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Properties

Label 867.6.a.u
Level $867$
Weight $6$
Character orbit 867.a
Self dual yes
Analytic conductor $139.053$
Analytic rank $1$
Dimension $28$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,6,Mod(1,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 867.a (trivial)

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: yes
Analytic conductor: \(139.052771778\)
Analytic rank: \(1\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9} - 800 q^{10} - 1132 q^{11} + 3456 q^{12} - 828 q^{13} - 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} - 9216 q^{20} - 3528 q^{21} - 664 q^{22} - 4548 q^{23} + 9904 q^{25} + 20412 q^{27} - 36936 q^{28} - 36680 q^{29} - 7200 q^{30} - 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} - 42360 q^{37} - 1984 q^{38} - 7452 q^{39} - 81000 q^{40} - 6060 q^{41} - 14112 q^{42} - 50044 q^{43} - 56960 q^{44} - 19764 q^{45} - 51488 q^{46} + 16552 q^{47} + 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} + 9248 q^{56} - 67788 q^{57} - 95256 q^{58} - 89832 q^{59} - 82944 q^{60} - 154104 q^{61} - 106624 q^{62} - 31752 q^{63} + 93720 q^{64} - 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} + 24504 q^{71} - 58184 q^{73} - 308992 q^{74} + 89136 q^{75} - 602544 q^{76} - 240576 q^{77} - 136080 q^{79} - 317440 q^{80} + 183708 q^{81} - 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} - 145152 q^{88} - 120880 q^{89} - 64800 q^{90} - 290152 q^{91} - 491616 q^{92} - 69192 q^{93} - 799168 q^{94} - 543084 q^{95} - 409328 q^{97} + 193072 q^{98} - 91692 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.9873 9.00000 88.7215 4.69803 −98.8860 −7.69579 −623.219 81.0000 −51.6188
1.2 −9.63043 9.00000 60.7452 −14.5139 −86.6739 39.9641 −276.829 81.0000 139.775
1.3 −9.48495 9.00000 57.9642 −38.7969 −85.3645 −235.518 −246.269 81.0000 367.987
1.4 −9.14605 9.00000 51.6502 111.291 −82.3145 23.2196 −179.722 81.0000 −1017.87
1.5 −8.53096 9.00000 40.7773 −62.0793 −76.7787 −151.712 −74.8788 81.0000 529.596
1.6 −6.54127 9.00000 10.7882 17.0260 −58.8714 −82.3266 138.752 81.0000 −111.372
1.7 −6.13922 9.00000 5.69003 −76.4764 −55.2530 112.843 161.523 81.0000 469.505
1.8 −5.43762 9.00000 −2.43230 −32.4834 −48.9386 230.948 187.230 81.0000 176.632
1.9 −4.87620 9.00000 −8.22272 −88.7638 −43.8858 118.568 196.134 81.0000 432.830
1.10 −4.85553 9.00000 −8.42383 82.3928 −43.6998 35.3683 196.279 81.0000 −400.061
1.11 −3.92923 9.00000 −16.5612 −77.2813 −35.3630 −239.397 190.808 81.0000 303.656
1.12 −2.68843 9.00000 −24.7723 41.7848 −24.1959 18.3667 152.629 81.0000 −112.336
1.13 −1.50376 9.00000 −29.7387 75.9621 −13.5339 126.404 92.8403 81.0000 −114.229
1.14 0.614853 9.00000 −31.6220 24.2891 5.53368 30.9562 −39.1182 81.0000 14.9342
1.15 0.803842 9.00000 −31.3538 37.1951 7.23458 −118.502 −50.9265 81.0000 29.8990
1.16 1.77098 9.00000 −28.8636 −66.3668 15.9388 199.014 −107.788 81.0000 −117.534
1.17 2.14764 9.00000 −27.3876 22.9740 19.3288 −41.5643 −127.544 81.0000 49.3400
1.18 2.41208 9.00000 −26.1818 −49.1460 21.7088 −82.6237 −140.340 81.0000 −118.544
1.19 3.74189 9.00000 −17.9983 −104.629 33.6770 −143.317 −187.088 81.0000 −391.510
1.20 5.50033 9.00000 −1.74633 −45.6942 49.5030 98.6843 −185.616 81.0000 −251.333
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.6.a.u 28
17.b even 2 1 867.6.a.t 28
17.e odd 16 2 51.6.h.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.6.h.a ✓ 56 17.e odd 16 2
867.6.a.t 28 17.b even 2 1
867.6.a.u 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(867))\):

\( T_{2}^{28} - 640 T_{2}^{26} + 180224 T_{2}^{24} - 29441284 T_{2}^{22} + 34944 T_{2}^{21} + \cdots - 11\!\cdots\!96 \) Copy content Toggle raw display
\( T_{5}^{28} + 244 T_{5}^{27} - 18934 T_{5}^{26} - 8872652 T_{5}^{25} - 139972777 T_{5}^{24} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display