LMFDB - Embedded newform 2009.4.a.g.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,4,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, 4, names="a")

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

chi := DirichletCharacter("2009.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
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:	$118.534837202$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 112 x^{17} + 431 x^{16} + 5147 x^{15} - 18874 x^{14} - 125634 x^{13} + \cdots - 4783968 $ x^19 - 4x^18 - 112x^17 + 431x^16 + 5147x^15 - 18874x^14 - 125634x^13 + 435758x^12 + 1750454x^11 - 5749720x^10 - 13679019x^9 + 43655645x^8 + 52025410x^7 - 179063110x^6 - 42432043x^5 + 318217154x^4 - 170747148x^3 - 42883392x^2 + 42900480x - 4783968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.09684$ 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)

$f(q)$	$=$	$q+2.09684 q^{2} +7.20675 q^{3} -3.60327 q^{4} +16.1298 q^{5} +15.1114 q^{6} -24.3302 q^{8} +24.9373 q^{9} +O(q^{10})$	\(q+2.09684 q^{2} +7.20675 q^{3} -3.60327 q^{4} +16.1298 q^{5} +15.1114 q^{6} -24.3302 q^{8} +24.9373 q^{9} +33.8216 q^{10} -54.7610 q^{11} -25.9678 q^{12} +16.1054 q^{13} +116.243 q^{15} -22.1904 q^{16} -14.2008 q^{17} +52.2895 q^{18} -110.714 q^{19} -58.1200 q^{20} -114.825 q^{22} -113.769 q^{23} -175.342 q^{24} +135.171 q^{25} +33.7705 q^{26} -14.8655 q^{27} -125.201 q^{29} +243.744 q^{30} -121.044 q^{31} +148.112 q^{32} -394.649 q^{33} -29.7767 q^{34} -89.8557 q^{36} +259.259 q^{37} -232.150 q^{38} +116.068 q^{39} -392.441 q^{40} -41.0000 q^{41} -436.321 q^{43} +197.318 q^{44} +402.233 q^{45} -238.555 q^{46} -185.599 q^{47} -159.920 q^{48} +283.431 q^{50} -102.341 q^{51} -58.0322 q^{52} -480.635 q^{53} -31.1706 q^{54} -883.283 q^{55} -797.891 q^{57} -262.527 q^{58} +565.540 q^{59} -418.856 q^{60} +454.466 q^{61} -253.809 q^{62} +488.090 q^{64} +259.778 q^{65} -827.515 q^{66} +584.545 q^{67} +51.1691 q^{68} -819.904 q^{69} -690.467 q^{71} -606.729 q^{72} -4.41350 q^{73} +543.625 q^{74} +974.140 q^{75} +398.933 q^{76} +243.376 q^{78} +792.031 q^{79} -357.926 q^{80} -780.439 q^{81} -85.9704 q^{82} +393.316 q^{83} -229.055 q^{85} -914.894 q^{86} -902.296 q^{87} +1332.34 q^{88} -936.747 q^{89} +843.419 q^{90} +409.940 q^{92} -872.331 q^{93} -389.171 q^{94} -1785.80 q^{95} +1067.41 q^{96} +1.31027 q^{97} -1365.59 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{2} - 13 q^{3} + 88 q^{4} - 26 q^{5} - 30 q^{6} + 51 q^{8} + 206 q^{9}+O(q^{10}) $ 19 * q + 4 * q^2 - 13 * q^3 + 88 * q^4 - 26 * q^5 - 30 * q^6 + 51 * q^8 + 206 * q^9	\( 19 q + 4 q^{2} - 13 q^{3} + 88 q^{4} - 26 q^{5} - 30 q^{6} + 51 q^{8} + 206 q^{9} - 108 q^{10} - 4 q^{11} - 52 q^{12} - 239 q^{13} - 144 q^{15} + 484 q^{16} - 117 q^{17} - 193 q^{18} - 323 q^{19} - 519 q^{20} + 105 q^{22} + 121 q^{23} + 127 q^{24} + 989 q^{25} - 134 q^{26} - 328 q^{27} + 72 q^{29} + 311 q^{30} - 872 q^{31} + 422 q^{32} - 746 q^{33} - 639 q^{34} + 552 q^{36} + 419 q^{37} - 106 q^{38} + 420 q^{39} - 1463 q^{40} - 779 q^{41} + 1231 q^{43} - 183 q^{44} - 1638 q^{45} - 630 q^{46} + 163 q^{47} - 1694 q^{48} + 791 q^{50} - 1020 q^{51} - 2737 q^{52} - 308 q^{53} - 932 q^{54} - 1668 q^{55} + 3346 q^{57} + 495 q^{58} - 902 q^{59} - 1080 q^{60} - 1390 q^{61} - 946 q^{62} + 2239 q^{64} - 1500 q^{65} - 3336 q^{66} + 86 q^{67} - 849 q^{68} - 1459 q^{69} + 784 q^{71} - 2013 q^{72} - 4270 q^{73} - 2009 q^{74} + 1841 q^{75} - 155 q^{76} - 1633 q^{78} - 392 q^{79} - 1160 q^{80} + 451 q^{81} - 164 q^{82} - 112 q^{83} + 2740 q^{85} - 6521 q^{86} - 1868 q^{87} - 1540 q^{88} - 687 q^{89} + 7589 q^{90} - 3552 q^{92} - 3908 q^{93} + 1705 q^{94} - 3636 q^{95} + 7802 q^{96} - 577 q^{97} + 1040 q^{99}+O(q^{100}) \) 19 * q + 4 * q^2 - 13 * q^3 + 88 * q^4 - 26 * q^5 - 30 * q^6 + 51 * q^8 + 206 * q^9 - 108 * q^10 - 4 * q^11 - 52 * q^12 - 239 * q^13 - 144 * q^15 + 484 * q^16 - 117 * q^17 - 193 * q^18 - 323 * q^19 - 519 * q^20 + 105 * q^22 + 121 * q^23 + 127 * q^24 + 989 * q^25 - 134 * q^26 - 328 * q^27 + 72 * q^29 + 311 * q^30 - 872 * q^31 + 422 * q^32 - 746 * q^33 - 639 * q^34 + 552 * q^36 + 419 * q^37 - 106 * q^38 + 420 * q^39 - 1463 * q^40 - 779 * q^41 + 1231 * q^43 - 183 * q^44 - 1638 * q^45 - 630 * q^46 + 163 * q^47 - 1694 * q^48 + 791 * q^50 - 1020 * q^51 - 2737 * q^52 - 308 * q^53 - 932 * q^54 - 1668 * q^55 + 3346 * q^57 + 495 * q^58 - 902 * q^59 - 1080 * q^60 - 1390 * q^61 - 946 * q^62 + 2239 * q^64 - 1500 * q^65 - 3336 * q^66 + 86 * q^67 - 849 * q^68 - 1459 * q^69 + 784 * q^71 - 2013 * q^72 - 4270 * q^73 - 2009 * q^74 + 1841 * q^75 - 155 * q^76 - 1633 * q^78 - 392 * q^79 - 1160 * q^80 + 451 * q^81 - 164 * q^82 - 112 * q^83 + 2740 * q^85 - 6521 * q^86 - 1868 * q^87 - 1540 * q^88 - 687 * q^89 + 7589 * q^90 - 3552 * q^92 - 3908 * q^93 + 1705 * q^94 - 3636 * q^95 + 7802 * q^96 - 577 * q^97 + 1040 * 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 $

$2$	2.09684	0.741345	0.370672	−	0.928764i	$-0.379127\pi$
$2$	2.09684	0.741345	0.370672	+	0.928764i	$0.379127\pi$
$3$	7.20675	1.38694	0.693470	−	0.720485i	$-0.256081\pi$
$3$	7.20675	1.38694	0.693470	+	0.720485i	$0.256081\pi$
$4$	−3.60327	−0.450408
$5$	16.1298	1.44269	0.721347	−	0.692574i	$-0.243523\pi$
$5$	16.1298	1.44269	0.721347	+	0.692574i	$0.243523\pi$
$6$	15.1114	1.02820
$7$	0	0
$7$	0	0
$8$	−24.3302	−1.07525
$9$	24.9373	0.923603
$10$	33.8216	1.06953
$11$	−54.7610	−1.50100	−0.750502	−	0.660868i	$-0.770188\pi$
$11$	−54.7610	−1.50100	−0.750502	+	0.660868i	$0.770188\pi$
$12$	−25.9678	−0.624689
$13$	16.1054	0.343604	0.171802	−	0.985132i	$-0.445041\pi$
$13$	16.1054	0.343604	0.171802	+	0.985132i	$0.445041\pi$
$14$	0	0
$15$	116.243	2.00093
$16$	−22.1904	−0.346724
$17$	−14.2008	−0.202599	−0.101300	−	0.994856i	$-0.532300\pi$
$17$	−14.2008	−0.202599	−0.101300	+	0.994856i	$0.532300\pi$
$18$	52.2895	0.684708
$19$	−110.714	−1.33682	−0.668411	−	0.743792i	$-0.733025\pi$
