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Properties

Label 320.6.c.k
Level $320$
Weight $6$
Character orbit 320.c
Analytic conductor $51.323$
Analytic rank $0$
Dimension $12$
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] = [320,6,Mod(129,320)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (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: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 165x^{10} + 9528x^{8} + 254984x^{6} + 3245664x^{4} + 15975501x^{2} + 588289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{44}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 160)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{7} + \beta_{5} + 5) q^{5} + \beta_{9} q^{7} + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots - 142) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{7} + \beta_{5} + 5) q^{5} + \beta_{9} q^{7} + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots - 142) q^{9}+ \cdots + (69 \beta_{11} + 69 \beta_{10} + \cdots + 141 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 60 q^{5} - 1676 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 60 q^{5} - 1676 q^{9} - 688 q^{21} + 1500 q^{25} + 21304 q^{29} - 109672 q^{41} + 75140 q^{45} - 32348 q^{49} - 64552 q^{61} - 160 q^{65} + 166352 q^{69} - 173764 q^{81} - 151360 q^{85} - 3720 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 165x^{10} + 9528x^{8} + 254984x^{6} + 3245664x^{4} + 15975501x^{2} + 588289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 170020 \nu^{10} - 24454244 \nu^{8} - 1102611568 \nu^{6} - 20045402992 \nu^{4} + \cdots + 2177766131 ) / 200642913 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 411520 \nu^{10} - 60010112 \nu^{8} - 2780210368 \nu^{6} - 52857554368 \nu^{4} + \cdots - 34380714880 ) / 66880971 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5323964 \nu^{10} - 786315016 \nu^{8} - 37088343272 \nu^{6} - 711962887784 \nu^{4} + \cdots - 96708608876 ) / 200642913 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1310906 \nu^{11} - 195210058 \nu^{9} - 9277834112 \nu^{7} - 175028468600 \nu^{5} + \cdots + 2576633734216 \nu ) / 51297704757 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1957360 \nu^{11} + 286234304 \nu^{9} + 13567743616 \nu^{7} + 284453317984 \nu^{5} + \cdots + 8621360914288 \nu ) / 51297704757 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67875062 \nu^{11} - 10019691676 \nu^{9} - 471366569900 \nu^{7} - 8947051389500 \nu^{5} + \cdots + 38667851889478 \nu ) / 153893114271 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 85161215 \nu^{11} + 484184090 \nu^{10} + 12477922798 \nu^{9} + 70984599790 \nu^{8} + \cdots + 17382114505340 ) / 153893114271 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 85161215 \nu^{11} - 484184090 \nu^{10} + 12477922798 \nu^{9} - 70984599790 \nu^{8} + \cdots - 17382114505340 ) / 153893114271 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 175955038 \nu^{11} + 25169463050 \nu^{9} + 1119493454860 \nu^{7} + 19636262498968 \nu^{5} + \cdots - 173990789836964 \nu ) / 153893114271 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2541138896 \nu^{11} - 3284674432 \nu^{10} - 372595783468 \nu^{9} - 475746157328 \nu^{8} + \cdots - 240122345851648 ) / 153893114271 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2541138896 \nu^{11} - 3284674432 \nu^{10} + 372595783468 \nu^{9} - 475746157328 \nu^{8} + \cdots - 240122345851648 ) / 153893114271 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} - \beta_{7} - 5\beta_{6} - 2\beta_{5} + 40\beta_{4} ) / 160 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{10} - 48\beta_{8} + 48\beta_{7} + 8\beta_{3} + 5\beta_{2} + 120\beta _1 - 8840 ) / 320 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16 \beta_{11} - 16 \beta_{10} - 40 \beta_{9} - 130 \beta_{8} - 130 \beta_{7} + 910 \beta_{6} + \cdots - 3576 \beta_{4} ) / 320 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 80 \beta_{11} - 80 \beta_{10} + 9816 \beta_{8} - 9816 \beta_{7} - 1520 \beta_{3} - 995 \beta_{2} + \cdots + 876040 ) / 