LMFDB - Embedded newform 9075.2.a.ci.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
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:	no (minimal twist has level 1815)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$2$	−2.16425	−1.53035	−0.765177	−	0.643820i	$-0.777349\pi$
$2$	−2.16425	−1.53035	−0.765177	+	0.643820i	$0.777349\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.68397	1.34198
$5$	0	0
$5$	0	0
$6$	2.16425	0.883551
$7$	−0.480279	−0.181528	−0.0907642	−	0.995872i	$-0.528931\pi$
$7$	−0.480279	−0.181528	−0.0907642	+	0.995872i	$0.528931\pi$
$8$	−1.48028	−0.523358
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−2.68397	−0.774795
$13$	−3.36794	−0.934098	−0.467049	−	0.884231i	$-0.654683\pi$
$13$	−3.36794	−0.934098	−0.467049	+	0.884231i	$0.654683\pi$
$14$	1.03944	0.277803
$15$	0	0
$16$	−2.16425	−0.541062
$17$	−4.84822	−1.17587	−0.587933	−	0.808910i	$-0.700058\pi$
$17$	−4.84822	−1.17587	−0.587933	+	0.808910i	$0.700058\pi$
$18$	−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	$0.567999\pi$
$20$	0	0
$21$	0.480279	0.104805
$22$	0	0
$23$	−3.48028	−0.725688	−0.362844	−	0.931850i	$-0.618194\pi$
$23$	−3.48028	−0.725688	−0.362844	+	0.931850i	$0.618194\pi$
$24$	1.48028	0.302161
$25$	0	0
$26$	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	$0.781386\pi$
$30$	0	0
$31$	−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	0
$36$	2.68397	0.447328
$37$	−5.44084	−0.894468	−0.447234	−	0.894417i	$-0.647591\pi$
$37$	−5.44084	−0.894468	−0.447234	+	0.894417i	$0.647591\pi$
$38$	4.00000	0.648886
$39$	3.36794	0.539302
$40$	0	0
$41$	−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	$-0.607064\pi$
$43$	−4.32850	−0.660089	−0.330045	+	0.943965i	$0.607064\pi$
$44$	0	0
$45$	0	0
$46$	7.53219	1.11056
$47$	3.17671	0.463371	0.231686	−	0.972791i	$-0.425576\pi$
$47$	3.17671	0.463371	0.231686	+	0.972791i	$0.425576\pi$
$48$	2.16425	0.312382
$49$	−6.76933	−0.967047
$50$	0	0
$51$	4.84822	0.678886
$52$	−9.03944	−1.25355
$53$	−13.5052	−1.85508	−0.927542	−	0.373720i	$-0.878082\pi$
$53$	−13.5052	−1.85508	−0.927542	+	0.373720i	$0.878082\pi$
$54$	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	$0.0627900\pi$
$60$	0	0
$61$	−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
$65$	0	0
$66$	0	0
$67$	−8.17671	−0.998944	−0.499472	−	0.866330i	$-0.666473\pi$
$67$	−8.17671	−0.998944	−0.499472	+	0.866330i	$0.666473\pi$
$68$	−13.0125	−1.57799
$69$	3.48028	0.418976
$70$	0	0
$71$	−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	$-0.611186\pi$
$73$	−5.84822	−0.684482	−0.342241	+	0.939612i	$0.611186\pi$
$74$	11.7753	1.36885
$75$	0	0
$76$	−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	$0.820530\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	14.4074	1.59103
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	1.28905	0.140647
$85$	0	0
$86$	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	$0.402817\pi$
$90$	0	0
$91$	1.61755	0.169565
$92$	−9.34096	−0.973862
$93$	2.36794	0.245544
$94$	−6.87519	−0.709122
$95$	0	0
$96$	−7.64453	−0.780216
$97$	8.17671	0.830219	0.415110	−	0.909771i	$-0.363743\pi$
$97$	8.17671	0.830219	0.415110	+	0.909771i	$0.363743\pi$
$98$	14.6505	1.47993
$99$	0	0
$100$	0	0
$101$	−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	$-0.145936\pi$
$103$	18.2016	1.79346	0.896731	+	0.442577i	$0.145936\pi$
