Properties

Label 7605.2.a.bh
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta + 1) q^{7} + (\beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta + 1) q^{7} + (\beta + 4) q^{8} + \beta q^{10} + (\beta + 3) q^{11} + (2 \beta + 4) q^{14} + 3 \beta q^{16} + (\beta - 3) q^{17} + (2 \beta - 4) q^{19} + (\beta + 2) q^{20} + (4 \beta + 4) q^{22} + (\beta + 3) q^{23} + q^{25} + (4 \beta + 6) q^{28} + ( - 4 \beta + 2) q^{29} - 6 q^{31} + (\beta + 4) q^{32} + ( - 2 \beta + 4) q^{34} + (\beta + 1) q^{35} + ( - \beta - 3) q^{37} + ( - 2 \beta + 8) q^{38} + (\beta + 4) q^{40} + ( - \beta + 3) q^{41} + ( - 2 \beta - 2) q^{43} + (6 \beta + 10) q^{44} + (4 \beta + 4) q^{46} - 4 q^{47} + (3 \beta - 2) q^{49} + \beta q^{50} + (5 \beta - 3) q^{53} + (\beta + 3) q^{55} + (6 \beta + 8) q^{56} + ( - 2 \beta - 16) q^{58} + 4 q^{59} + (\beta - 11) q^{61} - 6 \beta q^{62} + ( - \beta + 4) q^{64} + ( - 2 \beta + 8) q^{67} - 2 q^{68} + (2 \beta + 4) q^{70} + ( - 3 \beta - 1) q^{71} + ( - 2 \beta - 6) q^{73} + ( - 4 \beta - 4) q^{74} + 2 \beta q^{76} + (5 \beta + 7) q^{77} + ( - \beta - 3) q^{79} + 3 \beta q^{80} + (2 \beta - 4) q^{82} + ( - 6 \beta + 2) q^{83} + (\beta - 3) q^{85} + ( - 4 \beta - 8) q^{86} + (8 \beta + 16) q^{88} + ( - 3 \beta + 9) q^{89} + (6 \beta + 10) q^{92} - 4 \beta q^{94} + (2 \beta - 4) q^{95} + ( - 7 \beta + 3) q^{97} + (\beta + 12) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{7} + 9 q^{8} + q^{10} + 7 q^{11} + 10 q^{14} + 3 q^{16} - 5 q^{17} - 6 q^{19} + 5 q^{20} + 12 q^{22} + 7 q^{23} + 2 q^{25} + 16 q^{28} - 12 q^{31} + 9 q^{32} + 6 q^{34} + 3 q^{35} - 7 q^{37} + 14 q^{38} + 9 q^{40} + 5 q^{41} - 6 q^{43} + 26 q^{44} + 12 q^{46} - 8 q^{47} - q^{49} + q^{50} - q^{53} + 7 q^{55} + 22 q^{56} - 34 q^{58} + 8 q^{59} - 21 q^{61} - 6 q^{62} + 7 q^{64} + 14 q^{67} - 4 q^{68} + 10 q^{70} - 5 q^{71} - 14 q^{73} - 12 q^{74} + 2 q^{76} + 19 q^{77} - 7 q^{79} + 3 q^{80} - 6 q^{82} - 2 q^{83} - 5 q^{85} - 20 q^{86} + 40 q^{88} + 15 q^{89} + 26 q^{92} - 4 q^{94} - 6 q^{95} - q^{97} + 25 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 0.438447 1.00000 0 −0.561553 2.43845 0 −1.56155
1.2 2.56155 0 4.56155 1.00000 0 3.56155 6.56155 0 2.56155
\(n\): 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.bh 2
3.b odd 2 1 2535.2.a.p 2
13.b even 2 1 7605.2.a.bc 2
13.d odd 4 2 585.2.b.e 4
39.d odd 2 1 2535.2.a.q 2
39.f even 4 2 195.2.b.c 4
156.l odd 4 2 3120.2.g.n 4
195.j odd 4 2 975.2.h.e 4
195.n even 4 2 975.2.b.f 4
195.u odd 4 2 975.2.h.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.c 4 39.f even 4 2
585.2.b.e 4 13.d odd 4 2
975.2.b.f 4 195.n even 4 2
975.2.h.e 4 195.j odd 4 2
975.2.h.g 4 195.u odd 4 2
2535.2.a.p 2 3.b odd 2 1
2535.2.a.q 2 39.d odd 2 1
3120.2.g.n 4 156.l odd 4 2
7605.2.a.bc 2 13.b even 2 1
7605.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 68 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + T - 106 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 21T + 106 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$79$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 152 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 18 \) Copy content Toggle raw display
$97$ \( T^{2} + T - 208 \) Copy content Toggle raw display
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