# Properties

 Label 325.2.n.e Level $325$ Weight $2$ Character orbit 325.n Analytic conductor $2.595$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(101,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("325.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.n (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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 16x^{8} + 84x^{6} + 163x^{4} + 118x^{2} + 27$$ x^10 + 16*x^8 + 84*x^6 + 163*x^4 + 118*x^2 + 27 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + (\beta_{9} - \beta_{8} - \beta_{2} - 1) q^{3} + (\beta_{9} - \beta_{7} + \cdots - 2 \beta_{2}) q^{4}+ \cdots + ( - \beta_{7} - \beta_{5} + 2 \beta_{2}) q^{9}+O(q^{10})$$ q - b3 * q^2 + (b9 - b8 - b2 - 1) * q^3 + (b9 - b7 + b6 + b4 - 2*b2) * q^4 + (b7 - b5 + b3 + b2 + b1 + 2) * q^6 + (b7 - b5) * q^7 + (-b5 - b4 + 2*b2 + b1 + 1) * q^8 + (-b7 - b5 + 2*b2) * q^9 $$q - \beta_{3} q^{2} + (\beta_{9} - \beta_{8} - \beta_{2} - 1) q^{3} + (\beta_{9} - \beta_{7} + \cdots - 2 \beta_{2}) q^{4}+ \cdots + ( - 4 \beta_{9} + 2 \beta_{8} + \cdots - 5 \beta_1) q^{99}+O(q^{100})$$ q - b3 * q^2 + (b9 - b8 - b2 - 1) * q^3 + (b9 - b7 + b6 + b4 - 2*b2) * q^4 + (b7 - b5 + b3 + b2 + b1 + 2) * q^6 + (b7 - b5) * q^7 + (-b5 - b4 + 2*b2 + b1 + 1) * q^8 + (-b7 - b5 + 2*b2) * q^9 + (-b6 - b4 + b3 + b2 - 1) * q^11 + (-b7 - b6 + b5 - b4 - 2*b3 - b1 - 4) * q^12 + (-b9 + b8 - b6 - b5 + b4 + b3 + 1) * q^13 + (-b8 - b7 - b6 - 2) * q^14 + (b9 - b8 - 2*b7 + b6 + 2*b5 + b4 - 4*b2 - 4) * q^16 + (-b9 + b6 + b5 + b4 - 2*b2) * q^17 + (2*b9 - b8 - 3*b7 + 3*b6 + 2*b5 + 2*b4 - 4*b2 - 2) * q^18 + (b9 - 2*b8 + b7 + b6 - b4 + 2*b3 + 2*b1) * q^19 + (-2*b7 + 2*b6 + b5 + b4 - 2*b1) * q^21 + (b7 + b5 + b3 + 2*b2 - b1) * q^22 + (-b9 + b8 - b3 + 3*b2 - 2*b1 + 3) * q^23 + (2*b9 + 2*b8 + b6 + b4 + 2*b3 - 4*b2 + 4) * q^24 + (-b9 + b8 - b6 - b5 + b4 - 3*b3 + 3*b2 - b1) * q^26 + (b8 + b5 - b4 + 3) * q^27 + (b9 + b8 + b6 + b4 + b3 - 3*b2 + 3) * q^28 + (2*b9 - 2*b8 + b7 - b6 - b5 + b2 + 1) * q^29 + (2*b9 - b8 - b5 - b4 - 4*b2 - b1 - 2) * q^31 + (-b9 + 2*b8 + b7 - b6 - 2*b5 + b4 + 2*b3 + b2 + 2*b1 + 2) * q^32 + (-b9 + 2*b8 - b6 - b5 + b4 + b3 + b1) * q^33 + (-2*b9 + b8 + b7 - b6 - b5 - b4 + 2*b2 + b1 + 1) * q^34 + (-b9 + b8 + 2*b7 - b6 - 2*b5 - b4 + 3*b3 + b2 + 6*b1 + 1) * q^36 + (-2*b9 - 2*b8 - b6 - b4 + 3*b3 + b2 - 1) * q^37 + (-4*b8 + 3*b5 - 3*b4 + 2*b3 + b1 - 9) * q^38 + (-2*b9 - 2*b7 + 2*b6 + 3*b5 + b4 - b3 - 2*b2 - 2*b1 - 1) * q^39 + (b9 + b8 + 2*b6 + 2*b4 - b3 - 2*b2 + 2) * q^41 + (-2*b9 + 2*b8 + b7 - b5 - b4 + b3 + 8*b2 + 2*b1 + 8) * q^42 + (2*b7 - 2*b6 - 2*b4 + b3 - b1) * q^43 + (-4*b9 + 2*b8 + 3*b7 - 3*b6 - b5 - b4 + 8*b2 - 2*b1 + 4) * q^44 + (-b9 + 2*b8 - b6 - b5 + b4 - 3*b3 + 3*b2 - 3*b1 + 6) * q^46 + (2*b9 - b8 + b7 - b6 - b5 - b4 - 4*b2 - 2) * q^47 + (-3*b9 + 2*b7 - 3*b6 - b5 - 3*b4 - 2*b3 + 8*b2 + 2*b1) * q^48 + (-b9 + b8 - 2*b7 + b6 + 2*b5 + b4 + b3 - b2 + 2*b1 - 1) * q^49 + (-4*b8 + b7 + b6 + 2*b5 - 2*b4 - 2*b3 - b1) * q^51 + (2*b9 - b7 + 2*b6 + 3*b4 - 2*b3 - 7*b2 - 2*b1 + 4) * q^52 + (b7 + b6 - 2*b5 + 2*b4 + 4) * q^53 + (b7 - b6 + b5 - 2*b4 - 2*b3 + b2 - 1) * q^54 + (2*b7 - 2*b6 - 2*b4 - 3*b3 + 5*b2 + 3*b1) * q^56 + (4*b9 - 2*b8 - 3*b7 + 3*b6 + b5 + b4 - 3*b1) * q^57 + (2*b7 - b6 - 3*b5 + b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^58 + (-2*b9 + 4*b8 + b7 - b5 - b3 - 2*b2 - b1 - 4) * q^59 + (-3*b9 + b7 - b6 - b4 + 3*b3 - b2 - 3*b1) * q^61 + (b9 - b8 + 3*b3 + 3*b2 + 6*b1 + 3) * q^62 + (b9 + b8 + 2*b7 - b6 + 2*b5 - 3*b4 - 5*b3) * q^63 + (-b5 + b4 - 4*b3 - 2*b1 - 3) * q^64 + (-b7 - b6 - 2*b3 - b1 - 5) * q^66 + (-b7 - b6 - b5 - b3) * q^67 + (4*b9 - 4*b8 - 2*b7 + 2*b6 + 2*b5 - b3 - 5*b2 - 2*b1 - 5) * q^68 + (2*b9 + 2*b7 - 2*b6 - 2*b4 - b3 - 4*b2 + b1) * q^69 + (-3*b7 - b6 + 2*b5 + b4 + 2*b3 - 3*b2 + 2*b1 - 6) * q^71 + (2*b9 - 4*b8 - 2*b7 + 2*b5 - b3 - 11*b2 - b1 - 22) * q^72 + (-2*b9 + b8 - b7 + b6 + 3*b5 + 3*b4 - 8*b2 + b1 - 4) * q^73 + (-2*b9 + b7 + 2*b6 + 3*b5 + 2*b4 + b3 + 4*b2 - b1) * q^74 + (b9 + b8 + 2*b7 + b6 + 2*b5 - b4 + 8*b3) * q^76 + (b7 + b6 - 2*b3 - b1 - 4) * q^77 + (-3*b9 + 4*b8 + 2*b7 + b6 - b5 + 2*b3 + 4*b2 + 2*b1 + 10) * q^78 + (-3*b8 + 2*b7 + 2*b6 + 2*b5 - 2*b4 - 4*b3 - 2*b1) * q^79 + (3*b9 - 3*b8 + 2*b7 - 2*b5 - 2*b4 + b3 - 3*b2 + 2*b1 - 3) * q^81 + (-b9 - 3*b6 - 3*b5 - 3*b4 - 2*b3 + 3*b2 + 2*b1) * q^82 + (-b5 - b4 + 2*b2 + 4*b1 + 1) * q^83 + (2*b9 - 4*b8 - 3*b7 - b6 + 2*b5 + b4 - 3*b3 - 8*b2 - 3*b1 - 16) * q^84 + (-2*b9 + b8 + 4*b7 - 4*b6 - b5 - b4 + 8*b2 - 2*b1 + 4) * q^86 + (2*b9 - 4*b7 + 2*b6 - 2*b5 + 2*b4 + b3 + 9*b2 - b1) * q^87 + (-3*b9 + 3*b8 + b7 - 2*b6 - b5 + b4 - 4*b3 - 8*b1) * q^88 + (2*b9 + 2*b8 - b7 - b6 - b5 - b3 - 2*b2 + 2) * q^89 + (-2*b9 + 3*b8 + 2*b7 + 2*b6 - b4 - 2*b3 + 3*b2 - 2*b1 + 5) * q^91 + (2*b8 - b7 - b6 - 4*b5 + 4*b4 - 4*b3 - 2*b1 + 5) * q^92 + (-2*b7 - 2*b6 - 2*b5 + 5*b2 - 5) * q^93 + (b9 - b8 + b7 - b6 - b5 + 2*b3 + b2 + 4*b1 + 1) * q^94 + (-2*b9 + b8 - 5*b7 + 5*b6 + 5*b5 + 5*b4 - 10*b2 - 5*b1 - 5) * q^96 + (-b9 + 2*b8 - b7 + 2*b6 + 3*b5 - 2*b4 - b2 - 2) * q^97 + (2*b6 + 2*b5 - 2*b4 + b3 - 3*b2 + b1 - 6) * q^98 + (-4*b9 + 2*b8 + b7 - b6 - 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 3 q^{3} + 6 q^{4} + 9 q^{6} - 6 q^{7} - 8 q^{9}+O(q^{10})$$ 10 * q - 3 * q^3 + 6 * q^4 + 9 * q^6 - 6 * q^7 - 8 * q^9 $$10 q - 3 q^{3} + 6 q^{4} + 9 q^{6} - 6 q^{7} - 8 q^{9} - 9 q^{11} - 28 q^{12} + 8 q^{13} - 8 q^{14} - 12 q^{16} + 8 q^{17} - 12 q^{22} + 13 q^{23} + 42 q^{24} - 17 q^{26} + 30 q^{27} + 33 q^{28} + 7 q^{29} + 3 q^{32} - 6 q^{33} - 3 q^{36} + 3 q^{37} - 62 q^{38} + 8 q^{39} + 12 q^{41} + 32 q^{42} + 4 q^{43} + 39 q^{46} - 26 q^{48} - q^{49} + 16 q^{51} + 61 q^{52} + 24 q^{53} - 9 q^{54} - 21 q^{56} + 18 q^{58} - 48 q^{59} + 13 q^{61} + 17 q^{62} - 34 q^{64} - 42 q^{66} + 6 q^{67} - 13 q^{68} + 20 q^{69} - 27 q^{71} - 141 q^{72} - 26 q^{74} - 12 q^{76} - 48 q^{77} + 56 q^{78} + 4 q^{79} - 17 q^{81} - q^{82} - 90 q^{84} - 49 q^{87} - 6 q^{88} + 24 q^{89} + 13 q^{91} + 34 q^{92} - 63 q^{93} + 5 q^{94} - 15 q^{97} - 45 q^{98}+O(q^{100})$$ 10 * q - 3 * q^3 + 6 * q^4 + 9 * q^6 - 6 * q^7 - 8 * q^9 - 9 * q^11 - 28 * q^12 + 8 * q^13 - 8 * q^14 - 12 * q^16 + 8 * q^17 - 12 * q^22 + 13 * q^23 + 42 * q^24 - 17 * q^26 + 30 * q^27 + 33 * q^28 + 7 * q^29 + 3 * q^32 - 6 * q^33 - 3 * q^36 + 3 * q^37 - 62 * q^38 + 8 * q^39 + 12 * q^41 + 32 * q^42 + 4 * q^43 + 39 * q^46 - 26 * q^48 - q^49 + 16 * q^51 + 61 * q^52 + 24 * q^53 - 9 * q^54 - 21 * q^56 + 18 * q^58 - 48 * q^59 + 13 * q^61 + 17 * q^62 - 34 * q^64 - 42 * q^66 + 6 * q^67 - 13 * q^68 + 20 * q^69 - 27 * q^71 - 141 * q^72 - 26 * q^74 - 12 * q^76 - 48 * q^77 + 56 * q^78 + 4 * q^79 - 17 * q^81 - q^82 - 90 * q^84 - 49 * q^87 - 6 * q^88 + 24 * q^89 + 13 * q^91 + 34 * q^92 - 63 * q^93 + 5 * q^94 - 15 * q^97 - 45 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 16x^{8} + 84x^{6} + 163x^{4} + 118x^{2} + 27$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{9} + 13\nu^{7} + 45\nu^{5} + 13\nu^{3} - 41\nu - 15 ) / 30$$ (v^9 + 13*v^7 + 45*v^5 + 13*v^3 - 41*v - 15) / 30 $$\beta_{3}$$ $$=$$ $$( \nu^{8} + 13\nu^{6} + 50\nu^{4} + 53\nu^{2} - 5\nu + 9 ) / 10$$ (v^8 + 13*v^6 + 50*v^4 + 53*v^2 - 5*v + 9) / 10 $$\beta_{4}$$ $$=$$ $$( \nu^{9} + 6\nu^{8} + 13\nu^{7} + 93\nu^{6} + 45\nu^{5} + 450\nu^{4} + 28\nu^{3} + 678\nu^{2} + 34\nu + 219 ) / 30$$ (v^9 + 6*v^8 + 13*v^7 + 93*v^6 + 45*v^5 + 450*v^4 + 28*v^3 + 678*v^2 + 34*v + 219) / 30 $$\beta_{5}$$ $$=$$ $$( \nu^{9} - 6\nu^{8} + 13\nu^{7} - 93\nu^{6} + 45\nu^{5} - 450\nu^{4} + 28\nu^{3} - 678\nu^{2} + 34\nu - 219 ) / 30$$ (v^9 - 6*v^8 + 13*v^7 - 93*v^6 + 45*v^5 - 450*v^4 + 28*v^3 - 678*v^2 + 34*v - 219) / 30 $$\beta_{6}$$ $$=$$ $$( 2\nu^{9} + 31\nu^{7} + 150\nu^{5} + 5\nu^{4} + 226\nu^{3} + 35\nu^{2} + 68\nu + 20 ) / 10$$ (2*v^9 + 31*v^7 + 150*v^5 + 5*v^4 + 226*v^3 + 35*v^2 + 68*v + 20) / 10 $$\beta_{7}$$ $$=$$ $$( -2\nu^{9} - 31\nu^{7} - 150\nu^{5} + 5\nu^{4} - 226\nu^{3} + 35\nu^{2} - 68\nu + 20 ) / 10$$ (-2*v^9 - 31*v^7 - 150*v^5 + 5*v^4 - 226*v^3 + 35*v^2 - 68*v + 20) / 10 $$\beta_{8}$$ $$=$$ $$( -2\nu^{8} - 31\nu^{6} - 150\nu^{4} - 231\nu^{2} - 93 ) / 5$$ (-2*v^8 - 31*v^6 - 150*v^4 - 231*v^2 - 93) / 5 $$\beta_{9}$$ $$=$$ $$( -4\nu^{9} - 2\nu^{8} - 62\nu^{7} - 31\nu^{6} - 305\nu^{5} - 150\nu^{4} - 497\nu^{3} - 231\nu^{2} - 211\nu - 93 ) / 10$$ (-4*v^9 - 2*v^8 - 62*v^7 - 31*v^6 - 305*v^5 - 150*v^4 - 497*v^3 - 231*v^2 - 211*v - 93) / 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{8} + \beta_{5} - \beta_{4} - 4$$ -b8 + b5 - b4 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 2\beta_{2} - 5\beta _1 - 1$$ b5 + b4 - 2*b2 - 5*b1 - 1 $$\nu^{4}$$ $$=$$ $$7\beta_{8} + \beta_{7} + \beta_{6} - 7\beta_{5} + 