# Properties

 Label 390.2.bn.a Level $390$ Weight $2$ Character orbit 390.bn Analytic conductor $3.114$ Analytic rank $0$ Dimension $8$ 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.bn (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{4} q^{2} + \zeta_{24} q^{3} + (\zeta_{24}^{4} - 1) q^{4} + (2 \zeta_{24}^{5} + \zeta_{24}^{3} - 2 \zeta_{24}) q^{5} + \zeta_{24}^{5} q^{6} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{2} - 2 \zeta_{24} - 1) q^{7} - q^{8} + \zeta_{24}^{2} q^{9}+O(q^{10})$$ q + z^4 * q^2 + z * q^3 + (z^4 - 1) * q^4 + (2*z^5 + z^3 - 2*z) * q^5 + z^5 * q^6 + (-2*z^7 + 2*z^5 - z^4 - z^2 - 2*z - 1) * q^7 - q^8 + z^2 * q^9 $$q + \zeta_{24}^{4} q^{2} + \zeta_{24} q^{3} + (\zeta_{24}^{4} - 1) q^{4} + (2 \zeta_{24}^{5} + \zeta_{24}^{3} - 2 \zeta_{24}) q^{5} + \zeta_{24}^{5} q^{6} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{2} - 2 \zeta_{24} - 1) q^{7} - q^{8} + \zeta_{24}^{2} q^{9} + (\zeta_{24}^{7} - 2 \zeta_{24}) q^{10} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{2} - \zeta_{24} - 2) q^{11} + (\zeta_{24}^{5} - \zeta_{24}) q^{12} + (3 \zeta_{24}^{5} + 2 \zeta_{24}^{3} - 3 \zeta_{24}) q^{13} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 2 \zeta_{24} + 1) q^{14} + (2 \zeta_{24}^{6} + \zeta_{24}^{4} - 2 \zeta_{24}^{2}) q^{15} - \zeta_{24}^{4} q^{16} + (2 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + \zeta_{24}^{4} - \zeta_{24}^{2} + 1) q^{17} + \zeta_{24}^{6} q^{18} + (\zeta_{24}^{7} - 4 \zeta_{24}^{6} - \zeta_{24}^{5} + 2 \zeta_{24}^{4} - \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 2 \zeta_{24} + 2) q^{19} + (\zeta_{24}^{7} - 2 \zeta_{24}^{5} - \zeta_{24}^{3}) q^{20} + (2 \zeta_{24}^{6} - \zeta_{24}^{5} - 2 \zeta_{24}^{4} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} - \zeta_{24} + 2) q^{21} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} - 2 \zeta_{24}) q^{22} + ( - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{5} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 2 \zeta_{24} + 1) q^{23} - \zeta_{24} q^{24} + ( - 3 \zeta_{24}^{6} - 4) q^{25} + (2 \zeta_{24}^{7} - 3 \zeta_{24}) q^{26} + \zeta_{24}^{3} q^{27} + ( - \zeta_{24}^{6} - 2 \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} + \zeta_{24}^{2} + 2) q^{28} + ( - \zeta_{24}^{7} - 2 \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + \cdots + 4) q^{29} + \cdots + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{4} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 2 \zeta_{24}) q^{99} +O(q^{100})$$ q + z^4 * q^2 + z * q^3 + (z^4 - 1) * q^4 + (2*z^5 + z^3 - 2*z) * q^5 + z^5 * q^6 + (-2*z^7 + 2*z^5 - z^4 - z^2 - 2*z - 1) * q^7 - q^8 + z^2 * q^9 + (z^7 - 2*z) * q^10 + (2*z^7 + 2*z^5 + 2*z^2 - z - 2) * q^11 + (z^5 - z) * q^12 + (3*z^5 + 2*z^3 - 3*z) * q^13 + (-2*z^7 - z^6 - 2*z^4 + 2*z^3 - 2*z + 1) * q^14 + (2*z^6 + z^4 - 2*z^2) * q^15 - z^4 * q^16 + (2*z^6 + 4*z^5 + z^4 - z^2 + 1) * q^17 + z^6 * q^18 + (z^7 - 4*z^6 - z^5 + 2*z^4 - z^3 + 2*z^2 + 2*z + 2) * q^19 + (z^7 - 2*z^5 - z^3) * q^20 + (2*z^6 - z^5 - 2*z^4 - z^3 - 2*z^2 - z + 2) * q^21 + (2*z^7 + 2*z^6 + z^5 - 2*z^4 - 2*z^3 - 2*z) * q^22 + (-3*z^6 - 4*z^5 - 3*z^4 + 2*z^2 + 2*z + 1) * q^23 - z * q^24 + (-3*z^6 - 4) * q^25 + (2*z^7 - 3*z) * q^26 + z^3 * q^27 + (-z^6 - 2*z^5 - z^4 + 2*z^3 + z^2 + 2) * q^28 + (-z^7 - 2*z^6 + 3*z^5 - 2*z^4 - 3*z^3 + 2*z^2 + z + 4) * q^29 + (z^4 - 2*z^2 - 1) * q^30 + (-4*z^7 + 2*z^6 + 3*z^4 + 2*z^3 - 3*z^2 - 2) * q^31 + (-z^4 + 1) * q^32 + (2*z^6 + 2*z^4 + 2*z^3 - z^2 - 2*z - 2) * q^33 + (z^6 + 4*z^5 + 2*z^4 - 2*z^2 - 4*z - 1) * q^34 + (-3*z^7 - 6*z^6 - 3*z^5 + 4*z^4 + z^3 + 2*z^2 + 4*z - 2) * q^35 + (z^6 - z^2) * q^36 + (2*z^7 + 2*z^6 - z^5 - 2*z^4 + z^3 - 2*z^2 - 2*z + 4) * q^37 + (-2*z^6 + z^5 + 4*z^4 - z^3 + 4*z^2 + z - 2) * q^38 + (3*z^6 + 2*z^4 - 3*z^2) * q^39 + (-2*z^5 - z^3 + 2*z) * q^40 + (z^6 - z^4 + 3*z^2 - 4*z + 4) * q^41 + (-z^7 - 2*z^5 - 2*z^2 + z + 2) * q^42 + (-z^7 + 2*z^6 - 2*z^5 + z^3 - 2*z^2 + 4*z - 2) * q^43 + (2*z^6 - z^5 - 2*z^4 - 2*z^3 - 2*z^2 - z + 2) * q^44 + (2*z^7 + z^5 - 2*z^3) * q^45 + (-z^6 - 2*z^5 - 2*z^4 + 3*z^2 + 4*z + 3) * q^46 + (-2*z^7 + 4*z^6 - 2*z^5 - 2*z^4 + 1) * q^47 - z^5 * q^48 + (4*z^7 - 2*z^6 + 5*z^4 - 2*z^2 + 4*z) * q^49 + (-3*z^6 - 4*z^4 + 3*z^2) * q^50 + (2*z^7 + 4*z^6 + z^5 - z^3 + z) * q^51 + (2*z^7 - 3*z^5 - 2*z^3) * q^52 + (-2*z^6 - 8*z^4 + 8*z^2 + 2) * q^53 + z^7 * q^54 + (4*z^7 - 2*z^5 - 3*z^4 - 6*z^3 - 4*z^2 + 4*z - 2) * q^55 + (2*z^7 - 2*z^5 + z^4 + z^2 + 2*z + 1) * q^56 + (-4*z^7 - z^6 + 2*z^5 + 2*z^3 + 2*z^2 + 2*z - 1) * q^57 + (-4*z^7 + 4*z^5 + 2*z^4 + z^3 + 2*z^2 - 3*z + 2) * q^58 + (2*z^7 - 2*z^6 + 3*z^5 + 4*z^4 + 2*z^3 + 6*z^2 - 6) * q^59 + (-2*z^6 - 1) * q^60 + (-3*z^7 + 8*z^6 - 3*z^5 + 2*z^4 - z^3 - 4*z^2 + 4*z - 2) * q^61 + (-2*z^7 - z^6 + z^4 + 4*z^3 - 2*z^2 - 3) * q^62 + (2*z^7 - z^6 - 2*z^5 - z^4 - 2*z^3 - z^2 + 2*z) * q^63 + q^64 + (-4*z^6 - 7) * q^65 + (2*z^7 + z^6 - 2*z^5 - 2*z^2 - 2) * q^66 + (3*z^7 - 4*z^4 + 3*z) * q^67 + (-z^6 + z^4 - z^2 - 4*z - 2) * q^68 + (-3*z^7 - 4*z^6 - 3*z^5 + 2*z^3 + 2*z^2 + z) * q^69 + (-2*z^7 - 4*z^6 + z^5 + 2*z^4 + 3*z^3 + 6*z^2 + 3*z - 4) * q^70 + (-2*z^7 + 4*z^6 + 4*z^5 + 6*z^4 - 2*z^3 + 2*z^2 - 2) * q^71 - z^2 * q^72 + (-2*z^7 + 5*z^6 + 2*z^5 - 10*z^2 - 1) * q^73 + (3*z^7 - 3*z^5 + 2*z^4 - 2*z^3 - 2*z^2 + z + 2) * q^74 + (-3*z^7 - 4*z) * q^75 + (-z^7 + 2*z^6 + 2*z^5 + 2*z^4 + 2*z^2 - z - 4) * q^76 + (2*z^7 - 4*z^6 - 13*z^5 - 2*z^4 - z^3 + 2*z^2 + 13*z + 4) * q^77 + (2*z^4 - 3*z^2 - 2) * q^78 + (2*z^7 - 3*z^6 - 2*z^5 - 8*z^4 - 4*z^3 + 4*z + 4) * q^79 + (-z^7 + 2*z) * q^80 + z^4 * q^81 + (4*z^6 - 4*z^5 + 3*z^4 - z^2 + 1) * q^82 + (-4*z^7 + 4*z^6 - 7*z^5 - 8*z^4 - 3*z^3 + 3*z + 4) * q^83 + (-z^7 - 2*z^6 - z^5 + 2*z^4 + z^3 + 2*z) * q^84 + (-z^7 + 3*z^5 + 4*z^4 - z^3 - 8*z^2 - 6*z - 4) * q^85 + (2*z^5 - 2*z^4 + z^3 - 2*z^2 + 2*z) * q^86 + (-2*z^7 + 3*z^6 - 2*z^5 - 4*z^4 + 2*z^3 + z^2 + 4*z + 1) * q^87 + (-2*z^7 - 2*z^5 - 2*z^2 + z + 2) * q^88 + (-2*z^7 - 5*z^6 + 5*z^4 + 4*z^3 + 3*z^2 - 4*z - 2) * q^89 + (z^5 - 2*z^3 - z) * q^90 + (-5*z^7 - 10*z^6 - 5*z^5 + 6*z^4 + z^3 + 4*z^2 + 6*z - 4) * q^91 + (2*z^6 + 2*z^5 + z^4 + z^2 + 2*z + 2) * q^92 + (2*z^7 + 3*z^5 - 2*z^4 - 3*z^3 - 2*z + 4) * q^93 + (-2*z^7 + 4*z^6 - 2*z^5 - z^4 + 2*z^3 - 4*z^2 + 2*z + 2) * q^94 + (6*z^7 + 4*z^6 + 2*z^5 - z^4 + 6*z^3 - 3*z^2 - 4*z + 3) * q^95 + (-z^5 + z) * q^96 + (2*z^7 + 2*z^6 + 2*z^5 + z^4 + 2*z^3 - z^2 - 4*z - 1) * q^97 + (4*z^7 - 4*z^6 + 4*z^5 + 5*z^4 - 4*z^3 + 2*z^2 - 5) * q^98 + (2*z^7 + 2*z^5 + 2*z^4 - z^3 - 2*z^2 - 2*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 4 q^{4} - 12 q^{7} - 8 q^{8}+O(q^{10})$$ 8 * q + 4 * q^2 - 4 * q^4 - 12 * q^7 - 8 * q^8 $$8 q + 4 q^{2} - 4 q^{4} - 12 q^{7} - 8 q^{8} - 16 q^{11} + 4 q^{15} - 4 q^{16} + 12 q^{17} + 24 q^{19} + 8 q^{21} - 8 q^{22} - 4 q^{23} - 32 q^{25} + 12 q^{28} + 24 q^{29} - 4 q^{30} - 4 q^{31} + 4 q^{32} - 8 