LMFDB - Embedded newform 2009.2.a.p.1.7

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 10x^{5} - x^{4} + 30x^{3} + 7x^{2} - 25x - 11 $ x^7 - 10x^5 - x^4 + 30x^3 + 7x^2 - 25x - 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.25969$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10619 q^{2} +1.25969 q^{3} +2.43602 q^{4} -4.35435 q^{5} +2.65313 q^{6} +0.918336 q^{8} -1.41319 q^{9} +O(q^{10})$	\(q+2.10619 q^{2} +1.25969 q^{3} +2.43602 q^{4} -4.35435 q^{5} +2.65313 q^{6} +0.918336 q^{8} -1.41319 q^{9} -9.17108 q^{10} +3.59430 q^{11} +3.06862 q^{12} -3.32330 q^{13} -5.48512 q^{15} -2.93785 q^{16} +0.740538 q^{17} -2.97644 q^{18} -2.53221 q^{19} -10.6073 q^{20} +7.57026 q^{22} -3.70722 q^{23} +1.15682 q^{24} +13.9604 q^{25} -6.99949 q^{26} -5.55924 q^{27} -5.72652 q^{29} -11.5527 q^{30} -8.25016 q^{31} -8.02433 q^{32} +4.52769 q^{33} +1.55971 q^{34} -3.44255 q^{36} -7.68545 q^{37} -5.33331 q^{38} -4.18632 q^{39} -3.99876 q^{40} +1.00000 q^{41} +11.5870 q^{43} +8.75578 q^{44} +6.15352 q^{45} -7.80810 q^{46} +5.63934 q^{47} -3.70077 q^{48} +29.4032 q^{50} +0.932846 q^{51} -8.09563 q^{52} +0.0701455 q^{53} -11.7088 q^{54} -15.6509 q^{55} -3.18980 q^{57} -12.0611 q^{58} -2.31532 q^{59} -13.3619 q^{60} -0.289030 q^{61} -17.3764 q^{62} -11.0250 q^{64} +14.4708 q^{65} +9.53616 q^{66} +1.48459 q^{67} +1.80396 q^{68} -4.66994 q^{69} -14.1072 q^{71} -1.29778 q^{72} -10.5766 q^{73} -16.1870 q^{74} +17.5857 q^{75} -6.16852 q^{76} -8.81717 q^{78} +8.44424 q^{79} +12.7924 q^{80} -2.76333 q^{81} +2.10619 q^{82} +12.6206 q^{83} -3.22456 q^{85} +24.4045 q^{86} -7.21362 q^{87} +3.30077 q^{88} +17.5858 q^{89} +12.9605 q^{90} -9.03087 q^{92} -10.3926 q^{93} +11.8775 q^{94} +11.0262 q^{95} -10.1081 q^{96} +2.10865 q^{97} -5.07942 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) $ 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9	$ 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) $ 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9 - 8 * q^10 - 12 * q^12 - q^13 - 4 * q^15 + 5 * q^16 + 11 * q^17 + 16 * q^18 - 9 * q^19 - 12 * q^20 + 3 * q^23 + 23 * q^24 + 31 * q^25 - 10 * q^26 - 22 * q^27 - 14 * q^29 + 6 * q^30 - 34 * q^31 - 20 * q^32 - 12 * q^33 - 15 * q^34 + q^36 + 7 * q^37 + 39 * q^38 - 22 * q^39 - 50 * q^40 + 7 * q^41 - 3 * q^43 + 26 * q^44 + 4 * q^45 - 8 * q^46 - 17 * q^47 - 17 * q^48 + 11 * q^50 + 8 * q^51 + 25 * q^52 + 24 * q^53 - 68 * q^54 - 48 * q^55 + 22 * q^57 - 38 * q^58 - 4 * q^59 - 6 * q^60 - 16 * q^61 - 24 * q^62 + 8 * q^64 - 6 * q^65 + 12 * q^66 - 24 * q^67 + 10 * q^68 - 35 * q^69 - 12 * q^71 - 11 * q^72 - 14 * q^73 - 6 * q^74 + 19 * q^75 - 42 * q^76 - 29 * q^78 - 8 * q^79 + 92 * q^80 + 15 * q^81 - q^82 - 14 * q^83 + 16 * q^85 + 35 * q^86 + 20 * q^87 + 22 * q^88 + 19 * q^89 - 24 * q^90 - 10 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^95 + 25 * q^96 - 23 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.10619	1.48930	0.744649	−	0.667456i	$-0.232617\pi$
$2$	2.10619	1.48930	0.744649	+	0.667456i	$0.232617\pi$
$3$	1.25969	0.727281	0.363640	−	0.931539i	$-0.381534\pi$
$3$	1.25969	0.727281	0.363640	+	0.931539i	$0.381534\pi$
$4$	2.43602	1.21801
$5$	−4.35435	−1.94733	−0.973663	−	0.227991i	$-0.926784\pi$
$5$	−4.35435	−1.94733	−0.973663	+	0.227991i	$0.926784\pi$
$6$	2.65313	1.08314
$7$	0	0
$7$	0	0
$8$	0.918336	0.324681
$9$	−1.41319	−0.471063
$10$	−9.17108	−2.90015
$11$	3.59430	1.08372	0.541861	−	0.840468i	$-0.317720\pi$
$11$	3.59430	1.08372	0.541861	+	0.840468i	$0.317720\pi$
$12$	3.06862	0.885835
$13$	−3.32330	−0.921718	−0.460859	−	0.887473i	$-0.652459\pi$
$13$	−3.32330	−0.921718	−0.460859	+	0.887473i	$0.652459\pi$
$14$	0	0
$15$	−5.48512	−1.41625
$16$	−2.93785	−0.734463
$17$	0.740538	0.179607	0.0898034	−	0.995960i	$-0.471376\pi$
$17$	0.740538	0.179607	0.0898034	+	0.995960i	$0.471376\pi$
$18$	−2.97644	−0.701553
$19$	−2.53221	−0.580930	−0.290465	−	0.956886i	$-0.593810\pi$
$19$	−2.53221	−0.580930	−0.290465	+	0.956886i	$0.593810\pi$
$20$	−10.6073	−2.37186
$21$	0	0
$22$	7.57026	1.61399
$23$	−3.70722	−0.773010	−0.386505	−	0.922287i	$-0.626318\pi$
$23$	−3.70722	−0.773010	−0.386505	+	0.922287i	$0.626318\pi$
$24$	1.15682	0.236134
$25$	13.9604	2.79208
$26$	−6.99949	−1.37271
$27$	−5.55924	−1.06988
$28$	0	0
$29$	−5.72652	−1.06339	−0.531694	−	0.846937i	$-0.678444\pi$
$29$	−5.72652	−1.06339	−0.531694	+	0.846937i	$0.678444\pi$
$30$	−11.5527	−2.10922
$31$	−8.25016	−1.48177	−0.740886	−	0.671630i	$-0.765594\pi$
$31$	−8.25016	−1.48177	−0.740886	+	0.671630i	$0.765594\pi$
$32$	−8.02433	−1.41851
$33$	4.52769	0.788170
$34$	1.55971	0.267488
$35$	0	0
$36$	−3.44255	−0.573759
$37$	−7.68545	−1.26348	−0.631740	−	0.775180i	$-0.717659\pi$
$37$	−7.68545	−1.26348	−0.631740	+	0.775180i	$0.717659\pi$
$38$	−5.33331	−0.865178
$39$	−4.18632	−0.670348
$40$	−3.99876	−0.632259
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	11.5870	1.76701	0.883503	−	0.468425i	$-0.155178\pi$
$43$	11.5870	1.76701	0.883503	+	0.468425i	$0.155178\pi$
$44$	8.75578	1.31998
$45$	6.15352	0.917313
$46$	−7.80810	−1.15124
$47$	5.63934	0.822582	0.411291	−	0.911504i	$-0.365078\pi$
$47$	5.63934	0.822582	0.411291	+	0.911504i	$0.365078\pi$
$48$	−3.70077	−0.534161
$49$	0	0
$50$	29.4032	4.15824
$51$	0.932846	0.130625
$52$	−8.09563	−1.12266
$53$	0.0701455	0.00963523	0.00481761	−	0.999988i	$-0.498467\pi$
$53$	0.0701455	0.00963523	0.00481761	+	0.999988i	$0.498467\pi$
$54$	−11.7088	−1.59336
$55$	−15.6509	−2.11036
$56$	0	0
$57$	−3.18980	−0.422499
$58$	−12.0611	−1.58370
