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Properties

Label 189.6.h.a
Level $189$
Weight $6$
Character orbit 189.h
Analytic conductor $30.313$
Analytic rank $0$
Dimension $76$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,6,Mod(37,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 189.h (of order \(3\), degree \(2\), not 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: \(30.3125419447\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 2 q^{2} + 1150 q^{4} - 101 q^{5} + 28 q^{7} + 72 q^{8} - 66 q^{10} - 191 q^{11} + 179 q^{13} - 416 q^{14} + 16318 q^{16} - 2043 q^{17} - 64 q^{19} - 2782 q^{20} - 66 q^{22} - 2604 q^{23} - 18749 q^{25}+ \cdots + 282479 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −10.9172 0 87.1846 −40.9498 − 70.9272i 0 −107.267 + 72.8067i −602.459 0 447.056 + 774.324i
37.2 −10.3855 0 75.8581 39.4589 + 68.3448i 0 50.3371 + 119.470i −455.487 0 −409.800 − 709.794i
37.3 −9.81701 0 64.3736 −0.453286 − 0.785115i 0 −96.2340 − 86.8678i −317.812 0 4.44991 + 7.70748i
37.4 −9.72907 0 62.6549 −16.5597 − 28.6823i 0 88.9463 − 94.3163i −298.243 0 161.111 + 279.052i
37.5 −9.14504 0 51.6318 41.8963 + 72.5666i 0 74.4422 − 106.138i −179.533 0 −383.144 − 663.624i
37.6 −8.81400 0 45.6865 5.67709 + 9.83300i 0 42.0784 + 122.623i −120.633 0 −50.0378 − 86.6681i
37.7 −8.33464 0 37.4662 −50.6971 − 87.8099i 0 127.276 − 24.6517i −45.5588 0 422.542 + 731.864i
37.8 −7.07030 0 17.9891 16.4872 + 28.5567i 0 −69.0783 + 109.705i 99.0612 0 −116.569 − 201.904i
37.9 −6.94033 0 16.1682 5.68653 + 9.84936i 0 −100.728 − 81.6139i 109.878 0 −39.4664 − 68.3578i
37.10 −5.61825 0 −0.435257 −9.51260 − 16.4763i 0 93.8122 + 89.4778i 182.229 0 53.4442 + 92.5680i
37.11 −5.32824 0 −3.60982 −38.2058 − 66.1743i 0 −123.522 − 39.3605i 189.738 0 203.570 + 352.593i
37.12 −4.91629 0 −7.83005 21.1613 + 36.6524i 0 108.906 − 70.3317i 195.816 0 −104.035 − 180.194i
37.13 −4.55945 0 −11.2114 53.5656 + 92.7784i 0 −99.4660 − 83.1476i 197.020 0 −244.230 − 423.019i
37.14 −4.36774 0 −12.9228 −43.9378 − 76.1025i 0 −66.2807 + 111.418i 196.211 0 191.909 + 332.396i
37.15 −2.93872 0 −23.3639 17.7468 + 30.7383i 0 −81.7771 + 100.596i 162.699 0 −52.1529 − 90.3315i
37.16 −2.69455 0 −24.7394 −27.4913 − 47.6164i 0 37.5886 − 124.073i 152.887 0 74.0768 + 128.305i
37.17 −1.98517 0 −28.0591 −16.9202 − 29.3067i 0 123.901 − 38.1504i 119.227 0 33.5895 + 58.1787i
37.18 −0.975729 0 −31.0480 20.6627 + 35.7889i 0 110.650 + 67.5548i 61.5177 0 −20.1612 − 34.9203i
37.19 0.744106 0 −31.4463 48.6549 + 84.2728i 0 116.701 − 56.4623i −47.2108 0 36.2044 + 62.7079i
37.20 0.964960 0 −31.0689 −33.8862 − 58.6927i 0 −51.0811 − 119.154i −60.8589 0 −32.6989 − 56.6361i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.6.h.a 76
3.b odd 2 1 63.6.h.a yes 76
7.c even 3 1 189.6.g.a 76
9.c even 3 1 189.6.g.a 76
9.d odd 6 1 63.6.g.a ✓ 76
21.h odd 6 1 63.6.g.a ✓ 76
63.h even 3 1 inner 189.6.h.a 76
63.j odd 6 1 63.6.h.a yes 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.g.a ✓ 76 9.d odd 6 1
63.6.g.a ✓ 76 21.h odd 6 1
63.6.h.a yes 76 3.b odd 2 1
63.6.h.a yes 76 63.j odd 6 1
189.6.g.a 76 7.c even 3 1
189.6.g.a 76 9.c even 3 1
189.6.h.a 76 1.a even 1 1 trivial
189.6.h.a 76 63.h even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(189, [\chi])\).