# Properties

 Label 520.2.bu.b Level $520$ Weight $2$ Character orbit 520.bu Analytic conductor $4.152$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(121,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("520.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.bu (of order $$6$$, 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: $$4.15222090511$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 22x^{14} + 183x^{12} + 730x^{10} + 1485x^{8} + 1552x^{6} + 812x^{4} + 192x^{2} + 16$$ x^16 + 22*x^14 + 183*x^12 + 730*x^10 + 1485*x^8 + 1552*x^6 + 812*x^4 + 192*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( - \beta_{10} + \beta_{9}) q^{5} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{7}+ \cdots + ( - \beta_{15} - \beta_{13} + \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \cdots - 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + (-b10 + b9) * q^5 + (b14 + b13 - b12 - b10 + 2*b9 - b8 - 2*b7 - 2*b5 - b4 - b3 + b2) * q^7 + (-b15 - b13 + b12 + 2*b10 - 2*b9 + b8 + b7 + b6 + 2*b5 + b3 - 2*b2 + b1 - 1) * q^9 $$q + \beta_{2} q^{3} + ( - \beta_{10} + \beta_{9}) q^{5} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{7}+ \cdots + ( - \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{7} - 6 \beta_{6} + \cdots + 2) q^{99}+O(q^{100})$$ q + b2 * q^3 + (-b10 + b9) * q^5 + (b14 + b13 - b12 - b10 + 2*b9 - b8 - 2*b7 - 2*b5 - b4 - b3 + b2) * q^7 + (-b15 - b13 + b12 + 2*b10 - 2*b9 + b8 + b7 + b6 + 2*b5 + b3 - 2*b2 + b1 - 1) * q^9 + (-b15 - b9 + b8 + b7 + b6 + b5 - b4 - b3 - b2) * q^11 + (b15 + b14 + b13 - b10 + b9 - b8 - b7 + b6 - b5 - b4 + b3) * q^13 + (-b12 + b10) * q^15 + (-b15 - b14 - 2*b13 + 2*b12 + b10 - 2*b9 + b8 + b7 - b6 + 3*b5 + b4 + 2*b3 - b2 - b1 + 1) * q^17 + (b11 - 2*b9 + b6 - b4 + b1 + 2) * q^19 + (b15 + 2*b14 - b12 - b11 + b10 + b7 - 2*b5 + 2*b2 - 1) * q^21 + (-2*b13 - b11 - b6 + b3 - b2 - 1) * q^23 - q^25 + (b14 + 2*b11 - b10 + b9 - b7 - b6 - 2*b5 - b4 - b3 + b2 + 2) * q^27 + (-b15 - 2*b14 - b13 + b12 + b11 + b10 - b9 + b8 + 3*b7 + 3*b5 + b3 - 2*b2 - 2*b1 + 1) * q^29 + (2*b15 + b14 + 2*b13 - 2*b12 - b11 - 3*b10 + 5*b9 - 2*b7 - 4*b5 - b4 - b3 + 3*b2 - 2*b1) * q^31 + (-b15 - 4*b14 - 2*b13 + 3*b12 + 4*b11 + 4*b10 - 7*b9 + 2*b8 + 7*b7 + 9*b5 + b4 + 3*b3 - 7*b2 + 4) * q^33 + (b13 + b6 - b4 - b2) * q^35 + (-2*b15 - 3*b13 + 3*b10 - 2*b9 + 2*b8 + 2*b7 + 2*b5 + b4 + b3 - b2 + b1 - 2) * q^37 + (-b15 - b13 + 2*b11 - b10 + b9 - b7 - 3*b6 + b5 + b4 - b3 + 3*b2 + 3*b1 - 2) * q^39 + (2*b14 + b13 - b12 - 3*b10 + 2*b9 - 2*b8 - 2*b7 + 2*b6 - 4*b5 - b4 - b3 + b2 + b1 - 2) * q^41 + (b15 + b14 - 2*b12 - b11 + 3*b10 + b8 - b7 - b5 + b2 - b1) * q^43 + (b15 - b13 + b12 - b11 - 2*b9 + b7 + b5 + b4 + b3 + b1) * q^45 + (-b14 + 4*b13 + 2*b11 + 2*b10 - 2*b9 + 3*b6 - 3*b4 - b3 - 3*b2 - 4*b1 + 4) * q^47 + (b14 + 2*b13 - b12 + b11 - b10 + b9 - b8 - 3*b7 + b6 - 2*b5 - 2*b4 - b3 - 2*b2 - 3*b1 + 1) * q^49 + (-3*b15 + 3*b14 - 3*b12 + 2*b10 + 5*b9 - 3*b7 - 2*b6 - 6*b5 - 3*b4 - 3*b3 + 3*b2 - 5) * q^51 + (2*b13 - 2*b12 - 2*b11 + b10 + b9 - 2*b8 - b7 - b6) * q^53 + (-b14 - 2*b13 + b12 + 2*b10 - 3*b9 + b8 + 3*b7 + 2*b5 + 2*b3 - 2*b2 - b1) * q^55 + (-2*b15 - b14 - 2*b13 + 2*b12 + b11 + b10 - 3*b9 + 2*b7 - 3*b6 + 4*b5 + 2*b4 + 3*b2 + 6*b1 - 2) * q^57 + (2*b15 + 3*b14 + b13 - b12 - 4*b11 - 3*b10 + 6*b9 - 4*b7 - b6 - 7*b5 + b4 - b3 + 4*b2 + b1 - 3) * q^59 + (b14 + b13 - b12 + b11 + b10 + 2*b6 - b5 - 3*b3 + b1) * q^61 + (2*b15 - 4*b14 - b13 + 6*b12 - 5*b10 - 2*b9 + 2*b8 + 2*b7 + 2*b6 + 6*b5 + 3*b4 + 3*b3 - 4*b2 + b1 - 2) * q^63 + (b15 - b14 + b13 + b5 + b4) * q^65 + (b15 + b14 + 2*b13 + b12 + b11 - 3*b10 + 2*b9 - 2*b8 - 2*b7 + 4*b6 - 3*b5 - b4 - 2*b3 + 1) * q^67 + (b14 + b13 - b12 - 3*b10 + 2*b9 - 2*b8 - b6 - 3*b5 - b4 - b3 + 2*b2 - 5*b1 + 5) * q^69 + (-2*b15 - 2*b14 - b13 + 2*b10 - 5*b9 + b8 + 2*b7 + 3*b5 + b4 + 4*b1 + 4) * q^71 + (-2*b15 - 2*b14 - 4*b13 + 2*b12 - 2*b9 + 2*b7 - 2*b6 + 4*b5 + b4 + 3*b3 - 2*b2 - 4*b1 + 2) * q^73 - b2 * q^75 + (2*b15 + 2*b12 - 3*b11 - 5*b10 - 3*b9 - 2*b6 + b4 + b3) * q^77 + (2*b15 - b14 + 2*b12 - 2*b11 + b10 + b9 + b7 + 2*b5 + b4 + b3 - b2 + 2) * q^79 + (3*b15 + 4*b14 - 2*b12 - 5*b11 - 4*b10 + 7*b9 - b8 - 5*b7 - 2*b6 - 7*b5 + 3*b4 - 2*b3 + 9*b2 - 5) * q^81 + (-b14 - 2*b13 - b11 + b10 - b9 - 2*b6 + b4 + b3 + b2 + 2*b1 - 2) * q^83 + (-b11 + b9 - b8 + b5 + b4 + b2 + b1) * q^85 + (b15 - 2*b13 - b12 - b11 + 4*b10 - b9 - b7 + b6 - b5 + 2*b4 + b3 - b2 + 2*b1 - 3) * q^87 + (b15 + 3*b14 + 2*b13 - 5*b12 - 2*b11 + 3*b10 + 4*b9 - 4*b8 - 4*b7 - b6 - 7*b5 - b4 + b3 + 3*b2 - b1) * q^89 + (-b15 - 2*b14 + 2*b13 + 2*b12 + b11 - 5*b10 + 2*b9 + b8 + b7 + 2*b6 + 2*b5 - b4 - b3 - b2 + 1) * q^91 + (b15 - 3*b14 - b12 - b10 - 2*b9 + 2*b8 + 2*b7 + 5*b6 + 5*b5 + b4 + b3 - 5*b2 + 2*b1 - 4) * q^93 + (b15 - b12 - b11 - 2*b10 + 2*b9 - b7 - b5 + b3 - 2*b1 + 1) * q^95 + (-b15 + b14 + 2*b13 - 2*b12 - b10 + 4*b9 - b8 - 3*b7 + b6 - 3*b5 - b4 - 2*b3 + 3*b2 + 3*b1 + 3) * q^97 + (-b15 - 2*b14 - 2*b13 + b12 - b10 - 2*b7 - 6*b6 + 2*b5 + 6*b4 - 4*b3 + 8*b2 - 4*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{3} + 6 q^{7} - 16 q^{9}+O(q^{10})$$ 16 * q - 4 * q^3 + 6 * q^7 - 16 * q^9 $$16 q - 4 q^{3} + 6 q^{7} - 16 q^{9} - 6 q^{11} - 2 q^{13} + 4 q^{17} + 30 q^{19} + 