LMFDB - Embedded newform 3840.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	3840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +3.62721 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +3.62721 q^{7} +1.00000 q^{9} -6.20555 q^{11} -0.578337 q^{13} -1.00000 q^{15} +1.42166 q^{17} -5.62721 q^{19} -3.62721 q^{21} +5.62721 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -2.57834 q^{31} +6.20555 q^{33} +3.62721 q^{35} +7.83276 q^{37} +0.578337 q^{39} -5.25443 q^{41} +7.25443 q^{43} +1.00000 q^{45} +6.78389 q^{47} +6.15667 q^{49} -1.42166 q^{51} +2.00000 q^{53} -6.20555 q^{55} +5.62721 q^{57} +2.20555 q^{59} +12.4111 q^{61} +3.62721 q^{63} -0.578337 q^{65} +4.00000 q^{67} -5.62721 q^{69} -8.41110 q^{71} +6.00000 q^{73} -1.00000 q^{75} -22.5089 q^{77} +5.42166 q^{79} +1.00000 q^{81} +3.25443 q^{83} +1.42166 q^{85} -2.00000 q^{87} +13.2544 q^{89} -2.09775 q^{91} +2.57834 q^{93} -5.62721 q^{95} +4.84333 q^{97} -6.20555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 4 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 6 q^{31} + 4 q^{33} - 2 q^{35} - 4 q^{37} + 10 q^{41} - 4 q^{43} + 3 q^{45} + 4 q^{47} + 15 q^{49} - 6 q^{51} + 6 q^{53} - 4 q^{55} + 4 q^{57} - 8 q^{59} + 8 q^{61} - 2 q^{63} + 12 q^{67} - 4 q^{69} + 4 q^{71} + 18 q^{73} - 3 q^{75} - 16 q^{77} + 18 q^{79} + 3 q^{81} - 16 q^{83} + 6 q^{85} - 6 q^{87} + 14 q^{89} + 16 q^{91} + 6 q^{93} - 4 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 4 * q^11 - 3 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 + 4 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 6 * q^31 + 4 * q^33 - 2 * q^35 - 4 * q^37 + 10 * q^41 - 4 * q^43 + 3 * q^45 + 4 * q^47 + 15 * q^49 - 6 * q^51 + 6 * q^53 - 4 * q^55 + 4 * q^57 - 8 * q^59 + 8 * q^61 - 2 * q^63 + 12 * q^67 - 4 * q^69 + 4 * q^71 + 18 * q^73 - 3 * q^75 - 16 * q^77 + 18 * q^79 + 3 * q^81 - 16 * q^83 + 6 * q^85 - 6 * q^87 + 14 * q^89 + 16 * q^91 + 6 * q^93 - 4 * q^95 + 18 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.62721	1.37096	0.685479	−	0.728093i	$-0.259593\pi$
$7$	3.62721	1.37096	0.685479	+	0.728093i	$0.259593\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.20555	−1.87104	−0.935522	−	0.353269i	$-0.885070\pi$
$11$	−6.20555	−1.87104	−0.935522	+	0.353269i	$0.885070\pi$
$12$	0	0
$13$	−0.578337	−0.160402	−0.0802009	−	0.996779i	$-0.525556\pi$
$13$	−0.578337	−0.160402	−0.0802009	+	0.996779i	$0.525556\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.42166	0.344804	0.172402	−	0.985027i	$-0.444847\pi$
$17$	1.42166	0.344804	0.172402	+	0.985027i	$0.444847\pi$
$18$	0	0
$19$	−5.62721	−1.29097	−0.645486	−	0.763772i	$-0.723345\pi$
$19$	−5.62721	−1.29097	−0.645486	+	0.763772i	$0.723345\pi$
$20$	0	0
$21$	−3.62721	−0.791523
$22$	0	0
$23$	5.62721	1.17336	0.586678	−	0.809821i	$-0.300436\pi$
$23$	5.62721	1.17336	0.586678	+	0.809821i	$0.300436\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−2.57834	−0.463083	−0.231542	−	0.972825i	$-0.574377\pi$
$31$	−2.57834	−0.463083	−0.231542	+	0.972825i	$0.574377\pi$
$32$	0	0
$33$	6.20555	1.08025
$34$	0	0
$35$	3.62721	0.613111
$36$	0	0
$37$	7.83276	1.28770	0.643849	−	0.765152i	$-0.277336\pi$
$37$	7.83276	1.28770	0.643849	+	0.765152i	$0.277336\pi$
$38$	0	0
$39$	0.578337	0.0926081
$40$	0	0
$41$	−5.25443	−0.820603	−0.410302	−	0.911950i	$-0.634577\pi$
$41$	−5.25443	−0.820603	−0.410302	+	0.911950i	$0.634577\pi$
$42$	0	0
$43$	7.25443	1.10629	0.553145	−	0.833085i	$-0.313428\pi$
$43$	7.25443	1.10629	0.553145	+	0.833085i	$0.313428\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	6.78389	0.989532	0.494766	−	0.869026i	$-0.335254\pi$
$47$	6.78389	0.989532	0.494766	+	0.869026i	$0.335254\pi$
$48$	0	0
$49$	6.15667	0.879525
$50$	0	0
$51$	−1.42166	−0.199073
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−6.20555	−0.836756
$56$	0	0
$57$	5.62721	0.745343
$58$	0	0
$59$	2.20555	0.287138	0.143569	−	0.989640i	$-0.454142\pi$
$59$	2.20555	0.287138	0.143569	+	0.989640i	$0.454142\pi$
$60$	0	0
$61$	12.4111	1.58908	0.794539	−	0.607213i	$-0.207712\pi$
$61$	12.4111	1.58908	0.794539	+	0.607213i	$0.207712\pi$
$62$	0	0
$63$	3.62721	0.456986
$64$	0	0
$65$	−0.578337	−0.0717339
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−5.62721	−0.677437
$70$	0	0
$71$	−8.41110	−0.998214	−0.499107	−	0.866540i	$-0.666339\pi$
$71$	−8.41110	−0.998214	−0.499107	+	0.866540i	$0.666339\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−22.5089	−2.56512
$78$	0	0
$79$	5.42166	0.609985	0.304992	−	0.952355i	$-0.401346\pi$
$79$	5.42166	0.609985	0.304992	+	0.952355i	$0.401346\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.25443	0.357220	0.178610	−	0.983920i	$-0.442840\pi$
$83$	3.25443	0.357220	0.178610	+	0.983920i	$0.442840\pi$
$84$	0	0
$85$	1.42166	0.154201
