# Properties

 Label 475.2.e.f Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(26,475)]

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([0, 2]))

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

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

chi := DirichletCharacter("475.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.e (of order $$3$$, 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: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} - 552 x^{3} + 1107 x^{2} + 33 x + 1$$ x^12 - 3*x^11 + 17*x^10 - 18*x^9 + 109*x^8 - 93*x^7 + 484*x^6 - 147*x^5 + 1009*x^4 - 552*x^3 + 1107*x^2 + 33*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{8} + \beta_{7}) q^{2} + (\beta_{9} + \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{10} - \beta_{8}) q^{4} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{6} + (\beta_{7} + \beta_{4} + 1) q^{7} + (\beta_{6} + 1) q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_1 - 1) q^{9}+O(q^{10})$$ q + (b8 + b7) * q^2 + (b9 + b2 - b1 + 1) * q^3 + (b10 - b8) * q^4 + (-b8 + b3 + b1) * q^6 + (b7 + b4 + 1) * q^7 + (b6 + 1) * q^8 + (-b11 - b9 - b8 + b1 - 1) * q^9 $$q + (\beta_{8} + \beta_{7}) q^{2} + (\beta_{9} + \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{10} - \beta_{8}) q^{4} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{6} + (\beta_{7} + \beta_{4} + 1) q^{7} + (\beta_{6} + 1) q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_1 - 1) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2}) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{12} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{3} + \beta_1 - 1) q^{13} + ( - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3}) q^{14} + (2 \beta_{10} - \beta_{9} + 2 \beta_{6}) q^{16} + ( - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{2}) q^{18} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - 1) q^{19} + ( - 2 \beta_{10} + \beta_{9} - 3 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{21} + (\beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{22} + (3 \beta_{10} + \beta_{9} + \beta_{3} + 1) q^{23} + (\beta_{11} + 2 \beta_{9} - \beta_{7} - \beta_{4} + \beta_{2} - \beta_1) q^{24} + ( - \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{2} + 2) q^{26} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + 3) q^{27} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_1 - 1) q^{28} + ( - 2 \beta_{11} - \beta_{10} - \beta_{8} + \beta_{3} + \beta_1) q^{29} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{31} + ( - 2 \beta_{10} + 3 \beta_{8}) q^{32} + ( - \beta_{10} - 4 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 2 \beta_{3}) q^{33}+ \cdots + (\beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 6 \beta_{8} + 2 \beta_{3} + 2 \beta_1 + 4) q^{99}+O(q^{100})$$ q + (b8 + b7) * q^2 + (b9 + b2 - b1 + 1) * q^3 + (b10 - b8) * q^4 + (-b8 + b3 + b1) * q^6 + (b7 + b4 + 1) * q^7 + (b6 + 1) * q^8 + (-b11 - b9 - b8 + b1 - 1) * q^9 + (-b7 + b6 + b5 + b2) * q^11 + (-b6 + b5 + b4 - b2 + 1) * q^12 + (-b10 - b9 - b8 + b3 + b1 - 1) * q^13 + (-b9 + b6 - b5 - b3) * q^14 + (2*b10 - b9 + 2*b6) * q^16 + (-b9 - b6 + b5 + b3 - b2 + b1 - 1) * q^17 + (-2*b7 - b6 + 2*b5 - b2) * q^18 + (b11 - b10 + b9 + b8 - b7 - b5 - b4 - 1) * q^19 + (-2*b10 + b9 - 3*b6 + b5 + b3 + b2 - b1 + 1) * q^21 + (b11 + b9 + 2*b8 + b7 - b6 + b5 - b4 + b3 + b2 - b1) * q^22 + (3*b10 + b9 + b3 + 1) * q^23 + (b11 + 2*b9 - b7 - b4 + b2 - b1) * q^24 + (-b6 + b5 + b4 - 3*b2 + 2) * q^26 + (-b7 + 2*b6 + b4 + 3) * q^27 + (b11 - b10 - b9 + b8 + 2*b1 - 1) * q^28 + (-2*b11 - b10 - b8 + b3 + b1) * q^29 + (2*b7 + b6 - b5 + b4 + b2 + 1) * q^31 + (-2*b10 + 3*b8) * q^32 + (-b10 - 4*b9 + 3*b8 + 3*b7 + b6 - 2*b5 - 2*b3) * q^33 + (b11 + b10 + b9 + b8 - b3 - 3*b1 + 1) * q^34 + (3*b9 + 2*b8 + 2*b7 + b6 - b5 - b3 + 3*b2 - 3*b1 + 3) * q^36 + (2*b6 + 2*b2) * q^37 + (-b11 - b10 + b9 - 2*b8 + 2*b7 - 2*b6 + b4 + b3 - 2*b2 + 2*b1 - 1) * q^38 + (-6*b7 - b6 + 2*b5 - b4 - 2*b2 - 2) * q^39 + (b11 - 2*b10 + 3*b8 + 2*b7 - 2*b6 - b4 - b2 + b1 - 2) * q^41 + (b11 + b10 + 3*b9 - 3*b8 + b3 - b1 + 3) * q^42 + (-b11 + 3*b9 + b7 + b4 + b2 - b1 + 2) * q^43 + (b11 - b10 - 4*b9 - b8 + b1 - 4) * q^44 + (-b7 - b6 + b4 - 2*b2 - 3) * q^46 + (-2*b11 - 2*b10 + 2*b9 - b8 + b3 + 2) * q^47 + (2*b11 + 3*b9 - b1 + 3) * q^48 + (3*b7 - 2*b6 - b5 + b4 + 2) * q^49 + (-b10 + 2*b9 + 4*b8 - b3 - b1 + 2) * q^51 + (b11 - 2*b10 + 3*b8 + 2*b7 - 2*b5 - b4 - 2*b3 + 3*b2 - 3*b1 + 2) * q^52 + (-b11 + 2*b10 - b9 + b8 - b3 - 4*b1 - 1) * q^53 + (-2*b10 + b9 + 6*b8 + 6*b7 - b6 - b5 - b3) * q^54 + (b7 - b5 - 2*b2) * q^56 + (3*b10 - b9 - 3*b8 - 3*b7 + b6 - b4 + b2 + 2*b1 - 3) * q^57 + (b7 - 3*b6 + 3*b5 + b4 - 3*b2) * q^58 + (-2*b11 + 2*b10 + 4*b9 - 5*b8 - 3*b7 + 3*b6 - b5 + 2*b4 - b3 + 2*b2 - 2*b1 + 4) * q^59 + (b11 + b10 - 5*b9 - 2*b3 + b1 - 5) * q^61 + (-b11 - 4*b9 - 2*b8 - b7 + b4 - 3*b2 + 3*b1 - 2) * q^62 + (-b11 - b10 - 8*b9 + 2*b8 - b3 + b1 - 8) * q^63 + (5*b7 + b6 - 2) * q^64 + (-2*b11 + b10 - 7*b9 + 4*b1 - 7) * q^66 + (5*b10 + b9 - 3*b8 - 2*b1 + 1) * q^67 + (2*b6 - 2*b5 - b4 + 3*b2 + 1) * q^68 + (-b7 + b5 + 2*b4 + b2 + 3) * q^69 + (b11 + b10 + b9 - b8 - 2*b7 - b6 + 2*b5 - b4 + 2*b3 - 4*b2 + 4*b1 - 5) * q^71 + (-b11 - b10 - 3*b9 - b8 - b3 + 3*b1 - 3) * q^72 + (2*b10 + b9 - 4*b8 - 4*b7 + 3*b6 - b5 - b3 - b2 + b1 - 1) * q^73 + (-2*b9 - 2*b6 + 2*b5 + 2*b3 - 2*b2 + 2*b1 - 2) * q^74 + (b10 - b9 - 2*b8 - 