Properties

Label 2925.2.a.v
Level $2925$
Weight $2$
Character orbit 2925.a
Self dual yes
Analytic conductor $23.356$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.3562425912\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−2.41421 0 3.82843 0 0 2.82843 −4.41421 0 0
1.2 0.414214 0 −1.82843 0 0 −2.82843 −1.58579 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2925.2.a.v 2
3.b odd 2 1 975.2.a.l 2
5.b even 2 1 117.2.a.c 2
5.c odd 4 2 2925.2.c.u 4
15.d odd 2 1 39.2.a.b 2
15.e even 4 2 975.2.c.h 4
20.d odd 2 1 1872.2.a.w 2
35.c odd 2 1 5733.2.a.u 2
40.e odd 2 1 7488.2.a.co 2
40.f even 2 1 7488.2.a.cl 2
45.h odd 6 2 1053.2.e.m 4
45.j even 6 2 1053.2.e.e 4
60.h even 2 1 624.2.a.k 2
65.d even 2 1 1521.2.a.f 2
65.g odd 4 2 1521.2.b.j 4
105.g even 2 1 1911.2.a.h 2
120.i odd 2 1 2496.2.a.bf 2
120.m even 2 1 2496.2.a.bi 2
165.d even 2 1 4719.2.a.p 2
195.e odd 2 1 507.2.a.h 2
195.n even 4 2 507.2.b.e 4
195.x odd 6 2 507.2.e.h 4
195.y odd 6 2 507.2.e.d 4
195.bh even 12 4 507.2.j.f 8
780.d even 2 1 8112.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 15.d odd 2 1
117.2.a.c 2 5.b even 2 1
507.2.a.h 2 195.e odd 2 1
507.2.b.e 4 195.n even 4 2
507.2.e.d 4 195.y odd 6 2
507.2.e.h 4 195.x odd 6 2
507.2.j.f 8 195.bh even 12 4
624.2.a.k 2 60.h even 2 1
975.2.a.l 2 3.b odd 2 1
975.2.c.h 4 15.e even 4 2
1053.2.e.e 4 45.j even 6 2
1053.2.e.m 4 45.h odd 6 2
1521.2.a.f 2 65.d even 2 1
1521.2.b.j 4 65.g odd 4 2
1872.2.a.w 2 20.d odd 2 1
1911.2.a.h 2 105.g even 2 1
2496.2.a.bf 2 120.i odd 2 1
2496.2.a.bi 2 120.m even 2 1
2925.2.a.v 2 1.a even 1 1 trivial
2925.2.c.u 4 5.c odd 4 2
4719.2.a.p 2 165.d even 2 1
5733.2.a.u 2 35.c odd 2 1
7488.2.a.cl 2 40.f even 2 1
7488.2.a.co 2 40.e odd 2 1
8112.2.a.bm 2 780.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2925))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{7}^{2} - 8 \)
\( T_{11} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -28 - 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( 56 + 16 T + T^{2} \)
$43$ \( -16 + 8 T + T^{2} \)
$47$ \( 4 + 12 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( -28 + 4 T + T^{2} \)
$61$ \( -124 - 4 T + T^{2} \)
$67$ \( 8 + 8 T + T^{2} \)
$71$ \( ( 2 + T )^{2} \)
$73$ \( 4 + 12 T + T^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -28 + 4 T + T^{2} \)
$89$ \( 136 + 24 T + T^{2} \)
$97$ \( -28 - 4 T + T^{2} \)
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