# Properties

 Label 7098.2.a.bl Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + (-b - 1) * q^5 - q^6 + q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + (\beta + 1) q^{10} + (\beta - 1) q^{11} + q^{12} - q^{14} + ( - \beta - 1) q^{15} + q^{16} + ( - 3 \beta + 1) q^{17} - q^{18} + (3 \beta - 3) q^{19} + ( - \beta - 1) q^{20} + q^{21} + ( - \beta + 1) q^{22} + ( - \beta - 3) q^{23} - q^{24} + 3 \beta q^{25} + q^{27} + q^{28} + (3 \beta - 1) q^{29} + (\beta + 1) q^{30} + ( - 4 \beta + 4) q^{31} - q^{32} + (\beta - 1) q^{33} + (3 \beta - 1) q^{34} + ( - \beta - 1) q^{35} + q^{36} + (\beta + 5) q^{37} + ( - 3 \beta + 3) q^{38} + (\beta + 1) q^{40} + (2 \beta - 4) q^{41} - q^{42} + (\beta - 9) q^{43} + (\beta - 1) q^{44} + ( - \beta - 1) q^{45} + (\beta + 3) q^{46} + q^{48} + q^{49} - 3 \beta q^{50} + ( - 3 \beta + 1) q^{51} + (4 \beta + 2) q^{53} - q^{54} + ( - \beta - 3) q^{55} - q^{56} + (3 \beta - 3) q^{57} + ( - 3 \beta + 1) q^{58} + ( - 4 \beta + 8) q^{59} + ( - \beta - 1) q^{60} + (3 \beta - 1) q^{61} + (4 \beta - 4) q^{62} + q^{63} + q^{64} + ( - \beta + 1) q^{66} + (2 \beta + 2) q^{67} + ( - 3 \beta + 1) q^{68} + ( - \beta - 3) q^{69} + (\beta + 1) q^{70} - 8 q^{71} - q^{72} + (\beta + 1) q^{73} + ( - \beta - 5) q^{74} + 3 \beta q^{75} + (3 \beta - 3) q^{76} + (\beta - 1) q^{77} + ( - 2 \beta - 6) q^{79} + ( - \beta - 1) q^{80} + q^{81} + ( - 2 \beta + 4) q^{82} + (2 \beta - 14) q^{83} + q^{84} + (5 \beta + 11) q^{85} + ( - \beta + 9) q^{86} + (3 \beta - 1) q^{87} + ( - \beta + 1) q^{88} - 10 q^{89} + (\beta + 1) q^{90} + ( - \beta - 3) q^{92} + ( - 4 \beta + 4) q^{93} + ( - 3 \beta - 9) q^{95} - q^{96} + ( - 8 \beta + 6) q^{97} - q^{98} + (\beta - 1) q^{99} +O(q^{100})$$ q - q^2 + q^3 + q^4 + (-b - 1) * q^5 - q^6 + q^7 - q^8 + q^9 + (b + 1) * q^10 + (b - 1) * q^11 + q^12 - q^14 + (-b - 1) * q^15 + q^16 + (-3*b + 1) * q^17 - q^18 + (3*b - 3) * q^19 + (-b - 1) * q^20 + q^21 + (-b + 1) * q^22 + (-b - 3) * q^23 - q^24 + 3*b * q^25 + q^27 + q^28 + (3*b - 1) * q^29 + (b + 1) * q^30 + (-4*b + 4) * q^31 - q^32 + (b - 1) * q^33 + (3*b - 1) * q^34 + (-b - 1) * q^35 + q^36 + (b + 5) * q^37 + (-3*b + 3) * q^38 + (b + 1) * q^40 + (2*b - 4) * q^41 - q^42 + (b - 9) * q^43 + (b - 1) * q^44 + (-b - 1) * q^45 + (b + 3) * q^46 + q^48 + q^49 - 3*b * q^50 + (-3*b + 1) * q^51 + (4*b + 2) * q^53 - q^54 + (-b - 3) * q^55 - q^56 + (3*b - 3) * q^57 + (-3*b + 1) * q^58 + (-4*b + 8) * q^59 + (-b - 1) * q^60 + (3*b - 1) * q^61 + (4*b - 4) * q^62 + q^63 + q^64 + (-b + 1) * q^66 + (2*b + 2) * q^67 + (-3*b + 1) * q^68 + (-b - 3) * q^69 + (b + 1) * q^70 - 8 * q^71 - q^72 + (b + 1) * q^73 + (-b - 5) * q^74 + 3*b * q^75 + (3*b - 3) * q^76 + (b - 1) * q^77 + (-2*b - 6) * q^79 + (-b - 1) * q^80 + q^81 + (-2*b + 4) * q^82 + (2*b - 14) * q^83 + q^84 + (5*b + 11) * q^85 + (-b + 9) * q^86 + (3*b - 1) * q^87 + (-b + 1) * q^88 - 10 * q^89 + (b + 1) * q^90 + (-b - 3) * q^92 + (-4*b + 4) * q^93 + (-3*b - 9) * q^95 - q^96 + (-8*b + 6) * q^97 - q^98 + (b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - q^{11} + 