LMFDB - Embedded newform 6010.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6010,2,Mod(1,6010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6010, 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("6010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6010 = 2 \cdot 5 \cdot 601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6010.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:	$47.9900916148$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 9 x^{14} + 75 x^{13} - 178 x^{12} - 232 x^{11} + 872 x^{10} + 228 x^{9} - 1986 x^{8} + \cdots - 10 $ x^16 - 8x^15 + 9x^14 + 75x^13 - 178x^12 - 232x^11 + 872x^10 + 228x^9 - 1986x^8 + 164x^7 + 2332x^6 - 440x^5 - 1344x^4 + 244x^3 + 295x^2 - 41*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.45194$ of defining polynomial
Character	$\chi$	$=$	6010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.45194 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.45194 q^{6} +1.53125 q^{7} +1.00000 q^{8} +3.01199 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.45194 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.45194 q^{6} +1.53125 q^{7} +1.00000 q^{8} +3.01199 q^{9} +1.00000 q^{10} -0.396039 q^{11} -2.45194 q^{12} -1.09276 q^{13} +1.53125 q^{14} -2.45194 q^{15} +1.00000 q^{16} +3.63794 q^{17} +3.01199 q^{18} -2.31419 q^{19} +1.00000 q^{20} -3.75453 q^{21} -0.396039 q^{22} -7.59249 q^{23} -2.45194 q^{24} +1.00000 q^{25} -1.09276 q^{26} -0.0293880 q^{27} +1.53125 q^{28} -0.633968 q^{29} -2.45194 q^{30} -1.51909 q^{31} +1.00000 q^{32} +0.971063 q^{33} +3.63794 q^{34} +1.53125 q^{35} +3.01199 q^{36} -9.28972 q^{37} -2.31419 q^{38} +2.67939 q^{39} +1.00000 q^{40} -8.32601 q^{41} -3.75453 q^{42} +4.30278 q^{43} -0.396039 q^{44} +3.01199 q^{45} -7.59249 q^{46} +1.40081 q^{47} -2.45194 q^{48} -4.65526 q^{49} +1.00000 q^{50} -8.92000 q^{51} -1.09276 q^{52} -3.05413 q^{53} -0.0293880 q^{54} -0.396039 q^{55} +1.53125 q^{56} +5.67424 q^{57} -0.633968 q^{58} -3.82943 q^{59} -2.45194 q^{60} -1.88634 q^{61} -1.51909 q^{62} +4.61211 q^{63} +1.00000 q^{64} -1.09276 q^{65} +0.971063 q^{66} +0.761251 q^{67} +3.63794 q^{68} +18.6163 q^{69} +1.53125 q^{70} +10.9195 q^{71} +3.01199 q^{72} -9.16888 q^{73} -9.28972 q^{74} -2.45194 q^{75} -2.31419 q^{76} -0.606437 q^{77} +2.67939 q^{78} +12.0604 q^{79} +1.00000 q^{80} -8.96390 q^{81} -8.32601 q^{82} +5.18730 q^{83} -3.75453 q^{84} +3.63794 q^{85} +4.30278 q^{86} +1.55445 q^{87} -0.396039 q^{88} +4.52465 q^{89} +3.01199 q^{90} -1.67330 q^{91} -7.59249 q^{92} +3.72471 q^{93} +1.40081 q^{94} -2.31419 q^{95} -2.45194 q^{96} -3.00998 q^{97} -4.65526 q^{98} -1.19287 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} - 8 q^{3} + 16 q^{4} + 16 q^{5} - 8 q^{6} - 10 q^{7} + 16 q^{8} - 2 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 - 8 * q^3 + 16 * q^4 + 16 * q^5 - 8 * q^6 - 10 * q^7 + 16 * q^8 - 2 * q^9	\( 16 q + 16 q^{2} - 8 q^{3} + 16 q^{4} + 16 q^{5} - 8 q^{6} - 10 q^{7} + 16 q^{8} - 2 q^{9} + 16 q^{10} - 14 q^{11} - 8 q^{12} - 20 q^{13} - 10 q^{14} - 8 q^{15} + 16 q^{16} - 27 q^{17} - 2 q^{18} - 17 q^{19} + 16 q^{20} - 12 q^{21} - 14 q^{22} - 9 q^{23} - 8 q^{24} + 16 q^{25} - 20 q^{26} - 11 q^{27} - 10 q^{28} - 23 q^{29} - 8 q^{30} - 21 q^{31} + 16 q^{32} - 9 q^{33} - 27 q^{34} - 10 q^{35} - 2 q^{36} - 16 q^{37} - 17 q^{38} - 6 q^{39} + 16 q^{40} - 35 q^{41} - 12 q^{42} + 3 q^{43} - 14 q^{44} - 2 q^{45} - 9 q^{46} - 25 q^{47} - 8 q^{48} - 24 q^{49} + 16 q^{50} - q^{51} - 20 q^{52} - 39 q^{53} - 11 q^{54} - 14 q^{55} - 10 q^{56} - 6 q^{57} - 23 q^{58} - 32 q^{59} - 8 q^{60} - 38 q^{61} - 21 q^{62} + q^{63} + 16 q^{64} - 20 q^{65} - 9 q^{66} + 5 q^{67} - 27 q^{68} - 25 q^{69} - 10 q^{70} - 16 q^{71} - 2 q^{72} - 17 q^{73} - 16 q^{74} - 8 q^{75} - 17 q^{76} - 34 q^{77} - 6 q^{78} - 40 q^{79} + 16 q^{80} - 28 q^{81} - 35 q^{82} - 22 q^{83} - 12 q^{84} - 27 q^{85} + 3 q^{86} + 10 q^{87} - 14 q^{88} - 46 q^{89} - 2 q^{90} - q^{91} - 9 q^{92} + 14 q^{93} - 25 q^{94} - 17 q^{95} - 8 q^{96} - 21 q^{97} - 24 q^{98} + 15 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 - 8 * q^3 + 16 * q^4 + 16 * q^5 - 8 * q^6 - 10 * q^7 + 16 * q^8 - 2 * q^9 + 16 * q^10 - 14 * q^11 - 8 * q^12 - 20 * q^13 - 10 * q^14 - 8 * q^15 + 16 * q^16 - 27 * q^17 - 2 * q^18 - 17 * q^19 + 16 * q^20 - 12 * q^21 - 14 * q^22 - 9 * q^23 - 8 * q^24 + 16 * q^25 - 20 * q^26 - 11 * q^27 - 10 * q^28 - 23 * q^29 - 8 * q^30 - 21 * q^31 + 16 * q^32 - 9 * q^33 - 27 * q^34 - 10 * q^35 - 2 * q^36 - 16 * q^37 - 17 * q^38 - 6 * q^39 + 16 * q^40 - 35 * q^41 - 12 * q^42 + 3 * q^43 - 14 * q^44 - 2 * q^45 - 9 * q^46 - 25 * q^47 - 8 * q^48 - 24 * q^49 + 16 * q^50 - q^51 - 20 * q^52 - 39 * q^53 - 11 * q^54 - 14 * q^55 - 10 * q^56 - 6 * q^57 - 23 * q^58 - 32 * q^59 - 8 * q^60 - 38 * q^61 - 21 * q^62 + q^63 + 16 * q^64 - 20 * q^65 - 9 * q^66 + 5 * q^67 - 27 * q^68 - 25 * q^69 - 10 * q^70 - 16 * q^71 - 2 * q^72 - 17 * q^73 - 16 * q^74 - 8 * q^75 - 17 * q^76 - 34 * q^77 - 6 * q^78 - 40 * q^79 + 16 * q^80 - 28 * q^81 - 35 * q^82 - 22 * q^83 - 12 * q^84 - 27 * q^85 + 3 * q^86 + 10 * q^87 - 14 * q^88 - 46 * q^89 - 2 * q^90 - q^91 - 9 * q^92 + 14 * q^93 - 25 * q^94 - 17 * q^95 - 8 * q^96 - 21 * q^97 - 24 * q^98 + 15 * 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
$3$	−2.45194	−1.41563	−0.707813	−	0.706400i	$-0.750318\pi$
$3$	−2.45194	−1.41563	−0.707813	+	0.706400i	$0.750318\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.45194	−1.00100
$7$	1.53125	0.578759	0.289380	−	0.957214i	$-0.406551\pi$
$7$	1.53125	0.578759	0.289380	+	0.957214i	$0.406551\pi$
$8$	1.00000	0.353553
$9$	3.01199	1.00400
$10$	1.00000	0.316228
$11$	−0.396039	−0.119410	−0.0597052	−	0.998216i	$-0.519016\pi$
$11$	−0.396039	−0.119410	−0.0597052	+	0.998216i	$0.519016\pi$
$12$	−2.45194	−0.707813
$13$	−1.09276	−0.303078	−0.151539	−	0.988451i	$-0.548423\pi$
$13$	−1.09276	−0.303078	−0.151539	+	0.988451i	$0.548423\pi$
$14$	1.53125	0.409245
$15$	−2.45194	−0.633087
$16$	1.00000	0.250000
$17$	3.63794	0.882331	0.441166	−	0.897426i	$-0.354565\pi$
$17$	3.63794	0.882331	0.441166	+	0.897426i	$0.354565\pi$
$18$	3.01199	0.709932
$19$	−2.31419	−0.530911	−0.265456	−	0.964123i	$-0.585522\pi$
$19$	−2.31419	−0.530911	−0.265456	+	0.964123i	$0.585522\pi$
$20$	1.00000	0.223607
$21$	−3.75453	−0.819306
$22$	−0.396039	−0.0844359
