Properties

Label 26.3.d.a
Level 26
Weight 3
Character orbit 26.d
Analytic conductor 0.708
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 26 = 2 \cdot 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 26.d (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.708448687337\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( 1 + i ) q^{2} \) \( + 2 i q^{4} \) \( + ( -3 - 3 i ) q^{5} \) \( + ( 2 - 2 i ) q^{7} \) \( + ( -2 + 2 i ) q^{8} \) \( -9 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + i ) q^{2} \) \( + 2 i q^{4} \) \( + ( -3 - 3 i ) q^{5} \) \( + ( 2 - 2 i ) q^{7} \) \( + ( -2 + 2 i ) q^{8} \) \( -9 q^{9} \) \( -6 i q^{10} \) \( + ( 6 - 6 i ) q^{11} \) \( + 13 i q^{13} \) \( + 4 q^{14} \) \( -4 q^{16} \) \( + 6 i q^{17} \) \( + ( -9 - 9 i ) q^{18} \) \( + ( 26 + 26 i ) q^{19} \) \( + ( 6 - 6 i ) q^{20} \) \( + 12 q^{22} \) \( -24 i q^{23} \) \( -7 i q^{25} \) \( + ( -13 + 13 i ) q^{26} \) \( + ( 4 + 4 i ) q^{28} \) \( -48 q^{29} \) \( + ( -14 - 14 i ) q^{31} \) \( + ( -4 - 4 i ) q^{32} \) \( + ( -6 + 6 i ) q^{34} \) \( -12 q^{35} \) \( -18 i q^{36} \) \( + ( 37 - 37 i ) q^{37} \) \( + 52 i q^{38} \) \( + 12 q^{40} \) \( + ( -9 - 9 i ) q^{41} \) \( + 36 i q^{43} \) \( + ( 12 + 12 i ) q^{44} \) \( + ( 27 + 27 i ) q^{45} \) \( + ( 24 - 24 i ) q^{46} \) \( + ( 42 - 42 i ) q^{47} \) \( + 41 i q^{49} \) \( + ( 7 - 7 i ) q^{50} \) \( -26 q^{52} \) \( + 30 q^{53} \) \( -36 q^{55} \) \( + 8 i q^{56} \) \( + ( -48 - 48 i ) q^{58} \) \( + ( -54 + 54 i ) q^{59} \) \( -18 q^{61} \) \( -28 i q^{62} \) \( + ( -18 + 18 i ) q^{63} \) \( -8 i q^{64} \) \( + ( 39 - 39 i ) q^{65} \) \( + ( -22 - 22 i ) q^{67} \) \( -12 q^{68} \) \( + ( -12 - 12 i ) q^{70} \) \( + ( 6 + 6 i ) q^{71} \) \( + ( 18 - 18 i ) q^{72} \) \( + ( 17 - 17 i ) q^{73} \) \( + 74 q^{74} \) \( + ( -52 + 52 i ) q^{76} \) \( -24 i q^{77} \) \( -108 q^{79} \) \( + ( 12 + 12 i ) q^{80} \) \( + 81 q^{81} \) \( -18 i q^{82} \) \( + ( 78 + 78 i ) q^{83} \) \( + ( 18 - 18 i ) q^{85} \) \( + ( -36 + 36 i ) q^{86} \) \( + 24 i q^{88} \) \( + ( -9 + 9 i ) q^{89} \) \( + 54 i q^{90} \) \( + ( 26 + 26 i ) q^{91} \) \( + 48 q^{92} \) \( + 84 q^{94} \) \( -156 i q^{95} \) \( + ( -47 - 47 i ) q^{97} \) \( + ( -41 + 41 i ) q^{98} \) \( + ( -54 + 54 i ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 8q^{14} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 18q^{18} \) \(\mathstrut +\mathstrut 52q^{19} \) \(\mathstrut +\mathstrut 12q^{20} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 26q^{26} \) \(\mathstrut +\mathstrut 8q^{28} \) \(\mathstrut -\mathstrut 96q^{29} \) \(\mathstrut -\mathstrut 28q^{31} \) \(\mathstrut -\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 12q^{34} \) \(\mathstrut -\mathstrut 24q^{35} \) \(\mathstrut +\mathstrut 74q^{37} \) \(\mathstrut +\mathstrut 24q^{40} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut +\mathstrut 24q^{44} \) \(\mathstrut +\mathstrut 54q^{45} \) \(\mathstrut +\mathstrut 48q^{46} \) \(\mathstrut +\mathstrut 84q^{47} \) \(\mathstrut +\mathstrut 14q^{50} \) \(\mathstrut -\mathstrut 52q^{52} \) \(\mathstrut +\mathstrut 60q^{53} \) \(\mathstrut -\mathstrut 72q^{55} \) \(\mathstrut -\mathstrut 96q^{58} \) \(\mathstrut -\mathstrut 108q^{59} \) \(\mathstrut -\mathstrut 36q^{61} \) \(\mathstrut -\mathstrut 36q^{63} \) \(\mathstrut +\mathstrut 78q^{65} \) \(\mathstrut -\mathstrut 44q^{67} \) \(\mathstrut -\mathstrut 24q^{68} \) \(\mathstrut -\mathstrut 24q^{70} \) \(\mathstrut +\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 36q^{72} \) \(\mathstrut +\mathstrut 34q^{73} \) \(\mathstrut +\mathstrut 148q^{74} \) \(\mathstrut -\mathstrut 104q^{76} \) \(\mathstrut -\mathstrut 216q^{79} \) \(\mathstrut +\mathstrut 24q^{80} \) \(\mathstrut +\mathstrut 162q^{81} \) \(\mathstrut +\mathstrut 156q^{83} \) \(\mathstrut +\mathstrut 36q^{85} \) \(\mathstrut -\mathstrut 72q^{86} \) \(\mathstrut -\mathstrut 18q^{89} \) \(\mathstrut +\mathstrut 52q^{91} \) \(\mathstrut +\mathstrut 96q^{92} \) \(\mathstrut +\mathstrut 168q^{94} \) \(\mathstrut -\mathstrut 94q^{97} \) \(\mathstrut -\mathstrut 82q^{98} \) \(\mathstrut -\mathstrut 108q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(15\)
\(\chi(n)\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.00000i
1.00000i
1.00000 1.00000i 0 2.00000i −3.00000 + 3.00000i 0 2.00000 + 2.00000i −2.00000 2.00000i −9.00000 6.00000i
21.1 1.00000 + 1.00000i 0 2.00000i −3.00000 3.00000i 0 2.00000 2.00000i −2.00000 + 2.00000i −9.00000 6.00000i
\(n\): 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.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(26, [\chi])\).