LMFDB - Embedded newform 507.4.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.107680$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.447278 q^{2} +3.00000 q^{3} -7.79994 q^{4} +1.93073 q^{5} +1.34183 q^{6} -8.14537 q^{7} -7.06697 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.447278 q^{2} +3.00000 q^{3} -7.79994 q^{4} +1.93073 q^{5} +1.34183 q^{6} -8.14537 q^{7} -7.06697 q^{8} +9.00000 q^{9} +0.863573 q^{10} +8.40842 q^{11} -23.3998 q^{12} -3.64325 q^{14} +5.79219 q^{15} +59.2386 q^{16} -52.1271 q^{17} +4.02550 q^{18} +48.8304 q^{19} -15.0596 q^{20} -24.4361 q^{21} +3.76090 q^{22} +88.9229 q^{23} -21.2009 q^{24} -121.272 q^{25} +27.0000 q^{27} +63.5334 q^{28} +191.979 q^{29} +2.59072 q^{30} +115.257 q^{31} +83.0319 q^{32} +25.2252 q^{33} -23.3153 q^{34} -15.7265 q^{35} -70.1995 q^{36} -136.716 q^{37} +21.8408 q^{38} -13.6444 q^{40} +436.077 q^{41} -10.9297 q^{42} +202.048 q^{43} -65.5852 q^{44} +17.3766 q^{45} +39.7733 q^{46} +618.160 q^{47} +177.716 q^{48} -276.653 q^{49} -54.2425 q^{50} -156.381 q^{51} -453.170 q^{53} +12.0765 q^{54} +16.2344 q^{55} +57.5631 q^{56} +146.491 q^{57} +85.8679 q^{58} +500.044 q^{59} -45.1787 q^{60} +480.502 q^{61} +51.5522 q^{62} -73.3083 q^{63} -436.771 q^{64} +11.2827 q^{66} +886.769 q^{67} +406.589 q^{68} +266.769 q^{69} -7.03412 q^{70} +123.732 q^{71} -63.6027 q^{72} +673.168 q^{73} -61.1501 q^{74} -363.817 q^{75} -380.875 q^{76} -68.4897 q^{77} -681.298 q^{79} +114.374 q^{80} +81.0000 q^{81} +195.048 q^{82} -939.418 q^{83} +190.600 q^{84} -100.643 q^{85} +90.3716 q^{86} +575.936 q^{87} -59.4220 q^{88} -754.979 q^{89} +7.77215 q^{90} -693.594 q^{92} +345.772 q^{93} +276.489 q^{94} +94.2783 q^{95} +249.096 q^{96} +1051.10 q^{97} -123.741 q^{98} +75.6757 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * 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 $

$2$	0.447278	0.158137	0.0790684	−	0.996869i	$-0.474805\pi$
$2$	0.447278	0.158137	0.0790684	+	0.996869i	$0.474805\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.79994	−0.974993
$5$	1.93073	0.172690	0.0863448	−	0.996265i	$-0.472481\pi$
$5$	1.93073	0.172690	0.0863448	+	0.996265i	$0.472481\pi$
$6$	1.34183	0.0913003
$7$	−8.14537	−0.439809	−0.219904	−	0.975521i	$-0.570575\pi$
$7$	−8.14537	−0.439809	−0.219904	+	0.975521i	$0.570575\pi$
$8$	−7.06697	−0.312319
$9$	9.00000	0.333333
$10$	0.863573	0.0273086
$11$	8.40842	0.230476	0.115238	−	0.993338i	$-0.463237\pi$
$11$	8.40842	0.230476	0.115238	+	0.993338i	$0.463237\pi$
$12$	−23.3998	−0.562912
$13$	0	0
$13$	0	0
$14$	−3.64325	−0.0695499
$15$	5.79219	0.0997024
$16$	59.2386	0.925604
$17$	−52.1271	−0.743688	−0.371844	−	0.928295i	$-0.621274\pi$
$17$	−52.1271	−0.743688	−0.371844	+	0.928295i	$0.621274\pi$
$18$	4.02550	0.0527122
$19$	48.8304	0.589604	0.294802	−	0.955558i	$-0.404746\pi$
$19$	48.8304	0.589604	0.294802	+	0.955558i	$0.404746\pi$
$20$	−15.0596	−0.168371
$21$	−24.4361	−0.253924
$22$	3.76090	0.0364467
$23$	88.9229	0.806162	0.403081	−	0.915164i	$-0.367939\pi$
$23$	88.9229	0.806162	0.403081	+	0.915164i	$0.367939\pi$
$24$	−21.2009	−0.180317
$25$	−121.272	−0.970178
$26$	0	0
$27$	27.0000	0.192450
$28$	63.5334	0.428810
$29$	191.979	1.22930	0.614648	−	0.788802i	$-0.289298\pi$
$29$	191.979	1.22930	0.614648	+	0.788802i	$0.289298\pi$
$30$	2.59072	0.0157666
$31$	115.257	0.667769	0.333885	−	0.942614i	$-0.391640\pi$
$31$	115.257	0.667769	0.333885	+	0.942614i	$0.391640\pi$
$32$	83.0319	0.458691
$33$	25.2252	0.133065
$34$	−23.3153	−0.117604
$35$	−15.7265	−0.0759504
$36$	−70.1995	−0.324998
$37$	−136.716	−0.607459	−0.303729	−	0.952758i	$-0.598232\pi$
$37$	−136.716	−0.607459	−0.303729	+	0.952758i	$0.598232\pi$
$38$	21.8408	0.0932380
$39$	0	0
$40$	−13.6444	−0.0539342
$41$	436.077	1.66107	0.830535	−	0.556967i	$-0.188035\pi$
$41$	436.077	1.66107	0.830535	+	0.556967i	$0.188035\pi$
$42$	−10.9297	−0.0401547
$43$	202.048	0.716559	0.358279	−	0.933614i	$-0.383364\pi$
$43$	202.048	0.716559	0.358279	+	0.933614i	$0.383364\pi$
$44$	−65.5852	−0.224712
$45$	17.3766	0.0575632
$46$	39.7733	0.127484
$47$	618.160	1.91847	0.959233	−	0.282617i	$-0.0912024\pi$
$47$	618.160	1.91847	0.959233	+	0.282617i	$0.0912024\pi$
$48$	177.716	0.534398
$49$	−276.653	−0.806568
$50$	−54.2425	−0.153421
$51$	−156.381	−0.429368
$52$	0	0
$53$	−453.170	−1.17448	−0.587242	−	0.809411i	$-0.699786\pi$
$53$	−453.170	−1.17448	−0.587242	+	0.809411i	$0.699786\pi$
$54$	12.0765	0.0304334
$55$	16.2344	0.0398008
$56$	57.5631	0.137361
$57$	146.491	0.340408
$58$	85.8679	0.194397
$59$	500.044	1.10339	0.551697	−	0.834045i	$-0.313981\pi$
$59$	500.044	1.10339	0.551697	+	0.834045i	$0.313981\pi$
$60$	−45.1787	−0.0972091
$61$	480.502	1.00856	0.504279	−	0.863541i	$-0.331758\pi$
$61$	480.502	1.00856	0.504279	+	0.863541i	$0.331758\pi$
$62$	51.5522	0.105599
$63$	−73.3083	−0.146603
$64$	−436.771	−0.853068
$65$	0	0
$66$	11.2827	0.0210425
$67$	886.769	1.61696	0.808478	−	0.588526i	$-0.200292\pi$
$67$	886.769	1.61696	0.808478	+	0.588526i	$0.200292\pi$
$68$	406.589	0.725090
$69$	266.769	0.465438
$70$	−7.03412	−0.0120105
$71$	123.732	0.206821	0.103410	−	0.994639i	$-0.467025\pi$
$71$	123.732	0.206821	0.103410	+	0.994639i	$0.467025\pi$
$72$	−63.6027	−0.104106
$73$	673.168	1.07929	0.539646	−	0.841892i	$-0.318558\pi$
$73$	673.168	1.07929	0.539646	+	0.841892i	$0.318558\pi$
$74$	−61.1501	−0.0960616
$75$	−363.817	−0.560133
$76$	−380.875	−0.574859
$77$	−68.4897	−0.101365
$78$	0	0