$23$	−7.59249	−1.58314	−0.791572	−	0.611076i	$-0.790737\pi$
$23$	−7.59249	−1.58314	−0.791572	+	0.611076i	$0.790737\pi$
$24$	−2.45194	−0.500499
$25$	1.00000	0.200000
$26$	−1.09276	−0.214309
$27$	−0.0293880	−0.00565572
$28$	1.53125	0.289380
$29$	−0.633968	−0.117725	−0.0588624	−	0.998266i	$-0.518747\pi$
$29$	−0.633968	−0.117725	−0.0588624	+	0.998266i	$0.518747\pi$
$30$	−2.45194	−0.447660
$31$	−1.51909	−0.272837	−0.136418	−	0.990651i	$-0.543559\pi$
$31$	−1.51909	−0.272837	−0.136418	+	0.990651i	$0.543559\pi$
$32$	1.00000	0.176777
$33$	0.971063	0.169040
$34$	3.63794	0.623902
$35$	1.53125	0.258829
$36$	3.01199	0.501998
$37$	−9.28972	−1.52722	−0.763610	−	0.645678i	$-0.776575\pi$
$37$	−9.28972	−1.52722	−0.763610	+	0.645678i	$0.776575\pi$
$38$	−2.31419	−0.375411
$39$	2.67939	0.429045
$40$	1.00000	0.158114
$41$	−8.32601	−1.30030	−0.650152	−	0.759804i	$-0.725295\pi$
$41$	−8.32601	−1.30030	−0.650152	+	0.759804i	$0.725295\pi$
$42$	−3.75453	−0.579337
$43$	4.30278	0.656167	0.328084	−	0.944649i	$-0.393597\pi$
$43$	4.30278	0.656167	0.328084	+	0.944649i	$0.393597\pi$
$44$	−0.396039	−0.0597052
$45$	3.01199	0.449000
$46$	−7.59249	−1.11945
$47$	1.40081	0.204329	0.102164	−	0.994768i	$-0.467423\pi$
$47$	1.40081	0.204329	0.102164	+	0.994768i	$0.467423\pi$
$48$	−2.45194	−0.353906
$49$	−4.65526	−0.665038
$50$	1.00000	0.141421
$51$	−8.92000	−1.24905
$52$	−1.09276	−0.151539
$53$	−3.05413	−0.419517	−0.209758	−	0.977753i	$-0.567268\pi$
$53$	−3.05413	−0.419517	−0.209758	+	0.977753i	$0.567268\pi$
$54$	−0.0293880	−0.00399920
$55$	−0.396039	−0.0534020
$56$	1.53125	0.204622
$57$	5.67424	0.751571
$58$	−0.633968	−0.0832440
$59$	−3.82943	−0.498550	−0.249275	−	0.968433i	$-0.580192\pi$
$59$	−3.82943	−0.498550	−0.249275	+	0.968433i	$0.580192\pi$
$60$	−2.45194	−0.316543
$61$	−1.88634	−0.241521	−0.120760	−	0.992682i	$-0.538533\pi$
$61$	−1.88634	−0.241521	−0.120760	+	0.992682i	$0.538533\pi$
$62$	−1.51909	−0.192925
$63$	4.61211	0.581072
$64$	1.00000	0.125000
$65$	−1.09276	−0.135541
$66$	0.971063	0.119530
$67$	0.761251	0.0930017	0.0465008	−	0.998918i	$-0.485193\pi$
$67$	0.761251	0.0930017	0.0465008	+	0.998918i	$0.485193\pi$
$68$	3.63794	0.441166
$69$	18.6163	2.24114
$70$	1.53125	0.183020
$71$	10.9195	1.29591	0.647955	−	0.761679i	$-0.275625\pi$
$71$	10.9195	1.29591	0.647955	+	0.761679i	$0.275625\pi$
$72$	3.01199	0.354966
$73$	−9.16888	−1.07314	−0.536568	−	0.843857i	$-0.680280\pi$
$73$	−9.16888	−1.07314	−0.536568	+	0.843857i	$0.680280\pi$
$74$	−9.28972	−1.07991
$75$	−2.45194	−0.283125
$76$	−2.31419	−0.265456
$77$	−0.606437	−0.0691099
$78$	2.67939	0.303381
$79$	12.0604	1.35690	0.678448	−	0.734649i	$-0.262653\pi$
$79$	12.0604	1.35690	0.678448	+	0.734649i	$0.262653\pi$
$80$	1.00000	0.111803
$81$	−8.96390	−0.995989
$82$	−8.32601	−0.919454
$83$	5.18730	0.569381	0.284690	−	0.958620i	$-0.408109\pi$
$83$	5.18730	0.569381	0.284690	+	0.958620i	$0.408109\pi$
$84$	−3.75453	−0.409653
$85$	3.63794	0.394591
$86$	4.30278	0.463980
$87$	1.55445	0.166654
$88$	−0.396039	−0.0422179
$89$	4.52465	0.479612	0.239806	−	0.970821i	$-0.422916\pi$
$89$	4.52465	0.479612	0.239806	+	0.970821i	$0.422916\pi$
$90$	3.01199	0.317491
$91$	−1.67330	−0.175409
$92$	−7.59249	−0.791572
$93$	3.72471	0.386235
$94$	1.40081	0.144482
$95$	−2.31419	−0.237431
$96$	−2.45194	−0.250250
$97$	−3.00998	−0.305617	−0.152809	−	0.988256i	$-0.548832\pi$
$97$	−3.00998	−0.305617	−0.152809	+	0.988256i	$0.548832\pi$
$98$	−4.65526	−0.470253
$99$	−1.19287	−0.119887
$100$	1.00000	0.100000
$101$	0.532308	0.0529667	0.0264833	−	0.999649i	$-0.491569\pi$
$101$	0.532308	0.0529667	0.0264833	+	0.999649i	$0.491569\pi$
$102$	−8.92000	−0.883212
$103$	6.08972	0.600038	0.300019	−	0.953933i	$-0.403007\pi$
$103$	6.08972	0.600038	0.300019	+	0.953933i	$0.403007\pi$
$104$	−1.09276	−0.107154
$105$	−3.75453	−0.366405
$106$	−3.05413	−0.296643
$107$	13.6173	1.31644	0.658219	−	0.752826i	$-0.271310\pi$
$107$	13.6173	1.31644	0.658219	+	0.752826i	$0.271310\pi$
$108$	−0.0293880	−0.00282786
$109$	−9.80852	−0.939486	−0.469743	−	0.882803i	$-0.655653\pi$
$109$	−9.80852	−0.939486	−0.469743	+	0.882803i	$0.655653\pi$
$110$	−0.396039	−0.0377609
$111$	22.7778	2.16197
$112$	1.53125	0.144690
$113$	−20.8087	−1.95752	−0.978758	−	0.205021i	$-0.934274\pi$
$113$	−20.8087	−1.95752	−0.978758	+	0.205021i	$0.934274\pi$
$114$	5.67424	0.531441
$115$	−7.59249	−0.708004
$116$	−0.633968	−0.0588624
$117$	−3.29139	−0.304289
$118$	−3.82943	−0.352528
$119$	5.57062	0.510657
$120$	−2.45194	−0.223830
$121$	−10.8432	−0.985741
$122$	−1.88634	−0.170781
$123$	20.4148	1.84074
$124$	−1.51909	−0.136418
$125$	1.00000	0.0894427
$126$	4.61211	0.410880
$127$	−9.69057	−0.859899	−0.429949	−	0.902853i	$-0.641469\pi$
$127$	−9.69057	−0.859899	−0.429949	+	0.902853i	$0.641469\pi$
$128$	1.00000	0.0883883
$129$	−10.5501	−0.928887
$130$	−1.09276	−0.0958418
$131$	−21.5290	−1.88099	−0.940497	−	0.339803i	$-0.889640\pi$
$131$	−21.5290	−1.88099	−0.940497	+	0.339803i	$0.889640\pi$
$132$	0.971063	0.0845202
$133$	−3.54361	−0.307270
$134$	0.761251	0.0657621
$135$	−0.0293880	−0.00252932
$136$	3.63794	0.311951
$137$	−12.2619	−1.04760	−0.523801	−	0.851841i	$-0.675486\pi$
$137$	−12.2619	−1.04760	−0.523801	+	0.851841i	$0.675486\pi$
$138$	18.6163	1.58472
$139$	6.79180	0.576073	0.288036	−	0.957619i	$-0.406998\pi$
$139$	6.79180	0.576073	0.288036	+	0.957619i	$0.406998\pi$
$140$	1.53125	0.129415
$141$	−3.43469	−0.289253
$142$	10.9195	0.916347
$143$	0.432778	0.0361907
$144$	3.01199	0.250999
$145$	−0.633968	−0.0526481
$146$	−9.16888	−0.758822
$147$	11.4144	0.941444
$148$	−9.28972	−0.763610
$149$	1.03273	0.0846047	0.0423023	−	0.999105i	$-0.486531\pi$
$149$	1.03273	0.0846047	0.0423023	+	0.999105i	$0.486531\pi$
$150$	−2.45194	−0.200200
$151$	−4.34453	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$151$	−4.34453	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$152$	−2.31419	−0.187705
$153$	10.9574	0.885856
$154$	−0.606437	−0.0488681
$155$	−1.51909	−0.122016
$156$	2.67939	0.214523
$157$	5.37740	0.429163	0.214582	−	0.976706i	$-0.431161\pi$
$157$	5.37740	0.429163	0.214582	+	0.976706i	$0.431161\pi$
$158$	12.0604	0.959470
$159$	7.48852	0.593878
$160$	1.00000	0.0790569
$161$	−11.6260	−0.916260
$162$	−8.96390	−0.704270
$163$	14.9767	1.17306	0.586531	−	0.809927i	$-0.300493\pi$
$163$	14.9767	1.17306	0.586531	+	0.809927i	$0.300493\pi$
