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Properties

Label 89.6.c.a
Level $89$
Weight $6$
Character orbit 89.c
Analytic conductor $14.274$
Analytic rank $0$
Dimension $74$
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] = [89,6,Mod(34,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.34");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 89.c (of order \(4\), degree \(2\), 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: \(14.2741599634\)
Analytic rank: \(0\)
Dimension: \(74\)
Relative dimension: \(37\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 74 q + 4 q^{2} + 22 q^{3} + 1180 q^{4} + 22 q^{6} - 2 q^{7} + 252 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q + 4 q^{2} + 22 q^{3} + 1180 q^{4} + 22 q^{6} - 2 q^{7} + 252 q^{8} - 932 q^{11} + 334 q^{12} + 1176 q^{13} + 964 q^{14} - 586 q^{15} + 17292 q^{16} + 542 q^{19} + 1708 q^{22} - 4782 q^{23} - 12380 q^{24} - 27026 q^{25} + 17712 q^{26} - 17372 q^{27} - 21622 q^{28} - 18212 q^{29} + 8464 q^{30} - 12276 q^{31} + 10108 q^{32} + 5654 q^{33} + 6840 q^{35} - 8606 q^{37} - 13760 q^{38} + 8752 q^{39} - 14428 q^{41} - 44876 q^{43} - 41028 q^{44} + 108764 q^{45} - 13010 q^{46} - 18560 q^{48} + 110068 q^{50} + 31762 q^{51} + 39320 q^{52} - 106856 q^{54} + 17256 q^{56} - 175852 q^{57} + 205736 q^{58} - 47538 q^{59} + 22672 q^{60} - 20336 q^{61} + 498 q^{62} + 70820 q^{63} + 254028 q^{64} + 180274 q^{65} - 12822 q^{66} + 15000 q^{67} + 234082 q^{70} + 168636 q^{73} + 68498 q^{74} - 177578 q^{75} - 169166 q^{76} - 82404 q^{77} + 56512 q^{78} - 508046 q^{81} + 152926 q^{82} - 401110 q^{83} + 33204 q^{85} + 355942 q^{86} - 894860 q^{87} - 292980 q^{88} - 125904 q^{89} - 185936 q^{90} - 101396 q^{91} - 176824 q^{92} - 105576 q^{93} - 226134 q^{95} - 900726 q^{96} + 495728 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −11.1652 12.9086 + 12.9086i 92.6620 51.3368i −144.127 − 144.127i −38.1861 − 38.1861i −677.304 90.2614i − 573.186i
34.2 −10.1318 −8.88786 − 8.88786i 70.6540 − 31.9907i 90.0503 + 90.0503i 110.515 + 110.515i −391.636 − 85.0118i 324.125i
34.3 −10.1230 −2.74151 − 2.74151i 70.4757 − 74.3546i 27.7524 + 27.7524i −121.852 − 121.852i −389.490 − 227.968i 752.694i
34.4 −9.31444 −10.5560 − 10.5560i 54.7588 91.9156i 98.3231 + 98.3231i 4.82417 + 4.82417i −211.986 − 20.1424i − 856.142i
34.5 −8.73716 −20.7191 − 20.7191i 44.3380 − 24.7452i 181.026 + 181.026i −130.971 − 130.971i −107.800 615.563i 216.203i
34.6 −8.33444 5.03297 + 5.03297i 37.4628 − 25.6804i −41.9469 − 41.9469i 79.2999 + 79.2999i −45.5295 − 192.339i 214.031i
34.7 −8.23319 16.0215 + 16.0215i 35.7853 11.4927i −131.908 − 131.908i 122.032 + 122.032i −31.1654 270.380i − 94.6214i
34.8 −8.17932 19.6369 + 19.6369i 34.9013 − 99.3531i −160.616 − 160.616i −76.1107 − 76.1107i −23.7311 528.213i 812.641i
34.9 −6.92880 8.64309 + 8.64309i 16.0083 48.3047i −59.8862 − 59.8862i −120.452 − 120.452i 110.804 − 93.5940i − 334.694i
34.10 −5.56471 −4.45821 − 4.45821i −1.03401 − 16.0987i 24.8087 + 24.8087i −67.5418 − 67.5418i 183.825 − 203.249i 89.5848i
34.11 −5.27752 −17.4351 − 17.4351i −4.14778 − 72.5592i 92.0140 + 92.0140i 151.321 + 151.321i 190.771 364.964i 382.933i
34.12 −5.08215 −14.2757 − 14.2757i −6.17175 27.4125i 72.5514 + 72.5514i 12.0772 + 12.0772i 193.995 164.593i − 139.314i
34.13 −4.26104 1.21850 + 1.21850i −13.8435 − 98.8112i −5.19207 − 5.19207i 23.4297 + 23.4297i 195.341 − 240.031i 421.038i
34.14 −3.79238 3.54149 + 3.54149i −17.6178 84.9208i −13.4307 − 13.4307i 146.015 + 146.015i 188.170 − 217.916i − 322.052i
34.15 −2.68206 21.2053 + 21.2053i −24.8065 57.1882i −56.8740 − 56.8740i −34.7134 − 34.7134i 152.359 656.331i − 153.382i
34.16 −1.36016 12.7284 + 12.7284i −30.1500 − 35.3046i −17.3127 − 17.3127i −22.6307 − 22.6307i 84.5341 81.0259i 48.0201i
34.17 −1.35957 −6.45473 − 6.45473i −30.1516 7.99433i 8.77564 + 8.77564i −99.5277 − 99.5277i 84.4993 − 159.673i − 10.8688i
34.18 −1.22065 11.2674 + 11.2674i −30.5100 − 47.1280i −13.7536 − 13.7536i 62.8269 + 62.8269i 76.3029 10.9088i 57.5268i
34.19 −0.469944 −17.4069 − 17.4069i −31.7792 65.5409i 8.18025 + 8.18025i 47.5080 + 47.5080i 29.9726 362.997i − 30.8006i
34.20 1.05214 −14.9069 − 14.9069i −30.8930 − 85.8974i −15.6842 − 15.6842i −118.190 − 118.190i −66.1722 201.434i − 90.3760i
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.37
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.6.c.a ✓ 74
89.c even 4 1 inner 89.6.c.a ✓ 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.6.c.a ✓ 74 1.a even 1 1 trivial
89.6.c.a ✓ 74 89.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(89, [\chi])\).