640 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 835 \beta_{11} + 835 \beta_{10} + 1930 \beta_{9} + 9416 \beta_{8} + 9416 \beta_{7} + \cdots + 112850 \beta_{4} ) / 160 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12696 \beta_{11} + 12696 \beta_{10} - 839160 \beta_{8} + 839160 \beta_{7} + 128112 \beta_{3} + \cdots - 58009960 ) / 640 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 142620 \beta_{11} - 142620 \beta_{10} - 319200 \beta_{9} - 1729042 \beta_{8} + \cdots - 16637360 \beta_{4} ) / 320 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 586649 \beta_{11} - 586649 \beta_{10} + 34087152 \beta_{8} - 34087152 \beta_{7} - 5190248 \beta_{3} + \cdots + 2171360360 ) / 320 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5783906 \beta_{11} + 5783906 \beta_{10} + 12748460 \beta_{9} + 71743835 \beta_{8} + \cdots + 645384276 \beta_{4} ) / 160 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 24339950 \beta_{11} + 24339950 \beta_{10} - 1362127056 \beta_{8} + 1362127056 \beta_{7} + \cdots - 84593817040 ) / 160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 461870614 \beta_{11} - 461870614 \beta_{10} - 1010984020 \beta_{9} - 5773585979 \beta_{8} + \cdots - 50835708524 \beta_{4} ) / 160 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
− 8.90339i
− 4.23250i
− 4.32684i
− 5.17887i
− 0.192622i
− 4.71554i
4.71554i
0.192622i
5.17887i
4.32684i
4.23250i
8.90339i
0 − 26.2718i 0 −48.8634 − 27.1545i 0 − 23.9306i 0 −447.206 0
129.2 0 − 26.2718i 0 −48.8634 + 27.1545i 0 − 23.9306i 0 −447.206 0
129.3 0 − 19.0114i 0 46.9000 − 30.4203i 0 − 91.0286i 0 −118.434 0
129.4 0 − 19.0114i 0 46.9000 + 30.4203i 0 − 91.0286i 0 −118.434 0
129.5 0 − 9.81633i 0 16.9634 − 53.2658i 0 222.821i 0 146.640 0
129.6 0 − 9.81633i 0 16.9634 + 53.2658i 0 222.821i 0 146.640 0
129.7 0 9.81633i 0 16.9634 − 53.2658i 0 − 222.821i 0 146.640 0
129.8 0 9.81633i 0 16.9634 + 53.2658i 0 − 222.821i 0 146.640 0
129.9 0 19.0114i 0 46.9000 − 30.4203i 0 91.0286i 0 −118.434 0
129.10 0 19.0114i 0 46.9000 + 30.4203i 0 91.0286i 0 −118.434 0
129.11 0 26.2718i 0 −48.8634 − 27.1545i 0 23.9306i 0 −447.206 0
129.12 0 26.2718i 0 −48.8634 + 27.1545i 0 23.9306i 0 −447.206 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.12
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
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.c.k 12
4.b odd 2 1 inner 320.6.c.k 12
5.b even 2 1 inner 320.6.c.k 12
8.b even 2 1 160.6.c.c ✓ 12
8.d odd 2 1 160.6.c.c ✓ 12
20.d odd 2 1 inner 320.6.c.k 12
40.e odd 2 1 160.6.c.c ✓ 12
40.f even 2 1 160.6.c.c ✓ 12
40.i odd 4 1 800.6.a.z 6
40.i odd 4 1 800.6.a.ba 6
40.k even 4 1 800.6.a.z 6
40.k even 4 1 800.6.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.c ✓ 12 8.b even 2 1
160.6.c.c ✓ 12 8.d odd 2 1
160.6.c.c ✓ 12 40.e odd 2 1
160.6.c.c ✓ 12 40.f even 2 1
320.6.c.k 12 1.a even 1 1 trivial
320.6.c.k 12 4.b odd 2 1 inner
320.6.c.k 12 5.b even 2 1 inner
320.6.c.k 12 20.d odd 2 1 inner
800.6.a.z 6 40.i odd 4 1
800.6.a.z 6 40.k even 4 1
800.6.a.ba 6 40.i odd 4 1
800.6.a.ba 6 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{6} + 1148T_{3}^{4} + 350800T_{3}^{2} + 24038400 \) Copy content Toggle raw display
\( T_{11}^{6} - 746112T_{11}^{4} + 138141388800T_{11}^{2} - 6940560693657600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 1148 T^{4} + \cdots + 24038400)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 30 T^{5} + \cdots + 30517578125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 58508 T^{4} + \cdots + 235600358400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 29\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 5326 T^{2} + \cdots + 85122490200)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 27418 T^{2} + \cdots + 49588433400)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 1837262215800)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 60\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 221085541919000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
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