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Properties

Label 324.5.d.f
Level $324$
Weight $5$
Character orbit 324.d
Analytic conductor $33.492$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,5,Mod(163,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.163");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 36)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + q^{2} + q^{4} + 2 q^{5} + 61 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + q^{2} + q^{4} + 2 q^{5} + 61 q^{8} + 14 q^{10} + 2 q^{13} - 252 q^{14} + q^{16} - 28 q^{17} + 140 q^{20} + 33 q^{22} + 1752 q^{25} + 548 q^{26} - 258 q^{28} - 526 q^{29} + 121 q^{32} - 385 q^{34} - 4 q^{37} - 1395 q^{38} + 2276 q^{40} + 2762 q^{41} + 3357 q^{44} + 1788 q^{46} - 3428 q^{49} - 6375 q^{50} - 1438 q^{52} - 5044 q^{53} + 7506 q^{56} + 4064 q^{58} + 2 q^{61} - 9162 q^{62} + 4513 q^{64} + 2014 q^{65} + 11405 q^{68} - 3666 q^{70} - 1708 q^{73} - 14620 q^{74} - 1581 q^{76} + 3942 q^{77} + 22760 q^{80} - 4243 q^{82} + 1252 q^{85} - 22113 q^{86} - 1995 q^{88} + 6524 q^{89} + 30294 q^{92} - 7524 q^{94} - 5638 q^{97} - 46469 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −3.92469 − 0.772521i 0 14.8064 + 6.06382i 21.1512 0 44.6184i −53.4262 − 35.2369i 0 −83.0119 − 16.3397i
163.2 −3.92469 + 0.772521i 0 14.8064 − 6.06382i 21.1512 0 − 44.6184i −53.4262 + 35.2369i 0 −83.0119 + 16.3397i
163.3 −3.49753 − 1.94095i 0 8.46540 + 13.5771i −33.2277 0 46.1602i −3.25547 − 63.9171i 0 116.215 + 64.4935i
163.4 −3.49753 + 1.94095i 0 8.46540 − 13.5771i −33.2277 0 − 46.1602i −3.25547 + 63.9171i 0 116.215 − 64.4935i
163.5 −3.19553 − 2.40595i 0 4.42285 + 15.3766i −2.03090 0 − 23.1347i 22.8618 − 59.7774i 0 6.48981 + 4.88624i
163.6 −3.19553 + 2.40595i 0 4.42285 − 15.3766i −2.03090 0 23.1347i 22.8618 + 59.7774i 0 6.48981 − 4.88624i
163.7 −1.75474 − 3.59457i 0 −9.84180 + 12.6150i 46.6931 0 − 60.5483i 62.6153 + 13.2409i 0 −81.9341 − 167.841i
163.8 −1.75474 + 3.59457i 0 −9.84180 − 12.6150i 46.6931 0 60.5483i 62.6153 − 13.2409i 0 −81.9341 + 167.841i
163.9 −1.16626 − 3.82620i 0 −13.2797 + 8.92473i −39.0789 0 − 12.2052i 49.6354 + 40.4021i 0 45.5763 + 149.524i
163.10 −1.16626 + 3.82620i 0 −13.2797 − 8.92473i −39.0789 0 12.2052i 49.6354 − 40.4021i 0 45.5763 − 149.524i
163.11 −0.328300 − 3.98650i 0 −15.7844 + 2.61754i 5.66182 0 52.1456i 15.6169 + 62.0654i 0 −1.85878 − 22.5709i
163.12 −0.328300 + 3.98650i 0 −15.7844 − 2.61754i 5.66182 0 − 52.1456i 15.6169 − 62.0654i 0 −1.85878 + 22.5709i
163.13 1.34653 − 3.76654i 0 −12.3737 − 10.1435i 11.0316 0 − 11.9152i −54.8676 + 32.9476i 0 14.8544 − 41.5509i
163.14 1.34653 + 3.76654i 0 −12.3737 + 10.1435i 11.0316 0 11.9152i −54.8676 − 32.9476i 0 14.8544 + 41.5509i
163.15 2.43802 − 3.17113i 0 −4.11208 − 15.4626i −22.1491 0 − 95.5959i −59.0591 − 24.6582i 0 −54.0000 + 70.2376i
163.16 2.43802 + 3.17113i 0 −4.11208 + 15.4626i −22.1491 0 95.5959i −59.0591 + 24.6582i 0 −54.0000 − 70.2376i
163.17 2.82463 − 2.83222i 0 −0.0429591 − 15.9999i 11.7888 0 58.3756i −45.4367 − 45.0722i 0 33.2988 − 33.3884i
163.18 2.82463 + 2.83222i 0 −0.0429591 + 15.9999i 11.7888 0 − 58.3756i −45.4367 + 45.0722i 0 33.2988 + 33.3884i
163.19 3.76125 − 1.36126i 0 12.2940 − 10.2400i −28.6092 0 25.6487i 32.3013 − 55.2506i 0 −107.606 + 38.9445i
163.20 3.76125 + 1.36126i 0 12.2940 + 10.2400i −28.6092 0 − 25.6487i 32.3013 + 55.2506i 0 −107.606 − 38.9445i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.5.d.f 22
3.b odd 2 1 324.5.d.e 22
4.b odd 2 1 inner 324.5.d.f 22
9.c even 3 2 36.5.f.a ✓ 44
9.d odd 6 2 108.5.f.a 44
12.b even 2 1 324.5.d.e 22
36.f odd 6 2 36.5.f.a ✓ 44
36.h even 6 2 108.5.f.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.f.a ✓ 44 9.c even 3 2
36.5.f.a ✓ 44 36.f odd 6 2
108.5.f.a 44 9.d odd 6 2
108.5.f.a 44 36.h even 6 2
324.5.d.e 22 3.b odd 2 1
324.5.d.e 22 12.b even 2 1
324.5.d.f 22 1.a even 1 1 trivial
324.5.d.f 22 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{11} - T_{5}^{10} - 3875 T_{5}^{9} + 491 T_{5}^{8} + 4952203 T_{5}^{7} - 10600531 T_{5}^{6} + \cdots + 36175408350208 \) acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display