$59$	−2.31532	−0.301429	−0.150715	−	0.988577i	$-0.548157\pi$
$59$	−2.31532	−0.301429	−0.150715	+	0.988577i	$0.548157\pi$
$60$	−13.3619	−1.72501
$61$	−0.289030	−0.0370065	−0.0185033	−	0.999829i	$-0.505890\pi$
$61$	−0.289030	−0.0370065	−0.0185033	+	0.999829i	$0.505890\pi$
$62$	−17.3764	−2.20680
$63$	0	0
$64$	−11.0250	−1.37813
$65$	14.4708	1.79489
$66$	9.53616	1.17382
$67$	1.48459	0.181372	0.0906860	−	0.995880i	$-0.471094\pi$
$67$	1.48459	0.181372	0.0906860	+	0.995880i	$0.471094\pi$
$68$	1.80396	0.218763
$69$	−4.66994	−0.562195
$70$	0	0
$71$	−14.1072	−1.67422	−0.837111	−	0.547033i	$-0.815757\pi$
$71$	−14.1072	−1.67422	−0.837111	+	0.547033i	$0.815757\pi$
$72$	−1.29778	−0.152945
$73$	−10.5766	−1.23789	−0.618947	−	0.785433i	$-0.712441\pi$
$73$	−10.5766	−1.23789	−0.618947	+	0.785433i	$0.712441\pi$
$74$	−16.1870	−1.88170
$75$	17.5857	2.03063
$76$	−6.16852	−0.707578
$77$	0	0
$78$	−8.81717	−0.998348
$79$	8.44424	0.950052	0.475026	−	0.879972i	$-0.342439\pi$
$79$	8.44424	0.950052	0.475026	+	0.879972i	$0.342439\pi$
$80$	12.7924	1.43024
$81$	−2.76333	−0.307037
$82$	2.10619	0.232589
$83$	12.6206	1.38529	0.692644	−	0.721280i	$-0.256446\pi$
$83$	12.6206	1.38529	0.692644	+	0.721280i	$0.256446\pi$
$84$	0	0
$85$	−3.22456	−0.349753
$86$	24.4045	2.63160
$87$	−7.21362	−0.773381
$88$	3.30077	0.351864
$89$	17.5858	1.86409	0.932047	−	0.362337i	$-0.118021\pi$
$89$	17.5858	1.86409	0.932047	+	0.362337i	$0.118021\pi$
$90$	12.9605	1.36615
$91$	0	0
$92$	−9.03087	−0.941533
$93$	−10.3926	−1.07766
$94$	11.8775	1.22507
$95$	11.0262	1.13126
$96$	−10.1081	−1.03166
$97$	2.10865	0.214101	0.107051	−	0.994254i	$-0.465859\pi$
$97$	2.10865	0.214101	0.107051	+	0.994254i	$0.465859\pi$
$98$	0	0
$99$	−5.07942	−0.510501
$100$	34.0078	3.40078
$101$	6.11462	0.608427	0.304213	−	0.952604i	$-0.401606\pi$
$101$	6.11462	0.608427	0.304213	+	0.952604i	$0.401606\pi$
$102$	1.96475	0.194539
$103$	−6.95784	−0.685577	−0.342788	−	0.939413i	$-0.611371\pi$
$103$	−6.95784	−0.685577	−0.342788	+	0.939413i	$0.611371\pi$
$104$	−3.05191	−0.299264
$105$	0	0
$106$	0.147739	0.0143497
$107$	3.93007	0.379935	0.189967	−	0.981790i	$-0.439162\pi$
$107$	3.93007	0.379935	0.189967	+	0.981790i	$0.439162\pi$
$108$	−13.5424	−1.30312
$109$	12.7072	1.21713	0.608567	−	0.793503i	$-0.291745\pi$
$109$	12.7072	1.21713	0.608567	+	0.793503i	$0.291745\pi$
$110$	−32.9636	−3.14296
$111$	−9.68126	−0.918904
$112$	0	0
$113$	−5.89978	−0.555004	−0.277502	−	0.960725i	$-0.589507\pi$
$113$	−5.89978	−0.555004	−0.277502	+	0.960725i	$0.589507\pi$
$114$	−6.71831	−0.629227
$115$	16.1426	1.50530
$116$	−13.9499	−1.29522
$117$	4.69645	0.434187
$118$	−4.87650	−0.448918
$119$	0	0
$120$	−5.03718	−0.459830
$121$	1.91900	0.174454
$122$	−0.608752	−0.0551138
$123$	1.25969	0.113582
$124$	−20.0975	−1.80481
$125$	−39.0167	−3.48976
$126$	0	0
$127$	1.93650	0.171837	0.0859185	−	0.996302i	$-0.472618\pi$
$127$	1.93650	0.171837	0.0859185	+	0.996302i	$0.472618\pi$
$128$	−7.17210	−0.633930
$129$	14.5960	1.28511
$130$	30.4783	2.67312
$131$	−7.30510	−0.638249	−0.319125	−	0.947713i	$-0.603389\pi$
$131$	−7.30510	−0.638249	−0.319125	+	0.947713i	$0.603389\pi$
$132$	11.0295	0.959999
$133$	0	0
$134$	3.12683	0.270117
$135$	24.2069	2.08340
$136$	0.680062	0.0583148
$137$	−16.8656	−1.44092	−0.720462	−	0.693494i	$-0.756070\pi$
$137$	−16.8656	−1.44092	−0.720462	+	0.693494i	$0.756070\pi$
$138$	−9.83577	−0.837276
$139$	−5.83514	−0.494930	−0.247465	−	0.968897i	$-0.579598\pi$
$139$	−5.83514	−0.494930	−0.247465	+	0.968897i	$0.579598\pi$
$140$	0	0
$141$	7.10380	0.598248
$142$	−29.7125	−2.49342
$143$	−11.9449	−0.998887
$144$	4.15174	0.345978
$145$	24.9353	2.07076
$146$	−22.2762	−1.84359
$147$	0	0
$148$	−18.7219	−1.53893
$149$	−6.67607	−0.546925	−0.273463	−	0.961883i	$-0.588169\pi$
$149$	−6.67607	−0.546925	−0.273463	+	0.961883i	$0.588169\pi$
$150$	37.0388	3.02421
$151$	−3.31677	−0.269915	−0.134957	−	0.990851i	$-0.543090\pi$
$151$	−3.31677	−0.269915	−0.134957	+	0.990851i	$0.543090\pi$
$152$	−2.32542	−0.188617
$153$	−1.04652	−0.0846061
$154$	0	0
$155$	35.9241	2.88549
$156$	−10.1980	−0.816490
$157$	−7.34054	−0.585839	−0.292919	−	0.956137i	$-0.594627\pi$
$157$	−7.34054	−0.585839	−0.292919	+	0.956137i	$0.594627\pi$
$158$	17.7851	1.41491
$159$	0.0883614	0.00700751
$160$	34.9408	2.76231
$161$	0	0
$162$	−5.82009	−0.457270
$163$	−1.69251	−0.132567	−0.0662837	−	0.997801i	$-0.521114\pi$
$163$	−1.69251	−0.132567	−0.0662837	+	0.997801i	$0.521114\pi$
$164$	2.43602	0.190221
$165$	−19.7152	−1.53482
$166$	26.5813	2.06311
$167$	20.4174	1.57994	0.789972	−	0.613143i	$-0.210095\pi$
$167$	20.4174	1.57994	0.789972	+	0.613143i	$0.210095\pi$
$168$	0	0
$169$	−1.95566	−0.150435
$170$	−6.79153	−0.520887
$171$	3.57850	0.273654
$172$	28.2262	2.15223
$173$	−14.5966	−1.10976	−0.554878	−	0.831932i	$-0.687235\pi$
$173$	−14.5966	−1.10976	−0.554878	+	0.831932i	$0.687235\pi$
$174$	−15.1932	−1.15179
$175$	0	0
$176$	−10.5595	−0.795954
$177$	−2.91658	−0.219224
$178$	37.0390	2.77619
$179$	20.1577	1.50665	0.753327	−	0.657646i	$-0.228448\pi$
$179$	20.1577	1.50665	0.753327	+	0.657646i	$0.228448\pi$
$180$	14.9901	1.11730
$181$	4.24595	0.315599	0.157799	−	0.987471i	$-0.449560\pi$
$181$	4.24595	0.315599	0.157799	+	0.987471i	$0.449560\pi$
$182$	0	0
$183$	−0.364088	−0.0269141
$184$	−3.40448	−0.250981
$185$	33.4652	2.46041
$186$	−21.8888	−1.60496
$187$	2.66172	0.194644
$188$	13.7375	1.00191
$189$	0	0
$190$	23.2231	1.68478
$191$	−6.28298	−0.454621	−0.227310	−	0.973822i	$-0.572993\pi$
$191$	−6.28298	−0.454621	−0.227310	+	0.973822i	$0.572993\pi$
