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Properties

Label 98.4.c.h
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{3} + 4 \beta_{2} q^{4} -14 \beta_{1} q^{5} + 10 \beta_{3} q^{6} -8 q^{8} + ( -23 - 23 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{3} + 4 \beta_{2} q^{4} -14 \beta_{1} q^{5} + 10 \beta_{3} q^{6} -8 q^{8} + ( -23 - 23 \beta_{2} ) q^{9} + ( -28 \beta_{1} - 28 \beta_{3} ) q^{10} -14 \beta_{2} q^{11} -20 \beta_{1} q^{12} + 36 \beta_{3} q^{13} + 140 q^{15} + ( -16 - 16 \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} -46 \beta_{2} q^{18} -\beta_{1} q^{19} -56 \beta_{3} q^{20} + 28 q^{22} + ( -140 - 140 \beta_{2} ) q^{23} + ( -40 \beta_{1} - 40 \beta_{3} ) q^{24} + 267 \beta_{2} q^{25} -72 \beta_{1} q^{26} + 20 \beta_{3} q^{27} -286 q^{29} + ( 280 + 280 \beta_{2} ) q^{30} + ( 66 \beta_{1} + 66 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + 70 \beta_{1} q^{33} -2 \beta_{3} q^{34} + 92 q^{36} + ( 38 + 38 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{38} -360 \beta_{2} q^{39} + 112 \beta_{1} q^{40} -89 \beta_{3} q^{41} -34 q^{43} + ( 56 + 56 \beta_{2} ) q^{44} + ( 322 \beta_{1} + 322 \beta_{3} ) q^{45} -280 \beta_{2} q^{46} + 370 \beta_{1} q^{47} -80 \beta_{3} q^{48} -534 q^{50} + ( 10 + 10 \beta_{2} ) q^{51} + ( -144 \beta_{1} - 144 \beta_{3} ) q^{52} -74 \beta_{2} q^{53} -40 \beta_{1} q^{54} + 196 \beta_{3} q^{55} + 10 q^{57} + ( -572 - 572 \beta_{2} ) q^{58} + ( -307 \beta_{1} - 307 \beta_{3} ) q^{59} + 560 \beta_{2} q^{60} + 10 \beta_{1} q^{61} + 132 \beta_{3} q^{62} + 64 q^{64} + ( 1008 + 1008 \beta_{2} ) q^{65} + ( 140 \beta_{1} + 140 \beta_{3} ) q^{66} + 684 \beta_{2} q^{67} + 4 \beta_{1} q^{68} -700 \beta_{3} q^{69} + 588 q^{71} + ( 184 + 184 \beta_{2} ) q^{72} + ( 191 \beta_{1} + 191 \beta_{3} ) q^{73} + 76 \beta_{2} q^{74} -1335 \beta_{1} q^{75} -4 \beta_{3} q^{76} + 720 q^{78} + ( -1220 - 1220 \beta_{2} ) q^{79} + ( 224 \beta_{1} + 224 \beta_{3} ) q^{80} -821 \beta_{2} q^{81} + 178 \beta_{1} q^{82} + 299 \beta_{3} q^{83} -28 q^{85} + ( -68 - 68 \beta_{2} ) q^{86} + ( -1430 \beta_{1} - 1430 \beta_{3} ) q^{87} + 112 \beta_{2} q^{88} + 437 \beta_{1} q^{89} + 644 \beta_{3} q^{90} + 560 q^{92} + ( -660 - 660 \beta_{2} ) q^{93} + ( 740 \beta_{1} + 740 \beta_{3} ) q^{94} + 28 \beta_{2} q^{95} + 160 \beta_{1} q^{96} + 1049 \beta_{3} q^{97} -322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} - 46q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} - 46q^{9} + 28q^{11} + 560q^{15} - 32q^{16} + 92q^{18} + 112q^{22} - 280q^{23} - 534q^{25} - 1144q^{29} + 560q^{30} + 64q^{32} + 368q^{36} + 76q^{37} + 720q^{39} - 136q^{43} + 112q^{44} + 560q^{46} - 2136q^{50} + 20q^{51} + 148q^{53} + 40q^{57} - 1144q^{58} - 1120q^{60} + 256q^{64} + 2016q^{65} - 1368q^{67} + 2352q^{71} + 368q^{72} - 152q^{74} + 2880q^{78} - 2440q^{79} + 1642q^{81} - 112q^{85} - 136q^{86} - 224q^{88} + 2240q^{92} - 1320q^{93} - 56q^{95} - 1288q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

