# Properties

 Label 176.2.q.b Level $176$ Weight $2$ Character orbit 176.q Analytic conductor $1.405$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(63,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 3]))

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

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

chi := DirichletCharacter("176.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.q (of order $$10$$, degree $$4$$, 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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.4526322734619140625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} - 96 x^{7} + 56 x^{6} + 48 x^{5} + 48 x^{4} - 192 x^{3} + 448 x^{2} - 384 x + 256$$ x^16 - 3*x^15 + 7*x^14 - 6*x^13 + 3*x^12 + 6*x^11 + 14*x^10 - 48*x^9 + 113*x^8 - 96*x^7 + 56*x^6 + 48*x^5 + 48*x^4 - 192*x^3 + 448*x^2 - 384*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_{2}) q^{3} + (\beta_{13} + \beta_{8} - \beta_{3} + \beta_1 + 1) q^{5} + (2 \beta_{14} - 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} + \beta_{2}) q^{7} + (\beta_{15} - \beta_{13} + 2 \beta_{3} + 2) q^{9}+O(q^{10})$$ q + (b11 - b9 - b6 + b4 + b2) * q^3 + (b13 + b8 - b3 + b1 + 1) * q^5 + (2*b14 - 2*b12 - b11 - b10 + b6 - b4 + b2) * q^7 + (b15 - b13 + 2*b3 + 2) * q^9 $$q + (\beta_{11} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_{2}) q^{3} + (\beta_{13} + \beta_{8} - \beta_{3} + \beta_1 + 1) q^{5} + (2 \beta_{14} - 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} + \beta_{2}) q^{7} + (\beta_{15} - \beta_{13} + 2 \beta_{3} + 2) q^{9} + ( - \beta_{11} - 2 \beta_{10} + \beta_{4}) q^{11} + (\beta_{8} + 2 \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_1 + 2) q^{13} + ( - 2 \beta_{12} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{6} + \beta_{4} + \beta_{2}) q^{15} + (\beta_{13} - \beta_{8} + \beta_{7} + \beta_{5} + 1) q^{17} + (\beta_{14} + 3 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + \beta_{9} + 2 \beta_{6} - \beta_{4} - \beta_{2}) q^{19} + ( - \beta_{15} - \beta_{8} - 3 \beta_{7} - 2 \beta_{3} - 2) q^{21} - 2 \beta_{12} q^{23} + (\beta_{8} - 3 \beta_{5} - 3) q^{25} + ( - 3 \beta_{14} + 4 \beta_{12} + 4 \beta_{11} + 4 \beta_{10} + 2 \beta_{9}) q^{27} + (\beta_{15} - 2 \beta_{13} - 2 \beta_{8} - 4 \beta_{5} - \beta_{3} - 4) q^{29} + ( - 3 \beta_{14} - \beta_{12} - \beta_{10} - 4 \beta_{6} - 3 \beta_{2}) q^{31} + ( - 2 \beta_{15} - 3 \beta_{7} - 3 \beta_{3} + \beta_1 - 5) q^{33} + ( - \beta_{14} + 5 \beta_{12} + \beta_{11} + 2 \beta_{10} + 4 \beta_{9} - \beta_{6} - 