# Properties

 Label 605.2.g.f Level $605$ Weight $2$ Character orbit 605.g Analytic conductor $4.831$ 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] = [605,2,Mod(81,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("605.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.g (of order $$5$$, degree $$4$$, 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: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.64000000.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16$$ x^8 + 2*x^6 + 4*x^4 + 8*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{2} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{3} + (\beta_{6} + 2 \beta_1) q^{4} - \beta_{2} q^{5} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{6} - 2 \beta_{6} q^{7} + ( - \beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_1) q^{8} + ( - 5 \beta_{6} - 5 \beta_{4} - 5 \beta_{2} - 5) q^{9}+O(q^{10})$$ q + (-b6 - b4 + b3 - b2 - 1) * q^2 + (2*b7 + 2*b5 + 2*b3 + 2*b1) * q^3 + (b6 + 2*b1) * q^4 - b2 * q^5 + (-2*b7 - 4*b2) * q^6 - 2*b6 * q^7 + (-b7 - b5 + 3*b4 - b3 - b1) * q^8 + (-5*b6 - 5*b4 - 5*b2 - 5) * q^9 $$q + ( - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{2} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{3} + (\beta_{6} + 2 \beta_1) q^{4} - \beta_{2} q^{5} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{6} - 2 \beta_{6} q^{7} + ( - \beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_1) q^{8} + ( - 5 \beta_{6} - 5 \beta_{4} - 5 \beta_{2} - 5) q^{9} + ( - \beta_{5} - 1) q^{10} + ( - 2 \beta_{5} - 8) q^{12} + (4 \beta_{6} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{13} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{14} + 2 \beta_1 q^{15} + 3 \beta_{2} q^{16} + (2 \beta_{7} + 4 \beta_{2}) q^{17} + (5 \beta_{6} + 5 \beta_1) q^{18} + (\beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 1) q^{20} + 4 \beta_{5} q^{21} - 2 \beta_{5} q^{23} + (4 \beta_{6} + 4 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 4) q^{24} + \beta_{4} q^{25} - 2 \beta_1 q^{26} - 4 \beta_{7} q^{27} + ( - 4 \beta_{7} - 2 \beta_{2}) q^{28} + (2 \beta_{6} - 4 \beta_1) q^{29} + ( - 2 \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{30} + (\beta_{5} - 3) q^{32} + (6 \beta_{5} + 8) q^{34} + ( - 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} - 2) q^{35} + ( - 10 \beta_{7} - 10 \beta_{5} + 5 \beta_{4} - 10 \beta_{3} - 10 \beta_1) q^{36} + ( - 2 \beta_{6} - 4 \beta_1) q^{37} + (8 \beta_{7} - 8 \beta_{2}) q^{39} + ( - 3 \beta_{6} - \beta_1) q^{40} + 6 \beta_{4} q^{41} + ( - 8 \beta_{6} - 8 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} - 