Properties

Label 240.3.bg.c
Level $240$
Weight $3$
Character orbit 240.bg
Analytic conductor $6.540$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.bg (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{5} + ( 3 - 3 \beta_{2} - 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{5} + ( 3 - 3 \beta_{2} - 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 4 + \beta_{1} - \beta_{3} ) q^{11} + ( 12 + 2 \beta_{1} + 12 \beta_{2} ) q^{13} + ( -6 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{15} + ( -6 + 6 \beta_{2} + 14 \beta_{3} ) q^{17} + ( 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{19} + ( 6 + 3 \beta_{1} - 3 \beta_{3} ) q^{21} + ( 14 - 6 \beta_{1} + 14 \beta_{2} ) q^{23} + ( 4 - 14 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{25} + 3 \beta_{3} q^{27} + ( 9 \beta_{1} - 22 \beta_{2} + 9 \beta_{3} ) q^{29} + ( 20 - 10 \beta_{1} + 10 \beta_{3} ) q^{31} + ( 3 + 4 \beta_{1} + 3 \beta_{2} ) q^{33} + ( 6 + 9 \beta_{1} + 18 \beta_{2} + 7 \beta_{3} ) q^{35} + ( 28 - 28 \beta_{2} - 6 \beta_{3} ) q^{37} + ( 12 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} ) q^{39} + ( -14 + 14 \beta_{1} - 14 \beta_{3} ) q^{41} + ( 2 - 28 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -9 - 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{45} + ( 4 - 4 \beta_{2} - 38 \beta_{3} ) q^{47} + ( -12 \beta_{1} + 19 \beta_{2} - 12 \beta_{3} ) q^{49} + ( -42 - 6 \beta_{1} + 6 \beta_{3} ) q^{51} + ( -30 + 20 \beta_{1} - 30 \beta_{2} ) q^{53} + ( -5 + 15 \beta_{2} + 10 \beta_{3} ) q^{55} + ( -18 + 18 \beta_{2} + 2 \beta_{3} ) q^{57} + ( 21 \beta_{1} + 4 \beta_{2} + 21 \beta_{3} ) q^{59} + ( -6 + 22 \beta_{1} - 22 \beta_{3} ) q^{61} + ( 9 + 6 \beta_{1} + 9 \beta_{2} ) q^{63} + ( -36 - 34 \beta_{1} + 42 \beta_{2} + 18 \beta_{3} ) q^{65} + ( 2 - 2 \beta_{2} - 68 \beta_{3} ) q^{67} + ( 14 \beta_{1} - 18 \beta_{2} + 14 \beta_{3} ) q^{69} + ( -68 + 8 \beta_{1} - 8 \beta_{3} ) q^{71} + ( 27 - 40 \beta_{1} + 27 \beta_{2} ) q^{73} + ( 6 + 4 \beta_{1} - 42 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 18 - 18 \beta_{2} - 14 \beta_{3} ) q^{77} + ( -6 \beta_{1} - 112 \beta_{2} - 6 \beta_{3} ) q^{79} -9 q^{81} + ( -68 - 22 \beta_{1} - 68 \beta_{2} ) q^{83} + ( 18 - 48 \beta_{1} - 96 \beta_{2} - 4 \beta_{3} ) q^{85} + ( -27 + 27 \beta_{2} - 22 \beta_{3} ) q^{87} + ( 4 \beta_{1} + 62 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 84 + 30 \beta_{1} - 30 \beta_{3} ) q^{91} + ( -30 + 20 \beta_{1} - 30 \beta_{2} ) q^{93} + ( -24 - 16 \beta_{1} - 52 \beta_{2} + 22 \beta_{3} ) q^{95} + ( -37 + 37 \beta_{2} ) q^{97} + ( 3 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 12q^{7} + O(q^{10}) \) \( 4q + 4q^{5} + 12q^{7} + 16q^{11} + 48q^{13} - 24q^{15} - 24q^{17} + 24q^{21} + 56q^{23} + 16q^{25} + 80q^{31} + 12q^{33} + 24q^{35} + 112q^{37} - 56q^{41} + 8q^{43} - 36q^{45} + 16q^{47} - 168q^{51} - 120q^{53} - 20q^{55} - 72q^{57} - 24q^{61} + 36q^{63} - 144q^{65} + 8q^{67} - 272q^{71} + 108q^{73} + 24q^{75} + 72q^{77} - 36q^{81} - 272q^{83} + 72q^{85} - 108q^{87} + 336q^{91} - 120q^{93} - 96q^{95} - 148q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 4.67423 1.77526i 0 0.550510 + 0.550510i 0 3.00000i 0
