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Properties

Label 335.2.c.a
Level $335$
Weight $2$
Character orbit 335.c
Analytic conductor $2.675$
Analytic rank $0$
Dimension $32$
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] = [335,2,Mod(269,335)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(335, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("335.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 335 = 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 335.c (of order \(2\), degree \(1\), 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: \(2.67498846771\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 34 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 34 q^{4} - 32 q^{9} + 8 q^{10} + 4 q^{11} - 4 q^{14} + 2 q^{15} + 30 q^{16} - 4 q^{19} - 18 q^{20} - 16 q^{21} + 28 q^{24} - 4 q^{25} + 8 q^{26} + 12 q^{29} - 6 q^{30} - 28 q^{34} - 2 q^{35} + 50 q^{36} + 24 q^{39} - 2 q^{40} + 16 q^{41} + 8 q^{44} - 16 q^{45} - 32 q^{46} + 8 q^{49} - 12 q^{50} + 12 q^{51} - 4 q^{54} - 52 q^{55} + 4 q^{56} + 8 q^{59} - 6 q^{60} - 8 q^{61} - 66 q^{64} - 14 q^{65} + 52 q^{66} - 40 q^{69} + 34 q^{70} - 8 q^{71} - 12 q^{74} - 18 q^{75} + 52 q^{76} + 36 q^{79} + 44 q^{80} + 8 q^{81} - 14 q^{84} + 18 q^{85} + 6 q^{86} + 52 q^{89} - 4 q^{90} + 12 q^{91} - 20 q^{95} - 86 q^{96} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 − 2.77079i − 2.55374i −5.67727 2.21627 − 0.296886i −7.07589 0.0699520i 10.1889i −3.52161 −0.822609 − 6.14082i
269.2 − 2.69484i 1.01920i −5.26216 −0.888582 + 2.05193i 2.74659 0.980605i 8.79100i 1.96122 5.52963 + 2.39459i
269.3 − 2.45545i 2.29476i −4.02922 −0.269019 − 2.21983i 5.63467 − 1.29043i 4.98266i −2.26594 −5.45067 + 0.660562i
269.4 − 2.14774i − 2.85755i −2.61279 −2.18688 + 0.466423i −6.13729 1.24960i 1.31612i −5.16561 1.00176 + 4.69685i
269.5 − 2.07922i 0.227863i −2.32316 2.22225 + 0.248169i 0.473778 2.59066i 0.671930i 2.94808 0.515999 − 4.62056i
269.6 − 2.05256i − 1.12640i −2.21302 0.324659 − 2.21237i −2.31201 − 2.14569i 0.437233i 1.73122 −4.54104 − 0.666383i
269.7 − 1.93637i − 1.26216i −1.74953 −0.458794 + 2.18849i −2.44401 − 4.21095i − 0.485003i 1.40695 4.23774 + 0.888394i
269.8 − 1.91846i 3.36335i −1.68048 1.71715 + 1.43227i 6.45244 1.63623i − 0.612985i −8.31212 2.74776 − 3.29427i
269.9 − 1.64551i 1.89995i −0.707696 −2.20662 + 0.361682i 3.12639 − 3.99393i − 2.12650i −0.609819 0.595150 + 3.63102i
269.10 − 1.09236i − 0.598616i 0.806746 −0.486680 + 2.18246i −0.653905 4.52852i − 3.06598i 2.64166 2.38404 + 0.531631i
269.11 − 0.964275i 1.11119i 1.07017 1.69214 − 1.46173i 1.07149 − 0.972995i − 2.96049i 1.76525 −1.40951 − 1.63169i
269.12 − 0.948466i − 2.83343i 1.10041 1.83124 + 1.28318i −2.68742 − 0.473827i − 2.94064i −5.02835 1.21705 − 1.73687i
269.13 − 0.933303i 2.21443i 1.12895 −1.03102 − 1.98418i 2.06674 4.60360i − 2.92025i −1.90372 −1.85184 + 0.962257i
269.14 − 0.809225i − 0.662149i 1.34516 −2.06040 − 0.868755i −0.535827 − 0.0170165i − 2.70698i 2.56156 −0.703018 + 1.66733i
269.15 − 0.414598i 1.68109i 1.82811 1.11943 + 1.93569i 0.696978 − 3.14198i − 1.58713i 0.173920 0.802532 − 0.464112i
269.16 − 0.155588i − 2.71711i 1.97579 −1.53513 − 1.62585i −0.422749 − 1.84017i − 0.618584i −4.38270 −0.252961 + 0.238847i
269.17 0.155588i 2.71711i 1.97579 −1.53513 + 1.62585i −0.422749 1.84017i 0.618584i −4.38270 −0.252961 − 0.238847i
269.18 0.414598i − 1.68109i 1.82811 1.11943 − 1.93569i 0.696978 3.14198i 1.58713i 0.173920 0.802532 + 0.464112i
269.19 0.809225i 0.662149i 1.34516 −2.06040 + 0.868755i −0.535827 0.0170165i 2.70698i 2.56156 −0.703018 − 1.66733i
269.20 0.933303i − 2.21443i 1.12895 −1.03102 + 1.98418i 2.06674 − 4.60360i 2.92025i −1.90372 −1.85184 − 0.962257i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 335.2.c.a ✓ 32
5.b even 2 1 inner 335.2.c.a ✓ 32
5.c odd 4 1 1675.2.a.o 16
5.c odd 4 1 1675.2.a.p 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
335.2.c.a ✓ 32 1.a even 1 1 trivial
335.2.c.a ✓ 32 5.b even 2 1 inner
1675.2.a.o 16 5.c odd 4 1
1675.2.a.p 16 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(335, [\chi])\).