↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(159,320)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.159");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.e (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: 8.0.691798081536.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + ( - \beta_{6} - \beta_{4}) q^{5} + \beta_{2} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + ( - \beta_{6} - \beta_{4}) q^{5} + \beta_{2} q^{7} + 3 q^{9} - \beta_{7} q^{11} + 4 \beta_{6} q^{13} + ( - \beta_{2} - 3 \beta_1) q^{15} + \beta_{3} q^{17} - 3 \beta_{7} q^{19} + ( - 3 \beta_{6} - 6 \beta_{4}) q^{21} - \beta_{2} q^{23} + ( - \beta_{3} - 13) q^{25} - 12 \beta_{5} q^{27} + ( - 4 \beta_{6} - 8 \beta_{4}) q^{29} - 2 \beta_1 q^{31} - \beta_{3} q^{33} + ( - 3 \beta_{7} - 19 \beta_{5}) q^{35} + 3 \beta_{6} q^{37} + 24 \beta_1 q^{39} - 24 q^{41} + 23 \beta_{5} q^{43} + ( - 3 \beta_{6} - 3 \beta_{4}) q^{45} - \beta_{2} q^{47} + 65 q^{49} - 6 \beta_{7} q^{51} + 10 \beta_{6} q^{53} + (2 \beta_{2} - 19 \beta_1) q^{55} - 3 \beta_{3} q^{57} + 5 \beta_{7} q^{59} + (3 \beta_{6} + 6 \beta_{4}) q^{61} + 3 \beta_{2} q^{63} + (4 \beta_{3} - 48) q^{65} - 3 \beta_{5} q^{67} + (3 \beta_{6} + 6 \beta_{4}) q^{69} - 42 \beta_1 q^{71} + 5 \beta_{3} q^{73} + (6 \beta_{7} + 13 \beta_{5}) q^{75} - 19 \beta_{6} q^{77} + 50 \beta_1 q^{79} - 45 q^{81} - 7 \beta_{5} q^{83} + ( - 13 \beta_{6} + 12 \beta_{4}) q^{85} - 8 \beta_{2} q^{87} + 150 q^{89} + 24 \beta_{7} q^{91} + 2 \beta_{6} q^{93} + (6 \beta_{2} - 57 \beta_1) q^{95} - \beta_{3} q^{97} - 3 \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 104 q^{25} - 192 q^{41} + 520 q^{49} - 384 q^{65} - 360 q^{81} + 1200 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{6} + 12\nu^{5} - 110\nu^{4} + 200\nu^{3} - 886\nu^{2} + 788\nu - 280 ) / 975 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 15\nu^{4} - 25\nu^{3} + 9\nu^{2} + 3\nu - 730 ) / 75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} + 42\nu^{4} - 74\nu^{3} + 378\nu^{2} - 342\nu + 88 ) / 39 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\nu^{7} - 119\nu^{6} + 1077\nu^{5} - 2395\nu^{4} + 11891\nu^{3} - 15501\nu^{2} + 65243\nu - 30115 ) / 19695 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -34\nu^{7} + 119\nu^{6} - 1077\nu^{5} + 2395\nu^{4} - 11891\nu^{3} + 15501\nu^{2} - 25853\nu + 10420 ) / 19695 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\nu^{7} - 154\nu^{6} + 966\nu^{5} - 2030\nu^{4} + 7546\nu^{3} - 9366\nu^{2} - 13526\nu + 8260 ) / 7575 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -16\nu^{7} + 56\nu^{6} - 388\nu^{5} + 830\nu^{4} - 2744\nu^{3} + 3314\nu^{2} + 4588\nu - 2820 ) / 1515 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 6\beta _1 - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18\beta_{7} + 25\beta_{6} - 42\beta_{5} - 16\beta_{4} + 3\beta_{3} - 6\beta_{2} + 18\beta _1 - 56 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{7} + 25\beta_{6} - 44\beta_{5} - 18\beta_{4} - 14\beta_{3} + 4\beta_{2} - 162\beta _1 + 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -240\beta_{7} - 475\beta_{6} - 266\beta_{5} - 136\beta_{4} - 75\beta_{3} + 30\beta_{2} - 840\beta _1 + 224 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -405\beta_{7} - 775\beta_{6} - 288\beta_{5} - 158\beta_{4} + 126\beta_{3} + 228\beta_{2} + 1341\beta _1 + 2322 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 252\beta_{7} + 1100\beta_{6} + 2262\beta_{5} + 1261\beta_{4} + 287\beta_{3} + 371\beta_{2} + 3087\beta _1 + 3851 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
