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Properties

Label 7400.2.a.be
Level $7400$
Weight $2$
Character orbit 7400.a
Self dual yes
Analytic conductor $59.089$
Analytic rank $0$
Dimension $16$
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] = [7400,2,Mod(1,7400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7400, 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("7400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7400.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: \(59.0892974957\)
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: \( x^{16} - 2 x^{15} - 33 x^{14} + 66 x^{13} + 404 x^{12} - 796 x^{11} - 2273 x^{10} + 4284 x^{9} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1480)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{8} + 1) q^{7} + (\beta_{2} + 1) q^{9} + \beta_{7} q^{11} - \beta_{14} q^{13} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{11} + 1) q^{19} + ( - \beta_{15} - \beta_{13} + \cdots + \beta_1) q^{21}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} + 9 q^{7} + 22 q^{9} + 3 q^{11} - 4 q^{13} - 9 q^{17} + 14 q^{19} + 8 q^{21} + 18 q^{23} - 4 q^{27} + q^{29} + 23 q^{31} - 10 q^{33} + 16 q^{37} + 23 q^{41} - 11 q^{43} + 32 q^{47} + 45 q^{49}+ \cdots - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 33 x^{14} + 66 x^{13} + 404 x^{12} - 796 x^{11} - 2273 x^{10} + 4284 x^{9} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 734038251140 \nu^{15} + 22507424628655 \nu^{14} - 66747757257083 \nu^{13} + \cdots - 71\!\cdots\!90 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1978876789657 \nu^{15} - 50267633335272 \nu^{14} + 27826535434397 \nu^{13} + \cdots + 12\!\cdots\!12 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1615550965631 \nu^{15} - 10335920720187 \nu^{14} + 81254903921620 \nu^{13} + \cdots + 48\!\cdots\!56 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7125784976077 \nu^{15} + 1317200650744 \nu^{14} + 256975318051897 \nu^{13} + \cdots + 43\!\cdots\!24 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14399954587895 \nu^{15} + 14637956723222 \nu^{14} + 503531838899579 \nu^{13} + \cdots - 22\!\cdots\!72 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30079433014147 \nu^{15} + 40227434856318 \nu^{14} + \cdots + 52\!\cdots\!88 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 44720099371065 \nu^{15} + 92885284952754 \nu^{14} + \cdots - 54\!\cdots\!52 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23630524916430 \nu^{15} + 60147992099071 \nu^{14} + 756186101280111 \nu^{13} + \cdots - 39\!\cdots\!26 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30499877110179 \nu^{15} - 55597545410478 \nu^{14} + \cdots + 887908579904830 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 75348438992069 \nu^{15} + 128912803883594 \nu^{14} + \cdots + 17\!\cdots\!56 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38227715756425 \nu^{15} - 76049503836810 \nu^{14} + \cdots + 34\!\cdots\!40 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 66124921197320 \nu^{15} - 135734026778415 \nu^{14} + \cdots + 55\!\cdots\!84 ) / 548023221091718 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 132436574713783 \nu^{15} + 260852429120424 \nu^{14} + \cdots - 88\!\cdots\!08 ) / 10\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} - \beta_{8} + 2\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{7} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{15} - 12 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} - 15 \beta_{14} - 16 \beta_{13} + 20 \beta_{12} + 32 \beta_{11} - 21 \beta_{10} + \cdots + 312 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 148 \beta_{15} - 117 \beta_{14} - 44 \beta_{13} - 60 \beta_{12} - 63 \beta_{11} - 7 \beta_{10} + \cdots - 213 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 56 \beta_{15} - 176 \beta_{14} - 189 \beta_{13} + 300 \beta_{12} + 421 \beta_{11} - 310 \beta_{10} + \cdots + 3131 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1661 \beta_{15} - 1067 \beta_{14} - 701 \beta_{13} - 876 \beta_{12} - 962 \beta_{11} + 65 \beta_{10} + \cdots - 2691 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1048 \beta_{15} - 1927 \beta_{14} - 1954 \beta_{13} + 4035 \beta_{12} + 5239 \beta_{11} - 4068 \beta_{10} + \cdots + 32571 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18718 \beta_{15} - 9349 \beta_{14} - 9865 \beta_{13} - 11457 \beta_{12} - 13063 \beta_{11} + \cdots - 33456 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16574 \beta_{15} - 20652 \beta_{14} - 18433 \beta_{13} + 51360 \beta_{12} + 63623 \beta_{11} + \cdots + 347406 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 212203 \beta_{15} - 78720 \beta_{14} - 130171 \beta_{13} - 142701 \beta_{12} - 167732 \beta_{11} + \cdots - 413744 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 239591 \beta_{15} - 220497 \beta_{14} - 158584 \beta_{13} + 633529 \beta_{12} + 762358 \beta_{11} + \cdots + 3774216 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2418060 \beta_{15} - 627694 \beta_{14} - 1651946 \beta_{13} - 1735219 \beta_{12} - 2089599 \beta_{11} + \cdots - 5102286 \) 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
−3.43243
−3.04373
−2.70502
−1.47399
−1.16696
−0.902602
−0.189778
−0.100725
0.336056
0.444459
1.18682
1.75598
2.38176
2.53062
3.10845
3.27108
0 −3.43243 0 0 0 4.96522 0 8.78154 0
1.2 0 −3.04373 0 0 0 −3.50721 0 6.26427 0
1.3 0 −2.70502 0 0 0 0.533645 0 4.31715 0
1.4 0 −1.47399 0 0 0 0.907297 0 −0.827364 0
1.5 0 −1.16696 0 0 0 −2.23151 0 −1.63819 0
1.6 0 −0.902602 0 0 0 −1.32204 0 −2.18531 0
1.7 0 −0.189778 0 0 0 2.89459 0 −2.96398 0
1.8 0 −0.100725 0 0 0 3.98340 0 −2.98985 0
1.9 0 0.336056 0 0 0 3.02156 0 −2.88707 0
1.10 0 0.444459 0 0 0 −4.10506 0 −2.80246 0
1.11 0 1.18682 0 0 0 0.0116605 0 −1.59145 0
1.12 0 1.75598 0 0 0 −4.69529 0 0.0834815 0
1.13 0 2.38176 0 0 0 5.05491 0 2.67277 0
1.14 0 2.53062 0 0 0 −0.665840 0 3.40405 0
1.15 0 3.10845 0 0 0 3.70694 0 6.66248 0
1.16 0 3.27108 0 0 0 0.447728 0 7.69993 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(37\) \( -1 \)