$104$	4.98549	0.488867
$105$	0	0
$106$	29.2286	2.83893
$107$	9.91259	0.958286	0.479143	−	0.877737i	$-0.340947\pi$
$107$	9.91259	0.958286	0.479143	+	0.877737i	$0.340947\pi$
$108$	−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	$-0.368348\pi$
$113$	8.54465	0.803813	0.401907	+	0.915681i	$0.368348\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−22.3534	−2.07546
$117$	−3.36794	−0.311366
$118$	−32.6030	−3.00135
$119$	2.32850	0.213453
$120$	0	0
$121$	0	0
$122$	25.8856	2.34357
$123$	6.65699	0.600241
$124$	−6.35547	−0.570738
$125$	0	0
$126$	1.03944	0.0926009
$127$	0.255598	0.0226806	0.0113403	−	0.999936i	$-0.496390\pi$
$127$	0.255598	0.0226806	0.0113403	+	0.999936i	$0.496390\pi$
$128$	11.1497	0.985507
$129$	4.32850	0.381103
$130$	0	0
$131$	−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
$135$	0	0
$136$	7.17671	0.615398
$137$	−0.927102	−0.0792077	−0.0396038	−	0.999215i	$-0.512610\pi$
$137$	−0.927102	−0.0792077	−0.0396038	+	0.999215i	$0.512610\pi$
$138$	−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	$0.613289\pi$
$140$	0	0
$141$	−3.17671	−0.267527
$142$	26.7359	2.24362
$143$	0	0
$144$	−2.16425	−0.180354
$145$	0	0
$146$	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	$0.727924\pi$
$150$	0	0
$151$	−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
$155$	0	0
$156$	9.03944	0.723735
$157$	−2.25560	−0.180016	−0.0900081	−	0.995941i	$-0.528689\pi$
$157$	−2.25560	−0.180016	−0.0900081	+	0.995941i	$0.528689\pi$
$158$	32.5177	2.58697
$159$	13.5052	1.07103
$160$	0	0
$161$	1.67150	0.131733
$162$	−2.16425	−0.170039
$163$	−0.255598	−0.0200200	−0.0100100	−	0.999950i	$-0.503186\pi$
$163$	−0.255598	−0.0200200	−0.0100100	+	0.999950i	$0.503186\pi$
$164$	−17.8672	−1.39519
$165$	0	0
$166$	−8.65699	−0.671913
$167$	16.1373	1.24874	0.624370	−	0.781129i	$-0.285356\pi$
$167$	16.1373	1.24874	0.624370	+	0.781129i	$0.285356\pi$
$168$	−0.710947	−0.0548507
$169$	−1.65699	−0.127461
$170$	0	0
$171$	−1.84822	−0.141337
$172$	−11.6175	−0.885830
$173$	−9.39287	−0.714127	−0.357063	−	0.934080i	$-0.616222\pi$
$173$	−9.39287	−0.714127	−0.357063	+	0.934080i	$0.616222\pi$
$174$	−18.0249	−1.36647
$175$	0	0
$176$	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	$0.386149\pi$
$180$	0	0
$181$	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
$185$	0	0
$186$	−5.12481	−0.375769
$187$	0	0
$188$	8.52620	0.621837
$189$	0.480279	0.0349351
$190$	0	0
$191$	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	$-0.572285\pi$
$193$	−6.25560	−0.450288	−0.225144	+	0.974326i	$0.572285\pi$
$194$	−17.6964	−1.27053
$195$	0	0
$196$	−18.1687	−1.29776
$197$	−5.03944	−0.359045	−0.179523	−	0.983754i	$-0.557455\pi$
$197$	−5.03944	−0.359045	−0.179523	+	0.983754i	$0.557455\pi$
$198$	0	0
$199$	−14.5926	−1.03444	−0.517222	−	0.855851i	$-0.673034\pi$
$199$	−14.5926	−1.03444	−0.517222	+	0.855851i	$0.673034\pi$
$200$	0	0
$201$	8.17671	0.576741
$202$	5.92112	0.416608
$203$	4.00000	0.280745
$204$	13.0125	0.911055
$205$	0	0
$206$	−39.3929	−2.74463
$207$	−3.48028	−0.241896
$208$	7.28905	0.505405
$209$	0	0
$210$	0	0
$211$	−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
$215$	0	0
$216$	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	$-0.615309\pi$
$223$	−10.5841	−0.708763	−0.354382	+	0.935101i	$0.615309\pi$
$224$	−3.67150	−0.245313
$225$	0	0
$226$	−18.4927	−1.23012
