LMFDB - Embedded newform 2009.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278 q^{3} -2.00000 q^{4} +2.30278 q^{5} -1.30278 q^{9} +O(q^{10})$	$q+1.30278 q^{3} -2.00000 q^{4} +2.30278 q^{5} -1.30278 q^{9} -0.697224 q^{11} -2.60555 q^{12} -1.30278 q^{13} +3.00000 q^{15} +4.00000 q^{16} -6.90833 q^{17} -3.60555 q^{19} -4.60555 q^{20} +1.60555 q^{23} +0.302776 q^{25} -5.60555 q^{27} -1.60555 q^{29} -1.30278 q^{31} -0.908327 q^{33} +2.60555 q^{36} -7.21110 q^{37} -1.69722 q^{39} +1.00000 q^{41} -2.39445 q^{43} +1.39445 q^{44} -3.00000 q^{45} +9.00000 q^{47} +5.21110 q^{48} -9.00000 q^{51} +2.60555 q^{52} -0.908327 q^{53} -1.60555 q^{55} -4.69722 q^{57} -7.60555 q^{59} -6.00000 q^{60} +7.21110 q^{61} -8.00000 q^{64} -3.00000 q^{65} +0.605551 q^{67} +13.8167 q^{68} +2.09167 q^{69} -10.8167 q^{71} -5.69722 q^{73} +0.394449 q^{75} +7.21110 q^{76} +11.4222 q^{79} +9.21110 q^{80} -3.39445 q^{81} -4.39445 q^{83} -15.9083 q^{85} -2.09167 q^{87} -7.60555 q^{89} -3.21110 q^{92} -1.69722 q^{93} -8.30278 q^{95} -5.69722 q^{97} +0.908327 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9}+O(q^{10}) $ 2 * q - q^3 - 4 * q^4 + q^5 + q^9	$ 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9} - 5 q^{11} + 2 q^{12} + q^{13} + 6 q^{15} + 8 q^{16} - 3 q^{17} - 2 q^{20} - 4 q^{23} - 3 q^{25} - 4 q^{27} + 4 q^{29} + q^{31} + 9 q^{33} - 2 q^{36} - 7 q^{39} + 2 q^{41} - 12 q^{43} + 10 q^{44} - 6 q^{45} + 18 q^{47} - 4 q^{48} - 18 q^{51} - 2 q^{52} + 9 q^{53} + 4 q^{55} - 13 q^{57} - 8 q^{59} - 12 q^{60} - 16 q^{64} - 6 q^{65} - 6 q^{67} + 6 q^{68} + 15 q^{69} - 15 q^{73} + 8 q^{75} - 6 q^{79} + 4 q^{80} - 14 q^{81} - 16 q^{83} - 21 q^{85} - 15 q^{87} - 8 q^{89} + 8 q^{92} - 7 q^{93} - 13 q^{95} - 15 q^{97} - 9 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^4 + q^5 + q^9 - 5 * q^11 + 2 * q^12 + q^13 + 6 * q^15 + 8 * q^16 - 3 * q^17 - 2 * q^20 - 4 * q^23 - 3 * q^25 - 4 * q^27 + 4 * q^29 + q^31 + 9 * q^33 - 2 * q^36 - 7 * q^39 + 2 * q^41 - 12 * q^43 + 10 * q^44 - 6 * q^45 + 18 * q^47 - 4 * q^48 - 18 * q^51 - 2 * q^52 + 9 * q^53 + 4 * q^55 - 13 * q^57 - 8 * q^59 - 12 * q^60 - 16 * q^64 - 6 * q^65 - 6 * q^67 + 6 * q^68 + 15 * q^69 - 15 * q^73 + 8 * q^75 - 6 * q^79 + 4 * q^80 - 14 * q^81 - 16 * q^83 - 21 * q^85 - 15 * q^87 - 8 * q^89 + 8 * q^92 - 7 * q^93 - 13 * q^95 - 15 * q^97 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.30278	0.752158	0.376079	−	0.926588i	$-0.377272\pi$
$3$	1.30278	0.752158	0.376079	+	0.926588i	$0.377272\pi$
$4$	−2.00000	−1.00000
$5$	2.30278	1.02983	0.514916	−	0.857240i	$-0.327823\pi$
$5$	2.30278	1.02983	0.514916	+	0.857240i	$0.327823\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	−0.697224	−0.210221	−0.105111	−	0.994461i	$-0.533520\pi$
$11$	−0.697224	−0.210221	−0.105111	+	0.994461i	$0.533520\pi$
$12$	−2.60555	−0.752158
$13$	−1.30278	−0.361325	−0.180662	−	0.983545i	$-0.557824\pi$
$13$	−1.30278	−0.361325	−0.180662	+	0.983545i	$0.557824\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	4.00000	1.00000
$17$	−6.90833	−1.67552	−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833	−1.67552	−0.837758	+	0.546042i	$0.816134\pi$
$18$	0	0
$19$	−3.60555	−0.827170	−0.413585	−	0.910465i	$-0.635724\pi$
$19$	−3.60555	−0.827170	−0.413585	+	0.910465i	$0.635724\pi$
$20$	−4.60555	−1.02983
$21$	0	0
$22$	0	0
$23$	1.60555	0.334781	0.167390	−	0.985891i	$-0.446466\pi$
$23$	1.60555	0.334781	0.167390	+	0.985891i	$0.446466\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	−5.60555	−1.07879
$28$	0	0
$29$	−1.60555	−0.298143	−0.149072	−	0.988826i	$-0.547629\pi$
$29$	−1.60555	−0.298143	−0.149072	+	0.988826i	$0.547629\pi$
$30$	0	0
$31$	−1.30278	−0.233985	−0.116993	−	0.993133i	$-0.537325\pi$
$31$	−1.30278	−0.233985	−0.116993	+	0.993133i	$0.537325\pi$
$32$	0	0
$33$	−0.908327	−0.158119
$34$	0	0
$35$	0	0
$36$	2.60555	0.434259
$37$	−7.21110	−1.18550	−0.592749	−	0.805387i	$-0.701957\pi$
$37$	−7.21110	−1.18550	−0.592749	+	0.805387i	$0.701957\pi$
$38$	0	0
$39$	−1.69722	−0.271773
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−2.39445	−0.365150	−0.182575	−	0.983192i	$-0.558443\pi$
$43$	−2.39445	−0.365150	−0.182575	+	0.983192i	$0.558443\pi$
$44$	1.39445	0.210221
$45$	−3.00000	−0.447214
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	5.21110	0.752158
$49$	0	0
$50$	0	0
$51$	−9.00000	−1.26025
$52$	2.60555	0.361325
$53$	−0.908327	−0.124768	−0.0623841	−	0.998052i	$-0.519870\pi$
$53$	−0.908327	−0.124768	−0.0623841	+	0.998052i	$0.519870\pi$
$54$	0	0
$55$	−1.60555	−0.216492
$56$	0	0
$57$	−4.69722	−0.622163
$58$	0	0
$59$	−7.60555	−0.990158	−0.495079	−	0.868848i	$-0.664861\pi$
$59$	−7.60555	−0.990158	−0.495079	+	0.868848i	$0.664861\pi$
$60$	−6.00000	−0.774597
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−3.00000	−0.372104
$66$	0	0
$67$	0.605551	0.0739799	0.0369899	−	0.999316i	$-0.488223\pi$
