LMFDB - Embedded newform 5780.2.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(1,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5780.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.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:	$46.1535323683$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 206x^{8} - 16x^{7} - 776x^{6} + 152x^{5} + 1226x^{4} - 384x^{3} - 588x^{2} + 200x + 34 $ x^12 - 24x^10 + 206x^8 - 16x^7 - 776x^6 + 152x^5 + 1226x^4 - 384x^3 - 588x^2 + 200*x + 34
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.70227$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70227 q^{3} -1.00000 q^{5} -4.39180 q^{7} -0.102270 q^{9} +O(q^{10})$	$q-1.70227 q^{3} -1.00000 q^{5} -4.39180 q^{7} -0.102270 q^{9} -3.38441 q^{11} -1.84670 q^{13} +1.70227 q^{15} +2.94345 q^{19} +7.47604 q^{21} +5.11453 q^{23} +1.00000 q^{25} +5.28091 q^{27} +5.90363 q^{29} +1.21678 q^{31} +5.76119 q^{33} +4.39180 q^{35} -4.98270 q^{37} +3.14359 q^{39} -1.86776 q^{41} +9.24170 q^{43} +0.102270 q^{45} +11.3010 q^{47} +12.2879 q^{49} +0.123979 q^{53} +3.38441 q^{55} -5.01055 q^{57} -1.21387 q^{59} -10.0520 q^{61} +0.449148 q^{63} +1.84670 q^{65} +11.0327 q^{67} -8.70632 q^{69} -14.5300 q^{71} +4.51647 q^{73} -1.70227 q^{75} +14.8637 q^{77} -7.26817 q^{79} -8.68273 q^{81} +3.62145 q^{83} -10.0496 q^{87} -13.6845 q^{89} +8.11034 q^{91} -2.07128 q^{93} -2.94345 q^{95} +14.7004 q^{97} +0.346123 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.70227	−0.982807	−0.491404	−	0.870932i	$-0.663516\pi$
$3$	−1.70227	−0.982807	−0.491404	+	0.870932i	$0.663516\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.39180	−1.65994	−0.829972	−	0.557805i	$-0.811644\pi$
$7$	−4.39180	−1.65994	−0.829972	+	0.557805i	$0.811644\pi$
$8$	0	0
$9$	−0.102270	−0.0340899
$10$	0	0
$11$	−3.38441	−1.02044	−0.510219	−	0.860044i	$-0.670436\pi$
$11$	−3.38441	−1.02044	−0.510219	+	0.860044i	$0.670436\pi$
$12$	0	0
$13$	−1.84670	−0.512183	−0.256091	−	0.966653i	$-0.582435\pi$
$13$	−1.84670	−0.512183	−0.256091	+	0.966653i	$0.582435\pi$
$14$	0	0
$15$	1.70227	0.439525
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.94345	0.675274	0.337637	−	0.941276i	$-0.390372\pi$
$19$	2.94345	0.675274	0.337637	+	0.941276i	$0.390372\pi$
$20$	0	0
$21$	7.47604	1.63141
$22$	0	0
$23$	5.11453	1.06645	0.533227	−	0.845972i	$-0.320979\pi$
$23$	5.11453	1.06645	0.533227	+	0.845972i	$0.320979\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.28091	1.01631
$28$	0	0
$29$	5.90363	1.09628	0.548138	−	0.836388i	$-0.315337\pi$
$29$	5.90363	1.09628	0.548138	+	0.836388i	$0.315337\pi$
$30$	0	0
$31$	1.21678	0.218539	0.109270	−	0.994012i	$-0.465149\pi$
$31$	1.21678	0.218539	0.109270	+	0.994012i	$0.465149\pi$
$32$	0	0
$33$	5.76119	1.00289
$34$	0	0
$35$	4.39180	0.742350
$36$	0	0
$37$	−4.98270	−0.819151	−0.409576	−	0.912276i	$-0.634323\pi$
$37$	−4.98270	−0.819151	−0.409576	+	0.912276i	$0.634323\pi$
$38$	0	0
$39$	3.14359	0.503377
$40$	0	0
$41$	−1.86776	−0.291695	−0.145847	−	0.989307i	$-0.546591\pi$
$41$	−1.86776	−0.291695	−0.145847	+	0.989307i	$0.546591\pi$
$42$	0	0
$43$	9.24170	1.40935	0.704673	−	0.709532i	$-0.251094\pi$
$43$	9.24170	1.40935	0.704673	+	0.709532i	$0.251094\pi$
$44$	0	0
$45$	0.102270	0.0152455
$46$	0	0
$47$	11.3010	1.64842	0.824209	−	0.566285i	$-0.191620\pi$
$47$	11.3010	1.64842	0.824209	+	0.566285i	$0.191620\pi$
$48$	0	0
$49$	12.2879	1.75542
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.123979	0.0170299	0.00851494	−	0.999964i	$-0.497290\pi$
$53$	0.123979	0.0170299	0.00851494	+	0.999964i	$0.497290\pi$
$54$	0	0
$55$	3.38441	0.456354
$56$	0	0
$57$	−5.01055	−0.663664
$58$	0	0
$59$	−1.21387	−0.158032	−0.0790159	−	0.996873i	$-0.525178\pi$
$59$	−1.21387	−0.158032	−0.0790159	+	0.996873i	$0.525178\pi$
$60$	0	0
$61$	−10.0520	−1.28702	−0.643511	−	0.765437i	$-0.722523\pi$
$61$	−10.0520	−1.28702	−0.643511	+	0.765437i	$0.722523\pi$
$62$	0	0
$63$	0.449148	0.0565874
$64$	0	0
$65$	1.84670	0.229055
$66$	0	0
$67$	11.0327	1.34785	0.673927	−	0.738798i	$-0.264606\pi$
$67$	11.0327	1.34785	0.673927	+	0.738798i	$0.264606\pi$
$68$	0	0
$69$	−8.70632	−1.04812
$70$	0	0
$71$	−14.5300	−1.72439	−0.862194	−	0.506578i	$-0.830910\pi$
$71$	−14.5300	−1.72439	−0.862194	+	0.506578i	$0.830910\pi$
$72$	0	0
$73$	4.51647	0.528613	0.264307	−	0.964439i	$-0.414857\pi$
$73$	4.51647	0.528613	0.264307	+	0.964439i	$0.414857\pi$
$74$	0	0
$75$	−1.70227	−0.196561
$76$	0	0
$77$	14.8637	1.69387
$78$	0	0
$79$	−7.26817	−0.817733	−0.408866	−	0.912594i	$-0.634076\pi$
