# Properties

 Label 1078.2.a.x Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ 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] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.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: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta + 2) q^{5} + (\beta + 1) q^{6} + q^{8} + 2 \beta q^{9}+O(q^{10})$$ q + q^2 + (b + 1) * q^3 + q^4 + (-b + 2) * q^5 + (b + 1) * q^6 + q^8 + 2*b * q^9 $$q + q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta + 2) q^{5} + (\beta + 1) q^{6} + q^{8} + 2 \beta q^{9} + ( - \beta + 2) q^{10} + q^{11} + (\beta + 1) q^{12} + (2 \beta + 1) q^{13} + \beta q^{15} + q^{16} + ( - 4 \beta + 2) q^{17} + 2 \beta q^{18} + ( - \beta + 2) q^{19} + ( - \beta + 2) q^{20} + q^{22} + ( - 3 \beta - 2) q^{23} + (\beta + 1) q^{24} + ( - 4 \beta + 1) q^{25} + (2 \beta + 1) q^{26} + ( - \beta + 1) q^{27} + (4 \beta - 3) q^{29} + \beta q^{30} + 4 q^{31} + q^{32} + (\beta + 1) q^{33} + ( - 4 \beta + 2) q^{34} + 2 \beta q^{36} + ( - \beta - 8) q^{37} + ( - \beta + 2) q^{38} + (3 \beta + 5) q^{39} + ( - \beta + 2) q^{40} + (\beta + 4) q^{41} - 4 \beta q^{43} + q^{44} + (4 \beta - 4) q^{45} + ( - 3 \beta - 2) q^{46} + (6 \beta + 2) q^{47} + (\beta + 1) q^{48} + ( - 4 \beta + 1) q^{50} + ( - 2 \beta - 6) q^{51} + (2 \beta + 1) q^{52} + (7 \beta - 2) q^{53} + ( - \beta + 1) q^{54} + ( - \beta + 2) q^{55} + \beta q^{57} + (4 \beta - 3) q^{58} + ( - \beta + 7) q^{59} + \beta q^{60} + ( - 2 \beta - 9) q^{61} + 4 q^{62} + q^{64} + (3 \beta - 2) q^{65} + (\beta + 1) q^{66} + ( - 3 \beta + 7) q^{67} + ( - 4 \beta + 2) q^{68} + ( - 5 \beta - 8) q^{69} + ( - 5 \beta - 4) q^{71} + 2 \beta q^{72} + (\beta + 8) q^{73} + ( - \beta - 8) q^{74} + ( - 3 \beta - 7) q^{75} + ( - \beta + 2) q^{76} + (3 \beta + 5) q^{78} + ( - 3 \beta - 9) q^{79} + ( - \beta + 2) q^{80} + ( - 6 \beta - 1) q^{81} + (\beta + 4) q^{82} + (10 \beta - 2) q^{83} + ( - 10 \beta + 12) q^{85} - 4 \beta q^{86} + (\beta + 5) q^{87} + q^{88} + ( - 6 \beta - 4) q^{89} + (4 \beta - 4) q^{90} + ( - 3 \beta - 2) q^{92} + (4 \beta + 4) q^{93} + (6 \beta + 2) q^{94} + ( - 4 \beta + 6) q^{95} + (\beta + 1) q^{96} + (2 \beta + 1) q^{97} + 2 \beta q^{99} +O(q^{100})$$ q + q^2 + (b + 1) * q^3 + q^4 + (-b + 2) * q^5 + (b + 1) * q^6 + q^8 + 2*b * q^9 + (-b + 2) * q^10 + q^11 + (b + 1) * q^12 + (2*b + 1) * q^13 + b * q^15 + q^16 + (-4*b + 2) * q^17 + 2*b * q^18 + (-b + 2) * q^19 + (-b + 2) * q^20 + q^22 + (-3*b - 2) * q^23 + (b + 1) * q^24 + (-4*b + 1) * q^25 + (2*b + 1) * q^26 + (-b + 1) * q^27 + (4*b - 3) * q^29 + b * q^30 + 4 * q^31 + q^32 + (b + 1) * q^33 + (-4*b + 2) * q^34 + 2*b * q^36 + (-b - 8) * q^37 + (-b + 2) * q^38 + (3*b + 5) * q^39 + (-b + 2) * q^40 + (b + 4) * q^41 - 4*b * q^43 + q^44 + (4*b - 4) * q^45 + (-3*b - 2) * q^46 + (6*b + 2) * q^47 + (b + 1) * q^48 + (-4*b + 1) * q^50 + (-2*b - 6) * q^51 + (2*b + 1) * q^52 + (7*b - 2) * q^53 + (-b + 1) * q^54 + (-b + 2) * q^55 + b * q^57 + (4*b - 3) * q^58 + (-b + 7) * q^59 + b * q^60 + (-2*b - 9) * q^61 + 4 * q^62 + q^64 + (3*b - 2) * q^65 + (b + 1) * q^66 + (-3*b + 7) * q^67 + (-4*b + 2) * q^68 + (-5*b - 8) * q^69 + (-5*b - 4) * q^71 + 2*b * q^72 + (b + 8) * q^73 + (-b - 8) * q^74 + (-3*b - 7) * q^75 + (-b + 2) * q^76 + (3*b + 5) * q^78 + (-3*b - 9) * q^79 + (-b + 2) * q^80 + (-6*b - 1) * q^81 + (b + 4) * q^82 + (10*b - 2) * q^83 + (-10*b + 12) * q^85 - 4*b * q^86 + (b + 5) * q^87 + q^88 + (-6*b - 4) * q^89 + (4*b - 4) * q^90 + (-3*b - 2) * q^92 + (4*b + 4) * q^93 + (6*b + 2) * q^94 + (-4*b + 6) * q^95 + (b + 1) * q^96 + (2*b + 1) * q^97 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} - 6 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} - 16 q^{37} + 4 q^{38} + 10 q^{39} + 4 q^{40} + 8 q^{41} + 2 q^{44} - 8 q^{45} - 4 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{58} + 14 q^{59} - 18 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} + 2 q^{66} + 14 q^{67} + 4 q^{68} - 16 q^{69} - 8 q^{71} + 16 q^{73} - 16 q^{74} - 14 q^{75} + 4 q^{76} + 10 q^{78} - 18 q^{79} + 4 q^{80} - 2 q^{81} + 8 q^{82} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 2 q^{88} - 8 q^{89} - 8 q^{90} - 4 q^{92} + 8 q^{93} + 4 q^{94} + 12 q^{95} + 2 q^{96} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 4 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 - 6 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^33 + 4 * q^34 - 16 * q^37 + 4 * q^38 + 10 * q^39 + 4 * q^40 + 8 * q^41 + 2 * q^44 - 8 * q^45 - 4 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^58 + 14 * q^59 - 18 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 + 2 * q^66 + 14 * q^67 + 4 * q^68 - 16 * q^69 - 8 * q^71 + 16 * q^73 - 16 * q^74 - 14 * q^75 + 4 * q^76 + 10 * q^78 - 18 * q^79 + 4 * q^80 - 2 * q^81 + 8 * q^82 - 4 * q^83 + 24 * q^85 + 10 * q^87 + 2 * q^88 - 8 * q^89 - 8 * q^90 - 4 * q^92 + 8 * q^93 + 4 * q^94 + 12 * q^95 + 2 * q^96 + 2 * q^97

