# Properties

 Label 1045.6.a.a Level $1045$ Weight $6$ Character orbit 1045.a Self dual yes Analytic conductor $167.601$ Analytic rank $1$ Dimension $35$ CM no Inner twists $1$

# Related objects

## 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: $$1$$ Dimension: $$35$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$35 q - 4 q^{2} - 27 q^{3} + 520 q^{4} - 875 q^{5} - 291 q^{6} + 117 q^{7} - 498 q^{8} + 2046 q^{9}+O(q^{10})$$ 35 * q - 4 * q^2 - 27 * q^3 + 520 * q^4 - 875 * q^5 - 291 * q^6 + 117 * q^7 - 498 * q^8 + 2046 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$35 q - 4 q^{2} - 27 q^{3} + 520 q^{4} - 875 q^{5} - 291 q^{6} + 117 q^{7} - 498 q^{8} + 2046 q^{9} + 100 q^{10} + 4235 q^{11} - 568 q^{12} - 717 q^{13} - 2585 q^{14} + 675 q^{15} + 3356 q^{16} - 3349 q^{17} - 5533 q^{18} + 12635 q^{19} - 13000 q^{20} + 289 q^{21} - 484 q^{22} - 820 q^{23} - 21748 q^{24} + 21875 q^{25} - 6267 q^{26} - 13650 q^{27} - 6487 q^{28} - 13357 q^{29} + 7275 q^{30} - 15341 q^{31} - 16405 q^{32} - 3267 q^{33} - 1255 q^{34} - 2925 q^{35} + 23487 q^{36} - 511 q^{37} - 1444 q^{38} - 33584 q^{39} + 12450 q^{40} - 36855 q^{41} + 16330 q^{42} + 10991 q^{43} + 62920 q^{44} - 51150 q^{45} - 20443 q^{46} - 33594 q^{47} + 36221 q^{48} + 23422 q^{49} - 2500 q^{50} - 53530 q^{51} + 89382 q^{52} + 13103 q^{53} + 65776 q^{54} - 105875 q^{55} + 130911 q^{56} - 9747 q^{57} + 127808 q^{58} - 161139 q^{59} + 14200 q^{60} - 91587 q^{61} + 131818 q^{62} + 16590 q^{63} - 23186 q^{64} + 17925 q^{65} - 35211 q^{66} + 39210 q^{67} + 26300 q^{68} - 23174 q^{69} + 64625 q^{70} - 167772 q^{71} + 135820 q^{72} - 5106 q^{73} - 256965 q^{74} - 16875 q^{75} + 187720 q^{76} + 14157 q^{77} + 492812 q^{78} - 156897 q^{79} - 83900 q^{80} + 31279 q^{81} + 46818 q^{82} - 185627 q^{83} + 165864 q^{84} + 83725 q^{85} - 159946 q^{86} - 112092 q^{87} - 60258 q^{88} - 144420 q^{89} + 138325 q^{90} - 442480 q^{91} - 205876 q^{92} + 125910 q^{93} - 110044 q^{94} - 315875 q^{95} - 554286 q^{96} + 41200 q^{97} + 41052 q^{98} + 247566 q^{99}+O(q^{100})$$ 35 * q - 4 * q^2 - 27 * q^3 + 520 * q^4 - 875 * q^5 - 291 * q^6 + 117 * q^7 - 498 * q^8 + 2046 * q^9 + 100 * q^10 + 4235 * q^11 - 568 * q^12 - 717 * q^13 - 2585 * q^14 + 675 * q^15 + 3356 * q^16 - 3349 * q^17 - 5533 * q^18 + 12635 * q^19 - 13000 * q^20 + 289 * q^21 - 484 * q^22 - 820 * q^23 - 21748 * q^24 + 21875 * q^25 - 6267 * q^26 - 13650 * q^27 - 6487 * q^28 - 13357 * q^29 + 7275 * q^30 - 15341 * q^31 - 16405 * q^32 - 3267 * q^33 - 1255 * q^34 - 2925 * q^35 + 23487 * q^36 - 511 * q^37 - 1444 * q^38 - 33584 * q^39 + 12450 * q^40 - 36855 * q^41 + 16330 * q^42 + 10991 * q^43 + 62920 * q^44 - 51150 * q^45 - 20443 * q^46 - 33594 * q^47 + 36221 * q^48 + 23422 * q^49 - 2500 * q^50 - 53530 * q^51 + 89382 * q^52 + 13103 * q^53 + 65776 * q^54 - 105875 * q^55 + 130911 * q^56 - 9747 * q^57 + 127808 * q^58 - 161139 * q^59 + 14200 * q^60 - 91587 * q^61 + 131818 * q^62 + 16590 * q^63 - 23186 * q^64 + 17925 * q^65 - 35211 * q^66 + 39210 * q^67 + 26300 * q^68 - 23174 * q^69 + 64625 * q^70 - 167772 * q^71 + 135820 * q^72 - 5106 * q^73 - 256965 * q^74 - 16875 * q^75 + 187720 * q^76 + 14157 * q^77 + 492812 * q^78 - 156897 * q^79 - 83900 * q^80 + 31279 * q^81 + 46818 * q^82 - 185627 * q^83 + 165864 * q^84 + 83725 * q^85 - 159946 * q^86 - 112092 * q^87 - 60258 * q^88 - 144420 * q^89 + 138325 * q^90 - 442480 * q^91 - 205876 * q^92 + 125910 * q^93 - 110044 * q^94 - 315875 * q^95 - 554286 * q^96 + 41200 * q^97 + 41052 * q^98 + 247566 * 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 −11.2768 20.0827 95.1666 −25.0000 −226.469 −33.0605 −712.318 160.315 281.920
1.2 −10.0372 −21.3301 68.7460 −25.0000 214.096 45.5863 −368.828 211.975 250.931
1.3 −9.69226 4.57537 61.9400 −25.0000 −44.3457 180.537 −290.186 −222.066 242.307
1.4 −9.31795 4.33678 54.8242 −25.0000 −40.4099 −224.649 −212.674 −224.192 232.949
1.5 −9.20757 −16.1697 52.7793 −25.0000 148.883 −66.9325 −191.327 18.4588 230.189
1.6 −8.58946 14.9980 41.7788 −25.0000 −128.825 −101.857 −83.9948 −18.0599 214.737
1.7 −7.58272 22.1703 25.4976 −25.0000 −168.111 53.6476 49.3055 248.522 189.568
1.8 −7.13459 −23.4497 18.9024 −25.0000 167.304 −10.1832 93.4458 306.890 178.365
1.9 −6.89804 25.8209 15.5830 −25.0000 −178.113 221.117 113.245 423.717 172.451
1.10 −6.12909 −3.52111 5.56569 −25.0000 21.5812 66.3590 162.018 −230.602 153.227
1.11 −5.42281 −30.3382 −2.59316 −25.0000 164.518 20.9364 187.592 677.408 135.570
1.12 −5.16335 0.950399 −5.33984 −25.0000 −4.90724 −84.8257 192.799 −242.097 129.084
1.13 −3.68873 −20.7447 −18.3933 −25.0000 76.5215 188.389 185.887 187.342 92.2182
1.14 −3.41781 −5.73624 −20.3185 −25.0000 19.6054 236.087 178.815 −210.096 85.4453
1.15 −2.54765 26.9200 −25.5095 −25.0000 −68.5826 6.38524 146.514 481.684 63.6912
1.16 −1.44052 10.2879 −29.9249 −25.0000 −14.8199 14.1538 89.2038 −137.159 36.0129
1.17 −0.635479 −27.7922 −31.5962 −25.0000 17.6614 19.6289 40.4140 529.409 15.8870
1.18 −0.259047 8.53779 −31.9329 −25.0000 −2.21168 −211.963 16.5616 −170.106 6.47616
1.19 1.05998 −6.05236 −30.8764 −25.0000 −6.41538 −213.822 −66.6477 −206.369 −26.4995
1.20 1.69647 12.5434 −29.1220 −25.0000 21.2794 107.740 −103.691 −85.6632 −42.4117
See all 35 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.35 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.a 35

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.a 35 1.a even 1 1 trivial