LMFDB - Embedded newform 5445.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
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			1.3
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.16425 q^{2} +2.68397 q^{4} -1.00000 q^{5} -0.480279 q^{7} +1.48028 q^{8} +O(q^{10})$	$q+2.16425 q^{2} +2.68397 q^{4} -1.00000 q^{5} -0.480279 q^{7} +1.48028 q^{8} -2.16425 q^{10} -3.36794 q^{13} -1.03944 q^{14} -2.16425 q^{16} +4.84822 q^{17} +1.84822 q^{19} -2.68397 q^{20} -3.48028 q^{23} +1.00000 q^{25} -7.28905 q^{26} -1.28905 q^{28} -8.32850 q^{29} -2.36794 q^{31} -7.64453 q^{32} +10.4927 q^{34} +0.480279 q^{35} +5.44084 q^{37} +4.00000 q^{38} -1.48028 q^{40} -6.65699 q^{41} -4.32850 q^{43} -7.53219 q^{46} +3.17671 q^{47} -6.76933 q^{49} +2.16425 q^{50} -9.03944 q^{52} -13.5052 q^{53} -0.710947 q^{56} -18.0249 q^{58} -15.0644 q^{59} +11.9606 q^{61} -5.12481 q^{62} -12.2162 q^{64} +3.36794 q^{65} +8.17671 q^{67} +13.0125 q^{68} +1.03944 q^{70} +12.3534 q^{71} -5.84822 q^{73} +11.7753 q^{74} +4.96056 q^{76} +15.0249 q^{79} +2.16425 q^{80} -14.4074 q^{82} -4.00000 q^{83} -4.84822 q^{85} -9.36794 q^{86} -5.67150 q^{89} +1.61755 q^{91} -9.34096 q^{92} +6.87519 q^{94} -1.84822 q^{95} -8.17671 q^{97} -14.6505 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7	$ 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + q^{10} - 4 q^{13} - 12 q^{14} + q^{16} + 4 q^{17} - 5 q^{19} - 5 q^{20} - 6 q^{23} + 3 q^{25} + 2 q^{26} + 20 q^{28} - 10 q^{29} - q^{31} - 11 q^{32} + 9 q^{34} - 3 q^{35} + 3 q^{37} + 12 q^{38} + 10 q^{41} + 2 q^{43} - 9 q^{46} - 16 q^{47} + 8 q^{49} - q^{50} - 36 q^{52} - 26 q^{56} - 18 q^{58} - 18 q^{59} + 27 q^{61} + q^{62} - 20 q^{64} + 4 q^{65} - q^{67} + 21 q^{68} + 12 q^{70} - 14 q^{71} - 7 q^{73} + 32 q^{74} + 6 q^{76} + 9 q^{79} - q^{80} - 46 q^{82} - 12 q^{83} - 4 q^{85} - 22 q^{86} - 32 q^{89} - 34 q^{91} + 5 q^{92} + 37 q^{94} + 5 q^{95} + q^{97} - 57 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + q^10 - 4 * q^13 - 12 * q^14 + q^16 + 4 * q^17 - 5 * q^19 - 5 * q^20 - 6 * q^23 + 3 * q^25 + 2 * q^26 + 20 * q^28 - 10 * q^29 - q^31 - 11 * q^32 + 9 * q^34 - 3 * q^35 + 3 * q^37 + 12 * q^38 + 10 * q^41 + 2 * q^43 - 9 * q^46 - 16 * q^47 + 8 * q^49 - q^50 - 36 * q^52 - 26 * q^56 - 18 * q^58 - 18 * q^59 + 27 * q^61 + q^62 - 20 * q^64 + 4 * q^65 - q^67 + 21 * q^68 + 12 * q^70 - 14 * q^71 - 7 * q^73 + 32 * q^74 + 6 * q^76 + 9 * q^79 - q^80 - 46 * q^82 - 12 * q^83 - 4 * q^85 - 22 * q^86 - 32 * q^89 - 34 * q^91 + 5 * q^92 + 37 * q^94 + 5 * q^95 + 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	$-0.222651\pi$
$2$	2.16425	1.53035	0.765177	+	0.643820i	$0.222651\pi$
$3$	0	0
$3$	0	0
$4$	2.68397	1.34198
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−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
$9$	0	0
$10$	−2.16425	−0.684395
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−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	$-0.299942\pi$
$17$	4.84822	1.17587	0.587933	+	0.808910i	$0.299942\pi$
$18$	0	0
$19$	1.84822	0.424010	0.212005	−	0.977269i	$-0.432001\pi$
