LMFDB - Embedded newform 4010.2.a.m.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + \cdots + 104 $ x^20 - 4x^19 - 35x^18 + 154x^17 + 460x^16 - 2392x^15 - 2591x^14 + 19157x^13 + 2776x^12 - 83577x^11 + 34362x^10 + 190617x^9 - 150697x^8 - 189098x^7 + 211360x^6 + 37365x^5 - 68876x^4 - 8984x^3 + 7920x^2 + 1972*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$3.13315$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.13315 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.13315 q^{6} +3.12083 q^{7} -1.00000 q^{8} +6.81662 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.13315 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.13315 q^{6} +3.12083 q^{7} -1.00000 q^{8} +6.81662 q^{9} -1.00000 q^{10} -2.67249 q^{11} +3.13315 q^{12} +2.42291 q^{13} -3.12083 q^{14} +3.13315 q^{15} +1.00000 q^{16} -2.85678 q^{17} -6.81662 q^{18} +3.73304 q^{19} +1.00000 q^{20} +9.77801 q^{21} +2.67249 q^{22} -1.03314 q^{23} -3.13315 q^{24} +1.00000 q^{25} -2.42291 q^{26} +11.9581 q^{27} +3.12083 q^{28} -0.520613 q^{29} -3.13315 q^{30} +7.07212 q^{31} -1.00000 q^{32} -8.37332 q^{33} +2.85678 q^{34} +3.12083 q^{35} +6.81662 q^{36} -6.84683 q^{37} -3.73304 q^{38} +7.59133 q^{39} -1.00000 q^{40} +2.43071 q^{41} -9.77801 q^{42} -6.36906 q^{43} -2.67249 q^{44} +6.81662 q^{45} +1.03314 q^{46} +0.242392 q^{47} +3.13315 q^{48} +2.73955 q^{49} -1.00000 q^{50} -8.95073 q^{51} +2.42291 q^{52} +5.11504 q^{53} -11.9581 q^{54} -2.67249 q^{55} -3.12083 q^{56} +11.6962 q^{57} +0.520613 q^{58} -11.7090 q^{59} +3.13315 q^{60} +12.7736 q^{61} -7.07212 q^{62} +21.2735 q^{63} +1.00000 q^{64} +2.42291 q^{65} +8.37332 q^{66} +9.68194 q^{67} -2.85678 q^{68} -3.23699 q^{69} -3.12083 q^{70} +2.28354 q^{71} -6.81662 q^{72} +0.735162 q^{73} +6.84683 q^{74} +3.13315 q^{75} +3.73304 q^{76} -8.34039 q^{77} -7.59133 q^{78} -8.11979 q^{79} +1.00000 q^{80} +17.0165 q^{81} -2.43071 q^{82} -5.02541 q^{83} +9.77801 q^{84} -2.85678 q^{85} +6.36906 q^{86} -1.63116 q^{87} +2.67249 q^{88} -5.39616 q^{89} -6.81662 q^{90} +7.56147 q^{91} -1.03314 q^{92} +22.1580 q^{93} -0.242392 q^{94} +3.73304 q^{95} -3.13315 q^{96} +6.12154 q^{97} -2.73955 q^{98} -18.2174 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) $ 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 - 9 * q^13 - 11 * q^14 + 4 * q^15 + 20 * q^16 - 11 * q^17 - 26 * q^18 + 17 * q^19 + 20 * q^20 - 2 * q^21 - 10 * q^22 - 3 * q^23 - 4 * q^24 + 20 * q^25 + 9 * q^26 - 2 * q^27 + 11 * q^28 + 6 * q^29 - 4 * q^30 + 28 * q^31 - 20 * q^32 + 2 * q^33 + 11 * q^34 + 11 * q^35 + 26 * q^36 + 33 * q^37 - 17 * q^38 + 36 * q^39 - 20 * q^40 + 32 * q^41 + 2 * q^42 + 30 * q^43 + 10 * q^44 + 26 * q^45 + 3 * q^46 + 13 * q^47 + 4 * q^48 + 43 * q^49 - 20 * q^50 + 43 * q^51 - 9 * q^52 + 2 * q^54 + 10 * q^55 - 11 * q^56 + 19 * q^57 - 6 * q^58 + 52 * q^59 + 4 * q^60 + 25 * q^61 - 28 * q^62 + 16 * q^63 + 20 * q^64 - 9 * q^65 - 2 * q^66 + 40 * q^67 - 11 * q^68 + 39 * q^69 - 11 * q^70 + 25 * q^71 - 26 * q^72 - 5 * q^73 - 33 * q^74 + 4 * q^75 + 17 * q^76 - 9 * q^77 - 36 * q^78 + 40 * q^79 + 20 * q^80 + 48 * q^81 - 32 * q^82 + 10 * q^83 - 2 * q^84 - 11 * q^85 - 30 * q^86 + 10 * q^87 - 10 * q^88 + 17 * q^89 - 26 * q^90 + 88 * q^91 - 3 * q^92 - 4 * q^93 - 13 * q^94 + 17 * q^95 - 4 * q^96 - 2 * q^97 - 43 * q^98 + 50 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.13315	1.80892	0.904462	−	0.426554i	$-0.140272\pi$
$3$	3.13315	1.80892	0.904462	+	0.426554i	$0.140272\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.13315	−1.27910
$7$	3.12083	1.17956	0.589781	−	0.807564i	$-0.299214\pi$
$7$	3.12083	1.17956	0.589781	+	0.807564i	$0.299214\pi$
$8$	−1.00000	−0.353553
$9$	6.81662	2.27221
$10$	−1.00000	−0.316228
$11$	−2.67249	−0.805787	−0.402894	−	0.915247i	$-0.631996\pi$
$11$	−2.67249	−0.805787	−0.402894	+	0.915247i	$0.631996\pi$
$12$	3.13315	0.904462
$13$	2.42291	0.671993	0.335997	−	0.941863i	$-0.390927\pi$
$13$	2.42291	0.671993	0.335997	+	0.941863i	$0.390927\pi$
$14$	−3.12083	−0.834076
$15$	3.13315	0.808976
$16$	1.00000	0.250000
$17$	−2.85678	−0.692872	−0.346436	−	0.938074i	$-0.612608\pi$
$17$	−2.85678	−0.692872	−0.346436	+	0.938074i	$0.612608\pi$
$18$	−6.81662	−1.60669
$19$	3.73304	0.856418	0.428209	−	0.903680i	$-0.359145\pi$
$19$	3.73304	0.856418	0.428209	+	0.903680i	$0.359145\pi$
$20$	1.00000	0.223607
$21$	9.77801	2.13374
$22$	2.67249	0.569778
$23$	−1.03314	−0.215425	−0.107713	−	0.994182i	$-0.534353\pi$
$23$	−1.03314	−0.215425	−0.107713	+	0.994182i	$0.534353\pi$
$24$	−3.13315	−0.639551
$25$	1.00000	0.200000
$26$	−2.42291	−0.475171
$27$	11.9581	2.30133
$28$	3.12083	0.589781
$29$	−0.520613	−0.0966754	−0.0483377	−	0.998831i	$-0.515392\pi$
$29$	−0.520613	−0.0966754	−0.0483377	+	0.998831i	$0.515392\pi$
$30$	−3.13315	−0.572032
$31$	7.07212	1.27019	0.635095	−	0.772434i	$-0.280961\pi$
$31$	7.07212	1.27019	0.635095	+	0.772434i	$0.280961\pi$
$32$	−1.00000	−0.176777
$33$	−8.37332	−1.45761
$34$	2.85678	0.489934
$35$	3.12083	0.527516
$36$	6.81662	1.13610
$37$	−6.84683	−1.12561	−0.562806	−	0.826589i	$-0.690278\pi$
$37$	−6.84683	−1.12561	−0.562806	+	0.826589i	$0.690278\pi$
$38$	−3.73304	−0.605579
$39$	7.59133	1.21559
$40$	−1.00000	−0.158114
$41$	2.43071	0.379613	0.189806	−	0.981822i	$-0.439214\pi$
$41$	2.43071	0.379613	0.189806	+	0.981822i	$0.439214\pi$
$42$	−9.77801	−1.50878
$43$	−6.36906	−0.971273	−0.485636	−	0.874161i	$-0.661412\pi$
$43$	−6.36906	−0.971273	−0.485636	+	0.874161i	$0.661412\pi$
$44$	−2.67249	−0.402894
$45$	6.81662	1.01616
$46$	1.03314	0.152329
$47$	0.242392	0.0353565	0.0176782	−	0.999844i	$-0.494373\pi$
$47$	0.242392	0.0353565	0.0176782	+	0.999844i	$0.494373\pi$
$48$	3.13315	0.452231
$49$	2.73955	0.391364
$50$	−1.00000	−0.141421
$51$	−8.95073	−1.25335
$52$	2.42291	0.335997
$53$	5.11504	0.702605	0.351302	−	0.936262i	$-0.385739\pi$