$192$	−13.8881	−1.00229
$193$	−6.49397	−0.467446	−0.233723	−	0.972303i	$-0.575091\pi$
$193$	−6.49397	−0.467446	−0.233723	+	0.972303i	$0.575091\pi$
$194$	4.44121	0.318861
$195$	18.2287	1.30539
$196$	0	0
$197$	−10.7402	−0.765210	−0.382605	−	0.923912i	$-0.624973\pi$
$197$	−10.7402	−0.765210	−0.382605	+	0.923912i	$0.624973\pi$
$198$	−10.6982	−0.760289
$199$	6.35562	0.450538	0.225269	−	0.974297i	$-0.427674\pi$
$199$	6.35562	0.450538	0.225269	+	0.974297i	$0.427674\pi$
$200$	12.8203	0.906534
$201$	1.87012	0.131908
$202$	12.8785	0.906129
$203$	0	0
$204$	2.27243	0.159102
$205$	−4.35435	−0.304121
$206$	−14.6545	−1.02103
$207$	5.23901	0.364136
$208$	9.76337	0.676968
$209$	−9.10154	−0.629567
$210$	0	0
$211$	−2.46234	−0.169514	−0.0847571	−	0.996402i	$-0.527011\pi$
$211$	−2.46234	−0.169514	−0.0847571	+	0.996402i	$0.527011\pi$
$212$	0.170876	0.0117358
$213$	−17.7707	−1.21763
$214$	8.27747	0.565836
$215$	−50.4541	−3.44094
$216$	−5.10524	−0.347368
$217$	0	0
$218$	26.7638	1.81267
$219$	−13.3232	−0.900297
$220$	−38.1258	−2.57044
$221$	−2.46103	−0.165547
$222$	−20.3905	−1.36852
$223$	14.5377	0.973514	0.486757	−	0.873537i	$-0.338180\pi$
$223$	14.5377	0.973514	0.486757	+	0.873537i	$0.338180\pi$
$224$	0	0
$225$	−19.7287	−1.31524
$226$	−12.4260	−0.826567
$227$	14.1044	0.936141	0.468070	−	0.883691i	$-0.344949\pi$
$227$	14.1044	0.936141	0.468070	+	0.883691i	$0.344949\pi$
$228$	−7.77041	−0.514608
$229$	24.9111	1.64617	0.823085	−	0.567919i	$-0.192251\pi$
$229$	24.9111	1.64617	0.823085	+	0.567919i	$0.192251\pi$
$230$	33.9992	2.24184
$231$	0	0
$232$	−5.25886	−0.345261
$233$	−24.8270	−1.62647	−0.813234	−	0.581937i	$-0.802295\pi$
$233$	−24.8270	−1.62647	−0.813234	+	0.581937i	$0.802295\pi$
$234$	9.89160	0.646634
$235$	−24.5557	−1.60184
$236$	−5.64017	−0.367144
$237$	10.6371	0.690954
$238$	0	0
$239$	28.0200	1.81246	0.906230	−	0.422785i	$-0.138947\pi$
$239$	28.0200	1.81246	0.906230	+	0.422785i	$0.138947\pi$
$240$	16.1145	1.04019
$241$	−17.6689	−1.13816	−0.569078	−	0.822283i	$-0.692700\pi$
$241$	−17.6689	−1.13816	−0.569078	+	0.822283i	$0.692700\pi$
$242$	4.04176	0.259814
$243$	13.1968	0.846573
$244$	−0.704083	−0.0450743
$245$	0	0
$246$	2.65313	0.169158
$247$	8.41531	0.535454
$248$	−7.57642	−0.481103
$249$	15.8980	1.00749
$250$	−82.1765	−5.19730
$251$	−10.4453	−0.659303	−0.329651	−	0.944103i	$-0.606931\pi$
$251$	−10.4453	−0.659303	−0.329651	+	0.944103i	$0.606931\pi$
$252$	0	0
$253$	−13.3249	−0.837728
$254$	4.07864	0.255916
$255$	−4.06194	−0.254369
$256$	6.94429	0.434018
$257$	21.7605	1.35738	0.678692	−	0.734423i	$-0.262547\pi$
$257$	21.7605	1.35738	0.678692	+	0.734423i	$0.262547\pi$
$258$	30.7420	1.91391
$259$	0	0
$260$	35.2512	2.18619
$261$	8.09265	0.500922
$262$	−15.3859	−0.950544
$263$	−0.827922	−0.0510519	−0.0255259	−	0.999674i	$-0.508126\pi$
$263$	−0.827922	−0.0510519	−0.0255259	+	0.999674i	$0.508126\pi$
$264$	4.15794	0.255904
$265$	−0.305438	−0.0187629
$266$	0	0
$267$	22.1526	1.35572
$268$	3.61650	0.220913
$269$	19.9912	1.21889	0.609443	−	0.792830i	$-0.291393\pi$
$269$	19.9912	1.21889	0.609443	+	0.792830i	$0.291393\pi$
$270$	50.9842	3.10280
$271$	−22.5635	−1.37064	−0.685319	−	0.728243i	$-0.740337\pi$
$271$	−22.5635	−1.37064	−0.685319	+	0.728243i	$0.740337\pi$
$272$	−2.17559	−0.131915
$273$	0	0
$274$	−35.5221	−2.14597
$275$	50.1779	3.02584
$276$	−11.3761	−0.684759
$277$	−14.7237	−0.884661	−0.442331	−	0.896852i	$-0.645848\pi$
$277$	−14.7237	−0.884661	−0.442331	+	0.896852i	$0.645848\pi$
$278$	−12.2899	−0.737099
$279$	11.6590	0.698008
$280$	0	0
$281$	−30.6497	−1.82841	−0.914203	−	0.405258i	$-0.867182\pi$
$281$	−30.6497	−1.82841	−0.914203	+	0.405258i	$0.867182\pi$
$282$	14.9619	0.890970
$283$	−2.61001	−0.155149	−0.0775746	−	0.996987i	$-0.524718\pi$
$283$	−2.61001	−0.155149	−0.0775746	+	0.996987i	$0.524718\pi$
$284$	−34.3655	−2.03922
$285$	13.8895	0.822743
$286$	−25.1583	−1.48764
$287$	0	0
$288$	11.3399	0.668210
$289$	−16.4516	−0.967741
$290$	52.5183	3.08398
$291$	2.65624	0.155712
$292$	−25.7647	−1.50777
$293$	−23.8553	−1.39364	−0.696821	−	0.717245i	$-0.745403\pi$
$293$	−23.8553	−1.39364	−0.696821	+	0.717245i	$0.745403\pi$
$294$	0	0
$295$	10.0817	0.586981
$296$	−7.05782	−0.410227
$297$	−19.9816	−1.15945
$298$	−14.0611	−0.814535
$299$	12.3202	0.712497
$300$	42.8392	2.47332
$301$	0	0
$302$	−6.98573	−0.401984
$303$	7.70250	0.442497
$304$	7.43927	0.426671
$305$	1.25854	0.0720638
$306$	−2.20416	−0.126004
$307$	5.03182	0.287181	0.143591	−	0.989637i	$-0.454135\pi$
$307$	5.03182	0.287181	0.143591	+	0.989637i	$0.454135\pi$
$308$	0	0
$309$	−8.76471	−0.498607
$310$	75.6629	4.29736
$311$	22.7857	1.29206	0.646029	−	0.763313i	$-0.276429\pi$
$311$	22.7857	1.29206	0.646029	+	0.763313i	$0.276429\pi$
$312$	−3.84445	−0.217649
$313$	−6.48172	−0.366368	−0.183184	−	0.983079i	$-0.558640\pi$
$313$	−6.48172	−0.366368	−0.183184	+	0.983079i	$0.558640\pi$
$314$	−15.4605	−0.872488
$315$	0	0
$316$	20.5703	1.15717
$317$	−9.08376	−0.510195	−0.255097	−	0.966915i	$-0.582108\pi$
$317$	−9.08376	−0.510195	−0.255097	+	0.966915i	$0.582108\pi$
$318$	0.186106	0.0104363
$319$	−20.5828	−1.15242
$320$	48.0069	2.68367
$321$	4.95066	0.276319
$322$	0	0
$323$	−1.87520	−0.104339
$324$	−6.73153	−0.373974
$325$	−46.3946	−2.57351
$326$	−3.56473	−0.197432
$327$	16.0071	0.885197
$328$	0.918336	0.0507066
$329$	0	0
$330$	−41.5238	−2.28581
$331$	−31.2707	−1.71879	−0.859396	−	0.511310i	$-0.829160\pi$
$331$	−31.2707	−1.71879	−0.859396	+	0.511310i	$0.829160\pi$
$332$	30.7439	1.68729
$333$	10.8610	0.595178