$19$	1.84822	0.424010	0.212005	+	0.977269i	$0.432001\pi$
$20$	−2.68397	−0.600154
$21$	0	0
$22$	0	0
$23$	−3.48028	−0.725688	−0.362844	−	0.931850i	$-0.618194\pi$
$23$	−3.48028	−0.725688	−0.362844	+	0.931850i	$0.618194\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.28905	−1.42950
$27$	0	0
$28$	−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
$35$	0.480279	0.0811819
$36$	0	0
$37$	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
$39$	0	0
$40$	−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	$0.674002\pi$
$42$	0	0
$43$	−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	$0.425576\pi$
$48$	0	0
$49$	−6.76933	−0.967047
$50$	2.16425	0.306071
$51$	0	0
$52$	−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	$0.878082\pi$
$54$	0	0
$55$	0	0
$56$	−0.710947	−0.0950042
$57$	0	0
$58$	−18.0249	−2.36679
$59$	−15.0644	−1.96121	−0.980607	−	0.195984i	$-0.937210\pi$
$59$	−15.0644	−1.96121	−0.980607	+	0.195984i	$0.937210\pi$
$60$	0	0
$61$	11.9606	1.53139	0.765696	−	0.643202i	$-0.222395\pi$
$61$	11.9606	1.53139	0.765696	+	0.643202i	$0.222395\pi$
$62$	−5.12481	−0.650851
$63$	0	0
$64$	−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
$69$	0	0
$70$	1.03944	0.124237
$71$	12.3534	1.46608	0.733041	−	0.680184i	$-0.238100\pi$
$71$	12.3534	1.46608	0.733041	+	0.680184i	$0.238100\pi$
$72$	0	0
$73$	−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
$78$	0	0
$79$	15.0249	1.69044	0.845218	−	0.534421i	$-0.179470\pi$
$79$	15.0249	1.69044	0.845218	+	0.534421i	$0.179470\pi$
$80$	2.16425	0.241970
$81$	0	0
$82$	−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	$0.570452\pi$
$84$	0	0
$85$	−4.84822	−0.525863
$86$	−9.36794	−1.01017
$87$	0	0
$88$	0	0
$89$	−5.67150	−0.601178	−0.300589	−	0.953754i	$-0.597183\pi$
$89$	−5.67150	−0.601178	−0.300589	+	0.953754i	$0.597183\pi$
$90$	0	0
$91$	1.61755	0.169565
$92$	−9.34096	−0.973862
$93$	0	0
$94$	6.87519	0.709122
$95$	−1.84822	−0.189623
$96$	0	0
$97$	−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	$0.543462\pi$
$102$	0	0
$103$	−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	0
$106$	−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	$0.659053\pi$
$108$	0	0
$109$	−14.5841	−1.39690	−0.698451	−	0.715657i	$-0.746127\pi$
$109$	−14.5841	−1.39690	−0.698451	+	0.715657i	$0.746127\pi$
$110$	0	0
$111$	0	0
$112$	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	$0.368348\pi$
$114$	0	0
$115$	3.48028	0.324538
$116$	−22.3534	−2.07546
$117$	0	0
$118$	−32.6030	−3.00135
$119$	−2.32850	−0.213453
$120$	0	0
$121$	0	0
$122$	25.8856	2.34357
$123$	0	0
$124$	−6.35547	−0.570738
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	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
$129$	0	0
$130$	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
$135$	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	$0.512610\pi$
$138$	0	0
$139$	8.21616	0.696885	0.348443	−	0.937330i	$-0.386711\pi$
$139$	8.21616	0.696885	0.348443	+	0.937330i	$0.386711\pi$
$140$	1.28905	0.108945
$141$	0	0
$142$	26.7359	2.24362
$143$	0	0
$144$	0	0
$145$	8.32850	0.691644
$146$	−12.6570	−1.04750
$147$	0	0
