Properties

Label 480.2.b.b
Level $480$
Weight $2$
Character orbit 480.b
Analytic conductor $3.833$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(431,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1649659456.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + \beta_{2} q^{7} + ( - \beta_{5} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + \beta_{2} q^{7} + ( - \beta_{5} - \beta_{3}) q^{9} + \beta_{4} q^{11} + (\beta_{7} + \beta_{3} + \beta_{2} + \beta_1) q^{13} - \beta_1 q^{15} + (\beta_{5} + \beta_{2} - \beta_1) q^{17} + (\beta_{7} + \beta_{6} - \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 + 1) q^{21} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3}) q^{23} + q^{25} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{27} + ( - \beta_{7} - \beta_{5} + \beta_{3} - 2 \beta_1) q^{29} + (2 \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{31} + ( - \beta_{6} + \beta_{4} - \beta_{2}) q^{33} + \beta_{2} q^{35} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{39} + (\beta_{7} + \beta_{5} + \beta_{3}) q^{41} + (\beta_{5} + \beta_1) q^{43} + ( - \beta_{5} - \beta_{3}) q^{45} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 4) q^{47} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_1 - 3) q^{49} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_1 - 2) q^{51} + (2 \beta_{5} + 2 \beta_1 - 2) q^{53} + \beta_{4} q^{55} + ( - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{57} + ( - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{59} + ( - \beta_{7} - 3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{61} + (\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 4) q^{63} + (\beta_{7} + \beta_{3} + \beta_{2} + \beta_1) q^{65} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{3} - \beta_1 + 4) q^{67} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{69} + (2 \beta_{6} - 4) q^{71} + (2 \beta_{6} - 2) q^{73} - \beta_1 q^{75} + (2 \beta_{7} - 4 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{77} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{79} + ( - \beta_{7} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{81} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{83} + (\beta_{5} + \beta_{2} - \beta_1) q^{85} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{3} - \beta_{2} + 4) q^{87} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2}) q^{89} + (2 \beta_{6} - 4) q^{91} + ( - 3 \beta_{7} - 4 \beta_{5} - \beta_{3} - 3 \beta_{2} - \beta_1) q^{93} + (\beta_{7} + \beta_{6} - \beta_{3} + \beta_1) q^{95} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2) q^{97} + (2 \beta_{7} - 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 12 q^{27} - 4 q^{33} + 16 q^{39} + 28 q^{47} - 16 q^{49} - 20 q^{51} - 16 q^{53} - 4 q^{57} - 28 q^{63} + 32 q^{67} - 20 q^{69} - 24 q^{71} - 8 q^{73} + 8 q^{81} + 36 q^{87} - 24 q^{91} + 4 q^{95} + 8 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + \nu^{6} - 4\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - \nu^{5} + 2\nu^{4} - 2\nu^{3} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{6} + 8\nu^{3} + 4\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 4\nu^{2} + 8\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 4\nu^{3} + 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} + 8\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + \nu^{5} - 4\nu^{4} + 2\nu^{3} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} - 2\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - 9\beta_{5} - 3\beta_{4} - 5\beta_{3} - 5\beta_{2} - 2\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{7} + \beta_{6} + 5\beta_{5} - \beta_{4} + \beta_{3} - 7\beta_{2} + 10\beta _1 + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} + 3\beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta_{3} + 11\beta_{2} + 14\beta _1 + 20 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.814732 1.15595i
0.814732 + 1.15595i
−0.578647 + 1.29041i
