LMFDB - Embedded newform 4010.2.a.i.1.3

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 12x^{8} + 34x^{7} + 46x^{6} - 104x^{5} - 90x^{4} + 89x^{3} + 82x^{2} + 12x - 1 $ x^10 - 3x^9 - 12x^8 + 34x^7 + 46x^6 - 104x^5 - 90x^4 + 89x^3 + 82x^2 + 12*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.54628$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.74362 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.74362 q^{6} +2.79885 q^{7} -1.00000 q^{8} +0.0402221 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.74362 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.74362 q^{6} +2.79885 q^{7} -1.00000 q^{8} +0.0402221 q^{9} -1.00000 q^{10} +2.90568 q^{11} -1.74362 q^{12} +3.29780 q^{13} -2.79885 q^{14} -1.74362 q^{15} +1.00000 q^{16} -4.26503 q^{17} -0.0402221 q^{18} -0.133159 q^{19} +1.00000 q^{20} -4.88013 q^{21} -2.90568 q^{22} -1.71130 q^{23} +1.74362 q^{24} +1.00000 q^{25} -3.29780 q^{26} +5.16074 q^{27} +2.79885 q^{28} -2.64609 q^{29} +1.74362 q^{30} -9.13104 q^{31} -1.00000 q^{32} -5.06642 q^{33} +4.26503 q^{34} +2.79885 q^{35} +0.0402221 q^{36} -11.8979 q^{37} +0.133159 q^{38} -5.75012 q^{39} -1.00000 q^{40} -5.26369 q^{41} +4.88013 q^{42} -4.56873 q^{43} +2.90568 q^{44} +0.0402221 q^{45} +1.71130 q^{46} +6.49550 q^{47} -1.74362 q^{48} +0.833534 q^{49} -1.00000 q^{50} +7.43660 q^{51} +3.29780 q^{52} +4.03808 q^{53} -5.16074 q^{54} +2.90568 q^{55} -2.79885 q^{56} +0.232179 q^{57} +2.64609 q^{58} -6.96517 q^{59} -1.74362 q^{60} -5.75525 q^{61} +9.13104 q^{62} +0.112575 q^{63} +1.00000 q^{64} +3.29780 q^{65} +5.06642 q^{66} -1.15003 q^{67} -4.26503 q^{68} +2.98386 q^{69} -2.79885 q^{70} -0.443856 q^{71} -0.0402221 q^{72} -0.955975 q^{73} +11.8979 q^{74} -1.74362 q^{75} -0.133159 q^{76} +8.13256 q^{77} +5.75012 q^{78} -14.5373 q^{79} +1.00000 q^{80} -9.11905 q^{81} +5.26369 q^{82} +16.8564 q^{83} -4.88013 q^{84} -4.26503 q^{85} +4.56873 q^{86} +4.61378 q^{87} -2.90568 q^{88} -13.2086 q^{89} -0.0402221 q^{90} +9.23003 q^{91} -1.71130 q^{92} +15.9211 q^{93} -6.49550 q^{94} -0.133159 q^{95} +1.74362 q^{96} -8.57086 q^{97} -0.833534 q^{98} +0.116873 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{2} - 4 q^{3} + 10 q^{4} + 10 q^{5} + 4 q^{6} - 3 q^{7} - 10 q^{8} + 6 q^{9}+O(q^{10}) $ 10 * q - 10 * q^2 - 4 * q^3 + 10 * q^4 + 10 * q^5 + 4 * q^6 - 3 * q^7 - 10 * q^8 + 6 * q^9	\( 10 q - 10 q^{2} - 4 q^{3} + 10 q^{4} + 10 q^{5} + 4 q^{6} - 3 q^{7} - 10 q^{8} + 6 q^{9} - 10 q^{10} - 11 q^{11} - 4 q^{12} + 6 q^{13} + 3 q^{14} - 4 q^{15} + 10 q^{16} + 9 q^{17} - 6 q^{18} - 13 q^{19} + 10 q^{20} - 24 q^{21} + 11 q^{22} - 3 q^{23} + 4 q^{24} + 10 q^{25} - 6 q^{26} - 10 q^{27} - 3 q^{28} - 4 q^{29} + 4 q^{30} - 17 q^{31} - 10 q^{32} - 2 q^{33} - 9 q^{34} - 3 q^{35} + 6 q^{36} - 15 q^{37} + 13 q^{38} - 6 q^{39} - 10 q^{40} - 11 q^{41} + 24 q^{42} - 11 q^{43} - 11 q^{44} + 6 q^{45} + 3 q^{46} + 3 q^{47} - 4 q^{48} - 5 q^{49} - 10 q^{50} - 21 q^{51} + 6 q^{52} + 25 q^{53} + 10 q^{54} - 11 q^{55} + 3 q^{56} + 31 q^{57} + 4 q^{58} - 46 q^{59} - 4 q^{60} - 54 q^{61} + 17 q^{62} - 6 q^{63} + 10 q^{64} + 6 q^{65} + 2 q^{66} - 26 q^{67} + 9 q^{68} - 9 q^{69} + 3 q^{70} - 16 q^{71} - 6 q^{72} + 4 q^{73} + 15 q^{74} - 4 q^{75} - 13 q^{76} + 11 q^{77} + 6 q^{78} - 19 q^{79} + 10 q^{80} - 6 q^{81} + 11 q^{82} + 19 q^{83} - 24 q^{84} + 9 q^{85} + 11 q^{86} + 28 q^{87} + 11 q^{88} - 30 q^{89} - 6 q^{90} - 38 q^{91} - 3 q^{92} - 18 q^{93} - 3 q^{94} - 13 q^{95} + 4 q^{96} - 16 q^{97} + 5 q^{98} - 59 q^{99}+O(q^{100}) \) 10 * q - 10 * q^2 - 4 * q^3 + 10 * q^4 + 10 * q^5 + 4 * q^6 - 3 * q^7 - 10 * q^8 + 6 * q^9 - 10 * q^10 - 11 * q^11 - 4 * q^12 + 6 * q^13 + 3 * q^14 - 4 * q^15 + 10 * q^16 + 9 * q^17 - 6 * q^18 - 13 * q^19 + 10 * q^20 - 24 * q^21 + 11 * q^22 - 3 * q^23 + 4 * q^24 + 10 * q^25 - 6 * q^26 - 10 * q^27 - 3 * q^28 - 4 * q^29 + 4 * q^30 - 17 * q^31 - 10 * q^32 - 2 * q^33 - 9 * q^34 - 3 * q^35 + 6 * q^36 - 15 * q^37 + 13 * q^38 - 6 * q^39 - 10 * q^40 - 11 * q^41 + 24 * q^42 - 11 * q^43 - 11 * q^44 + 6 * q^45 + 3 * q^46 + 3 * q^47 - 4 * q^48 - 5 * q^49 - 10 * q^50 - 21 * q^51 + 6 * q^52 + 25 * q^53 + 10 * q^54 - 11 * q^55 + 3 * q^56 + 31 * q^57 + 4 * q^58 - 46 * q^59 - 4 * q^60 - 54 * q^61 + 17 * q^62 - 6 * q^63 + 10 * q^64 + 6 * q^65 + 2 * q^66 - 26 * q^67 + 9 * q^68 - 9 * q^69 + 3 * q^70 - 16 * q^71 - 6 * q^72 + 4 * q^73 + 15 * q^74 - 4 * q^75 - 13 * q^76 + 11 * q^77 + 6 * q^78 - 19 * q^79 + 10 * q^80 - 6 * q^81 + 11 * q^82 + 19 * q^83 - 24 * q^84 + 9 * q^85 + 11 * q^86 + 28 * q^87 + 11 * q^88 - 30 * q^89 - 6 * q^90 - 38 * q^91 - 3 * q^92 - 18 * q^93 - 3 * q^94 - 13 * q^95 + 4 * q^96 - 16 * q^97 + 5 * q^98 - 59 * 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
$3$	−1.74362	−1.00668	−0.503341	−	0.864088i	$-0.667896\pi$
$3$	−1.74362	−1.00668	−0.503341	+	0.864088i	$0.667896\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.74362	0.711831
$7$	2.79885	1.05786	0.528932	−	0.848664i	$-0.322593\pi$
$7$	2.79885	1.05786	0.528932	+	0.848664i	$0.322593\pi$
$8$	−1.00000	−0.353553
$9$	0.0402221	0.0134074
$10$	−1.00000	−0.316228
$11$	2.90568	0.876097	0.438048	−	0.898951i	$-0.355670\pi$
$11$	2.90568	0.876097	0.438048	+	0.898951i	$0.355670\pi$
$12$	−1.74362	−0.503341
$13$	3.29780	0.914645	0.457322	−	0.889301i	$-0.348809\pi$
$13$	3.29780	0.914645	0.457322	+	0.889301i	$0.348809\pi$
$14$	−2.79885	−0.748023
$15$	−1.74362	−0.450202
$16$	1.00000	0.250000
$17$	−4.26503	−1.03442	−0.517210	−	0.855858i	$-0.673030\pi$
$17$	−4.26503	−1.03442	−0.517210	+	0.855858i	$0.673030\pi$
$18$	−0.0402221	−0.00948044
$19$	−0.133159	−0.0305488	−0.0152744	−	0.999883i	$-0.504862\pi$
$19$	−0.133159	−0.0305488	−0.0152744	+	0.999883i	$0.504862\pi$
$20$	1.00000	0.223607
$21$	−4.88013	−1.06493
$22$	−2.90568	−0.619494
$23$	−1.71130	−0.356831	−0.178415	−	0.983955i	$-0.557097\pi$
$23$	−1.71130	−0.356831	−0.178415	+	0.983955i	$0.557097\pi$
$24$	1.74362	0.355916
$25$	1.00000	0.200000
$26$	−3.29780	−0.646751
$27$	5.16074	0.993184
$28$	2.79885	0.528932
$29$	−2.64609	−0.491366	−0.245683	−	0.969350i	$-0.579012\pi$
$29$	−2.64609	−0.491366	−0.245683	+	0.969350i	$0.579012\pi$
$30$	1.74362	0.318341
$31$	−9.13104	−1.63998	−0.819992	−	0.572375i	$-0.806022\pi$
