	−9.32805	−1.19433	−0.597167	+	0.802117i	$0.703707\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.725630	0.0900033
$66$	0	0
$67$	8.56185	1.04600	0.522998	−	0.852334i	$-0.324813\pi$
$67$	8.56185	1.04600	0.522998	+	0.852334i	$0.324813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.72091	−0.322912	−0.161456	−	0.986880i	$-0.551619\pi$
$71$	−2.72091	−0.322912	−0.161456	+	0.986880i	$0.551619\pi$
$72$	0	0
$73$	13.2828	1.55463	0.777315	−	0.629112i	$-0.216581\pi$
$73$	13.2828	1.55463	0.777315	+	0.629112i	$0.216581\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.08499	0.123646
$78$	0	0
$79$	−13.0825	−1.47189	−0.735945	−	0.677041i	$-0.763262\pi$
$79$	−13.0825	−1.47189	−0.735945	+	0.677041i	$0.763262\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.2960	−1.78871	−0.894357	−	0.447354i	$-0.852366\pi$
$83$	−16.2960	−1.78871	−0.894357	+	0.447354i	$0.852366\pi$
$84$	0	0
$85$	−0.737339	−0.0799757
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.1978	1.50496	0.752480	−	0.658615i	$-0.228857\pi$
$89$	14.1978	1.50496	0.752480	+	0.658615i	$0.228857\pi$
$90$	0	0
$91$	−3.20566	−0.336045
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.75440	0.179998
$96$	0	0
$97$	18.6778	1.89644	0.948222	−	0.317610i	$-0.102880\pi$
$97$	18.6778	1.89644	0.948222	+	0.317610i	$0.102880\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.28617	0.227482	0.113741	−	0.993510i	$-0.463717\pi$
$101$	2.28617	0.227482	0.113741	+	0.993510i	$0.463717\pi$
$102$	0	0
$103$	−3.40597	−0.335600	−0.167800	−	0.985821i	$-0.553666\pi$
$103$	−3.40597	−0.335600	−0.167800	+	0.985821i	$0.553666\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.95549	0.189044	0.0945221	−	0.995523i	$-0.469868\pi$
$107$	1.95549	0.189044	0.0945221	+	0.995523i	$0.469868\pi$
$108$	0	0
$109$	13.7052	1.31272	0.656362	−	0.754446i	$-0.272094\pi$
$109$	13.7052	1.31272	0.656362	+	0.754446i	$0.272094\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.39023	−0.224854	−0.112427	−	0.993660i	$-0.535862\pi$
$113$	−2.39023	−0.224854	−0.112427	+	0.993660i	$0.535862\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.25739	0.298605
$120$	0	0
$121$	−10.9397	−0.994517
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.14403	−0.367723	−0.183861	−	0.982952i	$-0.558860\pi$
$127$	−4.14403	−0.367723	−0.183861	+	0.982952i	$0.558860\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.86966	0.774945	0.387473	−	0.921881i	$-0.373348\pi$
$131$	8.86966	0.774945	0.387473	+	0.921881i	$0.373348\pi$
$132$	0	0
$133$	−7.75054	−0.672057
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.92185	0.847681	0.423840	−	0.905737i	$-0.360682\pi$
$137$	9.92185	0.847681	0.423840	+	0.905737i	$0.360682\pi$
$138$	0	0
$139$	−10.0654	−0.853735	−0.426868	−	0.904314i	$-0.640383\pi$
$139$	−10.0654	−0.853735	−0.426868	+	0.904314i	$0.640383\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.178213	−0.0149029
$144$	0	0
$145$	−10.2258	−0.849210
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.1715	0.915208	0.457604	−	0.889156i	$-0.348708\pi$
$149$	11.1715	0.915208	0.457604	+	0.889156i	$0.348708\pi$
$150$	0	0
$151$	−10.6914	−0.870057	−0.435029	−	0.900417i	$-0.643262\pi$
$151$	−10.6914	−0.870057	−0.435029	+	0.900417i	$0.643262\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.46297	−0.278152
$156$	0	0
$157$	21.6541	1.72818	0.864092	−	0.503334i	$-0.167893\pi$
$157$	21.6541	1.72818	0.864092	+	0.503334i	$0.167893\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.41777	−0.348169
$162$	0	0
$163$	4.37720	0.342849	0.171424	−	0.985197i	$-0.445163\pi$
$163$	4.37720	0.342849	0.171424	+	0.985197i	$0.445163\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.29928	−0.487453	−0.243726	−	0.969844i	$-0.578370\pi$
$167$	−6.29928	−0.487453	−0.243726	+	0.969844i	$0.578370\pi$
$168$	0	0
$169$	−12.4735	−0.959497
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.90390	0.524894	0.262447	−	0.964946i	$-0.415470\pi$
$173$	6.90390	0.524894	0.262447	+	0.964946i	$0.415470\pi$
$174$	0	0
