LMFDB - Embedded newform 9009.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9009,2,Mod(1,9009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9009 = 3^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9009.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:	$71.9372271810$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23301.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2x + 1 $ x^4 - x^3 - 6x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.82757$ of defining polynomial
Character	$\chi$	$=$	9009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35366 q^{2} -0.167605 q^{4} -0.473915 q^{5} +1.00000 q^{7} -2.93420 q^{8} +O(q^{10})$	\(q+1.35366 q^{2} -0.167605 q^{4} -0.473915 q^{5} +1.00000 q^{7} -2.93420 q^{8} -0.641520 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.35366 q^{14} -3.63670 q^{16} +1.18123 q^{17} +2.64152 q^{19} +0.0794304 q^{20} +1.35366 q^{22} -0.986370 q^{23} -4.77540 q^{25} -1.35366 q^{26} -0.167605 q^{28} +5.28786 q^{29} +0.0521702 q^{31} +0.945545 q^{32} +1.59899 q^{34} -0.473915 q^{35} -3.53972 q^{37} +3.57572 q^{38} +1.39056 q^{40} +0.353660 q^{41} +0.0473498 q^{43} -0.167605 q^{44} -1.33521 q^{46} -2.98637 q^{47} +1.00000 q^{49} -6.46427 q^{50} +0.167605 q^{52} +6.84120 q^{53} -0.473915 q^{55} -2.93420 q^{56} +7.15796 q^{58} -7.94301 q^{59} +13.5822 q^{61} +0.0706207 q^{62} +8.55335 q^{64} +0.473915 q^{65} +8.99036 q^{67} -0.197981 q^{68} -0.641520 q^{70} -9.31030 q^{71} -1.11543 q^{73} -4.79157 q^{74} -0.442731 q^{76} +1.00000 q^{77} -0.181235 q^{79} +1.72349 q^{80} +0.478735 q^{82} -1.48754 q^{83} -0.559805 q^{85} +0.0640955 q^{86} -2.93420 q^{88} -17.0312 q^{89} -1.00000 q^{91} +0.165320 q^{92} -4.04253 q^{94} -1.25186 q^{95} -5.00881 q^{97} +1.35366 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8} + 13 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{14} + 20 q^{16} - 9 q^{17} - 5 q^{19} + 4 q^{20} + 2 q^{22} - 9 q^{23} + 5 q^{25} - 2 q^{26} + 8 q^{28} + 9 q^{29} + 14 q^{31} + 16 q^{32} + 15 q^{34} + 5 q^{35} - 16 q^{37} - 10 q^{38} + 30 q^{40} - 2 q^{41} - 5 q^{43} + 8 q^{44} + 12 q^{46} - 17 q^{47} + 4 q^{49} + 19 q^{50} - 8 q^{52} + 12 q^{53} + 5 q^{55} - 3 q^{56} - 18 q^{58} + q^{59} + 32 q^{61} + 28 q^{62} + 31 q^{64} - 5 q^{65} - 2 q^{67} - 18 q^{68} + 13 q^{70} + 4 q^{71} + 18 q^{73} - 35 q^{74} - 40 q^{76} + 4 q^{77} + 13 q^{79} + 40 q^{80} + 14 q^{82} + 6 q^{83} - 9 q^{85} + 26 q^{86} - 3 q^{88} - 15 q^{89} - 4 q^{91} + 18 q^{92} + 8 q^{94} - 19 q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8 + 13 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^14 + 20 * q^16 - 9 * q^17 - 5 * q^19 + 4 * q^20 + 2 * q^22 - 9 * q^23 + 5 * q^25 - 2 * q^26 + 8 * q^28 + 9 * q^29 + 14 * q^31 + 16 * q^32 + 15 * q^34 + 5 * q^35 - 16 * q^37 - 10 * q^38 + 30 * q^40 - 2 * q^41 - 5 * q^43 + 8 * q^44 + 12 * q^46 - 17 * q^47 + 4 * q^49 + 19 * q^50 - 8 * q^52 + 12 * q^53 + 5 * q^55 - 3 * q^56 - 18 * q^58 + q^59 + 32 * q^61 + 28 * q^62 + 31 * q^64 - 5 * q^65 - 2 * q^67 - 18 * q^68 + 13 * q^70 + 4 * q^71 + 18 * q^73 - 35 * q^74 - 40 * q^76 + 4 * q^77 + 13 * q^79 + 40 * q^80 + 14 * q^82 + 6 * q^83 - 9 * q^85 + 26 * q^86 - 3 * q^88 - 15 * q^89 - 4 * q^91 + 18 * q^92 + 8 * q^94 - 19 * q^95 + 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.35366	0.957182	0.478591	−	0.878038i	$-0.341148\pi$
$2$	1.35366	0.957182	0.478591	+	0.878038i	$0.341148\pi$
$3$	0	0
$3$	0	0
$4$	−0.167605	−0.0838024
$5$	−0.473915	−0.211941	−0.105971	−	0.994369i	$-0.533795\pi$
$5$	−0.473915	−0.211941	−0.105971	+	0.994369i	$0.533795\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.93420	−1.03740
$9$	0	0
$10$	−0.641520	−0.202866
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.35366	0.361781
$15$	0	0
$16$	−3.63670	−0.909175
$17$	1.18123	0.286492	0.143246	−	0.989687i	$-0.454246\pi$
$17$	1.18123	0.286492	0.143246	+	0.989687i	$0.454246\pi$
$18$	0	0
$19$	2.64152	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$19$	2.64152	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$20$	0.0794304	0.0177612
$21$	0	0
$22$	1.35366	0.288601
$23$	−0.986370	−0.205672	−0.102836	−	0.994698i	$-0.532792\pi$
$23$	−0.986370	−0.205672	−0.102836	+	0.994698i	$0.532792\pi$
$24$	0	0
$25$	−4.77540	−0.955081
$26$	−1.35366	−0.265475
$27$	0	0
$28$	−0.167605	−0.0316743
$29$	5.28786	0.981931	0.490965	−	0.871179i	$-0.336644\pi$
$29$	5.28786	0.981931	0.490965	+	0.871179i	$0.336644\pi$
$30$	0	0
$31$	0.0521702	0.00937004	0.00468502	−	0.999989i	$-0.498509\pi$
$31$	0.0521702	0.00937004	0.00468502	+	0.999989i	$0.498509\pi$
$32$	0.945545	0.167150
$33$	0	0
$34$	1.59899	0.274225
$35$	−0.473915	−0.0801062
$36$	0	0
$37$	−3.53972	−0.581926	−0.290963	−	0.956734i	$-0.593976\pi$
$37$	−3.53972	−0.581926	−0.290963	+	0.956734i	$0.593976\pi$
$38$	3.57572	0.580058
$39$	0	0
$40$	1.39056	0.219867
$41$	0.353660	0.0552324	0.0276162	−	0.999619i	$-0.491208\pi$
$41$	0.353660	0.0552324	0.0276162	+	0.999619i	$0.491208\pi$
$42$	0	0
$43$	0.0473498	0.00722077	0.00361039	−	0.999993i	$-0.498851\pi$
$43$	0.0473498	0.00722077	0.00361039	+	0.999993i	$0.498851\pi$
$44$	−0.167605	−0.0252674
$45$	0	0
$46$	−1.33521	−0.196866
$47$	−2.98637	−0.435607	−0.217803	−	0.975993i	$-0.569889\pi$
$47$	−2.98637	−0.435607	−0.217803	+	0.975993i	$0.569889\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.46427	−0.914186