$31$	−9.13104	−1.63998	−0.819992	+	0.572375i	$0.806022\pi$
$32$	−1.00000	−0.176777
$33$	−5.06642	−0.881950
$34$	4.26503	0.731446
$35$	2.79885	0.473091
$36$	0.0402221	0.00670368
$37$	−11.8979	−1.95600	−0.978001	−	0.208598i	$-0.933110\pi$
$37$	−11.8979	−1.95600	−0.978001	+	0.208598i	$0.933110\pi$
$38$	0.133159	0.0216012
$39$	−5.75012	−0.920756
$40$	−1.00000	−0.158114
$41$	−5.26369	−0.822050	−0.411025	−	0.911624i	$-0.634829\pi$
$41$	−5.26369	−0.822050	−0.411025	+	0.911624i	$0.634829\pi$
$42$	4.88013	0.753021
$43$	−4.56873	−0.696724	−0.348362	−	0.937360i	$-0.613262\pi$
$43$	−4.56873	−0.696724	−0.348362	+	0.937360i	$0.613262\pi$
$44$	2.90568	0.438048
$45$	0.0402221	0.00599596
$46$	1.71130	0.252317
$47$	6.49550	0.947466	0.473733	−	0.880668i	$-0.342906\pi$
$47$	6.49550	0.947466	0.473733	+	0.880668i	$0.342906\pi$
$48$	−1.74362	−0.251670
$49$	0.833534	0.119076
$50$	−1.00000	−0.141421
$51$	7.43660	1.04133
$52$	3.29780	0.457322
$53$	4.03808	0.554673	0.277336	−	0.960773i	$-0.410548\pi$
$53$	4.03808	0.554673	0.277336	+	0.960773i	$0.410548\pi$
$54$	−5.16074	−0.702287
$55$	2.90568	0.391802
$56$	−2.79885	−0.374011
$57$	0.232179	0.0307529
$58$	2.64609	0.347448
$59$	−6.96517	−0.906788	−0.453394	−	0.891310i	$-0.649787\pi$
$59$	−6.96517	−0.906788	−0.453394	+	0.891310i	$0.649787\pi$
$60$	−1.74362	−0.225101
$61$	−5.75525	−0.736885	−0.368442	−	0.929651i	$-0.620109\pi$
$61$	−5.75525	−0.736885	−0.368442	+	0.929651i	$0.620109\pi$
$62$	9.13104	1.15964
$63$	0.112575	0.0141832
$64$	1.00000	0.125000
$65$	3.29780	0.409041
$66$	5.06642	0.623633
$67$	−1.15003	−0.140499	−0.0702494	−	0.997529i	$-0.522380\pi$
$67$	−1.15003	−0.140499	−0.0702494	+	0.997529i	$0.522380\pi$
$68$	−4.26503	−0.517210
$69$	2.98386	0.359215
$70$	−2.79885	−0.334526
$71$	−0.443856	−0.0526760	−0.0263380	−	0.999653i	$-0.508385\pi$
$71$	−0.443856	−0.0526760	−0.0263380	+	0.999653i	$0.508385\pi$
$72$	−0.0402221	−0.00474022
$73$	−0.955975	−0.111888	−0.0559442	−	0.998434i	$-0.517817\pi$
$73$	−0.955975	−0.111888	−0.0559442	+	0.998434i	$0.517817\pi$
$74$	11.8979	1.38310
$75$	−1.74362	−0.201336
$76$	−0.133159	−0.0152744
$77$	8.13256	0.926791
$78$	5.75012	0.651073
$79$	−14.5373	−1.63557	−0.817785	−	0.575523i	$-0.804798\pi$
$79$	−14.5373	−1.63557	−0.817785	+	0.575523i	$0.804798\pi$
$80$	1.00000	0.111803
$81$	−9.11905	−1.01323
$82$	5.26369	0.581277
$83$	16.8564	1.85023	0.925115	−	0.379688i	$-0.123969\pi$
$83$	16.8564	1.85023	0.925115	+	0.379688i	$0.123969\pi$
$84$	−4.88013	−0.532466
$85$	−4.26503	−0.462607
$86$	4.56873	0.492658
$87$	4.61378	0.494649
$88$	−2.90568	−0.309747
$89$	−13.2086	−1.40011	−0.700054	−	0.714090i	$-0.746841\pi$
$89$	−13.2086	−1.40011	−0.700054	+	0.714090i	$0.746841\pi$
$90$	−0.0402221	−0.00423978
$91$	9.23003	0.967570
$92$	−1.71130	−0.178415
$93$	15.9211	1.65094
$94$	−6.49550	−0.669960
$95$	−0.133159	−0.0136618
$96$	1.74362	0.177958
$97$	−8.57086	−0.870239	−0.435120	−	0.900373i	$-0.643294\pi$
$97$	−8.57086	−0.870239	−0.435120	+	0.900373i	$0.643294\pi$
$98$	−0.833534	−0.0841997
$99$	0.116873	0.0117462
$100$	1.00000	0.100000
$101$	13.3505	1.32842	0.664211	−	0.747545i	$-0.268768\pi$
$101$	13.3505	1.32842	0.664211	+	0.747545i	$0.268768\pi$
$102$	−7.43660	−0.736333
$103$	−16.0085	−1.57737	−0.788684	−	0.614799i	$-0.789237\pi$
$103$	−16.0085	−1.57737	−0.788684	+	0.614799i	$0.789237\pi$
$104$	−3.29780	−0.323376
$105$	−4.88013	−0.476252
$106$	−4.03808	−0.392213
$107$	−10.8925	−1.05302	−0.526511	−	0.850168i	$-0.676500\pi$
$107$	−10.8925	−1.05302	−0.526511	+	0.850168i	$0.676500\pi$
$108$	5.16074	0.496592
$109$	10.3384	0.990235	0.495117	−	0.868826i	$-0.335125\pi$
$109$	10.3384	0.990235	0.495117	+	0.868826i	$0.335125\pi$
$110$	−2.90568	−0.277046
$111$	20.7455	1.96907
$112$	2.79885	0.264466
$113$	11.1906	1.05272	0.526360	−	0.850262i	$-0.323557\pi$
$113$	11.1906	1.05272	0.526360	+	0.850262i	$0.323557\pi$
$114$	−0.232179	−0.0217456
$115$	−1.71130	−0.159580
$116$	−2.64609	−0.245683
$117$	0.132644	0.0122630
$118$	6.96517	0.641196
$119$	−11.9371	−1.09428
$120$	1.74362	0.159170
$121$	−2.55700	−0.232455
$122$	5.75525	0.521056
$123$	9.17789	0.827542
$124$	−9.13104	−0.819992
$125$	1.00000	0.0894427
$126$	−0.112575	−0.0100290
$127$	−0.0819637	−0.00727310	−0.00363655	−	0.999993i	$-0.501158\pi$
$127$	−0.0819637	−0.00727310	−0.00363655	+	0.999993i	$0.501158\pi$
$128$	−1.00000	−0.0883883
$129$	7.96614	0.701379
$130$	−3.29780	−0.289236
$131$	3.94983	0.345098	0.172549	−	0.985001i	$-0.444800\pi$
$131$	3.94983	0.345098	0.172549	+	0.985001i	$0.444800\pi$
$132$	−5.06642	−0.440975
$133$	−0.372691	−0.0323164
$134$	1.15003	0.0993477
$135$	5.16074	0.444166
$136$	4.26503	0.365723
$137$	17.4396	1.48996	0.744981	−	0.667085i	$-0.232458\pi$
$137$	17.4396	1.48996	0.744981	+	0.667085i	$0.232458\pi$
$138$	−2.98386	−0.254003
$139$	−6.57556	−0.557732	−0.278866	−	0.960330i	$-0.589958\pi$
$139$	−6.57556	−0.557732	−0.278866	+	0.960330i	$0.589958\pi$
$140$	2.79885	0.236546
$141$	−11.3257	−0.953797
$142$	0.443856	0.0372476
$143$	9.58236	0.801317
$144$	0.0402221	0.00335184
$145$	−2.64609	−0.219746
$146$	0.955975	0.0791170
$147$	−1.45337	−0.119872
$148$	−11.8979	−0.978001
$149$	6.75074	0.553042	0.276521	−	0.961008i	$-0.410818\pi$
$149$	6.75074	0.553042	0.276521	+	0.961008i	$0.410818\pi$
$150$	1.74362	0.142366
$151$	9.75498	0.793849	0.396925	−	0.917851i	$-0.370077\pi$
$151$	9.75498	0.793849	0.396925	+	0.917851i	$0.370077\pi$
$152$	0.133159	0.0108006
$153$	−0.171548	−0.0138689
$154$	−8.13256	−0.655340
$155$	−9.13104	−0.733423
$156$	−5.75012	−0.460378
$157$	16.8599	1.34557	0.672784	−	0.739839i	$-0.265098\pi$
$157$	16.8599	1.34557	0.672784	+	0.739839i	$0.265098\pi$
$158$	14.5373	1.15652
$159$	−7.04089	−0.558379
$160$	−1.00000	−0.0790569
$161$	−4.78967	−0.377478
$162$	9.11905	0.716460
$163$	1.37041	0.107339	0.0536693	−	0.998559i	$-0.482908\pi$
$163$	1.37041	0.107339	0.0536693	+	0.998559i	$0.482908\pi$
$164$	−5.26369	−0.411025
$165$	−5.06642	−0.394420
$166$	−16.8564	−1.30831
$167$	−16.5346	−1.27949	−0.639743	−	0.768589i	$-0.720959\pi$
$167$	−16.5346	−1.27949	−0.639743	+	0.768589i	$0.720959\pi$
$168$	4.88013	0.376510
$169$	−2.12453	−0.163425
$170$	4.26503	0.327113
$171$	−0.00535593	−0.000409579 0
$172$	−4.56873	−0.348362
$173$	16.2885	1.23840	0.619198	−	0.785235i	$-0.287458\pi$
