LMFDB - Embedded newform 2009.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$16.0419457661$
Analytic rank:	$0$
Dimension:	$7$
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			1.3
Root			$1.28109$ 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)

$f(q)$	$=$	$q-1.35880 q^{2} -0.281095 q^{3} -0.153673 q^{4} +3.77273 q^{5} +0.381951 q^{6} +2.92640 q^{8} -2.92099 q^{9} +O(q^{10})$	\(q-1.35880 q^{2} -0.281095 q^{3} -0.153673 q^{4} +3.77273 q^{5} +0.381951 q^{6} +2.92640 q^{8} -2.92099 q^{9} -5.12637 q^{10} -1.53021 q^{11} +0.0431968 q^{12} -0.587073 q^{13} -1.06049 q^{15} -3.66904 q^{16} -6.56030 q^{17} +3.96902 q^{18} -0.782214 q^{19} -0.579768 q^{20} +2.07924 q^{22} +3.72301 q^{23} -0.822597 q^{24} +9.23349 q^{25} +0.797713 q^{26} +1.66436 q^{27} +7.82177 q^{29} +1.44100 q^{30} +5.80820 q^{31} -0.867331 q^{32} +0.430133 q^{33} +8.91411 q^{34} +0.448877 q^{36} +8.38915 q^{37} +1.06287 q^{38} +0.165023 q^{39} +11.0405 q^{40} -1.00000 q^{41} -9.36609 q^{43} +0.235152 q^{44} -11.0201 q^{45} -5.05881 q^{46} +11.9226 q^{47} +1.03135 q^{48} -12.5464 q^{50} +1.84407 q^{51} +0.0902175 q^{52} +1.92403 q^{53} -2.26152 q^{54} -5.77306 q^{55} +0.219876 q^{57} -10.6282 q^{58} +4.28746 q^{59} +0.162970 q^{60} -8.46897 q^{61} -7.89217 q^{62} +8.51660 q^{64} -2.21487 q^{65} -0.584463 q^{66} -11.5220 q^{67} +1.00814 q^{68} -1.04652 q^{69} +8.18700 q^{71} -8.54798 q^{72} -1.05736 q^{73} -11.3991 q^{74} -2.59549 q^{75} +0.120205 q^{76} -0.224233 q^{78} +7.19339 q^{79} -13.8423 q^{80} +8.29511 q^{81} +1.35880 q^{82} +8.44048 q^{83} -24.7502 q^{85} +12.7266 q^{86} -2.19866 q^{87} -4.47800 q^{88} +7.83130 q^{89} +14.9741 q^{90} -0.572127 q^{92} -1.63266 q^{93} -16.2004 q^{94} -2.95108 q^{95} +0.243802 q^{96} +13.2606 q^{97} +4.46971 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.35880	−0.960814	−0.480407	−	0.877046i	$-0.659511\pi$
$2$	−1.35880	−0.960814	−0.480407	+	0.877046i	$0.659511\pi$
$3$	−0.281095	−0.162290	−0.0811451	−	0.996702i	$-0.525858\pi$
$3$	−0.281095	−0.162290	−0.0811451	+	0.996702i	$0.525858\pi$
$4$	−0.153673	−0.0768366
$5$	3.77273	1.68722	0.843608	−	0.536960i	$-0.180427\pi$
$5$	3.77273	1.68722	0.843608	+	0.536960i	$0.180427\pi$
$6$	0.381951	0.155931
$7$	0	0
$7$	0	0
$8$	2.92640	1.03464
$9$	−2.92099	−0.973662
$10$	−5.12637	−1.62110
$11$	−1.53021	−0.461375	−0.230687	−	0.973028i	$-0.574097\pi$
$11$	−1.53021	−0.461375	−0.230687	+	0.973028i	$0.574097\pi$
$12$	0.0431968	0.0124698
$13$	−0.587073	−0.162825	−0.0814124	−	0.996680i	$-0.525943\pi$
$13$	−0.587073	−0.162825	−0.0814124	+	0.996680i	$0.525943\pi$
$14$	0	0
$15$	−1.06049	−0.273819
$16$	−3.66904	−0.917260
$17$	−6.56030	−1.59111	−0.795553	−	0.605884i	$-0.792820\pi$
$17$	−6.56030	−1.59111	−0.795553	+	0.605884i	$0.792820\pi$
$18$	3.96902	0.935508
$19$	−0.782214	−0.179452	−0.0897261	−	0.995966i	$-0.528599\pi$
$19$	−0.782214	−0.179452	−0.0897261	+	0.995966i	$0.528599\pi$
$20$	−0.579768	−0.129640
$21$	0	0
$22$	2.07924	0.443295
$23$	3.72301	0.776301	0.388151	−	0.921596i	$-0.373114\pi$
$23$	3.72301	0.776301	0.388151	+	0.921596i	$0.373114\pi$
$24$	−0.822597	−0.167912
$25$	9.23349	1.84670
$26$	0.797713	0.156444
$27$	1.66436	0.320306
$28$	0	0
$29$	7.82177	1.45247	0.726233	−	0.687449i	$-0.241269\pi$
$29$	7.82177	1.45247	0.726233	+	0.687449i	$0.241269\pi$
$30$	1.44100	0.263089
$31$	5.80820	1.04318	0.521592	−	0.853195i	$-0.325338\pi$
$31$	5.80820	1.04318	0.521592	+	0.853195i	$0.325338\pi$
$32$	−0.867331	−0.153324
$33$	0.430133	0.0748766
$34$	8.91411	1.52876
$35$	0	0
$36$	0.448877	0.0748129
$37$	8.38915	1.37917	0.689584	−	0.724206i	$-0.257793\pi$
$37$	8.38915	1.37917	0.689584	+	0.724206i	$0.257793\pi$
$38$	1.06287	0.172420
$39$	0.165023	0.0264249
$40$	11.0405	1.74566
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−9.36609	−1.42831	−0.714157	−	0.699985i	$-0.753190\pi$
$43$	−9.36609	−1.42831	−0.714157	+	0.699985i	$0.753190\pi$
$44$	0.235152	0.0354505
$45$	−11.0201	−1.64278
$46$	−5.05881	−0.745881
$47$	11.9226	1.73909	0.869547	−	0.493851i	$-0.164411\pi$
$47$	11.9226	1.73909	0.869547	+	0.493851i	$0.164411\pi$
$48$	1.03135	0.148862
$49$	0	0
$50$	−12.5464	−1.77433
$51$	1.84407	0.258221
$52$	0.0902175	0.0125109
$53$	1.92403	0.264286	0.132143	−	0.991231i	$-0.457814\pi$
$53$	1.92403	0.264286	0.132143	+	0.991231i	$0.457814\pi$
$54$	−2.26152	−0.307754
$55$	−5.77306	−0.778439
$56$	0	0
$57$	0.219876	0.0291233
$58$	−10.6282	−1.39555
$59$	4.28746	0.558180	0.279090	−	0.960265i	$-0.409967\pi$
$59$	4.28746	0.558180	0.279090	+	0.960265i	$0.409967\pi$
$60$	0.162970	0.0210393
$61$	−8.46897	−1.08434	−0.542170	−	0.840269i	$-0.682397\pi$
$61$	−8.46897	−1.08434	−0.542170	+	0.840269i	$0.682397\pi$
$62$	−7.89217	−1.00231
$63$	0	0
$64$	8.51660	1.06458
$65$	−2.21487	−0.274721
$66$	−0.584463	−0.0719425
$67$	−11.5220	−1.40764	−0.703818	−	0.710381i	$-0.748523\pi$
$67$	−11.5220	−1.40764	−0.703818	+	0.710381i	$0.748523\pi$
$68$	1.00814	0.122255
$69$	−1.04652	−0.125986
$70$	0	0
$71$	8.18700	0.971618	0.485809	−	0.874065i	$-0.338525\pi$
$71$	8.18700	0.971618	0.485809	+	0.874065i	$0.338525\pi$
$72$	−8.54798	−1.00739
$73$	−1.05736	−0.123755	−0.0618774	−	0.998084i	$-0.519709\pi$
$73$	−1.05736	−0.123755	−0.0618774	+	0.998084i	$0.519709\pi$
$74$	−11.3991	−1.32512
$75$	−2.59549	−0.299701
$76$	0.120205	0.0137885