$51$	0	0
$52$	0.167605	0.0232426
$53$	6.84120	0.939712	0.469856	−	0.882743i	$-0.344306\pi$
$53$	6.84120	0.939712	0.469856	+	0.882743i	$0.344306\pi$
$54$	0	0
$55$	−0.473915	−0.0639027
$56$	−2.93420	−0.392099
$57$	0	0
$58$	7.15796	0.939887
$59$	−7.94301	−1.03409	−0.517046	−	0.855958i	$-0.672968\pi$
$59$	−7.94301	−1.03409	−0.517046	+	0.855958i	$0.672968\pi$
$60$	0	0
$61$	13.5822	1.73903	0.869514	−	0.493908i	$-0.164432\pi$
$61$	13.5822	1.73903	0.869514	+	0.493908i	$0.164432\pi$
$62$	0.0706207	0.00896884
$63$	0	0
$64$	8.55335	1.06917
$65$	0.473915	0.0587819
$66$	0	0
$67$	8.99036	1.09835	0.549174	−	0.835708i	$-0.314943\pi$
$67$	8.99036	1.09835	0.549174	+	0.835708i	$0.314943\pi$
$68$	−0.197981	−0.0240087
$69$	0	0
$70$	−0.641520	−0.0766763
$71$	−9.31030	−1.10493	−0.552465	−	0.833536i	$-0.686313\pi$
$71$	−9.31030	−1.10493	−0.552465	+	0.833536i	$0.686313\pi$
$72$	0	0
$73$	−1.11543	−0.130552	−0.0652759	−	0.997867i	$-0.520793\pi$
$73$	−1.11543	−0.130552	−0.0652759	+	0.997867i	$0.520793\pi$
$74$	−4.79157	−0.557009
$75$	0	0
$76$	−0.442731	−0.0507847
$77$	1.00000	0.113961
$78$	0	0
$79$	−0.181235	−0.0203905	−0.0101953	−	0.999948i	$-0.503245\pi$
$79$	−0.181235	−0.0203905	−0.0101953	+	0.999948i	$0.503245\pi$
$80$	1.72349	0.192692
$81$	0	0
$82$	0.478735	0.0528675
$83$	−1.48754	−0.163279	−0.0816396	−	0.996662i	$-0.526016\pi$
$83$	−1.48754	−0.163279	−0.0816396	+	0.996662i	$0.526016\pi$
$84$	0	0
$85$	−0.559805	−0.0607194
$86$	0.0640955	0.00691160
$87$	0	0
$88$	−2.93420	−0.312787
$89$	−17.0312	−1.80531	−0.902654	−	0.430366i	$-0.858384\pi$
$89$	−17.0312	−1.80531	−0.902654	+	0.430366i	$0.858384\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0.165320	0.0172358
$93$	0	0
$94$	−4.04253	−0.416955
$95$	−1.25186	−0.128438
$96$	0	0
$97$	−5.00881	−0.508568	−0.254284	−	0.967130i	$-0.581840\pi$
$97$	−5.00881	−0.508568	−0.254284	+	0.967130i	$0.581840\pi$
$98$	1.35366	0.136740
$99$	0	0
$100$	0.800380	0.0800380
$101$	11.8500	1.17912	0.589560	−	0.807724i	$-0.299301\pi$
$101$	11.8500	1.17912	0.589560	+	0.807724i	$0.299301\pi$
$102$	0	0
$103$	13.4739	1.32762	0.663812	−	0.747899i	$-0.268937\pi$
$103$	13.4739	1.32762	0.663812	+	0.747899i	$0.268937\pi$
$104$	2.93420	0.287722
$105$	0	0
$106$	9.26067	0.899475
$107$	20.5749	1.98905	0.994525	−	0.104501i	$-0.0333246\pi$
$107$	20.5749	1.98905	0.994525	+	0.104501i	$0.0333246\pi$
$108$	0	0
$109$	8.06979	0.772946	0.386473	−	0.922301i	$-0.373693\pi$
$109$	8.06979	0.772946	0.386473	+	0.922301i	$0.373693\pi$
$110$	−0.641520	−0.0611665
$111$	0	0
$112$	−3.63670	−0.343636
$113$	10.4370	0.981832	0.490916	−	0.871207i	$-0.336662\pi$
$113$	10.4370	0.981832	0.490916	+	0.871207i	$0.336662\pi$
$114$	0	0
$115$	0.467455	0.0435904
$116$	−0.886270	−0.0822881
$117$	0	0
$118$	−10.7521	−0.989814
$119$	1.18123	0.108284
$120$	0	0
$121$	1.00000	0.0909091
$122$	18.3857	1.66457
$123$	0	0
$124$	−0.00874397	−0.000785232 0
$125$	4.63271	0.414362
$126$	0	0
$127$	8.08335	0.717282	0.358641	−	0.933476i	$-0.383240\pi$
$127$	8.08335	0.717282	0.358641	+	0.933476i	$0.383240\pi$
$128$	9.68723	0.856238
$129$	0	0
$130$	0.641520	0.0562650
$131$	0.566910	0.0495311	0.0247656	−	0.999693i	$-0.492116\pi$
$131$	0.566910	0.0495311	0.0247656	+	0.999693i	$0.492116\pi$
$132$	0	0
$133$	2.64152	0.229049
$134$	12.1699	1.05132
$135$	0	0
$136$	−3.46598	−0.297205
$137$	−14.9342	−1.27591	−0.637957	−	0.770072i	$-0.720220\pi$
$137$	−14.9342	−1.27591	−0.637957	+	0.770072i	$0.720220\pi$
$138$	0	0
$139$	21.0905	1.78887	0.894437	−	0.447193i	$-0.147576\pi$
$139$	21.0905	1.78887	0.894437	+	0.447193i	$0.147576\pi$
$140$	0.0794304	0.00671309
$141$	0	0
$142$	−12.6030	−1.05762
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−2.50600	−0.208112
$146$	−1.50992	−0.124962
$147$	0	0
$148$	0.593273	0.0487667
$149$	−6.32393	−0.518076	−0.259038	−	0.965867i	$-0.583406\pi$
$149$	−6.32393	−0.518076	−0.259038	+	0.965867i	$0.583406\pi$
$150$	0	0
$151$	−1.06808	−0.0869195	−0.0434598	−	0.999055i	$-0.513838\pi$
$151$	−1.06808	−0.0869195	−0.0434598	+	0.999055i	$0.513838\pi$
$152$	−7.75075	−0.628669
$153$	0	0
$154$	1.35366	0.109081
$155$	−0.0247242	−0.00198590
$156$	0	0
$157$	−5.85484	−0.467267	−0.233633	−	0.972325i	$-0.575062\pi$
$157$	−5.85484	−0.467267	−0.233633	+	0.972325i	$0.575062\pi$
$158$	−0.245330	−0.0195174
$159$	0	0
$160$	−0.448108	−0.0354261
$161$	−0.986370	−0.0777368
$162$	0	0
$163$	1.01591	0.0795726	0.0397863	−	0.999208i	$-0.487332\pi$
$163$	1.01591	0.0795726	0.0397863	+	0.999208i	$0.487332\pi$
$164$	−0.0592751	−0.00462861
$165$	0	0
$166$	−2.01363	−0.156288
$167$	−11.1475	−0.862621	−0.431310	−	0.902204i	$-0.641949\pi$
$167$	−11.1475	−0.862621	−0.431310	+	0.902204i	$0.641949\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.757785	−0.0581195
$171$	0	0
$172$	−0.00793605	−0.000605118 0
$173$	6.70960	0.510122	0.255061	−	0.966925i	$-0.417904\pi$
$173$	6.70960	0.510122	0.255061	+	0.966925i	$0.417904\pi$
$174$	0	0
$175$	−4.77540	−0.360987
$176$	−3.63670	−0.274127
$177$	0	0
$178$	−23.0545	−1.72801
$179$	10.4787	0.783217	0.391609	−	0.920132i	$-0.371919\pi$
$179$	10.4787	0.783217	0.391609	+	0.920132i	$0.371919\pi$
$180$	0	0
$181$	−1.73622	−0.129052	−0.0645261	−	0.997916i	$-0.520554\pi$
$181$	−1.73622	−0.129052	−0.0645261	+	0.997916i	$0.520554\pi$
