LMFDB - Embedded newform 2009.2.a.s.1.12

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + \cdots - 464 $ x^17 - 3x^16 - 25x^15 + 77x^14 + 247x^13 - 790x^12 - 1231x^11 + 4173x^10 + 3251x^9 - 12183x^8 - 4259x^7 + 19567x^6 + 2029x^5 - 16136x^4 + 299x^3 + 5775x^2 - 312x - 464
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			1.12
Root			$1.20954$ 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)

$f(q)$	$=$	$q+1.20954 q^{2} +0.150404 q^{3} -0.537006 q^{4} +3.32743 q^{5} +0.181921 q^{6} -3.06862 q^{8} -2.97738 q^{9} +O(q^{10})$	\(q+1.20954 q^{2} +0.150404 q^{3} -0.537006 q^{4} +3.32743 q^{5} +0.181921 q^{6} -3.06862 q^{8} -2.97738 q^{9} +4.02467 q^{10} +2.93942 q^{11} -0.0807680 q^{12} +1.58375 q^{13} +0.500459 q^{15} -2.63761 q^{16} +1.28348 q^{17} -3.60127 q^{18} +2.64008 q^{19} -1.78685 q^{20} +3.55535 q^{22} +7.50919 q^{23} -0.461533 q^{24} +6.07177 q^{25} +1.91562 q^{26} -0.899024 q^{27} -0.257756 q^{29} +0.605327 q^{30} -7.18760 q^{31} +2.94693 q^{32} +0.442101 q^{33} +1.55243 q^{34} +1.59887 q^{36} +5.94655 q^{37} +3.19329 q^{38} +0.238203 q^{39} -10.2106 q^{40} +1.00000 q^{41} +6.23289 q^{43} -1.57848 q^{44} -9.90701 q^{45} +9.08269 q^{46} +11.6338 q^{47} -0.396708 q^{48} +7.34407 q^{50} +0.193041 q^{51} -0.850483 q^{52} +6.89872 q^{53} -1.08741 q^{54} +9.78069 q^{55} +0.397079 q^{57} -0.311767 q^{58} -9.78449 q^{59} -0.268750 q^{60} +4.88177 q^{61} -8.69371 q^{62} +8.83966 q^{64} +5.26982 q^{65} +0.534740 q^{66} -6.14237 q^{67} -0.689238 q^{68} +1.12941 q^{69} -9.76957 q^{71} +9.13644 q^{72} -4.95646 q^{73} +7.19260 q^{74} +0.913220 q^{75} -1.41774 q^{76} +0.288117 q^{78} -7.81553 q^{79} -8.77647 q^{80} +8.79692 q^{81} +1.20954 q^{82} +16.0489 q^{83} +4.27070 q^{85} +7.53895 q^{86} -0.0387677 q^{87} -9.01994 q^{88} -7.63944 q^{89} -11.9830 q^{90} -4.03248 q^{92} -1.08105 q^{93} +14.0715 q^{94} +8.78466 q^{95} +0.443231 q^{96} -3.42888 q^{97} -8.75175 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9}+O(q^{10}) $ 17 * q + 3 * q^2 + q^3 + 25 * q^4 - q^5 + 2 * q^6 + 9 * q^8 + 26 * q^9	$ 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9} - 2 q^{10} + 15 q^{11} + 4 q^{12} - 5 q^{13} + 24 q^{15} + 33 q^{16} + 4 q^{17} + 10 q^{18} + 5 q^{19} - 26 q^{20} + 16 q^{22} + 12 q^{23} + 16 q^{24} + 24 q^{25} + 31 q^{26} - 11 q^{27} + 14 q^{29} - 33 q^{30} - 3 q^{31} + 16 q^{32} + 4 q^{33} - 24 q^{34} + 57 q^{36} + 24 q^{37} + 45 q^{39} + 36 q^{40} + 17 q^{41} + 14 q^{43} - 9 q^{44} - 21 q^{45} + 44 q^{46} + 19 q^{47} - 60 q^{48} - 4 q^{50} + 2 q^{51} - 25 q^{52} + 4 q^{53} + 68 q^{54} + 9 q^{55} - 12 q^{57} - q^{58} - 27 q^{59} + 66 q^{60} - q^{61} - 23 q^{62} + 75 q^{64} + 22 q^{65} - 16 q^{66} + 49 q^{67} + 45 q^{68} + 12 q^{69} + 40 q^{71} - 23 q^{72} - 14 q^{73} + 33 q^{74} + 27 q^{75} - 9 q^{76} - 12 q^{78} + 61 q^{79} - 82 q^{80} + 53 q^{81} + 3 q^{82} - 18 q^{83} - 13 q^{85} - 4 q^{86} - 17 q^{87} + 74 q^{88} + 18 q^{89} - 20 q^{90} + 28 q^{92} - 36 q^{93} - 5 q^{94} + 20 q^{95} + 148 q^{96} + 26 q^{97} + 38 q^{99}+O(q^{100}) $ 17 * q + 3 * q^2 + q^3 + 25 * q^4 - q^5 + 2 * q^6 + 9 * q^8 + 26 * q^9 - 2 * q^10 + 15 * q^11 + 4 * q^12 - 5 * q^13 + 24 * q^15 + 33 * q^16 + 4 * q^17 + 10 * q^18 + 5 * q^19 - 26 * q^20 + 16 * q^22 + 12 * q^23 + 16 * q^24 + 24 * q^25 + 31 * q^26 - 11 * q^27 + 14 * q^29 - 33 * q^30 - 3 * q^31 + 16 * q^32 + 4 * q^33 - 24 * q^34 + 57 * q^36 + 24 * q^37 + 45 * q^39 + 36 * q^40 + 17 * q^41 + 14 * q^43 - 9 * q^44 - 21 * q^45 + 44 * q^46 + 19 * q^47 - 60 * q^48 - 4 * q^50 + 2 * q^51 - 25 * q^52 + 4 * q^53 + 68 * q^54 + 9 * q^55 - 12 * q^57 - q^58 - 27 * q^59 + 66 * q^60 - q^61 - 23 * q^62 + 75 * q^64 + 22 * q^65 - 16 * q^66 + 49 * q^67 + 45 * q^68 + 12 * q^69 + 40 * q^71 - 23 * q^72 - 14 * q^73 + 33 * q^74 + 27 * q^75 - 9 * q^76 - 12 * q^78 + 61 * q^79 - 82 * q^80 + 53 * q^81 + 3 * q^82 - 18 * q^83 - 13 * q^85 - 4 * q^86 - 17 * q^87 + 74 * q^88 + 18 * q^89 - 20 * q^90 + 28 * q^92 - 36 * q^93 - 5 * q^94 + 20 * q^95 + 148 * q^96 + 26 * q^97 + 38 * 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 $

$2$	1.20954	0.855276	0.427638	−	0.903950i	$-0.359346\pi$
$2$	1.20954	0.855276	0.427638	+	0.903950i	$0.359346\pi$
$3$	0.150404	0.0868360	0.0434180	−	0.999057i	$-0.486175\pi$
$3$	0.150404	0.0868360	0.0434180	+	0.999057i	$0.486175\pi$
$4$	−0.537006	−0.268503
$5$	3.32743	1.48807	0.744035	−	0.668140i	$-0.232909\pi$
$5$	3.32743	1.48807	0.744035	+	0.668140i	$0.232909\pi$
$6$	0.181921	0.0742687
$7$	0	0
$7$	0	0
$8$	−3.06862	−1.08492
$9$	−2.97738	−0.992460
$10$	4.02467	1.27271
$11$	2.93942	0.886267	0.443134	−	0.896456i	$-0.353867\pi$
$11$	2.93942	0.886267	0.443134	+	0.896456i	$0.353867\pi$
$12$	−0.0807680	−0.0233157
$13$	1.58375	0.439254	0.219627	−	0.975584i	$-0.429516\pi$
$13$	1.58375	0.439254	0.219627	+	0.975584i	$0.429516\pi$
$14$	0	0
$15$	0.500459	0.129218
$16$	−2.63761	−0.659403
$17$	1.28348	0.311290	0.155645	−	0.987813i	$-0.450254\pi$
$17$	1.28348	0.311290	0.155645	+	0.987813i	$0.450254\pi$
$18$	−3.60127	−0.848827
$19$	2.64008	0.605675	0.302838	−	0.953042i	$-0.402066\pi$
$19$	2.64008	0.605675	0.302838	+	0.953042i	$0.402066\pi$
$20$	−1.78685	−0.399551
$21$	0	0
$22$	3.55535	0.758003
$23$	7.50919	1.56577	0.782887	−	0.622164i	$-0.213746\pi$
$23$	7.50919	1.56577	0.782887	+	0.622164i	$0.213746\pi$
$24$	−0.461533	−0.0942101
$25$	6.07177	1.21435
$26$	1.91562	0.375683
$27$	−0.899024	−0.173017
$28$	0	0
$29$	−0.257756	−0.0478641	−0.0239321	−	0.999714i	$-0.507619\pi$
$29$	−0.257756	−0.0478641	−0.0239321	+	0.999714i	$0.507619\pi$
$30$	0.605327	0.110517
$31$	−7.18760	−1.29093	−0.645465	−	0.763790i	$-0.723337\pi$
$31$	−7.18760	−1.29093	−0.645465	+	0.763790i	$0.723337\pi$
$32$	2.94693	0.520948
$33$	0.442101	0.0769599
$34$	1.55243	0.266239
$35$	0	0
$36$	1.59887	0.266478
$37$	5.94655	0.977606	0.488803	−	0.872394i	$-0.337434\pi$
