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Properties

Label 84.9.p.b
Level $84$
Weight $9$
Character orbit 84.p
Analytic conductor $34.220$
Analytic rank $0$
Dimension $40$
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] = [84,9,Mod(53,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.p (of order \(6\), 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: \(34.2198032451\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 0 −78.9384 + 18.1585i 0 −705.317 + 407.215i 0 −2163.85 − 1040.46i 0 5901.54 − 2866.81i 0
53.2 0 −77.6406 − 23.0855i 0 726.008 − 419.161i 0 −1188.37 + 2086.29i 0 5495.12 + 3584.74i 0
53.3 0 −72.4880 − 36.1454i 0 −737.233 + 425.642i 0 2359.93 − 442.194i 0 3948.02 + 5240.22i 0
53.4 0 −71.6341 + 37.8094i 0 −304.163 + 175.609i 0 517.537 + 2344.56i 0 3701.90 − 5416.89i 0
53.5 0 −71.4876 − 38.0858i 0 357.784 − 206.567i 0 −24.5961 − 2400.87i 0 3659.95 + 5445.32i 0
53.6 0 −67.0937 + 45.3810i 0 701.327 − 404.911i 0 1952.36 − 1397.53i 0 2442.13 − 6089.56i 0
53.7 0 −34.9991 − 73.0484i 0 −282.056 + 162.845i 0 −2400.75 + 34.7040i 0 −4111.13 + 5113.25i 0
53.8 0 −17.6795 − 79.0470i 0 128.664 − 74.2840i 0 2032.57 + 1278.06i 0 −5935.87 + 2795.03i 0
53.9 0 −5.75427 + 80.7953i 0 −701.327 + 404.911i 0 1952.36 − 1397.53i 0 −6494.78 − 929.836i 0
53.10 0 3.07318 + 80.9417i 0 304.163 − 175.609i 0 517.537 + 2344.56i 0 −6542.11 + 497.497i 0
53.11 0 21.2854 − 78.1533i 0 609.234 − 351.741i 0 162.484 − 2395.50i 0 −5654.86 − 3327.05i 0
53.12 0 23.7435 + 77.4419i 0 705.317 − 407.215i 0 −2163.85 − 1040.46i 0 −5433.50 + 3677.48i 0
53.13 0 28.5930 − 75.7855i 0 −918.968 + 530.566i 0 −1255.82 + 2046.39i 0 −4925.88 − 4333.87i 0
53.14 0 51.3357 − 62.6550i 0 918.968 − 530.566i 0 −1255.82 + 2046.39i 0 −1290.30 − 6432.87i 0
53.15 0 57.0400 − 57.5103i 0 −609.234 + 351.741i 0 162.484 − 2395.50i 0 −53.8743 − 6560.78i 0
53.16 0 58.8129 + 55.6960i 0 −726.008 + 419.161i 0 −1188.37 + 2086.29i 0 356.917 + 6551.28i 0
53.17 0 67.5468 + 44.7037i 0 737.233 − 425.642i 0 2359.93 − 442.194i 0 2564.15 + 6039.19i 0
53.18 0 68.7270 + 42.8672i 0 −357.784 + 206.567i 0 −24.5961 − 2400.87i 0 2885.81 + 5892.27i 0
53.19 0 77.2965 − 24.2126i 0 −128.664 + 74.2840i 0 2032.57 + 1278.06i 0 5388.50 − 3743.10i 0
53.20 0 80.7613 − 6.21408i 0 282.056 − 162.845i 0 −2400.75 + 34.7040i 0 6483.77 − 1003.71i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.9.p.b ✓ 40
3.b odd 2 1 inner 84.9.p.b ✓ 40
7.c even 3 1 inner 84.9.p.b ✓ 40
21.h odd 6 1 inner 84.9.p.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.p.b ✓ 40 1.a even 1 1 trivial
84.9.p.b ✓ 40 3.b odd 2 1 inner
84.9.p.b ✓ 40 7.c even 3 1 inner
84.9.p.b ✓ 40 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} - 4789646 T_{5}^{38} + 13199357691382 T_{5}^{36} + \cdots + 37\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display