$164$	−8.32601	−0.650152
$165$	0.971063	0.0755972
$166$	5.18730	0.402613
$167$	−2.41704	−0.187037	−0.0935183	−	0.995618i	$-0.529811\pi$
$167$	−2.41704	−0.187037	−0.0935183	+	0.995618i	$0.529811\pi$
$168$	−3.75453	−0.289669
$169$	−11.8059	−0.908144
$170$	3.63794	0.279018
$171$	−6.97030	−0.533032
$172$	4.30278	0.328084
$173$	−11.4142	−0.867808	−0.433904	−	0.900959i	$-0.642864\pi$
$173$	−11.4142	−0.867808	−0.433904	+	0.900959i	$0.642864\pi$
$174$	1.55445	0.117842
$175$	1.53125	0.115752
$176$	−0.396039	−0.0298526
$177$	9.38952	0.705760
$178$	4.52465	0.339137
$179$	−1.20874	−0.0903458	−0.0451729	−	0.998979i	$-0.514384\pi$
$179$	−1.20874	−0.0903458	−0.0451729	+	0.998979i	$0.514384\pi$
$180$	3.01199	0.224500
$181$	14.9579	1.11182	0.555908	−	0.831244i	$-0.312371\pi$
$181$	14.9579	1.11182	0.555908	+	0.831244i	$0.312371\pi$
$182$	−1.67330	−0.124033
$183$	4.62518	0.341903
$184$	−7.59249	−0.559726
$185$	−9.28972	−0.682993
$186$	3.72471	0.273109
$187$	−1.44077	−0.105360
$188$	1.40081	0.102164
$189$	−0.0450005	−0.00327330
$190$	−2.31419	−0.167889
$191$	−9.94895	−0.719881	−0.359940	−	0.932975i	$-0.617203\pi$
$191$	−9.94895	−0.719881	−0.359940	+	0.932975i	$0.617203\pi$
$192$	−2.45194	−0.176953
$193$	25.9678	1.86920	0.934602	−	0.355694i	$-0.115756\pi$
$193$	25.9678	1.86920	0.934602	+	0.355694i	$0.115756\pi$
$194$	−3.00998	−0.216104
$195$	2.67939	0.191875
$196$	−4.65526	−0.332519
$197$	23.4003	1.66720	0.833602	−	0.552366i	$-0.186275\pi$
$197$	23.4003	1.66720	0.833602	+	0.552366i	$0.186275\pi$
$198$	−1.19287	−0.0847732
$199$	−5.85957	−0.415374	−0.207687	−	0.978195i	$-0.566594\pi$
$199$	−5.85957	−0.415374	−0.207687	+	0.978195i	$0.566594\pi$
$200$	1.00000	0.0707107
$201$	−1.86654	−0.131656
$202$	0.532308	0.0374531
$203$	−0.970765	−0.0681344
$204$	−8.92000	−0.624525
$205$	−8.32601	−0.581514
$206$	6.08972	0.424291
$207$	−22.8685	−1.58947
$208$	−1.09276	−0.0757696
$209$	0.916510	0.0633963
$210$	−3.75453	−0.259087
$211$	−18.4089	−1.26732	−0.633660	−	0.773612i	$-0.718448\pi$
$211$	−18.4089	−1.26732	−0.633660	+	0.773612i	$0.718448\pi$
$212$	−3.05413	−0.209758
$213$	−26.7740	−1.83452
$214$	13.6173	0.930863
$215$	4.30278	0.293447
$216$	−0.0293880	−0.00199960
$217$	−2.32611	−0.157907
$218$	−9.80852	−0.664317
$219$	22.4815	1.51916
$220$	−0.396039	−0.0267010
$221$	−3.97542	−0.267415
$222$	22.7778	1.52874
$223$	−20.6115	−1.38025	−0.690124	−	0.723691i	$-0.742444\pi$
$223$	−20.6115	−1.38025	−0.690124	+	0.723691i	$0.742444\pi$
$224$	1.53125	0.102311
$225$	3.01199	0.200799
$226$	−20.8087	−1.38417
$227$	1.48231	0.0983846	0.0491923	−	0.998789i	$-0.484335\pi$
$227$	1.48231	0.0983846	0.0491923	+	0.998789i	$0.484335\pi$
$228$	5.67424	0.375786
$229$	16.8575	1.11397	0.556987	−	0.830521i	$-0.311957\pi$
$229$	16.8575	1.11397	0.556987	+	0.830521i	$0.311957\pi$
$230$	−7.59249	−0.500634
$231$	1.48694	0.0978337
$232$	−0.633968	−0.0416220
$233$	9.60557	0.629281	0.314641	−	0.949211i	$-0.398116\pi$
$233$	9.60557	0.629281	0.314641	+	0.949211i	$0.398116\pi$
$234$	−3.29139	−0.215165
$235$	1.40081	0.0913786
$236$	−3.82943	−0.249275
$237$	−29.5712	−1.92086
$238$	5.57062	0.361089
$239$	16.5353	1.06958	0.534789	−	0.844986i	$-0.320391\pi$
$239$	16.5353	1.06958	0.534789	+	0.844986i	$0.320391\pi$
$240$	−2.45194	−0.158272
$241$	−12.5932	−0.811197	−0.405599	−	0.914051i	$-0.632937\pi$
$241$	−12.5932	−0.811197	−0.405599	+	0.914051i	$0.632937\pi$
$242$	−10.8432	−0.697024
$243$	22.0671	1.41560
$244$	−1.88634	−0.120760
$245$	−4.65526	−0.297414
$246$	20.4148	1.30160
$247$	2.52886	0.160908
$248$	−1.51909	−0.0964623
$249$	−12.7189	−0.806030
$250$	1.00000	0.0632456
$251$	−16.6158	−1.04878	−0.524389	−	0.851479i	$-0.675706\pi$
$251$	−16.6158	−1.04878	−0.524389	+	0.851479i	$0.675706\pi$
$252$	4.61211	0.290536
$253$	3.00693	0.189044
$254$	−9.69057	−0.608040
$255$	−8.92000	−0.558592
$256$	1.00000	0.0625000
$257$	−9.19197	−0.573379	−0.286690	−	0.958024i	$-0.592555\pi$
$257$	−9.19197	−0.573379	−0.286690	+	0.958024i	$0.592555\pi$
$258$	−10.5501	−0.656822
$259$	−14.2249	−0.883893
$260$	−1.09276	−0.0677704
$261$	−1.90950	−0.118195
$262$	−21.5290	−1.33006
$263$	−2.49743	−0.153998	−0.0769990	−	0.997031i	$-0.524534\pi$
$263$	−2.49743	−0.153998	−0.0769990	+	0.997031i	$0.524534\pi$
$264$	0.971063	0.0597648
$265$	−3.05413	−0.187614
$266$	−3.54361	−0.217273
$267$	−11.0942	−0.678951
$268$	0.761251	0.0465008
$269$	−3.88751	−0.237026	−0.118513	−	0.992953i	$-0.537813\pi$
$269$	−3.88751	−0.237026	−0.118513	+	0.992953i	$0.537813\pi$
$270$	−0.0293880	−0.00178850
$271$	−30.2193	−1.83569	−0.917846	−	0.396936i	$-0.870073\pi$
$271$	−30.2193	−1.83569	−0.917846	+	0.396936i	$0.870073\pi$
$272$	3.63794	0.220583
$273$	4.10282	0.248314
$274$	−12.2619	−0.740767
$275$	−0.396039	−0.0238821
$276$	18.6163	1.12057
$277$	−20.8024	−1.24990	−0.624949	−	0.780666i	$-0.714880\pi$
$277$	−20.8024	−1.24990	−0.624949	+	0.780666i	$0.714880\pi$
$278$	6.79180	0.407345
$279$	−4.57548	−0.273927
$280$	1.53125	0.0915099
$281$	−14.8183	−0.883986	−0.441993	−	0.897019i	$-0.645728\pi$
$281$	−14.8183	−0.883986	−0.441993	+	0.897019i	$0.645728\pi$
$282$	−3.43469	−0.204533
$283$	16.1605	0.960640	0.480320	−	0.877093i	$-0.340521\pi$
$283$	16.1605	0.960640	0.480320	+	0.877093i	$0.340521\pi$
$284$	10.9195	0.647955
$285$	5.67424	0.336113
$286$	0.432778	0.0255907
$287$	−12.7492	−0.752563
$288$	3.01199	0.177483
$289$	−3.76536	−0.221492
$290$	−0.633968	−0.0372279
$291$	7.38027	0.432639
$292$	−9.16888	−0.536568
$293$	−23.7155	−1.38548	−0.692738	−	0.721189i	$-0.743596\pi$
$293$	−23.7155	−1.38548	−0.692738	+	0.721189i	$0.743596\pi$
$294$	11.4144	0.665701
$295$	−3.82943	−0.222958
$296$	−9.28972	−0.539954
$297$	0.0116388	0.000675352 0
$298$	1.03273	0.0598245
$299$	8.29681	0.479817
$300$	−2.45194	−0.141563
$301$	6.58864	0.379763
$302$	−4.34453	−0.250000
$303$	−1.30519	−0.0749809
$304$	−2.31419	−0.132728
$305$	−1.88634	−0.108011
$306$	10.9574	0.626395
$307$	8.00442	0.456837	0.228418	−	0.973563i	$-0.426645\pi$
$307$	8.00442	0.456837	0.228418	+	0.973563i	$0.426645\pi$
$308$	−0.606437	−0.0345549
$309$	−14.9316	−0.849428
$310$	−1.51909	−0.0862785
$311$	−0.0537748	−0.00304929	−0.00152464	−	0.999999i	$-0.500485\pi$
$311$	−0.0537748	−0.00304929	−0.00152464	+	0.999999i	$0.500485\pi$
$312$	2.67939	0.151690
$313$	−1.40462	−0.0793941	−0.0396970	−	0.999212i	$-0.512639\pi$
