LMFDB - Embedded newform 2009.2.a.o.1.4

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:	$6$
Coefficient field:	6.6.185257757.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 $ x^6 - x^5 - 10x^4 + 10x^3 + 23x^2 - 24x + 5
Coefficient ring:	$\Z[a_1, a_2]$
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			1.4
Root			$0.306800$ 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)

$f(q)$	$=$	$q-0.306800 q^{2} +1.50512 q^{3} -1.90587 q^{4} -0.333855 q^{5} -0.461772 q^{6} +1.19832 q^{8} -0.734606 q^{9} +O(q^{10})$	\(q-0.306800 q^{2} +1.50512 q^{3} -1.90587 q^{4} -0.333855 q^{5} -0.461772 q^{6} +1.19832 q^{8} -0.734606 q^{9} +0.102427 q^{10} +5.36678 q^{11} -2.86857 q^{12} -6.79231 q^{13} -0.502492 q^{15} +3.44410 q^{16} +1.40075 q^{17} +0.225377 q^{18} +2.64242 q^{19} +0.636285 q^{20} -1.64653 q^{22} +2.49488 q^{23} +1.80362 q^{24} -4.88854 q^{25} +2.08388 q^{26} -5.62104 q^{27} +8.50214 q^{29} +0.154165 q^{30} -1.13536 q^{31} -3.45330 q^{32} +8.07767 q^{33} -0.429751 q^{34} +1.40007 q^{36} +10.2811 q^{37} -0.810696 q^{38} -10.2233 q^{39} -0.400066 q^{40} -1.00000 q^{41} -4.95666 q^{43} -10.2284 q^{44} +0.245252 q^{45} -0.765429 q^{46} +8.99982 q^{47} +5.18380 q^{48} +1.49981 q^{50} +2.10830 q^{51} +12.9453 q^{52} +6.86429 q^{53} +1.72454 q^{54} -1.79173 q^{55} +3.97717 q^{57} -2.60846 q^{58} +11.9369 q^{59} +0.957687 q^{60} -1.72627 q^{61} +0.348328 q^{62} -5.82873 q^{64} +2.26764 q^{65} -2.47823 q^{66} +3.32361 q^{67} -2.66965 q^{68} +3.75510 q^{69} -0.907819 q^{71} -0.880295 q^{72} -12.7532 q^{73} -3.15425 q^{74} -7.35785 q^{75} -5.03613 q^{76} +3.13650 q^{78} +10.9710 q^{79} -1.14983 q^{80} -6.25654 q^{81} +0.306800 q^{82} +4.90393 q^{83} -0.467647 q^{85} +1.52071 q^{86} +12.7968 q^{87} +6.43114 q^{88} +11.5506 q^{89} -0.0752432 q^{90} -4.75492 q^{92} -1.70885 q^{93} -2.76115 q^{94} -0.882186 q^{95} -5.19763 q^{96} -7.42196 q^{97} -3.94247 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 4 q^{3} + 9 q^{4} + q^{5} + 4 q^{6} + 3 q^{8} + 14 q^{9}+O(q^{10}) $ 6 * q - q^2 + 4 * q^3 + 9 * q^4 + q^5 + 4 * q^6 + 3 * q^8 + 14 * q^9	$ 6 q - q^{2} + 4 q^{3} + 9 q^{4} + q^{5} + 4 q^{6} + 3 q^{8} + 14 q^{9} - 10 q^{10} + 6 q^{11} - 9 q^{12} - 7 q^{13} - 13 q^{15} + 7 q^{16} - 7 q^{17} - 5 q^{18} - 2 q^{19} - 11 q^{20} + 15 q^{22} + 20 q^{23} + 36 q^{24} + 29 q^{25} + 43 q^{26} - 2 q^{27} - 9 q^{29} + 13 q^{30} + 27 q^{31} - 10 q^{32} - 17 q^{33} - 6 q^{34} + 29 q^{36} + 19 q^{37} + 23 q^{38} + q^{39} - 23 q^{40} - 6 q^{41} + 19 q^{43} + 21 q^{44} + 35 q^{45} - 8 q^{46} + 19 q^{47} + 9 q^{48} - 58 q^{50} - 19 q^{51} + 5 q^{53} + 37 q^{54} - 3 q^{55} + 37 q^{57} + 13 q^{58} + 7 q^{59} - 110 q^{60} + 12 q^{61} - 37 q^{64} - 13 q^{65} + 54 q^{66} + 27 q^{67} - 31 q^{68} - 16 q^{69} - 6 q^{71} + 5 q^{72} - 52 q^{73} - 14 q^{74} + 46 q^{75} - 13 q^{76} - 45 q^{78} + 26 q^{80} - 22 q^{81} + q^{82} - 12 q^{83} + 25 q^{85} - 10 q^{86} - 42 q^{87} - 2 q^{88} + 38 q^{89} - 93 q^{90} + 45 q^{92} + 33 q^{93} + 8 q^{94} + q^{95} + 12 q^{96} - 8 q^{97} + q^{99}+O(q^{100}) $ 6 * q - q^2 + 4 * q^3 + 9 * q^4 + q^5 + 4 * q^6 + 3 * q^8 + 14 * q^9 - 10 * q^10 + 6 * q^11 - 9 * q^12 - 7 * q^13 - 13 * q^15 + 7 * q^16 - 7 * q^17 - 5 * q^18 - 2 * q^19 - 11 * q^20 + 15 * q^22 + 20 * q^23 + 36 * q^24 + 29 * q^25 + 43 * q^26 - 2 * q^27 - 9 * q^29 + 13 * q^30 + 27 * q^31 - 10 * q^32 - 17 * q^33 - 6 * q^34 + 29 * q^36 + 19 * q^37 + 23 * q^38 + q^39 - 23 * q^40 - 6 * q^41 + 19 * q^43 + 21 * q^44 + 35 * q^45 - 8 * q^46 + 19 * q^47 + 9 * q^48 - 58 * q^50 - 19 * q^51 + 5 * q^53 + 37 * q^54 - 3 * q^55 + 37 * q^57 + 13 * q^58 + 7 * q^59 - 110 * q^60 + 12 * q^61 - 37 * q^64 - 13 * q^65 + 54 * q^66 + 27 * q^67 - 31 * q^68 - 16 * q^69 - 6 * q^71 + 5 * q^72 - 52 * q^73 - 14 * q^74 + 46 * q^75 - 13 * q^76 - 45 * q^78 + 26 * q^80 - 22 * q^81 + q^82 - 12 * q^83 + 25 * q^85 - 10 * q^86 - 42 * q^87 - 2 * q^88 + 38 * q^89 - 93 * q^90 + 45 * q^92 + 33 * q^93 + 8 * q^94 + q^95 + 12 * q^96 - 8 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.306800	−0.216940	−0.108470	−	0.994100i	$-0.534595\pi$
$2$	−0.306800	−0.216940	−0.108470	+	0.994100i	$0.534595\pi$
$3$	1.50512	0.868983	0.434491	−	0.900676i	$-0.356928\pi$
$3$	1.50512	0.868983	0.434491	+	0.900676i	$0.356928\pi$
$4$	−1.90587	−0.952937
$5$	−0.333855	−0.149304	−0.0746522	−	0.997210i	$-0.523785\pi$
$5$	−0.333855	−0.149304	−0.0746522	+	0.997210i	$0.523785\pi$
$6$	−0.461772	−0.188518
$7$	0	0
$7$	0	0
$8$	1.19832	0.423671
$9$	−0.734606	−0.244869
$10$	0.102427	0.0323902
$11$	5.36678	1.61815	0.809073	−	0.587708i	$-0.199970\pi$
$11$	5.36678	1.61815	0.809073	+	0.587708i	$0.199970\pi$
$12$	−2.86857	−0.828086
$13$	−6.79231	−1.88385	−0.941924	−	0.335827i	$-0.890984\pi$
$13$	−6.79231	−1.88385	−0.941924	+	0.335827i	$0.890984\pi$
$14$	0	0
$15$	−0.502492	−0.129743
$16$	3.44410	0.861025
$17$	1.40075	0.339732	0.169866	−	0.985467i	$-0.445667\pi$
$17$	1.40075	0.339732	0.169866	+	0.985467i	$0.445667\pi$
$18$	0.225377	0.0531219
$19$	2.64242	0.606214	0.303107	−	0.952957i	$-0.401976\pi$
$19$	2.64242	0.606214	0.303107	+	0.952957i	$0.401976\pi$
$20$	0.636285	0.142278
$21$	0	0
$22$	−1.64653	−0.351041
$23$	2.49488	0.520218	0.260109	−	0.965579i	$-0.416242\pi$
$23$	2.49488	0.520218	0.260109	+	0.965579i	$0.416242\pi$
$24$	1.80362	0.368163
$25$	−4.88854	−0.977708
$26$	2.08388	0.408683
$27$	−5.62104	−1.08177
$28$	0	0
$29$	8.50214	1.57881	0.789404	−	0.613874i	$-0.210390\pi$
$29$	8.50214	1.57881	0.789404	+	0.613874i	$0.210390\pi$
$30$	0.154165	0.0281465
$31$	−1.13536	−0.203916	−0.101958	−	0.994789i	$-0.532511\pi$
$31$	−1.13536	−0.203916	−0.101958	+	0.994789i	$0.532511\pi$
$32$	−3.45330	−0.610462
$33$	8.07767	1.40614
$34$	−0.429751	−0.0737016
$35$	0	0
$36$	1.40007	0.233344
$37$	10.2811	1.69021	0.845104	−	0.534602i	$-0.179538\pi$