3*b7 - b4 - b3 - 2*b2 - 2*b1) * q^76 + (-3*b7 - 6*b6 + b5 - 2*b4 - b2 - 4) * q^77 + (2*b11 - 3*b10 + 12*b9 + 7*b8 + 5*b7 - 2*b6 - b5 - 2*b4 - b3 + 6*b2 - 6*b1 + 4) * q^78 + (b11 + 3*b10 - 2*b9 + 5*b8 + 4*b7 + 3*b6 - b4 - b2 + b1 - 2) * q^79 + (-b11 - 2*b10 + 2*b9 - b8 - b6 - b5 + b4 - b3 + 2*b2 - 2*b1 + 3) * q^81 + (3*b10 - 3*b9 - 5*b8 - b1 - 3) * q^82 + (-2*b7 + 4*b6 - 3*b5 - b2 - 5) * q^83 + (-2*b6 + b4 + b2 + 7) * q^84 + (-b9 - 3*b8 + b1 - 1) * q^86 + (-6*b7 + 5*b6 - b4 - 3*b2 - 2) * q^87 + (-2*b7 - b6 + 2*b5 - 2*b4 + b2 + 1) * q^88 + (-b11 + 3*b10 - 4*b9 + 4*b8 + b3 + 2*b1 - 4) * q^89 + (-b11 + 2*b10 + 5*b9 - 4*b8 + 2*b3 - 3*b1 + 5) * q^91 + (4*b10 + 6*b9 - b8 - b7 + 5*b6 - b5 - b3 + 2*b2 - 2*b1 + 2) * q^92 + (-b10 - b9 - 5*b8 - 5*b7 - 4*b6 + 3*b5 + 3*b3 - b2 + b1 - 1) * q^93 + (4*b7 - 2*b6 + 2*b5 + b4 - 2*b2 + 2) * q^94 + (b7 + 3*b6 - 3*b5 - 2*b4 + 3*b2 - 1) * q^96 + (-b11 - 6*b10 - b9 + b8 + 2*b7 - 4*b6 - 2*b5 + b4 - 2*b3 - 3*b2 + 3*b1 - 2) * q^97 + (-b11 + b10 - 3*b9 - 4*b8 - 3*b7 + 2*b6 - b5 + b4 - b3 - 2*b2 + 2*b1 - 1) * q^98 + (b11 - 4*b10 + 4*b9 - 6*b8 + 2*b3 + 2*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{6} + 4 q^{7} + 12 q^{8} - 7 q^{9}+O(q^{10})$$ 12 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 + q^6 + 4 * q^7 + 12 * q^8 - 7 * q^9 $$12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{6} + 4 q^{7} + 12 q^{8} - 7 q^{9} - 2 q^{11} + 14 q^{12} - 5 q^{13} + 6 q^{14} + 6 q^{16} + 3 q^{17} + 14 q^{18} - 6 q^{19} - 3 q^{21} - 9 q^{22} + 6 q^{23} - 11 q^{24} + 38 q^{26} + 36 q^{27} + 4 q^{28} - 3 q^{29} - 6 q^{31} + 6 q^{32} + 18 q^{33} + q^{34} - 13 q^{36} - 12 q^{37} - 18 q^{38} + 16 q^{39} - 11 q^{41} + 11 q^{42} - 13 q^{43} - 21 q^{44} - 24 q^{46} + 6 q^{47} + 19 q^{48} + 8 q^{49} + 17 q^{51} + q^{52} - 18 q^{53} - 18 q^{54} + 8 q^{56} - 20 q^{57} + 10 q^{58} - 4 q^{59} - 25 q^{61} + 21 q^{62} - 43 q^{63} - 44 q^{64} - 34 q^{66} - 6 q^{67} - 2 q^{68} + 26 q^{69} - 18 q^{71} - 13 q^{72} - q^{73} + 6 q^{74} + 24 q^{76} - 22 q^{77} - 72 q^{78} - 3 q^{79} - 2 q^{81} - 31 q^{82} - 46 q^{83} + 74 q^{84} - 9 q^{86} + 22 q^{87} + 22 q^{88} - 12 q^{89} + 11 q^{91} - 28 q^{92} + 13 q^{93} + 16 q^{94} - 26 q^{96} - 3 q^{97} + 22 q^{98} + 20 q^{99}+O(q^{100})$$ 12 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 + q^6 + 4 * q^7 + 12 * q^8 - 7 * q^9 - 2 * q^11 + 14 * q^12 - 5 * q^13 + 6 * q^14 + 6 * q^16 + 3 * q^17 + 14 * q^18 - 6 * q^19 - 3 * q^21 - 9 * q^22 + 6 * q^23 - 11 * q^24 + 38 * q^26 + 36 * q^27 + 4 * q^28 - 3 * q^29 - 6 * q^31 + 6 * q^32 + 18 * q^33 + q^34 - 13 * q^36 - 12 * q^37 - 18 * q^38 + 16 * q^39 - 11 * q^41 + 11 * q^42 - 13 * q^43 - 21 * q^44 - 24 * q^46 + 6 * q^47 + 19 * q^48 + 8 * q^49 + 17 * q^51 + q^52 - 18 * q^53 - 18 * q^54 + 8 * q^56 - 20 * q^57 + 10 * q^58 - 4 * q^59 - 25 * q^61 + 21 * q^62 - 43 * q^63 - 44 * q^64 - 34 * q^66 - 6 * q^67 - 2 * q^68 + 26 * q^69 - 18 * q^71 - 13 * q^72 - q^73 + 6 * q^74 + 24 * q^76 - 22 * q^77 - 72 * q^78 - 3 * q^79 - 2 * q^81 - 31 * q^82 - 46 * q^83 + 74 * q^84 - 9 * q^86 + 22 * q^87 + 22 * q^88 - 12 * q^89 + 11 * q^91 - 28 * q^92 + 13 * q^93 + 16 * q^94 - 26 * q^96 - 3 * q^97 + 22 * q^98 + 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} - 552 x^{3} + 1107 x^{2} + 33 x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 3959208 \nu^{11} - 46922925 \nu^{10} + 38785859 \nu^{9} - 621660414 \nu^{8} - 801763109 \nu^{7} - 3678052623 \nu^{6} + \cdots - 68499350729 ) / 66494854340$$ (-3959208*v^11 - 46922925*v^10 + 38785859*v^9 - 621660414*v^8 - 801763109*v^7 - 3678052623*v^6 - 4177401796*v^5 - 15342668397*v^4 - 38817354653*v^3 - 24355803798*v^2 - 726241735*v - 68499350729) / 66494854340 $$\beta_{3}$$ $$=$$ $$( 690858 \nu^{11} - 11701549 \nu^{10} + 76199247 \nu^{9} - 313859676 \nu^{8} + 833713615 \nu^{7} - 1701731499 \nu^{6} + 2957264154 \nu^{5} + \cdots - 6178545 ) / 1955731010$$ (690858*v^11 - 11701549*v^10 + 76199247*v^9 - 313859676*v^8 + 833713615*v^7 - 1701731499*v^6 + 2957264154*v^5 - 5508756455*v^4 + 7520604435*v^3 - 6717081048*v^2 - 1547540653*v - 6178545) / 1955731010 $$\beta_{4}$$ $$=$$ $$( - 54841341 \nu^{11} + 153015320 \nu^{10} - 731712017 \nu^{9} + 251450977 \nu^{8} - 3244495858 \nu^{7} + 1416907499 \nu^{6} + \cdots - 263970961763 ) / 66494854340$$ (-54841341*v^11 + 153015320*v^10 - 731712017*v^9 + 251450977*v^8 - 3244495858*v^7 + 1416907499*v^6 - 11747270177*v^5 - 19479845384*v^4 + 12276068039*v^3 - 38482449021*v^2 - 1147600790*v - 263970961763) / 66494854340 $$\beta_{5}$$ $$=$$ $$( 134211789 \nu^{11} - 998274384 \nu^{10} + 3072233475 \nu^{9} - 7500316929 \nu^{8} + 1833052570 \nu^{7} - 27189880401 \nu^{6} + \cdots - 148724948799 ) / 66494854340$$ (134211789*v^11 - 998274384*v^10 + 3072233475*v^9 - 7500316929*v^8 + 1833052570*v^7 - 27189880401*v^6 - 7077581935*v^5 - 102294574896*v^4 - 204345695921*v^3 - 138016601091*v^2 - 4114960306*v - 148724948799) / 66494854340 $$\beta_{6}$$ $$=$$ $$( - 5051595 \nu^{11} + 9282581 \nu^{10} - 64765518 \nu^{9} - 29949491 \nu^{8} - 345971017 \nu^{7} - 412414770 \nu^{6} - 1358444363 \nu^{5} + \cdots - 4335298296 ) / 1955731010$$ (-5051595*v^11 + 9282581*v^10 - 64765518*v^9 - 29949491*v^8 - 345971017*v^7 - 412414770*v^6 - 1358444363*v^5 - 2951210507*v^4 - 2448832100*v^3 - 5335902045*v^2 - 159117251*v - 4335298296) / 1955731010 $$\beta_{7}$$ $$=$$ $$( - 6178545 \nu^{11} + 17844777 \nu^{10} - 93333716 \nu^{9} + 35014563 \nu^{8} - 