# Properties

 Label 1859.4.a.q Level $1859$ Weight $4$ Character orbit 1859.a Self dual yes Analytic conductor $109.685$ Analytic rank $0$ Dimension $51$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1859.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.684550701$$ Analytic rank: $$0$$ Dimension: $$51$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$51 q + 21 q^{3} + 234 q^{4} + 41 q^{5} - 73 q^{6} + 4 q^{7} - 21 q^{8} + 594 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$51 q + 21 q^{3} + 234 q^{4} + 41 q^{5} - 73 q^{6} + 4 q^{7} - 21 q^{8} + 594 q^{9} + 212 q^{10} + 561 q^{11} + 209 q^{12} + 280 q^{14} + 284 q^{15} + 1246 q^{16} + 164 q^{17} - 189 q^{18} + 26 q^{19} + 438 q^{20} + 134 q^{21} + 373 q^{23} - 354 q^{24} + 2048 q^{25} + 1470 q^{27} - 1245 q^{28} + 898 q^{29} + 427 q^{30} + 767 q^{31} + 1127 q^{32} + 231 q^{33} + 206 q^{34} + 54 q^{35} + 3415 q^{36} + 395 q^{37} + 1577 q^{38} + 3253 q^{40} - 354 q^{41} + 942 q^{42} + 484 q^{43} + 2574 q^{44} + 1452 q^{45} - 2117 q^{46} + 1925 q^{47} + 1780 q^{48} + 4535 q^{49} - 1093 q^{50} + 230 q^{51} + 1387 q^{53} - 5271 q^{54} + 451 q^{55} + 2568 q^{56} - 5738 q^{57} + 3695 q^{58} + 1145 q^{59} - 1590 q^{60} + 5382 q^{61} - 395 q^{62} + 710 q^{63} + 9839 q^{64} - 803 q^{66} - 210 q^{67} + 1742 q^{68} + 7028 q^{69} - 6747 q^{70} + 3693 q^{71} - 12481 q^{72} + 968 q^{73} + 1735 q^{74} - 727 q^{75} - 2801 q^{76} + 44 q^{77} + 4234 q^{79} + 2390 q^{80} + 7743 q^{81} + 4770 q^{82} - 2798 q^{83} + 14821 q^{84} - 1802 q^{85} + 6558 q^{86} + 1896 q^{87} - 231 q^{88} + 3927 q^{89} + 1927 q^{90} + 1984 q^{92} - 1332 q^{93} + 7590 q^{94} + 4944 q^{95} - 7280 q^{96} + 3913 q^{97} - 15201 q^{98} + 6534 q^{99} + O(q^{100})$$

## 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 −5.57916 9.06032 23.1271 −7.74998 −50.5490 23.9301 −84.3964 55.0894 43.2384
1.2 −5.41392 −5.45021 21.3105 −17.8801 29.5070 −29.5931 −72.0621 2.70474 96.8014
1.3 −5.33739 −5.24916 20.4878 17.7171 28.0168 −32.5164 −66.6520 0.553673 −94.5632
1.4 −5.05441 0.801365 17.5470 −17.5680 −4.05042 1.71020 −48.2547 −26.3578 88.7960
1.5 −4.95360 8.81730 16.5381 19.6953 −43.6773 6.57864 −42.2943 50.7448 −97.5628
1.6 −4.84162 9.88636 15.4413 −0.786780 −47.8660 −26.4479 −36.0278 70.7402 3.80929
1.7 −4.54788 −9.03091 12.6832 5.41468 41.0715 26.3763 −21.2987 54.5573 −24.6253
1.8 −4.52468 7.83415 12.4727 −15.8043 −35.4471 −30.6432 −20.2377 34.3740 71.5092
1.9 −4.24016 −4.67403 9.97893 −7.09125 19.8186 −21.9298 −8.39097 −5.15349 30.0680
1.10 −4.08285 −3.72122 8.66965 2.40264 15.1932 5.44341 −2.73408 −13.1525 −9.80961
1.11 −3.69645 1.78714 5.66374 8.08739 −6.60608 3.93004 8.63587 −23.8061 −29.8946
1.12 −3.57359 −4.93113 4.77054 6.17910 17.6218 19.8272 11.5408 −2.68401 −22.0816
1.13 −3.39599 6.27073 3.53273 −11.8750 −21.2953 −17.2790 15.1708 12.3221 40.3274
1.14 −3.26686 5.88118 2.67240 20.5347 −19.2130 32.9664 17.4046 7.58824 −67.0842
1.15 −2.96025 1.91558 0.763092 5.98319 −5.67060 31.3583 21.4231 −23.3305 −17.7117
1.16 −2.92218 −6.70384 0.539146 12.5727 19.5898 −15.7135 21.8020 17.9414 −36.7396
1.17 −2.41499 1.53640 −2.16780 15.9482 −3.71040 −30.4066 24.5552 −24.6395 −38.5149
1.18 −1.90937 −0.199217 −4.35431 −18.1265 0.380379 33.8392 23.5889 −26.9603 34.6102
1.19 −1.77976 −6.74491 −4.83246 −21.4141 12.0043 −5.98846 22.8387 18.4938 38.1119
1.20 −1.66790 7.00149 −5.21812 −13.7691 −11.6778 14.0493 22.0465 22.0209 22.9654
See all 51 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.51 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$
$$13$$ $$-1$$

