LMFDB - Embedded newform 2009.2.a.t.1.8

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			1.8
Root			$0.928350$ 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)

$f(q)$	$=$	$q-0.928350 q^{2} -2.79733 q^{3} -1.13817 q^{4} +3.35114 q^{5} +2.59691 q^{6} +2.91332 q^{8} +4.82508 q^{9} +O(q^{10})$	\(q-0.928350 q^{2} -2.79733 q^{3} -1.13817 q^{4} +3.35114 q^{5} +2.59691 q^{6} +2.91332 q^{8} +4.82508 q^{9} -3.11103 q^{10} -0.213023 q^{11} +3.18383 q^{12} -5.00897 q^{13} -9.37425 q^{15} -0.428245 q^{16} +2.11080 q^{17} -4.47936 q^{18} +1.21832 q^{19} -3.81415 q^{20} +0.197760 q^{22} -7.54545 q^{23} -8.14952 q^{24} +6.23011 q^{25} +4.65008 q^{26} -5.10536 q^{27} +1.66679 q^{29} +8.70258 q^{30} -8.48574 q^{31} -5.42907 q^{32} +0.595898 q^{33} -1.95956 q^{34} -5.49174 q^{36} +7.74783 q^{37} -1.13103 q^{38} +14.0118 q^{39} +9.76292 q^{40} -1.00000 q^{41} +3.54077 q^{43} +0.242456 q^{44} +16.1695 q^{45} +7.00481 q^{46} +7.99292 q^{47} +1.19794 q^{48} -5.78372 q^{50} -5.90461 q^{51} +5.70104 q^{52} +6.41670 q^{53} +4.73956 q^{54} -0.713870 q^{55} -3.40805 q^{57} -1.54737 q^{58} -8.78829 q^{59} +10.6695 q^{60} -10.2563 q^{61} +7.87774 q^{62} +5.89657 q^{64} -16.7857 q^{65} -0.553202 q^{66} +0.727259 q^{67} -2.40244 q^{68} +21.1071 q^{69} +3.91355 q^{71} +14.0570 q^{72} +12.0206 q^{73} -7.19270 q^{74} -17.4277 q^{75} -1.38665 q^{76} -13.0078 q^{78} +6.07849 q^{79} -1.43511 q^{80} -0.193845 q^{81} +0.928350 q^{82} -13.6638 q^{83} +7.07357 q^{85} -3.28707 q^{86} -4.66258 q^{87} -0.620605 q^{88} -14.2081 q^{89} -15.0110 q^{90} +8.58797 q^{92} +23.7375 q^{93} -7.42023 q^{94} +4.08276 q^{95} +15.1869 q^{96} +13.0513 q^{97} -1.02786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.928350	−0.656443	−0.328221	−	0.944601i	$-0.606449\pi$
$2$	−0.928350	−0.656443	−0.328221	+	0.944601i	$0.606449\pi$
$3$	−2.79733	−1.61504	−0.807521	−	0.589839i	$-0.799191\pi$
$3$	−2.79733	−1.61504	−0.807521	+	0.589839i	$0.799191\pi$
$4$	−1.13817	−0.569083
$5$	3.35114	1.49867	0.749337	−	0.662189i	$-0.230372\pi$
$5$	3.35114	1.49867	0.749337	+	0.662189i	$0.230372\pi$
$6$	2.59691	1.06018
$7$	0	0
$7$	0	0
$8$	2.91332	1.03001
$9$	4.82508	1.60836
$10$	−3.11103	−0.983793
$11$	−0.213023	−0.0642290	−0.0321145	−	0.999484i	$-0.510224\pi$
$11$	−0.213023	−0.0642290	−0.0321145	+	0.999484i	$0.510224\pi$
$12$	3.18383	0.919093
$13$	−5.00897	−1.38924	−0.694619	−	0.719378i	$-0.744427\pi$
$13$	−5.00897	−1.38924	−0.694619	+	0.719378i	$0.744427\pi$
$14$	0	0
$15$	−9.37425	−2.42042
$16$	−0.428245	−0.107061
$17$	2.11080	0.511944	0.255972	−	0.966684i	$-0.417605\pi$
$17$	2.11080	0.511944	0.255972	+	0.966684i	$0.417605\pi$
$18$	−4.47936	−1.05580
$19$	1.21832	0.279502	0.139751	−	0.990187i	$-0.455370\pi$
$19$	1.21832	0.279502	0.139751	+	0.990187i	$0.455370\pi$
$20$	−3.81415	−0.852870
$21$	0	0
$22$	0.197760	0.0421626
$23$	−7.54545	−1.57333	−0.786667	−	0.617378i	$-0.788195\pi$
$23$	−7.54545	−1.57333	−0.786667	+	0.617378i	$0.788195\pi$
$24$	−8.14952	−1.66351
$25$	6.23011	1.24602
$26$	4.65008	0.911955
$27$	−5.10536	−0.982527
$28$	0	0
$29$	1.66679	0.309516	0.154758	−	0.987952i	$-0.450540\pi$
$29$	1.66679	0.309516	0.154758	+	0.987952i	$0.450540\pi$
$30$	8.70258	1.58887
$31$	−8.48574	−1.52408	−0.762042	−	0.647527i	$-0.775803\pi$
$31$	−8.48574	−1.52408	−0.762042	+	0.647527i	$0.775803\pi$
$32$	−5.42907	−0.959733
$33$	0.595898	0.103732
$34$	−1.95956	−0.336062
$35$	0	0
$36$	−5.49174	−0.915291
$37$	7.74783	1.27374	0.636868	−	0.770973i	$-0.280230\pi$
$37$	7.74783	1.27374	0.636868	+	0.770973i	$0.280230\pi$
$38$	−1.13103	−0.183477
$39$	14.0118	2.24368
$40$	9.76292	1.54365
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	3.54077	0.539962	0.269981	−	0.962866i	$-0.412983\pi$
$43$	3.54077	0.539962	0.269981	+	0.962866i	$0.412983\pi$
$44$	0.242456	0.0365516
$45$	16.1695	2.41041
$46$	7.00481	1.03280
$47$	7.99292	1.16589	0.582944	−	0.812513i	$-0.301901\pi$
$47$	7.99292	1.16589	0.582944	+	0.812513i	$0.301901\pi$
$48$	1.19794	0.172908
$49$	0	0
$50$	−5.78372	−0.817942
$51$	−5.90461	−0.826811
$52$	5.70104	0.790592
$53$	6.41670	0.881401	0.440701	−	0.897654i	$-0.354730\pi$
$53$	6.41670	0.881401	0.440701	+	0.897654i	$0.354730\pi$
$54$	4.73956	0.644972
$55$	−0.713870	−0.0962583
$56$	0	0
$57$	−3.40805	−0.451407
$58$	−1.54737	−0.203179
$59$	−8.78829	−1.14414	−0.572069	−	0.820205i	$-0.693859\pi$
$59$	−8.78829	−1.14414	−0.572069	+	0.820205i	$0.693859\pi$
$60$	10.6695	1.37742
$61$	−10.2563	−1.31318	−0.656591	−	0.754247i	$-0.728002\pi$
$61$	−10.2563	−1.31318	−0.656591	+	0.754247i	$0.728002\pi$
$62$	7.87774	1.00047
$63$	0	0
$64$	5.89657	0.737071
$65$	−16.7857	−2.08201
$66$	−0.553202	−0.0680944
$67$	0.727259	0.0888488	0.0444244	−	0.999013i	$-0.485855\pi$
$67$	0.727259	0.0888488	0.0444244	+	0.999013i	$0.485855\pi$
$68$	−2.40244	−0.291339
$69$	21.1071	2.54100
$70$	0	0
$71$	3.91355	0.464453	0.232226	−	0.972662i	$-0.425399\pi$
$71$	3.91355	0.464453	0.232226	+	0.972662i	$0.425399\pi$
$72$	14.0570	1.65663
$73$	12.0206	1.40691	0.703453	−	0.710742i	$-0.251640\pi$
$73$	12.0206	1.40691	0.703453	+	0.710742i	$0.251640\pi$
$74$	−7.19270	−0.836134
$75$	−17.4277	−2.01238
$76$	−1.38665	−0.159060
$77$	0	0
$78$	−13.0078	−1.47285
$79$	6.07849	0.683884	0.341942	−	0.939721i	$-0.388915\pi$
