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Properties

Label 1045.4.a.d
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
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: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 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 −5.35702 −10.2482 20.6977 5.00000 54.9000 −17.5818 −68.0218 78.0263 −26.7851
1.2 −4.84890 −0.335233 15.5118 5.00000 1.62551 −24.3429 −36.4242 −26.8876 −24.2445
1.3 −4.73877 1.86006 14.4559 5.00000 −8.81437 1.31072 −30.5930 −23.5402 −23.6938
1.4 −4.71998 6.85295 14.2783 5.00000 −32.3458 15.9495 −29.6333 19.9630 −23.5999
1.5 −3.16156 −8.60870 1.99549 5.00000 27.2170 32.5634 18.9837 47.1098 −15.8078
1.6 −3.11441 −0.179144 1.69958 5.00000 0.557928 16.3837 19.6221 −26.9679 −15.5721
1.7 −2.86531 −5.40234 0.209982 5.00000 15.4794 −24.6494 22.3208 2.18531 −14.3265
1.8 −2.52988 7.31543 −1.59971 5.00000 −18.5072 2.79284 24.2861 26.5155 −12.6494
1.9 −2.22182 6.66604 −3.06353 5.00000 −14.8107 −18.2528 24.5811 17.4361 −11.1091
1.10 −1.15290 −8.73139 −6.67082 5.00000 10.0664 10.3417 16.9140 49.2372 −5.76450
1.11 −0.0295893 −3.21150 −7.99912 5.00000 0.0950261 7.73734 0.473403 −16.6862 −0.147946
1.12 0.0768470 2.42732 −7.99409 5.00000 0.186533 −11.4153 −1.22910 −21.1081 0.384235
1.13 1.04826 7.96396 −6.90116 5.00000 8.34827 −11.0379 −15.6202 36.4247 5.24128
1.14 1.41332 3.57743 −6.00252 5.00000 5.05605 30.1984 −19.7901 −14.2020 7.06661
1.15 1.45119 −8.75662 −5.89406 5.00000 −12.7075 −31.7430 −20.1629 49.6783 7.25593
1.16 2.01284 −4.64592 −3.94849 5.00000 −9.35147 7.18147 −24.0504 −5.41547 10.0642
1.17 3.65397 3.36009 5.35150 5.00000 12.2777 −1.36266 −9.67753 −15.7098 18.2699
1.18 3.68170 2.35610 5.55493 5.00000 8.67444 −1.20638 −9.00201 −21.4488 18.4085
1.19 3.88965 −9.46724 7.12938 5.00000 −36.8243 13.8479 −3.38640 62.6286 19.4483
1.20 4.17470 5.44396 9.42811 5.00000 22.7269 −28.9642 5.96192 2.63670 20.8735
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.d ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.d ✓ 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 4 T_{2}^{21} - 117 T_{2}^{20} - 442 T_{2}^{19} + 5863 T_{2}^{18} + 20719 T_{2}^{17} + \cdots - 4958464 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display