8) q^{42} - 6 q^{43} - 5 q^{45} + (4 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 4) q^{46} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{47} - 6 \beta_1 q^{48} - 3 \beta_{2} q^{49} + (\beta_{7} + \beta_{2}) q^{50} + ( - 8 \beta_{6} - 8 \beta_1) q^{51} + (6 \beta_{7} + 6 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 6 \beta_1) q^{52} + ( - 6 \beta_{6} - 6 \beta_{4} + 4 \beta_{3} - 6 \beta_{2} - 6) q^{53} + ( - 4 \beta_{5} - 8) q^{54} + ( - 2 \beta_{5} - 6) q^{56} + (2 \beta_{7} + 2 \beta_{5} - 6 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{58} + ( - 4 \beta_{6} + 4 \beta_1) q^{59} + (2 \beta_{7} + 8 \beta_{2}) q^{60} + ( - 8 \beta_{7} + 2 \beta_{2}) q^{61} - 10 \beta_{4} q^{63} + (7 \beta_{6} + 7 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} + 7) q^{64} + ( - 2 \beta_{5} + 4) q^{65} + (6 \beta_{5} + 4) q^{67} + ( - 12 \beta_{6} - 12 \beta_{4} + 10 \beta_{3} - 12 \beta_{2} - 12) q^{68} + 8 \beta_{4} q^{69} + (2 \beta_{6} + 2 \beta_1) q^{70} + 8 \beta_{7} q^{71} + (5 \beta_{7} + 15 \beta_{2}) q^{72} + ( - 4 \beta_{6} + 2 \beta_1) q^{73} + (6 \beta_{7} + 6 \beta_{5} - 10 \beta_{4} + 6 \beta_{3} + 6 \beta_1) q^{74} - 2 \beta_{3} q^{75} + 8 q^{78} + ( - 4 \beta_{6} - 4 \beta_{4} - 4 \beta_{2} - 4) q^{79} - 3 \beta_{4} q^{80} + \beta_{6} q^{81} + (6 \beta_{7} + 6 \beta_{2}) q^{82} - 6 \beta_{2} q^{83} + (16 \beta_{6} + 4 \beta_1) q^{84} + (2 \beta_{7} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{85} + (6 \beta_{6} + 6 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} + 6) q^{86} + ( - 4 \beta_{5} + 16) q^{87} + ( - 8 \beta_{5} - 2) q^{89} + (5 \beta_{6} + 5 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 5) q^{90} + (4 \beta_{7} + 4 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} + 4 \beta_1) q^{91} + ( - 8 \beta_{6} - 2 \beta_1) q^{92} + (2 \beta_{7} + 4 \beta_{2}) q^{94} + ( - 6 \beta_{7} - 6 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 6 \beta_1) q^{96} + (2 \beta_{6} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 2) q^{97} + ( - 3 \beta_{5} - 3) q^{98}+O(q^{100})$$ q + (-b6 - b4 + b3 - b2 - 1) * q^2 + (2*b7 + 2*b5 + 2*b3 + 2*b1) * q^3 + (b6 + 2*b1) * q^4 - b2 * q^5 + (-2*b7 - 4*b2) * q^6 - 2*b6 * q^7 + (-b7 - b5 + 3*b4 - b3 - b1) * q^8 + (-5*b6 - 5*b4 - 5*b2 - 5) * q^9 + (-b5 - 1) * q^10 + (-2*b5 - 8) * q^12 + (4*b6 + 4*b4 + 2*b3 + 4*b2 + 4) * q^13 + (2*b7 + 2*b5 - 2*b4 + 2*b3 + 2*b1) * q^14 + 2*b1 * q^15 + 3*b2 * q^16 + (2*b7 + 4*b2) * q^17 + (5*b6 + 5*b1) * q^18 + (b6 + b4 - 2*b3 + b2 + 1) * q^20 + 4*b5 * q^21 - 2*b5 * q^23 + (4*b6 + 4*b4 - 6*b3 + 4*b2 + 4) * q^24 + b4 * q^25 - 2*b1 * q^26 - 4*b7 * q^27 + (-4*b7 - 2*b2) * q^28 + (2*b6 - 4*b1) * q^29 + (-2*b7 - 2*b5 + 4*b4 - 2*b3 - 2*b1) * q^30 + (b5 - 3) * q^32 + (6*b5 + 8) * q^34 + (-2*b6 - 2*b4 - 2*b2 - 2) * q^35 + (-10*b7 - 10*b5 + 5*b4 - 10*b3 - 10*b1) * q^36 + (-2*b6 - 4*b1) * q^37 + (8*b7 - 8*b2) * q^39 + (-3*b6 - b1) * q^40 + 6*b4 * q^41 + (-8*b6 - 8*b4 + 4*b3 - 8*b2 - 8) * q^42 - 6 * q^43 - 5 * q^45 + (4*b6 + 4*b4 - 2*b3 + 4*b2 + 4) * q^46 + (-2*b7 - 2*b5 - 2*b3 - 2*b1) * q^47 - 6*b1 * q^48 - 3*b2 * q^49 + (b7 + b2) * q^50 + (-8*b6 - 8*b1) * q^51 + (6*b7 + 6*b5 + 4*b4 + 6*b3 + 6*b1) * q^52 + (-6*b6 - 6*b4 + 4*b3 - 6*b2 - 6) * q^53 + (-4*b5 - 8) * q^54 + (-2*b5 - 6) * q^56 + (2*b7 + 2*b5 - 6*b4 + 2*b3 + 2*b1) * q^58 + (-4*b6 + 4*b1) * q^59 + (2*b7 + 8*b2) * q^60 + (-8*b7 + 2*b2) * q^61 - 10*b4 * q^63 + (7*b6 + 7*b4 - 2*b3 + 7*b2 + 7) * q^64 + (-2*b5 + 4) * q^65 + (6*b5 + 4) * q^67 + (-12*b6 - 12*b4 + 10*b3 - 12*b2 - 12) * q^68 + 8*b4 * q^69 + (2*b6 + 2*b1) * q^70 + 8*b7 * q^71 + (5*b7 + 15*b2) * q^72 + (-4*b6 + 2*b1) * q^73 + (6*b7 + 6*b5 - 10*b4 + 6*b3 + 6*b1) * q^74 - 2*b3 * q^75 + 8 * q^78 + (-4*b6 - 4*b4 - 4*b2 - 4) * q^79 - 3*b4 * q^80 + b6 * q^81 + (6*b7 + 6*b2) * q^82 - 6*b2 * q^83 + (16*b6 + 4*b1) * q^84 + (2*b7 + 2*b5 - 4*b4 + 2*b3 + 2*b1) * q^85 + (6*b6 + 6*b4 - 6*b3 + 6*b2 + 6) * q^86 + (-4*b5 + 16) * q^87 + (-8*b5 - 2) * q^89 + (5*b6 + 5*b4 - 5*b3 + 5*b2 + 5) * q^90 + (4*b7 + 4*b5 + 8*b4 + 4*b3 + 4*b1) * q^91 + (-8*b6 - 2*b1) * q^92 + (2*b7 + 4*b2) * q^94 + (-6*b7 - 6*b5 - 4*b4 - 6*b3 - 6*b1) * q^96 + (2*b6 + 2*b4 + 4*b3 + 2*b2 + 2) * q^97 + (-3*b5 - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} - 10 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 8 * q^6 + 4 * q^7 - 6 * q^8 - 10 * q^9 $$8 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} - 10 q^{9} - 8 q^{10} - 64 q^{12} + 8 q^{13} + 4 q^{14} - 6 q^{16} - 8 q^{17} - 10 q^{18} + 2 q^{20} + 8 q^{24} - 2 q^{25} + 4 q^{28} - 4 q^{29} - 8 q^{30} - 24 q^{32} + 64 q^{34} - 4 q^{35} - 10 q^{36} + 4 q^{37} + 16 q^{39} + 6 q^{40} - 12 q^{41} - 16 q^{42} - 48 q^{43} - 40 q^{45} + 8 q^{46} + 6 q^{49} - 2 q^{50} + 16 q^{51} - 8 q^{52} - 12 q^{53} - 64 q^{54} - 48 q^{56} + 12 q^{58} + 8 q^{59} - 16 q^{60} - 4 q^{61} + 20 q^{63} + 14 q^{64} + 32 q^{65} + 32 q^{67} - 24 q^{68} - 16 q^{69} - 4 q^{70} - 30 q^{72} + 8 q^{73} + 20 q^{74} + 64 q^{78} - 8 q^{79} + 6 q^{80} - 2 q^{81} - 12 q^{82} + 12 q^{83} - 32 q^{84} + 8 q^{85} + 12 q^{86} + 128 q^{87} - 16 q^{89} + 10 q^{90} - 16 q^{91} + 16 q^{92} - 8 q^{94} + 8 q^{96} + 4 q^{97} - 24 q^{98}+O(q^{100})$$ 8 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 