Properties

Label 98.4.c.f
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: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 14 \zeta_{6} q^{5} + 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} + ( - 28 \zeta_{6} + 28) q^{10} + ( - 28 \zeta_{6} + 28) q^{11} + 32 \zeta_{6} q^{12} - 18 q^{13} - 112 q^{15} - 16 \zeta_{6} q^{16} + ( - 74 \zeta_{6} + 74) q^{17} + ( - 74 \zeta_{6} + 74) q^{18} + 80 \zeta_{6} q^{19} + 56 q^{20} + 56 q^{22} + 112 \zeta_{6} q^{23} + (64 \zeta_{6} - 64) q^{24} + (71 \zeta_{6} - 71) q^{25} - 36 \zeta_{6} q^{26} - 80 q^{27} + 190 q^{29} - 224 \zeta_{6} q^{30} + ( - 72 \zeta_{6} + 72) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 224 \zeta_{6} q^{33} + 148 q^{34} + 148 q^{36} + 346 \zeta_{6} q^{37} + (160 \zeta_{6} - 160) q^{38} + (144 \zeta_{6} - 144) q^{39} + 112 \zeta_{6} q^{40} - 162 q^{41} - 412 q^{43} + 112 \zeta_{6} q^{44} + (518 \zeta_{6} - 518) q^{45} + (224 \zeta_{6} - 224) q^{46} + 24 \zeta_{6} q^{47} - 128 q^{48} - 142 q^{50} - 592 \zeta_{6} q^{51} + ( - 72 \zeta_{6} + 72) q^{52} + (318 \zeta_{6} - 318) q^{53} - 160 \zeta_{6} q^{54} - 392 q^{55} + 640 q^{57} + 380 \zeta_{6} q^{58} + (200 \zeta_{6} - 200) q^{59} + ( - 448 \zeta_{6} + 448) q^{60} - 198 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} + 252 \zeta_{6} q^{65} + ( - 448 \zeta_{6} + 448) q^{66} + ( - 716 \zeta_{6} + 716) q^{67} + 296 \zeta_{6} q^{68} + 896 q^{69} + 392 q^{71} + 296 \zeta_{6} q^{72} + ( - 538 \zeta_{6} + 538) q^{73} + (692 \zeta_{6} - 692) q^{74} + 568 \zeta_{6} q^{75} - 320 q^{76} - 288 q^{78} - 240 \zeta_{6} q^{79} + (224 \zeta_{6} - 224) q^{80} + ( - 359 \zeta_{6} + 359) q^{81} - 324 \zeta_{6} q^{82} + 1072 q^{83} - 1036 q^{85} - 824 \zeta_{6} q^{86} + ( - 1520 \zeta_{6} + 1520) q^{87} + (224 \zeta_{6} - 224) q^{88} + 810 \zeta_{6} q^{89} - 1036 q^{90} - 448 q^{92} - 576 \zeta_{6} q^{93} + (48 \zeta_{6} - 48) q^{94} + ( - 1120 \zeta_{6} + 1120) q^{95} - 256 \zeta_{6} q^{96} - 1354 q^{97} - 1036 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 8 q^{3} - 4 q^{4} - 14 q^{5} + 32 q^{6} - 16 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 8 q^{3} - 4 q^{4} - 14 q^{5} + 32 q^{6} - 16 q^{8} - 37 q^{9} + 28 q^{10} + 28 q^{11} + 32 q^{12} - 36 q^{13} - 224 q^{15} - 16 q^{16} + 74 q^{17} + 74 q^{18} + 80 q^{19} + 112 q^{20} + 112 q^{22} + 112 q^{23} - 64 q^{24} - 71 q^{25} - 36 q^{26} - 160 q^{27} + 380 q^{29} - 224 q^{30} + 72 q^{31} + 32 q^{32} - 224 q^{33} + 296 q^{34} + 296 q^{36} + 346 q^{37} - 160 q^{38} - 144 q^{39} + 112 q^{40} - 324 q^{41} - 824 q^{43} + 112 q^{44} - 518 q^{45} - 224 q^{46} + 24 q^{47} - 256 q^{48} - 284 q^{50} - 592 q^{51} + 72 q^{52} - 318 q^{53} - 160 q^{54} - 784 q^{55} + 1280 q^{57} + 380 q^{58} - 200 q^{59} + 448 q^{60} - 198 q^{61} + 288 q^{62} + 128 q^{64} + 252 q^{65} + 448 q^{66} + 716 q^{67} + 296 q^{68} + 1792 q^{69} + 784 q^{71} + 296 q^{72} + 538 q^{73} - 692 q^{74} + 568 q^{75} - 640 q^{76} - 576 q^{78} - 240 q^{79} - 224 q^{80} + 359 q^{81} - 324 q^{82} + 2144 q^{83} - 2072 q^{85} - 824 q^{86} + 1520 q^{87} - 224 q^{88} + 810 q^{89} - 2072 q^{90} - 896 q^{92} - 576 q^{93} - 48 q^{94} + 1120 q^{95} - 256 q^{96} - 2708 q^{97} - 2072 q^{99}+O(q^{100}) \) 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 4.00000 6.92820i −2.00000 + 3.46410i −7.00000 12.1244i 16.0000 0 −8.00000 −18.5000 32.0429i 14.0000 24.2487i
