Properties

Label 2025.2.b.h
Level $2025$
Weight $2$
Character orbit 2025.b
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 405)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (2 \beta_1 - 2) q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{7} + (2 \beta_{3} - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (2 \beta_1 - 2) q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{7} + (2 \beta_{3} - 2 \beta_{2}) q^{8} + (\beta_1 + 4) q^{11} - 2 \beta_{3} q^{13} - 2 \beta_1 q^{14} + ( - 4 \beta_1 + 8) q^{16} + (\beta_{3} + \beta_{2}) q^{17} + ( - 2 \beta_1 - 1) q^{19} + ( - 3 \beta_{3} - \beta_{2}) q^{22} + ( - 2 \beta_{3} - \beta_{2}) q^{23} + (4 \beta_1 - 8) q^{26} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{28} + (3 \beta_1 - 2) q^{29} - 3 q^{31} - 8 \beta_{3} q^{32} + 2 q^{34} + ( - \beta_{3} - \beta_{2}) q^{37} + ( - \beta_{3} + 2 \beta_{2}) q^{38} + (3 \beta_1 + 2) q^{41} + ( - 3 \beta_{3} + \beta_{2}) q^{43} + (6 \beta_1 - 2) q^{44} + (2 \beta_1 - 6) q^{46} + (\beta_{3} + 4 \beta_{2}) q^{47} + ( - 6 \beta_1 - 5) q^{49} + (8 \beta_{3} - 4 \beta_{2}) q^{52} + (\beta_{3} + 3 \beta_{2}) q^{53} - 12 q^{56} + (5 \beta_{3} - 3 \beta_{2}) q^{58} + (\beta_1 - 10) q^{59} + 4 q^{61} + 3 \beta_{3} q^{62} + (8 \beta_1 - 16) q^{64} + (2 \beta_{3} + \beta_{2}) q^{67} + 2 \beta_{2} q^{68} + (\beta_1 + 2) q^{71} + (5 \beta_{3} + 2 \beta_{2}) q^{73} - 2 q^{74} + (2 \beta_1 - 10) q^{76} + ( - 7 \beta_{3} - 11 \beta_{2}) q^{77} + ( - 2 \beta_1 - 12) q^{79} + (\beta_{3} - 3 \beta_{2}) q^{82} + 3 \beta_{3} q^{83} + (8 \beta_1 - 14) q^{86} + (2 \beta_{3} - 8 \beta_{2}) q^{88} + 3 \beta_1 q^{89} - 4 \beta_1 q^{91} + (4 \beta_{3} - 4 \beta_{2}) q^{92} + (6 \beta_1 - 4) q^{94} + ( - 5 \beta_{3} - 3 \beta_{2}) q^{97} + ( - \beta_{3} + 6 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{11} + 32 q^{16} - 4 q^{19} - 32 q^{26} - 8 q^{29} - 12 q^{31} + 8 q^{34} + 8 q^{41} - 8 q^{44} - 24 q^{46} - 20 q^{49} - 48 q^{56} - 40 q^{59} + 16 q^{61} - 64 q^{64} + 8 q^{71} - 8 q^{74} - 40 q^{76} - 48 q^{79} - 56 q^{86} - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
2.73205i 0 −5.46410 0 0 1.26795i 9.46410i 0 0
649.2 0.732051i 0 1.46410 0 0 4.73205i 2.53590i 0 0
649.3 0.732051i 0 1.46410 0 0 4.73205i 2.53590i 0 0
649.4 2.73205i 0 −5.46410 0 0 1.26795i 9.46410i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.2.b.h 4
3.b odd 2 1 2025.2.b.g 4
5.b even 2 1 inner 2025.2.b.h 4
5.c odd 4 1 405.2.a.h yes 2
5.c odd 4 1 2025.2.a.g 2
15.d odd 2 1 2025.2.b.g 4
15.e even 4 1 405.2.a.g 2
15.e even 4 1 2025.2.a.m 2
20.e even 4 1 6480.2.a.bi 2
45.k odd 12 2 405.2.e.i 4
45.l even 12 2 405.2.e.l 4
60.l odd 4 1 6480.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 15.e even 4 1
405.2.a.h yes 2 5.c odd 4 1
405.2.e.i 4 45.k odd 12 2
405.2.e.l 4 45.l even 12 2
2025.2.a.g 2 5.c odd 4 1
2025.2.a.m 2 15.e even 4 1
2025.2.b.g 4 3.b odd 2 1
2025.2.b.g 4 15.d odd 2 1
2025.2.b.h 4 1.a even 1 1 trivial
2025.2.b.h 4 5.b even 2 1 inner
6480.2.a.bi 2 20.e even 4 1
6480.2.a.br 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2025, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 8T_{11} + 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T + 13)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 11)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 23)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$53$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$59$ \( (T^{2} + 20 T + 97)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$79$ \( (T^{2} + 24 T + 132)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
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