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Properties

Label 98.4.c.g
Level $98$
Weight $4$
Character orbit 98.c
Analytic conductor $5.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.78218718056\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
Defining polynomial: \(x^{4} + 22 x^{2} + 484\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -2 - 2 \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + 4 \beta_{2} q^{4} + \beta_{1} q^{5} -2 \beta_{3} q^{6} + 8 q^{8} + ( -61 - 61 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -2 - 2 \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + 4 \beta_{2} q^{4} + \beta_{1} q^{5} -2 \beta_{3} q^{6} + 8 q^{8} + ( -61 - 61 \beta_{2} ) q^{9} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{10} + 20 \beta_{2} q^{11} -4 \beta_{1} q^{12} + 7 \beta_{3} q^{13} -88 q^{15} + ( -16 - 16 \beta_{2} ) q^{16} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{17} + 122 \beta_{2} q^{18} + \beta_{1} q^{19} + 4 \beta_{3} q^{20} + 40 q^{22} + ( -48 - 48 \beta_{2} ) q^{23} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{24} -37 \beta_{2} q^{25} + 14 \beta_{1} q^{26} -34 \beta_{3} q^{27} -166 q^{29} + ( 176 + 176 \beta_{2} ) q^{30} + ( 22 \beta_{1} + 22 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} -20 \beta_{1} q^{33} + 12 \beta_{3} q^{34} + 244 q^{36} + ( 78 + 78 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{38} -616 \beta_{2} q^{39} + 8 \beta_{1} q^{40} + 42 \beta_{3} q^{41} + 436 q^{43} + ( -80 - 80 \beta_{2} ) q^{44} + ( -61 \beta_{1} - 61 \beta_{3} ) q^{45} + 96 \beta_{2} q^{46} + 22 \beta_{1} q^{47} -16 \beta_{3} q^{48} -74 q^{50} + ( 528 + 528 \beta_{2} ) q^{51} + ( -28 \beta_{1} - 28 \beta_{3} ) q^{52} + 62 \beta_{2} q^{53} -68 \beta_{1} q^{54} + 20 \beta_{3} q^{55} -88 q^{57} + ( 332 + 332 \beta_{2} ) q^{58} + ( 71 \beta_{1} + 71 \beta_{3} ) q^{59} -352 \beta_{2} q^{60} + 29 \beta_{1} q^{61} -44 \beta_{3} q^{62} + 64 q^{64} + ( -616 - 616 \beta_{2} ) q^{65} + ( 40 \beta_{1} + 40 \beta_{3} ) q^{66} + 580 \beta_{2} q^{67} + 24 \beta_{1} q^{68} -48 \beta_{3} q^{69} -544 q^{71} + ( -488 - 488 \beta_{2} ) q^{72} + ( 64 \beta_{1} + 64 \beta_{3} ) q^{73} -156 \beta_{2} q^{74} + 37 \beta_{1} q^{75} + 4 \beta_{3} q^{76} -1232 q^{78} + ( 680 + 680 \beta_{2} ) q^{79} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{80} + 1345 \beta_{2} q^{81} + 84 \beta_{1} q^{82} + 21 \beta_{3} q^{83} + 528 q^{85} + ( -872 - 872 \beta_{2} ) q^{86} + ( -166 \beta_{1} - 166 \beta_{3} ) q^{87} + 160 \beta_{2} q^{88} -160 \beta_{1} q^{89} + 122 \beta_{3} q^{90} + 192 q^{92} + ( -1936 - 1936 \beta_{2} ) q^{93} + ( -44 \beta_{1} - 44 \beta_{3} ) q^{94} + 88 \beta_{2} q^{95} -32 \beta_{1} q^{96} -70 \beta_{3} q^{97} + 1220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 8q^{4} + 32q^{8} - 122q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 8q^{4} + 32q^{8} - 122q^{9} - 40q^{11} - 352q^{15} - 32q^{16} - 244q^{18} + 160q^{22} - 96q^{23} + 74q^{25} - 664q^{29} + 352q^{30} - 64q^{32} + 976q^{36} + 156q^{37} + 1232q^{39} + 1744q^{43} - 160q^{44} - 192q^{46} - 296q^{50} + 1056q^{51} - 124q^{53} - 352q^{57} + 664q^{58} + 704q^{60} + 256q^{64} - 1232q^{65} - 1160q^{67} - 2176q^{71} - 976q^{72} + 312q^{74} - 4928q^{78} + 1360q^{79} - 2690q^{81} + 2112q^{85} - 1744q^{86} - 320q^{88} + 768q^{92} - 3872q^{93} - 176q^{95} + 4880q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 22 x^{2} + 484\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/22\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/11\)

