Properties

 Label 387.2.h.d Level 387 Weight 2 Character orbit 387.h Analytic conductor 3.090 Analytic rank 0 Dimension 4 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 387.h (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.09021055822$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} -3 \beta_{2} q^{4} -2 \beta_{1} q^{5} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} -3 \beta_{2} q^{4} -2 \beta_{1} q^{5} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} ) q^{8} + ( -2 - 4 \beta_{1} - 2 \beta_{3} ) q^{10} + ( 3 + \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( 5 - 3 \beta_{2} ) q^{16} + ( 5 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{17} -2 \beta_{1} q^{19} + ( -6 - 6 \beta_{1} - 6 \beta_{3} ) q^{20} + ( 2 - \beta_{2} ) q^{22} + ( 5 + \beta_{1} + 5 \beta_{3} ) q^{23} + ( 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{25} + ( 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{26} + ( -3 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{28} -3 \beta_{3} q^{29} + ( 6 - 3 \beta_{2} ) q^{32} + ( 7 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 4 + 2 \beta_{2} ) q^{35} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{37} + ( -2 - 4 \beta_{1} - 2 \beta_{3} ) q^{38} + ( -8 - 10 \beta_{1} - 8 \beta_{3} ) q^{40} + ( -7 - 4 \beta_{2} ) q^{41} + ( -7 - \beta_{3} ) q^{43} + ( -3 - 6 \beta_{2} ) q^{44} + ( 6 + 7 \beta_{1} + 6 \beta_{3} ) q^{46} + ( -3 + 3 \beta_{2} ) q^{47} + ( -6 + 8 \beta_{1} - 6 \beta_{3} ) q^{49} + ( 7 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{50} + ( 9 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{52} + ( -2 - \beta_{1} - 2 \beta_{3} ) q^{53} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{55} + ( -2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{56} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{58} + ( -2 - 5 \beta_{2} ) q^{59} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{61} + ( -1 - 6 \beta_{2} ) q^{64} + ( 2 - 6 \beta_{2} ) q^{65} + ( 2 - 3 \beta_{1} + 2 \beta_{3} ) q^{67} + ( 6 \beta_{1} + 6 \beta_{2} + 15 \beta_{3} ) q^{68} + 2 q^{70} + ( -11 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} ) q^{71} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{73} -3 \beta_{1} q^{74} + ( -6 - 6 \beta_{1} - 6 \beta_{3} ) q^{76} + ( 7 \beta_{1} + 7 \beta_{2} - 11 \beta_{3} ) q^{77} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{79} + ( -6 - 16 \beta_{1} - 6 \beta_{3} ) q^{80} + ( -3 - \beta_{2} ) q^{82} + ( 8 - 13 \beta_{1} + 8 \beta_{3} ) q^{83} + ( 10 - 4 \beta_{2} ) q^{85} + ( -7 - \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{86} + ( -1 - 7 \beta_{2} ) q^{88} + 3 \beta_{1} q^{89} + ( 4 - 3 \beta_{1} + 4 \beta_{3} ) q^{91} + ( 3 + 18 \beta_{1} + 3 \beta_{3} ) q^{92} + ( -6 + 9 \beta_{2} ) q^{94} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( -8 - 2 \beta_{2} ) q^{97} + ( 2 + 10 \beta_{1} + 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{2} + 6q^{4} - 2q^{5} + 4q^{7} + 12q^{8} + O(q^{10})$$ $$4q + 6q^{2} + 6q^{4} - 2q^{5} + 4q^{7} + 12q^{8} - 8q^{10} + 10q^{11} - 5q^{13} + q^{14} + 26q^{16} + q^{17} - 2q^{19} - 18q^{20} + 10q^{22} + 11q^{23} - 2q^{25} - 10q^{26} - 9q^{28} + 6q^{29} + 30q^{32} - 11q^{34} + 12q^{35} + 3q^{37} - 8q^{38} - 26q^{40} - 20q^{41} - 26q^{43} + 19q^{46} - 18q^{47} - 4q^{49} - 13q^{50} - 15q^{52} - 5q^{53} - 8q^{56} + 9q^{58} + 2q^{59} + q^{61} + 8q^{64} + 20q^{65} + q^{67} - 36q^{68} + 8q^{70} + 3q^{71} + 3q^{73} - 3q^{74} - 18q^{76} + 15q^{77} + 5q^{79} - 28q^{80} - 10q^{82} + 3q^{83} + 48q^{85} - 39q^{86} + 10q^{88} + 3q^{89} + 5q^{91} + 24q^{92} - 42q^{94} - 12q^{95} - 28q^{97} + 14q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$173$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