$67$	0.605551	0.0739799	0.0369899	+	0.999316i	$0.488223\pi$
$68$	13.8167	1.67552
$69$	2.09167	0.251808
$70$	0	0
$71$	−10.8167	−1.28370	−0.641850	−	0.766830i	$-0.721833\pi$
$71$	−10.8167	−1.28370	−0.641850	+	0.766830i	$0.721833\pi$
$72$	0	0
$73$	−5.69722	−0.666810	−0.333405	−	0.942784i	$-0.608198\pi$
$73$	−5.69722	−0.666810	−0.333405	+	0.942784i	$0.608198\pi$
$74$	0	0
$75$	0.394449	0.0455470
$76$	7.21110	0.827170
$77$	0	0
$78$	0	0
$79$	11.4222	1.28510	0.642549	−	0.766244i	$-0.277877\pi$
$79$	11.4222	1.28510	0.642549	+	0.766244i	$0.277877\pi$
$80$	9.21110	1.02983
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−4.39445	−0.482353	−0.241177	−	0.970481i	$-0.577533\pi$
$83$	−4.39445	−0.482353	−0.241177	+	0.970481i	$0.577533\pi$
$84$	0	0
$85$	−15.9083	−1.72550
$86$	0	0
$87$	−2.09167	−0.224251
$88$	0	0
$89$	−7.60555	−0.806187	−0.403093	−	0.915159i	$-0.632065\pi$
$89$	−7.60555	−0.806187	−0.403093	+	0.915159i	$0.632065\pi$
$90$	0	0
$91$	0	0
$92$	−3.21110	−0.334781
$93$	−1.69722	−0.175994
$94$	0	0
$95$	−8.30278	−0.851847
$96$	0	0
$97$	−5.69722	−0.578465	−0.289233	−	0.957259i	$-0.593400\pi$
$97$	−5.69722	−0.578465	−0.289233	+	0.957259i	$0.593400\pi$
$98$	0	0
$99$	0.908327	0.0912903
$100$	−0.605551	−0.0605551
$101$	4.39445	0.437264	0.218632	−	0.975807i	$-0.429841\pi$
$101$	4.39445	0.437264	0.218632	+	0.975807i	$0.429841\pi$
$102$	0	0
$103$	16.2111	1.59733	0.798664	−	0.601778i	$-0.205541\pi$
$103$	16.2111	1.59733	0.798664	+	0.601778i	$0.205541\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.90833	0.377832	0.188916	−	0.981993i	$-0.439503\pi$
$107$	3.90833	0.377832	0.188916	+	0.981993i	$0.439503\pi$
$108$	11.2111	1.07879
$109$	2.69722	0.258347	0.129174	−	0.991622i	$-0.458768\pi$
$109$	2.69722	0.258347	0.129174	+	0.991622i	$0.458768\pi$
$110$	0	0
$111$	−9.39445	−0.891682
$112$	0	0
$113$	13.8167	1.29976	0.649881	−	0.760036i	$-0.274819\pi$
$113$	13.8167	1.29976	0.649881	+	0.760036i	$0.274819\pi$
$114$	0	0
$115$	3.69722	0.344768
$116$	3.21110	0.298143
$117$	1.69722	0.156908
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.5139	−0.955807
$122$	0	0
$123$	1.30278	0.117467
$124$	2.60555	0.233985
$125$	−10.8167	−0.967471
$126$	0	0
$127$	−18.7250	−1.66157	−0.830787	−	0.556591i	$-0.812109\pi$
$127$	−18.7250	−1.66157	−0.830787	+	0.556591i	$0.812109\pi$
$128$	0	0
$129$	−3.11943	−0.274650
$130$	0	0
$131$	−5.30278	−0.463306	−0.231653	−	0.972799i	$-0.574413\pi$
$131$	−5.30278	−0.463306	−0.231653	+	0.972799i	$0.574413\pi$
$132$	1.81665	0.158119
$133$	0	0
$134$	0	0
$135$	−12.9083	−1.11097
$136$	0	0
$137$	−22.8167	−1.94936	−0.974679	−	0.223608i	$-0.928216\pi$
$137$	−22.8167	−1.94936	−0.974679	+	0.223608i	$0.928216\pi$
$138$	0	0
$139$	−5.90833	−0.501138	−0.250569	−	0.968099i	$-0.580618\pi$
$139$	−5.90833	−0.501138	−0.250569	+	0.968099i	$0.580618\pi$
$140$	0	0
$141$	11.7250	0.987422
$142$	0	0
$143$	0.908327	0.0759581
$144$	−5.21110	−0.434259
$145$	−3.69722	−0.307038
$146$	0	0
$147$	0	0
$148$	14.4222	1.18550
$149$	19.8167	1.62344	0.811722	−	0.584044i	$-0.198531\pi$
$149$	19.8167	1.62344	0.811722	+	0.584044i	$0.198531\pi$
$150$	0	0
$151$	−14.1194	−1.14902	−0.574511	−	0.818497i	$-0.694808\pi$
$151$	−14.1194	−1.14902	−0.574511	+	0.818497i	$0.694808\pi$
$152$	0	0
$153$	9.00000	0.727607
$154$	0	0
$155$	−3.00000	−0.240966
$156$	3.39445	0.271773
$157$	16.2111	1.29379	0.646893	−	0.762580i	$-0.276068\pi$
$157$	16.2111	1.29379	0.646893	+	0.762580i	$0.276068\pi$
$158$	0	0
$159$	−1.18335	−0.0938455
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.7250	1.31000	0.655001	−	0.755628i	$-0.272668\pi$
$163$	16.7250	1.31000	0.655001	+	0.755628i	$0.272668\pi$
$164$	−2.00000	−0.156174
$165$	−2.09167	−0.162837
$166$	0	0
$167$	16.8167	1.30131	0.650656	−	0.759373i	$-0.274494\pi$
$167$	16.8167	1.30131	0.650656	+	0.759373i	$0.274494\pi$
$168$	0	0
$169$	−11.3028	−0.869444
$170$	0	0
$171$	4.69722	0.359206
$172$	4.78890	0.365150
$173$	−2.51388	−0.191127	−0.0955633	−	0.995423i	$-0.530465\pi$
$173$	−2.51388	−0.191127	−0.0955633	+	0.995423i	$0.530465\pi$
$174$	0	0
$175$	0	0
$176$	−2.78890	−0.210221
$177$	−9.90833	−0.744755
$178$	0	0
$179$	−17.3028	−1.29327	−0.646635	−	0.762799i	$-0.723824\pi$
$179$	−17.3028	−1.29327	−0.646635	+	0.762799i	$0.723824\pi$
$180$	6.00000	0.447214
$181$	15.0278	1.11700	0.558502	−	0.829503i	$-0.311376\pi$
$181$	15.0278	1.11700	0.558502	+	0.829503i	$0.311376\pi$
$182$	0	0
$183$	9.39445	0.694458
$184$	0	0
$185$	−16.6056	−1.22086
$186$	0	0
$187$	4.81665	0.352229
$188$	−18.0000	−1.31278
$189$	0	0
$190$	0	0
$191$	−15.6972	−1.13581	−0.567906	−	0.823094i	$-0.692246\pi$
$191$	−15.6972	−1.13581	−0.567906	+	0.823094i	$0.692246\pi$
$192$	−10.4222	−0.752158
$193$	−14.6056	−1.05133	−0.525665	−	0.850691i	$-0.676184\pi$
$193$	−14.6056	−1.05133	−0.525665	+	0.850691i	$0.676184\pi$
$194$	0	0
