# Properties

 Label 975.2.bb.k Level $975$ Weight $2$ Character orbit 975.bb Analytic conductor $7.785$ 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] = [975,2,Mod(724,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("975.724");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.bb (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: $$7.78541419707$$ 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: $$x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49$$ x^12 - 15*x^10 + 90*x^8 - 247*x^6 + 270*x^4 + 21*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} - \beta_{6} q^{3} + ( - \beta_{9} - \beta_{7} - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 1) q^{4} + ( - \beta_{9} - \beta_{7}) q^{6} + ( - \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_1) q^{7} + ( - 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b8 * q^2 - b6 * q^3 + (-b9 - b7 - b4 - 2*b3 + 3*b2 + 1) * q^4 + (-b9 - b7) * q^6 + (-b11 - 2*b10 - b8 + 2*b6 - b5 - b1) * q^7 + (-2*b11 - 4*b10 + 2*b6 - 2*b5 - 2*b1) * q^8 + b2 * q^9 $$q + \beta_{8} q^{2} - \beta_{6} q^{3} + ( - \beta_{9} - \beta_{7} - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 1) q^{4} + ( - \beta_{9} - \beta_{7}) q^{6} + ( - \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_1) q^{7} + ( - 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{8} + \beta_{2} q^{9} + 2 \beta_{9} q^{11} + ( - \beta_{11} - 2 \beta_{10} + \beta_{5}) q^{12} + ( - \beta_{11} - 3 \beta_{10} + 4 \beta_{6} - 2 \beta_{5} - \beta_1) q^{13} + (2 \beta_{7} - \beta_{4} + \beta_{3} - 4) q^{14} + ( - 2 \beta_{9} + 4 \beta_{2} - 4) q^{16} + (\beta_{11} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} + 3 \beta_1) q^{17} + ( - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{5} - \beta_1) q^{18} + ( - \beta_{9} - \beta_{7} - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + 1) q^{19} + (\beta_{7} - 1) q^{21} + (2 \beta_{11} + 10 \beta_{10} + 2 \beta_{8} - 8 \beta_{6} + 4 \beta_1) q^{22} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{5} - 4 \beta_1) q^{23} + ( - 2 \beta_{9} + 2 \beta_{2} - 2) q^{24} + (2 \beta_{7} - 3 \beta_{4} - \beta_{3} + 6 \beta_{2} - 4) q^{26} - \beta_{10} q^{27} + ( - \beta_{10} - 5 \beta_{8} + 4 \beta_{6} - \beta_{5} - 2 \beta_1) q^{28} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{29} + ( - \beta_{7} + 3 \beta_{4} - 3 \beta_{3} + 1) q^{31} + ( - 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{8} + 4 \beta_{6} - 4 \beta_1) q^{32} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{33} + ( - 3 \beta_{7} - \beta_{4} + \beta_{3}) q^{34} + ( - \beta_{9} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 1) q^{36} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - 4 \beta_1) q^{37} + ( - 6 \beta_{11} - 8 \beta_{10} + 6 \beta_{6} - 6 \beta_{5} - 6 \beta_1) q^{38} + ( - \beta_{9} + \beta_{4} - \beta_{2} - 2) q^{39} + (2 \beta_{9} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{41} + ( - \beta_{10} - 2 \beta_{8} + 4 \beta_{6} - \beta_{5} - 2 \beta_1) q^{42} + (3 \beta_{10} - 2 \beta_{6} - \beta_{5} + \beta_1) q^{43} + ( - 8 \beta_{7} + 4) q^{44} + (4 \beta_{9} + 4 \beta_{7} - 4 \beta_{2}) q^{46} + (3 \beta_{11} - \beta_{10} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_1) q^{47} + ( - 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{8} + 6 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{48} + (3 \beta_{9} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{49} + ( - \beta_{7} + 2 \beta_{4} - 2 \beta_{3}) q^{51} + ( - 4 \beta_{11} - 5 \beta_{10} - 7 \beta_{8} + 6 \beta_{6} - 7 \beta_{5} - 8 \beta_1) q^{52} + (2 \beta_{11} + 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{53} - \beta_{9} q^{54} + (6 \beta_{9} + 6 \beta_{7} + 2 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} - 2) q^{56} + ( - \beta_{11} - 4 \beta_{10} + \beta_{5}) q^{57} + (\beta_{11} + 3 \beta_{10} + \beta_{8} - 2 \beta_{6} + 2 \beta_1) q^{58} + (2 \beta_{9} + 2 \beta_{7} + \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 1) q^{59} + ( - \beta_{9} - \beta_{7} + \beta_{4} + 2 \beta_{3} - 1) q^{61} + ( - 2 \beta_{10} + 5 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_1) q^{62} + ( - \beta_{8} + \beta_{6}) q^{63} + (4 \beta_{7} + 4) q^{64} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 8) q^{66} + (3 \beta_{10} - 3 \beta_{6} + 3 \beta_{5} + 6 \beta_1) q^{67} - 14 \beta_{6} q^{68} + (2 \beta_{9} + 2 \beta_{7} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2) q^{69} + ( - 2 \beta_{9} - 2 \beta_{7} - 3 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} + 3) q^{71} + ( - 2 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{72} + (3 \beta_{11} - 4 \beta_{10} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_1) q^{73} + (2 \beta_{9} + 2 \beta_{7} - 4 \beta_{2}) q^{74} + ( - 6 \beta_{9} - 8 \beta_{4} - 4 \beta_{3} + 20 \beta_{2} - 12) q^{76} + ( - 4 \beta_{11} - 10 \beta_{10} + 2 \beta_{6} - 2 \beta_1) q^{77} + ( - 5 \beta_{10} - 2 \beta_{8} + 4 \beta_{6} + \beta_{5} - 2 \beta_1) q^{78} + ( - \beta_{7} + 3 \beta_{4} - 3 \beta_{3} + 1) q^{79} + (\beta_{2} - 1) q^{81} + (3 \beta_{11} + 9 \beta_{10} + 3 \beta_{8} - 