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Properties

Label 285.2.k.d
Level $285$
Weight $2$
Character orbit 285.k
Analytic conductor $2.276$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [285,2,Mod(77,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("285.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.k (of order \(4\), 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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} + 4 q^{6} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} + 4 q^{6} + 4 q^{7} - 4 q^{10} - 18 q^{12} - 8 q^{13} - 8 q^{15} - 84 q^{16} + 8 q^{21} + 40 q^{22} - 20 q^{25} - 14 q^{27} + 36 q^{28} + 28 q^{30} - 28 q^{33} + 92 q^{36} - 4 q^{37} - 20 q^{40} - 100 q^{42} + 16 q^{43} + 28 q^{45} - 24 q^{46} - 58 q^{48} + 32 q^{51} + 148 q^{52} - 72 q^{55} - 2 q^{57} - 12 q^{58} + 58 q^{60} - 112 q^{61} - 64 q^{63} + 92 q^{66} - 8 q^{67} + 8 q^{70} - 88 q^{72} + 76 q^{73} + 80 q^{75} + 36 q^{76} - 36 q^{78} + 4 q^{81} + 20 q^{82} - 28 q^{85} - 4 q^{87} + 140 q^{88} + 76 q^{90} - 24 q^{91} - 48 q^{93} + 32 q^{96} + 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.91639 + 1.91639i −0.520717 − 1.65192i − 5.34506i 0.478119 − 2.18435i 4.16362 + 2.16783i −1.54943 − 1.54943i 6.41043 + 6.41043i −2.45771 + 1.72037i 3.26980 + 5.10232i
77.2 −1.87817 + 1.87817i 1.11989 + 1.32130i − 5.05506i −1.67041 + 1.48652i −4.58498 − 0.378288i 2.61406 + 2.61406i 5.73793 + 5.73793i −0.491688 + 2.95943i 0.345374 − 5.92925i
77.3 −1.53257 + 1.53257i 1.34142 − 1.09572i − 2.69753i 2.05989 + 0.869965i −0.376555 + 3.73508i 1.55058 + 1.55058i 1.06901 + 1.06901i 0.598810 − 2.93963i −4.49021 + 1.82365i
77.4 −1.43829 + 1.43829i −1.69247 − 0.368185i − 2.13738i −2.18737 + 0.464149i 2.96382 − 1.90470i −1.23221 − 1.23221i 0.197591 + 0.197591i 2.72888 + 1.24628i 2.47849 − 3.81366i
77.5 −0.885404 + 0.885404i −0.193756 + 1.72118i 0.432118i −0.370079 + 2.20523i −1.35239 − 1.69549i 0.957875 + 0.957875i −2.15341 − 2.15341i −2.92492 − 0.666976i −1.62485 − 2.28019i
77.6 −0.637774 + 0.637774i −1.73195 − 0.0191055i 1.18649i 1.18060 + 1.89899i 1.11677 − 1.09240i 2.03027 + 2.03027i −2.03226 − 2.03226i 2.99927 + 0.0661793i −1.96409 − 0.458173i
77.7 −0.631220 + 0.631220i −1.41501 + 0.998873i 1.20312i 1.34140 − 1.78903i 0.262674 − 1.52369i −3.10938 − 3.10938i −2.02187 − 2.02187i 1.00451 − 2.82683i 0.282552 + 1.97599i
77.8 −0.538497 + 0.538497i 1.54795 + 0.777077i 1.42004i 1.01787 − 1.99096i −1.25202 + 0.415113i 1.88175 + 1.88175i −1.84168 − 1.84168i 1.79230 + 2.40575i 0.524007 + 1.62025i
