Properties

Label 7605.2.a.bb
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 1) q^{4} - q^{5} + q^{7} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 1) q^{4} - q^{5} + q^{7} - 3 q^{8} + \beta q^{10} + (2 \beta - 3) q^{11} - \beta q^{14} + (\beta - 2) q^{16} + ( - 2 \beta - 3) q^{17} + (2 \beta + 1) q^{19} + ( - \beta - 1) q^{20} + (\beta - 6) q^{22} + 3 q^{23} + q^{25} + (\beta + 1) q^{28} + (4 \beta - 3) q^{29} + 4 q^{31} + (\beta + 3) q^{32} + (5 \beta + 6) q^{34} - q^{35} + ( - 2 \beta + 1) q^{37} + ( - 3 \beta - 6) q^{38} + 3 q^{40} + 3 q^{41} + ( - 4 \beta - 1) q^{43} + (\beta + 3) q^{44} - 3 \beta q^{46} + 4 \beta q^{47} - 6 q^{49} - \beta q^{50} + (4 \beta - 6) q^{53} + ( - 2 \beta + 3) q^{55} - 3 q^{56} + ( - \beta - 12) q^{58} + ( - 6 \beta + 3) q^{59} - q^{61} - 4 \beta q^{62} + ( - 6 \beta + 1) q^{64} + 7 q^{67} + ( - 7 \beta - 9) q^{68} + \beta q^{70} + (6 \beta - 9) q^{71} + ( - 4 \beta + 10) q^{73} + (\beta + 6) q^{74} + (5 \beta + 7) q^{76} + (2 \beta - 3) q^{77} + (4 \beta - 4) q^{79} + ( - \beta + 2) q^{80} - 3 \beta q^{82} - 4 \beta q^{83} + (2 \beta + 3) q^{85} + (5 \beta + 12) q^{86} + ( - 6 \beta + 9) q^{88} + ( - 4 \beta + 3) q^{89} + (3 \beta + 3) q^{92} + ( - 4 \beta - 12) q^{94} + ( - 2 \beta - 1) q^{95} + ( - 2 \beta + 13) q^{97} + 6 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - 4 q^{11} - q^{14} - 3 q^{16} - 8 q^{17} + 4 q^{19} - 3 q^{20} - 11 q^{22} + 6 q^{23} + 2 q^{25} + 3 q^{28} - 2 q^{29} + 8 q^{31} + 7 q^{32} + 17 q^{34} - 2 q^{35} - 15 q^{38} + 6 q^{40} + 6 q^{41} - 6 q^{43} + 7 q^{44} - 3 q^{46} + 4 q^{47} - 12 q^{49} - q^{50} - 8 q^{53} + 4 q^{55} - 6 q^{56} - 25 q^{58} - 2 q^{61} - 4 q^{62} - 4 q^{64} + 14 q^{67} - 25 q^{68} + q^{70} - 12 q^{71} + 16 q^{73} + 13 q^{74} + 19 q^{76} - 4 q^{77} - 4 q^{79} + 3 q^{80} - 3 q^{82} - 4 q^{83} + 8 q^{85} + 29 q^{86} + 12 q^{88} + 2 q^{89} + 9 q^{92} - 28 q^{94} - 4 q^{95} + 24 q^{97} + 6 q^{98}+O(q^{100}) \) 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
2.30278
−1.30278
−2.30278 0 3.30278 −1.00000 0 1.00000 −3.00000 0 2.30278
1.2 1.30278 0 −0.302776 −1.00000 0 1.00000 −3.00000 0 −1.30278
\(n\): 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.bb 2
3.b odd 2 1 845.2.a.f 2
13.b even 2 1 7605.2.a.bg 2
13.e even 6 2 585.2.j.d 4
15.d odd 2 1 4225.2.a.t 2
39.d odd 2 1 845.2.a.c 2
39.f even 4 2 845.2.c.d 4
39.h odd 6 2 65.2.e.b 4
39.i odd 6 2 845.2.e.d 4
39.k even 12 4 845.2.m.d 8
156.r even 6 2 1040.2.q.o 4
195.e odd 2 1 4225.2.a.x 2
195.y odd 6 2 325.2.e.a 4
195.bf even 12 4 325.2.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b 4 39.h odd 6 2
325.2.e.a 4 195.y odd 6 2
325.2.o.b 8 195.bf even 12 4
585.2.j.d 4 13.e even 6 2
845.2.a.c 2 39.d odd 2 1
845.2.a.f 2 3.b odd 2 1
845.2.c.d 4 39.f even 4 2
845.2.e.d 4 39.i odd 6 2
845.2.m.d 8 39.k even 12 4
1040.2.q.o 4 156.r even 6 2
4225.2.a.t 2 15.d odd 2 1
4225.2.a.x 2 195.e odd 2 1
7605.2.a.bb 2 1.a even 1 1 trivial
7605.2.a.bg 2 13.b even 2 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 13 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 43 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 117 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T - 7)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 81 \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 12 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$89$ \( T^{2} - 2T - 51 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 131 \) Copy content Toggle raw display
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