$175$	−4.41777	−0.333952
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.27569	−0.244837	−0.122418	−	0.992479i	$-0.539065\pi$
$179$	−3.27569	−0.244837	−0.122418	+	0.992479i	$0.539065\pi$
$180$	0	0
$181$	0.828526	0.0615838	0.0307919	−	0.999526i	$-0.490197\pi$
$181$	0.828526	0.0615838	0.0307919	+	0.999526i	$0.490197\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.273677	0.0201212
$186$	0	0
$187$	0.181089	0.0132425
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.79434	0.636336	0.318168	−	0.948034i	$-0.396932\pi$
$191$	8.79434	0.636336	0.318168	+	0.948034i	$0.396932\pi$
$192$	0	0
$193$	−5.30469	−0.381840	−0.190920	−	0.981606i	$-0.561147\pi$
$193$	−5.30469	−0.381840	−0.190920	+	0.981606i	$0.561147\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.2703	1.23046	0.615228	−	0.788349i	$-0.289064\pi$
$197$	17.2703	1.23046	0.615228	+	0.788349i	$0.289064\pi$
$198$	0	0
$199$	0.0989687	0.00701571	0.00350785	−	0.999994i	$-0.498883\pi$
$199$	0.0989687	0.00701571	0.00350785	+	0.999994i	$0.498883\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	45.1754	3.17069
$204$	0	0
$205$	6.36154	0.444310
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.430877	−0.0298044
$210$	0	0
$211$	−6.76070	−0.465426	−0.232713	−	0.972546i	$-0.574760\pi$
$211$	−6.76070	−0.465426	−0.232713	+	0.972546i	$0.574760\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.02877	0.0701617
$216$	0	0
$217$	15.2986	1.03854
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.535035	−0.0359904
$222$	0	0
$223$	−16.1550	−1.08182	−0.540908	−	0.841082i	$-0.681919\pi$
$223$	−16.1550	−1.08182	−0.540908	+	0.841082i	$0.681919\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.06484	−0.402538	−0.201269	−	0.979536i	$-0.564507\pi$
$227$	−6.06484	−0.402538	−0.201269	+	0.979536i	$0.564507\pi$
$228$	0	0
$229$	24.0582	1.58981	0.794905	−	0.606733i	$-0.207520\pi$
$229$	24.0582	1.58981	0.794905	+	0.606733i	$0.207520\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.2539	0.802781	0.401391	−	0.915907i	$-0.368527\pi$
$233$	12.2539	0.802781	0.401391	+	0.915907i	$0.368527\pi$
$234$	0	0
$235$	3.89849	0.254309
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.4590	1.19401	0.597005	−	0.802237i	$-0.296357\pi$
$239$	18.4590	1.19401	0.597005	+	0.802237i	$0.296357\pi$
$240$	0	0
$241$	8.62544	0.555614	0.277807	−	0.960637i	$-0.410392\pi$
$241$	8.62544	0.555614	0.277807	+	0.960637i	$0.410392\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.5166	0.799659
$246$	0	0
$247$	1.27305	0.0810020
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.4513	0.722797	0.361399	−	0.932411i	$-0.382299\pi$
$251$	11.4513	0.722797	0.361399	+	0.932411i	$0.382299\pi$
$252$	0	0
$253$	−0.245598	−0.0154406
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.6765	1.53928	0.769638	−	0.638480i	$-0.220437\pi$
$257$	24.6765	1.53928	0.769638	+	0.638480i	$0.220437\pi$
$258$	0	0
$259$	−1.20904	−0.0751263
$260$	0	0
$261$	0	0
$262$	0	0
$263$	31.1427	1.92034	0.960169	−	0.279420i	$-0.0901420\pi$
$263$	31.1427	1.92034	0.960169	+	0.279420i	$0.0901420\pi$
$264$	0	0
$265$	−5.12696	−0.314947
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.0065	1.76856	0.884278	−	0.466961i	$-0.154651\pi$
$269$	29.0065	1.76856	0.884278	+	0.466961i	$0.154651\pi$
$270$	0	0
$271$	−12.5881	−0.764670	−0.382335	−	0.924024i	$-0.624880\pi$
$271$	−12.5881	−0.764670	−0.382335	+	0.924024i	$0.624880\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.245598	−0.0148101
$276$	0	0
$277$	20.4773	1.23036	0.615180	−	0.788387i	$-0.289083\pi$
$277$	20.4773	1.23036	0.615180	+	0.788387i	$0.289083\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.22925	−0.0733310	−0.0366655	−	0.999328i	$-0.511674\pi$
$281$	−1.22925	−0.0733310	−0.0366655	+	0.999328i	$0.511674\pi$
$282$	0	0
$283$	−3.67264	−0.218316	−0.109158	−	0.994024i	$-0.534815\pi$
$283$	−3.67264	−0.218316	−0.109158	+	0.994024i	$0.534815\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.1038	−1.65892
$288$	0	0
$289$	−16.4563	−0.968019
