↑

Properties

Label 36.4.h.b
Level $36$
Weight $4$
Character orbit 36.h
Analytic conductor $2.124$
Analytic rank $0$
Dimension $24$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,4,Mod(11,36)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("36.11"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.12406876021\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{4} - 72 q^{5} + 60 q^{6} - 84 q^{9} + 96 q^{10} + 216 q^{12} - 216 q^{13} - 36 q^{14} - 72 q^{16} + 276 q^{18} - 540 q^{20} + 384 q^{21} - 192 q^{22} - 168 q^{24} + 252 q^{25} - 672 q^{28}+ \cdots + 588 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.81402 − 0.285097i −0.828122 − 5.12974i 7.83744 + 1.60454i −14.4924 + 8.36717i 0.867881 + 14.6713i −16.7175 − 9.65186i −21.5973 − 6.74964i −25.6284 + 8.49610i 43.1673 − 19.4137i
11.2 −2.52436 + 1.27578i 5.18398 + 0.355390i 4.74478 − 6.44105i −1.23846 + 0.715028i −13.5396 + 5.71649i 23.8818 + 13.7882i −3.76017 + 22.3128i 26.7474 + 3.68468i 2.21411 − 3.38499i
11.3 −1.93145 + 2.06627i −2.72340 + 4.42528i −0.538974 − 7.98182i −4.71466 + 2.72201i −3.88372 − 14.1745i −20.9358 − 12.0873i 17.5336 + 14.3029i −12.1662 − 24.1036i 3.48173 − 14.9992i
11.4 −1.65391 − 2.29447i 0.828122 + 5.12974i −2.52915 + 7.58969i −14.4924 + 8.36717i 10.4004 − 10.3842i 16.7175 + 9.65186i 21.5973 − 6.74964i −25.6284 + 8.49610i 43.1673 + 19.4137i
11.5 −1.38284 + 2.46734i −1.42987 − 4.99554i −4.17551 − 6.82387i 14.6499 − 8.45813i 14.3030 + 3.38006i −3.08966 − 1.78382i 22.6108 − 0.866066i −22.9109 + 14.2860i 0.610574 + 47.8425i
11.6 −0.157323 − 2.82405i −5.18398 − 0.355390i −7.95050 + 0.888573i −1.23846 + 0.715028i −0.188082 + 14.6957i −23.8818 − 13.7882i 3.76017 + 22.3128i 26.7474 + 3.68468i 2.21411 + 3.38499i
11.7 0.664105 + 2.74936i −5.00415 − 1.39947i −7.11793 + 3.65173i −14.2911 + 8.25096i 0.524375 − 14.6876i 19.2620 + 11.1209i −14.7670 − 17.1446i 23.0829 + 14.0063i −32.1756 − 33.8118i
11.8 0.823719 − 2.70582i 2.72340 − 4.42528i −6.64298 − 4.45768i −4.71466 + 2.72201i −9.73072 − 11.0142i 20.9358 + 12.0873i −17.5336 + 14.3029i −12.1662 − 24.1036i 3.48173 + 14.9992i
11.9 1.44536 − 2.43124i 1.42987 + 4.99554i −3.82189 − 7.02803i 14.6499 − 8.45813i 14.2121 + 3.74398i 3.08966 + 1.78382i −22.6108 − 0.866066i −22.9109 + 14.2860i 0.610574 − 47.8425i
11.10 2.15223 + 1.83518i 2.90476 − 4.30841i 1.26420 + 7.89948i 2.08666 − 1.20474i 14.1584 − 3.94193i −2.30362 − 1.33000i −11.7761 + 19.3216i −10.1248 − 25.0298i 6.70190 + 1.23654i
11.11 2.66543 + 0.946295i −2.90476 + 4.30841i 6.20905 + 5.04457i 2.08666 − 1.20474i −11.8195 + 8.73501i 2.30362 + 1.33000i 11.7761 + 19.3216i −10.1248 − 25.0298i 6.70190 − 1.23654i
