↑

Properties

Label 320.4.n.j
Level $320$
Weight $4$
Character orbit 320.n
Analytic conductor $18.881$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(63,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.63");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.n (of order \(4\), degree \(2\), 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: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 23x^{10} + 396x^{8} - 2987x^{6} + 16861x^{4} - 4788x^{2} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + ( - 2 \beta_{7} + \beta_{6} + \cdots - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{4} - \beta_1 - 1) q^{5} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + ( - 2 \beta_{7} + \beta_{6} + \cdots - 3 \beta_1) q^{9}+ \cdots + ( - 25 \beta_{11} + \cdots + 53 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{5} + 188 q^{13} - 172 q^{17} - 688 q^{21} + 468 q^{25} - 1328 q^{33} + 1140 q^{37} - 16 q^{41} - 2044 q^{45} + 884 q^{53} - 256 q^{57} - 1952 q^{61} + 3468 q^{65} - 4324 q^{73} + 3152 q^{77} - 2012 q^{81} + 476 q^{85} - 1008 q^{93} + 3644 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 23x^{10} + 396x^{8} - 2987x^{6} + 16861x^{4} - 4788x^{2} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 266\nu^{11} - 6073\nu^{9} + 103680\nu^{7} - 766030\nu^{5} + 4224683\nu^{3} - 59616\nu ) / 608040 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3770 \nu^{11} - 4277 \nu^{10} - 86940 \nu^{9} + 94788 \nu^{8} + 1496880 \nu^{7} - 1647864 \nu^{6} + \cdots + 10405584 ) / 5472360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3770 \nu^{11} + 4277 \nu^{10} - 86940 \nu^{9} - 94788 \nu^{8} + 1496880 \nu^{7} + 1647864 \nu^{6} + \cdots - 10405584 ) / 5472360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50939 \nu^{11} + 6256 \nu^{10} + 1155942 \nu^{9} - 107712 \nu^{8} - 19854720 \nu^{7} + \cdots + 136180440 ) / 16417080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58387 \nu^{11} - 322 \nu^{10} - 1323171 \nu^{9} + 5544 \nu^{8} + 22757760 \nu^{7} + \cdots + 117929520 ) / 16417080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58387 \nu^{11} - 322 \nu^{10} + 1323171 \nu^{9} + 5544 \nu^{8} - 22757760 \nu^{7} + \cdots + 117929520 ) / 16417080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50939 \nu^{11} - 6256 \nu^{10} + 1155942 \nu^{9} + 107712 \nu^{8} - 19854720 \nu^{7} + \cdots - 136180440 ) / 16417080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -18041\nu^{11} + 424296\nu^{9} - 7352856\nu^{7} + 57147259\nu^{5} - 326059164\nu^{3} + 180738864\nu ) / 4104270 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 48560 \nu^{11} - 54791 \nu^{10} - 1113660 \nu^{9} + 1324044 \nu^{8} + 19174320 \nu^{7} + \cdots + 133445232 ) / 16417080 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -42091\nu^{10} + 946764\nu^{8} - 16348392\nu^{6} + 120983753\nu^{4} - 695392740\nu^{2} + 103442832 ) / 8208540 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 48560 \nu^{11} + 54791 \nu^{10} - 1113660 \nu^{9} - 1324044 \nu^{8} + 19174320 \nu^{7} + \cdots - 133445232 ) / 16417080 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} + 2\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - 6 \beta_{10} + \beta_{9} + \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{4} + \cdots + 62 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} - 5\beta_{6} + 5\beta_{5} + 7\beta_{4} + 18\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{11} - 66 \beta_{10} + 7 \beta_{9} - 7 \beta_{7} + 41 \beta_{6} + 41 \beta_{5} + 7 \beta_{4} + \cdots - 706 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 109 \beta_{11} + 109 \beta_{9} - 30 \beta_{8} + 177 \beta_{7} - 109 \beta_{6} + 109 \beta_{5} + \cdots + 730 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{7} + 136\beta_{6} + 136\beta_{5} + 7\beta_{4} - 2070 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1213 \beta_{11} + 1213 \beta_{9} - 690 \beta_{8} - 2245 \beta_{7} + 1213 \beta_{6} - 1213 \beta_{5} + \cdots - 12074 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 251 \beta_{11} + 8826 \beta_{10} + 251 \beta_{9} + 251 \beta_{7} + 7167 \beta_{6} + 7167 \beta_{5} + \cdots - 98774 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -14299\beta_{7} + 6881\beta_{6} - 6881\beta_{5} - 14299\beta_{4} - 90954\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 9245 \beta_{11} + 104826 \beta_{10} + 9245 \beta_{9} - 9245 \beta_{7} - 93965 \beta_{6} + \cdots + 1195978 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 159121 \beta_{11} - 159121 \beta_{9} + 182550 \beta_{8} - 365541 \beta_{7} + 159121 \beta_{6} + \cdots - 2604322 \beta_1 ) / 16 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
