LMFDB - Embedded newform 2009.4.a.h.1.16

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:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.116020 q^{2} +5.21300 q^{3} -7.98654 q^{4} +11.3743 q^{5} -0.604813 q^{6} +1.85476 q^{8} +0.175418 q^{9} +O(q^{10})$	\(q-0.116020 q^{2} +5.21300 q^{3} -7.98654 q^{4} +11.3743 q^{5} -0.604813 q^{6} +1.85476 q^{8} +0.175418 q^{9} -1.31965 q^{10} -0.732584 q^{11} -41.6339 q^{12} -5.18872 q^{13} +59.2945 q^{15} +63.6771 q^{16} -61.6069 q^{17} -0.0203520 q^{18} +58.5729 q^{19} -90.8417 q^{20} +0.0849944 q^{22} +12.6434 q^{23} +9.66886 q^{24} +4.37576 q^{25} +0.601996 q^{26} -139.837 q^{27} -152.380 q^{29} -6.87935 q^{30} +246.600 q^{31} -22.2259 q^{32} -3.81897 q^{33} +7.14763 q^{34} -1.40098 q^{36} -242.121 q^{37} -6.79563 q^{38} -27.0488 q^{39} +21.0967 q^{40} -41.0000 q^{41} -53.0553 q^{43} +5.85081 q^{44} +1.99526 q^{45} -1.46689 q^{46} +53.1723 q^{47} +331.949 q^{48} -0.507676 q^{50} -321.157 q^{51} +41.4400 q^{52} -415.720 q^{53} +16.2238 q^{54} -8.33267 q^{55} +305.341 q^{57} +17.6791 q^{58} +566.889 q^{59} -473.558 q^{60} +412.515 q^{61} -28.6105 q^{62} -506.838 q^{64} -59.0184 q^{65} +0.443076 q^{66} -1079.17 q^{67} +492.026 q^{68} +65.9101 q^{69} +393.115 q^{71} +0.325357 q^{72} -16.2065 q^{73} +28.0909 q^{74} +22.8109 q^{75} -467.795 q^{76} +3.13821 q^{78} -324.479 q^{79} +724.286 q^{80} -733.706 q^{81} +4.75682 q^{82} -413.445 q^{83} -700.738 q^{85} +6.15547 q^{86} -794.355 q^{87} -1.35877 q^{88} -471.998 q^{89} -0.231490 q^{90} -100.977 q^{92} +1285.53 q^{93} -6.16905 q^{94} +666.229 q^{95} -115.864 q^{96} -857.433 q^{97} -0.128508 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} - 12 q^{3} + 103 q^{4} - 20 q^{5} - 36 q^{6} - 9 q^{8} + 318 q^{9}+O(q^{10}) $ 30 * q + q^2 - 12 * q^3 + 103 * q^4 - 20 * q^5 - 36 * q^6 - 9 * q^8 + 318 * q^9	\( 30 q + q^{2} - 12 q^{3} + 103 q^{4} - 20 q^{5} - 36 q^{6} - 9 q^{8} + 318 q^{9} - 80 q^{10} + 38 q^{11} + 83 q^{12} - 78 q^{13} + 24 q^{15} + 287 q^{16} - 260 q^{17} - 185 q^{18} - 336 q^{19} - 240 q^{20} - 160 q^{22} - 90 q^{23} - 1112 q^{24} + 606 q^{25} + 55 q^{26} - 432 q^{27} + 130 q^{29} - 674 q^{30} - 1320 q^{31} - 331 q^{32} - 152 q^{33} - 816 q^{34} + 983 q^{36} - 4 q^{37} - 396 q^{38} - 248 q^{39} - 934 q^{40} - 1230 q^{41} - 214 q^{43} + 926 q^{44} - 804 q^{45} - 248 q^{46} - 2262 q^{47} + 568 q^{48} - 543 q^{50} + 204 q^{51} - 650 q^{52} - 522 q^{53} - 3253 q^{54} - 1328 q^{55} - 160 q^{57} + 888 q^{58} - 656 q^{59} + 994 q^{60} - 4300 q^{61} + 728 q^{62} + 1637 q^{64} + 1848 q^{65} + 744 q^{66} + 1642 q^{67} - 4860 q^{68} + 1556 q^{69} - 980 q^{71} - 2248 q^{72} - 1112 q^{73} + 1609 q^{74} - 6916 q^{75} - 3096 q^{76} + 343 q^{78} + 2068 q^{79} + 2440 q^{80} + 3130 q^{81} - 41 q^{82} - 356 q^{83} + 788 q^{85} - 514 q^{86} - 820 q^{87} - 1130 q^{88} - 5560 q^{89} - 2160 q^{90} + 1573 q^{92} + 124 q^{93} + 2377 q^{94} + 580 q^{95} - 9857 q^{96} - 3828 q^{97} - 2870 q^{99}+O(q^{100}) \) 30 * q + q^2 - 12 * q^3 + 103 * q^4 - 20 * q^5 - 36 * q^6 - 9 * q^8 + 318 * q^9 - 80 * q^10 + 38 * q^11 + 83 * q^12 - 78 * q^13 + 24 * q^15 + 287 * q^16 - 260 * q^17 - 185 * q^18 - 336 * q^19 - 240 * q^20 - 160 * q^22 - 90 * q^23 - 1112 * q^24 + 606 * q^25 + 55 * q^26 - 432 * q^27 + 130 * q^29 - 674 * q^30 - 1320 * q^31 - 331 * q^32 - 152 * q^33 - 816 * q^34 + 983 * q^36 - 4 * q^37 - 396 * q^38 - 248 * q^39 - 934 * q^40 - 1230 * q^41 - 214 * q^43 + 926 * q^44 - 804 * q^45 - 248 * q^46 - 2262 * q^47 + 568 * q^48 - 543 * q^50 + 204 * q^51 - 650 * q^52 - 522 * q^53 - 3253 * q^54 - 1328 * q^55 - 160 * q^57 + 888 * q^58 - 656 * q^59 + 994 * q^60 - 4300 * q^61 + 728 * q^62 + 1637 * q^64 + 1848 * q^65 + 744 * q^66 + 1642 * q^67 - 4860 * q^68 + 1556 * q^69 - 980 * q^71 - 2248 * q^72 - 1112 * q^73 + 1609 * q^74 - 6916 * q^75 - 3096 * q^76 + 343 * q^78 + 2068 * q^79 + 2440 * q^80 + 3130 * q^81 - 41 * q^82 - 356 * q^83 + 788 * q^85 - 514 * q^86 - 820 * q^87 - 1130 * q^88 - 5560 * q^89 - 2160 * q^90 + 1573 * q^92 + 124 * q^93 + 2377 * q^94 + 580 * q^95 - 9857 * q^96 - 3828 * q^97 - 2870 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.116020	−0.0410192	−0.0205096	−	0.999790i	$-0.506529\pi$
$2$	−0.116020	−0.0410192	−0.0205096	+	0.999790i	$0.506529\pi$
$3$	5.21300	1.00324	0.501622	−	0.865087i	$-0.332737\pi$
$3$	5.21300	1.00324	0.501622	+	0.865087i	$0.332737\pi$
$4$	−7.98654	−0.998317
$5$	11.3743	1.01735	0.508676	−	0.860958i	$-0.330135\pi$
$5$	11.3743	1.01735	0.508676	+	0.860958i	$0.330135\pi$
$6$	−0.604813	−0.0411523
$7$	0	0
$7$	0	0
$8$	1.85476	0.0819695
$9$	0.175418	0.00649695
$10$	−1.31965	−0.0417310
$11$	−0.732584	−0.0200802	−0.0100401	−	0.999950i	$-0.503196\pi$
$11$	−0.732584	−0.0200802	−0.0100401	+	0.999950i	$0.503196\pi$
$12$	−41.6339	−1.00156
$13$	−5.18872	−0.110699	−0.0553497	−	0.998467i	$-0.517627\pi$
$13$	−5.18872	−0.110699	−0.0553497	+	0.998467i	$0.517627\pi$
$14$	0	0
$15$	59.2945	1.02065
$16$	63.6771	0.994955
$17$	−61.6069	−0.878933	−0.439467	−	0.898259i	$-0.644832\pi$
$17$	−61.6069	−0.878933	−0.439467	+	0.898259i	$0.644832\pi$
$18$	−0.0203520	−0.000266500 0
$19$	58.5729	0.707240	0.353620	−	0.935389i	$-0.384951\pi$
$19$	58.5729	0.707240	0.353620	+	0.935389i	$0.384951\pi$
$20$	−90.8417	−1.01564
$21$	0	0
$22$	0.0849944	0.000823676 0
$23$	12.6434	0.114623	0.0573116	−	0.998356i	$-0.481747\pi$
$23$	12.6434	0.114623	0.0573116	+	0.998356i	$0.481747\pi$
$24$	9.66886	0.0822353
$25$	4.37576	0.0350061
$26$	0.601996	0.00454081
$27$	−139.837	−0.996725
$28$	0	0
$29$	−152.380	−0.975730	−0.487865	−	0.872919i	$-0.662224\pi$
$29$	−152.380	−0.975730	−0.487865	+	0.872919i	$0.662224\pi$
$30$	−6.87935	−0.0418664
$31$	246.600	1.42873	0.714366	−	0.699772i	$-0.246715\pi$
$31$	246.600	1.42873	0.714366	+	0.699772i	$0.246715\pi$
$32$	−22.2259	−0.122782
$33$	−3.81897	−0.0201454
$34$	7.14763	0.0360532
$35$	0	0
$36$	−1.40098	−0.00648602
$37$	−242.121	−1.07580	−0.537899	−	0.843010i	$-0.680782\pi$
