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Properties

Label 8004.2.a.h
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{5} + (\beta_{12} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{5} + (\beta_{12} - 1) q^{7} + q^{9} + \beta_{10} q^{11} - \beta_{11} q^{13} - \beta_1 q^{15} + ( - \beta_{12} - \beta_{11} + \beta_{3}) q^{17} + ( - \beta_{11} - \beta_{9} + \beta_{5} + \cdots - 1) q^{19}+ \cdots + \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1690006925 \nu^{12} + 14190017491 \nu^{11} + 29382303177 \nu^{10} - 452564017120 \nu^{9} + \cdots + 30728510682372 ) / 1564335784584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13787660437 \nu^{12} + 39779102519 \nu^{11} + 446589014949 \nu^{10} + \cdots - 34093521863244 ) / 1564335784584 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15536082095 \nu^{12} + 45388236577 \nu^{11} + 513140254587 \nu^{10} + \cdots - 52850941100892 ) / 1564335784584 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 905712243 \nu^{12} - 4656709097 \nu^{11} - 23117172527 \nu^{10} + 141573490824 \nu^{9} + \cdots - 557666645400 ) / 86907543588 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1019705775 \nu^{12} + 3494592671 \nu^{11} + 31563088001 \nu^{10} - 106568815746 \nu^{9} + \cdots - 2052309240216 ) / 86907543588 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2437175963 \nu^{12} + 8549688593 \nu^{11} + 79092677443 \nu^{10} - 269326838404 \nu^{9} + \cdots - 11343802257396 ) / 173815087176 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9734759813 \nu^{12} - 29112317203 \nu^{11} - 328022585121 \nu^{10} + 908769962656 \nu^{9} + \cdots + 34162657809876 ) / 521445261528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5847934169 \nu^{12} + 19644634477 \nu^{11} + 194857600527 \nu^{10} - 623292257566 \nu^{9} + \cdots - 28467417835332 ) / 260722630764 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41709748121 \nu^{12} + 127653995371 \nu^{11} + 1373816531673 \nu^{10} + \cdots - 119261517833700 ) / 1564335784584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30303637499 \nu^{12} - 106595365123 \nu^{11} - 972434668869 \nu^{10} + 3320160658246 \nu^{9} + \cdots + 106105578701832 ) / 782167892292 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20460244711 \nu^{12} + 65987984381 \nu^{11} + 668494464807 \nu^{10} + \cdots - 70761852857268 ) / 521445261528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{12} - \beta_{11} - 2 \beta_{9} + 3 \beta_{8} - 3 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 18 \beta_{12} - \beta_{11} + 24 \beta_{10} + 19 \beta_{9} + 22 \beta_{8} - 4 \beta_{7} - 7 \beta_{4} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 66 \beta_{12} - 18 \beta_{11} + 10 \beta_{10} - 45 \beta_{9} + 73 \beta_{8} + 10 \beta_{7} - 76 \beta_{6} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 296 \beta_{12} - 39 \beta_{11} + 465 \beta_{10} + 312 \beta_{9} + 440 \beta_{8} - 82 \beta_{7} + \cdots + 944 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1219 \beta_{12} - 317 \beta_{11} + 374 \beta_{10} - 812 \beta_{9} + 1556 \beta_{8} + 251 \beta_{7} + \cdots - 656 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4765 \beta_{12} - 982 \beta_{11} + 8657 \beta_{10} + 5067 \beta_{9} + 8645 \beta_{8} - 1396 \beta_{7} + \cdots + 15162 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21176 \beta_{12} - 5934 \beta_{11} + 10269 \beta_{10} - 13604 \beta_{9} + 31979 \beta_{8} + 4777 \beta_{7} + \cdots - 7476 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 76136 \beta_{12} - 21554 \beta_{11} + 160366 \beta_{10} + 83086 \beta_{9} + 169068 \beta_{8} + \cdots + 257569 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 358520 \beta_{12} - 114667 \beta_{11} + 249828 \beta_{10} - 219382 \beta_{9} + 647100 \beta_{8} + \cdots - 43483 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1210903 \beta_{12} - 447830 \beta_{11} + 2982387 \beta_{10} + 1377739 \beta_{9} + 3304019 \beta_{8} + \cdots + 4510744 \) 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
−4.03097
−2.79682
−1.91772
−1.42689
−1.03716
−0.672529
1.23293
1.59293
1.79014
2.19606
2.69157
2.93518
4.44328
0 −1.00000 0 −4.03097 0 −3.90486 0 1.00000 0
1.2 0 −1.00000 0 −2.79682 0 0.517096 0 1.00000 0
1.3 0 −1.00000 0 −1.91772 0 −2.53447 0 1.00000 0
1.4 0 −1.00000 0 −1.42689 0 −0.237900 0 1.00000 0
1.5 0 −1.00000 0 −1.03716 0 0.840656 0 1.00000 0
1.6 0 −1.00000 0 −0.672529 0 −0.215981 0 1.00000 0
1.7 0 −1.00000 0 1.23293 0 −4.84147 0 1.00000 0
1.8 0 −1.00000 0 1.59293 0 3.02509 0 1.00000 0
1.9 0 −1.00000 0 1.79014 0 −5.03804 0 1.00000 0
1.10 0 −1.00000 0 2.19606 0 4.05334 0 1.00000 0
1.11 0 −1.00000 0 2.69157 0 −2.34023 0 1.00000 0
1.12 0 −1.00000 0 2.93518 0 2.71487 0 1.00000 0
1.13 0 −1.00000 0 4.44328 0 −0.0380915 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.h ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.h ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5}^{13} - 5 T_{5}^{12} - 27 T_{5}^{11} + 158 T_{5}^{10} + 180 T_{5}^{9} - 1652 T_{5}^{8} + \cdots - 5832 \) Copy content Toggle raw display
\( T_{7}^{13} + 8 T_{7}^{12} - 23 T_{7}^{11} - 283 T_{7}^{10} + 20 T_{7}^{9} + 3318 T_{7}^{8} + 2079 T_{7}^{7} + \cdots - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} \) Copy content Toggle raw display
$3$ \( (T + 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 5 T^{12} + \cdots - 5832 \) Copy content Toggle raw display
$7$ \( T^{13} + 8 T^{12} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{13} - T^{12} + \cdots - 2952 \) Copy content Toggle raw display
$13$ \( T^{13} - T^{12} + \cdots + 42444 \) Copy content Toggle raw display
$17$ \( T^{13} + 2 T^{12} + \cdots + 165888 \) Copy content Toggle raw display
$19$ \( T^{13} + 10 T^{12} + \cdots + 82944 \) Copy content Toggle raw display
$23$ \( (T + 1)^{13} \) Copy content Toggle raw display
$29$ \( (T + 1)^{13} \) Copy content Toggle raw display
$31$ \( T^{13} + 26 T^{12} + \cdots + 383488 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 244323648 \) Copy content Toggle raw display
$41$ \( T^{13} - 21 T^{12} + \cdots - 1000544 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 602399232 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 3908373504 \) Copy content Toggle raw display
$53$ \( T^{13} - 7 T^{12} + \cdots - 322272 \) Copy content Toggle raw display
$59$ \( T^{13} + 11 T^{12} + \cdots - 35188992 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 1147566898944 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 5721469584 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 1588002816 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 3998247936 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 514676970624 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 5858592768 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 9383738112 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 431792790048 \) Copy content Toggle raw display
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