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Properties

Label 320.3.g.a
Level $320$
Weight $3$
Character orbit 320.g
Analytic conductor $8.719$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(31,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.g (of order \(2\), degree \(1\), 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: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{6} q^{5} + ( - \beta_{5} - 3 \beta_{4}) q^{7} + ( - \beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{6} q^{5} + ( - \beta_{5} - 3 \beta_{4}) q^{7} + ( - \beta_{7} - 1) q^{9} + ( - 3 \beta_{2} - \beta_1) q^{11} + (4 \beta_{6} + 3 \beta_{3}) q^{13} + ( - \beta_{5} + 2 \beta_{4}) q^{15} + ( - 3 \beta_{7} + 4) q^{17} + ( - \beta_{2} + 7 \beta_1) q^{19} + 2 \beta_{6} q^{21} + ( - 5 \beta_{5} - \beta_{4}) q^{23} - 5 q^{25} + (2 \beta_{2} - 2 \beta_1) q^{27} + (8 \beta_{6} - 6 \beta_{3}) q^{29} + ( - 14 \beta_{5} - 4 \beta_{4}) q^{31} + ( - 3 \beta_{7} + 22) q^{33} + 5 \beta_1 q^{35} + (14 \beta_{6} - 8 \beta_{3}) q^{37} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{39} + (\beta_{7} - 62) q^{41} + (15 \beta_{2} + 12 \beta_1) q^{43} + (\beta_{6} + 5 \beta_{3}) q^{45} + (5 \beta_{5} - 7 \beta_{4}) q^{47} + ( - 5 \beta_{7} + 9) q^{49} + ( - 28 \beta_{2} - 6 \beta_1) q^{51} + (4 \beta_{6} + 9 \beta_{3}) q^{53} + ( - 4 \beta_{5} + 3 \beta_{4}) q^{55} + ( - \beta_{7} + 22) q^{57} + (7 \beta_{2} - 25 \beta_1) q^{59} + ( - 18 \beta_{6} - 10 \beta_{3}) q^{61} + (11 \beta_{5} + 23 \beta_{4}) q^{63} + (3 \beta_{7} + 20) q^{65} + ( - 13 \beta_{2} + 4 \beta_1) q^{67} + ( - 18 \beta_{6} + 14 \beta_{3}) q^{69} + (6 \beta_{5} + 4 \beta_{4}) q^{71} + (\beta_{7} + 40) q^{73} + 5 \beta_{2} q^{75} + ( - 2 \beta_{6} - 5 \beta_{3}) q^{77} + (4 \beta_{5} - 30 \beta_{4}) q^{79} + (11 \beta_{7} - 11) q^{81} + (23 \beta_{2} + 42 \beta_1) q^{83} + ( - 4 \beta_{6} + 15 \beta_{3}) q^{85} + (20 \beta_{5} - 28 \beta_{4}) q^{87} + (10 \beta_{7} + 26) q^{89} + ( - 6 \beta_{2} - 44 \beta_1) q^{91} + ( - 48 \beta_{6} + 38 \beta_{3}) q^{93} + (6 \beta_{5} + 23 \beta_{4}) q^{95} + (\beta_{7} - 80) q^{97} + ( - 19 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 32 q^{17} - 40 q^{25} + 176 q^{33} - 496 q^{41} + 72 q^{49} + 176 q^{57} + 160 q^{65} + 320 q^{73} - 88 q^{81} + 208 q^{89} - 640 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + \nu^{6} + 8\nu^{5} - 20\nu^{3} + 14\nu + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - \nu^{6} + 8\nu^{5} - 20\nu^{3} + 14\nu - 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} - 8\nu^{5} + 24\nu^{4} + 20\nu^{3} - 56\nu^{2} - \nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{4} - \beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} - \beta _1 + 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{5} + 3\beta_{4} + 7\beta_{3} - 3\beta_{2} + 3\beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} + 10\beta_{6} + 11\beta_{4} + 11\beta_{2} + 11\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{2} + 2\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 26\beta_{6} - 29\beta_{4} + 29\beta_{2} + 29\beta_1 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.40126 + 0.809017i
−1.40126 − 0.809017i
1.40126 + 0.809017i
1.40126 − 0.809017i
0.535233 + 0.309017i
0.535233 − 0.309017i
−0.535233 + 0.309017i
−0.535233 − 0.309017i
0 −3.96812 0 − 2.23607i 0 − 1.12702i 0 6.74597 0
