Properties

Label 2925.2.c.p
Level $2925$
Weight $2$
Character orbit 2925.c
Analytic conductor $23.356$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (3 \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (3 \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + \beta_{3} q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{3} - 2) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + (\beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{3} + 2) q^{19} + 4 \beta_{2} q^{22} + (5 \beta_{2} + \beta_1) q^{23} + (\beta_{3} - 1) q^{26} - 2 \beta_{2} q^{28} + ( - 2 \beta_{3} + 4) q^{29} + 6 q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} - 4 q^{34} + (3 \beta_{2} - 3 \beta_1) q^{37} + ( - 8 \beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{3} + 2) q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{3} + 4) q^{44} - 4 \beta_{3} q^{46} + ( - 6 \beta_{2} + 2 \beta_1) q^{47} + ( - 5 \beta_{3} - 1) q^{49} + (2 \beta_{2} - \beta_1) q^{52} + (3 \beta_{2} + 3 \beta_1) q^{53} + ( - 2 \beta_{3} - 6) q^{56} + ( - 8 \beta_{2} + 4 \beta_1) q^{58} - 12 q^{59} + (3 \beta_{3} - 2) q^{61} + 6 \beta_1 q^{62} + ( - \beta_{3} - 3) q^{64} + ( - 4 \beta_{2} - 6 \beta_1) q^{67} + (2 \beta_{2} - 2 \beta_1) q^{68} + (\beta_{3} + 12) q^{71} - 6 \beta_{2} q^{73} + ( - 6 \beta_{3} + 18) q^{74} + (6 \beta_{3} - 14) q^{76} + (7 \beta_{2} + 3 \beta_1) q^{77} + 3 \beta_{3} q^{79} + ( - 4 \beta_{2} + 2 \beta_1) q^{82} + (4 \beta_{2} + 8 \beta_1) q^{83} + 8 q^{86} + 4 \beta_1 q^{88} + (3 \beta_{3} - 6) q^{89} + (\beta_{3} + 2) q^{91} + ( - 6 \beta_{2} + 2 \beta_1) q^{92} + (8 \beta_{3} - 16) q^{94} + (\beta_{2} + 7 \beta_1) q^{97} + ( - 20 \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2224.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 0 −4.56155 0 0 0.438447i 6.56155i 0 0
2224.2 1.56155i 0 −0.438447 0 0 4.56155i 2.43845i 0 0
2224.3 1.56155i 0 −0.438447 0 0 4.56155i 2.43845i 0 0
2224.4 2.56155i 0 −4.56155 0 0 0.438447i 6.56155i 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 2925.2.c.p 4
3.b odd 2 1 2925.2.c.o 4
5.b even 2 1 inner 2925.2.c.p 4
5.c odd 4 1 585.2.a.l yes 2
5.c odd 4 1 2925.2.a.x 2
15.d odd 2 1 2925.2.c.o 4
15.e even 4 1 585.2.a.j 2
15.e even 4 1 2925.2.a.bc 2
20.e even 4 1 9360.2.a.cw 2
60.l odd 4 1 9360.2.a.cl 2
65.h odd 4 1 7605.2.a.bd 2
195.s even 4 1 7605.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.j 2 15.e even 4 1
585.2.a.l yes 2 5.c odd 4 1
2925.2.a.x 2 5.c odd 4 1
2925.2.a.bc 2 15.e even 4 1
2925.2.c.o 4 3.b odd 2 1
2925.2.c.o 4 15.d odd 2 1
2925.2.c.p 4 1.a even 1 1 trivial
2925.2.c.p 4 5.b even 2 1 inner
7605.2.a.bd 2 65.h odd 4 1
7605.2.a.bi 2 195.s even 4 1
9360.2.a.cl 2 60.l odd 4 1
9360.2.a.cw 2 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 \(S_{2}^{\mathrm{new}}(2925, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 21T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 49T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 117T^{2} + 324 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 132T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 81T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T + 12)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 308 T^{2} + 23104 \) Copy content Toggle raw display
$71$ \( (T^{2} - 25 T + 152)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 3 T - 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 429 T^{2} + 40804 \) Copy content Toggle raw display
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