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Properties

Label 4034.2.a.b
Level $4034$
Weight $2$
Character orbit 4034.a
Self dual yes
Analytic conductor $32.212$
Analytic rank $1$
Dimension $35$
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: \(35\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 35 q - 35 q^{2} - 6 q^{3} + 35 q^{4} + 6 q^{5} + 6 q^{6} - 14 q^{7} - 35 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q - 35 q^{2} - 6 q^{3} + 35 q^{4} + 6 q^{5} + 6 q^{6} - 14 q^{7} - 35 q^{8} + 23 q^{9} - 6 q^{10} - 9 q^{11} - 6 q^{12} - 7 q^{13} + 14 q^{14} - 19 q^{15} + 35 q^{16} + 17 q^{17} - 23 q^{18} - 25 q^{19} + 6 q^{20} - 15 q^{21} + 9 q^{22} - 12 q^{23} + 6 q^{24} + 7 q^{25} + 7 q^{26} - 27 q^{27} - 14 q^{28} - 13 q^{29} + 19 q^{30} - 69 q^{31} - 35 q^{32} + q^{33} - 17 q^{34} - 4 q^{35} + 23 q^{36} - 22 q^{37} + 25 q^{38} - 38 q^{39} - 6 q^{40} + 15 q^{42} - 32 q^{43} - 9 q^{44} + 9 q^{45} + 12 q^{46} - 18 q^{47} - 6 q^{48} - 19 q^{49} - 7 q^{50} - 21 q^{51} - 7 q^{52} + 20 q^{53} + 27 q^{54} - 54 q^{55} + 14 q^{56} + 28 q^{57} + 13 q^{58} - 21 q^{59} - 19 q^{60} - 67 q^{61} + 69 q^{62} - 28 q^{63} + 35 q^{64} + 22 q^{65} - q^{66} - 18 q^{67} + 17 q^{68} - 42 q^{69} + 4 q^{70} - 36 q^{71} - 23 q^{72} - 18 q^{73} + 22 q^{74} - 49 q^{75} - 25 q^{76} + 20 q^{77} + 38 q^{78} - 92 q^{79} + 6 q^{80} - 25 q^{81} + 42 q^{83} - 15 q^{84} - 29 q^{85} + 32 q^{86} - 40 q^{87} + 9 q^{88} - 8 q^{89} - 9 q^{90} - 89 q^{91} - 12 q^{92} - q^{93} + 18 q^{94} - 62 q^{95} + 6 q^{96} - 40 q^{97} + 19 q^{98} - 64 q^{99}+O(q^{100}) \) 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 −3.27997 1.00000 2.11941 3.27997 −0.124927 −1.00000 7.75819 −2.11941
1.2 −1.00000 −3.25702 1.00000 −1.18968 3.25702 1.65832 −1.00000 7.60820 1.18968
1.3 −1.00000 −2.93683 1.00000 0.0231797 2.93683 −4.16598 −1.00000 5.62497 −0.0231797
1.4 −1.00000 −2.83579 1.00000 4.39396 2.83579 −1.33504 −1.00000 5.04172 −4.39396
1.5 −1.00000 −2.39986 1.00000 −1.11547 2.39986 2.17590 −1.00000 2.75933 1.11547
1.6 −1.00000 −2.23091 1.00000 1.94904 2.23091 1.55699 −1.00000 1.97697 −1.94904
1.7 −1.00000 −2.01951 1.00000 3.52220 2.01951 −0.0194784 −1.00000 1.07840 −3.52220
1.8 −1.00000 −1.96432 1.00000 −0.678471 1.96432 4.37365 −1.00000 0.858567 0.678471
1.9 −1.00000 −1.95232 1.00000 −1.60299 1.95232 −2.46247 −1.00000 0.811553 1.60299
1.10 −1.00000 −1.90524 1.00000 −3.49980 1.90524 −2.88392 −1.00000 0.629935 3.49980
1.11 −1.00000 −1.74374 1.00000 −3.34943 1.74374 0.122108 −1.00000 0.0406325 3.34943
1.12 −1.00000 −1.52925 1.00000 3.51675 1.52925 −1.62140 −1.00000 −0.661396 −3.51675
1.13 −1.00000 −1.35256 1.00000 −0.395568 1.35256 1.50560 −1.00000 −1.17057 0.395568
1.14 −1.00000 −1.19422 1.00000 1.77357 1.19422 2.63831 −1.00000 −1.57384 −1.77357
1.15 −1.00000 −0.841269 1.00000 −1.88225 0.841269 −4.21612 −1.00000 −2.29227 1.88225
1.16 −1.00000 −0.715791 1.00000 2.28149 0.715791 −4.66835 −1.00000 −2.48764 −2.28149
1.17 −1.00000 −0.237603 1.00000 3.33211 0.237603 4.34322 −1.00000 −2.94355 −3.33211
1.18 −1.00000 0.00791248 1.00000 −2.05271 −0.00791248 −2.16347 −1.00000 −2.99994 2.05271
1.19 −1.00000 0.0555653 1.00000 0.675723 −0.0555653 −0.0958620 −1.00000 −2.99691 −0.675723
1.20 −1.00000 0.280261 1.00000 0.751535 −0.280261 1.30988 −1.00000 −2.92145 −0.751535
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
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.b ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4034.2.a.b ✓ 35 1.a even 1 1 trivial