# Properties

 Label 555.2.c.c Level $555$ Weight $2$ Character orbit 555.c Analytic conductor $4.432$ Analytic rank $0$ Dimension $26$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [555,2,Mod(334,555)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(555, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("555.334");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$555 = 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 555.c (of order $$2$$, degree $$1$$, 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: $$4.43169731218$$ Analytic rank: $$0$$ Dimension: $$26$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$26 q - 32 q^{4} - 2 q^{5} - 26 q^{9}+O(q^{10})$$ 26 * q - 32 * q^4 - 2 * q^5 - 26 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$26 q - 32 q^{4} - 2 q^{5} - 26 q^{9} + 20 q^{11} - 16 q^{14} + 44 q^{16} - 28 q^{19} + 4 q^{20} + 16 q^{21} + 10 q^{25} + 32 q^{26} - 46 q^{29} - 4 q^{30} + 32 q^{31} + 20 q^{34} - 12 q^{35} + 32 q^{36} - 22 q^{39} - 36 q^{40} + 56 q^{41} - 20 q^{44} + 2 q^{45} + 36 q^{46} - 74 q^{49} - 60 q^{50} + 4 q^{51} - 8 q^{55} + 72 q^{56} + 10 q^{59} - 24 q^{60} + 36 q^{61} - 32 q^{64} - 36 q^{65} + 28 q^{66} + 4 q^{69} - 92 q^{70} + 36 q^{71} + 116 q^{76} - 56 q^{79} - 96 q^{80} + 26 q^{81} - 8 q^{84} - 24 q^{85} + 108 q^{86} - 58 q^{89} + 60 q^{91} - 4 q^{94} - 4 q^{95} - 20 q^{96} - 20 q^{99}+O(q^{100})$$ 26 * q - 32 * q^4 - 2 * q^5 - 26 * q^9 + 20 * q^11 - 16 * q^14 + 44 * q^16 - 28 * q^19 + 4 * q^20 + 16 * q^21 + 10 * q^25 + 32 * q^26 - 46 * q^29 - 4 * q^30 + 32 * q^31 + 20 * q^34 - 12 * q^35 + 32 * q^36 - 22 * q^39 - 36 * q^40 + 56 * q^41 - 20 * q^44 + 2 * q^45 + 36 * q^46 - 74 * q^49 - 60 * q^50 + 4 * q^51 - 8 * q^55 + 72 * q^56 + 10 * q^59 - 24 * q^60 + 36 * q^61 - 32 * q^64 - 36 * q^65 + 28 * q^66 + 4 * q^69 - 92 * q^70 + 36 * q^71 + 116 * q^76 - 56 * q^79 - 96 * q^80 + 26 * q^81 - 8 * q^84 - 24 * q^85 + 108 * q^86 - 58 * q^89 + 60 * q^91 - 4 * q^94 - 4 * q^95 - 20 * q^96 - 20 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
334.1 2.77939i 1.00000i −5.72501 −2.22462 + 0.225945i −2.77939 0.0319162i 10.3533i −1.00000 0.627991 + 6.18310i
334.2 2.55509i 1.00000i −4.52851 2.13933 0.650593i 2.55509 2.23683i 6.46057i −1.00000 −1.66233 5.46619i
334.3 2.40245i 1.00000i −3.77179 −1.28557 + 1.82956i 2.40245 3.30864i 4.25663i −1.00000 4.39544 + 3.08853i
334.4 2.36117i 1.00000i −3.57513 1.66180 + 1.49613i −2.36117 3.86072i 3.71916i −1.00000 3.53262 3.92381i
334.5 2.32415i 1.00000i −3.40166 −2.03612 0.924245i 2.32415 3.18077i 3.25767i −1.00000 −2.14808 + 4.73224i
334.6 1.82554i 1.00000i −1.33258 1.97176 1.05459i −1.82554 4.15424i 1.21840i −1.00000 −1.92520 3.59951i
334.7 1.59785i 1.00000i −0.553137 −1.43895 + 1.71155i −1.59785 4.47204i 2.31188i −1.00000 2.73481 + 2.29923i
334.8 1.46114i 1.00000i −0.134925 1.43182 1.71753i 1.46114 3.04927i 2.72513i −1.00000 −2.50954 2.09208i
334.9 1.33085i 1.00000i 0.228850 1.74423 1.39916i −1.33085 3.76351i 2.96625i −1.00000 −1.86207 2.32130i
334.10 0.919689i 1.00000i 1.15417 −0.388610 2.20204i 0.919689 0.259618i 2.90086i −1.00000 −2.02519 + 0.357401i
334.11 0.550826i 1.00000i 1.69659 −0.808939 + 2.08461i 0.550826 2.51774i 2.03618i −1.00000 1.14826 + 0.445584i
334.12 0.214651i 1.00000i 1.95393 −2.20564 0.367653i −0.214651 4.41925i 0.848712i −1.00000 −0.0789169 + 0.473441i
334.13 0.103900i 1.00000i 1.98920 0.439510 2.19245i −0.103900 0.609052i 0.414478i −1.00000 −0.227795 0.0456650i
334.14 0.103900i 1.00000i 1.98920 0.439510 + 2.19245i −0.103900 0.609052i 0.414478i −1.00000 −0.227795 + 0.0456650i
334.15 0.214651i 1.00000i 1.95393 −2.20564 + 0.367653i −0.214651 4.41925i 0.848712i −1.00000 −0.0789169 0.473441i
334.16 0.550826i 1.00000i 1.69659 −0.808939 2.08461i 0.550826 2.51774i 2.03618i −1.00000 1.14826 0.445584i
334.17 0.919689i 1.00000i 1.15417 −0.388610 + 2.20204i 0.919689 0.259618i 2.90086i −1.00000 −2.02519 0.357401i
334.18 1.33085i 1.00000i 0.228850 1.74423 + 1.39916i −1.33085 3.76351i 2.96625i −1.00000 −1.86207 + 2.32130i
334.19 1.46114i 1.00000i −0.134925 1.43182 + 1.71753i 1.46114 3.04927i 2.72513i −1.00000 −2.50954 + 2.09208i
334.20 1.59785i 1.00000i −0.553137 −1.43895 1.71155i −1.59785 4.47204i 2.31188i −1.00000 2.73481 2.29923i
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 334.26 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.c.c 26
3.b odd 2 1 1665.2.c.f 26
5.b even 2 1 inner 555.2.c.c 26
5.c odd 4 1 2775.2.a.bg 13
5.c odd 4 1 2775.2.a.bh 13
15.d odd 2 1 1665.2.c.f 26
15.e even 4 1 8325.2.a.cu 13
15.e even 4 1 8325.2.a.cv 13

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.c 26 1.a even 1 1 trivial
555.2.c.c 26 5.b even 2 1 inner
1665.2.c.f 26 3.b odd 2 1
1665.2.c.f 26 15.d odd 2 1
2775.2.a.bg 13 5.c odd 4 1
2775.2.a.bh 13 5.c odd 4 1
8325.2.a.cu 13 15.e even 4 1
8325.2.a.cv 13 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{26} + 42 T_{2}^{24} + 771 T_{2}^{22} + 8130 T_{2}^{20} + 54435 T_{2}^{18} + 241528 T_{2}^{16} + 719422 T_{2}^{14} + 1425812 T_{2}^{12} + 1821683 T_{2}^{10} + 1407938 T_{2}^{8} + 583595 T_{2}^{6} + \cdots + 36$$ acting on $$S_{2}^{\mathrm{new}}(555, [\chi])$$.