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	13.2544	1.40497	0.702483	−	0.711700i	$-0.252075\pi$
$89$	13.2544	1.40497	0.702483	+	0.711700i	$0.252075\pi$
$90$	0	0
$91$	−2.09775	−0.219904
$92$	0	0
$93$	2.57834	0.267361
$94$	0	0
$95$	−5.62721	−0.577340
$96$	0	0
$97$	4.84333	0.491765	0.245883	−	0.969300i	$-0.420922\pi$
$97$	4.84333	0.491765	0.245883	+	0.969300i	$0.420922\pi$
$98$	0	0
$99$	−6.20555	−0.623681
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−2.47054	−0.243429	−0.121715	−	0.992565i	$-0.538839\pi$
$103$	−2.47054	−0.243429	−0.121715	+	0.992565i	$0.538839\pi$
$104$	0	0
$105$	−3.62721	−0.353980
$106$	0	0
$107$	−14.0978	−1.36288	−0.681441	−	0.731873i	$-0.738646\pi$
$107$	−14.0978	−1.36288	−0.681441	+	0.731873i	$0.738646\pi$
$108$	0	0
$109$	7.25443	0.694848	0.347424	−	0.937708i	$-0.387056\pi$
$109$	7.25443	0.694848	0.347424	+	0.937708i	$0.387056\pi$
$110$	0	0
$111$	−7.83276	−0.743453
$112$	0	0
$113$	−9.08719	−0.854851	−0.427425	−	0.904051i	$-0.640579\pi$
$113$	−9.08719	−0.854851	−0.427425	+	0.904051i	$0.640579\pi$
$114$	0	0
$115$	5.62721	0.524740
$116$	0	0
$117$	−0.578337	−0.0534673
$118$	0	0
$119$	5.15667	0.472712
$120$	0	0
$121$	27.5089	2.50080
$122$	0	0
$123$	5.25443	0.473776
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.4705	−0.929110	−0.464555	−	0.885544i	$-0.653786\pi$
$127$	−10.4705	−0.929110	−0.464555	+	0.885544i	$0.653786\pi$
$128$	0	0
$129$	−7.25443	−0.638717
$130$	0	0
$131$	−13.4600	−1.17600	−0.588002	−	0.808860i	$-0.700085\pi$
$131$	−13.4600	−1.17600	−0.588002	+	0.808860i	$0.700085\pi$
$132$	0	0
$133$	−20.4111	−1.76987
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	10.5783	0.903768	0.451884	−	0.892077i	$-0.350752\pi$
$137$	10.5783	0.903768	0.451884	+	0.892077i	$0.350752\pi$
$138$	0	0
$139$	12.4705	1.05774	0.528869	−	0.848704i	$-0.322616\pi$
$139$	12.4705	1.05774	0.528869	+	0.848704i	$0.322616\pi$
$140$	0	0
$141$	−6.78389	−0.571306
$142$	0	0
$143$	3.58890	0.300119
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−6.15667	−0.507794
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−12.6761	−1.03157	−0.515783	−	0.856719i	$-0.672499\pi$
$151$	−12.6761	−1.03157	−0.515783	+	0.856719i	$0.672499\pi$
$152$	0	0
$153$	1.42166	0.114935
$154$	0	0
$155$	−2.57834	−0.207097
$156$	0	0
$157$	−1.32391	−0.105660	−0.0528298	−	0.998604i	$-0.516824\pi$
$157$	−1.32391	−0.105660	−0.0528298	+	0.998604i	$0.516824\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	20.4111	1.60862
$162$	0	0
$163$	15.2544	1.19482	0.597409	−	0.801936i	$-0.296197\pi$
$163$	15.2544	1.19482	0.597409	+	0.801936i	$0.296197\pi$
$164$	0	0
$165$	6.20555	0.483101
$166$	0	0
$167$	−10.7839	−0.834482	−0.417241	−	0.908796i	$-0.637003\pi$
$167$	−10.7839	−0.834482	−0.417241	+	0.908796i	$0.637003\pi$
$168$	0	0
$169$	−12.6655	−0.974271
$170$	0	0
$171$	−5.62721	−0.430324
$172$	0	0
$173$	13.6655	1.03897	0.519485	−	0.854479i	$-0.326124\pi$
$173$	13.6655	1.03897	0.519485	+	0.854479i	$0.326124\pi$
$174$	0	0
$175$	3.62721	0.274192
$176$	0	0
$177$	−2.20555	−0.165779
$178$	0	0
$179$	−9.04888	−0.676345	−0.338172	−	0.941084i	$-0.609809\pi$
$179$	−9.04888	−0.676345	−0.338172	+	0.941084i	$0.609809\pi$
$180$	0	0
$181$	23.2544	1.72849	0.864244	−	0.503073i	$-0.167797\pi$
$181$	23.2544	1.72849	0.864244	+	0.503073i	$0.167797\pi$
$182$	0	0
$183$	−12.4111	−0.917455
$184$	0	0
$185$	7.83276	0.575876
$186$	0	0
$187$	−8.82220	−0.645143
$188$	0	0
$189$	−3.62721	−0.263841
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	25.6655	1.84745	0.923723	−	0.383062i	$-0.125131\pi$
$193$	25.6655	1.84745	0.923723	+	0.383062i	$0.125131\pi$
$194$	0	0
$195$	0.578337	0.0414156
$196$	0	0
$197$	−15.1567	−1.07987	−0.539934	−	0.841707i	$-0.681551\pi$
$197$	−15.1567	−1.07987	−0.539934	+	0.841707i	$0.681551\pi$
$198$	0	0
$199$	20.6761	1.46569	0.732845	−	0.680396i	$-0.238192\pi$
$199$	20.6761	1.46569	0.732845	+	0.680396i	$0.238192\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	7.25443	0.509161
$204$	0	0
$205$	−5.25443	−0.366985
$206$	0	0
$207$	5.62721	0.391118
$208$	0	0
$209$	34.9200	2.41546
$210$	0	0
$211$	−2.03831	−0.140323	−0.0701616	−	0.997536i	$-0.522352\pi$
$211$	−2.03831	−0.140323	−0.0701616	+	0.997536i	$0.522352\pi$
$212$	0	0
$213$	8.41110	0.576319
$214$	0	0
$215$	7.25443	0.494748
$216$	0	0
$217$	−9.35218	−0.634867
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−0.822200	−0.0553072
$222$	0	0
$223$	−7.21611	−0.483227	−0.241613	−	0.970373i	$-0.577677\pi$
$223$	−7.21611	−0.483227	−0.241613	+	0.970373i	$0.577677\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	1.15667	0.0767712	0.0383856	−	0.999263i	$-0.487778\pi$
$227$	1.15667	0.0767712	0.0383856	+	0.999263i	$0.487778\pi$
$228$	0	0
$229$	−14.0978	−0.931606	−0.465803	−	0.884889i	$-0.654234\pi$
$229$	−14.0978	−0.931606	−0.465803	+	0.884889i	$0.654234\pi$