$53$	5.11504	0.702605	0.351302	+	0.936262i	$0.385739\pi$
$54$	−11.9581	−1.62728
$55$	−2.67249	−0.360359
$56$	−3.12083	−0.417038
$57$	11.6962	1.54920
$58$	0.520613	0.0683598
$59$	−11.7090	−1.52439	−0.762193	−	0.647350i	$-0.775877\pi$
$59$	−11.7090	−1.52439	−0.762193	+	0.647350i	$0.775877\pi$
$60$	3.13315	0.404488
$61$	12.7736	1.63549	0.817747	−	0.575578i	$-0.195223\pi$
$61$	12.7736	1.63549	0.817747	+	0.575578i	$0.195223\pi$
$62$	−7.07212	−0.898160
$63$	21.2735	2.68021
$64$	1.00000	0.125000
$65$	2.42291	0.300525
$66$	8.37332	1.03068
$67$	9.68194	1.18284	0.591419	−	0.806365i	$-0.298568\pi$
$67$	9.68194	1.18284	0.591419	+	0.806365i	$0.298568\pi$
$68$	−2.85678	−0.346436
$69$	−3.23699	−0.389688
$70$	−3.12083	−0.373010
$71$	2.28354	0.271006	0.135503	−	0.990777i	$-0.456735\pi$
$71$	2.28354	0.271006	0.135503	+	0.990777i	$0.456735\pi$
$72$	−6.81662	−0.803347
$73$	0.735162	0.0860442	0.0430221	−	0.999074i	$-0.486301\pi$
$73$	0.735162	0.0860442	0.0430221	+	0.999074i	$0.486301\pi$
$74$	6.84683	0.795928
$75$	3.13315	0.361785
$76$	3.73304	0.428209
$77$	−8.34039	−0.950475
$78$	−7.59133	−0.859549
$79$	−8.11979	−0.913547	−0.456774	−	0.889583i	$-0.650995\pi$
$79$	−8.11979	−0.913547	−0.456774	+	0.889583i	$0.650995\pi$
$80$	1.00000	0.111803
$81$	17.0165	1.89072
$82$	−2.43071	−0.268427
$83$	−5.02541	−0.551610	−0.275805	−	0.961214i	$-0.588944\pi$
$83$	−5.02541	−0.551610	−0.275805	+	0.961214i	$0.588944\pi$
$84$	9.77801	1.06687
$85$	−2.85678	−0.309862
$86$	6.36906	0.686794
$87$	−1.63116	−0.174879
$88$	2.67249	0.284889
$89$	−5.39616	−0.571992	−0.285996	−	0.958231i	$-0.592324\pi$
$89$	−5.39616	−0.571992	−0.285996	+	0.958231i	$0.592324\pi$
$90$	−6.81662	−0.718535
$91$	7.56147	0.792657
$92$	−1.03314	−0.107713
$93$	22.1580	2.29768
$94$	−0.242392	−0.0250008
$95$	3.73304	0.383002
$96$	−3.13315	−0.319776
$97$	6.12154	0.621548	0.310774	−	0.950484i	$-0.399412\pi$
$97$	6.12154	0.621548	0.310774	+	0.950484i	$0.399412\pi$
$98$	−2.73955	−0.276736
$99$	−18.2174	−1.83092
$100$	1.00000	0.100000
$101$	−6.39920	−0.636744	−0.318372	−	0.947966i	$-0.603136\pi$
$101$	−6.39920	−0.636744	−0.318372	+	0.947966i	$0.603136\pi$
$102$	8.95073	0.886254
$103$	1.56823	0.154522	0.0772612	−	0.997011i	$-0.475382\pi$
$103$	1.56823	0.154522	0.0772612	+	0.997011i	$0.475382\pi$
$104$	−2.42291	−0.237586
$105$	9.77801	0.954236
$106$	−5.11504	−0.496817
$107$	5.57565	0.539018	0.269509	−	0.962998i	$-0.413139\pi$
$107$	5.57565	0.539018	0.269509	+	0.962998i	$0.413139\pi$
$108$	11.9581	1.15066
$109$	1.27551	0.122172	0.0610858	−	0.998133i	$-0.480544\pi$
$109$	1.27551	0.122172	0.0610858	+	0.998133i	$0.480544\pi$
$110$	2.67249	0.254812
$111$	−21.4521	−2.03615
$112$	3.12083	0.294890
$113$	4.97877	0.468363	0.234182	−	0.972193i	$-0.424759\pi$
$113$	4.97877	0.468363	0.234182	+	0.972193i	$0.424759\pi$
$114$	−11.6962	−1.09545
$115$	−1.03314	−0.0963412
$116$	−0.520613	−0.0483377
$117$	16.5160	1.52691
$118$	11.7090	1.07790
$119$	−8.91552	−0.817284
$120$	−3.13315	−0.286016
$121$	−3.85778	−0.350707
$122$	−12.7736	−1.15647
$123$	7.61577	0.686691
$124$	7.07212	0.635095
$125$	1.00000	0.0894427
$126$	−21.2735	−1.89519
$127$	11.0010	0.976185	0.488093	−	0.872792i	$-0.337693\pi$
$127$	11.0010	0.976185	0.488093	+	0.872792i	$0.337693\pi$
$128$	−1.00000	−0.0883883
$129$	−19.9552	−1.75696
$130$	−2.42291	−0.212503
$131$	−9.07086	−0.792525	−0.396263	−	0.918137i	$-0.629693\pi$
$131$	−9.07086	−0.792525	−0.396263	+	0.918137i	$0.629693\pi$
$132$	−8.37332	−0.728804
$133$	11.6502	1.01020
$134$	−9.68194	−0.836392
$135$	11.9581	1.02919
$136$	2.85678	0.244967
$137$	−15.8817	−1.35687	−0.678433	−	0.734662i	$-0.737341\pi$
$137$	−15.8817	−1.35687	−0.678433	+	0.734662i	$0.737341\pi$
$138$	3.23699	0.275551
$139$	−2.86616	−0.243104	−0.121552	−	0.992585i	$-0.538787\pi$
$139$	−2.86616	−0.243104	−0.121552	+	0.992585i	$0.538787\pi$
$140$	3.12083	0.263758
$141$	0.759450	0.0639572
$142$	−2.28354	−0.191630
$143$	−6.47520	−0.541484
$144$	6.81662	0.568052
$145$	−0.520613	−0.0432346
$146$	−0.735162	−0.0608424
$147$	8.58342	0.707949
$148$	−6.84683	−0.562806
$149$	18.5892	1.52288	0.761442	−	0.648234i	$-0.224492\pi$
$149$	18.5892	1.52288	0.761442	+	0.648234i	$0.224492\pi$
$150$	−3.13315	−0.255821
$151$	−10.2775	−0.836369	−0.418184	−	0.908362i	$-0.637333\pi$
$151$	−10.2775	−0.836369	−0.418184	+	0.908362i	$0.637333\pi$
$152$	−3.73304	−0.302789
$153$	−19.4736	−1.57435
$154$	8.34039	0.672087
$155$	7.07212	0.568046
$156$	7.59133	0.607793
$157$	−17.6046	−1.40500	−0.702498	−	0.711686i	$-0.747932\pi$
$157$	−17.6046	−1.40500	−0.702498	+	0.711686i	$0.747932\pi$
$158$	8.11979	0.645976
$159$	16.0262	1.27096
$160$	−1.00000	−0.0790569
$161$	−3.22426	−0.254107
$162$	−17.0165	−1.33694
$163$	12.0570	0.944373	0.472187	−	0.881499i	$-0.343465\pi$
$163$	12.0570	0.944373	0.472187	+	0.881499i	$0.343465\pi$
$164$	2.43071	0.189806
$165$	−8.37332	−0.651862
$166$	5.02541	0.390047
$167$	−2.42388	−0.187565	−0.0937827	−	0.995593i	$-0.529896\pi$
$167$	−2.42388	−0.187565	−0.0937827	+	0.995593i	$0.529896\pi$
$168$	−9.77801	−0.754390
$169$	−7.12952	−0.548425
$170$	2.85678	0.219105
$171$	25.4467	1.94596
$172$	−6.36906	−0.485636
$173$	−8.11033	−0.616617	−0.308308	−	0.951286i	$-0.599763\pi$
$173$	−8.11033	−0.616617	−0.308308	+	0.951286i	$0.599763\pi$
$174$	1.63116	0.123658
$175$	3.12083	0.235912
$176$	−2.67249	−0.201447
$177$	−36.6861	−2.75750
$178$	5.39616	0.404459
$179$	2.07836	0.155344	0.0776719	−	0.996979i	$-0.475251\pi$
$179$	2.07836	0.155344	0.0776719	+	0.996979i	$0.475251\pi$
$180$	6.81662	0.508081
$181$	5.04998	0.375362	0.187681	−	0.982230i	$-0.439903\pi$
$181$	5.04998	0.375362	0.187681	+	0.982230i	$0.439903\pi$
$182$	−7.56147	−0.560493
$183$	40.0216	2.95848
$184$	1.03314	0.0761644
$185$	−6.84683	−0.503389
$186$	−22.1580	−1.62470
$187$	7.63473	0.558307
$188$	0.242392	0.0176782
$189$	37.3190	2.71456
