# Properties

 Label 150.3.b.b Level $150$ Weight $3$ Character orbit 150.b Analytic conductor $4.087$ 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] = [150,3,Mod(149,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("150.149");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.b (of order $$2$$, degree $$1$$, 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: $$4.08720396540$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 7x^{4} + 1$$ x^8 + 7*x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - \beta_{3} q^{3} + 2 q^{4} + ( - \beta_{4} - 1) q^{6} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2) q^{9}+O(q^{10})$$ q + b2 * q^2 - b3 * q^3 + 2 * q^4 + (-b4 - 1) * q^6 + (b7 - 2*b3 + b2 + 2*b1) * q^7 + 2*b2 * q^8 + (-b6 + 2*b5 + b4 + 2) * q^9 $$q + \beta_{2} q^{2} - \beta_{3} q^{3} + 2 q^{4} + ( - \beta_{4} - 1) q^{6} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2) q^{9} - 3 \beta_{5} q^{11} - 2 \beta_{3} q^{12} + 5 \beta_1 q^{13} + ( - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4}) q^{14} + 4 q^{16} + ( - \beta_{7} - 4 \beta_{3} + 14 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{7} + 2 \beta_{3} + \beta_{2} - 5 \beta_1) q^{18} + ( - 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - 8) q^{19} + (\beta_{6} + 10 \beta_{5} + 3 \beta_{4} - 13) q^{21} + 6 \beta_1 q^{22} + ( - \beta_{7} - 4 \beta_{3} - \beta_{2} + 2 \beta_1) q^{23} + ( - 2 \beta_{4} - 2) q^{24} - 5 \beta_{5} q^{26} + (\beta_{7} - \beta_{3} - 17 \beta_{2} + 4 \beta_1) q^{27} + (2 \beta_{7} - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{28} + (4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4}) q^{29} + 8 q^{31} + 4 \beta_{2} q^{32} + ( - 3 \beta_{7} + 6 \beta_{2} - 3 \beta_1) q^{33} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} + 24) q^{34} + ( - 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4) q^{36} + ( - 4 \beta_{7} + 8 \beta_{3} - 4 \beta_{2} - 15 \beta_1) q^{37} + (2 \beta_{7} + 8 \beta_{3} - 12 \beta_{2} - 4 \beta_1) q^{38} + (5 \beta_{6} + 5 \beta_{5} + 10) q^{39} + (2 \beta_{6} - 10 \beta_{5} + 2 \beta_{4}) q^{41} + ( - \beta_{7} + 6 \beta_{3} - 16 \beta_{2} - 19 \beta_1) q^{42} + ( - 3 \beta_{7} + 6 \beta_{3} - 3 \beta_{2} - 10 \beta_1) q^{43} - 6 \beta_{5} q^{44} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} - 6) q^{46} + (5 \beta_{7} + 20 \beta_{3} + 5 \beta_{2} - 10 \beta_1) q^{47} - 4 \beta_{3} q^{48} + (4 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} - 45) q^{49} + ( - 3 \beta_{6} + 6 \beta_{5} - 12 \beta_{4} + 18) q^{51} + 10 \beta_1 q^{52} + (\beta_{7} + 4 \beta_{3} - 14 \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{6} - 