$313$	−1.40462	−0.0793941	−0.0396970	+	0.999212i	$0.512639\pi$
$314$	5.37740	0.303464
$315$	4.61211	0.259863
$316$	12.0604	0.678448
$317$	22.2009	1.24693	0.623464	−	0.781852i	$-0.285725\pi$
$317$	22.2009	1.24693	0.623464	+	0.781852i	$0.285725\pi$
$318$	7.48852	0.419936
$319$	0.251076	0.0140576
$320$	1.00000	0.0559017
$321$	−33.3888	−1.86358
$322$	−11.6260	−0.647893
$323$	−8.41889	−0.468439
$324$	−8.96390	−0.497994
$325$	−1.09276	−0.0606157
$326$	14.9767	0.829480
$327$	24.0499	1.32996
$328$	−8.32601	−0.459727
$329$	2.14499	0.118257
$330$	0.971063	0.0534553
$331$	−9.64406	−0.530086	−0.265043	−	0.964237i	$-0.585386\pi$
$331$	−9.64406	−0.530086	−0.265043	+	0.964237i	$0.585386\pi$
$332$	5.18730	0.284690
$333$	−27.9805	−1.53332
$334$	−2.41704	−0.132255
$335$	0.761251	0.0415916
$336$	−3.75453	−0.204827
$337$	21.3371	1.16230	0.581152	−	0.813795i	$-0.302602\pi$
$337$	21.3371	1.16230	0.581152	+	0.813795i	$0.302602\pi$
$338$	−11.8059	−0.642154
$339$	51.0215	2.77111
$340$	3.63794	0.197295
$341$	0.601620	0.0325795
$342$	−6.97030	−0.376911
$343$	−17.8472	−0.963656
$344$	4.30278	0.231990
$345$	18.6163	1.00227
$346$	−11.4142	−0.613633
$347$	−8.03328	−0.431249	−0.215625	−	0.976476i	$-0.569179\pi$
$347$	−8.03328	−0.431249	−0.215625	+	0.976476i	$0.569179\pi$
$348$	1.55445	0.0833271
$349$	−29.8503	−1.59785	−0.798925	−	0.601431i	$-0.794597\pi$
$349$	−29.8503	−1.59785	−0.798925	+	0.601431i	$0.794597\pi$
$350$	1.53125	0.0818489
$351$	0.0321142	0.00171413
$352$	−0.396039	−0.0211090
$353$	−17.5750	−0.935421	−0.467711	−	0.883882i	$-0.654921\pi$
$353$	−17.5750	−0.935421	−0.467711	+	0.883882i	$0.654921\pi$
$354$	9.38952	0.499047
$355$	10.9195	0.579549
$356$	4.52465	0.239806
$357$	−13.6588	−0.722900
$358$	−1.20874	−0.0638841
$359$	−0.327073	−0.0172623	−0.00863113	−	0.999963i	$-0.502747\pi$
$359$	−0.327073	−0.0172623	−0.00863113	+	0.999963i	$0.502747\pi$
$360$	3.01199	0.158746
$361$	−13.6445	−0.718133
$362$	14.9579	0.786172
$363$	26.5867	1.39544
$364$	−1.67330	−0.0877047
$365$	−9.16888	−0.479921
$366$	4.62518	0.241762
$367$	2.12995	0.111183	0.0555913	−	0.998454i	$-0.482296\pi$
$367$	2.12995	0.111183	0.0555913	+	0.998454i	$0.482296\pi$
$368$	−7.59249	−0.395786
$369$	−25.0778	−1.30550
$370$	−9.28972	−0.482949
$371$	−4.67664	−0.242799
$372$	3.72471	0.193117
$373$	14.4437	0.747866	0.373933	−	0.927456i	$-0.378009\pi$
$373$	14.4437	0.747866	0.373933	+	0.927456i	$0.378009\pi$
$374$	−1.44077	−0.0745004
$375$	−2.45194	−0.126617
$376$	1.40081	0.0722411
$377$	0.692777	0.0356798
$378$	−0.0450005	−0.00231457
$379$	−0.670819	−0.0344576	−0.0172288	−	0.999852i	$-0.505484\pi$
$379$	−0.670819	−0.0344576	−0.0172288	+	0.999852i	$0.505484\pi$
$380$	−2.31419	−0.118715
$381$	23.7606	1.21729
$382$	−9.94895	−0.509033
$383$	16.8074	0.858820	0.429410	−	0.903110i	$-0.358722\pi$
$383$	16.8074	0.858820	0.429410	+	0.903110i	$0.358722\pi$
$384$	−2.45194	−0.125125
$385$	−0.606437	−0.0309069
$386$	25.9678	1.32173
$387$	12.9599	0.658789
$388$	−3.00998	−0.152809
$389$	−12.7529	−0.646599	−0.323299	−	0.946297i	$-0.604792\pi$
$389$	−12.7529	−0.646599	−0.323299	+	0.946297i	$0.604792\pi$
$390$	2.67939	0.135676
$391$	−27.6211	−1.39686
$392$	−4.65526	−0.235126
$393$	52.7876	2.66278
$394$	23.4003	1.17889
$395$	12.0604	0.606822
$396$	−1.19287	−0.0599437
$397$	−25.8713	−1.29845	−0.649223	−	0.760598i	$-0.724906\pi$
$397$	−25.8713	−1.29845	−0.649223	+	0.760598i	$0.724906\pi$
$398$	−5.85957	−0.293714
$399$	8.68870	0.434979
$400$	1.00000	0.0500000
$401$	−28.6031	−1.42837	−0.714186	−	0.699956i	$-0.753203\pi$
$401$	−28.6031	−1.42837	−0.714186	+	0.699956i	$0.753203\pi$
$402$	−1.86654	−0.0930945
$403$	1.66001	0.0826909
$404$	0.532308	0.0264833
$405$	−8.96390	−0.445420
$406$	−0.970765	−0.0481783
$407$	3.67909	0.182366
$408$	−8.92000	−0.441606
$409$	14.6563	0.724705	0.362352	−	0.932041i	$-0.381974\pi$
$409$	14.6563	0.724705	0.362352	+	0.932041i	$0.381974\pi$
$410$	−8.32601	−0.411192
$411$	30.0653	1.48301
$412$	6.08972	0.300019
$413$	−5.86383	−0.288540
$414$	−22.8685	−1.12392
$415$	5.18730	0.254635
$416$	−1.09276	−0.0535772
$417$	−16.6530	−0.815503
$418$	0.916510	0.0448279
$419$	−17.9226	−0.875577	−0.437789	−	0.899078i	$-0.644238\pi$
$419$	−17.9226	−0.875577	−0.437789	+	0.899078i	$0.644238\pi$
$420$	−3.75453	−0.183202
$421$	−22.5700	−1.09999	−0.549997	−	0.835167i	$-0.685371\pi$
$421$	−22.5700	−1.09999	−0.549997	+	0.835167i	$0.685371\pi$
$422$	−18.4089	−0.896130
$423$	4.21921	0.205145
$424$	−3.05413	−0.148322
$425$	3.63794	0.176466
$426$	−26.7740	−1.29720
$427$	−2.88846	−0.139782
$428$	13.6173	0.658219
$429$	−1.06114	−0.0512325
$430$	4.30278	0.207498
$431$	−32.5987	−1.57022	−0.785112	−	0.619354i	$-0.787395\pi$
$431$	−32.5987	−1.57022	−0.785112	+	0.619354i	$0.787395\pi$
$432$	−0.0293880	−0.00141393
$433$	15.0639	0.723927	0.361963	−	0.932192i	$-0.382107\pi$
$433$	15.0639	0.723927	0.361963	+	0.932192i	$0.382107\pi$
$434$	−2.32611	−0.111657
$435$	1.55445	0.0745301
$436$	−9.80852	−0.469743
$437$	17.5705	0.840509
$438$	22.4815	1.07421
$439$	−14.3218	−0.683541	−0.341771	−	0.939783i	$-0.611027\pi$
$439$	−14.3218	−0.683541	−0.341771	+	0.939783i	$0.611027\pi$
$440$	−0.396039	−0.0188804
$441$	−14.0216	−0.667695
$442$	−3.97542	−0.189091
$443$	−27.0727	−1.28626	−0.643131	−	0.765756i	$-0.722365\pi$
$443$	−27.0727	−1.28626	−0.643131	+	0.765756i	$0.722365\pi$
$444$	22.7778	1.08099
$445$	4.52465	0.214489
$446$	−20.6115	−0.975983
$447$	−2.53219	−0.119769
$448$	1.53125	0.0723449
$449$	−15.6995	−0.740906	−0.370453	−	0.928851i	$-0.620798\pi$
$449$	−15.6995	−0.740906	−0.370453	+	0.928851i	$0.620798\pi$
$450$	3.01199	0.141986
$451$	3.29743	0.155270
$452$	−20.8087	−0.978758
$453$	10.6525	0.500499
$454$	1.48231	0.0695685
$455$	−1.67330	−0.0784455
$456$	5.67424	0.265721
$457$	−12.6625	−0.592325	−0.296162	−	0.955138i	$-0.595707\pi$
$457$	−12.6625	−0.592325	−0.296162	+	0.955138i	$0.595707\pi$
$458$	16.8575	0.787699
$459$	−0.106912	−0.00499022
$460$	−7.59249	−0.354002
$461$	−4.40611	−0.205213	−0.102607	−	0.994722i	$-0.532718\pi$
$461$	−4.40611	−0.205213	−0.102607	+	0.994722i	$0.532718\pi$
$462$	1.48694	0.0691789
$463$	−0.442044	−0.0205435	−0.0102718	−	0.999947i	$-0.503270\pi$
$463$	−0.442044	−0.0205435	−0.0102718	+	0.999947i	$0.503270\pi$
$464$	−0.633968	−0.0294312
$465$	3.72471	0.172729
$466$	9.60557	0.444969
$467$	9.22001	0.426651	0.213326	−	0.976981i	$-0.431570\pi$
