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Properties

Label 8040.2.a.x
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8040,2,Mod(1,8040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8040.a (trivial)

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: yes
Analytic conductor: \(64.1997232251\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 31x^{6} + 49x^{5} + 260x^{4} - 133x^{3} - 501x^{2} - 141x + 35 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} - \beta_{4} q^{11} + (\beta_{2} + \beta_1 + 1) q^{13} + q^{15} + ( - \beta_{5} + 2) q^{17} + (\beta_{6} + 1) q^{19} + ( - \beta_1 + 1) q^{21} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 2) q^{23}+ \cdots - \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{5} + 5 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{5} + 5 q^{7} + 8 q^{9} + 3 q^{11} + 13 q^{13} + 8 q^{15} + 14 q^{17} + 11 q^{19} + 5 q^{21} + 20 q^{23} + 8 q^{25} + 8 q^{27} - 15 q^{29} + 3 q^{31} + 3 q^{33} + 5 q^{35} + 20 q^{37} + 13 q^{39} - 5 q^{41} - q^{43} + 8 q^{45} + 19 q^{47} + 17 q^{49} + 14 q^{51} - 5 q^{53} + 3 q^{55} + 11 q^{57} + 15 q^{59} + 23 q^{61} + 5 q^{63} + 13 q^{65} + 8 q^{67} + 20 q^{69} - 26 q^{71} + 38 q^{73} + 8 q^{75} + 27 q^{77} - 3 q^{79} + 8 q^{81} + 30 q^{83} + 14 q^{85} - 15 q^{87} - 12 q^{89} - 27 q^{91} + 3 q^{93} + 11 q^{95} + 7 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 31x^{6} + 49x^{5} + 260x^{4} - 133x^{3} - 501x^{2} - 141x + 35 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 309 \nu^{7} + 6052 \nu^{6} - 7955 \nu^{5} - 159060 \nu^{4} + 198046 \nu^{3} + 1135931 \nu^{2} + \cdots - 1372685 ) / 263902 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1772 \nu^{7} - 2252 \nu^{6} - 69251 \nu^{5} + 23082 \nu^{4} + 713302 \nu^{3} - 23348 \nu^{2} + \cdots - 71698 ) / 131951 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14143 \nu^{7} + 55802 \nu^{6} + 380631 \nu^{5} - 1074672 \nu^{4} - 2419824 \nu^{3} + \cdots - 2651615 ) / 527804 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23005 \nu^{7} - 80766 \nu^{6} - 685413 \nu^{5} + 1496008 \nu^{4} + 5553756 \nu^{3} - 5594857 \nu^{2} + \cdots - 1173127 ) / 527804 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11695 \nu^{7} + 39883 \nu^{6} + 342448 \nu^{5} - 702333 \nu^{4} - 2639835 \nu^{3} + 2519085 \nu^{2} + \cdots - 84540 ) / 131951 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12004 \nu^{7} - 45935 \nu^{6} - 334493 \nu^{5} + 861393 \nu^{4} + 2441789 \nu^{3} - 3523065 \nu^{2} + \cdots + 533568 ) / 131951 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{2} + 3\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{7} + 3\beta_{6} - 3\beta_{5} + 3\beta_{4} + 2\beta_{3} + 4\beta_{2} + 20\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 35\beta_{7} + 32\beta_{6} - 10\beta_{5} + 11\beta_{3} + 51\beta_{2} + 97\beta _1 + 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 204\beta_{7} + 134\beta_{6} - 126\beta_{5} + 74\beta_{4} + 77\beta_{3} + 206\beta_{2} + 571\beta _1 + 466 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1196\beta_{7} + 980\beta_{6} - 549\beta_{5} + 91\beta_{4} + 464\beta_{3} + 1545\beta_{2} + 3136\beta _1 + 3317 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7037 \beta_{7} + 4871 \beta_{6} - 4284 \beta_{5} + 1800 \beta_{4} + 2725 \beta_{3} + 7766 \beta_{2} + \cdots + 16062 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.74623
3.37343
1.85184
0.157530
−0.575082
−1.03998
−2.74787
−3.76610
0 1.00000 0 1.00000 0 −4.74623 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −2.37343 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 −0.851844 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 0.842470 0 1.00000 0
1.5 0 1.00000 0 1.00000 0 1.57508 0 1.00000 0
1.6 0 1.00000 0 1.00000 0 2.03998 0 1.00000 0
1.7 0 1.00000 0 1.00000 0 3.74787 0 1.00000 0
1.8 0 1.00000 0 1.00000 0 4.76610 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8040.2.a.x ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.x ✓ 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{8} - 5T_{7}^{7} - 24T_{7}^{6} + 144T_{7}^{5} + 5T_{7}^{4} - 728T_{7}^{3} + 650T_{7}^{2} + 456T_{7} - 464 \) Copy content Toggle raw display
\( T_{11}^{8} - 3T_{11}^{7} - 54T_{11}^{6} + 150T_{11}^{5} + 953T_{11}^{4} - 2206T_{11}^{3} - 6198T_{11}^{2} + 7888T_{11} + 15800 \) Copy content Toggle raw display
\( T_{13}^{8} - 13T_{13}^{7} + 29T_{13}^{6} + 241T_{13}^{5} - 1133T_{13}^{4} + 390T_{13}^{3} + 3726T_{13}^{2} - 3072T_{13} - 344 \) Copy content Toggle raw display
\( T_{17}^{8} - 14T_{17}^{7} - 6T_{17}^{6} + 743T_{17}^{5} - 1679T_{17}^{4} - 10624T_{17}^{3} + 32698T_{17}^{2} + 20656T_{17} - 86480 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 5 T^{7} + \cdots - 464 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 15800 \) Copy content Toggle raw display
$13$ \( T^{8} - 13 T^{7} + \cdots - 344 \) Copy content Toggle raw display
$17$ \( T^{8} - 14 T^{7} + \cdots - 86480 \) Copy content Toggle raw display
$19$ \( T^{8} - 11 T^{7} + \cdots - 121072 \) Copy content Toggle raw display
$23$ \( T^{8} - 20 T^{7} + \cdots + 384928 \) Copy content Toggle raw display
$29$ \( T^{8} + 15 T^{7} + \cdots - 2000 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + \cdots + 13996 \) Copy content Toggle raw display
$37$ \( T^{8} - 20 T^{7} + \cdots + 345736 \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{7} + \cdots - 251776 \) Copy content Toggle raw display
$43$ \( T^{8} + T^{7} + \cdots + 896 \) Copy content Toggle raw display
$47$ \( T^{8} - 19 T^{7} + \cdots + 13040 \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{7} + \cdots + 69664 \) Copy content Toggle raw display
$59$ \( T^{8} - 15 T^{7} + \cdots + 558880 \) Copy content Toggle raw display
$61$ \( T^{8} - 23 T^{7} + \cdots + 59428 \) Copy content Toggle raw display
$67$ \( (T - 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 26 T^{7} + \cdots - 25000 \) Copy content Toggle raw display
$73$ \( T^{8} - 38 T^{7} + \cdots + 29936 \) Copy content Toggle raw display
$79$ \( T^{8} + 3 T^{7} + \cdots + 13700 \) Copy content Toggle raw display
$83$ \( T^{8} - 30 T^{7} + \cdots + 1793248 \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + \cdots - 12575792 \) Copy content Toggle raw display
$97$ \( T^{8} - 7 T^{7} + \cdots - 1008992 \) Copy content Toggle raw display
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