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Properties

Label 2005.4.a.a
Level $2005$
Weight $4$
Character orbit 2005.a
Self dual yes
Analytic conductor $118.299$
Analytic rank $1$
Dimension $96$
CM no
Inner twists $1$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 27 q^{2} - 42 q^{3} + 355 q^{4} + 480 q^{5} - 20 q^{6} - 274 q^{7} - 303 q^{8} + 728 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 27 q^{2} - 42 q^{3} + 355 q^{4} + 480 q^{5} - 20 q^{6} - 274 q^{7} - 303 q^{8} + 728 q^{9} - 135 q^{10} - 458 q^{11} - 456 q^{12} - 284 q^{13} - 204 q^{14} - 210 q^{15} + 1071 q^{16} - 532 q^{17} - 792 q^{18} - 406 q^{19} + 1775 q^{20} - 318 q^{21} - 514 q^{22} - 1230 q^{23} + 11 q^{24} + 2400 q^{25} - 817 q^{26} - 1368 q^{27} - 2326 q^{28} - 908 q^{29} - 100 q^{30} - 116 q^{31} - 2476 q^{32} - 1180 q^{33} + 28 q^{34} - 1370 q^{35} + 1735 q^{36} - 1320 q^{37} - 1401 q^{38} - 1606 q^{39} - 1515 q^{40} - 1002 q^{41} - 476 q^{42} - 3458 q^{43} - 3062 q^{44} + 3640 q^{45} - 1139 q^{46} - 3250 q^{47} - 4024 q^{48} + 3408 q^{49} - 675 q^{50} - 4350 q^{51} - 3611 q^{52} - 2724 q^{53} - 848 q^{54} - 2290 q^{55} - 1984 q^{56} - 2934 q^{57} - 2637 q^{58} - 3676 q^{59} - 2280 q^{60} - 1028 q^{61} - 5281 q^{62} - 8888 q^{63} + 1519 q^{64} - 1420 q^{65} - 3362 q^{66} - 5154 q^{67} - 6516 q^{68} - 1524 q^{69} - 1020 q^{70} - 2464 q^{71} - 9093 q^{72} - 5632 q^{73} - 5821 q^{74} - 1050 q^{75} - 4187 q^{76} - 4884 q^{77} - 9184 q^{78} - 3942 q^{79} + 5355 q^{80} + 2152 q^{81} - 5558 q^{82} - 12844 q^{83} - 5100 q^{84} - 2660 q^{85} - 4211 q^{86} - 11766 q^{87} - 6954 q^{88} - 2488 q^{89} - 3960 q^{90} - 3744 q^{91} - 12563 q^{92} - 6516 q^{93} - 3724 q^{94} - 2030 q^{95} - 3345 q^{96} - 7396 q^{97} - 11796 q^{98} - 13762 q^{99}+O(q^{100}) \) 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 −5.51388 7.49628 22.4028 5.00000 −41.3336 19.9421 −79.4155 29.1942 −27.5694
1.2 −5.50179 5.24315 22.2697 5.00000 −28.8467 −32.6046 −78.5086 0.490639 −27.5089
1.3 −5.47990 −1.63001 22.0293 5.00000 8.93227 −25.5815 −76.8792 −24.3431 −27.3995
1.4 −5.43846 −7.58372 21.5769 5.00000 41.2438 12.1155 −73.8374 30.5128 −27.1923
1.5 −5.37453 −9.76819 20.8856 5.00000 52.4994 −8.62860 −69.2542 68.4175 −26.8727
1.6 −5.21424 6.51632 19.1883 5.00000 −33.9776 −23.1304 −58.3382 15.4624 −26.0712
1.7 −5.13523 −3.07459 18.3706 5.00000 15.7888 35.6512 −53.2555 −17.5469 −25.6762
1.8 −5.13327 −7.36607 18.3505 5.00000 37.8120 −6.06060 −53.1317 27.2590 −25.6663
1.9 −4.99510 3.89726 16.9510 5.00000 −19.4672 19.0550 −44.7111 −11.8114 −24.9755
1.10 −4.85301 −2.54125 15.5517 5.00000 12.3327 −2.84631 −36.6483 −20.5420 −24.2650
1.11 −4.64396 −8.87378 13.5663 5.00000 41.2094 −32.4012 −25.8498 51.7439 −23.2198
1.12 −4.52537 −7.14772 12.4790 5.00000 32.3461 14.4067 −20.2691 24.0898 −22.6269
1.13 −4.48397 −1.22477 12.1060 5.00000 5.49181 18.2139 −18.4110 −25.4999 −22.4198
1.14 −4.41053 −2.00199 11.4528 5.00000 8.82983 2.87040 −15.2286 −22.9920 −22.0527
1.15 −4.31653 −5.38535 10.6324 5.00000 23.2460 −27.9373 −11.3629 2.00199 −21.5826
1.16 −4.20586 3.58158 9.68929 5.00000 −15.0636 −7.85378 −7.10491 −14.1723 −21.0293
1.17 −4.15223 8.55178 9.24098 5.00000 −35.5089 −8.20695 −5.15283 46.1329 −20.7611
1.18 −4.12579 5.28373 9.02213 5.00000 −21.7996 21.7013 −4.21710 0.917795 −20.6289
1.19 −4.08009 1.72135 8.64713 5.00000 −7.02325 4.45362 −2.64034 −24.0370 −20.4004
1.20 −4.05875 5.80575 8.47348 5.00000 −23.5641 −24.4166 −1.92174 6.70679 −20.2938
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.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.4.a.a ✓ 96 1.a even 1 1 trivial