LMFDB - Embedded newform 2009.2.a.t.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.0419457661$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + \cdots + 49 $ x^20 - 2x^19 - 27x^18 + 52x^17 + 302x^16 - 552x^15 - 1820x^14 + 3090x^13 + 6449x^12 - 9852x^11 - 13797x^10 + 18080x^9 + 17721x^8 - 18446x^7 - 13352x^6 + 9524x^5 + 5690x^4 - 1972x^3 - 1179x^2 + 42*x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-2.11567$ 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)

$f(q)$	$=$	$q+2.11567 q^{2} -1.23352 q^{3} +2.47606 q^{4} -0.585449 q^{5} -2.60971 q^{6} +1.00718 q^{8} -1.47844 q^{9} +O(q^{10})$	\(q+2.11567 q^{2} -1.23352 q^{3} +2.47606 q^{4} -0.585449 q^{5} -2.60971 q^{6} +1.00718 q^{8} -1.47844 q^{9} -1.23862 q^{10} +4.77087 q^{11} -3.05425 q^{12} -2.83079 q^{13} +0.722160 q^{15} -2.82126 q^{16} -6.00129 q^{17} -3.12789 q^{18} -1.43428 q^{19} -1.44960 q^{20} +10.0936 q^{22} +1.19566 q^{23} -1.24237 q^{24} -4.65725 q^{25} -5.98902 q^{26} +5.52422 q^{27} -5.56162 q^{29} +1.52785 q^{30} -4.50146 q^{31} -7.98320 q^{32} -5.88494 q^{33} -12.6967 q^{34} -3.66070 q^{36} +7.14099 q^{37} -3.03447 q^{38} +3.49182 q^{39} -0.589650 q^{40} -1.00000 q^{41} +2.52839 q^{43} +11.8129 q^{44} +0.865552 q^{45} +2.52963 q^{46} +5.98089 q^{47} +3.48007 q^{48} -9.85320 q^{50} +7.40268 q^{51} -7.00920 q^{52} -10.7530 q^{53} +11.6874 q^{54} -2.79310 q^{55} +1.76921 q^{57} -11.7665 q^{58} -2.24030 q^{59} +1.78811 q^{60} -8.42708 q^{61} -9.52359 q^{62} -11.2473 q^{64} +1.65728 q^{65} -12.4506 q^{66} -15.6745 q^{67} -14.8595 q^{68} -1.47487 q^{69} +2.03697 q^{71} -1.48905 q^{72} -6.17349 q^{73} +15.1080 q^{74} +5.74479 q^{75} -3.55137 q^{76} +7.38755 q^{78} +9.99268 q^{79} +1.65170 q^{80} -2.37889 q^{81} -2.11567 q^{82} +0.987320 q^{83} +3.51345 q^{85} +5.34924 q^{86} +6.86034 q^{87} +4.80511 q^{88} -8.84173 q^{89} +1.83122 q^{90} +2.96053 q^{92} +5.55262 q^{93} +12.6536 q^{94} +0.839701 q^{95} +9.84740 q^{96} +15.0087 q^{97} -7.05345 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9	$ 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9} - 16 q^{10} - 6 q^{12} - 12 q^{13} + 14 q^{16} - 8 q^{17} - 18 q^{18} - 36 q^{19} - 24 q^{20} - 8 q^{22} - 12 q^{23} - 36 q^{24} + 20 q^{25} + 22 q^{26} - 32 q^{27} + 4 q^{29} + 28 q^{30} - 80 q^{31} + 6 q^{32} + 12 q^{33} - 48 q^{34} + 26 q^{36} + 4 q^{37} - 12 q^{38} - 28 q^{39} + 4 q^{40} - 20 q^{41} - 20 q^{44} - 40 q^{45} + 8 q^{46} - 32 q^{47} - 16 q^{48} + 6 q^{50} - 20 q^{51} - 36 q^{52} + 4 q^{53} + 50 q^{54} - 64 q^{55} - 4 q^{57} - 32 q^{59} + 20 q^{60} - 44 q^{61} + 8 q^{62} - 30 q^{64} - 8 q^{65} - 32 q^{66} - 4 q^{67} + 48 q^{68} + 24 q^{69} + 8 q^{71} - 8 q^{72} - 48 q^{73} - 38 q^{74} - 24 q^{75} - 84 q^{76} + 30 q^{78} - 4 q^{79} - 56 q^{80} + 2 q^{82} - 8 q^{83} - 12 q^{85} - 24 q^{86} - 40 q^{87} - 48 q^{88} - 20 q^{89} - 48 q^{90} - 50 q^{92} + 48 q^{93} - 26 q^{94} + 20 q^{95} - 70 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) $ 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9 - 16 * q^10 - 6 * q^12 - 12 * q^13 + 14 * q^16 - 8 * q^17 - 18 * q^18 - 36 * q^19 - 24 * q^20 - 8 * q^22 - 12 * q^23 - 36 * q^24 + 20 * q^25 + 22 * q^26 - 32 * q^27 + 4 * q^29 + 28 * q^30 - 80 * q^31 + 6 * q^32 + 12 * q^33 - 48 * q^34 + 26 * q^36 + 4 * q^37 - 12 * q^38 - 28 * q^39 + 4 * q^40 - 20 * q^41 - 20 * q^44 - 40 * q^45 + 8 * q^46 - 32 * q^47 - 16 * q^48 + 6 * q^50 - 20 * q^51 - 36 * q^52 + 4 * q^53 + 50 * q^54 - 64 * q^55 - 4 * q^57 - 32 * q^59 + 20 * q^60 - 44 * q^61 + 8 * q^62 - 30 * q^64 - 8 * q^65 - 32 * q^66 - 4 * q^67 + 48 * q^68 + 24 * q^69 + 8 * q^71 - 8 * q^72 - 48 * q^73 - 38 * q^74 - 24 * q^75 - 84 * q^76 + 30 * q^78 - 4 * q^79 - 56 * q^80 + 2 * q^82 - 8 * q^83 - 12 * q^85 - 24 * q^86 - 40 * q^87 - 48 * q^88 - 20 * q^89 - 48 * q^90 - 50 * q^92 + 48 * q^93 - 26 * q^94 + 20 * q^95 - 70 * q^96 - 8 * q^97 - 8 * 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 $

$2$	2.11567	1.49600	0.748002	−	0.663697i	$-0.231013\pi$
$2$	2.11567	1.49600	0.748002	+	0.663697i	$0.231013\pi$
$3$	−1.23352	−0.712170	−0.356085	−	0.934454i	$-0.615889\pi$
$3$	−1.23352	−0.712170	−0.356085	+	0.934454i	$0.615889\pi$
$4$	2.47606	1.23803
$5$	−0.585449	−0.261821	−0.130910	−	0.991394i	$-0.541790\pi$
$5$	−0.585449	−0.261821	−0.130910	+	0.991394i	$0.541790\pi$
$6$	−2.60971	−1.06541
$7$	0	0
$7$	0	0
$8$	1.00718	0.356090
$9$	−1.47844	−0.492813
$10$	−1.23862	−0.391685
$11$	4.77087	1.43847	0.719236	−	0.694766i	$-0.244492\pi$
$11$	4.77087	1.43847	0.719236	+	0.694766i	$0.244492\pi$
$12$	−3.05425	−0.881687
$13$	−2.83079	−0.785120	−0.392560	−	0.919726i	$-0.628411\pi$
$13$	−2.83079	−0.785120	−0.392560	+	0.919726i	$0.628411\pi$
$14$	0	0
$15$	0.722160	0.186461
$16$	−2.82126	−0.705315
$17$	−6.00129	−1.45553	−0.727763	−	0.685829i	$-0.759440\pi$
$17$	−6.00129	−1.45553	−0.727763	+	0.685829i	$0.759440\pi$
$18$	−3.12789	−0.737251
$19$	−1.43428	−0.329048	−0.164524	−	0.986373i	$-0.552609\pi$
$19$	−1.43428	−0.329048	−0.164524	+	0.986373i	$0.552609\pi$
$20$	−1.44960	−0.324141
$21$	0	0
$22$	10.0936	2.15196
$23$	1.19566	0.249313	0.124656	−	0.992200i	$-0.460217\pi$
$23$	1.19566	0.249313	0.124656	+	0.992200i	$0.460217\pi$
$24$	−1.24237	−0.253597
$25$	−4.65725	−0.931450
$26$	−5.98902	−1.17454
$27$	5.52422	1.06314
$28$	0	0
$29$	−5.56162	−1.03277	−0.516383	−	0.856358i	$-0.672722\pi$
$29$	−5.56162	−1.03277	−0.516383	+	0.856358i	$0.672722\pi$
$30$	1.52785	0.278946
$31$	−4.50146	−0.808486	−0.404243	−	0.914652i	$-0.632465\pi$
$31$	−4.50146	−0.808486	−0.404243	+	0.914652i	$0.632465\pi$
$32$	−7.98320	−1.41124
$33$	−5.88494	−1.02444
$34$	−12.6967	−2.17747
$35$	0	0
$36$	−3.66070	−0.610117
$37$	7.14099	1.17397	0.586986	−	0.809597i	$-0.300314\pi$
