Properties

Label 26.2.c.a
Level 26
Weight 2
Character orbit 26.c
Analytic conductor 0.208
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 26 = 2 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 26.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.207611045255\)
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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} - q^{5} -4 \zeta_{6} q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} - q^{5} -4 \zeta_{6} q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} + ( 3 + \zeta_{6} ) q^{13} + 4 q^{14} + ( -1 + \zeta_{6} ) q^{16} -3 \zeta_{6} q^{17} -3 q^{18} + \zeta_{6} q^{20} -4 \zeta_{6} q^{22} + ( 4 - 4 \zeta_{6} ) q^{23} -4 q^{25} + ( -4 + 3 \zeta_{6} ) q^{26} + ( -4 + 4 \zeta_{6} ) q^{28} + ( 1 - \zeta_{6} ) q^{29} + 4 q^{31} -\zeta_{6} q^{32} + 3 q^{34} + 4 \zeta_{6} q^{35} + ( 3 - 3 \zeta_{6} ) q^{36} + ( -3 + 3 \zeta_{6} ) q^{37} - q^{40} + ( 9 - 9 \zeta_{6} ) q^{41} + 8 \zeta_{6} q^{43} + 4 q^{44} -3 \zeta_{6} q^{45} + 4 \zeta_{6} q^{46} -8 q^{47} + ( -9 + 9 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} + ( 1 - 4 \zeta_{6} ) q^{52} -9 q^{53} + ( 4 - 4 \zeta_{6} ) q^{55} -4 \zeta_{6} q^{56} + \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} -7 \zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} + ( 12 - 12 \zeta_{6} ) q^{63} + q^{64} + ( -3 - \zeta_{6} ) q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} + ( -3 + 3 \zeta_{6} ) q^{68} -4 q^{70} + 8 \zeta_{6} q^{71} + 3 \zeta_{6} q^{72} + 11 q^{73} -3 \zeta_{6} q^{74} + 16 q^{77} -4 q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + 9 \zeta_{6} q^{82} + 3 \zeta_{6} q^{85} -8 q^{86} + ( -4 + 4 \zeta_{6} ) q^{88} + ( 6 - 6 \zeta_{6} ) q^{89} + 3 q^{90} + ( 4 - 16 \zeta_{6} ) q^{91} -4 q^{92} + ( 8 - 8 \zeta_{6} ) q^{94} -2 \zeta_{6} q^{97} -9 \zeta_{6} q^{98} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 2q^{5} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 2q^{5} - 4q^{7} + 2q^{8} + 3q^{9} + q^{10} - 4q^{11} + 7q^{13} + 8q^{14} - q^{16} - 3q^{17} - 6q^{18} + q^{20} - 4q^{22} + 4q^{23} - 8q^{25} - 5q^{26} - 4q^{28} + q^{29} + 8q^{31} - q^{32} + 6q^{34} + 4q^{35} + 3q^{36} - 3q^{37} - 2q^{40} + 9q^{41} + 8q^{43} + 8q^{44} - 3q^{45} + 4q^{46} - 16q^{47} - 9q^{49} + 4q^{50} - 2q^{52} - 18q^{53} + 4q^{55} - 4q^{56} + q^{58} + 4q^{59} - 7q^{61} - 4q^{62} + 12q^{63} + 2q^{64} - 7q^{65} - 4q^{67} - 3q^{68} - 8q^{70} + 8q^{71} + 3q^{72} + 22q^{73} - 3q^{74} + 32q^{77} - 8q^{79} + q^{80} - 9q^{81} + 9q^{82} + 3q^{85} - 16q^{86} - 4q^{88} + 6q^{89} + 6q^{90} - 8q^{91} - 8q^{92} + 8q^{94} - 2q^{97} - 9q^{98} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −2.00000 + 3.46410i 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
