# Properties

 Label 1859.4.a.b Level $1859$ Weight $4$ Character orbit 1859.a Self dual yes Analytic conductor $109.685$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.297133.1 Defining polynomial: $$x^{4} - 19 x^{2} - 2 x + 52$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{4} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{5} + ( 4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( 4 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{7} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{8} + ( -11 - 3 \beta_{1} + 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{4} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{5} + ( 4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( 4 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{7} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{8} + ( -11 - 3 \beta_{1} + 3 \beta_{3} ) q^{9} + ( -22 - 7 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{10} + 11 q^{11} + ( 4 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( -8 - \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{14} + ( -4 - 12 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{15} + ( -38 - 3 \beta_{1} - 10 \beta_{2} ) q^{16} + ( -10 + 8 \beta_{1} + \beta_{2} - 7 \beta_{3} ) q^{17} + ( 18 + 11 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{18} + ( 7 - 6 \beta_{1} - 13 \beta_{2} + 10 \beta_{3} ) q^{19} + ( 46 + 15 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} ) q^{20} + ( 43 - 13 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} ) q^{21} -11 \beta_{1} q^{22} + ( -38 + 2 \beta_{1} - 16 \beta_{2} + \beta_{3} ) q^{23} + ( 40 - 12 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} ) q^{24} + ( -35 + 29 \beta_{1} + 22 \beta_{2} + 16 \beta_{3} ) q^{25} + ( 5 + 15 \beta_{1} + 9 \beta_{2} + 29 \beta_{3} ) q^{27} + ( 18 - 13 \beta_{1} + 16 \beta_{2} + 14 \beta_{3} ) q^{28} + ( -46 + 23 \beta_{1} + 23 \beta_{2} + 19 \beta_{3} ) q^{29} + ( 80 + 20 \beta_{1} + 15 \beta_{2} + 17 \beta_{3} ) q^{30} + ( 100 + 24 \beta_{1} + 52 \beta_{2} - 10 \beta_{3} ) q^{31} + ( 6 + 27 \beta_{1} + 22 \beta_{2} + 52 \beta_{3} ) q^{32} + ( -11 - 11 \beta_{3} ) q^{33} + ( -48 + 8 \beta_{1} + 5 \beta_{2} - 9 \beta_{3} ) q^{34} + ( 34 - 11 \beta_{1} + 18 \beta_{3} ) q^{35} + ( -46 - 5 \beta_{1} - 31 \beta_{2} - 15 \beta_{3} ) q^{36} + ( 88 - 36 \beta_{1} + 19 \beta_{2} + 25 \beta_{3} ) q^{37} + ( -32 + 2 \beta_{1} - 18 \beta_{2} + 36 \beta_{3} ) q^{38} + ( 34 - 23 \beta_{1} - 6 \beta_{2} + 20 \beta_{3} ) q^{40} + ( 38 - 72 \beta_{1} - 46 \beta_{2} + 25 \beta_{3} ) q^{41} + ( 82 - 28 \beta_{1} + 11 \beta_{2} + 19 \beta_{3} ) q^{42} + ( -88 - 25 \beta_{1} - 100 \beta_{2} - 32 \beta_{3} ) q^{43} + ( 22 + 11 \beta_{1} + 22 \beta_{2} ) q^{44} + ( -82 - 27 \beta_{1} - 25 \beta_{2} - 21 \beta_{3} ) q^{45} + ( -88 + 51 \beta_{1} - 7 \beta_{2} + 33 \beta_{3} ) q^{46} + ( -170 - 32 \beta_{1} - 119 \beta_{2} - 21 \beta_{3} ) q^{47} + ( 36 \beta_{1} - \beta_{2} + 25 \beta_{3} ) q^{48} + ( -124 - 25 \beta_{1} - 65 \beta_{2} - 18 \beta_{3} ) q^{49} + ( -266 - 32 \beta_{1} - 106 \beta_{2} - 28 \beta_{3} ) q^{50} + ( 88 - 40 \beta_{1} - 23 \beta_{2} + 33 \beta_{3} ) q^{51} + ( 103 - 62 \beta_{1} + 69 \beta_{2} + 18 \beta_{3} ) q^{53} + ( -230 - 58 \beta_{1} - 117 \beta_{2} + 11 \beta_{3} ) q^{54} + ( 22 + 33 \beta_{1} + 22 \beta_{2} ) q^{55} + ( 202 - 27 \beta_{1} - 80 \beta_{2} - 2 \beta_{3} ) q^{56} + ( -198 + 81 \beta_{1} + 5 \beta_{2} - 46 \beta_{3} ) q^{57} + ( -214 - 19 \beta_{1} - 103 \beta_{2} - 27 \beta_{3} ) q^{58} + ( 132 - 17 \beta_{1} - 13 \beta_{2} + 61 \beta_{3} ) q^{59} + ( -176 - 36 \beta_{1} - 35 \beta_{2} - 37 \beta_{3} ) q^{60} + ( -218 - 7 \beta_{1} - 71 \beta_{2} + 7 \beta_{3} ) q^{61} + ( 8 - 166 \beta_{1} - 18 \beta_{2} - 114 \beta_{3} ) q^{62} + ( -209 + 8 \beta_{1} + 59 \beta_{2} + 35 \beta_{3} ) q^{63} + ( -86 - 83 \beta_{1} - 130 \beta_{2} + 8 \beta_{3} ) q^{64} + ( 44 + 22 \beta_{1} + 33 \beta_{2} - 11 \beta_{3} ) q^{66} + ( 224 - 113 \beta_{1} + 33 \beta_{2} + 43 \beta_{3} ) q^{67} + ( 56 - 20 \beta_{1} + 3 \beta_{2} + 37 \beta_{3} ) q^{68} + ( -65 + 47 \beta_{1} - 22 \beta_{2} + 22 \beta_{3} ) q^{69} + ( 38 - 41 \beta_{1} - 32 \beta_{2} + 18 \beta_{3} ) q^{70} + ( -180 + 112 \beta_{1} + 61 \beta_{2} + 29 \beta_{3} ) q^{71} + ( -158 + 9 \beta_{1} + 79 \beta_{2} + 23 \beta_{3} ) q^{72} + ( 343 + 67 \beta_{1} + 66 \beta_{2} + 155 \beta_{3} ) q^{73} + ( 336 - 96 \beta_{1} - 3 \beta_{2} - 13 \beta_{3} ) q^{74} + ( -211 - 76 \beta_{1} - 65 \beta_{2} + 54 \beta_{3} ) q^{75} + ( -292 + 60 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{76} + ( 44 + 22 \beta_{1} - 11 \beta_{2} - 44 \beta_{3} ) q^{77} + ( -46 - 142 \beta_{1} + 94 \beta_{2} - 50 \beta_{3} ) q^{79} + ( -242 - 145 \beta_{1} - 94 \beta_{2} - 32 \beta_{3} ) q^{80} + ( -158 + 111 \beta_{1} - 36 \beta_{2} - 120 \beta_{3} ) q^{81} + ( 436 + 55 \beta_{1} + 69 \beta_{2} + 117 \beta_{3} ) q^{82} + ( -110 + 13 \beta_{1} - 3 \beta_{2} - 206 \beta_{3} ) q^{83} + ( -96 + 20 \beta_{1} + 55 \beta_{2} - 43 \beta_{3} ) q^{84} + ( 152 - 36 \beta_{1} - 7 \beta_{2} + 55 \beta_{3} ) q^{85} + ( -22 + 245 \beta_{1} + 146 \beta_{2} + 168 \beta_{3} ) q^{86} + ( -216 - 58 \beta_{1} - 46 \beta_{2} + 54 \beta_{3} ) q^{87} + ( -22 + 33 \beta_{1} - 22 \beta_{2} - 44 \beta_{3} ) q^{88} + ( -644 - 20 \beta_{1} - 24 \beta_{2} - 148 \beta_{3} ) q^{89} + ( 254 + 155 \beta_{1} + 117 \beta_{2} + 29 \beta_{3} ) q^{90} + ( -366 - 5 \beta_{1} - 73 \beta_{2} + 39 \beta_{3} ) q^{92} + ( 214 - 234 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -72 + 342 \beta_{1} + 127 \beta_{2} + 217 \beta_{3} ) q^{94} + ( -228 + 56 \beta_{1} + 28 \beta_{2} - 80 \beta_{3} ) q^{95} + ( -784 + 36 \beta_{1} - 59 \beta_{2} - 61 \beta_{3} ) q^{96} + ( 564 - 84 \beta_{1} - 248 \beta_{2} + 8 \beta_{3} ) q^{97} + ( 62 + 232 \beta_{1} + 104 \beta_{2} + 112 \beta_{3} ) q^{98} + ( -121 - 33 \beta_{1} + 33 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 6 q^{4} + 6 q^{5} + 13 q^{6} + 17 q^{7} - 6 q^{8} - 44 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} + 6 q^{4} + 6 q^{5} + 13 q^{6} + 17 q^{7} - 6 q^{8} - 44 q^{9} - 82 q^{10} + 44 q^{11} + 17 q^{12} - 40 q^{14} - 9 q^{15} - 142 q^{16} - 41 q^{17} + 75 q^{18} + 41 q^{19} + 174 q^{20} + 179 q^{21} - 136 q^{23} + 171 q^{24} - 162 q^{25} + 11 q^{27} + 56 q^{28} - 207 q^{29} + 305 q^{30} + 348 q^{31} + 2 q^{32} - 44 q^{33} - 197 q^{34} + 136 q^{35} - 153 q^{36} + 333 q^{37} - 110 q^{38} + 142 q^{40} + 198 q^{41} + 317 q^{42} - 252 q^{43} + 66 q^{44} - 303 q^{45} - 345 q^{46} - 561 q^{47} + q^{48} - 431 q^{49} - 958 q^{50} + 375 q^{51} + 343 q^{53} - 803 q^{54} + 66 q^{55} + 888 q^{56} - 797 q^{57} - 753 q^{58} + 541 q^{59} - 669 q^{60} - 