5 \beta_{5} - \beta_{4} - 35) q^{54} + ( - 4 \beta_{6} - 6 \beta_{5} - 4 \beta_{4}) q^{56} + ( - 3 \beta_{7} + 8 \beta_{3} - 30 \beta_{2} + 15 \beta_1) q^{57} + ( - 4 \beta_{7} + 8 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{58} + ( - 4 \beta_{6} - 22 \beta_{5} - 4 \beta_{4}) q^{59} + ( - 4 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 16) q^{61} + 8 \beta_{2} q^{62} + (7 \beta_{7} + 8 \beta_{3} - 11 \beta_{2} + 28 \beta_1) q^{63} + 8 q^{64} + (6 \beta_{6} + 6 \beta_{5} + 12) q^{66} + (3 \beta_{7} - 6 \beta_{3} + 3 \beta_{2} - 38 \beta_1) q^{67} + ( - 2 \beta_{7} - 8 \beta_{3} + 28 \beta_{2} + 4 \beta_1) q^{68} + ( - 3 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} + 33) q^{69} + ( - 4 \beta_{6} - 19 \beta_{5} - 4 \beta_{4}) q^{71} + (2 \beta_{7} + 4 \beta_{3} + 2 \beta_{2} - 10 \beta_1) q^{72} + (4 \beta_{7} - 8 \beta_{3} + 4 \beta_{2} - 21 \beta_1) q^{73} + (8 \beta_{6} + 19 \beta_{5} + 8 \beta_{4}) q^{74} + ( - 4 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 16) q^{76} + ( - 6 \beta_{7} - 24 \beta_{3} + 12 \beta_1) q^{77} + ( - 5 \beta_{7} + 10 \beta_{2} - 5 \beta_1) q^{78} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} + 28) q^{79} + (4 \beta_{6} + 10 \beta_{5} + 20 \beta_{4} + 7) q^{81} + ( - 2 \beta_{7} + 4 \beta_{3} - 2 \beta_{2} + 22 \beta_1) q^{82} + ( - \beta_{7} - 4 \beta_{3} + 11 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{6} + 20 \beta_{5} + 6 \beta_{4} - 26) q^{84} + (6 \beta_{6} + 13 \beta_{5} + 6 \beta_{4}) q^{86} + ( - 12 \beta_{3} + 36 \beta_{2} + 36 \beta_1) q^{87} + 12 \beta_1 q^{88} + ( - 4 \beta_{6} + 8 \beta_{5} - 4 \beta_{4}) q^{89} + (10 \beta_{6} - 5 \beta_{5} - 20 \beta_{4} - 20) q^{91} + ( - 2 \beta_{7} - 8 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{92} - 8 \beta_{3} q^{93} + ( - 10 \beta_{6} + 5 \beta_{5} + 20 \beta_{4} + 30) q^{94} + ( - 4 \beta_{4} - 4) q^{96} + (4 \beta_{7} - 8 \beta_{3} + 4 \beta_{2} + 41 \beta_1) q^{97} + ( - 4 \beta_{7} - 16 \beta_{3} - 37 \beta_{2} + 8 \beta_1) q^{98} + ( - 6 \beta_{6} - 15 \beta_{5} - 12 \beta_{4} + 48) q^{99}+O(q^{100})$$ q + b2 * q^2 - b3 * q^3 + 2 * q^4 + (-b4 - 1) * q^6 + (b7 - 2*b3 + b2 + 2*b1) * q^7 + 2*b2 * q^8 + (-b6 + 2*b5 + b4 + 2) * q^9 - 3*b5 * q^11 - 2*b3 * q^12 + 5*b1 * q^13 + (-2*b6 - 3*b5 - 2*b4) * q^14 + 4 * q^16 + (-b7 - 4*b3 + 14*b2 + 2*b1) * q^17 + (b7 + 2*b3 + b2 - 5*b1) * q^18 + (-2*b6 + b5 + 4*b4 - 8) * q^19 + (b6 + 10*b5 + 3*b4 - 13) * q^21 + 6*b1 * q^22 + (-b7 - 4*b3 - b2 + 2*b1) * q^23 + (-2*b4 - 2) * q^24 - 5*b5 * q^26 + (b7 - b3 - 17*b2 + 4*b1) * q^27 + (2*b7 - 4*b3 + 2*b2 + 4*b1) * q^28 + (4*b6 + 4*b5 + 4*b4) * q^29 + 8 * q^31 + 4*b2 * q^32 + (-3*b7 + 6*b2 - 3*b1) * q^33 + (2*b6 - b5 - 4*b4 + 24) * q^34 + (-2*b6 + 4*b5 + 2*b4 + 4) * q^36 + (-4*b7 + 8*b3 - 4*b2 - 15*b1) * q^37 + (2*b7 + 8*b3 - 12*b2 - 4*b1) * q^38 + (5*b6 + 5*b5 + 10) * q^39 + (2*b6 - 10*b5 + 2*b4) * q^41 + (-b7 + 6*b3 - 16*b2 - 19*b1) * q^42 + (-3*b7 + 6*b3 - 3*b2 - 10*b1) * q^43 - 6*b5 * q^44 + (2*b6 - b5 - 4*b4 - 6) * q^46 + (5*b7 + 20*b3 + 5*b2 - 10*b1) * q^47 - 4*b3 * q^48 + (4*b6 - 2*b5 - 8*b4 - 45) * q^49 + (-3*b6 + 6*b5 - 12*b4 + 18) * q^51 + 10*b1 * q^52 + (b7 + 4*b3 - 14*b2 - 2*b1) * q^53 + (-2*b6 - 5*b5 - b4 - 35) * q^54 + (-4*b6 - 6*b5 - 4*b4) * q^56 + (-3*b7 + 8*b3 - 30*b2 + 15*b1) * q^57 + (-4*b7 + 8*b3 - 4*b2 - 4*b1) * q^58 + (-4*b6 - 22*b5 - 4*b4) * q^59 + (-4*b6 + 2*b5 + 8*b4 - 16) * q^61 + 8*b2 * q^62 + (7*b7 + 8*b3 - 11*b2 + 28*b1) * q^63 + 8 * q^64 + (6*b6 + 6*b5 + 12) * q^66 + (3*b7 - 6*b3 + 3*b2 - 38*b1) * q^67 + (-2*b7 - 8*b3 + 28*b2 + 4*b1) * q^68 + (-3*b6 + 6*b5 + 3*b4 + 33) * q^69 + (-4*b6 - 19*b5 - 4*b4) * q^71 + (2*b7 + 4*b3 + 2*b2 - 10*b1) * q^72 + (4*b7 - 8*b3 + 4*b2 - 21*b1) * q^73 + (8*b6 + 19*b5 + 8*b4) * q^74 + (-4*b6 + 2*b5 + 8*b4 - 16) * q^76 + (-6*b7 - 24*b3 + 12*b1) * q^77 + (-5*b7 + 10*b2 - 5*b1) * q^78 + (2*b6 - b5 - 4*b4 + 28) * q^79 + (4*b6 + 10*b5 + 20*b4 + 7) * q^81 + (-2*b7 + 4*b3 - 2*b2 + 22*b1) * q^82 + (-b7 - 4*b3 + 11*b2 + 2*b1) * q^83 + (2*b6 + 20*b5 + 6*b4 - 26) * q^84 + (6*b6 + 13*b5 + 6*b4) * q^86 + (-12*b3 + 36*b2 + 36*b1) * q^87 + 12*b1 * q^88 + (-4*b6 + 8*b5 - 4*b4) * q^89 + (10*b6 - 5*b5 - 20*b4 - 20) * q^91 + (-2*b7 - 8*b3 - 2*b2 + 4*b1) * q^92 - 8*b3 * q^93 + (-10*b6 + 5*b5 + 20*b4 + 30) * q^94 + (-4*b4 - 4) * q^96 + (4*b7 - 8*b3 + 4*b2 + 41*b1) * q^97 + (-4*b7 - 16*b3 - 37*b2 + 8*b1) * q^98 + (-6*b6 - 15*b5 - 12*b4 + 48) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{4} - 8 q^{6} + 16 q^{9}+O(q^{10})$$ 8 * q + 16 * q^4 - 8 * q^6 + 16 * q^9 $$8 q + 16 q^{4} - 8 q^{6} + 16 q^{9} + 32 q^{16} - 64 q^{19} - 104 q^{21} - 16 q^{24} + 64 q^{31} + 192 q^{34} + 32 q^{36} + 80 q^{39} - 48 q^{46} - 360 q^{49} + 144 q^{51} - 280 q^{54} - 128 q^{61} + 64 q^{64} + 96 q^{66} + 264 q^{69} - 128 q^{76} + 224 q^{79} + 56 q^{81} - 208 q^{84} - 160 q^{91} + 240 q^{94} - 32 q^{96} + 384 q^{99}+O(q^{100})$$ 8 * q + 16 * q^4 - 8 * q^6 + 16 * q^9 + 32 * q^16 - 64 * q^19 - 104 * q^21 - 16 * q^24 + 64 * q^31 + 192 * q^34 + 32 * q^36 + 80 * q^39 - 48 * q^46 - 360 * q^49 + 144 * q^51 - 280 * q^54 - 128 * q^61 + 64 * q^64 + 96 * q^66 + 264 * q^69 - 128 * q^76 + 224 * q^79 + 56 * q^81 - 208 * q^84 - 160 * q^91 + 240 * q^94 - 32 * q^96 + 384 