LMFDB - Embedded newform 6042.2.a.bb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6042,2,Mod(1,6042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6042.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,9,-9,9,-5,-9,-5,9,9,-5,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.2456129013$
Analytic rank:	$1$
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} - 4x^{8} - 16x^{7} + 76x^{6} + 30x^{5} - 366x^{4} + 300x^{3} + 101x^{2} - 106x + 16 $ x^9 - 4x^8 - 16x^7 + 76x^6 + 30x^5 - 366x^4 + 300x^3 + 101x^2 - 106x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.25110$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.51610 q^{5} -1.00000 q^{6} -4.25110 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.51610 q^{10} +1.96359 q^{11} -1.00000 q^{12} -2.77674 q^{13} -4.25110 q^{14} +2.51610 q^{15} +1.00000 q^{16} +6.83822 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.51610 q^{20} +4.25110 q^{21} +1.96359 q^{22} +6.37856 q^{23} -1.00000 q^{24} +1.33076 q^{25} -2.77674 q^{26} -1.00000 q^{27} -4.25110 q^{28} +5.73064 q^{29} +2.51610 q^{30} -0.117223 q^{31} +1.00000 q^{32} -1.96359 q^{33} +6.83822 q^{34} +10.6962 q^{35} +1.00000 q^{36} -5.48789 q^{37} -1.00000 q^{38} +2.77674 q^{39} -2.51610 q^{40} -9.92819 q^{41} +4.25110 q^{42} -1.57694 q^{43} +1.96359 q^{44} -2.51610 q^{45} +6.37856 q^{46} +0.666390 q^{47} -1.00000 q^{48} +11.0719 q^{49} +1.33076 q^{50} -6.83822 q^{51} -2.77674 q^{52} +1.00000 q^{53} -1.00000 q^{54} -4.94060 q^{55} -4.25110 q^{56} +1.00000 q^{57} +5.73064 q^{58} +3.13796 q^{59} +2.51610 q^{60} -0.415900 q^{61} -0.117223 q^{62} -4.25110 q^{63} +1.00000 q^{64} +6.98655 q^{65} -1.96359 q^{66} +7.65377 q^{67} +6.83822 q^{68} -6.37856 q^{69} +10.6962 q^{70} +9.28244 q^{71} +1.00000 q^{72} -0.563587 q^{73} -5.48789 q^{74} -1.33076 q^{75} -1.00000 q^{76} -8.34745 q^{77} +2.77674 q^{78} -16.0290 q^{79} -2.51610 q^{80} +1.00000 q^{81} -9.92819 q^{82} +0.307939 q^{83} +4.25110 q^{84} -17.2056 q^{85} -1.57694 q^{86} -5.73064 q^{87} +1.96359 q^{88} +10.6809 q^{89} -2.51610 q^{90} +11.8042 q^{91} +6.37856 q^{92} +0.117223 q^{93} +0.666390 q^{94} +2.51610 q^{95} -1.00000 q^{96} -8.44389 q^{97} +11.0719 q^{98} +1.96359 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20}+ \cdots + 3 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9 - 5 * q^10 + 3 * q^11 - 9 * q^12 - 4 * q^13 - 5 * q^14 + 5 * q^15 + 9 * q^16 - 16 * q^17 + 9 * q^18 - 9 * q^19 - 5 * q^20 + 5 * q^21 + 3 * q^22 - 9 * q^24 + 8 * q^25 - 4 * q^26 - 9 * q^27 - 5 * q^28 - 2 * q^29 + 5 * q^30 - 10 * q^31 + 9 * q^32 - 3 * q^33 - 16 * q^34 - q^35 + 9 * q^36 - 14 * q^37 - 9 * q^38 + 4 * q^39 - 5 * q^40 - 21 * q^41 + 5 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 - 8 * q^47 - 9 * q^48 - 14 * q^49 + 8 * q^50 + 16 * q^51 - 4 * q^52 + 9 * q^53 - 9 * q^54 - 14 * q^55 - 5 * q^56 + 9 * q^57 - 2 * q^58 - 12 * q^59 + 5 * q^60 - 13 * q^61 - 10 * q^62 - 5 * q^63 + 9 * q^64 - 13 * q^65 - 3 * q^66 + 22 * q^67 - 16 * q^68 - q^70 - 13 * q^71 + 9 * q^72 - 17 * q^73 - 14 * q^74 - 8 * q^75 - 9 * q^76 - 25 * q^77 + 4 * q^78 - 5 * q^80 + 9 * q^81 - 21 * q^82 - 24 * q^83 + 5 * q^84 - 16 * q^85 - 4 * q^86 + 2 * q^87 + 3 * q^88 - 23 * q^89 - 5 * q^90 - 11 * q^91 + 10 * q^93 - 8 * q^94 + 5 * q^95 - 9 * q^96 - 29 * q^97 - 14 * q^98 + 3 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.51610	−1.12523	−0.562617	−	0.826718i	$-0.690205\pi$
$5$	−2.51610	−1.12523	−0.562617	+	0.826718i	$0.690205\pi$
$6$	−1.00000	−0.408248
$7$	−4.25110	−1.60677	−0.803383	−	0.595462i	$-0.796969\pi$
$7$	−4.25110	−1.60677	−0.803383	+	0.595462i	$0.796969\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.51610	−0.795661
$11$	1.96359	0.592046	0.296023	−	0.955181i	$-0.404339\pi$
$11$	1.96359	0.592046	0.296023	+	0.955181i	$0.404339\pi$
$12$	−1.00000	−0.288675
$13$	−2.77674	−0.770129	−0.385064	−	0.922890i	$-0.625821\pi$
$13$	−2.77674	−0.770129	−0.385064	+	0.922890i	$0.625821\pi$
$14$	−4.25110	−1.13616
$15$	2.51610	0.649654
$16$	1.00000	0.250000
$17$	6.83822	1.65851	0.829256	−	0.558870i	$-0.188765\pi$
$17$	6.83822	1.65851	0.829256	+	0.558870i	$0.188765\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.51610	−0.562617
$21$	4.25110	0.927667
$22$	1.96359	0.418640
$23$	6.37856	1.33002	0.665011	−	0.746833i	$-0.268427\pi$
$23$	6.37856	1.33002	0.665011	+	0.746833i	$0.268427\pi$
$24$	−1.00000	−0.204124
$25$	1.33076	0.266152
$26$	−2.77674	−0.544563
$27$	−1.00000	−0.192450
$28$	−4.25110	−0.803383
$29$	5.73064	1.06415	0.532076	−	0.846696i	$-0.321412\pi$
$29$	5.73064	1.06415	0.532076	+	0.846696i	$0.321412\pi$
$30$	2.51610	0.459375
$31$	−0.117223	−0.0210539	−0.0105269	−	0.999945i	$-0.503351\pi$
$31$	−0.117223	−0.0210539	−0.0105269	+	0.999945i	$0.503351\pi$
$32$	1.00000	0.176777
$33$	−1.96359	−0.341818
$34$	6.83822	1.17274
$35$	10.6962	1.80799
$36$	1.00000	0.166667
$37$	−5.48789	−0.902204	−0.451102	−	0.892472i	$-0.648969\pi$
$37$	−5.48789	−0.902204	−0.451102	+	0.892472i	$0.648969\pi$
$38$	−1.00000	−0.162221
$39$	2.77674	0.444634
$40$	−2.51610	−0.397830
$41$	−9.92819	−1.55052	−0.775261	−	0.631640i	$-0.782382\pi$
$41$	−9.92819	−1.55052	−0.775261	+	0.631640i	$0.782382\pi$
$42$	4.25110	0.655960
$43$	−1.57694	−0.240481	−0.120241	−	0.992745i	$-0.538367\pi$
$43$	−1.57694	−0.240481	−0.120241	+	0.992745i	$0.538367\pi$
$44$	1.96359	0.296023
$45$	−2.51610	−0.375078
$46$	6.37856	0.940468
$47$	0.666390	0.0972030	0.0486015	−	0.998818i	$-0.484524\pi$
$47$	0.666390	0.0972030	0.0486015	+	0.998818i	$0.484524\pi$
$48$	−1.00000	−0.144338
$49$	11.0719	1.58170
$50$	1.33076	0.188198
$51$	−6.83822	−0.957542
$52$	−2.77674	−0.385064
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	−4.94060	−0.666191
$56$	−4.25110	−0.568078
$57$	1.00000	0.132453
$58$	5.73064	0.752470
$59$	3.13796	0.408528	0.204264	−	0.978916i	$-0.434520\pi$
