Properties

Label 7600.2.a.cf
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 4 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 1 - \beta_{1} ) q^{7} + ( 2 - \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 1 - \beta_{1} ) q^{7} + ( 2 - \beta_{3} ) q^{9} + ( -1 - \beta_{1} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} ) q^{13} + ( -2 - 2 \beta_{2} ) q^{17} - q^{19} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{21} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{27} + ( 2 + 2 \beta_{2} ) q^{29} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{31} + ( -2 - \beta_{1} + \beta_{3} ) q^{33} + ( 1 - \beta_{2} - \beta_{3} ) q^{37} + ( 3 - \beta_{1} ) q^{39} + ( 4 - \beta_{1} + \beta_{3} ) q^{41} + ( 1 - \beta_{1} ) q^{43} + ( -3 + \beta_{1} - 2 \beta_{3} ) q^{47} + ( 5 - 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 10 + 2 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 3 + \beta_{2} + \beta_{3} ) q^{53} + \beta_{2} q^{57} + ( -\beta_{1} - \beta_{3} ) q^{59} + ( 5 + 2 \beta_{1} - \beta_{3} ) q^{61} + ( 7 + \beta_{1} + 4 \beta_{2} ) q^{63} + ( -6 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{67} + ( 8 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{71} + ( -6 + 2 \beta_{2} ) q^{73} + ( 10 + 2 \beta_{3} ) q^{77} + ( 4 - \beta_{1} - \beta_{3} ) q^{79} + ( 4 + 2 \beta_{1} - \beta_{3} ) q^{81} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{83} + ( -10 - 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 8 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 10 + 2 \beta_{1} - 4 \beta_{3} ) q^{93} + ( -7 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} + ( 3 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{7} + 8q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{7} + 8q^{9} - 4q^{11} - 2q^{13} - 4q^{17} - 4q^{19} - 4q^{21} - 8q^{23} - 4q^{27} + 4q^{29} - 4q^{31} - 8q^{33} + 6q^{37} + 12q^{39} + 16q^{41} + 4q^{43} - 12q^{47} + 20q^{49} + 36q^{51} + 10q^{53} - 2q^{57} + 20q^{61} + 20q^{63} - 18q^{67} + 28q^{69} + 20q^{71} - 28q^{73} + 40q^{77} + 16q^{79} + 16q^{81} - 36q^{87} + 4q^{89} + 36q^{91} + 40q^{93} - 30q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 2 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 11\)\()/2\)

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
−0.552409
2.78165
−1.51658
1.28734
0 −2.87834 0 0 0 3.10482 0 5.28487 0
1.2 0 0.296842 0 0 0 −3.56331 0 −2.91188 0
1.3 0 1.53844 0 0 0 5.03316 0 −0.633188 0
1.4 0 3.04306 0 0 0 −0.574672 0 6.26020 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 7600.2.a.cf 4
4.b odd 2 1 475.2.a.i 4
5.b even 2 1 1520.2.a.t 4
12.b even 2 1 4275.2.a.bo 4
20.d odd 2 1 95.2.a.b 4
20.e even 4 2 475.2.b.e 8
40.e odd 2 1 6080.2.a.cc 4
40.f even 2 1 6080.2.a.ch 4
60.h even 2 1 855.2.a.m 4
76.d even 2 1 9025.2.a.bf 4
140.c even 2 1 4655.2.a.y 4
380.d even 2 1 1805.2.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 20.d odd 2 1
475.2.a.i 4 4.b odd 2 1
475.2.b.e 8 20.e even 4 2
855.2.a.m 4 60.h even 2 1
1520.2.a.t 4 5.b even 2 1
1805.2.a.p 4 380.d even 2 1
4275.2.a.bo 4 12.b even 2 1
4655.2.a.y 4 140.c even 2 1
6080.2.a.cc 4 40.e odd 2 1
6080.2.a.ch 4 40.f even 2 1
7600.2.a.cf 4 1.a even 1 1 trivial
9025.2.a.bf 4 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{4} - 2 T_{3}^{3} - 8 T_{3}^{2} + 16 T_{3} - 4 \)
\( T_{7}^{4} - 4 T_{7}^{3} - 16 T_{7}^{2} + 48 T_{7} + 32 \)
\( T_{11}^{4} + 4 T_{11}^{3} - 16 T_{11}^{2} - 32 T_{11} + 48 \)
\( T_{13}^{4} + 2 T_{13}^{3} - 24 T_{13}^{2} - 32 T_{13} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( -4 + 16 T - 8 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 32 + 48 T - 16 T^{2} - 4 T^{3} + T^{4} \)
$11$ \( 48 - 32 T - 16 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( 20 - 32 T - 24 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( 48 - 16 T - 32 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 288 - 176 T - 24 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 48 + 16 T - 32 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( -640 - 512 T - 80 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( 4 + 40 T - 24 T^{2} - 6 T^{3} + T^{4} \)
$41$ \( -240 + 32 T + 56 T^{2} - 16 T^{3} + T^{4} \)
$43$ \( 32 + 48 T - 16 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 1056 - 656 T - 64 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( -348 + 184 T - 10 T^{3} + T^{4} \)
$59$ \( -192 + 224 T - 64 T^{2} + T^{4} \)
$61$ \( -2656 + 688 T + 56 T^{2} - 20 T^{3} + T^{4} \)
$67$ \( -1076 - 488 T + 8 T^{2} + 18 T^{3} + T^{4} \)
$71$ \( -4224 + 1024 T + 32 T^{2} - 20 T^{3} + T^{4} \)
$73$ \( 176 + 784 T + 256 T^{2} + 28 T^{3} + T^{4} \)
$79$ \( -1856 + 480 T + 32 T^{2} - 16 T^{3} + T^{4} \)
$83$ \( 480 - 112 T - 72 T^{2} + T^{4} \)
$89$ \( 240 - 176 T - 144 T^{2} - 4 T^{3} + T^{4} \)
$97$ \( -1388 + 8 T + 224 T^{2} + 30 T^{3} + T^{4} \)
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