$173$	16.2885	1.23840	0.619198	+	0.785235i	$0.287458\pi$
$174$	−4.61378	−0.349770
$175$	2.79885	0.211573
$176$	2.90568	0.219024
$177$	12.1446	0.912846
$178$	13.2086	0.990026
$179$	−4.16912	−0.311614	−0.155807	−	0.987787i	$-0.549798\pi$
$179$	−4.16912	−0.311614	−0.155807	+	0.987787i	$0.549798\pi$
$180$	0.0402221	0.00299798
$181$	0.834483	0.0620266	0.0310133	−	0.999519i	$-0.490127\pi$
$181$	0.834483	0.0620266	0.0310133	+	0.999519i	$0.490127\pi$
$182$	−9.23003	−0.684175
$183$	10.0350	0.741808
$184$	1.71130	0.126159
$185$	−11.8979	−0.874751
$186$	−15.9211	−1.16739
$187$	−12.3928	−0.906253
$188$	6.49550	0.473733
$189$	14.4441	1.05065
$190$	0.133159	0.00966037
$191$	17.5122	1.26714	0.633569	−	0.773686i	$-0.281589\pi$
$191$	17.5122	1.26714	0.633569	+	0.773686i	$0.281589\pi$
$192$	−1.74362	−0.125835
$193$	−3.27577	−0.235795	−0.117897	−	0.993026i	$-0.537615\pi$
$193$	−3.27577	−0.235795	−0.117897	+	0.993026i	$0.537615\pi$
$194$	8.57086	0.615352
$195$	−5.75012	−0.411774
$196$	0.833534	0.0595382
$197$	−16.3680	−1.16617	−0.583087	−	0.812410i	$-0.698155\pi$
$197$	−16.3680	−1.16617	−0.583087	+	0.812410i	$0.698155\pi$
$198$	−0.116873	−0.00830578
$199$	5.25004	0.372166	0.186083	−	0.982534i	$-0.440421\pi$
$199$	5.25004	0.372166	0.186083	+	0.982534i	$0.440421\pi$
$200$	−1.00000	−0.0707107
$201$	2.00522	0.141438
$202$	−13.3505	−0.939336
$203$	−7.40599	−0.519798
$204$	7.43660	0.520666
$205$	−5.26369	−0.367632
$206$	16.0085	1.11537
$207$	−0.0688321	−0.00478416
$208$	3.29780	0.228661
$209$	−0.386918	−0.0267637
$210$	4.88013	0.336761
$211$	−18.6119	−1.28130	−0.640648	−	0.767835i	$-0.721334\pi$
$211$	−18.6119	−1.28130	−0.640648	+	0.767835i	$0.721334\pi$
$212$	4.03808	0.277336
$213$	0.773917	0.0530280
$214$	10.8925	0.744599
$215$	−4.56873	−0.311584
$216$	−5.16074	−0.351144
$217$	−25.5564	−1.73488
$218$	−10.3384	−0.700202
$219$	1.66686	0.112636
$220$	2.90568	0.195901
$221$	−14.0652	−0.946127
$222$	−20.7455	−1.39234
$223$	−5.61665	−0.376118	−0.188059	−	0.982158i	$-0.560220\pi$
$223$	−5.61665	−0.376118	−0.188059	+	0.982158i	$0.560220\pi$
$224$	−2.79885	−0.187006
$225$	0.0402221	0.00268147
$226$	−11.1906	−0.744385
$227$	−19.0382	−1.26361	−0.631806	−	0.775127i	$-0.717686\pi$
$227$	−19.0382	−1.26361	−0.631806	+	0.775127i	$0.717686\pi$
$228$	0.232179	0.0153764
$229$	−18.2529	−1.20618	−0.603092	−	0.797672i	$-0.706065\pi$
$229$	−18.2529	−1.20618	−0.603092	+	0.797672i	$0.706065\pi$
$230$	1.71130	0.112840
$231$	−14.1801	−0.932983
$232$	2.64609	0.173724
$233$	17.9389	1.17522	0.587609	−	0.809145i	$-0.300069\pi$
$233$	17.9389	1.17522	0.587609	+	0.809145i	$0.300069\pi$
$234$	−0.132644	−0.00867123
$235$	6.49550	0.423720
$236$	−6.96517	−0.453394
$237$	25.3475	1.64650
$238$	11.9371	0.773770
$239$	−27.5765	−1.78378	−0.891889	−	0.452254i	$-0.850620\pi$
$239$	−27.5765	−1.78378	−0.891889	+	0.452254i	$0.850620\pi$
$240$	−1.74362	−0.112550
$241$	9.60993	0.619030	0.309515	−	0.950895i	$-0.399833\pi$
$241$	9.60993	0.619030	0.309515	+	0.950895i	$0.399833\pi$
$242$	2.55700	0.164370
$243$	0.417972	0.0268129
$244$	−5.75525	−0.368442
$245$	0.833534	0.0532525
$246$	−9.17789	−0.585161
$247$	−0.439131	−0.0279413
$248$	9.13104	0.579822
$249$	−29.3912	−1.86259
$250$	−1.00000	−0.0632456
$251$	−27.8466	−1.75766	−0.878832	−	0.477131i	$-0.841677\pi$
$251$	−27.8466	−1.75766	−0.878832	+	0.477131i	$0.841677\pi$
$252$	0.112575	0.00709159
$253$	−4.97250	−0.312618
$254$	0.0819637	0.00514286
$255$	7.43660	0.465698
$256$	1.00000	0.0625000
$257$	1.46595	0.0914436	0.0457218	−	0.998954i	$-0.485441\pi$
$257$	1.46595	0.0914436	0.0457218	+	0.998954i	$0.485441\pi$
$258$	−7.96614	−0.495950
$259$	−33.3004	−2.06919
$260$	3.29780	0.204521
$261$	−0.106431	−0.00658792
$262$	−3.94983	−0.244021
$263$	24.0201	1.48114	0.740570	−	0.671979i	$-0.234556\pi$
$263$	24.0201	1.48114	0.740570	+	0.671979i	$0.234556\pi$
$264$	5.06642	0.311816
$265$	4.03808	0.248057
$266$	0.372691	0.0228512
$267$	23.0308	1.40946
$268$	−1.15003	−0.0702494
$269$	15.2572	0.930250	0.465125	−	0.885245i	$-0.346009\pi$
$269$	15.2572	0.930250	0.465125	+	0.885245i	$0.346009\pi$
$270$	−5.16074	−0.314072
$271$	−7.93860	−0.482235	−0.241118	−	0.970496i	$-0.577514\pi$
$271$	−7.93860	−0.482235	−0.241118	+	0.970496i	$0.577514\pi$
$272$	−4.26503	−0.258605
$273$	−16.0937	−0.974034
$274$	−17.4396	−1.05356
$275$	2.90568	0.175219
$276$	2.98386	0.179607
$277$	4.90642	0.294798	0.147399	−	0.989077i	$-0.452910\pi$
$277$	4.90642	0.294798	0.147399	+	0.989077i	$0.452910\pi$
$278$	6.57556	0.394376
$279$	−0.367270	−0.0219879
$280$	−2.79885	−0.167263
$281$	−29.6156	−1.76672	−0.883360	−	0.468695i	$-0.844724\pi$
$281$	−29.6156	−1.76672	−0.883360	+	0.468695i	$0.844724\pi$
$282$	11.3257	0.674436
$283$	2.32408	0.138152	0.0690762	−	0.997611i	$-0.477995\pi$
$283$	2.32408	0.138152	0.0690762	+	0.997611i	$0.477995\pi$
$284$	−0.443856	−0.0263380
$285$	0.232179	0.0137531
$286$	−9.58236	−0.566617
$287$	−14.7322	−0.869617
$288$	−0.0402221	−0.00237011
$289$	1.19045	0.0700267
$290$	2.64609	0.155384
$291$	14.9444	0.876054
$292$	−0.955975	−0.0559442
$293$	−24.9104	−1.45528	−0.727641	−	0.685958i	$-0.759384\pi$
$293$	−24.9104	−1.45528	−0.727641	+	0.685958i	$0.759384\pi$
$294$	1.45337	0.0847622
$295$	−6.96517	−0.405528
$296$	11.8979	0.691551
$297$	14.9955	0.870126
$298$	−6.75074	−0.391060
$299$	−5.64352	−0.326373
$300$	−1.74362	−0.100668
$301$	−12.7872	−0.737039
$302$	−9.75498	−0.561336
$303$	−23.2782	−1.33730
$304$	−0.133159	−0.00763719
$305$	−5.75525	−0.329545
$306$	0.171548	0.00980677
$307$	21.2022	1.21008	0.605038	−	0.796197i	$-0.293158\pi$
$307$	21.2022	1.21008	0.605038	+	0.796197i	$0.293158\pi$
$308$	8.13256	0.463396
$309$	27.9128	1.58791
$310$	9.13104	0.518608
$311$	5.00671	0.283904	0.141952	−	0.989874i	$-0.454662\pi$
$311$	5.00671	0.283904	0.141952	+	0.989874i	$0.454662\pi$
$312$	5.75012	0.325536
$313$	23.1915	1.31086	0.655431	−	0.755255i	$-0.272487\pi$
$313$	23.1915	1.31086	0.655431	+	0.755255i	$0.272487\pi$
$314$	−16.8599	−0.951460
$315$	0.112575	0.00634291
$316$	−14.5373	−0.817785
$317$	16.0766	0.902952	0.451476	−	0.892283i	$-0.350898\pi$
$317$	16.0766	0.902952	0.451476	+	0.892283i	$0.350898\pi$
$318$	7.04089	0.394833
$319$	−7.68869	−0.430484
$320$	1.00000	0.0559017
