# Properties

 Label 675.2.k.b Level $675$ Weight $2$ Character orbit 675.k Analytic conductor $5.390$ 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] = [675,2,Mod(199,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, 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("675.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.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: $$5.38990213644$$ 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: $$2^{6}\cdot 3^{2}$$ 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.

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

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

 $$\beta_{1}$$ $$=$$ $$( 473 \nu^{11} - 516 \nu^{10} + 1118 \nu^{9} - 3014 \nu^{8} - 4564 \nu^{7} - 5676 \nu^{6} - 2470 \nu^{5} + 66640 \nu^{4} + 113100 \nu^{3} + 96048 \nu^{2} + 54216 \nu - 194196 ) / 51972$$ (473*v^11 - 516*v^10 + 1118*v^9 - 3014*v^8 - 4564*v^7 - 5676*v^6 - 2470*v^5 + 66640*v^4 + 113100*v^3 + 96048*v^2 + 54216*v - 194196) / 51972 $$\beta_{2}$$ $$=$$ $$( - 748 \nu^{11} + 816 \nu^{10} - 1768 \nu^{9} + 4162 \nu^{8} + 6311 \nu^{7} + 8976 \nu^{6} + 7532 \nu^{5} - 93902 \nu^{4} - 164352 \nu^{3} - 141012 \nu^{2} - 80298 \nu + 98004 ) / 51972$$ (-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 $$\beta_{3}$$ $$=$$ $$( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} - 11380 \nu^{5} - 57708 \nu^{4} - 98118 \nu^{3} + 132080 \nu^{2} + \cdots - 15984 ) / 51972$$ (-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 $$\beta_{4}$$ $$=$$ $$( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} - 11380 \nu^{5} - 57708 \nu^{4} - 98118 \nu^{3} + 97432 \nu^{2} - 49284 \nu - 15984 ) / 34648$$ (-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 + 97432*v^2 - 49284*v - 15984) / 34648 $$\beta_{5}$$ $$=$$ $$( - 2327 \nu^{11} + 5669 \nu^{10} - 7159 \nu^{9} + 5370 \nu^{8} + 34543 \nu^{7} - 32710 \nu^{6} - 24718 \nu^{5} - 137484 \nu^{4} - 236286 \nu^{3} + 157488 \nu^{2} + \cdots - 38664 ) / 51972$$ (-2327*v^11 + 5669*v^10 - 7159*v^9 + 5370*v^8 + 34543*v^7 - 32710*v^6 - 24718*v^5 - 137484*v^4 - 236286*v^3 + 157488*v^2 - 119214*v - 38664) / 51972 $$\beta_{6}$$ $$=$$ $$( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + 10408 \nu^{4} + 27758 \nu^{3} + 22776 \nu^{2} + 12468 \nu + 5172 ) / 1704$$ (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 + 5172) / 1704 $$\beta_{7}$$ $$=$$ $$( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} - 12648 \nu^{5} - 151968 \nu^{4} - 420176 \nu^{3} - 400856 \nu^{2} + \cdots - 105120 ) / 12993$$ (-1688*v^11 + 1054*v^10 - 840*v^9 + 684*v^8 + 26378*v^7 + 24587*v^6 - 12648*v^5 - 151968*v^4 - 420176*v^3 - 