$37$	5.94655	0.977606	0.488803	+	0.872394i	$0.337434\pi$
$38$	3.19329	0.518019
$39$	0.238203	0.0381430
$40$	−10.2106	−1.61444
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	6.23289	0.950507	0.475253	−	0.879849i	$-0.342356\pi$
$43$	6.23289	0.950507	0.475253	+	0.879849i	$0.342356\pi$
$44$	−1.57848	−0.237965
$45$	−9.90701	−1.47685
$46$	9.08269	1.33917
$47$	11.6338	1.69696	0.848480	−	0.529228i	$-0.177518\pi$
$47$	11.6338	1.69696	0.848480	+	0.529228i	$0.177518\pi$
$48$	−0.396708	−0.0572599
$49$	0	0
$50$	7.34407	1.03861
$51$	0.193041	0.0270312
$52$	−0.850483	−0.117941
$53$	6.89872	0.947612	0.473806	−	0.880629i	$-0.342880\pi$
$53$	6.89872	0.947612	0.473806	+	0.880629i	$0.342880\pi$
$54$	−1.08741	−0.147977
$55$	9.78069	1.31883
$56$	0	0
$57$	0.397079	0.0525944
$58$	−0.311767	−0.0409370
$59$	−9.78449	−1.27383	−0.636916	−	0.770933i	$-0.719790\pi$
$59$	−9.78449	−1.27383	−0.636916	+	0.770933i	$0.719790\pi$
$60$	−0.268750	−0.0346954
$61$	4.88177	0.625047	0.312523	−	0.949910i	$-0.398826\pi$
$61$	4.88177	0.625047	0.312523	+	0.949910i	$0.398826\pi$
$62$	−8.69371	−1.10410
$63$	0	0
$64$	8.83966	1.10496
$65$	5.26982	0.653640
$66$	0.534740	0.0658219
$67$	−6.14237	−0.750410	−0.375205	−	0.926942i	$-0.622428\pi$
$67$	−6.14237	−0.750410	−0.375205	+	0.926942i	$0.622428\pi$
$68$	−0.689238	−0.0835824
$69$	1.12941	0.135966
$70$	0	0
$71$	−9.76957	−1.15944	−0.579718	−	0.814817i	$-0.696837\pi$
$71$	−9.76957	−1.15944	−0.579718	+	0.814817i	$0.696837\pi$
$72$	9.13644	1.07674
$73$	−4.95646	−0.580110	−0.290055	−	0.957010i	$-0.593674\pi$
$73$	−4.95646	−0.580110	−0.290055	+	0.957010i	$0.593674\pi$
$74$	7.19260	0.836123
$75$	0.913220	0.105450
$76$	−1.41774	−0.162625
$77$	0	0
$78$	0.288117	0.0326228
$79$	−7.81553	−0.879316	−0.439658	−	0.898165i	$-0.644900\pi$
$79$	−7.81553	−0.879316	−0.439658	+	0.898165i	$0.644900\pi$
$80$	−8.77647	−0.981239
$81$	8.79692	0.977435
$82$	1.20954	0.133572
$83$	16.0489	1.76160	0.880799	−	0.473490i	$-0.157006\pi$
$83$	16.0489	1.76160	0.880799	+	0.473490i	$0.157006\pi$
$84$	0	0
$85$	4.27070	0.463222
$86$	7.53895	0.812946
$87$	−0.0387677	−0.00415633
$88$	−9.01994	−0.961529
$89$	−7.63944	−0.809779	−0.404889	−	0.914366i	$-0.632690\pi$
$89$	−7.63944	−0.809779	−0.404889	+	0.914366i	$0.632690\pi$
$90$	−11.9830	−1.26311
$91$	0	0
$92$	−4.03248	−0.420415
$93$	−1.08105	−0.112099
$94$	14.0715	1.45137
$95$	8.78466	0.901287
$96$	0.443231	0.0452370
$97$	−3.42888	−0.348150	−0.174075	−	0.984732i	$-0.555693\pi$
$97$	−3.42888	−0.348150	−0.174075	+	0.984732i	$0.555693\pi$
$98$	0	0
$99$	−8.75175	−0.879584
$100$	−3.26057	−0.326057
$101$	−14.4980	−1.44261	−0.721305	−	0.692618i	$-0.756457\pi$
$101$	−14.4980	−1.44261	−0.721305	+	0.692618i	$0.756457\pi$
$102$	0.233492	0.0231191
$103$	−8.62746	−0.850089	−0.425044	−	0.905172i	$-0.639742\pi$
$103$	−8.62746	−0.850089	−0.425044	+	0.905172i	$0.639742\pi$
$104$	−4.85993	−0.476555
$105$	0	0
$106$	8.34430	0.810470
$107$	8.43286	0.815236	0.407618	−	0.913153i	$-0.366360\pi$
$107$	8.43286	0.815236	0.407618	+	0.913153i	$0.366360\pi$
$108$	0.482781	0.0464556
$109$	2.96943	0.284420	0.142210	−	0.989837i	$-0.454579\pi$
$109$	2.96943	0.284420	0.142210	+	0.989837i	$0.454579\pi$
$110$	11.8302	1.12796
$111$	0.894386	0.0848914
$112$	0	0
$113$	7.83550	0.737102	0.368551	−	0.929607i	$-0.379854\pi$
$113$	7.83550	0.737102	0.368551	+	0.929607i	$0.379854\pi$
$114$	0.480284	0.0449827
$115$	24.9863	2.32998
$116$	0.138417	0.0128517
$117$	−4.71543	−0.435941
$118$	−11.8348	−1.08948
$119$	0	0
$120$	−1.53572	−0.140191
$121$	−2.35984	−0.214531
$122$	5.90471	0.534588
$123$	0.150404	0.0135615
$124$	3.85978	0.346619
$125$	3.56623	0.318974
$126$	0	0
$127$	4.90332	0.435099	0.217550	−	0.976049i	$-0.430194\pi$
$127$	4.90332	0.435099	0.217550	+	0.976049i	$0.430194\pi$
$128$	4.79810	0.424096
$129$	0.937454	0.0825382
$130$	6.37407	0.559043
$131$	−1.62929	−0.142352	−0.0711761	−	0.997464i	$-0.522675\pi$
$131$	−1.62929	−0.142352	−0.0711761	+	0.997464i	$0.522675\pi$
$132$	−0.237411	−0.0206639
$133$	0	0
$134$	−7.42946	−0.641807
$135$	−2.99144	−0.257462
$136$	−3.93852	−0.337725
$137$	15.2417	1.30219	0.651095	−	0.758996i	$-0.274310\pi$
$137$	15.2417	1.30219	0.651095	+	0.758996i	$0.274310\pi$
$138$	1.36608	0.116288
$139$	−9.89872	−0.839599	−0.419799	−	0.907617i	$-0.637900\pi$
$139$	−9.89872	−0.839599	−0.419799	+	0.907617i	$0.637900\pi$
$140$	0	0
$141$	1.74977	0.147357
$142$	−11.8167	−0.991637
$143$	4.65530	0.389296
$144$	7.85317	0.654431
$145$	−0.857665	−0.0712252
$146$	−5.99505	−0.496154
$147$	0	0
$148$	−3.19333	−0.262490
$149$	−18.9386	−1.55151	−0.775757	−	0.631032i	$-0.782632\pi$
$149$	−18.9386	−1.55151	−0.775757	+	0.631032i	$0.782632\pi$
$150$	1.10458	0.0901885
$151$	0.379041	0.0308459	0.0154230	−	0.999881i	$-0.495091\pi$
$151$	0.379041	0.0308459	0.0154230	+	0.999881i	$0.495091\pi$
$152$	−8.10138	−0.657109
$153$	−3.82142	−0.308943
$154$	0	0
$155$	−23.9162	−1.92100
$156$	−0.127916	−0.0102415
$157$	−9.43348	−0.752873	−0.376437	−	0.926442i	$-0.622851\pi$
$157$	−9.43348	−0.752873	−0.376437	+	0.926442i	$0.622851\pi$
$158$	−9.45322	−0.752058
$159$	1.03760	0.0822868
$160$	9.80569	0.775208
$161$	0	0
$162$	10.6403	0.835977
$163$	4.33522	0.339560	0.169780	−	0.985482i	$-0.445694\pi$
$163$	4.33522	0.339560	0.169780	+	0.985482i	$0.445694\pi$
$164$	−0.537006	−0.0419331
$165$	1.47106	0.114522
$166$	19.4119	1.50665
$167$	−8.85191	−0.684982	−0.342491	−	0.939521i	$-0.611271\pi$
$167$	−8.85191	−0.684982	−0.342491	+	0.939521i	$0.611271\pi$
$168$	0	0
$169$	−10.4917	−0.807056
$170$	5.16559	0.396183
$171$	−7.86051	−0.601108
$172$	−3.34710	−0.255214
$173$	20.2364	1.53855	0.769274	−	0.638919i	$-0.220618\pi$
$173$	20.2364	1.53855	0.769274	+	0.638919i	$0.220618\pi$
$174$	−0.0468911	−0.00355481
$175$	0	0
$176$	−7.75304	−0.584407
$177$	−1.47163	−0.110614
$178$	−9.24023	−0.692585
