# Properties

 Label 475.2.j.c Level $475$ Weight $2$ Character orbit 475.j Analytic conductor $3.793$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(49,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("475.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.j (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: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: 16.0.1387535264013605949997056.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 11x^{14} + 82x^{12} - 337x^{10} + 1006x^{8} - 1596x^{6} + 1765x^{4} - 414x^{2} + 81$$ x^16 - 11*x^14 + 82*x^12 - 337*x^10 + 1006*x^8 - 1596*x^6 + 1765*x^4 - 414*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{2} + (\beta_{13} + \beta_{7}) q^{3} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{4} + (\beta_{9} + \beta_{6} - \beta_1) q^{6} + ( - \beta_{15} + \beta_{8}) q^{7} + ( - 2 \beta_{15} + \beta_{14} + \cdots + 2 \beta_{8}) q^{8}+ \cdots + (\beta_{9} - \beta_{4} - 2 \beta_1) q^{9}+O(q^{10})$$ q + b14 * q^2 + (b13 + b7) * q^3 + (-b9 - b6 - b4) * q^4 + (b9 + b6 - b1) * q^6 + (-b15 + b8) * q^7 + (-2*b15 + b14 - b13 + b10 + 2*b8) * q^8 + (b9 - b4 - 2*b1) * q^9 $$q + \beta_{14} q^{2} + (\beta_{13} + \beta_{7}) q^{3} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{4} + (\beta_{9} + \beta_{6} - \beta_1) q^{6} + ( - \beta_{15} + \beta_{8}) q^{7} + ( - 2 \beta_{15} + \beta_{14} + \cdots + 2 \beta_{8}) q^{8}+ \cdots + (5 \beta_{9} - 12 \beta_{4} - 5 \beta_1) q^{99}+O(q^{100})$$ q + b14 * q^2 + (b13 + b7) * q^3 + (-b9 - b6 - b4) * q^4 + (b9 + b6 - b1) * q^6 + (-b15 + b8) * q^7 + (-2*b15 + b14 - b13 + b10 + 2*b8) * q^8 + (b9 - b4 - 2*b1) * q^9 + (b5 + 2*b3) * q^11 + (b15 - 2*b14 - 2*b10 - b8) * q^12 + (-b11 + b8 - b7) * q^13 + (-2*b9 - b6 + b5 - b3 + 2*b2 - b1) * q^14 + (-4*b9 - 2*b6 + 2*b5 - b4 - b3 + 4*b2 - b1 - 1) * q^16 + (b14 - 2*b13 - b12 - 2*b7) * q^17 + (-b15 + 2*b13 - b12 - b11 + b8) * q^18 + (2*b6 - 2*b5 - b3 - b2 - b1 - 1) * q^19 + (b9 + b4 + b3 - b2 + b1 + 1) * q^21 + (-b15 - b14 + 3*b13 + b12 + 3*b7) * q^22 + (-b11 - b10 + b8 + b7) * q^23 + (2*b9 + b6 - b5 + 6*b4 + 3*b3 - 2*b2 + 3*b1 + 6) * q^24 + (-2*b3 - b2) * q^26 + (-b15 + 3*b14 + b13 - 2*b12 - 2*b11 + 3*b10 + b8) * q^27 + (b11 + 3*b10 + 2*b8) * q^28 + (2*b9 + 2*b6 + b1) * q^29 + (b5 - b3 - 3*b2 - 1) * q^31 + (2*b11 + 5*b10 + 3*b8 - b7) * q^32 + (3*b14 + 2*b13 - 2*b12 + 2*b7) * q^33 + (-b9 - 3*b6 - 3*b4 + 2*b1) * q^34 + (b6 - b5 + 