LMFDB - Embedded newform 2009.2.a.t.1.17

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			1.17
Root			$-1.89507$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.89507 q^{2} +1.57790 q^{3} +1.59127 q^{4} -3.12194 q^{5} +2.99022 q^{6} -0.774564 q^{8} -0.510238 q^{9} +O(q^{10})$	\(q+1.89507 q^{2} +1.57790 q^{3} +1.59127 q^{4} -3.12194 q^{5} +2.99022 q^{6} -0.774564 q^{8} -0.510238 q^{9} -5.91628 q^{10} -2.47721 q^{11} +2.51087 q^{12} +3.58819 q^{13} -4.92610 q^{15} -4.65040 q^{16} -7.11446 q^{17} -0.966935 q^{18} -5.99670 q^{19} -4.96786 q^{20} -4.69447 q^{22} +9.03548 q^{23} -1.22218 q^{24} +4.74649 q^{25} +6.79985 q^{26} -5.53880 q^{27} +9.03999 q^{29} -9.33528 q^{30} -8.92613 q^{31} -7.26368 q^{32} -3.90878 q^{33} -13.4824 q^{34} -0.811929 q^{36} -3.61418 q^{37} -11.3641 q^{38} +5.66180 q^{39} +2.41814 q^{40} -1.00000 q^{41} -8.37070 q^{43} -3.94191 q^{44} +1.59293 q^{45} +17.1228 q^{46} -1.53886 q^{47} -7.33785 q^{48} +8.99492 q^{50} -11.2259 q^{51} +5.70979 q^{52} -0.436569 q^{53} -10.4964 q^{54} +7.73369 q^{55} -9.46217 q^{57} +17.1314 q^{58} -9.21554 q^{59} -7.83877 q^{60} +11.5476 q^{61} -16.9156 q^{62} -4.46435 q^{64} -11.2021 q^{65} -7.40739 q^{66} +2.71201 q^{67} -11.3211 q^{68} +14.2571 q^{69} +4.30671 q^{71} +0.395212 q^{72} +6.16745 q^{73} -6.84911 q^{74} +7.48948 q^{75} -9.54238 q^{76} +10.7295 q^{78} +0.388867 q^{79} +14.5182 q^{80} -7.20894 q^{81} -1.89507 q^{82} +7.05134 q^{83} +22.2109 q^{85} -15.8630 q^{86} +14.2642 q^{87} +1.91875 q^{88} -11.6241 q^{89} +3.01871 q^{90} +14.3779 q^{92} -14.0845 q^{93} -2.91623 q^{94} +18.7213 q^{95} -11.4613 q^{96} +10.4689 q^{97} +1.26397 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.89507	1.34001	0.670007	−	0.742355i	$-0.266291\pi$
$2$	1.89507	1.34001	0.670007	+	0.742355i	$0.266291\pi$
$3$	1.57790	0.911000	0.455500	−	0.890236i	$-0.349461\pi$
$3$	1.57790	0.911000	0.455500	+	0.890236i	$0.349461\pi$
$4$	1.59127	0.795637
$5$	−3.12194	−1.39617	−0.698086	−	0.716013i	$-0.745965\pi$
$5$	−3.12194	−1.39617	−0.698086	+	0.716013i	$0.745965\pi$
$6$	2.99022	1.22075
$7$	0	0
$7$	0	0
$8$	−0.774564	−0.273850
$9$	−0.510238	−0.170079
$10$	−5.91628	−1.87089
$11$	−2.47721	−0.746906	−0.373453	−	0.927649i	$-0.621826\pi$
$11$	−2.47721	−0.746906	−0.373453	+	0.927649i	$0.621826\pi$
$12$	2.51087	0.724825
$13$	3.58819	0.995185	0.497592	−	0.867411i	$-0.334218\pi$
$13$	3.58819	0.995185	0.497592	+	0.867411i	$0.334218\pi$
$14$	0	0
$15$	−4.92610	−1.27191
$16$	−4.65040	−1.16260
$17$	−7.11446	−1.72551	−0.862755	−	0.505622i	$-0.831263\pi$
$17$	−7.11446	−1.72551	−0.862755	+	0.505622i	$0.831263\pi$
$18$	−0.966935	−0.227909
$19$	−5.99670	−1.37574	−0.687868	−	0.725836i	$-0.741453\pi$
$19$	−5.99670	−1.37574	−0.687868	+	0.725836i	$0.741453\pi$
$20$	−4.96786	−1.11085
$21$	0	0
$22$	−4.69447	−1.00086
$23$	9.03548	1.88403	0.942014	−	0.335574i	$-0.108930\pi$
$23$	9.03548	1.88403	0.942014	+	0.335574i	$0.108930\pi$
$24$	−1.22218	−0.249477
$25$	4.74649	0.949299
$26$	6.79985	1.33356
$27$	−5.53880	−1.06594
$28$	0	0
$29$	9.03999	1.67868	0.839342	−	0.543604i	$-0.182941\pi$
$29$	9.03999	1.67868	0.839342	+	0.543604i	$0.182941\pi$
$30$	−9.33528	−1.70438
$31$	−8.92613	−1.60318	−0.801590	−	0.597874i	$-0.796012\pi$
$31$	−8.92613	−1.60318	−0.801590	+	0.597874i	$0.796012\pi$
$32$	−7.26368	−1.28405
$33$	−3.90878	−0.680431
$34$	−13.4824	−2.31221
$35$	0	0
$36$	−0.811929	−0.135321
$37$	−3.61418	−0.594168	−0.297084	−	0.954851i	$-0.596014\pi$
$37$	−3.61418	−0.594168	−0.297084	+	0.954851i	$0.596014\pi$
$38$	−11.3641	−1.84351
$39$	5.66180	0.906613
$40$	2.41814	0.382341
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−8.37070	−1.27652	−0.638260	−	0.769821i	$-0.720346\pi$
$43$	−8.37070	−1.27652	−0.638260	+	0.769821i	$0.720346\pi$
$44$	−3.94191	−0.594266
$45$	1.59293	0.237460
$46$	17.1228	2.52462
$47$	−1.53886	−0.224465	−0.112233	−	0.993682i	$-0.535800\pi$
$47$	−1.53886	−0.224465	−0.112233	+	0.993682i	$0.535800\pi$
$48$	−7.33785	−1.05913
$49$	0	0
$50$	8.99492	1.27207
$51$	−11.2259	−1.57194
$52$	5.70979	0.791805
$53$	−0.436569	−0.0599673	−0.0299837	−	0.999550i	$-0.509546\pi$
$53$	−0.436569	−0.0599673	−0.0299837	+	0.999550i	$0.509546\pi$
$54$	−10.4964	−1.42838
$55$	7.73369	1.04281
$56$	0	0
$57$	−9.46217	−1.25330
$58$	17.1314	2.24946
$59$	−9.21554	−1.19976	−0.599880	−	0.800090i	$-0.704785\pi$
$59$	−9.21554	−1.19976	−0.599880	+	0.800090i	$0.704785\pi$
$60$	−7.83877	−1.01198
$61$	11.5476	1.47851	0.739257	−	0.673423i	$-0.235177\pi$
$61$	11.5476	1.47851	0.739257	+	0.673423i	$0.235177\pi$
$62$	−16.9156	−2.14828
$63$	0	0
$64$	−4.46435	−0.558044
$65$	−11.2021	−1.38945
$66$	−7.40739	−0.911787
$67$	2.71201	0.331325	0.165662	−	0.986183i	$-0.447024\pi$
$67$	2.71201	0.331325	0.165662	+	0.986183i	$0.447024\pi$
$68$	−11.3211	−1.37288
$69$	14.2571	1.71635
$70$	0	0
$71$	4.30671	0.511113	0.255556	−	0.966794i	$-0.417741\pi$
$71$	4.30671	0.511113	0.255556	+	0.966794i	$0.417741\pi$
$72$	0.395212	0.0465762
$73$	6.16745	0.721846	0.360923	−	0.932596i	$-0.382462\pi$
$73$	6.16745	0.721846	0.360923	+	0.932596i	$0.382462\pi$
$74$	−6.84911	−0.796193
$75$	7.48948	0.864811
$76$	−9.54238	−1.09459
$77$	0	0
$78$	10.7295	1.21487
$79$	0.388867	0.0437510	0.0218755	−	0.999761i	$-0.493036\pi$