$195$	−3.90833	−0.279881
$196$	0	0
$197$	−6.90833	−0.492198	−0.246099	−	0.969245i	$-0.579149\pi$
$197$	−6.90833	−0.492198	−0.246099	+	0.969245i	$0.579149\pi$
$198$	0	0
$199$	3.78890	0.268588	0.134294	−	0.990942i	$-0.457123\pi$
$199$	3.78890	0.268588	0.134294	+	0.990942i	$0.457123\pi$
$200$	0	0
$201$	0.788897	0.0556445
$202$	0	0
$203$	0	0
$204$	18.0000	1.26025
$205$	2.30278	0.160833
$206$	0	0
$207$	−2.09167	−0.145381
$208$	−5.21110	−0.361325
$209$	2.51388	0.173889
$210$	0	0
$211$	15.1194	1.04086	0.520432	−	0.853903i	$-0.325771\pi$
$211$	15.1194	1.04086	0.520432	+	0.853903i	$0.325771\pi$
$212$	1.81665	0.124768
$213$	−14.0917	−0.965546
$214$	0	0
$215$	−5.51388	−0.376043
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.42221	−0.501546
$220$	3.21110	0.216492
$221$	9.00000	0.605406
$222$	0	0
$223$	9.51388	0.637096	0.318548	−	0.947907i	$-0.396805\pi$
$223$	9.51388	0.637096	0.318548	+	0.947907i	$0.396805\pi$
$224$	0	0
$225$	−0.394449	−0.0262966
$226$	0	0
$227$	−27.2111	−1.80606	−0.903032	−	0.429573i	$-0.858664\pi$
$227$	−27.2111	−1.80606	−0.903032	+	0.429573i	$0.858664\pi$
$228$	9.39445	0.622163
$229$	−28.7250	−1.89820	−0.949100	−	0.314975i	$-0.898004\pi$
$229$	−28.7250	−1.89820	−0.949100	+	0.314975i	$0.898004\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.3305	−1.06985	−0.534924	−	0.844900i	$-0.679660\pi$
$233$	−16.3305	−1.06985	−0.534924	+	0.844900i	$0.679660\pi$
$234$	0	0
$235$	20.7250	1.35195
$236$	15.2111	0.990158
$237$	14.8806	0.966597
$238$	0	0
$239$	1.18335	0.0765443	0.0382722	−	0.999267i	$-0.487815\pi$
$239$	1.18335	0.0765443	0.0382722	+	0.999267i	$0.487815\pi$
$240$	12.0000	0.774597
$241$	17.8167	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$241$	17.8167	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$242$	0	0
$243$	12.3944	0.795104
$244$	−14.4222	−0.923287
$245$	0	0
$246$	0	0
$247$	4.69722	0.298877
$248$	0	0
$249$	−5.72498	−0.362806
$250$	0	0
$251$	11.0917	0.700100	0.350050	−	0.936731i	$-0.386165\pi$
$251$	11.0917	0.700100	0.350050	+	0.936731i	$0.386165\pi$
$252$	0	0
$253$	−1.11943	−0.0703779
$254$	0	0
$255$	−20.7250	−1.29785
$256$	16.0000	1.00000
$257$	−5.09167	−0.317610	−0.158805	−	0.987310i	$-0.550764\pi$
$257$	−5.09167	−0.317610	−0.158805	+	0.987310i	$0.550764\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	2.09167	0.129471
$262$	0	0
$263$	12.6972	0.782944	0.391472	−	0.920190i	$-0.371966\pi$
$263$	12.6972	0.782944	0.391472	+	0.920190i	$0.371966\pi$
$264$	0	0
$265$	−2.09167	−0.128490
$266$	0	0
$267$	−9.90833	−0.606380
$268$	−1.21110	−0.0739799
$269$	−3.69722	−0.225424	−0.112712	−	0.993628i	$-0.535954\pi$
$269$	−3.69722	−0.225424	−0.112712	+	0.993628i	$0.535954\pi$
$270$	0	0
$271$	−3.18335	−0.193375	−0.0966873	−	0.995315i	$-0.530825\pi$
$271$	−3.18335	−0.193375	−0.0966873	+	0.995315i	$0.530825\pi$
$272$	−27.6333	−1.67552
$273$	0	0
$274$	0	0
$275$	−0.211103	−0.0127300
$276$	−4.18335	−0.251808
$277$	25.7250	1.54566	0.772832	−	0.634610i	$-0.218839\pi$
$277$	25.7250	1.54566	0.772832	+	0.634610i	$0.218839\pi$
$278$	0	0
$279$	1.69722	0.101610
$280$	0	0
$281$	11.5139	0.686860	0.343430	−	0.939178i	$-0.388411\pi$
$281$	11.5139	0.686860	0.343430	+	0.939178i	$0.388411\pi$
$282$	0	0
$283$	19.4222	1.15453	0.577265	−	0.816557i	$-0.304120\pi$
$283$	19.4222	1.15453	0.577265	+	0.816557i	$0.304120\pi$
$284$	21.6333	1.28370
$285$	−10.8167	−0.640723
$286$	0	0
$287$	0	0
$288$	0	0
$289$	30.7250	1.80735
$290$	0	0
$291$	−7.42221	−0.435097
$292$	11.3944	0.666810
$293$	30.2111	1.76495	0.882476	−	0.470358i	$-0.155875\pi$
$293$	30.2111	1.76495	0.882476	+	0.470358i	$0.155875\pi$
$294$	0	0
$295$	−17.5139	−1.01970
$296$	0	0
$297$	3.90833	0.226784
$298$	0	0
$299$	−2.09167	−0.120965
$300$	−0.788897	−0.0455470
$301$	0	0
$302$	0	0
$303$	5.72498	0.328892
$304$	−14.4222	−0.827170
$305$	16.6056	0.950831
$306$	0	0
$307$	−21.5416	−1.22945	−0.614723	−	0.788743i	$-0.710732\pi$
$307$	−21.5416	−1.22945	−0.614723	+	0.788743i	$0.710732\pi$
$308$	0	0
$309$	21.1194	1.20144
$310$	0	0
$311$	−1.18335	−0.0671014	−0.0335507	−	0.999437i	$-0.510682\pi$
$311$	−1.18335	−0.0671014	−0.0335507	+	0.999437i	$0.510682\pi$
$312$	0	0
$313$	12.7250	0.719258	0.359629	−	0.933095i	$-0.382903\pi$
$313$	12.7250	0.719258	0.359629	+	0.933095i	$0.382903\pi$
$314$	0	0
$315$	0	0
$316$	−22.8444	−1.28510
$317$	−5.09167	−0.285977	−0.142988	−	0.989724i	$-0.545671\pi$
$317$	−5.09167	−0.285977	−0.142988	+	0.989724i	$0.545671\pi$
$318$	0	0
$319$	1.11943	0.0626760
$320$	−18.4222	−1.02983
$321$	5.09167	0.284189
$322$	0	0
$323$	24.9083	1.38594
$324$	6.78890	0.377161
$325$	−0.394449	−0.0218801
$326$	0	0
$327$	3.51388	0.194318
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−36.0278	−1.98026	−0.990132	−	0.140136i	$-0.955246\pi$
$331$	−36.0278	−1.98026	−0.990132	+	0.140136i	$0.955246\pi$
$332$	8.78890	0.482353
