LMFDB - Embedded newform 8037.2.a.t.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.17517 q^{2} -0.618977 q^{4} +0.0159136 q^{5} -0.631968 q^{7} +3.07774 q^{8} +O(q^{10})$	\(q-1.17517 q^{2} -0.618977 q^{4} +0.0159136 q^{5} -0.631968 q^{7} +3.07774 q^{8} -0.0187011 q^{10} -2.39034 q^{11} -6.35191 q^{13} +0.742669 q^{14} -2.37891 q^{16} +0.874570 q^{17} +1.00000 q^{19} -0.00985012 q^{20} +2.80906 q^{22} -8.15968 q^{23} -4.99975 q^{25} +7.46457 q^{26} +0.391173 q^{28} -7.32435 q^{29} -0.949906 q^{31} -3.35986 q^{32} -1.02777 q^{34} -0.0100569 q^{35} -4.80302 q^{37} -1.17517 q^{38} +0.0489778 q^{40} -2.52342 q^{41} +2.63706 q^{43} +1.47957 q^{44} +9.58900 q^{46} +1.00000 q^{47} -6.60062 q^{49} +5.87555 q^{50} +3.93168 q^{52} -4.32807 q^{53} -0.0380389 q^{55} -1.94503 q^{56} +8.60736 q^{58} -4.56619 q^{59} +9.48283 q^{61} +1.11630 q^{62} +8.70623 q^{64} -0.101081 q^{65} +9.91919 q^{67} -0.541338 q^{68} +0.0118185 q^{70} -3.44443 q^{71} +14.3710 q^{73} +5.64436 q^{74} -0.618977 q^{76} +1.51062 q^{77} -3.15765 q^{79} -0.0378570 q^{80} +2.96544 q^{82} -5.55523 q^{83} +0.0139175 q^{85} -3.09899 q^{86} -7.35686 q^{88} -8.93461 q^{89} +4.01420 q^{91} +5.05065 q^{92} -1.17517 q^{94} +0.0159136 q^{95} -4.30850 q^{97} +7.75684 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) $ 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8	$ 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) $ 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8 + 7 * q^10 + 3 * q^11 + 11 * q^13 - 9 * q^14 + 40 * q^16 + 6 * q^17 + 24 * q^19 + 17 * q^20 + 15 * q^22 + 19 * q^23 + 54 * q^25 + q^26 + 26 * q^28 - 32 * q^29 + 12 * q^31 - 30 * q^32 + 38 * q^34 + 35 * q^35 + 14 * q^37 - 2 * q^38 + 47 * q^40 - 30 * q^41 + 10 * q^43 - q^44 - 3 * q^46 + 24 * q^47 + 56 * q^49 - 44 * q^50 - 10 * q^53 + 24 * q^55 + 6 * q^56 + 41 * q^58 - 11 * q^59 + 10 * q^61 - 2 * q^62 + 61 * q^64 - 35 * q^65 + 12 * q^67 + 22 * q^68 - 20 * q^70 - 44 * q^71 + 33 * q^73 - 36 * q^74 + 32 * q^76 + 23 * q^77 + 50 * q^79 + 13 * q^80 + 5 * q^82 + 72 * q^83 + 33 * q^85 - 44 * q^86 + 38 * q^88 - 48 * q^89 + 47 * q^91 + 29 * q^92 - 2 * q^94 + 2 * q^95 + 50 * q^97 - 22 * q^98

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 $

$2$	−1.17517	−0.830970	−0.415485	−	0.909600i	$-0.636388\pi$
$2$	−1.17517	−0.830970	−0.415485	+	0.909600i	$0.636388\pi$
$3$	0	0
$3$	0	0
$4$	−0.618977	−0.309488
$5$	0.0159136	0.00711676	0.00355838	−	0.999994i	$-0.498867\pi$
$5$	0.0159136	0.00711676	0.00355838	+	0.999994i	$0.498867\pi$
$6$	0	0
$7$	−0.631968	−0.238861	−0.119431	−	0.992843i	$-0.538107\pi$
$7$	−0.631968	−0.238861	−0.119431	+	0.992843i	$0.538107\pi$
$8$	3.07774	1.08815
$9$	0	0
$10$	−0.0187011	−0.00591381
$11$	−2.39034	−0.720716	−0.360358	−	0.932814i	$-0.617345\pi$
$11$	−2.39034	−0.720716	−0.360358	+	0.932814i	$0.617345\pi$
$12$	0	0
$13$	−6.35191	−1.76170	−0.880851	−	0.473393i	$-0.843029\pi$
$13$	−6.35191	−1.76170	−0.880851	+	0.473393i	$0.843029\pi$
$14$	0.742669	0.198487
$15$	0	0
$16$	−2.37891	−0.594728
$17$	0.874570	0.212114	0.106057	−	0.994360i	$-0.466177\pi$
$17$	0.874570	0.212114	0.106057	+	0.994360i	$0.466177\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.00985012	−0.00220255
$21$	0	0
$22$	2.80906	0.598893
$23$	−8.15968	−1.70141	−0.850705	−	0.525643i	$-0.823825\pi$
$23$	−8.15968	−1.70141	−0.850705	+	0.525643i	$0.823825\pi$
$24$	0	0
$25$	−4.99975	−0.999949
$26$	7.46457	1.46392
$27$	0	0
$28$	0.391173	0.0739248
$29$	−7.32435	−1.36010	−0.680049	−	0.733166i	$-0.738042\pi$
$29$	−7.32435	−1.36010	−0.680049	+	0.733166i	$0.738042\pi$
$30$	0	0
$31$	−0.949906	−0.170608	−0.0853041	−	0.996355i	$-0.527186\pi$
$31$	−0.949906	−0.170608	−0.0853041	+	0.996355i	$0.527186\pi$
$32$	−3.35986	−0.593944
$33$	0	0
$34$	−1.02777	−0.176261
$35$	−0.0100569	−0.00169992
$36$	0	0
$37$	−4.80302	−0.789611	−0.394806	−	0.918765i	$-0.629188\pi$
$37$	−4.80302	−0.789611	−0.394806	+	0.918765i	$0.629188\pi$
$38$	−1.17517	−0.190638
$39$	0	0
$40$	0.0489778	0.00774407
$41$	−2.52342	−0.394092	−0.197046	−	0.980394i	$-0.563135\pi$
$41$	−2.52342	−0.394092	−0.197046	+	0.980394i	$0.563135\pi$
$42$	0	0
$43$	2.63706	0.402147	0.201074	−	0.979576i	$-0.435557\pi$
$43$	2.63706	0.402147	0.201074	+	0.979576i	$0.435557\pi$
$44$	1.47957	0.223053
$45$	0	0
$46$	9.58900	1.41382
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−6.60062	−0.942945
$50$	5.87555	0.830928
$51$	0	0
$52$	3.93168	0.545227
$53$	−4.32807	−0.594506	−0.297253	−	0.954799i	$-0.596070\pi$
$53$	−4.32807	−0.594506	−0.297253	+	0.954799i	$0.596070\pi$
$54$	0	0
$55$	−0.0380389	−0.00512916
$56$	−1.94503	−0.259916
$57$	0	0
$58$	8.60736	1.13020
$59$	−4.56619	−0.594467	−0.297234	−	0.954805i	$-0.596064\pi$
$59$	−4.56619	−0.594467	−0.297234	+	0.954805i	$0.596064\pi$
$60$	0	0
$61$	9.48283	1.21415	0.607076	−	0.794644i	$-0.292342\pi$
$61$	9.48283	1.21415	0.607076	+	0.794644i	$0.292342\pi$
$62$	1.11630	0.141770
$63$	0	0
$64$	8.70623	1.08828
$65$	−0.101081	−0.0125376
$66$	0	0
$67$	9.91919	1.21182	0.605911	−	0.795532i	$-0.292809\pi$
$67$	9.91919	1.21182	0.605911	+	0.795532i	$0.292809\pi$
