Properties

Label 72.5.j.a
Level $72$
Weight $5$
Character orbit 72.j
Analytic conductor $7.443$
Analytic rank $0$
Dimension $92$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,5,Mod(5,72)] 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("72.5"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.44263734204\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 3 q^{2} - q^{4} - 19 q^{6} - 2 q^{7} - 4 q^{9} + 28 q^{10} + 62 q^{12} + 852 q^{14} + 158 q^{15} - q^{16} - 506 q^{18} + 1950 q^{20} + 31 q^{22} - 6 q^{23} - 1415 q^{24} - 4752 q^{25} + 508 q^{28}+ \cdots - 2 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −3.99936 − 0.0715673i −8.43375 − 3.14194i 15.9898 + 0.572447i −5.94208 + 10.2920i 33.5048 + 13.1693i 21.5744 + 37.3679i −63.9078 − 3.43376i 61.2564 + 52.9967i 24.5011 − 40.7361i
5.2 −3.96631 − 0.518044i −4.87156 + 7.56756i 15.4633 + 4.10945i 9.54377 − 16.5303i 23.2425 − 27.4916i −6.73634 − 11.6677i −59.2032 − 24.3100i −33.5358 − 73.7316i −46.4170 + 60.6202i
5.3 −3.95490 − 0.598961i 0.933380 − 8.95147i 15.2825 + 4.73766i −12.1310 + 21.0114i −9.05300 + 34.8431i −22.4004 − 38.7986i −57.6031 − 27.8906i −79.2576 − 16.7102i 60.5617 − 75.8321i
5.4 −3.93003 + 0.744918i 8.05757 + 4.00943i 14.8902 − 5.85509i 13.0982 − 22.6868i −34.6532 − 9.75495i −43.7982 − 75.8606i −54.1573 + 34.1026i 48.8489 + 64.6126i −34.5765 + 98.9167i
5.5 −3.75717 + 1.37247i 8.39079 − 3.25495i 12.2327 − 10.3132i −10.8201 + 18.7410i −27.0583 + 23.7455i 18.2392 + 31.5913i −31.8056 + 55.5374i 59.8107 − 54.6231i 14.9316 − 85.2634i
5.6 −3.63985 − 1.65877i 4.81678 + 7.60254i 10.4970 + 12.0753i −5.01037 + 8.67822i −4.92146 − 35.6620i 23.4688 + 40.6492i −18.1772 − 61.3644i −34.5973 + 73.2395i 32.6322 − 23.2763i
5.7 −3.58461 − 1.77498i 7.77894 − 4.52637i 9.69890 + 12.7252i 19.2792 − 33.3925i −35.9187 + 2.41784i 36.2264 + 62.7460i −12.1798 − 62.8303i 40.0239 − 70.4208i −128.379 + 85.4792i
5.8 −3.48346 + 1.96609i −0.573879 − 8.98168i 8.26898 − 13.6976i 16.4840 − 28.5511i 19.6579 + 30.1590i 2.21331 + 3.83357i −1.87399 + 63.9726i −80.3413 + 10.3088i −1.28728 + 131.866i
5.9 −3.44441 + 2.03372i 0.573879 + 8.98168i 7.72797 − 14.0099i −16.4840 + 28.5511i −20.2429 − 29.7695i 2.21331 + 3.83357i 1.87399 + 63.9726i −80.3413 + 10.3088i −1.28728 − 131.866i
5.10 −3.09807 − 2.53021i −7.96182 − 4.19635i 3.19606 + 15.6775i 20.1379 − 34.8798i 14.0486 + 33.1457i −36.9510 − 64.0010i 29.7659 − 56.6568i 45.7812 + 66.8213i −150.642 + 57.1069i
5.11 −3.06718 + 2.56757i −8.39079 + 3.25495i 2.81517 − 15.7504i 10.8201 − 18.7410i 17.3787 − 31.5274i 18.2392 + 31.5913i 31.8056 + 55.5374i 59.8107 − 54.6231i 14.9316 + 85.2634i
