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

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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: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 7 \zeta_{6} q^{5} -2 q^{6} -8 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 7 \zeta_{6} q^{5} -2 q^{6} -8 q^{8} + 26 \zeta_{6} q^{9} + ( -14 + 14 \zeta_{6} ) q^{10} + ( -35 + 35 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{12} -66 q^{13} -7 q^{15} -16 \zeta_{6} q^{16} + ( 59 - 59 \zeta_{6} ) q^{17} + ( -52 + 52 \zeta_{6} ) q^{18} + 137 \zeta_{6} q^{19} -28 q^{20} -70 q^{22} + 7 \zeta_{6} q^{23} + ( 8 - 8 \zeta_{6} ) q^{24} + ( 76 - 76 \zeta_{6} ) q^{25} -132 \zeta_{6} q^{26} -53 q^{27} + 106 q^{29} -14 \zeta_{6} q^{30} + ( 75 - 75 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -35 \zeta_{6} q^{33} + 118 q^{34} -104 q^{36} -11 \zeta_{6} q^{37} + ( -274 + 274 \zeta_{6} ) q^{38} + ( 66 - 66 \zeta_{6} ) q^{39} -56 \zeta_{6} q^{40} + 498 q^{41} + 260 q^{43} -140 \zeta_{6} q^{44} + ( -182 + 182 \zeta_{6} ) q^{45} + ( -14 + 14 \zeta_{6} ) q^{46} -171 \zeta_{6} q^{47} + 16 q^{48} + 152 q^{50} + 59 \zeta_{6} q^{51} + ( 264 - 264 \zeta_{6} ) q^{52} + ( 417 - 417 \zeta_{6} ) q^{53} -106 \zeta_{6} q^{54} -245 q^{55} -137 q^{57} + 212 \zeta_{6} q^{58} + ( -17 + 17 \zeta_{6} ) q^{59} + ( 28 - 28 \zeta_{6} ) q^{60} + 51 \zeta_{6} q^{61} + 150 q^{62} + 64 q^{64} -462 \zeta_{6} q^{65} + ( 70 - 70 \zeta_{6} ) q^{66} + ( -439 + 439 \zeta_{6} ) q^{67} + 236 \zeta_{6} q^{68} -7 q^{69} -784 q^{71} -208 \zeta_{6} q^{72} + ( 295 - 295 \zeta_{6} ) q^{73} + ( 22 - 22 \zeta_{6} ) q^{74} + 76 \zeta_{6} q^{75} -548 q^{76} + 132 q^{78} + 495 \zeta_{6} q^{79} + ( 112 - 112 \zeta_{6} ) q^{80} + ( -649 + 649 \zeta_{6} ) q^{81} + 996 \zeta_{6} q^{82} -932 q^{83} + 413 q^{85} + 520 \zeta_{6} q^{86} + ( -106 + 106 \zeta_{6} ) q^{87} + ( 280 - 280 \zeta_{6} ) q^{88} -873 \zeta_{6} q^{89} -364 q^{90} -28 q^{92} + 75 \zeta_{6} q^{93} + ( 342 - 342 \zeta_{6} ) q^{94} + ( -959 + 959 \zeta_{6} ) q^{95} + 32 \zeta_{6} q^{96} + 290 q^{97} -910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} - 4q^{4} + 7q^{5} - 4q^{6} - 16q^{8} + 26q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} - 4q^{4} + 7q^{5} - 4q^{6} - 16q^{8} + 26q^{9} - 14q^{10} - 35q^{11} - 4q^{12} - 132q^{13} - 14q^{15} - 16q^{16} + 59q^{17} - 52q^{18} + 137q^{19} - 56q^{20} - 140q^{22} + 7q^{23} + 8q^{24} + 76q^{25} - 132q^{26} - 106q^{27} + 212q^{29} - 14q^{30} + 75q^{31} + 32q^{32} - 35q^{33} + 236q^{34} - 208q^{36} - 11q^{37} - 274q^{38} + 66q^{39} - 56q^{40} + 996q^{41} + 520q^{43} - 140q^{44} - 182q^{45} - 14q^{46} - 171q^{47} + 32q^{48} + 304q^{50} + 59q^{51} + 264q^{52} + 417q^{53} - 106q^{54} - 490q^{55} - 274q^{57} + 212q^{58} - 17q^{59} + 28q^{60} + 51q^{61} + 300q^{62} + 128q^{64} - 462q^{65} + 70q^{66} - 439q^{67} + 236q^{68} - 14q^{69} - 1568q^{71} - 208q^{72} + 295q^{73} + 22q^{74} + 76q^{75} - 1096q^{76} + 264q^{78} + 495q^{79} + 112q^{80} - 649q^{81} + 996q^{82} - 1864q^{83} + 826q^{85} + 520q^{86} - 106q^{87} + 280q^{88} - 873q^{89} - 728q^{90} - 56q^{92} + 75q^{93} + 342q^{94} - 959q^{95} + 32q^{96} + 580q^{97} - 1820q^{99} + O(q^{100}) \)

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
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 \(S_{4}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ \( 1 + T - 26 T^{2} + 27 T^{3} + 729 T^{4} \)
$5$ \( 1 - 7 T - 76 T^{2} - 875 T^{3} + 15625 T^{4} \)
$7$ 1
$11$ \( 1 + 35 T - 106 T^{2} + 46585 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 + 66 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 59 T - 1432 T^{2} - 289867 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 137 T + 11910 T^{2} - 939683 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 7 T - 12118 T^{2} - 85169 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 106 T + 24389 T^{2} )^{2} \)
$31$ \( 1 - 75 T - 24166 T^{2} - 2234325 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 11 T - 50532 T^{2} + 557183 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 - 498 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 260 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 171 T - 74582 T^{2} + 17753733 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 417 T + 25012 T^{2} - 62081709 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 17 T - 205090 T^{2} + 3491443 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 51 T - 224380 T^{2} - 11576031 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 439 T - 108042 T^{2} + 132034957 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 784 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 295 T - 301992 T^{2} - 114760015 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 495 T - 248014 T^{2} - 244054305 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 + 932 T + 571787 T^{2} )^{2} \)
$89$ \( 1 + 873 T + 57160 T^{2} + 615437937 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 - 290 T + 912673 T^{2} )^{2} \)
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