$59$	3.13796	0.408528	0.204264	+	0.978916i	$0.434520\pi$
$60$	2.51610	0.324827
$61$	−0.415900	−0.0532505	−0.0266252	−	0.999645i	$-0.508476\pi$
$61$	−0.415900	−0.0532505	−0.0266252	+	0.999645i	$0.508476\pi$
$62$	−0.117223	−0.0148873
$63$	−4.25110	−0.535589
$64$	1.00000	0.125000
$65$	6.98655	0.866575
$66$	−1.96359	−0.241702
$67$	7.65377	0.935057	0.467529	−	0.883978i	$-0.345144\pi$
$67$	7.65377	0.935057	0.467529	+	0.883978i	$0.345144\pi$
$68$	6.83822	0.829256
$69$	−6.37856	−0.767889
$70$	10.6962	1.27844
$71$	9.28244	1.10162	0.550811	−	0.834630i	$-0.314318\pi$
$71$	9.28244	1.10162	0.550811	+	0.834630i	$0.314318\pi$
$72$	1.00000	0.117851
$73$	−0.563587	−0.0659628	−0.0329814	−	0.999456i	$-0.510500\pi$
$73$	−0.563587	−0.0659628	−0.0329814	+	0.999456i	$0.510500\pi$
$74$	−5.48789	−0.637954
$75$	−1.33076	−0.153663
$76$	−1.00000	−0.114708
$77$	−8.34745	−0.951280
$78$	2.77674	0.314404
$79$	−16.0290	−1.80340	−0.901700	−	0.432363i	$-0.857680\pi$
$79$	−16.0290	−1.80340	−0.901700	+	0.432363i	$0.857680\pi$
$80$	−2.51610	−0.281309
$81$	1.00000	0.111111
$82$	−9.92819	−1.09639
$83$	0.307939	0.0338007	0.0169003	−	0.999857i	$-0.494620\pi$
$83$	0.307939	0.0338007	0.0169003	+	0.999857i	$0.494620\pi$
$84$	4.25110	0.463834
$85$	−17.2056	−1.86621
$86$	−1.57694	−0.170046
$87$	−5.73064	−0.614389
$88$	1.96359	0.209320
$89$	10.6809	1.13217	0.566086	−	0.824346i	$-0.308457\pi$
$89$	10.6809	1.13217	0.566086	+	0.824346i	$0.308457\pi$
$90$	−2.51610	−0.265220
$91$	11.8042	1.23742
$92$	6.37856	0.665011
$93$	0.117223	0.0121555
$94$	0.666390	0.0687329
$95$	2.51610	0.258146
$96$	−1.00000	−0.102062
$97$	−8.44389	−0.857348	−0.428674	−	0.903459i	$-0.641019\pi$
$97$	−8.44389	−0.857348	−0.428674	+	0.903459i	$0.641019\pi$
$98$	11.0719	1.11843
$99$	1.96359	0.197349
$100$	1.33076	0.133076
$101$	4.79119	0.476741	0.238371	−	0.971174i	$-0.423387\pi$
$101$	4.79119	0.476741	0.238371	+	0.971174i	$0.423387\pi$
$102$	−6.83822	−0.677084
$103$	−9.79253	−0.964887	−0.482443	−	0.875927i	$-0.660251\pi$
$103$	−9.79253	−0.964887	−0.482443	+	0.875927i	$0.660251\pi$
$104$	−2.77674	−0.272282
$105$	−10.6962	−1.04384
$106$	1.00000	0.0971286
$107$	−15.2769	−1.47688	−0.738438	−	0.674322i	$-0.764436\pi$
$107$	−15.2769	−1.47688	−0.738438	+	0.674322i	$0.764436\pi$
$108$	−1.00000	−0.0962250
$109$	7.91690	0.758302	0.379151	−	0.925335i	$-0.376216\pi$
$109$	7.91690	0.758302	0.379151	+	0.925335i	$0.376216\pi$
$110$	−4.94060	−0.471068
$111$	5.48789	0.520888
$112$	−4.25110	−0.401692
$113$	2.86186	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$113$	2.86186	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$114$	1.00000	0.0936586
$115$	−16.0491	−1.49659
$116$	5.73064	0.532076
$117$	−2.77674	−0.256710
$118$	3.13796	0.288873
$119$	−29.0700	−2.66484
$120$	2.51610	0.229688
$121$	−7.14429	−0.649481
$122$	−0.415900	−0.0376538
$123$	9.92819	0.895195
$124$	−0.117223	−0.0105269
$125$	9.23217	0.825751
$126$	−4.25110	−0.378719
$127$	−18.0636	−1.60289	−0.801444	−	0.598069i	$-0.795935\pi$
$127$	−18.0636	−1.60289	−0.801444	+	0.598069i	$0.795935\pi$
$128$	1.00000	0.0883883
$129$	1.57694	0.138842
$130$	6.98655	0.612761
$131$	−18.2116	−1.59115	−0.795576	−	0.605854i	$-0.792832\pi$
$131$	−18.2116	−1.59115	−0.795576	+	0.605854i	$0.792832\pi$
$132$	−1.96359	−0.170909
$133$	4.25110	0.368618
$134$	7.65377	0.661185
$135$	2.51610	0.216551
$136$	6.83822	0.586372
$137$	−18.2145	−1.55617	−0.778084	−	0.628160i	$-0.783808\pi$
$137$	−18.2145	−1.55617	−0.778084	+	0.628160i	$0.783808\pi$
$138$	−6.37856	−0.542979
$139$	7.07490	0.600085	0.300042	−	0.953926i	$-0.402999\pi$
$139$	7.07490	0.600085	0.300042	+	0.953926i	$0.402999\pi$
$140$	10.6962	0.903994
$141$	−0.666390	−0.0561202
$142$	9.28244	0.778965
$143$	−5.45239	−0.455952
$144$	1.00000	0.0833333
$145$	−14.4189	−1.19742
$146$	−0.563587	−0.0466428
$147$	−11.0719	−0.913194
$148$	−5.48789	−0.451102
$149$	−16.8298	−1.37875	−0.689375	−	0.724405i	$-0.742115\pi$
$149$	−16.8298	−1.37875	−0.689375	+	0.724405i	$0.742115\pi$
$150$	−1.33076	−0.108656
$151$	9.22370	0.750614	0.375307	−	0.926901i	$-0.377537\pi$
$151$	9.22370	0.750614	0.375307	+	0.926901i	$0.377537\pi$
$152$	−1.00000	−0.0811107
$153$	6.83822	0.552837
$154$	−8.34745	−0.672657
$155$	0.294945	0.0236906
$156$	2.77674	0.222317
$157$	−18.0398	−1.43973	−0.719865	−	0.694114i	$-0.755796\pi$
$157$	−18.0398	−1.43973	−0.719865	+	0.694114i	$0.755796\pi$
$158$	−16.0290	−1.27520
$159$	−1.00000	−0.0793052
$160$	−2.51610	−0.198915
$161$	−27.1159	−2.13704
$162$	1.00000	0.0785674
$163$	−0.0245472	−0.00192269	−0.000961343	−	1.00000i	$-0.500306\pi$
$163$	−0.0245472	−0.00192269	−0.000961343		1.00000i	$0.500306\pi$
$164$	−9.92819	−0.775261
$165$	4.94060	0.384625
$166$	0.307939	0.0239007
$167$	14.1169	1.09239	0.546197	−	0.837657i	$-0.316075\pi$
$167$	14.1169	1.09239	0.546197	+	0.837657i	$0.316075\pi$
$168$	4.25110	0.327980
$169$	−5.28972	−0.406902
$170$	−17.2056	−1.31961
$171$	−1.00000	−0.0764719
$172$	−1.57694	−0.120241
$173$	−22.0740	−1.67826	−0.839129	−	0.543933i	$-0.816935\pi$
$173$	−22.0740	−1.67826	−0.839129	+	0.543933i	$0.816935\pi$
$174$	−5.73064	−0.434439
$175$	−5.65721	−0.427645
$176$	1.96359	0.148012
$177$	−3.13796	−0.235864
$178$	10.6809	0.800567
$179$	18.7893	1.40438	0.702189	−	0.711991i	$-0.252206\pi$
$179$	18.7893	1.40438	0.702189	+	0.711991i	$0.252206\pi$
$180$	−2.51610	−0.187539
$181$	−11.4778	−0.853136	−0.426568	−	0.904455i	$-0.640278\pi$
$181$	−11.4778	−0.853136	−0.426568	+	0.904455i	$0.640278\pi$
$182$	11.8042	0.874986
$183$	0.415900	0.0307442
$184$	6.37856	0.470234
$185$	13.8081	1.01519
$186$	0.117223	0.00859521
$187$	13.4275	0.981915
$188$	0.666390	0.0486015
$189$	4.25110	0.309222
$190$	2.51610	0.182537
$191$	−1.50328	−0.108774	−0.0543869	−	0.998520i	$-0.517320\pi$
$191$	−1.50328	−0.108774	−0.0543869	+	0.998520i	$0.517320\pi$