$148$	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	$-0.405674\pi$
$151$	7.17671	0.584033	0.292016	+	0.956413i	$0.405674\pi$
$152$	2.73588	0.221909
$153$	0	0
$154$	0	0
$155$	2.36794	0.190197
$156$	0	0
$157$	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
$159$	0	0
$160$	7.64453	0.604353
$161$	1.67150	0.131733
$162$	0	0
$163$	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	0
$169$	−1.65699	−0.127461
$170$	−10.4927	−0.804757
$171$	0	0
$172$	−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	$0.383778\pi$
$174$	0	0
$175$	−0.480279	−0.0363057
$176$	0	0
$177$	0	0
$178$	−12.2745	−0.920016
$179$	−9.36794	−0.700193	−0.350096	−	0.936714i	$-0.613851\pi$
$179$	−9.36794	−0.700193	−0.350096	+	0.936714i	$0.613851\pi$
$180$	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
$183$	0	0
$184$	−5.15178	−0.379794
$185$	−5.44084	−0.400018
$186$	0	0
$187$	0	0
$188$	8.52620	0.621837
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−19.2891	−1.39571	−0.697853	−	0.716241i	$-0.745861\pi$
$191$	−19.2891	−1.39571	−0.697853	+	0.716241i	$0.745861\pi$
$192$	0	0
$193$	−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	$-0.442545\pi$
$197$	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
$201$	0	0
$202$	−5.92112	−0.416608
$203$	4.00000	0.280745
$204$	0	0
$205$	6.65699	0.464944
$206$	−39.3929	−2.74463
$207$	0	0
$208$	7.28905	0.505405
$209$	0	0
$210$	0	0
$211$	19.2496	1.32520	0.662599	−	0.748974i	$-0.269453\pi$
$211$	19.2496	1.32520	0.662599	+	0.748974i	$0.269453\pi$
$212$	−36.2476	−2.48949
$213$	0	0
$214$	−21.4533	−1.46652
$215$	4.32850	0.295201
$216$	0	0
$217$	1.13727	0.0772030
$218$	−31.5636	−2.13776
$219$	0	0
$220$	0	0
$221$	−16.3285	−1.09837
$222$	0	0
$223$	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
$225$	0	0
$226$	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	$0.857821\pi$
$228$	0	0
$229$	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	$0.274164\pi$
$234$	0	0
$235$	−3.17671	−0.207226
$236$	−40.4323	−2.63192
$237$	0	0
$238$	−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	$-0.425490\pi$
$241$	7.20164	0.463899	0.231949	+	0.972728i	$0.425490\pi$
$242$	0	0
$243$	0	0
$244$	32.1018	2.05511
$245$	6.76933	0.432477
$246$	0	0
$247$	−6.22468	−0.396067
$248$	−3.50521	−0.222581
$249$	0	0
$250$	−2.16425	−0.136879
$251$	3.36794	0.212582	0.106291	−	0.994335i	$-0.466102\pi$
$251$	3.36794	0.212582	0.106291	+	0.994335i	$0.466102\pi$
$252$	0	0
$253$	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	$0.382334\pi$
$258$	0	0
$259$	−2.61312	−0.162371
$260$	9.03944	0.560602
$261$	0	0
$262$	−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
$267$	0	0
$268$	21.9460	1.34057
$269$	1.47175	0.0897344	0.0448672	−	0.998993i	$-0.485714\pi$
$269$	1.47175	0.0897344	0.0448672	+	0.998993i	$0.485714\pi$
$270$	0	0
$271$	−5.70496	−0.346552	−0.173276	−	0.984873i	$-0.555435\pi$
$271$	−5.70496	−0.346552	−0.173276	+	0.984873i	$0.555435\pi$
$272$	−10.4927	−0.636216
$273$	0	0
$274$	−2.00648	−0.121216
$275$	0	0
$276$	0	0
$277$	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
$279$	0	0
$280$	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	$0.554108\pi$