$179$	−23.2744	−1.73961	−0.869804	−	0.493398i	$-0.835755\pi$
$179$	−23.2744	−1.73961	−0.869804	+	0.493398i	$0.835755\pi$
$180$	5.32012	0.396538
$181$	0.632828	0.0470377	0.0235189	−	0.999723i	$-0.492513\pi$
$181$	0.632828	0.0470377	0.0235189	+	0.999723i	$0.492513\pi$
$182$	0	0
$183$	0.734240	0.0542766
$184$	−23.0428	−1.69874
$185$	19.7867	1.45475
$186$	−1.30757	−0.0958758
$187$	3.77269	0.275886
$188$	−6.24740	−0.455639
$189$	0	0
$190$	10.6254	0.770849
$191$	23.0008	1.66428	0.832139	−	0.554568i	$-0.187116\pi$
$191$	23.0008	1.66428	0.832139	+	0.554568i	$0.187116\pi$
$192$	1.32952	0.0959501
$193$	−8.88498	−0.639555	−0.319778	−	0.947493i	$-0.603608\pi$
$193$	−8.88498	−0.639555	−0.319778	+	0.947493i	$0.603608\pi$
$194$	−4.14737	−0.297764
$195$	0.792603	0.0567595
$196$	0	0
$197$	23.4404	1.67006	0.835028	−	0.550207i	$-0.185451\pi$
$197$	23.4404	1.67006	0.835028	+	0.550207i	$0.185451\pi$
$198$	−10.5856	−0.752287
$199$	19.6087	1.39002	0.695011	−	0.718999i	$-0.255399\pi$
$199$	19.6087	1.39002	0.695011	+	0.718999i	$0.255399\pi$
$200$	−18.6319	−1.31748
$201$	−0.923839	−0.0651626
$202$	−17.5360	−1.23383
$203$	0	0
$204$	−0.103664	−0.00725796
$205$	3.32743	0.232398
$206$	−10.4353	−0.727061
$207$	−22.3577	−1.55397
$208$	−4.17732	−0.289645
$209$	7.76028	0.536790
$210$	0	0
$211$	21.6422	1.48991	0.744954	−	0.667116i	$-0.232472\pi$
$211$	21.6422	1.48991	0.744954	+	0.667116i	$0.232472\pi$
$212$	−3.70465	−0.254437
$213$	−1.46939	−0.100681
$214$	10.1999	0.697252
$215$	20.7395	1.41442
$216$	2.75876	0.187710
$217$	0	0
$218$	3.59165	0.243258
$219$	−0.745473	−0.0503744
$220$	−5.25229	−0.354109
$221$	2.03272	0.136735
$222$	1.08180	0.0726056
$223$	−28.6820	−1.92069	−0.960345	−	0.278813i	$-0.910059\pi$
$223$	−28.6820	−1.92069	−0.960345	+	0.278813i	$0.910059\pi$
$224$	0	0
$225$	−18.0780	−1.20520
$226$	9.47738	0.630426
$227$	17.1604	1.13898	0.569488	−	0.821999i	$-0.307141\pi$
$227$	17.1604	1.13898	0.569488	+	0.821999i	$0.307141\pi$
$228$	−0.213234	−0.0141217
$229$	−14.9394	−0.987225	−0.493612	−	0.869682i	$-0.664324\pi$
$229$	−14.9394	−0.987225	−0.493612	+	0.869682i	$0.664324\pi$
$230$	30.2220	1.99278
$231$	0	0
$232$	0.790955	0.0519288
$233$	−14.1981	−0.930150	−0.465075	−	0.885271i	$-0.653973\pi$
$233$	−14.1981	−0.930150	−0.465075	+	0.885271i	$0.653973\pi$
$234$	−5.70351	−0.372850
$235$	38.7105	2.52520
$236$	5.25433	0.342027
$237$	−1.17549	−0.0763562
$238$	0	0
$239$	−1.09744	−0.0709874	−0.0354937	−	0.999370i	$-0.511300\pi$
$239$	−1.09744	−0.0709874	−0.0354937	+	0.999370i	$0.511300\pi$
$240$	−1.32002	−0.0852068
$241$	−21.6661	−1.39563	−0.697817	−	0.716276i	$-0.745845\pi$
$241$	−21.6661	−1.39563	−0.697817	+	0.716276i	$0.745845\pi$
$242$	−2.85433	−0.183483
$243$	4.02017	0.257894
$244$	−2.62154	−0.167827
$245$	0	0
$246$	0.181921	0.0115988
$247$	4.18122	0.266045
$248$	22.0560	1.40056
$249$	2.41383	0.152970
$250$	4.31351	0.272811
$251$	4.75831	0.300342	0.150171	−	0.988660i	$-0.452018\pi$
$251$	4.75831	0.300342	0.150171	+	0.988660i	$0.452018\pi$
$252$	0	0
$253$	22.0726	1.38769
$254$	5.93077	0.372130
$255$	0.642331	0.0402243
$256$	−11.8758	−0.742239
$257$	−1.36362	−0.0850602	−0.0425301	−	0.999095i	$-0.513542\pi$
$257$	−1.36362	−0.0850602	−0.0425301	+	0.999095i	$0.513542\pi$
$258$	1.13389	0.0705929
$259$	0	0
$260$	−2.82992	−0.175504
$261$	0.767438	0.0475032
$262$	−1.97070	−0.121750
$263$	0.257118	0.0158545	0.00792727	−	0.999969i	$-0.497477\pi$
$263$	0.257118	0.0158545	0.00792727	+	0.999969i	$0.497477\pi$
$264$	−1.35664	−0.0834953
$265$	22.9550	1.41011
$266$	0	0
$267$	−1.14900	−0.0703179
$268$	3.29849	0.201487
$269$	−9.79827	−0.597411	−0.298706	−	0.954345i	$-0.596555\pi$
$269$	−9.79827	−0.597411	−0.298706	+	0.954345i	$0.596555\pi$
$270$	−3.61827	−0.220201
$271$	−14.5579	−0.884329	−0.442165	−	0.896934i	$-0.645789\pi$
$271$	−14.5579	−0.884329	−0.442165	+	0.896934i	$0.645789\pi$
$272$	−3.38533	−0.205266
$273$	0	0
$274$	18.4355	1.11373
$275$	17.8475	1.07624
$276$	−0.606502	−0.0365071
$277$	−16.4859	−0.990542	−0.495271	−	0.868739i	$-0.664931\pi$
$277$	−16.4859	−0.990542	−0.495271	+	0.868739i	$0.664931\pi$
$278$	−11.9729	−0.718089
$279$	21.4002	1.28120
$280$	0	0
$281$	9.18691	0.548045	0.274023	−	0.961723i	$-0.411646\pi$
$281$	9.18691	0.548045	0.274023	+	0.961723i	$0.411646\pi$
$282$	2.11642	0.126031
$283$	1.94662	0.115714	0.0578572	−	0.998325i	$-0.481573\pi$
$283$	1.94662	0.115714	0.0578572	+	0.998325i	$0.481573\pi$
$284$	5.24632	0.311312
$285$	1.32125	0.0782642
$286$	5.63079	0.332955
$287$	0	0
$288$	−8.77412	−0.517020
$289$	−15.3527	−0.903098
$290$	−1.03738	−0.0609172
$291$	−0.515718	−0.0302319
$292$	2.66165	0.155761
$293$	14.3423	0.837888	0.418944	−	0.908012i	$-0.362400\pi$
$293$	14.3423	0.837888	0.418944	+	0.908012i	$0.362400\pi$
$294$	0	0
$295$	−32.5572	−1.89555
$296$	−18.2477	−1.06062
$297$	−2.64260	−0.153339
$298$	−22.9071	−1.32697
$299$	11.8927	0.687772
$300$	−0.490405	−0.0283135
$301$	0	0
$302$	0.458466	0.0263818
$303$	−2.18057	−0.125270
$304$	−6.96350	−0.399384
$305$	16.2437	0.930114
$306$	−4.62217	−0.264232
$307$	−1.79234	−0.102294	−0.0511472	−	0.998691i	$-0.516288\pi$
$307$	−1.79234	−0.102294	−0.0511472	+	0.998691i	$0.516288\pi$
$308$	0	0
$309$	−1.29761	−0.0738183
$310$	−28.9277	−1.64298
$311$	10.9797	0.622600	0.311300	−	0.950312i	$-0.399236\pi$
$311$	10.9797	0.622600	0.311300	+	0.950312i	$0.399236\pi$
$312$	−0.730954	−0.0413821
$313$	−20.0008	−1.13051	−0.565256	−	0.824916i	$-0.691223\pi$
$313$	−20.0008	−1.13051	−0.565256	+	0.824916i	$0.691223\pi$
$314$	−11.4102	−0.643915
$315$	0	0
$316$	4.19698	0.236099
$317$	−33.3833	−1.87499	−0.937495	−	0.347998i	$-0.886862\pi$
$317$	−33.3833	−1.87499	−0.937495	+	0.347998i	$0.886862\pi$
$318$	1.25502	0.0703780
$319$	−0.757653	−0.0424204
$320$	29.4133	1.64426
$321$	1.26834	0.0707918
$322$	0	0
$323$	3.38849	0.188541
$324$	−4.72400	−0.262444
$325$	9.61617	0.533409
$326$	5.24363	0.290418
