# Properties

 Label 740.2.i.a Level $740$ Weight $2$ Character orbit 740.i Analytic conductor $5.909$ Analytic rank $0$ Dimension $14$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(121,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("740.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.i (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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} + 24x^{12} + 204x^{10} + 727x^{8} + 1008x^{6} + 426x^{4} + 64x^{2} + 3$$ x^14 + 24*x^12 + 204*x^10 + 727*x^8 + 1008*x^6 + 426*x^4 + 64*x^2 + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + (\beta_{10} - 1) q^{5} + ( - \beta_{10} + \beta_{6} + \beta_{5} + 1) q^{7} + ( - \beta_{13} + \beta_{11} + \cdots + \beta_{6}) q^{9}+O(q^{10})$$ q + b7 * q^3 + (b10 - 1) * q^5 + (-b10 + b6 + b5 + 1) * q^7 + (-b13 + b11 - 2*b10 + b9 + b6) * q^9 $$q + \beta_{7} q^{3} + (\beta_{10} - 1) q^{5} + ( - \beta_{10} + \beta_{6} + \beta_{5} + 1) q^{7} + ( - \beta_{13} + \beta_{11} + \cdots + \beta_{6}) q^{9}+ \cdots + ( - 2 \beta_{13} + 2 \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100})$$ q + b7 * q^3 + (b10 - 1) * q^5 + (-b10 + b6 + b5 + 1) * q^7 + (-b13 + b11 - 2*b10 + b9 + b6) * q^9 + (-b8 + 1) * q^11 + (-b13 - b12 - b10 - b8 - b7 + 1) * q^13 + (-b7 + b1) * q^15 + (-b12 + b11 + b9 - b2) * q^17 + (-b13 - b12 - b10 - b8 - b7 - b3 + 1) * q^19 + (b13 + b12 + 2*b7 + 2*b4 + 2*b3 + b2 - 2*b1) * q^21 + (-b9 + b5 + b4 + b1 + 1) * q^23 - b10 * q^25 + (-2*b8 + b5 + 3*b4 - 3*b1) * q^27 + (b9 + b8 - b5 + b2 + b1) * q^29 + (b9 + b8 - b4 + b2 + b1 - 1) * q^31 + (b13 - b12 + b11 - 2*b10 + b8 + 3*b7 + b3 + 2) * q^33 + (b10 - b6) * q^35 + (b12 - b11 + b10 - b8 - b7 + b5 + b4 - b3 + b2 + 1) * q^37 + (b13 - 2*b12 + 3*b10 + 2*b7 - 2*b6 + b4 + b3 - 2*b2 - 2*b1) * q^39 + (b13 + b11 + b8 + b7 - b6 - b5 + 2*b3) * q^41 + (b9 - b8 + b4 - 2*b2 - b1 - 1) * q^43 + (-b9 - b8 + b5 + 2) * q^45 + (b9 - b8 + 2*b5 - b1) * q^47 + (-b13 - 2*b10 + b7 + b6 + b4 + b3 - b1) * q^49 + (-b9 + b8 + 2*b5 - b2 + 2) * q^51 + (b13 + 2*b10 - b7 + b1) * q^53 + (b13 + b10 + b8 - 1) * q^55 + (b13 - 2*b12 - b11 + 4*b10 - b9 + b7 - 4*b6 - 2*b2 - b1) * q^57 + (b13 - b12 + 2*b11 - 3*b10 + 2*b9 + b7 + 2*b6 + b4 + b3 - b2 - b1) * q^59 + (-b13 + b12 + b11 - b10 - b8 - b7 + 1) * q^61 + (3*b9 + 2*b8 - 4*b5 + b4 + 2*b2 - b1 - 7) * q^63 + (b13 + b12 + b10 + b7 + b2 - b1) * q^65 + (b13 + b12 - 2*b11 - b10 + b8 - 2*b7 + 2*b6 + 2*b5 - 3*b3 + 1) * q^67 + (b12 - b11 - 2*b10 + 2*b7 - 2*b6 - 2*b5 + b3 + 2) * q^69 + (-b13 - b12 + 2*b11 + b10 - b8 - 2*b7 + b6 + b5 - 1) * q^71 + (b9 + b8 - b5 - b4 - b1 - 3) * q^73 - b1 * q^75 + (b12 + b11 - b10 + b7 + 3*b6 + 3*b5 + 1) * q^77 + (-b13 + 2*b12 - b8 + b7 + b6 + b5 - b3) * q^79 + (3*b13 - b12 - b11 + 8*b10 + 3*b8 + 2*b7 - 6*b6 - 6*b5 + b3 - 8) * q^81 + (-b13 - b12 - 2*b11 + b10 - 2*b9 + b7 - 3*b6 - b2 - b1) * q^83 + (-b9 + b2) * q^85 + (-4*b13 - b12 + b11 - 5*b10 - 4*b8 - 5*b7 + b6 + b5 - 3*b3 + 5) * q^87 + (-b13 - b12 - 3*b11 + 2*b10 - 3*b9 - 3*b7 - b6 - b2 + 3*b1) * q^89 + (b12 - b11 - b10 - b9 - 3*b7 + 2*b6 - 2*b4 - 2*b3 + b2 + 3*b1) * q^91 + (-3*b13 + 2*b11 - 6*b10 - 3*b8 - 4*b7 + 3*b6 + 3*b5 + 6) * q^93 + (b13 + b12 + b10 + b7 + b4 + b3 + b2 - b1) * q^95 + (-b8 - 2*b5 + 2*b4 + b1 - 5) * q^97 + (-2*b13 + 2*b11 - 9*b10 + 2*b9 + b7 + 3*b6 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 7 q^{5} + 4 q^{7} - 13 q^{9}+O(q^{10})$$ 14 * q - 7 * q^5 + 4 * q^7 - 13 * q^9 $$14 q - 7 q^{5} + 4 q^{7} - 13 q^{9} + 14 q^{11} + 4 q^{13} + q^{17} + 4 q^{19} - 3 q^{21} + 12 q^{23} - 7 q^{25} - 6 q^{27} - 4 q^{29} - 24 q^{31} + 13 q^{33} + 4 q^{35} + 10 q^{37} + 21 q^{39} + 5 q^{41} - 6 q^{43} + 26 q^{45} - 16 q^{47} - 11 q^{49} + 26 q^{51} + 14 q^{53} - 7 q^{55} + 24 q^{57} - 16 q^{59} + 12 q^{61} - 98 q^{63} + 4 q^{65} + 21 q^{69} - 9 q^{71} - 40 q^{73} + 3 q^{77} + 3 q^{79} - 43 q^{81} + 5 q^{83} - 2 q^{85} + 31 q^{87} + 20 q^{89} - 2 q^{91} + 37 q^{93} + 4 q^{95} - 58 q^{97} - 58 q^{99}+O(q^{100})$$ 14 * q - 7 * q^5 + 4 * q^7 - 13 * q^9 + 14 * q^11 + 4 * q^13 + q^17 + 4 * q^19 - 3 * q^21 + 12 * q^23 - 7 * q^25 - 6 * q^27 - 4 * q^29 - 24 * q^31 + 13 * q^33 + 4 * q^35 + 10 * q^37 + 21 * q^39 + 5 * q^41 - 6 * q^43 + 26 * q^45 - 16 * q^47 - 11 * q^49 + 26 * q^51 + 14 * q^53 - 7 * q^55 + 24 * q^57 - 16 * q^59 + 12 * q^61 - 98 * q^63 + 4 * q^65 + 21 * q^69 - 9 * q^71 - 40 * q^73 + 3 * q^77 + 3 * q^79 - 43 * q^81 + 5 * q^83 - 2 * q^85 + 31 * q^87 + 20 * q^89 - 2 * q^91 + 37 * q^93 + 4 * q^95 - 58 * q^97 - 58 