$79$	6.07849	0.683884	0.341942	+	0.939721i	$0.388915\pi$
$80$	−1.43511	−0.160450
$81$	−0.193845	−0.0215383
$82$	0.928350	0.102519
$83$	−13.6638	−1.49979	−0.749897	−	0.661554i	$-0.769897\pi$
$83$	−13.6638	−1.49979	−0.749897	+	0.661554i	$0.769897\pi$
$84$	0	0
$85$	7.07357	0.767237
$86$	−3.28707	−0.354454
$87$	−4.66258	−0.499881
$88$	−0.620605	−0.0661567
$89$	−14.2081	−1.50605	−0.753027	−	0.657990i	$-0.771407\pi$
$89$	−14.2081	−1.50605	−0.753027	+	0.657990i	$0.771407\pi$
$90$	−15.0110	−1.58229
$91$	0	0
$92$	8.58797	0.895358
$93$	23.7375	2.46146
$94$	−7.42023	−0.765338
$95$	4.08276	0.418882
$96$	15.1869	1.55001
$97$	13.0513	1.32516	0.662579	−	0.748992i	$-0.269462\pi$
$97$	13.0513	1.32516	0.662579	+	0.748992i	$0.269462\pi$
$98$	0	0
$99$	−1.02786	−0.103303
$100$	−7.09090	−0.709090
$101$	−15.4114	−1.53349	−0.766747	−	0.641950i	$-0.778126\pi$
$101$	−15.4114	−1.53349	−0.766747	+	0.641950i	$0.778126\pi$
$102$	5.48154	0.542754
$103$	−8.92932	−0.879832	−0.439916	−	0.898039i	$-0.644992\pi$
$103$	−8.92932	−0.879832	−0.439916	+	0.898039i	$0.644992\pi$
$104$	−14.5927	−1.43093
$105$	0	0
$106$	−5.95694	−0.578589
$107$	8.66116	0.837306	0.418653	−	0.908146i	$-0.362502\pi$
$107$	8.66116	0.837306	0.418653	+	0.908146i	$0.362502\pi$
$108$	5.81075	0.559139
$109$	9.67524	0.926720	0.463360	−	0.886170i	$-0.346644\pi$
$109$	9.67524	0.926720	0.463360	+	0.886170i	$0.346644\pi$
$110$	0.662722	0.0631880
$111$	−21.6733	−2.05714
$112$	0	0
$113$	2.53869	0.238820	0.119410	−	0.992845i	$-0.461900\pi$
$113$	2.53869	0.238820	0.119410	+	0.992845i	$0.461900\pi$
$114$	3.16386	0.296323
$115$	−25.2858	−2.35791
$116$	−1.89709	−0.176140
$117$	−24.1687	−2.23440
$118$	8.15861	0.751061
$119$	0	0
$120$	−27.3102	−2.49306
$121$	−10.9546	−0.995875
$122$	9.52142	0.862029
$123$	2.79733	0.252227
$124$	9.65819	0.867331
$125$	4.12227	0.368707
$126$	0	0
$127$	−6.96966	−0.618457	−0.309228	−	0.950988i	$-0.600071\pi$
$127$	−6.96966	−0.618457	−0.309228	+	0.950988i	$0.600071\pi$
$128$	5.38406	0.475889
$129$	−9.90471	−0.872061
$130$	15.5830	1.36672
$131$	−11.5194	−1.00646	−0.503229	−	0.864153i	$-0.667855\pi$
$131$	−11.5194	−1.00646	−0.503229	+	0.864153i	$0.667855\pi$
$132$	−0.678231	−0.0590324
$133$	0	0
$134$	−0.675151	−0.0583241
$135$	−17.1087	−1.47249
$136$	6.14942	0.527309
$137$	−22.4713	−1.91986	−0.959928	−	0.280248i	$-0.909583\pi$
$137$	−22.4713	−1.91986	−0.959928	+	0.280248i	$0.909583\pi$
$138$	−19.5948	−1.66802
$139$	−19.7137	−1.67209	−0.836046	−	0.548659i	$-0.815138\pi$
$139$	−19.7137	−1.67209	−0.836046	+	0.548659i	$0.815138\pi$
$140$	0	0
$141$	−22.3589	−1.88296
$142$	−3.63314	−0.304887
$143$	1.06703	0.0892294
$144$	−2.06632	−0.172193
$145$	5.58565	0.463863
$146$	−11.1593	−0.923553
$147$	0	0
$148$	−8.81832	−0.724861
$149$	11.3470	0.929583	0.464791	−	0.885420i	$-0.346129\pi$
$149$	11.3470	0.929583	0.464791	+	0.885420i	$0.346129\pi$
$150$	16.1790	1.32101
$151$	−1.96596	−0.159987	−0.0799937	−	0.996795i	$-0.525490\pi$
$151$	−1.96596	−0.159987	−0.0799937	+	0.996795i	$0.525490\pi$
$152$	3.54935	0.287891
$153$	10.1848	0.823390
$154$	0	0
$155$	−28.4369	−2.28411
$156$	−15.9477	−1.27684
$157$	−19.9793	−1.59452	−0.797261	−	0.603635i	$-0.793718\pi$
$157$	−19.9793	−1.59452	−0.797261	+	0.603635i	$0.793718\pi$
$158$	−5.64297	−0.448930
$159$	−17.9496	−1.42350
$160$	−18.1936	−1.43833
$161$	0	0
$162$	0.179956	0.0141386
$163$	7.55741	0.591942	0.295971	−	0.955197i	$-0.404357\pi$
$163$	7.55741	0.591942	0.295971	+	0.955197i	$0.404357\pi$
$164$	1.13817	0.0888759
$165$	1.99693	0.155461
$166$	12.6848	0.984529
$167$	11.6052	0.898036	0.449018	−	0.893523i	$-0.351774\pi$
$167$	11.6052	0.898036	0.449018	+	0.893523i	$0.351774\pi$
$168$	0	0
$169$	12.0898	0.929983
$170$	−6.56675	−0.503647
$171$	5.87849	0.449540
$172$	−4.02998	−0.307283
$173$	13.3467	1.01473	0.507364	−	0.861732i	$-0.330620\pi$
$173$	13.3467	1.01473	0.507364	+	0.861732i	$0.330620\pi$
$174$	4.32850	0.328143
$175$	0	0
$176$	0.0912262	0.00687644
$177$	24.5838	1.84783
$178$	13.1901	0.988638
$179$	−22.6130	−1.69017	−0.845087	−	0.534629i	$-0.820451\pi$
$179$	−22.6130	−1.69017	−0.845087	+	0.534629i	$0.820451\pi$
$180$	−18.4036	−1.37172
$181$	24.9397	1.85376	0.926878	−	0.375363i	$-0.122482\pi$
$181$	24.9397	1.85376	0.926878	+	0.375363i	$0.122482\pi$
$182$	0	0
$183$	28.6903	2.12084
$184$	−21.9823	−1.62055
$185$	25.9640	1.90891
$186$	−22.0367	−1.61581
$187$	−0.449650	−0.0328816
$188$	−9.09727	−0.663487
$189$	0	0
$190$	−3.79023	−0.274972
$191$	3.96178	0.286665	0.143332	−	0.989675i	$-0.454218\pi$
$191$	3.96178	0.286665	0.143332	+	0.989675i	$0.454218\pi$
$192$	−16.4947	−1.19040
$193$	1.92686	0.138699	0.0693493	−	0.997592i	$-0.477908\pi$
$193$	1.92686	0.138699	0.0693493	+	0.997592i	$0.477908\pi$
$194$	−12.1162	−0.869890
$195$	46.9553	3.36254
$196$	0	0
$197$	−25.7779	−1.83660	−0.918300	−	0.395886i	$-0.870437\pi$
$197$	−25.7779	−1.83660	−0.918300	+	0.395886i	$0.870437\pi$
$198$	0.954209	0.0678127
$199$	2.22255	0.157553	0.0787763	−	0.996892i	$-0.474899\pi$
$199$	2.22255	0.157553	0.0787763	+	0.996892i	$0.474899\pi$
$200$	18.1503	1.28342
$201$	−2.03439	−0.143495
$202$	14.3072	1.00665
$203$	0	0
$204$	6.72043	0.470524
$205$	−3.35114	−0.234053
$206$	8.28954	0.577559
$207$	−36.4074	−2.53049