$79$	−7.26817	−0.817733	−0.408866	+	0.912594i	$0.634076\pi$
$80$	0	0
$81$	−8.68273	−0.964748
$82$	0	0
$83$	3.62145	0.397506	0.198753	−	0.980050i	$-0.436311\pi$
$83$	3.62145	0.397506	0.198753	+	0.980050i	$0.436311\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−10.0496	−1.07743
$88$	0	0
$89$	−13.6845	−1.45056	−0.725278	−	0.688456i	$-0.758289\pi$
$89$	−13.6845	−1.45056	−0.725278	+	0.688456i	$0.758289\pi$
$90$	0	0
$91$	8.11034	0.850194
$92$	0	0
$93$	−2.07128	−0.214782
$94$	0	0
$95$	−2.94345	−0.301992
$96$	0	0
$97$	14.7004	1.49260	0.746300	−	0.665610i	$-0.231829\pi$
$97$	14.7004	1.49260	0.746300	+	0.665610i	$0.231829\pi$
$98$	0	0
$99$	0.346123	0.0347867
$100$	0	0
$101$	−14.2710	−1.42002	−0.710010	−	0.704192i	$-0.751310\pi$
$101$	−14.2710	−1.42002	−0.710010	+	0.704192i	$0.751310\pi$
$102$	0	0
$103$	−12.9344	−1.27447	−0.637234	−	0.770670i	$-0.719922\pi$
$103$	−12.9344	−1.27447	−0.637234	+	0.770670i	$0.719922\pi$
$104$	0	0
$105$	−7.47604	−0.729587
$106$	0	0
$107$	0.944716	0.0913291	0.0456646	−	0.998957i	$-0.485459\pi$
$107$	0.944716	0.0913291	0.0456646	+	0.998957i	$0.485459\pi$
$108$	0	0
$109$	−9.83828	−0.942336	−0.471168	−	0.882043i	$-0.656167\pi$
$109$	−9.83828	−0.942336	−0.471168	+	0.882043i	$0.656167\pi$
$110$	0	0
$111$	8.48191	0.805068
$112$	0	0
$113$	1.61361	0.151796	0.0758980	−	0.997116i	$-0.475818\pi$
$113$	1.61361	0.151796	0.0758980	+	0.997116i	$0.475818\pi$
$114$	0	0
$115$	−5.11453	−0.476932
$116$	0	0
$117$	0.188862	0.0174603
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.454252	0.0412956
$122$	0	0
$123$	3.17943	0.286680
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.4236	1.36862	0.684310	−	0.729191i	$-0.260104\pi$
$127$	15.4236	1.36862	0.684310	+	0.729191i	$0.260104\pi$
$128$	0	0
$129$	−15.7319	−1.38512
$130$	0	0
$131$	−0.0943652	−0.00824473	−0.00412236	−	0.999992i	$-0.501312\pi$
$131$	−0.0943652	−0.00824473	−0.00412236	+	0.999992i	$0.501312\pi$
$132$	0	0
$133$	−12.9270	−1.12092
$134$	0	0
$135$	−5.28091	−0.454508
$136$	0	0
$137$	−3.60342	−0.307861	−0.153930	−	0.988082i	$-0.549193\pi$
$137$	−3.60342	−0.307861	−0.153930	+	0.988082i	$0.549193\pi$
$138$	0	0
$139$	11.8291	1.00333	0.501667	−	0.865061i	$-0.332720\pi$
$139$	11.8291	1.00333	0.501667	+	0.865061i	$0.332720\pi$
$140$	0	0
$141$	−19.2374	−1.62008
$142$	0	0
$143$	6.25000	0.522651
$144$	0	0
$145$	−5.90363	−0.490270
$146$	0	0
$147$	−20.9174	−1.72523
$148$	0	0
$149$	−19.3619	−1.58619	−0.793094	−	0.609100i	$-0.791531\pi$
$149$	−19.3619	−1.58619	−0.793094	+	0.609100i	$0.791531\pi$
$150$	0	0
$151$	18.0122	1.46581	0.732906	−	0.680330i	$-0.238164\pi$
$151$	18.0122	1.46581	0.732906	+	0.680330i	$0.238164\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.21678	−0.0977338
$156$	0	0
$157$	−14.5166	−1.15855	−0.579275	−	0.815132i	$-0.696664\pi$
$157$	−14.5166	−1.15855	−0.579275	+	0.815132i	$0.696664\pi$
$158$	0	0
$159$	−0.211047	−0.0167371
$160$	0	0
$161$	−22.4620	−1.77025
$162$	0	0
$163$	−22.8172	−1.78718	−0.893592	−	0.448881i	$-0.851823\pi$
$163$	−22.8172	−1.78718	−0.893592	+	0.448881i	$0.851823\pi$
$164$	0	0
$165$	−5.76119	−0.448508
$166$	0	0
$167$	−0.341680	−0.0264400	−0.0132200	−	0.999913i	$-0.504208\pi$
$167$	−0.341680	−0.0264400	−0.0132200	+	0.999913i	$0.504208\pi$
$168$	0	0
$169$	−9.58970	−0.737669
$170$	0	0
$171$	−0.301026	−0.0230200
$172$	0	0
$173$	5.41329	0.411565	0.205782	−	0.978598i	$-0.434026\pi$
$173$	5.41329	0.411565	0.205782	+	0.978598i	$0.434026\pi$
$174$	0	0
$175$	−4.39180	−0.331989
$176$	0	0
$177$	2.06633	0.155315
$178$	0	0
$179$	14.0076	1.04697	0.523487	−	0.852034i	$-0.324631\pi$
$179$	14.0076	1.04697	0.523487	+	0.852034i	$0.324631\pi$
$180$	0	0
$181$	24.8537	1.84736	0.923681	−	0.383162i	$-0.125165\pi$
$181$	24.8537	1.84736	0.923681	+	0.383162i	$0.125165\pi$
$182$	0	0
$183$	17.1112	1.26489
$184$	0	0
$185$	4.98270	0.366335
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−23.1927	−1.68702
$190$	0	0
$191$	22.5455	1.63134	0.815668	−	0.578521i	$-0.196370\pi$
$191$	22.5455	1.63134	0.815668	+	0.578521i	$0.196370\pi$
$192$	0	0
$193$	−6.08750	−0.438188	−0.219094	−	0.975704i	$-0.570310\pi$
$193$	−6.08750	−0.438188	−0.219094	+	0.975704i	$0.570310\pi$
$194$	0	0
$195$	−3.14359	−0.225117
$196$	0	0
$197$	−4.15577	−0.296086	−0.148043	−	0.988981i	$-0.547297\pi$
$197$	−4.15577	−0.296086	−0.148043	+	0.988981i	$0.547297\pi$
$198$	0	0
$199$	−1.49355	−0.105875	−0.0529374	−	0.998598i	$-0.516858\pi$
$199$	−1.49355	−0.105875	−0.0529374	+	0.998598i	$0.516858\pi$
$200$	0	0
$201$	−18.7806	−1.32468
$202$	0	0