7\beta_{4} + 24$$ 7*b8 + b7 + b6 - 7*b5 + 7*b4 + 24 $$\nu^{5}$$ $$=$$ $$-2\beta_{9} + \beta_{8} + 2\beta_{7} - 2\beta_{6} - 9\beta_{5} - 9\beta_{4} + 18\beta_{2} + 30\beta _1 + 9$$ -2*b9 + b8 + 2*b7 - 2*b6 - 9*b5 - 9*b4 + 18*b2 + 30*b1 + 9 $$\nu^{6}$$ $$=$$ $$-46\beta_{8} - 10\beta_{7} - 10\beta_{6} + 45\beta_{5} - 45\beta_{4} - 4\beta_{3} - 2\beta _1 - 155$$ -46*b8 - 10*b7 - 10*b6 + 45*b5 - 45*b4 - 4*b3 - 2*b1 - 155 $$\nu^{7}$$ $$=$$ $$24\beta_{9} - 12\beta_{8} - 25\beta_{7} + 25\beta_{6} + 68\beta_{5} + 68\beta_{4} - 148\beta_{2} - 190\beta _1 - 74$$ 24*b9 - 12*b8 - 25*b7 + 25*b6 + 68*b5 + 68*b4 - 148*b2 - 190*b1 - 74 $$\nu^{8}$$ $$=$$ $$301\beta_{8} + 80\beta_{7} + 80\beta_{6} - 288\beta_{5} + 288\beta_{4} + 62\beta_{3} + 31\beta _1 + 1018$$ 301*b8 + 80*b7 + 80*b6 - 288*b5 + 288*b4 + 62*b3 + 31*b1 + 1018 $$\nu^{9}$$ $$=$$ $$- 222 \beta_{9} + 111 \beta_{8} + 235 \beta_{7} - 235 \beta_{6} - 492 \beta_{5} - 492 \beta_{4} + \cdots + 585$$ -222*b9 + 111*b8 + 235*b7 - 235*b6 - 492*b5 - 492*b4 + 1170*b2 + 1226*b1 + 585

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 2.63252i 1.36551i − 0.666048i − 0.887996i − 2.44399i − 2.63252i − 1.36551i 0.666048i 0.887996i 2.44399i
−2.27983 + 1.31626i −1.34953 2.33745i 2.46508 4.26964i 0 6.15337 + 3.55265i 2.37354 + 1.37037i 7.71370i −2.14244 + 3.71081i 0
101.2 −1.18257 + 0.682755i 0.199959 + 0.346339i −0.0676905 + 0.117243i 0 −0.472929 0.273046i −3.15550 1.82183i 2.91589i 1.42003 2.45957i 0
101.3 0.576815 0.333024i 1.24146 + 2.15028i −0.778190 + 1.34786i 0 1.43219 + 0.826874i 0.509002 + 0.293872i 2.36872i −1.58246 + 2.74090i 0
101.4 0.769027 0.443998i −1.53097 2.65171i −0.605732 + 1.04916i 0 −2.35471 1.35949i −3.08568 1.78152i 2.85177i −3.18771 + 5.52128i 0
101.5 2.11655 1.22199i −0.0609297 0.105533i 1.98653 3.44078i 0 −0.257922 0.148911i 0.358632 + 0.207056i 4.82215i 1.49258 2.58522i 0
251.1 −2.27983 1.31626i −1.34953 + 2.33745i 2.46508 + 4.26964i 0 6.15337 3.55265i 2.37354 1.37037i 7.71370i −2.14244 3.71081i 0
251.2 −1.18257 0.682755i 0.199959 0.346339i −0.0676905 0.117243i 0 −0.472929 + 0.273046i −3.15550 + 1.82183i 2.91589i 1.42003 + 2.45957i 0
