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Properties

Label 98.3.d.b
Level $98$
Weight $3$
Character orbit 98.d
Analytic conductor $2.670$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,3,Mod(19,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 98.d (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(2.67030659073\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (2 \beta_{3} + 2 \beta_1) q^{3} + ( - 2 \beta_{4} - 2) q^{4} + ( - \beta_{7} + 4 \beta_{5} - \beta_{3}) q^{5} + 4 \beta_{7} q^{6} + 2 \beta_{2} q^{8} + ( - 8 \beta_{6} - 7 \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (2 \beta_{3} + 2 \beta_1) q^{3} + ( - 2 \beta_{4} - 2) q^{4} + ( - \beta_{7} + 4 \beta_{5} - \beta_{3}) q^{5} + 4 \beta_{7} q^{6} + 2 \beta_{2} q^{8} + ( - 8 \beta_{6} - 7 \beta_{4}) q^{9} + ( - 3 \beta_{3} + 5 \beta_1) q^{10} + (4 \beta_{6} - 12 \beta_{4} + \cdots - 12) q^{11}+ \cdots + ( - 68 \beta_{2} - 20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 28 q^{9} - 48 q^{11} + 96 q^{15} - 16 q^{16} + 64 q^{18} - 64 q^{22} - 80 q^{23} + 36 q^{25} - 128 q^{29} + 32 q^{30} - 112 q^{36} + 128 q^{37} + 80 q^{39} - 352 q^{43} - 96 q^{44} - 96 q^{46} + 368 q^{50} - 144 q^{51} + 96 q^{53} + 640 q^{57} + 104 q^{58} - 96 q^{60} + 64 q^{64} - 256 q^{67} + 64 q^{71} + 128 q^{72} + 24 q^{74} - 256 q^{78} + 192 q^{79} - 132 q^{81} - 576 q^{85} - 64 q^{86} + 64 q^{88} + 320 q^{92} - 192 q^{93} + 80 q^{95} - 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 34\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} - 28\nu^{3} + 16\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{7} + 21\nu^{5} - 70\nu^{3} + 6\nu ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 3\beta_{5} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} - 6\beta_{4} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 10\beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{3} - 34\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.662827 + 0.382683i
0.662827 − 0.382683i
−1.60021 + 0.923880i
1.60021 − 0.923880i
−0.662827 − 0.382683i
0.662827 + 0.382683i
−1.60021 − 0.923880i
1.60021 + 0.923880i
−0.707107 + 1.22474i −4.52607 + 2.61313i −1.00000 − 1.73205i −1.05110 − 0.606854i − 7.39104i 0 2.82843 9.15685 − 15.8601i 1.48648 − 0.858221i
19.2 −0.707107 + 1.22474i 4.52607 − 2.61313i −1.00000 − 1.73205i 1.05110 + 0.606854i 7.39104i 0 2.82843 9.15685 − 15.8601i −1.48648 + 0.858221i
19.3 0.707107 − 1.22474i −1.87476 + 1.08239i −1.00000 − 1.73205i −7.06365 − 4.07820i 3.06147i 0 −2.82843 −2.15685 + 3.73578i −9.98951 + 5.76745i
19.4 0.707107 − 1.22474i 1.87476 − 1.08239i −1.00000 − 1.73205i 7.06365 + 4.07820i − 3.06147i 0 −2.82843 −2.15685 + 3.73578i 9.98951 − 5.76745i
31.1 −0.707107 − 1.22474i −4.52607 − 2.61313i −1.00000 + 1.73205i −1.05110 + 0.606854i 7.39104i 0 2.82843 9.15685 + 15.8601i 1.48648 + 0.858221i
31.2 −0.707107 − 1.22474i 4.52607 + 2.61313i −1.00000 + 1.73205i 1.05110 − 0.606854i − 7.39104i 0 2.82843 9.15685 + 15.8601i −1.48648 − 0.858221i
31.3 0.707107 + 1.22474i −1.87476 − 1.08239i −1.00000 + 1.73205i −7.06365 + 4.07820i − 3.06147i 0 −2.82843 −2.15685 − 3.73578i −9.98951 − 5.76745i
31.4 0.707107 + 1.22474i 1.87476 + 1.08239i −1.00000 + 1.73205i 7.06365 − 4.07820i 3.06147i 0 −2.82843 −2.15685 − 3.73578i 9.98951 + 5.76745i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.3.d.b 8
3.b odd 2 1 882.3.n.j 8
4.b odd 2 1 784.3.s.j 8
7.b odd 2 1 inner 98.3.d.b 8
7.c even 3 1 98.3.b.a ✓ 4
7.c even 3 1 inner 98.3.d.b 8
7.d odd 6 1 98.3.b.a ✓ 4
7.d odd 6 1 inner 98.3.d.b 8
21.c even 2 1 882.3.n.j 8
21.g even 6 1 882.3.c.a 4
21.g even 6 1 882.3.n.j 8
21.h odd 6 1 882.3.c.a 4
21.h odd 6 1 882.3.n.j 8
28.d even 2 1 784.3.s.j 8
28.f even 6 1 784.3.c.b 4
28.f even 6 1 784.3.s.j 8
28.g odd 6 1 784.3.c.b 4
28.g odd 6 1 784.3.s.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.3.b.a ✓ 4 7.c even 3 1
98.3.b.a ✓ 4 7.d odd 6 1
98.3.d.b 8 1.a even 1 1 trivial
98.3.d.b 8 7.b odd 2 1 inner
98.3.d.b 8 7.c even 3 1 inner
98.3.d.b 8 7.d odd 6 1 inner
784.3.c.b 4 28.f even 6 1
784.3.c.b 4 28.g odd 6 1
784.3.s.j 8 4.b odd 2 1
784.3.s.j 8 28.d even 2 1
784.3.s.j 8 28.f even 6 1
784.3.s.j 8 28.g odd 6 1
882.3.c.a 4 21.g even 6 1
882.3.c.a 4 21.h odd 6 1
882.3.n.j 8 3.b odd 2 1
882.3.n.j 8 21.c even 2 1
882.3.n.j 8 21.g even 6 1
882.3.n.j 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 32T_{3}^{6} + 896T_{3}^{4} - 4096T_{3}^{2} + 16384 \) acting on \(S_{3}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 32 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$5$ \( T^{8} - 68 T^{6} + \cdots + 9604 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{3} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 68 T^{2} + 98)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 324 T^{6} + \cdots + 172186884 \) Copy content Toggle raw display
$19$ \( T^{8} - 928 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$23$ \( (T^{4} + 40 T^{3} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32 T - 82)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} - 320 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} + \cdots + 1012036)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1028 T^{2} + 164738)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 88 T + 1808)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20466865537024 \) Copy content Toggle raw display
$53$ \( (T^{4} - 48 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 1368408064 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{4} + 128 T^{3} + \cdots + 8667136)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T - 3136)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{4} - 96 T^{3} + \cdots + 2262016)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2848 T^{2} + 1812608)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 72564506884 \) Copy content Toggle raw display
$97$ \( (T^{4} + 26500 T^{2} + 54100802)^{2} \) Copy content Toggle raw display
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