# Properties

 Label 189.2.s.b Level $189$ Weight $2$ Character orbit 189.s Analytic conductor $1.509$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(17,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 1]))

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

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

chi := DirichletCharacter("189.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.s (of order $$6$$, degree $$2$$, not 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: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: 10.0.288778218147.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9$$ x^10 - x^9 + 7*x^8 - 4*x^7 + 34*x^6 - 19*x^5 + 64*x^4 - x^3 + 64*x^2 - 21*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} + \cdots + 29709 ) / 72795$$ (-339*v^9 + 1348*v^8 - 4381*v^7 + 7882*v^6 - 19883*v^5 + 36059*v^4 - 75410*v^3 + 44484*v^2 - 15165*v + 29709) / 72795 $$\beta_{3}$$ $$=$$ $$( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + \cdots + 98232 ) / 72795$$ (658*v^9 + 2394*v^8 + 4352*v^7 + 10326*v^6 + 25351*v^5 + 51907*v^4 + 47450*v^3 + 30472*v^2 + 130790*v + 98232) / 72795 $$\beta_{4}$$ $$=$$ $$( - 4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} + \cdots - 398583 ) / 218385$$ (-4192*v^9 - 796*v^8 - 21678*v^7 - 20279*v^6 - 85319*v^5 - 118353*v^4 - 2560*v^3 - 414508*v^2 + 81750*v - 398583) / 218385 $$\beta_{5}$$ $$=$$ $$( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + \cdots - 336546 ) / 218385$$ (8236*v^9 - 9272*v^8 + 54399*v^7 - 28438*v^6 + 233822*v^5 - 150966*v^4 + 361225*v^3 + 82264*v^2 + 31515*v - 336546) / 218385 $$\beta_{6}$$ $$=$$ $$( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + \cdots - 54156 ) / 72795$$ (3301*v^9 - 2962*v^8 + 21759*v^7 - 8823*v^6 + 104352*v^5 - 42836*v^4 + 175205*v^3 + 72109*v^2 + 166780*v - 54156) / 72795 $$\beta_{7}$$ $$=$$ $$( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} + \cdots + 11514 ) / 14559$$ (-840*v^9 + 248*v^8 - 5659*v^7 - 998*v^6 - 27923*v^5 - 3072*v^4 - 51488*v^3 - 30640*v^2 - 51320*v + 11514) / 14559 $$\beta_{8}$$ $$=$$ $$( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + \cdots + 17856 ) / 43677$$ (3085*v^9 - 1373*v^8 + 17808*v^7 + 1181*v^6 + 84554*v^5 + 5736*v^4 + 111910*v^3 + 124546*v^2 + 106440*v + 17856) / 43677 $$\beta_{9}$$ $$=$$ $$( - 18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} + \cdots + 178101 ) / 218385$$ (-18476*v^9 + 18997*v^8 - 128469*v^7 + 65033*v^6 - 601717*v^5 + 295851*v^4 - 1019855*v^3 - 222374*v^2 - 668685*v + 178101) / 218385
