# Properties

 Label 845.2.o.b Level $845$ Weight $2$ Character orbit 845.o Analytic conductor $6.747$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(258,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([9, 7]))

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

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

chi := DirichletCharacter("845.258");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.o (of order $$12$$, degree $$4$$, 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: $$6.74735897080$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{2} q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - 2 \zeta_{12}^{3} + 1) q^{5} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} - 2 \zeta_{12} q^{7} + 3 q^{8} - \zeta_{12} q^{9} +O(q^{10})$$ q + z^2 * q^2 + (z^3 - z^2 - z) * q^3 + (-z^2 + 1) * q^4 + (-2*z^3 + 1) * q^5 + (-z^2 - z + 1) * q^6 - 2*z * q^7 + 3 * q^8 - z * q^9 $$q + \zeta_{12}^{2} q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - 2 \zeta_{12}^{3} + 1) q^{5} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} - 2 \zeta_{12} q^{7} + 3 q^{8} - \zeta_{12} q^{9} + ( - 2 \zeta_{12}^{3} + \cdots + 2 \zeta_{12}) q^{10} + \cdots + (\zeta_{12}^{3} - 1) q^{99} +O(q^{100})$$ q + z^2 * q^2 + (z^3 - z^2 - z) * q^3 + (-z^2 + 1) * q^4 + (-2*z^3 + 1) * q^5 + (-z^2 - z + 1) * q^6 - 2*z * q^7 + 3 * q^8 - z * q^9 + (-2*z^3 + z^2 + 2*z) * q^10 + (-z^3 - z^2 + z) * q^11 + (z^3 - 1) * q^12 - 2*z^3 * q^14 + (3*z^3 + z^2 - 3*z) * q^15 + z^2 * q^16 + (z^2 + z - 1) * q^17 - z^3 * q^18 + (5*z^2 - 5*z - 5) * q^19 + (-z^2 - 2*z + 1) * q^20 + (2*z^3 + 2) * q^21 + (-z^2 + z + 1) * q^22 + (-3*z^3 - 3*z^2 + 3*z) * q^23 + (3*z^3 - 3*z^2 - 3*z) * q^24 + (-4*z^3 - 3) * q^25 + (4*z^3 + 4) * q^27 + (2*z^3 - 2*z) * q^28 + (z^2 - 3*z - 1) * q^30 + (5*z^3 - 5) * q^31 + (-5*z^2 + 5) * q^32 + (2*z^2 - 2) * q^33 + (z^3 - 1) * q^34 + (4*z^2 - 2*z - 4) * q^35 + (z^3 - z) * q^36 + (-5*z^3 - 5) * q^38 + (-6*z^3 + 3) * q^40 + (7*z^3 - 7*z^2 - 7*z) * q^41 + (2*z^3 + 2*z^2 - 2*z) * q^42 + (-z^2 + z + 1) * q^43 + (-z^3 - 1) * q^44 + (2*z^2 - z - 2) * q^45 + (-3*z^2 + 3*z + 3) * q^46 - 6*z^3 * q^47 + (-z^2 - z + 1) * q^48 - 3*z^2 * q^49 + (-4*z^3 - 3*z^2 + 4*z) * q^50 - 2*z^3 * q^51 + (-5*z^3 + 5) * q^53 + (4*z^3 + 4*z^2 - 4*z) * q^54 + (z^3 - 3*z^2 - z) * q^55 - 6*z * q^56 + 10 * q^57 + (7*z^2 + 7*z - 7) * q^59 + (3*z^3 + 1) * q^60 + (-14*z^2 + 14) * q^61 + (5*z^3 - 5*z^2 - 5*z) * q^62 + 2*z^2 * q^63 + 7 * q^64 - 2 * q^66 - 4*z^2 * q^67 + (-z^3 + z^2 + z) * q^68 + (6*z^2 - 6) * q^69 + (-2*z^3 - 4) * q^70 + (-z^2 - z + 1) * q^71 - 3*z * q^72 + 10 * q^73 + (z^3 + 7*z^2 - z) * q^75 + (5*z^3 + 5*z^2 - 5*z) * q^76 + (2*z^3 - 2) * q^77 - 2*z^3 * q^79 + (-2*z^3 + z^2 + 2*z) * q^80 - 5*z^2 * q^81 + (-7*z^2 - 7*z + 7) * q^82 + 6*z^3 * q^83 + (-2*z^2 + 2*z + 2) * q^84 + (-z^2 + 3*z + 1) * q^85 + (z^3 + 1) * q^86 + (-3*z^3 - 3*z^2 + 3*z) * q^88 + (-5*z^3 + 5*z^2 + 5*z) * q^89 + (-z^3 - 2) * q^90 + (-3*z^3 - 3) * q^92 + (-10*z^3 + 10*z) * q^93 + (-6*z^3 + 6*z) * q^94 + (15*z^2 + 5*z - 15) * q^95 + (5*z^3 - 5) * q^96 + (-2*z^2 + 2) * q^97 + (-3*z^2 + 3) * q^98 + (z^3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 12 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 12 * q^8 $$4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 12 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{12} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 10 q^{19} + 2 q^{20} + 8 q^{21} + 2 q^{22} - 6 q^{23} - 6 q^{24} - 12 q^{25} + 16 q^{27} - 2 q^{30} - 20 q^{31} + 10 q^{32} - 4 q^{33} - 4 q^{34} - 8 q^{35} - 20 q^{38} + 12 q^{40} - 14 q^{41} + 4 q^{42} + 2 q^{43} - 4 q^{44} - 4 q^{45} + 6 q^{46} + 2 q^{48} - 6 q^{49} - 6 q^{50} + 20 q^{53} + 8 q^{54} - 6 q^{55} + 40 q^{57} - 14 q^{59} + 4 q^{60} + 28 q^{61} - 10 q^{62} + 4 q^{63} + 28 q^{64} - 8 q^{66} - 8 q^{67} + 2 q^{68} - 12 q^{69} - 16 q^{70} + 2 q^{71} + 40 q^{73} + 14 q^{75} + 10 q^{76} - 8 q^{77} + 2 q^{80} - 10 q^{81} + 14 q^{82} + 4 q^{84} + 2 q^{85} + 4 q^{86} - 6 q^{88} + 10 q^{89} - 8 q^{90} - 12 q^{92} - 30 q^{95} - 20 q^{96} + 4 q^{97} + 6 q^{98} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 12 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^12 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 10 * q^19 + 2 * q^20 + 8 * q^21 + 2 * q^22 - 6 * q^23 - 6 * q^24 - 12 * q^25 + 16 * q^27 - 2 * q^30 - 20 * q^31 + 10 * q^32 - 4 * q^33 - 4 * q^34 - 8 * q^35 - 20 * q^38 + 12 * q^40 - 14 * q^41 + 4 * q^42 + 2 * q^43 - 4 * q^44 - 4 * q^45 + 6 * q^46 + 2 * q^48 - 6 * q^49 - 6 * q^50 + 20 * q^53 + 8 * q^54 - 6 * q^55 + 40 * q^57 - 14 * q^59 + 4 * q^60 + 28 * q^61 - 10 * q^62 + 4 * q^63 + 28 * q^64 - 8 * q^66 - 8 * q^67 + 2 * q^68 - 12 * q^69 - 16 * q^70 + 2 * q^71 + 40 * q^73 + 14 * q^75 + 10 * q^76 - 8 * q^77 + 2 * q^80 - 10 * q^81 + 14 * q^82 + 4 * q^84 + 2 * q^85 + 4 * q^86 - 6 * q^88 + 10 * q^89 - 8 * q^90 - 12 * q^92 - 30 * q^95 - 20 * q^96 + 4 * q^97 + 6 * q^98 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$\zeta_{12}$$ $$\zeta_{12}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
258.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0.500000 + 0.866025i 0.366025 1.36603i 0.500000 0.866025i 1.00000 + 2.00000i 1.36603 0.366025i 1.73205 + 1.00000i 3.00000 0.866025 + 0.500000i −1.23205 + 1.86603i
357.1 0.500000 0.866025i 0.366025 + 1.36603i 0.500000 + 0.866025i 1.00000 2.00000i 1.36603 + 0.366025i 1.73205 1.00000i 3.00000 0.866025 0.500000i −1.23205 1.86603i
488.1 0.500000 0.866025i −1.36603 + 0.366025i 0.500000 + 0.866025i 1.00000 + 2.00000i −0.366025 + 1.36603i −1.73205 + 1.00000i 3.00000 −0.866025 + 0.500000i 2.23205 + 0.133975i
587.1 0.500000 + 0.866025i −1.36603 0.366025i 0.500000 0.866025i 1.00000 2.00000i −0.366025 1.36603i −1.73205 1.00000i 3.00000 −0.866025 0.500000i 2.23205 0.133975i
 $$n$$: 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.c even 3 1 inner