$182$	−1.35366	−0.100340
$183$	0	0
$184$	2.89421	0.213364
$185$	1.67752	0.123334
$186$	0	0
$187$	1.18123	0.0863804
$188$	0.500530	0.0365049
$189$	0	0
$190$	−1.69459	−0.122938
$191$	−7.81311	−0.565337	−0.282669	−	0.959218i	$-0.591220\pi$
$191$	−7.81311	−0.565337	−0.282669	+	0.959218i	$0.591220\pi$
$192$	0	0
$193$	21.3729	1.53846	0.769228	−	0.638974i	$-0.220641\pi$
$193$	21.3729	1.53846	0.769228	+	0.638974i	$0.220641\pi$
$194$	−6.78023	−0.486792
$195$	0	0
$196$	−0.167605	−0.0119718
$197$	8.77457	0.625162	0.312581	−	0.949891i	$-0.398806\pi$
$197$	8.77457	0.625162	0.312581	+	0.949891i	$0.398806\pi$
$198$	0	0
$199$	16.7201	1.18526	0.592629	−	0.805476i	$-0.298090\pi$
$199$	16.7201	1.18526	0.592629	+	0.805476i	$0.298090\pi$
$200$	14.0120	0.990797
$201$	0	0
$202$	16.0409	1.12863
$203$	5.28786	0.371135
$204$	0	0
$205$	−0.167605	−0.0117060
$206$	18.2391	1.27078
$207$	0	0
$208$	3.63670	0.252160
$209$	2.64152	0.182718
$210$	0	0
$211$	23.6166	1.62583	0.812917	−	0.582379i	$-0.197878\pi$
$211$	23.6166	1.62583	0.812917	+	0.582379i	$0.197878\pi$
$212$	−1.14662	−0.0787501
$213$	0	0
$214$	27.8514	1.90388
$215$	−0.0224398	−0.00153038
$216$	0	0
$217$	0.0521702	0.00354154
$218$	10.9238	0.739850
$219$	0	0
$220$	0.0794304	0.00535519
$221$	−1.18123	−0.0794585
$222$	0	0
$223$	7.24279	0.485013	0.242507	−	0.970150i	$-0.422030\pi$
$223$	7.24279	0.485013	0.242507	+	0.970150i	$0.422030\pi$
$224$	0.945545	0.0631769
$225$	0	0
$226$	14.1282	0.939792
$227$	18.2711	1.21270	0.606348	−	0.795199i	$-0.292634\pi$
$227$	18.2711	1.21270	0.606348	+	0.795199i	$0.292634\pi$
$228$	0	0
$229$	12.3729	0.817625	0.408813	−	0.912618i	$-0.365943\pi$
$229$	12.3729	0.817625	0.408813	+	0.912618i	$0.365943\pi$
$230$	0.632776	0.0417240
$231$	0	0
$232$	−15.5156	−1.01865
$233$	−12.7770	−0.837052	−0.418526	−	0.908205i	$-0.637453\pi$
$233$	−12.7770	−0.837052	−0.418526	+	0.908205i	$0.637453\pi$
$234$	0	0
$235$	1.41529	0.0923230
$236$	1.33129	0.0866593
$237$	0	0
$238$	1.59899	0.103647
$239$	−19.9615	−1.29120	−0.645600	−	0.763676i	$-0.723393\pi$
$239$	−19.9615	−1.29120	−0.645600	+	0.763676i	$0.723393\pi$
$240$	0	0
$241$	16.0425	1.03339	0.516695	−	0.856169i	$-0.327162\pi$
$241$	16.0425	1.03339	0.516695	+	0.856169i	$0.327162\pi$
$242$	1.35366	0.0870166
$243$	0	0
$244$	−2.27645	−0.145735
$245$	−0.473915	−0.0302773
$246$	0	0
$247$	−2.64152	−0.168076
$248$	−0.153078	−0.00972045
$249$	0	0
$250$	6.27111	0.396620
$251$	11.1883	0.706202	0.353101	−	0.935585i	$-0.385127\pi$
$251$	11.1883	0.706202	0.353101	+	0.935585i	$0.385127\pi$
$252$	0	0
$253$	−0.986370	−0.0620125
$254$	10.9421	0.686569
$255$	0	0
$256$	−3.99347	−0.249592
$257$	7.13871	0.445300	0.222650	−	0.974898i	$-0.428529\pi$
$257$	7.13871	0.445300	0.222650	+	0.974898i	$0.428529\pi$
$258$	0	0
$259$	−3.53972	−0.219947
$260$	−0.0794304	−0.00492606
$261$	0	0
$262$	0.767403	0.0474103
$263$	−5.97274	−0.368295	−0.184147	−	0.982899i	$-0.558952\pi$
$263$	−5.97274	−0.368295	−0.184147	+	0.982899i	$0.558952\pi$
$264$	0	0
$265$	−3.24215	−0.199164
$266$	3.57572	0.219241
$267$	0	0
$268$	−1.50683	−0.0920441
$269$	−15.2799	−0.931633	−0.465817	−	0.884881i	$-0.654239\pi$
$269$	−15.2799	−0.931633	−0.465817	+	0.884881i	$0.654239\pi$
$270$	0	0
$271$	5.71613	0.347230	0.173615	−	0.984814i	$-0.444455\pi$
$271$	5.71613	0.347230	0.173615	+	0.984814i	$0.444455\pi$
$272$	−4.29580	−0.260471
$273$	0	0
$274$	−20.2158	−1.22128
$275$	−4.77540	−0.287968
$276$	0	0
$277$	5.42174	0.325761	0.162881	−	0.986646i	$-0.447921\pi$
$277$	5.42174	0.325761	0.162881	+	0.986646i	$0.447921\pi$
$278$	28.5494	1.71228
$279$	0	0
$280$	1.39056	0.0831019
$281$	−15.4257	−0.920222	−0.460111	−	0.887861i	$-0.652190\pi$
$281$	−15.4257	−0.920222	−0.460111	+	0.887861i	$0.652190\pi$
$282$	0	0
$283$	5.59417	0.332539	0.166269	−	0.986080i	$-0.446828\pi$
$283$	5.59417	0.332539	0.166269	+	0.986080i	$0.446828\pi$
$284$	1.56045	0.0925957
$285$	0	0
$286$	−1.35366	−0.0800436
$287$	0.353660	0.0208759
$288$	0	0
$289$	−15.6047	−0.917923
$290$	−3.39227	−0.199201
$291$	0	0
$292$	0.186952	0.0109405
$293$	5.29503	0.309339	0.154669	−	0.987966i	$-0.450569\pi$
$293$	5.29503	0.309339	0.154669	+	0.987966i	$0.450569\pi$
$294$	0	0
$295$	3.76431	0.219167
$296$	10.3862	0.603687
$297$	0	0
$298$	−8.56045	−0.495894
$299$	0.986370	0.0570432
$300$	0	0
$301$	0.0473498	0.00272920
$302$	−1.44582	−0.0831978
$303$	0	0
$304$	−9.60641	−0.550966
$305$	−6.43683	−0.368572
$306$	0	0
$307$	5.06410	0.289023	0.144512	−	0.989503i	$-0.453839\pi$
$307$	5.06410	0.289023	0.144512	+	0.989503i	$0.453839\pi$
$308$	−0.167605	−0.00955017
$309$	0	0
$310$	−0.0334682	−0.00190087
$311$	−26.5781	−1.50710	−0.753552	−	0.657388i	$-0.771661\pi$
$311$	−26.5781	−1.50710	−0.753552	+	0.657388i	$0.771661\pi$
$312$	0	0
$313$	28.3696	1.60354	0.801772	−	0.597631i	$-0.203891\pi$
$313$	28.3696	1.60354	0.801772	+	0.597631i	$0.203891\pi$
$314$	−7.92546	−0.447259
$315$	0	0
$316$	0.0303758	0.00170877
$317$	34.1772	1.91958	0.959792	−	0.280712i	$-0.0905706\pi$
$317$	34.1772	1.91958	0.959792	+	0.280712i	$0.0905706\pi$
$318$	0	0
$319$	5.28786	0.296063
$320$	−4.05356	−0.226601
$321$	0	0
$322$	−1.33521	−0.0744083
$323$	3.12025	0.173616
$324$	0	0
$325$	4.77540	0.264892
$326$	1.37520	0.0761654
$327$	0	0
$328$	−1.03771	−0.0572979
$329$	−2.98637	−0.164644
$330$	0	0
