# Properties

 Label 1323.2.h.g Level $1323$ Weight $2$ Character orbit 1323.h Analytic conductor $10.564$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(226,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 2]))

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

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

chi := DirichletCharacter("1323.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.h (of order $$3$$, 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: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9$$ x^12 - 7*x^10 + 37*x^8 - 78*x^6 + 123*x^4 - 36*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{5}$$ Twist minimal: no (minimal twist has level 441) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{5} + \beta_1 + 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{5} - \beta_1 - 2) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b5 + b1 + 1) * q^4 + b2 * q^5 + (-b5 - b1 - 2) * q^8 $$q - \beta_1 q^{2} + (\beta_{5} + \beta_1 + 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{5} - \beta_1 - 2) q^{8} + ( - \beta_{10} - 2 \beta_{2}) q^{10} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{11} + (\beta_{11} - \beta_{9}) q^{13} + ( - \beta_{5} + 2 \beta_1) q^{16} + (\beta_{11} - \beta_{10} - \beta_{2}) q^{17} + ( - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}) q^{19} + (\beta_{11} + 2 \beta_{10} + 3 \beta_{2}) q^{20} + \beta_{4} q^{22} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} + 1) q^{23} + (3 \beta_{6} + \beta_{4} - 3 \beta_1) q^{25} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8}) q^{26} + ( - 2 \beta_{7} + \beta_{6} + 4 \beta_{4} + 4) q^{29} + ( - 2 \beta_{8} + \beta_{3}) q^{31} + (\beta_1 - 3) q^{32} + (2 \beta_{11} + 3 \beta_{2}) q^{34} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{37} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_{2}) q^{38} + ( - \beta_{11} - 2 \beta_{10} - 4 \beta_{2}) q^{40} + (\beta_{11} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{3} + \beta_{2}) q^{41} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2) q^{43} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{44} + (\beta_{7} - 4 \beta_{6} - \beta_{4} - 1) q^{46} + ( - 2 \beta_{9} - \beta_{8} + \beta_{3}) q^{47} + (3 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + 4 \beta_1) q^{50} + ( - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}) q^{52} + (2 \beta_{7} - 3 \beta_{6} + 5 \beta_{4} + 5) q^{53} + ( - \beta_{9} - \beta_{8} - \beta_{3}) q^{55} + (\beta_{7} - 7 \beta_{6} - \beta_{4} - 1) q^{58} + (\beta_{9} - \beta_{8} - 2 \beta_{3}) q^{59} + ( - 2 \beta_{9} + \beta_{8} - 2 \beta_{3}) q^{61} + (2 \beta_{9} + \beta_{8} - 4 \beta_{3}) q^{62} + (\beta_{5} - 2 \beta_1 - 3) q^{64} - 3 \beta_{5} q^{65} + ( - 3 \beta_{5} + 3 \beta_1 - 2) q^{67} + ( - 3 \beta_{10} - 4 \beta_{2}) q^{68} + ( - \beta_1 - 