Properties

Label 7448.2.a.bo
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $1$
Dimension $7$
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 14 x^{5} + 13 x^{4} + 50 x^{3} - 53 x^{2} - 25 x + 28\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{4} + \beta_{5} ) q^{9} + ( \beta_{1} - \beta_{3} ) q^{11} + ( -1 + \beta_{2} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{15} + ( -1 + \beta_{3} + \beta_{6} ) q^{17} - q^{19} + ( -\beta_{2} - \beta_{4} + \beta_{6} ) q^{23} + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{25} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{27} + ( -\beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{29} + ( 1 - \beta_{2} + \beta_{4} ) q^{31} + ( -3 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{33} + ( -1 - 2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{39} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{41} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{43} + ( -3 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{45} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{47} + ( 1 + \beta_{3} + \beta_{4} - \beta_{6} ) q^{51} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{53} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{55} + \beta_{1} q^{57} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{59} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{61} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{65} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{67} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{69} + ( -1 + 3 \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{71} + ( -7 - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{73} + ( -6 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{75} + ( \beta_{3} - \beta_{5} ) q^{79} + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{81} + ( 3 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{83} + ( -2 + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{85} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{87} + ( -6 - \beta_{1} - \beta_{4} - 2 \beta_{6} ) q^{89} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{93} + \beta_{5} q^{95} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{97} + ( 6 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - q^{3} - q^{5} + 8q^{9} + O(q^{10}) \) \( 7q - q^{3} - q^{5} + 8q^{9} + 3q^{11} - 6q^{13} - 4q^{15} - 10q^{17} - 7q^{19} - 2q^{23} + 8q^{25} + 2q^{27} + 3q^{29} + 6q^{31} - 21q^{33} - 7q^{37} - 2q^{39} - 9q^{41} + q^{43} - 24q^{45} - 15q^{47} + 6q^{51} + 5q^{53} + 6q^{55} + q^{57} + 9q^{59} - 13q^{61} - 6q^{65} - 2q^{67} + 20q^{69} - q^{71} - 42q^{73} - 40q^{75} - 3q^{79} - q^{81} + 12q^{83} - 16q^{85} + 2q^{87} - 41q^{89} - 2q^{93} + q^{95} - 5q^{97} + 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 14 x^{5} + 13 x^{4} + 50 x^{3} - 53 x^{2} - 25 x + 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 7 \nu^{3} + 12 \nu^{2} - \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 14 \nu^{2} + 13 \nu - 12 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 13 \nu^{4} + 2 \nu^{3} - 39 \nu^{2} + 16 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 13 \nu^{4} - 2 \nu^{3} + 41 \nu^{2} - 24 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 15 \nu^{4} - 57 \nu^{2} + 8 \nu + 34 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 10 \beta_{5} + 9 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(2 \beta_{6} + 15 \beta_{5} + 13 \beta_{4} - 9 \beta_{3} + 11 \beta_{2} + 56 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(13 \beta_{6} + 93 \beta_{5} + 78 \beta_{4} - 15 \beta_{3} + 15 \beta_{2} + 53 \beta_{1} + 196\)