$77$	0	0
$78$	−0.224233	−0.0253894
$79$	7.19339	0.809319	0.404660	−	0.914467i	$-0.367390\pi$
$79$	7.19339	0.809319	0.404660	+	0.914467i	$0.367390\pi$
$80$	−13.8423	−1.54761
$81$	8.29511	0.921679
$82$	1.35880	0.150054
$83$	8.44048	0.926463	0.463231	−	0.886237i	$-0.346690\pi$
$83$	8.44048	0.926463	0.463231	+	0.886237i	$0.346690\pi$
$84$	0	0
$85$	−24.7502	−2.68454
$86$	12.7266	1.37234
$87$	−2.19866	−0.235721
$88$	−4.47800	−0.477357
$89$	7.83130	0.830116	0.415058	−	0.909795i	$-0.363761\pi$
$89$	7.83130	0.830116	0.415058	+	0.909795i	$0.363761\pi$
$90$	14.9741	1.57840
$91$	0	0
$92$	−0.572127	−0.0596484
$93$	−1.63266	−0.169299
$94$	−16.2004	−1.67095
$95$	−2.95108	−0.302775
$96$	0.243802	0.0248830
$97$	13.2606	1.34641	0.673204	−	0.739457i	$-0.264917\pi$
$97$	13.2606	1.34641	0.673204	+	0.739457i	$0.264917\pi$
$98$	0	0
$99$	4.46971	0.449223
$100$	−1.41894	−0.141894
$101$	−6.44946	−0.641746	−0.320873	−	0.947122i	$-0.603976\pi$
$101$	−6.44946	−0.641746	−0.320873	+	0.947122i	$0.603976\pi$
$102$	−2.50571	−0.248102
$103$	18.0323	1.77678	0.888388	−	0.459094i	$-0.151826\pi$
$103$	18.0323	1.77678	0.888388	+	0.459094i	$0.151826\pi$
$104$	−1.71801	−0.168465
$105$	0	0
$106$	−2.61436	−0.253929
$107$	13.0048	1.25722	0.628609	−	0.777721i	$-0.283625\pi$
$107$	13.0048	1.25722	0.628609	+	0.777721i	$0.283625\pi$
$108$	−0.255767	−0.0246112
$109$	−10.5419	−1.00973	−0.504866	−	0.863198i	$-0.668458\pi$
$109$	−10.5419	−1.00973	−0.504866	+	0.863198i	$0.668458\pi$
$110$	7.84441	0.747935
$111$	−2.35815	−0.223825
$112$	0	0
$113$	−0.782350	−0.0735973	−0.0367987	−	0.999323i	$-0.511716\pi$
$113$	−0.782350	−0.0735973	−0.0367987	+	0.999323i	$0.511716\pi$
$114$	−0.298767	−0.0279821
$115$	14.0459	1.30979
$116$	−1.20200	−0.111603
$117$	1.71483	0.158536
$118$	−5.82578	−0.536307
$119$	0	0
$120$	−3.10343	−0.283304
$121$	−8.65847	−0.787133
$122$	11.5076	1.04185
$123$	0.281095	0.0253455
$124$	−0.892566	−0.0801548
$125$	15.9718	1.42856
$126$	0	0
$127$	−3.17842	−0.282039	−0.141020	−	0.990007i	$-0.545038\pi$
$127$	−3.17842	−0.282039	−0.141020	+	0.990007i	$0.545038\pi$
$128$	−9.83766	−0.869535
$129$	2.63276	0.231801
$130$	3.00956	0.263955
$131$	−14.7349	−1.28739	−0.643695	−	0.765282i	$-0.722599\pi$
$131$	−14.7349	−1.28739	−0.643695	+	0.765282i	$0.722599\pi$
$132$	−0.0661000	−0.00575326
$133$	0	0
$134$	15.6560	1.35248
$135$	6.27917	0.540425
$136$	−19.1981	−1.64622
$137$	7.83949	0.669772	0.334886	−	0.942259i	$-0.391302\pi$
$137$	7.83949	0.669772	0.334886	+	0.942259i	$0.391302\pi$
$138$	1.42201	0.121049
$139$	10.9730	0.930716	0.465358	−	0.885123i	$-0.345926\pi$
$139$	10.9730	0.930716	0.465358	+	0.885123i	$0.345926\pi$
$140$	0	0
$141$	−3.35139	−0.282238
$142$	−11.1245	−0.933544
$143$	0.898344	0.0751233
$144$	10.7172	0.893101
$145$	29.5094	2.45062
$146$	1.43674	0.118905
$147$	0	0
$148$	−1.28919	−0.105971
$149$	−6.34397	−0.519718	−0.259859	−	0.965647i	$-0.583676\pi$
$149$	−6.34397	−0.519718	−0.259859	+	0.965647i	$0.583676\pi$
$150$	3.52674	0.287957
$151$	−0.708131	−0.0576269	−0.0288134	−	0.999585i	$-0.509173\pi$
$151$	−0.708131	−0.0576269	−0.0288134	+	0.999585i	$0.509173\pi$
$152$	−2.28907	−0.185668
$153$	19.1625	1.54920
$154$	0	0
$155$	21.9128	1.76008
$156$	−0.0253597	−0.00203040
$157$	12.2054	0.974096	0.487048	−	0.873375i	$-0.338074\pi$
$157$	12.2054	0.974096	0.487048	+	0.873375i	$0.338074\pi$
$158$	−9.77435	−0.777605
$159$	−0.540834	−0.0428909
$160$	−3.27221	−0.258691
$161$	0	0
$162$	−11.2714	−0.885562
$163$	1.01447	0.0794595	0.0397297	−	0.999210i	$-0.487350\pi$
$163$	1.01447	0.0794595	0.0397297	+	0.999210i	$0.487350\pi$
$164$	0.153673	0.0119999
$165$	1.62278	0.126333
$166$	−11.4689	−0.890158
$167$	−5.31016	−0.410913	−0.205456	−	0.978666i	$-0.565868\pi$
$167$	−5.31016	−0.410913	−0.205456	+	0.978666i	$0.565868\pi$
$168$	0	0
$169$	−12.6553	−0.973488
$170$	33.6305	2.57934
$171$	2.28484	0.174726
$172$	1.43932	0.109747
$173$	−2.51787	−0.191430	−0.0957151	−	0.995409i	$-0.530514\pi$
$173$	−2.51787	−0.191430	−0.0957151	+	0.995409i	$0.530514\pi$
$174$	2.98753	0.226484
$175$	0	0
$176$	5.61439	0.423200
$177$	−1.20518	−0.0905871
$178$	−10.6411	−0.797587
$179$	13.8227	1.03316	0.516580	−	0.856239i	$-0.327205\pi$
$179$	13.8227	1.03316	0.516580	+	0.856239i	$0.327205\pi$
$180$	1.69349	0.126226
$181$	−3.90137	−0.289987	−0.144993	−	0.989433i	$-0.546316\pi$
$181$	−3.90137	−0.289987	−0.144993	+	0.989433i	$0.546316\pi$
$182$	0	0
$183$	2.38058	0.175978
$184$	10.8950	0.803192
$185$	31.6500	2.32695
$186$	2.21845	0.162664
$187$	10.0386	0.734096
$188$	−1.83219	−0.133626
$189$	0	0
$190$	4.00992	0.290910
$191$	−15.2024	−1.10001	−0.550003	−	0.835163i	$-0.685373\pi$
$191$	−15.2024	−1.10001	−0.550003	+	0.835163i	$0.685373\pi$
$192$	−2.39397	−0.172770
$193$	14.7029	1.05834	0.529168	−	0.848517i	$-0.322504\pi$
$193$	14.7029	1.05834	0.529168	+	0.848517i	$0.322504\pi$
$194$	−18.0184	−1.29365
$195$	0.622588	0.0445845
$196$	0	0
$197$	11.4673	0.817010	0.408505	−	0.912756i	$-0.366050\pi$
$197$	11.4673	0.817010	0.408505	+	0.912756i	$0.366050\pi$
$198$	−6.07343	−0.431620
$199$	16.6817	1.18253	0.591266	−	0.806476i	$-0.298628\pi$
$199$	16.6817	1.18253	0.591266	+	0.806476i	$0.298628\pi$
$200$	27.0209	1.91067
$201$	3.23877	0.228445
$202$	8.76351	0.616598
$203$	0	0
$204$	−0.283384	−0.0198408
$205$	−3.77273	−0.263499
