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Properties

Label 320.8.c.m
Level $320$
Weight $8$
Character orbit 320.c
Analytic conductor $99.963$
Analytic rank $0$
Dimension $16$
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] = [320,8,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, 8, names="a")
 
//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

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: \(99.9632081549\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 1543 x^{14} + 63372 x^{13} - 906041 x^{12} + 9657224 x^{11} - 3362386879 x^{10} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{84}\cdot 3^{4}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} - 10) q^{5} + (\beta_{10} - 8 \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \cdots - 88) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} - 10) q^{5} + (\beta_{10} - 8 \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \cdots - 88) q^{9}+ \cdots + (120 \beta_{12} + 82 \beta_{11} + \cdots + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 168 q^{5} - 1408 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 168 q^{5} - 1408 q^{9} + 295504 q^{21} - 306288 q^{25} + 936864 q^{29} + 170160 q^{41} - 704808 q^{45} + 2768512 q^{49} - 8362256 q^{61} + 3609792 q^{65} - 5509424 q^{69} - 1752800 q^{81} + 7869824 q^{85} - 38719200 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 1543 x^{14} + 63372 x^{13} - 906041 x^{12} + 9657224 x^{11} - 3362386879 x^{10} + \cdots + 12\!\cdots\!80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23\!\cdots\!18 \nu^{15} + \cdots - 19\!\cdots\!40 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!32 \nu^{15} + \cdots + 34\!\cdots\!60 ) / 54\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37\!\cdots\!16 \nu^{15} + \cdots + 13\!\cdots\!80 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 60\!\cdots\!96 \nu^{15} + \cdots - 43\!\cdots\!80 ) / 39\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!58 \nu^{15} + \cdots + 16\!\cdots\!40 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\!\cdots\!58 \nu^{15} + \cdots - 16\!\cdots\!15 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!18 \nu^{15} + \cdots - 18\!\cdots\!15 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!44 \nu^{15} + \cdots - 61\!\cdots\!20 ) / 28\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 70\!\cdots\!74 \nu^{15} + \cdots + 50\!\cdots\!95 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!06 \nu^{15} + \cdots - 74\!\cdots\!80 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!36 \nu^{15} + \cdots - 99\!\cdots\!80 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 34\!\cdots\!98 \nu^{15} + \cdots + 67\!\cdots\!40 ) / 28\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!94 \nu^{15} + \cdots + 15\!\cdots\!95 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!72 \nu^{15} + \cdots - 10\!\cdots\!60 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 68\!\cdots\!92 \nu^{15} + \cdots + 50\!\cdots\!60 ) / 24\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{15} - 24 \beta_{14} + 24 \beta_{13} + 3 \beta_{12} - 42 \beta_{11} - 609 \beta_{10} + \cdots + 10128 ) / 61440 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12 \beta_{15} - 408 \beta_{14} - 1032 \beta_{13} - 59 \beta_{12} - 4326 \beta_{11} - 4759 \beta_{10} + \cdots - 5924592 ) / 30720 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3388 \beta_{15} + 21128 \beta_{14} - 168 \beta_{13} + 10051 \beta_{12} + 111318 \beta_{11} + \cdots - 383276656 ) / 30720 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18300 \beta_{15} + 88639 \beta_{14} + 91443 \beta_{13} - 8471 \beta_{12} + 407280 \beta_{11} + \cdots + 939119758 ) / 1920 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5437396 \beta_{15} + 23370120 \beta_{14} + 6702072 \beta_{13} - 14239871 \beta_{12} + \cdots + 906760797264 ) / 30720 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8786526 \beta_{15} - 557445798 \beta_{14} - 286290174 \beta_{13} - 53102321 \beta_{12} + \cdots + 4526552594916 ) / 3840 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4537158646 \beta_{15} - 32518432412 \beta_{14} + 27656268 \beta_{13} + 8845040268 \beta_{12} + \cdots - 508610575130504 ) / 7680 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 94051002798 \beta_{15} + 470472621016 \beta_{14} + 131183500716 \beta_{13} + 232895167295 \beta_{12} + \cdots - 16\!