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Properties

Label 98.7.b.b
Level $98$
Weight $7$
Character orbit 98.b
Analytic conductor $22.545$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,7,Mod(97,98)] 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("98.97"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,256,0,0,0,0,-7392] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} - 4574 x^{6} + 10300 x^{5} + 7143777 x^{4} - 6367456 x^{3} - 4336152944 x^{2} + \cdots + 895644970318 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 \beta_1 q^{2} + (\beta_{4} - \beta_{2}) q^{3} + 32 q^{4} + ( - 11 \beta_{6} - 2 \beta_{4} + \cdots - 3 \beta_{2}) q^{5} + (4 \beta_{6} - 8 \beta_{4} - 4 \beta_{3}) q^{6} + 128 \beta_1 q^{8} + ( - 3 \beta_{7} - 174 \beta_1 - 924) q^{9}+ \cdots + (1860 \beta_{7} + 3705 \beta_{5} + \cdots + 1368900) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 256 q^{4} - 7392 q^{9} - 648 q^{11} - 28632 q^{15} + 8192 q^{16} - 11232 q^{18} - 20896 q^{22} - 5128 q^{23} - 78096 q^{25} - 35296 q^{29} + 86208 q^{30} - 236544 q^{36} + 238208 q^{37} - 544920 q^{39}+ \cdots + 10951200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} - 4574 x^{6} + 10300 x^{5} + 7143777 x^{4} - 6367456 x^{3} - 4336152944 x^{2} + \cdots + 895644970318 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 234984317442 \nu^{7} + 35786357369516 \nu^{6} + 698644857780452 \nu^{5} + \cdots - 83\!\cdots\!40 ) / 11\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 45\!\cdots\!92 \nu^{7} + \cdots - 21\!\cdots\!80 ) / 16\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\!\cdots\!83 \nu^{7} + \cdots - 33\!\cdots\!04 ) / 16\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!14 \nu^{7} + \cdots - 53\!\cdots\!70 ) / 23\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!06 \nu^{7} + \cdots + 11\!\cdots\!90 ) / 16\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!80 \nu^{7} + \cdots - 89\!\cdots\!08 ) / 23\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\!\cdots\!64 \nu^{7} + \cdots + 21\!\cdots\!71 ) / 16\!\cdots\!23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{5} - 2\beta_{4} - 7\beta _1 + 7 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 16\beta_{6} + \beta_{5} - 4\beta_{4} + 28\beta_{3} - 28\beta_{2} - 6019\beta _1 + 16037 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2001 \beta_{7} + 2626 \beta_{6} + 1143 \beta_{5} - 9486 \beta_{4} + 84 \beta_{3} - 126 \beta_{2} + \cdots + 42091 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10273 \beta_{7} + 63104 \beta_{6} + 3995 \beta_{5} - 55072 \beta_{4} + 88368 \beta_{3} + \cdots + 23443063 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3619433 \beta_{7} + 10137788 \beta_{6} + 1667017 \beta_{5} - 26868680 \beta_{4} + 501340 \beta_{3} + \cdots + 126824187 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 26569311 \beta_{7} + 192475664 \beta_{6} + 10796967 \beta_{5} - 231143348 \beta_{4} + \cdots + 38311755965 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6345646175 \beta_{7} + 27014245188 \beta_{6} + 2717706433 \beta_{5} - 65540317558 \beta_{4} + \cdots + 310474436125 ) / 14 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−40.6924 + 1.84776i
43.1066 − 1.84776i
43.1066 + 1.84776i
−40.6924 − 1.84776i
22.9862 + 0.765367i
−23.4004 − 0.765367i
−23.4004 + 0.765367i
22.9862 − 0.765367i
−5.65685 − 51.4700i 32.0000 − 190.235i 291.158i 0 −181.019 −1920.16 1076.13i
97.2 −5.65685 − 12.6670i 32.0000 − 92.8795i 71.6555i 0 −181.019 568.546 525.406i
97.3 −5.65685 12.6670i 32.0000 92.8795i − 71.6555i 0 −181.019 568.546 − 525.406i
97.4 −5.65685 51.4700i 32.0000 190.235i − 291.158i 0 −181.019 −1920.16 − 1076.13i
97.5 5.65685 − 50.8921i 32.0000 86.1422i − 287.889i 0 181.019 −1861.00 487.294i
97.6 5.65685 − 34.8194i 32.0000 − 222.062i − 196.968i 0 181.019 −483.387 − 1256.17i
97.7 5.65685 34.8194i 32.0000 222.062i 196.968i 0 181.019 −483.387 1256.17i
97.8 5.65685 50.8921i 32.0000 − 86.1422i 287.889i 0 181.019 −1861.00 − 487.294i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

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 98.7.b.b ✓ 8
7.b odd 2 1 inner 98.7.b.b ✓ 8
7.c even 3 2 98.7.d.d 16
7.d odd 6 2 98.7.d.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.7.b.b ✓ 8 1.a even 1 1 trivial
98.7.b.b ✓ 8 7.b odd 2 1 inner
98.7.d.d 16 7.c even 3 2
98.7.d.d 16 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6612T_{3}^{6} + 14248386T_{3}^{4} + 10438687296T_{3}^{2} + 1334745817344 \) acting on \(S_{7}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 1334745817344 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 1956539497696)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 17\!\cdots\!92)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 54\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 62\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 95\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 93\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 78\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 80\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 82\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
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