$467$	9.22001	0.426651	0.213326	+	0.976981i	$0.431570\pi$
$468$	−3.29139	−0.152145
$469$	1.16567	0.0538256
$470$	1.40081	0.0646144
$471$	−13.1850	−0.607534
$472$	−3.82943	−0.176264
$473$	−1.70407	−0.0783532
$474$	−29.5712	−1.35825
$475$	−2.31419	−0.106182
$476$	5.57062	0.255329
$477$	−9.19899	−0.421193
$478$	16.5353	0.756305
$479$	−37.4477	−1.71103	−0.855514	−	0.517779i	$-0.826759\pi$
$479$	−37.4477	−1.71103	−0.855514	+	0.517779i	$0.826759\pi$
$480$	−2.45194	−0.111915
$481$	10.1515	0.462867
$482$	−12.5932	−0.573603
$483$	28.5063	1.29708
$484$	−10.8432	−0.492871
$485$	−3.00998	−0.136676
$486$	22.0671	1.00098
$487$	−31.5684	−1.43050	−0.715250	−	0.698869i	$-0.753687\pi$
$487$	−31.5684	−1.43050	−0.715250	+	0.698869i	$0.753687\pi$
$488$	−1.88634	−0.0853905
$489$	−36.7218	−1.66062
$490$	−4.65526	−0.210303
$491$	33.9877	1.53384	0.766921	−	0.641741i	$-0.221788\pi$
$491$	33.9877	1.53384	0.766921	+	0.641741i	$0.221788\pi$
$492$	20.4148	0.920371
$493$	−2.30634	−0.103872
$494$	2.52886	0.113779
$495$	−1.19287	−0.0536153
$496$	−1.51909	−0.0682092
$497$	16.7206	0.750020
$498$	−12.7189	−0.569949
$499$	21.0596	0.942756	0.471378	−	0.881931i	$-0.343757\pi$
$499$	21.0596	0.942756	0.471378	+	0.881931i	$0.343757\pi$
$500$	1.00000	0.0447214
$501$	5.92644	0.264774
$502$	−16.6158	−0.741598
$503$	24.9741	1.11354	0.556770	−	0.830667i	$-0.312040\pi$
$503$	24.9741	1.11354	0.556770	+	0.830667i	$0.312040\pi$
$504$	4.61211	0.205440
$505$	0.532308	0.0236874
$506$	3.00693	0.133674
$507$	28.9472	1.28559
$508$	−9.69057	−0.429949
$509$	16.0776	0.712629	0.356314	−	0.934366i	$-0.384033\pi$
$509$	16.0776	0.712629	0.356314	+	0.934366i	$0.384033\pi$
$510$	−8.92000	−0.394984
$511$	−14.0399	−0.621088
$512$	1.00000	0.0441942
$513$	0.0680093	0.00300269
$514$	−9.19197	−0.405440
$515$	6.08972	0.268345
$516$	−10.5501	−0.464444
$517$	−0.554775	−0.0243990
$518$	−14.2249	−0.625007
$519$	27.9870	1.22849
$520$	−1.09276	−0.0479209
$521$	−5.18769	−0.227277	−0.113638	−	0.993522i	$-0.536251\pi$
$521$	−5.18769	−0.227277	−0.113638	+	0.993522i	$0.536251\pi$
$522$	−1.90950	−0.0835766
$523$	−19.3734	−0.847141	−0.423570	−	0.905863i	$-0.639223\pi$
$523$	−19.3734	−0.847141	−0.423570	+	0.905863i	$0.639223\pi$
$524$	−21.5290	−0.940497
$525$	−3.75453	−0.163861
$526$	−2.49743	−0.108893
$527$	−5.52637	−0.240732
$528$	0.971063	0.0422601
$529$	34.6460	1.50635
$530$	−3.05413	−0.132663
$531$	−11.5342	−0.500541
$532$	−3.54361	−0.153635
$533$	9.09836	0.394094
$534$	−11.0942	−0.480091
$535$	13.6173	0.588729
$536$	0.761251	0.0328811
$537$	2.96376	0.127896
$538$	−3.88751	−0.167602
$539$	1.84367	0.0794124
$540$	−0.0293880	−0.00126466
$541$	19.8067	0.851557	0.425778	−	0.904827i	$-0.360000\pi$
$541$	19.8067	0.851557	0.425778	+	0.904827i	$0.360000\pi$
$542$	−30.2193	−1.29803
$543$	−36.6759	−1.57391
$544$	3.63794	0.155976
$545$	−9.80852	−0.420151
$546$	4.10282	0.175585
$547$	2.93473	0.125480	0.0627399	−	0.998030i	$-0.480016\pi$
$547$	2.93473	0.125480	0.0627399	+	0.998030i	$0.480016\pi$
$548$	−12.2619	−0.523801
$549$	−5.68162	−0.242486
$550$	−0.396039	−0.0168872
$551$	1.46712	0.0625014
$552$	18.6163	0.792362
$553$	18.4675	0.785316
$554$	−20.8024	−0.883811
$555$	22.7778	0.966863
$556$	6.79180	0.288036
$557$	−25.6017	−1.08478	−0.542389	−	0.840128i	$-0.682480\pi$
$557$	−25.6017	−1.08478	−0.542389	+	0.840128i	$0.682480\pi$
$558$	−4.57548	−0.193695
$559$	−4.70192	−0.198870
$560$	1.53125	0.0647073
$561$	3.53267	0.149150
$562$	−14.8183	−0.625072
$563$	11.7753	0.496271	0.248135	−	0.968725i	$-0.420182\pi$
$563$	11.7753	0.496271	0.248135	+	0.968725i	$0.420182\pi$
$564$	−3.43469	−0.144627
$565$	−20.8087	−0.875427
$566$	16.1605	0.679275
$567$	−13.7260	−0.576438
$568$	10.9195	0.458173
$569$	−8.94237	−0.374884	−0.187442	−	0.982276i	$-0.560020\pi$
$569$	−8.94237	−0.374884	−0.187442	+	0.982276i	$0.560020\pi$
$570$	5.67424	0.237668
$571$	35.7012	1.49405	0.747025	−	0.664796i	$-0.231482\pi$
$571$	35.7012	1.49405	0.747025	+	0.664796i	$0.231482\pi$
$572$	0.432778	0.0180953
$573$	24.3942	1.01908
$574$	−12.7492	−0.532142
$575$	−7.59249	−0.316629
$576$	3.01199	0.125499
$577$	24.9421	1.03835	0.519176	−	0.854668i	$-0.326239\pi$
$577$	24.9421	1.03835	0.519176	+	0.854668i	$0.326239\pi$
$578$	−3.76536	−0.156618
$579$	−63.6714	−2.64609
$580$	−0.633968	−0.0263241
$581$	7.94308	0.329534
$582$	7.38027	0.305922
$583$	1.20956	0.0500947
$584$	−9.16888	−0.379411
$585$	−3.29139	−0.136082
$586$	−23.7155	−0.979680
$587$	−17.1312	−0.707081	−0.353541	−	0.935419i	$-0.615022\pi$
$587$	−17.1312	−0.707081	−0.353541	+	0.935419i	$0.615022\pi$
$588$	11.4144	0.470722
$589$	3.51546	0.144852
$590$	−3.82943	−0.157655
$591$	−57.3760	−2.36014
$592$	−9.28972	−0.381805
$593$	12.8189	0.526407	0.263204	−	0.964740i	$-0.415221\pi$
$593$	12.8189	0.526407	0.263204	+	0.964740i	$0.415221\pi$
$594$	0.0116388	0.000477546 0
$595$	5.57062	0.228373
$596$	1.03273	0.0423023
$597$	14.3673	0.588014
$598$	8.29681	0.339282
$599$	32.3387	1.32132	0.660662	−	0.750684i	$-0.270276\pi$
$599$	32.3387	1.32132	0.660662	+	0.750684i	$0.270276\pi$
$600$	−2.45194	−0.100100
$601$	−1.00000	−0.0407909
$601$	−1.00000	−0.0407909
$602$	6.58864	0.268533
$603$	2.29288	0.0933732
$604$	−4.34453	−0.176777
$605$	−10.8432	−0.440837
$606$	−1.30519	−0.0530195
$607$	20.5926	0.835829	0.417914	−	0.908486i	$-0.362761\pi$
$607$	20.5926	0.835829	0.417914	+	0.908486i	$0.362761\pi$
$608$	−2.31419	−0.0938527
$609$	2.38025	0.0964527
$610$	−1.88634	−0.0763756
$611$	−1.53075	−0.0619276
$612$	10.9574	0.442928
$613$	−7.15441	−0.288964	−0.144482	−	0.989507i	$-0.546152\pi$
$613$	−7.15441	−0.288964	−0.144482	+	0.989507i	$0.546152\pi$
$614$	8.00442	0.323032
$615$	20.4148	0.823205
$616$	−0.606437	−0.0244340
$617$	24.3727	0.981208	0.490604	−	0.871383i	$-0.336776\pi$
$617$	24.3727	0.981208	0.490604	+	0.871383i	$0.336776\pi$
$618$	−14.9316	−0.600637
$619$	4.67772	0.188014	0.0940068	−	0.995572i	$-0.470032\pi$
$619$	4.67772	0.188014	0.0940068	+	0.995572i	$0.470032\pi$
$620$	−1.51909	−0.0610081
$621$	0.223128	0.00895383
$622$	−0.0537748	−0.00215617
$623$	6.92839	0.277580
$624$	2.67939	0.107261
$625$	1.00000	0.0400000
$626$	−1.40462	−0.0561401
$627$	−2.24722	−0.0897454
$628$	5.37740	0.214582
$629$	−33.7955	−1.34751
$630$	4.61211	0.183751
$631$	0.155312	0.00618288	0.00309144	−	0.999995i	$-0.499016\pi$
