Properties

Label 98.2.c.a
Level 98
Weight 2
Character orbit 98.c
Analytic conductor 0.783
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -2 q^{6} - q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -2 q^{6} - q^{8} -\zeta_{6} q^{9} -2 \zeta_{6} q^{12} + 4 q^{13} -\zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} + ( 2 - 2 \zeta_{6} ) q^{24} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} -4 q^{27} -6 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 6 q^{34} + q^{36} -2 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + ( -8 + 8 \zeta_{6} ) q^{39} -6 q^{41} + 8 q^{43} -12 \zeta_{6} q^{47} + 2 q^{48} + 5 q^{50} + 12 \zeta_{6} q^{51} + ( -4 + 4 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} -4 \zeta_{6} q^{54} -4 q^{57} -6 \zeta_{6} q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -4 q^{62} + q^{64} + ( 4 - 4 \zeta_{6} ) q^{67} + 6 \zeta_{6} q^{68} + \zeta_{6} q^{72} + ( 2 - 2 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + 10 \zeta_{6} q^{75} -2 q^{76} -8 q^{78} -8 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -6 \zeta_{6} q^{82} + 6 q^{83} + 8 \zeta_{6} q^{86} + ( 12 - 12 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( 12 - 12 \zeta_{6} ) q^{94} + 2 \zeta_{6} q^{96} + 10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 2q^{3} - q^{4} - 4q^{6} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} - 2q^{3} - q^{4} - 4q^{6} - 2q^{8} - q^{9} - 2q^{12} + 8q^{13} - q^{16} + 6q^{17} + q^{18} + 2q^{19} + 2q^{24} + 5q^{25} + 4q^{26} - 8q^{27} - 12q^{29} - 4q^{31} + q^{32} + 12q^{34} + 2q^{36} - 2q^{37} - 2q^{38} - 8q^{39} - 12q^{41} + 16q^{43} - 12q^{47} + 4q^{48} + 10q^{50} + 12q^{51} - 4q^{52} - 6q^{53} - 4q^{54} - 8q^{57} - 6q^{58} - 6q^{59} + 8q^{61} - 8q^{62} + 2q^{64} + 4q^{67} + 6q^{68} + q^{72} + 2q^{73} + 2q^{74} + 10q^{75} - 4q^{76} - 16q^{78} - 8q^{79} + 11q^{81} - 6q^{82} + 12q^{83} + 8q^{86} + 12q^{87} - 6q^{89} - 8q^{93} + 12q^{94} + 2q^{96} + 20q^{97} + O(q^{100}) \)

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
0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i 0 −2.00000 0 −1.00000 −0.500000 0.866025i 0
79.1 0.500000 0.866025i −1.00000 1.73205i −0.500000 0.866025i 0 −2.00000 0 −1.00000 −0.500000 + 0.866025i 0
\(n\): 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 98.2.c.a 2
3.b odd 2 1 882.2.g.d 2
4.b odd 2 1 784.2.i.i 2
7.b odd 2 1 98.2.c.b 2
7.c even 3 1 98.2.a.a 1
7.c even 3 1 inner 98.2.c.a 2
7.d odd 6 1 14.2.a.a 1
7.d odd 6 1 98.2.c.b 2
21.c even 2 1 882.2.g.c 2
21.g even 6 1 126.2.a.b 1
21.g even 6 1 882.2.g.c 2
21.h odd 6 1 882.2.a.i 1
21.h odd 6 1 882.2.g.d 2
28.d even 2 1 784.2.i.c 2
28.f even 6 1 112.2.a.c 1
28.f even 6 1 784.2.i.c 2
28.g odd 6 1 784.2.a.b 1
28.g odd 6 1 784.2.i.i 2
35.i odd 6 1 350.2.a.f 1
35.j even 6 1 2450.2.a.t 1
35.k even 12 2 350.2.c.d 2
35.l odd 12 2 2450.2.c.c 2
56.j odd 6 1 448.2.a.g 1
56.k odd 6 1 3136.2.a.z 1
56.m even 6 1 448.2.a.a 1
56.p even 6 1 3136.2.a.e 1
63.i even 6 1 1134.2.f.f 2
63.k odd 6 1 1134.2.f.l 2
63.s even 6 1 1134.2.f.f 2
63.t odd 6 1 1134.2.f.l 2
77.i even 6 1 1694.2.a.e 1
84.j odd 6 1 1008.2.a.h 1
84.n even 6 1 7056.2.a.bd 1
91.s odd 6 1 2366.2.a.j 1
91.bb even 12 2 2366.2.d.b 2
105.p even 6 1 3150.2.a.i 1
105.w odd 12 2 3150.2.g.j 2
112.v even 12 2 1792.2.b.g 2
112.x odd 12 2 1792.2.b.c 2
119.h odd 6 1 4046.2.a.f 1
133.o even 6 1 5054.2.a.c 1
140.s even 6 1 2800.2.a.g 1
140.x odd 12 2 2800.2.g.h 2
161.g even 6 1 7406.2.a.a 1
168.ba even 6 1 4032.2.a.w 1
168.be odd 6 1 4032.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 7.d odd 6 1
98.2.a.a 1 7.c even 3 1
98.2.c.a 2 1.a even 1 1 trivial
98.2.c.a 2 7.c even 3 1 inner
98.2.c.b 2 7.b odd 2 1
98.2.c.b 2 7.d odd 6 1
112.2.a.c 1 28.f even 6 1
126.2.a.b 1 21.g even 6 1
350.2.a.f 1 35.i odd 6 1
350.2.c.d 2 35.k even 12 2
448.2.a.a 1 56.m even 6 1
448.2.a.g 1 56.j odd 6 1
784.2.a.b 1 28.g odd 6 1
784.2.i.c 2 28.d even 2 1
784.2.i.c 2 28.f even 6 1
784.2.i.i 2 4.b odd 2 1
784.2.i.i 2 28.g odd 6 1
882.2.a.i 1 21.h odd 6 1
882.2.g.c 2 21.c even 2 1
882.2.g.c 2 21.g even 6 1
882.2.g.d 2 3.b odd 2 1
882.2.g.d 2 21.h odd 6 1
1008.2.a.h 1 84.j odd 6 1
1134.2.f.f 2 63.i even 6 1
1134.2.f.f 2 63.s even 6 1
1134.2.f.l 2 63.k odd 6 1
1134.2.f.l 2 63.t odd 6 1
1694.2.a.e 1 77.i even 6 1
1792.2.b.c 2 112.x odd 12 2
1792.2.b.g 2 112.v even 12 2
2366.2.a.j 1 91.s odd 6 1
2366.2.d.b 2 91.bb even 12 2
2450.2.a.t 1 35.j even 6 1
2450.2.c.c 2 35.l odd 12 2
2800.2.a.g 1 140.s even 6 1
2800.2.g.h 2 140.x odd 12 2
3136.2.a.e 1 56.p even 6 1
3136.2.a.z 1 56.k odd 6 1
3150.2.a.i 1 105.p even 6 1
3150.2.g.j 2 105.w odd 12 2
4032.2.a.r 1 168.be odd 6 1
4032.2.a.w 1 168.ba even 6 1
4046.2.a.f 1 119.h odd 6 1
5054.2.a.c 1 133.o even 6 1
7056.2.a.bd 1 84.n even 6 1
7406.2.a.a 1 161.g even 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 \(S_{2}^{\mathrm{new}}(98, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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