# Properties

 Label 225.4.k.b Level $225$ Weight $4$ Character orbit 225.k Analytic conductor $13.275$ 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] = [225,4,Mod(49,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ x^8 + 5*x^6 + 16*x^4 + 45*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 9) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{3} + ( - 3 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 3) q^{4} + ( - 3 \beta_{6} - 18 \beta_{3} + 3 \beta_{2} - 9) q^{6} + ( - 3 \beta_{7} + 2 \beta_1) q^{7} + (16 \beta_{5} - \beta_{4} + 16 \beta_1) q^{8} + ( - 6 \beta_{6} - 24 \beta_{3} + 3 \beta_{2} - 27) q^{9}+O(q^{10})$$ q + (-b7 + b1) * q^2 + (-b7 - b5 + b4) * q^3 + (-3*b6 - 4*b3 + 3*b2 - 3) * q^4 + (-3*b6 - 18*b3 + 3*b2 - 9) * q^6 + (-3*b7 + 2*b1) * q^7 + (16*b5 - b4 + 16*b1) * q^8 + (-6*b6 - 24*b3 + 3*b2 - 27) * q^9 $$q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{3} + ( - 3 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 3) q^{4} + ( - 3 \beta_{6} - 18 \beta_{3} + 3 \beta_{2} - 9) q^{6} + ( - 3 \beta_{7} + 2 \beta_1) q^{7} + (16 \beta_{5} - \beta_{4} + 16 \beta_1) q^{8} + ( - 6 \beta_{6} - 24 \beta_{3} + 3 \beta_{2} - 27) q^{9} + ( - 29 \beta_{3} - 8 \beta_{2} - 29) q^{11} + ( - 2 \beta_{7} + 52 \beta_{5} + 5 \beta_{4} + 30 \beta_1) q^{12} + (15 \beta_{7} + 13 \beta_{5} + 15 \beta_{4} + 15 \beta_1) q^{13} + ( - 8 \beta_{6} - 34 \beta_{3} + 8 \beta_{2} - 8) q^{14} + (8 \beta_{3} - 9 \beta_{2} + 8) q^{16} + ( - 54 \beta_{5} - 9 \beta_{4} - 54 \beta_1) q^{17} + (18 \beta_{7} + 72 \beta_{5} + 27 \beta_{4} + 72 \beta_1) q^{18} + ( - 27 \beta_{6} + 25) q^{19} + ( - 8 \beta_{6} - 53 \beta_{3} + 7 \beta_{2} - 27) q^{21} + (21 \beta_{7} - 6 \beta_{5} + 21 \beta_{4} + 21 \beta_1) q^{22} + (19 \beta_{7} - 7 \beta_{5} + 19 \beta_{4} + 19 \beta_1) q^{23} + (18 \beta_{6} - 24 \beta_{3} + 15 \beta_{2} + 18) q^{24} + (28 \beta_{6} - 118) q^{26} + (36 \beta_{7} + 117 \beta_{5} + 18 \beta_{4} + 135 \beta_1) q^{27} + (92 \beta_{5} + 18 \beta_{4} + 92 \beta_1) q^{28} + ( - 25 \beta_{3} - \beta_{2} - 25) q^{29} + (3 \beta_{6} + 23 \beta_{3} - 3 \beta_{2} + 3) q^{31} + ( - 9 \beta_{7} - 216 \beta_{5} - 9 \beta_{4} - 9 \beta_1) q^{32} + (66 \beta_{7} - 6 \beta_{5} + 21 \beta_{4} + 159 \beta_1) q^{33} + (180 \beta_{3} - 63 \beta_{2} + 180) q^{34} + (60 \beta_{6} - 156 \beta_{3} + 15 \beta_{2} - 108) q^{36} + ( - 2 \beta_{5} - 54 \beta_{4} - 2 \beta_1) q^{37} + ( - 79 \beta_{7} + 241 \beta_1) q^{38} + (11 \beta_{6} + 107 \beta_{3} + 17 \beta_{2} - 135) q^{39} + (98 \beta_{6} + 115 \beta_{3} - 98 \beta_{2} + 98) q^{41} + (18 \beta_{7} + 162 \beta_{5} + 60 \beta_{4} + 150 \beta_1) q^{42} + ( - 6 \beta_{7} - 47 \beta_1) q^{43} + (79 \beta_{6} + 155) q^{44} + (12 \beta_{6} - 126) q^{46} + ( - 91 \beta_{7} + 154 \beta_1) q^{47} + ( - 7 \beta_{7} - 88 \beta_{5} - 17 \beta_{4} + 57 \beta_1) q^{48} + ( - 21 \beta_{6} + 246 \beta_{3} + 21 \beta_{2} - 21) q^{49} + ( - 36 \beta_{6} + 180 \beta_{3} - 63 \beta_{2} + 162) q^{51} + (54 \beta_{7} - 358 \beta_1) q^{52} + ( - 54 \beta_{5} - 162 \beta_{4} - 54 \beta_1) q^{53} + (54 \beta_{6} - 54 \beta_{3} + 81 \beta_{2} - 324) q^{54} + ( - 56 \beta_{3} + 46 \beta_{2} - 56) q^{56} + ( - 79 \beta_{7} + 164 \beta_{5} - 2 \beta_{4} + 405 \beta_1) q^{57} + (24 \beta_{7} + 42 \beta_{5} + 24 \beta_{4} + 24 \beta_1) q^{58} + (136 \beta_{6} - 331 \beta_{3} - 136 \beta_{2} + 136) q^{59} + ( - 167 \beta_{3} - 105 \beta_{2} - 167) q^{61} + ( - 70 \beta_{5} - 26 \beta_{4} - 70 \beta_1) q^{62} + (57 \beta_{7} + 192 \beta_{5} + 78 \beta_{4} + 216 \beta_1) q^{63} + ( - 153 \beta_{6} + 287) q^{64} + ( - 33 \beta_{6} + 174 \beta_{3} + 48 \beta_{2} - 189) q^{66} + (66 \beta_{7} + 527 \beta_{5} + 66 \beta_{4} + 66 \beta_1) q^{67} + ( - 171 \beta_{7} - 432 \beta_{5} - 171 \beta_{4} - 171 \beta_1) q^{68} + ( - 33 \beta_{6} + 159 \beta_{3} + 45 \beta_{2} - 171) q^{69} + ( - 144 \beta_{6} + 612) q^{71} + (27 \beta_{7} + 99 \beta_{4} - 333 \beta_1) q^{72} + ( - 106 \beta_{5} + 243 \beta_{4} - 106 \beta_1) q^{73} + (436 \beta_{3} - 56 \beta_{2} + 436) q^{74} + ( - 183 \beta_{6} - 856 \beta_{3} + 183 \beta_{2} - 183) q^{76} + (71 \beta_{7} - 47 \beta_{5} + 71 \beta_{4} + 71 \beta_1) q^{77} + (174 \beta_{7} - 78 \beta_{5} - 90 \beta_{4} - 420 \beta_1) q^{78} + ( - 247 \beta_{3} - 309 \beta_{2} - 247) q^{79} + (135 \beta_{6} + 216 \beta_{3} + 135 \beta_{2}) q^{81} + ( - 1014 \beta_{5} - 213 \beta_{4} - 1014 \beta_1) q^{82} + ( - 107 \beta_{7} - 460 \beta_1) q^{83} + (56 \beta_{6} - 328 \beta_{3} + 110 \beta_{2} - 324) q^{84} + (35 \beta_{6} + 34 \beta_{3} - 35 \beta_{2} + 35) q^{86} + (51 \beta_{7} + 42 \beta_{5} + 24 \beta_{4} + 84 \beta_1) q^{87} + ( - 165 \beta_{7} - 693 \beta_1) q^{88} + (72 \beta_{6} + 234) q^{89} + (69 \beta_{6} - 356) q^{91} + ( - 