$327$	0.446615	0.0246979
$328$	−3.06862	−0.169436
$329$	0	0
$330$	1.77931	0.0979477
$331$	29.3896	1.61540	0.807700	−	0.589594i	$-0.200712\pi$
$331$	29.3896	1.61540	0.807700	+	0.589594i	$0.200712\pi$
$332$	−8.61836	−0.472994
$333$	−17.7051	−0.970234
$334$	−10.7068	−0.585848
$335$	−20.4383	−1.11666
$336$	0	0
$337$	−28.3258	−1.54301	−0.771503	−	0.636226i	$-0.780495\pi$
$337$	−28.3258	−1.54301	−0.771503	+	0.636226i	$0.780495\pi$
$338$	−12.6902	−0.690256
$339$	1.17849	0.0640070
$340$	−2.29339	−0.124376
$341$	−21.1273	−1.14411
$342$	−9.50762	−0.514113
$343$	0	0
$344$	−19.1264	−1.03122
$345$	3.75804	0.202326
$346$	24.4768	1.31588
$347$	−16.4954	−0.885521	−0.442760	−	0.896640i	$-0.646001\pi$
$347$	−16.4954	−0.885521	−0.442760	+	0.896640i	$0.646001\pi$
$348$	0.0208185	0.00111599
$349$	−26.7298	−1.43081	−0.715407	−	0.698708i	$-0.753759\pi$
$349$	−26.7298	−1.43081	−0.715407	+	0.698708i	$0.753759\pi$
$350$	0	0
$351$	−1.42383	−0.0759984
$352$	8.66224	0.461699
$353$	−28.0645	−1.49372	−0.746862	−	0.664980i	$-0.768440\pi$
$353$	−28.0645	−1.49372	−0.746862	+	0.664980i	$0.768440\pi$
$354$	−1.78000	−0.0946059
$355$	−32.5075	−1.72532
$356$	4.10242	0.217428
$357$	0	0
$358$	−28.1513	−1.48784
$359$	−34.3670	−1.81382	−0.906912	−	0.421321i	$-0.861567\pi$
$359$	−34.3670	−1.81382	−0.906912	+	0.421321i	$0.861567\pi$
$360$	30.4008	1.60226
$361$	−12.0300	−0.633158
$362$	0.765432	0.0402302
$363$	−0.354930	−0.0186290
$364$	0	0
$365$	−16.4923	−0.863245
$366$	0.888095	0.0464214
$367$	−2.44875	−0.127824	−0.0639119	−	0.997956i	$-0.520358\pi$
$367$	−2.44875	−0.127824	−0.0639119	+	0.997956i	$0.520358\pi$
$368$	−19.8063	−1.03248
$369$	−2.97738	−0.154996
$370$	23.9329	1.24421
$371$	0	0
$372$	0.580528	0.0300990
$373$	0.489825	0.0253622	0.0126811	−	0.999920i	$-0.495963\pi$
$373$	0.489825	0.0253622	0.0126811	+	0.999920i	$0.495963\pi$
$374$	4.56323	0.235959
$375$	0.536377	0.0276984
$376$	−35.6996	−1.84107
$377$	−0.408222	−0.0210245
$378$	0	0
$379$	−21.7119	−1.11527	−0.557633	−	0.830088i	$-0.688290\pi$
$379$	−21.7119	−1.11527	−0.557633	+	0.830088i	$0.688290\pi$
$380$	−4.71741	−0.241998
$381$	0.737480	0.0377823
$382$	27.8204	1.42342
$383$	−18.8548	−0.963434	−0.481717	−	0.876327i	$-0.659987\pi$
$383$	−18.8548	−0.963434	−0.481717	+	0.876327i	$0.659987\pi$
$384$	0.721655	0.0368268
$385$	0	0
$386$	−10.7468	−0.546996
$387$	−18.5577	−0.943339
$388$	1.84133	0.0934791
$389$	5.41870	0.274739	0.137370	−	0.990520i	$-0.456135\pi$
$389$	5.41870	0.274739	0.137370	+	0.990520i	$0.456135\pi$
$390$	0.958688	0.0485450
$391$	9.63792	0.487410
$392$	0	0
$393$	−0.245053	−0.0123613
$394$	28.3521	1.42836
$395$	−26.0056	−1.30848
$396$	4.69974	0.236171
$397$	−1.97993	−0.0993697	−0.0496849	−	0.998765i	$-0.515822\pi$
$397$	−1.97993	−0.0993697	−0.0496849	+	0.998765i	$0.515822\pi$
$398$	23.7175	1.18885
$399$	0	0
$400$	−16.0150	−0.800749
$401$	30.8109	1.53862	0.769310	−	0.638875i	$-0.220600\pi$
$401$	30.8109	1.53862	0.769310	+	0.638875i	$0.220600\pi$
$402$	−1.11742	−0.0557320
$403$	−11.3834	−0.567046
$404$	7.78553	0.387345
$405$	29.2711	1.45449
$406$	0	0
$407$	17.4794	0.866420
$408$	−0.592370	−0.0293267
$409$	−15.5784	−0.770304	−0.385152	−	0.922853i	$-0.625851\pi$
$409$	−15.5784	−0.770304	−0.385152	+	0.922853i	$0.625851\pi$
$410$	4.02467	0.198764
$411$	2.29242	0.113077
$412$	4.63300	0.228251
$413$	0	0
$414$	−27.0426	−1.32907
$415$	53.4016	2.62138
$416$	4.66720	0.228828
$417$	−1.48881	−0.0729074
$418$	9.38639	0.459103
$419$	3.22178	0.157394	0.0786970	−	0.996899i	$-0.474924\pi$
$419$	3.22178	0.157394	0.0786970	+	0.996899i	$0.474924\pi$
$420$	0	0
$421$	12.6274	0.615423	0.307712	−	0.951480i	$-0.400437\pi$
$421$	12.6274	0.615423	0.307712	+	0.951480i	$0.400437\pi$
$422$	26.1771	1.27428
$423$	−34.6381	−1.68416
$424$	−21.1695	−1.02808
$425$	7.79301	0.378017
$426$	−1.77729	−0.0861098
$427$	0	0
$428$	−4.52850	−0.218893
$429$	0.700178	0.0338049
$430$	25.0853	1.20972
$431$	21.3262	1.02725	0.513624	−	0.858016i	$-0.328303\pi$
$431$	21.3262	1.02725	0.513624	+	0.858016i	$0.328303\pi$
$432$	2.37128	0.114088
$433$	−37.7215	−1.81278	−0.906390	−	0.422443i	$-0.861173\pi$
$433$	−37.7215	−1.81278	−0.906390	+	0.422443i	$0.861173\pi$
$434$	0	0
$435$	−0.128997	−0.00618491
$436$	−1.59460	−0.0763676
$437$	19.8248	0.948350
$438$	−0.901682	−0.0430840
$439$	−9.45391	−0.451210	−0.225605	−	0.974219i	$-0.572436\pi$
$439$	−9.45391	−0.451210	−0.225605	+	0.974219i	$0.572436\pi$
$440$	−30.0132	−1.43082
$441$	0	0
$442$	2.45866	0.116947
$443$	15.1182	0.718286	0.359143	−	0.933283i	$-0.383069\pi$
$443$	15.1182	0.718286	0.359143	+	0.933283i	$0.383069\pi$
$444$	−0.480290	−0.0227936
$445$	−25.4197	−1.20501
$446$	−34.6922	−1.64272
$447$	−2.84845	−0.134727
$448$	0	0
$449$	28.1933	1.33052	0.665262	−	0.746610i	$-0.268320\pi$
$449$	28.1933	1.33052	0.665262	+	0.746610i	$0.268320\pi$
$450$	−21.8661	−1.03078
$451$	2.93942	0.138412
$452$	−4.20771	−0.197914
$453$	0.0570094	0.00267854
$454$	20.7563	0.974140
$455$	0	0
$456$	−1.21848	−0.0570607
$457$	30.9384	1.44724	0.723618	−	0.690201i	$-0.242478\pi$
$457$	30.9384	1.44724	0.723618	+	0.690201i	$0.242478\pi$
$458$	−18.0699	−0.844350
$459$	−1.15388	−0.0538586
$460$	−13.4178	−0.625607
$461$	−7.88860	−0.367409	−0.183704	−	0.982982i	$-0.558809\pi$
$461$	−7.88860	−0.367409	−0.183704	+	0.982982i	$0.558809\pi$
$462$	0	0
$463$	0.191611	0.00890491	0.00445246	−	0.999990i	$-0.498583\pi$
$463$	0.191611	0.00890491	0.00445246	+	0.999990i	$0.498583\pi$
$464$	0.679861	0.0315618
$465$	−3.59710	−0.166812
$466$	−17.1733	−0.795535
$467$	7.49574	0.346862	0.173431	−	0.984846i	$-0.444515\pi$
$467$	7.49574	0.346862	0.173431	+	0.984846i	$0.444515\pi$
$468$	2.53221	0.117051
$469$	0	0
$470$	46.8220	2.15974
$471$	−1.41884	−0.0653765
$472$	30.0248	1.38201
$473$	18.3211	0.842403
$474$	−1.42180	−0.0653057
$475$	16.0299	0.735504
$476$	0	0
$477$	−20.5401	−0.940467
