# Properties

 Label 225.2.k.b Level $225$ Weight $2$ Character orbit 225.k Analytic conductor $1.797$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,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, 2, names="a")

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

chi := DirichletCharacter("225.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 45) 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} + \beta_{6} + \beta_1) q^{2} + \beta_{6} q^{3} + ( - \beta_{11} + 2 \beta_{8} + 2) q^{4} + (2 \beta_{8} + \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{11} + \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10})$$ q + (-b7 + b6 + b1) * q^2 + b6 * q^3 + (-b11 + 2*b8 + 2) * q^4 + (2*b8 + b5 - b2) * q^6 + (-b10 + b7 - b6 - b1) * q^7 + (-2*b9 - b7 + 2*b6 + b1) * q^8 + (b11 + b5 + b4 + 1) * q^9 $$q + ( - \beta_{7} + \beta_{6} + \beta_1) q^{2} + \beta_{6} q^{3} + ( - \beta_{11} + 2 \beta_{8} + 2) q^{4} + (2 \beta_{8} + \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{11} + 2 \beta_{5} - \beta_{4} - 4) q^{99}+O(q^{100})$$ q + (-b7 + b6 + b1) * q^2 + b6 * q^3 + (-b11 + 2*b8 + 2) * q^4 + (2*b8 + b5 - b2) * q^6 + (-b10 + b7 - b6 - b1) * q^7 + (-2*b9 - b7 + 2*b6 + b1) * q^8 + (b11 + b5 + b4 + 1) * q^9 + (b11 - b8 + b2) * q^11 + (2*b10 - b9 + 2*b7 + b3 - b1) * q^12 + (b6 + b3 - b1) * q^13 + (-3*b8 - 2*b5 - b4 - 3) * q^14 + (-b11 + 2*b8 - b5 - 2*b4 - b2) * q^16 + (-b9 - b7 + b6) * q^17 + (b10 + b9 + b7 - b6 + 2*b3 + b1) * q^18 + (b2 - 1) * q^19 + (-2*b8 - 2*b5 - b4 - 1) * q^21 + (2*b9 - 2*b7 - b6 + b1) * q^22 + (-b9 + b7 - 2*b6 + 2*b1) * q^23 + (b11 - 4*b8 + 2*b5 + b4 - b2 - 5) * q^24 + (-b2 - 1) * q^26 + (-b10 - b9 + 2*b6 + b3 + 2*b1) * q^27 + (-b10 + 2*b9 - b7 - 2*b6 - b3 - 3*b1) * q^28 + (b11 + 2*b8 - b5 - 2*b4 + b2) * q^29 + (b11 - 3*b8 + 4*b5 + 2*b4 - 3) * q^31 + (b9 - b7 - 3*b3) * q^32 + (-b10 - b9 - b7 + b6 - 2*b3 + 2*b1) * q^33 + (-2*b11 + 2*b8 - b5 - 2*b4 - 2*b2) * q^34 + (b11 - 3*b8 + b5 + 4*b4 + 3*b2 + 1) * q^36 + (b10 - b9 + b7 + b6 + b3 + 2*b1) * q^37 + (-b9 - b7 + b1) * q^38 + (-2*b8 + b4 - b2 + 1) * q^39 + (-2*b11 + 5*b8 - 2*b5 - b4 + 5) * q^41 + (-3*b10 + 3*b9 - 2*b7 - 2*b6 - 3*b3 - 3*b1) * q^42 + (b10 + b9 + 3*b7 - 2*b6 - 3*b1) * q^43 + (-b5 + b4 + 2*b2 + 8) * q^44 + (-2*b5 + 2*b4 + 3*b2) * q^46 + (-3*b10 + b9 + 4*b7 - 3*b6 - 4*b1) * q^47 + (b10 + 4*b9 + b7 - 3*b6 - b3 - 2*b1) * q^48 + (b11 - b8 + 4*b5 + 2*b4 - 1) * q^49 + (-4*b8 + b5 - b2 - 3) * q^51 + (2*b10 - b9 + b7 - 2*b6 - b1) * q^52 + (2*b9 - 2*b6 - 2*b1) * q^53 + (-2*b11 + 9*b8 - 2*b5 + b4 + 4) * q^54 - 3*b8 * q^56 + (b10 - 2*b9 - b6 - b3 + b1) * q^57 + (b9 - b7 - 3*b3) * q^58 + (b11 - b8 + 2*b5 + b4 - 1) * q^59 + (-2*b11 + b8 - b5 - 2*b4 - 2*b2) * q^61 + (6*b10 + 3*b9 + 3*b7 - 3*b6 + 6*b3) * q^62 + (-3*b10 + 3*b9 - b7 - b6 - 3*b3 - 3*b1) * q^63 + (b5 - b4 + 5) * q^64 + (-3*b11 + 7*b8 - b5 - 3*b4 - 2*b2 + 3) * q^66 + (b9 - b7 + 3*b6 - b3 - 3*b1) * q^67 + (-2*b9 + 2*b7 + b6 - 3*b3 - b1) * q^68 + (-b11 + 4*b8 - 2*b5 - b4 + b2 - 4) * q^69 + (-b5 + b4 + 4*b2 - 2) * q^71 + (-5*b7 + b6 + 3*b3 + 3*b1) * q^72 + (2*b10 - 4*b9 + 4*b6 + 2*b3 + 4*b1) * q^73 + (b11 + 5*b8 + 2*b5 + 4*b4 + b2) * q^74 + (4*b8 - 2*b5 - b4 + 4) * q^76 + (-2*b9 + 2*b7 - b6 + b1) * q^77 + (-b10 + 2*b9 - b6 + b3 - b1) * q^78 + (-2*b11 - 2*b5 - 4*b4 - 2*b2) * q^79 + (b11 + 4*b8 + b4 - 2*b2 - 2) * q^81 + (-3*b10 - 8*b9 - 3*b7 + 8*b6 - 3*b3 + 5*b1) * q^82 + (-3*b10 + 3*b7 - 3*b6 - 3*b1) * q^83 + (-2*b11 - 5*b8 - 2*b5 - 4*b4 - b2 + 2) * q^84 + (5*b11 - 11*b8 + 4*b5 + 2*b4 - 11) * q^86 + (-b10 + 2*b9 + b7 - 3*b6 - 5*b3 + 2*b1) * q^87 + (-3*b10 - b9 - 5*b7 + 4*b6 + 5*b1) * q^88 + 3 * q^89 + (-b2 + 3) * q^91 + (-6*b10 - b9 - 2*b7 + b6 + 2*b1) * q^92 + (4*b10 - 5*b9 - 5*b7 + 4*b6 + 5*b3 + 7*b1) * q^93 + (2*b11 - 11*b8 - 4*b5 - 2*b4 - 11) * q^94 + (2*b11 - b8 + 3*b5 - b4 + 2*b2 + 2) * q^96 + (2*b10 + 4*b9 + 2*b7 + 2*b6 - 2*b1) * q^97 + (6*b10 + b9 + 3*b7 - b6 + 6*b3 + 2*b1) * q^98 + (2*b11 + 2*b5 - b4 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 10 q^{4} - 8 q^{6} + 14 q^{9}+O(q^{10})$$ 12 * q + 10 * q^4 - 8 * q^6 + 14 * q^9 $$12 q + 10 q^{4} - 8 q^{6} + 14 q^{9} + 4 q^{11} - 18 q^{14} - 10 q^{16} - 16 q^{19} - 30 q^{24} - 8 q^{26} - 14 q^{29} - 16 q^{31} - 8 q^{34} + 20 q^{36} + 28 q^{39} + 26 q^{41} + 88 q^{44} - 12 q^{46} - 4 q^{49} - 8 q^{51} - 10 q^{54} + 18 q^{56} - 4 q^{59} - 2 q^{61} + 60 q^{64} - 4 q^{66} - 78 q^{69} - 40 q^{71} - 32 q^{74} + 24 q^{76} + 4 q^{79} - 38 q^{81} + 54 q^{84} - 56 q^{86} + 36 q^{89} + 40 q^{91} - 62 q^{94} + 26 q^{96} - 44 q^{99}+O(q^{100})$$ 12 * q + 10 * q^4 - 8 * q^6 + 14 * q^9 + 4 * q^11 - 18 * q^14 - 10 * q^16 - 16 * q^19 - 30 * q^24 - 8 * q^26 - 14 * q^29 - 16 * q^31 - 8 * q^34 + 20 * q^36 + 28 * q^39 + 26 * q^41 + 88 * q^44 - 12 * q^46 - 4 * q^49 - 8 * q^51 - 10 * q^54 + 18 * q^56 - 4 * q^59 - 2 * q^61 + 60 * q^64 - 4 * q^66 - 78 * q^69 - 40 * q^71 - 32 * q^74 + 24 * q^76 + 4 * q^79 - 38 * q^81 + 54 * q^84 - 56 * q^86 + 36 * q^89 + 40 * q^91 - 62 * q^94 + 26 * q^96 - 44 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( 