$230$	0	0
$231$	22.5089	1.48097
$232$	0	0
$233$	14.5783	0.955059	0.477529	−	0.878616i	$-0.341532\pi$
$233$	14.5783	0.955059	0.477529	+	0.878616i	$0.341532\pi$
$234$	0	0
$235$	6.78389	0.442532
$236$	0	0
$237$	−5.42166	−0.352175
$238$	0	0
$239$	19.2544	1.24547	0.622733	−	0.782435i	$-0.286022\pi$
$239$	19.2544	1.24547	0.622733	+	0.782435i	$0.286022\pi$
$240$	0	0
$241$	−13.6655	−0.880274	−0.440137	−	0.897931i	$-0.645070\pi$
$241$	−13.6655	−0.880274	−0.440137	+	0.897931i	$0.645070\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.15667	0.393335
$246$	0	0
$247$	3.25443	0.207074
$248$	0	0
$249$	−3.25443	−0.206241
$250$	0	0
$251$	7.14663	0.451091	0.225546	−	0.974233i	$-0.427584\pi$
$251$	7.14663	0.451091	0.225546	+	0.974233i	$0.427584\pi$
$252$	0	0
$253$	−34.9200	−2.19540
$254$	0	0
$255$	−1.42166	−0.0890280
$256$	0	0
$257$	7.73501	0.482497	0.241248	−	0.970463i	$-0.422443\pi$
$257$	7.73501	0.482497	0.241248	+	0.970463i	$0.422443\pi$
$258$	0	0
$259$	28.4111	1.76538
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	18.7839	1.15826	0.579132	−	0.815234i	$-0.303392\pi$
$263$	18.7839	1.15826	0.579132	+	0.815234i	$0.303392\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−13.2544	−0.811158
$268$	0	0
$269$	8.50885	0.518794	0.259397	−	0.965771i	$-0.416476\pi$
$269$	8.50885	0.518794	0.259397	+	0.965771i	$0.416476\pi$
$270$	0	0
$271$	30.9894	1.88247	0.941237	−	0.337746i	$-0.109665\pi$
$271$	30.9894	1.88247	0.941237	+	0.337746i	$0.109665\pi$
$272$	0	0
$273$	2.09775	0.126962
$274$	0	0
$275$	−6.20555	−0.374209
$276$	0	0
$277$	−9.51941	−0.571966	−0.285983	−	0.958235i	$-0.592320\pi$
$277$	−9.51941	−0.571966	−0.285983	+	0.958235i	$0.592320\pi$
$278$	0	0
$279$	−2.57834	−0.154361
$280$	0	0
$281$	−13.6655	−0.815217	−0.407608	−	0.913157i	$-0.633637\pi$
$281$	−13.6655	−0.815217	−0.407608	+	0.913157i	$0.633637\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	5.62721	0.333327
$286$	0	0
$287$	−19.0589	−1.12501
$288$	0	0
$289$	−14.9789	−0.881110
$290$	0	0
$291$	−4.84333	−0.283921
$292$	0	0
$293$	4.31335	0.251989	0.125994	−	0.992031i	$-0.459788\pi$
$293$	4.31335	0.251989	0.125994	+	0.992031i	$0.459788\pi$
$294$	0	0
$295$	2.20555	0.128412
$296$	0	0
$297$	6.20555	0.360083
$298$	0	0
$299$	−3.25443	−0.188208
$300$	0	0
$301$	26.3133	1.51668
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	12.4111	0.710658
$306$	0	0
$307$	25.5678	1.45923	0.729615	−	0.683858i	$-0.239699\pi$
$307$	25.5678	1.45923	0.729615	+	0.683858i	$0.239699\pi$
$308$	0	0
$309$	2.47054	0.140544
$310$	0	0
$311$	20.0766	1.13844	0.569221	−	0.822185i	$-0.307245\pi$
$311$	20.0766	1.13844	0.569221	+	0.822185i	$0.307245\pi$
$312$	0	0
$313$	−7.15667	−0.404519	−0.202260	−	0.979332i	$-0.564828\pi$
$313$	−7.15667	−0.404519	−0.202260	+	0.979332i	$0.564828\pi$
$314$	0	0
$315$	3.62721	0.204370
$316$	0	0
$317$	24.1744	1.35777	0.678884	−	0.734245i	$-0.262464\pi$
$317$	24.1744	1.35777	0.678884	+	0.734245i	$0.262464\pi$
$318$	0	0
$319$	−12.4111	−0.694888
$320$	0	0
$321$	14.0978	0.786860
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−0.578337	−0.0320804
$326$	0	0
$327$	−7.25443	−0.401171
$328$	0	0
$329$	24.6066	1.35661
$330$	0	0
$331$	27.1950	1.49477	0.747386	−	0.664390i	$-0.231309\pi$
$331$	27.1950	1.49477	0.747386	+	0.664390i	$0.231309\pi$
$332$	0	0
$333$	7.83276	0.429233
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	22.8222	1.24320	0.621602	−	0.783333i	$-0.286482\pi$
$337$	22.8222	1.24320	0.621602	+	0.783333i	$0.286482\pi$
$338$	0	0
$339$	9.08719	0.493548
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	−3.05892	−0.165166
$344$	0	0
$345$	−5.62721	−0.302959
$346$	0	0
$347$	23.6655	1.27043	0.635216	−	0.772335i	$-0.280911\pi$
$347$	23.6655	1.27043	0.635216	+	0.772335i	$0.280911\pi$
$348$	0	0
$349$	−34.9200	−1.86922	−0.934611	−	0.355671i	$-0.884252\pi$
$349$	−34.9200	−1.86922	−0.934611	+	0.355671i	$0.884252\pi$
$350$	0	0
$351$	0.578337	0.0308694
$352$	0	0
$353$	−15.9305	−0.847896	−0.423948	−	0.905687i	$-0.639356\pi$
$353$	−15.9305	−0.847896	−0.423948	+	0.905687i	$0.639356\pi$
$354$	0	0
$355$	−8.41110	−0.446415
$356$	0	0
$357$	−5.15667	−0.272920
$358$	0	0
$359$	−8.41110	−0.443921	−0.221960	−	0.975056i	$-0.571246\pi$
$359$	−8.41110	−0.443921	−0.221960	+	0.975056i	$0.571246\pi$
$360$	0	0
$361$	12.6655	0.666607
$362$	0	0
$363$	−27.5089	−1.44384
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−24.4494	−1.27625	−0.638124	−	0.769933i	$-0.720289\pi$
$367$	−24.4494	−1.27625	−0.638124	+	0.769933i	$0.720289\pi$
$368$	0	0
$369$	−5.25443	−0.273534
$370$	0	0
$371$	7.25443	0.376631
$372$	0	0
$373$	0.167237	0.00865920	0.00432960	−	0.999991i	$-0.498622\pi$
$373$	0.167237	0.00865920	0.00432960	+	0.999991i	$0.498622\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−1.15667	−0.0595718
$378$	0	0
