↑

Properties

Label 9800.2.a.db
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, 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("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.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: \(78.2533939809\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 22x^{8} - 16x^{7} + 146x^{6} + 200x^{5} - 206x^{4} - 440x^{3} - 124x^{2} + 72x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1960)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( - \beta_{8} - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + ( - \beta_{8} - \beta_{4} + 1) q^{9} + (\beta_1 - 1) q^{11} + (\beta_{6} - \beta_{5} - \beta_{3}) q^{13} + (\beta_{9} - \beta_{7} + \beta_{3}) q^{17} + (\beta_{9} - \beta_{7} + \cdots + \beta_{5}) q^{19}+ \cdots + (6 \beta_{8} + 2 \beta_{4} + \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 14 q^{9} - 12 q^{11} - 16 q^{23} + 12 q^{29} - 36 q^{37} - 20 q^{39} - 24 q^{43} - 36 q^{51} - 8 q^{53} - 16 q^{57} - 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} - 48 q^{93} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 22x^{8} - 16x^{7} + 146x^{6} + 200x^{5} - 206x^{4} - 440x^{3} - 124x^{2} + 72x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 134 \nu^{9} - 1139 \nu^{8} - 724 \nu^{7} + 18925 \nu^{6} - 5572 \nu^{5} - 98852 \nu^{4} + \cdots - 15000 ) / 5966 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 96 \nu^{9} - 2167 \nu^{8} + 7108 \nu^{7} + 39557 \nu^{6} - 71340 \nu^{5} - 236118 \nu^{4} + \cdots - 73312 ) / 2983 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 424 \nu^{9} + 621 \nu^{8} + 8524 \nu^{7} - 6010 \nu^{6} - 55564 \nu^{5} - 296 \nu^{4} + \cdots - 5608 ) / 5966 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1151 \nu^{9} - 2326 \nu^{8} - 20466 \nu^{7} + 24293 \nu^{6} + 115180 \nu^{5} - 26944 \nu^{4} + \cdots - 4982 ) / 5966 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1431 \nu^{9} - 1260 \nu^{8} + 36226 \nu^{7} + 40122 \nu^{6} - 272544 \nu^{5} - 355976 \nu^{4} + \cdots - 135264 ) / 5966 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1450 \nu^{9} + 393 \nu^{8} + 33034 \nu^{7} + 10881 \nu^{6} - 234088 \nu^{5} - 185508 \nu^{4} + \cdots - 49346 ) / 5966 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2516 \nu^{9} - 3488 \nu^{8} - 49568 \nu^{7} + 26095 \nu^{6} + 314068 \nu^{5} + 94736 \nu^{4} + \cdots + 61982 ) / 5966 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2773 \nu^{9} - 4181 \nu^{8} - 53806 \nu^{7} + 34142 \nu^{6} + 338376 \nu^{5} + 76736 \nu^{4} + \cdots + 84236 ) / 5966 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2773 \nu^{9} - 4181 \nu^{8} - 53806 \nu^{7} + 34142 \nu^{6} + 338376 \nu^{5} + 76736 \nu^{4} + \cdots + 84236 ) / 5966 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{9} + \beta_{8} + 4\beta_{7} - \beta_{4} + \beta_{2} - 3\beta _1 + 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{9} + 4\beta_{8} + \beta_{7} - \beta_{5} + 2\beta_{3} + \beta_{2} - \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -28\beta_{9} + 10\beta_{8} + 24\beta_{7} + 4\beta_{6} - 4\beta_{5} - 6\beta_{4} + 7\beta_{2} - 16\beta _1 + 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 111 \beta_{9} + 81 \beta_{8} + 32 \beta_{7} + 2 \beta_{6} - 27 \beta_{5} - 3 \beta_{4} + 45 \beta_{3} + \cdots + 155 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 176 \beta_{9} + 75 \beta_{8} + 130 \beta_{7} + 28 \beta_{6} - 34 \beta_{5} - 29 \beta_{4} + \cdots + 391 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 648 \beta_{9} + 444 \beta_{8} + 223 \beta_{7} + 31 \beta_{6} - 164 \beta_{5} - 22 \beta_{4} + \cdots + 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4304 \beta_{9} + 2055 \beta_{8} + 2784 \beta_{7} + 672 \beta_{6} - 944 \beta_{5} - 523 \beta_{4} + \cdots + 8839 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7702 \beta_{9} + 5048 \beta_{8} + 2902 \beta_{7} + 586 \beta_{6} - 1990 \beta_{5} - 234 \beta_{4} + \cdots + 12906 \) 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
