Properties

Label 13.2.e.a
Level 13
Weight 2
Character orbit 13.e
Analytic conductor 0.104
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.103805522628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( 4 - 2 \zeta_{6} ) q^{6} + ( -1 + 2 \zeta_{6} ) q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( 4 - 2 \zeta_{6} ) q^{6} + ( -1 + 2 \zeta_{6} ) q^{8} -\zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{10} -2 q^{12} + ( -1 - 3 \zeta_{6} ) q^{13} + ( 2 + 2 \zeta_{6} ) q^{15} + ( 5 - 5 \zeta_{6} ) q^{16} + 3 \zeta_{6} q^{17} + ( -1 + 2 \zeta_{6} ) q^{18} + ( -4 + 2 \zeta_{6} ) q^{19} + ( 2 - \zeta_{6} ) q^{20} + ( 6 - 6 \zeta_{6} ) q^{23} + ( -2 - 2 \zeta_{6} ) q^{24} + 2 q^{25} + ( -2 + 7 \zeta_{6} ) q^{26} -4 q^{27} + ( -3 + 3 \zeta_{6} ) q^{29} -6 \zeta_{6} q^{30} + ( -2 + 4 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{32} + ( 3 - 6 \zeta_{6} ) q^{34} + ( 1 - \zeta_{6} ) q^{36} + ( 5 + 5 \zeta_{6} ) q^{37} + 6 q^{38} + ( 8 - 2 \zeta_{6} ) q^{39} + 3 q^{40} + ( -3 - 3 \zeta_{6} ) q^{41} -8 \zeta_{6} q^{43} + ( -2 + \zeta_{6} ) q^{45} + ( -12 + 6 \zeta_{6} ) q^{46} + ( 2 - 4 \zeta_{6} ) q^{47} + 10 \zeta_{6} q^{48} + ( -7 + 7 \zeta_{6} ) q^{49} + ( -2 - 2 \zeta_{6} ) q^{50} -6 q^{51} + ( 3 - 4 \zeta_{6} ) q^{52} -3 q^{53} + ( 4 + 4 \zeta_{6} ) q^{54} + ( 4 - 8 \zeta_{6} ) q^{57} + ( 6 - 3 \zeta_{6} ) q^{58} + ( 8 - 4 \zeta_{6} ) q^{59} + ( -2 + 4 \zeta_{6} ) q^{60} -\zeta_{6} q^{61} + ( 6 - 6 \zeta_{6} ) q^{62} - q^{64} + ( -7 + 5 \zeta_{6} ) q^{65} + ( 2 + 2 \zeta_{6} ) q^{67} + ( -3 + 3 \zeta_{6} ) q^{68} + 12 \zeta_{6} q^{69} + ( 4 - 2 \zeta_{6} ) q^{71} + ( 2 - \zeta_{6} ) q^{72} + ( -1 + 2 \zeta_{6} ) q^{73} -15 \zeta_{6} q^{74} + ( -4 + 4 \zeta_{6} ) q^{75} + ( -2 - 2 \zeta_{6} ) q^{76} + ( -10 - 4 \zeta_{6} ) q^{78} + 4 q^{79} + ( -5 - 5 \zeta_{6} ) q^{80} + ( 11 - 11 \zeta_{6} ) q^{81} + 9 \zeta_{6} q^{82} + ( -8 + 16 \zeta_{6} ) q^{83} + ( 6 - 3 \zeta_{6} ) q^{85} + ( -8 + 16 \zeta_{6} ) q^{86} -6 \zeta_{6} q^{87} + ( -4 - 4 \zeta_{6} ) q^{89} + 3 q^{90} + 6 q^{92} + ( -4 - 4 \zeta_{6} ) q^{93} + ( -6 + 6 \zeta_{6} ) q^{94} + 6 \zeta_{6} q^{95} + ( 6 - 12 \zeta_{6} ) q^{96} + ( 8 - 4 \zeta_{6} ) q^{97} + ( 14 - 7 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} - 2q^{3} + q^{4} + 6q^{6} - q^{9} + O(q^{10}) \) \( 2q - 3q^{2} - 2q^{3} + q^{4} + 6q^{6} - q^{9} - 3q^{10} - 4q^{12} - 5q^{13} + 6q^{15} + 5q^{16} + 3q^{17} - 6q^{19} + 3q^{20} + 6q^{23} - 6q^{24} + 4q^{25} + 3q^{26} - 8q^{27} - 3q^{29} - 6q^{30} - 9q^{32} + q^{36} + 15q^{37} + 12q^{38} + 14q^{39} + 6q^{40} - 9q^{41} - 8q^{43} - 3q^{45} - 18q^{46} + 10q^{48} - 7q^{49} - 6q^{50} - 12q^{51} + 2q^{52} - 6q^{53} + 12q^{54} + 9q^{58} + 12q^{59} - q^{61} + 6q^{62} - 2q^{64} - 9q^{65} + 6q^{67} - 3q^{68} + 12q^{69} + 6q^{71} + 3q^{72} - 15q^{74} - 4q^{75} - 6q^{76} - 24q^{78} + 8q^{79} - 15q^{80} + 11q^{81} + 9q^{82} + 9q^{85} - 6q^{87} - 12q^{89} + 6q^{90} + 12q^{92} - 12q^{93} - 6q^{94} + 6q^{95} + 12q^{97} + 21q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i 1.73205i 3.00000 1.73205i 0 1.73205i −0.500000 0.866025i −1.50000 + 2.59808i
