# Properties

 Label 1045.4.a.i Level $1045$ Weight $4$ Character orbit 1045.a Self dual yes Analytic conductor $61.657$ Analytic rank $0$ Dimension $25$ CM no Inner twists $1$

# Related objects

## 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: $$25$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10})$$ 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9} - 10 q^{10} + 275 q^{11} + 44 q^{12} + 53 q^{13} - 51 q^{14} - 45 q^{15} + 438 q^{16} + 153 q^{17} + 9 q^{18} + 475 q^{19} - 610 q^{20} + 259 q^{21} + 22 q^{22} - 7 q^{23} + 186 q^{24} + 625 q^{25} + 543 q^{26} + 495 q^{27} - 525 q^{28} + 169 q^{29} - 55 q^{30} + 102 q^{31} + 327 q^{32} + 99 q^{33} - 879 q^{34} + 75 q^{35} + 2293 q^{36} - 46 q^{37} + 38 q^{38} + 233 q^{39} - 300 q^{40} + 1190 q^{41} - 684 q^{42} - 408 q^{43} + 1342 q^{44} - 1500 q^{45} + 757 q^{46} + 1068 q^{47} + 715 q^{48} + 1930 q^{49} + 50 q^{50} + 1655 q^{51} - 94 q^{52} + 143 q^{53} + 1970 q^{54} - 1375 q^{55} - 1397 q^{56} + 171 q^{57} + 1366 q^{58} + 2945 q^{59} - 220 q^{60} + 1160 q^{61} + 194 q^{62} + 1804 q^{63} + 3000 q^{64} - 265 q^{65} + 121 q^{66} - 353 q^{67} + 5452 q^{68} + 3289 q^{69} + 255 q^{70} + 230 q^{71} + 196 q^{72} + 1357 q^{73} + 4379 q^{74} + 225 q^{75} + 2318 q^{76} - 165 q^{77} + 2008 q^{78} + 1266 q^{79} - 2190 q^{80} + 1709 q^{81} + 1010 q^{82} + 3856 q^{83} + 9354 q^{84} - 765 q^{85} + 6746 q^{86} + 3113 q^{87} + 660 q^{88} + 3562 q^{89} - 45 q^{90} - 833 q^{91} + 4276 q^{92} + 1312 q^{93} + 5124 q^{94} - 2375 q^{95} + 3828 q^{96} - 914 q^{97} + 2478 q^{98} + 3300 q^{99}+O(q^{100})$$ 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9 - 10 * q^10 + 275 * q^11 + 44 * q^12 + 53 * q^13 - 51 * q^14 - 45 * q^15 + 438 * q^16 + 153 * q^17 + 9 * q^18 + 475 * q^19 - 610 * q^20 + 259 * q^21 + 22 * q^22 - 7 * q^23 + 186 * q^24 + 625 * q^25 + 543 * q^26 + 495 * q^27 - 525 * q^28 + 169 * q^29 - 55 * q^30 + 102 * q^31 + 327 * q^32 + 99 * q^33 - 879 * q^34 + 75 * q^35 + 2293 * q^36 - 46 * q^37 + 38 * q^38 + 233 * q^39 - 300 * q^40 + 1190 * q^41 - 684 * q^42 - 408 * q^43 + 1342 * q^44 - 1500 * q^45 + 757 * q^46 + 1068 * q^47 + 715 * q^48 + 1930 * q^49 + 50 * q^50 + 1655 * q^51 - 94 * q^52 + 143 * q^53 + 1970 * q^54 - 1375 * q^55 - 1397 * q^56 + 171 * q^57 + 1366 * q^58 + 2945 * q^59 - 220 * q^60 + 1160 * q^61 + 194 * q^62 + 1804 * q^63 + 3000 * q^64 - 265 * q^65 + 121 * q^66 - 353 * q^67 + 5452 * q^68 + 3289 * q^69 + 255 * q^70 + 230 * q^71 + 196 * q^72 + 1357 * q^73 + 4379 * q^74 + 225 * q^75 + 2318 * q^76 - 165 * q^77 + 