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Properties

Label 320.5.b.d
Level $320$
Weight $5$
Character orbit 320.b
Analytic conductor $33.078$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,5,Mod(191,320)]
 
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, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.191");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.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: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.246034965625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 112x^{2} - 192x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + (\beta_{4} - 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + (\beta_{4} - 41) q^{9} + (\beta_{7} - \beta_1) q^{11} + (\beta_{6} - \beta_{2} - 44) q^{13} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{15} + ( - 3 \beta_{4} - 10 \beta_{2} - 6) q^{17} + (\beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_1) q^{19}+ \cdots + (21 \beta_{7} - 42 \beta_{5} + \cdots + 139 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 328 q^{9} - 352 q^{13} - 48 q^{17} - 16 q^{21} + 1000 q^{25} - 1200 q^{29} - 1120 q^{33} + 5728 q^{37} + 4896 q^{41} + 400 q^{45} - 5768 q^{49} - 2592 q^{53} + 3840 q^{57} - 7936 q^{61} - 1200 q^{65} + 2256 q^{69} - 14448 q^{73} - 2400 q^{77} - 936 q^{81} - 11200 q^{85} + 23760 q^{89} - 11360 q^{93} - 4368 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 112x^{2} - 192x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 11\nu^{6} + 15\nu^{5} - 29\nu^{4} + 41\nu^{3} - 76\nu^{2} + 256 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 23\nu^{5} + 69\nu^{4} - 97\nu^{3} + 228\nu^{2} - 384\nu + 480 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{7} - 55\nu^{6} + 171\nu^{5} - 273\nu^{4} + 621\nu^{3} - 1036\nu^{2} + 960\nu - 896 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{7} + 11\nu^{6} - 63\nu^{5} + 125\nu^{4} - 377\nu^{3} + 612\nu^{2} - 1152\nu + 1120 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 35\nu^{6} - 71\nu^{5} + 197\nu^{4} - 433\nu^{3} + 748\nu^{2} - 1408\nu + 1280 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -43\nu^{7} + 65\nu^{6} - 29\nu^{5} + 215\nu^{4} - 523\nu^{3} + 780\nu^{2} + 1920\nu + 928 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\nu^{7} - 53\nu^{6} + 113\nu^{5} - 419\nu^{4} + 855\nu^{3} - 532\nu^{2} + 1408\nu - 2816 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{5} + \beta_{4} + 3\beta_{2} - 4\beta _1 + 48 ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} + 5\beta_{2} + 4\beta _1 - 40 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + \beta_{5} - 4\beta_{4} + 19\beta_{2} - 12\beta _1 + 216 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{7} + \beta_{6} - 10\beta_{5} - 3\beta_{4} + 16\beta_{3} + 107\beta_{2} - 76\beta _1 - 272 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{6} + 12\beta_{5} + \beta_{4} + 24\beta_{3} + \beta_{2} - 576 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{7} - 5\beta_{6} + 46\beta_{5} - 9\beta_{4} + 32\beta_{3} - 199\beta_{2} - 380\beta _1 + 4720 ) / 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - 37\beta_{5} + 56\beta_{4} + 48\beta_{3} + 43\beta_{2} - 348\beta _1 + 2376 ) / 64 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−1.02661 − 1.71641i
1.95003 − 0.444269i
−0.641015 + 1.89449i
1.21760 + 1.58665i
1.21760 − 1.58665i
−0.641015 − 1.89449i
1.95003 + 0.444269i
−1.02661 + 1.71641i
0 − 15.5779i 0 −11.1803 0 37.6230i 0 −161.671 0
191.2 0 − 12.9912i 0 11.1803 0 − 78.0345i 0 −87.7712 0
191.3 0 − 8.14153i 0 11.1803 0 63.6032i 0 14.7155 0
191.4 0 − 3.20523i 0 −11.1803 0 − 30.6227i 0 70.7265 0
191.5 0 3.20523i 0 −11.1803 0 30.6227i 0 70.7265 0
191.6 0 8.14153i 0 11.1803 0 − 63.6032i 0 14.7155 0
191.7 0 12.9912i 0 11.1803 0 78.0345i 0 −87.7712 0
191.8 0 15.5779i 0 −11.1803 0 − 37.6230i 0 −161.671 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.5.b.d 8
4.b odd 2 1 inner 320.5.b.d 8
8.b even 2 1 20.5.b.a ✓ 8
8.d odd 2 1 20.5.b.a ✓ 8
24.f even 2 1 180.5.c.a 8
24.h odd 2 1 180.5.c.a 8
40.e odd 2 1 100.5.b.c 8
40.f even 2 1 100.5.b.c 8
40.i odd 4 2 100.5.d.c 16
40.k even 4 2 100.5.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.b.a ✓ 8 8.b even 2 1
20.5.b.a ✓ 8 8.d odd 2 1
100.5.b.c 8 40.e odd 2 1
100.5.b.c 8 40.f even 2 1
100.5.d.c 16 40.i odd 4 2
100.5.d.c 16 40.k even 4 2
180.5.c.a 8 24.f even 2 1
180.5.c.a 8 24.h odd 2 1
320.5.b.d 8 1.a even 1 1 trivial
320.5.b.d 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 488T_{3}^{6} + 73136T_{3}^{4} + 3415680T_{3}^{2} + 27889920 \) acting on \(S_{5}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 488 T^{6} + \cdots + 27889920 \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 32698357408000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 204994355200000 \) Copy content Toggle raw display
$13$ \( (T^{4} + 176 T^{3} + \cdots - 8015600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{3} + \cdots + 688291600)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( (T^{4} + 600 T^{3} + \cdots - 98595968624)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 6304245167600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 1586334915856)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 43\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296 T^{3} + \cdots + 486559363600)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 17262940540816)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 39753982895600)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 26555598339856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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