# Properties

 Label 1045.6.a.g Level $1045$ Weight $6$ Character orbit 1045.a Self dual yes Analytic conductor $167.601$ Analytic rank $0$ Dimension $39$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1045.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$167.601091705$$ Analytic rank: $$0$$ Dimension: $$39$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$39 q + 12 q^{2} + 27 q^{3} + 670 q^{4} - 975 q^{5} + 69 q^{6} - 251 q^{7} + 270 q^{8} + 3666 q^{9}+O(q^{10})$$ 39 * q + 12 * q^2 + 27 * q^3 + 670 * q^4 - 975 * q^5 + 69 * q^6 - 251 * q^7 + 270 * q^8 + 3666 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$39 q + 12 q^{2} + 27 q^{3} + 670 q^{4} - 975 q^{5} + 69 q^{6} - 251 q^{7} + 270 q^{8} + 3666 q^{9} - 300 q^{10} - 4719 q^{11} + 872 q^{12} - 717 q^{13} + 2647 q^{14} - 675 q^{15} + 15078 q^{16} + 2155 q^{17} + 3293 q^{18} + 14079 q^{19} - 16750 q^{20} + 6891 q^{21} - 1452 q^{22} + 1296 q^{23} - 3138 q^{24} + 24375 q^{25} + 2585 q^{26} + 3846 q^{27} - 20901 q^{28} - 7635 q^{29} - 1725 q^{30} + 9595 q^{31} - 43 q^{32} - 3267 q^{33} - 28383 q^{34} + 6275 q^{35} + 80573 q^{36} - 28965 q^{37} + 4332 q^{38} + 21358 q^{39} - 6750 q^{40} + 19627 q^{41} + 12478 q^{42} + 3711 q^{43} - 81070 q^{44} - 91650 q^{45} + 37459 q^{46} + 38450 q^{47} - 12259 q^{48} + 78758 q^{49} + 7500 q^{50} - 68114 q^{51} - 50322 q^{52} + 20631 q^{53} - 124732 q^{54} + 117975 q^{55} + 94893 q^{56} + 9747 q^{57} - 148140 q^{58} + 170405 q^{59} - 21800 q^{60} - 22345 q^{61} + 246390 q^{62} - 151688 q^{63} + 340312 q^{64} + 17925 q^{65} - 8349 q^{66} - 16408 q^{67} - 32276 q^{68} + 93536 q^{69} - 66175 q^{70} + 119338 q^{71} + 135668 q^{72} - 141606 q^{73} + 71843 q^{74} + 16875 q^{75} + 241870 q^{76} + 30371 q^{77} + 708290 q^{78} + 14727 q^{79} - 376950 q^{80} + 441659 q^{81} - 219870 q^{82} + 384909 q^{83} + 877024 q^{84} - 53875 q^{85} + 250290 q^{86} - 77038 q^{87} - 32670 q^{88} + 443394 q^{89} - 82325 q^{90} - 207088 q^{91} - 237112 q^{92} + 396718 q^{93} + 409516 q^{94} - 351975 q^{95} + 100332 q^{96} - 152942 q^{97} + 895680 q^{98} - 443586 q^{99}+O(q^{100})$$ 39 * q + 12 * q^2 + 27 * q^3 + 670 * q^4 - 975 * q^5 + 69 * q^6 - 251 * q^7 + 270 * q^8 + 3666 * q^9 - 300 * q^10 - 4719 * q^11 + 872 * q^12 - 717 * q^13 + 2647 * q^14 - 675 * q^15 + 15078 * q^16 + 2155 * q^17 + 3293 * q^18 + 14079 * q^19 - 16750 * q^20 + 6891 * q^21 - 1452 * q^22 + 1296 * q^23 - 3138 * q^24 + 24375 * q^25 + 2585 * q^26 + 3846 * q^27 - 20901 * q^28 - 7635 * q^29 - 1725 * q^30 + 9595 * q^31 - 43 * q^32 - 3267 * q^33 - 28383 * q^34 + 6275 * q^35 + 80573 * q^36 - 28965 * q^37 + 4332 * q^38 + 21358 * q^39 - 6750 * q^40 + 19627 * q^41 + 12478 * q^42 + 3711 * q^43 - 81070 * q^44 - 91650 * q^45 + 37459 * q^46 + 38450 * q^47 - 12259 * q^48 + 78758 * q^49 + 7500 * q^50 - 68114 * q^51 - 50322 * q^52 + 20631 * q^53 - 124732 * q^54 + 117975 * q^55 + 94893 * q^56 + 9747 * q^57 - 148140 * q^58 + 170405 * q^59 - 21800 * q^60 - 22345 * q^61 + 246390 * q^62 - 151688 * q^63 + 340312 * q^64 + 17925 * q^65 - 8349 * q^66 - 16408 * q^67 - 32276 * q^68 + 93536 * q^69 - 66175 * q^70 + 119338 * q^71 + 135668 * q^72 - 141606 * q^73 + 71843 * q^74 + 16875 * q^75 + 241870 * q^76 + 30371 * q^77 + 708290 * q^78 + 14727 * q^79 - 376950 * q^80 + 441659 * q^81 - 219870 * q^82 + 384909 * q^83 + 877024 * q^84 - 53875 * q^85 + 250290 * q^86 - 77038 * q^87 - 32670 * q^88 + 443394 * q^89 - 82325 * q^90 - 207088 * q^91 - 237112 * q^92 + 396718 * q^93 + 409516 * q^94 - 351975 * q^95 + 100332 * q^96 - 152942 * q^97 + 895680 * q^98 - 443586 * q^99

