# Properties

 Label 390.2.bb.c Level $390$ Weight $2$ Character orbit 390.bb Analytic conductor $3.114$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(121,390)]

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, 0, 1]))

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

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

chi := DirichletCharacter("390.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.bb (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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.17284886784.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704$$ x^8 - 2*x^7 + 2*x^6 + 30*x^5 + 185*x^4 + 36*x^3 + 8*x^2 + 208*x + 2704 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632 ) / 20561424$$ (-2225*v^7 + 27304*v^6 - 47714*v^5 - 21770*v^4 + 204731*v^3 + 5488658*v^2 + 258164*v - 320632) / 20561424 $$\beta_{3}$$ $$=$$ $$( - 1777 \nu^{7} - 125666 \nu^{6} + 800522 \nu^{5} - 2370014 \nu^{4} - 784473 \nu^{3} - 5277648 \nu^{2} + 34618460 \nu - 62031216 ) / 6853808$$ (-1777*v^7 - 125666*v^6 + 800522*v^5 - 2370014*v^4 - 784473*v^3 - 5277648*v^2 + 34618460*v - 62031216) / 6853808 $$\beta_{4}$$ $$=$$ $$( - 299 \nu^{7} + 7801 \nu^{6} - 39200 \nu^{5} + 84154 \nu^{4} + 80915 \nu^{3} + 360425 \nu^{2} - 1640788 \nu + 2274116 ) / 395412$$ (-299*v^7 + 7801*v^6 - 39200*v^5 + 84154*v^4 + 80915*v^3 + 360425*v^2 - 1640788*v + 2274116) / 395412 $$\beta_{5}$$ $$=$$ $$( 43733 \nu^{7} - 71918 \nu^{6} - 318186 \nu^{5} + 3350390 \nu^{4} + 3714597 \nu^{3} - 2633192 \nu^{2} - 18392236 \nu + 94417440 ) / 20561424$$ (43733*v^7 - 71918*v^6 - 318186*v^5 + 3350390*v^4 + 3714597*v^3 - 2633192*v^2 - 18392236*v + 94417440) / 20561424 $$\beta_{6}$$ $$=$$ $$( -151\nu^{7} + 822\nu^{6} - 2018\nu^{5} - 2554\nu^{4} - 7135\nu^{3} + 37828\nu^{2} - 46500\nu - 3328 ) / 51792$$ (-151*v^7 + 822*v^6 - 2018*v^5 - 2554*v^4 - 7135*v^3 + 37828*v^2 - 46500*v - 3328) / 51792 $$\beta_{7}$$ $$=$$ $$( - 123974 \nu^{7} + 362933 \nu^{6} - 209286 \nu^{5} - 5793110 \nu^{4} - 10547568 \nu^{3} + 7562729 \nu^{2} + 3493156 \nu - 117816036 ) / 10280712$$ (-123974*v^7 + 362933*v^6 - 209286*v^5 - 5793110*v^4 - 10547568*v^3 + 7562729*v^2 + 3493156*v - 117816036) / 10280712
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} - 4\beta_{6} + \beta_{5} + \beta_{4} + 9\beta_{2} + \beta _1 + 1$$ b7 - 4*b6 + b5 + b4 + 9*b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$6\beta_{7} - 10\beta_{6} + 26\beta_{5} + 10\beta_{4} + 10\beta_{3} + 18\beta_{2} + 3\beta _1 - 18$$ 6*b7 - 10*b6 + 26*b5 + 10*b4 + 10*b3 + 18*b2 + 3*b1 - 18 $$\nu^{4}$$ $$=$$ $$23\beta_{7} + 139\beta_{5} + 5\beta_{4} + 28\beta_{3} + 67\beta_{2} - 5\beta _1 - 149$$ 23*b7 + 139*b5 + 5*b4 + 28*b3 + 67*b2 - 5*b1 - 149 $$\nu^{5}$$ $$=$$ $$250\beta_{6} + 322\beta_{5} - 72\beta_{4} + 72\beta_{3} - 76\beta_{2} - 149\beta _1 - 398$$ 250*b6 + 322*b5 - 72*b4 + 72*b3 - 76*b2 - 149*b1 - 398 $$\nu^{6}$$ $$=$$ $$-471\beta_{7} + 1748\beta_{6} - 619\beta_{5} - 619\beta_{4} - 148\beta_{3} - 2095\beta_{2} - 619\beta _1 - 255$$ -471*b7 + 1748*b6 - 619*b5 - 619*b4 - 148*b3 - 2095*b2 - 619*b1 - 255 $$\nu^{7}$$ $$=$$ $$- 2986 \beta_{7} + 5302 \beta_{6} - 11002 \beta_{5} - 2714 \beta_{4} - 2714 \beta_{3} - 10118 \beta_{2} - 1493 \beta _1 + 7530$$ -2986*b7 + 5302*b6 - 11002*b5 - 2714*b4 - 2714*b3 - 10118*b2 - 1493*b1 + 7530