$227$	27.1767	1.80378	0.901891	−	0.431964i	$-0.142179\pi$
$227$	27.1767	1.80378	0.901891	+	0.431964i	$0.142179\pi$
$228$	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	$0.257394\pi$
$230$	0	0
$231$	0	0
$232$	12.3285	0.809405
$233$	−19.8877	−1.30288	−0.651442	−	0.758698i	$-0.725836\pi$
$233$	−19.8877	−1.30288	−0.651442	+	0.758698i	$0.725836\pi$
$234$	7.28905	0.476500
$235$	0	0
$236$	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	$0.369450\pi$
$240$	0	0
$241$	−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
$245$	0	0
$246$	−14.4074	−0.918581
$247$	6.22468	0.396067
$248$	3.50521	0.222581
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−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
$255$	0	0
$256$	0.301518	0.0188449
$257$	11.5841	0.722596	0.361298	−	0.932450i	$-0.382334\pi$
$257$	11.5841	0.722596	0.361298	+	0.932450i	$0.382334\pi$
$258$	−9.36794	−0.583222
$259$	2.61312	0.162371
$260$	0	0
$261$	−8.32850	−0.515521
$262$	1.59262	0.0983924
$263$	−28.5696	−1.76168	−0.880838	−	0.473418i	$-0.843020\pi$
$263$	−28.5696	−1.76168	−0.880838	+	0.473418i	$0.843020\pi$
$264$	0	0
$265$	0	0
$266$	−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	$0.514286\pi$
$270$	0	0
$271$	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	$-0.233432\pi$
$277$	24.7299	1.48588	0.742938	+	0.669361i	$0.233432\pi$
$278$	17.7818	1.06648
$279$	−2.36794	−0.141765
$280$	0	0
$281$	−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	$-0.728399\pi$
$283$	−22.1228	−1.31506	−0.657531	+	0.753428i	$0.728399\pi$
$284$	−33.1562	−1.96746
$285$	0	0
$286$	0	0
$287$	3.19721	0.188725
$288$	7.64453	0.450458
$289$	6.50521	0.382659
$290$	0	0
$291$	−8.17671	−0.479327
$292$	−15.6964	−0.918564
$293$	12.8482	0.750601	0.375300	−	0.926903i	$-0.377540\pi$
$293$	12.8482	0.750601	0.375300	+	0.926903i	$0.377540\pi$
$294$	−14.6505	−0.854435
$295$	0	0
$296$	8.05395	0.468127
$297$	0	0
$298$	34.6819	2.00907
$299$	11.7214	0.677864
$300$	0	0
$301$	2.07888	0.119825
$302$	15.5322	0.893777
$303$	2.73588	0.157172
$304$	4.00000	0.229416
$305$	0	0
$306$	10.4927	0.599830
$307$	8.50521	0.485418	0.242709	−	0.970099i	$-0.421964\pi$
$307$	8.50521	0.485418	0.242709	+	0.970099i	$0.421964\pi$
$308$	0	0
$309$	−18.2016	−1.03546
$310$	0	0
$311$	−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	$-0.691486\pi$
$313$	−20.0249	−1.13188	−0.565938	+	0.824448i	$0.691486\pi$
$314$	4.88167	0.275489
$315$	0	0
$316$	−40.3264	−2.26854
$317$	23.8088	1.33723	0.668617	−	0.743607i	$-0.266887\pi$
$317$	23.8088	1.33723	0.668617	+	0.743607i	$0.266887\pi$
$318$	−29.2286	−1.63906
$319$	0	0
$320$	0	0
$321$	−9.91259	−0.553267
$322$	−3.61755	−0.201598
$323$	8.96056	0.498579
$324$	2.68397	0.149109
$325$	0	0
$326$	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
$335$	0	0
$336$	−1.03944	−0.0567062
$337$	3.03346	0.165243	0.0826214	−	0.996581i	$-0.473671\pi$
$337$	3.03346	0.165243	0.0826214	+	0.996581i	$0.473671\pi$
$338$	3.58614	0.195060
$339$	−8.54465	−0.464082
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	6.61312	0.357075
$344$	6.40738	0.345463
$345$	0	0
$346$	20.3285	1.09287
$347$	−34.0584	−1.82835	−0.914175	−	0.405320i	$-0.867160\pi$
$347$	−34.0584	−1.82835	−0.914175	+	0.405320i	$0.867160\pi$
$348$	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	$0.679902\pi$
$350$	0	0
