Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 7 \zeta_{6} q^{5} - 2 q^{6} - 8 q^{8} + 26 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 7 \zeta_{6} q^{5} - 2 q^{6} - 8 q^{8} + 26 \zeta_{6} q^{9} + (14 \zeta_{6} - 14) q^{10} + (35 \zeta_{6} - 35) q^{11} - 4 \zeta_{6} q^{12} - 66 q^{13} - 7 q^{15} - 16 \zeta_{6} q^{16} + ( - 59 \zeta_{6} + 59) q^{17} + (52 \zeta_{6} - 52) q^{18} + 137 \zeta_{6} q^{19} - 28 q^{20} - 70 q^{22} + 7 \zeta_{6} q^{23} + ( - 8 \zeta_{6} + 8) q^{24} + ( - 76 \zeta_{6} + 76) q^{25} - 132 \zeta_{6} q^{26} - 53 q^{27} + 106 q^{29} - 14 \zeta_{6} q^{30} + ( - 75 \zeta_{6} + 75) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 35 \zeta_{6} q^{33} + 118 q^{34} - 104 q^{36} - 11 \zeta_{6} q^{37} + (274 \zeta_{6} - 274) q^{38} + ( - 66 \zeta_{6} + 66) q^{39} - 56 \zeta_{6} q^{40} + 498 q^{41} + 260 q^{43} - 140 \zeta_{6} q^{44} + (182 \zeta_{6} - 182) q^{45} + (14 \zeta_{6} - 14) q^{46} - 171 \zeta_{6} q^{47} + 16 q^{48} + 152 q^{50} + 59 \zeta_{6} q^{51} + ( - 264 \zeta_{6} + 264) q^{52} + ( - 417 \zeta_{6} + 417) q^{53} - 106 \zeta_{6} q^{54} - 245 q^{55} - 137 q^{57} + 212 \zeta_{6} q^{58} + (17 \zeta_{6} - 17) q^{59} + ( - 28 \zeta_{6} + 28) q^{60} + 51 \zeta_{6} q^{61} + 150 q^{62} + 64 q^{64} - 462 \zeta_{6} q^{65} + ( - 70 \zeta_{6} + 70) q^{66} + (439 \zeta_{6} - 439) q^{67} + 236 \zeta_{6} q^{68} - 7 q^{69} - 784 q^{71} - 208 \zeta_{6} q^{72} + ( - 295 \zeta_{6} + 295) q^{73} + ( - 22 \zeta_{6} + 22) q^{74} + 76 \zeta_{6} q^{75} - 548 q^{76} + 132 q^{78} + 495 \zeta_{6} q^{79} + ( - 112 \zeta_{6} + 112) q^{80} + (649 \zeta_{6} - 649) q^{81} + 996 \zeta_{6} q^{82} - 932 q^{83} + 413 q^{85} + 520 \zeta_{6} q^{86} + (106 \zeta_{6} - 106) q^{87} + ( - 280 \zeta_{6} + 280) q^{88} - 873 \zeta_{6} q^{89} - 364 q^{90} - 28 q^{92} + 75 \zeta_{6} q^{93} + ( - 342 \zeta_{6} + 342) q^{94} + (959 \zeta_{6} - 959) q^{95} + 32 \zeta_{6} q^{96} + 290 q^{97} - 910 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} - 4 q^{4} + 7 q^{5} - 4 q^{6} - 16 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} - 4 q^{4} + 7 q^{5} - 4 q^{6} - 16 q^{8} + 26 q^{9} - 14 q^{10} - 35 q^{11} - 4 q^{12} - 132 q^{13} - 14 q^{15} - 16 q^{16} + 59 q^{17} - 52 q^{18} + 137 q^{19} - 56 q^{20} - 140 q^{22} + 7 q^{23} + 8 q^{24} + 76 q^{25} - 132 q^{26} - 106 q^{27} + 212 q^{29} - 14 q^{30} + 75 q^{31} + 32 q^{32} - 35 q^{33} + 236 q^{34} - 208 q^{36} - 11 q^{37} - 274 q^{38} + 66 q^{39} - 56 q^{40} + 996 q^{41} + 520 q^{43} - 140 q^{44} - 182 q^{45} - 14 q^{46} - 171 q^{47} + 32 q^{48} + 304 q^{50} + 59 q^{51} + 264 q^{52} + 417 q^{53} - 106 q^{54} - 490 q^{55} - 274 q^{57} + 212 q^{58} - 17 q^{59} + 28 q^{60} + 51 q^{61} + 300 q^{62} + 128 q^{64} - 462 q^{65} + 70 q^{66} - 439 q^{67} + 236 q^{68} - 14 q^{69} - 1568 