$37$	−242.121	−1.07580	−0.537899	+	0.843010i	$0.680782\pi$
$38$	−6.79563	−0.0290104
$39$	−27.0488	−0.111059
$40$	21.0967	0.0833919
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	−53.0553	−0.188159	−0.0940797	−	0.995565i	$-0.529991\pi$
$43$	−53.0553	−0.188159	−0.0940797	+	0.995565i	$0.529991\pi$
$44$	5.85081	0.0200464
$45$	1.99526	0.00660969
$46$	−1.46689	−0.00470176
$47$	53.1723	0.165021	0.0825104	−	0.996590i	$-0.473706\pi$
$47$	53.1723	0.165021	0.0825104	+	0.996590i	$0.473706\pi$
$48$	331.949	0.998182
$49$	0	0
$50$	−0.507676	−0.00143592
$51$	−321.157	−0.881784
$52$	41.4400	0.110513
$53$	−415.720	−1.07743	−0.538713	−	0.842490i	$-0.681089\pi$
$53$	−415.720	−1.07743	−0.538713	+	0.842490i	$0.681089\pi$
$54$	16.2238	0.0408849
$55$	−8.33267	−0.0204287
$56$	0	0
$57$	305.341	0.709533
$58$	17.6791	0.0400237
$59$	566.889	1.25089	0.625446	−	0.780267i	$-0.284917\pi$
$59$	566.889	1.25089	0.625446	+	0.780267i	$0.284917\pi$
$60$	−473.558	−1.01893
$61$	412.515	0.865854	0.432927	−	0.901429i	$-0.357481\pi$
$61$	412.515	0.865854	0.432927	+	0.901429i	$0.357481\pi$
$62$	−28.6105	−0.0586055
$63$	0	0
$64$	−506.838	−0.989919
$65$	−59.0184	−0.112620
$66$	0.443076	0.000826347 0
$67$	−1079.17	−1.96779	−0.983893	−	0.178758i	$-0.942792\pi$
$67$	−1079.17	−1.96779	−0.983893	+	0.178758i	$0.942792\pi$
$68$	492.026	0.877454
$69$	65.9101	0.114995
$70$	0	0
$71$	393.115	0.657101	0.328550	−	0.944486i	$-0.393440\pi$
$71$	393.115	0.657101	0.328550	+	0.944486i	$0.393440\pi$
$72$	0.325357	0.000532552 0
$73$	−16.2065	−0.0259840	−0.0129920	−	0.999916i	$-0.504136\pi$
$73$	−16.2065	−0.0259840	−0.0129920	+	0.999916i	$0.504136\pi$
$74$	28.0909	0.0441284
$75$	22.8109	0.0351197
$76$	−467.795	−0.706050
$77$	0	0
$78$	3.13821	0.00455554
$79$	−324.479	−0.462110	−0.231055	−	0.972941i	$-0.574218\pi$
$79$	−324.479	−0.462110	−0.231055	+	0.972941i	$0.574218\pi$
$80$	724.286	1.01222
$81$	−733.706	−1.00645
$82$	4.75682	0.00640613
$83$	−413.445	−0.546765	−0.273383	−	0.961905i	$-0.588143\pi$
$83$	−413.445	−0.546765	−0.273383	+	0.961905i	$0.588143\pi$
$84$	0	0
$85$	−700.738	−0.894185
$86$	6.15547	0.00771815
$87$	−794.355	−0.978895
$88$	−1.35877	−0.00164597
$89$	−471.998	−0.562154	−0.281077	−	0.959685i	$-0.590692\pi$
$89$	−471.998	−0.562154	−0.281077	+	0.959685i	$0.590692\pi$
$90$	−0.231490	−0.000271124 0
$91$	0	0
$92$	−100.977	−0.114430
$93$	1285.53	1.43337
$94$	−6.16905	−0.00676903
$95$	666.229	0.719512
$96$	−115.864	−0.123180
$97$	−857.433	−0.897516	−0.448758	−	0.893653i	$-0.648134\pi$
$97$	−857.433	−0.897516	−0.448758	+	0.893653i	$0.648134\pi$
$98$	0	0
$99$	−0.128508	−0.000130460 0
$100$	−34.9472	−0.0349472
$101$	576.045	0.567511	0.283755	−	0.958897i	$-0.408420\pi$
$101$	576.045	0.567511	0.283755	+	0.958897i	$0.408420\pi$
$102$	37.2606	0.0361701
$103$	82.5251	0.0789460	0.0394730	−	0.999221i	$-0.487432\pi$
$103$	82.5251	0.0789460	0.0394730	+	0.999221i	$0.487432\pi$
$104$	−9.62383	−0.00907398
$105$	0	0
$106$	48.2318	0.0441952
$107$	−1083.99	−0.979375	−0.489687	−	0.871898i	$-0.662889\pi$
$107$	−1083.99	−0.979375	−0.489687	+	0.871898i	$0.662889\pi$
$108$	1116.81	0.995048
$109$	−2046.16	−1.79804	−0.899021	−	0.437905i	$-0.855721\pi$
$109$	−2046.16	−1.79804	−0.899021	+	0.437905i	$0.855721\pi$
$110$	0.966756	0.000837969 0
$111$	−1262.18	−1.07929
$112$	0	0
$113$	122.893	0.102308	0.0511539	−	0.998691i	$-0.483710\pi$
$113$	122.893	0.102308	0.0511539	+	0.998691i	$0.483710\pi$
$114$	−35.4256	−0.0291045
$115$	143.810	0.116612
$116$	1216.99	0.974088
$117$	−0.910194	−0.000719209 0
$118$	−65.7704	−0.0513107
$119$	0	0
$120$	109.977	0.0836623
$121$	−1330.46	−0.999597
$122$	−47.8599	−0.0355167
$123$	−213.733	−0.156680
$124$	−1969.48	−1.42633
$125$	−1372.02	−0.981739
$126$	0	0
$127$	906.630	0.633468	0.316734	−	0.948514i	$-0.397414\pi$
$127$	906.630	0.633468	0.316734	+	0.948514i	$0.397414\pi$
$128$	236.610	0.163388
$129$	−276.577	−0.188770
$130$	6.84731	0.00461960
$131$	1955.61	1.30429	0.652145	−	0.758094i	$-0.273869\pi$
$131$	1955.61	1.30429	0.652145	+	0.758094i	$0.273869\pi$
$132$	30.5003	0.0201115
$133$	0	0
$134$	125.205	0.0807171
$135$	−1590.55	−1.01402
$136$	−114.266	−0.0720457
$137$	808.536	0.504218	0.252109	−	0.967699i	$-0.418876\pi$
$137$	808.536	0.504218	0.252109	+	0.967699i	$0.418876\pi$
$138$	−7.64689	−0.00471700
$139$	−53.3583	−0.0325596	−0.0162798	−	0.999867i	$-0.505182\pi$
$139$	−53.3583	−0.0325596	−0.0162798	+	0.999867i	$0.505182\pi$
$140$	0	0
$141$	277.187	0.165556
$142$	−45.6092	−0.0269538
$143$	3.80118	0.00222287
$144$	11.1701	0.00646417
$145$	−1733.22	−0.992661
$146$	1.88028	0.00106584
$147$	0	0
$148$	1933.71	1.07399
$149$	−2871.57	−1.57885	−0.789424	−	0.613848i	$-0.789621\pi$
$149$	−2871.57	−1.57885	−0.789424	+	0.613848i	$0.789621\pi$
$150$	−2.64652	−0.00144058
$151$	1788.84	0.964063	0.482031	−	0.876154i	$-0.339899\pi$
$151$	1788.84	0.964063	0.482031	+	0.876154i	$0.339899\pi$
$152$	108.639	0.0579721
$153$	−10.8069	−0.00571039
$154$	0	0
$155$	2804.92	1.45352
$156$	216.027	0.110872
$157$	1697.23	0.862765	0.431382	−	0.902169i	$-0.358026\pi$
$157$	1697.23	0.862765	0.431382	+	0.902169i	$0.358026\pi$
$158$	37.6460	0.0189554
$159$	−2167.15	−1.08092
$160$	−252.805	−0.124912
$161$	0	0
$162$	85.1245	0.0412840
$163$	−1875.03	−0.901003	−0.450502	−	0.892776i	$-0.648755\pi$
$163$	−1875.03	−0.901003	−0.450502	+	0.892776i	$0.648755\pi$
$164$	327.448	0.155911
$165$	−43.4382	−0.0204949
$166$	47.9679	0.0224279
$167$	88.1579	0.0408495	0.0204247	−	0.999791i	$-0.493498\pi$
$167$	88.1579	0.0408495	0.0204247	+	0.999791i	$0.493498\pi$
$168$	0	0
$169$	−2170.08	−0.987746
$170$	81.2996	0.0366788
$171$	10.2747	0.00459490
$172$	423.728	0.187843
$173$	−1666.80	−0.732512	−0.366256	−	0.930514i	$-0.619361\pi$
$173$	−1666.80	−0.732512	−0.366256	+	0.930514i	$0.619361\pi$
$174$	92.1611	0.0401535
$175$	0	0
$176$	−46.6489	−0.0199789
$177$	2955.19	1.25495
$178$	54.7612	0.0230591