$334$	43.0028	2.35301
$335$	−6.46445	−0.353191
$336$	0	0
$337$	−32.4602	−1.76822	−0.884109	−	0.467281i	$-0.845234\pi$
$337$	−32.4602	−1.76822	−0.884109	+	0.467281i	$0.845234\pi$
$338$	−4.11899	−0.224043
$339$	−7.43187	−0.403644
$340$	−7.85510	−0.426002
$341$	−29.6536	−1.60583
$342$	7.53698	0.407553
$343$	0	0
$344$	10.6408	0.573713
$345$	20.3346	1.09478
$346$	−30.7431	−1.65276
$347$	14.1444	0.759311	0.379655	−	0.925128i	$-0.376043\pi$
$347$	14.1444	0.759311	0.379655	+	0.925128i	$0.376043\pi$
$348$	−17.5725	−0.941985
$349$	−30.3361	−1.62385	−0.811926	−	0.583760i	$-0.801581\pi$
$349$	−30.3361	−1.62385	−0.811926	+	0.583760i	$0.801581\pi$
$350$	0	0
$351$	18.4750	0.986124
$352$	−28.8419	−1.53728
$353$	19.3853	1.03178	0.515889	−	0.856656i	$-0.327462\pi$
$353$	19.3853	1.03178	0.515889	+	0.856656i	$0.327462\pi$
$354$	−6.14286	−0.326489
$355$	61.4279	3.26026
$356$	42.8394	2.27048
$357$	0	0
$358$	42.4558	2.24386
$359$	−1.65789	−0.0875002	−0.0437501	−	0.999043i	$-0.513931\pi$
$359$	−1.65789	−0.0875002	−0.0437501	+	0.999043i	$0.513931\pi$
$360$	5.65100	0.297834
$361$	−12.5879	−0.662521
$362$	8.94275	0.470020
$363$	2.41733	0.126877
$364$	0	0
$365$	46.0541	2.41058
$366$	−0.766836	−0.0400832
$367$	−14.9459	−0.780168	−0.390084	−	0.920779i	$-0.627554\pi$
$367$	−14.9459	−0.780168	−0.390084	+	0.920779i	$0.627554\pi$
$368$	10.8913	0.567747
$369$	−1.41319	−0.0735677
$370$	70.4838	3.66428
$371$	0	0
$372$	−25.3166	−1.31261
$373$	−11.0259	−0.570897	−0.285449	−	0.958394i	$-0.592143\pi$
$373$	−11.0259	−0.570897	−0.285449	+	0.958394i	$0.592143\pi$
$374$	5.60607	0.289883
$375$	−49.1489	−2.53804
$376$	5.17880	0.267076
$377$	19.0309	0.980143
$378$	0	0
$379$	−5.68502	−0.292020	−0.146010	−	0.989283i	$-0.546643\pi$
$379$	−5.68502	−0.292020	−0.146010	+	0.989283i	$0.546643\pi$
$380$	26.8599	1.37788
$381$	2.43939	0.124974
$382$	−13.2331	−0.677066
$383$	−15.7424	−0.804398	−0.402199	−	0.915552i	$-0.631754\pi$
$383$	−15.7424	−0.804398	−0.402199	+	0.915552i	$0.631754\pi$
$384$	−9.03460	−0.461045
$385$	0	0
$386$	−13.6775	−0.696167
$387$	−16.3747	−0.832371
$388$	5.13672	0.260777
$389$	28.1615	1.42784	0.713922	−	0.700225i	$-0.246917\pi$
$389$	28.1615	1.42784	0.713922	+	0.700225i	$0.246917\pi$
$390$	38.3931	1.94411
$391$	−2.74534	−0.138838
$392$	0	0
$393$	−9.20214	−0.464187
$394$	−22.6209	−1.13963
$395$	−36.7692	−1.85006
$396$	−12.3736	−0.621795
$397$	−18.5355	−0.930268	−0.465134	−	0.885240i	$-0.653994\pi$
$397$	−18.5355	−0.930268	−0.465134	+	0.885240i	$0.653994\pi$
$398$	13.3861	0.670985
$399$	0	0
$400$	−41.0136	−2.05068
$401$	−9.01389	−0.450132	−0.225066	−	0.974344i	$-0.572260\pi$
$401$	−9.01389	−0.450132	−0.225066	+	0.974344i	$0.572260\pi$
$402$	3.93883	0.196451
$403$	27.4178	1.36578
$404$	14.8953	0.741070
$405$	12.0325	0.597901
$406$	0	0
$407$	−27.6238	−1.36926
$408$	0.856665	0.0424113
$409$	−13.4896	−0.667018	−0.333509	−	0.942747i	$-0.608233\pi$
$409$	−13.4896	−0.667018	−0.333509	+	0.942747i	$0.608233\pi$
$410$	−9.17108	−0.452927
$411$	−21.2454	−1.04796
$412$	−16.9494	−0.835039
$413$	0	0
$414$	11.0343	0.542307
$415$	−54.9544	−2.69761
$416$	26.6673	1.30747
$417$	−7.35045	−0.359953
$418$	−19.1695	−0.937612
$419$	−27.6136	−1.34901	−0.674506	−	0.738270i	$-0.735643\pi$
$419$	−27.6136	−1.34901	−0.674506	+	0.738270i	$0.735643\pi$
$420$	0	0
$421$	7.34435	0.357942	0.178971	−	0.983854i	$-0.442723\pi$
$421$	7.34435	0.357942	0.178971	+	0.983854i	$0.442723\pi$
$422$	−5.18613	−0.252457
$423$	−7.96945	−0.387488
$424$	0.0644171	0.00312837
$425$	10.3382	0.501476
$426$	−37.4284	−1.81341
$427$	0	0
$428$	9.57373	0.462764
$429$	−15.0469	−0.726471
$430$	−106.266	−5.12458
$431$	−26.1083	−1.25759	−0.628797	−	0.777570i	$-0.716452\pi$
$431$	−26.1083	−1.25759	−0.628797	+	0.777570i	$0.716452\pi$
$432$	16.3322	0.785784
$433$	−29.2289	−1.40465	−0.702325	−	0.711856i	$-0.747855\pi$
$433$	−29.2289	−1.40465	−0.702325	+	0.711856i	$0.747855\pi$
$434$	0	0
$435$	31.4106	1.50603
$436$	30.9551	1.48248
$437$	9.38749	0.449064
$438$	−28.0611	−1.34081
$439$	15.6801	0.748368	0.374184	−	0.927354i	$-0.377923\pi$
$439$	15.6801	0.748368	0.374184	+	0.927354i	$0.377923\pi$
$440$	−14.3727	−0.685193
$441$	0	0
$442$	−5.18339	−0.246549
$443$	11.8734	0.564123	0.282062	−	0.959396i	$-0.408982\pi$
$443$	11.8734	0.564123	0.282062	+	0.959396i	$0.408982\pi$
$444$	−23.5837	−1.11923
$445$	−76.5749	−3.63000
$446$	30.6190	1.44985
$447$	−8.40977	−0.397768
$448$	0	0
$449$	−10.0145	−0.472614	−0.236307	−	0.971678i	$-0.575937\pi$
$449$	−10.0145	−0.472614	−0.236307	+	0.971678i	$0.575937\pi$
$450$	−41.5522	−1.95879
$451$	3.59430	0.169249
$452$	−14.3720	−0.676001
$453$	−4.17809	−0.196304
$454$	29.7064	1.39419
$455$	0	0
$456$	−2.92930	−0.137177
$457$	−15.3679	−0.718879	−0.359439	−	0.933168i	$-0.617032\pi$
$457$	−15.3679	−0.718879	−0.359439	+	0.933168i	$0.617032\pi$
$458$	52.4673	2.45164
$459$	−4.11682	−0.192157
$460$	39.3236	1.83347
$461$	3.10047	0.144403	0.0722015	−	0.997390i	$-0.476998\pi$
$461$	3.10047	0.144403	0.0722015	+	0.997390i	$0.476998\pi$
$462$	0	0
$463$	−4.35814	−0.202540	−0.101270	−	0.994859i	$-0.532291\pi$
$463$	−4.35814	−0.202540	−0.101270	+	0.994859i	$0.532291\pi$
$464$	16.8237	0.781018
$465$	45.2532	2.09856
$466$	−52.2902	−2.42230
$467$	42.8985	1.98511	0.992553	−	0.121814i	$-0.0388712\pi$
$467$	42.8985	1.98511	0.992553	+	0.121814i	$0.0388712\pi$
$468$	11.4406	0.528844
$469$	0	0
$470$	−51.7188	−2.38561
$471$	−9.24678	−0.426069
$472$	−2.12624	−0.0978683
$473$	41.6473	1.91494
$474$	22.4037	1.02904
$475$	−35.3507	−1.62200