$37$	10.2811	1.69021	0.845104	+	0.534602i	$0.179538\pi$
$38$	−0.810696	−0.131512
$39$	−10.2233	−1.63703
$40$	−0.400066	−0.0632559
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−4.95666	−0.755884	−0.377942	−	0.925829i	$-0.623368\pi$
$43$	−4.95666	−0.755884	−0.377942	+	0.925829i	$0.623368\pi$
$44$	−10.2284	−1.54199
$45$	0.245252	0.0365599
$46$	−0.765429	−0.112856
$47$	8.99982	1.31276	0.656380	−	0.754431i	$-0.272087\pi$
$47$	8.99982	1.31276	0.656380	+	0.754431i	$0.272087\pi$
$48$	5.18380	0.748216
$49$	0	0
$50$	1.49981	0.212104
$51$	2.10830	0.295221
$52$	12.9453	1.79519
$53$	6.86429	0.942883	0.471442	−	0.881897i	$-0.343734\pi$
$53$	6.86429	0.942883	0.471442	+	0.881897i	$0.343734\pi$
$54$	1.72454	0.234680
$55$	−1.79173	−0.241596
$56$	0	0
$57$	3.97717	0.526789
$58$	−2.60846	−0.342507
$59$	11.9369	1.55405	0.777023	−	0.629472i	$-0.216729\pi$
$59$	11.9369	1.55405	0.777023	+	0.629472i	$0.216729\pi$
$60$	0.957687	0.123637
$61$	−1.72627	−0.221027	−0.110513	−	0.993875i	$-0.535249\pi$
$61$	−1.72627	−0.221027	−0.110513	+	0.993875i	$0.535249\pi$
$62$	0.348328	0.0442376
$63$	0	0
$64$	−5.82873	−0.728591
$65$	2.26764	0.281267
$66$	−2.47823	−0.305049
$67$	3.32361	0.406044	0.203022	−	0.979174i	$-0.434924\pi$
$67$	3.32361	0.406044	0.203022	+	0.979174i	$0.434924\pi$
$68$	−2.66965	−0.323743
$69$	3.75510	0.452060
$70$	0	0
$71$	−0.907819	−0.107738	−0.0538691	−	0.998548i	$-0.517155\pi$
$71$	−0.907819	−0.107738	−0.0538691	+	0.998548i	$0.517155\pi$
$72$	−0.880295	−0.103744
$73$	−12.7532	−1.49265	−0.746324	−	0.665583i	$-0.768183\pi$
$73$	−12.7532	−1.49265	−0.746324	+	0.665583i	$0.768183\pi$
$74$	−3.15425	−0.366675
$75$	−7.35785	−0.849612
$76$	−5.03613	−0.577683
$77$	0	0
$78$	3.13650	0.355138
$79$	10.9710	1.23434	0.617168	−	0.786832i	$-0.288280\pi$
$79$	10.9710	1.23434	0.617168	+	0.786832i	$0.288280\pi$
$80$	−1.14983	−0.128555
$81$	−6.25654	−0.695171
$82$	0.306800	0.0338804
$83$	4.90393	0.538276	0.269138	−	0.963102i	$-0.413261\pi$
$83$	4.90393	0.538276	0.269138	+	0.963102i	$0.413261\pi$
$84$	0	0
$85$	−0.467647	−0.0507235
$86$	1.52071	0.163982
$87$	12.7968	1.37196
$88$	6.43114	0.685562
$89$	11.5506	1.22436	0.612182	−	0.790717i	$-0.290292\pi$
$89$	11.5506	1.22436	0.612182	+	0.790717i	$0.290292\pi$
$90$	−0.0752432	−0.00793133
$91$	0	0
$92$	−4.75492	−0.495735
$93$	−1.70885	−0.177200
$94$	−2.76115	−0.284791
$95$	−0.882186	−0.0905103
$96$	−5.19763	−0.530481
$97$	−7.42196	−0.753586	−0.376793	−	0.926298i	$-0.622973\pi$
$97$	−7.42196	−0.753586	−0.376793	+	0.926298i	$0.622973\pi$
$98$	0	0
$99$	−3.94247	−0.396233
$100$	9.31694	0.931694
$101$	14.8629	1.47891	0.739456	−	0.673205i	$-0.235083\pi$
$101$	14.8629	1.47891	0.739456	+	0.673205i	$0.235083\pi$
$102$	−0.646827	−0.0640455
$103$	−15.3241	−1.50993	−0.754966	−	0.655764i	$-0.772347\pi$
$103$	−15.3241	−1.50993	−0.754966	+	0.655764i	$0.772347\pi$
$104$	−8.13938	−0.798131
$105$	0	0
$106$	−2.10597	−0.204549
$107$	11.0421	1.06748	0.533742	−	0.845647i	$-0.320785\pi$
$107$	11.0421	1.06748	0.533742	+	0.845647i	$0.320785\pi$
$108$	10.7130	1.03086
$109$	19.8981	1.90589	0.952946	−	0.303140i	$-0.0980349\pi$
$109$	19.8981	1.90589	0.952946	+	0.303140i	$0.0980349\pi$
$110$	0.549702	0.0524120
$111$	15.4744	1.46876
$112$	0	0
$113$	9.11906	0.857849	0.428925	−	0.903340i	$-0.358893\pi$
$113$	9.11906	0.857849	0.428925	+	0.903340i	$0.358893\pi$
$114$	−1.22020	−0.114282
$115$	−0.832926	−0.0776708
$116$	−16.2040	−1.50450
$117$	4.98967	0.461295
$118$	−3.66223	−0.337136
$119$	0	0
$120$	−0.602148	−0.0549683
$121$	17.8024	1.61840
$122$	0.529621	0.0479496
$123$	−1.50512	−0.135712
$124$	2.16385	0.194319
$125$	3.30134	0.295280
$126$	0	0
$127$	−17.4056	−1.54449	−0.772247	−	0.635323i	$-0.780867\pi$
$127$	−17.4056	−1.54449	−0.772247	+	0.635323i	$0.780867\pi$
$128$	8.69485	0.768523
$129$	−7.46039	−0.656850
$130$	−0.695713	−0.0610181
$131$	5.83566	0.509864	0.254932	−	0.966959i	$-0.417947\pi$
$131$	5.83566	0.509864	0.254932	+	0.966959i	$0.417947\pi$
$132$	−15.3950	−1.33996
$133$	0	0
$134$	−1.01968	−0.0880873
$135$	1.87661	0.161513
$136$	1.67855	0.143935
$137$	10.7780	0.920824	0.460412	−	0.887705i	$-0.347702\pi$
$137$	10.7780	0.920824	0.460412	+	0.887705i	$0.347702\pi$
$138$	−1.15206	−0.0980702
$139$	8.37703	0.710530	0.355265	−	0.934766i	$-0.384391\pi$
$139$	8.37703	0.710530	0.355265	+	0.934766i	$0.384391\pi$
$140$	0	0
$141$	13.5458	1.14077
$142$	0.278519	0.0233728
$143$	−36.4529	−3.04834
$144$	−2.53006	−0.210838
$145$	−2.83848	−0.235723
$146$	3.91268	0.323816
$147$	0	0
$148$	−19.5945	−1.61066
$149$	−13.9455	−1.14246	−0.571232	−	0.820789i	$-0.693534\pi$
$149$	−13.9455	−1.14246	−0.571232	+	0.820789i	$0.693534\pi$
$150$	2.25739	0.184315
$151$	−6.08548	−0.495229	−0.247614	−	0.968859i	$-0.579647\pi$
$151$	−6.08548	−0.495229	−0.247614	+	0.968859i	$0.579647\pi$
$152$	3.16648	0.256835
$153$	−1.02900	−0.0831897
$154$	0	0
$155$	0.379044	0.0304456
$156$	19.4842	1.55999
$157$	−1.42037	−0.113358	−0.0566788	−	0.998392i	$-0.518051\pi$
$157$	−1.42037	−0.113358	−0.0566788	+	0.998392i	$0.518051\pi$
$158$	−3.36591	−0.267777
$159$	10.3316	0.819349
$160$	1.15290	0.0911447
$161$	0	0
$162$	1.91951	0.150811
$163$	−7.27285	−0.569653	−0.284827	−	0.958579i	$-0.591936\pi$
$163$	−7.27285	−0.569653	−0.284827	+	0.958579i	$0.591936\pi$
$164$	1.90587	0.148824
$165$	−2.69677	−0.209943
$166$	−1.50453	−0.116774
$167$	4.28895	0.331889	0.165945	−	0.986135i	$-0.446933\pi$
$167$	4.28895	0.331889	0.165945	+	0.986135i	$0.446933\pi$
$168$	0	0
$169$	33.1354	2.54888
$170$	0.143474	0.0110040
$171$	−1.94114	−0.148443
$172$	9.44678	0.720310
$173$	1.47044	0.111795	0.0558977	−	0.998437i	$-0.482198\pi$
$173$	1.47044	0.111795	0.0558977	+	0.998437i	$0.482198\pi$
$174$	−3.92605	−0.297633
$175$	0	0
$176$	18.4838	1.39327
$177$	17.9664	1.35044
$178$	−3.54374	−0.265614
$179$	−0.125462	−0.00937749	−0.00468874	−	0.999989i	$-0.501492\pi$