$79$	0.388867	0.0437510	0.0218755	+	0.999761i	$0.493036\pi$
$80$	14.5182	1.62319
$81$	−7.20894	−0.800993
$82$	−1.89507	−0.209275
$83$	7.05134	0.773985	0.386993	−	0.922083i	$-0.373514\pi$
$83$	7.05134	0.773985	0.386993	+	0.922083i	$0.373514\pi$
$84$	0	0
$85$	22.2109	2.40911
$86$	−15.8630	−1.71055
$87$	14.2642	1.52928
$88$	1.91875	0.204540
$89$	−11.6241	−1.23215	−0.616074	−	0.787688i	$-0.711278\pi$
$89$	−11.6241	−1.23215	−0.616074	+	0.787688i	$0.711278\pi$
$90$	3.01871	0.318200
$91$	0	0
$92$	14.3779	1.49900
$93$	−14.0845	−1.46050
$94$	−2.91623	−0.300787
$95$	18.7213	1.92077
$96$	−11.4613	−1.16977
$97$	10.4689	1.06295	0.531476	−	0.847073i	$-0.321638\pi$
$97$	10.4689	1.06295	0.531476	+	0.847073i	$0.321638\pi$
$98$	0	0
$99$	1.26397	0.127033
$100$	7.55297	0.755297
$101$	9.43867	0.939183	0.469591	−	0.882884i	$-0.344401\pi$
$101$	9.43867	0.939183	0.469591	+	0.882884i	$0.344401\pi$
$102$	−21.2738	−2.10642
$103$	−7.20731	−0.710157	−0.355079	−	0.934836i	$-0.615546\pi$
$103$	−7.20731	−0.710157	−0.355079	+	0.934836i	$0.615546\pi$
$104$	−2.77928	−0.272531
$105$	0	0
$106$	−0.827326	−0.0803570
$107$	−7.17570	−0.693702	−0.346851	−	0.937920i	$-0.612749\pi$
$107$	−7.17570	−0.693702	−0.346851	+	0.937920i	$0.612749\pi$
$108$	−8.81374	−0.848103
$109$	−12.8300	−1.22889	−0.614447	−	0.788958i	$-0.710621\pi$
$109$	−12.8300	−1.22889	−0.614447	+	0.788958i	$0.710621\pi$
$110$	14.6558	1.39738
$111$	−5.70281	−0.541287
$112$	0	0
$113$	15.8789	1.49376	0.746879	−	0.664960i	$-0.231552\pi$
$113$	15.8789	1.49376	0.746879	+	0.664960i	$0.231552\pi$
$114$	−17.9314	−1.67943
$115$	−28.2082	−2.63043
$116$	14.3851	1.33562
$117$	−1.83083	−0.169260
$118$	−17.4640	−1.60770
$119$	0	0
$120$	3.81558	0.348313
$121$	−4.86345	−0.442131
$122$	21.8834	1.98123
$123$	−1.57790	−0.142274
$124$	−14.2039	−1.27555
$125$	0.791427	0.0707874
$126$	0	0
$127$	−19.7032	−1.74838	−0.874189	−	0.485585i	$-0.838607\pi$
$127$	−19.7032	−1.74838	−0.874189	+	0.485585i	$0.838607\pi$
$128$	6.06711	0.536262
$129$	−13.2081	−1.16291
$130$	−21.2287	−1.86188
$131$	−8.98260	−0.784814	−0.392407	−	0.919792i	$-0.628357\pi$
$131$	−8.98260	−0.784814	−0.392407	+	0.919792i	$0.628357\pi$
$132$	−6.21994	−0.541376
$133$	0	0
$134$	5.13943	0.443979
$135$	17.2918	1.48824
$136$	5.51060	0.472530
$137$	8.94745	0.764432	0.382216	−	0.924073i	$-0.375161\pi$
$137$	8.94745	0.764432	0.382216	+	0.924073i	$0.375161\pi$
$138$	27.0181	2.29993
$139$	4.85149	0.411498	0.205749	−	0.978605i	$-0.434037\pi$
$139$	4.85149	0.411498	0.205749	+	0.978605i	$0.434037\pi$
$140$	0	0
$141$	−2.42816	−0.204488
$142$	8.16150	0.684898
$143$	−8.88869	−0.743309
$144$	2.37281	0.197734
$145$	−28.2223	−2.34373
$146$	11.6877	0.967283
$147$	0	0
$148$	−5.75115	−0.472742
$149$	−1.14539	−0.0938337	−0.0469168	−	0.998899i	$-0.514940\pi$
$149$	−1.14539	−0.0938337	−0.0469168	+	0.998899i	$0.514940\pi$
$150$	14.1931	1.15886
$151$	14.5501	1.18407	0.592035	−	0.805912i	$-0.298325\pi$
$151$	14.5501	1.18407	0.592035	+	0.805912i	$0.298325\pi$
$152$	4.64482	0.376745
$153$	3.63007	0.293474
$154$	0	0
$155$	27.8668	2.23832
$156$	9.00947	0.721335
$157$	−6.23465	−0.497579	−0.248789	−	0.968558i	$-0.580033\pi$
$157$	−6.23465	−0.497579	−0.248789	+	0.968558i	$0.580033\pi$
$158$	0.736929	0.0586269
$159$	−0.688861	−0.0546302
$160$	22.6767	1.79275
$161$	0	0
$162$	−13.6614	−1.07334
$163$	8.73385	0.684088	0.342044	−	0.939684i	$-0.388881\pi$
$163$	8.73385	0.684088	0.342044	+	0.939684i	$0.388881\pi$
$164$	−1.59127	−0.124258
$165$	12.2030	0.950000
$166$	13.3628	1.03715
$167$	−7.37954	−0.571046	−0.285523	−	0.958372i	$-0.592167\pi$
$167$	−7.37954	−0.571046	−0.285523	+	0.958372i	$0.592167\pi$
$168$	0	0
$169$	−0.124898	−0.00960750
$170$	42.0911	3.22824
$171$	3.05975	0.233985
$172$	−13.3201	−1.01565
$173$	21.1685	1.60941	0.804705	−	0.593675i	$-0.202324\pi$
$173$	21.1685	1.60941	0.804705	+	0.593675i	$0.202324\pi$
$174$	27.0315	2.04926
$175$	0	0
$176$	11.5200	0.868352
$177$	−14.5412	−1.09298
$178$	−22.0284	−1.65110
$179$	−5.13375	−0.383715	−0.191857	−	0.981423i	$-0.561451\pi$
$179$	−5.13375	−0.383715	−0.191857	+	0.981423i	$0.561451\pi$
$180$	2.53479	0.188932
$181$	−4.56104	−0.339019	−0.169510	−	0.985529i	$-0.554218\pi$
$181$	−4.56104	−0.339019	−0.169510	+	0.985529i	$0.554218\pi$
$182$	0	0
$183$	18.2209	1.34693
$184$	−6.99855	−0.515940
$185$	11.2833	0.829561
$186$	−26.6911	−1.95708
$187$	17.6240	1.28879
$188$	−2.44874	−0.178593
$189$	0	0
$190$	35.4781	2.57385
$191$	3.47482	0.251429	0.125715	−	0.992066i	$-0.459878\pi$
$191$	3.47482	0.251429	0.125715	+	0.992066i	$0.459878\pi$
$192$	−7.04429	−0.508378
$193$	−16.2855	−1.17225	−0.586127	−	0.810219i	$-0.699348\pi$
$193$	−16.2855	−1.17225	−0.586127	+	0.810219i	$0.699348\pi$
$194$	19.8392	1.42437
$195$	−17.6758	−1.26579
$196$	0	0
$197$	−6.44475	−0.459169	−0.229585	−	0.973289i	$-0.573737\pi$
$197$	−6.44475	−0.459169	−0.229585	+	0.973289i	$0.573737\pi$
$198$	2.39530	0.170226
$199$	−1.35506	−0.0960575	−0.0480287	−	0.998846i	$-0.515294\pi$
$199$	−1.35506	−0.0960575	−0.0480287	+	0.998846i	$0.515294\pi$
$200$	−3.67646	−0.259965
$201$	4.27927	0.301837
$202$	17.8869	1.25852
$203$	0	0
$204$	−17.8635	−1.25069
$205$	3.12194	0.218046
$206$	−13.6583	−0.951621
$207$	−4.61025	−0.320434