2 q^{12} - 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + q^{22} - 7 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{27} + 2 q^{28} + q^{29} + 3 q^{30} + 4 q^{31} - 2 q^{32} - q^{33} + q^{34} - 3 q^{35} + 2 q^{36} + 11 q^{37} + 3 q^{38} + 3 q^{40} - 6 q^{41} - 2 q^{42} - 17 q^{43} - q^{44} - 3 q^{45} + 7 q^{46} + 2 q^{48} + 2 q^{49} - 3 q^{50} - q^{51} + 8 q^{53} - 2 q^{54} - 7 q^{55} - 2 q^{56} - 3 q^{57} - q^{58} + 12 q^{59} - 3 q^{60} + q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} + q^{66} + 6 q^{67} - q^{68} - 7 q^{69} + 3 q^{70} - 16 q^{71} - 2 q^{72} + 3 q^{73} - 11 q^{74} + 3 q^{75} - 3 q^{76} - q^{77} - 14 q^{79} - 3 q^{80} + 2 q^{81} + 6 q^{82} - 26 q^{83} + 2 q^{84} + 27 q^{85} + 17 q^{86} + q^{87} + q^{88} - 20 q^{89} + 3 q^{90} - 7 q^{92} + 4 q^{93} - 21 q^{95} - 2 q^{96} + 4 q^{97} - 2 q^{98} - q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 3 * q^10 - q^11 + 2 * q^12 - 2 * q^14 - 3 * q^15 + 2 * q^16 - q^17 - 2 * q^18 - 3 * q^19 - 3 * q^20 + 2 * q^21 + q^22 - 7 * q^23 - 2 * q^24 + 3 * q^25 + 2 * q^27 + 2 * q^28 + q^29 + 3 * q^30 + 4 * q^31 - 2 * q^32 - q^33 + q^34 - 3 * q^35 + 2 * q^36 + 11 * q^37 + 3 * q^38 + 3 * q^40 - 6 * q^41 - 2 * q^42 - 17 * q^43 - q^44 - 3 * q^45 + 7 * q^46 + 2 * q^48 + 2 * q^49 - 3 * q^50 - q^51 + 8 * q^53 - 2 * q^54 - 7 * q^55 - 2 * q^56 - 3 * q^57 - q^58 + 12 * q^59 - 3 * q^60 + q^61 - 4 * q^62 + 2 * q^63 + 2 * q^64 + q^66 + 6 * q^67 - q^68 - 7 * q^69 + 3 * q^70 - 16 * q^71 - 2 * q^72 + 3 * q^73 - 11 * q^74 + 3 * q^75 - 3 * q^76 - q^77 - 14 * q^79 - 3 * q^80 + 2 * q^81 + 6 * q^82 - 26 * q^83 + 2 * q^84 + 27 * q^85 + 17 * q^86 + q^87 + q^88 - 20 * q^89 + 3 * q^90 - 7 * q^92 + 4 * q^93 - 21 * q^95 - 2 * q^96 + 4 * q^97 - 2 * q^98 - 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
 2.56155 −1.56155
−1.00000 1.00000 1.00000 −3.56155 −1.00000 1.00000 −1.00000 1.00000 3.56155
1.2 −1.00000 1.00000 1.00000 0.561553 −1.00000 1.00000 −1.00000 1.00000 −0.561553
 $$n$$: 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.bl 2
13.b even 2 1 546.2.a.j 2
39.d odd 2 1 1638.2.a.u 2
52.b odd 2 1 4368.2.a.be 2
91.b odd 2 1 3822.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.j 2 13.b even 2 1
1638.2.a.u 2 39.d odd 2 1
3822.2.a.bo 2 91.b odd 2 1
4368.2.a.be 2 52.b odd 2 1
7098.2.a.bl 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{11}^{2} + T_{11} - 4$$ T11^2 + T11 - 4 $$T_{17}^{2} + T_{17} - 38$$ T17^2 + T17 - 38

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + T - 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} + T - 38$$
$19$ $$T^{2} + 3T - 36$$
$23$ $$T^{2} + 7T + 8$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - 4T - 64$$
$37$ $$T^{2} - 11T + 26$$
$41$ $$T^{2} + 6T - 8$$
$43$ $$T^{2} + 17T + 68$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$T^{2} - 12T - 32$$
$61$ $$T^{2} - T - 38$$
$67$ $$T^{2} - 6T - 8$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 3T - 2$$
$79$ $$T^{2} + 14T + 32$$
$83$ $$T^{2} + 26T + 152$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 4T - 268$$