$179$	−0.125462	−0.00937749	−0.00468874	+	0.999989i	$0.501492\pi$
$180$	−0.467418	−0.0348393
$181$	9.01874	0.670357	0.335179	−	0.942155i	$-0.391203\pi$
$181$	9.01874	0.670357	0.335179	+	0.942155i	$0.391203\pi$
$182$	0	0
$183$	−2.59825	−0.192068
$184$	2.98967	0.220401
$185$	−3.43240	−0.252355
$186$	0.524276	0.0384418
$187$	7.51753	0.549736
$188$	−17.1525	−1.25098
$189$	0	0
$190$	0.270655	0.0196354
$191$	−4.66382	−0.337462	−0.168731	−	0.985662i	$-0.553967\pi$
$191$	−4.66382	−0.337462	−0.168731	+	0.985662i	$0.553967\pi$
$192$	−8.77296	−0.633134
$193$	−17.2818	−1.24397	−0.621987	−	0.783028i	$-0.713674\pi$
$193$	−17.2818	−1.24397	−0.621987	+	0.783028i	$0.713674\pi$
$194$	2.27706	0.163483
$195$	3.41308	0.244416
$196$	0	0
$197$	−8.59288	−0.612217	−0.306109	−	0.951997i	$-0.599027\pi$
$197$	−8.59288	−0.612217	−0.306109	+	0.951997i	$0.599027\pi$
$198$	1.20955	0.0859590
$199$	16.6515	1.18039	0.590197	−	0.807260i	$-0.299050\pi$
$199$	16.6515	1.18039	0.590197	+	0.807260i	$0.299050\pi$
$200$	−5.85805	−0.414227
$201$	5.00244	0.352845
$202$	−4.55993	−0.320836
$203$	0	0
$204$	−4.01816	−0.281327
$205$	0.333855	0.0233174
$206$	4.70145	0.327565
$207$	−1.83275	−0.127385
$208$	−23.3934	−1.62204
$209$	14.1813	0.980943
$210$	0	0
$211$	−3.49751	−0.240778	−0.120389	−	0.992727i	$-0.538414\pi$
$211$	−3.49751	−0.240778	−0.120389	+	0.992727i	$0.538414\pi$
$212$	−13.0825	−0.898508
$213$	−1.36638	−0.0936227
$214$	−3.38773	−0.231580
$215$	1.65481	0.112857
$216$	−6.73582	−0.458314
$217$	0	0
$218$	−6.10474	−0.413465
$219$	−19.1951	−1.29709
$220$	3.41480	0.230226
$221$	−9.51433	−0.640003
$222$	−4.74754	−0.318634
$223$	−20.9946	−1.40590	−0.702950	−	0.711239i	$-0.748134\pi$
$223$	−20.9946	−1.40590	−0.702950	+	0.711239i	$0.748134\pi$
$224$	0	0
$225$	3.59115	0.239410
$226$	−2.79773	−0.186102
$227$	−23.7408	−1.57573	−0.787866	−	0.615847i	$-0.788814\pi$
$227$	−23.7408	−1.57573	−0.787866	+	0.615847i	$0.788814\pi$
$228$	−7.57999	−0.501997
$229$	5.34742	0.353367	0.176684	−	0.984268i	$-0.443463\pi$
$229$	5.34742	0.353367	0.176684	+	0.984268i	$0.443463\pi$
$230$	0.255542	0.0168499
$231$	0	0
$232$	10.1883	0.668895
$233$	−18.7258	−1.22677	−0.613383	−	0.789786i	$-0.710192\pi$
$233$	−18.7258	−1.22677	−0.613383	+	0.789786i	$0.710192\pi$
$234$	−1.53083	−0.100074
$235$	−3.00463	−0.196001
$236$	−22.7501	−1.48091
$237$	16.5127	1.07262
$238$	0	0
$239$	−25.2700	−1.63458	−0.817291	−	0.576225i	$-0.804525\pi$
$239$	−25.2700	−1.63458	−0.817291	+	0.576225i	$0.804525\pi$
$240$	−1.73063	−0.111712
$241$	13.7336	0.884661	0.442331	−	0.896852i	$-0.354152\pi$
$241$	13.7336	0.884661	0.442331	+	0.896852i	$0.354152\pi$
$242$	−5.46177	−0.351096
$243$	7.44626	0.477678
$244$	3.29006	0.210624
$245$	0	0
$246$	0.461772	0.0294415
$247$	−17.9482	−1.14201
$248$	−1.36052	−0.0863933
$249$	7.38101	0.467753
$250$	−1.01285	−0.0640583
$251$	−26.4800	−1.67140	−0.835701	−	0.549185i	$-0.814938\pi$
$251$	−26.4800	−1.67140	−0.835701	+	0.549185i	$0.814938\pi$
$252$	0	0
$253$	13.3895	0.841789
$254$	5.34003	0.335063
$255$	−0.703867	−0.0440778
$256$	8.98988	0.561868
$257$	20.2549	1.26347	0.631733	−	0.775186i	$-0.282344\pi$
$257$	20.2549	1.26347	0.631733	+	0.775186i	$0.282344\pi$
$258$	2.28885	0.142497
$259$	0	0
$260$	−4.32184	−0.268029
$261$	−6.24572	−0.386600
$262$	−1.79038	−0.110610
$263$	11.9620	0.737608	0.368804	−	0.929507i	$-0.379767\pi$
$263$	11.9620	0.737608	0.368804	+	0.929507i	$0.379767\pi$
$264$	9.67965	0.595741
$265$	−2.29168	−0.140777
$266$	0	0
$267$	17.3851	1.06395
$268$	−6.33438	−0.386934
$269$	11.2325	0.684859	0.342429	−	0.939544i	$-0.388750\pi$
$269$	11.2325	0.684859	0.342429	+	0.939544i	$0.388750\pi$
$270$	−0.575744	−0.0350387
$271$	−11.8175	−0.717864	−0.358932	−	0.933364i	$-0.616859\pi$
$271$	−11.8175	−0.717864	−0.358932	+	0.933364i	$0.616859\pi$
$272$	4.82433	0.292518
$273$	0	0
$274$	−3.30668	−0.199764
$275$	−26.2357	−1.58208
$276$	−7.15674	−0.430785
$277$	14.4718	0.869529	0.434764	−	0.900544i	$-0.356832\pi$
$277$	14.4718	0.869529	0.434764	+	0.900544i	$0.356832\pi$
$278$	−2.57007	−0.154143
$279$	0.834039	0.0499326
$280$	0	0
$281$	−4.77157	−0.284648	−0.142324	−	0.989820i	$-0.545458\pi$
$281$	−4.77157	−0.284648	−0.142324	+	0.989820i	$0.545458\pi$
$282$	−4.15587	−0.247478
$283$	6.12204	0.363918	0.181959	−	0.983306i	$-0.441756\pi$
$283$	6.12204	0.363918	0.181959	+	0.983306i	$0.441756\pi$
$284$	1.73019	0.102668
$285$	−1.32780	−0.0786520
$286$	11.1837	0.661308
$287$	0	0
$288$	2.53681	0.149483
$289$	−15.0379	−0.884582
$290$	0.870846	0.0511378
$291$	−11.1710	−0.654853
$292$	24.3060	1.42240
$293$	20.0296	1.17014	0.585070	−	0.810983i	$-0.301067\pi$
$293$	20.0296	1.17014	0.585070	+	0.810983i	$0.301067\pi$
$294$	0	0
$295$	−3.98518	−0.232026
$296$	12.3201	0.716092
$297$	−30.1669	−1.75046
$298$	4.27849	0.247846
$299$	−16.9460	−0.980011
$300$	14.0231	0.809626
$301$	0	0
$302$	1.86702	0.107435
$303$	22.3705	1.28515
$304$	9.10078	0.521965
$305$	0.576324	0.0330002
$306$	0.315697	0.0180472
$307$	5.95010	0.339590	0.169795	−	0.985479i	$-0.445689\pi$
$307$	5.95010	0.339590	0.169795	+	0.985479i	$0.445689\pi$
$308$	0	0
$309$	−23.0647	−1.31211
$310$	−0.116291	−0.00660487
$311$	17.9083	1.01549	0.507744	−	0.861508i	$-0.330480\pi$
$311$	17.9083	1.01549	0.507744	+	0.861508i	$0.330480\pi$
$312$	−12.2508	−0.693563
$313$	−1.69266	−0.0956746	−0.0478373	−	0.998855i	$-0.515233\pi$
$313$	−1.69266	−0.0956746	−0.0478373	+	0.998855i	$0.515233\pi$
$314$	0.435768	0.0245918
$315$	0	0
$316$	−20.9094	−1.17624
$317$	27.4139	1.53972	0.769859	−	0.638214i	$-0.220326\pi$
$317$	27.4139	1.53972	0.769859	+	0.638214i	$0.220326\pi$
$318$	−3.16974	−0.177750
$319$	45.6292	2.55474
$320$	1.94595	0.108782
$321$	16.6198	0.927625
$322$	0	0
$323$	3.70138	0.205950
$324$	11.9242	0.662454
$325$	33.2045	1.84185
$326$	2.23131	0.123581
$327$	29.9491	1.65619
$328$	−1.19832	−0.0661663