2*b4 + b3 + b1 + 2) * q^36 + (b15 - 2*b14 + b12 + b11 - 2*b10 - b8) * q^37 + (-b15 + b12 + 2*b11 - 2*b10 - b8 - 3*b7) * q^38 + (-3*b3 - 2*b2 - 5) * q^39 + (-2*b9 + 3*b6 - 3*b5 + 3*b4 + 2*b3 + 2*b2 + 2*b1 + 3) * q^41 + (-b11 - 2*b10 - 2*b8 + b7) * q^42 + (-b15 - b14 - 4*b13 - 2*b12 - 4*b7) * q^43 + (b6 + 3*b4) * q^44 + (b5 + 3) * q^46 + (-2*b11 - 2*b10 + 3*b8 - 2*b7) * q^47 + (-b11 - 5*b10 - 4*b8 + 2*b7) * q^48 + (b5 + b3 + b2 + 3) * q^49 + (-b9 + b6 + 6*b4 + b1) * q^51 + (-b15 + b14 - b12) * q^52 + (3*b10 + 2*b8 - 2*b7) * q^53 + (-3*b6 + 3*b5 - 9*b4 - 2*b3 - 2*b1 - 9) * q^54 + (3*b5 + 5*b2 - 9) * q^56 + (b15 - b14 - 2*b13 - b11 + 3*b10 - 3*b7) * q^57 + (5*b15 - 4*b14 + b13 - 4*b10 - 5*b8) * q^58 + (5*b3 + 5*b1) * q^59 + (4*b9 - 2*b6 - b4) * q^61 + (-b15 + b14 + 4*b12) * q^62 + (b11 - 2*b8 + b7) * q^63 + (3*b5 - 2*b3 + 6*b2 - 13) * q^64 + (3*b9 - b6 - 9*b4 - 2*b1) * q^66 + (-2*b10 + 2*b7) * q^67 + (-2*b15 + 5*b14 - b13 + 5*b10 + 2*b8) * q^68 + (b5 + 2*b3 + b2 + 3) * q^69 + (3*b9 - 6*b4 - b3 - 3*b2 - b1 - 6) * q^71 + (3*b11 - 3*b10 - 4*b8 + 4*b7) * q^72 + (-b15 + b14 + 3*b13 + 3*b12 + 3*b7) * q^73 + (2*b9 + 3*b6 - 3*b5 + 6*b4 + b3 - 2*b2 + b1 + 6) * q^74 + (-2*b9 - b6 - 2*b5 - 2*b4 - 3*b2 - b1 + 6) * q^76 + (2*b15 + b14 - 3*b13 - 2*b12 - 2*b11 + b10 - 2*b8) * q^77 + (b15 - 3*b14 - 3*b13 + 2*b12 - 3*b7) * q^78 + (-2*b9 + b6 - b5 + 5*b4 + b3 + 2*b2 + b1 + 5) * q^79 + (2*b9 + 3*b6 - 3*b5 - 4*b4 - 2*b3 - 2*b2 - 2*b1 - 4) * q^81 + (5*b11 - 4*b10 - 3*b8 - b7) * q^82 + (2*b15 - b14 + 2*b13 + 2*b12 + 2*b11 - b10 - 2*b8) * q^83 + (-3*b5 + b3 - 5*b2 + 4) * q^84 + (-b9 - 4*b6 + 3*b4 + 3*b1) * q^86 + (-2*b15 + 3*b14 + b13 + b12 + b11 + 3*b10 + 2*b8) * q^87 + (3*b15 - 2*b14 - 5*b13 - b12 - b11 - 2*b10 - 3*b8) * q^88 + (-2*b9 + 3*b6 - 3*b4 - 3*b1) * q^89 + (-b6 + b4 - 2*b1) * q^91 + (-b15 - b13 - b12 - b7) * q^92 + (3*b15 - 3*b14 - 5*b13 + b12 - 5*b7) * q^93 + (-b5 - 5*b3 - 2*b2 + 6) * q^94 + (-5*b5 - 9*b2 + 3) * q^96 + (4*b15 + 5*b13 + 4*b12 + 5*b7) * q^97 + (b15 + b14 + 2*b13 + 2*b7) * q^98 + (5*b9 - 12*b4 - 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 10 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10})$$ 16 * q + 10 * q^4 - 4 * q^6 + 2 * q^9 $$16 q + 10 q^{4} - 4 q^{6} + 2 q^{9} - 8 q^{11} - 2 q^{14} - 14 q^{16} - 10 q^{19} + 8 q^{21} + 46 q^{24} + 12 q^{26} - 2 q^{29} + 30 q^{34} + 14 q^{36} - 60 q^{39} + 16 q^{41} - 24 q^{44} + 48 q^{46} + 40 q^{49} - 44 q^{51} - 