Properties

Label 98.2.c.c
Level $98$
Weight $2$
Character orbit 98.c
Analytic conductor $0.783$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} + \beta_{3} q^{6} + q^{8} -\beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} + \beta_{3} q^{6} + q^{8} -\beta_{2} q^{9} -2 \beta_{1} q^{10} + ( 2 + 2 \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{3} ) q^{12} -4 q^{15} + \beta_{2} q^{16} + \beta_{1} q^{17} + ( 1 + \beta_{2} ) q^{18} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{19} -2 \beta_{3} q^{20} -2 q^{22} -4 \beta_{2} q^{23} + \beta_{1} q^{24} + ( -3 - 3 \beta_{2} ) q^{25} -4 \beta_{3} q^{27} + 2 q^{29} -4 \beta_{2} q^{30} -6 \beta_{1} q^{31} + ( -1 - \beta_{2} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{33} + \beta_{3} q^{34} - q^{36} + 10 \beta_{2} q^{37} + 5 \beta_{1} q^{38} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{40} + 7 \beta_{3} q^{41} + 2 q^{43} -2 \beta_{2} q^{44} + 2 \beta_{1} q^{45} + ( 4 + 4 \beta_{2} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + \beta_{3} q^{48} + 3 q^{50} + 2 \beta_{2} q^{51} + ( 2 + 2 \beta_{2} ) q^{53} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{54} + 4 \beta_{3} q^{55} + 10 q^{57} + 2 \beta_{2} q^{58} + \beta_{1} q^{59} + ( 4 + 4 \beta_{2} ) q^{60} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{61} -6 \beta_{3} q^{62} + q^{64} -2 \beta_{1} q^{66} + ( -12 - 12 \beta_{2} ) q^{67} + ( -\beta_{1} - \beta_{3} ) q^{68} -4 \beta_{3} q^{69} -12 q^{71} -\beta_{2} q^{72} + \beta_{1} q^{73} + ( -10 - 10 \beta_{2} ) q^{74} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{75} + 5 \beta_{3} q^{76} -4 \beta_{2} q^{79} -2 \beta_{1} q^{80} + ( 5 + 5 \beta_{2} ) q^{81} + ( -7 \beta_{1} - 7 \beta_{3} ) q^{82} -7 \beta_{3} q^{83} -4 q^{85} + 2 \beta_{2} q^{86} + 2 \beta_{1} q^{87} + ( 2 + 2 \beta_{2} ) q^{88} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{89} + 2 \beta_{3} q^{90} -4 q^{92} -12 \beta_{2} q^{93} -2 \beta_{1} q^{94} + ( 20 + 20 \beta_{2} ) q^{95} + ( -\beta_{1} - \beta_{3} ) q^{96} -7 \beta_{3} q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 4q^{8} + 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 4q^{8} + 2q^{9} + 4q^{11} - 16q^{15} - 2q^{16} + 2q^{18} - 8q^{22} + 8q^{23} - 6q^{25} + 8q^{29} + 8q^{30} - 2q^{32} - 4q^{36} - 20q^{37} + 8q^{43} + 4q^{44} + 8q^{46} + 12q^{50} - 4q^{51} + 4q^{53} + 40q^{57} - 4q^{58} + 8q^{60} + 4q^{64} - 24q^{67} - 48q^{71} + 2q^{72} - 20q^{74} + 8q^{79} + 10q^{81} - 16q^{85} - 4q^{86} + 4q^{88} - 16q^{92} + 24q^{93} + 40q^{95} + 8q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.500000 0.866025i −0.707107 + 1.22474i −0.500000 + 0.866025i 1.41421 + 2.44949i 1.41421 0 1.00000 0.500000 + 0.866025i 1.41421 2.44949i
67.2 −0.500000 0.866025i 0.707107 1.22474i −0.500000 + 0.866025i −1.41421 2.44949i −1.41421 0 1.00000 0.500000 + 0.866025i −1.41421 + 2.44949i
