# Properties

 Label 2320.2.d.j Level $2320$ Weight $2$ Character orbit 2320.d Analytic conductor $18.525$ 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] = [2320,2,Mod(929,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2320.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.d (of order $$2$$, degree $$1$$, 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: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 33x^{14} + 432x^{12} + 2881x^{10} + 10441x^{8} + 20304x^{6} + 19480x^{4} + 7344x^{2} + 400$$ x^16 + 33*x^14 + 432*x^12 + 2881*x^10 + 10441*x^8 + 20304*x^6 + 19480*x^4 + 7344*x^2 + 400 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 1160) 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{8} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 - b8 * q^5 - b13 * q^7 + (b10 + b9 - 1) * q^9 $$q + \beta_1 q^{3} - \beta_{8} q^{5} - \beta_{13} q^{7} + (\beta_{10} + \beta_{9} - 1) q^{9} + (\beta_{9} - \beta_{2} - 1) q^{11} + (\beta_{13} + \beta_{11} + \cdots + \beta_{5}) q^{13}+ \cdots + ( - \beta_{15} - \beta_{14} + 2 \beta_{10} + \cdots + 4) q^{99}+O(q^{100})$$ q + b1 * q^3 - b8 * q^5 - b13 * q^7 + (b10 + b9 - 1) * q^9 + (b9 - b2 - 1) * q^11 + (b13 + b11 - b8 + b6 + b5) * q^13 + (-b14 + b13 + b12 - b9 - b8 + b6 + b5 + b3 + b2 + 1) * q^15 + (b13 - b11) * q^17 + (b15 + b9 - b2 + 1) * q^19 + (b15 + b10 + b7) * q^21 + (b12 - b11 - b1) * q^23 + (b14 + b11 + b5 + b4 + b1 + 1) * q^25 + (b13 + b12 + 2*b11 - b8 + b6 + b5 + b4) * q^27 - q^29 + (b14 + b10 + 2*b9 + b8 + b5 - 2) * q^31 + (-2*b13 - b12 - b6 + b4 - b1) * q^33 + (-b14 - b13 - b10 - b9 + b4 + b2 - 1) * q^35 + (b15 - b14 + b13 - 2*b8 + b6 + 2*b5 + 2*b3 + b2 + b1 + 1) * q^37 + (-b15 + b14 - b10 - b9 + b8 + b5 - b2 + 1) * q^39 + (-2*b7 - 2) * q^41 + (b15 - b14 - b13 + b12 - b8 - b6 + b5 + b4 + 2*b3 + b2 - b1 + 1) * q^43 + (b15 - 2*b10 + b8 + b6 + b5 - b4 - b2 - 1) * q^45 + (b13 + b12 + b11 - b8 + b5 - b4) * q^47 + (b15 + b9 + b8 + b7 + b5 + 1) * q^49 + (-b15 - b14 - b10 - b7 + b2 - 1) * q^51 + (b15 - b14 + 2*b13 + b12 - b11 + 2*b6 - b4 + 2*b3 + b2 - 2*b1 + 1) * q^53 + (-b15 + b14 - b11 + b9 + b8 - b7 + b6 + b5 - b4 - b3 - 2) * q^55 + (-b12 - b8 - b6 + b5 + b4 + b1) * q^57 + (-b15 + b14 + b9 + b8 + b7 + b5 - b2 - 1) * q^59 + (b15 + b9 + b8 - b7 + b5 - 2) * q^61 + (2*b13 + b12 - b11 - b8 + b5 - b4 - b1) * q^63 + (-b15 + b14 - b12 + b11 - b9 - b7 - b6 + b5 + 2*b1 - 2) * q^65 + (3*b13 + 2*b11 - b8 + b5 - b4 + 2*b1) * q^67 + (2*b15 - 2*b14 - 3*b10 - b9 + b8 + b5 + 2) * q^69 + (-b14 + 2*b10 + 2*b7 + b2 + 1) * q^71 + (-3*b13 - b12 - b11 + b8 - b6 - b5 - b4 - b1) * q^73 + (-b15 - 2*b11 - 2*b9 + b7 - 2*b6 - b3 + 2*b1 - 2) * q^75 + (2*b13 + b12 - 2*b11 - b8 + b6 + b5 - b4 + b1) * q^77 + (-b15 + 2*b14 + b8 - b7 + b5 - b2 + 5) * q^79 + (b15 - b10 + 2*b8 + b7 + 2*b5 - 2*b2 - 1) * q^81 + (-b15 + b14 - b13 - b11 - b6 - 2*b3 - b2 - b1 - 1) * q^83 + (b15 + b14 + b13 + b12 + b10 + b9 - b4 - b2 + 1) * q^85 - b1 * q^87 + (2*b15 - 2*b14 - 2*b10 - 2*b7 + 4) * q^89 + (b15 - b10 - b7 + 2) * q^91 + (-b15 + b14 - b13 + 2*b11 - b8 - 2*b6 + b5 + 2*b4 - 2*b3 - b2 - 3*b1 - 1) * q^93 + (b11 + b9 - b8 + b6 + b5 - b4 + b2 + b1 - 1) * q^95 + (-b15 + b14 - b11 - 2*b3 - b2 - 3*b1 - 1) * q^97 + (-b15 - b14 + 2*b10 - b9 - 2*b8 + 2*b7 - 2*b5 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{5} - 18 q^{9}+O(q^{10})$$ 16 * q - 2 * q^5 - 18 * q^9 $$16 q - 2 q^{5} - 18 q^{9} - 8 q^{11} + 10 q^{15} + 16 q^{19} - 12 q^{21} + 10 q^{25} - 16 q^{29} - 34 q^{31} - 10 q^{35} + 26 q^{39} - 36 q^{41} - 4 q^{45} + 18 q^{49} - 20 q^{55} - 2 q^{59} - 34 q^{61} - 36 q^{65} + 50 q^{69} + 12 q^{71} - 24 q^{75} + 78 q^{79} + 2 q^{85} + 72 q^{89} + 28 q^{91} - 16 q^{95} + 60 q^{99}+O(q^{100})$$ 16 * q - 2 * q^5 - 18 * q^9 - 8 * q^11 + 10 * q^15 + 16 * q^19 - 12 * q^21 + 10 * q^25 - 16 * q^29 - 34 * q^31 - 10 * q^35 + 26 * q^39 - 36 * q^41 - 4 * q^45 + 18 * q^49 - 20 * q^55 - 2 * q^59 - 34 * q^61 - 36 * q^65 + 50 * q^69 + 12 * q^71 - 24 * q^75 + 78 * q^79 + 2 * q^85 + 72 * q^89 + 28 * q^91 - 16 * q^95 + 60 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 33x^{14} + 432x^{12} + 2881x^{10} + 10441x^{8} + 20304x^{6} + 19480x^{4} + 7344x^{2} + 400$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 11\nu^{14} + 361\nu^{12} + 4638\nu^{10} + 29535\nu^{8} + 96409\nu^{6} + 146846\nu^{4} + 70748\nu^{2} - 6504 ) / 2432$$ (11*v^14 + 361*v^12 + 4638*v^10 + 29535*v^8 + 96409*v^6 + 146846*v^4 + 70748*v^2 - 6504) / 2432 $$\beta_{3}$$ $$=$$ $$( 44 \nu^{15} - 5 \nu^{14} + 1292 \nu^{13} + 95 \nu^{12} + 15208 \nu^{11} + 4110 \nu^{10} + \cdots + 20920 ) / 12160$$ (44*v^15 - 5*v^14 + 1292*v^13 + 95*v^12 + 15208*v^11 + 4110*v^10 + 95644*v^9 + 39015*v^8 + 355084*v^7 + 148095*v^6 + 767656*v^5 + 243470*v^4 + 830800*v^3 + 170900*v^2 + 286496*v + 20920) / 12160 $$\beta_{4}$$ $$=$$ $$( - 11 \nu^{15} - 361 \nu^{13} - 4638 \nu^{11} - 29535 \nu^{9} - 96409 \nu^{7} - 146846 \nu^{5} + \cdots + 21096 \nu ) / 2432$$ (-11*v^15 - 361*v^13 - 4638*v^11 - 29535*v^9 - 96409*v^7 - 146846*v^5 - 68316*v^3 + 21096*v) / 2432 $$\beta_{5}$$ $$=$$ $$( - 33 \nu^{15} + 85 \nu^{14} - 969 \nu^{13} + 2375 \nu^{12} - 11026 \nu^{11} + 25130 \nu^{10} + \cdots + 15240 ) / 12160$$ (-33*v^15 + 85*v^14 - 969*v^13 + 2375*v^12 - 11026*v^11 + 25130*v^10 - 62613*v^9 + 127905*v^8 - 190313*v^7 + 324215*v^6 - 312402*v^5 + 370090*v^4 - 278060*v^3 + 138500*v^2 - 119112*v + 15240) / 12160 $$\beta_{6}$$ $$=$$ $$( - 17 \nu^{15} - 741 \nu^{13} - 11904 \nu^{11} - 90637 \nu^{9} - 347677 \nu^{7} - 653328 \nu^{5} + \cdots - 119328 \nu ) / 6080$$ (-17*v^15 - 741*v^13 - 11904*v^11 - 90637*v^9 - 347677*v^7 - 653328*v^5 - 526260*v^3 - 119328*v) / 6080 $$\beta_{7}$$ $$=$$ $$( - 23 \nu^{14} - 513 \nu^{12} - 3514 \nu^{10} - 4603 \nu^{8} + 34591 \nu^{6} + 121702 \nu^{4} + \cdots + 15672 ) / 2432$$ (-23*v^14 - 513*v^12 - 3514*v^10 - 4603*v^8 + 34591*v^6 + 121702*v^4 + 113844*v^2 + 15672) / 2432 $$\beta_{8}$$ $$=$$ $$( 33 \nu^{15} + 85 \nu^{14} + 969 \nu^{13} + 2375 \nu^{12} + 11026 \nu^{11} + 25130 \nu^{10} + \cdots + 15240 ) / 12160$$ (33*v^15 + 85*v^14 + 969*v^13 + 2375*v^12 + 11026*v^11 + 25130*v^10 + 62613*v^9 + 127905*v^8 + 190313*v^7 + 324215*v^6 + 312402*v^5 + 370090*v^4 + 278060*v^3 + 138500*v^2 + 119112*v + 15240) / 12160 $$\beta_{9}$$ $$=$$ $$( 35 \nu^{14} + 969 \nu^{12} + 10218 \nu^{10} + 52935 \nu^{8} + 144489 \nu^{6} + 201866 \nu^{4} + \cdots + 13768 ) / 2432$$ (35*v^14 + 969*v^12 + 10218*v^10 + 52935*v^8 + 144489*v^6 + 201866*v^4 + 117740*v^2 + 13768) / 2432 $$\beta_{10}$$ $$=$$ $$( - 35 \nu^{14} - 969 \nu^{12} - 10218 \nu^{10} - 52935 \nu^{8} - 144489 \nu^{6} - 201866 \nu^{4} + \cdots - 4040 ) / 2432$$ (-35*v^14 - 969*v^12 - 10218*v^10 - 52935*v^8 - 144489*v^6 - 201866*v^4 - 115308*v^2 - 4040) / 2432 $$\beta_{11}$$ $$=$$ $$( 23 \nu^{15} + 684 \nu^{13} + 7941 \nu^{11} + 46403 \nu^{9} + 145168 \nu^{7} + 233997 \nu^{5} + \cdots + 33652 \nu ) / 3040$$ (23*v^15 + 684*v^13 + 7941*v^11 + 46403*v^9 + 145168*v^7 + 233997*v^5 + 165380*v^3 + 33652*v) / 3040 $$\beta_{12}$$ $$=$$ $$( 83 \nu^{15} + 2869 \nu^{13} + 38326 \nu^{11} + 249303 \nu^{9} + 820453 \nu^{7} + 1290102 \nu^{5} + \cdots + 107512 \nu ) / 12160$$ (83*v^15 + 2869*v^13 + 38326*v^11 + 249303*v^9 + 820453*v^7 + 1290102*v^5 + 802700*v^3 + 107512*v) / 12160 $$\beta_{13}$$ $$=$$ $$( - 28 \nu^{15} - 779 \nu^{13} - 8201 \nu^{11} - 41588 \nu^{9} - 105943 \nu^{7} - 124097 \nu^{5} + \cdots + 16908 \nu ) / 3040$$ (-28*v^15 - 779*v^13 - 8201*v^11 - 41588*v^9 - 105943*v^7 - 124097*v^5 - 40840*v^3 + 16908*v) / 3040 $$\beta_{14}$$ $$=$$ $$( - 49 \nu^{14} - 1235 \nu^{12} - 11250 \nu^{10} - 46445 \nu^{8} - 89987 \nu^{6} - 79282 \nu^{4} + \cdots - 16296 ) / 2432$$ (-49*v^14 - 1235*v^12 - 11250*v^10 - 46445*v^8 - 89987*v^6 - 79282*v^4 - 37460*v^2 - 16296) / 2432 $$\beta_{15}$$ $$=$$ $$( - 29 \nu^{14} - 817 \nu^{12} - 8766 \nu^{10} - 45793 \nu^{8} - 122817 \nu^{6} - 161758 \nu^{4} + \cdots - 10296 ) / 1216$$ (-29*v^14 - 817*v^12 - 8766*v^10 - 45793*v^8 - 122817*v^6 - 161758*v^4 - 88284*v^2 - 10296) / 1216
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + \beta_{9} - 4$$ b10 + b9 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{13} + \beta_{12} + 2\beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - 6\beta_1$$ b13 + b12 + 2*b11 - b8 + b6 + b5 + b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{15} - 10\beta_{10} - 9\beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{5} - 2\beta_{2} + 26$$ b15 - 10*b10 - 9*b9 + 2*b8 + b7 + 2*b5 - 2*b2 + 26 $$\nu^{5}$$ $$=$$ $$- 2 \beta_{15} + 2 \beta_{14} - 14 \beta_{13} - 16 \beta_{12} - 23 \beta_{11} + 14 \beta_{8} - 17 \beta_{6} + \cdots - 2$$ -2*b15 + 2*b14 - 14*b13 - 16*b12 - 23*b11 + 14*b8 - 17*b6 - 14*b5 - 10*b4 - 4*b3 - 2*b2 + 47*b1 - 2 $$\nu^{6}$$ $$=$$ $$- 21 \beta_{15} - 4 \beta_{14} + 99 \beta_{10} + 74 \beta_{9} - 35 \beta_{8} - 13 \beta_{7} + \cdots - 202$$ -21*b15 - 4*b14 + 99*b10 + 74*b9 - 35*b8 - 13*b7 - 35*b5 + 32*b2 - 202 $$\nu^{7}$$ $$=$$ $$35 \beta_{15} - 35 \beta_{14} + 167 \beta_{13} + 197 \beta_{12} + 233 \beta_{11} - 151 \beta_{8} + \cdots + 35$$ 35*b15 - 35*b14 + 167*b13 + 197*b12 + 233*b11 - 151*b8 + 218*b6 + 151*b5 + 87*b4 + 70*b3 + 35*b2 - 417*b1 + 35 $$\nu^{8}$$ $$=$$ $$294 \beta_{15} + 86 \beta_{14} - 995 \beta_{10} - 605 \beta_{9} + 450 \beta_{8} + 138 \beta_{7} + \cdots + 1724$$ 294*b15 + 86*b14 - 995*b10 - 605*b9 + 450*b8 + 138*b7 + 450*b5 - 410*b2 + 1724 $$\nu^{9}$$ $$=$$ $$- 450 \beta_{15} + 450 \beta_{14} - 1883 \beta_{13} - 2219 \beta_{12} - 2340 \beta_{11} + 1535 \beta_{8} + \cdots - 450$$ -450*b15 + 450*b14 - 1883*b13 - 2219*b12 - 2340*b11 + 1535*b8 - 2529*b6 - 1535*b5 - 743*b4 - 900*b3 - 450*b2 + 3938*b1 - 450 $$\nu^{10}$$ $$=$$ $$- 3549 \beta_{15} - 1272 \beta_{14} + 10102 \beta_{10} + 5065 \beta_{9} - 5198 \beta_{8} - 1389 \beta_{7} + \cdots - 15622$$ -3549*b15 - 1272*b14 + 10102*b10 + 5065*b9 - 5198*b8 - 1389*b7 - 5198*b5 + 4810*b2 - 15622 $$\nu^{11}$$ $$=$$ $$5198 \beta_{15} - 5198 \beta_{14} + 20586 \beta_{13} + 24036 \beta_{12} + 23683 \beta_{11} - 15450 \beta_{8} + \cdots + 5198$$ 5198*b15 - 5198*b14 + 20586*b13 + 24036*b12 + 23683*b11 - 15450*b8 + 27969*b6 + 15450*b5 + 6454*b4 + 10396*b3 + 5198*b2 - 38487*b1 + 5198 $$\nu^{12}$$ $$=$$ $$40005 \beta_{15} + 16108 \beta_{14} - 103203 \beta_{10} - 43930 \beta_{9} + 57203 \beta_{8} + \cdots + 147530$$ 40005*b15 + 16108*b14 - 103203*b10 - 43930*b9 + 57203*b8 + 13837*b7 + 57203*b5 - 53784*b2 + 147530 $$\nu^{13}$$ $$=$$ $$- 57203 \beta_{15} + 57203 \beta_{14} - 220711 \beta_{13} - 255285 \beta_{12} - 241481 \beta_{11} + \cdots - 57203$$ -57203*b15 + 57203*b14 - 220711*b13 - 255285*b12 - 241481*b11 + 156007*b8 - 301338*b6 - 156007*b5 - 57767*b4 - 114406*b3 - 57203*b2 + 383881*b1 - 57203 $$\nu^{14}$$ $$=$$ $$- 435190 \beta_{15} - 188166 \beta_{14} + 1058523 \beta_{10} + 395685 \beta_{9} - 613826 \beta_{8} + \cdots - 1434156$$ -435190*b15 - 188166*b14 + 1058523*b10 + 395685*b9 - 613826*b8 - 138394*b7 - 613826*b5 + 584330*b2 - 1434156 $$\nu^{15}$$ $$=$$ $$613826 \beta_{15} - 613826 \beta_{14} + 2336403 \beta_{13} + 2682339 \beta_{12} + 2474756 \beta_{11} + \cdots + 613826$$ 613826*b15 - 613826*b14 + 2336403*b13 + 2682339*b12 + 2474756*b11 - 1584311*b8 + 3197065*b6 + 1584311*b5 + 534079*b4 + 1227652*b3 + 613826*b2 - 3877762*b1 + 613826

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$581$$ $$1857$$ $$2031$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
929.1
 − 3.21506i − 2.60186i − 2.59166i − 2.04061i − 1.61606i − 1.25178i − 0.877146i − 0.254777i 0.254777i 0.877146i 1.25178i 1.61606i 2.04061i 2.59166i 2.60186i 3.21506i
0 3.21506i 0 −2.23110 0.148957i 0 0.627864i 0 −7.33659 0
929.2 0 2.60186i 0 0.149706 + 2.23105i 0 3.57384i 0 −3.76968 0
929.3 0 2.59166i 0 2.07683 + 0.828709i 0 2.97644i 0 −3.71672 0
929.4 0 2.04061i 0 1.92659 1.13502i 0 3.45115i 0 −1.16410 0
929.5 0 1.61606i 0 1.41675 1.72998i 0 1.06577i 0 0.388359 0
929.6 0 1.25178i 0 −2.17253 + 0.529275i 0 0.133263i 0 1.43304 0
929.7 0 0.877146i 0 −1.53115 + 1.62959i 0 3.05899i 0 2.23061 0
929.8 0 0.254777i 0 −0.635094 + 2.14398i 0 1.59775i 0 2.93509 0
929.9 0 0.254777i 0 −0.635094 2.14398i 0 1.59775i 0 2.93509 0
929.10 0 0.877146i 0 −1.53115 1.62959i 0 3.05899i 0 2.23061 0
929.11 0 1.25178i 0 −2.17253 0.529275i 0 0.133263i 0 1.43304 0
