LMFDB - Embedded newform 1815.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1815,2,Mod(1,1815)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1815.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1815, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1815.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,3,5,3,-1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.16425 q^{2} +1.00000 q^{3} +2.68397 q^{4} +1.00000 q^{5} +2.16425 q^{6} +0.480279 q^{7} +1.48028 q^{8} +1.00000 q^{9} +2.16425 q^{10} +2.68397 q^{12} +3.36794 q^{13} +1.03944 q^{14} +1.00000 q^{15} -2.16425 q^{16} +4.84822 q^{17} +2.16425 q^{18} -1.84822 q^{19} +2.68397 q^{20} +0.480279 q^{21} +3.48028 q^{23} +1.48028 q^{24} +1.00000 q^{25} +7.28905 q^{26} +1.00000 q^{27} +1.28905 q^{28} -8.32850 q^{29} +2.16425 q^{30} -2.36794 q^{31} -7.64453 q^{32} +10.4927 q^{34} +0.480279 q^{35} +2.68397 q^{36} +5.44084 q^{37} -4.00000 q^{38} +3.36794 q^{39} +1.48028 q^{40} -6.65699 q^{41} +1.03944 q^{42} +4.32850 q^{43} +1.00000 q^{45} +7.53219 q^{46} -3.17671 q^{47} -2.16425 q^{48} -6.76933 q^{49} +2.16425 q^{50} +4.84822 q^{51} +9.03944 q^{52} +13.5052 q^{53} +2.16425 q^{54} +0.710947 q^{56} -1.84822 q^{57} -18.0249 q^{58} +15.0644 q^{59} +2.68397 q^{60} -11.9606 q^{61} -5.12481 q^{62} +0.480279 q^{63} -12.2162 q^{64} +3.36794 q^{65} +8.17671 q^{67} +13.0125 q^{68} +3.48028 q^{69} +1.03944 q^{70} -12.3534 q^{71} +1.48028 q^{72} +5.84822 q^{73} +11.7753 q^{74} +1.00000 q^{75} -4.96056 q^{76} +7.28905 q^{78} -15.0249 q^{79} -2.16425 q^{80} +1.00000 q^{81} -14.4074 q^{82} -4.00000 q^{83} +1.28905 q^{84} +4.84822 q^{85} +9.36794 q^{86} -8.32850 q^{87} +5.67150 q^{89} +2.16425 q^{90} +1.61755 q^{91} +9.34096 q^{92} -2.36794 q^{93} -6.87519 q^{94} -1.84822 q^{95} -7.64453 q^{96} -8.17671 q^{97} -14.6505 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9} - q^{10} + 5 q^{12} + 4 q^{13} + 12 q^{14} + 3 q^{15} + q^{16} + 4 q^{17} - q^{18} + 5 q^{19} + 5 q^{20} - 3 q^{21} + 6 q^{23}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 3 * q^7 + 3 * q^9 - q^10 + 5 * q^12 + 4 * q^13 + 12 * q^14 + 3 * q^15 + q^16 + 4 * q^17 - q^18 + 5 * q^19 + 5 * q^20 - 3 * q^21 + 6 * q^23 + 3 * q^25 - 2 * q^26 + 3 * q^27 - 20 * q^28 - 10 * q^29 - q^30 - q^31 - 11 * q^32 + 9 * q^34 - 3 * q^35 + 5 * q^36 + 3 * q^37 - 12 * q^38 + 4 * q^39 + 10 * q^41 + 12 * q^42 - 2 * q^43 + 3 * q^45 + 9 * q^46 + 16 * q^47 + q^48 + 8 * q^49 - q^50 + 4 * q^51 + 36 * q^52 - q^54 + 26 * q^56 + 5 * q^57 - 18 * q^58 + 18 * q^59 + 5 * q^60 - 27 * q^61 + q^62 - 3 * q^63 - 20 * q^64 + 4 * q^65 - q^67 + 21 * q^68 + 6 * q^69 + 12 * q^70 + 14 * q^71 + 7 * q^73 + 32 * q^74 + 3 * q^75 - 6 * q^76 - 2 * q^78 - 9 * q^79 + q^80 + 3 * q^81 - 46 * q^82 - 12 * q^83 - 20 * q^84 + 4 * q^85 + 22 * q^86 - 10 * q^87 + 32 * q^89 - q^90 - 34 * q^91 - 5 * q^92 - q^93 - 37 * q^94 + 5 * q^95 - 11 * q^96 + q^97 - 57 * q^98

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 $

$2$	2.16425	1.53035	0.765177	−	0.643820i	$-0.222651\pi$
$2$	2.16425	1.53035	0.765177	+	0.643820i	$0.222651\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.68397	1.34198
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.16425	0.883551
$7$	0.480279	0.181528	0.0907642	−	0.995872i	$-0.471069\pi$
$7$	0.480279	0.181528	0.0907642	+	0.995872i	$0.471069\pi$
$8$	1.48028	0.523358
$9$	1.00000	0.333333
$10$	2.16425	0.684395
$11$	0	0
$11$	0	0
$12$	2.68397	0.774795
$13$	3.36794	0.934098	0.467049	−	0.884231i	$-0.345317\pi$
$13$	3.36794	0.934098	0.467049	+	0.884231i	$0.345317\pi$
$14$	1.03944	0.277803
$15$	1.00000	0.258199
$16$	−2.16425	−0.541062
$17$	4.84822	1.17587	0.587933	−	0.808910i	$-0.299942\pi$
$17$	4.84822	1.17587	0.587933	+	0.808910i	$0.299942\pi$
$18$	2.16425	0.510118
$19$	−1.84822	−0.424010	−0.212005	−	0.977269i	$-0.567999\pi$
$19$	−1.84822	−0.424010	−0.212005	+	0.977269i	$0.567999\pi$
$20$	2.68397	0.600154
$21$	0.480279	0.104805
$22$	0	0
$23$	3.48028	0.725688	0.362844	−	0.931850i	$-0.381806\pi$
$23$	3.48028	0.725688	0.362844	+	0.931850i	$0.381806\pi$
$24$	1.48028	0.302161
$25$	1.00000	0.200000
$26$	7.28905	1.42950
$27$	1.00000	0.192450
$28$	1.28905	0.243608
$29$	−8.32850	−1.54656	−0.773281	−	0.634063i	$-0.781386\pi$
$29$	−8.32850	−1.54656	−0.773281	+	0.634063i	$0.781386\pi$
$30$	2.16425	0.395136
$31$	−2.36794	−0.425294	−0.212647	−	0.977129i	$-0.568208\pi$
$31$	−2.36794	−0.425294	−0.212647	+	0.977129i	$0.568208\pi$
$32$	−7.64453	−1.35137
$33$	0	0
$34$	10.4927	1.79949
$35$	0.480279	0.0811819
$36$	2.68397	0.447328
$37$	5.44084	0.894468	0.447234	−	0.894417i	$-0.352409\pi$
$37$	5.44084	0.894468	0.447234	+	0.894417i	$0.352409\pi$
$38$	−4.00000	−0.648886
$39$	3.36794	0.539302
$40$	1.48028	0.234053
$41$	−6.65699	−1.03965	−0.519824	−	0.854274i	$-0.674002\pi$
$41$	−6.65699	−1.03965	−0.519824	+	0.854274i	$0.674002\pi$
$42$	1.03944	0.160389
$43$	4.32850	0.660089	0.330045	−	0.943965i	$-0.392936\pi$
$43$	4.32850	0.660089	0.330045	+	0.943965i	$0.392936\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	7.53219	1.11056
$47$	−3.17671	−0.463371	−0.231686	−	0.972791i	$-0.574424\pi$
$47$	−3.17671	−0.463371	−0.231686	+	0.972791i	$0.574424\pi$
$48$	−2.16425	−0.312382
$49$	−6.76933	−0.967047
$50$	2.16425	0.306071
$51$	4.84822	0.678886
$52$	9.03944	1.25355
$53$	13.5052	1.85508	0.927542	−	0.373720i	$-0.121918\pi$
$53$	13.5052	1.85508	0.927542	+	0.373720i	$0.121918\pi$
$54$	2.16425	0.294517
$55$	0	0
$56$	0.710947	0.0950042
$57$	−1.84822	−0.244802
$58$	−18.0249	−2.36679
$59$	15.0644	1.96121	0.980607	−	0.195984i	$-0.0627900\pi$
$59$	15.0644	1.96121	0.980607	+	0.195984i	$0.0627900\pi$