$190$	−3.73304	−0.270823
$191$	22.6878	1.64163	0.820816	−	0.571192i	$-0.193519\pi$
$191$	22.6878	1.64163	0.820816	+	0.571192i	$0.193519\pi$
$192$	3.13315	0.226116
$193$	−7.06188	−0.508325	−0.254163	−	0.967161i	$-0.581800\pi$
$193$	−7.06188	−0.508325	−0.254163	+	0.967161i	$0.581800\pi$
$194$	−6.12154	−0.439501
$195$	7.59133	0.543626
$196$	2.73955	0.195682
$197$	−20.3289	−1.44838	−0.724188	−	0.689602i	$-0.757785\pi$
$197$	−20.3289	−1.44838	−0.724188	+	0.689602i	$0.757785\pi$
$198$	18.2174	1.29465
$199$	−8.86241	−0.628239	−0.314120	−	0.949383i	$-0.601709\pi$
$199$	−8.86241	−0.628239	−0.314120	+	0.949383i	$0.601709\pi$
$200$	−1.00000	−0.0707107
$201$	30.3350	2.13966
$202$	6.39920	0.450246
$203$	−1.62474	−0.114035
$204$	−8.95073	−0.626676
$205$	2.43071	0.169768
$206$	−1.56823	−0.109264
$207$	−7.04255	−0.489491
$208$	2.42291	0.167998
$209$	−9.97652	−0.690091
$210$	−9.77801	−0.674747
$211$	19.0262	1.30982	0.654910	−	0.755707i	$-0.272706\pi$
$211$	19.0262	1.30982	0.654910	+	0.755707i	$0.272706\pi$
$212$	5.11504	0.351302
$213$	7.15467	0.490230
$214$	−5.57565	−0.381143
$215$	−6.36906	−0.434366
$216$	−11.9581	−0.813642
$217$	22.0709	1.49827
$218$	−1.27551	−0.0863883
$219$	2.30337	0.155647
$220$	−2.67249	−0.180179
$221$	−6.92172	−0.465605
$222$	21.4521	1.43977
$223$	3.52530	0.236071	0.118036	−	0.993009i	$-0.462340\pi$
$223$	3.52530	0.236071	0.118036	+	0.993009i	$0.462340\pi$
$224$	−3.12083	−0.208519
$225$	6.81662	0.454442
$226$	−4.97877	−0.331183
$227$	1.51894	0.100815	0.0504077	−	0.998729i	$-0.483948\pi$
$227$	1.51894	0.100815	0.0504077	+	0.998729i	$0.483948\pi$
$228$	11.6962	0.774598
$229$	−13.6501	−0.902021	−0.451011	−	0.892519i	$-0.648936\pi$
$229$	−13.6501	−0.902021	−0.451011	+	0.892519i	$0.648936\pi$
$230$	1.03314	0.0681235
$231$	−26.1317	−1.71934
$232$	0.520613	0.0341799
$233$	22.6719	1.48529	0.742643	−	0.669687i	$-0.233572\pi$
$233$	22.6719	1.48529	0.742643	+	0.669687i	$0.233572\pi$
$234$	−16.5160	−1.07969
$235$	0.242392	0.0158119
$236$	−11.7090	−0.762193
$237$	−25.4405	−1.65254
$238$	8.91552	0.577907
$239$	−12.4020	−0.802218	−0.401109	−	0.916030i	$-0.631375\pi$
$239$	−12.4020	−0.802218	−0.401109	+	0.916030i	$0.631375\pi$
$240$	3.13315	0.202244
$241$	−2.14862	−0.138405	−0.0692023	−	0.997603i	$-0.522045\pi$
$241$	−2.14862	−0.138405	−0.0692023	+	0.997603i	$0.522045\pi$
$242$	3.85778	0.247987
$243$	17.4411	1.11884
$244$	12.7736	0.817747
$245$	2.73955	0.175023
$246$	−7.61577	−0.485564
$247$	9.04481	0.575507
$248$	−7.07212	−0.449080
$249$	−15.7454	−0.997821
$250$	−1.00000	−0.0632456
$251$	0.0624655	0.00394279	0.00197139	−	0.999998i	$-0.499372\pi$
$251$	0.0624655	0.00394279	0.00197139	+	0.999998i	$0.499372\pi$
$252$	21.2735	1.34010
$253$	2.76107	0.173587
$254$	−11.0010	−0.690267
$255$	−8.95073	−0.560516
$256$	1.00000	0.0625000
$257$	−8.48794	−0.529463	−0.264732	−	0.964322i	$-0.585283\pi$
$257$	−8.48794	−0.529463	−0.264732	+	0.964322i	$0.585283\pi$
$258$	19.9552	1.24236
$259$	−21.3678	−1.32773
$260$	2.42291	0.150262
$261$	−3.54882	−0.219667
$262$	9.07086	0.560400
$263$	−18.8670	−1.16339	−0.581695	−	0.813407i	$-0.697610\pi$
$263$	−18.8670	−1.16339	−0.581695	+	0.813407i	$0.697610\pi$
$264$	8.37332	0.515342
$265$	5.11504	0.314214
$266$	−11.6502	−0.714317
$267$	−16.9070	−1.03469
$268$	9.68194	0.591419
$269$	−1.74916	−0.106648	−0.0533242	−	0.998577i	$-0.516982\pi$
$269$	−1.74916	−0.106648	−0.0533242	+	0.998577i	$0.516982\pi$
$270$	−11.9581	−0.727744
$271$	−13.5454	−0.822823	−0.411412	−	0.911450i	$-0.634964\pi$
$271$	−13.5454	−0.822823	−0.411412	+	0.911450i	$0.634964\pi$
$272$	−2.85678	−0.173218
$273$	23.6912	1.43386
$274$	15.8817	0.959449
$275$	−2.67249	−0.161157
$276$	−3.23699	−0.194844
$277$	−3.24797	−0.195152	−0.0975759	−	0.995228i	$-0.531109\pi$
$277$	−3.24797	−0.195152	−0.0975759	+	0.995228i	$0.531109\pi$
$278$	2.86616	0.171901
$279$	48.2080	2.88614
$280$	−3.12083	−0.186505
$281$	12.0729	0.720209	0.360104	−	0.932912i	$-0.382741\pi$
$281$	12.0729	0.720209	0.360104	+	0.932912i	$0.382741\pi$
$282$	−0.759450	−0.0452246
$283$	1.88804	0.112233	0.0561163	−	0.998424i	$-0.482128\pi$
$283$	1.88804	0.112233	0.0561163	+	0.998424i	$0.482128\pi$
$284$	2.28354	0.135503
$285$	11.6962	0.692821
$286$	6.47520	0.382887
$287$	7.58581	0.447776
$288$	−6.81662	−0.401673
$289$	−8.83879	−0.519929
$290$	0.520613	0.0305714
$291$	19.1797	1.12433
$292$	0.735162	0.0430221
$293$	−11.2510	−0.657291	−0.328646	−	0.944453i	$-0.606592\pi$
$293$	−11.2510	−0.657291	−0.328646	+	0.944453i	$0.606592\pi$
$294$	−8.58342	−0.500595
$295$	−11.7090	−0.681726
$296$	6.84683	0.397964
$297$	−31.9578	−1.85438
$298$	−18.5892	−1.07684
$299$	−2.50321	−0.144764
$300$	3.13315	0.180892
$301$	−19.8767	−1.14568
$302$	10.2775	0.591402
$303$	−20.0496	−1.15182
$304$	3.73304	0.214104
$305$	12.7736	0.731415
$306$	19.4736	1.11323
$307$	−17.1973	−0.981500	−0.490750	−	0.871300i	$-0.663277\pi$
$307$	−17.1973	−0.981500	−0.490750	+	0.871300i	$0.663277\pi$
$308$	−8.34039	−0.475238
$309$	4.91350	0.279519
$310$	−7.07212	−0.401670
$311$	−20.4510	−1.15967	−0.579834	−	0.814734i	$-0.696883\pi$
$311$	−20.4510	−1.15967	−0.579834	+	0.814734i	$0.696883\pi$
$312$	−7.59133	−0.429774
$313$	31.6299	1.78783	0.893915	−	0.448237i	$-0.147948\pi$
$313$	31.6299	1.78783	0.893915	+	0.448237i	$0.147948\pi$
$314$	17.6046	0.993482
$315$	21.2735	1.19863
$316$	−8.11979	−0.456774
$317$	27.2996	1.53330	0.766650	−	0.642066i	$-0.221922\pi$
$317$	27.2996	1.53330	0.766650	+	0.642066i	$0.221922\pi$
$318$	−16.0262	−0.898704
$319$	1.39133	0.0778998
$320$	1.00000	0.0559017
$321$	17.4693	0.975043
$322$	3.22426	0.179681
$323$	−10.6645	−0.593388
$324$	17.0165	0.945361
$325$	2.42291	0.134399
$326$	−12.0570	−0.667773
$327$	3.99636	0.220999
$328$	−2.43071	−0.134213
$329$	0.756463	0.0417051
$330$	8.37332	0.460936
$331$	−22.8926	−1.25829	−0.629146	−	0.777287i	$-0.716595\pi$