5.12 −2.83732 − 2.81950i −7.20486 + 5.39351i 0.100792 + 15.9997i −21.1127 + 36.5683i 35.6495 + 5.01101i −16.9047 − 29.2798i 44.8252 − 45.6804i 22.8201 − 77.7190i 163.008 − 44.2286i
5.13 −2.81177 − 2.84499i 8.34560 − 3.36912i −0.187929 + 15.9989i −8.74932 + 15.1543i −33.0510 − 14.2700i −30.7371 − 53.2382i 46.0451 − 44.4505i 58.2981 − 56.2346i 67.7148 − 17.7186i
5.14 −2.61013 + 3.03104i −8.05757 − 4.00943i −2.37444 − 15.8228i −13.0982 + 22.6868i 33.1841 − 13.9577i −43.7982 − 75.8606i 54.1573 + 34.1026i 48.8489 + 64.6126i −34.5765 − 98.9167i
5.15 −2.21293 − 3.33211i −3.11153 − 8.44502i −6.20585 + 14.7475i −5.93817 + 10.2852i −21.2541 + 29.0562i 29.0886 + 50.3830i 62.8732 − 11.9566i −61.6367 + 52.5539i 47.4122 − 2.97388i
5.16 −1.93770 + 3.49933i 8.43375 + 3.14194i −8.49063 − 13.5613i 5.94208 − 10.2920i −27.3368 + 23.4244i 21.5744 + 37.3679i 63.9078 − 3.43376i 61.2564 + 52.9967i 24.5011 + 40.7361i
5.17 −1.67951 − 3.63032i 5.83929 + 6.84855i −10.3585 + 12.1943i 3.78105 − 6.54897i 15.0553 − 32.7007i −14.0251 − 24.2922i 61.6666 + 17.1240i −12.8053 + 79.9814i −30.1252 − 2.72734i
5.18 −1.53452 + 3.69395i 4.87156 − 7.56756i −11.2905 − 11.3369i −9.54377 + 16.5303i 20.4787 + 29.6078i −6.73634 − 11.6677i 59.2032 − 24.3100i −33.5358 − 73.7316i −46.4170 − 60.6202i
5.19 −1.45874 + 3.72453i −0.933380 + 8.95147i −11.7442 − 10.8662i 12.1310 − 21.0114i −31.9784 − 16.5342i −22.4004 − 38.7986i 57.6031 − 27.8906i −79.2576 − 16.7102i 60.5617 + 75.8321i
5.20 −1.38904 − 3.75108i −4.45965 + 7.81739i −12.1411 + 10.4208i 15.7860 − 27.3421i 35.5182 + 5.86981i 22.5676 + 39.0883i 55.9537 + 31.0674i −41.2231 − 69.7256i −124.490 − 21.2351i
See all 92 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.d odd 6 1 inner
72.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.5.j.a ✓ 92
3.b odd 2 1 216.5.j.a 92
4.b odd 2 1 288.5.n.a 92
8.b even 2 1 inner 72.5.j.a ✓ 92
8.d odd 2 1 288.5.n.a 92
9.c even 3 1 216.5.j.a 92
9.d odd 6 1 inner 72.5.j.a ✓ 92
12.b even 2 1 864.5.n.a 92
24.f even 2 1 864.5.n.a 92
24.h odd 2 1 216.5.j.a 92
36.f odd 6 1 864.5.n.a 92
36.h even 6 1 288.5.n.a 92
72.j odd 6 1 inner 72.5.j.a ✓ 92
72.l even 6 1 288.5.n.a 92
72.n even 6 1 216.5.j.a 92
72.p odd 6 1 864.5.n.a 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.5.j.a ✓ 92 1.a even 1 1 trivial
72.5.j.a ✓ 92 8.b even 2 1 inner
72.5.j.a ✓ 92 9.d odd 6 1 inner
72.5.j.a ✓ 92 72.j odd 6 1 inner
216.5.j.a 92 3.b odd 2 1
216.5.j.a 92 9.c even 3 1
216.5.j.a 92 24.h odd 2 1
216.5.j.a 92 72.n even 6 1
288.5.n.a 92 4.b odd 2 1
288.5.n.a 92 8.d odd 2 1
288.5.n.a 92 36.h even 6 1
288.5.n.a 92 72.l even 6 1
864.5.n.a 92 12.b even 2 1
864.5.n.a 92 24.f even 2 1
864.5.n.a 92 36.f odd 6 1
864.5.n.a 92 72.p odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(72, [\chi])\).