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 3.17270 + 3.17270i −1.80668 − 1.80668i 1.33404 − 1.33404i −1.70006 + 1.70006i 3.17270 − 3.17270i −1.80668 + 1.80668i 1.33404 + 1.33404i −1.70006 − 1.70006i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i −2.01141 + 1.16129i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i 1.14539 0.661290i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.3 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i −3.15637 + 1.82233i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i 4.02239 2.32233i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i −2.01141 1.16129i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i 1.14539 + 0.661290i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.3 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i −3.15637 1.82233i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i 4.02239 + 2.32233i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.4 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.e 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.bb.c 8
3.b odd 2 1 1170.2.bs.f 8
5.b even 2 1 1950.2.bc.g 8
5.c odd 4 1 1950.2.y.j 8
5.c odd 4 1 1950.2.y.k 8
13.c even 3 1 5070.2.b.ba 8
13.e even 6 1 inner 390.2.bb.c 8
13.e even 6 1 5070.2.b.ba 8
13.f odd 12 1 5070.2.a.bz 4
13.f odd 12 1 5070.2.a.ca 4
39.h odd 6 1 1170.2.bs.f 8
65.l even 6 1 1950.2.bc.g 8
65.r odd 12 1 1950.2.y.j 8
65.r odd 12 1 1950.2.y.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 1.a even 1 1 trivial
390.2.bb.c 8 13.e even 6 1 inner
1170.2.bs.f 8 3.b odd 2 1
1170.2.bs.f 8 39.h odd 6 1
1950.2.y.j 8 5.c odd 4 1
1950.2.y.j 8 65.r odd 12 1
1950.2.y.k 8 5.c odd 4 1
1950.2.y.k 8 65.r odd 12 1
1950.2.bc.g 8 5.b even 2 1
1950.2.bc.g 8 65.l even 6 1
5070.2.a.bz 4 13.f odd 12 1
5070.2.a.ca 4 13.f odd 12 1
5070.2.b.ba 8 13.c even 3 1
5070.2.b.ba 8 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 21T_{7}^{6} + 389T_{7}^{4} + 504T_{7}^{3} - 900T_{7}^{2} - 1248T_{7} + 2704$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{4}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} - 21 T^{6} + 389 T^{4} + \cdots + 2704$$
$11$ $$T^{8} - 6 T^{7} - 18 T^{6} + \cdots + 32761$$
$13$ $$T^{8} + 12 T^{7} + 45 T^{6} + \cdots + 28561$$
$17$ $$(T^{2} - 4 T + 16)^{4}$$
$19$ $$T^{8} + 6 T^{7} - 45 T^{6} + \cdots + 219024$$
$23$ $$T^{8} - 4 T^{7} + 59 T^{6} + \cdots + 2704$$
$29$ $$T^{8} + 8 T^{7} + 103 T^{6} + \cdots + 141376$$
$31$ $$T^{8} + 174 T^{6} + 8777 T^{4} + \cdots + 913936$$
$37$ $$T^{8} - 30 T^{7} + 366 T^{6} + \cdots + 644809$$
$41$ $$T^{8} - 84 T^{6} + 6224 T^{4} + \cdots + 692224$$
$43$ $$T^{8} - 14 T^{7} + 239 T^{6} + \cdots + 327184$$
$47$ $$T^{8} + 228 T^{6} + 16934 T^{4} + \cdots + 1216609$$
$53$ $$(T^{4} - 8 T^{3} - 75 T^{2} + 4 T + 52)^{2}$$
$59$ $$T^{8} - 24 T^{7} + 239 T^{6} + \cdots + 80656$$
$61$ $$(T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16)^{2}$$
$67$ $$T^{8} - 24 T^{7} + 164 T^{6} + \cdots + 123904$$
$71$ $$T^{8} + 12 T^{7} - 156 T^{6} + \cdots + 25240576$$
$73$ $$T^{8} + 392 T^{6} + \cdots + 20647936$$
$79$ $$(T^{4} + 10 T^{3} - 147 T^{2} - 860 T + 3508)^{2}$$
$83$ $$T^{8} + 456 T^{6} + \cdots + 25240576$$
$89$ $$T^{8} - 42 T^{7} + 759 T^{6} + \cdots + 1008016$$
$97$ $$T^{8} + 24 T^{7} + 20 T^{6} + \cdots + 29246464$$