$329$	0	0
$330$	0.827369	0.0455451
$331$	−10.2877	−0.565461	−0.282731	−	0.959199i	$-0.591240\pi$
$331$	−10.2877	−0.565461	−0.282731	+	0.959199i	$0.591240\pi$
$332$	−9.34627	−0.512943
$333$	−7.55258	−0.413879
$334$	−1.31585	−0.0720002
$335$	−1.10960	−0.0606241
$336$	0	0
$337$	−4.60125	−0.250646	−0.125323	−	0.992116i	$-0.539997\pi$
$337$	−4.60125	−0.250646	−0.125323	+	0.992116i	$0.539997\pi$
$338$	−10.1660	−0.552955
$339$	13.7253	0.745456
$340$	0.891277	0.0483363
$341$	−6.09321	−0.329966
$342$	0.595542	0.0322032
$343$	0	0
$344$	−5.93968	−0.320246
$345$	−1.25366	−0.0674946
$346$	−0.451131	−0.0242529
$347$	−7.50283	−0.402773	−0.201386	−	0.979512i	$-0.564545\pi$
$347$	−7.50283	−0.402773	−0.201386	+	0.979512i	$0.564545\pi$
$348$	−24.3890	−1.30739
$349$	−22.7698	−1.21884	−0.609419	−	0.792848i	$-0.708597\pi$
$349$	−22.7698	−1.21884	−0.609419	+	0.792848i	$0.708597\pi$
$350$	0	0
$351$	38.1798	2.03789
$352$	−18.5331	−0.987817
$353$	−23.3999	−1.24545	−0.622726	−	0.782440i	$-0.713975\pi$
$353$	−23.3999	−1.24545	−0.622726	+	0.782440i	$0.713975\pi$
$354$	−5.51211	−0.292965
$355$	0.303080	0.0160858
$356$	−22.0140	−1.16674
$357$	0	0
$358$	0.0384918	0.00203436
$359$	−18.7370	−0.988901	−0.494451	−	0.869206i	$-0.664631\pi$
$359$	−18.7370	−0.988901	−0.494451	+	0.869206i	$0.664631\pi$
$360$	0.293890	0.0154894
$361$	−12.0176	−0.632505
$362$	−2.76695	−0.145428
$363$	26.7948	1.40636
$364$	0	0
$365$	4.25771	0.222859
$366$	0.797144	0.0416674
$367$	−7.12448	−0.371895	−0.185947	−	0.982560i	$-0.559535\pi$
$367$	−7.12448	−0.371895	−0.185947	+	0.982560i	$0.559535\pi$
$368$	8.59261	0.447921
$369$	0.734606	0.0382420
$370$	1.05306	0.0547461
$371$	0	0
$372$	3.25685	0.168860
$373$	−5.51396	−0.285502	−0.142751	−	0.989759i	$-0.545595\pi$
$373$	−5.51396	−0.285502	−0.142751	+	0.989759i	$0.545595\pi$
$374$	−2.30638	−0.119260
$375$	4.96892	0.256594
$376$	10.7847	0.556178
$377$	−57.7492	−2.97423
$378$	0	0
$379$	0.286604	0.0147218	0.00736092	−	0.999973i	$-0.497657\pi$
$379$	0.286604	0.0147218	0.00736092	+	0.999973i	$0.497657\pi$
$380$	1.68133	0.0862506
$381$	−26.1975	−1.34214
$382$	1.43086	0.0732092
$383$	9.93579	0.507695	0.253848	−	0.967244i	$-0.418304\pi$
$383$	9.93579	0.507695	0.253848	+	0.967244i	$0.418304\pi$
$384$	13.0868	0.667834
$385$	0	0
$386$	5.30207	0.269868
$387$	3.64119	0.185092
$388$	14.1453	0.718120
$389$	−8.60242	−0.436160	−0.218080	−	0.975931i	$-0.569979\pi$
$389$	−8.60242	−0.436160	−0.218080	+	0.975931i	$0.569979\pi$
$390$	−1.04713	−0.0530237
$391$	3.49470	0.176735
$392$	0	0
$393$	8.78338	0.443063
$394$	2.63630	0.132815
$395$	−3.66272	−0.184292
$396$	7.51385	0.377585
$397$	−12.3332	−0.618983	−0.309492	−	0.950902i	$-0.600159\pi$
$397$	−12.3332	−0.618983	−0.309492	+	0.950902i	$0.600159\pi$
$398$	−5.10868	−0.256075
$399$	0	0
$400$	−16.8366	−0.841832
$401$	−3.40012	−0.169794	−0.0848969	−	0.996390i	$-0.527056\pi$
$401$	−3.40012	−0.169794	−0.0848969	+	0.996390i	$0.527056\pi$
$402$	−1.53475	−0.0765463
$403$	7.71169	0.384147
$404$	−28.3268	−1.40931
$405$	2.08877	0.103792
$406$	0	0
$407$	55.1766	2.73500
$408$	2.52643	0.125077
$409$	−20.3679	−1.00713	−0.503563	−	0.863958i	$-0.667978\pi$
$409$	−20.3679	−1.00713	−0.503563	+	0.863958i	$0.667978\pi$
$410$	−0.102427	−0.00505849
$411$	16.2222	0.800181
$412$	29.2059	1.43887
$413$	0	0
$414$	0.562288	0.0276350
$415$	−1.63720	−0.0803670
$416$	23.4558	1.15002
$417$	12.6085	0.617439
$418$	−4.35083	−0.212806
$419$	22.7693	1.11235	0.556177	−	0.831064i	$-0.312268\pi$
$419$	22.7693	1.11235	0.556177	+	0.831064i	$0.312268\pi$
$420$	0	0
$421$	16.7047	0.814135	0.407067	−	0.913398i	$-0.366551\pi$
$421$	16.7047	0.814135	0.407067	+	0.913398i	$0.366551\pi$
$422$	1.07304	0.0522346
$423$	−6.61132	−0.321453
$424$	8.22564	0.399472
$425$	−6.84763	−0.332159
$426$	0.419205	0.0203106
$427$	0	0
$428$	−21.0449	−1.01724
$429$	−54.8660	−2.64896
$430$	−0.507695	−0.0244832
$431$	27.2639	1.31325	0.656627	−	0.754216i	$-0.271983\pi$
$431$	27.2639	1.31325	0.656627	+	0.754216i	$0.271983\pi$
$432$	−19.3594	−0.931431
$433$	5.85443	0.281346	0.140673	−	0.990056i	$-0.455073\pi$
$433$	5.85443	0.281346	0.140673	+	0.990056i	$0.455073\pi$
$434$	0	0
$435$	−4.27226	−0.204839
$436$	−37.9233	−1.81619
$437$	6.59252	0.315363
$438$	5.88906	0.281390
$439$	4.72841	0.225675	0.112837	−	0.993613i	$-0.464006\pi$
$439$	4.72841	0.225675	0.112837	+	0.993613i	$0.464006\pi$
$440$	−2.14707	−0.102357
$441$	0	0
$442$	2.91900	0.138843
$443$	−0.0474597	−0.00225488	−0.00112744	−	0.999999i	$-0.500359\pi$
$443$	−0.0474597	−0.00225488	−0.00112744	+	0.999999i	$0.500359\pi$
$444$	−29.4922	−1.39964
$445$	−3.85623	−0.182803
$446$	6.44114	0.304997
$447$	−20.9897	−0.992781
$448$	0	0
$449$	−15.7131	−0.741546	−0.370773	−	0.928724i	$-0.620907\pi$
$449$	−15.7131	−0.741546	−0.370773	+	0.928724i	$0.620907\pi$
$450$	−1.10177	−0.0519377
$451$	−5.36678	−0.252712
$452$	−17.3798	−0.817476
$453$	−9.15939	−0.430346
$454$	7.28368	0.341840
$455$	0	0
$456$	4.76594	0.223185
$457$	20.3006	0.949624	0.474812	−	0.880087i	$-0.342516\pi$
$457$	20.3006	0.949624	0.474812	+	0.880087i	$0.342516\pi$
$458$	−1.64059	−0.0766597
$459$	−7.87368	−0.367512
$460$	1.58745	0.0740154
$461$	18.2345	0.849267	0.424633	−	0.905365i	$-0.360403\pi$
$461$	18.2345	0.849267	0.424633	+	0.905365i	$0.360403\pi$
$462$	0	0
$463$	−22.6565	−1.05293	−0.526467	−	0.850195i	$-0.676484\pi$
$463$	−22.6565	−1.05293	−0.526467	+	0.850195i	$0.676484\pi$
$464$	29.2822	1.35939
$465$	0.570508	0.0264567
$466$	5.74507	0.266135
$467$	−38.4889	−1.78105	−0.890526	−	0.454932i	$-0.849664\pi$
$467$	−38.4889	−1.78105	−0.890526	+	0.454932i	$0.849664\pi$
$468$	−9.50968	−0.439585
$469$	0	0
$470$	0.921822	0.0425205
$471$	−2.13783	−0.0985058
$472$	14.3042	0.658404
$473$	−26.6013	−1.22313
$474$	−5.06611	−0.232694
$475$	−12.9176	−0.592700
$476$	0	0
$477$	−5.04255	−0.230882
$478$	7.75285	0.354607