79.1 1.00000 1.73205i 4.00000 + 6.92820i −2.00000 3.46410i −7.00000 + 12.1244i 16.0000 0 −8.00000 −18.5000 + 32.0429i 14.0000 + 24.2487i
\(n\): 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.f 2
3.b odd 2 1 882.4.g.k 2
7.b odd 2 1 98.4.c.d 2
7.c even 3 1 98.4.a.a 1
7.c even 3 1 inner 98.4.c.f 2
7.d odd 6 1 14.4.a.a 1
7.d odd 6 1 98.4.c.d 2
21.c even 2 1 882.4.g.b 2
21.g even 6 1 126.4.a.h 1
21.g even 6 1 882.4.g.b 2
21.h odd 6 1 882.4.a.i 1
21.h odd 6 1 882.4.g.k 2
28.f even 6 1 112.4.a.a 1
28.g odd 6 1 784.4.a.s 1
35.i odd 6 1 350.4.a.l 1
35.j even 6 1 2450.4.a.bo 1
35.k even 12 2 350.4.c.b 2
56.j odd 6 1 448.4.a.b 1
56.m even 6 1 448.4.a.o 1
77.i even 6 1 1694.4.a.g 1
84.j odd 6 1 1008.4.a.s 1
91.s odd 6 1 2366.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 7.d odd 6 1
98.4.a.a 1 7.c even 3 1
98.4.c.d 2 7.b odd 2 1
98.4.c.d 2 7.d odd 6 1
98.4.c.f 2 1.a even 1 1 trivial
98.4.c.f 2 7.c even 3 1 inner
112.4.a.a 1 28.f even 6 1
126.4.a.h 1 21.g even 6 1
350.4.a.l 1 35.i odd 6 1
350.4.c.b 2 35.k even 12 2
448.4.a.b 1 56.j odd 6 1
448.4.a.o 1 56.m even 6 1
784.4.a.s 1 28.g odd 6 1
882.4.a.i 1 21.h odd 6 1
882.4.g.b 2 21.c even 2 1
882.4.g.b 2 21.g even 6 1
882.4.g.k 2 3.b odd 2 1
882.4.g.k 2 21.h odd 6 1
1008.4.a.s 1 84.j odd 6 1
1694.4.a.g 1 77.i even 6 1
2366.4.a.h 1 91.s odd 6 1
2450.4.a.bo 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8T_{3} + 64 \) 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
$3$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$5$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 28T + 784 \) Copy content Toggle raw display
$13$ \( (T + 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 74T + 5476 \) Copy content Toggle raw display
$19$ \( T^{2} - 80T + 6400 \) Copy content Toggle raw display
$23$ \( T^{2} - 112T + 12544 \) Copy content Toggle raw display
$29$ \( (T - 190)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$37$ \( T^{2} - 346T + 119716 \) Copy content Toggle raw display
$41$ \( (T + 162)^{2} \) Copy content Toggle raw display
$43$ \( (T + 412)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 24T + 576 \) Copy content Toggle raw display
$53$ \( T^{2} + 318T + 101124 \) Copy content Toggle raw display
$59$ \( T^{2} + 200T + 40000 \) Copy content Toggle raw display
$61$ \( T^{2} + 198T + 39204 \) Copy content Toggle raw display
$67$ \( T^{2} - 716T + 512656 \) Copy content Toggle raw display
$71$ \( (T - 392)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 538T + 289444 \) Copy content Toggle raw display
$79$ \( T^{2} + 240T + 57600 \) Copy content Toggle raw display
$83$ \( (T - 1072)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 810T + 656100 \) Copy content Toggle raw display
$97$ \( (T + 1354)^{2} \) Copy content Toggle raw display
show more
show less