68 q^{54} - 164 q^{56} - 10 q^{59} - 224 q^{64} + 62 q^{66} + 36 q^{69} - 40 q^{71} + 50 q^{74} + 126 q^{76} + 34 q^{79} - 24 q^{81} + 80 q^{84} - 16 q^{86} + 22 q^{89} - 12 q^{91} + 124 q^{94} + 84 q^{96} + 76 q^{99}+O(q^{100})$$ 16 * q + 10 * q^4 - 4 * q^6 + 2 * q^9 - 8 * q^11 - 2 * q^14 - 14 * q^16 - 10 * q^19 + 8 * q^21 + 46 * q^24 + 12 * q^26 - 2 * q^29 + 30 * q^34 + 14 * q^36 - 60 * q^39 + 16 * q^41 - 24 * q^44 + 48 * q^46 + 40 * q^49 - 44 * q^51 - 68 * q^54 - 164 * q^56 - 10 * q^59 - 224 * q^64 + 62 * q^66 + 36 * q^69 - 40 * q^71 + 50 * q^74 + 126 * q^76 + 34 * q^79 - 24 * q^81 + 80 * q^84 - 16 * q^86 + 22 * q^89 - 12 * q^91 + 124 * q^94 + 84 * q^96 + 76 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 11x^{14} + 82x^{12} - 337x^{10} + 1006x^{8} - 1596x^{6} + 1765x^{4} - 414x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( - 17 \nu^{14} - 281 \nu^{12} + 3853 \nu^{10} - 26683 \nu^{8} + 96538 \nu^{6} - 201735 \nu^{4} + \cdots - 46638 ) / 73569$$ (-17*v^14 - 281*v^12 + 3853*v^10 - 26683*v^8 + 96538*v^6 - 201735*v^4 + 198007*v^2 - 46638) / 73569 $$\beta_{2}$$ $$=$$ $$( - 105339 \nu^{14} + 1015482 \nu^{12} - 7295401 \nu^{10} + 25578470 \nu^{8} - 71187724 \nu^{6} + \cdots - 284972250 ) / 104051089$$ (-105339*v^14 + 1015482*v^12 - 7295401*v^10 + 25578470*v^8 - 71187724*v^6 + 51321701*v^4 - 12081573*v^2 - 284972250) / 104051089 $$\beta_{3}$$ $$=$$ $$( 261339 \nu^{14} - 2789005 \nu^{12} + 18099401 \nu^{10} - 63458470 \nu^{8} + 140708376 \nu^{6} + \cdots + 45708276 ) / 104051089$$ (261339*v^14 - 2789005*v^12 + 18099401*v^10 - 63458470*v^8 + 140708376*v^6 - 127325701*v^4 + 29973573*v^2 + 45708276) / 104051089 $$\beta_{4}$$ $$=$$ $$( 3020113 \nu^{14} - 32273192 \nu^{12} + 238509928 \nu^{10} - 952119472 \nu^{8} + \cdots - 1141592625 ) / 936459801$$ (3020113*v^14 - 32273192*v^12 + 238509928*v^10 - 952119472*v^8 + 2808027448*v^6 - 4179410832*v^4 + 4868604136*v^2 - 1141592625) / 936459801 $$\beta_{5}$$ $$=$$ $$( - 383448 \nu^{14} + 3735013 \nu^{12} - 26556232 \nu^{10} + 93109040 \nu^{8} - 239139277 \nu^{6} + \cdots - 184780641 ) / 104051089$$ (-383448*v^14 + 3735013*v^12 - 26556232*v^10 + 93109040*v^8 - 239139277*v^6 + 186817832*v^4 - 43978536*v^2 - 184780641) / 104051089 $$\beta_{6}$$ $$=$$ $$( 1440028 \nu^{14} - 17040962 \nu^{12} + 129078913 \nu^{10} - 568442422 \nu^{8} + 1740211588 \nu^{6} + \cdots - 733877370 ) / 312153267$$ (1440028*v^14 - 17040962*v^12 + 