Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 98.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.67030659073\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{2} - \beta_{3} ) q^{3} + 2 q^{4} + ( \beta_{2} - 2 \beta_{3} ) q^{5} + ( 2 \beta_{2} + \beta_{3} ) q^{6} + 2 \beta_{1} q^{8} + 6 \beta_{1} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{2} - \beta_{3} ) q^{3} + 2 q^{4} + ( \beta_{2} - 2 \beta_{3} ) q^{5} + ( 2 \beta_{2} + \beta_{3} ) q^{6} + 2 \beta_{1} q^{8} + 6 \beta_{1} q^{9} + ( 4 \beta_{2} - \beta_{3} ) q^{10} + ( -9 + 3 \beta_{1} ) q^{11} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{12} + ( -6 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -9 + 3 \beta_{1} ) q^{15} + 4 q^{16} + ( -5 \beta_{2} - 2 \beta_{3} ) q^{17} + 12 q^{18} + ( -\beta_{2} + \beta_{3} ) q^{19} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{20} + ( 6 - 9 \beta_{1} ) q^{22} + ( -15 - 9 \beta_{1} ) q^{23} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{24} + ( -2 - 12 \beta_{1} ) q^{25} + ( -4 \beta_{2} + 6 \beta_{3} ) q^{26} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{27} + ( 12 - 6 \beta_{1} ) q^{29} + ( 6 - 9 \beta_{1} ) q^{30} + ( -7 \beta_{2} + 15 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( 15 \beta_{2} + 12 \beta_{3} ) q^{33} + ( 4 \beta_{2} + 5 \beta_{3} ) q^{34} + 12 \beta_{1} q^{36} + ( 31 - 24 \beta_{1} ) q^{37} + ( -2 \beta_{2} + \beta_{3} ) q^{38} + ( -6 + 12 \beta_{1} ) q^{39} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{40} + ( 2 \beta_{2} - 10 \beta_{3} ) q^{41} + ( -2 - 6 \beta_{1} ) q^{43} + ( -18 + 6 \beta_{1} ) q^{44} + ( 24 \beta_{2} - 6 \beta_{3} ) q^{45} + ( -18 - 15 \beta_{1} ) q^{46} + ( -29 \beta_{2} + \beta_{3} ) q^{47} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{48} + ( -24 - 2 \beta_{1} ) q^{50} + ( -27 + 21 \beta_{1} ) q^{51} + ( -12 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 39 + 12 \beta_{1} ) q^{53} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{54} + ( 3 \beta_{2} + 15 \beta_{3} ) q^{55} + 3 q^{57} + ( -12 + 12 \beta_{1} ) q^{58} + ( -13 \beta_{2} - 25 \beta_{3} ) q^{59} + ( -18 + 6 \beta_{1} ) q^{60} + ( 7 \beta_{2} - 32 \beta_{3} ) q^{61} + ( -30 \beta_{2} + 7 \beta_{3} ) q^{62} + 8 q^{64} + ( 42 + 42 \beta_{1} ) q^{65} + ( -24 \beta_{2} - 15 \beta_{3} ) q^{66} + ( 29 - 45 \beta_{1} ) q^{67} + ( -10 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -3 \beta_{2} + 6 \beta_{3} ) q^{69} + ( -6 + 30 \beta_{1} ) q^{71} + 24 q^{72} + ( 53 \beta_{2} + 16 \beta_{3} ) q^{73} + ( -48 + 31 \beta_{1} ) q^{74} + ( -22 \beta_{2} - 10 \beta_{3} ) q^{75} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 24 - 6 \beta_{1} ) q^{78} + ( -55 + 15 \beta_{1} ) q^{79} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{80} + ( -9 + 54 \beta_{1} ) q^{81} + ( 20 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 68 \beta_{2} - 4 \beta_{3} ) q^{83} + ( -9 + 24 \beta_{1} ) q^{85} + ( -12 - 2 \beta_{1} ) q^{86} + ( -24 \beta_{2} - 18 \beta_{3} ) q^{87} + ( 12 - 18 \beta_{1} ) q^{88} + ( 63 \beta_{2} + 24 \beta_{3} ) q^{89} + ( 12 \beta_{2} - 24 \beta_{3} ) q^{90} + ( -30 - 18 \beta_{1} ) q^{92} + ( 69 - 24 \beta_{1} ) q^{93} + ( -2 \beta_{2} + 29 \beta_{3} ) q^{94} + ( 15 + 9 \beta_{1} ) q^{95} + ( 8 \beta_{2} + 4 \beta_{3} ) q^{96} + ( -22 \beta_{2} + 26 \beta_{3} ) q^{97} + ( 36 - 54 