↑

Properties

Label 4013.2.a.c
Level $4013$
Weight $2$
Character orbit 4013.a
Self dual yes
Analytic conductor $32.044$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4013,2,Mod(1,4013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4013 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4013.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: \(32.0439663311\)
Analytic rank: \(0\)
Dimension: \(176\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 11 q^{2} + 53 q^{3} + 191 q^{4} + 5 q^{5} + 19 q^{6} + 46 q^{7} + 36 q^{8} + 193 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q + 11 q^{2} + 53 q^{3} + 191 q^{4} + 5 q^{5} + 19 q^{6} + 46 q^{7} + 36 q^{8} + 193 q^{9} + 43 q^{10} + 18 q^{11} + 95 q^{12} + 95 q^{13} + 2 q^{14} + 36 q^{15} + 225 q^{16} + 35 q^{17} + 46 q^{18} + 127 q^{19} + 4 q^{20} + 32 q^{21} + 60 q^{22} + 35 q^{23} + 26 q^{24} + 207 q^{25} + 19 q^{26} + 191 q^{27} + 87 q^{28} + 16 q^{29} + 28 q^{30} + 93 q^{31} + 73 q^{32} + 70 q^{33} + 45 q^{34} + 73 q^{35} + 206 q^{36} + 64 q^{37} + 35 q^{38} + 72 q^{39} + 139 q^{40} + 19 q^{41} + 35 q^{42} + 261 q^{43} + 11 q^{44} + 12 q^{45} + 58 q^{46} + 40 q^{47} + 130 q^{48} + 234 q^{49} - 14 q^{50} + 76 q^{51} + 263 q^{52} + 17 q^{53} + 28 q^{54} + 170 q^{55} - 10 q^{56} + 60 q^{57} + 52 q^{58} + 69 q^{59} + 37 q^{60} + 110 q^{61} + 71 q^{62} + 101 q^{63} + 250 q^{64} - q^{65} + 43 q^{66} + 190 q^{67} + 48 q^{68} + 45 q^{69} + 14 q^{70} + 9 q^{71} + 98 q^{72} + 182 q^{73} - 23 q^{74} + 219 q^{75} + 197 q^{76} + 25 q^{77} - 26 q^{78} + 105 q^{79} + 20 q^{80} + 236 q^{81} + 107 q^{82} + 130 q^{83} + 38 q^{84} + 73 q^{85} - 24 q^{86} + 171 q^{87} + 165 q^{88} + 40 q^{89} + 45 q^{90} + 182 q^{91} - 4 q^{92} + 23 q^{93} + 98 q^{94} + 30 q^{95} - 2 q^{96} + 168 q^{97} + 82 q^{98} + 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.79865 3.21303 5.83245 0.484406 −8.99216 2.55250 −10.7257 7.32359 −1.35568
1.2 −2.76303 −0.741544 5.63435 1.49530 2.04891 0.791604 −10.0418 −2.45011 −4.13156
1.3 −2.71973 2.24397 5.39695 −3.35299 −6.10299 −2.43693 −9.23880 2.03538 9.11924
1.4 −2.69276 −0.212283 5.25096 −3.83789 0.571627 2.56921 −8.75407 −2.95494 10.3345
1.5 −2.69142 −1.48564 5.24374 2.79908 3.99849 −0.286125 −8.73026 −0.792861 −7.53350
1.6 −2.65007 −1.98668 5.02285 −4.07833 5.26483 0.415430 −8.01074 0.946892 10.8079
1.7 −2.63729 1.99604 4.95530 −2.90741 −5.26413 3.48548 −7.79400 0.984162 7.66769
1.8 −2.62564 0.652884 4.89398 −2.04233 −1.71424 −2.37529 −7.59855 −2.57374 5.36241
1.9 −2.61853 0.324920 4.85671 3.20033 −0.850813 −1.73793 −7.48039 −2.89443 −8.38017
1.10 −2.61670 −2.37899 4.84710 −0.969862 6.22508 5.09491 −7.45000 2.65957 2.53783
1.11 −2.59980 1.10973 4.75898 2.90792 −2.88507 4.77007 −7.17280 −1.76851 −7.56002
1.12 −2.57761 −2.77945 4.64406 −1.52510 7.16432 −0.558594 −6.81534 4.72533 3.93111
1.13 −2.53190 1.30897 4.41050 −0.932957 −3.31419 0.385559 −6.10313 −1.28658 2.36215
1.14 −2.43054 3.05350 3.90754 −3.60333 −7.42166 −4.38981 −4.63635 6.32386 8.75804
1.15 −2.42541 2.84636 3.88263 −0.259504 −6.90359 3.45507 −4.56617 5.10174 0.629405
1.16 −2.39764 −0.316927 3.74866 −2.45861 0.759875 1.72271 −4.19265 −2.89956 5.89485
1.17 −2.39761 0.688358 3.74852 1.63631 −1.65041 −2.07745 −4.19226 −2.52616 −3.92323
1.18 −2.37116 2.70615 3.62242 4.34995 −6.41674 1.61584 −3.84703 4.32327 −10.3145
1.19 −2.28450 2.00074 3.21893 3.40444 −4.57068 0.591134 −2.78465 1.00294 −7.77744
1.20 −2.28329 3.29602 3.21343 1.63708 −7.52578 −3.76844 −2.77062 7.86375 −3.73794
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.176
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4013\) \(-1\)

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 4013.2.a.c ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4013.2.a.c ✓ 176 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{176} - 11 T_{2}^{175} - 211 T_{2}^{174} + 2738 T_{2}^{173} + 20639 T_{2}^{172} + \cdots + 14852737934123 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4013))\). Copy content Toggle raw display