79.1 −0.500000 + 0.866025i −0.707107 1.22474i −0.500000 0.866025i 1.41421 2.44949i 1.41421 0 1.00000 0.500000 0.866025i 1.41421 + 2.44949i
79.2 −0.500000 + 0.866025i 0.707107 + 1.22474i −0.500000 0.866025i −1.41421 + 2.44949i −1.41421 0 1.00000 0.500000 0.866025i −1.41421 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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 98.2.c.c 4
3.b odd 2 1 882.2.g.l 4
4.b odd 2 1 784.2.i.m 4
7.b odd 2 1 inner 98.2.c.c 4
7.c even 3 1 98.2.a.b 2
7.c even 3 1 inner 98.2.c.c 4
7.d odd 6 1 98.2.a.b 2
7.d odd 6 1 inner 98.2.c.c 4
21.c even 2 1 882.2.g.l 4
21.g even 6 1 882.2.a.n 2
21.g even 6 1 882.2.g.l 4
21.h odd 6 1 882.2.a.n 2
21.h odd 6 1 882.2.g.l 4
28.d even 2 1 784.2.i.m 4
28.f even 6 1 784.2.a.l 2
28.f even 6 1 784.2.i.m 4
28.g odd 6 1 784.2.a.l 2
28.g odd 6 1 784.2.i.m 4
35.i odd 6 1 2450.2.a.bj 2
35.j even 6 1 2450.2.a.bj 2
35.k even 12 2 2450.2.c.v 4
35.l odd 12 2 2450.2.c.v 4
56.j odd 6 1 3136.2.a.bn 2
56.k odd 6 1 3136.2.a.bm 2
56.m even 6 1 3136.2.a.bm 2
56.p even 6 1 3136.2.a.bn 2
84.j odd 6 1 7056.2.a.cl 2
84.n even 6 1 7056.2.a.cl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 7.c even 3 1
98.2.a.b 2 7.d odd 6 1
98.2.c.c 4 1.a even 1 1 trivial
98.2.c.c 4 7.b odd 2 1 inner
98.2.c.c 4 7.c even 3 1 inner
98.2.c.c 4 7.d odd 6 1 inner
784.2.a.l 2 28.f even 6 1
784.2.a.l 2 28.g odd 6 1
784.2.i.m 4 4.b odd 2 1
784.2.i.m 4 28.d even 2 1
784.2.i.m 4 28.f even 6 1
784.2.i.m 4 28.g odd 6 1
882.2.a.n 2 21.g even 6 1
882.2.a.n 2 21.h odd 6 1
882.2.g.l 4 3.b odd 2 1
882.2.g.l 4 21.c even 2 1
882.2.g.l 4 21.g even 6 1
882.2.g.l 4 21.h odd 6 1
2450.2.a.bj 2 35.i odd 6 1
2450.2.a.bj 2 35.j even 6 1
2450.2.c.v 4 35.k even 12 2
2450.2.c.v 4 35.l odd 12 2
3136.2.a.bm 2 56.k odd 6 1
3136.2.a.bm 2 56.m even 6 1
3136.2.a.bn 2 56.j odd 6 1
3136.2.a.bn 2 56.p even 6 1
7056.2.a.cl 2 84.j odd 6 1
7056.2.a.cl 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 4 + 2 T^{2} + T^{4} \)
$5$ \( 64 + 8 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 - 2 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 2500 + 50 T^{2} + T^{4} \)
$23$ \( ( 16 - 4 T + T^{2} )^{2} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( 5184 + 72 T^{2} + T^{4} \)
$37$ \( ( 100 + 10 T + T^{2} )^{2} \)
$41$ \( ( -98 + T^{2} )^{2} \)
$43$ \( ( -2 + T )^{4} \)
$47$ \( 64 + 8 T^{2} + T^{4} \)
$53$ \( ( 4 - 2 T + T^{2} )^{2} \)
$59$ \( 4 + 2 T^{2} + T^{4} \)
$61$ \( 64 + 8 T^{2} + T^{4} \)
$67$ \( ( 144 + 12 T + T^{2} )^{2} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( 4 + 2 T^{2} + T^{4} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( -98 + T^{2} )^{2} \)
$89$ \( 2500 + 50 T^{2} + T^{4} \)
$97$ \( ( -98 + T^{2} )^{2} \)
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