1330 \nu^{11} + 3816 \nu^{10} - 5925 \nu^{9} + 4140 \nu^{8} - 21730 \nu^{7} - 93918 \nu^{6} + \cdots + 195120 ) / 103944$$ (1330*v^11 + 3816*v^10 - 5925*v^9 + 4140*v^8 - 21730*v^7 - 93918*v^6 + 6180*v^5 + 198780*v^4 + 694990*v^3 + 1107384*v^2 + 507900*v + 195120) / 103944 $$\beta_{2}$$ $$=$$ $$( - 748 \nu^{11} + 816 \nu^{10} - 1768 \nu^{9} + 4162 \nu^{8} + 6311 \nu^{7} + 8976 \nu^{6} + \cdots + 98004 ) / 51972$$ (-748*v^11 + 816*v^10 - 1768*v^9 + 4162*v^8 + 6311*v^7 + 8976*v^6 + 7532*v^5 - 93902*v^4 - 164352*v^3 - 141012*v^2 - 80298*v + 98004) / 51972 $$\beta_{3}$$ $$=$$ $$( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} + \cdots - 15984 ) / 51972$$ (-962*v^11 + 2392*v^10 - 2887*v^9 + 2220*v^8 + 13990*v^7 - 14442*v^6 - 11380*v^5 - 57708*v^4 - 98118*v^3 + 132080*v^2 - 49284*v - 15984) / 51972 $$\beta_{4}$$ $$=$$ $$( - 1833 \nu^{11} + 1380 \nu^{10} - 718 \nu^{9} + 112 \nu^{8} + 30103 \nu^{7} + 21996 \nu^{6} + \cdots - 25344 ) / 51972$$ (-1833*v^11 + 1380*v^10 - 718*v^9 + 112*v^8 + 30103*v^7 + 21996*v^6 - 25222*v^5 - 158690*v^4 - 443420*v^3 - 362580*v^2 - 197850*v - 25344) / 51972 $$\beta_{5}$$ $$=$$ $$( - 1054 \nu^{11} + 840 \nu^{10} - 684 \nu^{9} + 630 \nu^{8} + 15925 \nu^{7} + 12648 \nu^{6} + \cdots - 60768 ) / 25986$$ (-1054*v^11 + 840*v^10 - 684*v^9 + 630*v^8 + 15925*v^7 + 12648*v^6 - 10080*v^5 - 92976*v^4 - 247336*v^3 - 203772*v^2 - 111966*v - 60768) / 25986 $$\beta_{6}$$ $$=$$ $$( 5984 \nu^{11} - 7522 \nu^{10} + 8393 \nu^{9} - 6600 \nu^{8} - 90816 \nu^{7} - 28498 \nu^{6} + \cdots + 272448 ) / 103944$$ (5984*v^11 - 7522*v^10 + 8393*v^9 - 6600*v^8 - 90816*v^7 - 28498*v^6 + 55616*v^5 + 473748*v^4 + 1167562*v^3 + 792408*v^2 + 746328*v + 272448) / 103944 $$\beta_{7}$$ $$=$$ $$( - 6823 \nu^{11} - 186 \nu^{10} + 2391 \nu^{9} - 2376 \nu^{8} + 113464 \nu^{7} + 159834 \nu^{6} + \cdots - 548280 ) / 103944$$ (-6823*v^11 - 186*v^10 + 2391*v^9 - 2376*v^8 + 113464*v^7 + 159834*v^6 - 23754*v^5 - 683700*v^4 - 2089522*v^3 - 2446620*v^2 - 1454952*v - 548280) / 103944 $$\beta_{8}$$ $$=$$ $$( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + \cdots + 3468 ) / 1704$$ (118*v^11 - 90*v^10 + 53*v^9 - 32*v^8 - 1866*v^7 - 1416*v^6 + 1364*v^5 + 10408*v^4 + 27758*v^3 + 22776*v^2 + 12468*v + 3468) / 1704 $$\beta_{9}$$ $$=$$ $$( - 9709 \nu^{11} + 6990 \nu^{10} - 6270 \nu^{9} + 4284 \nu^{8} + 155434 \nu^{7} + 116508 \nu^{6} + \cdots - 596232 ) / 103944$$ (-9709*v^11 + 6990*v^10 - 6270*v^9 + 4284*v^8 + 155434*v^7 + 116508*v^6 - 57894*v^5 - 856824*v^4 - 2383876*v^3 - 2154324*v^2 - 1602804*v - 596232) / 103944 $$\beta_{10}$$ $$=$$ $$( - 5790 \nu^{11} + 1824 \nu^{10} - 473 \nu^{9} + 516 \nu^{8} + 91522 \nu^{7} + 112790 \nu^{6} + \cdots - 404496 ) / 51972$$ (-5790*v^11 + 1824*v^10 - 473*v^9 + 516*v^8 + 91522*v^7 + 112790*v^6 - 39212*v^5 - 550164*v^4 - 1582586*v^3 - 1735504*v^2 - 1080204*v - 404496) / 51972 $$\beta_{11}$$ $$=$$ $$( - 17172 \nu^{11} + 12498 \nu^{10} - 4217 \nu^{9} - 3016 \nu^{8} + 280340 \nu^{7} + 206064 \nu^{6} + \cdots - 737076 ) / 103944$$ (-17172*v^11 + 12498*v^10 - 4217*v^9 - 3016*v^8 + 280340*v^7 + 206064*v^6 - 219272*v^5 - 1528612*v^4 - 3992950*v^3 - 3266544*v^2 - 1783248*v - 737076) / 103944
 $$\nu$$ $$=$$ $$( -\beta_{9} + \beta_{7} + \beta_{5} + \beta_1 ) / 2$$ (-b9 + b7 + b5 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -2\beta_{9} + 2\beta_{7} + 3\beta_{3} ) / 2$$ (-2*b9 + 2*b7 + 3*b3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{10} - 4\beta_{9} - 4\beta_{8} + 4\beta_{7} - 4\beta_{6} - 4\beta_{5} - 4\beta_{4} + \beta_{3} - 2 ) / 2$$ (-b10 - 4*b9 - 4*b8 + 4*b7 - 4*b6 - 4*b5 - 4*b4 + b3 - 2) / 2 $$\nu^{4}$$ $$=$$ $$-2\beta_{11} - 10\beta_{8} - 3\beta_{5} - 6\beta_{4} - 2\beta_{2}$$ -2*b11 - 10*b8 - 3*b5 - 6*b4 - 2*b2 $$\nu^{5}$$ $$=$$ $$( - \beta_{11} + 14 \beta_{10} - 2 \beta_{9} - 13 \beta_{8} - 2 \beta_{7} + 20 \beta_{6} - 19 \beta_{4} + \cdots + 13 ) / 2$$ (-b11 + 14*b10 - 2*b9 - 13*b8 - 2*b7 + 20*b6 - 19*b4 + 7*b3 - 2*b2 + 20*b1 + 13) / 2 $$\nu^{6}$$ $$=$$ $$27\beta_{10} - 24\beta_{9} + 8\beta_{7} + 24\beta_{6} + 27\beta_{3} + 32\beta_1$$ 27*b10 - 24*b9 + 8*b7 + 24*b6 + 27*b3 + 32*b1 $$\nu^{7}$$ $$=$$ $$- 4 \beta_{11} + 20 \beta_{10} - 51 \beta_{9} - 36 \beta_{8} + 43 \beta_{7} + 8 \beta_{6} - 47 \beta_{5} + \cdots - 72$$ -4*b11 + 20*b10 - 51*b9 - 36*b8 + 43*b7 + 8*b6 - 47*b5 + 40*b3 + 4*b2 + 43*b1 - 72 $$\nu^{8}$$ $$=$$ $$-35\beta_{11} - 223\beta_{8} - 166\beta_{5} - 83\beta_{4} - 223$$ -35*b11 - 223*b8 - 166*b5 - 83*b4 - 223 $$\nu^{9}$$ $$=$$ $$- 48 \beta_{11} + 107 \beta_{10} + 212 \beta_{9} - 380 \beta_{8} - 260 \beta_{7} + 212 \beta_{6} + \cdots - 190$$ -48*b11 + 107*b10 + 212*b9 - 380*b8 - 260*b7 + 212*b6 - 236*b5 - 236*b4 - 107*b3 - 24*b2 + 48*b1 - 190 $$\nu^{10}$$ $$=$$ $$636\beta_{10} + 262\beta_{9} - 590\beta_{7} + 852\beta_{6} + 590\beta_1$$ 636*b10 + 262*b9 - 590*b7 + 852*b6 + 590*b1 $$\nu^{11}$$ $$=$$ $$131 \beta_{11} + 1114 \beta_{10} - 262 \beta_{9} + 983 \beta_{8} - 262 \beta_{7} + 1324 \beta_{6} + \cdots - 983$$ 131*b11 + 1114*b10 - 262*b9 + 983*b8 - 262*b7 + 1324*b6 + 1193*b4 + 557*b3 + 262*b2 + 1324*b1 - 983

