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: \( x^{4} + 22x^{2} + 484 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} + 8 q^{8} + ( - 61 \beta_{2} - 61) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} + 8 q^{8} + ( - 61 \beta_{2} - 61) q^{9} + ( - 2 \beta_{3} - 2 \beta_1) q^{10} + 20 \beta_{2} q^{11} - 4 \beta_1 q^{12} + 7 \beta_{3} q^{13} - 88 q^{15} + ( - 16 \beta_{2} - 16) q^{16} + ( - 6 \beta_{3} - 6 \beta_1) q^{17} + 122 \beta_{2} q^{18} + \beta_1 q^{19} + 4 \beta_{3} q^{20} + 40 q^{22} + ( - 48 \beta_{2} - 48) q^{23} + (8 \beta_{3} + 8 \beta_1) q^{24} - 37 \beta_{2} q^{25} + 14 \beta_1 q^{26} - 34 \beta_{3} q^{27} - 166 q^{29} + (176 \beta_{2} + 176) q^{30} + (22 \beta_{3} + 22 \beta_1) q^{31} + 32 \beta_{2} q^{32} - 20 \beta_1 q^{33} + 12 \beta_{3} q^{34} + 244 q^{36} + (78 \beta_{2} + 78) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} - 616 \beta_{2} q^{39} + 8 \beta_1 q^{40} + 42 \beta_{3} q^{41} + 436 q^{43} + ( - 80 \beta_{2} - 80) q^{44} + ( - 61 \beta_{3} - 61 \beta_1) q^{45} + 96 \beta_{2} q^{46} + 22 \beta_1 q^{47} - 16 \beta_{3} q^{48} - 74 q^{50} + (528 \beta_{2} + 528) q^{51} + ( - 28 \beta_{3} - 28 \beta_1) q^{52} + 62 \beta_{2} q^{53} - 68 \beta_1 q^{54} + 20 \beta_{3} q^{55} - 88 q^{57} + (332 \beta_{2} + 332) q^{58} + (71 \beta_{3} + 71 \beta_1) q^{59} - 352 \beta_{2} q^{60} + 29 \beta_1 q^{61} - 44 \beta_{3} q^{62} + 64 q^{64} + ( - 616 \beta_{2} - 616) q^{65} + (40 \beta_{3} + 40 \beta_1) q^{66} + 580 \beta_{2} q^{67} + 24 \beta_1 q^{68} - 48 \beta_{3} q^{69} - 544 q^{71} + ( - 488 \beta_{2} - 488) q^{72} + (64 \beta_{3} + 64 \beta_1) q^{73} - 156 \beta_{2} q^{74} + 37 \beta_1 q^{75} + 4 \beta_{3} q^{76} - 1232 q^{78} + (680 \beta_{2} + 680) q^{79} + ( - 16 \beta_{3} - 16 \beta_1) q^{80} + 1345 \beta_{2} q^{81} + 84 \beta_1 q^{82} + 21 \beta_{3} q^{83} + 528 q^{85} + ( - 872 \beta_{2} - 872) q^{86} + ( - 166 \beta_{3} - 166 \beta_1) q^{87} + 160 \beta_{2} q^{88} - 160 \beta_1 q^{89} + 122 \beta_{3} q^{90} + 192 q^{92} + ( - 1936 \beta_{2} - 1936) q^{93} + ( - 44 \beta_{3} - 44 \beta_1) q^{94} + 88 \beta_{2} q^{95} - 32 \beta_1 q^{96} - 70 \beta_{3} q^{97} + 1220 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 122 q^{9} - 40 q^{11} - 352 q^{15} - 32 q^{16} - 244 q^{18} + 160 q^{22} - 96 q^{23} + 74 q^{25} - 664 q^{29} + 352 q^{30} - 64 q^{32} + 976 q^{36} + 156 q^{37} + 1232 q^{39} + 1744 q^{43} - 160 q^{44} - 192 q^{46} - 296 q^{50} + 1056 q^{51} - 124 q^{53} - 352 q^{57} + 664 q^{58} + 704 q^{60} + 256 q^{64} - 1232 q^{65} - 1160 q^{67} - 2176 q^{71} - 976 q^{72} + 312 q^{74} - 4928 q^{78} + 1360 q^{79} - 2690 q^{81} + 2112 q^{85} - 1744 q^{86} - 320 q^{88} + 768 q^{92} - 3872 q^{93} - 176 q^{95} + 4880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{3} \) Copy content Toggle raw display

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 \( T_{3}^{4} + 88T_{3}^{2} + 7744 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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