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 7400.2.a.be 16
5.b even 2 1 7400.2.a.bd 16
5.c odd 4 2 1480.2.d.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.d.c ✓ 32 5.c odd 4 2
7400.2.a.bd 16 5.b even 2 1
7400.2.a.be 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7400))\):

\( T_{3}^{16} - 2 T_{3}^{15} - 33 T_{3}^{14} + 66 T_{3}^{13} + 404 T_{3}^{12} - 796 T_{3}^{11} - 2273 T_{3}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{16} - 9 T_{7}^{15} - 38 T_{7}^{14} + 522 T_{7}^{13} + 41 T_{7}^{12} - 10867 T_{7}^{11} + \cdots + 1088 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 9 T^{15} + \cdots + 1088 \) Copy content Toggle raw display
$11$ \( T^{16} - 3 T^{15} + \cdots - 23763968 \) Copy content Toggle raw display
$13$ \( T^{16} + 4 T^{15} + \cdots + 29707264 \) Copy content Toggle raw display
$17$ \( T^{16} + 9 T^{15} + \cdots + 97800192 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 538505216 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 174358528 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 49220009984 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 314427792 \) Copy content Toggle raw display
$37$ \( (T - 1)^{16} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 31001520832 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 374194176 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 7578127104 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 442116241408 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 63898386432 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 990990761984 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 147435008 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 20551437811712 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 313851217482752 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 2447424256 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 2485440892672 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 205458112512 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 1415820529664 \) Copy content Toggle raw display
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