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 - \beta_{8}$$ $$-1$$

## 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
 −0.180407 + 0.673288i 2.17840 + 0.583700i −0.403293 + 1.50511i −1.50511 − 0.403293i 0.583700 − 2.17840i −0.673288 − 0.180407i −0.180407 − 0.673288i 2.17840 − 0.583700i −0.403293 − 1.50511i −1.50511 + 0.403293i 0.583700 + 2.17840i −0.673288 + 0.180407i
−2.17731 + 1.25707i 1.19154 + 1.25707i 2.16044 3.74200i 0 −4.17458 1.23917i 0.445256 0.257068i 5.83502i −0.160442 + 2.99571i 0
49.2 −1.80664 + 1.04307i −1.38276 + 1.04307i 1.17597 2.03684i 0 1.41016 3.32675i 3.53869 2.04307i 0.734191i 0.824030 2.88461i 0
49.3 −0.495361 + 0.285997i −1.70828 + 0.285997i −0.836412 + 1.44871i 0 0.764419 0.630233i −1.23669 + 0.714003i 2.10083i 2.83641 0.977122i 0
49.4 0.495361 0.285997i 1.70828 0.285997i −0.836412 + 1.44871i 0 0.764419 0.630233i 1.23669 0.714003i 2.10083i 2.83641 0.977122i 0
49.5 1.80664 1.04307i 1.38276 1.04307i 1.17597 2.03684i 0 1.41016 3.32675i −3.53869 + 2.04307i 0.734191i 0.824030 2.88461i 0
49.6 2.17731 1.25707i −1.19154 1.25707i 2.16044 3.74200i 0 −4.17458 1.23917i −0.445256 + 0.257068i 5.83502i −0.160442 + 2.99571i 0
124.1 −2.17731 1.25707i 1.19154 1.25707i 2.16044 + 3.74200i 0 −4.17458 + 1.23917i 0.445256 + 0.257068i 5.83502i −0.160442 2.99571i 0
124.2 −1.80664 1.04307i −1.38276 1.04307i 1.17597 + 2.03684i 0 1.41016 + 3.32675i 3.53869 + 2.04307i 0.734191i 0.824030 + 2.88461i 0
124.3 −0.495361 0.285997i −1.70828 0.285997i −0.836412 1.44871i 0 0.764419 + 0.630233i −1.23669 0.714003i 2.10083i 2.83641 + 0.977122i 0
124.4 0.495361 + 0.285997i 1.70828 + 0.285997i −0.836412 1.44871i 0 0.764419 + 0.630233i 1.23669 + 0.714003i 2.10083i 2.83641 + 0.977122i 0
124.5 1.80664 + 1.04307i 1.38276 + 1.04307i 1.17597 + 2.03684i 0 1.41016 + 3.32675i −3.53869 2.04307i 0.734191i 0.824030 + 2.88461i 0
124.6 2.17731 + 1.25707i −1.19154 + 1.25707i 2.16044 + 3.74200i 0 −4.17458 + 1.23917i −0.445256 0.257068i 5.83502i −0.160442 2.99571i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.6 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 225.2.k.b 12
3.b odd 2 1 675.2.k.b 12
5.b even 2 1 inner 225.2.k.b 12
5.c odd 4 1 45.2.e.b 6
5.c odd 4 1 225.2.e.b 6
9.c even 3 1 inner 225.2.k.b 12
9.c even 3 1 2025.2.b.l 6
9.d odd 6 1 675.2.k.b 12
9.d odd 6 1 2025.2.b.m 6
15.d odd 2 1 675.2.k.b 12
15.e even 4 1 135.2.e.b 6
15.e even 4 1 675.2.e.b 6
20.e even 4 1 720.2.q.i 6
45.h odd 6 1 675.2.k.b 12