−0.724745 + 3.40419i
−0.724745 − 0.954705i
1.72474 + 3.40419i
1.72474 − 0.954705i
−0.724745 + 0.954705i
−0.724745 − 3.40419i
1.72474 + 0.954705i
1.72474 − 3.40419i
0 − 2.44949i 0 −2.44949 − 4.35890i 0 10.6771 0 3.00000 0
159.2 0 − 2.44949i 0 −2.44949 + 4.35890i 0 −10.6771 0 3.00000 0
159.3 0 − 2.44949i 0 2.44949 − 4.35890i 0 10.6771 0 3.00000 0
159.4 0 − 2.44949i 0 2.44949 + 4.35890i 0 −10.6771 0 3.00000 0
159.5 0 2.44949i 0 −2.44949 − 4.35890i 0 −10.6771 0 3.00000 0
159.6 0 2.44949i 0 −2.44949 + 4.35890i 0 10.6771 0 3.00000 0
159.7 0 2.44949i 0 2.44949 − 4.35890i 0 −10.6771 0 3.00000 0
159.8 0 2.44949i 0 2.44949 + 4.35890i 0 10.6771 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.e.a ✓ 8
4.b odd 2 1 inner 320.3.e.a ✓ 8
5.b even 2 1 inner 320.3.e.a ✓ 8
5.c odd 4 2 1600.3.g.h 8
8.b even 2 1 inner 320.3.e.a ✓ 8
8.d odd 2 1 inner 320.3.e.a ✓ 8
16.e even 4 2 1280.3.h.i 8
16.f odd 4 2 1280.3.h.i 8
20.d odd 2 1 inner 320.3.e.a ✓ 8
20.e even 4 2 1600.3.g.h 8
40.e odd 2 1 inner 320.3.e.a ✓ 8
40.f even 2 1 inner 320.3.e.a ✓ 8
40.i odd 4 2 1600.3.g.h 8
40.k even 4 2 1600.3.g.h 8
80.k odd 4 2 1280.3.h.i 8
80.q even 4 2 1280.3.h.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.e.a ✓ 8 1.a even 1 1 trivial
320.3.e.a ✓ 8 4.b odd 2 1 inner
320.3.e.a ✓ 8 5.b even 2 1 inner
320.3.e.a ✓ 8 8.b even 2 1 inner
320.3.e.a ✓ 8 8.d odd 2 1 inner
320.3.e.a ✓ 8 20.d odd 2 1 inner
320.3.e.a ✓ 8 40.e odd 2 1 inner
320.3.e.a ✓ 8 40.f even 2 1 inner
1280.3.h.i 8 16.e even 4 2
1280.3.h.i 8 16.f odd 4 2
1280.3.h.i 8 80.k odd 4 2
1280.3.h.i 8 80.q even 4 2
1600.3.g.h 8 5.c odd 4 2
1600.3.g.h 8 20.e even 4 2
1600.3.g.h 8 40.i odd 4 2
1600.3.g.h 8 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 6 \) 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
$3$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 26 T^{2} + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 114)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 76)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 384)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 456)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 684)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 114)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1216)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 216)^{4} \) Copy content Toggle raw display
$41$ \( (T + 24)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3174)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 114)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2400)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 1900)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 684)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 7056)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11400)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 10000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 294)^{4} \) Copy content Toggle raw display
$89$ \( (T - 150)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 456)^{4} \) Copy content Toggle raw display
show more
show less