$68$	−0.541338	−0.0656469
$69$	0	0
$70$	0.0118185	0.00141258
$71$	−3.44443	−0.408778	−0.204389	−	0.978890i	$-0.565521\pi$
$71$	−3.44443	−0.408778	−0.204389	+	0.978890i	$0.565521\pi$
$72$	0	0
$73$	14.3710	1.68200	0.841001	−	0.541033i	$-0.181967\pi$
$73$	14.3710	1.68200	0.841001	+	0.541033i	$0.181967\pi$
$74$	5.64436	0.656144
$75$	0	0
$76$	−0.618977	−0.0710015
$77$	1.51062	0.172151
$78$	0	0
$79$	−3.15765	−0.355264	−0.177632	−	0.984097i	$-0.556844\pi$
$79$	−3.15765	−0.355264	−0.177632	+	0.984097i	$0.556844\pi$
$80$	−0.0378570	−0.00423254
$81$	0	0
$82$	2.96544	0.327479
$83$	−5.55523	−0.609765	−0.304883	−	0.952390i	$-0.598617\pi$
$83$	−5.55523	−0.609765	−0.304883	+	0.952390i	$0.598617\pi$
$84$	0	0
$85$	0.0139175	0.00150957
$86$	−3.09899	−0.334172
$87$	0	0
$88$	−7.35686	−0.784244
$89$	−8.93461	−0.947066	−0.473533	−	0.880776i	$-0.657022\pi$
$89$	−8.93461	−0.947066	−0.473533	+	0.880776i	$0.657022\pi$
$90$	0	0
$91$	4.01420	0.420803
$92$	5.05065	0.526567
$93$	0	0
$94$	−1.17517	−0.121209
$95$	0.0159136	0.00163270
$96$	0	0
$97$	−4.30850	−0.437462	−0.218731	−	0.975785i	$-0.570192\pi$
$97$	−4.30850	−0.437462	−0.218731	+	0.975785i	$0.570192\pi$
$98$	7.75684	0.783559
$99$	0	0
$100$	3.09473	0.309473
$101$	15.5694	1.54922	0.774609	−	0.632440i	$-0.217947\pi$
$101$	15.5694	1.54922	0.774609	+	0.632440i	$0.217947\pi$
$102$	0	0
$103$	−5.59770	−0.551558	−0.275779	−	0.961221i	$-0.588936\pi$
$103$	−5.59770	−0.551558	−0.275779	+	0.961221i	$0.588936\pi$
$104$	−19.5495	−1.91699
$105$	0	0
$106$	5.08622	0.494017
$107$	−10.6871	−1.03316	−0.516578	−	0.856240i	$-0.672794\pi$
$107$	−10.6871	−1.03316	−0.516578	+	0.856240i	$0.672794\pi$
$108$	0	0
$109$	8.39308	0.803911	0.401955	−	0.915659i	$-0.368331\pi$
$109$	8.39308	0.803911	0.401955	+	0.915659i	$0.368331\pi$
$110$	0.0447021	0.00426218
$111$	0	0
$112$	1.50340	0.142058
$113$	−8.35115	−0.785611	−0.392805	−	0.919622i	$-0.628495\pi$
$113$	−8.35115	−0.785611	−0.392805	+	0.919622i	$0.628495\pi$
$114$	0	0
$115$	−0.129849	−0.0121085
$116$	4.53361	0.420935
$117$	0	0
$118$	5.36605	0.493985
$119$	−0.552700	−0.0506659
$120$	0	0
$121$	−5.28626	−0.480569
$122$	−11.1439	−1.00892
$123$	0	0
$124$	0.587970	0.0528013
$125$	−0.159132	−0.0142332
$126$	0	0
$127$	−0.727385	−0.0645449	−0.0322725	−	0.999479i	$-0.510274\pi$
$127$	−0.727385	−0.0645449	−0.0322725	+	0.999479i	$0.510274\pi$
$128$	−3.51158	−0.310383
$129$	0	0
$130$	0.118788	0.0104184
$131$	−0.888944	−0.0776674	−0.0388337	−	0.999246i	$-0.512364\pi$
$131$	−0.888944	−0.0776674	−0.0388337	+	0.999246i	$0.512364\pi$
$132$	0	0
$133$	−0.631968	−0.0547986
$134$	−11.6567	−1.00699
$135$	0	0
$136$	2.69170	0.230811
$137$	−4.18464	−0.357518	−0.178759	−	0.983893i	$-0.557208\pi$
$137$	−4.18464	−0.357518	−0.178759	+	0.983893i	$0.557208\pi$
$138$	0	0
$139$	−14.6374	−1.24153	−0.620764	−	0.783997i	$-0.713178\pi$
$139$	−14.6374	−1.24153	−0.620764	+	0.783997i	$0.713178\pi$
$140$	0.00622496	0.000526105 0
$141$	0	0
$142$	4.04779	0.339683
$143$	15.1832	1.26969
$144$	0	0
$145$	−0.116557	−0.00967949
$146$	−16.8884	−1.39769
$147$	0	0
$148$	2.97296	0.244376
$149$	5.19794	0.425832	0.212916	−	0.977071i	$-0.431704\pi$
$149$	5.19794	0.425832	0.212916	+	0.977071i	$0.431704\pi$
$150$	0	0
$151$	19.7981	1.61114	0.805571	−	0.592499i	$-0.201858\pi$
$151$	19.7981	1.61114	0.805571	+	0.592499i	$0.201858\pi$
$152$	3.07774	0.249638
$153$	0	0
$154$	−1.77523	−0.143052
$155$	−0.0151164	−0.00121418
$156$	0	0
$157$	−13.9074	−1.10993	−0.554967	−	0.831873i	$-0.687269\pi$
$157$	−13.9074	−1.10993	−0.554967	+	0.831873i	$0.687269\pi$
$158$	3.71078	0.295213
$159$	0	0
$160$	−0.0534673	−0.00422696
$161$	5.15665	0.406401
$162$	0	0
$163$	15.6338	1.22454	0.612268	−	0.790650i	$-0.290257\pi$
$163$	15.6338	1.22454	0.612268	+	0.790650i	$0.290257\pi$
$164$	1.56194	0.121967
$165$	0	0
$166$	6.52833	0.506697
$167$	−4.93107	−0.381578	−0.190789	−	0.981631i	$-0.561105\pi$
$167$	−4.93107	−0.381578	−0.190789	+	0.981631i	$0.561105\pi$
$168$	0	0
$169$	27.3467	2.10360
$170$	−0.0163554	−0.00125440
$171$	0	0
$172$	−1.63228	−0.124460
$173$	0.887071	0.0674427	0.0337214	−	0.999431i	$-0.489264\pi$
$173$	0.887071	0.0674427	0.0337214	+	0.999431i	$0.489264\pi$
$174$	0	0
$175$	3.15968	0.238849
$176$	5.68642	0.428630
$177$	0	0
$178$	10.4997	0.786984
$179$	−19.1789	−1.43350	−0.716749	−	0.697331i	$-0.754371\pi$
$179$	−19.1789	−1.43350	−0.716749	+	0.697331i	$0.754371\pi$
$180$	0	0
$181$	21.9301	1.63005	0.815026	−	0.579424i	$-0.196723\pi$
$181$	21.9301	1.63005	0.815026	+	0.579424i	$0.196723\pi$
$182$	−4.71737	−0.349675
$183$	0	0
$184$	−25.1134	−1.85138
$185$	−0.0764331	−0.00561947
$186$	0	0
$187$	−2.09052	−0.152874
$188$	−0.618977	−0.0451435
$189$	0	0
$190$	−0.0187011	−0.00135672
$191$	−23.1074	−1.67199	−0.835996	−	0.548735i	$-0.815109\pi$
$191$	−23.1074	−1.67199	−0.835996	+	0.548735i	$0.815109\pi$
$192$	0	0
$193$	17.3498	1.24887	0.624434	−	0.781078i	$-0.285330\pi$
$193$	17.3498	1.24887	0.624434	+	0.781078i	$0.285330\pi$
$194$	5.06321	0.363518
$195$	0	0
$196$	4.08563	0.291831
$197$	4.48744	0.319717	0.159858	−	0.987140i	$-0.448896\pi$