$478$	−1.32740	−0.0607138
$479$	−3.82060	−0.174568	−0.0872838	−	0.996183i	$-0.527819\pi$
$479$	−3.82060	−0.174568	−0.0872838	+	0.996183i	$0.527819\pi$
$480$	1.47482	0.0673159
$481$	9.41785	0.429417
$482$	−26.2060	−1.19365
$483$	0	0
$484$	1.26725	0.0576021
$485$	−11.4093	−0.518071
$486$	4.86256	0.220570
$487$	20.2027	0.915473	0.457737	−	0.889088i	$-0.348660\pi$
$487$	20.2027	0.915473	0.457737	+	0.889088i	$0.348660\pi$
$488$	−14.9803	−0.678126
$489$	0.652035	0.0294861
$490$	0	0
$491$	6.06794	0.273842	0.136921	−	0.990582i	$-0.456279\pi$
$491$	6.06794	0.273842	0.136921	+	0.990582i	$0.456279\pi$
$492$	−0.0807680	−0.00364130
$493$	−0.330826	−0.0148996
$494$	5.05737	0.227542
$495$	−29.1208	−1.30888
$496$	18.9581	0.851244
$497$	0	0
$498$	2.91963	0.130832
$499$	21.3645	0.956406	0.478203	−	0.878249i	$-0.341288\pi$
$499$	21.3645	0.956406	0.478203	+	0.878249i	$0.341288\pi$
$500$	−1.91509	−0.0856453
$501$	−1.33137	−0.0594810
$502$	5.75538	0.256875
$503$	−2.26565	−0.101020	−0.0505101	−	0.998724i	$-0.516085\pi$
$503$	−2.26565	−0.101020	−0.0505101	+	0.998724i	$0.516085\pi$
$504$	0	0
$505$	−48.2412	−2.14670
$506$	26.6978	1.18686
$507$	−1.57800	−0.0700815
$508$	−2.63311	−0.116825
$509$	−26.9845	−1.19607	−0.598034	−	0.801471i	$-0.704051\pi$
$509$	−26.9845	−1.19607	−0.598034	+	0.801471i	$0.704051\pi$
$510$	0.776927	0.0344029
$511$	0	0
$512$	−23.9605	−1.05891
$513$	−2.37349	−0.104792
$514$	−1.64936	−0.0727500
$515$	−28.7072	−1.26499
$516$	−0.503418	−0.0221617
$517$	34.1965	1.50396
$518$	0	0
$519$	3.04365	0.133601
$520$	−16.1711	−0.709147
$521$	−0.137421	−0.00602054	−0.00301027	−	0.999995i	$-0.500958\pi$
$521$	−0.137421	−0.00602054	−0.00301027	+	0.999995i	$0.500958\pi$
$522$	0.928249	0.0406284
$523$	4.55527	0.199188	0.0995941	−	0.995028i	$-0.468246\pi$
$523$	4.55527	0.199188	0.0995941	+	0.995028i	$0.468246\pi$
$524$	0.874940	0.0382219
$525$	0	0
$526$	0.310995	0.0135600
$527$	−9.22516	−0.401854
$528$	−1.16609	−0.0507476
$529$	33.3879	1.45165
$530$	27.7650	1.20604
$531$	29.1321	1.26423
$532$	0	0
$533$	1.58375	0.0685999
$534$	−1.38977	−0.0601413
$535$	28.0597	1.21313
$536$	18.8486	0.814134
$537$	−3.50056	−0.151061
$538$	−11.8514	−0.510952
$539$	0	0
$540$	1.60642	0.0691292
$541$	−13.6947	−0.588781	−0.294390	−	0.955685i	$-0.595117\pi$
$541$	−13.6947	−0.588781	−0.294390	+	0.955685i	$0.595117\pi$
$542$	−17.6084	−0.756346
$543$	0.0951800	0.00408457
$544$	3.78233	0.162166
$545$	9.88056	0.423237
$546$	0	0
$547$	40.9516	1.75097	0.875483	−	0.483249i	$-0.160543\pi$
$547$	40.9516	1.75097	0.875483	+	0.483249i	$0.160543\pi$
$548$	−8.18490	−0.349642
$549$	−14.5349	−0.620334
$550$	21.5873	0.920484
$551$	−0.680496	−0.0289901
$552$	−3.46574	−0.147512
$553$	0	0
$554$	−19.9404	−0.847187
$555$	2.97600	0.126324
$556$	5.31567	0.225435
$557$	10.0115	0.424201	0.212101	−	0.977248i	$-0.431970\pi$
$557$	10.0115	0.424201	0.212101	+	0.977248i	$0.431970\pi$
$558$	25.8845	1.09578
$559$	9.87135	0.417513
$560$	0	0
$561$	0.567429	0.0239569
$562$	11.1120	0.468730
$563$	39.1317	1.64920	0.824602	−	0.565714i	$-0.191399\pi$
$563$	39.1317	1.64920	0.824602	+	0.565714i	$0.191399\pi$
$564$	−0.939636	−0.0395658
$565$	26.0721	1.09686
$566$	2.35452	0.0989677
$567$	0	0
$568$	29.9791	1.25789
$569$	−19.2481	−0.806921	−0.403461	−	0.914997i	$-0.632193\pi$
$569$	−19.2481	−0.806921	−0.403461	+	0.914997i	$0.632193\pi$
$570$	1.59811	0.0669375
$571$	−17.1600	−0.718122	−0.359061	−	0.933314i	$-0.616903\pi$
$571$	−17.1600	−0.718122	−0.359061	+	0.933314i	$0.616903\pi$
$572$	−2.49992	−0.104527
$573$	3.45941	0.144519
$574$	0	0
$575$	45.5941	1.90140
$576$	−26.3190	−1.09663
$577$	−23.4192	−0.974954	−0.487477	−	0.873136i	$-0.662083\pi$
$577$	−23.4192	−0.974954	−0.487477	+	0.873136i	$0.662083\pi$
$578$	−18.5697	−0.772398
$579$	−1.33634	−0.0555364
$580$	0.460571	0.0191242
$581$	0	0
$582$	−0.623783	−0.0258566
$583$	20.2782	0.839837
$584$	15.2095	0.629373
$585$	−15.6902	−0.648711
$586$	17.3477	0.716626
$587$	45.9873	1.89810	0.949049	−	0.315127i	$-0.102047\pi$
$587$	45.9873	1.89810	0.949049	+	0.315127i	$0.102047\pi$
$588$	0	0
$589$	−18.9758	−0.781884
$590$	−39.3793	−1.62122
$591$	3.52553	0.145021
$592$	−15.6847	−0.644637
$593$	27.3545	1.12331	0.561657	−	0.827370i	$-0.310164\pi$
$593$	27.3545	1.12331	0.561657	+	0.827370i	$0.310164\pi$
$594$	−3.19634	−0.131148
$595$	0	0
$596$	10.1702	0.416586
$597$	2.94923	0.120704
$598$	14.3847	0.588235
$599$	10.4968	0.428890	0.214445	−	0.976736i	$-0.431206\pi$
$599$	10.4968	0.428890	0.214445	+	0.976736i	$0.431206\pi$
$600$	−2.80232	−0.114404
$601$	−31.8880	−1.30074	−0.650369	−	0.759618i	$-0.725386\pi$
$601$	−31.8880	−1.30074	−0.650369	+	0.759618i	$0.725386\pi$
$602$	0	0
$603$	18.2882	0.744751
$604$	−0.203547	−0.00828222
$605$	−7.85219	−0.319237
$606$	−2.63749	−0.107141
$607$	−11.6404	−0.472471	−0.236235	−	0.971696i	$-0.575914\pi$
$607$	−11.6404	−0.472471	−0.236235	+	0.971696i	$0.575914\pi$
$608$	7.78011	0.315525
$609$	0	0
$610$	19.6475	0.795504
$611$	18.4250	0.745396
$612$	2.05212	0.0829521
$613$	13.0920	0.528779	0.264390	−	0.964416i	$-0.414829\pi$
$613$	13.0920	0.528779	0.264390	+	0.964416i	$0.414829\pi$
$614$	−2.16792	−0.0874900
$615$	0.500459	0.0201805
$616$	0	0
$617$	2.94711	0.118646	0.0593232	−	0.998239i	$-0.481106\pi$
$617$	2.94711	0.118646	0.0593232	+	0.998239i	$0.481106\pi$
$618$	−1.56951	−0.0631350
$619$	−7.10394	−0.285531	−0.142766	−	0.989757i	$-0.545600\pi$
$619$	−7.10394	−0.285531	−0.142766	+	0.989757i	$0.545600\pi$
$620$	12.8431	0.515793
$621$	−6.75094	−0.270906
$622$	13.2804	0.532495
$623$	0	0
$624$	−0.628288	−0.0251516
$625$	−18.4925	−0.739699
$626$	−24.1918	−0.966900
$627$	1.16718	0.0466127
$628$	5.06583	0.202149
$629$	7.63229	0.304319
$630$	0	0
$631$	35.7244	1.42217	0.711083	−	0.703108i	$-0.248205\pi$
$631$	35.7244	1.42217	0.711083	+	0.703108i	$0.248205\pi$
$632$	23.9829	0.953987
