Properties

Label 98.4.c.a
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 9 \zeta_{6} q^{5} + 10 q^{6} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 9 \zeta_{6} q^{5} + 10 q^{6} + 8 q^{8} + 2 \zeta_{6} q^{9} + (18 \zeta_{6} - 18) q^{10} + ( - 57 \zeta_{6} + 57) q^{11} - 20 \zeta_{6} q^{12} + 70 q^{13} + 45 q^{15} - 16 \zeta_{6} q^{16} + ( - 51 \zeta_{6} + 51) q^{17} + ( - 4 \zeta_{6} + 4) q^{18} + 5 \zeta_{6} q^{19} + 36 q^{20} - 114 q^{22} - 69 \zeta_{6} q^{23} + (40 \zeta_{6} - 40) q^{24} + ( - 44 \zeta_{6} + 44) q^{25} - 140 \zeta_{6} q^{26} - 145 q^{27} + 114 q^{29} - 90 \zeta_{6} q^{30} + ( - 23 \zeta_{6} + 23) q^{31} + (32 \zeta_{6} - 32) q^{32} + 285 \zeta_{6} q^{33} - 102 q^{34} - 8 q^{36} + 253 \zeta_{6} q^{37} + ( - 10 \zeta_{6} + 10) q^{38} + (350 \zeta_{6} - 350) q^{39} - 72 \zeta_{6} q^{40} + 42 q^{41} - 124 q^{43} + 228 \zeta_{6} q^{44} + ( - 18 \zeta_{6} + 18) q^{45} + (138 \zeta_{6} - 138) q^{46} + 201 \zeta_{6} q^{47} + 80 q^{48} - 88 q^{50} + 255 \zeta_{6} q^{51} + (280 \zeta_{6} - 280) q^{52} + ( - 393 \zeta_{6} + 393) q^{53} + 290 \zeta_{6} q^{54} - 513 q^{55} - 25 q^{57} - 228 \zeta_{6} q^{58} + ( - 219 \zeta_{6} + 219) q^{59} + (180 \zeta_{6} - 180) q^{60} - 709 \zeta_{6} q^{61} - 46 q^{62} + 64 q^{64} - 630 \zeta_{6} q^{65} + ( - 570 \zeta_{6} + 570) q^{66} + (419 \zeta_{6} - 419) q^{67} + 204 \zeta_{6} q^{68} + 345 q^{69} - 96 q^{71} + 16 \zeta_{6} q^{72} + (313 \zeta_{6} - 313) q^{73} + ( - 506 \zeta_{6} + 506) q^{74} + 220 \zeta_{6} q^{75} - 20 q^{76} + 700 q^{78} - 461 \zeta_{6} q^{79} + (144 \zeta_{6} - 144) q^{80} + ( - 671 \zeta_{6} + 671) q^{81} - 84 \zeta_{6} q^{82} + 588 q^{83} - 459 q^{85} + 248 \zeta_{6} q^{86} + (570 \zeta_{6} - 570) q^{87} + ( - 456 \zeta_{6} + 456) q^{88} - 1017 \zeta_{6} q^{89} - 36 q^{90} + 276 q^{92} + 115 \zeta_{6} q^{93} + ( - 402 \zeta_{6} + 402) q^{94} + ( - 45 \zeta_{6} + 45) q^{95} - 160 \zeta_{6} q^{96} + 1834 q^{97} + 114 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 9 q^{5} + 20 q^{6} + 16 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 9 q^{5} + 20 q^{6} + 16 q^{8} + 2 q^{9} - 18 q^{10} + 57 q^{11} - 20 q^{12} + 140 q^{13} + 90 q^{15} - 16 q^{16} + 51 q^{17} + 4 q^{18} + 5 q^{19} + 72 q^{20} - 228 q^{22} - 69 q^{23} - 40 q^{24} + 44 q^{25} - 140 q^{26} - 290 q^{27} + 228 q^{29} - 90 q^{30} + 23 q^{31} - 32 q^{32} + 285 q^{33} - 204 q^{34} - 16 q^{36} + 253 q^{37} + 10 q^{38} - 350 q^{39} - 72 q^{40} + 84 q^{41} - 248 q^{43} + 228 q^{44} + 18 q^{45} - 138 q^{46} + 201 q^{47} + 160 q^{48} - 176 q^{50} + 255 q^{51} - 280 q^{52} + 393 q^{53} + 290 q^{54} - 1026 q^{55} - 50 q^{57} - 228 q^{58} + 219 q^{59} - 180 q^{60} - 709 q^{61} - 92 q^{62} + 128 q^{64} - 630 q^{65} + 570 q^{66} - 419 q^{67} + 204 q^{68} + 690 q^{69} - 192 q^{71} + 16 q^{72} - 313 q^{73} + 506 q^{74} + 220 