$208$	−16.6865	−1.15700
$209$	14.8551	1.02755
$210$	0	0
$211$	−19.0138	−1.30896	−0.654482	−	0.756078i	$-0.727113\pi$
$211$	−19.0138	−1.30896	−0.654482	+	0.756078i	$0.727113\pi$
$212$	−0.694700	−0.0477122
$213$	6.79555	0.465623
$214$	−13.5984	−0.929570
$215$	26.1328	1.78224
$216$	4.29015	0.291908
$217$	0	0
$218$	−24.3138	−1.64674
$219$	9.73161	0.657601
$220$	12.3064	0.829698
$221$	−25.5280	−1.71720
$222$	−10.8072	−0.725332
$223$	−2.62540	−0.175810	−0.0879049	−	0.996129i	$-0.528017\pi$
$223$	−2.62540	−0.175810	−0.0879049	+	0.996129i	$0.528017\pi$
$224$	0	0
$225$	−2.42184	−0.161456
$226$	30.0915	2.00166
$227$	−7.07709	−0.469723	−0.234862	−	0.972029i	$-0.575464\pi$
$227$	−7.07709	−0.469723	−0.234862	+	0.972029i	$0.575464\pi$
$228$	−15.0569	−0.997168
$229$	−6.78303	−0.448235	−0.224118	−	0.974562i	$-0.571950\pi$
$229$	−6.78303	−0.448235	−0.224118	+	0.974562i	$0.571950\pi$
$230$	−53.4564	−3.52481
$231$	0	0
$232$	−7.00204	−0.459707
$233$	−7.98179	−0.522904	−0.261452	−	0.965216i	$-0.584201\pi$
$233$	−7.98179	−0.522904	−0.261452	+	0.965216i	$0.584201\pi$
$234$	−3.46955	−0.226811
$235$	4.80421	0.313392
$236$	−14.6644	−0.954574
$237$	0.613593	0.0398572
$238$	0	0
$239$	13.2781	0.858887	0.429444	−	0.903094i	$-0.358710\pi$
$239$	13.2781	0.858887	0.429444	+	0.903094i	$0.358710\pi$
$240$	22.9083	1.47872
$241$	9.73257	0.626930	0.313465	−	0.949600i	$-0.398510\pi$
$241$	9.73257	0.626930	0.313465	+	0.949600i	$0.398510\pi$
$242$	−9.21655	−0.592462
$243$	5.24142	0.336237
$244$	18.3753	1.17636
$245$	0	0
$246$	−2.99022	−0.190649
$247$	−21.5173	−1.36911
$248$	6.91385	0.439030
$249$	11.1263	0.705100
$250$	1.49981	0.0948561
$251$	10.0364	0.633492	0.316746	−	0.948510i	$-0.397410\pi$
$251$	10.0364	0.633492	0.316746	+	0.948510i	$0.397410\pi$
$252$	0	0
$253$	−22.3828	−1.40719
$254$	−37.3389	−2.34285
$255$	35.0465	2.19470
$256$	20.4263	1.27664
$257$	−23.3460	−1.45628	−0.728142	−	0.685426i	$-0.759616\pi$
$257$	−23.3460	−1.45628	−0.728142	+	0.685426i	$0.759616\pi$
$258$	−25.0302	−1.55831
$259$	0	0
$260$	−17.8256	−1.10550
$261$	−4.61255	−0.285510
$262$	−17.0226	−1.05166
$263$	7.46153	0.460098	0.230049	−	0.973179i	$-0.426111\pi$
$263$	7.46153	0.460098	0.230049	+	0.973179i	$0.426111\pi$
$264$	3.02760	0.186336
$265$	1.36294	0.0837247
$266$	0	0
$267$	−18.3416	−1.12249
$268$	4.31555	0.263614
$269$	−2.41427	−0.147200	−0.0736002	−	0.997288i	$-0.523449\pi$
$269$	−2.41427	−0.147200	−0.0736002	+	0.997288i	$0.523449\pi$
$270$	32.7691	1.99426
$271$	−17.7171	−1.07624	−0.538118	−	0.842870i	$-0.680864\pi$
$271$	−17.7171	−1.07624	−0.538118	+	0.842870i	$0.680864\pi$
$272$	33.0851	2.00608
$273$	0	0
$274$	16.9560	1.02435
$275$	−11.7580	−0.709037
$276$	22.6869	1.36559
$277$	13.2764	0.797703	0.398852	−	0.917015i	$-0.369409\pi$
$277$	13.2764	0.797703	0.398852	+	0.917015i	$0.369409\pi$
$278$	9.19388	0.551413
$279$	4.55445	0.272668
$280$	0	0
$281$	−5.58617	−0.333243	−0.166621	−	0.986021i	$-0.553286\pi$
$281$	−5.58617	−0.333243	−0.166621	+	0.986021i	$0.553286\pi$
$282$	−4.60152	−0.274017
$283$	−18.4441	−1.09639	−0.548194	−	0.836351i	$-0.684684\pi$
$283$	−18.4441	−1.09639	−0.548194	+	0.836351i	$0.684684\pi$
$284$	6.85315	0.406660
$285$	29.5403	1.74982
$286$	−16.8446	−0.996045
$287$	0	0
$288$	3.70621	0.218390
$289$	33.6156	1.97739
$290$	−53.4831	−3.14063
$291$	16.5188	0.968349
$292$	9.81410	0.574327
$293$	−25.2343	−1.47420	−0.737101	−	0.675783i	$-0.763806\pi$
$293$	−25.2343	−1.47420	−0.737101	+	0.675783i	$0.763806\pi$
$294$	0	0
$295$	28.7703	1.67507
$296$	2.79941	0.162713
$297$	13.7207	0.796159
$298$	−2.17058	−0.125738
$299$	32.4210	1.87496
$300$	11.9178	0.688075
$301$	0	0
$302$	27.5734	1.58667
$303$	14.8933	0.855595
$304$	27.8870	1.59943
$305$	−36.0508	−2.06426
$306$	6.87923	0.393259
$307$	6.32587	0.361037	0.180518	−	0.983572i	$-0.442222\pi$
$307$	6.32587	0.361037	0.180518	+	0.983572i	$0.442222\pi$
$308$	0	0
$309$	−11.3724	−0.646953
$310$	52.8094	2.99937
$311$	13.5024	0.765649	0.382824	−	0.923821i	$-0.374951\pi$
$311$	13.5024	0.765649	0.382824	+	0.923821i	$0.374951\pi$
$312$	−4.38542	−0.248276
$313$	−5.10715	−0.288673	−0.144337	−	0.989529i	$-0.546105\pi$
$313$	−5.10715	−0.288673	−0.144337	+	0.989529i	$0.546105\pi$
$314$	−11.8151	−0.666763
$315$	0	0
$316$	0.618794	0.0348099
$317$	10.1152	0.568123	0.284062	−	0.958806i	$-0.408318\pi$
$317$	10.1152	0.568123	0.284062	+	0.958806i	$0.408318\pi$
$318$	−1.30544	−0.0732052
$319$	−22.3939	−1.25382
$320$	13.9374	0.779126
$321$	−11.3225	−0.631962
$322$	0	0
$323$	42.6633	2.37385
$324$	−11.4714	−0.637300
$325$	17.0313	0.944728
$326$	16.5512	0.916687
$327$	−20.2445	−1.11952
$328$	0.774564	0.0427681
$329$	0	0
$330$	23.1254	1.27301
$331$	−18.3598	−1.00915	−0.504574	−	0.863369i	$-0.668350\pi$
$331$	−18.3598	−1.00915	−0.504574	+	0.863369i	$0.668350\pi$
$332$	11.2206	0.615811
$333$	1.84409	0.101056
$334$	−13.9847	−0.765209
$335$	−8.46672	−0.462586
$336$	0	0
$337$	−32.5700	−1.77420	−0.887099	−	0.461578i	$-0.847283\pi$
$337$	−32.5700	−1.77420	−0.887099	+	0.461578i	$0.847283\pi$
$338$	−0.236689	−0.0128742
$339$	25.0552	1.36081
$340$	35.3436	1.91678
$341$	22.1119	1.19742
$342$	5.79842	0.313543
$343$	0	0
$344$	6.48364	0.349575
$345$	−44.5097	−2.39632
$346$	40.1157	2.15663
