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Properties

Label 804.2.y.b
Level $804$
Weight $2$
Character orbit 804.y
Analytic conductor $6.420$
Analytic rank $0$
Dimension $120$
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: \(120\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 12 q^{3} - 2 q^{5} + q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 12 q^{3} - 2 q^{5} + q^{7} - 12 q^{9} + 11 q^{11} + 2 q^{13} - 9 q^{15} + 48 q^{17} - 4 q^{19} - q^{21} + 22 q^{23} - 42 q^{25} + 12 q^{27} - q^{29} + 27 q^{31} + 17 q^{35} - 8 q^{37} - 2 q^{39} - 58 q^{41} - 17 q^{43} - 2 q^{45} - 84 q^{47} + 101 q^{49} - 26 q^{51} + 28 q^{53} - 9 q^{55} + 26 q^{57} + 34 q^{59} + 16 q^{61} + 12 q^{63} + 144 q^{65} + 23 q^{67} + 11 q^{69} + 173 q^{71} - 2 q^{73} + 42 q^{75} + 128 q^{77} + 31 q^{79} - 12 q^{81} + 47 q^{83} - 75 q^{85} - 10 q^{87} - 67 q^{89} + 16 q^{91} + 6 q^{93} - 79 q^{95} + 10 q^{97}+O(q^{100}) \) 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 −0.597458 − 4.15542i 0 3.22256 + 0.307717i 0 0.415415 + 0.909632i 0
49.2 0 −0.841254 − 0.540641i 0 −0.355125 − 2.46995i 0 1.47905 + 0.141233i 0 0.415415 + 0.909632i 0
49.3 0 −0.841254 − 0.540641i 0 −0.166640 − 1.15901i 0 −3.32963 − 0.317941i 0 0.415415 + 0.909632i 0
49.4 0 −0.841254 − 0.540641i 0 −0.102535 − 0.713146i 0 −0.326230 − 0.0311512i 0 0.415415 + 0.909632i 0
49.5 0 −0.841254 − 0.540641i 0 0.370568 + 2.57736i 0 5.06200 + 0.483363i 0 0.415415 + 0.909632i 0
49.6 0 −0.841254 − 0.540641i 0 0.567849 + 3.94948i 0 −4.15279 − 0.396544i 0 0.415415 + 0.909632i 0
73.1 0 −0.415415 + 0.909632i 0 −3.13336 − 0.920039i 0 −2.56141 + 0.493671i 0 −0.654861 − 0.755750i 0
73.2 0 −0.415415 + 0.909632i 0 −1.98933 − 0.584119i 0 1.45645 − 0.280708i 0 −0.654861 − 0.755750i 0
73.3 0 −0.415415 + 0.909632i 0 0.443767 + 0.130302i 0 4.44269 − 0.856259i 0 −0.654861 − 0.755750i 0
73.4 0 −0.415415 + 0.909632i 0 1.03511 + 0.303934i 0 −4.65735 + 0.897630i 0 −0.654861 − 0.755750i 0
73.5 0 −0.415415 + 0.909632i 0 2.10734 + 0.618772i 0 0.474282 − 0.0914103i 0 −0.654861 − 0.755750i 0
73.6 0 −0.415415 + 0.909632i 0 3.42078 + 1.00443i 0 −0.977853 + 0.188466i 0 −0.654861 − 0.755750i 0
121.1 0 0.654861 − 0.755750i 0 −2.40809 + 1.54759i 0 −2.21597 − 0.887140i 0 −0.142315 − 0.989821i 0
121.2 0 0.654861 − 0.755750i 0 −2.07356 + 1.33260i 0 4.33337 + 1.73482i 0 −0.142315 − 0.989821i 0
121.3 0 0.654861 − 0.755750i 0 −1.40224 + 0.901162i 0 −1.99913 − 0.800331i 0 −0.142315 − 0.989821i 0
121.4 0 0.654861 − 0.755750i 0 −0.192294 + 0.123580i 0 −3.39999 − 1.36115i 0 −0.142315 − 0.989821i 0
121.5 0 0.654861 − 0.755750i 0 2.24994 − 1.44595i 0 2.21379 + 0.886268i 0 −0.142315 − 0.989821i 0
121.6 0 0.654861 − 0.755750i 0 2.26427 − 1.45516i 0 −0.275857 − 0.110437i 0 −0.142315 − 0.989821i 0
157.1 0 0.142315 − 0.989821i 0 −1.67835 − 3.67507i 0 2.54620 − 2.42780i 0 −0.959493 − 0.281733i 0
157.2 0 0.142315 − 0.989821i 0 −1.19502 − 2.61673i 0 −1.34977 + 1.28701i 0 −0.959493 − 0.281733i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
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.b ✓ 120
67.g even 33 1 inner 804.2.y.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.y.b ✓ 120 1.a even 1 1 trivial
804.2.y.b ✓ 120 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{120} + 2 T_{5}^{119} + 53 T_{5}^{118} + 107 T_{5}^{117} + 1616 T_{5}^{116} + \cdots + 18\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\). Copy content Toggle raw display