$203$	−25.9276	−1.81976
$204$	0	0
$205$	1.86776	0.130450
$206$	0	0
$207$	−0.523062	−0.0363553
$208$	0	0
$209$	−9.96185	−0.689076
$210$	0	0
$211$	1.01993	0.0702147	0.0351073	−	0.999384i	$-0.488823\pi$
$211$	1.01993	0.0702147	0.0351073	+	0.999384i	$0.488823\pi$
$212$	0	0
$213$	24.7339	1.69474
$214$	0	0
$215$	−9.24170	−0.630279
$216$	0	0
$217$	−5.34384	−0.362763
$218$	0	0
$219$	−7.68827	−0.519525
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.2912	−0.957010	−0.478505	−	0.878085i	$-0.658821\pi$
$223$	−14.2912	−0.957010	−0.478505	+	0.878085i	$0.658821\pi$
$224$	0	0
$225$	−0.102270	−0.00681798
$226$	0	0
$227$	2.30354	0.152892	0.0764458	−	0.997074i	$-0.475643\pi$
$227$	2.30354	0.152892	0.0764458	+	0.997074i	$0.475643\pi$
$228$	0	0
$229$	24.2371	1.60164	0.800818	−	0.598908i	$-0.204399\pi$
$229$	24.2371	1.60164	0.800818	+	0.598908i	$0.204399\pi$
$230$	0	0
$231$	−25.3020	−1.66475
$232$	0	0
$233$	28.7912	1.88618	0.943088	−	0.332543i	$-0.107907\pi$
$233$	28.7912	1.88618	0.943088	+	0.332543i	$0.107907\pi$
$234$	0	0
$235$	−11.3010	−0.737195
$236$	0	0
$237$	12.3724	0.803674
$238$	0	0
$239$	−8.87485	−0.574066	−0.287033	−	0.957921i	$-0.592669\pi$
$239$	−8.87485	−0.574066	−0.287033	+	0.957921i	$0.592669\pi$
$240$	0	0
$241$	−2.50845	−0.161584	−0.0807918	−	0.996731i	$-0.525745\pi$
$241$	−2.50845	−0.161584	−0.0807918	+	0.996731i	$0.525745\pi$
$242$	0	0
$243$	−1.06235	−0.0681498
$244$	0	0
$245$	−12.2879	−0.785045
$246$	0	0
$247$	−5.43567	−0.345863
$248$	0	0
$249$	−6.16469	−0.390672
$250$	0	0
$251$	22.1496	1.39807	0.699035	−	0.715088i	$-0.253613\pi$
$251$	22.1496	1.39807	0.699035	+	0.715088i	$0.253613\pi$
$252$	0	0
$253$	−17.3097	−1.08825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.51606	−0.406461	−0.203230	−	0.979131i	$-0.565144\pi$
$257$	−6.51606	−0.406461	−0.203230	+	0.979131i	$0.565144\pi$
$258$	0	0
$259$	21.8830	1.35975
$260$	0	0
$261$	−0.603763	−0.0373720
$262$	0	0
$263$	−5.13499	−0.316637	−0.158319	−	0.987388i	$-0.550607\pi$
$263$	−5.13499	−0.316637	−0.158319	+	0.987388i	$0.550607\pi$
$264$	0	0
$265$	−0.123979	−0.00761599
$266$	0	0
$267$	23.2948	1.42562
$268$	0	0
$269$	10.8545	0.661811	0.330905	−	0.943664i	$-0.392646\pi$
$269$	10.8545	0.661811	0.330905	+	0.943664i	$0.392646\pi$
$270$	0	0
$271$	−8.40983	−0.510861	−0.255430	−	0.966827i	$-0.582217\pi$
$271$	−8.40983	−0.510861	−0.255430	+	0.966827i	$0.582217\pi$
$272$	0	0
$273$	−13.8060	−0.835577
$274$	0	0
$275$	−3.38441	−0.204088
$276$	0	0
$277$	31.5190	1.89379	0.946897	−	0.321538i	$-0.104200\pi$
$277$	31.5190	1.89379	0.946897	+	0.321538i	$0.104200\pi$
$278$	0	0
$279$	−0.124439	−0.00744999
$280$	0	0
$281$	3.44721	0.205643	0.102822	−	0.994700i	$-0.467213\pi$
$281$	3.44721	0.205643	0.102822	+	0.994700i	$0.467213\pi$
$282$	0	0
$283$	30.8693	1.83499	0.917495	−	0.397747i	$-0.130208\pi$
$283$	30.8693	1.83499	0.917495	+	0.397747i	$0.130208\pi$
$284$	0	0
$285$	5.01055	0.296800
$286$	0	0
$287$	8.20282	0.484197
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−25.0241	−1.46694
$292$	0	0
$293$	11.3397	0.662473	0.331236	−	0.943548i	$-0.392534\pi$
$293$	11.3397	0.662473	0.331236	+	0.943548i	$0.392534\pi$
$294$	0	0
$295$	1.21387	0.0706740
$296$	0	0
$297$	−17.8728	−1.03708
$298$	0	0
$299$	−9.44500	−0.546219
$300$	0	0
$301$	−40.5877	−2.33944
$302$	0	0
$303$	24.2932	1.39561
$304$	0	0
$305$	10.0520	0.575574
$306$	0	0
$307$	−23.1509	−1.32129	−0.660647	−	0.750697i	$-0.729718\pi$
$307$	−23.1509	−1.32129	−0.660647	+	0.750697i	$0.729718\pi$
$308$	0	0
$309$	22.0179	1.25256
$310$	0	0
$311$	32.3288	1.83320	0.916598	−	0.399809i	$-0.130924\pi$
$311$	32.3288	1.83320	0.916598	+	0.399809i	$0.130924\pi$
$312$	0	0
$313$	−9.02885	−0.510341	−0.255171	−	0.966896i	$-0.582132\pi$
$313$	−9.02885	−0.510341	−0.255171	+	0.966896i	$0.582132\pi$
$314$	0	0
$315$	−0.449148	−0.0253066
$316$	0	0
$317$	−24.8937	−1.39817	−0.699084	−	0.715039i	$-0.746409\pi$
$317$	−24.8937	−1.39817	−0.699084	+	0.715039i	$0.746409\pi$
$318$	0	0
$319$	−19.9803	−1.11868
$320$	0	0
$321$	−1.60816	−0.0897589
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.84670	−0.102437
$326$	0	0
$327$	16.7474	0.926135
$328$	0	0
$329$	−49.6317	−2.73628
$330$	0	0
$331$	−10.9447	−0.601576	−0.300788	−	0.953691i	$-0.597250\pi$
$331$	−10.9447	−0.601576	−0.300788	+	0.953691i	$0.597250\pi$
$332$	0	0
$333$	0.509580	0.0279248
$334$	0	0
$335$	−11.0327	−0.602779
$336$	0	0
$337$	−22.1233	−1.20513	−0.602566	−	0.798069i	$-0.705855\pi$
$337$	−22.1233	−1.20513	−0.602566	+	0.798069i	$0.705855\pi$
$338$	0	0