400856*v^2 - 282372*v - 105120) / 12993 $$\beta_{8}$$ $$=$$ $$( 14137 \nu^{11} - 3520 \nu^{10} + 77 \nu^{9} - 84 \nu^{8} - 226010 \nu^{7} - 282250 \nu^{6} + 85550 \nu^{5} + 1356228 \nu^{4} + 3952074 \nu^{3} + 4346412 \nu^{2} + \cdots + 1015848 ) / 103944$$ (14137*v^11 - 3520*v^10 + 77*v^9 - 84*v^8 - 226010*v^7 - 282250*v^6 + 85550*v^5 + 1356228*v^4 + 3952074*v^3 + 4346412*v^2 + 2709180*v + 1015848) / 103944 $$\beta_{9}$$ $$=$$ $$( - 17172 \nu^{11} + 12498 \nu^{10} - 4217 \nu^{9} - 3016 \nu^{8} + 280340 \nu^{7} + 206064 \nu^{6} - 219272 \nu^{5} - 1528612 \nu^{4} - 3992950 \nu^{3} + \cdots - 737076 ) / 103944$$ (-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 $$\beta_{10}$$ $$=$$ $$( 11258 \nu^{11} - 7163 \nu^{10} + 6136 \nu^{9} - 4254 \nu^{8} - 179857 \nu^{7} - 152420 \nu^{6} + 68632 \nu^{5} + 1007136 \nu^{4} + 2820480 \nu^{3} + 2696676 \nu^{2} + \cdots + 708480 ) / 51972$$ (11258*v^11 - 7163*v^10 + 6136*v^9 - 4254*v^8 - 179857*v^7 - 152420*v^6 + 68632*v^5 + 1007136*v^4 + 2820480*v^3 + 2696676*v^2 + 1902042*v + 708480) / 51972 $$\beta_{11}$$ $$=$$ $$( 29270 \nu^{11} - 21978 \nu^{10} + 11125 \nu^{9} - 2248 \nu^{8} - 467946 \nu^{7} - 351240 \nu^{6} + 350356 \nu^{5} + 2589800 \nu^{4} + 6858478 \nu^{3} + \cdots + 1273908 ) / 103944$$ (29270*v^11 - 21978*v^10 + 11125*v^9 - 2248*v^8 - 467946*v^7 - 351240*v^6 + 350356*v^5 + 2589800*v^4 + 6858478*v^3 + 5621880*v^2 + 3074676*v + 1273908) / 103944
 $$\nu$$ $$=$$ $$( -\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + \beta_{5} - 4\beta_{4} + \beta_{2} + \beta_1 ) / 6$$ (-b11 - b10 - b9 + 2*b8 + b5 - 4*b4 + b2 + b1) / 6 $$\nu^{2}$$ $$=$$ $$( -2\beta_{4} + 3\beta_{3} ) / 2$$ (-2*b4 + 3*b3) / 2 $$\nu^{3}$$ $$=$$ $$( 8 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 3 \beta_{7} - 12 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} + 4 \beta _1 + 6 ) / 6$$ (8*b11 + 4*b10 + 8*b9 - 8*b8 - 3*b7 - 12*b6 + 8*b5 - 8*b4 + 6*b3 + 4*b2 + 4*b1 + 6) / 6 $$\nu^{4}$$ $$=$$ $$3\beta_{11} + \beta_{9} - 10\beta_{6} + \beta_{2} + 3\beta _1 + 10$$ 3*b11 + b9 - 10*b6 + b2 + 3*b1 + 10 $$\nu^{5}$$ $$=$$ $$( 19 \beta_{11} - 32 \beta_{10} + 16 \beta_{9} + 76 \beta_{8} + 42 \beta_{7} - 39 \beta_{6} - 16 \beta_{5} - 38 \beta_{4} - 21 \beta_{3} + 32 \beta_{2} + 38 \beta _1 + 78 ) / 6$$ (19*b11 - 32*b10 + 16*b9 + 76*b8 + 42*b7 - 39*b6 - 16*b5 - 38*b4 - 21*b3 + 32*b2 + 38*b1 + 78) / 6 $$\nu^{6}$$ $$=$$ $$-8\beta_{10} + 32\beta_{8} + 27\beta_{7} - 32\beta_{4}$$ -8*b10 + 32*b8 + 27*b7 - 32*b4 $$\nu^{7}$$ $$=$$ $$( 47 \beta_{11} - 35 \beta_{10} + 35 \beta_{9} + 94 \beta_{8} + 60 \beta_{7} - 108 \beta_{6} + 35 \beta_{5} - 188 \beta_{4} + 60 \beta_{3} - 35 \beta_{2} - 47 \beta _1 - 108 ) / 3$$ (47*b11 - 35*b10 + 35*b9 + 94*b8 + 60*b7 - 108*b6 + 35*b5 - 188*b4 + 60*b3 - 35*b2 - 47*b1 - 108) / 3 $$\nu^{8}$$ $$=$$ $$83\beta_{11} + 48\beta_{9} - 223\beta_{6}$$ 83*b11 + 48*b9 - 223*b6 $$\nu^{9}$$ $$=$$ $$( 472 \beta_{11} - 164 \beta_{10} + 328 \beta_{9} + 472 \beta_{8} + 321 \beta_{7} - 1140 \beta_{6} - 328 \beta_{5} + 472 \beta_{4} - 642 \beta_{3} + 164 \beta_{2} + 236 \beta _1 + 570 ) / 3$$ (472*b11 - 164*b10 + 328*b9 + 472*b8 + 321*b7 - 1140*b6 - 328*b5 + 472*b4 - 642*b3 + 164*b2 + 236*b1 + 570) / 3 $$\nu^{10}$$ $$=$$ $$-262\beta_{10} + 852\beta_{8} + 636\beta_{7} - 262\beta_{5} - 636\beta_{3}$$ -262*b10 + 852*b8 + 636*b7 - 262*b5 - 636*b3 $$\nu^{11}$$ $$=$$ $$( - 1193 \beta_{11} - 1600 \beta_{10} - 800 \beta_{9} + 4772 \beta_{8} + 3342 \beta_{7} + 2949 \beta_{6} - 800 \beta_{5} - 2386 \beta_{4} - 1671 \beta_{3} - 1600 \beta_{2} - 2386 \beta _1 - 5898 ) / 3$$ (-1193*b11 - 1600*b10 - 800*b9 + 4772*b8 + 3342*b7 + 2949*b6 - 800*b5 - 2386*b4 - 1671*b3 - 1600*b2 - 2386*b1 - 5898) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −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 0 2.16044 3.74200i 0 0 −0.445256 + 0.257068i 5.83502i 0 0
199.2 −1.80664 + 1.04307i 0 1.17597 2.03684i 0 0 −3.53869 + 2.04307i 0.734191i 0 0
199.3 −0.495361 + 0.285997i 0 −0.836412 + 1.44871i 0 0 1.23669 0.714003i 2.10083i 0 0
199.4 0.495361 0.285997i 0 −0.836412 + 1.44871i 0 0 −1.23669 + 0.714003i 2.10083i 0 0
199.5 1.80664 1.04307i 0 1.17597 2.03684i 0 0 3.53869 2.04307i 0.734191i 0 0
199.6 2.17731 1.25707i 0 2.16044 3.74200i 0 0 0.445256 0.257068i 5.83502i 0 0
424.1 −2.17731 1.25707i 0 2.16044 + 3.74200i 0 0 −0.445256 0.257068i 5.83502i 0 0
424.2 −1.80664 1.04307i 0 1.17597 + 2.03684i 0 0 −3.53869 2.04307i 0.734191i 0 0
424.3 −0.495361 0.285997i 0 −0.836412 1.44871i 0 0 1.23669 + 0.714003i 2.10083i 0 0
424.4 0.495361 + 0.285997i 0 −0.836412 1.44871i 0 0 −1.23669 0.714003i 2.10083i 0 0
424.5 1.80664 + 1.04307i 0 1.17597 + 2.03684i 0 0 3.53869 + 2.04307i 0.734191i 0 0
424.6 2.17731 + 1.25707i 0 2.16044 + 3.74200i 0 0 0.445256 + 0.257068i 5.83502i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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 675.2.k.b 12
3.b odd 2 1 225.2.k.b 12
5.b even 2 1 inner 675.2.k.b 12
5.c odd 4 1 135.2.e.b 6
5.c odd 4 1 675.2.e.b 6
9.c even 3 1 inner 675.2.k.b 12
9.c even 3 1 2025.2.b.m 6
9.d odd 6 1 225.2.k.b 12
9.d odd 6 1 2025.2.b.l 6
15.d odd 2 1 225.2.k.b 12
15.e even 4 1 45.2.e.b 6
15.e even 4 1 225.2.e.b 6
20.e even 4 1 2160.2.q.k 6
45.h odd 6 1 225.2.k.b 12
45.h odd 6 1 2025.2.b.l 6
45.j even 6 1 inner 675.2.k.b 12
45.j even 6 1 2025.2.b.m 6