$379$	−7.72496	−0.396805	−0.198402	−	0.980121i	$-0.563575\pi$
$379$	−7.72496	−0.396805	−0.198402	+	0.980121i	$0.563575\pi$
$380$	0	0
$381$	10.4705	0.536422
$382$	0	0
$383$	−1.62721	−0.0831467	−0.0415734	−	0.999135i	$-0.513237\pi$
$383$	−1.62721	−0.0831467	−0.0415734	+	0.999135i	$0.513237\pi$
$384$	0	0
$385$	−22.5089	−1.14716
$386$	0	0
$387$	7.25443	0.368763
$388$	0	0
$389$	−12.3133	−0.624312	−0.312156	−	0.950031i	$-0.601051\pi$
$389$	−12.3133	−0.624312	−0.312156	+	0.950031i	$0.601051\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	13.4600	0.678966
$394$	0	0
$395$	5.42166	0.272793
$396$	0	0
$397$	−19.0872	−0.957959	−0.478979	−	0.877826i	$-0.658993\pi$
$397$	−19.0872	−0.957959	−0.478979	+	0.877826i	$0.658993\pi$
$398$	0	0
$399$	20.4111	1.02183
$400$	0	0
$401$	−14.4111	−0.719656	−0.359828	−	0.933019i	$-0.617165\pi$
$401$	−14.4111	−0.719656	−0.359828	+	0.933019i	$0.617165\pi$
$402$	0	0
$403$	1.49115	0.0742794
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−48.6066	−2.40934
$408$	0	0
$409$	8.31335	0.411069	0.205534	−	0.978650i	$-0.434107\pi$
$409$	8.31335	0.411069	0.205534	+	0.978650i	$0.434107\pi$
$410$	0	0
$411$	−10.5783	−0.521791
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	3.25443	0.159753
$416$	0	0
$417$	−12.4705	−0.610685
$418$	0	0
$419$	−7.36222	−0.359668	−0.179834	−	0.983697i	$-0.557556\pi$
$419$	−7.36222	−0.359668	−0.179834	+	0.983697i	$0.557556\pi$
$420$	0	0
$421$	30.0978	1.46687	0.733437	−	0.679757i	$-0.237915\pi$
$421$	30.0978	1.46687	0.733437	+	0.679757i	$0.237915\pi$
$422$	0	0
$423$	6.78389	0.329844
$424$	0	0
$425$	1.42166	0.0689608
$426$	0	0
$427$	45.0177	2.17856
$428$	0	0
$429$	−3.58890	−0.173274
$430$	0	0
$431$	8.41110	0.405148	0.202574	−	0.979267i	$-0.435069\pi$
$431$	8.41110	0.405148	0.202574	+	0.979267i	$0.435069\pi$
$432$	0	0
$433$	−4.31335	−0.207286	−0.103643	−	0.994615i	$-0.533050\pi$
$433$	−4.31335	−0.207286	−0.103643	+	0.994615i	$0.533050\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	−31.6655	−1.51477
$438$	0	0
$439$	−9.83276	−0.469292	−0.234646	−	0.972081i	$-0.575393\pi$
$439$	−9.83276	−0.469292	−0.234646	+	0.972081i	$0.575393\pi$
$440$	0	0
$441$	6.15667	0.293175
$442$	0	0
$443$	−21.3522	−1.01447	−0.507236	−	0.861807i	$-0.669333\pi$
$443$	−21.3522	−1.01447	−0.507236	+	0.861807i	$0.669333\pi$
$444$	0	0
$445$	13.2544	0.628320
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	−20.3133	−0.958646	−0.479323	−	0.877639i	$-0.659118\pi$
$449$	−20.3133	−0.958646	−0.479323	+	0.877639i	$0.659118\pi$
$450$	0	0
$451$	32.6066	1.53539
$452$	0	0
$453$	12.6761	0.595575
$454$	0	0
$455$	−2.09775	−0.0983441
$456$	0	0
$457$	3.35218	0.156808	0.0784041	−	0.996922i	$-0.475018\pi$
$457$	3.35218	0.156808	0.0784041	+	0.996922i	$0.475018\pi$
$458$	0	0
$459$	−1.42166	−0.0663575
$460$	0	0
$461$	28.5089	1.32779	0.663895	−	0.747826i	$-0.268902\pi$
$461$	28.5089	1.32779	0.663895	+	0.747826i	$0.268902\pi$
$462$	0	0
$463$	23.6272	1.09805	0.549025	−	0.835806i	$-0.314999\pi$
$463$	23.6272	1.09805	0.549025	+	0.835806i	$0.314999\pi$
$464$	0	0
$465$	2.57834	0.119568
$466$	0	0
$467$	−29.5678	−1.36823	−0.684117	−	0.729373i	$-0.739812\pi$
$467$	−29.5678	−1.36823	−0.684117	+	0.729373i	$0.739812\pi$
$468$	0	0
$469$	14.5089	0.669957
$470$	0	0
$471$	1.32391	0.0610026
$472$	0	0
$473$	−45.0177	−2.06992
$474$	0	0
$475$	−5.62721	−0.258194
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−22.0978	−1.00967	−0.504836	−	0.863215i	$-0.668447\pi$
$479$	−22.0978	−1.00967	−0.504836	+	0.863215i	$0.668447\pi$
$480$	0	0
$481$	−4.52998	−0.206549
$482$	0	0
$483$	−20.4111	−0.928737
$484$	0	0
$485$	4.84333	0.219924
$486$	0	0
$487$	−4.03831	−0.182993	−0.0914967	−	0.995805i	$-0.529165\pi$
$487$	−4.03831	−0.182993	−0.0914967	+	0.995805i	$0.529165\pi$
$488$	0	0
$489$	−15.2544	−0.689829
$490$	0	0
$491$	−18.2056	−0.821605	−0.410802	−	0.911724i	$-0.634751\pi$
$491$	−18.2056	−0.821605	−0.410802	+	0.911724i	$0.634751\pi$
$492$	0	0
$493$	2.84333	0.128057
$494$	0	0
$495$	−6.20555	−0.278919
$496$	0	0
$497$	−30.5089	−1.36851
$498$	0	0
$499$	0.0594386	0.00266084	0.00133042	−	0.999999i	$-0.499577\pi$
$499$	0.0594386	0.00266084	0.00133042	+	0.999999i	$0.499577\pi$
$500$	0	0
$501$	10.7839	0.481789
$502$	0	0
$503$	−2.03831	−0.0908839	−0.0454419	−	0.998967i	$-0.514470\pi$
$503$	−2.03831	−0.0908839	−0.0454419	+	0.998967i	$0.514470\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	12.6655	0.562496
$508$	0	0
$509$	−40.7044	−1.80419	−0.902094	−	0.431539i	$-0.857971\pi$
$509$	−40.7044	−1.80419	−0.902094	+	0.431539i	$0.857971\pi$
$510$	0	0
$511$	21.7633	0.962751
$512$	0	0
$513$	5.62721	0.248448
$514$	0	0
$515$	−2.47054	−0.108865
$516$	0	0
$517$	−42.0978	−1.85146
$518$	0	0
$519$	−13.6655	−0.599850
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−35.3311	−1.54492	−0.772460	−	0.635064i	$-0.780974\pi$