$179$	1120.48	0.467871	0.233935	−	0.972252i	$-0.424840\pi$
$179$	1120.48	0.467871	0.233935	+	0.972252i	$0.424840\pi$
$180$	−15.9352	−0.00659857
$181$	−385.515	−0.158315	−0.0791577	−	0.996862i	$-0.525223\pi$
$181$	−385.515	−0.158315	−0.0791577	+	0.996862i	$0.525223\pi$
$182$	0	0
$183$	2150.44	0.868662
$184$	23.4505	0.00939560
$185$	−2753.97	−1.09447
$186$	−149.147	−0.0587956
$187$	45.1322	0.0176492
$188$	−424.663	−0.164743
$189$	0	0
$190$	−77.2958	−0.0295138
$191$	−4.45475	−0.00168762	−0.000843808	−	1.00000i	$-0.500269\pi$
$191$	−4.45475	−0.00168762	−0.000843808		1.00000i	$0.500269\pi$
$192$	−2642.15	−0.993129
$193$	1388.65	0.517914	0.258957	−	0.965889i	$-0.416621\pi$
$193$	1388.65	0.517914	0.258957	+	0.965889i	$0.416621\pi$
$194$	99.4793	0.0368154
$195$	−307.663	−0.112986
$196$	0	0
$197$	−1879.72	−0.679821	−0.339911	−	0.940458i	$-0.610397\pi$
$197$	−1879.72	−0.679821	−0.339911	+	0.940458i	$0.610397\pi$
$198$	0.0149095	5.35138e−6 0
$199$	4553.45	1.62204	0.811019	−	0.585019i	$-0.198913\pi$
$199$	4553.45	1.62204	0.811019	+	0.585019i	$0.198913\pi$
$200$	8.11598	0.00286943
$201$	−5625.72	−1.97417
$202$	−66.8327	−0.0232789
$203$	0	0
$204$	2564.93	0.880300
$205$	−466.348	−0.158884
$206$	−9.57456	−0.00323831
$207$	2.21788	0.000744701 0
$208$	−330.403	−0.110141
$209$	−42.9096	−0.0142015
$210$	0	0
$211$	−1078.95	−0.352030	−0.176015	−	0.984388i	$-0.556321\pi$
$211$	−1078.95	−0.352030	−0.176015	+	0.984388i	$0.556321\pi$
$212$	3320.16	1.07561
$213$	2049.31	0.659232
$214$	125.764	0.0401732
$215$	−603.469	−0.191424
$216$	−259.363	−0.0817010
$217$	0	0
$218$	237.395	0.0737544
$219$	−84.4846	−0.0260682
$220$	66.5492	0.0203943
$221$	319.661	0.0972974
$222$	146.438	0.0442715
$223$	−2601.86	−0.781315	−0.390658	−	0.920536i	$-0.627752\pi$
$223$	−2601.86	−0.781315	−0.390658	+	0.920536i	$0.627752\pi$
$224$	0	0
$225$	0.767587	0.000227433 0
$226$	−14.2580	−0.00419659
$227$	113.011	0.0330432	0.0165216	−	0.999864i	$-0.494741\pi$
$227$	113.011	0.0330432	0.0165216	+	0.999864i	$0.494741\pi$
$228$	−2438.62	−0.708340
$229$	−3357.58	−0.968887	−0.484443	−	0.874823i	$-0.660978\pi$
$229$	−3357.58	−0.968887	−0.484443	+	0.874823i	$0.660978\pi$
$230$	−16.6849	−0.00478334
$231$	0	0
$232$	−282.627	−0.0799801
$233$	−1824.76	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$233$	−1824.76	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$234$	0.105601	2.95014e−5 0
$235$	604.800	0.167884
$236$	−4527.48	−1.24879
$237$	−1691.51	−0.463609
$238$	0	0
$239$	−5404.41	−1.46269	−0.731344	−	0.682009i	$-0.761107\pi$
$239$	−5404.41	−1.46269	−0.731344	+	0.682009i	$0.761107\pi$
$240$	3775.70	1.01550
$241$	−600.225	−0.160431	−0.0802155	−	0.996778i	$-0.525561\pi$
$241$	−600.225	−0.160431	−0.0802155	+	0.996778i	$0.525561\pi$
$242$	154.360	0.0410027
$243$	−49.2201	−0.0129937
$244$	−3294.56	−0.864397
$245$	0	0
$246$	24.7973	0.00642691
$247$	−303.919	−0.0782911
$248$	457.384	0.117112
$249$	−2155.29	−0.548539
$250$	159.182	0.0402702
$251$	471.035	0.118452	0.0592260	−	0.998245i	$-0.481137\pi$
$251$	471.035	0.118452	0.0592260	+	0.998245i	$0.481137\pi$
$252$	0	0
$253$	−9.26236	−0.00230166
$254$	−105.187	−0.0259844
$255$	−3652.95	−0.897085
$256$	4027.26	0.983217
$257$	−2089.81	−0.507232	−0.253616	−	0.967305i	$-0.581620\pi$
$257$	−2089.81	−0.507232	−0.253616	+	0.967305i	$0.581620\pi$
$258$	32.0885	0.00774319
$259$	0	0
$260$	471.352	0.112431
$261$	−26.7301	−0.00633927
$262$	−226.889	−0.0535010
$263$	3757.80	0.881048	0.440524	−	0.897741i	$-0.354793\pi$
$263$	3757.80	0.881048	0.440524	+	0.897741i	$0.354793\pi$
$264$	−7.08326	−0.00165130
$265$	−4728.54	−1.09612
$266$	0	0
$267$	−2460.53	−0.563977
$268$	8618.84	1.96448
$269$	−4747.84	−1.07614	−0.538068	−	0.842901i	$-0.680846\pi$
$269$	−4747.84	−1.07614	−0.538068	+	0.842901i	$0.680846\pi$
$270$	184.536	0.0415944
$271$	−5516.26	−1.23649	−0.618246	−	0.785985i	$-0.712156\pi$
$271$	−5516.26	−1.23649	−0.618246	+	0.785985i	$0.712156\pi$
$272$	−3922.95	−0.874499
$273$	0	0
$274$	−93.8063	−0.0206827
$275$	−3.20562	−0.000702931 0
$276$	−526.394	−0.114801
$277$	2060.35	0.446911	0.223455	−	0.974714i	$-0.428266\pi$
$277$	2060.35	0.446911	0.223455	+	0.974714i	$0.428266\pi$
$278$	6.19062	0.00133557
$279$	43.2580	0.00928240
$280$	0	0
$281$	−6891.22	−1.46297	−0.731487	−	0.681855i	$-0.761173\pi$
$281$	−6891.22	−1.46297	−0.731487	+	0.681855i	$0.761173\pi$
$282$	−32.1593	−0.00679098
$283$	3336.53	0.700835	0.350417	−	0.936594i	$-0.386040\pi$
$283$	3336.53	0.700835	0.350417	+	0.936594i	$0.386040\pi$
$284$	−3139.63	−0.655995
$285$	3473.05	0.721846
$286$	−0.441013	−9.11805e−5 0
$287$	0	0
$288$	−3.89881	−0.000797707 0
$289$	−1117.59	−0.227477
$290$	201.088	0.0407182
$291$	−4469.80	−0.900427
$292$	129.434	0.0259402
$293$	−1570.43	−0.313124	−0.156562	−	0.987668i	$-0.550041\pi$
$293$	−1570.43	−0.313124	−0.156562	+	0.987668i	$0.550041\pi$
$294$	0	0
$295$	6447.99	1.27260
$296$	−449.076	−0.0881825
$297$	102.442	0.0200145
$298$	333.160	0.0647632
$299$	−65.6032	−0.0126887
$300$	−182.180	−0.0350606
$301$	0	0
$302$	−207.541	−0.0395451
$303$	3002.92	0.569351
$304$	3729.76	0.703672
$305$	4692.08	0.880878
$306$	1.25382	0.000234236 0
$307$	−3445.42	−0.640522	−0.320261	−	0.947329i	$-0.603771\pi$
$307$	−3445.42	−0.640522	−0.320261	+	0.947329i	$0.603771\pi$
$308$	0	0
$309$	430.204	0.0792021
$310$	−325.426	−0.0596225
$311$	4522.68	0.824623	0.412311	−	0.911043i	$-0.364722\pi$
$311$	4522.68	0.824623	0.412311	+	0.911043i	$0.364722\pi$
$312$	−50.1690	−0.00910341
$313$	−7462.76	−1.34767	−0.673834	−	0.738883i	$-0.735354\pi$
$313$	−7462.76	−1.34767	−0.673834	+	0.738883i	$0.735354\pi$
$314$	−196.913	−0.0353900
$315$	0	0
$316$	2591.46	0.461333
$317$	614.982	0.108962	0.0544808	−	0.998515i	$-0.482650\pi$
$317$	614.982	0.108962	0.0544808	+	0.998515i	$0.482650\pi$
$318$	251.433	0.0443385
$319$	111.631	0.0195929
$320$	−5764.96	−1.00710
$321$	−5650.84	−0.982551
$322$	0	0
$323$	−3608.50	−0.621616
$324$	5859.77	1.00476
$325$	−22.7046	−0.00387516