$208$	2.14507	0.148734
$209$	−0.259531	−0.0179521
$210$	0	0
$211$	−17.4514	−1.20141	−0.600703	−	0.799473i	$-0.705112\pi$
$211$	−17.4514	−1.20141	−0.600703	+	0.799473i	$0.705112\pi$
$212$	−7.30327	−0.501591
$213$	−10.9475	−0.750110
$214$	−8.04059	−0.549643
$215$	11.8656	0.809226
$216$	−14.8735	−1.01202
$217$	0	0
$218$	−8.98201	−0.608338
$219$	−33.6257	−2.27221
$220$	0.812503	0.0547790
$221$	−10.5729	−0.711212
$222$	20.1204	1.35039
$223$	−25.4204	−1.70228	−0.851139	−	0.524940i	$-0.824088\pi$
$223$	−25.4204	−1.70228	−0.851139	+	0.524940i	$0.824088\pi$
$224$	0	0
$225$	30.0608	2.00405
$226$	−2.35679	−0.156772
$227$	−16.6990	−1.10835	−0.554175	−	0.832400i	$-0.686966\pi$
$227$	−16.6990	−1.10835	−0.554175	+	0.832400i	$0.686966\pi$
$228$	3.87893	0.256888
$229$	−11.6228	−0.768059	−0.384030	−	0.923321i	$-0.625464\pi$
$229$	−11.6228	−0.768059	−0.384030	+	0.923321i	$0.625464\pi$
$230$	23.4741	1.54784
$231$	0	0
$232$	4.85590	0.318805
$233$	−16.9966	−1.11348	−0.556741	−	0.830686i	$-0.687948\pi$
$233$	−16.9966	−1.11348	−0.556741	+	0.830686i	$0.687948\pi$
$234$	22.4370	1.46675
$235$	26.7854	1.74728
$236$	10.0025	0.651110
$237$	−17.0036	−1.10450
$238$	0	0
$239$	17.1928	1.11211	0.556056	−	0.831145i	$-0.312314\pi$
$239$	17.1928	1.11211	0.556056	+	0.831145i	$0.312314\pi$
$240$	4.01447	0.259133
$241$	−15.5596	−1.00228	−0.501140	−	0.865366i	$-0.667086\pi$
$241$	−15.5596	−1.00228	−0.501140	+	0.865366i	$0.667086\pi$
$242$	10.1697	0.653735
$243$	15.8583	1.01731
$244$	11.6734	0.747310
$245$	0	0
$246$	−2.59691	−0.165573
$247$	−6.10253	−0.388295
$248$	−24.7217	−1.56983
$249$	38.2222	2.42223
$250$	−3.82691	−0.242035
$251$	0.255136	0.0161040	0.00805201	−	0.999968i	$-0.497437\pi$
$251$	0.255136	0.0161040	0.00805201	+	0.999968i	$0.497437\pi$
$252$	0	0
$253$	1.60736	0.101054
$254$	6.47028	0.405981
$255$	−19.7871	−1.23912
$256$	−16.7914	−1.04946
$257$	1.60836	0.100327	0.0501634	−	0.998741i	$-0.484026\pi$
$257$	1.60836	0.100327	0.0501634	+	0.998741i	$0.484026\pi$
$258$	9.19504	0.572458
$259$	0	0
$260$	19.1050	1.18484
$261$	8.04241	0.497813
$262$	10.6941	0.660682
$263$	−14.1955	−0.875332	−0.437666	−	0.899138i	$-0.644195\pi$
$263$	−14.1955	−0.875332	−0.437666	+	0.899138i	$0.644195\pi$
$264$	1.73604	0.106846
$265$	21.5032	1.32093
$266$	0	0
$267$	39.7448	2.43234
$268$	−0.827741	−0.0505624
$269$	13.8844	0.846546	0.423273	−	0.906002i	$-0.360881\pi$
$269$	13.8844	0.846546	0.423273	+	0.906002i	$0.360881\pi$
$270$	15.8829	0.966603
$271$	15.9163	0.966844	0.483422	−	0.875388i	$-0.339394\pi$
$271$	15.9163	0.966844	0.483422	+	0.875388i	$0.339394\pi$
$272$	−0.903939	−0.0548093
$273$	0	0
$274$	20.8613	1.26027
$275$	−1.32716	−0.0800307
$276$	−24.0234	−1.44604
$277$	0.891025	0.0535365	0.0267683	−	0.999642i	$-0.491478\pi$
$277$	0.891025	0.0535365	0.0267683	+	0.999642i	$0.491478\pi$
$278$	18.3012	1.09763
$279$	−40.9444	−2.45128
$280$	0	0
$281$	13.1116	0.782174	0.391087	−	0.920354i	$-0.372099\pi$
$281$	13.1116	0.782174	0.391087	+	0.920354i	$0.372099\pi$
$282$	20.7569	1.23605
$283$	16.6776	0.991378	0.495689	−	0.868500i	$-0.334916\pi$
$283$	16.6776	0.991378	0.495689	+	0.868500i	$0.334916\pi$
$284$	−4.45427	−0.264312
$285$	−11.4208	−0.676512
$286$	−0.990576	−0.0585740
$287$	0	0
$288$	−26.1957	−1.54360
$289$	−12.5445	−0.737914
$290$	−5.18544	−0.304499
$291$	−36.5088	−2.14019
$292$	−13.6815	−0.800647
$293$	−1.62730	−0.0950678	−0.0475339	−	0.998870i	$-0.515136\pi$
$293$	−1.62730	−0.0950678	−0.0475339	+	0.998870i	$0.515136\pi$
$294$	0	0
$295$	−29.4508	−1.71469
$296$	22.5719	1.31196
$297$	1.08756	0.0631067
$298$	−10.5340	−0.610218
$299$	37.7949	2.18574
$300$	19.8356	1.14521
$301$	0	0
$302$	1.82510	0.105023
$303$	43.1109	2.47666
$304$	−0.521740	−0.0299238
$305$	−34.3702	−1.96803
$306$	−9.45503	−0.540508
$307$	8.30009	0.473711	0.236856	−	0.971545i	$-0.423883\pi$
$307$	8.30009	0.473711	0.236856	+	0.971545i	$0.423883\pi$
$308$	0	0
$309$	24.9783	1.42097
$310$	26.3994	1.49938
$311$	0.157416	0.00892624	0.00446312	−	0.999990i	$-0.498579\pi$
$311$	0.157416	0.00892624	0.00446312	+	0.999990i	$0.498579\pi$
$312$	40.8207	2.31102
$313$	−13.3874	−0.756702	−0.378351	−	0.925662i	$-0.623509\pi$
$313$	−13.3874	−0.756702	−0.378351	+	0.925662i	$0.623509\pi$
$314$	18.5478	1.04671
$315$	0	0
$316$	−6.91833	−0.389187
$317$	32.4429	1.82218	0.911088	−	0.412211i	$-0.135243\pi$
$317$	32.4429	1.82218	0.911088	+	0.412211i	$0.135243\pi$
$318$	16.6636	0.934446
$319$	−0.355066	−0.0198799
$320$	19.7602	1.10463
$321$	−24.2282	−1.35228
$322$	0	0
$323$	2.57163	0.143089
$324$	0.220627	0.0122571
$325$	−31.2064	−1.73102
$326$	−7.01592	−0.388576
$327$	−27.0649	−1.49669
$328$	−2.91332	−0.160861
$329$	0	0
$330$	−1.85385	−0.102051
$331$	−20.0148	−1.10011	−0.550057	−	0.835127i	$-0.685394\pi$
$331$	−20.0148	−1.10011	−0.550057	+	0.835127i	$0.685394\pi$
$332$	15.5517	0.853508
$333$	37.3839	2.04862
$334$	−10.7737	−0.589509
$335$	2.43714	0.133155
$336$	0	0
$337$	−3.20111	−0.174376	−0.0871878	−	0.996192i	$-0.527788\pi$
$337$	−3.20111	−0.174376	−0.0871878	+	0.996192i	$0.527788\pi$
$338$	−11.2235	−0.610481
$339$	−7.10157	−0.385704
$340$	−8.05090	−0.436621
$341$	1.80766	0.0978904
$342$	−5.45730	−0.295097
$343$	0	0
$344$	10.3154	0.556168
$345$	70.7329	3.80813