8 \beta_{6} + 2 \beta_{5} + 4 \beta_1) q^{82} + (2 \beta_{11} + 6 \beta_{10} - 2 \beta_{5}) q^{83} + (5 \beta_{9} + 5 \beta_{7} + \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 1) q^{84} + ( - 4 \beta_{7} - \beta_{4} + \beta_{3} - 2) q^{86} + (2 \beta_{10} - 3 \beta_{6} + \beta_{5} - \beta_1) q^{87} + (4 \beta_{10} + 8 \beta_{8} - 16 \beta_{6} + 4 \beta_{5} + 8 \beta_1) q^{88} + ( - 2 \beta_{4} - \beta_{3} - 9 \beta_{2} + 11) q^{89} + (3 \beta_{9} + \beta_{7} + \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + 2) q^{91} + (4 \beta_{11} + 20 \beta_{10} - 4 \beta_{6} + 4 \beta_{5} + 4 \beta_1) q^{92} + (3 \beta_{10} + \beta_{8} - \beta_{6} + 3 \beta_{5} + 6 \beta_1) q^{93} + ( - \beta_{9} + 6 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} + 9) q^{94} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} - 4) q^{96} + (4 \beta_{11} - \beta_{10} + 4 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 5 \beta_1) q^{97} + (2 \beta_{11} + 12 \beta_{10} + 2 \beta_{8} - 10 \beta_{6} + 4 \beta_1) q^{98} - 2 \beta_{7} q^{99}+O(q^{100})$$ q + b8 * q^2 - b6 * q^3 + (-b9 - b7 - b4 - 2*b3 + 3*b2 + 1) * q^4 + (-b9 - b7) * q^6 + (-b11 - 2*b10 - b8 + 2*b6 - b5 - b1) * q^7 + (-2*b11 - 4*b10 + 2*b6 - 2*b5 - 2*b1) * q^8 + b2 * q^9 + 2*b9 * q^11 + (-b11 - 2*b10 + b5) * q^12 + (-b11 - 3*b10 + 4*b6 - 2*b5 - b1) * q^13 + (2*b7 - b4 + b3 - 4) * q^14 + (-2*b9 + 4*b2 - 4) * q^16 + (b11 + b10 + b8 + b6 - b5 + 3*b1) * q^17 + (-b11 - b10 + b6 - b5 - b1) * q^18 + (-b9 - b7 - b4 - 2*b3 + 5*b2 + 1) * q^19 + (b7 - 1) * q^21 + (2*b11 + 10*b10 + 2*b8 - 8*b6 + 4*b1) * q^22 + (-2*b10 - 2*b8 - 2*b5 - 4*b1) * q^23 + (-2*b9 + 2*b2 - 2) * q^24 + (2*b7 - 3*b4 - b3 + 6*b2 - 4) * q^26 - b10 * q^27 + (-b10 - 5*b8 + 4*b6 - b5 - 2*b1) * q^28 + (-2*b4 - b3 - b2 + 3) * q^29 + (-b7 + 3*b4 - 3*b3 + 1) * q^31 + (-2*b11 - 6*b10 - 2*b8 + 4*b6 - 4*b1) * q^32 + (2*b11 + 2*b10 + 2*b8 - 2*b6 + 2*b5 + 2*b1) * q^33 + (-3*b7 - b4 + b3) * q^34 + (-b9 - 2*b4 - b3 + 3*b2 - 1) * q^36 + (-2*b10 - 2*b8 - 2*b6 - 2*b5 - 4*b1) * q^37 + (-6*b11 - 8*b10 + 6*b6 - 6*b5 - 6*b1) * q^38 + (-b9 + b4 - b2 - 2) * q^39 + (2*b9 + 2*b4 + b3 - b2 - 1) * q^41 + (-b10 - 2*b8 + 4*b6 - b5 - 2*b1) * q^42 + (3*b10 - 2*b6 - b5 + b1) * q^43 + (-8*b7 + 4) * q^44 + (4*b9 + 4*b7 - 4*b2) * q^46 + (3*b11 - b10 - 3*b6 + 3*b5 + 3*b1) * q^47 + (-2*b11 - 6*b10 - 2*b8 + 6*b6 - 2*b5 - 2*b1) * q^48 + (3*b9 + 2*b4 + b3 + b2 - 3) * q^49 + (-b7 + 2*b4 - 2*b3) * q^51 + (-4*b11 - 5*b10 - 7*b8 + 6*b6 - 7*b5 - 8*b1) * q^52 + (2*b11 + 2*b10 - 2*b6 + 2*b5 + 2*b1) * q^53 - b9 * q^54 + (6*b9 + 6*b7 + 