$331$	−22.8926	−1.25829	−0.629146	+	0.777287i	$0.716595\pi$
$332$	−5.02541	−0.275805
$333$	−46.6723	−2.55762
$334$	2.42388	0.132629
$335$	9.68194	0.528981
$336$	9.77801	0.533434
$337$	−29.2246	−1.59196	−0.795982	−	0.605320i	$-0.793045\pi$
$337$	−29.2246	−1.59196	−0.795982	+	0.605320i	$0.793045\pi$
$338$	7.12952	0.387795
$339$	15.5992	0.847233
$340$	−2.85678	−0.154931
$341$	−18.9002	−1.02350
$342$	−25.4467	−1.37600
$343$	−13.2961	−0.717923
$344$	6.36906	0.343397
$345$	−3.23699	−0.174274
$346$	8.11033	0.436014
$347$	−35.2311	−1.89131	−0.945653	−	0.325179i	$-0.894576\pi$
$347$	−35.2311	−1.89131	−0.945653	+	0.325179i	$0.894576\pi$
$348$	−1.63116	−0.0874393
$349$	10.6892	0.572180	0.286090	−	0.958203i	$-0.407644\pi$
$349$	10.6892	0.572180	0.286090	+	0.958203i	$0.407644\pi$
$350$	−3.12083	−0.166815
$351$	28.9732	1.54648
$352$	2.67249	0.142444
$353$	18.8743	1.00458	0.502290	−	0.864699i	$-0.332491\pi$
$353$	18.8743	1.00458	0.502290	+	0.864699i	$0.332491\pi$
$354$	36.6861	1.94985
$355$	2.28354	0.121198
$356$	−5.39616	−0.285996
$357$	−27.9337	−1.47841
$358$	−2.07836	−0.109845
$359$	−0.933001	−0.0492419	−0.0246210	−	0.999697i	$-0.507838\pi$
$359$	−0.933001	−0.0492419	−0.0246210	+	0.999697i	$0.507838\pi$
$360$	−6.81662	−0.359268
$361$	−5.06442	−0.266548
$362$	−5.04998	−0.265421
$363$	−12.0870	−0.634403
$364$	7.56147	0.396329
$365$	0.735162	0.0384801
$366$	−40.0216	−2.09196
$367$	17.5809	0.917714	0.458857	−	0.888510i	$-0.348259\pi$
$367$	17.5809	0.917714	0.458857	+	0.888510i	$0.348259\pi$
$368$	−1.03314	−0.0538564
$369$	16.5692	0.862559
$370$	6.84683	0.355950
$371$	15.9631	0.828765
$372$	22.1580	1.14884
$373$	−15.3491	−0.794746	−0.397373	−	0.917657i	$-0.630078\pi$
$373$	−15.3491	−0.794746	−0.397373	+	0.917657i	$0.630078\pi$
$374$	−7.63473	−0.394783
$375$	3.13315	0.161795
$376$	−0.242392	−0.0125004
$377$	−1.26140	−0.0649652
$378$	−37.3190	−1.91948
$379$	−21.5490	−1.10690	−0.553448	−	0.832884i	$-0.686688\pi$
$379$	−21.5490	−1.10690	−0.553448	+	0.832884i	$0.686688\pi$
$380$	3.73304	0.191501
$381$	34.4679	1.76585
$382$	−22.6878	−1.16081
$383$	−13.6902	−0.699535	−0.349767	−	0.936837i	$-0.613739\pi$
$383$	−13.6902	−0.699535	−0.349767	+	0.936837i	$0.613739\pi$
$384$	−3.13315	−0.159888
$385$	−8.34039	−0.425065
$386$	7.06188	0.359440
$387$	−43.4155	−2.20693
$388$	6.12154	0.310774
$389$	33.0231	1.67434	0.837168	−	0.546945i	$-0.184209\pi$
$389$	33.0231	1.67434	0.837168	+	0.546945i	$0.184209\pi$
$390$	−7.59133	−0.384402
$391$	2.95147	0.149262
$392$	−2.73955	−0.138368
$393$	−28.4204	−1.43362
$394$	20.3289	1.02416
$395$	−8.11979	−0.408551
$396$	−18.2174	−0.915458
$397$	5.70794	0.286473	0.143237	−	0.989688i	$-0.454249\pi$
$397$	5.70794	0.286473	0.143237	+	0.989688i	$0.454249\pi$
$398$	8.86241	0.444232
$399$	36.5017	1.82737
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−30.3350	−1.51297
$403$	17.1351	0.853560
$404$	−6.39920	−0.318372
$405$	17.0165	0.845556
$406$	1.62474	0.0806346
$407$	18.2981	0.907003
$408$	8.95073	0.443127
$409$	12.9448	0.640078	0.320039	−	0.947404i	$-0.396304\pi$
$409$	12.9448	0.640078	0.320039	+	0.947404i	$0.396304\pi$
$410$	−2.43071	−0.120044
$411$	−49.7598	−2.45447
$412$	1.56823	0.0772612
$413$	−36.5418	−1.79811
$414$	7.04255	0.346123
$415$	−5.02541	−0.246688
$416$	−2.42291	−0.118793
$417$	−8.98010	−0.439758
$418$	9.97652	0.487968
$419$	20.3642	0.994856	0.497428	−	0.867505i	$-0.334278\pi$
$419$	20.3642	0.994856	0.497428	+	0.867505i	$0.334278\pi$
$420$	9.77801	0.477118
$421$	−9.67447	−0.471505	−0.235752	−	0.971813i	$-0.575755\pi$
$421$	−9.67447	−0.471505	−0.235752	+	0.971813i	$0.575755\pi$
$422$	−19.0262	−0.926183
$423$	1.65229	0.0803373
$424$	−5.11504	−0.248408
$425$	−2.85678	−0.138574
$426$	−7.15467	−0.346645
$427$	39.8642	1.92916
$428$	5.57565	0.269509
$429$	−20.2878	−0.979503
$430$	6.36906	0.307143
$431$	−36.1693	−1.74222	−0.871108	−	0.491092i	$-0.836598\pi$
$431$	−36.1693	−1.74222	−0.871108	+	0.491092i	$0.836598\pi$
$432$	11.9581	0.575332
$433$	−20.0718	−0.964588	−0.482294	−	0.876010i	$-0.660196\pi$
$433$	−20.0718	−0.964588	−0.482294	+	0.876010i	$0.660196\pi$
$434$	−22.0709	−1.05944
$435$	−1.63116	−0.0782081
$436$	1.27551	0.0610858
$437$	−3.85677	−0.184494
$438$	−2.30337	−0.110059
$439$	11.0460	0.527197	0.263599	−	0.964632i	$-0.415091\pi$
$439$	11.0460	0.527197	0.263599	+	0.964632i	$0.415091\pi$
$440$	2.67249	0.127406
$441$	18.6745	0.889261
$442$	6.92172	0.329233
$443$	33.2080	1.57776	0.788881	−	0.614546i	$-0.210661\pi$
$443$	33.2080	1.57776	0.788881	+	0.614546i	$0.210661\pi$
$444$	−21.4521	−1.01807
$445$	−5.39616	−0.255803
$446$	−3.52530	−0.166928
$447$	58.2426	2.75478
$448$	3.12083	0.147445
$449$	22.1237	1.04408	0.522040	−	0.852921i	$-0.325171\pi$
$449$	22.1237	1.04408	0.522040	+	0.852921i	$0.325171\pi$
$450$	−6.81662	−0.321339
$451$	−6.49605	−0.305887
$452$	4.97877	0.234182
$453$	−32.2008	−1.51293
$454$	−1.51894	−0.0712872
$455$	7.56147	0.354487
$456$	−11.6962	−0.547723
$457$	1.30636	0.0611089	0.0305544	−	0.999533i	$-0.490273\pi$
$457$	1.30636	0.0611089	0.0305544	+	0.999533i	$0.490273\pi$
$458$	13.6501	0.637825
$459$	−34.1616	−1.59453
$460$	−1.03314	−0.0481706
$461$	9.09890	0.423778	0.211889	−	0.977294i	$-0.432038\pi$
$461$	9.09890	0.423778	0.211889	+	0.977294i	$0.432038\pi$
$462$	26.1317	1.21576
$463$	31.0434	1.44271	0.721354	−	0.692567i	$-0.243520\pi$
$463$	31.0434	1.44271	0.721354	+	0.692567i	$0.243520\pi$
$464$	−0.520613	−0.0241689
$465$	22.1580	1.02755
$466$	−22.6719	−1.05026
$467$	−2.63348	−0.121863	−0.0609314	−	0.998142i	$-0.519407\pi$
$467$	−2.63348	−0.121863	−0.0609314	+	0.998142i	$0.519407\pi$
$468$	16.5160	0.763454
$469$	30.2156	1.39523
$470$	−0.242392	−0.0111807
$471$	−55.1577	−2.54153
$472$	11.7090	0.538952
$473$	17.0213	0.782639
$474$	25.4405	1.16852
$475$	3.73304	0.171284
