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Properties

Label 232.4.o.a
Level $232$
Weight $4$
Character orbit 232.o
Analytic conductor $13.688$
Analytic rank $0$
Dimension $528$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,4,Mod(5,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 7, 11]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.o (of order \(14\), degree \(6\), 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: \(13.6884431213\)
Analytic rank: \(0\)
Dimension: \(528\)
Relative dimension: \(88\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 7 q^{2} + 5 q^{4} + 41 q^{6} - 38 q^{7} - 28 q^{8} - 766 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 528 q - 7 q^{2} + 5 q^{4} + 41 q^{6} - 38 q^{7} - 28 q^{8} - 766 q^{9} - 7 q^{10} - 7 q^{14} - 14 q^{15} - 155 q^{16} + 826 q^{18} + 21 q^{20} + 402 q^{22} - 614 q^{23} + 153 q^{24} + 2034 q^{25} - 7 q^{26} + 840 q^{28} + 378 q^{30} - 14 q^{31} + 238 q^{32} + 98 q^{33} + 269 q^{34} + 948 q^{36} + 1108 q^{38} + 2086 q^{39} - 7 q^{40} + 2986 q^{42} - 882 q^{44} - 14 q^{47} - 4396 q^{48} - 3734 q^{49} - 56 q^{50} - 1492 q^{52} - 1418 q^{54} - 14 q^{55} + 2345 q^{56} + 84 q^{57} - 3927 q^{58} + 3402 q^{60} - 871 q^{62} + 210 q^{63} + 2114 q^{64} - 600 q^{65} - 3955 q^{66} - 2156 q^{68} + 498 q^{71} - 9135 q^{72} + 2576 q^{73} - 518 q^{74} + 5243 q^{76} - 2091 q^{78} + 2086 q^{79} + 3474 q^{80} - 4502 q^{81} - 3832 q^{82} + 2583 q^{84} + 796 q^{86} - 1342 q^{87} - 10 q^{88} - 14 q^{89} - 8008 q^{90} + 3496 q^{92} + 3944 q^{94} - 14 q^{95} - 1648 q^{96} + 3164 q^{97} + 6678 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.82003 + 0.217821i 1.61196 + 7.06244i 7.90511 − 1.22852i 3.64677 − 2.90821i −6.08410 − 19.5652i −4.02149 − 17.6193i −22.0250 + 5.18635i −22.9535 + 11.0538i −9.65054 + 8.99556i
5.2 −2.81978 + 0.220999i −0.700106 − 3.06736i 7.90232 − 1.24634i 5.00974 − 3.99514i 2.65203 + 8.49457i −0.0793850 − 0.347808i −22.0074 + 5.26080i 15.4076 − 7.41990i −13.2435 + 12.3726i
5.3 −2.81099 + 0.313561i −0.352368 − 1.54383i 7.80336 − 1.76284i 0.408717 − 0.325941i 1.47459 + 4.22919i 7.22862 + 31.6706i −21.3824 + 7.40216i 22.0669 − 10.6269i −1.04670 + 1.04438i
5.4 −2.79039 − 0.462275i −1.14212 − 5.00394i 7.57260 + 2.57986i −12.2351 + 9.75720i 0.873762 + 14.4909i −1.21693 − 5.33170i −19.9380 − 10.6994i 0.591207 − 0.284710i 38.6514 − 21.5705i
5.5 −2.77642 − 0.539896i −1.75561 − 7.69184i 7.41702 + 2.99796i 10.9392 − 8.72370i 0.721524 + 22.3036i 0.861400 + 3.77404i −18.9742 − 12.3280i −31.7560 + 15.2929i −35.0816 + 18.3146i
5.6 −2.74634 + 0.676489i 1.20561 + 5.28214i 7.08473 − 3.71573i −12.2064 + 9.73427i −6.88432 − 13.6909i −1.69935 − 7.44533i −16.9434 + 14.9974i −2.12130 + 1.02156i 26.9377 − 34.9911i
5.7 −2.69280 + 0.865356i −1.38368 − 6.06232i 6.50232 − 4.66046i −10.5510 + 8.41417i 8.97204 + 15.1272i 0.716771 + 3.14038i −13.4765 + 18.1765i −10.5109 + 5.06180i 21.1305 − 31.7881i
