# Properties

 Label 7098.2.a.bn Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + 2 \beta q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + 2*b * q^5 - q^6 + q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + 2 \beta q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - 2 \beta q^{10} + 4 q^{11} + q^{12} - q^{14} + 2 \beta q^{15} + q^{16} + ( - 2 \beta + 3) q^{17} - q^{18} + ( - 2 \beta + 4) q^{19} + 2 \beta q^{20} + q^{21} - 4 q^{22} + ( - \beta + 4) q^{23} - q^{24} + 7 q^{25} + q^{27} + q^{28} + 8 q^{29} - 2 \beta q^{30} + (2 \beta - 3) q^{31} - q^{32} + 4 q^{33} + (2 \beta - 3) q^{34} + 2 \beta q^{35} + q^{36} + (4 \beta + 2) q^{37} + (2 \beta - 4) q^{38} - 2 \beta q^{40} - 4 \beta q^{41} - q^{42} + ( - 2 \beta + 1) q^{43} + 4 q^{44} + 2 \beta q^{45} + (\beta - 4) q^{46} + ( - 2 \beta - 4) q^{47} + q^{48} + q^{49} - 7 q^{50} + ( - 2 \beta + 3) q^{51} + ( - \beta - 10) q^{53} - q^{54} + 8 \beta q^{55} - q^{56} + ( - 2 \beta + 4) q^{57} - 8 q^{58} + (\beta + 8) q^{59} + 2 \beta q^{60} + 3 \beta q^{61} + ( - 2 \beta + 3) q^{62} + q^{63} + q^{64} - 4 q^{66} + (3 \beta - 2) q^{67} + ( - 2 \beta + 3) q^{68} + ( - \beta + 4) q^{69} - 2 \beta q^{70} + ( - 4 \beta - 5) q^{71} - q^{72} + (4 \beta - 4) q^{73} + ( - 4 \beta - 2) q^{74} + 7 q^{75} + ( - 2 \beta + 4) q^{76} + 4 q^{77} + (2 \beta - 6) q^{79} + 2 \beta q^{80} + q^{81} + 4 \beta q^{82} - \beta q^{83} + q^{84} + (6 \beta - 12) q^{85} + (2 \beta - 1) q^{86} + 8 q^{87} - 4 q^{88} + (\beta - 10) q^{89} - 2 \beta q^{90} + ( - \beta + 4) q^{92} + (2 \beta - 3) q^{93} + (2 \beta + 4) q^{94} + (8 \beta - 12) q^{95} - q^{96} + ( - 6 \beta + 2) q^{97} - q^{98} + 4 q^{99} +O(q^{100})$$ q - q^2 + q^3 + q^4 + 2*b * q^5 - q^6 + q^7 - q^8 + q^9 - 2*b * q^10 + 4 * q^11 + q^12 - q^14 + 2*b * q^15 + q^16 + (-2*b + 3) * q^17 - q^18 + (-2*b + 4) * q^19 + 2*b * q^20 + q^21 - 4 * q^22 + (-b + 4) * q^23 - q^24 + 7 * q^25 + q^27 + q^28 + 8 * q^29 - 2*b * q^30 + (2*b - 3) * q^31 - q^32 + 4 * q^33 + (2*b - 3) * q^34 + 2*b * q^35 + q^36 + (4*b + 2) * q^37 + (2*b - 4) * q^38 - 2*b * q^40 - 4*b * q^41 - q^42 + (-2*b + 1) * q^43 + 4 * q^44 + 2*b * q^45 + (b - 4) * q^46 + (-2*b - 4) * q^47 + q^48 + q^49 - 7 * q^50 + (-2*b + 3) * q^51 + (-b - 10) * q^53 - q^54 + 8*b * q^55 - q^56 + (-2*b + 4) * q^57 - 8 * q^58 + (b + 8) * q^59 + 2*b * q^60 + 3*b * q^61 + (-2*b + 3) * q^62 + q^63 + q^64 - 4 * q^66 + (3*b - 2) * q^67 + (-2*b + 3) * q^68 + (-b + 4) * q^69 - 2*b * q^70 + (-4*b - 5) * q^71 - q^72 + (4*b - 4) * q^73 + (-4*b - 2) * q^74 + 7 * q^75 + (-2*b + 4) * q^76 + 4 * q^77 + (2*b - 6) * q^79 + 2*b * q^80 + q^81 + 4*b * q^82 - b * q^83 + q^84 + (6*b - 12) * q^85 + (2*b - 1) * q^86 + 8 * q^87 - 4 * q^88 + (b - 10) * q^89 - 2*b * q^90 + (-b + 4) * q^92 + (2*b - 3) * q^93 + (2*b + 4) * q^94 + (8*b - 12) * q^95 - q^96 + (-6*b + 2) * q^97 - q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 8 q^{11} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{21} - 8 q^{22} + 8 q^{23} - 2 q^{24} + 14 q^{25} + 2 q^{27} + 2 q^{28} + 16 q^{29} - 6 q^{31} - 2 q^{32} + 8 q^{33} - 6 q^{34} + 2 q^{36} + 4 q^{37} - 8 q^{38} - 2 q^{42} + 2 q^{43} + 8 q^{44} - 8 q^{46} - 8 q^{47} + 2 q^{48} + 2 q^{49} - 14 q^{50} + 6 q^{51} - 20 q^{53} - 2 q^{54} - 2 q^{56} + 8 q^{57} - 16 q^{58} + 16 q^{59} + 6 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{66} - 4 q^{67} + 6 q^{68} + 8 q^{69} - 10 q^{71} - 2 q^{72} - 8 q^{73} - 4 q^{74} + 14 q^{75} + 8 q^{76} + 8 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{84} - 24 q^{85} - 2 q^{86} + 16 q^{87} - 8 q^{88} - 20 q^{89} + 8 q^{92} - 6 q^{93} + 8 q^{94} - 24 q^{95} - 2 q^{96} + 4 q^{97} - 2 q^{98} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 8 * q^11 + 2 * q^12 - 2 * q^14 + 2 * q^16 + 6 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^21 - 8 * q^22 + 8 * q^23 - 2 * q^24 + 14 * q^25 + 2 * q^27 + 2 * q^28 + 16 * q^29 - 6 * q^31 - 2 * q^32 + 8 * q^33 - 6 * q^34 + 2 * q^36 + 4 * q^37 - 8 * q^38 - 2 * q^42 + 2 * q^43 + 8 * q^44 - 8 * q^46 - 8 * q^47 + 2 * q^48 + 2 * q^49 - 14 * q^50 + 6 * q^51 - 20 * q^53 - 2 * q^54 - 2 * q^56 + 8 * q^57 - 16 * q^58 + 16 * q^59 + 6 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^66 - 4 * q^67 + 6 * q^68 + 8 * q^69 - 10 * q^71 - 2 * q^72 - 8 * q^73 - 4 * q^74 + 14 * q^75 + 8 * q^76 + 8 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^84 - 24 * q^85 - 2 * q^86 + 16 * q^87 - 8 * q^88 - 20 * q^89 + 8 * q^92 - 6 * q^93 + 8 * q^94 - 24 * q^95 - 2 * q^96 + 4 * q^97 - 2 * q^98 + 8 * 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
 −1.73205 1.73205
−1.00000 1.00000 1.00000 −3.46410 −1.00000 1.00000 −1.00000 1.00000 3.46410
1.2 −1.00000 1.00000 1.00000 3.46410 −1.00000 1.00000 −1.00000 1.00000 −3.46410
 $$n$$: 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.bn 2
13.b even 2 1 7098.2.a.bz 2
13.f odd 12 2 546.2.s.c 4
39.k even 12 2 1638.2.bj.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 13.f odd 12 2
1638.2.bj.e 4 39.k even 12 2
7098.2.a.bn 2 1.a even 1 1 trivial
7098.2.a.bz 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{11} - 4$$ T11 - 4 $$T_{17}^{2} - 6T_{17} - 3$$ T17^2 - 6*T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 6T - 3$$
$19$ $$T^{2} - 8T + 4$$
$23$ $$T^{2} - 8T + 13$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} + 6T - 3$$
$37$ $$T^{2} - 4T - 44$$
$41$ $$T^{2} - 48$$
$43$ $$T^{2} - 2T - 11$$
$47$ $$T^{2} + 8T + 4$$
$53$ $$T^{2} + 20T + 97$$
$59$ $$T^{2} - 16T + 61$$
$61$ $$T^{2} - 27$$
$67$ $$T^{2} + 4T - 23$$
$71$ $$T^{2} + 10T - 23$$
$73$ $$T^{2} + 8T - 32$$
$79$ $$T^{2} + 12T + 24$$
$83$ $$T^{2} - 3$$
$89$ $$T^{2} + 20T + 97$$
$97$ $$T^{2} - 4T - 104$$