# Properties

 Label 725.2.e.b Level $725$ Weight $2$ Character orbit 725.e Analytic conductor $5.789$ 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] = [725,2,Mod(157,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("725.157");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.e (of order $$4$$, degree $$2$$, 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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 28x^{14} + 288x^{12} + 1372x^{10} + 3184x^{8} + 3696x^{6} + 2076x^{4} + 504x^{2} + 36$$ x^16 + 28*x^14 + 288*x^12 + 1372*x^10 + 3184*x^8 + 3696*x^6 + 2076*x^4 + 504*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + \beta_{9} q^{3} - \beta_{2} q^{4} + (\beta_{14} - \beta_{12} + \cdots + \beta_{4}) q^{6}+ \cdots + (\beta_{5} + \beta_{2} + 2) q^{9}+O(q^{10})$$ q + b7 * q^2 + b9 * q^3 - b2 * q^4 + (b14 - b12 + b6 + b4) * q^6 - b11 * q^7 + (-b15 + b11 - b10 + b7 + b1) * q^8 + (b5 + b2 + 2) * q^9 $$q + \beta_{7} q^{2} + \beta_{9} q^{3} - \beta_{2} q^{4} + (\beta_{14} - \beta_{12} + \cdots + \beta_{4}) q^{6}+ \cdots + ( - 3 \beta_{14} + \beta_{13} + \cdots + 5) q^{99}+O(q^{100})$$ q + b7 * q^2 + b9 * q^3 - b2 * q^4 + (b14 - b12 + b6 + b4) * q^6 - b11 * q^7 + (-b15 + b11 - b10 + b7 + b1) * q^8 + (b5 + b2 + 2) * q^9 + (-b14 - b13 + b5) * q^11 + (-2*b11 - 2*b10 + b3 + 2*b1) * q^12 + (b15 + b7 + b3 + b1) * q^13 + (b12 - b6 - b4 - b2 - 1) * q^14 + (-b14 - b12 + b5 - b2) * q^16 + (-b15 + 2*b7) * q^17 + (3*b15 - 2*b11 + 2*b10 - b8 + 3*b7 - 2*b1) * q^18 + (b13 - b12 + b6 + b5 + 1) * q^19 + (b6 + 2*b4 - 2*b2 - 1) * q^21 + (2*b15 + 3*b10 - b7 - 2*b3 - 2*b1) * q^22 + (-2*b15 - b10 + b9 + b8 - b7 + 2*b3 + 2*b1) * q^23 + (-b13 - 2*b6 - 2*b4) * q^24 + (-b13 - 2*b6 - b5 - b4 - b2 - 2) * q^26 + (3*b11 + 3*b10 - 2*b3 - b1) * q^27 + (-b15 + b11 + b9 - b8 - b7 - b3 - b1) * q^28 + (b13 - b12 - b5 - 2*b4 + b2) * q^29 + (-b14 - b13 - 2*b6 + b5 + b4 - b2 + 2) * q^31 + (-b15 + b11 - b10 + b8 - b7 + b1) * q^32 + (-2*b15 + 2*b11 - b7 - 2*b3 - b1) * q^33 + (b5 - 2*b2 - 4) * q^34 + (b14 + b12 - b5 - 4*b2 - 5) * q^36 + (-3*b11 - 3*b10 + 3*b3 - b1) * q^37 + (2*b15 - 3*b11 + 2*b3) * q^38 + (4*b14 + b13 + 2*b6 - b5 - 2) * q^39 + (-b13 - b5) * q^41 + (-2*b15 - 4*b10 - 3*b7 + 2*b3 + 7*b1) * q^42 + (3*b11 + 3*b10 - 3*b9 - 2*b3 - 3*b1) * q^43 + (b14 + b6 + 2*b4 - 2*b2 - 1) * q^44 + (b14 - 2*b13 - 2*b6 + 2*b5 - b4 + b2 + 2) * q^46 + (2*b11 + 2*b10 + b9 + 2*b3 - 2*b1) * q^47 + (-b11 - b10 - b9 - 2*b3 - 3*b1) * q^48 + (-b14 - b13 + b12 + 2*b6 - b4) * q^49 + (b14 - b13 - b12 + b6 + 3*b4) * q^51 + (-b15 + 4*b11 - b9 + b8 - b7 - b3 - b1) * q^52 + (b9 + b8 - 3*b7 + 3*b1) * q^53 + (3*b14 + 2*b13 - 3*b12 + 2*b6 + 4*b4) * q^54 + (b13 - b12 + 2*b6 + b5 + b4 + b2 + 2) * q^56 + (2*b15 + 2*b10 + b9 + b8 + b7 - 2*b3 - 3*b1) * q^57 + (-b15 + b10 + 2*b8 - 4*b1) * q^58 + (3*b13 - b6 - b4) * q^59 + (-2*b14 - 2*b6 + 2) * q^61 + (b15 + b10 - b3 - b1) * q^62 + (2*b15 - 5*b11 - b9 + b8 + 2*b3) * q^63 + (-2*b14 - 2*b12 + 3*b5 + 3) * q^64 + (2*b13 - 2*b12 + 4*b6 + 2*b5 + 3*b4 + 3*b2 + 4) * q^66 + (-b15 + b10 - 3*b7 + b3 + 2*b1) * q^67 + (-2*b15 + b11 - b10 - b8 - 2*b7 + b1) * q^68 + (-b13 + 6*b12 + b6 - b5 + 1) * q^69 + (2*b14 - b13 - 2*b12 - b6 + b4) * q^71 + (2*b11 - 2*b10 - 3*b8 - b7 + 2*b1) * q^72 + (2*b11 - 2*b10 + 3*b8 - 2*b7 + 2*b1) * q^73 + (-3*b14 - 3*b13 + 3*b12 + 2*b6 - 2*b4) * q^74 + (b12 - b6 - 3*b4 - 3*b2 - 1) * q^76 + (3*b15 - 3*b11 + 3*b10 - b8 - 3*b1) * q^77 + (-2*b15 - 6*b10 - 3*b9 - 3*b8 + 2*b7 + 2*b3 + 4*b1) * q^78 + (-b13 - b12 - 2*b6 - b5 - 2*b4 - 2*b2 - 2) * q^79 + (-b5 + 5*b2) * q^81 + (-2*b15 + 2*b11 - b9 + b8 - 2*b3) * q^82 + (3*b15 + 7*b10 - 2*b9 - 2*b8 - 2*b7 - 3*b3 - 5*b1) * q^83 + (-4*b14 - 2*b13 - 8*b6 + 2*b5 - 3*b4 + 3*b2 + 8) * q^84 + (2*b13 + 3*b6 + 3*b4) * q^86 + (4*b11 - 2*b10 - 2*b9 + b7 - b3 - 3*b1) * q^87 + (2*b15 + b10 - b9 - b8 - 4*b7 - 2*b3 + 3*b1) * q^88 + (b13 + 2*b12 - 3*b6 + b5 - 3) * q^89 + (-b14 + b12 + 2*b6 - 4*b4) * q^91 + (b15 + 3*b10 - b9 - b8 + 2*b7 - b3 - 5*b1) * q^92 + (-b15 - 2*b11 + 2*b9 - 2*b8 - b7 - b3 - b1) * q^93 + (3*b14 - 2*b13 - 3*b12 + 5*b6 + 5*b4) * q^94 + (-2*b14 + 2*b12 + b6 - 3*b4) * q^96 + (-b11 - b10 + b9 - 2*b3 - 5*b1) * q^97 + (3*b11 + 3*b10 + b9 - 3*b3) * q^98 + (-3*b14 + b13 - 5*b6 - b5 - 3*b4 + 3*b2 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 8 q^{4} + 40 q^{9}+O(q^{10})$$ 16 * q - 8 * q^4 + 40 * q^9 $$16 q - 8 q^{4} + 40 q^{9} - 4 q^{11} - 20 q^{14} - 16 q^{16} + 12 q^{19} - 32 q^{21} - 40 q^{26} + 4 q^{29} + 20 q^{31} - 80 q^{34} - 104 q^{36} - 16 q^{39} - 28 q^{44} + 44 q^{46} + 36 q^{56} + 24 q^{61} + 32 q^{64} + 80 q^{66} + 40 q^{69} - 36 q^{76} - 52 q^{79} + 40 q^{81} + 