LMFDB - Embedded newform 4003.2.a.c.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$0$
Dimension:	$179$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45370 q^{2} -0.415301 q^{3} +4.02066 q^{4} -3.18084 q^{5} +1.01903 q^{6} -2.85630 q^{7} -4.95811 q^{8} -2.82752 q^{9} +O(q^{10})$	\(q-2.45370 q^{2} -0.415301 q^{3} +4.02066 q^{4} -3.18084 q^{5} +1.01903 q^{6} -2.85630 q^{7} -4.95811 q^{8} -2.82752 q^{9} +7.80484 q^{10} +1.47266 q^{11} -1.66979 q^{12} +3.83328 q^{13} +7.00851 q^{14} +1.32101 q^{15} +4.12441 q^{16} +3.98568 q^{17} +6.93791 q^{18} -4.57320 q^{19} -12.7891 q^{20} +1.18622 q^{21} -3.61347 q^{22} +0.0487036 q^{23} +2.05911 q^{24} +5.11776 q^{25} -9.40574 q^{26} +2.42018 q^{27} -11.4842 q^{28} +7.06083 q^{29} -3.24136 q^{30} -1.88706 q^{31} -0.203855 q^{32} -0.611598 q^{33} -9.77968 q^{34} +9.08543 q^{35} -11.3685 q^{36} -3.06475 q^{37} +11.2213 q^{38} -1.59197 q^{39} +15.7710 q^{40} +1.59606 q^{41} -2.91064 q^{42} -9.79157 q^{43} +5.92107 q^{44} +8.99391 q^{45} -0.119504 q^{46} -12.2176 q^{47} -1.71287 q^{48} +1.15844 q^{49} -12.5575 q^{50} -1.65526 q^{51} +15.4123 q^{52} +1.66410 q^{53} -5.93840 q^{54} -4.68430 q^{55} +14.1618 q^{56} +1.89926 q^{57} -17.3252 q^{58} -9.23933 q^{59} +5.31133 q^{60} -1.01299 q^{61} +4.63028 q^{62} +8.07625 q^{63} -7.74862 q^{64} -12.1931 q^{65} +1.50068 q^{66} -10.5900 q^{67} +16.0251 q^{68} -0.0202267 q^{69} -22.2930 q^{70} -11.1767 q^{71} +14.0192 q^{72} +3.38359 q^{73} +7.51998 q^{74} -2.12541 q^{75} -18.3873 q^{76} -4.20636 q^{77} +3.90621 q^{78} -5.99683 q^{79} -13.1191 q^{80} +7.47747 q^{81} -3.91626 q^{82} -17.3073 q^{83} +4.76941 q^{84} -12.6778 q^{85} +24.0256 q^{86} -2.93237 q^{87} -7.30161 q^{88} -6.21452 q^{89} -22.0684 q^{90} -10.9490 q^{91} +0.195821 q^{92} +0.783696 q^{93} +29.9783 q^{94} +14.5466 q^{95} +0.0846611 q^{96} -3.11759 q^{97} -2.84247 q^{98} -4.16398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9}+O(q^{10}) $ 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9	\( 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9} + 9 q^{10} + 46 q^{11} + 33 q^{12} + 47 q^{13} + 22 q^{14} + 36 q^{15} + 222 q^{16} + 103 q^{17} + 43 q^{18} + 12 q^{19} + 102 q^{20} + 50 q^{21} + 39 q^{22} + 121 q^{23} - 3 q^{24} + 246 q^{25} + 52 q^{26} + 49 q^{27} + 41 q^{28} + 138 q^{29} + 28 q^{30} + 5 q^{31} + 137 q^{32} + 63 q^{33} + 2 q^{34} + 72 q^{35} + 279 q^{36} + 118 q^{37} + 123 q^{38} + q^{39} + 9 q^{40} + 50 q^{41} + 48 q^{42} + 48 q^{43} + 108 q^{44} + 158 q^{45} + 13 q^{46} + 85 q^{47} + 50 q^{48} + 230 q^{49} + 78 q^{50} + 15 q^{51} + 41 q^{52} + 399 q^{53} - 5 q^{54} + 24 q^{55} + 53 q^{56} + 45 q^{57} + 27 q^{58} + 48 q^{59} + 66 q^{60} + 46 q^{61} + 81 q^{62} + 78 q^{63} + 252 q^{64} + 153 q^{65} + 6 q^{66} + 70 q^{67} + 240 q^{68} + 120 q^{69} - 31 q^{70} + 86 q^{71} + 89 q^{72} + 45 q^{73} + 68 q^{74} + 17 q^{75} - 13 q^{76} + 362 q^{77} + 69 q^{78} + 31 q^{79} + 169 q^{80} + 303 q^{81} + 25 q^{82} + 106 q^{83} + 13 q^{84} + 115 q^{85} + 95 q^{86} + 32 q^{87} + 83 q^{88} + 105 q^{89} - 38 q^{90} + 3 q^{91} + 310 q^{92} + 298 q^{93} - 17 q^{94} + 102 q^{95} - 82 q^{96} + 34 q^{97} + 81 q^{98} + 58 q^{99}+O(q^{100}) \) 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9 + 9 * q^10 + 46 * q^11 + 33 * q^12 + 47 * q^13 + 22 * q^14 + 36 * q^15 + 222 * q^16 + 103 * q^17 + 43 * q^18 + 12 * q^19 + 102 * q^20 + 50 * q^21 + 39 * q^22 + 121 * q^23 - 3 * q^24 + 246 * q^25 + 52 * q^26 + 49 * q^27 + 41 * q^28 + 138 * q^29 + 28 * q^30 + 5 * q^31 + 137 * q^32 + 63 * q^33 + 2 * q^34 + 72 * q^35 + 279 * q^36 + 118 * q^37 + 123 * q^38 + q^39 + 9 * q^40 + 50 * q^41 + 48 * q^42 + 48 * q^43 + 108 * q^44 + 158 * q^45 + 13 * q^46 + 85 * q^47 + 50 * q^48 + 230 * q^49 + 78 * q^50 + 15 * q^51 + 41 * q^52 + 399 * q^53 - 5 * q^54 + 24 * q^55 + 53 * q^56 + 45 * q^57 + 27 * q^58 + 48 * q^59 + 66 * q^60 + 46 * q^61 + 81 * q^62 + 78 * q^63 + 252 * q^64 + 153 * q^65 + 6 * q^66 + 70 * q^67 + 240 * q^68 + 120 * q^69 - 31 * q^70 + 86 * q^71 + 89 * q^72 + 45 * q^73 + 68 * q^74 + 17 * q^75 - 13 * q^76 + 362 * q^77 + 69 * q^78 + 31 * q^79 + 169 * q^80 + 303 * q^81 + 25 * q^82 + 106 * q^83 + 13 * q^84 + 115 * q^85 + 95 * q^86 + 32 * q^87 + 83 * q^88 + 105 * q^89 - 38 * q^90 + 3 * q^91 + 310 * q^92 + 298 * q^93 - 17 * q^94 + 102 * q^95 - 82 * q^96 + 34 * q^97 + 81 * q^98 + 58 * 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 $

$2$	−2.45370	−1.73503	−0.867515	−	0.497410i	$-0.834284\pi$
$2$	−2.45370	−1.73503	−0.867515	+	0.497410i	$0.834284\pi$
$3$	−0.415301	−0.239774	−0.119887	−	0.992788i	$-0.538253\pi$
$3$	−0.415301	−0.239774	−0.119887	+	0.992788i	$0.538253\pi$
$4$	4.02066	2.01033
$5$	−3.18084	−1.42252	−0.711258	−	0.702931i	$-0.751874\pi$
$5$	−3.18084	−1.42252	−0.711258	+	0.702931i	$0.751874\pi$
$6$	1.01903	0.416016
$7$	−2.85630	−1.07958	−0.539790	−	0.841800i	$-0.681496\pi$
$7$	−2.85630	−1.07958	−0.539790	+	0.841800i	$0.681496\pi$
$8$	−4.95811	−1.75296
$9$	−2.82752	−0.942508
$10$	7.80484	2.46811
$11$	1.47266	0.444024	0.222012	−	0.975044i	$-0.428738\pi$
$11$	1.47266	0.444024	0.222012	+	0.975044i	$0.428738\pi$
$12$	−1.66979	−0.482026
$13$	3.83328	1.06316	0.531581	−	0.847008i	$-0.321598\pi$
$13$	3.83328	1.06316	0.531581	+	0.847008i	$0.321598\pi$
$14$	7.00851	1.87310
$15$	1.32101	0.341083
$16$	4.12441	1.03110
$17$	3.98568	0.966669	0.483335	−	0.875436i	$-0.339425\pi$
$17$	3.98568	0.966669	0.483335	+	0.875436i	$0.339425\pi$
$18$	6.93791	1.63528
$19$	−4.57320	−1.04917	−0.524583	−	0.851360i	$-0.675779\pi$
$19$	−4.57320	−1.04917	−0.524583	+	0.851360i	$0.675779\pi$
$20$	−12.7891	−2.85973
$21$	1.18622	0.258855
$22$	−3.61347	−0.770395
$23$	0.0487036	0.0101554	0.00507770	−	0.999987i	$-0.498384\pi$
$23$	0.0487036	0.0101554	0.00507770	+	0.999987i	$0.498384\pi$
$24$	2.05911	0.420314
$25$	5.11776	1.02355
$26$	−9.40574	−1.84462
$27$	2.42018	0.465763
$28$	−11.4842	−2.17031
$29$	7.06083	1.31116	0.655582	−	0.755124i	$-0.272423\pi$
$29$	7.06083	1.31116	0.655582	+	0.755124i	$0.272423\pi$
$30$	−3.24136	−0.591789
$31$	−1.88706	−0.338925	−0.169463	−	0.985537i	$-0.554203\pi$
$31$	−1.88706	−0.338925	−0.169463	+	0.985537i	$0.554203\pi$