$339$	−2.74681	−0.149186
$340$	0	0
$341$	−4.11807	−0.223006
$342$	0	0
$343$	−23.2234	−1.25395
$344$	0	0
$345$	8.70632	0.468733
$346$	0	0
$347$	9.37524	0.503289	0.251645	−	0.967820i	$-0.419029\pi$
$347$	9.37524	0.503289	0.251645	+	0.967820i	$0.419029\pi$
$348$	0	0
$349$	−15.8987	−0.851038	−0.425519	−	0.904950i	$-0.639908\pi$
$349$	−15.8987	−0.851038	−0.425519	+	0.904950i	$0.639908\pi$
$350$	0	0
$351$	−9.75225	−0.520537
$352$	0	0
$353$	2.31276	0.123096	0.0615479	−	0.998104i	$-0.480396\pi$
$353$	2.31276	0.123096	0.0615479	+	0.998104i	$0.480396\pi$
$354$	0	0
$355$	14.5300	0.771170
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.3232	−0.861505	−0.430753	−	0.902470i	$-0.641752\pi$
$359$	−16.3232	−0.861505	−0.430753	+	0.902470i	$0.641752\pi$
$360$	0	0
$361$	−10.3361	−0.544005
$362$	0	0
$363$	−0.773260	−0.0405856
$364$	0	0
$365$	−4.51647	−0.236403
$366$	0	0
$367$	14.7150	0.768115	0.384057	−	0.923309i	$-0.374526\pi$
$367$	14.7150	0.768115	0.384057	+	0.923309i	$0.374526\pi$
$368$	0	0
$369$	0.191015	0.00994386
$370$	0	0
$371$	−0.544492	−0.0282686
$372$	0	0
$373$	17.2542	0.893390	0.446695	−	0.894686i	$-0.352601\pi$
$373$	17.2542	0.893390	0.446695	+	0.894686i	$0.352601\pi$
$374$	0	0
$375$	1.70227	0.0879050
$376$	0	0
$377$	−10.9022	−0.561494
$378$	0	0
$379$	−28.0648	−1.44159	−0.720797	−	0.693146i	$-0.756224\pi$
$379$	−28.0648	−1.44159	−0.720797	+	0.693146i	$0.756224\pi$
$380$	0	0
$381$	−26.2551	−1.34509
$382$	0	0
$383$	−14.8835	−0.760510	−0.380255	−	0.924882i	$-0.624164\pi$
$383$	−14.8835	−0.760510	−0.380255	+	0.924882i	$0.624164\pi$
$384$	0	0
$385$	−14.8637	−0.757522
$386$	0	0
$387$	−0.945147	−0.0480445
$388$	0	0
$389$	21.8809	1.10941	0.554704	−	0.832048i	$-0.312831\pi$
$389$	21.8809	1.10941	0.554704	+	0.832048i	$0.312831\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0.160635	0.00810298
$394$	0	0
$395$	7.26817	0.365701
$396$	0	0
$397$	−0.670968	−0.0336749	−0.0168374	−	0.999858i	$-0.505360\pi$
$397$	−0.670968	−0.0336749	−0.0168374	+	0.999858i	$0.505360\pi$
$398$	0	0
$399$	22.0053	1.10165
$400$	0	0
$401$	8.35855	0.417406	0.208703	−	0.977979i	$-0.433076\pi$
$401$	8.35855	0.417406	0.208703	+	0.977979i	$0.433076\pi$
$402$	0	0
$403$	−2.24702	−0.111932
$404$	0	0
$405$	8.68273	0.431448
$406$	0	0
$407$	16.8635	0.835894
$408$	0	0
$409$	15.7292	0.777760	0.388880	−	0.921288i	$-0.372862\pi$
$409$	15.7292	0.777760	0.388880	+	0.921288i	$0.372862\pi$
$410$	0	0
$411$	6.13400	0.302568
$412$	0	0
$413$	5.33105	0.262324
$414$	0	0
$415$	−3.62145	−0.177770
$416$	0	0
$417$	−20.1364	−0.986084
$418$	0	0
$419$	−35.9588	−1.75670	−0.878352	−	0.478015i	$-0.841356\pi$
$419$	−35.9588	−1.75670	−0.878352	+	0.478015i	$0.841356\pi$
$420$	0	0
$421$	−2.72519	−0.132818	−0.0664088	−	0.997793i	$-0.521154\pi$
$421$	−2.72519	−0.132818	−0.0664088	+	0.997793i	$0.521154\pi$
$422$	0	0
$423$	−1.15575	−0.0561945
$424$	0	0
$425$	0	0
$426$	0	0
$427$	44.1462	2.13639
$428$	0	0
$429$	−10.6392	−0.513665
$430$	0	0
$431$	−9.13750	−0.440138	−0.220069	−	0.975484i	$-0.570628\pi$
$431$	−9.13750	−0.440138	−0.220069	+	0.975484i	$0.570628\pi$
$432$	0	0
$433$	−30.7506	−1.47778	−0.738890	−	0.673826i	$-0.764650\pi$
$433$	−30.7506	−1.47778	−0.738890	+	0.673826i	$0.764650\pi$
$434$	0	0
$435$	10.0496	0.481841
$436$	0	0
$437$	15.0544	0.720148
$438$	0	0
$439$	−33.3494	−1.59168	−0.795839	−	0.605508i	$-0.792970\pi$
$439$	−33.3494	−1.59168	−0.795839	+	0.605508i	$0.792970\pi$
$440$	0	0
$441$	−1.25668	−0.0598420
$442$	0	0
$443$	−11.3143	−0.537558	−0.268779	−	0.963202i	$-0.586620\pi$
$443$	−11.3143	−0.537558	−0.268779	+	0.963202i	$0.586620\pi$
$444$	0	0
$445$	13.6845	0.648708
$446$	0	0
$447$	32.9592	1.55892
$448$	0	0
$449$	8.37571	0.395274	0.197637	−	0.980275i	$-0.436673\pi$
$449$	8.37571	0.395274	0.197637	+	0.980275i	$0.436673\pi$
$450$	0	0
$451$	6.32127	0.297657
$452$	0	0
$453$	−30.6617	−1.44061
$454$	0	0
$455$	−8.11034	−0.380219
$456$	0	0
$457$	−3.49973	−0.163710	−0.0818552	−	0.996644i	$-0.526084\pi$
$457$	−3.49973	−0.163710	−0.0818552	+	0.996644i	$0.526084\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.10153	0.144453	0.0722263	−	0.997388i	$-0.476990\pi$
$461$	3.10153	0.144453	0.0722263	+	0.997388i	$0.476990\pi$
$462$	0	0
$463$	−3.71866	−0.172821	−0.0864103	−	0.996260i	$-0.527540\pi$
$463$	−3.71866	−0.172821	−0.0864103	+	0.996260i	$0.527540\pi$
$464$	0	0
$465$	2.07128	0.0960535
$466$	0	0
$467$	−4.96884	−0.229930	−0.114965	−	0.993370i	$-0.536676\pi$
$467$	−4.96884	−0.229930	−0.114965	+	0.993370i	$0.536676\pi$
$468$	0	0
$469$	−48.4533	−2.23736