6 q^{23} - 16 q^{25} + 44 q^{27} - 16 q^{29} + 24 q^{33} - 6 q^{35} - 24 q^{37} - 8 q^{39} - 24 q^{41} + 6 q^{43} + 12 q^{45} - 4 q^{49} - 40 q^{51} + 4 q^{53} - 6 q^{55} + 12 q^{59} - 2 q^{61} - 60 q^{63} - 10 q^{65} - 6 q^{67} + 52 q^{69} + 72 q^{71} + 4 q^{75} + 32 q^{77} + 36 q^{79} - 28 q^{81} - 22 q^{87} + 24 q^{89} - 22 q^{91} - 96 q^{93} + 10 q^{95} + 60 q^{97}+O(q^{100})$$ 16 * q - 4 * q^3 + 6 * q^7 - 16 * q^9 - 6 * q^11 - 2 * q^13 + 4 * q^17 + 30 * q^19 + 6 * q^23 - 16 * q^25 + 44 * q^27 - 16 * q^29 + 24 * q^33 - 6 * q^35 - 24 * q^37 - 8 * q^39 - 24 * q^41 + 6 * q^43 + 12 * q^45 - 4 * q^49 - 40 * q^51 + 4 * q^53 - 6 * q^55 + 12 * q^59 - 2 * q^61 - 60 * q^63 - 10 * q^65 - 6 * q^67 + 52 * q^69 + 72 * q^71 + 4 * q^75 + 32 * q^77 + 36 * q^79 - 28 * q^81 - 22 * q^87 + 24 * q^89 - 22 * q^91 - 96 * q^93 + 10 * q^95 + 60 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 22x^{14} + 183x^{12} + 730x^{10} + 1485x^{8} + 1552x^{6} + 812x^{4} + 192x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( - 18 \nu^{15} - 385 \nu^{13} - 3058 \nu^{11} - 11251 \nu^{9} - 19650 \nu^{7} - 15009 \nu^{5} - 3626 \nu^{3} + 148 \nu + 104 ) / 208$$ (-18*v^15 - 385*v^13 - 3058*v^11 - 11251*v^9 - 19650*v^7 - 15009*v^5 - 3626*v^3 + 148*v + 104) / 208 $$\beta_{2}$$ $$=$$ $$( 4 \nu^{15} + 45 \nu^{14} + 113 \nu^{13} + 956 \nu^{12} + 1240 \nu^{11} + 7515 \nu^{10} + 6627 \nu^{9} + 27224 \nu^{8} + 17644 \nu^{7} + 46681 \nu^{6} + 21141 \nu^{5} + 36086 \nu^{4} + \cdots + 904 ) / 208$$ (4*v^15 + 45*v^14 + 113*v^13 + 956*v^12 + 1240*v^11 + 7515*v^10 + 6627*v^9 + 27224*v^8 + 17644*v^7 + 46681*v^6 + 21141*v^5 + 36086*v^4 + 8106*v^3 + 11444*v^2 - 108*v + 904) / 208 $$\beta_{3}$$ $$=$$ $$( - 11 \nu^{15} + 12 \nu^{14} - 288 \nu^{13} + 248 \nu^{12} - 2981 \nu^{11} + 1848 \nu^{10} - 15504 \nu^{9} + 5932 \nu^{8} - 42411 \nu^{7} + 7120 \nu^{6} - 57946 \nu^{5} - 472 \nu^{4} + \cdots - 1312 ) / 208$$ (-11*v^15 + 12*v^14 - 288*v^13 + 248*v^12 - 2981*v^11 + 1848*v^10 - 15504*v^9 + 5932*v^8 - 42411*v^7 + 7120*v^6 - 57946*v^5 - 472*v^4 - 33556*v^3 - 4568*v^2 - 5488*v - 1312) / 208 $$\beta_{4}$$ $$=$$ $$( 11 \nu^{15} + 12 \nu^{14} + 288 \nu^{13} + 248 \nu^{12} + 2981 \nu^{11} + 1848 \nu^{10} + 15504 \nu^{9} + 5932 \nu^{8} + 42411 \nu^{7} + 7120 \nu^{6} + 57946 \nu^{5} - 472 \nu^{4} + \cdots - 1312 ) / 208$$ (11*v^15 + 12*v^14 + 288*v^13 + 248*v^12 + 2981*v^11 + 1848*v^10 + 15504*v^9 + 5932*v^8 + 42411*v^7 + 7120*v^6 + 57946*v^5 - 472*v^4 + 33556*v^3 - 4568*v^2 + 5488*v - 1312) / 208 $$\beta_{5}$$ $$=$$ $$( 45 \nu^{14} + 956 \nu^{12} + 7515 \nu^{10} + 27224 \nu^{8} + 46681 \nu^{6} + 36086 \nu^{4} + 11548 \nu^{2} + 104 \nu + 1216 ) / 104$$ (45*v^14 + 956*v^12 + 7515*v^10 + 27224*v^8 + 46681*v^6 + 36086*v^4 + 11548*v^2 + 104*v + 1216) / 104 $$\beta_{6}$$ $$=$$ $$( 45\nu^{14} + 956\nu^{12} + 7515\nu^{10} + 27224\nu^{8} + 46681\nu^{6} + 36086\nu^{4} + 11444\nu^{2} + 904 ) / 104$$ (45*v^14 + 956*v^12 + 7515*v^10 + 27224*v^8 + 46681*v^6 + 36086*v^4 + 11444*v^2 + 904) / 104 $$\beta_{7}$$ $$=$$ $$( -\nu^{15} - 22\nu^{13} - 183\nu^{11} - 730\nu^{9} - 1485\nu^{7} - 1552\nu^{5} - 812\nu^{3} - 192\nu ) / 8$$ (-v^15 - 22*v^13 - 183*v^11 - 730*v^9 - 1485*v^7 - 1552*v^5 - 812*v^3 - 192*v) / 8 $$\beta_{8}$$ $$=$$ $$( 39 \nu^{15} - 34 \nu^{14} + 806 \nu^{13} - 720 \nu^{12} + 6045 \nu^{11} - 5626 \nu^{10} + 20046 \nu^{9} - 20144 \nu^{8} + 28379 \nu^{7} - 33754 \nu^{6} + 12896 \nu^{5} - 25096 \nu^{4} + \cdots - 928 ) / 208$$ (39*v^15 - 34*v^14 + 806*v^13 - 720*v^12 + 6045*v^11 - 5626*v^10 + 20046*v^9 - 20144*v^8 + 28379*v^7 - 33754*v^6 + 12896*v^5 - 25096*v^4 - 832*v^3 - 7944*v^2 - 520*v - 928) / 208 $$\beta_{9}$$ $$=$$ $$( - 76 \nu^{15} + 39 \nu^{14} - 1627 \nu^{13} + 806 \nu^{12} - 12952 \nu^{11} + 6045 \nu^{10} - 47965 \nu^{9} + 20046 \nu^{8} - 85636 \nu^{7} + 28379 \nu^{6} - 71271 \nu^{5} + 12896 \nu^{4} + \cdots - 624 ) / 208$$ (-76*v^15 + 39*v^14 - 1627*v^13 + 806*v^12 - 12952*v^11 + 6045*v^10 - 47965*v^9 + 20046*v^8 - 85636*v^7 + 28379*v^6 - 71271*v^5 + 12896*v^4 - 25626*v^3 - 832*v^2 - 3044*v - 624) / 208 $$\beta_{10}$$ $$=$$ $$( 76 \nu^{15} + 39 \nu^{14} + 1627 \nu^{13} + 806 \nu^{12} + 12952 \nu^{11} + 6045 \nu^{10} + 47965 \nu^{9} + 20046 \nu^{8} + 85636 \nu^{7} + 28379 \nu^{6} + 71271 \nu^{5} + 12896 \nu^{4} + \cdots - 624 ) / 208$$ (76*v^15 + 39*v^14 + 1627*v^13 + 806*v^12 + 12952*v^11 + 6045*v^10 + 47965*v^9 + 20046*v^8 + 85636*v^7 + 28379*v^6 + 71271*v^5 + 12896*v^4 + 25626*v^3 - 832*v^2 + 3044*v - 624) / 208 $$\beta_{11}$$ $$=$$ $$( -76\nu^{14} - 1627\nu^{12} - 12952\nu^{10} - 47965\nu^{8} - 85636\nu^{6} - 71271\nu^{4} - 25626\nu^{2} - 3044 ) / 52$$ (-76*v^14 - 1627*v^12 - 12952*v^10 - 47965*v^8 - 85636*v^6 - 71271*v^4 - 25626*v^2 - 3044) / 52 $$\beta_{12}$$ $$=$$ $$( - 107 \nu^{15} - 15 \nu^{14} - 2298 \nu^{13} - 284 \nu^{12} - 18389 \nu^{11} - 1777 \nu^{10} - 68706 \nu^{9} - 3476 \nu^{8} - 124591 \nu^{7} + 3801 \nu^{6} - 106456 \nu^{5} + \cdots + 2160 ) / 208$$ (-107*v^15 - 15*v^14 - 2298*v^13 - 284*v^12 - 18389*v^11 - 1777*v^10 - 68706*v^9 - 3476*v^8 - 124591*v^7 + 3801*v^6 - 106456*v^5 + 17698*v^4 - 39704*v^3 + 13380*v^2 - 4768*v + 2160) / 208 $$\beta_{13}$$ $$=$$ $$( - 81 \nu^{15} + 119 \nu^{14} - 