2 \beta_{7} - 430 \beta_1) q^{92} + ( - 17 \beta_{7} - 71 \beta_{5} - 43 \beta_{4} - 87 \beta_1) q^{93} + ( - 336 \beta_{6} - 1218 \beta_{3} + 336 \beta_{2} - 336) q^{94} + ( - 423 \beta_{6} + 144 \beta_{3} + 198 \beta_{2} + 81) q^{96} + ( - 102 \beta_{7} + 317 \beta_1) q^{97} + ( - 324 \beta_{5} - 225 \beta_{4} - 324 \beta_1) q^{98} + ( - 81 \beta_{6} + 801 \beta_{3} + 279 \beta_{2} + 216) q^{99}+O(q^{100})$$ q + (-b7 + b1) * q^2 + (-b7 - b5 + b4) * q^3 + (-3*b6 - 4*b3 + 3*b2 - 3) * q^4 + (-3*b6 - 18*b3 + 3*b2 - 9) * q^6 + (-3*b7 + 2*b1) * q^7 + (16*b5 - b4 + 16*b1) * q^8 + (-6*b6 - 24*b3 + 3*b2 - 27) * q^9 + (-29*b3 - 8*b2 - 29) * q^11 + (-2*b7 + 52*b5 + 5*b4 + 30*b1) * q^12 + (15*b7 + 13*b5 + 15*b4 + 15*b1) * q^13 + (-8*b6 - 34*b3 + 8*b2 - 8) * q^14 + (8*b3 - 9*b2 + 8) * q^16 + (-54*b5 - 9*b4 - 54*b1) * q^17 + (18*b7 + 72*b5 + 27*b4 + 72*b1) * q^18 + (-27*b6 + 25) * q^19 + (-8*b6 - 53*b3 + 7*b2 - 27) * q^21 + (21*b7 - 6*b5 + 21*b4 + 21*b1) * q^22 + (19*b7 - 7*b5 + 19*b4 + 19*b1) * q^23 + (18*b6 - 24*b3 + 15*b2 + 18) * q^24 + (28*b6 - 118) * q^26 + (36*b7 + 117*b5 + 18*b4 + 135*b1) * q^27 + (92*b5 + 18*b4 + 92*b1) * q^28 + (-25*b3 - b2 - 25) * q^29 + (3*b6 + 23*b3 - 3*b2 + 3) * q^31 + (-9*b7 - 216*b5 - 9*b4 - 9*b1) * q^32 + (66*b7 - 6*b5 + 21*b4 + 159*b1) * q^33 + (180*b3 - 63*b2 + 180) * q^34 + (60*b6 - 156*b3 + 15*b2 - 108) * q^36 + (-2*b5 - 54*b4 - 2*b1) * q^37 + (-79*b7 + 241*b1) * q^38 + (11*b6 + 107*b3 + 17*b2 - 135) * q^39 + (98*b6 + 115*b3 - 98*b2 + 98) * q^41 + (18*b7 + 162*b5 + 60*b4 + 150*b1) * q^42 + (-6*b7 - 47*b1) * q^43 + (79*b6 + 155) * q^44 + (12*b6 - 126) * q^46 + (-91*b7 + 154*b1) * q^47 + (-7*b7 - 88*b5 - 17*b4 + 57*b1) * q^48 + (-21*b6 + 246*b3 + 21*b2 - 21) * q^49 + (-36*b6 + 180*b3 - 63*b2 + 162) * q^51 + (54*b7 - 358*b1) * q^52 + (-54*b5 - 162*b4 - 54*b1) * q^53 + (54*b6 - 54*b3 + 81*b2 - 324) * q^54 + (-56*b3 + 46*b2 - 56) * q^56 + (-79*b7 + 164*b5 - 2*b4 + 405*b1) * q^57 + (24*b7 + 42*b5 + 24*b4 + 24*b1) * q^58 + (136*b6 - 331*b3 - 136*b2 + 136) * q^59 + (-167*b3 - 105*b2 - 167) * q^61 + (-70*b5 - 26*b4 - 70*b1) * q^62 + (57*b7 + 192*b5 + 78*b4 + 216*b1) * q^63 + (-153*b6 + 287) * q^64 + (-33*b6 + 174*b3 + 48*b2 - 189) * q^66 + (66*b7 + 527*b5 + 66*b4 + 66*b1) * q^67 + (-171*b7 - 432*b5 - 171*b4 - 171*b1) * q^68 + (-33*b6 + 159*b3 + 45*b2 - 171) * q^69 + (-144*b6 + 612) * q^71 + (27*b7 + 99*b4 - 