$331$	−14.6222	−0.803711	−0.401855	−	0.915703i	$-0.631635\pi$
$331$	−14.6222	−0.803711	−0.401855	+	0.915703i	$0.631635\pi$
$332$	0.249320	0.0136832
$333$	0	0
$334$	−15.0899	−0.825685
$335$	−4.26067	−0.232785
$336$	0	0
$337$	5.85059	0.318702	0.159351	−	0.987222i	$-0.449060\pi$
$337$	5.85059	0.318702	0.159351	+	0.987222i	$0.449060\pi$
$338$	1.35366	0.0736294
$339$	0	0
$340$	0.0938259	0.00508843
$341$	0.0521702	0.00282517
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.138934	−0.00749080
$345$	0	0
$346$	9.08252	0.488279
$347$	22.8628	1.22734	0.613670	−	0.789563i	$-0.289693\pi$
$347$	22.8628	1.22734	0.613670	+	0.789563i	$0.289693\pi$
$348$	0	0
$349$	32.5049	1.73995	0.869974	−	0.493098i	$-0.164135\pi$
$349$	32.5049	1.73995	0.869974	+	0.493098i	$0.164135\pi$
$350$	−6.46427	−0.345530
$351$	0	0
$352$	0.945545	0.0503977
$353$	−12.0674	−0.642285	−0.321142	−	0.947031i	$-0.604067\pi$
$353$	−12.0674	−0.642285	−0.321142	+	0.947031i	$0.604067\pi$
$354$	0	0
$355$	4.41229	0.234180
$356$	2.85452	0.151289
$357$	0	0
$358$	14.1846	0.749682
$359$	3.91909	0.206842	0.103421	−	0.994638i	$-0.467021\pi$
$359$	3.91909	0.206842	0.103421	+	0.994638i	$0.467021\pi$
$360$	0	0
$361$	−12.0224	−0.632757
$362$	−2.35025	−0.123526
$363$	0	0
$364$	0.167605	0.00878487
$365$	0.528621	0.0276693
$366$	0	0
$367$	−16.3095	−0.851347	−0.425674	−	0.904877i	$-0.639963\pi$
$367$	−16.3095	−0.851347	−0.425674	+	0.904877i	$0.639963\pi$
$368$	3.58713	0.186992
$369$	0	0
$370$	2.27080	0.118053
$371$	6.84120	0.355178
$372$	0	0
$373$	24.8556	1.28698	0.643488	−	0.765456i	$-0.277487\pi$
$373$	24.8556	1.28698	0.643488	+	0.765456i	$0.277487\pi$
$374$	1.59899	0.0826818
$375$	0	0
$376$	8.76261	0.451897
$377$	−5.28786	−0.272339
$378$	0	0
$379$	−3.15227	−0.161921	−0.0809606	−	0.996717i	$-0.525799\pi$
$379$	−3.15227	−0.161921	−0.0809606	+	0.996717i	$0.525799\pi$
$380$	0.209817	0.0107634
$381$	0	0
$382$	−10.5763	−0.541131
$383$	16.3664	0.836284	0.418142	−	0.908382i	$-0.362682\pi$
$383$	16.3664	0.836284	0.418142	+	0.908382i	$0.362682\pi$
$384$	0	0
$385$	−0.473915	−0.0241529
$386$	28.9317	1.47258
$387$	0	0
$388$	0.839500	0.0426192
$389$	−20.7383	−1.05147	−0.525737	−	0.850647i	$-0.676210\pi$
$389$	−20.7383	−1.05147	−0.525737	+	0.850647i	$0.676210\pi$
$390$	0	0
$391$	−1.16513	−0.0589234
$392$	−2.93420	−0.148199
$393$	0	0
$394$	11.8778	0.598394
$395$	0.0858899	0.00432159
$396$	0	0
$397$	−31.1217	−1.56195	−0.780977	−	0.624560i	$-0.785278\pi$
$397$	−31.1217	−1.56195	−0.780977	+	0.624560i	$0.785278\pi$
$398$	22.6334	1.13451
$399$	0	0
$400$	17.3667	0.868336
$401$	18.0995	0.903847	0.451923	−	0.892057i	$-0.350738\pi$
$401$	18.0995	0.903847	0.451923	+	0.892057i	$0.350738\pi$
$402$	0	0
$403$	−0.0521702	−0.00259878
$404$	−1.98612	−0.0988131
$405$	0	0
$406$	7.15796	0.355244
$407$	−3.53972	−0.175457
$408$	0	0
$409$	−0.425920	−0.0210604	−0.0105302	−	0.999945i	$-0.503352\pi$
$409$	−0.425920	−0.0210604	−0.0105302	+	0.999945i	$0.503352\pi$
$410$	−0.226880	−0.0112048
$411$	0	0
$412$	−2.25829	−0.111258
$413$	−7.94301	−0.390850
$414$	0	0
$415$	0.704970	0.0346056
$416$	−0.945545	−0.0463592
$417$	0	0
$418$	3.57572	0.174894
$419$	4.16513	0.203480	0.101740	−	0.994811i	$-0.467559\pi$
$419$	4.16513	0.203480	0.101740	+	0.994811i	$0.467559\pi$
$420$	0	0
$421$	−13.7610	−0.670671	−0.335335	−	0.942099i	$-0.608850\pi$
$421$	−13.7610	−0.670671	−0.335335	+	0.942099i	$0.608850\pi$
$422$	31.9689	1.55622
$423$	0	0
$424$	−20.0735	−0.974853
$425$	−5.64087	−0.273623
$426$	0	0
$427$	13.5822	0.657291
$428$	−3.44845	−0.166687
$429$	0	0
$430$	−0.0303758	−0.00146485
$431$	19.9214	0.959579	0.479789	−	0.877384i	$-0.340713\pi$
$431$	19.9214	0.959579	0.479789	+	0.877384i	$0.340713\pi$
$432$	0	0
$433$	38.2295	1.83719	0.918595	−	0.395200i	$-0.129325\pi$
$433$	38.2295	1.83719	0.918595	+	0.395200i	$0.129325\pi$
$434$	0.0706207	0.00338990
$435$	0	0
$436$	−1.35253	−0.0647747
$437$	−2.60552	−0.124639
$438$	0	0
$439$	21.5927	1.03056	0.515282	−	0.857021i	$-0.327687\pi$
$439$	21.5927	1.03056	0.515282	+	0.857021i	$0.327687\pi$
$440$	1.39056	0.0662924
$441$	0	0
$442$	−1.59899	−0.0760562
$443$	−23.9638	−1.13855	−0.569277	−	0.822145i	$-0.692777\pi$
$443$	−23.9638	−1.13855	−0.569277	+	0.822145i	$0.692777\pi$
$444$	0	0
$445$	8.07136	0.382619
$446$	9.80428	0.464246
$447$	0	0
$448$	8.55335	0.404108
$449$	17.7232	0.836411	0.418206	−	0.908352i	$-0.362659\pi$
$449$	17.7232	0.836411	0.418206	+	0.908352i	$0.362659\pi$
$450$	0	0
$451$	0.353660	0.0166532
$452$	−1.74929	−0.0822798
$453$	0	0
$454$	24.7329	1.16077
$455$	0.473915	0.0222175
$456$	0	0
$457$	14.4056	0.673865	0.336932	−	0.941529i	$-0.390611\pi$
$457$	14.4056	0.673865	0.336932	+	0.941529i	$0.390611\pi$
$458$	16.7487	0.782616
$459$	0	0
$460$	−0.0783477	−0.00365298
$461$	12.6237	0.587945	0.293972	−	0.955814i	$-0.405023\pi$
$461$	12.6237	0.587945	0.293972	+	0.955814i	$0.405023\pi$
$462$	0	0
$463$	−15.7488	−0.731908	−0.365954	−	0.930633i	$-0.619257\pi$
$463$	−15.7488	−0.731908	−0.365954	+	0.930633i	$0.619257\pi$
$464$	−19.2304	−0.892747
$465$	0	0
$466$	−17.2958	−0.801211
$467$	−10.9390	−0.506197	−0.253099	−	0.967440i	$-0.581450\pi$
$467$	−10.9390	−0.506197	−0.253099	+	0.967440i	$0.581450\pi$
$468$	0	0
$469$	8.99036	0.415136
$470$	1.91581	0.0883700
$471$	0	0
$472$	23.3064	1.07276
$473$	0.0473498	0.00217715
$474$	0	0