136 q^{84} - 40 q^{89} + 92 q^{99}+O(q^{100})$$ 16 * q - 8 * q^4 + 40 * q^9 - 4 * q^11 - 20 * q^14 - 16 * q^16 + 12 * q^19 - 32 * q^21 - 40 * q^26 + 4 * q^29 + 20 * q^31 - 80 * q^34 - 104 * q^36 - 16 * q^39 - 28 * q^44 + 44 * q^46 + 36 * q^56 + 24 * q^61 + 32 * q^64 + 80 * q^66 + 40 * q^69 - 36 * q^76 - 52 * q^79 + 40 * q^81 + 136 * q^84 - 40 * q^89 + 92 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 28x^{14} + 288x^{12} + 1372x^{10} + 3184x^{8} + 3696x^{6} + 2076x^{4} + 504x^{2} + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( - 30 \nu^{14} - 863 \nu^{12} - 9206 \nu^{10} - 45753 \nu^{8} - 108620 \nu^{6} - 113912 \nu^{4} + \cdots - 1362 ) / 1596$$ (-30*v^14 - 863*v^12 - 9206*v^10 - 45753*v^8 - 108620*v^6 - 113912*v^4 - 41766*v^2 - 1362) / 1596 $$\beta_{2}$$ $$=$$ $$( -30\nu^{14} - 863\nu^{12} - 9206\nu^{10} - 45753\nu^{8} - 108620\nu^{6} - 113912\nu^{4} - 42564\nu^{2} - 3756 ) / 798$$ (-30*v^14 - 863*v^12 - 9206*v^10 - 45753*v^8 - 108620*v^6 - 113912*v^4 - 42564*v^2 - 3756) / 798 $$\beta_{3}$$ $$=$$ $$( 90 \nu^{14} + 2380 \nu^{12} + 22165 \nu^{10} + 87612 \nu^{8} + 137266 \nu^{6} + 65590 \nu^{4} + \cdots - 6972 ) / 1596$$ (90*v^14 + 2380*v^12 + 22165*v^10 + 87612*v^8 + 137266*v^6 + 65590*v^4 - 11958*v^2 - 6972) / 1596 $$\beta_{4}$$ $$=$$ $$( - 313 \nu^{15} - 8446 \nu^{13} - 81570 \nu^{11} - 346642 \nu^{9} - 643474 \nu^{7} - 486936 \nu^{5} + \cdots + 720 \nu ) / 9576$$ (-313*v^15 - 8446*v^13 - 81570*v^11 - 346642*v^9 - 643474*v^7 - 486936*v^5 - 89400*v^3 + 720*v) / 9576 $$\beta_{5}$$ $$=$$ $$( - 66 \nu^{14} - 1834 \nu^{12} - 18604 \nu^{10} - 86247 \nu^{8} - 188956 \nu^{6} - 194488 \nu^{4} + \cdots - 10794 ) / 798$$ (-66*v^14 - 1834*v^12 - 18604*v^10 - 86247*v^8 - 188956*v^6 - 194488*v^4 - 84384*v^2 - 10794) / 798 $$\beta_{6}$$ $$=$$ $$( 493 \nu^{15} + 13624 \nu^{13} + 136806 \nu^{11} + 621160 \nu^{9} + 1295194 \nu^{7} + 1170408 \nu^{5} + \cdots - 2124 \nu ) / 9576$$ (493*v^15 + 13624*v^13 + 136806*v^11 + 621160*v^9 + 1295194*v^7 + 1170408*v^5 + 339996*v^3 - 2124*v) / 9576 $$\beta_{7}$$ $$=$$ $$( - 493 \nu^{15} - 13624 \nu^{13} - 136806 \nu^{11} - 621160 \nu^{9} - 1295194 \nu^{7} + \cdots + 11700 \nu ) / 9576$$ (-493*v^15 - 13624*v^13 - 136806*v^11 - 621160*v^9 - 1295194*v^7 - 1170408*v^5 - 339996*v^3 + 11700*v) / 9576 $$\beta_{8}$$ $$=$$ $$( 349 \nu^{15} + 