$321$	18.9925	1.06006
$322$	4.78967	0.266918
$323$	0.567927	0.0316003
$324$	−9.11905	−0.506614
$325$	3.29780	0.182929
$326$	−1.37041	−0.0758999
$327$	−18.0262	−0.996851
$328$	5.26369	0.290639
$329$	18.1799	1.00229
$330$	5.06642	0.278897
$331$	3.88676	0.213636	0.106818	−	0.994279i	$-0.465934\pi$
$331$	3.88676	0.213636	0.106818	+	0.994279i	$0.465934\pi$
$332$	16.8564	0.925115
$333$	−0.478559	−0.0262249
$334$	16.5346	0.904733
$335$	−1.15003	−0.0628330
$336$	−4.88013	−0.266233
$337$	9.32872	0.508168	0.254084	−	0.967182i	$-0.418226\pi$
$337$	9.32872	0.508168	0.254084	+	0.967182i	$0.418226\pi$
$338$	2.12453	0.115559
$339$	−19.5121	−1.05975
$340$	−4.26503	−0.231304
$341$	−26.5319	−1.43678
$342$	0.00535593	0.000289616 0
$343$	−17.2590	−0.931897
$344$	4.56873	0.246329
$345$	2.98386	0.160646
$346$	−16.2885	−0.875678
$347$	−34.2218	−1.83712	−0.918560	−	0.395282i	$-0.870647\pi$
$347$	−34.2218	−1.83712	−0.918560	+	0.395282i	$0.870647\pi$
$348$	4.61378	0.247324
$349$	−33.0131	−1.76715	−0.883576	−	0.468288i	$-0.844871\pi$
$349$	−33.0131	−1.76715	−0.883576	+	0.468288i	$0.844871\pi$
$350$	−2.79885	−0.149605
$351$	17.0191	0.908411
$352$	−2.90568	−0.154873
$353$	11.6475	0.619934	0.309967	−	0.950747i	$-0.399682\pi$
$353$	11.6475	0.619934	0.309967	+	0.950747i	$0.399682\pi$
$354$	−12.1446	−0.645480
$355$	−0.443856	−0.0235574
$356$	−13.2086	−0.700054
$357$	20.8139	1.10159
$358$	4.16912	0.220345
$359$	−11.8552	−0.625695	−0.312848	−	0.949803i	$-0.601283\pi$
$359$	−11.8552	−0.625695	−0.312848	+	0.949803i	$0.601283\pi$
$360$	−0.0402221	−0.00211989
$361$	−18.9823	−0.999067
$362$	−0.834483	−0.0438594
$363$	4.45845	0.234008
$364$	9.23003	0.483785
$365$	−0.955975	−0.0500380
$366$	−10.0350	−0.524538
$367$	−21.3941	−1.11676	−0.558382	−	0.829584i	$-0.688578\pi$
$367$	−21.3941	−1.11676	−0.558382	+	0.829584i	$0.688578\pi$
$368$	−1.71130	−0.0892077
$369$	−0.211717	−0.0110215
$370$	11.8979	0.618542
$371$	11.3020	0.586768
$372$	15.9211	0.825471
$373$	−22.1828	−1.14858	−0.574291	−	0.818651i	$-0.694722\pi$
$373$	−22.1828	−1.14858	−0.574291	+	0.818651i	$0.694722\pi$
$374$	12.3928	0.640817
$375$	−1.74362	−0.0900403
$376$	−6.49550	−0.334980
$377$	−8.72626	−0.449425
$378$	−14.4441	−0.742925
$379$	21.7483	1.11714	0.558569	−	0.829458i	$-0.311351\pi$
$379$	21.7483	1.11714	0.558569	+	0.829458i	$0.311351\pi$
$380$	−0.133159	−0.00683091
$381$	0.142914	0.00732169
$382$	−17.5122	−0.896002
$383$	15.3907	0.786429	0.393214	−	0.919447i	$-0.371363\pi$
$383$	15.3907	0.786429	0.393214	+	0.919447i	$0.371363\pi$
$384$	1.74362	0.0889789
$385$	8.13256	0.414474
$386$	3.27577	0.166732
$387$	−0.183764	−0.00934124
$388$	−8.57086	−0.435120
$389$	21.0759	1.06859	0.534296	−	0.845297i	$-0.320577\pi$
$389$	21.0759	1.06859	0.534296	+	0.845297i	$0.320577\pi$
$390$	5.75012	0.291168
$391$	7.29874	0.369113
$392$	−0.833534	−0.0420998
$393$	−6.88701	−0.347404
$394$	16.3680	0.824610
$395$	−14.5373	−0.731450
$396$	0.116873	0.00587308
$397$	−35.7748	−1.79549	−0.897744	−	0.440518i	$-0.854795\pi$
$397$	−35.7748	−1.79549	−0.897744	+	0.440518i	$0.854795\pi$
$398$	−5.25004	−0.263161
$399$	0.649833	0.0325324
$400$	1.00000	0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−2.00522	−0.100011
$403$	−30.1123	−1.50000
$404$	13.3505	0.664211
$405$	−9.11905	−0.453129
$406$	7.40599	0.367553
$407$	−34.5715	−1.71365
$408$	−7.43660	−0.368167
$409$	−33.6239	−1.66259	−0.831297	−	0.555828i	$-0.812401\pi$
$409$	−33.6239	−1.66259	−0.831297	+	0.555828i	$0.812401\pi$
$410$	5.26369	0.259955
$411$	−30.4080	−1.49992
$412$	−16.0085	−0.788684
$413$	−19.4944	−0.959258
$414$	0.0688321	0.00338291
$415$	16.8564	0.827448
$416$	−3.29780	−0.161688
$417$	11.4653	0.561458
$418$	0.386918	0.0189248
$419$	−5.91594	−0.289012	−0.144506	−	0.989504i	$-0.546159\pi$
$419$	−5.91594	−0.289012	−0.144506	+	0.989504i	$0.546159\pi$
$420$	−4.88013	−0.238126
$421$	−15.9075	−0.775284	−0.387642	−	0.921810i	$-0.626710\pi$
$421$	−15.9075	−0.775284	−0.387642	+	0.921810i	$0.626710\pi$
$422$	18.6119	0.906013
$423$	0.261263	0.0127030
$424$	−4.03808	−0.196106
$425$	−4.26503	−0.206884
$426$	−0.773917	−0.0374964
$427$	−16.1081	−0.779524
$428$	−10.8925	−0.526511
$429$	−16.7080	−0.806671
$430$	4.56873	0.220324
$431$	21.9256	1.05612	0.528059	−	0.849208i	$-0.322920\pi$
$431$	21.9256	1.05612	0.528059	+	0.849208i	$0.322920\pi$
$432$	5.16074	0.248296
$433$	31.5732	1.51731	0.758656	−	0.651491i	$-0.225856\pi$
$433$	31.5732	1.51731	0.758656	+	0.651491i	$0.225856\pi$
$434$	25.5564	1.22675
$435$	4.61378	0.221214
$436$	10.3384	0.495117
$437$	0.227875	0.0109007
$438$	−1.66686	−0.0796456
$439$	5.12336	0.244525	0.122262	−	0.992498i	$-0.460985\pi$
$439$	5.12336	0.244525	0.122262	+	0.992498i	$0.460985\pi$
$440$	−2.90568	−0.138523
$441$	0.0335265	0.00159650
$442$	14.0652	0.669013
$443$	−35.2770	−1.67606	−0.838030	−	0.545625i	$-0.816292\pi$
$443$	−35.2770	−1.67606	−0.838030	+	0.545625i	$0.816292\pi$
$444$	20.7455	0.984536
$445$	−13.2086	−0.626148
$446$	5.61665	0.265956
$447$	−11.7708	−0.556737
$448$	2.79885	0.132233
$449$	−13.7311	−0.648009	−0.324005	−	0.946056i	$-0.605029\pi$
$449$	−13.7311	−0.648009	−0.324005	+	0.946056i	$0.605029\pi$
$450$	−0.0402221	−0.00189609
$451$	−15.2946	−0.720195
$452$	11.1906	0.526360
$453$	−17.0090	−0.799153
$454$	19.0382	0.893508
$455$	9.23003	0.432710
$456$	−0.232179	−0.0108728
$457$	32.3369	1.51265	0.756327	−	0.654194i	$-0.226992\pi$
$457$	32.3369	1.51265	0.756327	+	0.654194i	$0.226992\pi$
$458$	18.2529	0.852900
$459$	−22.0107	−1.02737
$460$	−1.71130	−0.0797898
$461$	0.219374	0.0102173	0.00510864	−	0.999987i	$-0.498374\pi$
$461$	0.219374	0.0102173	0.00510864	+	0.999987i	$0.498374\pi$
$462$	14.1801	0.659719
$463$	−7.36374	−0.342222	−0.171111	−	0.985252i	$-0.554736\pi$
$463$	−7.36374	−0.342222	−0.171111	+	0.985252i	$0.554736\pi$
$464$	−2.64609	−0.122841
$465$	15.9211	0.738323
$466$	−17.9389	−0.831005
$467$	−13.1656	−0.609232	−0.304616	−	0.952475i	$-0.598528\pi$
$467$	−13.1656	−0.609232	−0.304616	+	0.952475i	$0.598528\pi$
$468$	0.132644	0.00613149
$469$	−3.21876	−0.148629
$470$	−6.49550	−0.299615
$471$	−29.3974	−1.35456
$472$	6.96517	0.320598
$473$	−13.2753	−0.610398
$474$	−25.3475	−1.16425
$475$	−0.133159	−0.00610975
$476$	−11.9371	−0.547138
