Properties

 Label 1520.2.q.j Level $1520$ Weight $2$ Character orbit 1520.q Analytic conductor $12.137$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.q (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.3518667.1 Defining polynomial: $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{9}+O(q^{10})$$ q + b1 * q^3 - b3 * q^5 + (b2 - 1) * q^7 + (-b5 + b4 - b3 + b2 - 2) * q^9 $$q + \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{9} + ( - \beta_{2} + 2) q^{11} + (5 \beta_{3} + 5) q^{13} + ( - \beta_{2} + \beta_1) q^{15} + ( - 3 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{5} - 4 \beta_{3} - \beta_1) q^{21} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{3} - 1) q^{25} + (\beta_{4} + \beta_{2} + 2) q^{27} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{29} + ( - \beta_{4} - \beta_{2} + 1) q^{31} + (\beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{33} + (\beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{4} + 3 \beta_{2} - 1) q^{37} + 5 \beta_{2} q^{39} + (2 \beta_{5} - \beta_{3} + \beta_1) q^{41} + (3 \beta_{5} - \beta_{3} - \beta_1) q^{43} + (\beta_{4} + \beta_{2} - 2) q^{45} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{47} + ( - \beta_{4} - 3 \beta_{2} - 1) q^{49} + ( - 5 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{51} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 3) q^{53} + ( - 2 \beta_{3} - \beta_1) q^{55} + (3 \beta_{4} + 7 \beta_{3} + 2 \beta_1 - 1) q^{57} + ( - 2 \beta_{5} - \beta_{3} + \beta_1) q^{59} + ( - 4 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{61} + (4 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 4) q^{63} + 5 q^{65} + (2 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 2 \beta_1 - 6) q^{67} + (5 \beta_{4} + 2 \beta_{2} - 4) q^{69} + ( - \beta_{5} + 11 \beta_{3} + 3 \beta_1) q^{71} + (3 \beta_{5} - 9 \beta_{3} - 2 \beta_1) q^{73} - \beta_{2} q^{75} + (\beta_{4} + 4 \beta_{2} - 7) q^{77} + (4 \beta_{5} + 6 \beta_{3} - 2 \beta_1) q^{79} + ( - 2 \beta_{5} - 6 \beta_{3} + \beta_1) q^{81} + (3 \beta_{4} + 3 \beta_{2} - 1) q^{83} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{85} + (\beta_{4} + 2 \beta_{2} - 5) q^{87} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 6) q^{89} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 5) q^{91} + ( - \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{93} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{95} + (4 \beta_{5} - 3 \beta_1) q^{97} + ( - \beta_{5} + \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 6) q^{99}+O(q^{100})$$ q + b1 * q^3 - b3 * q^5 + (b2 - 1) * q^7 + (-b5 + b4 - b3 + b2 - 2) * q^9 + (-b2 + 2) * q^11 + (5*b3 + 5) * q^13 + (-b2 + b1) * q^15 + (-3*b5 + 2*b3 + 2*b1) * q^17 + (-2*b5 + b4 + b3 - b2 + b1) * q^19 + (-b5 - 4*b3 - b1) * q^21 + (-3*b5 + 3*b4 + 3*b3 + b2 + 2*b1) * q^23 + (-b3 - 1) * q^25 + (b4 + b2 + 2) * q^27 + (b3 - b2 + b1 + 1) * q^29 + (-b4 - b2 + 1) * q^31 + (b5 + 4*b3 + 2*b1) * q^33 + (b3 + b1) * q^35 + (-2*b4 + 3*b2 - 1) * q^37 + 5*b2 * q^39 + (2*b5 - b3 + b1) * q^41 + (3*b5 - b3 - b1) * q^43 + (b4 + b2 - 2) * q^45 + (2*b5 - 2*b4 + b3 - b2 - b1 + 3) * q^47 + (-b4 - 3*b2 - 1) * q^49 + (-5*b5 + 5*b4 + b3 + b2 + 4*b1 - 4) * q^51 + (-b5 + b4 - 2*b3 - 3*b2 + 4*b1 - 3) * q^53 + (-2*b3 - b1) * q^55 + (3*b4 + 7*b3 + 2*b1 - 1) * q^57 + (-2*b5 - b3 + b1) * q^59 + (-4*b5 + 4*b4 + 5*b3 + 2*b2 + 2*b1 + 1) * q^61 + (4*b3 - 3*b2 + 3*b1 + 4) * q^63 + 5 * q^65 + (2*b5 - 2*b4 - 8*b3 - 2*b1 - 6) * q^67 + (5*b4 + 2*b2 - 4) * q^69 + (-b5 + 11*b3 + 3*b1) * q^71 + (3*b5 - 9*b3 - 2*b1) * q^73 - b2 * q^75 + (b4 + 4*b2 - 7) * q^77 + (4*b5 + 6*b3 - 2*b1) * q^79 + (-2*b5 - 6*b3 + b1) * q^81 + (3*b4 + 3*b2 - 1) * q^83 + (-3*b5 + 3*b4 + 2*b3 + b2 + 2*b1 - 1) * q^85 + (b4 + 2*b2 - 5) * q^87 + (4*b5 - 4*b4 + 2*b3 - 4*b1 + 6) * q^89 + (-5*b3 + 5*b2 - 5*b1 - 5) * q^91 + (-b5 + 3*b3 + 2*b1) * q^93 + (-b5 + 2*b4 + b2 - b1 - 1) * q^95 + (4*b5 - 3*b1) * q^97 + (-b5 + b4 - 5*b3 + 4*b2 - 3*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 6 * q + q^3 + 3 * q^5 - 4 * q^7 - 4 * q^9 $$6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9} + 10 q^{11} + 15 q^{13} - q^{15} - q^{17} + 12 q^{21} + 4 q^{23} - 3 q^{25} + 16 q^{27} + 2 q^{29} + 2 q^{31} - 11 q^{33} - 2 q^{35} - 4 q^{37} + 10 q^{39} + 2 q^{41} - q^{43} - 8 q^{45} + 6 q^{47} - 14 q^{49} - 6 q^{51} - 11 q^{53} + 5 q^{55} - 19 q^{57} + 6 q^{59} + 9 q^{61} + 9 q^{63} + 30 q^{65} - 20 q^{67} - 10 q^{69} - 29 q^{71} + 22 q^{73} - 2 q^{75} - 32 q^{77} - 24 q^{79} + 21 q^{81} + 6 q^{83} + q^{85} - 24 q^{87} + 14 q^{89} - 10 q^{91} - 6 q^{93} - 7 q^{97} - 13 q^{99}+O(q^{100})$$ 6 * q + q^3 + 3 * q^5 - 4 * q^7 - 4 * q^9 + 10 * q^11 + 15 * q^13 - q^15 - q^17 + 12 * q^21 + 4 * q^23 - 3 * q^25 + 16 * q^27 + 2 * q^29 + 2 * q^31 - 11 * q^33 - 2 * q^35 - 4 * q^37 + 10 * q^39 + 2 * q^41 - q^43 - 8 * q^45 + 6 * q^47 - 14 * q^49 - 6 * q^51 - 11 * q^53 + 5 * q^55 - 19 * q^57 + 6 * q^59 + 9 * q^61 + 9 * q^63 + 30 * q^65 - 20 * q^67 - 10 * q^69 - 29 * q^71 + 22 * q^73 - 2 * q^75 - 32 * q^77 - 24 * q^79 + 21 * q^81 + 6 * q^83 + q^85 - 24 * q^87 + 14 * q^89 - 10 * q^91 - 6 * q^93 - 7 * q^97 - 13 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259$$ (-v^5 + 7*v^4 - 49*v^3 + 43*v^2 - 42*v + 294) / 259 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259$$ (-6*v^5 + 5*v^4 - 35*v^3 - v^2 - 215*v - 49) / 259 $$\beta_{4}$$ $$=$$ $$( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259$$ (-5*v^5 + 35*v^4 + 14*v^3 + 215*v^2 - 210*v + 952) / 259 $$\beta_{5}$$ $$=$$ $$( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259$$ (18*v^5 + 22*v^4 + 105*v^3 + 3*v^2 + 608*v + 147) / 259
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5$$ -b5 + b4 - 4*b3 + b2 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 5\beta_{2} + 2$$ b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$7\beta_{5} + 21\beta_{3} + \beta_1$$ 7*b5 + 21*b3 + b1 $$\nu^{5}$$ $$=$$ $$6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19$$ 6*b5 - 6*b4 - 25*b3 + 29*b2 - 35*b1 - 19

