Properties

Label 2013.4.a.d
Level $2013$
Weight $4$
Character orbit 2013.a
Self dual yes
Analytic conductor $118.771$
Analytic rank $1$
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] = [2013,4,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(118.770844842\)
Analytic rank: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 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 −5.57224 −3.00000 23.0498 −16.5315 16.7167 −30.4189 −83.8611 9.00000 92.1176
1.2 −5.34535 −3.00000 20.5728 5.60041 16.0360 11.9095 −67.2058 9.00000 −29.9362
1.3 −5.20750 −3.00000 19.1180 13.8219 15.6225 −32.6173 −57.8972 9.00000 −71.9777
1.4 −4.94279 −3.00000 16.4312 0.994625 14.8284 23.9386 −41.6735 9.00000 −4.91622
1.5 −4.85102 −3.00000 15.5324 −6.76910 14.5530 −14.4227 −36.5396 9.00000 32.8370
1.6 −4.27346 −3.00000 10.2624 −10.6639 12.8204 −5.91849 −9.66837 9.00000 45.5715
1.7 −4.00724 −3.00000 8.05801 20.5241 12.0217 −10.4521 −0.232470 9.00000 −82.2450
1.8 −3.99170 −3.00000 7.93363 2.24192 11.9751 2.67076 0.264912 9.00000 −8.94905
1.9 −3.73288 −3.00000 5.93441 −18.1575 11.1986 21.4866 7.71060 9.00000 67.7799
1.10 −2.84685 −3.00000 0.104547 −9.28768 8.54055 6.39681 22.4772 9.00000 26.4406
1.11 −2.75819 −3.00000 −0.392374 6.83862 8.27458 −1.91645 23.1478 9.00000 −18.8622
1.12 −2.70850 −3.00000 −0.664006 8.28724 8.12551 −34.8516 23.4665 9.00000 −22.4460
1.13 −2.29940 −3.00000 −2.71276 −16.8558 6.89820 −2.53212 24.6329 9.00000 38.7581
1.14 −1.94841 −3.00000 −4.20370 15.7553 5.84523 3.79858 23.7778 9.00000 −30.6978
1.15 −1.80549 −3.00000 −4.74021 −6.98611 5.41646 21.1508 23.0023 9.00000 12.6133
1.16 −1.54125 −3.00000 −5.62456 −4.20975 4.62374 −32.1268 20.9988 9.00000 6.48826
1.17 −1.30641 −3.00000 −6.29329 −3.39016 3.91923 25.4059 18.6729 9.00000 4.42894
1.18 −0.441122 −3.00000 −7.80541 14.6691 1.32336 −18.4288 6.97211 9.00000 −6.47085
1.19 0.197712 −3.00000 −7.96091 13.9720 −0.593137 33.5020 −3.15567 9.00000 2.76244
1.20 0.250375 −3.00000 −7.93731 9.04215 −0.751125 10.1810 −3.99031 9.00000 2.26393
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
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(1\)

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 2013.4.a.d 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.d 37 1.a even 1 1 trivial