$79$	−681.298	−0.970279	−0.485139	−	0.874437i	$-0.661231\pi$
$79$	−681.298	−0.970279	−0.485139	+	0.874437i	$0.661231\pi$
$80$	114.374	0.159842
$81$	81.0000	0.111111
$82$	195.048	0.262676
$83$	−939.418	−1.24234	−0.621172	−	0.783675i	$-0.713343\pi$
$83$	−939.418	−1.24234	−0.621172	+	0.783675i	$0.713343\pi$
$84$	190.600	0.247574
$85$	−100.643	−0.128427
$86$	90.3716	0.113314
$87$	575.936	0.709734
$88$	−59.4220	−0.0719819
$89$	−754.979	−0.899187	−0.449593	−	0.893233i	$-0.648431\pi$
$89$	−754.979	−0.899187	−0.449593	+	0.893233i	$0.648431\pi$
$90$	7.77215	0.00910286
$91$	0	0
$92$	−693.594	−0.786002
$93$	345.772	0.385537
$94$	276.489	0.303380
$95$	94.2783	0.101818
$96$	249.096	0.264825
$97$	1051.10	1.10024	0.550118	−	0.835087i	$-0.314583\pi$
$97$	1051.10	1.10024	0.550118	+	0.835087i	$0.314583\pi$
$98$	−123.741	−0.127548
$99$	75.6757	0.0768252
$100$	945.917	0.945917
$101$	599.873	0.590986	0.295493	−	0.955345i	$-0.404516\pi$
$101$	599.873	0.590986	0.295493	+	0.955345i	$0.404516\pi$
$102$	−69.9460	−0.0678989
$103$	−293.312	−0.280591	−0.140295	−	0.990110i	$-0.544805\pi$
$103$	−293.312	−0.280591	−0.140295	+	0.990110i	$0.544805\pi$
$104$	0	0
$105$	−47.1795	−0.0438500
$106$	−202.693	−0.185729
$107$	1533.69	1.38568	0.692839	−	0.721092i	$-0.256360\pi$
$107$	1533.69	1.38568	0.692839	+	0.721092i	$0.256360\pi$
$108$	−210.598	−0.187637
$109$	590.718	0.519088	0.259544	−	0.965731i	$-0.416428\pi$
$109$	590.718	0.519088	0.259544	+	0.965731i	$0.416428\pi$
$110$	7.26128	0.00629396
$111$	−410.148	−0.350717
$112$	−482.521	−0.407089
$113$	−653.785	−0.544274	−0.272137	−	0.962259i	$-0.587730\pi$
$113$	−653.785	−0.544274	−0.272137	+	0.962259i	$0.587730\pi$
$114$	65.5224	0.0538310
$115$	171.686	0.139216
$116$	−1497.42	−1.19855
$117$	0	0
$118$	223.659	0.174487
$119$	424.595	0.327080
$120$	−40.9332	−0.0311389
$121$	−1260.30	−0.946881
$122$	214.918	0.159490
$123$	1308.23	0.959019
$124$	−899.002	−0.651070
$125$	−475.485	−0.340229
$126$	−32.7892	−0.0231833
$127$	−141.203	−0.0986595	−0.0493297	−	0.998783i	$-0.515709\pi$
$127$	−141.203	−0.0986595	−0.0493297	+	0.998783i	$0.515709\pi$
$128$	−859.613	−0.593592
$129$	606.144	0.413705
$130$	0	0
$131$	−731.910	−0.488147	−0.244074	−	0.969757i	$-0.578484\pi$
$131$	−731.910	−0.488147	−0.244074	+	0.969757i	$0.578484\pi$
$132$	−196.755	−0.129738
$133$	−397.742	−0.259313
$134$	396.633	0.255700
$135$	52.1297	0.0332341
$136$	368.381	0.232268
$137$	1762.27	1.09898	0.549492	−	0.835499i	$-0.314821\pi$
$137$	1762.27	1.09898	0.549492	+	0.835499i	$0.314821\pi$
$138$	119.320	0.0736028
$139$	−664.776	−0.405652	−0.202826	−	0.979215i	$-0.565013\pi$
$139$	−664.776	−0.405652	−0.202826	+	0.979215i	$0.565013\pi$
$140$	122.666	0.0740511
$141$	1854.48	1.10763
$142$	55.3425	0.0327059
$143$	0	0
$144$	533.148	0.308535
$145$	370.659	0.212286
$146$	301.094	0.170676
$147$	−829.959	−0.465672
$148$	1066.38	0.592268
$149$	−3300.71	−1.81480	−0.907399	−	0.420270i	$-0.861936\pi$
$149$	−3300.71	−1.81480	−0.907399	+	0.420270i	$0.861936\pi$
$150$	−162.727	−0.0885776
$151$	1464.15	0.789079	0.394540	−	0.918879i	$-0.370904\pi$
$151$	1464.15	0.789079	0.394540	+	0.918879i	$0.370904\pi$
$152$	−345.083	−0.184144
$153$	−469.144	−0.247896
$154$	−30.6339	−0.0160296
$155$	222.531	0.115317
$156$	0	0
$157$	1535.32	0.780456	0.390228	−	0.920718i	$-0.372396\pi$
$157$	1535.32	0.780456	0.390228	+	0.920718i	$0.372396\pi$
$158$	−304.730	−0.153437
$159$	−1359.51	−0.678089
$160$	160.312	0.0792111
$161$	−724.310	−0.354557
$162$	36.2295	0.0175707
$163$	793.554	0.381325	0.190663	−	0.981656i	$-0.438936\pi$
$163$	793.554	0.381325	0.190663	+	0.981656i	$0.438936\pi$
$164$	−3401.38	−1.61953
$165$	48.7031	0.0229790
$166$	−420.181	−0.196460
$167$	−1336.87	−0.619460	−0.309730	−	0.950825i	$-0.600239\pi$
$167$	−1336.87	−0.619460	−0.309730	+	0.950825i	$0.600239\pi$
$168$	172.689	0.0793052
$169$	0	0
$170$	−45.0156	−0.0203090
$171$	439.474	0.196535
$172$	−1575.96	−0.698640
$173$	568.460	0.249822	0.124911	−	0.992168i	$-0.460135\pi$
$173$	568.460	0.249822	0.124911	+	0.992168i	$0.460135\pi$
$174$	257.604	0.112235
$175$	987.808	0.426693
$176$	498.103	0.213329
$177$	1500.13	0.637044
$178$	−337.686	−0.142194
$179$	1546.00	0.645552	0.322776	−	0.946475i	$-0.395384\pi$
$179$	1546.00	0.645552	0.322776	+	0.946475i	$0.395384\pi$
$180$	−135.536	−0.0561237
$181$	3408.96	1.39992	0.699960	−	0.714182i	$-0.253201\pi$
$181$	3408.96	1.39992	0.699960	+	0.714182i	$0.253201\pi$
$182$	0	0
$183$	1441.51	0.582291
$184$	−628.416	−0.251780
$185$	−263.962	−0.104902
$186$	154.656	0.0609675
$187$	−438.307	−0.171402
$188$	−4821.61	−1.87049
$189$	−219.925	−0.0846412
$190$	42.1686	0.0161012
$191$	−3464.71	−1.31255	−0.656277	−	0.754520i	$-0.727870\pi$
$191$	−3464.71	−1.31255	−0.656277	+	0.754520i	$0.727870\pi$
$192$	−1310.31	−0.492519
$193$	4652.24	1.73510	0.867552	−	0.497346i	$-0.165692\pi$
$193$	4652.24	1.73510	0.867552	+	0.497346i	$0.165692\pi$
$194$	470.134	0.173988
$195$	0	0
$196$	2157.88	0.786398
$197$	−2870.98	−1.03832	−0.519160	−	0.854677i	$-0.673755\pi$
$197$	−2870.98	−1.03832	−0.519160	+	0.854677i	$0.673755\pi$
$198$	33.8481	0.0121489
$199$	25.0330	0.00891730	0.00445865	−	0.999990i	$-0.498581\pi$
$199$	25.0330	0.00891730	0.00445865	+	0.999990i	$0.498581\pi$
$200$	857.028	0.303005
$201$	2660.31	0.933550
$202$	268.310	0.0934565
$203$	−1563.74	−0.540655
$204$	1219.77	0.418631
$205$	841.947	0.286849
$206$	−131.192	−0.0443717
$207$	800.307	0.268721
$208$	0	0