$192$	−1.00000	−0.0721688
$193$	10.5502	0.759417	0.379709	−	0.925106i	$-0.376024\pi$
$193$	10.5502	0.759417	0.379709	+	0.925106i	$0.376024\pi$
$194$	−8.44389	−0.606236
$195$	−6.98655	−0.500318
$196$	11.0719	0.790849
$197$	−5.16113	−0.367715	−0.183858	−	0.982953i	$-0.558859\pi$
$197$	−5.16113	−0.367715	−0.183858	+	0.982953i	$0.558859\pi$
$198$	1.96359	0.139547
$199$	−21.8391	−1.54814	−0.774068	−	0.633103i	$-0.781781\pi$
$199$	−21.8391	−1.54814	−0.774068	+	0.633103i	$0.781781\pi$
$200$	1.33076	0.0940991
$201$	−7.65377	−0.539856
$202$	4.79119	0.337107
$203$	−24.3615	−1.70985
$204$	−6.83822	−0.478771
$205$	24.9803	1.74470
$206$	−9.79253	−0.682278
$207$	6.37856	0.443341
$208$	−2.77674	−0.192532
$209$	−1.96359	−0.135825
$210$	−10.6962	−0.738108
$211$	18.8235	1.29586	0.647932	−	0.761699i	$-0.275634\pi$
$211$	18.8235	1.29586	0.647932	+	0.761699i	$0.275634\pi$
$212$	1.00000	0.0686803
$213$	−9.28244	−0.636022
$214$	−15.2769	−1.04431
$215$	3.96774	0.270598
$216$	−1.00000	−0.0680414
$217$	0.498327	0.0338287
$218$	7.91690	0.536200
$219$	0.563587	0.0380837
$220$	−4.94060	−0.333095
$221$	−18.9879	−1.27727
$222$	5.48789	0.368323
$223$	−22.7495	−1.52342	−0.761711	−	0.647917i	$-0.775640\pi$
$223$	−22.7495	−1.52342	−0.761711	+	0.647917i	$0.775640\pi$
$224$	−4.25110	−0.284039
$225$	1.33076	0.0887175
$226$	2.86186	0.190368
$227$	7.21237	0.478701	0.239351	−	0.970933i	$-0.423065\pi$
$227$	7.21237	0.478701	0.239351	+	0.970933i	$0.423065\pi$
$228$	1.00000	0.0662266
$229$	3.30012	0.218078	0.109039	−	0.994037i	$-0.465223\pi$
$229$	3.30012	0.218078	0.109039	+	0.994037i	$0.465223\pi$
$230$	−16.0491	−1.05825
$231$	8.34745	0.549222
$232$	5.73064	0.376235
$233$	−7.84552	−0.513977	−0.256989	−	0.966414i	$-0.582730\pi$
$233$	−7.84552	−0.513977	−0.256989	+	0.966414i	$0.582730\pi$
$234$	−2.77674	−0.181521
$235$	−1.67671	−0.109376
$236$	3.13796	0.204264
$237$	16.0290	1.04119
$238$	−29.0700	−1.88433
$239$	1.41077	0.0912550	0.0456275	−	0.998959i	$-0.485471\pi$
$239$	1.41077	0.0912550	0.0456275	+	0.998959i	$0.485471\pi$
$240$	2.51610	0.162414
$241$	3.80810	0.245301	0.122651	−	0.992450i	$-0.460861\pi$
$241$	3.80810	0.245301	0.122651	+	0.992450i	$0.460861\pi$
$242$	−7.14429	−0.459253
$243$	−1.00000	−0.0641500
$244$	−0.415900	−0.0266252
$245$	−27.8580	−1.77978
$246$	9.92819	0.632998
$247$	2.77674	0.176680
$248$	−0.117223	−0.00744367
$249$	−0.307939	−0.0195148
$250$	9.23217	0.583894
$251$	28.9995	1.83043	0.915215	−	0.402965i	$-0.132020\pi$
$251$	28.9995	1.83043	0.915215	+	0.402965i	$0.132020\pi$
$252$	−4.25110	−0.267794
$253$	12.5249	0.787435
$254$	−18.0636	−1.13341
$255$	17.2056	1.07746
$256$	1.00000	0.0625000
$257$	−22.4357	−1.39950	−0.699751	−	0.714387i	$-0.746706\pi$
$257$	−22.4357	−1.39950	−0.699751	+	0.714387i	$0.746706\pi$
$258$	1.57694	0.0981761
$259$	23.3296	1.44963
$260$	6.98655	0.433288
$261$	5.73064	0.354718
$262$	−18.2116	−1.12511
$263$	2.03837	0.125691	0.0628456	−	0.998023i	$-0.479982\pi$
$263$	2.03837	0.125691	0.0628456	+	0.998023i	$0.479982\pi$
$264$	−1.96359	−0.120851
$265$	−2.51610	−0.154563
$266$	4.25110	0.260652
$267$	−10.6809	−0.653660
$268$	7.65377	0.467529
$269$	−24.6140	−1.50074	−0.750370	−	0.661018i	$-0.770125\pi$
$269$	−24.6140	−1.50074	−0.750370	+	0.661018i	$0.770125\pi$
$270$	2.51610	0.153125
$271$	−26.9494	−1.63706	−0.818531	−	0.574462i	$-0.805211\pi$
$271$	−26.9494	−1.63706	−0.818531	+	0.574462i	$0.805211\pi$
$272$	6.83822	0.414628
$273$	−11.8042	−0.714423
$274$	−18.2145	−1.10038
$275$	2.61308	0.157574
$276$	−6.37856	−0.383944
$277$	−22.7108	−1.36456	−0.682281	−	0.731090i	$-0.739012\pi$
$277$	−22.7108	−1.36456	−0.682281	+	0.731090i	$0.739012\pi$
$278$	7.07490	0.424324
$279$	−0.117223	−0.00701796
$280$	10.6962	0.639221
$281$	3.42468	0.204299	0.102150	−	0.994769i	$-0.467428\pi$
$281$	3.42468	0.204299	0.102150	+	0.994769i	$0.467428\pi$
$282$	−0.666390	−0.0396830
$283$	−9.90360	−0.588708	−0.294354	−	0.955696i	$-0.595104\pi$
$283$	−9.90360	−0.588708	−0.294354	+	0.955696i	$0.595104\pi$
$284$	9.28244	0.550811
$285$	−2.51610	−0.149041
$286$	−5.45239	−0.322407
$287$	42.2058	2.49133
$288$	1.00000	0.0589256
$289$	29.7612	1.75066
$290$	−14.4189	−0.846705
$291$	8.44389	0.494990
$292$	−0.563587	−0.0329814
$293$	3.52895	0.206163	0.103082	−	0.994673i	$-0.467130\pi$
$293$	3.52895	0.206163	0.103082	+	0.994673i	$0.467130\pi$
$294$	−11.0719	−0.645726
$295$	−7.89544	−0.459690
$296$	−5.48789	−0.318977
$297$	−1.96359	−0.113939
$298$	−16.8298	−0.974924
$299$	−17.7116	−1.02429
$300$	−1.33076	−0.0768316
$301$	6.70374	0.386397
$302$	9.22370	0.530764
$303$	−4.79119	−0.275247
$304$	−1.00000	−0.0573539
$305$	1.04645	0.0599193
$306$	6.83822	0.390915
$307$	28.6666	1.63609	0.818046	−	0.575153i	$-0.195058\pi$
$307$	28.6666	1.63609	0.818046	+	0.575153i	$0.195058\pi$
$308$	−8.34745	−0.475640
$309$	9.79253	0.557078
$310$	0.294945	0.0167517
$311$	−0.461697	−0.0261804	−0.0130902	−	0.999914i	$-0.504167\pi$
$311$	−0.461697	−0.0261804	−0.0130902	+	0.999914i	$0.504167\pi$
$312$	2.77674	0.157202
$313$	−22.6577	−1.28069	−0.640343	−	0.768089i	$-0.721208\pi$
$313$	−22.6577	−1.28069	−0.640343	+	0.768089i	$0.721208\pi$
$314$	−18.0398	−1.01804
$315$	10.6962	0.602663
$316$	−16.0290	−0.901700
$317$	23.2735	1.30717	0.653586	−	0.756852i	$-0.273264\pi$
$317$	23.2735	1.30717	0.653586	+	0.756852i	$0.273264\pi$
$318$	−1.00000	−0.0560772
$319$	11.2527	0.630028
$320$	−2.51610	−0.140654
$321$	15.2769	0.852674
$322$	−27.1159	−1.51111
$323$	−6.83822	−0.380489
$324$	1.00000	0.0555556
$325$	−3.69518	−0.204972
$326$	−0.0245472	−0.00135954
$327$	−7.91690	−0.437806
$328$	−9.92819	−0.548193
$329$	−2.83290	−0.156183
$330$	4.94060	0.271971
$331$	5.39248	0.296397	0.148199	−	0.988958i	$-0.452652\pi$
$331$	5.39248	0.296397	0.148199	+	0.988958i	$0.452652\pi$
$332$	0.307939	0.0169003
$333$	−5.48789	−0.300735
