# Properties

 Label 390.2.b.c Level $390$ Weight $2$ Character orbit 390.b Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} - \beta_1 q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 - q^3 - q^4 + b1 * q^5 - b1 * q^6 + (b2 - b1) * q^7 - b1 * q^8 + q^9 $$q + \beta_1 q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} - \beta_1 q^{8} + q^{9} - q^{10} + q^{12} - \beta_{3} q^{13} + ( - \beta_{3} + 1) q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_{3} + 1) q^{17} + \beta_1 q^{18} + (\beta_{2} - \beta_1) q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1) q^{21} + ( - \beta_{3} - 5) q^{23} + \beta_1 q^{24} - q^{25} - \beta_{2} q^{26} - q^{27} + ( - \beta_{2} + \beta_1) q^{28} + ( - \beta_{3} + 1) q^{29} + q^{30} + 6 \beta_1 q^{31} + \beta_1 q^{32} + ( - \beta_{2} + \beta_1) q^{34} + ( - \beta_{3} + 1) q^{35} - q^{36} + (2 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{3} + 1) q^{38} + \beta_{3} q^{39} + q^{40} + ( - 2 \beta_{2} - 4 \beta_1) q^{41} + (\beta_{3} - 1) q^{42} + 8 q^{43} + \beta_1 q^{45} + ( - \beta_{2} - 5 \beta_1) q^{46} + (2 \beta_{2} - 2 \beta_1) q^{47} - q^{48} + (2 \beta_{3} - 7) q^{49} - \beta_1 q^{50} + (\beta_{3} - 1) q^{51} + \beta_{3} q^{52} + 6 q^{53} - \beta_1 q^{54} + (\beta_{3} - 1) q^{56} + ( - \beta_{2} + \beta_1) q^{57} + ( - \beta_{2} + \beta_1) q^{58} + (2 \beta_{2} - 2 \beta_1) q^{59} + \beta_1 q^{60} + (2 \beta_{3} - 4) q^{61} - 6 q^{62} + (\beta_{2} - \beta_1) q^{63} - q^{64} - \beta_{2} q^{65} + (2 \beta_{2} + 4 \beta_1) q^{67} + (\beta_{3} - 1) q^{68} + (\beta_{3} + 5) q^{69} + ( - \beta_{2} + \beta_1) q^{70} + ( - 2 \beta_{2} + 2 \beta_1) q^{71} - \beta_1 q^{72} + ( - \beta_{2} - 5 \beta_1) q^{73} + ( - 2 \beta_{3} + 2) q^{74} + q^{75} + ( - \beta_{2} + \beta_1) q^{76} + \beta_{2} q^{78} + 4 \beta_{3} q^{79} + \beta_1 q^{80} + q^{81} + (2 \beta_{3} + 4) q^{82} + (2 \beta_{2} + 10 \beta_1) q^{83} + (\beta_{2} - \beta_1) q^{84} + ( - \beta_{2} + \beta_1) q^{85} + 8 \beta_1 q^{86} + (\beta_{3} - 1) q^{87} + ( - 2 \beta_{2} + 8 \beta_1) q^{89} - q^{90} + (\beta_{2} - 13 \beta_1) q^{91} + (\beta_{3} + 5) q^{92} - 6 \beta_1 q^{93} + ( - 2 \beta_{3} + 2) q^{94} + ( - \beta_{3} + 1) q^{95} - \beta_1 q^{96} + (\beta_{2} + 5 \beta_1) q^{97} + (2 \beta_{2} - 7 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 - q^3 - q^4 + b1 * q^5 - b1 * q^6 + (b2 - b1) * q^7 - b1 * q^8 + q^9 - q^10 + q^12 - b3 * q^13 + (-b3 + 1) * q^14 - b1 * q^15 + q^16 + (-b3 + 1) * q^17 + b1 * q^18 + (b2 - b1) * q^19 - b1 * q^20 + (-b2 + b1) * q^21 + (-b3 - 5) * q^23 + b1 * q^24 - q^25 - b2 * q^26 - q^27 + (-b2 + b1) * q^28 + (-b3 + 1) * q^29 + q^30 + 6*b1 * q^31 + b1 * q^32 + (-b2 + b1) * q^34 + (-b3 + 1) * q^35 - q^36 + (2*b2 - 2*b1) * q^37 + (-b3 + 1) * q^38 + b3 * q^39 + q^40 + (-2*b2 - 4*b1) * q^41 + (b3 - 1) * q^42 + 8 * q^43 + b1 * q^45 + (-b2 - 5*b1) * q^46 + (2*b2 - 2*b1) * q^47 - q^48 + (2*b3 - 7) * q^49 - b1 * q^50 + (b3 - 1) * q^51 + b3 * q^52 + 6 * q^53 - b1 * q^54 + (b3 - 1) * q^56 + (-b2 + b1) * q^57 + (-b2 + b1) * q^58 + (2*b2 - 2*b1) * q^59 + b1 * q^60 + (2*b3 - 4) * q^61 - 6 * q^62 + (b2 - b1) * q^63 - q^64 - b2 * q^65 + (2*b2 + 4*b1) * q^67 + (b3 - 1) * q^68 + (b3 + 5) * q^69 + (-b2 + b1) * q^70 + (-2*b2 + 2*b1) * q^71 - b1 * q^72 + (-b2 - 5*b1) * q^73 + (-2*b3 + 2) * q^74 + q^75 + (-b2 + b1) * q^76 + b2 * q^78 + 4*b3 * q^79 + b1 * q^80 + q^81 + (2*b3 + 4) * q^82 + (2*b2 + 10*b1) * q^83 + (b2 - b1) * q^84 + (-b2 + b1) * q^85 + 8*b1 * q^86 + (b3 - 1) * q^87 + (-2*b2 + 8*b1) * q^89 - q^90 + (b2 - 13*b1) * q^91 + (b3 + 5) * q^92 - 6*b1 * q^93 + (-2*b3 + 2) * q^94 + (-b3 + 1) * q^95 - b1 * q^96 + (b2 + 5*b1) * q^97 + (2*b2 - 7*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 4 q^{17} - 20 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} + 4 q^{38} + 4 q^{40} - 4 q^{42} + 32 q^{43} - 4 q^{48} - 28 q^{49} - 4 q^{51} + 24 q^{53} - 4 q^{56} - 16 q^{61} - 24 q^{62} - 4 q^{64} - 4 q^{68} + 20 q^{69} + 8 q^{74} + 4 q^{75} + 4 q^{81} + 16 q^{82} - 4 q^{87} - 4 q^{90} + 20 q^{92} + 8 q^{94} + 4 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 4 * q^14 + 4 * q^16 + 4 * q^17 - 20 * q^23 - 4 * q^25 - 4 * q^27 + 4 * q^29 + 4 * q^30 + 4 * q^35 - 4 * q^36 + 4 * q^38 + 4 * q^40 - 4 * q^42 + 32 * q^43 - 4 * q^48 - 28 * q^49 - 4 * q^51 + 24 * q^53 - 4 * q^56 - 16 * q^61 - 24 * q^62 - 4 * q^64 - 4 * q^68 + 20 * q^69 + 8 * q^74 + 4 * q^75 + 4 * q^81 + 16 * q^82 - 4 * q^87 - 4 * q^90 + 20 * q^92 + 8 * q^94 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 10\nu ) / 3$$ (v^3 + 10*v) / 3 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 7$$ 2*v^2 + 7
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 7 ) / 2$$ (b3 - 7) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{2} + 5\beta_1$$ -2*b2 + 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 − 1.30278i 2.30278i − 2.30278i 1.30278i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.60555i 1.00000i 1.00000 −1.00000
181.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 4.60555i 1.00000i 1.00000 −1.00000
181.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 4.60555i 1.00000i 1.00000 −1.00000
181.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.60555i 1.00000i 1.00000 −1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.b.c 4
3.b odd 2 1 1170.2.b.d 4
4.b odd 2 1 3120.2.g.q 4
5.b even 2 1 1950.2.b.k 4
5.c odd 4 1 1950.2.f.m 4
5.c odd 4 1 1950.2.f.n 4
13.b even 2 1 inner 390.2.b.c 4
13.d odd 4 1 5070.2.a.z 2
13.d odd 4 1 5070.2.a.bf 2
39.d odd 2 1 1170.2.b.d 4
52.b odd 2 1 3120.2.g.q 4
65.d even 2 1 1950.2.b.k 4
65.h odd 4 1 1950.2.f.m 4
65.h odd 4 1 1950.2.f.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c 4 1.a even 1 1 trivial
390.2.b.c 4 13.b even 2 1 inner
1170.2.b.d 4 3.b odd 2 1
1170.2.b.d 4 39.d odd 2 1
1950.2.b.k 4 5.b even 2 1
1950.2.b.k 4 65.d even 2 1
1950.2.f.m 4 5.c odd 4 1
1950.2.f.m 4 65.h odd 4 1
1950.2.f.n 4 5.c odd 4 1
1950.2.f.n 4 65.h odd 4 1
3120.2.g.q 4 4.b odd 2 1
3120.2.g.q 4 52.b odd 2 1
5070.2.a.z 2 13.d odd 4 1
5070.2.a.bf 2 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{4} + 28T_{7}^{2} + 144$$ T7^4 + 28*T7^2 + 144 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T + 1)^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} + 28T^{2} + 144$$
$11$ $$T^{4}$$
$13$ $$(T^{2} - 13)^{2}$$
$17$ $$(T^{2} - 2 T - 12)^{2}$$
$19$ $$T^{4} + 28T^{2} + 144$$
$23$ $$(T^{2} + 10 T + 12)^{2}$$
$29$ $$(T^{2} - 2 T - 12)^{2}$$
$31$ $$(T^{2} + 36)^{2}$$
$37$ $$T^{4} + 112T^{2} + 2304$$
$41$ $$T^{4} + 136T^{2} + 1296$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4} + 112T^{2} + 2304$$
$53$ $$(T - 6)^{4}$$
$59$ $$T^{4} + 112T^{2} + 2304$$
$61$ $$(T^{2} + 8 T - 36)^{2}$$
$67$ $$T^{4} + 136T^{2} + 1296$$
$71$ $$T^{4} + 112T^{2} + 2304$$
$73$ $$T^{4} + 76T^{2} + 144$$
$79$ $$(T^{2} - 208)^{2}$$
$83$ $$T^{4} + 304T^{2} + 2304$$
$89$ $$T^{4} + 232T^{2} + 144$$
$97$ $$T^{4} + 76T^{2} + 144$$