$346$	−12.3904	−0.666110
$347$	6.12096	0.328590	0.164295	−	0.986411i	$-0.447465\pi$
$347$	6.12096	0.328590	0.164295	+	0.986411i	$0.447465\pi$
$348$	5.30679	0.284474
$349$	−36.3598	−1.94629	−0.973147	−	0.230183i	$-0.926067\pi$
$349$	−36.3598	−1.94629	−0.973147	+	0.230183i	$0.926067\pi$
$350$	0	0
$351$	25.5726	1.36496
$352$	1.15652	0.0616427
$353$	16.2502	0.864913	0.432456	−	0.901655i	$-0.357647\pi$
$353$	16.2502	0.864913	0.432456	+	0.901655i	$0.357647\pi$
$354$	−22.8224	−1.21299
$355$	13.1148	0.696063
$356$	16.1712	0.857070
$357$	0	0
$358$	20.9928	1.10950
$359$	−18.2251	−0.961883	−0.480942	−	0.876753i	$-0.659705\pi$
$359$	−18.2251	−0.961883	−0.480942	+	0.876753i	$0.659705\pi$
$360$	47.1069	2.48275
$361$	−17.5157	−0.921879
$362$	−23.1528	−1.21688
$363$	30.6437	1.60838
$364$	0	0
$365$	40.2827	2.10849
$366$	−26.6346	−1.39221
$367$	−19.8542	−1.03638	−0.518189	−	0.855266i	$-0.673394\pi$
$367$	−19.8542	−1.03638	−0.518189	+	0.855266i	$0.673394\pi$
$368$	3.23130	0.168443
$369$	−4.82508	−0.251184
$370$	−24.1037	−1.25309
$371$	0	0
$372$	−27.0172	−1.40078
$373$	−18.9596	−0.981691	−0.490845	−	0.871247i	$-0.663312\pi$
$373$	−18.9596	−0.981691	−0.490845	+	0.871247i	$0.663312\pi$
$374$	0.417432	0.0215849
$375$	−11.5314	−0.595477
$376$	23.2859	1.20088
$377$	−8.34892	−0.429991
$378$	0	0
$379$	26.4927	1.36084	0.680420	−	0.732822i	$-0.261797\pi$
$379$	26.4927	1.36084	0.680420	+	0.732822i	$0.261797\pi$
$380$	−4.64686	−0.238379
$381$	19.4965	0.998834
$382$	−3.67792	−0.188179
$383$	−4.78200	−0.244349	−0.122174	−	0.992509i	$-0.538987\pi$
$383$	−4.78200	−0.244349	−0.122174	+	0.992509i	$0.538987\pi$
$384$	−15.0610	−0.768580
$385$	0	0
$386$	−1.78880	−0.0910477
$387$	17.0845	0.868453
$388$	−14.8545	−0.754125
$389$	−18.4864	−0.937297	−0.468649	−	0.883385i	$-0.655259\pi$
$389$	−18.4864	−0.937297	−0.468649	+	0.883385i	$0.655259\pi$
$390$	−43.5910	−2.20731
$391$	−15.9269	−0.805459
$392$	0	0
$393$	32.2237	1.62547
$394$	23.9309	1.20562
$395$	20.3698	1.02492
$396$	1.16987	0.0587882
$397$	−7.32932	−0.367848	−0.183924	−	0.982940i	$-0.558880\pi$
$397$	−7.32932	−0.367848	−0.183924	+	0.982940i	$0.558880\pi$
$398$	−2.06331	−0.103424
$399$	0	0
$400$	−2.66801	−0.133401
$401$	−20.7464	−1.03603	−0.518013	−	0.855373i	$-0.673328\pi$
$401$	−20.7464	−1.03603	−0.518013	+	0.855373i	$0.673328\pi$
$402$	1.88862	0.0941959
$403$	42.5048	2.11732
$404$	17.5408	0.872685
$405$	−0.649599	−0.0322789
$406$	0	0
$407$	−1.65047	−0.0818107
$408$	−17.2020	−0.851626
$409$	22.2182	1.09862	0.549309	−	0.835619i	$-0.314891\pi$
$409$	22.2182	1.09862	0.549309	+	0.835619i	$0.314891\pi$
$410$	3.11103	0.153643
$411$	62.8598	3.10065
$412$	10.1631	0.500698
$413$	0	0
$414$	33.7988	1.66112
$415$	−45.7892	−2.24770
$416$	27.1941	1.33330
$417$	55.1457	2.70050
$418$	0.240935	0.0117845
$419$	12.9754	0.633891	0.316946	−	0.948444i	$-0.397343\pi$
$419$	12.9754	0.633891	0.316946	+	0.948444i	$0.397343\pi$
$420$	0	0
$421$	4.74986	0.231494	0.115747	−	0.993279i	$-0.463074\pi$
$421$	4.74986	0.231494	0.115747	+	0.993279i	$0.463074\pi$
$422$	16.2010	0.788654
$423$	38.5665	1.87517
$424$	18.6939	0.907855
$425$	13.1505	0.637893
$426$	10.1631	0.492404
$427$	0	0
$428$	−9.85784	−0.476497
$429$	−2.98483	−0.144109
$430$	−11.0154	−0.531211
$431$	−15.3617	−0.739946	−0.369973	−	0.929043i	$-0.620633\pi$
$431$	−15.3617	−0.739946	−0.369973	+	0.929043i	$0.620633\pi$
$432$	2.18634	0.105191
$433$	7.36716	0.354043	0.177021	−	0.984207i	$-0.443354\pi$
$433$	7.36716	0.354043	0.177021	+	0.984207i	$0.443354\pi$
$434$	0	0
$435$	−15.6249	−0.749158
$436$	−11.0120	−0.527381
$437$	−9.19277	−0.439750
$438$	31.2164	1.49158
$439$	−24.7483	−1.18117	−0.590585	−	0.806975i	$-0.701103\pi$
$439$	−24.7483	−1.18117	−0.590585	+	0.806975i	$0.701103\pi$
$440$	−2.07973	−0.0991473
$441$	0	0
$442$	9.81538	0.466870
$443$	−20.5194	−0.974908	−0.487454	−	0.873149i	$-0.662074\pi$
$443$	−20.5194	−0.974908	−0.487454	+	0.873149i	$0.662074\pi$
$444$	24.6678	1.17068
$445$	−47.6132	−2.25708
$446$	23.5991	1.11745
$447$	−31.7414	−1.50131
$448$	0	0
$449$	6.31955	0.298238	0.149119	−	0.988819i	$-0.452356\pi$
$449$	6.31955	0.298238	0.149119	+	0.988819i	$0.452356\pi$
$450$	−27.9069	−1.31555
$451$	0.213023	0.0100309
$452$	−2.88945	−0.135908
$453$	5.49944	0.258386
$454$	15.5025	0.727568
$455$	0	0
$456$	−9.92873	−0.464955
$457$	−13.8770	−0.649141	−0.324570	−	0.945862i	$-0.605220\pi$
$457$	−13.8770	−0.649141	−0.324570	+	0.945862i	$0.605220\pi$
$458$	10.7901	0.504187
$459$	−10.7764	−0.502998
$460$	28.7795	1.34185
$461$	−3.52429	−0.164142	−0.0820712	−	0.996626i	$-0.526153\pi$
$461$	−3.52429	−0.164142	−0.0820712	+	0.996626i	$0.526153\pi$
$462$	0	0
$463$	−5.56453	−0.258606	−0.129303	−	0.991605i	$-0.541274\pi$
$463$	−5.56453	−0.258606	−0.129303	+	0.991605i	$0.541274\pi$
$464$	−0.713796	−0.0331371
$465$	79.5475	3.68893
$466$	15.7788	0.730937
$467$	16.0538	0.742882	0.371441	−	0.928457i	$-0.378864\pi$
$467$	16.0538	0.742882	0.371441	+	0.928457i	$0.378864\pi$
$468$	27.5080	1.27156
$469$	0	0
$470$	−24.8662	−1.14699
$471$	55.8888	2.57522
$472$	−25.6031	−1.17848
$473$	−0.754266	−0.0346812
$474$	15.7853	0.725041
$475$	7.59027	0.348266
$476$	0	0
$477$	30.9611	1.41761
$478$	−15.9610	−0.730038
$479$	−39.1087	−1.78692	−0.893462	−	0.449139i	$-0.851731\pi$