$326$	217.541	0.0369585
$327$	−10666.6	−1.80387
$328$	−76.0451	−0.0128015
$329$	0	0
$330$	5.03970	0.000840686 0
$331$	−604.974	−0.100460	−0.0502302	−	0.998738i	$-0.515995\pi$
$331$	−604.974	−0.100460	−0.0502302	+	0.998738i	$0.515995\pi$
$332$	3302.00	0.545845
$333$	−42.4724	−0.00698940
$334$	−10.2281	−0.00167561
$335$	−12274.9	−2.00193
$336$	0	0
$337$	−3961.43	−0.640335	−0.320168	−	0.947361i	$-0.603739\pi$
$337$	−3961.43	−0.640335	−0.320168	+	0.947361i	$0.603739\pi$
$338$	251.772	0.0405166
$339$	640.641	0.102640
$340$	5596.47	0.892680
$341$	−180.655	−0.0286893
$342$	−1.19207	−0.000188479 0
$343$	0	0
$344$	−98.4046	−0.0154233
$345$	749.685	0.116990
$346$	193.382	0.0300471
$347$	5410.78	0.837078	0.418539	−	0.908199i	$-0.362542\pi$
$347$	5410.78	0.837078	0.418539	+	0.908199i	$0.362542\pi$
$348$	6344.15	0.977247
$349$	11258.8	1.72685	0.863423	−	0.504481i	$-0.168316\pi$
$349$	11258.8	1.72685	0.863423	+	0.504481i	$0.168316\pi$
$350$	0	0
$351$	725.574	0.110337
$352$	16.2823	0.00246549
$353$	3415.60	0.514997	0.257499	−	0.966279i	$-0.417102\pi$
$353$	3415.60	0.514997	0.257499	+	0.966279i	$0.417102\pi$
$354$	−342.861	−0.0514771
$355$	4471.42	0.668503
$356$	3769.63	0.561208
$357$	0	0
$358$	−129.998	−0.0191917
$359$	−4344.91	−0.638762	−0.319381	−	0.947626i	$-0.603475\pi$
$359$	−4344.91	−0.638762	−0.319381	+	0.947626i	$0.603475\pi$
$360$	3.70073	0.000541793 0
$361$	−3428.21	−0.499812
$362$	44.7274	0.00649398
$363$	−6935.71	−1.00284
$364$	0	0
$365$	−184.338	−0.0264348
$366$	−249.494	−0.0356319
$367$	6013.19	0.855275	0.427637	−	0.903950i	$-0.359346\pi$
$367$	6013.19	0.855275	0.427637	+	0.903950i	$0.359346\pi$
$368$	805.096	0.114045
$369$	−7.19213	−0.00101465
$370$	319.516	0.0448941
$371$	0	0
$372$	−10266.9	−1.43095
$373$	8197.98	1.13800	0.569002	−	0.822336i	$-0.307330\pi$
$373$	8197.98	1.13800	0.569002	+	0.822336i	$0.307330\pi$
$374$	−5.23624	−0.000723956 0
$375$	−7152.36	−0.984923
$376$	98.6217	0.0135267
$377$	790.655	0.108013
$378$	0	0
$379$	−2592.38	−0.351350	−0.175675	−	0.984448i	$-0.556211\pi$
$379$	−2592.38	−0.351350	−0.175675	+	0.984448i	$0.556211\pi$
$380$	−5320.86	−0.718301
$381$	4726.27	0.635522
$382$	0.516840	6.92247e−5 0
$383$	9593.90	1.27996	0.639981	−	0.768391i	$-0.278942\pi$
$383$	9593.90	1.27996	0.639981	+	0.768391i	$0.278942\pi$
$384$	1233.45	0.163917
$385$	0	0
$386$	−161.112	−0.0212445
$387$	−9.30683	−0.00122246
$388$	6847.92	0.896006
$389$	−10732.2	−1.39883	−0.699413	−	0.714718i	$-0.746555\pi$
$389$	−10732.2	−1.39883	−0.699413	+	0.714718i	$0.746555\pi$
$390$	35.6950	0.00463459
$391$	−778.921	−0.100746
$392$	0	0
$393$	10194.6	1.30852
$394$	218.085	0.0278857
$395$	−3690.73	−0.470129
$396$	1.02634	0.000130241 0
$397$	−1934.33	−0.244538	−0.122269	−	0.992497i	$-0.539017\pi$
$397$	−1934.33	−0.244538	−0.122269	+	0.992497i	$0.539017\pi$
$398$	−528.291	−0.0665348
$399$	0	0
$400$	278.636	0.0348295
$401$	1774.62	0.220998	0.110499	−	0.993876i	$-0.464755\pi$
$401$	1774.62	0.220998	0.110499	+	0.993876i	$0.464755\pi$
$402$	652.696	0.0809789
$403$	−1279.54	−0.158160
$404$	−4600.60	−0.566556
$405$	−8345.42	−1.02392
$406$	0	0
$407$	177.374	0.0216023
$408$	−595.668	−0.0722793
$409$	−13348.2	−1.61375	−0.806877	−	0.590719i	$-0.798844\pi$
$409$	−13348.2	−1.61375	−0.806877	+	0.590719i	$0.798844\pi$
$410$	54.1057	0.00651729
$411$	4214.90	0.505854
$412$	−659.090	−0.0788132
$413$	0	0
$414$	−0.257318	−3.05471e−5 0
$415$	−4702.67	−0.556253
$416$	115.324	0.0135919
$417$	−278.157	−0.0326652
$418$	4.97837	0.000582536 0
$419$	155.947	0.0181826	0.00909131	−	0.999959i	$-0.497106\pi$
$419$	155.947	0.0181826	0.00909131	+	0.999959i	$0.497106\pi$
$420$	0	0
$421$	4081.21	0.472461	0.236231	−	0.971697i	$-0.424088\pi$
$421$	4081.21	0.472461	0.236231	+	0.971697i	$0.424088\pi$
$422$	125.180	0.0144400
$423$	9.32736	0.00107213
$424$	−771.060	−0.0883160
$425$	−269.577	−0.0307680
$426$	−237.761	−0.0270412
$427$	0	0
$428$	8657.31	0.977727
$429$	19.8156	0.00223008
$430$	70.0144	0.00785208
$431$	2237.78	0.250093	0.125046	−	0.992151i	$-0.460092\pi$
$431$	2237.78	0.250093	0.125046	+	0.992151i	$0.460092\pi$
$432$	−8904.40	−0.991697
$433$	1124.30	0.124781	0.0623906	−	0.998052i	$-0.480128\pi$
$433$	1124.30	0.124781	0.0623906	+	0.998052i	$0.480128\pi$
$434$	0	0
$435$	−9035.27	−0.995881
$436$	16341.7	1.79502
$437$	740.561	0.0810660
$438$	9.80190	0.00106930
$439$	8662.57	0.941782	0.470891	−	0.882192i	$-0.343933\pi$
$439$	8662.57	0.941782	0.470891	+	0.882192i	$0.343933\pi$
$440$	−15.4551	−0.00167453
$441$	0	0
$442$	−37.0871	−0.00399107
$443$	6310.41	0.676787	0.338394	−	0.941005i	$-0.390116\pi$
$443$	6310.41	0.676787	0.338394	+	0.941005i	$0.390116\pi$
$444$	10080.4	1.07747
$445$	−5368.67	−0.571909
$446$	301.867	0.0320490
$447$	−14969.5	−1.58397
$448$	0	0
$449$	−12706.5	−1.33554	−0.667769	−	0.744369i	$-0.732750\pi$
$449$	−12706.5	−1.33554	−0.667769	+	0.744369i	$0.732750\pi$
$450$	−0.0890554	−9.32913e−6 0
$451$	30.0360	0.00313600
$452$	−981.488	−0.102136
$453$	9325.21	0.967189
$454$	−13.1115	−0.00135541
$455$	0	0
$456$	566.333	0.0581601
$457$	1831.06	0.187425	0.0937125	−	0.995599i	$-0.470127\pi$
$457$	1831.06	0.187425	0.0937125	+	0.995599i	$0.470127\pi$
$458$	389.546	0.0397430
$459$	8614.90	0.876055
$460$	−1148.55	−0.116416
$461$	−7898.77	−0.798009	−0.399005	−	0.916949i	$-0.630644\pi$
$461$	−7898.77	−0.798009	−0.399005	+	0.916949i	$0.630644\pi$
$462$	0	0
$463$	−4704.61	−0.472228	−0.236114	−	0.971725i	$-0.575874\pi$
$463$	−4704.61	−0.472228	−0.236114	+	0.971725i	$0.575874\pi$
$464$	−9703.09	−0.970808
$465$	14622.0	1.45824
$466$	211.708	0.0210455
$467$	−1782.65	−0.176641	−0.0883204	−	0.996092i	$-0.528150\pi$
$467$	−1782.65	−0.176641	−0.0883204	+	0.996092i	$0.528150\pi$
$468$	7.26930	0.000717999 0
$469$	0	0
$470$	−70.1689	−0.00688649
$471$	8847.69	0.865563
$472$	1051.44	0.102535
$473$	38.8675	0.00377828
$474$	196.249	0.0190169
$475$	256.301	0.0247577
$476$	0	0
$477$	−72.9246	−0.00699998