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.94294	0.464031	0.232016	−	0.972712i	$-0.425468\pi$
$293$	7.94294	0.464031	0.232016	+	0.972712i	$0.425468\pi$
$294$	0	0
$295$	3.69214	0.214964
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.725630	0.0419642
$300$	0	0
$301$	−4.54488	−0.261962
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.32805	−0.534123
$306$	0	0
$307$	24.4721	1.39669	0.698347	−	0.715759i	$-0.253919\pi$
$307$	24.4721	1.39669	0.698347	+	0.715759i	$0.253919\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.535097	−0.0303426	−0.0151713	−	0.999885i	$-0.504829\pi$
$311$	−0.535097	−0.0303426	−0.0151713	+	0.999885i	$0.504829\pi$
$312$	0	0
$313$	19.2521	1.08819	0.544096	−	0.839023i	$-0.316873\pi$
$313$	19.2521	1.08819	0.544096	+	0.839023i	$0.316873\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.7411	1.22110	0.610551	−	0.791977i	$-0.290948\pi$
$317$	21.7411	1.22110	0.610551	+	0.791977i	$0.290948\pi$
$318$	0	0
$319$	2.51145	0.140614
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.29359	−0.0719773
$324$	0	0
$325$	0.725630	0.0402507
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.2226	−0.949513
$330$	0	0
$331$	−12.4465	−0.684119	−0.342059	−	0.939678i	$-0.611124\pi$
$331$	−12.4465	−0.684119	−0.342059	+	0.939678i	$0.611124\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.56185	0.467784
$336$	0	0
$337$	0.540350	0.0294347	0.0147174	−	0.999892i	$-0.495315\pi$
$337$	0.540350	0.0294347	0.0147174	+	0.999892i	$0.495315\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.850498	0.0460570
$342$	0	0
$343$	−24.3713	−1.31592
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.94246	0.319008	0.159504	−	0.987197i	$-0.449011\pi$
$347$	5.94246	0.319008	0.159504	+	0.987197i	$0.449011\pi$
$348$	0	0
$349$	−6.74811	−0.361218	−0.180609	−	0.983555i	$-0.557807\pi$
$349$	−6.74811	−0.361218	−0.180609	+	0.983555i	$0.557807\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.31562	0.282922	0.141461	−	0.989944i	$-0.454820\pi$
$353$	5.31562	0.282922	0.141461	+	0.989944i	$0.454820\pi$
$354$	0	0
$355$	−2.72091	−0.144411
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.3858	0.759255	0.379627	−	0.925140i	$-0.376052\pi$
$359$	14.3858	0.759255	0.379627	+	0.925140i	$0.376052\pi$
$360$	0	0
$361$	−15.9221	−0.838004
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.2828	0.695251
$366$	0	0
$367$	−6.18466	−0.322836	−0.161418	−	0.986886i	$-0.551607\pi$
$367$	−6.18466	−0.322836	−0.161418	+	0.986886i	$0.551607\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	22.6497	1.17591
$372$	0	0
$373$	−33.8811	−1.75430	−0.877148	−	0.480220i	$-0.840557\pi$
$373$	−33.8811	−1.75430	−0.877148	+	0.480220i	$0.840557\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.42018	−0.382159
$378$	0	0
$379$	6.63112	0.340618	0.170309	−	0.985391i	$-0.445523\pi$
$379$	6.63112	0.340618	0.170309	+	0.985391i	$0.445523\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.4498	1.55591	0.777957	−	0.628318i	$-0.216256\pi$
$383$	30.4498	1.55591	0.777957	+	0.628318i	$0.216256\pi$
$384$	0	0
$385$	1.08499	0.0552964
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.5655	−0.839905	−0.419953	−	0.907546i	$-0.637953\pi$
$389$	−16.5655	−0.839905	−0.419953	+	0.907546i	$0.637953\pi$
$390$	0	0
$391$	−0.737339	−0.0372889
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.0825	−0.658250
$396$	0	0
$397$	−25.1543	−1.26246	−0.631229	−	0.775596i	$-0.717449\pi$
$397$	−25.1543	−1.26246	−0.631229	+	0.775596i	$0.717449\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.79559	−0.289418	−0.144709	−	0.989474i	$-0.546225\pi$
$401$	−5.79559	−0.289418	−0.144709	+	0.989474i	$0.546225\pi$
$402$	0	0
$403$	−2.51283	−0.125173
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.0672146	−0.00333170
$408$	0	0
$409$	0.360203	0.0178109	0.00890546	−	0.999960i	$-0.497165\pi$
$409$	0.360203	0.0178109	0.00890546	+	0.999960i	$0.497165\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−16.3110	−0.802611
$414$	0	0
$415$	−16.2960	−0.799937