$32$	−0.203855	−0.0360368
$33$	−0.611598	−0.106465
$34$	−9.77968	−1.67720
$35$	9.08543	1.53572
$36$	−11.3685	−1.89475
$37$	−3.06475	−0.503841	−0.251921	−	0.967748i	$-0.581062\pi$
$37$	−3.06475	−0.503841	−0.251921	+	0.967748i	$0.581062\pi$
$38$	11.2213	1.82033
$39$	−1.59197	−0.254919
$40$	15.7710	2.49361
$41$	1.59606	0.249263	0.124631	−	0.992203i	$-0.460225\pi$
$41$	1.59606	0.249263	0.124631	+	0.992203i	$0.460225\pi$
$42$	−2.91064	−0.449122
$43$	−9.79157	−1.49320	−0.746601	−	0.665273i	$-0.768315\pi$
$43$	−9.79157	−1.49320	−0.746601	+	0.665273i	$0.768315\pi$
$44$	5.92107	0.892635
$45$	8.99391	1.34073
$46$	−0.119504	−0.0176199
$47$	−12.2176	−1.78212	−0.891059	−	0.453887i	$-0.850037\pi$
$47$	−12.2176	−1.78212	−0.891059	+	0.453887i	$0.850037\pi$
$48$	−1.71287	−0.247232
$49$	1.15844	0.165491
$50$	−12.5575	−1.77589
$51$	−1.65526	−0.231782
$52$	15.4123	2.13731
$53$	1.66410	0.228581	0.114291	−	0.993447i	$-0.463541\pi$
$53$	1.66410	0.228581	0.114291	+	0.993447i	$0.463541\pi$
$54$	−5.93840	−0.808114
$55$	−4.68430	−0.631631
$56$	14.1618	1.89246
$57$	1.89926	0.251563
$58$	−17.3252	−2.27491
$59$	−9.23933	−1.20286	−0.601429	−	0.798926i	$-0.705402\pi$
$59$	−9.23933	−1.20286	−0.601429	+	0.798926i	$0.705402\pi$
$60$	5.31133	0.685689
$61$	−1.01299	−0.129701	−0.0648503	−	0.997895i	$-0.520657\pi$
$61$	−1.01299	−0.129701	−0.0648503	+	0.997895i	$0.520657\pi$
$62$	4.63028	0.588046
$63$	8.07625	1.01751
$64$	−7.74862	−0.968577
$65$	−12.1931	−1.51236
$66$	1.50068	0.184721
$67$	−10.5900	−1.29377	−0.646887	−	0.762586i	$-0.723929\pi$
$67$	−10.5900	−1.29377	−0.646887	+	0.762586i	$0.723929\pi$
$68$	16.0251	1.94333
$69$	−0.0202267	−0.00243500
$70$	−22.2930	−2.66452
$71$	−11.1767	−1.32644	−0.663218	−	0.748427i	$-0.730810\pi$
$71$	−11.1767	−1.32644	−0.663218	+	0.748427i	$0.730810\pi$
$72$	14.0192	1.65218
$73$	3.38359	0.396020	0.198010	−	0.980200i	$-0.436552\pi$
$73$	3.38359	0.396020	0.198010	+	0.980200i	$0.436552\pi$
$74$	7.51998	0.874180
$75$	−2.12541	−0.245421
$76$	−18.3873	−2.10917
$77$	−4.20636	−0.479359
$78$	3.90621	0.442292
$79$	−5.99683	−0.674696	−0.337348	−	0.941380i	$-0.609530\pi$
$79$	−5.99683	−0.674696	−0.337348	+	0.941380i	$0.609530\pi$
$80$	−13.1191	−1.46676
$81$	7.47747	0.830830
$82$	−3.91626	−0.432478
$83$	−17.3073	−1.89972	−0.949862	−	0.312669i	$-0.898777\pi$
$83$	−17.3073	−1.89972	−0.949862	+	0.312669i	$0.898777\pi$
$84$	4.76941	0.520385
$85$	−12.6778	−1.37510
$86$	24.0256	2.59075
$87$	−2.93237	−0.314383
$88$	−7.30161	−0.778355
$89$	−6.21452	−0.658737	−0.329369	−	0.944201i	$-0.606836\pi$
$89$	−6.21452	−0.658737	−0.329369	+	0.944201i	$0.606836\pi$
$90$	−22.0684	−2.32621
$91$	−10.9490	−1.14777
$92$	0.195821	0.0204157
$93$	0.783696	0.0812655
$94$	29.9783	3.09203
$95$	14.5466	1.49245
$96$	0.0846611	0.00864069
$97$	−3.11759	−0.316543	−0.158271	−	0.987396i	$-0.550592\pi$
$97$	−3.11759	−0.316543	−0.158271	+	0.987396i	$0.550592\pi$
$98$	−2.84247	−0.287133
$99$	−4.16398	−0.418496
$100$	20.5768	2.05768
$101$	11.4905	1.14334	0.571672	−	0.820482i	$-0.306295\pi$
$101$	11.4905	1.14334	0.571672	+	0.820482i	$0.306295\pi$
$102$	4.06151	0.402149
$103$	1.42647	0.140554	0.0702772	−	0.997528i	$-0.477612\pi$
$103$	1.42647	0.140554	0.0702772	+	0.997528i	$0.477612\pi$
$104$	−19.0058	−1.86368
$105$	−3.77319	−0.368226
$106$	−4.08320	−0.396595
$107$	9.89389	0.956479	0.478239	−	0.878230i	$-0.341275\pi$
$107$	9.89389	0.956479	0.478239	+	0.878230i	$0.341275\pi$
$108$	9.73072	0.936339
$109$	15.9376	1.52655	0.763273	−	0.646076i	$-0.223591\pi$
$109$	15.9376	1.52655	0.763273	+	0.646076i	$0.223591\pi$
$110$	11.4939	1.09590
$111$	1.27279	0.120808
$112$	−11.7805	−1.11316
$113$	12.7238	1.19695	0.598477	−	0.801140i	$-0.295773\pi$
$113$	12.7238	1.19695	0.598477	+	0.801140i	$0.295773\pi$
$114$	−4.66021	−0.436469
$115$	−0.154918	−0.0144462
$116$	28.3892	2.63587
$117$	−10.8387	−1.00204
$118$	22.6706	2.08700
$119$	−11.3843	−1.04360
$120$	−6.54970	−0.597903
$121$	−8.83127	−0.802843
$122$	2.48559	0.225034
$123$	−0.662846	−0.0597668
$124$	−7.58722	−0.681352
$125$	−0.374563	−0.0335019
$126$	−19.8167	−1.76542
$127$	8.13664	0.722010	0.361005	−	0.932564i	$-0.382434\pi$
$127$	8.13664	0.722010	0.361005	+	0.932564i	$0.382434\pi$
$128$	19.4205	1.71655
$129$	4.06645	0.358031
$130$	29.9182	2.62400
$131$	9.25580	0.808683	0.404342	−	0.914608i	$-0.367501\pi$
$131$	9.25580	0.808683	0.404342	+	0.914608i	$0.367501\pi$
$132$	−2.45903	−0.214031
$133$	13.0624	1.13266
$134$	25.9847	2.24474
$135$	−7.69820	−0.662556
$136$	−19.7614	−1.69453
$137$	12.3617	1.05613	0.528065	−	0.849204i	$-0.322918\pi$
$137$	12.3617	1.05613	0.528065	+	0.849204i	$0.322918\pi$
$138$	0.0496302	0.00422480
$139$	−16.8653	−1.43050	−0.715250	−	0.698869i	$-0.753687\pi$
$139$	−16.8653	−1.43050	−0.715250	+	0.698869i	$0.753687\pi$
$140$	36.5295	3.08730
$141$	5.07398	0.427306
$142$	27.4244	2.30141
$143$	5.64512	0.472069
$144$	−11.6619	−0.971822
$145$	−22.4594	−1.86515
$146$	−8.30234	−0.687106
$147$	−0.481101	−0.0396806
$148$	−12.3223	−1.01289
$149$	−16.4659	−1.34894	−0.674469	−	0.738303i	$-0.735627\pi$
$149$	−16.4659	−1.34894	−0.674469	+	0.738303i	$0.735627\pi$
$150$	5.21513	0.425813
$151$	−14.5433	−1.18352	−0.591758	−	0.806116i	$-0.701566\pi$
$151$	−14.5433	−1.18352	−0.591758	+	0.806116i	$0.701566\pi$
$152$	22.6745	1.83914
$153$	−11.2696	−0.911094
$154$	10.3212	0.831703
$155$	6.00243	0.482126
$156$	−6.40076	−0.512471
$157$	−16.4117	−1.30980	−0.654900	−	0.755716i	$-0.727289\pi$
$157$	−16.4117	−1.30980	−0.654900	+	0.755716i	$0.727289\pi$
$158$	14.7144	1.17062
$159$	−0.691101	−0.0548078
$160$	0.648430	0.0512629
$161$	−0.139112	−0.0109636
$162$	−18.3475	−1.44152
$163$	22.9306	1.79606	0.898032	−	0.439931i	$-0.144997\pi$
$163$	22.9306	1.79606	0.898032	+	0.439931i	$0.144997\pi$
$164$	6.41722	0.501101
$165$	1.94540	0.151449
$166$	42.4670	3.29608
$167$	9.95056	0.769997	0.384999	−	0.922917i	$-0.374202\pi$
$167$	9.95056	0.769997	0.384999	+	0.922917i	$0.374202\pi$
$168$	−5.88143	−0.453762
$169$	1.69405	0.130312
$170$	31.1076	2.38584
$171$	12.9309	0.988847
$172$	−39.3686	−3.00183
$173$	−1.17203	−0.0891079	−0.0445540	−	0.999007i	$-0.514187\pi$
