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Properties

Label 804.2.y.a
Level $804$
Weight $2$
Character orbit 804.y
Analytic conductor $6.420$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [804,2,Mod(49,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 0, 46]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.y (of order \(33\), degree \(20\), 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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(5\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q - 10 q^{3} + 2 q^{5} - 3 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q - 10 q^{3} + 2 q^{5} - 3 q^{7} - 10 q^{9} - 13 q^{11} - 3 q^{13} - 9 q^{15} - 44 q^{17} - 16 q^{19} - 3 q^{21} - 16 q^{23} + 28 q^{25} - 10 q^{27} - 7 q^{29} + 20 q^{31} - 2 q^{33} - 19 q^{35} - 22 q^{37} - 3 q^{39} - 14 q^{41} - 27 q^{43} + 2 q^{45} + 4 q^{47} - 92 q^{49} + 22 q^{51} + 8 q^{53} - 13 q^{55} + 17 q^{57} + 22 q^{59} + 17 q^{61} - 3 q^{63} + 56 q^{65} - 14 q^{67} + 17 q^{69} - q^{71} + 26 q^{73} + 28 q^{75} + 112 q^{77} + 69 q^{79} - 10 q^{81} + 15 q^{83} + 69 q^{85} + 4 q^{87} + 73 q^{89} - 40 q^{91} - 13 q^{93} + 59 q^{95} + 29 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 0.841254 + 0.540641i 0 −0.185691 − 1.29151i 0 −3.04666 − 0.290921i 0 0.415415 + 0.909632i 0
49.2 0 0.841254 + 0.540641i 0 −0.144003 − 1.00156i 0 −0.962860 − 0.0919419i 0 0.415415 + 0.909632i 0
49.3 0 0.841254 + 0.540641i 0 −0.138288 − 0.961813i 0 4.23021 + 0.403936i 0 0.415415 + 0.909632i 0
49.4 0 0.841254 + 0.540641i 0 0.237705 + 1.65327i 0 −3.35218 − 0.320095i 0 0.415415 + 0.909632i 0
49.5 0 0.841254 + 0.540641i 0 0.513618 + 3.57229i 0 −0.127258 − 0.0121517i 0 0.415415 + 0.909632i 0
73.1 0 0.415415 − 0.909632i 0 −4.17942 − 1.22719i 0 1.04002 − 0.200448i 0 −0.654861 − 0.755750i 0
73.2 0 0.415415 − 0.909632i 0 −1.28965 − 0.378675i 0 −0.228496 + 0.0440390i 0 −0.654861 − 0.755750i 0
73.3 0 0.415415 − 0.909632i 0 −0.620947 − 0.182327i 0 −2.64850 + 0.510456i 0 −0.654861 − 0.755750i 0
73.4 0 0.415415 − 0.909632i 0 0.921315 + 0.270523i 0 1.11642 − 0.215173i 0 −0.654861 − 0.755750i 0
73.5 0 0.415415 − 0.909632i 0 3.28440 + 0.964386i 0 2.82322 − 0.544130i 0 −0.654861 − 0.755750i 0
121.1 0 −0.654861 + 0.755750i 0 −2.40665 + 1.54666i 0 1.93826 + 0.775962i 0 −0.142315 − 0.989821i 0
121.2 0 −0.654861 + 0.755750i 0 −1.55544 + 0.999620i 0 −0.0528822 − 0.0211708i 0 −0.142315 − 0.989821i 0
121.3 0 −0.654861 + 0.755750i 0 −0.236859 + 0.152220i 0 −1.28374 − 0.513932i 0 −0.142315 − 0.989821i 0
121.4 0 −0.654861 + 0.755750i 0 2.57269 − 1.65337i 0 −1.89045 − 0.756823i 0 −0.142315 − 0.989821i 0
121.5 0 −0.654861 + 0.755750i 0 3.18825 − 2.04896i 0 4.41412 + 1.76715i 0 −0.142315 − 0.989821i 0
157.1 0 −0.142315 + 0.989821i 0 −0.926989 − 2.02982i 0 2.57416 − 2.45446i 0 −0.959493 − 0.281733i 0
157.2 0 −0.142315 + 0.989821i 0 −0.793528 − 1.73758i 0 −0.366312 + 0.349278i 0 −0.959493 − 0.281733i 0
157.3 0 −0.142315 + 0.989821i 0 0.336386 + 0.736583i 0 −0.860971 + 0.820934i 0 −0.959493 − 0.281733i 0
157.4 0 −0.142315 + 0.989821i 0 0.716491 + 1.56890i 0 0.876788 − 0.836015i 0 −0.959493 − 0.281733i 0
157.5 0 −0.142315 + 0.989821i 0 1.26894 + 2.77859i 0 −3.37248 + 3.21566i 0 −0.959493 − 0.281733i 0
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.y.a ✓ 100
67.g even 33 1 inner 804.2.y.a ✓ 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.y.a ✓ 100 1.a even 1 1 trivial
804.2.y.a ✓ 100 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{100} - 2 T_{5}^{99} + 13 T_{5}^{98} + 49 T_{5}^{97} + 40 T_{5}^{96} + 287 T_{5}^{95} + \cdots + 25\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\). Copy content Toggle raw display