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Properties

Label 98.5.d.b
Level $98$
Weight $5$
Character orbit 98.d
Analytic conductor $10.130$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,5,Mod(19,98)]
 
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([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 98.d (of order \(6\), 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: \(10.1302563822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta_{6} q^{2} + 3 \beta_{3} q^{3} + ( - 8 \beta_{4} - 8) q^{4} + ( - 19 \beta_{7} + 13 \beta_{5} - 19 \beta_{3}) q^{5} + (6 \beta_{7} + 6 \beta_{5} - 6 \beta_1) q^{6} + 16 \beta_{2} q^{8} + ( - 9 \beta_{6} + 63 \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{6} q^{2} + 3 \beta_{3} q^{3} + ( - 8 \beta_{4} - 8) q^{4} + ( - 19 \beta_{7} + 13 \beta_{5} - 19 \beta_{3}) q^{5} + (6 \beta_{7} + 6 \beta_{5} - 6 \beta_1) q^{6} + 16 \beta_{2} q^{8} + ( - 9 \beta_{6} + 63 \beta_{4}) q^{9} + (12 \beta_{3} + 64 \beta_1) q^{10} + (23 \beta_{6} + 162 \beta_{4} + \cdots + 162) q^{11}+ \cdots + (9 \beta_{2} - 9792) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} - 252 q^{9} + 648 q^{11} - 912 q^{15} - 256 q^{16} + 144 q^{18} - 736 q^{22} + 344 q^{23} + 1740 q^{25} - 9088 q^{29} + 288 q^{30} + 4032 q^{36} - 4672 q^{37} + 2280 q^{39} + 10736 q^{43} + 5184 q^{44} - 8448 q^{46} + 9664 q^{50} + 3240 q^{51} - 6168 q^{53} + 2208 q^{57} - 4384 q^{58} + 3648 q^{60} + 4096 q^{64} + 11424 q^{65} - 3376 q^{67} + 11264 q^{71} + 1152 q^{72} - 5952 q^{74} - 11904 q^{78} - 9360 q^{79} - 10692 q^{81} + 5136 q^{85} - 6512 q^{86} + 2944 q^{88} - 5504 q^{92} - 3408 q^{93} + 32632 q^{95} - 78336 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 34\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} - 28\nu^{3} + 16\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{7} + 21\nu^{5} - 70\nu^{3} + 6\nu ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 3\beta_{5} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} - 6\beta_{4} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 10\beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{3} - 34\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.662827 + 0.382683i
0.662827 − 0.382683i
1.60021 − 0.923880i
−1.60021 + 0.923880i
−0.662827 − 0.382683i
0.662827 + 0.382683i
1.60021 + 0.923880i
−1.60021 − 0.923880i
−1.41421 + 2.44949i −4.80062 + 2.77164i −4.00000 − 6.92820i 21.7872 + 12.5788i − 15.6788i 0 22.6274 −25.1360 + 43.5369i −61.6234 + 35.5783i
19.2 −1.41421 + 2.44949i 4.80062 − 2.77164i −4.00000 − 6.92820i −21.7872 − 12.5788i 15.6788i 0 22.6274 −25.1360 + 43.5369i 61.6234 − 35.5783i
19.3 1.41421 − 2.44949i −1.98848 + 1.14805i −4.00000 − 6.92820i 33.3964 + 19.2814i 6.49435i 0 −22.6274 −37.8640 + 65.5823i 94.4593 − 54.5361i
19.4 1.41421 − 2.44949i 1.98848 − 1.14805i −4.00000 − 6.92820i −33.3964 − 19.2814i − 6.49435i 0 −22.6274 −37.8640 + 65.5823i −94.4593 + 54.5361i
31.1 −1.41421 − 2.44949i −4.80062 − 2.77164i −4.00000 + 6.92820i 21.7872 − 12.5788i 15.6788i 0 22.6274 −25.1360 − 43.5369i −61.6234 − 35.5783i
31.2 −1.41421 − 2.44949i 4.80062 + 2.77164i −4.00000 + 6.92820i −21.7872 + 12.5788i − 15.6788i 0 22.6274 −25.1360 − 43.5369i 61.6234 + 35.5783i
31.3 1.41421 + 2.44949i −1.98848 − 1.14805i −4.00000 + 6.92820i 33.3964 − 19.2814i − 6.49435i 0 −22.6274 −37.8640 − 65.5823i 94.4593 + 54.5361i
31.4 1.41421 + 2.44949i 1.98848 + 1.14805i −4.00000 + 6.92820i −33.3964 + 19.2814i 6.49435i 0 −22.6274 −37.8640 − 65.5823i −94.4593 − 54.5361i
\(n\): e.g. 2-40 or 990-1000
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 98.5.d.b 8
7.b odd 2 1 inner 98.5.d.b 8
7.c even 3 1 98.5.b.c ✓ 4
7.c even 3 1 inner 98.5.d.b 8
7.d odd 6 1 98.5.b.c ✓ 4
7.d odd 6 1 inner 98.5.d.b 8
21.g even 6 1 882.5.c.d 4
21.h odd 6 1 882.5.c.d 4
28.f even 6 1 784.5.c.d 4
28.g odd 6 1 784.5.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.b.c ✓ 4 7.c even 3 1
98.5.b.c ✓ 4 7.d odd 6 1
98.5.d.b 8 1.a even 1 1 trivial
98.5.d.b 8 7.b odd 2 1 inner
98.5.d.b 8 7.c even 3 1 inner
98.5.d.b 8 7.d odd 6 1 inner
784.5.c.d 4 28.f even 6 1
784.5.c.d 4 28.g odd 6 1
882.5.c.d 4 21.g even 6 1
882.5.c.d 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 36T_{3}^{6} + 1134T_{3}^{4} - 5832T_{3}^{2} + 26244 \) acting on \(S_{5}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 36 T^{6} + \cdots + 26244 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 885842380864 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 324 T^{3} + \cdots + 634334596)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 39464 T^{2} + 14300552)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{4} - 172 T^{3} + \cdots + 302689229584)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2272 T + 1140344)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 1182560551936)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 10873453918082)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2684 T + 1469666)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 2276199829264)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{4} + 1688 T^{3} + \cdots + 359836818496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2816 T - 15493408)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 73\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 639511671490624)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 15\!\cdots\!78)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 43\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 10\!\cdots\!02)^{2} \) Copy content Toggle raw display
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