Properties

Label 2025.2.b.m
Level $2025$
Weight $2$
Character orbit 2025.b
Analytic conductor $16.170$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.5089536.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{5} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{5} + \beta_{2}) q^{8} + ( - \beta_{3} + 1) q^{11} + ( - \beta_{4} + \beta_{2}) q^{13} + (\beta_{3} - \beta_1 - 3) q^{14} + ( - \beta_1 + 2) q^{16} + \beta_{2} q^{17} + ( - \beta_{3} - 1) q^{19} + (2 \beta_{5} - \beta_{2}) q^{22} + (\beta_{5} + 2 \beta_{2}) q^{23} + ( - \beta_{3} + 1) q^{26} + ( - 3 \beta_{5} - \beta_{4} - \beta_{2}) q^{28} + ( - 2 \beta_{3} + \beta_1 - 2) q^{29} + (\beta_{3} - 2 \beta_1 + 3) q^{31} + ( - \beta_{5} - 3 \beta_{4}) q^{32} + ( - \beta_{3} - \beta_1 + 2) q^{34} + (2 \beta_{5} + \beta_{4} + \beta_{2}) q^{37} - \beta_{2} q^{38} + ( - \beta_{3} - \beta_1 + 5) q^{41} + (2 \beta_{5} - \beta_{4} - \beta_{2}) q^{43} + (\beta_{3} + \beta_1 - 8) q^{44} + ( - \beta_{3} - 2 \beta_1) q^{46} + ( - 3 \beta_{5} - 3 \beta_{4} + \beta_{2}) q^{47} + (\beta_{3} - 2 \beta_1 + 1) q^{49} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{52} + 2 \beta_{5} q^{53} + 3 q^{56} + (\beta_{5} + 3 \beta_{4}) q^{58} + (\beta_1 - 1) q^{59} + ( - \beta_{3} - \beta_1 + 1) q^{61} + ( - 6 \beta_{4} - 3 \beta_{2}) q^{62} + ( - \beta_{3} + \beta_1 + 5) q^{64} + (\beta_{5} + \beta_{4} + 3 \beta_{2}) q^{67} + (2 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{68} + (3 \beta_{3} + \beta_1 + 2) q^{71} + (4 \beta_{5} + 2 \beta_{4}) q^{73} + (\beta_{3} - 2 \beta_1 - 5) q^{74} + ( - \beta_{3} + \beta_1 - 4) q^{76} + (2 \beta_{5} + \beta_{2}) q^{77} - 2 \beta_1 q^{79} + (5 \beta_{5} - 3 \beta_{4} - 3 \beta_{2}) q^{82} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{83} + (3 \beta_{3} + 2 \beta_1 - 11) q^{86} + ( - 4 \beta_{5} + 3 \beta_{4} + \beta_{2}) q^{88} - 3 q^{89} + (\beta_{3} + 3) q^{91} + (\beta_{5} - 6 \beta_{4} - \beta_{2}) q^{92} + ( - 4 \beta_{3} + 2 \beta_1 + 11) q^{94} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2}) q^{97} + ( - 2 \beta_{5} - 6 \beta_{4} - 3 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 4 q^{11} - 18 q^{14} + 10 q^{16} - 8 q^{19} + 4 q^{26} - 14 q^{29} + 16 q^{31} + 8 q^{34} + 26 q^{41} - 44 q^{44} - 6 q^{46} + 4 q^{49} + 18 q^{56} - 4 q^{59} + 2 q^{61} + 30 q^{64} + 20 q^{71} - 32 q^{74} - 24 q^{76} - 4 q^{79} - 56 q^{86} - 18 q^{89} + 20 q^{91} + 62 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 226\nu - 138 ) / 131 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\nu^{5} - 56\nu^{4} + 14\nu^{3} + 308\nu^{2} + 772\nu - 534 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 23\nu^{2} + 386\nu - 267 ) / 131 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} + 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 4\beta_{3} + 4\beta_{2} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 3\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14\beta_{5} - 3\beta_{4} + 18\beta_{3} - 18\beta_{2} + 7\beta _1 - 31 ) / 2 \) 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
1.66044 + 1.66044i
0.675970 0.675970i
−1.33641 1.33641i
−1.33641 + 1.33641i
0.675970 + 0.675970i
1.66044 1.66044i
2.51414i 0 −4.32088 0 0 0.514137i 5.83502i 0 0
649.2 2.08613i 0 −2.35194 0 0 4.08613i 0.734191i 0 0