$470$	0	0
$471$	24.7112	1.13863
$472$	0	0
$473$	−31.2777	−1.43815
$474$	0	0
$475$	2.94345	0.135055
$476$	0	0
$477$	−0.0126793	−0.000580547 0
$478$	0	0
$479$	0.793644	0.0362625	0.0181313	−	0.999836i	$-0.494228\pi$
$479$	0.793644	0.0362625	0.0181313	+	0.999836i	$0.494228\pi$
$480$	0	0
$481$	9.20156	0.419555
$482$	0	0
$483$	38.2364	1.73982
$484$	0	0
$485$	−14.7004	−0.667511
$486$	0	0
$487$	−12.2726	−0.556122	−0.278061	−	0.960563i	$-0.589692\pi$
$487$	−12.2726	−0.556122	−0.278061	+	0.960563i	$0.589692\pi$
$488$	0	0
$489$	38.8411	1.75646
$490$	0	0
$491$	6.03448	0.272332	0.136166	−	0.990686i	$-0.456522\pi$
$491$	6.03448	0.272332	0.136166	+	0.990686i	$0.456522\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.346123	−0.0155571
$496$	0	0
$497$	63.8127	2.86239
$498$	0	0
$499$	20.2900	0.908305	0.454153	−	0.890924i	$-0.349942\pi$
$499$	20.2900	0.908305	0.454153	+	0.890924i	$0.349942\pi$
$500$	0	0
$501$	0.581632	0.0259854
$502$	0	0
$503$	19.7041	0.878564	0.439282	−	0.898349i	$-0.355233\pi$
$503$	19.7041	0.878564	0.439282	+	0.898349i	$0.355233\pi$
$504$	0	0
$505$	14.2710	0.635052
$506$	0	0
$507$	16.3243	0.724986
$508$	0	0
$509$	−16.0258	−0.710332	−0.355166	−	0.934803i	$-0.615576\pi$
$509$	−16.0258	−0.710332	−0.355166	+	0.934803i	$0.615576\pi$
$510$	0	0
$511$	−19.8355	−0.877469
$512$	0	0
$513$	15.5441	0.686288
$514$	0	0
$515$	12.9344	0.569960
$516$	0	0
$517$	−38.2472	−1.68211
$518$	0	0
$519$	−9.21489	−0.404489
$520$	0	0
$521$	−5.00943	−0.219467	−0.109734	−	0.993961i	$-0.535000\pi$
$521$	−5.00943	−0.219467	−0.109734	+	0.993961i	$0.535000\pi$
$522$	0	0
$523$	6.24363	0.273015	0.136507	−	0.990639i	$-0.456412\pi$
$523$	6.24363	0.273015	0.136507	+	0.990639i	$0.456412\pi$
$524$	0	0
$525$	7.47604	0.326281
$526$	0	0
$527$	0	0
$528$	0	0
$529$	3.15841	0.137322
$530$	0	0
$531$	0.124142	0.00538729
$532$	0	0
$533$	3.44919	0.149401
$534$	0	0
$535$	−0.944716	−0.0408436
$536$	0	0
$537$	−23.8447	−1.02897
$538$	0	0
$539$	−41.5873	−1.79129
$540$	0	0
$541$	−5.23216	−0.224948	−0.112474	−	0.993655i	$-0.535878\pi$
$541$	−5.23216	−0.224948	−0.112474	+	0.993655i	$0.535878\pi$
$542$	0	0
$543$	−42.3078	−1.81560
$544$	0	0
$545$	9.83828	0.421426
$546$	0	0
$547$	−18.9256	−0.809199	−0.404600	−	0.914494i	$-0.632589\pi$
$547$	−18.9256	−0.809199	−0.404600	+	0.914494i	$0.632589\pi$
$548$	0	0
$549$	1.02801	0.0438745
$550$	0	0
$551$	17.3770	0.740287
$552$	0	0
$553$	31.9203	1.35739
$554$	0	0
$555$	−8.48191	−0.360037
$556$	0	0
$557$	−6.11156	−0.258955	−0.129478	−	0.991582i	$-0.541330\pi$
$557$	−6.11156	−0.258955	−0.129478	+	0.991582i	$0.541330\pi$
$558$	0	0
$559$	−17.0667	−0.721843
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.0516	−0.423626	−0.211813	−	0.977310i	$-0.567937\pi$
$563$	−10.0516	−0.423626	−0.211813	+	0.977310i	$0.567937\pi$
$564$	0	0
$565$	−1.61361	−0.0678852
$566$	0	0
$567$	38.1328	1.60143
$568$	0	0
$569$	−18.2569	−0.765368	−0.382684	−	0.923879i	$-0.625000\pi$
$569$	−18.2569	−0.765368	−0.382684	+	0.923879i	$0.625000\pi$
$570$	0	0
$571$	26.4676	1.10763	0.553817	−	0.832638i	$-0.313171\pi$
$571$	26.4676	1.10763	0.553817	+	0.832638i	$0.313171\pi$
$572$	0	0
$573$	−38.3786	−1.60329
$574$	0	0
$575$	5.11453	0.213291
$576$	0	0
$577$	38.5542	1.60503	0.802517	−	0.596630i	$-0.203494\pi$
$577$	38.5542	1.60503	0.802517	+	0.596630i	$0.203494\pi$
$578$	0	0
$579$	10.3626	0.430655
$580$	0	0
$581$	−15.9047	−0.659838
$582$	0	0
$583$	−0.419597	−0.0173779
$584$	0	0
$585$	−0.188862	−0.00780847
$586$	0	0
$587$	−19.1814	−0.791700	−0.395850	−	0.918315i	$-0.629550\pi$
$587$	−19.1814	−0.791700	−0.395850	+	0.918315i	$0.629550\pi$
$588$	0	0
$589$	3.58152	0.147574
$590$	0	0
$591$	7.07424	0.290996
$592$	0	0
$593$	−33.3403	−1.36912	−0.684562	−	0.728955i	$-0.740006\pi$
$593$	−33.3403	−1.36912	−0.684562	+	0.728955i	$0.740006\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.54243	0.104055
$598$	0	0
$599$	−47.2875	−1.93211	−0.966057	−	0.258329i	$-0.916828\pi$
$599$	−47.2875	−1.93211	−0.966057	+	0.258329i	$0.916828\pi$
$600$	0	0
$601$	−23.9628	−0.977464	−0.488732	−	0.872434i	$-0.662540\pi$
$601$	−23.9628	−0.977464	−0.488732	+	0.872434i	$0.662540\pi$
$602$	0	0
$603$	−1.12831	−0.0459483
$604$	0	0
$605$	−0.454252	−0.0184680
$606$	0	0
$607$	−28.6867	−1.16436	−0.582179	−	0.813061i	$-0.697800\pi$
$607$	−28.6867	−1.16436	−0.582179	+	0.813061i	$0.697800\pi$
$608$	0	0
$609$	44.1358	1.78847
$610$	0	0
$611$	−20.8695	−0.844291
$612$	0	0
$613$	−31.0375	−1.25359	−0.626796	−	0.779183i	$-0.715634\pi$
$613$	−31.0375	−1.25359	−0.626796	+	0.779183i	$0.715634\pi$
