Properties

Label 4015.2.a.g
Level $4015$
Weight $2$
Character orbit 4015.a
Self dual yes
Analytic conductor $32.060$
Analytic rank $1$
Dimension $32$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, 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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(1\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9} + 5 q^{10} - 32 q^{11} - 24 q^{12} - q^{13} - 5 q^{14} + 7 q^{15} + 47 q^{16} - 30 q^{17} - 11 q^{18} + 16 q^{19} - 37 q^{20} + q^{21} + 5 q^{22} - 26 q^{23} - 21 q^{24} + 32 q^{25} - q^{26} - 31 q^{27} - 24 q^{28} - 10 q^{29} + 3 q^{30} - 2 q^{31} - 31 q^{32} + 7 q^{33} - 14 q^{34} + 38 q^{36} - 28 q^{37} - 63 q^{38} - 2 q^{39} + 18 q^{40} - 62 q^{41} - 9 q^{42} + 8 q^{43} - 37 q^{44} - 29 q^{45} + 19 q^{46} - 21 q^{47} - 79 q^{48} + 34 q^{49} - 5 q^{50} + 17 q^{51} + 15 q^{52} - 32 q^{53} + 5 q^{54} + 32 q^{55} - 52 q^{56} - 57 q^{57} + 4 q^{58} - 37 q^{59} + 24 q^{60} + 15 q^{61} - 22 q^{62} + 5 q^{63} + 70 q^{64} + q^{65} + 3 q^{66} - 42 q^{67} - 81 q^{68} - 8 q^{69} + 5 q^{70} - 40 q^{71} - 27 q^{72} - 32 q^{73} - 17 q^{74} - 7 q^{75} + 21 q^{76} - 105 q^{78} + 18 q^{79} - 47 q^{80} + 12 q^{81} - 70 q^{82} - 26 q^{83} + 22 q^{84} + 30 q^{85} - 45 q^{86} - 18 q^{87} + 18 q^{88} - 83 q^{89} + 11 q^{90} - 18 q^{91} - 73 q^{92} - 68 q^{93} + 56 q^{94} - 16 q^{95} - 35 q^{96} - 99 q^{97} - 61 q^{98} - 29 q^{99}+O(q^{100}) \) 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 −2.80101 2.07520 5.84563 −1.00000 −5.81265 2.07490 −10.7716 1.30645 2.80101
1.2 −2.75666 −2.97515 5.59919 −1.00000 8.20149 1.32848 −9.92176 5.85152 2.75666
1.3 −2.62713 −1.17329 4.90182 −1.00000 3.08238 3.02645 −7.62347 −1.62339 2.62713
1.4 −2.55668 0.570205 4.53663 −1.00000 −1.45783 −4.65776 −6.48536 −2.67487 2.55668
1.5 −2.31178 −1.68436 3.34432 −1.00000 3.89386 −2.49479 −3.10778 −0.162937 2.31178
1.6 −2.06358 2.59950 2.25838 −1.00000 −5.36430 −1.01430 −0.533185 3.75743 2.06358
1.7 −2.01115 −3.15685 2.04473 −1.00000 6.34891 −3.03904 −0.0899564 6.96573 2.01115
1.8 −1.96967 2.18511 1.87958 −1.00000 −4.30394 1.92639 0.237186 1.77472 1.96967
1.9 −1.75190 −0.429918 1.06914 −1.00000 0.753172 3.79529 1.63077 −2.81517 1.75190
1.10 −1.71308 1.21665 0.934626 −1.00000 −2.08421 −4.29501 1.82507 −1.51977 1.71308
1.11 −1.50747 −0.655267 0.272451 −1.00000 0.987792 −1.30717 2.60422 −2.57063 1.50747
1.12 −0.962434 2.32544 −1.07372 −1.00000 −2.23808 2.98112 2.95825 2.40766 0.962434
1.13 −0.960547 −2.83486 −1.07735 −1.00000 2.72301 4.60653 2.95594 5.03641 0.960547
1.14 −0.763072 0.889359 −1.41772 −1.00000 −0.678645 2.40954 2.60797 −2.20904 0.763072
1.15 −0.711670 −2.33630 −1.49353 −1.00000 1.66268 −3.28922 2.48624 2.45831 0.711670
1.16 −0.193228 −1.07555 −1.96266 −1.00000 0.207825 1.51348 0.765696 −1.84320 0.193228
1.17 0.0852596 0.824276 −1.99273 −1.00000 0.0702774 0.193385 −0.340419 −2.32057 −0.0852596
1.18 0.338336 −0.770870 −1.88553 −1.00000 −0.260813 −3.60696 −1.31461 −2.40576 −0.338336
1.19 0.467946 3.03596 −1.78103 −1.00000 1.42067 −2.71430 −1.76932 6.21706 −0.467946
1.20 0.472050 −2.02631 −1.77717 −1.00000 −0.956522 −4.07098 −1.78301 1.10595 −0.472050
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)
\(73\) \(1\)

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 4015.2.a.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4015.2.a.g 32 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 5 T_{2}^{31} - 38 T_{2}^{30} - 219 T_{2}^{29} + 602 T_{2}^{28} + 4267 T_{2}^{27} - 4936 T_{2}^{26} - 48848 T_{2}^{25} + 18739 T_{2}^{24} + 365631 T_{2}^{23} + 20864 T_{2}^{22} - 1885002 T_{2}^{21} - 613517 T_{2}^{20} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\). Copy content Toggle raw display