q^{33} + 24 q^{37} + 8 q^{39} + 28 q^{41} + 16 q^{42} - 16 q^{43} + 8 q^{44} + 16 q^{46} + 20 q^{49} - 16 q^{50} - 16 q^{53} - 28 q^{55} + 12 q^{56} - 8 q^{57} + 24 q^{58} - 32 q^{59} - 8 q^{60} - 8 q^{61} - 20 q^{62} - 4 q^{63} + 8 q^{64} - 56 q^{65} - 16 q^{66} - 16 q^{67} - 12 q^{68} - 24 q^{70} + 8 q^{71} - 8 q^{73} + 24 q^{74} - 24 q^{76} + 24 q^{77} - 8 q^{78} + 4 q^{81} + 20 q^{82} + 8 q^{84} - 16 q^{85} - 8 q^{86} - 8 q^{87} + 16 q^{88} + 4 q^{89} - 8 q^{91} + 20 q^{92} + 24 q^{93} + 12 q^{94} + 20 q^{95} - 4 q^{97} - 20 q^{98} + 8 q^{99}+O(q^{100})$$ 8 * q + 4 * q^2 - 4 * q^4 - 12 * q^7 - 8 * q^8 - 16 * q^11 + 4 * q^15 - 4 * q^16 + 12 * q^17 + 24 * q^19 + 8 * q^21 - 8 * q^22 - 4 * q^23 - 32 * q^25 + 12 * q^28 + 24 * q^29 - 4 * q^30 - 4 * q^31 + 4 * q^32 - 8 * q^33 + 24 * q^37 + 8 * q^39 + 28 * q^41 + 16 * q^42 - 16 * q^43 + 8 * q^44 + 16 * q^46 + 20 * q^49 - 16 * q^50 - 16 * q^53 - 28 * q^55 + 12 * q^56 - 8 * q^57 + 24 * q^58 - 32 * q^59 - 8 * q^60 - 8 * q^61 - 20 * q^62 - 4 * q^63 + 8 * q^64 - 56 * q^65 - 16 * q^66 - 16 * q^67 - 12 * q^68 - 24 * q^70 + 8 * q^71 - 8 * q^73 + 24 * q^74 - 24 * q^76 + 24 * q^77 - 8 * q^78 + 4 * q^81 + 20 * q^82 + 8 * q^84 - 16 * q^85 - 8 * q^86 - 8 * q^87 + 16 * q^88 + 4 * q^89 - 8 * q^91 + 20 * q^92 + 24 * q^93 + 12 * q^94 + 20 * q^95 - 4 * q^97 - 20 * q^98 + 8 * q^99

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$\zeta_{24}^{6}$$ $$\zeta_{24}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 0.707107 2.12132i −0.258819 0.965926i −1.46945 0.848387i −1.00000 0.866025 + 0.500000i 2.19067 0.448288i
67.2 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −0.707107 + 2.12132i 0.258819 + 0.965926i −3.26260 1.88366i −1.00000 0.866025 + 0.500000i −2.19067 + 0.448288i
97.1 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i −0.707107 + 2.12132i −0.965926 0.258819i −3.98004 + 2.29788i −1.00000 −0.866025 + 0.500000i 1.48356 + 1.67303i
97.2 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 0.707107 2.12132i 0.965926 + 0.258819i 2.71209 1.56583i −1.00000 −0.866025 + 0.500000i −1.48356 1.67303i
163.1 0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 0.707107 + 2.12132i −0.258819 + 0.965926i −1.46945 + 0.848387i −1.00000 0.866025 0.500000i 2.19067 + 0.448288i