$476$	0	0
$477$	−0.0991288	−0.00453880
$478$	59.0152	2.69929
$479$	6.87307	0.314039	0.157019	−	0.987596i	$-0.449812\pi$
$479$	6.87307	0.314039	0.157019	+	0.987596i	$0.449812\pi$
$480$	44.0145	2.00898
$481$	25.5411	1.16457
$482$	−37.2141	−1.69505
$483$	0	0
$484$	4.67471	0.212487
$485$	−9.18182	−0.416925
$486$	27.7949	1.26080
$487$	−15.9922	−0.724675	−0.362338	−	0.932047i	$-0.618021\pi$
$487$	−15.9922	−0.724675	−0.362338	+	0.932047i	$0.618021\pi$
$488$	−0.265427	−0.0120153
$489$	−2.13203	−0.0964136
$490$	0	0
$491$	28.6929	1.29489	0.647445	−	0.762112i	$-0.275837\pi$
$491$	28.6929	1.29489	0.647445	+	0.762112i	$0.275837\pi$
$492$	3.06862	0.138344
$493$	−4.24070	−0.190992
$494$	17.7242	0.797450
$495$	22.1176	0.994113
$496$	24.2377	1.08831
$497$	0	0
$498$	33.4841	1.50046
$499$	8.59068	0.384572	0.192286	−	0.981339i	$-0.438410\pi$
$499$	8.59068	0.384572	0.192286	+	0.981339i	$0.438410\pi$
$500$	−95.0455	−4.25056
$501$	25.7195	1.14906
$502$	−21.9998	−0.981899
$503$	4.86078	0.216731	0.108366	−	0.994111i	$-0.465438\pi$
$503$	4.86078	0.216731	0.108366	+	0.994111i	$0.465438\pi$
$504$	0	0
$505$	−26.6252	−1.18481
$506$	−28.0647	−1.24763
$507$	−2.46352	−0.109409
$508$	4.71736	0.209299
$509$	−28.5607	−1.26593	−0.632965	−	0.774181i	$-0.718162\pi$
$509$	−28.5607	−1.26593	−0.632965	+	0.774181i	$0.718162\pi$
$510$	−8.55520	−0.378831
$511$	0	0
$512$	28.9702	1.28031
$513$	14.0772	0.621523
$514$	45.8317	2.02155
$515$	30.2969	1.33504
$516$	35.5562	1.56528
$517$	20.2695	0.891450
$518$	0	0
$519$	−18.3871	−0.807104
$520$	13.2891	0.582765
$521$	−39.2475	−1.71947	−0.859733	−	0.510744i	$-0.829370\pi$
$521$	−39.2475	−1.71947	−0.859733	+	0.510744i	$0.829370\pi$
$522$	17.0446	0.746022
$523$	−26.1309	−1.14262	−0.571312	−	0.820733i	$-0.693566\pi$
$523$	−26.1309	−1.14262	−0.571312	+	0.820733i	$0.693566\pi$
$524$	−17.7954	−0.777394
$525$	0	0
$526$	−1.74376	−0.0760314
$527$	−6.10956	−0.266136
$528$	−13.3017	−0.578882
$529$	−9.25649	−0.402456
$530$	−0.643310	−0.0279436
$531$	3.27199	0.141992
$532$	0	0
$533$	−3.32330	−0.143948
$534$	46.6576	2.01907
$535$	−17.1129	−0.739857
$536$	1.36336	0.0588880
$537$	25.3923	1.09576
$538$	42.1052	1.81528
$539$	0	0
$540$	58.9684	2.53760
$541$	26.7705	1.15095	0.575476	−	0.817819i	$-0.304817\pi$
$541$	26.7705	1.15095	0.575476	+	0.817819i	$0.304817\pi$
$542$	−47.5230	−2.04129
$543$	5.34856	0.229529
$544$	−5.94232	−0.254775
$545$	−55.3318	−2.37016
$546$	0	0
$547$	21.8567	0.934525	0.467262	−	0.884119i	$-0.345240\pi$
$547$	21.8567	0.934525	0.467262	+	0.884119i	$0.345240\pi$
$548$	−41.0849	−1.75506
$549$	0.408454	0.0174324
$550$	105.684	4.50638
$551$	14.5008	0.617753
$552$	−4.28857	−0.182534
$553$	0	0
$554$	−31.0108	−1.31752
$555$	42.1556	1.78941
$556$	−14.2145	−0.602830
$557$	30.1664	1.27819	0.639096	−	0.769127i	$-0.279308\pi$
$557$	30.1664	1.27819	0.639096	+	0.769127i	$0.279308\pi$
$558$	24.5561	1.03954
$559$	−38.5072	−1.62868
$560$	0	0
$561$	3.35293	0.141561
$562$	−64.5539	−2.72304
$563$	−39.2819	−1.65553	−0.827767	−	0.561072i	$-0.810389\pi$
$563$	−39.2819	−1.65553	−0.827767	+	0.561072i	$0.810389\pi$
$564$	17.3050	0.728671
$565$	25.6897	1.08077
$566$	−5.49717	−0.231063
$567$	0	0
$568$	−12.9552	−0.543587
$569$	−40.0665	−1.67967	−0.839837	−	0.542838i	$-0.817350\pi$
$569$	−40.0665	−1.67967	−0.839837	+	0.542838i	$0.817350\pi$
$570$	29.2539	1.22531
$571$	−8.32357	−0.348330	−0.174165	−	0.984716i	$-0.555723\pi$
$571$	−8.32357	−0.348330	−0.174165	+	0.984716i	$0.555723\pi$
$572$	−29.0981	−1.21665
$573$	−7.91459	−0.330637
$574$	0	0
$575$	−51.7543	−2.15830
$576$	15.5804	0.649185
$577$	5.70342	0.237436	0.118718	−	0.992928i	$-0.462121\pi$
$577$	5.70342	0.237436	0.118718	+	0.992928i	$0.462121\pi$
$578$	−34.6501	−1.44126
$579$	−8.18037	−0.339965
$580$	60.7428	2.52221
$581$	0	0
$582$	5.59454	0.231901
$583$	0.252124	0.0104419
$584$	−9.71284	−0.401920
$585$	−20.4500	−0.845504
$586$	−50.2437	−2.07555
$587$	39.5941	1.63422	0.817112	−	0.576479i	$-0.195574\pi$
$587$	39.5941	1.63422	0.817112	+	0.576479i	$0.195574\pi$
$588$	0	0
$589$	20.8912	0.860806
$590$	21.2340	0.874190
$591$	−13.5293	−0.556522
$592$	22.5787	0.927979
$593$	−24.0235	−0.986527	−0.493263	−	0.869880i	$-0.664196\pi$
$593$	−24.0235	−0.986527	−0.493263	+	0.869880i	$0.664196\pi$
$594$	−42.0849	−1.72676
$595$	0	0
$596$	−16.2630	−0.666160
$597$	8.00609	0.327667
$598$	25.9487	1.06112
$599$	21.1624	0.864674	0.432337	−	0.901712i	$-0.357689\pi$
$599$	21.1624	0.864674	0.432337	+	0.901712i	$0.357689\pi$
$600$	16.1496	0.659305
$601$	38.7514	1.58070	0.790352	−	0.612653i	$-0.209898\pi$
$601$	38.7514	1.58070	0.790352	+	0.612653i	$0.209898\pi$
$602$	0	0
$603$	−2.09801	−0.0854376
$604$	−8.07971	−0.328759
$605$	−8.35599	−0.339719
$606$	16.2229	0.659010
$607$	25.6539	1.04126	0.520630	−	0.853782i	$-0.325697\pi$
$607$	25.6539	1.04126	0.520630	+	0.853782i	$0.325697\pi$
$608$	20.3193	0.824058
$609$	0	0
$610$	2.65072	0.107324
$611$	−18.7412	−0.758189
$612$	−2.54934	−0.103051
$613$	14.8689	0.600549	0.300275	−	0.953853i	$-0.402922\pi$
$613$	14.8689	0.600549	0.300275	+	0.953853i	$0.402922\pi$
$614$	10.5980	0.427699
$615$	−5.48512	−0.221182
$616$	0	0
$617$	6.04103	0.243203	0.121601	−	0.992579i	$-0.461197\pi$
$617$	6.04103	0.243203	0.121601	+	0.992579i	$0.461197\pi$
$618$	−18.4601	−0.742574
$619$	−33.6267	−1.35157	−0.675785	−	0.737098i	$-0.736195\pi$
$619$	−33.6267	−1.35157	−0.675785	+	0.737098i	$0.736195\pi$
$620$	87.5118	3.51456
$621$	20.6093	0.827024
$622$	47.9909	1.92426
$623$	0	0
$624$	12.2988	0.492346