$478$	627.020	0.0599984
$479$	−3942.79	−0.376097	−0.188049	−	0.982160i	$-0.560216\pi$
$479$	−3942.79	−0.376097	−0.188049	+	0.982160i	$0.560216\pi$
$480$	−1317.87	−0.125317
$481$	1256.30	0.119090
$482$	69.6381	0.00658076
$483$	0	0
$484$	10625.8	0.997915
$485$	−9752.74	−0.913091
$486$	5.71051	0.000532992 0
$487$	12494.6	1.16259	0.581297	−	0.813691i	$-0.302545\pi$
$487$	12494.6	1.16259	0.581297	+	0.813691i	$0.302545\pi$
$488$	765.114	0.0709736
$489$	−9774.53	−0.903925
$490$	0	0
$491$	−8976.29	−0.825040	−0.412520	−	0.910949i	$-0.635351\pi$
$491$	−8976.29	−0.825040	−0.412520	+	0.910949i	$0.635351\pi$
$492$	1706.99	0.156417
$493$	9387.63	0.857601
$494$	35.2606	0.00321144
$495$	−1.46170	−0.000132724 0
$496$	15702.8	1.42152
$497$	0	0
$498$	250.057	0.0225006
$499$	−13684.8	−1.22769	−0.613844	−	0.789427i	$-0.710378\pi$
$499$	−13684.8	−1.22769	−0.613844	+	0.789427i	$0.710378\pi$
$500$	10957.7	0.980087
$501$	459.567	0.0409819
$502$	−54.6494	−0.00485881
$503$	−6227.59	−0.552037	−0.276019	−	0.961152i	$-0.589015\pi$
$503$	−6227.59	−0.552037	−0.276019	+	0.961152i	$0.589015\pi$
$504$	0	0
$505$	6552.13	0.577359
$506$	1.07462	9.44123e−5 0
$507$	−11312.6	−0.990949
$508$	−7240.84	−0.632402
$509$	−166.757	−0.0145214	−0.00726068	−	0.999974i	$-0.502311\pi$
$509$	−166.757	−0.0145214	−0.00726068	+	0.999974i	$0.502311\pi$
$510$	423.815	0.0367977
$511$	0	0
$512$	−2360.12	−0.203718
$513$	−8190.64	−0.704924
$514$	242.459	0.0208063
$515$	938.669	0.0803160
$516$	2208.90	0.188452
$517$	−38.9532	−0.00331365
$518$	0	0
$519$	−8689.05	−0.734888
$520$	−109.465	−0.00923143
$521$	−13253.9	−1.11452	−0.557259	−	0.830339i	$-0.688147\pi$
$521$	−13253.9	−1.11452	−0.557259	+	0.830339i	$0.688147\pi$
$522$	3.10122	0.000260032 0
$523$	−21874.5	−1.82888	−0.914440	−	0.404722i	$-0.867368\pi$
$523$	−21874.5	−1.82888	−0.914440	+	0.404722i	$0.867368\pi$
$524$	−15618.5	−1.30210
$525$	0	0
$526$	−435.979	−0.0361399
$527$	−15192.3	−1.25576
$528$	−243.181	−0.0200437
$529$	−12007.1	−0.986862
$530$	548.605	0.0449621
$531$	99.4423	0.00812698
$532$	0	0
$533$	212.738	0.0172884
$534$	285.471	0.0231339
$535$	−12329.7	−0.996369
$536$	−2001.60	−0.161298
$537$	5841.09	0.469388
$538$	550.844	0.0441423
$539$	0	0
$540$	12703.0	1.01231
$541$	903.958	0.0718377	0.0359189	−	0.999355i	$-0.488564\pi$
$541$	903.958	0.0718377	0.0359189	+	0.999355i	$0.488564\pi$
$542$	639.997	0.0507199
$543$	−2009.69	−0.158829
$544$	1369.27	0.107917
$545$	−23273.7	−1.82924
$546$	0	0
$547$	22009.2	1.72037	0.860187	−	0.509978i	$-0.170347\pi$
$547$	22009.2	1.72037	0.860187	+	0.509978i	$0.170347\pi$
$548$	−6457.41	−0.503370
$549$	72.3623	0.00562541
$550$	0.371916	2.88337e−5 0
$551$	−8925.32	−0.690075
$552$	122.247	0.00942607
$553$	0	0
$554$	−239.041	−0.0183319
$555$	−14356.5	−1.09801
$556$	426.148	0.0325049
$557$	16240.3	1.23541	0.617705	−	0.786410i	$-0.288063\pi$
$557$	16240.3	1.23541	0.617705	+	0.786410i	$0.288063\pi$
$558$	−5.01879	−0.000380757 0
$559$	275.289	0.0208291
$560$	0	0
$561$	235.275	0.0177064
$562$	799.519	0.0600101
$563$	9858.58	0.737992	0.368996	−	0.929431i	$-0.379702\pi$
$563$	9858.58	0.737992	0.368996	+	0.929431i	$0.379702\pi$
$564$	−2213.77	−0.165277
$565$	1397.83	0.104083
$566$	−387.104	−0.0287477
$567$	0	0
$568$	729.133	0.0538622
$569$	−23659.2	−1.74314	−0.871569	−	0.490273i	$-0.836897\pi$
$569$	−23659.2	−1.74314	−0.871569	+	0.490273i	$0.836897\pi$
$570$	−402.944	−0.0296096
$571$	−8745.80	−0.640981	−0.320491	−	0.947252i	$-0.603848\pi$
$571$	−8745.80	−0.640981	−0.320491	+	0.947252i	$0.603848\pi$
$572$	−30.3583	−0.00221913
$573$	−23.2227	−0.00169309
$574$	0	0
$575$	55.3246	0.00401251
$576$	−88.9084	−0.00643145
$577$	15668.5	1.13048	0.565240	−	0.824926i	$-0.308783\pi$
$577$	15668.5	1.13048	0.565240	+	0.824926i	$0.308783\pi$
$578$	129.663	0.00933092
$579$	7239.06	0.519594
$580$	13842.4	0.990991
$581$	0	0
$582$	518.586	0.0369348
$583$	304.550	0.0216349
$584$	−30.0591	−0.00212989
$585$	−10.3529	−0.000731689 0
$586$	182.201	0.0128441
$587$	24799.2	1.74373	0.871866	−	0.489744i	$-0.162910\pi$
$587$	24799.2	1.74373	0.871866	+	0.489744i	$0.162910\pi$
$588$	0	0
$589$	14444.1	1.01046
$590$	−748.095	−0.0522010
$591$	−9799.01	−0.682026
$592$	−15417.6	−1.07037
$593$	21015.8	1.45534	0.727669	−	0.685928i	$-0.240604\pi$
$593$	21015.8	1.45534	0.727669	+	0.685928i	$0.240604\pi$
$594$	−11.8853	−0.000820978 0
$595$	0	0
$596$	22933.9	1.57619
$597$	23737.2	1.62730
$598$	7.61127	0.000520482 0
$599$	17733.3	1.20962	0.604809	−	0.796370i	$-0.293249\pi$
$599$	17733.3	1.20962	0.604809	+	0.796370i	$0.293249\pi$
$600$	42.3087	0.00287874
$601$	16862.1	1.14446	0.572230	−	0.820093i	$-0.306078\pi$
$601$	16862.1	1.14446	0.572230	+	0.820093i	$0.306078\pi$
$602$	0	0
$603$	−189.306	−0.0127846
$604$	−14286.6	−0.962441
$605$	−15133.2	−1.01694
$606$	−348.399	−0.0233544
$607$	−3378.76	−0.225930	−0.112965	−	0.993599i	$-0.536035\pi$
$607$	−3378.76	−0.225930	−0.112965	+	0.993599i	$0.536035\pi$
$608$	−1301.83	−0.0868361
$609$	0	0
$610$	−544.375	−0.0361330
$611$	−275.896	−0.0182677
$612$	86.3100	0.00570078
$613$	10066.8	0.663284	0.331642	−	0.943405i	$-0.392397\pi$
$613$	10066.8	0.663284	0.331642	+	0.943405i	$0.392397\pi$
$614$	399.737	0.0262737
$615$	−2431.08	−0.159399
$616$	0	0
$617$	7244.29	0.472681	0.236341	−	0.971670i	$-0.424052\pi$
$617$	7244.29	0.472681	0.236341	+	0.971670i	$0.424052\pi$
$618$	−49.9122	−0.00324881
$619$	19980.4	1.29739	0.648693	−	0.761050i	$-0.275316\pi$
$619$	19980.4	1.29739	0.648693	+	0.761050i	$0.275316\pi$
$620$	−22401.6	−1.45108
$621$	−1768.01	−0.114248
$622$	−524.721	−0.0338254
$623$	0	0
$624$	−1722.39	−0.110498
$625$	−16152.8	−1.03378
$626$	865.829	0.0552803
$627$	−223.688	−0.0142476
$628$	−13555.0	−0.861313
$629$	14916.3	0.945554
$630$	0	0
$631$	1932.26	0.121905	0.0609524	−	0.998141i	$-0.480586\pi$
$631$	1932.26	0.121905	0.0609524	+	0.998141i	$0.480586\pi$
$632$	−601.829	−0.0378789
$633$	−5624.59	−0.353171
$634$	−71.3501	−0.00446952