$633$	3.25507	0.129378
$634$	−40.3785	−1.60363
$635$	16.3154	0.647458
$636$	−0.557196	−0.0220942
$637$	0	0
$638$	−0.916413	−0.0362812
$639$	29.0877	1.15069
$640$	15.9653	0.631085
$641$	−44.9514	−1.77547	−0.887737	−	0.460351i	$-0.847724\pi$
$641$	−44.9514	−1.77547	−0.887737	+	0.460351i	$0.847724\pi$
$642$	1.53411	0.0605465
$643$	4.45310	0.175613	0.0878066	−	0.996138i	$-0.472014\pi$
$643$	4.45310	0.175613	0.0878066	+	0.996138i	$0.472014\pi$
$644$	0	0
$645$	3.11931	0.122823
$646$	4.09853	0.161254
$647$	−16.1379	−0.634448	−0.317224	−	0.948351i	$-0.602751\pi$
$647$	−16.1379	−0.634448	−0.317224	+	0.948351i	$0.602751\pi$
$648$	−26.9944	−1.06044
$649$	−28.7607	−1.12896
$650$	11.6312	0.456212
$651$	0	0
$652$	−2.32804	−0.0911729
$653$	−5.15543	−0.201748	−0.100874	−	0.994899i	$-0.532164\pi$
$653$	−5.15543	−0.201748	−0.100874	+	0.994899i	$0.532164\pi$
$654$	0.540200	0.0211235
$655$	−5.42136	−0.211830
$656$	−2.63761	−0.102982
$657$	14.7573	0.575736
$658$	0	0
$659$	−9.47882	−0.369243	−0.184621	−	0.982810i	$-0.559106\pi$
$659$	−9.47882	−0.369243	−0.184621	+	0.982810i	$0.559106\pi$
$660$	−0.789966	−0.0307494
$661$	18.3366	0.713211	0.356606	−	0.934255i	$-0.383934\pi$
$661$	18.3366	0.713211	0.356606	+	0.934255i	$0.383934\pi$
$662$	35.5480	1.38161
$663$	0.305730	0.0118736
$664$	−49.2480	−1.91119
$665$	0	0
$666$	−21.4151	−0.829818
$667$	−1.93554	−0.0749444
$668$	4.75353	0.183919
$669$	−4.31390	−0.166785
$670$	−24.7210	−0.955055
$671$	14.3496	0.553958
$672$	0	0
$673$	−27.4997	−1.06004	−0.530018	−	0.847986i	$-0.677815\pi$
$673$	−27.4997	−1.06004	−0.530018	+	0.847986i	$0.677815\pi$
$674$	−34.2613	−1.31970
$675$	−5.45866	−0.210104
$676$	5.63412	0.216697
$677$	−8.82906	−0.339328	−0.169664	−	0.985502i	$-0.554268\pi$
$677$	−8.82906	−0.339328	−0.169664	+	0.985502i	$0.554268\pi$
$678$	1.42544	0.0547436
$679$	0	0
$680$	−13.1051	−0.502559
$681$	2.58100	0.0989042
$682$	−25.5544	−0.978529
$683$	18.6321	0.712937	0.356468	−	0.934307i	$-0.383981\pi$
$683$	18.6321	0.712937	0.356468	+	0.934307i	$0.383981\pi$
$684$	4.22114	0.161399
$685$	50.7158	1.93775
$686$	0	0
$687$	−2.24695	−0.0857266
$688$	−16.4400	−0.626767
$689$	10.9259	0.416242
$690$	4.54552	0.173045
$691$	23.2475	0.884375	0.442187	−	0.896923i	$-0.354203\pi$
$691$	23.2475	0.884375	0.442187	+	0.896923i	$0.354203\pi$
$692$	−10.8671	−0.413104
$693$	0	0
$694$	−19.9519	−0.757365
$695$	−32.9373	−1.24938
$696$	0.118963	0.00450928
$697$	1.28348	0.0486154
$698$	−32.3309	−1.22374
$699$	−2.13546	−0.0807705
$700$	0	0
$701$	35.8177	1.35282	0.676408	−	0.736527i	$-0.263536\pi$
$701$	35.8177	1.35282	0.676408	+	0.736527i	$0.263536\pi$
$702$	−1.72218	−0.0649996
$703$	15.6993	0.592112
$704$	25.9834	0.979288
$705$	5.82223	0.219278
$706$	−33.9452	−1.27755
$707$	0	0
$708$	0.790273	0.0297003
$709$	39.1492	1.47028	0.735140	−	0.677915i	$-0.237116\pi$
$709$	39.1492	1.47028	0.735140	+	0.677915i	$0.237116\pi$
$710$	−39.3193	−1.47563
$711$	23.2698	0.872685
$712$	23.4425	0.878545
$713$	−53.9730	−2.02131
$714$	0	0
$715$	15.4902	0.579300
$716$	12.4985	0.467090
$717$	−0.165059	−0.00616426
$718$	−41.5684	−1.55132
$719$	31.8661	1.18841	0.594203	−	0.804315i	$-0.297468\pi$
$719$	31.8661	1.18841	0.594203	+	0.804315i	$0.297468\pi$
$720$	26.1309	0.973840
$721$	0	0
$722$	−14.5508	−0.541525
$723$	−3.25867	−0.121191
$724$	−0.339832	−0.0126298
$725$	−1.56504	−0.0581240
$726$	−0.429303	−0.0159329
$727$	−19.0831	−0.707751	−0.353876	−	0.935293i	$-0.615136\pi$
$727$	−19.0831	−0.707751	−0.353876	+	0.935293i	$0.615136\pi$
$728$	0	0
$729$	−25.7861	−0.955041
$730$	−19.9481	−0.738313
$731$	7.99981	0.295884
$732$	−0.394291	−0.0145734
$733$	49.7705	1.83832	0.919158	−	0.393889i	$-0.128871\pi$
$733$	49.7705	1.83832	0.919158	+	0.393889i	$0.128871\pi$
$734$	−2.96187	−0.109325
$735$	0	0
$736$	22.1290	0.815687
$737$	−18.0550	−0.665063
$738$	−3.60127	−0.132564
$739$	−13.5737	−0.499315	−0.249658	−	0.968334i	$-0.580318\pi$
$739$	−13.5737	−0.499315	−0.249658	+	0.968334i	$0.580318\pi$
$740$	−10.6256	−0.390604
$741$	0.628874	0.0231023
$742$	0	0
$743$	−35.3536	−1.29700	−0.648499	−	0.761216i	$-0.724603\pi$
$743$	−35.3536	−1.29700	−0.648499	+	0.761216i	$0.724603\pi$
$744$	3.31732	0.121619
$745$	−63.0169	−2.30876
$746$	0.592465	0.0216917
$747$	−47.7837	−1.74832
$748$	−2.02596	−0.0740763
$749$	0	0
$750$	0.648771	0.0236898
$751$	37.8705	1.38192	0.690958	−	0.722895i	$-0.257189\pi$
$751$	37.8705	1.38192	0.690958	+	0.722895i	$0.257189\pi$
$752$	−30.6854	−1.11898
$753$	0.715671	0.0260805
$754$	−0.493762	−0.0179817
$755$	1.26123	0.0459009
$756$	0	0
$757$	6.24629	0.227025	0.113513	−	0.993537i	$-0.463790\pi$
$757$	6.24629	0.227025	0.113513	+	0.993537i	$0.463790\pi$
$758$	−26.2615	−0.953860
$759$	3.31982	0.120502
$760$	−26.9568	−0.977825
$761$	17.1285	0.620909	0.310455	−	0.950588i	$-0.399519\pi$
$761$	17.1285	0.620909	0.310455	+	0.950588i	$0.399519\pi$
$762$	0.892014	0.0323143
$763$	0	0
$764$	−12.3515	−0.446863
$765$	−12.7155	−0.459729
$766$	−22.8057	−0.824002
$767$	−15.4962	−0.559535
$768$	−1.78618	−0.0644530
$769$	20.9857	0.756764	0.378382	−	0.925649i	$-0.376480\pi$
$769$	20.9857	0.756764	0.378382	+	0.925649i	$0.376480\pi$
$770$	0	0
$771$	−0.205094	−0.00738629
$772$	4.77129	0.171722
$773$	41.5409	1.49412	0.747061	−	0.664756i	$-0.231464\pi$
$773$	41.5409	1.49412	0.747061	+	0.664756i	$0.231464\pi$
$774$	−22.4463	−0.806816
$775$	−43.6414	−1.56765
$776$	10.5219	0.377714
$777$	0	0
$778$	6.55415	0.234978
$779$	2.64008	0.0945906
$780$	−0.425632	−0.0152401
$781$	−28.7168	−1.02757
$782$	11.6575	0.416870
$783$	0.231729	0.00828132
$784$	0	0
$785$	−31.3892	−1.12033
$786$	−0.296402	−0.0105723
$787$	32.8118	1.16961	0.584807	−	0.811173i	$-0.301170\pi$
$787$	32.8118	1.16961	0.584807	+	0.811173i	$0.301170\pi$
$788$	−12.5876	−0.448415
$789$	0.0386716	0.00137675
$790$	−31.4549	−1.11911
$791$	0	0
$792$	26.8558	0.954279