−0.578647 1.29041i
1.40014 + 0.199044i
1.40014 0.199044i
−1.13622 + 0.842022i
−1.13622 0.842022i
0 −1.48716 0.887900i 0 1.00000 0 0.797253i 0 1.42327 + 2.64089i 0
431.2 0 −1.48716 + 0.887900i 0 1.00000 0 0.797253i 0 1.42327 2.64089i 0
431.3 0 −0.751690 1.56044i 0 1.00000 0 4.28591i 0 −1.86993 + 2.34593i 0
431.4 0 −0.751690 + 1.56044i 0 1.00000 0 4.28591i 0 −1.86993 2.34593i 0
431.5 0 0.520627 1.65195i 0 1.00000 0 1.92736i 0 −2.45790 1.72010i 0
431.6 0 0.520627 + 1.65195i 0 1.00000 0 1.92736i 0 −2.45790 + 1.72010i 0
431.7 0 1.71822 0.218455i 0 1.00000 0 3.64426i 0 2.90455 0.750707i 0
431.8 0 1.71822 + 0.218455i 0 1.00000 0 3.64426i 0 2.90455 + 0.750707i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.b.b 8
3.b odd 2 1 480.2.b.a 8
4.b odd 2 1 120.2.b.a 8
5.b even 2 1 2400.2.b.f 8
5.c odd 4 2 2400.2.m.d 16
8.b even 2 1 120.2.b.b yes 8
8.d odd 2 1 480.2.b.a 8
12.b even 2 1 120.2.b.b yes 8
15.d odd 2 1 2400.2.b.e 8
15.e even 4 2 2400.2.m.c 16
20.d odd 2 1 600.2.b.f 8
20.e even 4 2 600.2.m.c 16
24.f even 2 1 inner 480.2.b.b 8
24.h odd 2 1 120.2.b.a 8
40.e odd 2 1 2400.2.b.e 8
40.f even 2 1 600.2.b.e 8
40.i odd 4 2 600.2.m.d 16
40.k even 4 2 2400.2.m.c 16
60.h even 2 1 600.2.b.e 8
60.l odd 4 2 600.2.m.d 16
120.i odd 2 1 600.2.b.f 8
120.m even 2 1 2400.2.b.f 8
120.q odd 4 2 2400.2.m.d 16
120.w even 4 2 600.2.m.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.b.a 8 4.b odd 2 1
120.2.b.a 8 24.h odd 2 1
120.2.b.b yes 8 8.b even 2 1
120.2.b.b yes 8 12.b even 2 1
480.2.b.a 8 3.b odd 2 1
480.2.b.a 8 8.d odd 2 1
480.2.b.b 8 1.a even 1 1 trivial
480.2.b.b 8 24.f even 2 1 inner
600.2.b.e 8 40.f even 2 1
600.2.b.e 8 60.h even 2 1
600.2.b.f 8 20.d odd 2 1
600.2.b.f 8 120.i odd 2 1
600.2.m.c 16 20.e even 4 2
600.2.m.c 16 120.w even 4 2
600.2.m.d 16 40.i odd 4 2
600.2.m.d 16 60.l odd 4 2
2400.2.b.e 8 15.d odd 2 1
2400.2.b.e 8 40.e odd 2 1
2400.2.b.f 8 5.b even 2 1
2400.2.b.f 8 120.m even 2 1
2400.2.m.c 16 15.e even 4 2
2400.2.m.c 16 40.k even 4 2
2400.2.m.d 16 5.c odd 4 2
2400.2.m.d 16 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{4} + 2T_{23}^{3} - 44T_{23}^{2} - 188T_{23} - 192 \) acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{5} - 2 T^{4} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 36 T^{6} + 384 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{8} + 48 T^{6} + 672 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + 880 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( T^{8} + 52 T^{6} + 816 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 36 T^{2} + 72 T - 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} - 44 T^{2} - 188 T - 192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 64 T^{2} + 112 T - 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 140 T^{6} + 4544 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$37$ \( T^{8} + 228 T^{6} + 17072 T^{4} + \cdots + 746496 \) Copy content Toggle raw display
$41$ \( T^{8} + 64 T^{6} + 1344 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} - 20 T^{2} + 788 T - 2208)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} - 24 T^{2} - 128 T + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 160 T^{6} + 672 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{8} + 208 T^{6} + 11392 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} - 52 T^{2} + 828 T + 2896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} - 64 T^{2} - 992 T - 2304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} - 112 T^{2} - 624 T - 656)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 108 T^{6} + 2048 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$83$ \( T^{8} + 152 T^{6} + 7952 T^{4} + \cdots + 1032256 \) Copy content Toggle raw display
$89$ \( T^{8} + 384 T^{6} + 45056 T^{4} + \cdots + 1048576 \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} - 176 T^{2} - 784 T - 784)^{2} \) Copy content Toggle raw display
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