\cdots\!44 ) / 3840 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3762881042872 \beta_{15} + 140757685439316 \beta_{14} + 22981400655948 \beta_{13} + \cdots + 18\!\cdots\!96 ) / 7680 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 85075939186748 \beta_{15} + 407456446995280 \beta_{14} + 52993072184944 \beta_{13} + \cdots + 28\!\cdots\!44 ) / 2560 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11\!\cdots\!20 \beta_{15} + \cdots + 12\!\cdots\!48 ) / 3840 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 17\!\cdots\!47 \beta_{15} + \cdots - 34\!\cdots\!92 ) / 1920 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 10\!\cdots\!08 \beta_{15} + \cdots - 99\!\cdots\!84 ) / 7680 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 14\!\cdots\!50 \beta_{15} + \cdots - 84\!\cdots\!64 ) / 3840 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 19\!\cdots\!48 \beta_{15} + \cdots + 12\!\cdots\!68 ) / 3840 \) 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
−12.3530 − 16.2523i
16.4459 + 46.8783i
−7.57409 − 18.5741i
−11.0390 + 32.1593i
−37.5379 − 6.62423i
26.5049 + 0.305110i
17.4642 − 0.164060i
9.08903 + 27.5240i
9.08903 − 27.5240i
17.4642 + 0.164060i
26.5049 − 0.305110i
−37.5379 + 6.62423i
−11.0390 − 32.1593i
−7.57409 + 18.5741i
16.4459 − 46.8783i
−12.3530 + 16.2523i
0 − 67.4377i 0 −142.754 − 240.304i 0 383.530i 0 −2360.85 0
129.2 0 − 67.4377i 0 −142.754 + 240.304i 0 383.530i 0 −2360.85 0
129.3 0 − 66.3965i 0 172.384 − 220.020i 0 649.626i 0 −2221.50 0
129.4 0 − 66.3965i 0 172.384 + 220.020i 0 649.626i 0 −2221.50 0
129.5 0 − 11.4278i 0 −216.513 − 176.769i 0 800.077i 0 2056.41 0
129.6 0 − 11.4278i 0 −216.513 + 176.769i 0 800.077i 0 2056.41 0
129.7 0 − 3.61340i 0 144.883 − 239.027i 0 − 1180.17i 0 2173.94 0
129.8 0 − 3.61340i 0 144.883 + 239.027i 0 − 1180.17i 0 2173.94 0
129.9 0 3.61340i 0 144.883 − 239.027i 0 1180.17i 0 2173.94 0
129.10 0 3.61340i 0 144.883 + 239.027i 0 1180.17i 0 2173.94 0
129.11 0 11.4278i 0 −216.513 − 176.769i 0 − 800.077i 0 2056.41 0
129.12 0 11.4278i 0 −216.513 + 176.769i 0 − 800.077i 0 2056.41 0
129.13 0 66.3965i 0 172.384 − 220.020i 0 − 649.626i 0 −2221.50 0
129.14 0 66.3965i 0 172.384 + 220.020i 0 − 649.626i 0 −2221.50 0
129.15 0 67.4377i 0 −142.754 − 240.304i 0 − 383.530i 0 −2360.85 0
129.16 0 67.4377i 0 −142.754 + 240.304i 0 − 383.530i 0 −2360.85 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.16
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 320.8.c.m 16
4.b odd 2 1 inner 320.8.c.m 16
5.b even 2 1 inner 320.8.c.m 16
8.b even 2 1 160.8.c.c ✓ 16
8.d odd 2 1 160.8.c.c ✓ 16
20.d odd 2 1 inner 320.8.c.m 16
40.e odd 2 1 160.8.c.c ✓ 16
40.f even 2 1 160.8.c.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.8.c.c ✓ 16 8.b even 2 1
160.8.c.c ✓ 16 8.d odd 2 1
160.8.c.c ✓ 16 40.e odd 2 1
160.8.c.c ✓ 16 40.f even 2 1
320.8.c.m 16 1.a even 1 1 trivial
320.8.c.m 16 4.b odd 2 1 inner
320.8.c.m 16 5.b even 2 1 inner
320.8.c.m 16 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{8} + 9100T_{3}^{6} + 21337488T_{3}^{4} + 2895367680T_{3}^{2} + 34186530816 \) Copy content Toggle raw display
\( T_{11}^{8} - 88700032 T_{11}^{6} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 9100 T^{6} + \cdots + 34186530816)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 37\!\cdots\!25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 55\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 22\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 18\!\cdots\!36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 12\!\cdots\!60)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 17\!\cdots\!64)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
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