# Properties

 Label 1078.2.e.q Level $1078$ Weight $2$ Character orbit 1078.e Analytic conductor $8.608$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(67,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1078.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.e (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - \beta_1 - 1) q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 1) q^{6} + q^{8} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1 + 1) * q^3 + (-b1 - 1) * q^4 + (b3 + b2 + b1) * q^5 + (b3 - 1) * q^6 + q^8 + (2*b3 + 2*b2 + 3*b1) * q^9 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - \beta_1 - 1) q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 1) q^{6} + q^{8} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_1 - 1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{12} + ( - \beta_{3} - 1) q^{13} + (2 \beta_{3} - 6) q^{15} + \beta_1 q^{16} + (2 \beta_{2} + 2 \beta_1 + 2) q^{17} + ( - 2 \beta_{2} - 3 \beta_1 - 3) q^{18} + (\beta_{3} + \beta_{2} - 5 \beta_1) q^{19} + ( - \beta_{3} + 1) q^{20} + q^{22} + 4 \beta_1 q^{23} + (\beta_{2} + \beta_1 + 1) q^{24} + ( - 2 \beta_{2} - \beta_1 - 1) q^{25} + (\beta_{3} + \beta_{2} - \beta_1) q^{26} + (2 \beta_{3} - 10) q^{27} + 2 \beta_{3} q^{29} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{30} + ( - 2 \beta_1 - 2) q^{31} + ( - \beta_1 - 1) q^{32} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{33} + (2 \beta_{3} - 2) q^{34} + ( - 2 \beta_{3} + 3) q^{36} + ( - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{2} + 5 \beta_1 + 5) q^{38} + (4 \beta_1 + 4) q^{39} + (\beta_{3} + \beta_{2} + \beta_1) q^{40} + ( - 2 \beta_{3} + 2) q^{41} + ( - 2 \beta_{3} - 6) q^{43} + \beta_1 q^{44} + ( - 5 \beta_{2} - 13 \beta_1 - 13) q^{45} + ( - 4 \beta_1 - 4) q^{46} - 2 \beta_1 q^{47} + (\beta_{3} - 1) q^{48} + ( - 2 \beta_{3} + 1) q^{50} + (4 \beta_{3} + 4 \beta_{2} + 12 \beta_1) q^{51} + ( - \beta_{2} + \beta_1 + 1) q^{52} + (2 \beta_{2} - 4 \beta_1 - 4) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1) q^{54} + ( - \beta_{3} + 1) q^{55} - 4 \beta_{3} q^{57} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{58} + ( - \beta_{2} - 5 \beta_1 - 5) q^{59} + (2 \beta_{2} + 6 \beta_1 + 6) q^{60} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{61} + 2 q^{62} + q^{64} + 4 \beta_1 q^{65} + (\beta_{2} + \beta_1 + 1) q^{66} + (6 \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{68} + (4 \beta_{3} - 4) q^{69} + (2 \beta_{3} + 2) q^{71} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{72} + (4 \beta_{2} - 4 \beta_1 - 4) q^{73} + (4 \beta_{2} + 2 \beta_1 + 2) q^{74} + ( - 3 \beta_{3} - 3 \beta_{2} - 11 \beta_1) q^{75} + ( - \beta_{3} - 5) q^{76} - 4 q^{78} + ( - \beta_{2} - \beta_1 - 1) q^{80} + ( - 6 \beta_{2} - 11 \beta_1 - 11) q^{81} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{82} + ( - 5 \beta_{3} - 1) q^{83} + (4 \beta_{3} - 12) q^{85} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{86} + ( - 2 \beta_{2} - 10 \beta_1 - 10) q^{87} + ( - \beta_1 - 1) q^{88} + 10 \beta_1 q^{89} + ( - 5 \beta_{3} + 13) q^{90} + 4 q^{92} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{93} + (2 \beta_1 + 2) q^{94} + 4 \beta_{2} q^{95} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{96} + (2 \beta_{3} + 8) q^{97} + ( - 2 \beta_{3} + 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1 + 1) * q^3 + (-b1 - 1) * q^4 + (b3 + b2 + b1) * q^5 + (b3 - 1) * q^6 + q^8 + (2*b3 + 2*b2 + 3*b1) * q^9 + (-b2 - b1 - 1) * q^10 + (-b1 - 1) * q^11 + (-b3 - b2 - b1) * q^12 + (-b3 - 1) * q^13 + (2*b3 - 6) * q^15 + b1 * q^16 + (2*b2 + 2*b1 + 2) * q^17 + (-2*b2 - 3*b1 - 3) * q^18 + (b3 + b2 - 5*b1) * q^19 + (-b3 + 1) * q^20 + q^22 + 4*b1 * q^23 + (b2 + b1 + 1) * q^24 + (-2*b2 - b1 - 1) * q^25 + (b3 + b2 - b1) * q^26 + (2*b3 - 10) * q^27 + 2*b3 * q^29 + (-2*b3 - 2*b2 - 6*b1) * q^30 + (-2*b1 - 2) * q^31 + (-b1 - 1) * q^32 + (-b3 - b2 - b1) * q^33 + (2*b3 - 2) * q^34 + (-2*b3 + 3) * q^36 + (-4*b3 - 4*b2 - 2*b1) * q^37 + (-b2 + 5*b1 + 5) * q^38 + (4*b1 + 4) * q^39 + (b3 + b2 + b1) * q^40 + (-2*b3 + 2) * q^41 + (-2*b3 - 6) * q^43 + b1 * q^44 + (-5*b2 - 13*b1 - 13) * q^45 + (-4*b1 - 4) * q^46 - 2*b1 * q^47 + (b3 - 1) * q^48 + (-2*b3 + 1) * q^50 + (4*b3 + 4*b2 + 12*b1) * q^51 + (-b2 + b1 + 1) * q^52 + (2*b2 - 4*b1 - 4) * q^53 + (-2*b3 - 2*b2 - 10*b1) * q^54 + (-b3 + 1) * q^55 - 4*b3 * q^57 + (-2*b3 - 2*b2) * q^58 + (-b2 - 5*b1 - 5) * q^59 + (2*b2 + 6*b1 + 6) * q^60 + (-b3 - b2 - 3*b1) * q^61 + 2 * q^62 + q^64 + 4*b1 * q^65 + (b2 + b1 + 1) * q^66 + (6*b2 + 2*b1 + 2) * q^67 + (-2*b3 - 2*b2 - 2*b1) * q^68 + (4*b3 - 4) * q^69 + (2*b3 + 2) * q^71 + (2*b3 + 2*b2 + 3*b1) * q^72 + (4*b2 - 4*b1 - 4) * q^73 + (4*b2 + 2*b1 + 2) * q^74 + (-3*b3 - 3*b2 - 11*b1) * q^75 + (-b3 - 5) * q^76 - 4 * q^78 + (-b2 - b1 - 1) * q^80 + (-6*b2 - 11*b1 - 11) * q^81 + (2*b3 + 2*b2 + 2*b1) * q^82 + (-5*b3 - 1) * q^83 + (4*b3 - 12) * q^85 + (2*b3 + 2*b2 - 6*b1) * q^86 + (-2*b2 - 10*b1 - 10) * q^87 + (-b1 - 1) * q^88 + 10*b1 * q^89 + (-5*b3 + 13) * q^90 + 4 * q^92 + (-2*b3 - 2*b2 - 2*b1) * q^93 + (2*b1 + 2) * q^94 + 4*b2 * q^95 + (-b3 - b2 - b1) * q^96 + (2*b3 + 8) * q^97 + (-2*b3 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^5 - 4 * q^6 + 4 * q^8 - 6 * q^9 $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} - 6 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} - 24 q^{15} - 2 q^{16} + 4 q^{17} - 6 q^{18} + 10 q^{19} + 4 q^{20} + 4 q^{22} - 8 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{26} - 40 q^{27} + 12 q^{30} - 4 q^{31} - 2 q^{32} + 2 q^{33} - 8 q^{34} + 12 q^{36} + 4 q^{37} + 10 q^{38} + 8 q^{39} - 2 q^{40} + 8 q^{41} - 24 q^{43} - 2 q^{44} - 26 q^{45} - 8 q^{46} + 4 q^{47} - 4 q^{48} + 4 q^{50} - 24 q^{51} + 2 q^{52} - 8 q^{53} + 20 q^{54} + 4 q^{55} - 10 q^{59} + 12 q^{60} + 6 q^{61} + 8 q^{62} + 4 q^{64} - 8 q^{65} + 2 q^{66} + 4 q^{67} + 4 q^{68} - 16 q^{69} + 8 q^{71} - 6 q^{72} - 8 q^{73} + 4 q^{74} + 22 q^{75} - 20 q^{76} - 16 q^{78} - 2 q^{80} - 22 q^{81} - 4 q^{82} - 4 q^{83} - 48 q^{85} + 12 q^{86} - 20 q^{87} - 2 q^{88} - 20 q^{89} + 52 q^{90} + 16 q^{92} + 4 q^{93} + 4 q^{94} + 2 q^{96} + 32 q^{97} + 12 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^5 - 4 * q^6 + 4 * q^8 - 6 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 - 4 * q^13 - 24 * q^15 - 2 * q^16 + 4 * q^17 - 6 * q^18 + 10 * q^19 + 4 * q^20 + 4 * q^22 - 8 * q^23 + 2 * q^24 - 2 * q^25 + 2 * q^26 - 40 * q^27 + 12 * q^30 - 4 * q^31 - 2 * q^32 + 2 * q^33 - 8 * q^34 + 12 * q^36 + 4 * q^37 + 10 * q^38 + 8 * q^39 - 2 * q^40 + 8 * q^41 - 24 * q^43 - 2 * q^44 - 26 * q^45 - 8 * q^46 + 4 * q^47 - 4 * q^48 + 4 * q^50 - 24 * q^51 + 2 * q^52 - 8 * q^53 + 20 * q^54 + 4 * q^55 - 10 * q^59 + 12 * q^60 + 6 * q^61 + 8 * q^62 + 4 * q^64 - 8 * q^65 + 2 * q^66 + 4 * q^67 + 4 * q^68 - 16 * q^69 + 8 * q^71 - 6 * q^72 - 8 * q^73 + 4 * q^74 + 22 * q^75 - 20 * q^76 - 16 * q^78 - 2 * q^80 - 22 * q^81 - 4 * q^82 - 4 * q^83 - 48 * q^85 + 12 * q^86 - 20 * q^87 - 2 * q^88 - 20 * q^89 + 52 * q^90 + 16 * q^92 + 4 * q^93 + 4 * q^94 + 2 * q^96 + 32 * q^97 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2$$ (v^3 - 2*v^2 + 6*v - 1) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$ v^3 + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2$$ (b3 + b2 + 3*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
−0.500000 0.866025i −0.618034 + 1.07047i −0.500000 + 0.866025i 0.618034 + 1.07047i 1.23607 0 1.00000 0.736068 + 1.27491i 0.618034 1.07047i
67.2 −0.500000 0.866025i 1.61803 2.80252i −0.500000 + 0.866025i −1.61803 2.80252i −3.23607 0 1.00000 −3.73607 6.47106i −1.61803 + 2.80252i
177.1 −0.500000 + 0.866025i −0.618034 1.07047i −0.500000 0.866025i 0.618034 1.07047i 1.23607 0 1.00000 0.736068 1.27491i 0.618034 + 1.07047i
177.2 −0.500000 + 0.866025i 1.61803 + 2.80252i −0.500000 0.866025i −1.61803 + 2.80252i −3.23607 0 1.00000 −3.73607 + 6.47106i −1.61803 2.80252i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.q 4
7.b odd 2 1 1078.2.e.n 4
7.c even 3 1 154.2.a.d 2
7.c even 3 1 inner 1078.2.e.q 4
7.d odd 6 1 1078.2.a.w 2
7.d odd 6 1 1078.2.e.n 4
21.g even 6 1 9702.2.a.cu 2
21.h odd 6 1 1386.2.a.m 2
28.f even 6 1 8624.2.a.bf 2
28.g odd 6 1 1232.2.a.p 2
35.j even 6 1 3850.2.a.bj 2
35.l odd 12 2 3850.2.c.q 4
56.k odd 6 1 4928.2.a.bk 2
56.p even 6 1 4928.2.a.bt 2
77.h odd 6 1 1694.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 7.c even 3 1
1078.2.a.w 2 7.d odd 6 1
1078.2.e.n 4 7.b odd 2 1
1078.2.e.n 4 7.d odd 6 1
1078.2.e.q 4 1.a even 1 1 trivial
1078.2.e.q 4 7.c even 3 1 inner
1232.2.a.p 2 28.g odd 6 1
1386.2.a.m 2 21.h odd 6 1
1694.2.a.l 2 77.h odd 6 1
3850.2.a.bj 2 35.j even 6 1
3850.2.c.q 4 35.l odd 12 2
4928.2.a.bk 2 56.k odd 6 1
4928.2.a.bt 2 56.p even 6 1
8624.2.a.bf 2 28.f even 6 1
9702.2.a.cu 2 21.g even 6 1

## Hecke kernels

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

 $$T_{3}^{4} - 2T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 16$$ T3^4 - 2*T3^3 + 8*T3^2 + 8*T3 + 16 $$T_{5}^{4} + 2T_{5}^{3} + 8T_{5}^{2} - 8T_{5} + 16$$ T5^4 + 2*T5^3 + 8*T5^2 - 8*T5 + 16 $$T_{13}^{2} + 2T_{13} - 4$$ T13^2 + 2*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16$$
$5$ $$T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + T + 1)^{2}$$
$13$ $$(T^{2} + 2 T - 4)^{2}$$
$17$ $$T^{4} - 4 T^{3} + 32 T^{2} + 64 T + 256$$
$19$ $$T^{4} - 10 T^{3} + 80 T^{2} + \cdots + 400$$
$23$ $$(T^{2} + 4 T + 16)^{2}$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T^{2} + 2 T + 4)^{2}$$
$37$ $$T^{4} - 4 T^{3} + 92 T^{2} + \cdots + 5776$$
$41$ $$(T^{2} - 4 T - 16)^{2}$$
$43$ $$(T^{2} + 12 T + 16)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} + 8 T^{3} + 68 T^{2} - 32 T + 16$$
$59$ $$T^{4} + 10 T^{3} + 80 T^{2} + \cdots + 400$$
$61$ $$T^{4} - 6 T^{3} + 32 T^{2} - 24 T + 16$$
$67$ $$T^{4} - 4 T^{3} + 192 T^{2} + \cdots + 30976$$
$71$ $$(T^{2} - 4 T - 16)^{2}$$
$73$ $$T^{4} + 8 T^{3} + 128 T^{2} + \cdots + 4096$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 2 T - 124)^{2}$$
$89$ $$(T^{2} + 10 T + 100)^{2}$$
$97$ $$(T^{2} - 16 T + 44)^{2}$$