$479$	−7.16791	−0.327510	−0.163755	−	0.986501i	$-0.552361\pi$
$479$	−7.16791	−0.327510	−0.163755	+	0.986501i	$0.552361\pi$
$480$	1.73525	0.0792032
$481$	−69.8326	−3.18409
$482$	−4.21348	−0.191919
$483$	0	0
$484$	−33.9291	−1.54223
$485$	2.47786	0.112514
$486$	−2.28451	−0.103628
$487$	33.8331	1.53312	0.766562	−	0.642170i	$-0.221966\pi$
$487$	33.8331	1.53312	0.766562	+	0.642170i	$0.221966\pi$
$488$	−2.06863	−0.0936425
$489$	−10.9465	−0.495019
$490$	0	0
$491$	−17.9615	−0.810589	−0.405295	−	0.914186i	$-0.632831\pi$
$491$	−17.9615	−0.810589	−0.405295	+	0.914186i	$0.632831\pi$
$492$	2.86857	0.129325
$493$	11.9094	0.536372
$494$	5.50650	0.247749
$495$	1.31621	0.0591593
$496$	−3.91028	−0.175577
$497$	0	0
$498$	−2.26450	−0.101474
$499$	−17.8312	−0.798234	−0.399117	−	0.916900i	$-0.630683\pi$
$499$	−17.8312	−0.798234	−0.399117	+	0.916900i	$0.630683\pi$
$500$	−6.29193	−0.281384
$501$	6.45540	0.288406
$502$	8.12406	0.362595
$503$	19.4629	0.867807	0.433903	−	0.900959i	$-0.357136\pi$
$503$	19.4629	0.867807	0.433903	+	0.900959i	$0.357136\pi$
$504$	0	0
$505$	−4.96204	−0.220808
$506$	−4.10789	−0.182618
$507$	49.8729	2.21493
$508$	33.1728	1.47180
$509$	−11.6060	−0.514428	−0.257214	−	0.966354i	$-0.582805\pi$
$509$	−11.6060	−0.514428	−0.257214	+	0.966354i	$0.582805\pi$
$510$	0.215946	0.00956226
$511$	0	0
$512$	−20.1478	−0.890415
$513$	−14.8532	−0.655784
$514$	−6.21420	−0.274097
$515$	5.11603	0.225439
$516$	14.2186	0.625937
$517$	48.3001	2.12424
$518$	0	0
$519$	2.21319	0.0971482
$520$	2.71737	0.119165
$521$	−38.1115	−1.66970	−0.834848	−	0.550480i	$-0.814445\pi$
$521$	−38.1115	−1.66970	−0.834848	+	0.550480i	$0.814445\pi$
$522$	1.91619	0.0838693
$523$	−1.24317	−0.0543600	−0.0271800	−	0.999631i	$-0.508653\pi$
$523$	−1.24317	−0.0543600	−0.0271800	+	0.999631i	$0.508653\pi$
$524$	−11.1220	−0.485868
$525$	0	0
$526$	−3.66994	−0.160017
$527$	−1.59035	−0.0692768
$528$	27.8203	1.21072
$529$	−16.7756	−0.729373
$530$	0.703087	0.0305401
$531$	−8.76888	−0.380537
$532$	0	0
$533$	6.79231	0.294208
$534$	−5.33376	−0.230814
$535$	−3.68647	−0.159380
$536$	3.98276	0.172029
$537$	−0.188836	−0.00814888
$538$	−3.44614	−0.148574
$539$	0	0
$540$	−3.57658	−0.153912
$541$	4.63538	0.199290	0.0996452	−	0.995023i	$-0.468229\pi$
$541$	4.63538	0.199290	0.0996452	+	0.995023i	$0.468229\pi$
$542$	3.62562	0.155734
$543$	13.5743	0.582529
$544$	−4.83721	−0.207394
$545$	−6.64307	−0.284558
$546$	0	0
$547$	1.40215	0.0599518	0.0299759	−	0.999551i	$-0.490457\pi$
$547$	1.40215	0.0599518	0.0299759	+	0.999551i	$0.490457\pi$
$548$	−20.5415	−0.877487
$549$	1.26813	0.0541224
$550$	8.04913	0.343216
$551$	22.4663	0.957095
$552$	4.49982	0.191525
$553$	0	0
$554$	−4.43996	−0.188636
$555$	−5.16619	−0.219293
$556$	−15.9656	−0.677091
$557$	−16.0241	−0.678961	−0.339481	−	0.940613i	$-0.610251\pi$
$557$	−16.0241	−0.678961	−0.339481	+	0.940613i	$0.610251\pi$
$558$	−0.255883	−0.0108324
$559$	33.6672	1.42397
$560$	0	0
$561$	11.3148	0.477711
$562$	1.46392	0.0617517
$563$	−16.8191	−0.708840	−0.354420	−	0.935086i	$-0.615322\pi$
$563$	−16.8191	−0.708840	−0.354420	+	0.935086i	$0.615322\pi$
$564$	−25.8167	−1.08708
$565$	−3.04444	−0.128081
$566$	−1.87824	−0.0789485
$567$	0	0
$568$	−1.08786	−0.0456456
$569$	−18.9838	−0.795844	−0.397922	−	0.917419i	$-0.630269\pi$
$569$	−18.9838	−0.795844	−0.397922	+	0.917419i	$0.630269\pi$
$570$	0.407369	0.0170628
$571$	16.1497	0.675844	0.337922	−	0.941174i	$-0.390276\pi$
$571$	16.1497	0.675844	0.337922	+	0.941174i	$0.390276\pi$
$572$	69.4745	2.90488
$573$	−7.01962	−0.293249
$574$	0	0
$575$	−12.1963	−0.508621
$576$	4.28182	0.178409
$577$	−5.74140	−0.239018	−0.119509	−	0.992833i	$-0.538132\pi$
$577$	−5.74140	−0.239018	−0.119509	+	0.992833i	$0.538132\pi$
$578$	4.61363	0.191902
$579$	−26.0113	−1.08099
$580$	5.40978	0.224629
$581$	0	0
$582$	3.42725	0.142064
$583$	36.8392	1.52572
$584$	−15.2824	−0.632391
$585$	−1.66582	−0.0688733
$586$	−6.14508	−0.253851
$587$	18.4924	0.763264	0.381632	−	0.924314i	$-0.375362\pi$
$587$	18.4924	0.763264	0.381632	+	0.924314i	$0.375362\pi$
$588$	0	0
$589$	−3.00009	−0.123617
$590$	1.22265	0.0503358
$591$	−12.9333	−0.532006
$592$	35.4093	1.45531
$593$	−15.9503	−0.655002	−0.327501	−	0.944851i	$-0.606207\pi$
$593$	−15.9503	−0.655002	−0.327501	+	0.944851i	$0.606207\pi$
$594$	9.25521	0.379746
$595$	0	0
$596$	26.5784	1.08870
$597$	25.0625	1.02574
$598$	5.19903	0.212604
$599$	9.95944	0.406932	0.203466	−	0.979082i	$-0.434779\pi$
$599$	9.95944	0.406932	0.203466	+	0.979082i	$0.434779\pi$
$600$	−8.81708	−0.359956
$601$	26.2678	1.07149	0.535743	−	0.844381i	$-0.320031\pi$
$601$	26.2678	1.07149	0.535743	+	0.844381i	$0.320031\pi$
$602$	0	0
$603$	−2.44154	−0.0994273
$604$	11.5981	0.471922
$605$	−5.94341	−0.241634
$606$	−6.86326	−0.278801
$607$	−6.91881	−0.280826	−0.140413	−	0.990093i	$-0.544843\pi$
$607$	−6.91881	−0.280826	−0.140413	+	0.990093i	$0.544843\pi$
$608$	−9.12507	−0.370071
$609$	0	0
$610$	−0.176816	−0.00715908
$611$	−61.1296	−2.47304
$612$	1.96114	0.0792745
$613$	−46.1436	−1.86372	−0.931861	−	0.362817i	$-0.881815\pi$
$613$	−46.1436	−1.86372	−0.931861	+	0.362817i	$0.881815\pi$
$614$	−1.82549	−0.0736709
$615$	0.502492	0.0202624
$616$	0	0
$617$	−10.6123	−0.427237	−0.213618	−	0.976917i	$-0.568525\pi$
$617$	−10.6123	−0.427237	−0.213618	+	0.976917i	$0.568525\pi$
$618$	7.07625	0.284649
$619$	−8.29763	−0.333510	−0.166755	−	0.985998i	$-0.553329\pi$
$619$	−8.29763	−0.333510	−0.166755	+	0.985998i	$0.553329\pi$
$620$	−0.722410	−0.0290127
$621$	−14.0238	−0.562756
$622$	−5.49428	−0.220301
$623$	0	0
$624$	−35.2099	−1.40953
$625$	23.3405	0.933622
$626$	0.519307	0.0207557
$627$	21.3446	0.852422
$628$	2.70704	0.108023
$629$	14.4013	0.574218
$630$	0	0
$631$	−30.1153	−1.19887	−0.599435	−	0.800423i	$-0.704608\pi$
$631$	−30.1153	−1.19887	−0.599435	+	0.800423i	$0.704608\pi$
$632$	13.1468	0.522952