$631$	0.155312	0.00618288	0.00309144	+	0.999995i	$0.499016\pi$
$632$	12.0604	0.479735
$633$	45.1374	1.79405
$634$	22.2009	0.881711
$635$	−9.69057	−0.384559
$636$	7.48852	0.296939
$637$	5.08711	0.201558
$638$	0.251076	0.00994020
$639$	32.8895	1.30109
$640$	1.00000	0.0395285
$641$	48.8605	1.92988	0.964938	−	0.262479i	$-0.0845400\pi$
$641$	48.8605	1.92988	0.964938	+	0.262479i	$0.0845400\pi$
$642$	−33.3888	−1.31775
$643$	1.16722	0.0460305	0.0230153	−	0.999735i	$-0.492673\pi$
$643$	1.16722	0.0460305	0.0230153	+	0.999735i	$0.492673\pi$
$644$	−11.6260	−0.458130
$645$	−10.5501	−0.415411
$646$	−8.41889	−0.331237
$647$	24.0041	0.943699	0.471849	−	0.881679i	$-0.343587\pi$
$647$	24.0041	0.943699	0.471849	+	0.881679i	$0.343587\pi$
$648$	−8.96390	−0.352135
$649$	1.51661	0.0595320
$650$	−1.09276	−0.0428617
$651$	5.70348	0.223537
$652$	14.9767	0.586531
$653$	13.4784	0.527450	0.263725	−	0.964598i	$-0.415049\pi$
$653$	13.4784	0.527450	0.263725	+	0.964598i	$0.415049\pi$
$654$	24.0499	0.940424
$655$	−21.5290	−0.841206
$656$	−8.32601	−0.325076
$657$	−27.6165	−1.07742
$658$	2.14499	0.0836205
$659$	1.40035	0.0545499	0.0272750	−	0.999628i	$-0.491317\pi$
$659$	1.40035	0.0545499	0.0272750	+	0.999628i	$0.491317\pi$
$660$	0.971063	0.0377986
$661$	21.4629	0.834811	0.417406	−	0.908720i	$-0.362939\pi$
$661$	21.4629	0.834811	0.417406	+	0.908720i	$0.362939\pi$
$662$	−9.64406	−0.374827
$663$	9.74746	0.378560
$664$	5.18730	0.201306
$665$	−3.54361	−0.137415
$666$	−27.9805	−1.08422
$667$	4.81340	0.186375
$668$	−2.41704	−0.0935183
$669$	50.5381	1.95391
$670$	0.761251	0.0294097
$671$	0.747064	0.0288401
$672$	−3.75453	−0.144834
$673$	37.2551	1.43608	0.718038	−	0.696004i	$-0.245040\pi$
$673$	37.2551	1.43608	0.718038	+	0.696004i	$0.245040\pi$
$674$	21.3371	0.821874
$675$	−0.0293880	−0.00113114
$676$	−11.8059	−0.454072
$677$	−36.9326	−1.41943	−0.709717	−	0.704487i	$-0.751177\pi$
$677$	−36.9326	−1.41943	−0.709717	+	0.704487i	$0.751177\pi$
$678$	51.0215	1.95947
$679$	−4.60904	−0.176879
$680$	3.63794	0.139509
$681$	−3.63454	−0.139276
$682$	0.601620	0.0230372
$683$	50.0879	1.91656	0.958280	−	0.285830i	$-0.0922692\pi$
$683$	50.0879	1.91656	0.958280	+	0.285830i	$0.0922692\pi$
$684$	−6.97030	−0.266516
$685$	−12.2619	−0.468502
$686$	−17.8472	−0.681408
$687$	−41.3335	−1.57697
$688$	4.30278	0.164042
$689$	3.33744	0.127146
$690$	18.6163	0.708710
$691$	−42.7100	−1.62477	−0.812383	−	0.583124i	$-0.801830\pi$
$691$	−42.7100	−1.62477	−0.812383	+	0.583124i	$0.801830\pi$
$692$	−11.4142	−0.433904
$693$	−1.82658	−0.0693860
$694$	−8.03328	−0.304939
$695$	6.79180	0.257628
$696$	1.55445	0.0589212
$697$	−30.2896	−1.14730
$698$	−29.8503	−1.12985
$699$	−23.5522	−0.890827
$700$	1.53125	0.0578759
$701$	20.5609	0.776573	0.388287	−	0.921539i	$-0.373067\pi$
$701$	20.5609	0.776573	0.388287	+	0.921539i	$0.373067\pi$
$702$	0.0321142	0.00121207
$703$	21.4981	0.810818
$704$	−0.396039	−0.0149263
$705$	−3.43469	−0.129358
$706$	−17.5750	−0.661443
$707$	0.815099	0.0306549
$708$	9.38952	0.352880
$709$	23.4133	0.879305	0.439653	−	0.898168i	$-0.355102\pi$
$709$	23.4133	0.879305	0.439653	+	0.898168i	$0.355102\pi$
$710$	10.9195	0.409803
$711$	36.3256	1.36232
$712$	4.52465	0.169568
$713$	11.5337	0.431940
$714$	−13.6588	−0.511167
$715$	0.432778	0.0161850
$716$	−1.20874	−0.0451729
$717$	−40.5434	−1.51412
$718$	−0.327073	−0.0122063
$719$	0.106141	0.00395838	0.00197919	−	0.999998i	$-0.499370\pi$
$719$	0.106141	0.00395838	0.00197919	+	0.999998i	$0.499370\pi$
$720$	3.01199	0.112250
$721$	9.32490	0.347277
$722$	−13.6445	−0.507797
$723$	30.8776	1.14835
$724$	14.9579	0.555908
$725$	−0.633968	−0.0235450
$726$	26.5867	0.986725
$727$	−24.5064	−0.908891	−0.454446	−	0.890774i	$-0.650163\pi$
$727$	−24.5064	−0.908891	−0.454446	+	0.890774i	$0.650163\pi$
$728$	−1.67330	−0.0620166
$729$	−27.2153	−1.00797
$730$	−9.16888	−0.339356
$731$	15.6533	0.578957
$732$	4.62518	0.170951
$733$	21.9169	0.809517	0.404759	−	0.914424i	$-0.367356\pi$
$733$	21.9169	0.809517	0.404759	+	0.914424i	$0.367356\pi$
$734$	2.12995	0.0786180
$735$	11.4144	0.421027
$736$	−7.59249	−0.279863
$737$	−0.301486	−0.0111054
$738$	−25.0778	−0.923127
$739$	0.152216	0.00559937	0.00279968	−	0.999996i	$-0.499109\pi$
$739$	0.152216	0.00559937	0.00279968	+	0.999996i	$0.499109\pi$
$740$	−9.28972	−0.341497
$741$	−6.20060	−0.227785
$742$	−4.67664	−0.171685
$743$	31.6241	1.16018	0.580089	−	0.814553i	$-0.303018\pi$
$743$	31.6241	1.16018	0.580089	+	0.814553i	$0.303018\pi$
$744$	3.72471	0.136555
$745$	1.03273	0.0378364
$746$	14.4437	0.528821
$747$	15.6241	0.571655
$748$	−1.44077	−0.0526798
$749$	20.8516	0.761901
$750$	−2.45194	−0.0895320
$751$	37.8318	1.38050	0.690251	−	0.723570i	$-0.257500\pi$
$751$	37.8318	1.38050	0.690251	+	0.723570i	$0.257500\pi$
$752$	1.40081	0.0510822
$753$	40.7408	1.48468
$754$	0.692777	0.0252295
$755$	−4.34453	−0.158114
$756$	−0.0450005	−0.00163665
$757$	44.2639	1.60880	0.804400	−	0.594088i	$-0.202487\pi$
$757$	44.2639	1.60880	0.804400	+	0.594088i	$0.202487\pi$
$758$	−0.670819	−0.0243652
$759$	−7.37279	−0.267615
$760$	−2.31419	−0.0839444
$761$	9.80467	0.355419	0.177710	−	0.984083i	$-0.443131\pi$
$761$	9.80467	0.355419	0.177710	+	0.984083i	$0.443131\pi$
$762$	23.7606	0.860757
$763$	−15.0193	−0.543736
$764$	−9.94895	−0.359940
$765$	10.9574	0.396167
$766$	16.8074	0.607277
$767$	4.18467	0.151100
$768$	−2.45194	−0.0884766
$769$	−16.5291	−0.596053	−0.298027	−	0.954558i	$-0.596328\pi$
$769$	−16.5291	−0.596053	−0.298027	+	0.954558i	$0.596328\pi$
$770$	−0.606437	−0.0218545
$771$	22.5381	0.811690
$772$	25.9678	0.934602
$773$	−5.81158	−0.209028	−0.104514	−	0.994523i	$-0.533329\pi$
$773$	−5.81158	−0.209028	−0.104514	+	0.994523i	$0.533329\pi$
$774$	12.9599	0.465834
$775$	−1.51909	−0.0545673
$776$	−3.00998	−0.108052
$777$	34.8785	1.25126
$778$	−12.7529	−0.457214
$779$	19.2679	0.690346
$780$	2.67939	0.0959375
$781$	−4.32457	−0.154745
$782$	−27.6211	−0.987728
$783$	0.0186310	0.000665819 0
$784$	−4.65526	−0.166259
$785$	5.37740	0.191928
$786$	52.7876	1.88287
$787$	9.72019	0.346487	0.173244	−	0.984879i	$-0.444575\pi$
$787$	9.72019	0.346487	0.173244	+	0.984879i	$0.444575\pi$
$788$	23.4003	0.833602
$789$	6.12353	0.218004
$790$	12.0604	0.429088
$791$	−31.8634	−1.13293
$792$	−1.19287	−0.0423866
$793$	2.06132	0.0731997
$794$	−25.8713	−0.918140
$795$	7.48852	0.265591
$796$	−5.85957	−0.207687