163.2 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i −0.707107 2.12132i 0.258819 0.965926i −3.26260 + 1.88366i −1.00000 0.866025 0.500000i −2.19067 0.448288i
193.1 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i −0.707107 2.12132i −0.965926 + 0.258819i −3.98004 2.29788i −1.00000 −0.866025 0.500000i 1.48356 1.67303i
193.2 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i 0.707107 + 2.12132i 0.965926 0.258819i 2.71209 + 1.56583i −1.00000 −0.866025 0.500000i −1.48356 + 1.67303i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bn.a yes 8
5.c odd 4 1 390.2.bd.a 8
13.f odd 12 1 390.2.bd.a 8
65.o even 12 1 inner 390.2.bn.a yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.a 8 5.c odd 4 1
390.2.bd.a 8 13.f odd 12 1
390.2.bn.a yes 8 1.a even 1 1 trivial
390.2.bn.a yes 8 65.o even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 12T_{7}^{7} + 48T_{7}^{6} - 388T_{7}^{4} + 4800T_{7}^{2} + 11040T_{7} + 8464$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{4}$$
$3$ $$T^{8} - T^{4} + 1$$
$5$ $$(T^{4} + 8 T^{2} + 25)^{2}$$
$7$ $$T^{8} + 12 T^{7} + 48 T^{6} + \cdots + 8464$$
$11$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 36481$$
$13$ $$(T^{4} + 24 T^{2} + 169)^{2}$$
$17$ $$T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 10000$$
$19$ $$T^{8} - 24 T^{7} + 300 T^{6} + \cdots + 150544$$
$23$ $$T^{8} + 4 T^{7} + 38 T^{6} + \cdots + 5041$$
$29$ $$T^{8} - 24 T^{7} + 204 T^{6} + \cdots + 2211169$$
$31$ $$T^{8} + 4 T^{7} + 8 T^{6} - 92 T^{5} + \cdots + 529$$
$37$ $$T^{8} - 24 T^{7} + 228 T^{6} + \cdots + 529$$
$41$ $$T^{8} - 28 T^{7} + 344 T^{6} + \cdots + 8464$$
$43$ $$T^{8} + 16 T^{7} + 80 T^{6} + \cdots + 625$$
$47$ $$T^{8} + 108 T^{6} + 3830 T^{4} + \cdots + 96721$$
$53$ $$(T^{4} + 8 T^{3} + 32 T^{2} - 704 T + 7744)^{2}$$
$59$ $$T^{8} + 32 T^{7} + 464 T^{6} + \cdots + 3411409$$
$61$ $$T^{8} + 8 T^{7} + 188 T^{6} + \cdots + 234256$$
$67$ $$(T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4)^{2}$$
$71$ $$T^{8} - 8 T^{7} + 128 T^{6} + \cdots + 37552384$$
$73$ $$(T^{4} + 4 T^{3} - 160 T^{2} - 568 T + 4036)^{2}$$
$79$ $$T^{8} + 324 T^{6} + 34470 T^{4} + \cdots + 7834401$$
$83$ $$T^{8} + 552 T^{6} + \cdots + 238764304$$
$89$ $$T^{8} - 4 T^{7} + 200 T^{6} + \cdots + 10000$$
$97$ $$T^{8} + 4 T^{7} + 64 T^{6} + \cdots + 150544$$