$19$	−110.714	−1.33682	−0.668411	+	0.743792i	$0.733025\pi$
$20$	−58.1200	−0.649801
$21$	0	0
$22$	−114.825	−1.11276
$23$	−113.769	−1.03141	−0.515705	−	0.856766i	$-0.672470\pi$
$23$	−113.769	−1.03141	−0.515705	+	0.856766i	$0.672470\pi$
$24$	−175.342	−1.49131
$25$	135.171	1.08136
$26$	33.7705	0.254729
$27$	−14.8655	−0.105958
$28$	0	0
$29$	−125.201	−0.801701	−0.400851	−	0.916143i	$-0.631285\pi$
$29$	−125.201	−0.801701	−0.400851	+	0.916143i	$0.631285\pi$
$30$	243.744	1.48338
$31$	−121.044	−0.701293	−0.350646	−	0.936508i	$-0.614038\pi$
$31$	−121.044	−0.701293	−0.350646	+	0.936508i	$0.614038\pi$
$32$	148.112	0.818210
$33$	−394.649	−2.08180
$34$	−29.7767	−0.150196
$35$	0	0
$36$	−89.8557	−0.415998
$37$	259.259	1.15194	0.575972	−	0.817469i	$-0.304624\pi$
$37$	259.259	1.15194	0.575972	+	0.817469i	$0.304624\pi$
$38$	−232.150	−0.991046
$39$	116.068	0.476558
$40$	−392.441	−1.55126
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	−436.321	−1.54740	−0.773701	−	0.633551i	$-0.781597\pi$
$43$	−436.321	−1.54740	−0.773701	+	0.633551i	$0.781597\pi$
$44$	197.318	0.676065
$45$	402.233	1.33248
$46$	−238.555	−0.764631
$47$	−185.599	−0.576008	−0.288004	−	0.957629i	$-0.592992\pi$
$47$	−185.599	−0.576008	−0.288004	+	0.957629i	$0.592992\pi$
$48$	−159.920	−0.480886
$49$	0	0
$50$	283.431	0.801663
$51$	−102.341	−0.280993
$52$	−58.0322	−0.154762
$53$	−480.635	−1.24567	−0.622833	−	0.782355i	$-0.714018\pi$
$53$	−480.635	−1.24567	−0.622833	+	0.782355i	$0.714018\pi$
$54$	−31.1706	−0.0785514
$55$	−883.283	−2.16549
$56$	0	0
$57$	−797.891	−1.85409
$58$	−262.527	−0.594337
$59$	565.540	1.24792	0.623958	−	0.781458i	$-0.285524\pi$
$59$	565.540	1.24792	0.623958	+	0.781458i	$0.285524\pi$
$60$	−418.856	−0.901235
$61$	454.466	0.953909	0.476955	−	0.878928i	$-0.341741\pi$
$61$	454.466	0.953909	0.476955	+	0.878928i	$0.341741\pi$
$62$	−253.809	−0.519900
$63$	0	0
$64$	488.090	0.953300
$65$	259.778	0.495715
$66$	−827.515	−1.54333
$67$	584.545	1.06587	0.532937	−	0.846155i	$-0.321088\pi$
$67$	584.545	1.06587	0.532937	+	0.846155i	$0.321088\pi$
$68$	51.1691	0.0912524
$69$	−819.904	−1.43051
$70$	0	0
$71$	−690.467	−1.15413	−0.577066	−	0.816697i	$-0.695803\pi$
$71$	−690.467	−1.15413	−0.577066	+	0.816697i	$0.695803\pi$
$72$	−606.729	−0.993106
$73$	−4.41350	−0.00707618	−0.00353809	−	0.999994i	$-0.501126\pi$
$73$	−4.41350	−0.00707618	−0.00353809	+	0.999994i	$0.501126\pi$
$74$	543.625	0.853988
$75$	974.140	1.49979
$76$	398.933	0.602116
$77$	0	0
$78$	243.376	0.353293
$79$	792.031	1.12798	0.563990	−	0.825782i	$-0.309266\pi$
$79$	792.031	1.12798	0.563990	+	0.825782i	$0.309266\pi$
$80$	−357.926	−0.500217
$81$	−780.439	−1.07056
$82$	−85.9704	−0.115779
$83$	393.316	0.520145	0.260072	−	0.965589i	$-0.416254\pi$
$83$	393.316	0.520145	0.260072	+	0.965589i	$0.416254\pi$
$84$	0	0
$85$	−229.055	−0.292289
$86$	−914.894	−1.14716
$87$	−902.296	−1.11191
$88$	1332.34	1.61396
$89$	−936.747	−1.11567	−0.557837	−	0.829950i	$-0.688369\pi$
$89$	−936.747	−1.11567	−0.557837	+	0.829950i	$0.688369\pi$
$90$	843.419	0.987824
$91$	0	0
$92$	409.940	0.464556
$93$	−872.331	−0.972651
$94$	−389.171	−0.427020
$95$	−1785.80	−1.92863
$96$	1067.41	1.13481
$97$	1.31027	0.00137153	0.000685764	−	1.00000i	$-0.499782\pi$
$97$	1.31027	0.00137153	0.000685764		1.00000i	$0.499782\pi$
$98$	0	0
$99$	−1365.59	−1.38633
$100$	−487.055	−0.487055
$101$	17.5797	0.0173192	0.00865962	−	0.999963i	$-0.497244\pi$
$101$	17.5797	0.0173192	0.00865962	+	0.999963i	$0.497244\pi$
$102$	−214.593	−0.208313
$103$	−773.530	−0.739982	−0.369991	−	0.929035i	$-0.620639\pi$
$103$	−773.530	−0.739982	−0.369991	+	0.929035i	$0.620639\pi$
$104$	−391.848	−0.369461
$105$	0	0
$106$	−1007.81	−0.923468
$107$	−244.139	−0.220577	−0.110289	−	0.993900i	$-0.535178\pi$
$107$	−244.139	−0.220577	−0.110289	+	0.993900i	$0.535178\pi$
$108$	53.5643	0.0477243
$109$	−322.822	−0.283676	−0.141838	−	0.989890i	$-0.545301\pi$
$109$	−322.822	−0.283676	−0.141838	+	0.989890i	$0.545301\pi$
$110$	−1852.10	−1.60537
$111$	1868.42	1.59768
$112$	0	0
$113$	1806.14	1.50361	0.751804	−	0.659387i	$-0.229184\pi$
$113$	1806.14	1.50361	0.751804	+	0.659387i	$0.229184\pi$
$114$	−1673.05	−1.37452
$115$	−1835.07	−1.48801
$116$	451.134	0.361093
$117$	401.626	0.317353
$118$	1185.85	0.925136
$119$	0	0
$120$	−2828.23	−2.15150
$121$	1667.76	1.25301
$122$	952.943	0.707175
$123$	−295.477	−0.216604
$124$	436.152	0.315868
$125$	164.048	0.117384
$126$	0	0
$127$	−1592.55	−1.11273	−0.556364	−	0.830939i	$-0.687804\pi$
$127$	−1592.55	−1.11273	−0.556364	+	0.830939i	$0.687804\pi$
$128$	−161.450	−0.111486
$129$	−3144.46	−2.14615
$130$	544.712	0.367495
$131$	−2423.48	−1.61634	−0.808170	−	0.588950i	$-0.799542\pi$
$131$	−2423.48	−1.61634	−0.808170	+	0.588950i	$0.799542\pi$
$132$	1422.02	0.937661
$133$	0	0
$134$	1225.70	0.790180
$135$	−239.777	−0.152865
$136$	345.507	0.217845
$137$	2110.28	1.31601	0.658006	−	0.753013i	$-0.271400\pi$
$137$	2110.28	1.31601	0.658006	+	0.753013i	$0.271400\pi$
$138$	−1719.21	−1.06050
$139$	283.449	0.172963	0.0864814	−	0.996253i	$-0.472438\pi$
$139$	283.449	0.172963	0.0864814	+	0.996253i	$0.472438\pi$
$140$	0	0
$141$	−1337.56	−0.798889
$142$	−1447.80	−0.855610
$143$	−881.950	−0.515751
$144$	−553.367	−0.320236
$145$	−2019.47	−1.15661
$146$	−9.25441	−0.00524589
$147$	0	0
$148$	−934.180	−0.518845
$149$	2529.58	1.39082	0.695408	−	0.718615i	$-0.255224\pi$
$149$	2529.58	1.39082	0.695408	+	0.718615i	$0.255224\pi$
$150$	2042.62	1.11186
$151$	695.345	0.374744	0.187372	−	0.982289i	$-0.440003\pi$
$151$	695.345	0.374744	0.187372	+	0.982289i	$0.440003\pi$
$152$	2693.70	1.43742
$153$	−354.128	−0.187121
$154$	0	0
$155$	−1952.41	−1.01175
$156$	−418.224	−0.214646
$157$	−342.895	−0.174305	−0.0871527	−	0.996195i	$-0.527777\pi$
$157$	−342.895	−0.174305	−0.0871527	+	0.996195i	$0.527777\pi$
$158$	1660.76	0.836222
$159$	−3463.82	−1.72766
$160$	2389.01	1.18043
$161$	0	0
$162$	−1636.45	−0.793654
