Properties

 Label 3040.1.cn.a Level $3040$ Weight $1$ Character orbit 3040.cn Analytic conductor $1.517$ Analytic rank $0$ Dimension $32$ Projective image $D_{32}$ CM discriminant -95 Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,1,Mod(189,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 3, 4, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3040.189");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3040.cn (of order $$8$$, degree $$4$$, 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: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{64})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{32} + 1$$ x^32 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{32}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{32} + \cdots)$$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{64}^{27} q^{2} + ( - \zeta_{64}^{31} - \zeta_{64}^{25}) q^{3} - \zeta_{64}^{22} q^{4} + \zeta_{64}^{20} q^{5} + ( - \zeta_{64}^{26} - \zeta_{64}^{20}) q^{6} - \zeta_{64}^{17} q^{8} + ( - \zeta_{64}^{30} + \cdots - \zeta_{64}^{18}) q^{9} +O(q^{10})$$ q - z^27 * q^2 + (-z^31 - z^25) * q^3 - z^22 * q^4 + z^20 * q^5 + (-z^26 - z^20) * q^6 - z^17 * q^8 + (-z^30 - z^24 - z^18) * q^9 $$q - \zeta_{64}^{27} q^{2} + ( - \zeta_{64}^{31} - \zeta_{64}^{25}) q^{3} - \zeta_{64}^{22} q^{4} + \zeta_{64}^{20} q^{5} + ( - \zeta_{64}^{26} - \zeta_{64}^{20}) q^{6} - \zeta_{64}^{17} q^{8} + ( - \zeta_{64}^{30} + \cdots - \zeta_{64}^{18}) q^{9} + \cdots + (\zeta_{64}^{30} + \zeta_{64}^{26} + \cdots - 1) q^{99} +O(q^{100})$$ q - z^27 * q^2 + (-z^31 - z^25) * q^3 - z^22 * q^4 + z^20 * q^5 + (-z^26 - z^20) * q^6 - z^17 * q^8 + (-z^30 - z^24 - z^18) * q^9 + z^15 * q^10 + (-z^6 - z^2) * q^11 + (-z^21 - z^15) * q^12 + (z^21 + z^3) * q^13 + (z^19 + z^13) * q^15 - z^12 * q^16 + (-z^25 - z^19 - z^13) * q^18 + z^12 * q^19 + z^10 * q^20 + (z^29 - z) * q^22 + (-z^16 - z^10) * q^24 - z^8 * q^25 + (-z^30 + z^16) * q^26 + (-z^29 - z^23 - z^17 - z^11) * q^27 + (z^14 + z^8) * q^30 - z^7 * q^32 + (z^31 + z^27 - z^5 - z) * q^33 + (-z^20 - z^14 - z^8) * q^36 + (z^31 + z^9) * q^37 + z^7 * q^38 + (-z^28 + z^20 + z^14 + z^2) * q^39 + z^5 * q^40 + (z^28 + z^24) * q^44 + (z^18 + z^12 + z^6) * q^45 + (-z^11 - z^5) * q^48 + z^16 * q^49 - z^3 * q^50 + (-z^25 + z^11) * q^52 + (z^23 - z^17) * q^53 + (-z^24 - z^18 - z^12 - z^6) * q^54 + (-z^26 - z^22) * q^55 + (z^11 + z^5) * q^57 + (z^9 + z^3) * q^60 + (-z^14 + z^10) * q^61 - z^2 * q^64 + (z^23 - z^9) * q^65 + (z^28 + z^26 + z^22 - 1) * q^66 + (z^29 + z^27) * q^67 + (-z^15 - z^9 - z^3) * q^72 + (z^26 + z^4) * q^74 + (-z^7 - z) * q^75 + z^2 * q^76 + (-z^29 - z^23 + z^15 + z^9) * q^78 + q^80 + (-z^28 - z^22 - z^16 - z^10 - z^4) * q^81 + (z^23 + z^19) * q^88 + (z^13 + z^7 + z) * q^90 - q^95 + (-z^6 - 1) * q^96 + (-z^25 + z^7) * q^97 + z^11 * q^98 + (z^30 + z^26 + z^24 + z^20 - z^4 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$32 q+O(q^{10})$$ 32 * q $$32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100})$$ 32 * q - 32 * q^66 + 32 * q^80 - 32 * q^95 - 32 * q^96 - 32 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1217$$ $$1921$$ $$2661$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{64}^{8}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