9721 \nu^{13} + 99081 \nu^{11} + 464029 \nu^{9} + 1039552 \nu^{7} + 1115358 \nu^{5} + \cdots + 56934 \nu ) / 4788$$ (349*v^15 + 9721*v^13 + 99081*v^11 + 464029*v^9 + 1039552*v^7 + 1115358*v^5 + 501492*v^3 + 56934*v) / 4788 $$\beta_{9}$$ $$=$$ $$( - 109 \nu^{14} - 2969 \nu^{12} - 29157 \nu^{10} - 128006 \nu^{8} - 255446 \nu^{6} - 229940 \nu^{4} + \cdots - 7240 ) / 532$$ (-109*v^14 - 2969*v^12 - 29157*v^10 - 128006*v^8 - 255446*v^6 - 229940*v^4 - 81446*v^2 - 7240) / 532 $$\beta_{10}$$ $$=$$ $$( 1441 \nu^{15} + 21 \nu^{14} + 39295 \nu^{13} + 264 \nu^{12} + 386592 \nu^{11} - 2520 \nu^{10} + \cdots - 16272 ) / 9576$$ (1441*v^15 + 21*v^14 + 39295*v^13 + 264*v^12 + 386592*v^11 - 2520*v^10 + 1702360*v^9 - 50982*v^8 + 3415408*v^7 - 249018*v^6 + 3101394*v^5 - 425160*v^4 + 1124184*v^3 - 235620*v^2 + 114696*v - 16272) / 9576 $$\beta_{11}$$ $$=$$ $$( - 1441 \nu^{15} + 201 \nu^{14} - 39295 \nu^{13} + 5442 \nu^{12} - 386592 \nu^{11} + 52716 \nu^{10} + \cdots - 8100 ) / 9576$$ (-1441*v^15 + 201*v^14 - 39295*v^13 + 5442*v^12 - 386592*v^11 + 52716*v^10 - 1702360*v^9 + 223536*v^8 - 3415408*v^7 + 402702*v^6 - 3101394*v^5 + 258312*v^4 - 1124184*v^3 + 14976*v^2 - 114696*v - 8100) / 9576 $$\beta_{12}$$ $$=$$ $$( 1420 \nu^{15} - 1212 \nu^{14} + 39031 \nu^{13} - 33330 \nu^{12} + 389112 \nu^{11} - 332448 \nu^{10} + \cdots - 95472 ) / 9576$$ (1420*v^15 - 1212*v^14 + 39031*v^13 - 33330*v^12 + 389112*v^11 - 332448*v^10 + 1753342*v^9 - 1498236*v^8 + 3664426*v^7 - 3125280*v^6 + 3526554*v^5 - 2975964*v^4 + 1359804*v^3 - 1107360*v^2 + 121392*v - 95472) / 9576 $$\beta_{13}$$ $$=$$ $$( 316 \nu^{15} + 8766 \nu^{13} + 88715 \nu^{11} + 410047 \nu^{9} + 895332 \nu^{7} + 919808 \nu^{5} + \cdots + 54786 \nu ) / 1596$$ (316*v^15 + 8766*v^13 + 88715*v^11 + 410047*v^9 + 895332*v^7 + 919808*v^5 + 400248*v^3 + 54786*v) / 1596 $$\beta_{14}$$ $$=$$ $$( - 1420 \nu^{15} - 1212 \nu^{14} - 39031 \nu^{13} - 33330 \nu^{12} - 389112 \nu^{11} - 332448 \nu^{10} + \cdots - 95472 ) / 9576$$ (-1420*v^15 - 1212*v^14 - 39031*v^13 - 33330*v^12 - 389112*v^11 - 332448*v^10 - 1753342*v^9 - 1498236*v^8 - 3664426*v^7 - 3125280*v^6 - 3526554*v^5 - 2975964*v^4 - 1359804*v^3 - 1107360*v^2 - 121392*v - 95472) / 9576 $$\beta_{15}$$ $$=$$ $$( - 406 \nu^{15} - 11146 \nu^{13} - 110880 \nu^{11} - 497659 \nu^{9} - 1032598 \nu^{7} + \cdots - 47814 \nu ) / 1596$$ (-406*v^15 - 11146*v^13 - 110880*v^11 - 497659*v^9 - 1032598*v^7 - 985398*v^5 - 388290*v^3 - 47814*v) / 1596
