Properties

Label 2240.2.l.a
Level $2240$
Weight $2$
Character orbit 2240.l
Analytic conductor $17.886$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 3 x^{10} + 11 x^{8} + 22 x^{6} + 99 x^{4} + 243 x^{2} + 729\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} -\beta_{4} q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{4} - \beta_{5} - \beta_{10} ) q^{9} +O(q^{10})\) \( q + \beta_{9} q^{3} -\beta_{4} q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{4} - \beta_{5} - \beta_{10} ) q^{9} + ( -\beta_{2} - \beta_{7} - \beta_{8} ) q^{11} + ( 2 + \beta_{10} ) q^{13} + ( -\beta_{3} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{15} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{17} + ( -\beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{19} + \beta_{1} q^{21} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{23} + ( -2 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{25} + ( -\beta_{2} + \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{27} + ( -\beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{29} + 2 \beta_{9} q^{31} + ( \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{33} + \beta_{7} q^{35} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{10} ) q^{37} + ( \beta_{2} - \beta_{7} + \beta_{9} ) q^{39} + 2 q^{41} + ( -2 \beta_{2} + 2 \beta_{7} + 2 \beta_{11} ) q^{43} + ( -2 - 2 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} + \beta_{10} ) q^{45} + ( -\beta_{2} + 2 \beta_{3} - \beta_{7} + \beta_{8} ) q^{47} - q^{49} + ( -\beta_{2} + 2 \beta_{3} - \beta_{7} - \beta_{8} ) q^{51} + 2 \beta_{10} q^{53} + ( -\beta_{2} - 4 \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{55} + 2 \beta_{6} q^{57} + ( \beta_{2} - 2 \beta_{3} + \beta_{7} - 2 \beta_{8} ) q^{59} + ( 4 \beta_{1} - \beta_{4} - \beta_{5} ) q^{61} + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{63} + ( -1 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{65} + ( -2 \beta_{2} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{67} + ( -2 \beta_{1} - 4 \beta_{6} ) q^{69} + ( 2 \beta_{9} - 2 \beta_{11} ) q^{71} + ( -2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{75} + ( -\beta_{4} + \beta_{5} - \beta_{10} ) q^{77} + ( -\beta_{2} + \beta_{7} - 5 \beta_{9} ) q^{79} + ( -1 + 4 \beta_{4} - 4 \beta_{5} ) q^{81} + ( -\beta_{2} + \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{83} + ( -6 - \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{10} ) q^{85} + ( -5 \beta_{2} - 6 \beta_{3} - 5 \beta_{7} + \beta_{8} ) q^{87} + ( 6 + 2 \beta_{10} ) q^{89} + ( 2 \beta_{3} - \beta_{8} ) q^{91} + ( 8 + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{10} ) q^{93} + ( \beta_{2} - 3 \beta_{3} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{95} + ( -\beta_{1} - 3 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -4 \beta_{2} - 2 \beta_{3} - 4 \beta_{7} + 2 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{5} + 24q^{9} + O(q^{10}) \) \( 12q - 4q^{5} + 24q^{9} + 20q^{13} - 12q^{25} + 24q^{37} + 24q^{41} - 48q^{45} - 12q^{49} - 8q^{53} - 12q^{65} - 4q^{77} + 20q^{81} - 56q^{85} + 64q^{89} + 120q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 3 x^{10} + 11 x^{8} + 22 x^{6} + 99 x^{4} + 243 x^{2} + 729\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -5 \nu^{10} - 51 \nu^{8} - 82 \nu^{6} - 587 \nu^{4} - 1773 \nu^{2} - 3888 \)\()/2268\)