$60$	2.68397	0.346499
$61$	−11.9606	−1.53139	−0.765696	−	0.643202i	$-0.777605\pi$
$61$	−11.9606	−1.53139	−0.765696	+	0.643202i	$0.777605\pi$
$62$	−5.12481	−0.650851
$63$	0.480279	0.0605094
$64$	−12.2162	−1.52702
$65$	3.36794	0.417741
$66$	0	0
$67$	8.17671	0.998944	0.499472	−	0.866330i	$-0.333527\pi$
$67$	8.17671	0.998944	0.499472	+	0.866330i	$0.333527\pi$
$68$	13.0125	1.57799
$69$	3.48028	0.418976
$70$	1.03944	0.124237
$71$	−12.3534	−1.46608	−0.733041	−	0.680184i	$-0.761900\pi$
$71$	−12.3534	−1.46608	−0.733041	+	0.680184i	$0.761900\pi$
$72$	1.48028	0.174453
$73$	5.84822	0.684482	0.342241	−	0.939612i	$-0.388814\pi$
$73$	5.84822	0.684482	0.342241	+	0.939612i	$0.388814\pi$
$74$	11.7753	1.36885
$75$	1.00000	0.115470
$76$	−4.96056	−0.569015
$77$	0	0
$78$	7.28905	0.825323
$79$	−15.0249	−1.69044	−0.845218	−	0.534421i	$-0.820530\pi$
$79$	−15.0249	−1.69044	−0.845218	+	0.534421i	$0.820530\pi$
$80$	−2.16425	−0.241970
$81$	1.00000	0.111111
$82$	−14.4074	−1.59103
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	1.28905	0.140647
$85$	4.84822	0.525863
$86$	9.36794	1.01017
$87$	−8.32850	−0.892908
$88$	0	0
$89$	5.67150	0.601178	0.300589	−	0.953754i	$-0.402817\pi$
$89$	5.67150	0.601178	0.300589	+	0.953754i	$0.402817\pi$
$90$	2.16425	0.228132
$91$	1.61755	0.169565
$92$	9.34096	0.973862
$93$	−2.36794	−0.245544
$94$	−6.87519	−0.709122
$95$	−1.84822	−0.189623
$96$	−7.64453	−0.780216
$97$	−8.17671	−0.830219	−0.415110	−	0.909771i	$-0.636257\pi$
$97$	−8.17671	−0.830219	−0.415110	+	0.909771i	$0.636257\pi$
$98$	−14.6505	−1.47993
$99$	0	0
$100$	2.68397	0.268397
$101$	−2.73588	−0.272230	−0.136115	−	0.990693i	$-0.543462\pi$
$101$	−2.73588	−0.272230	−0.136115	+	0.990693i	$0.543462\pi$
$102$	10.4927	1.03894
$103$	−18.2016	−1.79346	−0.896731	−	0.442577i	$-0.854064\pi$
$103$	−18.2016	−1.79346	−0.896731	+	0.442577i	$0.854064\pi$
$104$	4.98549	0.488867
$105$	0.480279	0.0468704
$106$	29.2286	2.83893
$107$	−9.91259	−0.958286	−0.479143	−	0.877737i	$-0.659053\pi$
$107$	−9.91259	−0.958286	−0.479143	+	0.877737i	$0.659053\pi$
$108$	2.68397	0.258265
$109$	14.5841	1.39690	0.698451	−	0.715657i	$-0.253873\pi$
$109$	14.5841	1.39690	0.698451	+	0.715657i	$0.253873\pi$
$110$	0	0
$111$	5.44084	0.516421
$112$	−1.03944	−0.0982181
$113$	−8.54465	−0.803813	−0.401907	−	0.915681i	$-0.631652\pi$
$113$	−8.54465	−0.803813	−0.401907	+	0.915681i	$0.631652\pi$
$114$	−4.00000	−0.374634
$115$	3.48028	0.324538
$116$	−22.3534	−2.07546
$117$	3.36794	0.311366
$118$	32.6030	3.00135
$119$	2.32850	0.213453
$120$	1.48028	0.135130
$121$	0	0
$122$	−25.8856	−2.34357
$123$	−6.65699	−0.600241
$124$	−6.35547	−0.570738
$125$	1.00000	0.0894427
$126$	1.03944	0.0926009
$127$	−0.255598	−0.0226806	−0.0113403	−	0.999936i	$-0.503610\pi$
$127$	−0.255598	−0.0226806	−0.0113403	+	0.999936i	$0.503610\pi$
$128$	−11.1497	−0.985507
$129$	4.32850	0.381103
$130$	7.28905	0.639292
$131$	−0.735877	−0.0642938	−0.0321469	−	0.999483i	$-0.510234\pi$
$131$	−0.735877	−0.0642938	−0.0321469	+	0.999483i	$0.510234\pi$
$132$	0	0
$133$	−0.887659	−0.0769698
$134$	17.6964	1.52874
$135$	1.00000	0.0860663
$136$	7.17671	0.615398
$137$	0.927102	0.0792077	0.0396038	−	0.999215i	$-0.487390\pi$
$137$	0.927102	0.0792077	0.0396038	+	0.999215i	$0.487390\pi$
$138$	7.53219	0.641182
$139$	−8.21616	−0.696885	−0.348443	−	0.937330i	$-0.613289\pi$
$139$	−8.21616	−0.696885	−0.348443	+	0.937330i	$0.613289\pi$
$140$	1.28905	0.108945
$141$	−3.17671	−0.267527
$142$	−26.7359	−2.24362
$143$	0	0
$144$	−2.16425	−0.180354
$145$	−8.32850	−0.691644
$146$	12.6570	1.04750
$147$	−6.76933	−0.558325
$148$	14.6030	1.20036
$149$	−16.0249	−1.31281	−0.656407	−	0.754407i	$-0.727924\pi$
$149$	−16.0249	−1.31281	−0.656407	+	0.754407i	$0.727924\pi$
$150$	2.16425	0.176710
$151$	−7.17671	−0.584033	−0.292016	−	0.956413i	$-0.594326\pi$
$151$	−7.17671	−0.584033	−0.292016	+	0.956413i	$0.594326\pi$
$152$	−2.73588	−0.221909
$153$	4.84822	0.391955
$154$	0	0
$155$	−2.36794	−0.190197
$156$	9.03944	0.723735
$157$	2.25560	0.180016	0.0900081	−	0.995941i	$-0.471311\pi$
$157$	2.25560	0.180016	0.0900081	+	0.995941i	$0.471311\pi$
$158$	−32.5177	−2.58697
$159$	13.5052	1.07103
$160$	−7.64453	−0.604353
$161$	1.67150	0.131733
$162$	2.16425	0.170039
$163$	0.255598	0.0200200	0.0100100	−	0.999950i	$-0.496814\pi$
$163$	0.255598	0.0200200	0.0100100	+	0.999950i	$0.496814\pi$
$164$	−17.8672	−1.39519
$165$	0	0
$166$	−8.65699	−0.671913
$167$	−16.1373	−1.24874	−0.624370	−	0.781129i	$-0.714644\pi$
$167$	−16.1373	−1.24874	−0.624370	+	0.781129i	$0.714644\pi$
$168$	0.710947	0.0548507
$169$	−1.65699	−0.127461
$170$	10.4927	0.804757
$171$	−1.84822	−0.141337
$172$	11.6175	0.885830
$173$	9.39287	0.714127	0.357063	−	0.934080i	$-0.383778\pi$
$173$	9.39287	0.714127	0.357063	+	0.934080i	$0.383778\pi$
$174$	−18.0249	−1.36647
$175$	0.480279	0.0363057
$176$	0	0
$177$	15.0644	1.13231
$178$	12.2745	0.920016
$179$	9.36794	0.700193	0.350096	−	0.936714i	$-0.386149\pi$
$179$	9.36794	0.700193	0.350096	+	0.936714i	$0.386149\pi$
$180$	2.68397	0.200051
$181$	17.7693	1.32078	0.660392	−	0.750921i	$-0.270390\pi$
$181$	17.7693	1.32078	0.660392	+	0.750921i	$0.270390\pi$
$182$	3.50078	0.259495
$183$	−11.9606	−0.884150
$184$	5.15178	0.379794
$185$	5.44084	0.400018
$186$	−5.12481	−0.375769
$187$	0	0
$188$	−8.52620	−0.621837
$189$	0.480279	0.0349351
$190$	−4.00000	−0.290191
$191$	19.2891	1.39571	0.697853	−	0.716241i	$-0.254139\pi$
$191$	19.2891	1.39571	0.697853	+	0.716241i	$0.254139\pi$
$192$	−12.2162	−0.881625
$193$	6.25560	0.450288	0.225144	−	0.974326i	$-0.427715\pi$
$193$	6.25560	0.450288	0.225144	+	0.974326i	$0.427715\pi$
