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Properties

Label 52.4.f.b
Level $52$
Weight $4$
Character orbit 52.f
Analytic conductor $3.068$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [52,4,Mod(31,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 52.f (of order \(4\), degree \(2\), 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: \(3.06809932030\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 20 q^{5} - 16 q^{6} - 40 q^{8} - 332 q^{9} - 96 q^{13} + 204 q^{14} + 52 q^{16} + 330 q^{18} + 500 q^{20} + 224 q^{21} - 48 q^{22} - 496 q^{24} + 570 q^{26} - 356 q^{28} + 560 q^{29} + 292 q^{32}+ \cdots + 3226 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.82787 − 0.0560904i 6.97823i 7.99371 + 0.317233i 3.73459 − 3.73459i 0.391412 − 19.7335i −18.7167 + 18.7167i −22.5874 − 1.34546i −21.6957 −10.7704 + 10.3515i
31.2 −2.81956 − 0.223782i − 1.00320i 7.89984 + 1.26193i −4.98300 + 4.98300i −0.224497 + 2.82857i 16.5820 − 16.5820i −21.9917 − 5.32593i 25.9936 15.1650 − 12.9348i
31.3 −2.50920 − 1.30534i − 9.31194i 4.59218 + 6.55072i 11.9568 − 11.9568i −12.1552 + 23.3655i −6.41108 + 6.41108i −2.97178 − 22.4314i −59.7122 −45.6098 + 14.3944i
31.4 −2.25470 + 1.70773i − 8.10067i 2.16733 − 7.70082i −10.8445 + 10.8445i 13.8337 + 18.2646i −18.4065 + 18.4065i 8.26423 + 21.0642i −38.6209 5.93160 − 42.9704i
31.5 −1.84904 − 2.14034i − 1.29511i −1.16213 + 7.91514i −10.6631 + 10.6631i −2.77198 + 2.39471i −7.71781 + 7.71781i 19.0899 − 12.1480i 25.3227 42.5392 + 3.10621i
31.6 −1.70773 + 2.25470i 8.10067i −2.16733 − 7.70082i −10.8445 + 10.8445i −18.2646 − 13.8337i 18.4065 − 18.4065i 21.0642 + 8.26423i −38.6209 −5.93160 − 42.9704i
31.7 −1.42308 − 2.44435i 3.58200i −3.94969 + 6.95701i 10.6687 − 10.6687i 8.75566 − 5.09747i 6.84080 − 6.84080i 22.6261 − 0.245959i 14.1693 −41.2606 − 10.8957i
31.8 0.0560904 + 2.82787i − 6.97823i −7.99371 + 0.317233i 3.73459 − 3.73459i 19.7335 − 0.391412i 18.7167 − 18.7167i −1.34546 − 22.5874i −21.6957 10.7704 + 10.3515i
31.9 0.223782 + 2.81956i 1.00320i −7.89984 + 1.26193i −4.98300 + 4.98300i −2.82857 + 0.224497i −16.5820 + 16.5820i −5.32593 − 21.9917i 25.9936 −15.1650 − 12.9348i
31.10 0.289217 − 2.81360i 7.82965i −7.83271 − 1.62748i −8.33093 + 8.33093i 22.0295 + 2.26447i −11.3199 + 11.3199i −6.84443 + 21.5674i −34.3035 21.0305 + 25.8494i
31.11 0.340566 − 2.80785i − 6.14077i −7.76803 − 1.91251i −1.41066 + 1.41066i −17.2423 − 2.09133i 3.71079 − 3.71079i −8.01557 + 21.1601i −10.7090 3.48049 + 4.44133i
31.12 1.30534 + 2.50920i 9.31194i −4.59218 + 6.55072i 11.9568 − 11.9568i −23.3655 + 12.1552i 6.41108 − 6.41108i −22.4314 − 2.97178i −59.7122 45.6098 + 14.3944i
31.13 1.74069 − 2.22935i 3.23176i −1.93998 − 7.76122i 4.87198 − 4.87198i 7.20471 + 5.62550i 22.0805 − 22.0805i −20.6794 − 9.18500i 16.5557 −2.38072 − 19.3420i
31.14 2.14034 + 1.84904i 1.29511i 1.16213 + 7.91514i −10.6631 + 10.6631i −2.39471 + 2.77198i 7.71781 − 7.71781i −12.1480 + 19.0899i 25.3227 −42.5392 + 3.10621i
31.15 2.22935 − 1.74069i − 3.23176i 1.93998 − 7.76122i 4.87198 − 4.87198i −5.62550 − 7.20471i −22.0805 + 22.0805i −9.18500 − 20.6794i 16.5557 2.38072 − 19.3420i
31.16 2.44435 + 1.42308i − 3.58200i 3.94969 + 6.95701i 10.6687 − 10.6687i 5.09747 − 8.75566i −6.84080 + 6.84080i −0.245959 + 22.6261i 14.1693 41.2606 − 10.8957i
31.17 2.80785 − 0.340566i 6.14077i 7.76803 − 1.91251i −1.41066 + 1.41066i 2.09133 + 17.2423i −3.71079 + 3.71079i 21.1601 − 8.01557i −10.7090 −3.48049 + 4.44133i
31.18 2.81360 − 0.289217i − 7.82965i 7.83271 − 1.62748i −8.33093 + 8.33093i −2.26447 − 22.0295i 11.3199 − 11.3199i 21.5674 − 6.84443i −34.3035 −21.0305 + 25.8494i
47.1 −2.82787 + 0.0560904i − 6.97823i 7.99371 − 0.317233i 3.73459 + 3.73459i 0.391412 + 19.7335i −18.7167 − 18.7167i −22.5874 + 1.34546i −21.6957 −10.7704 − 10.3515i
47.2 −2.81956 + 0.223782i 1.00320i 7.89984 − 1.26193i −4.98300 − 4.98300i −0.224497 − 2.82857i 16.5820 + 16.5820i −21.9917 + 5.32593i 25.9936 15.1650 + 12.9348i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.4.f.b ✓ 36
4.b odd 2 1 inner 52.4.f.b ✓ 36
13.d odd 4 1 inner 52.4.f.b ✓ 36
52.f even 4 1 inner 52.4.f.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.4.f.b ✓ 36 1.a even 1 1 trivial
52.4.f.b ✓ 36 4.b odd 2 1 inner
52.4.f.b ✓ 36 13.d odd 4 1 inner
52.4.f.b ✓ 36 52.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 326 T_{3}^{16} + 43311 T_{3}^{14} + 3016492 T_{3}^{12} + 117803647 T_{3}^{10} + \cdots + 144896061440 \) acting on \(S_{4}^{\mathrm{new}}(52, [\chi])\). Copy content Toggle raw display