# Properties

 Label 300.2.e.e Level $300$ Weight $2$ Character orbit 300.e Analytic conductor $2.396$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(251,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("300.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.e (of order $$2$$, degree $$1$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4521217600.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16$$ x^8 + x^6 - 2*x^4 + 4*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 2\nu^{6} - 3\nu^{5} + 2\nu^{4} + 2\nu^{3} + 4\nu^{2} + 12\nu + 8 ) / 16$$ (v^7 + 2*v^6 - 3*v^5 + 2*v^4 + 2*v^3 + 4*v^2 + 12*v + 8) / 16 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} + 3\nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} - 12\nu + 8 ) / 16$$ (-v^7 + 2*v^6 + 3*v^5 + 2*v^4 - 2*v^3 + 4*v^2 - 12*v + 8) / 16 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 2\nu^{6} + 3\nu^{5} + 6\nu^{4} + 6\nu^{3} + 12\nu^{2} + 4\nu - 8 ) / 16$$ (-v^7 - 2*v^6 + 3*v^5 + 6*v^4 + 6*v^3 + 12*v^2 + 4*v - 8) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + \nu^{5} + 6\nu^{3} + 4\nu ) / 8$$ (v^7 + v^5 + 6*v^3 + 4*v) / 8 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + \nu^{5} - 2\nu^{3} + 4\nu ) / 8$$ (v^7 + v^5 - 2*v^3 + 4*v) / 8 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} + 3\nu^{5} - 6\nu^{4} + 6\nu^{3} - 12\nu^{2} + 4\nu + 8 ) / 16$$ (-v^7 + 2*v^6 + 3*v^5 - 6*v^4 + 6*v^3 - 12*v^2 + 4*v + 8) / 16
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2$$ (b7 + b6 - b5 + b4 - b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta_1$$ b1 $$\nu^{3}$$ $$=$$ $$-\beta_{6} + \beta_{5}$$ -b6 + b5 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1$$ -b7 + b4 + b3 + b2 - 2*b1 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2}$$ b7 + 2*b6 + b4 + b3 - b2 $$\nu^{6}$$ $$=$$ $$\beta_{7} - \beta_{4} + 3\beta_{3} + 3\beta_{2} - 4$$ b7 - b4 + 3*b3 + 3*b2 - 4 $$\nu^{7}$$ $$=$$ $$-3\beta_{7} + 2\beta_{6} + 4\beta_{5} - 3\beta_{4} + \beta_{3} - \beta_{2}$$ -3*b7 + 2*b6 + 4*b5 - 3*b4 + b3 - b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.29437 − 0.569745i 1.29437 + 0.569745i 0.273147 − 1.38758i 0.273147 + 1.38758i −0.273147 − 1.38758i −0.273147 + 1.38758i −1.29437 − 0.569745i −1.29437 + 0.569745i
−1.29437 0.569745i −0.908080 1.47492i 1.35078 + 1.47492i 0 0.335062 + 2.42647i 2.50967i −0.908080 2.67869i −1.35078 + 2.67869i 0
251.2 −1.29437 + 0.569745i −0.908080 + 1.47492i 1.35078 1.47492i 0 0.335062 2.42647i 2.50967i −0.908080 + 2.67869i −1.35078 2.67869i 0