$477$	0.162420	0.00743670
$478$	27.5765	1.26132
$479$	−28.8129	−1.31650	−0.658249	−	0.752801i	$-0.728702\pi$
$479$	−28.8129	−1.31650	−0.658249	+	0.752801i	$0.728702\pi$
$480$	1.74362	0.0795851
$481$	−39.2369	−1.78905
$482$	−9.60993	−0.437720
$483$	8.35137	0.380001
$484$	−2.55700	−0.116227
$485$	−8.57086	−0.389183
$486$	−0.417972	−0.0189596
$487$	−2.51157	−0.113810	−0.0569051	−	0.998380i	$-0.518123\pi$
$487$	−2.51157	−0.113810	−0.0569051	+	0.998380i	$0.518123\pi$
$488$	5.75525	0.260528
$489$	−2.38947	−0.108056
$490$	−0.833534	−0.0376552
$491$	−11.0188	−0.497272	−0.248636	−	0.968597i	$-0.579982\pi$
$491$	−11.0188	−0.497272	−0.248636	+	0.968597i	$0.579982\pi$
$492$	9.17789	0.413771
$493$	11.2856	0.508279
$494$	0.439131	0.0197575
$495$	0.116873	0.00525304
$496$	−9.13104	−0.409996
$497$	−1.24228	−0.0557241
$498$	29.3912	1.31705
$499$	18.0283	0.807058	0.403529	−	0.914967i	$-0.367783\pi$
$499$	18.0283	0.807058	0.403529	+	0.914967i	$0.367783\pi$
$500$	1.00000	0.0447214
$501$	28.8301	1.28803
$502$	27.8466	1.24286
$503$	30.9924	1.38188	0.690941	−	0.722911i	$-0.257196\pi$
$503$	30.9924	1.38188	0.690941	+	0.722911i	$0.257196\pi$
$504$	−0.112575	−0.00501451
$505$	13.3505	0.594088
$506$	4.97250	0.221055
$507$	3.70438	0.164517
$508$	−0.0819637	−0.00363655
$509$	10.7138	0.474880	0.237440	−	0.971402i	$-0.423692\pi$
$509$	10.7138	0.474880	0.237440	+	0.971402i	$0.423692\pi$
$510$	−7.43660	−0.329298
$511$	−2.67562	−0.118363
$512$	−1.00000	−0.0441942
$513$	−0.687198	−0.0303406
$514$	−1.46595	−0.0646604
$515$	−16.0085	−0.705420
$516$	7.96614	0.350690
$517$	18.8739	0.830072
$518$	33.3004	1.46313
$519$	−28.4011	−1.24667
$520$	−3.29780	−0.144618
$521$	37.4207	1.63943	0.819715	−	0.572772i	$-0.194132\pi$
$521$	37.4207	1.63943	0.819715	+	0.572772i	$0.194132\pi$
$522$	0.106431	0.00465837
$523$	18.2035	0.795982	0.397991	−	0.917389i	$-0.369708\pi$
$523$	18.2035	0.795982	0.397991	+	0.917389i	$0.369708\pi$
$524$	3.94983	0.172549
$525$	−4.88013	−0.212986
$526$	−24.0201	−1.04732
$527$	38.9441	1.69643
$528$	−5.06642	−0.220488
$529$	−20.0715	−0.872672
$530$	−4.03808	−0.175403
$531$	−0.280154	−0.0121576
$532$	−0.372691	−0.0161582
$533$	−17.3586	−0.751883
$534$	−23.0308	−0.996641
$535$	−10.8925	−0.470926
$536$	1.15003	0.0496738
$537$	7.26937	0.313696
$538$	−15.2572	−0.657786
$539$	2.42199	0.104322
$540$	5.16074	0.222083
$541$	12.4127	0.533662	0.266831	−	0.963743i	$-0.414023\pi$
$541$	12.4127	0.533662	0.266831	+	0.963743i	$0.414023\pi$
$542$	7.93860	0.340992
$543$	−1.45502	−0.0624410
$544$	4.26503	0.182862
$545$	10.3384	0.442846
$546$	16.0937	0.688746
$547$	36.2153	1.54845	0.774227	−	0.632908i	$-0.218139\pi$
$547$	36.2153	1.54845	0.774227	+	0.632908i	$0.218139\pi$
$548$	17.4396	0.744981
$549$	−0.231488	−0.00987969
$550$	−2.90568	−0.123899
$551$	0.352350	0.0150106
$552$	−2.98386	−0.127002
$553$	−40.6876	−1.73021
$554$	−4.90642	−0.208454
$555$	20.7455	0.880596
$556$	−6.57556	−0.278866
$557$	10.1921	0.431854	0.215927	−	0.976410i	$-0.430723\pi$
$557$	10.1921	0.431854	0.215927	+	0.976410i	$0.430723\pi$
$558$	0.367270	0.0155478
$559$	−15.0667	−0.637255
$560$	2.79885	0.118273
$561$	21.6084	0.912308
$562$	29.6156	1.24926
$563$	0.952127	0.0401274	0.0200637	−	0.999799i	$-0.493613\pi$
$563$	0.952127	0.0401274	0.0200637	+	0.999799i	$0.493613\pi$
$564$	−11.3257	−0.476898
$565$	11.1906	0.470790
$566$	−2.32408	−0.0976885
$567$	−25.5228	−1.07186
$568$	0.443856	0.0186238
$569$	−1.70621	−0.0715282	−0.0357641	−	0.999360i	$-0.511386\pi$
$569$	−1.70621	−0.0715282	−0.0357641	+	0.999360i	$0.511386\pi$
$570$	−0.232179	−0.00972491
$571$	1.32147	0.0553017	0.0276508	−	0.999618i	$-0.491197\pi$
$571$	1.32147	0.0553017	0.0276508	+	0.999618i	$0.491197\pi$
$572$	9.58236	0.400659
$573$	−30.5347	−1.27560
$574$	14.7322	0.614912
$575$	−1.71130	−0.0713662
$576$	0.0402221	0.00167592
$577$	−23.1428	−0.963448	−0.481724	−	0.876323i	$-0.659989\pi$
$577$	−23.1428	−0.963448	−0.481724	+	0.876323i	$0.659989\pi$
$578$	−1.19045	−0.0495164
$579$	5.71170	0.237370
$580$	−2.64609	−0.109873
$581$	47.1784	1.95729
$582$	−14.9444	−0.619464
$583$	11.7334	0.485947
$584$	0.955975	0.0395585
$585$	0.132644	0.00548417
$586$	24.9104	1.02904
$587$	37.1775	1.53448	0.767240	−	0.641360i	$-0.221629\pi$
$587$	37.1775	1.53448	0.767240	+	0.641360i	$0.221629\pi$
$588$	−1.45337	−0.0599360
$589$	1.21588	0.0500995
$590$	6.96517	0.286751
$591$	28.5397	1.17397
$592$	−11.8979	−0.489001
$593$	20.3523	0.835769	0.417885	−	0.908500i	$-0.362772\pi$
$593$	20.3523	0.835769	0.417885	+	0.908500i	$0.362772\pi$
$594$	−14.9955	−0.615272
$595$	−11.9371	−0.489375
$596$	6.75074	0.276521
$597$	−9.15410	−0.374652
$598$	5.64352	0.230781
$599$	−41.5364	−1.69713	−0.848566	−	0.529089i	$-0.822534\pi$
$599$	−41.5364	−1.69713	−0.848566	+	0.529089i	$0.822534\pi$
$600$	1.74362	0.0711831
$601$	26.1358	1.06610	0.533050	−	0.846084i	$-0.321046\pi$
$601$	26.1358	1.06610	0.533050	+	0.846084i	$0.321046\pi$
$602$	12.7872	0.521166
$603$	−0.0462567	−0.00188372
$604$	9.75498	0.396925
$605$	−2.55700	−0.103957
$606$	23.2782	0.945612
$607$	−6.25162	−0.253745	−0.126873	−	0.991919i	$-0.540494\pi$
$607$	−6.25162	−0.253745	−0.126873	+	0.991919i	$0.540494\pi$
$608$	0.133159	0.00540031
$609$	12.9133	0.523271
$610$	5.75525	0.233023
$611$	21.4209	0.866595
$612$	−0.171548	−0.00693443
$613$	4.38076	0.176937	0.0884685	−	0.996079i	$-0.471803\pi$
$613$	4.38076	0.176937	0.0884685	+	0.996079i	$0.471803\pi$
$614$	−21.2022	−0.855653
$615$	9.17789	0.370088
$616$	−8.13256	−0.327670
$617$	−35.3261	−1.42217	−0.711087	−	0.703104i	$-0.751797\pi$
$617$	−35.3261	−1.42217	−0.711087	+	0.703104i	$0.751797\pi$
$618$	−27.9128	−1.12282
$619$	−13.5824	−0.545922	−0.272961	−	0.962025i	$-0.588003\pi$
$619$	−13.5824	−0.545922	−0.272961	+	0.962025i	$0.588003\pi$
$620$	−9.13104	−0.366712
$621$	−8.83157	−0.354399
$622$	−5.00671	−0.200751
$623$	−36.9688	−1.48112
$624$	−5.75012	−0.230189
$625$	1.00000	0.0400000
$626$	−23.1915	−0.926920
$627$	0.674639	0.0269425
$628$	16.8599	0.672784
$629$	50.7449	2.02333
$630$	−0.112575	−0.00448511
$631$	8.62253	0.343257	0.171629	−	0.985162i	$-0.445097\pi$
$631$	8.62253	0.343257	0.171629	+	0.985162i	$0.445097\pi$
$632$	14.5373	0.578262
$633$	32.4521	1.28986
$634$	−16.0766	−0.638483