$479$	−39.1087	−1.78692	−0.893462	+	0.449139i	$0.851731\pi$
$480$	50.8935	2.32296
$481$	−38.8086	−1.76952
$482$	14.4447	0.657940
$483$	0	0
$484$	12.4682	0.566735
$485$	43.7367	1.98598
$486$	−14.7221	−0.667807
$487$	−21.1878	−0.960112	−0.480056	−	0.877238i	$-0.659384\pi$
$487$	−21.1878	−0.960112	−0.480056	+	0.877238i	$0.659384\pi$
$488$	−29.8798	−1.35260
$489$	−21.1406	−0.956011
$490$	0	0
$491$	8.80565	0.397394	0.198697	−	0.980061i	$-0.436329\pi$
$491$	8.80565	0.397394	0.198697	+	0.980061i	$0.436329\pi$
$492$	−3.18383	−0.143538
$493$	3.51826	0.158455
$494$	5.66529	0.254893
$495$	−3.44448	−0.154818
$496$	3.63398	0.163170
$497$	0	0
$498$	−35.4835	−1.59006
$499$	−22.7575	−1.01876	−0.509382	−	0.860541i	$-0.670126\pi$
$499$	−22.7575	−1.01876	−0.509382	+	0.860541i	$0.670126\pi$
$500$	−4.69183	−0.209825
$501$	−32.4636	−1.45037
$502$	−0.236855	−0.0105714
$503$	15.5778	0.694578	0.347289	−	0.937758i	$-0.387102\pi$
$503$	15.5778	0.694578	0.347289	+	0.937758i	$0.387102\pi$
$504$	0	0
$505$	−51.6458	−2.29821
$506$	−1.49219	−0.0663359
$507$	−33.8192	−1.50196
$508$	7.93263	0.351953
$509$	−8.00848	−0.354970	−0.177485	−	0.984124i	$-0.556796\pi$
$509$	−8.00848	−0.354970	−0.177485	+	0.984124i	$0.556796\pi$
$510$	18.3694	0.813410
$511$	0	0
$512$	4.82020	0.213025
$513$	−6.21996	−0.274618
$514$	−1.49312	−0.0658588
$515$	−29.9234	−1.31858
$516$	11.2732	0.496275
$517$	−1.70268	−0.0748838
$518$	0	0
$519$	−37.3350	−1.63883
$520$	−48.9022	−2.14450
$521$	5.96444	0.261307	0.130654	−	0.991428i	$-0.458292\pi$
$521$	5.96444	0.261307	0.130654	+	0.991428i	$0.458292\pi$
$522$	−7.46617	−0.326785
$523$	−9.61460	−0.420417	−0.210208	−	0.977657i	$-0.567414\pi$
$523$	−9.61460	−0.420417	−0.210208	+	0.977657i	$0.567414\pi$
$524$	13.1110	0.572758
$525$	0	0
$526$	13.1784	0.574605
$527$	−17.9117	−0.780246
$528$	−0.255190	−0.0111057
$529$	33.9337	1.47538
$530$	−19.9625	−0.867116
$531$	−42.4042	−1.84019
$532$	0	0
$533$	5.00897	0.216963
$534$	−36.8970	−1.59669
$535$	29.0247	1.25485
$536$	2.11874	0.0915154
$537$	63.2561	2.72970
$538$	−12.8896	−0.555709
$539$	0	0
$540$	19.4726	0.837967
$541$	−13.6461	−0.586691	−0.293346	−	0.956006i	$-0.594769\pi$
$541$	−13.6461	−0.586691	−0.293346	+	0.956006i	$0.594769\pi$
$542$	−14.7759	−0.634677
$543$	−69.7648	−2.99389
$544$	−11.4597	−0.491330
$545$	32.4230	1.38885
$546$	0	0
$547$	−2.87328	−0.122852	−0.0614262	−	0.998112i	$-0.519565\pi$
$547$	−2.87328	−0.122852	−0.0614262	+	0.998112i	$0.519565\pi$
$548$	25.5761	1.09256
$549$	−49.4874	−2.11207
$550$	1.23207	0.0525356
$551$	2.03069	0.0865102
$552$	61.4918	2.61726
$553$	0	0
$554$	−0.827183	−0.0351436
$555$	−72.6301	−3.08297
$556$	22.4374	0.951559
$557$	16.3276	0.691823	0.345911	−	0.938267i	$-0.387570\pi$
$557$	16.3276	0.691823	0.345911	+	0.938267i	$0.387570\pi$
$558$	38.0107	1.60912
$559$	−17.7356	−0.750136
$560$	0	0
$561$	1.25782	0.0531052
$562$	−12.1722	−0.513452
$563$	27.3560	1.15292	0.576459	−	0.817126i	$-0.304434\pi$
$563$	27.3560	1.15292	0.576459	+	0.817126i	$0.304434\pi$
$564$	25.4481	1.07156
$565$	8.50750	0.357913
$566$	−15.4826	−0.650782
$567$	0	0
$568$	11.4014	0.478392
$569$	−5.83582	−0.244650	−0.122325	−	0.992490i	$-0.539035\pi$
$569$	−5.83582	−0.244650	−0.122325	+	0.992490i	$0.539035\pi$
$570$	10.6025	0.444091
$571$	10.5953	0.443401	0.221700	−	0.975115i	$-0.428839\pi$
$571$	10.5953	0.443401	0.221700	+	0.975115i	$0.428839\pi$
$572$	−1.21446	−0.0507789
$573$	−11.0824	−0.462975
$574$	0	0
$575$	−47.0090	−1.96041
$576$	28.4514	1.18548
$577$	−7.00274	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$577$	−7.00274	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$578$	11.6457	0.484398
$579$	−5.39008	−0.224004
$580$	−6.35740	−0.263977
$581$	0	0
$582$	33.8930	1.40491
$583$	−1.36691	−0.0566115
$584$	35.0199	1.44913
$585$	−80.9925	−3.34863
$586$	1.51070	0.0624065
$587$	29.8962	1.23395	0.616974	−	0.786983i	$-0.288358\pi$
$587$	29.8962	1.23395	0.616974	+	0.786983i	$0.288358\pi$
$588$	0	0
$589$	−10.3384	−0.425985
$590$	27.3406	1.12560
$591$	72.1094	2.96618
$592$	−3.31797	−0.136368
$593$	−18.6873	−0.767395	−0.383698	−	0.923459i	$-0.625350\pi$
$593$	−18.6873	−0.767395	−0.383698	+	0.923459i	$0.625350\pi$
$594$	−1.00964	−0.0414259
$595$	0	0
$596$	−12.9148	−0.529010
$597$	−6.21723	−0.254454
$598$	−35.0869	−1.43481
$599$	−18.9893	−0.775882	−0.387941	−	0.921684i	$-0.626814\pi$
$599$	−18.9893	−0.775882	−0.387941	+	0.921684i	$0.626814\pi$
$600$	−50.7724	−2.07278
$601$	−21.2130	−0.865298	−0.432649	−	0.901563i	$-0.642421\pi$
$601$	−21.2130	−0.865298	−0.432649	+	0.901563i	$0.642421\pi$
$602$	0	0
$603$	3.50908	0.142901
$604$	2.23759	0.0910462
$605$	−36.7104	−1.49249
$606$	−40.0220	−1.62578
$607$	−35.2077	−1.42904	−0.714518	−	0.699617i	$-0.753354\pi$
$607$	−35.2077	−1.42904	−0.714518	+	0.699617i	$0.753354\pi$
$608$	−6.61435	−0.268247
$609$	0	0
$610$	31.9076	1.29190
$611$	−40.0363	−1.61970
$612$	−11.5920	−0.468577
$613$	17.2390	0.696278	0.348139	−	0.937443i	$-0.386814\pi$
$613$	17.2390	0.696278	0.348139	+	0.937443i	$0.386814\pi$
$614$	−7.70539	−0.310964
$615$	9.37425	0.378006
$616$	0	0
$617$	21.6614	0.872055	0.436027	−	0.899933i	$-0.356385\pi$
$617$	21.6614	0.872055	0.436027	+	0.899933i	$0.356385\pi$
$618$	−23.1886	−0.932783