$347$	27.9803	1.50206	0.751031	−	0.660267i	$-0.229557\pi$
$347$	27.9803	1.50206	0.751031	+	0.660267i	$0.229557\pi$
$348$	22.6982	1.21675
$349$	10.0218	0.536455	0.268228	−	0.963356i	$-0.413562\pi$
$349$	10.0218	0.536455	0.268228	+	0.963356i	$0.413562\pi$
$350$	0	0
$351$	−19.8743	−1.06081
$352$	17.9936	0.959064
$353$	−3.92033	−0.208658	−0.104329	−	0.994543i	$-0.533269\pi$
$353$	−3.92033	−0.208658	−0.104329	+	0.994543i	$0.533269\pi$
$354$	−27.5565	−1.46461
$355$	−13.4453	−0.713602
$356$	−18.4971	−0.980343
$357$	0	0
$358$	−9.72879	−0.514183
$359$	−19.1431	−1.01034	−0.505168	−	0.863021i	$-0.668569\pi$
$359$	−19.1431	−1.01034	−0.505168	+	0.863021i	$0.668569\pi$
$360$	−1.23383	−0.0650284
$361$	16.9604	0.892651
$362$	−8.64347	−0.454291
$363$	−7.67402	−0.402782
$364$	0	0
$365$	−19.2544	−1.00782
$366$	34.5298	1.80490
$367$	−12.5790	−0.656616	−0.328308	−	0.944571i	$-0.606478\pi$
$367$	−12.5790	−0.656616	−0.328308	+	0.944571i	$0.606478\pi$
$368$	−42.0186	−2.19037
$369$	0.510238	0.0265620
$370$	21.3825	1.11162
$371$	0	0
$372$	−22.4123	−1.16202
$373$	−35.4026	−1.83308	−0.916540	−	0.399943i	$-0.869030\pi$
$373$	−35.4026	−1.83308	−0.916540	+	0.399943i	$0.869030\pi$
$374$	33.3986	1.72700
$375$	1.24879	0.0644873
$376$	1.19194	0.0614697
$377$	32.4372	1.67060
$378$	0	0
$379$	−18.1951	−0.934619	−0.467310	−	0.884094i	$-0.654777\pi$
$379$	−18.1951	−0.934619	−0.467310	+	0.884094i	$0.654777\pi$
$380$	29.7907	1.52823
$381$	−31.0897	−1.59277
$382$	6.58501	0.336919
$383$	−33.3543	−1.70432	−0.852162	−	0.523277i	$-0.824709\pi$
$383$	−33.3543	−1.70432	−0.852162	+	0.523277i	$0.824709\pi$
$384$	9.57328	0.488535
$385$	0	0
$386$	−30.8620	−1.57084
$387$	4.27105	0.217110
$388$	16.6588	0.845724
$389$	4.26120	0.216052	0.108026	−	0.994148i	$-0.465547\pi$
$389$	4.26120	0.216052	0.108026	+	0.994148i	$0.465547\pi$
$390$	−33.4968	−1.69617
$391$	−64.2826	−3.25091
$392$	0	0
$393$	−14.1736	−0.714965
$394$	−12.2132	−0.615293
$395$	−1.21402	−0.0610840
$396$	2.01132	0.101072
$397$	2.13534	0.107170	0.0535849	−	0.998563i	$-0.482935\pi$
$397$	2.13534	0.107170	0.0535849	+	0.998563i	$0.482935\pi$
$398$	−2.56792	−0.128718
$399$	0	0
$400$	−22.0731	−1.10365
$401$	−3.75522	−0.187527	−0.0937634	−	0.995595i	$-0.529890\pi$
$401$	−3.75522	−0.187527	−0.0937634	+	0.995595i	$0.529890\pi$
$402$	8.10950	0.404465
$403$	−32.0286	−1.59546
$404$	15.0195	0.747248
$405$	22.5059	1.11833
$406$	0	0
$407$	8.95308	0.443788
$408$	8.69517	0.430475
$409$	17.3950	0.860127	0.430063	−	0.902799i	$-0.358491\pi$
$409$	17.3950	0.860127	0.430063	+	0.902799i	$0.358491\pi$
$410$	5.91628	0.292184
$411$	14.1182	0.696398
$412$	−11.4688	−0.565027
$413$	0	0
$414$	−8.73672	−0.429387
$415$	−22.0138	−1.08062
$416$	−26.0634	−1.27787
$417$	7.65515	0.374874
$418$	28.1513	1.37693
$419$	11.4568	0.559701	0.279850	−	0.960044i	$-0.409715\pi$
$419$	11.4568	0.559701	0.279850	+	0.960044i	$0.409715\pi$
$420$	0	0
$421$	−0.481034	−0.0234441	−0.0117221	−	0.999931i	$-0.503731\pi$
$421$	−0.481034	−0.0234441	−0.0117221	+	0.999931i	$0.503731\pi$
$422$	−36.0324	−1.75403
$423$	0.785184	0.0381769
$424$	0.338150	0.0164220
$425$	−33.7688	−1.63803
$426$	12.8780	0.623942
$427$	0	0
$428$	−11.4185	−0.551934
$429$	−14.0254	−0.677155
$430$	49.5234	2.38823
$431$	5.92032	0.285172	0.142586	−	0.989782i	$-0.454458\pi$
$431$	5.92032	0.285172	0.142586	+	0.989782i	$0.454458\pi$
$432$	25.7576	1.23926
$433$	6.92900	0.332987	0.166493	−	0.986043i	$-0.446756\pi$
$433$	6.92900	0.332987	0.166493	+	0.986043i	$0.446756\pi$
$434$	0	0
$435$	−44.5319	−2.13514
$436$	−20.4161	−0.977754
$437$	−54.1830	−2.59193
$438$	18.4420	0.881195
$439$	27.3202	1.30392	0.651960	−	0.758253i	$-0.273947\pi$
$439$	27.3202	1.30392	0.651960	+	0.758253i	$0.273947\pi$
$440$	−5.99023	−0.285573
$441$	0	0
$442$	−48.3773	−2.30107
$443$	32.2590	1.53267	0.766335	−	0.642441i	$-0.222078\pi$
$443$	32.2590	1.53267	0.766335	+	0.642441i	$0.222078\pi$
$444$	−9.07473	−0.430668
$445$	36.2896	1.72029
$446$	−4.97531	−0.235588
$447$	−1.80730	−0.0854824
$448$	0	0
$449$	−17.0930	−0.806670	−0.403335	−	0.915052i	$-0.632149\pi$
$449$	−17.0930	−0.806670	−0.403335	+	0.915052i	$0.632149\pi$
$450$	−4.58955	−0.216354
$451$	2.47721	0.116647
$452$	25.2676	1.18849
$453$	22.9586	1.07869
$454$	−13.4116	−0.629435
$455$	0	0
$456$	7.32906	0.343214
$457$	−13.2781	−0.621125	−0.310563	−	0.950553i	$-0.600517\pi$
$457$	−13.2781	−0.621125	−0.310563	+	0.950553i	$0.600517\pi$
$458$	−12.8543	−0.600641
$459$	39.4056	1.83929
$460$	−44.8870	−2.09287
$461$	10.7809	0.502115	0.251058	−	0.967972i	$-0.419222\pi$
$461$	10.7809	0.502115	0.251058	+	0.967972i	$0.419222\pi$
$462$	0	0
$463$	−25.7876	−1.19845	−0.599226	−	0.800580i	$-0.704525\pi$
$463$	−25.7876	−1.19845	−0.599226	+	0.800580i	$0.704525\pi$
$464$	−42.0395	−1.95164
$465$	43.9710	2.03911
$466$	−15.1260	−0.700699
$467$	−24.5678	−1.13686	−0.568430	−	0.822731i	$-0.692449\pi$
$467$	−24.5678	−1.13686	−0.568430	+	0.822731i	$0.692449\pi$
$468$	−2.91335	−0.134670
$469$	0	0
$470$	9.10430	0.419950
$471$	−9.83763	−0.453294
$472$	7.13802	0.328554
$473$	20.7360	0.953441
$474$	1.16280	0.0534091
$475$	−28.4633	−1.30599
$476$	0	0
$477$	0.222754	0.0101992
$478$	25.1628	1.15092
$479$	−10.6687	−0.487464	−0.243732	−	0.969843i	$-0.578372\pi$
