# Properties

 Label 390.2.x.b Level $390$ Weight $2$ Character orbit 390.x Analytic conductor $3.114$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(49,390)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("390.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.x (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197$$ x^12 - 2*x^11 - 8*x^10 + 34*x^9 + 8*x^8 - 134*x^7 + 98*x^6 + 154*x^5 + 104*x^4 + 190*x^3 - 1196*x^2 - 338*x + 2197 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + 1) q^{2} - \beta_{4} q^{3} + \beta_{6} q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{5}+ \cdots - \beta_{6} q^{9}+O(q^{10})$$ q + (b6 + 1) * q^2 - b4 * q^3 + b6 * q^4 + (-b11 + b10 + b9 - b7 + b5 - 2*b4 + b3) * q^5 + (-b7 - b4) * q^6 + (b9 + b7 - b4 - b2) * q^7 - q^8 - b6 * q^9 $$q + (\beta_{6} + 1) q^{2} - \beta_{4} q^{3} + \beta_{6} q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{5}+ \cdots + (\beta_{11} - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100})$$ q + (b6 + 1) * q^2 - b4 * q^3 + b6 * q^4 + (-b11 + b10 + b9 - b7 + b5 - 2*b4 + b3) * q^5 + (-b7 - b4) * q^6 + (b9 + b7 - b4 - b2) * q^7 - q^8 - b6 * q^9 + (-b11 + b9 - b7 + b5 - b4 + b1) * q^10 + (b9 - b8 + b5 - b2) * q^11 - b7 * q^12 + (-b11 + b5 + 2*b1) * q^13 + (-b10 - b4 - b3 - b2) * q^14 + (b10 + b9 - b8 + b5 - b4 + b1) * q^15 + (-b6 - 1) * q^16 + (-b11 + 2*b9 - 3*b8 - b7 + 2*b5 - 3*b4 - 2*b3 - b2 + 3*b1 - 4) * q^17 + q^18 + (-2*b11 - b9 + 2*b8 - b7 - b4 + b2 - 2*b1) * q^19 + (-b10 + b4 - b3 + b1) * q^20 + (-b11 + b10 + b9 - b6 + b5 - b4 + b1 - 1) * q^21 + (-b11 + b9 - b8 + b7 + b6 + b5 - b3 + b1) * q^22 + (-2*b10 - 2*b9 + b7 - 3*b5 - b3 + b2 + b1 - 1) * q^23 + b4 * q^24 + (2*b11 + b10 + b9 - 2*b7 + b6 + b5 - 2*b4 + 2*b3 - b2 - b1 + 1) * q^25 + (b11 + b5 - b2 + b1) * q^26 + b7 * q^27 + (-b10 - b9 - b7 - b3) * q^28 + (-2*b10 - 3*b9 + b8 + b7 + b6 + 2*b4 - b3 - b1 + 2) * q^29 + (b10 + b9 + b6 - b4 + 1) * q^30 + (3*b11 - 2*b10 - b9 - b8 - b7 + b6 - 3*b5 + 2*b4 - b3 - b2 - 2*b1 + 1) * q^31 - b6 * q^32 + (b10 + b6 + b5 - b4) * q^33 + (-b11 + b10 + b9 - b8 - b7 - b6 + b5 - b4 - 2*b3 + 2*b1 - 2) * q^34 + (2*b10 + b9 - b7 - 2*b6 + 2*b5 + b4 + 2*b3 + b2 - 2) * q^35 + (b6 + 1) * q^36 + (b10 + 2*b9 + b8 + 3*b6 - 2*b5 + 2*b3 + 2*b2 - b1 + 4) * q^37 + (-2*b11 - b10 - 2*b9 + 2*b8 - 2*b7 - 2*b6 + b4 + b3 + 3*b2) * q^38 + (b11 - 2*b5) * q^39 + (b11 - b10 - b9 + b7 - b5 + 2*b4 - b3) * q^40 + (-b10 + b9 - 2*b8 + b6 + b5 - 2*b4 - b2 + b1 - 2) * q^41 + (b8 - b6 + 1) * q^42 + (b11 - 2*b9 + b8 + 2*b7 + b6 - b5 + 3*b4 + b3 - b2 - b1 + 4) * q^43 + (-b11 + b7 + b6 - b3 + b2 + b1) * q^44 + (b10 - b4 + b3 - b1) * q^45 + (b11 - b9 - b8 - 2*b7 - b6 - 2*b4 - 2*b2 + b1 - 2) * q^46 + (-b11 + b10 - b9 + b8 + 3*b7 + b6 - b5 + 5*b4 - 2*b1 - 1) * q^47 + (b7 + b4) * q^48 + (-4*b11 + 3*b10 + 3*b9 + b7 + 3*b6 + 2*b5 - 2*b4 + b3 + 2*b2 + 2*b1 + 1) * q^49 + (b11 + 2*b9 - b8 - b7 + b6 - b4 - b2 - b1 - 1) * q^50 + (2*b10 - b9 + 2*b8 - b5 + 2*b4 + 2*b3 + 2*b2 - 2*b1 + 3) * q^51 + (2*b11 - b2 - b1) * q^52 + (-b11 - 2*b10 - 5*b9 + b8 - b7 - 5*b6 + b5 + 2*b4 - b3 + 3*b2 - 2) * q^53 - b4 * q^54 + (2*b11 - b10 - b9 + 2*b8 - 2*b7 - 4*b6 - b5 + 2*b3 - b2 - 3*b1 + 1) * q^55 + (-b9 - b7 + b4 + b2) * q^56 + (b11 - b10 + b9 - b6 + b5 + b4 + b1 + 1) * q^57 + (-2*b9 - b7 + b6 + b5 + b3 + 2*b2) * q^58 + (3*b11 - 2*b10 - b8 + 6*b7 - 3*b6 - b5 + 7*b4 - b3 - b2 + b1 - 4) * q^59 + (b8 + b6 - b5 - b1 + 1) * q^60 + (b11 - b8 + 3*b7 - 2*b5 - b4 - 2*b3 - b2 - b1) * q^61 + (-2*b8 + b7 + 2*b6 - b5 + 2*b4 - b3 - b2 - b1 - 1) * q^62 + (b10 + b9 + b7 + b3) * q^63 + q^64 + (4*b9 - 4*b8 + b7 + 3*b6 + b5 - b3 - 3*b2 + 3*b1 - 5) * q^65 + (b10 + b8 - b7 - b4 + b3 + b2 - b1) * q^66 + (4*b11 - 2*b10 - 2*b9 + 2*b8 - 2*b7 - 2*b6 - 4*b5 + 2*b4 + 2*b3 - 4*b1 + 2) * q^67 + (b10 - b9 + 2*b8 - b6 - b5 + 2*b4 + b2 - b1 + 2) * q^68 + (-b11 - 2*b10 - 2*b9 + b8 - 3*b6 - 3*b5 + 3*b4 - b3 + 3*b2 + b1) * q^69 + (b10 + 2*b9 + 2*b7 - 2*b6 + b4 + b3 + b2) * q^70 + (-2*b10 - 4*b9 + 4*b8 + 6*b7 - 2*b6 - 2*b5 + 8*b4 - 4*b1) * q^71 + b6 * q^72 + (-b11 + b10 - 3*b9 + 5*b8 - b7 + b6 - 3*b5 - 3*b4 - 6*b1) * q^73 + (-2*b10 + b9 + 2*b7 + 3*b6 - 3*b5 + b4 - b3 - b2) * q^74 + (b10 + b9 - 2*b8 - b7 + b6 + 2*b5 - 2*b4 - b3 - 3*b2) * q^75 + (-b10 - b9 - b7 - 2*b6 + 2*b4 + b3 + 2*b2 + 2*b1) * q^76 + (2*b11 - b10 - 2*b9 - 2*b7 - 2*b6 + b4 + b3 - b2 - 2*b1) * q^77 + (b11 - b5 - 2*b2 - b1) * q^78 + (-b11 + 4*b10 - b9 + 3*b8 - 3*b7 + b6 - b5 - 4*b4 + 3*b3 + 3*b2 - 4*b1 + 3) * q^79 + (b11 - b9 + b7 - b5 + b4 - b1) * q^80 + (-b6 - 1) * q^81 + (-b11 + 2*b9 - 3*b8 - b7 + 2*b5 - 3*b4 - 2*b3 - b2 + 3*b1 - 4) * q^82 + (2*b10 - 2*b9 - 2*b5 + 2*b4 + 2*b3 + 2*b2 - 6) * q^83 + (b11 - b10 - b9 + b8 - b5 + b4 - b1 + 2) * q^84 + (5*b11 - b10 + b9 - 3*b8 + b7 + b6 - 3*b5 + 5*b4 - 2*b3 - 4*b2 - 2*b1 + 4) * q^85 + (b11 + b10 + b7 + 3*b6 + 2*b5 - b4 + 2*b3 - 2*b2 - b1 + 2) * q^86 + (3*b11 - 3*b10 - 2*b9 + b8 - b7 - 2*b6 - 2*b5 + b4 - b3 - b2 - b1) * q^87 + (-b9 + b8 - b5 + b2) * q^88 + (-5*b10 - b9 - 4*b8 + b7 - 2*b6 - 2*b5 - b3 + 4*b1 - 2) * q^89 + (-b11 + b10 + b9 - b7 + b5 - 2*b4 + b3) * q^90 + (-b10 + 3*b9 + 5*b7 - 2*b5 + 4*b4 - b3 + 2*b1) * q^91 + (b11 + 2*b10 + b9 - b8 - 3*b7 - b6 + 3*b5 - 2*b4 + b3 - 3*b2 - 1) * q^92 + (-2*b11 - b10 - 2*b9 + b8 - b7 + 2*b6 + b5 - b3 + b2 + 2) * q^93 + (-2*b11 + b10 - b9 + 2*b8 + 4*b7 - 2*b6 + b5 + 2*b4 + 2*b3 + b2 - b1 - 1) * q^94 + (2*b11 - 2*b10 - b9 + 2*b8 + 5*b7 + 2*b6 - 2*b5 + b4 + 2*b3 + 3*b2 - 2*b1 + 6) * q^95 + b7 * q^96 + (-2*b11 + 4*b10 + 2*b9 + 2*b8 - 8*b6 + 4*b5 - 4*b4 + 2*b1) * q^97 + (-2*b11 + b9 + 2*b8 + b7 + b6 - b4 + b2 + 2*b1) * q^98 + (b11 - b7 - b6 + b3 - b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{2} - 6 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} + 6 q^{9}+O(q^{10})$$ 12 * q + 6 * q^2 - 6 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 + 6 * q^9 $$12 q + 6 q^{2} - 6 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} + 6 q^{9} + 4 q^{10} + 6 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{15} - 6 q^{16} - 18 q^{17} + 12 q^{18} - 6 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} - 10 q^{25} - 2 q^{26} + 2 q^{28} + 14 q^{29} + 6 q^{30} + 6 q^{32} - 6 q^{33} - 22 q^{35} + 6 q^{36} + 12 q^{37} - 2 q^{39} - 2 q^{40} - 18 q^{41} + 12 q^{42} + 36 q^{43} - 2 q^{45} - 6 q^{46} - 16 q^{47} + 8 q^{49} - 20 q^{50} + 16 q^{51} - 10 q^{52} + 8 q^{55} - 2 q^{56} + 8 q^{57} - 14 q^{58} - 36 q^{59} + 10 q^{61} - 6 q^{62} - 2 q^{63} + 12 q^{64} - 44 q^{65} - 12 q^{66} - 4 q^{67} + 18 q^{68} + 16 q^{69} + 4 q^{70} - 12 q^{71} - 6 q^{72} - 28 q^{73} - 12 q^{74} + 16 q^{75} + 6 q^{76} + 2 q^{78} + 4 q^{79} - 4 q^{80} - 6 q^{81} - 18 q^{82} - 72 q^{83} + 12 q^{84} + 48 q^{85} - 6 q^{87} - 6 q^{88} + 18 q^{89} + 2 q^{90} + 2 q^{91} + 16 q^{93} - 8 q^{94} + 18 q^{95} + 48 q^{97} - 8 q^{98}+O(q^{100})$$ 12 * q + 6 * q^2 - 6 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 + 6 * q^9 + 4 * q^10 + 6 * q^11 + 8 * q^13 + 4 * q^14 + 6 * q^15 - 6 * q^16 - 18 * q^17 + 12 * q^18 - 6 * q^19 + 2 * q^20 + 6 * q^22 - 6 * q^23 - 10 * q^25 - 2 * q^26 + 2 * q^28 + 14 * q^29 + 6 * q^30 + 6 * q^32 - 6 * q^33 - 22 * q^35 + 6 * q^36 + 12 * q^37 - 2 * q^39 - 2 * q^40 - 18 * q^41 + 12 * q^42 + 36 * q^43 - 2 * q^45 - 6 * q^46 - 16 * q^47 + 8 * q^49 - 20 * q^50 + 16 * q^51 - 10 * q^52 + 8 * q^55 - 2 * q^56 + 8 * q^57 - 14 * q^58 - 36 * q^59 + 10 * q^61 - 6 * q^62 - 2 * q^63 + 12 * q^64 - 44 * q^65 - 12 * q^66 - 4 * q^67 + 18 * q^68 + 16 * q^69 + 4 * q^70 - 12 * q^71 - 6 * q^72 - 28 * q^73 - 12 * q^74 + 16 * q^75 + 6 * q^76 + 2 * q^78 + 4 * q^79 - 4 * q^80 - 6 * q^81 - 18 * q^82 - 72 * q^83 + 12 * q^84 + 48 * q^85 - 6 * q^87 - 6 * q^88 + 18 * q^89 + 2 * q^90 + 2 * q^91 + 16 * q^93 - 8 * q^94 + 18 * q^95 + 48 * q^97 - 8 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 203419 \nu^{11} - 163110633 \nu^{10} + 591783880 \nu^{9} + 97338749 \nu^{8} + \cdots + 81183629852 ) / 63907274600$$ (-203419*v^11 - 163110633*v^10 + 591783880*v^9 + 97338749*v^8 - 4513282461*v^7 + 6489146722*v^6 + 4655211661*v^5 - 5740233327*v^4 - 8165110694*v^3 - 33307875431*v^2 + 64277898945*v + 81183629852) / 63907274600 $$\beta_{3}$$ $$=$$ $$( 60968787 \nu^{11} - 2097063441 \nu^{10} + 2300362460 \nu^{9} + 17147379373 \nu^{8} + \cdots + 1269272187404 ) / 830794569800$$ (60968787*v^11 - 2097063441*v^10 + 2300362460*v^9 + 17147379373*v^8 - 55879543947*v^7 - 372787906*v^6 + 135156205247*v^5 - 272345430479*v^4 + 414518069862*v^3 - 554153957987*v^2 - 833086363785*v + 1269272187404) / 830794569800 $$\beta_{4}$$ $$=$$ $$( - 120243408 \nu^{11} + 454030419 \nu^{10} + 2051209685 \nu^{9} - 7036618932 \nu^{8} + \cdots + 618192439839 ) / 830794569800$$ (-120243408*v^11 + 454030419*v^10 + 2051209685*v^9 - 7036618932*v^8 + 2513656023*v^7 + 29967292429*v^6 - 18436259198*v^5 + 35262141211*v^4 + 8309942667*v^3 + 28687118858*v^2 + 98318215515*v + 618192439839) / 830794569800 $$\beta_{5}$$ $$=$$ $$( 16426431 \nu^{11} + 83789417 \nu^{10} - 226795620 \nu^{9} + 267354099 \nu^{8} + \cdots + 20321135952 ) / 63907274600$$ (16426431*v^11 + 83789417*v^10 - 226795620*v^9 + 267354099*v^8 + 1065744289*v^7 - 511723478*v^6 + 4136894311*v^5 + 1601173623*v^4 + 3964105106*v^3 - 3499453881*v^2 + 44426935995*v + 20321135952) / 63907274600 $$\beta_{6}$$ $$=$$ $$( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200$$ (4036*v^11 - 37023*v^10 + 57555*v^9 + 233294*v^8 - 817691*v^7 + 35557*v^6 + 1844066*v^5 - 940887*v^4 + 77961*v^3 - 7481036*v^2 - 4439955*v + 11867687) / 11796200 $$\beta_{7}$$ $$=$$ $$( - 20638 \nu^{11} + 28887 \nu^{10} - 18833 \nu^{9} - 91862 \nu^{8} - 291763 \nu^{7} + \cdots + 21764327 ) / 47610004$$ (-20638*v^11 + 28887*v^10 - 18833*v^9 - 91862*v^8 - 291763*v^7 - 1390803*v^6 + 3193024*v^5 - 2792113*v^4 - 7615595*v^3 - 12957312*v^2 + 2476201*v + 21764327) / 47610004 $$\beta_{8}$$ $$=$$ $$( 5665538 \nu^{11} - 7145579 \nu^{10} - 9226575 \nu^{9} + 110201552 \nu^{8} - 247897203 \nu^{7} + \cdots - 40106339 ) / 12781454920$$ (5665538*v^11 - 7145579*v^10 - 9226575*v^9 + 110201552*v^8 - 247897203*v^7 + 32851611*v^6 + 1496635908*v^5 - 1228577871*v^4 - 1963928377*v^3 + 3156976042*v^2 + 8732837145*v - 40106339) / 12781454920 $$\beta_{9}$$ $$=$$ $$( - 680246389 \nu^{11} - 845264698 \nu^{10} + 11881898255 \nu^{9} - 20291622981 \nu^{8} + \cdots + 1108736562037 ) / 830794569800$$ (-680246389*v^11 - 845264698*v^10 + 11881898255*v^9 - 20291622981*v^8 - 56431077566*v^7 + 138431742257*v^6 - 46377269759*v^5 - 85505629812*v^4 - 58320816489*v^3 - 633066289511*v^2 + 550784663520*v + 1108736562037) / 830794569800 $$\beta_{10}$$ $$=$$ $$( - 1202455216 \nu^{11} + 4870077513 \nu^{10} - 1280213355 \nu^{9} - 35058222714 \nu^{8} + \cdots - 1182431604497 ) / 830794569800$$ (-1202455216*v^11 + 4870077513*v^10 - 1280213355*v^9 - 35058222714*v^8 + 75140023171*v^7 - 22116401667*v^6 - 93428783546*v^5 + 206023103197*v^4 - 574822497041*v^3 + 859741244016*v^2 + 783010043855*v - 1182431604497) / 830794569800 $$\beta_{11}$$ $$=$$ $$( - 78524401 \nu^{11} + 161812093 \nu^{10} + 556127095 \nu^{9} - 2253760279 \nu^{8} + \cdots + 16572567908 ) / 31953637300$$ (-78524401*v^11 + 161812093*v^10 + 556127095*v^9 - 2253760279*v^8 - 695335944*v^7 + 7168384238*v^6 - 1904719281*v^5 - 3795941083*v^4 - 26424465301*v^3 - 15605355549*v^2 + 99935137380*v + 16572567908) / 31953637300
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + 2\beta_{10} + 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{5} - 3\beta_{4} + \beta_{3} + 2$$ -b11 + 2*b10 + 2*b9 + b8 - 2*b7 + b5 - 3*b4 + b3 + 2 $$\nu^{3}$$ $$=$$ $$- 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \cdots - 5$$ -2*b11 + b10 + 3*b9 + b8 + 2*b7 - b6 + b5 + 4*b4 + b3 - b2 + 3*b1 - 5 $$\nu^{4}$$ $$=$$ $$-7\beta_{11} + 7\beta_{10} + 12\beta_{9} + \beta_{8} + 2\beta_{7} + 9\beta_{5} - 6\beta_{4} + \beta_{2} - 3$$ -7*b11 + 7*b10 + 12*b9 + b8 + 2*b7 + 9*b5 - 6*b4 + b2 - 3 $$\nu^{5}$$ $$=$$ $$- 4 \beta_{11} - 5 \beta_{10} - 8 \beta_{9} + 4 \beta_{8} + 29 \beta_{7} - 12 \beta_{6} - 7 \beta_{5} + \cdots - 42$$ -4*b11 - 5*b10 - 8*b9 + 4*b8 + 29*b7 - 12*b6 - 7*b5 + 47*b4 - b3 + 12*b2 - 42 $$\nu^{6}$$ $$=$$ $$- 6 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 24 \beta_{6} + 18 \beta_{5} - 20 \beta_{4} + \cdots - 7$$ -6*b11 + 8*b10 - 2*b9 - 10*b8 - 24*b6 + 18*b5 - 20*b4 + 6*b3 + 38*b2 - 32*b1 - 7 $$\nu^{7}$$ $$=$$ $$50 \beta_{11} - 70 \beta_{10} - 162 \beta_{9} - 50 \beta_{8} + 54 \beta_{7} - 36 \beta_{6} - 90 \beta_{5} + \cdots - 104$$ 50*b11 - 70*b10 - 162*b9 - 50*b8 + 54*b7 - 36*b6 - 90*b5 + 202*b4 - 8*b3 + 76*b2 - 11*b1 - 104 $$\nu^{8}$$ $$=$$ $$81 \beta_{11} + 80 \beta_{10} - 90 \beta_{9} - 155 \beta_{8} - 382 \beta_{7} + 114 \beta_{6} + \cdots + 388$$ 81*b11 + 80*b10 - 90*b9 - 155*b8 - 382*b7 + 114*b6 + 113*b5 - 491*b4 + 73*b3 + 64*b2 - 112*b1 + 388 $$\nu^{9}$$ $$=$$ $$404 \beta_{11} - 223 \beta_{10} - 627 \beta_{9} - 213 \beta_{8} - 540 \beta_{7} + 459 \beta_{6} + \cdots + 249$$ 404*b11 - 223*b10 - 627*b9 - 213*b8 - 540*b7 + 459*b6 - 365*b5 + 328*b4 + 7*b3 - 231*b2 + 163*b1 + 249 $$\nu^{10}$$ $$=$$ $$327 \beta_{11} + 649 \beta_{10} + 866 \beta_{9} - 57 \beta_{8} - 2616 \beta_{7} + 1480 \beta_{6} + \cdots + 2981$$ 327*b11 + 649*b10 + 866*b9 - 57*b8 - 2616*b7 + 1480*b6 + 1181*b5 - 3528*b4 + 248*b3 - 1421*b2 - 202*b1 + 2981 $$\nu^{11}$$ $$=$$ $$1120 \beta_{11} - 2139 \beta_{10} - 614 \beta_{9} + 1672 \beta_{8} + 195 \beta_{7} + 1784 \beta_{6} + \cdots - 288$$ 1120*b11 - 2139*b10 - 614*b9 + 1672*b8 + 195*b7 + 1784*b6 - 1755*b5 + 3931*b4 - 1427*b3 - 3608*b2 + 1392*b1 - 288

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{6}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.40719 − 0.536449i −1.44229 + 0.433312i −0.330925 + 1.46916i 2.00607 − 1.30680i 1.75374 + 1.62986i −2.39378 + 0.0429626i 1.40719 + 0.536449i −1.44229 − 0.433312i −0.330925 − 1.46916i 2.00607 + 1.30680i 1.75374 − 1.62986i −2.39378 − 0.0429626i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −2.03420 0.928463i −0.866025 0.500000i −1.40247 + 2.42916i −1.00000 0.500000 0.866025i −0.213026 2.22590i
49.2 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0.230377 + 2.22417i −0.866025 0.500000i −0.432713 + 0.749482i −1.00000 0.500000 0.866025i −1.81100 + 1.31160i
49.3 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0.571769 2.16173i −0.866025 0.500000i 0.603137 1.04466i −1.00000 0.500000 0.866025i 2.15800 0.585699i
49.4 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i −1.26873 1.84128i 0.866025 + 0.500000i 2.17283 3.76344i −1.00000 0.500000 0.866025i 0.960230 2.01940i
49.5 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 1.40066 + 1.74303i 0.866025 + 0.500000i −0.763837 + 1.32301i −1.00000 0.500000 0.866025i −0.809179 + 2.08452i
49.6 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 2.10012 0.767774i 0.866025 + 0.500000i 0.823063 1.42559i −1.00000 0.500000 0.866025i 1.71497 + 1.43487i
199.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i −2.03420 + 0.928463i −0.866025 + 0.500000i −1.40247 2.42916i −1.00000 0.500000 + 0.866025i −0.213026 + 2.22590i
199.2 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0.230377 2.22417i −0.866025 + 0.500000i −0.432713 0.749482i −1.00000 0.500000 + 0.866025i −1.81100 1.31160i
199.3 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0.571769 + 2.16173i −0.866025 + 0.500000i 0.603137 + 1.04466i −1.00000 0.500000 + 0.866025i 2.15800 + 0.585699i
