# Properties

 Label 1849.4.a.b Level $1849$ Weight $4$ Character orbit 1849.a Self dual yes Analytic conductor $109.095$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.094531601$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.45868.1 Defining polynomial: $$x^{4} - x^{3} - 10 x^{2} + 11 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} ) q^{2} + ( 3 - \beta_{2} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{4} + ( 6 + 3 \beta_{1} + \beta_{3} ) q^{5} + ( -7 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( 6 - 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 19 + \beta_{1} - 11 \beta_{2} - 8 \beta_{3} ) q^{8} + ( -1 + 5 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{3} ) q^{2} + ( 3 - \beta_{2} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{4} + ( 6 + 3 \beta_{1} + \beta_{3} ) q^{5} + ( -7 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( 6 - 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 19 + \beta_{1} - 11 \beta_{2} - 8 \beta_{3} ) q^{8} + ( -1 + 5 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} ) q^{9} + ( -2 - \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{10} + ( -12 - 19 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{11} + ( -17 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{12} + ( -2 + 7 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{13} + ( 30 + 6 \beta_{1} - 14 \beta_{2} - 8 \beta_{3} ) q^{14} + ( 25 + 11 \beta_{1} - 19 \beta_{2} + 8 \beta_{3} ) q^{15} + ( 53 - 11 \beta_{1} + \beta_{2} - 22 \beta_{3} ) q^{16} + ( -65 + 15 \beta_{1} + 38 \beta_{2} + 5 \beta_{3} ) q^{17} + ( -77 - 17 \beta_{1} + 26 \beta_{2} + 12 \beta_{3} ) q^{18} + ( -30 + 21 \beta_{1} + 17 \beta_{3} ) q^{19} + ( -14 - 15 \beta_{1} - 10 \beta_{2} - 9 \beta_{3} ) q^{20} + ( -22 - 40 \beta_{1} + 38 \beta_{2} - 6 \beta_{3} ) q^{21} + ( -50 - \beta_{1} - \beta_{2} + 35 \beta_{3} ) q^{22} + ( -20 - 12 \beta_{1} - 11 \beta_{2} + 29 \beta_{3} ) q^{23} + ( 53 + 20 \beta_{1} - 23 \beta_{2} + 25 \beta_{3} ) q^{24} + ( -36 + 34 \beta_{1} + 9 \beta_{2} + 19 \beta_{3} ) q^{25} + ( 12 + \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{26} + ( 51 + 59 \beta_{1} - 45 \beta_{2} + 6 \beta_{3} ) q^{27} + ( 18 + 58 \beta_{1} - 50 \beta_{2} - 52 \beta_{3} ) q^{28} + ( 9 - 42 \beta_{1} + 105 \beta_{2} + 57 \beta_{3} ) q^{29} + ( -77 - 27 \beta_{1} + 35 \beta_{2} - 20 \beta_{3} ) q^{30} + ( 15 + 79 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{31} + ( 79 + 15 \beta_{1} + 11 \beta_{2} - 54 \beta_{3} ) q^{32} + ( 8 - 60 \beta_{1} + 60 \beta_{2} - 10 \beta_{3} ) q^{33} + ( -29 + 33 \beta_{1} + 30 \beta_{2} + 118 \beta_{3} ) q^{34} + ( -96 - 14 \beta_{1} + 24 \beta_{2} - 30 \beta_{3} ) q^{35} + ( -113 - 26 \beta_{1} + 99 \beta_{2} + 83 \beta_{3} ) q^{36} + ( 115 + 22 \beta_{1} - 33 \beta_{2} - 7 \beta_{3} ) q^{37} + ( -166 - 17 \beta_{1} + 72 \beta_{2} + 81 \beta_{3} ) q^{38} + ( -26 + 20 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} ) q^{39} + ( 54 + 7 \beta_{1} - 90 \beta_{2} + \beta_{3} ) q^{40} + ( -104 - 66 \beta_{1} - 9 \beta_{2} - 95 \beta_{3} ) q^{41} + ( 102 + 44 \beta_{1} - 58 \beta_{2} + 42 \beta_{3} ) q^{42} + ( -236 + 116 \beta_{1} + 64 \beta_{2} + 106 \beta_{3} ) q^{44} + ( 115 + 25 \beta_{1} - 147 \beta_{2} + 52 \beta_{3} ) q^{45} + ( -274 - 40 \beta_{1} + 75 \beta_{2} + 96 \beta_{3} ) q^{46} + ( 17 + 98 \beta_{1} - 147 \beta_{2} - 113 \beta_{3} ) q^{47} + ( -57 - 80 \beta_{1} + 71 \beta_{2} + 7 \beta_{3} ) q^{48} + ( 93 - 140 \beta_{1} + 4 \beta_{2} + 64 \beta_{3} ) q^{49} + ( -170 - 10 \beta_{1} + 91 \beta_{2} + 102 \beta_{3} ) q^{50} + ( -426 - 59 \beta_{1} + 114 \beta_{2} - 131 \beta_{3} ) q^{51} + ( 70 - 50 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} ) q^{52} + ( -306 - 99 \beta_{1} + 103 \beta_{2} + 52 \beta_{3} ) q^{53} + ( -87 - 51 \beta_{1} + 77 \beta_{2} - 78 \beta_{3} ) q^{54} + ( -304 - 78 \beta_{1} - 66 \beta_{2} - 50 \beta_{3} ) q^{55} + ( 94 - 46 \beta_{1} + 14 \beta_{2} - 160 \beta_{3} ) q^{56} + ( 59 + 97 \beta_{1} - 101 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -237 + 48 \beta_{1} + 129 \beta_{2} + 267 \beta_{3} ) q^{58} + ( 222 - 94 \beta_{1} + 22 \beta_{2} + 12 \beta_{3} ) q^{59} + ( -47 - 33 \beta_{1} + 65 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -98 + 50 \beta_{1} - 30 \beta_{2} - 86 \beta_{3} ) q^{61} + ( 7 - 7 \beta_{1} + 76 \beta_{2} - 26 \beta_{3} ) q^{62} + ( -514 - 30 \beta_{1} + 172 \beta_{2} - 56 \beta_{3} ) q^{63} + ( 109 + 153 \beta_{1} - 155 \beta_{2} - 54 \beta_{3} ) q^{64} + ( 76 + 12 \beta_{1} + 34 \beta_{2} + 12 \beta_{3} ) q^{65} + ( 208 + 70 \beta_{1} - 90 \beta_{2} + 22 \beta_{3} ) q^{66} + ( -26 + 89 \beta_{1} + 125 \beta_{2} - 14 \beta_{3} ) q^{67} + ( -393 - 208 \beta_{1} + 83 \beta_{2} + 373 \beta_{3} ) q^{68} + ( 319 + 55 \beta_{1} - 93 \beta_{2} + 60 \beta_{3} ) q^{69} + ( 192 + 54 \beta_{1} - 104 \beta_{2} + 30 \beta_{3} ) q^{70} + ( -50 - 92 \beta_{1} - 176 \beta_{2} - 116 \beta_{3} ) q^{71} + ( 37 + 152 \beta_{1} + 15 \beta_{2} + 365 \beta_{3} ) q^{72} + ( -122 + 94 \beta_{1} - 234 \beta_{2} - 130 \beta_{3} ) q^{73} + ( 105 - 26 \beta_{1} + \beta_{2} - 169 \beta_{3} ) q^{74} + ( -15 + 113 \beta_{1} - 115 \beta_{2} - 16 \beta_{3} ) q^{75} + ( -430 - 177 \beta_{1} + 226 \beta_{2} + 345 \beta_{3} ) q^{76} + ( 548 - 282 \beta_{1} + 134 \beta_{2} + 242 \beta_{3} ) q^{77} + ( -22 - 10 \beta_{1} + 8 \beta_{2} ) q^{78} + ( 320 - 29 \beta_{1} - 156 \beta_{2} + 135 \beta_{3} ) q^{79} + ( -22 + 29 \beta_{1} + 90 \beta_{2} - 69 \beta_{3} ) q^{80} + ( 496 + 189 \beta_{1} - 117 \beta_{2} - 36 \beta_{3} ) q^{81} + ( 638 + 86 \beta_{1} - 351 \beta_{2} - 190 \beta_{3} ) q^{82} + ( -212 - 41 \beta_{1} - 91 \beta_{2} - 276 \beta_{3} ) q^{83} + ( -174 + 220 \beta_{1} - 134 \beta_{2} + 14 \beta_{3} ) q^{84} + ( -87 - 39 \beta_{1} + 539 \beta_{2} - 38 \beta_{3} ) q^{85} + ( -96 - 327 \beta_{1} + 204 \beta_{2} - 87 \beta_{3} ) q^{87} + ( -556 - 34 \beta_{1} + 442 \beta_{2} + 338 \beta_{3} ) q^{88} + ( 222 - 198 \beta_{1} + 48 \beta_{2} - 228 \beta_{3} ) q^{89} + ( -595 - 199 \beta_{1} + 181 \beta_{2} - 106 \beta_{3} ) q^{90} + ( -232 + 154 \beta_{1} - 78 \beta_{2} - 70 \beta_{3} ) q^{91} + ( -732 + 75 \beta_{1} + 336 \beta_{2} + 405 \beta_{3} ) q^{92} + ( 12 + 259 \beta_{1} - 272 \beta_{2} + 29 \beta_{3} ) q^{93} + ( 627 - 34 \beta_{1} - 241 \beta_{2} - 503 \beta_{3} ) q^{94} + ( 271 + 62 \beta_{1} + 3 \beta_{2} + 101 \beta_{3} ) q^{95} + ( -395 - 96 \beta_{1} + 125 \beta_{2} - 51 \beta_{3} ) q^{96} + ( -468 + 170 \beta_{1} - 63 \beta_{2} - 241 \beta_{3} ) q^{97} + ( -411 - 60 \beta_{1} + 52 \beta_{2} + 103 \beta_{3} ) q^{98} + ( -112 + 133 \beta_{1} + 257 \beta_{2} - 294 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 11q^{3} + 2q^{4} + 27q^{5} - 27q^{6} + 20q^{7} + 66q^{8} - 9q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 11q^{3} + 2q^{4} + 27q^{5} - 27q^{6} + 20q^{7} + 66q^{8} - 9q^{9} - 3q^{10} - 62q^{11} - 61q^{12} - 2q^{13} + 112q^{14} + 92q^{15} + 202q^{16} - 207q^{17} - 299q^{18} - 99q^{19} - 81q^{20} - 90q^{21} - 202q^{22} - 103q^{23} + 209q^{24} - 101q^{25} + 50q^{26} + 218q^{27} + 80q^{28} + 99q^{29} - 300q^{30} + 131q^{31} + 342q^{32} + 32q^{33} - 53q^{34} - 374q^{35} - 379q^{36} + 449q^{37} - 609q^{38} - 98q^{39} + 133q^{40} - 491q^{41} + 394q^{42} - 764q^{44} + 338q^{45} - 1061q^{46} + 19q^{47} - 237q^{48} + 236q^{49} - 599q^{50} - 1649q^{51} + 224q^{52} - 1220q^{53} - 322q^{54} - 1360q^{55} + 344q^{56} + 232q^{57} - 771q^{58} + 816q^{59} - 156q^{60} - 372q^{61} + 97q^{62} - 1914q^{63} + 434q^{64} + 350q^{65} + 812q^{66} + 110q^{67} - 1697q^{68} + 1238q^{69} + 718q^{70} - 468q^{71} + 315q^{72} - 628q^{73} + 395q^{74} - 62q^{75} - 1671q^{76} + 2044q^{77} - 90q^{78} + 1095q^{79} + 31q^{80} + 2056q^{81} + 2287q^{82} - 980q^{83} - 610q^{84} + 152q^{85} - 507q^{87} - 1816q^{88} + 738q^{89} - 2398q^{90} - 852q^{91} - 2517q^{92} + 35q^{93} + 2233q^{94} + 1149q^{95} - 1551q^{96} - 1765q^{97} - 1652q^{98} - 58q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10 x^{2} + 11 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} - 8 \nu - 2$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{3} + \nu^{2} + 19 \nu - 11$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} - \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 9 \beta_{1} - 3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.21390 −3.18808 3.05867 −0.0844804
−2.32005 9.13635 −2.61739 12.9617 −21.1967 −21.6165 24.6328 56.4728 −30.0718
1.2 −0.132290 3.71054 −7.98250 −2.43196 −0.490867 30.9271 2.11433 −13.2319 0.321724
1.3 1.25341 −1.08716 −6.42897 14.9226 −1.36266 −2.62749 −18.0854 −25.8181 18.7041
1.4 5.19893 −0.759721 19.0289 1.54763 −3.94973 13.3169 57.3382 −26.4228 8.04602
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$-1$$

## 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 1849.4.a.b 4
43.b odd 2 1 43.4.a.a 4
129.d even 2 1 387.4.a.e 4
172.d even 2 1 688.4.a.f 4
215.d odd 2 1 1075.4.a.a 4
301.c even 2 1 2107.4.a.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.a.a 4 43.b odd 2 1
387.4.a.e 4 129.d even 2 1
688.4.a.f 4 172.d even 2 1
1075.4.a.a 4 215.d odd 2 1
1849.4.a.b 4 1.a even 1 1 trivial
2107.4.a.b 4 301.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4 T_{2}^{3} - 9 T_{2}^{2} + 14 T_{2} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1849))$$.