# Properties

 Label 1078.2.a.w 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: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta - 1) q^{5} + (\beta + 1) q^{6} + q^{8} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + q^2 + (b + 1) * q^3 + q^4 + (-b - 1) * q^5 + (b + 1) * q^6 + q^8 + (2*b + 3) * q^9 $$q + q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta - 1) q^{5} + (\beta + 1) q^{6} + q^{8} + (2 \beta + 3) q^{9} + ( - \beta - 1) q^{10} + q^{11} + (\beta + 1) q^{12} + ( - \beta + 1) q^{13} + ( - 2 \beta - 6) q^{15} + q^{16} + (2 \beta + 2) q^{17} + (2 \beta + 3) q^{18} + ( - \beta + 5) q^{19} + ( - \beta - 1) q^{20} + q^{22} + 4 q^{23} + (\beta + 1) q^{24} + (2 \beta + 1) q^{25} + ( - \beta + 1) q^{26} + (2 \beta + 10) q^{27} - 2 \beta q^{29} + ( - 2 \beta - 6) q^{30} - 2 q^{31} + q^{32} + (\beta + 1) q^{33} + (2 \beta + 2) q^{34} + (2 \beta + 3) q^{36} + ( - 4 \beta - 2) q^{37} + ( - \beta + 5) q^{38} - 4 q^{39} + ( - \beta - 1) q^{40} + ( - 2 \beta - 2) q^{41} + (2 \beta - 6) q^{43} + q^{44} + ( - 5 \beta - 13) q^{45} + 4 q^{46} + 2 q^{47} + (\beta + 1) q^{48} + (2 \beta + 1) q^{50} + (4 \beta + 12) q^{51} + ( - \beta + 1) q^{52} + ( - 2 \beta + 4) q^{53} + (2 \beta + 10) q^{54} + ( - \beta - 1) q^{55} + 4 \beta q^{57} - 2 \beta q^{58} + ( - \beta - 5) q^{59} + ( - 2 \beta - 6) q^{60} + (\beta + 3) q^{61} - 2 q^{62} + q^{64} + 4 q^{65} + (\beta + 1) q^{66} + ( - 6 \beta - 2) q^{67} + (2 \beta + 2) q^{68} + (4 \beta + 4) q^{69} + ( - 2 \beta + 2) q^{71} + (2 \beta + 3) q^{72} + (4 \beta - 4) q^{73} + ( - 4 \beta - 2) q^{74} + (3 \beta + 11) q^{75} + ( - \beta + 5) q^{76} - 4 q^{78} + ( - \beta - 1) q^{80} + (6 \beta + 11) q^{81} + ( - 2 \beta - 2) q^{82} + ( - 5 \beta + 1) q^{83} + ( - 4 \beta - 12) q^{85} + (2 \beta - 6) q^{86} + ( - 2 \beta - 10) q^{87} + q^{88} - 10 q^{89} + ( - 5 \beta - 13) q^{90} + 4 q^{92} + ( - 2 \beta - 2) q^{93} + 2 q^{94} - 4 \beta q^{95} + (\beta + 1) q^{96} + (2 \beta - 8) q^{97} + (2 \beta + 3) q^{99} +O(q^{100})$$ q + q^2 + (b + 1) * q^3 + q^4 + (-b - 1) * q^5 + (b + 1) * q^6 + q^8 + (2*b + 3) * q^9 + (-b - 1) * q^10 + q^11 + (b + 1) * q^12 + (-b + 1) * q^13 + (-2*b - 6) * q^15 + q^16 + (2*b + 2) * q^17 + (2*b + 3) * q^18 + (-b + 5) * q^19 + (-b - 1) * q^20 + q^22 + 4 * q^23 + (b + 1) * q^24 + (2*b + 1) * q^25 + (-b + 1) * q^26 + (2*b + 10) * q^27 - 2*b * q^29 + (-2*b - 6) * q^30 - 2 * q^31 + q^32 + (b + 1) * q^33 + (2*b + 2) * q^34 + (2*b + 3) * q^36 + (-4*b - 2) * q^37 + (-b + 5) * q^38 - 4 * q^39 + (-b - 1) * q^40 + (-2*b - 2) * q^41 + (2*b - 6) * q^43 + q^44 + (-5*b - 13) * q^45 + 4 * q^46 + 2 * q^47 + (b + 1) * q^48 + (2*b + 1) * q^50 + (4*b + 12) * q^51 + (-b + 1) * q^52 + (-2*b + 4) * q^53 + (2*b + 10) * q^54 + (-b - 1) * q^55 + 4*b * q^57 - 2*b * q^58 + (-b - 5) * q^59 + (-2*b - 6) * q^60 + (b + 3) * q^61 - 2 * q^62 + q^64 + 4 * q^65 + (b + 1) * q^66 + (-6*b - 2) * q^67 + (2*b + 2) * q^68 + (4*b + 4) * q^69 + (-2*b + 2) * q^71 + (2*b + 3) * q^72 + (4*b - 4) * q^73 + (-4*b - 2) * q^74 + (3*b + 11) * q^75 + (-b + 5) * q^76 - 4 * q^78 + (-b - 1) * q^80 + (6*b + 11) * q^81 + (-2*b - 2) * q^82 + (-5*b + 1) * q^83 + (-4*b - 12) * q^85 + (2*b - 6) * q^86 + (-2*b - 10) * q^87 + q^88 - 10 * q^89 + (-5*b - 13) * q^90 + 4 * q^92 + (-2*b - 2) * q^93 + 2 * q^94 - 4*b * q^95 + (b + 1) * q^96 + (2*b - 8) * q^97 + (2*b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 6 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} - 12 q^{15} + 2 q^{16} + 4 q^{17} + 6 q^{18} + 10 q^{19} - 2 q^{20} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 20 q^{27} - 12 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} + 6 q^{36} - 4 q^{37} + 10 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 12 q^{43} + 2 q^{44} - 26 q^{45} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} + 24 q^{51} + 2 q^{52} + 8 q^{53} + 20 q^{54} - 2 q^{55} - 10 q^{59} - 12 q^{60} + 6 q^{61} - 4 q^{62} + 2 q^{64} + 8 q^{65} + 2 q^{66} - 4 q^{67} + 4 q^{68} + 8 q^{69} + 4 q^{71} + 6 q^{72} - 8 q^{73} - 4 q^{74} + 22 q^{75} + 10 q^{76} - 8 q^{78} - 2 q^{80} + 22 q^{81} - 4 q^{82} + 2 q^{83} - 24 q^{85} - 12 q^{86} - 20 q^{87} + 2 q^{88} - 20 q^{89} - 26 q^{90} + 8 q^{92} - 4 q^{93} + 4 q^{94} + 2 q^{96} - 16 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 6 * q^9 - 2 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 - 12 * q^15 + 2 * q^16 + 4 * q^17 + 6 * q^18 + 10 * q^19 - 2 * q^20 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 20 * q^27 - 12 * q^30 - 4 * q^31 + 2 * q^32 + 2 * q^33 + 4 * q^34 + 6 * q^36 - 4 * q^37 + 10 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 - 12 * q^43 + 2 * q^44 - 26 * q^45 + 8 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^50 + 24 * q^51 + 2 * q^52 + 8 * q^53 + 20 * q^54 - 2 * q^55 - 10 * q^59 - 12 * q^60 + 6 * q^61 - 4 * q^62 + 2 * q^64 + 8 * q^65 + 2 * q^66 - 4 * q^67 + 4 * q^68 + 8 * q^69 + 4 * q^71 + 6 * q^72 - 8 * q^73 - 4 * q^74 + 22 * q^75 + 10 * q^76 - 8 * q^78 - 2 * q^80 + 22 * q^81 - 4 * q^82 + 2 * q^83 - 24 * q^85 - 12 * q^86 - 20 * q^87 + 2 * q^88 - 20 * q^89 - 26 * q^90 + 8 * q^92 - 4 * q^93 + 4 * q^94 + 2 * q^96 - 16 * q^97 + 6 * 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
 −0.618034 1.61803
1.00000 −1.23607 1.00000 1.23607 −1.23607 0 1.00000 −1.47214 1.23607
1.2 1.00000 3.23607 1.00000 −3.23607 3.23607 0 1.00000 7.47214 −3.23607
 $$n$$: 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.w 2
3.b odd 2 1 9702.2.a.cu 2
4.b odd 2 1 8624.2.a.bf 2
7.b odd 2 1 154.2.a.d 2
7.c even 3 2 1078.2.e.n 4
7.d odd 6 2 1078.2.e.q 4
21.c even 2 1 1386.2.a.m 2
28.d even 2 1 1232.2.a.p 2
35.c odd 2 1 3850.2.a.bj 2
35.f even 4 2 3850.2.c.q 4
56.e even 2 1 4928.2.a.bk 2
56.h odd 2 1 4928.2.a.bt 2
77.b even 2 1 1694.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 7.b odd 2 1
1078.2.a.w 2 1.a even 1 1 trivial
1078.2.e.n 4 7.c even 3 2
1078.2.e.q 4 7.d odd 6 2
1232.2.a.p 2 28.d even 2 1
1386.2.a.m 2 21.c even 2 1
1694.2.a.l 2 77.b even 2 1
3850.2.a.bj 2 35.c odd 2 1
3850.2.c.q 4 35.f even 4 2
4928.2.a.bk 2 56.e even 2 1
4928.2.a.bt 2 56.h odd 2 1
8624.2.a.bf 2 4.b odd 2 1
9702.2.a.cu 2 3.b odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2} + 2T - 4$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} - 4T - 16$$
$19$ $$T^{2} - 10T + 20$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} + 4T - 16$$
$43$ $$T^{2} + 12T + 16$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 10T + 20$$
$61$ $$T^{2} - 6T + 4$$
$67$ $$T^{2} + 4T - 176$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$T^{2} + 8T - 64$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 2T - 124$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 16T + 44$$