$476$	−8.91552	−0.408642
$477$	34.8673	1.59646
$478$	12.4020	0.567254
$479$	25.4735	1.16391	0.581957	−	0.813219i	$-0.302287\pi$
$479$	25.4735	1.16391	0.581957	+	0.813219i	$0.302287\pi$
$480$	−3.13315	−0.143008
$481$	−16.5892	−0.756404
$482$	2.14862	0.0978668
$483$	−10.1021	−0.459661
$484$	−3.85778	−0.175354
$485$	6.12154	0.277965
$486$	−17.4411	−0.791142
$487$	2.44423	0.110759	0.0553793	−	0.998465i	$-0.482363\pi$
$487$	2.44423	0.110759	0.0553793	+	0.998465i	$0.482363\pi$
$488$	−12.7736	−0.578234
$489$	37.7762	1.70830
$490$	−2.73955	−0.123760
$491$	27.8680	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$491$	27.8680	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$492$	7.61577	0.343345
$493$	1.48728	0.0669836
$494$	−9.04481	−0.406945
$495$	−18.2174	−0.818811
$496$	7.07212	0.317548
$497$	7.12653	0.319668
$498$	15.7454	0.705566
$499$	27.8971	1.24884	0.624422	−	0.781087i	$-0.285335\pi$
$499$	27.8971	1.24884	0.624422	+	0.781087i	$0.285335\pi$
$500$	1.00000	0.0447214
$501$	−7.59437	−0.339292
$502$	−0.0624655	−0.00278797
$503$	−23.4359	−1.04496	−0.522478	−	0.852653i	$-0.674992\pi$
$503$	−23.4359	−1.04496	−0.522478	+	0.852653i	$0.674992\pi$
$504$	−21.2735	−0.947597
$505$	−6.39920	−0.284761
$506$	−2.76107	−0.122745
$507$	−22.3379	−0.992059
$508$	11.0010	0.488093
$509$	2.49284	0.110493	0.0552465	−	0.998473i	$-0.482406\pi$
$509$	2.49284	0.110493	0.0552465	+	0.998473i	$0.482406\pi$
$510$	8.95073	0.396345
$511$	2.29431	0.101494
$512$	−1.00000	−0.0441942
$513$	44.6399	1.97090
$514$	8.48794	0.374387
$515$	1.56823	0.0691045
$516$	−19.9552	−0.878480
$517$	−0.647791	−0.0284898
$518$	21.3678	0.938845
$519$	−25.4109	−1.11541
$520$	−2.42291	−0.106251
$521$	8.64549	0.378766	0.189383	−	0.981903i	$-0.439351\pi$
$521$	8.64549	0.378766	0.189383	+	0.981903i	$0.439351\pi$
$522$	3.54882	0.155328
$523$	−21.8082	−0.953605	−0.476803	−	0.879010i	$-0.658204\pi$
$523$	−21.8082	−0.953605	−0.476803	+	0.879010i	$0.658204\pi$
$524$	−9.07086	−0.396263
$525$	9.77801	0.426747
$526$	18.8670	0.822641
$527$	−20.2035	−0.880079
$528$	−8.37332	−0.364402
$529$	−21.9326	−0.953592
$530$	−5.11504	−0.222183
$531$	−79.8160	−3.46372
$532$	11.6502	0.505099
$533$	5.88938	0.255097
$534$	16.9070	0.731637
$535$	5.57565	0.241056
$536$	−9.68194	−0.418196
$537$	6.51181	0.281005
$538$	1.74916	0.0754118
$539$	−7.32143	−0.315356
$540$	11.9581	0.514593
$541$	5.22585	0.224677	0.112338	−	0.993670i	$-0.464166\pi$
$541$	5.22585	0.224677	0.112338	+	0.993670i	$0.464166\pi$
$542$	13.5454	0.581824
$543$	15.8223	0.679001
$544$	2.85678	0.122484
$545$	1.27551	0.0546368
$546$	−23.6912	−1.01389
$547$	−31.3273	−1.33946	−0.669728	−	0.742606i	$-0.733589\pi$
$547$	−31.3273	−1.33946	−0.669728	+	0.742606i	$0.733589\pi$
$548$	−15.8817	−0.678433
$549$	87.0729	3.71618
$550$	2.67249	0.113956
$551$	−1.94347	−0.0827945
$552$	3.23699	0.137776
$553$	−25.3404	−1.07759
$554$	3.24797	0.137993
$555$	−21.4521	−0.910592
$556$	−2.86616	−0.121552
$557$	−16.3760	−0.693873	−0.346936	−	0.937889i	$-0.612778\pi$
$557$	−16.3760	−0.693873	−0.346936	+	0.937889i	$0.612778\pi$
$558$	−48.2080	−2.04081
$559$	−15.4316	−0.652689
$560$	3.12083	0.131879
$561$	23.9208	1.00994
$562$	−12.0729	−0.509265
$563$	31.4652	1.32610	0.663049	−	0.748576i	$-0.269262\pi$
$563$	31.4652	1.32610	0.663049	+	0.748576i	$0.269262\pi$
$564$	0.759450	0.0319786
$565$	4.97877	0.209458
$566$	−1.88804	−0.0793604
$567$	53.1055	2.23022
$568$	−2.28354	−0.0958152
$569$	29.4663	1.23529	0.617645	−	0.786457i	$-0.288087\pi$
$569$	29.4663	1.23529	0.617645	+	0.786457i	$0.288087\pi$
$570$	−11.6962	−0.489899
$571$	−10.5248	−0.440451	−0.220225	−	0.975449i	$-0.570679\pi$
$571$	−10.5248	−0.440451	−0.220225	+	0.975449i	$0.570679\pi$
$572$	−6.47520	−0.270742
$573$	71.0843	2.96959
$574$	−7.58581	−0.316626
$575$	−1.03314	−0.0430851
$576$	6.81662	0.284026
$577$	8.05942	0.335518	0.167759	−	0.985828i	$-0.446347\pi$
$577$	8.05942	0.335518	0.167759	+	0.985828i	$0.446347\pi$
$578$	8.83879	0.367645
$579$	−22.1259	−0.919522
$580$	−0.520613	−0.0216173
$581$	−15.6834	−0.650658
$582$	−19.1797	−0.795024
$583$	−13.6699	−0.566150
$584$	−0.735162	−0.0304212
$585$	16.5160	0.682854
$586$	11.2510	0.464775
$587$	2.91058	0.120132	0.0600662	−	0.998194i	$-0.480869\pi$
$587$	2.91058	0.120132	0.0600662	+	0.998194i	$0.480869\pi$
$588$	8.58342	0.353974
$589$	26.4005	1.08781
$590$	11.7090	0.482053
$591$	−63.6936	−2.62000
$592$	−6.84683	−0.281403
$593$	0.659977	0.0271020	0.0135510	−	0.999908i	$-0.495686\pi$
$593$	0.659977	0.0271020	0.0135510	+	0.999908i	$0.495686\pi$
$594$	31.9578	1.31125
$595$	−8.91552	−0.365501
$596$	18.5892	0.761442
$597$	−27.7672	−1.13644
$598$	2.50321	0.102364
$599$	−8.70741	−0.355775	−0.177888	−	0.984051i	$-0.556926\pi$
$599$	−8.70741	−0.355775	−0.177888	+	0.984051i	$0.556926\pi$
$600$	−3.13315	−0.127910
$601$	14.0633	0.573654	0.286827	−	0.957982i	$-0.407400\pi$
$601$	14.0633	0.573654	0.286827	+	0.957982i	$0.407400\pi$
$602$	19.8767	0.810115
$603$	65.9982	2.68765
$604$	−10.2775	−0.418184
$605$	−3.85778	−0.156841
$606$	20.0496	0.814461
$607$	−19.8463	−0.805538	−0.402769	−	0.915302i	$-0.631952\pi$
$607$	−19.8463	−0.805538	−0.402769	+	0.915302i	$0.631952\pi$
$608$	−3.73304	−0.151395
$609$	−5.09056	−0.206280
$610$	−12.7736	−0.517188
$611$	0.587293	0.0237593
$612$	−19.4736	−0.787174
$613$	−7.69878	−0.310951	−0.155475	−	0.987840i	$-0.549691\pi$
$613$	−7.69878	−0.310951	−0.155475	+	0.987840i	$0.549691\pi$
$614$	17.1973	0.694025
$615$	7.61577	0.307097
$616$	8.34039	0.336044
$617$	37.2210	1.49846	0.749231	−	0.662309i	$-0.230423\pi$
$617$	37.2210	1.49846	0.749231	+	0.662309i	$0.230423\pi$
$618$	−4.91350	−0.197650
$619$	24.1197	0.969451	0.484726	−	0.874666i	$-0.338919\pi$
$619$	24.1197	0.969451	0.484726	+	0.874666i	$0.338919\pi$
$620$	7.07212	0.284023
$621$	−12.3544	−0.495765
$622$	20.4510	0.820009