$479$	−10.6687	−0.487464	−0.243732	+	0.969843i	$0.578372\pi$
$480$	35.7816	1.63320
$481$	−12.9684	−0.591307
$482$	18.4439	0.840095
$483$	0	0
$484$	−7.73907	−0.351776
$485$	−32.6831	−1.48406
$486$	9.93283	0.450563
$487$	−7.74207	−0.350827	−0.175413	−	0.984495i	$-0.556126\pi$
$487$	−7.74207	−0.350827	−0.175413	+	0.984495i	$0.556126\pi$
$488$	−8.94432	−0.404890
$489$	13.7811	0.623204
$490$	0	0
$491$	−2.97531	−0.134274	−0.0671370	−	0.997744i	$-0.521386\pi$
$491$	−2.97531	−0.134274	−0.0671370	+	0.997744i	$0.521386\pi$
$492$	−2.51087	−0.113199
$493$	−64.3146	−2.89659
$494$	−40.7767	−1.83463
$495$	−3.94602	−0.177361
$496$	41.5100	1.86385
$497$	0	0
$498$	21.0851	0.944844
$499$	1.45634	0.0651947	0.0325973	−	0.999469i	$-0.489622\pi$
$499$	1.45634	0.0651947	0.0325973	+	0.999469i	$0.489622\pi$
$500$	1.25938	0.0563211
$501$	−11.6442	−0.520223
$502$	19.0196	0.848887
$503$	37.5010	1.67209	0.836043	−	0.548664i	$-0.184863\pi$
$503$	37.5010	1.67209	0.836043	+	0.548664i	$0.184863\pi$
$504$	0	0
$505$	−29.4669	−1.31126
$506$	−42.4168	−1.88566
$507$	−0.197076	−0.00875243
$508$	−31.3532	−1.39107
$509$	−6.21143	−0.275317	−0.137658	−	0.990480i	$-0.543958\pi$
$509$	−6.21143	−0.275317	−0.137658	+	0.990480i	$0.543958\pi$
$510$	66.4155	2.94093
$511$	0	0
$512$	26.5749	1.17446
$513$	33.2145	1.46646
$514$	−44.2422	−1.95144
$515$	22.5008	0.991503
$516$	−21.0177	−0.925254
$517$	3.81207	0.167655
$518$	0	0
$519$	33.4017	1.46617
$520$	8.67674	0.380500
$521$	−33.0896	−1.44968	−0.724841	−	0.688917i	$-0.758087\pi$
$521$	−33.0896	−1.44968	−0.724841	+	0.688917i	$0.758087\pi$
$522$	−8.74108	−0.382587
$523$	29.5269	1.29112	0.645560	−	0.763710i	$-0.276624\pi$
$523$	29.5269	1.29112	0.645560	+	0.763710i	$0.276624\pi$
$524$	−14.2938	−0.624427
$525$	0	0
$526$	14.1401	0.616537
$527$	63.5046	2.76630
$528$	18.1774	0.791069
$529$	58.6399	2.54956
$530$	2.58286	0.112192
$531$	4.70212	0.204055
$532$	0	0
$533$	−3.58819	−0.155422
$534$	−34.7585	−1.50415
$535$	22.4021	0.968527
$536$	−2.10062	−0.0907331
$537$	−8.10053	−0.349564
$538$	−4.57519	−0.197251
$539$	0	0
$540$	27.5160	1.18410
$541$	−29.5888	−1.27212	−0.636060	−	0.771639i	$-0.719437\pi$
$541$	−29.5888	−1.27212	−0.636060	+	0.771639i	$0.719437\pi$
$542$	−33.5750	−1.44217
$543$	−7.19686	−0.308847
$544$	51.6772	2.21564
$545$	40.0546	1.71575
$546$	0	0
$547$	−37.8516	−1.61842	−0.809208	−	0.587522i	$-0.800103\pi$
$547$	−37.8516	−1.61842	−0.809208	+	0.587522i	$0.800103\pi$
$548$	14.2378	0.608210
$549$	−5.89201	−0.251465
$550$	−22.2823	−0.950119
$551$	−54.2101	−2.30943
$552$	−11.0430	−0.470021
$553$	0	0
$554$	25.1597	1.06893
$555$	17.8038	0.755730
$556$	7.72004	0.327403
$557$	−44.0069	−1.86463	−0.932315	−	0.361647i	$-0.882215\pi$
$557$	−44.0069	−1.86463	−0.932315	+	0.361647i	$0.882215\pi$
$558$	8.63099	0.365379
$559$	−30.0357	−1.27037
$560$	0	0
$561$	27.8089	1.17409
$562$	−10.5862	−0.446550
$563$	−23.5580	−0.992852	−0.496426	−	0.868079i	$-0.665355\pi$
$563$	−23.5580	−0.992852	−0.496426	+	0.868079i	$0.665355\pi$
$564$	−3.86386	−0.162698
$565$	−49.5728	−2.08554
$566$	−34.9528	−1.46917
$567$	0	0
$568$	−3.33582	−0.139968
$569$	31.6342	1.32618	0.663088	−	0.748542i	$-0.269246\pi$
$569$	31.6342	1.32618	0.663088	+	0.748542i	$0.269246\pi$
$570$	55.9808	2.34478
$571$	8.77077	0.367045	0.183523	−	0.983015i	$-0.441250\pi$
$571$	8.77077	0.367045	0.183523	+	0.983015i	$0.441250\pi$
$572$	−14.1443	−0.591404
$573$	5.48291	0.229052
$574$	0	0
$575$	42.8869	1.78851
$576$	2.27788	0.0949119
$577$	29.3406	1.22147	0.610733	−	0.791837i	$-0.290875\pi$
$577$	29.3406	1.22147	0.610733	+	0.791837i	$0.290875\pi$
$578$	63.7037	2.64973
$579$	−25.6968	−1.06792
$580$	−44.9094	−1.86476
$581$	0	0
$582$	31.3042	1.29760
$583$	1.08147	0.0447899
$584$	−4.77708	−0.197677
$585$	5.71574	0.236317
$586$	−47.8206	−1.97545
$587$	0.497739	0.0205439	0.0102719	−	0.999947i	$-0.496730\pi$
$587$	0.497739	0.0205439	0.0102719	+	0.999947i	$0.496730\pi$
$588$	0	0
$589$	53.5273	2.20555
$590$	54.5217	2.24462
$591$	−10.1692	−0.418303
$592$	16.8074	0.690779
$593$	15.2791	0.627437	0.313719	−	0.949516i	$-0.398425\pi$
$593$	15.2791	0.627437	0.313719	+	0.949516i	$0.398425\pi$
$594$	26.0017	1.06686
$595$	0	0
$596$	−1.82262	−0.0746575
$597$	−2.13814	−0.0875083
$598$	61.4399	2.51247
$599$	34.9568	1.42830	0.714148	−	0.699995i	$-0.246814\pi$
$599$	34.9568	1.42830	0.714148	+	0.699995i	$0.246814\pi$
$600$	−5.80108	−0.236828
$601$	−45.0759	−1.83868	−0.919342	−	0.393459i	$-0.871278\pi$
$601$	−45.0759	−1.83868	−0.919342	+	0.393459i	$0.871278\pi$
$602$	0	0
$603$	−1.38377	−0.0563515
$604$	23.1532	0.942089
$605$	15.1834	0.617292
$606$	28.2237	1.14651
$607$	−6.55822	−0.266190	−0.133095	−	0.991103i	$-0.542492\pi$
$607$	−6.55822	−0.266190	−0.133095	+	0.991103i	$0.542492\pi$
$608$	43.5581	1.76651
$609$	0	0
$610$	−68.3186	−2.76614
$611$	−5.52171	−0.223384
$612$	5.77644	0.233499
$613$	−10.4603	−0.422489	−0.211245	−	0.977433i	$-0.567752\pi$
$613$	−10.4603	−0.422489	−0.211245	+	0.977433i	$0.567752\pi$
$614$	11.9879	0.483794
$615$	4.92610	0.198639
$616$	0	0
$617$	−13.1861	−0.530852	−0.265426	−	0.964131i	$-0.585513\pi$
$617$	−13.1861	−0.530852	−0.265426	+	0.964131i	$0.585513\pi$
$618$	−21.5514	−0.866926
$619$	−23.2046	−0.932670	−0.466335	−	0.884608i	$-0.654426\pi$