$282$	0	0
$283$	−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
$288$	0	0
$289$	6.50521	0.382659
$290$	18.0249	1.05846
$291$	0	0
$292$	−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	$0.622460\pi$
$294$	0	0
$295$	15.0644	0.877082
$296$	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
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−11.9606	−0.684860
$306$	0	0
$307$	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
$309$	0	0
$310$	5.12481	0.291069
$311$	6.32850	0.358856	0.179428	−	0.983771i	$-0.442575\pi$
$311$	6.32850	0.358856	0.179428	+	0.983771i	$0.442575\pi$
$312$	0	0
$313$	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	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	$0.266887\pi$
$318$	0	0
$319$	0	0
$320$	12.2162	0.682904
$321$	0	0
$322$	3.61755	0.201598
$323$	8.96056	0.498579
$324$	0	0
$325$	−3.36794	−0.186820
$326$	0.553177	0.0306376
$327$	0	0
$328$	−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
$333$	0	0
$334$	−34.9251	−1.91101
$335$	−8.17671	−0.446742
$336$	0	0
$337$	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
$339$	0	0
$340$	−13.0125	−0.705700
$341$	0	0
$342$	0	0
$343$	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	$-0.132840\pi$
$347$	34.0584	1.82835	0.914175	+	0.405320i	$0.132840\pi$
$348$	0	0
$349$	20.0104	1.07113	0.535567	−	0.844493i	$-0.320098\pi$
$349$	20.0104	1.07113	0.535567	+	0.844493i	$0.320098\pi$
$350$	−1.03944	−0.0555605
$351$	0	0
$352$	0	0
$353$	−30.9770	−1.64874	−0.824369	−	0.566053i	$-0.808470\pi$
$353$	−30.9770	−1.64874	−0.824369	+	0.566053i	$0.808470\pi$
$354$	0	0
$355$	−12.3534	−0.655652
$356$	−15.2221	−0.806772
$357$	0	0
$358$	−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
$365$	5.84822	0.306110
$366$	0	0
$367$	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
$369$	0	0
$370$	−11.7753	−0.612170
$371$	6.48627	0.336750
$372$	0	0
$373$	−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
$378$	0	0
$379$	−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	0
$382$	−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	$0.464931\pi$
$384$	0	0
$385$	0	0
$386$	−13.5387	−0.689100
$387$	0	0
$388$	−21.9460	−1.11414
$389$	−35.4178	−1.79575	−0.897877	−	0.440247i	$-0.854891\pi$
$389$	−35.4178	−1.79575	−0.897877	+	0.440247i	$0.854891\pi$
$390$	0	0
$391$	−16.8731	−0.853312
$392$	−10.0205	−0.506112
$393$	0	0
$394$	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	0
$400$	−2.16425	−0.108212
$401$	3.08930	0.154272	0.0771362	−	0.997021i	$-0.475422\pi$
$401$	3.08930	0.154272	0.0771362	+	0.997021i	$0.475422\pi$
$402$	0	0
$403$	7.97507	0.397267
$404$	−7.34301	−0.365328
$405$	0	0
$406$	8.65699	0.429639
$407$	0	0
$408$	0	0
$409$	24.9211	1.23227	0.616135	−	0.787641i	$-0.288698\pi$
$409$	24.9211	1.23227	0.616135	+	0.787641i	$0.288698\pi$
$410$	14.4074	0.711530
$411$	0	0
$412$	−48.8526	−2.40680
$413$	7.23510	0.356016
$414$	0	0
$415$	4.00000	0.196352
$416$	25.7463	1.26232
$417$	0	0
$418$	0	0
$419$	4.22468	0.206389	0.103195	−	0.994661i	$-0.467094\pi$
$419$	4.22468	0.206389	0.103195	+	0.994661i	$0.467094\pi$
