# Properties

 Label 1859.4.a.a Level $1859$ Weight $4$ Character orbit 1859.a Self dual yes Analytic conductor $109.685$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( -1 + 4 \beta ) q^{3} + ( -4 - 2 \beta ) q^{4} + ( -1 + 8 \beta ) q^{5} + ( 13 - 5 \beta ) q^{6} + ( -10 - 4 \beta ) q^{7} + ( 6 - 10 \beta ) q^{8} + ( 22 - 8 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( -1 + 4 \beta ) q^{3} + ( -4 - 2 \beta ) q^{4} + ( -1 + 8 \beta ) q^{5} + ( 13 - 5 \beta ) q^{6} + ( -10 - 4 \beta ) q^{7} + ( 6 - 10 \beta ) q^{8} + ( 22 - 8 \beta ) q^{9} + ( 25 - 9 \beta ) q^{10} + 11 q^{11} + ( -20 - 14 \beta ) q^{12} + ( -2 - 6 \beta ) q^{14} + ( 97 - 12 \beta ) q^{15} + ( -4 + 32 \beta ) q^{16} + ( -62 - 12 \beta ) q^{17} + ( -46 + 30 \beta ) q^{18} + ( -36 + 60 \beta ) q^{19} + ( -44 - 30 \beta ) q^{20} + ( -38 - 36 \beta ) q^{21} + ( -11 + 11 \beta ) q^{22} + ( -49 + 36 \beta ) q^{23} + ( -126 + 34 \beta ) q^{24} + ( 68 - 16 \beta ) q^{25} + ( -91 - 12 \beta ) q^{27} + ( 64 + 36 \beta ) q^{28} + ( 72 + 56 \beta ) q^{29} + ( -133 + 109 \beta ) q^{30} + ( 17 + 28 \beta ) q^{31} + ( 52 + 44 \beta ) q^{32} + ( -11 + 44 \beta ) q^{33} + ( 26 - 50 \beta ) q^{34} + ( -86 - 76 \beta ) q^{35} + ( -40 - 12 \beta ) q^{36} + ( -27 - 8 \beta ) q^{37} + ( 216 - 96 \beta ) q^{38} + ( -246 + 58 \beta ) q^{40} + ( -268 - 4 \beta ) q^{41} + ( -70 - 2 \beta ) q^{42} + ( -30 + 16 \beta ) q^{43} + ( -44 - 22 \beta ) q^{44} + ( -214 + 184 \beta ) q^{45} + ( 157 - 85 \beta ) q^{46} + ( 136 - 120 \beta ) q^{47} + ( 388 - 48 \beta ) q^{48} + ( -195 + 80 \beta ) q^{49} + ( -116 + 84 \beta ) q^{50} + ( -82 - 236 \beta ) q^{51} + ( -246 + 56 \beta ) q^{53} + ( 55 - 79 \beta ) q^{54} + ( -11 + 88 \beta ) q^{55} + ( 60 + 76 \beta ) q^{56} + ( 756 - 204 \beta ) q^{57} + ( 96 + 16 \beta ) q^{58} + ( -317 - 132 \beta ) q^{59} + ( -316 - 146 \beta ) q^{60} + ( 420 - 184 \beta ) q^{61} + ( 67 - 11 \beta ) q^{62} + ( -124 - 8 \beta ) q^{63} + ( 112 - 248 \beta ) q^{64} + ( 143 - 55 \beta ) q^{66} + ( -377 - 20 \beta ) q^{67} + ( 320 + 172 \beta ) q^{68} + ( 481 - 232 \beta ) q^{69} + ( -142 - 10 \beta ) q^{70} + ( 339 + 76 \beta ) q^{71} + ( 372 - 268 \beta ) q^{72} + ( 200 - 468 \beta ) q^{73} + ( 3 - 19 \beta ) q^{74} + ( -260 + 288 \beta ) q^{75} + ( -216 - 168 \beta ) q^{76} + ( -110 - 44 \beta ) q^{77} + ( 158 - 656 \beta ) q^{79} + ( 772 - 64 \beta ) q^{80} + ( -647 - 136 \beta ) q^{81} + ( 256 - 264 \beta ) q^{82} + ( -234 + 120 \beta ) q^{83} + ( 368 + 220 \beta ) q^{84} + ( -226 - 484 \beta ) q^{85} + ( 78 - 46 \beta ) q^{86} + ( 600 + 232 \beta ) q^{87} + ( 66 - 110 \beta ) q^{88} + ( 921 - 328 \beta ) q^{89} + ( 766 - 398 \beta ) q^{90} + ( -20 - 46 \beta ) q^{92} + ( 319 + 40 \beta ) q^{93} + ( -496 + 256 \beta ) q^{94} + ( 1476 - 348 \beta ) q^{95} + ( 476 + 164 \beta ) q^{96} + ( -1097 + 144 \beta ) q^{97} + ( 435 - 275 \beta ) q^{98} + ( 242 - 88 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 26q^{6} - 20q^{7} + 12q^{8} + 44q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 