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Properties

Label 1045.4.a.e
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $0$
Dimension $22$
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] = [1045,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(61.6569959560\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 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 −5.28344 5.20066 19.9148 5.00000 −27.4774 6.49388 −62.9511 0.0468967 −26.4172
1.2 −4.73543 −6.01363 14.4243 5.00000 28.4771 1.72878 −30.4218 9.16373 −23.6772
1.3 −4.22352 0.642817 9.83811 5.00000 −2.71495 −28.8154 −7.76327 −26.5868 −21.1176
1.4 −4.20431 0.957996 9.67621 5.00000 −4.02771 35.2276 −7.04730 −26.0822 −21.0215
1.5 −3.10270 7.55803 1.62677 5.00000 −23.4503 17.4690 19.7742 30.1238 −15.5135
1.6 −2.53824 −3.76357 −1.55736 5.00000 9.55282 −9.64472 24.2588 −12.8356 −12.6912
1.7 −1.63750 1.39513 −5.31860 5.00000 −2.28453 2.22178 21.8092 −25.0536 −8.18749
1.8 −0.933736 5.15873 −7.12814 5.00000 −4.81690 −23.0928 14.1257 −0.387467 −4.66868
1.9 −0.905089 −8.62521 −7.18081 5.00000 7.80658 3.26066 13.7400 47.3942 −4.52545
1.10 0.0230482 4.90691 −7.99947 5.00000 0.113095 35.2639 −0.368758 −2.92227 0.115241
1.11 0.520930 10.1782 −7.72863 5.00000 5.30216 12.5313 −8.19352 76.5967 2.60465
1.12 0.620206 0.467922 −7.61534 5.00000 0.290208 −15.6454 −9.68474 −26.7810 3.10103
1.13 1.45750 −8.28435 −5.87568 5.00000 −12.0745 3.32162 −20.2239 41.6304 7.28752
1.14 2.35488 −1.20591 −2.45454 5.00000 −2.83978 20.7607 −24.6192 −25.5458 11.7744
1.15 2.70597 6.22305 −0.677713 5.00000 16.8394 −21.7619 −23.4817 11.7264 13.5299
1.16 2.94017 −3.08997 0.644599 5.00000 −9.08503 21.8209 −21.6261 −17.4521 14.7008
1.17 4.12095 −6.27512 8.98226 5.00000 −25.8595 −21.3904 4.04785 12.3772 20.6048
1.18 4.30475 8.78392 10.5309 5.00000 37.8126 4.68368 10.8947 50.1573 21.5237
1.19 4.50809 6.83061 12.3229 5.00000 30.7930 25.7280 19.4881 19.6572 22.5405
1.20 5.07818 −9.17350 17.7879 5.00000 −46.5847 27.6942 49.7048 57.1531 25.3909
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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 1045.4.a.e ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.e ✓ 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 12 T_{2}^{21} - 64 T_{2}^{20} + 1242 T_{2}^{19} + 113 T_{2}^{18} - 51909 T_{2}^{17} + 90761 T_{2}^{16} + 1119302 T_{2}^{15} - 3160473 T_{2}^{14} - 13106574 T_{2}^{13} + 49480690 T_{2}^{12} + \cdots - 11927552 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display