Properties

Label 7600.2.a.ch
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 10 x^{4} + 16 x^{3} + 15 x^{2} - 14 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3800)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{4} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{4} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{13} + ( -3 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{17} + q^{19} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{23} + ( -1 - 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{27} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} ) q^{29} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{31} + ( -\beta_{2} + \beta_{4} ) q^{33} + ( -2 + 3 \beta_{1} - \beta_{5} ) q^{37} + ( -4 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{39} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{43} + ( -1 - \beta_{2} + \beta_{4} ) q^{47} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{49} + ( -6 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{51} + ( 1 - 3 \beta_{3} - \beta_{5} ) q^{53} -\beta_{1} q^{57} + ( -1 + 4 \beta_{1} - \beta_{5} ) q^{59} + ( 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} + ( -3 - 6 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{63} + ( 2 - \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{67} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{69} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{73} + ( -6 + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{77} + ( 3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} ) q^{79} + ( 5 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( -5 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{83} + ( 5 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{87} + ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{89} + ( -7 + \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{91} + ( 7 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{3} - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 6q - 2q^{3} - 2q^{7} + 6q^{9} - 3q^{11} - q^{13} - 14q^{17} + 6q^{19} + 15q^{21} - 12q^{23} - 8q^{27} + 9q^{29} - 5q^{31} + 2q^{33} - 8q^{37} - 12q^{39} + 3q^{41} - 15q^{43} - 4q^{47} + 22q^{49} - 33q^{51} + 13q^{53} - 2q^{57} + 9q^{61} - 21q^{63} + 3q^{67} - 11q^{69} - 19q^{71} + 3q^{73} - 36q^{77} + 16q^{79} + 26q^{81} - 31q^{83} + 25q^{87} + 14q^{89} - 42q^{91} + 39q^{93} - 11q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 10 x^{4} + 16 x^{3} + 15 x^{2} - 14 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 9 \nu^{2} + 6 \nu + 5 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 15 \nu^{2} + 6 \nu - 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 10 \nu^{3} + 15 \nu^{2} + 15 \nu - 7 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + 9 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 12 \beta_{1} + 32\)
\(\nu^{5}\)\(=\)\(-22 \beta_{5} + 24 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 84 \beta_{1} + 21\)

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
3.26143
1.93590
0.848258
−0.185519
−1.08999
−2.77008
0 −3.26143 0 0 0 −4.07225 0 7.63693 0
1.2 0 −1.93590 0 0 0 1.24708 0 0.747704 0
1.3 0 −0.848258 0 0 0 −1.74484 0 −2.28046 0
1.4 0 0.185519 0 0 0 4.45651 0 −2.96558 0
1.5 0 1.08999 0 0 0 −4.19727 0 −1.81192 0
1.6 0 2.77008 0 0 0 2.31077 0 4.67334 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.ch 6
4.b odd 2 1 3800.2.a.bc yes 6
5.b even 2 1 7600.2.a.cl 6
20.d odd 2 1 3800.2.a.ba 6
20.e even 4 2 3800.2.d.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 20.d odd 2 1
3800.2.a.bc yes 6 4.b odd 2 1
3800.2.d.q 12 20.e even 4 2
7600.2.a.ch 6 1.a even 1 1 trivial
7600.2.a.cl 6 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{6} + 2 T_{3}^{5} - 10 T_{3}^{4} - 16 T_{3}^{3} + 15 T_{3}^{2} + 14 T_{3} - 3 \)
\( T_{7}^{6} + 2 T_{7}^{5} - 30 T_{7}^{4} - 48 T_{7}^{3} + 223 T_{7}^{2} + 154 T_{7} - 383 \)
\( T_{11}^{6} + 3 T_{11}^{5} - 40 T_{11}^{4} - 139 T_{11}^{3} + 10 T_{11}^{2} + 208 T_{11} - 88 \)
\( T_{13}^{6} + T_{13}^{5} - 65 T_{13}^{4} - 15 T_{13}^{3} + 883 T_{13}^{2} - 1621 T_{13} + 825 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( -3 + 14 T + 15 T^{2} - 16 T^{3} - 10 T^{4} + 2 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( -383 + 154 T + 223 T^{2} - 48 T^{3} - 30 T^{4} + 2 T^{5} + T^{6} \)
$11$ \( -88 + 208 T + 10 T^{2} - 139 T^{3} - 40 T^{4} + 3 T^{5} + T^{6} \)
$13$ \( 825 - 1621 T + 883 T^{2} - 15 T^{3} - 65 T^{4} + T^{5} + T^{6} \)
$17$ \( 3147 + 1000 T - 989 T^{2} - 342 T^{3} + 22 T^{4} + 14 T^{5} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( -2487 + 3914 T - 659 T^{2} - 602 T^{3} - 26 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( 21951 - 12285 T + 315 T^{2} + 705 T^{3} - 67 T^{4} - 9 T^{5} + T^{6} \)
$31$ \( 11000 + 13500 T + 3220 T^{2} - 599 T^{3} - 136 T^{4} + 5 T^{5} + T^{6} \)
$37$ \( -7365 + 6089 T + 582 T^{2} - 737 T^{3} - 90 T^{4} + 8 T^{5} + T^{6} \)
$41$ \( -113472 - 13200 T + 8392 T^{2} + 469 T^{3} - 180 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( 6120 - 1776 T - 3142 T^{2} - 695 T^{3} + 13 T^{4} + 15 T^{5} + T^{6} \)
$47$ \( 1791 + 2115 T + 452 T^{2} - 191 T^{3} - 52 T^{4} + 4 T^{5} + T^{6} \)
$53$ \( 9285 - 43846 T + 11490 T^{2} + 1891 T^{3} - 192 T^{4} - 13 T^{5} + T^{6} \)
$59$ \( -32328 + 728 T + 5938 T^{2} - 255 T^{3} - 193 T^{4} + T^{6} \)
$61$ \( 12424 + 17080 T + 6598 T^{2} + 443 T^{3} - 132 T^{4} - 9 T^{5} + T^{6} \)
$67$ \( -136033 - 3443 T + 9859 T^{2} + 185 T^{3} - 199 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( -27576 - 44984 T - 22606 T^{2} - 3491 T^{3} - 75 T^{4} + 19 T^{5} + T^{6} \)
$73$ \( -741033 - 51021 T + 27869 T^{2} + 821 T^{3} - 305 T^{4} - 3 T^{5} + T^{6} \)
$79$ \( -553536 - 208816 T + 15944 T^{2} + 4505 T^{3} - 285 T^{4} - 16 T^{5} + T^{6} \)
$83$ \( -71160 - 41588 T - 5676 T^{2} + 937 T^{3} + 326 T^{4} + 31 T^{5} + T^{6} \)
$89$ \( 3240 - 1836 T - 1030 T^{2} + 727 T^{3} - 67 T^{4} - 14 T^{5} + T^{6} \)
$97$ \( -288792 + 44292 T + 13744 T^{2} - 1473 T^{3} - 208 T^{4} + 11 T^{5} + T^{6} \)
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