# Properties

 Label 98.4.a.b Level $98$ Weight $4$ Character orbit 98.a Self dual yes Analytic conductor $5.782$ Analytic rank $0$ 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] = [98,4,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, 4, names="a")

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$5.78218718056$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - q^{3} + 4 q^{4} + 7 q^{5} + 2 q^{6} - 8 q^{8} - 26 q^{9}+O(q^{10})$$ q - 2 * q^2 - q^3 + 4 * q^4 + 7 * q^5 + 2 * q^6 - 8 * q^8 - 26 * q^9 $$q - 2 q^{2} - q^{3} + 4 q^{4} + 7 q^{5} + 2 q^{6} - 8 q^{8} - 26 q^{9} - 14 q^{10} + 35 q^{11} - 4 q^{12} + 66 q^{13} - 7 q^{15} + 16 q^{16} + 59 q^{17} + 52 q^{18} + 137 q^{19} + 28 q^{20} - 70 q^{22} - 7 q^{23} + 8 q^{24} - 76 q^{25} - 132 q^{26} + 53 q^{27} + 106 q^{29} + 14 q^{30} + 75 q^{31} - 32 q^{32} - 35 q^{33} - 118 q^{34} - 104 q^{36} + 11 q^{37} - 274 q^{38} - 66 q^{39} - 56 q^{40} - 498 q^{41} + 260 q^{43} + 140 q^{44} - 182 q^{45} + 14 q^{46} - 171 q^{47} - 16 q^{48} + 152 q^{50} - 59 q^{51} + 264 q^{52} - 417 q^{53} - 106 q^{54} + 245 q^{55} - 137 q^{57} - 212 q^{58} - 17 q^{59} - 28 q^{60} + 51 q^{61} - 150 q^{62} + 64 q^{64} + 462 q^{65} + 70 q^{66} + 439 q^{67} + 236 q^{68} + 7 q^{69} - 784 q^{71} + 208 q^{72} + 295 q^{73} - 22 q^{74} + 76 q^{75} + 548 q^{76} + 132 q^{78} - 495 q^{79} + 112 q^{80} + 649 q^{81} + 996 q^{82} + 932 q^{83} + 413 q^{85} - 520 q^{86} - 106 q^{87} - 280 q^{88} - 873 q^{89} + 364 q^{90} - 28 q^{92} - 75 q^{93} + 342 q^{94} + 959 q^{95} + 32 q^{96} - 290 q^{97} - 910 q^{99}+O(q^{100})$$ q - 2 * q^2 - q^3 + 4 * q^4 + 7 * q^5 + 2 * q^6 - 8 * q^8 - 26 * q^9 - 14 * q^10 + 35 * q^11 - 4 * q^12 + 66 * q^13 - 7 * q^15 + 16 * q^16 + 59 * q^17 + 52 * q^18 + 137 * q^19 + 28 * q^20 - 70 * q^22 - 7 * q^23 + 8 * q^24 - 76 * q^25 - 132 * q^26 + 53 * q^27 + 106 * q^29 + 14 * q^30 + 75 * q^31 - 32 * q^32 - 35 * q^33 - 118 * q^34 - 104 * q^36 + 11 * q^37 - 274 * q^38 - 66 * q^39 - 56 * q^40 - 498 * q^41 + 260 * q^43 + 140 * q^44 - 182 * q^45 + 14 * q^46 - 171 * q^47 - 16 * q^48 + 152 * q^50 - 59 * q^51 + 264 * q^52 - 417 * q^53 - 106 * q^54 + 245 * q^55 - 137 * q^57 - 212 * q^58 - 17 * q^59 - 28 * q^60 + 51 * q^61 - 150 * q^62 + 64 * q^64 + 462 * q^65 + 70 * q^66 + 439 * q^67 + 236 * q^68 + 7 * q^69 - 784 * q^71 + 208 * q^72 + 295 * q^73 - 22 * q^74 + 76 * q^75 + 548 * q^76 + 132 * q^78 - 495 * q^79 + 112 * q^80 + 649 * q^81 + 996 * q^82 + 932 * q^83 + 413 * q^85 - 520 * q^86 - 106 * q^87 - 280 * q^88 - 873 * q^89 + 364 * q^90 - 28 * q^92 - 75 * q^93 + 342 * q^94 + 959 * q^95 + 32 * q^96 - 290 * q^97 - 910 * 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
−2.00000 −1.00000 4.00000 7.00000 2.00000 0 −8.00000 −26.0000 −14.0000
 $$n$$: 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 98.4.a.b 1
3.b odd 2 1 882.4.a.k 1
4.b odd 2 1 784.4.a.l 1
5.b even 2 1 2450.4.a.bh 1
7.b odd 2 1 98.4.a.c 1
7.c even 3 2 14.4.c.b 2
7.d odd 6 2 98.4.c.e 2
21.c even 2 1 882.4.a.p 1
21.g even 6 2 882.4.g.d 2
21.h odd 6 2 126.4.g.c 2
28.d even 2 1 784.4.a.j 1
28.g odd 6 2 112.4.i.b 2
35.c odd 2 1 2450.4.a.bf 1
35.j even 6 2 350.4.e.b 2
35.l odd 12 4 350.4.j.d 4
56.k odd 6 2 448.4.i.d 2
56.p even 6 2 448.4.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 7.c even 3 2
98.4.a.b 1 1.a even 1 1 trivial
98.4.a.c 1 7.b odd 2 1
98.4.c.e 2 7.d odd 6 2
112.4.i.b 2 28.g odd 6 2
126.4.g.c 2 21.h odd 6 2
350.4.e.b 2 35.j even 6 2
350.4.j.d 4 35.l odd 12 4
448.4.i.c 2 56.p even 6 2
448.4.i.d 2 56.k odd 6 2
784.4.a.j 1 28.d even 2 1
784.4.a.l 1 4.b odd 2 1
882.4.a.k 1 3.b odd 2 1
882.4.a.p 1 21.c even 2 1
882.4.g.d 2 21.g even 6 2
2450.4.a.bf 1 35.c odd 2 1
2450.4.a.bh 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 1$$
$5$ $$T - 7$$
$7$ $$T$$
$11$ $$T - 35$$
$13$ $$T - 66$$
$17$ $$T - 59$$
$19$ $$T - 137$$
$23$ $$T + 7$$
$29$ $$T - 106$$
$31$ $$T - 75$$
$37$ $$T - 11$$
$41$ $$T + 498$$
$43$ $$T - 260$$
$47$ $$T + 171$$
$53$ $$T + 417$$
$59$ $$T + 17$$
$61$ $$T - 51$$
$67$ $$T - 439$$
$71$ $$T + 784$$
$73$ $$T - 295$$
$79$ $$T + 495$$
$83$ $$T - 932$$
$89$ $$T + 873$$
$97$ $$T + 290$$