↑

Properties

Label 2005.4.a.b
Level $2005$
Weight $4$
Character orbit 2005.a
Self dual yes
Analytic conductor $118.299$
Analytic rank $1$
Dimension $96$
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] = [2005,4,Mod(1,2005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2005.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2005 = 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2005.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: \(118.298829562\)
Analytic rank: \(1\)
Dimension: \(96\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 9 q^{2} - 12 q^{3} + 355 q^{4} - 480 q^{5} - 40 q^{6} + 86 q^{7} - 135 q^{8} + 712 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 9 q^{2} - 12 q^{3} + 355 q^{4} - 480 q^{5} - 40 q^{6} + 86 q^{7} - 135 q^{8} + 712 q^{9} + 45 q^{10} - 422 q^{11} - 78 q^{12} + 106 q^{13} - 204 q^{14} + 60 q^{15} + 1215 q^{16} - 214 q^{17} - 254 q^{18} - 510 q^{19} - 1775 q^{20} - 530 q^{21} + 182 q^{22} - 384 q^{23} - 577 q^{24} + 2400 q^{25} - 549 q^{26} - 288 q^{27} + 506 q^{28} - 952 q^{29} + 200 q^{30} - 400 q^{31} - 1294 q^{32} + 20 q^{33} - 304 q^{34} - 430 q^{35} + 1447 q^{36} + 10 q^{37} - 585 q^{38} - 834 q^{39} + 675 q^{40} - 674 q^{41} - 870 q^{42} - 1054 q^{43} - 3554 q^{44} - 3560 q^{45} - 855 q^{46} - 126 q^{47} - 584 q^{48} + 3536 q^{49} - 225 q^{50} - 3766 q^{51} + 859 q^{52} - 2534 q^{53} - 1528 q^{54} + 2110 q^{55} - 3412 q^{56} + 266 q^{57} + 759 q^{58} - 5260 q^{59} + 390 q^{60} - 1388 q^{61} - 1793 q^{62} + 1380 q^{63} + 2011 q^{64} - 530 q^{65} - 3106 q^{66} - 1592 q^{67} - 2308 q^{68} - 448 q^{69} + 1020 q^{70} - 3796 q^{71} - 3955 q^{72} + 2276 q^{73} - 5293 q^{74} - 300 q^{75} - 4399 q^{76} - 2420 q^{77} - 1850 q^{78} - 4246 q^{79} - 6075 q^{80} + 1416 q^{81} + 2608 q^{82} - 5752 q^{83} - 3460 q^{84} + 1070 q^{85} - 3851 q^{86} + 210 q^{87} + 2104 q^{88} - 3856 q^{89} + 1270 q^{90} - 4032 q^{91} - 4899 q^{92} + 1040 q^{93} - 4200 q^{94} + 2550 q^{95} - 4641 q^{96} + 1330 q^{97} - 3824 q^{98} - 9226 q^{99}+O(q^{100}) \) 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 −5.52779 1.61731 22.5565 −5.00000 −8.94017 −6.30665 −80.4650 −24.3843 27.6389
1.2 −5.35771 −6.89589 20.7050 −5.00000 36.9462 13.0731 −68.0698 20.5533 26.7885
1.3 −5.31151 4.97478 20.2121 −5.00000 −26.4236 −11.7865 −64.8648 −2.25159 26.5575
1.4 −5.28349 −3.25112 19.9152 −5.00000 17.1772 21.0515 −62.9540 −16.4302 26.4174
1.5 −5.24525 8.64235 19.5126 −5.00000 −45.3313 13.3294 −60.3866 47.6902 26.2262
1.6 −5.13213 0.887459 18.3388 −5.00000 −4.55456 36.4957 −53.0601 −26.2124 25.6607
1.7 −5.12639 −8.89021 18.2798 −5.00000 45.5747 21.7039 −52.6985 52.0359 25.6319
1.8 −5.12635 6.40604 18.2795 −5.00000 −32.8396 −23.9955 −52.6961 14.0374 25.6317
1.9 −5.08466 −3.39622 17.8538 −5.00000 17.2686 −20.2810 −50.1030 −15.4657 25.4233
1.10 −4.78114 3.99211 14.8593 −5.00000 −19.0868 24.0899 −32.7950 −11.0630 23.9057
1.11 −4.68906 −7.26986 13.9873 −5.00000 34.0888 −18.0584 −28.0747 25.8508 23.4453
1.12 −4.55408 −5.34279 12.7397 −5.00000 24.3315 −14.2605 −21.5850 1.54544 22.7704
1.13 −4.49004 0.830924 12.1605 −5.00000 −3.73089 −32.6767 −18.6808 −26.3096 22.4502
1.14 −4.45342 10.0560 11.8330 −5.00000 −44.7836 12.4079 −17.0698 74.1233 22.2671
1.15 −4.33112 7.75188 10.7586 −5.00000 −33.5743 1.17635 −11.9478 33.0917 21.6556
1.16 −4.22815 −5.50466 9.87723 −5.00000 23.2745 8.61621 −7.93721 3.30128 21.1407
1.17 −4.13580 −6.28590 9.10484 −5.00000 25.9972 3.61276 −4.56939 12.5126 20.6790
1.18 −3.85415 −1.98110 6.85443 −5.00000 7.63545 2.90389 4.41518 −23.0752 19.2707
1.19 −3.83628 4.68685 6.71706 −5.00000 −17.9801 15.2537 4.92170 −5.03342 19.1814
1.20 −3.72165 1.10148 5.85064 −5.00000 −4.09933 −29.5493 7.99915 −25.7867 18.6082
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.96
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(401\) \(-1\)

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 2005.4.a.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.4.a.b ✓ 96 1.a even 1 1 trivial