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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 2 q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 2 q^{6} - q^{8} - \zeta_{6} q^{9} - 2 \zeta_{6} q^{12} + 4 q^{13} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - \zeta_{6} + 1) q^{18} + 2 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 2) q^{24} + ( - 5 \zeta_{6} + 5) q^{25} + 4 \zeta_{6} q^{26} - 4 q^{27} - 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 q^{34} + q^{36} - 2 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{38} + (8 \zeta_{6} - 8) q^{39} - 6 q^{41} + 8 q^{43} - 12 \zeta_{6} q^{47} + 2 q^{48} + 5 q^{50} + 12 \zeta_{6} q^{51} + (4 \zeta_{6} - 4) q^{52} + (6 \zeta_{6} - 6) q^{53} - 4 \zeta_{6} q^{54} - 4 q^{57} - 6 \zeta_{6} q^{58} + (6 \zeta_{6} - 6) q^{59} + 8 \zeta_{6} q^{61} - 4 q^{62} + q^{64} + ( - 4 \zeta_{6} + 4) q^{67} + 6 \zeta_{6} q^{68} + \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 10 \zeta_{6} q^{75} - 2 q^{76} - 8 q^{78} - 8 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 6 \zeta_{6} q^{82} + 6 q^{83} + 8 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{87} - 6 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{94} + 2 \zeta_{6} q^{96} + 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 4 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 4 q^{6} - 2 q^{8} - q^{9} - 2 q^{12} + 8 q^{13} - q^{16} + 6 q^{17} + q^{18} + 2 q^{19} + 2 q^{24} + 5 q^{25} + 4 q^{26} - 8 q^{27} - 12 q^{29} - 4 q^{31} + q^{32} + 12 q^{34} + 2 q^{36} - 2 q^{37} - 2 q^{38} - 8 q^{39} - 12 q^{41} + 16 q^{43} - 12 q^{47} + 4 q^{48} + 10 q^{50} + 12 q^{51} - 4 q^{52} - 6 q^{53} - 4 q^{54} - 8 q^{57} - 6 q^{58} - 6 q^{59} + 8 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{67} + 6 q^{68} + q^{72} + 2 q^{73} + 2 q^{74} + 10 q^{75} - 4 q^{76} - 16 q^{78} - 8 q^{79} + 11 q^{81} - 6 q^{82} + 12 q^{83} + 8 q^{86} + 12 q^{87} - 6 q^{89} - 8 q^{93} + 12 q^{94} + 2 q^{96} + 20 q^{97}+O(q^{100}) \) 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
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 \( T_{3}^{2} + 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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