$333$	9.39445	0.514813
$334$	0	0
$335$	1.39445	0.0761869
$336$	0	0
$337$	16.7250	0.911068	0.455534	−	0.890218i	$-0.349448\pi$
$337$	16.7250	0.911068	0.455534	+	0.890218i	$0.349448\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	31.8167	1.72550
$341$	0.908327	0.0491887
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.81665	0.259320
$346$	0	0
$347$	−31.8167	−1.70801	−0.854004	−	0.520267i	$-0.825832\pi$
$347$	−31.8167	−1.70801	−0.854004	+	0.520267i	$0.825832\pi$
$348$	4.18335	0.224251
$349$	26.8167	1.43546	0.717731	−	0.696320i	$-0.245181\pi$
$349$	26.8167	1.43546	0.717731	+	0.696320i	$0.245181\pi$
$350$	0	0
$351$	7.30278	0.389793
$352$	0	0
$353$	−18.6333	−0.991751	−0.495875	−	0.868394i	$-0.665153\pi$
$353$	−18.6333	−0.991751	−0.495875	+	0.868394i	$0.665153\pi$
$354$	0	0
$355$	−24.9083	−1.32200
$356$	15.2111	0.806187
$357$	0	0
$358$	0	0
$359$	15.6333	0.825094	0.412547	−	0.910936i	$-0.364639\pi$
$359$	15.6333	0.825094	0.412547	+	0.910936i	$0.364639\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	0	0
$363$	−13.6972	−0.718918
$364$	0	0
$365$	−13.1194	−0.686702
$366$	0	0
$367$	31.2111	1.62921	0.814603	−	0.580019i	$-0.196955\pi$
$367$	31.2111	1.62921	0.814603	+	0.580019i	$0.196955\pi$
$368$	6.42221	0.334781
$369$	−1.30278	−0.0678198
$370$	0	0
$371$	0	0
$372$	3.39445	0.175994
$373$	−17.3305	−0.897341	−0.448670	−	0.893697i	$-0.648102\pi$
$373$	−17.3305	−0.897341	−0.448670	+	0.893697i	$0.648102\pi$
$374$	0	0
$375$	−14.0917	−0.727691
$376$	0	0
$377$	2.09167	0.107727
$378$	0	0
$379$	−22.2111	−1.14091	−0.570454	−	0.821330i	$-0.693233\pi$
$379$	−22.2111	−1.14091	−0.570454	+	0.821330i	$0.693233\pi$
$380$	16.6056	0.851847
$381$	−24.3944	−1.24977
$382$	0	0
$383$	9.21110	0.470665	0.235333	−	0.971915i	$-0.424382\pi$
$383$	9.21110	0.470665	0.235333	+	0.971915i	$0.424382\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.11943	0.158570
$388$	11.3944	0.578465
$389$	25.1194	1.27361	0.636803	−	0.771027i	$-0.280257\pi$
$389$	25.1194	1.27361	0.636803	+	0.771027i	$0.280257\pi$
$390$	0	0
$391$	−11.0917	−0.560930
$392$	0	0
$393$	−6.90833	−0.348479
$394$	0	0
$395$	26.3028	1.32344
$396$	−1.81665	−0.0912903
$397$	4.27502	0.214557	0.107279	−	0.994229i	$-0.465786\pi$
$397$	4.27502	0.214557	0.107279	+	0.994229i	$0.465786\pi$
$398$	0	0
$399$	0	0
$400$	1.21110	0.0605551
$401$	13.6056	0.679429	0.339714	−	0.940529i	$-0.389670\pi$
$401$	13.6056	0.679429	0.339714	+	0.940529i	$0.389670\pi$
$402$	0	0
$403$	1.69722	0.0845448
$404$	−8.78890	−0.437264
$405$	−7.81665	−0.388413
$406$	0	0
$407$	5.02776	0.249217
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	−29.7250	−1.46623
$412$	−32.4222	−1.59733
$413$	0	0
$414$	0	0
$415$	−10.1194	−0.496743
$416$	0	0
$417$	−7.69722	−0.376935
$418$	0	0
$419$	4.88057	0.238431	0.119216	−	0.992868i	$-0.461962\pi$
$419$	4.88057	0.238431	0.119216	+	0.992868i	$0.461962\pi$
$420$	0	0
$421$	14.9083	0.726587	0.363294	−	0.931675i	$-0.381652\pi$
$421$	14.9083	0.726587	0.363294	+	0.931675i	$0.381652\pi$
$422$	0	0
$423$	−11.7250	−0.570088
$424$	0	0
$425$	−2.09167	−0.101461
$426$	0	0
$427$	0	0
$428$	−7.81665	−0.377832
$429$	1.18335	0.0571325
$430$	0	0
$431$	23.5139	1.13262	0.566312	−	0.824191i	$-0.308370\pi$
$431$	23.5139	1.13262	0.566312	+	0.824191i	$0.308370\pi$
$432$	−22.4222	−1.07879
$433$	−27.3944	−1.31649	−0.658247	−	0.752802i	$-0.728702\pi$
$433$	−27.3944	−1.31649	−0.658247	+	0.752802i	$0.728702\pi$
$434$	0	0
$435$	−4.81665	−0.230941
$436$	−5.39445	−0.258347
$437$	−5.78890	−0.276921
$438$	0	0
$439$	−22.7889	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$439$	−22.7889	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.1194	−0.908392	−0.454196	−	0.890902i	$-0.650073\pi$
$443$	−19.1194	−0.908392	−0.454196	+	0.890902i	$0.650073\pi$
$444$	18.7889	0.891682
$445$	−17.5139	−0.830237
$446$	0	0
$447$	25.8167	1.22109
$448$	0	0
$449$	−3.63331	−0.171466	−0.0857332	−	0.996318i	$-0.527323\pi$
$449$	−3.63331	−0.171466	−0.0857332	+	0.996318i	$0.527323\pi$
$450$	0	0
$451$	−0.697224	−0.0328310
$452$	−27.6333	−1.29976
$453$	−18.3944	−0.864247
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.0278	0.749747	0.374873	−	0.927076i	$-0.377686\pi$
$457$	16.0278	0.749747	0.374873	+	0.927076i	$0.377686\pi$
$458$	0	0
$459$	38.7250	1.80753
$460$	−7.39445	−0.344768
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	−18.3028	−0.850602	−0.425301	−	0.905052i	$-0.639832\pi$
$463$	−18.3028	−0.850602	−0.425301	+	0.905052i	$0.639832\pi$
$464$	−6.42221	−0.298143
$465$	−3.90833	−0.181244
$466$	0	0
$467$	22.5416	1.04310	0.521551	−	0.853220i	$-0.325354\pi$
$467$	22.5416	1.04310	0.521551	+	0.853220i	$0.325354\pi$
$468$	−3.39445	−0.156908
$469$	0	0
$470$	0	0
$471$	21.1194	0.973132
$472$	0	0
$473$	1.66947	0.0767622
$474$	0	0
$475$	−1.09167	−0.0500894
$476$	0	0
$477$	1.18335	0.0541817
$478$	0	0
