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Properties

Label 363.2.j.a
Level $363$
Weight $2$
Character orbit 363.j
Analytic conductor $2.899$
Analytic rank $0$
Dimension $420$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(32,363)]
 
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([11, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.j (of order \(22\), 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: \(420\)
Relative dimension: \(42\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 420 q - 22 q^{3} - 58 q^{4} + 11 q^{6} - 22 q^{7} - 18 q^{9} - 44 q^{10} - 5 q^{12} - 44 q^{13} + 36 q^{15} - 70 q^{16} + 11 q^{18} - 22 q^{19} - 44 q^{21} + 22 q^{22} - 66 q^{24} + 24 q^{25} - 10 q^{27}+ \cdots + 99 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.387754 + 2.69689i −0.434595 − 1.67664i −5.20386 − 1.52799i 0.517216 − 0.236205i 4.69023 − 0.521929i 2.68392 + 4.17627i 3.87494 − 8.48493i −2.62225 + 1.45732i 0.436465 + 1.48646i
32.2 −0.365833 + 2.54443i −0.176717 + 1.72301i −4.42128 − 1.29820i −2.49959 + 1.14152i −4.31943 − 1.07998i 0.379383 + 0.590331i 2.78491 − 6.09811i −2.93754 − 0.608971i −1.99009 − 6.77762i
32.3 −0.361342 + 2.51319i 1.32607 − 1.11424i −4.26657 − 1.25278i −3.13442 + 1.43144i 2.32114 + 3.73529i −1.50991 − 2.34946i 2.58065 − 5.65084i 0.516928 − 2.95513i −2.46489 − 8.39464i
32.4 −0.358461 + 2.49315i −1.73073 + 0.0675632i −4.16833 − 1.22393i 1.25213 − 0.571829i 0.451955 − 4.33920i −1.20271 − 1.87145i 2.45295 − 5.37121i 2.99087 − 0.233868i 0.976817 + 3.32673i
32.5 −0.316621 + 2.20215i 1.63910 + 0.559788i −2.83022 − 0.831029i −0.318850 + 0.145614i −1.75171 + 3.43229i −0.363417 − 0.565488i 0.877733 − 1.92197i 2.37327 + 1.83509i −0.219709 − 0.748259i
32.6 −0.311426 + 2.16602i 0.202912 − 1.72012i −2.67566 − 0.785646i 1.74484 − 0.796842i 3.66263 + 0.975202i −2.13578 − 3.32334i 0.716897 − 1.56979i −2.91765 − 0.698066i 1.18258 + 4.02751i
32.7 −0.308207 + 2.14363i −0.853328 + 1.50726i −2.58117 − 0.757901i 3.98329 − 1.81911i −2.96801 − 2.29377i 1.74278 + 2.71182i 0.620889 − 1.35956i −1.54366 − 2.57237i 2.67181 + 9.09936i
32.8 −0.266170 + 1.85125i 1.32649 + 1.11374i −1.43731 − 0.422032i −0.671943 + 0.306866i −2.41490 + 2.15922i 2.09860 + 3.26549i −0.390038 + 0.854064i 0.519146 + 2.95474i −0.389236 − 1.32562i
32.9 −0.265003 + 1.84314i −1.46622 − 0.922059i −1.40795 − 0.413412i 0.0491386 − 0.0224408i 2.08804 − 2.45810i 0.549764 + 0.855450i −0.411996 + 0.902145i 1.29961 + 2.70389i 0.0283397 + 0.0965161i
32.10 −0.257411 + 1.79033i −1.68358 + 0.406907i −1.22005 − 0.358238i −3.36319 + 1.53592i −0.295129 − 3.11890i 1.56775 + 2.43946i −0.547339 + 1.19851i 2.66885 − 1.37012i −1.88408 − 6.41659i
32.11 −0.254274 + 1.76851i −0.803137 + 1.53459i −1.14400 − 0.335909i −0.290233 + 0.132545i −2.50973 − 1.81057i −2.20243 − 3.42705i −0.599495 + 1.31271i −1.70994 − 2.46497i −0.160609 − 0.546985i
32.12 −0.213105 + 1.48218i 1.43375 − 0.971775i −0.232453 − 0.0682543i 2.11238 − 0.964690i 1.13480 + 2.33217i 1.03013 + 1.60291i −1.09340 + 2.39421i 1.11130 − 2.78658i 0.979685 + 3.33650i
32.13 −0.209582 + 1.45768i 0.609230 − 1.62137i −0.161914 − 0.0475423i −2.60528 + 1.18979i 2.23575 + 1.22787i 1.45075 + 2.25741i −1.12030 + 2.45312i −2.25768 − 1.97557i −1.18831 − 4.04702i
32.14 −0.193551 + 1.34618i 1.09302 + 1.34362i 0.144252 + 0.0423561i 3.40713 − 1.55598i −2.02031 + 1.21134i −2.14471 − 3.33724i −1.21489 + 2.66023i −0.610623 + 2.93720i 1.43518 + 4.88776i
32.15 −0.154971 + 1.07784i −0.801675 − 1.53536i 0.781254 + 0.229397i −1.70392 + 0.778155i 1.77911 − 0.626146i −1.90542 − 2.96490i −1.27304 + 2.78757i −1.71463 + 2.46171i −0.574672 − 1.95715i
32.16 −0.124578 + 0.866456i −1.03984 − 1.38518i 1.18376 + 0.347583i 2.78717 − 1.27286i 1.32974 − 0.728412i 0.384572 + 0.598405i −1.17592 + 2.57490i −0.837471 + 2.88074i 0.755657 + 2.57353i
32.17 −0.0828482 + 0.576222i 0.253347 + 1.71342i 1.59382 + 0.467987i 0.422364 − 0.192887i −1.00830 + 0.00403037i 0.481188 + 0.748744i −0.885375 + 1.93870i −2.87163 + 0.868182i 0.0761537 + 0.259356i
32.18 −0.0793462 + 0.551865i 1.63436 − 0.573475i 1.62073 + 0.475888i −0.495271 + 0.226183i 0.186801 + 0.947448i −1.05566 − 1.64263i −0.854445 + 1.87097i 2.34225 − 1.87453i −0.0855244 − 0.291270i
32.19 −0.0650821 + 0.452656i 0.599616 + 1.62495i 1.71832 + 0.504546i −3.22999 + 1.47508i −0.774567 + 0.165664i −0.550400 − 0.856439i −0.720165 + 1.57694i −2.28092 + 1.94869i −0.457491 − 1.55807i
32.20 −0.0577953 + 0.401975i −1.37405 + 1.05451i 1.76074 + 0.517001i 0.766302 − 0.349959i −0.344473 − 0.613279i 1.94086 + 3.02004i −0.646991 + 1.41671i 0.776017 − 2.89790i 0.0963859 + 0.328260i
See next 80 embeddings (of 420 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
121.f odd 22 1 inner
363.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.j.a ✓ 420
3.b odd 2 1 inner 363.2.j.a ✓ 420
121.f odd 22 1 inner 363.2.j.a ✓ 420
363.j even 22 1 inner 363.2.j.a ✓ 420
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.j.a ✓ 420 1.a even 1 1 trivial
363.2.j.a ✓ 420 3.b odd 2 1 inner
363.2.j.a ✓ 420 121.f odd 22 1 inner
363.2.j.a ✓ 420 363.j even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(363, [\chi])\).