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Properties

Label 87.4.k.a
Level $87$
Weight $4$
Character orbit 87.k
Analytic conductor $5.133$
Analytic rank $0$
Dimension $336$
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] = [87,4,Mod(2,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.2");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 87.k (of order \(28\), degree \(12\), 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: \(5.13316617050\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 12 q^{3} - 28 q^{4} - 14 q^{6} - 20 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 12 q^{3} - 28 q^{4} - 14 q^{6} - 20 q^{7} - 14 q^{9} - 16 q^{10} + 28 q^{12} - 28 q^{13} + 86 q^{15} + 1164 q^{16} + 56 q^{18} + 194 q^{21} - 28 q^{22} - 1226 q^{24} - 1020 q^{25} - 936 q^{27} - 864 q^{30} - 204 q^{31} - 350 q^{33} - 28 q^{34} + 394 q^{36} - 696 q^{37} + 2330 q^{39} + 620 q^{40} - 14 q^{42} + 576 q^{43} - 162 q^{45} + 5320 q^{46} + 1960 q^{48} - 2212 q^{49} - 14 q^{51} - 3400 q^{52} - 1698 q^{54} - 6568 q^{55} - 6300 q^{58} + 2312 q^{60} - 3000 q^{61} - 14 q^{63} - 3808 q^{64} + 1804 q^{66} + 1820 q^{67} + 898 q^{69} + 6944 q^{70} - 11732 q^{72} + 7236 q^{73} + 2052 q^{75} - 5420 q^{76} - 642 q^{78} + 2604 q^{79} + 10126 q^{81} + 7900 q^{82} + 5576 q^{84} + 6020 q^{85} + 6956 q^{87} + 11512 q^{88} + 4128 q^{90} + 5516 q^{91} + 2450 q^{93} + 7624 q^{94} + 10332 q^{96} - 4788 q^{97} + 8626 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.612662 + 5.43752i 1.15656 + 5.06580i −21.3919 − 4.88256i 6.48097 + 8.12687i −28.2540 + 3.18520i 2.13894 + 9.37133i 25.1969 − 72.0086i −24.3247 + 11.7178i −48.1607 + 30.2614i
2.2 −0.597167 + 5.30000i 4.06093 − 3.24174i −19.9340 − 4.54980i −13.3979 − 16.8004i 14.7562 + 23.4588i 2.56694 + 11.2465i 21.9254 − 62.6593i 5.98229 − 26.3289i 97.0429 − 60.9761i
2.3 −0.534178 + 4.74096i −5.12961 − 0.828907i −14.3919 − 3.28486i −0.952155 − 1.19396i 6.66994 − 23.8765i −2.46438 − 10.7972i 10.6553 − 30.4510i 25.6258 + 8.50394i 6.16916 − 3.87634i
2.4 −0.498912 + 4.42796i −0.698437 − 5.14900i −11.5585 − 2.63816i 8.94685 + 11.2190i 23.1480 − 0.523759i 3.45842 + 15.1523i 5.67459 − 16.2171i −26.0244 + 7.19250i −54.1410 + 34.0190i
2.5 −0.450339 + 3.99687i 4.96833 − 1.52174i −7.97271 − 1.81972i 7.66696 + 9.61407i 3.84476 + 20.5430i −2.69897 − 11.8250i 0.236122 − 0.674798i 22.3686 − 15.1210i −41.8789 + 26.3142i
2.6 −0.428100 + 3.79949i −1.09978 + 5.07843i −6.45347 − 1.47296i −9.01590 − 11.3056i −18.8247 − 6.35268i −5.12301 − 22.4454i −1.74343 + 4.98244i −24.5810 − 11.1703i 46.8152 − 29.4160i
2.7 −0.341479 + 3.03071i 4.17295 + 3.09620i −1.26916 − 0.289678i −3.62286 − 4.54293i −10.8087 + 11.5897i 3.20775 + 14.0541i −6.74718 + 19.2823i 7.82706 + 25.8406i 15.0054 − 9.42853i