45.h odd 6 1 2025.2.b.m 6
45.j even 6 1 inner 225.2.k.b 12
45.j even 6 1 2025.2.b.l 6
45.k odd 12 1 45.2.e.b 6
45.k odd 12 1 225.2.e.b 6
45.k odd 12 1 405.2.a.j 3
45.k odd 12 1 2025.2.a.n 3
45.l even 12 1 135.2.e.b 6
45.l even 12 1 405.2.a.i 3
45.l even 12 1 675.2.e.b 6
45.l even 12 1 2025.2.a.o 3
60.l odd 4 1 2160.2.q.k 6
180.v odd 12 1 2160.2.q.k 6
180.v odd 12 1 6480.2.a.bs 3
180.x even 12 1 720.2.q.i 6
180.x even 12 1 6480.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 5.c odd 4 1
45.2.e.b 6 45.k odd 12 1
135.2.e.b 6 15.e even 4 1
135.2.e.b 6 45.l even 12 1
225.2.e.b 6 5.c odd 4 1
225.2.e.b 6 45.k odd 12 1
225.2.k.b 12 1.a even 1 1 trivial
225.2.k.b 12 5.b even 2 1 inner
225.2.k.b 12 9.c even 3 1 inner
225.2.k.b 12 45.j even 6 1 inner
405.2.a.i 3 45.l even 12 1
405.2.a.j 3 45.k odd 12 1
675.2.e.b 6 15.e even 4 1
675.2.e.b 6 45.l even 12 1
675.2.k.b 12 3.b odd 2 1
675.2.k.b 12 9.d odd 6 1
675.2.k.b 12 15.d odd 2 1
675.2.k.b 12 45.h odd 6 1
720.2.q.i 6 20.e even 4 1
720.2.q.i 6 180.x even 12 1
2025.2.a.n 3 45.k odd 12 1
2025.2.a.o 3 45.l even 12 1
2025.2.b.l 6 9.c even 3 1
2025.2.b.l 6 45.j even 6 1
2025.2.b.m 6 9.d odd 6 1
2025.2.b.m 6 45.h odd 6 1
2160.2.q.k 6 60.l odd 4 1
2160.2.q.k 6 180.v odd 12 1
6480.2.a.bs 3 180.v odd 12 1
6480.2.a.bv 3 180.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 11T_{2}^{10} + 90T_{2}^{8} - 323T_{2}^{6} + 862T_{2}^{4} - 279T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 11 T^{10} + \cdots + 81$$
$3$ $$T^{12} - 7 T^{10} + \cdots + 729$$
$5$ $$T^{12}$$
$7$ $$T^{12} - 19 T^{10} + \cdots + 81$$
$11$ $$(T^{6} - 2 T^{5} + \cdots + 144)^{2}$$
$13$ $$T^{12} - 24 T^{10} + \cdots + 256$$
$17$ $$(T^{6} + 20 T^{4} + \cdots + 144)^{2}$$
$19$ $$(T^{3} + 4 T^{2} - 4 T - 4)^{4}$$
$23$ $$T^{12} + \cdots + 187388721$$
$29$ $$(T^{6} + 7 T^{5} + \cdots + 2601)^{2}$$
$31$ $$(T^{6} + 8 T^{5} + \cdots + 219024)^{2}$$
$37$ $$(T^{6} + 60 T^{4} + \cdots + 16)^{2}$$
$41$ $$(T^{6} - 13 T^{5} + 150 T^{4} + \cdots + 9)^{2}$$
$43$ $$T^{12} - 108 T^{10} + \cdots + 256$$
$47$ $$T^{12} + \cdots + 18539817921$$
$53$ $$(T^{6} + 44 T^{4} + \cdots + 576)^{2}$$
$59$ $$(T^{6} + 2 T^{5} + \cdots + 576)^{2}$$
$61$ $$(T^{6} + T^{5} + 38 T^{4} + \cdots + 5041)^{2}$$
$67$ $$T^{12} + \cdots + 66074188401$$
$71$ $$(T^{3} + 10 T^{2} + \cdots - 708)^{4}$$
$73$ $$(T^{6} + 192 T^{4} + \cdots + 16384)^{2}$$
$79$ $$(T^{6} - 2 T^{5} + \cdots + 576)^{2}$$
$83$ $$T^{12} - 171 T^{10} + \cdots + 43046721$$
$89$ $$(T - 3)^{12}$$
$97$ $$T^{12} + \cdots + 2891414573056$$