$194$	−17.6964	−1.27053
$195$	3.36794	0.241183
$196$	−18.1687	−1.29776
$197$	5.03944	0.359045	0.179523	−	0.983754i	$-0.442545\pi$
$197$	5.03944	0.359045	0.179523	+	0.983754i	$0.442545\pi$
$198$	0	0
$199$	−14.5926	−1.03444	−0.517222	−	0.855851i	$-0.673034\pi$
$199$	−14.5926	−1.03444	−0.517222	+	0.855851i	$0.673034\pi$
$200$	1.48028	0.104672
$201$	8.17671	0.576741
$202$	−5.92112	−0.416608
$203$	−4.00000	−0.280745
$204$	13.0125	0.911055
$205$	−6.65699	−0.464944
$206$	−39.3929	−2.74463
$207$	3.48028	0.241896
$208$	−7.28905	−0.505405
$209$	0	0
$210$	1.03944	0.0717283
$211$	−19.2496	−1.32520	−0.662599	−	0.748974i	$-0.730547\pi$
$211$	−19.2496	−1.32520	−0.662599	+	0.748974i	$0.730547\pi$
$212$	36.2476	2.48949
$213$	−12.3534	−0.846443
$214$	−21.4533	−1.46652
$215$	4.32850	0.295201
$216$	1.48028	0.100720
$217$	−1.13727	−0.0772030
$218$	31.5636	2.13776
$219$	5.84822	0.395186
$220$	0	0
$221$	16.3285	1.09837
$222$	11.7753	0.790308
$223$	10.5841	0.708763	0.354382	−	0.935101i	$-0.384691\pi$
$223$	10.5841	0.708763	0.354382	+	0.935101i	$0.384691\pi$
$224$	−3.67150	−0.245313
$225$	1.00000	0.0666667
$226$	−18.4927	−1.23012
$227$	−27.1767	−1.80378	−0.901891	−	0.431964i	$-0.857821\pi$
$227$	−27.1767	−1.80378	−0.901891	+	0.431964i	$0.857821\pi$
$228$	−4.96056	−0.328521
$229$	20.8981	1.38098	0.690492	−	0.723340i	$-0.257394\pi$
$229$	20.8981	1.38098	0.690492	+	0.723340i	$0.257394\pi$
$230$	7.53219	0.496658
$231$	0	0
$232$	−12.3285	−0.809405
$233$	19.8877	1.30288	0.651442	−	0.758698i	$-0.274164\pi$
$233$	19.8877	1.30288	0.651442	+	0.758698i	$0.274164\pi$
$234$	7.28905	0.476500
$235$	−3.17671	−0.207226
$236$	40.4323	2.63192
$237$	−15.0249	−0.975974
$238$	5.03944	0.326659
$239$	12.3285	0.797464	0.398732	−	0.917067i	$-0.369450\pi$
$239$	12.3285	0.797464	0.398732	+	0.917067i	$0.369450\pi$
$240$	−2.16425	−0.139702
$241$	−7.20164	−0.463899	−0.231949	−	0.972728i	$-0.574510\pi$
$241$	−7.20164	−0.463899	−0.231949	+	0.972728i	$0.574510\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−32.1018	−2.05511
$245$	−6.76933	−0.432477
$246$	−14.4074	−0.918581
$247$	−6.22468	−0.396067
$248$	−3.50521	−0.222581
$249$	−4.00000	−0.253490
$250$	2.16425	0.136879
$251$	−3.36794	−0.212582	−0.106291	−	0.994335i	$-0.533898\pi$
$251$	−3.36794	−0.212582	−0.106291	+	0.994335i	$0.533898\pi$
$252$	1.28905	0.0812027
$253$	0	0
$254$	−0.553177	−0.0347094
$255$	4.84822	0.303607
$256$	0.301518	0.0188449
$257$	−11.5841	−0.722596	−0.361298	−	0.932450i	$-0.617666\pi$
$257$	−11.5841	−0.722596	−0.361298	+	0.932450i	$0.617666\pi$
$258$	9.36794	0.583222
$259$	2.61312	0.162371
$260$	9.03944	0.560602
$261$	−8.32850	−0.515521
$262$	−1.59262	−0.0983924
$263$	28.5696	1.76168	0.880838	−	0.473418i	$-0.156980\pi$
$263$	28.5696	1.76168	0.880838	+	0.473418i	$0.156980\pi$
$264$	0	0
$265$	13.5052	0.829618
$266$	−1.92112	−0.117791
$267$	5.67150	0.347090
$268$	21.9460	1.34057
$269$	−1.47175	−0.0897344	−0.0448672	−	0.998993i	$-0.514286\pi$
$269$	−1.47175	−0.0897344	−0.0448672	+	0.998993i	$0.514286\pi$
$270$	2.16425	0.131712
$271$	5.70496	0.346552	0.173276	−	0.984873i	$-0.444565\pi$
$271$	5.70496	0.346552	0.173276	+	0.984873i	$0.444565\pi$
$272$	−10.4927	−0.636216
$273$	1.61755	0.0978985
$274$	2.00648	0.121216
$275$	0	0
$276$	9.34096	0.562260
$277$	−24.7299	−1.48588	−0.742938	−	0.669361i	$-0.766568\pi$
$277$	−24.7299	−1.48588	−0.742938	+	0.669361i	$0.766568\pi$
$278$	−17.7818	−1.06648
$279$	−2.36794	−0.141765
$280$	0.710947	0.0424872
$281$	−5.67150	−0.338334	−0.169167	−	0.985587i	$-0.554108\pi$
$281$	−5.67150	−0.338334	−0.169167	+	0.985587i	$0.554108\pi$
$282$	−6.87519	−0.409412
$283$	22.1228	1.31506	0.657531	−	0.753428i	$-0.271601\pi$
$283$	22.1228	1.31506	0.657531	+	0.753428i	$0.271601\pi$
$284$	−33.1562	−1.96746
$285$	−1.84822	−0.109479
$286$	0	0
$287$	−3.19721	−0.188725
$288$	−7.64453	−0.450458
$289$	6.50521	0.382659
$290$	−18.0249	−1.05846
$291$	−8.17671	−0.479327
$292$	15.6964	0.918564
$293$	−12.8482	−0.750601	−0.375300	−	0.926903i	$-0.622460\pi$
$293$	−12.8482	−0.750601	−0.375300	+	0.926903i	$0.622460\pi$
$294$	−14.6505	−0.854435
$295$	15.0644	0.877082
$296$	8.05395	0.468127
$297$	0	0
$298$	−34.6819	−2.00907
$299$	11.7214	0.677864
$300$	2.68397	0.154959
$301$	2.07888	0.119825
$302$	−15.5322	−0.893777
$303$	−2.73588	−0.157172
$304$	4.00000	0.229416
$305$	−11.9606	−0.684860
$306$	10.4927	0.599830
$307$	−8.50521	−0.485418	−0.242709	−	0.970099i	$-0.578036\pi$
$307$	−8.50521	−0.485418	−0.242709	+	0.970099i	$0.578036\pi$
$308$	0	0
$309$	−18.2016	−1.03546
$310$	−5.12481	−0.291069
$311$	−6.32850	−0.358856	−0.179428	−	0.983771i	$-0.557425\pi$
$311$	−6.32850	−0.358856	−0.179428	+	0.983771i	$0.557425\pi$
$312$	4.98549	0.282248
$313$	20.0249	1.13188	0.565938	−	0.824448i	$-0.308514\pi$
$313$	20.0249	1.13188	0.565938	+	0.824448i	$0.308514\pi$
$314$	4.88167	0.275489
$315$	0.480279	0.0270606
$316$	−40.3264	−2.26854
$317$	−23.8088	−1.33723	−0.668617	−	0.743607i	$-0.733113\pi$
$317$	−23.8088	−1.33723	−0.668617	+	0.743607i	$0.733113\pi$
$318$	29.2286	1.63906
$319$	0	0
$320$	−12.2162	−0.682904
$321$	−9.91259	−0.553267
$322$	3.61755	0.201598
$323$	−8.96056	−0.498579
$324$	2.68397	0.149109
$325$	3.36794	0.186820
$326$	0.553177	0.0306376
$327$	14.5841	0.806502
$328$	−9.85420	−0.544107
$329$	−1.52571	−0.0841150
$330$	0	0
$331$	−15.3285	−0.842530	−0.421265	−	0.906938i	$-0.638414\pi$
$331$	−15.3285	−0.842530	−0.421265	+	0.906938i	$0.638414\pi$
$332$	−10.7359	−0.589208
$333$	5.44084	0.298156
$334$	−34.9251	−1.91101
$335$	8.17671	0.446742
$336$	−1.03944	−0.0567062
$337$	−3.03346	−0.165243	−0.0826214	−	0.996581i	$-0.526329\pi$