$197$	4.48744	0.319717	0.159858	+	0.987140i	$0.448896\pi$
$198$	0	0
$199$	−15.1812	−1.07617	−0.538085	−	0.842891i	$-0.680852\pi$
$199$	−15.1812	−1.07617	−0.538085	+	0.842891i	$0.680852\pi$
$200$	−15.3879	−1.08809
$201$	0	0
$202$	−18.2967	−1.28735
$203$	4.62876	0.324875
$204$	0	0
$205$	−0.0401566	−0.00280466
$206$	6.57825	0.458328
$207$	0	0
$208$	15.1106	1.04773
$209$	−2.39034	−0.165344
$210$	0	0
$211$	−20.7626	−1.42936	−0.714678	−	0.699454i	$-0.753427\pi$
$211$	−20.7626	−1.42936	−0.714678	+	0.699454i	$0.753427\pi$
$212$	2.67898	0.183993
$213$	0	0
$214$	12.5591	0.858522
$215$	0.0419649	0.00286199
$216$	0	0
$217$	0.600310	0.0407517
$218$	−9.86329	−0.668026
$219$	0	0
$220$	0.0235452	0.00158742
$221$	−5.55519	−0.373682
$222$	0	0
$223$	11.5021	0.770240	0.385120	−	0.922867i	$-0.374160\pi$
$223$	11.5021	0.770240	0.385120	+	0.922867i	$0.374160\pi$
$224$	2.12332	0.141870
$225$	0	0
$226$	9.81402	0.652819
$227$	8.89059	0.590089	0.295045	−	0.955484i	$-0.404666\pi$
$227$	8.89059	0.590089	0.295045	+	0.955484i	$0.404666\pi$
$228$	0	0
$229$	11.5775	0.765063	0.382532	−	0.923942i	$-0.375052\pi$
$229$	11.5775	0.765063	0.382532	+	0.923942i	$0.375052\pi$
$230$	0.152595	0.0100618
$231$	0	0
$232$	−22.5425	−1.47999
$233$	22.6197	1.48187	0.740934	−	0.671578i	$-0.234383\pi$
$233$	22.6197	1.48187	0.740934	+	0.671578i	$0.234383\pi$
$234$	0	0
$235$	0.0159136	0.00103809
$236$	2.82637	0.183981
$237$	0	0
$238$	0.649516	0.0421019
$239$	20.1366	1.30253	0.651263	−	0.758852i	$-0.274240\pi$
$239$	20.1366	1.30253	0.651263	+	0.758852i	$0.274240\pi$
$240$	0	0
$241$	9.23254	0.594720	0.297360	−	0.954765i	$-0.403894\pi$
$241$	9.23254	0.594720	0.297360	+	0.954765i	$0.403894\pi$
$242$	6.21225	0.399338
$243$	0	0
$244$	−5.86965	−0.375766
$245$	−0.105039	−0.00671071
$246$	0	0
$247$	−6.35191	−0.404162
$248$	−2.92357	−0.185647
$249$	0	0
$250$	0.187007	0.0118273
$251$	−13.7956	−0.870772	−0.435386	−	0.900244i	$-0.643388\pi$
$251$	−13.7956	−0.870772	−0.435386	+	0.900244i	$0.643388\pi$
$252$	0	0
$253$	19.5044	1.22623
$254$	0.854800	0.0536349
$255$	0	0
$256$	−13.2858	−0.830360
$257$	23.2579	1.45079	0.725393	−	0.688335i	$-0.241658\pi$
$257$	23.2579	1.45079	0.725393	+	0.688335i	$0.241658\pi$
$258$	0	0
$259$	3.03535	0.188608
$260$	0.0625671	0.00388025
$261$	0	0
$262$	1.04466	0.0645393
$263$	8.75569	0.539899	0.269950	−	0.962874i	$-0.412993\pi$
$263$	8.75569	0.539899	0.269950	+	0.962874i	$0.412993\pi$
$264$	0	0
$265$	−0.0688750	−0.00423096
$266$	0.742669	0.0455360
$267$	0	0
$268$	−6.13975	−0.375045
$269$	−3.56456	−0.217335	−0.108668	−	0.994078i	$-0.534658\pi$
$269$	−3.56456	−0.217335	−0.108668	+	0.994078i	$0.534658\pi$
$270$	0	0
$271$	−2.72732	−0.165673	−0.0828366	−	0.996563i	$-0.526398\pi$
$271$	−2.72732	−0.165673	−0.0828366	+	0.996563i	$0.526398\pi$
$272$	−2.08053	−0.126150
$273$	0	0
$274$	4.91766	0.297087
$275$	11.9511	0.720679
$276$	0	0
$277$	−32.3246	−1.94220	−0.971099	−	0.238679i	$-0.923286\pi$
$277$	−32.3246	−1.94220	−0.971099	+	0.238679i	$0.923286\pi$
$278$	17.2014	1.03167
$279$	0	0
$280$	−0.0309524	−0.00184976
$281$	−18.5955	−1.10932	−0.554659	−	0.832078i	$-0.687151\pi$
$281$	−18.5955	−1.10932	−0.554659	+	0.832078i	$0.687151\pi$
$282$	0	0
$283$	−21.3344	−1.26820	−0.634099	−	0.773252i	$-0.718629\pi$
$283$	−21.3344	−1.26820	−0.634099	+	0.773252i	$0.718629\pi$
$284$	2.13202	0.126512
$285$	0	0
$286$	−17.8429	−1.05507
$287$	1.59472	0.0941333
$288$	0	0
$289$	−16.2351	−0.955008
$290$	0.136974	0.00804337
$291$	0	0
$292$	−8.89534	−0.520560
$293$	−26.8623	−1.56931	−0.784656	−	0.619932i	$-0.787160\pi$
$293$	−26.8623	−1.56931	−0.784656	+	0.619932i	$0.787160\pi$
$294$	0	0
$295$	−0.0726643	−0.00423068
$296$	−14.7825	−0.859212
$297$	0	0
$298$	−6.10846	−0.353853
$299$	51.8295	2.99738
$300$	0	0
$301$	−1.66653	−0.0960575
$302$	−23.2661	−1.33881
$303$	0	0
$304$	−2.37891	−0.136440
$305$	0.150906	0.00864083
$306$	0	0
$307$	1.43100	0.0816714	0.0408357	−	0.999166i	$-0.486998\pi$
$307$	1.43100	0.0816714	0.0408357	+	0.999166i	$0.486998\pi$
$308$	−0.935039	−0.0532788
$309$	0	0
$310$	0.0177643	0.00100895
$311$	−12.9510	−0.734386	−0.367193	−	0.930145i	$-0.619681\pi$
$311$	−12.9510	−0.734386	−0.367193	+	0.930145i	$0.619681\pi$
$312$	0	0
$313$	−27.2117	−1.53809	−0.769047	−	0.639192i	$-0.779269\pi$
$313$	−27.2117	−1.53809	−0.769047	+	0.639192i	$0.779269\pi$
$314$	16.3436	0.922321
$315$	0	0
$316$	1.95451	0.109950
$317$	3.07043	0.172453	0.0862263	−	0.996276i	$-0.472519\pi$
$317$	3.07043	0.172453	0.0862263	+	0.996276i	$0.472519\pi$
$318$	0	0
$319$	17.5077	0.980244
$320$	0.138547	0.00774502
$321$	0	0
$322$	−6.05994	−0.337707
$323$	0.874570	0.0486624
$324$	0	0
$325$	31.7579	1.76161
$326$	−18.3724	−1.01755
$327$	0	0
$328$	−7.76643	−0.428829
$329$	−0.631968	−0.0348415
$330$	0	0
$331$	16.0700	0.883288	0.441644	−	0.897190i	$-0.354395\pi$
$331$	16.0700	0.883288	0.441644	+	0.897190i	$0.354395\pi$
$332$	3.43856	0.188715
$333$	0	0
$334$	5.79484	0.317080
$335$	0.157850	0.00862425
$336$	0	0
$337$	25.8043	1.40565	0.702824	−	0.711364i	$-0.251922\pi$
$337$	25.8043	1.40565	0.702824	+	0.711364i	$0.251922\pi$