$173$	−1.17203	−0.0891079	−0.0445540	+	0.999007i	$0.514187\pi$
$174$	7.19517	0.545464
$175$	−14.6178	−1.10500
$176$	6.07385	0.457834
$177$	3.83710	0.288414
$178$	15.2486	1.14293
$179$	−11.0931	−0.829137	−0.414569	−	0.910018i	$-0.636068\pi$
$179$	−11.0931	−0.829137	−0.414569	+	0.910018i	$0.636068\pi$
$180$	36.1615	2.69532
$181$	−14.6027	−1.08541	−0.542706	−	0.839922i	$-0.682600\pi$
$181$	−14.6027	−1.08541	−0.542706	+	0.839922i	$0.682600\pi$
$182$	26.8656	1.99141
$183$	0.420697	0.0310989
$184$	−0.241478	−0.0178020
$185$	9.74848	0.716722
$186$	−1.92296	−0.140998
$187$	5.86955	0.429224
$188$	−49.1228	−3.58265
$189$	−6.91275	−0.502829
$190$	−35.6932	−2.58945
$191$	26.1706	1.89364	0.946818	−	0.321769i	$-0.104277\pi$
$191$	26.1706	1.89364	0.946818	+	0.321769i	$0.104277\pi$
$192$	3.21801	0.232240
$193$	12.7562	0.918210	0.459105	−	0.888382i	$-0.348170\pi$
$193$	12.7562	0.918210	0.459105	+	0.888382i	$0.348170\pi$
$194$	7.64963	0.549212
$195$	5.06379	0.362626
$196$	4.65770	0.332693
$197$	7.48480	0.533270	0.266635	−	0.963798i	$-0.414088\pi$
$197$	7.48480	0.533270	0.266635	+	0.963798i	$0.414088\pi$
$198$	10.2172	0.726104
$199$	−12.7764	−0.905692	−0.452846	−	0.891589i	$-0.649591\pi$
$199$	−12.7764	−0.905692	−0.452846	+	0.891589i	$0.649591\pi$
$200$	−25.3744	−1.79424
$201$	4.39804	0.310214
$202$	−28.1942	−1.98374
$203$	−20.1678	−1.41550
$204$	−6.65523	−0.465959
$205$	−5.07682	−0.354580
$206$	−3.50014	−0.243866
$207$	−0.137711	−0.00957155
$208$	15.8100	1.09623
$209$	−6.73478	−0.465854
$210$	9.25829	0.638883
$211$	8.56374	0.589552	0.294776	−	0.955566i	$-0.404755\pi$
$211$	8.56374	0.589552	0.294776	+	0.955566i	$0.404755\pi$
$212$	6.69077	0.459524
$213$	4.64171	0.318045
$214$	−24.2767	−1.65952
$215$	31.1455	2.12410
$216$	−11.9995	−0.816463
$217$	5.38999	0.365897
$218$	−39.1062	−2.64861
$219$	−1.40521	−0.0949553
$220$	−18.8340	−1.26979
$221$	15.2782	1.02773
$222$	−3.12306	−0.209606
$223$	−17.4201	−1.16654	−0.583268	−	0.812279i	$-0.698226\pi$
$223$	−17.4201	−1.16654	−0.583268	+	0.812279i	$0.698226\pi$
$224$	0.582270	0.0389045
$225$	−14.4706	−0.964706
$226$	−31.2204	−2.07675
$227$	9.44268	0.626732	0.313366	−	0.949632i	$-0.398543\pi$
$227$	9.44268	0.626732	0.313366	+	0.949632i	$0.398543\pi$
$228$	7.63627	0.505725
$229$	−26.6402	−1.76043	−0.880215	−	0.474574i	$-0.842602\pi$
$229$	−26.6402	−1.76043	−0.880215	+	0.474574i	$0.842602\pi$
$230$	0.380124	0.0250646
$231$	1.74691	0.114938
$232$	−35.0084	−2.29841
$233$	20.0451	1.31320	0.656600	−	0.754239i	$-0.271994\pi$
$233$	20.0451	1.31320	0.656600	+	0.754239i	$0.271994\pi$
$234$	26.5950	1.73857
$235$	38.8622	2.53509
$236$	−37.1482	−2.41814
$237$	2.49049	0.161775
$238$	27.9337	1.81067
$239$	5.83490	0.377429	0.188714	−	0.982032i	$-0.439568\pi$
$239$	5.83490	0.377429	0.188714	+	0.982032i	$0.439568\pi$
$240$	5.44837	0.351691
$241$	−1.21367	−0.0781797	−0.0390898	−	0.999236i	$-0.512446\pi$
$241$	−1.21367	−0.0781797	−0.0390898	+	0.999236i	$0.512446\pi$
$242$	21.6693	1.39296
$243$	−10.3659	−0.664975
$244$	−4.07291	−0.260741
$245$	−3.68481	−0.235414
$246$	1.62643	0.103697
$247$	−17.5304	−1.11543
$248$	9.35623	0.594121
$249$	7.18775	0.455505
$250$	0.919067	0.0581269
$251$	14.5934	0.921128	0.460564	−	0.887626i	$-0.347647\pi$
$251$	14.5934	0.921128	0.460564	+	0.887626i	$0.347647\pi$
$252$	32.4719	2.04554
$253$	0.0717238	0.00450924
$254$	−19.9649	−1.25271
$255$	5.26511	0.329714
$256$	−32.1550	−2.00969
$257$	−3.11554	−0.194342	−0.0971709	−	0.995268i	$-0.530979\pi$
$257$	−3.11554	−0.194342	−0.0971709	+	0.995268i	$0.530979\pi$
$258$	−9.97787	−0.621195
$259$	8.75383	0.543937
$260$	−49.0242	−3.04035
$261$	−19.9647	−1.23578
$262$	−22.7110	−1.40309
$263$	−21.5673	−1.32990	−0.664950	−	0.746888i	$-0.731547\pi$
$263$	−21.5673	−1.32990	−0.664950	+	0.746888i	$0.731547\pi$
$264$	3.03237	0.186629
$265$	−5.29322	−0.325160
$266$	−32.0514	−1.96519
$267$	2.58090	0.157948
$268$	−42.5788	−2.60091
$269$	−3.82904	−0.233461	−0.116730	−	0.993164i	$-0.537241\pi$
$269$	−3.82904	−0.233461	−0.116730	+	0.993164i	$0.537241\pi$
$270$	18.8891	1.14955
$271$	−12.5093	−0.759888	−0.379944	−	0.925009i	$-0.624057\pi$
$271$	−12.5093	−0.759888	−0.379944	+	0.925009i	$0.624057\pi$
$272$	16.4386	0.996734
$273$	4.54713	0.275205
$274$	−30.3319	−1.83242
$275$	7.53672	0.454481
$276$	−0.0813246	−0.00489516
$277$	−8.80554	−0.529073	−0.264537	−	0.964376i	$-0.585219\pi$
$277$	−8.80554	−0.529073	−0.264537	+	0.964376i	$0.585219\pi$
$278$	41.3826	2.48196
$279$	5.33570	0.319440
$280$	−45.0466	−2.69205
$281$	20.5445	1.22558	0.612790	−	0.790246i	$-0.290047\pi$
$281$	20.5445	1.22558	0.612790	+	0.790246i	$0.290047\pi$
$282$	−12.4500	−0.741389
$283$	11.8855	0.706522	0.353261	−	0.935525i	$-0.385073\pi$
$283$	11.8855	0.706522	0.353261	+	0.935525i	$0.385073\pi$
$284$	−44.9379	−2.66658
$285$	−6.04124	−0.357852
$286$	−13.8515	−0.819054
$287$	−4.55882	−0.269099
$288$	0.576404	0.0339650
$289$	−1.11436	−0.0655507
$290$	55.1087	3.23609
$291$	1.29474	0.0758988
$292$	13.6043	0.796131
$293$	14.4312	0.843078	0.421539	−	0.906810i	$-0.361490\pi$
$293$	14.4312	0.843078	0.421539	+	0.906810i	$0.361490\pi$
$294$	1.18048	0.0688470
$295$	29.3888	1.71108
$296$	15.1954	0.883212
$297$	3.56410	0.206810
$298$	40.4024	2.34045
$299$	0.186695	0.0107968
$300$	−8.54556	−0.493378
$301$	27.9677	1.61203
$302$	35.6849	2.05344
$303$	−4.77200	−0.274144
$304$	−18.8618	−1.08180
$305$	3.22217	0.184501
$306$	27.6523	1.58078
$307$	17.1739	0.980165	0.490082	−	0.871676i	$-0.336967\pi$
$307$	17.1739	0.980165	0.490082	+	0.871676i	$0.336967\pi$
$308$	−16.9124	−0.963671
$309$	−0.592415	−0.0337013
$310$	−14.7282	−0.836504
$311$	−23.7296	−1.34558	−0.672791	−	0.739832i	$-0.734905\pi$
$311$	−23.7296	−1.34558	−0.672791	+	0.739832i	$0.734905\pi$
$312$	7.89314	0.446861
$313$	−25.7050	−1.45293	−0.726467	−	0.687202i	$-0.758839\pi$
$313$	−25.7050	−1.45293	−0.726467	+	0.687202i	$0.758839\pi$
$314$	40.2696	2.27254
$315$	−25.6893	−1.44743
$316$	−24.1112	−1.35636
$317$	−8.02788	−0.450891	−0.225445	−	0.974256i	$-0.572384\pi$
$317$	−8.02788	−0.450891	−0.225445	+	0.974256i	$0.572384\pi$
$318$	1.69576	0.0950933
$319$	10.3982	0.582188
$320$	24.6471	1.37782
$321$	−4.10895	−0.229339