$416$	0	0
$417$	0	0
$418$	0	0
$419$	29.3599	1.43432	0.717162	−	0.696907i	$-0.245441\pi$
$419$	29.3599	1.43432	0.717162	+	0.696907i	$0.245441\pi$
$420$	0	0
$421$	−18.8355	−0.917985	−0.458992	−	0.888440i	$-0.651789\pi$
$421$	−18.8355	−0.917985	−0.458992	+	0.888440i	$0.651789\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.737339	−0.0357662
$426$	0	0
$427$	41.2091	1.99425
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.92732	0.237341	0.118670	−	0.992934i	$-0.462137\pi$
$431$	4.92732	0.237341	0.118670	+	0.992934i	$0.462137\pi$
$432$	0	0
$433$	15.3671	0.738497	0.369248	−	0.929331i	$-0.379615\pi$
$433$	15.3671	0.738497	0.369248	+	0.929331i	$0.379615\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.75440	0.0839244
$438$	0	0
$439$	16.3652	0.781066	0.390533	−	0.920589i	$-0.372291\pi$
$439$	16.3652	0.781066	0.390533	+	0.920589i	$0.372291\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.9633	−0.615903	−0.307952	−	0.951402i	$-0.599644\pi$
$443$	−12.9633	−0.615903	−0.307952	+	0.951402i	$0.599644\pi$
$444$	0	0
$445$	14.1978	0.673039
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−25.2506	−1.19165	−0.595824	−	0.803115i	$-0.703175\pi$
$449$	−25.2506	−1.19165	−0.595824	+	0.803115i	$0.703175\pi$
$450$	0	0
$451$	−1.56238	−0.0735697
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.20566	−0.150284
$456$	0	0
$457$	13.3241	0.623277	0.311639	−	0.950201i	$-0.399122\pi$
$457$	13.3241	0.623277	0.311639	+	0.950201i	$0.399122\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.8825	−0.926019	−0.463010	−	0.886353i	$-0.653230\pi$
$461$	−19.8825	−0.926019	−0.463010	+	0.886353i	$0.653230\pi$
$462$	0	0
$463$	1.65157	0.0767549	0.0383774	−	0.999263i	$-0.487781\pi$
$463$	1.65157	0.0767549	0.0383774	+	0.999263i	$0.487781\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.28348	−0.0593922	−0.0296961	−	0.999559i	$-0.509454\pi$
$467$	−1.28348	−0.0593922	−0.0296961	+	0.999559i	$0.509454\pi$
$468$	0	0
$469$	−37.8243	−1.74656
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.252664	−0.0116175
$474$	0	0
$475$	1.75440	0.0804975
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.0595	−1.46484	−0.732419	−	0.680855i	$-0.761609\pi$
$479$	−32.0595	−1.46484	−0.732419	+	0.680855i	$0.761609\pi$
$480$	0	0
$481$	0.198588	0.00905486
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.6778	0.848115
$486$	0	0
$487$	−8.95626	−0.405847	−0.202923	−	0.979195i	$-0.565044\pi$
$487$	−8.95626	−0.405847	−0.202923	+	0.979195i	$0.565044\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.2930	0.825552	0.412776	−	0.910833i	$-0.364559\pi$
$491$	18.2930	0.825552	0.412776	+	0.910833i	$0.364559\pi$
$492$	0	0
$493$	7.53992	0.339581
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0203	0.539186
$498$	0	0
$499$	−42.2920	−1.89325	−0.946626	−	0.322335i	$-0.895532\pi$
$499$	−42.2920	−1.89325	−0.946626	+	0.322335i	$0.895532\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.96067	0.399537	0.199768	−	0.979843i	$-0.435981\pi$
$503$	8.96067	0.399537	0.199768	+	0.979843i	$0.435981\pi$
$504$	0	0
$505$	2.28617	0.101733
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.5698	0.956064	0.478032	−	0.878342i	$-0.341350\pi$
$509$	21.5698	0.956064	0.478032	+	0.878342i	$0.341350\pi$
$510$	0	0
$511$	−58.6801	−2.59586
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.40597	−0.150085
$516$	0	0
$517$	−0.957461	−0.0421091
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.90097	−0.170905	−0.0854523	−	0.996342i	$-0.527234\pi$
$521$	−3.90097	−0.170905	−0.0854523	+	0.996342i	$0.527234\pi$
$522$	0	0
$523$	36.3684	1.59028	0.795140	−	0.606426i	$-0.207397\pi$
$523$	36.3684	1.59028	0.795140	+	0.606426i	$0.207397\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.55338	0.111227
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.61613	0.199947
$534$	0	0
$535$	1.95549	0.0845432
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.07406	−0.132409
$540$	0	0
$541$	13.0728	0.562045	0.281023	−	0.959701i	$-0.409326\pi$
