# Properties

 Label 300.3.c.e Level $300$ Weight $3$ Character orbit 300.c Analytic conductor $8.174$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,3,Mod(151,300)]

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, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("300.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.c (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: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4069419264.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 7x^{6} + 50x^{4} - 84x^{3} + 55x^{2} - 12x + 1$$ x^8 - 7*x^6 + 50*x^4 - 84*x^3 + 55*x^2 - 12*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} - \beta_{5} q^{3} + (\beta_{7} - 1) q^{4} + ( - \beta_{2} - 1) q^{6} + (\beta_{7} + \beta_{6} + \beta_{5}) q^{7} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 3) q^{8} - 3 q^{9}+O(q^{10})$$ q + b4 * q^2 - b5 * q^3 + (b7 - 1) * q^4 + (-b2 - 1) * q^6 + (b7 + b6 + b5) * q^7 + (b6 - 2*b5 - b4 - b3 - 3) * q^8 - 3 * q^9 $$q + \beta_{4} q^{2} - \beta_{5} q^{3} + (\beta_{7} - 1) q^{4} + ( - \beta_{2} - 1) q^{6} + (\beta_{7} + \beta_{6} + \beta_{5}) q^{7} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 3) q^{8} - 3 q^{9} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{5} - \beta_{4} + \beta_1) q^{12} + (\beta_{7} - \beta_{5} - \beta_{4} - 4 \beta_{3} + \beta_1 - 2) q^{13} + (\beta_{7} - 3 \beta_{5} + \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{14} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + \cdots + 3) q^{16}+ \cdots + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 3) q^{99}+O(q^{100})$$ q + b4 * q^2 - b5 * q^3 + (b7 - 1) * q^4 + (-b2 - 1) * q^6 + (b7 + b6 + b5) * q^7 + (b6 - 2*b5 - b4 - b3 - 3) * q^8 - 3 * q^9 + (-b6 - b5 + b4 - 2*b3 - 2*b2 - b1 - 1) * q^11 + (b5 - b4 + b1) * q^12 + (b7 - b5 - b4 - 4*b3 + b1 - 2) * q^13 + (b7 - 3*b5 + b4 - 4*b3 - b2 + b1 + 2) * q^14 + (-b7 - 2*b6 - 3*b5 - 3*b4 - 2*b3 - 4*b2 + b1 + 3) * q^16 + (b7 + 2*b6 + 2*b5 + 6*b4 - b3 + 5*b2 - 2*b1 + 2) * q^17 - 3*b4 * q^18 + (-2*b7 + b6 - 4*b5 - 7*b4 + 2*b3 + 2*b2 - b1 - 1) * q^19 + (-b5 - b4 - 3*b3 - b2 + b1 + 2) * q^21 + (-3*b7 - 3*b5 - 5*b4 + 8*b3 + 2*b2 + b1 + 1) * q^22 + (4*b7 + b6 - 5*b5 + 5*b4 - 4*b3 - 4*b2 - b1 - 1) * q^23 + (-b6 + 2*b5 + b4 - 3*b3 - 5) * q^24 + (-4*b7 - 4*b6 - 8*b5 - 5*b4 - 2*b2 - 10) * q^26 + 3*b5 * q^27 + (-2*b7 - 4*b6 - 7*b5 - b4 - 4*b2 + b1 - 14) * q^28 + (-5*b7 - 2*b5 - 2*b4 - b3 - 7*b2 + 2*b1 - 6) * q^29 + (3*b7 + 3*b6 - 7*b5 - 4*b4 - 4*b3 - 4*b2 - 4*b1 - 4) * q^31 + (-6*b7 - 2*b6 + 6*b5 + 6*b3 + 2*b1 - 12) * q^32 + (3*b7 - 2*b6 - 4*b4 + 3*b3 + b2) * q^33 + (5*b7 - 15*b5 + b4 - 3*b1 + 15) * q^34 + (-3*b7 + 3) * q^36 + (8*b7 + 2*b5 + 2*b4 - 2*b3 + 10*b2 - 2*b1 + 24) * q^37 + (-5*b7 + 3*b5 + b4 - 5*b2 - b1 + 20) * q^38 + (-3*b7 - 4*b6 + 2*b5 + 3*b4 + b1 + 1) * q^39 + (9*b7 - 2*b6 - 2*b5 - 6*b4 - 9*b3 + 5*b2 + 2*b1 - 4) * q^41 + (-3*b7 - 4*b6 - 3*b5 - 2*b2 + b1 - 7) * q^42 + (-4*b7 - 3*b6 + 8*b5 - 5*b4 - 2*b3 - 2*b2 - 3*b1 - 3) * q^43 + (4*b7 + 4*b6 + 22*b5 + 10*b4 - 8*b3 - 4*b2 - 2*b1 + 4) * q^44 + (b7 - 15*b5 - 5*b4 - 6*b2 + 5*b1 - 11) * q^46 + (10*b7 + 4*b6 + 12*b5 + 10*b4 - 8*b3 - 8*b2 - 2*b1 - 2) * q^47 + (-3*b7 - 2*b6 - 5*b5 - 9*b4 + 6*b3 + 4*b2 - b1 - 3) * q^48 + (-2*b7 - 4*b6 - 8*b4 + 14*b3 - 6*b2 + 2) * q^49 + (6*b7 - b6 + b5 + 15*b4 - 6*b3 - 6*b2 + b1 + 1) * q^51 + (-9*b7 + 14*b5 - 14*b4 + 16*b3 - 2*b1 - 7) * q^52 + (-5*b7 + 4*b6 + 8*b4 - 7*b3 - b2 + 38) * q^53 + (3*b2 + 3) * q^54 + (-4*b7 + b6 + 14*b5 - 17*b4 + 11*b3 - 23) * q^56 + (3*b7 + 2*b6 + 2*b5 + 6*b4 - 3*b3 + 7*b2 - 2*b1 - 7) * q^57 + (-b7 - 8*b6 + 19*b5 - 5*b4 - 4*b2 + 7*b1 + 1) * q^58 + (-4*b6 + 2*b5 + 12*b4 + 4*b1 + 4) * q^59 + (-b7 + 8*b6 + 7*b5 + 23*b4 - 2*b3 + 14*b2 - 7*b1 + 24) * q^61 + (-9*b7 - 21*b5 - 9*b4 + 4*b3 - 9*b2 + 7*b1 + 28) * q^62 + (-3*b7 - 3*b6 - 3*b5) * q^63 + (6*b7 + 30*b5 - 6*b4 + 8*b2 - 2*b1 + 2) * q^64 + (-3*b7 + 8*b6 + b5 + b4 + 4*b2 - 3*b1 - 13) * q^66 + (12*b7 - b6 - 14*b5 + 25*b4 - 14*b3 - 14*b2 - b1 - 1) * q^67 + (-2*b7 + 8*b6 - 16*b5 + 12*b4 + 4*b3 - 12*b2 - 10) * q^68 + (3*b7 - 4*b6 - 4*b5 - 12*b4 - 3*b3 - 5*b2 + 4*b1 - 16) * q^69 + (2*b7 - 3*b6 - b5 + 15*b4 + 5*b1 + 5) * q^71 + (-3*b6 + 6*b5 + 3*b4 + 3*b3 + 9) * q^72 + (4*b7 + 4*b6 + 10*b5 + 18*b4 + 14*b3 + 18*b2 - 10*b1 - 20) * q^73 + (-2*b7 + 8*b6 - 34*b5 + 20*b4 + 4*b2 - 10*b1 - 10) * q^74 + (-4*b6 + 13*b5 + 19*b4 + 8*b3 + 4*b2 + 5*b1 + 24) * q^76 + (b7 + 4*b6 - 2*b5 + 6*b4 - 19*b3 + 3*b2 + 2*b1 + 46) * q^77 + (12*b5 - 2*b4 + 12*b3 + 9*b2 - 4*b1 - 15) * q^78 + (-10*b7 - 6*b5 - 18*b4 + 12*b3 + 12*b2 + 2*b1 + 2) * q^79 + 9 * q^81 + (-15*b7 - 35*b5 - 13*b4 - 7*b1 - 45) * q^82 + (20*b7 + 12*b6 + 18*b5 + 20*b4 - 4*b3 - 4*b2 + 4*b1 + 4) * q^83 + (-3*b7 + 14*b5 - 10*b4 + 12*b3 + 4*b2 - 2*b1 - 17) * q^84 + (-13*b7 + 3*b5 - 11*b4 + 24*b3 + 17*b2 - b1 + 34) * q^86 + (-6*b7 - b6 - b5 - 15*b4 - 5*b1 - 5) * q^87 + (4*b7 - 6*b6 - 28*b5 - 2*b4 - 2*b3 + 16*b2 + 8*b1 + 30) * q^88 + (-8*b7 + 4*b6 + 8*b4 - 4*b3 - 4*b2 + 16) * q^89 + (-2*b7 + 15*b6 + 48*b5 - 37*b4 + 14*b3 + 14*b2 - 3*b1 - 3) * q^91 + (-4*b6 + 14*b5 - 6*b4 - 16*b3 - 20*b2 + 6*b1 - 48) * q^92 + (12*b7 - 4*b6 - 3*b5 - 11*b4 - 9*b3 + 5*b2 + 3*b1 - 12) * q^93 + (4*b7 - 36*b5 - 8*b4 - 8*b3 + 6*b2 + 12*b1 + 2) * q^94 + (-6*b7 + 6*b6 + 14*b5 + 6*b3 - 6*b1 + 12) * q^96 + (12*b7 - 8*b6 - 8*b5 - 24*b4 - 12*b3 - 4*b2 + 8*b1 - 35) * q^97 + (2*b7 + 16*b6 + 42*b5 + 12*b4 + 8*b2 + 2*b1 - 2) * q^98 + (3*b6 + 3*b5 - 3*b4 + 6*b3 + 6*b2 + 3*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 8 q^{4} - 6 q^{6} - 20 q^{8} - 24 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 8 * q^4 - 6 * q^6 - 20 * q^8 - 24 * q^9 $$8 q - 2 q^{2} - 8 q^{4} - 6 q^{6} - 20 q^{8} - 24 q^{9} - 8 q^{13} + 22 q^{14} + 40 q^{16} + 6 q^{18} + 24 q^{21} - 4 q^{22} - 36 q^{24} - 66 q^{26} - 104 q^{28} - 32 q^{29} - 112 q^{32} + 124 q^{34} + 24 q^{36} + 176 q^{37} + 170 q^{38} - 16 q^{41} - 54 q^{42} + 40 q^{44} - 76 q^{46} - 24 q^{48} + 16 q^{49} - 56 q^{52} + 304 q^{53} + 18 q^{54} - 172 q^{56} - 72 q^{57} + 12 q^{58} + 136 q^{61} + 238 q^{62} + 16 q^{64} - 108 q^{66} - 88 q^{68} - 96 q^{69} + 60 q^{72} - 240 q^{73} - 108 q^{74} + 120 q^{76} + 384 q^{77} - 150 q^{78} + 72 q^{81} - 320 q^{82} - 144 q^{84} + 214 q^{86} + 200 q^{88} + 128 q^{89} - 312 q^{92} - 72 q^{93} + 12 q^{94} + 96 q^{96} - 216 q^{97} - 60 q^{98}+O(q^{100})$$ 8 * q - 2 * q^2 - 8 * q^4 - 6 * q^6 - 20 * q^8 - 24 * q^9 - 8 * q^13 + 22 * q^14 + 40 * q^16 + 6 * q^18 + 24 * q^21 - 4 * q^22 - 36 * q^24 - 66 * q^26 - 104 * q^28 - 32 * q^29 - 112 * q^32 + 124 * q^34 + 24 * q^36 + 176 * q^37 + 170 * q^38 - 16 * q^41 - 54 * q^42 + 40 * q^44 - 76 * q^46 - 24 * q^48 + 16 * q^49 - 56 * q^52 + 304 * q^53 + 18 * q^54 - 172 * q^56 - 72 * q^57 + 12 * q^58 + 136 * q^61 + 238 * q^62 + 16 * q^64 - 108 * q^66 - 88 * q^68 - 96 * q^69 + 60 * q^72 - 240 * q^73 - 108 * q^74 + 120 * q^76 + 384 * q^77 - 150 * q^78 + 72 * q^81 - 320 * q^82 - 144 * q^84 + 214 * q^86 + 200 * q^88 + 128 * q^89 - 312 * q^92 - 72 * q^93 + 12 * q^94 + 96 * q^96 - 216 * q^97 - 60 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7x^{6} + 50x^{4} - 84x^{3} + 55x^{2} - 12x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -73\nu^{7} - 599\nu^{6} - 16\nu^{5} + 3897\nu^{4} + 291\nu^{3} - 20680\nu^{2} + 20620\nu - 5669 ) / 515$$ (-73*v^7 - 599*v^6 - 16*v^5 + 3897*v^4 + 291*v^3 - 20680*v^2 + 20620*v - 5669) / 515 $$\beta_{2}$$ $$=$$ $$( -291\nu^{7} + 32\nu^{6} + 1968\nu^{5} - 381\nu^{4} - 14163\nu^{3} + 27350\nu^{2} - 21000\nu + 3582 ) / 515$$ (-291*v^7 + 32*v^6 + 1968*v^5 - 381*v^4 - 14163*v^3 + 27350*v^2 - 21000*v + 3582) / 515 $$\beta_{3}$$ $$=$$ $$( 347\nu^{7} - 24\nu^{6} - 2506\nu^{5} + 157\nu^{4} + 17961\nu^{3} - 30040\nu^{2} + 17810\nu - 1399 ) / 515$$ (347*v^7 - 24*v^6 - 2506*v^5 + 157*v^4 + 17961*v^3 - 30040*v^2 + 17810*v - 1399) / 515 $$\beta_{4}$$ $$=$$ $$( 347\nu^{7} - 24\nu^{6} - 2506\nu^{5} + 157\nu^{4} + 17961\nu^{3} - 30040\nu^{2} + 16780\nu - 1399 ) / 515$$ (347*v^7 - 24*v^6 - 2506*v^5 + 157*v^4 + 17961*v^3 - 30040*v^2 + 16780*v - 1399) / 515 $$\beta_{5}$$ $$=$$ $$( -486\nu^{7} - 143\nu^{6} + 3308\nu^{5} + 914\nu^{4} - 23728\nu^{3} + 34050\nu^{2} - 19070\nu + 2372 ) / 515$$ (-486*v^7 - 143*v^6 + 3308*v^5 + 914*v^4 - 23728*v^3 + 34050*v^2 - 19070*v + 2372) / 515 $$\beta_{6}$$ $$=$$ $$( 924\nu^{7} + 132\nu^{6} - 6302\nu^{5} - 606\nu^{4} + 45672\nu^{3} - 72710\nu^{2} + 44700\nu - 5953 ) / 515$$ (924*v^7 + 132*v^6 - 6302*v^5 - 606*v^4 + 45672*v^3 - 72710*v^2 + 44700*v - 5953) / 515 $$\beta_{7}$$ $$=$$ $$( 1134\nu^{7} + 162\nu^{6} - 8062\nu^{5} - 1446\nu^{4} + 57082\nu^{3} - 85630\nu^{2} + 45870\nu - 5363 ) / 515$$ (1134*v^7 + 162*v^6 - 8062*v^5 - 1446*v^4 + 57082*v^3 - 85630*v^2 + 45870*v - 5363) / 515
