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Properties

Label 2001.4.a.h
Level $2001$
Weight $4$
Character orbit 2001.a
Self dual yes
Analytic conductor $118.063$
Analytic rank $0$
Dimension $44$
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] = [2001,4,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2001.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(118.062821921\)
Analytic rank: \(0\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 6 q^{2} + 132 q^{3} + 210 q^{4} + 15 q^{5} + 18 q^{6} + 78 q^{7} + 12 q^{8} + 396 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 6 q^{2} + 132 q^{3} + 210 q^{4} + 15 q^{5} + 18 q^{6} + 78 q^{7} + 12 q^{8} + 396 q^{9} + 214 q^{10} + 111 q^{11} + 630 q^{12} + 275 q^{13} + 104 q^{14} + 45 q^{15} + 1062 q^{16} - 58 q^{17} + 54 q^{18} + 331 q^{19} + 287 q^{20} + 234 q^{21} + 285 q^{22} + 1012 q^{23} + 36 q^{24} + 1903 q^{25} + 1084 q^{26} + 1188 q^{27} + 222 q^{28} - 1276 q^{29} + 642 q^{30} + 1394 q^{31} + 42 q^{32} + 333 q^{33} + 373 q^{34} + 567 q^{35} + 1890 q^{36} + 1229 q^{37} + 733 q^{38} + 825 q^{39} + 2483 q^{40} - 107 q^{41} + 312 q^{42} + 1165 q^{43} + 1639 q^{44} + 135 q^{45} + 138 q^{46} + 964 q^{47} + 3186 q^{48} + 4264 q^{49} + 495 q^{50} - 174 q^{51} + 2679 q^{52} - 380 q^{53} + 162 q^{54} + 1260 q^{55} + 2229 q^{56} + 993 q^{57} - 174 q^{58} + 897 q^{59} + 861 q^{60} + 2584 q^{61} + 3034 q^{62} + 702 q^{63} + 6866 q^{64} - 286 q^{65} + 855 q^{66} + 2277 q^{67} - 1554 q^{68} + 3036 q^{69} + 689 q^{70} + 4304 q^{71} + 108 q^{72} + 4712 q^{73} - 1005 q^{74} + 5709 q^{75} + 2877 q^{76} + 919 q^{77} + 3252 q^{78} + 3864 q^{79} + 2593 q^{80} + 3564 q^{81} + 3297 q^{82} - 540 q^{83} + 666 q^{84} + 6537 q^{85} + 3789 q^{86} - 3828 q^{87} + 1707 q^{88} - 331 q^{89} + 1926 q^{90} + 4311 q^{91} + 4830 q^{92} + 4182 q^{93} + 6189 q^{94} + 3267 q^{95} + 126 q^{96} + 5572 q^{97} + 2588 q^{98} + 999 q^{99}+O(q^{100}) \) 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 −5.57518 3.00000 23.0826 −14.6563 −16.7255 −14.0051 −84.0883 9.00000 81.7114
1.2 −5.27629 3.00000 19.8392 −7.51271 −15.8289 20.4014 −62.4673 9.00000 39.6392
1.3 −5.17547 3.00000 18.7854 16.2940 −15.5264 −25.7915 −55.8197 9.00000 −84.3288
1.4 −5.12770 3.00000 18.2933 −6.55578 −15.3831 17.3973 −52.7811 9.00000 33.6161
1.5 −5.03566 3.00000 17.3579 13.6126 −15.1070 24.5658 −47.1232 9.00000 −68.5482
1.6 −4.70747 3.00000 14.1603 −20.4476 −14.1224 −29.3817 −28.9995 9.00000 96.2567
1.7 −4.54386 3.00000 12.6466 13.4410 −13.6316 1.08944 −21.1137 9.00000 −61.0740
1.8 −4.01706 3.00000 8.13673 0.375609 −12.0512 −23.3454 −0.549272 9.00000 −1.50884
1.9 −3.63285 3.00000 5.19757 −13.8371 −10.8985 −0.384852 10.1808 9.00000 50.2680
1.10 −3.24407 3.00000 2.52397 −19.5917 −9.73220 36.4327 17.7646 9.00000 63.5566
1.11 −3.19635 3.00000 2.21665 9.22135 −9.58905 −13.6162 18.4856 9.00000 −29.4747
1.12 −3.00964 3.00000 1.05794 17.7761 −9.02892 14.5114 20.8931 9.00000 −53.4996
1.13 −2.95952 3.00000 0.758772 4.73795 −8.87857 27.1051 21.4306 9.00000 −14.0221
1.14 −2.77019 3.00000 −0.326072 −1.57049 −8.31056 −15.6447 23.0648 9.00000 4.35056
1.15 −2.18892 3.00000 −3.20863 10.5230 −6.56676 30.7082 24.5348 9.00000 −23.0341
1.16 −1.70995 3.00000 −5.07606 −9.43326 −5.12986 −3.20578 22.3595 9.00000 16.1304
1.17 −1.61295 3.00000 −5.39841 −4.25187 −4.83884 −1.08998 21.6109 9.00000 6.85803
1.18 −1.29985 3.00000 −6.31038 −11.8310 −3.89956 −23.8904 18.6014 9.00000 15.3786
1.19 −0.903337 3.00000 −7.18398 −21.3096 −2.71001 8.89985 13.7163 9.00000 19.2497
1.20 −0.481918 3.00000 −7.76776 20.6702 −1.44575 −14.3569 7.59876 9.00000 −9.96135
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

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 2001.4.a.h ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.4.a.h ✓ 44 1.a even 1 1 trivial