$206$	−24.5022	−1.70715
$207$	−10.8749	−0.755855
$208$	2.15399	0.149353
$209$	1.19695	0.0827947
$210$	0	0
$211$	9.81295	0.675551	0.337776	−	0.941227i	$-0.390325\pi$
$211$	9.81295	0.675551	0.337776	+	0.941227i	$0.390325\pi$
$212$	−0.295672	−0.0203068
$213$	−2.30132	−0.157684
$214$	−17.6708	−1.20795
$215$	−35.3357	−2.40988
$216$	4.87058	0.331401
$217$	0	0
$218$	14.3243	0.970164
$219$	0.297219	0.0200842
$220$	0.887164	0.0598126
$221$	3.85138	0.259072
$222$	3.20424	0.215055
$223$	7.71539	0.516661	0.258330	−	0.966057i	$-0.416828\pi$
$223$	7.71539	0.516661	0.258330	+	0.966057i	$0.416828\pi$
$224$	0	0
$225$	−26.9709	−1.79806
$226$	1.06305	0.0707133
$227$	26.8002	1.77879	0.889396	−	0.457137i	$-0.151125\pi$
$227$	26.8002	1.77879	0.889396	+	0.457137i	$0.151125\pi$
$228$	−0.0337891	−0.00223774
$229$	10.8206	0.715049	0.357524	−	0.933904i	$-0.383621\pi$
$229$	10.8206	0.715049	0.357524	+	0.933904i	$0.383621\pi$
$230$	−19.0855	−1.25846
$231$	0	0
$232$	22.8896	1.50278
$233$	−9.09982	−0.596149	−0.298075	−	0.954543i	$-0.596344\pi$
$233$	−9.09982	−0.596149	−0.298075	+	0.954543i	$0.596344\pi$
$234$	−2.33011	−0.152324
$235$	44.9808	2.93423
$236$	−0.658868	−0.0428886
$237$	−2.02202	−0.131345
$238$	0	0
$239$	−3.68484	−0.238353	−0.119176	−	0.992873i	$-0.538025\pi$
$239$	−3.68484	−0.238353	−0.119176	+	0.992873i	$0.538025\pi$
$240$	3.89100	0.251163
$241$	8.01651	0.516389	0.258194	−	0.966093i	$-0.416873\pi$
$241$	8.01651	0.516389	0.258194	+	0.966093i	$0.416873\pi$
$242$	11.7651	0.756289
$243$	−7.32479	−0.469885
$244$	1.30145	0.0833170
$245$	0	0
$246$	−0.381951	−0.0243523
$247$	0.459217	0.0292193
$248$	16.9971	1.07932
$249$	−2.37257	−0.150356
$250$	−21.7024	−1.37258
$251$	−1.60958	−0.101596	−0.0507979	−	0.998709i	$-0.516176\pi$
$251$	−1.60958	−0.101596	−0.0507979	+	0.998709i	$0.516176\pi$
$252$	0	0
$253$	−5.69697	−0.358166
$254$	4.31882	0.270987
$255$	6.95716	0.435675
$256$	−3.66583	−0.229114
$257$	−20.1668	−1.25797	−0.628986	−	0.777417i	$-0.716530\pi$
$257$	−20.1668	−1.25797	−0.628986	+	0.777417i	$0.716530\pi$
$258$	−3.57738	−0.222718
$259$	0	0
$260$	0.340366	0.0211086
$261$	−22.8473	−1.41421
$262$	20.0217	1.23694
$263$	−24.8290	−1.53102	−0.765512	−	0.643422i	$-0.777514\pi$
$263$	−24.8290	−1.53102	−0.765512	+	0.643422i	$0.777514\pi$
$264$	1.25874	0.0774703
$265$	7.25884	0.445907
$266$	0	0
$267$	−2.20134	−0.134720
$268$	1.77062	0.108158
$269$	30.7089	1.87235	0.936177	−	0.351530i	$-0.114338\pi$
$269$	30.7089	1.87235	0.936177	+	0.351530i	$0.114338\pi$
$270$	−8.53212	−0.519248
$271$	18.2934	1.11125	0.555624	−	0.831434i	$-0.312480\pi$
$271$	18.2934	1.11125	0.555624	+	0.831434i	$0.312480\pi$
$272$	24.0700	1.45946
$273$	0	0
$274$	−10.6523	−0.643527
$275$	−14.1291	−0.852020
$276$	0.160822	0.00968034
$277$	1.06650	0.0640798	0.0320399	−	0.999487i	$-0.489800\pi$
$277$	1.06650	0.0640798	0.0320399	+	0.999487i	$0.489800\pi$
$278$	−14.9100	−0.894245
$279$	−16.9657	−1.01571
$280$	0	0
$281$	1.32567	0.0790828	0.0395414	−	0.999218i	$-0.487410\pi$
$281$	1.32567	0.0790828	0.0395414	+	0.999218i	$0.487410\pi$
$282$	4.55385	0.271178
$283$	−21.5009	−1.27809	−0.639047	−	0.769168i	$-0.720671\pi$
$283$	−21.5009	−1.27809	−0.639047	+	0.769168i	$0.720671\pi$
$284$	−1.25812	−0.0746558
$285$	0.829534	0.0491373
$286$	−1.22067	−0.0721795
$287$	0	0
$288$	2.53346	0.149286
$289$	26.0375	1.53162
$290$	−40.0973	−2.35459
$291$	−3.72748	−0.218509
$292$	0.162488	0.00950890
$293$	−19.4483	−1.13618	−0.568092	−	0.822965i	$-0.692318\pi$
$293$	−19.4483	−1.13618	−0.568092	+	0.822965i	$0.692318\pi$
$294$	0	0
$295$	16.1754	0.941770
$296$	24.5500	1.42694
$297$	−2.54681	−0.147781
$298$	8.62017	0.499353
$299$	−2.18568	−0.126401
$300$	0.398857	0.0230280
$301$	0	0
$302$	0.962205	0.0553687
$303$	1.81291	0.104149
$304$	2.86997	0.164604
$305$	−31.9511	−1.82952
$306$	−26.0380	−1.48849
$307$	28.5378	1.62874	0.814369	−	0.580348i	$-0.197083\pi$
$307$	28.5378	1.62874	0.814369	+	0.580348i	$0.197083\pi$
$308$	0	0
$309$	−5.06879	−0.288353
$310$	−29.7750	−1.69111
$311$	9.63544	0.546375	0.273188	−	0.961961i	$-0.411922\pi$
$311$	9.63544	0.546375	0.273188	+	0.961961i	$0.411922\pi$
$312$	0.482925	0.0273402
$313$	−33.6028	−1.89934	−0.949671	−	0.313248i	$-0.898583\pi$
$313$	−33.6028	−1.89934	−0.949671	+	0.313248i	$0.898583\pi$
$314$	−16.5846	−0.935925
$315$	0	0
$316$	−1.10543	−0.0621854
$317$	−30.9417	−1.73786	−0.868931	−	0.494934i	$-0.835192\pi$
$317$	−30.9417	−1.73786	−0.868931	+	0.494934i	$0.835192\pi$
$318$	0.734884	0.0412102
$319$	−11.9689	−0.670131
$320$	32.1308	1.79617
$321$	−3.65557	−0.204034
$322$	0	0
$323$	5.13156	0.285528
$324$	−1.27474	−0.0708187
$325$	−5.42073	−0.300688
$326$	−1.37846	−0.0763458
$327$	2.96328	0.163870
$328$	−2.92640	−0.161584
$329$	0	0
$330$	−2.20502	−0.121382
$331$	−10.0239	−0.550962	−0.275481	−	0.961306i	$-0.588837\pi$
$331$	−10.0239	−0.550962	−0.275481	+	0.961306i	$0.588837\pi$
$332$	−1.29708	−0.0711863
$333$	−24.5046	−1.34284
$334$	7.21543	0.394811
$335$	−43.4694	−2.37498
$336$	0	0
$337$	5.00576	0.272681	0.136341	−	0.990662i	$-0.456466\pi$
$337$	5.00576	0.272681	0.136341	+	0.990662i	$0.456466\pi$
$338$	17.1960	0.935341
$339$	0.219915	0.0119441
$340$	3.80345	0.206271
$341$	−8.88775	−0.481299
$342$	−3.10463	−0.167879
$343$	0	0
$344$	−27.4089	−1.47779
$345$	−3.94823	−0.212566