−1.84294
0.422497
−0.609577
3.50994
1.61569
−0.432569
−1.92302
−1.30040
3.39389
−2.83351
0 −3.30674 0 0 0 0 0 7.93455 0
1.2 0 −2.69859 0 0 0 0 0 4.28239 0
1.3 0 −1.35056 0 0 0 0 0 −1.17599 0
1.4 0 −1.12212 0 0 0 0 0 −1.74084 0
1.5 0 −0.836591 0 0 0 0 0 −2.30012 0
1.6 0 0.836591 0 0 0 0 0 −2.30012 0
1.7 0 1.12212 0 0 0 0 0 −1.74084 0
1.8 0 1.35056 0 0 0 0 0 −1.17599 0
1.9 0 2.69859 0 0 0 0 0 4.28239 0
1.10 0 3.30674 0 0 0 0 0 7.93455 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.db 10
5.b even 2 1 9800.2.a.dc 10
5.c odd 4 2 1960.2.g.g ✓ 20
7.b odd 2 1 inner 9800.2.a.db 10
35.c odd 2 1 9800.2.a.dc 10
35.f even 4 2 1960.2.g.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.g.g ✓ 20 5.c odd 4 2
1960.2.g.g ✓ 20 35.f even 4 2
9800.2.a.db 10 1.a even 1 1 trivial
9800.2.a.db 10 7.b odd 2 1 inner
9800.2.a.dc 10 5.b even 2 1
9800.2.a.dc 10 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{10} - 22T_{3}^{8} + 153T_{3}^{6} - 384T_{3}^{4} + 384T_{3}^{2} - 128 \) Copy content Toggle raw display
\( T_{11}^{5} + 6T_{11}^{4} - 15T_{11}^{3} - 76T_{11}^{2} + 112T_{11} - 32 \) Copy content Toggle raw display
\( T_{13}^{10} - 92T_{13}^{8} + 2873T_{13}^{6} - 34134T_{13}^{4} + 110844T_{13}^{2} - 95048 \) Copy content Toggle raw display
\( T_{19}^{10} - 112T_{19}^{8} + 3776T_{19}^{6} - 39040T_{19}^{4} + 153600T_{19}^{2} - 204800 \) Copy content Toggle raw display
\( T_{23}^{5} + 8T_{23}^{4} - 70T_{23}^{3} - 528T_{23}^{2} + 1056T_{23} + 7232 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 22 T^{8} + \cdots - 128 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} + 6 T^{4} - 15 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 92 T^{8} + \cdots - 95048 \) Copy content Toggle raw display
$17$ \( T^{10} - 76 T^{8} + \cdots - 245000 \) Copy content Toggle raw display
$19$ \( T^{10} - 112 T^{8} + \cdots - 204800 \) Copy content Toggle raw display
$23$ \( (T^{5} + 8 T^{4} + \cdots + 7232)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 6 T^{4} + \cdots - 2848)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 164 T^{8} + \cdots - 6422528 \) Copy content Toggle raw display
$37$ \( (T^{5} + 18 T^{4} + \cdots - 976)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 170 T^{8} + \cdots - 3135008 \) Copy content Toggle raw display
$43$ \( (T^{5} + 12 T^{4} + \cdots + 2560)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 150 T^{8} + \cdots - 131072 \) Copy content Toggle raw display
$53$ \( (T^{5} + 4 T^{4} + \cdots - 3904)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 685388288 \) Copy content Toggle raw display
$61$ \( T^{10} - 230 T^{8} + \cdots - 8000000 \) Copy content Toggle raw display
$67$ \( (T^{5} + 20 T^{4} + \cdots - 11648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 4 T^{4} + \cdots - 640)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 150 T^{8} + \cdots - 8388608 \) Copy content Toggle raw display
$79$ \( (T^{5} + 2 T^{4} + \cdots - 2144)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 416 T^{8} + \cdots - 52428800 \) Copy content Toggle raw display
$89$ \( T^{10} - 214 T^{8} + \cdots - 51200 \) Copy content Toggle raw display
$97$ \( T^{10} - 428 T^{8} + \cdots - 41332232 \) Copy content Toggle raw display
show more
show less