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Properties

Label 98.7.b.c
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,-1512] 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: \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + 32 q^{4} + (\beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{5} + ( - 3 \beta_{5} - \beta_{4} + \cdots - 4 \beta_1) q^{6} + 32 \beta_{2} q^{8} + ( - \beta_{7} + \beta_{6} + 53 \beta_{2} - 189) q^{9}+ \cdots + ( - 693 \beta_{7} + 993 \beta_{6} + \cdots - 578241) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 256 q^{4} - 1512 q^{9} + 2712 q^{11} + 27144 q^{15} + 8192 q^{16} + 13632 q^{18} + 25248 q^{22} + 8256 q^{23} - 9328 q^{25} - 30312 q^{29} - 19296 q^{30} - 48384 q^{36} + 12248 q^{37} - 201528 q^{39}+ \cdots - 4625928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10737582 \nu^{7} - 884171859 \nu^{6} + 3763944460 \nu^{5} - 189328804101 \nu^{4} + \cdots - 25\!\cdots\!75 ) / 235754430743125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14893 \nu^{7} - 236741 \nu^{6} + 3320715 \nu^{5} + 8132426 \nu^{4} + 331237078 \nu^{3} + \cdots + 846958238000 ) / 106489495000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13703534741 \nu^{7} + 783676536967 \nu^{6} - 17549027818005 \nu^{5} + \cdots + 15\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4720226743 \nu^{7} + 273735261539 \nu^{6} + 1627505361655 \nu^{5} + 80948867089876 \nu^{4} + \cdots + 71\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16436625417 \nu^{7} - 325838189779 \nu^{6} + 5277118070185 \nu^{5} - 79313151390756 \nu^{4} + \cdots - 68\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 189405903441 \nu^{7} - 597326060217 \nu^{6} + 42232124128455 \nu^{5} + \cdots + 11\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53284369571 \nu^{7} - 114528049627 \nu^{6} + 11880897421605 \nu^{5} + 29096299771222 \nu^{4} + \cdots + 26\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 33\beta_{5} + 9\beta_{4} + 9\beta_{3} - 11\beta_{2} - 6\beta _1 + 42 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 3\beta_{6} + 75\beta_{5} - 54\beta_{4} - 12\beta_{3} + 379\beta_{2} - 849\beta _1 - 5943 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 143\beta_{7} - 217\beta_{6} + 2846\beta_{2} - 13377 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 427 \beta_{7} - 1137 \beta_{6} - 24453 \beta_{5} + 15486 \beta_{4} + 576 \beta_{3} + 111421 \beta_{2} + \cdots - 973497 ) / 84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23747 \beta_{7} + 45681 \beta_{6} - 554595 \beta_{5} + 55908 \beta_{4} - 404706 \beta_{3} + \cdots + 4513761 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -130311\beta_{7} + 329029\beta_{6} - 26161377\beta_{2} + 178784949 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4421299 \beta_{7} + 9613689 \beta_{6} + 116235321 \beta_{5} - 23388042 \beta_{4} + 85652148 \beta_{3} + \cdots + 1260231609 ) / 84 \) 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
−6.30576 − 10.9219i
7.51287 − 13.0127i
7.51287 + 13.0127i
−6.30576 + 10.9219i
−4.86132 + 8.42006i
4.65421 + 8.06134i
4.65421 − 8.06134i
−4.86132 − 8.42006i
−5.65685 − 42.4218i 32.0000 187.462i 239.974i 0 −181.019 −1070.61 − 1060.44i
97.2 −5.65685 − 25.2754i 32.0000 − 12.4115i 142.979i 0 −181.019 90.1550 70.2099i
97.3 −5.65685 25.2754i 32.0000 12.4115i − 142.979i 0 −181.019 90.1550 − 70.2099i
97.4 −5.65685 42.4218i 32.0000 − 187.462i − 239.974i 0 −181.019 −1070.61 1060.44i
97.5 5.65685 − 31.8814i 32.0000 129.137i − 180.349i 0 181.019 −287.425 730.512i
97.6 5.65685 − 14.7350i 32.0000 123.254i − 83.3537i 0 181.019 511.880 697.228i
97.7 5.65685 14.7350i 32.0000 − 123.254i 83.3537i 0 181.019 511.880 − 697.228i
97.8 5.65685 31.8814i 32.0000 − 129.137i 180.349i 0 181.019 −287.425 − 730.512i
\(n\): e.g. 2-40 or 990-1000
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.c 8
7.b odd 2 1 inner 98.7.b.c 8
7.c even 3 1 14.7.d.a ✓ 8
7.c even 3 1 98.7.d.c 8
7.d odd 6 1 14.7.d.a ✓ 8
7.d odd 6 1 98.7.d.c 8
21.g even 6 1 126.7.n.c 8
21.h odd 6 1 126.7.n.c 8
28.f even 6 1 112.7.s.c 8
28.g odd 6 1 112.7.s.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.d.a ✓ 8 7.c even 3 1
14.7.d.a ✓ 8 7.d odd 6 1
98.7.b.c 8 1.a even 1 1 trivial
98.7.b.c 8 7.b odd 2 1 inner
98.7.d.c 8 7.c even 3 1
98.7.d.c 8 7.d odd 6 1
112.7.s.c 8 28.f even 6 1
112.7.s.c 8 28.g odd 6 1
126.7.n.c 8 21.g even 6 1
126.7.n.c 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 3672T_{3}^{6} + 4378302T_{3}^{4} + 1956305304T_{3}^{2} + 253716712209 \) 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
$3$ \( T^{8} + \cdots + 253716712209 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 2395921328703)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 10\!\cdots\!03)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 15\!\cdots\!69)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 71\!\cdots\!97)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 70\!\cdots\!13)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 56\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 79\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 91\!\cdots\!75)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
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