$37$	7.14099	1.17397	0.586986	+	0.809597i	$0.300314\pi$
$38$	−3.03447	−0.492256
$39$	3.49182	0.559139
$40$	−0.589650	−0.0932319
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	2.52839	0.385576	0.192788	−	0.981240i	$-0.438247\pi$
$43$	2.52839	0.385576	0.192788	+	0.981240i	$0.438247\pi$
$44$	11.8129	1.78087
$45$	0.865552	0.129029
$46$	2.52963	0.372973
$47$	5.98089	0.872403	0.436201	−	0.899849i	$-0.356324\pi$
$47$	5.98089	0.872403	0.436201	+	0.899849i	$0.356324\pi$
$48$	3.48007	0.502304
$49$	0	0
$50$	−9.85320	−1.39345
$51$	7.40268	1.03658
$52$	−7.00920	−0.972001
$53$	−10.7530	−1.47703	−0.738517	−	0.674235i	$-0.764473\pi$
$53$	−10.7530	−1.47703	−0.738517	+	0.674235i	$0.764473\pi$
$54$	11.6874	1.59046
$55$	−2.79310	−0.376622
$56$	0	0
$57$	1.76921	0.234338
$58$	−11.7665	−1.54502
$59$	−2.24030	−0.291662	−0.145831	−	0.989310i	$-0.546586\pi$
$59$	−2.24030	−0.291662	−0.145831	+	0.989310i	$0.546586\pi$
$60$	1.78811	0.230844
$61$	−8.42708	−1.07898	−0.539489	−	0.841993i	$-0.681382\pi$
$61$	−8.42708	−1.07898	−0.539489	+	0.841993i	$0.681382\pi$
$62$	−9.52359	−1.20950
$63$	0	0
$64$	−11.2473	−1.40591
$65$	1.65728	0.205561
$66$	−12.4506	−1.53256
$67$	−15.6745	−1.91495	−0.957475	−	0.288518i	$-0.906838\pi$
$67$	−15.6745	−1.91495	−0.957475	+	0.288518i	$0.906838\pi$
$68$	−14.8595	−1.80198
$69$	−1.47487	−0.177553
$70$	0	0
$71$	2.03697	0.241744	0.120872	−	0.992668i	$-0.461431\pi$
$71$	2.03697	0.241744	0.120872	+	0.992668i	$0.461431\pi$
$72$	−1.48905	−0.175486
$73$	−6.17349	−0.722553	−0.361276	−	0.932459i	$-0.617659\pi$
$73$	−6.17349	−0.722553	−0.361276	+	0.932459i	$0.617659\pi$
$74$	15.1080	1.75627
$75$	5.74479	0.663351
$76$	−3.55137	−0.407370
$77$	0	0
$78$	7.38755	0.836475
$79$	9.99268	1.12426	0.562132	−	0.827047i	$-0.309981\pi$
$79$	9.99268	1.12426	0.562132	+	0.827047i	$0.309981\pi$
$80$	1.65170	0.184666
$81$	−2.37889	−0.264321
$82$	−2.11567	−0.233637
$83$	0.987320	0.108372	0.0541862	−	0.998531i	$-0.482744\pi$
$83$	0.987320	0.108372	0.0541862	+	0.998531i	$0.482744\pi$
$84$	0	0
$85$	3.51345	0.381087
$86$	5.34924	0.576823
$87$	6.86034	0.735506
$88$	4.80511	0.512226
$89$	−8.84173	−0.937222	−0.468611	−	0.883405i	$-0.655245\pi$
$89$	−8.84173	−0.937222	−0.468611	+	0.883405i	$0.655245\pi$
$90$	1.83122	0.193028
$91$	0	0
$92$	2.96053	0.308656
$93$	5.55262	0.575780
$94$	12.6536	1.30512
$95$	0.839701	0.0861515
$96$	9.84740	1.00505
$97$	15.0087	1.52391	0.761953	−	0.647632i	$-0.224241\pi$
$97$	15.0087	1.52391	0.761953	+	0.647632i	$0.224241\pi$
$98$	0	0
$99$	−7.05345	−0.708898
$100$	−11.5316	−1.15316
$101$	−9.32332	−0.927705	−0.463852	−	0.885912i	$-0.653533\pi$
$101$	−9.32332	−0.927705	−0.463852	+	0.885912i	$0.653533\pi$
$102$	15.6616	1.55073
$103$	7.33173	0.722417	0.361209	−	0.932485i	$-0.382364\pi$
$103$	7.33173	0.722417	0.361209	+	0.932485i	$0.382364\pi$
$104$	−2.85110	−0.279574
$105$	0	0
$106$	−22.7497	−2.20965
$107$	−3.58171	−0.346257	−0.173129	−	0.984899i	$-0.555388\pi$
$107$	−3.58171	−0.346257	−0.173129	+	0.984899i	$0.555388\pi$
$108$	13.6783	1.31619
$109$	−12.7703	−1.22317	−0.611587	−	0.791177i	$-0.709468\pi$
$109$	−12.7703	−1.22317	−0.611587	+	0.791177i	$0.709468\pi$
$110$	−5.90928	−0.563428
$111$	−8.80852	−0.836068
$112$	0	0
$113$	−6.06708	−0.570743	−0.285371	−	0.958417i	$-0.592117\pi$
$113$	−6.06708	−0.570743	−0.285371	+	0.958417i	$0.592117\pi$
$114$	3.74307	0.350570
$115$	−0.699999	−0.0652753
$116$	−13.7709	−1.27859
$117$	4.18516	0.386918
$118$	−4.73972	−0.436327
$119$	0	0
$120$	0.727342	0.0663970
$121$	11.7612	1.06920
$122$	−17.8289	−1.61415
$123$	1.23352	0.111222
$124$	−11.1459	−1.00093
$125$	5.65383	0.505694
$126$	0	0
$127$	−10.4102	−0.923752	−0.461876	−	0.886945i	$-0.652823\pi$
$127$	−10.4102	−0.923752	−0.461876	+	0.886945i	$0.652823\pi$
$128$	−7.82915	−0.692006
$129$	−3.11881	−0.274596
$130$	3.50627	0.307520
$131$	2.78609	0.243421	0.121711	−	0.992566i	$-0.461162\pi$
$131$	2.78609	0.243421	0.121711	+	0.992566i	$0.461162\pi$
$132$	−14.5714	−1.26828
$133$	0	0
$134$	−33.1621	−2.86477
$135$	−3.23415	−0.278351
$136$	−6.04435	−0.518299
$137$	7.60252	0.649527	0.324764	−	0.945795i	$-0.394715\pi$
$137$	7.60252	0.649527	0.324764	+	0.945795i	$0.394715\pi$
$138$	−3.12033	−0.265620
$139$	21.7010	1.84066	0.920329	−	0.391146i	$-0.127921\pi$
$139$	21.7010	1.84066	0.920329	+	0.391146i	$0.127921\pi$
$140$	0	0
$141$	−7.37752	−0.621299
$142$	4.30955	0.361650
$143$	−13.5053	−1.12937
$144$	4.17106	0.347589
$145$	3.25604	0.270400
$146$	−13.0611	−1.08094
$147$	0	0
$148$	17.6815	1.45341
$149$	−1.49636	−0.122587	−0.0612934	−	0.998120i	$-0.519523\pi$
$149$	−1.49636	−0.122587	−0.0612934	+	0.998120i	$0.519523\pi$
$150$	12.1541	0.992376
$151$	7.11631	0.579117	0.289559	−	0.957160i	$-0.406491\pi$
$151$	7.11631	0.579117	0.289559	+	0.957160i	$0.406491\pi$
$152$	−1.44458	−0.117171
$153$	8.87255	0.717303
$154$	0	0
$155$	2.63537	0.211678
$156$	8.64595	0.692230
$157$	11.8998	0.949704	0.474852	−	0.880066i	$-0.342502\pi$
$157$	11.8998	0.949704	0.474852	+	0.880066i	$0.342502\pi$
$158$	21.1412	1.68190
$159$	13.2639	1.05190
$160$	4.67376	0.369493
$161$	0	0
$162$	−5.03295	−0.395426
$163$	−6.83976	−0.535731	−0.267866	−	0.963456i	$-0.586318\pi$
$163$	−6.83976	−0.535731	−0.267866	+	0.963456i	$0.586318\pi$
$164$	−2.47606	−0.193347
$165$	3.44533	0.268219
$166$	2.08884	0.162126
$167$	9.41169	0.728298	0.364149	−	0.931341i	$-0.381360\pi$
$167$	9.41169	0.728298	0.364149	+	0.931341i	$0.381360\pi$
$168$	0	0
$169$	−4.98662	−0.383586
$170$	7.43329	0.570108
$171$	2.12050	0.162159
$172$	6.26043	0.477354
$173$	12.4605	0.947351	0.473676	−	0.880699i	$-0.342927\pi$
$173$	12.4605	0.947351	0.473676	+	0.880699i	$0.342927\pi$
$174$	14.5142	1.10032
$175$	0	0
$176$	−13.4599	−1.01458
$177$	2.76344	0.207713
$178$	−18.7062	−1.40209