199.4 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i −1.26873 + 1.84128i 0.866025 0.500000i 2.17283 + 3.76344i −1.00000 0.500000 + 0.866025i 0.960230 + 2.01940i
199.5 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 1.40066 1.74303i 0.866025 0.500000i −0.763837 1.32301i −1.00000 0.500000 + 0.866025i −0.809179 2.08452i
199.6 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 2.10012 + 0.767774i 0.866025 0.500000i 0.823063 + 1.42559i −1.00000 0.500000 + 0.866025i 1.71497 1.43487i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.6 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.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.x.b yes 12
3.b odd 2 1 1170.2.bj.c 12
5.b even 2 1 390.2.x.a 12
5.c odd 4 1 1950.2.bc.i 12
5.c odd 4 1 1950.2.bc.j 12
13.e even 6 1 390.2.x.a 12
15.d odd 2 1 1170.2.bj.d 12
39.h odd 6 1 1170.2.bj.d 12
65.l even 6 1 inner 390.2.x.b yes 12
65.r odd 12 1 1950.2.bc.i 12
65.r odd 12 1 1950.2.bc.j 12
195.y odd 6 1 1170.2.bj.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 5.b even 2 1
390.2.x.a 12 13.e even 6 1
390.2.x.b yes 12 1.a even 1 1 trivial
390.2.x.b yes 12 65.l even 6 1 inner
1170.2.bj.c 12 3.b odd 2 1
1170.2.bj.c 12 195.y odd 6 1
1170.2.bj.d 12 15.d odd 2 1
1170.2.bj.d 12 39.h odd 6 1
1950.2.bc.i 12 5.c odd 4 1
1950.2.bc.i 12 65.r odd 12 1
1950.2.bc.j 12 5.c odd 4 1
1950.2.bc.j 12 65.r odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} - 2 T_{7}^{11} + 19 T_{7}^{10} + 6 T_{7}^{9} + 205 T_{7}^{8} - 20 T_{7}^{7} + 708 T_{7}^{6} + \cdots + 1024$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{6}$$
$3$ $$(T^{4} - T^{2} + 1)^{3}$$
$5$ $$T^{12} - 2 T^{11} + \cdots + 15625$$
$7$ $$T^{12} - 2 T^{11} + \cdots + 1024$$
$11$ $$T^{12} - 6 T^{11} + \cdots + 16$$
$13$ $$T^{12} - 8 T^{11} + \cdots + 4826809$$
$17$ $$T^{12} + 18 T^{11} + \cdots + 65536$$
$19$ $$T^{12} + 6 T^{11} + \cdots + 1982464$$
$23$ $$T^{12} + \cdots + 190660864$$
$29$ $$T^{12} - 14 T^{11} + \cdots + 21904$$
$31$ $$T^{12} + \cdots + 177209344$$
$37$ $$T^{12} + \cdots + 227195329$$
$41$ $$T^{12} + 18 T^{11} + \cdots + 65536$$
$43$ $$T^{12} + \cdots + 349241344$$
$47$ $$(T^{6} + 8 T^{5} + \cdots + 5956)^{2}$$
$53$ $$T^{12} + \cdots + 2473271824$$
$59$ $$T^{12} + \cdots + 4983230464$$
$61$ $$T^{12} - 10 T^{11} + \cdots + 89718784$$
$67$ $$T^{12} + 4 T^{11} + \cdots + 83759104$$
$71$ $$T^{12} + 12 T^{11} + \cdots + 4194304$$
$73$ $$(T^{6} + 14 T^{5} + \cdots - 230528)^{2}$$
$79$ $$(T^{6} - 2 T^{5} + \cdots - 29312)^{2}$$
$83$ $$(T^{6} + 36 T^{5} + \cdots - 6912)^{2}$$
$89$ $$T^{12} + \cdots + 1341001056256$$
$97$ $$T^{12} + \cdots + 415519473664$$