q^{71} - 208 q^{72} + 295 q^{73} + 22 q^{74} + 76 q^{75} - 1096 q^{76} + 264 q^{78} + 495 q^{79} + 112 q^{80} - 649 q^{81} + 996 q^{82} - 1864 q^{83} + 826 q^{85} + 520 q^{86} - 106 q^{87} + 280 q^{88} - 873 q^{89} - 728 q^{90} - 56 q^{92} + 75 q^{93} + 342 q^{94} - 959 q^{95} + 32 q^{96} + 580 q^{97} - 1820 q^{99}+O(q^{100}) \) 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 −0.500000 + 0.866025i −2.00000 + 3.46410i 3.50000 + 6.06218i −2.00000 0 −8.00000 13.0000 + 22.5167i −7.00000 + 12.1244i
79.1 1.00000 1.73205i −0.500000 0.866025i −2.00000 3.46410i 3.50000 6.06218i −2.00000 0 −8.00000 13.0000 22.5167i −7.00000 12.1244i
\(n\): 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.e 2
3.b odd 2 1 882.4.g.d 2
7.b odd 2 1 14.4.c.b 2
7.c even 3 1 98.4.a.c 1
7.c even 3 1 inner 98.4.c.e 2
7.d odd 6 1 14.4.c.b 2
7.d odd 6 1 98.4.a.b 1
21.c even 2 1 126.4.g.c 2
21.g even 6 1 126.4.g.c 2
21.g even 6 1 882.4.a.k 1
21.h odd 6 1 882.4.a.p 1
21.h odd 6 1 882.4.g.d 2
28.d even 2 1 112.4.i.b 2
28.f even 6 1 112.4.i.b 2
28.f even 6 1 784.4.a.l 1
28.g odd 6 1 784.4.a.j 1
35.c odd 2 1 350.4.e.b 2
35.f even 4 2 350.4.j.d 4
35.i odd 6 1 350.4.e.b 2
35.i odd 6 1 2450.4.a.bh 1
35.j even 6 1 2450.4.a.bf 1
35.k even 12 2 350.4.j.d 4
56.e even 2 1 448.4.i.d 2
56.h odd 2 1 448.4.i.c 2
56.j odd 6 1 448.4.i.c 2
56.m even 6 1 448.4.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 7.b odd 2 1
14.4.c.b 2 7.d odd 6 1
98.4.a.b 1 7.d odd 6 1
98.4.a.c 1 7.c even 3 1
98.4.c.e 2 1.a even 1 1 trivial
98.4.c.e 2 7.c even 3 1 inner
112.4.i.b 2 28.d even 2 1
112.4.i.b 2 28.f even 6 1
126.4.g.c 2 21.c even 2 1
126.4.g.c 2 21.g even 6 1
350.4.e.b 2 35.c odd 2 1
350.4.e.b 2 35.i odd 6 1
350.4.j.d 4 35.f even 4 2
350.4.j.d 4 35.k even 12 2
448.4.i.c 2 56.h odd 2 1
448.4.i.c 2 56.j odd 6 1
448.4.i.d 2 56.e even 2 1
448.4.i.d 2 56.m even 6 1
784.4.a.j 1 28.g odd 6 1
784.4.a.l 1 28.f even 6 1
882.4.a.k 1 21.g even 6 1
882.4.a.p 1 21.h odd 6 1
882.4.g.d 2 3.b odd 2 1
882.4.g.d 2 21.h odd 6 1
2450.4.a.bf 1 35.j even 6 1
2450.4.a.bh 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) 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
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T + 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} - 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$29$ \( (T - 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T - 498)^{2} \) Copy content Toggle raw display
$43$ \( (T - 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} - 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} + 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T + 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T + 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T - 290)^{2} \) Copy content Toggle raw display
show more
show less