 $$\nu$$ $$=$$ $$\beta_{7} + \beta_{6}$$ b7 + b6 $$\nu^{2}$$ $$=$$ $$-\beta_{2} + 2\beta _1 - 3$$ -b2 + 2*b1 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{11} - \beta_{10} - 6\beta_{7} - 7\beta_{6} - 3\beta_{4} + \beta_1$$ -b15 + b11 - b10 - 6*b7 - 7*b6 - 3*b4 + b1 $$\nu^{4}$$ $$=$$ $$-\beta_{14} - \beta_{12} + 4\beta_{11} + 4\beta_{10} + \beta_{5} - 4\beta_{3} + 11\beta_{2} - 20\beta _1 + 21$$ -b14 - b12 + 4*b11 + 4*b10 + b5 - 4*b3 + 11*b2 - 20*b1 + 21 $$\nu^{5}$$ $$=$$ $$17 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} - 17 \beta_{11} + 17 \beta_{10} + \cdots - 17 \beta_1$$ 17*b15 + 5*b14 + 5*b13 - 5*b12 - 17*b11 + 17*b10 + b8 + 46*b7 + 61*b6 + 35*b4 - 17*b1 $$\nu^{6}$$ $$=$$ $$23 \beta_{14} + 23 \beta_{12} - 62 \beta_{11} - 62 \beta_{10} - 6 \beta_{9} - 22 \beta_{5} + 62 \beta_{3} + \cdots - 188$$ 23*b14 + 23*b12 - 62*b11 - 62*b10 - 6*b9 - 22*b5 + 62*b3 - 116*b2 + 194*b1 - 188 $$\nu^{7}$$ $$=$$ $$- 222 \beta_{15} - 91 \beta_{14} - 84 \beta_{13} + 91 \beta_{12} + 223 \beta_{11} - 223 \beta_{10} + \cdots + 223 \beta_1$$ -222*b15 - 91*b14 - 84*b13 + 91*b12 + 223*b11 - 223*b10 - 30*b8 - 390*b7 - 582*b6 - 378*b4 + 223*b1 $$\nu^{8}$$ $$=$$ $$- 344 \beta_{14} - 344 \beta_{12} + 776 \beta_{11} + 776 \beta_{10} + 128 \beta_{9} + 306 \beta_{5} + \cdots + 1838$$ -344*b14 - 344*b12 + 776*b11 + 776*b10 + 128*b9 + 306*b5 - 768*b3 + 1244*b2 - 1944*b1 + 1838 $$\nu^{9}$$ $$=$$ $$2624 \beta_{15} + 1248 \beta_{14} + 1074 \beta_{13} - 1248 \beta_{12} - 2670 \beta_{11} + \cdots - 2670 \beta_1$$ 2624*b15 + 1248*b14 + 1074*b13 - 1248*b12 - 2670*b11 + 2670*b10 + 510*b8 + 3562*b7 + 5854*b6 + 4092*b4 - 2670*b1 $$\nu^{10}$$ $$=$$ $$4428 \beta_{14} + 4428 \beta_{12} - 9084 \beta_{11} - 9084 \beta_{10} - 1932 \beta_{9} - 3698 \beta_{5} + \cdots - 18828$$ 4428*b14 + 4428*b12 - 9084*b11 - 9084*b10 - 1932*b9 - 3698*b5 + 8864*b3 - 13504*b2 + 20096*b1 - 18828 $$\nu^{11}$$ $$=$$ $$- 29764 \beta_{15} - 15444 \beta_{14} - 12562 \beta_{13} + 15444 \beta_{12} + 30714 \beta_{11} + \cdots + 30714 \beta_1$$ -29764*b15 - 15444*b14 - 12562*b13 + 15444*b12 + 30714*b11 - 30714*b10 - 7090*b8 - 34488*b7 - 60952*b6 - 44616*b4 + 30714*b1 $$\nu^{12}$$ $$=$$ $$- 