Properties

Label 2025.2.b.b
Level $2025$
Weight $2$
Character orbit 2025.b
Analytic conductor $16.170$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 2 q^{4} + 5 q^{11} - 2 \beta q^{13} - 4 q^{16} - 2 \beta q^{17} + 5 q^{19} + 5 \beta q^{22} - 3 \beta q^{23} + 8 q^{26} + 5 q^{29} - 9 q^{31} - 4 \beta q^{32} + 8 q^{34} - 5 \beta q^{37} + 5 \beta q^{38} + 7 q^{41} + \beta q^{43} - 10 q^{44} + 12 q^{46} + \beta q^{47} + 7 q^{49} + 4 \beta q^{52} - 4 \beta q^{53} + 5 \beta q^{58} + q^{59} - 2 q^{61} - 9 \beta q^{62} + 8 q^{64} + 3 \beta q^{67} + 4 \beta q^{68} + q^{71} + 4 \beta q^{73} + 20 q^{74} - 10 q^{76} - 12 q^{79} + 7 \beta q^{82} - 3 \beta q^{83} - 4 q^{86} + 9 q^{89} + 6 \beta q^{92} - 4 q^{94} + 7 \beta q^{97} + 7 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 10 q^{11} - 8 q^{16} + 10 q^{19} + 16 q^{26} + 10 q^{29} - 18 q^{31} + 16 q^{34} + 14 q^{41} - 20 q^{44} + 24 q^{46} + 14 q^{49} + 2 q^{59} - 4 q^{61} + 16 q^{64} + 2 q^{71} + 40 q^{74} - 20 q^{76} - 24 q^{79} - 8 q^{86} + 18 q^{89} - 8 q^{94}+O(q^{100}) \) 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
1.00000i
1.00000i
2.00000i 0 −2.00000 0 0 0 0 0 0
649.2 2.00000i 0 −2.00000 0 0 0 0 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.b 2
3.b odd 2 1 2025.2.b.a 2
5.b even 2 1 inner 2025.2.b.b 2
5.c odd 4 1 405.2.a.f yes 1
5.c odd 4 1 2025.2.a.a 1
15.d odd 2 1 2025.2.b.a 2
15.e even 4 1 405.2.a.a 1
15.e even 4 1 2025.2.a.f 1
20.e even 4 1 6480.2.a.r 1
45.k odd 12 2 405.2.e.a 2
45.l even 12 2 405.2.e.g 2
60.l odd 4 1 6480.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.a 1 15.e even 4 1
405.2.a.f yes 1 5.c odd 4 1
405.2.e.a 2 45.k odd 12 2
405.2.e.g 2 45.l even 12 2
2025.2.a.a 1 5.c odd 4 1
2025.2.a.f 1 15.e even 4 1
2025.2.b.a 2 3.b odd 2 1
2025.2.b.a 2 15.d odd 2 1
2025.2.b.b 2 1.a even 1 1 trivial
2025.2.b.b 2 5.b even 2 1 inner
6480.2.a.f 1 60.l odd 4 1
6480.2.a.r 1 20.e even 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 9)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T - 1)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 36 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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