$334$	14.1169	0.772440
$335$	−19.2577	−1.05216
$336$	4.25110	0.231917
$337$	5.36502	0.292251	0.146126	−	0.989266i	$-0.453320\pi$
$337$	5.36502	0.292251	0.146126	+	0.989266i	$0.453320\pi$
$338$	−5.28972	−0.287723
$339$	−2.86186	−0.155435
$340$	−17.2056	−0.933107
$341$	−0.230179	−0.0124649
$342$	−1.00000	−0.0540738
$343$	−17.3100	−0.934654
$344$	−1.57694	−0.0850230
$345$	16.0491	0.864055
$346$	−22.0740	−1.18671
$347$	11.7607	0.631349	0.315675	−	0.948867i	$-0.397769\pi$
$347$	11.7607	0.631349	0.315675	+	0.948867i	$0.397769\pi$
$348$	−5.73064	−0.307195
$349$	2.82073	0.150990	0.0754950	−	0.997146i	$-0.475946\pi$
$349$	2.82073	0.150990	0.0754950	+	0.997146i	$0.475946\pi$
$350$	−5.65721	−0.302390
$351$	2.77674	0.148211
$352$	1.96359	0.104660
$353$	−19.2115	−1.02253	−0.511264	−	0.859424i	$-0.670822\pi$
$353$	−19.2115	−1.02253	−0.511264	+	0.859424i	$0.670822\pi$
$354$	−3.13796	−0.166781
$355$	−23.3555	−1.23958
$356$	10.6809	0.566086
$357$	29.0700	1.53855
$358$	18.7893	0.993045
$359$	−16.7903	−0.886156	−0.443078	−	0.896483i	$-0.646114\pi$
$359$	−16.7903	−0.886156	−0.443078	+	0.896483i	$0.646114\pi$
$360$	−2.51610	−0.132610
$361$	1.00000	0.0526316
$362$	−11.4778	−0.603258
$363$	7.14429	0.374978
$364$	11.8042	0.618709
$365$	1.41804	0.0742236
$366$	0.415900	0.0217394
$367$	5.76787	0.301080	0.150540	−	0.988604i	$-0.451899\pi$
$367$	5.76787	0.301080	0.150540	+	0.988604i	$0.451899\pi$
$368$	6.37856	0.332506
$369$	−9.92819	−0.516841
$370$	13.8081	0.717848
$371$	−4.25110	−0.220706
$372$	0.117223	0.00607773
$373$	32.1422	1.66426	0.832130	−	0.554581i	$-0.187122\pi$
$373$	32.1422	1.66426	0.832130	+	0.554581i	$0.187122\pi$
$374$	13.4275	0.694319
$375$	−9.23217	−0.476747
$376$	0.666390	0.0343665
$377$	−15.9125	−0.819535
$378$	4.25110	0.218653
$379$	27.4613	1.41059	0.705296	−	0.708913i	$-0.250814\pi$
$379$	27.4613	1.41059	0.705296	+	0.708913i	$0.250814\pi$
$380$	2.51610	0.129073
$381$	18.0636	0.925428
$382$	−1.50328	−0.0769146
$383$	30.2728	1.54687	0.773435	−	0.633876i	$-0.218537\pi$
$383$	30.2728	1.54687	0.773435	+	0.633876i	$0.218537\pi$
$384$	−1.00000	−0.0510310
$385$	21.0030	1.07041
$386$	10.5502	0.536989
$387$	−1.57694	−0.0801604
$388$	−8.44389	−0.428674
$389$	−17.8415	−0.904597	−0.452299	−	0.891867i	$-0.649396\pi$
$389$	−17.8415	−0.904597	−0.452299	+	0.891867i	$0.649396\pi$
$390$	−6.98655	−0.353778
$391$	43.6180	2.20586
$392$	11.0719	0.559215
$393$	18.2116	0.918652
$394$	−5.16113	−0.260014
$395$	40.3305	2.02925
$396$	1.96359	0.0986744
$397$	13.9044	0.697841	0.348920	−	0.937152i	$-0.386548\pi$
$397$	13.9044	0.697841	0.348920	+	0.937152i	$0.386548\pi$
$398$	−21.8391	−1.09470
$399$	−4.25110	−0.212821
$400$	1.33076	0.0665381
$401$	−33.3661	−1.66622	−0.833112	−	0.553104i	$-0.813443\pi$
$401$	−33.3661	−1.66622	−0.833112	+	0.553104i	$0.813443\pi$
$402$	−7.65377	−0.381736
$403$	0.325498	0.0162142
$404$	4.79119	0.238371
$405$	−2.51610	−0.125026
$406$	−24.3615	−1.20904
$407$	−10.7760	−0.534146
$408$	−6.83822	−0.338542
$409$	18.2552	0.902660	0.451330	−	0.892357i	$-0.350950\pi$
$409$	18.2552	0.902660	0.451330	+	0.892357i	$0.350950\pi$
$410$	24.9803	1.23369
$411$	18.2145	0.898454
$412$	−9.79253	−0.482443
$413$	−13.3398	−0.656409
$414$	6.37856	0.313489
$415$	−0.774805	−0.0380337
$416$	−2.77674	−0.136141
$417$	−7.07490	−0.346459
$418$	−1.96359	−0.0960426
$419$	−20.4441	−0.998761	−0.499381	−	0.866383i	$-0.666439\pi$
$419$	−20.4441	−0.998761	−0.499381	+	0.866383i	$0.666439\pi$
$420$	−10.6962	−0.521921
$421$	−0.374365	−0.0182454	−0.00912272	−	0.999958i	$-0.502904\pi$
$421$	−0.374365	−0.0182454	−0.00912272	+	0.999958i	$0.502904\pi$
$422$	18.8235	0.916314
$423$	0.666390	0.0324010
$424$	1.00000	0.0485643
$425$	9.10004	0.441417
$426$	−9.28244	−0.449736
$427$	1.76803	0.0855611
$428$	−15.2769	−0.738438
$429$	5.45239	0.263244
$430$	3.96774	0.191342
$431$	−19.6831	−0.948099	−0.474050	−	0.880498i	$-0.657208\pi$
$431$	−19.6831	−0.948099	−0.474050	+	0.880498i	$0.657208\pi$
$432$	−1.00000	−0.0481125
$433$	−16.9190	−0.813073	−0.406537	−	0.913634i	$-0.633264\pi$
$433$	−16.9190	−0.813073	−0.406537	+	0.913634i	$0.633264\pi$
$434$	0.498327	0.0239205
$435$	14.4189	0.691332
$436$	7.91690	0.379151
$437$	−6.37856	−0.305128
$438$	0.563587	0.0269292
$439$	−22.5619	−1.07682	−0.538411	−	0.842682i	$-0.680975\pi$
$439$	−22.5619	−1.07682	−0.538411	+	0.842682i	$0.680975\pi$
$440$	−4.94060	−0.235534
$441$	11.0719	0.527233
$442$	−18.9879	−0.903164
$443$	−13.7676	−0.654120	−0.327060	−	0.945004i	$-0.606058\pi$
$443$	−13.7676	−0.654120	−0.327060	+	0.945004i	$0.606058\pi$
$444$	5.48789	0.260444
$445$	−26.8742	−1.27396
$446$	−22.7495	−1.07722
$447$	16.8298	0.796022
$448$	−4.25110	−0.200846
$449$	−6.70165	−0.316270	−0.158135	−	0.987417i	$-0.550548\pi$
$449$	−6.70165	−0.316270	−0.158135	+	0.987417i	$0.550548\pi$
$450$	1.33076	0.0627327
$451$	−19.4949	−0.917981
$452$	2.86186	0.134611
$453$	−9.22370	−0.433367
$454$	7.21237	0.338493
$455$	−29.7006	−1.39238
$456$	1.00000	0.0468293
$457$	23.6382	1.10575	0.552873	−	0.833265i	$-0.313531\pi$
$457$	23.6382	1.10575	0.552873	+	0.833265i	$0.313531\pi$
$458$	3.30012	0.154205
$459$	−6.83822	−0.319181
$460$	−16.0491	−0.748293
$461$	−30.6242	−1.42631	−0.713155	−	0.701007i	$-0.752734\pi$
$461$	−30.6242	−1.42631	−0.713155	+	0.701007i	$0.752734\pi$
$462$	8.34745	0.388358
$463$	12.0260	0.558898	0.279449	−	0.960161i	$-0.409848\pi$
$463$	12.0260	0.558898	0.279449	+	0.960161i	$0.409848\pi$
$464$	5.73064	0.266038
$465$	−0.294945	−0.0136777
$466$	−7.84552	−0.363437
$467$	−28.5051	−1.31906	−0.659528	−	0.751680i	$-0.729244\pi$
$467$	−28.5051	−1.31906	−0.659528	+	0.751680i	$0.729244\pi$
$468$	−2.77674	−0.128355
$469$	−32.5370	−1.50242
$470$	−1.67671	−0.0773406
$471$	18.0398	0.831228
$472$	3.13796	0.144437
$473$	−3.09647	−0.142376
$474$	16.0290	0.736235
$475$	−1.33076	−0.0610595