3 \beta_{4} + \cdots - 3 \beta_{2}) q^{35}+ \cdots + (\beta_{14} - 5 \beta_{12} - 6 \beta_{10} - \beta_{9} - 5 \beta_{6} + 3 \beta_{4} - 3 \beta_{2}) q^{99}+O(q^{100})$$ q + (b11 - b9 - b6 + b4 + b2) * q^3 + (b13 + b8 - b3 + b1 + 1) * q^5 + (2*b14 - 2*b12 - b11 - b10 + b6 - b4 + b2) * q^7 + (b15 - b13 + 2*b3 + 2) * q^9 + (-b11 - 2*b10 + b4) * q^11 + (b8 + 2*b7 + 2*b5 + b3 - b1 + 2) * q^13 + (-2*b12 - 2*b11 - b10 - 2*b9 - b6 + b4 + b2) * q^15 + (b13 - b8 + b7 + b5 + 1) * q^17 + (b14 + 3*b12 + 2*b11 + 2*b10 + b9 + 2*b6 - b4 - b2) * q^19 + (-b15 - b8 - 3*b7 - 2*b3 - 2) * q^21 - 2*b12 * q^23 + (b8 - 3*b5 - 3) * q^25 + (-3*b14 + 4*b12 + 4*b11 + 4*b10 + 2*b9) * q^27 + (b15 - 2*b13 - 2*b8 - 4*b5 - b3 - 4) * q^29 + (-3*b14 - b12 - b10 - 4*b6 - 3*b2) * q^31 + (-2*b15 - 3*b7 - 3*b3 + b1 - 5) * q^33 + (-b14 + 5*b12 + b11 + 2*b10 + 4*b9 - b6 - 3*b4 - 3*b2) * q^35 + (-b15 + 3*b3) * q^37 + (-b14 + b12 - 5*b11 + 5*b10 + 4*b6 - 4*b4 - 3*b2) * q^39 + (-b15 + 4*b5 - b1 - 4) * q^41 + (-3*b14 + 2*b12 + 2*b11 - 2*b10 + 2*b9 + b6 + b4) * q^43 + (b15 + b13 - b8 + b7 - 3) * q^45 + (2*b14 - 2*b12 - b11 + 2*b6 + 2*b4 + b2) * q^47 + (-b13 - b8 - b7 - 4*b5 - 3*b3 - 3*b1 - 1) * q^49 + (2*b14 - b12 + b11 - 2*b10 - b9 - 2*b6 + 4*b4) * q^51 + (2*b15 - 2*b13 - 3*b8 + 2*b5 + 3*b3 - 3*b1) * q^53 + (2*b14 - 2*b12 - 2*b11 - 2*b10 - 2*b9 + 4*b6 - 4*b4 + b2) * q^55 + (b15 + b13 - b8 + 8*b7 + 3*b5 + 2*b3 + b1 + 5) * q^57 + (-b12 - 2*b11 + 3*b10 - b9 + 2*b6 - 3*b4 - 3*b2) * q^59 + (-2*b15 - b13 + 3*b8 - 2*b5 + 5*b3 + b1 + 3) * q^61 + (3*b14 - 3*b12 + b11 - 6*b10 + 2*b9 - 5*b6 + 3*b4 + 2*b2) * q^63 + (b13 + 2*b8 - 4*b7 + 4*b5 - 2*b3 + 2*b1 + 1) * q^65 + (b14 - 3*b10 - 5*b9 - 2*b6 + 7*b4 + 5*b2) * q^67 + (-4*b7 - 2*b5 - 4*b3 - 2) * q^69 + (-4*b14 + 2*b12 + 2*b11 + 2*b10 + 4*b9 + b2) * q^71 + (-b15 + b13 + 2*b8 - 4*b7 - 6*b5 - 4*b3 + b1 - 2) * q^73 + (-b14 - 6*b11 + 2*b10 + 5*b6 - 6*b4 - b2) * q^75 + (b15 + 2*b13 + 2*b8 + 6*b7 + 10*b5 + 5*b3 + 2*b1 + 8) * q^77 + (3*b14 + 3*b12 + 3*b10 - 6*b4 - 3*b2) * q^79 + (4*b15 + 4*b7 + 4*b3 + 4) * q^81 + (-b14 - 5*b12 + b11 - b10 - 3*b9 - 8*b6 + 2*b4) * q^83 + (2*b13 + b8 + 5*b7 - 4*b5 - 5*b3 + 2*b1 + 6) * q^85 + (-2*b14 + 2*b12 + 6*b11 - 3*b10 + 2*b9 - b6 + 3*b4 + 4*b2) * q^87 + (-b15 - 2*b13 + b8 - 2*b7 - b5 + 2) * q^89 + (5*b14 - 5*b12 + 4*b11 - 4*b9 + b6 + b4 + 9*b2) * q^91 + (-b13 - b8 - 11*b7 - 6*b5 - 5*b3 - 1) * q^93 + (4*b14 + 2*b12 + 6*b11 + b10 - 6*b9 - 3*b6 + 11*b4 + 5*b2) * q^95 + (-4*b15 + 4*b13 + 4*b8 + 4*b5 + 5*b3 + 4*b1 + 9) * q^97 + (b14 - 5*b12 - 6*b10 - b9 - 