$179$	8.11135	0.606271	0.303135	−	0.952947i	$-0.401967\pi$
$179$	8.11135	0.606271	0.303135	+	0.952947i	$0.401967\pi$
$180$	2.14315	0.159741
$181$	21.8320	1.62276	0.811381	−	0.584517i	$-0.198716\pi$
$181$	21.8320	1.62276	0.811381	+	0.584517i	$0.198716\pi$
$182$	0	0
$183$	10.3949	0.768416
$184$	1.20424	0.0887779
$185$	−4.18069	−0.307370
$186$	11.7475	0.861368
$187$	−28.6314	−2.09373
$188$	14.8090	1.08006
$189$	0	0
$190$	1.77653	0.128883
$191$	−8.31641	−0.601754	−0.300877	−	0.953663i	$-0.597279\pi$
$191$	−8.31641	−0.601754	−0.300877	+	0.953663i	$0.597279\pi$
$192$	13.8737	1.00125
$193$	−7.95439	−0.572569	−0.286285	−	0.958145i	$-0.592420\pi$
$193$	−7.95439	−0.572569	−0.286285	+	0.958145i	$0.592420\pi$
$194$	31.7535	2.27977
$195$	−2.04429	−0.146394
$196$	0	0
$197$	0.609693	0.0434388	0.0217194	−	0.999764i	$-0.493086\pi$
$197$	0.609693	0.0434388	0.0217194	+	0.999764i	$0.493086\pi$
$198$	−14.9228	−1.06051
$199$	23.4466	1.66209	0.831044	−	0.556207i	$-0.187744\pi$
$199$	23.4466	1.66209	0.831044	+	0.556207i	$0.187744\pi$
$200$	−4.69067	−0.331680
$201$	19.3348	1.36377
$202$	−19.7251	−1.38785
$203$	0	0
$204$	18.3294	1.28332
$205$	0.585449	0.0408895
$206$	15.5115	1.08074
$207$	−1.76772	−0.122865
$208$	7.98640	0.553757
$209$	−6.84279	−0.473326
$210$	0	0
$211$	25.1280	1.72988	0.864941	−	0.501874i	$-0.167356\pi$
$211$	25.1280	1.72988	0.864941	+	0.501874i	$0.167356\pi$
$212$	−26.6249	−1.82861
$213$	−2.51263	−0.172163
$214$	−7.57772	−0.518002
$215$	−1.48024	−0.100952
$216$	5.56386	0.378573
$217$	0	0
$218$	−27.0177	−1.82987
$219$	7.61510	0.514581
$220$	−6.91588	−0.466268
$221$	16.9884	1.14276
$222$	−18.6359	−1.25076
$223$	−10.6661	−0.714256	−0.357128	−	0.934055i	$-0.616244\pi$
$223$	−10.6661	−0.714256	−0.357128	+	0.934055i	$0.616244\pi$
$224$	0	0
$225$	6.88547	0.459031
$226$	−12.8359	−0.853833
$227$	23.5612	1.56381	0.781905	−	0.623398i	$-0.214248\pi$
$227$	23.5612	1.56381	0.781905	+	0.623398i	$0.214248\pi$
$228$	4.38067	0.290117
$229$	−16.7878	−1.10937	−0.554684	−	0.832061i	$-0.687161\pi$
$229$	−16.7878	−1.10937	−0.554684	+	0.832061i	$0.687161\pi$
$230$	−1.48097	−0.0976521
$231$	0	0
$232$	−5.60153	−0.367758
$233$	10.5779	0.692983	0.346491	−	0.938053i	$-0.387373\pi$
$233$	10.5779	0.692983	0.346491	+	0.938053i	$0.387373\pi$
$234$	8.85441	0.578831
$235$	−3.50151	−0.228413
$236$	−5.54710	−0.361085
$237$	−12.3261	−0.800668
$238$	0	0
$239$	−11.2347	−0.726715	−0.363358	−	0.931650i	$-0.618370\pi$
$239$	−11.2347	−0.726715	−0.363358	+	0.931650i	$0.618370\pi$
$240$	−2.03740	−0.131514
$241$	2.02745	0.130599	0.0652997	−	0.997866i	$-0.479200\pi$
$241$	2.02745	0.130599	0.0652997	+	0.997866i	$0.479200\pi$
$242$	24.8828	1.59953
$243$	−13.6383	−0.874896
$244$	−20.8659	−1.33580
$245$	0	0
$246$	2.60971	0.166389
$247$	4.06016	0.258342
$248$	−4.53376	−0.287894
$249$	−1.21787	−0.0771796
$250$	11.9616	0.756520
$251$	−5.52265	−0.348587	−0.174293	−	0.984694i	$-0.555764\pi$
$251$	−5.52265	−0.348587	−0.174293	+	0.984694i	$0.555764\pi$
$252$	0	0
$253$	5.70435	0.358629
$254$	−22.0244	−1.38194
$255$	−4.33389	−0.271399
$256$	5.93070	0.370669
$257$	−12.3221	−0.768634	−0.384317	−	0.923201i	$-0.625563\pi$
$257$	−12.3221	−0.768634	−0.384317	+	0.923201i	$0.625563\pi$
$258$	−6.59836	−0.410796
$259$	0	0
$260$	4.10353	0.254490
$261$	8.22252	0.508961
$262$	5.89444	0.364159
$263$	22.3230	1.37650	0.688248	−	0.725475i	$-0.258380\pi$
$263$	22.3230	1.37650	0.688248	+	0.725475i	$0.258380\pi$
$264$	−5.92717	−0.364792
$265$	6.29531	0.386718
$266$	0	0
$267$	10.9064	0.667461
$268$	−38.8110	−2.37076
$269$	−5.44867	−0.332211	−0.166106	−	0.986108i	$-0.553119\pi$
$269$	−5.44867	−0.332211	−0.166106	+	0.986108i	$0.553119\pi$
$270$	−6.84240	−0.416415
$271$	−7.45275	−0.452722	−0.226361	−	0.974043i	$-0.572683\pi$
$271$	−7.45275	−0.452722	−0.226361	+	0.974043i	$0.572683\pi$
$272$	16.9312	1.02660
$273$	0	0
$274$	16.0844	0.971695
$275$	−22.2191	−1.33986
$276$	−3.65185	−0.219816
$277$	13.1565	0.790499	0.395250	−	0.918574i	$-0.370658\pi$
$277$	13.1565	0.790499	0.395250	+	0.918574i	$0.370658\pi$
$278$	45.9122	2.75363
$279$	6.65514	0.398433
$280$	0	0
$281$	−17.4036	−1.03821	−0.519107	−	0.854709i	$-0.673735\pi$
$281$	−17.4036	−1.03821	−0.519107	+	0.854709i	$0.673735\pi$
$282$	−15.6084	−0.929466
$283$	−30.6226	−1.82032	−0.910162	−	0.414252i	$-0.864043\pi$
$283$	−30.6226	−1.82032	−0.910162	+	0.414252i	$0.864043\pi$
$284$	5.04365	0.299285
$285$	−1.03578	−0.0613545
$286$	−28.5728	−1.68955
$287$	0	0
$288$	11.8027	0.695480
$289$	19.0155	1.11856
$290$	6.88871	0.404519
$291$	−18.5135	−1.08528
$292$	−15.2859	−0.894540
$293$	−33.0367	−1.93003	−0.965013	−	0.262203i	$-0.915551\pi$
$293$	−33.0367	−1.93003	−0.965013	+	0.262203i	$0.915551\pi$
$294$	0	0
$295$	1.31158	0.0763631
$296$	7.19223	0.418040
$297$	26.3554	1.52929
$298$	−3.16581	−0.183390
$299$	−3.38467	−0.195741
$300$	14.2244	0.821247
$301$	0	0
$302$	15.0558	0.866362
$303$	11.5005	0.660684
$304$	4.04649	0.232082
$305$	4.93363	0.282499
$306$	18.7714	1.07309
$307$	−7.00620	−0.399865	−0.199932	−	0.979810i	$-0.564072\pi$
$307$	−7.00620	−0.399865	−0.199932	+	0.979810i	$0.564072\pi$
$308$	0	0
$309$	−9.04381	−0.514484
$310$	5.57558	0.316672
$311$	−23.2704	−1.31954	−0.659771	−	0.751467i	$-0.729347\pi$
$311$	−23.2704	−1.31954	−0.659771	+	0.751467i	$0.729347\pi$
$312$	3.51688	0.199104
$313$	11.4649	0.648035	0.324018	−	0.946051i	$-0.394966\pi$
$313$	11.4649	0.648035	0.324018	+	0.946051i	$0.394966\pi$
$314$	25.1759	1.42076
$315$	0	0
$316$	24.7424	1.39187
$317$	26.1624	1.46943	0.734714	−	0.678377i	$-0.237316\pi$
$317$	26.1624	1.46943	0.734714	+	0.678377i	$0.237316\pi$
$318$	28.0621	1.57365
$319$	−26.5338	−1.48561
$320$	6.58472	0.368097
$321$	4.41810	0.246594
$322$	0	0
$323$	8.60756	0.478937