11.12 2.71307 − 0.799546i 5.00415 + 1.39947i 6.72145 − 4.33844i −14.2911 + 8.25096i 14.6955 − 0.204180i −19.2620 − 11.1209i 14.7670 − 17.1446i 23.0829 + 14.0063i −32.1756 + 33.8118i
23.1 −2.81402 + 0.285097i −0.828122 + 5.12974i 7.83744 − 1.60454i −14.4924 − 8.36717i 0.867881 − 14.6713i −16.7175 + 9.65186i −21.5973 + 6.74964i −25.6284 − 8.49610i 43.1673 + 19.4137i
23.2 −2.52436 − 1.27578i 5.18398 − 0.355390i 4.74478 + 6.44105i −1.23846 − 0.715028i −13.5396 − 5.71649i 23.8818 − 13.7882i −3.76017 − 22.3128i 26.7474 − 3.68468i 2.21411 + 3.38499i
23.3 −1.93145 − 2.06627i −2.72340 − 4.42528i −0.538974 + 7.98182i −4.71466 − 2.72201i −3.88372 + 14.1745i −20.9358 + 12.0873i 17.5336 − 14.3029i −12.1662 + 24.1036i 3.48173 + 14.9992i
23.4 −1.65391 + 2.29447i 0.828122 − 5.12974i −2.52915 − 7.58969i −14.4924 − 8.36717i 10.4004 + 10.3842i 16.7175 − 9.65186i 21.5973 + 6.74964i −25.6284 − 8.49610i 43.1673 − 19.4137i
23.5 −1.38284 − 2.46734i −1.42987 + 4.99554i −4.17551 + 6.82387i 14.6499 + 8.45813i 14.3030 − 3.38006i −3.08966 + 1.78382i 22.6108 + 0.866066i −22.9109 − 14.2860i 0.610574 − 47.8425i
23.6 −0.157323 + 2.82405i −5.18398 + 0.355390i −7.95050 − 0.888573i −1.23846 − 0.715028i −0.188082 − 14.6957i −23.8818 + 13.7882i 3.76017 − 22.3128i 26.7474 − 3.68468i 2.21411 − 3.38499i
23.7 0.664105 − 2.74936i −5.00415 + 1.39947i −7.11793 − 3.65173i −14.2911 − 8.25096i 0.524375 + 14.6876i 19.2620 − 11.1209i −14.7670 + 17.1446i 23.0829 − 14.0063i −32.1756 + 33.8118i
23.8 0.823719 + 2.70582i 2.72340 + 4.42528i −6.64298 + 4.45768i −4.71466 − 2.72201i −9.73072 + 11.0142i 20.9358 − 12.0873i −17.5336 − 14.3029i −12.1662 + 24.1036i 3.48173 − 14.9992i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.4.h.b ✓ 24
3.b odd 2 1 108.4.h.b 24
4.b odd 2 1 inner 36.4.h.b ✓ 24
9.c even 3 1 108.4.h.b 24
9.c even 3 1 324.4.b.c 24
9.d odd 6 1 inner 36.4.h.b ✓ 24
9.d odd 6 1 324.4.b.c 24
12.b even 2 1 108.4.h.b 24
36.f odd 6 1 108.4.h.b 24
36.f odd 6 1 324.4.b.c 24
36.h even 6 1 inner 36.4.h.b ✓ 24
36.h even 6 1 324.4.b.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.b ✓ 24 1.a even 1 1 trivial
36.4.h.b ✓ 24 4.b odd 2 1 inner
36.4.h.b ✓ 24 9.d odd 6 1 inner
36.4.h.b ✓ 24 36.h even 6 1 inner
108.4.h.b 24 3.b odd 2 1
108.4.h.b 24 9.c even 3 1
108.4.h.b 24 12.b even 2 1
108.4.h.b 24 36.f odd 6 1
324.4.b.c 24 9.c even 3 1
324.4.b.c 24 9.d odd 6 1
324.4.b.c 24 36.f odd 6 1
324.4.b.c 24 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 36 T_{5}^{11} + 210 T_{5}^{10} - 7992 T_{5}^{9} - 59073 T_{5}^{8} + 2269332 T_{5}^{7} + \cdots + 7678666384 \) acting on \(S_{4}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display