$541$	13.0728	0.562045	0.281023	+	0.959701i	$0.409326\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.7052	0.587068
$546$	0	0
$547$	13.7971	0.589924	0.294962	−	0.955509i	$-0.404693\pi$
$547$	13.7971	0.589924	0.294962	+	0.955509i	$0.404693\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17.9402	−0.764280
$552$	0	0
$553$	57.7952	2.45770
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.70874	0.241887	0.120943	−	0.992659i	$-0.461408\pi$
$557$	5.70874	0.241887	0.120943	+	0.992659i	$0.461408\pi$
$558$	0	0
$559$	0.746508	0.0315739
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.2976	−1.06617	−0.533084	−	0.846063i	$-0.678967\pi$
$563$	−25.2976	−1.06617	−0.533084	+	0.846063i	$0.678967\pi$
$564$	0	0
$565$	−2.39023	−0.100558
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.2692	−0.472429	−0.236214	−	0.971701i	$-0.575907\pi$
$569$	−11.2692	−0.472429	−0.236214	+	0.971701i	$0.575907\pi$
$570$	0	0
$571$	−14.8808	−0.622740	−0.311370	−	0.950289i	$-0.600788\pi$
$571$	−14.8808	−0.622740	−0.311370	+	0.950289i	$0.600788\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	32.1743	1.33944	0.669718	−	0.742616i	$-0.266415\pi$
$577$	32.1743	1.33944	0.669718	+	0.742616i	$0.266415\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	71.9917	2.98672
$582$	0	0
$583$	1.25917	0.0521495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.8027	0.693520	0.346760	−	0.937954i	$-0.387282\pi$
$587$	16.8027	0.693520	0.346760	+	0.937954i	$0.387282\pi$
$588$	0	0
$589$	−6.07544	−0.250334
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.3158	−0.464684	−0.232342	−	0.972634i	$-0.574639\pi$
$593$	−11.3158	−0.464684	−0.232342	+	0.972634i	$0.574639\pi$
$594$	0	0
$595$	3.25739	0.133540
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.7326	1.66429	0.832144	−	0.554560i	$-0.187113\pi$
$599$	40.7326	1.66429	0.832144	+	0.554560i	$0.187113\pi$
$600$	0	0
$601$	14.4452	0.589234	0.294617	−	0.955615i	$-0.404808\pi$
$601$	14.4452	0.589234	0.294617	+	0.955615i	$0.404808\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.9397	−0.444761
$606$	0	0
$607$	32.8828	1.33467	0.667336	−	0.744757i	$-0.267435\pi$
$607$	32.8828	1.33467	0.667336	+	0.744757i	$0.267435\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.82886	0.114443
$612$	0	0
$613$	−13.0141	−0.525634	−0.262817	−	0.964846i	$-0.584651\pi$
$613$	−13.0141	−0.525634	−0.262817	+	0.964846i	$0.584651\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.9737	0.804109	0.402055	−	0.915616i	$-0.368296\pi$
$617$	19.9737	0.804109	0.402055	+	0.915616i	$0.368296\pi$
$618$	0	0
$619$	29.4274	1.18279	0.591394	−	0.806382i	$-0.298578\pi$
$619$	29.4274	1.18279	0.591394	+	0.806382i	$0.298578\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−62.7224	−2.51292
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.201793	−0.00804602
$630$	0	0
$631$	35.6892	1.42076	0.710382	−	0.703817i	$-0.248522\pi$
$631$	35.6892	1.42076	0.710382	+	0.703817i	$0.248522\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.14403	−0.164451
$636$	0	0
$637$	9.08245	0.359860
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.0690	−1.26665	−0.633324	−	0.773887i	$-0.718310\pi$
$641$	−32.0690	−1.26665	−0.633324	+	0.773887i	$0.718310\pi$
$642$	0	0
$643$	−24.9651	−0.984528	−0.492264	−	0.870446i	$-0.663831\pi$
$643$	−24.9651	−0.984528	−0.492264	+	0.870446i	$0.663831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.4886	−0.608919	−0.304460	−	0.952525i	$-0.598476\pi$
$647$	−15.4886	−0.608919	−0.304460	+	0.952525i	$0.598476\pi$
$648$	0	0
$649$	−0.906780	−0.0355942
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.4293	−0.447264	−0.223632	−	0.974674i	$-0.571791\pi$
$653$	−11.4293	−0.447264	−0.223632	+	0.974674i	$0.571791\pi$
$654$	0	0
$655$	8.86966	0.346566
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.1213	−1.13441	−0.567203	−	0.823578i	$-0.691974\pi$
$659$	−29.1213	−1.13441	−0.567203	+	0.823578i	$0.691974\pi$
$660$	0	0
$661$	19.4829	0.757799	0.378899	−	0.925438i	$-0.376303\pi$
