# Properties

 Label 325.2.m.c Level $325$ Weight $2$ Character orbit 325.m Analytic conductor $2.595$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(49,325)]

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([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("325.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.m (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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.22581504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 8$$ (v^7 - 2*v^6 + v^5 + 4*v^4 - 3*v^3 - 2*v^2 + 8*v - 8) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 8 ) / 8$$ (-v^7 + 2*v^6 - v^5 - 4*v^4 + 3*v^3 + 10*v^2 - 16*v + 8) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 18\nu - 16 ) / 4$$ (v^7 - 3*v^6 + 3*v^5 + 3*v^4 - 7*v^3 - 3*v^2 + 18*v - 16) / 4 $$\beta_{5}$$ $$=$$ $$( 2\nu^{7} - 5\nu^{6} + 2\nu^{5} + 7\nu^{4} - 8\nu^{3} - 9\nu^{2} + 28\nu - 20 ) / 4$$ (2*v^7 - 5*v^6 + 2*v^5 + 7*v^4 - 8*v^3 - 9*v^2 + 28*v - 20) / 4 $$\beta_{6}$$ $$=$$ $$( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4$$ (-3*v^7 + 7*v^6 - 3*v^5 - 11*v^4 + 15*v^3 + 11*v^2 - 40*v + 32) / 4 $$\beta_{7}$$ $$=$$ $$( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8$$ (7*v^7 - 20*v^6 + 11*v^5 + 30*v^4 - 45*v^3 - 28*v^2 + 116*v - 88) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_1$$ b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 2\beta_{2} + \beta _1 - 1$$ b6 + b5 + 2*b2 + b1 - 1 $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4\beta_{2} + \beta _1 - 1$$ b7 + b6 - b5 - b4 - b3 + 4*b2 + b1 - 1 $$\nu^{5}$$ $$=$$ $$\beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} + 1$$ b6 - b5 + 2*b4 - 2*b3 + 4*b2 + 1 $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 3\beta_{6} - 5\beta_{5} + \beta_{4} - 4\beta_{3} + 3\beta_{2} + 4\beta _1 - 1$$ -b7 - 3*b6 - 5*b5 + b4 - 4*b3 + 3*b2 + 4*b1 - 1 $$\nu^{7}$$ $$=$$ $$-6\beta_{7} - 8\beta_{6} - 2\beta_{5} + 4\beta_{4} + 2\beta_{2} + \beta _1 + 6$$ -6*b7 - 8*b6 - 2*b5 + 4*b4 + 2*b2 + b1 + 6

