Properties

 Label 7098.2.a.z 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 3 q^{11} + q^{12} + q^{14} - q^{15} + q^{16} + 5 q^{17} + q^{18} - q^{19} - q^{20} + q^{21} - 3 q^{22} + 3 q^{23} + q^{24} - 4 q^{25} + q^{27} + q^{28} + 5 q^{29} - q^{30} - 4 q^{31} + q^{32} - 3 q^{33} + 5 q^{34} - q^{35} + q^{36} + 5 q^{37} - q^{38} - q^{40} + 8 q^{41} + q^{42} - q^{43} - 3 q^{44} - q^{45} + 3 q^{46} - 8 q^{47} + q^{48} + q^{49} - 4 q^{50} + 5 q^{51} + 6 q^{53} + q^{54} + 3 q^{55} + q^{56} - q^{57} + 5 q^{58} - q^{60} + 13 q^{61} - 4 q^{62} + q^{63} + q^{64} - 3 q^{66} + 10 q^{67} + 5 q^{68} + 3 q^{69} - q^{70} - 8 q^{71} + q^{72} + 15 q^{73} + 5 q^{74} - 4 q^{75} - q^{76} - 3 q^{77} + 6 q^{79} - q^{80} + q^{81} + 8 q^{82} + 2 q^{83} + q^{84} - 5 q^{85} - q^{86} + 5 q^{87} - 3 q^{88} + 2 q^{89} - q^{90} + 3 q^{92} - 4 q^{93} - 8 q^{94} + q^{95} + q^{96} + 2 q^{97} + q^{98} - 3 q^{99}+O(q^{100})$$ 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 + 5 * q^17 + q^18 - q^19 - q^20 + q^21 - 3 * q^22 + 3 * q^23 + q^24 - 4 * q^25 + q^27 + q^28 + 5 * q^29 - q^30 - 4 * q^31 + q^32 - 3 * q^33 + 5 * q^34 - q^35 + q^36 + 5 * q^37 - q^38 - q^40 + 8 * q^41 + q^42 - q^43 - 3 * q^44 - q^45 + 3 * q^46 - 8 * q^47 + q^48 + q^49 - 4 * q^50 + 5 * q^51 + 6 * q^53 + q^54 + 3 * q^55 + q^56 - q^57 + 5 * q^58 - q^60 + 13 * q^61 - 4 * q^62 + q^63 + q^64 - 3 * q^66 + 10 * q^67 + 5 * q^68 + 3 * q^69 - q^70 - 8 * q^71 + q^72 + 15 * q^73 + 5 * q^74 - 4 * q^75 - q^76 - 3 * q^77 + 6 * q^79 - q^80 + q^81 + 8 * q^82 + 2 * q^83 + q^84 - 5 * q^85 - q^86 + 5 * q^87 - 3 * q^88 + 2 * q^89 - q^90 + 3 * q^92 - 4 * q^93 - 8 * q^94 + q^95 + q^96 + 2 * 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.z 1
13.b even 2 1 546.2.a.c 1
39.d odd 2 1 1638.2.a.o 1
52.b odd 2 1 4368.2.a.h 1
91.b odd 2 1 3822.2.a.e 1

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

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 $$T_{17} - 5$$ T17 - 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T - 5$$
$19$ $$T + 1$$
$23$ $$T - 3$$
$29$ $$T - 5$$
$31$ $$T + 4$$
$37$ $$T - 5$$
$41$ $$T - 8$$
$43$ $$T + 1$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T - 13$$
$67$ $$T - 10$$
$71$ $$T + 8$$
$73$ $$T - 15$$
$79$ $$T - 6$$
$83$ $$T - 2$$
$89$ $$T - 2$$
$97$ $$T - 2$$