77.9 −0.0580413 + 0.0580413i 0.605290 − 1.62284i 1.99326i −1.89399 − 1.18861i 0.0590602 + 0.129324i −2.14350 − 2.14350i −0.231774 − 0.231774i −2.26725 − 1.96458i 0.178918 − 0.0409414i
77.10 0.0580413 − 0.0580413i 1.62284 − 0.605290i 1.99326i 1.89399 + 1.18861i 0.0590602 − 0.129324i −2.14350 − 2.14350i 0.231774 + 0.231774i 2.26725 − 1.96458i 0.178918 − 0.0409414i
77.11 0.538497 − 0.538497i −0.777077 − 1.54795i 1.42004i −1.01787 + 1.99096i −1.25202 − 0.415113i 1.88175 + 1.88175i 1.84168 + 1.84168i −1.79230 + 2.40575i 0.524007 + 1.62025i
77.12 0.631220 − 0.631220i −0.998873 + 1.41501i 1.20312i −1.34140 + 1.78903i 0.262674 + 1.52369i −3.10938 − 3.10938i 2.02187 + 2.02187i −1.00451 − 2.82683i 0.282552 + 1.97599i
77.13 0.637774 − 0.637774i 0.0191055 + 1.73195i 1.18649i −1.18060 − 1.89899i 1.11677 + 1.09240i 2.03027 + 2.03027i 2.03226 + 2.03226i −2.99927 + 0.0661793i −1.96409 − 0.458173i
77.14 0.885404 − 0.885404i −1.72118 + 0.193756i 0.432118i 0.370079 − 2.20523i −1.35239 + 1.69549i 0.957875 + 0.957875i 2.15341 + 2.15341i 2.92492 − 0.666976i −1.62485 − 2.28019i
77.15 1.43829 − 1.43829i 0.368185 + 1.69247i − 2.13738i 2.18737 − 0.464149i 2.96382 + 1.90470i −1.23221 − 1.23221i −0.197591 − 0.197591i −2.72888 + 1.24628i 2.47849 − 3.81366i
77.16 1.53257 − 1.53257i 1.09572 − 1.34142i − 2.69753i −2.05989 − 0.869965i −0.376555 − 3.73508i 1.55058 + 1.55058i −1.06901 − 1.06901i −0.598810 − 2.93963i −4.49021 + 1.82365i
77.17 1.87817 − 1.87817i −1.32130 − 1.11989i − 5.05506i 1.67041 − 1.48652i −4.58498 + 0.378288i 2.61406 + 2.61406i −5.73793 − 5.73793i 0.491688 + 2.95943i 0.345374 − 5.92925i
77.18 1.91639 − 1.91639i 1.65192 + 0.520717i − 5.34506i −0.478119 + 2.18435i 4.16362 − 2.16783i −1.54943 − 1.54943i −6.41043 − 6.41043i 2.45771 + 1.72037i 3.26980 + 5.10232i
248.1 −1.91639 − 1.91639i −0.520717 + 1.65192i 5.34506i 0.478119 + 2.18435i 4.16362 − 2.16783i −1.54943 + 1.54943i 6.41043 − 6.41043i −2.45771 − 1.72037i 3.26980 − 5.10232i
248.2 −1.87817 − 1.87817i 1.11989 − 1.32130i 5.05506i −1.67041 − 1.48652i −4.58498 + 0.378288i 2.61406 − 2.61406i 5.73793 − 5.73793i −0.491688 − 2.95943i 0.345374 + 5.92925i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.k.d ✓ 36
3.b odd 2 1 inner 285.2.k.d ✓ 36
5.c odd 4 1 inner 285.2.k.d ✓ 36
15.e even 4 1 inner 285.2.k.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.k.d ✓ 36 1.a even 1 1 trivial
285.2.k.d ✓ 36 3.b odd 2 1 inner
285.2.k.d ✓ 36 5.c odd 4 1 inner
285.2.k.d ✓ 36 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 147 T_{2}^{32} + 7717 T_{2}^{28} + 174263 T_{2}^{24} + 1640199 T_{2}^{20} + 4869629 T_{2}^{16} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display