$623$	−16.8405	−0.674699
$624$	7.59133	0.303896
$625$	1.00000	0.0400000
$626$	−31.6299	−1.26419
$627$	−31.2579	−1.24832
$628$	−17.6046	−0.702498
$629$	19.5599	0.779904
$630$	−21.2735	−0.847556
$631$	17.6452	0.702445	0.351223	−	0.936292i	$-0.385766\pi$
$631$	17.6452	0.702445	0.351223	+	0.936292i	$0.385766\pi$
$632$	8.11979	0.322988
$633$	59.6121	2.36937
$634$	−27.2996	−1.08421
$635$	11.0010	0.436563
$636$	16.0262	0.635479
$637$	6.63768	0.262994
$638$	−1.39133	−0.0550835
$639$	15.5660	0.615783
$640$	−1.00000	−0.0395285
$641$	−10.7640	−0.425153	−0.212577	−	0.977144i	$-0.568186\pi$
$641$	−10.7640	−0.425153	−0.212577	+	0.977144i	$0.568186\pi$
$642$	−17.4693	−0.689460
$643$	3.08672	0.121728	0.0608642	−	0.998146i	$-0.480614\pi$
$643$	3.08672	0.121728	0.0608642	+	0.998146i	$0.480614\pi$
$644$	−3.22426	−0.127054
$645$	−19.9552	−0.785736
$646$	10.6645	0.419588
$647$	25.5600	1.00487	0.502433	−	0.864616i	$-0.332438\pi$
$647$	25.5600	1.00487	0.502433	+	0.864616i	$0.332438\pi$
$648$	−17.0165	−0.668471
$649$	31.2923	1.22833
$650$	−2.42291	−0.0950342
$651$	69.1513	2.71025
$652$	12.0570	0.472187
$653$	−14.2862	−0.559062	−0.279531	−	0.960137i	$-0.590179\pi$
$653$	−14.2862	−0.559062	−0.279531	+	0.960137i	$0.590179\pi$
$654$	−3.99636	−0.156270
$655$	−9.07086	−0.354428
$656$	2.43071	0.0949032
$657$	5.01132	0.195510
$658$	−0.756463	−0.0294900
$659$	−29.3918	−1.14494	−0.572471	−	0.819925i	$-0.694015\pi$
$659$	−29.3918	−1.14494	−0.572471	+	0.819925i	$0.694015\pi$
$660$	−8.37332	−0.325931
$661$	41.7635	1.62441	0.812205	−	0.583372i	$-0.198267\pi$
$661$	41.7635	1.62441	0.812205	+	0.583372i	$0.198267\pi$
$662$	22.8926	0.889747
$663$	−21.6868	−0.842245
$664$	5.02541	0.195024
$665$	11.6502	0.451774
$666$	46.6723	1.80851
$667$	0.537868	0.0208263
$668$	−2.42388	−0.0937827
$669$	11.0453	0.427036
$670$	−9.68194	−0.374046
$671$	−34.1374	−1.31786
$672$	−9.77801	−0.377195
$673$	−18.0142	−0.694395	−0.347197	−	0.937792i	$-0.612867\pi$
$673$	−18.0142	−0.694395	−0.347197	+	0.937792i	$0.612867\pi$
$674$	29.2246	1.12569
$675$	11.9581	0.460266
$676$	−7.12952	−0.274212
$677$	−22.2267	−0.854243	−0.427121	−	0.904194i	$-0.640472\pi$
$677$	−22.2267	−0.854243	−0.427121	+	0.904194i	$0.640472\pi$
$678$	−15.5992	−0.599084
$679$	19.1043	0.733154
$680$	2.85678	0.109553
$681$	4.75905	0.182367
$682$	18.9002	0.723726
$683$	11.3353	0.433732	0.216866	−	0.976201i	$-0.430417\pi$
$683$	11.3353	0.433732	0.216866	+	0.976201i	$0.430417\pi$
$684$	25.4467	0.972980
$685$	−15.8817	−0.606809
$686$	13.2961	0.507648
$687$	−42.7677	−1.63169
$688$	−6.36906	−0.242818
$689$	12.3933	0.472146
$690$	3.23699	0.123230
$691$	−8.36262	−0.318129	−0.159065	−	0.987268i	$-0.550848\pi$
$691$	−8.36262	−0.318129	−0.159065	+	0.987268i	$0.550848\pi$
$692$	−8.11033	−0.308308
$693$	−56.8533	−2.15968
$694$	35.2311	1.33735
$695$	−2.86616	−0.108720
$696$	1.63116	0.0618289
$697$	−6.94400	−0.263023
$698$	−10.6892	−0.404592
$699$	71.0345	2.68677
$700$	3.12083	0.117956
$701$	9.46045	0.357316	0.178658	−	0.983911i	$-0.442824\pi$
$701$	9.46045	0.357316	0.178658	+	0.983911i	$0.442824\pi$
$702$	−28.9732	−1.09352
$703$	−25.5595	−0.963994
$704$	−2.67249	−0.100723
$705$	0.759450	0.0286025
$706$	−18.8743	−0.710345
$707$	−19.9708	−0.751079
$708$	−36.6861	−1.37875
$709$	−16.2829	−0.611517	−0.305759	−	0.952109i	$-0.598910\pi$
$709$	−16.2829	−0.611517	−0.305759	+	0.952109i	$0.598910\pi$
$710$	−2.28354	−0.0856997
$711$	−55.3495	−2.07577
$712$	5.39616	0.202230
$713$	−7.30652	−0.273631
$714$	27.9337	1.04539
$715$	−6.47520	−0.242159
$716$	2.07836	0.0776719
$717$	−38.8573	−1.45115
$718$	0.933001	0.0348193
$719$	−2.77474	−0.103480	−0.0517401	−	0.998661i	$-0.516477\pi$
$719$	−2.77474	−0.103480	−0.0517401	+	0.998661i	$0.516477\pi$
$720$	6.81662	0.254041
$721$	4.89418	0.182269
$722$	5.06442	0.188478
$723$	−6.73194	−0.250363
$724$	5.04998	0.187681
$725$	−0.520613	−0.0193351
$726$	12.0870	0.448590
$727$	−32.7419	−1.21433	−0.607164	−	0.794576i	$-0.707693\pi$
$727$	−32.7419	−1.21433	−0.607164	+	0.794576i	$0.707693\pi$
$728$	−7.56147	−0.280247
$729$	3.59594	0.133183
$730$	−0.735162	−0.0272096
$731$	18.1950	0.672967
$732$	40.0216	1.47924
$733$	34.4567	1.27269	0.636343	−	0.771406i	$-0.280446\pi$
$733$	34.4567	1.27269	0.636343	+	0.771406i	$0.280446\pi$
$734$	−17.5809	−0.648922
$735$	8.58342	0.316604
$736$	1.03314	0.0380822
$737$	−25.8749	−0.953115
$738$	−16.5692	−0.609921
$739$	33.1364	1.21894	0.609471	−	0.792808i	$-0.291382\pi$
$739$	33.1364	1.21894	0.609471	+	0.792808i	$0.291382\pi$
$740$	−6.84683	−0.251694
$741$	28.3387	1.04105
$742$	−15.9631	−0.586025
$743$	−38.5371	−1.41379	−0.706895	−	0.707319i	$-0.749905\pi$
$743$	−38.5371	−1.41379	−0.706895	+	0.707319i	$0.749905\pi$
$744$	−22.1580	−0.812352
$745$	18.5892	0.681054
$746$	15.3491	0.561970
$747$	−34.2563	−1.25337
$748$	7.63473	0.279153
$749$	17.4006	0.635805
$750$	−3.13315	−0.114406
$751$	5.48153	0.200024	0.100012	−	0.994986i	$-0.468112\pi$
$751$	5.48153	0.200024	0.100012	+	0.994986i	$0.468112\pi$
$752$	0.242392	0.00883912
$753$	0.195714	0.00713221
$754$	1.26140	0.0459374
$755$	−10.2775	−0.374035
$756$	37.3190	1.35728
$757$	8.63250	0.313754	0.156877	−	0.987618i	$-0.449857\pi$
$757$	8.63250	0.313754	0.156877	+	0.987618i	$0.449857\pi$
$758$	21.5490	0.782694
$759$	8.65085	0.314006
$760$	−3.73304	−0.135412
$761$	−31.5715	−1.14446	−0.572232	−	0.820092i	$-0.693922\pi$
$761$	−31.5715	−1.14446	−0.572232	+	0.820092i	$0.693922\pi$
$762$	−34.4679	−1.24864
$763$	3.98064	0.144109
$764$	22.6878	0.820816
$765$	−19.4736	−0.704070
$766$	13.6902	0.494646
$767$	−28.3699	−1.02438
$768$	3.13315	0.113058
$769$	18.8117	0.678367	0.339183	−	0.940720i	$-0.389849\pi$
$769$	18.8117	0.678367	0.339183	+	0.940720i	$0.389849\pi$
$770$	8.34039	0.300567
$771$	−26.5940	−0.957759
$772$	−7.06188	−0.254163
$773$	11.3852	0.409496	0.204748	−	0.978815i	$-0.434363\pi$