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 −1.25351 − 2.17114i 0.610938 + 1.05818i 1.14257 + 1.97899i −1.25351 + 2.17114i 0.610938 − 1.05818i 1.14257 − 1.97899i
0 −1.25351 2.17114i 0 0.500000 + 0.866025i 0 −3.50702 0 −1.64257 + 2.84502i 0
881.2 0 0.610938 + 1.05818i 0 0.500000 + 0.866025i 0 0.221876 0 0.753509 1.30512i 0
881.3 0 1.14257 + 1.97899i 0 0.500000 + 0.866025i 0 1.28514 0 −1.11094 + 1.92420i 0
961.1 0 −1.25351 + 2.17114i 0 0.500000 0.866025i 0 −3.50702 0 −1.64257 2.84502i 0
961.2 0 0.610938 1.05818i 0 0.500000 0.866025i 0 0.221876 0 0.753509 + 1.30512i 0
961.3 0 1.14257 1.97899i 0 0.500000 0.866025i 0 1.28514 0 −1.11094 1.92420i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.q.j 6
4.b odd 2 1 95.2.e.b 6
12.b even 2 1 855.2.k.g 6
19.c even 3 1 inner 1520.2.q.j 6
20.d odd 2 1 475.2.e.d 6
20.e even 4 2 475.2.j.b 12
76.f even 6 1 1805.2.a.g 3
76.g odd 6 1 95.2.e.b 6
76.g odd 6 1 1805.2.a.h 3
228.m even 6 1 855.2.k.g 6
380.p odd 6 1 475.2.e.d 6
380.p odd 6 1 9025.2.a.z 3
380.s even 6 1 9025.2.a.ba 3
380.v even 12 2 475.2.j.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 4.b odd 2 1
95.2.e.b 6 76.g odd 6 1
475.2.e.d 6 20.d odd 2 1
475.2.e.d 6 380.p odd 6 1
475.2.j.b 12 20.e even 4 2
475.2.j.b 12 380.v even 12 2
855.2.k.g 6 12.b even 2 1
855.2.k.g 6 228.m even 6 1
1520.2.q.j 6 1.a even 1 1 trivial
1520.2.q.j 6 19.c even 3 1 inner
1805.2.a.g 3 76.f even 6 1
1805.2.a.h 3 76.g odd 6 1
9025.2.a.z 3 380.p odd 6 1
9025.2.a.ba 3 380.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{6} - T_{3}^{5} + 7T_{3}^{4} - 8T_{3}^{3} + 43T_{3}^{2} - 42T_{3} + 49$$ T3^6 - T3^5 + 7*T3^4 - 8*T3^3 + 43*T3^2 - 42*T3 + 49 $$T_{7}^{3} + 2T_{7}^{2} - 5T_{7} + 1$$ T7^3 + 2*T7^2 - 5*T7 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - T^{5} + 7 T^{4} - 8 T^{3} + \cdots + 49$$
$5$ $$(T^{2} - T + 1)^{3}$$
$7$ $$(T^{3} + 2 T^{2} - 5 T + 1)^{2}$$
$11$ $$(T^{3} - 5 T^{2} + 2 T + 1)^{2}$$
$13$ $$(T^{2} - 5 T + 25)^{3}$$
$17$ $$T^{6} + T^{5} + 45 T^{4} - 30 T^{3} + \cdots + 49$$
$19$ $$T^{6} - 133T^{3} + 6859$$
$23$ $$T^{6} - 4 T^{5} + 55 T^{4} + \cdots + 2401$$
$29$ $$T^{6} - 2 T^{5} + 9 T^{4} + 12 T^{3} + \cdots + 1$$
$31$ $$(T^{3} - T^{2} - 6 T + 7)^{2}$$
$37$ $$(T^{3} + 2 T^{2} - 119 T - 227)^{2}$$
$41$ $$T^{6} - 2 T^{5} + 47 T^{4} + \cdots + 1369$$
$43$ $$T^{6} + T^{5} + 45 T^{4} + 198 T^{3} + \cdots + 14641$$
$47$ $$T^{6} - 6 T^{5} + 43 T^{4} + \cdots + 2401$$
$53$ $$T^{6} + 11 T^{5} + 163 T^{4} + \cdots + 96721$$
$59$ $$T^{6} - 6 T^{5} + 43 T^{4} + \cdots + 2401$$
$61$ $$T^{6} - 9 T^{5} + 130 T^{4} + \cdots + 2401$$
$67$ $$T^{6} + 20 T^{5} + 292 T^{4} + \cdots + 7744$$
$71$ $$T^{6} + 29 T^{5} + 605 T^{4} + \cdots + 218089$$
$73$ $$T^{6} - 22 T^{5} + 367 T^{4} + \cdots + 5929$$
$79$ $$T^{6} + 24 T^{5} + 460 T^{4} + \cdots + 61504$$
$83$ $$(T^{3} - 3 T^{2} - 54 T - 77)^{2}$$
$89$ $$T^{6} - 14 T^{5} + 232 T^{4} + \cdots + 3136$$
$97$ $$T^{6} + 7 T^{5} + 115 T^{4} + \cdots + 14641$$