$337$	−3.03346	−0.165243	−0.0826214	+	0.996581i	$0.526329\pi$
$338$	−3.58614	−0.195060
$339$	−8.54465	−0.464082
$340$	13.0125	0.705700
$341$	0	0
$342$	−4.00000	−0.216295
$343$	−6.61312	−0.357075
$344$	6.40738	0.345463
$345$	3.48028	0.187372
$346$	20.3285	1.09287
$347$	34.0584	1.82835	0.914175	−	0.405320i	$-0.132840\pi$
$347$	34.0584	1.82835	0.914175	+	0.405320i	$0.132840\pi$
$348$	−22.3534	−1.19827
$349$	−20.0104	−1.07113	−0.535567	−	0.844493i	$-0.679902\pi$
$349$	−20.0104	−1.07113	−0.535567	+	0.844493i	$0.679902\pi$
$350$	1.03944	0.0555605
$351$	3.36794	0.179767
$352$	0	0
$353$	30.9770	1.64874	0.824369	−	0.566053i	$-0.191530\pi$
$353$	30.9770	1.64874	0.824369	+	0.566053i	$0.191530\pi$
$354$	32.6030	1.73283
$355$	−12.3534	−0.655652
$356$	15.2221	0.806772
$357$	2.32850	0.123237
$358$	20.2745	1.07154
$359$	29.8252	1.57411	0.787056	−	0.616881i	$-0.211604\pi$
$359$	29.8252	1.57411	0.787056	+	0.616881i	$0.211604\pi$
$360$	1.48028	0.0780175
$361$	−15.5841	−0.820215
$362$	38.4572	2.02127
$363$	0	0
$364$	4.34145	0.227554
$365$	5.84822	0.306110
$366$	−25.8856	−1.35306
$367$	25.4178	1.32680	0.663399	−	0.748266i	$-0.269113\pi$
$367$	25.4178	1.32680	0.663399	+	0.748266i	$0.269113\pi$
$368$	−7.53219	−0.392642
$369$	−6.65699	−0.346549
$370$	11.7753	0.612170
$371$	6.48627	0.336750
$372$	−6.35547	−0.329516
$373$	30.8088	1.59522	0.797609	−	0.603175i	$-0.206098\pi$
$373$	30.8088	1.59522	0.797609	+	0.603175i	$0.206098\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−4.70242	−0.242509
$377$	−28.0499	−1.44464
$378$	1.03944	0.0534631
$379$	−29.5301	−1.51686	−0.758431	−	0.651754i	$-0.774034\pi$
$379$	−29.5301	−1.51686	−0.758431	+	0.651754i	$0.774034\pi$
$380$	−4.96056	−0.254471
$381$	−0.255598	−0.0130947
$382$	41.7463	2.13593
$383$	−4.30357	−0.219902	−0.109951	−	0.993937i	$-0.535069\pi$
$383$	−4.30357	−0.219902	−0.109951	+	0.993937i	$0.535069\pi$
$384$	−11.1497	−0.568983
$385$	0	0
$386$	13.5387	0.689100
$387$	4.32850	0.220030
$388$	−21.9460	−1.11414
$389$	35.4178	1.79575	0.897877	−	0.440247i	$-0.145109\pi$
$389$	35.4178	1.79575	0.897877	+	0.440247i	$0.145109\pi$
$390$	7.28905	0.369096
$391$	16.8731	0.853312
$392$	−10.0205	−0.506112
$393$	−0.735877	−0.0371201
$394$	10.9066	0.549467
$395$	−15.0249	−0.755986
$396$	0	0
$397$	−18.0729	−0.907053	−0.453526	−	0.891243i	$-0.649834\pi$
$397$	−18.0729	−0.907053	−0.453526	+	0.891243i	$0.649834\pi$
$398$	−31.5820	−1.58306
$399$	−0.887659	−0.0444386
$400$	−2.16425	−0.108212
$401$	−3.08930	−0.154272	−0.0771362	−	0.997021i	$-0.524578\pi$
$401$	−3.08930	−0.154272	−0.0771362	+	0.997021i	$0.524578\pi$
$402$	17.6964	0.882618
$403$	−7.97507	−0.397267
$404$	−7.34301	−0.365328
$405$	1.00000	0.0496904
$406$	−8.65699	−0.429639
$407$	0	0
$408$	7.17671	0.355300
$409$	−24.9211	−1.23227	−0.616135	−	0.787641i	$-0.711302\pi$
$409$	−24.9211	−1.23227	−0.616135	+	0.787641i	$0.711302\pi$
$410$	−14.4074	−0.711530
$411$	0.927102	0.0457306
$412$	−48.8526	−2.40680
$413$	7.23510	0.356016
$414$	7.53219	0.370187
$415$	−4.00000	−0.196352
$416$	−25.7463	−1.26232
$417$	−8.21616	−0.402347
$418$	0	0
$419$	−4.22468	−0.206389	−0.103195	−	0.994661i	$-0.532906\pi$
$419$	−4.22468	−0.206389	−0.103195	+	0.994661i	$0.532906\pi$
$420$	1.28905	0.0628994
$421$	−21.7299	−1.05905	−0.529525	−	0.848294i	$-0.677630\pi$
$421$	−21.7299	−1.05905	−0.529525	+	0.848294i	$0.677630\pi$
$422$	−41.6609	−2.02802
$423$	−3.17671	−0.154457
$424$	19.9915	0.970872
$425$	4.84822	0.235173
$426$	−26.7359	−1.29536
$427$	−5.74440	−0.277991
$428$	−26.6051	−1.28601
$429$	0	0
$430$	9.36794	0.451762
$431$	−9.89619	−0.476682	−0.238341	−	0.971181i	$-0.576604\pi$
$431$	−9.89619	−0.476682	−0.238341	+	0.971181i	$0.576604\pi$
$432$	−2.16425	−0.104127
$433$	39.3449	1.89080	0.945398	−	0.325919i	$-0.105674\pi$
$433$	39.3449	1.89080	0.945398	+	0.325919i	$0.105674\pi$
$434$	−2.46134	−0.118148
$435$	−8.32850	−0.399321
$436$	39.1433	1.87462
$437$	−6.43231	−0.307699
$438$	12.6570	0.604774
$439$	−3.71095	−0.177114	−0.0885569	−	0.996071i	$-0.528226\pi$
$439$	−3.71095	−0.177114	−0.0885569	+	0.996071i	$0.528226\pi$
$440$	0	0
$441$	−6.76933	−0.322349
$442$	35.3389	1.68090
$443$	4.96056	0.235683	0.117842	−	0.993032i	$-0.462402\pi$
$443$	4.96056	0.235683	0.117842	+	0.993032i	$0.462402\pi$
$444$	14.6030	0.693029
$445$	5.67150	0.268855
$446$	22.9066	1.08466
$447$	−16.0249	−0.757953
$448$	−5.86716	−0.277197
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	2.16425	0.102024
$451$	0	0
$452$	−22.9336	−1.07870
$453$	−7.17671	−0.337191
$454$	−58.8171	−2.76043
$455$	1.61755	0.0758319
$456$	−2.73588	−0.128119
$457$	−12.7359	−0.595759	−0.297880	−	0.954603i	$-0.596279\pi$
$457$	−12.7359	−0.595759	−0.297880	+	0.954603i	$0.596279\pi$
$458$	45.2286	2.11339
$459$	4.84822	0.226295
$460$	9.34096	0.435525
$461$	40.0249	1.86415	0.932073	−	0.362269i	$-0.117998\pi$
$461$	40.0249	1.86415	0.932073	+	0.362269i	$0.117998\pi$
$462$	0	0
$463$	−25.5387	−1.18688	−0.593441	−	0.804877i	$-0.702231\pi$
$463$	−25.5387	−1.18688	−0.593441	+	0.804877i	$0.702231\pi$
$464$	18.0249	0.836786
$465$	−2.36794	−0.109811
$466$	43.0418	1.99387
$467$	−6.59861	−0.305347	−0.152674	−	0.988277i	$-0.548788\pi$
$467$	−6.59861	−0.305347	−0.152674	+	0.988277i	$0.548788\pi$
$468$	9.03944	0.417848
$469$	3.92710	0.181337
$470$	−6.87519	−0.317129
$471$	2.25560	0.103932
$472$	22.2995	1.02642
$473$	0	0
$474$	−32.5177	−1.49359
$475$	−1.84822	−0.0848020
$476$	6.24961	0.286450
$477$	13.5052	0.618361
$478$	26.6819	1.22040
$479$	36.8277	1.68270	0.841351	−	0.540490i	$-0.181761\pi$
$479$	36.8277	1.68270	0.841351	+	0.540490i	$0.181761\pi$