$773$	11.3852	0.409496	0.204748	+	0.978815i	$0.434363\pi$
$774$	43.4155	1.56054
$775$	7.07212	0.254038
$776$	−6.12154	−0.219750
$777$	−66.9484	−2.40176
$778$	−33.0231	−1.18393
$779$	9.07392	0.325107
$780$	7.59133	0.271813
$781$	−6.10275	−0.218373
$782$	−2.95147	−0.105544
$783$	−6.22552	−0.222482
$784$	2.73955	0.0978411
$785$	−17.6046	−0.628333
$786$	28.4204	1.01372
$787$	0.931759	0.0332136	0.0166068	−	0.999862i	$-0.494714\pi$
$787$	0.931759	0.0332136	0.0166068	+	0.999862i	$0.494714\pi$
$788$	−20.3289	−0.724188
$789$	−59.1132	−2.10449
$790$	8.11979	0.288889
$791$	15.5379	0.552463
$792$	18.2174	0.647327
$793$	30.9493	1.09904
$794$	−5.70794	−0.202567
$795$	16.0262	0.568390
$796$	−8.86241	−0.314120
$797$	−24.1948	−0.857024	−0.428512	−	0.903536i	$-0.640962\pi$
$797$	−24.1948	−0.857024	−0.428512	+	0.903536i	$0.640962\pi$
$798$	−36.5017	−1.29215
$799$	−0.692461	−0.0244975
$800$	−1.00000	−0.0353553
$801$	−36.7836	−1.29968
$802$	1.00000	0.0353112
$803$	−1.96472	−0.0693333
$804$	30.3350	1.06983
$805$	−3.22426	−0.113640
$806$	−17.1351	−0.603558
$807$	−5.48039	−0.192919
$808$	6.39920	0.225123
$809$	−37.7849	−1.32845	−0.664224	−	0.747534i	$-0.731238\pi$
$809$	−37.7849	−1.32845	−0.664224	+	0.747534i	$0.731238\pi$
$810$	−17.0165	−0.597899
$811$	38.2466	1.34302	0.671509	−	0.740996i	$-0.265646\pi$
$811$	38.2466	1.34302	0.671509	+	0.740996i	$0.265646\pi$
$812$	−1.62474	−0.0570173
$813$	−42.4397	−1.48843
$814$	−18.2981	−0.641348
$815$	12.0570	0.422337
$816$	−8.95073	−0.313338
$817$	−23.7760	−0.831816
$818$	−12.9448	−0.452604
$819$	51.5437	1.80108
$820$	2.43071	0.0848840
$821$	−18.9115	−0.660014	−0.330007	−	0.943978i	$-0.607051\pi$
$821$	−18.9115	−0.660014	−0.330007	+	0.943978i	$0.607051\pi$
$822$	49.7598	1.73557
$823$	40.8839	1.42512	0.712562	−	0.701609i	$-0.247535\pi$
$823$	40.8839	1.42512	0.712562	+	0.701609i	$0.247535\pi$
$824$	−1.56823	−0.0546319
$825$	−8.37332	−0.291522
$826$	36.5418	1.27145
$827$	−7.44212	−0.258788	−0.129394	−	0.991593i	$-0.541303\pi$
$827$	−7.44212	−0.258788	−0.129394	+	0.991593i	$0.541303\pi$
$828$	−7.04255	−0.244746
$829$	35.5779	1.23567	0.617836	−	0.786307i	$-0.288010\pi$
$829$	35.5779	1.23567	0.617836	+	0.786307i	$0.288010\pi$
$830$	5.02541	0.174434
$831$	−10.1764	−0.353015
$832$	2.42291	0.0839992
$833$	−7.82630	−0.271165
$834$	8.98010	0.310956
$835$	−2.42388	−0.0838818
$836$	−9.97652	−0.345045
$837$	84.5688	2.92313
$838$	−20.3642	−0.703469
$839$	6.04045	0.208540	0.104270	−	0.994549i	$-0.466749\pi$
$839$	6.04045	0.208540	0.104270	+	0.994549i	$0.466749\pi$
$840$	−9.77801	−0.337373
$841$	−28.7290	−0.990654
$842$	9.67447	0.333404
$843$	37.8262	1.30280
$844$	19.0262	0.654910
$845$	−7.12952	−0.245263
$846$	−1.65229	−0.0568070
$847$	−12.0395	−0.413680
$848$	5.11504	0.175651
$849$	5.91553	0.203020
$850$	2.85678	0.0979868
$851$	7.07376	0.242485
$852$	7.15467	0.245115
$853$	−15.2475	−0.522063	−0.261032	−	0.965330i	$-0.584063\pi$
$853$	−15.2475	−0.522063	−0.261032	+	0.965330i	$0.584063\pi$
$854$	−39.8642	−1.36413
$855$	25.4467	0.870260
$856$	−5.57565	−0.190572
$857$	−49.9270	−1.70547	−0.852737	−	0.522341i	$-0.825059\pi$
$857$	−49.9270	−1.70547	−0.852737	+	0.522341i	$0.825059\pi$
$858$	20.2878	0.692613
$859$	12.2746	0.418804	0.209402	−	0.977830i	$-0.432848\pi$
$859$	12.2746	0.418804	0.209402	+	0.977830i	$0.432848\pi$
$860$	−6.36906	−0.217183
$861$	23.7675	0.809994
$862$	36.1693	1.23193
$863$	0.356865	0.0121478	0.00607392	−	0.999982i	$-0.498067\pi$
$863$	0.356865	0.0121478	0.00607392	+	0.999982i	$0.498067\pi$
$864$	−11.9581	−0.406821
$865$	−8.11033	−0.275759
$866$	20.0718	0.682066
$867$	−27.6933	−0.940512
$868$	22.0709	0.749134
$869$	21.7001	0.736125
$870$	1.63116	0.0553014
$871$	23.4584	0.794859
$872$	−1.27551	−0.0431941
$873$	41.7282	1.41229
$874$	3.85677	0.130457
$875$	3.12083	0.105503
$876$	2.30337	0.0778237
$877$	−49.7510	−1.67997	−0.839986	−	0.542608i	$-0.817437\pi$
$877$	−49.7510	−1.67997	−0.839986	+	0.542608i	$0.817437\pi$
$878$	−11.0460	−0.372785
$879$	−35.2511	−1.18899
$880$	−2.67249	−0.0900897
$881$	37.9399	1.27823	0.639115	−	0.769112i	$-0.279301\pi$
$881$	37.9399	1.27823	0.639115	+	0.769112i	$0.279301\pi$
$882$	−18.6745	−0.628803
$883$	8.31663	0.279877	0.139939	−	0.990160i	$-0.455310\pi$
$883$	8.31663	0.279877	0.139939	+	0.990160i	$0.455310\pi$
$884$	−6.92172	−0.232803
$885$	−36.6861	−1.23319
$886$	−33.2080	−1.11565
$887$	−27.9568	−0.938699	−0.469349	−	0.883012i	$-0.655512\pi$
$887$	−27.9568	−0.938699	−0.469349	+	0.883012i	$0.655512\pi$
$888$	21.4521	0.719886
$889$	34.3323	1.15147
$890$	5.39616	0.180880
$891$	−45.4765	−1.52352
$892$	3.52530	0.118036
$893$	0.904858	0.0302799
$894$	−58.2426	−1.94792
$895$	2.07836	0.0694719
$896$	−3.12083	−0.104259
$897$	−7.84294	−0.261868
$898$	−22.1237	−0.738276
$899$	−3.68184	−0.122796
$900$	6.81662	0.227221
$901$	−14.6126	−0.486815
$902$	6.49605	0.216295
$903$	−62.2768	−2.07244
$904$	−4.97877	−0.165591
$905$	5.04998	0.167867
$906$	32.2008	1.06980
$907$	46.9099	1.55762	0.778808	−	0.627262i	$-0.215825\pi$
$907$	46.9099	1.55762	0.778808	+	0.627262i	$0.215825\pi$
$908$	1.51894	0.0504077
$909$	−43.6209	−1.44682
$910$	−7.56147	−0.250660
$911$	−52.8396	−1.75065	−0.875327	−	0.483532i	$-0.839354\pi$
$911$	−52.8396	−1.75065	−0.875327	+	0.483532i	$0.839354\pi$
$912$	11.6962	0.387299
$913$	13.4304	0.444480
$914$	−1.30636	−0.0432105
$915$	40.0216	1.32307
$916$	−13.6501	−0.451011
$917$	−28.3086	−0.934832
$918$	34.1616	1.12750
$919$	−27.7244	−0.914542	−0.457271	−	0.889327i	$-0.651173\pi$
$919$	−27.7244	−0.914542	−0.457271	+	0.889327i	$0.651173\pi$
$920$	1.03314	0.0340617
$921$	−53.8816	−1.77546
$922$	−9.09890	−0.299656
$923$	5.53280	0.182114
$924$	−26.1317	−0.859669
$925$	−6.84683	−0.225122
$926$	−31.0434	−1.02015
$927$	10.6900	0.351107
$928$	0.520613	0.0170900
