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Properties

Label 806.2.bg.a
Level $806$
Weight $2$
Character orbit 806.bg
Analytic conductor $6.436$
Analytic rank $0$
Dimension $152$
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] = [806,2,Mod(37,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([7, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bg (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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q + 12 q^{7} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q + 12 q^{7} + 80 q^{9} + 12 q^{11} + 4 q^{12} + 4 q^{14} + 76 q^{16} - 24 q^{17} - 32 q^{18} - 4 q^{19} - 8 q^{21} + 12 q^{22} - 12 q^{28} + 12 q^{29} + 36 q^{31} + 28 q^{33} - 24 q^{34} - 4 q^{35} + 4 q^{37} + 72 q^{38} + 20 q^{39} + 12 q^{41} - 32 q^{43} + 48 q^{45} + 24 q^{46} - 8 q^{47} - 24 q^{51} - 4 q^{52} - 12 q^{53} - 32 q^{57} + 12 q^{58} + 24 q^{59} + 12 q^{61} - 36 q^{62} + 4 q^{63} + 132 q^{65} + 32 q^{66} - 52 q^{67} + 48 q^{70} + 8 q^{71} + 32 q^{72} + 4 q^{73} + 184 q^{75} + 28 q^{76} - 96 q^{77} + 12 q^{78} + 12 q^{79} - 60 q^{81} + 96 q^{83} + 8 q^{84} + 24 q^{85} - 48 q^{86} - 128 q^{87} + 12 q^{89} + 44 q^{91} - 28 q^{93} - 8 q^{94} - 124 q^{97} - 8 q^{98} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.258819 + 0.965926i −2.78210 − 1.60625i −0.866025 − 0.500000i −0.701020 + 0.187838i 2.27158 − 2.27158i −0.938932 + 0.938932i 0.707107 − 0.707107i 3.66007 + 6.33943i − 0.725749i
37.2 −0.258819 + 0.965926i −2.72436 − 1.57291i −0.866025 − 0.500000i 2.26153 − 0.605976i 2.22443 − 2.22443i 2.75818 − 2.75818i 0.707107 − 0.707107i 3.44809 + 5.97226i 2.34131i
37.3 −0.258819 + 0.965926i −2.37283 − 1.36996i −0.866025 − 0.500000i 1.00527 − 0.269362i 1.93741 − 1.93741i −3.39925 + 3.39925i 0.707107 − 0.707107i 2.25356 + 3.90327i 1.04073i
37.4 −0.258819 + 0.965926i −2.19153 − 1.26528i −0.866025 − 0.500000i −3.34281 + 0.895702i 1.78937 − 1.78937i 3.04291 − 3.04291i 0.707107 − 0.707107i 1.70186 + 2.94771i − 3.46073i
37.5 −0.258819 + 0.965926i −1.46861 − 0.847900i −0.866025 − 0.500000i −3.37075 + 0.903191i 1.19911 − 1.19911i −2.51540 + 2.51540i 0.707107 − 0.707107i −0.0621321 − 0.107616i − 3.48966i
37.6 −0.258819 + 0.965926i −1.34485 − 0.776451i −0.866025 − 0.500000i 0.803265 − 0.215234i 1.09807 − 1.09807i −0.501959 + 0.501959i 0.707107 − 0.707107i −0.294248 − 0.509652i 0.831601i
37.7 −0.258819 + 0.965926i −1.06993 − 0.617724i −0.866025 − 0.500000i −1.75382 + 0.469935i 0.873593 − 0.873593i 1.65566 − 1.65566i 0.707107 − 0.707107i −0.736835 − 1.27624i − 1.81569i
37.8 −0.258819 + 0.965926i −0.976653 − 0.563871i −0.866025 − 0.500000i 2.81194 − 0.753457i 0.797434 − 0.797434i 1.87822 − 1.87822i 0.707107 − 0.707107i −0.864099 − 1.49666i 2.91114i