$480$	−7.64453	−0.348923
$481$	18.3244	0.835521
$482$	−15.5861	−0.709929
$483$	1.67150	0.0760561
$484$	0	0
$485$	−8.17671	−0.371285
$486$	2.16425	0.0981723
$487$	−6.40738	−0.290346	−0.145173	−	0.989406i	$-0.546374\pi$
$487$	−6.40738	−0.290346	−0.145173	+	0.989406i	$0.546374\pi$
$488$	−17.7050	−0.801466
$489$	0.255598	0.0115585
$490$	−14.6505	−0.661843
$491$	2.43231	0.109769	0.0548843	−	0.998493i	$-0.482521\pi$
$491$	2.43231	0.109769	0.0548843	+	0.998493i	$0.482521\pi$
$492$	−17.8672	−0.805514
$493$	−40.3784	−1.81855
$494$	−13.4718	−0.606123
$495$	0	0
$496$	5.12481	0.230111
$497$	−5.93309	−0.266135
$498$	−8.65699	−0.387929
$499$	−36.2805	−1.62414	−0.812070	−	0.583560i	$-0.801659\pi$
$499$	−36.2805	−1.62414	−0.812070	+	0.583560i	$0.801659\pi$
$500$	2.68397	0.120031
$501$	−16.1373	−0.720960
$502$	−7.28905	−0.325326
$503$	11.9915	0.534673	0.267337	−	0.963603i	$-0.413856\pi$
$503$	11.9915	0.534673	0.267337	+	0.963603i	$0.413856\pi$
$504$	0.710947	0.0316681
$505$	−2.73588	−0.121745
$506$	0	0
$507$	−1.65699	−0.0735896
$508$	−0.686016	−0.0304371
$509$	17.2102	0.762827	0.381414	−	0.924404i	$-0.375437\pi$
$509$	17.2102	0.762827	0.381414	+	0.924404i	$0.375437\pi$
$510$	10.4927	0.464627
$511$	2.80877	0.124253
$512$	22.9520	1.01435
$513$	−1.84822	−0.0816008
$514$	−25.0709	−1.10583
$515$	−18.2016	−0.802060
$516$	11.6175	0.511434
$517$	0	0
$518$	5.65544	0.248486
$519$	9.39287	0.412301
$520$	4.98549	0.218628
$521$	17.4348	0.763835	0.381917	−	0.924196i	$-0.375264\pi$
$521$	17.4348	0.763835	0.381917	+	0.924196i	$0.375264\pi$
$522$	−18.0249	−0.788930
$523$	18.5592	0.811536	0.405768	−	0.913976i	$-0.367004\pi$
$523$	18.5592	0.811536	0.405768	+	0.913976i	$0.367004\pi$
$524$	−1.97507	−0.0862813
$525$	0.480279	0.0209611
$526$	61.8317	2.69599
$527$	−11.4803	−0.500089
$528$	0	0
$529$	−10.8877	−0.473376
$530$	29.2286	1.26961
$531$	15.0644	0.653738
$532$	−2.38245	−0.103292
$533$	−22.4203	−0.971133
$534$	12.2745	0.531171
$535$	−9.91259	−0.428559
$536$	12.1038	0.522805
$537$	9.36794	0.404256
$538$	−3.18524	−0.137325
$539$	0	0
$540$	2.68397	0.115500
$541$	31.9710	1.37454	0.687270	−	0.726402i	$-0.258809\pi$
$541$	31.9710	1.37454	0.687270	+	0.726402i	$0.258809\pi$
$542$	12.3469	0.530347
$543$	17.7693	0.762555
$544$	−37.0623	−1.58903
$545$	14.5841	0.624714
$546$	3.50078	0.149819
$547$	−15.7214	−0.672197	−0.336098	−	0.941827i	$-0.609108\pi$
$547$	−15.7214	−0.672197	−0.336098	+	0.941827i	$0.609108\pi$
$548$	2.48831	0.106295
$549$	−11.9606	−0.510464
$550$	0	0
$551$	15.3929	0.655758
$552$	5.15178	0.219274
$553$	−7.21616	−0.306862
$554$	−53.5216	−2.27392
$555$	5.44084	0.230951
$556$	−22.0519	−0.935209
$557$	14.1622	0.600072	0.300036	−	0.953928i	$-0.403001\pi$
$557$	14.1622	0.600072	0.300036	+	0.953928i	$0.403001\pi$
$558$	−5.12481	−0.216950
$559$	14.5781	0.616588
$560$	−1.03944	−0.0439245
$561$	0	0
$562$	−12.2745	−0.517770
$563$	−1.47175	−0.0620270	−0.0310135	−	0.999519i	$-0.509873\pi$
$563$	−1.47175	−0.0620270	−0.0310135	+	0.999519i	$0.509873\pi$
$564$	−8.52620	−0.359018
$565$	−8.54465	−0.359476
$566$	47.8791	2.01251
$567$	0.480279	0.0201698
$568$	−18.2865	−0.767285
$569$	10.2496	0.429686	0.214843	−	0.976649i	$-0.431076\pi$
$569$	10.2496	0.429686	0.214843	+	0.976649i	$0.431076\pi$
$570$	−4.00000	−0.167542
$571$	28.4178	1.18925	0.594624	−	0.804004i	$-0.297301\pi$
$571$	28.4178	1.18925	0.594624	+	0.804004i	$0.297301\pi$
$572$	0	0
$573$	19.2891	0.805812
$574$	−6.91956	−0.288817
$575$	3.48028	0.145138
$576$	−12.2162	−0.509006
$577$	34.0439	1.41726	0.708632	−	0.705578i	$-0.249312\pi$
$577$	34.0439	1.41726	0.708632	+	0.705578i	$0.249312\pi$
$578$	14.0789	0.585604
$579$	6.25560	0.259974
$580$	−22.3534	−0.928175
$581$	−1.92112	−0.0797013
$582$	−17.6964	−0.733541
$583$	0	0
$584$	8.65699	0.358229
$585$	3.36794	0.139247
$586$	−27.8067	−1.14869
$587$	26.9520	1.11243	0.556215	−	0.831039i	$-0.312253\pi$
$587$	26.9520	1.11243	0.556215	+	0.831039i	$0.312253\pi$
$588$	−18.1687	−0.749264
$589$	4.37646	0.180329
$590$	32.6030	1.34225
$591$	5.03944	0.207295
$592$	−11.7753	−0.483963
$593$	7.31398	0.300349	0.150175	−	0.988659i	$-0.452016\pi$
$593$	7.31398	0.300349	0.150175	+	0.988659i	$0.452016\pi$
$594$	0	0
$595$	2.32850	0.0954590
$596$	−43.0104	−1.76178
$597$	−14.5926	−0.597236
$598$	25.3679	1.03737
$599$	−32.0918	−1.31124	−0.655619	−	0.755092i	$-0.727592\pi$
$599$	−32.0918	−1.31124	−0.655619	+	0.755092i	$0.727592\pi$
$600$	1.48028	0.0604321
$601$	−7.39885	−0.301806	−0.150903	−	0.988549i	$-0.548218\pi$
$601$	−7.39885	−0.301806	−0.150903	+	0.988549i	$0.548218\pi$
$602$	4.49922	0.183375
$603$	8.17671	0.332981
$604$	−19.2621	−0.783763
$605$	0	0
$606$	−5.92112	−0.240529
$607$	23.2891	0.945274	0.472637	−	0.881257i	$-0.343302\pi$
$607$	23.2891	0.945274	0.472637	+	0.881257i	$0.343302\pi$
$608$	14.1287	0.572996
$609$	−4.00000	−0.162088
$610$	−25.8856	−1.04808
$611$	−10.6990	−0.432834
$612$	13.0125	0.525998
$613$	−22.5052	−0.908977	−0.454488	−	0.890753i	$-0.650178\pi$
$613$	−22.5052	−0.908977	−0.454488	+	0.890753i	$0.650178\pi$
$614$	−18.4074	−0.742861
$615$	−6.65699	−0.268436
$616$	0	0
$617$	26.7069	1.07518	0.537589	−	0.843207i	$-0.319335\pi$
$617$	26.7069	1.07518	0.537589	+	0.843207i	$0.319335\pi$
$618$	−39.3929	−1.58461
$619$	32.4992	1.30625	0.653127	−	0.757248i	$-0.273457\pi$
$619$	32.4992	1.30625	0.653127	+	0.757248i	$0.273457\pi$
$620$	−6.35547	−0.255242
$621$	3.48028	0.139659
$622$	−13.6964	−0.549177
$623$	2.72390	0.109131
$624$	−7.28905	−0.291796
$625$	1.00000	0.0400000
$626$	43.3389	1.73217
$627$	0	0
$628$	6.05395	0.241579
$629$	26.3784	1.05177
$630$	1.03944	0.0414124