$351$	3.36794	0.179767
$352$	0	0
$353$	−30.9770	−1.64874	−0.824369	−	0.566053i	$-0.808470\pi$
$353$	−30.9770	−1.64874	−0.824369	+	0.566053i	$0.808470\pi$
$354$	32.6030	1.73283
$355$	0	0
$356$	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	$0.211604\pi$
$360$	0	0
$361$	−15.5841	−0.820215
$362$	−38.4572	−2.02127
$363$	0	0
$364$	4.34145	0.227554
$365$	0	0
$366$	−25.8856	−1.35306
$367$	−25.4178	−1.32680	−0.663399	−	0.748266i	$-0.730887\pi$
$367$	−25.4178	−1.32680	−0.663399	+	0.748266i	$0.730887\pi$
$368$	7.53219	0.392642
$369$	−6.65699	−0.346549
$370$	0	0
$371$	6.48627	0.336750
$372$	6.35547	0.329516
$373$	−30.8088	−1.59522	−0.797609	−	0.603175i	$-0.793902\pi$
$373$	−30.8088	−1.59522	−0.797609	+	0.603175i	$0.793902\pi$
$374$	0	0
$375$	0	0
$376$	−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	$0.774034\pi$
$380$	0	0
$381$	−0.255598	−0.0130947
$382$	−41.7463	−2.13593
$383$	4.30357	0.219902	0.109951	−	0.993937i	$-0.464931\pi$
$383$	4.30357	0.219902	0.109951	+	0.993937i	$0.464931\pi$
$384$	−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	$0.145109\pi$
$390$	0	0
$391$	16.8731	0.853312
$392$	10.0205	0.506112
$393$	0.735877	0.0371201
$394$	10.9066	0.549467
$395$	0	0
$396$	0	0
$397$	18.0729	0.907053	0.453526	−	0.891243i	$-0.350166\pi$
$397$	18.0729	0.907053	0.453526	+	0.891243i	$0.350166\pi$
$398$	31.5820	1.58306
$399$	−0.887659	−0.0444386
$400$	0	0
$401$	−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
$405$	0	0
$406$	−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	$0.711302\pi$
$410$	0	0
$411$	0.927102	0.0457306
$412$	48.8526	2.40680
$413$	−7.23510	−0.356016
$414$	7.53219	0.370187
$415$	0	0
$416$	−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	$0.532906\pi$
$420$	0	0
$421$	−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
$425$	0	0
$426$	−26.7359	−1.29536
$427$	5.74440	0.277991
$428$	26.6051	1.28601
$429$	0	0
$430$	0	0
$431$	−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	$-0.894326\pi$
$433$	−39.3449	−1.89080	−0.945398	+	0.325919i	$0.894326\pi$
$434$	−2.46134	−0.118148
$435$	0	0
$436$	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	$-0.537598\pi$
$443$	−4.96056	−0.235683	−0.117842	+	0.993032i	$0.537598\pi$
$444$	14.6030	0.693029
$445$	0	0
$446$	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	$0.326259\pi$
$450$	0	0
$451$	0	0
$452$	22.9336	1.07870
$453$	7.17671	0.337191
$454$	−58.8171	−2.76043
$455$	0	0
$456$	−2.73588	−0.128119
$457$	12.7359	0.595759	0.297880	−	0.954603i	$-0.403721\pi$
$457$	12.7359	0.595759	0.297880	+	0.954603i	$0.403721\pi$
$458$	−45.2286	−2.11339
$459$	4.84822	0.226295
$460$	0	0
$461$	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	$-0.297769\pi$
$463$	25.5387	1.18688	0.593441	+	0.804877i	$0.297769\pi$
$464$	18.0249	0.836786
$465$	0	0
$466$	43.0418	1.99387
$467$	6.59861	0.305347	0.152674	−	0.988277i	$-0.451212\pi$
$467$	6.59861	0.305347	0.152674	+	0.988277i	$0.451212\pi$
$468$	−9.03944	−0.417848
$469$	3.92710	0.181337
$470$	0	0
$471$	2.25560	0.103932
$472$	−22.2995	−1.02642
$473$	0	0
$474$	−32.5177	−1.49359
$475$	0	0
$476$	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	$0.181761\pi$
$480$	0	0
$481$	18.3244	0.835521
$482$	15.5861	0.709929
$483$	−1.67150	−0.0760561
$484$	0	0
$485$	0	0
$486$	2.16425	0.0981723
$487$	6.40738	0.290346	0.145173	−	0.989406i	$-0.453626\pi$