$661$	19.4829	0.757799	0.378899	+	0.925438i	$0.376303\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.75054	−0.300553
$666$	0	0
$667$	−10.2258	−0.395946
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.29095	0.0884411
$672$	0	0
$673$	−47.1173	−1.81624	−0.908120	−	0.418710i	$-0.862482\pi$
$673$	−47.1173	−1.81624	−0.908120	+	0.418710i	$0.862482\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.5617	−0.751817	−0.375909	−	0.926657i	$-0.622669\pi$
$677$	−19.5617	−0.751817	−0.375909	+	0.926657i	$0.622669\pi$
$678$	0	0
$679$	−82.5141	−3.16660
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.0384	0.996332	0.498166	−	0.867082i	$-0.334007\pi$
$683$	26.0384	0.996332	0.498166	+	0.867082i	$0.334007\pi$
$684$	0	0
$685$	9.92185	0.379094
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.72028	−0.141731
$690$	0	0
$691$	46.0969	1.75361	0.876804	−	0.480849i	$-0.159671\pi$
$691$	46.0969	1.75361	0.876804	+	0.480849i	$0.159671\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.0654	−0.381802
$696$	0	0
$697$	−4.69062	−0.177670
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.32214	−0.125475	−0.0627377	−	0.998030i	$-0.519983\pi$
$701$	−3.32214	−0.125475	−0.0627377	+	0.998030i	$0.519983\pi$
$702$	0	0
$703$	0.480140	0.0181088
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.0997	−0.379840
$708$	0	0
$709$	14.5408	0.546090	0.273045	−	0.962001i	$-0.411969\pi$
$709$	14.5408	0.546090	0.273045	+	0.962001i	$0.411969\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.46297	−0.129689
$714$	0	0
$715$	−0.178213	−0.00666479
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.9144	−0.891856	−0.445928	−	0.895069i	$-0.647126\pi$
$719$	−23.9144	−0.891856	−0.445928	+	0.895069i	$0.647126\pi$
$720$	0	0
$721$	15.0468	0.560371
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.2258	−0.379778
$726$	0	0
$727$	24.2960	0.901089	0.450544	−	0.892754i	$-0.351230\pi$
$727$	24.2960	0.901089	0.450544	+	0.892754i	$0.351230\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.758555	−0.0280562
$732$	0	0
$733$	−23.9750	−0.885537	−0.442768	−	0.896636i	$-0.646004\pi$
$733$	−23.9750	−0.885537	−0.442768	+	0.896636i	$0.646004\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.10277	−0.0774566
$738$	0	0
$739$	28.7736	1.05845	0.529227	−	0.848480i	$-0.322482\pi$
$739$	28.7736	1.05845	0.529227	+	0.848480i	$0.322482\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.2923	1.07463	0.537315	−	0.843381i	$-0.319439\pi$
$743$	29.2923	1.07463	0.537315	+	0.843381i	$0.319439\pi$
$744$	0	0
$745$	11.1715	0.409293
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.63889	−0.315658
$750$	0	0
$751$	−12.2197	−0.445905	−0.222952	−	0.974829i	$-0.571569\pi$
$751$	−12.2197	−0.445905	−0.222952	+	0.974829i	$0.571569\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.6914	−0.389101
$756$	0	0
$757$	−4.27120	−0.155239	−0.0776197	−	0.996983i	$-0.524732\pi$
$757$	−4.27120	−0.155239	−0.0776197	+	0.996983i	$0.524732\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.7938	−0.608775	−0.304388	−	0.952548i	$-0.598452\pi$
$761$	−16.7938	−0.608775	−0.304388	+	0.952548i	$0.598452\pi$
$762$	0	0
$763$	−60.5466	−2.19193
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.67912	0.0967375
$768$	0	0
$769$	−11.2226	−0.404698	−0.202349	−	0.979313i	$-0.564857\pi$
$769$	−11.2226	−0.404698	−0.202349	+	0.979313i	$0.564857\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.7835	−0.999303	−0.499652	−	0.866226i	$-0.666539\pi$
$773$	−27.7835	−0.999303	−0.499652	+	0.866226i	$0.666539\pi$
$774$	0	0
$775$	−3.46297	−0.124394
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.1607	0.399874
$780$	0	0
$781$	0.668249	0.0239118
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.6541	0.772867
$786$	0	0
$787$	−45.2896	−1.61440	−0.807200	−	0.590277i	$-0.799018\pi$
$787$	−45.2896	−1.61440	−0.807200	+	0.590277i	$0.799018\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.5595	0.375452
$792$	0	0
$793$	−6.76871	−0.240364
