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 \(3\) and degree \(2\))

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: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(26, [\chi])\).