6) q^{71} + ( - \beta_{10} + \beta_{2}) q^{73} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 7 \beta_{4} - \beta_1) q^{74} + (2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{3} + \beta_{2}) q^{76} + (3 \beta_{5} + 3 \beta_1 - 1) q^{79} + ( - \beta_{11} + \beta_{10} + 4 \beta_{2}) q^{80} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 4 \beta_{3} - 4 \beta_{2}) q^{82} + (\beta_{11} - \beta_{10} - 3 \beta_{2}) q^{83} + ( - 6 \beta_{6} - 3 \beta_{4} + 6 \beta_1) q^{85} + (2 \beta_{7} + \beta_{6} - 7 \beta_{4} - 7) q^{86} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_1) q^{88} + (\beta_{11} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{3} + 3 \beta_{2}) q^{89} + (4 \beta_{6} + 9 \beta_{4} + 9) q^{92} + ( - \beta_{9} + 3 \beta_{8} - 3 \beta_{3}) q^{94} + (3 \beta_{5} - 9) q^{95} + ( - \beta_{11} - \beta_{10} - 2 \beta_{2}) q^{97}+O(q^{100})$$ q - b1 * q^2 + (b5 + b1 + 1) * q^4 + b2 * q^5 + (-b5 - b1 - 2) * q^8 + (-b10 - 2*b2) * q^10 + (-b7 - b5 - b4) * q^11 + (b11 - b9) * q^13 + (-b5 + 2*b1) * q^16 + (b11 - b10 - b2) * q^17 + (-b10 + b8 + b3 + b2) * q^19 + (b11 + 2*b10 + 3*b2) * q^20 + b4 * q^22 + (-2*b7 + b6 + b4 + 1) * q^23 + (3*b6 + b4 - 3*b1) * q^25 + (b11 - b10 - b9 + b8) * q^26 + (-2*b7 + b6 + 4*b4 + 4) * q^29 + (-2*b8 + b3) * q^31 + (b1 - 3) * q^32 + (2*b11 + 3*b2) * q^34 + (b7 + 2*b6 + b5 - 2*b4 - 2*b1) * q^37 + (b11 - b10 - b9 + b8 - b3 - b2) * q^38 + (-b11 - 2*b10 - 4*b2) * q^40 + (b11 + 2*b10 - b9 - 2*b8 + b3 + b2) * q^41 + (b7 + 2*b6 - 2*b4 - 2) * q^43 + (2*b7 - b6 + 2*b5 + 2*b4 + b1) * q^44 + (b7 - 4*b6 - b4 - 1) * q^46 + (-2*b9 - b8 + b3) * q^47 + (3*b7 - 4*b6 + 3*b5 - 9*b4 + 4*b1) * q^50 + (-b10 + b8 + b3 + b2) * q^52 + (2*b7 - 3*b6 + 5*b4 + 5) * q^53 + (-b9 - b8 - b3) * q^55 + (b7 - 7*b6 - b4 - 1) * q^58 + (b9 - b8 - 2*b3) * q^59 + (-2*b9 + b8 - 2*b3) * q^61 + (2*b9 + b8 - 4*b3) * q^62 + (b5 - 2*b1 - 3) * q^64 - 3*b5 * q^65 + (-3*b5 + 3*b1 - 2) * q^67 + (-3*b10 - 4*b2) * q^68 + (-b1 - 6) * q^71 + (-b10 + b2) * q^73 + (2*b7 + b6 + 2*b5 - 7*b4 - b1) * q^74 + (2*b11 + 2*b10 - 2*b9 - 2*b8 + b3 + b2) * q^76 + (3*b5 + 3*b1 - 1) * q^79 + (-b11 + b10 + 4*b2) * q^80 + (-b11 - 2*b10 + b9 + 2*b8 - 4*b3 - 4*b2) * q^82 + (b11 - b10 - 3*b2) * q^83 + (-6*b6 - 3*b4 + 6*b1) * q^85 + (2*b7 + b6 - 7*b4 - 7) * q^86 + (-b7 + b6 - b5 - b4 - b1) * q^88 + (b11 + 2*b10 - b9 - 2*b8 + 3*b3 + 3*b2) * q^89 + (4*b6 + 9*b4 + 9) * q^92 + (-b9 + 3*b8 - 3*b3) * q^94 + (3*b5 - 9) * q^95 + (-b11 - b10 - 2*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{2} + 12 q^{4} - 24 q^{8}+O(q^{10})$$ 12 * q - 4 * q^2 + 12 * q^4 - 24 * q^8 $$12 q - 4 q^{2} + 12 q^{4} - 24 q^{8} + 8 q^{11} + 12 q^{16} - 6 q^{22} + 4 q^{23} - 12 q^{25} + 22 q^{29} - 32 q^{32} + 6 q^{37} - 6 q^{43} - 14 q^{44} - 12 q^{46} + 56 q^{50} + 28 q^{53} - 18 q^{58} - 48 q^{64} + 12 q^{65} - 76 q^{71} + 36 q^{74} - 12 