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Properties

Label 363.2.m.b
Level $363$
Weight $2$
Character orbit 363.m
Analytic conductor $2.899$
Analytic rank $0$
Dimension $440$
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(4,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.m (of order \(55\), degree \(40\), 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: \(440\)
Relative dimension: \(11\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q + 3 q^{2} - 110 q^{3} + 7 q^{4} + q^{5} + 3 q^{6} + 3 q^{7} + 5 q^{8} - 110 q^{9} - 13 q^{10} - 20 q^{11} - 38 q^{12} - 13 q^{13} - 34 q^{14} - 4 q^{15} - 3 q^{16} - 2 q^{17} - 2 q^{18} + 10 q^{19}+ \cdots - 15 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.76262 − 0.157973i 0.309017 − 0.951057i 5.62017 + 0.644855i −1.50308 − 2.84865i −1.00394 + 2.57860i −1.68962 − 0.714040i −9.97133 − 1.72561i −0.809017 − 0.587785i 3.70243 + 8.10719i
4.2 −2.16144 − 0.123596i 0.309017 − 0.951057i 2.66960 + 0.306308i 1.04679 + 1.98389i −0.785470 + 2.01746i 4.13263 + 1.74646i −1.46580 − 0.253667i −0.809017 − 0.587785i −2.01738 − 4.41744i
4.3 −1.89926 − 0.108603i 0.309017 − 0.951057i 1.60842 + 0.184549i 0.610127 + 1.15632i −0.690191 + 1.77274i −2.44208 − 1.03203i 0.714239 + 0.123604i −0.809017 − 0.587785i −1.03321 − 2.26241i
4.4 −1.60187 − 0.0915982i 0.309017 − 0.951057i 0.570630 + 0.0654737i −1.42300 − 2.69689i −0.582120 + 1.49516i −0.631217 − 0.266755i 2.25389 + 0.390052i −0.809017 − 0.587785i 2.03243 + 4.45041i
4.5 −0.133309 − 0.00762291i 0.309017 − 0.951057i −1.96925 − 0.225950i 0.618248 + 1.17171i −0.0484447 + 0.124429i −3.55860 − 1.50388i 0.523940 + 0.0906714i −0.809017 − 0.587785i −0.0734864 − 0.160913i
4.6 0.253301 + 0.0144843i 0.309017 − 0.951057i −1.92301 − 0.220645i 0.851566 + 1.61390i 0.0920495 − 0.236427i 2.44186 + 1.03194i −0.983901 − 0.170271i −0.809017 − 0.587785i 0.192326 + 0.421135i
4.7 0.718602 + 0.0410911i 0.309017 − 0.951057i −1.47226 − 0.168927i −1.11463 − 2.11246i 0.261140 − 0.670733i 0.776409 + 0.328113i −2.46950 − 0.427363i −0.809017 − 0.587785i −0.714171 − 1.56382i
4.8 1.70429 + 0.0974551i 0.309017 − 0.951057i 0.908160 + 0.104202i −1.52620 − 2.89248i 0.619341 − 1.59077i −2.06439 − 0.872419i −1.82654 − 0.316095i −0.809017 − 0.587785i −2.31922 − 5.07837i
4.9 1.85734 + 0.106207i 0.309017 − 0.951057i 1.45149 + 0.166542i 1.41316 + 2.67824i 0.674960 − 1.73362i 2.13291 + 0.901375i −0.988042 − 0.170987i −0.809017 − 0.587785i 2.34028 + 5.12450i
4.10 2.20895 + 0.126313i 0.309017 − 0.951057i 2.87656 + 0.330055i −0.205284 − 0.389057i 0.802735 − 2.06181i 1.81373 + 0.766487i 1.95219 + 0.337840i −0.809017 − 0.587785i −0.404321 − 0.885340i
4.11 2.65688 + 0.151926i 0.309017 − 0.951057i 5.04897 + 0.579315i 0.113625 + 0.215343i 0.965511 − 2.47990i −2.63737 − 1.11456i 8.08200 + 1.39865i −0.809017 − 0.587785i 0.269171 + 0.589403i
16.1 −2.27448 − 0.260972i −0.809017 − 0.587785i 3.15711 + 0.734155i −1.54808 + 2.26399i 1.68669 + 1.54803i 2.04619 + 2.10548i −2.67667 − 0.955034i 0.309017 + 0.951057i 4.11190 − 4.74538i
16.2 −2.09801 − 0.240724i −0.809017 − 0.587785i 2.39568 + 0.557092i −0.401344 + 0.586947i 1.55583 + 1.42793i −3.13041 − 3.22111i −0.914137 − 0.326163i 0.309017 + 0.951057i 0.983317 − 1.13481i
16.3 −1.76698 − 0.202742i −0.809017 − 0.587785i 1.13309 + 0.263488i 1.23964 − 1.81292i 1.31035 + 1.20263i 0.373520 + 0.384344i 1.40156 + 0.500075i 0.309017 + 0.951057i −2.55797 + 2.95206i
16.4 −0.619472 − 0.0710778i −0.809017 − 0.587785i −1.56933 − 0.364932i 1.40786 − 2.05892i 0.459385 + 0.421620i 2.71195 + 2.79053i 2.12077 + 0.756688i 0.309017 + 0.951057i −1.01847 + 1.17538i
16.5 −0.511920 − 0.0587373i −0.809017 − 0.587785i −1.68941 − 0.392856i −2.08546 + 3.04988i 0.379627 + 0.348418i −1.38226 − 1.42231i 1.81239 + 0.646660i 0.309017 + 0.951057i 1.24673 − 1.43880i
16.6 0.163465 + 0.0187558i −0.809017 − 0.587785i −1.92165 − 0.446862i −0.218207 + 0.319118i −0.121221 − 0.111256i 0.717419 + 0.738207i −0.615678 − 0.219674i 0.309017 + 0.951057i −0.0416545 + 0.0480719i
16.7 0.987264 + 0.113278i −0.809017 − 0.587785i −0.986166 − 0.229323i −1.18196 + 1.72857i −0.732130 − 0.671943i −0.0954404 − 0.0982058i −2.81953 − 1.00601i 0.309017 + 0.951057i −1.36272 + 1.57266i
16.8 1.46026 + 0.167549i −0.809017 − 0.587785i 0.156250 + 0.0363344i 2.05290 − 3.00227i −1.08289 − 0.993867i −1.18215 − 1.21641i −2.54664 − 0.908638i 0.309017 + 0.951057i 3.50079 − 4.04013i
16.9 2.05705 + 0.236024i −0.809017 − 0.587785i 2.22770 + 0.518030i −2.33388 + 3.41320i −1.52545 − 1.40005i 2.70226 + 2.78056i 0.559962 + 0.199794i 0.309017 + 0.951057i −5.60650 + 6.47025i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.m.b ✓ 440
121.g even 55 1 inner 363.2.m.b ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.m.b ✓ 440 1.a even 1 1 trivial
363.2.m.b ✓ 440 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{440} - 3 T_{2}^{439} - 10 T_{2}^{438} + 40 T_{2}^{437} + T_{2}^{436} - 74 T_{2}^{435} + \cdots + 12\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display