$420$	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
$423$	0	0
$424$	−19.9915	−0.970872
$425$	4.84822	0.235173
$426$	0	0
$427$	−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	$0.576604\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	−39.1433	−1.87462
$437$	−6.43231	−0.307699
$438$	0	0
$439$	3.71095	0.177114	0.0885569	−	0.996071i	$-0.471774\pi$
$439$	3.71095	0.177114	0.0885569	+	0.996071i	$0.471774\pi$
$440$	0	0
$441$	0	0
$442$	−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	$0.537598\pi$
$444$	0	0
$445$	5.67150	0.268855
$446$	22.9066	1.08466
$447$	0	0
$448$	5.86716	0.277197
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	0	0
$452$	22.9336	1.07870
$453$	0	0
$454$	−58.8171	−2.76043
$455$	−1.61755	−0.0758319
$456$	0	0
$457$	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
$459$	0	0
$460$	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
$465$	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	$0.451212\pi$
$468$	0	0
$469$	−3.92710	−0.181337
$470$	−6.87519	−0.317129
$471$	0	0
$472$	−22.2995	−1.02642
$473$	0	0
$474$	0	0
$475$	1.84822	0.0848020
$476$	−6.24961	−0.286450
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	8.17671	0.371285
$486$	0	0
$487$	−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	0
$490$	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	$0.482521\pi$
$492$	0	0
$493$	−40.3784	−1.81855
$494$	−13.4718	−0.606123
$495$	0	0
$496$	5.12481	0.230111
$497$	−5.93309	−0.266135
$498$	0	0
$499$	−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
$501$	0	0
$502$	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	0
$505$	2.73588	0.121745
$506$	0	0
$507$	0	0
$508$	0.686016	0.0304371
$509$	−17.2102	−0.762827	−0.381414	−	0.924404i	$-0.624563\pi$
$509$	−17.2102	−0.762827	−0.381414	+	0.924404i	$0.624563\pi$
$510$	0	0
$511$	2.80877	0.124253
$512$	22.9520	1.01435
$513$	0	0
$514$	25.0709	1.10583
$515$	18.2016	0.802060
$516$	0	0
$517$	0	0
$518$	−5.65544	−0.248486
$519$	0	0
$520$	4.98549	0.218628
$521$	−17.4348	−0.763835	−0.381917	−	0.924196i	$-0.624736\pi$
$521$	−17.4348	−0.763835	−0.381917	+	0.924196i	$0.624736\pi$
$522$	0	0
$523$	−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
$530$	29.2286	1.26961
$531$	0	0
$532$	−2.38245	−0.103292
$533$	22.4203	0.971133
$534$	0	0
$535$	9.91259	0.428559
$536$	12.1038	0.522805
$537$	0	0
$538$	3.18524	0.137325
$539$	0	0
$540$	0	0
$541$	−31.9710	−1.37454	−0.687270	−	0.726402i	$-0.741191\pi$
$541$	−31.9710	−1.37454	−0.687270	+	0.726402i	$0.741191\pi$
$542$	−12.3469	−0.530347
$543$	0	0
$544$	−37.0623	−1.58903
$545$	14.5841	0.624714
$546$	0	0
$547$	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
$549$	0	0
$550$	0	0
$551$	−15.3929	−0.655758
$552$	0	0
$553$	−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	$-0.403001\pi$
$557$	14.1622	0.600072	0.300036	+	0.953928i	$0.403001\pi$
$558$	0	0
$559$	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	$0.509873\pi$
$564$	0	0
$565$	−8.54465	−0.359476
$566$	−47.8791	−2.01251
$567$	0	0
$568$	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	$-0.702699\pi$