$338$	−32.1371	−1.74803
$339$	0	0
$340$	−0.00861462	−0.000467193 0
$341$	2.27060	0.122960
$342$	0	0
$343$	8.59515	0.464095
$344$	8.11618	0.437595
$345$	0	0
$346$	−1.04246	−0.0560429
$347$	24.2498	1.30180	0.650898	−	0.759165i	$-0.274393\pi$
$347$	24.2498	1.30180	0.650898	+	0.759165i	$0.274393\pi$
$348$	0	0
$349$	15.4890	0.829109	0.414555	−	0.910024i	$-0.363937\pi$
$349$	15.4890	0.829109	0.414555	+	0.910024i	$0.363937\pi$
$350$	−3.71316	−0.198477
$351$	0	0
$352$	8.03121	0.428065
$353$	−15.1734	−0.807600	−0.403800	−	0.914847i	$-0.632311\pi$
$353$	−15.1734	−0.807600	−0.403800	+	0.914847i	$0.632311\pi$
$354$	0	0
$355$	−0.0548131	−0.00290918
$356$	5.53032	0.293106
$357$	0	0
$358$	22.5385	1.19119
$359$	12.4637	0.657808	0.328904	−	0.944363i	$-0.393321\pi$
$359$	12.4637	0.657808	0.328904	+	0.944363i	$0.393321\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−25.7716	−1.35452
$363$	0	0
$364$	−2.48470	−0.130234
$365$	0.228694	0.0119704
$366$	0	0
$367$	−26.8927	−1.40379	−0.701893	−	0.712283i	$-0.747661\pi$
$367$	−26.8927	−1.40379	−0.701893	+	0.712283i	$0.747661\pi$
$368$	19.4112	1.01188
$369$	0	0
$370$	0.0898219	0.00466962
$371$	2.73520	0.142005
$372$	0	0
$373$	−16.6417	−0.861675	−0.430838	−	0.902429i	$-0.641782\pi$
$373$	−16.6417	−0.861675	−0.430838	+	0.902429i	$0.641782\pi$
$374$	2.45672	0.127034
$375$	0	0
$376$	3.07774	0.158722
$377$	46.5236	2.39609
$378$	0	0
$379$	−28.9438	−1.48674	−0.743371	−	0.668880i	$-0.766774\pi$
$379$	−28.9438	−1.48674	−0.743371	+	0.668880i	$0.766774\pi$
$380$	−0.00985012	−0.000505301 0
$381$	0	0
$382$	27.1551	1.38938
$383$	14.4192	0.736788	0.368394	−	0.929670i	$-0.379908\pi$
$383$	14.4192	0.736788	0.368394	+	0.929670i	$0.379908\pi$
$384$	0	0
$385$	0.0240393	0.00122516
$386$	−20.3890	−1.03777
$387$	0	0
$388$	2.66686	0.135389
$389$	−6.92854	−0.351291	−0.175645	−	0.984453i	$-0.556201\pi$
$389$	−6.92854	−0.351291	−0.175645	+	0.984453i	$0.556201\pi$
$390$	0	0
$391$	−7.13621	−0.360893
$392$	−20.3150	−1.02606
$393$	0	0
$394$	−5.27350	−0.265675
$395$	−0.0502495	−0.00252833
$396$	0	0
$397$	2.38684	0.119792	0.0598961	−	0.998205i	$-0.480923\pi$
$397$	2.38684	0.119792	0.0598961	+	0.998205i	$0.480923\pi$
$398$	17.8405	0.894265
$399$	0	0
$400$	11.8940	0.594698
$401$	−18.8629	−0.941968	−0.470984	−	0.882142i	$-0.656101\pi$
$401$	−18.8629	−0.941968	−0.470984	+	0.882142i	$0.656101\pi$
$402$	0	0
$403$	6.03372	0.300561
$404$	−9.63713	−0.479465
$405$	0	0
$406$	−5.43957	−0.269961
$407$	11.4809	0.569085
$408$	0	0
$409$	−2.16281	−0.106944	−0.0534720	−	0.998569i	$-0.517029\pi$
$409$	−2.16281	−0.106944	−0.0534720	+	0.998569i	$0.517029\pi$
$410$	0.0471908	0.00233059
$411$	0	0
$412$	3.46485	0.170701
$413$	2.88568	0.141995
$414$	0	0
$415$	−0.0884034	−0.00433955
$416$	21.3415	1.04635
$417$	0	0
$418$	2.80906	0.137396
$419$	−32.8229	−1.60350	−0.801751	−	0.597658i	$-0.796098\pi$
$419$	−32.8229	−1.60350	−0.801751	+	0.597658i	$0.796098\pi$
$420$	0	0
$421$	0.272161	0.0132643	0.00663215	−	0.999978i	$-0.497889\pi$
$421$	0.272161	0.0132643	0.00663215	+	0.999978i	$0.497889\pi$
$422$	24.3996	1.18775
$423$	0	0
$424$	−13.3207	−0.646909
$425$	−4.37263	−0.212104
$426$	0	0
$427$	−5.99284	−0.290014
$428$	6.61504	0.319750
$429$	0	0
$430$	−0.0493159	−0.00237822
$431$	−14.8273	−0.714206	−0.357103	−	0.934065i	$-0.616236\pi$
$431$	−14.8273	−0.714206	−0.357103	+	0.934065i	$0.616236\pi$
$432$	0	0
$433$	−3.54163	−0.170200	−0.0851000	−	0.996372i	$-0.527121\pi$
$433$	−3.54163	−0.170200	−0.0851000	+	0.996372i	$0.527121\pi$
$434$	−0.705466	−0.0338635
$435$	0	0
$436$	−5.19512	−0.248801
$437$	−8.15968	−0.390330
$438$	0	0
$439$	5.63562	0.268973	0.134487	−	0.990915i	$-0.457061\pi$
$439$	5.63562	0.268973	0.134487	+	0.990915i	$0.457061\pi$
$440$	−0.117074	−0.00558127
$441$	0	0
$442$	6.52829	0.310519
$443$	25.0145	1.18848	0.594238	−	0.804289i	$-0.297454\pi$
$443$	25.0145	1.18848	0.594238	+	0.804289i	$0.297454\pi$
$444$	0	0
$445$	−0.142181	−0.00674004
$446$	−13.5170	−0.640046
$447$	0	0
$448$	−5.50206	−0.259948
$449$	−4.81452	−0.227211	−0.113606	−	0.993526i	$-0.536240\pi$
$449$	−4.81452	−0.227211	−0.113606	+	0.993526i	$0.536240\pi$
$450$	0	0
$451$	6.03184	0.284028
$452$	5.16917	0.243137
$453$	0	0
$454$	−10.4479	−0.490346
$455$	0.0638802	0.00299475
$456$	0	0
$457$	30.8809	1.44455	0.722273	−	0.691608i	$-0.243097\pi$
$457$	30.8809	1.44455	0.722273	+	0.691608i	$0.243097\pi$
$458$	−13.6055	−0.635745
$459$	0	0
$460$	0.0803738	0.00374745
$461$	−6.92384	−0.322475	−0.161238	−	0.986916i	$-0.551549\pi$
$461$	−6.92384	−0.322475	−0.161238	+	0.986916i	$0.551549\pi$
$462$	0	0
$463$	40.2282	1.86956	0.934780	−	0.355226i	$-0.115596\pi$
$463$	40.2282	1.86956	0.934780	+	0.355226i	$0.115596\pi$
$464$	17.4240	0.808889
$465$	0	0
$466$	−26.5820	−1.23139
$467$	9.21768	0.426543	0.213272	−	0.976993i	$-0.431588\pi$
$467$	9.21768	0.426543	0.213272	+	0.976993i	$0.431588\pi$
$468$	0	0
$469$	−6.26861	−0.289458
$470$	−0.0187011	−0.000862619 0
$471$	0	0
$472$	−14.0535	−0.646867
$473$	−6.30347	−0.289834
$474$	0	0
$475$	−4.99975	−0.229404
$476$	0.342108	0.0156805
$477$	0	0
$478$	−23.6639	−1.08236