129078913*v^10 - 568442422*v^8 + 1740211588*v^6 - 3097432050*v^4 + 3126614206*v^2 - 733877370) / 312153267 $$\beta_{7}$$ $$=$$ $$( - 750126 \nu^{15} + 7539500 \nu^{13} - 51951034 \nu^{11} + 182145980 \nu^{9} + \cdots - 931665523 \nu ) / 312153267$$ (-750126*v^15 + 7539500*v^13 - 51951034*v^11 + 182145980*v^9 - 451035377*v^7 + 365465234*v^5 - 86033682*v^3 - 931665523*v) / 312153267 $$\beta_{8}$$ $$=$$ $$( - 1066143 \nu^{15} + 10585946 \nu^{13} - 73837237 \nu^{11} + 258881390 \nu^{9} + \cdots - 1786582273 \nu ) / 312153267$$ (-1066143*v^15 + 10585946*v^13 - 73837237*v^11 + 258881390*v^9 - 664598549*v^7 + 519430337*v^5 - 122278401*v^3 - 1786582273*v) / 312153267 $$\beta_{9}$$ $$=$$ $$( 2704096 \nu^{14} - 29226746 \nu^{12} + 216623725 \nu^{10} - 875384062 \nu^{8} + \cdots - 1060049574 ) / 312153267$$ (2704096*v^14 - 29226746*v^12 + 216623725*v^10 - 875384062*v^8 + 2594464276*v^6 - 4025445729*v^4 + 4520206150*v^2 - 1060049574) / 312153267 $$\beta_{10}$$ $$=$$ $$( - 488787 \nu^{15} + 4750495 \nu^{13} - 33851633 \nu^{11} + 118687510 \nu^{9} + \cdots - 573803980 \nu ) / 104051089$$ (-488787*v^15 + 4750495*v^13 - 33851633*v^11 + 118687510*v^9 - 310327001*v^7 + 238139533*v^5 - 56060109*v^3 - 573803980*v) / 104051089 $$\beta_{11}$$ $$=$$ $$( - 1500252 \nu^{15} + 15079000 \nu^{13} - 103902068 \nu^{11} + 364291960 \nu^{9} + \cdots - 1239024512 \nu ) / 312153267$$ (-1500252*v^15 + 15079000*v^13 - 103902068*v^11 + 364291960*v^9 - 902070754*v^7 + 730930468*v^5 - 172067364*v^3 - 1239024512*v) / 312153267 $$\beta_{12}$$ $$=$$ $$( - 30472184 \nu^{15} + 349306330 \nu^{13} - 2649325358 \nu^{11} + 11246493662 \nu^{9} + \cdots + 14234057118 \nu ) / 2809379403$$ (-30472184*v^15 + 349306330*v^13 - 2649325358*v^11 + 11246493662*v^9 - 34081774484*v^7 + 56231857968*v^5 - 60658992614*v^3 + 14234057118*v) / 2809379403 $$\beta_{13}$$ $$=$$ $$( 31047565 \nu^{15} - 339328241 \nu^{13} + 2507751769 \nu^{11} - 10118919067 \nu^{9} + \cdots - 2156816727 \nu ) / 2809379403$$ (31047565*v^15 - 339328241*v^13 + 2507751769*v^11 - 10118919067*v^9 + 29524287979*v^7 - 43943348586*v^5 + 45709611853*v^3 - 2156816727*v) / 2809379403 $$\beta_{14}$$ $$=$$ $$( - 15452485 \nu^{15} + 171076316 \nu^{13} - 1275617842 \nu^{11} + 5283598924 \nu^{9} + \cdots + 6523373457 \nu ) / 936459801$$ (-15452485*v^15 + 171076316*v^13 - 1275617842*v^11 + 5283598924*v^9 - 15812055040*v^7 + 25548044169*v^5 - 27809065204*v^3 + 6523373457*v) / 936459801 $$\beta_{15}$$ $$=$$ $$( 48633295 \nu^{15} - 534513455 \nu^{13} + 3989805988 \nu^{11} - 16358061805 \nu^{9} + \cdots - 20082252600 \nu ) / 2809379403$$ (48633295*v^15 - 534513455*v^13 + 3989805988*v^11 - 16358061805*v^9 + 48815148070*v^7 - 76883173041*v^5 + 85617164065*v^3 - 20082252600*v) / 2809379403