$635$	−0.0819637	−0.00325263
$636$	−7.04089	−0.279189
$637$	2.74883	0.108913
$638$	7.68869	0.304398
$639$	−0.0178528	−0.000706247 0
$640$	−1.00000	−0.0395285
$641$	4.17080	0.164737	0.0823683	−	0.996602i	$-0.473752\pi$
$641$	4.17080	0.164737	0.0823683	+	0.996602i	$0.473752\pi$
$642$	−18.9925	−0.749574
$643$	31.3917	1.23797	0.618984	−	0.785403i	$-0.287544\pi$
$643$	31.3917	1.23797	0.618984	+	0.785403i	$0.287544\pi$
$644$	−4.78967	−0.188739
$645$	7.96614	0.313666
$646$	−0.567927	−0.0223448
$647$	16.0835	0.632309	0.316154	−	0.948708i	$-0.397608\pi$
$647$	16.0835	0.632309	0.316154	+	0.948708i	$0.397608\pi$
$648$	9.11905	0.358230
$649$	−20.2386	−0.794434
$650$	−3.29780	−0.129350
$651$	44.5607	1.74647
$652$	1.37041	0.0536693
$653$	−14.8296	−0.580328	−0.290164	−	0.956977i	$-0.593710\pi$
$653$	−14.8296	−0.580328	−0.290164	+	0.956977i	$0.593710\pi$
$654$	18.0262	0.704880
$655$	3.94983	0.154332
$656$	−5.26369	−0.205512
$657$	−0.0384513	−0.00150013
$658$	−18.1799	−0.708726
$659$	−45.1202	−1.75763	−0.878817	−	0.477159i	$-0.841667\pi$
$659$	−45.1202	−1.75763	−0.878817	+	0.477159i	$0.841667\pi$
$660$	−5.06642	−0.197210
$661$	22.9207	0.891513	0.445757	−	0.895154i	$-0.352935\pi$
$661$	22.9207	0.891513	0.445757	+	0.895154i	$0.352935\pi$
$662$	−3.88676	−0.151063
$663$	24.5244	0.952449
$664$	−16.8564	−0.654155
$665$	−0.372691	−0.0144523
$666$	0.478559	0.0185438
$667$	4.52825	0.175335
$668$	−16.5346	−0.639743
$669$	9.79331	0.378631
$670$	1.15003	0.0444296
$671$	−16.7229	−0.645582
$672$	4.88013	0.188255
$673$	22.9161	0.883352	0.441676	−	0.897175i	$-0.354384\pi$
$673$	22.9161	0.883352	0.441676	+	0.897175i	$0.354384\pi$
$674$	−9.32872	−0.359329
$675$	5.16074	0.198637
$676$	−2.12453	−0.0817127
$677$	−22.9572	−0.882317	−0.441158	−	0.897429i	$-0.645432\pi$
$677$	−22.9572	−0.882317	−0.441158	+	0.897429i	$0.645432\pi$
$678$	19.5121	0.749358
$679$	−23.9885	−0.920595
$680$	4.26503	0.163556
$681$	33.1955	1.27205
$682$	26.5319	1.01596
$683$	3.83965	0.146920	0.0734600	−	0.997298i	$-0.476596\pi$
$683$	3.83965	0.146920	0.0734600	+	0.997298i	$0.476596\pi$
$684$	−0.00535593	−0.000204789 0
$685$	17.4396	0.666331
$686$	17.2590	0.658951
$687$	31.8261	1.21424
$688$	−4.56873	−0.174181
$689$	13.3168	0.507328
$690$	−2.98386	−0.113594
$691$	−22.2790	−0.847533	−0.423767	−	0.905771i	$-0.639292\pi$
$691$	−22.2790	−0.847533	−0.423767	+	0.905771i	$0.639292\pi$
$692$	16.2885	0.619198
$693$	0.327109	0.0124258
$694$	34.2218	1.29904
$695$	−6.57556	−0.249425
$696$	−4.61378	−0.174885
$697$	22.4498	0.850346
$698$	33.0131	1.24956
$699$	−31.2787	−1.18307
$700$	2.79885	0.105786
$701$	45.8891	1.73321	0.866603	−	0.498997i	$-0.166298\pi$
$701$	45.8891	1.73321	0.866603	+	0.498997i	$0.166298\pi$
$702$	−17.0191	−0.642343
$703$	1.58431	0.0597535
$704$	2.90568	0.109512
$705$	−11.3257	−0.426551
$706$	−11.6475	−0.438359
$707$	37.3659	1.40529
$708$	12.1446	0.456423
$709$	24.5503	0.922006	0.461003	−	0.887398i	$-0.347490\pi$
$709$	24.5503	0.922006	0.461003	+	0.887398i	$0.347490\pi$
$710$	0.443856	0.0166576
$711$	−0.584720	−0.0219287
$712$	13.2086	0.495013
$713$	15.6260	0.585197
$714$	−20.8139	−0.778940
$715$	9.58236	0.358360
$716$	−4.16912	−0.155807
$717$	48.0831	1.79570
$718$	11.8552	0.442433
$719$	−19.2846	−0.719193	−0.359597	−	0.933108i	$-0.617086\pi$
$719$	−19.2846	−0.719193	−0.359597	+	0.933108i	$0.617086\pi$
$720$	0.0402221	0.00149899
$721$	−44.8054	−1.66864
$722$	18.9823	0.706447
$723$	−16.7561	−0.623166
$724$	0.834483	0.0310133
$725$	−2.64609	−0.0982732
$726$	−4.45845	−0.165468
$727$	6.33098	0.234803	0.117402	−	0.993085i	$-0.462544\pi$
$727$	6.33098	0.234803	0.117402	+	0.993085i	$0.462544\pi$
$728$	−9.23003	−0.342088
$729$	26.6284	0.986236
$730$	0.955975	0.0353822
$731$	19.4857	0.720706
$732$	10.0350	0.370904
$733$	39.5562	1.46104	0.730521	−	0.682890i	$-0.239277\pi$
$733$	39.5562	1.46104	0.730521	+	0.682890i	$0.239277\pi$
$734$	21.3941	0.789671
$735$	−1.45337	−0.0536083
$736$	1.71130	0.0630794
$737$	−3.34163	−0.123091
$738$	0.211717	0.00779340
$739$	−2.00932	−0.0739141	−0.0369571	−	0.999317i	$-0.511766\pi$
$739$	−2.00932	−0.0739141	−0.0369571	+	0.999317i	$0.511766\pi$
$740$	−11.8979	−0.437376
$741$	0.765680	0.0281279
$742$	−11.3020	−0.414908
$743$	−39.4729	−1.44812	−0.724060	−	0.689737i	$-0.757726\pi$
$743$	−39.4729	−1.44812	−0.724060	+	0.689737i	$0.757726\pi$
$744$	−15.9211	−0.583696
$745$	6.75074	0.247328
$746$	22.1828	0.812170
$747$	0.678000	0.0248067
$748$	−12.3928	−0.453126
$749$	−30.4865	−1.11395
$750$	1.74362	0.0636681
$751$	0.749562	0.0273519	0.0136760	−	0.999906i	$-0.495647\pi$
$751$	0.749562	0.0273519	0.0136760	+	0.999906i	$0.495647\pi$
$752$	6.49550	0.236867
$753$	48.5541	1.76941
$754$	8.72626	0.317792
$755$	9.75498	0.355020
$756$	14.4441	0.525327
$757$	13.3021	0.483472	0.241736	−	0.970342i	$-0.422283\pi$
$757$	13.3021	0.483472	0.241736	+	0.970342i	$0.422283\pi$
$758$	−21.7483	−0.789935
$759$	8.67016	0.314707
$760$	0.133159	0.00483018
$761$	−30.1643	−1.09345	−0.546727	−	0.837311i	$-0.684126\pi$
$761$	−30.1643	−1.09345	−0.546727	+	0.837311i	$0.684126\pi$
$762$	−0.142914	−0.00517722
$763$	28.9355	1.04753
$764$	17.5122	0.633569
$765$	−0.171548	−0.00620234
$766$	−15.3907	−0.556089
$767$	−22.9697	−0.829388
$768$	−1.74362	−0.0629176
$769$	−31.2458	−1.12675	−0.563377	−	0.826200i	$-0.690498\pi$
$769$	−31.2458	−1.12675	−0.563377	+	0.826200i	$0.690498\pi$
$770$	−8.13256	−0.293077
$771$	−2.55607	−0.0920546
$772$	−3.27577	−0.117897
$773$	5.49037	0.197475	0.0987375	−	0.995114i	$-0.468520\pi$
$773$	5.49037	0.197475	0.0987375	+	0.995114i	$0.468520\pi$
$774$	0.183764	0.00660525
$775$	−9.13104	−0.327997
$776$	8.57086	0.307676
$777$	58.0633	2.08301
$778$	−21.0759	−0.755609
$779$	0.700907	0.0251126
$780$	−5.75012	−0.205887
$781$	−1.28971	−0.0461493
$782$	−7.29874	−0.261003
$783$	−13.6558	−0.488017
$784$	0.833534	0.0297691
$785$	16.8599	0.601756
$786$	6.88701	0.245651
$787$	−19.4370	−0.692854	−0.346427	−	0.938077i	$-0.612605\pi$
$787$	−19.4370	−0.692854	−0.346427	+	0.938077i	$0.612605\pi$
$788$	−16.3680	−0.583087
$789$	−41.8819	−1.49104
$790$	14.5373	0.517213
$791$	31.3206	1.11363
$792$	−0.116873	−0.00415289
$793$	−18.9797	−0.673988
$794$	35.7748	1.26960
$795$	−7.04089	−0.249715
$796$	5.25004	0.186083