$476$	−29.0700	−1.33242
$477$	1.00000	0.0457869
$478$	1.41077	0.0645270
$479$	−15.9510	−0.728818	−0.364409	−	0.931239i	$-0.618729\pi$
$479$	−15.9510	−0.728818	−0.364409	+	0.931239i	$0.618729\pi$
$480$	2.51610	0.114844
$481$	15.2384	0.694813
$482$	3.80810	0.173454
$483$	27.1159	1.23382
$484$	−7.14429	−0.324741
$485$	21.2457	0.964717
$486$	−1.00000	−0.0453609
$487$	18.8436	0.853884	0.426942	−	0.904279i	$-0.359591\pi$
$487$	18.8436	0.853884	0.426942	+	0.904279i	$0.359591\pi$
$488$	−0.415900	−0.0188269
$489$	0.0245472	0.00111006
$490$	−27.8580	−1.25850
$491$	−12.6466	−0.570732	−0.285366	−	0.958419i	$-0.592115\pi$
$491$	−12.6466	−0.570732	−0.285366	+	0.958419i	$0.592115\pi$
$492$	9.92819	0.447597
$493$	39.1873	1.76491
$494$	2.77674	0.124931
$495$	−4.94060	−0.222064
$496$	−0.117223	−0.00526347
$497$	−39.4606	−1.77005
$498$	−0.307939	−0.0137991
$499$	22.4656	1.00570	0.502850	−	0.864374i	$-0.332285\pi$
$499$	22.4656	1.00570	0.502850	+	0.864374i	$0.332285\pi$
$500$	9.23217	0.412875
$501$	−14.1169	−0.630694
$502$	28.9995	1.29431
$503$	−25.3832	−1.13178	−0.565891	−	0.824480i	$-0.691468\pi$
$503$	−25.3832	−1.13178	−0.565891	+	0.824480i	$0.691468\pi$
$504$	−4.25110	−0.189359
$505$	−12.0551	−0.536445
$506$	12.5249	0.556800
$507$	5.28972	0.234925
$508$	−18.0636	−0.801444
$509$	22.7504	1.00839	0.504197	−	0.863588i	$-0.331788\pi$
$509$	22.7504	1.00839	0.504197	+	0.863588i	$0.331788\pi$
$510$	17.2056	0.761879
$511$	2.39587	0.105987
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	−22.4357	−0.989598
$515$	24.6390	1.08572
$516$	1.57694	0.0694210
$517$	1.30852	0.0575487
$518$	23.3296	1.02504
$519$	22.0740	0.968943
$520$	6.98655	0.306381
$521$	29.1879	1.27874	0.639372	−	0.768898i	$-0.279194\pi$
$521$	29.1879	1.27874	0.639372	+	0.768898i	$0.279194\pi$
$522$	5.73064	0.250823
$523$	−25.5569	−1.11752	−0.558762	−	0.829328i	$-0.688724\pi$
$523$	−25.5569	−1.11752	−0.558762	+	0.829328i	$0.688724\pi$
$524$	−18.2116	−0.795576
$525$	5.65721	0.246901
$526$	2.03837	0.0888771
$527$	−0.801597	−0.0349181
$528$	−1.96359	−0.0854545
$529$	17.6861	0.768959
$530$	−2.51610	−0.109292
$531$	3.13796	0.136176
$532$	4.25110	0.184309
$533$	27.5680	1.19410
$534$	−10.6809	−0.462207
$535$	38.4383	1.66183
$536$	7.65377	0.330593
$537$	−18.7893	−0.810818
$538$	−24.6140	−1.06118
$539$	21.7407	0.936439
$540$	2.51610	0.108276
$541$	13.0945	0.562978	0.281489	−	0.959565i	$-0.409172\pi$
$541$	13.0945	0.562978	0.281489	+	0.959565i	$0.409172\pi$
$542$	−26.9494	−1.15758
$543$	11.4778	0.492558
$544$	6.83822	0.293186
$545$	−19.9197	−0.853267
$546$	−11.8042	−0.505173
$547$	−36.5701	−1.56362	−0.781812	−	0.623515i	$-0.785704\pi$
$547$	−36.5701	−1.56362	−0.781812	+	0.623515i	$0.785704\pi$
$548$	−18.2145	−0.778084
$549$	−0.415900	−0.0177502
$550$	2.61308	0.111422
$551$	−5.73064	−0.244133
$552$	−6.37856	−0.271490
$553$	68.1408	2.89764
$554$	−22.7108	−0.964891
$555$	−13.8081	−0.586121
$556$	7.07490	0.300042
$557$	29.6306	1.25549	0.627744	−	0.778420i	$-0.283978\pi$
$557$	29.6306	1.25549	0.627744	+	0.778420i	$0.283978\pi$
$558$	−0.117223	−0.00496245
$559$	4.37876	0.185202
$560$	10.6962	0.451997
$561$	−13.4275	−0.566909
$562$	3.42468	0.144461
$563$	3.50188	0.147586	0.0737932	−	0.997274i	$-0.476490\pi$
$563$	3.50188	0.147586	0.0737932	+	0.997274i	$0.476490\pi$
$564$	−0.666390	−0.0280601
$565$	−7.20072	−0.302937
$566$	−9.90360	−0.416279
$567$	−4.25110	−0.178530
$568$	9.28244	0.389482
$569$	9.14241	0.383270	0.191635	−	0.981466i	$-0.438621\pi$
$569$	9.14241	0.383270	0.191635	+	0.981466i	$0.438621\pi$
$570$	−2.51610	−0.105388
$571$	27.8557	1.16572	0.582862	−	0.812571i	$-0.301933\pi$
$571$	27.8557	1.16572	0.582862	+	0.812571i	$0.301933\pi$
$572$	−5.45239	−0.227976
$573$	1.50328	0.0628005
$574$	42.2058	1.76164
$575$	8.48835	0.353989
$576$	1.00000	0.0416667
$577$	−8.06279	−0.335658	−0.167829	−	0.985816i	$-0.553676\pi$
$577$	−8.06279	−0.335658	−0.167829	+	0.985816i	$0.553676\pi$
$578$	29.7612	1.23790
$579$	−10.5502	−0.438450
$580$	−14.4189	−0.598711
$581$	−1.30908	−0.0543098
$582$	8.44389	0.350011
$583$	1.96359	0.0813238
$584$	−0.563587	−0.0233214
$585$	6.98655	0.288858
$586$	3.52895	0.145779
$587$	15.4235	0.636596	0.318298	−	0.947991i	$-0.396889\pi$
$587$	15.4235	0.636596	0.318298	+	0.947991i	$0.396889\pi$
$588$	−11.0719	−0.456597
$589$	0.117223	0.00483009
$590$	−7.89544	−0.325050
$591$	5.16113	0.212301
$592$	−5.48789	−0.225551
$593$	0.610048	0.0250517	0.0125258	−	0.999922i	$-0.496013\pi$
$593$	0.610048	0.0250517	0.0125258	+	0.999922i	$0.496013\pi$
$594$	−1.96359	−0.0805673
$595$	73.1430	2.99857
$596$	−16.8298	−0.689375
$597$	21.8391	0.893816
$598$	−17.7116	−0.724281
$599$	−0.916684	−0.0374547	−0.0187273	−	0.999825i	$-0.505961\pi$
$599$	−0.916684	−0.0374547	−0.0187273	+	0.999825i	$0.505961\pi$
$600$	−1.33076	−0.0543281
$601$	3.30078	0.134642	0.0673208	−	0.997731i	$-0.478555\pi$
$601$	3.30078	0.134642	0.0673208	+	0.997731i	$0.478555\pi$
$602$	6.70374	0.273224
$603$	7.65377	0.311686
$604$	9.22370	0.375307
$605$	17.9758	0.730819
$606$	−4.79119	−0.194629
$607$	−37.2085	−1.51025	−0.755123	−	0.655583i	$-0.772423\pi$
$607$	−37.2085	−1.51025	−0.755123	+	0.655583i	$0.772423\pi$
$608$	−1.00000	−0.0405554
$609$	24.3615	0.987180
$610$	1.04645	0.0423693
$611$	−1.85039	−0.0748589
$612$	6.83822	0.276419
$613$	−37.8515	−1.52881	−0.764403	−	0.644738i	$-0.776966\pi$
$613$	−37.8515	−1.52881	−0.764403	+	0.644738i	$0.776966\pi$
$614$	28.6666	1.15689
$615$	−24.9803	−1.00730
$616$	−8.34745	−0.336328
$617$	−1.91601	−0.0771357	−0.0385678	−	0.999256i	$-0.512280\pi$
$617$	−1.91601	−0.0771357	−0.0385678	+	0.999256i	$0.512280\pi$
$618$	9.79253	0.393913
$619$	−20.0309	−0.805109	−0.402554	−	0.915396i	$-0.631878\pi$
$619$	−20.0309	−0.805109	−0.402554	+	0.915396i	$0.631878\pi$
$620$	0.294945	0.0118453
$621$	−6.37856	−0.255963
$622$	−0.461697	−0.0185124