$633$	−5.26418	−0.209232
$634$	−8.41059	−0.334027
$635$	5.81093	0.230600
$636$	−19.6907	−0.780788
$637$	0	0
$638$	−13.9990	−0.554227
$639$	0.666889	0.0263817
$640$	−2.90282	−0.114744
$641$	7.79316	0.307811	0.153906	−	0.988086i	$-0.450815\pi$
$641$	7.79316	0.307811	0.153906	+	0.988086i	$0.450815\pi$
$642$	−5.09895	−0.201239
$643$	6.51291	0.256844	0.128422	−	0.991720i	$-0.459009\pi$
$643$	6.51291	0.256844	0.128422	+	0.991720i	$0.459009\pi$
$644$	0	0
$645$	2.49069	0.0980706
$646$	−1.13558	−0.0446789
$647$	−12.5924	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$647$	−12.5924	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$648$	−7.49735	−0.294524
$649$	64.0625	2.51468
$650$	−10.1871	−0.399572
$651$	0	0
$652$	13.8611	0.542844
$653$	−23.5504	−0.921599	−0.460800	−	0.887504i	$-0.652437\pi$
$653$	−23.5504	−0.921599	−0.460800	+	0.887504i	$0.652437\pi$
$654$	−9.18838	−0.359294
$655$	−1.94826	−0.0761249
$656$	−3.44410	−0.134470
$657$	9.36856	0.365502
$658$	0	0
$659$	24.4826	0.953706	0.476853	−	0.878983i	$-0.341777\pi$
$659$	24.4826	0.953706	0.476853	+	0.878983i	$0.341777\pi$
$660$	5.13970	0.200063
$661$	0.193864	0.00754045	0.00377023	−	0.999993i	$-0.498800\pi$
$661$	0.193864	0.00754045	0.00377023	+	0.999993i	$0.498800\pi$
$662$	3.15626	0.122671
$663$	−14.3202	−0.556152
$664$	5.87649	0.228052
$665$	0	0
$666$	2.31713	0.0897871
$667$	21.2118	0.821324
$668$	−8.17421	−0.316270
$669$	−31.5994	−1.22170
$670$	0.340426	0.0131518
$671$	−9.26453	−0.357653
$672$	0	0
$673$	−14.8193	−0.571243	−0.285622	−	0.958342i	$-0.592200\pi$
$673$	−14.8193	−0.571243	−0.285622	+	0.958342i	$0.592200\pi$
$674$	1.41166	0.0543753
$675$	27.4787	1.05766
$676$	−63.1520	−2.42892
$677$	23.8251	0.915674	0.457837	−	0.889036i	$-0.348624\pi$
$677$	23.8251	0.915674	0.457837	+	0.889036i	$0.348624\pi$
$678$	−4.21093	−0.161720
$679$	0	0
$680$	−0.560392	−0.0214901
$681$	−35.7328	−1.36928
$682$	1.86940	0.0715830
$683$	12.0613	0.461514	0.230757	−	0.973011i	$-0.425880\pi$
$683$	12.0613	0.461514	0.230757	+	0.973011i	$0.425880\pi$
$684$	3.69957	0.141456
$685$	−3.59828	−0.137483
$686$	0	0
$687$	8.04852	0.307070
$688$	−17.0713	−0.650836
$689$	−46.6244	−1.77625
$690$	0.384622	0.0146423
$691$	6.78963	0.258290	0.129145	−	0.991626i	$-0.458777\pi$
$691$	6.78963	0.258290	0.129145	+	0.991626i	$0.458777\pi$
$692$	−2.80247	−0.106534
$693$	0	0
$694$	2.30187	0.0873778
$695$	−2.79671	−0.106085
$696$	15.3347	0.581259
$697$	−1.40075	−0.0530572
$698$	6.98577	0.264415
$699$	−28.1846	−1.06604
$700$	0	0
$701$	2.74195	0.103562	0.0517809	−	0.998658i	$-0.483510\pi$
$701$	2.74195	0.103562	0.0517809	+	0.998658i	$0.483510\pi$
$702$	−11.7136	−0.442100
$703$	27.1671	1.02463
$704$	−31.2815	−1.17897
$705$	−4.52234	−0.170321
$706$	7.17909	0.270189
$707$	0	0
$708$	−34.2418	−1.28688
$709$	37.5281	1.40940	0.704698	−	0.709507i	$-0.251082\pi$
$709$	37.5281	1.40940	0.704698	+	0.709507i	$0.251082\pi$
$710$	−0.0929848	−0.00348966
$711$	−8.05937	−0.302250
$712$	13.8414	0.518728
$713$	−2.83258	−0.106081
$714$	0	0
$715$	12.1700	0.455131
$716$	0.239115	0.00893615
$717$	−38.0345	−1.42042
$718$	5.74851	0.214533
$719$	−4.62968	−0.172658	−0.0863289	−	0.996267i	$-0.527514\pi$
$719$	−4.62968	−0.172658	−0.0863289	+	0.996267i	$0.527514\pi$
$720$	0.844671	0.0314790
$721$	0	0
$722$	3.68700	0.137216
$723$	20.6708	0.768756
$724$	−17.1886	−0.638808
$725$	−41.5631	−1.54361
$726$	−8.22064	−0.305096
$727$	−5.89676	−0.218699	−0.109349	−	0.994003i	$-0.534877\pi$
$727$	−5.89676	−0.218699	−0.109349	+	0.994003i	$0.534877\pi$
$728$	0	0
$729$	29.9772	1.11026
$730$	−1.30627	−0.0483471
$731$	−6.94305	−0.256798
$732$	4.95194	0.183029
$733$	14.6106	0.539654	0.269827	−	0.962909i	$-0.413033\pi$
$733$	14.6106	0.539654	0.269827	+	0.962909i	$0.413033\pi$
$734$	2.18579	0.0806791
$735$	0	0
$736$	−8.61555	−0.317573
$737$	17.8371	0.657038
$738$	−0.225377	−0.00829625
$739$	21.5160	0.791480	0.395740	−	0.918363i	$-0.370488\pi$
$739$	21.5160	0.791480	0.395740	+	0.918363i	$0.370488\pi$
$740$	6.54173	0.240479
$741$	−27.0142	−0.992391
$742$	0	0
$743$	9.13830	0.335252	0.167626	−	0.985851i	$-0.446390\pi$
$743$	9.13830	0.335252	0.167626	+	0.985851i	$0.446390\pi$
$744$	−2.04775	−0.0750743
$745$	4.65578	0.170575
$746$	1.69168	0.0619370
$747$	−3.60245	−0.131807
$748$	−14.3275	−0.523864
$749$	0	0
$750$	−1.52446	−0.0556656
$751$	−37.8459	−1.38102	−0.690509	−	0.723324i	$-0.742613\pi$
$751$	−37.8459	−1.38102	−0.690509	+	0.723324i	$0.742613\pi$
$752$	30.9963	1.13032
$753$	−39.8556	−1.45242
$754$	17.7175	0.645231
$755$	2.03166	0.0739398
$756$	0	0
$757$	0.674866	0.0245284	0.0122642	−	0.999925i	$-0.496096\pi$
$757$	0.674866	0.0245284	0.0122642	+	0.999925i	$0.496096\pi$
$758$	−0.0879300	−0.00319376
$759$	20.1528	0.731500
$760$	−1.05714	−0.0383466
$761$	−10.5510	−0.382472	−0.191236	−	0.981544i	$-0.561250\pi$
$761$	−10.5510	−0.382472	−0.191236	+	0.981544i	$0.561250\pi$
$762$	8.03740	0.291164
$763$	0	0
$764$	8.88865	0.321580
$765$	0.343536	0.0124206
$766$	−3.04830	−0.110140
$767$	−81.0788	−2.92759
$768$	13.5309	0.488253
$769$	21.1928	0.764234	0.382117	−	0.924114i	$-0.375195\pi$
$769$	21.1928	0.764234	0.382117	+	0.924114i	$0.375195\pi$
$770$	0	0
$771$	30.4861	1.09793
$772$	32.9370	1.18543
$773$	53.3444	1.91866	0.959332	−	0.282279i	$-0.0910905\pi$
$773$	53.3444	1.91866	0.959332	+	0.282279i	$0.0910905\pi$
$774$	−1.11712	−0.0401540
$775$	5.55024	0.199370
$776$	−8.89390	−0.319272
$777$	0	0
$778$	2.63922	0.0946208
$779$	−2.64242	−0.0946747
$780$	−6.50490	−0.232913
$781$	−4.87207	−0.174336
$782$	−1.07217	−0.0383409
$783$	−47.7909	−1.70791
$784$	0	0
$785$	0.474196	0.0169248
$786$	−2.69474	−0.0961183
$787$	−19.5237	−0.695944	−0.347972	−	0.937505i	$-0.613129\pi$
$787$	−19.5237	−0.695944	−0.347972	+	0.937505i	$0.613129\pi$
$788$	16.3769	0.583404
$789$	18.0043	0.640969
$790$	1.12372	0.0399803
$791$	0	0