$571$	−28.4178	−1.18925	−0.594624	+	0.804004i	$0.702699\pi$
$572$	0	0
$573$	0	0
$574$	6.91956	0.288817
$575$	−3.48028	−0.145138
$576$	0	0
$577$	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
$579$	0	0
$580$	22.3534	0.928175
$581$	1.92112	0.0797013
$582$	0	0
$583$	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	$0.687747\pi$
$588$	0	0
$589$	−4.37646	−0.180329
$590$	32.6030	1.34225
$591$	0	0
$592$	−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
$597$	0	0
$598$	25.3679	1.03737
$599$	32.0918	1.31124	0.655619	−	0.755092i	$-0.272408\pi$
$599$	32.0918	1.31124	0.655619	+	0.755092i	$0.272408\pi$
$600$	0	0
$601$	7.39885	0.301806	0.150903	−	0.988549i	$-0.451782\pi$
$601$	7.39885	0.301806	0.150903	+	0.988549i	$0.451782\pi$
$602$	4.49922	0.183375
$603$	0	0
$604$	19.2621	0.783763
$605$	0	0
$606$	0	0
$607$	−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
$609$	0	0
$610$	−25.8856	−1.04808
$611$	−10.6990	−0.432834
$612$	0	0
$613$	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	$0.680665\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	13.6964	0.549177
$623$	2.72390	0.109131
$624$	0	0
$625$	1.00000	0.0400000
$626$	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
$633$	0	0
$634$	51.5281	2.04644
$635$	−0.255598	−0.0101431
$636$	0	0
$637$	22.7987	0.903317
$638$	0	0
$639$	0	0
$640$	11.1497	0.440732
$641$	−36.4572	−1.43997	−0.719987	−	0.693987i	$-0.755852\pi$
$641$	−36.4572	−1.43997	−0.719987	+	0.693987i	$0.755852\pi$
$642$	0	0
$643$	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
$645$	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	$0.378035\pi$
$648$	0	0
$649$	0	0
$650$	−7.28905	−0.285900
$651$	0	0
$652$	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	$0.471760\pi$
$654$	0	0
$655$	0.735877	0.0287531
$656$	14.4074	0.562514
$657$	0	0
$658$	−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
$663$	0	0
$664$	−5.92112	−0.229784
$665$	0.887659	0.0344220
$666$	0	0
$667$	28.9855	1.12232
$668$	−43.3119	−1.67579
$669$	0	0
$670$	−17.6964	−0.683673
$671$	0	0
$672$	0	0
$673$	−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	$-0.349709\pi$
$677$	23.6674	0.909612	0.454806	+	0.890590i	$0.349709\pi$
$678$	0	0
$679$	3.92710	0.150708
$680$	−7.17671	−0.275214
$681$	0	0
$682$	0	0
$683$	−17.7753	−0.680154	−0.340077	−	0.940398i	$-0.610453\pi$
$683$	−17.7753	−0.680154	−0.340077	+	0.940398i	$0.610453\pi$
$684$	0	0
$685$	0.927102	0.0354227
$686$	14.3124	0.546451
$687$	0	0
$688$	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
$695$	−8.21616	−0.311657
$696$	0	0
$697$	−32.2745	−1.22249
$698$	43.3075	1.63921
$699$	0	0
$700$	−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	$0.522563\pi$
$702$	0	0
$703$	10.0558	0.379263
$704$	0	0
$705$	0	0
$706$	−67.0418	−2.52315
$707$	1.31398	0.0494174
$708$	0	0
$709$	−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
$711$	0	0
$712$	−8.39541	−0.314631
$713$	8.24109	0.308631
$714$	0	0
$715$	0	0
$716$	−25.1433	−0.939648
$717$	0	0
$718$	64.5491	2.40895
$719$	15.8252	0.590180	0.295090	−	0.955470i	$-0.404650\pi$
$719$	15.8252	0.590180	0.295090	+	0.955470i	$0.404650\pi$