q^{79} + 30 q^{85} - 36 q^{86} + 6 q^{88} + 62 q^{92} - 120 q^{95}+O(q^{100})$$ 12 * q - 4 * q^2 + 12 * q^4 - 24 * q^8 + 8 * q^11 + 12 * q^16 - 6 * q^22 + 4 * q^23 - 12 * q^25 + 22 * q^29 - 32 * q^32 + 6 * q^37 - 6 * q^43 - 14 * q^44 - 12 * q^46 + 56 * q^50 + 28 * q^53 - 18 * q^58 - 48 * q^64 + 12 * q^65 - 76 * q^71 + 36 * q^74 - 12 * q^79 + 30 * q^85 - 36 * q^86 + 6 * q^88 + 62 * q^92 - 120 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -49\nu^{10} + 259\nu^{8} - 1369\nu^{6} + 861\nu^{4} - 252\nu^{2} - 7266 ) / 4299$$ (-49*v^10 + 259*v^8 - 1369*v^6 + 861*v^4 - 252*v^2 - 7266) / 4299 $$\beta_{2}$$ $$=$$ $$( -148\nu^{11} + 987\nu^{9} - 5217\nu^{7} + 10175\nu^{5} - 17343\nu^{3} - 3522\nu ) / 4299$$ (-148*v^11 + 987*v^9 - 5217*v^7 + 10175*v^5 - 17343*v^3 - 3522*v) / 4299 $$\beta_{3}$$ $$=$$ $$( -148\nu^{11} + 987\nu^{9} - 5217\nu^{7} + 10175\nu^{5} - 17343\nu^{3} + 9375\nu ) / 4299$$ (-148*v^11 + 987*v^9 - 5217*v^7 + 10175*v^5 - 17343*v^3 + 9375*v) / 4299 $$\beta_{4}$$ $$=$$ $$( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 5076 ) / 4299$$ (148*v^10 - 987*v^8 + 5217*v^6 - 10175*v^4 + 17343*v^2 - 5076) / 4299 $$\beta_{5}$$ $$=$$ $$( 161\nu^{10} - 851\nu^{8} + 3884\nu^{6} - 2829\nu^{4} + 828\nu^{2} + 6678 ) / 4299$$ (161*v^10 - 851*v^8 + 3884*v^6 - 2829*v^4 + 828*v^2 + 6678) / 4299 $$\beta_{6}$$ $$=$$ $$( -296\nu^{10} + 1974\nu^{8} - 10434\nu^{6} + 20350\nu^{4} - 30387\nu^{2} + 1554 ) / 4299$$ (-296*v^10 + 1974*v^8 - 10434*v^6 + 20350*v^4 - 30387*v^2 + 1554) / 4299 $$\beta_{7}$$ $$=$$ $$( -120\nu^{10} + 839\nu^{8} - 4230\nu^{6} + 8250\nu^{4} - 10034\nu^{2} + 630 ) / 1433$$ (-120*v^10 + 839*v^8 - 4230*v^6 + 8250*v^4 - 10034*v^2 + 630) / 1433 $$\beta_{8}$$ $$=$$ $$( -494\nu^{11} + 3430\nu^{9} - 18130\nu^{7} + 38978\nu^{5} - 64569\nu^{3} + 34836\nu ) / 4299$$ (-494*v^11 + 3430*v^9 - 18130*v^7 + 38978*v^5 - 64569*v^3 + 34836*v) / 4299 $$\beta_{9}$$ $$=$$ $$( 532\nu^{11} - 4245\nu^{9} + 23052\nu^{7} - 58070\nu^{5} + 93015\nu^{3} - 50082\nu ) / 4299$$ (532*v^11 - 4245*v^9 + 23052*v^7 - 58070*v^5 + 93015*v^3 - 50082*v) / 4299 $$\beta_{10}$$ $$=$$ $$( 641\nu^{11} - 4207\nu^{9} + 22237\nu^{7} - 41561\nu^{5} + 65325\nu^{3} + 12756\nu ) / 4299$$ (641*v^11 - 4207*v^9 + 22237*v^7 - 41561*v^5 + 65325*v^3 + 12756*v) / 4299 $$\beta_{11}$$ $$=$$ $$( -1162\nu^{11} + 7575\nu^{9} - 38811\nu^{7} + 69140\nu^{5} - 96255\nu^{3} - 17544\nu ) / 4299$$ (-1162*v^11 + 7575*v^9 - 38811*v^7 + 69140*v^5 - 96255*v^3 - 17544*v) / 4299