$625$	100.091	4.00363
$626$	−13.6517	−0.545632
$627$	−11.4651	−0.457872
$628$	−17.8817	−0.713557
$629$	−5.69136	−0.226930
$630$	0	0
$631$	−11.1390	−0.443436	−0.221718	−	0.975111i	$-0.571166\pi$
$631$	−11.1390	−0.443436	−0.221718	+	0.975111i	$0.571166\pi$
$632$	7.75465	0.308463
$633$	−3.10177	−0.123284
$634$	−19.1321	−0.759832
$635$	−8.43222	−0.334623
$636$	0.215250	0.00853522
$637$	0	0
$638$	−43.3512	−1.71629
$639$	19.9362	0.788664
$640$	31.2299	1.23447
$641$	36.6505	1.44761	0.723803	−	0.690007i	$-0.242392\pi$
$641$	36.6505	1.44761	0.723803	+	0.690007i	$0.242392\pi$
$642$	10.4270	0.411522
$643$	−14.2324	−0.561269	−0.280635	−	0.959815i	$-0.590545\pi$
$643$	−14.2324	−0.561269	−0.280635	+	0.959815i	$0.590545\pi$
$644$	0	0
$645$	−63.5563	−2.50253
$646$	−3.94952	−0.155392
$647$	−11.9169	−0.468500	−0.234250	−	0.972176i	$-0.575263\pi$
$647$	−11.9169	−0.468500	−0.234250	+	0.972176i	$0.575263\pi$
$648$	−2.53767	−0.0996890
$649$	−8.32196	−0.326666
$650$	−97.7157	−3.83272
$651$	0	0
$652$	−4.12298	−0.161468
$653$	−17.0674	−0.667901	−0.333950	−	0.942591i	$-0.608382\pi$
$653$	−17.0674	−0.667901	−0.333950	+	0.942591i	$0.608382\pi$
$654$	33.7140	1.31832
$655$	31.8090	1.24288
$656$	−2.93785	−0.114704
$657$	14.9467	0.583126
$658$	0	0
$659$	26.8205	1.04478	0.522389	−	0.852707i	$-0.325041\pi$
$659$	26.8205	1.04478	0.522389	+	0.852707i	$0.325041\pi$
$660$	−48.0265	−1.86943
$661$	6.03410	0.234699	0.117350	−	0.993091i	$-0.462560\pi$
$661$	6.03410	0.234699	0.117350	+	0.993091i	$0.462560\pi$
$662$	−65.8619	−2.55979
$663$	−3.10013	−0.120399
$664$	11.5899	0.449776
$665$	0	0
$666$	22.8753	0.886398
$667$	21.2295	0.822009
$668$	49.7371	1.92439
$669$	18.3129	0.708018
$670$	−13.6153	−0.526006
$671$	−1.03886	−0.0401048
$672$	0	0
$673$	38.7403	1.49333	0.746665	−	0.665201i	$-0.231654\pi$
$673$	38.7403	1.49333	0.746665	+	0.665201i	$0.231654\pi$
$674$	−68.3671	−2.63340
$675$	−77.6092	−2.98718
$676$	−4.76403	−0.183232
$677$	46.5558	1.78928	0.894642	−	0.446784i	$-0.147431\pi$
$677$	46.5558	1.78928	0.894642	+	0.446784i	$0.147431\pi$
$678$	−15.6529	−0.601146
$679$	0	0
$680$	−2.96123	−0.113558
$681$	17.7671	0.680837
$682$	−62.4559	−2.39156
$683$	42.7857	1.63715	0.818575	−	0.574400i	$-0.194764\pi$
$683$	42.7857	1.63715	0.818575	+	0.574400i	$0.194764\pi$
$684$	8.71728	0.333314
$685$	73.4387	2.80595
$686$	0	0
$687$	31.3801	1.19723
$688$	−34.0410	−1.29780
$689$	−0.233115	−0.00888096
$690$	42.8284	1.63045
$691$	−42.3190	−1.60989	−0.804946	−	0.593349i	$-0.797806\pi$
$691$	−42.3190	−1.60989	−0.804946	+	0.593349i	$0.797806\pi$
$692$	−35.5575	−1.35169
$693$	0	0
$694$	29.7907	1.13084
$695$	25.4083	0.963791
$696$	−6.62452	−0.251102
$697$	0.740538	0.0280499
$698$	−63.8934	−2.41840
$699$	−31.2742	−1.18290
$700$	0	0
$701$	12.3774	0.467487	0.233744	−	0.972298i	$-0.424902\pi$
$701$	12.3774	0.467487	0.233744	+	0.972298i	$0.424902\pi$
$702$	38.9118	1.46863
$703$	19.4612	0.733993
$704$	−39.6273	−1.49351
$705$	−30.9325	−1.16498
$706$	40.8291	1.53662
$707$	0	0
$708$	−7.10485	−0.267016
$709$	21.4871	0.806963	0.403482	−	0.914988i	$-0.367800\pi$
$709$	21.4871	0.806963	0.403482	+	0.914988i	$0.367800\pi$
$710$	129.379	4.85549
$711$	−11.9333	−0.447534
$712$	16.1497	0.605235
$713$	30.5852	1.14542
$714$	0	0
$715$	52.0125	1.94516
$716$	49.1044	1.83512
$717$	35.2964	1.31817
$718$	−3.49183	−0.130314
$719$	−26.6690	−0.994587	−0.497293	−	0.867582i	$-0.665673\pi$
$719$	−26.6690	−0.994587	−0.497293	+	0.867582i	$0.665673\pi$
$720$	−18.0781	−0.673732
$721$	0	0
$722$	−26.5124	−0.986691
$723$	−22.2573	−0.827759
$724$	10.3432	0.384402
$725$	−79.9444	−2.96906
$726$	5.09135	0.188958
$727$	−3.21407	−0.119203	−0.0596017	−	0.998222i	$-0.518983\pi$
$727$	−3.21407	−0.119203	−0.0596017	+	0.998222i	$0.518983\pi$
$728$	0	0
$729$	24.9138	0.922734
$730$	96.9986	3.59008
$731$	8.58064	0.317366
$732$	−0.886924	−0.0327817
$733$	−10.4404	−0.385623	−0.192812	−	0.981236i	$-0.561761\pi$
$733$	−10.4404	−0.385623	−0.192812	+	0.981236i	$0.561761\pi$
$734$	−31.4788	−1.16190
$735$	0	0
$736$	29.7480	1.09653
$737$	5.33608	0.196557
$738$	−2.97644	−0.109564
$739$	−17.2518	−0.634617	−0.317308	−	0.948322i	$-0.602779\pi$
$739$	−17.2518	−0.634617	−0.317308	+	0.948322i	$0.602779\pi$
$740$	81.5217	2.99680
$741$	10.6007	0.389425
$742$	0	0
$743$	13.7042	0.502758	0.251379	−	0.967889i	$-0.419116\pi$
$743$	13.7042	0.502758	0.251379	+	0.967889i	$0.419116\pi$
$744$	−9.54391	−0.349897
$745$	29.0700	1.06504
$746$	−23.2225	−0.850237
$747$	−17.8352	−0.652558
$748$	6.48399	0.237078
$749$	0	0
$750$	−103.517	−3.77990
$751$	5.53248	0.201883	0.100942	−	0.994892i	$-0.467814\pi$
$751$	5.53248	0.201883	0.100942	+	0.994892i	$0.467814\pi$
$752$	−16.5675	−0.604156
$753$	−13.1578	−0.479498
$754$	40.0827	1.45973
$755$	14.4424	0.525612
$756$	0	0
$757$	27.4953	0.999332	0.499666	−	0.866218i	$-0.333456\pi$
$757$	27.4953	0.999332	0.499666	+	0.866218i	$0.333456\pi$
$758$	−11.9737	−0.434904
$759$	−16.7852	−0.609263
$760$	10.1257	0.367298
$761$	18.9868	0.688270	0.344135	−	0.938920i	$-0.388172\pi$
$761$	18.9868	0.688270	0.344135	+	0.938920i	$0.388172\pi$
$762$	5.13781	0.186123
$763$	0	0
$764$	−15.3055	−0.553732
$765$	4.55692	0.164756
$766$	−33.1564	−1.19799
$767$	7.69452	0.277833
$768$	8.74763	0.315653
$769$	−14.2359	−0.513358	−0.256679	−	0.966497i	$-0.582628\pi$
$769$	−14.2359	−0.513358	−0.256679	+	0.966497i	$0.582628\pi$
$770$	0	0
$771$	27.4114	0.987199
$772$	−15.8194	−0.569354
$773$	−25.6161	−0.921347	−0.460674	−	0.887570i	$-0.652392\pi$
$773$	−25.6161	−0.921347	−0.460674	+	0.887570i	$0.652392\pi$
