# Properties

 Label 740.2.d.a Level $740$ Weight $2$ Character orbit 740.d Analytic conductor $5.909$ Analytic rank $0$ Dimension $18$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(149,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("740.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.d (of order $$2$$, degree $$1$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144$$ x^18 + 26*x^16 + 283*x^14 + 1674*x^12 + 5841*x^10 + 12196*x^8 + 14736*x^6 + 9408*x^4 + 2592*x^2 + 144 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{13} q^{3} + \beta_{10} q^{5} - \beta_{8} q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 2) q^{9}+O(q^{10})$$ q - b13 * q^3 + b10 * q^5 - b8 * q^7 + (-b11 + b10 + b7 - b6 - 2) * q^9 $$q - \beta_{13} q^{3} + \beta_{10} q^{5} - \beta_{8} q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 2) q^{9}+ \cdots + (\beta_{17} + \beta_{16} + \beta_{11} + \cdots + 2) q^{99}+O(q^{100})$$ q - b13 * q^3 + b10 * q^5 - b8 * q^7 + (-b11 + b10 + b7 - b6 - 2) * q^9 + (-b12 + b3 - b2) * q^11 + (-b15 + b11 + b10 + b8 - b5 + b1) * q^13 + (b17 + b16 + b14 + b12 + b11 + b5 - b3) * q^15 + (-b15 - b14 - b13 + b8 + 2*b5 - b4) * q^17 + (b9 + b6 - b3 + b2 + 1) * q^19 + (b12 - b11 + b10 + b9 + b7 + b3 - 2) * q^21 + (b17 - b16 - b15 + b14 - 2*b5 - 2*b1) * q^23 + (-b16 - b11 - b8 + b7 + b4 - b3 - b2 - b1 - 1) * q^25 + (2*b17 - 2*b16 + b15 + 2*b13 - b11 - b10) * q^27 + (2*b17 + 2*b16 + b11 - b10 - 3*b9 - b7 + b6 + b3 + 2*b2 + 2) * q^29 + (b17 + b16 + b11 - b10 - b7 + b3 + 1) * q^31 + (b17 - b16 + b15 + b13 - b8 - 4*b5 + 2*b4) * q^33 + (2*b17 + b13 - b12 + 2*b11 - b10 - b9 - b7 + b6 + b4 - b3 + b2 + b1 + 2) * q^35 + b5 * q^37 + (b17 + b16 + b11 - b10 + b9 - b7 - b6 - 2*b3 - b2) * q^39 + (b17 + b16 + b11 - b10 - b9 + b6 + b2 + 1) * q^41 + (b17 - b16 + b15 - 2*b14 + b13 - 2*b11 - 2*b10 - b5 - b4 + b1) * q^43 + (2*b15 - 2*b14 - b12 - b11 - 3*b10 + b8 - b7 + 4*b5 - 2*b4 - b3 - b2 + b1 + 1) * q^45 + (b17 - b16 + 2*b15 - 2*b14 - b11 - b10 + b8 + 2*b5 + 2*b1) * q^47 + (-2*b17 - 2*b16 - b11 + b10 + 2*b9 + b7 + b6) * q^49 + (b17 + b16 - 2*b11 + 2*b10 - 2*b6 + 5*b3 - b2 - 4) * q^51 + (-b17 + b16 + b15 - b13 + b8 - 2*b5 - 2*b4) * q^53 + (-b15 + b14 - b13 - b12 + b10 - b7 - 2*b6 + 2*b5 + b4 + b3 - b2 - b1 - 1) * q^55 + (-b15 + b14 - b13 + b8 + 2*b5 - b4 - 