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Properties

Label 87.5.h.a
Level $87$
Weight $5$
Character orbit 87.h
Analytic conductor $8.993$
Analytic rank $0$
Dimension $228$
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] = [87,5,Mod(5,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 11]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 87.h (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: \(8.99318678829\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 7 q^{3} - 298 q^{4} + 253 q^{6} + 70 q^{7} - 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q - 7 q^{3} - 298 q^{4} + 253 q^{6} + 70 q^{7} - 41 q^{9} - 14 q^{10} + 560 q^{13} - 7 q^{15} - 1498 q^{16} - 1722 q^{18} - 14 q^{19} - 3157 q^{21} - 1084 q^{22} + 3875 q^{24} + 56 q^{25} + 371 q^{27} - 2500 q^{28} - 8102 q^{30} - 14 q^{31} - 3503 q^{33} - 6616 q^{34} - 6791 q^{36} - 14 q^{37} + 11319 q^{39} + 21490 q^{40} - 5048 q^{42} + 5376 q^{43} - 51 q^{45} + 21714 q^{48} + 15952 q^{49} + 6478 q^{51} + 18772 q^{52} + 5721 q^{54} + 23562 q^{55} + 4470 q^{57} - 15610 q^{58} - 4270 q^{60} - 25466 q^{61} + 11921 q^{63} - 68310 q^{64} - 7 q^{66} - 7662 q^{67} - 27335 q^{69} + 50050 q^{72} + 53452 q^{73} - 18564 q^{76} - 51173 q^{78} - 94920 q^{79} - 36645 q^{81} + 22758 q^{82} - 68439 q^{84} + 7854 q^{85} - 44849 q^{87} + 4324 q^{88} + 143563 q^{90} + 79468 q^{91} + 19421 q^{93} + 22360 q^{94} - 55258 q^{96} - 105252 q^{97}+O(q^{100}) \) 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 −4.84636 − 6.07714i −0.0630893 + 8.99978i −9.88410 + 43.3051i 6.17647 − 4.92557i 54.9986 − 43.2327i 7.38534 + 32.3573i 199.022 − 95.8438i −80.9920 − 1.13558i −59.8668 − 13.6642i
5.2 −4.64187 − 5.82072i −6.85126 − 5.83611i −8.77349 + 38.4392i −6.45122 + 5.14468i −2.16770 + 66.9697i −1.08395 − 4.74909i 157.146 − 75.6775i 12.8796 + 79.9695i 59.8914 + 13.6698i
5.3 −4.45315 − 5.58408i 7.66573 − 4.71557i −7.79100 + 34.1346i −30.7329 + 24.5087i −60.4687 − 21.8069i 1.07892 + 4.72706i 122.345 − 58.9182i 36.5268 − 72.2966i 273.717 + 62.4740i
5.4 −3.93802 − 4.93812i 8.99751 + 0.211854i −5.31669 + 23.2939i 31.0400 − 24.7536i −34.3862 − 45.2650i 13.0809 + 57.3112i 44.9158 − 21.6303i 80.9102 + 3.81232i −244.472 − 55.7991i
5.5 −3.86613 − 4.84798i 2.21666 − 8.72275i −4.99556 + 21.8870i 19.0555 − 15.1963i −50.8576 + 22.9770i −6.41339 − 28.0989i 36.0334 − 17.3528i −71.1728 − 38.6708i −147.342 − 33.6299i
5.6 −3.71513 − 4.65863i −7.95393 + 4.21129i −4.34028 + 19.0160i 24.0467 − 19.1766i 49.1688 + 21.4089i −17.2589 − 75.6163i 18.8168 − 9.06171i 45.5300 − 66.9927i −178.673 − 40.7810i
5.7 −3.46377 − 4.34343i −7.49274 + 4.98586i −3.30735 + 14.4904i −33.4963 + 26.7124i 47.6089 + 15.2743i 4.07883 + 17.8705i −5.69063 + 2.74046i 31.2823 − 74.7156i 232.047 + 52.9632i