$774$	−34.4881	−1.23965
$775$	−115.176	−4.13723
$776$	1.93645	0.0695145
$777$	0	0
$778$	59.3133	2.12649
$779$	−2.53221	−0.0907260
$780$	44.4055	1.58997
$781$	−50.7057	−1.81439
$782$	−5.78219	−0.206771
$783$	31.8351	1.13769
$784$	0	0
$785$	31.9633	1.14082
$786$	−19.3814	−0.691312
$787$	−28.2897	−1.00842	−0.504210	−	0.863581i	$-0.668216\pi$
$787$	−28.2897	−1.00842	−0.504210	+	0.863581i	$0.668216\pi$
$788$	−26.1634	−0.932033
$789$	−1.04292	−0.0371290
$790$	−77.4428	−2.75529
$791$	0	0
$792$	−4.66462	−0.165750
$793$	0.960535	0.0341096
$794$	−39.0391	−1.38545
$795$	−0.384757	−0.0136459
$796$	15.4824	0.548759
$797$	2.55854	0.0906281	0.0453141	−	0.998973i	$-0.485571\pi$
$797$	2.55854	0.0906281	0.0453141	+	0.998973i	$0.485571\pi$
$798$	0	0
$799$	4.17614	0.147741
$800$	−112.023	−3.96061
$801$	−24.8521	−0.878106
$802$	−18.9849	−0.670381
$803$	−38.0154	−1.34153
$804$	4.55566	0.160666
$805$	0	0
$806$	57.7469	2.03405
$807$	25.1827	0.886472
$808$	5.61527	0.197544
$809$	−14.4954	−0.509633	−0.254816	−	0.966989i	$-0.582015\pi$
$809$	−14.4954	−0.509633	−0.254816	+	0.966989i	$0.582015\pi$
$810$	25.3427	0.890453
$811$	−0.673450	−0.0236480	−0.0118240	−	0.999930i	$-0.503764\pi$
$811$	−0.673450	−0.0236480	−0.0118240	+	0.999930i	$0.503764\pi$
$812$	0	0
$813$	−28.4230	−0.996838
$814$	−58.1809	−2.03924
$815$	7.36977	0.258152
$816$	−2.74056	−0.0959389
$817$	−29.3409	−1.02651
$818$	−28.4116	−0.993388
$819$	0	0
$820$	−10.6073	−0.370422
$821$	42.9684	1.49961	0.749804	−	0.661660i	$-0.230148\pi$
$821$	42.9684	1.49961	0.749804	+	0.661660i	$0.230148\pi$
$822$	−44.7467	−1.56072
$823$	−14.8325	−0.517028	−0.258514	−	0.966007i	$-0.583233\pi$
$823$	−14.8325	−0.517028	−0.258514	+	0.966007i	$0.583233\pi$
$824$	−6.38964	−0.222594
$825$	63.2084	2.20063
$826$	0	0
$827$	−8.46975	−0.294522	−0.147261	−	0.989098i	$-0.547046\pi$
$827$	−8.46975	−0.294522	−0.147261	+	0.989098i	$0.547046\pi$
$828$	12.7623	0.443521
$829$	39.7121	1.37926	0.689629	−	0.724163i	$-0.257774\pi$
$829$	39.7121	1.37926	0.689629	+	0.724163i	$0.257774\pi$
$830$	−115.744	−4.01754
$831$	−18.5473	−0.643397
$832$	36.6395	1.27025
$833$	0	0
$834$	−15.4814	−0.536078
$835$	−88.9045	−3.07667
$836$	−22.1715	−0.766818
$837$	45.8646	1.58531
$838$	−58.1593	−2.00908
$839$	49.1299	1.69615	0.848076	−	0.529874i	$-0.177761\pi$
$839$	49.1299	1.69615	0.848076	+	0.529874i	$0.177761\pi$
$840$	0	0
$841$	3.79298	0.130792
$842$	15.4686	0.533082
$843$	−38.6090	−1.32976
$844$	−5.99829	−0.206470
$845$	8.51564	0.292947
$846$	−16.7851	−0.577085
$847$	0	0
$848$	−0.206077	−0.00707672
$849$	−3.28780	−0.112837
$850$	21.7742	0.746848
$851$	28.4917	0.976682
$852$	−43.2898	−1.48308
$853$	−1.86593	−0.0638882	−0.0319441	−	0.999490i	$-0.510170\pi$
$853$	−1.86593	−0.0638882	−0.0319441	+	0.999490i	$0.510170\pi$
$854$	0	0
$855$	−15.5820	−0.532894
$856$	3.60913	0.123357
$857$	6.18827	0.211387	0.105694	−	0.994399i	$-0.466294\pi$
$857$	6.18827	0.211387	0.105694	+	0.994399i	$0.466294\pi$
$858$	−31.6916	−1.08193
$859$	−38.6170	−1.31760	−0.658798	−	0.752320i	$-0.728935\pi$
$859$	−38.6170	−1.31760	−0.658798	+	0.752320i	$0.728935\pi$
$860$	−122.907	−4.19109
$861$	0	0
$862$	−54.9890	−1.87293
$863$	37.7365	1.28457	0.642283	−	0.766468i	$-0.277987\pi$
$863$	37.7365	1.28457	0.642283	+	0.766468i	$0.277987\pi$
$864$	44.6092	1.51763
$865$	63.5586	2.16106
$866$	−61.5614	−2.09194
$867$	−20.7239	−0.703820
$868$	0	0
$869$	30.3512	1.02959
$870$	66.1566	2.24292
$871$	−4.93376	−0.167174
$872$	11.6695	0.395180
$873$	−2.97992	−0.100855
$874$	19.7718	0.668791
$875$	0	0
$876$	−32.4555	−1.09657
$877$	1.08029	0.0364787	0.0182394	−	0.999834i	$-0.494194\pi$
$877$	1.08029	0.0364787	0.0182394	+	0.999834i	$0.494194\pi$
$878$	33.0251	1.11454
$879$	−30.0502	−1.01357
$880$	45.9799	1.54998
$881$	−21.9302	−0.738846	−0.369423	−	0.929261i	$-0.620445\pi$
$881$	−21.9302	−0.738846	−0.369423	+	0.929261i	$0.620445\pi$
$882$	0	0
$883$	−48.1701	−1.62105	−0.810527	−	0.585702i	$-0.800819\pi$
$883$	−48.1701	−1.62105	−0.810527	+	0.585702i	$0.800819\pi$
$884$	−5.99512	−0.201638
$885$	12.6998	0.426900
$886$	25.0076	0.840148
$887$	−18.6773	−0.627124	−0.313562	−	0.949568i	$-0.601522\pi$
$887$	−18.6773	−0.627124	−0.313562	+	0.949568i	$0.601522\pi$
$888$	−8.89064	−0.298350
$889$	0	0
$890$	−161.281	−5.40615
$891$	−9.93225	−0.332743
$892$	35.4140	1.18575
$893$	−14.2800	−0.477862
$894$	−17.7125	−0.592396
$895$	−87.7736	−2.93395
$896$	0	0
$897$	15.5196	0.518185
$898$	−21.0924	−0.703863
$899$	47.2447	1.57570
$900$	−48.0594	−1.60198
$901$	0.0519454	0.00173055
$902$	7.57026	0.252062
$903$	0	0
$904$	−5.41798	−0.180199
$905$	−18.4883	−0.614573
$906$	−8.79983	−0.292355
$907$	13.2202	0.438970	0.219485	−	0.975616i	$-0.429562\pi$
$907$	13.2202	0.438970	0.219485	+	0.975616i	$0.429562\pi$
$908$	34.3585	1.14023
$909$	−8.64110	−0.286607
$910$	0	0
$911$	4.36801	0.144719	0.0723593	−	0.997379i	$-0.476947\pi$
$911$	4.36801	0.144719	0.0723593	+	0.997379i	$0.476947\pi$
$912$	9.37115	0.310310
$913$	45.3621	1.50127
$914$	−32.3676	−1.07063
$915$	1.58537	0.0524106
$916$	60.6838	2.00505
$917$	0	0
$918$	−8.67080	−0.286179
$919$	22.6548	0.747313	0.373657	−	0.927567i	$-0.378104\pi$
$919$	22.6548	0.747313	0.373657	+	0.927567i	$0.378104\pi$
$920$	14.8243	0.488742
$921$	6.33852	0.208861
$922$	6.53016	0.215059
$923$	46.8826	1.54316
$924$	0	0
$925$	−107.292	−3.52774
$926$	−9.17906	−0.301643
$927$	9.83274	0.322950
$928$	45.9515	1.50843
$929$	7.76388	0.254725	0.127362	−	0.991856i	$-0.459349\pi$