$209$	410.587	0.135889
$210$	−21.1024	−0.00693429
$211$	−3605.91	−1.17650	−0.588249	−	0.808679i	$-0.700183\pi$
$211$	−3605.91	−1.17650	−0.588249	+	0.808679i	$0.700183\pi$
$212$	3534.70	1.14511
$213$	371.195	0.119408
$214$	685.987	0.219127
$215$	390.100	0.123742
$216$	−190.808	−0.0601058
$217$	−938.815	−0.293691
$218$	264.215	0.0820868
$219$	2019.50	0.623130
$220$	−126.627	−0.0388054
$221$	0	0
$222$	−183.450	−0.0554612
$223$	−4304.37	−1.29256	−0.646282	−	0.763098i	$-0.723677\pi$
$223$	−4304.37	−1.29256	−0.646282	+	0.763098i	$0.723677\pi$
$224$	−676.326	−0.201736
$225$	−1091.45	−0.323393
$226$	−292.424	−0.0860697
$227$	−5475.19	−1.60089	−0.800443	−	0.599409i	$-0.795402\pi$
$227$	−5475.19	−1.60089	−0.800443	+	0.599409i	$0.795402\pi$
$228$	−1142.62	−0.331895
$229$	378.108	0.109110	0.0545548	−	0.998511i	$-0.482626\pi$
$229$	378.108	0.109110	0.0545548	+	0.998511i	$0.482626\pi$
$230$	76.7914	0.0220151
$231$	−205.469	−0.0585232
$232$	−1356.71	−0.383932
$233$	2547.96	0.716405	0.358202	−	0.933644i	$-0.383390\pi$
$233$	2547.96	0.716405	0.358202	+	0.933644i	$0.383390\pi$
$234$	0	0
$235$	1193.50	0.331299
$236$	−3900.32	−1.07580
$237$	−2043.89	−0.560191
$238$	189.912	0.0517234
$239$	−6313.91	−1.70884	−0.854420	−	0.519583i	$-0.826087\pi$
$239$	−6313.91	−1.70884	−0.854420	+	0.519583i	$0.826087\pi$
$240$	343.121	0.0922849
$241$	−6763.73	−1.80784	−0.903921	−	0.427699i	$-0.859324\pi$
$241$	−6763.73	−1.80784	−0.903921	+	0.427699i	$0.859324\pi$
$242$	−563.704	−0.149737
$243$	243.000	0.0641500
$244$	−3747.89	−0.983336
$245$	−534.142	−0.139286
$246$	585.144	0.151656
$247$	0	0
$248$	−814.521	−0.208557
$249$	−2818.25	−0.717267
$250$	−212.674	−0.0538027
$251$	−3416.00	−0.859028	−0.429514	−	0.903060i	$-0.641315\pi$
$251$	−3416.00	−0.859028	−0.429514	+	0.903060i	$0.641315\pi$
$252$	571.801	0.142937
$253$	747.701	0.185801
$254$	−63.1571	−0.0156017
$255$	−301.930	−0.0741474
$256$	3109.68	0.759199
$257$	7002.36	1.69959	0.849796	−	0.527112i	$-0.176725\pi$
$257$	7002.36	1.69959	0.849796	+	0.527112i	$0.176725\pi$
$258$	271.115	0.0654220
$259$	1113.60	0.267166
$260$	0	0
$261$	1727.81	0.409765
$262$	−327.367	−0.0771940
$263$	3369.76	0.790071	0.395035	−	0.918666i	$-0.370732\pi$
$263$	3369.76	0.790071	0.395035	+	0.918666i	$0.370732\pi$
$264$	−178.266	−0.0415588
$265$	−874.948	−0.202821
$266$	−177.901	−0.0410069
$267$	−2264.94	−0.519146
$268$	−6916.75	−1.57652
$269$	−7801.70	−1.76832	−0.884160	−	0.467185i	$-0.845268\pi$
$269$	−7801.70	−1.76832	−0.884160	+	0.467185i	$0.845268\pi$
$270$	23.3165	0.00525554
$271$	4410.05	0.988528	0.494264	−	0.869312i	$-0.335438\pi$
$271$	4410.05	0.988528	0.494264	+	0.869312i	$0.335438\pi$
$272$	−3087.94	−0.688360
$273$	0	0
$274$	788.225	0.173790
$275$	−1019.71	−0.223603
$276$	−2080.78	−0.453798
$277$	−4636.44	−1.00569	−0.502846	−	0.864376i	$-0.667714\pi$
$277$	−4636.44	−1.00569	−0.502846	+	0.864376i	$0.667714\pi$
$278$	−297.340	−0.0641485
$279$	1037.32	0.222590
$280$	111.139	0.0237207
$281$	1090.50	0.231508	0.115754	−	0.993278i	$-0.463072\pi$
$281$	1090.50	0.231508	0.115754	+	0.993278i	$0.463072\pi$
$282$	829.468	0.175156
$283$	1140.52	0.239565	0.119782	−	0.992800i	$-0.461780\pi$
$283$	1140.52	0.239565	0.119782	+	0.992800i	$0.461780\pi$
$284$	−965.101	−0.201649
$285$	282.835	0.0587849
$286$	0	0
$287$	−3552.01	−0.730553
$288$	747.287	0.152897
$289$	−2195.76	−0.446929
$290$	165.788	0.0335703
$291$	3153.30	0.635222
$292$	−5250.67	−1.05230
$293$	−335.079	−0.0668106	−0.0334053	−	0.999442i	$-0.510635\pi$
$293$	−335.079	−0.0668106	−0.0334053	+	0.999442i	$0.510635\pi$
$294$	−371.223	−0.0736399
$295$	965.449	0.190545
$296$	966.168	0.189721
$297$	227.027	0.0443551
$298$	−1476.34	−0.286986
$299$	0	0
$300$	2837.75	0.546125
$301$	−1645.76	−0.315149
$302$	654.883	0.124782
$303$	1799.62	0.341206
$304$	2892.65	0.545739
$305$	927.719	0.174167
$306$	−209.838	−0.0392014
$307$	2540.04	0.472207	0.236104	−	0.971728i	$-0.424130\pi$
$307$	2540.04	0.472207	0.236104	+	0.971728i	$0.424130\pi$
$308$	534.215	0.0988303
$309$	−879.935	−0.161999
$310$	99.5332	0.0182358
$311$	2376.36	0.433283	0.216642	−	0.976251i	$-0.430490\pi$
$311$	2376.36	0.433283	0.216642	+	0.976251i	$0.430490\pi$
$312$	0	0
$313$	1315.78	0.237612	0.118806	−	0.992917i	$-0.462093\pi$
$313$	1315.78	0.237612	0.118806	+	0.992917i	$0.462093\pi$
$314$	686.713	0.123419
$315$	−141.538	−0.0253168
$316$	5314.09	0.946015
$317$	−6043.13	−1.07071	−0.535357	−	0.844626i	$-0.679823\pi$
$317$	−6043.13	−1.07071	−0.535357	+	0.844626i	$0.679823\pi$
$318$	−608.079	−0.107231
$319$	1614.24	0.283323
$320$	−843.286	−0.147316
$321$	4601.08	0.800022
$322$	−323.968	−0.0560685
$323$	−2545.39	−0.438481
$324$	−631.795	−0.108333
$325$	0	0
$326$	354.940	0.0603015
$327$	1772.16	0.299695
$328$	−3081.75	−0.518784
$329$	−5035.14	−0.843758
$330$	21.7838	0.00363382
$331$	8168.18	1.35639	0.678193	−	0.734884i	$-0.262763\pi$
$331$	8168.18	1.35639	0.678193	+	0.734884i	$0.262763\pi$
$332$	7327.40	1.21128
$333$	−1230.44	−0.202486
$334$	−597.951	−0.0979594
$335$	1712.11	0.279232
$336$	−1447.56	−0.235033
$337$	3076.32	0.497263	0.248631	−	0.968598i	$-0.420019\pi$
$337$	3076.32	0.497263	0.248631	+	0.968598i	$0.420019\pi$
$338$	0	0
$339$	−1961.36	−0.314237
$340$	785.012	0.125215
$341$	969.133	0.153905
$342$	196.567	0.0310793
$343$	5047.30	0.794544
$344$	−1427.87	−0.223795
$345$	515.058	0.0803762
$346$	254.260	0.0395061
$347$	−8754.07	−1.35430	−0.677152	−	0.735844i	$-0.736786\pi$