$619$	−19.8892	−0.799416	−0.399708	−	0.916643i	$-0.630888\pi$
$619$	−19.8892	−0.799416	−0.399708	+	0.916643i	$0.630888\pi$
$620$	32.3659	1.29985
$621$	38.5222	1.54584
$622$	−0.146137	−0.00585956
$623$	0	0
$624$	−6.00047	−0.240211
$625$	−17.3363	−0.693451
$626$	12.4282	0.496732
$627$	0.725995	0.0289934
$628$	22.7398	0.907415
$629$	16.3541	0.652081
$630$	0	0
$631$	−28.5292	−1.13573	−0.567865	−	0.823122i	$-0.692230\pi$
$631$	−28.5292	−1.13573	−0.567865	+	0.823122i	$0.692230\pi$
$632$	17.7086	0.704409
$633$	48.8175	1.94032
$634$	−30.1184	−1.19615
$635$	−23.3563	−0.926865
$636$	20.4297	0.810090
$637$	0	0
$638$	0.329626	0.0130500
$639$	18.8832	0.747007
$640$	18.0427	0.713202
$641$	8.47489	0.334738	0.167369	−	0.985894i	$-0.446473\pi$
$641$	8.47489	0.334738	0.167369	+	0.985894i	$0.446473\pi$
$642$	22.4922	0.887697
$643$	−19.9388	−0.786309	−0.393155	−	0.919472i	$-0.628616\pi$
$643$	−19.9388	−0.786309	−0.393155	+	0.919472i	$0.628616\pi$
$644$	0	0
$645$	−33.1920	−1.30693
$646$	−2.38737	−0.0939299
$647$	−18.2089	−0.715867	−0.357933	−	0.933747i	$-0.616518\pi$
$647$	−18.2089	−0.715867	−0.357933	+	0.933747i	$0.616518\pi$
$648$	−0.564731	−0.0221847
$649$	1.87211	0.0734868
$650$	28.9705	1.13632
$651$	0	0
$652$	−8.60158	−0.336864
$653$	34.2135	1.33888	0.669439	−	0.742867i	$-0.266534\pi$
$653$	34.2135	1.33888	0.669439	+	0.742867i	$0.266534\pi$
$654$	25.1257	0.982492
$655$	−38.6032	−1.50835
$656$	0.428245	0.0167202
$657$	58.0004	2.26281
$658$	0	0
$659$	3.42703	0.133498	0.0667491	−	0.997770i	$-0.478737\pi$
$659$	3.42703	0.133498	0.0667491	+	0.997770i	$0.478737\pi$
$660$	−2.27284	−0.0884703
$661$	37.8346	1.47160	0.735798	−	0.677201i	$-0.236807\pi$
$661$	37.8346	1.47160	0.735798	+	0.677201i	$0.236807\pi$
$662$	18.5808	0.722162
$663$	29.5760	1.14864
$664$	−39.8069	−1.54481
$665$	0	0
$666$	−34.7053	−1.34480
$667$	−12.5767	−0.486972
$668$	−13.2086	−0.511057
$669$	71.1095	2.74925
$670$	−2.26252	−0.0874088
$671$	2.18483	0.0843444
$672$	0	0
$673$	24.1061	0.929222	0.464611	−	0.885515i	$-0.346194\pi$
$673$	24.1061	0.929222	0.464611	+	0.885515i	$0.346194\pi$
$674$	2.97175	0.114468
$675$	−31.8069	−1.22425
$676$	−13.7602	−0.529238
$677$	0.831150	0.0319437	0.0159719	−	0.999872i	$-0.494916\pi$
$677$	0.831150	0.0319437	0.0159719	+	0.999872i	$0.494916\pi$
$678$	6.59274	0.253193
$679$	0	0
$680$	20.6076	0.790264
$681$	46.7126	1.79003
$682$	−1.67814	−0.0642594
$683$	1.39137	0.0532393	0.0266197	−	0.999646i	$-0.491526\pi$
$683$	1.39137	0.0532393	0.0266197	+	0.999646i	$0.491526\pi$
$684$	−6.69070	−0.255825
$685$	−75.3045	−2.87724
$686$	0	0
$687$	32.5130	1.24045
$688$	−1.51632	−0.0578090
$689$	−32.1410	−1.22448
$690$	−65.6649	−2.49982
$691$	16.4243	0.624809	0.312404	−	0.949949i	$-0.398866\pi$
$691$	16.4243	0.624809	0.312404	+	0.949949i	$0.398866\pi$
$692$	−15.1907	−0.577464
$693$	0	0
$694$	−5.68239	−0.215701
$695$	−66.0632	−2.50592
$696$	−13.5836	−0.514884
$697$	−2.11080	−0.0799522
$698$	33.7546	1.27763
$699$	47.5451	1.79832
$700$	0	0
$701$	15.7745	0.595796	0.297898	−	0.954598i	$-0.403714\pi$
$701$	15.7745	0.595796	0.297898	+	0.954598i	$0.403714\pi$
$702$	−23.7403	−0.896020
$703$	9.43934	0.356011
$704$	−1.25611	−0.0473413
$705$	−74.9276	−2.82194
$706$	−15.0859	−0.567766
$707$	0	0
$708$	−27.9804	−1.05157
$709$	11.7867	0.442657	0.221329	−	0.975199i	$-0.428961\pi$
$709$	11.7867	0.442657	0.221329	+	0.975199i	$0.428961\pi$
$710$	−12.1752	−0.456925
$711$	29.3292	1.09993
$712$	−41.3926	−1.55125
$713$	64.0287	2.39789
$714$	0	0
$715$	3.57576	0.133726
$716$	25.7373	0.961850
$717$	−48.0941	−1.79611
$718$	16.9193	0.631421
$719$	−42.9303	−1.60103	−0.800516	−	0.599311i	$-0.795441\pi$
$719$	−42.9303	−1.60103	−0.800516	+	0.599311i	$0.795441\pi$
$720$	−6.92451	−0.258061
$721$	0	0
$722$	16.2607	0.605160
$723$	43.5253	1.61873
$724$	−28.3856	−1.05494
$725$	10.3843	0.385663
$726$	−28.4481	−1.05581
$727$	26.9295	0.998761	0.499381	−	0.866383i	$-0.333561\pi$
$727$	26.9295	0.998761	0.499381	+	0.866383i	$0.333561\pi$
$728$	0	0
$729$	−43.7795	−1.62146
$730$	−37.3964	−1.38410
$731$	7.47384	0.276430
$732$	−32.6543	−1.20694
$733$	−8.48657	−0.313459	−0.156729	−	0.987642i	$-0.550095\pi$
$733$	−8.48657	−0.313459	−0.156729	+	0.987642i	$0.550095\pi$
$734$	18.4316	0.680323
$735$	0	0
$736$	40.9648	1.50998
$737$	−0.154923	−0.00570667
$738$	4.47936	0.164888
$739$	28.1651	1.03607	0.518035	−	0.855360i	$-0.326664\pi$
$739$	28.1651	1.03607	0.518035	+	0.855360i	$0.326664\pi$
$740$	−29.5514	−1.08633
$741$	17.0708	0.627112
$742$	0	0
$743$	−1.52995	−0.0561286	−0.0280643	−	0.999606i	$-0.508934\pi$
$743$	−1.52995	−0.0561286	−0.0280643	+	0.999606i	$0.508934\pi$
$744$	69.1548	2.53534
$745$	38.0253	1.39314
$746$	17.6011	0.644424
$747$	−65.9288	−2.41221
$748$	0.511776	0.0187124
$749$	0	0
$750$	10.7051	0.390896
$751$	30.5076	1.11324	0.556619	−	0.830768i	$-0.312098\pi$
$751$	30.5076	1.11324	0.556619	+	0.830768i	$0.312098\pi$
$752$	−3.42293	−0.124821
$753$	−0.713700	−0.0260087
$754$	7.75072	0.282264
$755$	−6.58819	−0.239769
$756$	0	0
$757$	−44.9072	−1.63218	−0.816091	−	0.577924i	$-0.803863\pi$
$757$	−44.9072	−1.63218	−0.816091	+	0.577924i	$0.803863\pi$
$758$	−24.5945	−0.893314
$759$	−4.49631	−0.163206
$760$	11.8944	0.431454
