Properties

Label 26.2.b.a
Level 26
Weight 2
Character orbit 26.b
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + i q^{2} \) \(- q^{3}\) \(- q^{4}\) \( -3 i q^{5} \) \( -i q^{6} \) \( + 3 i q^{7} \) \( -i q^{8} \) \( -2 q^{9} \) \(+O(q^{10})\) \( q\) \( + i q^{2} \) \(- q^{3}\) \(- q^{4}\) \( -3 i q^{5} \) \( -i q^{6} \) \( + 3 i q^{7} \) \( -i q^{8} \) \( -2 q^{9} \) \( + 3 q^{10} \) \(+ q^{12}\) \( + ( 2 + 3 i ) q^{13} \) \( -3 q^{14} \) \( + 3 i q^{15} \) \(+ q^{16}\) \( + 3 q^{17} \) \( -2 i q^{18} \) \( -6 i q^{19} \) \( + 3 i q^{20} \) \( -3 i q^{21} \) \( -6 q^{23} \) \( + i q^{24} \) \( -4 q^{25} \) \( + ( -3 + 2 i ) q^{26} \) \( + 5 q^{27} \) \( -3 i q^{28} \) \( -3 q^{30} \) \( + i q^{32} \) \( + 3 i q^{34} \) \( + 9 q^{35} \) \( + 2 q^{36} \) \( + 3 i q^{37} \) \( + 6 q^{38} \) \( + ( -2 - 3 i ) q^{39} \) \( -3 q^{40} \) \( + 3 q^{42} \) \(- q^{43}\) \( + 6 i q^{45} \) \( -6 i q^{46} \) \( + 3 i q^{47} \) \(- q^{48}\) \( -2 q^{49} \) \( -4 i q^{50} \) \( -3 q^{51} \) \( + ( -2 - 3 i ) q^{52} \) \( -6 q^{53} \) \( + 5 i q^{54} \) \( + 3 q^{56} \) \( + 6 i q^{57} \) \( -6 i q^{59} \) \( -3 i q^{60} \) \( -8 q^{61} \) \( -6 i q^{63} \) \(- q^{64}\) \( + ( 9 - 6 i ) q^{65} \) \( -12 i q^{67} \) \( -3 q^{68} \) \( + 6 q^{69} \) \( + 9 i q^{70} \) \( + 15 i q^{71} \) \( + 2 i q^{72} \) \( + 6 i q^{73} \) \( -3 q^{74} \) \( + 4 q^{75} \) \( + 6 i q^{76} \) \( + ( 3 - 2 i ) q^{78} \) \( + 10 q^{79} \) \( -3 i q^{80} \) \(+ q^{81}\) \( + 6 i q^{83} \) \( + 3 i q^{84} \) \( -9 i q^{85} \) \( -i q^{86} \) \( -6 i q^{89} \) \( -6 q^{90} \) \( + ( -9 + 6 i ) q^{91} \) \( + 6 q^{92} \) \( -3 q^{94} \) \( -18 q^{95} \) \( -i q^{96} \) \( -12 i q^{97} \) \( -2 i q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 6q^{10} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut +\mathstrut 10q^{27} \) \(\mathstrut -\mathstrut 6q^{30} \) \(\mathstrut +\mathstrut 18q^{35} \) \(\mathstrut +\mathstrut 4q^{36} \) \(\mathstrut +\mathstrut 12q^{38} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 6q^{40} \) \(\mathstrut +\mathstrut 6q^{42} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 6q^{51} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 6q^{56} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 18q^{65} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut +\mathstrut 12q^{69} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 8q^{75} \) \(\mathstrut +\mathstrut 6q^{78} \) \(\mathstrut +\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 12q^{90} \) \(\mathstrut -\mathstrut 18q^{91} \) \(\mathstrut +\mathstrut 12q^{92} \) \(\mathstrut -\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 36q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(15\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
25.2 1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
\(n\): 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.b Even 1 yes

Hecke kernels

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