$347$	−8754.07	−1.35430	−0.677152	+	0.735844i	$0.736786\pi$
$348$	−4492.27	−0.691985
$349$	1064.97	0.163343	0.0816715	−	0.996659i	$-0.473974\pi$
$349$	1064.97	0.163343	0.0816715	+	0.996659i	$0.473974\pi$
$350$	441.825	0.0674758
$351$	0	0
$352$	698.167	0.105717
$353$	5047.62	0.761070	0.380535	−	0.924767i	$-0.375740\pi$
$353$	5047.62	0.761070	0.380535	+	0.924767i	$0.375740\pi$
$354$	670.977	0.100740
$355$	238.893	0.0357158
$356$	5888.79	0.876700
$357$	1273.78	0.188840
$358$	691.494	0.102085
$359$	8152.39	1.19851	0.599257	−	0.800557i	$-0.295463\pi$
$359$	8152.39	1.19851	0.599257	+	0.800557i	$0.295463\pi$
$360$	−122.800	−0.0179781
$361$	−4474.59	−0.652367
$362$	1524.75	0.221379
$363$	−3780.90	−0.546682
$364$	0	0
$365$	1299.71	0.186383
$366$	644.754	0.0920816
$367$	11636.3	1.65507	0.827537	−	0.561412i	$-0.189742\pi$
$367$	11636.3	1.65507	0.827537	+	0.561412i	$0.189742\pi$
$368$	5267.67	0.746186
$369$	3924.70	0.553690
$370$	−118.064	−0.0165888
$371$	3691.24	0.516548
$372$	−2697.00	−0.375896
$373$	4720.59	0.655289	0.327644	−	0.944801i	$-0.393745\pi$
$373$	4720.59	0.655289	0.327644	+	0.944801i	$0.393745\pi$
$374$	−196.045	−0.0271049
$375$	−1426.45	−0.196431
$376$	−4368.52	−0.599173
$377$	0	0
$378$	−98.3677	−0.0133849
$379$	12933.7	1.75293	0.876464	−	0.481467i	$-0.159896\pi$
$379$	12933.7	1.75293	0.876464	+	0.481467i	$0.159896\pi$
$380$	−735.365	−0.0992722
$381$	−423.609	−0.0569611
$382$	−1549.69	−0.207563
$383$	554.945	0.0740376	0.0370188	−	0.999315i	$-0.488214\pi$
$383$	554.945	0.0740376	0.0370188	+	0.999315i	$0.488214\pi$
$384$	−2578.84	−0.342711
$385$	−132.235	−0.0175047
$386$	2080.84	0.274384
$387$	1818.43	0.238853
$388$	−8198.51	−1.07272
$389$	5091.35	0.663604	0.331802	−	0.943349i	$-0.392343\pi$
$389$	5091.35	0.663604	0.331802	+	0.943349i	$0.392343\pi$
$390$	0	0
$391$	−4635.30	−0.599532
$392$	1955.10	0.251907
$393$	−2195.73	−0.281832
$394$	−1284.13	−0.164196
$395$	−1315.40	−0.167557
$396$	−590.266	−0.0749040
$397$	10254.6	1.29638	0.648188	−	0.761480i	$-0.275527\pi$
$397$	10254.6	1.29638	0.648188	+	0.761480i	$0.275527\pi$
$398$	11.1967	0.00141015
$399$	−1193.23	−0.149714
$400$	−7184.00	−0.898001
$401$	505.816	0.0629906	0.0314953	−	0.999504i	$-0.489973\pi$
$401$	505.816	0.0629906	0.0314953	+	0.999504i	$0.489973\pi$
$402$	1189.90	0.147629
$403$	0	0
$404$	−4678.97	−0.576207
$405$	156.389	0.0191877
$406$	−699.426	−0.0854974
$407$	−1149.57	−0.140005
$408$	1105.14	0.134100
$409$	2219.21	0.268295	0.134148	−	0.990961i	$-0.457170\pi$
$409$	2219.21	0.268295	0.134148	+	0.990961i	$0.457170\pi$
$410$	376.585	0.0453614
$411$	5286.81	0.634499
$412$	2287.82	0.273574
$413$	−4073.04	−0.485282
$414$	357.960	0.0424946
$415$	−1813.76	−0.214540
$416$	0	0
$417$	−1994.33	−0.234203
$418$	183.646	0.0214891
$419$	10249.3	1.19501	0.597506	−	0.801864i	$-0.296158\pi$
$419$	10249.3	1.19501	0.597506	+	0.801864i	$0.296158\pi$
$420$	367.997	0.0427534
$421$	97.9194	0.0113356	0.00566781	−	0.999984i	$-0.498196\pi$
$421$	97.9194	0.0113356	0.00566781	+	0.999984i	$0.498196\pi$
$422$	−1612.85	−0.186048
$423$	5563.44	0.639489
$424$	3202.54	0.366814
$425$	6321.58	0.721510
$426$	166.028	0.0188828
$427$	−3913.87	−0.443572
$428$	−11962.7	−1.35103
$429$	0	0
$430$	174.483	0.0195682
$431$	11727.2	1.31062	0.655310	−	0.755360i	$-0.272538\pi$
$431$	11727.2	1.31062	0.655310	+	0.755360i	$0.272538\pi$
$432$	1599.44	0.178133
$433$	6945.13	0.770812	0.385406	−	0.922747i	$-0.374061\pi$
$433$	6945.13	0.770812	0.385406	+	0.922747i	$0.374061\pi$
$434$	−419.911	−0.0464433
$435$	1111.98	0.122564
$436$	−4607.57	−0.506107
$437$	4342.15	0.475316
$438$	903.281	0.0985398
$439$	11446.3	1.24442	0.622210	−	0.782851i	$-0.286235\pi$
$439$	11446.3	1.24442	0.622210	+	0.782851i	$0.286235\pi$
$440$	−114.728	−0.0124305
$441$	−2489.88	−0.268856
$442$	0	0
$443$	−10513.6	−1.12758	−0.563789	−	0.825919i	$-0.690657\pi$
$443$	−10513.6	−1.12758	−0.563789	+	0.825919i	$0.690657\pi$
$444$	3199.13	0.341946
$445$	−1457.66	−0.155280
$446$	−1925.25	−0.204402
$447$	−9902.14	−1.04777
$448$	3557.66	0.375187
$449$	−9862.37	−1.03660	−0.518300	−	0.855199i	$-0.673435\pi$
$449$	−9862.37	−1.03660	−0.518300	+	0.855199i	$0.673435\pi$
$450$	−488.182	−0.0511403
$451$	3666.72	0.382836
$452$	5099.49	0.530663
$453$	4392.45	0.455575
$454$	−2448.93	−0.253159
$455$	0	0
$456$	−1035.25	−0.106316
$457$	−5990.48	−0.613179	−0.306590	−	0.951842i	$-0.599188\pi$
$457$	−5990.48	−0.613179	−0.306590	+	0.951842i	$0.599188\pi$
$458$	169.120	0.0172542
$459$	−1407.43	−0.143123
$460$	−1339.14	−0.135734
$461$	16579.5	1.67502	0.837510	−	0.546423i	$-0.184011\pi$
$461$	16579.5	1.67502	0.837510	+	0.546423i	$0.184011\pi$
$462$	−91.9018	−0.00925467
$463$	3606.65	0.362020	0.181010	−	0.983481i	$-0.442063\pi$
$463$	3606.65	0.362020	0.181010	+	0.983481i	$0.442063\pi$
$464$	11372.6	1.13784
$465$	667.593	0.0665782
$466$	1139.65	0.113290
$467$	−6789.61	−0.672775	−0.336387	−	0.941724i	$-0.609205\pi$
$467$	−6789.61	−0.672775	−0.336387	+	0.941724i	$0.609205\pi$
$468$	0	0
$469$	−7223.06	−0.711152
$470$	533.826	0.0523906
$471$	4605.95	0.450596
$472$	−3533.80	−0.344611
$473$	1698.90	0.165149
$474$	−914.189	−0.0885867
$475$	−5921.78	−0.572021
$476$	−3311.82	−0.318901
$477$	−4078.53	−0.391495
$478$	−2824.07	−0.270230
$479$	−12688.3	−1.21032	−0.605161	−	0.796103i	$-0.706891\pi$
$479$	−12688.3	−1.21032	−0.605161	+	0.796103i	$0.706891\pi$
$480$	480.936	0.0457326
$481$	0	0
$482$	−3025.27	−0.285886