2*b4 + 4*b3 - 12*b2 - 2) * q^56 + (-b11 - 4*b10 + b5) * q^57 + (b11 + 3*b10 + b8 - 2*b6 + 2*b1) * q^58 + (2*b9 + 2*b7 + b4 + 2*b3 - 3*b2 - 1) * q^59 + (-b9 - b7 + b4 + 2*b3 - 1) * q^61 + (-2*b10 + 5*b8 + 2*b6 - 2*b5 - 4*b1) * q^62 + (-b8 + b6) * q^63 + (4*b7 + 4) * q^64 + (-2*b7 + 2*b4 - 2*b3 + 8) * q^66 + (3*b10 - 3*b6 + 3*b5 + 6*b1) * q^67 - 14*b6 * q^68 + (2*b9 + 2*b7 + 2*b4 + 4*b3 - 2*b2 - 2) * q^69 + (-2*b9 - 2*b7 - 3*b4 - 6*b3 + 7*b2 + 3) * q^71 + (-2*b11 - 4*b10 - 2*b8 + 4*b6 - 2*b5 - 2*b1) * q^72 + (3*b11 - 4*b10 - 3*b6 + 3*b5 + 3*b1) * q^73 + (2*b9 + 2*b7 - 4*b2) * q^74 + (-6*b9 - 8*b4 - 4*b3 + 20*b2 - 12) * q^76 + (-4*b11 - 10*b10 + 2*b6 - 2*b1) * q^77 + (-5*b10 - 2*b8 + 4*b6 + b5 - 2*b1) * q^78 + (-b7 + 3*b4 - 3*b3 + 1) * q^79 + (b2 - 1) * q^81 + (3*b11 + 9*b10 + 3*b8 - 8*b6 + 2*b5 + 4*b1) * q^82 + (2*b11 + 6*b10 - 2*b5) * q^83 + (5*b9 + 5*b7 + b4 + 2*b3 - 5*b2 - 1) * q^84 + (-4*b7 - b4 + b3 - 2) * q^86 + (2*b10 - 3*b6 + b5 - b1) * q^87 + (4*b10 + 8*b8 - 16*b6 + 4*b5 + 8*b1) * q^88 + (-2*b4 - b3 - 9*b2 + 11) * q^89 + (3*b9 + b7 + b4 + 2*b3 - 8*b2 + 2) * q^91 + (4*b11 + 20*b10 - 4*b6 + 4*b5 + 4*b1) * q^92 + (3*b10 + b8 - b6 + 3*b5 + 6*b1) * q^93 + (-b9 + 6*b4 + 3*b3 - 15*b2 + 9) * q^94 + (2*b7 - 2*b4 + 2*b3 - 4) * q^96 + (4*b11 - b10 + 4*b8 + 2*b6 + 3*b5 + 5*b1) * q^97 + (2*b11 + 12*b10 + 2*b8 - 10*b6 + 4*b1) * q^98 - 2*b7 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 12 q^{4} + 6 q^{9}+O(q^{10})$$ 12 * q + 12 * q^4 + 6 * q^9 $$12 q + 12 q^{4} + 6 q^{9} - 48 q^{14} - 24 q^{16} + 24 q^{19} - 12 q^{21} - 12 q^{24} - 36 q^{26} + 12 q^{29} + 12 q^{31} - 12 q^{36} - 24 q^{39} + 48 q^{44} - 24 q^{46} - 12 q^{49} - 60 q^{56} - 12 q^{59} + 6 q^{61} + 48 q^{64} + 96 q^{66} + 24 q^{71} - 24 q^{74} - 96 q^{76} + 12 q^{79} - 6 q^{81} - 24 q^{84} - 24 q^{86} + 60 q^{89} - 6 q^{91} + 72 q^{94} - 48 q^{96}+O(q^{100})$$ 12 * q + 12 * q^4 + 6 * q^9 - 48 * q^14 - 24 * q^16 + 24 * q^19 - 12 * q^21 - 12 * q^24 - 36 * q^26 + 12 * q^29 + 12 * q^31 - 12 * q^36 - 24 * q^39 + 48 * q^44 - 24 * q^46 - 12 * q^49 - 60 * q^56 - 12 * q^59 + 6 * q^61 + 48 * q^64 + 96 * q^66 + 24 * q^71 - 24 * q^74 - 96 * q^76 + 12 * q^79 - 6 * q^81 - 24 * q^84 - 24 * q^86 + 60 * q^89 - 6 * q^91 + 72 * q^94 - 48 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 2\nu^{10} - 83\nu^{8} + 906\nu^{6} - 3874\nu^{4} + 5950\nu^{2} + 1505 ) / 2947$$ (2*v^10 - 83*v^8 + 906*v^6 - 3874*v^4 + 5950*v^2 + 1505) / 2947 $$\beta_{3}$$ $$=$$ $$( -25\nu^{10} + 406\nu^{8} - 2063\nu^{6} + 2536\nu^{4} + 3931\nu^{2} + 3290 ) / 5894$$ (-25*v^10 + 406*v^8 - 2063*v^6 + 2536*v^4 + 3931*v^2 + 3290) / 5894 $$\beta_{4}$$ $$=$$ $$( -29\nu^{10} + 151\nu^{8} + 335\nu^{6} - 3609\nu^{4} + 2977\nu^{2} + 3227 ) / 5894$$ (-29*v^10 + 151*v^8 + 335*v^6 - 3609*v^4 + 2977*v^2 + 3227) / 5894 $$\beta_{5}$$ $$=$$ $$( 2\nu^{11} - 83\nu^{9} + 906\nu^{7} - 3874\nu^{5} + 5950\nu^{3} - 1442\nu ) / 2947$$ (2*v^11 - 83*v^9 + 906*v^7 - 3874*v^5 + 5950*v^3 - 1442*v) / 2947 $$\beta_{6}$$ $$=$$ $$( -40\nu^{11} + 397\nu^{9} + 404\nu^{7} - 17245\nu^{5} + 61188\nu^{3} - 56623\nu ) / 41258$$ (-40*v^11 + 397*v^9 + 404*v^7 - 17245*v^5 + 61188*v^3 - 56623*v) / 41258 $$\beta_{7}$$ $$=$$ $$( -58\nu^{10} + 723\nu^{8} - 3540\nu^{6} + 6675\nu^{4} + 902\nu^{2} - 14175 ) / 5894$$ (-58*v^10 + 723*v^8 - 3540*v^6 + 6675*v^4 + 902*v^2 - 14175) / 5894 $$\beta_{8}$$ $$=$$ $$( -55\nu^{11} + 1651\nu^{9} - 15653\nu^{7} + 69487\nu^{5} - 156047\nu^{3} + 151641\nu ) / 41258$$ (-55*v^11 + 1651*v^9 - 15653*v^7 + 69487*v^5 - 156047*v^3 + 151641*v) / 41258 $$\beta_{9}$$ $$=$$ $$( -89\nu^{10} + 1378\nu^{8} - 8321\nu^{6} + 23780\nu^{4} - 30699\nu^{2} + 8176 ) / 5894$$ (-89*v^10 + 1378*v^8 - 8321*v^6 + 23780*v^4 - 30699*v^2 + 8176) / 5894 $$\beta_{10}$$ $$=$$ $$( -501\nu^{11} + 7109\nu^{9} - 40029\nu^{7} + 98967\nu^{5} - 88545\nu^{3} - 45465\nu ) / 41258$$ (-501*v^11 + 7109*v^9 - 40029*v^7 + 98967*v^5 - 88545*v^3 - 45465*v) / 41258 $$\beta_{11}$$ $$=$$ $$( -332\nu^{11} + 5358\nu^{9} - 33779\nu^{7} + 94100\nu^{5} - 82129\nu^{3} - 75957\nu ) / 20629$$ (-332*v^11 + 5358*v^9 - 33779*v^7 + 94100*v^5 - 82129*v^3 - 75957*v) / 20629
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + 3$$ b7 - b4 - b3 + b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{10} - \beta_{8} - 2\beta_{6} + \beta_{5} + 4\beta_1$$ b11 - b10 - b8 - 2*b6 + b5 + 4*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{9} + 4\beta_{7} - 6\beta_{4} - 8\beta_{3} + 9\beta_{2} + 10$$ 2*b9 + 4*b7 - 6*b4 - 8*b3 + 9*b2 + 10 $$\nu^{5}$$ $$=$$ $$8\beta_{11} - 8\beta_{10} - 4\beta_{8} - 18\beta_{6} + 13\beta_{5} + 17\beta_1$$ 8*b11 - 8*b10 - 4*b8 - 18*b6 + 13*b5 + 17*b1 $$\nu^{6}$$ $$=$$ $$17\beta_{9} + 13\beta_{7} - 33\beta_{4} - 42\beta_{3} + 65\beta_{2} + 16$$ 17*b9 + 13*b7 - 33*b4 - 42*b3 + 65*b2 + 16 $$\nu^{7}$$ $$=$$ $$50\beta_{11} - 53\beta_{10} - 5\beta_{8} - 90\beta_{6} + 99\beta_{5} + 69\beta_1$$ 50*b11 - 53*b10 - 5*b8 - 90*b6 + 99*b5 + 69*b1 $$\nu^{8}$$ $$=$$ $$104\beta_{9} + 24\beta_{7} - 172\beta_{4} - 168\beta_{3} + 365\beta_{2} - 85$$ 104*b9 + 24*b7 - 172*b4 - 168*b3 + 365*b2 - 85 $$\nu^{9}$$ $$=$$ $$276\beta_{11} - 320\beta_{10} + 84\beta_{8} - 288\beta_{6} + 573\beta_{5} + 