$487$	6.40738	0.290346	0.145173	+	0.989406i	$0.453626\pi$
$488$	17.7050	0.801466
$489$	0.255598	0.0115585
$490$	0	0
$491$	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	$0.801659\pi$
$500$	0	0
$501$	−16.1373	−0.720960
$502$	7.28905	0.325326
$503$	−11.9915	−0.534673	−0.267337	−	0.963603i	$-0.586144\pi$
$503$	−11.9915	−0.534673	−0.267337	+	0.963603i	$0.586144\pi$
$504$	0.710947	0.0316681
$505$	0	0
$506$	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	$0.375437\pi$
$510$	0	0
$511$	2.80877	0.124253
$512$	−22.9520	−1.01435
$513$	1.84822	0.0816008
$514$	−25.0709	−1.10583
$515$	0	0
$516$	11.6175	0.511434
$517$	0	0
$518$	−5.65544	−0.248486
$519$	9.39287	0.412301
$520$	0	0
$521$	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	$-0.632996\pi$
$523$	−18.5592	−0.811536	−0.405768	+	0.913976i	$0.632996\pi$
$524$	−1.97507	−0.0862813
$525$	0	0
$526$	61.8317	2.69599
$527$	11.4803	0.500089
$528$	0	0
$529$	−10.8877	−0.473376
$530$	0	0
$531$	15.0644	0.653738
$532$	2.38245	0.103292
$533$	22.4203	0.971133
$534$	12.2745	0.531171
$535$	0	0
$536$	12.1038	0.522805
$537$	−9.36794	−0.404256
$538$	3.18524	0.137325
$539$	0	0
$540$	0	0
$541$	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
$545$	0	0
$546$	3.50078	0.149819
$547$	15.7214	0.672197	0.336098	−	0.941827i	$-0.390892\pi$
$547$	15.7214	0.672197	0.336098	+	0.941827i	$0.390892\pi$
$548$	−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
$555$	0	0
$556$	−22.0519	−0.935209
$557$	−14.1622	−0.600072	−0.300036	−	0.953928i	$-0.596999\pi$
$557$	−14.1622	−0.600072	−0.300036	+	0.953928i	$0.596999\pi$
$558$	5.12481	0.216950
$559$	14.5781	0.616588
$560$	0	0
$561$	0	0
$562$	12.2745	0.517770
$563$	1.47175	0.0620270	0.0310135	−	0.999519i	$-0.490127\pi$
$563$	1.47175	0.0620270	0.0310135	+	0.999519i	$0.490127\pi$
$564$	−8.52620	−0.359018
$565$	0	0
$566$	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	$0.431076\pi$
$570$	0	0
$571$	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
$575$	0	0
$576$	−12.2162	−0.509006
$577$	−34.0439	−1.41726	−0.708632	−	0.705578i	$-0.750688\pi$
$577$	−34.0439	−1.41726	−0.708632	+	0.705578i	$0.750688\pi$
$578$	−14.0789	−0.585604
$579$	6.25560	0.259974
$580$	0	0
$581$	−1.92112	−0.0797013
$582$	17.6964	0.733541
$583$	0	0
$584$	8.65699	0.358229
$585$	0	0
$586$	−27.8067	−1.14869
$587$	−26.9520	−1.11243	−0.556215	−	0.831039i	$-0.687747\pi$
$587$	−26.9520	−1.11243	−0.556215	+	0.831039i	$0.687747\pi$
$588$	18.1687	0.749264
$589$	4.37646	0.180329
$590$	0	0
$591$	5.03944	0.207295
$592$	11.7753	0.483963
$593$	−7.31398	−0.300349	−0.150175	−	0.988659i	$-0.547984\pi$
$593$	−7.31398	−0.300349	−0.150175	+	0.988659i	$0.547984\pi$
$594$	0	0
$595$	0	0
$596$	−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	$0.727592\pi$
$600$	0	0
$601$	−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	$-0.656698\pi$
$607$	−23.2891	−0.945274	−0.472637	+	0.881257i	$0.656698\pi$
$608$	−14.1287	−0.572996
$609$	−4.00000	−0.162088
$610$	0	0
$611$	−10.6990	−0.432834
$612$	−13.0125	−0.525998
$613$	22.5052	0.908977	0.454488	−	0.890753i	$-0.349822\pi$
$613$	22.5052	0.908977	0.454488	+	0.890753i	$0.349822\pi$
$614$	−18.4074	−0.742861
$615$	0	0
$616$	0	0
$617$	−26.7069	−1.07518	−0.537589	−	0.843207i	$-0.680665\pi$
$617$	−26.7069	−1.07518	−0.537589	+	0.843207i	$0.680665\pi$
