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Properties

Label 93.4.c.b
Level $93$
Weight $4$
Character orbit 93.c
Analytic conductor $5.487$
Analytic rank $0$
Dimension $28$
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] = [93,4,Mod(92,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.92");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 93.c (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: \(5.48717763053\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 144 q^{4} - 64 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 144 q^{4} - 64 q^{7} + 20 q^{9} - 8 q^{10} + 248 q^{16} + 88 q^{18} + 308 q^{19} - 840 q^{25} - 420 q^{28} + 328 q^{31} - 180 q^{33} + 540 q^{36} + 1152 q^{39} + 44 q^{40} - 1568 q^{45} - 84 q^{49} + 1620 q^{51} - 200 q^{63} + 148 q^{64} + 2052 q^{66} - 748 q^{67} - 288 q^{69} - 2068 q^{70} - 628 q^{72} + 4560 q^{76} - 3312 q^{78} + 164 q^{81} - 5396 q^{82} + 5688 q^{87} - 1312 q^{90} - 1368 q^{93} - 932 q^{94} - 988 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
92.1 − 5.24508i −5.08240 + 1.08131i −19.5109 − 7.60471i 5.67158 + 26.6576i −11.1491 60.3756i 24.6615 − 10.9913i −39.8873
92.2 − 5.24508i 5.08240 − 1.08131i −19.5109 − 7.60471i −5.67158 − 26.6576i −11.1491 60.3756i 24.6615 − 10.9913i −39.8873
92.3 − 4.83642i −2.52655 − 4.54054i −15.3909 17.4129i −21.9600 + 12.2195i 2.07413 35.7456i −14.2331 + 22.9438i 84.2160
92.4 − 4.83642i 2.52655 + 4.54054i −15.3909 17.4129i 21.9600 − 12.2195i 2.07413 35.7456i −14.2331 + 22.9438i 84.2160
92.5 − 4.20796i −0.517353 − 5.17033i −9.70695 − 15.2575i −21.7566 + 2.17700i 24.2010 7.18279i −26.4647 + 5.34977i −64.2028
92.6 − 4.20796i 0.517353 + 5.17033i −9.70695 − 15.2575i 21.7566 − 2.17700i 24.2010 7.18279i −26.4647 + 5.34977i −64.2028
92.7 − 3.25182i −1.88621 + 4.84172i −2.57431 2.11610i 15.7444 + 6.13360i −32.4015 − 17.6433i −19.8845 − 18.2650i 6.88116
92.8 − 3.25182i 1.88621 − 4.84172i −2.57431 2.11610i −15.7444 − 6.13360i −32.4015 − 17.6433i −19.8845 − 18.2650i 6.88116
92.9 − 2.97231i −4.89717 + 1.73716i −0.834616 8.81124i 5.16337 + 14.5559i 20.5950 − 21.2977i 20.9646 − 17.0143i 26.1897
92.10 − 2.97231i 4.89717 − 1.73716i −0.834616 8.81124i −5.16337 − 14.5559i 20.5950 − 21.2977i 20.9646 − 17.0143i 26.1897
92.11 − 1.95427i −4.50778 − 2.58456i 4.18081 − 11.2496i −5.05094 + 8.80943i −8.06707 − 23.8046i 13.6401 + 23.3012i −21.9848
92.12 − 1.95427i 4.50778 + 2.58456i 4.18081 − 11.2496i 5.05094 − 8.80943i −8.06707 − 23.8046i 13.6401 + 23.3012i −21.9848
92.13 − 0.403871i −4.08142 − 3.21589i 7.83689 16.8077i −1.29881 + 1.64837i −11.2524 − 6.39606i 6.31606 + 26.2509i 6.78815
92.14 − 0.403871i 4.08142 + 3.21589i 7.83689 16.8077i 1.29881 − 1.64837i −11.2524 − 6.39606i 6.31606 + 26.2509i 6.78815
92.15 0.403871i −4.08142 + 3.21589i 7.83689 − 16.8077i −1.29881 − 1.64837i −11.2524 6.39606i 6.31606 − 26.2509i 6.78815
92.16 0.403871i 4.08142 − 3.21589i 7.83689 − 16.8077i 1.29881 + 1.64837i −11.2524 6.39606i 6.31606 − 26.2509i 6.78815
92.17 1.95427i −4.50778 + 2.58456i 4.18081 11.2496i −5.05094 − 8.80943i −8.06707 23.8046i 13.6401 − 23.3012i −21.9848
92.18 1.95427i 4.50778 − 2.58456i 4.18081 11.2496i 5.05094 + 8.80943i −8.06707 23.8046i 13.6401 − 23.3012i −21.9848
92.19 2.97231i −4.89717 − 1.73716i −0.834616 − 8.81124i 5.16337 − 14.5559i 20.5950 21.2977i 20.9646 + 17.0143i 26.1897
92.20 2.97231i 4.89717 + 1.73716i −0.834616 − 8.81124i −5.16337 + 14.5559i 20.5950 21.2977i 20.9646 + 17.0143i 26.1897
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 92.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.4.c.b ✓ 28
3.b odd 2 1 inner 93.4.c.b ✓ 28
31.b odd 2 1 inner 93.4.c.b ✓ 28
93.c even 2 1 inner 93.4.c.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.c.b ✓ 28 1.a even 1 1 trivial
93.4.c.b ✓ 28 3.b odd 2 1 inner
93.4.c.b ✓ 28 31.b odd 2 1 inner
93.4.c.b ✓ 28 93.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 92T_{2}^{12} + 3321T_{2}^{10} + 59669T_{2}^{8} + 557626T_{2}^{6} + 2549667T_{2}^{4} + 4466736T_{2}^{2} + 663120 \) acting on \(S_{4}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display