359601729 \nu^{7} - 259108930 \nu^{6} - 1288684281 \nu^{5} + \cdots - 612082342 ) / 1955731010$$ (-6178545*v^11 + 17844777*v^10 - 93333716*v^9 + 35014563*v^8 - 359601729*v^7 - 259108930*v^6 - 1288684281*v^5 - 2049018039*v^4 - 725395450*v^3 - 4110047595*v^2 - 122568267*v - 612082342) / 1955731010 $$\beta_{8}$$ $$=$$ $$( - 469619124 \nu^{11} + 1742271771 \nu^{10} - 9648336987 \nu^{9} + 16279393858 \nu^{8} - 67720087321 \nu^{7} + 86418671699 \nu^{6} + \cdots + 402309009 ) / 66494854340$$ (-469619124*v^11 + 1742271771*v^10 - 9648336987*v^9 + 16279393858*v^8 - 67720087321*v^7 + 86418671699*v^6 - 294644252580*v^5 + 212867920667*v^4 - 647754164847*v^3 + 432662257370*v^2 - 680489253663*v + 402309009) / 66494854340 $$\beta_{9}$$ $$=$$ $$( - 2004496389 \nu^{11} + 6017448375 \nu^{10} - 34029515688 \nu^{9} + 36042149143 \nu^{8} - 217868445987 \nu^{7} + \cdots - 65422139102 ) / 66494854340$$ (-2004496389*v^11 + 6017448375*v^10 - 34029515688*v^9 + 36042149143*v^8 - 217868445987*v^7 + 187219927286*v^6 - 966498199653*v^5 + 298838370979*v^4 - 2007194188104*v^3 + 1145299361381*v^2 - 2194621698825*v - 65422139102) / 66494854340 $$\beta_{10}$$ $$=$$ $$( - 4220156417 \nu^{11} + 12953281828 \nu^{10} - 72904583371 \nu^{9} + 81775705089 \nu^{8} - 469415474758 \nu^{7} + \cdots + 2372026047 ) / 66494854340$$ (-4220156417*v^11 + 12953281828*v^10 - 72904583371*v^9 + 81775705089*v^8 - 469415474758*v^7 + 426177732417*v^6 - 2086126950865*v^5 + 755327449736*v^4 - 4359372249951*v^3 + 2534882432535*v^2 - 4859631701974*v + 2372026047) / 66494854340 $$\beta_{11}$$ $$=$$ $$( 6424307742 \nu^{11} - 19688524501 \nu^{10} + 111043957893 \nu^{9} - 124776050724 \nu^{8} + 717279166315 \nu^{7} - 650339598681 \nu^{6} + \cdots - 3616495515 ) / 66494854340$$ (6424307742*v^11 - 19688524501*v^10 + 111043957893*v^9 - 124776050724*v^8 + 717279166315*v^7 - 650339598681*v^6 + 3178214179566*v^5 - 1144205547385*v^4 + 6642795442545*v^3 - 3864903739992*v^2 + 7195985653273*v - 3616495515) / 66494854340
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + 3\beta_{9} + \beta_{8} - \beta_{4} - \beta_{2} + \beta _1 - 2$$ b11 + 3*b9 + b8 - b4 - b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} - 2\beta_{4} - 6\beta_{2} - 10$$ -b7 + 2*b6 - 2*b4 - 6*b2 - 10 $$\nu^{4}$$ $$=$$ $$-10\beta_{11} - 6\beta_{10} - 18\beta_{9} - 6\beta_{8} + \beta_{3} - 11\beta _1 - 18$$ -10*b11 - 6*b10 - 18*b9 - 6*b8 + b3 - 11*b1 - 18 $$\nu^{5}$$ $$=$$ $$- 28 \beta_{11} - 27 \beta_{10} - 31 \beta_{9} - 8 \beta_{8} + 20 \beta_{7} - 31 \beta_{6} + 4 \beta_{5} + 28 \beta_{4} + 4 \beta_{3} + 45 \beta_{2} - 45 \beta _1 + 73$$ -28*b11 - 27*b10 - 31*b9 - 8*b8 + 20*b7 - 31*b6 + 4*b5 + 28*b4 + 4*b3 + 45*b2 - 45*b1 + 73 $$\nu^{6}$$ $$=$$ $$71\beta_{7} - 107\beta_{6} + 20\beta_{5} + 104\beta_{4} + 112\beta_{2} + 358$$ 71*b7 - 107*b6 + 20*b5 + 104*b4 + 112*b2 + 358 $$\nu^{7}$$ $$=$$ $$323\beta_{11} + 315\beta_{10} + 359\beta_{9} + 52\beta_{8} - 71\beta_{3} + 391\beta _1 + 359$$ 323*b11 + 315*b10 + 359*b9 + 52*b8 - 71*b3 + 391*b1 + 359 $$\nu^{8}$$ $$=$$ $$1100 \beta_{11} + 1032 \beta_{10} + 1307 \beta_{9} + 178 \beta_{8} - 922 \beta_{7} + 1303 \beta_{6} - 271 \beta_{5} - 1100 \beta_{4} - 271 \beta_{3} - 1125 \beta_{2} + 1125 \beta _1 - 2225$$ 1100*b11 + 1032*b10 + 1307*b9 + 178*b8 - 922*b7 + 1303*b6 - 271*b5 - 1100*b4 - 271*b3 - 1125*b2 + 1125*b1 - 2225 $$\nu^{9}$$ $$=$$ $$-3216\beta_{7} + 4425\beta_{6} - 922\beta_{5} - 3528\beta_{4} - 3710\beta_{2} - 11087$$ -3216*b7 + 4425*b6 - 922*b5 - 3528*b4 - 3710*b2 - 11087 $$\nu^{10}$$ $$=$$ $$-11663\beta_{11} - 11481\beta_{10} - 12949\beta_{9} - 944\beta_{8} + 3216\beta_{3} - 11399\beta _1 - 12949$$ -11663*b11 - 11481*b10 - 12949*b9 - 944*b8 + 3216*b3 - 11399*b1 - 12949 $$\nu^{11}$$ $$=$$ $$- 37759 \beta_{11} - 38023 \beta_{10} - 40447 \beta_{9} - 1751 \beta_{8} + 36008 \beta_{7} - 48742 \beta_{6} + 10719 \beta_{5} + 37759 \beta_{4} + 10719 \beta_{3} + 36955 \beta_{2} - 36955 \beta _1 + 74714$$ -37759*b11 - 38023*b10 - 40447*b9 - 1751*b8 + 36008*b7 - 48742*b6 + 10719*b5 + 37759*b4 + 10719*b3 + 36955*b2 - 36955*b1 + 74714

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −0.975939 + 1.69038i 1.20634 − 2.08945i −0.0149173 + 0.0258375i 1.62208 − 2.80952i −0.928369 + 1.60798i 0.590804 − 1.02330i −0.975939 − 1.69038i 1.20634 + 2.08945i −0.0149173 − 0.0258375i 1.62208 + 2.80952i −0.928369 − 1.60798i 0.590804 + 1.02330i
−1.08504 1.87935i −1.47594 2.55640i −1.35464 + 2.34630i 0 −3.20292 + 5.54761i 0.591620 1.53919 −2.85679 + 4.94811i 0
26.2 −1.08504 1.87935i 0.706345 + 1.22342i −1.35464 + 2.34630i 0 1.53283 2.65494i −1.76171 1.53919 0.502155 0.869757i 0
26.3 −0.155554 0.269427i −0.514917 0.891863i 0.951606 1.64823i 0 −0.160195 + 0.277466i −3.28038 −1.21432 0.969720 1.67960i 0
26.4 −0.155554 0.269427i 1.12208 + 1.94349i 0.951606 1.64823i 0 0.349087 0.604636i 3.96928 −1.21432 −1.01811 + 1.76343i 0
26.5 0.740597 + 1.28275i −1.42837 2.47401i −0.0969683 + 0.167954i 0 2.11569 3.66449i 3.78541 2.67513 −2.58048 + 4.46952i 0
26.6 0.740597 + 1.28275i 0.0908038 + 0.157277i −0.0969683 + 0.167954i 0 −0.134498 + 0.232958i −1.30422 2.67513 1.48351 2.56951i 0
201.1 −1.08504 + 1.87935i −1.47594 + 2.55640i −1.35464 2.34630i 0 −3.20292 5.54761i 0.591620 1.53919 −2.85679 4.94811i 0
201.2 −1.08504 + 1.87935i 0.706345 1.22342i −1.35464 2.34630i 0 1.53283 + 2.65494i −1.76171 1.53919 0.502155 + 0.869757i 0