$618$	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	$0.273457\pi$
$620$	0	0
$621$	3.48028	0.139659
$622$	13.6964	0.549177
$623$	−2.72390	−0.109131
$624$	−7.28905	−0.291796
$625$	0	0
$626$	43.3389	1.73217
$627$	0	0
$628$	−6.05395	−0.241579
$629$	26.3784	1.05177
$630$	0	0
$631$	−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	0
$636$	36.2476	1.43731
$637$	22.7987	0.903317
$638$	0	0
$639$	−12.3534	−0.488694
$640$	0	0
$641$	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	$-0.670023\pi$
$643$	−25.8192	−1.01821	−0.509105	+	0.860705i	$0.670023\pi$
$644$	4.48627	0.176784
$645$	0	0
$646$	−19.3929	−0.763002
$647$	19.0189	0.747712	0.373856	−	0.927487i	$-0.378035\pi$
$647$	19.0189	0.747712	0.373856	+	0.927487i	$0.378035\pi$
$648$	−1.48028	−0.0581508
$649$	0	0
$650$	0	0
$651$	−1.13727	−0.0445731
$652$	−0.686016	−0.0268665
$653$	4.52825	0.177204	0.0886020	−	0.996067i	$-0.471760\pi$
$653$	4.52825	0.177204	0.0886020	+	0.996067i	$0.471760\pi$
$654$	31.5636	1.23423
$655$	0	0
$656$	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	0
$666$	11.7753	0.456284
$667$	28.9855	1.12232
$668$	43.3119	1.67579
$669$	10.5841	0.409205
$670$	0	0
$671$	0	0
$672$	3.67150	0.141631
$673$	−25.6485	−0.988676	−0.494338	−	0.869270i	$-0.664589\pi$
$673$	−25.6485	−0.988676	−0.494338	+	0.869270i	$0.664589\pi$
$674$	−6.56515	−0.252880
$675$	0	0
$676$	−4.44731	−0.171051
$677$	−23.6674	−0.909612	−0.454806	−	0.890590i	$-0.650291\pi$
$677$	−23.6674	−0.909612	−0.454806	+	0.890590i	$0.650291\pi$
$678$	18.4927	0.710210
$679$	−3.92710	−0.150708
$680$	0	0
$681$	−27.1767	−1.04141
$682$	0	0
$683$	−17.7753	−0.680154	−0.340077	−	0.940398i	$-0.610453\pi$
$683$	−17.7753	−0.680154	−0.340077	+	0.940398i	$0.610453\pi$
$684$	−4.96056	−0.189672
$685$	0	0
$686$	−14.3124	−0.546451
$687$	−20.8981	−0.797311
$688$	9.36794	0.357149
$689$	45.4847	1.73283
$690$	0	0
$691$	−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
$695$	0	0
$696$	−12.3285	−0.467310
$697$	32.2745	1.22249
$698$	43.3075	1.63921
$699$	19.8877	0.752220
$700$	0	0
$701$	−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
$705$	0	0
$706$	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	$0.531615\pi$
$710$	0	0
$711$	−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	$0.595350\pi$
$720$	0	0
$721$	−8.74186	−0.325564
$722$	33.7278	1.25522
$723$	7.20164	0.267832
$724$	47.6923	1.77247
$725$	0	0
$726$	0	0
$727$	−13.6964	−0.507973	−0.253986	−	0.967208i	$-0.581742\pi$
$727$	−13.6964	−0.507973	−0.253986	+	0.967208i	$0.581742\pi$
$728$	−2.39442	−0.0887433
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.9855	0.776176
$732$	32.1018	1.18652
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	55.0104	2.03047
$735$	0	0
$736$	−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	$0.611897\pi$
$740$	0	0
$741$	−6.22468	−0.228669
$742$	−14.0379	−0.515347
$743$	−7.55916	−0.277319	−0.138659	−	0.990340i	$-0.544279\pi$
$743$	−7.55916	−0.277319	−0.138659	+	0.990340i	$0.544279\pi$
$744$	−3.50521	−0.128507
$745$	0	0
$746$	66.6778	2.44125
$747$	4.00000	0.146352
$748$	0	0
$749$	−4.76081	−0.173956
$750$	0	0
$751$	−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
$755$	0	0
$756$	1.28905	0.0468824
$757$	47.8981	1.74089	0.870443	−	0.492270i	$-0.163833\pi$
$757$	47.8981	1.74089	0.870443	+	0.492270i	$0.163833\pi$
