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Properties

Label 363.4.d.d
Level $363$
Weight $4$
Character orbit 363.d
Analytic conductor $21.418$
Analytic rank $0$
Dimension $40$
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] = [363,4,Mod(362,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.362");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.d (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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 33)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{3} + 132 q^{4} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{3} + 132 q^{4} - 60 q^{9} - 90 q^{12} - 212 q^{15} + 460 q^{16} + 296 q^{25} + 280 q^{27} - 320 q^{31} - 188 q^{34} - 2254 q^{36} + 796 q^{37} + 1240 q^{42} - 1662 q^{45} - 486 q^{48} + 1536 q^{49} + 2268 q^{58} - 6636 q^{60} - 1512 q^{64} + 4556 q^{67} - 1140 q^{69} - 4276 q^{70} + 3462 q^{75} - 7640 q^{78} + 3572 q^{81} + 9408 q^{82} - 816 q^{91} - 11516 q^{93} - 7916 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1 −5.40291 −0.843368 + 5.12725i 21.1915 7.39188i 4.55664 − 27.7021i − 14.1673i −71.2724 −25.5775 − 8.64832i − 39.9377i
362.2 −5.40291 −0.843368 − 5.12725i 21.1915 − 7.39188i 4.55664 + 27.7021i 14.1673i −71.2724 −25.5775 + 8.64832i 39.9377i
362.3 −4.73579 −3.47601 + 3.86230i 14.4277 − 3.60263i 16.4617 − 18.2910i 29.9863i −30.4401 −2.83468 − 26.8508i 17.0613i
362.4 −4.73579 −3.47601 − 3.86230i 14.4277 3.60263i 16.4617 + 18.2910i − 29.9863i −30.4401 −2.83468 + 26.8508i − 17.0613i
362.5 −4.13050 2.49192 + 4.55964i 9.06107 0.648224i −10.2929 − 18.8336i − 14.4739i −4.38274 −14.5806 + 22.7245i − 2.67749i
362.6 −4.13050 2.49192 − 4.55964i 9.06107 − 0.648224i −10.2929 + 18.8336i 14.4739i −4.38274 −14.5806 − 22.7245i 2.67749i
362.7 −4.08863 5.14531 + 0.725123i 8.71694 6.78488i −21.0373 − 2.96476i − 18.8082i −2.93129 25.9484 + 7.46197i − 27.7409i
362.8 −4.08863 5.14531 − 0.725123i 8.71694 − 6.78488i −21.0373 + 2.96476i 18.8082i −2.93129 25.9484 − 7.46197i 27.7409i
362.9 −3.67560 −1.47344 + 4.98287i 5.51006 17.2142i 5.41579 − 18.3150i 16.8899i 9.15204 −22.6579 − 14.6839i − 63.2724i
362.10 −3.67560 −1.47344 − 4.98287i 5.51006 − 17.2142i 5.41579 + 18.3150i − 16.8899i 9.15204 −22.6579 + 14.6839i 63.2724i
362.11 −2.85553 −4.97590 + 1.49680i 0.154061 5.17068i 14.2088 − 4.27416i 16.0801i 22.4043 22.5192 − 14.8959i − 14.7651i
362.12 −2.85553 −4.97590 − 1.49680i 0.154061 − 5.17068i 14.2088 + 4.27416i − 16.0801i 22.4043 22.5192 + 14.8959i 14.7651i
362.13 −1.60481 −4.98378 − 1.47035i −5.42459 18.2497i 7.99801 + 2.35963i − 4.32675i 21.5439 22.6761 + 14.6558i − 29.2872i
362.14 −1.60481 −4.98378 + 1.47035i −5.42459 − 18.2497i 7.99801 − 2.35963i 4.32675i 21.5439 22.6761 − 14.6558i 29.2872i
362.15 −1.45083 4.65440 − 2.31010i −5.89509 − 15.3838i −6.75274 + 3.35155i 13.5090i 20.1594 16.3269 − 21.5042i 22.3193i
362.16 −1.45083 4.65440 + 2.31010i −5.89509 15.3838i −6.75274 − 3.35155i − 13.5090i 20.1594 16.3269 + 21.5042i − 22.3193i
362.17 −1.05192 −1.20902 + 5.05354i −6.89347 − 12.2895i 1.27179 − 5.31590i 22.4006i 15.6667 −24.0765 − 12.2197i 12.9275i
362.18 −1.05192 −1.20902 − 5.05354i −6.89347 12.2895i 1.27179 + 5.31590i − 22.4006i 15.6667 −24.0765 + 12.2197i − 12.9275i
362.19 −0.389720 2.66989 − 4.45777i −7.84812 4.25677i −1.04051 + 1.73728i − 11.6631i 6.17632 −12.7434 − 23.8035i − 1.65895i
362.20 −0.389720 2.66989 + 4.45777i −7.84812 − 4.25677i −1.04051 − 1.73728i 11.6631i 6.17632 −12.7434 + 23.8035i 1.65895i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.d.d 40
3.b odd 2 1 inner 363.4.d.d 40
11.b odd 2 1 inner 363.4.d.d 40
11.c even 5 1 33.4.f.a ✓ 40
11.d odd 10 1 33.4.f.a ✓ 40
33.d even 2 1 inner 363.4.d.d 40
33.f even 10 1 33.4.f.a ✓ 40
33.h odd 10 1 33.4.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.f.a ✓ 40 11.c even 5 1
33.4.f.a ✓ 40 11.d odd 10 1
33.4.f.a ✓ 40 33.f even 10 1
33.4.f.a ✓ 40 33.h odd 10 1
363.4.d.d 40 1.a even 1 1 trivial
363.4.d.d 40 3.b odd 2 1 inner
363.4.d.d 40 11.b odd 2 1 inner
363.4.d.d 40 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 113 T_{2}^{18} + 5291 T_{2}^{16} - 133194 T_{2}^{14} + 1954379 T_{2}^{12} - 16940617 T_{2}^{10} + \cdots + 18740480 \) acting on \(S_{4}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display