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{3} ) / 2$$ (-b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - 2\beta_{6} - 7\beta_{5} + \beta_{4} - 2\beta_{3} + \beta _1 + 7 ) / 4$$ (-b7 - 2*b6 - 7*b5 + b4 - 2*b3 + b1 + 7) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + 4\beta_{6} + 6\beta_{5} - 6\beta_{4} + 3\beta_{3} + 3\beta_{2} ) / 2$$ (b7 + 4*b6 + 6*b5 - 6*b4 + 3*b3 + 3*b2) / 2 $$\nu^{4}$$ $$=$$ $$( -7\beta_{7} - 11\beta_{6} - 22\beta_{5} + 11\beta_{4} + 7\beta_{3} - 4\beta_{2} - 22 ) / 2$$ (-7*b7 - 11*b6 - 22*b5 + 11*b4 + 7*b3 - 4*b2 - 22) / 2 $$\nu^{5}$$ $$=$$ $$( 19\beta_{7} + 58\beta_{6} + 55\beta_{5} - 19\beta_{4} - 88\beta_{3} + 36\beta_{2} + 11\beta _1 + 195 ) / 4$$ (19*b7 + 58*b6 + 55*b5 - 19*b4 - 88*b3 + 36*b2 + 11*b1 + 195) / 4 $$\nu^{6}$$ $$=$$ $$( -25\beta_{7} - 26\beta_{6} + 29\beta_{5} - 23\beta_{4} + 190\beta_{3} - 22\beta_{2} - 29\beta _1 - 407 ) / 2$$ (-25*b7 - 26*b6 + 29*b5 - 23*b4 + 190*b3 - 22*b2 - 29*b1 - 407) / 2 $$\nu^{7}$$ $$=$$ $$( -25\beta_{7} - 81\beta_{6} - 403\beta_{5} + 330\beta_{4} - 598\beta_{3} - 25\beta_{2} + 81\beta _1 + 997 ) / 2$$ (-25*b7 - 81*b6 - 403*b5 + 330*b4 - 598*b3 - 25*b2 + 81*b1 + 997) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 1.65359 − 0.954702i 1.65359 + 0.954702i −2.65095 + 1.53053i −2.65095 − 1.53053i 0.845613 + 0.488215i 0.845613 − 0.488215i 0.151747 − 0.0876113i 0.151747 + 0.0876113i
−1.97650 0.305673i 1.73205i 3.81313 + 1.20833i 0 −0.529441 + 3.42340i 0.329898i −7.16731 3.55383i −3.00000 0
151.2 −1.97650 + 0.305673i 1.73205i 3.81313 1.20833i 0 −0.529441 3.42340i 0.329898i −7.16731 + 3.55383i −3.00000 0
151.3 −0.534079 1.92737i 1.73205i −3.42952 + 2.05874i 0 −3.33830 + 0.925051i 11.9716i 5.79958 + 5.51043i −3.00000 0
151.4 −0.534079 + 1.92737i 1.73205i −3.42952 2.05874i 0 −3.33830 0.925051i 11.9716i 5.79958 5.51043i −3.00000 0
151.5 0.177680 1.99209i 1.73205i −3.93686 0.707911i 0 3.45040 + 0.307751i 1.19501i −2.10973 + 7.71680i −3.00000 0
151.6 0.177680 + 1.99209i 1.73205i −3.93686 + 0.707911i 0 3.45040 0.307751i 1.19501i −2.10973 7.71680i −3.00000 0
151.7 1.33290 1.49110i 1.73205i −0.446749 3.97497i 0 −2.58266 2.30865i 6.56834i −6.52255 4.63210i −3.00000 0
151.8 1.33290 + 1.49110i 1.73205i −0.446749 + 3.97497i 0 −2.58266 + 2.30865i 6.56834i −6.52255 + 4.63210i −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.c.e 8
3.b odd 2 1 900.3.c.t 8
4.b odd 2 1 inner 300.3.c.e 8
5.b even 2 1 300.3.c.g yes 8
5.c odd 4 2 300.3.f.c 16
12.b even 2 1 900.3.c.t 8