$346$	3.42127	0.183929
$347$	−11.0497	−0.593177	−0.296589	−	0.955005i	$-0.595849\pi$
$347$	−11.0497	−0.593177	−0.296589	+	0.955005i	$0.595849\pi$
$348$	0.337875	0.0181120
$349$	10.7923	0.577698	0.288849	−	0.957375i	$-0.406727\pi$
$349$	10.7923	0.577698	0.288849	+	0.957375i	$0.406727\pi$
$350$	0	0
$351$	−0.977100	−0.0521538
$352$	1.32720	0.0707398
$353$	−21.4959	−1.14411	−0.572057	−	0.820214i	$-0.693854\pi$
$353$	−21.4959	−1.14411	−0.572057	+	0.820214i	$0.693854\pi$
$354$	1.63760	0.0870373
$355$	30.8873	1.63933
$356$	−1.20346	−0.0637833
$357$	0	0
$358$	−18.7823	−0.992674
$359$	−9.86289	−0.520543	−0.260272	−	0.965535i	$-0.583812\pi$
$359$	−9.86289	−0.520543	−0.260272	+	0.965535i	$0.583812\pi$
$360$	−32.2492	−1.69968
$361$	−18.3881	−0.967797
$362$	5.30117	0.278623
$363$	2.43385	0.127744
$364$	0	0
$365$	−3.98914	−0.208801
$366$	−3.23473	−0.169082
$367$	27.1347	1.41642	0.708210	−	0.706001i	$-0.249503\pi$
$367$	27.1347	1.41642	0.708210	+	0.706001i	$0.249503\pi$
$368$	−13.6599	−0.712070
$369$	2.92099	0.152060
$370$	−43.0059	−2.23577
$371$	0	0
$372$	0.250896	0.0130083
$373$	−2.74716	−0.142242	−0.0711212	−	0.997468i	$-0.522658\pi$
$373$	−2.74716	−0.142242	−0.0711212	+	0.997468i	$0.522658\pi$
$374$	−13.6404	−0.705330
$375$	−4.48959	−0.231842
$376$	34.8904	1.79934
$377$	−4.59195	−0.236498
$378$	0	0
$379$	29.3984	1.51010	0.755048	−	0.655670i	$-0.227614\pi$
$379$	29.3984	1.51010	0.755048	+	0.655670i	$0.227614\pi$
$380$	0.453502	0.0232642
$381$	0.893437	0.0457722
$382$	20.6569	1.05690
$383$	−6.69629	−0.342164	−0.171082	−	0.985257i	$-0.554726\pi$
$383$	−6.69629	−0.342164	−0.171082	+	0.985257i	$0.554726\pi$
$384$	2.76532	0.141117
$385$	0	0
$386$	−19.9782	−1.01686
$387$	27.3582	1.39070
$388$	−2.03780	−0.103453
$389$	0.478863	0.0242793	0.0121397	−	0.999926i	$-0.496136\pi$
$389$	0.478863	0.0242793	0.0121397	+	0.999926i	$0.496136\pi$
$390$	−0.845970	−0.0428374
$391$	−24.4241	−1.23518
$392$	0	0
$393$	4.14189	0.208931
$394$	−15.5817	−0.784995
$395$	27.1387	1.36550
$396$	−0.686875	−0.0345168
$397$	−24.2293	−1.21603	−0.608016	−	0.793925i	$-0.708034\pi$
$397$	−24.2293	−1.21603	−0.608016	+	0.793925i	$0.708034\pi$
$398$	−22.6670	−1.13619
$399$	0	0
$400$	−33.8780	−1.69390
$401$	23.2843	1.16276	0.581381	−	0.813631i	$-0.302512\pi$
$401$	23.2843	1.16276	0.581381	+	0.813631i	$0.302512\pi$
$402$	−4.40083	−0.219493
$403$	−3.40984	−0.169856
$404$	0.991110	0.0493096
$405$	31.2952	1.55507
$406$	0	0
$407$	−12.8371	−0.636313
$408$	5.39648	0.267166
$409$	30.6511	1.51560	0.757800	−	0.652487i	$-0.226274\pi$
$409$	30.6511	1.51560	0.757800	+	0.652487i	$0.226274\pi$
$410$	5.12637	0.253173
$411$	−2.20364	−0.108697
$412$	−2.77108	−0.136521
$413$	0	0
$414$	14.7767	0.726236
$415$	31.8436	1.56314
$416$	0.509187	0.0249649
$417$	−3.08445	−0.151046
$418$	−1.62641	−0.0795503
$419$	5.54908	0.271090	0.135545	−	0.990771i	$-0.456721\pi$
$419$	5.54908	0.271090	0.135545	+	0.990771i	$0.456721\pi$
$420$	0	0
$421$	11.8783	0.578913	0.289457	−	0.957191i	$-0.406525\pi$
$421$	11.8783	0.578913	0.289457	+	0.957191i	$0.406525\pi$
$422$	−13.3338	−0.649079
$423$	−34.8258	−1.69329
$424$	5.63048	0.273440
$425$	−60.5745	−2.93829
$426$	3.12703	0.151505
$427$	0	0
$428$	−1.99849	−0.0966004
$429$	−0.252520	−0.0121918
$430$	48.0140	2.31544
$431$	−15.8059	−0.761345	−0.380673	−	0.924710i	$-0.624307\pi$
$431$	−15.8059	−0.761345	−0.380673	+	0.924710i	$0.624307\pi$
$432$	−6.10659	−0.293804
$433$	−17.8949	−0.859976	−0.429988	−	0.902835i	$-0.641482\pi$
$433$	−17.8949	−0.859976	−0.429988	+	0.902835i	$0.641482\pi$
$434$	0	0
$435$	−8.29495	−0.397712
$436$	1.62001	0.0775844
$437$	−2.91219	−0.139309
$438$	−0.403860	−0.0192972
$439$	39.1919	1.87053	0.935263	−	0.353955i	$-0.115163\pi$
$439$	39.1919	1.87053	0.935263	+	0.353955i	$0.115163\pi$
$440$	−16.8943	−0.805404
$441$	0	0
$442$	−5.23324	−0.248920
$443$	37.8749	1.79949	0.899746	−	0.436414i	$-0.143752\pi$
$443$	37.8749	1.79949	0.899746	+	0.436414i	$0.143752\pi$
$444$	0.362384	0.0171980
$445$	29.5454	1.40058
$446$	−10.4836	−0.496415
$447$	1.78326	0.0843452
$448$	0	0
$449$	22.7250	1.07246	0.536230	−	0.844072i	$-0.319848\pi$
$449$	22.7250	1.07246	0.536230	+	0.844072i	$0.319848\pi$
$450$	36.6479	1.72760
$451$	1.53021	0.0720546
$452$	0.120226	0.00565497
$453$	0.199052	0.00935227
$454$	−36.4160	−1.70909
$455$	0	0
$456$	0.643447	0.0301322
$457$	−24.3084	−1.13710	−0.568550	−	0.822649i	$-0.692495\pi$
$457$	−24.3084	−1.13710	−0.568550	+	0.822649i	$0.692495\pi$
$458$	−14.7031	−0.687029
$459$	−10.9187	−0.509641
$460$	−2.15848	−0.100640
$461$	27.0463	1.25967	0.629837	−	0.776727i	$-0.283122\pi$
$461$	27.0463	1.25967	0.629837	+	0.776727i	$0.283122\pi$
$462$	0	0
$463$	9.77747	0.454398	0.227199	−	0.973848i	$-0.427043\pi$
$463$	9.77747	0.454398	0.227199	+	0.973848i	$0.427043\pi$
$464$	−28.6984	−1.33229
$465$	−6.15957	−0.285643
$466$	12.3648	0.572788
$467$	−25.1283	−1.16280	−0.581400	−	0.813618i	$-0.697495\pi$
$467$	−25.1283	−1.16280	−0.581400	+	0.813618i	$0.697495\pi$
$468$	−0.263524	−0.0121814
$469$	0	0
$470$	−61.1198	−2.81925
$471$	−3.43087	−0.158086
$472$	12.5468	0.577515
$473$	14.3320	0.658988
$474$	2.74752	0.126198
$475$	−7.22256	−0.331394
$476$	0	0
$477$	−5.62006	−0.257325
$478$	5.00695	0.229012
$479$	−35.1189	−1.60462	−0.802311	−	0.596906i	$-0.796397\pi$