## Character values

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

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{6}$$

## 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
 0.665665 − 1.24775i −1.27597 + 0.609843i 1.40994 − 0.109843i 1.20036 + 0.747754i 0.665665 + 1.24775i −1.27597 − 0.609843i 1.40994 + 0.109843i 1.20036 − 0.747754i
−0.747754 1.29515i 0.0820885 0.0473938i −0.118272 + 0.204852i 0 −0.122764 0.0708778i −2.41342 + 4.18016i −2.63726 −1.49551 + 2.59030i 0
49.2 −0.109843 0.190254i 1.38581 0.800098i 0.975869 1.69025i 0 −0.304444 0.175771i 0.166123 0.287734i −0.868145 −0.219687 + 0.380509i 0
49.3 0.609843 + 1.05628i −2.01978 + 1.16612i 0.256182 0.443720i 0 −2.46350 1.42231i −1.80010 + 3.11786i 3.06430 1.21969 2.11256i 0
49.4 1.24775 + 2.16117i −2.44811 + 1.41342i −2.11378 + 3.66117i 0 −6.10929 3.52720i −0.952606 + 1.64996i −5.55889 2.49551 4.32235i 0
199.1 −0.747754 + 1.29515i 0.0820885 + 0.0473938i −0.118272 0.204852i 0 −0.122764 + 0.0708778i −2.41342 4.18016i −2.63726 −1.49551 2.59030i 0
199.2 −0.109843 + 0.190254i 1.38581 + 0.800098i 0.975869 + 1.69025i 0 −0.304444 + 0.175771i 0.166123 + 0.287734i −0.868145 −0.219687 0.380509i 0
199.3 0.609843 1.05628i −2.01978 1.16612i 0.256182 + 0.443720i 0 −2.46350 + 1.42231i −1.80010 3.11786i 3.06430 1.21969 + 2.11256i 0
199.4 1.24775 2.16117i −2.44811 1.41342i −2.11378 3.66117i 0 −6.10929 + 3.52720i −0.952606 1.64996i −5.55889 2.49551 + 4.32235i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 325.2.m.c 8
5.b even 2 1 325.2.m.b 8
5.c odd 4 1 65.2.m.a 8
5.c odd 4 1 325.2.n.d 8
13.e even 6 1 325.2.m.b 8
15.e even 4 1 585.2.bu.c 8
20.e even 4 1 1040.2.da.b 8
65.f even 4 1 845.2.e.m 8
65.h odd 4 1 845.2.m.g 8
65.k even 4 1 845.2.e.n 8
65.l even 6 1 inner 325.2.m.c 8
65.o even 12 1 845.2.a.l 4
65.o even 12 1 845.2.e.n 8
65.o even 12 1 4225.2.a.bi 4
65.q odd 12 1 845.2.c.g 8
65.q odd 12 1 845.2.m.g 8
65.r odd 12 1 65.2.m.a 8
65.r odd 12 1 325.2.n.d 8
65.r odd 12 1 845.2.c.g 8
65.t even 12 1 845.2.a.m 4
65.t even 12 1 845.2.e.m 8
65.t even 12 1 4225.2.a.bl 4
195.bc odd 12 1 7605.2.a.cf 4
195.bf even 12 1 585.2.bu.c 8
195.bn odd 12 1 7605.2.a.cj 4
260.bg even 12 1 1040.2.da.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 5.c odd 4 1
65.2.m.a 8 65.r odd 12 1
325.2.m.b 8 5.b even 2 1
325.2.m.b 8 13.e even 6 1
325.2.m.c 8 1.a even 1 1 trivial
325.2.m.c 8 65.l even 6 1 inner
325.2.n.d 8 5.c odd 4 1
325.2.n.d 8 65.r odd 12 1
585.2.bu.c 8 15.e even 4 1
585.2.bu.c 8 195.bf even 12 1
845.2.a.l 4 65.o even 12 1
845.2.a.m 4 65.t even 12 1
845.2.c.g 8 65.q odd 12 1
845.2.c.g 8 65.r odd 12 1
845.2.e.m 8 65.f even 4 1
845.2.e.m 8 65.t even 12 1
845.2.e.n 8 65.k even 4 1
845.2.e.n 8 65.o even 12 1
845.2.m.g 8 65.h odd 4 1
845.2.m.g 8 65.q odd 12 1
1040.2.da.b 8 20.e even 4 1
1040.2.da.b 8 260.bg even 12 1
4225.2.a.bi 4 65.o even 12 1
4225.2.a.bl 4 65.t even 12 1
7605.2.a.cf 4 195.bc odd 12 1
7605.2.a.cj 4 195.bn odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 2T_{2}^{7} + 7T_{2}^{6} - 2T_{2}^{5} + 16T_{2}^{4} - 8T_{2}^{3} + 19T_{2}^{2} + 4T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{7} + \cdots + 1$$
$3$ $$T^{8} + 6 T^{7} + \cdots + 1$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 10 T^{7} + \cdots + 121$$
$11$ $$T^{8} - 30 T^{6} + \cdots + 1089$$
$13$ $$T^{8} + 8 T^{7} + \cdots + 28561$$
$17$ $$T^{8} + 18 T^{7} + \cdots + 169$$
$19$ $$(T^{4} + 6 T^{3} + \cdots + 169)^{2}$$
$23$ $$T^{8} + 6 T^{7} + \cdots + 89401$$
$29$ $$T^{8} - 8 T^{7} + \cdots + 1$$
$31$ $$(T^{4} + 32 T^{2} + 64)^{2}$$
$37$ $$T^{8} - 2 T^{7} + \cdots + 1$$
$41$ $$(T^{4} - 6 T^{3} + 11 T^{2} + \cdots + 1)^{2}$$
$43$ $$T^{8} - 18 T^{7} + \cdots + 169$$
$47$ $$(T^{4} - 8 T^{3} + \cdots - 1328)^{2}$$
$53$ $$T^{8} + 72 T^{6} + \cdots + 2304$$
$59$ $$T^{8} - 12 T^{7} + \cdots + 9$$
$61$ $$T^{8} + 28 T^{7} + \cdots + 1590121$$
$67$ $$T^{8} - 30 T^{7} + \cdots + 7667361$$
$71$ $$T^{8} - 218 T^{6} + \cdots + 109767529$$
$73$ $$(T^{4} - 8 T^{3} + \cdots - 1712)^{2}$$
$79$ $$(T^{4} - 8 T^{3} + \cdots + 4432)^{2}$$
$83$ $$(T^{4} + 12 T^{3} + \cdots - 192)^{2}$$
$89$ $$T^{8} + 24 T^{7} + \cdots + 78375609$$
$97$ $$T^{8} - 2 T^{7} + \cdots + 196249$$