$324$	−5.89027	−0.327237
$325$	13.1837	0.731300
$326$	−14.4707	−0.801456
$327$	15.7524	0.871108
$328$	−1.00718	−0.0556120
$329$	0	0
$330$	7.28919	0.401256
$331$	−24.1423	−1.32698	−0.663489	−	0.748186i	$-0.730925\pi$
$331$	−24.1423	−1.32698	−0.663489	+	0.748186i	$0.730925\pi$
$332$	2.44466	0.134168
$333$	−10.5575	−0.578549
$334$	19.9120	1.08954
$335$	9.17664	0.501373
$336$	0	0
$337$	7.52527	0.409928	0.204964	−	0.978770i	$-0.434292\pi$
$337$	7.52527	0.409928	0.204964	+	0.978770i	$0.434292\pi$
$338$	−10.5500	−0.573846
$339$	7.48383	0.406466
$340$	8.69949	0.471796
$341$	−21.4759	−1.16298
$342$	4.48629	0.242591
$343$	0	0
$344$	2.54653	0.137300
$345$	0.863460	0.0464871
$346$	26.3622	1.41724
$347$	−7.57639	−0.406722	−0.203361	−	0.979104i	$-0.565187\pi$
$347$	−7.57639	−0.406722	−0.203361	+	0.979104i	$0.565187\pi$
$348$	16.9866	0.910576
$349$	−17.5936	−0.941765	−0.470882	−	0.882196i	$-0.656064\pi$
$349$	−17.5936	−0.941765	−0.470882	+	0.882196i	$0.656064\pi$
$350$	0	0
$351$	−15.6379	−0.834691
$352$	−38.0868	−2.03004
$353$	20.2500	1.07780	0.538900	−	0.842370i	$-0.318840\pi$
$353$	20.2500	1.07780	0.538900	+	0.842370i	$0.318840\pi$
$354$	5.84652	0.310739
$355$	−1.19254	−0.0632935
$356$	−21.8926	−1.16031
$357$	0	0
$358$	17.1609	0.906984
$359$	11.7375	0.619482	0.309741	−	0.950821i	$-0.399758\pi$
$359$	11.7375	0.619482	0.309741	+	0.950821i	$0.399758\pi$
$360$	0.871763	0.0459459
$361$	−16.9428	−0.891728
$362$	46.1894	2.42766
$363$	−14.5076	−0.761453
$364$	0	0
$365$	3.61427	0.189179
$366$	21.9922	1.14955
$367$	−31.5417	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$367$	−31.5417	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$368$	−3.37327	−0.175844
$369$	1.47844	0.0769645
$370$	−8.84495	−0.459827
$371$	0	0
$372$	13.7486	0.712831
$373$	−7.54821	−0.390831	−0.195416	−	0.980721i	$-0.562606\pi$
$373$	−7.54821	−0.390831	−0.195416	+	0.980721i	$0.562606\pi$
$374$	−60.5745	−3.13223
$375$	−6.97408	−0.360140
$376$	6.02381	0.310654
$377$	15.7438	0.810846
$378$	0	0
$379$	−28.6602	−1.47218	−0.736088	−	0.676886i	$-0.763329\pi$
$379$	−28.6602	−1.47218	−0.736088	+	0.676886i	$0.763329\pi$
$380$	2.07915	0.106658
$381$	12.8411	0.657868
$382$	−17.5948	−0.900227
$383$	15.6994	0.802202	0.401101	−	0.916034i	$-0.368628\pi$
$383$	15.6994	0.802202	0.401101	+	0.916034i	$0.368628\pi$
$384$	9.65738	0.492826
$385$	0	0
$386$	−16.8289	−0.856566
$387$	−3.73807	−0.190017
$388$	37.1625	1.88664
$389$	−25.8450	−1.31039	−0.655196	−	0.755459i	$-0.727414\pi$
$389$	−25.8450	−1.31039	−0.655196	+	0.755459i	$0.727414\pi$
$390$	−4.32503	−0.219006
$391$	−7.17551	−0.362881
$392$	0	0
$393$	−3.43668	−0.173358
$394$	1.28991	0.0649846
$395$	−5.85020	−0.294356
$396$	−17.4647	−0.877636
$397$	4.87804	0.244822	0.122411	−	0.992480i	$-0.460937\pi$
$397$	4.87804	0.244822	0.122411	+	0.992480i	$0.460937\pi$
$398$	49.6053	2.48649
$399$	0	0
$400$	13.1393	0.656966
$401$	30.1936	1.50780	0.753899	−	0.656990i	$-0.228171\pi$
$401$	30.1936	1.50780	0.753899	+	0.656990i	$0.228171\pi$
$402$	40.9060	2.04021
$403$	12.7427	0.634759
$404$	−23.0851	−1.14852
$405$	1.39272	0.0692049
$406$	0	0
$407$	34.0687	1.68872
$408$	7.45580	0.369117
$409$	−31.2548	−1.54545	−0.772725	−	0.634741i	$-0.781107\pi$
$409$	−31.2548	−1.54545	−0.772725	+	0.634741i	$0.781107\pi$
$410$	1.23862	0.0611709
$411$	−9.37782	−0.462574
$412$	18.1538	0.894373
$413$	0	0
$414$	−3.73990	−0.183806
$415$	−0.578026	−0.0283742
$416$	22.5988	1.10800
$417$	−26.7685	−1.31086
$418$	−14.4771	−0.708097
$419$	28.5274	1.39365	0.696827	−	0.717239i	$-0.254594\pi$
$419$	28.5274	1.39365	0.696827	+	0.717239i	$0.254594\pi$
$420$	0	0
$421$	23.2283	1.13208	0.566039	−	0.824378i	$-0.308475\pi$
$421$	23.2283	1.13208	0.566039	+	0.824378i	$0.308475\pi$
$422$	53.1625	2.58791
$423$	−8.84239	−0.429932
$424$	−10.8301	−0.525957
$425$	27.9495	1.35575
$426$	−5.31590	−0.257556
$427$	0	0
$428$	−8.86852	−0.428676
$429$	16.6590	0.804306
$430$	−3.13171	−0.151024
$431$	12.0476	0.580311	0.290155	−	0.956980i	$-0.406293\pi$
$431$	12.0476	0.580311	0.290155	+	0.956980i	$0.406293\pi$
$432$	−15.5853	−0.749847
$433$	12.7944	0.614860	0.307430	−	0.951571i	$-0.400531\pi$
$433$	12.7944	0.614860	0.307430	+	0.951571i	$0.400531\pi$
$434$	0	0
$435$	−4.01638	−0.192571
$436$	−31.6200	−1.51432
$437$	−1.71492	−0.0820358
$438$	16.1110	0.769815
$439$	−31.6091	−1.50862	−0.754310	−	0.656518i	$-0.772029\pi$
$439$	−31.6091	−1.50862	−0.754310	+	0.656518i	$0.772029\pi$
$440$	−2.81314	−0.134111
$441$	0	0
$442$	35.9418	1.70958
$443$	−10.9402	−0.519782	−0.259891	−	0.965638i	$-0.583687\pi$
$443$	−10.9402	−0.519782	−0.259891	+	0.965638i	$0.583687\pi$
$444$	−21.8104	−1.03507
$445$	5.17638	0.245384
$446$	−22.5660	−1.06853
$447$	1.84578	0.0873026
$448$	0	0
$449$	−24.3547	−1.14937	−0.574684	−	0.818375i	$-0.694875\pi$
$449$	−24.3547	−1.14937	−0.574684	+	0.818375i	$0.694875\pi$
$450$	14.5674	0.686712
$451$	−4.77087	−0.224652
$452$	−15.0224	−0.706595
$453$	−8.77808	−0.412430
$454$	49.8476	2.33947
$455$	0	0
$456$	1.78191	0.0834455
$457$	33.6288	1.57309	0.786545	−	0.617533i	$-0.211868\pi$
$457$	33.6288	1.57309	0.786545	+	0.617533i	$0.211868\pi$
$458$	−35.5174	−1.65962
$459$	−33.1525	−1.54742
$460$	−1.73324	−0.0808126
$461$	−27.0874	−1.26159	−0.630793	−	0.775951i	$-0.717270\pi$
$461$	−27.0874	−1.26159	−0.630793	+	0.775951i	$0.717270\pi$
$462$	0	0
$463$	−9.00060	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$463$	−9.00060	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$464$	15.6908	0.728426
$465$	−3.25077	−0.150751
$466$	22.3794	1.03671
$467$	−10.7438	−0.497164	−0.248582	−	0.968611i	$-0.579965\pi$
$467$	−10.7438	−0.497164	−0.248582	+	0.968611i	$0.579965\pi$
$468$	10.3627	0.479015
$469$	0	0
$470$	−7.40803	−0.341707
$471$	−14.6785	−0.676351