$619$	−23.2046	−0.932670	−0.466335	+	0.884608i	$0.654426\pi$
$620$	44.3437	1.78089
$621$	−50.0457	−2.00826
$622$	25.5879	1.02598
$623$	0	0
$624$	−26.3296	−1.05403
$625$	−26.2033	−1.04813
$626$	−9.67839	−0.386826
$627$	23.4398	0.936094
$628$	−9.92103	−0.395892
$629$	25.7130	1.02524
$630$	0	0
$631$	−25.7919	−1.02676	−0.513379	−	0.858162i	$-0.671607\pi$
$631$	−25.7919	−1.02676	−0.513379	+	0.858162i	$0.671607\pi$
$632$	−0.301203	−0.0119812
$633$	−30.0018	−1.19247
$634$	19.1689	0.761293
$635$	61.5122	2.44104
$636$	−1.09617	−0.0434658
$637$	0	0
$638$	−42.4379	−1.68013
$639$	−2.19745	−0.0869298
$640$	−18.9411	−0.748715
$641$	−18.3270	−0.723873	−0.361936	−	0.932203i	$-0.617884\pi$
$641$	−18.3270	−0.723873	−0.361936	+	0.932203i	$0.617884\pi$
$642$	−21.4569	−0.846838
$643$	32.9938	1.30115	0.650575	−	0.759442i	$-0.274528\pi$
$643$	32.9938	1.30115	0.650575	+	0.759442i	$0.274528\pi$
$644$	0	0
$645$	41.2349	1.62362
$646$	80.8497	3.18099
$647$	23.7648	0.934290	0.467145	−	0.884181i	$-0.345283\pi$
$647$	23.7648	0.934290	0.467145	+	0.884181i	$0.345283\pi$
$648$	5.58378	0.219352
$649$	22.8288	0.896109
$650$	32.2755	1.26595
$651$	0	0
$652$	13.8979	0.544285
$653$	2.32882	0.0911336	0.0455668	−	0.998961i	$-0.485491\pi$
$653$	2.32882	0.0911336	0.0455668	+	0.998961i	$0.485491\pi$
$654$	−38.3646	−1.50018
$655$	28.0431	1.09574
$656$	4.65040	0.181567
$657$	−3.14687	−0.122771
$658$	0	0
$659$	−10.0113	−0.389985	−0.194992	−	0.980805i	$-0.562468\pi$
$659$	−10.0113	−0.389985	−0.194992	+	0.980805i	$0.562468\pi$
$660$	19.4183	0.755855
$661$	22.8365	0.888235	0.444118	−	0.895969i	$-0.353517\pi$
$661$	22.8365	0.888235	0.444118	+	0.895969i	$0.353517\pi$
$662$	−34.7931	−1.35227
$663$	−40.2806	−1.56437
$664$	−5.46171	−0.211956
$665$	0	0
$666$	3.49468	0.135416
$667$	81.6806	3.16269
$668$	−11.7429	−0.454345
$669$	−4.14262	−0.160163
$670$	−16.0450	−0.619872
$671$	−28.6057	−1.10431
$672$	0	0
$673$	−6.59421	−0.254188	−0.127094	−	0.991891i	$-0.540565\pi$
$673$	−6.59421	−0.254188	−0.127094	+	0.991891i	$0.540565\pi$
$674$	−61.7222	−2.37745
$675$	−26.2899	−1.01190
$676$	−0.198746	−0.00764408
$677$	30.6496	1.17796	0.588980	−	0.808147i	$-0.299530\pi$
$677$	30.6496	1.17796	0.588980	+	0.808147i	$0.299530\pi$
$678$	47.4813	1.82351
$679$	0	0
$680$	−17.2038	−0.659734
$681$	−11.1669	−0.427918
$682$	41.9034	1.60457
$683$	0.229687	0.00878873	0.00439437	−	0.999990i	$-0.498601\pi$
$683$	0.229687	0.00878873	0.00439437	+	0.999990i	$0.498601\pi$
$684$	4.86889	0.186167
$685$	−27.9334	−1.06728
$686$	0	0
$687$	−10.7029	−0.408342
$688$	38.9271	1.48408
$689$	−1.56649	−0.0596785
$690$	−84.3487	−3.21110
$691$	1.61668	0.0615013	0.0307506	−	0.999527i	$-0.490210\pi$
$691$	1.61668	0.0615013	0.0307506	+	0.999527i	$0.490210\pi$
$692$	33.6848	1.28051
$693$	0	0
$694$	53.0246	2.01278
$695$	−15.1460	−0.574522
$696$	−11.0485	−0.418793
$697$	7.11446	0.269479
$698$	18.9920	0.718858
$699$	−12.5944	−0.476366
$700$	0	0
$701$	28.0273	1.05858	0.529288	−	0.848442i	$-0.322459\pi$
$701$	28.0273	1.05858	0.529288	+	0.848442i	$0.322459\pi$
$702$	−37.6630	−1.42150
$703$	21.6732	0.817418
$704$	11.0591	0.416807
$705$	7.58056	0.285500
$706$	−7.42928	−0.279605
$707$	0	0
$708$	−23.1390	−0.869616
$709$	48.9214	1.83728	0.918642	−	0.395091i	$-0.129287\pi$
$709$	48.9214	1.83728	0.918642	+	0.395091i	$0.129287\pi$
$710$	−25.4797	−0.956236
$711$	−0.198415	−0.00744115
$712$	9.00358	0.337423
$713$	−80.6518	−3.02043
$714$	0	0
$715$	27.7499	1.03779
$716$	−8.16920	−0.305297
$717$	20.9514	0.782446
$718$	−36.2775	−1.35386
$719$	32.5456	1.21375	0.606874	−	0.794798i	$-0.292423\pi$
$719$	32.5456	1.21375	0.606874	+	0.794798i	$0.292423\pi$
$720$	−7.40777	−0.276071
$721$	0	0
$722$	32.1410	1.19616
$723$	15.3570	0.571133
$724$	−7.25786	−0.269736
$725$	42.9082	1.59357
$726$	−14.5428	−0.539733
$727$	−24.4090	−0.905278	−0.452639	−	0.891694i	$-0.649517\pi$
$727$	−24.4090	−0.905278	−0.452639	+	0.891694i	$0.649517\pi$
$728$	0	0
$729$	29.8972	1.10731
$730$	−36.4883	−1.35049
$731$	59.5531	2.20265
$732$	28.9944	1.07166
$733$	−23.0042	−0.849680	−0.424840	−	0.905268i	$-0.639670\pi$
$733$	−23.0042	−0.849680	−0.424840	+	0.905268i	$0.639670\pi$
$734$	−23.8380	−0.879875
$735$	0	0
$736$	−65.6308	−2.41918
$737$	−6.71821	−0.247468
$738$	0.966935	0.0355934
$739$	6.28473	0.231188	0.115594	−	0.993297i	$-0.463123\pi$
$739$	6.28473	0.231188	0.115594	+	0.993297i	$0.463123\pi$
$740$	17.9547	0.660029
$741$	−33.9521	−1.24726
$742$	0	0
$743$	−0.355429	−0.0130394	−0.00651971	−	0.999979i	$-0.502075\pi$
$743$	−0.355429	−0.0130394	−0.00651971	+	0.999979i	$0.502075\pi$
$744$	10.9094	0.399956
$745$	3.57582	0.131008
$746$	−67.0903	−2.45635
$747$	−3.59787	−0.131639
$748$	28.0446	1.02541
$749$	0	0
$750$	2.36654	0.0864139
$751$	16.0914	0.587183	0.293592	−	0.955931i	$-0.405149\pi$
$751$	16.0914	0.587183	0.293592	+	0.955931i	$0.405149\pi$
$752$	7.15629	0.260963
$753$	15.8364	0.577111
$754$	61.4706	2.23863
$755$	−45.4245	−1.65317
$756$	0	0
$757$	−5.45992	−0.198444	−0.0992221	−	0.995065i	$-0.531635\pi$
$757$	−5.45992	−0.198444	−0.0992221	+	0.995065i	$0.531635\pi$
$758$	−34.4809	−1.25240
$759$	−35.3177	−1.28195
$760$	−14.5008	−0.526001
$761$	25.1724	0.912497	0.456249	−	0.889852i	$-0.349193\pi$