$635$	10312.3	0.644460
$636$	17308.0	1.07910
$637$	0	0
$638$	−12.9514	−0.000803685 0
$639$	68.9593	0.00426915
$640$	2691.29	0.166223
$641$	−14848.3	−0.914935	−0.457468	−	0.889226i	$-0.651243\pi$
$641$	−14848.3	−0.914935	−0.457468	+	0.889226i	$0.651243\pi$
$642$	655.610	0.0403035
$643$	23982.2	1.47087	0.735433	−	0.677598i	$-0.236979\pi$
$643$	23982.2	1.47087	0.735433	+	0.677598i	$0.236979\pi$
$644$	0	0
$645$	−3145.89	−0.192045
$646$	418.657	0.0254982
$647$	−22562.3	−1.37097	−0.685483	−	0.728089i	$-0.740409\pi$
$647$	−22562.3	−1.37097	−0.685483	+	0.728089i	$0.740409\pi$
$648$	−1360.85	−0.0824986
$649$	−415.294	−0.0251182
$650$	2.63419	0.000158956 0
$651$	0	0
$652$	14975.0	0.899487
$653$	18956.1	1.13600	0.568001	−	0.823028i	$-0.307717\pi$
$653$	18956.1	1.13600	0.568001	+	0.823028i	$0.307717\pi$
$654$	1237.54	0.0739936
$655$	22243.7	1.32692
$656$	−2610.76	−0.155386
$657$	−2.84291	−0.000168816 0
$658$	0	0
$659$	3964.70	0.234359	0.117180	−	0.993111i	$-0.462615\pi$
$659$	3964.70	0.234359	0.117180	+	0.993111i	$0.462615\pi$
$660$	346.921	0.0204604
$661$	−11950.0	−0.703176	−0.351588	−	0.936155i	$-0.614358\pi$
$661$	−11950.0	−0.703176	−0.351588	+	0.936155i	$0.614358\pi$
$662$	70.1890	0.00412081
$663$	1666.39	0.0976130
$664$	−766.841	−0.0448181
$665$	0	0
$666$	4.92764	0.000286700 0
$667$	−1926.60	−0.111841
$668$	−704.076	−0.0407807
$669$	−13563.5	−0.783849
$670$	1424.13	0.0821177
$671$	−302.202	−0.0173865
$672$	0	0
$673$	−386.588	−0.0221425	−0.0110712	−	0.999939i	$-0.503524\pi$
$673$	−386.588	−0.0221425	−0.0110712	+	0.999939i	$0.503524\pi$
$674$	459.605	0.0262661
$675$	−611.892	−0.0348915
$676$	17331.4	0.986084
$677$	−5099.88	−0.289519	−0.144760	−	0.989467i	$-0.546241\pi$
$677$	−5099.88	−0.289519	−0.144760	+	0.989467i	$0.546241\pi$
$678$	−74.3271	−0.00421020
$679$	0	0
$680$	−1299.70	−0.0732959
$681$	589.127	0.0331504
$682$	20.9596	0.00117681
$683$	21997.9	1.23240	0.616199	−	0.787591i	$-0.288672\pi$
$683$	21997.9	1.23240	0.616199	+	0.787591i	$0.288672\pi$
$684$	−82.0595	−0.00458717
$685$	9196.57	0.512968
$686$	0	0
$687$	−17503.1	−0.972029
$688$	−3378.41	−0.187210
$689$	2157.06	0.119270
$690$	−86.9784	−0.00479886
$691$	−13181.4	−0.725677	−0.362839	−	0.931852i	$-0.618192\pi$
$691$	−13181.4	−0.725677	−0.362839	+	0.931852i	$0.618192\pi$
$692$	13312.0	0.731280
$693$	0	0
$694$	−627.759	−0.0343363
$695$	−606.916	−0.0331246
$696$	−1473.34	−0.0802395
$697$	2525.88	0.137266
$698$	−1306.24	−0.0708339
$699$	−9512.47	−0.514728
$700$	0	0
$701$	5950.35	0.320602	0.160301	−	0.987068i	$-0.448754\pi$
$701$	5950.35	0.320602	0.160301	+	0.987068i	$0.448754\pi$
$702$	−84.1811	−0.00452594
$703$	−14181.8	−0.760846
$704$	371.302	0.0198778
$705$	3152.83	0.168429
$706$	−396.278	−0.0211248
$707$	0	0
$708$	−23601.8	−1.25284
$709$	1277.17	0.0676517	0.0338258	−	0.999428i	$-0.489231\pi$
$709$	1277.17	0.0676517	0.0338258	+	0.999428i	$0.489231\pi$
$710$	−518.774	−0.0274215
$711$	−56.9193	−0.00300231
$712$	−875.443	−0.0460795
$713$	3117.87	0.163766
$714$	0	0
$715$	43.2359	0.00226144
$716$	−8948.79	−0.467084
$717$	−28173.2	−1.46743
$718$	504.096	0.0262015
$719$	−9727.06	−0.504532	−0.252266	−	0.967658i	$-0.581176\pi$
$719$	−9727.06	−0.504532	−0.252266	+	0.967658i	$0.581176\pi$
$720$	127.053	0.00657634
$721$	0	0
$722$	397.741	0.0205019
$723$	−3128.97	−0.160951
$724$	3078.93	0.158049
$725$	−666.777	−0.0341565
$726$	804.681	0.0411357
$727$	−11238.7	−0.573344	−0.286672	−	0.958029i	$-0.592549\pi$
$727$	−11238.7	−0.573344	−0.286672	+	0.958029i	$0.592549\pi$
$728$	0	0
$729$	19553.5	0.993419
$730$	21.3869	0.00108434
$731$	3268.57	0.165379
$732$	−17174.6	−0.867200
$733$	2646.49	0.133356	0.0666782	−	0.997775i	$-0.478760\pi$
$733$	2646.49	0.133356	0.0666782	+	0.997775i	$0.478760\pi$
$734$	−697.650	−0.0350827
$735$	0	0
$736$	−281.011	−0.0140736
$737$	790.583	0.0395136
$738$	0.834430	4.16203e−5 0
$739$	−14020.2	−0.697892	−0.348946	−	0.937143i	$-0.613460\pi$
$739$	−14020.2	−0.697892	−0.348946	+	0.937143i	$0.613460\pi$
$740$	21994.7	1.09262
$741$	−1584.33	−0.0785450
$742$	0	0
$743$	−32289.4	−1.59433	−0.797163	−	0.603765i	$-0.793667\pi$
$743$	−32289.4	−1.59433	−0.797163	+	0.603765i	$0.793667\pi$
$744$	2384.34	0.117492
$745$	−32662.3	−1.60625
$746$	−951.129	−0.0466800
$747$	−72.5256	−0.00355231
$748$	−360.450	−0.0176195
$749$	0	0
$750$	829.816	0.0404008
$751$	20294.3	0.986085	0.493042	−	0.870005i	$-0.335885\pi$
$751$	20294.3	0.986085	0.493042	+	0.870005i	$0.335885\pi$
$752$	3385.86	0.164188
$753$	2455.51	0.118836
$754$	−91.7318	−0.00443060
$755$	20346.8	0.980792
$756$	0	0
$757$	−10796.1	−0.518348	−0.259174	−	0.965831i	$-0.583450\pi$
$757$	−10796.1	−0.518348	−0.259174	+	0.965831i	$0.583450\pi$
$758$	300.768	0.0144121
$759$	−48.2847	−0.00230912
$760$	1235.69	0.0589780
$761$	13835.7	0.659057	0.329529	−	0.944146i	$-0.393110\pi$
$761$	13835.7	0.659057	0.329529	+	0.944146i	$0.393110\pi$
$762$	−548.341	−0.0260686
$763$	0	0
$764$	35.5781	0.00168478
$765$	−122.922	−0.00580947
$766$	−1113.08	−0.0525031
$767$	−2941.43	−0.138473
$768$	20994.1	0.986405
$769$	−5863.49	−0.274958	−0.137479	−	0.990505i	$-0.543900\pi$
$769$	−5863.49	−0.274958	−0.137479	+	0.990505i	$0.543900\pi$
$770$	0	0
$771$	−10894.2	−0.508877
$772$	−11090.5	−0.517043
$773$	−7880.07	−0.366658	−0.183329	−	0.983052i	$-0.558687\pi$
$773$	−7880.07	−0.366658	−0.183329	+	0.983052i	$0.558687\pi$
$774$	1.07978	5.01445e−5 0
$775$	1079.06	0.0500144
$776$	−1590.33	−0.0735690
$777$	0	0
$778$	1245.15	0.0573788
$779$	−2401.49	−0.110452
$780$	2457.16	0.112796
$781$	−287.990	−0.0131947
$782$	90.3704	0.00413253
$783$	21308.2	0.972535
$784$	0	0
$785$	19304.9	0.877736
$786$	−1182.77	−0.0536745
$787$	9007.09	0.407965	0.203982	−	0.978975i	$-0.434611\pi$
$787$	9007.09	0.407965	0.203982	+	0.978975i	$0.434611\pi$
$788$	15012.5	0.678677
$789$	19589.4	0.883906
$790$	428.199	0.0192843
$791$	0	0
$792$	−0.238352	−1.06938e−5 0
$793$	−2140.42	−0.0958496
$794$	224.421	0.0100307