$163$	−577.178	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$163$	−577.178	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$164$	147.734	0.0703419
$165$	−6365.60	−3.00340
$166$	824.720	0.385606
$167$	2484.95	1.15145	0.575723	−	0.817645i	$-0.304721\pi$
$167$	2484.95	1.15145	0.575723	+	0.817645i	$0.304721\pi$
$168$	0	0
$169$	−1937.61	−0.881937
$170$	−480.292	−0.216687
$171$	−2760.92	−1.23469
$172$	1572.18	0.696962
$173$	2400.26	1.05485	0.527423	−	0.849603i	$-0.323158\pi$
$173$	2400.26	1.05485	0.527423	+	0.849603i	$0.323158\pi$
$174$	−1891.97	−0.824309
$175$	0	0
$176$	1215.16	0.520435
$177$	4075.71	1.73079
$178$	−1964.21	−0.827099
$179$	4161.78	1.73780	0.868900	−	0.494988i	$-0.164827\pi$
$179$	4161.78	1.73780	0.868900	+	0.494988i	$0.164827\pi$
$180$	−1449.35	−0.600158
$181$	−3277.30	−1.34586	−0.672928	−	0.739708i	$-0.734964\pi$
$181$	−3277.30	−1.34586	−0.672928	+	0.739708i	$0.734964\pi$
$182$	0	0
$183$	3275.23	1.32301
$184$	2768.02	1.10903
$185$	4181.80	1.66190
$186$	−1829.14	−0.721070
$187$	777.647	0.304103
$188$	668.762	0.259439
$189$	0	0
$190$	−3744.54	−1.42978
$191$	940.243	0.356197	0.178098	−	0.984013i	$-0.443005\pi$
$191$	940.243	0.356197	0.178098	+	0.984013i	$0.443005\pi$
$192$	3517.54	1.32217
$193$	−1254.08	−0.467724	−0.233862	−	0.972270i	$-0.575136\pi$
$193$	−1254.08	−0.467724	−0.233862	+	0.972270i	$0.575136\pi$
$194$	2.74743	0.00101677
$195$	1872.15	0.687527
$196$	0	0
$197$	3320.14	1.20076	0.600381	−	0.799714i	$-0.295016\pi$
$197$	3320.14	1.20076	0.600381	+	0.799714i	$0.295016\pi$
$198$	−2863.42	−1.02775
$199$	4338.23	1.54537	0.772685	−	0.634790i	$-0.218913\pi$
$199$	4338.23	1.54537	0.772685	+	0.634790i	$0.218913\pi$
$200$	−3288.72	−1.16274
$201$	4212.67	1.47830
$202$	36.8617	0.0128395
$203$	0	0
$204$	368.763	0.126562
$205$	−661.322	−0.225311
$206$	−1621.97	−0.548582
$207$	−2837.09	−0.952614
$208$	−357.386	−0.119136
$209$	6062.83	2.00658
$210$	0	0
$211$	−5393.12	−1.75961	−0.879805	−	0.475334i	$-0.842327\pi$
$211$	−5393.12	−1.75961	−0.879805	+	0.475334i	$0.842327\pi$
$212$	1731.86	0.561058
$213$	−4976.03	−1.60071
$214$	−511.919	−0.163524
$215$	−7037.77	−2.23243
$216$	361.680	0.113932
$217$	0	0
$218$	−676.905	−0.210302
$219$	−31.8070	−0.00981424
$220$	3182.70	0.975354
$221$	−228.710	−0.0696139
$222$	3917.77	1.18443
$223$	−3279.09	−0.984683	−0.492341	−	0.870402i	$-0.663859\pi$
$223$	−3279.09	−0.984683	−0.492341	+	0.870402i	$0.663859\pi$
$224$	0	0
$225$	3370.79	0.998751
$226$	3787.19	1.11469
$227$	1452.67	0.424745	0.212373	−	0.977189i	$-0.431881\pi$
$227$	1452.67	0.424745	0.212373	+	0.977189i	$0.431881\pi$
$228$	2875.01	0.835099
$229$	2576.09	0.743374	0.371687	−	0.928358i	$-0.378779\pi$
$229$	2576.09	0.743374	0.371687	+	0.928358i	$0.378779\pi$
$230$	−3847.85	−1.10313
$231$	0	0
$232$	3046.17	0.862031
$233$	−2671.87	−0.751244	−0.375622	−	0.926773i	$-0.622571\pi$
$233$	−2671.87	−0.751244	−0.375622	+	0.926773i	$0.622571\pi$
$234$	842.145	0.235268
$235$	−2993.67	−0.831003
$236$	−2037.79	−0.562072
$237$	5707.97	1.56444
$238$	0	0
$239$	−1749.00	−0.473361	−0.236681	−	0.971587i	$-0.576060\pi$
$239$	−1749.00	−0.473361	−0.236681	+	0.971587i	$0.576060\pi$
$240$	−2579.48	−0.693771
$241$	−4093.60	−1.09416	−0.547079	−	0.837081i	$-0.684260\pi$
$241$	−4093.60	−1.09416	−0.547079	+	0.837081i	$0.684260\pi$
$242$	3497.03	0.928915
$243$	−5223.06	−1.37885
$244$	−1637.56	−0.429648
$245$	0	0
$246$	−619.567	−0.160578
$247$	−1783.11	−0.459337
$248$	2945.01	0.754067
$249$	2834.53	0.721409
$250$	343.983	0.0870216
$251$	938.887	0.236104	0.118052	−	0.993007i	$-0.462335\pi$
$251$	938.887	0.236104	0.118052	+	0.993007i	$0.462335\pi$
$252$	0	0
$253$	6230.09	1.54815
$254$	−3339.33	−0.824915
$255$	−1650.75	−0.405387
$256$	−4243.25	−1.03595
$257$	−7612.11	−1.84759	−0.923795	−	0.382888i	$-0.874929\pi$
$257$	−7612.11	−1.84759	−0.923795	+	0.382888i	$0.874929\pi$
$258$	−6593.42	−1.59104
$259$	0	0
$260$	−936.048	−0.223274
$261$	−3122.18	−0.740454
$262$	−5081.65	−1.19826
$263$	1736.80	0.407208	0.203604	−	0.979053i	$-0.434734\pi$
$263$	1736.80	0.407208	0.203604	+	0.979053i	$0.434734\pi$
$264$	9601.87	2.23846
$265$	−7752.55	−1.79711
$266$	0	0
$267$	−6750.91	−1.54737
$268$	−2106.27	−0.480078
$269$	−5208.37	−1.18052	−0.590260	−	0.807213i	$-0.700975\pi$
$269$	−5208.37	−1.18052	−0.590260	+	0.807213i	$0.700975\pi$
$270$	−502.775	−0.113326
$271$	−7012.13	−1.57180	−0.785898	−	0.618356i	$-0.787799\pi$
$271$	−7012.13	−1.57180	−0.785898	+	0.618356i	$0.787799\pi$
$272$	315.120	0.0702461
$273$	0	0
$274$	4424.92	0.975618
$275$	−7402.07	−1.62313
$276$	2954.33	0.644311
$277$	4769.05	1.03446	0.517229	−	0.855847i	$-0.326964\pi$
$277$	4769.05	1.03446	0.517229	+	0.855847i	$0.326964\pi$
$278$	594.347	0.128225
$279$	−3018.50	−0.647716
$280$	0	0
$281$	8603.76	1.82654	0.913270	−	0.407356i	$-0.133549\pi$
$281$	8603.76	1.82654	0.913270	+	0.407356i	$0.133549\pi$
$282$	−2804.66	−0.592252
$283$	−1370.29	−0.287829	−0.143914	−	0.989590i	$-0.545969\pi$
$283$	−1370.29	−0.287829	−0.143914	+	0.989590i	$0.545969\pi$
$284$	2487.94	0.519831
$285$	−12869.8	−2.67489
$286$	−1849.31	−0.382349
$287$	0	0
$288$	3693.51	0.755701
$289$	−4711.34	−0.958954
$290$	−4234.51	−0.857446
$291$	9.44282	0.00190223
$292$	15.9030	0.00318717
$293$	1166.83	0.232651	0.116325	−	0.993211i	$-0.462888\pi$
$293$	1166.83	0.232651	0.116325	+	0.993211i	$0.462888\pi$
$294$	0	0
$295$	9122.05	1.80036
$296$	−6307.82	−1.23863
$297$	814.049	0.159043
$298$	5304.13	1.03107
$299$	−1832.30	−0.354397
$300$	−3510.09	−0.675517
$301$	0	0
$302$	1458.03	0.277815
$303$	126.692	0.0240207
$304$	2456.79	0.463509
$305$	7330.45	1.37620
$306$	−742.550	−0.138721
$307$	−2474.09	−0.459948	−0.229974	−	0.973197i	$-0.573864\pi$
$307$	−2474.09	−0.459948	−0.229974	+	0.973197i	$0.573864\pi$
$308$	0	0
$309$	−5574.64	−1.02631
$310$	−4093.89	−0.750056
$311$	6891.25	1.25649	0.628243	−	0.778017i	$-0.283774\pi$
$311$	6891.25	1.25649	0.628243	+	0.778017i	$0.283774\pi$
$312$	−2823.95	−0.512420