251.3 −0.273147 1.38758i 1.55737 0.758030i −1.85078 + 0.758030i 0 −1.47722 1.95392i 3.56393i 1.55737 + 2.36106i 1.85078 2.36106i 0
251.4 −0.273147 + 1.38758i 1.55737 + 0.758030i −1.85078 0.758030i 0 −1.47722 + 1.95392i 3.56393i 1.55737 2.36106i 1.85078 + 2.36106i 0
251.5 0.273147 1.38758i −1.55737 + 0.758030i −1.85078 0.758030i 0 0.626440 + 2.36803i 3.56393i −1.55737 + 2.36106i 1.85078 2.36106i 0
251.6 0.273147 + 1.38758i −1.55737 0.758030i −1.85078 + 0.758030i 0 0.626440 2.36803i 3.56393i −1.55737 2.36106i 1.85078 + 2.36106i 0
251.7 1.29437 0.569745i 0.908080 + 1.47492i 1.35078 1.47492i 0 2.01572 + 1.39172i 2.50967i 0.908080 2.67869i −1.35078 + 2.67869i 0
251.8 1.29437 + 0.569745i 0.908080 1.47492i 1.35078 + 1.47492i 0 2.01572 1.39172i 2.50967i 0.908080 + 2.67869i −1.35078 2.67869i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.e.e yes 8
3.b odd 2 1 inner 300.2.e.e yes 8
4.b odd 2 1 inner 300.2.e.e yes 8
5.b even 2 1 300.2.e.d 8
5.c odd 4 2 300.2.h.c 16
12.b even 2 1 inner 300.2.e.e yes 8
15.d odd 2 1 300.2.e.d 8
15.e even 4 2 300.2.h.c 16
20.d odd 2 1 300.2.e.d 8
20.e even 4 2 300.2.h.c 16
60.h even 2 1 300.2.e.d 8
60.l odd 4 2 300.2.h.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.e.d 8 5.b even 2 1
300.2.e.d 8 15.d odd 2 1
300.2.e.d 8 20.d odd 2 1
300.2.e.d 8 60.h even 2 1
300.2.e.e yes 8 1.a even 1 1 trivial
300.2.e.e yes 8 3.b odd 2 1 inner
300.2.e.e yes 8 4.b odd 2 1 inner
300.2.e.e yes 8 12.b even 2 1 inner
300.2.h.c 16 5.c odd 4 2
300.2.h.c 16 15.e even 4 2
300.2.h.c 16 20.e even 4 2
300.2.h.c 16 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{4} + 19T_{7}^{2} + 80$$ T7^4 + 19*T7^2 + 80 $$T_{13}^{2} + T_{13} - 10$$ T13^2 + T13 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{6} - 2 T^{4} + 4 T^{2} + \cdots + 16$$
$3$ $$T^{8} - T^{6} + 8 T^{4} - 9 T^{2} + \cdots + 81$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 19 T^{2} + 80)^{2}$$
$11$ $$(T^{4} - 29 T^{2} + 200)^{2}$$
$13$ $$(T^{2} + T - 10)^{4}$$
$17$ $$(T^{4} + 59 T^{2} + 40)^{2}$$
$19$ $$(T^{4} + 26 T^{2} + 5)^{2}$$
$23$ $$(T^{4} - 28 T^{2} + 32)^{2}$$
$29$ $$(T^{4} + 36 T^{2} + 160)^{2}$$
$31$ $$(T^{4} + 55 T^{2} + 500)^{2}$$
$37$ $$(T^{2} + 2 T - 40)^{4}$$
$41$ $$(T^{4} + 115 T^{2} + 1000)^{2}$$
$43$ $$(T^{4} + 71 T^{2} + 20)^{2}$$
$47$ $$(T^{4} - 52 T^{2} + 512)^{2}$$
$53$ $$(T^{4} + 36 T^{2} + 160)^{2}$$
$59$ $$T^{8}$$
$61$ $$(T^{2} + 5 T - 4)^{4}$$
$67$ $$(T^{4} + 34 T^{2} + 125)^{2}$$
$71$ $$(T^{4} - 284 T^{2} + 20000)^{2}$$
$73$ $$(T^{2} - 9 T + 10)^{4}$$
$79$ $$(T^{4} + 76 T^{2} + 1280)^{2}$$
$83$ $$(T^{4} - 13 T^{2} + 32)^{2}$$
$89$ $$(T^{4} + 51 T^{2} + 640)^{2}$$
$97$ $$(T^{2} - 3 T - 90)^{4}$$