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Properties

Label 320.8.c.k
Level $320$
Weight $8$
Character orbit 320.c
Analytic conductor $99.963$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,8,Mod(129,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.129"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,744,0,0,0,-8648,0,-9120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(99.9632081549\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 196x^{6} + 7674x^{4} + 75204x^{2} + 18225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{38}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1 + 93) q^{5} + ( - \beta_{7} + \beta_{3} + \cdots - 3 \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + \cdots - 1081) q^{9} + ( - \beta_{6} - \beta_{4} + \cdots - 1140) q^{11}+ \cdots + (3725 \beta_{6} - 44 \beta_{5} + \cdots + 2580308) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 744 q^{5} - 8648 q^{9} - 9120 q^{11} + 36320 q^{15} + 61600 q^{19} + 30304 q^{21} - 254424 q^{25} - 35760 q^{29} + 519168 q^{31} + 162144 q^{35} + 1136192 q^{39} - 852336 q^{41} - 180392 q^{45} - 4310376 q^{49}+ \cdots + 20642464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 196x^{6} + 7674x^{4} + 75204x^{2} + 18225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -241\nu^{7} - 47839\nu^{5} - 1942269\nu^{3} - 19327851\nu ) / 121230 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 544\nu^{7} + 106912\nu^{5} + 4276896\nu^{3} + 47739168\nu ) / 181845 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1034 \nu^{7} + 666 \nu^{6} - 194654 \nu^{5} + 115200 \nu^{4} - 6546126 \nu^{3} + 2212434 \nu^{2} + \cdots - 16016940 ) / 181845 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1034 \nu^{7} - 20070 \nu^{6} + 194654 \nu^{5} - 3366720 \nu^{4} + 6546126 \nu^{3} + \cdots - 116845740 ) / 181845 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1034 \nu^{7} - 15714 \nu^{6} + 194654 \nu^{5} - 3032640 \nu^{4} + 6546126 \nu^{3} + \cdots - 674861220 ) / 181845 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3413 \nu^{7} + 10656 \nu^{6} + 635099 \nu^{5} + 1843200 \nu^{4} + 20357697 \nu^{3} + \cdots - 256271040 ) / 363690 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5984 \nu^{7} + 666 \nu^{6} - 1143704 \nu^{5} + 115200 \nu^{4} - 40661076 \nu^{3} + \cdots - 16016940 ) / 181845 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + \beta_{6} - 11\beta_{3} - 5\beta_{2} - 21\beta_1 ) / 640 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -11\beta_{6} - \beta_{5} - 3\beta_{4} - 26\beta_{3} + 11\beta _1 - 15680 ) / 320 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -303\beta_{7} - 117\beta_{6} + 1239\beta_{3} + 2085\beta_{2} + 4057\beta_1 ) / 640 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1628\beta_{6} + \beta_{5} + 558\beta_{4} + 3815\beta_{3} - 1628\beta _1 + 1845440 ) / 320 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 41793\beta_{7} + 15531\beta_{6} - 166041\beta_{3} - 336105\beta_{2} - 649351\beta_1 ) / 640 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -236321\beta_{6} + 3149\beta_{5} - 86553\beta_{4} - 556046\beta_{3} + 236321\beta _1 - 259426880 ) / 320 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6094653\beta_{7} - 2220207\beta_{6} + 23856309\beta_{3} + 50315085\beta_{2} + 97563547\beta_1 ) / 640 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.498606i
12.1404i
− 5.79010i
3.85173i
− 3.85173i
5.79010i
− 12.1404i
− 0.498606i
0 − 77.5195i 0 184.066 + 210.344i 0 1384.40i 0 −3822.28 0
129.2 0 − 65.0953i 0 −61.1163 + 272.745i 0 − 1567.21i 0 −2050.39 0
129.3 0 − 52.5319i 0 14.9454 − 279.109i 0 338.895i 0 −572.598 0
129.4 0 − 8.10762i 0 234.105 − 152.708i 0 − 980.714i 0 2121.27 0
129.5 0 8.10762i 0 234.105 + 152.708i 0 980.714i 0 2121.27 0
129.6 0 52.5319i 0 14.9454 + 279.109i 0 − 338.895i 0 −572.598 0
129.7 0 65.0953i 0 −61.1163 − 272.745i 0 1567.21i 0 −2050.39 0
129.8 0 77.5195i 0 184.066 − 210.344i 0 − 1384.40i 0 −3822.28 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.8.c.k 8
4.b odd 2 1 320.8.c.l 8
5.b even 2 1 inner 320.8.c.k 8
8.b even 2 1 40.8.c.b ✓ 8
8.d odd 2 1 80.8.c.e 8
20.d odd 2 1 320.8.c.l 8
24.h odd 2 1 360.8.f.b 8
40.e odd 2 1 80.8.c.e 8
40.f even 2 1 40.8.c.b ✓ 8
40.i odd 4 1 200.8.a.q 4
40.i odd 4 1 200.8.a.r 4
40.k even 4 1 400.8.a.bj 4
40.k even 4 1 400.8.a.bl 4
120.i odd 2 1 360.8.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.c.b ✓ 8 8.b even 2 1
40.8.c.b ✓ 8 40.f even 2 1
80.8.c.e 8 8.d odd 2 1
80.8.c.e 8 40.e odd 2 1
200.8.a.q 4 40.i odd 4 1
200.8.a.r 4 40.i odd 4 1
320.8.c.k 8 1.a even 1 1 trivial
320.8.c.k 8 5.b even 2 1 inner
320.8.c.l 8 4.b odd 2 1
320.8.c.l 8 20.d odd 2 1
360.8.f.b 8 24.h odd 2 1
360.8.f.b 8 120.i odd 2 1
400.8.a.bj 4 40.k even 4 1
400.8.a.bl 4 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 \(S_{8}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{8} + 13072T_{3}^{6} + 54595296T_{3}^{4} + 73802016000T_{3}^{2} + 4619060640000 \) Copy content Toggle raw display
\( T_{11}^{4} + 4560T_{11}^{3} - 21371552T_{11}^{2} - 6659746560T_{11} + 6076238622976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 4619060640000 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 6076238622976)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 50\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 35\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 36\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 79\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
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