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Properties

Label 1336.4.a.a
Level $1336$
Weight $4$
Character orbit 1336.a
Self dual yes
Analytic conductor $78.827$
Analytic rank $1$
Dimension $29$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1336,4,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1336.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1336.a (trivial)

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: yes
Analytic conductor: \(78.8265517677\)
Analytic rank: \(1\)
Dimension: \(29\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 9 q^{3} + 12 q^{5} - 82 q^{7} + 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - 9 q^{3} + 12 q^{5} - 82 q^{7} + 206 q^{9} - 96 q^{11} + 99 q^{13} - 102 q^{15} - 150 q^{17} - 215 q^{19} - 123 q^{21} - 401 q^{23} + 713 q^{25} - 546 q^{27} - 647 q^{29} - 267 q^{31} - 316 q^{33} - 819 q^{35} + 1095 q^{37} - 720 q^{39} - 648 q^{41} - 589 q^{43} + 1192 q^{45} - 991 q^{47} + 865 q^{49} - 1404 q^{51} - 39 q^{53} - 1687 q^{55} - 411 q^{57} - 1014 q^{59} + 290 q^{61} - 2714 q^{63} - 2249 q^{65} - 2183 q^{67} - 177 q^{69} - 2763 q^{71} - 582 q^{73} - 1971 q^{75} - 465 q^{77} - 1630 q^{79} + 413 q^{81} - 1731 q^{83} - 110 q^{85} - 4204 q^{87} - 4857 q^{89} - 4150 q^{91} - 343 q^{93} - 6832 q^{95} - 3448 q^{97} - 7567 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −9.91770 0 18.7264 0 −7.09412 0 71.3607 0
1.2 0 −9.52701 0 −1.09838 0 −30.9883 0 63.7638 0
1.3 0 −8.56216 0 4.26658 0 30.8391 0 46.3106 0
1.4 0 −8.46648 0 9.24173 0 13.9709 0 44.6813 0
1.5 0 −7.17186 0 −19.8687 0 −14.3697 0 24.4355 0
1.6 0 −6.62821 0 −6.20223 0 −1.28095 0 16.9331 0
1.7 0 −5.51183 0 10.9992 0 −24.9833 0 3.38022 0
1.8 0 −5.07733 0 17.2750 0 −15.6278 0 −1.22074 0
1.9 0 −4.89907 0 −12.1467 0 31.2121 0 −2.99908 0
1.10 0 −3.39859 0 −17.9886 0 −13.5585 0 −15.4496 0
1.11 0 −2.70350 0 −12.9211 0 −33.6442 0 −19.6911 0
1.12 0 −2.45610 0 −12.7811 0 5.69432 0 −20.9676 0
1.13 0 −1.41762 0 3.42632 0 9.85385 0 −24.9904 0
1.14 0 −1.35842 0 18.5805 0 −20.2367 0 −25.1547 0
1.15 0 −1.00048 0 −7.81175 0 8.99990 0 −25.9990 0
1.16 0 0.112792 0 5.64261 0 21.0567 0 −26.9873 0
1.17 0 0.352167 0 11.5300 0 16.5105 0 −26.8760 0
1.18 0 2.25098 0 −2.48081 0 −9.96878 0 −21.9331 0
1.19 0 2.28813 0 16.2612 0 8.59596 0 −21.7644 0
1.20 0 3.64490 0 11.4553 0 −3.39781 0 −13.7147 0
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(167\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1336.4.a.a ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.4.a.a ✓ 29 1.a even 1 1 trivial