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Properties

Label 1045.6.a.d
Level $1045$
Weight $6$
Character orbit 1045.a
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
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] = [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: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 4 q^{2} + 27 q^{3} + 616 q^{4} - 925 q^{5} + 141 q^{6} - 79 q^{7} + 72 q^{8} + 3140 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 4 q^{2} + 27 q^{3} + 616 q^{4} - 925 q^{5} + 141 q^{6} - 79 q^{7} + 72 q^{8} + 3140 q^{9} - 100 q^{10} + 4477 q^{11} + 872 q^{12} + 719 q^{13} - 625 q^{14} - 675 q^{15} + 6940 q^{16} + 119 q^{17} - 4237 q^{18} - 13357 q^{19} - 15400 q^{20} + 2905 q^{21} + 484 q^{22} - 1252 q^{23} + 5884 q^{24} + 23125 q^{25} + 13201 q^{26} + 9918 q^{27} + 15461 q^{28} + 13221 q^{29} - 3525 q^{30} + 6419 q^{31} + 13173 q^{32} + 3267 q^{33} + 35415 q^{34} + 1975 q^{35} + 80543 q^{36} + 9037 q^{37} - 1444 q^{38} - 6184 q^{39} - 1800 q^{40} + 52577 q^{41} - 28578 q^{42} + 963 q^{43} + 74536 q^{44} - 78500 q^{45} - 10531 q^{46} + 49346 q^{47} + 80107 q^{48} + 70288 q^{49} + 2500 q^{50} + 140786 q^{51} + 165062 q^{52} - 34457 q^{53} + 34216 q^{54} - 111925 q^{55} - 64095 q^{56} - 9747 q^{57} - 126140 q^{58} + 56521 q^{59} - 21800 q^{60} + 6613 q^{61} + 494 q^{62} - 125618 q^{63} - 140426 q^{64} - 17975 q^{65} + 17061 q^{66} - 43534 q^{67} - 138520 q^{68} + 34618 q^{69} + 15625 q^{70} + 95986 q^{71} - 42192 q^{72} + 109218 q^{73} - 182005 q^{74} + 16875 q^{75} - 222376 q^{76} - 9559 q^{77} - 369624 q^{78} + 64943 q^{79} - 173500 q^{80} + 388941 q^{81} - 126926 q^{82} + 109741 q^{83} - 112886 q^{84} - 2975 q^{85} + 43866 q^{86} + 142492 q^{87} + 8712 q^{88} - 119092 q^{89} + 105925 q^{90} + 349320 q^{91} + 433396 q^{92} - 108630 q^{93} + 196160 q^{94} + 333925 q^{95} + 376630 q^{96} + 68774 q^{97} + 310926 q^{98} + 379940 q^{99}+O(q^{100}) \) 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 −10.9267 0.166886 87.3933 −25.0000 −1.82352 16.8566 −605.267 −242.972 273.168
1.2 −9.72609 −28.7123 62.5968 −25.0000 279.258 −28.8219 −297.587 581.397 243.152
1.3 −9.71240 14.3516 62.3307 −25.0000 −139.389 −67.8346 −294.584 −37.0305 242.810
1.4 −9.62194 30.2604 60.5817 −25.0000 −291.163 83.2794 −275.011 672.690 240.548
1.5 −9.18127 5.05214 52.2956 −25.0000 −46.3850 173.574 −186.340 −217.476 229.532
1.6 −9.13266 −22.4746 51.4055 −25.0000 205.253 200.793 −177.224 262.109 228.317
1.7 −7.96108 25.0041 31.3788 −25.0000 −199.060 −85.0628 4.94545 382.206 199.027
1.8 −7.36955 −2.61962 22.3102 −25.0000 19.3054 −105.754 71.4093 −236.138 184.239
1.9 −7.19884 −12.8812 19.8234 −25.0000 92.7298 73.5738 87.6577 −77.0744 179.971
1.10 −7.16342 −23.9676 19.3145 −25.0000 171.690 −233.748 90.8713 331.447 179.085
1.11 −5.60239 14.2277 −0.613205 −25.0000 −79.7094 −144.706 182.712 −40.5711 140.060
1.12 −4.82991 17.2638 −8.67197 −25.0000 −83.3827 225.887 196.442 55.0394 120.748
1.13 −4.40815 −14.4025 −12.5682 −25.0000 63.4884 −75.0406 196.463 −35.5679 110.204
1.14 −3.01425 4.06006 −22.9143 −25.0000 −12.2380 −121.519 165.525 −226.516 75.3562
1.15 −2.49234 20.1221 −25.7882 −25.0000 −50.1511 −99.5533 144.028 161.900 62.3085
1.16 −2.37290 −13.1216 −26.3693 −25.0000 31.1362 161.256 138.505 −70.8249 59.3226
1.17 −0.603954 −13.9085 −31.6352 −25.0000 8.40008 −27.7104 38.4328 −49.5544 15.0989
1.18 0.242878 −26.7520 −31.9410 −25.0000 −6.49745 −185.794 −15.5298 472.668 −6.07194
1.19 0.506691 30.1195 −31.7433 −25.0000 15.2613 184.693 −32.2981 664.181 −12.6673
1.20 0.872903 −19.2132 −31.2380 −25.0000 −16.7713 14.4713 −55.2007 126.149 −21.8226
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
\(5\) \(1\)
\(11\) \(-1\)
\(19\) \(1\)

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.d ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.d ✓ 37 1.a even 1 1 trivial