Properties

Label 240.4.f.f
Level $240$
Weight $4$
Character orbit 240.f
Analytic conductor $14.160$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \(x^{4} + 21 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + \beta_{2} ) q^{7} -9 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + \beta_{2} ) q^{7} -9 q^{9} + ( 20 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{11} + ( 3 \beta_{1} + 13 \beta_{2} ) q^{13} + ( -14 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{15} + ( 5 \beta_{1} - 17 \beta_{2} ) q^{17} + ( 32 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{19} + ( 12 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{21} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{23} + ( 61 + 9 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} ) q^{25} + 9 \beta_{2} q^{27} + ( 166 + 7 \beta_{1} + 7 \beta_{2} - 14 \beta_{3} ) q^{29} + ( -28 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{31} + ( 9 \beta_{1} - 21 \beta_{2} ) q^{33} + ( 80 + 5 \beta_{1} - 55 \beta_{2} - 10 \beta_{3} ) q^{35} + ( 27 \beta_{1} - 51 \beta_{2} ) q^{37} + ( 120 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -202 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 48 \beta_{1} - 20 \beta_{2} ) q^{43} + ( -9 + 9 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{45} + ( 46 \beta_{1} + 30 \beta_{2} ) q^{47} + ( -41 - 6 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{49} + ( -148 + 5 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} ) q^{51} + ( 41 \beta_{1} + 67 \beta_{2} ) q^{53} + ( 204 - 9 \beta_{1} - 23 \beta_{2} + 24 \beta_{3} ) q^{55} + ( -36 \beta_{1} - 28 \beta_{2} ) q^{57} + ( -92 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{59} + ( 154 - 16 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} ) q^{61} + ( -27 \beta_{1} - 9 \beta_{2} ) q^{63} + ( 248 - 43 \beta_{1} - 31 \beta_{2} - 22 \beta_{3} ) q^{65} + ( -30 \beta_{1} + 122 \beta_{2} ) q^{67} + ( 76 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{69} + ( 48 + 30 \beta_{1} + 30 \beta_{2} - 60 \beta_{3} ) q^{71} + ( -54 \beta_{1} + 222 \beta_{2} ) q^{73} + ( 156 + 39 \beta_{1} - 52 \beta_{2} - 24 \beta_{3} ) q^{75} + ( 54 \beta_{1} - 102 \beta_{2} ) q^{77} + ( 140 - 50 \beta_{1} - 50 \beta_{2} + 100 \beta_{3} ) q^{79} + 81 q^{81} + ( -104 \beta_{1} + 164 \beta_{2} ) q^{83} + ( -128 + 83 \beta_{1} - 129 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -63 \beta_{1} - 159 \beta_{2} ) q^{87} + ( -582 - 24 \beta_{1} - 24 \beta_{2} + 48 \beta_{3} ) q^{89} + ( -528 - 42 \beta_{1} - 42 \beta_{2} + 84 \beta_{3} ) q^{91} + ( 18 \beta_{1} + 26 \beta_{2} ) q^{93} + ( -704 - 76 \beta_{1} - 132 \beta_{2} + 16 \beta_{3} ) q^{95} + ( -120 \beta_{1} + 128 \beta_{2} ) q^{97} + ( -180 + 9 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} - 36 q^{9} + O(q^{10}) \) \( 4 q + 6 q^{5} - 36 q^{9} + 84 q^{11} - 54 q^{15} + 112 q^{19} + 36 q^{21} + 256 q^{25} + 636 q^{29} - 104 q^{31} + 300 q^{35} + 468 q^{39} - 816 q^{41} - 54 q^{45} - 140 q^{49} - 612 q^{51} + 864 q^{55} - 372 q^{59} + 680 q^{61} + 948 q^{65} + 288 q^{69} + 72 q^{71} + 