26q^{6} - 20q^{7} + 12q^{8} + 44q^{9} + 50q^{10} + 22q^{11} - 40q^{12} - 4q^{14} + 194q^{15} - 8q^{16} - 124q^{17} - 92q^{18} - 72q^{19} - 88q^{20} - 76q^{21} - 22q^{22} - 98q^{23} - 252q^{24} + 136q^{25} - 182q^{27} + 128q^{28} + 144q^{29} - 266q^{30} + 34q^{31} + 104q^{32} - 22q^{33} + 52q^{34} - 172q^{35} - 80q^{36} - 54q^{37} + 432q^{38} - 492q^{40} - 536q^{41} - 140q^{42} - 60q^{43} - 88q^{44} - 428q^{45} + 314q^{46} + 272q^{47} + 776q^{48} - 390q^{49} - 232q^{50} - 164q^{51} - 492q^{53} + 110q^{54} - 22q^{55} + 120q^{56} + 1512q^{57} + 192q^{58} - 634q^{59} - 632q^{60} + 840q^{61} + 134q^{62} - 248q^{63} + 224q^{64} + 286q^{66} - 754q^{67} + 640q^{68} + 962q^{69} - 284q^{70} + 678q^{71} + 744q^{72} + 400q^{73} + 6q^{74} - 520q^{75} - 432q^{76} - 220q^{77} + 316q^{79} + 1544q^{80} - 1294q^{81} + 512q^{82} - 468q^{83} + 736q^{84} - 452q^{85} + 156q^{86} + 1200q^{87} + 132q^{88} + 1842q^{89} + 1532q^{90} - 40q^{92} + 638q^{93} - 992q^{94} + 2952q^{95} + 952q^{96} - 2194q^{97} + 870q^{98} + 484q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−2.73205 −7.92820 −0.535898 −14.8564 21.6603 −3.07180 23.3205 35.8564 40.5885
1.2 0.732051 5.92820 −7.46410 12.8564 4.33975 −16.9282 −11.3205 8.14359 9.41154
 $$n$$: 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.a 2
13.b even 2 1 11.4.a.a 2
39.d odd 2 1 99.4.a.c 2
52.b odd 2 1 176.4.a.i 2
65.d even 2 1 275.4.a.b 2
65.h odd 4 2 275.4.b.c 4
91.b odd 2 1 539.4.a.e 2
104.e even 2 1 704.4.a.p 2
104.h odd 2 1 704.4.a.n 2
143.d odd 2 1 121.4.a.c 2
143.l odd 10 4 121.4.c.f 8
143.n even 10 4 121.4.c.c 8
156.h even 2 1 1584.4.a.bc 2
195.e odd 2 1 2475.4.a.q 2
429.e even 2 1 1089.4.a.v 2
572.b even 2 1 1936.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 13.b even 2 1
99.4.a.c 2 39.d odd 2 1
121.4.a.c 2 143.d odd 2 1
121.4.c.c 8 143.n even 10 4
121.4.c.f 8 143.l odd 10 4
176.4.a.i 2 52.b odd 2 1
275.4.a.b 2 65.d even 2 1
275.4.b.c 4 65.h odd 4 2
539.4.a.e 2 91.b odd 2 1
704.4.a.n 2 104.h odd 2 1
704.4.a.p 2 104.e even 2 1
1089.4.a.v 2 429.e even 2 1
1584.4.a.bc 2 156.h even 2 1
1859.4.a.a 2 1.a even 1 1 trivial
1936.4.a.w 2 572.b even 2 1
2475.4.a.q 2 195.e odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + 2 T + T^{2}$$
$3$ $$-47 + 2 T + T^{2}$$
$5$ $$-191 + 2 T + T^{2}$$
$7$ $$52 + 20 T + T^{2}$$
$11$ $$( -11 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$3412 + 124 T + T^{2}$$
$19$ $$-9504 + 72 T + T^{2}$$
$23$ $$-1487 + 98 T + T^{2}$$
$29$ $$-4224 - 144 T + T^{2}$$
$31$ $$-2063 - 34 T + T^{2}$$
$37$ $$537 + 54 T + T^{2}$$
$41$ $$71776 + 536 T + T^{2}$$
$43$ $$132 + 60 T + T^{2}$$
$47$ $$-24704 - 272 T + T^{2}$$
$53$ $$51108 + 492 T + T^{2}$$
$59$ $$48217 + 634 T + T^{2}$$
$61$ $$74832 - 840 T + T^{2}$$
$67$ $$140929 + 754 T + T^{2}$$
$71$ $$97593 - 678 T + T^{2}$$
$73$ $$-617072 - 400 T + T^{2}$$
$79$ $$-1266044 - 316 T + T^{2}$$
$83$ $$11556 + 468 T + T^{2}$$
$89$ $$525489 - 1842 T + T^{2}$$
$97$ $$1141201 + 2194 T + T^{2}$$
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