2008 * q^78 + 1266 * q^79 - 2190 * q^80 + 1709 * q^81 + 1010 * q^82 + 3856 * q^83 + 9354 * q^84 - 765 * q^85 + 6746 * q^86 + 3113 * q^87 + 660 * q^88 + 3562 * q^89 - 45 * q^90 - 833 * q^91 + 4276 * q^92 + 1312 * q^93 + 5124 * q^94 - 2375 * q^95 + 3828 * q^96 - 914 * q^97 + 2478 * q^98 + 3300 * 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 −5.27434 4.83101 19.8186 −5.00000 −25.4803 −13.3199 −62.3354 −3.66139 26.3717
1.2 −5.23688 −8.48150 19.4249 −5.00000 44.4166 −12.7409 −59.8310 44.9358 26.1844
1.3 −4.81539 9.01446 15.1880 −5.00000 −43.4082 26.8150 −34.6132 54.2605 24.0770
1.4 −4.50858 −2.01806 12.3273 −5.00000 9.09858 15.1423 −19.5101 −22.9274 22.5429
1.5 −3.75267 −6.96028 6.08254 −5.00000 26.1196 −29.7997 7.19560 21.4455 18.7634
1.6 −3.65423 −3.03858 5.35338 −5.00000 11.1037 26.1615 9.67137 −17.7670 18.2711
1.7 −2.74955 4.78923 −0.439974 −5.00000 −13.1682 −9.83055 23.2061 −4.06331 13.7478
1.8 −2.74566 4.96249 −0.461347 −5.00000 −13.6253 −34.3482 23.2320 −2.37368 13.7283
1.9 −2.51141 8.82864 −1.69283 −5.00000 −22.1723 6.85719 24.3426 50.9449 12.5570
1.10 −1.72202 −1.27869 −5.03464 −5.00000 2.20194 3.57098 22.4459 −25.3649 8.61011
1.11 −1.64212 −7.74531 −5.30343 −5.00000 12.7188 −5.31024 21.8459 32.9899 8.21062
1.12 −0.333805 −8.69029 −7.88857 −5.00000 2.90086 18.9477 5.30368 48.5211 1.66902
1.13 0.109799 6.64848 −7.98794 −5.00000 0.729999 28.5341 −1.75547 17.2022 −0.548997
1.14 0.197347 −0.963676 −7.96105 −5.00000 −0.190178 0.480073 −3.14986 −26.0713 −0.986733
1.15 1.70323 9.22981 −5.09901 −5.00000 15.7205 −12.5263 −22.3106 58.1894 −8.51615
1.16 1.75191 3.91466 −4.93083 −5.00000 6.85812 −21.2936 −22.6536 −11.6754 −8.75953
1.17 2.33606 −1.07169 −2.54281 −5.00000 −2.50353 23.6079 −24.6287 −25.8515 −11.6803
1.18 3.11750 0.0197889 1.71881 −5.00000 0.0616921 −19.0525 −19.5816 −26.9996 −15.5875
1.19 3.49473 −5.86270 4.21313 −5.00000 −20.4885 −34.6786 −13.2341 7.37127 −17.4736
1.20 3.81190 −9.32002 6.53062 −5.00000 −35.5270 25.5875 −5.60115 59.8628 −19.0595
See all 25 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.25 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 1045.4.a.i 25

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.i 25 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{25} - 2 T_{2}^{24} - 159 T_{2}^{23} + 290 T_{2}^{22} + 11059 T_{2}^{21} - 18125 T_{2}^{20} - 442742 T_{2}^{19} + 639787 T_{2}^{18} + 11296856 T_{2}^{17} - 14043714 T_{2}^{16} - 192160271 T_{2}^{15} + \cdots - 2492006400$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1045))$$.