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7x^{4} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{6} - 16\nu^{2} ) / 3$$ (-2*v^6 - 16*v^2) / 3 $$\beta_{2}$$ $$=$$ $$( -2\nu^{7} + \nu^{5} - 13\nu^{3} + 5\nu ) / 3$$ (-2*v^7 + v^5 - 13*v^3 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} - \nu^{6} - 2\nu^{4} - 21\nu^{3} - 8\nu^{2} - 3\nu - 7 ) / 3$$ (-3*v^7 - v^6 - 2*v^4 - 21*v^3 - 8*v^2 - 3*v - 7) / 3 $$\beta_{4}$$ $$=$$ $$( 2\nu^{7} - 3\nu^{6} - 2\nu^{5} + 16\nu^{3} - 18\nu^{2} - 16\nu ) / 3$$ (2*v^7 - 3*v^6 - 2*v^5 + 16*v^3 - 18*v^2 - 16*v) / 3 $$\beta_{5}$$ $$=$$ $$( 4\nu^{7} + 2\nu^{5} + 26\nu^{3} + 10\nu ) / 3$$ (4*v^7 + 2*v^5 + 26*v^3 + 10*v) / 3 $$\beta_{6}$$ $$=$$ $$-2\nu^{7} - 2\nu^{6} - 14\nu^{3} - 12\nu^{2} + 2\nu$$ -2*v^7 - 2*v^6 - 14*v^3 - 12*v^2 + 2*v $$\beta_{7}$$ $$=$$ $$( 8\nu^{7} + 2\nu^{5} - 4\nu^{4} + 58\nu^{3} + 22\nu - 14 ) / 3$$ (8*v^7 + 2*v^5 - 4*v^4 + 58*v^3 + 22*v - 14) / 3
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 2\beta_{2} + \beta_1 ) / 12$$ (b7 + b6 - 2*b5 - 2*b4 - 2*b3 - 2*b2 + b1) / 12 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 12$$ (2*b6 + 2*b5 + 2*b4 - 9*b1) / 12 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} - 5\beta_{5} + 4\beta_{4} - 4\beta_{3} + 14\beta_{2} + 2\beta_1 ) / 12$$ (2*b7 - 2*b6 - 5*b5 + 4*b4 - 4*b3 + 14*b2 + 2*b1) / 12 $$\nu^{4}$$ $$=$$ $$( -\beta_{7} - 4\beta_{3} + 2\beta_{2} + 2\beta _1 - 14 ) / 4$$ (-b7 - 4*b3 + 2*b2 + 2*b1 - 14) / 4 $$\nu^{5}$$ $$=$$ $$( -5\beta_{7} - 5\beta_{6} + 19\beta_{5} + 10\beta_{4} + 10\beta_{3} + 28\beta_{2} - 5\beta_1 ) / 12$$ (-5*b7 - 5*b6 + 19*b5 + 10*b4 + 10*b3 + 28*b2 - 5*b1) / 12 $$\nu^{6}$$ $$=$$ $$( -8\beta_{6} - 8\beta_{5} - 8\beta_{4} + 27\beta_1 ) / 6$$ (-8*b6 - 8*b5 - 8*b4 + 27*b1) / 6 $$\nu^{7}$$ $$=$$ $$( -13\beta_{7} + 13\beta_{6} + 37\beta_{5} - 26\beta_{4} + 26\beta_{3} - 100\beta_{2} - 13\beta_1 ) / 12$$ (-13*b7 + 13*b6 + 37*b5 - 26*b4 + 26*b3 - 100*b2 - 13*b1) / 12

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 1.14412 − 1.14412i 1.14412 + 1.14412i −0.437016 − 0.437016i −0.437016 + 0.437016i −1.14412 − 1.14412i −1.14412 + 1.14412i 0.437016 − 0.437016i 0.437016 + 0.437016i
−1.41421 −1.52896 2.58114i 2.00000 0 2.16228 + 3.65028i 7.48683i −2.82843 −4.32456 + 7.89292i 0
149.2 −1.41421 −1.52896 + 2.58114i 2.00000 0 2.16228 3.65028i 7.48683i −2.82843 −4.32456 7.89292i 0
149.3 −1.41421 2.94317 0.581139i 2.00000 0 −4.16228 + 0.821854i 11.4868i −2.82843 8.32456 3.42079i 0
149.4 −1.41421 2.94317 + 0.581139i 2.00000 0 −4.16228 0.821854i 11.4868i −2.82843 8.32456 + 3.42079i 0
149.5 1.41421 −2.94317 0.581139i 2.00000 0 −4.16228 0.821854i 11.4868i 2.82843 8.32456 + 3.42079i 0