$758$	63.9105	2.32134
$759$	0	0
$760$	0	0
$761$	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
$765$	0	0
$766$	−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	$-0.596463\pi$
$773$	−16.5945	−0.596863	−0.298432	+	0.954431i	$0.596463\pi$
$774$	9.36794	0.336724
$775$	0	0
$776$	−12.1038	−0.434502
$777$	−2.61312	−0.0937451
$778$	−76.6529	−2.74814
$779$	12.3036	0.440821
$780$	0	0
$781$	0	0
$782$	−36.5177	−1.30587
$783$	8.32850	0.297636
$784$	14.6505	0.523233
$785$	0	0
$786$	−1.59262	−0.0568068
$787$	−0.157770	−0.00562388	−0.00281194	−	0.999996i	$-0.500895\pi$
$787$	−0.157770	−0.00562388	−0.00281194	+	0.999996i	$0.500895\pi$
$788$	−13.5257	−0.481833
$789$	28.5696	1.01710
$790$	0	0
$791$	−4.10381	−0.145915
$792$	0	0
$793$	40.2824	1.43047
$794$	−39.1142	−1.38811
$795$	0	0
$796$	−39.1661	−1.38821
$797$	−26.0997	−0.924500	−0.462250	−	0.886750i	$-0.652958\pi$
$797$	−26.0997	−0.924500	−0.462250	+	0.886750i	$0.652958\pi$
$798$	1.92112	0.0680067
$799$	−15.4014	−0.544862
$800$	0	0
$801$	5.67150	0.200393
$802$	6.68602	0.236091
$803$	0	0
$804$	21.9460	0.773977
$805$	0	0
$806$	−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	$0.534901\pi$
$810$	0	0
$811$	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	0
$816$	−10.4927	−0.367320
$817$	8.00000	0.279885
$818$	53.9355	1.88581
$819$	1.61755	0.0565217
$820$	0	0
$821$	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	$-0.102808\pi$
$823$	54.4093	1.89659	0.948294	+	0.317393i	$0.102808\pi$
$824$	−26.9435	−0.938622
$825$	0	0
$826$	15.6585	0.544831
$827$	−15.5506	−0.540749	−0.270374	−	0.962755i	$-0.587147\pi$
$827$	−15.5506	−0.540749	−0.270374	+	0.962755i	$0.587147\pi$
$828$	−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	$0.319570\pi$
$830$	0	0
$831$	−24.7299	−0.857870
$832$	41.1433	1.42639
$833$	32.8192	1.13712
$834$	−17.7818	−0.615733
$835$	0	0
$836$	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	0
$841$	40.3638	1.39186
$842$	47.0289	1.62072
$843$	5.67150	0.195337
$844$	−51.6654	−1.77840
$845$	0	0
$846$	−6.87519	−0.236374
$847$	0	0
$848$	29.2286	1.00371
$849$	22.1228	0.759251
$850$	0	0
$851$	18.9356	0.649105
$852$	33.1562	1.13591
$853$	55.5945	1.90352	0.951760	−	0.306844i	$-0.0992729\pi$
$853$	55.5945	1.90352	0.951760	+	0.306844i	$0.0992729\pi$
$854$	−12.4323	−0.425425
$855$	0	0
$856$	−14.6734	−0.501526
$857$	15.2016	0.519278	0.259639	−	0.965706i	$-0.416396\pi$
$857$	15.2016	0.519278	0.259639	+	0.965706i	$0.416396\pi$
$858$	0	0
$859$	−40.1228	−1.36897	−0.684485	−	0.729027i	$-0.739973\pi$
$859$	−40.1228	−1.36897	−0.684485	+	0.729027i	$0.739973\pi$
$860$	0	0
$861$	−3.19721	−0.108961
$862$	21.4178	0.729493
$863$	15.5506	0.529350	0.264675	−	0.964338i	$-0.414735\pi$
$863$	15.5506	0.529350	0.264675	+	0.964338i	$0.414735\pi$
$864$	−7.64453	−0.260072
$865$	0	0
$866$	85.1521	2.89359
$867$	−6.50521	−0.220928
$868$	3.05240	0.103605
$869$	0	0
$870$	0	0
$871$	27.5387	0.933112
$872$	−21.5885	−0.731080
$873$	8.17671	0.276740
$874$	−13.9211	−0.470889
$875$	0	0
$876$	15.6964	0.530333
$877$	4.75891	0.160697	0.0803486	−	0.996767i	$-0.474397\pi$
$877$	4.75891	0.160697	0.0803486	+	0.996767i	$0.474397\pi$
$878$	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	$-0.831887\pi$
$883$	−51.3329	−1.72749	−0.863745	+	0.503929i	$0.831887\pi$
$884$	43.8252	1.47400
$885$	0	0
$886$	10.7359	0.360679