$472$	−2.25637	−0.103858
$473$	12.0626	0.554640
$474$	−26.0780	−1.19780
$475$	6.67982	0.306491
$476$	0	0
$477$	15.8976	0.727902
$478$	−23.7690	−1.08717
$479$	11.3800	0.519967	0.259984	−	0.965613i	$-0.416283\pi$
$479$	11.3800	0.519967	0.259984	+	0.965613i	$0.416283\pi$
$480$	−5.76515	−0.263142
$481$	−20.2147	−0.921709
$482$	4.28941	0.195377
$483$	0	0
$484$	29.1214	1.32370
$485$	−8.78685	−0.398990
$486$	−28.8541	−1.30885
$487$	−13.8806	−0.628990	−0.314495	−	0.949259i	$-0.601835\pi$
$487$	−13.8806	−0.628990	−0.314495	+	0.949259i	$0.601835\pi$
$488$	−8.48755	−0.384213
$489$	8.43695	0.381532
$490$	0	0
$491$	−23.4423	−1.05793	−0.528967	−	0.848642i	$-0.677421\pi$
$491$	−23.4423	−1.05793	−0.528967	+	0.848642i	$0.677421\pi$
$492$	3.05425	0.137696
$493$	33.3769	1.50322
$494$	8.58996	0.386481
$495$	4.12943	0.185604
$496$	12.6998	0.570237
$497$	0	0
$498$	−2.57662	−0.115461
$499$	8.93170	0.399838	0.199919	−	0.979812i	$-0.435932\pi$
$499$	8.93170	0.399838	0.199919	+	0.979812i	$0.435932\pi$
$500$	13.9992	0.626063
$501$	−11.6095	−0.518672
$502$	−11.6841	−0.521487
$503$	−20.6071	−0.918823	−0.459412	−	0.888224i	$-0.651940\pi$
$503$	−20.6071	−0.918823	−0.459412	+	0.888224i	$0.651940\pi$
$504$	0	0
$505$	5.45833	0.242892
$506$	12.0685	0.536511
$507$	6.15107	0.273179
$508$	−25.7761	−1.14363
$509$	1.41941	0.0629143	0.0314572	−	0.999505i	$-0.489985\pi$
$509$	1.41941	0.0629143	0.0314572	+	0.999505i	$0.489985\pi$
$510$	−9.16908	−0.406014
$511$	0	0
$512$	28.2057	1.24653
$513$	−7.92331	−0.349823
$514$	−26.0696	−1.14988
$515$	−4.29236	−0.189144
$516$	−7.72234	−0.339957
$517$	28.5341	1.25493
$518$	0	0
$519$	−15.3702	−0.674675
$520$	1.66918	0.0731982
$521$	31.0731	1.36134	0.680668	−	0.732592i	$-0.261690\pi$
$521$	31.0731	1.36134	0.680668	+	0.732592i	$0.261690\pi$
$522$	17.3961	0.761408
$523$	−31.3241	−1.36971	−0.684854	−	0.728680i	$-0.740134\pi$
$523$	−31.3241	−1.36971	−0.684854	+	0.728680i	$0.740134\pi$
$524$	6.89850	0.301363
$525$	0	0
$526$	47.2281	2.05924
$527$	27.0145	1.17677
$528$	16.6030	0.722551
$529$	−21.5704	−0.937843
$530$	13.3188	0.578532
$531$	3.31214	0.143735
$532$	0	0
$533$	2.83079	0.122615
$534$	23.0744	0.998525
$535$	2.09691	0.0906573
$536$	−15.7870	−0.681895
$537$	−10.0055	−0.431768
$538$	−11.5276	−0.496989
$539$	0	0
$540$	−8.00794	−0.344607
$541$	−37.8390	−1.62682	−0.813412	−	0.581688i	$-0.802393\pi$
$541$	−37.8390	−1.62682	−0.813412	+	0.581688i	$0.802393\pi$
$542$	−15.7675	−0.677274
$543$	−26.9301	−1.15568
$544$	47.9095	2.05410
$545$	7.47636	0.320252
$546$	0	0
$547$	44.0001	1.88131	0.940654	−	0.339367i	$-0.110213\pi$
$547$	44.0001	1.88131	0.940654	+	0.339367i	$0.110213\pi$
$548$	18.8243	0.804133
$549$	12.4589	0.531734
$550$	−47.0083	−2.00444
$551$	7.97694	0.339829
$552$	−1.48545	−0.0632250
$553$	0	0
$554$	27.8349	1.18259
$555$	5.15694	0.218900
$556$	53.7329	2.27878
$557$	−8.16812	−0.346094	−0.173047	−	0.984914i	$-0.555361\pi$
$557$	−8.16812	−0.346094	−0.173047	+	0.984914i	$0.555361\pi$
$558$	14.0801	0.596057
$559$	−7.15734	−0.302723
$560$	0	0
$561$	35.3172	1.49109
$562$	−36.8204	−1.55317
$563$	−15.5764	−0.656465	−0.328233	−	0.944597i	$-0.606453\pi$
$563$	−15.5764	−0.656465	−0.328233	+	0.944597i	$0.606453\pi$
$564$	−18.2672	−0.769186
$565$	3.55197	0.149432
$566$	−64.7873	−2.72321
$567$	0	0
$568$	2.05159	0.0860826
$569$	36.5465	1.53211	0.766054	−	0.642776i	$-0.222218\pi$
$569$	36.5465	1.53211	0.766054	+	0.642776i	$0.222218\pi$
$570$	−2.19138	−0.0917866
$571$	1.09770	0.0459374	0.0229687	−	0.999736i	$-0.492688\pi$
$571$	1.09770	0.0459374	0.0229687	+	0.999736i	$0.492688\pi$
$572$	−33.4400	−1.39820
$573$	10.2584	0.428552
$574$	0	0
$575$	−5.56850	−0.232222
$576$	16.6285	0.692852
$577$	21.6446	0.901078	0.450539	−	0.892757i	$-0.351232\pi$
$577$	21.6446	0.901078	0.450539	+	0.892757i	$0.351232\pi$
$578$	40.2304	1.67336
$579$	9.81186	0.407767
$580$	8.06214	0.334762
$581$	0	0
$582$	−39.1684	−1.62358
$583$	−51.3010	−2.12467
$584$	−6.21779	−0.257294
$585$	−2.45020	−0.101303
$586$	−69.8947	−2.88733
$587$	20.0228	0.826427	0.413214	−	0.910634i	$-0.364406\pi$
$587$	20.0228	0.826427	0.413214	+	0.910634i	$0.364406\pi$
$588$	0	0
$589$	6.45637	0.266030
$590$	2.77487	0.114239
$591$	−0.752065	−0.0309358
$592$	−20.1466	−0.828020
$593$	−2.66297	−0.109355	−0.0546775	−	0.998504i	$-0.517413\pi$
$593$	−2.66297	−0.109355	−0.0546775	+	0.998504i	$0.517413\pi$
$594$	55.7592	2.28783
$595$	0	0
$596$	−3.70507	−0.151766
$597$	−28.9218	−1.18369
$598$	−7.16084	−0.292829
$599$	−20.1904	−0.824959	−0.412480	−	0.910967i	$-0.635337\pi$
$599$	−20.1904	−0.824959	−0.412480	+	0.910967i	$0.635337\pi$
$600$	5.78601	0.236213
$601$	−17.7316	−0.723288	−0.361644	−	0.932316i	$-0.617784\pi$
$601$	−17.7316	−0.723288	−0.361644	+	0.932316i	$0.617784\pi$
$602$	0	0
$603$	23.1739	0.943713
$604$	17.6204	0.716963
$605$	−6.88559	−0.279939
$606$	24.3312	0.988386
$607$	3.14569	0.127679	0.0638397	−	0.997960i	$-0.479665\pi$
$607$	3.14569	0.127679	0.0638397	+	0.997960i	$0.479665\pi$
$608$	11.4502	0.464367
$609$	0	0
$610$	10.4379	0.422619
$611$	−16.9307	−0.684941
$612$	21.9689	0.888041
$613$	34.3584	1.38772	0.693862	−	0.720108i	$-0.255908\pi$
$613$	34.3584	1.38772	0.693862	+	0.720108i	$0.255908\pi$
$614$	−14.8228	−0.598199
$615$	−0.722160	−0.0291203
$616$	0	0
$617$	−35.8384	−1.44280	−0.721401	−	0.692518i	$-0.756501\pi$
$617$	−35.8384	−1.44280	−0.721401	+	0.692518i	$0.756501\pi$
$618$	−19.1337	−0.769670
$619$	−39.4123	−1.58411	−0.792056	−	0.610448i	$-0.790989\pi$
$619$	−39.4123	−1.58411	−0.792056	+	0.610448i	$0.790989\pi$
$620$	6.52533	0.262064
$621$	6.60511	0.265054
$622$	−49.2324	−1.97404
$623$	0	0
$624$	−9.85135	−0.394369
$625$	19.9762	0.799049
$626$	24.2560	0.969463
$627$	8.44068	0.337088
$628$	29.4644	1.17576