$479$	10.9725	0.501347	0.250673	−	0.968072i	$-0.419348\pi$
$479$	10.9725	0.501347	0.250673	+	0.968072i	$0.419348\pi$
$480$	0	0
$481$	30.5083	1.39106
$482$	−10.8498	−0.494195
$483$	0	0
$484$	3.27207	0.148731
$485$	−0.0685635	−0.00311331
$486$	0	0
$487$	0.487892	0.0221085	0.0110543	−	0.999939i	$-0.496481\pi$
$487$	0.487892	0.0221085	0.0110543	+	0.999939i	$0.496481\pi$
$488$	29.1857	1.32117
$489$	0	0
$490$	0.123439	0.00557640
$491$	41.4877	1.87231	0.936157	−	0.351582i	$-0.114356\pi$
$491$	41.4877	1.87231	0.936157	+	0.351582i	$0.114356\pi$
$492$	0	0
$493$	−6.40566	−0.288496
$494$	7.46457	0.335847
$495$	0	0
$496$	2.25975	0.101466
$497$	2.17677	0.0976414
$498$	0	0
$499$	−17.3289	−0.775747	−0.387874	−	0.921713i	$-0.626790\pi$
$499$	−17.3289	−0.775747	−0.387874	+	0.921713i	$0.626790\pi$
$500$	0.0984987	0.00440500
$501$	0	0
$502$	16.2122	0.723586
$503$	28.8355	1.28571	0.642856	−	0.765987i	$-0.277749\pi$
$503$	28.8355	1.28571	0.642856	+	0.765987i	$0.277749\pi$
$504$	0	0
$505$	0.247765	0.0110254
$506$	−22.9210	−1.01896
$507$	0	0
$508$	0.450234	0.0199759
$509$	27.4364	1.21610	0.608048	−	0.793901i	$-0.291953\pi$
$509$	27.4364	1.21610	0.608048	+	0.793901i	$0.291953\pi$
$510$	0	0
$511$	−9.08203	−0.401765
$512$	22.6362	1.00039
$513$	0	0
$514$	−27.3319	−1.20556
$515$	−0.0890794	−0.00392531
$516$	0	0
$517$	−2.39034	−0.105127
$518$	−3.56705	−0.156727
$519$	0	0
$520$	−0.311103	−0.0136428
$521$	20.9820	0.919237	0.459618	−	0.888117i	$-0.347986\pi$
$521$	20.9820	0.919237	0.459618	+	0.888117i	$0.347986\pi$
$522$	0	0
$523$	−4.60461	−0.201346	−0.100673	−	0.994920i	$-0.532100\pi$
$523$	−4.60461	−0.201346	−0.100673	+	0.994920i	$0.532100\pi$
$524$	0.550236	0.0240372
$525$	0	0
$526$	−10.2894	−0.448640
$527$	−0.830759	−0.0361884
$528$	0	0
$529$	43.5803	1.89480
$530$	0.0809398	0.00351580
$531$	0	0
$532$	0.391173	0.0169595
$533$	16.0285	0.694273
$534$	0	0
$535$	−0.170069	−0.00735273
$536$	30.5287	1.31864
$537$	0	0
$538$	4.18896	0.180599
$539$	15.7777	0.679595
$540$	0	0
$541$	8.07419	0.347136	0.173568	−	0.984822i	$-0.444470\pi$
$541$	8.07419	0.347136	0.173568	+	0.984822i	$0.444470\pi$
$542$	3.20507	0.137669
$543$	0	0
$544$	−2.93843	−0.125984
$545$	0.133564	0.00572124
$546$	0	0
$547$	19.6795	0.841433	0.420716	−	0.907192i	$-0.361779\pi$
$547$	19.6795	0.841433	0.420716	+	0.907192i	$0.361779\pi$
$548$	2.59019	0.110648
$549$	0	0
$550$	−14.0446	−0.598863
$551$	−7.32435	−0.312028
$552$	0	0
$553$	1.99553	0.0848588
$554$	37.9869	1.61391
$555$	0	0
$556$	9.06022	0.384239
$557$	−17.2999	−0.733022	−0.366511	−	0.930414i	$-0.619448\pi$
$557$	−17.2999	−0.733022	−0.366511	+	0.930414i	$0.619448\pi$
$558$	0	0
$559$	−16.7503	−0.708464
$560$	0.0239244	0.00101099
$561$	0	0
$562$	21.8529	0.921810
$563$	14.9006	0.627983	0.313992	−	0.949426i	$-0.398334\pi$
$563$	14.9006	0.627983	0.313992	+	0.949426i	$0.398334\pi$
$564$	0	0
$565$	−0.132897	−0.00559100
$566$	25.0715	1.05383
$567$	0	0
$568$	−10.6011	−0.444811
$569$	−28.6412	−1.20070	−0.600350	−	0.799737i	$-0.704972\pi$
$569$	−28.6412	−1.20070	−0.600350	+	0.799737i	$0.704972\pi$
$570$	0	0
$571$	16.8648	0.705768	0.352884	−	0.935667i	$-0.385201\pi$
$571$	16.8648	0.705768	0.352884	+	0.935667i	$0.385201\pi$
$572$	−9.39808	−0.392953
$573$	0	0
$574$	−1.87407	−0.0782220
$575$	40.7963	1.70132
$576$	0	0
$577$	6.35968	0.264757	0.132378	−	0.991199i	$-0.457739\pi$
$577$	6.35968	0.264757	0.132378	+	0.991199i	$0.457739\pi$
$578$	19.0790	0.793583
$579$	0	0
$580$	0.0721458	0.00299569
$581$	3.51072	0.145649
$582$	0	0
$583$	10.3456	0.428470
$584$	44.2303	1.83026
$585$	0	0
$586$	31.5677	1.30405
$587$	−8.05301	−0.332383	−0.166192	−	0.986093i	$-0.553147\pi$
$587$	−8.05301	−0.332383	−0.166192	+	0.986093i	$0.553147\pi$
$588$	0	0
$589$	−0.949906	−0.0391402
$590$	0.0853929	0.00351557
$591$	0	0
$592$	11.4260	0.469604
$593$	−22.1625	−0.910104	−0.455052	−	0.890465i	$-0.650379\pi$
$593$	−22.1625	−0.910104	−0.455052	+	0.890465i	$0.650379\pi$
$594$	0	0
$595$	−0.00879542	−0.000360577 0
$596$	−3.21740	−0.131790
$597$	0	0
$598$	−60.9085	−2.49073
$599$	−31.0119	−1.26711	−0.633556	−	0.773697i	$-0.718406\pi$
$599$	−31.0119	−1.26711	−0.633556	+	0.773697i	$0.718406\pi$
$600$	0	0
$601$	36.1110	1.47300	0.736499	−	0.676439i	$-0.236478\pi$
$601$	36.1110	1.47300	0.736499	+	0.676439i	$0.236478\pi$
$602$	1.95846	0.0798209
$603$	0	0
$604$	−12.2545	−0.498630
$605$	−0.0841232	−0.00342009
$606$	0	0
$607$	−23.5634	−0.956408	−0.478204	−	0.878249i	$-0.658712\pi$
$607$	−23.5634	−0.956408	−0.478204	+	0.878249i	$0.658712\pi$
$608$	−3.35986	−0.136260
$609$	0	0
$610$	−0.177340	−0.00718027
$611$	−6.35191	−0.256971
$612$	0	0
$613$	−26.1644	−1.05677	−0.528385	−	0.849005i	$-0.677202\pi$
$613$	−26.1644	−1.05677	−0.528385	+	0.849005i	$0.677202\pi$
$614$	−1.68167	−0.0678665
$615$	0	0
$616$	4.64930	0.187326
$617$	21.1405	0.851083	0.425542	−	0.904939i	$-0.360084\pi$
$617$	21.1405	0.851083	0.425542	+	0.904939i	$0.360084\pi$
$618$	0	0
$619$	17.6242	0.708375	0.354187	−	0.935174i	$-0.384758\pi$
$619$	17.6242	0.708375	0.354187	+	0.935174i	$0.384758\pi$
$620$	0.00935670	0.000375774 0
