Properties

Label 7605.2.a.ba
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 1) q^{4} - q^{5} + ( - 2 \beta + 3) q^{7} + (2 \beta - 1) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 1) q^{4} - q^{5} + ( - 2 \beta + 3) q^{7} + (2 \beta - 1) q^{8} + \beta q^{10} + ( - 2 \beta - 1) q^{11} + ( - \beta + 2) q^{14} - 3 \beta q^{16} + ( - 4 \beta + 1) q^{17} + ( - 2 \beta + 3) q^{19} + ( - \beta + 1) q^{20} + (3 \beta + 2) q^{22} + ( - 2 \beta - 5) q^{23} + q^{25} + (3 \beta - 5) q^{28} + ( - 4 \beta + 5) q^{29} + ( - \beta + 5) q^{32} + (3 \beta + 4) q^{34} + (2 \beta - 3) q^{35} + 3 q^{37} + ( - \beta + 2) q^{38} + ( - 2 \beta + 1) q^{40} + (8 \beta - 7) q^{41} + (2 \beta - 5) q^{43} + ( - \beta - 1) q^{44} + (7 \beta + 2) q^{46} - 8 \beta q^{47} + ( - 8 \beta + 6) q^{49} - \beta q^{50} - 6 q^{53} + (2 \beta + 1) q^{55} + (4 \beta - 7) q^{56} + ( - \beta + 4) q^{58} + ( - 6 \beta - 3) q^{59} + ( - 12 \beta + 7) q^{61} + (2 \beta + 1) q^{64} + (6 \beta + 1) q^{67} + (\beta - 5) q^{68} + (\beta - 2) q^{70} + ( - 2 \beta + 5) q^{71} - 6 q^{73} - 3 \beta q^{74} + (3 \beta - 5) q^{76} + q^{77} + 3 \beta q^{80} + ( - \beta - 8) q^{82} + (8 \beta - 4) q^{83} + (4 \beta - 1) q^{85} + (3 \beta - 2) q^{86} + ( - 4 \beta - 3) q^{88} + 9 q^{89} + ( - 5 \beta + 3) q^{92} + (8 \beta + 8) q^{94} + (2 \beta - 3) q^{95} + (4 \beta - 1) q^{97} + (2 \beta + 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7} + q^{10} - 4 q^{11} + 3 q^{14} - 3 q^{16} - 2 q^{17} + 4 q^{19} + q^{20} + 7 q^{22} - 12 q^{23} + 2 q^{25} - 7 q^{28} + 6 q^{29} + 9 q^{32} + 11 q^{34} - 4 q^{35} + 6 q^{37} + 3 q^{38} - 6 q^{41} - 8 q^{43} - 3 q^{44} + 11 q^{46} - 8 q^{47} + 4 q^{49} - q^{50} - 12 q^{53} + 4 q^{55} - 10 q^{56} + 7 q^{58} - 12 q^{59} + 2 q^{61} + 4 q^{64} + 8 q^{67} - 9 q^{68} - 3 q^{70} + 8 q^{71} - 12 q^{73} - 3 q^{74} - 7 q^{76} + 2 q^{77} + 3 q^{80} - 17 q^{82} + 2 q^{85} - q^{86} - 10 q^{88} + 18 q^{89} + q^{92} + 24 q^{94} - 4 q^{95} + 2 q^{97} + 18 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−1.61803 0 0.618034 −1.00000 0 −0.236068 2.23607 0 1.61803
1.2 0.618034 0 −1.61803 −1.00000 0 4.23607 −2.23607 0 −0.618034
\(n\): 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.ba 2
3.b odd 2 1 845.2.a.e 2
13.b even 2 1 7605.2.a.bf 2
13.c even 3 2 585.2.j.e 4
15.d odd 2 1 4225.2.a.u 2
39.d odd 2 1 845.2.a.b 2
39.f even 4 2 845.2.c.c 4
39.h odd 6 2 845.2.e.g 4
39.i odd 6 2 65.2.e.a 4
39.k even 12 4 845.2.m.e 8
156.p even 6 2 1040.2.q.n 4
195.e odd 2 1 4225.2.a.y 2
195.x odd 6 2 325.2.e.b 4
195.bl even 12 4 325.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 39.i odd 6 2
325.2.e.b 4 195.x odd 6 2
325.2.o.a 8 195.bl even 12 4
585.2.j.e 4 13.c even 3 2
845.2.a.b 2 39.d odd 2 1
845.2.a.e 2 3.b odd 2 1
845.2.c.c 4 39.f even 4 2
845.2.e.g 4 39.h odd 6 2
845.2.m.e 8 39.k even 12 4
1040.2.q.n 4 156.p even 6 2
4225.2.a.u 2 15.d odd 2 1
4225.2.a.y 2 195.e odd 2 1
7605.2.a.ba 2 1.a even 1 1 trivial
7605.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 71 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 179 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 29 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T + 11 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 80 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
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