$797$	−8.35740	−0.296034	−0.148017	−	0.988985i	$-0.547289\pi$
$797$	−8.35740	−0.296034	−0.148017	+	0.988985i	$0.547289\pi$
$798$	−0.649833	−0.0230038
$799$	−27.7035	−0.980079
$800$	−1.00000	−0.0353553
$801$	−0.531278	−0.0187718
$802$	−1.00000	−0.0353112
$803$	−2.77776	−0.0980250
$804$	2.00522	0.0707188
$805$	−4.78967	−0.168814
$806$	30.1123	1.06066
$807$	−26.6029	−0.936465
$808$	−13.3505	−0.469668
$809$	46.6245	1.63923	0.819615	−	0.572914i	$-0.194187\pi$
$809$	46.6245	1.63923	0.819615	+	0.572914i	$0.194187\pi$
$810$	9.11905	0.320411
$811$	25.6122	0.899368	0.449684	−	0.893188i	$-0.351537\pi$
$811$	25.6122	0.899368	0.449684	+	0.893188i	$0.351537\pi$
$812$	−7.40599	−0.259899
$813$	13.8419	0.485457
$814$	34.5715	1.21173
$815$	1.37041	0.0480033
$816$	7.43660	0.260333
$817$	0.608367	0.0212841
$818$	33.6239	1.17563
$819$	0.371251	0.0129726
$820$	−5.26369	−0.183816
$821$	27.8303	0.971285	0.485643	−	0.874157i	$-0.338586\pi$
$821$	27.8303	0.971285	0.485643	+	0.874157i	$0.338586\pi$
$822$	30.4080	1.06060
$823$	−3.07396	−0.107151	−0.0535757	−	0.998564i	$-0.517062\pi$
$823$	−3.07396	−0.107151	−0.0535757	+	0.998564i	$0.517062\pi$
$824$	16.0085	0.557684
$825$	−5.06642	−0.176390
$826$	19.4944	0.678298
$827$	39.3794	1.36935	0.684677	−	0.728846i	$-0.259943\pi$
$827$	39.3794	1.36935	0.684677	+	0.728846i	$0.259943\pi$
$828$	−0.0688321	−0.00239208
$829$	−35.6968	−1.23980	−0.619900	−	0.784680i	$-0.712827\pi$
$829$	−35.6968	−1.23980	−0.619900	+	0.784680i	$0.712827\pi$
$830$	−16.8564	−0.585094
$831$	−8.55495	−0.296768
$832$	3.29780	0.114331
$833$	−3.55505	−0.123175
$834$	−11.4653	−0.397011
$835$	−16.5346	−0.572203
$836$	−0.386918	−0.0133818
$837$	−47.1229	−1.62881
$838$	5.91594	0.204363
$839$	−40.1386	−1.38574	−0.692869	−	0.721063i	$-0.743654\pi$
$839$	−40.1386	−1.38574	−0.692869	+	0.721063i	$0.743654\pi$
$840$	4.88013	0.168381
$841$	−21.9982	−0.758560
$842$	15.9075	0.548209
$843$	51.6385	1.77852
$844$	−18.6119	−0.640648
$845$	−2.12453	−0.0730860
$846$	−0.261263	−0.00898240
$847$	−7.15665	−0.245905
$848$	4.03808	0.138668
$849$	−4.05232	−0.139075
$850$	4.26503	0.146289
$851$	20.3609	0.697962
$852$	0.773917	0.0265140
$853$	−29.0724	−0.995421	−0.497711	−	0.867343i	$-0.665826\pi$
$853$	−29.0724	−0.995421	−0.497711	+	0.867343i	$0.665826\pi$
$854$	16.1081	0.551207
$855$	−0.00535593	−0.000183169 0
$856$	10.8925	0.372299
$857$	51.8740	1.77198	0.885990	−	0.463704i	$-0.153480\pi$
$857$	51.8740	1.77198	0.885990	+	0.463704i	$0.153480\pi$
$858$	16.7080	0.570403
$859$	27.4459	0.936443	0.468222	−	0.883611i	$-0.344895\pi$
$859$	27.4459	0.936443	0.468222	+	0.883611i	$0.344895\pi$
$860$	−4.56873	−0.155792
$861$	25.6875	0.875427
$862$	−21.9256	−0.746788
$863$	−44.1100	−1.50152	−0.750761	−	0.660574i	$-0.770313\pi$
$863$	−44.1100	−1.50152	−0.750761	+	0.660574i	$0.770313\pi$
$864$	−5.16074	−0.175572
$865$	16.2885	0.553827
$866$	−31.5732	−1.07290
$867$	−2.07570	−0.0704946
$868$	−25.5564	−0.867440
$869$	−42.2407	−1.43292
$870$	−4.61378	−0.156422
$871$	−3.79257	−0.128506
$872$	−10.3384	−0.350101
$873$	−0.344738	−0.0116676
$874$	−0.227875	−0.00770799
$875$	2.79885	0.0946182
$876$	1.66686	0.0563180
$877$	13.0104	0.439331	0.219666	−	0.975575i	$-0.429503\pi$
$877$	13.0104	0.439331	0.219666	+	0.975575i	$0.429503\pi$
$878$	−5.12336	−0.172905
$879$	43.4344	1.46501
$880$	2.90568	0.0979506
$881$	47.0400	1.58482	0.792408	−	0.609991i	$-0.208827\pi$
$881$	47.0400	1.58482	0.792408	+	0.609991i	$0.208827\pi$
$882$	−0.0335265	−0.00112890
$883$	−52.3454	−1.76156	−0.880782	−	0.473522i	$-0.842982\pi$
$883$	−52.3454	−1.76156	−0.880782	+	0.473522i	$0.842982\pi$
$884$	−14.0652	−0.473064
$885$	12.1446	0.408237
$886$	35.2770	1.18515
$887$	29.3491	0.985447	0.492723	−	0.870186i	$-0.336001\pi$
$887$	29.3491	0.985447	0.492723	+	0.870186i	$0.336001\pi$
$888$	−20.7455	−0.696172
$889$	−0.229404	−0.00769395
$890$	13.2086	0.442753
$891$	−26.4971	−0.887685
$892$	−5.61665	−0.188059
$893$	−0.864934	−0.0289439
$894$	11.7708	0.393673
$895$	−4.16912	−0.139358
$896$	−2.79885	−0.0935029
$897$	9.84018	0.328554
$898$	13.7311	0.458212
$899$	24.1615	0.805832
$900$	0.0402221	0.00134074
$901$	−17.2225	−0.573765
$902$	15.2946	0.509255
$903$	22.2960	0.741964
$904$	−11.1906	−0.372192
$905$	0.834483	0.0277392
$906$	17.0090	0.565087
$907$	−10.7838	−0.358069	−0.179034	−	0.983843i	$-0.557297\pi$
$907$	−10.7838	−0.358069	−0.179034	+	0.983843i	$0.557297\pi$
$908$	−19.0382	−0.631806
$909$	0.536984	0.0178106
$910$	−9.23003	−0.305972
$911$	24.3801	0.807747	0.403874	−	0.914815i	$-0.367664\pi$
$911$	24.3801	0.807747	0.403874	+	0.914815i	$0.367664\pi$
$912$	0.232179	0.00768822
$913$	48.9793	1.62098
$914$	−32.3369	−1.06961
$915$	10.0350	0.331747
$916$	−18.2529	−0.603092
$917$	11.0550	0.365067
$918$	22.0107	0.726461
$919$	−37.3006	−1.23043	−0.615216	−	0.788358i	$-0.710931\pi$
$919$	−37.3006	−1.23043	−0.615216	+	0.788358i	$0.710931\pi$
$920$	1.71130	0.0564199
$921$	−36.9687	−1.21816
$922$	−0.219374	−0.00722470
$923$	−1.46375	−0.0481798
$924$	−14.1801	−0.466492
$925$	−11.8979	−0.391201
$926$	7.36374	0.241988
$927$	−0.643897	−0.0211483
$928$	2.64609	0.0868620
$929$	42.9751	1.40997	0.704984	−	0.709223i	$-0.250954\pi$
$929$	42.9751	1.40997	0.704984	+	0.709223i	$0.250954\pi$
$930$	−15.9211	−0.522073
$931$	−0.110993	−0.00363763
$932$	17.9389	0.587609
$933$	−8.72981	−0.285801
$934$	13.1656	0.430792
$935$	−12.3928	−0.405289
$936$	−0.132644	−0.00433562
$937$	11.9642	0.390853	0.195426	−	0.980718i	$-0.437391\pi$
$937$	11.9642	0.390853	0.195426	+	0.980718i	$0.437391\pi$
$938$	3.21876	0.105096
$939$	−40.4373	−1.31962
$940$	6.49550	0.211860
$941$	−6.91846	−0.225535	−0.112768	−	0.993621i	$-0.535972\pi$
$941$	−6.91846	−0.225535	−0.112768	+	0.993621i	$0.535972\pi$
$942$	29.3974	0.957817
$943$	9.00775	0.293333
$944$	−6.96517	−0.226697
$945$	14.4441	0.469867
$946$	13.2753	0.431616
$947$	−9.80580	−0.318646	−0.159323	−	0.987227i	$-0.550931\pi$
$947$	−9.80580	−0.318646	−0.159323	+	0.987227i	$0.550931\pi$
$948$	25.3475	0.823249
$949$	−3.15261	−0.102338
$950$	0.133159	0.00432025
$951$	−28.0315	−0.908985
$952$	11.9371	0.386885
$953$	22.4361	0.726776	0.363388	−	0.931638i	$-0.381620\pi$
$953$	22.4361	0.726776	0.363388	+	0.931638i	$0.381620\pi$
$954$	−0.162420	−0.00525854