$483$	−2172.93	−0.204703
$484$	9830.26	0.923202
$485$	2029.39	0.189999
$486$	108.689	0.0101445
$487$	−1217.28	−0.113265	−0.0566325	−	0.998395i	$-0.518036\pi$
$487$	−1217.28	−0.113265	−0.0566325	+	0.998395i	$0.518036\pi$
$488$	−3395.69	−0.314991
$489$	2380.66	0.220158
$490$	−238.910	−0.0220262
$491$	−10496.3	−0.964747	−0.482374	−	0.875966i	$-0.660225\pi$
$491$	−10496.3	−0.964747	−0.482374	+	0.875966i	$0.660225\pi$
$492$	−10204.1	−0.935037
$493$	−10007.3	−0.914211
$494$	0	0
$495$	146.109	0.0132669
$496$	6827.69	0.618090
$497$	−1007.84	−0.0909615
$498$	−1260.54	−0.113426
$499$	−10516.5	−0.943450	−0.471725	−	0.881746i	$-0.656368\pi$
$499$	−10516.5	−0.943450	−0.471725	+	0.881746i	$0.656368\pi$
$500$	3708.75	0.331721
$501$	−4010.60	−0.357645
$502$	−1527.90	−0.135844
$503$	13400.9	1.18790	0.593952	−	0.804500i	$-0.297567\pi$
$503$	13400.9	1.18790	0.593952	+	0.804500i	$0.297567\pi$
$504$	518.068	0.0457869
$505$	1158.19	0.102057
$506$	334.430	0.0293819
$507$	0	0
$508$	1101.38	0.0961923
$509$	4683.90	0.407878	0.203939	−	0.978984i	$-0.434626\pi$
$509$	4683.90	0.407878	0.203939	+	0.978984i	$0.434626\pi$
$510$	−135.047	−0.0117254
$511$	−5483.20	−0.474682
$512$	8267.80	0.713649
$513$	1318.42	0.113469
$514$	3132.00	0.268768
$515$	−566.305	−0.0484551
$516$	−4727.89	−0.403360
$517$	5197.75	0.442160
$518$	498.090	0.0422487
$519$	1705.38	0.144235
$520$	0	0
$521$	15034.6	1.26426	0.632128	−	0.774864i	$-0.282182\pi$
$521$	15034.6	1.26426	0.632128	+	0.774864i	$0.282182\pi$
$522$	772.811	0.0647989
$523$	8009.33	0.669643	0.334822	−	0.942282i	$-0.391324\pi$
$523$	8009.33	0.669643	0.334822	+	0.942282i	$0.391324\pi$
$524$	5708.86	0.475940
$525$	2963.42	0.246351
$526$	1507.22	0.124939
$527$	−6008.04	−0.496612
$528$	1494.31	0.123166
$529$	−4259.71	−0.350103
$530$	−391.345	−0.0320735
$531$	4500.40	0.367798
$532$	3102.36	0.252828
$533$	0	0
$534$	−1013.06	−0.0820960
$535$	2961.14	0.239292
$536$	−6266.77	−0.505006
$537$	4638.01	0.372710
$538$	−3489.53	−0.279636
$539$	−2326.21	−0.185894
$540$	−406.608	−0.0324030
$541$	20833.2	1.65561	0.827807	−	0.561013i	$-0.189588\pi$
$541$	20833.2	1.65561	0.827807	+	0.561013i	$0.189588\pi$
$542$	1972.52	0.156323
$543$	10226.9	0.808245
$544$	−4328.22	−0.341123
$545$	1140.52	0.0896411
$546$	0	0
$547$	−14247.2	−1.11365	−0.556825	−	0.830630i	$-0.687981\pi$
$547$	−14247.2	−1.11365	−0.556825	+	0.830630i	$0.687981\pi$
$548$	−13745.6	−1.07150
$549$	4324.52	0.336186
$550$	−456.093	−0.0353598
$551$	9374.41	0.724797
$552$	−1885.25	−0.145365
$553$	5549.43	0.426737
$554$	−2073.78	−0.159037
$555$	−791.885	−0.0605651
$556$	5185.22	0.395508
$557$	2007.94	0.152745	0.0763725	−	0.997079i	$-0.475666\pi$
$557$	2007.94	0.152745	0.0763725	+	0.997079i	$0.475666\pi$
$558$	463.969	0.0351996
$559$	0	0
$560$	−931.616	−0.0703000
$561$	−1314.92	−0.0989589
$562$	487.758	0.0366100
$563$	10562.1	0.790659	0.395330	−	0.918539i	$-0.370630\pi$
$563$	10562.1	0.790659	0.395330	+	0.918539i	$0.370630\pi$
$564$	−14464.8	−1.07993
$565$	−1262.28	−0.0939905
$566$	510.129	0.0378840
$567$	−659.775	−0.0488676
$568$	−874.409	−0.0645940
$569$	−23207.2	−1.70983	−0.854916	−	0.518766i	$-0.826392\pi$
$569$	−23207.2	−1.70983	−0.854916	+	0.518766i	$0.826392\pi$
$570$	126.506	0.00929605
$571$	7987.78	0.585426	0.292713	−	0.956200i	$-0.405442\pi$
$571$	7987.78	0.585426	0.292713	+	0.956200i	$0.405442\pi$
$572$	0	0
$573$	−10394.1	−0.757803
$574$	−1588.74	−0.115527
$575$	−10783.9	−0.782120
$576$	−3930.94	−0.284356
$577$	3224.41	0.232641	0.116321	−	0.993212i	$-0.462890\pi$
$577$	3224.41	0.232641	0.116321	+	0.993212i	$0.462890\pi$
$578$	−982.116	−0.0706759
$579$	13956.7	1.00176
$580$	−2891.12	−0.206978
$581$	7651.90	0.546393
$582$	1410.40	0.100452
$583$	−3810.44	−0.270690
$584$	−4757.26	−0.337084
$585$	0	0
$586$	−149.873	−0.0105652
$587$	−661.034	−0.0464800	−0.0232400	−	0.999730i	$-0.507398\pi$
$587$	−661.034	−0.0464800	−0.0232400	+	0.999730i	$0.507398\pi$
$588$	6473.63	0.454027
$589$	5628.07	0.393719
$590$	431.824	0.0301321
$591$	−8612.94	−0.599474
$592$	−8098.87	−0.562266
$593$	−10172.5	−0.704443	−0.352221	−	0.935917i	$-0.614574\pi$
$593$	−10172.5	−0.704443	−0.352221	+	0.935917i	$0.614574\pi$
$594$	101.544	0.00701417
$595$	819.777	0.0564834
$596$	25745.4	1.76942
$597$	75.0990	0.00514840
$598$	0	0
$599$	23462.2	1.60040	0.800200	−	0.599733i	$-0.204726\pi$
$599$	23462.2	1.60040	0.800200	+	0.599733i	$0.204726\pi$
$600$	2571.08	0.174940
$601$	−10482.6	−0.711474	−0.355737	−	0.934586i	$-0.615770\pi$
$601$	−10482.6	−0.711474	−0.355737	+	0.934586i	$0.615770\pi$
$602$	−736.111	−0.0498366
$603$	7980.92	0.538986
$604$	−11420.3	−0.769347
$605$	−2433.29	−0.163516
$606$	804.930	0.0539572
$607$	−24949.8	−1.66834	−0.834169	−	0.551508i	$-0.814053\pi$
$607$	−24949.8	−1.66834	−0.834169	+	0.551508i	$0.814053\pi$
$608$	4054.48	0.270446
$609$	−4691.21	−0.312147
$610$	414.949	0.0275423
$611$	0	0
$612$	3659.30	0.241697
$613$	8238.86	0.542846	0.271423	−	0.962460i	$-0.412506\pi$
$613$	8238.86	0.542846	0.271423	+	0.962460i	$0.412506\pi$
$614$	1136.10	0.0746733
$615$	2525.84	0.165613
$616$	484.014	0.0316583
$617$	5248.85	0.342481	0.171241	−	0.985229i	$-0.445222\pi$
$617$	5248.85	0.342481	0.171241	+	0.985229i	$0.445222\pi$
$618$	−393.576	−0.0256180
$619$	−17820.9	−1.15716	−0.578580	−	0.815626i	$-0.696393\pi$
$619$	−17820.9	−1.15716	−0.578580	+	0.815626i	$0.696393\pi$
$620$	−1735.73	−0.112433
$621$	2400.92	0.155146
$622$	1062.89	0.0685180
