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Properties

Label 252.8.e.a
Level $252$
Weight $8$
Character orbit 252.e
Analytic conductor $78.721$
Analytic rank $0$
Dimension $84$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(71,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.71");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.e (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: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 104 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 104 q^{4} - 5816 q^{10} + 16684 q^{16} - 201700 q^{22} - 1312500 q^{25} + 39788 q^{28} - 400048 q^{34} - 165544 q^{37} - 1477624 q^{40} + 1616724 q^{46} - 9882516 q^{49} - 3744592 q^{52} + 12053212 q^{58} - 8764504 q^{61} - 5645200 q^{64} + 8997576 q^{70} - 38069208 q^{76} + 24696392 q^{82} - 29977064 q^{85} - 48692036 q^{88} + 73638456 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −11.2902 − 0.728783i 0 126.938 + 16.4562i 507.809i 0 343.000i −1421.16 − 278.305i 0 370.082 − 5733.27i
71.2 −11.2902 + 0.728783i 0 126.938 − 16.4562i − 507.809i 0 − 343.000i −1421.16 + 278.305i 0 370.082 + 5733.27i
71.3 −11.2162 − 1.48214i 0 123.607 + 33.2480i 80.1172i 0 − 343.000i −1337.12 − 556.119i 0 118.745 − 898.610i
71.4 −11.2162 + 1.48214i 0 123.607 − 33.2480i − 80.1172i 0 343.000i −1337.12 + 556.119i 0 118.745 + 898.610i
71.5 −11.2131 − 1.50530i 0 123.468 + 33.7582i − 353.743i 0 343.000i −1333.65 − 564.392i 0 −532.489 + 3966.56i
71.6 −11.2131 + 1.50530i 0 123.468 − 33.7582i 353.743i 0 − 343.000i −1333.65 + 564.392i 0 −532.489 − 3966.56i
71.7 −10.8566 − 3.18337i 0 107.732 + 69.1213i 512.680i 0 343.000i −949.569 − 1093.38i 0 1632.05 − 5565.97i
71.8 −10.8566 + 3.18337i 0 107.732 − 69.1213i − 512.680i 0 − 343.000i −949.569 + 1093.38i 0 1632.05 + 5565.97i
71.9 −10.6314 − 3.86955i 0 98.0532 + 82.2774i − 192.805i 0 − 343.000i −724.067 − 1254.14i 0 −746.066 + 2049.78i
71.10 −10.6314 + 3.86955i 0 98.0532 − 82.2774i 192.805i 0 343.000i −724.067 + 1254.14i 0 −746.066 − 2049.78i
71.11 −10.2476 − 4.79436i 0 82.0282 + 98.2617i 145.328i 0 − 343.000i −369.493 − 1400.22i 0 696.754 − 1489.27i
71.12 −10.2476 + 4.79436i 0 82.0282 − 98.2617i − 145.328i 0 343.000i −369.493 + 1400.22i 0 696.754 + 1489.27i
71.13 −9.96852 − 5.35058i 0 70.7426 + 106.675i 147.762i 0 343.000i −134.428 − 1441.90i 0 790.614 − 1472.97i
71.14 −9.96852 + 5.35058i 0 70.7426 − 106.675i − 147.762i 0 − 343.000i −134.428 + 1441.90i 0 790.614 + 1472.97i
71.15 −9.67940 − 5.85741i 0 59.3814 + 113.392i − 181.049i 0 343.000i 89.4107 − 1445.39i 0 −1060.48 + 1752.44i
71.16 −9.67940 + 5.85741i 0 59.3814 − 113.392i 181.049i 0 − 343.000i 89.4107 + 1445.39i 0 −1060.48 − 1752.44i
71.17 −9.44210 − 6.23271i 0 50.3066 + 117.700i 254.394i 0 − 343.000i 258.590 − 1424.88i 0 1585.56 − 2402.01i
71.18 −9.44210 + 6.23271i 0 50.3066 − 117.700i − 254.394i 0 343.000i 258.590 + 1424.88i 0 1585.56 + 2402.01i
71.19 −8.97566 − 6.88749i 0 33.1249 + 123.640i − 498.542i 0 343.000i 554.249 − 1337.89i 0 −3433.70 + 4474.74i
71.20 −8.97566 + 6.88749i 0 33.1249 − 123.640i 498.542i 0 − 343.000i 554.249 + 1337.89i 0 −3433.70 − 4474.74i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.84
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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.e.a ✓ 84
3.b odd 2 1 inner 252.8.e.a ✓ 84
4.b odd 2 1 inner 252.8.e.a ✓ 84
12.b even 2 1 inner 252.8.e.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.e.a ✓ 84 1.a even 1 1 trivial
252.8.e.a ✓ 84 3.b odd 2 1 inner
252.8.e.a ✓ 84 4.b odd 2 1 inner
252.8.e.a ✓ 84 12.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(252, [\chi])\).