Properties

Label 4033.2.a.f
Level $4033$
Weight $2$
Character orbit 4033.a
Self dual yes
Analytic conductor $32.204$
Analytic rank $0$
Dimension $85$
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] = [4033,2,Mod(1,4033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4033 = 37 \cdot 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4033.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: \(32.2036671352\)
Analytic rank: \(0\)
Dimension: \(85\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 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 −2.76864 3.12213 5.66539 −1.61337 −8.64407 −3.34335 −10.1482 6.74771 4.46684
1.2 −2.68964 −0.345637 5.23416 3.86855 0.929638 −2.16399 −8.69874 −2.88054 −10.4050
1.3 −2.66672 −1.32046 5.11137 0.282486 3.52128 −2.11170 −8.29715 −1.25639 −0.753309
1.4 −2.62495 2.49737 4.89034 0.915854 −6.55547 2.77253 −7.58698 3.23688 −2.40407
1.5 −2.49324 −1.70202 4.21624 −1.58478 4.24353 2.99306 −5.52560 −0.103143 3.95124
1.6 −2.42751 0.768452 3.89281 −1.80408 −1.86543 −1.49541 −4.59481 −2.40948 4.37943
1.7 −2.36110 2.47312 3.57478 1.50101 −5.83928 4.30200 −3.71820 3.11634 −3.54403
1.8 −2.31084 0.814794 3.33999 −0.287417 −1.88286 −2.20640 −3.09651 −2.33611 0.664174
1.9 −2.27621 −2.60153 3.18111 2.30412 5.92161 1.78008 −2.68845 3.76795 −5.24464
1.10 −2.26706 −2.69699 3.13957 −3.90758 6.11425 −4.79951 −2.58348 4.27377 8.85872
1.11 −2.15921 −0.135675 2.66221 −3.14294 0.292951 0.215452 −1.42985 −2.98159 6.78628
1.12 −2.11247 2.70421 2.46253 2.42834 −5.71257 1.36575 −0.977087 4.31276 −5.12979
1.13 −2.03156 3.25300 2.12723 1.52354 −6.60865 −4.26037 −0.258468 7.58200 −3.09515
1.14 −1.99373 −2.36120 1.97494 −0.776635 4.70759 −0.440394 0.0499623 2.57527 1.54840
1.15 −1.96262 −1.03474 1.85187 3.48986 2.03080 −2.58926 0.290726 −1.92931 −6.84926
1.16 −1.92648 −3.23973 1.71134 1.34988 6.24129 3.96616 0.556094 7.49587 −2.60052
1.17 −1.88550 3.14155 1.55513 −2.47668 −5.92341 4.66231 0.838810 6.86935 4.66980
1.18 −1.82466 0.0328358 1.32938 −0.181740 −0.0599140 4.41961 1.22366 −2.99892 0.331613
1.19 −1.67574 1.31893 0.808091 −4.46158 −2.21019 0.153470 1.99732 −1.26041 7.47643
1.20 −1.66354 1.93600 0.767368 −2.27919 −3.22061 0.280246 2.05053 0.748091 3.79153
See all 85 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.85
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(-1\)
\(109\) \(1\)

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 4033.2.a.f 85
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4033.2.a.f 85 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{85} - 11 T_{2}^{84} - 71 T_{2}^{83} + 1200 T_{2}^{82} + 1241 T_{2}^{81} - 61991 T_{2}^{80} + 59762 T_{2}^{79} + 2012021 T_{2}^{78} - 4372344 T_{2}^{77} - 45842717 T_{2}^{76} + 144738316 T_{2}^{75} + \cdots + 90944 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\). Copy content Toggle raw display