9.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 0 −2.00000 3.46410i 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.2.c.a 2
3.b odd 2 1 234.2.h.c 2
4.b odd 2 1 208.2.i.b 2
5.b even 2 1 650.2.e.c 2
5.c odd 4 2 650.2.o.c 4
7.b odd 2 1 1274.2.g.a 2
7.c even 3 1 1274.2.e.n 2
7.c even 3 1 1274.2.h.b 2
7.d odd 6 1 1274.2.e.m 2
7.d odd 6 1 1274.2.h.a 2
8.b even 2 1 832.2.i.e 2
8.d odd 2 1 832.2.i.f 2
12.b even 2 1 1872.2.t.k 2
13.b even 2 1 338.2.c.e 2
13.c even 3 1 inner 26.2.c.a 2
13.c even 3 1 338.2.a.e 1
13.d odd 4 2 338.2.e.b 4
13.e even 6 1 338.2.a.c 1
13.e even 6 1 338.2.c.e 2
13.f odd 12 2 338.2.b.b 2
13.f odd 12 2 338.2.e.b 4
39.h odd 6 1 3042.2.a.k 1
39.i odd 6 1 234.2.h.c 2
39.i odd 6 1 3042.2.a.e 1
39.k even 12 2 3042.2.b.e 2
52.i odd 6 1 2704.2.a.i 1
52.j odd 6 1 208.2.i.b 2
52.j odd 6 1 2704.2.a.h 1
52.l even 12 2 2704.2.f.g 2
65.l even 6 1 8450.2.a.s 1
65.n even 6 1 650.2.e.c 2
65.n even 6 1 8450.2.a.f 1
65.q odd 12 2 650.2.o.c 4
91.g even 3 1 1274.2.e.n 2
91.h even 3 1 1274.2.h.b 2
91.m odd 6 1 1274.2.e.m 2
91.n odd 6 1 1274.2.g.a 2
91.v odd 6 1 1274.2.h.a 2
104.n odd 6 1 832.2.i.f 2
104.r even 6 1 832.2.i.e 2
156.p even 6 1 1872.2.t.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a 2 1.a even 1 1 trivial
26.2.c.a 2 13.c even 3 1 inner
208.2.i.b 2 4.b odd 2 1
208.2.i.b 2 52.j odd 6 1
234.2.h.c 2 3.b odd 2 1
234.2.h.c 2 39.i odd 6 1
338.2.a.c 1 13.e even 6 1
338.2.a.e 1 13.c even 3 1
338.2.b.b 2 13.f odd 12 2
338.2.c.e 2 13.b even 2 1
338.2.c.e 2 13.e even 6 1
338.2.e.b 4 13.d odd 4 2
338.2.e.b 4 13.f odd 12 2
650.2.e.c 2 5.b even 2 1
650.2.e.c 2 65.n even 6 1
650.2.o.c 4 5.c odd 4 2
650.2.o.c 4 65.q odd 12 2
832.2.i.e 2 8.b even 2 1
832.2.i.e 2 104.r even 6 1
832.2.i.f 2 8.d odd 2 1
832.2.i.f 2 104.n odd 6 1
1274.2.e.m 2 7.d odd 6 1
1274.2.e.m 2 91.m odd 6 1
1274.2.e.n 2 7.c even 3 1
1274.2.e.n 2 91.g even 3 1
1274.2.g.a 2 7.b odd 2 1
1274.2.g.a 2 91.n odd 6 1
1274.2.h.a 2 7.d odd 6 1
1274.2.h.a 2 91.v odd 6 1
1274.2.h.b 2 7.c even 3 1
1274.2.h.b 2 91.h even 3 1
1872.2.t.k 2 12.b even 2 1
1872.2.t.k 2 156.p even 6 1
2704.2.a.h 1 52.j odd 6 1
2704.2.a.i 1 52.i odd 6 1
2704.2.f.g 2 52.l even 12 2
3042.2.a.e 1 39.i odd 6 1
3042.2.a.k 1 39.h odd 6 1
3042.2.b.e 2 39.k even 12 2
8450.2.a.f 1 65.n even 6 1
8450.2.a.s 1 65.l even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(26, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \)
$5$ \( ( 1 + T + 5 T^{2} )^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 - 7 T + 13 T^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 9 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 8 T - 7 T^{2} - 568 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - 11 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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