1778 \nu^{13} + 2546 \nu^{12} - 14723 \nu^{11} + 20237 \nu^{10} - 58150 \nu^{9} + 74638 \nu^{8} - 115517 \nu^{7} + 131711 \nu^{6} - 113736 \nu^{5} + \cdots + 3768 ) / 208$$ (-81*v^15 + 119*v^14 - 1778*v^13 + 2546*v^12 - 14723*v^11 + 20237*v^10 - 58150*v^9 + 74638*v^8 - 115517*v^7 + 131711*v^6 - 113736*v^5 + 105984*v^4 - 51092*v^3 + 35240*v^2 - 7784*v + 3768) / 208 $$\beta_{14}$$ $$=$$ $$( 122 \nu^{15} - 45 \nu^{14} + 2569 \nu^{13} - 956 \nu^{12} + 19906 \nu^{11} - 7515 \nu^{10} + 70323 \nu^{9} - 27224 \nu^{8} + 115018 \nu^{7} - 46681 \nu^{6} + 81049 \nu^{5} - 36086 \nu^{4} + \cdots - 904 ) / 208$$ (122*v^15 - 45*v^14 + 2569*v^13 - 956*v^12 + 19906*v^11 - 7515*v^10 + 70323*v^9 - 27224*v^8 + 115018*v^7 - 46681*v^6 + 81049*v^5 - 36086*v^4 + 22034*v^3 - 11444*v^2 + 1620*v - 904) / 208 $$\beta_{15}$$ $$=$$ $$( 183 \nu^{15} - 178 \nu^{14} + 3925 \nu^{13} - 3774 \nu^{12} + 31341 \nu^{11} - 29570 \nu^{10} + 116671 \nu^{9} - 106486 \nu^{8} + 210227 \nu^{7} - 180398 \nu^{6} + 177727 \nu^{5} + \cdots - 4320 ) / 208$$ (183*v^15 - 178*v^14 + 3925*v^13 - 3774*v^12 + 31341*v^11 - 29570*v^10 + 116671*v^9 - 106486*v^8 + 210227*v^7 - 180398*v^6 + 177727*v^5 - 135938*v^4 + 65330*v^3 - 42100*v^2 + 7812*v - 4320) / 208
 $$\nu$$ $$=$$ $$( \beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{2} ) / 2$$ (b14 - b10 + b9 - b7 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{14} + \beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{2} - 6 ) / 2$$ (-b14 + b10 - b9 + b7 - 2*b6 + 2*b5 - b2 - 6) / 2 $$\nu^{3}$$ $$=$$ $$- \beta_{15} - 4 \beta_{14} + \beta_{12} + \beta_{11} + 5 \beta_{10} - 6 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} - 4 \beta_{2} - 2 \beta _1 + 2$$ -b15 - 4*b14 + b12 + b11 + 5*b10 - 6*b9 + 3*b7 + 2*b5 - 4*b2 - 2*b1 + 2 $$\nu^{4}$$ $$=$$ $$( 2 \beta_{15} + 13 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 11 \beta_{10} + 17 \beta_{9} - 4 \beta_{8} - 15 \beta_{7} + 14 \beta_{6} - 26 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 13 \beta_{2} + 30 ) / 2$$ (2*b15 + 13*b14 + 4*b13 - 2*b12 - 2*b11 - 11*b10 + 17*b9 - 4*b8 - 15*b7 + 14*b6 - 26*b5 - 2*b4 - 2*b3 + 13*b2 + 30) / 2 $$\nu^{5}$$ $$=$$ $$( 24 \beta_{15} + 59 \beta_{14} + 8 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} - 83 \beta_{10} + 107 \beta_{9} - 51 \beta_{7} + 2 \beta_{6} - 48 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 63 \beta_{2} + 36 \beta _1 - 38 ) / 2$$ (24*b15 + 59*b14 + 8*b13 - 24*b12 - 20*b11 - 83*b10 + 107*b9 - 51*b7 + 2*b6 - 48*b5 - 4*b4 - 4*b3 + 63*b2 + 36*b1 - 38) / 2 $$\nu^{6}$$ $$=$$ $$- 15 \beta_{15} - 59 \beta_{14} - 24 \beta_{13} + 9 \beta_{12} + 12 \beta_{11} + 46 \beta_{10} - 87 \beta_{9} + 24 \beta_{8} + 71 \beta_{7} - 50 \beta_{6} + 118 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} - 59 \beta_{2} - 90$$ -15*b15 - 59*b14 - 24*b13 + 9*b12 + 12*b11 + 46*b10 - 87*b9 + 24*b8 + 71*b7 - 50*b6 + 118*b5 + 11*b4 + 11*b3 - 59*b2 - 90 $$\nu^{7}$$ $$=$$ $$( - 222 \beta_{15} - 435 \beta_{14} - 120 \beta_{13} + 222 \beta_{12} + 162 \beta_{11} + 649 \beta_{10} - 871 \beta_{9} + 425 \beta_{7} - 30 \beta_{6} + 444 \beta_{5} + 56 \beta_{4} + 64 \beta_{3} - 495 \beta_{2} - 296 \beta _1 + 310 ) / 2$$ (-222*b15 - 435*b14 - 120*b13 + 222*b12 + 162*b11 + 649*b10 - 871*b9 + 425*b7 - 30*b6 + 444*b5 + 56*b4 + 64*b3 - 495*b2 - 296*b1 + 310) / 2 $$\nu^{8}$$ $$=$$ $$( 324 \beta_{15} + 991 \beta_{14} + 468 \beta_{13} - 144 \beta_{12} - 222 \beta_{11} - 739 \beta_{10} + 1567 \beta_{9} - 468 \beta_{8} - 1225 \beta_{7} + 750 \beta_{6} - 1982 \beta_{5} - 208 \beta_{4} - 208 \beta_{3} + \cdots + 1178 ) / 2$$ (324*b15 + 991*b14 + 468*b13 - 144*b12 - 222*b11 - 739*b10 + 1567*b9 - 468*b8 - 1225*b7 + 750*b6 - 1982*b5 - 208*b4 - 208*b3 + 991*b2 + 1178) / 2 $$\nu^{9}$$ $$=$$ $$941 \beta_{15} + 1615 \beta_{14} + 640 \beta_{13} - 941 \beta_{12} - 621 \beta_{11} - 2486 \beta_{10} + 3427 \beta_{9} - 1722 \beta_{7} + 158 \beta_{6} - 1882 \beta_{5} - 288 \beta_{4} - 352 \beta_{3} + 1939 \beta_{2} + \cdots - 1230$$ 941*b15 + 1615*b14 + 640*b13 - 941*b12 - 621*b11 - 2486*b10 + 3427*b9 - 1722*b7 + 158*b6 - 1882*b5 - 288*b4 - 352*b3 + 1939*b2 + 1218*b1 - 1230 $$\nu^{10}$$ $$=$$ $$( - 3082 \beta_{15} - 8089 \beta_{14} - 4236 \beta_{13} + 1154 \beta_{12} + 1882 \beta_{11} + 5891 \beta_{10} - 13369 \beta_{9} + 4236 \beta_{8} + 10207 \beta_{7} - 5778 \beta_{6} + 16178 \beta_{5} + \cdots - 8118 ) / 2$$ (-3082*b15 - 8089*b14 - 4236*b13 + 1154*b12 + 1882*b11 + 5891*b10 - 13369*b9 + 4236*b8 + 10207*b7 - 5778*b6 + 16178*b5 + 1858*b4 + 1858*b3 - 8089*b2 - 8118) / 2 $$\nu^{11}$$ $$=$$ $$( - 15364 \beta_{15} - 24169 \beta_{14} - 11968 \beta_{13} + 15364 \beta_{12} + 9380 \beta_{11} + 37877 \beta_{10} - 53241 \beta_{9} + 27457 \beta_{7} - 2902 \beta_{6} + 30728 \beta_{5} + 5280 \beta_{4} + \cdots + 19394 ) / 2$$ (-15364*b15 - 24169*b14 - 11968*b13 + 15364*b12 + 9380*b11 + 37877*b10 - 53241*b9 + 27457*b7 - 2902*b6 + 30728*b5 + 5280*b4 + 6688*b3 - 30333*b2 - 20028*b1 + 19394) / 2 $$\nu^{12}$$ $$=$$ $$13721 \beta_{15} + 32540 \beta_{14} + 18376 \beta_{13} - 4655 \beta_{12} - 7682 \beta_{11} - 23407 \beta_{10} + 55394 \beta_{9} - 18376 \beta_{8} - 41728 \beta_{7} + 22526 \beta_{6} - 65080 \beta_{5} + \cdots + 28932$$ 13721*b15 + 32540*b14 + 18376*b13 - 4655*b12 - 7682*b11 - 23407*b10 + 55394*b9 - 18376*b8 - 41728*b7 + 22526*b6 - 65080*b5 - 7999*b4 - 7999*b3 + 32540*b2 + 28932 $$\nu^{13}$$ $$=$$ $$( 123118 \beta_{15} + 182167 \beta_{14} + 104792 \beta_{13} - 123118 \beta_{12} - 70722 \beta_{11} - 288649 \beta_{10} + 411767 \beta_{9} - 216953 \beta_{7} + 24954 \beta_{6} - 246236 \beta_{5} + \cdots - 152526 ) / 2$$ (123118*b15 + 182167*b14 + 104792*b13 - 123118*b12 - 70722*b11 - 288649*b10 + 411767*b9 - 216953*b7 + 24954*b6 - 246236*b5 - 45708*b4 - 59084*b3 + 237051*b2 + 163608*b1 - 152526) / 2 $$\nu^{14}$$ $$=$$ $$( - 234792 \beta_{15} - 519021 \beta_{14} - 309908 \beta_{13} + 75116 \beta_{12} + 123118 \beta_{11} + 370949 \beta_{10} - 901885 \beta_{9} + 309908 \beta_{8} + 673975 \beta_{7} + \cdots - 422094 ) / 2$$ (-234792*b15 - 519021*b14 - 309908*b13 + 75116*b12 + 123118*b11 + 370949*b10 - 901885*b9 + 309908*b8 + 673975*b7 - 352890*b6 + 1038042*b5 + 134200*b4 + 134200*b3 - 519021*b2 - 422094) / 2 $$\nu^{15}$$ $$=$$ $$- 488399 \beta_{15} - 690968 \beta_{14} - 441948 \beta_{13} + 488399 \beta_{12} + 267425 \beta_{11} + 1102735 \beta_{10} - 1591134 \beta_{9} + 852893 \beta_{7} - 103578 \beta_{6} + \cdots + 598824$$ -488399*b15 - 690968*b14 - 441948*b13 + 488399*b12 + 267425*b11 + 1102735*b10 - 1591134*b9 + 852893*b7 - 103578*b6 + 976798*b5 + 191432*b4 + 250516*b3 - 925760*b2 - 662798*b1 + 598824

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$261$$ $$391$$ $$417$$ $$\chi(n)$$ $$1 - \beta_{1}$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 − 0.551543i 0.956612i − 0.432338i 0.897932i 1.44614i 2.44974i − 2.79253i − 1.97402i 0.551543i − 0.956612i 0.432338i − 0.897932i − 1.44614i − 2.44974i 2.79253i 1.97402i
0 −1.62367 + 2.81228i 0 1.00000i 0 4.00640 2.31309i 0 −3.77262 6.53437i 0
121.2 0 −1.52075 + 2.63402i 0 1.00000i 0 −2.67664 + 1.54536i 0 −3.12538 5.41331i 0
121.3 0 −1.19037 + 2.06179i 0 1.00000i 0 3.14022 1.81301i 0 −1.33398 2.31052i 0
121.4 0 −0.647893 + 1.12218i 0 1.00000i 0 −1.06291 + 0.613670i 0 0.660468 + 1.14396i 0
121.5 0 0.268727 0.465448i 0 1.00000i 0 −0.331682 + 0.191497i 0 1.35557 + 2.34792i 0
121.6 0 0.275748 0.477609i 0 1.00000i 0 −1.57306 + 0.908206i 0 1.34793 + 2.33468i 0
121.7 0 1.00284 1.73697i 0 1.00000i 0 1.48627 0.858099i 0 −0.511370 0.885719i 0
121.8 0 1.43538 2.48615i 0 1.00000i 0 0.0113994 0.00658143i 0 −2.62062 4.53905i 0
361.1 0 −1.62367 2.81228i 0 1.00000i 0 4.00640 + 2.31309i 0 −3.77262 + 6.53437i 0
361.2 0 −1.52075 2.63402i 0 1.00000i 0 −2.67664 1.54536i 0 −3.12538 + 5.41331i 0