q^{75} - 40 q^{76} + 1400 q^{78} - 461 q^{79} - 144 q^{80} + 671 q^{81} - 84 q^{82} + 1176 q^{83} - 918 q^{85} + 248 q^{86} - 570 q^{87} + 456 q^{88} - 1017 q^{89} - 72 q^{90} + 552 q^{92} + 115 q^{93} + 402 q^{94} + 45 q^{95} - 160 q^{96} + 3668 q^{97} + 228 q^{99}+O(q^{100}) \) 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 −2.50000 + 4.33013i −2.00000 + 3.46410i −4.50000 7.79423i 10.0000 0 8.00000 1.00000 + 1.73205i −9.00000 + 15.5885i
79.1 −1.00000 + 1.73205i −2.50000 4.33013i −2.00000 3.46410i −4.50000 + 7.79423i 10.0000 0 8.00000 1.00000 1.73205i −9.00000 15.5885i
\(n\): 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.a 2
3.b odd 2 1 882.4.g.u 2
7.b odd 2 1 14.4.c.a 2
7.c even 3 1 98.4.a.f 1
7.c even 3 1 inner 98.4.c.a 2
7.d odd 6 1 14.4.c.a 2
7.d odd 6 1 98.4.a.d 1
21.c even 2 1 126.4.g.d 2
21.g even 6 1 126.4.g.d 2
21.g even 6 1 882.4.a.f 1
21.h odd 6 1 882.4.a.c 1
21.h odd 6 1 882.4.g.u 2
28.d even 2 1 112.4.i.a 2
28.f even 6 1 112.4.i.a 2
28.f even 6 1 784.4.a.p 1
28.g odd 6 1 784.4.a.c 1
35.c odd 2 1 350.4.e.e 2
35.f even 4 2 350.4.j.b 4
35.i odd 6 1 350.4.e.e 2
35.i odd 6 1 2450.4.a.q 1
35.j even 6 1 2450.4.a.d 1
35.k even 12 2 350.4.j.b 4
56.e even 2 1 448.4.i.e 2
56.h odd 2 1 448.4.i.b 2
56.j odd 6 1 448.4.i.b 2
56.m even 6 1 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 7.b odd 2 1
14.4.c.a 2 7.d odd 6 1
98.4.a.d 1 7.d odd 6 1
98.4.a.f 1 7.c even 3 1
98.4.c.a 2 1.a even 1 1 trivial
98.4.c.a 2 7.c even 3 1 inner
112.4.i.a 2 28.d even 2 1
112.4.i.a 2 28.f even 6 1
126.4.g.d 2 21.c even 2 1
126.4.g.d 2 21.g even 6 1
350.4.e.e 2 35.c odd 2 1
350.4.e.e 2 35.i odd 6 1
350.4.j.b 4 35.f even 4 2
350.4.j.b 4 35.k even 12 2
448.4.i.b 2 56.h odd 2 1
448.4.i.b 2 56.j odd 6 1
448.4.i.e 2 56.e even 2 1
448.4.i.e 2 56.m even 6 1
784.4.a.c 1 28.g odd 6 1
784.4.a.p 1 28.f even 6 1
882.4.a.c 1 21.h odd 6 1
882.4.a.f 1 21.g even 6 1
882.4.g.u 2 3.b odd 2 1
882.4.g.u 2 21.h odd 6 1
2450.4.a.d 1 35.j even 6 1
2450.4.a.q 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} + 25 \) 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
$3$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T - 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T - 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} - 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T - 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} - 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} - 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} + 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} + 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T + 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T - 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T - 1834)^{2} \) Copy content Toggle raw display
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