$631$	−23.1767	−0.922650	−0.461325	−	0.887231i	$-0.652626\pi$
$631$	−23.1767	−0.922650	−0.461325	+	0.887231i	$0.652626\pi$
$632$	−22.2411	−0.884703
$633$	−19.2496	−0.765103
$634$	−51.5281	−2.04644
$635$	−0.255598	−0.0101431
$636$	36.2476	1.43731
$637$	−22.7987	−0.903317
$638$	0	0
$639$	−12.3534	−0.488694
$640$	−11.1497	−0.440732
$641$	36.4572	1.43997	0.719987	−	0.693987i	$-0.244148\pi$
$641$	36.4572	1.43997	0.719987	+	0.693987i	$0.244148\pi$
$642$	−21.4533	−0.846694
$643$	25.8192	1.01821	0.509105	−	0.860705i	$-0.329977\pi$
$643$	25.8192	1.01821	0.509105	+	0.860705i	$0.329977\pi$
$644$	4.48627	0.176784
$645$	4.32850	0.170434
$646$	−19.3929	−0.763002
$647$	−19.0189	−0.747712	−0.373856	−	0.927487i	$-0.621965\pi$
$647$	−19.0189	−0.747712	−0.373856	+	0.927487i	$0.621965\pi$
$648$	1.48028	0.0581508
$649$	0	0
$650$	7.28905	0.285900
$651$	−1.13727	−0.0445731
$652$	0.686016	0.0268665
$653$	−4.52825	−0.177204	−0.0886020	−	0.996067i	$-0.528240\pi$
$653$	−4.52825	−0.177204	−0.0886020	+	0.996067i	$0.528240\pi$
$654$	31.5636	1.23423
$655$	−0.735877	−0.0287531
$656$	14.4074	0.562514
$657$	5.84822	0.228161
$658$	−3.30201	−0.128726
$659$	−10.9356	−0.425992	−0.212996	−	0.977053i	$-0.568322\pi$
$659$	−10.9356	−0.425992	−0.212996	+	0.977053i	$0.568322\pi$
$660$	0	0
$661$	−4.50521	−0.175232	−0.0876162	−	0.996154i	$-0.527925\pi$
$661$	−4.50521	−0.175232	−0.0876162	+	0.996154i	$0.527925\pi$
$662$	−33.1747	−1.28937
$663$	16.3285	0.634146
$664$	−5.92112	−0.229784
$665$	−0.887659	−0.0344220
$666$	11.7753	0.456284
$667$	−28.9855	−1.12232
$668$	−43.3119	−1.67579
$669$	10.5841	0.409205
$670$	17.6964	0.683673
$671$	0	0
$672$	−3.67150	−0.141631
$673$	25.6485	0.988676	0.494338	−	0.869270i	$-0.335411\pi$
$673$	25.6485	0.988676	0.494338	+	0.869270i	$0.335411\pi$
$674$	−6.56515	−0.252880
$675$	1.00000	0.0384900
$676$	−4.44731	−0.171051
$677$	23.6674	0.909612	0.454806	−	0.890590i	$-0.349709\pi$
$677$	23.6674	0.909612	0.454806	+	0.890590i	$0.349709\pi$
$678$	−18.4927	−0.710210
$679$	−3.92710	−0.150708
$680$	7.17671	0.275214
$681$	−27.1767	−1.04141
$682$	0	0
$683$	17.7753	0.680154	0.340077	−	0.940398i	$-0.389547\pi$
$683$	17.7753	0.680154	0.340077	+	0.940398i	$0.389547\pi$
$684$	−4.96056	−0.189672
$685$	0.927102	0.0354227
$686$	−14.3124	−0.546451
$687$	20.8981	0.797311
$688$	−9.36794	−0.357149
$689$	45.4847	1.73283
$690$	7.53219	0.286745
$691$	−8.44682	−0.321332	−0.160666	−	0.987009i	$-0.551364\pi$
$691$	−8.44682	−0.321332	−0.160666	+	0.987009i	$0.551364\pi$
$692$	25.2102	0.958347
$693$	0	0
$694$	73.7108	2.79802
$695$	−8.21616	−0.311657
$696$	−12.3285	−0.467310
$697$	−32.2745	−1.22249
$698$	−43.3075	−1.63921
$699$	19.8877	0.752220
$700$	1.28905	0.0487216
$701$	−3.75039	−0.141650	−0.0708251	−	0.997489i	$-0.522563\pi$
$701$	−3.75039	−0.141650	−0.0708251	+	0.997489i	$0.522563\pi$
$702$	7.28905	0.275108
$703$	−10.0558	−0.379263
$704$	0	0
$705$	−3.17671	−0.119642
$706$	67.0418	2.52315
$707$	−1.31398	−0.0494174
$708$	40.4323	1.51954
$709$	−5.28053	−0.198314	−0.0991572	−	0.995072i	$-0.531615\pi$
$709$	−5.28053	−0.198314	−0.0991572	+	0.995072i	$0.531615\pi$
$710$	−26.7359	−1.00338
$711$	−15.0249	−0.563479
$712$	8.39541	0.314631
$713$	−8.24109	−0.308631
$714$	5.03944	0.188596
$715$	0	0
$716$	25.1433	0.939648
$717$	12.3285	0.460416
$718$	64.5491	2.40895
$719$	−15.8252	−0.590180	−0.295090	−	0.955470i	$-0.595350\pi$
$719$	−15.8252	−0.590180	−0.295090	+	0.955470i	$0.595350\pi$
$720$	−2.16425	−0.0806568
$721$	−8.74186	−0.325564
$722$	−33.7278	−1.25522
$723$	−7.20164	−0.267832
$724$	47.6923	1.77247
$725$	−8.32850	−0.309313
$726$	0	0
$727$	13.6964	0.507973	0.253986	−	0.967208i	$-0.418258\pi$
$727$	13.6964	0.507973	0.253986	+	0.967208i	$0.418258\pi$
$728$	2.39442	0.0887433
$729$	1.00000	0.0370370
$730$	12.6570	0.468456
$731$	20.9855	0.776176
$732$	−32.1018	−1.18652
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	55.0104	2.03047
$735$	−6.76933	−0.249691
$736$	−26.6051	−0.980676
$737$	0	0
$738$	−14.4074	−0.530343
$739$	−18.7214	−0.688677	−0.344338	−	0.938846i	$-0.611897\pi$
$739$	−18.7214	−0.688677	−0.344338	+	0.938846i	$0.611897\pi$
$740$	14.6030	0.536818
$741$	−6.22468	−0.228669
$742$	14.0379	0.515347
$743$	7.55916	0.277319	0.138659	−	0.990340i	$-0.455721\pi$
$743$	7.55916	0.277319	0.138659	+	0.990340i	$0.455721\pi$
$744$	−3.50521	−0.128507
$745$	−16.0249	−0.587108
$746$	66.6778	2.44125
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−4.76081	−0.173956
$750$	2.16425	0.0790272
$751$	−21.1707	−0.772531	−0.386265	−	0.922388i	$-0.626235\pi$
$751$	−21.1707	−0.772531	−0.386265	+	0.922388i	$0.626235\pi$
$752$	6.87519	0.250713
$753$	−3.36794	−0.122734
$754$	−60.7069	−2.21081
$755$	−7.17671	−0.261187
$756$	1.28905	0.0468824
$757$	−47.8981	−1.74089	−0.870443	−	0.492270i	$-0.836167\pi$
$757$	−47.8981	−1.74089	−0.870443	+	0.492270i	$0.836167\pi$
$758$	−63.9105	−2.32134
$759$	0	0
$760$	−2.73588	−0.0992407
$761$	10.1038	0.366263	0.183132	−	0.983088i	$-0.441377\pi$
$761$	10.1038	0.366263	0.183132	+	0.983088i	$0.441377\pi$
$762$	−0.553177	−0.0200395
$763$	7.00443	0.253577
$764$	51.7712	1.87302
$765$	4.84822	0.175288
$766$	−9.31398	−0.336528
$767$	50.7359	1.83197
$768$	0.301518	0.0108801
$769$	24.7463	0.892374	0.446187	−	0.894940i	$-0.352782\pi$
$769$	24.7463	0.892374	0.446187	+	0.894940i	$0.352782\pi$
$770$	0	0
$771$	−11.5841	−0.417191
$772$	16.7898	0.604279
$773$	16.5945	0.596863	0.298432	−	0.954431i	$-0.403537\pi$
$773$	16.5945	0.596863	0.298432	+	0.954431i	$0.403537\pi$
$774$	9.36794	0.336724
$775$	−2.36794	−0.0850589
$776$	−12.1038	−0.434502
$777$	2.61312	0.0937451
$778$	76.6529	2.74814