$479$	37.7527	1.72497	0.862483	−	0.506086i	$-0.168908\pi$
$479$	37.7527	1.72497	0.862483	+	0.506086i	$0.168908\pi$
$480$	0	0
$481$	9.39445	0.428350
$482$	0	0
$483$	0	0
$484$	21.0278	0.955807
$485$	−13.1194	−0.595723
$486$	0	0
$487$	−28.6972	−1.30040	−0.650198	−	0.759765i	$-0.725314\pi$
$487$	−28.6972	−1.30040	−0.650198	+	0.759765i	$0.725314\pi$
$488$	0	0
$489$	21.7889	0.985328
$490$	0	0
$491$	13.1194	0.592072	0.296036	−	0.955177i	$-0.404335\pi$
$491$	13.1194	0.592072	0.296036	+	0.955177i	$0.404335\pi$
$492$	−2.60555	−0.117467
$493$	11.0917	0.499544
$494$	0	0
$495$	2.09167	0.0940137
$496$	−5.21110	−0.233985
$497$	0	0
$498$	0	0
$499$	−40.8444	−1.82845	−0.914223	−	0.405210i	$-0.867198\pi$
$499$	−40.8444	−1.82845	−0.914223	+	0.405210i	$0.867198\pi$
$500$	21.6333	0.967471
$501$	21.9083	0.978792
$502$	0	0
$503$	−7.18335	−0.320290	−0.160145	−	0.987094i	$-0.551196\pi$
$503$	−7.18335	−0.320290	−0.160145	+	0.987094i	$0.551196\pi$
$504$	0	0
$505$	10.1194	0.450309
$506$	0	0
$507$	−14.7250	−0.653959
$508$	37.4500	1.66157
$509$	8.72498	0.386728	0.193364	−	0.981127i	$-0.438060\pi$
$509$	8.72498	0.386728	0.193364	+	0.981127i	$0.438060\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	20.2111	0.892342
$514$	0	0
$515$	37.3305	1.64498
$516$	6.23886	0.274650
$517$	−6.27502	−0.275975
$518$	0	0
$519$	−3.27502	−0.143757
$520$	0	0
$521$	28.0555	1.22913	0.614567	−	0.788865i	$-0.289331\pi$
$521$	28.0555	1.22913	0.614567	+	0.788865i	$0.289331\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	10.6056	0.463306
$525$	0	0
$526$	0	0
$527$	9.00000	0.392046
$528$	−3.63331	−0.158119
$529$	−20.4222	−0.887922
$530$	0	0
$531$	9.90833	0.429985
$532$	0	0
$533$	−1.30278	−0.0564295
$534$	0	0
$535$	9.00000	0.389104
$536$	0	0
$537$	−22.5416	−0.972743
$538$	0	0
$539$	0	0
$540$	25.8167	1.11097
$541$	−37.2111	−1.59983	−0.799915	−	0.600113i	$-0.795122\pi$
$541$	−37.2111	−1.59983	−0.799915	+	0.600113i	$0.795122\pi$
$542$	0	0
$543$	19.5778	0.840164
$544$	0	0
$545$	6.21110	0.266054
$546$	0	0
$547$	−0.724981	−0.0309979	−0.0154990	−	0.999880i	$-0.504934\pi$
$547$	−0.724981	−0.0309979	−0.0154990	+	0.999880i	$0.504934\pi$
$548$	45.6333	1.94936
$549$	−9.39445	−0.400945
$550$	0	0
$551$	5.78890	0.246615
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−21.6333	−0.918283
$556$	11.8167	0.501138
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	0	0
$559$	3.11943	0.131938
$560$	0	0
$561$	6.27502	0.264932
$562$	0	0
$563$	−24.2111	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$563$	−24.2111	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$564$	−23.4500	−0.987422
$565$	31.8167	1.33854
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.2389	0.596924	0.298462	−	0.954422i	$-0.403526\pi$
$569$	14.2389	0.596924	0.298462	+	0.954422i	$0.403526\pi$
$570$	0	0
$571$	−12.0278	−0.503346	−0.251673	−	0.967812i	$-0.580981\pi$
$571$	−12.0278	−0.503346	−0.251673	+	0.967812i	$0.580981\pi$
$572$	−1.81665	−0.0759581
$573$	−20.4500	−0.854309
$574$	0	0
$575$	0.486122	0.0202727
$576$	10.4222	0.434259
$577$	−42.0555	−1.75079	−0.875397	−	0.483405i	$-0.839400\pi$
$577$	−42.0555	−1.75079	−0.875397	+	0.483405i	$0.839400\pi$
$578$	0	0
$579$	−19.0278	−0.790767
$580$	7.39445	0.307038
$581$	0	0
$582$	0	0
$583$	0.633308	0.0262289
$584$	0	0
$585$	3.90833	0.161589
$586$	0	0
$587$	29.4500	1.21553	0.607765	−	0.794117i	$-0.292066\pi$
$587$	29.4500	1.21553	0.607765	+	0.794117i	$0.292066\pi$
$588$	0	0
$589$	4.69722	0.193546
$590$	0	0
$591$	−9.00000	−0.370211
$592$	−28.8444	−1.18550
$593$	−2.78890	−0.114526	−0.0572631	−	0.998359i	$-0.518237\pi$
$593$	−2.78890	−0.114526	−0.0572631	+	0.998359i	$0.518237\pi$
$594$	0	0
$595$	0	0
$596$	−39.6333	−1.62344
$597$	4.93608	0.202020
$598$	0	0
$599$	−13.6056	−0.555908	−0.277954	−	0.960594i	$-0.589656\pi$
$599$	−13.6056	−0.555908	−0.277954	+	0.960594i	$0.589656\pi$
$600$	0	0
$601$	46.1472	1.88238	0.941191	−	0.337874i	$-0.109708\pi$
$601$	46.1472	1.88238	0.941191	+	0.337874i	$0.109708\pi$
$602$	0	0
$603$	−0.788897	−0.0321264
$604$	28.2389	1.14902
$605$	−24.2111	−0.984321
$606$	0	0
$607$	17.8167	0.723156	0.361578	−	0.932342i	$-0.382238\pi$
$607$	17.8167	0.723156	0.361578	+	0.932342i	$0.382238\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.7250	−0.474342
$612$	−18.0000	−0.727607
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	0	0
$615$	3.00000	0.120972
$616$	0	0
$617$	37.5416	1.51137	0.755685	−	0.654936i	$-0.227304\pi$
$617$	37.5416	1.51137	0.755685	+	0.654936i	$0.227304\pi$
$618$	0	0
$619$	−8.48612	−0.341086	−0.170543	−	0.985350i	$-0.554552\pi$
$619$	−8.48612	−0.341086	−0.170543	+	0.985350i	$0.554552\pi$
$620$	6.00000	0.240966
$621$	−9.00000	−0.361158
$622$	0	0
$623$	0	0
$624$	−6.78890	−0.271773
$625$	−26.4222	−1.05689
$626$	0	0
$627$	3.27502	0.130792
$628$	−32.4222	−1.29379
$629$	49.8167	1.98632
$630$	0	0