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
3.11867
1.84812
1.13946
0.824576
−0.779807
−2.52291
−2.62810
0 −3.11867 0 −2.20110 0 0 0 6.72607 0
1.2 0 −1.84812 0 4.19961 0 0 0 0.415553 0
1.3 0 −1.13946 0 −3.27400 0 0 0 −1.70164 0
1.4 0 −0.824576 0 1.46996 0 0 0 −2.32007 0
1.5 0 0.779807 0 1.35094 0 0 0 −2.39190 0
1.6 0 2.52291 0 −0.138247 0 0 0 3.36506 0
1.7 0 2.62810 0 −2.40717 0 0 0 3.90693 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bo 7
7.b odd 2 1 7448.2.a.bp yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7448.2.a.bo 7 1.a even 1 1 trivial
7448.2.a.bp yes 7 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3}^{7} + T_{3}^{6} - 14 T_{3}^{5} - 13 T_{3}^{4} + 50 T_{3}^{3} + 53 T_{3}^{2} - 25 T_{3} - 28 \)
\( T_{5}^{7} + T_{5}^{6} - 21 T_{5}^{5} - 28 T_{5}^{4} + 91 T_{5}^{3} + 83 T_{5}^{2} - 135 T_{5} - 20 \)
\( T_{11}^{7} - 3 T_{11}^{6} - 32 T_{11}^{5} + 101 T_{11}^{4} + 128 T_{11}^{3} - 327 T_{11}^{2} + 19 T_{11} + 100 \)
\( T_{13}^{7} + 6 T_{13}^{6} - 24 T_{13}^{5} - 102 T_{13}^{4} + 271 T_{13}^{3} + 100 T_{13}^{2} - 272 T_{13} - 128 \)
\( T_{17}^{7} + 10 T_{17}^{6} - 7 T_{17}^{5} - 258 T_{17}^{4} - 341 T_{17}^{3} + 882 T_{17}^{2} + 1772 T_{17} + 640 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( -28 - 25 T + 53 T^{2} + 50 T^{3} - 13 T^{4} - 14 T^{5} + T^{6} + T^{7} \)
$5$ \( -20 - 135 T + 83 T^{2} + 91 T^{3} - 28 T^{4} - 21 T^{5} + T^{6} + T^{7} \)
$7$ \( T^{7} \)
$11$ \( 100 + 19 T - 327 T^{2} + 128 T^{3} + 101 T^{4} - 32 T^{5} - 3 T^{6} + T^{7} \)
$13$ \( -128 - 272 T + 100 T^{2} + 271 T^{3} - 102 T^{4} - 24 T^{5} + 6 T^{6} + T^{7} \)
$17$ \( 640 + 1772 T + 882 T^{2} - 341 T^{3} - 258 T^{4} - 7 T^{5} + 10 T^{6} + T^{7} \)
$19$ \( ( 1 + T )^{7} \)
$23$ \( -20480 - 2340 T + 6338 T^{2} + 1553 T^{3} - 244 T^{4} - 78 T^{5} + 2 T^{6} + T^{7} \)
$29$ \( 2630 - 3541 T - 6307 T^{2} + 2202 T^{3} + 265 T^{4} - 96 T^{5} - 3 T^{6} + T^{7} \)
$31$ \( 800 - 3876 T - 1970 T^{2} + 1659 T^{3} + 276 T^{4} - 75 T^{5} - 6 T^{6} + T^{7} \)
$37$ \( 51278 - 58421 T + 12499 T^{2} + 3997 T^{3} - 808 T^{4} - 133 T^{5} + 7 T^{6} + T^{7} \)
$41$ \( 20228 - 21525 T + 2537 T^{2} + 2422 T^{3} - 369 T^{4} - 86 T^{5} + 9 T^{6} + T^{7} \)
$43$ \( 1280 - 10304 T + 1184 T^{2} + 3600 T^{3} + 40 T^{4} - 119 T^{5} - T^{6} + T^{7} \)
$47$ \( -368216 + 125633 T + 58785 T^{2} - 3505 T^{3} - 2308 T^{4} - 105 T^{5} + 15 T^{6} + T^{7} \)
$53$ \( 650 - 4145 T - 4001 T^{2} + 9268 T^{3} + 761 T^{4} - 214 T^{5} - 5 T^{6} + T^{7} \)
$59$ \( 1337012 - 180079 T - 120567 T^{2} + 14381 T^{3} + 1988 T^{4} - 227 T^{5} - 9 T^{6} + T^{7} \)
$61$ \( 144140 - 119927 T + 17427 T^{2} + 6383 T^{3} - 1152 T^{4} - 129 T^{5} + 13 T^{6} + T^{7} \)
$67$ \( -56000 + 13828 T + 33972 T^{2} + 7021 T^{3} - 612 T^{4} - 173 T^{5} + 2 T^{6} + T^{7} \)
$71$ \( 15700 - 14915 T - 3715 T^{2} + 6161 T^{3} - 810 T^{4} - 235 T^{5} + T^{6} + T^{7} \)
$73$ \( 37216 - 1964 T - 21198 T^{2} - 1777 T^{3} + 2718 T^{4} + 587 T^{5} + 42 T^{6} + T^{7} \)
$79$ \( -448 - 2672 T + 704 T^{2} + 604 T^{3} - 96 T^{4} - 45 T^{5} + 3 T^{6} + T^{7} \)
$83$ \( 19760 - 104036 T - 71912 T^{2} + 2631 T^{3} + 3188 T^{4} - 245 T^{5} - 12 T^{6} + T^{7} \)
$89$ \( -98048 - 224320 T - 135808 T^{2} - 20264 T^{3} + 1340 T^{4} + 533 T^{5} + 41 T^{6} + T^{7} \)
$97$ \( 1035860 - 661021 T + 26923 T^{2} + 24285 T^{3} - 1052 T^{4} - 293 T^{5} + 5 T^{6} + T^{7} \)
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