5*b6 + 3*b4 - 3*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{5} + 22 q^{9}+O(q^{10})$$ 16 * q + 6 * q^5 + 22 * q^9 $$16 q + 6 q^{5} + 22 q^{9} + 10 q^{13} + 10 q^{17} - 42 q^{25} - 30 q^{29} - 48 q^{33} - 6 q^{37} - 70 q^{41} - 56 q^{45} + 38 q^{49} + 6 q^{53} + 20 q^{57} + 30 q^{61} + 8 q^{69} + 10 q^{73} + 10 q^{77} + 8 q^{81} + 90 q^{85} + 52 q^{89} + 82 q^{93} + 76 q^{97}+O(q^{100})$$ 16 * q + 6 * q^5 + 22 * q^9 + 10 * q^13 + 10 * q^17 - 42 * q^25 - 30 * q^29 - 48 * q^33 - 6 * q^37 - 70 * q^41 - 56 * q^45 + 38 * q^49 + 6 * q^53 + 20 * q^57 + 30 * q^61 + 8 * q^69 + 10 * q^73 + 10 * q^77 + 8 * q^81 + 90 * q^85 + 52 * q^89 + 82 * q^93 + 76 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} - 96 x^{7} + 56 x^{6} + 48 x^{5} + 48 x^{4} - 192 x^{3} + 448 x^{2} - 384 x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( 5 \nu^{15} - 271 \nu^{14} + 335 \nu^{13} + 26 \nu^{12} - 2081 \nu^{11} + 5050 \nu^{10} - 5150 \nu^{9} + 540 \nu^{8} + 3509 \nu^{7} + 2264 \nu^{6} - 14492 \nu^{5} + 41100 \nu^{4} + \cdots + 17472 ) / 15296$$ (5*v^15 - 271*v^14 + 335*v^13 + 26*v^12 - 2081*v^11 + 5050*v^10 - 5150*v^9 + 540*v^8 + 3509*v^7 + 2264*v^6 - 14492*v^5 + 41100*v^4 - 35296*v^3 + 40272*v^2 - 11552*v + 17472) / 15296 $$\beta_{2}$$ $$=$$ $$( 45 \nu^{15} + 907 \nu^{14} - 331 \nu^{13} - 1200 \nu^{12} + 7083 \nu^{11} - 6652 \nu^{10} - 4286 \nu^{9} + 27804 \nu^{8} - 33427 \nu^{7} - 1134 \nu^{6} + 59816 \nu^{5} - 66992 \nu^{4} + \cdots + 111360 ) / 30592$$ (45*v^15 + 907*v^14 - 331*v^13 - 1200*v^12 + 7083*v^11 - 6652*v^10 - 4286*v^9 + 27804*v^8 - 33427*v^7 - 1134*v^6 + 59816*v^5 - 66992*v^4 - 27040*v^3 + 159776*v^2 - 218688*v + 111360) / 30592 $$\beta_{3}$$ $$=$$ $$( - 25 \nu^{15} - 79 \nu^{14} + 237 \nu^{13} - 130 \nu^{12} - 589 \nu^{11} + 2952 \nu^{10} - 5320 \nu^{9} + 5904 \nu^{8} - 1293 \nu^{7} + 630 \nu^{6} - 2586 \nu^{5} + 22984 \nu^{4} + \cdots + 19712 ) / 15296$$ (-25*v^15 - 79*v^14 + 237*v^13 - 130*v^12 - 589*v^11 + 2952*v^10 - 5320*v^9 + 5904*v^8 - 1293*v^7 + 630*v^6 - 2586*v^5 + 22984*v^4 - 35752*v^3 + 51024*v^2 - 34016*v + 19712) / 15296 $$\beta_{4}$$ $$=$$ $$( - 79 \nu^{15} + 649 \nu^{14} - 2425 \nu^{13} + 6090 \nu^{12} - 8993 \nu^{11} + 6250 \nu^{10} + 6802 \nu^{9} - 19048 \nu^{8} + 6889 \nu^{7} + 38988 \nu^{6} - 80388 \nu^{5} + \cdots - 113920 ) / 30592$$ (-79*v^15 + 649*v^14 - 2425*v^13 + 6090*v^12 - 8993*v^11 + 6250*v^10 + 6802*v^9 - 19048*v^8 + 6889*v^7 + 38988*v^6 - 80388*v^5 + 59016*v^4 + 44496*v^3 - 179712*v^2 + 211584*v - 113920) / 30592 $$\beta_{5}$$ $$=$$ $$( - 103 \nu^{15} + 