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{8} + 3\beta_{6} - \beta_{4} - \beta_1$$ b9 - b8 + 3*b6 - b4 - b1 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} - 3\beta_{2}$$ b9 + b8 - 3*b2 $$\nu^{4}$$ $$=$$ $$-5\beta_{9} + 5\beta_{8} + \beta_{7} - 12\beta_{6} - 5\beta_{5} - \beta_{3} - 5\beta_{2} + 5\beta _1 - 12$$ -5*b9 + 5*b8 + b7 - 12*b6 - 5*b5 - b3 - 5*b2 + 5*b1 - 12 $$\nu^{5}$$ $$=$$ $$-5\beta_{9} - \beta_{8} - \beta_{7} - 7\beta_{5} + 4\beta_{4} - 2\beta_{3} + 11\beta_{2} - 11\beta_1$$ -5*b9 - b8 - b7 - 7*b5 + 4*b4 - 2*b3 + 11*b2 - 11*b1 $$\nu^{6}$$ $$=$$ $$6\beta_{9} - 8\beta_{8} - 14\beta_{7} + 16\beta_{5} + 9\beta_{4} - 7\beta_{3} + 22\beta_{2} + 51$$ 6*b9 - 8*b8 - 14*b7 + 16*b5 + 9*b4 - 7*b3 + 22*b2 + 51 $$\nu^{7}$$ $$=$$ $$\beta_{9} - 31\beta_{8} - 8\beta_{7} + 31\beta_{5} - 30\beta_{4} + 8\beta_{3} + \beta_{2} + 43\beta_1$$ b9 - 31*b8 - 8*b7 + 31*b5 - 30*b4 + 8*b3 + b2 + 43*b1 $$\nu^{8}$$ $$=$$ $$75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + \cdots - 112 \beta_1$$ 75*b9 - 66*b8 + 38*b7 + 222*b6 + 47*b5 - 37*b4 + 76*b3 + 8*b2 - 112*b1 $$\nu^{9}$$ $$=$$ $$95\beta_{9} + 189\beta_{8} + 94\beta_{7} + 37\beta_{5} + 84\beta_{4} + 47\beta_{3} - 194\beta_{2} - 3$$ 95*b9 + 189*b8 + 94*b7 + 37*b5 + 84*b4 + 47*b3 - 194*b2 - 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$-\beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.827154 − 1.43267i −1.04536 + 1.81062i −0.539982 + 0.935277i 0.187540 − 0.324828i 1.07065 − 1.85442i 0.827154 + 1.43267i −1.04536 − 1.81062i −0.539982 − 0.935277i 0.187540 + 0.324828i 1.07065 + 1.85442i
−1.81474 1.04774i 0 1.19552 + 2.07070i 2.08983 0 −0.879217 + 2.49539i 0.819421i 0 −3.79250 2.18960i
17.2 −1.30778 0.755047i 0 0.140193 + 0.242822i 0.775876 0 2.05881 1.66171i 2.59678i 0 −1.01468 0.585823i
17.3 0.254498 + 0.146935i 0 −0.956820 1.65726i −3.06027 0 −1.22581 2.34465i 1.15010i 0 −0.778834 0.449660i
17.4 0.621951 + 0.359083i 0 −0.742118 1.28539i 1.44755 0 2.19442 + 1.47801i 2.50226i 0 0.900304 + 0.519791i
17.5 2.24607 + 1.29677i 0 2.36322 + 4.09323i −1.25299 0 −0.648211 2.56512i 7.07116i 0 −2.81429 1.62483i
89.1 −1.81474 + 1.04774i 0 1.19552 2.07070i 2.08983 0 −0.879217 2.49539i 0.819421i 0 −3.79250 + 2.18960i
89.2 −1.30778 + 0.755047i 0 0.140193 0.242822i 0.775876 0 2.05881 + 1.66171i 2.59678i 0 −1.01468 + 0.585823i
89.3 0.254498 0.146935i 0 −0.956820 + 1.65726i −3.06027 0 −1.22581 + 2.34465i 1.15010i 0 −0.778834 + 0.449660i
89.4 0.621951 0.359083i 0 −0.742118 + 1.28539i 1.44755 0 2.19442 1.47801i 2.50226i 0 0.900304 0.519791i
89.5 2.24607 1.29677i 0 2.36322 4.09323i −1.25299 0 −0.648211 + 2.56512i 7.07116i 0 −2.81429 + 1.62483i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.s.b 10
3.b odd 2 1 63.2.s.b yes 10
4.b odd 2 1 3024.2.df.b 10
7.b odd 2 1 1323.2.s.b 10
7.c even 3 1 1323.2.i.b 10
7.c even 3 1 1323.2.o.d 10
7.d odd 6 1 189.2.i.b 10
7.d odd 6 1 1323.2.o.c 10
9.c even 3 1 63.2.i.b 10
9.c even 3 1 567.2.p.d 10
9.d odd 6 1 189.2.i.b 10
9.d odd 6 1 567.2.p.c 10
12.b even 2 1 1008.2.df.b 10
21.c even 2 1 441.2.s.b 10
21.g even 6 1 63.2.i.b 10
21.g even 6 1 441.2.o.d 10
