# Properties

 Label 7098.2.a.o Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 - q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 3 q^{11} + q^{12} + q^{14} + q^{15} + q^{16} - 7 q^{17} - q^{18} - 3 q^{19} + q^{20} - q^{21} - 3 q^{22} - q^{23} - q^{24} - 4 q^{25} + q^{27} - q^{28} - q^{29} - q^{30} + 8 q^{31} - q^{32} + 3 q^{33} + 7 q^{34} - q^{35} + q^{36} - q^{37} + 3 q^{38} - q^{40} - 4 q^{41} + q^{42} + 5 q^{43} + 3 q^{44} + q^{45} + q^{46} + q^{48} + q^{49} + 4 q^{50} - 7 q^{51} - 6 q^{53} - q^{54} + 3 q^{55} + q^{56} - 3 q^{57} + q^{58} + 10 q^{59} + q^{60} - 13 q^{61} - 8 q^{62} - q^{63} + q^{64} - 3 q^{66} + 8 q^{67} - 7 q^{68} - q^{69} + q^{70} - 6 q^{71} - q^{72} - 13 q^{73} + q^{74} - 4 q^{75} - 3 q^{76} - 3 q^{77} - 12 q^{79} + q^{80} + q^{81} + 4 q^{82} - 2 q^{83} - q^{84} - 7 q^{85} - 5 q^{86} - q^{87} - 3 q^{88} + 12 q^{89} - q^{90} - q^{92} + 8 q^{93} - 3 q^{95} - q^{96} - 6 q^{97} - q^{98} + 3 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 - q^7 - q^8 + q^9 - q^10 + 3 * q^11 + q^12 + q^14 + q^15 + q^16 - 7 * q^17 - q^18 - 3 * q^19 + q^20 - q^21 - 3 * q^22 - q^23 - q^24 - 4 * q^25 + q^27 - q^28 - q^29 - q^30 + 8 * q^31 - q^32 + 3 * q^33 + 7 * q^34 - q^35 + q^36 - q^37 + 3 * q^38 - q^40 - 4 * q^41 + q^42 + 5 * q^43 + 3 * q^44 + q^45 + q^46 + q^48 + q^49 + 4 * q^50 - 7 * q^51 - 6 * q^53 - q^54 + 3 * q^55 + q^56 - 3 * q^57 + q^58 + 10 * q^59 + q^60 - 13 * q^61 - 8 * q^62 - q^63 + q^64 - 3 * q^66 + 8 * q^67 - 7 * q^68 - q^69 + q^70 - 6 * q^71 - q^72 - 13 * q^73 + q^74 - 4 * q^75 - 3 * q^76 - 3 * q^77 - 12 * q^79 + q^80 + q^81 + 4 * q^82 - 2 * q^83 - q^84 - 7 * q^85 - 5 * q^86 - q^87 - 3 * q^88 + 12 * q^89 - q^90 - q^92 + 8 * q^93 - 3 * q^95 - q^96 - 6 * q^97 - q^98 + 3 * 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
−1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 −1.00000
 $$n$$: 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.o 1
13.b even 2 1 7098.2.a.y 1
13.d odd 4 2 546.2.c.c 2
39.f even 4 2 1638.2.c.e 2
52.f even 4 2 4368.2.h.h 2
91.i even 4 2 3822.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.c 2 13.d odd 4 2
1638.2.c.e 2 39.f even 4 2
3822.2.c.b 2 91.i even 4 2
4368.2.h.h 2 52.f even 4 2
7098.2.a.o 1 1.a even 1 1 trivial
7098.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{11} - 3$$ T11 - 3 $$T_{17} + 7$$ T17 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T$$
$17$ $$T + 7$$
$19$ $$T + 3$$
$23$ $$T + 1$$
$29$ $$T + 1$$
$31$ $$T - 8$$
$37$ $$T + 1$$
$41$ $$T + 4$$
$43$ $$T - 5$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 10$$
$61$ $$T + 13$$
$67$ $$T - 8$$
$71$ $$T + 6$$
$73$ $$T + 13$$
$79$ $$T + 12$$
$83$ $$T + 2$$
$89$ $$T - 12$$
$97$ $$T + 6$$