# Properties

 Label 285.2.k.c Level $285$ Weight $2$ Character orbit 285.k Analytic conductor $2.276$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(77,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("285.77");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.k (of order $$4$$, degree $$2$$, 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: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 2 q^{3} - 12 q^{6}+O(q^{10})$$ 28 * q - 2 * q^3 - 12 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 2 q^{3} - 12 q^{6} - 8 q^{10} + 34 q^{12} + 8 q^{13} - 14 q^{15} - 20 q^{16} - 24 q^{18} - 4 q^{21} - 32 q^{22} + 8 q^{25} + 22 q^{27} - 28 q^{28} + 12 q^{30} + 72 q^{31} - 84 q^{36} - 12 q^{37} + 20 q^{40} + 48 q^{42} - 12 q^{43} - 52 q^{45} + 8 q^{46} + 46 q^{48} + 28 q^{51} - 76 q^{52} + 104 q^{55} + 2 q^{57} - 60 q^{58} - 22 q^{60} + 96 q^{61} + 56 q^{63} - 28 q^{66} - 72 q^{67} + 68 q^{70} + 20 q^{72} - 72 q^{73} + 2 q^{75} - 36 q^{76} + 76 q^{78} - 100 q^{81} - 116 q^{82} - 44 q^{85} + 4 q^{87} + 60 q^{88} - 36 q^{90} - 80 q^{91} + 52 q^{93} - 80 q^{96}+O(q^{100})$$ 28 * q - 2 * q^3 - 12 * q^6 - 8 * q^10 + 34 * q^12 + 8 * q^13 - 14 * q^15 - 20 * q^16 - 24 * q^18 - 4 * q^21 - 32 * q^22 + 8 * q^25 + 22 * q^27 - 28 * q^28 + 12 * q^30 + 72 * q^31 - 84 * q^36 - 12 * q^37 + 20 * q^40 + 48 * q^42 - 12 * q^43 - 52 * q^45 + 8 * q^46 + 46 * q^48 + 28 * q^51 - 76 * q^52 + 104 * q^55 + 2 * q^57 - 60 * q^58 - 22 * q^60 + 96 * q^61 + 56 * q^63 - 28 * q^66 - 72 * q^67 + 68 * q^70 + 20 * q^72 - 72 * q^73 + 2 * q^75 - 36 * q^76 + 76 * q^78 - 100 * q^81 - 116 * q^82 - 44 * q^85 + 4 * q^87 + 60 * q^88 - 36 * q^90 - 80 * q^91 + 52 * q^93 - 80 * q^96

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1 −1.82223 + 1.82223i −1.28293 + 1.16366i 4.64103i 1.53042 + 1.63028i 0.217337 4.45823i −1.83721 1.83721i 4.81255 + 4.81255i 0.291804 2.98577i −5.75952 0.181957i
77.2 −1.60748 + 1.60748i −1.46845 + 0.918505i 3.16797i 0.294779 2.21655i 0.884027 3.83698i 3.56650 + 3.56650i 1.87749 + 1.87749i 1.31270 2.69756i 3.08921 + 4.03691i
77.3 −1.53127 + 1.53127i 1.71677 0.229557i 2.68957i −1.36334 1.77237i −2.27733 + 2.98035i −1.27504 1.27504i 1.05592 + 1.05592i 2.89461 0.788193i 4.80162 + 0.626344i
77.4 −1.48042 + 1.48042i 1.05938 + 1.37030i 2.38331i 2.23587 0.0297943i −3.59695 0.460300i −1.93250 1.93250i 0.567468 + 0.567468i −0.755445 + 2.90333i −3.26593 + 3.35414i
77.5 −0.913256 + 0.913256i −0.951938 1.44700i 0.331928i 2.10088 0.765698i 2.19085 + 0.452119i 0.194936 + 0.194936i −2.12965 2.12965i −1.18763 + 2.75491i −1.21936 + 2.61792i
77.6 −0.473364 + 0.473364i 1.62852 0.589855i 1.55185i −1.83078 + 1.28384i −0.491666 + 1.05010i 1.99868 + 1.99868i −1.68132 1.68132i 2.30414 1.92118i 0.258900 1.47435i