$631$	14.8444	0.590947	0.295473	−	0.955351i	$-0.404523\pi$
$631$	14.8444	0.590947	0.295473	+	0.955351i	$0.404523\pi$
$632$	0	0
$633$	19.6972	0.782894
$634$	0	0
$635$	−43.1194	−1.71114
$636$	2.36669	0.0938455
$637$	0	0
$638$	0	0
$639$	14.0917	0.557458
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	−7.18335	−0.282844
$646$	0	0
$647$	−27.6333	−1.08638	−0.543189	−	0.839611i	$-0.682783\pi$
$647$	−27.6333	−1.08638	−0.543189	+	0.839611i	$0.682783\pi$
$648$	0	0
$649$	5.30278	0.208152
$650$	0	0
$651$	0	0
$652$	−33.4500	−1.31000
$653$	16.1833	0.633303	0.316652	−	0.948542i	$-0.397441\pi$
$653$	16.1833	0.633303	0.316652	+	0.948542i	$0.397441\pi$
$654$	0	0
$655$	−12.2111	−0.477127
$656$	4.00000	0.156174
$657$	7.42221	0.289568
$658$	0	0
$659$	20.2389	0.788394	0.394197	−	0.919026i	$-0.371023\pi$
$659$	20.2389	0.788394	0.394197	+	0.919026i	$0.371023\pi$
$660$	4.18335	0.162837
$661$	−28.7889	−1.11976	−0.559879	−	0.828574i	$-0.689152\pi$
$661$	−28.7889	−1.11976	−0.559879	+	0.828574i	$0.689152\pi$
$662$	0	0
$663$	11.7250	0.455361
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.57779	−0.0998126
$668$	−33.6333	−1.30131
$669$	12.3944	0.479197
$670$	0	0
$671$	−5.02776	−0.194094
$672$	0	0
$673$	20.8444	0.803493	0.401746	−	0.915751i	$-0.368403\pi$
$673$	20.8444	0.803493	0.401746	+	0.915751i	$0.368403\pi$
$674$	0	0
$675$	−1.69722	−0.0653262
$676$	22.6056	0.869444
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−35.4500	−1.35845
$682$	0	0
$683$	42.2111	1.61516	0.807581	−	0.589756i	$-0.200776\pi$
$683$	42.2111	1.61516	0.807581	+	0.589756i	$0.200776\pi$
$684$	−9.39445	−0.359206
$685$	−52.5416	−2.00751
$686$	0	0
$687$	−37.4222	−1.42775
$688$	−9.57779	−0.365150
$689$	1.18335	0.0450819
$690$	0	0
$691$	21.3028	0.810396	0.405198	−	0.914229i	$-0.367203\pi$
$691$	21.3028	0.810396	0.405198	+	0.914229i	$0.367203\pi$
$692$	5.02776	0.191127
$693$	0	0
$694$	0	0
$695$	−13.6056	−0.516088
$696$	0	0
$697$	−6.90833	−0.261672
$698$	0	0
$699$	−21.2750	−0.804695
$700$	0	0
$701$	42.8444	1.61821	0.809106	−	0.587663i	$-0.199952\pi$
$701$	42.8444	1.61821	0.809106	+	0.587663i	$0.199952\pi$
$702$	0	0
$703$	26.0000	0.980609
$704$	5.57779	0.210221
$705$	27.0000	1.01688
$706$	0	0
$707$	0	0
$708$	19.8167	0.744755
$709$	−29.6056	−1.11186	−0.555930	−	0.831229i	$-0.687638\pi$
$709$	−29.6056	−1.11186	−0.555930	+	0.831229i	$0.687638\pi$
$710$	0	0
$711$	−14.8806	−0.558065
$712$	0	0
$713$	−2.09167	−0.0783338
$714$	0	0
$715$	2.09167	0.0782241
$716$	34.6056	1.29327
$717$	1.54163	0.0575734
$718$	0	0
$719$	4.88057	0.182015	0.0910073	−	0.995850i	$-0.470991\pi$
$719$	4.88057	0.182015	0.0910073	+	0.995850i	$0.470991\pi$
$720$	−12.0000	−0.447214
$721$	0	0
$722$	0	0
$723$	23.2111	0.863230
$724$	−30.0555	−1.11700
$725$	−0.486122	−0.0180541
$726$	0	0
$727$	−12.5416	−0.465143	−0.232572	−	0.972579i	$-0.574714\pi$
$727$	−12.5416	−0.465143	−0.232572	+	0.972579i	$0.574714\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	16.5416	0.611814
$732$	−18.7889	−0.694458
$733$	−1.23886	−0.0457583	−0.0228791	−	0.999738i	$-0.507283\pi$
$733$	−1.23886	−0.0457583	−0.0228791	+	0.999738i	$0.507283\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.422205	−0.0155521
$738$	0	0
$739$	−16.2111	−0.596335	−0.298168	−	0.954514i	$-0.596375\pi$
$739$	−16.2111	−0.596335	−0.298168	+	0.954514i	$0.596375\pi$
$740$	33.2111	1.22086
$741$	6.11943	0.224803
$742$	0	0
$743$	−42.3583	−1.55397	−0.776987	−	0.629516i	$-0.783253\pi$
$743$	−42.3583	−1.55397	−0.776987	+	0.629516i	$0.783253\pi$
$744$	0	0
$745$	45.6333	1.67188
$746$	0	0
$747$	5.72498	0.209466
$748$	−9.63331	−0.352229
$749$	0	0
$750$	0	0
$751$	1.93608	0.0706487	0.0353243	−	0.999376i	$-0.488754\pi$
$751$	1.93608	0.0706487	0.0353243	+	0.999376i	$0.488754\pi$
$752$	36.0000	1.31278
$753$	14.4500	0.526586
$754$	0	0
$755$	−32.5139	−1.18330
$756$	0	0
$757$	30.0555	1.09239	0.546193	−	0.837659i	$-0.316076\pi$
$757$	30.0555	1.09239	0.546193	+	0.837659i	$0.316076\pi$
$758$	0	0
$759$	−1.45837	−0.0529353
$760$	0	0
$761$	−17.0917	−0.619573	−0.309786	−	0.950806i	$-0.600258\pi$
$761$	−17.0917	−0.619573	−0.309786	+	0.950806i	$0.600258\pi$
$762$	0	0
$763$	0	0
$764$	31.3944	1.13581
$765$	20.7250	0.749313
$766$	0	0
$767$	9.90833	0.357769
$768$	20.8444	0.752158
$769$	30.0278	1.08283	0.541414	−	0.840756i	$-0.317889\pi$
$769$	30.0278	1.08283	0.541414	+	0.840756i	$0.317889\pi$
$770$	0	0
$771$	−6.63331	−0.238893
$772$	29.2111	1.05133
$773$	−24.2111	−0.870813	−0.435406	−	0.900234i	$-0.643395\pi$
$773$	−24.2111	−0.870813	−0.435406	+	0.900234i	$0.643395\pi$
$774$	0	0
$775$	−0.394449	−0.0141690
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.60555	−0.129182
$780$	7.81665	0.279881
$781$	7.54163	0.269861
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	37.3305	1.33238
$786$	0	0
$787$	−3.18335	−0.113474	−0.0567370	−	0.998389i	$-0.518070\pi$