$623$	6149.58	0.395470
$624$	0	0
$625$	14241.0	0.911424
$626$	588.522	0.0375752
$627$	1231.76	0.0784557
$628$	−11975.4	−0.760938
$629$	7126.62	0.451760
$630$	−63.3071	−0.00400351
$631$	25445.0	1.60531	0.802653	−	0.596446i	$-0.203421\pi$
$631$	25445.0	1.60531	0.802653	+	0.596446i	$0.203421\pi$
$632$	4814.71	0.303036
$633$	−10817.7	−0.679252
$634$	−2702.96	−0.169319
$635$	−272.625	−0.0170375
$636$	10604.1	0.661132
$637$	0	0
$638$	722.013	0.0448037
$639$	1113.59	0.0689402
$640$	−1659.68	−0.102507
$641$	−18701.0	−1.15233	−0.576167	−	0.817332i	$-0.695452\pi$
$641$	−18701.0	−1.15233	−0.576167	+	0.817332i	$0.695452\pi$
$642$	2057.96	0.126513
$643$	−28465.9	−1.74586	−0.872928	−	0.487849i	$-0.837782\pi$
$643$	−28465.9	−1.74586	−0.872928	+	0.487849i	$0.837782\pi$
$644$	5649.58	0.345690
$645$	1170.30	0.0714426
$646$	−1138.50	−0.0693399
$647$	−10924.6	−0.663818	−0.331909	−	0.943311i	$-0.607693\pi$
$647$	−10924.6	−0.663818	−0.331909	+	0.943311i	$0.607693\pi$
$648$	−572.425	−0.0347021
$649$	4204.58	0.254305
$650$	0	0
$651$	−2816.44	−0.169562
$652$	−6189.68	−0.371789
$653$	−27110.9	−1.62470	−0.812351	−	0.583169i	$-0.801813\pi$
$653$	−27110.9	−1.62470	−0.812351	+	0.583169i	$0.801813\pi$
$654$	792.646	0.0473929
$655$	−1413.12	−0.0842980
$656$	25832.6	1.53749
$657$	6058.51	0.359764
$658$	−2252.11	−0.133429
$659$	1727.08	0.102090	0.0510452	−	0.998696i	$-0.483745\pi$
$659$	1727.08	0.102090	0.0510452	+	0.998696i	$0.483745\pi$
$660$	−379.881	−0.0224043
$661$	20651.1	1.21518	0.607592	−	0.794249i	$-0.292136\pi$
$661$	20651.1	1.21518	0.607592	+	0.794249i	$0.292136\pi$
$662$	3653.45	0.214494
$663$	0	0
$664$	6638.84	0.388007
$665$	−767.932	−0.0447806
$666$	−550.351	−0.0320205
$667$	17071.3	0.991010
$668$	10427.5	0.603969
$669$	−12913.1	−0.746263
$670$	765.790	0.0441568
$671$	4040.26	0.232448
$672$	−2028.98	−0.116472
$673$	10858.0	0.621907	0.310954	−	0.950425i	$-0.399352\pi$
$673$	10858.0	0.621907	0.310954	+	0.950425i	$0.399352\pi$
$674$	1375.97	0.0786355
$675$	−3274.35	−0.186711
$676$	0	0
$677$	29132.9	1.65387	0.826933	−	0.562301i	$-0.190084\pi$
$677$	29132.9	1.65387	0.826933	+	0.562301i	$0.190084\pi$
$678$	−877.272	−0.0496924
$679$	−8561.59	−0.483894
$680$	711.244	0.0401102
$681$	−16425.6	−0.924272
$682$	433.472	0.0243380
$683$	−9443.65	−0.529065	−0.264532	−	0.964377i	$-0.585218\pi$
$683$	−9443.65	−0.529065	−0.264532	+	0.964377i	$0.585218\pi$
$684$	−3427.87	−0.191620
$685$	3402.46	0.189783
$686$	2257.55	0.125647
$687$	1134.33	0.0629945
$688$	11969.0	0.663249
$689$	0	0
$690$	230.374	0.0127104
$691$	−23700.1	−1.30477	−0.652384	−	0.757889i	$-0.726231\pi$
$691$	−23700.1	−1.30477	−0.652384	+	0.757889i	$0.726231\pi$
$692$	−4433.96	−0.243575
$693$	−616.407	−0.0337884
$694$	−3915.51	−0.214165
$695$	−1283.50	−0.0700519
$696$	−4070.12	−0.221663
$697$	−22731.5	−1.23532
$698$	476.340	0.0258305
$699$	7643.88	0.413616
$700$	−7704.84	−0.416022
$701$	21643.1	1.16612	0.583059	−	0.812430i	$-0.301856\pi$
$701$	21643.1	1.16612	0.583059	+	0.812430i	$0.301856\pi$
$702$	0	0
$703$	−6675.91	−0.358160
$704$	−3672.55	−0.196611
$705$	3580.50	0.191276
$706$	2257.69	0.120353
$707$	−4886.18	−0.259921
$708$	−11700.9	−0.621114
$709$	−28965.3	−1.53429	−0.767146	−	0.641472i	$-0.778324\pi$
$709$	−28965.3	−1.53429	−0.767146	+	0.641472i	$0.778324\pi$
$710$	106.851	0.00564798
$711$	−6131.68	−0.323426
$712$	5335.41	0.280833
$713$	10249.0	0.538330
$714$	569.736	0.0298625
$715$	0	0
$716$	−12058.7	−0.629408
$717$	−18941.7	−0.986599
$718$	3646.39	0.189529
$719$	−21256.5	−1.10255	−0.551276	−	0.834323i	$-0.685859\pi$
$719$	−21256.5	−1.10255	−0.551276	+	0.834323i	$0.685859\pi$
$720$	1029.36	0.0532807
$721$	2389.13	0.123406
$722$	−2001.39	−0.103163
$723$	−20291.2	−1.04376
$724$	−26589.7	−1.36491
$725$	−23281.7	−1.19264
$726$	−1691.11	−0.0864505
$727$	−11868.9	−0.605494	−0.302747	−	0.953071i	$-0.597904\pi$
$727$	−11868.9	−0.605494	−0.302747	+	0.953071i	$0.597904\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	581.330	0.0294739
$731$	−10532.2	−0.532896
$732$	−11243.7	−0.567729
$733$	27296.4	1.37546	0.687732	−	0.725965i	$-0.258607\pi$
$733$	27296.4	1.37546	0.687732	+	0.725965i	$0.258607\pi$
$734$	5204.68	0.261728
$735$	−1602.43	−0.0804168
$736$	7383.44	0.369779
$737$	7456.32	0.372669
$738$	1755.43	0.0875587
$739$	−26424.6	−1.31535	−0.657674	−	0.753302i	$-0.728460\pi$
$739$	−26424.6	−1.31535	−0.657674	+	0.753302i	$0.728460\pi$
$740$	2058.89	0.102279
$741$	0	0
$742$	1651.01	0.0816853
$743$	−25974.8	−1.28253	−0.641267	−	0.767318i	$-0.721591\pi$
$743$	−25974.8	−1.28253	−0.641267	+	0.767318i	$0.721591\pi$
$744$	−2443.56	−0.120410
$745$	−6372.78	−0.313397
$746$	2111.42	0.103625
$747$	−8454.76	−0.414114
$748$	3418.77	0.167116
$749$	−12492.5	−0.609433
$750$	−638.022	−0.0310630
$751$	20234.8	0.983195	0.491598	−	0.870823i	$-0.336413\pi$
$751$	20234.8	0.983195	0.491598	+	0.870823i	$0.336413\pi$
$752$	36619.0	1.77574
$753$	−10248.0	−0.495960
$754$	0	0
$755$	2826.88	0.136266
$756$	1715.40	0.0825246
$757$	−3363.48	−0.161490	−0.0807449	−	0.996735i	$-0.525730\pi$
$757$	−3363.48	−0.161490	−0.0807449	+	0.996735i	$0.525730\pi$
$758$	5784.97	0.277202
$759$	2243.10	0.107272
$760$	−666.262	−0.0317998
$761$	−21311.3	−1.01516	−0.507579	−	0.861605i	$-0.669459\pi$
$761$	−21311.3	−1.01516	−0.507579	+	0.861605i	$0.669459\pi$
$762$	−189.471	−0.00900764
$763$	−4811.62	−0.228299
$764$	27024.5	1.27973
$765$	−905.790	−0.0428090
$766$	248.215	0.0117081
$767$	0	0