## 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 1859.4.a.q yes 51
13.b even 2 1 1859.4.a.p 51

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.4.a.p 51 13.b even 2 1
1859.4.a.q yes 51 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$15\!\cdots\!92$$$$T_{2}^{37} +$$$$37\!\cdots\!28$$$$T_{2}^{36} +$$$$34\!\cdots\!15$$$$T_{2}^{35} -$$$$10\!\cdots\!48$$$$T_{2}^{34} -$$$$65\!\cdots\!85$$$$T_{2}^{33} +$$$$22\!\cdots\!53$$$$T_{2}^{32} +$$$$99\!\cdots\!01$$$$T_{2}^{31} -$$$$39\!\cdots\!86$$$$T_{2}^{30} -$$$$12\!\cdots\!32$$$$T_{2}^{29} +$$$$55\!\cdots\!26$$$$T_{2}^{28} +$$$$12\!\cdots\!17$$$$T_{2}^{27} -$$$$62\!\cdots\!04$$$$T_{2}^{26} -$$$$10\!\cdots\!58$$$$T_{2}^{25} +$$$$55\!\cdots\!71$$$$T_{2}^{24} +$$$$66\!\cdots\!26$$$$T_{2}^{23} -$$$$39\!\cdots\!84$$$$T_{2}^{22} -$$$$35\!\cdots\!44$$$$T_{2}^{21} +$$$$22\!\cdots\!56$$$$T_{2}^{20} +$$$$14\!\cdots\!84$$$$T_{2}^{19} -$$$$96\!\cdots\!96$$$$T_{2}^{18} -$$$$48\!\cdots\!64$$$$T_{2}^{17} +$$$$32\!\cdots\!16$$$$T_{2}^{16} +$$$$12\!\cdots\!68$$$$T_{2}^{15} -$$$$81\!\cdots\!44$$$$T_{2}^{14} -$$$$22\!\cdots\!04$$$$T_{2}^{13} +$$$$14\!\cdots\!80$$$$T_{2}^{12} +$$$$30\!\cdots\!24$$$$T_{2}^{11} -$$$$18\!\cdots\!52$$$$T_{2}^{10} -$$$$28\!\cdots\!28$$$$T_{2}^{9} +$$$$15\!\cdots\!24$$$$T_{2}^{8} +$$$$17\!\cdots\!12$$$$T_{2}^{7} -$$$$76\!\cdots\!12$$$$T_{2}^{6} -$$$$58\!\cdots\!20$$$$T_{2}^{5} +$$$$16\!\cdots\!00$$$$T_{2}^{4} +$$$$89\!\cdots\!68$$$$T_{2}^{3} -$$$$43\!\cdots\!92$$$$T_{2}^{2} -$$$$20\!\cdots\!56$$$$T_{2} +$$$$10\!\cdots\!24$$">$$T_{2}^{51} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1859))$$.