$479$	−35.1189	−1.60462	−0.802311	+	0.596906i	$0.796397\pi$
$480$	0.919800	0.0419829
$481$	−4.92505	−0.224563
$482$	−10.8928	−0.496153
$483$	0	0
$484$	1.33057	0.0604807
$485$	50.0286	2.27168
$486$	9.95289	0.451472
$487$	−19.5496	−0.885876	−0.442938	−	0.896552i	$-0.646064\pi$
$487$	−19.5496	−0.885876	−0.442938	+	0.896552i	$0.646064\pi$
$488$	−24.7836	−1.12190
$489$	−0.285162	−0.0128955
$490$	0	0
$491$	10.2980	0.464742	0.232371	−	0.972627i	$-0.425352\pi$
$491$	10.2980	0.464742	0.232371	+	0.972627i	$0.425352\pi$
$492$	−0.0431968	−0.00194746
$493$	−51.3132	−2.31103
$494$	−0.623982	−0.0280743
$495$	16.8630	0.757936
$496$	−21.3105	−0.956871
$497$	0	0
$498$	3.22384	0.144464
$499$	−21.2111	−0.949541	−0.474770	−	0.880110i	$-0.657469\pi$
$499$	−21.2111	−0.949541	−0.474770	+	0.880110i	$0.657469\pi$
$500$	−2.45444	−0.109766
$501$	1.49266	0.0666871
$502$	2.18709	0.0976146
$503$	44.6725	1.99185	0.995923	−	0.0902029i	$-0.0287516\pi$
$503$	44.6725	1.99185	0.995923	+	0.0902029i	$0.0287516\pi$
$504$	0	0
$505$	−24.3321	−1.08276
$506$	7.74103	0.344131
$507$	3.55735	0.157988
$508$	0.488438	0.0216709
$509$	−23.3618	−1.03549	−0.517747	−	0.855534i	$-0.673229\pi$
$509$	−23.3618	−1.03549	−0.517747	+	0.855534i	$0.673229\pi$
$510$	−9.45337	−0.418602
$511$	0	0
$512$	24.6564	1.08967
$513$	−1.30188	−0.0574796
$514$	27.4026	1.20868
$515$	68.0310	2.99780
$516$	−0.404585	−0.0178108
$517$	−18.2441	−0.802374
$518$	0	0
$519$	0.707761	0.0310672
$520$	−6.48160	−0.284237
$521$	2.33762	0.102413	0.0512064	−	0.998688i	$-0.483693\pi$
$521$	2.33762	0.102413	0.0512064	+	0.998688i	$0.483693\pi$
$522$	31.0448	1.35879
$523$	−1.57772	−0.0689888	−0.0344944	−	0.999405i	$-0.510982\pi$
$523$	−1.57772	−0.0689888	−0.0344944	+	0.999405i	$0.510982\pi$
$524$	2.26435	0.0989187
$525$	0	0
$526$	33.7376	1.47103
$527$	−38.1036	−1.65982
$528$	−1.57818	−0.0686813
$529$	−9.13920	−0.397357
$530$	−9.86328	−0.428433
$531$	−12.5236	−0.543478
$532$	0	0
$533$	0.587073	0.0254290
$534$	2.99117	0.129441
$535$	49.0635	2.12120
$536$	−33.7180	−1.45640
$537$	−3.88550	−0.167672
$538$	−41.7271	−1.79898
$539$	0	0
$540$	−0.964941	−0.0415245
$541$	−5.19622	−0.223403	−0.111702	−	0.993742i	$-0.535630\pi$
$541$	−5.19622	−0.223403	−0.111702	+	0.993742i	$0.535630\pi$
$542$	−24.8571	−1.06770
$543$	1.09666	0.0470620
$544$	5.68995	0.243955
$545$	−39.7718	−1.70364
$546$	0	0
$547$	10.6657	0.456031	0.228016	−	0.973657i	$-0.426776\pi$
$547$	10.6657	0.456031	0.228016	+	0.973657i	$0.426776\pi$
$548$	−1.20472	−0.0514631
$549$	24.7377	1.05578
$550$	19.1986	0.818632
$551$	−6.11830	−0.260648
$552$	−3.06253	−0.130350
$553$	0	0
$554$	−1.44916	−0.0615688
$555$	−8.89665	−0.377642
$556$	−1.68625	−0.0715131
$557$	−25.3290	−1.07322	−0.536611	−	0.843830i	$-0.680296\pi$
$557$	−25.3290	−1.07322	−0.536611	+	0.843830i	$0.680296\pi$
$558$	23.0529	0.975907
$559$	5.49858	0.232565
$560$	0	0
$561$	−2.82180	−0.119137
$562$	−1.80131	−0.0759839
$563$	−2.64057	−0.111287	−0.0556433	−	0.998451i	$-0.517721\pi$
$563$	−2.64057	−0.111287	−0.0556433	+	0.998451i	$0.517721\pi$
$564$	0.515019	0.0216862
$565$	−2.95160	−0.124175
$566$	29.2153	1.22801
$567$	0	0
$568$	23.9585	1.00527
$569$	−29.7217	−1.24600	−0.622998	−	0.782223i	$-0.714086\pi$
$569$	−29.7217	−1.24600	−0.622998	+	0.782223i	$0.714086\pi$
$570$	−1.12717	−0.0472118
$571$	14.4005	0.602640	0.301320	−	0.953523i	$-0.402573\pi$
$571$	14.4005	0.602640	0.301320	+	0.953523i	$0.402573\pi$
$572$	−0.138051	−0.00577222
$573$	4.27331	0.178520
$574$	0	0
$575$	34.3764	1.43359
$576$	−24.8769	−1.03654
$577$	−7.51404	−0.312813	−0.156407	−	0.987693i	$-0.549991\pi$
$577$	−7.51404	−0.312813	−0.156407	+	0.987693i	$0.549991\pi$
$578$	−35.3797	−1.47160
$579$	−4.13290	−0.171758
$580$	−4.53481	−0.188298
$581$	0	0
$582$	5.06489	0.209946
$583$	−2.94416	−0.121935
$584$	−3.09426	−0.128042
$585$	6.46960	0.267485
$586$	26.4263	1.09166
$587$	−45.6741	−1.88517	−0.942585	−	0.333965i	$-0.891613\pi$
$587$	−45.6741	−1.88517	−0.942585	+	0.333965i	$0.891613\pi$
$588$	0	0
$589$	−4.54326	−0.187202
$590$	−21.9791	−0.904865
$591$	−3.22339	−0.132593
$592$	−30.7801	−1.26506
$593$	24.2450	0.995624	0.497812	−	0.867285i	$-0.334137\pi$
$593$	24.2450	0.995624	0.497812	+	0.867285i	$0.334137\pi$
$594$	3.46060	0.141990
$595$	0	0
$596$	0.974899	0.0399334
$597$	−4.68913	−0.191913
$598$	2.96989	0.121448
$599$	6.56723	0.268330	0.134165	−	0.990959i	$-0.457165\pi$
$599$	6.56723	0.268330	0.134165	+	0.990959i	$0.457165\pi$
$600$	−7.59544	−0.310082
$601$	−22.8645	−0.932661	−0.466331	−	0.884610i	$-0.654424\pi$
$601$	−22.8645	−0.932661	−0.466331	+	0.884610i	$0.654424\pi$
$602$	0	0
$603$	33.6556	1.37056
$604$	0.108821	0.00442785
$605$	−32.6661	−1.32806
$606$	−2.46338	−0.100068
$607$	2.13050	0.0864744	0.0432372	−	0.999065i	$-0.486233\pi$
$607$	2.13050	0.0864744	0.0432372	+	0.999065i	$0.486233\pi$
$608$	0.678439	0.0275143
$609$	0	0
$610$	43.4151	1.75782
$611$	−6.99946	−0.283168
$612$	−2.94477	−0.119035
$613$	−13.9256	−0.562448	−0.281224	−	0.959642i	$-0.590740\pi$
$613$	−13.9256	−0.562448	−0.281224	+	0.959642i	$0.590740\pi$
$614$	−38.7770	−1.56491
$615$	1.06049	0.0427633
$616$	0	0
$617$	−34.8843	−1.40439	−0.702195	−	0.711985i	$-0.747796\pi$
$617$	−34.8843	−1.40439	−0.702195	+	0.711985i	$0.747796\pi$
$618$	6.88745	0.277054
$619$	−31.8727	−1.28107	−0.640537	−	0.767928i	$-0.721288\pi$