\(\beta_{2}\)\(=\)\((\)\( -10 \nu^{11} - 39 \nu^{9} + 25 \nu^{7} - 1048 \nu^{5} - 1593 \nu^{3} - 2673 \nu \)\()/6804\)
\(\beta_{3}\)\(=\)\((\)\( -23 \nu^{11} + 30 \nu^{9} - 37 \nu^{7} + 97 \nu^{5} + 954 \nu^{3} + 2187 \nu \)\()/13608\)
\(\beta_{4}\)\(=\)\((\)\( -23 \nu^{10} - 96 \nu^{8} - 415 \nu^{6} - 155 \nu^{4} - 2952 \nu^{2} - 3483 \)\()/4536\)
\(\beta_{5}\)\(=\)\((\)\( -23 \nu^{10} + 30 \nu^{8} - 37 \nu^{6} + 97 \nu^{4} - 3582 \nu^{2} - 2349 \)\()/4536\)
\(\beta_{6}\)\(=\)\((\)\( 10 \nu^{10} + 39 \nu^{8} - 25 \nu^{6} - 86 \nu^{4} + 459 \nu^{2} + 1539 \)\()/1134\)
\(\beta_{7}\)\(=\)\((\)\( 25 \nu^{11} + 129 \nu^{9} + 32 \nu^{7} + 415 \nu^{5} + 423 \nu^{3} + 4698 \nu \)\()/6804\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} + 3 \nu^{9} + 11 \nu^{7} + 22 \nu^{5} + 99 \nu^{3} + 486 \nu \)\()/243\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{11} + 3 \nu^{9} + 11 \nu^{7} + 22 \nu^{5} + 99 \nu^{3} \)\()/243\)
\(\beta_{10}\)\(=\)\((\)\( 4 \nu^{10} + 3 \nu^{8} + 17 \nu^{6} + 70 \nu^{4} + 117 \nu^{2} + 243 \)\()/324\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{11} + 7 \nu^{7} + 11 \nu^{5} - 96 \nu^{3} - 45 \nu \)\()/216\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + \beta_{8}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} - 2 \beta_{5} - \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{9} - \beta_{7} + 3 \beta_{3}\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{10} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 5 \beta_{1} - 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(4 \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{7} - 8 \beta_{3} - 14 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(\beta_{10} - 5 \beta_{6} + 3 \beta_{5} - 11 \beta_{4} + 4 \beta_{1} + 6\)
\(\nu^{7}\)\(=\)\((\)\(16 \beta_{11} + 21 \beta_{9} + 11 \beta_{8} - 14 \beta_{7} - 12 \beta_{3} + 18 \beta_{2}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-17 \beta_{10} + 34 \beta_{6} + 40 \beta_{5} - 14 \beta_{4} - 19 \beta_{1} - 56\)\()/2\)
\(\nu^{9}\)\(=\)\(3 \beta_{11} + 15 \beta_{9} - 20 \beta_{8} + 54 \beta_{7} + 77 \beta_{3} + 23 \beta_{2}\)
\(\nu^{10}\)\(=\)\((\)\(143 \beta_{10} + 52 \beta_{6} - 32 \beta_{5} + 34 \beta_{4} + 97 \beta_{1} - 2\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-84 \beta_{11} - 11 \beta_{9} + 21 \beta_{8} + 72 \beta_{7} - 748 \beta_{3} - 28 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1569.1
1.55635 0.760115i
1.55635 + 0.760115i
1.02957 1.39283i
1.02957 + 1.39283i
0.517456 1.65295i
0.517456 + 1.65295i
−0.517456 + 1.65295i
−0.517456 1.65295i
−1.02957 + 1.39283i
