Properties

Label 240.3.c.c
Level $240$
Weight $3$
Character orbit 240.c
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-17})\)
Defining polynomial: \( x^{4} + 16x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2}) q^{5} + ( - \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2}) q^{5} + ( - \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} - 8) q^{9} + 4 \beta_{3} q^{11} + ( - 2 \beta_{3} + 9 \beta_{2} + \beta_1 - 2) q^{15} + 8 \beta_{2} q^{17} - 12 q^{19} + ( - \beta_{3} + 17) q^{21} + 17 \beta_{2} q^{23} + ( - 4 \beta_{2} - 8 \beta_1 - 9) q^{25} + (9 \beta_{2} - 7 \beta_1) q^{27} + 32 q^{31} + (36 \beta_{2} + 4 \beta_1) q^{33} + (4 \beta_{3} - 17 \beta_{2}) q^{35} + (4 \beta_{2} + 8 \beta_1) q^{37} + 14 \beta_{3} q^{41} + (7 \beta_{2} + 14 \beta_1) q^{43} + ( - 8 \beta_{3} - 18 \beta_{2} - 4 \beta_1 - 17) q^{45} - 25 \beta_{2} q^{47} + 15 q^{49} + ( - 8 \beta_{3} - 8) q^{51} - 48 \beta_{2} q^{53} + ( - 8 \beta_{2} - 16 \beta_1 - 68) q^{55} - 12 \beta_1 q^{57} + 4 \beta_{3} q^{59} - 16 q^{61} + ( - 9 \beta_{2} + 16 \beta_1) q^{63} + (\beta_{2} + 2 \beta_1) q^{67} + ( - 17 \beta_{3} - 17) q^{69} + (20 \beta_{2} + 40 \beta_1) q^{73} + ( - 4 \beta_{3} - 9 \beta_1 + 68) q^{75} - 68 \beta_{2} q^{77} + 72 q^{79} + ( - 16 \beta_{3} + 47) q^{81} - 31 \beta_{2} q^{83} + ( - 8 \beta_{2} - 16 \beta_1 + 32) q^{85} + 16 \beta_{3} q^{89} + 32 \beta_1 q^{93} + ( - 12 \beta_{3} - 24 \beta_{2}) q^{95} + ( - 28 \beta_{2} - 56 \beta_1) q^{97} + ( - 32 \beta_{3} - 68) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{9} - 8 q^{15} - 48 q^{19} + 68 q^{21} - 36 q^{25} + 128 q^{31} - 68 q^{45} + 60 q^{49} - 32 q^{51} - 272 q^{55} - 64 q^{61} - 68 q^{69} + 272 q^{75} + 288 q^{79} + 188 q^{81} + 128 q^{85} - 272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 16x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 7\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{2} - 7\beta_1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.707107 2.91548i
−0.707107 + 2.91548i
0.707107 2.91548i
0.707107 + 2.91548i
0 −0.707107 2.91548i 0 2.82843 + 4.12311i 0 5.83095i 0 −8.00000 + 4.12311i 0
209.2 0 −0.707107 + 2.91548i 0 2.82843 4.12311i 0 5.83095i 0 −8.00000 4.12311i 0
209.3 0 0.707107 2.91548i 0 −2.82843 4.12311i 0 5.83095i 0 −8.00000 4.12311i 0
209.4 0 0.707107 + 2.91548i 0 −2.82843 + 4.12311i 0 5.83095i 0 −8.00000 + 4.12311i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.c.c 4
3.b odd 2 1 inner 240.3.c.c 4
4.b odd 2 1 30.3.b.a 4
5.b even 2 1 inner 240.3.c.c 4
5.c odd 4 2 1200.3.l.t 4
8.b even 2 1 960.3.c.e 4
8.d odd 2 1 960.3.c.f 4
12.b even 2 1 30.3.b.a 4
15.d odd 2 1 inner 240.3.c.c 4
15.e even 4 2 1200.3.l.t 4
20.d odd 2 1 30.3.b.a 4
20.e even 4 2 150.3.d.d 4
24.f even 2 1 960.3.c.f 4
24.h odd 2 1 960.3.c.e 4
36.f odd 6 2 810.3.j.c 8
36.h even 6 2 810.3.j.c 8
40.e odd 2 1 960.3.c.f 4
40.f even 2 1 960.3.c.e 4
60.h even 2 1 30.3.b.a 4
60.l odd 4 2 150.3.d.d 4
120.i odd 2 1 960.3.c.e 4
120.m even 2 1 960.3.c.f 4
180.n even 6 2 810.3.j.c 8
180.p odd 6 2 810.3.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 4.b odd 2 1
30.3.b.a 4 12.b even 2 1
30.3.b.a 4 20.d odd 2 1
30.3.b.a 4 60.h even 2 1
150.3.d.d 4 20.e even 4 2
150.3.d.d 4 60.l odd 4 2
240.3.c.c 4 1.a even 1 1 trivial
240.3.c.c 4 3.b odd 2 1 inner
240.3.c.c 4 5.b even 2 1 inner
240.3.c.c 4 15.d odd 2 1 inner
810.3.j.c 8 36.f odd 6 2
810.3.j.c 8 36.h even 6 2
810.3.j.c 8 180.n even 6 2
810.3.j.c 8 180.p odd 6 2
960.3.c.e 4 8.b even 2 1
960.3.c.e 4 24.h odd 2 1
960.3.c.e 4 40.f even 2 1
960.3.c.e 4 120.i odd 2 1
960.3.c.f 4 8.d odd 2 1
960.3.c.f 4 24.f even 2 1
960.3.c.f 4 40.e odd 2 1
960.3.c.f 4 120.m even 2 1
1200.3.l.t 4 5.c odd 4 2
1200.3.l.t 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} + 34 \) Copy content Toggle raw display
\( T_{17}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 16T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 18T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 34)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$19$ \( (T + 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 578)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 544)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3332)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1666)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1250)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4608)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$61$ \( (T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 34)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 13600)^{2} \) Copy content Toggle raw display
$79$ \( (T - 72)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1922)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4352)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 26656)^{2} \) Copy content Toggle raw display
show more
show less