$619$	−31.8727	−1.28107	−0.640537	+	0.767928i	$0.721288\pi$
$620$	−3.36741	−0.135238
$621$	6.19642	0.248654
$622$	−13.0926	−0.524965
$623$	0	0
$624$	−0.605477	−0.0242385
$625$	14.0899	0.563594
$626$	45.6594	1.82492
$627$	−0.336456	−0.0134368
$628$	−1.87564	−0.0748463
$629$	−55.0354	−2.19440
$630$	0	0
$631$	9.96929	0.396871	0.198436	−	0.980114i	$-0.436414\pi$
$631$	9.96929	0.396871	0.198436	+	0.980114i	$0.436414\pi$
$632$	21.0508	0.837354
$633$	−2.75837	−0.109635
$634$	42.0435	1.66976
$635$	−11.9913	−0.475861
$636$	0.0831118	0.00329560
$637$	0	0
$638$	16.2633	0.643871
$639$	−23.9141	−0.946027
$640$	−37.1148	−1.46709
$641$	38.8172	1.53319	0.766594	−	0.642133i	$-0.221950\pi$
$641$	38.8172	1.53319	0.766594	+	0.642133i	$0.221950\pi$
$642$	4.96718	0.196039
$643$	−17.9051	−0.706110	−0.353055	−	0.935603i	$-0.614857\pi$
$643$	−17.9051	−0.706110	−0.353055	+	0.935603i	$0.614857\pi$
$644$	0	0
$645$	9.93268	0.391099
$646$	−6.97274	−0.274339
$647$	−6.22800	−0.244848	−0.122424	−	0.992478i	$-0.539067\pi$
$647$	−6.22800	−0.244848	−0.122424	+	0.992478i	$0.539067\pi$
$648$	24.2748	0.953606
$649$	−6.56070	−0.257530
$650$	7.36567	0.288905
$651$	0	0
$652$	−0.155897	−0.00610540
$653$	−44.2209	−1.73050	−0.865248	−	0.501344i	$-0.832839\pi$
$653$	−44.2209	−1.73050	−0.865248	+	0.501344i	$0.832839\pi$
$654$	−4.02649	−0.157448
$655$	−55.5906	−2.17211
$656$	3.66904	0.143252
$657$	3.08854	0.120495
$658$	0	0
$659$	−25.3675	−0.988178	−0.494089	−	0.869411i	$-0.664498\pi$
$659$	−25.3675	−0.988178	−0.494089	+	0.869411i	$0.664498\pi$
$660$	−0.249377	−0.00970700
$661$	−27.2889	−1.06141	−0.530707	−	0.847556i	$-0.678073\pi$
$661$	−27.2889	−1.06141	−0.530707	+	0.847556i	$0.678073\pi$
$662$	13.6204	0.529372
$663$	−1.08260	−0.0420448
$664$	24.7002	0.958555
$665$	0	0
$666$	33.2967	1.29022
$667$	29.1205	1.12755
$668$	0.816030	0.0315731
$669$	−2.16876	−0.0838489
$670$	59.0660	2.28192
$671$	12.9593	0.500287
$672$	0	0
$673$	2.43120	0.0937158	0.0468579	−	0.998902i	$-0.485079\pi$
$673$	2.43120	0.0937158	0.0468579	+	0.998902i	$0.485079\pi$
$674$	−6.80181	−0.261996
$675$	15.3678	0.591508
$676$	1.94479	0.0747995
$677$	−3.98914	−0.153315	−0.0766576	−	0.997057i	$-0.524425\pi$
$677$	−3.98914	−0.153315	−0.0766576	+	0.997057i	$0.524425\pi$
$678$	−0.298819	−0.0114761
$679$	0	0
$680$	−72.4292	−2.77753
$681$	−7.53340	−0.288681
$682$	12.0766	0.462439
$683$	29.9721	1.14685	0.573426	−	0.819257i	$-0.305614\pi$
$683$	29.9721	1.14685	0.573426	+	0.819257i	$0.305614\pi$
$684$	−0.351118	−0.0134253
$685$	29.5763	1.13005
$686$	0	0
$687$	−3.04163	−0.116045
$688$	34.3645	1.31014
$689$	−1.12955	−0.0430323
$690$	5.36484	0.204236
$691$	−13.4964	−0.513425	−0.256713	−	0.966488i	$-0.582639\pi$
$691$	−13.4964	−0.513425	−0.256713	+	0.966488i	$0.582639\pi$
$692$	0.386929	0.0147089
$693$	0	0
$694$	15.0142	0.569933
$695$	41.3981	1.57032
$696$	−6.43416	−0.243886
$697$	6.56030	0.248489
$698$	−14.6645	−0.555061
$699$	2.55791	0.0967491
$700$	0	0
$701$	−42.7642	−1.61518	−0.807591	−	0.589743i	$-0.799229\pi$
$701$	−42.7642	−1.61518	−0.807591	+	0.589743i	$0.799229\pi$
$702$	1.32768	0.0501101
$703$	−6.56211	−0.247495
$704$	−13.0322	−0.491168
$705$	−12.6439	−0.476196
$706$	29.2086	1.09928
$707$	0	0
$708$	0.185204	0.00696040
$709$	−9.91620	−0.372411	−0.186205	−	0.982511i	$-0.559619\pi$
$709$	−9.91620	−0.372411	−0.186205	+	0.982511i	$0.559619\pi$
$710$	−41.9696	−1.57509
$711$	−21.0118	−0.788003
$712$	22.9175	0.858871
$713$	21.6240	0.809825
$714$	0	0
$715$	3.38921	0.126749
$716$	−2.12418	−0.0793845
$717$	1.03579	0.0386823
$718$	13.4017	0.500145
$719$	17.3198	0.645919	0.322960	−	0.946413i	$-0.395322\pi$
$719$	17.3198	0.645919	0.322960	+	0.946413i	$0.395322\pi$
$720$	40.4331	1.50685
$721$	0	0
$722$	24.9857	0.929873
$723$	−2.25340	−0.0838048
$724$	0.599537	0.0222816
$725$	72.2222	2.68227
$726$	−3.30711	−0.122738
$727$	40.4385	1.49978	0.749891	−	0.661561i	$-0.230106\pi$
$727$	40.4385	1.49978	0.749891	+	0.661561i	$0.230106\pi$
$728$	0	0
$729$	−22.8264	−0.845422
$730$	5.42042	0.200619
$731$	61.4443	2.27260
$732$	−0.365832	−0.0135215
$733$	5.25881	0.194238	0.0971192	−	0.995273i	$-0.469037\pi$
$733$	5.25881	0.194238	0.0971192	+	0.995273i	$0.469037\pi$
$734$	−36.8706	−1.36092
$735$	0	0
$736$	−3.22908	−0.119026
$737$	17.6310	0.649447
$738$	−3.96902	−0.146102
$739$	35.8773	1.31977	0.659884	−	0.751368i	$-0.270605\pi$
$739$	35.8773	1.31977	0.659884	+	0.751368i	$0.270605\pi$
$740$	−4.86376	−0.178795
$741$	−0.129084	−0.00474200
$742$	0	0
$743$	−30.2113	−1.10835	−0.554173	−	0.832402i	$-0.686965\pi$
$743$	−30.2113	−1.10835	−0.554173	+	0.832402i	$0.686965\pi$
$744$	−4.77781	−0.175163
$745$	−23.9341	−0.876877
$746$	3.73282	0.136668
$747$	−24.6545	−0.902061
$748$	−1.54267	−0.0564055
$749$	0	0
$750$	6.10044	0.222757
$751$	41.2217	1.50420	0.752101	−	0.659047i	$-0.229040\pi$
$751$	41.2217	1.50420	0.752101	+	0.659047i	$0.229040\pi$
$752$	−43.7446	−1.59520
$753$	0.452444	0.0164880
$754$	6.23953	0.227230
$755$	−2.67159	−0.0972290
$756$	0	0
$757$	−29.1279	−1.05867	−0.529335	−	0.848413i	$-0.677559\pi$
$757$	−29.1279	−1.05867	−0.529335	+	0.848413i	$0.677559\pi$
$758$	−39.9465	−1.45092
$759$	1.60139	0.0581268
$760$	−8.63606	−0.313263
$761$	−10.2126	−0.370207	−0.185104	−	0.982719i	$-0.559262\pi$