10.1 −1.50000 + 0.866025i −1.00000 1.73205i 0.500000 0.866025i 1.73205i 3.00000 + 1.73205i 0 1.73205i −0.500000 + 0.866025i −1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.2.e.a 2
3.b odd 2 1 117.2.q.c 2
4.b odd 2 1 208.2.w.b 2
5.b even 2 1 325.2.n.a 2
5.c odd 4 2 325.2.m.a 4
7.b odd 2 1 637.2.q.a 2
7.c even 3 1 637.2.k.a 2
7.c even 3 1 637.2.u.c 2
7.d odd 6 1 637.2.k.c 2
7.d odd 6 1 637.2.u.b 2
8.b even 2 1 832.2.w.d 2
8.d odd 2 1 832.2.w.a 2
12.b even 2 1 1872.2.by.d 2
13.b even 2 1 169.2.e.a 2
13.c even 3 1 169.2.b.a 2
13.c even 3 1 169.2.e.a 2
13.d odd 4 2 169.2.c.a 4
13.e even 6 1 inner 13.2.e.a 2
13.e even 6 1 169.2.b.a 2
13.f odd 12 2 169.2.a.a 2
13.f odd 12 2 169.2.c.a 4
39.h odd 6 1 117.2.q.c 2
39.h odd 6 1 1521.2.b.a 2
39.i odd 6 1 1521.2.b.a 2
39.k even 12 2 1521.2.a.k 2
52.i odd 6 1 208.2.w.b 2
52.i odd 6 1 2704.2.f.b 2
52.j odd 6 1 2704.2.f.b 2
52.l even 12 2 2704.2.a.o 2
65.l even 6 1 325.2.n.a 2
65.r odd 12 2 325.2.m.a 4
65.s odd 12 2 4225.2.a.v 2
91.k even 6 1 637.2.u.c 2
91.l odd 6 1 637.2.u.b 2
91.p odd 6 1 637.2.k.c 2
91.t odd 6 1 637.2.q.a 2
91.u even 6 1 637.2.k.a 2
91.bc even 12 2 8281.2.a.q 2
104.p odd 6 1 832.2.w.a 2
104.s even 6 1 832.2.w.d 2
156.r even 6 1 1872.2.by.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 1.a even 1 1 trivial
13.2.e.a 2 13.e even 6 1 inner
117.2.q.c 2 3.b odd 2 1
117.2.q.c 2 39.h odd 6 1
169.2.a.a 2 13.f odd 12 2
169.2.b.a 2 13.c even 3 1
169.2.b.a 2 13.e even 6 1
169.2.c.a 4 13.d odd 4 2
169.2.c.a 4 13.f odd 12 2
169.2.e.a 2 13.b even 2 1
169.2.e.a 2 13.c even 3 1
208.2.w.b 2 4.b odd 2 1
208.2.w.b 2 52.i odd 6 1
325.2.m.a 4 5.c odd 4 2
325.2.m.a 4 65.r odd 12 2
325.2.n.a 2 5.b even 2 1
325.2.n.a 2 65.l even 6 1
637.2.k.a 2 7.c even 3 1
637.2.k.a 2 91.u even 6 1
637.2.k.c 2 7.d odd 6 1
637.2.k.c 2 91.p odd 6 1
637.2.q.a 2 7.b odd 2 1
637.2.q.a 2 91.t odd 6 1
637.2.u.b 2 7.d odd 6 1
637.2.u.b 2 91.l odd 6 1
637.2.u.c 2 7.c even 3 1
637.2.u.c 2 91.k even 6 1
832.2.w.a 2 8.d odd 2 1
832.2.w.a 2 104.p odd 6 1
832.2.w.d 2 8.b even 2 1
832.2.w.d 2 104.s even 6 1
1521.2.a.k 2 39.k even 12 2
1521.2.b.a 2 39.h odd 6 1
1521.2.b.a 2 39.i odd 6 1
1872.2.by.d 2 12.b even 2 1
1872.2.by.d 2 156.r even 6 1
2704.2.a.o 2 52.l even 12 2
2704.2.f.b 2 52.i odd 6 1
2704.2.f.b 2 52.j odd 6 1
4225.2.a.v 2 65.s odd 12 2
8281.2.a.q 2 91.bc even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(13, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \)
$3$ \( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( 1 - 7 T^{2} + 25 T^{4} \)
$7$ \( 1 + 7 T^{2} + 49 T^{4} \)
$11$ \( 1 + 11 T^{2} + 121 T^{4} \)
$13$ \( 1 + 5 T + 13 T^{2} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 - 50 T^{2} + 961 T^{4} \)
$37$ \( 1 - 15 T + 112 T^{2} - 555 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \)
$47$ \( 1 - 82 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 + 3 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 12 T + 107 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( 1 - 6 T + 83 T^{2} - 426 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 26 T^{2} + 6889 T^{4} \)
$89$ \( 1 + 12 T + 137 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 12 T + 145 T^{2} - 1164 T^{3} + 9409 T^{4} \)
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