$792$	−4.72435	−0.167873
$793$	11.7254	0.416380
$794$	3.78381	0.134282
$795$	−3.44925	−0.122332
$796$	−31.7356	−1.12484
$797$	1.37202	0.0485993	0.0242996	−	0.999705i	$-0.492264\pi$
$797$	1.37202	0.0485993	0.0242996	+	0.999705i	$0.492264\pi$
$798$	0	0
$799$	12.6065	0.445986
$800$	16.8816	0.596854
$801$	−8.48516	−0.299808
$802$	1.04316	0.0368351
$803$	−68.4436	−2.41532
$804$	−9.53402	−0.336239
$805$	0	0
$806$	−2.36595	−0.0833370
$807$	16.9063	0.595131
$808$	17.8105	0.626572
$809$	2.90209	0.102032	0.0510160	−	0.998698i	$-0.483754\pi$
$809$	2.90209	0.102032	0.0510160	+	0.998698i	$0.483754\pi$
$810$	−0.640836	−0.0225167
$811$	−25.6648	−0.901214	−0.450607	−	0.892722i	$-0.648792\pi$
$811$	−25.6648	−0.901214	−0.450607	+	0.892722i	$0.648792\pi$
$812$	0	0
$813$	−17.7868	−0.623812
$814$	−16.9282	−0.593333
$815$	2.42807	0.0850517
$816$	7.26121	0.254193
$817$	−13.0976	−0.458227
$818$	6.24887	0.218487
$819$	0	0
$820$	−0.636285	−0.0222200
$821$	44.4170	1.55016	0.775082	−	0.631861i	$-0.217709\pi$
$821$	44.4170	1.55016	0.775082	+	0.631861i	$0.217709\pi$
$822$	−4.97696	−0.173592
$823$	45.7425	1.59448	0.797242	−	0.603660i	$-0.206292\pi$
$823$	45.7425	1.59448	0.797242	+	0.603660i	$0.206292\pi$
$824$	−18.3633	−0.639714
$825$	−39.4880	−1.37480
$826$	0	0
$827$	−15.8042	−0.549566	−0.274783	−	0.961506i	$-0.588606\pi$
$827$	−15.8042	−0.549566	−0.274783	+	0.961506i	$0.588606\pi$
$828$	3.49299	0.121390
$829$	−36.8011	−1.27816	−0.639078	−	0.769142i	$-0.720684\pi$
$829$	−36.8011	−1.27816	−0.639078	+	0.769142i	$0.720684\pi$
$830$	0.502293	0.0174348
$831$	21.7819	0.755606
$832$	39.5905	1.37256
$833$	0	0
$834$	−3.86828	−0.133947
$835$	−1.43189	−0.0495525
$836$	−27.0278	−0.934776
$837$	6.38188	0.220590
$838$	−6.98563	−0.241315
$839$	−1.85627	−0.0640857	−0.0320428	−	0.999486i	$-0.510201\pi$
$839$	−1.85627	−0.0640857	−0.0320428	+	0.999486i	$0.510201\pi$
$840$	0	0
$841$	43.2864	1.49263
$842$	−5.12499	−0.176619
$843$	−7.18180	−0.247354
$844$	6.66581	0.229447
$845$	−11.0624	−0.380559
$846$	2.02835	0.0697363
$847$	0	0
$848$	23.6413	0.811846
$849$	9.21442	0.316238
$850$	2.10085	0.0720587
$851$	25.6502	0.879276
$852$	2.60414	0.0892165
$853$	10.3089	0.352968	0.176484	−	0.984303i	$-0.443528\pi$
$853$	10.3089	0.352968	0.176484	+	0.984303i	$0.443528\pi$
$854$	0	0
$855$	0.648059	0.0221631
$856$	13.2320	0.452262
$857$	46.7471	1.59685	0.798426	−	0.602093i	$-0.205666\pi$
$857$	46.7471	1.59685	0.798426	+	0.602093i	$0.205666\pi$
$858$	16.8329	0.574666
$859$	−7.91458	−0.270042	−0.135021	−	0.990843i	$-0.543110\pi$
$859$	−7.91458	−0.270042	−0.135021	+	0.990843i	$0.543110\pi$
$860$	−3.15385	−0.107545
$861$	0	0
$862$	−8.36455	−0.284898
$863$	5.79263	0.197183	0.0985917	−	0.995128i	$-0.468566\pi$
$863$	5.79263	0.197183	0.0985917	+	0.995128i	$0.468566\pi$
$864$	19.4111	0.660380
$865$	−0.490913	−0.0166915
$866$	−1.79614	−0.0610353
$867$	−22.6339	−0.768687
$868$	0	0
$869$	58.8791	1.99734
$870$	1.31073	0.0444379
$871$	−22.5750	−0.764924
$872$	23.8443	0.807471
$873$	5.45221	0.184529
$874$	−2.02259	−0.0684150
$875$	0	0
$876$	36.5834	1.23604
$877$	−31.9200	−1.07786	−0.538931	−	0.842350i	$-0.681172\pi$
$877$	−31.9200	−1.07786	−0.538931	+	0.842350i	$0.681172\pi$
$878$	−1.45068	−0.0489579
$879$	30.1470	1.01683
$880$	−6.17089	−0.208021
$881$	−39.6254	−1.33501	−0.667507	−	0.744604i	$-0.732638\pi$
$881$	−39.6254	−1.33501	−0.667507	+	0.744604i	$0.732638\pi$
$882$	0	0
$883$	−4.87657	−0.164110	−0.0820548	−	0.996628i	$-0.526148\pi$
$883$	−4.87657	−0.164110	−0.0820548	+	0.996628i	$0.526148\pi$
$884$	18.1331	0.609883
$885$	−5.99818	−0.201627
$886$	0.0145606	0.000489174 0
$887$	56.0263	1.88118	0.940590	−	0.339544i	$-0.110273\pi$
$887$	56.0263	1.88118	0.940590	+	0.339544i	$0.110273\pi$
$888$	18.5433	0.622272
$889$	0	0
$890$	1.18309	0.0396574
$891$	−33.5775	−1.12489
$892$	40.0130	1.33973
$893$	23.7814	0.795813
$894$	6.43966	0.215374
$895$	0.0418861	0.00140010
$896$	0	0
$897$	−25.5058	−0.851613
$898$	4.82077	0.160871
$899$	−9.65296	−0.321944
$900$	−6.84428	−0.228143
$901$	9.61516	0.320328
$902$	1.64653	0.0548235
$903$	0	0
$904$	10.9276	0.363446
$905$	−3.01095	−0.100087
$906$	2.81010	0.0933594
$907$	−15.5703	−0.517004	−0.258502	−	0.966011i	$-0.583229\pi$
$907$	−15.5703	−0.517004	−0.258502	+	0.966011i	$0.583229\pi$
$908$	45.2470	1.50157
$909$	−10.9184	−0.362139
$910$	0	0
$911$	4.04882	0.134143	0.0670716	−	0.997748i	$-0.478634\pi$
$911$	4.04882	0.134143	0.0670716	+	0.997748i	$0.478634\pi$
$912$	13.6978	0.453579
$913$	26.3183	0.871010
$914$	−6.22824	−0.206012
$915$	0.867438	0.0286766
$916$	−10.1915	−0.336737
$917$	0	0
$918$	2.41565	0.0797282
$919$	29.7034	0.979826	0.489913	−	0.871771i	$-0.337028\pi$
$919$	29.7034	0.979826	0.489913	+	0.871771i	$0.337028\pi$
$920$	−0.998115	−0.0329069
$921$	8.95564	0.295098
$922$	−5.59436	−0.184240
$923$	6.16618	0.202962
$924$	0	0
$925$	−50.2597	−1.65253
$926$	6.95100	0.228424
$927$	11.2572	0.369735
$928$	−29.3604	−0.963803
$929$	43.0141	1.41125	0.705624	−	0.708587i	$-0.250667\pi$
$929$	43.0141	1.41125	0.705624	+	0.708587i	$0.250667\pi$
$930$	−0.175032	−0.00573952
$931$	0	0
$932$	35.6889	1.16903
$933$	26.9542	0.882442
$934$	11.8084	0.386382
$935$	−2.50976	−0.0820780
$936$	5.97923	0.195437
$937$	−30.4657	−0.995270	−0.497635	−	0.867386i	$-0.665798\pi$
$937$	−30.4657	−0.995270	−0.497635	+	0.867386i	$0.665798\pi$
$938$	0	0
$939$	−2.54766	−0.0831396
$940$	5.72645	0.186776
$941$	−29.2636	−0.953965	−0.476983	−	0.878913i	$-0.658269\pi$
$941$	−29.2636	−0.953965	−0.476983	+	0.878913i	$0.658269\pi$
$942$	0.655885	0.0213699
$943$	−2.49488	−0.0812444
$944$	41.1118	1.33807
$945$	0	0
$946$	8.16130	0.265347
$947$	15.9593	0.518606	0.259303	−	0.965796i	$-0.416507\pi$
$947$	15.9593	0.518606	0.259303	+	0.965796i	$0.416507\pi$
$948$	−31.4712	−1.02214
$949$	86.6236	2.81192
$950$	3.96312	0.128581
$951$	41.2613	1.33799