8 * q^6 + 4 * q^7 - 6 * q^8 - 10 * q^9 - 8 * q^10 - 64 * q^12 + 8 * q^13 + 4 * q^14 - 6 * q^16 - 8 * q^17 - 10 * q^18 + 2 * q^20 + 8 * q^24 - 2 * q^25 + 4 * q^28 - 4 * q^29 - 8 * q^30 - 24 * q^32 + 64 * q^34 - 4 * q^35 - 10 * q^36 + 4 * q^37 + 16 * q^39 + 6 * q^40 - 12 * q^41 - 16 * q^42 - 48 * q^43 - 40 * q^45 + 8 * q^46 + 6 * q^49 - 2 * q^50 + 16 * q^51 - 8 * q^52 - 12 * q^53 - 64 * q^54 - 48 * q^56 + 12 * q^58 + 8 * q^59 - 16 * q^60 - 4 * q^61 + 20 * q^63 + 14 * q^64 + 32 * q^65 + 32 * q^67 - 24 * q^68 - 16 * q^69 - 4 * q^70 - 30 * q^72 + 8 * q^73 + 20 * q^74 + 64 * q^78 - 8 * q^79 + 6 * q^80 - 2 * q^81 - 12 * q^82 + 12 * q^83 - 32 * q^84 + 8 * q^85 + 12 * q^86 + 128 * q^87 - 16 * q^89 + 10 * q^90 - 16 * q^91 + 16 * q^92 - 8 * q^94 + 8 * q^96 + 4 * q^97 - 24 * q^98

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.437016 + 1.34500i −0.437016 − 1.34500i 1.14412 + 0.831254i −1.14412 − 0.831254i 0.437016 − 1.34500i −0.437016 + 1.34500i 1.14412 − 0.831254i −1.14412 + 0.831254i
−1.95314 1.41904i −0.874032 + 2.68999i 1.18305 + 3.64105i 0.809017 0.587785i 5.52431 4.01365i −0.618034 1.90211i 1.36407 4.19817i −4.04508 2.93893i −2.41421
81.2 0.335106 + 0.243469i 0.874032 2.68999i −0.565015 1.73894i 0.809017 0.587785i 0.947822 0.688633i −0.618034 1.90211i 0.490035 1.50817i −4.04508 2.93893i 0.414214
251.1 −0.127999 + 0.393941i −2.28825 + 1.66251i 1.47923 + 1.07472i −0.309017 0.951057i −0.362036 1.11423i 1.61803 + 1.17557i −1.28293 + 0.932102i 1.54508 4.75528i 0.414214
251.2 0.746033 2.29605i 2.28825 1.66251i −3.09726 2.25029i −0.309017 0.951057i −2.11010 6.49422i 1.61803 + 1.17557i −3.57117 + 2.59461i 1.54508 4.75528i −2.41421
366.1 −1.95314 + 1.41904i −0.874032 2.68999i 1.18305 3.64105i 0.809017 + 0.587785i 5.52431 + 4.01365i −0.618034 + 1.90211i 1.36407 + 4.19817i −4.04508 + 2.93893i −2.41421
366.2 0.335106 0.243469i 0.874032 + 2.68999i −0.565015 + 1.73894i 0.809017 + 0.587785i 0.947822 + 0.688633i −0.618034 + 1.90211i 0.490035 + 1.50817i −4.04508 + 2.93893i 0.414214
511.1 −0.127999 0.393941i −2.28825 1.66251i 1.47923 1.07472i −0.309017 + 0.951057i −0.362036 + 1.11423i 1.61803 1.17557i −1.28293 0.932102i 1.54508 + 4.75528i 0.414214
511.2 0.746033 + 2.29605i 2.28825 + 1.66251i −3.09726 + 2.25029i −0.309017 + 0.951057i −2.11010 + 6.49422i 1.61803 1.17557i −3.57117 2.59461i 1.54508 + 4.75528i −2.41421
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.f 8
11.b odd 2 1 605.2.g.l 8
11.c even 5 1 55.2.a.b 2
11.c even 5 3 inner 605.2.g.f 8
11.d odd 10 1 605.2.a.d 2
11.d odd 10 3 605.2.g.l 8
33.f even 10 1 5445.2.a.y 2
33.h odd 10 1 495.2.a.b 2