240\beta_1$$ 276*b11 - 320*b10 + 84*b8 - 288*b6 + 573*b5 + 240*b1 $$\nu^{10}$$ $$=$$ $$489\beta_{9} - 120\beta_{7} - 836\beta_{4} - 467\beta_{3} + 1634\beta_{2} - 1083$$ 489*b9 - 120*b7 - 836*b4 - 467*b3 + 1634*b2 - 1083 $$\nu^{11}$$ $$=$$ $$1325\beta_{11} - 1792\beta_{10} + 978\beta_{8} - 98\beta_{6} + 2612\beta_{5} + 453\beta_1$$ 1325*b11 - 1792*b10 + 978*b8 - 98*b6 + 2612*b5 + 453*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
724.1
 −0.385124 − 0.500000i 2.23871 + 0.500000i −1.75780 + 0.500000i 1.75780 − 0.500000i −2.23871 − 0.500000i 0.385124 + 0.500000i −0.385124 + 0.500000i 2.23871 − 0.500000i −1.75780 − 0.500000i 1.75780 + 0.500000i −2.23871 + 0.500000i 0.385124 − 0.500000i
−2.25312 1.30084i −0.866025 0.500000i 2.38437 + 4.12985i 0 1.30084 + 2.25312i 3.11915 1.80084i 7.20336i 0.500000 + 0.866025i 0
724.2 −1.95878 1.13090i 0.866025 + 0.500000i 1.55787 + 2.69832i 0 −1.13090 1.95878i 1.09275 0.630901i 2.52360i 0.500000 + 0.866025i 0
724.3 −0.294342 0.169938i 0.866025 + 0.500000i −0.942242 1.63201i 0 −0.169938 0.294342i −0.571683 + 0.330062i 1.32025i 0.500000 + 0.866025i 0
724.4 0.294342 + 0.169938i −0.866025 0.500000i −0.942242 1.63201i 0 −0.169938 0.294342i 0.571683 0.330062i 1.32025i 0.500000 + 0.866025i 0
724.5 1.95878 + 1.13090i −0.866025 0.500000i 1.55787 + 2.69832i 0 −1.13090 1.95878i −1.09275 + 0.630901i 2.52360i 0.500000 + 0.866025i 0
724.6 2.25312 + 1.30084i 0.866025 + 0.500000i 2.38437 + 4.12985i 0 1.30084 + 2.25312i −3.11915 + 1.80084i 7.20336i 0.500000 + 0.866025i 0
874.1 −2.25312 + 1.30084i −0.866025 + 0.500000i 2.38437 4.12985i 0 1.30084 2.25312i 3.11915 + 1.80084i 7.20336i 0.500000 0.866025i 0
874.2 −1.95878 + 1.13090i 0.866025 0.500000i 1.55787 2.69832i 0 −1.13090 + 1.95878i 1.09275 + 0.630901i 2.52360i 0.500000 0.866025i 0
874.3 −0.294342 + 0.169938i 0.866025 0.500000i −0.942242 + 1.63201i 0 −0.169938 + 0.294342i −0.571683 0.330062i 1.32025i 0.500000 0.866025i 0
874.4 0.294342 0.169938i −0.866025 + 0.500000i −0.942242 + 1.63201i 0 −0.169938 + 0.294342i 0.571683 + 0.330062i 1.32025i 0.500000 0.866025i 0
874.5 1.95878 1.13090i −0.866025 + 0.500000i 1.55787 2.69832i 0 −1.13090 + 1.95878i −1.09275 0.630901i 2.52360i 0.500000 0.866025i 0
874.6 2.25312 1.30084i 0.866025 0.500000i 2.38437 4.12985i 0 1.30084 2.25312i −3.11915 1.80084i 7.20336i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 874.6 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
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bb.k 12
5.b even 2 1 inner 975.2.bb.k 12
5.c odd 4 1 195.2.i.d 6
5.c odd 4 1 975.2.i.l 6