2*b1) * q^57 + (3*b17 + 3*b16 + 2*b11 - 2*b10 - b9 - 2*b7 + b6 - 2*b3 + 2*b2 + 3) * q^59 + (b11 - b10 - b9 - b7 + b6 + b3 + 2*b2 + 2) * q^61 + (b17 - b16 + b15 + 2*b14 + 2*b13 - b8 - 4*b5) * q^63 + (-3*b17 - b15 - b13 - 2*b11 + 3*b10 + 2*b9 + 2*b7 - b6 + 2*b5 - b4 - b3 - b2 + b1 - 3) * q^65 + (-b17 + b16 + b15 + 2*b14 - 2*b13 - b8 - 2*b1) * q^67 + (2*b17 + 2*b16 + 3*b11 - 3*b10 - b9 - 3*b7 - b6 - b3 - 2*b2 + 2) * q^69 + (b17 + b16 + 3*b12 - b11 + b10 + 2*b7 + b3 - b2 - 4) * q^71 + (-b17 + b16 - b15 + 2*b14 + b13 + 2*b11 + 2*b10 - 3*b8 - 6*b5 + 2*b4) * q^73 + (2*b17 - b16 - 2*b14 + b13 + 2*b12 - 3*b11 + b8 + b7 + 4*b5 - b4 + b3 + b2 + b1 - 2) * q^75 + (-b15 + 2*b14 + b13 + b11 + b10 - b8 + 2*b5 + 2*b4) * q^77 + (-3*b17 - 3*b16 - 4*b11 + 4*b10 + 2*b9 + 3*b7 - 4*b6 + b3 - 4*b2 - 7) * q^79 + (-2*b12 + 3*b11 - 3*b10 - b9 - 3*b7 + b2 + 5) * q^81 + (2*b17 - 2*b16 + 2*b15 - 4*b14 + b13 - 2*b11 - 2*b10 + 2*b8 - 2*b4 + 2*b1) * q^83 + (-2*b17 - b16 - 2*b15 + 2*b14 - b12 + 2*b11 + b10 + b8 + b6 + 2*b5 - b4 - 2*b3 - b1 + 1) * q^85 + (-2*b15 + 3*b14 + b13 + 3*b11 + 3*b10 - 3*b5 + b4 + b1) * q^87 + (-2*b17 - 2*b16 + 2*b12 - 2*b11 + 2*b10 + b9 + 3*b7 + b6 + b3 - 2) * q^89 + (b17 + b16 + 3*b11 - 3*b10 - b9 - b7 + b6 - 2*b3 + b2 + 4) * q^91 + (-b17 + b16 + 2*b15 - b13 - b11 - b10 - b8 - 4*b5 + b4) * q^93 + (-b17 - 2*b16 + b15 + b12 - b11 + 3*b9 + b7 + b6 + 2*b5 + b4 - 2*b3 + b2 - b1 + 1) * q^95 + (-b17 + b16 - b15 + b14 + 2*b11 + 2*b10 - 2*b8 + 4*b4) * q^97 + (b17 + b16 + b11 - b10 + 2*b6 - 4*b3 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 2 q^{5} - 18 q^{9}+O(q^{10})$$ 18 * q + 2 * q^5 - 18 * q^9 $$18 q + 2 q^{5} - 18 q^{9} + 6 q^{15} + 4 q^{19} - 16 q^{21} - 2 q^{25} - 4 q^{29} + 8 q^{31} - 2 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{45} + 6 q^{49} - 40 q^{51} - 6 q^{55} + 8 q^{59} - 12 q^{65} + 28 q^{69} - 24 q^{71} - 16 q^{75} - 24 q^{79} + 34 q^{81} + 36 q^{91} + 24 q^{95} + 16 q^{99}+O(q^{100})$$ 18 * q + 2 * q^5 - 18 * q^9 + 6 * q^15 + 4 * q^19 - 16 * q^21 - 2 * q^25 - 4 * q^29 + 8 * q^31 - 2 * q^35 + 8 * q^39 - 4 * q^41 + 8 * q^45 + 6 * q^49 - 40 * q^51 - 6 * q^55 + 8 * q^59 - 12 * q^65 + 28 * q^69 - 24 * q^71 - 16 * q^75 - 24 * q^79 + 34 * q^81 + 36 * q^91 + 24 * q^95 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{17} - 20\nu^{15} - 148\nu^{13} - 