−1.02957 1.39283i
−1.55635 + 0.760115i
−1.55635 0.760115i
0 −3.11270 0 −2.08433 0.809661i 0 1.00000i 0 6.68890 0
1569.2 0 −3.11270 0 −2.08433 + 0.809661i 0 1.00000i 0 6.68890 0
1569.3 0 −2.05914 0 1.27280 1.83847i 0 1.00000i 0 1.24006 0
1569.4 0 −2.05914 0 1.27280 + 1.83847i 0 1.00000i 0 1.24006 0
1569.5 0 −1.03491 0 −0.188470 2.22811i 0 1.00000i 0 −1.92896 0
1569.6 0 −1.03491 0 −0.188470 + 2.22811i 0 1.00000i 0 −1.92896 0
1569.7 0 1.03491 0 −0.188470 2.22811i 0 1.00000i 0 −1.92896 0
1569.8 0 1.03491 0 −0.188470 + 2.22811i 0 1.00000i 0 −1.92896 0
1569.9 0 2.05914 0 1.27280 1.83847i 0 1.00000i 0 1.24006 0
1569.10 0 2.05914 0 1.27280 + 1.83847i 0 1.00000i 0 1.24006 0
1569.11 0 3.11270 0 −2.08433 0.809661i 0 1.00000i 0 6.68890 0
1569.12 0 3.11270 0 −2.08433 + 0.809661i 0 1.00000i 0 6.68890 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1569.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.l.a 12
4.b odd 2 1 inner 2240.2.l.a 12
5.b even 2 1 2240.2.l.b yes 12
8.b even 2 1 2240.2.l.b yes 12
8.d odd 2 1 2240.2.l.b yes 12
20.d odd 2 1 2240.2.l.b yes 12
40.e odd 2 1 inner 2240.2.l.a 12
40.f even 2 1 inner 2240.2.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.l.a 12 1.a even 1 1 trivial
2240.2.l.a 12 4.b odd 2 1 inner
2240.2.l.a 12 40.e odd 2 1 inner
2240.2.l.a 12 40.f even 2 1 inner
2240.2.l.b yes 12 5.b even 2 1
2240.2.l.b yes 12 8.b even 2 1
2240.2.l.b yes 12 8.d odd 2 1
2240.2.l.b yes 12 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{6} - 15 T_{3}^{4} + 56 T_{3}^{2} - 44 \)
\( T_{13}^{3} - 5 T_{13}^{2} - 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( -44 + 56 T^{2} - 15 T^{4} + T^{6} )^{2} \)
$5$ \( ( 125 + 50 T + 25 T^{2} + 16 T^{3} + 5 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$7$ \( ( 1 + T^{2} )^{6} \)
$11$ \( ( 2704 + 680 T^{2} + 49 T^{4} + T^{6} )^{2} \)
$13$ \( ( 2 - 2 T - 5 T^{2} + T^{3} )^{4} \)
$17$ \( ( 176 + 152 T^{2} + 35 T^{4} + T^{6} )^{2} \)
$19$ \( ( 256 + 164 T^{2} + 28 T^{4} + T^{6} )^{2} \)
$23$ \( ( 3136 + 2160 T^{2} + 92 T^{4} + T^{6} )^{2} \)
$29$ \( ( 85184 + 6160 T^{2} + 139 T^{4} + T^{6} )^{2} \)
$31$ \( ( -2816 + 896 T^{2} - 60 T^{4} + T^{6} )^{2} \)
$37$ \( ( 32 - 64 T - 6 T^{2} + T^{3} )^{4} \)
$41$ \( ( -2 + T )^{12} \)
$43$ \( ( -45056 + 12048 T^{2} - 216 T^{4} + T^{6} )^{2} \)
$47$ \( ( 256 + 352 T^{2} + 65 T^{4} + T^{6} )^{2} \)
$53$ \( ( -112 - 40 T + 2 T^{2} + T^{3} )^{4} \)
$59$ \( ( 38416 + 4116 T^{2} + 128 T^{4} + T^{6} )^{2} \)
$61$ \( ( 180224 + 12644 T^{2} + 212 T^{4} + T^{6} )^{2} \)
$67$ \( ( -1301696 + 36976 T^{2} - 340 T^{4} + T^{6} )^{2} \)
$71$ \( ( -704 + 2608 T^{2} - 196 T^{4} + T^{6} )^{2} \)
$73$ \( ( 45056 + 12048 T^{2} + 216 T^{4} + T^{6} )^{2} \)
$79$ \( ( -790064 + 30616 T^{2} - 331 T^{4} + T^{6} )^{2} \)
$83$ \( ( -240944 + 15156 T^{2} - 232 T^{4} + T^{6} )^{2} \)
$89$ \( ( -16 + 44 T - 16 T^{2} + T^{3} )^{4} \)
$97$ \( ( 1655984 + 46616 T^{2} + 387 T^{4} + T^{6} )^{2} \)
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