$629$	−42.8551	−1.70875
$630$	0	0
$631$	9.63071	0.383393	0.191696	−	0.981454i	$-0.438601\pi$
$631$	9.63071	0.383393	0.191696	+	0.981454i	$0.438601\pi$
$632$	10.0644	0.400340
$633$	−30.9957	−1.23197
$634$	55.3511	2.19827
$635$	6.09461	0.241857
$636$	32.8423	1.30228
$637$	0	0
$638$	−56.1366	−2.22247
$639$	−3.01154	−0.119135
$640$	4.58357	0.181181
$641$	−26.0918	−1.03056	−0.515282	−	0.857020i	$-0.672313\pi$
$641$	−26.0918	−1.03056	−0.515282	+	0.857020i	$0.672313\pi$
$642$	9.34723	0.368906
$643$	−28.8475	−1.13764	−0.568818	−	0.822463i	$-0.692599\pi$
$643$	−28.8475	−1.13764	−0.568818	+	0.822463i	$0.692599\pi$
$644$	0	0
$645$	1.82590	0.0718948
$646$	18.2107	0.716492
$647$	−29.3372	−1.15336	−0.576682	−	0.816969i	$-0.695653\pi$
$647$	−29.3372	−1.15336	−0.576682	+	0.816969i	$0.695653\pi$
$648$	−2.39596	−0.0941223
$649$	−10.6882	−0.419547
$650$	27.8924	1.09403
$651$	0	0
$652$	−16.9356	−0.663250
$653$	18.5416	0.725589	0.362794	−	0.931869i	$-0.381823\pi$
$653$	18.5416	0.725589	0.362794	+	0.931869i	$0.381823\pi$
$654$	33.3268	1.30318
$655$	−1.63111	−0.0637328
$656$	2.82126	0.110152
$657$	9.12714	0.356084
$658$	0	0
$659$	21.7205	0.846110	0.423055	−	0.906104i	$-0.360958\pi$
$659$	21.7205	0.846110	0.423055	+	0.906104i	$0.360958\pi$
$660$	8.53084	0.332062
$661$	−45.0636	−1.75277	−0.876386	−	0.481610i	$-0.840052\pi$
$661$	−45.0636	−1.75277	−0.876386	+	0.481610i	$0.840052\pi$
$662$	−51.0770	−1.98517
$663$	−20.9554	−0.813842
$664$	0.994405	0.0385904
$665$	0	0
$666$	−22.3362	−0.865511
$667$	−6.64981	−0.257482
$668$	23.3039	0.901653
$669$	13.1568	0.508672
$670$	19.4147	0.750057
$671$	−40.2045	−1.55208
$672$	0	0
$673$	−43.3004	−1.66911	−0.834553	−	0.550928i	$-0.814274\pi$
$673$	−43.3004	−1.66911	−0.834553	+	0.550928i	$0.814274\pi$
$674$	15.9210	0.613254
$675$	−25.7277	−0.990259
$676$	−12.3471	−0.474890
$677$	43.7236	1.68043	0.840217	−	0.542251i	$-0.182428\pi$
$677$	43.7236	1.68043	0.840217	+	0.542251i	$0.182428\pi$
$678$	15.8333	0.608075
$679$	0	0
$680$	3.53866	0.135701
$681$	−29.0631	−1.11370
$682$	−45.4358	−1.73983
$683$	0.327183	0.0125193	0.00625966	−	0.999980i	$-0.498007\pi$
$683$	0.327183	0.0125193	0.00625966	+	0.999980i	$0.498007\pi$
$684$	5.25049	0.200757
$685$	−4.45089	−0.170060
$686$	0	0
$687$	20.7080	0.790059
$688$	−7.13325	−0.271952
$689$	30.4394	1.15965
$690$	1.82680	0.0695449
$691$	0.425463	0.0161854	0.00809268	−	0.999967i	$-0.497424\pi$
$691$	0.425463	0.0161854	0.00809268	+	0.999967i	$0.497424\pi$
$692$	30.8528	1.17285
$693$	0	0
$694$	−16.0291	−0.608458
$695$	−12.7048	−0.481922
$696$	6.90957	0.261906
$697$	6.00129	0.227315
$698$	−37.2223	−1.40888
$699$	−13.0480	−0.493522
$700$	0	0
$701$	−29.1181	−1.09977	−0.549887	−	0.835239i	$-0.685329\pi$
$701$	−29.1181	−1.09977	−0.549887	+	0.835239i	$0.685329\pi$
$702$	−33.0847	−1.24870
$703$	−10.2422	−0.386292
$704$	−53.6594	−2.02236
$705$	4.31916	0.162669
$706$	42.8424	1.61239
$707$	0	0
$708$	6.84243	0.257154
$709$	−5.46503	−0.205244	−0.102622	−	0.994720i	$-0.532723\pi$
$709$	−5.46503	−0.205244	−0.102622	+	0.994720i	$0.532723\pi$
$710$	−2.52302	−0.0946874
$711$	−14.7736	−0.554053
$712$	−8.90518	−0.333736
$713$	−5.38222	−0.201566
$714$	0	0
$715$	7.90669	0.295693
$716$	20.0842	0.750580
$717$	13.8582	0.517545
$718$	24.8327	0.926748
$719$	−38.3658	−1.43080	−0.715401	−	0.698714i	$-0.753756\pi$
$719$	−38.3658	−1.43080	−0.715401	+	0.698714i	$0.753756\pi$
$720$	−2.44195	−0.0910060
$721$	0	0
$722$	−35.8454	−1.33403
$723$	−2.50089	−0.0930090
$724$	54.0573	2.00902
$725$	25.9018	0.961970
$726$	−30.6933	−1.13914
$727$	−22.5433	−0.836087	−0.418043	−	0.908427i	$-0.637284\pi$
$727$	−22.5433	−0.836087	−0.418043	+	0.908427i	$0.637284\pi$
$728$	0	0
$729$	23.9597	0.887396
$730$	7.64659	0.283013
$731$	−15.1736	−0.561216
$732$	25.7384	0.951320
$733$	17.9636	0.663501	0.331750	−	0.943367i	$-0.392361\pi$
$733$	17.9636	0.663501	0.331750	+	0.943367i	$0.392361\pi$
$734$	−66.7318	−2.46312
$735$	0	0
$736$	−9.54522	−0.351841
$737$	−74.7812	−2.75460
$738$	3.12789	0.115139
$739$	19.8359	0.729674	0.364837	−	0.931071i	$-0.381125\pi$
$739$	19.8359	0.729674	0.364837	+	0.931071i	$0.381125\pi$
$740$	−10.3516	−0.380533
$741$	−5.00827	−0.183983
$742$	0	0
$743$	33.4473	1.22706	0.613531	−	0.789671i	$-0.289749\pi$
$743$	33.4473	1.22706	0.613531	+	0.789671i	$0.289749\pi$
$744$	5.59246	0.205030
$745$	0.876044	0.0320958
$746$	−15.9695	−0.584685
$747$	−1.45969	−0.0534074
$748$	−70.8929	−2.59210
$749$	0	0
$750$	−14.7549	−0.538771
$751$	−12.0815	−0.440859	−0.220429	−	0.975403i	$-0.570746\pi$
$751$	−12.0815	−0.440859	−0.220429	+	0.975403i	$0.570746\pi$
$752$	−16.8737	−0.615319
$753$	6.81227	0.248253
$754$	33.3086	1.21303
$755$	−4.16624	−0.151625
$756$	0	0
$757$	39.0296	1.41855	0.709277	−	0.704930i	$-0.249022\pi$
$757$	39.0296	1.41855	0.709277	+	0.704930i	$0.249022\pi$
$758$	−60.6355	−2.20238
$759$	−7.03640	−0.255405
$760$	0.845726	0.0306777
$761$	−34.0106	−1.23288	−0.616442	−	0.787401i	$-0.711426\pi$
$761$	−34.0106	−1.23288	−0.616442	+	0.787401i	$0.711426\pi$
$762$	27.1675	0.984174
$763$	0	0
$764$	−20.5919	−0.744989
$765$	−5.19442	−0.187805
$766$	33.2147	1.20010
$767$	6.34181	0.228990
$768$	−7.31561	−0.263979
$769$	13.9708	0.503798	0.251899	−	0.967754i	$-0.418945\pi$
$769$	13.9708	0.503798	0.251899	+	0.967754i	$0.418945\pi$
$770$	0	0
$771$	15.1995	0.547398
$772$	−19.6955	−0.708857
$773$	−7.40300	−0.266267	−0.133134	−	0.991098i	$-0.542504\pi$
$773$	−7.40300	−0.266267	−0.133134	+	0.991098i	$0.542504\pi$
$774$	−7.90853	−0.284266
$775$	20.9644	0.753064
$776$	15.1164	0.542648
$777$	0	0
$778$	−54.6794	−1.96035
$779$	1.43428	0.0513886
$780$	−5.06176	−0.181240
$781$	9.71811	0.347741
$782$	−15.1810	−0.542872
$783$	−30.7236	−1.09797
$784$	0	0
$785$	−6.96670	−0.248652