201.3 −0.155554 + 0.269427i −0.514917 + 0.891863i 0.951606 + 1.64823i 0 −0.160195 0.277466i −3.28038 −1.21432 0.969720 + 1.67960i 0
201.4 −0.155554 + 0.269427i 1.12208 1.94349i 0.951606 + 1.64823i 0 0.349087 + 0.604636i 3.96928 −1.21432 −1.01811 1.76343i 0
201.5 0.740597 1.28275i −1.42837 + 2.47401i −0.0969683 0.167954i 0 2.11569 + 3.66449i 3.78541 2.67513 −2.58048 4.46952i 0
201.6 0.740597 1.28275i 0.0908038 0.157277i −0.0969683 0.167954i 0 −0.134498 0.232958i −1.30422 2.67513 1.48351 + 2.56951i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.f 12
5.b even 2 1 475.2.e.h yes 12
5.c odd 4 2 475.2.j.d 24
19.c even 3 1 inner 475.2.e.f 12
19.c even 3 1 9025.2.a.bz 6
19.d odd 6 1 9025.2.a.bs 6
95.h odd 6 1 9025.2.a.by 6
95.i even 6 1 475.2.e.h yes 12
95.i even 6 1 9025.2.a.br 6
95.m odd 12 2 475.2.j.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.e.f 12 1.a even 1 1 trivial
475.2.e.f 12 19.c even 3 1 inner
475.2.e.h yes 12 5.b even 2 1
475.2.e.h yes 12 95.i even 6 1
475.2.j.d 24 5.c odd 4 2
475.2.j.d 24 95.m odd 12 2
9025.2.a.br 6 95.i even 6 1
9025.2.a.bs 6 19.d odd 6 1
9025.2.a.by 6 95.h odd 6 1
9025.2.a.bz 6 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{5} + 4T_{2}^{4} - T_{2}^{3} + 10T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + T^{5} + 4 T^{4} - T^{3} + 10 T^{2} + \cdots + 1)^{2}$$
$3$ $$T^{12} + 3 T^{11} + 17 T^{10} + 18 T^{9} + \cdots + 25$$
$5$ $$T^{12}$$
$7$ $$(T^{6} - 2 T^{5} - 21 T^{4} + 20 T^{3} + \cdots - 67)^{2}$$
$11$ $$(T^{6} + T^{5} - 46 T^{4} + 9 T^{3} + 522 T^{2} + \cdots + 247)^{2}$$
$13$ $$T^{12} + 5 T^{11} + 61 T^{10} + \cdots + 100489$$
$17$ $$T^{12} - 3 T^{11} + 33 T^{10} - 34 T^{9} + \cdots + 1$$
$19$ $$T^{12} + 6 T^{11} + 15 T^{10} + \cdots + 47045881$$
$23$ $$T^{12} - 6 T^{11} + 93 T^{10} + \cdots + 1151329$$
$29$ $$T^{12} + 3 T^{11} + 117 T^{10} + \cdots + 163353961$$
$31$ $$(T^{6} + 3 T^{5} - 64 T^{4} - 189 T^{3} + \cdots - 631)^{2}$$
$37$ $$(T^{6} + 6 T^{5} - 32 T^{4} - 168 T^{3} + \cdots - 64)^{2}$$
$41$ $$T^{12} + 11 T^{11} + 145 T^{10} + \cdots + 1739761$$
$43$ $$T^{12} + 13 T^{11} + 131 T^{10} + \cdots + 829921$$
$47$ $$T^{12} - 6 T^{11} + 167 T^{10} + \cdots + 537266041$$
$53$ $$T^{12} + 18 T^{11} + \cdots + 116868943321$$
$59$ $$T^{12} + 4 T^{11} + 253 T^{10} + \cdots + 134397649$$
$61$ $$T^{12} + 25 T^{11} + \cdots + 305935081$$
$67$ $$T^{12} + 6 T^{11} + 200 T^{10} + \cdots + 145926400$$
$71$ $$T^{12} + 18 T^{11} + \cdots + 135885649$$
$73$ $$T^{12} + T^{11} + 167 T^{10} + \cdots + 4699788025$$
$79$ $$T^{12} + 3 T^{11} + \cdots + 106504975201$$
$83$ $$(T^{6} + 23 T^{5} - 74 T^{4} + \cdots + 378053)^{2}$$
$89$ $$T^{12} + 12 T^{11} + \cdots + 122226452881$$
$97$ $$T^{12} + 3 T^{11} + \cdots + 25482056161$$