13.c even 3 1 inner 975.2.bb.k 12
15.e even 4 1 585.2.j.f 6
65.n even 6 1 inner 975.2.bb.k 12
65.q odd 12 1 195.2.i.d 6
65.q odd 12 1 975.2.i.l 6
65.q odd 12 1 2535.2.a.bb 3
65.r odd 12 1 2535.2.a.ba 3
195.bf even 12 1 7605.2.a.bw 3
195.bl even 12 1 585.2.j.f 6
195.bl even 12 1 7605.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 5.c odd 4 1
195.2.i.d 6 65.q odd 12 1
585.2.j.f 6 15.e even 4 1
585.2.j.f 6 195.bl even 12 1
975.2.i.l 6 5.c odd 4 1
975.2.i.l 6 65.q odd 12 1
975.2.bb.k 12 1.a even 1 1 trivial
975.2.bb.k 12 5.b even 2 1 inner
975.2.bb.k 12 13.c even 3 1 inner
975.2.bb.k 12 65.n even 6 1 inner
2535.2.a.ba 3 65.r odd 12 1
2535.2.a.bb 3 65.q odd 12 1
7605.2.a.bv 3 195.bl even 12 1
7605.2.a.bw 3 195.bf even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{12} - 12T_{2}^{10} + 108T_{2}^{8} - 424T_{2}^{6} + 1248T_{2}^{4} - 144T_{2}^{2} + 16$$ T2^12 - 12*T2^10 + 108*T2^8 - 424*T2^6 + 1248*T2^4 - 144*T2^2 + 16 $$T_{7}^{12} - 15T_{7}^{10} + 198T_{7}^{8} - 387T_{7}^{6} + 594T_{7}^{4} - 243T_{7}^{2} + 81$$ T7^12 - 15*T7^10 + 198*T7^8 - 387*T7^6 + 594*T7^4 - 243*T7^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 12 T^{10} + 108 T^{8} + \cdots + 16$$
$3$ $$(T^{4} - T^{2} + 1)^{3}$$
$5$ $$T^{12}$$
$7$ $$T^{12} - 15 T^{10} + 198 T^{8} + \cdots + 81$$
$11$ $$(T^{6} + 24 T^{4} + 32 T^{3} + 576 T^{2} + \cdots + 256)^{2}$$
$13$ $$T^{12} + 21 T^{10} + 378 T^{8} + \cdots + 4826809$$
$17$ $$T^{12} - 84 T^{10} + 5292 T^{8} + \cdots + 92236816$$
$19$ $$(T^{6} - 12 T^{5} + 108 T^{4} - 424 T^{3} + \cdots + 16)^{2}$$
$23$ $$T^{12} - 96 T^{10} + 6912 T^{8} + \cdots + 84934656$$
$29$ $$(T^{6} - 6 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 196)^{2}$$
$31$ $$(T^{3} - 3 T^{2} - 93 T + 363)^{4}$$
$37$ $$T^{12} - 108 T^{10} + 10464 T^{8} + \cdots + 4096$$
$41$ $$(T^{6} + 24 T^{4} + 52 T^{3} + 576 T^{2} + \cdots + 676)^{2}$$
$43$ $$T^{12} - 51 T^{10} + 2178 T^{8} + \cdots + 14641$$
$47$ $$(T^{6} + 156 T^{4} + 4980 T^{2} + \cdots + 42436)^{2}$$
$53$ $$(T^{6} + 48 T^{4} + 576 T^{2} + 256)^{2}$$
$59$ $$(T^{6} + 6 T^{5} + 48 T^{4} - 44 T^{3} + \cdots + 196)^{2}$$
$61$ $$(T^{6} - 3 T^{5} + 30 T^{4} - 71 T^{3} + \cdots + 4489)^{2}$$
$67$ $$T^{12} - 243 T^{10} + \cdots + 15178486401$$
$71$ $$(T^{6} - 12 T^{5} + 192 T^{4} + \cdots + 465124)^{2}$$
$73$ $$(T^{6} + 255 T^{4} + 12387 T^{2} + \cdots + 7921)^{2}$$
$79$ $$(T^{3} - 3 T^{2} - 93 T + 363)^{4}$$
$83$ $$(T^{6} + 204 T^{4} + 9648 T^{2} + \cdots + 28224)^{2}$$
$89$ $$(T^{6} - 30 T^{5} + 612 T^{4} + \cdots + 777924)^{2}$$
$97$ $$T^{12} - 243 T^{10} + \cdots + 120354180241$$