$621$	0	0
$622$	15.2197	0.610253
$623$	5.64638	0.226218
$624$	0	0
$625$	24.9962	0.999848
$626$	31.9783	1.27811
$627$	0	0
$628$	8.60837	0.343511
$629$	−4.20057	−0.167488
$630$	0	0
$631$	37.4514	1.49092	0.745458	−	0.666553i	$-0.232231\pi$
$631$	37.4514	1.49092	0.745458	+	0.666553i	$0.232231\pi$
$632$	−9.71844	−0.386579
$633$	0	0
$634$	−3.60827	−0.143303
$635$	−0.0115753	−0.000459351 0
$636$	0	0
$637$	41.9265	1.66119
$638$	−20.5745	−0.814554
$639$	0	0
$640$	−0.0558817	−0.00220892
$641$	−24.0724	−0.950801	−0.475401	−	0.879769i	$-0.657697\pi$
$641$	−24.0724	−0.950801	−0.475401	+	0.879769i	$0.657697\pi$
$642$	0	0
$643$	−19.8865	−0.784248	−0.392124	−	0.919912i	$-0.628260\pi$
$643$	−19.8865	−0.784248	−0.392124	+	0.919912i	$0.628260\pi$
$644$	−3.19185	−0.125776
$645$	0	0
$646$	−1.02777	−0.0404370
$647$	25.4529	1.00066	0.500329	−	0.865836i	$-0.333213\pi$
$647$	25.4529	1.00066	0.500329	+	0.865836i	$0.333213\pi$
$648$	0	0
$649$	10.9148	0.428442
$650$	−37.3210	−1.46385
$651$	0	0
$652$	−9.67698	−0.378980
$653$	21.3755	0.836489	0.418244	−	0.908335i	$-0.362646\pi$
$653$	21.3755	0.836489	0.418244	+	0.908335i	$0.362646\pi$
$654$	0	0
$655$	−0.0141463	−0.000552740 0
$656$	6.00300	0.234378
$657$	0	0
$658$	0.742669	0.0289523
$659$	32.1739	1.25332	0.626659	−	0.779293i	$-0.284422\pi$
$659$	32.1739	1.25332	0.626659	+	0.779293i	$0.284422\pi$
$660$	0	0
$661$	2.27410	0.0884521	0.0442261	−	0.999022i	$-0.485918\pi$
$661$	2.27410	0.0884521	0.0442261	+	0.999022i	$0.485918\pi$
$662$	−18.8850	−0.733986
$663$	0	0
$664$	−17.0975	−0.663514
$665$	−0.0100569	−0.000389988 0
$666$	0	0
$667$	59.7644	2.31409
$668$	3.05222	0.118094
$669$	0	0
$670$	−0.185500	−0.00716649
$671$	−22.6672	−0.875059
$672$	0	0
$673$	40.3913	1.55697	0.778485	−	0.627663i	$-0.215988\pi$
$673$	40.3913	1.55697	0.778485	+	0.627663i	$0.215988\pi$
$674$	−30.3244	−1.16805
$675$	0	0
$676$	−16.9270	−0.651039
$677$	−6.09740	−0.234342	−0.117171	−	0.993112i	$-0.537383\pi$
$677$	−6.09740	−0.234342	−0.117171	+	0.993112i	$0.537383\pi$
$678$	0	0
$679$	2.72283	0.104493
$680$	0.0428345	0.00164263
$681$	0	0
$682$	−2.66834	−0.102176
$683$	21.5943	0.826283	0.413142	−	0.910667i	$-0.364431\pi$
$683$	21.5943	0.826283	0.413142	+	0.910667i	$0.364431\pi$
$684$	0	0
$685$	−0.0665925	−0.00254437
$686$	−10.1008	−0.385649
$687$	0	0
$688$	−6.27333	−0.239168
$689$	27.4915	1.04734
$690$	0	0
$691$	3.25929	0.123989	0.0619946	−	0.998076i	$-0.480254\pi$
$691$	3.25929	0.123989	0.0619946	+	0.998076i	$0.480254\pi$
$692$	−0.549076	−0.0208727
$693$	0	0
$694$	−28.4976	−1.08175
$695$	−0.232933	−0.00883566
$696$	0	0
$697$	−2.20691	−0.0835925
$698$	−18.2022	−0.688965
$699$	0	0
$700$	−1.95577	−0.0739211
$701$	−24.3733	−0.920569	−0.460284	−	0.887772i	$-0.652253\pi$
$701$	−24.3733	−0.920569	−0.460284	+	0.887772i	$0.652253\pi$
$702$	0	0
$703$	−4.80302	−0.181149
$704$	−20.8109	−0.784339
$705$	0	0
$706$	17.8314	0.671092
$707$	−9.83939	−0.370048
$708$	0	0
$709$	24.2438	0.910495	0.455248	−	0.890365i	$-0.349551\pi$
$709$	24.2438	0.910495	0.455248	+	0.890365i	$0.349551\pi$
$710$	0.0644147	0.00241744
$711$	0	0
$712$	−27.4984	−1.03055
$713$	7.75093	0.290275
$714$	0	0
$715$	0.241619	0.00903605
$716$	11.8713	0.443651
$717$	0	0
$718$	−14.6469	−0.546619
$719$	−35.2713	−1.31540	−0.657699	−	0.753281i	$-0.728470\pi$
$719$	−35.2713	−1.31540	−0.657699	+	0.753281i	$0.728470\pi$
$720$	0	0
$721$	3.53757	0.131746
$722$	−1.17517	−0.0437353
$723$	0	0
$724$	−13.5742	−0.504482
$725$	36.6199	1.36003
$726$	0	0
$727$	−22.8956	−0.849151	−0.424576	−	0.905392i	$-0.639577\pi$
$727$	−22.8956	−0.849151	−0.424576	+	0.905392i	$0.639577\pi$
$728$	12.3547	0.457895
$729$	0	0
$730$	−0.268754	−0.00994705
$731$	2.30629	0.0853012
$732$	0	0
$733$	5.76684	0.213003	0.106502	−	0.994313i	$-0.466035\pi$
$733$	5.76684	0.213003	0.106502	+	0.994313i	$0.466035\pi$
$734$	31.6034	1.16650
$735$	0	0
$736$	27.4153	1.01054
$737$	−23.7103	−0.873380
$738$	0	0
$739$	−18.9625	−0.697546	−0.348773	−	0.937207i	$-0.613402\pi$
$739$	−18.9625	−0.697546	−0.348773	+	0.937207i	$0.613402\pi$
$740$	0.0473103	0.00173916
$741$	0	0
$742$	−3.21432	−0.118002
$743$	−7.21885	−0.264834	−0.132417	−	0.991194i	$-0.542274\pi$
$743$	−7.21885	−0.264834	−0.132417	+	0.991194i	$0.542274\pi$
$744$	0	0
$745$	0.0827177	0.00303054
$746$	19.5568	0.716027
$747$	0	0
$748$	1.29398	0.0473128
$749$	6.75387	0.246781
$750$	0	0
$751$	14.8699	0.542610	0.271305	−	0.962493i	$-0.412545\pi$
$751$	14.8699	0.542610	0.271305	+	0.962493i	$0.412545\pi$
$752$	−2.37891	−0.0867501
$753$	0	0
$754$	−54.6731	−1.99108
$755$	0.315057	0.0114661
$756$	0	0
$757$	−37.0231	−1.34563	−0.672813	−	0.739812i	$-0.734914\pi$
$757$	−37.0231	−1.34563	−0.672813	+	0.739812i	$0.734914\pi$
$758$	34.0138	1.23544
$759$	0	0
$760$	0.0489778	0.00177661
$761$	−30.6330	−1.11045	−0.555223	−	0.831701i	$-0.687367\pi$
$761$	−30.6330	−1.11045	−0.555223	+	0.831701i	$0.687367\pi$
$762$	0	0
$763$	−5.30415	−0.192023
$764$	14.3029	0.517462
$765$	0	0
$766$	−16.9450	−0.612249
$767$	29.0040	1.04727
$768$	0	0
$769$	29.0159	1.04634	0.523171	−	0.852228i	$-0.324749\pi$
