# Properties

 Label 5070.2.b.o Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ 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] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, 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: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} - 2 \beta_1 q^{7} - \beta_1 q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 - q^3 - q^4 + b1 * q^5 - b1 * q^6 - 2*b1 * q^7 - b1 * q^8 + q^9 $$q + \beta_1 q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} - 2 \beta_1 q^{7} - \beta_1 q^{8} + q^{9} - q^{10} + (2 \beta_{2} - 3 \beta_1) q^{11} + q^{12} + 2 q^{14} - \beta_1 q^{15} + q^{16} - 4 q^{17} + \beta_1 q^{18} + ( - 2 \beta_{2} + 4 \beta_1) q^{19} - \beta_1 q^{20} + 2 \beta_1 q^{21} + ( - 2 \beta_{3} + 3) q^{22} + ( - \beta_{3} + 2) q^{23} + \beta_1 q^{24} - q^{25} - q^{27} + 2 \beta_1 q^{28} + (\beta_{3} + 2) q^{29} + q^{30} + \beta_{2} q^{31} + \beta_1 q^{32} + ( - 2 \beta_{2} + 3 \beta_1) q^{33} - 4 \beta_1 q^{34} + 2 q^{35} - q^{36} + (3 \beta_{2} - 4 \beta_1) q^{37} + (2 \beta_{3} - 4) q^{38} + q^{40} - 2 \beta_1 q^{41} - 2 q^{42} + ( - 4 \beta_{3} + 5) q^{43} + ( - 2 \beta_{2} + 3 \beta_1) q^{44} + \beta_1 q^{45} + ( - \beta_{2} + 2 \beta_1) q^{46} + ( - 2 \beta_{2} - 7 \beta_1) q^{47} - q^{48} + 3 q^{49} - \beta_1 q^{50} + 4 q^{51} + ( - 4 \beta_{3} - 6) q^{53} - \beta_1 q^{54} + ( - 2 \beta_{3} + 3) q^{55} - 2 q^{56} + (2 \beta_{2} - 4 \beta_1) q^{57} + (\beta_{2} + 2 \beta_1) q^{58} + ( - 2 \beta_{2} + 5 \beta_1) q^{59} + \beta_1 q^{60} + 6 \beta_{3} q^{61} - \beta_{3} q^{62} - 2 \beta_1 q^{63} - q^{64} + (2 \beta_{3} - 3) q^{66} + ( - 2 \beta_{2} + 8 \beta_1) q^{67} + 4 q^{68} + (\beta_{3} - 2) q^{69} + 2 \beta_1 q^{70} + (6 \beta_{2} - 2 \beta_1) q^{71} - \beta_1 q^{72} + 2 \beta_1 q^{73} + ( - 3 \beta_{3} + 4) q^{74} + q^{75} + (2 \beta_{2} - 4 \beta_1) q^{76} + (4 \beta_{3} - 6) q^{77} + (4 \beta_{3} - 7) q^{79} + \beta_1 q^{80} + q^{81} + 2 q^{82} + ( - 4 \beta_{2} + 2 \beta_1) q^{83} - 2 \beta_1 q^{84} - 4 \beta_1 q^{85} + ( - 4 \beta_{2} + 5 \beta_1) q^{86} + ( - \beta_{3} - 2) q^{87} + (2 \beta_{3} - 3) q^{88} + (2 \beta_{2} + 4 \beta_1) q^{89} - q^{90} + (\beta_{3} - 2) q^{92} - \beta_{2} q^{93} + (2 \beta_{3} + 7) q^{94} + (2 \beta_{3} - 4) q^{95} - \beta_1 q^{96} + (2 \beta_{2} + 4 \beta_1) q^{97} + 3 \beta_1 q^{98} + (2 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 - q^3 - q^4 + b1 * q^5 - b1 * q^6 - 2*b1 * q^7 - b1 * q^8 + q^9 - q^10 + (2*b2 - 3*b1) * q^11 + q^12 + 2 * q^14 - b1 * q^15 + q^16 - 4 * q^17 + b1 * q^18 + (-2*b2 + 4*b1) * q^19 - b1 * q^20 + 2*b1 * q^21 + (-2*b3 + 3) * q^22 + (-b3 + 2) * q^23 + b1 * q^24 - q^25 - q^27 + 2*b1 * q^28 + (b3 + 2) * q^29 + q^30 + b2 * q^31 + b1 * q^32 + (-2*b2 + 3*b1) * q^33 - 4*b1 * q^34 + 2 * q^35 - q^36 + (3*b2 - 4*b1) * q^37 + (2*b3 - 4) * q^38 + q^40 - 2*b1 * q^41 - 2 * q^42 + (-4*b3 + 5) * q^43 + (-2*b2 + 3*b1) * q^44 + b1 * q^45 + (-b2 + 2*b1) * q^46 + (-2*b2 - 7*b1) * q^47 - q^48 + 3 * q^49 - b1 * q^50 + 4 * q^51 + (-4*b3 - 6) * q^53 - b1 * q^54 + (-2*b3 + 3) * q^55 - 2 * q^56 + (2*b2 - 4*b1) * q^57 + (b2 + 2*b1) * q^58 + (-2*b2 + 5*b1) * q^59 + b1 * q^60 + 6*b3 * q^61 - b3 * q^62 - 2*b1 * q^63 - q^64 + (2*b3 - 3) * q^66 + (-2*b2 + 8*b1) * q^67 + 4 * q^68 + (b3 - 2) * q^69 + 2*b1 * q^70 + (6*b2 - 2*b1) * q^71 - b1 * q^72 + 2*b1 * q^73 + (-3*b3 + 4) * q^74 + q^75 + (2*b2 - 4*b1) * q^76 + (4*b3 - 6) * q^77 + (4*b3 - 7) * q^79 + b1 * q^80 + q^81 + 2 * q^82 + (-4*b2 + 2*b1) * q^83 - 2*b1 * q^84 - 4*b1 * q^85 + (-4*b2 + 5*b1) * q^86 + (-b3 - 2) * q^87 + (2*b3 - 3) * q^88 + (2*b2 + 4*b1) * q^89 - q^90 + (b3 - 2) * q^92 - b2 * q^93 + (2*b3 + 7) * q^94 + (2*b3 - 4) * q^95 - b1 * q^96 + (2*b2 + 4*b1) * q^97 + 3*b1 * q^98 + (2*b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 8 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} + 8 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 4 q^{36} - 16 q^{38} + 4 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{48} + 12 q^{49} + 16 q^{51} - 24 q^{53} + 12 q^{55} - 8 q^{56} - 4 q^{64} - 12 q^{66} + 16 q^{68} - 8 q^{69} + 16 q^{74} + 4 q^{75} - 24 q^{77} - 28 q^{79} + 4 q^{81} + 8 q^{82} - 8 q^{87} - 12 q^{88} - 4 q^{90} - 8 q^{92} + 28 q^{94} - 16 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 8 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 + 8 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 + 4 * q^30 + 8 * q^35 - 4 * q^36 - 16 * q^38 + 4 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^48 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 12 * q^55 - 8 * q^56 - 4 * q^64 - 12 * q^66 + 16 * q^68 - 8 * q^69 + 16 * q^74 + 4 * q^75 - 24 * q^77 - 28 * q^79 + 4 * q^81 + 8 * q^82 - 8 * q^87 - 12 * q^88 - 4 * q^90 - 8 * q^92 + 28 * q^94 - 16 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 −1.00000
1351.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 −1.00000
1351.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 −1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.o 4
13.b even 2 1 inner 5070.2.b.o 4
13.c even 3 1 390.2.bb.b 4
13.d odd 4 1 5070.2.a.y 2
13.d odd 4 1 5070.2.a.bg 2
13.e even 6 1 390.2.bb.b 4
39.h odd 6 1 1170.2.bs.e 4
39.i odd 6 1 1170.2.bs.e 4
65.l even 6 1 1950.2.bc.b 4
65.n even 6 1 1950.2.bc.b 4
65.q odd 12 1 1950.2.y.c 4
65.q odd 12 1 1950.2.y.f 4
65.r odd 12 1 1950.2.y.c 4
65.r odd 12 1 1950.2.y.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 13.c even 3 1
390.2.bb.b 4 13.e even 6 1
1170.2.bs.e 4 39.h odd 6 1
1170.2.bs.e 4 39.i odd 6 1
1950.2.y.c 4 65.q odd 12 1
1950.2.y.c 4 65.r odd 12 1
1950.2.y.f 4 65.q odd 12 1
1950.2.y.f 4 65.r odd 12 1
1950.2.bc.b 4 65.l even 6 1
1950.2.bc.b 4 65.n even 6 1
5070.2.a.y 2 13.d odd 4 1
5070.2.a.bg 2 13.d odd 4 1
5070.2.b.o 4 1.a even 1 1 trivial
5070.2.b.o 4 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{4} + 42T_{11}^{2} + 9$$ T11^4 + 42*T11^2 + 9 $$T_{17} + 4$$ T17 + 4 $$T_{31}^{2} + 3$$ T31^2 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T + 1)^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$T^{4} + 42T^{2} + 9$$
$13$ $$T^{4}$$
$17$ $$(T + 4)^{4}$$
$19$ $$T^{4} + 56T^{2} + 16$$
$23$ $$(T^{2} - 4 T + 1)^{2}$$
$29$ $$(T^{2} - 4 T + 1)^{2}$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$T^{4} + 86T^{2} + 121$$
$41$ $$(T^{2} + 4)^{2}$$
$43$ $$(T^{2} - 10 T - 23)^{2}$$
$47$ $$T^{4} + 122T^{2} + 1369$$
$53$ $$(T^{2} + 12 T - 12)^{2}$$
$59$ $$T^{4} + 74T^{2} + 169$$
$61$ $$(T^{2} - 108)^{2}$$
$67$ $$T^{4} + 152T^{2} + 2704$$
$71$ $$T^{4} + 224 T^{2} + 10816$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$(T^{2} + 14 T + 1)^{2}$$
$83$ $$T^{4} + 104T^{2} + 1936$$
$89$ $$T^{4} + 56T^{2} + 16$$
$97$ $$T^{4} + 56T^{2} + 16$$