$475$	−12.6143	−0.578785
$476$	−0.197981	−0.00907442
$477$	0	0
$478$	−27.0210	−1.23591
$479$	2.35913	0.107791	0.0538956	−	0.998547i	$-0.482836\pi$
$479$	2.35913	0.107791	0.0538956	+	0.998547i	$0.482836\pi$
$480$	0	0
$481$	3.53972	0.161397
$482$	21.7161	0.989143
$483$	0	0
$484$	−0.167605	−0.00761840
$485$	2.37375	0.107786
$486$	0	0
$487$	−15.4915	−0.701985	−0.350993	−	0.936378i	$-0.614156\pi$
$487$	−15.4915	−0.701985	−0.350993	+	0.936378i	$0.614156\pi$
$488$	−39.8530	−1.80406
$489$	0	0
$490$	−0.641520	−0.0289809
$491$	−33.6501	−1.51861	−0.759305	−	0.650735i	$-0.774461\pi$
$491$	−33.6501	−1.51861	−0.759305	+	0.650735i	$0.774461\pi$
$492$	0	0
$493$	6.24620	0.281315
$494$	−3.57572	−0.160879
$495$	0	0
$496$	−0.189727	−0.00851901
$497$	−9.31030	−0.417624
$498$	0	0
$499$	0.679994	0.0304407	0.0152204	−	0.999884i	$-0.495155\pi$
$499$	0.679994	0.0304407	0.0152204	+	0.999884i	$0.495155\pi$
$500$	−0.776464	−0.0347245
$501$	0	0
$502$	15.1452	0.675964
$503$	−19.4626	−0.867793	−0.433897	−	0.900963i	$-0.642862\pi$
$503$	−19.4626	−0.867793	−0.433897	+	0.900963i	$0.642862\pi$
$504$	0	0
$505$	−5.61590	−0.249904
$506$	−1.33521	−0.0593573
$507$	0	0
$508$	−1.35481	−0.0601099
$509$	15.3208	0.679083	0.339542	−	0.940591i	$-0.389728\pi$
$509$	15.3208	0.679083	0.339542	+	0.940591i	$0.389728\pi$
$510$	0	0
$511$	−1.11543	−0.0493439
$512$	−24.7803	−1.09514
$513$	0	0
$514$	9.66338	0.426233
$515$	−6.38549	−0.281378
$516$	0	0
$517$	−2.98637	−0.131340
$518$	−4.79157	−0.210530
$519$	0	0
$520$	−1.39056	−0.0609801
$521$	25.4651	1.11565	0.557823	−	0.829960i	$-0.311637\pi$
$521$	25.4651	1.11565	0.557823	+	0.829960i	$0.311637\pi$
$522$	0	0
$523$	−8.05680	−0.352299	−0.176150	−	0.984363i	$-0.556364\pi$
$523$	−8.05680	−0.352299	−0.176150	+	0.984363i	$0.556364\pi$
$524$	−0.0950167	−0.00415083
$525$	0	0
$526$	−8.08506	−0.352525
$527$	0.0616252	0.00268444
$528$	0	0
$529$	−22.0271	−0.957699
$530$	−4.38877	−0.190636
$531$	0	0
$532$	−0.442731	−0.0191948
$533$	−0.353660	−0.0153187
$534$	0	0
$535$	−9.75075	−0.421562
$536$	−26.3795	−1.13942
$537$	0	0
$538$	−20.6838	−0.891743
$539$	1.00000	0.0430730
$540$	0	0
$541$	37.4875	1.61171	0.805857	−	0.592110i	$-0.201705\pi$
$541$	37.4875	1.61171	0.805857	+	0.592110i	$0.201705\pi$
$542$	7.73770	0.332362
$543$	0	0
$544$	1.11691	0.0478872
$545$	−3.82439	−0.163819
$546$	0	0
$547$	−41.6968	−1.78283	−0.891414	−	0.453189i	$-0.850286\pi$
$547$	−41.6968	−1.78283	−0.891414	+	0.453189i	$0.850286\pi$
$548$	2.50304	0.106925
$549$	0	0
$550$	−6.46427	−0.275638
$551$	13.9680	0.595056
$552$	0	0
$553$	−0.181235	−0.00770689
$554$	7.33920	0.311813
$555$	0	0
$556$	−3.53487	−0.149912
$557$	−33.6785	−1.42700	−0.713501	−	0.700654i	$-0.752892\pi$
$557$	−33.6785	−1.42700	−0.713501	+	0.700654i	$0.752892\pi$
$558$	0	0
$559$	−0.0473498	−0.00200268
$560$	1.72349	0.0728306
$561$	0	0
$562$	−20.8812	−0.880820
$563$	1.55399	0.0654929	0.0327464	−	0.999464i	$-0.489575\pi$
$563$	1.55399	0.0654929	0.0327464	+	0.999464i	$0.489575\pi$
$564$	0	0
$565$	−4.94626	−0.208091
$566$	7.57260	0.318300
$567$	0	0
$568$	27.3183	1.14625
$569$	11.8653	0.497421	0.248711	−	0.968578i	$-0.419993\pi$
$569$	11.8653	0.497421	0.248711	+	0.968578i	$0.419993\pi$
$570$	0	0
$571$	11.7587	0.492085	0.246042	−	0.969259i	$-0.420870\pi$
$571$	11.7587	0.492085	0.246042	+	0.969259i	$0.420870\pi$
$572$	0.167605	0.00700791
$573$	0	0
$574$	0.478735	0.0199820
$575$	4.71032	0.196434
$576$	0	0
$577$	−23.8250	−0.991849	−0.495925	−	0.868366i	$-0.665171\pi$
$577$	−23.8250	−0.991849	−0.495925	+	0.868366i	$0.665171\pi$
$578$	−21.1234	−0.878619
$579$	0	0
$580$	0.420017	0.0174402
$581$	−1.48754	−0.0617138
$582$	0	0
$583$	6.84120	0.283334
$584$	3.27291	0.135434
$585$	0	0
$586$	7.16767	0.296094
$587$	12.3650	0.510358	0.255179	−	0.966894i	$-0.417866\pi$
$587$	12.3650	0.510358	0.255179	+	0.966894i	$0.417866\pi$
$588$	0	0
$589$	0.137809	0.00567830
$590$	5.09560	0.209782
$591$	0	0
$592$	12.8729	0.529072
$593$	19.6141	0.805453	0.402727	−	0.915320i	$-0.368062\pi$
$593$	19.6141	0.805453	0.402727	+	0.915320i	$0.368062\pi$
$594$	0	0
$595$	−0.559805	−0.0229498
$596$	1.05992	0.0434160
$597$	0	0
$598$	1.33521	0.0546008
$599$	−1.48272	−0.0605825	−0.0302912	−	0.999541i	$-0.509643\pi$
$599$	−1.48272	−0.0605825	−0.0302912	+	0.999541i	$0.509643\pi$
$600$	0	0
$601$	7.06972	0.288380	0.144190	−	0.989550i	$-0.453942\pi$
$601$	7.06972	0.288380	0.144190	+	0.989550i	$0.453942\pi$
$602$	0.0640955	0.00261234
$603$	0	0
$604$	0.179016	0.00728406
$605$	−0.473915	−0.0192674
$606$	0	0
$607$	−11.8348	−0.480361	−0.240181	−	0.970728i	$-0.577207\pi$
$607$	−11.8348	−0.480361	−0.240181	+	0.970728i	$0.577207\pi$
$608$	2.49768	0.101294
$609$	0	0
$610$	−8.71328	−0.352790
$611$	2.98637	0.120816
$612$	0	0
$613$	−5.79697	−0.234137	−0.117069	−	0.993124i	$-0.537350\pi$
$613$	−5.79697	−0.234137	−0.117069	+	0.993124i	$0.537350\pi$
$614$	6.85506	0.276648
$615$	0	0
$616$	−2.93420	−0.118222
$617$	−30.7443	−1.23772	−0.618859	−	0.785502i	$-0.712405\pi$
$617$	−30.7443	−1.23772	−0.618859	+	0.785502i	$0.712405\pi$
$618$	0	0
$619$	11.4635	0.460756	0.230378	−	0.973101i	$-0.426004\pi$
$619$	11.4635	0.460756	0.230378	+	0.973101i	$0.426004\pi$
$620$	0.00414390	0.000166423 0
$621$	0	0
$622$	−35.9777	−1.44257
$623$	−17.0312	−0.682343
$624$	0	0
$625$	21.6815	0.867261
$626$	38.4028	1.53488