77.7 −0.0307908 + 0.0307908i −1.45499 0.939687i 1.99810i −1.20318 1.88477i 0.0737341 0.0158665i −0.715376 0.715376i −0.123105 0.123105i 1.23398 + 2.73447i 0.0950807 + 0.0209865i
77.8 0.0307908 0.0307908i 0.939687 + 1.45499i 1.99810i 1.20318 + 1.88477i 0.0737341 + 0.0158665i −0.715376 0.715376i 0.123105 + 0.123105i −1.23398 + 2.73447i 0.0950807 + 0.0209865i
77.9 0.473364 0.473364i 0.589855 1.62852i 1.55185i 1.83078 1.28384i −0.491666 1.05010i 1.99868 + 1.99868i 1.68132 + 1.68132i −2.30414 1.92118i 0.258900 1.47435i
77.10 0.913256 0.913256i 1.44700 + 0.951938i 0.331928i −2.10088 + 0.765698i 2.19085 0.452119i 0.194936 + 0.194936i 2.12965 + 2.12965i 1.18763 + 2.75491i −1.21936 + 2.61792i
77.11 1.48042 1.48042i −1.37030 1.05938i 2.38331i −2.23587 + 0.0297943i −3.59695 + 0.460300i −1.93250 1.93250i −0.567468 0.567468i 0.755445 + 2.90333i −3.26593 + 3.35414i
77.12 1.53127 1.53127i 0.229557 1.71677i 2.68957i 1.36334 + 1.77237i −2.27733 2.98035i −1.27504 1.27504i −1.05592 1.05592i −2.89461 0.788193i 4.80162 + 0.626344i
77.13 1.60748 1.60748i −0.918505 + 1.46845i 3.16797i −0.294779 + 2.21655i 0.884027 + 3.83698i 3.56650 + 3.56650i −1.87749 1.87749i −1.31270 2.69756i 3.08921 + 4.03691i
77.14 1.82223 1.82223i −1.16366 + 1.28293i 4.64103i −1.53042 1.63028i 0.217337 + 4.45823i −1.83721 1.83721i −4.81255 4.81255i −0.291804 2.98577i −5.75952 0.181957i
248.1 −1.82223 1.82223i −1.28293 1.16366i 4.64103i 1.53042 1.63028i 0.217337 + 4.45823i −1.83721 + 1.83721i 4.81255 4.81255i 0.291804 + 2.98577i −5.75952 + 0.181957i
248.2 −1.60748 1.60748i −1.46845 0.918505i 3.16797i 0.294779 + 2.21655i 0.884027 + 3.83698i 3.56650 3.56650i 1.87749 1.87749i 1.31270 + 2.69756i 3.08921 4.03691i
248.3 −1.53127 1.53127i 1.71677 + 0.229557i 2.68957i −1.36334 + 1.77237i −2.27733 2.98035i −1.27504 + 1.27504i 1.05592 1.05592i 2.89461 + 0.788193i 4.80162 0.626344i
248.4 −1.48042 1.48042i 1.05938 1.37030i 2.38331i 2.23587 + 0.0297943i −3.59695 + 0.460300i −1.93250 + 1.93250i 0.567468 0.567468i −0.755445 2.90333i −3.26593 3.35414i
248.5 −0.913256 0.913256i −0.951938 + 1.44700i 0.331928i 2.10088 + 0.765698i 2.19085 0.452119i 0.194936 0.194936i −2.12965 + 2.12965i −1.18763 2.75491i −1.21936 2.61792i
248.6 −0.473364 0.473364i 1.62852 + 0.589855i 1.55185i −1.83078 1.28384i −0.491666 1.05010i 1.99868 1.99868i −1.68132 + 1.68132i 2.30414 + 1.92118i 0.258900 + 1.47435i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.k.c 28
3.b odd 2 1 inner 285.2.k.c 28
5.c odd 4 1 inner 285.2.k.c 28
15.e even 4 1 inner 285.2.k.c 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.k.c 28 1.a even 1 1 trivial
285.2.k.c 28 3.b odd 2 1 inner
285.2.k.c 28 5.c odd 4 1 inner
285.2.k.c 28 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 115T_{2}^{24} + 4853T_{2}^{20} + 91999T_{2}^{16} + 734303T_{2}^{12} + 1528685T_{2}^{8} + 278139T_{2}^{4} + 1$$ acting on $$S_{2}^{\mathrm{new}}(285, [\chi])$$.