Properties

Label 7605.2.a.cj
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} - q^{5} + (\beta_{3} - \beta_{2} - 3) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} - q^{5} + (\beta_{3} - \beta_{2} - 3) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} - \beta_1 q^{10} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + (\beta_{2} - 3 \beta_1 + 1) q^{14} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{16} + (\beta_{3} - 2 \beta_{2}) q^{17} + ( - \beta_{2} - 4) q^{19} + ( - \beta_{2} - \beta_1) q^{20} + ( - \beta_{3} + \beta_1 + 3) q^{22} + ( - \beta_{3} - 2 \beta_1 + 4) q^{23} + q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{29} + (2 \beta_{2} - 2) q^{31} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{32} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{34} + ( - \beta_{3} + \beta_{2} + 3) q^{35} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{37} + ( - \beta_{3} - 5 \beta_1 + 1) q^{38} + ( - \beta_{3} - \beta_{2} - 1) q^{40} + ( - \beta_{2} + 2) q^{41} + ( - \beta_{3} + 2 \beta_{2}) q^{43} + (3 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{44} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{46} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 6) q^{49} + \beta_1 q^{50} + (2 \beta_1 + 2) q^{53} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{55} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 3) q^{56} + ( - 2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 2) q^{58} + (\beta_{2} - 2 \beta_1 + 4) q^{59} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 7) q^{61} + (2 \beta_{3} - 2) q^{62} + ( - 4 \beta_{3} + 2 \beta_{2} + 3) q^{64} + ( - \beta_{3} + \beta_{2} - 7) q^{67} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{68} + ( - \beta_{2} + 3 \beta_1 - 1) q^{70} + (6 \beta_{3} - 3 \beta_{2} - 2) q^{71} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{73} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{74} + ( - \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 2) q^{76} + (3 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 4) q^{77} + (2 \beta_{3} - 6 \beta_1) q^{79} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{80} + ( - \beta_{3} + \beta_1 + 1) q^{82} + (2 \beta_{3} + 2 \beta_{2} - 4) q^{83} + ( - \beta_{3} + 2 \beta_{2}) q^{85} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{86} + (3 \beta_{3} + 2 \beta_{2} - 6) q^{88} + (2 \beta_{3} + \beta_{2} - 8 \beta_1) q^{89} + ( - 2 \beta_{3} - 3 \beta_1 - 3) q^{92} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 6) q^{94} + (\beta_{2} + 4) q^{95} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{97} + ( - 6 \beta_{2} + 4 \beta_1 - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 10 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 10 q^{7} + 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 16 q^{19} - 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} - 8 q^{28} - 8 q^{29} - 8 q^{31} + 4 q^{32} + 4 q^{34} + 10 q^{35} + 2 q^{37} - 8 q^{38} - 6 q^{40} + 8 q^{41} - 2 q^{43} + 12 q^{44} - 16 q^{46} + 8 q^{47} + 12 q^{49} + 2 q^{50} + 12 q^{53} - 12 q^{56} - 22 q^{58} + 12 q^{59} + 28 q^{61} - 4 q^{62} + 4 q^{64} - 30 q^{67} - 14 q^{68} + 2 q^{70} + 4 q^{71} + 8 q^{73} + 10 q^{74} - 20 q^{76} - 18 q^{77} - 8 q^{79} - 2 q^{80} + 4 q^{82} - 12 q^{83} - 2 q^{85} - 4 q^{86} - 18 q^{88} - 12 q^{89} - 22 q^{92} - 32 q^{94} + 16 q^{95} - 2 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) 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
−1.49551
−0.219687
1.21969
2.49551
−1.49551 0 0.236543 −1.00000 0 −4.82684 2.63726 0 1.49551
1.2 −0.219687 0 −1.95174 −1.00000 0 0.332247 0.868145 0 0.219687
1.3 1.21969 0 −0.512364 −1.00000 0 −3.60020 −3.06430 0 −1.21969
1.4 2.49551 0 4.22756 −1.00000 0 −1.90521 5.55889 0 −2.49551
\(n\): 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.cj 4
3.b odd 2 1 845.2.a.l 4
13.b even 2 1 7605.2.a.cf 4
13.f odd 12 2 585.2.bu.c 8
15.d odd 2 1 4225.2.a.bl 4
39.d odd 2 1 845.2.a.m 4
39.f even 4 2 845.2.c.g 8
39.h odd 6 2 845.2.e.m 8
39.i odd 6 2 845.2.e.n 8
39.k even 12 2 65.2.m.a 8
39.k even 12 2 845.2.m.g 8
156.v odd 12 2 1040.2.da.b 8
195.e odd 2 1 4225.2.a.bi 4
195.bc odd 12 2 325.2.m.b 8
195.bh even 12 2 325.2.n.d 8
195.bn odd 12 2 325.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 39.k even 12 2
325.2.m.b 8 195.bc odd 12 2
325.2.m.c 8 195.bn odd 12 2
325.2.n.d 8 195.bh even 12 2
585.2.bu.c 8 13.f odd 12 2
845.2.a.l 4 3.b odd 2 1
845.2.a.m 4 39.d odd 2 1
845.2.c.g 8 39.f even 4 2
845.2.e.m 8 39.h odd 6 2
845.2.e.n 8 39.i odd 6 2
845.2.m.g 8 39.k even 12 2
1040.2.da.b 8 156.v odd 12 2
4225.2.a.bi 4 195.e odd 2 1
4225.2.a.bl 4 15.d odd 2 1
7605.2.a.cf 4 13.b even 2 1
7605.2.a.cj 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 10T_{7}^{3} + 30T_{7}^{2} + 22T_{7} - 11 \) Copy content Toggle raw display
\( T_{11}^{4} - 30T_{11}^{2} + 33 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 3 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + 30 T^{2} + 22 T - 11 \) Copy content Toggle raw display
$11$ \( T^{4} - 30T^{2} + 33 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 18 T^{2} + 10 T + 13 \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T + 13)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + 6 T^{2} + 146 T - 299 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} - 18 T^{2} - 40 T + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} - 54 T^{2} - 38 T + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} - 18 T^{2} - 10 T + 13 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} - 72 T^{2} + \cdots - 1328 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 36 T^{2} - 48 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 30 T^{2} - 12 T - 3 \) Copy content Toggle raw display
$61$ \( T^{4} - 28 T^{3} + 258 T^{2} + \cdots + 1261 \) Copy content Toggle raw display
$67$ \( T^{4} + 30 T^{3} + 330 T^{2} + \cdots + 2769 \) Copy content Toggle raw display
$71$ \( T^{4} - 4 T^{3} - 210 T^{2} + \cdots + 10477 \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} - 84 T^{2} + \cdots - 1712 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 132 T^{2} + \cdots + 4432 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} - 24 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} - 234 T^{2} + \cdots + 8853 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} - 90 T^{2} + 374 T - 443 \) Copy content Toggle raw display
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