37.9 −0.258819 + 0.965926i −0.580469 − 0.335134i −0.866025 − 0.500000i −2.47425 + 0.662974i 0.473951 − 0.473951i −0.893199 + 0.893199i 0.707107 − 0.707107i −1.27537 − 2.20901i − 2.56154i
37.10 −0.258819 + 0.965926i −0.383819 − 0.221598i −0.866025 − 0.500000i 2.32802 − 0.623790i 0.313387 − 0.313387i 2.20062 − 2.20062i 0.707107 − 0.707107i −1.40179 − 2.42797i 2.41014i
37.11 −0.258819 + 0.965926i 0.228163 + 0.131730i −0.866025 − 0.500000i 2.12110 − 0.568348i −0.186294 + 0.186294i −1.87568 + 1.87568i 0.707107 − 0.707107i −1.46529 − 2.53796i 2.19593i
37.12 −0.258819 + 0.965926i 0.679748 + 0.392453i −0.866025 − 0.500000i 0.370280 − 0.0992163i −0.555012 + 0.555012i −1.12076 + 1.12076i 0.707107 − 0.707107i −1.19196 − 2.06454i 0.383343i
37.13 −0.258819 + 0.965926i 1.06401 + 0.614308i −0.866025 − 0.500000i −2.12740 + 0.570034i −0.868762 + 0.868762i 1.50784 − 1.50784i 0.707107 − 0.707107i −0.745252 − 1.29081i − 2.20244i
37.14 −0.258819 + 0.965926i 1.49857 + 0.865202i −0.866025 − 0.500000i 3.63689 − 0.974502i −1.22358 + 1.22358i 1.50844 − 1.50844i 0.707107 − 0.707107i −0.00285144 − 0.00493885i 3.76519i
37.15 −0.258819 + 0.965926i 1.91490 + 1.10557i −0.866025 − 0.500000i −1.85824 + 0.497915i −1.56351 + 1.56351i 1.66442 − 1.66442i 0.707107 − 0.707107i 0.944565 + 1.63603i − 1.92380i
37.16 −0.258819 + 0.965926i 1.95726 + 1.13003i −0.866025 − 0.500000i −0.321923 + 0.0862589i −1.59810 + 1.59810i −2.68179 + 2.68179i 0.707107 − 0.707107i 1.05392 + 1.82543i − 0.333279i
37.17 −0.258819 + 0.965926i 2.12148 + 1.22484i −0.866025 − 0.500000i 0.625984 − 0.167732i −1.73218 + 1.73218i 2.58646 − 2.58646i 0.707107 − 0.707107i 1.50046 + 2.59887i 0.648066i
37.18 −0.258819 + 0.965926i 2.66627 + 1.53937i −0.866025 − 0.500000i 4.07349 − 1.09149i −2.17700 + 2.17700i −1.73659 + 1.73659i 0.707107 − 0.707107i 3.23934 + 5.61070i 4.21718i
37.19 −0.258819 + 0.965926i 2.89871 + 1.67357i −0.866025 − 0.500000i −4.08756 + 1.09526i −2.36679 + 2.36679i −0.0660392 + 0.0660392i 0.707107 − 0.707107i 4.10168 + 7.10432i − 4.23175i
37.20 0.258819 − 0.965926i −2.64314 − 1.52602i −0.866025 − 0.500000i −2.42522 + 0.649836i −2.15812 + 2.15812i 1.96589 − 1.96589i −0.707107 + 0.707107i 3.15747 + 5.46889i 2.51077i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bg.a yes 152
13.f odd 12 1 806.2.ba.a ✓ 152
31.e odd 6 1 806.2.ba.a ✓ 152
403.bf even 12 1 inner 806.2.bg.a yes 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.ba.a ✓ 152 13.f odd 12 1
806.2.ba.a ✓ 152 31.e odd 6 1
806.2.bg.a yes 152 1.a even 1 1 trivial
806.2.bg.a yes 152 403.bf even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).