5.8 −2.68679 − 0.883840i 0.957403 + 4.19466i 6.43765 + 4.74938i −0.846064 + 0.674714i 1.13507 − 12.1163i 0.373905 + 1.63819i −13.0989 − 18.4504i 7.64763 − 3.68291i 2.86953 − 1.06503i
5.9 −2.67889 − 0.907495i 0.0922057 + 0.403980i 6.35291 + 4.86216i 9.57271 − 7.63398i 0.119600 − 1.16589i −5.17627 − 22.6787i −12.6064 − 18.7904i 24.1715 − 11.6404i −32.5720 + 11.7634i
5.10 −2.64387 − 1.00496i 0.884551 + 3.87547i 5.98010 + 5.31398i −13.4272 + 10.7078i 1.55606 − 11.1352i 5.93063 + 25.9838i −10.4703 − 20.0592i 10.0893 − 4.85876i 46.2607 − 14.8163i
5.11 −2.63629 − 1.02467i 2.16297 + 9.47659i 5.90010 + 5.40267i 5.65159 − 4.50699i 4.00817 − 27.1994i 3.79065 + 16.6079i −10.0184 − 20.2887i −60.8011 + 29.2803i −19.5174 + 6.09074i
5.12 −2.63112 + 1.03788i 1.25801 + 5.51171i 5.84561 − 5.46158i 15.8882 − 12.6704i −9.03048 − 13.1963i 3.23241 + 14.1621i −9.71203 + 20.4371i −4.47017 + 2.15272i −28.6535 + 49.8276i
5.13 −2.61176 + 1.08569i −1.99935 − 8.75970i 5.64256 − 5.67111i −2.62310 + 2.09185i 14.7321 + 20.7076i −7.04128 − 30.8499i −8.57995 + 20.9376i −48.4089 + 23.3125i 4.57981 − 8.31129i
5.14 −2.53885 + 1.24668i −0.295943 − 1.29661i 4.89155 − 6.33030i 9.62722 − 7.67745i 2.36782 + 2.92296i −7.01745 − 30.7455i −4.52705 + 22.1699i 22.7325 − 10.9474i −14.8707 + 31.4940i
5.15 −2.42154 − 1.46155i −0.539523 − 2.36380i 3.72772 + 7.07843i −9.56069 + 7.62440i −2.14835 + 6.51259i −5.30344 − 23.2359i 1.31868 − 22.5890i 19.0297 − 9.16421i 34.2951 − 4.48932i
5.16 −2.36811 − 1.54663i −1.97425 − 8.64975i 3.21588 + 7.32517i 0.213915 − 0.170592i −8.70272 + 23.5370i −1.55828 − 6.82725i 3.71379 − 22.3206i −46.5943 + 22.4387i −0.770416 + 0.0731320i
5.17 −2.29689 + 1.65054i 0.295031 + 1.29261i 2.55142 − 7.58223i −6.51562 + 5.19604i −2.81117 − 2.48203i 0.828590 + 3.63029i 6.65448 + 21.6268i 22.7424 − 10.9521i 6.38940 − 22.6890i
5.18 −2.26638 − 1.69219i 0.0472863 + 0.207175i 2.27299 + 7.67030i 16.2175 − 12.9330i 0.243410 − 0.549555i 5.54237 + 24.2827i 7.82813 − 21.2302i 24.2855 − 11.6953i −58.6401 + 1.86812i
5.19 −2.21344 + 1.76088i 2.00311 + 8.77622i 1.79864 − 7.79519i −6.00699 + 4.79041i −19.8876 − 15.8984i 5.84411 + 25.6047i 9.74518 + 20.4213i −48.6834 + 23.4447i 4.86079 − 21.1809i
5.20 −2.20926 + 1.76611i −2.00311 − 8.77622i 1.76169 − 7.80362i 6.00699 − 4.79041i 19.9252 + 15.8513i 5.84411 + 25.6047i 9.89003 + 20.3516i −48.6834 + 23.4447i −4.81061 + 21.1923i
See next 80 embeddings (of 528 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
29.e even 14 1 inner
232.o even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.4.o.a ✓ 528
8.b even 2 1 inner 232.4.o.a ✓ 528
29.e even 14 1 inner 232.4.o.a ✓ 528
232.o even 14 1 inner 232.4.o.a ✓ 528
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.o.a ✓ 528 1.a even 1 1 trivial
232.4.o.a ✓ 528 8.b even 2 1 inner
232.4.o.a ✓ 528 29.e even 14 1 inner
232.4.o.a ✓ 528 232.o even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).