333*b1) * q^72 + (-106*b5 + 243*b4 - 106*b1) * q^73 + (436*b3 - 56*b2 + 436) * q^74 + (-183*b6 - 856*b3 + 183*b2 - 183) * q^76 + (71*b7 - 47*b5 + 71*b4 + 71*b1) * q^77 + (174*b7 - 78*b5 - 90*b4 - 420*b1) * q^78 + (-247*b3 - 309*b2 - 247) * q^79 + (135*b6 + 216*b3 + 135*b2) * q^81 + (-1014*b5 - 213*b4 - 1014*b1) * q^82 + (-107*b7 - 460*b1) * q^83 + (56*b6 - 328*b3 + 110*b2 - 324) * q^84 + (35*b6 + 34*b3 - 35*b2 + 35) * q^86 + (51*b7 + 42*b5 + 24*b4 + 84*b1) * q^87 + (-165*b7 - 693*b1) * q^88 + (72*b6 + 234) * q^89 + (69*b6 - 356) * q^91 + (-2*b7 - 430*b1) * q^92 + (-17*b7 - 71*b5 - 43*b4 - 87*b1) * q^93 + (-336*b6 - 1218*b3 + 336*b2 - 336) * q^94 + (-423*b6 + 144*b3 + 198*b2 + 81) * q^96 + (-102*b7 + 317*b1) * q^97 + (-324*b5 - 225*b4 - 324*b1) * q^98 + (-81*b6 + 801*b3 + 279*b2 + 216) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 10 q^{4} + 18 q^{6} - 90 q^{9}+O(q^{10})$$ 8 * q + 10 * q^4 + 18 * q^6 - 90 * q^9 $$8 q + 10 q^{4} + 18 q^{6} - 90 q^{9} - 132 q^{11} + 120 q^{14} + 14 q^{16} + 308 q^{19} + 42 q^{21} + 198 q^{24} - 1056 q^{26} - 102 q^{29} - 86 q^{31} + 594 q^{34} - 450 q^{36} - 1518 q^{39} - 264 q^{41} + 924 q^{44} - 1056 q^{46} - 1026 q^{49} + 594 q^{51} - 2430 q^{54} - 132 q^{56} + 1596 q^{59} - 878 q^{61} + 2908 q^{64} - 1980 q^{66} - 1782 q^{69} + 5472 q^{71} + 1632 q^{74} + 3058 q^{76} - 1606 q^{79} - 1134 q^{81} - 1284 q^{84} - 66 q^{86} + 1584 q^{89} - 3124 q^{91} + 4200 q^{94} + 2160 q^{96} - 594 q^{99}+O(q^{100})$$ 8 * q + 10 * q^4 + 18 * q^6 - 90 * q^9 - 132 * q^11 + 120 * q^14 + 14 * q^16 + 308 * q^19 + 42 * q^21 + 198 * q^24 - 1056 * q^26 - 102 * q^29 - 86 * q^31 + 594 * q^34 - 450 * q^36 - 1518 * q^39 - 264 * q^41 + 924 * q^44 - 1056 * q^46 - 1026 * q^49 + 594 * q^51 - 2430 * q^54 - 132 * q^56 + 1596 * q^59 - 878 * q^61 + 2908 * q^64 - 1980 * q^66 - 1782 * q^69 + 5472 * q^71 + 1632 * q^74 + 3058 * q^76 - 1606 * q^79 - 1134 * q^81 - 1284 * q^84 - 66 * q^86 + 1584 * q^89 - 3124 * q^91 + 4200 * q^94 + 2160 * q^96 - 594 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432$$ (v^7 + 32*v^5 + 16*v^3 + 45*v) / 432 $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48$$ (v^6 - 16*v^4 - 32*v^2 - 51) / 48 $$\beta_{3}$$ $$=$$ $$( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144$$ (-5*v^6 - 16*v^4 - 80*v^2 - 225) / 144 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72$$ (v^7 + 8*v^5 + 40*v^3 + 165*v) / 72 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 48$$ (v^7 + 13*v) / 48 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9$$ (-v^6 - 5*v^4 - 7*v^2 - 27) / 9 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18$$ (v^7 + 2*v^5 + 10*v^3 - 3*v) / 18