$313$	7094.60	1.28118	0.640592	−	0.767882i	$-0.278689\pi$
$313$	7094.60	1.28118	0.640592	+	0.767882i	$0.278689\pi$
$314$	−718.995	−0.129220
$315$	0	0
$316$	−2853.90	−0.508051
$317$	−8334.09	−1.47662	−0.738311	−	0.674461i	$-0.764376\pi$
$317$	−8334.09	−1.47662	−0.738311	+	0.674461i	$0.764376\pi$
$318$	−7263.07	−1.28079
$319$	6856.15	1.20336
$320$	7872.79	1.37532
$321$	−1759.45	−0.305928
$322$	0	0
$323$	1572.23	0.270839
$324$	2812.13	0.482189
$325$	2176.98	0.371561
$326$	−1210.25	−0.205612
$327$	−2326.50	−0.393442
$328$	997.537	0.167926
$329$	0	0
$330$	−13347.6	−2.22656
$331$	−7074.81	−1.17482	−0.587412	−	0.809288i	$-0.699853\pi$
$331$	−7074.81	−1.17482	−0.587412	+	0.809288i	$0.699853\pi$
$332$	−1417.22	−0.234277
$333$	6465.22	1.06394
$334$	5210.55	0.853618
$335$	9428.59	1.53773
$336$	0	0
$337$	9885.96	1.59799	0.798995	−	0.601338i	$-0.205366\pi$
$337$	9885.96	1.59799	0.798995	+	0.601338i	$0.205366\pi$
$338$	−4062.87	−0.653819
$339$	13016.4	2.08541
$340$	825.347	0.131649
$341$	6628.46	1.05264
$342$	−5789.20	−0.915333
$343$	0	0
$344$	10615.8	1.66385
$345$	−13224.9	−2.06378
$346$	5032.96	0.782004
$347$	−6888.27	−1.06565	−0.532827	−	0.846224i	$-0.678870\pi$
$347$	−6888.27	−1.06565	−0.532827	+	0.846224i	$0.678870\pi$
$348$	3251.21	0.500814
$349$	4341.43	0.665878	0.332939	−	0.942948i	$-0.391960\pi$
$349$	4341.43	0.665878	0.332939	+	0.942948i	$0.391960\pi$
$350$	0	0
$351$	−239.415	−0.0364076
$352$	−8110.75	−1.22814
$353$	−5760.39	−0.868540	−0.434270	−	0.900783i	$-0.642994\pi$
$353$	−5760.39	−0.868540	−0.434270	+	0.900783i	$0.642994\pi$
$354$	8546.10	1.28311
$355$	−11137.1	−1.66506
$356$	3375.35	0.502509
$357$	0	0
$358$	8726.59	1.28831
$359$	−9343.03	−1.37356	−0.686778	−	0.726868i	$-0.740975\pi$
$359$	−9343.03	−1.37356	−0.686778	+	0.726868i	$0.740975\pi$
$360$	−9786.41	−1.43275
$361$	5398.68	0.787095
$362$	−6871.97	−0.997743
$363$	12019.1	1.73786
$364$	0	0
$365$	−71.1889	−0.0102088
$366$	6867.62	0.980810
$367$	6429.29	0.914459	0.457229	−	0.889349i	$-0.348842\pi$
$367$	6429.29	0.914459	0.457229	+	0.889349i	$0.348842\pi$
$368$	2524.57	0.357615
$369$	−1022.43	−0.144243
$370$	8768.56	1.23204
$371$	0	0
$372$	3143.24	0.438090
$373$	9506.44	1.31964	0.659819	−	0.751425i	$-0.270633\pi$
$373$	9506.44	1.31964	0.659819	+	0.751425i	$0.270633\pi$
$374$	1630.60	0.225445
$375$	1182.26	0.162804
$376$	4515.65	0.619354
$377$	−2016.43	−0.275467
$378$	0	0
$379$	−9781.04	−1.32564	−0.662821	−	0.748778i	$-0.730641\pi$
$379$	−9781.04	−1.32564	−0.662821	+	0.748778i	$0.730641\pi$
$380$	6434.72	0.868669
$381$	−11477.1	−1.54329
$382$	1971.54	0.264065
$383$	−7409.97	−0.988595	−0.494297	−	0.869293i	$-0.664575\pi$
$383$	−7409.97	−0.988595	−0.494297	+	0.869293i	$0.664575\pi$
$384$	−1163.53	−0.154625
$385$	0	0
$386$	−2629.60	−0.346744
$387$	−10880.7	−1.42919
$388$	−4.72126	−0.000617747 0
$389$	−14908.9	−1.94322	−0.971611	−	0.236583i	$-0.923972\pi$
$389$	−14908.9	−1.94322	−0.971611	+	0.236583i	$0.923972\pi$
$390$	3925.60	0.509694
$391$	1615.60	0.208963
$392$	0	0
$393$	−17465.4	−2.24177
$394$	6961.80	0.890179
$395$	12775.3	1.62733
$396$	4920.58	0.624415
$397$	−10325.3	−1.30533	−0.652663	−	0.757649i	$-0.726348\pi$
$397$	−10325.3	−1.30533	−0.652663	+	0.757649i	$0.726348\pi$
$398$	9096.56	1.14565
$399$	0	0
$400$	−2999.48	−0.374935
$401$	14457.5	1.80043	0.900213	−	0.435449i	$-0.143410\pi$
$401$	14457.5	1.80043	0.900213	+	0.435449i	$0.143410\pi$
$402$	8833.29	1.09593
$403$	−1949.46	−0.240967
$404$	−63.3442	−0.00780073
$405$	−12588.3	−1.54449
$406$	0	0
$407$	−14197.3	−1.72907
$408$	2489.98	0.302139
$409$	−9184.72	−1.11040	−0.555202	−	0.831716i	$-0.687359\pi$
$409$	−9184.72	−1.11040	−0.555202	+	0.831716i	$0.687359\pi$
$410$	−1386.69	−0.167033
$411$	15208.3	1.82523
$412$	2787.23	0.333294
$413$	0	0
$414$	−5948.91	−0.706215
$415$	6344.10	0.750409
$416$	2385.41	0.281140
$417$	2042.75	0.239889
$418$	12712.8	1.48756
$419$	11759.9	1.37114	0.685571	−	0.728006i	$-0.259553\pi$
$419$	11759.9	1.37114	0.685571	+	0.728006i	$0.259553\pi$
$420$	0	0
$421$	334.564	0.0387308	0.0193654	−	0.999812i	$-0.493835\pi$
$421$	334.564	0.0387308	0.0193654	+	0.999812i	$0.493835\pi$
$422$	−11308.5	−1.30448
$423$	−4628.33	−0.532003
$424$	11693.9	1.33940
$425$	−1919.52	−0.219084
$426$	−10433.9	−1.18668
$427$	0	0
$428$	879.696	0.0993498
$429$	−6355.99	−0.715315
$430$	−14757.1	−1.65500
$431$	−1373.80	−0.153535	−0.0767675	−	0.997049i	$-0.524460\pi$
$431$	−1373.80	−0.153535	−0.0767675	+	0.997049i	$0.524460\pi$
$432$	329.871	0.0367382
$433$	−6074.96	−0.674236	−0.337118	−	0.941462i	$-0.609452\pi$
$433$	−6074.96	−0.674236	−0.337118	+	0.941462i	$0.609452\pi$
$434$	0	0
$435$	−14553.9	−1.60415
$436$	1163.21	0.127770
$437$	12595.9	1.37881
$438$	−66.6942	−0.00727574
$439$	5894.68	0.640861	0.320430	−	0.947272i	$-0.396173\pi$
$439$	5894.68	0.640861	0.320430	+	0.947272i	$0.396173\pi$
$440$	21490.4	2.32845
$441$	0	0
$442$	−479.567	−0.0516079
$443$	−10295.1	−1.10414	−0.552070	−	0.833798i	$-0.686162\pi$
$443$	−10295.1	−1.10414	−0.552070	+	0.833798i	$0.686162\pi$
$444$	−6732.40	−0.719608
$445$	−15109.6	−1.60958
$446$	−6875.73	−0.729989
$447$	18230.1	1.92898
$448$	0	0
$449$	7557.56	0.794350	0.397175	−	0.917743i	$-0.369991\pi$
$449$	7557.56	0.794350	0.397175	+	0.917743i	$0.369991\pi$
$450$	7067.99	0.740419
$451$	2245.20	0.234418
$452$	−6508.01	−0.677237
$453$	5011.18	0.519748
$454$	3046.02	0.314882
$455$	0	0
$456$	19412.8	1.99362
$457$	−7341.15	−0.751432	−0.375716	−	0.926735i	$-0.622603\pi$
$457$	−7341.15	−0.751432	−0.375716	+	0.926735i	$0.622603\pi$
$458$	5401.64	0.551097
$459$	211.101	0.0214670
$460$	6612.24	0.670212
$461$	−13278.8	−1.34155	−0.670774	−	0.741662i	$-0.734038\pi$
$461$	−13278.8	−1.34155	−0.670774	+	0.741662i	$0.734038\pi$
$462$	0	0
$463$	16338.2	1.63996	0.819980	−	0.572392i	$-0.193984\pi$
$463$	16338.2	1.63996	0.819980	+	0.572392i	$0.193984\pi$
$464$	2778.26	0.277969
$465$	−14070.5	−1.40324
$466$	−5602.48	−0.556931