$720$	0	0
$721$	8.74186	0.325564
$722$	−33.7278	−1.25522
$723$	0	0
$724$	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
$729$	0	0
$730$	12.6570	0.468456
$731$	−20.9855	−0.776176
$732$	0	0
$733$	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
$738$	0	0
$739$	18.7214	0.688677	0.344338	−	0.938846i	$-0.388103\pi$
$739$	18.7214	0.688677	0.344338	+	0.938846i	$0.388103\pi$
$740$	−14.6030	−0.536818
$741$	0	0
$742$	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	$0.455721\pi$
$744$	0	0
$745$	16.0249	0.587108
$746$	−66.6778	−2.44125
$747$	0	0
$748$	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
$753$	0	0
$754$	60.7069	2.21081
$755$	−7.17671	−0.261187
$756$	0	0
$757$	−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	0
$763$	7.00443	0.253577
$764$	−51.7712	−1.87302
$765$	0	0
$766$	9.31398	0.336528
$767$	50.7359	1.83197
$768$	0	0
$769$	−24.7463	−0.892374	−0.446187	−	0.894940i	$-0.647218\pi$
$769$	−24.7463	−0.892374	−0.446187	+	0.894940i	$0.647218\pi$
$770$	0	0
$771$	0	0
$772$	−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	$0.596463\pi$
$774$	0	0
$775$	−2.36794	−0.0850589
$776$	−12.1038	−0.434502
$777$	0	0
$778$	−76.6529	−2.74814
$779$	−12.3036	−0.440821
$780$	0	0
$781$	0	0
$782$	−36.5177	−1.30587
$783$	0	0
$784$	14.6505	0.523233
$785$	−2.25560	−0.0805057
$786$	0	0
$787$	−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
$789$	0	0
$790$	−32.5177	−1.15693
$791$	−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	$0.652958\pi$
$798$	0	0
$799$	15.4014	0.544862
$800$	−7.64453	−0.270275
$801$	0	0
$802$	6.68602	0.236091
$803$	0	0
$804$	0	0
$805$	−1.67150	−0.0589128
$806$	17.2600	0.607959
$807$	0	0
$808$	−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	$-0.738333\pi$
$811$	−38.7712	−1.36144	−0.680721	+	0.732543i	$0.738333\pi$
$812$	10.7359	0.376755
$813$	0	0
$814$	0	0
$815$	−0.255598	−0.00895320
$816$	0	0
$817$	−8.00000	−0.279885
$818$	53.9355	1.88581
$819$	0	0
$820$	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	$0.381850\pi$
$822$	0	0
$823$	−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	$0.412853\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	41.1433	1.42639
$833$	−32.8192	−1.13712
$834$	0	0
$835$	16.1373	0.558453
$836$	0	0
$837$	0	0
$838$	9.14326	0.315849
$839$	−6.89365	−0.237995	−0.118998	−	0.992895i	$-0.537968\pi$
$839$	−6.89365	−0.237995	−0.118998	+	0.992895i	$0.537968\pi$
$840$	0	0
$841$	40.3638	1.39186
$842$	−47.0289	−1.62072
$843$	0	0
$844$	51.6654	1.77840
$845$	1.65699	0.0570022
$846$	0	0
$847$	0	0
$848$	29.2286	1.00371
$849$	0	0
$850$	10.4927	0.359898
$851$	−18.9356	−0.649105
$852$	0	0
$853$	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	$-0.583604\pi$
$857$	−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
$861$	0	0
$862$	−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	$0.414735\pi$
$864$	0	0
$865$	−9.39287	−0.319367
$866$	85.1521	2.89359
$867$	0	0
$868$	3.05240	0.103605
$869$	0	0
$870$	0	0
$871$	−27.5387	−0.933112
$872$	−21.5885	−0.731080
$873$	0	0
$874$	−13.9211	−0.470889
$875$	0.480279	0.0162364
$876$	0	0
