Properties

Label 7605.2.a.g
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7262307372\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + q^{5} + 3q^{8} + O(q^{10}) \) \( q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} - 4q^{11} - q^{16} - 2q^{17} - 4q^{19} - q^{20} + 4q^{22} + q^{25} + 2q^{29} - 5q^{32} + 2q^{34} + 10q^{37} + 4q^{38} + 3q^{40} + 10q^{41} + 4q^{43} + 4q^{44} + 8q^{47} - 7q^{49} - q^{50} + 10q^{53} - 4q^{55} - 2q^{58} - 4q^{59} - 2q^{61} + 7q^{64} - 12q^{67} + 2q^{68} - 8q^{71} - 10q^{73} - 10q^{74} + 4q^{76} - q^{80} - 10q^{82} + 12q^{83} - 2q^{85} - 4q^{86} - 12q^{88} - 6q^{89} - 8q^{94} - 4q^{95} - 2q^{97} + 7q^{98} + O(q^{100}) \)

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
0
−1.00000 0 −1.00000 1.00000 0 0 3.00000 0 −1.00000
\(n\): 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 7605.2.a.g 1
3.b odd 2 1 2535.2.a.j 1
13.b even 2 1 45.2.a.a 1
39.d odd 2 1 15.2.a.a 1
52.b odd 2 1 720.2.a.c 1
65.d even 2 1 225.2.a.b 1
65.h odd 4 2 225.2.b.b 2
91.b odd 2 1 2205.2.a.i 1
104.e even 2 1 2880.2.a.y 1
104.h odd 2 1 2880.2.a.bc 1
117.n odd 6 2 405.2.e.f 2
117.t even 6 2 405.2.e.c 2
143.d odd 2 1 5445.2.a.c 1
156.h even 2 1 240.2.a.d 1
195.e odd 2 1 75.2.a.b 1
195.s even 4 2 75.2.b.b 2
260.g odd 2 1 3600.2.a.u 1
260.p even 4 2 3600.2.f.e 2
273.g even 2 1 735.2.a.c 1
273.w odd 6 2 735.2.i.e 2
273.ba even 6 2 735.2.i.d 2
312.b odd 2 1 960.2.a.l 1
312.h even 2 1 960.2.a.a 1
429.e even 2 1 1815.2.a.d 1
624.v even 4 2 3840.2.k.r 2
624.bi odd 4 2 3840.2.k.m 2
663.g odd 2 1 4335.2.a.c 1
741.d even 2 1 5415.2.a.j 1
780.d even 2 1 1200.2.a.e 1
780.w odd 4 2 1200.2.f.h 2
897.g even 2 1 7935.2.a.d 1
1365.g even 2 1 3675.2.a.j 1
1560.n even 2 1 4800.2.a.bz 1
1560.y odd 2 1 4800.2.a.t 1
1560.bq even 4 2 4800.2.f.bf 2
1560.cs odd 4 2 4800.2.f.c 2
2145.k even 2 1 9075.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 39.d odd 2 1
45.2.a.a 1 13.b even 2 1
75.2.a.b 1 195.e odd 2 1
75.2.b.b 2 195.s even 4 2
225.2.a.b 1 65.d even 2 1
225.2.b.b 2 65.h odd 4 2
240.2.a.d 1 156.h even 2 1
405.2.e.c 2 117.t even 6 2
405.2.e.f 2 117.n odd 6 2
720.2.a.c 1 52.b odd 2 1
735.2.a.c 1 273.g even 2 1
735.2.i.d 2 273.ba even 6 2
735.2.i.e 2 273.w odd 6 2
960.2.a.a 1 312.h even 2 1
960.2.a.l 1 312.b odd 2 1
1200.2.a.e 1 780.d even 2 1
1200.2.f.h 2 780.w odd 4 2
1815.2.a.d 1 429.e even 2 1
2205.2.a.i 1 91.b odd 2 1
2535.2.a.j 1 3.b odd 2 1
2880.2.a.y 1 104.e even 2 1
2880.2.a.bc 1 104.h odd 2 1
3600.2.a.u 1 260.g odd 2 1
3600.2.f.e 2 260.p even 4 2
3675.2.a.j 1 1365.g even 2 1
3840.2.k.m 2 624.bi odd 4 2
3840.2.k.r 2 624.v even 4 2
4335.2.a.c 1 663.g odd 2 1
4800.2.a.t 1 1560.y odd 2 1
4800.2.a.bz 1 1560.n even 2 1
4800.2.f.c 2 1560.cs odd 4 2
4800.2.f.bf 2 1560.bq even 4 2
5415.2.a.j 1 741.d even 2 1
5445.2.a.c 1 143.d odd 2 1
7605.2.a.g 1 1.a even 1 1 trivial
7935.2.a.d 1 897.g even 2 1
9075.2.a.g 1 2145.k even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2} + 1 \)
\( T_{7} \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( T \)
$17$ \( 2 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( -2 + T \)
$31$ \( T \)
$37$ \( -10 + T \)
$41$ \( -10 + T \)
$43$ \( -4 + T \)
$47$ \( -8 + T \)
$53$ \( -10 + T \)
$59$ \( 4 + T \)
$61$ \( 2 + T \)
$67$ \( 12 + T \)
$71$ \( 8 + T \)
$73$ \( 10 + T \)
$79$ \( T \)
$83$ \( -12 + T \)
$89$ \( 6 + T \)
$97$ \( 2 + T \)
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