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Properties

Label 4029.2.a.l
Level $4029$
Weight $2$
Character orbit 4029.a
Self dual yes
Analytic conductor $32.172$
Analytic rank $0$
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] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, 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("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
Analytic rank: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{2} - 32 q^{3} + 41 q^{4} - q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{2} - 32 q^{3} + 41 q^{4} - q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 32 q^{9} + 17 q^{10} + 8 q^{11} - 41 q^{12} + 17 q^{13} + q^{14} + q^{15} + 55 q^{16} - 32 q^{17} - q^{18} + 48 q^{19} - 7 q^{20} - 4 q^{21} - 4 q^{22} - 19 q^{23} + 3 q^{24} + 63 q^{25} + 27 q^{26} - 32 q^{27} + 17 q^{28} - 15 q^{29} - 17 q^{30} + 20 q^{31} + 13 q^{32} - 8 q^{33} + q^{34} + 22 q^{35} + 41 q^{36} + 6 q^{37} + 11 q^{38} - 17 q^{39} + 47 q^{40} + q^{41} - q^{42} + 40 q^{43} + 22 q^{44} - q^{45} + 5 q^{46} - 5 q^{47} - 55 q^{48} + 88 q^{49} + 17 q^{50} + 32 q^{51} + 23 q^{52} - 34 q^{53} + q^{54} + 48 q^{55} - 48 q^{57} - 9 q^{58} + 41 q^{59} + 7 q^{60} + 20 q^{61} + 15 q^{62} + 4 q^{63} + 93 q^{64} - 58 q^{65} + 4 q^{66} + 52 q^{67} - 41 q^{68} + 19 q^{69} + 25 q^{70} + q^{71} - 3 q^{72} + 19 q^{73} + 12 q^{74} - 63 q^{75} + 128 q^{76} - 20 q^{77} - 27 q^{78} + 32 q^{79} - 16 q^{80} + 32 q^{81} - 5 q^{82} + 31 q^{83} - 17 q^{84} + q^{85} - 62 q^{86} + 15 q^{87} + 35 q^{88} + 18 q^{89} + 17 q^{90} + 48 q^{91} - 75 q^{92} - 20 q^{93} + 29 q^{94} + 5 q^{95} - 13 q^{96} + 17 q^{97} + 30 q^{98} + 8 q^{99}+O(q^{100}) \) 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 −2.77670 −1.00000 5.71004 −0.322981 2.77670 −2.70395 −10.3017 1.00000 0.896820
1.2 −2.65510 −1.00000 5.04957 −3.36488 2.65510 2.92058 −8.09693 1.00000 8.93412
1.3 −2.57711 −1.00000 4.64151 2.80201 2.57711 −2.75409 −6.80748 1.00000 −7.22109
1.4 −2.50856 −1.00000 4.29286 −2.69434 2.50856 3.74400 −5.75176 1.00000 6.75889
1.5 −2.27866 −1.00000 3.19228 −3.13196 2.27866 −0.0802156 −2.71679 1.00000 7.13667
1.6 −2.16757 −1.00000 2.69838 2.47527 2.16757 5.12027 −1.51379 1.00000 −5.36532
1.7 −2.03180 −1.00000 2.12821 1.66964 2.03180 −1.52463 −0.260503 1.00000 −3.39237
1.8 −1.67650 −1.00000 0.810659 −3.83953 1.67650 −4.72769 1.99393 1.00000 6.43697
1.9 −1.61915 −1.00000 0.621643 1.62143 1.61915 3.15810 2.23177 1.00000 −2.62534
1.10 −1.60231 −1.00000 0.567397 −0.489899 1.60231 −0.574101 2.29547 1.00000 0.784971
1.11 −1.34067 −1.00000 −0.202611 1.16053 1.34067 −4.05917 2.95297 1.00000 −1.55588
1.12 −1.17408 −1.00000 −0.621542 3.86856 1.17408 3.06255 3.07789 1.00000 −4.54199
1.13 −0.857766 −1.00000 −1.26424 −4.41756 0.857766 0.00991750 2.79995 1.00000 3.78923
1.14 −0.510062 −1.00000 −1.73984 −1.60726 0.510062 −0.772958 1.90755 1.00000 0.819801
1.15 −0.249798 −1.00000 −1.93760 0.601573 0.249798 −2.69175 0.983604 1.00000 −0.150272
1.16 −0.0772262 −1.00000 −1.99404 3.97384 0.0772262 0.969209 0.308444 1.00000 −0.306884
1.17 −0.0717307 −1.00000 −1.99485 −1.92367 0.0717307 3.89464 0.286554 1.00000 0.137987
1.18 0.403182 −1.00000 −1.83744 2.23661 −0.403182 −0.778796 −1.54719 1.00000 0.901761
1.19 0.437930 −1.00000 −1.80822 −1.50297 −0.437930 −4.33772 −1.66773 1.00000 −0.658196
1.20 0.937640 −1.00000 −1.12083 0.347083 −0.937640 4.09662 −2.92622 1.00000 0.325439
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
\(3\) \(1\)
\(17\) \(1\)
\(79\) \(-1\)

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 4029.2.a.l ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.l ✓ 32 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\( T_{2}^{32} + T_{2}^{31} - 52 T_{2}^{30} - 50 T_{2}^{29} + 1215 T_{2}^{28} + 1115 T_{2}^{27} - 16869 T_{2}^{26} + \cdots - 544 \) Copy content Toggle raw display
\( T_{5}^{32} + T_{5}^{31} - 111 T_{5}^{30} - 107 T_{5}^{29} + 5499 T_{5}^{28} + 5015 T_{5}^{27} + \cdots - 143498044 \) Copy content Toggle raw display