$322$	0.341340	0.0190221
$323$	−18.2273	−1.01420
$324$	30.0644	1.67024
$325$	19.6178	1.08820
$326$	−56.2649	−3.11622
$327$	−6.61891	−0.366027
$328$	−7.91344	−0.436947
$329$	34.8971	1.92394
$330$	−4.77342	−0.262768
$331$	5.24402	0.288237	0.144119	−	0.989560i	$-0.453965\pi$
$331$	5.24402	0.288237	0.144119	+	0.989560i	$0.453965\pi$
$332$	−69.5869	−3.81908
$333$	8.66565	0.474875
$334$	−24.4157	−1.33597
$335$	33.6851	1.84041
$336$	4.89247	0.266906
$337$	17.6574	0.961861	0.480930	−	0.876759i	$-0.340299\pi$
$337$	17.6574	0.961861	0.480930	+	0.876759i	$0.340299\pi$
$338$	−4.15670	−0.226095
$339$	−5.28421	−0.286999
$340$	−50.9732	−2.76441
$341$	−2.77899	−0.150491
$342$	−31.7285	−1.71568
$343$	16.6852	0.900918
$344$	48.5477	2.61752
$345$	0.0643378	0.00346383
$346$	2.87582	0.154605
$347$	−12.9881	−0.697238	−0.348619	−	0.937265i	$-0.613349\pi$
$347$	−12.9881	−0.697238	−0.348619	+	0.937265i	$0.613349\pi$
$348$	−11.7901	−0.632014
$349$	16.4846	0.882401	0.441201	−	0.897408i	$-0.354553\pi$
$349$	16.4846	0.882401	0.441201	+	0.897408i	$0.354553\pi$
$350$	35.8678	1.91722
$351$	9.27722	0.495182
$352$	−0.300209	−0.0160012
$353$	13.8669	0.738063	0.369032	−	0.929417i	$-0.379689\pi$
$353$	13.8669	0.738063	0.369032	+	0.929417i	$0.379689\pi$
$354$	−9.41512	−0.500408
$355$	35.5515	1.88688
$356$	−24.9865	−1.32428
$357$	4.72791	0.250227
$358$	27.2192	1.43858
$359$	−32.4774	−1.71409	−0.857047	−	0.515238i	$-0.827703\pi$
$359$	−32.4774	−1.71409	−0.857047	+	0.515238i	$0.827703\pi$
$360$	−44.5928	−2.35025
$361$	1.91420	0.100747
$362$	35.8308	1.88322
$363$	3.66764	0.192501
$364$	−44.0222	−2.30739
$365$	−10.7627	−0.563344
$366$	−1.03227	−0.0539575
$367$	8.16012	0.425955	0.212978	−	0.977057i	$-0.431684\pi$
$367$	8.16012	0.425955	0.212978	+	0.977057i	$0.431684\pi$
$368$	0.200873	0.0104712
$369$	−4.51290	−0.234932
$370$	−23.9199	−1.24354
$371$	−4.75315	−0.246771
$372$	3.15098	0.163371
$373$	24.1101	1.24837	0.624187	−	0.781275i	$-0.285430\pi$
$373$	24.1101	1.24837	0.624187	+	0.781275i	$0.285430\pi$
$374$	−14.4021	−0.744717
$375$	0.155556	0.00803290
$376$	60.5761	3.12398
$377$	27.0662	1.39398
$378$	16.9618	0.872423
$379$	11.5728	0.594454	0.297227	−	0.954807i	$-0.403938\pi$
$379$	11.5728	0.594454	0.297227	+	0.954807i	$0.403938\pi$
$380$	58.4871	3.00033
$381$	−3.37916	−0.173119
$382$	−64.2148	−3.28552
$383$	16.0663	0.820951	0.410476	−	0.911872i	$-0.365363\pi$
$383$	16.0663	0.820951	0.410476	+	0.911872i	$0.365363\pi$
$384$	−8.06536	−0.411584
$385$	13.3798	0.681896
$386$	−31.2999	−1.59312
$387$	27.6859	1.40735
$388$	−12.5348	−0.636356
$389$	21.8451	1.10759	0.553796	−	0.832652i	$-0.313179\pi$
$389$	21.8451	1.10759	0.553796	+	0.832652i	$0.313179\pi$
$390$	−12.4251	−0.629167
$391$	0.194117	0.00981691
$392$	−5.74367	−0.290099
$393$	−3.84394	−0.193901
$394$	−18.3655	−0.925240
$395$	19.0750	0.959765
$396$	−16.7420	−0.841316
$397$	−9.07170	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$397$	−9.07170	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$398$	31.3494	1.57140
$399$	−5.42484	−0.271582
$400$	21.1077	1.05539
$401$	−25.5092	−1.27387	−0.636934	−	0.770918i	$-0.719798\pi$
$401$	−25.5092	−1.27387	−0.636934	+	0.770918i	$0.719798\pi$
$402$	−10.7915	−0.538230
$403$	−7.23362	−0.360332
$404$	46.1993	2.29850
$405$	−23.7847	−1.18187
$406$	49.4859	2.45594
$407$	−4.51333	−0.223718
$408$	8.20695	0.406304
$409$	−7.55183	−0.373414	−0.186707	−	0.982416i	$-0.559782\pi$
$409$	−7.55183	−0.373414	−0.186707	+	0.982416i	$0.559782\pi$
$410$	12.4570	0.615207
$411$	−5.13382	−0.253233
$412$	5.73536	0.282561
$413$	26.3903	1.29858
$414$	0.337901	0.0166069
$415$	55.0518	2.70239
$416$	−0.781433	−0.0383129
$417$	7.00420	0.342997
$418$	16.5252	0.808272
$419$	1.89237	0.0924485	0.0462243	−	0.998931i	$-0.485281\pi$
$419$	1.89237	0.0924485	0.0462243	+	0.998931i	$0.485281\pi$
$420$	−15.1707	−0.740256
$421$	−9.42961	−0.459571	−0.229786	−	0.973241i	$-0.573803\pi$
$421$	−9.42961	−0.459571	−0.229786	+	0.973241i	$0.573803\pi$
$422$	−21.0129	−1.02289
$423$	34.5455	1.67966
$424$	−8.25077	−0.400693
$425$	20.3977	0.989435
$426$	−11.3894	−0.551818
$427$	2.89341	0.140022
$428$	39.7800	1.92284
$429$	−2.34443	−0.113190
$430$	−76.4217	−3.68538
$431$	3.05460	0.147135	0.0735674	−	0.997290i	$-0.476562\pi$
$431$	3.05460	0.147135	0.0735674	+	0.997290i	$0.476562\pi$
$432$	9.98180	0.480249
$433$	14.5869	0.701001	0.350501	−	0.936563i	$-0.386011\pi$
$433$	14.5869	0.701001	0.350501	+	0.936563i	$0.386011\pi$
$434$	−13.2255	−0.634842
$435$	9.32741	0.447215
$436$	64.0798	3.06887
$437$	−0.222731	−0.0106547
$438$	3.44797	0.164750
$439$	37.0215	1.76694	0.883471	−	0.468486i	$-0.155200\pi$
$439$	37.0215	1.76694	0.883471	+	0.468486i	$0.155200\pi$
$440$	23.2253	1.10722
$441$	−3.27552	−0.155977
$442$	−37.4883	−1.78313
$443$	−28.9365	−1.37481	−0.687407	−	0.726272i	$-0.741251\pi$
$443$	−28.9365	−1.37481	−0.687407	+	0.726272i	$0.741251\pi$
$444$	5.11747	0.242865
$445$	19.7674	0.937064
$446$	42.7438	2.02398
$447$	6.83830	0.323441
$448$	22.1324	1.04566
$449$	−29.4580	−1.39021	−0.695104	−	0.718909i	$-0.744642\pi$
$449$	−29.4580	−1.39021	−0.695104	+	0.718909i	$0.744642\pi$
$450$	35.5065	1.67379
$451$	2.35046	0.110679
$452$	51.1581	2.40628
$453$	6.03984	0.283776
$454$	−23.1695	−1.08740
$455$	34.8270	1.63272
$456$	−9.41672	−0.440979
$457$	22.4801	1.05157	0.525787	−	0.850616i	$-0.323771\pi$
$457$	22.4801	1.05157	0.525787	+	0.850616i	$0.323771\pi$
$458$	65.3671	3.05440
$459$	9.64605	0.450239
$460$	−0.622875	−0.0290417
$461$	6.96743	0.324505	0.162253	−	0.986749i	$-0.448124\pi$
$461$	6.96743	0.324505	0.162253	+	0.986749i	$0.448124\pi$
$462$	−4.28639	−0.199421
$463$	−10.2555	−0.476615	−0.238308	−	0.971190i	$-0.576593\pi$
$463$	−10.2555	−0.476615	−0.238308	+	0.971190i	$0.576593\pi$
$464$	29.1217	1.35194
$465$	−2.49281	−0.115601
$466$	−49.1848	−2.27844
$467$	4.05232	0.187519	0.0937595	−	0.995595i	$-0.470112\pi$
$467$	4.05232	0.187519	0.0937595	+	0.995595i	$0.470112\pi$
$468$	−43.5788	−2.01443
$469$	30.2482	1.39673
$470$	−95.3564	−4.39846
$471$	6.81581	0.314056
$472$	45.8096	2.10856
$473$	−14.4197	−0.663017
$474$	−6.11092	−0.280684
$475$	−23.4045	−1.07387
$476$	−45.7724	−2.09797