65.k even 4 1 inner
65.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.o.b 4
5.c odd 4 1 845.2.t.b 4
13.b even 2 1 845.2.o.a 4
13.c even 3 1 845.2.k.a 2
13.c even 3 1 inner 845.2.o.b 4
13.d odd 4 1 845.2.t.a 4
13.d odd 4 1 845.2.t.b 4
13.e even 6 1 65.2.k.a yes 2
13.e even 6 1 845.2.o.a 4
13.f odd 12 1 65.2.f.a 2
13.f odd 12 1 845.2.f.a 2
13.f odd 12 1 845.2.t.a 4
13.f odd 12 1 845.2.t.b 4
39.h odd 6 1 585.2.w.b 2
39.k even 12 1 585.2.n.c 2
52.i odd 6 1 1040.2.bg.a 2
52.l even 12 1 1040.2.cd.b 2
65.f even 4 1 845.2.o.a 4
65.h odd 4 1 845.2.t.a 4
65.k even 4 1 inner 845.2.o.b 4
65.l even 6 1 325.2.k.a 2
65.o even 12 1 325.2.k.a 2
65.o even 12 1 845.2.k.a 2
65.o even 12 1 inner 845.2.o.b 4
65.q odd 12 1 845.2.f.a 2
65.q odd 12 1 845.2.t.b 4
65.r odd 12 1 65.2.f.a 2
65.r odd 12 1 325.2.f.a 2
65.r odd 12 1 845.2.t.a 4
65.s odd 12 1 325.2.f.a 2
65.t even 12 1 65.2.k.a yes 2
65.t even 12 1 845.2.o.a 4
195.bc odd 12 1 585.2.w.b 2
195.bf even 12 1 585.2.n.c 2
260.bg even 12 1 1040.2.cd.b 2
260.bl odd 12 1 1040.2.bg.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a 2 13.f odd 12 1
65.2.f.a 2 65.r odd 12 1
65.2.k.a yes 2 13.e even 6 1
65.2.k.a yes 2 65.t even 12 1
325.2.f.a 2 65.r odd 12 1
325.2.f.a 2 65.s odd 12 1
325.2.k.a 2 65.l even 6 1
325.2.k.a 2 65.o even 12 1
585.2.n.c 2 39.k even 12 1
585.2.n.c 2 195.bf even 12 1
585.2.w.b 2 39.h odd 6 1
585.2.w.b 2 195.bc odd 12 1
845.2.f.a 2 13.f odd 12 1
845.2.f.a 2 65.q odd 12 1
845.2.k.a 2 13.c even 3 1
845.2.k.a 2 65.o even 12 1
845.2.o.a 4 13.b even 2 1
845.2.o.a 4 13.e even 6 1
845.2.o.a 4 65.f even 4 1
845.2.o.a 4 65.t even 12 1
845.2.o.b 4 1.a even 1 1 trivial
845.2.o.b 4 13.c even 3 1 inner
845.2.o.b 4 65.k even 4 1 inner
845.2.o.b 4 65.o even 12 1 inner
845.2.t.a 4 13.d odd 4 1
845.2.t.a 4 13.f odd 12 1
845.2.t.a 4 65.h odd 4 1
845.2.t.a 4 65.r odd 12 1
845.2.t.b 4 5.c odd 4 1
845.2.t.b 4 13.d odd 4 1
845.2.t.b 4 13.f odd 12 1
845.2.t.b 4 65.q odd 12 1
1040.2.bg.a 2 52.i odd 6 1
1040.2.bg.a 2 260.bl odd 12 1
1040.2.cd.b 2 52.l even 12 1
1040.2.cd.b 2 260.bg even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} + 4T_{3} + 4$$ T3^4 + 2*T3^3 + 2*T3^2 + 4*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$5$ $$(T^{2} - 2 T + 5)^{2}$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$19$ $$T^{4} + 10 T^{3} + \cdots + 2500$$
$23$ $$T^{4} + 6 T^{3} + \cdots + 324$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 10 T + 50)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4} + 14 T^{3} + \cdots + 9604$$
$43$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$(T^{2} - 10 T + 50)^{2}$$
$59$ $$T^{4} + 14 T^{3} + \cdots + 9604$$
$61$ $$(T^{2} - 14 T + 196)^{2}$$
$67$ $$(T^{2} + 4 T + 16)^{2}$$
$71$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$73$ $$(T - 10)^{4}$$
$79$ $$(T^{2} + 4)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$T^{4} - 10 T^{3} + \cdots + 2500$$
$97$ $$(T^{2} - 2 T + 4)^{2}$$