$795$	−24649.9	−1.09968
$796$	−36366.3	−1.61931
$797$	−11820.9	−0.525368	−0.262684	−	0.964882i	$-0.584608\pi$
$797$	−11820.9	−0.525368	−0.262684	+	0.964882i	$0.584608\pi$
$798$	0	0
$799$	−3275.78	−0.145042
$800$	−97.2552	−0.00429811
$801$	−82.7969	−0.00365229
$802$	−205.891	−0.00906517
$803$	11.8726	0.000521764 0
$804$	44930.0	1.97085
$805$	0	0
$806$	148.452	0.00648760
$807$	−24750.5	−1.07963
$808$	1068.42	0.0465186
$809$	29485.4	1.28140	0.640699	−	0.767792i	$-0.278645\pi$
$809$	29485.4	1.28140	0.640699	+	0.767792i	$0.278645\pi$
$810$	968.235	0.0420004
$811$	−11104.4	−0.480800	−0.240400	−	0.970674i	$-0.577279\pi$
$811$	−11104.4	−0.480800	−0.240400	+	0.970674i	$0.577279\pi$
$812$	0	0
$813$	−28756.3	−1.24050
$814$	−20.5790	−0.000886108 0
$815$	−21327.2	−0.916638
$816$	−20450.4	−0.877335
$817$	−3107.60	−0.133074
$818$	1548.66	0.0661950
$819$	0	0
$820$	3724.51	0.158616
$821$	41677.5	1.77169	0.885844	−	0.463983i	$-0.153580\pi$
$821$	41677.5	1.77169	0.885844	+	0.463983i	$0.153580\pi$
$822$	−489.013	−0.0207497
$823$	−13681.6	−0.579479	−0.289739	−	0.957106i	$-0.593569\pi$
$823$	−13681.6	−0.579479	−0.289739	+	0.957106i	$0.593569\pi$
$824$	153.064	0.00647117
$825$	−16.7109	−0.000705211 0
$826$	0	0
$827$	−25514.2	−1.07281	−0.536406	−	0.843960i	$-0.680219\pi$
$827$	−25514.2	−1.07281	−0.536406	+	0.843960i	$0.680219\pi$
$828$	−17.7132	−0.000743448 0
$829$	−18232.5	−0.763859	−0.381930	−	0.924191i	$-0.624740\pi$
$829$	−18232.5	−0.763859	−0.381930	+	0.924191i	$0.624740\pi$
$830$	545.604	0.0228171
$831$	10740.6	0.448360
$832$	2629.84	0.109583
$833$	0	0
$834$	32.2718	0.00133990
$835$	1002.74	0.0415583
$836$	342.699	0.0141776
$837$	−34483.7	−1.42405
$838$	−18.0930	−0.000745837 0
$839$	28581.3	1.17608	0.588042	−	0.808830i	$-0.299899\pi$
$839$	28581.3	1.17608	0.588042	+	0.808830i	$0.299899\pi$
$840$	0	0
$841$	−1169.47	−0.0479509
$842$	−473.502	−0.0193800
$843$	−35924.0	−1.46772
$844$	8617.11	0.351437
$845$	−24683.2	−1.00489
$846$	−1.08216	−4.39780e−5 0
$847$	0	0
$848$	−26471.9	−1.07199
$849$	17393.4	0.703108
$850$	31.2763	0.00126208
$851$	−3061.24	−0.123311
$852$	−16366.9	−0.658123
$853$	36538.2	1.46664	0.733320	−	0.679884i	$-0.237970\pi$
$853$	36538.2	1.46664	0.733320	+	0.679884i	$0.237970\pi$
$854$	0	0
$855$	116.868	0.00467463
$856$	−2010.54	−0.0802788
$857$	44408.8	1.77010	0.885049	−	0.465497i	$-0.154124\pi$
$857$	44408.8	1.77010	0.885049	+	0.465497i	$0.154124\pi$
$858$	−2.29900	−9.14762e−5 0
$859$	24743.9	0.982831	0.491416	−	0.870925i	$-0.336480\pi$
$859$	24743.9	0.982831	0.491416	+	0.870925i	$0.336480\pi$
$860$	4819.63	0.191102
$861$	0	0
$862$	−259.627	−0.0102586
$863$	31992.9	1.26194	0.630968	−	0.775809i	$-0.282658\pi$
$863$	31992.9	1.26194	0.630968	+	0.775809i	$0.282658\pi$
$864$	3107.99	0.122380
$865$	−18958.8	−0.745223
$866$	−130.441	−0.00511843
$867$	−5826.02	−0.228214
$868$	0	0
$869$	237.708	0.00927928
$870$	1048.27	0.0408503
$871$	5599.52	0.217833
$872$	−3795.13	−0.147385
$873$	−150.409	−0.00583112
$874$	−85.9199	−0.00332527
$875$	0	0
$876$	674.740	0.0260244
$877$	37265.4	1.43485	0.717425	−	0.696636i	$-0.245321\pi$
$877$	37265.4	1.43485	0.717425	+	0.696636i	$0.245321\pi$
$878$	−1005.03	−0.0386312
$879$	−8186.65	−0.314140
$880$	−530.600	−0.0203256
$881$	−38356.1	−1.46680	−0.733399	−	0.679798i	$-0.762067\pi$
$881$	−38356.1	−1.46680	−0.733399	+	0.679798i	$0.762067\pi$
$882$	0	0
$883$	17705.1	0.674774	0.337387	−	0.941366i	$-0.390457\pi$
$883$	17705.1	0.674774	0.337387	+	0.941366i	$0.390457\pi$
$884$	−2552.99	−0.0971337
$885$	33613.4	1.27673
$886$	−732.134	−0.0277613
$887$	44834.0	1.69716	0.848578	−	0.529070i	$-0.177459\pi$
$887$	44834.0	1.69716	0.848578	+	0.529070i	$0.177459\pi$
$888$	−2341.04	−0.0884685
$889$	0	0
$890$	622.873	0.0234593
$891$	537.501	0.0202098
$892$	20779.8	0.780001
$893$	3114.46	0.116709
$894$	1736.76	0.0649732
$895$	12744.8	0.475990
$896$	0	0
$897$	−341.990	−0.0127299
$898$	1474.21	0.0547827
$899$	−37576.8	−1.39406
$900$	−6.13036	−0.000227050 0
$901$	25611.2	0.946985
$902$	−3.48477	−0.000128637 0
$903$	0	0
$904$	227.936	0.00838612
$905$	−4384.98	−0.161063
$906$	−1081.91	−0.0396734
$907$	33197.9	1.21535	0.607673	−	0.794187i	$-0.292103\pi$
$907$	33197.9	1.21535	0.607673	+	0.794187i	$0.292103\pi$
$908$	−902.567	−0.0329876
$909$	101.048	0.00368709
$910$	0	0
$911$	37924.4	1.37924	0.689622	−	0.724170i	$-0.257777\pi$
$911$	37924.4	1.37924	0.689622	+	0.724170i	$0.257777\pi$
$912$	19443.2	0.705954
$913$	302.884	0.0109792
$914$	−212.439	−0.00768803
$915$	24459.9	0.883735
$916$	26815.4	0.967257
$917$	0	0
$918$	−999.500	−0.0359351
$919$	5600.65	0.201032	0.100516	−	0.994935i	$-0.467951\pi$
$919$	5600.65	0.201032	0.100516	+	0.994935i	$0.467951\pi$
$920$	266.734	0.00955864
$921$	−17961.0	−0.642599
$922$	916.414	0.0327337
$923$	−2039.76	−0.0727407
$924$	0	0
$925$	−1059.47	−0.0376595
$926$	545.828	0.0193704
$927$	14.4764	0.000512909 0
$928$	3386.77	0.119802
$929$	8232.50	0.290742	0.145371	−	0.989377i	$-0.453562\pi$
$929$	8232.50	0.290742	0.145371	+	0.989377i	$0.453562\pi$
$930$	−1696.45	−0.0598158
$931$	0	0
$932$	14573.5	0.512200
$933$	23576.8	0.827297
$934$	206.823	0.00724567
$935$	513.350	0.0179554
$936$	−1.68819	−5.89532e−5 0
$937$	48128.5	1.67800	0.839002	−	0.544128i	$-0.183139\pi$
$937$	48128.5	1.67800	0.839002	+	0.544128i	$0.183139\pi$
$938$	0	0
$939$	−38903.4	−1.35204
$940$	−4830.26	−0.167602
$941$	−8782.64	−0.304257	−0.152129	−	0.988361i	$-0.548613\pi$
$941$	−8782.64	−0.304257	−0.152129	+	0.988361i	$0.548613\pi$
$942$	−1026.51	−0.0355047
$943$	−518.380	−0.0179011
$944$	36097.8	1.24458
$945$	0	0
$946$	−4.50940	−0.000154982 0
$947$	10896.1	0.373893	0.186947	−	0.982370i	$-0.440141\pi$
$947$	10896.1	0.373893	0.186947	+	0.982370i	$0.440141\pi$
$948$	13509.3	0.462829
$949$	84.0911	0.00287641
$950$	−29.7361	−0.00101554
$951$	3205.90	0.109315
$952$	0	0
$953$	−29225.7	−0.993404	−0.496702	−	0.867921i	$-0.665456\pi$
$953$	−29225.7	−0.993404	−0.496702	+	0.867921i	$0.665456\pi$