$768$	9329.04	0.438324
$769$	−34989.6	−1.64078	−0.820388	−	0.571807i	$-0.806242\pi$
$769$	−34989.6	−1.64078	−0.820388	+	0.571807i	$0.806242\pi$
$770$	−59.1458	−0.00276814
$771$	21007.1	0.981260
$772$	−36287.2	−1.69171
$773$	20034.1	0.932181	0.466091	−	0.884737i	$-0.345662\pi$
$773$	20034.1	0.932181	0.466091	+	0.884737i	$0.345662\pi$
$774$	813.345	0.0377714
$775$	−13977.5	−0.647855
$776$	−7428.09	−0.343625
$777$	3340.81	0.154248
$778$	2277.25	0.104940
$779$	21293.9	0.979373
$780$	0	0
$781$	1040.39	0.0476671
$782$	−2073.27	−0.0948081
$783$	5183.43	0.236578
$784$	−16388.5	−0.746563
$785$	2964.28	0.134777
$786$	−982.102	−0.0445680
$787$	5591.98	0.253282	0.126641	−	0.991949i	$-0.459580\pi$
$787$	5591.98	0.253282	0.126641	+	0.991949i	$0.459580\pi$
$788$	22393.5	1.01235
$789$	10109.3	0.456147
$790$	−588.351	−0.0264969
$791$	5325.32	0.239376
$792$	−534.798	−0.0239940
$793$	0	0
$794$	4586.64	0.205005
$795$	−2624.84	−0.117099
$796$	−195.256	−0.00869430
$797$	−12738.6	−0.566152	−0.283076	−	0.959098i	$-0.591355\pi$
$797$	−12738.6	−0.566152	−0.283076	+	0.959098i	$0.591355\pi$
$798$	−533.704	−0.0236753
$799$	−32222.9	−1.42674
$800$	−10069.5	−0.445012
$801$	−6794.81	−0.299729
$802$	226.240	0.00996113
$803$	5660.28	0.248751
$804$	−20750.2	−0.910205
$805$	−1398.45	−0.0612283
$806$	0	0
$807$	−23405.1	−1.02094
$808$	−4239.28	−0.184576
$809$	−8806.48	−0.382719	−0.191359	−	0.981520i	$-0.561290\pi$
$809$	−8806.48	−0.382719	−0.191359	+	0.981520i	$0.561290\pi$
$810$	69.9494	0.00303429
$811$	−10565.8	−0.457478	−0.228739	−	0.973488i	$-0.573460\pi$
$811$	−10565.8	−0.457478	−0.228739	+	0.973488i	$0.573460\pi$
$812$	12197.1	0.527134
$813$	13230.1	0.570727
$814$	−514.176	−0.0221399
$815$	1532.14	0.0658509
$816$	−9263.82	−0.397425
$817$	9866.09	0.422486
$818$	992.604	0.0424273
$819$	0	0
$820$	−6567.14	−0.279676
$821$	422.966	0.0179800	0.00899002	−	0.999960i	$-0.497138\pi$
$821$	422.966	0.0179800	0.00899002	+	0.999960i	$0.497138\pi$
$822$	2364.67	0.100338
$823$	11148.8	0.472202	0.236101	−	0.971728i	$-0.424130\pi$
$823$	11148.8	0.472202	0.236101	+	0.971728i	$0.424130\pi$
$824$	2072.83	0.0876339
$825$	−3059.12	−0.129097
$826$	−1821.78	−0.0767409
$827$	−813.158	−0.0341914	−0.0170957	−	0.999854i	$-0.505442\pi$
$827$	−813.158	−0.0341914	−0.0170957	+	0.999854i	$0.505442\pi$
$828$	−6242.34	−0.262001
$829$	−28320.8	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$829$	−28320.8	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$830$	−811.255	−0.0339266
$831$	−13909.3	−0.580637
$832$	0	0
$833$	14421.1	0.599835
$834$	−892.020	−0.0370361
$835$	−2581.13	−0.106974
$836$	−3202.55	−0.132491
$837$	3111.95	0.128512
$838$	4584.28	0.188975
$839$	−22139.3	−0.911006	−0.455503	−	0.890234i	$-0.650541\pi$
$839$	−22139.3	−0.911006	−0.455503	+	0.890234i	$0.650541\pi$
$840$	333.416	0.0136952
$841$	12466.8	0.511166
$842$	43.7972	0.00179258
$843$	3271.50	0.133661
$844$	28125.9	1.14708
$845$	0	0
$846$	2488.41	0.101127
$847$	10265.6	0.416446
$848$	−26845.2	−1.08711
$849$	3421.56	0.138313
$850$	2827.50	0.114097
$851$	−12157.2	−0.489710
$852$	−2895.30	−0.116422
$853$	−14080.5	−0.565191	−0.282595	−	0.959239i	$-0.591195\pi$
$853$	−14080.5	−0.565191	−0.282595	+	0.959239i	$0.591195\pi$
$854$	−1750.59	−0.0701451
$855$	848.505	0.0339395
$856$	−10838.6	−0.432774
$857$	10907.7	0.434772	0.217386	−	0.976086i	$-0.430247\pi$
$857$	10907.7	0.434772	0.217386	+	0.976086i	$0.430247\pi$
$858$	0	0
$859$	7739.24	0.307403	0.153702	−	0.988117i	$-0.450881\pi$
$859$	7739.24	0.307403	0.153702	+	0.988117i	$0.450881\pi$
$860$	−3042.76	−0.120648
$861$	−10656.0	−0.421785
$862$	5245.30	0.207257
$863$	−29072.5	−1.14674	−0.573372	−	0.819295i	$-0.694365\pi$
$863$	−29072.5	−1.14674	−0.573372	+	0.819295i	$0.694365\pi$
$864$	2241.86	0.0882751
$865$	1097.54	0.0431417
$866$	3106.40	0.121894
$867$	−6587.28	−0.258034
$868$	7322.70	0.286346
$869$	−5728.64	−0.223626
$870$	497.363	0.0193818
$871$	0	0
$872$	−4174.59	−0.162121
$873$	9459.89	0.366746
$874$	1942.15	0.0751649
$875$	3873.00	0.149636
$876$	−15752.0	−0.607547
$877$	2391.96	0.0920988	0.0460494	−	0.998939i	$-0.485337\pi$
$877$	2391.96	0.0920988	0.0460494	+	0.998939i	$0.485337\pi$
$878$	5119.66	0.196788
$879$	−1005.24	−0.0385731
$880$	961.702	0.0368397
$881$	−11975.8	−0.457974	−0.228987	−	0.973430i	$-0.573541\pi$
$881$	−11975.8	−0.457974	−0.228987	+	0.973430i	$0.573541\pi$
$882$	−1113.67	−0.0425160
$883$	−44712.4	−1.70407	−0.852035	−	0.523485i	$-0.824632\pi$
$883$	−44712.4	−1.70407	−0.852035	+	0.523485i	$0.824632\pi$
$884$	0	0
$885$	2896.35	0.110011
$886$	−4702.51	−0.178311
$887$	17023.3	0.644403	0.322202	−	0.946671i	$-0.395577\pi$
$887$	17023.3	0.644403	0.322202	+	0.946671i	$0.395577\pi$
$888$	2898.51	0.109535
$889$	1150.15	0.0433913
$890$	−651.979	−0.0245555
$891$	681.082	0.0256084
$892$	33573.8	1.26024
$893$	30185.0	1.13113
$894$	−4429.01	−0.165692
$895$	2984.91	0.111480
$896$	7001.87	0.261067
$897$	0	0
$898$	−4411.22	−0.163925
$899$	22127.0	0.820886
$900$	8513.25	0.315306
$901$	23622.5	0.873449
$902$	1640.04	0.0605405
$903$	−4937.27	−0.181951
$904$	4620.28	0.169987
$905$	6581.77	0.241752
$906$	1964.65	0.0720432
$907$	2303.59	0.0843324	0.0421662	−	0.999111i	$-0.486574\pi$
$907$	2303.59	0.0843324	0.0421662	+	0.999111i	$0.486574\pi$
$908$	42706.2	1.56085
$909$	5398.85	0.196995
$910$	0	0
$911$	−12897.0	−0.469042	−0.234521	−	0.972111i	$-0.575352\pi$
$911$	−12897.0	−0.469042	−0.234521	+	0.972111i	$0.575352\pi$