## 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
 −1.41421 1.41421
1.00000 −0.414214 1.00000 3.41421 −0.414214 0 1.00000 −2.82843 3.41421
1.2 1.00000 2.41421 1.00000 0.585786 2.41421 0 1.00000 2.82843 0.585786
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 1078.2.a.x 2
3.b odd 2 1 9702.2.a.ch 2
4.b odd 2 1 8624.2.a.bh 2
7.b odd 2 1 1078.2.a.t 2
7.c even 3 2 1078.2.e.m 4
7.d odd 6 2 154.2.e.e 4
21.c even 2 1 9702.2.a.cx 2
21.g even 6 2 1386.2.k.t 4
28.d even 2 1 8624.2.a.cc 2
28.f even 6 2 1232.2.q.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 7.d odd 6 2
1078.2.a.t 2 7.b odd 2 1
1078.2.a.x 2 1.a even 1 1 trivial
1078.2.e.m 4 7.c even 3 2
1232.2.q.f 4 28.f even 6 2
1386.2.k.t 4 21.g even 6 2
8624.2.a.bh 2 4.b odd 2 1
8624.2.a.cc 2 28.d even 2 1
9702.2.a.ch 2 3.b odd 2 1
9702.2.a.cx 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1078))$$:

 $$T_{3}^{2} - 2T_{3} - 1$$ T3^2 - 2*T3 - 1 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{13}^{2} - 2T_{13} - 7$$ T13^2 - 2*T13 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 2T - 1$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 7$$
$17$ $$T^{2} - 4T - 28$$
$19$ $$T^{2} - 4T + 2$$
$23$ $$T^{2} + 4T - 14$$
$29$ $$T^{2} + 6T - 23$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 16T + 62$$
$41$ $$T^{2} - 8T + 14$$
$43$ $$T^{2} - 32$$
$47$ $$T^{2} - 4T - 68$$
$53$ $$T^{2} + 4T - 94$$
$59$ $$T^{2} - 14T + 47$$
$61$ $$T^{2} + 18T + 73$$
$67$ $$T^{2} - 14T + 31$$
$71$ $$T^{2} + 8T - 34$$
$73$ $$T^{2} - 16T + 62$$
$79$ $$T^{2} + 18T + 63$$
$83$ $$T^{2} + 4T - 196$$
$89$ $$T^{2} + 8T - 56$$
$97$ $$T^{2} - 2T - 7$$