$786$	−7.27088	−0.259344
$787$	40.3147	1.43706	0.718532	−	0.695494i	$-0.244814\pi$
$787$	40.3147	1.43706	0.718532	+	0.695494i	$0.244814\pi$
$788$	1.50963	0.0537785
$789$	−27.5358	−0.980300
$790$	−12.3771	−0.440357
$791$	0	0
$792$	−7.10406	−0.252432
$793$	23.8553	0.847127
$794$	10.3203	0.366254
$795$	−7.76536	−0.275409
$796$	58.0552	2.05771
$797$	49.5568	1.75539	0.877695	−	0.479219i	$-0.159080\pi$
$797$	49.5568	1.75539	0.877695	+	0.479219i	$0.159080\pi$
$798$	0	0
$799$	−35.8931	−1.26981
$800$	37.1798	1.31450
$801$	13.0720	0.461875
$802$	63.8797	2.25567
$803$	−29.4529	−1.03937
$804$	47.8740	1.68838
$805$	0	0
$806$	26.9593	0.949601
$807$	6.72102	0.236591
$808$	−9.39022	−0.330347
$809$	38.8519	1.36596	0.682980	−	0.730437i	$-0.260684\pi$
$809$	38.8519	1.36596	0.682980	+	0.730437i	$0.260684\pi$
$810$	2.94654	0.103531
$811$	−14.6663	−0.515004	−0.257502	−	0.966278i	$-0.582899\pi$
$811$	−14.6663	−0.515004	−0.257502	+	0.966278i	$0.582899\pi$
$812$	0	0
$813$	9.19308	0.322415
$814$	72.0782	2.52634
$815$	4.00433	0.140266
$816$	−20.8849	−0.731117
$817$	−3.62643	−0.126873
$818$	−66.1248	−2.31200
$819$	0	0
$820$	1.44960	0.0506224
$821$	4.41025	0.153919	0.0769594	−	0.997034i	$-0.475479\pi$
$821$	4.41025	0.153919	0.0769594	+	0.997034i	$0.475479\pi$
$822$	−19.8404	−0.692012
$823$	−29.0574	−1.01288	−0.506439	−	0.862276i	$-0.669038\pi$
$823$	−29.0574	−1.01288	−0.506439	+	0.862276i	$0.669038\pi$
$824$	7.38435	0.257246
$825$	27.4076	0.954212
$826$	0	0
$827$	−21.8438	−0.759584	−0.379792	−	0.925072i	$-0.624004\pi$
$827$	−21.8438	−0.759584	−0.379792	+	0.925072i	$0.624004\pi$
$828$	−4.37696	−0.152110
$829$	−45.0647	−1.56516	−0.782582	−	0.622548i	$-0.786098\pi$
$829$	−45.0647	−1.56516	−0.782582	+	0.622548i	$0.786098\pi$
$830$	−1.22291	−0.0424479
$831$	−16.2288	−0.562970
$832$	31.8388	1.10381
$833$	0	0
$834$	−56.6334	−1.96105
$835$	−5.51006	−0.190684
$836$	−16.9431	−0.585990
$837$	−24.8671	−0.859531
$838$	60.3545	2.08491
$839$	36.7818	1.26985	0.634925	−	0.772574i	$-0.281031\pi$
$839$	36.7818	1.26985	0.634925	+	0.772574i	$0.281031\pi$
$840$	0	0
$841$	1.93158	0.0666062
$842$	49.1434	1.69359
$843$	21.4677	0.739386
$844$	62.2183	2.14164
$845$	2.91941	0.100431
$846$	−18.7076	−0.643180
$847$	0	0
$848$	30.3369	1.04177
$849$	37.7734	1.29638
$850$	59.1319	2.02821
$851$	8.53821	0.292686
$852$	−6.22142	−0.213142
$853$	−13.4261	−0.459702	−0.229851	−	0.973226i	$-0.573824\pi$
$853$	−13.4261	−0.459702	−0.229851	+	0.973226i	$0.573824\pi$
$854$	0	0
$855$	−1.24145	−0.0424566
$856$	−3.60741	−0.123299
$857$	−13.7302	−0.469015	−0.234508	−	0.972114i	$-0.575348\pi$
$857$	−13.7302	−0.469015	−0.234508	+	0.972114i	$0.575348\pi$
$858$	35.2450	1.20325
$859$	−9.66544	−0.329780	−0.164890	−	0.986312i	$-0.552727\pi$
$859$	−9.66544	−0.329780	−0.164890	+	0.986312i	$0.552727\pi$
$860$	−3.66516	−0.124981
$861$	0	0
$862$	25.4887	0.868147
$863$	−55.8226	−1.90023	−0.950113	−	0.311907i	$-0.899032\pi$
$863$	−55.8226	−1.90023	−0.950113	+	0.311907i	$0.899032\pi$
$864$	−44.1010	−1.50035
$865$	−7.29497	−0.248036
$866$	27.0687	0.919833
$867$	−23.4559	−0.796603
$868$	0	0
$869$	47.6738	1.61722
$870$	−8.49733	−0.288086
$871$	44.3713	1.50347
$872$	−12.8619	−0.435560
$873$	−22.1895	−0.751001
$874$	−3.62820	−0.122726
$875$	0	0
$876$	18.8554	0.637065
$877$	−7.61906	−0.257278	−0.128639	−	0.991692i	$-0.541061\pi$
$877$	−7.61906	−0.257278	−0.128639	+	0.991692i	$0.541061\pi$
$878$	−66.8744	−2.25690
$879$	40.7513	1.37451
$880$	7.88007	0.265637
$881$	−28.3941	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$881$	−28.3941	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$882$	0	0
$883$	11.0978	0.373472	0.186736	−	0.982410i	$-0.440209\pi$
$883$	11.0978	0.373472	0.186736	+	0.982410i	$0.440209\pi$
$884$	42.0642	1.41477
$885$	−1.61785	−0.0543835
$886$	−23.1457	−0.777596
$887$	22.3698	0.751103	0.375551	−	0.926802i	$-0.377453\pi$
$887$	22.3698	0.751103	0.375551	+	0.926802i	$0.377453\pi$
$888$	−8.87173	−0.297716
$889$	0	0
$890$	10.9515	0.367096
$891$	−11.3494	−0.380219
$892$	−26.4099	−0.884269
$893$	−8.57831	−0.287062
$894$	3.90507	0.130605
$895$	−4.74878	−0.158734
$896$	0	0
$897$	4.17504	0.139401
$898$	−51.5264	−1.71946
$899$	25.0354	0.834977
$900$	17.0488	0.568293
$901$	64.5316	2.14986
$902$	−10.0936	−0.336080
$903$	0	0
$904$	−6.11061	−0.203236
$905$	−12.7815	−0.424873
$906$	−18.5715	−0.616997
$907$	49.6337	1.64806	0.824030	−	0.566546i	$-0.191721\pi$
$907$	49.6337	1.64806	0.824030	+	0.566546i	$0.191721\pi$
$908$	58.3388	1.93604
$909$	13.7840	0.457185
$910$	0	0
$911$	−22.3236	−0.739615	−0.369808	−	0.929108i	$-0.620576\pi$
$911$	−22.3236	−0.739615	−0.369808	+	0.929108i	$0.620576\pi$
$912$	−4.99141	−0.165282
$913$	4.71038	0.155891
$914$	71.1474	2.35335
$915$	−6.08570	−0.201187
$916$	−41.5675	−1.37343
$917$	0	0
$918$	−70.1396	−2.31495
$919$	−36.5947	−1.20715	−0.603574	−	0.797307i	$-0.706257\pi$
$919$	−36.5947	−1.20715	−0.603574	+	0.797307i	$0.706257\pi$
$920$	−0.705022	−0.0232439
$921$	8.64225	0.284772
$922$	−57.3080	−1.88734
$923$	−5.76623	−0.189798
$924$	0	0
$925$	−33.2574	−1.09350
$926$	−19.0423	−0.625769
$927$	−10.8395	−0.356017
$928$	44.3995	1.45749
$929$	37.7220	1.23762	0.618810	−	0.785541i	$-0.287615\pi$
$929$	37.7220	1.23762	0.618810	+	0.785541i	$0.287615\pi$
$930$	−6.87756	−0.225524
$931$	0	0
$932$	26.1915	0.857932
$933$	28.7044	0.939739
$934$	−22.7304	−0.743760
$935$	16.7622	0.548183
$936$	4.21519	0.137778
$937$	−32.2847	−1.05469	−0.527347	−	0.849650i	$-0.676813\pi$
$937$	−32.2847	−1.05469	−0.527347	+	0.849650i	$0.676813\pi$
$938$	0	0
$939$	−14.1421	−0.461511
$940$	−8.66993	−0.282782
$941$	34.4296	1.12237	0.561187	−	0.827689i	$-0.310345\pi$
$941$	34.4296	1.12237	0.561187	+	0.827689i	$0.310345\pi$
$942$	−31.0549	−1.01182