$614$	0	0
$615$	−3.17943	−0.128207
$616$	0	0
$617$	−15.1091	−0.608268	−0.304134	−	0.952629i	$-0.598367\pi$
$617$	−15.1091	−0.608268	−0.304134	+	0.952629i	$0.598367\pi$
$618$	0	0
$619$	15.7549	0.633243	0.316621	−	0.948552i	$-0.397452\pi$
$619$	15.7549	0.633243	0.316621	+	0.948552i	$0.397452\pi$
$620$	0	0
$621$	27.0094	1.08385
$622$	0	0
$623$	60.0996	2.40784
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.9578	0.677229
$628$	0	0
$629$	0	0
$630$	0	0
$631$	17.5705	0.699471	0.349735	−	0.936848i	$-0.386271\pi$
$631$	17.5705	0.699471	0.349735	+	0.936848i	$0.386271\pi$
$632$	0	0
$633$	−1.73619	−0.0690075
$634$	0	0
$635$	−15.4236	−0.612065
$636$	0	0
$637$	−22.6921	−0.899093
$638$	0	0
$639$	1.48598	0.0587843
$640$	0	0
$641$	41.4227	1.63610	0.818049	−	0.575149i	$-0.195056\pi$
$641$	41.4227	1.63610	0.818049	+	0.575149i	$0.195056\pi$
$642$	0	0
$643$	−21.8066	−0.859966	−0.429983	−	0.902837i	$-0.641481\pi$
$643$	−21.8066	−0.859966	−0.429983	+	0.902837i	$0.641481\pi$
$644$	0	0
$645$	15.7319	0.619443
$646$	0	0
$647$	24.5695	0.965927	0.482964	−	0.875640i	$-0.339560\pi$
$647$	24.5695	0.965927	0.482964	+	0.875640i	$0.339560\pi$
$648$	0	0
$649$	4.10822	0.161262
$650$	0	0
$651$	9.09667	0.356526
$652$	0	0
$653$	44.3404	1.73517	0.867587	−	0.497286i	$-0.165670\pi$
$653$	44.3404	1.73517	0.867587	+	0.497286i	$0.165670\pi$
$654$	0	0
$655$	0.0943652	0.00368715
$656$	0	0
$657$	−0.461899	−0.0180204
$658$	0	0
$659$	−31.6848	−1.23426	−0.617132	−	0.786860i	$-0.711706\pi$
$659$	−31.6848	−1.23426	−0.617132	+	0.786860i	$0.711706\pi$
$660$	0	0
$661$	−36.9094	−1.43561	−0.717804	−	0.696245i	$-0.754853\pi$
$661$	−36.9094	−1.43561	−0.717804	+	0.696245i	$0.754853\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.9270	0.501289
$666$	0	0
$667$	30.1943	1.16913
$668$	0	0
$669$	24.3275	0.940556
$670$	0	0
$671$	34.0200	1.31333
$672$	0	0
$673$	−25.7856	−0.993961	−0.496981	−	0.867762i	$-0.665558\pi$
$673$	−25.7856	−0.993961	−0.496981	+	0.867762i	$0.665558\pi$
$674$	0	0
$675$	5.28091	0.203262
$676$	0	0
$677$	15.9739	0.613929	0.306964	−	0.951721i	$-0.400687\pi$
$677$	15.9739	0.613929	0.306964	+	0.951721i	$0.400687\pi$
$678$	0	0
$679$	−64.5612	−2.47763
$680$	0	0
$681$	−3.92126	−0.150263
$682$	0	0
$683$	−10.1997	−0.390281	−0.195141	−	0.980775i	$-0.562516\pi$
$683$	−10.1997	−0.390281	−0.195141	+	0.980775i	$0.562516\pi$
$684$	0	0
$685$	3.60342	0.137680
$686$	0	0
$687$	−41.2582	−1.57410
$688$	0	0
$689$	−0.228953	−0.00872240
$690$	0	0
$691$	17.8358	0.678504	0.339252	−	0.940695i	$-0.389826\pi$
$691$	17.8358	0.678504	0.339252	+	0.940695i	$0.389826\pi$
$692$	0	0
$693$	−1.52010	−0.0577440
$694$	0	0
$695$	−11.8291	−0.448705
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−49.0105	−1.85375
$700$	0	0
$701$	−23.3651	−0.882486	−0.441243	−	0.897388i	$-0.645462\pi$
$701$	−23.3651	−0.882486	−0.441243	+	0.897388i	$0.645462\pi$
$702$	0	0
$703$	−14.6663	−0.553151
$704$	0	0
$705$	19.2374	0.724521
$706$	0	0
$707$	62.6755	2.35715
$708$	0	0
$709$	−15.1152	−0.567664	−0.283832	−	0.958874i	$-0.591606\pi$
$709$	−15.1152	−0.567664	−0.283832	+	0.958874i	$0.591606\pi$
$710$	0	0
$711$	0.743314	0.0278764
$712$	0	0
$713$	6.22324	0.233062
$714$	0	0
$715$	−6.25000	−0.233737
$716$	0	0
$717$	15.1074	0.564197
$718$	0	0
$719$	−21.9486	−0.818544	−0.409272	−	0.912412i	$-0.634217\pi$
$719$	−21.9486	−0.818544	−0.409272	+	0.912412i	$0.634217\pi$
$720$	0	0
$721$	56.8055	2.11555
$722$	0	0
$723$	4.27007	0.158805
$724$	0	0
$725$	5.90363	0.219255
$726$	0	0
$727$	31.2569	1.15925	0.579627	−	0.814882i	$-0.303198\pi$
$727$	31.2569	1.15925	0.579627	+	0.814882i	$0.303198\pi$
$728$	0	0
$729$	27.8566	1.03173
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−14.4381	−0.533285	−0.266642	−	0.963796i	$-0.585914\pi$
$733$	−14.4381	−0.533285	−0.266642	+	0.963796i	$0.585914\pi$
$734$	0	0
$735$	20.9174	0.771548
$736$	0	0
$737$	−37.3391	−1.37540
$738$	0	0
$739$	35.6182	1.31024	0.655119	−	0.755526i	$-0.272619\pi$
$739$	35.6182	1.31024	0.655119	+	0.755526i	$0.272619\pi$
$740$	0	0
$741$	9.25299	0.339917
$742$	0	0
$743$	24.7818	0.909155	0.454577	−	0.890707i	$-0.349790\pi$
$743$	24.7818	0.909155	0.454577	+	0.890707i	$0.349790\pi$
$744$	0	0
$745$	19.3619	0.709364
$746$	0	0
$747$	−0.370365	−0.0135509
$748$	0	0
$749$	−4.14900	−0.151601
$750$	0	0
$751$	21.5012	0.784589	0.392294	−	0.919840i	$-0.371681\pi$
$751$	21.5012	0.784589	0.392294	+	0.919840i	$0.371681\pi$
$752$	0	0
$753$	−37.7046	−1.37403
$754$	0	0
$755$	−18.0122	−0.655531
$756$	0	0
