↑

Properties

Label 2001.4.a.e
Level $2001$
Weight $4$
Character orbit 2001.a
Self dual yes
Analytic conductor $118.063$
Analytic rank $1$
Dimension $38$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 114 q^{3} + 138 q^{4} - 25 q^{5} - 34 q^{7} + 18 q^{8} + 342 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 114 q^{3} + 138 q^{4} - 25 q^{5} - 34 q^{7} + 18 q^{8} + 342 q^{9} - 156 q^{10} - 111 q^{11} - 414 q^{12} + 179 q^{13} - 142 q^{14} + 75 q^{15} + 470 q^{16} + 58 q^{17} + 25 q^{19} - 193 q^{20} + 102 q^{21} - 171 q^{22} - 874 q^{23} - 54 q^{24} + 749 q^{25} + 116 q^{26} - 1026 q^{27} - 576 q^{28} + 1102 q^{29} + 468 q^{30} - 846 q^{31} + 174 q^{32} + 333 q^{33} - 277 q^{34} - 943 q^{35} + 1242 q^{36} + 183 q^{37} + 27 q^{38} - 537 q^{39} - 1517 q^{40} - 115 q^{41} + 426 q^{42} - 585 q^{43} - 273 q^{44} - 225 q^{45} - 746 q^{47} - 1410 q^{48} + 1494 q^{49} + 833 q^{50} - 174 q^{51} + 1547 q^{52} - 470 q^{53} + 152 q^{55} - 1629 q^{56} - 75 q^{57} - 2409 q^{59} + 579 q^{60} + 812 q^{61} - 2390 q^{62} - 306 q^{63} + 966 q^{64} - 1410 q^{65} + 513 q^{66} - 549 q^{67} - 818 q^{68} + 2622 q^{69} + 745 q^{70} - 5612 q^{71} + 162 q^{72} + 2592 q^{73} - 1427 q^{74} - 2247 q^{75} + 1241 q^{76} - 431 q^{77} - 348 q^{78} - 3514 q^{79} - 1835 q^{80} + 3078 q^{81} + 2701 q^{82} - 1526 q^{83} + 1728 q^{84} + 3659 q^{85} - 4267 q^{86} - 3306 q^{87} + 451 q^{88} - 359 q^{89} - 1404 q^{90} - 5169 q^{91} - 3174 q^{92} + 2538 q^{93} + 33 q^{94} - 3929 q^{95} - 522 q^{96} + 3554 q^{97} + 1876 q^{98} - 999 q^{99}+O(q^{100}) \) 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 −5.41926 −3.00000 21.3684 −10.7075 16.2578 16.0266 −72.4468 9.00000 58.0266
1.2 −5.07433 −3.00000 17.7488 −0.846660 15.2230 5.06035 −49.4687 9.00000 4.29623
1.3 −5.01967 −3.00000 17.1971 15.2605 15.0590 12.9330 −46.1663 9.00000 −76.6027
1.4 −4.83398 −3.00000 15.3673 12.3301 14.5019 0.901160 −35.6134 9.00000 −59.6035
1.5 −4.74544 −3.00000 14.5192 11.8728 14.2363 −34.6809 −30.9364 9.00000 −56.3417
1.6 −4.07820 −3.00000 8.63169 −13.7488 12.2346 −25.9609 −2.57617 9.00000 56.0704
1.7 −3.68592 −3.00000 5.58603 2.87889 11.0578 −18.2236 8.89772 9.00000 −10.6114
1.8 −3.62305 −3.00000 5.12650 −15.0923 10.8692 21.3843 10.4108 9.00000 54.6803
1.9 −3.46459 −3.00000 4.00338 −3.25081 10.3938 −8.44443 13.8466 9.00000 11.2627
1.10 −3.24579 −3.00000 2.53514 −18.1117 9.73737 −0.626485 17.7378 9.00000 58.7868
1.11 −3.15610 −3.00000 1.96098 12.6714 9.46831 23.4721 19.0598 9.00000 −39.9922
1.12 −2.11226 −3.00000 −3.53834 20.6326 6.33679 −9.41843 24.3720 9.00000 −43.5815
1.13 −2.08842 −3.00000 −3.63851 3.77808 6.26526 −0.426456 24.3061 9.00000 −7.89020
1.14 −1.83239 −3.00000 −4.64236 −5.61321 5.49716 34.4110 23.1657 9.00000 10.2856
1.15 −1.68236 −3.00000 −5.16967 −4.43895 5.04707 −22.8925 22.1561 9.00000 7.46790
1.16 −1.24001 −3.00000 −6.46237 −11.4413 3.72003 7.26374 17.9335 9.00000 14.1873
1.17 −0.687927 −3.00000 −7.52676 4.58126 2.06378 −21.9023 10.6813 9.00000 −3.15158
1.18 −0.673060 −3.00000 −7.54699 4.84225 2.01918 11.9042 10.4641 9.00000 −3.25913
1.19 −0.576514 −3.00000 −7.66763 5.84325 1.72954 24.1370 9.03260 9.00000 −3.36871
1.20 0.269392 −3.00000 −7.92743 19.8852 −0.808177 −11.7053 −4.29073 9.00000 5.35692
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
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.e ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.4.a.e ✓ 38 1.a even 1 1 trivial