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Properties

Label 1045.6.a.g
Level $1045$
Weight $6$
Character orbit 1045.a
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
Dimension $39$
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: \(0\)
Dimension: \(39\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 39 q + 12 q^{2} + 27 q^{3} + 670 q^{4} - 975 q^{5} + 69 q^{6} - 251 q^{7} + 270 q^{8} + 3666 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 39 q + 12 q^{2} + 27 q^{3} + 670 q^{4} - 975 q^{5} + 69 q^{6} - 251 q^{7} + 270 q^{8} + 3666 q^{9} - 300 q^{10} - 4719 q^{11} + 872 q^{12} - 717 q^{13} + 2647 q^{14} - 675 q^{15} + 15078 q^{16} + 2155 q^{17} + 3293 q^{18} + 14079 q^{19} - 16750 q^{20} + 6891 q^{21} - 1452 q^{22} + 1296 q^{23} - 3138 q^{24} + 24375 q^{25} + 2585 q^{26} + 3846 q^{27} - 20901 q^{28} - 7635 q^{29} - 1725 q^{30} + 9595 q^{31} - 43 q^{32} - 3267 q^{33} - 28383 q^{34} + 6275 q^{35} + 80573 q^{36} - 28965 q^{37} + 4332 q^{38} + 21358 q^{39} - 6750 q^{40} + 19627 q^{41} + 12478 q^{42} + 3711 q^{43} - 81070 q^{44} - 91650 q^{45} + 37459 q^{46} + 38450 q^{47} - 12259 q^{48} + 78758 q^{49} + 7500 q^{50} - 68114 q^{51} - 50322 q^{52} + 20631 q^{53} - 124732 q^{54} + 117975 q^{55} + 94893 q^{56} + 9747 q^{57} - 148140 q^{58} + 170405 q^{59} - 21800 q^{60} - 22345 q^{61} + 246390 q^{62} - 151688 q^{63} + 340312 q^{64} + 17925 q^{65} - 8349 q^{66} - 16408 q^{67} - 32276 q^{68} + 93536 q^{69} - 66175 q^{70} + 119338 q^{71} + 135668 q^{72} - 141606 q^{73} + 71843 q^{74} + 16875 q^{75} + 241870 q^{76} + 30371 q^{77} + 708290 q^{78} + 14727 q^{79} - 376950 q^{80} + 441659 q^{81} - 219870 q^{82} + 384909 q^{83} + 877024 q^{84} - 53875 q^{85} + 250290 q^{86} - 77038 q^{87} - 32670 q^{88} + 443394 q^{89} - 82325 q^{90} - 207088 q^{91} - 237112 q^{92} + 396718 q^{93} + 409516 q^{94} - 351975 q^{95} + 100332 q^{96} - 152942 q^{97} + 895680 q^{98} - 443586 q^{99}+O(q^{100}) \) 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 −10.9132 −5.65221 87.0985 −25.0000 61.6839 60.6149 −601.302 −211.052 272.831
1.2 −10.6528 −25.7563 81.4826 −25.0000 274.378 −195.509 −527.129 420.388 266.321
1.3 −10.1686 30.3201 71.3997 −25.0000 −308.312 −123.753 −400.639 676.309 254.214
1.4 −9.86064 3.26128 65.2322 −25.0000 −32.1583 −108.556 −327.690 −232.364 246.516
1.5 −9.68916 21.3816 61.8799 −25.0000 −207.169 48.2370 −289.511 214.171 242.229
1.6 −8.68402 −13.3875 43.4122 −25.0000 116.258 −102.340 −99.1033 −63.7740 217.100
1.7 −8.55717 11.4773 41.2252 −25.0000 −98.2134 106.731 −78.9418 −111.271 213.929
1.8 −7.01607 −14.7319 17.2252 −25.0000 103.360 207.291 103.661 −25.9713 175.402
1.9 −6.68564 −22.1881 12.6977 −25.0000 148.342 35.5129 129.048 249.314 167.141
1.10 −6.61608 9.41089 11.7725 −25.0000 −62.2632 −82.7561 133.827 −154.435 165.402
1.11 −5.04716 16.9331 −6.52622 −25.0000 −85.4638 82.5359 194.448 43.7285 126.179
1.12 −4.54694 −4.48487 −11.3254 −25.0000 20.3924 16.9538 196.998 −222.886 113.673
1.13 −3.56187 −14.4846 −19.3131 −25.0000 51.5922 −241.062 182.771 −33.1968 89.0468
1.14 −2.78665 −26.7838 −24.2346 −25.0000 74.6370 −117.963 156.706 474.371 69.6662
1.15 −2.72815 13.3084 −24.5572 −25.0000 −36.3073 −136.165 154.296 −65.8862 68.2036
1.16 −2.63769 28.9603 −25.0426 −25.0000 −76.3883 −13.2455 150.461 595.699 65.9423
1.17 −2.21689 5.10751 −27.0854 −25.0000 −11.3228 167.053 130.986 −216.913 55.4222
1.18 −1.00103 20.9100 −30.9979 −25.0000 −20.9316 215.274 63.0630 194.229 25.0258
1.19 −0.429556 −22.9509 −31.8155 −25.0000 9.85872 −78.3412 27.4123 283.745 10.7389
1.20 0.325213 −11.1694 −31.8942 −25.0000 −3.63244 1.65378 −20.7792 −118.244 −8.13033
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
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.g ✓ 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.g ✓ 39 1.a even 1 1 trivial