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Properties

Label 1045.6.a.e
Level $1045$
Weight $6$
Character orbit 1045.a
Self dual yes
Analytic conductor $167.601$
Analytic rank $1$
Dimension $38$
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] = [1045,6,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(167.601091705\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9} - 600 q^{10} + 4598 q^{11} - 2008 q^{12} - 2663 q^{13} - 1565 q^{14} - 1575 q^{15} + 12390 q^{16} - 3311 q^{17} - 6383 q^{18} - 13718 q^{19} + 14850 q^{20} - 8179 q^{21} - 2904 q^{22} - 3412 q^{23} - 4100 q^{24} + 23750 q^{25} - 1399 q^{26} - 31596 q^{27} - 43653 q^{28} - 13633 q^{29} - 1675 q^{30} - 13789 q^{31} - 58603 q^{32} - 7623 q^{33} - 29149 q^{34} - 18225 q^{35} + 50641 q^{36} - 12103 q^{37} + 8664 q^{38} - 50960 q^{39} - 31800 q^{40} - 37885 q^{41} + 51100 q^{42} - 56119 q^{43} + 71874 q^{44} + 75725 q^{45} - 56291 q^{46} - 37532 q^{47} - 113895 q^{48} + 153501 q^{49} - 15000 q^{50} + 32882 q^{51} - 169554 q^{52} - 51511 q^{53} - 175060 q^{54} + 114950 q^{55} - 84247 q^{56} + 22743 q^{57} - 256962 q^{58} - 154267 q^{59} - 50200 q^{60} - 47165 q^{61} + 143002 q^{62} - 358780 q^{63} + 142292 q^{64} - 66575 q^{65} - 8107 q^{66} - 161712 q^{67} - 210188 q^{68} - 124602 q^{69} - 39125 q^{70} + 6118 q^{71} - 327878 q^{72} - 152182 q^{73} - 167349 q^{74} - 39375 q^{75} - 214434 q^{76} - 88209 q^{77} - 216594 q^{78} - 140433 q^{79} + 309750 q^{80} + 382874 q^{81} - 29842 q^{82} - 515287 q^{83} + 29222 q^{84} - 82775 q^{85} + 204974 q^{86} - 106764 q^{87} - 153912 q^{88} - 271610 q^{89} - 159575 q^{90} - 44332 q^{91} + 236348 q^{92} + 25202 q^{93} - 496224 q^{94} - 342950 q^{95} - 275218 q^{96} - 126390 q^{97} - 285506 q^{98} + 366509 q^{99}+O(q^{100}) \) 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 −11.0901 −27.5711 90.9897 25.0000 305.766 −40.7849 −654.200 517.168 −277.252
1.2 −10.8606 27.1939 85.9518 25.0000 −295.341 −206.298 −585.946 496.509 −271.514
1.3 −10.7913 −1.89630 84.4524 25.0000 20.4636 234.566 −566.030 −239.404 −269.783
1.4 −9.91265 15.6713 66.2607 25.0000 −155.344 −122.128 −339.614 2.59022 −247.816
1.5 −9.74865 −7.53353 63.0362 25.0000 73.4417 −163.789 −302.561 −186.246 −243.716
1.6 −9.21304 −14.8344 52.8801 25.0000 136.670 144.131 −192.369 −22.9413 −230.326
1.7 −8.88488 −24.9356 46.9410 25.0000 221.550 −188.162 −132.749 378.784 −222.122
1.8 −8.25643 −6.02271 36.1686 25.0000 49.7261 24.7368 −34.4178 −206.727 −206.411
1.9 −7.95052 14.0444 31.2108 25.0000 −111.660 100.860 6.27456 −45.7550 −198.763
1.10 −7.34806 21.4317 21.9939 25.0000 −157.482 84.2288 73.5250 216.320 −183.701
1.11 −6.08152 −12.1543 4.98491 25.0000 73.9166 37.2863 164.293 −95.2732 −152.038
1.12 −4.92584 16.7465 −7.73613 25.0000 −82.4904 −110.483 195.734 37.4445 −123.146
1.13 −4.85962 −16.0436 −8.38410 25.0000 77.9658 −94.2760 196.251 14.3969 −121.490
1.14 −3.63943 11.8452 −18.7545 25.0000 −43.1098 −246.643 184.718 −102.691 −90.9858
1.15 −3.39231 −29.7744 −20.4922 25.0000 101.004 10.0128 178.070 643.513 −84.8078
1.16 −2.67362 −11.1309 −24.8518 25.0000 29.7598 158.866 152.000 −119.103 −66.8405
1.17 −2.59417 30.0518 −25.2703 25.0000 −77.9597 −144.749 148.569 660.111 −64.8544
1.18 −1.87953 7.44103 −28.4674 25.0000 −13.9857 104.152 113.650 −187.631 −46.9883
1.19 −1.77527 9.46319 −28.8484 25.0000 −16.7997 232.548 108.022 −153.448 −44.3817
1.20 −0.836459 −16.2079 −31.3003 25.0000 13.5572 −208.079 52.9482 19.6951 −20.9115
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
\(5\) \(-1\)
\(11\) \(-1\)
\(19\) \(1\)

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 1045.6.a.e ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.e ✓ 38 1.a even 1 1 trivial