 $$\nu$$ $$=$$ $$( \beta_{11} - 2\beta_{7} ) / 2$$ (b11 - 2*b7) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{9} + 3\beta_{4} + \beta_{2} + 3$$ -b9 + 3*b4 + b2 + 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{15} + 8\beta_{13} + 3\beta_{12} + 3\beta_{11} + 2\beta_{8} ) / 2$$ (-2*b15 + 8*b13 + 3*b12 + 3*b11 + 2*b8) / 2 $$\nu^{4}$$ $$=$$ $$-5\beta_{9} + \beta_{6} + 12\beta_{4}$$ -5*b9 + b6 + 12*b4 $$\nu^{5}$$ $$=$$ $$( -12\beta_{15} - 2\beta_{14} + 34\beta_{13} + 11\beta_{12} + 34\beta_{7} ) / 2$$ (-12*b15 - 2*b14 + 34*b13 + 11*b12 + 34*b7) / 2 $$\nu^{6}$$ $$=$$ $$7\beta_{5} + \beta_{3} - 23\beta_{2} - 51$$ 7*b5 + b3 - 23*b2 - 51 $$\nu^{7}$$ $$=$$ $$( -47\beta_{11} + 16\beta_{10} - 60\beta_{8} + 148\beta_{7} ) / 2$$ (-47*b11 + 16*b10 - 60*b8 + 148*b7) / 2 $$\nu^{8}$$ $$=$$ $$104\beta_{9} - 38\beta_{6} + 38\beta_{5} - 222\beta_{4} + 11\beta_{3} - 104\beta_{2} + 11\beta _1 - 222$$ 104*b9 - 38*b6 + 38*b5 - 222*b4 + 11*b3 - 104*b2 + 11*b1 - 222 $$\nu^{9}$$ $$=$$ $$( 284\beta_{15} + 98\beta_{14} - 652\beta_{13} - 217\beta_{12} - 217\beta_{11} + 98\beta_{10} - 284\beta_{8} ) / 2$$ (284*b15 + 98*b14 - 652*b13 - 217*b12 - 217*b11 + 98*b10 - 284*b8) / 2 $$\nu^{10}$$ $$=$$ $$468\beta_{9} - 191\beta_{6} - 978\beta_{4} + 82\beta_1$$ 468*b9 - 191*b6 - 978*b4 + 82*b1 $$\nu^{11}$$ $$=$$ $$( 1318\beta_{15} + 546\beta_{14} - 2892\beta_{13} - 1033\beta_{12} - 2892\beta_{7} ) / 2$$ (1318*b15 + 546*b14 - 2892*b13 - 1033*b12 - 2892*b7) / 2 $$\nu^{12}$$ $$=$$ $$-932\beta_{5} - 519\beta_{3} + 2105\beta_{2} + 4338$$ -932*b5 - 519*b3 + 2105*b2 + 4338 $$\nu^{13}$$ $$=$$ $$( 4963\beta_{11} - 2902\beta_{10} + 6074\beta_{8} - 12886\beta_{7} ) / 2$$ (4963*b11 - 2902*b10 + 6074*b8 - 12886*b7) / 2 $$\nu^{14}$$ $$=$$ $$- 9480 \beta_{9} + 4488 \beta_{6} - 4488 \beta_{5} + 19329 \beta_{4} - 3008 \beta_{3} + 9480 \beta_{2} + \cdots + 19329$$ -9480*b9 + 4488*b6 - 4488*b5 + 19329*b4 - 3008*b3 + 9480*b2 - 3008*b1 + 19329 $$\nu^{15}$$ $$=$$ $$( - 27936 \beta_{15} - 14992 \beta_{14} + 57618 \beta_{13} + 23865 \beta_{12} + 23865 \beta_{11} + \cdots + 27936 \beta_{8} ) / 2$$ (-27936*b15 - 14992*b14 + 57618*b13 + 23865*b12 + 23865*b11 - 14992*b10 + 27936*b8) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$\beta_{4}$$