$477$	−4.70527	−0.215440
$478$	−14.3171	−0.654850
$479$	36.2095	1.65446	0.827228	−	0.561867i	$-0.189917\pi$
$479$	36.2095	1.65446	0.827228	+	0.561867i	$0.189917\pi$
$480$	−0.269294	−0.0122915
$481$	−11.7480	−0.535665
$482$	2.97800	0.135644
$483$	0.0577733	0.00262878
$484$	−35.5076	−1.61398
$485$	9.91655	0.450287
$486$	25.4349	1.15375
$487$	7.21245	0.326827	0.163414	−	0.986558i	$-0.447749\pi$
$487$	7.21245	0.326827	0.163414	+	0.986558i	$0.447749\pi$
$488$	5.02253	0.227359
$489$	−9.52310	−0.430650
$490$	9.04144	0.408451
$491$	−30.7216	−1.38645	−0.693223	−	0.720723i	$-0.743810\pi$
$491$	−30.7216	−1.38645	−0.693223	+	0.720723i	$0.743810\pi$
$492$	−2.66508	−0.120151
$493$	28.1422	1.26746
$494$	43.0144	1.93531
$495$	13.2450	0.595318
$496$	−7.78299	−0.349466
$497$	31.9241	1.43199
$498$	−17.6366	−0.790315
$499$	−2.36094	−0.105690	−0.0528451	−	0.998603i	$-0.516829\pi$
$499$	−2.36094	−0.105690	−0.0528451	+	0.998603i	$0.516829\pi$
$500$	−1.50599	−0.0673500
$501$	−4.13248	−0.184626
$502$	−35.8079	−1.59819
$503$	15.9751	0.712296	0.356148	−	0.934430i	$-0.384090\pi$
$503$	15.9751	0.712296	0.356148	+	0.934430i	$0.384090\pi$
$504$	−40.0430	−1.78366
$505$	−36.5493	−1.62642
$506$	−0.175989	−0.00782367
$507$	−0.703542	−0.0312454
$508$	32.7147	1.45148
$509$	18.4502	0.817792	0.408896	−	0.912581i	$-0.365914\pi$
$509$	18.4502	0.817792	0.408896	+	0.912581i	$0.365914\pi$
$510$	−12.9190	−0.572064
$511$	−9.66455	−0.427535
$512$	40.0577	1.77032
$513$	−11.0680	−0.488663
$514$	7.64460	0.337189
$515$	−4.53738	−0.199941
$516$	16.3498	0.719761
$517$	−17.9924	−0.791303
$518$	−21.4793	−0.943747
$519$	0.486746	0.0213658
$520$	60.4546	2.65111
$521$	40.0816	1.75601	0.878004	−	0.478653i	$-0.158875\pi$
$521$	40.0816	1.75601	0.878004	+	0.478653i	$0.158875\pi$
$522$	48.9874	2.14412
$523$	25.7630	1.12654	0.563269	−	0.826273i	$-0.309543\pi$
$523$	25.7630	1.12654	0.563269	+	0.826273i	$0.309543\pi$
$524$	37.2145	1.62572
$525$	6.07080	0.264952
$526$	52.9199	2.30742
$527$	−7.52120	−0.327629
$528$	−2.52248	−0.109777
$529$	−22.9976	−0.999897
$530$	12.9880	0.564163
$531$	26.1244	1.13370
$532$	52.5197	2.27702
$533$	6.11815	0.265006
$534$	−6.33275	−0.274045
$535$	−31.4709	−1.36061
$536$	52.5064	2.26793
$537$	4.60698	0.198806
$538$	9.39534	0.405062
$539$	1.70599	0.0734821
$540$	−30.9519	−1.33196
$541$	8.30184	0.356924	0.178462	−	0.983947i	$-0.442888\pi$
$541$	8.30184	0.356924	0.178462	+	0.983947i	$0.442888\pi$
$542$	30.6942	1.31843
$543$	6.06453	0.260254
$544$	−0.812500	−0.0348356
$545$	−50.6950	−2.17154
$546$	−11.1573	−0.477489
$547$	14.2221	0.608094	0.304047	−	0.952657i	$-0.401662\pi$
$547$	14.2221	0.608094	0.304047	+	0.952657i	$0.401662\pi$
$548$	49.7021	2.12317
$549$	2.86427	0.122244
$550$	−18.4929	−0.788539
$551$	−32.2906	−1.37563
$552$	0.100286	0.00426845
$553$	17.1287	0.728387
$554$	21.6062	0.917959
$555$	−4.04855	−0.171852
$556$	−67.8099	−2.87578
$557$	−40.0926	−1.69878	−0.849388	−	0.527768i	$-0.823029\pi$
$557$	−40.0926	−1.69878	−0.849388	+	0.527768i	$0.823029\pi$
$558$	−13.0922	−0.554238
$559$	−37.5339	−1.58751
$560$	37.4720	1.58348
$561$	−2.43763	−0.102917
$562$	−50.4101	−2.12642
$563$	3.96606	0.167149	0.0835747	−	0.996502i	$-0.473366\pi$
$563$	3.96606	0.167149	0.0835747	+	0.996502i	$0.473366\pi$
$564$	20.4008	0.859027
$565$	−40.4724	−1.70269
$566$	−29.1636	−1.22584
$567$	−21.3579	−0.896947
$568$	55.4155	2.32518
$569$	−0.313860	−0.0131577	−0.00657884	−	0.999978i	$-0.502094\pi$
$569$	−0.313860	−0.0131577	−0.00657884	+	0.999978i	$0.502094\pi$
$570$	14.8234	0.620884
$571$	−27.3508	−1.14459	−0.572297	−	0.820046i	$-0.693948\pi$
$571$	−27.3508	−1.14459	−0.572297	+	0.820046i	$0.693948\pi$
$572$	22.6971	0.949015
$573$	−10.8687	−0.454045
$574$	11.1860	0.466895
$575$	0.249253	0.0103946
$576$	21.9094	0.912892
$577$	32.3912	1.34846	0.674232	−	0.738520i	$-0.264475\pi$
$577$	32.3912	1.34846	0.674232	+	0.738520i	$0.264475\pi$
$578$	2.73432	0.113733
$579$	−5.29766	−0.220163
$580$	−90.3016	−3.74957
$581$	49.4348	2.05090
$582$	−3.17690	−0.131687
$583$	2.45065	0.101495
$584$	−16.7762	−0.694205
$585$	34.4762	1.42542
$586$	−35.4098	−1.46277
$587$	−43.1957	−1.78288	−0.891439	−	0.453141i	$-0.850303\pi$
$587$	−43.1957	−1.78288	−0.891439	+	0.453141i	$0.850303\pi$
$588$	−1.93435	−0.0797711
$589$	8.62989	0.355589
$590$	−72.1115	−2.96878
$591$	−3.10845	−0.127864
$592$	−12.6403	−0.519512
$593$	17.4547	0.716780	0.358390	−	0.933572i	$-0.383326\pi$
$593$	17.4547	0.716780	0.358390	+	0.933572i	$0.383326\pi$
$594$	−8.74525	−0.358822
$595$	36.2116	1.48453
$596$	−66.2038	−2.71181
$597$	5.30604	0.217162
$598$	−0.458093	−0.0187328
$599$	−3.40234	−0.139016	−0.0695079	−	0.997581i	$-0.522143\pi$
$599$	−3.40234	−0.139016	−0.0695079	+	0.997581i	$0.522143\pi$
$600$	10.5380	0.430213
$601$	7.96270	0.324805	0.162403	−	0.986725i	$-0.448076\pi$
$601$	7.96270	0.324805	0.162403	+	0.986725i	$0.448076\pi$
$602$	−68.6244	−2.79692
$603$	29.9435	1.21939
$604$	−58.4736	−2.37926
$605$	28.0909	1.14206
$606$	11.7091	0.475649
$607$	47.8410	1.94181	0.970904	−	0.239469i	$-0.0769732\pi$
$607$	47.8410	1.94181	0.970904	+	0.239469i	$0.0769732\pi$
$608$	0.932269	0.0378085
$609$	8.37573	0.339402
$610$	−7.90626	−0.320115
$611$	−46.8335	−1.89468
$612$	−45.3113	−1.83160
$613$	17.1970	0.694582	0.347291	−	0.937757i	$-0.387102\pi$
$613$	17.1970	0.694582	0.347291	+	0.937757i	$0.387102\pi$
$614$	−42.1396	−1.70062
$615$	2.10841	0.0850192
$616$	20.8556	0.840295
$617$	−25.6380	−1.03215	−0.516073	−	0.856545i	$-0.672607\pi$
$617$	−25.6380	−1.03215	−0.516073	+	0.856545i	$0.672607\pi$
$618$	1.45361	0.0584728
$619$	−27.3832	−1.10063	−0.550313	−	0.834959i	$-0.685492\pi$
$619$	−27.3832	−1.10063	−0.550313	+	0.834959i	$0.685492\pi$
$620$	24.1337	0.969234
$621$	0.117871	0.00473001
$622$	58.2254	2.33463
$623$	17.7505	0.711159
$624$	−6.56592	−0.262847
$625$	−24.3974	−0.975894
$626$	63.0725	2.52088
$627$	2.79696	0.111700
$628$	−65.9861	−2.63313
$629$	−12.2151	−0.487048
$630$	63.0339	2.51133
$631$	3.42616	0.136393	0.0681967	−	0.997672i	$-0.478275\pi$
$631$	3.42616	0.136393	0.0681967	+	0.997672i	$0.478275\pi$
$632$	29.7329	1.18271
$633$	−3.55653	−0.141359
$634$	19.6980	0.782309