$793$	7.73151	0.274554
$794$	−2.39481	−0.0849886
$795$	3.45253	0.122449
$796$	−10.5300	−0.373225
$797$	−26.3194	−0.932281	−0.466141	−	0.884711i	$-0.654356\pi$
$797$	−26.3194	−0.932281	−0.466141	+	0.884711i	$0.654356\pi$
$798$	0	0
$799$	14.9317	0.528247
$800$	17.8931	0.632615
$801$	22.7455	0.803673
$802$	37.2671	1.31595
$803$	−14.5691	−0.514132
$804$	0.496107	0.0174963
$805$	0	0
$806$	−13.7687	−0.484981
$807$	−1.47370	−0.0518768
$808$	44.4890	1.56512
$809$	−52.1309	−1.83282	−0.916412	−	0.400236i	$-0.868928\pi$
$809$	−52.1309	−1.83282	−0.916412	+	0.400236i	$0.868928\pi$
$810$	35.4047	1.24399
$811$	29.9997	1.05343	0.526715	−	0.850042i	$-0.323423\pi$
$811$	29.9997	1.05343	0.526715	+	0.850042i	$0.323423\pi$
$812$	0	0
$813$	−2.18957	−0.0767916
$814$	21.1420	0.741028
$815$	14.4251	0.505290
$816$	−0.509169	−0.0178245
$817$	16.4553	0.575698
$818$	−18.8428	−0.658822
$819$	0	0
$820$	−1.78685	−0.0623994
$821$	12.2038	0.425914	0.212957	−	0.977062i	$-0.431691\pi$
$821$	12.2038	0.425914	0.212957	+	0.977062i	$0.431691\pi$
$822$	2.77278	0.0967120
$823$	35.5471	1.23909	0.619547	−	0.784960i	$-0.287316\pi$
$823$	35.5471	1.23909	0.619547	+	0.784960i	$0.287316\pi$
$824$	26.4744	0.922279
$825$	2.68433	0.0934565
$826$	0	0
$827$	8.09277	0.281413	0.140707	−	0.990051i	$-0.455063\pi$
$827$	8.09277	0.281413	0.140707	+	0.990051i	$0.455063\pi$
$828$	12.0062	0.417245
$829$	−16.1319	−0.560284	−0.280142	−	0.959959i	$-0.590381\pi$
$829$	−16.1319	−0.560284	−0.280142	+	0.959959i	$0.590381\pi$
$830$	64.5916	2.24201
$831$	−2.47955	−0.0860147
$832$	13.9998	0.485357
$833$	0	0
$834$	−1.80078	−0.0623559
$835$	−29.4541	−1.01930
$836$	−4.16731	−0.144130
$837$	6.46182	0.223353
$838$	3.89688	0.134615
$839$	−32.9325	−1.13696	−0.568478	−	0.822698i	$-0.692468\pi$
$839$	−32.9325	−1.13696	−0.568478	+	0.822698i	$0.692468\pi$
$840$	0	0
$841$	−28.9336	−0.997709
$842$	15.2734	0.526357
$843$	1.38175	0.0475900
$844$	−11.6220	−0.400044
$845$	−34.9105	−1.20096
$846$	−41.8963	−1.44043
$847$	0	0
$848$	−18.1962	−0.624859
$849$	0.292780	0.0100482
$850$	9.42599	0.323309
$851$	44.6537	1.53071
$852$	0.789069	0.0270331
$853$	−32.2360	−1.10374	−0.551870	−	0.833930i	$-0.686085\pi$
$853$	−32.2360	−1.10374	−0.551870	+	0.833930i	$0.686085\pi$
$854$	0	0
$855$	−26.1553	−0.894491
$856$	−25.8772	−0.884466
$857$	−32.1580	−1.09850	−0.549249	−	0.835659i	$-0.685086\pi$
$857$	−32.1580	−1.09850	−0.549249	+	0.835659i	$0.685086\pi$
$858$	0.846895	0.0289125
$859$	21.1150	0.720435	0.360217	−	0.932868i	$-0.382702\pi$
$859$	21.1150	0.720435	0.360217	+	0.932868i	$0.382702\pi$
$860$	−11.1372	−0.379776
$861$	0	0
$862$	25.7950	0.878580
$863$	6.10615	0.207856	0.103928	−	0.994585i	$-0.466859\pi$
$863$	6.10615	0.207856	0.103928	+	0.994585i	$0.466859\pi$
$864$	−2.64936	−0.0901330
$865$	67.3353	2.28947
$866$	−45.6258	−1.55043
$867$	−2.30911	−0.0784214
$868$	0	0
$869$	−22.9731	−0.779308
$870$	−0.156027	−0.00528981
$871$	−9.72798	−0.329620
$872$	−9.11205	−0.308573
$873$	10.2091	0.345524
$874$	23.9790	0.811101
$875$	0	0
$876$	0.400323	0.0135257
$877$	−24.2130	−0.817616	−0.408808	−	0.912620i	$-0.634055\pi$
$877$	−24.2130	−0.817616	−0.408808	+	0.912620i	$0.634055\pi$
$878$	−11.4349	−0.385909
$879$	2.15715	0.0727589
$880$	−25.7977	−0.869640
$881$	−38.3269	−1.29127	−0.645634	−	0.763647i	$-0.723407\pi$
$881$	−38.3269	−1.29127	−0.645634	+	0.763647i	$0.723407\pi$
$882$	0	0
$883$	45.8351	1.54247	0.771237	−	0.636548i	$-0.219638\pi$
$883$	45.8351	1.54247	0.771237	+	0.636548i	$0.219638\pi$
$884$	−1.09158	−0.0367138
$885$	−4.89674	−0.164602
$886$	18.2861	0.614332
$887$	−41.3790	−1.38937	−0.694685	−	0.719314i	$-0.744456\pi$
$887$	−41.3790	−1.38937	−0.694685	+	0.719314i	$0.744456\pi$
$888$	−2.74453	−0.0921004
$889$	0	0
$890$	−30.7462	−1.03061
$891$	25.8578	0.866269
$892$	15.4024	0.515711
$893$	30.7140	1.02781
$894$	−3.44533	−0.115229
$895$	−77.4437	−2.58866
$896$	0	0
$897$	1.78871	0.0597233
$898$	34.1010	1.13797
$899$	1.85265	0.0617893
$900$	9.70797	0.323599
$901$	8.85439	0.294983
$902$	3.55535	0.118380
$903$	0	0
$904$	−24.0442	−0.799697
$905$	2.10569	0.0699954
$906$	0.0689553	0.00229089
$907$	39.2475	1.30319	0.651596	−	0.758566i	$-0.274100\pi$
$907$	39.2475	1.30319	0.651596	+	0.758566i	$0.274100\pi$
$908$	−9.21524	−0.305819
$909$	43.1662	1.43173
$910$	0	0
$911$	−26.3136	−0.871808	−0.435904	−	0.899993i	$-0.643571\pi$
$911$	−26.3136	−0.871808	−0.435904	+	0.899993i	$0.643571\pi$
$912$	−1.04734	−0.0346809
$913$	47.1745	1.56125
$914$	37.4213	1.23779
$915$	2.44313	0.0807674
$916$	8.02255	0.265073
$917$	0	0
$918$	−1.39567	−0.0460640
$919$	−7.79422	−0.257108	−0.128554	−	0.991703i	$-0.541034\pi$
$919$	−7.79422	−0.257108	−0.128554	+	0.991703i	$0.541034\pi$
$920$	−76.6733	−2.52784
$921$	−0.269576	−0.00888284
$922$	−9.54160	−0.314236
$923$	−15.4726	−0.509286
$924$	0	0
$925$	36.1061	1.18716
$926$	0.231762	0.00761616
$927$	25.6872	0.843679
$928$	−0.759589	−0.0249347
$929$	37.1799	1.21983	0.609916	−	0.792466i	$-0.291203\pi$
$929$	37.1799	1.21983	0.609916	+	0.792466i	$0.291203\pi$
$930$	−4.35085	−0.142670
$931$	0	0
$932$	7.62448	0.249748
$933$	1.65139	0.0540641
$934$	9.06642	0.296662
$935$	12.5533	0.410538
$936$	14.4698	0.472962
$937$	50.0449	1.63490	0.817448	−	0.576002i	$-0.195388\pi$
$937$	50.0449	1.63490	0.817448	+	0.576002i	$0.195388\pi$
$938$	0	0
$939$	−3.00821	−0.0981691
$940$	−20.7878	−0.678022
$941$	−43.3770	−1.41405	−0.707025	−	0.707189i	$-0.749963\pi$
$941$	−43.3770	−1.41405	−0.707025	+	0.707189i	$0.749963\pi$
$942$	−1.71614	−0.0559150
$943$	7.50919	0.244533
$944$	25.8077	0.839969
$945$	0	0
$946$	22.1601	0.720487
$947$	−4.43494	−0.144116	−0.0720581	−	0.997400i	$-0.522957\pi$
$947$	−4.43494	−0.144116	−0.0720581	+	0.997400i	$0.522957\pi$
$948$	0.631244	0.0205019
$949$	−7.84980	−0.254815
$950$	19.3889	0.629059
$951$	−5.02099	−0.162817
$952$	0	0