44.g even 10 1 9680.2.a.bn 2
44.h odd 10 1 880.2.a.m 2
55.h odd 10 1 3025.2.a.o 2
55.j even 10 1 275.2.a.c 2
55.k odd 20 2 275.2.b.d 4
77.j odd 10 1 2695.2.a.f 2
88.l odd 10 1 3520.2.a.bo 2
88.o even 10 1 3520.2.a.bn 2
132.o even 10 1 7920.2.a.ch 2
143.n even 10 1 9295.2.a.g 2
165.o odd 10 1 2475.2.a.x 2
165.v even 20 2 2475.2.c.l 4
220.n odd 10 1 4400.2.a.bn 2
220.v even 20 2 4400.2.b.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 11.c even 5 1
275.2.a.c 2 55.j even 10 1
275.2.b.d 4 55.k odd 20 2
495.2.a.b 2 33.h odd 10 1
605.2.a.d 2 11.d odd 10 1
605.2.g.f 8 1.a even 1 1 trivial
605.2.g.f 8 11.c even 5 3 inner
605.2.g.l 8 11.b odd 2 1
605.2.g.l 8 11.d odd 10 3
880.2.a.m 2 44.h odd 10 1
2475.2.a.x 2 165.o odd 10 1
2475.2.c.l 4 165.v even 20 2
2695.2.a.f 2 77.j odd 10 1
3025.2.a.o 2 55.h odd 10 1
3520.2.a.bn 2 88.o even 10 1
3520.2.a.bo 2 88.l odd 10 1
4400.2.a.bn 2 220.n odd 10 1
4400.2.b.q 4 220.v even 20 2
5445.2.a.y 2 33.f even 10 1
7920.2.a.ch 2 132.o even 10 1
9295.2.a.g 2 143.n even 10 1
9680.2.a.bn 2 44.g even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{8} + 2T_{2}^{7} + 5T_{2}^{6} + 12T_{2}^{5} + 29T_{2}^{4} - 12T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1$$ T2^8 + 2*T2^7 + 5*T2^6 + 12*T2^5 + 29*T2^4 - 12*T2^3 + 5*T2^2 - 2*T2 + 1 $$T_{3}^{8} + 8T_{3}^{6} + 64T_{3}^{4} + 512T_{3}^{2} + 4096$$ T3^8 + 8*T3^6 + 64*T3^4 + 512*T3^2 + 4096

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{7} + 5 T^{6} + 12 T^{5} + \cdots + 1$$
$3$ $$T^{8} + 8 T^{6} + 64 T^{4} + \cdots + 4096$$
$5$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$7$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2}$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 8 T^{7} + 56 T^{6} + \cdots + 4096$$
$17$ $$T^{8} + 8 T^{7} + 56 T^{6} + \cdots + 4096$$
$19$ $$T^{8}$$
$23$ $$(T^{2} - 8)^{4}$$
$29$ $$T^{8} + 4 T^{7} + 44 T^{6} + \cdots + 614656$$
$31$ $$T^{8}$$
$37$ $$T^{8} - 4 T^{7} + 44 T^{6} + \cdots + 614656$$
$41$ $$(T^{4} + 6 T^{3} + 36 T^{2} + 216 T + 1296)^{2}$$
$43$ $$(T + 6)^{8}$$
$47$ $$T^{8} + 8 T^{6} + 64 T^{4} + \cdots + 4096$$
$53$ $$T^{8} + 12 T^{7} + 140 T^{6} + \cdots + 256$$
$59$ $$T^{8} - 8 T^{7} + 80 T^{6} + \cdots + 65536$$
$61$ $$T^{8} + 4 T^{7} + 140 T^{6} + \cdots + 236421376$$
$67$ $$(T^{2} - 8 T - 56)^{4}$$
$71$ $$T^{8} + 128 T^{6} + \cdots + 268435456$$
$73$ $$T^{8} - 8 T^{7} + 56 T^{6} + \cdots + 4096$$
$79$ $$(T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2}$$
$83$ $$(T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2}$$
$89$ $$(T^{2} + 4 T - 124)^{4}$$
$97$ $$T^{8} - 4 T^{7} + 44 T^{6} + \cdots + 614656$$