468 \nu^{14} - 1404 \nu^{13} + 2189 \nu^{12} - 1681 \nu^{11} - 1977 \nu^{10} + 5710 \nu^{9} - 3954 \nu^{8} - 7851 \nu^{7} + 18991 \nu^{6} - 15702 \nu^{5} + \cdots - 5056 ) / 15296$$ (-103*v^15 + 468*v^14 - 1404*v^13 + 2189*v^12 - 1681*v^11 - 1977*v^10 + 5710*v^9 - 3954*v^8 - 7851*v^7 + 18991*v^6 - 15702*v^5 - 12072*v^4 + 48720*v^3 - 54096*v^2 + 36064*v - 5056) / 15296 $$\beta_{6}$$ $$=$$ $$( - 257 \nu^{15} + 2075 \nu^{14} - 2879 \nu^{13} + 4782 \nu^{12} + 5245 \nu^{11} - 4318 \nu^{10} + 13282 \nu^{9} + 34384 \nu^{8} - 37345 \nu^{7} + 80184 \nu^{6} + 41464 \nu^{5} + \cdots + 261376 ) / 30592$$ (-257*v^15 + 2075*v^14 - 2879*v^13 + 4782*v^12 + 5245*v^11 - 4318*v^10 + 13282*v^9 + 34384*v^8 - 37345*v^7 + 80184*v^6 + 41464*v^5 - 38976*v^4 + 117888*v^3 + 125760*v^2 - 129728*v + 261376) / 30592 $$\beta_{7}$$ $$=$$ $$( - 34 \nu^{15} - 117 \nu^{14} + 351 \nu^{13} - 1085 \nu^{12} + 958 \nu^{11} - 1119 \nu^{10} - 352 \nu^{9} - 2238 \nu^{8} + 3576 \nu^{7} - 10663 \nu^{6} + 7152 \nu^{5} - 9888 \nu^{4} + \cdots - 17856 ) / 3824$$ (-34*v^15 - 117*v^14 + 351*v^13 - 1085*v^12 + 958*v^11 - 1119*v^10 - 352*v^9 - 2238*v^8 + 3576*v^7 - 10663*v^6 + 7152*v^5 - 9888*v^4 + 726*v^3 - 12288*v^2 + 8192*v - 17856) / 3824 $$\beta_{8}$$ $$=$$ $$( 285 \nu^{15} - 1585 \nu^{14} + 5233 \nu^{13} - 10468 \nu^{12} + 12355 \nu^{11} - 2296 \nu^{10} - 18222 \nu^{9} + 26956 \nu^{8} + 8813 \nu^{7} - 76970 \nu^{6} + 111792 \nu^{5} + \cdots + 93440 ) / 30592$$ (285*v^15 - 1585*v^14 + 5233*v^13 - 10468*v^12 + 12355*v^11 - 2296*v^10 - 18222*v^9 + 26956*v^8 + 8813*v^7 - 76970*v^6 + 111792*v^5 - 34872*v^4 - 141936*v^3 + 287904*v^2 - 222528*v + 93440) / 30592 $$\beta_{9}$$ $$=$$ $$( 153 \nu^{15} - 310 \nu^{14} + 452 \nu^{13} + 461 \nu^{12} - 1443 \nu^{11} - 103 \nu^{10} + 3496 \nu^{9} - 11678 \nu^{8} + 7569 \nu^{7} - 3043 \nu^{6} - 14020 \nu^{5} - 2348 \nu^{4} + \cdots - 49664 ) / 15296$$ (153*v^15 - 310*v^14 + 452*v^13 + 461*v^12 - 1443*v^11 - 103*v^10 + 3496*v^9 - 11678*v^8 + 7569*v^7 - 3043*v^6 - 14020*v^5 - 2348*v^4 + 13224*v^3 - 63248*v^2 + 47264*v - 49664) / 15296 $$\beta_{10}$$ $$=$$ $$( 286 \nu^{15} - 731 \nu^{14} + 1715 \nu^{13} - 1333 \nu^{12} + 658 \nu^{11} + 1821 \nu^{10} + 824 \nu^{9} - 6874 \nu^{8} + 15442 \nu^{7} - 16337 \nu^{6} + 5550 \nu^{5} + 10632 \nu^{4} + \cdots - 31552 ) / 15296$$ (286*v^15 - 731*v^14 + 1715*v^13 - 1333*v^12 + 658*v^11 + 1821*v^10 + 824*v^9 - 6874*v^8 + 15442*v^7 - 16337*v^6 + 5550*v^5 + 10632*v^4 - 12096*v^3 - 16080*v^2 + 26016*v - 31552) / 15296 $$\beta_{11}$$ $$=$$ $$( - 201 \nu^{15} + 426 \nu^{14} - 561 \nu^{13} + 580 \nu^{12} - 902 \nu^{11} + 4203 \nu^{10} - 8787 \nu^{9} + 16054 \nu^{8} - 12671 \nu^{7} + 10371 \nu^{6} - 4549 \nu^{5} + 31774 \nu^{4} + \cdots + 53248 ) / 7648$$ (-201*v^15 + 426*v^14 - 561*v^13 + 580*v^12 - 902*v^11 + 4203*v^10 - 8787*v^9 + 16054*v^8 - 12671*v^7 + 10371*v^6 - 4549*v^5 + 31774*v^4 - 59268*v^3 + 102248*v^2 - 80912*v + 53248) / 7648 $$\beta_{12}$$ $$=$$ $$( 429 \nu^{15} - 1216 \nu^{14} + 2214 \nu^{13} - 2597 \nu^{12} + 509 \nu^{11} - 973 \nu^{10} + 4104 \nu^{9} - 18198 \nu^{8} + 24597 \nu^{7} - 26537 \nu^{6} + 3784 \nu^{5} - 7952 \nu^{4} + \cdots - 39680 ) / 15296$$ (429*v^15 - 1216*v^14 + 2214*v^13 - 2597*v^12 + 509*v^11 - 973*v^10 + 4104*v^9 - 18198*v^8 + 24597*v^7 - 26537*v^6 + 3784*v^5 - 7952*v^4 + 18184*v^3 - 64272*v^2 + 58144*v - 39680) / 15296 $$\beta_{13}$$ $$=$$ $$( - \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 14 \nu^{11} - 21 \nu^{10} + 15 \nu^{9} + 40 \nu^{8} - 51 \nu^{7} + 41 \nu^{6} + 109 \nu^{5} - 148 \nu^{4} + 184 \nu^{3} + 16 \nu^{2} - 64 \nu + 128 ) / 32$$ (-v^15 + 4*v^14 - 5*v^13 + 2*v^12 + 14*v^11 - 21*v^10 + 15*v^9 + 40*v^8 - 51*v^7 + 41*v^6 + 109*v^5 - 148*v^4 + 184*v^3 + 16*v^2 - 64*v + 128) / 32 $$\beta_{14}$$ $$=$$ $$( 909 \nu^{15} - 2615 \nu^{14} + 4499 \nu^{13} - 1774 \nu^{12} - 4721 \nu^{11} + 7022 \nu^{10} + 11126 \nu^{9} - 50008 \nu^{8} + 70837 \nu^{7} - 40784 \nu^{6} - 43312 \nu^{5} + \cdots - 197888 ) / 15296$$ (909*v^15 - 2615*v^14 + 4499*v^13 - 1774*v^12 - 4721*v^11 + 7022*v^10 + 11126*v^9 - 50008*v^8 + 70837*v^7 - 40784*v^6 - 43312*v^5 + 58200*v^4 + 29304*v^3 - 224832*v^2 + 272256*v - 197888) / 15296 $$\beta_{15}$$ $$=$$ $$( 1991 \nu^{15} - 5429 \nu^{14} + 11985 \nu^{13} - 9914 \nu^{12} + 8037 \nu^{11} + 442 \nu^{10} + 34306 \nu^{9} - 70816 \nu^{8} + 138423 \nu^{7} - 95392 \nu^{6} + 79432 \nu^{5} + \cdots - 161408 ) / 30592$$ (1991*v^15 - 5429*v^14 + 11985*v^13 - 9914*v^12 + 8037*v^11 + 442*v^10 + 34306*v^9 - 70816*v^8 + 138423*v^7 - 95392*v^6 + 79432*v^5 - 7392*v^4 + 162000*v^3 - 233280*v^2 + 354368*v - 161408) / 30592
 $$\nu$$ $$=$$ $$( \beta_{8} + \beta_{5} + \beta_{4} + 1 ) / 2$$ (b8 + b5 + b4 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{14} - \beta_{12} - 2\beta_{10} + \beta_{8} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b14 - b12 - 2*b10 + b8 + b5 - b3 + b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{14} - 2\beta_{9} + \beta_{7} + 4\beta_{5} + 4\beta_{3} + 2\beta_{2} ) / 2$$ (b14 - 2*b9 + b7 + 4*b5 + 4*b3 + 2*b2) / 2 $$\nu^{4}$$ $$=$$ $$( - \beta_{15} + 2 \beta_{13} + \beta_{12} + 3 \beta_{10} + \beta_{9} - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta _1 - 2 ) / 2$$ (-b15 + 2*b13 + b12 + 3*b10 + b9 - 2*b7 - b6 - 3*b4 - 2*b3 - 3*b2 + 2*b1 - 2) / 2 $$\nu^{5}$$ $$=$$ $$( 3 \beta_{15} - 5 \beta_{14} + \beta_{12} + 2 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - \beta_{2} + 1 ) / 2$$ (3*b15 - 5*b14 + b12 + 2*b10 + b9 - 3*b8 - b7 - 3*b6 - 4*b5 + 2*b4 - b2 + 1) / 2 $$\nu^{6}$$ $$=$$ $$( - 4 \beta_{14} + 4 \beta_{12} - \beta_{11} + 6 \beta_{9} - \beta_{7} + 2 \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - \beta_{3} - 5 \beta_{2} - 9 ) / 2$$ (-4*b14 + 4*b12 - b11 + 6*b9 - b7 + 2*b6 - 9*b5 + 3*b4 - b3 - 5*b2 - 9) / 2 $$\nu^{7}$$ $$=$$ $$( \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} - 12 \beta_{10} - 3 \beta_{8} + \beta_{6} - 11 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta _1 - 1 ) / 2$$ (b15 + 2*b14 - b13 - 2*b12 - b11 - 12*b10 - 3*b8 + b6 - 11*b5 - b4 + 2*b3 + 2*b2 - 3*b1 - 1) / 2 $$\nu^{8}$$ $$=$$ $$( \beta_{14} - \beta_{13} + 4 \beta_{12} + 4 \beta_{11} + 4 \beta_{10} - 8 \beta_{9} - \beta_{8} + \beta_{7} + 8 \beta_{5} + 9 \beta_{3} - \beta_{2} - 13 \beta _1 - 1 ) / 2$$ (b14 - b13 + 4*b12 + 4*b11 + 4*b10 - 8*b9 - b8 + b7 + 8*b5 + 9*b3 - b2 - 13*b1 - 1) / 2 $$\nu^{9}$$ $$=$$ $$( - \beta_{12} + 9 \beta_{11} + 8 \beta_{10} - \beta_{9} - 4 \beta_{7} - 2 \beta_{6} - 8 \beta_{4} - 39 \beta_{3} - 8 \beta_{2} - 4 ) / 2$$ (-b12 + 9*b11 + 8*b10 - b9 - 4*b7 - 2*b6 - 8*b4 - 39*b3 - 8*b2 - 4) / 2 $$\nu^{10}$$ $$=$$ $$( 6 \beta_{15} - 2 \beta_{14} - 15 \beta_{13} - 10 \beta_{12} + 8 \beta_{10} - 25 \beta_{9} - 6 \beta_{8} + 18 \beta_{7} + 6 \beta_{6} + 12 \beta_{5} + 19 \beta_{4} + 25 \beta_{2} + 10 ) / 2$$ (6*b15 - 2*b14 - 15*b13 - 10*b12 + 8*b10 - 25*b9 - 6*b8 + 18*b7 + 6*b6 + 12*b5 + 19*b4 + 25*b2 + 10) / 2 $$\nu^{11}$$ $$=$$ $$( - 27 \beta_{15} + 31 \beta_{14} - 31 \beta_{12} - 30 \beta_{11} - 4 \beta_{9} + 19 \beta_{8} + 3 \beta_{7} + 27 \beta_{6} - 9 \beta_{5} - 18 \beta_{4} + 3 \beta_{3} + \beta_{2} + 27 \beta _1 - 9 ) / 2$$ (-27*b15 + 31*b14 - 31*b12 - 30*b11 - 4*b9 + 19*b8 + 3*b7 + 27*b6 - 9*b5 - 18*b4 + 3*b3 + b2 + 27*b1 - 9) / 2 $$\nu^{12}$$ $$=$$ $$( 28 \beta_{14} - 62 \beta_{12} - 28 \beta_{11} - 31 \beta_{10} - 6 \beta_{6} + 17 \beta_{5} - 28 \beta_{4} + 80 \beta_{3} + 28 \beta_{2} + 80 ) / 2$$ (28*b14 - 62*b12 - 28*b11 - 31*b10 - 6*b6 + 17*b5 - 28*b4 + 80*b3 + 28*b2 + 80) / 2 $$\nu^{13}$$ $$=$$ $$( - 6 \beta_{14} + 26 \beta_{13} + 41 \beta_{12} + 41 \beta_{11} + 41 \beta_{10} + 88 \beta_{9} + 26 \beta_{8} + 