$952$	0	0
$953$	−14.9997	−0.485889	−0.242945	−	0.970040i	$-0.578113\pi$
$953$	−14.9997	−0.485889	−0.242945	+	0.970040i	$0.578113\pi$
$954$	1.54705	0.0500877
$955$	1.55704	0.0503846
$956$	48.1615	1.55765
$957$	68.6775	2.22003
$958$	2.19912	0.0710502
$959$	0	0
$960$	2.92889	0.0945296
$961$	−29.7110	−0.958418
$962$	21.4247	0.690759
$963$	−8.11162	−0.261393
$964$	−26.1746	−0.843026
$965$	5.76962	0.185731
$966$	0	0
$967$	19.8571	0.638560	0.319280	−	0.947660i	$-0.396559\pi$
$967$	19.8571	0.638560	0.319280	+	0.947660i	$0.396559\pi$
$968$	21.3330	0.685668
$969$	5.57103	0.178967
$970$	−0.760207	−0.0244088
$971$	51.9133	1.66598	0.832988	−	0.553291i	$-0.186628\pi$
$971$	51.9133	1.66598	0.832988	+	0.553291i	$0.186628\pi$
$972$	−14.1916	−0.455197
$973$	0	0
$974$	−10.3800	−0.332597
$975$	49.9768	1.60054
$976$	−5.94546	−0.190309
$977$	−38.4817	−1.23114	−0.615569	−	0.788083i	$-0.711074\pi$
$977$	−38.4817	−1.23114	−0.615569	+	0.788083i	$0.711074\pi$
$978$	3.35840	0.107390
$979$	61.9898	1.98120
$980$	0	0
$981$	−14.6173	−0.466693
$982$	5.51058	0.175850
$983$	42.7709	1.36418	0.682089	−	0.731269i	$-0.261071\pi$
$983$	42.7709	1.36418	0.682089	+	0.731269i	$0.261071\pi$
$984$	−1.80362	−0.0574974
$985$	2.86877	0.0914067
$986$	−3.65380	−0.116361
$987$	0	0
$988$	34.2069	1.08827
$989$	−12.3663	−0.393224
$990$	−0.403814	−0.0128341
$991$	17.8913	0.568335	0.284167	−	0.958775i	$-0.408283\pi$
$991$	17.8913	0.568335	0.284167	+	0.958775i	$0.408283\pi$
$992$	3.92072	0.124483
$993$	−15.4842	−0.491376
$994$	0	0
$995$	−5.55918	−0.176238
$996$	−14.0673	−0.445739
$997$	44.8202	1.41947	0.709735	−	0.704468i	$-0.248815\pi$
$997$	44.8202	1.41947	0.709735	+	0.704468i	$0.248815\pi$
$998$	5.47061	0.173169
$999$	−57.7907	−1.82842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.o.1.4		6
7.6	odd	2		287.2.a.f.1.4	✓	6
21.20	even	2		2583.2.a.t.1.3		6
28.27	even	2		4592.2.a.bg.1.3		6
35.34	odd	2		7175.2.a.p.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.f.1.4	✓	6	7.6	odd	2
2009.2.a.o.1.4		6	1.1	even	1	trivial
2583.2.a.t.1.3		6	21.20	even	2
4592.2.a.bg.1.3		6	28.27	even	2
7175.2.a.p.1.3		6	35.34	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)

\(f(q)\)	\(=\)	\(q-0.306800 q^{2} +1.50512 q^{3} -1.90587 q^{4} -0.333855 q^{5} -0.461772 q^{6} +1.19832 q^{8} -0.734606 q^{9} +O(q^{10})\)	\(q-0.306800 q^{2} +1.50512 q^{3} -1.90587 q^{4} -0.333855 q^{5} -0.461772 q^{6} +1.19832 q^{8} -0.734606 q^{9} +0.102427 q^{10} +5.36678 q^{11} -2.86857 q^{12} -6.79231 q^{13} -0.502492 q^{15} +3.44410 q^{16} +1.40075 q^{17} +0.225377 q^{18} +2.64242 q^{19} +0.636285 q^{20} -1.64653 q^{22} +2.49488 q^{23} +1.80362 q^{24} -4.88854 q^{25} +2.08388 q^{26} -5.62104 q^{27} +8.50214 q^{29} +0.154165 q^{30} -1.13536 q^{31} -3.45330 q^{32} +8.07767 q^{33} -0.429751 q^{34} +1.40007 q^{36} +10.2811 q^{37} -0.810696 q^{38} -10.2233 q^{39} -0.400066 q^{40} -1.00000 q^{41} -4.95666 q^{43} -10.2284 q^{44} +0.245252 q^{45} -0.765429 q^{46} +8.99982 q^{47} +5.18380 q^{48} +1.49981 q^{50} +2.10830 q^{51} +12.9453 q^{52} +6.86429 q^{53} +1.72454 q^{54} -1.79173 q^{55} +3.97717 q^{57} -2.60846 q^{58} +11.9369 q^{59} +0.957687 q^{60} -1.72627 q^{61} +0.348328 q^{62} -5.82873 q^{64} +2.26764 q^{65} -2.47823 q^{66} +3.32361 q^{67} -2.66965 q^{68} +3.75510 q^{69} -0.907819 q^{71} -0.880295 q^{72} -12.7532 q^{73} -3.15425 q^{74} -7.35785 q^{75} -5.03613 q^{76} +3.13650 q^{78} +10.9710 q^{79} -1.14983 q^{80} -6.25654 q^{81} +0.306800 q^{82} +4.90393 q^{83} -0.467647 q^{85} +1.52071 q^{86} +12.7968 q^{87} +6.43114 q^{88} +11.5506 q^{89} -0.0752432 q^{90} -4.75492 q^{92} -1.70885 q^{93} -2.76115 q^{94} -0.882186 q^{95} -5.19763 q^{96} -7.42196 q^{97} -3.94247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 4 q^{3} + 9 q^{4} + q^{5} + 4 q^{6} + 3 q^{8} + 14 q^{9}+O(q^{10}) \) 6 * q - q^2 + 4 * q^3 + 9 * q^4 + q^5 + 4 * q^6 + 3 * q^8 + 14 * q^9	\( 6 q - q^{2} + 4 q^{3} + 9 q^{4} + q^{5} + 4 q^{6} + 3 q^{8} + 14 q^{9} - 10 q^{10} + 6 q^{11} - 9 q^{12} - 7 q^{13} - 13 q^{15} + 7 q^{16} - 7 q^{17} - 5 q^{18} - 2 q^{19} - 11 q^{20} + 15 q^{22} + 20 q^{23} + 36 q^{24} + 29 q^{25} + 43 q^{26} - 2 q^{27} - 9 q^{29} + 13 q^{30} + 27 q^{31} - 10 q^{32} - 17 q^{33} - 6 q^{34} + 29 q^{36} + 19 q^{37} + 23 q^{38} + q^{39} - 23 q^{40} - 6 q^{41} + 19 q^{43} + 21 q^{44} + 35 q^{45} - 8 q^{46} + 19 q^{47} + 9 q^{48} - 58 q^{50} - 19 q^{51} + 5 q^{53} + 37 q^{54} - 3 q^{55} + 37 q^{57} + 13 q^{58} + 7 q^{59} - 110 q^{60} + 12 q^{61} - 37 q^{64} - 13 q^{65} + 54 q^{66} + 27 q^{67} - 31 q^{68} - 16 q^{69} - 6 q^{71} + 5 q^{72} - 52 q^{73} - 14 q^{74} + 46 q^{75} - 13 q^{76} - 45 q^{78} + 26 q^{80} - 22 q^{81} + q^{82} - 12 q^{83} + 25 q^{85} - 10 q^{86} - 42 q^{87} - 2 q^{88} + 38 q^{89} - 93 q^{90} + 45 q^{92} + 33 q^{93} + 8 q^{94} + q^{95} + 12 q^{96} - 8 q^{97} + q^{99}+O(q^{100}) \) 6 * q - q^2 + 4 * q^3 + 9 * q^4 + q^5 + 4 * q^6 + 3 * q^8 + 14 * q^9 - 10 * q^10 + 6 * q^11 - 9 * q^12 - 7 * q^13 - 13 * q^15 + 7 * q^16 - 7 * q^17 - 5 * q^18 - 2 * q^19 - 11 * q^20 + 15 * q^22 + 20 * q^23 + 36 * q^24 + 29 * q^25 + 43 * q^26 - 2 * q^27 - 9 * q^29 + 13 * q^30 + 27 * q^31 - 10 * q^32 - 17 * q^33 - 6 * q^34 + 29 * q^36 + 19 * q^37 + 23 * q^38 + q^39 - 23 * q^40 - 6 * q^41 + 19 * q^43 + 21 * q^44 + 35 * q^45 - 8 * q^46 + 19 * q^47 + 9 * q^48 - 58 * q^50 - 19 * q^51 + 5 * q^53 + 37 * q^54 - 3 * q^55 + 37 * q^57 + 13 * q^58 + 7 * q^59 - 110 * q^60 + 12 * q^61 - 37 * q^64 - 13 * q^65 + 54 * q^66 + 27 * q^67 - 31 * q^68 - 16 * q^69 - 6 * q^71 + 5 * q^72 - 52 * q^73 - 14 * q^74 + 46 * q^75 - 13 * q^76 - 45 * q^78 + 26 * q^80 - 22 * q^81 + q^82 - 12 * q^83 + 25 * q^85 - 10 * q^86 - 42 * q^87 - 2 * q^88 + 38 * q^89 - 93 * q^90 + 45 * q^92 + 33 * q^93 + 8 * q^94 + q^95 + 12 * q^96 - 8 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more