Properties

Label 84.5.g.a
Level $84$
Weight $5$
Character orbit 84.g
Analytic conductor $8.683$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,5,Mod(43,84)]
 
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, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.43");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 84.g (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: \(8.68307689904\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 34 q^{4} - 48 q^{5} + 36 q^{6} - 90 q^{8} - 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 34 q^{4} - 48 q^{5} + 36 q^{6} - 90 q^{8} - 648 q^{9} - 52 q^{10} - 360 q^{12} - 240 q^{13} - 294 q^{14} + 818 q^{16} + 1200 q^{17} + 162 q^{18} + 780 q^{20} + 2296 q^{22} - 1188 q^{24} + 1320 q^{25} - 2700 q^{26} - 98 q^{28} - 48 q^{29} + 1512 q^{30} + 414 q^{32} + 940 q^{34} - 918 q^{36} + 3088 q^{37} - 7200 q^{38} - 1436 q^{40} - 3600 q^{41} + 2472 q^{44} + 1296 q^{45} + 6600 q^{46} - 4176 q^{48} - 8232 q^{49} + 126 q^{50} + 3700 q^{52} - 6000 q^{53} - 972 q^{54} - 6174 q^{56} - 4588 q^{58} + 2304 q^{60} - 8432 q^{61} + 9408 q^{62} + 3226 q^{64} + 31584 q^{65} - 18504 q^{66} + 8700 q^{68} + 19584 q^{69} + 7644 q^{70} + 2430 q^{72} - 34320 q^{73} + 26340 q^{74} - 4464 q^{76} - 9408 q^{77} + 720 q^{78} + 24684 q^{80} + 17496 q^{81} + 3772 q^{82} - 15424 q^{85} - 2736 q^{86} - 12328 q^{88} + 46128 q^{89} + 1404 q^{90} - 81816 q^{92} + 7200 q^{93} - 70512 q^{94} + 6804 q^{96} - 61968 q^{97} + 2058 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −3.98722 0.319546i 5.19615i 15.7958 + 2.54820i 40.3533 −1.66041 + 20.7182i 18.5203i −62.1669 15.2077i −27.0000 −160.897 12.8947i
43.2 −3.98722 + 0.319546i 5.19615i 15.7958 2.54820i 40.3533 −1.66041 20.7182i 18.5203i −62.1669 + 15.2077i −27.0000 −160.897 + 12.8947i
43.3 −3.81725 1.19525i 5.19615i 13.1428 + 9.12512i −7.49303 −6.21069 + 19.8350i 18.5203i −39.2624 50.5417i −27.0000 28.6027 + 8.95603i
43.4 −3.81725 + 1.19525i 5.19615i 13.1428 9.12512i −7.49303 −6.21069 19.8350i 18.5203i −39.2624 + 50.5417i −27.0000 28.6027 8.95603i
43.5 −3.63258 1.67461i 5.19615i 10.3913 + 12.1664i −41.3815 8.70155 18.8755i 18.5203i −17.3735 61.5968i −27.0000 150.322 + 69.2981i
43.6 −3.63258 + 1.67461i 5.19615i 10.3913 12.1664i −41.3815 8.70155 + 18.8755i 18.5203i −17.3735 + 61.5968i −27.0000 150.322 69.2981i
43.7 −3.26473 2.31117i 5.19615i 5.31694 + 15.0907i 10.0790 12.0092 16.9640i 18.5203i 17.5189 61.5556i −27.0000 −32.9051 23.2943i
43.8 −3.26473 + 2.31117i 5.19615i 5.31694 15.0907i 10.0790 12.0092 + 16.9640i 18.5203i 17.5189 + 61.5556i −27.0000 −32.9051 + 23.2943i
43.9 −1.52999 3.69583i 5.19615i −11.3182 + 11.3092i −22.1164 −19.2041 + 7.95007i 18.5203i 59.1135 + 24.5273i −27.0000 33.8379 + 81.7383i
43.10 −1.52999 + 3.69583i 5.19615i −11.3182 11.3092i −22.1164 −19.2041 7.95007i 18.5203i 59.1135 24.5273i −27.0000 33.8379 81.7383i
43.11 −1.25775 3.79711i 5.19615i −12.8361 + 9.55162i 5.76271 19.7304 6.53545i 18.5203i 52.4132 + 36.7268i −27.0000 −7.24803 21.8817i
43.12 −1.25775 + 3.79711i 5.19615i −12.8361 9.55162i 5.76271 19.7304 + 6.53545i 18.5203i 52.4132 36.7268i −27.0000 −7.24803 + 21.8817i
43.13 0.113690 3.99838i 5.19615i −15.9741 0.909150i 19.7185 20.7762 + 0.590749i 18.5203i −5.45122 + 63.7674i −27.0000 2.24179 78.8421i
43.14 0.113690 + 3.99838i 5.19615i −15.9741 + 0.909150i 19.7185 20.7762 0.590749i 18.5203i −5.45122 63.7674i −27.0000 2.24179 + 78.8421i
43.15 1.64066 3.64804i 5.19615i −10.6165 11.9704i −17.1026 18.9558 + 8.52513i 18.5203i −61.0866 + 19.0899i −27.0000 −28.0596 + 62.3910i
43.16 1.64066 + 3.64804i 5.19615i −10.6165 + 11.9704i −17.1026 18.9558 8.52513i 18.5203i −61.0866 19.0899i −27.0000 −28.0596 62.3910i
43.17 1.76690 3.58860i 5.19615i −9.75616 12.6814i −22.5825 −18.6469 9.18106i 18.5203i −62.7466 + 12.6043i −27.0000 −39.9010 + 81.0398i
43.18 1.76690 + 3.58860i 5.19615i −9.75616 + 12.6814i −22.5825 −18.6469 + 9.18106i 18.5203i −62.7466 12.6043i −27.0000 −39.9010 81.0398i
43.19 3.20050 2.39933i 5.19615i 4.48640 15.3581i 28.1610 −12.4673 16.6303i 18.5203i −22.4906 59.9181i −27.0000 90.1292 67.5676i
43.20 3.20050 + 2.39933i 5.19615i 4.48640 + 15.3581i 28.1610 −12.4673 + 16.6303i 18.5203i −22.4906 + 59.9181i −27.0000 90.1292 + 67.5676i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.5.g.a 24
3.b odd 2 1 252.5.g.d 24
4.b odd 2 1 inner 84.5.g.a 24
8.b even 2 1 1344.5.m.e 24
8.d odd 2 1 1344.5.m.e 24
12.b even 2 1 252.5.g.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.5.g.a 24 1.a even 1 1 trivial
84.5.g.a 24 4.b odd 2 1 inner
252.5.g.d 24 3.b odd 2 1
252.5.g.d 24 12.b even 2 1
1344.5.m.e 24 8.b even 2 1
1344.5.m.e 24 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(84, [\chi])\).