$623$	−45.4056	−1.81914
$624$	2.77674	0.111159
$625$	−29.8829	−1.19532
$626$	−22.6577	−0.905582
$627$	1.96359	0.0784184
$628$	−18.0398	−0.719865
$629$	−37.5274	−1.49631
$630$	10.6962	0.426147
$631$	18.9725	0.755283	0.377642	−	0.925952i	$-0.376735\pi$
$631$	18.9725	0.755283	0.377642	+	0.925952i	$0.376735\pi$
$632$	−16.0290	−0.637598
$633$	−18.8235	−0.748167
$634$	23.2735	0.924310
$635$	45.4499	1.80363
$636$	−1.00000	−0.0396526
$637$	−30.7438	−1.21811
$638$	11.2527	0.445497
$639$	9.28244	0.367208
$640$	−2.51610	−0.0994576
$641$	6.94096	0.274152	0.137076	−	0.990561i	$-0.456230\pi$
$641$	6.94096	0.274152	0.137076	+	0.990561i	$0.456230\pi$
$642$	15.2769	0.602932
$643$	6.00715	0.236899	0.118449	−	0.992960i	$-0.462208\pi$
$643$	6.00715	0.236899	0.118449	+	0.992960i	$0.462208\pi$
$644$	−27.1159	−1.06852
$645$	−3.96774	−0.156230
$646$	−6.83822	−0.269046
$647$	23.6701	0.930569	0.465285	−	0.885161i	$-0.345952\pi$
$647$	23.6701	0.930569	0.465285	+	0.885161i	$0.345952\pi$
$648$	1.00000	0.0392837
$649$	6.16169	0.241868
$650$	−3.69518	−0.144937
$651$	−0.498327	−0.0195310
$652$	−0.0245472	−0.000961343 0
$653$	43.0932	1.68637	0.843183	−	0.537627i	$-0.180679\pi$
$653$	43.0932	1.68637	0.843183	+	0.537627i	$0.180679\pi$
$654$	−7.91690	−0.309575
$655$	45.8221	1.79042
$656$	−9.92819	−0.387631
$657$	−0.563587	−0.0219876
$658$	−2.83290	−0.110438
$659$	43.4810	1.69378	0.846890	−	0.531768i	$-0.178472\pi$
$659$	43.4810	1.69378	0.846890	+	0.531768i	$0.178472\pi$
$660$	4.94060	0.192313
$661$	−24.1652	−0.939919	−0.469959	−	0.882688i	$-0.655731\pi$
$661$	−24.1652	−0.939919	−0.469959	+	0.882688i	$0.655731\pi$
$662$	5.39248	0.209585
$663$	18.9879	0.737431
$664$	0.307939	0.0119503
$665$	−10.6962	−0.414781
$666$	−5.48789	−0.212651
$667$	36.5532	1.41535
$668$	14.1169	0.546197
$669$	22.7495	0.879548
$670$	−19.2577	−0.743988
$671$	−0.816658	−0.0315267
$672$	4.25110	0.163990
$673$	−6.57056	−0.253276	−0.126638	−	0.991949i	$-0.540419\pi$
$673$	−6.57056	−0.253276	−0.126638	+	0.991949i	$0.540419\pi$
$674$	5.36502	0.206653
$675$	−1.33076	−0.0512210
$676$	−5.28972	−0.203451
$677$	−47.1841	−1.81343	−0.906717	−	0.421740i	$-0.861420\pi$
$677$	−47.1841	−1.81343	−0.906717	+	0.421740i	$0.861420\pi$
$678$	−2.86186	−0.109909
$679$	35.8959	1.37756
$680$	−17.2056	−0.659806
$681$	−7.21237	−0.276378
$682$	−0.230179	−0.00881399
$683$	−4.37341	−0.167344	−0.0836719	−	0.996493i	$-0.526665\pi$
$683$	−4.37341	−0.167344	−0.0836719	+	0.996493i	$0.526665\pi$
$684$	−1.00000	−0.0382360
$685$	45.8295	1.75105
$686$	−17.3100	−0.660900
$687$	−3.30012	−0.125908
$688$	−1.57694	−0.0601203
$689$	−2.77674	−0.105785
$690$	16.0491	0.610979
$691$	−8.82808	−0.335836	−0.167918	−	0.985801i	$-0.553704\pi$
$691$	−8.82808	−0.335836	−0.167918	+	0.985801i	$0.553704\pi$
$692$	−22.0740	−0.839129
$693$	−8.34745	−0.317093
$694$	11.7607	0.446431
$695$	−17.8012	−0.675236
$696$	−5.73064	−0.217219
$697$	−67.8911	−2.57156
$698$	2.82073	0.106766
$699$	7.84552	0.296745
$700$	−5.65721	−0.213822
$701$	33.3377	1.25915	0.629574	−	0.776941i	$-0.283230\pi$
$701$	33.3377	1.25915	0.629574	+	0.776941i	$0.283230\pi$
$702$	2.77674	0.104801
$703$	5.48789	0.206980
$704$	1.96359	0.0740058
$705$	1.67671	0.0631484
$706$	−19.2115	−0.723036
$707$	−20.3678	−0.766012
$708$	−3.13796	−0.117932
$709$	22.1272	0.831004	0.415502	−	0.909592i	$-0.363606\pi$
$709$	22.1272	0.831004	0.415502	+	0.909592i	$0.363606\pi$
$710$	−23.3555	−0.876518
$711$	−16.0290	−0.601133
$712$	10.6809	0.400283
$713$	−0.747715	−0.0280021
$714$	29.0700	1.08792
$715$	13.7188	0.513053
$716$	18.7893	0.702189
$717$	−1.41077	−0.0526861
$718$	−16.7903	−0.626607
$719$	38.6599	1.44177	0.720886	−	0.693054i	$-0.243735\pi$
$719$	38.6599	1.44177	0.720886	+	0.693054i	$0.243735\pi$
$720$	−2.51610	−0.0937695
$721$	41.6291	1.55035
$722$	1.00000	0.0372161
$723$	−3.80810	−0.141625
$724$	−11.4778	−0.426568
$725$	7.62612	0.283227
$726$	7.14429	0.265150
$727$	24.8890	0.923083	0.461542	−	0.887119i	$-0.347296\pi$
$727$	24.8890	0.923083	0.461542	+	0.887119i	$0.347296\pi$
$728$	11.8042	0.437493
$729$	1.00000	0.0370370
$730$	1.41804	0.0524840
$731$	−10.7835	−0.398841
$732$	0.415900	0.0153721
$733$	−25.7848	−0.952383	−0.476192	−	0.879342i	$-0.657983\pi$
$733$	−25.7848	−0.952383	−0.476192	+	0.879342i	$0.657983\pi$
$734$	5.76787	0.212896
$735$	27.8580	1.02756
$736$	6.37856	0.235117
$737$	15.0289	0.553597
$738$	−9.92819	−0.365462
$739$	15.8842	0.584311	0.292155	−	0.956371i	$-0.405628\pi$
$739$	15.8842	0.584311	0.292155	+	0.956371i	$0.405628\pi$
$740$	13.8081	0.507595
$741$	−2.77674	−0.102006
$742$	−4.25110	−0.156063
$743$	15.0640	0.552644	0.276322	−	0.961065i	$-0.410884\pi$
$743$	15.0640	0.552644	0.276322	+	0.961065i	$0.410884\pi$
$744$	0.117223	0.00429761
$745$	42.3454	1.55142
$746$	32.1422	1.17681
$747$	0.307939	0.0112669
$748$	13.4275	0.490958
$749$	64.9438	2.37299
$750$	−9.23217	−0.337111
$751$	−26.4659	−0.965756	−0.482878	−	0.875688i	$-0.660409\pi$
$751$	−26.4659	−0.965756	−0.482878	+	0.875688i	$0.660409\pi$
$752$	0.666390	0.0243008
$753$	−28.9995	−1.05680
$754$	−15.9125	−0.579499
$755$	−23.2078	−0.844617
$756$	4.25110	0.154611
$757$	1.72136	0.0625637	0.0312819	−	0.999511i	$-0.490041\pi$
$757$	1.72136	0.0625637	0.0312819	+	0.999511i	$0.490041\pi$
$758$	27.4613	0.997439
$759$	−12.5249	−0.454626
$760$	2.51610	0.0912686
$761$	−36.3698	−1.31840	−0.659201	−	0.751967i	$-0.729105\pi$
$761$	−36.3698	−1.31840	−0.659201	+	0.751967i	$0.729105\pi$
$762$	18.0636	0.654377
$763$	−33.6556	−1.21841
$764$	−1.50328	−0.0543869
$765$	−17.2056	−0.622071
$766$	30.2728	1.09380
$767$	−8.71331	−0.314619
$768$	−1.00000	−0.0360844
$769$	−1.13835	−0.0410498	−0.0205249	−	0.999789i	$-0.506534\pi$
$769$	−1.13835	−0.0410498	−0.0205249	+	0.999789i	$0.506534\pi$
$770$	21.0030	0.756896
$771$	22.4357	0.808003
$772$	10.5502	0.379709