189.1
 0.773010 + 0.634393i −0.995185 + 0.0980171i −0.0980171 − 0.995185i 0.634393 − 0.773010i −0.634393 + 0.773010i 0.0980171 + 0.995185i 0.995185 − 0.0980171i −0.773010 − 0.634393i 0.773010 − 0.634393i −0.995185 − 0.0980171i −0.0980171 + 0.995185i 0.634393 + 0.773010i −0.634393 − 0.773010i 0.0980171 − 0.995185i 0.995185 + 0.0980171i −0.773010 + 0.634393i −0.290285 + 0.956940i 0.881921 − 0.471397i −0.471397 − 0.881921i −0.956940 − 0.290285i
−0.956940 + 0.290285i 0.871028 + 0.360791i 0.831470 0.555570i 0.382683 + 0.923880i −0.938254 0.0924099i 0 −0.634393 + 0.773010i −0.0785882 0.0785882i −0.634393 0.773010i
189.2 −0.881921 0.471397i −1.76820 0.732410i 0.555570 + 0.831470i −0.382683 0.923880i 1.21415 + 1.47945i 0 −0.0980171 0.995185i 1.88298 + 1.88298i −0.0980171 + 0.995185i
189.3 −0.471397 + 0.881921i 0.536376 + 0.222174i −0.555570 0.831470i −0.382683 0.923880i −0.448786 + 0.368309i 0 0.995185 0.0980171i −0.468769 0.468769i 0.995185 + 0.0980171i
189.4 −0.290285 0.956940i 1.62958 + 0.674993i −0.831470 + 0.555570i 0.382683 + 0.923880i 0.172887 1.75535i 0 0.773010 + 0.634393i 1.49280 + 1.49280i 0.773010 0.634393i
189.5 0.290285 + 0.956940i −1.62958 0.674993i −0.831470 + 0.555570i 0.382683 + 0.923880i 0.172887 1.75535i 0 −0.773010 0.634393i 1.49280 + 1.49280i −0.773010 + 0.634393i
189.6 0.471397 0.881921i −0.536376 0.222174i −0.555570 0.831470i −0.382683 0.923880i −0.448786 + 0.368309i 0 −0.995185 + 0.0980171i −0.468769 0.468769i −0.995185 0.0980171i
189.7 0.881921 + 0.471397i 1.76820 + 0.732410i 0.555570 + 0.831470i −0.382683 0.923880i 1.21415 + 1.47945i 0 0.0980171 + 0.995185i 1.88298 + 1.88298i 0.0980171 0.995185i
189.8 0.956940 0.290285i −0.871028 0.360791i 0.831470 0.555570i 0.382683 + 0.923880i −0.938254 0.0924099i 0 0.634393 0.773010i −0.0785882 0.0785882i 0.634393 + 0.773010i
949.1 −0.956940 0.290285i 0.871028 0.360791i 0.831470 + 0.555570i 0.382683 0.923880i −0.938254 + 0.0924099i 0 −0.634393 0.773010i −0.0785882 + 0.0785882i −0.634393 + 0.773010i
949.2 −0.881921 + 0.471397i −1.76820 + 0.732410i 0.555570 0.831470i −0.382683 + 0.923880i 1.21415 1.47945i 0 −0.0980171 + 0.995185i 1.88298 1.88298i −0.0980171 0.995185i
949.3 −0.471397 0.881921i 0.536376 0.222174i −0.555570 + 0.831470i −0.382683 + 0.923880i −0.448786 0.368309i 0 0.995185 + 0.0980171i −0.468769 + 0.468769i 0.995185 0.0980171i
949.4 −0.290285 + 0.956940i 1.62958 0.674993i −0.831470 0.555570i 0.382683 0.923880i 0.172887 + 1.75535i 0 0.773010 0.634393i 1.49280 1.49280i 0.773010 + 0.634393i
