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Properties

Label 98.8.c.n
Level $98$
Weight $8$
Character orbit 98.c
Analytic conductor $30.614$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,8,Mod(67,98)]
 
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([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), 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: \(30.6137324974\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} - 1790 x^{6} + 5384 x^{5} + 1207827 x^{4} - 2424632 x^{3} - 364080554 x^{2} + \cdots + 41373599428 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (8 \beta_1 + 8) q^{2} + (2 \beta_{3} + \beta_{2}) q^{3} + 64 \beta_1 q^{4} - 10 \beta_{6} q^{5} + ( - 16 \beta_{6} + 8 \beta_{5} + \cdots + 8 \beta_{2}) q^{6} - 512 q^{8} + (4 \beta_{7} - 1807 \beta_1 - 1807) q^{9}+ \cdots + ( - 22535 \beta_{4} - 15541232) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{2} - 256 q^{4} - 4096 q^{8} - 7228 q^{9} + 6272 q^{11} + 15680 q^{15} - 16384 q^{16} + 57824 q^{18} + 100352 q^{22} + 12544 q^{23} + 273300 q^{25} - 151472 q^{29} + 62720 q^{30} + 131072 q^{32}+ \cdots - 124329856 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} - 1790 x^{6} + 5384 x^{5} + 1207827 x^{4} - 2424632 x^{3} - 364080554 x^{2} + \cdots + 41373599428 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 4489\nu^{4} + 8988\nu^{3} + 2826029\nu^{2} - 2830524\nu - 552213858 ) / 6501586 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 4489\nu^{4} + 8988\nu^{3} + 6076822\nu^{2} - 6081317\nu - 2008569122 ) / 3250793 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 396 \nu^{7} - 1386 \nu^{6} - 745336 \nu^{5} + 1866805 \nu^{4} + 559592728 \nu^{3} + \cdots + 68636579158 ) / 16626877397 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1584 \nu^{7} + 5544 \nu^{6} + 2981344 \nu^{5} - 7467220 \nu^{4} - 2238370912 \nu^{3} + \cdots - 507322600190 ) / 16626877397 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1803 \nu^{6} + 5409 \nu^{5} + 2421437 \nu^{4} - 4851889 \nu^{3} - 1092935276 \nu^{2} + \cdots + 165736034072 ) / 6501586 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71682 \nu^{7} - 250887 \nu^{6} - 96159819 \nu^{5} + 241026765 \nu^{4} + 43352613941 \nu^{3} + \cdots + 3293624618256 ) / 33253754794 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 284992 \nu^{7} + 997472 \nu^{6} + 510562810 \nu^{5} - 1278900705 \nu^{4} + \cdots - 25850839394296 ) / 16626877397 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 4\beta_{3} + 14 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 4\beta_{3} + 28\beta_{2} - 56\beta _1 + 12558 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} + 8\beta_{6} + 451\beta_{4} + 5398\beta_{3} + 42\beta_{2} - 84\beta _1 + 18830 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{7} + 16 \beta_{6} - 112 \beta_{5} + 901 \beta_{4} + 10792 \beta_{3} + 25144 \beta_{2} + \cdots + 5543846 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9000 \beta_{7} + 36024 \beta_{6} - 280 \beta_{5} + 203831 \beta_{4} + 4031970 \beta_{3} + \cdots + 13828234 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 26970 \beta_{7} + 108032 \beta_{6} - 252224 \beta_{5} + 609241 \beta_{4} + 12068932 \beta_{3} + \cdots + 2406196618 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8498602 \beta_{7} + 56800772 \beta_{6} - 881804 \beta_{5} + 92279167 \beta_{4} + 2520100706 \beta_{3} + \cdots + 8373311310 ) / 28 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
21.0120 + 1.22474i
−20.0120 − 1.22474i
−21.4262 + 1.22474i
22.4262 − 1.22474i
21.0120 − 1.22474i
−20.0120 + 1.22474i
−21.4262 − 1.22474i
22.4262 + 1.22474i
4.00000 + 6.92820i −39.9078 + 69.1224i −32.0000 + 55.4256i −49.4975 − 85.7321i −638.525 0 −512.000 −2091.77 − 3623.05i 395.980 − 685.857i
67.2 4.00000 + 6.92820i −20.1088 + 34.8295i −32.0000 + 55.4256i 49.4975 + 85.7321i −321.741 0 −512.000 284.769 + 493.235i −395.980 + 685.857i
67.3 4.00000 + 6.92820i 20.1088 − 34.8295i −32.0000 + 55.4256i −49.4975 − 85.7321i 321.741 0 −512.000 284.769 + 493.235i 395.980 − 685.857i
67.4 4.00000 + 6.92820i 39.9078 − 69.1224i −32.0000 + 55.4256i 49.4975 + 85.7321i 638.525 0 −512.000 −2091.77 − 3623.05i −395.980 + 685.857i
79.1 4.00000 − 6.92820i −39.9078 − 69.1224i −32.0000 − 55.4256i −49.4975 + 85.7321i −638.525 0 −512.000 −2091.77 + 3623.05i 395.980 + 685.857i
79.2 4.00000 − 6.92820i −20.1088 − 34.8295i −32.0000 − 55.4256i 49.4975 − 85.7321i −321.741 0 −512.000 284.769 − 493.235i −395.980 − 685.857i
79.3 4.00000 − 6.92820i 20.1088 + 34.8295i −32.0000 − 55.4256i −49.4975 + 85.7321i 321.741 0 −512.000 284.769 − 493.235i 395.980 + 685.857i
79.4 4.00000 − 6.92820i 39.9078 + 69.1224i −32.0000 − 55.4256i 49.4975 − 85.7321i 638.525 0 −512.000 −2091.77 + 3623.05i −395.980 − 685.857i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
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 98.8.c.n 8
7.b odd 2 1 inner 98.8.c.n 8
7.c even 3 1 98.8.a.l ✓ 4
7.c even 3 1 inner 98.8.c.n 8
7.d odd 6 1 98.8.a.l ✓ 4
7.d odd 6 1 inner 98.8.c.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.8.a.l ✓ 4 7.c even 3 1
98.8.a.l ✓ 4 7.d odd 6 1
98.8.c.n 8 1.a even 1 1 trivial
98.8.c.n 8 7.b odd 2 1 inner
98.8.c.n 8 7.c even 3 1 inner
98.8.c.n 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 7988T_{3}^{6} + 53504044T_{3}^{4} + 82309150800T_{3}^{2} + 106174476810000 \) acting on \(S_{8}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 106174476810000 \) Copy content Toggle raw display
$5$ \( (T^{4} + 9800 T^{2} + 96040000)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 682988673938704)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 37868 T - 28920403868)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 76\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 77\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 1410880 T + 470168031964)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 45\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 66\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 2982923301456)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 33\!\cdots\!24)^{2} \) Copy content Toggle raw display
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