$523$	−35.3311	−1.54492	−0.772460	+	0.635064i	$0.780974\pi$
$524$	0	0
$525$	−3.62721	−0.158305
$526$	0	0
$527$	−3.66553	−0.159673
$528$	0	0
$529$	8.66553	0.376762
$530$	0	0
$531$	2.20555	0.0957127
$532$	0	0
$533$	3.03883	0.131626
$534$	0	0
$535$	−14.0978	−0.609499
$536$	0	0
$537$	9.04888	0.390488
$538$	0	0
$539$	−38.2056	−1.64563
$540$	0	0
$541$	3.05892	0.131513	0.0657567	−	0.997836i	$-0.479054\pi$
$541$	3.05892	0.131513	0.0657567	+	0.997836i	$0.479054\pi$
$542$	0	0
$543$	−23.2544	−0.997943
$544$	0	0
$545$	7.25443	0.310745
$546$	0	0
$547$	−32.0766	−1.37150	−0.685749	−	0.727838i	$-0.740525\pi$
$547$	−32.0766	−1.37150	−0.685749	+	0.727838i	$0.740525\pi$
$548$	0	0
$549$	12.4111	0.529693
$550$	0	0
$551$	−11.2544	−0.479455
$552$	0	0
$553$	19.6655	0.836263
$554$	0	0
$555$	−7.83276	−0.332482
$556$	0	0
$557$	−33.6655	−1.42645	−0.713227	−	0.700933i	$-0.752767\pi$
$557$	−33.6655	−1.42645	−0.713227	+	0.700933i	$0.752767\pi$
$558$	0	0
$559$	−4.19550	−0.177451
$560$	0	0
$561$	8.82220	0.372474
$562$	0	0
$563$	−5.35218	−0.225567	−0.112784	−	0.993620i	$-0.535977\pi$
$563$	−5.35218	−0.225567	−0.112784	+	0.993620i	$0.535977\pi$
$564$	0	0
$565$	−9.08719	−0.382301
$566$	0	0
$567$	3.62721	0.152329
$568$	0	0
$569$	−5.58890	−0.234299	−0.117149	−	0.993114i	$-0.537376\pi$
$569$	−5.58890	−0.234299	−0.117149	+	0.993114i	$0.537376\pi$
$570$	0	0
$571$	10.3728	0.434088	0.217044	−	0.976162i	$-0.430359\pi$
$571$	10.3728	0.434088	0.217044	+	0.976162i	$0.430359\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	5.62721	0.234671
$576$	0	0
$577$	21.6655	0.901948	0.450974	−	0.892537i	$-0.351077\pi$
$577$	21.6655	0.901948	0.450974	+	0.892537i	$0.351077\pi$
$578$	0	0
$579$	−25.6655	−1.06662
$580$	0	0
$581$	11.8045	0.489733
$582$	0	0
$583$	−12.4111	−0.514015
$584$	0	0
$585$	−0.578337	−0.0239113
$586$	0	0
$587$	−1.90225	−0.0785142	−0.0392571	−	0.999229i	$-0.512499\pi$
$587$	−1.90225	−0.0785142	−0.0392571	+	0.999229i	$0.512499\pi$
$588$	0	0
$589$	14.5089	0.597827
$590$	0	0
$591$	15.1567	0.623462
$592$	0	0
$593$	−2.57834	−0.105880	−0.0529398	−	0.998598i	$-0.516859\pi$
$593$	−2.57834	−0.105880	−0.0529398	+	0.998598i	$0.516859\pi$
$594$	0	0
$595$	5.15667	0.211403
$596$	0	0
$597$	−20.6761	−0.846216
$598$	0	0
$599$	−26.7244	−1.09193	−0.545966	−	0.837808i	$-0.683837\pi$
$599$	−26.7244	−1.09193	−0.545966	+	0.837808i	$0.683837\pi$
$600$	0	0
$601$	−33.3311	−1.35960	−0.679801	−	0.733397i	$-0.737934\pi$
$601$	−33.3311	−1.35960	−0.679801	+	0.733397i	$0.737934\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	27.5089	1.11839
$606$	0	0
$607$	−21.9406	−0.890540	−0.445270	−	0.895396i	$-0.646892\pi$
$607$	−21.9406	−0.890540	−0.445270	+	0.895396i	$0.646892\pi$
$608$	0	0
$609$	−7.25443	−0.293964
$610$	0	0
$611$	−3.92337	−0.158723
$612$	0	0
$613$	3.42166	0.138200	0.0690998	−	0.997610i	$-0.477987\pi$
$613$	3.42166	0.138200	0.0690998	+	0.997610i	$0.477987\pi$
$614$	0	0
$615$	5.25443	0.211879
$616$	0	0
$617$	−19.7350	−0.794502	−0.397251	−	0.917710i	$-0.630036\pi$
$617$	−19.7350	−0.794502	−0.397251	+	0.917710i	$0.630036\pi$
$618$	0	0
$619$	20.4705	0.822780	0.411390	−	0.911459i	$-0.365043\pi$
$619$	20.4705	0.822780	0.411390	+	0.911459i	$0.365043\pi$
$620$	0	0
$621$	−5.62721	−0.225812
$622$	0	0
$623$	48.0766	1.92615
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−34.9200	−1.39457
$628$	0	0
$629$	11.1355	0.444003
$630$	0	0
$631$	−1.08719	−0.0432803	−0.0216402	−	0.999766i	$-0.506889\pi$
$631$	−1.08719	−0.0432803	−0.0216402	+	0.999766i	$0.506889\pi$
$632$	0	0
$633$	2.03831	0.0810157
$634$	0	0
$635$	−10.4705	−0.415511
$636$	0	0
$637$	−3.56063	−0.141077
$638$	0	0
$639$	−8.41110	−0.332738
$640$	0	0
$641$	−27.9789	−1.10510	−0.552550	−	0.833480i	$-0.686345\pi$
$641$	−27.9789	−1.10510	−0.552550	+	0.833480i	$0.686345\pi$
$642$	0	0
$643$	−4.94108	−0.194857	−0.0974285	−	0.995243i	$-0.531062\pi$
$643$	−4.94108	−0.194857	−0.0974285	+	0.995243i	$0.531062\pi$
$644$	0	0
$645$	−7.25443	−0.285643
$646$	0	0
$647$	49.3694	1.94091	0.970455	−	0.241282i	$-0.0775679\pi$
$647$	49.3694	1.94091	0.970455	+	0.241282i	$0.0775679\pi$
$648$	0	0
$649$	−13.6867	−0.537248
$650$	0	0
$651$	9.35218	0.366541
$652$	0	0
$653$	−40.1744	−1.57214	−0.786072	−	0.618134i	$-0.787889\pi$
$653$	−40.1744	−1.57214	−0.786072	+	0.618134i	$0.787889\pi$
$654$	0	0
$655$	−13.4600	−0.525925
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	21.1255	0.822933	0.411466	−	0.911425i	$-0.365017\pi$
$659$	21.1255	0.822933	0.411466	+	0.911425i	$0.365017\pi$
$660$	0	0
$661$	−10.9200	−0.424737	−0.212368	−	0.977190i	$-0.568118\pi$
$661$	−10.9200	−0.424737	−0.212368	+	0.977190i	$0.568118\pi$
$662$	0	0
$663$	0.822200	0.0319316
$664$	0	0
$665$	−20.4111	−0.791509
$666$	0	0
$667$	11.2544	0.435773
$668$	0	0