$635$	−25.8814	−1.02707
$636$	−2.77868	−0.110182
$637$	4.44063	0.175944
$638$	−25.5141	−1.01011
$639$	31.6025	1.25018
$640$	−61.7736	−2.44182
$641$	0.940539	0.0371491	0.0185745	−	0.999827i	$-0.494087\pi$
$641$	0.940539	0.0371491	0.0185745	+	0.999827i	$0.494087\pi$
$642$	10.0821	0.397910
$643$	−21.0572	−0.830415	−0.415207	−	0.909727i	$-0.636291\pi$
$643$	−21.0572	−0.830415	−0.415207	+	0.909727i	$0.636291\pi$
$644$	−0.559322	−0.0220404
$645$	−12.9347	−0.509305
$646$	44.7245	1.75966
$647$	39.4010	1.54902	0.774508	−	0.632565i	$-0.217998\pi$
$647$	39.4010	1.54902	0.774508	+	0.632565i	$0.217998\pi$
$648$	−37.0741	−1.45641
$649$	−13.6064	−0.534098
$650$	−48.1363	−1.88806
$651$	−2.23847	−0.0877326
$652$	92.1962	3.61068
$653$	15.6535	0.612571	0.306285	−	0.951940i	$-0.400914\pi$
$653$	15.6535	0.612571	0.306285	+	0.951940i	$0.400914\pi$
$654$	16.2408	0.635067
$655$	−29.4412	−1.15036
$656$	6.58280	0.257015
$657$	−9.56720	−0.373252
$658$	−85.6271	−3.33809
$659$	−30.4412	−1.18582	−0.592911	−	0.805268i	$-0.702021\pi$
$659$	−30.4412	−1.18582	−0.592911	+	0.805268i	$0.702021\pi$
$660$	7.82178	0.304462
$661$	−11.9942	−0.466522	−0.233261	−	0.972414i	$-0.574940\pi$
$661$	−11.9942	−0.466522	−0.233261	+	0.972414i	$0.574940\pi$
$662$	−12.8673	−0.500100
$663$	−6.34507	−0.246422
$664$	85.8116	3.33013
$665$	−41.5495	−1.61122
$666$	−21.2629	−0.823922
$667$	0.343888	0.0133154
$668$	40.0079	1.54795
$669$	7.23459	0.279705
$670$	−82.6533	−3.19317
$671$	−1.49180	−0.0575902
$672$	−0.241817	−0.00932831
$673$	15.7547	0.607297	0.303649	−	0.952784i	$-0.401795\pi$
$673$	15.7547	0.607297	0.303649	+	0.952784i	$0.401795\pi$
$674$	−43.3261	−1.66886
$675$	12.3859	0.476733
$676$	6.81121	0.261970
$677$	−6.69488	−0.257305	−0.128653	−	0.991690i	$-0.541065\pi$
$677$	−6.69488	−0.257305	−0.128653	+	0.991690i	$0.541065\pi$
$678$	12.9659	0.497952
$679$	8.90476	0.341733
$680$	62.8580	2.41049
$681$	−3.92155	−0.150274
$682$	6.81883	0.261106
$683$	30.9121	1.18282	0.591410	−	0.806371i	$-0.298571\pi$
$683$	30.9121	1.18282	0.591410	+	0.806371i	$0.298571\pi$
$684$	51.9906	1.98791
$685$	−39.3205	−1.50236
$686$	−40.9406	−1.56312
$687$	11.0637	0.422106
$688$	−40.3844	−1.53964
$689$	6.37895	0.243018
$690$	−0.157866	−0.00600985
$691$	19.9693	0.759670	0.379835	−	0.925054i	$-0.375981\pi$
$691$	19.9693	0.759670	0.379835	+	0.925054i	$0.375981\pi$
$692$	−4.71235	−0.179137
$693$	11.8936	0.451800
$694$	31.8689	1.20973
$695$	53.6460	2.03491
$696$	14.5390	0.551100
$697$	6.36138	0.240955
$698$	−40.4484	−1.53099
$699$	−8.32476	−0.314871
$700$	−58.7734	−2.22143
$701$	−28.7748	−1.08681	−0.543405	−	0.839471i	$-0.682865\pi$
$701$	−28.7748	−1.08681	−0.543405	+	0.839471i	$0.682865\pi$
$702$	−22.7636	−0.859155
$703$	14.0157	0.528613
$704$	−11.4111	−0.430071
$705$	−16.1395	−0.607849
$706$	−34.0254	−1.28056
$707$	−32.8202	−1.23433
$708$	15.4277	0.579809
$709$	10.0280	0.376609	0.188305	−	0.982111i	$-0.439701\pi$
$709$	10.0280	0.376609	0.188305	+	0.982111i	$0.439701\pi$
$710$	−87.2327	−3.27379
$711$	16.9562	0.635906
$712$	30.8122	1.15474
$713$	−0.0919064	−0.00344192
$714$	−11.6009	−0.434152
$715$	−17.9562	−0.671526
$716$	−44.6016	−1.66684
$717$	−2.42324	−0.0904976
$718$	79.6900	2.97401
$719$	−16.6360	−0.620419	−0.310210	−	0.950668i	$-0.600399\pi$
$719$	−16.6360	−0.620419	−0.310210	+	0.950668i	$0.600399\pi$
$720$	37.0945	1.38243
$721$	−4.07443	−0.151740
$722$	−4.69688	−0.174800
$723$	0.504040	0.0187455
$724$	−58.7127	−2.18204
$725$	36.1356	1.34204
$726$	−8.99929	−0.333995
$727$	29.3994	1.09036	0.545182	−	0.838318i	$-0.316461\pi$
$727$	29.3994	1.09036	0.545182	+	0.838318i	$0.316461\pi$
$728$	54.2863	2.01199
$729$	−18.1274	−0.671386
$730$	26.4084	0.977420
$731$	−39.0261	−1.44343
$732$	1.69148	0.0625190
$733$	22.8461	0.843840	0.421920	−	0.906633i	$-0.361356\pi$
$733$	22.8461	0.843840	0.421920	+	0.906633i	$0.361356\pi$
$734$	−20.0225	−0.739045
$735$	1.53031	0.0564462
$736$	−0.00992846	−0.000365968 0
$737$	−15.5955	−0.574467
$738$	11.0733	0.407615
$739$	38.3670	1.41135	0.705677	−	0.708533i	$-0.250643\pi$
$739$	38.3670	1.41135	0.705677	+	0.708533i	$0.250643\pi$
$740$	39.1954	1.44085
$741$	7.28039	0.267452
$742$	11.6628	0.428156
$743$	40.3801	1.48140	0.740701	−	0.671834i	$-0.234493\pi$
$743$	40.3801	1.48140	0.740701	+	0.671834i	$0.234493\pi$
$744$	−3.88565	−0.142455
$745$	52.3754	1.91889
$746$	−59.1590	−2.16597
$747$	48.9369	1.79051
$748$	23.5995	0.862883
$749$	−28.2599	−1.03259
$750$	−0.381689	−0.0139373
$751$	29.2280	1.06655	0.533273	−	0.845943i	$-0.320962\pi$
$751$	29.2280	1.06655	0.533273	+	0.845943i	$0.320962\pi$
$752$	−50.3903	−1.83755
$753$	−6.06066	−0.220863
$754$	−66.4123	−2.41859
$755$	46.2599	1.68357
$756$	−27.7938	−1.01085
$757$	31.2717	1.13659	0.568294	−	0.822826i	$-0.307604\pi$
$757$	31.2717	1.13659	0.568294	+	0.822826i	$0.307604\pi$
$758$	−28.3962	−1.03140
$759$	−0.0297870	−0.00108120
$760$	−72.1238	−2.61621
$761$	−11.5473	−0.418588	−0.209294	−	0.977853i	$-0.567117\pi$
$761$	−11.5473	−0.418588	−0.209294	+	0.977853i	$0.567117\pi$
$762$	8.29145	0.300368
$763$	−45.5226	−1.64803
$764$	105.223	3.80684
$765$	35.8468	1.29605
$766$	−39.4220	−1.42438
$767$	−35.4170	−1.27883
$768$	13.3540	0.481871
$769$	−25.3576	−0.914418	−0.457209	−	0.889359i	$-0.651151\pi$
$769$	−25.3576	−0.914418	−0.457209	+	0.889359i	$0.651151\pi$
$770$	−32.8300	−1.18311
$771$	1.29389	0.0465982
$772$	51.2883	1.84591
$773$	−4.45434	−0.160211	−0.0801057	−	0.996786i	$-0.525526\pi$
$773$	−4.45434	−0.160211	−0.0801057	+	0.996786i	$0.525526\pi$
$774$	−67.9331	−2.44180
$775$	−9.65749	−0.346907
$776$	15.4573	0.554886
$777$	−3.63548	−0.130422
$778$	−53.6015	−1.92171
$779$	−7.29911	−0.261518
$780$	20.3598	0.728998
$781$	−16.4596	−0.588969
$782$	−0.476305	−0.0170326
$783$	17.0885	0.610692
$784$	4.77788	0.170638
$785$	52.2032	1.86321
$786$	9.43190	0.336425
$787$	29.7128	1.05915	0.529573	−	0.848264i	$-0.322352\pi$
$787$	29.7128	1.05915	0.529573	+	0.848264i	$0.322352\pi$
$788$	30.0939	1.07205
$789$	8.95694	0.318876
$790$	−46.8043	−1.66522
$791$	−36.3430	−1.29221
$792$	20.6455	0.733606
$793$	−3.88309	−0.137893
$794$	22.2593	0.789952
$795$	2.19828	0.0779650
$796$	−51.3694	−1.82074