459\nu^{11} - 273\nu^{9} + 1475\nu^{7} + 2595\nu^{5} + 714\nu^{3} - 252\nu ) / 36$$ (-v^17 - 20*v^15 - 148*v^13 - 459*v^11 - 273*v^9 + 1475*v^7 + 2595*v^5 + 714*v^3 - 252*v) / 36 $$\beta_{2}$$ $$=$$ $$( \nu^{16} + 20\nu^{14} + 151\nu^{12} + 510\nu^{10} + 591\nu^{8} - 620\nu^{6} - 1800\nu^{4} - 912\nu^{2} - 48 ) / 24$$ (v^16 + 20*v^14 + 151*v^12 + 510*v^10 + 591*v^8 - 620*v^6 - 1800*v^4 - 912*v^2 - 48) / 24 $$\beta_{3}$$ $$=$$ $$( \nu^{16} + 20\nu^{14} + 151\nu^{12} + 510\nu^{10} + 579\nu^{8} - 752\nu^{6} - 2244\nu^{4} - 1392\nu^{2} - 144 ) / 24$$ (v^16 + 20*v^14 + 151*v^12 + 510*v^10 + 579*v^8 - 752*v^6 - 2244*v^4 - 1392*v^2 - 144) / 24 $$\beta_{4}$$ $$=$$ $$( \nu^{17} + 26 \nu^{15} + 277 \nu^{13} + 1554 \nu^{11} + 4917 \nu^{9} + 8794 \nu^{7} + 8724 \nu^{5} + \cdots + 1368 \nu ) / 72$$ (v^17 + 26*v^15 + 277*v^13 + 1554*v^11 + 4917*v^9 + 8794*v^7 + 8724*v^5 + 4920*v^3 + 1368*v) / 72 $$\beta_{5}$$ $$=$$ $$( \nu^{17} + 26 \nu^{15} + 277 \nu^{13} + 1554 \nu^{11} + 4899 \nu^{9} + 8524 \nu^{7} + 7338 \nu^{5} + \cdots - 144 \nu ) / 72$$ (v^17 + 26*v^15 + 277*v^13 + 1554*v^11 + 4899*v^9 + 8524*v^7 + 7338*v^5 + 2184*v^3 - 144*v) / 72 $$\beta_{6}$$ $$=$$ $$( \nu^{16} + 23\nu^{14} + 214\nu^{12} + 1032\nu^{10} + 2742\nu^{8} + 3931\nu^{6} + 2778\nu^{4} + 852\nu^{2} + 96 ) / 12$$ (v^16 + 23*v^14 + 214*v^12 + 1032*v^10 + 2742*v^8 + 3931*v^6 + 2778*v^4 + 852*v^2 + 96) / 12 $$\beta_{7}$$ $$=$$ $$( \nu^{16} + 23\nu^{14} + 214\nu^{12} + 1035\nu^{10} + 2787\nu^{8} + 4162\nu^{6} + 3234\nu^{4} + 1092\nu^{2} + 60 ) / 12$$ (v^16 + 23*v^14 + 214*v^12 + 1035*v^10 + 2787*v^8 + 4162*v^6 + 3234*v^4 + 1092*v^2 + 60) / 12 $$\beta_{8}$$ $$=$$ $$( 2 \nu^{17} + 43 \nu^{15} + 365 \nu^{13} + 1542 \nu^{11} + 3327 \nu^{9} + 3251 \nu^{7} + 798 \nu^{5} + \cdots - 72 \nu ) / 36$$ (2*v^17 + 43*v^15 + 365*v^13 + 1542*v^11 + 3327*v^9 + 3251*v^7 + 798*v^5 - 258*v^3 - 72*v) / 36 $$\beta_{9}$$ $$=$$ $$( \nu^{16} + 22\nu^{14} + 193\nu^{12} + 858\nu^{10} + 2025\nu^{8} + 2414\nu^{6} + 1244\nu^{4} + 208\nu^{2} - 8 ) / 8$$ (v^16 + 22*v^14 + 193*v^12 + 858*v^10 + 2025*v^8 + 2414*v^6 + 1244*v^4 + 208*v^2 - 8) / 8 $$\beta_{10}$$ $$=$$ $$( \nu^{17} + 9 \nu^{16} + 20 \nu^{15} + 207 \nu^{14} + 148 \nu^{13} + 1935 \nu^{12} + 459 \nu^{11} + \cdots + 720 ) / 72$$ (v^17 + 9*v^16 + 20*v^15 + 207*v^14 + 148*v^13 + 1935*v^12 + 459*v^11 + 9450*v^10 + 264*v^9 + 25794*v^8 - 1592*v^7 + 38988*v^6 - 3090*v^5 + 30204*v^4 - 1416*v^3 + 9864*v^2 + 144*v + 720) / 72 $$\beta_{11}$$ $$=$$ $$( \nu^{17} - 9 \nu^{16} + 20 \nu^{15} - 207 \nu^{14} + 148 \nu^{13} - 1935 \nu^{12} + 459 \nu^{11} + \cdots - 720 ) / 72$$ (v^17 - 9*v^16 + 20*v^15 - 207*v^14 + 148*v^13 - 1935*v^12 + 459*v^11 - 9450*v^10 + 264*v^9 - 25794*v^8 - 1592*v^7 - 38988*v^6 - 3090*v^5 - 30204*v^4 - 1416*v^3 - 9864*v^2 + 144*v - 720) / 72 $$\beta_{12}$$ $$=$$ $$( -\nu^{16} - 23\nu^{14} - 214\nu^{12} - 1032\nu^{10} - 2745\nu^{8} - 3958\nu^{6} - 2829\nu^{4} - 804\nu^{2} - 24 ) / 6$$ (-v^16 - 23*v^14 - 214*v^12 - 1032*v^10 - 2745*v^8 - 3958*v^6 - 2829*v^4 - 804*v^2 - 24) / 6 $$\beta_{13}$$ $$=$$ $$( \nu^{17} + 24 \nu^{15} + 237 \nu^{13} + 1246 \nu^{11} + 3771 \nu^{9} + 6622 \nu^{7} + 6412 \nu^{5} + \cdots + 384 \nu ) / 24$$ (v^17 + 24*v^15 + 237*v^13 + 1246*v^11 + 3771*v^9 + 6622*v^7 + 6412*v^5 + 2940*v^3 + 384*v) / 24 $$\beta_{14}$$ $$=$$ $$( 7 \nu^{17} + 158 \nu^{15} + 1441 \nu^{13} + 6804 \nu^{11} + 17661 \nu^{9} + 24442 \nu^{7} + \cdots - 1152 \nu ) / 72$$ (7*v^17 + 158*v^15 + 1441*v^13 + 6804*v^11 + 17661*v^9 + 24442*v^7 + 15396*v^5 + 1824*v^3 - 1152*v) / 72 $$\beta_{15}$$ $$=$$ $$( 7 \nu^{17} + 158 \nu^{15} + 1441 \nu^{13} + 6822 \nu^{11} + 17967 \nu^{9} + 26368 \nu^{7} + \cdots + 1440 \nu ) / 72$$ (7*v^17 + 158*v^15 + 1441*v^13 + 6822*v^11 + 17967*v^9 + 26368*v^7 + 20832*v^5 + 8376*v^3 + 1440*v) / 72 $$\beta_{16}$$ $$=$$ $$( - 4 \nu^{17} + 9 \nu^{16} - 92 \nu^{15} + 207 \nu^{14} - 859 \nu^{13} + 1935 \nu^{12} - 4179 \nu^{11} + \cdots + 504 ) / 72$$ (-4*v^17 + 9*v^16 - 92*v^15 + 207*v^14 - 859*v^13 + 1935*v^12 - 4179*v^11 + 9450*v^10 - 11289*v^9 + 25794*v^8 - 16582*v^7 + 38988*v^6 - 11700*v^5 + 30204*v^4 - 2400*v^3 + 9792*v^2 + 432*v + 504) / 72 $$\beta_{17}$$ $$=$$ $$( 4 \nu^{17} + 9 \nu^{16} + 92 \nu^{15} + 207 \nu^{14} + 859 \nu^{13} + 1935 \nu^{12} + 4179 \nu^{11} + \cdots + 504 ) / 72$$ (4*v^17 + 9*v^16 + 92*v^15 + 207*v^14 + 859*v^13 + 1935*v^12 + 4179*v^11 + 9450*v^10 + 11289*v^9 + 25794*v^8 + 16582*v^7 + 38988*v^6 + 11700*v^5 + 30204*v^4 + 2400*v^3 + 9792*v^2 - 432*v + 504) / 72