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 24x^{12} + 204x^{10} + 727x^{8} + 1008x^{6} + 426x^{4} + 64x^{2} + 3$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{12} + 25\nu^{10} + 226\nu^{8} + 893\nu^{6} + 1496\nu^{4} + 863\nu^{2} + 60 ) / 27$$ (v^12 + 25*v^10 + 226*v^8 + 893*v^6 + 1496*v^4 + 863*v^2 + 60) / 27 $$\beta_{2}$$ $$=$$ $$( \nu^{12} + 25\nu^{10} + 226\nu^{8} + 893\nu^{6} + 1496\nu^{4} + 890\nu^{2} + 141 ) / 27$$ (v^12 + 25*v^10 + 226*v^8 + 893*v^6 + 1496*v^4 + 890*v^2 + 141) / 27 $$\beta_{3}$$ $$=$$ $$( \nu^{12} + 24\nu^{10} + 203\nu^{8} + 713\nu^{6} + 954\nu^{4} + 361\nu^{2} + 27\nu + 30 ) / 18$$ (v^12 + 24*v^10 + 203*v^8 + 713*v^6 + 954*v^4 + 361*v^2 + 27*v + 30) / 18 $$\beta_{4}$$ $$=$$ $$( -\nu^{12} - 24\nu^{10} - 203\nu^{8} - 713\nu^{6} - 954\nu^{4} - 361\nu^{2} - 30 ) / 9$$ (-v^12 - 24*v^10 - 203*v^8 - 713*v^6 - 954*v^4 - 361*v^2 - 30) / 9 $$\beta_{5}$$ $$=$$ $$( 13\nu^{12} + 310\nu^{10} + 2602\nu^{8} + 9017\nu^{6} + 11588\nu^{4} + 3680\nu^{2} + 303 ) / 27$$ (13*v^12 + 310*v^10 + 2602*v^8 + 9017*v^6 + 11588*v^4 + 3680*v^2 + 303) / 27 $$\beta_{6}$$ $$=$$ $$( 9 \nu^{13} - 13 \nu^{12} + 219 \nu^{11} - 310 \nu^{10} + 1905 \nu^{9} - 2602 \nu^{8} + 7092 \nu^{7} + \cdots - 303 ) / 54$$ (9*v^13 - 13*v^12 + 219*v^11 - 310*v^10 + 1905*v^9 - 2602*v^8 + 7092*v^7 - 9017*v^6 + 10806*v^5 - 11588*v^4 + 5637*v^3 - 3680*v^2 + 765*v - 303) / 54 $$\beta_{7}$$ $$=$$ $$( - 18 \nu^{13} + \nu^{12} - 432 \nu^{11} + 25 \nu^{10} - 3672 \nu^{9} + 226 \nu^{8} - 13077 \nu^{7} + \cdots + 60 ) / 54$$ (-18*v^13 + v^12 - 432*v^11 + 25*v^10 - 3672*v^9 + 226*v^8 - 13077*v^7 + 893*v^6 - 18009*v^5 + 1496*v^4 - 7101*v^3 + 863*v^2 - 540*v + 60) / 54 $$\beta_{8}$$ $$=$$ $$( -23\nu^{12} - 548\nu^{10} - 4595\nu^{8} - 15895\nu^{6} - 20287\nu^{4} - 6034\nu^{2} - 354 ) / 27$$ (-23*v^12 - 548*v^10 - 4595*v^8 - 15895*v^6 - 20287*v^4 - 6034*v^2 - 354) / 27 $$\beta_{9}$$ $$=$$ $$( 10\nu^{12} + 239\nu^{10} + 2016\nu^{8} + 7067\nu^{6} + 9367\nu^{4} + 3306\nu^{2} + 279 ) / 9$$ (10*v^12 + 239*v^10 + 2016*v^8 + 7067*v^6 + 9367*v^4 + 3306*v^2 + 279) / 9 $$\beta_{10}$$ $$=$$ $$( -10\nu^{13} - 239\nu^{11} - 2016\nu^{9} - 7067\nu^{7} - 9367\nu^{5} - 3306\nu^{3} - 279\nu + 9 ) / 18$$ (-10*v^13 - 239*v^11 - 2016*v^9 - 7067*v^7 - 9367*v^5 - 3306*v^3 - 279*v + 9) / 18 $$\beta_{11}$$ $$=$$ $$( - 9 \nu^{13} - 10 \nu^{12} - 216 \nu^{11} - 239 \nu^{10} - 1836 \nu^{9} - 2016 \nu^{8} - 6543 \nu^{7} + \cdots - 279 ) / 18$$ (-9*v^13 - 10*v^12 - 216*v^11 - 239*v^10 - 1836*v^9 - 2016*v^8 - 6543*v^7 - 7067*v^6 - 9072*v^5 - 9367*v^4 - 3834*v^3 - 3306*v^2 - 576*v - 279) / 18 $$\beta_{12}$$ $$=$$ $$( - 69 \nu^{13} - \nu^{12} - 1647 \nu^{11} - 25 \nu^{10} - 13863 \nu^{9} - 226 \nu^{8} - 48387 \nu^{7} + \cdots - 141 ) / 54$$ (-69*v^13 - v^12 - 1647*v^11 - 25*v^10 - 13863*v^9 - 226*v^8 - 48387*v^7 - 893*v^6 - 63432*v^5 - 1496*v^4 - 21570*v^3 - 890*v^2 - 1881*v - 141) / 54 $$\beta_{13}$$ $$=$$ $$( 135 \nu^{13} + 23 \nu^{12} + 3219 \nu^{11} + 548 \nu^{10} + 27039 \nu^{9} + 4595 \nu^{8} + 93933 \nu^{7} + \cdots + 354 ) / 54$$ (135*v^13 + 23*v^12 + 3219*v^11 + 548*v^10 + 27039*v^9 + 4595*v^8 + 93933*v^7 + 15895*v^6 + 121431*v^5 + 20287*v^4 + 38589*v^3 + 6034*v^2 + 2772*v + 354) / 54
 $$\nu$$ $$=$$ $$( \beta_{4} + 2\beta_{3} ) / 3$$ (b4 + 2*b3) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 - 3$$ b2 - b1 - 3 $$\nu^{3}$$ $$=$$ $$( 2 \beta_{13} + 4 \beta_{12} - 4 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots - 2 ) / 3$$ (2*b13 + 4*b12 - 4*b11 + 4*b10 - 2*b9 + b8 - 2*b7 - 2*b6 - b5 - 7*b4 - 14*b3 + 2*b2 + b1 - 2) / 3 $$\nu^{4}$$ $$=$$ $$\beta_{8} + 2\beta_{5} + \beta_{4} - 9\beta_{2} + 9\beta _1 + 21$$ b8 + 2*b5 + b4 - 9*b2 + 9*b1 + 21 $$\nu^{5}$$ $$=$$ $$( - 20 \beta_{13} - 46 \beta_{12} + 40 \beta_{11} - 28 \beta_{10} + 20 \beta_{9} - 10 \beta_{8} + \cdots + 14 ) / 3$$ (-20*b13 - 46*b12 + 40*b11 - 28*b10 + 20*b9 - 10*b8 + 26*b7 + 26*b6 + 13*b5 + 54*b4 + 108*b3 - 23*b2 - 13*b1 + 14) / 3 $$\nu^{6}$$ $$=$$ $$\beta_{9} - 11\beta_{8} - 25\beta_{5} - 15\beta_{4} + 76\beta_{2} - 79\beta _1 - 166$$ b9 - 11*b8 - 25*b5 - 15*b4 + 76*b2 - 79*b1 - 166 $$\nu^{7}$$ $$=$$ $$( 174 \beta_{13} + 438 \beta_{12} - 360 \beta_{11} + 168 \beta_{10} - 180 \beta_{9} + 87 \beta_{8} + \cdots - 84 ) / 3$$ (174*b13 + 438*b12 - 360*b11 + 168*b10 - 180*b9 + 87*b8 - 246*b7 - 264*b6 - 132*b5 - 437*b4 - 874*b3 + 219*b2 + 123*b1 - 84) / 3 $$\nu^{8}$$ $$=$$ $$-12\beta_{9} + 102\beta_{8} + 243\beta_{5} + 165\beta_{4} - 641\beta_{2} + 683\beta _1 + 1362$$ -12*b9 + 102*b8 + 243*b5 + 165*b4 - 641*b2 + 683*b1 + 1362 $$\nu^{9}$$ $$=$$ $$( - 1486 \beta_{13} - 3962 \beta_{12} + 3134 \beta_{11} - 938 \beta_{10} + 1567 \beta_{9} - 743 \beta_{8} + \cdots + 469 ) / 3$$ (-1486*b13 - 3962*b12 + 3134*b11 - 938*b10 + 1567*b9 - 743*b8 + 2116*b7 + 2422*b6 + 1211*b5 + 3623*b4 + 7246*b3 - 1981*b2 - 1058*b1 + 469) / 3 $$\nu^{10}$$ $$=$$ $$96\beta_{9} - 908\beta_{8} - 2173\beta_{5} - 1628\beta_{4} + 5439\beta_{2} - 5838\beta _1 - 11352$$ 96*b9 - 908*b8 - 2173*b5 - 1628*b4 + 5439*b2 - 5838*b1 - 11352 $$\nu^{11}$$ $$=$$ $$( 12694 \beta_{13} + 35156 \beta_{12} - 26900 \beta_{11} + 4610 \beta_{10} - 13450 \beta_{9} + 6347 \beta_{8} + \cdots - 2305 ) / 3$$ (12694*b13 + 35156*b12 - 26900*b11 + 4610*b10 - 13450*b9 + 6347*b8 - 17476*b7 - 21166*b6 - 10583*b5 - 30426*b4 - 60852*b3 + 17578*b2 + 8738*b1 - 2305) / 3 $$\nu^{12}$$ $$=$$ $$-581\beta_{9} + 7975\beta_{8} + 18740\beta_{5} + 15309\beta_{4} - 46376\beta_{2} + 49565\beta _1 + 95339$$ -581*b9 + 7975*b8 + 18740*b5 + 15309*b4 - 46376*b2 + 49565*b1 + 95339 $$\nu^{13}$$ $$=$$ $$( - 108702 \beta_{13} - 309258 \beta_{12} + 229362 \beta_{11} - 14904 \beta_{10} + 114681 \beta_{9} + \cdots + 7452 ) / 3$$ (-108702*b13 - 309258*b12 + 229362*b11 - 14904*b10 + 114681*b9 - 54351*b8 + 141246*b7 + 180468*b6 + 90234*b5 + 257317*b4 + 514634*b3 - 154629*b2 - 70623*b1 + 7452) / 3

## Character values

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

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-1 + \beta_{10}$$ $$1$$ $$1$$

## 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
 − 0.569716i − 0.401547i 2.12787i 0.300390i − 2.93925i − 1.39878i 2.88103i 0.569716i 0.401547i − 2.12787i − 0.300390i 2.93925i 1.39878i − 2.88103i
0 −1.67818 2.90670i 0 −0.500000 0.866025i 0 1.40373 + 2.43134i 0 −4.13261 + 7.15788i 0
121.2 0 −0.812038 1.40649i 0 −0.500000 0.866025i 0 0.0339191 + 0.0587497i 0 0.181190 0.313830i 0
121.3 0 −0.795627 1.37807i 0 −0.500000 0.866025i 0 −1.53626 2.66088i 0 0.233956 0.405224i 0
121.4 0 −0.117270 0.203117i 0 −0.500000 0.866025i 0 1.58956 + 2.75319i 0 1.47250 2.55044i 0
121.5 0 0.627876 + 1.08751i 0 −0.500000 0.866025i 0 −0.536763 0.929701i 0 0.711543 1.23243i 0
121.6 0 1.13042 + 1.95795i 0 −0.500000 0.866025i 0 −1.38845 2.40487i 0 −1.05570 + 1.82853i 0
