Properties

Label 2001.4.a.a
Level $2001$
Weight $4$
Character orbit 2001.a
Self dual yes
Analytic conductor $118.063$
Analytic rank $1$
Dimension $31$
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: \(1\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q - 10 q^{2} + 93 q^{3} + 74 q^{4} - 25 q^{5} - 30 q^{6} - 76 q^{7} - 117 q^{8} + 279 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q - 10 q^{2} + 93 q^{3} + 74 q^{4} - 25 q^{5} - 30 q^{6} - 76 q^{7} - 117 q^{8} + 279 q^{9} - 6 q^{10} - 109 q^{11} + 222 q^{12} - 317 q^{13} - 159 q^{14} - 75 q^{15} + 54 q^{16} - 180 q^{17} - 90 q^{18} - 217 q^{19} - 193 q^{20} - 228 q^{21} - 160 q^{22} + 713 q^{23} - 351 q^{24} + 208 q^{25} + 211 q^{26} + 837 q^{27} + 5 q^{28} + 899 q^{29} - 18 q^{30} - 622 q^{31} - 696 q^{32} - 327 q^{33} + 126 q^{34} - 627 q^{35} + 666 q^{36} - 539 q^{37} + 35 q^{38} - 951 q^{39} + 371 q^{40} - 459 q^{41} - 477 q^{42} - 719 q^{43} - 876 q^{44} - 225 q^{45} - 230 q^{46} - 1384 q^{47} + 162 q^{48} - 747 q^{49} - 2005 q^{50} - 540 q^{51} - 2190 q^{52} - 730 q^{53} - 270 q^{54} - 1408 q^{55} - 1107 q^{56} - 651 q^{57} - 290 q^{58} - 3765 q^{59} - 579 q^{60} - 1300 q^{61} - 70 q^{62} - 684 q^{63} - 813 q^{64} - 2490 q^{65} - 480 q^{66} - 2735 q^{67} - 2031 q^{68} + 2139 q^{69} - 2581 q^{70} - 4800 q^{71} - 1053 q^{72} - 1858 q^{73} - 801 q^{74} + 624 q^{75} - 2699 q^{76} - 1661 q^{77} + 633 q^{78} - 4466 q^{79} + 3587 q^{80} + 2511 q^{81} - 5615 q^{82} - 1234 q^{83} + 15 q^{84} - 3623 q^{85} - 1873 q^{86} + 2697 q^{87} - 6893 q^{88} - 455 q^{89} - 54 q^{90} - 2425 q^{91} + 1702 q^{92} - 1866 q^{93} - 3912 q^{94} - 1851 q^{95} - 2088 q^{96} - 2114 q^{97} + 1769 q^{98} - 981 q^{99}+O(q^{100}) \) 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 −5.54277 3.00000 22.7223 9.48245 −16.6283 −7.42459 −81.6022 9.00000 −52.5591
1.2 −4.89797 3.00000 15.9902 −7.63127 −14.6939 26.7143 −39.1355 9.00000 37.3778
1.3 −4.77986 3.00000 14.8471 6.64522 −14.3396 17.3119 −32.7283 9.00000 −31.7633
1.4 −4.75649 3.00000 14.6242 −4.17105 −14.2695 −30.3756 −31.5078 9.00000 19.8395
1.5 −4.48602 3.00000 12.1244 −19.6077 −13.4581 −5.45707 −18.5020 9.00000 87.9603
1.6 −3.66126 3.00000 5.40480 −8.70181 −10.9838 1.57002 9.50170 9.00000 31.8596
1.7 −3.56041 3.00000 4.67651 4.74657 −10.6812 4.84926 11.8330 9.00000 −16.8997
1.8 −3.36157 3.00000 3.30018 17.6011 −10.0847 3.08404 15.7988 9.00000 −59.1674
1.9 −3.23146 3.00000 2.44236 −12.0697 −9.69439 18.2309 17.9593 9.00000 39.0029
1.10 −2.59267 3.00000 −1.27808 −19.3273 −7.77800 −18.8270 24.0550 9.00000 50.1091
1.11 −2.13891 3.00000 −3.42505 19.6067 −6.41674 −3.28131 24.4372 9.00000 −41.9370
1.12 −1.89451 3.00000 −4.41082 −1.56986 −5.68354 −16.4152 23.5125 9.00000 2.97412
1.13 −1.62628 3.00000 −5.35521 10.1067 −4.87884 −33.4720 21.7193 9.00000 −16.4364
1.14 −1.12653 3.00000 −6.73092 7.04441 −3.37960 27.2935 16.5949 9.00000 −7.93577
1.15 −0.758503 3.00000 −7.42467 −11.2678 −2.27551 14.6463 11.6997 9.00000 8.54666
1.16 −0.322279 3.00000 −7.89614 14.3189 −0.966836 −0.273748 5.12299 9.00000 −4.61468
1.17 0.0801108 3.00000 −7.99358 −13.1587 0.240332 −10.8611 −1.28126 9.00000 −1.05416
1.18 0.414813 3.00000 −7.82793 −0.503347 1.24444 −9.79226 −6.56563 9.00000 −0.208795
1.19 1.17903 3.00000 −6.60988 5.70828 3.53710 −7.52643 −17.2255 9.00000 6.73025
1.20 1.27880 3.00000 −6.36468 11.8772 3.83639 −14.0141 −18.3695 9.00000 15.1885
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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.a 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.4.a.a 31 1.a even 1 1 trivial