$761$	−20.2331	−0.733448	−0.366724	−	0.930330i	$-0.619521\pi$
$761$	−20.2331	−0.733448	−0.366724	+	0.930330i	$0.619521\pi$
$762$	−18.0995	−0.655677
$763$	0	0
$764$	−4.50917	−0.163136
$765$	34.1305	1.23399
$766$	4.43937	0.160401
$767$	44.0203	1.58948
$768$	46.9713	1.69493
$769$	−4.75064	−0.171312	−0.0856562	−	0.996325i	$-0.527299\pi$
$769$	−4.75064	−0.171312	−0.0856562	+	0.996325i	$0.527299\pi$
$770$	0	0
$771$	−4.49913	−0.162032
$772$	−2.19309	−0.0789310
$773$	16.4925	0.593195	0.296598	−	0.955003i	$-0.404148\pi$
$773$	16.4925	0.593195	0.296598	+	0.955003i	$0.404148\pi$
$774$	−15.8604	−0.570089
$775$	−52.8671	−1.89904
$776$	38.0226	1.36493
$777$	0	0
$778$	17.1618	0.615282
$779$	−1.21832	−0.0436509
$780$	−53.4430	−1.91357
$781$	−0.833677	−0.0298313
$782$	14.7857	0.528737
$783$	−8.50958	−0.304107
$784$	0	0
$785$	−66.9533	−2.38967
$786$	−29.9149	−1.06703
$787$	3.76198	0.134100	0.0670500	−	0.997750i	$-0.478641\pi$
$787$	3.76198	0.134100	0.0670500	+	0.997750i	$0.478641\pi$
$788$	29.3395	1.04518
$789$	39.7096	1.41370
$790$	−18.9103	−0.672800
$791$	0	0
$792$	−2.99447	−0.106404
$793$	51.3734	1.82432
$794$	6.80417	0.241471
$795$	−60.1517	−2.13336
$796$	−2.52964	−0.0896606
$797$	−6.88710	−0.243954	−0.121977	−	0.992533i	$-0.538923\pi$
$797$	−6.88710	−0.243954	−0.121977	+	0.992533i	$0.538923\pi$
$798$	0	0
$799$	16.8714	0.596869
$800$	−33.8237	−1.19585
$801$	−68.5551	−2.42228
$802$	19.2599	0.680092
$803$	−2.56067	−0.0903642
$804$	2.31547	0.0816603
$805$	0	0
$806$	−39.4594	−1.38990
$807$	−38.8393	−1.36721
$808$	−44.8983	−1.57952
$809$	−20.0924	−0.706412	−0.353206	−	0.935546i	$-0.614909\pi$
$809$	−20.0924	−0.706412	−0.353206	+	0.935546i	$0.614909\pi$
$810$	0.603056	0.0211892
$811$	41.9779	1.47404	0.737021	−	0.675870i	$-0.236232\pi$
$811$	41.9779	1.47404	0.737021	+	0.675870i	$0.236232\pi$
$812$	0	0
$813$	−44.5231	−1.56149
$814$	1.53221	0.0537040
$815$	25.3259	0.887127
$816$	2.52862	0.0885194
$817$	4.31379	0.150920
$818$	−20.6262	−0.721180
$819$	0	0
$820$	3.81415	0.133196
$821$	8.31526	0.290205	0.145102	−	0.989417i	$-0.453649\pi$
$821$	8.31526	0.290205	0.145102	+	0.989417i	$0.453649\pi$
$822$	−58.3559	−2.03540
$823$	27.4626	0.957287	0.478644	−	0.878009i	$-0.341129\pi$
$823$	27.4626	0.957287	0.478644	+	0.878009i	$0.341129\pi$
$824$	−26.0139	−0.906239
$825$	3.71251	0.129253
$826$	0	0
$827$	−5.14395	−0.178873	−0.0894363	−	0.995993i	$-0.528507\pi$
$827$	−5.14395	−0.178873	−0.0894363	+	0.995993i	$0.528507\pi$
$828$	41.4376	1.44006
$829$	40.2338	1.39738	0.698689	−	0.715425i	$-0.253767\pi$
$829$	40.2338	1.39738	0.698689	+	0.715425i	$0.253767\pi$
$830$	42.5084	1.47549
$831$	−2.49250	−0.0864637
$832$	−29.5357	−1.02397
$833$	0	0
$834$	−51.1946	−1.77272
$835$	38.8905	1.34586
$836$	0.295389	0.0102163
$837$	43.3228	1.49745
$838$	−12.0457	−0.416113
$839$	53.2709	1.83912	0.919558	−	0.392955i	$-0.128547\pi$
$839$	53.2709	1.83912	0.919558	+	0.392955i	$0.128547\pi$
$840$	0	0
$841$	−26.2218	−0.904200
$842$	−4.40953	−0.151962
$843$	−36.6776	−1.26324
$844$	19.8626	0.683700
$845$	40.5145	1.39374
$846$	−35.8032	−1.23094
$847$	0	0
$848$	−2.74792	−0.0943639
$849$	−46.6527	−1.60112
$850$	−12.2083	−0.418740
$851$	−58.4608	−2.00401
$852$	12.4601	0.426875
$853$	−15.3414	−0.525280	−0.262640	−	0.964894i	$-0.584593\pi$
$853$	−15.3414	−0.525280	−0.262640	+	0.964894i	$0.584593\pi$
$854$	0	0
$855$	19.6996	0.673713
$856$	25.2327	0.862436
$857$	49.7610	1.69980	0.849901	−	0.526942i	$-0.176661\pi$
$857$	49.7610	1.69980	0.849901	+	0.526942i	$0.176661\pi$
$858$	2.77097	0.0945994
$859$	49.3028	1.68219	0.841095	−	0.540888i	$-0.181912\pi$
$859$	49.3028	1.68219	0.841095	+	0.540888i	$0.181912\pi$
$860$	−13.5050	−0.460517
$861$	0	0
$862$	14.2610	0.485732
$863$	38.3282	1.30471	0.652353	−	0.757915i	$-0.273782\pi$
$863$	38.3282	1.30471	0.652353	+	0.757915i	$0.273782\pi$
$864$	27.7174	0.942964
$865$	44.7264	1.52074
$866$	−6.83930	−0.232409
$867$	35.0912	1.19176
$868$	0	0
$869$	−1.29486	−0.0439251
$870$	14.5054	0.491779
$871$	−3.64282	−0.123432
$872$	28.1870	0.954534
$873$	62.9735	2.13133
$874$	8.53411	0.288671
$875$	0	0
$876$	38.2716	1.29308
$877$	22.3320	0.754099	0.377050	−	0.926193i	$-0.376939\pi$
$877$	22.3320	0.754099	0.377050	+	0.926193i	$0.376939\pi$
$878$	22.9751	0.775370
$879$	4.55210	0.153538
$880$	0.305711	0.0103055
$881$	−31.3529	−1.05631	−0.528154	−	0.849149i	$-0.677116\pi$
$881$	−31.3529	−1.05631	−0.528154	+	0.849149i	$0.677116\pi$
$882$	0	0
$883$	27.6378	0.930086	0.465043	−	0.885288i	$-0.346039\pi$
$883$	27.6378	0.930086	0.465043	+	0.885288i	$0.346039\pi$
$884$	12.0337	0.404739
$885$	82.3836	2.76929
$886$	19.0492	0.639971
$887$	−39.8708	−1.33873	−0.669365	−	0.742934i	$-0.733434\pi$
$887$	−39.8708	−1.33873	−0.669365	+	0.742934i	$0.733434\pi$
$888$	−63.1411	−2.11888
$889$	0	0
$890$	44.2017	1.48165
$891$	0.0412934	0.00138338
$892$	28.9327	0.968738
$893$	9.73794	0.325868
$894$	29.4671	0.985527
$895$	−75.7792	−2.53302
$896$	0	0
$897$	−105.725	−3.53005
$898$	−5.86676	−0.195776
$899$	−14.1440	−0.471728
$900$	−34.2142	−1.14047
$901$	13.5444	0.451228
$902$	−0.197760	−0.00658470
$903$	0	0
$904$	7.39601	0.245988
$905$	83.5764	2.77817
$906$	−5.10541	−0.169616
$907$	16.8303	0.558839	0.279420	−	0.960169i	$-0.409858\pi$
$907$	16.8303	0.558839	0.279420	+	0.960169i	$0.409858\pi$