## 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 −10.9132 −5.65221 87.0985 −25.0000 61.6839 60.6149 −601.302 −211.052 272.831
1.2 −10.6528 −25.7563 81.4826 −25.0000 274.378 −195.509 −527.129 420.388 266.321
1.3 −10.1686 30.3201 71.3997 −25.0000 −308.312 −123.753 −400.639 676.309 254.214
1.4 −9.86064 3.26128 65.2322 −25.0000 −32.1583 −108.556 −327.690 −232.364 246.516
1.5 −9.68916 21.3816 61.8799 −25.0000 −207.169 48.2370 −289.511 214.171 242.229
1.6 −8.68402 −13.3875 43.4122 −25.0000 116.258 −102.340 −99.1033 −63.7740 217.100
1.7 −8.55717 11.4773 41.2252 −25.0000 −98.2134 106.731 −78.9418 −111.271 213.929
1.8 −7.01607 −14.7319 17.2252 −25.0000 103.360 207.291 103.661 −25.9713 175.402
1.9 −6.68564 −22.1881 12.6977 −25.0000 148.342 35.5129 129.048 249.314 167.141
1.10 −6.61608 9.41089 11.7725 −25.0000 −62.2632 −82.7561 133.827 −154.435 165.402
1.11 −5.04716 16.9331 −6.52622 −25.0000 −85.4638 82.5359 194.448 43.7285 126.179
1.12 −4.54694 −4.48487 −11.3254 −25.0000 20.3924 16.9538 196.998 −222.886 113.673
1.13 −3.56187 −14.4846 −19.3131 −25.0000 51.5922 −241.062 182.771 −33.1968 89.0468
1.14 −2.78665 −26.7838 −24.2346 −25.0000 74.6370 −117.963 156.706 474.371 69.6662
1.15 −2.72815 13.3084 −24.5572 −25.0000 −36.3073 −136.165 154.296 −65.8862 68.2036
1.16 −2.63769 28.9603 −25.0426 −25.0000 −76.3883 −13.2455 150.461 595.699 65.9423
1.17 −2.21689 5.10751 −27.0854 −25.0000 −11.3228 167.053 130.986 −216.913 55.4222
1.18 −1.00103 20.9100 −30.9979 −25.0000 −20.9316 215.274 63.0630 194.229 25.0258
1.19 −0.429556 −22.9509 −31.8155 −25.0000 9.85872 −78.3412 27.4123 283.745 10.7389
1.20 0.325213 −11.1694 −31.8942 −25.0000 −3.63244 1.65378 −20.7792 −118.244 −8.13033
See all 39 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.39 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.g 39

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.g 39 1.a even 1 1 trivial