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Properties

Label 363.2.i.b
Level $363$
Weight $2$
Character orbit 363.i
Analytic conductor $2.899$
Analytic rank $0$
Dimension $110$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(34,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.i (of order \(11\), degree \(10\), 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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(110\)
Relative dimension: \(11\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 110 q - q^{2} - 110 q^{3} - 9 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + 110 q^{9} + 13 q^{10} + 10 q^{11} + 9 q^{12} + 24 q^{13} + 20 q^{14} - 2 q^{15} - 5 q^{16} + 2 q^{17} - q^{18} - 4 q^{20} + 4 q^{21}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −0.372443 − 2.59040i −1.00000 −4.65246 + 1.36609i −0.603037 + 1.32047i 0.372443 + 2.59040i 1.26395 + 0.812294i 3.09717 + 6.78185i 1.00000 3.64513 + 1.07031i
34.2 −0.333445 − 2.31916i −1.00000 −3.34834 + 0.983160i 0.934002 − 2.04518i 0.333445 + 2.31916i −0.690758 − 0.443923i 1.44995 + 3.17495i 1.00000 −5.05454 − 1.48415i
34.3 −0.234121 − 1.62835i −1.00000 −0.677711 + 0.198994i −0.779539 + 1.70695i 0.234121 + 1.62835i 2.23438 + 1.43595i −0.884093 − 1.93589i 1.00000 2.96201 + 0.869726i
34.4 −0.229572 − 1.59670i −1.00000 −0.577778 + 0.169651i −1.04500 + 2.28823i 0.229572 + 1.59670i −3.38186 − 2.17339i −0.936708 − 2.05111i 1.00000 3.89353 + 1.14324i
34.5 −0.118218 − 0.822222i −1.00000 1.25691 − 0.369063i 1.32647 − 2.90456i 0.118218 + 0.822222i 3.25231 + 2.09013i −1.14219 − 2.50105i 1.00000 −2.54501 − 0.747282i
34.6 3.16538e−5 0 0.000220157i −1.00000 1.91899 − 0.563465i 0.860954 − 1.88523i −3.16538e−5 0 0.000220157i −1.38748 − 0.891679i 0.000369588 0 0.000809285i 1.00000 0.000442298 0 0.000129870i
34.7 0.119152 + 0.828718i −1.00000 1.24641 − 0.365979i 0.0551712 − 0.120808i −0.119152 − 0.828718i −1.17983 − 0.758230i 1.14741 + 2.51247i 1.00000 0.106690 + 0.0313269i
34.8 0.134483 + 0.935349i −1.00000 1.06219 − 0.311888i −0.623940 + 1.36624i −0.134483 − 0.935349i 1.94368 + 1.24912i 1.21968 + 2.67072i 1.00000 −1.36182 − 0.399866i
34.9 0.296211 + 2.06019i −1.00000 −2.23765 + 0.657034i 0.454482 − 0.995176i −0.296211 − 2.06019i 3.71384 + 2.38674i −0.287163 − 0.628800i 1.00000 2.18487 + 0.641536i
34.10 0.319232 + 2.22031i −1.00000 −2.90888 + 0.854124i −1.74740 + 3.82626i −0.319232 − 2.22031i −2.15950 − 1.38783i −0.961360 − 2.10509i 1.00000 −9.05331 − 2.65829i
34.11 0.321558 + 2.23649i −1.00000 −2.97948 + 0.874855i 1.71191 − 3.74856i −0.321558 − 2.23649i −2.16567 − 1.39179i −1.03743 − 2.27166i 1.00000 8.93409 + 2.62328i
67.1 −2.49126 + 0.731501i −1.00000 3.98879 − 2.56344i 1.82020 + 2.10062i 2.49126 − 0.731501i −0.304924 − 0.667691i −4.66135 + 5.37948i 1.00000 −6.07119 − 3.90172i
67.2 −2.29718 + 0.674512i −1.00000 3.13954 − 2.01766i −2.16455 − 2.49802i 2.29718 − 0.674512i −1.83607 − 4.02043i −2.71547 + 3.13382i 1.00000 6.65729 + 4.27838i
67.3 −1.38165 + 0.405688i −1.00000 0.0618561 − 0.0397525i −0.813654 − 0.939007i 1.38165 − 0.405688i 1.03641 + 2.26941i 1.81663 − 2.09650i 1.00000 1.50513 + 0.967285i
67.4 −1.25314 + 0.367954i −1.00000 −0.247549 + 0.159090i 2.56030 + 2.95474i 1.25314 − 0.367954i 1.41216 + 3.09219i 1.96222 − 2.26452i 1.00000 −4.29561 − 2.76062i
67.5 −0.950338 + 0.279044i −1.00000 −0.857231 + 0.550909i −0.579522 − 0.668804i 0.950338 − 0.279044i −0.911908 − 1.99680i 1.95816 − 2.25983i 1.00000 0.737367 + 0.473877i
67.6 −0.0202291 + 0.00593980i −1.00000 −1.68213 + 1.08104i −1.16759 − 1.34747i 0.0202291 − 0.00593980i 0.759230 + 1.66248i 0.0552199 − 0.0637271i 1.00000 0.0316229 + 0.0203228i
67.7 0.621164 − 0.182390i −1.00000 −1.32993 + 0.854693i 0.973544 + 1.12353i −0.621164 + 0.182390i −0.181234 − 0.396846i −1.51811 + 1.75200i 1.00000 0.809652 + 0.520332i
67.8 1.48096 − 0.434848i −1.00000 0.321634 − 0.206702i 1.52565 + 1.76069i −1.48096 + 0.434848i 0.157288 + 0.344413i −1.63508 + 1.88699i 1.00000 3.02505 + 1.94408i
67.9 1.69108 − 0.496547i −1.00000 0.930702 − 0.598126i −1.35902 − 1.56839i −1.69108 + 0.496547i −2.05690 − 4.50398i −1.03146 + 1.19036i 1.00000 −3.07700 − 1.97747i
See next 80 embeddings (of 110 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.i.b ✓ 110
121.e even 11 1 inner 363.2.i.b ✓ 110
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.i.b ✓ 110 1.a even 1 1 trivial
363.2.i.b ✓ 110 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{110} + T_{2}^{109} + 16 T_{2}^{108} + 16 T_{2}^{107} + 165 T_{2}^{106} + 231 T_{2}^{105} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display