# Properties

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

# Related objects

## 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 −5.64970 −1.71534 23.9192 7.89488 9.69115 −11.5333 −89.9385 −24.0576 −44.6038
1.2 −5.53541 −5.57466 22.6408 −16.4961 30.8581 23.7730 −81.0427 4.07688 91.3127
1.3 −5.50285 4.91472 22.2813 −19.9510 −27.0449 −0.0210524 −78.5880 −2.84554 109.787
1.4 −5.02887 1.98348 17.2895 15.8951 −9.97465 8.39176 −46.7159 −23.0658 −79.9343
1.5 −4.99679 8.59946 16.9679 −10.1780 −42.9697 −9.94954 −44.8107 46.9506 50.8572
1.6 −4.81770 −9.24702 15.2103 −7.67520 44.5494 28.1687 −34.7370 58.5075 36.9768
1.7 −4.60628 7.07074 13.2178 −11.0597 −32.5698 −9.07530 −24.0348 22.9953 50.9440
1.8 −4.57982 0.436116 12.9748 3.32236 −1.99733 −25.3240 −22.7836 −26.8098 −15.2158
1.9 −3.95893 −9.52439 7.67316 16.7554 37.7064 −7.49508 1.29395 63.7139 −66.3336
1.10 −3.79290 −5.52551 6.38608 −13.9792 20.9577 −15.0455 6.12143 3.53129 53.0218
1.11 −3.74164 9.22560 5.99989 −0.452132 −34.5189 −22.7458 7.48368 58.1117 1.69172
1.12 −3.11673 −1.97531 1.71399 2.69678 6.15651 30.4006 19.5918 −23.0981 −8.40514
1.13 −3.06434 −3.83075 1.39021 −4.40166 11.7387 9.87023 20.2547 −12.3254 13.4882
1.14 −3.01034 0.463820 1.06212 13.2599 −1.39626 −19.7000 20.8853 −26.7849 −39.9167
1.15 −2.72755 0.736988 −0.560489 −18.0105 −2.01017 23.4852 23.3491 −26.4568 49.1244
1.16 −2.60372 8.44963 −1.22064 3.90395 −22.0005 31.4455 24.0080 44.3963 −10.1648
1.17 −2.52498 0.684748 −1.62445 14.7550 −1.72898 2.67947 24.3016 −26.5311 −37.2563
1.18 −2.33473 −7.99292 −2.54902 −16.6573 18.6613 −33.9647 24.6291 36.8868 38.8903
1.19 −2.01569 9.83385 −3.93701 16.4444 −19.8220 −5.22049 24.0613 69.7045 −33.1467
1.20 −1.58200 3.04819 −5.49728 −5.74523 −4.82223 −14.7260 21.3527 −17.7085 9.08894
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.p 51
13.b even 2 1 1859.4.a.q yes 51

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

## 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1859))$$.