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Properties

Label 1045.6.a.h
Level $1045$
Weight $6$
Character orbit 1045.a
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
Dimension $40$
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: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{2} + 63 q^{3} + 690 q^{4} + 1000 q^{5} + 365 q^{6} + 839 q^{7} + 846 q^{8} + 3555 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{2} + 63 q^{3} + 690 q^{4} + 1000 q^{5} + 365 q^{6} + 839 q^{7} + 846 q^{8} + 3555 q^{9} + 600 q^{10} + 4840 q^{11} + 2312 q^{12} + 2661 q^{13} + 395 q^{14} + 1575 q^{15} + 15974 q^{16} + 8249 q^{17} + 7225 q^{18} + 14440 q^{19} + 17250 q^{20} + 6845 q^{21} + 2904 q^{22} + 13948 q^{23} + 9740 q^{24} + 25000 q^{25} + 11581 q^{26} + 27864 q^{27} + 37879 q^{28} + 12965 q^{29} + 9125 q^{30} + 4411 q^{31} + 30751 q^{32} + 7623 q^{33} - 17739 q^{34} + 20975 q^{35} + 71345 q^{36} + 5729 q^{37} + 8664 q^{38} - 23560 q^{39} + 21150 q^{40} + 34059 q^{41} + 48528 q^{42} + 68593 q^{43} + 83490 q^{44} + 88875 q^{45} + 43829 q^{46} + 91592 q^{47} + 26539 q^{48} + 152447 q^{49} + 15000 q^{50} - 23170 q^{51} + 46798 q^{52} + 24361 q^{53} + 136436 q^{54} + 121000 q^{55} - 35393 q^{56} + 22743 q^{57} + 77722 q^{58} + 212881 q^{59} + 57800 q^{60} + 137627 q^{61} + 82606 q^{62} + 243832 q^{63} + 259580 q^{64} + 66525 q^{65} + 44165 q^{66} + 78752 q^{67} + 565000 q^{68} + 46134 q^{69} + 9875 q^{70} + 28888 q^{71} - 65574 q^{72} + 291074 q^{73} + 151963 q^{74} + 39375 q^{75} + 249090 q^{76} + 101519 q^{77} - 136222 q^{78} + 87079 q^{79} + 399350 q^{80} + 471360 q^{81} - 74882 q^{82} + 346989 q^{83} - 159196 q^{84} + 206225 q^{85} - 207742 q^{86} + 294612 q^{87} + 102366 q^{88} + 126718 q^{89} + 180625 q^{90} - 239900 q^{91} + 274196 q^{92} + 321654 q^{93} - 418108 q^{94} + 361000 q^{95} + 342154 q^{96} + 137404 q^{97} - 89356 q^{98} + 430155 q^{99}+O(q^{100}) \) 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 −10.7623 −11.3539 83.8277 25.0000 122.194 −59.3342 −557.787 −114.090 −269.058
1.2 −10.4622 24.7793 77.4578 25.0000 −259.246 156.281 −475.589 371.012 −261.555
1.3 −10.3994 4.60548 76.1474 25.0000 −47.8942 17.5294 −459.106 −221.790 −259.985
1.4 −10.1541 −26.5431 71.1059 25.0000 269.522 225.388 −397.085 461.539 −253.853
1.5 −8.55086 12.0717 41.1172 25.0000 −103.224 −150.646 −77.9596 −97.2734 −213.771
1.6 −8.31261 15.2747 37.0994 25.0000 −126.972 144.510 −42.3897 −9.68428 −207.815
1.7 −7.39269 −19.3511 22.6518 25.0000 143.057 −146.772 69.1080 131.466 −184.817
1.8 −7.16026 27.0461 19.2693 25.0000 −193.657 −115.417 91.1552 488.494 −179.006
1.9 −6.98489 −3.88370 16.7886 25.0000 27.1272 123.805 106.250 −227.917 −174.622
1.10 −6.34868 −22.4542 8.30576 25.0000 142.555 214.893 150.427 261.192 −158.717
1.11 −5.82679 −5.33310 1.95150 25.0000 31.0749 −202.310 175.086 −214.558 −145.670
1.12 −5.02898 3.76125 −6.70939 25.0000 −18.9153 170.122 194.669 −228.853 −125.724
1.13 −3.84457 26.7977 −17.2193 25.0000 −103.025 117.853 189.227 475.114 −96.1141
1.14 −2.87435 1.31331 −23.7381 25.0000 −3.77492 −104.963 160.211 −241.275 −71.8588
1.15 −2.77766 −22.7115 −24.2846 25.0000 63.0848 −147.613 156.340 272.811 −69.4415
1.16 −1.49444 1.90416 −29.7667 25.0000 −2.84565 61.7350 92.3065 −239.374 −37.3610
1.17 −1.40465 −17.9315 −30.0269 25.0000 25.1875 73.4146 87.1263 78.5376 −35.1163
1.18 −0.609213 20.7237 −31.6289 25.0000 −12.6251 −114.174 38.7636 186.471 −15.2303
1.19 −0.153758 −4.07857 −31.9764 25.0000 0.627114 −123.189 9.83689 −226.365 −3.84396
1.20 0.237841 14.0227 −31.9434 25.0000 3.33516 13.4367 −15.2083 −46.3651 5.94602
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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.h ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.h ✓ 40 1.a even 1 1 trivial