 $$\nu$$ $$=$$ $$( -\beta_{17} + \beta_{16} + 2\beta_{15} + 2\beta_{14} - 2\beta_{13} - \beta_{11} - \beta_{10} - 4\beta_{8} - 2\beta_1 ) / 6$$ (-b17 + b16 + 2*b15 + 2*b14 - 2*b13 - b11 - b10 - 4*b8 - 2*b1) / 6 $$\nu^{2}$$ $$=$$ $$( -\beta_{17} - \beta_{16} - \beta_{11} + \beta_{10} - 6 ) / 2$$ (-b17 - b16 - b11 + b10 - 6) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{17} - \beta_{16} - \beta_{15} - 2\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + 3\beta_{8} + 2\beta_1$$ b17 - b16 - b15 - 2*b14 + b13 + b11 + b10 + 3*b8 + 2*b1 $$\nu^{4}$$ $$=$$ $$( 7\beta_{17} + 7\beta_{16} + 7\beta_{11} - 7\beta_{10} - 2\beta_{9} + 2\beta_{6} + 2\beta_{2} + 28 ) / 2$$ (7*b17 + 7*b16 + 7*b11 - 7*b10 - 2*b9 + 2*b6 + 2*b2 + 28) / 2 $$\nu^{5}$$ $$=$$ $$( - 11 \beta_{17} + 11 \beta_{16} + 6 \beta_{15} + 24 \beta_{14} - 6 \beta_{13} - 9 \beta_{11} + \cdots - 22 \beta_1 ) / 2$$ (-11*b17 + 11*b16 + 6*b15 + 24*b14 - 6*b13 - 9*b11 - 9*b10 - 32*b8 - 4*b5 + 2*b4 - 22*b1) / 2 $$\nu^{6}$$ $$=$$ $$- 21 \beta_{17} - 21 \beta_{16} + \beta_{12} - 21 \beta_{11} + 21 \beta_{10} + 10 \beta_{9} - 8 \beta_{6} + \cdots - 72$$ -21*b17 - 21*b16 + b12 - 21*b11 + 21*b10 + 10*b9 - 8*b6 - b3 - 9*b2 - 72 $$\nu^{7}$$ $$=$$ $$( 59 \beta_{17} - 59 \beta_{16} - 16 \beta_{15} - 140 \beta_{14} + 16 \beta_{13} + 41 \beta_{11} + \cdots + 122 \beta_1 ) / 2$$ (59*b17 - 59*b16 - 16*b15 - 140*b14 + 16*b13 + 41*b11 + 41*b10 + 178*b8 + 40*b5 - 18*b4 + 122*b1) / 2 $$\nu^{8}$$ $$=$$ $$( 243 \beta_{17} + 243 \beta_{16} - 22 \beta_{12} + 243 \beta_{11} - 243 \beta_{10} - 146 \beta_{9} + \cdots + 772 ) / 2$$ (243*b17 + 243*b16 - 22*b12 + 243*b11 - 243*b10 - 146*b9 + 102*b6 + 18*b3 + 128*b2 + 772) / 2 $$\nu^{9}$$ $$=$$ $$- 157 \beta_{17} + 157 \beta_{16} + 13 \beta_{15} + 402 \beta_{14} - 13 \beta_{13} - 99 \beta_{11} + \cdots - 344 \beta_1$$ -157*b17 + 157*b16 + 13*b15 + 402*b14 - 13*b13 - 99*b11 - 99*b10 - 503*b8 - 150*b5 + 62*b4 - 344*b1 $$\nu^{10}$$ $$=$$ $$( - 1395 \beta_{17} - 1395 \beta_{16} + 176 \beta_{12} - 1395 \beta_{11} + 1395 \beta_{10} + 954 \beta_{9} + \cdots - 4244 ) / 2$$ (-1395*b17 - 1395*b16 + 176*b12 - 1395*b11 + 1395*b10 + 954*b9 + 8*b7 - 610*b6 - 116*b3 - 838*b2 - 4244) / 2 $$\nu^{11}$$ $$=$$ $$( 1667 \beta_{17} - 1667 \beta_{16} + 98 \beta_{15} - 4584 \beta_{14} - 90 \beta_{13} + 1017 \beta_{11} + \cdots + 3926 \beta_1 ) / 2$$ (1667*b17 - 1667*b16 + 98*b15 - 4584*b14 - 90*b13 + 1017*b11 + 1017*b10 + 5728*b8 + 2028*b5 - 786*b4 + 