$773$	19.4447	0.699379	0.349689	−	0.936866i	$-0.386287\pi$
$773$	19.4447	0.699379	0.349689	+	0.936866i	$0.386287\pi$
$774$	−1.57694	−0.0566820
$775$	−0.155996	−0.00560354
$776$	−8.44389	−0.303118
$777$	−23.3296	−0.836945
$778$	−17.8415	−0.639647
$779$	9.92819	0.355714
$780$	−6.98655	−0.250159
$781$	18.2269	0.652212
$782$	43.6180	1.55978
$783$	−5.73064	−0.204796
$784$	11.0719	0.395425
$785$	45.3898	1.62003
$786$	18.2116	0.649585
$787$	−42.4909	−1.51464	−0.757318	−	0.653046i	$-0.773491\pi$
$787$	−42.4909	−1.51464	−0.757318	+	0.653046i	$0.773491\pi$
$788$	−5.16113	−0.183858
$789$	−2.03837	−0.0725679
$790$	40.3305	1.43489
$791$	−12.1661	−0.432575
$792$	1.96359	0.0697733
$793$	1.15484	0.0410097
$794$	13.9044	0.493448
$795$	2.51610	0.0892369
$796$	−21.8391	−0.774068
$797$	−35.5146	−1.25799	−0.628995	−	0.777409i	$-0.716534\pi$
$797$	−35.5146	−1.25799	−0.628995	+	0.777409i	$0.716534\pi$
$798$	−4.25110	−0.150487
$799$	4.55692	0.161212
$800$	1.33076	0.0470495
$801$	10.6809	0.377391
$802$	−33.3661	−1.17820
$803$	−1.10666	−0.0390530
$804$	−7.65377	−0.269928
$805$	68.2264	2.40467
$806$	0.325498	0.0114652
$807$	24.6140	0.866452
$808$	4.79119	0.168553
$809$	−40.8919	−1.43768	−0.718841	−	0.695174i	$-0.755327\pi$
$809$	−40.8919	−1.43768	−0.718841	+	0.695174i	$0.755327\pi$
$810$	−2.51610	−0.0884068
$811$	15.7015	0.551355	0.275677	−	0.961250i	$-0.411098\pi$
$811$	15.7015	0.551355	0.275677	+	0.961250i	$0.411098\pi$
$812$	−24.3615	−0.854923
$813$	26.9494	0.945158
$814$	−10.7760	−0.377698
$815$	0.0617632	0.00216347
$816$	−6.83822	−0.239385
$817$	1.57694	0.0551702
$818$	18.2552	0.638277
$819$	11.8042	0.412472
$820$	24.9803	0.872351
$821$	−25.1876	−0.879054	−0.439527	−	0.898229i	$-0.644854\pi$
$821$	−25.1876	−0.879054	−0.439527	+	0.898229i	$0.644854\pi$
$822$	18.2145	0.635303
$823$	−27.0473	−0.942808	−0.471404	−	0.881917i	$-0.656253\pi$
$823$	−27.0473	−0.942808	−0.471404	+	0.881917i	$0.656253\pi$
$824$	−9.79253	−0.341139
$825$	−2.61308	−0.0909757
$826$	−13.3398	−0.464152
$827$	29.1801	1.01469	0.507346	−	0.861742i	$-0.330627\pi$
$827$	29.1801	1.01469	0.507346	+	0.861742i	$0.330627\pi$
$828$	6.37856	0.221670
$829$	−15.8490	−0.550458	−0.275229	−	0.961379i	$-0.588754\pi$
$829$	−15.8490	−0.550458	−0.275229	+	0.961379i	$0.588754\pi$
$830$	−0.774805	−0.0268939
$831$	22.7108	0.787830
$832$	−2.77674	−0.0962661
$833$	75.7120	2.62326
$834$	−7.07490	−0.244984
$835$	−35.5194	−1.22920
$836$	−1.96359	−0.0679124
$837$	0.117223	0.00405182
$838$	−20.4441	−0.706231
$839$	6.86851	0.237127	0.118564	−	0.992946i	$-0.462171\pi$
$839$	6.86851	0.237127	0.118564	+	0.992946i	$0.462171\pi$
$840$	−10.6962	−0.369054
$841$	3.84023	0.132422
$842$	−0.374365	−0.0129015
$843$	−3.42468	−0.117952
$844$	18.8235	0.647932
$845$	13.3095	0.457860
$846$	0.666390	0.0229110
$847$	30.3711	1.04356
$848$	1.00000	0.0343401
$849$	9.90360	0.339891
$850$	9.10004	0.312129
$851$	−35.0049	−1.19995
$852$	−9.28244	−0.318011
$853$	−22.6991	−0.777203	−0.388602	−	0.921406i	$-0.627042\pi$
$853$	−22.6991	−0.777203	−0.388602	+	0.921406i	$0.627042\pi$
$854$	1.76803	0.0605008
$855$	2.51610	0.0860488
$856$	−15.2769	−0.522154
$857$	16.4825	0.563033	0.281516	−	0.959556i	$-0.409163\pi$
$857$	16.4825	0.563033	0.281516	+	0.959556i	$0.409163\pi$
$858$	5.45239	0.186142
$859$	−32.1978	−1.09858	−0.549288	−	0.835633i	$-0.685101\pi$
$859$	−32.1978	−1.09858	−0.549288	+	0.835633i	$0.685101\pi$
$860$	3.96774	0.135299
$861$	−42.2058	−1.43837
$862$	−19.6831	−0.670407
$863$	−10.3252	−0.351473	−0.175737	−	0.984437i	$-0.556231\pi$
$863$	−10.3252	−0.351473	−0.175737	+	0.984437i	$0.556231\pi$
$864$	−1.00000	−0.0340207
$865$	55.5405	1.88843
$866$	−16.9190	−0.574930
$867$	−29.7612	−1.01074
$868$	0.498327	0.0169143
$869$	−31.4744	−1.06770
$870$	14.4189	0.488845
$871$	−21.2525	−0.720115
$872$	7.91690	0.268100
$873$	−8.44389	−0.285783
$874$	−6.37856	−0.215758
$875$	−39.2469	−1.32679
$876$	0.563587	0.0190418
$877$	−55.9240	−1.88842	−0.944210	−	0.329345i	$-0.893172\pi$
$877$	−55.9240	−1.88842	−0.944210	+	0.329345i	$0.893172\pi$
$878$	−22.5619	−0.761429
$879$	−3.52895	−0.119028
$880$	−4.94060	−0.166548
$881$	−40.3447	−1.35925	−0.679624	−	0.733561i	$-0.737857\pi$
$881$	−40.3447	−1.35925	−0.679624	+	0.733561i	$0.737857\pi$
$882$	11.0719	0.372810
$883$	18.6506	0.627643	0.313821	−	0.949482i	$-0.398391\pi$
$883$	18.6506	0.627643	0.313821	+	0.949482i	$0.398391\pi$
$884$	−18.9879	−0.638634
$885$	7.89544	0.265402
$886$	−13.7676	−0.462533
$887$	1.52230	0.0511140	0.0255570	−	0.999673i	$-0.491864\pi$
$887$	1.52230	0.0511140	0.0255570	+	0.999673i	$0.491864\pi$
$888$	5.48789	0.184162
$889$	76.7904	2.57547
$890$	−26.8742	−0.900825
$891$	1.96359	0.0657829
$892$	−22.7495	−0.761711
$893$	−0.666390	−0.0222999
$894$	16.8298	0.562872
$895$	−47.2757	−1.58025
$896$	−4.25110	−0.142019
$897$	17.7116	0.591373
$898$	−6.70165	−0.223637
$899$	−0.671763	−0.0224046
$900$	1.33076	0.0443587
$901$	6.83822	0.227814
$902$	−19.4949	−0.649111
$903$	−6.70374	−0.223087
$904$	2.86186	0.0951840
$905$	28.8792	0.959978
$906$	−9.22370	−0.306437
$907$	−9.52492	−0.316270	−0.158135	−	0.987418i	$-0.550548\pi$
$907$	−9.52492	−0.316270	−0.158135	+	0.987418i	$0.550548\pi$
$908$	7.21237	0.239351
$909$	4.79119	0.158914
$910$	−29.7006	−0.984564
$911$	−29.2804	−0.970104	−0.485052	−	0.874485i	$-0.661199\pi$
$911$	−29.2804	−0.970104	−0.485052	+	0.874485i	$0.661199\pi$
$912$	1.00000	0.0331133
$913$	0.604667	0.0200116
$914$	23.6382	0.781881
$915$	−1.04645	−0.0345944
$916$	3.30012	0.109039
$917$	77.4193	2.55661
$918$	−6.83822	−0.225695
$919$	−40.2105	−1.32642	−0.663212	−	0.748432i	$-0.730807\pi$
$919$	−40.2105	−1.32642	−0.663212	+	0.748432i	$0.730807\pi$
$920$	−16.0491	−0.529123
$921$	−28.6666	−0.944598
$922$	−30.6242	−1.00855
$923$	−25.7749	−0.848391
$924$	8.34745	0.274611
$925$	−7.30308	−0.240124
