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Properties

Label 8004.2.a.k
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $18$
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] = [8004,2,Mod(1,8004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.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: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + \beta_{7} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} + \beta_{7} q^{7} + q^{9} - \beta_{9} q^{11} + \beta_{8} q^{13} + \beta_1 q^{15} - \beta_{13} q^{17} + (\beta_{4} + 1) q^{19} + \beta_{7} q^{21} - q^{23} + (\beta_{2} + 3) q^{25} + q^{27} + q^{29} + (\beta_{14} - \beta_{8} - \beta_{6} + \cdots + 1) q^{31}+ \cdots - \beta_{9} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\!\cdots\!37 \nu^{17} + \cdots - 29\!\cdots\!36 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!71 \nu^{17} + \cdots - 52\!\cdots\!12 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19\!\cdots\!93 \nu^{17} + \cdots + 10\!\cdots\!04 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\!\cdots\!29 \nu^{17} + \cdots + 86\!\cdots\!88 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43\!\cdots\!31 \nu^{17} + \cdots - 39\!\cdots\!32 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44\!\cdots\!61 \nu^{17} + \cdots + 60\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51\!\cdots\!61 \nu^{17} + \cdots + 30\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!29 \nu^{17} + \cdots + 11\!\cdots\!88 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20\!\cdots\!32 \nu^{17} + \cdots - 30\!\cdots\!36 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 49\!\cdots\!31 \nu^{17} + \cdots + 25\!\cdots\!12 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 75\!\cdots\!14 \nu^{17} + \cdots - 70\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25\!\cdots\!13 \nu^{17} + \cdots + 16\!\cdots\!64 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{17} + \cdots - 11\!\cdots\!04 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 51\!\cdots\!47 \nu^{17} + \cdots + 89\!\cdots\!84 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 29\!\cdots\!83 \nu^{17} + \cdots - 14\!\cdots\!76 ) / 56\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + 2\beta_{2} + 11\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} + 2 \beta_{16} + 2 \beta_{15} + 24 \beta_{14} - 6 \beta_{13} + 17 \beta_{12} - 21 \beta_{11} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29 \beta_{16} + 42 \beta_{15} + 49 \beta_{14} - 9 \beta_{12} - 19 \beta_{11} - 42 \beta_{10} + \cdots + 1229 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 39 \beta_{17} + 72 \beta_{16} + 59 \beta_{15} + 481 \beta_{14} - 202 \beta_{13} + 238 \beta_{12} + \cdots + 1892 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 34 \beta_{17} + 637 \beta_{16} + 732 \beta_{15} + 1345 \beta_{14} - 460 \beta_{13} - 272 \beta_{12} + \cdots + 17949 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 967 \beta_{17} + 1816 \beta_{16} + 1311 \beta_{15} + 9196 \beta_{14} - 4954 \beta_{13} + 3130 \beta_{12} + \cdots + 37154 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1479 \beta_{17} + 12606 \beta_{16} + 12264 \beta_{15} + 30580 \beta_{14} - 16021 \beta_{13} + \cdots + 278596 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20278 \beta_{17} + 39789 \beta_{16} + 26568 \beta_{15} + 173546 \beta_{14} - 108389 \beta_{13} + \cdots + 706338 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 42563 \beta_{17} + 238053 \beta_{16} + 205200 \beta_{15} + 639849 \beta_{14} - 406287 \beta_{13} + \cdots + 4526448 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 395183 \beta_{17} + 812873 \beta_{16} + 518825 \beta_{15} + 3266881 \beta_{14} - 2248369 \beta_{13} + \cdots + 13272318 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1025003 \beta_{17} + 4403594 \beta_{16} + 3474899 \beta_{15} + 12834092 \beta_{14} - 9080592 \beta_{13} + \cdots + 76218350 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7452832 \beta_{17} + 15986248 \beta_{16} + 9963642 \beta_{15} + 61539929 \beta_{14} + \cdots + 248546557 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 22438351 \beta_{17} + 80865737 \beta_{16} + 59837840 \beta_{15} + 251519875 \beta_{14} + \cdots + 1320066906 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 138668160 \beta_{17} + 307761935 \beta_{16} + 189811401 \beta_{15} + 1161068731 \beta_{14} + \cdots + 4654580796 \) 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
−3.70317
−3.44612
−3.07005
−2.76979
−2.66067
−2.25234
−1.83157
0.140190
0.235801
0.767523
1.69179
1.90345
2.40801
2.45476
3.38212
3.50248
3.88351
4.36408
0 1.00000 0 −3.70317 0 −1.62014 0 1.00000 0
1.2 0 1.00000 0 −3.44612 0 4.67774 0 1.00000 0
1.3 0 1.00000 0 −3.07005 0 −0.993636 0 1.00000 0
1.4 0 1.00000 0 −2.76979 0 −3.16982 0 1.00000 0
1.5 0 1.00000 0 −2.66067 0 4.50596 0 1.00000 0
1.6 0 1.00000 0 −2.25234 0 3.02254 0 1.00000 0
1.7 0 1.00000 0 −1.83157 0 −3.10909 0 1.00000 0
1.8 0 1.00000 0 0.140190 0 −1.18735 0 1.00000 0
1.9 0 1.00000 0 0.235801 0 1.32081 0 1.00000 0
1.10 0 1.00000 0 0.767523 0 −4.22279 0 1.00000 0
1.11 0 1.00000 0 1.69179 0 3.04338 0 1.00000 0
1.12 0 1.00000 0 1.90345 0 2.67198 0 1.00000 0
1.13 0 1.00000 0 2.40801 0 3.93112 0 1.00000 0
1.14 0 1.00000 0 2.45476 0 0.345256 0 1.00000 0
1.15 0 1.00000 0 3.38212 0 5.02591 0 1.00000 0
1.16 0 1.00000 0 3.50248 0 −4.71048 0 1.00000 0
1.17 0 1.00000 0 3.88351 0 −3.85235 0 1.00000 0
1.18 0 1.00000 0 4.36408 0 0.320970 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

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 8004.2.a.k ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.k ✓ 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5}^{18} - 5 T_{5}^{17} - 55 T_{5}^{16} + 288 T_{5}^{15} + 1222 T_{5}^{14} - 6888 T_{5}^{13} + \cdots - 115488 \) Copy content Toggle raw display
\( T_{7}^{18} - 6 T_{7}^{17} - 77 T_{7}^{16} + 483 T_{7}^{15} + 2356 T_{7}^{14} - 15706 T_{7}^{13} + \cdots + 2162624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 5 T^{17} + \cdots - 115488 \) Copy content Toggle raw display
$7$ \( T^{18} - 6 T^{17} + \cdots + 2162624 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 467213184 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 4082087840 \) Copy content Toggle raw display
$17$ \( T^{18} - 7 T^{17} + \cdots + 33177600 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 138259290112 \) Copy content Toggle raw display
$23$ \( (T + 1)^{18} \) Copy content Toggle raw display
$29$ \( (T - 1)^{18} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 47539851264 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 88011603968 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 1065212050560 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 2872980099072 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 200448000 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 2582846976 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 1081728691200 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 7259060789248 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 134264805632 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 188938579968 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 149749399552 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 928864270080 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 21\!\cdots\!60 \) Copy content Toggle raw display
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