$797$	−24.3351	−0.861992	−0.430996	−	0.902354i	$-0.641838\pi$
$797$	−24.3351	−0.861992	−0.430996	+	0.902354i	$0.641838\pi$
$798$	8.68870	0.307576
$799$	5.09606	0.180286
$800$	1.00000	0.0353553
$801$	13.6282	0.481528
$802$	−28.6031	−1.01001
$803$	3.63124	0.128144
$804$	−1.86654	−0.0658278
$805$	−11.6260	−0.409764
$806$	1.66001	0.0584713
$807$	9.53192	0.335539
$808$	0.532308	0.0187265
$809$	−31.9916	−1.12477	−0.562383	−	0.826877i	$-0.690115\pi$
$809$	−31.9916	−1.12477	−0.562383	+	0.826877i	$0.690115\pi$
$810$	−8.96390	−0.314959
$811$	41.0016	1.43976	0.719881	−	0.694098i	$-0.244197\pi$
$811$	41.0016	1.43976	0.719881	+	0.694098i	$0.244197\pi$
$812$	−0.970765	−0.0340672
$813$	74.0958	2.59865
$814$	3.67909	0.128952
$815$	14.9767	0.524609
$816$	−8.92000	−0.312263
$817$	−9.95743	−0.348366
$818$	14.6563	0.512444
$819$	−5.03995	−0.176110
$820$	−8.32601	−0.290757
$821$	3.01881	0.105357	0.0526786	−	0.998612i	$-0.483224\pi$
$821$	3.01881	0.105357	0.0526786	+	0.998612i	$0.483224\pi$
$822$	30.0653	1.04865
$823$	22.0814	0.769711	0.384855	−	0.922977i	$-0.374251\pi$
$823$	22.0814	0.769711	0.384855	+	0.922977i	$0.374251\pi$
$824$	6.08972	0.212145
$825$	0.971063	0.0338081
$826$	−5.86383	−0.204029
$827$	4.29652	0.149405	0.0747024	−	0.997206i	$-0.476199\pi$
$827$	4.29652	0.149405	0.0747024	+	0.997206i	$0.476199\pi$
$828$	−22.8685	−0.794735
$829$	−24.2828	−0.843375	−0.421688	−	0.906741i	$-0.638562\pi$
$829$	−24.2828	−0.843375	−0.421688	+	0.906741i	$0.638562\pi$
$830$	5.18730	0.180054
$831$	51.0062	1.76939
$832$	−1.09276	−0.0378848
$833$	−16.9356	−0.586783
$834$	−16.6530	−0.576648
$835$	−2.41704	−0.0836453
$836$	0.916510	0.0316981
$837$	0.0446430	0.00154309
$838$	−17.9226	−0.619127
$839$	15.3535	0.530062	0.265031	−	0.964240i	$-0.414618\pi$
$839$	15.3535	0.530062	0.265031	+	0.964240i	$0.414618\pi$
$840$	−3.75453	−0.129544
$841$	−28.5981	−0.986141
$842$	−22.5700	−0.777813
$843$	36.3335	1.25139
$844$	−18.4089	−0.633660
$845$	−11.8059	−0.406134
$846$	4.21921	0.145060
$847$	−16.6036	−0.570507
$848$	−3.05413	−0.104879
$849$	−39.6244	−1.35991
$850$	3.63794	0.124780
$851$	70.5321	2.41781
$852$	−26.7740	−0.917261
$853$	56.1846	1.92372	0.961861	−	0.273537i	$-0.0881937\pi$
$853$	56.1846	1.92372	0.961861	+	0.273537i	$0.0881937\pi$
$854$	−2.88846	−0.0988411
$855$	−6.97030	−0.238379
$856$	13.6173	0.465431
$857$	−0.0467261	−0.00159613	−0.000798066	−	1.00000i	$-0.500254\pi$
$857$	−0.0467261	−0.00159613	−0.000798066		1.00000i	$0.500254\pi$
$858$	−1.06114	−0.0362268
$859$	22.7705	0.776918	0.388459	−	0.921466i	$-0.373008\pi$
$859$	22.7705	0.776918	0.388459	+	0.921466i	$0.373008\pi$
$860$	4.30278	0.146723
$861$	31.2603	1.06535
$862$	−32.5987	−1.11032
$863$	−7.02499	−0.239134	−0.119567	−	0.992826i	$-0.538151\pi$
$863$	−7.02499	−0.239134	−0.119567	+	0.992826i	$0.538151\pi$
$864$	−0.0293880	−0.000999800 0
$865$	−11.4142	−0.388095
$866$	15.0639	0.511894
$867$	9.23241	0.313549
$868$	−2.32611	−0.0789534
$869$	−4.77638	−0.162027
$870$	1.55445	0.0527007
$871$	−0.831868	−0.0281868
$872$	−9.80852	−0.332158
$873$	−9.06601	−0.306838
$874$	17.5705	0.594329
$875$	1.53125	0.0517658
$876$	22.4815	0.759580
$877$	18.7288	0.632427	0.316214	−	0.948688i	$-0.397588\pi$
$877$	18.7288	0.632427	0.316214	+	0.948688i	$0.397588\pi$
$878$	−14.3218	−0.483337
$879$	58.1490	1.96132
$880$	−0.396039	−0.0133505
$881$	14.8962	0.501867	0.250933	−	0.968004i	$-0.419263\pi$
$881$	14.8962	0.501867	0.250933	+	0.968004i	$0.419263\pi$
$882$	−14.0216	−0.472131
$883$	4.82170	0.162263	0.0811316	−	0.996703i	$-0.474147\pi$
$883$	4.82170	0.162263	0.0811316	+	0.996703i	$0.474147\pi$
$884$	−3.97542	−0.133708
$885$	9.38952	0.315625
$886$	−27.0727	−0.909525
$887$	−47.6008	−1.59828	−0.799140	−	0.601145i	$-0.794711\pi$
$887$	−47.6008	−1.59828	−0.799140	+	0.601145i	$0.794711\pi$
$888$	22.7778	0.764372
$889$	−14.8387	−0.497675
$890$	4.52465	0.151667
$891$	3.55006	0.118931
$892$	−20.6115	−0.690124
$893$	−3.24173	−0.108480
$894$	−2.53219	−0.0846891
$895$	−1.20874	−0.0404039
$896$	1.53125	0.0511556
$897$	−20.3432	−0.679241
$898$	−15.6995	−0.523900
$899$	0.963054	0.0321197
$900$	3.01199	0.100400
$901$	−11.1107	−0.370153
$902$	3.29743	0.109792
$903$	−16.1549	−0.537602
$904$	−20.8087	−0.692086
$905$	14.9579	0.497219
$906$	10.6525	0.353906
$907$	−3.99060	−0.132506	−0.0662529	−	0.997803i	$-0.521104\pi$
$907$	−3.99060	−0.132506	−0.0662529	+	0.997803i	$0.521104\pi$
$908$	1.48231	0.0491923
$909$	1.60330	0.0531783
$910$	−1.67330	−0.0554693
$911$	−42.2628	−1.40023	−0.700114	−	0.714031i	$-0.746867\pi$
$911$	−42.2628	−1.40023	−0.700114	+	0.714031i	$0.746867\pi$
$912$	5.67424	0.187893
$913$	−2.05438	−0.0679900
$914$	−12.6625	−0.418837
$915$	4.62518	0.152904
$916$	16.8575	0.556987
$917$	−32.9663	−1.08864
$918$	−0.106912	−0.00352862
$919$	14.7126	0.485323	0.242662	−	0.970111i	$-0.421980\pi$
$919$	14.7126	0.485323	0.242662	+	0.970111i	$0.421980\pi$
$920$	−7.59249	−0.250317
$921$	−19.6263	−0.646709
$922$	−4.40611	−0.145108
$923$	−11.9325	−0.392762
$924$	1.48694	0.0489169
$925$	−9.28972	−0.305444
$926$	−0.442044	−0.0145265
$927$	18.3421	0.602435
$928$	−0.633968	−0.0208110
$929$	−38.6413	−1.26778	−0.633890	−	0.773423i	$-0.718543\pi$
$929$	−38.6413	−1.26778	−0.633890	+	0.773423i	$0.718543\pi$
$930$	3.72471	0.122138
$931$	10.7732	0.353076
$932$	9.60557	0.314641
$933$	0.131852	0.00431665
$934$	9.22001	0.301688
$935$	−1.44077	−0.0471182
$936$	−3.29139	−0.107582
$937$	15.2972	0.499739	0.249870	−	0.968279i	$-0.419612\pi$
$937$	15.2972	0.499739	0.249870	+	0.968279i	$0.419612\pi$
$938$	1.16567	0.0380604
$939$	3.44405	0.112392
$940$	1.40081	0.0456893
$941$	10.9774	0.357855	0.178927	−	0.983862i	$-0.442737\pi$
$941$	10.9774	0.357855	0.178927	+	0.983862i	$0.442737\pi$
$942$	−13.1850	−0.429592
$943$	63.2152	2.05857
$944$	−3.82943	−0.124637
$945$	−0.0450005	−0.00146387
$946$	−1.70407	−0.0554041
$947$	−8.54937	−0.277817	−0.138909	−	0.990305i	$-0.544359\pi$
$947$	−8.54937	−0.277817	−0.138909	+	0.990305i	$0.544359\pi$
$948$	−29.5712	−0.960428
$949$	10.0194	0.325244
$950$	−2.31419	−0.0750822
$951$	−54.4352	−1.76518
$952$	5.57062	0.180545
$953$	−38.3295	−1.24161	−0.620806	−	0.783964i	$-0.713195\pi$
$953$	−38.3295	−1.24161	−0.620806	+	0.783964i	$0.713195\pi$
$954$	−9.19899	−0.297828
$955$	−9.94895	−0.321941
$956$	16.5353	0.534789
$957$	−0.615623	−0.0199003
$958$	−37.4477	−1.20988
