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Properties

Label 98.3.d.a
Level $98$
Weight $3$
Character orbit 98.d
Analytic conductor $2.670$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
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: \(2.67030659073\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{3} + 2 \beta_{2} q^{4} + (4 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{6} + 2 \beta_{3} q^{8} + 6 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{3} + 2 \beta_{2} q^{4} + (4 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{6} + 2 \beta_{3} q^{8} + 6 \beta_1 q^{9} + ( - \beta_{3} - 4 \beta_{2} + \beta_1 - 8) q^{10} + ( - 3 \beta_{3} - 9 \beta_{2} - 3 \beta_1) q^{11} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{12} + (2 \beta_{3} - 12 \beta_{2} + 4 \beta_1 - 6) q^{13} + (3 \beta_{3} - 9) q^{15} + ( - 4 \beta_{2} - 4) q^{16} + ( - 2 \beta_{3} + 5 \beta_{2} + \cdots + 10) q^{17}+ \cdots + ( - 54 \beta_{3} + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 4 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 4 q^{4} + 6 q^{5} - 24 q^{10} + 18 q^{11} - 12 q^{12} - 36 q^{15} - 8 q^{16} + 30 q^{17} - 24 q^{18} - 6 q^{19} + 24 q^{22} + 30 q^{23} - 24 q^{24} + 4 q^{25} - 24 q^{26} + 48 q^{29} - 12 q^{30} + 42 q^{31} + 90 q^{33} - 62 q^{37} + 12 q^{38} + 12 q^{39} + 48 q^{40} - 8 q^{43} + 36 q^{44} - 144 q^{45} + 36 q^{46} - 174 q^{47} - 96 q^{50} + 54 q^{51} + 72 q^{52} - 78 q^{53} + 36 q^{54} + 12 q^{57} + 24 q^{58} + 78 q^{59} + 36 q^{60} + 42 q^{61} + 32 q^{64} - 84 q^{65} + 144 q^{66} - 58 q^{67} - 60 q^{68} - 24 q^{71} - 48 q^{72} - 318 q^{73} + 96 q^{74} - 132 q^{75} + 96 q^{78} + 110 q^{79} - 24 q^{80} + 18 q^{81} + 120 q^{82} - 36 q^{85} + 24 q^{86} + 144 q^{87} - 24 q^{88} + 378 q^{89} - 120 q^{92} - 138 q^{93} + 12 q^{94} - 30 q^{95} + 48 q^{96} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i −0.621320 + 0.358719i −1.00000 − 1.73205i 5.74264 + 3.31552i − 1.01461i 0 2.82843 −4.24264 + 7.34847i −8.12132 + 4.68885i
19.2 0.707107 − 1.22474i 3.62132 − 2.09077i −1.00000 − 1.73205i −2.74264 − 1.58346i − 5.91359i 0 −2.82843 4.24264 − 7.34847i −3.87868 + 2.23936i
31.1 −0.707107 − 1.22474i −0.621320 − 0.358719i −1.00000 + 1.73205i 5.74264 − 3.31552i 1.01461i 0 2.82843 −4.24264 − 7.34847i −8.12132 − 4.68885i
31.2 0.707107 + 1.22474i 3.62132 + 2.09077i −1.00000 + 1.73205i −2.74264 + 1.58346i 5.91359i 0 −2.82843 4.24264 + 7.34847i −3.87868 − 2.23936i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 98.3.d.a 4
3.b odd 2 1 882.3.n.b 4
4.b odd 2 1 784.3.s.c 4
7.b odd 2 1 14.3.d.a ✓ 4
7.c even 3 1 14.3.d.a ✓ 4
7.c even 3 1 98.3.b.b 4
7.d odd 6 1 98.3.b.b 4
7.d odd 6 1 inner 98.3.d.a 4
21.c even 2 1 126.3.n.c 4
21.g even 6 1 882.3.c.f 4
21.g even 6 1 882.3.n.b 4