$467$	−8679.92	−0.860083	−0.430041	−	0.902809i	$-0.641501\pi$
$467$	−8679.92	−0.860083	−0.430041	+	0.902809i	$0.641501\pi$
$468$	−1447.17	−0.142939
$469$	0	0
$470$	−6277.25	−0.616059
$471$	−2471.16	−0.241751
$472$	−13759.7	−1.34182
$473$	23893.3	2.32266
$474$	11968.7	1.15979
$475$	−14965.3	−1.44559
$476$	0	0
$477$	−11985.7	−1.15050
$478$	−3667.37	−0.350924
$479$	2692.33	0.256818	0.128409	−	0.991721i	$-0.459013\pi$
$479$	2692.33	0.256818	0.128409	+	0.991721i	$0.459013\pi$
$480$	17217.0	1.63718
$481$	4175.49	0.395812
$482$	−8583.63	−0.811148
$483$	0	0
$484$	−6009.39	−0.564368
$485$	21.1345	0.00197869
$486$	−10951.9	−1.02220
$487$	−3213.44	−0.299003	−0.149502	−	0.988761i	$-0.547767\pi$
$487$	−3213.44	−0.299003	−0.149502	+	0.988761i	$0.547767\pi$
$488$	−11057.2	−1.02569
$489$	−4159.58	−0.384668
$490$	0	0
$491$	−3202.23	−0.294327	−0.147164	−	0.989112i	$-0.547014\pi$
$491$	−3202.23	−0.294327	−0.147164	+	0.989112i	$0.547014\pi$
$492$	1064.68	0.0975601
$493$	1777.96	0.162424
$494$	−3738.88	−0.340527
$495$	−22026.7	−2.00005
$496$	2686.00	0.243155
$497$	0	0
$498$	5943.55	0.534813
$499$	−15128.1	−1.35716	−0.678582	−	0.734525i	$-0.737405\pi$
$499$	−15128.1	−1.35716	−0.678582	+	0.734525i	$0.737405\pi$
$500$	−591.110	−0.0528705
$501$	17908.4	1.59699
$502$	1968.70	0.175034
$503$	8253.97	0.731662	0.365831	−	0.930681i	$-0.380785\pi$
$503$	8253.97	0.731662	0.365831	+	0.930681i	$0.380785\pi$
$504$	0	0
$505$	283.557	0.0249863
$506$	13063.5	1.14771
$507$	−13963.9	−1.22319
$508$	5738.40	0.501182
$509$	5229.64	0.455403	0.227701	−	0.973731i	$-0.426879\pi$
$509$	5229.64	0.455403	0.227701	+	0.973731i	$0.426879\pi$
$510$	−3461.35	−0.300531
$511$	0	0
$512$	−7605.82	−0.656509
$513$	1645.82	0.141647
$514$	−15961.4	−1.36970
$515$	−12476.9	−1.06757
$516$	11330.3	0.966645
$517$	10163.6	0.864590
$518$	0	0
$519$	17298.1	1.46301
$520$	−6320.44	−0.533018
$521$	5539.37	0.465804	0.232902	−	0.972500i	$-0.425178\pi$
$521$	5539.37	0.465804	0.232902	+	0.972500i	$0.425178\pi$
$522$	−6546.72	−0.548931
$523$	−22074.4	−1.84559	−0.922797	−	0.385286i	$-0.874103\pi$
$523$	−22074.4	−1.84559	−0.922797	+	0.385286i	$0.874103\pi$
$524$	8732.44	0.728013
$525$	0	0
$526$	3641.79	0.301882
$527$	1718.91	0.142081
$528$	8757.39	0.721812
$529$	776.360	0.0638086
$530$	−16255.8	−1.33228
$531$	14103.0	1.15258
$532$	0	0
$533$	−660.323	−0.0536619
$534$	−14155.6	−1.14714
$535$	−3937.91	−0.318225
$536$	−14222.1	−1.14608
$537$	29992.9	2.41022
$538$	−10921.1	−0.875172
$539$	0	0
$540$	863.982	0.0688516
$541$	12974.1	1.03105	0.515525	−	0.856874i	$-0.327597\pi$
$541$	12974.1	1.03105	0.515525	+	0.856874i	$0.327597\pi$
$542$	−14703.3	−1.16524
$543$	−23618.7	−1.86662
$544$	−2103.30	−0.165769
$545$	−5207.05	−0.409258
$546$	0	0
$547$	−563.254	−0.0440274	−0.0220137	−	0.999758i	$-0.507008\pi$
$547$	−563.254	−0.0440274	−0.0220137	+	0.999758i	$0.507008\pi$
$548$	−7603.91	−0.592743
$549$	11333.2	0.881033
$550$	−15520.9	−1.20330
$551$	13861.6	1.07173
$552$	19948.4	1.53815
$553$	0	0
$554$	9999.94	0.766889
$555$	30137.2	2.30496
$556$	−1021.34	−0.0779038
$557$	15050.9	1.14493	0.572465	−	0.819929i	$-0.305987\pi$
$557$	15050.9	1.14493	0.572465	+	0.819929i	$0.305987\pi$
$558$	−6329.31	−0.480181
$559$	−7027.14	−0.531693
$560$	0	0
$561$	5604.31	0.421772
$562$	18040.7	1.35409
$563$	5802.16	0.434337	0.217169	−	0.976134i	$-0.430318\pi$
$563$	5802.16	0.434337	0.217169	+	0.976134i	$0.430318\pi$
$564$	4819.60	0.359826
$565$	29132.7	2.16924
$566$	−2873.29	−0.213380
$567$	0	0
$568$	16799.2	1.24098
$569$	−2251.22	−0.165863	−0.0829316	−	0.996555i	$-0.526428\pi$
$569$	−2251.22	−0.165863	−0.0829316	+	0.996555i	$0.526428\pi$
$570$	−26986.0	−1.98301
$571$	−5531.03	−0.405370	−0.202685	−	0.979244i	$-0.564967\pi$
$571$	−5531.03	−0.405370	−0.202685	+	0.979244i	$0.564967\pi$
$572$	3177.90	0.232298
$573$	6776.10	0.494024
$574$	0	0
$575$	−15378.2	−1.11533
$576$	12171.6	0.880471
$577$	6743.38	0.486535	0.243267	−	0.969959i	$-0.421781\pi$
$577$	6743.38	0.486535	0.243267	+	0.969959i	$0.421781\pi$
$578$	−9878.92	−0.710915
$579$	−9037.85	−0.648705
$580$	7276.71	0.520946
$581$	0	0
$582$	19.8001	0.00141021
$583$	26320.0	1.86975
$584$	107.381	0.00760868
$585$	6478.15	0.457844
$586$	2446.65	0.172474
$587$	−3828.31	−0.269185	−0.134592	−	0.990901i	$-0.542973\pi$
$587$	−3828.31	−0.269185	−0.134592	+	0.990901i	$0.542973\pi$
$588$	0	0
$589$	13401.3	0.937504
$590$	19127.5	1.33469
$591$	23927.4	1.66539
$592$	−5753.05	−0.399407
$593$	8604.08	0.595830	0.297915	−	0.954592i	$-0.403709\pi$
$593$	8604.08	0.595830	0.297915	+	0.954592i	$0.403709\pi$
$594$	1706.93	0.117906
$595$	0	0
$596$	−9114.76	−0.626435
$597$	31264.5	2.14334
$598$	−3842.04	−0.262730
$599$	−21557.1	−1.47045	−0.735224	−	0.677824i	$-0.762923\pi$
$599$	−21557.1	−1.47045	−0.735224	+	0.677824i	$0.762923\pi$
$600$	−23701.0	−1.61265
$601$	13309.1	0.903310	0.451655	−	0.892193i	$-0.350834\pi$
$601$	13309.1	0.903310	0.451655	+	0.892193i	$0.350834\pi$
$602$	0	0
$603$	14577.0	0.984444
$604$	−2505.51	−0.168788
$605$	26900.7	1.80772
$606$	265.654	0.0178077
$607$	−11668.0	−0.780211	−0.390105	−	0.920770i	$-0.627561\pi$
$607$	−11668.0	−0.780211	−0.390105	+	0.920770i	$0.627561\pi$
$608$	−16398.1	−1.09380
$609$	0	0
$610$	15370.8	1.02024
$611$	−2989.15	−0.197918
$612$	1276.02	0.0842810
$613$	882.728	0.0581616	0.0290808	−	0.999577i	$-0.490742\pi$
$613$	882.728	0.0581616	0.0290808	+	0.999577i	$0.490742\pi$
$614$	−5187.77	−0.340980
$615$	−4765.98	−0.312493
$616$	0	0
$617$	−13499.1	−0.880796	−0.440398	−	0.897803i	$-0.645163\pi$
$617$	−13499.1	−0.880796	−0.440398	+	0.897803i	$0.645163\pi$
$618$	−11689.1	−0.760850
$619$	−3759.61	−0.244122	−0.122061	−	0.992523i	$-0.538950\pi$
$619$	−3759.61	−0.244122	−0.122061	+	0.992523i	$0.538950\pi$
$620$	7035.05	0.455701
$621$	1691.23	0.109286
$622$	14449.8	0.931489
$623$	0	0
$624$	−2575.59	−0.165234
$625$	−14250.2	−0.912016
$626$	14876.2	0.949798
$627$	43693.3	2.78300
