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Properties

Label 333.3.bb.c
Level $333$
Weight $3$
Character orbit 333.bb
Analytic conductor $9.074$
Analytic rank $0$
Dimension $24$
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] = [333,3,Mod(82,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.bb (of order \(12\), 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: \(9.07359280320\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 111)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 18 q^{4} - 8 q^{5} + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 18 q^{4} - 8 q^{5} + 36 q^{8} + 36 q^{10} + 4 q^{13} + 6 q^{14} + 26 q^{16} - 46 q^{17} - 156 q^{19} - 68 q^{20} + 32 q^{22} - 10 q^{23} - 90 q^{25} + 164 q^{26} - 48 q^{28} + 28 q^{29} + 120 q^{31} - 90 q^{32} + 46 q^{34} + 186 q^{35} + 56 q^{37} - 4 q^{38} + 216 q^{40} + 30 q^{41} + 250 q^{43} - 284 q^{44} - 18 q^{46} - 232 q^{47} - 164 q^{49} - 226 q^{50} - 488 q^{52} - 122 q^{53} - 250 q^{55} + 632 q^{56} - 360 q^{58} + 258 q^{59} + 108 q^{61} + 186 q^{62} - 162 q^{65} + 60 q^{67} - 214 q^{68} + 246 q^{70} - 174 q^{71} + 8 q^{74} - 498 q^{76} - 666 q^{77} - 104 q^{79} + 24 q^{80} + 1114 q^{82} + 26 q^{83} + 72 q^{86} - 334 q^{88} - 16 q^{89} + 20 q^{91} + 918 q^{92} + 400 q^{94} + 372 q^{95} - 278 q^{97} - 950 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −3.24096 − 0.868412i 0 6.28557 + 3.62898i −4.12843 + 1.10621i 0 −1.96785 + 3.40842i −7.72965 − 7.72965i 0 14.3407
82.2 −1.42776 − 0.382568i 0 −1.57196 − 0.907569i −4.56711 + 1.22375i 0 5.29119 − 9.16460i 6.07795 + 6.07795i 0 6.98891
82.3 −1.26888 − 0.339995i 0 −1.96965 − 1.13718i 6.74563 − 1.80749i 0 −3.45169 + 5.97850i 5.82814 + 5.82814i 0 −9.17391
82.4 1.49686 + 0.401082i 0 −1.38439 − 0.799276i 1.42641 − 0.382207i 0 −0.233753 + 0.404872i −6.13476 − 6.13476i 0 2.28844
82.5 2.77265 + 0.742930i 0 3.67155 + 2.11977i −5.87356 + 1.57382i 0 −4.19208 + 7.26090i 0.486192 + 0.486192i 0 −17.4546
82.6 3.53411 + 0.946963i 0 8.12913 + 4.69335i 1.79898 − 0.482035i 0 6.28624 − 10.8881i 13.9362 + 13.9362i 0 6.81426
199.1 −3.24096 + 0.868412i 0 6.28557 − 3.62898i −4.12843 − 1.10621i 0 −1.96785 − 3.40842i −7.72965 + 7.72965i 0 14.3407
199.2 −1.42776 + 0.382568i 0 −1.57196 + 0.907569i −4.56711 − 1.22375i 0 5.29119 + 9.16460i 6.07795 − 6.07795i 0 6.98891
199.3 −1.26888 + 0.339995i 0 −1.96965 + 1.13718i 6.74563 + 1.80749i 0 −3.45169 − 5.97850i 5.82814 − 5.82814i 0 −9.17391
199.4 1.49686 − 0.401082i 0 −1.38439 + 0.799276i 1.42641 + 0.382207i 0 −0.233753 − 0.404872i −6.13476 + 6.13476i 0 2.28844
199.5 2.77265 − 0.742930i 0 3.67155 − 2.11977i −5.87356 − 1.57382i 0 −4.19208 − 7.26090i 0.486192 − 0.486192i 0 −17.4546
199.6 3.53411 − 0.946963i 0 8.12913 − 4.69335i 1.79898 + 0.482035i 0 6.28624 + 10.8881i 13.9362 − 13.9362i 0 6.81426
208.1 −0.890494 − 3.32337i 0 −6.78769 + 3.91888i 0.668655 − 2.49545i 0 0.861050 + 1.49138i 9.33677 + 9.33677i 0 −8.88874
208.2 −0.368956 − 1.37696i 0 1.70421 − 0.983925i −2.49694 + 9.31872i 0 −4.83740 − 8.37862i −6.01563 − 6.01563i 0 13.7528
208.3 −0.154588 − 0.576931i 0 3.15515 − 1.82163i 1.83083 − 6.83276i 0 −0.127116 − 0.220172i −3.22807 − 3.22807i 0 −4.22506
208.4 0.236181 + 0.881438i 0 2.74295 − 1.58364i −0.925992 + 3.45585i 0 3.14793 + 5.45238i 4.62474 + 4.62474i 0 −3.26482
208.5 0.482343 + 1.80013i 0 0.456292 − 0.263440i 0.389542 − 1.45379i 0 −5.52162 − 9.56373i 5.96546 + 5.96546i 0 2.80490
208.6 0.829488 + 3.09569i 0 −5.43116 + 3.13568i 1.13198 − 4.22462i 0 4.74511 + 8.21876i −5.14737 − 5.14737i 0 14.0171
325.1 −0.890494 + 3.32337i 0 −6.78769 − 3.91888i 0.668655 + 2.49545i 0 0.861050 − 1.49138i 9.33677 − 9.33677i 0 −8.88874
325.2 −0.368956 + 1.37696i 0 1.70421 + 0.983925i −2.49694 − 9.31872i 0 −4.83740 + 8.37862i −6.01563 + 6.01563i 0 13.7528
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.3.bb.c 24
3.b odd 2 1 111.3.l.a ✓ 24
37.g odd 12 1 inner 333.3.bb.c 24
111.m even 12 1 111.3.l.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.l.a ✓ 24 3.b odd 2 1
111.3.l.a ✓ 24 111.m even 12 1
333.3.bb.c 24 1.a even 1 1 trivial
333.3.bb.c 24 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 4 T_{2}^{23} - T_{2}^{22} + 8 T_{2}^{21} - 114 T_{2}^{20} + 598 T_{2}^{19} - 35 T_{2}^{18} + \cdots + 2866249 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display