$627$	0	0
$628$	0.981298	0.0391581
$629$	−4.18123	−0.166717
$630$	0	0
$631$	−2.57007	−0.102313	−0.0511564	−	0.998691i	$-0.516291\pi$
$631$	−2.57007	−0.102313	−0.0511564	+	0.998691i	$0.516291\pi$
$632$	0.531779	0.0211530
$633$	0	0
$634$	46.2643	1.83739
$635$	−3.83082	−0.152022
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	7.15796	0.283387
$639$	0	0
$640$	−4.59092	−0.181472
$641$	13.7626	0.543590	0.271795	−	0.962355i	$-0.412383\pi$
$641$	13.7626	0.543590	0.271795	+	0.962355i	$0.412383\pi$
$642$	0	0
$643$	20.9175	0.824904	0.412452	−	0.910979i	$-0.364672\pi$
$643$	20.9175	0.824904	0.412452	+	0.910979i	$0.364672\pi$
$644$	0.165320	0.00651453
$645$	0	0
$646$	4.22376	0.166182
$647$	−40.5363	−1.59365	−0.796824	−	0.604211i	$-0.793488\pi$
$647$	−40.5363	−1.59365	−0.796824	+	0.604211i	$0.793488\pi$
$648$	0	0
$649$	−7.94301	−0.311790
$650$	6.46427	0.253550
$651$	0	0
$652$	−0.170272	−0.00666837
$653$	35.8595	1.40329	0.701645	−	0.712527i	$-0.252449\pi$
$653$	35.8595	1.40329	0.701645	+	0.712527i	$0.252449\pi$
$654$	0	0
$655$	−0.268667	−0.0104977
$656$	−1.28615	−0.0502159
$657$	0	0
$658$	−4.04253	−0.157594
$659$	−5.75779	−0.224291	−0.112146	−	0.993692i	$-0.535772\pi$
$659$	−5.75779	−0.224291	−0.112146	+	0.993692i	$0.535772\pi$
$660$	0	0
$661$	−17.1251	−0.666087	−0.333044	−	0.942911i	$-0.608076\pi$
$661$	−17.1251	−0.666087	−0.333044	+	0.942911i	$0.608076\pi$
$662$	−19.7935	−0.769298
$663$	0	0
$664$	4.36475	0.169385
$665$	−1.25186	−0.0485449
$666$	0	0
$667$	−5.21579	−0.201956
$668$	1.86838	0.0722896
$669$	0	0
$670$	−5.76749	−0.222818
$671$	13.5822	0.524337
$672$	0	0
$673$	19.9437	0.768771	0.384386	−	0.923173i	$-0.374413\pi$
$673$	19.9437	0.768771	0.384386	+	0.923173i	$0.374413\pi$
$674$	7.91972	0.305056
$675$	0	0
$676$	−0.167605	−0.00644634
$677$	−23.6637	−0.909470	−0.454735	−	0.890627i	$-0.650266\pi$
$677$	−23.6637	−0.909470	−0.454735	+	0.890627i	$0.650266\pi$
$678$	0	0
$679$	−5.00881	−0.192220
$680$	1.64258	0.0629900
$681$	0	0
$682$	0.0706207	0.00270421
$683$	13.3715	0.511645	0.255822	−	0.966724i	$-0.417654\pi$
$683$	13.3715	0.511645	0.255822	+	0.966724i	$0.417654\pi$
$684$	0	0
$685$	7.07754	0.270419
$686$	1.35366	0.0516830
$687$	0	0
$688$	−0.172197	−0.00656495
$689$	−6.84120	−0.260629
$690$	0	0
$691$	−15.8072	−0.601336	−0.300668	−	0.953729i	$-0.597210\pi$
$691$	−15.8072	−0.601336	−0.300668	+	0.953729i	$0.597210\pi$
$692$	−1.12456	−0.0427494
$693$	0	0
$694$	30.9485	1.17479
$695$	−9.99511	−0.379136
$696$	0	0
$697$	0.417755	0.0158236
$698$	44.0006	1.66545
$699$	0	0
$700$	0.800380	0.0302515
$701$	−22.2221	−0.839316	−0.419658	−	0.907682i	$-0.637850\pi$
$701$	−22.2221	−0.839316	−0.419658	+	0.907682i	$0.637850\pi$
$702$	0	0
$703$	−9.35023	−0.352650
$704$	8.55335	0.322366
$705$	0	0
$706$	−16.3352	−0.614784
$707$	11.8500	0.445666
$708$	0	0
$709$	−30.8556	−1.15881	−0.579403	−	0.815041i	$-0.696714\pi$
$709$	−30.8556	−1.15881	−0.579403	+	0.815041i	$0.696714\pi$
$710$	5.97274	0.224153
$711$	0	0
$712$	49.9731	1.87282
$713$	−0.0514591	−0.00192716
$714$	0	0
$715$	0.473915	0.0177234
$716$	−1.75629	−0.0656355
$717$	0	0
$718$	5.30512	0.197985
$719$	−28.9011	−1.07783	−0.538915	−	0.842360i	$-0.681165\pi$
$719$	−28.9011	−1.07783	−0.538915	+	0.842360i	$0.681165\pi$
$720$	0	0
$721$	13.4739	0.501795
$722$	−16.2742	−0.605663
$723$	0	0
$724$	0.290999	0.0108149
$725$	−25.2517	−0.937823
$726$	0	0
$727$	−47.9941	−1.78000	−0.890001	−	0.455958i	$-0.849297\pi$
$727$	−47.9941	−1.78000	−0.890001	+	0.455958i	$0.849297\pi$
$728$	2.93420	0.108749
$729$	0	0
$730$	0.715573	0.0264846
$731$	0.0559312	0.00206869
$732$	0	0
$733$	7.04742	0.260302	0.130151	−	0.991494i	$-0.458454\pi$
$733$	7.04742	0.260302	0.130151	+	0.991494i	$0.458454\pi$
$734$	−22.0775	−0.814894
$735$	0	0
$736$	−0.932657	−0.0343782
$737$	8.99036	0.331164
$738$	0	0
$739$	−18.4916	−0.680225	−0.340112	−	0.940385i	$-0.610465\pi$
$739$	−18.4916	−0.680225	−0.340112	+	0.940385i	$0.610465\pi$
$740$	−0.281161	−0.0103357
$741$	0	0
$742$	9.26067	0.339970
$743$	−24.0433	−0.882064	−0.441032	−	0.897491i	$-0.645388\pi$
$743$	−24.0433	−0.882064	−0.441032	+	0.897491i	$0.645388\pi$
$744$	0	0
$745$	2.99700	0.109802
$746$	33.6461	1.23187
$747$	0	0
$748$	−0.197981	−0.00723889
$749$	20.5749	0.751790
$750$	0	0
$751$	4.53091	0.165335	0.0826675	−	0.996577i	$-0.473656\pi$
$751$	4.53091	0.165335	0.0826675	+	0.996577i	$0.473656\pi$
$752$	10.8605	0.396043
$753$	0	0
$754$	−7.15796	−0.260678
$755$	0.506181	0.0184218
$756$	0	0
$757$	−34.7322	−1.26236	−0.631182	−	0.775635i	$-0.717430\pi$
$757$	−34.7322	−1.26236	−0.631182	+	0.775635i	$0.717430\pi$
$758$	−4.26710	−0.154988
$759$	0	0
$760$	3.67319	0.133241
$761$	34.3055	1.24357	0.621786	−	0.783187i	$-0.286407\pi$
$761$	34.3055	1.24357	0.621786	+	0.783187i	$0.286407\pi$
$762$	0	0
$763$	8.06979	0.292146
$764$	1.30951	0.0473766
$765$	0	0
$766$	22.1545	0.800476
$767$	7.94301	0.286805
$768$	0	0
$769$	37.6709	1.35845	0.679223	−	0.733932i	$-0.262317\pi$
$769$	37.6709	1.35845	0.679223	+	0.733932i	$0.262317\pi$
$770$	−0.641520	−0.0231188
$771$	0	0
$772$	−3.58220	−0.128926
$773$	29.0848	1.04611	0.523054	−	0.852300i	$-0.324793\pi$
$773$	29.0848	1.04611	0.523054	+	0.852300i	$0.324793\pi$
$774$	0	0
$775$	−0.249134	−0.00894915
$776$	14.6968	0.527586
$777$	0	0
$778$	−28.0726	−1.00645
$779$	0.934200	0.0334712