$955$	17.5122	0.566681
$956$	−27.5765	−0.891889
$957$	13.4062	0.433360
$958$	28.8129	0.930904
$959$	48.8106	1.57618
$960$	−1.74362	−0.0562752
$961$	52.3759	1.68955
$962$	39.2369	1.26505
$963$	−0.438121	−0.0141182
$964$	9.60993	0.309515
$965$	−3.27577	−0.105451
$966$	−8.35137	−0.268701
$967$	−52.4073	−1.68531	−0.842653	−	0.538457i	$-0.819008\pi$
$967$	−52.4073	−1.68531	−0.842653	+	0.538457i	$0.819008\pi$
$968$	2.55700	0.0821851
$969$	−0.990250	−0.0318114
$970$	8.57086	0.275194
$971$	−13.7733	−0.442005	−0.221002	−	0.975273i	$-0.570933\pi$
$971$	−13.7733	−0.442005	−0.221002	+	0.975273i	$0.570933\pi$
$972$	0.417972	0.0134065
$973$	−18.4040	−0.590004
$974$	2.51157	0.0804760
$975$	−5.75012	−0.184151
$976$	−5.75525	−0.184221
$977$	2.88069	0.0921616	0.0460808	−	0.998938i	$-0.485327\pi$
$977$	2.88069	0.0921616	0.0460808	+	0.998938i	$0.485327\pi$
$978$	2.38947	0.0764070
$979$	−38.3800	−1.22663
$980$	0.833534	0.0266263
$981$	0.415830	0.0132764
$982$	11.0188	0.351624
$983$	16.0515	0.511963	0.255981	−	0.966682i	$-0.417601\pi$
$983$	16.0515	0.511963	0.255981	+	0.966682i	$0.417601\pi$
$984$	−9.17789	−0.292580
$985$	−16.3680	−0.521529
$986$	−11.2856	−0.359408
$987$	−31.6989	−1.00899
$988$	−0.439131	−0.0139706
$989$	7.81846	0.248613
$990$	−0.116873	−0.00371446
$991$	28.2981	0.898920	0.449460	−	0.893301i	$-0.351616\pi$
$991$	28.2981	0.898920	0.449460	+	0.893301i	$0.351616\pi$
$992$	9.13104	0.289911
$993$	−6.77705	−0.215063
$994$	1.24228	0.0394029
$995$	5.25004	0.166438
$996$	−29.3912	−0.931296
$997$	−39.4637	−1.24983	−0.624913	−	0.780694i	$-0.714866\pi$
$997$	−39.4637	−1.24983	−0.624913	+	0.780694i	$0.714866\pi$
$998$	−18.0283	−0.570676
$999$	−61.4019	−1.94267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.i.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.i.1.3	✓	10	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.74362 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.74362 q^{6} +2.79885 q^{7} -1.00000 q^{8} +0.0402221 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.74362 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.74362 q^{6} +2.79885 q^{7} -1.00000 q^{8} +0.0402221 q^{9} -1.00000 q^{10} +2.90568 q^{11} -1.74362 q^{12} +3.29780 q^{13} -2.79885 q^{14} -1.74362 q^{15} +1.00000 q^{16} -4.26503 q^{17} -0.0402221 q^{18} -0.133159 q^{19} +1.00000 q^{20} -4.88013 q^{21} -2.90568 q^{22} -1.71130 q^{23} +1.74362 q^{24} +1.00000 q^{25} -3.29780 q^{26} +5.16074 q^{27} +2.79885 q^{28} -2.64609 q^{29} +1.74362 q^{30} -9.13104 q^{31} -1.00000 q^{32} -5.06642 q^{33} +4.26503 q^{34} +2.79885 q^{35} +0.0402221 q^{36} -11.8979 q^{37} +0.133159 q^{38} -5.75012 q^{39} -1.00000 q^{40} -5.26369 q^{41} +4.88013 q^{42} -4.56873 q^{43} +2.90568 q^{44} +0.0402221 q^{45} +1.71130 q^{46} +6.49550 q^{47} -1.74362 q^{48} +0.833534 q^{49} -1.00000 q^{50} +7.43660 q^{51} +3.29780 q^{52} +4.03808 q^{53} -5.16074 q^{54} +2.90568 q^{55} -2.79885 q^{56} +0.232179 q^{57} +2.64609 q^{58} -6.96517 q^{59} -1.74362 q^{60} -5.75525 q^{61} +9.13104 q^{62} +0.112575 q^{63} +1.00000 q^{64} +3.29780 q^{65} +5.06642 q^{66} -1.15003 q^{67} -4.26503 q^{68} +2.98386 q^{69} -2.79885 q^{70} -0.443856 q^{71} -0.0402221 q^{72} -0.955975 q^{73} +11.8979 q^{74} -1.74362 q^{75} -0.133159 q^{76} +8.13256 q^{77} +5.75012 q^{78} -14.5373 q^{79} +1.00000 q^{80} -9.11905 q^{81} +5.26369 q^{82} +16.8564 q^{83} -4.88013 q^{84} -4.26503 q^{85} +4.56873 q^{86} +4.61378 q^{87} -2.90568 q^{88} -13.2086 q^{89} -0.0402221 q^{90} +9.23003 q^{91} -1.71130 q^{92} +15.9211 q^{93} -6.49550 q^{94} -0.133159 q^{95} +1.74362 q^{96} -8.57086 q^{97} -0.833534 q^{98} +0.116873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} - 4 q^{3} + 10 q^{4} + 10 q^{5} + 4 q^{6} - 3 q^{7} - 10 q^{8} + 6 q^{9}+O(q^{10}) \) 10 * q - 10 * q^2 - 4 * q^3 + 10 * q^4 + 10 * q^5 + 4 * q^6 - 3 * q^7 - 10 * q^8 + 6 * q^9	\( 10 q - 10 q^{2} - 4 q^{3} + 10 q^{4} + 10 q^{5} + 4 q^{6} - 3 q^{7} - 10 q^{8} + 6 q^{9} - 10 q^{10} - 11 q^{11} - 4 q^{12} + 6 q^{13} + 3 q^{14} - 4 q^{15} + 10 q^{16} + 9 q^{17} - 6 q^{18} - 13 q^{19} + 10 q^{20} - 24 q^{21} + 11 q^{22} - 3 q^{23} + 4 q^{24} + 10 q^{25} - 6 q^{26} - 10 q^{27} - 3 q^{28} - 4 q^{29} + 4 q^{30} - 17 q^{31} - 10 q^{32} - 2 q^{33} - 9 q^{34} - 3 q^{35} + 6 q^{36} - 15 q^{37} + 13 q^{38} - 6 q^{39} - 10 q^{40} - 11 q^{41} + 24 q^{42} - 11 q^{43} - 11 q^{44} + 6 q^{45} + 3 q^{46} + 3 q^{47} - 4 q^{48} - 5 q^{49} - 10 q^{50} - 21 q^{51} + 6 q^{52} + 25 q^{53} + 10 q^{54} - 11 q^{55} + 3 q^{56} + 31 q^{57} + 4 q^{58} - 46 q^{59} - 4 q^{60} - 54 q^{61} + 17 q^{62} - 6 q^{63} + 10 q^{64} + 6 q^{65} + 2 q^{66} - 26 q^{67} + 9 q^{68} - 9 q^{69} + 3 q^{70} - 16 q^{71} - 6 q^{72} + 4 q^{73} + 15 q^{74} - 4 q^{75} - 13 q^{76} + 11 q^{77} + 6 q^{78} - 19 q^{79} + 10 q^{80} - 6 q^{81} + 11 q^{82} + 19 q^{83} - 24 q^{84} + 9 q^{85} + 11 q^{86} + 28 q^{87} + 11 q^{88} - 30 q^{89} - 6 q^{90} - 38 q^{91} - 3 q^{92} - 18 q^{93} - 3 q^{94} - 13 q^{95} + 4 q^{96} - 16 q^{97} + 5 q^{98} - 59 q^{99}+O(q^{100}) \) 10 * q - 10 * q^2 - 4 * q^3 + 10 * q^4 + 10 * q^5 + 4 * q^6 - 3 * q^7 - 10 * q^8 + 6 * q^9 - 10 * q^10 - 11 * q^11 - 4 * q^12 + 6 * q^13 + 3 * q^14 - 4 * q^15 + 10 * q^16 + 9 * q^17 - 6 * q^18 - 13 * q^19 + 10 * q^20 - 24 * q^21 + 11 * q^22 - 3 * q^23 + 4 * q^24 + 10 * q^25 - 6 * q^26 - 10 * q^27 - 3 * q^28 - 4 * q^29 + 4 * q^30 - 17 * q^31 - 10 * q^32 - 2 * q^33 - 9 * q^34 - 3 * q^35 + 6 * q^36 - 15 * q^37 + 13 * q^38 - 6 * q^39 - 10 * q^40 - 11 * q^41 + 24 * q^42 - 11 * q^43 - 11 * q^44 + 6 * q^45 + 3 * q^46 + 3 * q^47 - 4 * q^48 - 5 * q^49 - 10 * q^50 - 21 * q^51 + 6 * q^52 + 25 * q^53 + 10 * q^54 - 11 * q^55 + 3 * q^56 + 31 * q^57 + 4 * q^58 - 46 * q^59 - 4 * q^60 - 54 * q^61 + 17 * q^62 - 6 * q^63 + 10 * q^64 + 6 * q^65 + 2 * q^66 - 26 * q^67 + 9 * q^68 - 9 * q^69 + 3 * q^70 - 16 * q^71 - 6 * q^72 + 4 * q^73 + 15 * q^74 - 4 * q^75 - 13 * q^76 + 11 * q^77 + 6 * q^78 - 19 * q^79 + 10 * q^80 - 6 * q^81 + 11 * q^82 + 19 * q^83 - 24 * q^84 + 9 * q^85 + 11 * q^86 + 28 * q^87 + 11 * q^88 - 30 * q^89 - 6 * q^90 - 38 * q^91 - 3 * q^92 - 18 * q^93 - 3 * q^94 - 13 * q^95 + 4 * q^96 - 16 * q^97 + 5 * q^98 - 59 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more