 $$\nu$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3$$ (-b7 + 2*b5 + b4) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3$$ (2*b6 - 7*b3 - b2 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3$$ (4*b7 - 11*b5 + 2*b4 - 9*b1) / 3 $$\nu^{4}$$ $$=$$ $$( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3$$ (-5*b6 + 13*b3 - 5*b2) / 3 $$\nu^{5}$$ $$=$$ $$( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3$$ (-b7 - b5 - 2*b4 + 45*b1) / 3 $$\nu^{6}$$ $$=$$ $$( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3$$ (-16*b6 - 16*b3 + 32*b2 - 39) / 3 $$\nu^{7}$$ $$=$$ $$( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3$$ (13*b7 + 118*b5 - 13*b4) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 − 1.68614i −0.396143 − 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i 0.396143 + 1.68614i
−3.78651 + 2.18614i −3.78651 + 3.55842i 5.55842 9.62747i 0 6.55842 21.7518i −10.4935 + 6.05842i 13.6277i 1.67527 26.9480i 0
49.2 −1.18843 + 0.686141i −1.18843 + 5.05842i −3.05842 + 5.29734i 0 −2.05842 6.82701i −4.43132 + 2.55842i 19.3723i −24.1753 12.0232i 0
49.3 1.18843 0.686141i 1.18843 5.05842i −3.05842 + 5.29734i 0 −2.05842 6.82701i 4.43132 2.55842i 19.3723i −24.1753 12.0232i 0
49.4 3.78651 2.18614i 3.78651 3.55842i 5.55842 9.62747i 0 6.55842 21.7518i 10.4935 6.05842i 13.6277i 1.67527 26.9480i 0
124.1 −3.78651 2.18614i −3.78651 3.55842i 5.55842 + 9.62747i 0 6.55842 + 21.7518i −10.4935 6.05842i 13.6277i 1.67527 + 26.9480i 0
124.2 −1.18843 0.686141i −1.18843 5.05842i −3.05842 5.29734i 0 −2.05842 + 6.82701i −4.43132 2.55842i 19.3723i −24.1753 + 12.0232i 0
124.3 1.18843 + 0.686141i 1.18843 + 5.05842i −3.05842 5.29734i 0 −2.05842 + 6.82701i 4.43132 + 2.55842i 19.3723i −24.1753 + 12.0232i 0
124.4 3.78651 + 2.18614i 3.78651 + 3.55842i 5.55842 + 9.62747i 0 6.55842 + 21.7518i 10.4935 + 6.05842i 13.6277i 1.67527 + 26.9480i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.k.b 8
5.b even 2 1 inner 225.4.k.b 8
5.c odd 4 1 9.4.c.a 4
5.c odd 4 1 225.4.e.b 4
9.c even 3 1 inner 225.4.k.b 8
15.e even 4 1 27.4.c.a 4
20.e even 4 1 144.4.i.c 4
45.j even 6 1 inner 225.4.k.b 8