$761$	−10.2126	−0.370207	−0.185104	+	0.982719i	$0.559262\pi$
$762$	−1.21400	−0.0439785
$763$	0	0
$764$	2.33620	0.0845208
$765$	72.2951	2.61383
$766$	9.09889	0.328756
$767$	−2.51705	−0.0908855
$768$	1.03044	0.0371830
$769$	44.8525	1.61742	0.808711	−	0.588206i	$-0.200166\pi$
$769$	44.8525	1.61742	0.808711	+	0.588206i	$0.200166\pi$
$770$	0	0
$771$	5.66879	0.204157
$772$	−2.25944	−0.0813190
$773$	6.80825	0.244876	0.122438	−	0.992476i	$-0.460929\pi$
$773$	6.80825	0.244876	0.122438	+	0.992476i	$0.460929\pi$
$774$	−37.1742	−1.33620
$775$	53.6300	1.92645
$776$	38.8058	1.39305
$777$	0	0
$778$	−0.650677	−0.0233279
$779$	0.782214	0.0280257
$780$	−0.0956752	−0.00342572
$781$	−12.5278	−0.448280
$782$	33.1873	1.18678
$783$	13.0182	0.465234
$784$	0	0
$785$	46.0476	1.64351
$786$	−5.62799	−0.200744
$787$	56.0803	1.99905	0.999524	−	0.0308539i	$-0.00982267\pi$
$787$	56.0803	1.99905	0.999524	+	0.0308539i	$0.00982267\pi$
$788$	−1.76221	−0.0627763
$789$	6.97931	0.248470
$790$	−36.8760	−1.31199
$791$	0	0
$792$	13.0802	0.464784
$793$	4.97190	0.176557
$794$	32.9226	1.16838
$795$	−2.04042	−0.0723663
$796$	−2.56353	−0.0908618
$797$	48.1716	1.70632	0.853162	−	0.521646i	$-0.174682\pi$
$797$	48.1716	1.70632	0.853162	+	0.521646i	$0.174682\pi$
$798$	0	0
$799$	−78.2160	−2.76708
$800$	−8.00849	−0.283143
$801$	−22.8751	−0.808252
$802$	−31.6386	−1.11720
$803$	1.61798	0.0570973
$804$	−0.497713	−0.0175530
$805$	0	0
$806$	4.63328	0.163200
$807$	−8.63211	−0.303865
$808$	−18.8737	−0.663975
$809$	−42.4935	−1.49399	−0.746997	−	0.664828i	$-0.768505\pi$
$809$	−42.4935	−1.49399	−0.746997	+	0.664828i	$0.768505\pi$
$810$	−42.5238	−1.49413
$811$	2.95213	0.103663	0.0518317	−	0.998656i	$-0.483494\pi$
$811$	2.95213	0.103663	0.0518317	+	0.998656i	$0.483494\pi$
$812$	0	0
$813$	−5.14219	−0.180345
$814$	17.4431	0.611379
$815$	3.82732	0.134065
$816$	−6.76595	−0.236856
$817$	7.32628	0.256314
$818$	−41.6486	−1.45621
$819$	0	0
$820$	0.579768	0.0202464
$821$	−0.811655	−0.0283269	−0.0141635	−	0.999900i	$-0.504509\pi$
$821$	−0.811655	−0.0283269	−0.0141635	+	0.999900i	$0.504509\pi$
$822$	2.99430	0.104438
$823$	−4.07214	−0.141946	−0.0709729	−	0.997478i	$-0.522610\pi$
$823$	−4.07214	−0.141946	−0.0709729	+	0.997478i	$0.522610\pi$
$824$	52.7698	1.83832
$825$	3.97163	0.138274
$826$	0	0
$827$	−20.3822	−0.708758	−0.354379	−	0.935102i	$-0.615308\pi$
$827$	−20.3822	−0.708758	−0.354379	+	0.935102i	$0.615308\pi$
$828$	1.67117	0.0580773
$829$	29.3639	1.01985	0.509925	−	0.860219i	$-0.329673\pi$
$829$	29.3639	1.01985	0.509925	+	0.860219i	$0.329673\pi$
$830$	−43.2690	−1.50189
$831$	−0.299788	−0.0103995
$832$	−4.99987	−0.173339
$833$	0	0
$834$	4.19114	0.145127
$835$	−20.0338	−0.693298
$836$	−0.183939	−0.00636167
$837$	9.66693	0.334138
$838$	−7.54007	−0.260468
$839$	9.95705	0.343756	0.171878	−	0.985118i	$-0.445017\pi$
$839$	9.95705	0.343756	0.171878	+	0.985118i	$0.445017\pi$
$840$	0	0
$841$	32.1801	1.10966
$842$	−16.1402	−0.556228
$843$	−0.372639	−0.0128344
$844$	−1.50799	−0.0519071
$845$	−47.7452	−1.64248
$846$	47.3212	1.62694
$847$	0	0
$848$	−7.05933	−0.242418
$849$	6.04378	0.207422
$850$	82.3083	2.82315
$851$	31.2329	1.07065
$852$	0.353652	0.0121159
$853$	−12.4050	−0.424740	−0.212370	−	0.977189i	$-0.568118\pi$
$853$	−12.4050	−0.424740	−0.212370	+	0.977189i	$0.568118\pi$
$854$	0	0
$855$	8.62007	0.294800
$856$	38.0572	1.30077
$857$	16.1448	0.551494	0.275747	−	0.961230i	$-0.411075\pi$
$857$	16.1448	0.551494	0.275747	+	0.961230i	$0.411075\pi$
$858$	0.343123	0.0117140
$859$	43.8520	1.49621	0.748106	−	0.663580i	$-0.230964\pi$
$859$	43.8520	1.49621	0.748106	+	0.663580i	$0.230964\pi$
$860$	5.43015	0.185167
$861$	0	0
$862$	21.4770	0.731511
$863$	−29.4794	−1.00349	−0.501745	−	0.865016i	$-0.667308\pi$
$863$	−29.4794	−1.00349	−0.501745	+	0.865016i	$0.667308\pi$
$864$	−1.44355	−0.0491106
$865$	−9.49925	−0.322984
$866$	24.3156	0.826277
$867$	−7.31902	−0.248567
$868$	0	0
$869$	−11.0074	−0.373399
$870$	11.2711	0.382127
$871$	6.76425	0.229198
$872$	−30.8499	−1.04471
$873$	−38.7340	−1.31095
$874$	3.95707	0.133850
$875$	0	0
$876$	−0.0456746	−0.00154320
$877$	33.3689	1.12679	0.563393	−	0.826189i	$-0.309496\pi$
$877$	33.3689	1.12679	0.563393	+	0.826189i	$0.309496\pi$
$878$	−53.2537	−1.79723
$879$	5.46683	0.184391
$880$	21.1816	0.714030
$881$	35.4625	1.19476	0.597381	−	0.801958i	$-0.296208\pi$
$881$	35.4625	1.19476	0.597381	+	0.801958i	$0.296208\pi$
$882$	0	0
$883$	−55.1635	−1.85640	−0.928200	−	0.372082i	$-0.878644\pi$
$883$	−55.1635	−1.85640	−0.928200	+	0.372082i	$0.878644\pi$
$884$	−0.591854	−0.0199062
$885$	−4.54683	−0.152840
$886$	−51.4643	−1.72898
$887$	−10.4949	−0.352383	−0.176192	−	0.984356i	$-0.556378\pi$
$887$	−10.4949	−0.352383	−0.176192	+	0.984356i	$0.556378\pi$
$888$	−6.90089	−0.231579
$889$	0	0
$890$	−40.1461	−1.34570
$891$	−12.6932	−0.425240
$892$	−1.18565	−0.0396985
$893$	−9.32605	−0.312084
$894$	−2.42308	−0.0810400
$895$	52.1494	1.74316
$896$	0	0
$897$	0.614383	0.0205137
$898$	−30.8786	−1.03043
$899$	45.4304	1.51519
$900$	4.14470	0.138157
$901$	−12.6222	−0.420507
$902$	−2.07924	−0.0692311
$903$	0	0
$904$	−2.28947	−0.0761467
$905$	−14.7188	−0.489270
$906$	−0.270471	−0.00898579
$907$	12.5409	0.416415	0.208207	−	0.978085i	$-0.433237\pi$
$907$	12.5409	0.416415	0.208207	+	0.978085i	$0.433237\pi$
$908$	−4.11848	−0.136676
