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Properties

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 8 q^{2} + 63 q^{3} + 616 q^{4} + 950 q^{5} + 149 q^{6} + 275 q^{7} + 264 q^{8} + 3029 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 8 q^{2} + 63 q^{3} + 616 q^{4} + 950 q^{5} + 149 q^{6} + 275 q^{7} + 264 q^{8} + 3029 q^{9} + 200 q^{10} - 4598 q^{11} + 2312 q^{12} + 41 q^{13} + 23 q^{14} + 1575 q^{15} + 7196 q^{16} - 2431 q^{17} - 1689 q^{18} - 13718 q^{19} + 15400 q^{20} - 1577 q^{21} - 968 q^{22} + 9284 q^{23} + 7598 q^{24} + 23750 q^{25} + 13129 q^{26} + 9228 q^{27} - 1079 q^{28} - 559 q^{29} + 3725 q^{30} + 11147 q^{31} + 11051 q^{32} - 7623 q^{33} + 40895 q^{34} + 6875 q^{35} + 55887 q^{36} + 41579 q^{37} - 2888 q^{38} + 24982 q^{39} + 6600 q^{40} + 18597 q^{41} + 61360 q^{42} + 25353 q^{43} - 74536 q^{44} + 75725 q^{45} + 1611 q^{46} + 63516 q^{47} + 187737 q^{48} + 141609 q^{49} + 5000 q^{50} + 107546 q^{51} + 60018 q^{52} + 123045 q^{53} + 256696 q^{54} - 114950 q^{55} + 157335 q^{56} - 22743 q^{57} + 218938 q^{58} + 132925 q^{59} + 57800 q^{60} - 59107 q^{61} + 166982 q^{62} + 130582 q^{63} + 313126 q^{64} + 1025 q^{65} - 18029 q^{66} + 162534 q^{67} + 182980 q^{68} + 178552 q^{69} + 575 q^{70} + 157840 q^{71} + 98630 q^{72} - 63010 q^{73} + 122683 q^{74} + 39375 q^{75} - 222376 q^{76} - 33275 q^{77} + 277272 q^{78} - 16385 q^{79} + 179900 q^{80} + 290354 q^{81} + 362302 q^{82} + 138461 q^{83} + 446870 q^{84} - 60775 q^{85} + 643902 q^{86} + 291602 q^{87} - 31944 q^{88} + 224792 q^{89} - 42225 q^{90} + 498548 q^{91} + 581088 q^{92} + 134210 q^{93} + 35864 q^{94} - 342950 q^{95} + 377376 q^{96} + 292216 q^{97} - 58230 q^{98} - 366509 q^{99}+O(q^{100}) \) 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 −11.0683 7.73173 90.5062 25.0000 −85.5768 83.0891 −647.562 −183.220 −276.706
1.2 −10.4263 23.9600 76.7073 25.0000 −249.814 −14.0559 −466.130 331.082 −260.657
1.3 −10.3329 −11.3052 74.7684 25.0000 116.815 −198.142 −441.921 −115.192 −258.322
1.4 −9.16691 −28.2569 52.0323 25.0000 259.028 66.4323 −183.634 555.452 −229.173
1.5 −8.93899 3.69296 47.9055 25.0000 −33.0113 −46.2037 −142.179 −229.362 −223.475
1.6 −8.03656 17.6855 32.5863 25.0000 −142.131 252.455 −4.71219 69.7783 −200.914
1.7 −7.46340 17.5503 23.7024 25.0000 −130.985 −243.020 61.9286 65.0117 −186.585
1.8 −6.74504 27.3602 13.4956 25.0000 −184.545 −119.648 124.813 505.578 −168.626
1.9 −6.62298 −27.0099 11.8638 25.0000 178.886 116.446 133.361 486.535 −165.574
1.10 −6.60027 −22.9240 11.5635 25.0000 151.305 −137.012 134.886 282.511 −165.007
1.11 −6.07665 −12.6619 4.92570 25.0000 76.9417 227.711 164.521 −82.6774 −151.916
1.12 −5.63532 2.19407 −0.243132 25.0000 −12.3643 −35.7296 181.700 −238.186 −140.883
1.13 −4.27957 13.2431 −13.6853 25.0000 −56.6748 11.8777 195.513 −67.6203 −106.989
1.14 −4.07979 −6.58287 −15.3553 25.0000 26.8567 210.046 193.200 −199.666 −101.995
1.15 −3.39547 23.0537 −20.4708 25.0000 −78.2781 193.366 178.163 288.472 −84.8868
1.16 −2.29603 −17.4404 −26.7283 25.0000 40.0437 −42.9271 134.842 61.1692 −57.4007
1.17 −1.31861 2.12368 −30.2613 25.0000 −2.80029 −96.2869 82.0982 −238.490 −32.9652
1.18 −0.806491 22.8987 −31.3496 25.0000 −18.4676 −91.7329 51.0909 281.351 −20.1623
1.19 −0.202213 4.96200 −31.9591 25.0000 −1.00338 −55.2198 12.9334 −218.379 −5.05532
1.20 0.514635 −15.6824 −31.7352 25.0000 −8.07072 177.020 −32.8003 2.93807 12.8659
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.f ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.f ✓ 38 1.a even 1 1 trivial