$953$	31.9870	1.03616	0.518080	−	0.855332i	$-0.326647\pi$
$953$	31.9870	1.03616	0.518080	+	0.855332i	$0.326647\pi$
$954$	−24.8441	−0.804359
$955$	76.5334	2.47656
$956$	0.589330	0.0190603
$957$	−0.113954	−0.00368362
$958$	−4.62118	−0.149304
$959$	0	0
$960$	4.42389	0.142781
$961$	20.6616	0.666502
$962$	11.3913	0.367270
$963$	−25.1078	−0.809089
$964$	11.6348	0.374732
$965$	−29.5641	−0.951703
$966$	0	0
$967$	−26.4750	−0.851380	−0.425690	−	0.904869i	$-0.639969\pi$
$967$	−26.4750	−0.851380	−0.425690	+	0.904869i	$0.639969\pi$
$968$	7.24144	0.232749
$969$	0.509644	0.0163721
$970$	−13.8001	−0.443094
$971$	19.9699	0.640865	0.320432	−	0.947271i	$-0.396172\pi$
$971$	19.9699	0.640865	0.320432	+	0.947271i	$0.396172\pi$
$972$	−2.15885	−0.0692452
$973$	0	0
$974$	24.4361	0.782982
$975$	1.44631	0.0463191
$976$	−12.8762	−0.412158
$977$	0.608515	0.0194681	0.00973406	−	0.999953i	$-0.496902\pi$
$977$	0.608515	0.0194681	0.00973406	+	0.999953i	$0.496902\pi$
$978$	0.788665	0.0252187
$979$	−22.4555	−0.717680
$980$	0	0
$981$	−8.84112	−0.282275
$982$	7.33944	0.234211
$983$	−47.0091	−1.49936	−0.749680	−	0.661801i	$-0.769792\pi$
$983$	−47.0091	−1.49936	−0.749680	+	0.661801i	$0.769792\pi$
$984$	−0.461533	−0.0147131
$985$	77.9961	2.48516
$986$	−0.400148	−0.0127433
$987$	0	0
$988$	−2.24534	−0.0714338
$989$	46.8039	1.48828
$990$	−35.2229	−1.11946
$991$	43.4367	1.37981	0.689906	−	0.723899i	$-0.257652\pi$
$991$	43.4367	1.37981	0.689906	+	0.723899i	$0.257652\pi$
$992$	−21.1813	−0.672508
$993$	4.42033	0.140275
$994$	0	0
$995$	65.2465	2.06845
$996$	−1.29624	−0.0410729
$997$	44.6175	1.41305	0.706525	−	0.707688i	$-0.250262\pi$
$997$	44.6175	1.41305	0.706525	+	0.707688i	$0.250262\pi$
$998$	25.8413	0.817991
$999$	−5.34608	−0.169143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.s.1.12		17
7.2	even	3		287.2.e.d.165.6	✓	34
7.4	even	3		287.2.e.d.247.6	yes	34
7.6	odd	2		2009.2.a.r.1.12		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.e.d.165.6	✓	34	7.2	even	3
287.2.e.d.247.6	yes	34	7.4	even	3
2009.2.a.r.1.12		17	7.6	odd	2
2009.2.a.s.1.12		17	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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20954 q^{2} +0.150404 q^{3} -0.537006 q^{4} +3.32743 q^{5} +0.181921 q^{6} -3.06862 q^{8} -2.97738 q^{9} +O(q^{10})\)	\(q+1.20954 q^{2} +0.150404 q^{3} -0.537006 q^{4} +3.32743 q^{5} +0.181921 q^{6} -3.06862 q^{8} -2.97738 q^{9} +4.02467 q^{10} +2.93942 q^{11} -0.0807680 q^{12} +1.58375 q^{13} +0.500459 q^{15} -2.63761 q^{16} +1.28348 q^{17} -3.60127 q^{18} +2.64008 q^{19} -1.78685 q^{20} +3.55535 q^{22} +7.50919 q^{23} -0.461533 q^{24} +6.07177 q^{25} +1.91562 q^{26} -0.899024 q^{27} -0.257756 q^{29} +0.605327 q^{30} -7.18760 q^{31} +2.94693 q^{32} +0.442101 q^{33} +1.55243 q^{34} +1.59887 q^{36} +5.94655 q^{37} +3.19329 q^{38} +0.238203 q^{39} -10.2106 q^{40} +1.00000 q^{41} +6.23289 q^{43} -1.57848 q^{44} -9.90701 q^{45} +9.08269 q^{46} +11.6338 q^{47} -0.396708 q^{48} +7.34407 q^{50} +0.193041 q^{51} -0.850483 q^{52} +6.89872 q^{53} -1.08741 q^{54} +9.78069 q^{55} +0.397079 q^{57} -0.311767 q^{58} -9.78449 q^{59} -0.268750 q^{60} +4.88177 q^{61} -8.69371 q^{62} +8.83966 q^{64} +5.26982 q^{65} +0.534740 q^{66} -6.14237 q^{67} -0.689238 q^{68} +1.12941 q^{69} -9.76957 q^{71} +9.13644 q^{72} -4.95646 q^{73} +7.19260 q^{74} +0.913220 q^{75} -1.41774 q^{76} +0.288117 q^{78} -7.81553 q^{79} -8.77647 q^{80} +8.79692 q^{81} +1.20954 q^{82} +16.0489 q^{83} +4.27070 q^{85} +7.53895 q^{86} -0.0387677 q^{87} -9.01994 q^{88} -7.63944 q^{89} -11.9830 q^{90} -4.03248 q^{92} -1.08105 q^{93} +14.0715 q^{94} +8.78466 q^{95} +0.443231 q^{96} -3.42888 q^{97} -8.75175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9}+O(q^{10}) \) 17 * q + 3 * q^2 + q^3 + 25 * q^4 - q^5 + 2 * q^6 + 9 * q^8 + 26 * q^9	\( 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9} - 2 q^{10} + 15 q^{11} + 4 q^{12} - 5 q^{13} + 24 q^{15} + 33 q^{16} + 4 q^{17} + 10 q^{18} + 5 q^{19} - 26 q^{20} + 16 q^{22} + 12 q^{23} + 16 q^{24} + 24 q^{25} + 31 q^{26} - 11 q^{27} + 14 q^{29} - 33 q^{30} - 3 q^{31} + 16 q^{32} + 4 q^{33} - 24 q^{34} + 57 q^{36} + 24 q^{37} + 45 q^{39} + 36 q^{40} + 17 q^{41} + 14 q^{43} - 9 q^{44} - 21 q^{45} + 44 q^{46} + 19 q^{47} - 60 q^{48} - 4 q^{50} + 2 q^{51} - 25 q^{52} + 4 q^{53} + 68 q^{54} + 9 q^{55} - 12 q^{57} - q^{58} - 27 q^{59} + 66 q^{60} - q^{61} - 23 q^{62} + 75 q^{64} + 22 q^{65} - 16 q^{66} + 49 q^{67} + 45 q^{68} + 12 q^{69} + 40 q^{71} - 23 q^{72} - 14 q^{73} + 33 q^{74} + 27 q^{75} - 9 q^{76} - 12 q^{78} + 61 q^{79} - 82 q^{80} + 53 q^{81} + 3 q^{82} - 18 q^{83} - 13 q^{85} - 4 q^{86} - 17 q^{87} + 74 q^{88} + 18 q^{89} - 20 q^{90} + 28 q^{92} - 36 q^{93} - 5 q^{94} + 20 q^{95} + 148 q^{96} + 26 q^{97} + 38 q^{99}+O(q^{100}) \) 17 * q + 3 * q^2 + q^3 + 25 * q^4 - q^5 + 2 * q^6 + 9 * q^8 + 26 * q^9 - 2 * q^10 + 15 * q^11 + 4 * q^12 - 5 * q^13 + 24 * q^15 + 33 * q^16 + 4 * q^17 + 10 * q^18 + 5 * q^19 - 26 * q^20 + 16 * q^22 + 12 * q^23 + 16 * q^24 + 24 * q^25 + 31 * q^26 - 11 * q^27 + 14 * q^29 - 33 * q^30 - 3 * q^31 + 16 * q^32 + 4 * q^33 - 24 * q^34 + 57 * q^36 + 24 * q^37 + 45 * q^39 + 36 * q^40 + 17 * q^41 + 14 * q^43 - 9 * q^44 - 21 * q^45 + 44 * q^46 + 19 * q^47 - 60 * q^48 - 4 * q^50 + 2 * q^51 - 25 * q^52 + 4 * q^53 + 68 * q^54 + 9 * q^55 - 12 * q^57 - q^58 - 27 * q^59 + 66 * q^60 - q^61 - 23 * q^62 + 75 * q^64 + 22 * q^65 - 16 * q^66 + 49 * q^67 + 45 * q^68 + 12 * q^69 + 40 * q^71 - 23 * q^72 - 14 * q^73 + 33 * q^74 + 27 * q^75 - 9 * q^76 - 12 * q^78 + 61 * q^79 - 82 * q^80 + 53 * q^81 + 3 * q^82 - 18 * q^83 - 13 * q^85 - 4 * q^86 - 17 * q^87 + 74 * q^88 + 18 * q^89 - 20 * q^90 + 28 * q^92 - 36 * q^93 - 5 * q^94 + 20 * q^95 + 148 * q^96 + 26 * q^97 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

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