$909$	18.8388	0.624843
$910$	0	0
$911$	−45.5700	−1.50980	−0.754901	−	0.655839i	$-0.772315\pi$
$911$	−45.5700	−1.50980	−0.754901	+	0.655839i	$0.772315\pi$
$912$	−0.806735	−0.0267137
$913$	−12.9157	−0.427446
$914$	33.0302	1.09254
$915$	8.98129	0.296912
$916$	−1.66284	−0.0549419
$917$	0	0
$918$	14.8363	0.489670
$919$	52.3114	1.72559	0.862797	−	0.505550i	$-0.168711\pi$
$919$	52.3114	1.72559	0.862797	+	0.505550i	$0.168711\pi$
$920$	41.1040	1.35516
$921$	−8.02182	−0.264328
$922$	−36.7505	−1.21031
$923$	−4.80637	−0.158204
$924$	0	0
$925$	77.4611	2.54691
$926$	−13.2856	−0.436592
$927$	−52.6721	−1.72998
$928$	−6.78406	−0.222698
$929$	−11.0041	−0.361034	−0.180517	−	0.983572i	$-0.557777\pi$
$929$	−11.0041	−0.361034	−0.180517	+	0.983572i	$0.557777\pi$
$930$	8.36960	0.274450
$931$	0	0
$932$	1.39840	0.0458061
$933$	−2.70847	−0.0886714
$934$	34.1442	1.11723
$935$	37.8730	1.23858
$936$	5.01829	0.164028
$937$	−8.03703	−0.262558	−0.131279	−	0.991345i	$-0.541908\pi$
$937$	−8.03703	−0.262558	−0.131279	+	0.991345i	$0.541908\pi$
$938$	0	0
$939$	9.44557	0.308245
$940$	−6.91235	−0.225456
$941$	−23.7694	−0.774859	−0.387429	−	0.921899i	$-0.626637\pi$
$941$	−23.7694	−0.774859	−0.387429	+	0.921899i	$0.626637\pi$
$942$	4.66186	0.151891
$943$	−3.72301	−0.121238
$944$	−15.7308	−0.511996
$945$	0	0
$946$	−19.4743	−0.633165
$947$	−33.2635	−1.08092	−0.540459	−	0.841370i	$-0.681750\pi$
$947$	−33.2635	−1.08092	−0.540459	+	0.841370i	$0.681750\pi$
$948$	0.310731	0.0100921
$949$	0.620748	0.0201503
$950$	9.81399	0.318408
$951$	8.69756	0.282038
$952$	0	0
$953$	29.0945	0.942463	0.471232	−	0.882009i	$-0.343810\pi$
$953$	29.0945	0.942463	0.471232	+	0.882009i	$0.343810\pi$
$954$	7.63651	0.247241
$955$	−57.3545	−1.85595
$956$	0.566261	0.0183142
$957$	3.36440	0.108756
$958$	47.7194	1.54174
$959$	0	0
$960$	−9.03181	−0.291500
$961$	2.73524	0.0882335
$962$	6.69213	0.215763
$963$	−37.9867	−1.22411
$964$	−1.23192	−0.0396776
$965$	55.4700	1.78564
$966$	0	0
$967$	−52.2398	−1.67992	−0.839959	−	0.542650i	$-0.817421\pi$
$967$	−52.2398	−1.67992	−0.839959	+	0.542650i	$0.817421\pi$
$968$	−25.3382	−0.814399
$969$	−1.44245	−0.0463383
$970$	−67.9787	−2.18266
$971$	−1.75797	−0.0564159	−0.0282080	−	0.999602i	$-0.508980\pi$
$971$	−1.75797	−0.0564159	−0.0282080	+	0.999602i	$0.508980\pi$
$972$	1.12562	0.0361044
$973$	0	0
$974$	26.5639	0.851162
$975$	1.52374	0.0487987
$976$	31.0730	0.994621
$977$	−37.5188	−1.20033	−0.600167	−	0.799875i	$-0.704899\pi$
$977$	−37.5188	−1.20033	−0.600167	+	0.799875i	$0.704899\pi$
$978$	0.387478	0.0123902
$979$	−11.9835	−0.382995
$980$	0	0
$981$	30.7928	0.983137
$982$	−13.9929	−0.446531
$983$	−32.6951	−1.04281	−0.521405	−	0.853309i	$-0.674592\pi$
$983$	−32.6951	−1.04281	−0.521405	+	0.853309i	$0.674592\pi$
$984$	0.822597	0.0262234
$985$	43.2630	1.37847
$986$	69.7241	2.22047
$987$	0	0
$988$	−0.0705694	−0.00224511
$989$	−34.8700	−1.10880
$990$	−22.9134	−0.728236
$991$	−39.7198	−1.26174	−0.630870	−	0.775889i	$-0.717302\pi$
$991$	−39.7198	−1.26174	−0.630870	+	0.775889i	$0.717302\pi$
$992$	−5.03764	−0.159945
$993$	2.81766	0.0894158
$994$	0	0
$995$	62.9354	1.99519
$996$	0.364601	0.0115528
$997$	−8.11982	−0.257157	−0.128579	−	0.991699i	$-0.541041\pi$
$997$	−8.11982	−0.257157	−0.128579	+	0.991699i	$0.541041\pi$
$998$	28.8216	0.912332
$999$	13.9626	0.441756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.q.1.3	yes	7
7.6	odd	2		2009.2.a.p.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.p.1.3	✓	7	7.6	odd	2
2009.2.a.q.1.3	yes	7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.35880 q^{2} -0.281095 q^{3} -0.153673 q^{4} +3.77273 q^{5} +0.381951 q^{6} +2.92640 q^{8} -2.92099 q^{9} +O(q^{10})\)	\(q-1.35880 q^{2} -0.281095 q^{3} -0.153673 q^{4} +3.77273 q^{5} +0.381951 q^{6} +2.92640 q^{8} -2.92099 q^{9} -5.12637 q^{10} -1.53021 q^{11} +0.0431968 q^{12} -0.587073 q^{13} -1.06049 q^{15} -3.66904 q^{16} -6.56030 q^{17} +3.96902 q^{18} -0.782214 q^{19} -0.579768 q^{20} +2.07924 q^{22} +3.72301 q^{23} -0.822597 q^{24} +9.23349 q^{25} +0.797713 q^{26} +1.66436 q^{27} +7.82177 q^{29} +1.44100 q^{30} +5.80820 q^{31} -0.867331 q^{32} +0.430133 q^{33} +8.91411 q^{34} +0.448877 q^{36} +8.38915 q^{37} +1.06287 q^{38} +0.165023 q^{39} +11.0405 q^{40} -1.00000 q^{41} -9.36609 q^{43} +0.235152 q^{44} -11.0201 q^{45} -5.05881 q^{46} +11.9226 q^{47} +1.03135 q^{48} -12.5464 q^{50} +1.84407 q^{51} +0.0902175 q^{52} +1.92403 q^{53} -2.26152 q^{54} -5.77306 q^{55} +0.219876 q^{57} -10.6282 q^{58} +4.28746 q^{59} +0.162970 q^{60} -8.46897 q^{61} -7.89217 q^{62} +8.51660 q^{64} -2.21487 q^{65} -0.584463 q^{66} -11.5220 q^{67} +1.00814 q^{68} -1.04652 q^{69} +8.18700 q^{71} -8.54798 q^{72} -1.05736 q^{73} -11.3991 q^{74} -2.59549 q^{75} +0.120205 q^{76} -0.224233 q^{78} +7.19339 q^{79} -13.8423 q^{80} +8.29511 q^{81} +1.35880 q^{82} +8.44048 q^{83} -24.7502 q^{85} +12.7266 q^{86} -2.19866 q^{87} -4.47800 q^{88} +7.83130 q^{89} +14.9741 q^{90} -0.572127 q^{92} -1.63266 q^{93} -16.2004 q^{94} -2.95108 q^{95} +0.243802 q^{96} +13.2606 q^{97} +4.46971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 7 q^{3} + 9 q^{4} + 4 q^{5} - 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) 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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