5.8 −3.45324 − 4.33023i 4.08562 + 8.01921i −3.26566 + 14.3078i −10.7794 + 8.59626i 20.6164 − 45.3839i −17.3459 − 75.9976i −6.60819 + 3.18234i −47.6155 + 65.5268i 74.4475 + 16.9922i
5.9 −3.37914 − 4.23730i 7.86005 + 4.38401i −2.97584 + 13.0380i −8.34256 + 6.65297i −7.98382 − 48.1196i 2.43556 + 10.6709i −12.8261 + 6.17673i 42.5609 + 68.9171i 56.3813 + 12.8687i
5.10 −3.10594 − 3.89472i −8.27340 − 3.54273i −1.96168 + 8.59469i 14.5932 − 11.6377i 11.8987 + 43.2261i 16.5337 + 72.4387i −32.2446 + 15.5282i 55.8982 + 58.6208i −90.6514 − 20.6906i
5.11 −2.31161 − 2.89867i −2.74431 + 8.57139i 0.501599 − 2.19765i 16.3358 − 13.0274i 31.1894 − 11.8589i 11.3883 + 49.8954i −60.9758 + 29.3644i −65.9375 − 47.0452i −75.5240 − 17.2379i
5.12 −2.21069 − 2.77211i 0.110090 − 8.99933i 0.762855 − 3.34229i −15.5504 + 12.4010i −25.1905 + 19.5895i 13.2233 + 57.9349i −62.0642 + 29.8886i −80.9758 − 1.98146i 68.7542 + 15.6927i
5.13 −1.86558 − 2.33937i −7.16918 − 5.44085i 1.56810 − 6.87029i −19.6666 + 15.6836i 0.646564 + 26.9217i −16.5976 − 72.7190i −62.1310 + 29.9207i 21.7943 + 78.0129i 73.3792 + 16.7483i
5.14 −1.74166 − 2.18397i 7.93215 − 4.25218i 1.82398 − 7.99136i −1.62499 + 1.29589i −23.1018 − 9.91774i −10.0342 − 43.9626i −60.8980 + 29.3270i 44.8379 − 67.4579i 5.66036 + 1.29194i
5.15 −1.15986 − 1.45442i −1.57807 + 8.86057i 2.79028 − 12.2250i −4.43482 + 3.53665i 14.7173 − 7.98186i 2.86845 + 12.5675i −47.8334 + 23.0353i −76.0194 − 27.9652i 10.2876 + 2.34807i
5.16 −1.11500 − 1.39816i 6.56682 + 6.15442i 2.84870 − 12.4810i 33.2921 − 26.5495i 1.28290 − 16.0436i −6.73011 − 29.4866i −46.4061 + 22.3480i 5.24613 + 80.8299i −74.2410 − 16.9450i
5.17 −0.615476 − 0.771782i 8.48372 + 3.00443i 3.34350 − 14.6488i −22.6898 + 18.0945i −2.90276 − 8.39673i 16.5966 + 72.7146i −27.5938 + 13.2885i 62.9469 + 50.9774i 27.9300 + 6.37485i
5.18 −0.582461 − 0.730384i −8.96957 + 0.739525i 3.36614 − 14.7480i 15.8907 − 12.6724i 5.76456 + 6.12048i 1.15799 + 5.07350i −26.1992 + 12.6169i 79.9062 − 13.2664i −18.5115 − 4.22512i
5.19 −0.409239 − 0.513169i −2.02782 − 8.76858i 3.46447 − 15.1788i 30.3828 − 24.2295i −3.66990 + 4.62906i −8.91318 − 39.0512i −18.6690 + 8.99051i −72.7759 + 35.5621i −24.8677 − 5.67589i
5.20 0.409239 + 0.513169i 1.97754 + 8.78005i 3.46447 − 15.1788i −30.3828 + 24.2295i −3.69637 + 4.60795i −8.91318 − 39.0512i 18.6690 − 8.99051i −73.1786 + 34.7259i −24.8677 − 5.67589i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.e even 14 1 inner
87.h odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.5.h.a ✓ 228
3.b odd 2 1 inner 87.5.h.a ✓ 228
29.e even 14 1 inner 87.5.h.a ✓ 228
87.h odd 14 1 inner 87.5.h.a ✓ 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.5.h.a ✓ 228 1.a even 1 1 trivial
87.5.h.a ✓ 228 3.b odd 2 1 inner
87.5.h.a ✓ 228 29.e even 14 1 inner
87.5.h.a ✓ 228 87.h odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(87, [\chi])\).