Properties

Label 1045.4.a.f
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99} + O(q^{100}) \)

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 −5.44364 −3.90819 21.6332 −5.00000 21.2748 26.4600 −74.2143 −11.7261 27.2182
1.2 −4.98045 6.47284 16.8049 −5.00000 −32.2376 19.1655 −43.8523 14.8976 24.9023
1.3 −4.82291 8.55867 15.2605 −5.00000 −41.2777 −30.1370 −35.0167 46.2509 24.1146
1.4 −4.44156 −2.04104 11.7274 −5.00000 9.06537 −16.0963 −16.5556 −22.8342 22.2078
1.5 −3.76880 3.37173 6.20386 −5.00000 −12.7074 −10.1959 6.76930 −15.6315 18.8440
1.6 −3.46924 −9.89028 4.03563 −5.00000 34.3118 31.6510 13.7534 70.8177 17.3462
1.7 −3.41432 −6.29243 3.65755 −5.00000 21.4843 9.26458 14.8265 12.5947 17.0716
1.8 −2.74715 4.30519 −0.453153 −5.00000 −11.8270 27.4903 23.2221 −8.46533 13.7358
1.9 −2.04967 −4.19821 −3.79884 −5.00000 8.60496 −17.0006 24.1838 −9.37502 10.2484
1.10 −1.05934 2.12682 −6.87780 −5.00000 −2.25303 5.16798 15.7607 −22.4766 5.29670
1.11 −0.507452 7.53941 −7.74249 −5.00000 −3.82589 −10.8176 7.98856 29.8427 2.53726
1.12 −0.277460 −1.24426 −7.92302 −5.00000 0.345233 −34.3401 4.41800 −25.4518 1.38730
1.13 1.01119 −6.79853 −6.97749 −5.00000 −6.87463 1.66301 −15.1451 19.2201 −5.05597
1.14 1.33604 6.37251 −6.21499 −5.00000 8.51394 16.1837 −18.9918 13.6088 −6.68022
1.15 1.93160 7.20488 −4.26893 −5.00000 13.9169 −6.82167 −23.6986 24.9103 −9.65799
1.16 1.99974 −7.61217 −4.00104 −5.00000 −15.2224 −5.59893 −23.9990 30.9451 −9.99871
1.17 2.68859 −3.54772 −0.771496 −5.00000 −9.53836 27.8868 −23.5829 −14.4137 −13.4429
1.18 3.31281 3.48913 2.97471 −5.00000 11.5588 27.0446 −16.6478 −14.8260 −16.5641
1.19 3.55626 −9.55461 4.64698 −5.00000 −33.9787 −15.7816 −11.9242 64.2906 −17.7813
1.20 4.45595 5.90454 11.8555 −5.00000 26.3103 −31.8736 17.1798 7.86359 −22.2797
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.f 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.f 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\).