251.3 0.576815 + 0.333024i 1.24146 2.15028i −0.778190 1.34786i 0 1.43219 0.826874i 0.509002 0.293872i 2.36872i −1.58246 2.74090i 0
251.4 0.769027 + 0.443998i −1.53097 + 2.65171i −0.605732 1.04916i 0 −2.35471 + 1.35949i −3.08568 + 1.78152i 2.85177i −3.18771 5.52128i 0
251.5 2.11655 + 1.22199i −0.0609297 + 0.105533i 1.98653 + 3.44078i 0 −0.257922 + 0.148911i 0.358632 0.207056i 4.82215i 1.49258 + 2.58522i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 325.2.n.e 10
5.b even 2 1 325.2.n.f yes 10
5.c odd 4 2 325.2.m.d 20
13.e even 6 1 inner 325.2.n.e 10
13.f odd 12 2 4225.2.a.bv 10
65.l even 6 1 325.2.n.f yes 10
65.r odd 12 2 325.2.m.d 20
65.s odd 12 2 4225.2.a.bu 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.m.d 20 5.c odd 4 2
325.2.m.d 20 65.r odd 12 2
325.2.n.e 10 1.a even 1 1 trivial
325.2.n.e 10 13.e even 6 1 inner
325.2.n.f yes 10 5.b even 2 1
325.2.n.f yes 10 65.l even 6 1
4225.2.a.bu 10 65.s odd 12 2
4225.2.a.bv 10 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 8T_{2}^{8} + 54T_{2}^{6} - 15T_{2}^{5} - 77T_{2}^{4} + 30T_{2}^{3} + 91T_{2}^{2} - 90T_{2} + 27$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 8 T^{8} + \cdots + 27$$
$3$ $$T^{10} + 3 T^{9} + \cdots + 1$$
$5$ $$T^{10}$$
$7$ $$T^{10} + 6 T^{9} + \cdots + 75$$
$11$ $$T^{10} + 9 T^{9} + \cdots + 6075$$
$13$ $$T^{10} - 8 T^{9} + \cdots + 371293$$
$17$ $$T^{10} - 8 T^{9} + \cdots + 1521$$
$19$ $$T^{10} - 68 T^{8} + \cdots + 531723$$
$23$ $$T^{10} - 13 T^{9} + \cdots + 9$$
$29$ $$T^{10} - 7 T^{9} + \cdots + 1766241$$
$31$ $$T^{10} + 154 T^{8} + \cdots + 10546875$$
$37$ $$T^{10} + \cdots + 112914675$$
$41$ $$T^{10} - 12 T^{9} + \cdots + 45387$$
$43$ $$T^{10} - 4 T^{9} + \cdots + 525625$$
$47$ $$T^{10} + 160 T^{8} + \cdots + 771147$$
$53$ $$(T^{5} - 12 T^{4} + \cdots - 8469)^{2}$$
$59$ $$T^{10} + 48 T^{9} + \cdots + 3102867$$
$61$ $$T^{10} + \cdots + 2238709225$$
$67$ $$T^{10} - 6 T^{9} + \cdots + 16875$$
$71$ $$T^{10} + \cdots + 116251875$$
$73$ $$T^{10} + \cdots + 408916875$$
$79$ $$(T^{5} - 2 T^{4} + \cdots - 9475)^{2}$$
$83$ $$T^{10} + 393 T^{8} + \cdots + 243$$
$89$ $$T^{10} - 24 T^{9} + \cdots + 45139923$$
$97$ $$T^{10} + 15 T^{9} + \cdots + 5738067$$