$887$	20.1458	0.676430	0.338215	−	0.941069i	$-0.390177\pi$
$887$	20.1458	0.676430	0.338215	+	0.941069i	$0.390177\pi$
$888$	−8.05395	−0.270273
$889$	−0.122758	−0.00411718
$890$	0	0
$891$	0	0
$892$	−28.4074	−0.951149
$893$	−5.87126	−0.196474
$894$	−34.6819	−1.15994
$895$	0	0
$896$	−5.35498	−0.178897
$897$	−11.7214	−0.391365
$898$	−47.6135	−1.58888
$899$	19.7214	0.657744
$900$	0	0
$901$	65.4762	2.18133
$902$	0	0
$903$	−2.07888	−0.0691810
$904$	−12.6485	−0.420682
$905$	0	0
$906$	−15.5322	−0.516022
$907$	−4.20164	−0.139513	−0.0697566	−	0.997564i	$-0.522222\pi$
$907$	−4.20164	−0.139513	−0.0697566	+	0.997564i	$0.522222\pi$
$908$	72.9415	2.42065
$909$	−2.73588	−0.0907433
$910$	0	0
$911$	−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
$915$	0	0
$916$	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	$0.578811\pi$
$920$	0	0
$921$	−8.50521	−0.280256
$922$	−86.6239	−2.85281
$923$	41.6056	1.36946
$924$	0	0
$925$	0	0
$926$	−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	$0.0867943\pi$
$930$	0	0
$931$	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	$-0.308755\pi$
$937$	34.6090	1.13063	0.565314	+	0.824876i	$0.308755\pi$
$938$	−8.49922	−0.277509
$939$	20.0249	0.653489
$940$	0	0
$941$	−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	0
$946$	0	0
$947$	−17.0480	−0.553985	−0.276992	−	0.960872i	$-0.589338\pi$
$947$	−17.0480	−0.553985	−0.276992	+	0.960872i	$0.589338\pi$
$948$	40.3264	1.30974
$949$	19.6964	0.639373
$950$	0	0
$951$	−23.8088	−0.772052
$952$	−3.44682	−0.111712
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	29.2286	0.946312
$955$	0	0
$956$	33.0893	1.07018
$957$	0	0
$958$	−79.7043	−2.57513
$959$	0.445267	0.0143784
$960$	0	0
$961$	−25.3929	−0.819125
$962$	−39.6585	−1.27864
$963$	9.91259	0.319429
$964$	−19.3290	−0.622545
$965$	0	0
$966$	3.61755	0.116393
$967$	50.2016	1.61438	0.807188	−	0.590294i	$-0.200988\pi$
$967$	50.2016	1.61438	0.807188	+	0.590294i	$0.200988\pi$
$968$	0	0
$969$	−8.96056	−0.287855
$970$	0	0
$971$	−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
$975$	0	0
$976$	25.8856	0.828578
$977$	−19.5841	−0.626551	−0.313275	−	0.949662i	$-0.601426\pi$
$977$	−19.5841	−0.626551	−0.313275	+	0.949662i	$0.601426\pi$
$978$	−0.553177	−0.0176886
$979$	0	0
$980$	0	0
$981$	14.5841	0.465634
$982$	−5.26412	−0.167985
$983$	−13.4513	−0.429028	−0.214514	−	0.976721i	$-0.568817\pi$
$983$	−13.4513	−0.429028	−0.214514	+	0.976721i	$0.568817\pi$
$984$	−9.85420	−0.314141
$985$	0	0
$986$	−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
$995$	0	0
$996$	−10.7359	−0.340179
$997$	−17.8233	−0.564469	−0.282235	−	0.959345i	$-0.591076\pi$
$997$	−17.8233	−0.564469	−0.282235	+	0.959345i	$0.591076\pi$
$998$	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	9075.2.a.ci.1.1		3
5.4	even	2		1815.2.a.l.1.3	✓	3
11.10	odd	2		9075.2.a.ce.1.3		3
15.14	odd	2		5445.2.a.bc.1.1		3
55.54	odd	2		1815.2.a.n.1.1	yes	3
165.164	even	2		5445.2.a.ba.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.l.1.3	✓	3	5.4	even	2
1815.2.a.n.1.1	yes	3	55.54	odd	2
5445.2.a.ba.1.3		3	165.164	even	2
5445.2.a.bc.1.1		3	15.14	odd	2
9075.2.a.ce.1.3		3	11.10	odd	2
9075.2.a.ci.1.1		3	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

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

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