15.d odd 2 1 900.3.c.n 8
15.e even 4 2 900.3.f.h 16
20.d odd 2 1 300.3.c.g yes 8
20.e even 4 2 300.3.f.c 16
60.h even 2 1 900.3.c.n 8
60.l odd 4 2 900.3.f.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.c.e 8 1.a even 1 1 trivial
300.3.c.e 8 4.b odd 2 1 inner
300.3.c.g yes 8 5.b even 2 1
300.3.c.g yes 8 20.d odd 2 1
300.3.f.c 16 5.c odd 4 2
300.3.f.c 16 20.e even 4 2
900.3.c.n 8 15.d odd 2 1
900.3.c.n 8 60.h even 2 1
900.3.c.t 8 3.b odd 2 1
900.3.c.t 8 12.b even 2 1
900.3.f.h 16 15.e even 4 2
900.3.f.h 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 $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{8} + 188T_{7}^{6} + 6470T_{7}^{4} + 9532T_{7}^{2} + 961$$ T7^8 + 188*T7^6 + 6470*T7^4 + 9532*T7^2 + 961 $$T_{13}^{4} + 4T_{13}^{3} - 418T_{13}^{2} - 3916T_{13} - 1559$$ T13^4 + 4*T13^3 - 418*T13^2 - 3916*T13 - 1559

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{7} + 6 T^{6} + 16 T^{5} + \cdots + 256$$
$3$ $$(T^{2} + 3)^{4}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 188 T^{6} + 6470 T^{4} + \cdots + 961$$
$11$ $$T^{8} + 704 T^{6} + \cdots + 30824704$$
$13$ $$(T^{4} + 4 T^{3} - 418 T^{2} - 3916 T - 1559)^{2}$$
$17$ $$(T^{4} - 832 T^{2} - 1920 T + 132400)^{2}$$
$19$ $$T^{8} + 1132 T^{6} + \cdots + 1099651921$$
$23$ $$T^{8} + 1984 T^{6} + \cdots + 1611540736$$
$29$ $$(T^{4} + 16 T^{3} - 1600 T^{2} + \cdots + 93616)^{2}$$
$31$ $$T^{8} + 5660 T^{6} + \cdots + 819736484449$$
$37$ $$(T^{4} - 88 T^{3} - 712 T^{2} + \cdots - 4548464)^{2}$$
$41$ $$(T^{4} + 8 T^{3} - 4968 T^{2} + \cdots + 3504448)^{2}$$
$43$ $$T^{8} + 6892 T^{6} + \cdots + 974581609681$$
$47$ $$T^{8} + 12016 T^{6} + \cdots + 13610196640000$$
$53$ $$(T^{4} - 152 T^{3} + 4568 T^{2} + \cdots - 4946624)^{2}$$
$59$ $$T^{8} + 6192 T^{6} + \cdots + 195562066176$$
$61$ $$(T^{4} - 68 T^{3} - 8098 T^{2} + \cdots + 9745129)^{2}$$
$67$ $$T^{8} + 21548 T^{6} + \cdots + 23066205847441$$
$71$ $$T^{8} + 8816 T^{6} + \cdots + 11235904000000$$
$73$ $$(T^{4} + 120 T^{3} - 8584 T^{2} + \cdots - 43602032)^{2}$$
$79$ $$T^{8} + 14528 T^{6} + \cdots + 2278988775424$$
$83$ $$T^{8} + \cdots + 120362665464064$$
$89$ $$(T^{4} - 64 T^{3} - 3328 T^{2} + \cdots - 1507328)^{2}$$
$97$ $$(T^{4} + 108 T^{3} - 10986 T^{2} + \cdots + 15618033)^{2}$$