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.07523	0.144352	0.0721760	−	0.997392i	$-0.477006\pi$
$797$	4.07523	0.144352	0.0721760	+	0.997392i	$0.477006\pi$
$798$	0	0
$799$	−2.87451	−0.101693
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.26222	−0.115121
$804$	0	0
$805$	−4.41777	−0.155706
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.9172	0.629935	0.314967	−	0.949102i	$-0.398006\pi$
$809$	17.9172	0.629935	0.314967	+	0.949102i	$0.398006\pi$
$810$	0	0
$811$	−33.1654	−1.16459	−0.582297	−	0.812976i	$-0.697846\pi$
$811$	−33.1654	−1.16459	−0.582297	+	0.812976i	$0.697846\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.37720	0.153327
$816$	0	0
$817$	1.80488	0.0631448
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.2808	−0.777605	−0.388802	−	0.921321i	$-0.627111\pi$
$821$	−22.2808	−0.777605	−0.388802	+	0.921321i	$0.627111\pi$
$822$	0	0
$823$	−43.1993	−1.50583	−0.752916	−	0.658117i	$-0.771353\pi$
$823$	−43.1993	−1.50583	−0.752916	+	0.658117i	$0.771353\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.5318	0.783509	0.391754	−	0.920070i	$-0.371868\pi$
$827$	22.5318	0.783509	0.391754	+	0.920070i	$0.371868\pi$
$828$	0	0
$829$	−32.9217	−1.14342	−0.571710	−	0.820456i	$-0.693720\pi$
$829$	−32.9217	−1.14342	−0.571710	+	0.820456i	$0.693720\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.22902	−0.319767
$834$	0	0
$835$	−6.29928	−0.217995
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.2158	1.11221	0.556107	−	0.831111i	$-0.312295\pi$
$839$	32.2158	1.11221	0.556107	+	0.831111i	$0.312295\pi$
$840$	0	0
$841$	75.5679	2.60579
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.4735	−0.429100
$846$	0	0
$847$	48.3289	1.66060
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.273677	0.00938154
$852$	0	0
$853$	−18.2941	−0.626378	−0.313189	−	0.949691i	$-0.601397\pi$
$853$	−18.2941	−0.626378	−0.313189	+	0.949691i	$0.601397\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.56651	0.258467	0.129233	−	0.991614i	$-0.458748\pi$
$857$	7.56651	0.258467	0.129233	+	0.991614i	$0.458748\pi$
$858$	0	0
$859$	−19.0228	−0.649050	−0.324525	−	0.945877i	$-0.605204\pi$
$859$	−19.0228	−0.649050	−0.324525	+	0.945877i	$0.605204\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.2309	1.40352	0.701759	−	0.712415i	$-0.252398\pi$
$863$	41.2309	1.40352	0.701759	+	0.712415i	$0.252398\pi$
$864$	0	0
$865$	6.90390	0.234740
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.21302	0.108994
$870$	0	0
$871$	6.21273	0.210510
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.41777	−0.149348
$876$	0	0
$877$	39.9178	1.34793	0.673964	−	0.738764i	$-0.264590\pi$
$877$	39.9178	1.34793	0.673964	+	0.738764i	$0.264590\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.6579	1.30242	0.651209	−	0.758898i	$-0.274262\pi$
$881$	38.6579	1.30242	0.651209	+	0.758898i	$0.274262\pi$
$882$	0	0
$883$	39.8513	1.34110	0.670551	−	0.741864i	$-0.266058\pi$
$883$	39.8513	1.34110	0.670551	+	0.741864i	$0.266058\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.90143	0.0974204	0.0487102	−	0.998813i	$-0.484489\pi$
$887$	2.90143	0.0974204	0.0487102	+	0.998813i	$0.484489\pi$
$888$	0	0
$889$	18.3073	0.614008
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.83952	0.228876
$894$	0	0
$895$	−3.27569	−0.109494
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35.4118	1.18105
$900$	0	0
$901$	3.78031	0.125940
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.828526	0.0275411
$906$	0	0
$907$	−23.5967	−0.783516	−0.391758	−	0.920068i	$-0.628133\pi$
$907$	−23.5967	−0.783516	−0.391758	+	0.920068i	$0.628133\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.0893	0.963771	0.481886	−	0.876234i	$-0.339952\pi$
$911$	29.0893	0.963771	0.481886	+	0.876234i	$0.339952\pi$
$912$	0	0
$913$	4.00225	0.132455
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−39.1841	−1.29397
$918$	0	0
$919$	−21.1153	−0.696530	−0.348265	−	0.937396i	$-0.613229\pi$
$919$	−21.1153	−0.696530	−0.348265	+	0.937396i	$0.613229\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.97437	−0.0649872
