Properties

Label 98.4.c.b
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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.b 2
3.b odd 2 1 882.4.g.v 2
7.b odd 2 1 98.4.c.c 2
7.c even 3 1 98.4.a.e 1
7.c even 3 1 inner 98.4.c.b 2
7.d odd 6 1 14.4.a.b 1
7.d odd 6 1 98.4.c.c 2
21.c even 2 1 882.4.g.p 2
21.g even 6 1 126.4.a.d 1
21.g even 6 1 882.4.g.p 2
21.h odd 6 1 882.4.a.b 1
21.h odd 6 1 882.4.g.v 2
28.f even 6 1 112.4.a.e 1
28.g odd 6 1 784.4.a.h 1
35.i odd 6 1 350.4.a.f 1
35.j even 6 1 2450.4.a.i 1
35.k even 12 2 350.4.c.g 2
56.j odd 6 1 448.4.a.k 1
56.m even 6 1 448.4.a.g 1
77.i even 6 1 1694.4.a.b 1
84.j odd 6 1 1008.4.a.r 1
91.s odd 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.d odd 6 1
98.4.a.e 1 7.c even 3 1
98.4.c.b 2 1.a even 1 1 trivial
98.4.c.b 2 7.c even 3 1 inner
98.4.c.c 2 7.b odd 2 1
98.4.c.c 2 7.d odd 6 1
112.4.a.e 1 28.f even 6 1
126.4.a.d 1 21.g even 6 1
350.4.a.f 1 35.i odd 6 1
350.4.c.g 2 35.k even 12 2
448.4.a.g 1 56.m even 6 1
448.4.a.k 1 56.j odd 6 1
784.4.a.h 1 28.g odd 6 1
882.4.a.b 1 21.h odd 6 1
882.4.g.p 2 21.c even 2 1
882.4.g.p 2 21.g even 6 1
882.4.g.v 2 3.b odd 2 1
882.4.g.v 2 21.h odd 6 1
1008.4.a.r 1 84.j odd 6 1
1694.4.a.b 1 77.i even 6 1
2366.4.a.c 1 91.s odd 6 1
2450.4.a.i 1 35.j even 6 1

Hecke kernels

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

Hecke characteristic polynomials

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