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Properties

Label 232.4.m.a
Level $232$
Weight $4$
Character orbit 232.m
Analytic conductor $13.688$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,4,Mod(25,232)]
 
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, 0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.m (of order \(7\), 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: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q + q^{3} + 16 q^{5} - 71 q^{7} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + q^{3} + 16 q^{5} - 71 q^{7} - 76 q^{9} + 130 q^{11} - 39 q^{13} - 289 q^{15} - 102 q^{17} - 20 q^{19} - 107 q^{21} + 427 q^{23} - 579 q^{25} + 379 q^{27} + 29 q^{29} - 226 q^{31} + 1015 q^{33} + 300 q^{35} + 1523 q^{37} + 378 q^{39} - 192 q^{41} - 251 q^{43} - 1029 q^{45} - 881 q^{47} - 820 q^{49} - 809 q^{51} + 56 q^{53} + 1618 q^{55} + 1714 q^{57} - 158 q^{59} + 1063 q^{61} + 426 q^{63} - 311 q^{65} - 69 q^{67} - 308 q^{69} - 1925 q^{71} - 1935 q^{73} - 7430 q^{75} - 5284 q^{77} - 362 q^{79} + 969 q^{81} + 3377 q^{83} + 856 q^{85} + 2805 q^{87} + 2063 q^{89} + 729 q^{91} + 6228 q^{93} + 847 q^{95} + 3272 q^{97} + 3940 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −7.54041 + 3.63127i 0 −0.0664965 + 0.291340i 0 8.53056 − 4.10810i 0 26.8374 − 33.6530i 0
25.2 0 −6.58548 + 3.17140i 0 1.12848 − 4.94419i 0 −29.3259 + 14.1226i 0 16.4765 − 20.6609i 0
25.3 0 −5.29707 + 2.55093i 0 −2.15515 + 9.44232i 0 7.13978 − 3.43834i 0 4.71745 − 5.91550i 0
25.4 0 −2.76750 + 1.33276i 0 4.41052 − 19.3237i 0 17.5139 − 8.43426i 0 −10.9514 + 13.7326i 0
25.5 0 −2.25349 + 1.08522i 0 −4.80642 + 21.0583i 0 −6.13327 + 2.95363i 0 −12.9337 + 16.2184i 0
25.6 0 0.221330 − 0.106587i 0 2.41185 − 10.5670i 0 −17.9453 + 8.64201i 0 −16.7966 + 21.0623i 0
25.7 0 1.75352 − 0.844451i 0 −1.15927 + 5.07909i 0 20.2959 − 9.77401i 0 −14.4725 + 18.1479i 0
25.8 0 2.94273 − 1.41714i 0 −0.171636 + 0.751985i 0 −15.0983 + 7.27095i 0 −10.1829 + 12.7689i 0
25.9 0 6.91863 − 3.33184i 0 0.651218 − 2.85317i 0 19.0149 − 9.15710i 0 19.9321 − 24.9940i 0
25.10 0 7.47913 − 3.60176i 0 −4.04918 + 17.7406i 0 −15.6653 + 7.54400i 0 26.1305 − 32.7666i 0
25.11 0 8.23247 − 3.96455i 0 3.82301 − 16.7497i 0 −10.6000 + 5.10469i 0 35.2217 − 44.1666i 0
49.1 0 −5.76364 − 7.22737i 0 8.94769 − 4.30898i 0 −17.3016 − 21.6955i 0 −13.0073 + 56.9888i 0
49.2 0 −4.87909 − 6.11818i 0 −14.8626 + 7.15745i 0 6.98537 + 8.75937i 0 −7.61859 + 33.3792i 0
49.3 0 −3.95754 − 4.96259i 0 15.5382 − 7.48282i 0 18.3805 + 23.0484i 0 −2.95717 + 12.9562i 0
49.4 0 −2.86205 − 3.58890i 0 −1.07109 + 0.515811i 0 4.00699 + 5.02461i 0 1.31920 − 5.77978i 0
49.5 0 −1.24025 − 1.55523i 0 −5.44727 + 2.62327i 0 −20.4044 − 25.5863i 0 5.12756 − 22.4653i 0
49.6 0 0.0292917 + 0.0367307i 0 11.1718 − 5.38004i 0 −8.01664 − 10.0525i 0 6.00757 − 26.3209i 0
49.7 0 0.794986 + 0.996881i 0 −14.0522 + 6.76719i 0 0.235955 + 0.295879i 0 5.64630 − 24.7380i 0
49.8 0 1.85631 + 2.32774i 0 10.3987 − 5.00775i 0 0.398273 + 0.499418i 0 4.03559 − 17.6811i 0
49.9 0 2.71518 + 3.40473i 0 −8.38914 + 4.04000i 0 17.8992 + 22.4449i 0 1.78809 − 7.83412i 0
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.4.m.a ✓ 66
29.d even 7 1 inner 232.4.m.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.m.a ✓ 66 1.a even 1 1 trivial
232.4.m.a ✓ 66 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{66} - T_{3}^{65} + 187 T_{3}^{64} - 457 T_{3}^{63} + 23061 T_{3}^{62} - 62985 T_{3}^{61} + \cdots + 99\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(232, [\chi])\). Copy content Toggle raw display