$877$	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
$879$	0	0
$880$	0	0
$881$	28.0997	0.946704	0.473352	−	0.880873i	$-0.343044\pi$
$881$	28.0997	0.946704	0.473352	+	0.880873i	$0.343044\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	−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	$0.609823\pi$
$888$	0	0
$889$	−0.122758	−0.00411718
$890$	12.2745	0.411444
$891$	0	0
$892$	28.4074	0.951149
$893$	5.87126	0.196474
$894$	0	0
$895$	9.36794	0.313136
$896$	5.35498	0.178897
$897$	0	0
$898$	−47.6135	−1.58888
$899$	19.7214	0.657744
$900$	0	0
$901$	−65.4762	−2.18133
$902$	0	0
$903$	0	0
$904$	12.6485	0.420682
$905$	−17.7693	−0.590673
$906$	0	0
$907$	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
$909$	0	0
$910$	−3.50078	−0.116050
$911$	4.05395	0.134314	0.0671568	−	0.997742i	$-0.478607\pi$
$911$	4.05395	0.134314	0.0671568	+	0.997742i	$0.478607\pi$
$912$	0	0
$913$	0	0
$914$	27.5636	0.911723
$915$	0	0
$916$	56.0898	1.85326
$917$	0.353426	0.0116712
$918$	0	0
$919$	14.8586	0.490141	0.245071	−	0.969505i	$-0.421189\pi$
$919$	14.8586	0.490141	0.245071	+	0.969505i	$0.421189\pi$
$920$	5.15178	0.169849
$921$	0	0
$922$	86.6239	2.85281
$923$	−41.6056	−1.36946
$924$	0	0
$925$	5.44084	0.178894
$926$	−55.2720	−1.81635
$927$	0	0
$928$	63.6674	2.08999
$929$	−58.7069	−1.92611	−0.963055	−	0.269306i	$-0.913206\pi$
$929$	−58.7069	−1.92611	−0.963055	+	0.269306i	$0.913206\pi$
$930$	0	0
$931$	−12.5112	−0.410038
$932$	53.3779	1.74845
$933$	0	0
$934$	14.2810	0.467289
$935$	0	0
$936$	0	0
$937$	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
$939$	0	0
$940$	−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	$0.555858\pi$
$942$	0	0
$943$	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	$0.589338\pi$
$948$	0	0
$949$	19.6964	0.639373
$950$	4.00000	0.129777
$951$	0	0
$952$	−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	$0.199560\pi$
$954$	0	0
$955$	19.2891	0.624179
$956$	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
$963$	0	0
$964$	19.3290	0.622545
$965$	6.25560	0.201375
$966$	0	0
$967$	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
$969$	0	0
$970$	17.6964	0.568198
$971$	52.0748	1.67116	0.835580	−	0.549369i	$-0.185132\pi$
$971$	52.0748	1.67116	0.835580	+	0.549369i	$0.185132\pi$
$972$	0	0
$973$	−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	0
$979$	0	0
$980$	18.1687	0.580377
$981$	0	0
$982$	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	$0.568817\pi$
$984$	0	0
$985$	−5.03944	−0.160570
$986$	−87.3888	−2.78303
$987$	0	0
$988$	−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
$993$	0	0
$994$	−12.8407	−0.407281
$995$	14.5926	0.462617
$996$	0	0
$997$	−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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.l.1.3	✓	3	33.32	even	2
1815.2.a.n.1.1	yes	3	3.2	odd	2
5445.2.a.ba.1.3		3	1.1	even	1	trivial
5445.2.a.bc.1.1		3	11.10	odd	2
9075.2.a.ce.1.3		3	15.14	odd	2
9075.2.a.ci.1.1		3	165.164	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

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

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