$797$	36.7236	1.30082	0.650408	−	0.759585i	$-0.274598\pi$
$797$	36.7236	1.30082	0.650408	+	0.759585i	$0.274598\pi$
$798$	13.3110	0.471203
$799$	−48.6954	−1.72272
$800$	−1.04328	−0.0368855
$801$	17.5717	0.620865
$802$	62.5920	2.21020
$803$	4.98289	0.175842
$804$	17.6830	0.623632
$805$	0.442493	0.0155958
$806$	17.7492	0.625187
$807$	1.59021	0.0559779
$808$	−56.9710	−2.00423
$809$	2.17184	0.0763578	0.0381789	−	0.999271i	$-0.487844\pi$
$809$	2.17184	0.0763578	0.0381789	+	0.999271i	$0.487844\pi$
$810$	58.3605	2.05058
$811$	11.9110	0.418254	0.209127	−	0.977889i	$-0.432938\pi$
$811$	11.9110	0.418254	0.209127	+	0.977889i	$0.432938\pi$
$812$	−81.0881	−2.84563
$813$	5.19514	0.182202
$814$	11.0744	0.388157
$815$	−72.9386	−2.55493
$816$	−6.82695	−0.238991
$817$	44.7789	1.56661
$818$	18.5300	0.647885
$819$	30.9586	1.08178
$820$	−20.4122	−0.712824
$821$	−13.6014	−0.474693	−0.237347	−	0.971425i	$-0.576278\pi$
$821$	−13.6014	−0.474693	−0.237347	+	0.971425i	$0.576278\pi$
$822$	12.5969	0.439366
$823$	27.5095	0.958922	0.479461	−	0.877563i	$-0.340832\pi$
$823$	27.5095	0.958922	0.479461	+	0.877563i	$0.340832\pi$
$824$	−7.07260	−0.246386
$825$	−3.13001	−0.108973
$826$	−64.7539	−2.25308
$827$	35.5469	1.23609	0.618043	−	0.786144i	$-0.287926\pi$
$827$	35.5469	1.23609	0.618043	+	0.786144i	$0.287926\pi$
$828$	−0.553688	−0.0192420
$829$	−29.5771	−1.02726	−0.513628	−	0.858013i	$-0.671699\pi$
$829$	−29.5771	−1.02726	−0.513628	+	0.858013i	$0.671699\pi$
$830$	−135.081	−4.68873
$831$	3.65695	0.126858
$832$	−29.7026	−1.02975
$833$	4.61717	0.159975
$834$	−17.1862	−0.595110
$835$	−31.6512	−1.09533
$836$	−27.0783	−0.936522
$837$	−4.56701	−0.157859
$838$	−4.64333	−0.160401
$839$	4.37288	0.150969	0.0754844	−	0.997147i	$-0.475950\pi$
$839$	4.37288	0.150969	0.0754844	+	0.997147i	$0.475950\pi$
$840$	18.7079	0.645484
$841$	20.8553	0.719150
$842$	23.1375	0.797370
$843$	−8.53214	−0.293863
$844$	34.4319	1.18520
$845$	−5.38851	−0.185370
$846$	−84.7645	−2.91426
$847$	25.2247	0.866732
$848$	6.86341	0.235690
$849$	−4.93608	−0.169406
$850$	−50.0500	−1.71670
$851$	−0.149264	−0.00511671
$852$	18.6628	0.639376
$853$	−4.06987	−0.139350	−0.0696749	−	0.997570i	$-0.522196\pi$
$853$	−4.06987	−0.139350	−0.0696749	+	0.997570i	$0.522196\pi$
$854$	−7.09958	−0.242943
$855$	−41.1310	−1.40665
$856$	−49.0550	−1.67667
$857$	−52.4420	−1.79139	−0.895693	−	0.444673i	$-0.853320\pi$
$857$	−52.4420	−1.79139	−0.895693	+	0.444673i	$0.853320\pi$
$858$	5.75253	0.196388
$859$	18.9272	0.645788	0.322894	−	0.946435i	$-0.395344\pi$
$859$	18.9272	0.645788	0.322894	+	0.946435i	$0.395344\pi$
$860$	125.225	4.27015
$861$	1.89328	0.0645230
$862$	−7.49508	−0.255283
$863$	16.3931	0.558027	0.279014	−	0.960287i	$-0.409993\pi$
$863$	16.3931	0.558027	0.279014	+	0.960287i	$0.409993\pi$
$864$	−0.493365	−0.0167846
$865$	3.72805	0.126757
$866$	−35.7919	−1.21626
$867$	0.462796	0.0157174
$868$	21.6714	0.735574
$869$	−8.83129	−0.299581
$870$	−22.8867	−0.775932
$871$	−40.5944	−1.37549
$872$	−79.0205	−2.67597
$873$	8.81505	0.298344
$874$	0.546517	0.0184862
$875$	1.06986	0.0361680
$876$	−5.64988	−0.190892
$877$	−31.1690	−1.05250	−0.526251	−	0.850329i	$-0.676403\pi$
$877$	−31.1690	−1.05250	−0.526251	+	0.850329i	$0.676403\pi$
$878$	−90.8399	−3.06570
$879$	−5.99328	−0.202148
$880$	−19.3200	−0.651276
$881$	18.4490	0.621563	0.310781	−	0.950481i	$-0.399409\pi$
$881$	18.4490	0.621563	0.310781	+	0.950481i	$0.399409\pi$
$882$	8.03715	0.270625
$883$	−29.0624	−0.978029	−0.489015	−	0.872276i	$-0.662644\pi$
$883$	−29.0624	−0.978029	−0.489015	+	0.872276i	$0.662644\pi$
$884$	61.4286	2.06607
$885$	−12.2052	−0.410274
$886$	71.0016	2.38535
$887$	−21.9044	−0.735476	−0.367738	−	0.929929i	$-0.619868\pi$
$887$	−21.9044	−0.735476	−0.367738	+	0.929929i	$0.619868\pi$
$888$	−6.31065	−0.211772
$889$	−23.2407	−0.779467
$890$	−48.5033	−1.62584
$891$	11.0118	0.368909
$892$	−70.0404	−2.34513
$893$	55.8735	1.86974
$894$	−16.7792	−0.561179
$895$	35.2854	1.17946
$896$	−55.4708	−1.85315
$897$	−0.0775345	−0.00258880
$898$	72.2812	2.41205
$899$	−13.3242	−0.444386
$900$	−58.1813	−1.93938
$901$	6.63255	0.220962
$902$	−5.76732	−0.192031
$903$	−11.6150	−0.386523
$904$	−63.0860	−2.09821
$905$	46.4490	1.54402
$906$	−14.8200	−0.492361
$907$	−13.1867	−0.437857	−0.218928	−	0.975741i	$-0.570256\pi$
$907$	−13.1867	−0.437857	−0.218928	+	0.975741i	$0.570256\pi$
$908$	37.9658	1.25994
$909$	−32.4896	−1.07761
$910$	−85.4552	−2.83281
$911$	18.2306	0.604008	0.302004	−	0.953307i	$-0.402344\pi$
$911$	18.2306	0.604008	0.302004	+	0.953307i	$0.402344\pi$
$912$	7.83331	0.259387
$913$	−25.4878	−0.843523
$914$	−55.1595	−1.82451
$915$	−1.33817	−0.0442386
$916$	−107.111	−3.53905
$917$	−26.4373	−0.873038
$918$	−23.6686	−0.781179
$919$	−3.48482	−0.114954	−0.0574768	−	0.998347i	$-0.518306\pi$
$919$	−3.48482	−0.114954	−0.0574768	+	0.998347i	$0.518306\pi$
$920$	0.768102	0.0253236
$921$	−7.13233	−0.235018
$922$	−17.0960	−0.563027
$923$	−42.8436	−1.41021
$924$	7.02372	0.231063
$925$	−15.6846	−0.515708
$926$	25.1641	0.826942
$927$	−4.03338	−0.132474
$928$	−1.43938	−0.0472501
$929$	−36.1583	−1.18631	−0.593157	−	0.805087i	$-0.702119\pi$
$929$	−36.1583	−1.18631	−0.593157	+	0.805087i	$0.702119\pi$
$930$	6.11663	0.200572
$931$	−5.29778	−0.173628
$932$	80.5947	2.63997
$933$	9.85493	0.322636
$934$	−9.94319	−0.325351
$935$	−18.6701	−0.610578
$936$	53.7395	1.75653
$937$	−55.7418	−1.82100	−0.910502	−	0.413504i	$-0.864305\pi$
$937$	−55.7418	−1.82100	−0.910502	+	0.413504i	$0.864305\pi$
$938$	−74.2201	−2.42337
$939$	10.6753	0.348376
$940$	156.252	5.09637
$941$	36.3584	1.18525	0.592625	−	0.805478i	$-0.298091\pi$
$941$	36.3584	1.18525	0.592625	+	0.805478i	$0.298091\pi$
$942$	−16.7240	−0.544897
$943$	0.0777338	0.00253136
$944$	−38.1068	−1.24027
$945$	21.9884	0.715281
$946$	35.3816	1.15035
$947$	5.51942	0.179357	0.0896785	−	0.995971i	$-0.471416\pi$
$947$	5.51942	0.179357	0.0896785	+	0.995971i	$0.471416\pi$
$948$	10.0134	0.325221
$949$	12.9703	0.421033
$950$	57.4278	1.86320
$951$	3.33399	0.108112
$952$	56.4445	1.82938
$953$	0.837997	0.0271454	0.0135727	−	0.999908i	$-0.495680\pi$
$953$	0.837997	0.0271454	0.0135727	+	0.999908i	$0.495680\pi$
