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: \(\Q(\sqrt{-3}) \)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -9 \zeta_{6} q^{5} + 10 q^{6} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -9 \zeta_{6} q^{5} + 10 q^{6} + 8 q^{8} + 2 \zeta_{6} q^{9} + ( -18 + 18 \zeta_{6} ) q^{10} + ( 57 - 57 \zeta_{6} ) q^{11} -20 \zeta_{6} q^{12} + 70 q^{13} + 45 q^{15} -16 \zeta_{6} q^{16} + ( 51 - 51 \zeta_{6} ) q^{17} + ( 4 - 4 \zeta_{6} ) q^{18} + 5 \zeta_{6} q^{19} + 36 q^{20} -114 q^{22} -69 \zeta_{6} q^{23} + ( -40 + 40 \zeta_{6} ) q^{24} + ( 44 - 44 \zeta_{6} ) q^{25} -140 \zeta_{6} q^{26} -145 q^{27} + 114 q^{29} -90 \zeta_{6} q^{30} + ( 23 - 23 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 285 \zeta_{6} q^{33} -102 q^{34} -8 q^{36} + 253 \zeta_{6} q^{37} + ( 10 - 10 \zeta_{6} ) q^{38} + ( -350 + 350 \zeta_{6} ) q^{39} -72 \zeta_{6} q^{40} + 42 q^{41} -124 q^{43} + 228 \zeta_{6} q^{44} + ( 18 - 18 \zeta_{6} ) q^{45} + ( -138 + 138 \zeta_{6} ) q^{46} + 201 \zeta_{6} q^{47} + 80 q^{48} -88 q^{50} + 255 \zeta_{6} q^{51} + ( -280 + 280 \zeta_{6} ) q^{52} + ( 393 - 393 \zeta_{6} ) q^{53} + 290 \zeta_{6} q^{54} -513 q^{55} -25 q^{57} -228 \zeta_{6} q^{58} + ( 219 - 219 \zeta_{6} ) q^{59} + ( -180 + 180 \zeta_{6} ) q^{60} -709 \zeta_{6} q^{61} -46 q^{62} + 64 q^{64} -630 \zeta_{6} q^{65} + ( 570 - 570 \zeta_{6} ) q^{66} + ( -419 + 419 \zeta_{6} ) q^{67} + 204 \zeta_{6} q^{68} + 345 q^{69} -96 q^{71} + 16 \zeta_{6} q^{72} + ( -313 + 313 \zeta_{6} ) q^{73} + ( 506 - 506 \zeta_{6} ) q^{74} + 220 \zeta_{6} q^{75} -20 q^{76} + 700 q^{78} -461 \zeta_{6} q^{79} + ( -144 + 144 \zeta_{6} ) q^{80} + ( 671 - 671 \zeta_{6} ) q^{81} -84 \zeta_{6} q^{82} + 588 q^{83} -459 q^{85} + 248 \zeta_{6} q^{86} + ( -570 + 570 \zeta_{6} ) q^{87} + ( 456 - 456 \zeta_{6} ) q^{88} -1017 \zeta_{6} q^{89} -36 q^{90} + 276 q^{92} + 115 \zeta_{6} q^{93} + ( 402 - 402 \zeta_{6} ) q^{94} + ( 45 - 45 \zeta_{6} ) q^{95} -160 \zeta_{6} q^{96} + 1834 q^{97} + 114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 5q^{3} - 4q^{4} - 9q^{5} + 20q^{6} + 16q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 5q^{3} - 4q^{4} - 9q^{5} + 20q^{6} + 16q^{8} + 2q^{9} - 18q^{10} + 57q^{11} - 20q^{12} + 140q^{13} + 90q^{15} - 16q^{16} + 51q^{17} + 4q^{18} + 5q^{19} + 72q^{20} - 228q^{22} - 69q^{23} - 40q^{24} + 44q^{25} - 140q^{26} - 290q^{27} + 228q^{29} - 90q^{30} + 23q^{31} - 32q^{32} + 285q^{33} - 204q^{34} - 16q^{36} + 253q^{37} + 10q^{38} - 350q^{39} - 72q^{40} + 84q^{41} - 248q^{43} + 228q^{44} + 18q^{45} - 138q^{46} + 201q^{47} + 160q^{48} - 176q^{50} + 255q^{51} - 280q^{52} + 393q^{53} + 290q^{54} - 1026q^{55} - 50q^{57} - 228q^{58} + 219q^{59} - 180q^{60} - 709q^{61} - 92q^{62} + 128q^{64} - 630q^{65} + 570q^{66} - 419q^{67} + 204q^{68} + 690q^{69} - 192q^{71} + 16q^{72} - 313q^{73} + 506q^{74} + 220q^{75} - 40q^{76} + 1400q^{78} - 461q^{79} - 144q^{80} + 671q^{81} - 84q^{82} + 1176q^{83} - 918q^{85} + 248q^{86} - 570q^{87} + 456q^{88} - 1017q^{89} - 72q^{90} + 552q^{92} + 115q^{93} + 402q^{94} + 45q^{95} - 160q^{96} + 3668q^{97} + 228q^{99} + O(q^{100}) \)

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 −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 \( T_{3}^{2} + 5 T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} \)
$3$ \( 1 + 5 T - 2 T^{2} + 135 T^{3} + 729 T^{4} \)
$5$ \( 1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4} \)
$7$ \( \)
$11$ \( 1 - 57 T + 1918 T^{2} - 75867 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 70 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 51 T - 2312 T^{2} - 250563 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 5 T - 6834 T^{2} - 34295 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 69 T - 7406 T^{2} + 839523 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 114 T + 24389 T^{2} )^{2} \)
$31$ \( 1 - 23 T - 29262 T^{2} - 685193 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 253 T + 13356 T^{2} - 12815209 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 - 42 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 124 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 201 T - 63422 T^{2} - 20868423 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 393 T + 5572 T^{2} - 58508661 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 219 T - 157418 T^{2} - 44978001 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 709 T + 275700 T^{2} + 160929529 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 419 T - 125202 T^{2} + 126019697 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 96 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 313 T - 291048 T^{2} + 121762321 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 461 T - 280518 T^{2} + 227290979 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 - 588 T + 571787 T^{2} )^{2} \)
$89$ \( 1 + 1017 T + 329320 T^{2} + 716953473 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 - 1834 T + 912673 T^{2} )^{2} \)
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