$954$	8.46071	0.000287134 0
$955$	−50.6699	−0.00171690
$956$	43162.6	1.46023
$957$	581.932	0.0196564
$958$	457.442	0.0154272
$959$	0	0
$960$	−30052.7	−1.01036
$961$	31020.6	1.04128
$962$	−145.756	−0.00488499
$963$	−190.151	−0.00636295
$964$	4793.72	0.160161
$965$	15795.0	0.526902
$966$	0	0
$967$	−2703.96	−0.0899210	−0.0449605	−	0.998989i	$-0.514316\pi$
$967$	−2703.96	−0.0899210	−0.0449605	+	0.998989i	$0.514316\pi$
$968$	−2467.69	−0.0819364
$969$	−18811.1	−0.623632
$970$	1131.51	0.0374543
$971$	27050.2	0.894008	0.447004	−	0.894532i	$-0.352491\pi$
$971$	27050.2	0.894008	0.447004	+	0.894532i	$0.352491\pi$
$972$	393.098	0.0129718
$973$	0	0
$974$	−1449.62	−0.0476887
$975$	−118.359	−0.00388773
$976$	26267.7	0.861486
$977$	13803.1	0.451997	0.225999	−	0.974128i	$-0.427436\pi$
$977$	13803.1	0.451997	0.225999	+	0.974128i	$0.427436\pi$
$978$	1134.04	0.0370783
$979$	345.779	0.0112882
$980$	0	0
$981$	−358.933	−0.0116818
$982$	1041.43	0.0338425
$983$	−32032.2	−1.03934	−0.519668	−	0.854368i	$-0.673944\pi$
$983$	−32032.2	−1.03934	−0.519668	+	0.854368i	$0.673944\pi$
$984$	−396.423	−0.0128430
$985$	−21380.6	−0.691618
$986$	−1089.15	−0.0351782
$987$	0	0
$988$	2427.26	0.0781593
$989$	−670.799	−0.0215674
$990$	0.169586	5.44424e−6 0
$991$	44051.9	1.41206	0.706032	−	0.708180i	$-0.250483\pi$
$991$	44051.9	1.41206	0.706032	+	0.708180i	$0.250483\pi$
$992$	−5480.91	−0.175422
$993$	−3153.73	−0.100786
$994$	0	0
$995$	51792.5	1.65019
$996$	17213.3	0.547616
$997$	33388.9	1.06062	0.530310	−	0.847804i	$-0.322075\pi$
$997$	33388.9	1.06062	0.530310	+	0.847804i	$0.322075\pi$
$998$	1587.71	0.0503589
$999$	33857.4	1.07227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.4.a.h.1.16	✓	30
7.6	odd	2		2009.4.a.i.1.16	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.4.a.h.1.16	✓	30	1.1	even	1	trivial
2009.4.a.i.1.16	yes	30	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-0.116020 q^{2} +5.21300 q^{3} -7.98654 q^{4} +11.3743 q^{5} -0.604813 q^{6} +1.85476 q^{8} +0.175418 q^{9} +O(q^{10})\)	\(q-0.116020 q^{2} +5.21300 q^{3} -7.98654 q^{4} +11.3743 q^{5} -0.604813 q^{6} +1.85476 q^{8} +0.175418 q^{9} -1.31965 q^{10} -0.732584 q^{11} -41.6339 q^{12} -5.18872 q^{13} +59.2945 q^{15} +63.6771 q^{16} -61.6069 q^{17} -0.0203520 q^{18} +58.5729 q^{19} -90.8417 q^{20} +0.0849944 q^{22} +12.6434 q^{23} +9.66886 q^{24} +4.37576 q^{25} +0.601996 q^{26} -139.837 q^{27} -152.380 q^{29} -6.87935 q^{30} +246.600 q^{31} -22.2259 q^{32} -3.81897 q^{33} +7.14763 q^{34} -1.40098 q^{36} -242.121 q^{37} -6.79563 q^{38} -27.0488 q^{39} +21.0967 q^{40} -41.0000 q^{41} -53.0553 q^{43} +5.85081 q^{44} +1.99526 q^{45} -1.46689 q^{46} +53.1723 q^{47} +331.949 q^{48} -0.507676 q^{50} -321.157 q^{51} +41.4400 q^{52} -415.720 q^{53} +16.2238 q^{54} -8.33267 q^{55} +305.341 q^{57} +17.6791 q^{58} +566.889 q^{59} -473.558 q^{60} +412.515 q^{61} -28.6105 q^{62} -506.838 q^{64} -59.0184 q^{65} +0.443076 q^{66} -1079.17 q^{67} +492.026 q^{68} +65.9101 q^{69} +393.115 q^{71} +0.325357 q^{72} -16.2065 q^{73} +28.0909 q^{74} +22.8109 q^{75} -467.795 q^{76} +3.13821 q^{78} -324.479 q^{79} +724.286 q^{80} -733.706 q^{81} +4.75682 q^{82} -413.445 q^{83} -700.738 q^{85} +6.15547 q^{86} -794.355 q^{87} -1.35877 q^{88} -471.998 q^{89} -0.231490 q^{90} -100.977 q^{92} +1285.53 q^{93} -6.16905 q^{94} +666.229 q^{95} -115.864 q^{96} -857.433 q^{97} -0.128508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} - 12 q^{3} + 103 q^{4} - 20 q^{5} - 36 q^{6} - 9 q^{8} + 318 q^{9}+O(q^{10}) \) 30 * q + q^2 - 12 * q^3 + 103 * q^4 - 20 * q^5 - 36 * q^6 - 9 * q^8 + 318 * q^9	\( 30 q + q^{2} - 12 q^{3} + 103 q^{4} - 20 q^{5} - 36 q^{6} - 9 q^{8} + 318 q^{9} - 80 q^{10} + 38 q^{11} + 83 q^{12} - 78 q^{13} + 24 q^{15} + 287 q^{16} - 260 q^{17} - 185 q^{18} - 336 q^{19} - 240 q^{20} - 160 q^{22} - 90 q^{23} - 1112 q^{24} + 606 q^{25} + 55 q^{26} - 432 q^{27} + 130 q^{29} - 674 q^{30} - 1320 q^{31} - 331 q^{32} - 152 q^{33} - 816 q^{34} + 983 q^{36} - 4 q^{37} - 396 q^{38} - 248 q^{39} - 934 q^{40} - 1230 q^{41} - 214 q^{43} + 926 q^{44} - 804 q^{45} - 248 q^{46} - 2262 q^{47} + 568 q^{48} - 543 q^{50} + 204 q^{51} - 650 q^{52} - 522 q^{53} - 3253 q^{54} - 1328 q^{55} - 160 q^{57} + 888 q^{58} - 656 q^{59} + 994 q^{60} - 4300 q^{61} + 728 q^{62} + 1637 q^{64} + 1848 q^{65} + 744 q^{66} + 1642 q^{67} - 4860 q^{68} + 1556 q^{69} - 980 q^{71} - 2248 q^{72} - 1112 q^{73} + 1609 q^{74} - 6916 q^{75} - 3096 q^{76} + 343 q^{78} + 2068 q^{79} + 2440 q^{80} + 3130 q^{81} - 41 q^{82} - 356 q^{83} + 788 q^{85} - 514 q^{86} - 820 q^{87} - 1130 q^{88} - 5560 q^{89} - 2160 q^{90} + 1573 q^{92} + 124 q^{93} + 2377 q^{94} + 580 q^{95} - 9857 q^{96} - 3828 q^{97} - 2870 q^{99}+O(q^{100}) \) 30 * q + q^2 - 12 * q^3 + 103 * q^4 - 20 * q^5 - 36 * q^6 - 9 * q^8 + 318 * q^9 - 80 * q^10 + 38 * q^11 + 83 * q^12 - 78 * q^13 + 24 * q^15 + 287 * q^16 - 260 * q^17 - 185 * q^18 - 336 * q^19 - 240 * q^20 - 160 * q^22 - 90 * q^23 - 1112 * q^24 + 606 * q^25 + 55 * q^26 - 432 * q^27 + 130 * q^29 - 674 * q^30 - 1320 * q^31 - 331 * q^32 - 152 * q^33 - 816 * q^34 + 983 * q^36 - 4 * q^37 - 396 * q^38 - 248 * q^39 - 934 * q^40 - 1230 * q^41 - 214 * q^43 + 926 * q^44 - 804 * q^45 - 248 * q^46 - 2262 * q^47 + 568 * q^48 - 543 * q^50 + 204 * q^51 - 650 * q^52 - 522 * q^53 - 3253 * q^54 - 1328 * q^55 - 160 * q^57 + 888 * q^58 - 656 * q^59 + 994 * q^60 - 4300 * q^61 + 728 * q^62 + 1637 * q^64 + 1848 * q^65 + 744 * q^66 + 1642 * q^67 - 4860 * q^68 + 1556 * q^69 - 980 * q^71 - 2248 * q^72 - 1112 * q^73 + 1609 * q^74 - 6916 * q^75 - 3096 * q^76 + 343 * q^78 + 2068 * q^79 + 2440 * q^80 + 3130 * q^81 - 41 * q^82 - 356 * q^83 + 788 * q^85 - 514 * q^86 - 820 * q^87 - 1130 * q^88 - 5560 * q^89 - 2160 * q^90 + 1573 * q^92 + 124 * q^93 + 2377 * q^94 + 580 * q^95 - 9857 * q^96 - 3828 * q^97 - 2870 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more