$628$	1235.54	0.0785086
$629$	−3681.68	−0.233383
$630$	0	0
$631$	3102.41	0.195729	0.0978644	−	0.995200i	$-0.468799\pi$
$631$	3102.41	0.195729	0.0978644	+	0.995200i	$0.468799\pi$
$632$	−19270.2	−1.21286
$633$	−38866.9	−2.44048
$634$	−17475.2	−1.09469
$635$	−25687.6	−1.60532
$636$	12481.1	0.778154
$637$	0	0
$638$	14376.2	0.892102
$639$	−17218.4	−1.06596
$640$	−2604.15	−0.160841
$641$	−1715.22	−0.105690	−0.0528449	−	0.998603i	$-0.516829\pi$
$641$	−1715.22	−0.105690	−0.0528449	+	0.998603i	$0.516829\pi$
$642$	−3689.28	−0.226798
$643$	−9879.49	−0.605924	−0.302962	−	0.953003i	$-0.597975\pi$
$643$	−9879.49	−0.605924	−0.302962	+	0.953003i	$0.597975\pi$
$644$	0	0
$645$	−50719.4	−3.09624
$646$	3296.71	0.200785
$647$	−28716.4	−1.74491	−0.872456	−	0.488693i	$-0.837474\pi$
$647$	−28716.4	−1.74491	−0.872456	+	0.488693i	$0.837474\pi$
$648$	18988.2	1.15112
$649$	−30969.5	−1.87313
$650$	4564.78	0.275454
$651$	0	0
$652$	2079.73	0.124921
$653$	−8277.44	−0.496051	−0.248026	−	0.968753i	$-0.579782\pi$
$653$	−8277.44	−0.496051	−0.248026	+	0.968753i	$0.579782\pi$
$654$	−4878.29	−0.291676
$655$	−39090.3	−2.33188
$656$	909.804	0.0541492
$657$	−110.061	−0.00653559
$658$	0	0
$659$	637.149	0.0376628	0.0188314	−	0.999823i	$-0.494005\pi$
$659$	637.149	0.0376628	0.0188314	+	0.999823i	$0.494005\pi$
$660$	22937.0	1.35276
$661$	32133.7	1.89086	0.945428	−	0.325831i	$-0.105644\pi$
$661$	32133.7	1.89086	0.945428	+	0.325831i	$0.105644\pi$
$662$	−14834.7	−0.870950
$663$	−1648.25	−0.0965503
$664$	−9569.44	−0.559287
$665$	0	0
$666$	13556.5	0.788746
$667$	14244.0	0.826883
$668$	−8953.95	−0.518621
$669$	−23631.6	−1.36570
$670$	19770.2	1.13999
$671$	−24887.0	−1.43182
$672$	0	0
$673$	29852.0	1.70982	0.854911	−	0.518775i	$-0.173612\pi$
$673$	29852.0	1.70982	0.854911	+	0.518775i	$0.173612\pi$
$674$	20729.3	1.18466
$675$	−2009.38	−0.114579
$676$	6981.74	0.397231
$677$	−24825.7	−1.40935	−0.704675	−	0.709530i	$-0.748907\pi$
$677$	−24825.7	−1.40935	−0.704675	+	0.709530i	$0.748907\pi$
$678$	27293.3	1.54601
$679$	0	0
$680$	5572.96	0.314284
$681$	10469.0	0.589096
$682$	13898.8	0.780372
$683$	29366.6	1.64521	0.822607	−	0.568611i	$-0.192519\pi$
$683$	29366.6	1.64521	0.822607	+	0.568611i	$0.192519\pi$
$684$	9948.32	0.556116
$685$	34038.5	1.89860
$686$	0	0
$687$	18565.2	1.03102
$688$	9682.11	0.536522
$689$	−7740.84	−0.428015
$690$	−27730.5	−1.52997
$691$	24852.9	1.36823	0.684116	−	0.729373i	$-0.260188\pi$
$691$	24852.9	1.36823	0.684116	+	0.729373i	$0.260188\pi$
$692$	−8648.77	−0.475111
$693$	0	0
$694$	−14443.6	−0.790016
$695$	4571.97	0.249532
$696$	21953.0	1.19559
$697$	582.231	0.0316407
$698$	9103.28	0.493645
$699$	−19255.5	−1.04193
$700$	0	0
$701$	1484.47	0.0799825	0.0399912	−	0.999200i	$-0.487267\pi$
$701$	1484.47	0.0799825	0.0399912	+	0.999200i	$0.487267\pi$
$702$	−502.016	−0.0269905
$703$	−28703.7	−1.53995
$704$	−26728.2	−1.43091
$705$	−21574.7	−1.15255
$706$	−12078.6	−0.643888
$707$	0	0
$708$	−14685.9	−0.779560
$709$	34744.7	1.84043	0.920215	−	0.391412i	$-0.128013\pi$
$709$	34744.7	1.84043	0.920215	+	0.391412i	$0.128013\pi$
$710$	−23352.7	−1.23438
$711$	19751.1	1.04181
$712$	22791.2	1.19963
$713$	13771.0	0.723321
$714$	0	0
$715$	−14225.7	−0.744070
$716$	−14996.0	−0.782719
$717$	−12604.6	−0.656524
$718$	−19590.8	−1.01828
$719$	21885.5	1.13517	0.567587	−	0.823313i	$-0.307877\pi$
$719$	21885.5	1.13517	0.567587	+	0.823313i	$0.307877\pi$
$720$	−8925.70	−0.462002
$721$	0	0
$722$	11320.2	0.583508
$723$	−29501.6	−1.51753
$724$	11809.0	0.606185
$725$	−16923.5	−0.866931
$726$	25202.2	1.28835
$727$	5116.48	0.261018	0.130509	−	0.991447i	$-0.458339\pi$
$727$	5116.48	0.261018	0.130509	+	0.991447i	$0.458339\pi$
$728$	0	0
$729$	−16569.5	−0.841816
$730$	−149.272	−0.00756821
$731$	6196.08	0.313503
$732$	−11801.5	−0.595897
$733$	−7515.14	−0.378688	−0.189344	−	0.981911i	$-0.560636\pi$
$733$	−7515.14	−0.378688	−0.189344	+	0.981911i	$0.560636\pi$
$734$	13481.2	0.677929
$735$	0	0
$736$	−16850.5	−0.843911
$737$	−32010.2	−1.59988
$738$	−2143.87	−0.106933
$739$	11430.1	0.568964	0.284482	−	0.958681i	$-0.408178\pi$
$739$	11430.1	0.568964	0.284482	+	0.958681i	$0.408178\pi$
$740$	−15068.1	−0.748535
$741$	−12850.4	−0.637073
$742$	0	0
$743$	31750.5	1.56771	0.783857	−	0.620941i	$-0.213249\pi$
$743$	31750.5	1.56771	0.783857	+	0.620941i	$0.213249\pi$
$744$	21224.0	1.04585
$745$	40801.7	2.00652
$746$	19933.5	0.978306
$747$	9808.22	0.480407
$748$	−2802.07	−0.136970
$749$	0	0
$750$	2479.00	0.120694
$751$	−15496.4	−0.752958	−0.376479	−	0.926425i	$-0.622865\pi$
$751$	−15496.4	−0.752958	−0.376479	+	0.926425i	$0.622865\pi$
$752$	4118.50	0.199716
$753$	6766.33	0.327462
$754$	−4228.12	−0.204216
$755$	11215.8	0.540641
$756$	0	0
$757$	31991.5	1.53600	0.768000	−	0.640450i	$-0.221252\pi$
$757$	31991.5	1.53600	0.768000	+	0.640450i	$0.221252\pi$
$758$	−20509.3	−0.982757
$759$	44898.7	2.14719
$760$	43448.9	2.07376
$761$	−21997.1	−1.04782	−0.523912	−	0.851772i	$-0.675528\pi$
$761$	−21997.1	−1.04782	−0.523912	+	0.851772i	$0.675528\pi$
$762$	−24065.7	−1.14411
$763$	0	0
$764$	−3387.95	−0.160434
$765$	−5712.02	−0.269959
$766$	−15537.5	−0.732890
$767$	9108.28	0.428789
$768$	−30580.1	−1.43680
$769$	−28860.4	−1.35336	−0.676679	−	0.736279i	$-0.736581\pi$
$769$	−28860.4	−1.35336	−0.676679	+	0.736279i	$0.736581\pi$
$770$	0	0
$771$	−54858.6	−2.56250
$772$	4518.78	0.210667
$773$	21759.3	1.01246	0.506228	−	0.862400i	$-0.331040\pi$
$773$	21759.3	1.01246	0.506228	+	0.862400i	$0.331040\pi$
$774$	−22815.0	−1.05952
$775$	−16361.5	−0.758353
$776$	−31.8792	−0.00147474
$777$	0	0
$778$	−31261.7	−1.44060
$779$	4539.29	0.208777
$780$	−6745.87	−0.309668
$781$	37810.6	1.73236
$782$	3387.66	0.154914
$783$	1861.18	0.0849466
$784$	0	0
$785$	−5530.82	−0.251469
$786$	−36622.2	−1.66192
$787$	−9475.27	−0.429170	−0.214585	−	0.976705i	$-0.568840\pi$
$787$	−9475.27	−0.429170	−0.214585	+	0.976705i	$0.568840\pi$
$788$	−11963.3	−0.540833
$789$	12516.7	0.564774