$669$	7.21611	0.278991
$670$	0	0
$671$	−77.0177	−2.97324
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	30.4877	1.17174	0.585869	−	0.810406i	$-0.300753\pi$
$677$	30.4877	1.17174	0.585869	+	0.810406i	$0.300753\pi$
$678$	0	0
$679$	17.5678	0.674189
$680$	0	0
$681$	−1.15667	−0.0443239
$682$	0	0
$683$	−35.2544	−1.34897	−0.674487	−	0.738287i	$-0.735635\pi$
$683$	−35.2544	−1.34897	−0.674487	+	0.738287i	$0.735635\pi$
$684$	0	0
$685$	10.5783	0.404177
$686$	0	0
$687$	14.0978	0.537863
$688$	0	0
$689$	−1.15667	−0.0440658
$690$	0	0
$691$	28.1361	1.07035	0.535173	−	0.844742i	$-0.320246\pi$
$691$	28.1361	1.07035	0.535173	+	0.844742i	$0.320246\pi$
$692$	0	0
$693$	−22.5089	−0.855041
$694$	0	0
$695$	12.4705	0.473034
$696$	0	0
$697$	−7.47002	−0.282947
$698$	0	0
$699$	−14.5783	−0.551403
$700$	0	0
$701$	34.8222	1.31522	0.657608	−	0.753360i	$-0.271568\pi$
$701$	34.8222	1.31522	0.657608	+	0.753360i	$0.271568\pi$
$702$	0	0
$703$	−44.0766	−1.66238
$704$	0	0
$705$	−6.78389	−0.255496
$706$	0	0
$707$	7.25443	0.272831
$708$	0	0
$709$	−7.58890	−0.285007	−0.142504	−	0.989794i	$-0.545515\pi$
$709$	−7.58890	−0.285007	−0.142504	+	0.989794i	$0.545515\pi$
$710$	0	0
$711$	5.42166	0.203328
$712$	0	0
$713$	−14.5089	−0.543361
$714$	0	0
$715$	3.58890	0.134217
$716$	0	0
$717$	−19.2544	−0.719070
$718$	0	0
$719$	−3.66553	−0.136701	−0.0683505	−	0.997661i	$-0.521774\pi$
$719$	−3.66553	−0.136701	−0.0683505	+	0.997661i	$0.521774\pi$
$720$	0	0
$721$	−8.96117	−0.333731
$722$	0	0
$723$	13.6655	0.508226
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	36.1149	1.33943	0.669714	−	0.742619i	$-0.266416\pi$
$727$	36.1149	1.33943	0.669714	+	0.742619i	$0.266416\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.3133	0.381453
$732$	0	0
$733$	34.0071	1.25608	0.628041	−	0.778180i	$-0.283857\pi$
$733$	34.0071	1.25608	0.628041	+	0.778180i	$0.283857\pi$
$734$	0	0
$735$	−6.15667	−0.227092
$736$	0	0
$737$	−24.8222	−0.914338
$738$	0	0
$739$	52.0172	1.91348	0.956742	−	0.290939i	$-0.0939677\pi$
$739$	52.0172	1.91348	0.956742	+	0.290939i	$0.0939677\pi$
$740$	0	0
$741$	−3.25443	−0.119554
$742$	0	0
$743$	23.3139	0.855303	0.427651	−	0.903944i	$-0.359341\pi$
$743$	23.3139	0.855303	0.427651	+	0.903944i	$0.359341\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	3.25443	0.119073
$748$	0	0
$749$	−51.1355	−1.86845
$750$	0	0
$751$	11.1083	0.405348	0.202674	−	0.979246i	$-0.435037\pi$
$751$	11.1083	0.405348	0.202674	+	0.979246i	$0.435037\pi$
$752$	0	0
$753$	−7.14663	−0.260438
$754$	0	0
$755$	−12.6761	−0.461330
$756$	0	0
$757$	−13.3239	−0.484266	−0.242133	−	0.970243i	$-0.577847\pi$
$757$	−13.3239	−0.484266	−0.242133	+	0.970243i	$0.577847\pi$
$758$	0	0
$759$	34.9200	1.26751
$760$	0	0
$761$	17.1355	0.621163	0.310582	−	0.950547i	$-0.399476\pi$
$761$	17.1355	0.621163	0.310582	+	0.950547i	$0.399476\pi$
$762$	0	0
$763$	26.3133	0.952607
$764$	0	0
$765$	1.42166	0.0514003
$766$	0	0
$767$	−1.27555	−0.0460575
$768$	0	0
$769$	5.47002	0.197254	0.0986270	−	0.995124i	$-0.468555\pi$
$769$	5.47002	0.197254	0.0986270	+	0.995124i	$0.468555\pi$
$770$	0	0
$771$	−7.73501	−0.278570
$772$	0	0
$773$	3.15667	0.113538	0.0567688	−	0.998387i	$-0.481920\pi$
$773$	3.15667	0.113538	0.0567688	+	0.998387i	$0.481920\pi$
$774$	0	0
$775$	−2.57834	−0.0926166
$776$	0	0
$777$	−28.4111	−1.01924
$778$	0	0
$779$	29.5678	1.05938
$780$	0	0
$781$	52.1955	1.86770
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−1.32391	−0.0472524
$786$	0	0
$787$	31.1355	1.10986	0.554931	−	0.831896i	$-0.312745\pi$
$787$	31.1355	1.10986	0.554931	+	0.831896i	$0.312745\pi$
$788$	0	0
$789$	−18.7839	−0.668724
$790$	0	0
$791$	−32.9612	−1.17196
$792$	0	0
$793$	−7.17780	−0.254891
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	9.64440	0.341194
$800$	0	0
$801$	13.2544	0.468322
$802$	0	0
$803$	−37.2333	−1.31393
$804$	0	0
$805$	20.4111	0.719397
$806$	0	0
$807$	−8.50885	−0.299526
$808$	0	0
$809$	−29.0388	−1.02095	−0.510475	−	0.859892i	$-0.670531\pi$
$809$	−29.0388	−1.02095	−0.510475	+	0.859892i	$0.670531\pi$
$810$	0	0
$811$	−2.58838	−0.0908904	−0.0454452	−	0.998967i	$-0.514471\pi$
$811$	−2.58838	−0.0908904	−0.0454452	+	0.998967i	$0.514471\pi$
$812$	0	0
$813$	−30.9894	−1.08685
$814$	0	0
$815$	15.2544	0.534339
$816$	0	0
$817$	−40.8222	−1.42819
$818$	0	0
$819$	−2.09775	−0.0733014
$820$	0	0
$821$	38.1955	1.33303	0.666516	−	0.745491i	$-0.267785\pi$
$821$	38.1955	1.33303	0.666516	+	0.745491i	$0.267785\pi$
$822$	0	0
$823$	18.3517	0.639699	0.319849	−	0.947468i	$-0.396368\pi$
$823$	18.3517	0.639699	0.319849	+	0.947468i	$0.396368\pi$
$824$	0	0
$825$	6.20555	0.216050
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	−24.7456	−0.859449	−0.429725	−	0.902960i	$-0.641389\pi$
$829$	−24.7456	−0.859449	−0.429725	+	0.902960i	$0.641389\pi$
$830$	0	0
$831$	9.51941	0.330225
$832$	0	0
