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Properties

Label 476.3.s.a
Level $476$
Weight $3$
Character orbit 476.s
Analytic conductor $12.970$
Analytic rank $0$
Dimension $44$
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] = [476,3,Mod(341,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.341");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 476.s (of order \(6\), degree \(2\), 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: \(12.9700605836\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 6 q^{3} + 22 q^{7} + 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 6 q^{3} + 22 q^{7} + 88 q^{9} + 8 q^{11} + 16 q^{15} - 54 q^{19} - 16 q^{21} - 52 q^{23} + 150 q^{25} + 8 q^{29} + 78 q^{31} - 6 q^{33} + 10 q^{35} - 34 q^{37} - 60 q^{39} - 76 q^{43} - 72 q^{45} - 6 q^{47} - 56 q^{49} - 172 q^{53} - 64 q^{57} + 30 q^{59} + 444 q^{61} + 206 q^{63} - 54 q^{65} - 56 q^{67} + 204 q^{71} - 48 q^{73} + 132 q^{75} - 494 q^{77} - 16 q^{79} - 342 q^{81} + 594 q^{87} - 252 q^{89} + 284 q^{91} - 146 q^{93} + 148 q^{95} - 180 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 0 −4.95885 + 2.86300i 0 −7.68849 − 4.43895i 0 4.17226 + 5.62070i 0 11.8935 − 20.6001i 0
341.2 0 −4.95032 + 2.85807i 0 2.87495 + 1.65985i 0 −4.68127 + 5.20440i 0 11.8371 − 20.5025i 0
341.3 0 −4.26291 + 2.46119i 0 3.82411 + 2.20785i 0 6.67632 − 2.10397i 0 7.61492 − 13.1894i 0
341.4 0 −3.34426 + 1.93081i 0 6.37515 + 3.68069i 0 −6.40953 − 2.81388i 0 2.95607 − 5.12006i 0
341.5 0 −3.19262 + 1.84326i 0 −4.85650 − 2.80390i 0 −2.56019 − 6.51502i 0 2.29523 − 3.97546i 0
341.6 0 −3.14781 + 1.81739i 0 −1.12892 − 0.651780i 0 3.26437 − 6.19224i 0 2.10578 − 3.64733i 0
341.7 0 −2.50400 + 1.44568i 0 4.60744 + 2.66011i 0 5.50382 + 4.32527i 0 −0.319995 + 0.554248i 0
341.8 0 −2.25567 + 1.30231i 0 −5.00973 − 2.89237i 0 −6.80652 + 1.63441i 0 −1.10796 + 1.91905i 0
341.9 0 −1.92844 + 1.11339i 0 −1.23951 − 0.715633i 0 0.974006 + 6.93191i 0 −2.02074 + 3.50002i 0
341.10 0 −0.565149 + 0.326289i 0 1.39043 + 0.802767i 0 −5.00309 + 4.89583i 0 −4.28707 + 7.42543i 0
341.11 0 −0.354936 + 0.204922i 0 −3.41185 − 1.96983i 0 6.69953 − 2.02888i 0 −4.41601 + 7.64876i 0
341.12 0 −0.113042 + 0.0652646i 0 6.14108 + 3.54555i 0 −1.76537 − 6.77373i 0 −4.49148 + 7.77947i 0
341.13 0 0.557083 − 0.321632i 0 5.89169 + 3.40157i 0 3.87708 + 5.82823i 0 −4.29311 + 7.43588i 0
341.14 0 1.00421 − 0.579780i 0 −6.38618 − 3.68706i 0 6.83125 − 1.52774i 0 −3.82771 + 6.62979i 0
341.15 0 1.91865 − 1.10773i 0 −0.540464 − 0.312037i 0 −1.03671 − 6.92281i 0 −2.04587 + 3.54354i 0
341.16 0 2.37662 − 1.37214i 0 −2.65296 − 1.53169i 0 0.400909 + 6.98851i 0 −0.734465 + 1.27213i 0
341.17 0 2.58559 − 1.49279i 0 −8.32032 − 4.80374i 0 −6.17085 − 3.30463i 0 −0.0431496 + 0.0747374i 0
341.18 0 2.95906 − 1.70841i 0 6.58749 + 3.80329i 0 6.95862 − 0.760040i 0 1.33734 − 2.31634i 0
341.19 0 3.90995 − 2.25741i 0 2.01827 + 1.16525i 0 6.73594 − 1.90448i 0 5.69180 − 9.85849i 0
341.20 0 4.00062 − 2.30976i 0 7.29155 + 4.20978i 0 −5.51440 + 4.31178i 0 6.16996 − 10.6867i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.3.s.a ✓ 44
7.d odd 6 1 inner 476.3.s.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.3.s.a ✓ 44 1.a even 1 1 trivial
476.3.s.a ✓ 44 7.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(476, [\chi])\).