$769$	29.0159	1.04634	0.523171	+	0.852228i	$0.324749\pi$
$770$	−0.0282503	−0.00101807
$771$	0	0
$772$	−10.7391	−0.386510
$773$	−49.0247	−1.76329	−0.881647	−	0.471909i	$-0.843565\pi$
$773$	−49.0247	−1.76329	−0.881647	+	0.471909i	$0.843565\pi$
$774$	0	0
$775$	4.74929	0.170600
$776$	−13.2604	−0.476022
$777$	0	0
$778$	8.14221	0.291912
$779$	−2.52342	−0.0904109
$780$	0	0
$781$	8.23337	0.294613
$782$	8.38625	0.299892
$783$	0	0
$784$	15.7023	0.560796
$785$	−0.221317	−0.00789913
$786$	0	0
$787$	10.5453	0.375898	0.187949	−	0.982179i	$-0.439816\pi$
$787$	10.5453	0.375898	0.187949	+	0.982179i	$0.439816\pi$
$788$	−2.77762	−0.0989487
$789$	0	0
$790$	0.0590516	0.00210096
$791$	5.27766	0.187652
$792$	0	0
$793$	−60.2341	−2.13898
$794$	−2.80494	−0.0995437
$795$	0	0
$796$	9.39684	0.333062
$797$	−26.1995	−0.928034	−0.464017	−	0.885826i	$-0.653592\pi$
$797$	−26.1995	−0.928034	−0.464017	+	0.885826i	$0.653592\pi$
$798$	0	0
$799$	0.874570	0.0309400
$800$	16.7984	0.593914
$801$	0	0
$802$	22.1671	0.782747
$803$	−34.3517	−1.21225
$804$	0	0
$805$	0.0820607	0.00289226
$806$	−7.09064	−0.249757
$807$	0	0
$808$	47.9187	1.68578
$809$	8.51525	0.299380	0.149690	−	0.988733i	$-0.452172\pi$
$809$	8.51525	0.299380	0.149690	+	0.988733i	$0.452172\pi$
$810$	0	0
$811$	30.0648	1.05572	0.527859	−	0.849332i	$-0.322995\pi$
$811$	30.0648	1.05572	0.527859	+	0.849332i	$0.322995\pi$
$812$	−2.86509	−0.100545
$813$	0	0
$814$	−13.4920	−0.472893
$815$	0.248790	0.00871473
$816$	0	0
$817$	2.63706	0.0922589
$818$	2.54167	0.0888674
$819$	0	0
$820$	0.0248560	0.000868009 0
$821$	−9.78542	−0.341514	−0.170757	−	0.985313i	$-0.554621\pi$
$821$	−9.78542	−0.341514	−0.170757	+	0.985313i	$0.554621\pi$
$822$	0	0
$823$	26.6781	0.929941	0.464971	−	0.885326i	$-0.346065\pi$
$823$	26.6781	0.929941	0.464971	+	0.885326i	$0.346065\pi$
$824$	−17.2283	−0.600176
$825$	0	0
$826$	−3.39117	−0.117994
$827$	46.1275	1.60401	0.802005	−	0.597318i	$-0.203767\pi$
$827$	46.1275	1.60401	0.802005	+	0.597318i	$0.203767\pi$
$828$	0	0
$829$	1.40078	0.0486512	0.0243256	−	0.999704i	$-0.492256\pi$
$829$	1.40078	0.0486512	0.0243256	+	0.999704i	$0.492256\pi$
$830$	0.103889	0.00360604
$831$	0	0
$832$	−55.3012	−1.91722
$833$	−5.77270	−0.200012
$834$	0	0
$835$	−0.0784709	−0.00271560
$836$	1.47957	0.0511719
$837$	0	0
$838$	38.5724	1.33246
$839$	−37.3302	−1.28878	−0.644392	−	0.764696i	$-0.722889\pi$
$839$	−37.3302	−1.28878	−0.644392	+	0.764696i	$0.722889\pi$
$840$	0	0
$841$	24.6462	0.849868
$842$	−0.319835	−0.0110222
$843$	0	0
$844$	12.8516	0.442369
$845$	0.435184	0.0149708
$846$	0	0
$847$	3.34074	0.114789
$848$	10.2961	0.353570
$849$	0	0
$850$	5.13858	0.176252
$851$	39.1911	1.34345
$852$	0	0
$853$	14.6876	0.502893	0.251447	−	0.967871i	$-0.419094\pi$
$853$	14.6876	0.502893	0.251447	+	0.967871i	$0.419094\pi$
$854$	7.04261	0.240993
$855$	0	0
$856$	−32.8920	−1.12423
$857$	−48.3108	−1.65026	−0.825132	−	0.564940i	$-0.808899\pi$
$857$	−48.3108	−1.65026	−0.825132	+	0.564940i	$0.808899\pi$
$858$	0	0
$859$	47.2574	1.61240	0.806202	−	0.591641i	$-0.201520\pi$
$859$	47.2574	1.61240	0.806202	+	0.591641i	$0.201520\pi$
$860$	−0.0259753	−0.000885751 0
$861$	0	0
$862$	17.4246	0.593484
$863$	18.9045	0.643516	0.321758	−	0.946822i	$-0.395726\pi$
$863$	18.9045	0.643516	0.321758	+	0.946822i	$0.395726\pi$
$864$	0	0
$865$	0.0141165	0.000479974 0
$866$	4.16202	0.141431
$867$	0	0
$868$	−0.371578	−0.0126122
$869$	7.54787	0.256044
$870$	0	0
$871$	−63.0058	−2.13487
$872$	25.8317	0.874772
$873$	0	0
$874$	9.58900	0.324353
$875$	0.100566	0.00339975
$876$	0	0
$877$	−50.2933	−1.69828	−0.849141	−	0.528166i	$-0.822880\pi$
$877$	−50.2933	−1.69828	−0.849141	+	0.528166i	$0.822880\pi$
$878$	−6.62280	−0.223509
$879$	0	0
$880$	0.0904912	0.00305046
$881$	−38.5009	−1.29713	−0.648564	−	0.761160i	$-0.724630\pi$
$881$	−38.5009	−1.29713	−0.648564	+	0.761160i	$0.724630\pi$
$882$	0	0
$883$	20.1751	0.678945	0.339473	−	0.940616i	$-0.389751\pi$
$883$	20.1751	0.678945	0.339473	+	0.940616i	$0.389751\pi$
$884$	3.43853	0.115650
$885$	0	0
$886$	−29.3963	−0.987589
$887$	33.2329	1.11585	0.557926	−	0.829891i	$-0.311597\pi$
$887$	33.2329	1.11585	0.557926	+	0.829891i	$0.311597\pi$
$888$	0	0
$889$	0.459684	0.0154173
$890$	0.167087	0.00560078
$891$	0	0
$892$	−7.11955	−0.238380
$893$	1.00000	0.0334637
$894$	0	0
$895$	−0.305205	−0.0102019
$896$	2.21921	0.0741384
$897$	0	0
$898$	5.65788	0.188806
$899$	6.95745	0.232044
$900$	0	0
$901$	−3.78520	−0.126103
$902$	−7.08843	−0.236019
$903$	0	0
$904$	−25.7027	−0.854859
$905$	0.348986	0.0116007
$906$	0	0
$907$	−4.12262	−0.136889	−0.0684447	−	0.997655i	$-0.521804\pi$
$907$	−4.12262	−0.136889	−0.0684447	+	0.997655i	$0.521804\pi$
$908$	−5.50307	−0.182626
$909$	0	0
$910$	−0.0750701	−0.00248855
$911$	5.79988	0.192159	0.0960793	−	0.995374i	$-0.469370\pi$
$911$	5.79988	0.192159	0.0960793	+	0.995374i	$0.469370\pi$
$912$	0	0
$913$	13.2789	0.439467
$914$	−36.2902	−1.20037
$915$	0	0
$916$	−7.16621	−0.236778
$917$	0.561784	0.0185517
$918$	0	0
$919$	20.0562	0.661591	0.330796	−	0.943702i	$-0.392683\pi$
$919$	20.0562	0.661591	0.330796	+	0.943702i	$0.392683\pi$