6 \beta_{7} + 88 \beta_{5} + 62 \beta_{3} - 26 \beta_{2} - 21 \beta _1 + 26 ) / 2$$ (-6*b14 + 26*b13 + 41*b12 + 41*b11 + 41*b10 + 88*b9 + 26*b8 + 6*b7 + 88*b5 + 62*b3 - 26*b2 - 21*b1 + 26) / 2 $$\nu^{14}$$ $$=$$ $$( 20 \beta_{15} + 41 \beta_{13} + 83 \beta_{12} + 94 \beta_{11} - 52 \beta_{10} + 83 \beta_{9} + 9 \beta_{7} - 20 \beta_{6} + 52 \beta_{4} - 135 \beta_{3} + 52 \beta_{2} + 41 \beta _1 + 9 ) / 2$$ (20*b15 + 41*b13 + 83*b12 + 94*b11 - 52*b10 + 83*b9 + 9*b7 - 20*b6 + 52*b4 - 135*b3 + 52*b2 + 41*b1 + 9) / 2 $$\nu^{15}$$ $$=$$ $$( - 9 \beta_{14} + 135 \beta_{12} - 9 \beta_{10} - 135 \beta_{9} - 20 \beta_{7} - 18 \beta_{6} - 20 \beta_{5} + 153 \beta_{4} + 135 \beta_{2} - 91 ) / 2$$ (-9*b14 + 135*b12 - 9*b10 - 135*b9 - 20*b7 - 18*b6 - 20*b5 + 153*b4 + 135*b2 - 91) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 0.589191 − 1.28563i 0.818796 + 1.15307i 0.0153379 + 1.41413i −1.23234 − 0.693782i −1.23816 − 0.683337i 1.06903 − 0.925833i 1.21087 − 0.730613i 0.267278 + 1.38873i 0.589191 + 1.28563i 0.818796 − 1.15307i 0.0153379 − 1.41413i −1.23234 + 0.693782i −1.23816 + 0.683337i 1.06903 + 0.925833i 1.21087 + 0.730613i 0.267278 − 1.38873i
0 −2.94730 + 0.957636i 0 −0.540641 0.392798i 0 0.818875 2.52024i 0 5.34247 3.88153i 0
63.2 0 −1.30002 + 0.422403i 0 1.84966 + 1.34385i 0 −0.199199 + 0.613071i 0 −0.915415 + 0.665088i 0
63.3 0 1.30002 0.422403i 0 1.84966 + 1.34385i 0 0.199199 0.613071i 0 −0.915415 + 0.665088i 0
63.4 0 2.94730 0.957636i 0 −0.540641 0.392798i 0 −0.818875 + 2.52024i 0 5.34247 3.88153i 0
79.1 0 −0.930415 1.28061i 0 1.10004 + 3.38558i 0 2.06453 + 1.49997i 0 0.152771 0.470181i 0
79.2 0 −0.0876593 0.120653i 0 −0.909057 2.79779i 0 −3.71180 2.69678i 0 0.920178 2.83202i 0
79.3 0 0.0876593 + 0.120653i 0 −0.909057 2.79779i 0 3.71180 + 2.69678i 0 0.920178 2.83202i 0
79.4 0 0.930415 + 1.28061i 0 1.10004 + 3.38558i 0 −2.06453 1.49997i 0 0.152771 0.470181i 0
95.1 0 −2.94730 0.957636i 0 −0.540641 + 0.392798i 0 0.818875 + 2.52024i 0 5.34247 + 3.88153i 0
95.2 0 −1.30002 0.422403i 0 1.84966 1.34385i 0 −0.199199 0.613071i 0 −0.915415 0.665088i 0
95.3 0 1.30002 + 0.422403i 0 1.84966 1.34385i 0 0.199199 + 0.613071i 0 −0.915415 0.665088i 0
95.4 0 2.94730 + 0.957636i 0 −0.540641 + 0.392798i 0 −0.818875 2.52024i 0 5.34247 + 3.88153i 0
127.1 0 −0.930415 + 1.28061i 0 1.10004 3.38558i 0 2.06453 1.49997i 0 0.152771 + 0.470181i 0
127.2 0 −0.0876593 + 0.120653i 0 −0.909057 + 2.79779i 0 −3.71180 + 2.69678i 0 0.920178 + 2.83202i 0
127.3 0 0.0876593 0.120653i 0 −0.909057 + 2.79779i 0 3.71180 2.69678i 0 0.920178 + 2.83202i 0