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

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

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

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
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
1.00000 + 1.73205i −3.53553 + 6.12372i −2.00000 + 3.46410i −9.89949 − 17.1464i −14.1421 0 −8.00000 −11.5000 − 19.9186i 19.7990 − 34.2929i
67.2 1.00000 + 1.73205i 3.53553 − 6.12372i −2.00000 + 3.46410i 9.89949 + 17.1464i 14.1421 0 −8.00000 −11.5000 − 19.9186i −19.7990 + 34.2929i
79.1 1.00000 − 1.73205i −3.53553 − 6.12372i −2.00000 − 3.46410i −9.89949 + 17.1464i −14.1421 0 −8.00000 −11.5000 + 19.9186i 19.7990 + 34.2929i
79.2 1.00000 − 1.73205i 3.53553 + 6.12372i −2.00000 − 3.46410i 9.89949 − 17.1464i 14.1421 0 −8.00000 −11.5000 + 19.9186i −19.7990 − 34.2929i
\(n\): 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.h 4
3.b odd 2 1 882.4.g.ba 4
7.b odd 2 1 inner 98.4.c.h 4
7.c even 3 1 98.4.a.g ✓ 2
7.c even 3 1 inner 98.4.c.h 4
7.d odd 6 1 98.4.a.g ✓ 2
7.d odd 6 1 inner 98.4.c.h 4
21.c even 2 1 882.4.g.ba 4
21.g even 6 1 882.4.a.bg 2
21.g even 6 1 882.4.g.ba 4
21.h odd 6 1 882.4.a.bg 2
21.h odd 6 1 882.4.g.ba 4
28.f even 6 1 784.4.a.y 2
28.g odd 6 1 784.4.a.y 2
35.i odd 6 1 2450.4.a.bx 2
35.j even 6 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g ✓ 2 7.c even 3 1
98.4.a.g ✓ 2 7.d odd 6 1
98.4.c.h 4 1.a even 1 1 trivial
98.4.c.h 4 7.b odd 2 1 inner
98.4.c.h 4 7.c even 3 1 inner
98.4.c.h 4 7.d odd 6 1 inner
784.4.a.y 2 28.f even 6 1
784.4.a.y 2 28.g odd 6 1
882.4.a.bg 2 21.g even 6 1
882.4.a.bg 2 21.h odd 6 1
882.4.g.ba 4 3.b odd 2 1
882.4.g.ba 4 21.c even 2 1
882.4.g.ba 4 21.g even 6 1
882.4.g.ba 4 21.h odd 6 1
2450.4.a.bx 2 35.i odd 6 1
2450.4.a.bx 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 50 T_{3}^{2} + 2500 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( 2500 + 50 T^{2} + T^{4} \)
$5$ \( 153664 + 392 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 196 - 14 T + T^{2} )^{2} \)
$13$ \( ( -2592 + T^{2} )^{2} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 4 + 2 T^{2} + T^{4} \)
$23$ \( ( 19600 + 140 T + T^{2} )^{2} \)
$29$ \( ( 286 + T )^{4} \)
$31$ \( 75898944 + 8712 T^{2} + T^{4} \)
$37$ \( ( 1444 - 38 T + T^{2} )^{2} \)
$41$ \( ( -15842 + T^{2} )^{2} \)
$43$ \( ( 34 + T )^{4} \)
$47$ \( 74966440000 + 273800 T^{2} + T^{4} \)
$53$ \( ( 5476 - 74 T + T^{2} )^{2} \)
$59$ \( 35531496004 + 188498 T^{2} + T^{4} \)
$61$ \( 40000 + 200 T^{2} + T^{4} \)
$67$ \( ( 467856 + 684 T + T^{2} )^{2} \)
$71$ \( ( -588 + T )^{4} \)
$73$ \( 5323453444 + 72962 T^{2} + T^{4} \)
$79$ \( ( 1488400 + 1220 T + T^{2} )^{2} \)
$83$ \( ( -178802 + T^{2} )^{2} \)
$89$ \( 145876635844 + 381938 T^{2} + T^{4} \)
$97$ \( ( -2200802 + T^{2} )^{2} \)
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