Properties

Label 98.4.c.c
Level $98$
Weight $4$
Character orbit 98.c
Analytic conductor $5.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 12 \zeta_{6} q^{5} -4 q^{6} + 8 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 12 \zeta_{6} q^{5} -4 q^{6} + 8 q^{8} + 23 \zeta_{6} q^{9} + ( 24 - 24 \zeta_{6} ) q^{10} + ( -48 + 48 \zeta_{6} ) q^{11} + 8 \zeta_{6} q^{12} + 56 q^{13} + 24 q^{15} -16 \zeta_{6} q^{16} + ( 114 - 114 \zeta_{6} ) q^{17} + ( 46 - 46 \zeta_{6} ) q^{18} -2 \zeta_{6} q^{19} -48 q^{20} + 96 q^{22} + 120 \zeta_{6} q^{23} + ( 16 - 16 \zeta_{6} ) q^{24} + ( -19 + 19 \zeta_{6} ) q^{25} -112 \zeta_{6} q^{26} + 100 q^{27} -54 q^{29} -48 \zeta_{6} q^{30} + ( -236 + 236 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 96 \zeta_{6} q^{33} -228 q^{34} -92 q^{36} -146 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + ( 112 - 112 \zeta_{6} ) q^{39} + 96 \zeta_{6} q^{40} + 126 q^{41} -376 q^{43} -192 \zeta_{6} q^{44} + ( -276 + 276 \zeta_{6} ) q^{45} + ( 240 - 240 \zeta_{6} ) q^{46} + 12 \zeta_{6} q^{47} -32 q^{48} + 38 q^{50} -228 \zeta_{6} q^{51} + ( -224 + 224 \zeta_{6} ) q^{52} + ( -174 + 174 \zeta_{6} ) q^{53} -200 \zeta_{6} q^{54} -576 q^{55} -4 q^{57} + 108 \zeta_{6} q^{58} + ( -138 + 138 \zeta_{6} ) q^{59} + ( -96 + 96 \zeta_{6} ) q^{60} -380 \zeta_{6} q^{61} + 472 q^{62} + 64 q^{64} + 672 \zeta_{6} q^{65} + ( 192 - 192 \zeta_{6} ) q^{66} + ( 484 - 484 \zeta_{6} ) q^{67} + 456 \zeta_{6} q^{68} + 240 q^{69} + 576 q^{71} + 184 \zeta_{6} q^{72} + ( 1150 - 1150 \zeta_{6} ) q^{73} + ( -292 + 292 \zeta_{6} ) q^{74} + 38 \zeta_{6} q^{75} + 8 q^{76} -224 q^{78} -776 \zeta_{6} q^{79} + ( 192 - 192 \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} -252 \zeta_{6} q^{82} + 378 q^{83} + 1368 q^{85} + 752 \zeta_{6} q^{86} + ( -108 + 108 \zeta_{6} ) q^{87} + ( -384 + 384 \zeta_{6} ) q^{88} + 390 \zeta_{6} q^{89} + 552 q^{90} -480 q^{92} + 472 \zeta_{6} q^{93} + ( 24 - 24 \zeta_{6} ) q^{94} + ( 24 - 24 \zeta_{6} ) q^{95} + 64 \zeta_{6} q^{96} -1330 q^{97} -1104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} - 4q^{4} + 12q^{5} - 8q^{6} + 16q^{8} + 23q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} - 4q^{4} + 12q^{5} - 8q^{6} + 16q^{8} + 23q^{9} + 24q^{10} - 48q^{11} + 8q^{12} + 112q^{13} + 48q^{15} - 16q^{16} + 114q^{17} + 46q^{18} - 2q^{19} - 96q^{20} + 192q^{22} + 120q^{23} + 16q^{24} - 19q^{25} - 112q^{26} + 200q^{27} - 108q^{29} - 48q^{30} - 236q^{31} - 32q^{32} + 96q^{33} - 456q^{34} - 184q^{36} - 146q^{37} - 4q^{38} + 112q^{39} + 96q^{40} + 252q^{41} - 752q^{43} - 192q^{44} - 276q^{45} + 240q^{46} + 12q^{47} - 64q^{48} + 76q^{50} - 228q^{51} - 224q^{52} - 174q^{53} - 200q^{54} - 1152q^{55} - 8q^{57} + 108q^{58} - 138q^{59} - 96q^{60} - 