$920$	−0.399643	−0.0131758
$921$	0	0
$922$	8.13669	0.267968
$923$	21.8787	0.720146
$924$	0	0
$925$	24.0139	0.789571
$926$	−47.2749	−1.55355
$927$	0	0
$928$	24.6088	0.807823
$929$	−3.97806	−0.130516	−0.0652580	−	0.997868i	$-0.520787\pi$
$929$	−3.97806	−0.130516	−0.0652580	+	0.997868i	$0.520787\pi$
$930$	0	0
$931$	−6.60062	−0.216326
$932$	−14.0011	−0.458621
$933$	0	0
$934$	−10.8323	−0.354445
$935$	−0.0332676	−0.00108797
$936$	0	0
$937$	52.1773	1.70456	0.852279	−	0.523087i	$-0.175220\pi$
$937$	52.1773	1.70456	0.852279	+	0.523087i	$0.175220\pi$
$938$	7.36668	0.240531
$939$	0	0
$940$	−0.00985012	−0.000321276 0
$941$	−37.3770	−1.21845	−0.609227	−	0.792996i	$-0.708520\pi$
$941$	−37.3770	−1.21845	−0.609227	+	0.792996i	$0.708520\pi$
$942$	0	0
$943$	20.5903	0.670512
$944$	10.8626	0.353547
$945$	0	0
$946$	7.40765	0.240843
$947$	18.3441	0.596105	0.298052	−	0.954550i	$-0.403663\pi$
$947$	18.3441	0.596105	0.298052	+	0.954550i	$0.403663\pi$
$948$	0	0
$949$	−91.2835	−2.96319
$950$	5.87555	0.190628
$951$	0	0
$952$	−1.70107	−0.0551319
$953$	−49.2629	−1.59578	−0.797890	−	0.602803i	$-0.794051\pi$
$953$	−49.2629	−1.59578	−0.797890	+	0.602803i	$0.794051\pi$
$954$	0	0
$955$	−0.367721	−0.0118992
$956$	−12.4641	−0.403117
$957$	0	0
$958$	−12.8946	−0.416604
$959$	2.64456	0.0853972
$960$	0	0
$961$	−30.0977	−0.970893
$962$	−35.8525	−1.15593
$963$	0	0
$964$	−5.71473	−0.184059
$965$	0.276097	0.00888789
$966$	0	0
$967$	61.8100	1.98768	0.993838	−	0.110844i	$-0.0353554\pi$
$967$	61.8100	1.98768	0.993838	+	0.110844i	$0.0353554\pi$
$968$	−16.2697	−0.522929
$969$	0	0
$970$	0.0805737	0.00258707
$971$	28.6092	0.918112	0.459056	−	0.888407i	$-0.348188\pi$
$971$	28.6092	0.918112	0.459056	+	0.888407i	$0.348188\pi$
$972$	0	0
$973$	9.25037	0.296553
$974$	−0.573356	−0.0183715
$975$	0	0
$976$	−22.5588	−0.722091
$977$	47.9671	1.53460	0.767302	−	0.641286i	$-0.221599\pi$
$977$	47.9671	1.53460	0.767302	+	0.641286i	$0.221599\pi$
$978$	0	0
$979$	21.3568	0.682566
$980$	0.0650169	0.00207689
$981$	0	0
$982$	−48.7551	−1.55584
$983$	21.2225	0.676892	0.338446	−	0.940986i	$-0.390099\pi$
$983$	21.2225	0.676892	0.338446	+	0.940986i	$0.390099\pi$
$984$	0	0
$985$	0.0714111	0.00227535
$986$	7.52773	0.239732
$987$	0	0
$988$	3.93168	0.125084
$989$	−21.5175	−0.684218
$990$	0	0
$991$	−32.6078	−1.03582	−0.517911	−	0.855435i	$-0.673290\pi$
$991$	−32.6078	−1.03582	−0.517911	+	0.855435i	$0.673290\pi$
$992$	3.19155	0.101332
$993$	0	0
$994$	−2.55807	−0.0811371
$995$	−0.241588	−0.00765884
$996$	0	0
$997$	−33.8209	−1.07112	−0.535559	−	0.844498i	$-0.679899\pi$
$997$	−33.8209	−1.07112	−0.535559	+	0.844498i	$0.679899\pi$
$998$	20.3644	0.644623
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.t.1.10		24
3.2	odd	2		2679.2.a.o.1.15	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.o.1.15	✓	24	3.2	odd	2
8037.2.a.t.1.10		24	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.17517 q^{2} -0.618977 q^{4} +0.0159136 q^{5} -0.631968 q^{7} +3.07774 q^{8} +O(q^{10})\)	\(q-1.17517 q^{2} -0.618977 q^{4} +0.0159136 q^{5} -0.631968 q^{7} +3.07774 q^{8} -0.0187011 q^{10} -2.39034 q^{11} -6.35191 q^{13} +0.742669 q^{14} -2.37891 q^{16} +0.874570 q^{17} +1.00000 q^{19} -0.00985012 q^{20} +2.80906 q^{22} -8.15968 q^{23} -4.99975 q^{25} +7.46457 q^{26} +0.391173 q^{28} -7.32435 q^{29} -0.949906 q^{31} -3.35986 q^{32} -1.02777 q^{34} -0.0100569 q^{35} -4.80302 q^{37} -1.17517 q^{38} +0.0489778 q^{40} -2.52342 q^{41} +2.63706 q^{43} +1.47957 q^{44} +9.58900 q^{46} +1.00000 q^{47} -6.60062 q^{49} +5.87555 q^{50} +3.93168 q^{52} -4.32807 q^{53} -0.0380389 q^{55} -1.94503 q^{56} +8.60736 q^{58} -4.56619 q^{59} +9.48283 q^{61} +1.11630 q^{62} +8.70623 q^{64} -0.101081 q^{65} +9.91919 q^{67} -0.541338 q^{68} +0.0118185 q^{70} -3.44443 q^{71} +14.3710 q^{73} +5.64436 q^{74} -0.618977 q^{76} +1.51062 q^{77} -3.15765 q^{79} -0.0378570 q^{80} +2.96544 q^{82} -5.55523 q^{83} +0.0139175 q^{85} -3.09899 q^{86} -7.35686 q^{88} -8.93461 q^{89} +4.01420 q^{91} +5.05065 q^{92} -1.17517 q^{94} +0.0159136 q^{95} -4.30850 q^{97} +7.75684 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) \) 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8	\( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) \) 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8 + 7 * q^10 + 3 * q^11 + 11 * q^13 - 9 * q^14 + 40 * q^16 + 6 * q^17 + 24 * q^19 + 17 * q^20 + 15 * q^22 + 19 * q^23 + 54 * q^25 + q^26 + 26 * q^28 - 32 * q^29 + 12 * q^31 - 30 * q^32 + 38 * q^34 + 35 * q^35 + 14 * q^37 - 2 * q^38 + 47 * q^40 - 30 * q^41 + 10 * q^43 - q^44 - 3 * q^46 + 24 * q^47 + 56 * q^49 - 44 * q^50 - 10 * q^53 + 24 * q^55 + 6 * q^56 + 41 * q^58 - 11 * q^59 + 10 * q^61 - 2 * q^62 + 61 * q^64 - 35 * q^65 + 12 * q^67 + 22 * q^68 - 20 * q^70 - 44 * q^71 + 33 * q^73 - 36 * q^74 + 32 * q^76 + 23 * q^77 + 50 * q^79 + 13 * q^80 + 5 * q^82 + 72 * q^83 + 33 * q^85 - 44 * q^86 + 38 * q^88 - 48 * q^89 + 47 * q^91 + 29 * q^92 - 2 * q^94 + 2 * q^95 + 50 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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