## 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
 −0.426014 + 0.245959i −1.87040 + 1.07988i 1.77290 − 1.02359i 1.19454 − 0.689667i −1.19454 + 0.689667i −1.77290 + 1.02359i 1.87040 − 1.07988i 0.426014 − 0.245959i −0.426014 − 0.245959i −1.87040 − 1.07988i 1.77290 + 1.02359i 1.19454 + 0.689667i −1.19454 − 0.689667i −1.77290 − 1.02359i 1.87040 + 1.07988i 0.426014 + 0.245959i
−2.38851 1.37901i 1.29204 + 0.745959i 2.80333 + 4.85550i 0 −2.05737 3.56347i 2.84864i 9.94721i −0.387090 0.670459i 0
49.2 −1.44154 0.832272i 1.00438 + 0.579878i 0.385355 + 0.667454i 0 −0.965233 1.67183i 2.43525i 2.04621i −0.827483 1.43324i 0
49.3 −1.03136 0.595455i −2.63893 1.52359i −0.290867 0.503797i 0 1.81445 + 3.14272i 0.609175i 3.07461i 3.14263 + 5.44319i 0
49.4 −0.950409 0.548719i −0.328513 0.189667i −0.397815 0.689035i 0 0.208148 + 0.360522i 1.89307i 3.06803i −1.42805 2.47346i 0
49.5 0.950409 + 0.548719i 0.328513 + 0.189667i −0.397815 0.689035i 0 0.208148 + 0.360522i 1.89307i 3.06803i −1.42805 2.47346i 0
49.6 1.03136 + 0.595455i 2.63893 + 1.52359i −0.290867 0.503797i 0 1.81445 + 3.14272i 0.609175i 3.07461i 3.14263 + 5.44319i 0
49.7 1.44154 + 0.832272i −1.00438 0.579878i 0.385355 + 0.667454i 0 −0.965233 1.67183i 2.43525i 2.04621i −0.827483 1.43324i 0
49.8 2.38851 + 1.37901i −1.29204 0.745959i 2.80333 + 4.85550i 0 −2.05737 3.56347i 2.84864i 9.94721i −0.387090 0.670459i 0
349.1 −2.38851 + 1.37901i 1.29204 0.745959i 2.80333 4.85550i 0 −2.05737 + 3.56347i 2.84864i 9.94721i −0.387090 + 0.670459i 0
349.2 −1.44154 + 0.832272i 1.00438 0.579878i 0.385355 0.667454i 0 −0.965233 + 1.67183i 2.43525i 2.04621i −0.827483 + 1.43324i 0
349.3 −1.03136 + 0.595455i −2.63893 + 1.52359i −0.290867 + 0.503797i 0 1.81445 3.14272i 0.609175i 3.07461i 3.14263 5.44319i 0
349.4 −0.950409 + 0.548719i −0.328513 + 0.189667i −0.397815 + 0.689035i 0 0.208148 0.360522i 1.89307i 3.06803i −1.42805 + 2.47346i 0
349.5 0.950409 0.548719i 0.328513 0.189667i −0.397815 + 0.689035i 0 0.208148 0.360522i 1.89307i 3.06803i −1.42805 + 2.47346i 0
349.6 1.03136 0.595455i 2.63893 1.52359i −0.290867 + 0.503797i 0 1.81445 3.14272i 0.609175i 3.07461i 3.14263 5.44319i 0
349.7 1.44154 0.832272i −1.00438 + 0.579878i 0.385355 0.667454i 0 −0.965233 + 1.67183i 2.43525i 2.04621i −0.827483 + 1.43324i 0
