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Properties

Label 98.7.d.b
Level $98$
Weight $7$
Character orbit 98.d
Analytic conductor $22.545$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

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

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-128,0,0,0,0,-1092] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.5453001947\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.44930433024.14
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 77x^{4} + 230x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{4} q^{3} + (32 \beta_1 - 32) q^{4} + ( - 5 \beta_{7} + 5 \beta_{3}) q^{5} + ( - \beta_{3} + 5 \beta_{2}) q^{6} - 32 \beta_{6} q^{8} + (78 \beta_{5} - 273 \beta_1) q^{9} + ( - 25 \beta_{7} - 35 \beta_{4}) q^{10}+ \cdots + ( - 89856 \beta_{6} - 332982) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{4} - 1092 q^{9} + 4440 q^{11} - 8640 q^{15} - 4096 q^{16} + 9984 q^{18} + 3072 q^{22} - 40584 q^{23} + 40700 q^{25} - 36528 q^{29} + 42240 q^{30} + 69888 q^{36} + 23192 q^{37} - 208608 q^{39}+ \cdots - 2663856 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 77x^{4} + 230x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\nu^{6} + 77\nu^{4} + 770\nu^{2} + 2300 ) / 1771 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 32\nu^{7} - 462\nu^{5} - 1078\nu^{3} - 3266\nu ) / 1771 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64\nu^{7} + 2618\nu^{5} + 15554\nu^{3} + 74934\nu ) / 1771 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 106\nu^{7} + 462\nu^{5} + 1078\nu^{3} + 3128\nu ) / 1771 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -108\nu^{6} - 1540\nu^{4} - 8316\nu^{2} - 24840 ) / 1771 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{6} + 620 ) / 77 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -478\nu^{7} - 2618\nu^{5} - 15554\nu^{3} + 10488\nu ) / 1771 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{4} + \beta_{3} + 3\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 20\beta _1 - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} - 34\beta_{4} + 3\beta_{3} + 17\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} - 54\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{7} + 101\beta_{4} - 14\beta_{3} - 202\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -77\beta_{6} + 620 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{7} + 619\beta_{4} + \beta_{3} + 619\beta_{2} ) / 48 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{1}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
1.26631 − 2.19332i
−1.26631 + 2.19332i
0.946809 − 1.63992i
−0.946809 + 1.63992i
1.26631 + 2.19332i
−1.26631 − 2.19332i
0.946809 + 1.63992i
−0.946809 − 1.63992i
−2.82843 + 4.89898i −25.9408 + 14.9769i −16.0000 − 27.7128i 85.1967 + 49.1883i − 169.445i 0 181.019 84.1173 − 145.695i −481.945 + 278.251i
19.2 −2.82843 + 4.89898i 25.9408 − 14.9769i −16.0000 − 27.7128i −85.1967 − 49.1883i 169.445i 0 181.019 84.1173 − 145.695i 481.945 − 278.251i
19.3 2.82843 − 4.89898i −3.32777 + 1.92129i −16.0000 − 27.7128i −177.318 − 102.374i 21.7369i 0 −181.019 −357.117 + 618.545i −1003.06 + 579.117i
19.4 2.82843 − 4.89898i 3.32777 − 1.92129i −16.0000 − 27.7128i 177.318 + 102.374i − 21.7369i 0 −181.019 −357.117 + 618.545i 1003.06 − 579.117i
31.1 −2.82843 − 4.89898i −25.9408 − 14.9769i −16.0000 + 27.7128i 85.1967 − 49.1883i 169.445i 0 181.019 84.1173 + 145.695i −481.945 − 278.251i
31.2 −2.82843 − 4.89898i 25.9408 + 14.9769i −16.0000 + 27.7128i −85.1967 + 49.1883i − 169.445i 0 181.019 84.1173 + 145.695i 481.945 + 278.251i
31.3 2.82843 + 4.89898i −3.32777 − 1.92129i −16.0000 + 27.7128i −177.318 + 102.374i − 21.7369i 0 −181.019 −357.117 − 618.545i −1003.06 − 579.117i
31.4 2.82843 + 4.89898i 3.32777 + 1.92129i −16.0000 + 27.7128i 177.318 − 102.374i 21.7369i 0 −181.019 −357.117 − 618.545i 1003.06 + 579.117i
\(n\): e.g. 2-40 or 80-90
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 98.7.d.b 8
7.b odd 2 1 inner 98.7.d.b 8
7.c even 3 1 14.7.b.a ✓ 4
7.c even 3 1 inner 98.7.d.b 8
7.d odd 6 1 14.7.b.a ✓ 4
7.d odd 6 1 inner 98.7.d.b 8
21.g even 6 1 126.7.c.a 4
21.h odd 6 1 126.7.c.a 4
28.f even 6 1 112.7.c.c 4
28.g odd 6 1 112.7.c.c 4
35.i odd 6 1 350.7.b.a 4
35.j even 6 1 350.7.b.a 4
35.k even 12 2 350.7.d.a 8
35.l odd 12 2 350.7.d.a 8
56.j odd 6 1 448.7.c.h 4
56.k odd 6 1 448.7.c.e 4
56.m even 6 1 448.7.c.e 4
56.p even 6 1 448.7.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.b.a ✓ 4 7.c even 3 1
14.7.b.a ✓ 4 7.d odd 6 1
98.7.d.b 8 1.a even 1 1 trivial
98.7.d.b 8 7.b odd 2 1 inner
98.7.d.b 8 7.c even 3 1 inner
98.7.d.b 8 7.d odd 6 1 inner
112.7.c.c 4 28.f even 6 1
112.7.c.c 4 28.g odd 6 1
126.7.c.a 4 21.g even 6 1
126.7.c.a 4 21.h odd 6 1
350.7.b.a 4 35.i odd 6 1
350.7.b.a 4 35.j even 6 1
350.7.d.a 8 35.k even 12 2
350.7.d.a 8 35.l odd 12 2
448.7.c.e 4 56.k odd 6 1
448.7.c.e 4 56.m even 6 1
448.7.c.h 4 56.j odd 6 1
448.7.c.h 4 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 912T_{3}^{6} + 818496T_{3}^{4} - 12082176T_{3}^{2} + 175509504 \) acting on \(S_{7}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 912 T^{6} + \cdots + 175509504 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 1506736610064)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 26266196601792)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9132 T - 1357906716)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 75\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 22348 T - 10521464924)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 185366809042704)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 225804 T - 95443772508)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 72\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 21\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 39\!\cdots\!12)^{2} \) Copy content Toggle raw display
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