$757$	6.11757	0.222347	0.111173	−	0.993801i	$-0.464539\pi$
$757$	6.11757	0.222347	0.111173	+	0.993801i	$0.464539\pi$
$758$	0	0
$759$	29.4658	1.06954
$760$	0	0
$761$	−22.7201	−0.823604	−0.411802	−	0.911273i	$-0.635100\pi$
$761$	−22.7201	−0.823604	−0.411802	+	0.911273i	$0.635100\pi$
$762$	0	0
$763$	43.2078	1.56423
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.24165	0.0809411
$768$	0	0
$769$	−52.1040	−1.87892	−0.939459	−	0.342663i	$-0.888671\pi$
$769$	−52.1040	−1.87892	−0.939459	+	0.342663i	$0.888671\pi$
$770$	0	0
$771$	11.0921	0.399473
$772$	0	0
$773$	43.9831	1.58196	0.790980	−	0.611841i	$-0.209571\pi$
$773$	43.9831	1.58196	0.790980	+	0.611841i	$0.209571\pi$
$774$	0	0
$775$	1.21678	0.0437079
$776$	0	0
$777$	−37.2509	−1.33637
$778$	0	0
$779$	−5.49765	−0.196974
$780$	0	0
$781$	49.1754	1.75963
$782$	0	0
$783$	31.1765	1.11416
$784$	0	0
$785$	14.5166	0.518120
$786$	0	0
$787$	−2.33438	−0.0832116	−0.0416058	−	0.999134i	$-0.513247\pi$
$787$	−2.33438	−0.0832116	−0.0416058	+	0.999134i	$0.513247\pi$
$788$	0	0
$789$	8.74115	0.311193
$790$	0	0
$791$	−7.08666	−0.251973
$792$	0	0
$793$	18.5630	0.659190
$794$	0	0
$795$	0.211047	0.00748505
$796$	0	0
$797$	40.7460	1.44330	0.721648	−	0.692260i	$-0.243385\pi$
$797$	40.7460	1.44330	0.721648	+	0.692260i	$0.243385\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	1.39951	0.0494493
$802$	0	0
$803$	−15.2856	−0.539418
$804$	0	0
$805$	22.4620	0.791681
$806$	0	0
$807$	−18.4773	−0.650432
$808$	0	0
$809$	−45.1967	−1.58903	−0.794515	−	0.607244i	$-0.792275\pi$
$809$	−45.1967	−1.58903	−0.794515	+	0.607244i	$0.792275\pi$
$810$	0	0
$811$	28.0175	0.983827	0.491914	−	0.870644i	$-0.336298\pi$
$811$	28.0175	0.983827	0.491914	+	0.870644i	$0.336298\pi$
$812$	0	0
$813$	14.3158	0.502078
$814$	0	0
$815$	22.8172	0.799253
$816$	0	0
$817$	27.2025	0.951695
$818$	0	0
$819$	−0.829442	−0.0289831
$820$	0	0
$821$	−37.0287	−1.29231	−0.646156	−	0.763205i	$-0.723624\pi$
$821$	−37.0287	−1.29231	−0.646156	+	0.763205i	$0.723624\pi$
$822$	0	0
$823$	−39.3567	−1.37189	−0.685944	−	0.727655i	$-0.740610\pi$
$823$	−39.3567	−1.37189	−0.685944	+	0.727655i	$0.740610\pi$
$824$	0	0
$825$	5.76119	0.200579
$826$	0	0
$827$	49.5391	1.72264	0.861321	−	0.508061i	$-0.169638\pi$
$827$	49.5391	1.72264	0.861321	+	0.508061i	$0.169638\pi$
$828$	0	0
$829$	−22.3779	−0.777217	−0.388608	−	0.921403i	$-0.627044\pi$
$829$	−22.3779	−0.777217	−0.388608	+	0.921403i	$0.627044\pi$
$830$	0	0
$831$	−53.6539	−1.86123
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.341680	0.0118243
$836$	0	0
$837$	6.42568	0.222104
$838$	0	0
$839$	−44.7816	−1.54603	−0.773017	−	0.634385i	$-0.781253\pi$
$839$	−44.7816	−1.54603	−0.773017	+	0.634385i	$0.781253\pi$
$840$	0	0
$841$	5.85285	0.201822
$842$	0	0
$843$	−5.86809	−0.202108
$844$	0	0
$845$	9.58970	0.329896
$846$	0	0
$847$	−1.99498	−0.0685484
$848$	0	0
$849$	−52.5480	−1.80344
$850$	0	0
$851$	−25.4842	−0.873586
$852$	0	0
$853$	−4.73901	−0.162261	−0.0811304	−	0.996703i	$-0.525853\pi$
$853$	−4.73901	−0.162261	−0.0811304	+	0.996703i	$0.525853\pi$
$854$	0	0
$855$	0.301026	0.0102949
$856$	0	0
$857$	−50.7663	−1.73414	−0.867072	−	0.498183i	$-0.834001\pi$
$857$	−50.7663	−1.73414	−0.867072	+	0.498183i	$0.834001\pi$
$858$	0	0
$859$	−12.0387	−0.410754	−0.205377	−	0.978683i	$-0.565842\pi$
$859$	−12.0387	−0.410754	−0.205377	+	0.978683i	$0.565842\pi$
$860$	0	0
$861$	−13.9634	−0.475873
$862$	0	0
$863$	−25.3345	−0.862398	−0.431199	−	0.902257i	$-0.641909\pi$
$863$	−25.3345	−0.862398	−0.431199	+	0.902257i	$0.641909\pi$
$864$	0	0
$865$	−5.41329	−0.184057
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.5985	0.834446
$870$	0	0
$871$	−20.3740	−0.690348
$872$	0	0
$873$	−1.50341	−0.0508826
$874$	0	0
$875$	4.39180	0.148470
$876$	0	0
$877$	31.0341	1.04795	0.523974	−	0.851734i	$-0.324449\pi$
$877$	31.0341	1.04795	0.523974	+	0.851734i	$0.324449\pi$
$878$	0	0
$879$	−19.3033	−0.651083
$880$	0	0
$881$	51.1301	1.72262	0.861309	−	0.508082i	$-0.169645\pi$
$881$	51.1301	1.72262	0.861309	+	0.508082i	$0.169645\pi$
$882$	0	0
$883$	−16.4553	−0.553765	−0.276883	−	0.960904i	$-0.589301\pi$
$883$	−16.4553	−0.553765	−0.276883	+	0.960904i	$0.589301\pi$
$884$	0	0
$885$	−2.06633	−0.0694589
$886$	0	0
$887$	−22.6695	−0.761168	−0.380584	−	0.924746i	$-0.624277\pi$
$887$	−22.6695	−0.761168	−0.380584	+	0.924746i	$0.624277\pi$
$888$	0	0
$889$	−67.7372	−2.27183
$890$	0	0
$891$	29.3860	0.984466
$892$	0	0
$893$	33.2639	1.11313
$894$	0	0
$895$	−14.0076	−0.468221
$896$	0	0
$897$	16.0780	0.536828
$898$	0	0
$899$	7.18340	0.239580
$900$	0	0
$901$	0	0