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + O(q^{10}) \) \( 4 q + 8 q^{4} - 36 q^{11} - 36 q^{15} + 16 q^{16} + 48 q^{18} + 24 q^{22} - 60 q^{23} - 8 q^{25} + 48 q^{29} + 24 q^{30} + 124 q^{37} - 24 q^{39} - 8 q^{43} - 72 q^{44} - 72 q^{46} - 96 q^{50} - 108 q^{51} + 156 q^{53} + 12 q^{57} - 48 q^{58} - 72 q^{60} + 32 q^{64} + 168 q^{65} + 116 q^{67} - 24 q^{71} + 96 q^{72} - 192 q^{74} + 96 q^{78} - 220 q^{79} - 36 q^{81} - 36 q^{85} - 48 q^{86} + 48 q^{88} - 120 q^{92} + 276 q^{93} + 60 q^{95} + 144 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i
−1.41421 4.18154i 2.00000 3.16693i 5.91359i 0 −2.82843 −8.48528 4.47871i
97.2 −1.41421 4.18154i 2.00000 3.16693i 5.91359i 0 −2.82843 −8.48528 4.47871i
97.3 1.41421 0.717439i 2.00000 6.63103i 1.01461i 0 2.82843 8.48528 9.37769i
97.4 1.41421 0.717439i 2.00000 6.63103i 1.01461i 0 2.82843 8.48528 9.37769i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.3.b.b 4
3.b odd 2 1 882.3.c.f 4
4.b odd 2 1 784.3.c.e 4
7.b odd 2 1 inner 98.3.b.b 4
7.c even 3 1 14.3.d.a 4
7.c even 3 1 98.3.d.a 4
7.d odd 6 1 14.3.d.a 4
7.d odd 6 1 98.3.d.a 4
21.c even 2 1 882.3.c.f 4
21.g even 6 1 126.3.n.c 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.n.b 4
28.d even 2 1 784.3.c.e 4
28.f even 6 1 112.3.s.b 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.s.c 4
35.i odd 6 1 350.3.k.a 4
35.j even 6 1 350.3.k.a 4
35.k even 12 2 350.3.i.a 8
35.l odd 12 2 350.3.i.a 8
56.j odd 6 1 448.3.s.d 4
56.k odd 6 1 448.3.s.c 4
56.m even 6 1 448.3.s.c 4
56.p even 6 1 448.3.s.d 4
84.j odd 6 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.c even 3 1
14.3.d.a 4 7.d odd 6 1
98.3.b.b 4 1.a even 1 1 trivial
98.3.b.b 4 7.b odd 2 1 inner
98.3.d.a 4 7.c even 3 1
98.3.d.a 4 7.d odd 6 1
112.3.s.b 4 28.f even 6 1
112.3.s.b 4 28.g odd 6 1
126.3.n.c 4 21.g even 6 1
126.3.n.c 4 21.h odd 6 1
350.3.i.a 8 35.k even 12 2
350.3.i.a 8 35.l odd 12 2
350.3.k.a 4 35.i odd 6 1
350.3.k.a 4 35.j even 6 1
448.3.s.c 4 56.k odd 6 1
448.3.s.c 4 56.m even 6 1
448.3.s.d 4 56.j odd 6 1
448.3.s.d 4 56.p even 6 1
784.3.c.e 4 4.b odd 2 1
784.3.c.e 4 28.d even 2 1
784.3.s.c 4 28.f even 6 1
784.3.s.c 4 28.g odd 6 1
882.3.c.f 4 3.b odd 2 1
882.3.c.f 4 21.c even 2 1
882.3.n.b 4 21.g even 6 1
882.3.n.b 4 21.h odd 6 1
1008.3.cg.l 4 84.j odd 6 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 \( T_{3}^{4} + 18 T_{3}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( 9 + 18 T^{2} + T^{4} \)
$5$ \( 441 + 54 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 63 + 18 T + T^{2} )^{2} \)
$13$ \( 7056 + 264 T^{2} + T^{4} \)
$17$ \( 2601 + 198 T^{2} + T^{4} \)
$19$ \( 9 + 18 T^{2} + T^{4} \)
$23$ \( ( 63 + 30 T + T^{2} )^{2} \)
$29$ \( ( 72 - 24 T + T^{2} )^{2} \)
$31$ \( 1447209 + 2994 T^{2} + T^{4} \)
$37$ \( ( -191 - 62 T + T^{2} )^{2} \)
$41$ \( 345744 + 1224 T^{2} + T^{4} \)
$43$ \( ( -68 + 4 T + T^{2} )^{2} \)
$47$ \( 6335289 + 5058 T^{2} + T^{4} \)
$53$ \( ( 1233 - 78 T + T^{2} )^{2} \)
$59$ \( 10517049 + 8514 T^{2} + T^{4} \)
$61$ \( 35964009 + 12582 T^{2} + T^{4} \)
$67$ \( ( -3209 - 58 T + T^{2} )^{2} \)
$71$ \( ( -1764 + 12 T + T^{2} )^{2} \)
$73$ \( 47485881 + 19926 T^{2} + T^{4} \)
$79$ \( ( 2575 + 110 T + T^{2} )^{2} \)
$83$ \( 189778176 + 27936 T^{2} + T^{4} \)
$89$ \( 71419401 + 30726 T^{2} + T^{4} \)
$97$ \( 6780816 + 11016 T^{2} + T^{4} \)
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