$929$	7.76388	0.254725	0.127362	+	0.991856i	$0.459349\pi$
$930$	95.3115	3.12539
$931$	0	0
$932$	−60.4789	−1.98105
$933$	28.7028	0.939689
$934$	90.3522	2.95641
$935$	−11.5900	−0.379035
$936$	4.31292	0.140972
$937$	5.75669	0.188063	0.0940315	−	0.995569i	$-0.470025\pi$
$937$	5.75669	0.188063	0.0940315	+	0.995569i	$0.470025\pi$
$938$	0	0
$939$	−8.16493	−0.266453
$940$	−59.8181	−1.95105
$941$	−20.2050	−0.658665	−0.329332	−	0.944214i	$-0.606824\pi$
$941$	−20.2050	−0.658665	−0.329332	+	0.944214i	$0.606824\pi$
$942$	−19.4754	−0.634544
$943$	−3.70722	−0.120724
$944$	6.80207	0.221389
$945$	0	0
$946$	87.7169	2.85192
$947$	15.5551	0.505473	0.252736	−	0.967535i	$-0.418669\pi$
$947$	15.5551	0.505473	0.252736	+	0.967535i	$0.418669\pi$
$948$	25.9122	0.841589
$949$	35.1492	1.14099
$950$	−74.4552	−2.41564
$951$	−11.4427	−0.371055
$952$	0	0
$953$	11.5722	0.374861	0.187430	−	0.982278i	$-0.439984\pi$
$953$	11.5722	0.374861	0.187430	+	0.982278i	$0.439984\pi$
$954$	−0.208784	−0.00675962
$955$	27.3583	0.885295
$956$	68.2571	2.20759
$957$	−25.9279	−0.838130
$958$	14.4760	0.467697
$959$	0	0
$960$	60.4737	1.95178
$961$	37.0652	1.19565
$962$	53.7942	1.73440
$963$	−5.55394	−0.178973
$964$	−43.0419	−1.38629
$965$	28.2770	0.910270
$966$	0	0
$967$	9.57651	0.307960	0.153980	−	0.988074i	$-0.450791\pi$
$967$	9.57651	0.307960	0.153980	+	0.988074i	$0.450791\pi$
$968$	1.76228	0.0566419
$969$	−2.36217	−0.0758837
$970$	−19.3386	−0.620925
$971$	−14.5478	−0.466860	−0.233430	−	0.972374i	$-0.574995\pi$
$971$	−14.5478	−0.466860	−0.233430	+	0.972374i	$0.574995\pi$
$972$	32.1476	1.03113
$973$	0	0
$974$	−33.6825	−1.07926
$975$	−58.4427	−1.87166
$976$	0.849128	0.0271799
$977$	−8.39153	−0.268469	−0.134234	−	0.990950i	$-0.542858\pi$
$977$	−8.39153	−0.268469	−0.134234	+	0.990950i	$0.542858\pi$
$978$	−4.49045	−0.143589
$979$	63.2088	2.02016
$980$	0	0
$981$	−17.9577	−0.573346
$982$	60.4325	1.92848
$983$	−4.32914	−0.138078	−0.0690391	−	0.997614i	$-0.521993\pi$
$983$	−4.32914	−0.138078	−0.0690391	+	0.997614i	$0.521993\pi$
$984$	1.15682	0.0368779
$985$	46.7668	1.49011
$986$	−8.93170	−0.284443
$987$	0	0
$988$	20.4999	0.652187
$989$	−42.9557	−1.36591
$990$	46.5838	1.48053
$991$	31.1511	0.989548	0.494774	−	0.869022i	$-0.335251\pi$
$991$	31.1511	0.989548	0.494774	+	0.869022i	$0.335251\pi$
$992$	66.2020	2.10192
$993$	−39.3913	−1.25004
$994$	0	0
$995$	−27.6746	−0.877344
$996$	38.7278	1.22714
$997$	−5.25310	−0.166367	−0.0831837	−	0.996534i	$-0.526509\pi$
$997$	−5.25310	−0.166367	−0.0831837	+	0.996534i	$0.526509\pi$
$998$	18.0936	0.572742
$999$	42.7252	1.35177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.p.1.7	✓	7
7.6	odd	2		2009.2.a.q.1.7	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.p.1.7	✓	7	1.1	even	1	trivial
2009.2.a.q.1.7	yes	7	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.10619 q^{2} +1.25969 q^{3} +2.43602 q^{4} -4.35435 q^{5} +2.65313 q^{6} +0.918336 q^{8} -1.41319 q^{9} +O(q^{10})\)	\(q+2.10619 q^{2} +1.25969 q^{3} +2.43602 q^{4} -4.35435 q^{5} +2.65313 q^{6} +0.918336 q^{8} -1.41319 q^{9} -9.17108 q^{10} +3.59430 q^{11} +3.06862 q^{12} -3.32330 q^{13} -5.48512 q^{15} -2.93785 q^{16} +0.740538 q^{17} -2.97644 q^{18} -2.53221 q^{19} -10.6073 q^{20} +7.57026 q^{22} -3.70722 q^{23} +1.15682 q^{24} +13.9604 q^{25} -6.99949 q^{26} -5.55924 q^{27} -5.72652 q^{29} -11.5527 q^{30} -8.25016 q^{31} -8.02433 q^{32} +4.52769 q^{33} +1.55971 q^{34} -3.44255 q^{36} -7.68545 q^{37} -5.33331 q^{38} -4.18632 q^{39} -3.99876 q^{40} +1.00000 q^{41} +11.5870 q^{43} +8.75578 q^{44} +6.15352 q^{45} -7.80810 q^{46} +5.63934 q^{47} -3.70077 q^{48} +29.4032 q^{50} +0.932846 q^{51} -8.09563 q^{52} +0.0701455 q^{53} -11.7088 q^{54} -15.6509 q^{55} -3.18980 q^{57} -12.0611 q^{58} -2.31532 q^{59} -13.3619 q^{60} -0.289030 q^{61} -17.3764 q^{62} -11.0250 q^{64} +14.4708 q^{65} +9.53616 q^{66} +1.48459 q^{67} +1.80396 q^{68} -4.66994 q^{69} -14.1072 q^{71} -1.29778 q^{72} -10.5766 q^{73} -16.1870 q^{74} +17.5857 q^{75} -6.16852 q^{76} -8.81717 q^{78} +8.44424 q^{79} +12.7924 q^{80} -2.76333 q^{81} +2.10619 q^{82} +12.6206 q^{83} -3.22456 q^{85} +24.4045 q^{86} -7.21362 q^{87} +3.30077 q^{88} +17.5858 q^{89} +12.9605 q^{90} -9.03087 q^{92} -10.3926 q^{93} +11.8775 q^{94} +11.0262 q^{95} -10.1081 q^{96} +2.10865 q^{97} -5.07942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9	\( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) \) 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9 - 8 * q^10 - 12 * q^12 - q^13 - 4 * q^15 + 5 * q^16 + 11 * q^17 + 16 * q^18 - 9 * q^19 - 12 * q^20 + 3 * q^23 + 23 * q^24 + 31 * q^25 - 10 * q^26 - 22 * q^27 - 14 * q^29 + 6 * q^30 - 34 * q^31 - 20 * q^32 - 12 * q^33 - 15 * q^34 + q^36 + 7 * q^37 + 39 * q^38 - 22 * q^39 - 50 * q^40 + 7 * q^41 - 3 * q^43 + 26 * q^44 + 4 * q^45 - 8 * q^46 - 17 * q^47 - 17 * q^48 + 11 * q^50 + 8 * q^51 + 25 * q^52 + 24 * q^53 - 68 * q^54 - 48 * q^55 + 22 * q^57 - 38 * q^58 - 4 * q^59 - 6 * q^60 - 16 * q^61 - 24 * q^62 + 8 * q^64 - 6 * q^65 + 12 * q^66 - 24 * q^67 + 10 * q^68 - 35 * q^69 - 12 * q^71 - 11 * q^72 - 14 * q^73 - 6 * q^74 + 19 * q^75 - 42 * q^76 - 29 * q^78 - 8 * q^79 + 92 * q^80 + 15 * q^81 - q^82 - 14 * q^83 + 16 * q^85 + 35 * q^86 + 20 * q^87 + 22 * q^88 + 19 * q^89 - 24 * q^90 - 10 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^95 + 25 * q^96 - 23 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

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Database

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