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} ) / 3$$ (b3 - b2) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{4} + 2$$ b6 + 2*b4 + 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{10} + \beta_{8} - 4\beta_{3} - 8\beta_{2} ) / 3$$ (-2*b10 + b8 - 4*b3 - 8*b2) / 3 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 5\beta_{6} - \beta_{5} + 7\beta_{4} - 5\beta_1$$ -b7 + 5*b6 - b5 + 7*b4 - 5*b1 $$\nu^{5}$$ $$=$$ $$( -\beta_{11} - 6\beta_{10} + 2\beta_{9} + 12\beta_{8} - 34\beta_{3} - 17\beta_{2} ) / 3$$ (-b11 - 6*b10 + 2*b9 + 12*b8 - 34*b3 - 17*b2) / 3 $$\nu^{6}$$ $$=$$ $$-7\beta_{5} - 23\beta _1 - 28$$ -7*b5 - 23*b1 - 28 $$\nu^{7}$$ $$=$$ $$( 7\beta_{11} + 30\beta_{10} + 7\beta_{9} + 30\beta_{8} - 74\beta_{3} + 74\beta_{2} ) / 3$$ (7*b11 + 30*b10 + 7*b9 + 30*b8 - 74*b3 + 74*b2) / 3 $$\nu^{8}$$ $$=$$ $$37\beta_{7} - 104\beta_{6} - 118\beta_{4} - 118$$ 37*b7 - 104*b6 - 118*b4 - 118 $$\nu^{9}$$ $$=$$ $$( 74\beta_{11} + 282\beta_{10} - 37\beta_{9} - 141\beta_{8} + 326\beta_{3} + 652\beta_{2} ) / 3$$ (74*b11 + 282*b10 - 37*b9 - 141*b8 + 326*b3 + 652*b2) / 3 $$\nu^{10}$$ $$=$$ $$178\beta_{7} - 467\beta_{6} + 178\beta_{5} - 511\beta_{4} + 467\beta_1$$ 178*b7 - 467*b6 + 178*b5 - 511*b4 + 467*b1 $$\nu^{11}$$ $$=$$ $$( 178\beta_{11} + 645\beta_{10} - 356\beta_{9} - 1290\beta_{8} + 2890\beta_{3} + 1445\beta_{2} ) / 3$$ (178*b11 + 645*b10 - 356*b9 - 1290*b8 + 2890*b3 + 1445*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$\beta_{4}$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 1.82904 + 1.05600i −1.82904 − 1.05600i 1.29589 + 0.748185i −1.29589 − 0.748185i 0.474636 + 0.274031i −0.474636 − 0.274031i 1.82904 − 1.05600i −1.82904 + 1.05600i 1.29589 − 0.748185i −1.29589 + 0.748185i 0.474636 − 0.274031i −0.474636 + 0.274031i
−2.46050 0 4.05408 −1.82904 3.16799i 0 0 −5.05408 0 4.50036 + 7.79485i
226.2 −2.46050 0 4.05408 1.82904 + 3.16799i 0 0 −5.05408 0 −4.50036 7.79485i
226.3 −0.239123 0 −1.94282 −1.29589 2.24456i 0 0 0.942820 0 0.309879 + 0.536725i
226.4 −0.239123 0 −1.94282 1.29589 + 2.24456i 0 0 0.942820 0 −0.309879 0.536725i
226.5 1.69963 0 0.888736 −0.474636 0.822093i 0 0 −1.88874 0 −0.806704 1.39725i
226.6 1.69963 0 0.888736 0.474636 + 0.822093i 0 0 −1.88874 0 0.806704 + 1.39725i
802.1 −2.46050 0 4.05408 −1.82904 + 3.16799i 0 0 −5.05408 0 4.50036 7.79485i
802.2 −2.46050 0 4.05408 1.82904 3.16799i 0 0 −5.05408 0 −4.50036 + 7.79485i
802.3 −0.239123 0 −1.94282 −1.29589 + 2.24456i 0 0 0.942820 0 0.309879 0.536725i
802.4 −0.239123 0 −1.94282 1.29589 2.24456i 0 0 0.942820 0 −0.309879 + 0.536725i
802.5 1.69963 0 0.888736 −0.474636 + 0.822093i 0 0 −1.88874 0 −0.806704 + 1.39725i
802.6 1.69963 0 0.888736 0.474636 0.822093i 0 0 −1.88874 0 0.806704 1.39725i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.h even 3 1 inner
63.t odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.g 12
3.b odd 2 1 441.2.h.g 12
7.b odd 2 1 inner 1323.2.h.g 12
7.c even 3 1 1323.2.f.g 12
7.c even 3 1 1323.2.g.g 12
7.d odd 6 1 1323.2.f.g 12
7.d odd 6 1 1323.2.g.g 12
9.c even 3 1 1323.2.g.g 12
9.d odd 6 1 441.2.g.g 12
21.c even 2 1 441.2.h.g 12
21.g even 6 1 441.2.f.g 12
21.g even 6 1 441.2.g.g 12
21.h odd 6 1 441.2.f.g 12
21.h odd 6 1 441.2.g.g 12
63.g even 3 1 1323.2.f.g 12