$780$	0	0
$781$	−9.31030	−0.333149
$782$	−1.57720	−0.0564004
$783$	0	0
$784$	−3.63670	−0.129882
$785$	2.77469	0.0990331
$786$	0	0
$787$	−22.1978	−0.791266	−0.395633	−	0.918409i	$-0.629475\pi$
$787$	−22.1978	−0.791266	−0.395633	+	0.918409i	$0.629475\pi$
$788$	−1.47066	−0.0523901
$789$	0	0
$790$	0.116266	0.00413655
$791$	10.4370	0.371097
$792$	0	0
$793$	−13.5822	−0.482320
$794$	−42.1282	−1.49507
$795$	0	0
$796$	−2.80237	−0.0993274
$797$	4.82352	0.170858	0.0854289	−	0.996344i	$-0.472774\pi$
$797$	4.82352	0.170858	0.0854289	+	0.996344i	$0.472774\pi$
$798$	0	0
$799$	−3.52760	−0.124798
$800$	−4.51536	−0.159642
$801$	0	0
$802$	24.5006	0.865146
$803$	−1.11543	−0.0393628
$804$	0	0
$805$	0.467455	0.0164756
$806$	−0.0706207	−0.00248751
$807$	0	0
$808$	−34.7703	−1.22322
$809$	52.6654	1.85162	0.925808	−	0.377994i	$-0.123386\pi$
$809$	52.6654	1.85162	0.925808	+	0.377994i	$0.123386\pi$
$810$	0	0
$811$	55.0245	1.93217	0.966086	−	0.258220i	$-0.0831361\pi$
$811$	55.0245	1.93217	0.966086	+	0.258220i	$0.0831361\pi$
$812$	−0.886270	−0.0311020
$813$	0	0
$814$	−4.79157	−0.167944
$815$	−0.481457	−0.0168647
$816$	0	0
$817$	0.125075	0.00437583
$818$	−0.576551	−0.0201586
$819$	0	0
$820$	0.0280913	0.000980992 0
$821$	−4.65203	−0.162357	−0.0811786	−	0.996700i	$-0.525868\pi$
$821$	−4.65203	−0.162357	−0.0811786	+	0.996700i	$0.525868\pi$
$822$	0	0
$823$	19.5052	0.679910	0.339955	−	0.940442i	$-0.389588\pi$
$823$	19.5052	0.679910	0.339955	+	0.940442i	$0.389588\pi$
$824$	−39.5352	−1.37727
$825$	0	0
$826$	−10.7521	−0.374115
$827$	−44.8347	−1.55906	−0.779528	−	0.626367i	$-0.784541\pi$
$827$	−44.8347	−1.55906	−0.779528	+	0.626367i	$0.784541\pi$
$828$	0	0
$829$	−5.90643	−0.205139	−0.102569	−	0.994726i	$-0.532706\pi$
$829$	−5.90643	−0.205139	−0.102569	+	0.994726i	$0.532706\pi$
$830$	0.954289	0.0331239
$831$	0	0
$832$	−8.55335	−0.296534
$833$	1.18123	0.0409274
$834$	0	0
$835$	5.28297	0.182825
$836$	−0.442731	−0.0153122
$837$	0	0
$838$	5.63818	0.194768
$839$	35.4274	1.22309	0.611546	−	0.791209i	$-0.290548\pi$
$839$	35.4274	1.22309	0.611546	+	0.791209i	$0.290548\pi$
$840$	0	0
$841$	−1.03854	−0.0358117
$842$	−18.6277	−0.641954
$843$	0	0
$844$	−3.95826	−0.136249
$845$	−0.473915	−0.0163032
$846$	0	0
$847$	1.00000	0.0343604
$848$	−24.8794	−0.854362
$849$	0	0
$850$	−7.63583	−0.261907
$851$	3.49147	0.119686
$852$	0	0
$853$	−33.8898	−1.16037	−0.580183	−	0.814486i	$-0.697019\pi$
$853$	−33.8898	−1.16037	−0.580183	+	0.814486i	$0.697019\pi$
$854$	18.3857	0.629147
$855$	0	0
$856$	−60.3708	−2.06343
$857$	−44.2992	−1.51323	−0.756616	−	0.653860i	$-0.773149\pi$
$857$	−44.2992	−1.51323	−0.756616	+	0.653860i	$0.773149\pi$
$858$	0	0
$859$	8.35684	0.285132	0.142566	−	0.989785i	$-0.454465\pi$
$859$	8.35684	0.285132	0.142566	+	0.989785i	$0.454465\pi$
$860$	0.00376101	0.000128249 0
$861$	0	0
$862$	26.9668	0.918492
$863$	−6.61015	−0.225012	−0.112506	−	0.993651i	$-0.535888\pi$
$863$	−6.61015	−0.225012	−0.112506	+	0.993651i	$0.535888\pi$
$864$	0	0
$865$	−3.17978	−0.108116
$866$	51.7497	1.75853
$867$	0	0
$868$	−0.00874397	−0.000296790 0
$869$	−0.181235	−0.00614797
$870$	0	0
$871$	−8.99036	−0.304627
$872$	−23.6784	−0.801851
$873$	0	0
$874$	−3.52698	−0.119302
$875$	4.63271	0.156614
$876$	0	0
$877$	−46.8073	−1.58057	−0.790285	−	0.612739i	$-0.790068\pi$
$877$	−46.8073	−1.58057	−0.790285	+	0.612739i	$0.790068\pi$
$878$	29.2292	0.986437
$879$	0	0
$880$	1.72349	0.0580987
$881$	−12.5751	−0.423667	−0.211834	−	0.977306i	$-0.567943\pi$
$881$	−12.5751	−0.423667	−0.211834	+	0.977306i	$0.567943\pi$
$882$	0	0
$883$	−3.38256	−0.113832	−0.0569161	−	0.998379i	$-0.518127\pi$
$883$	−3.38256	−0.113832	−0.0569161	+	0.998379i	$0.518127\pi$
$884$	0.197981	0.00665881
$885$	0	0
$886$	−32.4389	−1.08980
$887$	33.0928	1.11115	0.555574	−	0.831467i	$-0.312499\pi$
$887$	33.0928	1.11115	0.555574	+	0.831467i	$0.312499\pi$
$888$	0	0
$889$	8.08335	0.271107
$890$	10.9259	0.366236
$891$	0	0
$892$	−1.21393	−0.0406453
$893$	−7.88855	−0.263980
$894$	0	0
$895$	−4.96603	−0.165996
$896$	9.68723	0.323628
$897$	0	0
$898$	23.9912	0.800598
$899$	0.275869	0.00920074
$900$	0	0
$901$	8.08107	0.269219
$902$	0.478735	0.0159401
$903$	0	0
$904$	−30.6243	−1.01855
$905$	0.822820	0.0273515
$906$	0	0
$907$	28.3535	0.941462	0.470731	−	0.882277i	$-0.343990\pi$
$907$	28.3535	0.941462	0.470731	+	0.882277i	$0.343990\pi$
$908$	−3.06233	−0.101627
$909$	0	0
$910$	0.641520	0.0212662
$911$	−2.92938	−0.0970547	−0.0485273	−	0.998822i	$-0.515453\pi$
$911$	−2.92938	−0.0970547	−0.0485273	+	0.998822i	$0.515453\pi$
$912$	0	0
$913$	−1.48754	−0.0492306
$914$	19.5003	0.645011
$915$	0	0
$916$	−2.07376	−0.0685189
$917$	0.566910	0.0187210
$918$	0	0
$919$	−33.4804	−1.10441	−0.552207	−	0.833707i	$-0.686214\pi$
$919$	−33.4804	−1.10441	−0.552207	+	0.833707i	$0.686214\pi$
$920$	−1.37161	−0.0452206
$921$	0	0
$922$	17.0882	0.562770
$923$	9.31030	0.306452
$924$	0	0
$925$	16.9036	0.555786
$926$	−21.3185	−0.700570
$927$	0	0
$928$	4.99991	0.164130
$929$	−57.6855	−1.89260	−0.946301	−	0.323288i	$-0.895212\pi$
$929$	−57.6855	−1.89260	−0.946301	+	0.323288i	$0.895212\pi$
$930$	0	0
$931$	2.64152	0.0865723
$932$	2.14149	0.0701469
$933$	0	0
$934$	−14.8077	−0.484523
$935$	−0.559805	−0.0183076
$936$	0	0
$937$	9.35919	0.305751	0.152876	−	0.988245i	$-0.451147\pi$
