Properties

Label 7605.2.a.bj
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} - q^{5} + \beta q^{7} + (2 \beta + 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} - q^{5} + \beta q^{7} + (2 \beta + 6) q^{8} + ( - \beta - 1) q^{10} + 2 q^{11} + (\beta + 3) q^{14} + (4 \beta + 8) q^{16} + (\beta + 1) q^{17} + ( - 2 \beta - 4) q^{19} + ( - 2 \beta - 2) q^{20} + (2 \beta + 2) q^{22} - 2 q^{23} + q^{25} + (2 \beta + 6) q^{28} + (\beta + 5) q^{29} + ( - 2 \beta + 1) q^{31} + (8 \beta + 8) q^{32} + (2 \beta + 4) q^{34} - \beta q^{35} + 6 \beta q^{37} + ( - 6 \beta - 10) q^{38} + ( - 2 \beta - 6) q^{40} + ( - \beta + 9) q^{41} + (3 \beta - 4) q^{43} + (4 \beta + 4) q^{44} + ( - 2 \beta - 2) q^{46} + (3 \beta + 5) q^{47} - 4 q^{49} + (\beta + 1) q^{50} + ( - 2 \beta + 6) q^{53} - 2 q^{55} + (6 \beta + 6) q^{56} + (6 \beta + 8) q^{58} + ( - 5 \beta + 3) q^{59} + ( - 6 \beta - 5) q^{61} + ( - \beta - 5) q^{62} + (8 \beta + 16) q^{64} + (3 \beta - 4) q^{67} + (4 \beta + 8) q^{68} + ( - \beta - 3) q^{70} + ( - \beta + 3) q^{71} - \beta q^{73} + (6 \beta + 18) q^{74} + ( - 12 \beta - 20) q^{76} + 2 \beta q^{77} - 11 q^{79} + ( - 4 \beta - 8) q^{80} + (8 \beta + 6) q^{82} + (4 \beta + 4) q^{83} + ( - \beta - 1) q^{85} + ( - \beta + 5) q^{86} + (4 \beta + 12) q^{88} + ( - \beta - 7) q^{89} + ( - 4 \beta - 4) q^{92} + (8 \beta + 14) q^{94} + (2 \beta + 4) q^{95} - 3 \beta q^{97} + ( - 4 \beta - 4) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8} - 2 q^{10} + 4 q^{11} + 6 q^{14} + 16 q^{16} + 2 q^{17} - 8 q^{19} - 4 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} + 12 q^{28} + 10 q^{29} + 2 q^{31} + 16 q^{32} + 8 q^{34} - 20 q^{38} - 12 q^{40} + 18 q^{41} - 8 q^{43} + 8 q^{44} - 4 q^{46} + 10 q^{47} - 8 q^{49} + 2 q^{50} + 12 q^{53} - 4 q^{55} + 12 q^{56} + 16 q^{58} + 6 q^{59} - 10 q^{61} - 10 q^{62} + 32 q^{64} - 8 q^{67} + 16 q^{68} - 6 q^{70} + 6 q^{71} + 36 q^{74} - 40 q^{76} - 22 q^{79} - 16 q^{80} + 12 q^{82} + 8 q^{83} - 2 q^{85} + 10 q^{86} + 24 q^{88} - 14 q^{89} - 8 q^{92} + 28 q^{94} + 8 q^{95} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−1.73205
1.73205
−0.732051 0 −1.46410 −1.00000 0 −1.73205 2.53590 0 0.732051
1.2 2.73205 0 5.46410 −1.00000 0 1.73205 9.46410 0 −2.73205
\(n\): 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.bj 2
3.b odd 2 1 2535.2.a.o 2
13.b even 2 1 7605.2.a.z 2
13.c even 3 2 585.2.j.c 4
39.d odd 2 1 2535.2.a.r 2
39.i odd 6 2 195.2.i.c 4
195.x odd 6 2 975.2.i.j 4
195.bl even 12 2 975.2.bb.a 4
195.bl even 12 2 975.2.bb.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.c 4 39.i odd 6 2
585.2.j.c 4 13.c even 3 2
975.2.i.j 4 195.x odd 6 2
975.2.bb.a 4 195.bl even 12 2
975.2.bb.h 4 195.bl even 12 2
2535.2.a.o 2 3.b odd 2 1
2535.2.a.r 2 39.d odd 2 1
7605.2.a.z 2 13.b even 2 1
7605.2.a.bj 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} - 108 \) Copy content Toggle raw display
$41$ \( T^{2} - 18T + 78 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 66 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T - 83 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$73$ \( T^{2} - 3 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 46 \) Copy content Toggle raw display
$97$ \( T^{2} - 27 \) Copy content Toggle raw display
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