$779$	12.3036	0.440821
$780$	9.03944	0.323664
$781$	0	0
$782$	36.5177	1.30587
$783$	−8.32850	−0.297636
$784$	14.6505	0.523233
$785$	2.25560	0.0805057
$786$	−1.59262	−0.0568068
$787$	0.157770	0.00562388	0.00281194	−	0.999996i	$-0.499105\pi$
$787$	0.157770	0.00562388	0.00281194	+	0.999996i	$0.499105\pi$
$788$	13.5257	0.481833
$789$	28.5696	1.01710
$790$	−32.5177	−1.15693
$791$	−4.10381	−0.145915
$792$	0	0
$793$	−40.2824	−1.43047
$794$	−39.1142	−1.38811
$795$	13.5052	0.478980
$796$	−39.1661	−1.38821
$797$	26.0997	0.924500	0.462250	−	0.886750i	$-0.347042\pi$
$797$	26.0997	0.924500	0.462250	+	0.886750i	$0.347042\pi$
$798$	−1.92112	−0.0680067
$799$	−15.4014	−0.544862
$800$	−7.64453	−0.270275
$801$	5.67150	0.200393
$802$	−6.68602	−0.236091
$803$	0	0
$804$	21.9460	0.773977
$805$	1.67150	0.0589128
$806$	−17.2600	−0.607959
$807$	−1.47175	−0.0518082
$808$	−4.04986	−0.142474
$809$	−6.22468	−0.218848	−0.109424	−	0.993995i	$-0.534901\pi$
$809$	−6.22468	−0.218848	−0.109424	+	0.993995i	$0.534901\pi$
$810$	2.16425	0.0760439
$811$	38.7712	1.36144	0.680721	−	0.732543i	$-0.261667\pi$
$811$	38.7712	1.36144	0.680721	+	0.732543i	$0.261667\pi$
$812$	−10.7359	−0.376755
$813$	5.70496	0.200082
$814$	0	0
$815$	0.255598	0.00895320
$816$	−10.4927	−0.367320
$817$	−8.00000	−0.279885
$818$	−53.9355	−1.88581
$819$	1.61755	0.0565217
$820$	−17.8672	−0.623948
$821$	20.7857	0.725427	0.362714	−	0.931901i	$-0.381850\pi$
$821$	20.7857	0.725427	0.362714	+	0.931901i	$0.381850\pi$
$822$	2.00648	0.0699840
$823$	−54.4093	−1.89659	−0.948294	−	0.317393i	$-0.897192\pi$
$823$	−54.4093	−1.89659	−0.948294	+	0.317393i	$0.897192\pi$
$824$	−26.9435	−0.938622
$825$	0	0
$826$	15.6585	0.544831
$827$	15.5506	0.540749	0.270374	−	0.962755i	$-0.412853\pi$
$827$	15.5506	0.540749	0.270374	+	0.962755i	$0.412853\pi$
$828$	9.34096	0.324621
$829$	30.9211	1.07393	0.536967	−	0.843603i	$-0.319570\pi$
$829$	30.9211	1.07393	0.536967	+	0.843603i	$0.319570\pi$
$830$	−8.65699	−0.300489
$831$	−24.7299	−0.857870
$832$	−41.1433	−1.42639
$833$	−32.8192	−1.13712
$834$	−17.7818	−0.615733
$835$	−16.1373	−0.558453
$836$	0	0
$837$	−2.36794	−0.0818479
$838$	−9.14326	−0.315849
$839$	6.89365	0.237995	0.118998	−	0.992895i	$-0.462032\pi$
$839$	6.89365	0.237995	0.118998	+	0.992895i	$0.462032\pi$
$840$	0.710947	0.0245300
$841$	40.3638	1.39186
$842$	−47.0289	−1.62072
$843$	−5.67150	−0.195337
$844$	−51.6654	−1.77840
$845$	−1.65699	−0.0570022
$846$	−6.87519	−0.236374
$847$	0	0
$848$	−29.2286	−1.00371
$849$	22.1228	0.759251
$850$	10.4927	0.359898
$851$	18.9356	0.649105
$852$	−33.1562	−1.13591
$853$	−55.5945	−1.90352	−0.951760	−	0.306844i	$-0.900727\pi$
$853$	−55.5945	−1.90352	−0.951760	+	0.306844i	$0.900727\pi$
$854$	−12.4323	−0.425425
$855$	−1.84822	−0.0632077
$856$	−14.6734	−0.501526
$857$	−15.2016	−0.519278	−0.259639	−	0.965706i	$-0.583604\pi$
$857$	−15.2016	−0.519278	−0.259639	+	0.965706i	$0.583604\pi$
$858$	0	0
$859$	−40.1228	−1.36897	−0.684485	−	0.729027i	$-0.739973\pi$
$859$	−40.1228	−1.36897	−0.684485	+	0.729027i	$0.739973\pi$
$860$	11.6175	0.396155
$861$	−3.19721	−0.108961
$862$	−21.4178	−0.729493
$863$	−15.5506	−0.529350	−0.264675	−	0.964338i	$-0.585265\pi$
$863$	−15.5506	−0.529350	−0.264675	+	0.964338i	$0.585265\pi$
$864$	−7.64453	−0.260072
$865$	9.39287	0.319367
$866$	85.1521	2.89359
$867$	6.50521	0.220928
$868$	−3.05240	−0.103605
$869$	0	0
$870$	−18.0249	−0.611102
$871$	27.5387	0.933112
$872$	21.5885	0.731080
$873$	−8.17671	−0.276740
$874$	−13.9211	−0.470889
$875$	0.480279	0.0162364
$876$	15.6964	0.530333
$877$	−4.75891	−0.160697	−0.0803486	−	0.996767i	$-0.525603\pi$
$877$	−4.75891	−0.160697	−0.0803486	+	0.996767i	$0.525603\pi$
$878$	−8.03141	−0.271047
$879$	−12.8482	−0.433360
$880$	0	0
$881$	−28.0997	−0.946704	−0.473352	−	0.880873i	$-0.656956\pi$
$881$	−28.0997	−0.946704	−0.473352	+	0.880873i	$0.656956\pi$
$882$	−14.6505	−0.493308
$883$	51.3329	1.72749	0.863745	−	0.503929i	$-0.168113\pi$
$883$	51.3329	1.72749	0.863745	+	0.503929i	$0.168113\pi$
$884$	43.8252	1.47400
$885$	15.0644	0.506383
$886$	10.7359	0.360679
$887$	−20.1458	−0.676430	−0.338215	−	0.941069i	$-0.609823\pi$
$887$	−20.1458	−0.676430	−0.338215	+	0.941069i	$0.609823\pi$
$888$	8.05395	0.270273
$889$	−0.122758	−0.00411718
$890$	12.2745	0.411444
$891$	0	0
$892$	28.4074	0.951149
$893$	5.87126	0.196474
$894$	−34.6819	−1.15994
$895$	9.36794	0.313136
$896$	−5.35498	−0.178897
$897$	11.7214	0.391365
$898$	47.6135	1.58888
$899$	19.7214	0.657744
$900$	2.68397	0.0894656
$901$	65.4762	2.18133
$902$	0	0
$903$	2.07888	0.0691810
$904$	−12.6485	−0.420682
$905$	17.7693	0.590673
$906$	−15.5322	−0.516022
$907$	4.20164	0.139513	0.0697566	−	0.997564i	$-0.477778\pi$
$907$	4.20164	0.139513	0.0697566	+	0.997564i	$0.477778\pi$
$908$	−72.9415	−2.42065
$909$	−2.73588	−0.0907433
$910$	3.50078	0.116050
$911$	−4.05395	−0.134314	−0.0671568	−	0.997742i	$-0.521393\pi$
$911$	−4.05395	−0.134314	−0.0671568	+	0.997742i	$0.521393\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−27.5636	−0.911723
$915$	−11.9606	−0.395404
$916$	56.0898	1.85326
$917$	−0.353426	−0.0116712
$918$	10.4927	0.346312
$919$	−14.8586	−0.490141	−0.245071	−	0.969505i	$-0.578811\pi$
$919$	−14.8586	−0.490141	−0.245071	+	0.969505i	$0.578811\pi$
$920$	5.15178	0.169849
$921$	−8.50521	−0.280256
$922$	86.6239	2.85281
$923$	−41.6056	−1.36946
$924$	0	0
$925$	5.44084	0.178894
$926$	−55.2720	−1.81635
$927$	−18.2016	−0.597820
$928$	63.6674	2.08999
$929$	58.7069	1.92611	0.963055	−	0.269306i	$-0.0867943\pi$
$929$	58.7069	1.92611	0.963055	+	0.269306i	$0.0867943\pi$
$930$	−5.12481	−0.168049
$931$	12.5112	0.410038
$932$	53.3779	1.74845
