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Properties

Label 2001.4.a.d
Level $2001$
Weight $4$
Character orbit 2001.a
Self dual yes
Analytic conductor $118.063$
Analytic rank $0$
Dimension $37$
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: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 4 q^{2} - 111 q^{3} + 146 q^{4} + 15 q^{5} - 12 q^{6} + 8 q^{7} + 3 q^{8} + 333 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 4 q^{2} - 111 q^{3} + 146 q^{4} + 15 q^{5} - 12 q^{6} + 8 q^{7} + 3 q^{8} + 333 q^{9} - 136 q^{10} + 109 q^{11} - 438 q^{12} - 269 q^{13} + 121 q^{14} - 45 q^{15} + 390 q^{16} + 180 q^{17} + 36 q^{18} + 117 q^{19} + 287 q^{20} - 24 q^{21} - 128 q^{22} - 851 q^{23} - 9 q^{24} + 490 q^{25} + 677 q^{26} - 999 q^{27} + 775 q^{28} - 1073 q^{29} + 408 q^{30} - 194 q^{31} + 668 q^{32} - 327 q^{33} + 972 q^{34} - 309 q^{35} + 1314 q^{36} - 565 q^{37} + 725 q^{38} + 807 q^{39} + 263 q^{40} + 521 q^{41} - 363 q^{42} - 61 q^{43} + 2242 q^{44} + 135 q^{45} - 92 q^{46} + 1142 q^{47} - 1170 q^{48} + 919 q^{49} + 1833 q^{50} - 540 q^{51} - 10 q^{52} - 120 q^{53} - 108 q^{54} - 996 q^{55} + 1707 q^{56} - 351 q^{57} - 116 q^{58} + 1073 q^{59} - 861 q^{60} - 428 q^{61} + 174 q^{62} + 72 q^{63} + 1479 q^{64} + 1410 q^{65} + 384 q^{66} + 175 q^{67} + 1483 q^{68} + 2553 q^{69} + 675 q^{70} + 2236 q^{71} + 27 q^{72} - 1058 q^{73} - 695 q^{74} - 1470 q^{75} + 1345 q^{76} - 1547 q^{77} - 2031 q^{78} + 1972 q^{79} - 2017 q^{80} + 2997 q^{81} + 2429 q^{82} - 832 q^{83} - 2325 q^{84} + 2299 q^{85} + 1527 q^{86} + 3219 q^{87} + 2579 q^{88} + 2817 q^{89} - 1224 q^{90} + 3175 q^{91} - 3358 q^{92} + 582 q^{93} + 1900 q^{94} + 8017 q^{95} - 2004 q^{96} + 912 q^{97} - 2565 q^{98} + 981 q^{99}+O(q^{100}) \) 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 −5.37336 −3.00000 20.8730 −5.43656 16.1201 5.91963 −69.1712 9.00000 29.2126
1.2 −5.24305 −3.00000 19.4896 11.9394 15.7292 0.00423859 −60.2407 9.00000 −62.5991
1.3 −4.98204 −3.00000 16.8207 6.31321 14.9461 −27.7568 −43.9453 9.00000 −31.4527
1.4 −4.65217 −3.00000 13.6426 −7.35673 13.9565 −5.22917 −26.2505 9.00000 34.2247
1.5 −4.40757 −3.00000 11.4267 −9.36502 13.2227 31.9822 −15.1033 9.00000 41.2770
1.6 −3.92246 −3.00000 7.38568 −2.06492 11.7674 5.51272 2.40965 9.00000 8.09958
1.7 −3.87951 −3.00000 7.05063 8.42923 11.6385 31.8700 3.68311 9.00000 −32.7013
1.8 −3.81938 −3.00000 6.58766 17.6451 11.4581 16.8603 5.39428 9.00000 −67.3932
1.9 −3.52007 −3.00000 4.39090 −15.0507 10.5602 −16.7576 12.7043 9.00000 52.9795
1.10 −3.45951 −3.00000 3.96821 18.1946 10.3785 −30.8151 13.9480 9.00000 −62.9444
1.11 −2.64188 −3.00000 −1.02048 4.63395 7.92563 7.87352 23.8310 9.00000 −12.2423
1.12 −2.21687 −3.00000 −3.08547 10.7596 6.65062 −26.2359 24.5751 9.00000 −23.8528
1.13 −2.17507 −3.00000 −3.26907 15.3323 6.52521 7.95319 24.5110 9.00000 −33.3489
1.14 −1.95818 −3.00000 −4.16554 −18.6254 5.87453 −1.49482 23.8223 9.00000 36.4718
1.15 −1.12477 −3.00000 −6.73490 −8.99407 3.37430 −26.8055 16.5733 9.00000 10.1162
1.16 −0.529130 −3.00000 −7.72002 11.9077 1.58739 −16.6855 8.31794 9.00000 −6.30071
1.17 −0.379327 −3.00000 −7.85611 −13.1850 1.13798 −14.3769 6.01465 9.00000 5.00142
1.18 −0.0682735 −3.00000 −7.99534 12.9829 0.204821 33.4985 1.09206 9.00000 −0.886392
1.19 0.166273 −3.00000 −7.97235 −4.23027 −0.498819 −0.0898903 −2.65577 9.00000 −0.703379
1.20 0.743169 −3.00000 −7.44770 −3.70226 −2.22951 13.2023 −11.4802 9.00000 −2.75140
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
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.d ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.4.a.d ✓ 37 1.a even 1 1 trivial