$924$	0	0
$925$	0.273677	0.00899846
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.96102	−0.228384	−0.114192	−	0.993459i	$-0.536428\pi$
$929$	−6.96102	−0.228384	−0.114192	+	0.993459i	$0.536428\pi$
$930$	0	0
$931$	21.9592	0.719685
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.181089	0.00592224
$936$	0	0
$937$	36.3321	1.18692	0.593459	−	0.804864i	$-0.297762\pi$
$937$	36.3321	1.18692	0.593459	+	0.804864i	$0.297762\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50.9998	1.66254	0.831272	−	0.555865i	$-0.187613\pi$
$941$	50.9998	1.66254	0.831272	+	0.555865i	$0.187613\pi$
$942$	0	0
$943$	6.36154	0.207160
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.02468	−0.0657933	−0.0328967	−	0.999459i	$-0.510473\pi$
$947$	−2.02468	−0.0657933	−0.0328967	+	0.999459i	$0.510473\pi$
$948$	0	0
$949$	9.63837	0.312875
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−40.7067	−1.31862	−0.659309	−	0.751872i	$-0.729151\pi$
$953$	−40.7067	−1.31862	−0.659309	+	0.751872i	$0.729151\pi$
$954$	0	0
$955$	8.79434	0.284578
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−43.8324	−1.41542
$960$	0	0
$961$	−19.0078	−0.613156
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.30469	−0.170764
$966$	0	0
$967$	−35.4889	−1.14125	−0.570623	−	0.821212i	$-0.693298\pi$
$967$	−35.4889	−1.14125	−0.570623	+	0.821212i	$0.693298\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.8380	−0.540357	−0.270179	−	0.962810i	$-0.587083\pi$
$971$	−16.8380	−0.540357	−0.270179	+	0.962810i	$0.587083\pi$
$972$	0	0
$973$	44.4665	1.42553
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.9820	1.11917	0.559586	−	0.828772i	$-0.310960\pi$
$977$	34.9820	1.11917	0.559586	+	0.828772i	$0.310960\pi$
$978$	0	0
$979$	−3.48694	−0.111443
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.4075	1.09743	0.548715	−	0.836010i	$-0.315117\pi$
$983$	34.4075	1.09743	0.548715	+	0.836010i	$0.315117\pi$
$984$	0	0
$985$	17.2703	0.550277
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.02877	0.0327131
$990$	0	0
$991$	−3.81411	−0.121159	−0.0605796	−	0.998163i	$-0.519295\pi$
$991$	−3.81411	−0.121159	−0.0605796	+	0.998163i	$0.519295\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.0989687	0.00313752
$996$	0	0
$997$	−32.6186	−1.03304	−0.516520	−	0.856275i	$-0.672773\pi$
$997$	−32.6186	−1.03304	−0.516520	+	0.856275i	$0.672773\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bu.1.1	yes	6
3.2	odd	2		8280.2.a.bt.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bt.1.1	✓	6	3.2	odd	2
8280.2.a.bu.1.1	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.41777 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.41777 q^{7} -0.245598 q^{11} +0.725630 q^{13} -0.737339 q^{17} +1.75440 q^{19} +1.00000 q^{23} +1.00000 q^{25} -10.2258 q^{29} -3.46297 q^{31} -4.41777 q^{35} +0.273677 q^{37} +6.36154 q^{41} +1.02877 q^{43} +3.89849 q^{47} +12.5166 q^{49} -5.12696 q^{53} -0.245598 q^{55} +3.69214 q^{59} -9.32805 q^{61} +0.725630 q^{65} +8.56185 q^{67} -2.72091 q^{71} +13.2828 q^{73} +1.08499 q^{77} -13.0825 q^{79} -16.2960 q^{83} -0.737339 q^{85} +14.1978 q^{89} -3.20566 q^{91} +1.75440 q^{95} +18.6778 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 2 q^{7}+O(q^{10}) \) 6 * q + 6 * q^5 - 2 * q^7	\( 6 q + 6 q^{5} - 2 q^{7} - 4 q^{11} + 4 q^{17} + 8 q^{19} + 6 q^{23} + 6 q^{25} + 18 q^{29} - 8 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 2 q^{59} - 8 q^{61} - 2 q^{67} + 2 q^{71} + 8 q^{73} + 16 q^{77} - 28 q^{79} + 24 q^{83} + 4 q^{85} + 4 q^{89} - 8 q^{91} + 8 q^{95} + 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^5 - 2 * q^7 - 4 * q^11 + 4 * q^17 + 8 * q^19 + 6 * q^23 + 6 * q^25 + 18 * q^29 - 8 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^41 + 8 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 2 * q^59 - 8 * q^61 - 2 * q^67 + 2 * q^71 + 8 * q^73 + 16 * q^77 - 28 * q^79 + 24 * q^83 + 4 * q^85 + 4 * q^89 - 8 * q^91 + 8 * q^95 + 24 * q^97

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