$959$	−18.7760	−0.606310
$960$	−2.45194	−0.0791359
$961$	−28.6924	−0.925560
$962$	10.1515	0.327297
$963$	41.0152	1.32170
$964$	−12.5932	−0.405599
$965$	25.9678	0.835934
$966$	28.5063	0.917174
$967$	58.2218	1.87229	0.936144	−	0.351618i	$-0.114368\pi$
$967$	58.2218	1.87229	0.936144	+	0.351618i	$0.114368\pi$
$968$	−10.8432	−0.348512
$969$	20.6426	0.663135
$970$	−3.00998	−0.0966446
$971$	−28.4335	−0.912473	−0.456237	−	0.889858i	$-0.650803\pi$
$971$	−28.4335	−0.912473	−0.456237	+	0.889858i	$0.650803\pi$
$972$	22.0671	0.707801
$973$	10.4000	0.333408
$974$	−31.5684	−1.01152
$975$	2.67939	0.0858091
$976$	−1.88634	−0.0603802
$977$	16.0515	0.513533	0.256766	−	0.966473i	$-0.417343\pi$
$977$	16.0515	0.513533	0.256766	+	0.966473i	$0.417343\pi$
$978$	−36.7218	−1.17423
$979$	−1.79194	−0.0572707
$980$	−4.65526	−0.148707
$981$	−29.5431	−0.943239
$982$	33.9877	1.08459
$983$	−41.7868	−1.33279	−0.666396	−	0.745598i	$-0.732164\pi$
$983$	−41.7868	−1.33279	−0.666396	+	0.745598i	$0.732164\pi$
$984$	20.4148	0.650801
$985$	23.4003	0.745596
$986$	−2.30634	−0.0734488
$987$	−5.25938	−0.167408
$988$	2.52886	0.0804538
$989$	−32.6688	−1.03881
$990$	−1.19287	−0.0379117
$991$	−7.71703	−0.245140	−0.122570	−	0.992460i	$-0.539114\pi$
$991$	−7.71703	−0.245140	−0.122570	+	0.992460i	$0.539114\pi$
$992$	−1.51909	−0.0482312
$993$	23.6466	0.750403
$994$	16.7206	0.530344
$995$	−5.85957	−0.185761
$996$	−12.7189	−0.403015
$997$	12.5044	0.396020	0.198010	−	0.980200i	$-0.436552\pi$
$997$	12.5044	0.396020	0.198010	+	0.980200i	$0.436552\pi$
$998$	21.0596	0.666629
$999$	0.273006	0.00863753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6010.2.a.c.1.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6010.2.a.c.1.2	✓	16	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:	\( N \)	\(=\)	\( 6010 = 2 \cdot 5 \cdot 601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6010.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.45194 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.45194 q^{6} +1.53125 q^{7} +1.00000 q^{8} +3.01199 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.45194 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.45194 q^{6} +1.53125 q^{7} +1.00000 q^{8} +3.01199 q^{9} +1.00000 q^{10} -0.396039 q^{11} -2.45194 q^{12} -1.09276 q^{13} +1.53125 q^{14} -2.45194 q^{15} +1.00000 q^{16} +3.63794 q^{17} +3.01199 q^{18} -2.31419 q^{19} +1.00000 q^{20} -3.75453 q^{21} -0.396039 q^{22} -7.59249 q^{23} -2.45194 q^{24} +1.00000 q^{25} -1.09276 q^{26} -0.0293880 q^{27} +1.53125 q^{28} -0.633968 q^{29} -2.45194 q^{30} -1.51909 q^{31} +1.00000 q^{32} +0.971063 q^{33} +3.63794 q^{34} +1.53125 q^{35} +3.01199 q^{36} -9.28972 q^{37} -2.31419 q^{38} +2.67939 q^{39} +1.00000 q^{40} -8.32601 q^{41} -3.75453 q^{42} +4.30278 q^{43} -0.396039 q^{44} +3.01199 q^{45} -7.59249 q^{46} +1.40081 q^{47} -2.45194 q^{48} -4.65526 q^{49} +1.00000 q^{50} -8.92000 q^{51} -1.09276 q^{52} -3.05413 q^{53} -0.0293880 q^{54} -0.396039 q^{55} +1.53125 q^{56} +5.67424 q^{57} -0.633968 q^{58} -3.82943 q^{59} -2.45194 q^{60} -1.88634 q^{61} -1.51909 q^{62} +4.61211 q^{63} +1.00000 q^{64} -1.09276 q^{65} +0.971063 q^{66} +0.761251 q^{67} +3.63794 q^{68} +18.6163 q^{69} +1.53125 q^{70} +10.9195 q^{71} +3.01199 q^{72} -9.16888 q^{73} -9.28972 q^{74} -2.45194 q^{75} -2.31419 q^{76} -0.606437 q^{77} +2.67939 q^{78} +12.0604 q^{79} +1.00000 q^{80} -8.96390 q^{81} -8.32601 q^{82} +5.18730 q^{83} -3.75453 q^{84} +3.63794 q^{85} +4.30278 q^{86} +1.55445 q^{87} -0.396039 q^{88} +4.52465 q^{89} +3.01199 q^{90} -1.67330 q^{91} -7.59249 q^{92} +3.72471 q^{93} +1.40081 q^{94} -2.31419 q^{95} -2.45194 q^{96} -3.00998 q^{97} -4.65526 q^{98} -1.19287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} - 8 q^{3} + 16 q^{4} + 16 q^{5} - 8 q^{6} - 10 q^{7} + 16 q^{8} - 2 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 - 8 * q^3 + 16 * q^4 + 16 * q^5 - 8 * q^6 - 10 * q^7 + 16 * q^8 - 2 * q^9	\( 16 q + 16 q^{2} - 8 q^{3} + 16 q^{4} + 16 q^{5} - 8 q^{6} - 10 q^{7} + 16 q^{8} - 2 q^{9} + 16 q^{10} - 14 q^{11} - 8 q^{12} - 20 q^{13} - 10 q^{14} - 8 q^{15} + 16 q^{16} - 27 q^{17} - 2 q^{18} - 17 q^{19} + 16 q^{20} - 12 q^{21} - 14 q^{22} - 9 q^{23} - 8 q^{24} + 16 q^{25} - 20 q^{26} - 11 q^{27} - 10 q^{28} - 23 q^{29} - 8 q^{30} - 21 q^{31} + 16 q^{32} - 9 q^{33} - 27 q^{34} - 10 q^{35} - 2 q^{36} - 16 q^{37} - 17 q^{38} - 6 q^{39} + 16 q^{40} - 35 q^{41} - 12 q^{42} + 3 q^{43} - 14 q^{44} - 2 q^{45} - 9 q^{46} - 25 q^{47} - 8 q^{48} - 24 q^{49} + 16 q^{50} - q^{51} - 20 q^{52} - 39 q^{53} - 11 q^{54} - 14 q^{55} - 10 q^{56} - 6 q^{57} - 23 q^{58} - 32 q^{59} - 8 q^{60} - 38 q^{61} - 21 q^{62} + q^{63} + 16 q^{64} - 20 q^{65} - 9 q^{66} + 5 q^{67} - 27 q^{68} - 25 q^{69} - 10 q^{70} - 16 q^{71} - 2 q^{72} - 17 q^{73} - 16 q^{74} - 8 q^{75} - 17 q^{76} - 34 q^{77} - 6 q^{78} - 40 q^{79} + 16 q^{80} - 28 q^{81} - 35 q^{82} - 22 q^{83} - 12 q^{84} - 27 q^{85} + 3 q^{86} + 10 q^{87} - 14 q^{88} - 46 q^{89} - 2 q^{90} - q^{91} - 9 q^{92} + 14 q^{93} - 25 q^{94} - 17 q^{95} - 8 q^{96} - 21 q^{97} - 24 q^{98} + 15 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 - 8 * q^3 + 16 * q^4 + 16 * q^5 - 8 * q^6 - 10 * q^7 + 16 * q^8 - 2 * q^9 + 16 * q^10 - 14 * q^11 - 8 * q^12 - 20 * q^13 - 10 * q^14 - 8 * q^15 + 16 * q^16 - 27 * q^17 - 2 * q^18 - 17 * q^19 + 16 * q^20 - 12 * q^21 - 14 * q^22 - 9 * q^23 - 8 * q^24 + 16 * q^25 - 20 * q^26 - 11 * q^27 - 10 * q^28 - 23 * q^29 - 8 * q^30 - 21 * q^31 + 16 * q^32 - 9 * q^33 - 27 * q^34 - 10 * q^35 - 2 * q^36 - 16 * q^37 - 17 * q^38 - 6 * q^39 + 16 * q^40 - 35 * q^41 - 12 * q^42 + 3 * q^43 - 14 * q^44 - 2 * q^45 - 9 * q^46 - 25 * q^47 - 8 * q^48 - 24 * q^49 + 16 * q^50 - q^51 - 20 * q^52 - 39 * q^53 - 11 * q^54 - 14 * q^55 - 10 * q^56 - 6 * q^57 - 23 * q^58 - 32 * q^59 - 8 * q^60 - 38 * q^61 - 21 * q^62 + q^63 + 16 * q^64 - 20 * q^65 - 9 * q^66 + 5 * q^67 - 27 * q^68 - 25 * q^69 - 10 * q^70 - 16 * q^71 - 2 * q^72 - 17 * q^73 - 16 * q^74 - 8 * q^75 - 17 * q^76 - 34 * q^77 - 6 * q^78 - 40 * q^79 + 16 * q^80 - 28 * q^81 - 35 * q^82 - 22 * q^83 - 12 * q^84 - 27 * q^85 + 3 * q^86 + 10 * q^87 - 14 * q^88 - 46 * q^89 - 2 * q^90 - q^91 - 9 * q^92 + 14 * q^93 - 25 * q^94 - 17 * q^95 - 8 * q^96 - 21 * q^97 - 24 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more