576 q^{75} + 760 q^{79} + 324 q^{81} - 508 q^{85} - 2232 q^{89} - 1944 q^{91} - 2784 q^{95} - 756 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 21 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 31 \nu \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{3} - 33 \nu \)\()/10\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 30 \nu^{2} - \nu + 320 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - \beta_{2} - \beta_{1} - 64\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-31 \beta_{2} - 33 \beta_{1}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
3.70156i
2.70156i
3.70156i
2.70156i
0 3.00000i 0 −8.10469 7.70156i 0 22.2094i 0 −9.00000 0
49.2 0 3.00000i 0 11.1047 1.29844i 0 16.2094i 0 −9.00000 0
49.3 0 3.00000i 0 −8.10469 + 7.70156i 0 22.2094i 0 −9.00000 0
49.4 0 3.00000i 0 11.1047 + 1.29844i 0 16.2094i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
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 240.4.f.f 4
3.b odd 2 1 720.4.f.j 4
4.b odd 2 1 15.4.b.a 4
5.b even 2 1 inner 240.4.f.f 4
5.c odd 4 1 1200.4.a.bn 2
5.c odd 4 1 1200.4.a.bt 2
8.b even 2 1 960.4.f.p 4
8.d odd 2 1 960.4.f.q 4
12.b even 2 1 45.4.b.b 4
15.d odd 2 1 720.4.f.j 4
20.d odd 2 1 15.4.b.a 4
20.e even 4 1 75.4.a.c 2
20.e even 4 1 75.4.a.f 2
40.e odd 2 1 960.4.f.q 4
40.f even 2 1 960.4.f.p 4
60.h even 2 1 45.4.b.b 4
60.l odd 4 1 225.4.a.i 2
60.l odd 4 1 225.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 4.b odd 2 1
15.4.b.a 4 20.d odd 2 1
45.4.b.b 4 12.b even 2 1
45.4.b.b 4 60.h even 2 1
75.4.a.c 2 20.e even 4 1
75.4.a.f 2 20.e even 4 1
225.4.a.i 2 60.l odd 4 1
225.4.a.o 2 60.l odd 4 1
240.4.f.f 4 1.a even 1 1 trivial
240.4.f.f 4 5.b even 2 1 inner
720.4.f.j 4 3.b odd 2 1
720.4.f.j 4 15.d odd 2 1
960.4.f.p 4 8.b even 2 1
960.4.f.p 4 40.f even 2 1
960.4.f.q 4 8.d odd 2 1
960.4.f.q 4 40.e odd 2 1
1200.4.a.bn 2 5.c odd 4 1
1200.4.a.bt 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{4} + 756 T_{7}^{2} + 129600 \)
\( T_{11}^{2} - 42 T_{11} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( 15625 - 750 T - 110 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( 129600 + 756 T^{2} + T^{4} \)
$11$ \( ( 72 - 42 T + T^{2} )^{2} \)
$13$ \( 1327104 + 3780 T^{2} + T^{4} \)
$17$ \( 2483776 + 7252 T^{2} + T^{4} \)
$19$ \( ( -5120 - 56 T + T^{2} )^{2} \)
$23$ \( 6400 + 2464 T^{2} + T^{4} \)
$29$ \( ( 7200 - 318 T + T^{2} )^{2} \)
$31$ \( ( -800 + 52 T + T^{2} )^{2} \)
$37$ \( 41990400 + 106596 T^{2} + T^{4} \)
$41$ \( ( 40140 + 408 T + T^{2} )^{2} \)
$43$ \( 8256266496 + 196128 T^{2} + T^{4} \)
$47$ \( 6186766336 + 189712 T^{2} + T^{4} \)
$53$ \( 813390400 + 218644 T^{2} + T^{4} \)
$59$ \( ( 8280 + 186 T + T^{2} )^{2} \)
$61$ \( ( -65564 - 340 T + T^{2} )^{2} \)
$67$ \( 9419867136 + 341712 T^{2} + T^{4} \)
$71$ \( ( -331776 - 36 T + T^{2} )^{2} \)
$73$ \( 104976000000 + 1126224 T^{2} + T^{4} \)
$79$ \( ( -886400 - 380 T + T^{2} )^{2} \)
$83$ \( 40558737664 + 1371040 T^{2} + T^{4} \)
$89$ \( ( 98820 + 1116 T + T^{2} )^{2} \)
$97$ \( 196199387136 + 1475712 T^{2} + T^{4} \)
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