$902$	0	0
$903$	69.0913	2.29922
$904$	0	0
$905$	−24.8537	−0.826165
$906$	0	0
$907$	14.4603	0.480147	0.240074	−	0.970755i	$-0.422828\pi$
$907$	14.4603	0.480147	0.240074	+	0.970755i	$0.422828\pi$
$908$	0	0
$909$	1.45949	0.0484084
$910$	0	0
$911$	−42.5336	−1.40920	−0.704600	−	0.709604i	$-0.748874\pi$
$911$	−42.5336	−1.40920	−0.704600	+	0.709604i	$0.748874\pi$
$912$	0	0
$913$	−12.2565	−0.405630
$914$	0	0
$915$	−17.1112	−0.565678
$916$	0	0
$917$	0.414433	0.0136858
$918$	0	0
$919$	−1.75021	−0.0577340	−0.0288670	−	0.999583i	$-0.509190\pi$
$919$	−1.75021	−0.0577340	−0.0288670	+	0.999583i	$0.509190\pi$
$920$	0	0
$921$	39.4092	1.29858
$922$	0	0
$923$	26.8325	0.883202
$924$	0	0
$925$	−4.98270	−0.163830
$926$	0	0
$927$	1.32280	0.0434465
$928$	0	0
$929$	29.3243	0.962098	0.481049	−	0.876694i	$-0.340256\pi$
$929$	29.3243	0.962098	0.481049	+	0.876694i	$0.340256\pi$
$930$	0	0
$931$	36.1688	1.18539
$932$	0	0
$933$	−55.0324	−1.80168
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.3087	0.859469	0.429734	−	0.902955i	$-0.358607\pi$
$937$	26.3087	0.859469	0.429734	+	0.902955i	$0.358607\pi$
$938$	0	0
$939$	15.3696	0.501567
$940$	0	0
$941$	26.9027	0.877004	0.438502	−	0.898730i	$-0.355509\pi$
$941$	26.9027	0.877004	0.438502	+	0.898730i	$0.355509\pi$
$942$	0	0
$943$	−9.55271	−0.311079
$944$	0	0
$945$	23.1927	0.754458
$946$	0	0
$947$	10.7719	0.350038	0.175019	−	0.984565i	$-0.444001\pi$
$947$	10.7719	0.350038	0.175019	+	0.984565i	$0.444001\pi$
$948$	0	0
$949$	−8.34058	−0.270747
$950$	0	0
$951$	42.3758	1.37413
$952$	0	0
$953$	39.6028	1.28286	0.641430	−	0.767181i	$-0.278341\pi$
$953$	39.6028	1.28286	0.641430	+	0.767181i	$0.278341\pi$
$954$	0	0
$955$	−22.5455	−0.729555
$956$	0	0
$957$	34.0119	1.09945
$958$	0	0
$959$	15.8255	0.511032
$960$	0	0
$961$	−29.5195	−0.952241
$962$	0	0
$963$	−0.0966159	−0.00311340
$964$	0	0
$965$	6.08750	0.195964
$966$	0	0
$967$	−9.83497	−0.316271	−0.158136	−	0.987417i	$-0.550548\pi$
$967$	−9.83497	−0.316271	−0.158136	+	0.987417i	$0.550548\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.8257	−1.05343	−0.526713	−	0.850043i	$-0.676576\pi$
$971$	−32.8257	−1.05343	−0.526713	+	0.850043i	$0.676576\pi$
$972$	0	0
$973$	−51.9512	−1.66548
$974$	0	0
$975$	3.14359	0.100675
$976$	0	0
$977$	−54.5327	−1.74466	−0.872328	−	0.488921i	$-0.837391\pi$
$977$	−54.5327	−1.74466	−0.872328	+	0.488921i	$0.837391\pi$
$978$	0	0
$979$	46.3140	1.48020
$980$	0	0
$981$	1.00616	0.0321242
$982$	0	0
$983$	−10.6174	−0.338641	−0.169321	−	0.985561i	$-0.554157\pi$
$983$	−10.6174	−0.338641	−0.169321	+	0.985561i	$0.554157\pi$
$984$	0	0
$985$	4.15577	0.132414
$986$	0	0
$987$	84.4866	2.68924
$988$	0	0
$989$	47.2670	1.50300
$990$	0	0
$991$	−60.1621	−1.91111	−0.955556	−	0.294809i	$-0.904744\pi$
$991$	−60.1621	−1.91111	−0.955556	+	0.294809i	$0.904744\pi$
$992$	0	0
$993$	18.6309	0.591233
$994$	0	0
$995$	1.49355	0.0473487
$996$	0	0
$997$	−24.6905	−0.781957	−0.390978	−	0.920400i	$-0.627863\pi$
$997$	−24.6905	−0.781957	−0.390978	+	0.920400i	$0.627863\pi$
$998$	0	0
$999$	−26.3132	−0.832512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.q.1.4		12
17.3	odd	16		340.2.u.a.281.2	yes	24
17.4	even	4		5780.2.c.j.5201.18		24
17.6	odd	16		340.2.u.a.121.2	✓	24
17.13	even	4		5780.2.c.j.5201.7		24
17.16	even	2		5780.2.a.r.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.u.a.121.2	✓	24	17.6	odd	16
340.2.u.a.281.2	yes	24	17.3	odd	16
5780.2.a.q.1.4		12	1.1	even	1	trivial
5780.2.a.r.1.9		12	17.16	even	2
5780.2.c.j.5201.7		24	17.13	even	4
5780.2.c.j.5201.18		24	17.4	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70227 q^{3} -1.00000 q^{5} -4.39180 q^{7} -0.102270 q^{9} +O(q^{10})\)	\(q-1.70227 q^{3} -1.00000 q^{5} -4.39180 q^{7} -0.102270 q^{9} -3.38441 q^{11} -1.84670 q^{13} +1.70227 q^{15} +2.94345 q^{19} +7.47604 q^{21} +5.11453 q^{23} +1.00000 q^{25} +5.28091 q^{27} +5.90363 q^{29} +1.21678 q^{31} +5.76119 q^{33} +4.39180 q^{35} -4.98270 q^{37} +3.14359 q^{39} -1.86776 q^{41} +9.24170 q^{43} +0.102270 q^{45} +11.3010 q^{47} +12.2879 q^{49} +0.123979 q^{53} +3.38441 q^{55} -5.01055 q^{57} -1.21387 q^{59} -10.0520 q^{61} +0.449148 q^{63} +1.84670 q^{65} +11.0327 q^{67} -8.70632 q^{69} -14.5300 q^{71} +4.51647 q^{73} -1.70227 q^{75} +14.8637 q^{77} -7.26817 q^{79} -8.68273 q^{81} +3.62145 q^{83} -10.0496 q^{87} -13.6845 q^{89} +8.11034 q^{91} -2.07128 q^{93} -2.94345 q^{95} +14.7004 q^{97} +0.346123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * q^99

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