63.h even 3 1 inner 1323.2.h.g 12
63.h even 3 1 3969.2.a.bd 6
63.i even 6 1 441.2.h.g 12
63.i even 6 1 3969.2.a.be 6
63.j odd 6 1 441.2.h.g 12
63.j odd 6 1 3969.2.a.be 6
63.k odd 6 1 1323.2.f.g 12
63.l odd 6 1 1323.2.g.g 12
63.n odd 6 1 441.2.f.g 12
63.o even 6 1 441.2.g.g 12
63.s even 6 1 441.2.f.g 12
63.t odd 6 1 inner 1323.2.h.g 12
63.t odd 6 1 3969.2.a.bd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 21.g even 6 1
441.2.f.g 12 21.h odd 6 1
441.2.f.g 12 63.n odd 6 1
441.2.f.g 12 63.s even 6 1
441.2.g.g 12 9.d odd 6 1
441.2.g.g 12 21.g even 6 1
441.2.g.g 12 21.h odd 6 1
441.2.g.g 12 63.o even 6 1
441.2.h.g 12 3.b odd 2 1
441.2.h.g 12 21.c even 2 1
441.2.h.g 12 63.i even 6 1
441.2.h.g 12 63.j odd 6 1
1323.2.f.g 12 7.c even 3 1
1323.2.f.g 12 7.d odd 6 1
1323.2.f.g 12 63.g even 3 1
1323.2.f.g 12 63.k odd 6 1
1323.2.g.g 12 7.c even 3 1
1323.2.g.g 12 7.d odd 6 1
1323.2.g.g 12 9.c even 3 1
1323.2.g.g 12 63.l odd 6 1
1323.2.h.g 12 1.a even 1 1 trivial
1323.2.h.g 12 7.b odd 2 1 inner
1323.2.h.g 12 63.h even 3 1 inner
1323.2.h.g 12 63.t odd 6 1 inner
3969.2.a.bd 6 63.h even 3 1
3969.2.a.bd 6 63.t odd 6 1
3969.2.a.be 6 63.i even 6 1
3969.2.a.be 6 63.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1$$ T2^3 + T2^2 - 4*T2 - 1 $$T_{5}^{12} + 21T_{5}^{10} + 333T_{5}^{8} + 2106T_{5}^{6} + 9963T_{5}^{4} + 8748T_{5}^{2} + 6561$$ T5^12 + 21*T5^10 + 333*T5^8 + 2106*T5^6 + 9963*T5^4 + 8748*T5^2 + 6561

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{3} + T^{2} - 4 T - 1)^{4}$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 21 T^{10} + 333 T^{8} + \cdots + 6561$$
$7$ $$T^{12}$$
$11$ $$(T^{6} - 4 T^{5} + 17 T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2}$$
$13$ $$T^{12} + 39 T^{10} + 1170 T^{8} + \cdots + 6561$$
$17$ $$T^{12} + 84 T^{10} + 5463 T^{8} + \cdots + 15752961$$
$19$ $$T^{12} + 75 T^{10} + 4329 T^{8} + \cdots + 15752961$$
$23$ $$(T^{6} - 2 T^{5} + 29 T^{4} - 68 T^{3} + \cdots + 3481)^{2}$$
$29$ $$(T^{6} - 11 T^{5} + 107 T^{4} - 332 T^{3} + \cdots + 7921)^{2}$$
$31$ $$(T^{6} - 129 T^{4} + 5508 T^{2} + \cdots - 77841)^{2}$$
$37$ $$(T^{6} - 3 T^{5} + 33 T^{4} + 126 T^{3} + \cdots + 729)^{2}$$
$41$ $$T^{12} + 162 T^{10} + \cdots + 43046721$$
$43$ $$(T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729)^{2}$$
$47$ $$(T^{6} - 183 T^{4} + 10584 T^{2} + \cdots - 194481)^{2}$$
$53$ $$(T^{6} - 14 T^{5} + 185 T^{4} + \cdots + 69169)^{2}$$
$59$ $$(T^{6} - 183 T^{4} + 9450 T^{2} + \cdots - 149769)^{2}$$
$61$ $$(T^{6} - 264 T^{4} + 21519 T^{2} + \cdots - 558009)^{2}$$
$67$ $$(T^{3} - 111 T + 353)^{4}$$
$71$ $$(T^{3} + 19 T^{2} + 116 T + 227)^{4}$$
$73$ $$T^{12} + 75 T^{10} + 4329 T^{8} + \cdots + 15752961$$
$79$ $$(T^{3} + 3 T^{2} - 78 T - 107)^{4}$$
$83$ $$T^{12} + 228 T^{10} + \cdots + 51769445841$$
$89$ $$T^{12} + 246 T^{10} + \cdots + 15752961$$
$97$ $$T^{12} + 111 T^{10} + \cdots + 96059601$$