$908$	19.0062	0.630743
$909$	−74.3613	−2.46641
$910$	0	0
$911$	−13.7627	−0.455978	−0.227989	−	0.973664i	$-0.573215\pi$
$911$	−13.7627	−0.455978	−0.227989	+	0.973664i	$0.573215\pi$
$912$	1.45948	0.0483282
$913$	2.91071	0.0963303
$914$	12.8828	0.426124
$915$	96.1449	3.17845
$916$	13.2287	0.437089
$917$	0	0
$918$	10.0043	0.330190
$919$	−8.00806	−0.264161	−0.132081	−	0.991239i	$-0.542166\pi$
$919$	−8.00806	−0.264161	−0.132081	+	0.991239i	$0.542166\pi$
$920$	−73.6656	−2.42868
$921$	−23.2181	−0.765063
$922$	3.27177	0.107750
$923$	−19.6028	−0.645235
$924$	0	0
$925$	48.2698	1.58710
$926$	5.16583	0.169760
$927$	−43.0847	−1.41509
$928$	−9.04914	−0.297053
$929$	4.61272	0.151339	0.0756693	−	0.997133i	$-0.475891\pi$
$929$	4.61272	0.151339	0.0756693	+	0.997133i	$0.475891\pi$
$930$	−73.8479	−2.42157
$931$	0	0
$932$	19.3449	0.633664
$933$	−0.440345	−0.0144163
$934$	−14.9036	−0.487659
$935$	−1.50684	−0.0492788
$936$	−70.4110	−2.30146
$937$	17.4245	0.569235	0.284618	−	0.958641i	$-0.408133\pi$
$937$	17.4245	0.569235	0.284618	+	0.958641i	$0.408133\pi$
$938$	0	0
$939$	37.4491	1.22211
$940$	−30.4862	−0.994350
$941$	−15.7178	−0.512385	−0.256193	−	0.966626i	$-0.582468\pi$
$941$	−15.7178	−0.512385	−0.256193	+	0.966626i	$0.582468\pi$
$942$	−51.8843	−1.69048
$943$	7.54545	0.245713
$944$	3.76354	0.122493
$945$	0	0
$946$	0.700223	0.0227662
$947$	−35.7603	−1.16205	−0.581027	−	0.813884i	$-0.697349\pi$
$947$	−35.7603	−1.16205	−0.581027	+	0.813884i	$0.697349\pi$
$948$	19.3529	0.628553
$949$	−60.2109	−1.95453
$950$	−7.04643	−0.228616
$951$	−90.7537	−2.94289
$952$	0	0
$953$	40.0014	1.29577	0.647886	−	0.761737i	$-0.275653\pi$
$953$	40.0014	1.29577	0.647886	+	0.761737i	$0.275653\pi$
$954$	−28.7427	−0.930580
$955$	13.2765	0.429617
$956$	−19.5683	−0.632884
$957$	0.993238	0.0321068
$958$	36.3066	1.17301
$959$	0	0
$960$	−55.2759	−1.78402
$961$	41.0079	1.32283
$962$	36.0280	1.16159
$963$	41.7908	1.34669
$964$	17.7094	0.570381
$965$	6.45718	0.207864
$966$	0	0
$967$	30.3732	0.976735	0.488367	−	0.872638i	$-0.337593\pi$
$967$	30.3732	0.976735	0.488367	+	0.872638i	$0.337593\pi$
$968$	−31.9143	−1.02576
$969$	−7.19371	−0.231095
$970$	−40.6029	−1.30368
$971$	−9.47278	−0.303996	−0.151998	−	0.988381i	$-0.548571\pi$
$971$	−9.47278	−0.303996	−0.151998	+	0.988381i	$0.548571\pi$
$972$	−18.0494	−0.578935
$973$	0	0
$974$	19.6697	0.630258
$975$	87.2948	2.79567
$976$	4.39220	0.140591
$977$	−44.0590	−1.40957	−0.704785	−	0.709421i	$-0.748957\pi$
$977$	−44.0590	−1.40957	−0.704785	+	0.709421i	$0.748957\pi$
$978$	19.6259	0.627566
$979$	3.02665	0.0967323
$980$	0	0
$981$	46.6838	1.49050
$982$	−8.17473	−0.260866
$983$	13.2195	0.421635	0.210818	−	0.977525i	$-0.432387\pi$
$983$	13.2195	0.421635	0.210818	+	0.977525i	$0.432387\pi$
$984$	8.14952	0.259797
$985$	−86.3852	−2.75246
$986$	−3.26618	−0.104016
$987$	0	0
$988$	6.94570	0.220972
$989$	−26.7167	−0.849540
$990$	3.19768	0.101629
$991$	−11.8824	−0.377456	−0.188728	−	0.982029i	$-0.560436\pi$
$991$	−11.8824	−0.377456	−0.188728	+	0.982029i	$0.560436\pi$
$992$	46.0697	1.46272
$993$	55.9881	1.77673
$994$	0	0
$995$	7.44808	0.236120
$996$	−43.5032	−1.37845
$997$	48.9936	1.55164	0.775821	−	0.630953i	$-0.217336\pi$
$997$	48.9936	1.55164	0.775821	+	0.630953i	$0.217336\pi$
$998$	21.1269	0.668760
$999$	−39.5554	−1.25148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.t.1.8	✓	20
7.6	odd	2		2009.2.a.u.1.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.8	✓	20	1.1	even	1	trivial
2009.2.a.u.1.8	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)

\(f(q)\)	\(=\)	\(q-0.928350 q^{2} -2.79733 q^{3} -1.13817 q^{4} +3.35114 q^{5} +2.59691 q^{6} +2.91332 q^{8} +4.82508 q^{9} +O(q^{10})\)	\(q-0.928350 q^{2} -2.79733 q^{3} -1.13817 q^{4} +3.35114 q^{5} +2.59691 q^{6} +2.91332 q^{8} +4.82508 q^{9} -3.11103 q^{10} -0.213023 q^{11} +3.18383 q^{12} -5.00897 q^{13} -9.37425 q^{15} -0.428245 q^{16} +2.11080 q^{17} -4.47936 q^{18} +1.21832 q^{19} -3.81415 q^{20} +0.197760 q^{22} -7.54545 q^{23} -8.14952 q^{24} +6.23011 q^{25} +4.65008 q^{26} -5.10536 q^{27} +1.66679 q^{29} +8.70258 q^{30} -8.48574 q^{31} -5.42907 q^{32} +0.595898 q^{33} -1.95956 q^{34} -5.49174 q^{36} +7.74783 q^{37} -1.13103 q^{38} +14.0118 q^{39} +9.76292 q^{40} -1.00000 q^{41} +3.54077 q^{43} +0.242456 q^{44} +16.1695 q^{45} +7.00481 q^{46} +7.99292 q^{47} +1.19794 q^{48} -5.78372 q^{50} -5.90461 q^{51} +5.70104 q^{52} +6.41670 q^{53} +4.73956 q^{54} -0.713870 q^{55} -3.40805 q^{57} -1.54737 q^{58} -8.78829 q^{59} +10.6695 q^{60} -10.2563 q^{61} +7.87774 q^{62} +5.89657 q^{64} -16.7857 q^{65} -0.553202 q^{66} +0.727259 q^{67} -2.40244 q^{68} +21.1071 q^{69} +3.91355 q^{71} +14.0570 q^{72} +12.0206 q^{73} -7.19270 q^{74} -17.4277 q^{75} -1.38665 q^{76} -13.0078 q^{78} +6.07849 q^{79} -1.43511 q^{80} -0.193845 q^{81} +0.928350 q^{82} -13.6638 q^{83} +7.07357 q^{85} -3.28707 q^{86} -4.66258 q^{87} -0.620605 q^{88} -14.2081 q^{89} -15.0110 q^{90} +8.58797 q^{92} +23.7375 q^{93} -7.42023 q^{94} +4.08276 q^{95} +15.1869 q^{96} +13.0513 q^{97} -1.02786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 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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