31.2 0 −3.96812 0 2.23607i 0 1.12702i 0 6.74597 0
31.3 0 −0.504017 0 − 2.23607i 0 − 8.87298i 0 −8.74597 0
31.4 0 −0.504017 0 2.23607i 0 8.87298i 0 −8.74597 0
31.5 0 0.504017 0 − 2.23607i 0 8.87298i 0 −8.74597 0
31.6 0 0.504017 0 2.23607i 0 − 8.87298i 0 −8.74597 0
31.7 0 3.96812 0 − 2.23607i 0 1.12702i 0 6.74597 0
31.8 0 3.96812 0 2.23607i 0 − 1.12702i 0 6.74597 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.g.a ✓ 8
3.b odd 2 1 2880.3.g.b 8
4.b odd 2 1 inner 320.3.g.a ✓ 8
5.b even 2 1 1600.3.g.i 8
5.c odd 4 1 1600.3.e.g 8
5.c odd 4 1 1600.3.e.l 8
8.b even 2 1 inner 320.3.g.a ✓ 8
8.d odd 2 1 inner 320.3.g.a ✓ 8
12.b even 2 1 2880.3.g.b 8
16.e even 4 2 1280.3.b.h 8
16.f odd 4 2 1280.3.b.h 8
20.d odd 2 1 1600.3.g.i 8
20.e even 4 1 1600.3.e.g 8
20.e even 4 1 1600.3.e.l 8
24.f even 2 1 2880.3.g.b 8
24.h odd 2 1 2880.3.g.b 8
40.e odd 2 1 1600.3.g.i 8
40.f even 2 1 1600.3.g.i 8
40.i odd 4 1 1600.3.e.g 8
40.i odd 4 1 1600.3.e.l 8
40.k even 4 1 1600.3.e.g 8
40.k even 4 1 1600.3.e.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.g.a ✓ 8 1.a even 1 1 trivial
320.3.g.a ✓ 8 4.b odd 2 1 inner
320.3.g.a ✓ 8 8.b even 2 1 inner
320.3.g.a ✓ 8 8.d odd 2 1 inner
1280.3.b.h 8 16.e even 4 2
1280.3.b.h 8 16.f odd 4 2
1600.3.e.g 8 5.c odd 4 1
1600.3.e.g 8 20.e even 4 1
1600.3.e.g 8 40.i odd 4 1
1600.3.e.g 8 40.k even 4 1
1600.3.e.l 8 5.c odd 4 1
1600.3.e.l 8 20.e even 4 1
1600.3.e.l 8 40.i odd 4 1
1600.3.e.l 8 40.k even 4 1
1600.3.g.i 8 5.b even 2 1
1600.3.g.i 8 20.d odd 2 1
1600.3.g.i 8 40.e odd 2 1
1600.3.g.i 8 40.f even 2 1
2880.3.g.b 8 3.b odd 2 1
2880.3.g.b 8 12.b even 2 1
2880.3.g.b 8 24.f even 2 1
2880.3.g.b 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 16T_{3}^{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 16 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 80 T^{2} + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 136 T^{2} + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 376 T^{2} + 784)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8 T - 524)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 856 T^{2} + 44944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 768 T^{2} + 133956)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1504 T^{2} + 12544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 5952 T^{2} + 8433216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3496 T^{2} + 44944)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 124 T + 3784)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4464 T^{2} + 4588164)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1472 T^{2} + 196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2104 T^{2} + 795664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 12184 T^{2} + 17205904)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 5640 T^{2} + 176400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 3376 T^{2} + 1444804)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1088 T^{2} + 287296)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 80 T + 1540)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8672 T^{2} + 14868736)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 28960 T^{2} + 118156900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 52 T - 5324)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 160 T + 6340)^{4} \) Copy content Toggle raw display
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