53248 \beta_{14} - 53248 \beta_{12} + 103336 \beta_{11} + 103336 \beta_{10} + 25416 \beta_{9} + \cdots + 198446$$ -53248*b14 - 53248*b12 + 103336*b11 + 103336*b10 + 25416*b9 + 42326*b5 - 99504*b3 + 147622*b2 - 212504*b1 + 198446 $$\nu^{13}$$ $$=$$ $$331778 \beta_{15} + 182000 \beta_{14} + 141830 \beta_{13} - 182000 \beta_{12} - 346532 \beta_{11} + \cdots - 346532 \beta_1$$ 331778*b15 + 182000*b14 + 141830*b13 - 182000*b12 - 346532*b11 + 346532*b10 + 89586*b8 + 348740*b7 + 648870*b6 + 488878*b4 - 346532*b1 $$\nu^{14}$$ $$=$$ $$618118 \beta_{14} + 618118 \beta_{12} - 1159240 \beta_{11} - 1159240 \beta_{10} - 311756 \beta_{9} + \cdots - 2129000$$ 618118*b14 + 618118*b12 - 1159240*b11 - 1159240*b10 - 311756*b9 - 473608*b5 + 1104316*b3 - 1620268*b2 + 2281728*b1 - 2129000 $$\nu^{15}$$ $$=$$ $$- 3671800 \beta_{15} - 2089114 \beta_{14} - 1577924 \beta_{13} + 2089114 \beta_{12} + \cdots + 3871234 \beta_1$$ -3671800*b15 - 2089114*b14 - 1577924*b13 + 2089114*b12 + 3871234*b11 - 3871234*b10 - 1074384*b8 - 3635520*b7 - 7004212*b6 - 5372992*b4 + 3871234*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-\beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 − 3.32267i − 2.66264i − 2.34301i − 1.19281i − 0.807187i 0.343006i 0.662643i 1.32267i − 1.32267i − 0.662643i − 0.343006i 0.807187i 1.19281i 2.34301i 2.66264i 3.32267i
2.32267i 3.11277 −3.39477 0 7.22993i −2.50245 2.50245i 3.23960i 6.68935 0
157.2 1.66264i −2.84124 −0.764383 0 4.72396i 0.890023 + 0.890023i 2.05439i 5.07263 0
157.3 1.34301i 0.842805 0.196335 0 1.13189i −0.720606 0.720606i 2.94969i −2.28968 0
157.4 0.192813i 1.87822 1.96282 0 0.362145i 1.55767 + 1.55767i 0.764084i 0.527707 0
157.5 0.192813i −1.87822 1.96282 0 0.362145i −1.55767 1.55767i 0.764084i 0.527707 0
157.6 1.34301i −0.842805 0.196335 0 1.13189i 0.720606 + 0.720606i 2.94969i −2.28968 0
157.7 1.66264i 2.84124 −0.764383 0 4.72396i −0.890023 0.890023i 2.05439i 5.07263 0
157.8 2.32267i −3.11277 −3.39477 0 7.22993i 2.50245 + 2.50245i 3.23960i 6.68935 0
568.1 2.32267i −3.11277 −3.39477 0 7.22993i 2.50245 2.50245i 3.23960i 6.68935 0
568.2 1.66264i 2.84124 −0.764383 0 4.72396i −0.890023 + 0.890023i 2.05439i 5.07263 0
568.3 1.34301i −0.842805 0.196335 0 1.13189i 0.720606 0.720606i 2.94969i −2.28968 0
568.4 0.192813i −1.87822 1.96282 0 0.362145i −1.55767 + 1.55767i 0.764084i 0.527707 0