$943$	−1.19566	−0.0389361
$944$	6.32046	0.205713
$945$	0	0
$946$	25.5205	0.829743
$947$	−19.2008	−0.623941	−0.311971	−	0.950092i	$-0.600989\pi$
$947$	−19.2008	−0.623941	−0.311971	+	0.950092i	$0.600989\pi$
$948$	−30.5202	−0.991249
$949$	17.4759	0.567291
$950$	14.1323	0.458512
$951$	−32.2718	−1.04648
$952$	0	0
$953$	−56.6738	−1.83584	−0.917922	−	0.396760i	$-0.870135\pi$
$953$	−56.6738	−1.83584	−0.917922	+	0.396760i	$0.870135\pi$
$954$	33.6341	1.08894
$955$	4.86884	0.157552
$956$	−27.8178	−0.899693
$957$	32.7298	1.05800
$958$	24.0764	0.777873
$959$	0	0
$960$	−8.12235	−0.262148
$961$	−10.7369	−0.346351
$962$	−42.7675	−1.37888
$963$	5.29535	0.170640
$964$	5.02007	0.161686
$965$	4.65689	0.149911
$966$	0	0
$967$	10.4461	0.335925	0.167963	−	0.985793i	$-0.446281\pi$
$967$	10.4461	0.335925	0.167963	+	0.985793i	$0.446281\pi$
$968$	11.8456	0.380732
$969$	−10.6176	−0.341085
$970$	−18.5901	−0.596891
$971$	−17.4061	−0.558587	−0.279294	−	0.960206i	$-0.590100\pi$
$971$	−17.4061	−0.558587	−0.279294	+	0.960206i	$0.590100\pi$
$972$	−33.7691	−1.08314
$973$	0	0
$974$	−29.3668	−0.940972
$975$	−16.2623	−0.520810
$976$	23.7750	0.761019
$977$	−40.6096	−1.29922	−0.649608	−	0.760269i	$-0.725067\pi$
$977$	−40.6096	−1.29922	−0.649608	+	0.760269i	$0.725067\pi$
$978$	17.8498	0.570773
$979$	−42.1828	−1.34817
$980$	0	0
$981$	18.8801	0.602796
$982$	−49.5961	−1.58267
$983$	−9.80926	−0.312867	−0.156433	−	0.987689i	$-0.550000\pi$
$983$	−9.80926	−0.312867	−0.156433	+	0.987689i	$0.550000\pi$
$984$	1.24237	0.0396052
$985$	−0.356944	−0.0113732
$986$	70.6144	2.24882
$987$	0	0
$988$	10.0532	0.319834
$989$	3.02310	0.0961290
$990$	8.73652	0.277665
$991$	20.7647	0.659612	0.329806	−	0.944049i	$-0.393017\pi$
$991$	20.7647	0.659612	0.329806	+	0.944049i	$0.393017\pi$
$992$	35.9361	1.14097
$993$	29.7798	0.945035
$994$	0	0
$995$	−13.7268	−0.435169
$996$	−3.01552	−0.0955505
$997$	−36.7284	−1.16320	−0.581600	−	0.813475i	$-0.697573\pi$
$997$	−36.7284	−1.16320	−0.581600	+	0.813475i	$0.697573\pi$
$998$	18.8965	0.598159
$999$	39.4484	1.24809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.t.1.18	✓	20
7.6	odd	2		2009.2.a.u.1.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.18	✓	20	1.1	even	1	trivial
2009.2.a.u.1.18	yes	20	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11567 q^{2} -1.23352 q^{3} +2.47606 q^{4} -0.585449 q^{5} -2.60971 q^{6} +1.00718 q^{8} -1.47844 q^{9} +O(q^{10})\)	\(q+2.11567 q^{2} -1.23352 q^{3} +2.47606 q^{4} -0.585449 q^{5} -2.60971 q^{6} +1.00718 q^{8} -1.47844 q^{9} -1.23862 q^{10} +4.77087 q^{11} -3.05425 q^{12} -2.83079 q^{13} +0.722160 q^{15} -2.82126 q^{16} -6.00129 q^{17} -3.12789 q^{18} -1.43428 q^{19} -1.44960 q^{20} +10.0936 q^{22} +1.19566 q^{23} -1.24237 q^{24} -4.65725 q^{25} -5.98902 q^{26} +5.52422 q^{27} -5.56162 q^{29} +1.52785 q^{30} -4.50146 q^{31} -7.98320 q^{32} -5.88494 q^{33} -12.6967 q^{34} -3.66070 q^{36} +7.14099 q^{37} -3.03447 q^{38} +3.49182 q^{39} -0.589650 q^{40} -1.00000 q^{41} +2.52839 q^{43} +11.8129 q^{44} +0.865552 q^{45} +2.52963 q^{46} +5.98089 q^{47} +3.48007 q^{48} -9.85320 q^{50} +7.40268 q^{51} -7.00920 q^{52} -10.7530 q^{53} +11.6874 q^{54} -2.79310 q^{55} +1.76921 q^{57} -11.7665 q^{58} -2.24030 q^{59} +1.78811 q^{60} -8.42708 q^{61} -9.52359 q^{62} -11.2473 q^{64} +1.65728 q^{65} -12.4506 q^{66} -15.6745 q^{67} -14.8595 q^{68} -1.47487 q^{69} +2.03697 q^{71} -1.48905 q^{72} -6.17349 q^{73} +15.1080 q^{74} +5.74479 q^{75} -3.55137 q^{76} +7.38755 q^{78} +9.99268 q^{79} +1.65170 q^{80} -2.37889 q^{81} -2.11567 q^{82} +0.987320 q^{83} +3.51345 q^{85} +5.34924 q^{86} +6.86034 q^{87} +4.80511 q^{88} -8.84173 q^{89} +1.83122 q^{90} +2.96053 q^{92} +5.55262 q^{93} +12.6536 q^{94} +0.839701 q^{95} +9.84740 q^{96} +15.0087 q^{97} -7.05345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9} - 16 q^{10} - 6 q^{12} - 12 q^{13} + 14 q^{16} - 8 q^{17} - 18 q^{18} - 36 q^{19} - 24 q^{20} - 8 q^{22} - 12 q^{23} - 36 q^{24} + 20 q^{25} + 22 q^{26} - 32 q^{27} + 4 q^{29} + 28 q^{30} - 80 q^{31} + 6 q^{32} + 12 q^{33} - 48 q^{34} + 26 q^{36} + 4 q^{37} - 12 q^{38} - 28 q^{39} + 4 q^{40} - 20 q^{41} - 20 q^{44} - 40 q^{45} + 8 q^{46} - 32 q^{47} - 16 q^{48} + 6 q^{50} - 20 q^{51} - 36 q^{52} + 4 q^{53} + 50 q^{54} - 64 q^{55} - 4 q^{57} - 32 q^{59} + 20 q^{60} - 44 q^{61} + 8 q^{62} - 30 q^{64} - 8 q^{65} - 32 q^{66} - 4 q^{67} + 48 q^{68} + 24 q^{69} + 8 q^{71} - 8 q^{72} - 48 q^{73} - 38 q^{74} - 24 q^{75} - 84 q^{76} + 30 q^{78} - 4 q^{79} - 56 q^{80} + 2 q^{82} - 8 q^{83} - 12 q^{85} - 24 q^{86} - 40 q^{87} - 48 q^{88} - 20 q^{89} - 48 q^{90} - 50 q^{92} + 48 q^{93} - 26 q^{94} + 20 q^{95} - 70 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9 - 16 * q^10 - 6 * q^12 - 12 * q^13 + 14 * q^16 - 8 * q^17 - 18 * q^18 - 36 * q^19 - 24 * q^20 - 8 * q^22 - 12 * q^23 - 36 * q^24 + 20 * q^25 + 22 * q^26 - 32 * q^27 + 4 * q^29 + 28 * q^30 - 80 * q^31 + 6 * q^32 + 12 * q^33 - 48 * q^34 + 26 * q^36 + 4 * q^37 - 12 * q^38 - 28 * q^39 + 4 * q^40 - 20 * q^41 - 20 * q^44 - 40 * q^45 + 8 * q^46 - 32 * q^47 - 16 * q^48 + 6 * q^50 - 20 * q^51 - 36 * q^52 + 4 * q^53 + 50 * q^54 - 64 * q^55 - 4 * q^57 - 32 * q^59 + 20 * q^60 - 44 * q^61 + 8 * q^62 - 30 * q^64 - 8 * q^65 - 32 * q^66 - 4 * q^67 + 48 * q^68 + 24 * q^69 + 8 * q^71 - 8 * q^72 - 48 * q^73 - 38 * q^74 - 24 * q^75 - 84 * q^76 + 30 * q^78 - 4 * q^79 - 56 * q^80 + 2 * q^82 - 8 * q^83 - 12 * q^85 - 24 * q^86 - 40 * q^87 - 48 * q^88 - 20 * q^89 - 48 * q^90 - 50 * q^92 + 48 * q^93 - 26 * q^94 + 20 * q^95 - 70 * q^96 - 8 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

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Database

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