$937$	9.35919	0.305751	0.152876	+	0.988245i	$0.451147\pi$
$938$	12.1699	0.397361
$939$	0	0
$940$	−0.237208	−0.00773689
$941$	58.2965	1.90041	0.950205	−	0.311625i	$-0.100873\pi$
$941$	58.2965	1.90041	0.950205	+	0.311625i	$0.100873\pi$
$942$	0	0
$943$	−0.348840	−0.0113598
$944$	28.8863	0.940170
$945$	0	0
$946$	0.0640955	0.00208392
$947$	3.89068	0.126430	0.0632150	−	0.998000i	$-0.479865\pi$
$947$	3.89068	0.126430	0.0632150	+	0.998000i	$0.479865\pi$
$948$	0	0
$949$	1.11543	0.0362085
$950$	−17.0755	−0.554003
$951$	0	0
$952$	−3.46598	−0.112333
$953$	−23.8932	−0.773976	−0.386988	−	0.922085i	$-0.626485\pi$
$953$	−23.8932	−0.773976	−0.386988	+	0.922085i	$0.626485\pi$
$954$	0	0
$955$	3.70275	0.119818
$956$	3.34563	0.108206
$957$	0	0
$958$	3.19345	0.103176
$959$	−14.9342	−0.482250
$960$	0	0
$961$	−30.9973	−0.999912
$962$	4.79157	0.154486
$963$	0	0
$964$	−2.68880	−0.0866005
$965$	−10.1289	−0.326062
$966$	0	0
$967$	16.2079	0.521211	0.260606	−	0.965445i	$-0.416078\pi$
$967$	16.2079	0.521211	0.260606	+	0.965445i	$0.416078\pi$
$968$	−2.93420	−0.0943088
$969$	0	0
$970$	3.21325	0.103171
$971$	47.8091	1.53427	0.767134	−	0.641487i	$-0.221682\pi$
$971$	47.8091	1.53427	0.767134	+	0.641487i	$0.221682\pi$
$972$	0	0
$973$	21.0905	0.676131
$974$	−20.9702	−0.671928
$975$	0	0
$976$	−49.3945	−1.58108
$977$	37.9893	1.21538	0.607692	−	0.794173i	$-0.292095\pi$
$977$	37.9893	1.21538	0.607692	+	0.794173i	$0.292095\pi$
$978$	0	0
$979$	−17.0312	−0.544321
$980$	0.0794304	0.00253731
$981$	0	0
$982$	−45.5509	−1.45359
$983$	12.1682	0.388105	0.194052	−	0.980991i	$-0.437837\pi$
$983$	12.1682	0.388105	0.194052	+	0.980991i	$0.437837\pi$
$984$	0	0
$985$	−4.15840	−0.132498
$986$	8.45524	0.269270
$987$	0	0
$988$	0.442731	0.0140852
$989$	−0.0467044	−0.00148511
$990$	0	0
$991$	−29.5541	−0.938817	−0.469408	−	0.882981i	$-0.655533\pi$
$991$	−29.5541	−0.938817	−0.469408	+	0.882981i	$0.655533\pi$
$992$	0.0493293	0.00156621
$993$	0	0
$994$	−12.6030	−0.399742
$995$	−7.92391	−0.251205
$996$	0	0
$997$	−19.4467	−0.615883	−0.307942	−	0.951405i	$-0.599640\pi$
$997$	−19.4467	−0.615883	−0.307942	+	0.951405i	$0.599640\pi$
$998$	0.920481	0.0291373
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9009.2.a.w.1.3		4
3.2	odd	2		1001.2.a.g.1.2	✓	4
21.20	even	2		7007.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1001.2.a.g.1.2	✓	4	3.2	odd	2
7007.2.a.j.1.2		4	21.20	even	2
9009.2.a.w.1.3		4	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:	\( N \)	\(=\)	\( 9009 = 3^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35366 q^{2} -0.167605 q^{4} -0.473915 q^{5} +1.00000 q^{7} -2.93420 q^{8} +O(q^{10})\)	\(q+1.35366 q^{2} -0.167605 q^{4} -0.473915 q^{5} +1.00000 q^{7} -2.93420 q^{8} -0.641520 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.35366 q^{14} -3.63670 q^{16} +1.18123 q^{17} +2.64152 q^{19} +0.0794304 q^{20} +1.35366 q^{22} -0.986370 q^{23} -4.77540 q^{25} -1.35366 q^{26} -0.167605 q^{28} +5.28786 q^{29} +0.0521702 q^{31} +0.945545 q^{32} +1.59899 q^{34} -0.473915 q^{35} -3.53972 q^{37} +3.57572 q^{38} +1.39056 q^{40} +0.353660 q^{41} +0.0473498 q^{43} -0.167605 q^{44} -1.33521 q^{46} -2.98637 q^{47} +1.00000 q^{49} -6.46427 q^{50} +0.167605 q^{52} +6.84120 q^{53} -0.473915 q^{55} -2.93420 q^{56} +7.15796 q^{58} -7.94301 q^{59} +13.5822 q^{61} +0.0706207 q^{62} +8.55335 q^{64} +0.473915 q^{65} +8.99036 q^{67} -0.197981 q^{68} -0.641520 q^{70} -9.31030 q^{71} -1.11543 q^{73} -4.79157 q^{74} -0.442731 q^{76} +1.00000 q^{77} -0.181235 q^{79} +1.72349 q^{80} +0.478735 q^{82} -1.48754 q^{83} -0.559805 q^{85} +0.0640955 q^{86} -2.93420 q^{88} -17.0312 q^{89} -1.00000 q^{91} +0.165320 q^{92} -4.04253 q^{94} -1.25186 q^{95} -5.00881 q^{97} +1.35366 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8} + 13 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{14} + 20 q^{16} - 9 q^{17} - 5 q^{19} + 4 q^{20} + 2 q^{22} - 9 q^{23} + 5 q^{25} - 2 q^{26} + 8 q^{28} + 9 q^{29} + 14 q^{31} + 16 q^{32} + 15 q^{34} + 5 q^{35} - 16 q^{37} - 10 q^{38} + 30 q^{40} - 2 q^{41} - 5 q^{43} + 8 q^{44} + 12 q^{46} - 17 q^{47} + 4 q^{49} + 19 q^{50} - 8 q^{52} + 12 q^{53} + 5 q^{55} - 3 q^{56} - 18 q^{58} + q^{59} + 32 q^{61} + 28 q^{62} + 31 q^{64} - 5 q^{65} - 2 q^{67} - 18 q^{68} + 13 q^{70} + 4 q^{71} + 18 q^{73} - 35 q^{74} - 40 q^{76} + 4 q^{77} + 13 q^{79} + 40 q^{80} + 14 q^{82} + 6 q^{83} - 9 q^{85} + 26 q^{86} - 3 q^{88} - 15 q^{89} - 4 q^{91} + 18 q^{92} + 8 q^{94} - 19 q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8 + 13 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^14 + 20 * q^16 - 9 * q^17 - 5 * q^19 + 4 * q^20 + 2 * q^22 - 9 * q^23 + 5 * q^25 - 2 * q^26 + 8 * q^28 + 9 * q^29 + 14 * q^31 + 16 * q^32 + 15 * q^34 + 5 * q^35 - 16 * q^37 - 10 * q^38 + 30 * q^40 - 2 * q^41 - 5 * q^43 + 8 * q^44 + 12 * q^46 - 17 * q^47 + 4 * q^49 + 19 * q^50 - 8 * q^52 + 12 * q^53 + 5 * q^55 - 3 * q^56 - 18 * q^58 + q^59 + 32 * q^61 + 28 * q^62 + 31 * q^64 - 5 * q^65 - 2 * q^67 - 18 * q^68 + 13 * q^70 + 4 * q^71 + 18 * q^73 - 35 * q^74 - 40 * q^76 + 4 * q^77 + 13 * q^79 + 40 * q^80 + 14 * q^82 + 6 * q^83 - 9 * q^85 + 26 * q^86 - 3 * q^88 - 15 * q^89 - 4 * q^91 + 18 * q^92 + 8 * q^94 - 19 * q^95 + 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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