# Properties

 Label 7098.2.a.k Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ 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] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.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: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - 2 * q^5 - q^6 + q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{12} - q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 4 q^{19} - 2 q^{20} + q^{21} + 6 q^{23} - q^{24} - q^{25} + q^{27} + q^{28} + 2 q^{30} - q^{32} - 2 q^{34} - 2 q^{35} + q^{36} - 2 q^{37} + 4 q^{38} + 2 q^{40} - q^{42} - 4 q^{43} - 2 q^{45} - 6 q^{46} + 8 q^{47} + q^{48} + q^{49} + q^{50} + 2 q^{51} + 4 q^{53} - q^{54} - q^{56} - 4 q^{57} - 6 q^{59} - 2 q^{60} + 12 q^{61} + q^{63} + q^{64} + 2 q^{67} + 2 q^{68} + 6 q^{69} + 2 q^{70} - q^{72} - 14 q^{73} + 2 q^{74} - q^{75} - 4 q^{76} - 2 q^{80} + q^{81} + 14 q^{83} + q^{84} - 4 q^{85} + 4 q^{86} + 4 q^{89} + 2 q^{90} + 6 q^{92} - 8 q^{94} + 8 q^{95} - q^{96} + 2 q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^3 + q^4 - 2 * q^5 - q^6 + q^7 - q^8 + q^9 + 2 * q^10 + q^12 - q^14 - 2 * q^15 + q^16 + 2 * q^17 - q^18 - 4 * q^19 - 2 * q^20 + q^21 + 6 * q^23 - q^24 - q^25 + q^27 + q^28 + 2 * q^30 - q^32 - 2 * q^34 - 2 * q^35 + q^36 - 2 * q^37 + 4 * q^38 + 2 * q^40 - q^42 - 4 * q^43 - 2 * q^45 - 6 * q^46 + 8 * q^47 + q^48 + q^49 + q^50 + 2 * q^51 + 4 * q^53 - q^54 - q^56 - 4 * q^57 - 6 * q^59 - 2 * q^60 + 12 * q^61 + q^63 + q^64 + 2 * q^67 + 2 * q^68 + 6 * q^69 + 2 * q^70 - q^72 - 14 * q^73 + 2 * q^74 - q^75 - 4 * q^76 - 2 * q^80 + q^81 + 14 * q^83 + q^84 - 4 * q^85 + 4 * q^86 + 4 * q^89 + 2 * q^90 + 6 * q^92 - 8 * q^94 + 8 * q^95 - q^96 + 2 * q^97 - q^98

## 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
−1.00000 1.00000 1.00000 −2.00000 −1.00000 1.00000 −1.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 7098.2.a.k 1
13.b even 2 1 7098.2.a.bc 1
13.d odd 4 2 546.2.c.b 2
39.f even 4 2 1638.2.c.b 2
52.f even 4 2 4368.2.h.f 2
91.i even 4 2 3822.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.b 2 13.d odd 4 2
1638.2.c.b 2 39.f even 4 2
3822.2.c.c 2 91.i even 4 2
4368.2.h.f 2 52.f even 4 2
7098.2.a.k 1 1.a even 1 1 trivial
7098.2.a.bc 1 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{11}$$ T11 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T - 6$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T - 4$$
$59$ $$T + 6$$
$61$ $$T - 12$$
$67$ $$T - 2$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T$$
$83$ $$T - 14$$
$89$ $$T - 4$$
$97$ $$T - 2$$