$833$	8.75272	0.303264
$834$	0	0
$835$	−10.7839	−0.373192
$836$	0	0
$837$	2.57834	0.0891204
$838$	0	0
$839$	53.4288	1.84457	0.922284	−	0.386514i	$-0.126321\pi$
$839$	53.4288	1.84457	0.922284	+	0.386514i	$0.126321\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	13.6655	0.470666
$844$	0	0
$845$	−12.6655	−0.435707
$846$	0	0
$847$	99.7805	3.42850
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	44.0766	1.51093
$852$	0	0
$853$	29.0661	0.995203	0.497602	−	0.867406i	$-0.334214\pi$
$853$	29.0661	0.995203	0.497602	+	0.867406i	$0.334214\pi$
$854$	0	0
$855$	−5.62721	−0.192447
$856$	0	0
$857$	10.2439	0.349924	0.174962	−	0.984575i	$-0.444020\pi$
$857$	10.2439	0.349924	0.174962	+	0.984575i	$0.444020\pi$
$858$	0	0
$859$	10.9794	0.374612	0.187306	−	0.982302i	$-0.440024\pi$
$859$	10.9794	0.374612	0.187306	+	0.982302i	$0.440024\pi$
$860$	0	0
$861$	19.0589	0.649526
$862$	0	0
$863$	38.8605	1.32283	0.661414	−	0.750021i	$-0.269957\pi$
$863$	38.8605	1.32283	0.661414	+	0.750021i	$0.269957\pi$
$864$	0	0
$865$	13.6655	0.464642
$866$	0	0
$867$	14.9789	0.508709
$868$	0	0
$869$	−33.6444	−1.14131
$870$	0	0
$871$	−2.31335	−0.0783848
$872$	0	0
$873$	4.84333	0.163922
$874$	0	0
$875$	3.62721	0.122622
$876$	0	0
$877$	−22.3416	−0.754423	−0.377211	−	0.926127i	$-0.623117\pi$
$877$	−22.3416	−0.754423	−0.377211	+	0.926127i	$0.623117\pi$
$878$	0	0
$879$	−4.31335	−0.145486
$880$	0	0
$881$	9.88112	0.332903	0.166452	−	0.986050i	$-0.446769\pi$
$881$	9.88112	0.332903	0.166452	+	0.986050i	$0.446769\pi$
$882$	0	0
$883$	10.6277	0.357652	0.178826	−	0.983881i	$-0.442770\pi$
$883$	10.6277	0.357652	0.178826	+	0.983881i	$0.442770\pi$
$884$	0	0
$885$	−2.20555	−0.0741388
$886$	0	0
$887$	−11.6061	−0.389694	−0.194847	−	0.980834i	$-0.562421\pi$
$887$	−11.6061	−0.389694	−0.194847	+	0.980834i	$0.562421\pi$
$888$	0	0
$889$	−37.9789	−1.27377
$890$	0	0
$891$	−6.20555	−0.207894
$892$	0	0
$893$	−38.1744	−1.27746
$894$	0	0
$895$	−9.04888	−0.302471
$896$	0	0
$897$	3.25443	0.108662
$898$	0	0
$899$	−5.15667	−0.171985
$900$	0	0
$901$	2.84333	0.0947249
$902$	0	0
$903$	−26.3133	−0.875653
$904$	0	0
$905$	23.2544	0.773003
$906$	0	0
$907$	−0.195504	−0.00649159	−0.00324580	−	0.999995i	$-0.501033\pi$
$907$	−0.195504	−0.00649159	−0.00324580	+	0.999995i	$0.501033\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	−7.88112	−0.261113	−0.130557	−	0.991441i	$-0.541676\pi$
$911$	−7.88112	−0.261113	−0.130557	+	0.991441i	$0.541676\pi$
$912$	0	0
$913$	−20.1955	−0.668374
$914$	0	0
$915$	−12.4111	−0.410298
$916$	0	0
$917$	−48.8222	−1.61225
$918$	0	0
$919$	−9.75614	−0.321825	−0.160913	−	0.986969i	$-0.551444\pi$
$919$	−9.75614	−0.321825	−0.160913	+	0.986969i	$0.551444\pi$
$920$	0	0
$921$	−25.5678	−0.842487
$922$	0	0
$923$	4.86445	0.160115
$924$	0	0
$925$	7.83276	0.257540
$926$	0	0
$927$	−2.47054	−0.0811431
$928$	0	0
$929$	6.82220	0.223829	0.111915	−	0.993718i	$-0.464302\pi$
$929$	6.82220	0.223829	0.111915	+	0.993718i	$0.464302\pi$
$930$	0	0
$931$	−34.6449	−1.13544
$932$	0	0
$933$	−20.0766	−0.657279
$934$	0	0
$935$	−8.82220	−0.288517
$936$	0	0
$937$	−57.5266	−1.87931	−0.939655	−	0.342123i	$-0.888854\pi$
$937$	−57.5266	−1.87931	−0.939655	+	0.342123i	$0.888854\pi$
$938$	0	0
$939$	7.15667	0.233549
$940$	0	0
$941$	−0.508852	−0.0165881	−0.00829405	−	0.999966i	$-0.502640\pi$
$941$	−0.508852	−0.0165881	−0.00829405	+	0.999966i	$0.502640\pi$
$942$	0	0
$943$	−29.5678	−0.962859
$944$	0	0
$945$	−3.62721	−0.117993
$946$	0	0
$947$	1.68665	0.0548088	0.0274044	−	0.999624i	$-0.491276\pi$
$947$	1.68665	0.0548088	0.0274044	+	0.999624i	$0.491276\pi$
$948$	0	0
$949$	−3.47002	−0.112642
$950$	0	0
$951$	−24.1744	−0.783908
$952$	0	0
$953$	−9.22616	−0.298865	−0.149432	−	0.988772i	$-0.547745\pi$
$953$	−9.22616	−0.298865	−0.149432	+	0.988772i	$0.547745\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	12.4111	0.401194
$958$	0	0
$959$	38.3699	1.23903
$960$	0	0
$961$	−24.3522	−0.785554
$962$	0	0
$963$	−14.0978	−0.454294
$964$	0	0
$965$	25.6655	0.826203
$966$	0	0
$967$	−12.2338	−0.393413	−0.196707	−	0.980462i	$-0.563025\pi$
$967$	−12.2338	−0.393413	−0.196707	+	0.980462i	$0.563025\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	−33.2444	−1.06686	−0.533431	−	0.845843i	$-0.679098\pi$
$971$	−33.2444	−1.06686	−0.533431	+	0.845843i	$0.679098\pi$
$972$	0	0
$973$	45.2333	1.45011
$974$	0	0
$975$	0.578337	0.0185216
$976$	0	0
$977$	7.93051	0.253720	0.126860	−	0.991921i	$-0.459510\pi$
$977$	7.93051	0.253720	0.126860	+	0.991921i	$0.459510\pi$
$978$	0	0
$979$	−82.2510	−2.62875
$980$	0	0
$981$	7.25443	0.231616
$982$	0	0
$983$	−41.8993	−1.33638	−0.668191	−	0.743990i	$-0.732931\pi$
$983$	−41.8993	−1.33638	−0.668191	+	0.743990i	$0.732931\pi$
$984$	0	0
$985$	−15.1567	−0.482932
$986$	0	0
$987$	−24.6066	−0.783237
$988$	0	0
$989$	40.8222	1.29807
$990$	0	0