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

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(22 \beta_{2}\)
\(\nu^{3}\)\(=\)\(11 \beta_{3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
2.34521 + 4.06202i
−2.34521 − 4.06202i
2.34521 − 4.06202i
−2.34521 + 4.06202i
−1.00000 − 1.73205i −4.69042 + 8.12404i −2.00000 + 3.46410i 4.69042 + 8.12404i 18.7617 0 8.00000 −30.5000 − 52.8275i 9.38083 − 16.2481i
67.2 −1.00000 − 1.73205i 4.69042 − 8.12404i −2.00000 + 3.46410i −4.69042 − 8.12404i −18.7617 0 8.00000 −30.5000 − 52.8275i −9.38083 + 16.2481i
79.1 −1.00000 + 1.73205i −4.69042 − 8.12404i −2.00000 − 3.46410i 4.69042 − 8.12404i 18.7617 0 8.00000 −30.5000 + 52.8275i 9.38083 + 16.2481i
79.2 −1.00000 + 1.73205i 4.69042 + 8.12404i −2.00000 − 3.46410i −4.69042 + 8.12404i −18.7617 0 8.00000 −30.5000 + 52.8275i −9.38083 − 16.2481i
\(n\): 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 98.4.c.g 4
3.b odd 2 1 882.4.g.bi 4
7.b odd 2 1 inner 98.4.c.g 4
7.c even 3 1 98.4.a.h ✓ 2
7.c even 3 1 inner 98.4.c.g 4
7.d odd 6 1 98.4.a.h ✓ 2
7.d odd 6 1 inner 98.4.c.g 4
21.c even 2 1 882.4.g.bi 4
21.g even 6 1 882.4.a.w 2
21.g even 6 1 882.4.g.bi 4
21.h odd 6 1 882.4.a.w 2
21.h odd 6 1 882.4.g.bi 4
28.f even 6 1 784.4.a.z 2
28.g odd 6 1 784.4.a.z 2
35.i odd 6 1 2450.4.a.bs 2
35.j even 6 1 2450.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h ✓ 2 7.c even 3 1
98.4.a.h ✓ 2 7.d odd 6 1
98.4.c.g 4 1.a even 1 1 trivial
98.4.c.g 4 7.b odd 2 1 inner
98.4.c.g 4 7.c even 3 1 inner
98.4.c.g 4 7.d odd 6 1 inner
784.4.a.z 2 28.f even 6 1
784.4.a.z 2 28.g odd 6 1
882.4.a.w 2 21.g even 6 1
882.4.a.w 2 21.h odd 6 1
882.4.g.bi 4 3.b odd 2 1
882.4.g.bi 4 21.c even 2 1
882.4.g.bi 4 21.g even 6 1
882.4.g.bi 4 21.h odd 6 1
2450.4.a.bs 2 35.i odd 6 1
2450.4.a.bs 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 88 T_{3}^{2} + 7744 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{2} \)
$3$ \( 7744 + 88 T^{2} + T^{4} \)
$5$ \( 7744 + 88 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 400 + 20 T + T^{2} )^{2} \)
$13$ \( ( -4312 + T^{2} )^{2} \)
$17$ \( 10036224 + 3168 T^{2} + T^{4} \)
$19$ \( 7744 + 88 T^{2} + T^{4} \)
$23$ \( ( 2304 + 48 T + T^{2} )^{2} \)
$29$ \( ( 166 + T )^{4} \)
$31$ \( 1814078464 + 42592 T^{2} + T^{4} \)
$37$ \( ( 6084 - 78 T + T^{2} )^{2} \)
$41$ \( ( -155232 + T^{2} )^{2} \)
$43$ \( ( -436 + T )^{4} \)
$47$ \( 1814078464 + 42592 T^{2} + T^{4} \)
$53$ \( ( 3844 + 62 T + T^{2} )^{2} \)
$59$ \( 196788057664 + 443608 T^{2} + T^{4} \)
$61$ \( 5477184064 + 74008 T^{2} + T^{4} \)
$67$ \( ( 336400 + 580 T + T^{2} )^{2} \)
$71$ \( ( 544 + T )^{4} \)
$73$ \( 129922760704 + 360448 T^{2} + T^{4} \)
$79$ \( ( 462400 - 680 T + T^{2} )^{2} \)
$83$ \( ( -38808 + T^{2} )^{2} \)
$89$ \( 5075107840000 + 2252800 T^{2} + T^{4} \)
$97$ \( ( -431200 + T^{2} )^{2} \)
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