21.h odd 6 1 441.2.i.b 10
21.h odd 6 1 441.2.o.c 10
28.f even 6 1 3024.2.ca.b 10
36.f odd 6 1 1008.2.ca.b 10
36.h even 6 1 3024.2.ca.b 10
63.g even 3 1 441.2.s.b 10
63.h even 3 1 441.2.o.d 10
63.i even 6 1 567.2.p.d 10
63.i even 6 1 1323.2.o.d 10
63.j odd 6 1 1323.2.o.c 10
63.k odd 6 1 63.2.s.b yes 10
63.l odd 6 1 441.2.i.b 10
63.n odd 6 1 1323.2.s.b 10
63.o even 6 1 1323.2.i.b 10
63.s even 6 1 inner 189.2.s.b 10
63.t odd 6 1 441.2.o.c 10
63.t odd 6 1 567.2.p.c 10
84.j odd 6 1 1008.2.ca.b 10
252.n even 6 1 1008.2.df.b 10
252.bn odd 6 1 3024.2.df.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b 10 9.c even 3 1
63.2.i.b 10 21.g even 6 1
63.2.s.b yes 10 3.b odd 2 1
63.2.s.b yes 10 63.k odd 6 1
189.2.i.b 10 7.d odd 6 1
189.2.i.b 10 9.d odd 6 1
189.2.s.b 10 1.a even 1 1 trivial
189.2.s.b 10 63.s even 6 1 inner
441.2.i.b 10 21.h odd 6 1
441.2.i.b 10 63.l odd 6 1
441.2.o.c 10 21.h odd 6 1
441.2.o.c 10 63.t odd 6 1
441.2.o.d 10 21.g even 6 1
441.2.o.d 10 63.h even 3 1
441.2.s.b 10 21.c even 2 1
441.2.s.b 10 63.g even 3 1
567.2.p.c 10 9.d odd 6 1
567.2.p.c 10 63.t odd 6 1
567.2.p.d 10 9.c even 3 1
567.2.p.d 10 63.i even 6 1
1008.2.ca.b 10 36.f odd 6 1
1008.2.ca.b 10 84.j odd 6 1
1008.2.df.b 10 12.b even 2 1
1008.2.df.b 10 252.n even 6 1
1323.2.i.b 10 7.c even 3 1
1323.2.i.b 10 63.o even 6 1
1323.2.o.c 10 7.d odd 6 1
1323.2.o.c 10 63.j odd 6 1
1323.2.o.d 10 7.c even 3 1
1323.2.o.d 10 63.i even 6 1
1323.2.s.b 10 7.b odd 2 1
1323.2.s.b 10 63.n odd 6 1
3024.2.ca.b 10 28.f even 6 1
3024.2.ca.b 10 36.h even 6 1
3024.2.df.b 10 4.b odd 2 1
3024.2.df.b 10 252.bn odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 7T_{2}^{8} + 42T_{2}^{6} + 24T_{2}^{5} - 46T_{2}^{4} - 21T_{2}^{3} + 52T_{2}^{2} - 21T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 7 T^{8} + \cdots + 3$$
$3$ $$T^{10}$$
$5$ $$(T^{5} - 9 T^{3} + 6 T^{2} + \cdots - 9)^{2}$$
$7$ $$T^{10} - 3 T^{9} + \cdots + 16807$$
$11$ $$T^{10} + 44 T^{8} + \cdots + 2883$$
$13$ $$T^{10} - 6 T^{9} + \cdots + 3267$$
$17$ $$T^{10} - 12 T^{9} + \cdots + 263169$$
$19$ $$T^{10} - 3 T^{9} + \cdots + 2187$$
$23$ $$T^{10} + 65 T^{8} + \cdots + 27$$
$29$ $$T^{10} - 15 T^{9} + \cdots + 186003$$
$31$ $$T^{10} + 9 T^{9} + \cdots + 16875$$
$37$ $$T^{10} - 6 T^{9} + \cdots + 369664$$
$41$ $$T^{10} - 9 T^{9} + \cdots + 40487769$$
$43$ $$T^{10} - 3 T^{9} + \cdots + 12243001$$
$47$ $$T^{10} + 15 T^{9} + \cdots + 321489$$
$53$ $$T^{10} - 9 T^{9} + \cdots + 871563$$
$59$ $$T^{10} - 18 T^{9} + \cdots + 4100625$$
$61$ $$T^{10} - 12 T^{9} + \cdots + 826875$$
$67$ $$T^{10} + 10 T^{9} + \cdots + 361$$
$71$ $$T^{10} + 359 T^{8} + \cdots + 46216875$$
$73$ $$T^{10} - 3 T^{9} + \cdots + 789507$$
$79$ $$T^{10} + \cdots + 1067655625$$
$83$ $$T^{10} + \cdots + 340734681$$
$89$ $$T^{10} + 24 T^{9} + \cdots + 32455809$$
$97$ $$T^{10} - 6 T^{9} + \cdots + 9687627$$