Properties

Label 1045.4.a.g
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $0$
Dimension $23$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(0\)
Dimension: \(23\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23q + 6q^{2} + 9q^{3} + 92q^{4} - 115q^{5} - 25q^{6} - 37q^{7} + 42q^{8} + 170q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 23q + 6q^{2} + 9q^{3} + 92q^{4} - 115q^{5} - 25q^{6} - 37q^{7} + 42q^{8} + 170q^{9} - 30q^{10} - 253q^{11} + 44q^{12} - 37q^{13} + 61q^{14} - 45q^{15} + 588q^{16} - 73q^{17} + 391q^{18} - 437q^{19} - 460q^{20} - 127q^{21} - 66q^{22} - 175q^{23} + 16q^{24} + 575q^{25} + 719q^{26} + 21q^{27} + 253q^{28} + 71q^{29} + 125q^{30} + 302q^{31} + 1107q^{32} - 99q^{33} + 1267q^{34} + 185q^{35} + 703q^{36} - 500q^{37} - 114q^{38} + 457q^{39} - 210q^{40} + 770q^{41} + 2596q^{42} - 902q^{43} - 1012q^{44} - 850q^{45} - 1101q^{46} + 356q^{47} + 1221q^{48} + 908q^{49} + 150q^{50} - 451q^{51} - 358q^{52} + 1327q^{53} + 2534q^{54} + 1265q^{55} + 3135q^{56} - 171q^{57} + 1014q^{58} + 3619q^{59} - 220q^{60} - 1432q^{61} + 1826q^{62} + 1658q^{63} + 4006q^{64} + 185q^{65} + 275q^{66} - 605q^{67} + 5128q^{68} + 3099q^{69} - 305q^{70} + 3230q^{71} + 2152q^{72} - 637q^{73} + 5063q^{74} + 225q^{75} - 1748q^{76} + 407q^{77} + 7230q^{78} + 2074q^{79} - 2940q^{80} + 2291q^{81} + 530q^{82} + 3882q^{83} + 5096q^{84} + 365q^{85} + 2262q^{86} - 27q^{87} - 462q^{88} - 210q^{89} - 1955q^{90} + 4133q^{91} - 6064q^{92} + 824q^{93} - 392q^{94} + 2185q^{95} + 2462q^{96} + 2032q^{97} + 7896q^{98} - 1870q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.41975 4.17361 21.3737 −5.00000 −22.6199 −25.0607 −72.4822 −9.58100 27.0988
1.2 −4.98043 −7.40260 16.8047 −5.00000 36.8681 8.03646 −43.8509 27.7985 24.9021
1.3 −4.38057 7.64668 11.1894 −5.00000 −33.4968 9.94553 −13.9715 31.4717 21.9029
1.4 −4.30610 2.30247 10.5425 −5.00000 −9.91467 14.9972 −10.9483 −21.6986 21.5305
1.5 −3.61172 −0.308289 5.04454 −5.00000 1.11345 −23.5260 10.6743 −26.9050 18.0586
1.6 −2.96023 −4.00535 0.762942 −5.00000 11.8567 6.39895 21.4233 −10.9572 14.8011
1.7 −2.48215 3.78622 −1.83894 −5.00000 −9.39797 1.80512 24.4217 −12.6645 12.4107
1.8 −1.58489 −6.85881 −5.48811 −5.00000 10.8705 9.66724 21.3772 20.0432 7.92447
1.9 −1.45938 5.64597 −5.87020 −5.00000 −8.23963 −17.6407 20.2419 4.87695 7.29692
1.10 −0.449859 −1.82696 −7.79763 −5.00000 0.821874 22.2872 7.10670 −23.6622 2.24929
1.11 −0.128788 9.96634 −7.98341 −5.00000 −1.28354 −35.2392 2.05847 72.3280 0.643939
1.12 0.666010 −10.2430 −7.55643 −5.00000 −6.82192 4.34789 −10.3607 77.9184 −3.33005
1.13 0.814185 3.88289 −7.33710 −5.00000 3.16139 17.3543 −12.4872 −11.9232 −4.07093
1.14 1.51379 1.98888 −5.70843 −5.00000 3.01075 −28.4418 −20.7517 −23.0444 −7.56896
1.15 2.28558 0.0269626 −2.77614 −5.00000 0.0616250 −20.0645 −24.6297 −26.9993 −11.4279
1.16 2.38912 −1.09640 −2.29208 −5.00000 −2.61943 −1.10283 −24.5891 −25.7979 −11.9456
1.17 2.47265 8.18859 −1.88602 −5.00000 20.2475 19.1708 −24.4446 40.0529 −12.3632
1.18 3.33800 −7.01943 3.14227 −5.00000 −23.4309 0.402575 −16.2151 22.2724 −16.6900
1.19 4.05796 −1.73694 8.46702 −5.00000 −7.04845 −13.5343 1.89515 −23.9830 −20.2898
1.20 4.39002 −6.92776 11.2723 −5.00000 −30.4130 −34.5356 14.3653 20.9939 −21.9501
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
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.g 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.g 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{23} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\).