Properties

Label 25.3.c.a
Level 25
Weight 3
Character orbit 25.c
Analytic conductor 0.681
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 25 = 5^{2} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 25.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.681200660901\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{3} q^{3} \) \( -\beta_{2} q^{4} \) \( -3 q^{6} \) \( -4 \beta_{1} q^{7} \) \( -5 \beta_{3} q^{8} \) \( + 6 \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{3} q^{3} \) \( -\beta_{2} q^{4} \) \( -3 q^{6} \) \( -4 \beta_{1} q^{7} \) \( -5 \beta_{3} q^{8} \) \( + 6 \beta_{2} q^{9} \) \( -3 q^{11} \) \( + \beta_{1} q^{12} \) \( + 6 \beta_{3} q^{13} \) \( -12 \beta_{2} q^{14} \) \( + 11 q^{16} \) \( + 11 \beta_{1} q^{17} \) \( + 6 \beta_{3} q^{18} \) \( + 5 \beta_{2} q^{19} \) \( + 12 q^{21} \) \( -3 \beta_{1} q^{22} \) \( -14 \beta_{3} q^{23} \) \( + 15 \beta_{2} q^{24} \) \( -18 q^{26} \) \( -15 \beta_{1} q^{27} \) \( + 4 \beta_{3} q^{28} \) \( -30 \beta_{2} q^{29} \) \( -38 q^{31} \) \( -9 \beta_{1} q^{32} \) \( -3 \beta_{3} q^{33} \) \( + 33 \beta_{2} q^{34} \) \( + 6 q^{36} \) \( + 16 \beta_{1} q^{37} \) \( + 5 \beta_{3} q^{38} \) \( -18 \beta_{2} q^{39} \) \( + 57 q^{41} \) \( + 12 \beta_{1} q^{42} \) \( -4 \beta_{3} q^{43} \) \( + 3 \beta_{2} q^{44} \) \( + 42 q^{46} \) \( + 6 \beta_{1} q^{47} \) \( + 11 \beta_{3} q^{48} \) \( -\beta_{2} q^{49} \) \( -33 q^{51} \) \( + 6 \beta_{1} q^{52} \) \( + 26 \beta_{3} q^{53} \) \( -45 \beta_{2} q^{54} \) \( -60 q^{56} \) \( -5 \beta_{1} q^{57} \) \( -30 \beta_{3} q^{58} \) \( + 90 \beta_{2} q^{59} \) \( -28 q^{61} \) \( -38 \beta_{1} q^{62} \) \( -24 \beta_{3} q^{63} \) \( -71 \beta_{2} q^{64} \) \( + 9 q^{66} \) \( -39 \beta_{1} q^{67} \) \( -11 \beta_{3} q^{68} \) \( + 42 \beta_{2} q^{69} \) \( + 42 q^{71} \) \( + 30 \beta_{1} q^{72} \) \( + 11 \beta_{3} q^{73} \) \( + 48 \beta_{2} q^{74} \) \( + 5 q^{76} \) \( + 12 \beta_{1} q^{77} \) \( -18 \beta_{3} q^{78} \) \( -80 \beta_{2} q^{79} \) \( -9 q^{81} \) \( + 57 \beta_{1} q^{82} \) \( + 91 \beta_{3} q^{83} \) \( -12 \beta_{2} q^{84} \) \( + 12 q^{86} \) \( + 30 \beta_{1} q^{87} \) \( + 15 \beta_{3} q^{88} \) \( -15 \beta_{2} q^{89} \) \( + 72 q^{91} \) \( -14 \beta_{1} q^{92} \) \( -38 \beta_{3} q^{93} \) \( + 18 \beta_{2} q^{94} \) \( + 27 q^{96} \) \( -44 \beta_{1} q^{97} \) \( -\beta_{3} q^{98} \) \( -18 \beta_{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 44q^{16} \) \(\mathstrut +\mathstrut 48q^{21} \) \(\mathstrut -\mathstrut 72q^{26} \) \(\mathstrut -\mathstrut 152q^{31} \) \(\mathstrut +\mathstrut 24q^{36} \) \(\mathstrut +\mathstrut 228q^{41} \) \(\mathstrut +\mathstrut 168q^{46} \) \(\mathstrut -\mathstrut 132q^{51} \) \(\mathstrut -\mathstrut 240q^{56} \) \(\mathstrut -\mathstrut 112q^{61} \) \(\mathstrut +\mathstrut 36q^{66} \) \(\mathstrut +\mathstrut 168q^{71} \) \(\mathstrut +\mathstrut 20q^{76} \) \(\mathstrut -\mathstrut 36q^{81} \) \(\mathstrut +\mathstrut 48q^{86} \) \(\mathstrut +\mathstrut 288q^{91} \) \(\mathstrut +\mathstrut 108q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{3}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i 1.22474 1.22474i 1.00000i 0 −3.00000 4.89898 + 4.89898i −6.12372 + 6.12372i 6.00000i 0
7.2 1.22474 + 1.22474i −1.22474 + 1.22474i 1.00000i 0 −3.00000 −4.89898 4.89898i 6.12372 6.12372i 6.00000i 0
18.1 −1.22474 + 1.22474i 1.22474 + 1.22474i 1.00000i 0 −3.00000 4.89898 4.89898i −6.12372 6.12372i 6.00000i 0
18.2 1.22474 1.22474i −1.22474 1.22474i 1.00000i 0 −3.00000 −4.89898 + 4.89898i 6.12372 + 6.12372i 6.00000i 0
\(n\): 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
5.b Even 1 yes
5.c Odd 2 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(25, [\chi])\).