568.5 0.192813i 1.87822 1.96282 0 0.362145i 1.55767 1.55767i 0.764084i 0.527707 0
568.6 1.34301i 0.842805 0.196335 0 1.13189i −0.720606 + 0.720606i 2.94969i −2.28968 0
568.7 1.66264i −2.84124 −0.764383 0 4.72396i 0.890023 0.890023i 2.05439i 5.07263 0
568.8 2.32267i 3.11277 −3.39477 0 7.22993i −2.50245 + 2.50245i 3.23960i 6.68935 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 157.8 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
145.e even 4 1 inner
145.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.e.b 16
5.b even 2 1 inner 725.2.e.b 16
5.c odd 4 2 725.2.j.b yes 16
29.c odd 4 1 725.2.j.b yes 16
145.e even 4 1 inner 725.2.e.b 16
145.f odd 4 1 725.2.j.b yes 16
145.j even 4 1 inner 725.2.e.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.e.b 16 1.a even 1 1 trivial
725.2.e.b 16 5.b even 2 1 inner
725.2.e.b 16 145.e even 4 1 inner
725.2.e.b 16 145.j even 4 1 inner
725.2.j.b yes 16 5.c odd 4 2
725.2.j.b yes 16 29.c odd 4 1
725.2.j.b yes 16 145.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 10T_{2}^{6} + 30T_{2}^{4} + 28T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 10 T^{6} + 30 T^{4} + \cdots + 1)^{2}$$
$3$ $$(T^{8} - 22 T^{6} + \cdots + 196)^{2}$$
$5$ $$T^{16}$$
$7$ $$T^{16} + 184 T^{12} + \cdots + 10000$$
$11$ $$(T^{8} + 2 T^{7} + \cdots + 2116)^{2}$$
$13$ $$T^{16} + \cdots + 723394816$$
$17$ $$(T^{8} + 58 T^{6} + \cdots + 9604)^{2}$$
$19$ $$(T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 36)^{2}$$
$23$ $$T^{16} + \cdots + 178506250000$$
$29$ $$(T^{8} - 2 T^{7} + \cdots + 707281)^{2}$$
$31$ $$(T^{8} - 10 T^{7} + \cdots + 196)^{2}$$
$37$ $$(T^{8} - 226 T^{6} + \cdots + 2149156)^{2}$$
$41$ $$(T^{8} + 72 T^{5} + \cdots + 576)^{2}$$
$43$ $$(T^{8} - 198 T^{6} + \cdots + 2965284)^{2}$$
$47$ $$(T^{8} - 414 T^{6} + \cdots + 104407524)^{2}$$
$53$ $$T^{16} + \cdots + 62500000000$$
$59$ $$(T^{8} + 204 T^{6} + \cdots + 7056)^{2}$$
$61$ $$(T^{8} - 12 T^{7} + \cdots + 451584)^{2}$$
$67$ $$T^{16} + \cdots + 1003875856$$
$71$ $$(T^{8} + 300 T^{6} + \cdots + 90000)^{2}$$
$73$ $$(T^{8} + 238 T^{6} + \cdots + 2365444)^{2}$$
$79$ $$(T^{8} + 26 T^{7} + \cdots + 3062500)^{2}$$
$83$ $$T^{16} + \cdots + 138458410000$$
$89$ $$(T^{8} + 20 T^{7} + \cdots + 8976016)^{2}$$
$97$ $$(T^{8} - 438 T^{6} + \cdots + 5560164)^{2}$$