97.2 0 1.22474 1.22474i 0 −2.67423 4.22474i 0 5.44949 + 5.44949i 0 3.00000i 0
193.1 0 −1.22474 1.22474i 0 4.67423 + 1.77526i 0 0.550510 0.550510i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 −2.67423 + 4.22474i 0 5.44949 5.44949i 0 3.00000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bg.c 4
3.b odd 2 1 720.3.bh.g 4
4.b odd 2 1 120.3.u.a 4
5.b even 2 1 1200.3.bg.e 4
5.c odd 4 1 inner 240.3.bg.c 4
5.c odd 4 1 1200.3.bg.e 4
8.b even 2 1 960.3.bg.d 4
8.d odd 2 1 960.3.bg.c 4
12.b even 2 1 360.3.v.b 4
15.e even 4 1 720.3.bh.g 4
20.d odd 2 1 600.3.u.e 4
20.e even 4 1 120.3.u.a 4
20.e even 4 1 600.3.u.e 4
40.i odd 4 1 960.3.bg.d 4
40.k even 4 1 960.3.bg.c 4
60.h even 2 1 1800.3.v.n 4
60.l odd 4 1 360.3.v.b 4
60.l odd 4 1 1800.3.v.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.a 4 4.b odd 2 1
120.3.u.a 4 20.e even 4 1
240.3.bg.c 4 1.a even 1 1 trivial
240.3.bg.c 4 5.c odd 4 1 inner
360.3.v.b 4 12.b even 2 1
360.3.v.b 4 60.l odd 4 1
600.3.u.e 4 20.d odd 2 1
600.3.u.e 4 20.e even 4 1
720.3.bh.g 4 3.b odd 2 1
720.3.bh.g 4 15.e even 4 1
960.3.bg.c 4 8.d odd 2 1
960.3.bg.c 4 40.k even 4 1
960.3.bg.d 4 8.b even 2 1
960.3.bg.d 4 40.i odd 4 1
1200.3.bg.e 4 5.b even 2 1
1200.3.bg.e 4 5.c odd 4 1
1800.3.v.n 4 60.h even 2 1
1800.3.v.n 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 12 T_{7}^{3} + 72 T_{7}^{2} - 72 T_{7} + 36 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( 625 - 100 T - 4 T^{3} + T^{4} \)
$7$ \( 36 - 72 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$11$ \( ( 10 - 8 T + T^{2} )^{2} \)
$13$ \( 76176 - 13248 T + 1152 T^{2} - 48 T^{3} + T^{4} \)
$17$ \( 266256 - 12384 T + 288 T^{2} + 24 T^{3} + T^{4} \)
$19$ \( 44944 + 440 T^{2} + T^{4} \)
$23$ \( 80656 - 15904 T + 1568 T^{2} - 56 T^{3} + T^{4} \)
$29$ \( 4 + 1940 T^{2} + T^{4} \)
$31$ \( ( -200 - 40 T + T^{2} )^{2} \)
$37$ \( 2131600 - 163520 T + 6272 T^{2} - 112 T^{3} + T^{4} \)
$41$ \( ( -980 + 28 T + T^{2} )^{2} \)
$43$ \( 5494336 + 18752 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( 18490000 + 68800 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( 360000 + 72000 T + 7200 T^{2} + 120 T^{3} + T^{4} \)
$59$ \( 6916900 + 5324 T^{2} + T^{4} \)
$61$ \( ( -2868 + 12 T + T^{2} )^{2} \)
$67$ \( 192210496 + 110912 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( 4240 + 136 T + T^{2} )^{2} \)
$73$ \( 11168964 + 360936 T + 5832 T^{2} - 108 T^{3} + T^{4} \)
$79$ \( 151979584 + 25520 T^{2} + T^{4} \)
$83$ \( 60777616 + 2120512 T + 36992 T^{2} + 272 T^{3} + T^{4} \)
$89$ \( 14047504 + 7880 T^{2} + T^{4} \)
$97$ \( ( 2738 + 74 T + T^{2} )^{2} \)
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