$929$	11.2559	0.369294	0.184647	−	0.982805i	$-0.440886\pi$
$929$	11.2559	0.369294	0.184647	+	0.982805i	$0.440886\pi$
$930$	−22.1580	−0.726590
$931$	10.2268	0.335171
$932$	22.6719	0.742643
$933$	−64.0760	−2.09775
$934$	2.63348	0.0861700
$935$	7.63473	0.249682
$936$	−16.5160	−0.539844
$937$	20.6699	0.675257	0.337629	−	0.941279i	$-0.390375\pi$
$937$	20.6699	0.675257	0.337629	+	0.941279i	$0.390375\pi$
$938$	−30.2156	−0.986576
$939$	99.1013	3.23405
$940$	0.242392	0.00790595
$941$	45.6578	1.48840	0.744202	−	0.667955i	$-0.232830\pi$
$941$	45.6578	1.48840	0.744202	+	0.667955i	$0.232830\pi$
$942$	55.1577	1.79713
$943$	−2.51127	−0.0817782
$944$	−11.7090	−0.381096
$945$	37.3190	1.21399
$946$	−17.0213	−0.553409
$947$	−1.59814	−0.0519326	−0.0259663	−	0.999663i	$-0.508266\pi$
$947$	−1.59814	−0.0519326	−0.0259663	+	0.999663i	$0.508266\pi$
$948$	−25.4405	−0.826269
$949$	1.78123	0.0578211
$950$	−3.73304	−0.121116
$951$	85.5338	2.77362
$952$	8.91552	0.288954
$953$	−41.0096	−1.32843	−0.664215	−	0.747541i	$-0.731234\pi$
$953$	−41.0096	−1.32843	−0.664215	+	0.747541i	$0.731234\pi$
$954$	−34.8673	−1.12887
$955$	22.6878	0.734161
$956$	−12.4020	−0.401109
$957$	4.35926	0.140915
$958$	−25.4735	−0.823012
$959$	−49.5640	−1.60051
$960$	3.13315	0.101122
$961$	19.0149	0.613384
$962$	16.5892	0.534858
$963$	38.0071	1.22476
$964$	−2.14862	−0.0692023
$965$	−7.06188	−0.227330
$966$	10.1021	0.325030
$967$	11.6919	0.375987	0.187994	−	0.982170i	$-0.439802\pi$
$967$	11.6919	0.375987	0.187994	+	0.982170i	$0.439802\pi$
$968$	3.85778	0.123994
$969$	−33.4134	−1.07339
$970$	−6.12154	−0.196551
$971$	−49.7961	−1.59803	−0.799017	−	0.601309i	$-0.794646\pi$
$971$	−49.7961	−1.59803	−0.799017	+	0.601309i	$0.794646\pi$
$972$	17.4411	0.559422
$973$	−8.94478	−0.286757
$974$	−2.44423	−0.0783181
$975$	7.59133	0.243117
$976$	12.7736	0.408873
$977$	−19.2363	−0.615424	−0.307712	−	0.951479i	$-0.599563\pi$
$977$	−19.2363	−0.615424	−0.307712	+	0.951479i	$0.599563\pi$
$978$	−37.7762	−1.20795
$979$	14.4212	0.460904
$980$	2.73955	0.0875117
$981$	8.69466	0.277599
$982$	−27.8680	−0.889303
$983$	−7.73632	−0.246750	−0.123375	−	0.992360i	$-0.539372\pi$
$983$	−7.73632	−0.246750	−0.123375	+	0.992360i	$0.539372\pi$
$984$	−7.61577	−0.242782
$985$	−20.3289	−0.647734
$986$	−1.48728	−0.0473646
$987$	2.37011	0.0754414
$988$	9.04481	0.287754
$989$	6.58016	0.209237
$990$	18.2174	0.578986
$991$	51.2392	1.62767	0.813833	−	0.581099i	$-0.197377\pi$
$991$	51.2392	1.62767	0.813833	+	0.581099i	$0.197377\pi$
$992$	−7.07212	−0.224540
$993$	−71.7260	−2.27616
$994$	−7.12653	−0.226040
$995$	−8.86241	−0.280957
$996$	−15.7454	−0.498911
$997$	−19.7174	−0.624458	−0.312229	−	0.950007i	$-0.601076\pi$
$997$	−19.7174	−0.624458	−0.312229	+	0.950007i	$0.601076\pi$
$998$	−27.8971	−0.883066
$999$	−81.8747	−2.59040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.m.1.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.m.1.19	✓	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.13315 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.13315 q^{6} +3.12083 q^{7} -1.00000 q^{8} +6.81662 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.13315 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.13315 q^{6} +3.12083 q^{7} -1.00000 q^{8} +6.81662 q^{9} -1.00000 q^{10} -2.67249 q^{11} +3.13315 q^{12} +2.42291 q^{13} -3.12083 q^{14} +3.13315 q^{15} +1.00000 q^{16} -2.85678 q^{17} -6.81662 q^{18} +3.73304 q^{19} +1.00000 q^{20} +9.77801 q^{21} +2.67249 q^{22} -1.03314 q^{23} -3.13315 q^{24} +1.00000 q^{25} -2.42291 q^{26} +11.9581 q^{27} +3.12083 q^{28} -0.520613 q^{29} -3.13315 q^{30} +7.07212 q^{31} -1.00000 q^{32} -8.37332 q^{33} +2.85678 q^{34} +3.12083 q^{35} +6.81662 q^{36} -6.84683 q^{37} -3.73304 q^{38} +7.59133 q^{39} -1.00000 q^{40} +2.43071 q^{41} -9.77801 q^{42} -6.36906 q^{43} -2.67249 q^{44} +6.81662 q^{45} +1.03314 q^{46} +0.242392 q^{47} +3.13315 q^{48} +2.73955 q^{49} -1.00000 q^{50} -8.95073 q^{51} +2.42291 q^{52} +5.11504 q^{53} -11.9581 q^{54} -2.67249 q^{55} -3.12083 q^{56} +11.6962 q^{57} +0.520613 q^{58} -11.7090 q^{59} +3.13315 q^{60} +12.7736 q^{61} -7.07212 q^{62} +21.2735 q^{63} +1.00000 q^{64} +2.42291 q^{65} +8.37332 q^{66} +9.68194 q^{67} -2.85678 q^{68} -3.23699 q^{69} -3.12083 q^{70} +2.28354 q^{71} -6.81662 q^{72} +0.735162 q^{73} +6.84683 q^{74} +3.13315 q^{75} +3.73304 q^{76} -8.34039 q^{77} -7.59133 q^{78} -8.11979 q^{79} +1.00000 q^{80} +17.0165 q^{81} -2.43071 q^{82} -5.02541 q^{83} +9.77801 q^{84} -2.85678 q^{85} +6.36906 q^{86} -1.63116 q^{87} +2.67249 q^{88} -5.39616 q^{89} -6.81662 q^{90} +7.56147 q^{91} -1.03314 q^{92} +22.1580 q^{93} -0.242392 q^{94} +3.73304 q^{95} -3.13315 q^{96} +6.12154 q^{97} -2.73955 q^{98} -18.2174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 - 9 * q^13 - 11 * q^14 + 4 * q^15 + 20 * q^16 - 11 * q^17 - 26 * q^18 + 17 * q^19 + 20 * q^20 - 2 * q^21 - 10 * q^22 - 3 * q^23 - 4 * q^24 + 20 * q^25 + 9 * q^26 - 2 * q^27 + 11 * q^28 + 6 * q^29 - 4 * q^30 + 28 * q^31 - 20 * q^32 + 2 * q^33 + 11 * q^34 + 11 * q^35 + 26 * q^36 + 33 * q^37 - 17 * q^38 + 36 * q^39 - 20 * q^40 + 32 * q^41 + 2 * q^42 + 30 * q^43 + 10 * q^44 + 26 * q^45 + 3 * q^46 + 13 * q^47 + 4 * q^48 + 43 * q^49 - 20 * q^50 + 43 * q^51 - 9 * q^52 + 2 * q^54 + 10 * q^55 - 11 * q^56 + 19 * q^57 - 6 * q^58 + 52 * q^59 + 4 * q^60 + 25 * q^61 - 28 * q^62 + 16 * q^63 + 20 * q^64 - 9 * q^65 - 2 * q^66 + 40 * q^67 - 11 * q^68 + 39 * q^69 - 11 * q^70 + 25 * q^71 - 26 * q^72 - 5 * q^73 - 33 * q^74 + 4 * q^75 + 17 * q^76 - 9 * q^77 - 36 * q^78 + 40 * q^79 + 20 * q^80 + 48 * q^81 - 32 * q^82 + 10 * q^83 - 2 * q^84 - 11 * q^85 - 30 * q^86 + 10 * q^87 - 10 * q^88 + 17 * q^89 - 26 * q^90 + 88 * q^91 - 3 * q^92 - 4 * q^93 - 13 * q^94 + 17 * q^95 - 4 * q^96 - 2 * q^97 - 43 * q^98 + 50 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more