45.k odd 12 1 9.4.c.a 4
45.k odd 12 1 81.4.a.d 2
45.k odd 12 1 225.4.e.b 4
45.k odd 12 1 2025.4.a.g 2
45.l even 12 1 27.4.c.a 4
45.l even 12 1 81.4.a.a 2
45.l even 12 1 2025.4.a.n 2
60.l odd 4 1 432.4.i.c 4
180.v odd 12 1 432.4.i.c 4
180.v odd 12 1 1296.4.a.i 2
180.x even 12 1 144.4.i.c 4
180.x even 12 1 1296.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 5.c odd 4 1
9.4.c.a 4 45.k odd 12 1
27.4.c.a 4 15.e even 4 1
27.4.c.a 4 45.l even 12 1
81.4.a.a 2 45.l even 12 1
81.4.a.d 2 45.k odd 12 1
144.4.i.c 4 20.e even 4 1
144.4.i.c 4 180.x even 12 1
225.4.e.b 4 5.c odd 4 1
225.4.e.b 4 45.k odd 12 1
225.4.k.b 8 1.a even 1 1 trivial
225.4.k.b 8 5.b even 2 1 inner
225.4.k.b 8 9.c even 3 1 inner
225.4.k.b 8 45.j even 6 1 inner
432.4.i.c 4 60.l odd 4 1
432.4.i.c 4 180.v odd 12 1
1296.4.a.i 2 180.v odd 12 1
1296.4.a.u 2 180.x even 12 1
2025.4.a.g 2 45.k odd 12 1
2025.4.a.n 2 45.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 21T_{2}^{6} + 405T_{2}^{4} - 756T_{2}^{2} + 1296$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 21 T^{6} + 405 T^{4} + \cdots + 1296$$
$3$ $$T^{8} + 45 T^{6} + 1296 T^{4} + \cdots + 531441$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 173 T^{6} + \cdots + 14776336$$
$11$ $$(T^{4} + 66 T^{3} + 3795 T^{2} + \cdots + 314721)^{2}$$
$13$ $$T^{8} - 3773 T^{6} + \cdots + 11117396444176$$
$17$ $$(T^{4} + 6237 T^{2} + 3175524)^{2}$$
$19$ $$(T^{2} - 77 T - 4532)^{4}$$
$23$ $$T^{8} - 6501 T^{6} + \cdots + 53618068974096$$
$29$ $$(T^{4} + 51 T^{3} + 1959 T^{2} + \cdots + 412164)^{2}$$
$31$ $$(T^{4} + 43 T^{3} + 1461 T^{2} + \cdots + 150544)^{2}$$
$37$ $$(T^{4} + 49364 T^{2} + \cdots + 549058624)^{2}$$
$41$ $$(T^{4} + 132 T^{3} + 92301 T^{2} + \cdots + 5606565129)^{2}$$
$43$ $$T^{8} - 4466 T^{6} + \cdots + 7216320515041$$
$47$ $$T^{8} - 216237 T^{6} + \cdots + 66\!\cdots\!76$$
$53$ $$(T^{4} + 434484 T^{2} + \cdots + 46562734656)^{2}$$
$59$ $$(T^{4} - 798 T^{3} + 630195 T^{2} + \cdots + 43678881)^{2}$$
$61$ $$(T^{4} + 439 T^{3} + 235497 T^{2} + \cdots + 1829786176)^{2}$$
$67$ $$T^{8} - 559946 T^{6} + \cdots + 18\!\cdots\!01$$
$71$ $$(T^{2} - 1368 T + 296784)^{4}$$
$73$ $$(T^{4} + 1077821 T^{2} + \cdots + 189571418404)^{2}$$
$79$ $$(T^{4} + 803 T^{3} + \cdots + 392522298256)^{2}$$
$83$ $$T^{8} - 519393 T^{6} + \cdots + 25\!\cdots\!36$$
$89$ $$(T^{2} - 396 T - 3564)^{4}$$
$97$ $$T^{8} - 442514 T^{6} + \cdots + 60\!\cdots\!61$$