Properties

Label 8037.2.a.r
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $0$
Dimension $23$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8037,2,Mod(1,8037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8037.a (trivial)

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: yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(0\)
Dimension: \(23\)
Twist minimal: no (minimal twist has level 893)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.71530 0 5.37287 −1.00168 0 2.38392 −9.15838 0 2.71986
1.2 −2.62457 0 4.88835 −0.357317 0 −2.31246 −7.58066 0 0.937801
1.3 −2.39494 0 3.73572 3.66484 0 3.11663 −4.15695 0 −8.77707
1.4 −2.37437 0 3.63763 1.61117 0 0.108357 −3.88834 0 −3.82552
1.5 −2.06457 0 2.26244 −3.06045 0 3.81585 −0.541831 0 6.31850
1.6 −1.73522 0 1.01098 −2.41866 0 1.62998 1.71616 0 4.19690
1.7 −1.67375 0 0.801455 −3.83185 0 −3.94220 2.00607 0 6.41357
1.8 −1.28005 0 −0.361479 4.05982 0 −2.17032 3.02280 0 −5.19677
1.9 −1.15471 0 −0.666651 −2.72924 0 1.99299 3.07920 0 3.15148
1.10 −0.772110 0 −1.40385 2.30868 0 3.13314 2.62814 0 −1.78256
1.11 −0.512932 0 −1.73690 0.842347 0 −1.17187 1.91678 0 −0.432067
1.12 0.102666 0 −1.98946 −0.421030 0 4.07188 −0.409581 0 −0.0432253
1.13 0.154056 0 −1.97627 −1.39588 0 −0.0231293 −0.612568 0 −0.215045
1.14 0.581855 0 −1.66145 2.54642 0 −4.55068 −2.13043 0 1.48165
1.15 1.00631 0 −0.987335 −4.09447 0 3.87614 −3.00619 0 −4.12032
1.16 1.23080 0 −0.485141 3.43966 0 5.19980 −3.05870 0 4.23352
1.17 1.45599 0 0.119916 −1.65894 0 2.91693 −2.73739 0 −2.41540
1.18 1.55024 0 0.403250 0.718559 0 −2.40005 −2.47535 0 1.11394
1.19 2.30976 0 3.33497 0.940736 0 4.70927 3.08346 0 2.17287
1.20 2.31786 0 3.37246 −4.06165 0 −2.19787 3.18116 0 −9.41433
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)
\(47\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8037.2.a.r 23
3.b odd 2 1 893.2.a.d 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
893.2.a.d 23 3.b odd 2 1
8037.2.a.r 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{23} + T_{2}^{22} - 38 T_{2}^{21} - 37 T_{2}^{20} + 622 T_{2}^{19} + 586 T_{2}^{18} - 5746 T_{2}^{17} - 5195 T_{2}^{16} + 32986 T_{2}^{15} + 28295 T_{2}^{14} - 122210 T_{2}^{13} - 97857 T_{2}^{12} + 294219 T_{2}^{11} + \cdots + 315 \) Copy content Toggle raw display
\( T_{5}^{23} + T_{5}^{22} - 80 T_{5}^{21} - 78 T_{5}^{20} + 2715 T_{5}^{19} + 2569 T_{5}^{18} - 51111 T_{5}^{17} - 46571 T_{5}^{16} + 586779 T_{5}^{15} + 508457 T_{5}^{14} - 4262394 T_{5}^{13} - 3446926 T_{5}^{12} + \cdots - 2408448 \) Copy content Toggle raw display