349.8 2.38851 1.37901i −1.29204 + 0.745959i 2.80333 4.85550i 0 −2.05737 + 3.56347i 2.84864i 9.94721i −0.387090 + 0.670459i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.c 16
5.b even 2 1 inner 475.2.j.c 16
5.c odd 4 1 95.2.e.c 8
5.c odd 4 1 475.2.e.e 8
15.e even 4 1 855.2.k.h 8
19.c even 3 1 inner 475.2.j.c 16
20.e even 4 1 1520.2.q.o 8
95.i even 6 1 inner 475.2.j.c 16
95.l even 12 1 1805.2.a.i 4
95.l even 12 1 9025.2.a.bp 4
95.m odd 12 1 95.2.e.c 8
95.m odd 12 1 475.2.e.e 8
95.m odd 12 1 1805.2.a.o 4
95.m odd 12 1 9025.2.a.bg 4
285.v even 12 1 855.2.k.h 8
380.v even 12 1 1520.2.q.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 5.c odd 4 1
95.2.e.c 8 95.m odd 12 1
475.2.e.e 8 5.c odd 4 1
475.2.e.e 8 95.m odd 12 1
475.2.j.c 16 1.a even 1 1 trivial
475.2.j.c 16 5.b even 2 1 inner
475.2.j.c 16 19.c even 3 1 inner
475.2.j.c 16 95.i even 6 1 inner
855.2.k.h 8 15.e even 4 1
855.2.k.h 8 285.v even 12 1
1520.2.q.o 8 20.e even 4 1
1520.2.q.o 8 380.v even 12 1
1805.2.a.i 4 95.l even 12 1
1805.2.a.o 4 95.m odd 12 1
9025.2.a.bg 4 95.m odd 12 1
9025.2.a.bp 4 95.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 13T_{2}^{14} + 119T_{2}^{12} - 504T_{2}^{10} + 1515T_{2}^{8} - 2714T_{2}^{6} + 3529T_{2}^{4} - 2628T_{2}^{2} + 1296$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 13 T^{14} + \cdots + 1296$$
$3$ $$T^{16} - 13 T^{14} + \cdots + 16$$
$5$ $$T^{16}$$
$7$ $$(T^{8} + 18 T^{6} + \cdots + 64)^{2}$$
$11$ $$(T^{4} + 2 T^{3} - 25 T^{2} + \cdots + 3)^{4}$$
$13$ $$T^{16} - 35 T^{14} + \cdots + 65536$$
$17$ $$T^{16} + \cdots + 136048896$$
$19$ $$(T^{8} + 5 T^{7} + \cdots + 130321)^{2}$$
$23$ $$T^{16} - 38 T^{14} + \cdots + 1296$$
$29$ $$(T^{8} + T^{7} + \cdots + 19881)^{2}$$
$31$ $$(T^{4} - 67 T^{2} + \cdots + 1063)^{4}$$
$37$ $$(T^{8} + 66 T^{6} + \cdots + 13924)^{2}$$
$41$ $$(T^{8} - 8 T^{7} + \cdots + 5008644)^{2}$$
$43$ $$T^{16} + \cdots + 397449550096$$
$47$ $$T^{16} + \cdots + 28770951188736$$
$53$ $$T^{16} - 197 T^{14} + \cdots + 8503056$$
$59$ $$(T^{8} + 5 T^{7} + \cdots + 3515625)^{2}$$
$61$ $$(T^{8} + 130 T^{6} + \cdots + 9296401)^{2}$$
$67$ $$T^{16} - 88 T^{14} + \cdots + 16777216$$
$71$ $$(T^{8} + 20 T^{7} + \cdots + 59049)^{2}$$
$73$ $$T^{16} + \cdots + 8874893813776$$
$79$ $$(T^{8} - 17 T^{7} + \cdots + 33856)^{2}$$
$83$ $$(T^{8} + 125 T^{6} + \cdots + 133956)^{2}$$
$89$ $$(T^{8} - 11 T^{7} + \cdots + 14561856)^{2}$$
$97$ $$T^{16} + \cdots + 30\!\cdots\!96$$