$991$	−35.1849	−1.11769	−0.558843	−	0.829273i	$-0.688755\pi$
$991$	−35.1849	−1.11769	−0.558843	+	0.829273i	$0.688755\pi$
$992$	0	0
$993$	−27.1950	−0.863007
$994$	0	0
$995$	20.6761	0.655476
$996$	0	0
$997$	−8.04836	−0.254894	−0.127447	−	0.991845i	$-0.540678\pi$
$997$	−8.04836	−0.254894	−0.127447	+	0.991845i	$0.540678\pi$
$998$	0	0
$999$	−7.83276	−0.247818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.a.bp.1.3		3
4.3	odd	2		3840.2.a.br.1.1		3
8.3	odd	2		3840.2.a.bo.1.1		3
8.5	even	2		3840.2.a.bq.1.3		3
16.3	odd	4		480.2.k.b.241.6		6
16.5	even	4		120.2.k.b.61.5	✓	6
16.11	odd	4		480.2.k.b.241.3		6
16.13	even	4		120.2.k.b.61.6	yes	6
48.5	odd	4		360.2.k.f.181.2		6
48.11	even	4		1440.2.k.f.721.3		6
48.29	odd	4		360.2.k.f.181.1		6
48.35	even	4		1440.2.k.f.721.6		6
80.3	even	4		2400.2.d.e.49.2		6
80.13	odd	4		600.2.d.f.349.3		6
80.19	odd	4		2400.2.k.c.1201.1		6
80.27	even	4		2400.2.d.e.49.5		6
80.29	even	4		600.2.k.c.301.1		6
80.37	odd	4		600.2.d.f.349.4		6
80.43	even	4		2400.2.d.f.49.2		6
80.53	odd	4		600.2.d.e.349.3		6
80.59	odd	4		2400.2.k.c.1201.4		6
80.67	even	4		2400.2.d.f.49.5		6
80.69	even	4		600.2.k.c.301.2		6
80.77	odd	4		600.2.d.e.349.4		6
240.29	odd	4		1800.2.k.p.901.6		6
240.53	even	4		1800.2.d.q.1549.4		6
240.59	even	4		7200.2.k.p.3601.1		6
240.77	even	4		1800.2.d.q.1549.3		6
240.83	odd	4		7200.2.d.r.2449.2		6
240.107	odd	4		7200.2.d.r.2449.5		6
240.149	odd	4		1800.2.k.p.901.5		6
240.173	even	4		1800.2.d.r.1549.4		6
240.179	even	4		7200.2.k.p.3601.2		6
240.197	even	4		1800.2.d.r.1549.3		6
240.203	odd	4		7200.2.d.q.2449.2		6
240.227	odd	4		7200.2.d.q.2449.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.5	✓	6	16.5	even	4
120.2.k.b.61.6	yes	6	16.13	even	4
360.2.k.f.181.1		6	48.29	odd	4
360.2.k.f.181.2		6	48.5	odd	4
480.2.k.b.241.3		6	16.11	odd	4
480.2.k.b.241.6		6	16.3	odd	4
600.2.d.e.349.3		6	80.53	odd	4
600.2.d.e.349.4		6	80.77	odd	4
600.2.d.f.349.3		6	80.13	odd	4
600.2.d.f.349.4		6	80.37	odd	4
600.2.k.c.301.1		6	80.29	even	4
600.2.k.c.301.2		6	80.69	even	4
1440.2.k.f.721.3		6	48.11	even	4
1440.2.k.f.721.6		6	48.35	even	4
1800.2.d.q.1549.3		6	240.77	even	4
1800.2.d.q.1549.4		6	240.53	even	4
1800.2.d.r.1549.3		6	240.197	even	4
1800.2.d.r.1549.4		6	240.173	even	4
1800.2.k.p.901.5		6	240.149	odd	4
1800.2.k.p.901.6		6	240.29	odd	4
2400.2.d.e.49.2		6	80.3	even	4
2400.2.d.e.49.5		6	80.27	even	4
2400.2.d.f.49.2		6	80.43	even	4
2400.2.d.f.49.5		6	80.67	even	4
2400.2.k.c.1201.1		6	80.19	odd	4
2400.2.k.c.1201.4		6	80.59	odd	4
3840.2.a.bo.1.1		3	8.3	odd	2
3840.2.a.bp.1.3		3	1.1	even	1	trivial
3840.2.a.bq.1.3		3	8.5	even	2
3840.2.a.br.1.1		3	4.3	odd	2
7200.2.d.q.2449.2		6	240.203	odd	4
7200.2.d.q.2449.5		6	240.227	odd	4
7200.2.d.r.2449.2		6	240.83	odd	4
7200.2.d.r.2449.5		6	240.107	odd	4
7200.2.k.p.3601.1		6	240.59	even	4
7200.2.k.p.3601.2		6	240.179	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.62721 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.62721 q^{7} +1.00000 q^{9} -6.20555 q^{11} -0.578337 q^{13} -1.00000 q^{15} +1.42166 q^{17} -5.62721 q^{19} -3.62721 q^{21} +5.62721 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -2.57834 q^{31} +6.20555 q^{33} +3.62721 q^{35} +7.83276 q^{37} +0.578337 q^{39} -5.25443 q^{41} +7.25443 q^{43} +1.00000 q^{45} +6.78389 q^{47} +6.15667 q^{49} -1.42166 q^{51} +2.00000 q^{53} -6.20555 q^{55} +5.62721 q^{57} +2.20555 q^{59} +12.4111 q^{61} +3.62721 q^{63} -0.578337 q^{65} +4.00000 q^{67} -5.62721 q^{69} -8.41110 q^{71} +6.00000 q^{73} -1.00000 q^{75} -22.5089 q^{77} +5.42166 q^{79} +1.00000 q^{81} +3.25443 q^{83} +1.42166 q^{85} -2.00000 q^{87} +13.2544 q^{89} -2.09775 q^{91} +2.57834 q^{93} -5.62721 q^{95} +4.84333 q^{97} -6.20555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 4 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 6 q^{31} + 4 q^{33} - 2 q^{35} - 4 q^{37} + 10 q^{41} - 4 q^{43} + 3 q^{45} + 4 q^{47} + 15 q^{49} - 6 q^{51} + 6 q^{53} - 4 q^{55} + 4 q^{57} - 8 q^{59} + 8 q^{61} - 2 q^{63} + 12 q^{67} - 4 q^{69} + 4 q^{71} + 18 q^{73} - 3 q^{75} - 16 q^{77} + 18 q^{79} + 3 q^{81} - 16 q^{83} + 6 q^{85} - 6 q^{87} + 14 q^{89} + 16 q^{91} + 6 q^{93} - 4 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 4 * q^11 - 3 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 + 4 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 6 * q^31 + 4 * q^33 - 2 * q^35 - 4 * q^37 + 10 * q^41 - 4 * q^43 + 3 * q^45 + 4 * q^47 + 15 * q^49 - 6 * q^51 + 6 * q^53 - 4 * q^55 + 4 * q^57 - 8 * q^59 + 8 * q^61 - 2 * q^63 + 12 * q^67 - 4 * q^69 + 4 * q^71 + 18 * q^73 - 3 * q^75 - 16 * q^77 + 18 * q^79 + 3 * q^81 - 16 * q^83 + 6 * q^85 - 6 * q^87 + 14 * q^89 + 16 * q^91 + 6 * q^93 - 4 * q^95 + 18 * q^97 - 4 * q^99

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