121.7 0 1.64482 + 2.84891i 0 −0.500000 0.866025i 0 2.43427 + 4.21628i 0 −3.91088 + 6.77384i 0
581.1 0 −1.67818 + 2.90670i 0 −0.500000 + 0.866025i 0 1.40373 2.43134i 0 −4.13261 7.15788i 0
581.2 0 −0.812038 + 1.40649i 0 −0.500000 + 0.866025i 0 0.0339191 0.0587497i 0 0.181190 + 0.313830i 0
581.3 0 −0.795627 + 1.37807i 0 −0.500000 + 0.866025i 0 −1.53626 + 2.66088i 0 0.233956 + 0.405224i 0
581.4 0 −0.117270 + 0.203117i 0 −0.500000 + 0.866025i 0 1.58956 2.75319i 0 1.47250 + 2.55044i 0
581.5 0 0.627876 1.08751i 0 −0.500000 + 0.866025i 0 −0.536763 + 0.929701i 0 0.711543 + 1.23243i 0
581.6 0 1.13042 1.95795i 0 −0.500000 + 0.866025i 0 −1.38845 + 2.40487i 0 −1.05570 1.82853i 0
581.7 0 1.64482 2.84891i 0 −0.500000 + 0.866025i 0 2.43427 4.21628i 0 −3.91088 6.77384i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.i.a 14
37.c even 3 1 inner 740.2.i.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.i.a 14 1.a even 1 1 trivial
740.2.i.a 14 37.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} + 17 T_{3}^{12} + 2 T_{3}^{11} + 216 T_{3}^{10} + 34 T_{3}^{9} + 1080 T_{3}^{8} + 486 T_{3}^{7} + \cdots + 361$$ acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$T^{14} + 17 T^{12} + \cdots + 361$$
$5$ $$(T^{2} + T + 1)^{7}$$
$7$ $$T^{14} - 4 T^{13} + \cdots + 729$$
$11$ $$(T^{7} - 7 T^{6} + \cdots + 171)^{2}$$
$13$ $$T^{14} - 4 T^{13} + \cdots + 6365529$$
$17$ $$T^{14} - T^{13} + \cdots + 50936769$$
$19$ $$T^{14} - 4 T^{13} + \cdots + 3651921$$
$23$ $$(T^{7} - 6 T^{6} + \cdots - 17403)^{2}$$
$29$ $$(T^{7} + 2 T^{6} + \cdots + 2439)^{2}$$
$31$ $$(T^{7} + 12 T^{6} + \cdots + 10623)^{2}$$
$37$ $$T^{14} + \cdots + 94931877133$$
$41$ $$T^{14} - 5 T^{13} + \cdots + 99620361$$
$43$ $$(T^{7} + 3 T^{6} + \cdots - 56313)^{2}$$
$47$ $$(T^{7} + 8 T^{6} + \cdots - 834219)^{2}$$
$53$ $$T^{14} - 14 T^{13} + \cdots + 729$$
$59$ $$T^{14} + \cdots + 307686681$$
$61$ $$T^{14} - 12 T^{13} + \cdots + 1521$$
$67$ $$T^{14} + \cdots + 1259819921889$$
$71$ $$T^{14} + \cdots + 13021320321$$
$73$ $$(T^{7} + 20 T^{6} + \cdots - 8451)^{2}$$
$79$ $$T^{14} + \cdots + 581726161$$
$83$ $$T^{14} + \cdots + 1719106455609$$
$89$ $$T^{14} + \cdots + 2708430524361$$
$97$ $$(T^{7} + 29 T^{6} + \cdots + 5722633)^{2}$$