$954$	11.5453	0.373794
$955$	−83.2445	−2.69373
$956$	23.4602	0.758757
$957$	−4.31839	−0.139594
$958$	−88.8475	−2.87053
$959$	−35.3086	−1.14018
$960$	−10.2360	−0.330365
$961$	−27.4390	−0.885130
$962$	28.8262	0.929395
$963$	−27.9752	−0.901489
$964$	−4.87978	−0.157167
$965$	−40.5754	−1.30617
$966$	−0.141759	−0.00456101
$967$	−16.0683	−0.516720	−0.258360	−	0.966049i	$-0.583182\pi$
$967$	−16.0683	−0.516720	−0.258360	+	0.966049i	$0.583182\pi$
$968$	43.7864	1.40735
$969$	7.56983	0.243178
$970$	−24.3323	−0.781262
$971$	52.6248	1.68881	0.844406	−	0.535704i	$-0.179954\pi$
$971$	52.6248	1.68881	0.844406	+	0.535704i	$0.179954\pi$
$972$	−41.6779	−1.33682
$973$	48.1724	1.54434
$974$	−17.6972	−0.567056
$975$	−8.14730	−0.260922
$976$	−4.17800	−0.133734
$977$	45.7366	1.46324	0.731622	−	0.681710i	$-0.238763\pi$
$977$	45.7366	1.46324	0.731622	+	0.681710i	$0.238763\pi$
$978$	23.3669	0.747190
$979$	−9.15187	−0.292495
$980$	−14.8154	−0.473260
$981$	−45.0640	−1.43878
$982$	75.3817	2.40553
$983$	11.6657	0.372079	0.186040	−	0.982542i	$-0.440435\pi$
$983$	11.6657	0.372079	0.186040	+	0.982542i	$0.440435\pi$
$984$	3.28646	0.104769
$985$	−23.8080	−0.758585
$986$	−69.0526	−2.19908
$987$	−14.4928	−0.461311
$988$	−70.4838	−2.24239
$989$	−0.476885	−0.0151641
$990$	−32.4993	−1.03289
$991$	29.2300	0.928521	0.464261	−	0.885699i	$-0.346320\pi$
$991$	29.2300	0.928521	0.464261	+	0.885699i	$0.346320\pi$
$992$	0.384685	0.0122138
$993$	−2.17785	−0.0691119
$994$	−78.3323	−2.48455
$995$	40.6396	1.28836
$996$	28.8995	0.915716
$997$	17.1860	0.544286	0.272143	−	0.962257i	$-0.412268\pi$
$997$	17.1860	0.544286	0.272143	+	0.962257i	$0.412268\pi$
$998$	5.79305	0.183376
$999$	−7.41723	−0.234671

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.c.1.13	✓	179

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.c.1.13	✓	179	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45370 q^{2} -0.415301 q^{3} +4.02066 q^{4} -3.18084 q^{5} +1.01903 q^{6} -2.85630 q^{7} -4.95811 q^{8} -2.82752 q^{9} +O(q^{10})\)	\(q-2.45370 q^{2} -0.415301 q^{3} +4.02066 q^{4} -3.18084 q^{5} +1.01903 q^{6} -2.85630 q^{7} -4.95811 q^{8} -2.82752 q^{9} +7.80484 q^{10} +1.47266 q^{11} -1.66979 q^{12} +3.83328 q^{13} +7.00851 q^{14} +1.32101 q^{15} +4.12441 q^{16} +3.98568 q^{17} +6.93791 q^{18} -4.57320 q^{19} -12.7891 q^{20} +1.18622 q^{21} -3.61347 q^{22} +0.0487036 q^{23} +2.05911 q^{24} +5.11776 q^{25} -9.40574 q^{26} +2.42018 q^{27} -11.4842 q^{28} +7.06083 q^{29} -3.24136 q^{30} -1.88706 q^{31} -0.203855 q^{32} -0.611598 q^{33} -9.77968 q^{34} +9.08543 q^{35} -11.3685 q^{36} -3.06475 q^{37} +11.2213 q^{38} -1.59197 q^{39} +15.7710 q^{40} +1.59606 q^{41} -2.91064 q^{42} -9.79157 q^{43} +5.92107 q^{44} +8.99391 q^{45} -0.119504 q^{46} -12.2176 q^{47} -1.71287 q^{48} +1.15844 q^{49} -12.5575 q^{50} -1.65526 q^{51} +15.4123 q^{52} +1.66410 q^{53} -5.93840 q^{54} -4.68430 q^{55} +14.1618 q^{56} +1.89926 q^{57} -17.3252 q^{58} -9.23933 q^{59} +5.31133 q^{60} -1.01299 q^{61} +4.63028 q^{62} +8.07625 q^{63} -7.74862 q^{64} -12.1931 q^{65} +1.50068 q^{66} -10.5900 q^{67} +16.0251 q^{68} -0.0202267 q^{69} -22.2930 q^{70} -11.1767 q^{71} +14.0192 q^{72} +3.38359 q^{73} +7.51998 q^{74} -2.12541 q^{75} -18.3873 q^{76} -4.20636 q^{77} +3.90621 q^{78} -5.99683 q^{79} -13.1191 q^{80} +7.47747 q^{81} -3.91626 q^{82} -17.3073 q^{83} +4.76941 q^{84} -12.6778 q^{85} +24.0256 q^{86} -2.93237 q^{87} -7.30161 q^{88} -6.21452 q^{89} -22.0684 q^{90} -10.9490 q^{91} +0.195821 q^{92} +0.783696 q^{93} +29.9783 q^{94} +14.5466 q^{95} +0.0846611 q^{96} -3.11759 q^{97} -2.84247 q^{98} -4.16398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9}+O(q^{10}) \) 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9	\( 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9} + 9 q^{10} + 46 q^{11} + 33 q^{12} + 47 q^{13} + 22 q^{14} + 36 q^{15} + 222 q^{16} + 103 q^{17} + 43 q^{18} + 12 q^{19} + 102 q^{20} + 50 q^{21} + 39 q^{22} + 121 q^{23} - 3 q^{24} + 246 q^{25} + 52 q^{26} + 49 q^{27} + 41 q^{28} + 138 q^{29} + 28 q^{30} + 5 q^{31} + 137 q^{32} + 63 q^{33} + 2 q^{34} + 72 q^{35} + 279 q^{36} + 118 q^{37} + 123 q^{38} + q^{39} + 9 q^{40} + 50 q^{41} + 48 q^{42} + 48 q^{43} + 108 q^{44} + 158 q^{45} + 13 q^{46} + 85 q^{47} + 50 q^{48} + 230 q^{49} + 78 q^{50} + 15 q^{51} + 41 q^{52} + 399 q^{53} - 5 q^{54} + 24 q^{55} + 53 q^{56} + 45 q^{57} + 27 q^{58} + 48 q^{59} + 66 q^{60} + 46 q^{61} + 81 q^{62} + 78 q^{63} + 252 q^{64} + 153 q^{65} + 6 q^{66} + 70 q^{67} + 240 q^{68} + 120 q^{69} - 31 q^{70} + 86 q^{71} + 89 q^{72} + 45 q^{73} + 68 q^{74} + 17 q^{75} - 13 q^{76} + 362 q^{77} + 69 q^{78} + 31 q^{79} + 169 q^{80} + 303 q^{81} + 25 q^{82} + 106 q^{83} + 13 q^{84} + 115 q^{85} + 95 q^{86} + 32 q^{87} + 83 q^{88} + 105 q^{89} - 38 q^{90} + 3 q^{91} + 310 q^{92} + 298 q^{93} - 17 q^{94} + 102 q^{95} - 82 q^{96} + 34 q^{97} + 81 q^{98} + 58 q^{99}+O(q^{100}) \) 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9 + 9 * q^10 + 46 * q^11 + 33 * q^12 + 47 * q^13 + 22 * q^14 + 36 * q^15 + 222 * q^16 + 103 * q^17 + 43 * q^18 + 12 * q^19 + 102 * q^20 + 50 * q^21 + 39 * q^22 + 121 * q^23 - 3 * q^24 + 246 * q^25 + 52 * q^26 + 49 * q^27 + 41 * q^28 + 138 * q^29 + 28 * q^30 + 5 * q^31 + 137 * q^32 + 63 * q^33 + 2 * q^34 + 72 * q^35 + 279 * q^36 + 118 * q^37 + 123 * q^38 + q^39 + 9 * q^40 + 50 * q^41 + 48 * q^42 + 48 * q^43 + 108 * q^44 + 158 * q^45 + 13 * q^46 + 85 * q^47 + 50 * q^48 + 230 * q^49 + 78 * q^50 + 15 * q^51 + 41 * q^52 + 399 * q^53 - 5 * q^54 + 24 * q^55 + 53 * q^56 + 45 * q^57 + 27 * q^58 + 48 * q^59 + 66 * q^60 + 46 * q^61 + 81 * q^62 + 78 * q^63 + 252 * q^64 + 153 * q^65 + 6 * q^66 + 70 * q^67 + 240 * q^68 + 120 * q^69 - 31 * q^70 + 86 * q^71 + 89 * q^72 + 45 * q^73 + 68 * q^74 + 17 * q^75 - 13 * q^76 + 362 * q^77 + 69 * q^78 + 31 * q^79 + 169 * q^80 + 303 * q^81 + 25 * q^82 + 106 * q^83 + 13 * q^84 + 115 * q^85 + 95 * q^86 + 32 * q^87 + 83 * q^88 + 105 * q^89 - 38 * q^90 + 3 * q^91 + 310 * q^92 + 298 * q^93 - 17 * q^94 + 102 * q^95 - 82 * q^96 + 34 * q^97 + 81 * q^98 + 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more