361.3 0 −1.19037 2.06179i 0 1.00000i 0 3.14022 + 1.81301i 0 −1.33398 + 2.31052i 0
361.4 0 −0.647893 1.12218i 0 1.00000i 0 −1.06291 0.613670i 0 0.660468 1.14396i 0
361.5 0 0.268727 + 0.465448i 0 1.00000i 0 −0.331682 0.191497i 0 1.35557 2.34792i 0
361.6 0 0.275748 + 0.477609i 0 1.00000i 0 −1.57306 0.908206i 0 1.34793 2.33468i 0
361.7 0 1.00284 + 1.73697i 0 1.00000i 0 1.48627 + 0.858099i 0 −0.511370 + 0.885719i 0
361.8 0 1.43538 + 2.48615i 0 1.00000i 0 0.0113994 + 0.00658143i 0 −2.62062 + 4.53905i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.bu.b 16
4.b odd 2 1 1040.2.da.f 16
13.e even 6 1 inner 520.2.bu.b 16
13.f odd 12 1 6760.2.a.bk 8
13.f odd 12 1 6760.2.a.bl 8
52.i odd 6 1 1040.2.da.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.bu.b 16 1.a even 1 1 trivial
520.2.bu.b 16 13.e even 6 1 inner
1040.2.da.f 16 4.b odd 2 1
1040.2.da.f 16 52.i odd 6 1
6760.2.a.bk 8 13.f odd 12 1
6760.2.a.bl 8 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + 4 T_{3}^{15} + 28 T_{3}^{14} + 60 T_{3}^{13} + 327 T_{3}^{12} + 574 T_{3}^{11} + 2508 T_{3}^{10} + 2550 T_{3}^{9} + 9913 T_{3}^{8} + 5782 T_{3}^{7} + 28018 T_{3}^{6} + 2252 T_{3}^{5} + 25368 T_{3}^{4} - 16680 T_{3}^{3} + \cdots + 2704$$ acting on $$S_{2}^{\mathrm{new}}(520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} + 4 T^{15} + 28 T^{14} + \cdots + 2704$$
$5$ $$(T^{2} + 1)^{8}$$
$7$ $$T^{16} - 6 T^{15} - 8 T^{14} + 120 T^{13} + \cdots + 1$$
$11$ $$T^{16} + 6 T^{15} - 64 T^{14} + \cdots + 38800441$$
$13$ $$T^{16} + 2 T^{15} + 38 T^{14} + \cdots + 815730721$$
$17$ $$T^{16} - 4 T^{15} + 104 T^{14} + \cdots + 63425296$$
$19$ $$T^{16} - 30 T^{15} + 388 T^{14} + \cdots + 6235009$$
$23$ $$T^{16} - 6 T^{15} + \cdots + 1235663104$$
$29$ $$T^{16} + 16 T^{15} + 202 T^{14} + \cdots + 4096$$
$31$ $$T^{16} + 288 T^{14} + \cdots + 38554107904$$
$37$ $$T^{16} + 24 T^{15} + 164 T^{14} + \cdots + 21557449$$
$41$ $$T^{16} + 24 T^{15} + \cdots + 1386221824$$
$43$ $$T^{16} - 6 T^{15} + \cdots + 143344017664$$
$47$ $$T^{16} + 388 T^{14} + \cdots + 1307674583296$$
$53$ $$(T^{8} - 2 T^{7} - 230 T^{6} + \cdots - 663344)^{2}$$
$59$ $$T^{16} - 12 T^{15} + \cdots + 3760434585856$$
$61$ $$T^{16} + 2 T^{15} + \cdots + 65554433296$$
$67$ $$T^{16} + 6 T^{15} + \cdots + 676131241984$$
$71$ $$T^{16} - 72 T^{15} + \cdots + 667763540224$$
$73$ $$T^{16} + 544 T^{14} + \cdots + 129784385536$$
$79$ $$(T^{8} - 18 T^{7} - 98 T^{6} + \cdots + 659776)^{2}$$
$83$ $$T^{16} + 176 T^{14} + \cdots + 89718784$$
$89$ $$T^{16} - 24 T^{15} + \cdots + 5504781135529$$
$97$ $$T^{16} - 60 T^{15} + \cdots + 2668426795024$$