$787$	−3.18335	−0.113474	−0.0567370	+	0.998389i	$0.518070\pi$
$788$	13.8167	0.492198
$789$	16.5416	0.588898
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.39445	−0.333607
$794$	0	0
$795$	−2.72498	−0.0966451
$796$	−7.57779	−0.268588
$797$	7.18335	0.254447	0.127224	−	0.991874i	$-0.459393\pi$
$797$	7.18335	0.254447	0.127224	+	0.991874i	$0.459393\pi$
$798$	0	0
$799$	−62.1749	−2.19959
$800$	0	0
$801$	9.90833	0.350094
$802$	0	0
$803$	3.97224	0.140177
$804$	−1.57779	−0.0556445
$805$	0	0
$806$	0	0
$807$	−4.81665	−0.169554
$808$	0	0
$809$	13.1833	0.463502	0.231751	−	0.972775i	$-0.425555\pi$
$809$	13.1833	0.463502	0.231751	+	0.972775i	$0.425555\pi$
$810$	0	0
$811$	29.3305	1.02993	0.514967	−	0.857210i	$-0.327804\pi$
$811$	29.3305	1.02993	0.514967	+	0.857210i	$0.327804\pi$
$812$	0	0
$813$	−4.14719	−0.145448
$814$	0	0
$815$	38.5139	1.34908
$816$	−36.0000	−1.26025
$817$	8.63331	0.302041
$818$	0	0
$819$	0	0
$820$	−4.60555	−0.160833
$821$	−40.7527	−1.42228	−0.711140	−	0.703050i	$-0.751821\pi$
$821$	−40.7527	−1.42228	−0.711140	+	0.703050i	$0.751821\pi$
$822$	0	0
$823$	−30.5139	−1.06365	−0.531823	−	0.846855i	$-0.678493\pi$
$823$	−30.5139	−1.06365	−0.531823	+	0.846855i	$0.678493\pi$
$824$	0	0
$825$	−0.275019	−0.00957494
$826$	0	0
$827$	−43.8167	−1.52365	−0.761827	−	0.647780i	$-0.775697\pi$
$827$	−43.8167	−1.52365	−0.761827	+	0.647780i	$0.775697\pi$
$828$	4.18335	0.145381
$829$	20.8167	0.722992	0.361496	−	0.932374i	$-0.382266\pi$
$829$	20.8167	0.722992	0.361496	+	0.932374i	$0.382266\pi$
$830$	0	0
$831$	33.5139	1.16258
$832$	10.4222	0.361325
$833$	0	0
$834$	0	0
$835$	38.7250	1.34013
$836$	−5.02776	−0.173889
$837$	7.30278	0.252421
$838$	0	0
$839$	−1.60555	−0.0554298	−0.0277149	−	0.999616i	$-0.508823\pi$
$839$	−1.60555	−0.0554298	−0.0277149	+	0.999616i	$0.508823\pi$
$840$	0	0
$841$	−26.4222	−0.911111
$842$	0	0
$843$	15.0000	0.516627
$844$	−30.2389	−1.04086
$845$	−26.0278	−0.895382
$846$	0	0
$847$	0	0
$848$	−3.63331	−0.124768
$849$	25.3028	0.868389
$850$	0	0
$851$	−11.5778	−0.396882
$852$	28.1833	0.965546
$853$	−34.0278	−1.16509	−0.582544	−	0.812799i	$-0.697943\pi$
$853$	−34.0278	−1.16509	−0.582544	+	0.812799i	$0.697943\pi$
$854$	0	0
$855$	10.8167	0.369922
$856$	0	0
$857$	−30.4861	−1.04139	−0.520693	−	0.853744i	$-0.674326\pi$
$857$	−30.4861	−1.04139	−0.520693	+	0.853744i	$0.674326\pi$
$858$	0	0
$859$	−34.5139	−1.17760	−0.588799	−	0.808279i	$-0.700399\pi$
$859$	−34.5139	−1.17760	−0.588799	+	0.808279i	$0.700399\pi$
$860$	11.0278	0.376043
$861$	0	0
$862$	0	0
$863$	39.8444	1.35632	0.678160	−	0.734915i	$-0.262778\pi$
$863$	39.8444	1.35632	0.678160	+	0.734915i	$0.262778\pi$
$864$	0	0
$865$	−5.78890	−0.196828
$866$	0	0
$867$	40.0278	1.35941
$868$	0	0
$869$	−7.96384	−0.270155
$870$	0	0
$871$	−0.788897	−0.0267308
$872$	0	0
$873$	7.42221	0.251204
$874$	0	0
$875$	0	0
$876$	14.8444	0.501546
$877$	19.2389	0.649650	0.324825	−	0.945774i	$-0.394695\pi$
$877$	19.2389	0.649650	0.324825	+	0.945774i	$0.394695\pi$
$878$	0	0
$879$	39.3583	1.32752
$880$	−6.42221	−0.216492
$881$	−35.0917	−1.18227	−0.591134	−	0.806573i	$-0.701320\pi$
$881$	−35.0917	−1.18227	−0.591134	+	0.806573i	$0.701320\pi$
$882$	0	0
$883$	−53.7527	−1.80892	−0.904462	−	0.426555i	$-0.859727\pi$
$883$	−53.7527	−1.80892	−0.904462	+	0.426555i	$0.859727\pi$
$884$	−18.0000	−0.605406
$885$	−22.8167	−0.766973
$886$	0	0
$887$	−2.78890	−0.0936420	−0.0468210	−	0.998903i	$-0.514909\pi$
$887$	−2.78890	−0.0936420	−0.0468210	+	0.998903i	$0.514909\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.36669	0.0792872
$892$	−19.0278	−0.637096
$893$	−32.4500	−1.08590
$894$	0	0
$895$	−39.8444	−1.33185
$896$	0	0
$897$	−2.72498	−0.0909845
$898$	0	0
$899$	2.09167	0.0697612
$900$	0.788897	0.0262966
$901$	6.27502	0.209051
$902$	0	0
$903$	0	0
$904$	0	0
$905$	34.6056	1.15033
$906$	0	0
$907$	−54.0278	−1.79396	−0.896981	−	0.442069i	$-0.854245\pi$
$907$	−54.0278	−1.79396	−0.896981	+	0.442069i	$0.854245\pi$
$908$	54.4222	1.80606
$909$	−5.72498	−0.189886
$910$	0	0
$911$	−10.3944	−0.344383	−0.172192	−	0.985063i	$-0.555085\pi$
$911$	−10.3944	−0.344383	−0.172192	+	0.985063i	$0.555085\pi$
$912$	−18.7889	−0.622163
$913$	3.06392	0.101401
$914$	0	0
$915$	21.6333	0.715175
$916$	57.4500	1.89820
$917$	0	0
$918$	0	0
$919$	−26.6056	−0.877636	−0.438818	−	0.898576i	$-0.644603\pi$
$919$	−26.6056	−0.877636	−0.438818	+	0.898576i	$0.644603\pi$
$920$	0	0
$921$	−28.0639	−0.924737
$922$	0	0
$923$	14.0917	0.463833
$924$	0	0
$925$	−2.18335	−0.0717880
$926$	0	0
$927$	−21.1194	−0.693653
$928$	0	0
$929$	18.9722	0.622459	0.311230	−	0.950335i	$-0.399259\pi$
$929$	18.9722	0.622459	0.311230	+	0.950335i	$0.399259\pi$
$930$	0	0
$931$	0	0
$932$	32.6611	1.06985
$933$	−1.54163	−0.0504708
$934$	0	0
$935$	11.0917	0.362736
$936$	0	0
$937$	−42.0555	−1.37389	−0.686947	−	0.726708i	$-0.741049\pi$