949.5 0.290285 0.956940i −1.62958 + 0.674993i −0.831470 0.555570i 0.382683 0.923880i 0.172887 + 1.75535i 0 −0.773010 + 0.634393i 1.49280 1.49280i −0.773010 0.634393i
949.6 0.471397 + 0.881921i −0.536376 + 0.222174i −0.555570 + 0.831470i −0.382683 + 0.923880i −0.448786 0.368309i 0 −0.995185 0.0980171i −0.468769 + 0.468769i −0.995185 + 0.0980171i
949.7 0.881921 0.471397i 1.76820 0.732410i 0.555570 0.831470i −0.382683 + 0.923880i 1.21415 1.47945i 0 0.0980171 0.995185i 1.88298 1.88298i 0.0980171 + 0.995185i
949.8 0.956940 + 0.290285i −0.871028 + 0.360791i 0.831470 + 0.555570i 0.382683 0.923880i −0.938254 + 0.0924099i 0 0.634393 + 0.773010i −0.0785882 + 0.0785882i 0.634393 0.773010i
1709.1 −0.995185 0.0980171i 0.591637 1.42834i 0.980785 + 0.195090i 0.923880 0.382683i −0.728789 + 1.36347i 0 −0.956940 0.290285i −0.983006 0.983006i −0.956940 + 0.290285i
1709.2 −0.773010 + 0.634393i −0.0750191 + 0.181112i 0.195090 0.980785i −0.923880 + 0.382683i −0.0569057 0.187593i 0 0.471397 + 0.881921i 0.679933 + 0.679933i 0.471397 0.881921i
1709.3 −0.634393 0.773010i −0.761681 + 1.83886i −0.195090 + 0.980785i −0.923880 + 0.382683i 1.90466 0.577774i 0 0.881921 0.471397i −2.09415 2.09415i 0.881921 + 0.471397i
1709.4 −0.0980171 + 0.995185i −0.485544 + 1.17221i −0.980785 0.195090i 0.923880 0.382683i −1.11897 0.598102i 0 0.290285 0.956940i −0.431207 0.431207i 0.290285 + 0.956940i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 189.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner
32.g even 8 1 inner
160.z even 8 1 inner
608.w odd 8 1 inner
3040.cn odd 8 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.cn.a 32
5.b even 2 1 inner 3040.1.cn.a 32
19.b odd 2 1 inner 3040.1.cn.a 32
32.g even 8 1 inner 3040.1.cn.a 32
95.d odd 2 1 CM 3040.1.cn.a 32
160.z even 8 1 inner 3040.1.cn.a 32
608.w odd 8 1 inner 3040.1.cn.a 32
3040.cn odd 8 1 inner 3040.1.cn.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.1.cn.a 32 1.a even 1 1 trivial
3040.1.cn.a 32 5.b even 2 1 inner
3040.1.cn.a 32 19.b odd 2 1 inner
3040.1.cn.a 32 32.g even 8 1 inner
3040.1.cn.a 32 95.d odd 2 1 CM
3040.1.cn.a 32 160.z even 8 1 inner
3040.1.cn.a 32 608.w odd 8 1 inner
3040.1.cn.a 32 3040.cn odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3040, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{32} + 1$$
$3$ $$T^{32} + 280 T^{24} + \cdots + 4$$
$5$ $$(T^{8} + 1)^{4}$$
$7$ $$T^{32}$$
$11$ $$(T^{16} + 16 T^{10} + \cdots + 4)^{2}$$
$13$ $$T^{32} + 280 T^{24} + \cdots + 4$$
$17$ $$T^{32}$$
$19$ $$(T^{8} + 1)^{4}$$
$23$ $$T^{32}$$
$29$ $$T^{32}$$
$31$ $$T^{32}$$
$37$ $$T^{32} + 280 T^{24} + \cdots + 4$$
$41$ $$T^{32}$$
$43$ $$T^{32}$$
$47$ $$T^{32}$$
$53$ $$T^{32} + 280 T^{24} + \cdots + 4$$
$59$ $$T^{32}$$
$61$ $$(T^{16} + 16 T^{10} + \cdots + 4)^{2}$$
$67$ $$T^{32} + 280 T^{24} + \cdots + 4$$
$71$ $$T^{32}$$
$73$ $$T^{32}$$
$79$ $$T^{32}$$
$83$ $$T^{32}$$
$89$ $$T^{32}$$
$97$ $$(T^{16} - 16 T^{14} + \cdots + 2)^{2}$$