Properties

Label 2240.2.g.l
Level $2240$
Weight $2$
Character orbit 2240.g
Analytic conductor $17.886$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} + ( -\beta_{2} - \beta_{5} ) q^{5} -\beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} + ( -\beta_{2} - \beta_{5} ) q^{5} -\beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{13} + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{15} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{19} -\beta_{3} q^{21} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{23} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( \beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{27} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{29} + ( 4 + 2 \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( -\beta_{1} - \beta_{3} ) q^{35} + 6 \beta_{4} q^{37} + ( -\beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{39} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 4 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{45} + ( 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{47} - q^{49} + ( -6 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{53} + ( -2 + \beta_{1} + 3 \beta_{2} - 4 \beta_{4} + \beta_{5} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 2 - 3 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -8 + \beta_{1} + \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{63} + ( -5 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 4 - 6 \beta_{3} ) q^{69} + ( 2 + 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{73} + ( 6 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{75} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{77} + ( -4 + \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{79} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{81} + ( -\beta_{1} + \beta_{2} + 6 \beta_{4} + 6 \beta_{5} ) q^{83} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -3 \beta_{1} + 3 \beta_{2} + 6 \beta_{4} + 3 \beta_{5} ) q^{87} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{91} + ( 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{4} - 2 \beta_{5} ) q^{93} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 7 \beta_{4} - 3 \beta_{5} ) q^{95} + ( -\beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} ) q^{97} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 8q^{9} + O(q^{10}) \) \( 6q - 8q^{9} - 14q^{11} - 18q^{15} + 8q^{19} + 2q^{21} - 10q^{25} + 6q^{29} + 20q^{31} + 2q^{35} + 10q^{39} + 36q^{41} + 28q^{45} - 6q^{49} - 42q^{51} - 12q^{55} + 12q^{59} - 48q^{61} - 22q^{65} + 36q^{69} + 8q^{71} + 40q^{75} - 34q^{79} + 30q^{81} - 14q^{85} - 10q^{91} - 4q^{95} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.32001 + 1.32001i
0.432320 0.432320i
−1.75233 + 1.75233i
−1.75233 1.75233i
0.432320 + 0.432320i
1.32001 1.32001i
0 3.12489i 0 −1.32001 1.80487i 0 1.00000i 0 −6.76491 0
449.2 0 1.76156i 0 −0.432320 2.19388i 0 1.00000i 0 −0.103084 0
449.3 0 0.363328i 0 1.75233 + 1.38900i 0 1.00000i 0 2.86799 0
449.4 0 0.363328i 0 1.75233 1.38900i 0 1.00000i 0 2.86799 0
449.5 0 1.76156i 0 −0.432320 + 2.19388i 0 1.00000i 0 −0.103084 0
449.6 0 3.12489i 0 −1.32001 + 1.80487i 0 1.00000i 0 −6.76491 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.g.l 6
4.b odd 2 1 2240.2.g.m 6
5.b even 2 1 inner 2240.2.g.l 6
8.b even 2 1 280.2.g.b 6
8.d odd 2 1 560.2.g.f 6
20.d odd 2 1 2240.2.g.m 6
24.f even 2 1 5040.2.t.y 6
24.h odd 2 1 2520.2.t.g 6
40.e odd 2 1 560.2.g.f 6
40.f even 2 1 280.2.g.b 6
40.i odd 4 1 1400.2.a.s 3
40.i odd 4 1 1400.2.a.t 3
40.k even 4 1 2800.2.a.bq 3
40.k even 4 1 2800.2.a.br 3
56.h odd 2 1 1960.2.g.c 6
120.i odd 2 1 2520.2.t.g 6
120.m even 2 1 5040.2.t.y 6
280.c odd 2 1 1960.2.g.c 6
280.s even 4 1 9800.2.a.cd 3
280.s even 4 1 9800.2.a.cg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 8.b even 2 1
280.2.g.b 6 40.f even 2 1
560.2.g.f 6 8.d odd 2 1
560.2.g.f 6 40.e odd 2 1
1400.2.a.s 3 40.i odd 4 1
1400.2.a.t 3 40.i odd 4 1
1960.2.g.c 6 56.h odd 2 1
1960.2.g.c 6 280.c odd 2 1
2240.2.g.l 6 1.a even 1 1 trivial
2240.2.g.l 6 5.b even 2 1 inner
2240.2.g.m 6 4.b odd 2 1
2240.2.g.m 6 20.d odd 2 1
2520.2.t.g 6 24.h odd 2 1
2520.2.t.g 6 120.i odd 2 1
2800.2.a.bq 3 40.k even 4 1
2800.2.a.br 3 40.k even 4 1
5040.2.t.y 6 24.f even 2 1
5040.2.t.y 6 120.m even 2 1
9800.2.a.cd 3 280.s even 4 1
9800.2.a.cg 3 280.s even 4 1

Hecke kernels

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

\( T_{3}^{6} + 13 T_{3}^{4} + 32 T_{3}^{2} + 4 \)
\( T_{11}^{3} + 7 T_{11}^{2} + 8 T_{11} - 8 \)
\( T_{19}^{3} - 4 T_{19}^{2} - 14 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 4 + 32 T^{2} + 13 T^{4} + T^{6} \)
$5$ \( 125 + 25 T^{2} - 8 T^{3} + 5 T^{4} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( -8 + 8 T + 7 T^{2} + T^{3} )^{2} \)
$13$ \( 11236 + 1544 T^{2} + 69 T^{4} + T^{6} \)
$17$ \( 400 + 536 T^{2} + 49 T^{4} + T^{6} \)
$19$ \( ( -8 - 14 T - 4 T^{2} + T^{3} )^{2} \)
$23$ \( 18496 + 2416 T^{2} + 92 T^{4} + T^{6} \)
$29$ \( ( 108 - 72 T - 3 T^{2} + T^{3} )^{2} \)
$31$ \( ( 80 + 8 T - 10 T^{2} + T^{3} )^{2} \)
$37$ \( ( 36 + T^{2} )^{3} \)
$41$ \( ( 88 + 68 T - 18 T^{2} + T^{3} )^{2} \)
$43$ \( 4096 + 1600 T^{2} + 80 T^{4} + T^{6} \)
$47$ \( 53824 + 6480 T^{2} + 177 T^{4} + T^{6} \)
$53$ \( 222784 + 11376 T^{2} + 188 T^{4} + T^{6} \)
$59$ \( ( -44 - 78 T - 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( 440 + 182 T + 24 T^{2} + T^{3} )^{2} \)
$67$ \( 262144 + 14336 T^{2} + 228 T^{4} + T^{6} \)
$71$ \( ( 64 - 20 T - 4 T^{2} + T^{3} )^{2} \)
$73$ \( 18496 + 2416 T^{2} + 92 T^{4} + T^{6} \)
$79$ \( ( -548 - 32 T + 17 T^{2} + T^{3} )^{2} \)
$83$ \( 678976 + 39940 T^{2} + 428 T^{4} + T^{6} \)
$89$ \( ( 464 - 172 T + T^{3} )^{2} \)
$97$ \( 1936 + 1048 T^{2} + 113 T^{4} + T^{6} \)
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