149.6 1.41421 −2.94317 + 0.581139i 2.00000 0 −4.16228 + 0.821854i 11.4868i 2.82843 8.32456 3.42079i 0
149.7 1.41421 1.52896 2.58114i 2.00000 0 2.16228 3.65028i 7.48683i 2.82843 −4.32456 7.89292i 0
149.8 1.41421 1.52896 + 2.58114i 2.00000 0 2.16228 + 3.65028i 7.48683i 2.82843 −4.32456 + 7.89292i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.8 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.b.b 8
3.b odd 2 1 inner 150.3.b.b 8
4.b odd 2 1 1200.3.c.k 8
5.b even 2 1 inner 150.3.b.b 8
5.c odd 4 1 30.3.d.a 4
5.c odd 4 1 150.3.d.c 4
12.b even 2 1 1200.3.c.k 8
15.d odd 2 1 inner 150.3.b.b 8
15.e even 4 1 30.3.d.a 4
15.e even 4 1 150.3.d.c 4
20.d odd 2 1 1200.3.c.k 8
20.e even 4 1 240.3.l.c 4
20.e even 4 1 1200.3.l.u 4
40.i odd 4 1 960.3.l.e 4
40.k even 4 1 960.3.l.f 4
45.k odd 12 2 810.3.h.a 8
45.l even 12 2 810.3.h.a 8
60.h even 2 1 1200.3.c.k 8
60.l odd 4 1 240.3.l.c 4
60.l odd 4 1 1200.3.l.u 4
120.q odd 4 1 960.3.l.f 4
120.w even 4 1 960.3.l.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 5.c odd 4 1
30.3.d.a 4 15.e even 4 1
150.3.b.b 8 1.a even 1 1 trivial
150.3.b.b 8 3.b odd 2 1 inner
150.3.b.b 8 5.b even 2 1 inner
150.3.b.b 8 15.d odd 2 1 inner
150.3.d.c 4 5.c odd 4 1
150.3.d.c 4 15.e even 4 1
240.3.l.c 4 20.e even 4 1
240.3.l.c 4 60.l odd 4 1
810.3.h.a 8 45.k odd 12 2
810.3.h.a 8 45.l even 12 2
960.3.l.e 4 40.i odd 4 1
960.3.l.e 4 120.w even 4 1
960.3.l.f 4 40.k even 4 1
960.3.l.f 4 120.q odd 4 1
1200.3.c.k 8 4.b odd 2 1
1200.3.c.k 8 12.b even 2 1
1200.3.c.k 8 20.d odd 2 1
1200.3.c.k 8 60.h even 2 1
1200.3.l.u 4 20.e even 4 1
1200.3.l.u 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 188T_{7}^{2} + 7396$$ acting on $$S_{3}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{4}$$
$3$ $$T^{8} - 8 T^{6} + 18 T^{4} + \cdots + 6561$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 188 T^{2} + 7396)^{2}$$
$11$ $$(T^{2} + 72)^{4}$$
$13$ $$(T^{2} + 100)^{4}$$
$17$ $$(T^{4} - 936 T^{2} + 11664)^{2}$$
$19$ $$(T^{2} + 16 T - 296)^{4}$$
$23$ $$(T^{4} - 396 T^{2} + 26244)^{2}$$
$29$ $$(T^{2} + 720)^{4}$$
$31$ $$(T - 8)^{8}$$
$37$ $$(T^{4} + 3848 T^{2} + 913936)^{2}$$
$41$ $$(T^{4} + 2664 T^{2} + 944784)^{2}$$
$43$ $$(T^{4} + 2012 T^{2} + 376996)^{2}$$
$47$ $$(T^{4} - 9900 T^{2} + 16402500)^{2}$$
$53$ $$(T^{4} - 936 T^{2} + 11664)^{2}$$
$59$ $$(T^{4} + 6624 T^{2} + 3504384)^{2}$$
$61$ $$(T^{2} + 32 T - 1184)^{4}$$
$67$ $$(T^{4} + 15068 T^{2} + 34975396)^{2}$$
$71$ $$(T^{4} + 5040 T^{2} + 1166400)^{2}$$
$73$ $$(T^{4} + 7880 T^{2} + 1123600)^{2}$$
$79$ $$(T^{2} - 56 T + 424)^{4}$$
$83$ $$(T^{4} - 684 T^{2} + 324)^{2}$$
$89$ $$(T^{4} + 3744 T^{2} + 186624)^{2}$$
$97$ $$(T^{4} + 13832 T^{2} + 16289296)^{2}$$