3.11366 + 1.79767i
−0.461926 + 0.266693i
−2.70956 − 1.56437i
2.70956 − 1.56437i
0.461926 + 0.266693i
−3.11366 + 1.79767i
3.11366 − 1.79767i
−0.461926 − 0.266693i
−2.70956 + 1.56437i
2.70956 + 1.56437i
0.461926 − 0.266693i
−3.11366 − 1.79767i
0 −6.07481 + 6.07481i 0 −5.85952 − 9.52187i 0 18.8345 + 18.8345i 0 − 46.8065i 0
63.2 0 −2.99189 + 2.99189i 0 −9.31566 + 6.18211i 0 −6.68730 − 6.68730i 0 9.09714i 0
63.3 0 −0.381190 + 0.381190i 0 11.1752 + 0.339759i 0 −22.0577 − 22.0577i 0 26.7094i 0
63.4 0 0.381190 − 0.381190i 0 11.1752 + 0.339759i 0 22.0577 + 22.0577i 0 26.7094i 0
63.5 0 2.99189 − 2.99189i 0 −9.31566 + 6.18211i 0 6.68730 + 6.68730i 0 9.09714i 0
63.6 0 6.07481 − 6.07481i 0 −5.85952 − 9.52187i 0 −18.8345 − 18.8345i 0 − 46.8065i 0
127.1 0 −6.07481 − 6.07481i 0 −5.85952 + 9.52187i 0 18.8345 − 18.8345i 0 46.8065i 0
127.2 0 −2.99189 − 2.99189i 0 −9.31566 − 6.18211i 0 −6.68730 + 6.68730i 0 − 9.09714i 0
127.3 0 −0.381190 − 0.381190i 0 11.1752 − 0.339759i 0 −22.0577 + 22.0577i 0 − 26.7094i 0
127.4 0 0.381190 + 0.381190i 0 11.1752 − 0.339759i 0 22.0577 − 22.0577i 0 − 26.7094i 0
127.5 0 2.99189 + 2.99189i 0 −9.31566 − 6.18211i 0 6.68730 − 6.68730i 0 − 9.09714i 0
127.6 0 6.07481 + 6.07481i 0 −5.85952 + 9.52187i 0 −18.8345 + 18.8345i 0 46.8065i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.n.j 12
4.b odd 2 1 inner 320.4.n.j 12
5.c odd 4 1 inner 320.4.n.j 12
8.b even 2 1 80.4.n.c ✓ 12
8.d odd 2 1 80.4.n.c ✓ 12
20.e even 4 1 inner 320.4.n.j 12
24.f even 2 1 720.4.x.g 12
24.h odd 2 1 720.4.x.g 12
40.e odd 2 1 400.4.n.f 12
40.f even 2 1 400.4.n.f 12
40.i odd 4 1 80.4.n.c ✓ 12
40.i odd 4 1 400.4.n.f 12
40.k even 4 1 80.4.n.c ✓ 12
40.k even 4 1 400.4.n.f 12
120.q odd 4 1 720.4.x.g 12
120.w even 4 1 720.4.x.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.4.n.c ✓ 12 8.b even 2 1
80.4.n.c ✓ 12 8.d odd 2 1
80.4.n.c ✓ 12 40.i odd 4 1
80.4.n.c ✓ 12 40.k even 4 1
320.4.n.j 12 1.a even 1 1 trivial
320.4.n.j 12 4.b odd 2 1 inner
320.4.n.j 12 5.c odd 4 1 inner
320.4.n.j 12 20.e even 4 1 inner
400.4.n.f 12 40.e odd 2 1
400.4.n.f 12 40.f even 2 1
400.4.n.f 12 40.i odd 4 1
400.4.n.f 12 40.k even 4 1
720.4.x.g 12 24.f even 2 1
720.4.x.g 12 24.h odd 2 1
720.4.x.g 12 120.q odd 4 1
720.4.x.g 12 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{12} + 5768T_{3}^{8} + 1746448T_{3}^{4} + 147456 \) Copy content Toggle raw display
\( T_{13}^{6} - 94T_{13}^{5} + 4418T_{13}^{4} + 51840T_{13}^{3} + 3755844T_{13}^{2} - 252583416T_{13} + 8493215112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 5768 T^{8} + \cdots + 147456 \) Copy content Toggle raw display
$5$ \( (T^{6} + 8 T^{5} + \cdots + 1953125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{6} + 5656 T^{4} + \cdots + 2209087488)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 94 T^{5} + \cdots + 8493215112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 86 T^{5} + \cdots + 13204425032)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 8224 T^{4} + \cdots - 949665792)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 5101382373376)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 85656 T^{4} + \cdots + 53341867008)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 570 T^{5} + \cdots + 146719445000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} + \cdots - 427488)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 806695711848968)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 57\!\cdots\!12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 488 T^{2} + \cdots + 9040176)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 22\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 63\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 52361196846408)^{2} \) Copy content Toggle raw display
show more
show less