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Properties

Label 4034.2.a.a
Level $4034$
Weight $2$
Character orbit 4034.a
Self dual yes
Analytic conductor $32.212$
Analytic rank $1$
Dimension $33$
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] = [4034,2,Mod(1,4034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4034, 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("4034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4034 = 2 \cdot 2017 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4034.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: \(32.2116521754\)
Analytic rank: \(1\)
Dimension: \(33\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q + 33 q^{2} - 14 q^{3} + 33 q^{4} - 22 q^{5} - 14 q^{6} - 12 q^{7} + 33 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q + 33 q^{2} - 14 q^{3} + 33 q^{4} - 22 q^{5} - 14 q^{6} - 12 q^{7} + 33 q^{8} + 17 q^{9} - 22 q^{10} - 19 q^{11} - 14 q^{12} - 29 q^{13} - 12 q^{14} - 5 q^{15} + 33 q^{16} - 47 q^{17} + 17 q^{18} - 35 q^{19} - 22 q^{20} - 31 q^{21} - 19 q^{22} - 2 q^{23} - 14 q^{24} + 13 q^{25} - 29 q^{26} - 47 q^{27} - 12 q^{28} - 29 q^{29} - 5 q^{30} - 53 q^{31} + 33 q^{32} - 23 q^{33} - 47 q^{34} - 14 q^{35} + 17 q^{36} - 42 q^{37} - 35 q^{38} - 22 q^{40} - 42 q^{41} - 31 q^{42} - 26 q^{43} - 19 q^{44} - 55 q^{45} - 2 q^{46} - 14 q^{48} - 21 q^{49} + 13 q^{50} - 13 q^{51} - 29 q^{52} - 40 q^{53} - 47 q^{54} - 34 q^{55} - 12 q^{56} - 30 q^{57} - 29 q^{58} - 45 q^{59} - 5 q^{60} - 93 q^{61} - 53 q^{62} + 4 q^{63} + 33 q^{64} - 26 q^{65} - 23 q^{66} - 28 q^{67} - 47 q^{68} - 60 q^{69} - 14 q^{70} + 4 q^{71} + 17 q^{72} - 52 q^{73} - 42 q^{74} - 41 q^{75} - 35 q^{76} - 38 q^{77} - 38 q^{79} - 22 q^{80} + 25 q^{81} - 42 q^{82} - 42 q^{83} - 31 q^{84} - 21 q^{85} - 26 q^{86} + 12 q^{87} - 19 q^{88} - 58 q^{89} - 55 q^{90} - 79 q^{91} - 2 q^{92} + 25 q^{93} + 16 q^{95} - 14 q^{96} - 64 q^{97} - 21 q^{98} - 38 q^{99}+O(q^{100}) \) 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 1.00000 −3.41599 1.00000 1.09379 −3.41599 4.04655 1.00000 8.66901 1.09379
1.2 1.00000 −3.31212 1.00000 −3.97668 −3.31212 0.855437 1.00000 7.97011 −3.97668
1.3 1.00000 −2.82298 1.00000 0.779916 −2.82298 −3.19115 1.00000 4.96923 0.779916
1.4 1.00000 −2.73960 1.00000 −3.73363 −2.73960 −3.98248 1.00000 4.50543 −3.73363
1.5 1.00000 −2.69679 1.00000 −2.92996 −2.69679 3.32033 1.00000 4.27265 −2.92996
1.6 1.00000 −2.68746 1.00000 1.22075 −2.68746 −1.79428 1.00000 4.22244 1.22075
1.7 1.00000 −2.63161 1.00000 1.62454 −2.63161 2.40683 1.00000 3.92538 1.62454
1.8 1.00000 −2.03082 1.00000 −0.733478 −2.03082 −1.85114 1.00000 1.12424 −0.733478
1.9 1.00000 −1.80963 1.00000 0.856694 −1.80963 0.353536 1.00000 0.274766 0.856694
1.10 1.00000 −1.73259 1.00000 3.77683 −1.73259 0.621551 1.00000 0.00185264 3.77683
1.11 1.00000 −1.72838 1.00000 −2.88558 −1.72838 −0.799535 1.00000 −0.0126910 −2.88558
1.12 1.00000 −1.65119 1.00000 −2.90851 −1.65119 0.175106 1.00000 −0.273565 −2.90851
1.13 1.00000 −1.42842 1.00000 2.01766 −1.42842 0.289927 1.00000 −0.959611 2.01766
1.14 1.00000 −0.945997 1.00000 −2.44770 −0.945997 2.03235 1.00000 −2.10509 −2.44770
1.15 1.00000 −0.523973 1.00000 1.37573 −0.523973 −3.92552 1.00000 −2.72545 1.37573
1.16 1.00000 −0.423572 1.00000 −0.487122 −0.423572 2.40385 1.00000 −2.82059 −0.487122
1.17 1.00000 −0.366256 1.00000 0.145528 −0.366256 3.13381 1.00000 −2.86586 0.145528
1.18 1.00000 −0.289410 1.00000 −3.81688 −0.289410 −2.88122 1.00000 −2.91624 −3.81688
1.19 1.00000 −0.264302 1.00000 2.82441 −0.264302 −2.96778 1.00000 −2.93014 2.82441
1.20 1.00000 −0.138152 1.00000 −3.80380 −0.138152 3.68849 1.00000 −2.98091 −3.80380
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2017\) \(-1\)

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 4034.2.a.a ✓ 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4034.2.a.a ✓ 33 1.a even 1 1 trivial