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Properties

Label 98.6.c.i
Level $98$
Weight $6$
Character orbit 98.c
Analytic conductor $15.718$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [98,6,Mod(67,98)] 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("98.67"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-16,0,-64,0,0,0,512,-712] 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: \(15.7176143417\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
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: \( x^{8} - 4x^{7} - 782x^{6} + 2360x^{5} + 232083x^{4} - 468104x^{3} - 30964778x^{2} + 31199224x + 1568558404 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 \beta_1 q^{2} + \beta_{5} q^{3} + ( - 16 \beta_1 - 16) q^{4} + ( - \beta_{3} - 3 \beta_{2}) q^{5} + ( - 4 \beta_{5} + 4 \beta_{2}) q^{6} + 64 q^{8} + ( - \beta_{7} - \beta_{4} + 178 \beta_1) q^{9}+ \cdots + ( - 377 \beta_{4} - 88625) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{2} - 64 q^{4} + 512 q^{8} - 712 q^{9} - 628 q^{11} + 10496 q^{15} - 1024 q^{16} - 2848 q^{18} + 5024 q^{22} - 2624 q^{23} - 4224 q^{25} + 35464 q^{29} - 20992 q^{30} - 4096 q^{32} + 22784 q^{36}+ \cdots - 709000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 782x^{6} + 2360x^{5} + 232083x^{4} - 468104x^{3} - 30964778x^{2} + 31199224x + 1568558404 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 1969\nu^{4} + 3948\nu^{3} + 544925\nu^{2} - 546900\nu - 47631234 ) / 1264018 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7938 \nu^{7} - 62968 \nu^{6} + 6811289 \nu^{5} + 72927312 \nu^{4} - 2342331110 \nu^{3} + \cdots + 7664781535922 ) / 57355448759 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2268 \nu^{7} + 7938 \nu^{6} + 1868296 \nu^{5} - 4690585 \nu^{4} - 618053896 \nu^{3} + \cdots - 33526714522 ) / 8193635537 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4536 \nu^{7} + 15876 \nu^{6} + 3736592 \nu^{5} - 9381170 \nu^{4} - 1236107792 \nu^{3} + \cdots - 124408877803 ) / 8193635537 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 636699 \nu^{7} - 33845076 \nu^{6} + 480976080 \nu^{5} + 20238060814 \nu^{4} + \cdots + 282957132345076 ) / 114710897518 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 222 \nu^{7} + 777 \nu^{6} + 128617 \nu^{5} - 323485 \nu^{4} - 24957999 \nu^{3} + \cdots - 815610796 ) / 20509726 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 717760 \nu^{7} + 2512160 \nu^{6} + 562362898 \nu^{5} - 1412187645 \nu^{4} + \cdots - 5511217271224 ) / 16387271074 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} + 7 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 12\beta_{3} - 28\beta_{2} - 28\beta _1 + 2751 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} - 4\beta_{6} + 199\beta_{4} - 1170\beta_{3} - 42\beta_{2} - 42\beta _1 + 4123 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{7} + 48 \beta_{6} - 112 \beta_{5} + 397 \beta_{4} + 3192 \beta_{3} - 11032 \beta_{2} + \cdots + 521059 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3960 \beta_{7} - 7792 \beta_{6} - 280 \beta_{5} + 39779 \beta_{4} - 382102 \beta_{3} - 27510 \beta_{2} + \cdots + 1295777 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11850 \beta_{7} + 31776 \beta_{6} - 111104 \beta_{5} + 118345 \beta_{4} + 489348 \beta_{3} + \cdots + 94519565 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1643698 \beta_{7} - 5316780 \beta_{6} - 387884 \beta_{5} + 7970803 \beta_{4} - 105317074 \beta_{3} + \cdots + 326288067 ) / 14 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
13.8730 − 1.22474i
−12.8730 + 1.22474i
−14.2872 − 1.22474i
15.2872 + 1.22474i
13.8730 + 1.22474i
−12.8730 − 1.22474i
−14.2872 + 1.22474i
15.2872 − 1.22474i
−2.00000 − 3.46410i −12.4310 + 21.5312i −8.00000 + 13.8564i −42.2428 − 73.1667i 99.4482 0 64.0000 −187.561 − 324.865i −168.971 + 292.667i
67.2 −2.00000 − 3.46410i −7.48128 + 12.9580i −8.00000 + 13.8564i −17.4941 − 30.3007i 59.8502 0 64.0000 9.56089 + 16.5600i −69.9764 + 121.203i
67.3 −2.00000 − 3.46410i 7.48128 − 12.9580i −8.00000 + 13.8564i 17.4941 + 30.3007i −59.8502 0 64.0000 9.56089 + 16.5600i 69.9764 − 121.203i
67.4 −2.00000 − 3.46410i 12.4310 − 21.5312i −8.00000 + 13.8564i 42.2428 + 73.1667i −99.4482 0 64.0000 −187.561 − 324.865i 168.971 − 292.667i
79.1 −2.00000 + 3.46410i −12.4310 − 21.5312i −8.00000 − 13.8564i −42.2428 + 73.1667i 99.4482 0 64.0000 −187.561 + 324.865i −168.971 − 292.667i
79.2 −2.00000 + 3.46410i −7.48128 − 12.9580i −8.00000 − 13.8564i −17.4941 + 30.3007i 59.8502 0 64.0000 9.56089 − 16.5600i −69.9764 − 121.203i
79.3 −2.00000 + 3.46410i 7.48128 + 12.9580i −8.00000 − 13.8564i 17.4941 − 30.3007i −59.8502 0 64.0000 9.56089 − 16.5600i 69.9764 + 121.203i
79.4 −2.00000 + 3.46410i 12.4310 + 21.5312i −8.00000 − 13.8564i 42.2428 − 73.1667i −99.4482 0 64.0000 −187.561 + 324.865i 168.971 + 292.667i
\(n\): 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 98.6.c.i 8
7.b odd 2 1 inner 98.6.c.i 8
7.c even 3 1 98.6.a.i ✓ 4
7.c even 3 1 inner 98.6.c.i 8
7.d odd 6 1 98.6.a.i ✓ 4
7.d odd 6 1 inner 98.6.c.i 8
21.g even 6 1 882.6.a.bv 4
21.h odd 6 1 882.6.a.bv 4
28.f even 6 1 784.6.a.bg 4
28.g odd 6 1 784.6.a.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.6.a.i ✓ 4 7.c even 3 1
98.6.a.i ✓ 4 7.d odd 6 1
98.6.c.i 8 1.a even 1 1 trivial
98.6.c.i 8 7.b odd 2 1 inner
98.6.c.i 8 7.c even 3 1 inner
98.6.c.i 8 7.d odd 6 1 inner
784.6.a.bg 4 28.f even 6 1
784.6.a.bg 4 28.g odd 6 1
882.6.a.bv 4 21.g even 6 1
882.6.a.bv 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 842T_{3}^{6} + 570580T_{3}^{4} + 116519328T_{3}^{2} + 19150131456 \) acting on \(S_{6}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 19150131456 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 76351525540096 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 314 T^{3} + \cdots + 105666604096)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 720594 T^{2} + 112021056)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 26677968765184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8866 T + 19301776)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 6812768176384)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4918 T - 22280072)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 28332 T - 3437738496)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 75\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 28\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 32780823506116)^{2} \) Copy content Toggle raw display
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