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Properties

Label 225.4.p.a
Level $225$
Weight $4$
Character orbit 225.p
Analytic conductor $13.275$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(32,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.32");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.p (of order \(12\), degree \(4\), 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 84 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 84 q^{6} + 324 q^{11} + 384 q^{16} - 888 q^{21} - 240 q^{31} + 2292 q^{36} - 1584 q^{41} + 936 q^{46} + 3468 q^{51} + 4716 q^{56} + 1644 q^{61} - 15984 q^{66} - 876 q^{76} + 6360 q^{81} - 4212 q^{86} + 2592 q^{91} + 372 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −5.13397 − 1.37564i 4.73932 + 2.13046i 17.5370 + 10.1250i 0 −21.4007 − 17.4573i −4.09967 + 15.3002i −46.0396 − 46.0396i 17.9223 + 20.1939i 0
32.2 −4.34020 − 1.16295i −5.13217 − 0.812923i 10.5566 + 6.09488i 0 21.3292 + 9.49671i 1.29219 − 4.82253i −13.3119 − 13.3119i 25.6783 + 8.34412i 0
32.3 −3.40243 − 0.911680i 2.54356 − 4.53104i 3.81720 + 2.20386i 0 −12.7852 + 13.0976i 8.63156 − 32.2134i 8.94749 + 8.94749i −14.0606 − 23.0500i 0
32.4 −2.86668 − 0.768124i −0.926498 + 5.11289i 0.699616 + 0.403924i 0 6.58330 − 13.9453i −0.804506 + 3.00246i 15.0931 + 15.0931i −25.2832 − 9.47416i 0
32.5 −1.10075 − 0.294946i 5.12492 − 0.857432i −5.80354 − 3.35067i 0 −5.89417 − 0.567753i −4.70619 + 17.5637i 11.8465 + 11.8465i 25.5296 − 8.78854i 0
32.6 −0.987897 − 0.264706i −2.85185 − 4.34361i −6.02233 − 3.47700i 0 1.66756 + 5.04594i −5.20649 + 19.4309i 10.8146 + 10.8146i −10.7339 + 24.7747i 0
32.7 0.987897 + 0.264706i 2.85185 + 4.34361i −6.02233 − 3.47700i 0 1.66756 + 5.04594i 5.20649 − 19.4309i −10.8146 − 10.8146i −10.7339 + 24.7747i 0
32.8 1.10075 + 0.294946i −5.12492 + 0.857432i −5.80354 − 3.35067i 0 −5.89417 − 0.567753i 4.70619 − 17.5637i −11.8465 − 11.8465i 25.5296 − 8.78854i 0
32.9 2.86668 + 0.768124i 0.926498 − 5.11289i 0.699616 + 0.403924i 0 6.58330 − 13.9453i 0.804506 − 3.00246i −15.0931 − 15.0931i −25.2832 − 9.47416i 0
32.10 3.40243 + 0.911680i −2.54356 + 4.53104i 3.81720 + 2.20386i 0 −12.7852 + 13.0976i −8.63156 + 32.2134i −8.94749 − 8.94749i −14.0606 − 23.0500i 0
32.11 4.34020 + 1.16295i 5.13217 + 0.812923i 10.5566 + 6.09488i 0 21.3292 + 9.49671i −1.29219 + 4.82253i 13.3119 + 13.3119i 25.6783 + 8.34412i 0
32.12 5.13397 + 1.37564i −4.73932 − 2.13046i 17.5370 + 10.1250i 0 −21.4007 − 17.4573i 4.09967 − 15.3002i 46.0396 + 46.0396i 17.9223 + 20.1939i 0
68.1 −1.37564 + 5.13397i −2.13046 + 4.73932i −17.5370 − 10.1250i 0 −21.4007 − 17.4573i 15.3002 + 4.09967i 46.0396 − 46.0396i −17.9223 − 20.1939i 0
68.2 −1.16295 + 4.34020i 0.812923 − 5.13217i −10.5566 − 6.09488i 0 21.3292 + 9.49671i −4.82253 − 1.29219i 13.3119 − 13.3119i −25.6783 − 8.34412i 0
68.3 −0.911680 + 3.40243i 4.53104 + 2.54356i −3.81720 − 2.20386i 0 −12.7852 + 13.0976i −32.2134 − 8.63156i −8.94749 + 8.94749i 14.0606 + 23.0500i 0
68.4 −0.768124 + 2.86668i −5.11289 − 0.926498i −0.699616 − 0.403924i 0 6.58330 − 13.9453i 3.00246 + 0.804506i −15.0931 + 15.0931i 25.2832 + 9.47416i 0
68.5 −0.294946 + 1.10075i 0.857432 + 5.12492i 5.80354 + 3.35067i 0 −5.89417 − 0.567753i 17.5637 + 4.70619i −11.8465 + 11.8465i −25.5296 + 8.78854i 0
68.6 −0.264706 + 0.987897i 4.34361 − 2.85185i 6.02233 + 3.47700i 0 1.66756 + 5.04594i 19.4309 + 5.20649i −10.8146 + 10.8146i 10.7339 − 24.7747i 0
68.7 0.264706 − 0.987897i −4.34361 + 2.85185i 6.02233 + 3.47700i 0 1.66756 + 5.04594i −19.4309 − 5.20649i 10.8146 − 10.8146i 10.7339 − 24.7747i 0
68.8 0.294946 − 1.10075i −0.857432 − 5.12492i 5.80354 + 3.35067i 0 −5.89417 − 0.567753i −17.5637 − 4.70619i 11.8465 − 11.8465i −25.5296 + 8.78854i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.p.a ✓ 48
5.b even 2 1 inner 225.4.p.a ✓ 48
5.c odd 4 2 inner 225.4.p.a ✓ 48
9.d odd 6 1 inner 225.4.p.a ✓ 48
45.h odd 6 1 inner 225.4.p.a ✓ 48
45.l even 12 2 inner 225.4.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.p.a ✓ 48 1.a even 1 1 trivial
225.4.p.a ✓ 48 5.b even 2 1 inner
225.4.p.a ✓ 48 5.c odd 4 2 inner
225.4.p.a ✓ 48 9.d odd 6 1 inner
225.4.p.a ✓ 48 45.h odd 6 1 inner
225.4.p.a ✓ 48 45.l even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 1440 T_{2}^{44} + 1453194 T_{2}^{40} - 710512992 T_{2}^{36} + 249100118667 T_{2}^{32} + \cdots + 51\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display