$926$	12.0260	0.395200
$927$	−9.79253	−0.321629
$928$	5.73064	0.188117
$929$	1.71699	0.0563327	0.0281663	−	0.999603i	$-0.491033\pi$
$929$	1.71699	0.0563327	0.0281663	+	0.999603i	$0.491033\pi$
$930$	−0.294945	−0.00967163
$931$	−11.0719	−0.362867
$932$	−7.84552	−0.256989
$933$	0.461697	0.0151153
$934$	−28.5051	−0.932714
$935$	−33.7849	−1.10488
$936$	−2.77674	−0.0907606
$937$	5.40914	0.176709	0.0883544	−	0.996089i	$-0.471839\pi$
$937$	5.40914	0.176709	0.0883544	+	0.996089i	$0.471839\pi$
$938$	−32.5370	−1.06237
$939$	22.6577	0.739405
$940$	−1.67671	−0.0546881
$941$	−60.1847	−1.96197	−0.980983	−	0.194093i	$-0.937824\pi$
$941$	−60.1847	−1.96197	−0.980983	+	0.194093i	$0.937824\pi$
$942$	18.0398	0.587767
$943$	−63.3276	−2.06223
$944$	3.13796	0.102132
$945$	−10.6962	−0.347948
$946$	−3.09647	−0.100675
$947$	−8.94861	−0.290791	−0.145395	−	0.989374i	$-0.546445\pi$
$947$	−8.94861	−0.290791	−0.145395	+	0.989374i	$0.546445\pi$
$948$	16.0290	0.520596
$949$	1.56493	0.0507999
$950$	−1.33076	−0.0431756
$951$	−23.2735	−0.754696
$952$	−29.0700	−0.942163
$953$	−21.9082	−0.709678	−0.354839	−	0.934928i	$-0.615464\pi$
$953$	−21.9082	−0.709678	−0.354839	+	0.934928i	$0.615464\pi$
$954$	1.00000	0.0323762
$955$	3.78241	0.122396
$956$	1.41077	0.0456275
$957$	−11.2527	−0.363747
$958$	−15.9510	−0.515352
$959$	77.4317	2.50040
$960$	2.51610	0.0812068
$961$	−30.9863	−0.999557
$962$	15.2384	0.491307
$963$	−15.2769	−0.492292
$964$	3.80810	0.122651
$965$	−26.5453	−0.854522
$966$	27.1159	0.872441
$967$	−20.1206	−0.647034	−0.323517	−	0.946222i	$-0.604865\pi$
$967$	−20.1206	−0.647034	−0.323517	+	0.946222i	$0.604865\pi$
$968$	−7.14429	−0.229626
$969$	6.83822	0.219675
$970$	21.2457	0.682158
$971$	15.5739	0.499789	0.249894	−	0.968273i	$-0.419604\pi$
$971$	15.5739	0.499789	0.249894	+	0.968273i	$0.419604\pi$
$972$	−1.00000	−0.0320750
$973$	−30.0761	−0.964196
$974$	18.8436	0.603787
$975$	3.69518	0.118340
$976$	−0.415900	−0.0133126
$977$	2.39326	0.0765671	0.0382836	−	0.999267i	$-0.487811\pi$
$977$	2.39326	0.0765671	0.0382836	+	0.999267i	$0.487811\pi$
$978$	0.0245472	0.000784933 0
$979$	20.9729	0.670298
$980$	−27.8580	−0.889891
$981$	7.91690	0.252767
$982$	−12.6466	−0.403569
$983$	26.8992	0.857951	0.428976	−	0.903316i	$-0.358875\pi$
$983$	26.8992	0.857951	0.428976	+	0.903316i	$0.358875\pi$
$984$	9.92819	0.316499
$985$	12.9859	0.413766
$986$	39.1873	1.24798
$987$	2.83290	0.0901721
$988$	2.77674	0.0883398
$989$	−10.0586	−0.319846
$990$	−4.94060	−0.157023
$991$	−33.1675	−1.05360	−0.526801	−	0.849989i	$-0.676609\pi$
$991$	−33.1675	−1.05360	−0.526801	+	0.849989i	$0.676609\pi$
$992$	−0.117223	−0.00372184
$993$	−5.39248	−0.171125
$994$	−39.4606	−1.25161
$995$	54.9495	1.74201
$996$	−0.307939	−0.00975741
$997$	47.5275	1.50521	0.752606	−	0.658472i	$-0.228797\pi$
$997$	47.5275	1.50521	0.752606	+	0.658472i	$0.228797\pi$
$998$	22.4656	0.711138
$999$	5.48789	0.173629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bb.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bb.1.3	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.51610 q^{5} -1.00000 q^{6} -4.25110 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.51610 q^{10} +1.96359 q^{11} -1.00000 q^{12} -2.77674 q^{13} -4.25110 q^{14} +2.51610 q^{15} +1.00000 q^{16} +6.83822 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.51610 q^{20} +4.25110 q^{21} +1.96359 q^{22} +6.37856 q^{23} -1.00000 q^{24} +1.33076 q^{25} -2.77674 q^{26} -1.00000 q^{27} -4.25110 q^{28} +5.73064 q^{29} +2.51610 q^{30} -0.117223 q^{31} +1.00000 q^{32} -1.96359 q^{33} +6.83822 q^{34} +10.6962 q^{35} +1.00000 q^{36} -5.48789 q^{37} -1.00000 q^{38} +2.77674 q^{39} -2.51610 q^{40} -9.92819 q^{41} +4.25110 q^{42} -1.57694 q^{43} +1.96359 q^{44} -2.51610 q^{45} +6.37856 q^{46} +0.666390 q^{47} -1.00000 q^{48} +11.0719 q^{49} +1.33076 q^{50} -6.83822 q^{51} -2.77674 q^{52} +1.00000 q^{53} -1.00000 q^{54} -4.94060 q^{55} -4.25110 q^{56} +1.00000 q^{57} +5.73064 q^{58} +3.13796 q^{59} +2.51610 q^{60} -0.415900 q^{61} -0.117223 q^{62} -4.25110 q^{63} +1.00000 q^{64} +6.98655 q^{65} -1.96359 q^{66} +7.65377 q^{67} +6.83822 q^{68} -6.37856 q^{69} +10.6962 q^{70} +9.28244 q^{71} +1.00000 q^{72} -0.563587 q^{73} -5.48789 q^{74} -1.33076 q^{75} -1.00000 q^{76} -8.34745 q^{77} +2.77674 q^{78} -16.0290 q^{79} -2.51610 q^{80} +1.00000 q^{81} -9.92819 q^{82} +0.307939 q^{83} +4.25110 q^{84} -17.2056 q^{85} -1.57694 q^{86} -5.73064 q^{87} +1.96359 q^{88} +10.6809 q^{89} -2.51610 q^{90} +11.8042 q^{91} +6.37856 q^{92} +0.117223 q^{93} +0.666390 q^{94} +2.51610 q^{95} -1.00000 q^{96} -8.44389 q^{97} +11.0719 q^{98} +1.96359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20}+ \cdots + 3 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9 - 5 * q^10 + 3 * q^11 - 9 * q^12 - 4 * q^13 - 5 * q^14 + 5 * q^15 + 9 * q^16 - 16 * q^17 + 9 * q^18 - 9 * q^19 - 5 * q^20 + 5 * q^21 + 3 * q^22 - 9 * q^24 + 8 * q^25 - 4 * q^26 - 9 * q^27 - 5 * q^28 - 2 * q^29 + 5 * q^30 - 10 * q^31 + 9 * q^32 - 3 * q^33 - 16 * q^34 - q^35 + 9 * q^36 - 14 * q^37 - 9 * q^38 + 4 * q^39 - 5 * q^40 - 21 * q^41 + 5 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 - 8 * q^47 - 9 * q^48 - 14 * q^49 + 8 * q^50 + 16 * q^51 - 4 * q^52 + 9 * q^53 - 9 * q^54 - 14 * q^55 - 5 * q^56 + 9 * q^57 - 2 * q^58 - 12 * q^59 + 5 * q^60 - 13 * q^61 - 10 * q^62 - 5 * q^63 + 9 * q^64 - 13 * q^65 - 3 * q^66 + 22 * q^67 - 16 * q^68 - q^70 - 13 * q^71 + 9 * q^72 - 17 * q^73 - 14 * q^74 - 8 * q^75 - 9 * q^76 - 25 * q^77 + 4 * q^78 - 5 * q^80 + 9 * q^81 - 21 * q^82 - 24 * q^83 + 5 * q^84 - 16 * q^85 - 4 * q^86 + 2 * q^87 + 3 * q^88 - 23 * q^89 - 5 * q^90 - 11 * q^91 + 10 * q^93 - 8 * q^94 + 5 * q^95 - 9 * q^96 - 29 * q^97 - 14 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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