Properties

Label 7448.2.a.bn
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.98211824.1
Defining polynomial: \(x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} -\beta_{3} q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} -\beta_{3} q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{11} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{17} - q^{19} + ( -2 \beta_{3} - \beta_{4} ) q^{23} + ( 2 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{25} + ( 1 - \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{27} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 + \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{31} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{33} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{37} + ( 6 - 5 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{39} + ( -4 - 2 \beta_{1} + 3 \beta_{5} ) q^{41} + ( \beta_{2} - \beta_{3} + \beta_{5} ) q^{43} + ( -1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{45} + ( 5 + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{47} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{51} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{53} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{55} + ( -1 + \beta_{1} ) q^{57} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{59} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{61} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{67} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{69} + ( 3 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{71} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{73} + ( 5 + 2 \beta_{2} - \beta_{4} ) q^{75} + ( -6 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{83} + ( 10 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{85} + ( 5 + 3 \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{87} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{89} + ( 2 - 6 \beta_{1} - 4 \beta_{2} - \beta_{4} ) q^{93} + \beta_{3} q^{95} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{97} + ( 8 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{3} + q^{5} + 3q^{9} + O(q^{10}) \) \( 6q + 3q^{3} + q^{5} + 3q^{9} + 3q^{11} + 6q^{13} - 8q^{15} - 2q^{17} - 6q^{19} + 2q^{23} + 5q^{25} + 6q^{27} - q^{29} + 14q^{31} + 19q^{33} - 7q^{37} + 22q^{39} - 21q^{41} + q^{43} - 4q^{45} + 23q^{47} + 2q^{51} - 15q^{53} + 2q^{55} - 3q^{57} + 25q^{59} + 21q^{61} + 6q^{65} + 2q^{67} - 20q^{69} + 13q^{71} - 2q^{73} + 24q^{75} - 17q^{79} - 2q^{81} + 4q^{83} + 40q^{85} + 30q^{87} - q^{89} + 6q^{93} - q^{95} - 13q^{97} + 41q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 5 \nu^{3} + 13 \nu^{2} + 3 \nu - 5 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 10 \nu^{2} + 19 \nu - 4 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} + 7 \nu^{3} - 17 \nu^{2} - 13 \nu + 11 \)\()/2\)
\(\beta_{5}\)\(=\)\( -\nu^{5} + 3 \nu^{4} + 6 \nu^{3} - 14 \nu^{2} - 9 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 7 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(11 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 13 \beta_{2} + 15 \beta_{1} + 20\)
\(\nu^{5}\)\(=\)\(30 \beta_{5} - 13 \beta_{4} + 6 \beta_{3} + 43 \beta_{2} + 64 \beta_{1} + 41\)

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
3.28578
2.15897
0.511631
0.129411
−1.04143
−2.04436
0 −2.28578 0 1.50233 0 0 0 2.22478 0
1.2 0 −1.15897 0 1.74190 0 0 0 −1.65679 0
1.3 0 0.488369 0 −3.51566 0 0 0 −2.76150 0
1.4 0 0.870589 0 0.696873 0 0 0 −2.24208 0
1.5 0 2.04143 0 3.17676 0 0 0 1.16742 0
1.6 0 3.04436 0 −2.60221 0 0 0 6.26815 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
\(2\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 7448.2.a.bn yes 6
7.b odd 2 1 7448.2.a.bm 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7448.2.a.bm 6 7.b odd 2 1
7448.2.a.bn yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3}^{6} - 3 T_{3}^{5} - 6 T_{3}^{4} + 19 T_{3}^{3} + 2 T_{3}^{2} - 19 T_{3} + 7 \)
\( T_{5}^{6} - T_{5}^{5} - 17 T_{5}^{4} + 24 T_{5}^{3} + 59 T_{5}^{2} - 123 T_{5} + 53 \)
\( T_{11}^{6} - 3 T_{11}^{5} - 28 T_{11}^{4} + 121 T_{11}^{3} + 4 T_{11}^{2} - 539 T_{11} + 575 \)
\( T_{13}^{6} - 6 T_{13}^{5} - 40 T_{13}^{4} + 194 T_{13}^{3} + 639 T_{13}^{2} - 1544 T_{13} - 3920 \)
\( T_{17}^{6} + 2 T_{17}^{5} - 55 T_{17}^{4} - 138 T_{17}^{3} + 91 T_{17}^{2} + 322 T_{17} + 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 7 - 19 T + 2 T^{2} + 19 T^{3} - 6 T^{4} - 3 T^{5} + T^{6} \)
$5$ \( 53 - 123 T + 59 T^{2} + 24 T^{3} - 17 T^{4} - T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 575 - 539 T + 4 T^{2} + 121 T^{3} - 28 T^{4} - 3 T^{5} + T^{6} \)
$13$ \( -3920 - 1544 T + 639 T^{2} + 194 T^{3} - 40 T^{4} - 6 T^{5} + T^{6} \)
$17$ \( 148 + 322 T + 91 T^{2} - 138 T^{3} - 55 T^{4} + 2 T^{5} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( -6860 - 1178 T + 1369 T^{2} + 140 T^{3} - 78 T^{4} - 2 T^{5} + T^{6} \)
$29$ \( 9521 - 8729 T + 2112 T^{2} + 147 T^{3} - 90 T^{4} + T^{5} + T^{6} \)
$31$ \( 3148 - 6410 T - 773 T^{2} + 596 T^{3} - 3 T^{4} - 14 T^{5} + T^{6} \)
$37$ \( -7 + 3 T + 93 T^{2} - 66 T^{3} - 27 T^{4} + 7 T^{5} + T^{6} \)
$41$ \( -43133 + 52561 T - 1954 T^{2} - 1921 T^{3} - 6 T^{4} + 21 T^{5} + T^{6} \)
$43$ \( -1472 - 224 T + 544 T^{2} + 52 T^{3} - 51 T^{4} - T^{5} + T^{6} \)
$47$ \( -4855 + 10983 T - 3633 T^{2} - 20 T^{3} + 159 T^{4} - 23 T^{5} + T^{6} \)
$53$ \( 4549 + 3249 T - 1382 T^{2} - 1021 T^{3} - 36 T^{4} + 15 T^{5} + T^{6} \)
$59$ \( -79 - 871 T + 1469 T^{2} - 856 T^{3} + 221 T^{4} - 25 T^{5} + T^{6} \)
$61$ \( -5867 - 2083 T + 2247 T^{2} + 1036 T^{3} + 11 T^{4} - 21 T^{5} + T^{6} \)
$67$ \( 4396 - 20032 T + 11549 T^{2} + 496 T^{3} - 253 T^{4} - 2 T^{5} + T^{6} \)
$71$ \( 1813 - 2249 T - 459 T^{2} + 398 T^{3} + T^{4} - 13 T^{5} + T^{6} \)
$73$ \( -71932 - 53374 T + 15735 T^{2} + 326 T^{3} - 245 T^{4} + 2 T^{5} + T^{6} \)
$79$ \( -35696 + 163568 T + 14268 T^{2} - 3908 T^{3} - 245 T^{4} + 17 T^{5} + T^{6} \)
$83$ \( -32732 + 1600 T + 4927 T^{2} + 180 T^{3} - 133 T^{4} - 4 T^{5} + T^{6} \)
$89$ \( -360640 + 90656 T + 35672 T^{2} - 840 T^{3} - 379 T^{4} + T^{5} + T^{6} \)
$97$ \( 74851 + 33099 T - 547 T^{2} - 1312 T^{3} - 69 T^{4} + 13 T^{5} + T^{6} \)
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