# Properties

 Label 1045.6.a.c Level $1045$ Weight $6$ Character orbit 1045.a Self dual yes Analytic conductor $167.601$ Analytic rank $1$ Dimension $37$ 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: $$37$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$37 q - 12 q^{2} - 27 q^{3} + 574 q^{4} - 925 q^{5} - 75 q^{6} + 337 q^{7} - 696 q^{8} + 3140 q^{9}+O(q^{10})$$ 37 * q - 12 * q^2 - 27 * q^3 + 574 * q^4 - 925 * q^5 - 75 * q^6 + 337 * q^7 - 696 * q^8 + 3140 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$37 q - 12 q^{2} - 27 q^{3} + 574 q^{4} - 925 q^{5} - 75 q^{6} + 337 q^{7} - 696 q^{8} + 3140 q^{9} + 300 q^{10} - 4477 q^{11} - 568 q^{12} + 719 q^{13} + 687 q^{14} + 675 q^{15} + 11494 q^{16} + 999 q^{17} - 595 q^{18} - 13357 q^{19} - 14350 q^{20} - 1077 q^{21} + 1452 q^{22} + 5096 q^{23} - 3154 q^{24} + 23125 q^{25} - 10395 q^{26} - 7578 q^{27} + 19863 q^{28} - 7969 q^{29} + 1875 q^{30} + 603 q^{31} - 27809 q^{32} + 3267 q^{33} - 24081 q^{34} - 8425 q^{35} + 59869 q^{36} + 7963 q^{37} + 4332 q^{38} + 86 q^{39} + 17400 q^{40} + 1475 q^{41} - 46542 q^{42} + 38059 q^{43} - 69454 q^{44} - 78500 q^{45} - 3413 q^{46} - 37658 q^{47} - 51317 q^{48} + 39188 q^{49} - 7500 q^{50} - 40262 q^{51} + 25358 q^{52} - 52545 q^{53} + 64732 q^{54} + 111925 q^{55} - 54173 q^{56} + 9747 q^{57} + 105808 q^{58} - 34039 q^{59} + 14200 q^{60} + 30023 q^{61} - 100198 q^{62} + 30376 q^{63} + 160888 q^{64} - 17975 q^{65} + 9075 q^{66} - 45284 q^{67} + 125176 q^{68} + 109244 q^{69} - 17175 q^{70} - 84020 q^{71} - 291176 q^{72} + 24542 q^{73} + 38795 q^{74} - 16875 q^{75} - 207214 q^{76} - 40777 q^{77} + 1042 q^{78} + 49303 q^{79} - 287350 q^{80} + 344453 q^{81} - 286030 q^{82} - 402155 q^{83} - 203270 q^{84} - 24975 q^{85} - 276426 q^{86} + 116994 q^{87} + 84216 q^{88} - 442930 q^{89} + 14875 q^{90} - 93040 q^{91} + 402160 q^{92} - 241950 q^{93} - 170720 q^{94} + 333925 q^{95} - 234384 q^{96} - 87732 q^{97} - 712662 q^{98} - 379940 q^{99}+O(q^{100})$$ 37 * q - 12 * q^2 - 27 * q^3 + 574 * q^4 - 925 * q^5 - 75 * q^6 + 337 * q^7 - 696 * q^8 + 3140 * q^9 + 300 * q^10 - 4477 * q^11 - 568 * q^12 + 719 * q^13 + 687 * q^14 + 675 * q^15 + 11494 * q^16 + 999 * q^17 - 595 * q^18 - 13357 * q^19 - 14350 * q^20 - 1077 * q^21 + 1452 * q^22 + 5096 * q^23 - 3154 * q^24 + 23125 * q^25 - 10395 * q^26 - 7578 * q^27 + 19863 * q^28 - 7969 * q^29 + 1875 * q^30 + 603 * q^31 - 27809 * q^32 + 3267 * q^33 - 24081 * q^34 - 8425 * q^35 + 59869 * q^36 + 7963 * q^37 + 4332 * q^38 + 86 * q^39 + 17400 * q^40 + 1475 * q^41 - 46542 * q^42 + 38059 * q^43 - 69454 * q^44 - 78500 * q^45 - 3413 * q^46 - 37658 * q^47 - 51317 * q^48 + 39188 * q^49 - 7500 * q^50 - 40262 * q^51 + 25358 * q^52 - 52545 * q^53 + 64732 * q^54 + 111925 * q^55 - 54173 * q^56 + 9747 * q^57 + 105808 * q^58 - 34039 * q^59 + 14200 * q^60 + 30023 * q^61 - 100198 * q^62 + 30376 * q^63 + 160888 * q^64 - 17975 * q^65 + 9075 * q^66 - 45284 * q^67 + 125176 * q^68 + 109244 * q^69 - 17175 * q^70 - 84020 * q^71 - 291176 * q^72 + 24542 * q^73 + 38795 * q^74 - 16875 * q^75 - 207214 * q^76 - 40777 * q^77 + 1042 * q^78 + 49303 * q^79 - 287350 * q^80 + 344453 * q^81 - 286030 * q^82 - 402155 * q^83 - 203270 * q^84 - 24975 * q^85 - 276426 * q^86 + 116994 * q^87 + 84216 * q^88 - 442930 * q^89 + 14875 * q^90 - 93040 * q^91 + 402160 * q^92 - 241950 * q^93 - 170720 * q^94 + 333925 * q^95 - 234384 * q^96 - 87732 * q^97 - 712662 * q^98 - 379940 * 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 −10.9276 −26.0746 87.4116 −25.0000 284.932 122.691 −605.513 436.887 273.189
1.2 −10.5955 22.4642 80.2653 −25.0000 −238.020 228.020 −511.396 261.641 264.888
1.3 −10.5199 0.539776 78.6693 −25.0000 −5.67841 3.26226 −490.959 −242.709 262.999
1.4 −10.2487 19.1450 73.0349 −25.0000 −196.210 −226.466 −420.553 123.531 256.216
1.5 −9.84120 −9.26217 64.8493 −25.0000 91.1509 −35.6928 −323.277 −157.212 246.030
1.6 −8.73290 −22.7868 44.2636 −25.0000 198.995 −46.5268 −107.097 276.237 218.323
1.7 −8.15394 5.08384 34.4868 −25.0000 −41.4534 169.957 −20.2770 −217.155 203.849
1.8 −7.04813 7.74381 17.6762 −25.0000 −54.5794 −183.718 100.956 −183.033 176.203
1.9 −6.88533 20.3717 15.4078 −25.0000 −140.266 93.5387 114.243 172.007 172.133
1.10 −6.52420 −27.2044 10.5652 −25.0000 177.487 −162.861 139.845 497.081 163.105
1.11 −5.93303 −1.91423 3.20088 −25.0000 11.3572 107.575 170.866 −239.336 148.326
1.12 −5.50202 28.7846 −1.72782 −25.0000 −158.373 5.95765 185.571 585.551 137.550
1.13 −4.33468 −21.6700 −13.2105 −25.0000 93.9326 25.9276 195.973 226.588 108.367
1.14 −3.96708 6.17610 −16.2623 −25.0000 −24.5011 98.7920 191.460 −204.856 99.1771
1.15 −3.53022 −7.27933 −19.5375 −25.0000 25.6976 −171.265 181.939 −190.011 88.2555
1.16 −2.23923 −26.8282 −26.9859 −25.0000 60.0744 193.889 132.083 476.753 55.9807
1.17 −1.08270 23.1648 −30.8278 −25.0000 −25.0805 −27.0095 68.0236 293.607 27.0675
1.18 −1.05078 5.26669 −30.8959 −25.0000 −5.53415 −191.487 66.0899 −215.262 26.2696
1.19 −0.312283 −15.4864 −31.9025 −25.0000 4.83615 −30.9789 19.9557 −3.17065 7.80708
1.20 0.814963 −1.30743 −31.3358 −25.0000 −1.06551 233.682 −51.6164 −241.291 −20.3741
See all 37 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.37 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.c 37

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.6.a.c 37 1.a even 1 1 trivial