# Properties

 Label 1859.4.a.n Level $1859$ Weight $4$ Character orbit 1859.a Self dual yes Analytic conductor $109.685$ Analytic rank $1$ Dimension $39$ 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: $$1$$ Dimension: $$39$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$39 q - 23 q^{3} + 114 q^{4} - 23 q^{5} - 77 q^{6} + 4 q^{7} + 21 q^{8} + 260 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$39 q - 23 q^{3} + 114 q^{4} - 23 q^{5} - 77 q^{6} + 4 q^{7} + 21 q^{8} + 260 q^{9} - 158 q^{10} + 429 q^{11} - 351 q^{12} - 176 q^{14} - 30 q^{15} + 230 q^{16} - 244 q^{17} - 21 q^{18} + 70 q^{19} - 366 q^{20} + 142 q^{21} - 47 q^{23} - 846 q^{24} + 322 q^{25} - 416 q^{27} - 1131 q^{28} - 838 q^{29} - 293 q^{30} - 507 q^{31} + 1433 q^{32} - 253 q^{33} - 166 q^{34} - 498 q^{35} + 815 q^{36} - 89 q^{37} + 81 q^{38} - 2917 q^{40} - 618 q^{41} - 318 q^{42} - 1064 q^{43} + 1254 q^{44} - 238 q^{45} + 1331 q^{46} - 1499 q^{47} - 1460 q^{48} - 413 q^{49} + 2459 q^{50} - 2350 q^{51} - 2745 q^{53} + 845 q^{54} - 253 q^{55} - 2904 q^{56} - 1450 q^{57} + 2509 q^{58} - 2285 q^{59} + 3566 q^{60} - 6218 q^{61} - 911 q^{62} + 1930 q^{63} + 67 q^{64} - 847 q^{66} - 546 q^{67} - 170 q^{68} - 5254 q^{69} + 2195 q^{70} + 263 q^{71} + 2393 q^{72} + 1148 q^{73} + 775 q^{74} - 5385 q^{75} + 7247 q^{76} + 44 q^{77} - 3666 q^{79} - 5594 q^{80} - 1901 q^{81} - 4414 q^{82} - 2722 q^{83} + 9971 q^{84} - 1858 q^{85} - 2478 q^{86} - 2284 q^{87} + 231 q^{88} - 13 q^{89} - 6771 q^{90} - 2232 q^{92} + 1082 q^{93} - 7330 q^{94} - 2352 q^{95} - 5770 q^{96} + 1197 q^{97} - 6813 q^{98} + 2860 q^{99} + O(q^{100})$$

## 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 −5.13931 −6.39990 18.4125 −3.19755 32.8911 11.5196 −53.5133 13.9588 16.4332
1.2 −5.01772 4.57830 17.1775 10.5311 −22.9726 16.1445 −46.0502 −6.03920 −52.8420
1.3 −4.78473 6.91979 14.8936 9.47124 −33.1093 13.6025 −32.9841 20.8835 −45.3173
1.4 −4.54302 0.739868 12.6390 5.27775 −3.36123 −32.8528 −21.0751 −26.4526 −23.9769
1.5 −4.48614 −3.10130 12.1255 −9.82658 13.9129 −8.62268 −18.5074 −17.3819 44.0834
1.6 −4.05479 −7.92582 8.44132 −7.77507 32.1375 −27.1534 −1.78944 35.8186 31.5263
1.7 −4.00081 −8.62277 8.00649 18.4684 34.4981 −15.5106 −0.0259569 47.3521 −73.8885
1.8 −3.98336 6.46359 7.86718 −1.70678 −25.7468 18.7420 0.529060 14.7780 6.79871
1.9 −3.79202 1.78265 6.37944 −15.2422 −6.75985 7.73177 6.14519 −23.8222 57.7988
1.10 −2.66272 7.39869 −0.909937 10.9244 −19.7006 −23.1620 23.7246 27.7406 −29.0887
1.11 −2.51874 1.77429 −1.65593 −18.5552 −4.46898 4.77905 24.3208 −23.8519 46.7358
1.12 −2.34716 −6.88614 −2.49082 4.99906 16.1629 27.5938 24.6237 20.4189 −11.7336
1.13 −2.30249 4.54839 −2.69855 −4.16672 −10.4726 −17.3608 24.6333 −6.31212 9.59382
1.14 −2.06825 −3.79252 −3.72233 11.8121 7.84389 −6.57089 24.2447 −12.6168 −24.4303
1.15 −1.49011 −8.91494 −5.77957 −12.8908 13.2843 33.8928 20.5331 52.4761 19.2087
1.16 −1.42955 −6.49934 −5.95638 0.801184 9.29115 16.7131 19.9514 15.2414 −1.14534
1.17 −1.40955 8.50422 −6.01318 −10.6119 −11.9871 26.4105 19.7522 45.3218 14.9579
1.18 −0.641292 −1.91302 −7.58874 −0.642358 1.22681 −23.2338 9.99694 −23.3404 0.411939
1.19 −0.150454 3.58277 −7.97736 14.8852 −0.539044 5.18902 2.40386 −14.1637 −2.23954
1.20 0.169295 6.82637 −7.97134 1.80831 1.15567 −16.1277 −2.70387 19.5993 0.306138
See all 39 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.39 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.n 39
13.b even 2 1 1859.4.a.o yes 39

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.4.a.n 39 1.a even 1 1 trivial
1859.4.a.o yes 39 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$55\!\cdots\!43$$$$T_{2}^{23} - 459171188888 T_{2}^{22} -$$$$51\!\cdots\!89$$$$T_{2}^{21} +$$$$68\!\cdots\!35$$$$T_{2}^{20} +$$$$36\!\cdots\!87$$$$T_{2}^{19} -$$$$68\!\cdots\!34$$$$T_{2}^{18} -$$$$19\!\cdots\!96$$$$T_{2}^{17} +$$$$48\!\cdots\!52$$$$T_{2}^{16} +$$$$78\!\cdots\!33$$$$T_{2}^{15} -$$$$24\!\cdots\!92$$$$T_{2}^{14} -$$$$23\!\cdots\!24$$$$T_{2}^{13} +$$$$85\!\cdots\!69$$$$T_{2}^{12} +$$$$49\!\cdots\!72$$$$T_{2}^{11} -$$$$21\!\cdots\!12$$$$T_{2}^{10} -$$$$71\!\cdots\!76$$$$T_{2}^{9} +$$$$35\!\cdots\!56$$$$T_{2}^{8} +$$$$65\!\cdots\!00$$$$T_{2}^{7} -$$$$35\!\cdots\!68$$$$T_{2}^{6} -$$$$33\!\cdots\!72$$$$T_{2}^{5} +$$$$19\!\cdots\!32$$$$T_{2}^{4} +$$$$66\!\cdots\!36$$$$T_{2}^{3} -$$$$41\!\cdots\!40$$$$T_{2}^{2} -$$$$88\!\cdots\!92$$$$T_{2} +$$$$91\!\cdots\!96$$">$$T_{2}^{39} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1859))$$.