45.k odd 12 1 135.2.e.b 6
45.k odd 12 1 405.2.a.i 3
45.k odd 12 1 675.2.e.b 6
45.k odd 12 1 2025.2.a.o 3
45.l even 12 1 45.2.e.b 6
45.l even 12 1 225.2.e.b 6
45.l even 12 1 405.2.a.j 3
45.l even 12 1 2025.2.a.n 3
60.l odd 4 1 720.2.q.i 6
180.v odd 12 1 720.2.q.i 6
180.v odd 12 1 6480.2.a.bv 3
180.x even 12 1 2160.2.q.k 6
180.x even 12 1 6480.2.a.bs 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 15.e even 4 1
45.2.e.b 6 45.l even 12 1
135.2.e.b 6 5.c odd 4 1
135.2.e.b 6 45.k odd 12 1
225.2.e.b 6 15.e even 4 1
225.2.e.b 6 45.l even 12 1
225.2.k.b 12 3.b odd 2 1
225.2.k.b 12 9.d odd 6 1
225.2.k.b 12 15.d odd 2 1
225.2.k.b 12 45.h odd 6 1
405.2.a.i 3 45.k odd 12 1
405.2.a.j 3 45.l even 12 1
675.2.e.b 6 5.c odd 4 1
675.2.e.b 6 45.k odd 12 1
675.2.k.b 12 1.a even 1 1 trivial
675.2.k.b 12 5.b even 2 1 inner
675.2.k.b 12 9.c even 3 1 inner
675.2.k.b 12 45.j even 6 1 inner
720.2.q.i 6 60.l odd 4 1
720.2.q.i 6 180.v odd 12 1
2025.2.a.n 3 45.l even 12 1
2025.2.a.o 3 45.k odd 12 1
2025.2.b.l 6 9.d odd 6 1
2025.2.b.l 6 45.h odd 6 1
2025.2.b.m 6 9.c even 3 1
2025.2.b.m 6 45.j even 6 1
2160.2.q.k 6 20.e even 4 1
2160.2.q.k 6 180.x even 12 1
6480.2.a.bs 3 180.x even 12 1
6480.2.a.bv 3 180.v odd 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 $$S_{2}^{\mathrm{new}}(675, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 11 T^{10} + 90 T^{8} - 323 T^{6} + \cdots + 81$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$T^{12} - 19 T^{10} + 322 T^{8} + \cdots + 81$$
$11$ $$(T^{6} + 2 T^{5} + 12 T^{4} + 8 T^{3} + \cdots + 144)^{2}$$
$13$ $$T^{12} - 24 T^{10} + 528 T^{8} + \cdots + 256$$
$17$ $$(T^{6} + 20 T^{4} + 112 T^{2} + 144)^{2}$$
$19$ $$(T^{3} + 4 T^{2} - 4 T - 4)^{4}$$
$23$ $$T^{12} - 75 T^{10} + \cdots + 187388721$$
$29$ $$(T^{6} - 7 T^{5} + 78 T^{4} + 101 T^{3} + \cdots + 2601)^{2}$$
$31$ $$(T^{6} + 8 T^{5} + 124 T^{4} + \cdots + 219024)^{2}$$
$37$ $$(T^{6} + 60 T^{4} + 192 T^{2} + 16)^{2}$$
$41$ $$(T^{6} + 13 T^{5} + 150 T^{4} + 241 T^{3} + \cdots + 9)^{2}$$
$43$ $$T^{12} - 108 T^{10} + 11568 T^{8} + \cdots + 256$$
$47$ $$T^{12} - 191 T^{10} + \cdots + 18539817921$$
$53$ $$(T^{6} + 44 T^{4} + 496 T^{2} + 576)^{2}$$
$59$ $$(T^{6} - 2 T^{5} + 24 T^{4} - 8 T^{3} + \cdots + 576)^{2}$$
$61$ $$(T^{6} + T^{5} + 38 T^{4} - 179 T^{3} + \cdots + 5041)^{2}$$
$67$ $$T^{12} - 199 T^{10} + \cdots + 66074188401$$
$71$ $$(T^{3} - 10 T^{2} - 92 T + 708)^{4}$$
$73$ $$(T^{6} + 192 T^{4} + 6144 T^{2} + \cdots + 16384)^{2}$$
$79$ $$(T^{6} - 2 T^{5} + 88 T^{4} + 216 T^{3} + \cdots + 576)^{2}$$
$83$ $$T^{12} - 171 T^{10} + \cdots + 43046721$$
$89$ $$(T + 3)^{12}$$
$97$ $$T^{12} - 396 T^{10} + \cdots + 2891414573056$$