127.4 0 0.930415 1.28061i 0 1.10004 3.38558i 0 −2.06453 + 1.49997i 0 0.152771 + 0.470181i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.q.b 16
4.b odd 2 1 inner 176.2.q.b 16
8.b even 2 1 704.2.u.b 16
8.d odd 2 1 704.2.u.b 16
11.c even 5 1 1936.2.e.f 16
11.d odd 10 1 inner 176.2.q.b 16
11.d odd 10 1 1936.2.e.f 16
44.g even 10 1 inner 176.2.q.b 16
44.g even 10 1 1936.2.e.f 16
44.h odd 10 1 1936.2.e.f 16
88.k even 10 1 704.2.u.b 16
88.p odd 10 1 704.2.u.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.q.b 16 1.a even 1 1 trivial
176.2.q.b 16 4.b odd 2 1 inner
176.2.q.b 16 11.d odd 10 1 inner
176.2.q.b 16 44.g even 10 1 inner
704.2.u.b 16 8.b even 2 1
704.2.u.b 16 8.d odd 2 1
704.2.u.b 16 88.k even 10 1
704.2.u.b 16 88.p odd 10 1
1936.2.e.f 16 11.c even 5 1
1936.2.e.f 16 11.d odd 10 1
1936.2.e.f 16 44.g even 10 1
1936.2.e.f 16 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - 17T_{3}^{14} + 120T_{3}^{12} - 227T_{3}^{10} + 699T_{3}^{8} - 1583T_{3}^{6} + 2000T_{3}^{4} + 27T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} - 17 T^{14} + 120 T^{12} - 227 T^{10} + \cdots + 1$$
$5$ $$(T^{8} - 3 T^{7} + 20 T^{6} - 38 T^{5} + \cdots + 256)^{2}$$
$7$ $$T^{16} - 5 T^{14} + 390 T^{12} + \cdots + 160000$$
$11$ $$T^{16} + 13 T^{14} + \cdots + 214358881$$
$13$ $$(T^{8} - 5 T^{7} + 10 T^{6} - 150 T^{5} + \cdots + 400)^{2}$$
$17$ $$(T^{8} - 5 T^{7} - 10 T^{6} + 25 T^{5} + \cdots + 9025)^{2}$$
$19$ $$T^{16} + 10 T^{14} + 2665 T^{12} + \cdots + 9150625$$
$23$ $$(T^{8} + 44 T^{6} + 512 T^{4} + 1216 T^{2} + \cdots + 256)^{2}$$
$29$ $$(T^{8} + 15 T^{7} + 30 T^{6} + \cdots + 336400)^{2}$$
$31$ $$T^{16} - 57 T^{14} + \cdots + 274760478976$$
$37$ $$(T^{8} + 3 T^{7} + 26 T^{6} + 154 T^{5} + \cdots + 16)^{2}$$
$41$ $$(T^{8} + 35 T^{7} + 530 T^{6} + \cdots + 42025)^{2}$$
$43$ $$(T^{8} - 235 T^{6} + 17805 T^{4} + \cdots + 5382400)^{2}$$
$47$ $$T^{16} - 73 T^{14} + \cdots + 3072924856576$$
$53$ $$(T^{8} - 3 T^{7} + 108 T^{6} + \cdots + 156816)^{2}$$
$59$ $$T^{16} - 78 T^{14} + \cdots + 1998607065841$$
$61$ $$(T^{8} - 15 T^{7} + 2650 T^{5} + \cdots + 70224400)^{2}$$
$67$ $$(T^{8} + 301 T^{6} + 21257 T^{4} + \cdots + 1547536)^{2}$$
$71$ $$T^{16} + 99 T^{14} + \cdots + 9508194263296$$
$73$ $$(T^{8} - 5 T^{7} - 10 T^{6} - 725 T^{5} + \cdots + 156025)^{2}$$
$79$ $$T^{16} + 315 T^{14} + \cdots + 6887475360000$$
$83$ $$T^{16} - 230 T^{14} + \cdots + 88223850625$$
$89$ $$(T^{4} - 13 T^{3} - T^{2} + 328 T - 164)^{4}$$
$97$ $$(T^{8} - 38 T^{7} + 781 T^{6} + \cdots + 11881)^{2}$$