3926*b1) / 2 $$\nu^{12}$$ $$=$$ $$4009 \beta_{17} + 4009 \beta_{16} - 621 \beta_{12} + 4005 \beta_{11} - 4005 \beta_{10} - 2966 \beta_{9} + \cdots + 11860$$ 4009*b17 + 4009*b16 - 621*b12 + 4005*b11 - 4005*b10 - 2966*b9 - 72*b7 + 1784*b6 + 329*b3 + 2637*b2 + 11860 $$\nu^{13}$$ $$=$$ $$( - 8843 \beta_{17} + 8843 \beta_{16} - 1528 \beta_{15} + 26052 \beta_{14} + 1392 \beta_{13} + \cdots - 22554 \beta_1 ) / 2$$ (-8843*b17 + 8843*b16 - 1528*b15 + 26052*b14 + 1392*b13 - 5505*b11 - 5505*b10 - 32738*b8 - 13064*b5 + 4818*b4 - 22554*b1) / 2 $$\nu^{14}$$ $$=$$ $$( - 46259 \beta_{17} - 46259 \beta_{16} + 8198 \beta_{12} - 46091 \beta_{11} + 46091 \beta_{10} + \cdots - 134036 ) / 2$$ (-46259*b17 - 46259*b16 + 8198*b12 - 46091*b11 + 46091*b10 + 35970*b9 + 1632*b7 - 20694*b6 - 3506*b3 - 32480*b2 - 134036) / 2 $$\nu^{15}$$ $$=$$ $$23445 \beta_{17} - 23445 \beta_{16} + 6331 \beta_{15} - 73934 \beta_{14} - 5599 \beta_{13} + \cdots + 65004 \beta_1$$ 23445*b17 - 23445*b16 + 6331*b15 - 73934*b14 - 5599*b13 + 15499*b11 + 15499*b10 + 93743*b8 + 41026*b5 - 14530*b4 + 65004*b1 $$\nu^{16}$$ $$=$$ $$( 267947 \beta_{17} + 267947 \beta_{16} - 51936 \beta_{12} + 265795 \beta_{11} - 265795 \beta_{10} + \cdots + 762932 ) / 2$$ (267947*b17 + 267947*b16 - 51936*b12 + 265795*b11 - 265795*b10 - 215122*b9 - 14976*b7 + 119610*b6 + 18044*b3 + 197446*b2 + 762932) / 2 $$\nu^{17}$$ $$=$$ $$( - 248475 \beta_{17} + 248475 \beta_{16} - 88802 \beta_{15} + 838984 \beta_{14} + 75978 \beta_{13} + \cdots - 750566 \beta_1 ) / 2$$ (-248475*b17 + 248475*b16 - 88802*b15 + 838984*b14 + 75978*b13 - 179337*b11 - 179337*b10 - 1074880*b8 - 507900*b5 + 173698*b4 - 750566*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 1.87047i − 0.807220i − 2.04568i 0.269568i 2.35298i − 0.981412i − 1.78123i − 2.40934i − 1.45428i 1.45428i 2.40934i 1.78123i 0.981412i − 2.35298i − 0.269568i 2.04568i 0.807220i 1.87047i
0 3.16661i 0 −1.81068 + 1.31203i 0 3.31356i 0 −7.02744 0
149.2 0 3.05716i 0 0.115303 2.23309i 0 0.448446i 0 −6.34624 0
149.3 0 2.45165i 0 0.873577 + 2.05836i 0 2.99321i 0 −3.01060 0
149.4 0 2.26639i 0 2.06173 + 0.865603i 0 0.375867i 0 −2.13650 0
149.5 0 1.51208i 0 −2.21246 + 0.324035i 0 1.74419i 0 0.713617 0
149.6 0 1.31739i 0 2.22789 0.191060i 0 3.90250i 0 1.26449 0
149.7 0 1.06683i 0 1.49239 1.66516i 0 0.712098i 0 1.86188 0