2.8 −0.316501 + 2.80902i 0.656519 − 5.15451i 0.00898647 + 0.00205110i −3.87749 − 4.86222i 14.2714 + 3.47558i −6.77049 − 29.6635i −7.47766 + 21.3699i −26.1380 − 6.76807i 14.8853 − 9.35306i
2.9 −0.310612 + 2.75676i −3.58097 + 3.76519i 0.296199 + 0.0676055i 7.92436 + 9.93684i −9.26741 − 11.0414i 2.66401 + 11.6718i −7.60846 + 21.7437i −1.35326 − 26.9661i −29.8548 + 18.7590i
2.10 −0.292041 + 2.59193i −3.13038 − 4.14738i 1.16660 + 0.266268i −8.18861 − 10.2682i 11.6639 − 6.90252i 7.29154 + 31.9463i −7.92267 + 22.6417i −7.40148 + 25.9657i 29.0059 − 18.2256i
2.11 −0.123032 + 1.09194i −5.09680 − 1.01125i 6.62223 + 1.51148i 3.60229 + 4.51713i 1.73130 − 5.44098i −2.99545 − 13.1239i −5.36861 + 15.3426i 24.9547 + 10.3083i −5.37563 + 3.37774i
2.12 −0.0834583 + 0.740713i 4.52204 − 2.55952i 7.25773 + 1.65653i 1.07095 + 1.34293i 1.51846 + 3.56315i 3.17814 + 13.9243i −3.80225 + 10.8662i 13.8977 − 23.1485i −1.08411 + 0.681188i
2.13 −0.0729778 + 0.647696i 2.87142 + 4.33070i 7.38524 + 1.68563i 11.0185 + 13.8167i −3.01452 + 1.54377i −6.59710 − 28.9038i −3.35293 + 9.58212i −10.5098 + 24.8705i −9.75314 + 6.12831i
2.14 −0.0456370 + 0.405040i −4.67140 + 2.27553i 7.63745 + 1.74320i −11.3426 − 14.2232i −0.708490 − 1.99595i 1.00079 + 4.38476i −2.13160 + 6.09175i 16.6440 − 21.2598i 6.27861 − 3.94511i
2.15 0.0456370 − 0.405040i 0.0233337 − 5.19610i 7.63745 + 1.74320i 11.3426 + 14.2232i −2.10356 − 0.246586i 1.00079 + 4.38476i 2.13160 − 6.09175i −26.9989 − 0.242488i 6.27861 − 3.94511i
2.16 0.0729778 − 0.647696i 5.14769 + 0.708045i 7.38524 + 1.68563i −11.0185 − 13.8167i 0.834265 − 3.28246i −6.59710 − 28.9038i 3.35293 − 9.58212i 25.9973 + 7.28959i −9.75314 + 6.12831i
2.17 0.0834583 − 0.740713i −0.344004 + 5.18475i 7.25773 + 1.65653i −1.07095 − 1.34293i 3.81170 + 0.687519i 3.17814 + 13.9243i 3.80225 − 10.8662i −26.7633 − 3.56715i −1.08411 + 0.681188i
2.18 0.123032 − 1.09194i −3.12253 − 4.15329i 6.62223 + 1.51148i −3.60229 − 4.51713i −4.91931 + 2.89862i −2.99545 − 13.1239i 5.36861 − 15.3426i −7.49965 + 25.9375i −5.37563 + 3.37774i
2.19 0.292041 − 2.59193i −5.09488 − 1.02089i 1.16660 + 0.266268i 8.18861 + 10.2682i −4.13400 + 12.9074i 7.29154 + 31.9463i 7.92267 − 22.6417i 24.9156 + 10.4026i 29.0059 − 18.2256i
2.20 0.310612 − 2.75676i 1.83859 − 4.86000i 0.296199 + 0.0676055i −7.92436 − 9.93684i −12.8267 − 6.57812i 2.66401 + 11.6718i 7.60846 − 21.7437i −20.2392 − 17.8711i −29.8548 + 18.7590i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.f odd 28 1 inner
87.k even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.4.k.a ✓ 336
3.b odd 2 1 inner 87.4.k.a ✓ 336
29.f odd 28 1 inner 87.4.k.a ✓ 336
87.k even 28 1 inner 87.4.k.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.4.k.a ✓ 336 1.a even 1 1 trivial
87.4.k.a ✓ 336 3.b odd 2 1 inner
87.4.k.a ✓ 336 29.f odd 28 1 inner
87.4.k.a ✓ 336 87.k even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(87, [\chi])\).