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Properties

Label 84.9.h.e
Level $84$
Weight $9$
Character orbit 84.h
Analytic conductor $34.220$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,9,Mod(83,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.83");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.h (of order \(2\), degree \(1\), 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: \(34.2198032451\)
Analytic rank: \(0\)
Dimension: \(120\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 1028 q^{4} - 26248 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 1028 q^{4} - 26248 q^{9} - 373500 q^{16} - 289340 q^{18} - 620200 q^{21} - 640592 q^{22} + 9327272 q^{25} + 54500 q^{28} - 1299380 q^{30} - 11227168 q^{36} - 8553936 q^{37} - 8996308 q^{42} - 26738344 q^{46} - 13574664 q^{49} - 50788176 q^{57} - 18533288 q^{58} - 24117668 q^{60} - 99498308 q^{64} + 10779752 q^{70} + 76835716 q^{72} - 13132860 q^{78} + 23514872 q^{81} - 171096188 q^{84} - 88829152 q^{85} + 184844592 q^{88} + 626423664 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −15.8393 − 2.26227i −53.5919 + 60.7364i 245.764 + 71.6654i −873.037 986.258 − 840.780i 817.708 + 2257.47i −3730.60 − 1691.11i −816.814 − 6509.96i 13828.3 + 1975.05i
83.2 −15.8393 − 2.26227i 53.5919 − 60.7364i 245.764 + 71.6654i 873.037 −986.258 + 840.780i −817.708 + 2257.47i −3730.60 − 1691.11i −816.814 − 6509.96i −13828.3 − 1975.05i
83.3 −15.8393 + 2.26227i −53.5919 − 60.7364i 245.764 − 71.6654i −873.037 986.258 + 840.780i 817.708 − 2257.47i −3730.60 + 1691.11i −816.814 + 6509.96i 13828.3 − 1975.05i
83.4 −15.8393 + 2.26227i 53.5919 + 60.7364i 245.764 − 71.6654i 873.037 −986.258 − 840.780i −817.708 − 2257.47i −3730.60 + 1691.11i −816.814 + 6509.96i −13828.3 + 1975.05i
83.5 −15.5666 − 3.69894i −6.21954 − 80.7609i 228.636 + 115.160i −832.715 −201.913 + 1280.17i −1274.55 + 2034.78i −3133.10 − 2638.35i −6483.63 + 1004.59i 12962.5 + 3080.16i
83.6 −15.5666 − 3.69894i 6.21954 + 80.7609i 228.636 + 115.160i 832.715 201.913 − 1280.17i 1274.55 + 2034.78i −3133.10 − 2638.35i −6483.63 + 1004.59i −12962.5 − 3080.16i
83.7 −15.5666 + 3.69894i −6.21954 + 80.7609i 228.636 − 115.160i −832.715 −201.913 − 1280.17i −1274.55 − 2034.78i −3133.10 + 2638.35i −6483.63 − 1004.59i 12962.5 − 3080.16i
83.8 −15.5666 + 3.69894i 6.21954 − 80.7609i 228.636 − 115.160i 832.715 201.913 + 1280.17i 1274.55 − 2034.78i −3133.10 + 2638.35i −6483.63 − 1004.59i −12962.5 + 3080.16i
83.9 −15.4621 − 4.11379i −70.5243 + 39.8412i 222.153 + 127.216i 445.440 1254.35 − 325.907i 907.809 − 2222.76i −2911.62 − 2880.92i 3386.35 − 5619.55i −6887.45 − 1832.45i
83.10 −15.4621 − 4.11379i 70.5243 − 39.8412i 222.153 + 127.216i −445.440 −1254.35 + 325.907i −907.809 − 2222.76i −2911.62 − 2880.92i 3386.35 − 5619.55i 6887.45 + 1832.45i
83.11 −15.4621 + 4.11379i −70.5243 − 39.8412i 222.153 − 127.216i 445.440 1254.35 + 325.907i 907.809 + 2222.76i −2911.62 + 2880.92i 3386.35 + 5619.55i −6887.45 + 1832.45i
83.12 −15.4621 + 4.11379i 70.5243 + 39.8412i 222.153 − 127.216i −445.440 −1254.35 − 325.907i −907.809 + 2222.76i −2911.62 + 2880.92i 3386.35 + 5619.55i 6887.45 − 1832.45i
83.13 −14.6697 − 6.38757i −24.3762 − 77.2451i 174.398 + 187.407i 165.741 −135.818 + 1288.86i 2400.65 + 41.1732i −1361.28 − 3863.18i −5372.60 + 3765.88i −2431.37 − 1058.68i
83.14 −14.6697 − 6.38757i 24.3762 + 77.2451i 174.398 + 187.407i −165.741 135.818 − 1288.86i −2400.65 + 41.1732i −1361.28 − 3863.18i −5372.60 + 3765.88i 2431.37 + 1058.68i
83.15 −14.6697 + 6.38757i −24.3762 + 77.2451i 174.398 − 187.407i 165.741 −135.818 − 1288.86i 2400.65 − 41.1732i −1361.28 + 3863.18i −5372.60 − 3765.88i −2431.37 + 1058.68i
83.16 −14.6697 + 6.38757i 24.3762 − 77.2451i 174.398 − 187.407i −165.741 135.818 + 1288.86i −2400.65 − 41.1732i −1361.28 + 3863.18i −5372.60 − 3765.88i 2431.37 − 1058.68i
83.17 −14.0419 − 7.66983i −54.1383 − 60.2498i 138.347 + 215.397i 889.283 298.096 + 1261.25i −2310.93 − 651.476i −290.594 − 4085.68i −699.086 + 6523.65i −12487.2 − 6820.65i
83.18 −14.0419 − 7.66983i 54.1383 + 60.2498i 138.347 + 215.397i −889.283 −298.096 − 1261.25i 2310.93 − 651.476i −290.594 − 4085.68i −699.086 + 6523.65i 12487.2 + 6820.65i
83.19 −14.0419 + 7.66983i −54.1383 + 60.2498i 138.347 − 215.397i 889.283 298.096 − 1261.25i −2310.93 + 651.476i −290.594 + 4085.68i −699.086 − 6523.65i −12487.2 + 6820.65i
83.20 −14.0419 + 7.66983i 54.1383 − 60.2498i 138.347 − 215.397i −889.283 −298.096 + 1261.25i 2310.93 + 651.476i −290.594 + 4085.68i −699.086 − 6523.65i 12487.2 − 6820.65i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.120
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.9.h.e ✓ 120
3.b odd 2 1 inner 84.9.h.e ✓ 120
4.b odd 2 1 inner 84.9.h.e ✓ 120
7.b odd 2 1 inner 84.9.h.e ✓ 120
12.b even 2 1 inner 84.9.h.e ✓ 120
21.c even 2 1 inner 84.9.h.e ✓ 120
28.d even 2 1 inner 84.9.h.e ✓ 120
84.h odd 2 1 inner 84.9.h.e ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.h.e ✓ 120 1.a even 1 1 trivial
84.9.h.e ✓ 120 3.b odd 2 1 inner
84.9.h.e ✓ 120 4.b odd 2 1 inner
84.9.h.e ✓ 120 7.b odd 2 1 inner
84.9.h.e ✓ 120 12.b even 2 1 inner
84.9.h.e ✓ 120 21.c even 2 1 inner
84.9.h.e ✓ 120 28.d even 2 1 inner
84.9.h.e ✓ 120 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\):

\( T_{5}^{30} - 7025284 T_{5}^{28} + 21775882117040 T_{5}^{26} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
\( T_{11}^{30} - 3582593188 T_{11}^{28} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display