# Properties

 Label 98.6.a.b Level $98$ Weight $6$ Character orbit 98.a Self dual yes Analytic conductor $15.718$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,6,Mod(1,98)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 98.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$15.7176143417$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} - 8 q^{3} + 16 q^{4} - 10 q^{5} - 32 q^{6} + 64 q^{8} - 179 q^{9}+O(q^{10})$$ q + 4 * q^2 - 8 * q^3 + 16 * q^4 - 10 * q^5 - 32 * q^6 + 64 * q^8 - 179 * q^9 $$q + 4 q^{2} - 8 q^{3} + 16 q^{4} - 10 q^{5} - 32 q^{6} + 64 q^{8} - 179 q^{9} - 40 q^{10} - 340 q^{11} - 128 q^{12} + 294 q^{13} + 80 q^{15} + 256 q^{16} - 1226 q^{17} - 716 q^{18} - 2432 q^{19} - 160 q^{20} - 1360 q^{22} + 2000 q^{23} - 512 q^{24} - 3025 q^{25} + 1176 q^{26} + 3376 q^{27} - 6746 q^{29} + 320 q^{30} - 8856 q^{31} + 1024 q^{32} + 2720 q^{33} - 4904 q^{34} - 2864 q^{36} + 9182 q^{37} - 9728 q^{38} - 2352 q^{39} - 640 q^{40} + 14574 q^{41} + 8108 q^{43} - 5440 q^{44} + 1790 q^{45} + 8000 q^{46} + 312 q^{47} - 2048 q^{48} - 12100 q^{50} + 9808 q^{51} + 4704 q^{52} - 14634 q^{53} + 13504 q^{54} + 3400 q^{55} + 19456 q^{57} - 26984 q^{58} + 27656 q^{59} + 1280 q^{60} - 34338 q^{61} - 35424 q^{62} + 4096 q^{64} - 2940 q^{65} + 10880 q^{66} + 12316 q^{67} - 19616 q^{68} - 16000 q^{69} + 36920 q^{71} - 11456 q^{72} + 61718 q^{73} + 36728 q^{74} + 24200 q^{75} - 38912 q^{76} - 9408 q^{78} - 64752 q^{79} - 2560 q^{80} + 16489 q^{81} + 58296 q^{82} + 77056 q^{83} + 12260 q^{85} + 32432 q^{86} + 53968 q^{87} - 21760 q^{88} + 8166 q^{89} + 7160 q^{90} + 32000 q^{92} + 70848 q^{93} + 1248 q^{94} + 24320 q^{95} - 8192 q^{96} - 20650 q^{97} + 60860 q^{99}+O(q^{100})$$ q + 4 * q^2 - 8 * q^3 + 16 * q^4 - 10 * q^5 - 32 * q^6 + 64 * q^8 - 179 * q^9 - 40 * q^10 - 340 * q^11 - 128 * q^12 + 294 * q^13 + 80 * q^15 + 256 * q^16 - 1226 * q^17 - 716 * q^18 - 2432 * q^19 - 160 * q^20 - 1360 * q^22 + 2000 * q^23 - 512 * q^24 - 3025 * q^25 + 1176 * q^26 + 3376 * q^27 - 6746 * q^29 + 320 * q^30 - 8856 * q^31 + 1024 * q^32 + 2720 * q^33 - 4904 * q^34 - 2864 * q^36 + 9182 * q^37 - 9728 * q^38 - 2352 * q^39 - 640 * q^40 + 14574 * q^41 + 8108 * q^43 - 5440 * q^44 + 1790 * q^45 + 8000 * q^46 + 312 * q^47 - 2048 * q^48 - 12100 * q^50 + 9808 * q^51 + 4704 * q^52 - 14634 * q^53 + 13504 * q^54 + 3400 * q^55 + 19456 * q^57 - 26984 * q^58 + 27656 * q^59 + 1280 * q^60 - 34338 * q^61 - 35424 * q^62 + 4096 * q^64 - 2940 * q^65 + 10880 * q^66 + 12316 * q^67 - 19616 * q^68 - 16000 * q^69 + 36920 * q^71 - 11456 * q^72 + 61718 * q^73 + 36728 * q^74 + 24200 * q^75 - 38912 * q^76 - 9408 * q^78 - 64752 * q^79 - 2560 * q^80 + 16489 * q^81 + 58296 * q^82 + 77056 * q^83 + 12260 * q^85 + 32432 * q^86 + 53968 * q^87 - 21760 * q^88 + 8166 * q^89 + 7160 * q^90 + 32000 * q^92 + 70848 * q^93 + 1248 * q^94 + 24320 * q^95 - 8192 * q^96 - 20650 * q^97 + 60860 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
4.00000 −8.00000 16.0000 −10.0000 −32.0000 0 64.0000 −179.000 −40.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.6.a.b 1
3.b odd 2 1 882.6.a.g 1
4.b odd 2 1 784.6.a.h 1
7.b odd 2 1 14.6.a.b 1
7.c even 3 2 98.6.c.b 2
7.d odd 6 2 98.6.c.a 2
21.c even 2 1 126.6.a.c 1
28.d even 2 1 112.6.a.d 1
35.c odd 2 1 350.6.a.b 1
35.f even 4 2 350.6.c.f 2
56.e even 2 1 448.6.a.k 1
56.h odd 2 1 448.6.a.f 1
84.h odd 2 1 1008.6.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 7.b odd 2 1
98.6.a.b 1 1.a even 1 1 trivial
98.6.c.a 2 7.d odd 6 2
98.6.c.b 2 7.c even 3 2
112.6.a.d 1 28.d even 2 1
126.6.a.c 1 21.c even 2 1
350.6.a.b 1 35.c odd 2 1
350.6.c.f 2 35.f even 4 2
448.6.a.f 1 56.h odd 2 1
448.6.a.k 1 56.e even 2 1
784.6.a.h 1 4.b odd 2 1
882.6.a.g 1 3.b odd 2 1
1008.6.a.n 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T + 8$$
$5$ $$T + 10$$
$7$ $$T$$
$11$ $$T + 340$$
$13$ $$T - 294$$
$17$ $$T + 1226$$
$19$ $$T + 2432$$
$23$ $$T - 2000$$
$29$ $$T + 6746$$
$31$ $$T + 8856$$
$37$ $$T - 9182$$
$41$ $$T - 14574$$
$43$ $$T - 8108$$
$47$ $$T - 312$$
$53$ $$T + 14634$$
$59$ $$T - 27656$$
$61$ $$T + 34338$$
$67$ $$T - 12316$$
$71$ $$T - 36920$$
$73$ $$T - 61718$$
$79$ $$T + 64752$$
$83$ $$T - 77056$$
$89$ $$T - 8166$$
$97$ $$T + 20650$$