801 q^{61} + 50 q^{62} - 895 q^{63} - 214 q^{64} + 143 q^{66} + 863 q^{67} + 221 q^{68} - 238 q^{69} + 184 q^{70} - 781 q^{71} - 711 q^{72} + 1306 q^{73} + 1347 q^{74} - 779 q^{75} - 1160 q^{76} + 187 q^{77} - 278 q^{79} - 874 q^{80} - 596 q^{81} + 1675 q^{82} - 437 q^{83} - 439 q^{84} + 615 q^{85} - 234 q^{86} - 818 q^{87} - 66 q^{88} - 2552 q^{89} + 899 q^{90} - 1391 q^{92} + 876 q^{93} - 415 q^{94} - 940 q^{95} - 3077 q^{96} + 2504 q^{97} + 144 q^{98} - 484 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 19 x^{2} - 2 x + 52$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 10$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 12 \nu + 8$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + \beta_{1} + 10$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} + 2$$

## 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
 4.03811 1.74344 −1.90566 −3.87589
−4.03811 −3.27078 8.30634 16.3826 13.2078 1.85898 −1.23702 −16.3020 −66.1546
1.2 −1.74344 1.66539 −4.96041 −1.47352 −2.90351 22.5004 22.5957 −24.2265 2.56900
1.3 1.90566 −6.07897 −4.36845 −8.17977 −11.5845 −17.8958 −23.5701 9.95391 −15.5879
1.4 3.87589 3.68437 7.02252 −0.729260 14.2802 10.5365 −3.78861 −13.4254 −2.82653
 $$n$$: e.g. 2-40 or 990-1000 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.b 4
13.b even 2 1 143.4.a.a 4
39.d odd 2 1 1287.4.a.b 4
52.b odd 2 1 2288.4.a.i 4
143.d odd 2 1 1573.4.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.a 4 13.b even 2 1
1287.4.a.b 4 39.d odd 2 1
1573.4.a.c 4 143.d odd 2 1
1859.4.a.b 4 1.a even 1 1 trivial
2288.4.a.i 4 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 19 T_{2}^{2} + 2 T_{2} + 52$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1859))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$52 + 2 T - 19 T^{2} + T^{4}$$
$3$ $$122 - 49 T - 24 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$-144 - 304 T - 151 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$-7887 + 4901 T - 326 T^{2} - 17 T^{3} + T^{4}$$
$11$ $$( -11 + T )^{4}$$
$13$ $$T^{4}$$
$17$ $$32696 + 3072 T - 1348 T^{2} + 41 T^{3} + T^{4}$$
$19$ $$1792256 + 131560 T - 7147 T^{2} - 41 T^{3} + T^{4}$$
$23$ $$-186204 - 180625 T + 276 T^{2} + 136 T^{3} + T^{4}$$
$29$ $$-98922752 - 2807024 T - 6000 T^{2} + 207 T^{3} + T^{4}$$
$31$ $$432658176 + 7956232 T - 15304 T^{2} - 348 T^{3} + T^{4}$$
$37$ $$-122208096 + 5983192 T - 562 T^{2} - 333 T^{3} + T^{4}$$
$41$ $$21770232 + 4184507 T - 94198 T^{2} - 198 T^{3} + T^{4}$$
$43$ $$845198188 - 38696306 T - 155331 T^{2} + 252 T^{3} + T^{4}$$
$47$ $$3989237104 - 74831588 T - 134344 T^{2} + 561 T^{3} + T^{4}$$
$53$ $$895612308 + 27025412 T - 176749 T^{2} - 343 T^{3} + T^{4}$$
$59$ $$50105064 + 13696184 T - 8404 T^{2} - 541 T^{3} + T^{4}$$
$61$ $$-107700832 - 8840308 T + 125050 T^{2} + 801 T^{3} + T^{4}$$
$67$ $$7976743328 + 89740612 T - 14114 T^{2} - 863 T^{3} + T^{4}$$
$71$ $$-24400890272 - 129421524 T - 19158 T^{2} + 781 T^{3} + T^{4}$$
$73$ $$74727204864 + 365499949 T - 144006 T^{2} - 1306 T^{3} + T^{4}$$
$79$ $$-98639841184 - 658012120 T - 1017308 T^{2} + 278 T^{3} + T^{4}$$
$83$ $$233243308272 - 146246408 T - 1159565 T^{2} + 437 T^{3} + T^{4}$$
$89$ $$-5112256512 + 218286400 T + 1797792 T^{2} + 2552 T^{3} + T^{4}$$
$97$ $$18716618752 + 536340864 T + 1145456 T^{2} - 2504 T^{3} + T^{4}$$
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