$761$	25.1724	0.912497	0.456249	+	0.889852i	$0.349193\pi$
$762$	−58.9170	−2.13434
$763$	0	0
$764$	5.52939	0.200046
$765$	−11.3329	−0.409740
$766$	−63.2086	−2.28382
$767$	−33.0671	−1.19398
$768$	32.2306	1.16302
$769$	14.8717	0.536286	0.268143	−	0.963379i	$-0.413590\pi$
$769$	14.8717	0.536286	0.268143	+	0.963379i	$0.413590\pi$
$770$	0	0
$771$	−36.8376	−1.32667
$772$	−25.9146	−0.932689
$773$	11.8929	0.427758	0.213879	−	0.976860i	$-0.431390\pi$
$773$	11.8929	0.427758	0.213879	+	0.976860i	$0.431390\pi$
$774$	8.09393	0.290930
$775$	−42.3678	−1.52190
$776$	−8.10880	−0.291089
$777$	0	0
$778$	8.07526	0.289512
$779$	5.99670	0.214854
$780$	−28.1270	−1.00711
$781$	−10.6686	−0.381753
$782$	−121.820	−4.35626
$783$	−50.0707	−1.78938
$784$	0	0
$785$	19.4642	0.694706
$786$	−26.8600	−0.958063
$787$	1.56890	0.0559254	0.0279627	−	0.999609i	$-0.491098\pi$
$787$	1.56890	0.0559254	0.0279627	+	0.999609i	$0.491098\pi$
$788$	−10.2554	−0.365332
$789$	11.7735	0.419149
$790$	−2.30065	−0.0818534
$791$	0	0
$792$	−0.979022	−0.0347880
$793$	41.4348	1.47139
$794$	4.04661	0.143609
$795$	2.15058	0.0762732
$796$	−2.15627	−0.0764269
$797$	−8.77673	−0.310888	−0.155444	−	0.987845i	$-0.549681\pi$
$797$	−8.77673	−0.310888	−0.155444	+	0.987845i	$0.549681\pi$
$798$	0	0
$799$	10.9481	0.387317
$800$	−34.4770	−1.21895
$801$	5.93105	0.209563
$802$	−7.11639	−0.251288
$803$	−15.2781	−0.539151
$804$	6.80949	0.240152
$805$	0	0
$806$	−60.6964	−2.13794
$807$	−3.80947	−0.134100
$808$	−7.31085	−0.257195
$809$	−16.4611	−0.578743	−0.289371	−	0.957217i	$-0.593446\pi$
$809$	−16.4611	−0.578743	−0.289371	+	0.957217i	$0.593446\pi$
$810$	42.6501	1.49857
$811$	19.6126	0.688692	0.344346	−	0.938843i	$-0.388101\pi$
$811$	19.6126	0.688692	0.344346	+	0.938843i	$0.388101\pi$
$812$	0	0
$813$	−27.9557	−0.980450
$814$	16.9667	0.594681
$815$	−27.2665	−0.955105
$816$	52.2049	1.82754
$817$	50.1966	1.75616
$818$	32.9646	1.15258
$819$	0	0
$820$	4.96786	0.173485
$821$	46.3721	1.61840	0.809199	−	0.587535i	$-0.199902\pi$
$821$	46.3721	1.61840	0.809199	+	0.587535i	$0.199902\pi$
$822$	26.7548	0.933182
$823$	−36.9837	−1.28917	−0.644586	−	0.764532i	$-0.722970\pi$
$823$	−36.9837	−1.28917	−0.644586	+	0.764532i	$0.722970\pi$
$824$	5.58252	0.194476
$825$	−18.5530	−0.645933
$826$	0	0
$827$	−7.16974	−0.249316	−0.124658	−	0.992200i	$-0.539783\pi$
$827$	−7.16974	−0.249316	−0.124658	+	0.992200i	$0.539783\pi$
$828$	−7.33617	−0.254949
$829$	10.5683	0.367053	0.183526	−	0.983015i	$-0.441249\pi$
$829$	10.5683	0.367053	0.183526	+	0.983015i	$0.441249\pi$
$830$	−41.7177	−1.44804
$831$	20.9488	0.726707
$832$	−16.0189	−0.555357
$833$	0	0
$834$	14.5070	0.502337
$835$	23.0385	0.797279
$836$	23.6385	0.817553
$837$	49.4400	1.70890
$838$	21.7114	0.750006
$839$	22.4068	0.773568	0.386784	−	0.922170i	$-0.373586\pi$
$839$	22.4068	0.773568	0.386784	+	0.922170i	$0.373586\pi$
$840$	0	0
$841$	52.7214	1.81798
$842$	−0.911590	−0.0314155
$843$	−8.81440	−0.303584
$844$	−30.2562	−1.04146
$845$	0.389922	0.0134137
$846$	1.48797	0.0511576
$847$	0	0
$848$	2.03022	0.0697179
$849$	−29.1029	−0.998808
$850$	−63.9940	−2.19498
$851$	−32.6559	−1.11943
$852$	10.8136	0.370467
$853$	−17.1768	−0.588122	−0.294061	−	0.955787i	$-0.595007\pi$
$853$	−17.1768	−0.588122	−0.294061	+	0.955787i	$0.595007\pi$
$854$	0	0
$855$	−9.55233	−0.326683
$856$	5.55804	0.189970
$857$	43.1901	1.47535	0.737673	−	0.675159i	$-0.235925\pi$
$857$	43.1901	1.47535	0.737673	+	0.675159i	$0.235925\pi$
$858$	−26.5791	−0.907397
$859$	−18.4603	−0.629858	−0.314929	−	0.949115i	$-0.601981\pi$
$859$	−18.4603	−0.629858	−0.314929	+	0.949115i	$0.601981\pi$
$860$	41.5845	1.41802
$861$	0	0
$862$	11.2194	0.382134
$863$	−11.8699	−0.404055	−0.202028	−	0.979380i	$-0.564753\pi$
$863$	−11.8699	−0.404055	−0.202028	+	0.979380i	$0.564753\pi$
$864$	40.2320	1.36872
$865$	−66.0867	−2.24701
$866$	13.1309	0.446207
$867$	53.0420	1.80140
$868$	0	0
$869$	−0.963305	−0.0326779
$870$	−84.3908	−2.86112
$871$	9.73120	0.329729
$872$	9.93768	0.336532
$873$	−5.34162	−0.180786
$874$	−102.680	−3.47322
$875$	0	0
$876$	15.4856	0.523212
$877$	−45.3383	−1.53097	−0.765483	−	0.643457i	$-0.777500\pi$
$877$	−45.3383	−1.53097	−0.765483	+	0.643457i	$0.777500\pi$
$878$	51.7735	1.74727
$879$	−39.8171	−1.34300
$880$	−35.9647	−1.21237
$881$	−32.8550	−1.10691	−0.553457	−	0.832878i	$-0.686692\pi$
$881$	−32.8550	−1.10691	−0.553457	+	0.832878i	$0.686692\pi$
$882$	0	0
$883$	24.6424	0.829282	0.414641	−	0.909985i	$-0.363907\pi$
$883$	24.6424	0.829282	0.414641	+	0.909985i	$0.363907\pi$
$884$	−40.6221	−1.36627
$885$	45.3966	1.52599
$886$	61.1328	2.05380
$887$	40.4767	1.35908	0.679538	−	0.733640i	$-0.262180\pi$
$887$	40.4767	1.35908	0.679538	+	0.733640i	$0.262180\pi$
$888$	4.41719	0.148231
$889$	0	0
$890$	68.7712	2.30522
$891$	17.8580	0.598267
$892$	−4.17773	−0.139881
$893$	9.22806	0.308805
$894$	−3.42496	−0.114548
$895$	16.0273	0.535732
$896$	0	0
$897$	51.1570	1.70808
$898$	−32.3924	−1.08095
$899$	−80.6921	−2.69123
$900$	−3.85382	−0.128461
$901$	3.10595	0.103474
$902$	4.69447	0.156309
$903$	0	0
$904$	−12.2992	−0.409065
$905$	14.2393	0.473330
$906$	43.5080	1.44546
$907$	−26.8838	−0.892661	−0.446331	−	0.894868i	$-0.647269\pi$
$907$	−26.8838	−0.892661	−0.446331	+	0.894868i	$0.647269\pi$