$790$	26787.7	1.20641
$791$	0	0
$792$	33225.0	1.49066
$793$	7319.38	0.327767
$794$	−21650.6	−0.967696
$795$	−55870.7	−2.49249
$796$	−15631.8	−0.696047
$797$	−12044.4	−0.535302	−0.267651	−	0.963516i	$-0.586247\pi$
$797$	−12044.4	−0.535302	−0.267651	+	0.963516i	$0.586247\pi$
$798$	0	0
$799$	2635.64	0.116699
$800$	20020.4	0.884783
$801$	−23359.9	−1.03044
$802$	30315.0	1.33474
$803$	241.688	0.0106214
$804$	−15179.4	−0.665840
$805$	0	0
$806$	−4087.71	−0.178639
$807$	−37535.4	−1.63731
$808$	−427.717	−0.0186225
$809$	−34910.7	−1.51718	−0.758588	−	0.651570i	$-0.774111\pi$
$809$	−34910.7	−1.51718	−0.758588	+	0.651570i	$0.774111\pi$
$810$	−26395.7	−1.14500
$811$	−40850.4	−1.76874	−0.884372	−	0.466782i	$-0.845413\pi$
$811$	−40850.4	−1.76874	−0.884372	+	0.466782i	$0.845413\pi$
$812$	0	0
$813$	−50534.7	−2.17999
$814$	−29769.4	−1.28184
$815$	−9309.77	−0.400131
$816$	2270.99	0.0974271
$817$	48307.0	2.06860
$818$	−19258.9	−0.823192
$819$	0	0
$820$	2382.92	0.101482
$821$	13901.0	0.590923	0.295462	−	0.955355i	$-0.404527\pi$
$821$	13901.0	0.590923	0.295462	+	0.955355i	$0.404527\pi$
$822$	31889.3	1.35312
$823$	−6005.97	−0.254380	−0.127190	−	0.991878i	$-0.540596\pi$
$823$	−6005.97	−0.254380	−0.127190	+	0.991878i	$0.540596\pi$
$824$	18820.1	0.795668
$825$	−53344.9	−2.25119
$826$	0	0
$827$	1372.28	0.0577013	0.0288506	−	0.999584i	$-0.490815\pi$
$827$	1372.28	0.0577013	0.0288506	+	0.999584i	$0.490815\pi$
$828$	10222.8	0.429065
$829$	−8446.16	−0.353857	−0.176928	−	0.984224i	$-0.556616\pi$
$829$	−8446.16	−0.353857	−0.176928	+	0.984224i	$0.556616\pi$
$830$	13302.6	0.556312
$831$	34369.4	1.43473
$832$	7860.90	0.327557
$833$	0	0
$834$	4283.31	0.177840
$835$	40081.8	1.66118
$836$	−21846.0	−0.903779
$837$	1799.37	0.0743076
$838$	24658.6	1.01649
$839$	−43976.7	−1.80959	−0.904794	−	0.425850i	$-0.859975\pi$
$839$	−43976.7	−1.80959	−0.904794	+	0.425850i	$0.859975\pi$
$840$	0	0
$841$	−8713.59	−0.357275
$842$	701.527	0.0287128
$843$	62005.2	2.53330
$844$	19432.9	0.792543
$845$	−31253.3	−1.27236
$846$	−9704.86	−0.394397
$847$	0	0
$848$	10665.5	0.431902
$849$	−9875.38	−0.399201
$850$	−4024.93	−0.162416
$851$	−29495.6	−1.18813
$852$	17929.9	0.720974
$853$	35342.0	1.41862	0.709312	−	0.704895i	$-0.249006\pi$
$853$	35342.0	1.41862	0.709312	+	0.704895i	$0.249006\pi$
$854$	0	0
$855$	−44533.0	−1.78128
$856$	5939.94	0.237176
$857$	4808.21	0.191651	0.0958257	−	0.995398i	$-0.469451\pi$
$857$	4808.21	0.191651	0.0958257	+	0.995398i	$0.469451\pi$
$858$	−13327.5	−0.530295
$859$	−21227.5	−0.843159	−0.421579	−	0.906792i	$-0.638524\pi$
$859$	−21227.5	−0.843159	−0.421579	+	0.906792i	$0.638524\pi$
$860$	25358.9	1.00550
$861$	0	0
$862$	−2880.64	−0.113822
$863$	40097.9	1.58163	0.790815	−	0.612055i	$-0.209657\pi$
$863$	40097.9	1.58163	0.790815	+	0.612055i	$0.209657\pi$
$864$	−2201.76	−0.0866959
$865$	38715.7	1.52182
$866$	−12738.2	−0.499841
$867$	−33953.5	−1.33001
$868$	0	0
$869$	−43372.3	−1.69310
$870$	−30517.1	−1.18923
$871$	9414.36	0.366238
$872$	7854.31	0.305023
$873$	32.6747	0.00126675
$874$	26411.5	1.02218
$875$	0	0
$876$	114.609	0.00442042
$877$	17755.2	0.683638	0.341819	−	0.939766i	$-0.388957\pi$
$877$	17755.2	0.683638	0.341819	+	0.939766i	$0.388957\pi$
$878$	12360.2	0.475099
$879$	8409.02	0.322673
$880$	19600.4	0.750828
$881$	−34274.2	−1.31070	−0.655350	−	0.755326i	$-0.727479\pi$
$881$	−34274.2	−1.31070	−0.655350	+	0.755326i	$0.727479\pi$
$882$	0	0
$883$	1753.38	0.0668243	0.0334122	−	0.999442i	$-0.489363\pi$
$883$	1753.38	0.0668243	0.0334122	+	0.999442i	$0.489363\pi$
$884$	824.101	0.0313547
$885$	65740.4	2.49699
$886$	−21587.1	−0.818548
$887$	39838.5	1.50806	0.754029	−	0.656841i	$-0.228108\pi$
$887$	39838.5	1.50806	0.754029	+	0.656841i	$0.228108\pi$
$888$	−45458.9	−1.71791
$889$	0	0
$890$	−31682.3	−1.19325
$891$	42737.6	1.60692
$892$	11815.4	0.443509
$893$	20548.5	0.770020
$894$	38225.6	1.43004
$895$	67128.7	2.50711
$896$	0	0
$897$	−13204.9	−0.491527
$898$	15847.0	0.588887
$899$	15154.8	0.562227
$900$	−12145.8	−0.449846
$901$	6825.38	0.252371
$902$	4707.82	0.173784
$903$	0	0
$904$	−43943.8	−1.61676
$905$	−52862.2	−1.94166
$906$	10507.6	0.385312
$907$	−34436.2	−1.26068	−0.630339	−	0.776320i	$-0.717084\pi$
$907$	−34436.2	−1.26068	−0.630339	+	0.776320i	$0.717084\pi$
$908$	−5234.36	−0.191309
$909$	438.389	0.0159961
$910$	0	0
$911$	−24021.3	−0.873613	−0.436806	−	0.899556i	$-0.643891\pi$
$911$	−24021.3	−0.873613	−0.436806	+	0.899556i	$0.643891\pi$
$912$	17705.5	0.642859
$913$	−21538.3	−0.780739
$914$	−15393.2	−0.557070
$915$	52828.8	1.90870
$916$	−9282.33	−0.334822
$917$	0	0
$918$	442.645	0.0159145
$919$	−5229.67	−0.187716	−0.0938579	−	0.995586i	$-0.529920\pi$
$919$	−5229.67	−0.187716	−0.0938579	+	0.995586i	$0.529920\pi$
$920$	44647.6	1.59999
$921$	−17830.2	−0.637920
$922$	−27843.4	−0.994549
$923$	−11120.3	−0.396564
$924$	0	0
$925$	35044.2	1.24567
$926$	34258.6	1.21578
$927$	−19289.7	−0.683450
$928$	−18543.8	−0.655960
$929$	−7801.06	−0.275505	−0.137753	−	0.990467i	$-0.543988\pi$
$929$	−7801.06	−0.275505	−0.137753	+	0.990467i	$0.543988\pi$
$930$	−29503.6	−1.04028
$931$	0	0
$932$	9627.45	0.338367
$933$	49663.6	1.74267
$934$	−18200.4	−0.637618
$935$	12543.3	0.438727
$936$	−9771.64	−0.341235
$937$	−34547.2	−1.20449	−0.602246	−	0.798311i	$-0.705727\pi$
$937$	−34547.2	−1.20449	−0.602246	+	0.798311i	$0.705727\pi$
$938$	0	0
$939$	51129.0	1.77692
$940$	10787.0	0.374290
$941$	51372.3	1.77969	0.889845	−	0.456263i	$-0.150812\pi$
$941$	51372.3	1.77969	0.889845	+	0.456263i	$0.150812\pi$
$942$	−5181.62	−0.179221
$943$	4664.52	0.161079
$944$	−12549.5	−0.432683
$945$	0	0
$946$	50100.5	1.72189
$947$	26662.8	0.914914	0.457457	−	0.889232i	$-0.348760\pi$
$947$	26662.8	0.914914	0.457457	+	0.889232i	$0.348760\pi$
$948$	−20567.3	−0.704637
$949$	−71.0814	−0.00243140
$950$	−31379.9	−1.07168
$951$	−60061.7	−2.04799
$952$	0	0
$953$	15336.0	0.521282	0.260641	−	0.965436i	$-0.416066\pi$
$953$	15336.0	0.521282	0.260641	+	0.965436i	$0.416066\pi$