$937$	−42.0555	−1.37389	−0.686947	+	0.726708i	$0.741049\pi$
$938$	0	0
$939$	16.5778	0.540996
$940$	−41.4500	−1.35195
$941$	−23.4500	−0.764447	−0.382223	−	0.924070i	$-0.624842\pi$
$941$	−23.4500	−0.764447	−0.382223	+	0.924070i	$0.624842\pi$
$942$	0	0
$943$	1.60555	0.0522839
$944$	−30.4222	−0.990158
$945$	0	0
$946$	0	0
$947$	−54.2111	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$947$	−54.2111	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$948$	−29.7611	−0.966597
$949$	7.42221	0.240935
$950$	0	0
$951$	−6.63331	−0.215100
$952$	0	0
$953$	34.5416	1.11891	0.559457	−	0.828860i	$-0.311010\pi$
$953$	34.5416	1.11891	0.559457	+	0.828860i	$0.311010\pi$
$954$	0	0
$955$	−36.1472	−1.16970
$956$	−2.36669	−0.0765443
$957$	1.45837	0.0471423
$958$	0	0
$959$	0	0
$960$	−24.0000	−0.774597
$961$	−29.3028	−0.945251
$962$	0	0
$963$	−5.09167	−0.164077
$964$	−35.6333	−1.14767
$965$	−33.6333	−1.08269
$966$	0	0
$967$	−34.4861	−1.10900	−0.554499	−	0.832184i	$-0.687090\pi$
$967$	−34.4861	−1.10900	−0.554499	+	0.832184i	$0.687090\pi$
$968$	0	0
$969$	32.4500	1.04244
$970$	0	0
$971$	−34.7527	−1.11527	−0.557634	−	0.830087i	$-0.688291\pi$
$971$	−34.7527	−1.11527	−0.557634	+	0.830087i	$0.688291\pi$
$972$	−24.7889	−0.795104
$973$	0	0
$974$	0	0
$975$	−0.513878	−0.0164573
$976$	28.8444	0.923287
$977$	−24.6333	−0.788089	−0.394045	−	0.919091i	$-0.628924\pi$
$977$	−24.6333	−0.788089	−0.394045	+	0.919091i	$0.628924\pi$
$978$	0	0
$979$	5.30278	0.169477
$980$	0	0
$981$	−3.51388	−0.112189
$982$	0	0
$983$	55.5416	1.77150	0.885752	−	0.464160i	$-0.153644\pi$
$983$	55.5416	1.77150	0.885752	+	0.464160i	$0.153644\pi$
$984$	0	0
$985$	−15.9083	−0.506881
$986$	0	0
$987$	0	0
$988$	−9.39445	−0.298877
$989$	−3.84441	−0.122245
$990$	0	0
$991$	−32.1194	−1.02031	−0.510154	−	0.860083i	$-0.670411\pi$
$991$	−32.1194	−1.02031	−0.510154	+	0.860083i	$0.670411\pi$
$992$	0	0
$993$	−46.9361	−1.48947
$994$	0	0
$995$	8.72498	0.276600
$996$	11.4500	0.362806
$997$	−18.0555	−0.571824	−0.285912	−	0.958256i	$-0.592296\pi$
$997$	−18.0555	−0.571824	−0.285912	+	0.958256i	$0.592296\pi$
$998$	0	0
$999$	40.4222	1.27890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.c.1.2		2
7.3	odd	6		287.2.e.b.247.2	yes	4
7.5	odd	6		287.2.e.b.165.2	✓	4
7.6	odd	2		2009.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.e.b.165.2	✓	4	7.5	odd	6
287.2.e.b.247.2	yes	4	7.3	odd	6
2009.2.a.c.1.2		2	1.1	even	1	trivial
2009.2.a.d.1.1		2	7.6	odd	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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{3} -2.00000 q^{4} +2.30278 q^{5} -1.30278 q^{9} +O(q^{10})\)	\(q+1.30278 q^{3} -2.00000 q^{4} +2.30278 q^{5} -1.30278 q^{9} -0.697224 q^{11} -2.60555 q^{12} -1.30278 q^{13} +3.00000 q^{15} +4.00000 q^{16} -6.90833 q^{17} -3.60555 q^{19} -4.60555 q^{20} +1.60555 q^{23} +0.302776 q^{25} -5.60555 q^{27} -1.60555 q^{29} -1.30278 q^{31} -0.908327 q^{33} +2.60555 q^{36} -7.21110 q^{37} -1.69722 q^{39} +1.00000 q^{41} -2.39445 q^{43} +1.39445 q^{44} -3.00000 q^{45} +9.00000 q^{47} +5.21110 q^{48} -9.00000 q^{51} +2.60555 q^{52} -0.908327 q^{53} -1.60555 q^{55} -4.69722 q^{57} -7.60555 q^{59} -6.00000 q^{60} +7.21110 q^{61} -8.00000 q^{64} -3.00000 q^{65} +0.605551 q^{67} +13.8167 q^{68} +2.09167 q^{69} -10.8167 q^{71} -5.69722 q^{73} +0.394449 q^{75} +7.21110 q^{76} +11.4222 q^{79} +9.21110 q^{80} -3.39445 q^{81} -4.39445 q^{83} -15.9083 q^{85} -2.09167 q^{87} -7.60555 q^{89} -3.21110 q^{92} -1.69722 q^{93} -8.30278 q^{95} -5.69722 q^{97} +0.908327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9}+O(q^{10}) \) 2 * q - q^3 - 4 * q^4 + q^5 + q^9	\( 2 q - q^{3} - 4 q^{4} + q^{5} + q^{9} - 5 q^{11} + 2 q^{12} + q^{13} + 6 q^{15} + 8 q^{16} - 3 q^{17} - 2 q^{20} - 4 q^{23} - 3 q^{25} - 4 q^{27} + 4 q^{29} + q^{31} + 9 q^{33} - 2 q^{36} - 7 q^{39} + 2 q^{41} - 12 q^{43} + 10 q^{44} - 6 q^{45} + 18 q^{47} - 4 q^{48} - 18 q^{51} - 2 q^{52} + 9 q^{53} + 4 q^{55} - 13 q^{57} - 8 q^{59} - 12 q^{60} - 16 q^{64} - 6 q^{65} - 6 q^{67} + 6 q^{68} + 15 q^{69} - 15 q^{73} + 8 q^{75} - 6 q^{79} + 4 q^{80} - 14 q^{81} - 16 q^{83} - 21 q^{85} - 15 q^{87} - 8 q^{89} + 8 q^{92} - 7 q^{93} - 13 q^{95} - 15 q^{97} - 9 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^4 + q^5 + q^9 - 5 * q^11 + 2 * q^12 + q^13 + 6 * q^15 + 8 * q^16 - 3 * q^17 - 2 * q^20 - 4 * q^23 - 3 * q^25 - 4 * q^27 + 4 * q^29 + q^31 + 9 * q^33 - 2 * q^36 - 7 * q^39 + 2 * q^41 - 12 * q^43 + 10 * q^44 - 6 * q^45 + 18 * q^47 - 4 * q^48 - 18 * q^51 - 2 * q^52 + 9 * q^53 + 4 * q^55 - 13 * q^57 - 8 * q^59 - 12 * q^60 - 16 * q^64 - 6 * q^65 - 6 * q^67 + 6 * q^68 + 15 * q^69 - 15 * q^73 + 8 * q^75 - 6 * q^79 + 4 * q^80 - 14 * q^81 - 16 * q^83 - 21 * q^85 - 15 * q^87 - 8 * q^89 + 8 * q^92 - 7 * q^93 - 13 * q^95 - 15 * q^97 - 9 * q^99

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