649.3 0.571993i 0 1.67282 0 0 1.42801i 2.10083i 0 0
649.4 0.571993i 0 1.67282 0 0 1.42801i 2.10083i 0 0
649.5 2.08613i 0 −2.35194 0 0 4.08613i 0.734191i 0 0
649.6 2.51414i 0 −4.32088 0 0 0.514137i 5.83502i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
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.m 6
3.b odd 2 1 2025.2.b.l 6
5.b even 2 1 inner 2025.2.b.m 6
5.c odd 4 1 405.2.a.i 3
5.c odd 4 1 2025.2.a.o 3
9.c even 3 2 675.2.k.b 12
9.d odd 6 2 225.2.k.b 12
15.d odd 2 1 2025.2.b.l 6
15.e even 4 1 405.2.a.j 3
15.e even 4 1 2025.2.a.n 3
20.e even 4 1 6480.2.a.bs 3
45.h odd 6 2 225.2.k.b 12
45.j even 6 2 675.2.k.b 12
45.k odd 12 2 135.2.e.b 6
45.k odd 12 2 675.2.e.b 6
45.l even 12 2 45.2.e.b 6
45.l even 12 2 225.2.e.b 6
60.l odd 4 1 6480.2.a.bv 3
180.v odd 12 2 720.2.q.i 6
180.x even 12 2 2160.2.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 45.l even 12 2
135.2.e.b 6 45.k odd 12 2
225.2.e.b 6 45.l even 12 2
225.2.k.b 12 9.d odd 6 2
225.2.k.b 12 45.h odd 6 2
405.2.a.i 3 5.c odd 4 1
405.2.a.j 3 15.e even 4 1
675.2.e.b 6 45.k odd 12 2
675.2.k.b 12 9.c even 3 2
675.2.k.b 12 45.j even 6 2
720.2.q.i 6 180.v odd 12 2
2025.2.a.n 3 15.e even 4 1
2025.2.a.o 3 5.c odd 4 1
2025.2.b.l 6 3.b odd 2 1
2025.2.b.l 6 15.d odd 2 1
2025.2.b.m 6 1.a even 1 1 trivial
2025.2.b.m 6 5.b even 2 1 inner
2160.2.q.k 6 180.x even 12 2
6480.2.a.bs 3 20.e even 4 1
6480.2.a.bv 3 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 \(S_{2}^{\mathrm{new}}(2025, [\chi])\):

\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 8T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + 31 T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 19 T^{4} + 39 T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 24 T^{4} + 48 T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{6} + 20 T^{4} + 112 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( (T^{3} + 4 T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 75 T^{4} + 1791 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$29$ \( (T^{3} + 7 T^{2} - 29 T - 51)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 60 T + 468)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 60 T^{4} + 192 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( (T^{3} - 13 T^{2} + 19 T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 108 T^{4} + 96 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{6} + 191 T^{4} + 9715 T^{2} + \cdots + 136161 \) Copy content Toggle raw display
$53$ \( T^{6} + 44 T^{4} + 496 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( (T^{3} + 2 T^{2} - 20 T - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - T^{2} - 37 T - 71)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 199 T^{4} + 12675 T^{2} + \cdots + 257049 \) Copy content Toggle raw display
$71$ \( (T^{3} - 10 T^{2} - 92 T + 708)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 192 T^{4} + 6144 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$79$ \( (T^{3} + 2 T^{2} - 84 T + 24)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 171 T^{4} + 3159 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$89$ \( (T + 3)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 396 T^{4} + 48240 T^{2} + \cdots + 1700416 \) Copy content Toggle raw display
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