208.1
 −0.309017 − 0.535233i 0.809017 + 1.40126i −0.309017 + 0.535233i 0.809017 − 1.40126i
0.381966 0 −1.85410 0.618034 + 1.07047i 0 2.11803 3.66854i −1.47214 0 0.236068 + 0.408882i
208.2 2.61803 0 4.85410 −1.61803 2.80252i 0 −0.118034 + 0.204441i 7.47214 0 −4.23607 7.33708i
307.1 0.381966 0 −1.85410 0.618034 1.07047i 0 2.11803 + 3.66854i −1.47214 0 0.236068 0.408882i
307.2 2.61803 0 4.85410 −1.61803 + 2.80252i 0 −0.118034 0.204441i 7.47214 0 −4.23607 + 7.33708i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.h.d 4
3.b odd 2 1 43.2.c.b 4
12.b even 2 1 688.2.i.e 4
43.c even 3 1 inner 387.2.h.d 4
129.f odd 6 1 43.2.c.b 4
129.f odd 6 1 1849.2.a.e 2
129.h even 6 1 1849.2.a.h 2
516.p even 6 1 688.2.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.b 4 3.b odd 2 1
43.2.c.b 4 129.f odd 6 1
387.2.h.d 4 1.a even 1 1 trivial
387.2.h.d 4 43.c even 3 1 inner
688.2.i.e 4 12.b even 2 1
688.2.i.e 4 516.p even 6 1
1849.2.a.e 2 129.f odd 6 1
1849.2.a.h 2 129.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(387, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} )^{2}$$
$3$ 1
$5$ $$1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 40 T^{5} - 50 T^{6} + 250 T^{7} + 625 T^{8}$$
$7$ $$1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 28 T^{5} + 147 T^{6} - 1372 T^{7} + 2401 T^{8}$$
$11$ $$( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 325 T^{5} - 1014 T^{6} + 10985 T^{7} + 28561 T^{8}$$
$17$ $$1 - T - 2 T^{2} + 31 T^{3} - 297 T^{4} + 527 T^{5} - 578 T^{6} - 4913 T^{7} + 83521 T^{8}$$
$19$ $$1 + 2 T - 30 T^{2} - 8 T^{3} + 719 T^{4} - 152 T^{5} - 10830 T^{6} + 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 - 11 T + 46 T^{2} - 319 T^{3} + 2313 T^{4} - 7337 T^{5} + 24334 T^{6} - 133837 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} + 961 T^{4} )^{2}$$
$37$ $$1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 999 T^{5} - 76664 T^{6} - 151959 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 10 T + 87 T^{2} + 410 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 13 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 9 T + 103 T^{2} + 423 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 + 5 T - 86 T^{2} + 25 T^{3} + 8187 T^{4} + 1325 T^{5} - 241574 T^{6} + 744385 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 - T + 87 T^{2} - 59 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 671 T^{5} - 409310 T^{6} - 226981 T^{7} + 13845841 T^{8}$$
$67$ $$1 - T - 122 T^{2} + 11 T^{3} + 10573 T^{4} + 737 T^{5} - 547658 T^{6} - 300763 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 3 T + 16 T^{2} + 447 T^{3} - 5631 T^{4} + 31737 T^{5} + 80656 T^{6} - 1073733 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 1971 T^{5} - 682112 T^{6} - 1167051 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 5 T - 138 T^{2} - 25 T^{3} + 18353 T^{4} - 1975 T^{5} - 861258 T^{6} - 2465195 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 3 T + 52 T^{2} + 627 T^{3} - 5787 T^{4} + 52041 T^{5} + 358228 T^{6} - 1715361 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 3 T - 160 T^{2} + 27 T^{3} + 19839 T^{4} + 2403 T^{5} - 1267360 T^{6} - 2114907 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 14 T + 238 T^{2} + 1358 T^{3} + 9409 T^{4} )^{2}$$