21.h odd 6 1 126.3.n.c 4
21.h odd 6 1 882.3.c.f 4
28.d even 2 1 112.3.s.b 4
28.f even 6 1 784.3.c.e 4
28.f even 6 1 784.3.s.c 4
28.g odd 6 1 112.3.s.b 4
28.g odd 6 1 784.3.c.e 4
35.c odd 2 1 350.3.k.a 4
35.f even 4 2 350.3.i.a 8
35.j even 6 1 350.3.k.a 4
35.l odd 12 2 350.3.i.a 8
56.e even 2 1 448.3.s.c 4
56.h odd 2 1 448.3.s.d 4
56.k odd 6 1 448.3.s.c 4
56.p even 6 1 448.3.s.d 4
84.h odd 2 1 1008.3.cg.l 4
84.n even 6 1 1008.3.cg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a ✓ 4 7.b odd 2 1
14.3.d.a ✓ 4 7.c even 3 1
98.3.b.b 4 7.c even 3 1
98.3.b.b 4 7.d odd 6 1
98.3.d.a 4 1.a even 1 1 trivial
98.3.d.a 4 7.d odd 6 1 inner
112.3.s.b 4 28.d even 2 1
112.3.s.b 4 28.g odd 6 1
126.3.n.c 4 21.c even 2 1
126.3.n.c 4 21.h odd 6 1
350.3.i.a 8 35.f even 4 2
350.3.i.a 8 35.l odd 12 2
350.3.k.a 4 35.c odd 2 1
350.3.k.a 4 35.j even 6 1
448.3.s.c 4 56.e even 2 1
448.3.s.c 4 56.k odd 6 1
448.3.s.d 4 56.h odd 2 1
448.3.s.d 4 56.p even 6 1
784.3.c.e 4 28.f even 6 1
784.3.c.e 4 28.g odd 6 1
784.3.s.c 4 4.b odd 2 1
784.3.s.c 4 28.f even 6 1
882.3.c.f 4 21.g even 6 1
882.3.c.f 4 21.h odd 6 1
882.3.n.b 4 3.b odd 2 1
882.3.n.b 4 21.g even 6 1
1008.3.cg.l 4 84.h odd 2 1
1008.3.cg.l 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6T_{3}^{3} + 9T_{3}^{2} + 18T_{3} + 9 \) acting on \(S_{3}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$13$ \( T^{4} + 264T^{2} + 7056 \) Copy content Toggle raw display
$17$ \( T^{4} - 30 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 30 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$29$ \( (T^{2} - 24 T + 72)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 42 T^{3} + \cdots + 1447209 \) Copy content Toggle raw display
$37$ \( T^{4} + 62 T^{3} + \cdots + 36481 \) Copy content Toggle raw display
$41$ \( T^{4} + 1224 T^{2} + 345744 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 174 T^{3} + \cdots + 6335289 \) Copy content Toggle raw display
$53$ \( T^{4} + 78 T^{3} + \cdots + 1520289 \) Copy content Toggle raw display
$59$ \( T^{4} - 78 T^{3} + \cdots + 10517049 \) Copy content Toggle raw display
$61$ \( T^{4} - 42 T^{3} + \cdots + 35964009 \) Copy content Toggle raw display
$67$ \( T^{4} + 58 T^{3} + \cdots + 10297681 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 1764)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 318 T^{3} + \cdots + 47485881 \) Copy content Toggle raw display
$79$ \( T^{4} - 110 T^{3} + \cdots + 6630625 \) Copy content Toggle raw display
$83$ \( T^{4} + 27936 T^{2} + 189778176 \) Copy content Toggle raw display
$89$ \( T^{4} - 378 T^{3} + \cdots + 71419401 \) Copy content Toggle raw display
$97$ \( T^{4} + 11016 T^{2} + 6780816 \) Copy content Toggle raw display
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