$908$	−11.2616	−0.373729
$909$	−4.81597	−0.159736
$910$	0	0
$911$	−15.7158	−0.520688	−0.260344	−	0.965516i	$-0.583836\pi$
$911$	−15.7158	−0.520688	−0.260344	+	0.965516i	$0.583836\pi$
$912$	44.0029	1.45708
$913$	−17.4676	−0.578094
$914$	−25.1629	−0.832316
$915$	−56.8844	−1.88054
$916$	−10.7937	−0.356632
$917$	0	0
$918$	74.6761	2.46468
$919$	−14.3753	−0.474197	−0.237098	−	0.971486i	$-0.576196\pi$
$919$	−14.3753	−0.474197	−0.237098	+	0.971486i	$0.576196\pi$
$920$	21.8490	0.720342
$921$	9.98158	0.328904
$922$	20.4305	0.672841
$923$	15.4533	0.508651
$924$	0	0
$925$	−17.1547	−0.564043
$926$	−48.8692	−1.60594
$927$	3.67745	0.120783
$928$	−65.6635	−2.15551
$929$	−20.4613	−0.671312	−0.335656	−	0.941985i	$-0.608958\pi$
$929$	−20.4613	−0.671312	−0.335656	+	0.941985i	$0.608958\pi$
$930$	83.3279	2.73243
$931$	0	0
$932$	−12.7012	−0.416042
$933$	21.3053	0.697506
$934$	−46.5575	−1.52341
$935$	−55.0210	−1.79938
$936$	1.41810	0.0463519
$937$	9.48803	0.309960	0.154980	−	0.987918i	$-0.450469\pi$
$937$	9.48803	0.309960	0.154980	+	0.987918i	$0.450469\pi$
$938$	0	0
$939$	−8.05856	−0.262981
$940$	7.64482	0.249347
$941$	7.95394	0.259291	0.129645	−	0.991560i	$-0.458616\pi$
$941$	7.95394	0.259291	0.129645	+	0.991560i	$0.458616\pi$
$942$	−18.6430	−0.607421
$943$	−9.03548	−0.294236
$944$	42.8559	1.39484
$945$	0	0
$946$	39.2960	1.27762
$947$	36.5486	1.18767	0.593835	−	0.804587i	$-0.297613\pi$
$947$	36.5486	1.18767	0.593835	+	0.804587i	$0.297613\pi$
$948$	0.976394	0.0317118
$949$	22.1300	0.718370
$950$	−53.9398	−1.75004
$951$	15.9607	0.517560
$952$	0	0
$953$	−48.5565	−1.57290	−0.786450	−	0.617655i	$-0.788083\pi$
$953$	−48.5565	−1.57290	−0.786450	+	0.617655i	$0.788083\pi$
$954$	0.422134	0.0136671
$955$	−10.8482	−0.351039
$956$	21.1290	0.683362
$957$	−35.3353	−1.14223
$958$	−20.2178	−0.653209
$959$	0	0
$960$	21.9918	0.709784
$961$	48.6757	1.57018
$962$	−24.5759	−0.792359
$963$	3.66132	0.117984
$964$	15.4872	0.498809
$965$	50.8422	1.63667
$966$	0	0
$967$	−2.81990	−0.0906817	−0.0453409	−	0.998972i	$-0.514437\pi$
$967$	−2.81990	−0.0906817	−0.0453409	+	0.998972i	$0.514437\pi$
$968$	3.76705	0.121078
$969$	67.3183	2.16257
$970$	−61.9367	−1.98867
$971$	−30.2345	−0.970272	−0.485136	−	0.874439i	$-0.661230\pi$
$971$	−30.2345	−0.970272	−0.485136	+	0.874439i	$0.661230\pi$
$972$	8.34053	0.267523
$973$	0	0
$974$	−14.6717	−0.470113
$975$	26.8737	0.860647
$976$	−53.7007	−1.71892
$977$	−22.6696	−0.725263	−0.362632	−	0.931933i	$-0.618122\pi$
$977$	−22.6696	−0.725263	−0.362632	+	0.931933i	$0.618122\pi$
$978$	26.1161	0.835101
$979$	28.7952	0.920299
$980$	0	0
$981$	6.54638	0.209010
$982$	−5.63841	−0.179929
$983$	9.70795	0.309636	0.154818	−	0.987943i	$-0.450521\pi$
$983$	9.70795	0.309636	0.154818	+	0.987943i	$0.450521\pi$
$984$	1.22218	0.0389617
$985$	20.1201	0.641079
$986$	−121.880	−3.88146
$987$	0	0
$988$	−34.2399	−1.08932
$989$	−75.6333	−2.40500
$990$	−7.47797	−0.237666
$991$	4.56981	0.145165	0.0725824	−	0.997362i	$-0.476876\pi$
$991$	4.56981	0.145165	0.0725824	+	0.997362i	$0.476876\pi$
$992$	64.8365	2.05856
$993$	−28.9699	−0.919333
$994$	0	0
$995$	4.23040	0.134113
$996$	17.7050	0.561004
$997$	−4.92320	−0.155919	−0.0779597	−	0.996957i	$-0.524841\pi$
$997$	−4.92320	−0.155919	−0.0779597	+	0.996957i	$0.524841\pi$
$998$	2.75986	0.0873618
$999$	20.0182	0.633348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.t.1.17	✓	20
7.6	odd	2		2009.2.a.u.1.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.17	✓	20	1.1	even	1	trivial
2009.2.a.u.1.17	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)

\(f(q)\)	\(=\)	\(q+1.89507 q^{2} +1.57790 q^{3} +1.59127 q^{4} -3.12194 q^{5} +2.99022 q^{6} -0.774564 q^{8} -0.510238 q^{9} +O(q^{10})\)	\(q+1.89507 q^{2} +1.57790 q^{3} +1.59127 q^{4} -3.12194 q^{5} +2.99022 q^{6} -0.774564 q^{8} -0.510238 q^{9} -5.91628 q^{10} -2.47721 q^{11} +2.51087 q^{12} +3.58819 q^{13} -4.92610 q^{15} -4.65040 q^{16} -7.11446 q^{17} -0.966935 q^{18} -5.99670 q^{19} -4.96786 q^{20} -4.69447 q^{22} +9.03548 q^{23} -1.22218 q^{24} +4.74649 q^{25} +6.79985 q^{26} -5.53880 q^{27} +9.03999 q^{29} -9.33528 q^{30} -8.92613 q^{31} -7.26368 q^{32} -3.90878 q^{33} -13.4824 q^{34} -0.811929 q^{36} -3.61418 q^{37} -11.3641 q^{38} +5.66180 q^{39} +2.41814 q^{40} -1.00000 q^{41} -8.37070 q^{43} -3.94191 q^{44} +1.59293 q^{45} +17.1228 q^{46} -1.53886 q^{47} -7.33785 q^{48} +8.99492 q^{50} -11.2259 q^{51} +5.70979 q^{52} -0.436569 q^{53} -10.4964 q^{54} +7.73369 q^{55} -9.46217 q^{57} +17.1314 q^{58} -9.21554 q^{59} -7.83877 q^{60} +11.5476 q^{61} -16.9156 q^{62} -4.46435 q^{64} -11.2021 q^{65} -7.40739 q^{66} +2.71201 q^{67} -11.3211 q^{68} +14.2571 q^{69} +4.30671 q^{71} +0.395212 q^{72} +6.16745 q^{73} -6.84911 q^{74} +7.48948 q^{75} -9.54238 q^{76} +10.7295 q^{78} +0.388867 q^{79} +14.5182 q^{80} -7.20894 q^{81} -1.89507 q^{82} +7.05134 q^{83} +22.2109 q^{85} -15.8630 q^{86} +14.2642 q^{87} +1.91875 q^{88} -11.6241 q^{89} +3.01871 q^{90} +14.3779 q^{92} -14.0845 q^{93} -2.91623 q^{94} +18.7213 q^{95} -11.4613 q^{96} +10.4689 q^{97} +1.26397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 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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