929.12 0 1.61606i 0 1.41675 + 1.72998i 0 1.06577i 0 0.388359 0
929.13 0 2.04061i 0 1.92659 + 1.13502i 0 3.45115i 0 −1.16410 0
929.14 0 2.59166i 0 2.07683 0.828709i 0 2.97644i 0 −3.71672 0
929.15 0 2.60186i 0 0.149706 2.23105i 0 3.57384i 0 −3.76968 0
929.16 0 3.21506i 0 −2.23110 + 0.148957i 0 0.627864i 0 −7.33659 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 929.16 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 2320.2.d.j 16
4.b odd 2 1 1160.2.d.c 16
5.b even 2 1 inner 2320.2.d.j 16
20.d odd 2 1 1160.2.d.c 16
20.e even 4 1 5800.2.a.be 8
20.e even 4 1 5800.2.a.bf 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.d.c 16 4.b odd 2 1
1160.2.d.c 16 20.d odd 2 1
2320.2.d.j 16 1.a even 1 1 trivial
2320.2.d.j 16 5.b even 2 1 inner
5800.2.a.be 8 20.e even 4 1
5800.2.a.bf 8 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2320, [\chi])$$:

 $$T_{3}^{16} + 33T_{3}^{14} + 432T_{3}^{12} + 2881T_{3}^{10} + 10441T_{3}^{8} + 20304T_{3}^{6} + 19480T_{3}^{4} + 7344T_{3}^{2} + 400$$ T3^16 + 33*T3^14 + 432*T3^12 + 2881*T3^10 + 10441*T3^8 + 20304*T3^6 + 19480*T3^4 + 7344*T3^2 + 400 $$T_{7}^{16} + 47T_{7}^{14} + 865T_{7}^{12} + 7816T_{7}^{10} + 35448T_{7}^{8} + 73872T_{7}^{6} + 61712T_{7}^{4} + 15488T_{7}^{2} + 256$$ T7^16 + 47*T7^14 + 865*T7^12 + 7816*T7^10 + 35448*T7^8 + 73872*T7^6 + 61712*T7^4 + 15488*T7^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} + 33 T^{14} + \cdots + 400$$
$5$ $$T^{16} + 2 T^{15} + \cdots + 390625$$
$7$ $$T^{16} + 47 T^{14} + \cdots + 256$$
$11$ $$(T^{8} + 4 T^{7} - 39 T^{6} + \cdots - 80)^{2}$$
$13$ $$T^{16} + 101 T^{14} + \cdots + 6400$$
$17$ $$T^{16} + 91 T^{14} + \cdots + 50176$$
$19$ $$(T^{8} - 8 T^{7} + \cdots - 5312)^{2}$$
$23$ $$T^{16} + 131 T^{14} + \cdots + 25887744$$
$29$ $$(T + 1)^{16}$$
$31$ $$(T^{8} + 17 T^{7} + \cdots + 84)^{2}$$
$37$ $$T^{16} + 280 T^{14} + \cdots + 26214400$$
$41$ $$(T^{8} + 18 T^{7} + \cdots - 434176)^{2}$$
$43$ $$T^{16} + \cdots + 2879442459664$$
$47$ $$T^{16} + \cdots + 26073206784$$
$53$ $$T^{16} + \cdots + 60390113536$$
$59$ $$(T^{8} + T^{7} + \cdots + 288128)^{2}$$
$61$ $$(T^{8} + 17 T^{7} + \cdots - 124928)^{2}$$
$67$ $$T^{16} + \cdots + 715418238976$$
$71$ $$(T^{8} - 6 T^{7} + \cdots + 240384)^{2}$$
$73$ $$T^{16} + \cdots + 747159699456$$
$79$ $$(T^{8} - 39 T^{7} + \cdots - 97924)^{2}$$
$83$ $$T^{16} + \cdots + 8947646464$$
$89$ $$(T^{8} - 36 T^{7} + \cdots - 2757376)^{2}$$
$97$ $$T^{16} + \cdots + 10983656505600$$