$912$	8677.95	0.315083
$913$	−7899.01	−0.286330
$914$	−2679.41	−0.0969662
$915$	2783.16	0.100556
$916$	−2949.22	−0.106381
$917$	5961.68	0.214691
$918$	−629.514	−0.0226330
$919$	16785.2	0.602496	0.301248	−	0.953546i	$-0.402597\pi$
$919$	16785.2	0.602496	0.301248	+	0.953546i	$0.402597\pi$
$920$	−1213.30	−0.0434797
$921$	7620.11	0.272629
$922$	7415.64	0.264882
$923$	0	0
$924$	1602.65	0.0570597
$925$	16579.9	0.589344
$926$	1613.18	0.0572487
$927$	−2639.81	−0.0935303
$928$	15940.4	0.563866
$929$	16348.0	0.577352	0.288676	−	0.957427i	$-0.406785\pi$
$929$	16348.0	0.577352	0.288676	+	0.957427i	$0.406785\pi$
$930$	298.600	0.0105285
$931$	−13509.1	−0.475556
$932$	−19873.9	−0.698489
$933$	7129.08	0.250156
$934$	−3036.85	−0.106390
$935$	−846.251	−0.0295993
$936$	0	0
$937$	−26247.5	−0.915121	−0.457560	−	0.889179i	$-0.651277\pi$
$937$	−26247.5	−0.915121	−0.457560	+	0.889179i	$0.651277\pi$
$938$	−3230.72	−0.112459
$939$	3947.35	0.137185
$940$	−9309.22	−0.323014
$941$	43838.4	1.51869	0.759347	−	0.650686i	$-0.225519\pi$
$941$	43838.4	1.51869	0.759347	+	0.650686i	$0.225519\pi$
$942$	2060.14	0.0712558
$943$	38777.3	1.33909
$944$	29621.9	1.02130
$945$	−424.615	−0.0146167
$946$	759.882	0.0261162
$947$	−45077.2	−1.54679	−0.773395	−	0.633924i	$-0.781443\pi$
$947$	−45077.2	−1.54679	−0.773395	+	0.633924i	$0.781443\pi$
$948$	15942.3	0.546182
$949$	0	0
$950$	−2648.68	−0.0904575
$951$	−18129.4	−0.618177
$952$	−3000.60	−0.102153
$953$	26479.1	0.900046	0.450023	−	0.893017i	$-0.351416\pi$
$953$	26479.1	0.900046	0.450023	+	0.893017i	$0.351416\pi$
$954$	−1824.24	−0.0619097
$955$	−6689.41	−0.226664
$956$	49248.1	1.66611
$957$	4842.71	0.163576
$958$	−5675.21	−0.191396
$959$	−14354.3	−0.483343
$960$	−2529.86	−0.0850529
$961$	−16506.7	−0.554084
$962$	0	0
$963$	13803.2	0.461893
$964$	52756.7	1.76263
$965$	8982.20	0.299635
$966$	−971.905	−0.0323711
$967$	−602.475	−0.0200355	−0.0100177	−	0.999950i	$-0.503189\pi$
$967$	−602.475	−0.0200355	−0.0100177	+	0.999950i	$0.503189\pi$
$968$	8906.49	0.295729
$969$	−7636.17	−0.253157
$970$	907.701	0.0300459
$971$	−11330.2	−0.374464	−0.187232	−	0.982316i	$-0.559952\pi$
$971$	−11330.2	−0.374464	−0.187232	+	0.982316i	$0.559952\pi$
$972$	−1895.39	−0.0625458
$973$	5414.85	0.178409
$974$	−544.461	−0.0179114
$975$	0	0
$976$	28464.3	0.933524
$977$	55633.6	1.82178	0.910888	−	0.412653i	$-0.135398\pi$
$977$	55633.6	1.82178	0.910888	+	0.412653i	$0.135398\pi$
$978$	1064.82	0.0348151
$979$	−6348.18	−0.207241
$980$	4166.27	0.135803
$981$	5316.47	0.173029
$982$	−4694.76	−0.152562
$983$	−6884.90	−0.223392	−0.111696	−	0.993742i	$-0.535628\pi$
$983$	−6884.90	−0.223392	−0.111696	+	0.993742i	$0.535628\pi$
$984$	−9245.24	−0.299520
$985$	−5543.08	−0.179307
$986$	−4476.05	−0.144570
$987$	−15105.4	−0.487144
$988$	0	0
$989$	17966.7	0.577662
$990$	65.3515	0.00209799
$991$	52646.1	1.68755	0.843773	−	0.536700i	$-0.180329\pi$
$991$	52646.1	1.68755	0.843773	+	0.536700i	$0.180329\pi$
$992$	9570.05	0.306300
$993$	24504.5	0.783109
$994$	−450.786	−0.0143844
$995$	48.3319	0.00153992
$996$	21982.2	0.699330
$997$	26100.7	0.829106	0.414553	−	0.910025i	$-0.363938\pi$
$997$	26100.7	0.829106	0.414553	+	0.910025i	$0.363938\pi$
$998$	−4703.78	−0.149194
$999$	−3691.33	−0.116906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.p.1.5	yes	9
3.2	odd	2		1521.4.a.bf.1.5		9
13.5	odd	4		507.4.b.k.337.8		18
13.8	odd	4		507.4.b.k.337.11		18
13.12	even	2		507.4.a.o.1.5	✓	9
39.38	odd	2		1521.4.a.bi.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.5	✓	9	13.12	even	2
507.4.a.p.1.5	yes	9	1.1	even	1	trivial
507.4.b.k.337.8		18	13.5	odd	4
507.4.b.k.337.11		18	13.8	odd	4
1521.4.a.bf.1.5		9	3.2	odd	2
1521.4.a.bi.1.5		9	39.38	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+0.447278 q^{2} +3.00000 q^{3} -7.79994 q^{4} +1.93073 q^{5} +1.34183 q^{6} -8.14537 q^{7} -7.06697 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.447278 q^{2} +3.00000 q^{3} -7.79994 q^{4} +1.93073 q^{5} +1.34183 q^{6} -8.14537 q^{7} -7.06697 q^{8} +9.00000 q^{9} +0.863573 q^{10} +8.40842 q^{11} -23.3998 q^{12} -3.64325 q^{14} +5.79219 q^{15} +59.2386 q^{16} -52.1271 q^{17} +4.02550 q^{18} +48.8304 q^{19} -15.0596 q^{20} -24.4361 q^{21} +3.76090 q^{22} +88.9229 q^{23} -21.2009 q^{24} -121.272 q^{25} +27.0000 q^{27} +63.5334 q^{28} +191.979 q^{29} +2.59072 q^{30} +115.257 q^{31} +83.0319 q^{32} +25.2252 q^{33} -23.3153 q^{34} -15.7265 q^{35} -70.1995 q^{36} -136.716 q^{37} +21.8408 q^{38} -13.6444 q^{40} +436.077 q^{41} -10.9297 q^{42} +202.048 q^{43} -65.5852 q^{44} +17.3766 q^{45} +39.7733 q^{46} +618.160 q^{47} +177.716 q^{48} -276.653 q^{49} -54.2425 q^{50} -156.381 q^{51} -453.170 q^{53} +12.0765 q^{54} +16.2344 q^{55} +57.5631 q^{56} +146.491 q^{57} +85.8679 q^{58} +500.044 q^{59} -45.1787 q^{60} +480.502 q^{61} +51.5522 q^{62} -73.3083 q^{63} -436.771 q^{64} +11.2827 q^{66} +886.769 q^{67} +406.589 q^{68} +266.769 q^{69} -7.03412 q^{70} +123.732 q^{71} -63.6027 q^{72} +673.168 q^{73} -61.1501 q^{74} -363.817 q^{75} -380.875 q^{76} -68.4897 q^{77} -681.298 q^{79} +114.374 q^{80} +81.0000 q^{81} +195.048 q^{82} -939.418 q^{83} +190.600 q^{84} -100.643 q^{85} +90.3716 q^{86} +575.936 q^{87} -59.4220 q^{88} -754.979 q^{89} +7.77215 q^{90} -693.594 q^{92} +345.772 q^{93} +276.489 q^{94} +94.2783 q^{95} +249.096 q^{96} +1051.10 q^{97} -123.741 q^{98} +75.6757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more