Properties

Label 4033.2.a.c
Level $4033$
Weight $2$
Character orbit 4033.a
Self dual yes
Analytic conductor $32.204$
Analytic rank $1$
Dimension $77$
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: \(1\)
Dimension: \(77\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 77 q - 9 q^{2} - 27 q^{3} + 73 q^{4} - 16 q^{5} - 2 q^{6} - 23 q^{7} - 24 q^{8} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 77 q - 9 q^{2} - 27 q^{3} + 73 q^{4} - 16 q^{5} - 2 q^{6} - 23 q^{7} - 24 q^{8} + 66 q^{9} - 11 q^{10} - 33 q^{11} - 52 q^{12} - 10 q^{13} - 18 q^{14} - 33 q^{15} + 53 q^{16} - 44 q^{17} - 27 q^{18} - 9 q^{19} - 45 q^{20} - 7 q^{21} - 3 q^{22} - 74 q^{23} + 21 q^{24} + 59 q^{25} - 47 q^{26} - 99 q^{27} - 49 q^{28} - 9 q^{29} - 39 q^{30} - 27 q^{31} - 47 q^{32} - 28 q^{33} - 23 q^{34} - 48 q^{35} + 77 q^{36} + 77 q^{37} - 66 q^{38} - 11 q^{39} - 2 q^{40} - 37 q^{41} - 24 q^{42} - 44 q^{43} - 54 q^{44} - 36 q^{45} - 41 q^{46} - 150 q^{47} - 135 q^{48} + 64 q^{49} + 4 q^{50} + 3 q^{51} - 57 q^{52} - 72 q^{53} + 21 q^{54} - 65 q^{55} - 92 q^{56} - 13 q^{57} - 12 q^{58} - 70 q^{59} - 22 q^{60} + 15 q^{61} - 86 q^{62} - 108 q^{63} + 10 q^{64} - 53 q^{65} - 55 q^{66} - 48 q^{67} - 70 q^{68} - 2 q^{69} + 11 q^{70} - 127 q^{71} - 12 q^{72} - 33 q^{73} - 9 q^{74} - 115 q^{75} - 24 q^{76} - 40 q^{77} + 81 q^{78} - 7 q^{79} - 62 q^{80} + 53 q^{81} - 68 q^{82} - 164 q^{83} + 7 q^{84} - 9 q^{85} - 50 q^{86} - 75 q^{87} - 82 q^{88} - 26 q^{89} + 23 q^{90} + 16 q^{91} - 117 q^{92} + 19 q^{93} + 23 q^{94} - 92 q^{95} - 35 q^{96} - 19 q^{97} - 10 q^{98} - 19 q^{99}+O(q^{100}) \) 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 −2.77942 −0.824598 5.72520 1.36321 2.29191 0.958419 −10.3539 −2.32004 −3.78894
1.2 −2.66405 1.68420 5.09718 0.183970 −4.48681 1.02981 −8.25104 −0.163454 −0.490107
1.3 −2.64480 0.673502 4.99498 −3.52022 −1.78128 −3.10314 −7.92112 −2.54639 9.31030
1.4 −2.64408 −3.27443 4.99114 −3.17028 8.65784 2.13489 −7.90879 7.72189 8.38246
1.5 −2.57358 −2.79713 4.62329 0.767586 7.19864 −3.79460 −6.75123 4.82396 −1.97544
1.6 −2.57274 −3.14788 4.61898 −2.11075 8.09868 −0.361357 −6.73794 6.90917 5.43041
1.7 −2.51099 0.0536632 4.30505 −1.45353 −0.134748 3.51496 −5.78795 −2.99712 3.64980
1.8 −2.50194 −1.32701 4.25970 0.884945 3.32011 3.50232 −5.65363 −1.23903 −2.21408
1.9 −2.37004 2.31466 3.61710 2.16488 −5.48585 −1.68976 −3.83259 2.35766 −5.13086
1.10 −2.36112 2.69902 3.57489 −1.67448 −6.37271 0.928256 −3.71850 4.28471 3.95366
1.11 −2.25607 0.265455 3.08984 2.86324 −0.598885 −5.00198 −2.45875 −2.92953 −6.45966
1.12 −2.25323 −3.09773 3.07703 4.21432 6.97988 −2.71387 −2.42679 6.59591 −9.49583
1.13 −2.16010 −1.27573 2.66602 −3.85876 2.75571 4.13321 −1.43868 −1.37250 8.33530
1.14 −2.03838 1.12284 2.15500 3.13166 −2.28878 0.892972 −0.315945 −1.73922 −6.38351
1.15 −1.98064 −1.34264 1.92294 −0.618382 2.65929 −3.38656 0.152622 −1.19732 1.22479
1.16 −1.93065 −1.88783 1.72740 2.91327 3.64473 1.86456 0.526302 0.563889 −5.62450
1.17 −1.80263 2.09678 1.24948 −2.41088 −3.77972 2.66635 1.35291 1.39648 4.34594
1.18 −1.78823 2.78240 1.19778 −2.10202 −4.97558 −2.70114 1.43455 4.74173 3.75890
1.19 −1.77580 −1.89592 1.15346 0.136329 3.36677 −1.80819 1.50328 0.594515 −0.242093
1.20 −1.67758 0.712654 0.814260 2.97956 −1.19553 2.13168 1.98917 −2.49212 −4.99844
See all 77 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.77
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.c 77
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4033.2.a.c 77 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{77} + 9 T_{2}^{76} - 73 T_{2}^{75} - 880 T_{2}^{74} + 2021 T_{2}^{73} + 40805 T_{2}^{72} - 10998 T_{2}^{71} - 1192879 T_{2}^{70} - 994027 T_{2}^{69} + 24646044 T_{2}^{68} + 39264137 T_{2}^{67} + \cdots + 5180112 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\). Copy content Toggle raw display