380q^{61} + 944q^{62} + 128q^{64} + 672q^{65} + 192q^{66} + 484q^{67} + 456q^{68} + 480q^{69} + 1152q^{71} + 184q^{72} + 1150q^{73} - 292q^{74} + 38q^{75} + 16q^{76} - 448q^{78} - 776q^{79} + 192q^{80} - 421q^{81} - 252q^{82} + 756q^{83} + 2736q^{85} + 752q^{86} - 108q^{87} - 384q^{88} + 390q^{89} + 1104q^{90} - 960q^{92} + 472q^{93} + 24q^{94} + 24q^{95} + 64q^{96} - 2660q^{97} - 2208q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 1.00000 1.73205i −2.00000 + 3.46410i 6.00000 + 10.3923i −4.00000 0 8.00000 11.5000 + 19.9186i 12.0000 20.7846i
79.1 −1.00000 + 1.73205i 1.00000 + 1.73205i −2.00000 3.46410i 6.00000 10.3923i −4.00000 0 8.00000 11.5000 19.9186i 12.0000 + 20.7846i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.c.c 2
3.b odd 2 1 882.4.g.p 2
7.b odd 2 1 98.4.c.b 2
7.c even 3 1 14.4.a.b 1
7.c even 3 1 inner 98.4.c.c 2
7.d odd 6 1 98.4.a.e 1
7.d odd 6 1 98.4.c.b 2
21.c even 2 1 882.4.g.v 2
21.g even 6 1 882.4.a.b 1
21.g even 6 1 882.4.g.v 2
21.h odd 6 1 126.4.a.d 1
21.h odd 6 1 882.4.g.p 2
28.f even 6 1 784.4.a.h 1
28.g odd 6 1 112.4.a.e 1
35.i odd 6 1 2450.4.a.i 1
35.j even 6 1 350.4.a.f 1
35.l odd 12 2 350.4.c.g 2
56.k odd 6 1 448.4.a.g 1
56.p even 6 1 448.4.a.k 1
77.h odd 6 1 1694.4.a.b 1
84.n even 6 1 1008.4.a.r 1
91.r even 6 1 2366.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 7.c even 3 1
98.4.a.e 1 7.d odd 6 1
98.4.c.b 2 7.b odd 2 1
98.4.c.b 2 7.d odd 6 1
98.4.c.c 2 1.a even 1 1 trivial
98.4.c.c 2 7.c even 3 1 inner
112.4.a.e 1 28.g odd 6 1
126.4.a.d 1 21.h odd 6 1
350.4.a.f 1 35.j even 6 1
350.4.c.g 2 35.l odd 12 2
448.4.a.g 1 56.k odd 6 1
448.4.a.k 1 56.p even 6 1
784.4.a.h 1 28.f even 6 1
882.4.a.b 1 21.g even 6 1
882.4.g.p 2 3.b odd 2 1
882.4.g.p 2 21.h odd 6 1
882.4.g.v 2 21.c even 2 1
882.4.g.v 2 21.g even 6 1
1008.4.a.r 1 84.n even 6 1
1694.4.a.b 1 77.h odd 6 1
2366.4.a.c 1 91.r even 6 1
2450.4.a.i 1 35.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( 4 - 2 T + T^{2} \)
$5$ \( 144 - 12 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 2304 + 48 T + T^{2} \)
$13$ \( ( -56 + T )^{2} \)
$17$ \( 12996 - 114 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 14400 - 120 T + T^{2} \)
$29$ \( ( 54 + T )^{2} \)
$31$ \( 55696 + 236 T + T^{2} \)
$37$ \( 21316 + 146 T + T^{2} \)
$41$ \( ( -126 + T )^{2} \)
$43$ \( ( 376 + T )^{2} \)
$47$ \( 144 - 12 T + T^{2} \)
$53$ \( 30276 + 174 T + T^{2} \)
$59$ \( 19044 + 138 T + T^{2} \)
$61$ \( 144400 + 380 T + T^{2} \)
$67$ \( 234256 - 484 T + T^{2} \)
$71$ \( ( -576 + T )^{2} \)
$73$ \( 1322500 - 1150 T + T^{2} \)
$79$ \( 602176 + 776 T + T^{2} \)
$83$ \( ( -378 + T )^{2} \)
$89$ \( 152100 - 390 T + T^{2} \)
$97$ \( ( 1330 + T )^{2} \)
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