149.8 0 0.430822i 0 −0.672341 + 2.13259i 0 3.62709i 0 2.81439 0
149.9 0 0.365498i 0 −1.07541 1.96049i 0 2.78997i 0 2.86641 0
149.10 0 0.365498i 0 −1.07541 + 1.96049i 0 2.78997i 0 2.86641 0
149.11 0 0.430822i 0 −0.672341 2.13259i 0 3.62709i 0 2.81439 0
149.12 0 1.06683i 0 1.49239 + 1.66516i 0 0.712098i 0 1.86188 0
149.13 0 1.31739i 0 2.22789 + 0.191060i 0 3.90250i 0 1.26449 0
149.14 0 1.51208i 0 −2.21246 0.324035i 0 1.74419i 0 0.713617 0
149.15 0 2.26639i 0 2.06173 0.865603i 0 0.375867i 0 −2.13650 0
149.16 0 2.45165i 0 0.873577 2.05836i 0 2.99321i 0 −3.01060 0
149.17 0 3.05716i 0 0.115303 + 2.23309i 0 0.448446i 0 −6.34624 0
149.18 0 3.16661i 0 −1.81068 1.31203i 0 3.31356i 0 −7.02744 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.18 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.d.a 18
3.b odd 2 1 6660.2.f.c 18
5.b even 2 1 inner 740.2.d.a 18
5.c odd 4 1 3700.2.a.o 9
5.c odd 4 1 3700.2.a.p 9
15.d odd 2 1 6660.2.f.c 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.d.a 18 1.a even 1 1 trivial
740.2.d.a 18 5.b even 2 1 inner
3700.2.a.o 9 5.c odd 4 1
3700.2.a.p 9 5.c odd 4 1
6660.2.f.c 18 3.b odd 2 1
6660.2.f.c 18 15.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$T^{18} + 36 T^{16} + \cdots + 324$$
$5$ $$T^{18} - 2 T^{17} + \cdots + 1953125$$
$7$ $$T^{18} + 60 T^{16} + \cdots + 6724$$
$11$ $$(T^{9} - 50 T^{7} + \cdots + 1152)^{2}$$
$13$ $$T^{18} + 116 T^{16} + \cdots + 9935104$$
$17$ $$T^{18} + 176 T^{16} + \cdots + 4665600$$
$19$ $$(T^{9} - 2 T^{8} + \cdots - 1368)^{2}$$
$23$ $$T^{18} + \cdots + 122242535424$$
$29$ $$(T^{9} + 2 T^{8} + \cdots + 236160)^{2}$$
$31$ $$(T^{9} - 4 T^{8} - 60 T^{7} + \cdots - 8)^{2}$$
$37$ $$(T^{2} + 1)^{9}$$
$41$ $$(T^{9} + 2 T^{8} + \cdots - 6128)^{2}$$
$43$ $$T^{18} + \cdots + 46269730816$$
$47$ $$T^{18} + \cdots + 561621343396$$
$53$ $$T^{18} + 420 T^{16} + \cdots + 70829056$$
$59$ $$(T^{9} - 4 T^{8} + \cdots - 1829376)^{2}$$
$61$ $$(T^{9} - 126 T^{7} + \cdots + 350208)^{2}$$
$67$ $$T^{18} + \cdots + 20250428416$$
$71$ $$(T^{9} + 12 T^{8} + \cdots + 50596992)^{2}$$
$73$ $$T^{18} + \cdots + 894480807568384$$
$79$ $$(T^{9} + 12 T^{8} + \cdots + 23212328)^{2}$$
$83$ $$T^{18} + \cdots + 32497272900$$
$89$ $$(T^{9} - 432 T^{7} + \cdots - 20536704)^{2}$$
$97$ $$T^{18} + \cdots + 84\!\cdots\!36$$
show more
show less