$933$	−6.32850	−0.207186
$934$	−14.2810	−0.467289
$935$	0	0
$936$	4.98549	0.162956
$937$	−34.6090	−1.13063	−0.565314	−	0.824876i	$-0.691245\pi$
$937$	−34.6090	−1.13063	−0.565314	+	0.824876i	$0.691245\pi$
$938$	8.49922	0.277509
$939$	20.0249	0.653489
$940$	−8.52620	−0.278094
$941$	−10.7109	−0.349167	−0.174583	−	0.984642i	$-0.555858\pi$
$941$	−10.7109	−0.349167	−0.174583	+	0.984642i	$0.555858\pi$
$942$	4.88167	0.159053
$943$	−23.1682	−0.754460
$944$	−32.6030	−1.06114
$945$	0.480279	0.0156235
$946$	0	0
$947$	17.0480	0.553985	0.276992	−	0.960872i	$-0.410662\pi$
$947$	17.0480	0.553985	0.276992	+	0.960872i	$0.410662\pi$
$948$	−40.3264	−1.30974
$949$	19.6964	0.639373
$950$	−4.00000	−0.129777
$951$	−23.8088	−0.772052
$952$	3.44682	0.111712
$953$	50.0000	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$954$	29.2286	0.946312
$955$	19.2891	0.624179
$956$	33.0893	1.07018
$957$	0	0
$958$	79.7043	2.57513
$959$	0.445267	0.0143784
$960$	−12.2162	−0.394275
$961$	−25.3929	−0.819125
$962$	39.6585	1.27864
$963$	−9.91259	−0.319429
$964$	−19.3290	−0.622545
$965$	6.25560	0.201375
$966$	3.61755	0.116393
$967$	−50.2016	−1.61438	−0.807188	−	0.590294i	$-0.799012\pi$
$967$	−50.2016	−1.61438	−0.807188	+	0.590294i	$0.799012\pi$
$968$	0	0
$969$	−8.96056	−0.287855
$970$	−17.6964	−0.568198
$971$	−52.0748	−1.67116	−0.835580	−	0.549369i	$-0.814868\pi$
$971$	−52.0748	−1.67116	−0.835580	+	0.549369i	$0.814868\pi$
$972$	2.68397	0.0860884
$973$	−3.94605	−0.126504
$974$	−13.8672	−0.444332
$975$	3.36794	0.107860
$976$	25.8856	0.828578
$977$	19.5841	0.626551	0.313275	−	0.949662i	$-0.398574\pi$
$977$	19.5841	0.626551	0.313275	+	0.949662i	$0.398574\pi$
$978$	0.553177	0.0176886
$979$	0	0
$980$	−18.1687	−0.580377
$981$	14.5841	0.465634
$982$	5.26412	0.167985
$983$	13.4513	0.429028	0.214514	−	0.976721i	$-0.431183\pi$
$983$	13.4513	0.429028	0.214514	+	0.976721i	$0.431183\pi$
$984$	−9.85420	−0.314141
$985$	5.03944	0.160570
$986$	−87.3888	−2.78303
$987$	−1.52571	−0.0485638
$988$	−16.7069	−0.531516
$989$	15.0644	0.479019
$990$	0	0
$991$	−36.4238	−1.15704	−0.578520	−	0.815668i	$-0.696369\pi$
$991$	−36.4238	−1.15704	−0.578520	+	0.815668i	$0.696369\pi$
$992$	18.1018	0.574732
$993$	−15.3285	−0.486435
$994$	−12.8407	−0.407281
$995$	−14.5926	−0.462617
$996$	−10.7359	−0.340179
$997$	17.8233	0.564469	0.282235	−	0.959345i	$-0.408924\pi$
$997$	17.8233	0.564469	0.282235	+	0.959345i	$0.408924\pi$
$998$	−78.5201	−2.48551
$999$	5.44084	0.172140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.2.a.l.1.3	✓	3
3.2	odd	2		5445.2.a.bc.1.1		3
5.4	even	2		9075.2.a.ci.1.1		3
11.10	odd	2		1815.2.a.n.1.1	yes	3
33.32	even	2		5445.2.a.ba.1.3		3
55.54	odd	2		9075.2.a.ce.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.l.1.3	✓	3	1.1	even	1	trivial
1815.2.a.n.1.1	yes	3	11.10	odd	2
5445.2.a.ba.1.3		3	33.32	even	2
5445.2.a.bc.1.1		3	3.2	odd	2
9075.2.a.ce.1.3		3	55.54	odd	2
9075.2.a.ci.1.1		3	5.4	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+2.16425 q^{2} +1.00000 q^{3} +2.68397 q^{4} +1.00000 q^{5} +2.16425 q^{6} +0.480279 q^{7} +1.48028 q^{8} +1.00000 q^{9} +2.16425 q^{10} +2.68397 q^{12} +3.36794 q^{13} +1.03944 q^{14} +1.00000 q^{15} -2.16425 q^{16} +4.84822 q^{17} +2.16425 q^{18} -1.84822 q^{19} +2.68397 q^{20} +0.480279 q^{21} +3.48028 q^{23} +1.48028 q^{24} +1.00000 q^{25} +7.28905 q^{26} +1.00000 q^{27} +1.28905 q^{28} -8.32850 q^{29} +2.16425 q^{30} -2.36794 q^{31} -7.64453 q^{32} +10.4927 q^{34} +0.480279 q^{35} +2.68397 q^{36} +5.44084 q^{37} -4.00000 q^{38} +3.36794 q^{39} +1.48028 q^{40} -6.65699 q^{41} +1.03944 q^{42} +4.32850 q^{43} +1.00000 q^{45} +7.53219 q^{46} -3.17671 q^{47} -2.16425 q^{48} -6.76933 q^{49} +2.16425 q^{50} +4.84822 q^{51} +9.03944 q^{52} +13.5052 q^{53} +2.16425 q^{54} +0.710947 q^{56} -1.84822 q^{57} -18.0249 q^{58} +15.0644 q^{59} +2.68397 q^{60} -11.9606 q^{61} -5.12481 q^{62} +0.480279 q^{63} -12.2162 q^{64} +3.36794 q^{65} +8.17671 q^{67} +13.0125 q^{68} +3.48028 q^{69} +1.03944 q^{70} -12.3534 q^{71} +1.48028 q^{72} +5.84822 q^{73} +11.7753 q^{74} +1.00000 q^{75} -4.96056 q^{76} +7.28905 q^{78} -15.0249 q^{79} -2.16425 q^{80} +1.00000 q^{81} -14.4074 q^{82} -4.00000 q^{83} +1.28905 q^{84} +4.84822 q^{85} +9.36794 q^{86} -8.32850 q^{87} +5.67150 q^{89} +2.16425 q^{90} +1.61755 q^{91} +9.34096 q^{92} -2.36794 q^{93} -6.87519 q^{94} -1.84822 q^{95} -7.64453 q^{96} -8.17671 q^{97} -14.6505 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9} - q^{10} + 5 q^{12} + 4 q^{13} + 12 q^{14} + 3 q^{15} + q^{16} + 4 q^{17} - q^{18} + 5 q^{19} + 5 q^{20} - 3 q^{21} + 6 q^{23}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 3 * q^7 + 3 * q^9 - q^10 + 5 * q^12 + 4 * q^13 + 12 * q^14 + 3 * q^15 + q^16 + 4 * q^17 - q^18 + 5 * q^19 + 5 * q^20 - 3 * q^21 + 6 * q^23 + 3 * q^25 - 2 * q^26 + 3 * q^27 - 20 * q^28 - 10 * q^29 - q^30 - q^31 - 11 * q^32 + 9 * q^34 - 3 * q^35 + 5 * q^36 + 3 * q^37 - 12 * q^38 + 4 * q^39 + 10 * q^41 + 12 * q^42 - 2 * q^43 + 3 * q^45 + 9 * q^46 + 16 * q^47 + q^48 + 8 * q^49 - q^50 + 4 * q^51 + 36 * q^52 - q^54 + 26 * q^56 + 5 * q^57 - 18 * q^58 + 18 * q^59 + 5 * q^60 - 27 * q^61 + q^62 - 3 * q^63 - 20 * q^64 + 4 * q^65 - q^67 + 21 * q^68 + 6 * q^69 + 12 * q^70 + 14 * q^71 + 7 * q^73 + 32 * q^74 + 3 * q^75 - 6 * q^76 - 2 * q^78 - 9 * q^79 + q^80 + 3 * q^81 - 46 * q^82 - 12 * q^83 - 20 * q^84 + 4 * q^85 + 22 * q^86 - 10 * q^87 + 32 * q^89 - q^90 - 34 * q^91 - 5 * q^92 - q^93 - 37 * q^94 + 5 * q^95 - 11 * q^96 + q^97 - 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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