$954$	−25132.1	−0.852917
$955$	15165.9	0.513883
$956$	6302.11	0.213206
$957$	49410.6	1.66898
$958$	5645.39	0.190391
$959$	0	0
$960$	56737.2	1.90749
$961$	−15139.4	−0.508188
$962$	8755.32	0.293433
$963$	−6088.15	−0.203726
$964$	14750.3	0.492818
$965$	−20228.1	−0.674782
$966$	0	0
$967$	−9211.85	−0.306342	−0.153171	−	0.988200i	$-0.548949\pi$
$967$	−9211.85	−0.306342	−0.153171	+	0.988200i	$0.548949\pi$
$968$	−40577.0	−1.34731
$969$	11330.7	0.375638
$970$	44.3156	0.00146689
$971$	−25093.3	−0.829332	−0.414666	−	0.909974i	$-0.636102\pi$
$971$	−25093.3	−0.829332	−0.414666	+	0.909974i	$0.636102\pi$
$972$	18820.1	0.621043
$973$	0	0
$974$	−6738.06	−0.221665
$975$	15689.0	0.515332
$976$	−10084.8	−0.330743
$977$	9589.57	0.314020	0.157010	−	0.987597i	$-0.449814\pi$
$977$	9589.57	0.314020	0.157010	+	0.987597i	$0.449814\pi$
$978$	−8721.97	−0.285172
$979$	51297.2	1.67463
$980$	0	0
$981$	−8050.29	−0.262004
$982$	−6714.56	−0.218198
$983$	46471.6	1.50785	0.753923	−	0.656963i	$-0.228159\pi$
$983$	46471.6	1.50785	0.753923	+	0.656963i	$0.228159\pi$
$984$	7189.01	0.232904
$985$	53553.2	1.73233
$986$	3728.09	0.120412
$987$	0	0
$988$	6425.00	0.206889
$989$	49639.7	1.59601
$990$	−46186.4	−1.48273
$991$	36711.0	1.17675	0.588377	−	0.808587i	$-0.299767\pi$
$991$	36711.0	1.17675	0.588377	+	0.808587i	$0.299767\pi$
$992$	−17928.0	−0.573805
$993$	−50986.4	−1.62941
$994$	0	0
$995$	69974.7	2.22950
$996$	−10213.6	−0.324929
$997$	30667.7	0.974177	0.487089	−	0.873352i	$-0.338059\pi$
$997$	30667.7	0.974177	0.487089	+	0.873352i	$0.338059\pi$
$998$	−31721.1	−1.00613
$999$	−3854.02	−0.122058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.4.a.g.1.13		19
7.6	odd	2		287.4.a.e.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.a.e.1.13	✓	19	7.6	odd	2
2009.4.a.g.1.13		19	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+2.09684 q^{2} +7.20675 q^{3} -3.60327 q^{4} +16.1298 q^{5} +15.1114 q^{6} -24.3302 q^{8} +24.9373 q^{9} +O(q^{10})\)	\(q+2.09684 q^{2} +7.20675 q^{3} -3.60327 q^{4} +16.1298 q^{5} +15.1114 q^{6} -24.3302 q^{8} +24.9373 q^{9} +33.8216 q^{10} -54.7610 q^{11} -25.9678 q^{12} +16.1054 q^{13} +116.243 q^{15} -22.1904 q^{16} -14.2008 q^{17} +52.2895 q^{18} -110.714 q^{19} -58.1200 q^{20} -114.825 q^{22} -113.769 q^{23} -175.342 q^{24} +135.171 q^{25} +33.7705 q^{26} -14.8655 q^{27} -125.201 q^{29} +243.744 q^{30} -121.044 q^{31} +148.112 q^{32} -394.649 q^{33} -29.7767 q^{34} -89.8557 q^{36} +259.259 q^{37} -232.150 q^{38} +116.068 q^{39} -392.441 q^{40} -41.0000 q^{41} -436.321 q^{43} +197.318 q^{44} +402.233 q^{45} -238.555 q^{46} -185.599 q^{47} -159.920 q^{48} +283.431 q^{50} -102.341 q^{51} -58.0322 q^{52} -480.635 q^{53} -31.1706 q^{54} -883.283 q^{55} -797.891 q^{57} -262.527 q^{58} +565.540 q^{59} -418.856 q^{60} +454.466 q^{61} -253.809 q^{62} +488.090 q^{64} +259.778 q^{65} -827.515 q^{66} +584.545 q^{67} +51.1691 q^{68} -819.904 q^{69} -690.467 q^{71} -606.729 q^{72} -4.41350 q^{73} +543.625 q^{74} +974.140 q^{75} +398.933 q^{76} +243.376 q^{78} +792.031 q^{79} -357.926 q^{80} -780.439 q^{81} -85.9704 q^{82} +393.316 q^{83} -229.055 q^{85} -914.894 q^{86} -902.296 q^{87} +1332.34 q^{88} -936.747 q^{89} +843.419 q^{90} +409.940 q^{92} -872.331 q^{93} -389.171 q^{94} -1785.80 q^{95} +1067.41 q^{96} +1.31027 q^{97} -1365.59 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{2} - 13 q^{3} + 88 q^{4} - 26 q^{5} - 30 q^{6} + 51 q^{8} + 206 q^{9}+O(q^{10}) \) 19 * q + 4 * q^2 - 13 * q^3 + 88 * q^4 - 26 * q^5 - 30 * q^6 + 51 * q^8 + 206 * q^9	\( 19 q + 4 q^{2} - 13 q^{3} + 88 q^{4} - 26 q^{5} - 30 q^{6} + 51 q^{8} + 206 q^{9} - 108 q^{10} - 4 q^{11} - 52 q^{12} - 239 q^{13} - 144 q^{15} + 484 q^{16} - 117 q^{17} - 193 q^{18} - 323 q^{19} - 519 q^{20} + 105 q^{22} + 121 q^{23} + 127 q^{24} + 989 q^{25} - 134 q^{26} - 328 q^{27} + 72 q^{29} + 311 q^{30} - 872 q^{31} + 422 q^{32} - 746 q^{33} - 639 q^{34} + 552 q^{36} + 419 q^{37} - 106 q^{38} + 420 q^{39} - 1463 q^{40} - 779 q^{41} + 1231 q^{43} - 183 q^{44} - 1638 q^{45} - 630 q^{46} + 163 q^{47} - 1694 q^{48} + 791 q^{50} - 1020 q^{51} - 2737 q^{52} - 308 q^{53} - 932 q^{54} - 1668 q^{55} + 3346 q^{57} + 495 q^{58} - 902 q^{59} - 1080 q^{60} - 1390 q^{61} - 946 q^{62} + 2239 q^{64} - 1500 q^{65} - 3336 q^{66} + 86 q^{67} - 849 q^{68} - 1459 q^{69} + 784 q^{71} - 2013 q^{72} - 4270 q^{73} - 2009 q^{74} + 1841 q^{75} - 155 q^{76} - 1633 q^{78} - 392 q^{79} - 1160 q^{80} + 451 q^{81} - 164 q^{82} - 112 q^{83} + 2740 q^{85} - 6521 q^{86} - 1868 q^{87} - 1540 q^{88} - 687 q^{89} + 7589 q^{90} - 3552 q^{92} - 3908 q^{93} + 1705 q^{94} - 3636 q^{95} + 7802 q^{96} - 577 q^{97} + 1040 q^{99}+O(q^{100}) \) 19 * q + 4 * q^2 - 13 * q^3 + 88 * q^4 - 26 * q^5 - 30 * q^6 + 51 * q^8 + 206 * q^9 - 108 * q^10 - 4 * q^11 - 52 * q^12 - 239 * q^13 - 144 * q^15 + 484 * q^16 - 117 * q^17 - 193 * q^18 - 323 * q^19 - 519 * q^20 + 105 * q^22 + 121 * q^23 + 127 * q^24 + 989 * q^25 - 134 * q^26 - 328 * q^27 + 72 * q^29 + 311 * q^30 - 872 * q^31 + 422 * q^32 - 746 * q^33 - 639 * q^34 + 552 * q^36 + 419 * q^37 - 106 * q^38 + 420 * q^39 - 1463 * q^40 - 779 * q^41 + 1231 * q^43 - 183 * q^44 - 1638 * q^45 - 630 * q^46 + 163 * q^47 - 1694 * q^48 + 791 * q^50 - 1020 * q^51 - 2737 * q^52 - 308 * q^53 - 932 * q^54 - 1668 * q^55 + 3346 * q^57 + 495 * q^58 - 902 * q^59 - 1080 * q^60 - 1390 * q^61 - 946 * q^62 + 2239 * q^64 - 1500 * q^65 - 3336 * q^66 + 86 * q^67 - 849 * q^68 - 1459 * q^69 + 784 * q^71 - 2013 * q^72 - 4270 * q^73 - 2009 * q^74 + 1841 * q^75 - 155 * q^76 - 1633 * q^78 - 392 * q^79 - 1160 * q^80 + 451 * q^81 - 164 * q^82 - 112 * q^83 + 2740 * q^85 - 6521 * q^86 - 1868 * q^87 - 1540 * q^88 - 687 * q^89 + 7589 * q^90 - 3552 * q^92 - 3908 * q^93 + 1705 * q^94 - 3636 * q^95 + 7802 * q^96 - 577 * q^97 + 1040 * q^99

Introduction

L-functions

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