# Properties

 Label 1840.2.e.d Level $1840$ Weight $2$ Character orbit 1840.e Analytic conductor $14.692$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.e (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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.527896576.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1$$ x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 7*x^4 - 10*x^3 + 8*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 115) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{6} - 1) q^{5} + (\beta_{7} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{5} - \beta_{2} + \cdots - 2) q^{9}+O(q^{10})$$ q + (b4 + b2 - b1) * q^3 + (-b6 - 1) * q^5 + (b7 - b6 - b4) * q^7 + (-b7 + b5 - b2 - b1 - 2) * q^9 $$q + (\beta_{4} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{6} - 1) q^{5} + (\beta_{7} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{5} - \beta_{2} + \cdots - 2) q^{9}+ \cdots + (2 \beta_{7} - 2 \beta_{5} + 4 \beta_{3} + \cdots + 12) q^{99}+O(q^{100})$$ q + (b4 + b2 - b1) * q^3 + (-b6 - 1) * q^5 + (b7 - b6 - b4) * q^7 + (-b7 + b5 - b2 - b1 - 2) * q^9 + (-b7 + b5 + b3 - b1 - 1) * q^11 + (-3*b4 + b2 - b1) * q^13 + (-b5 - 3*b4 - 2*b2 - 1) * q^15 + (b7 + b5 + 4*b4) * q^17 + (b3 - b1 - 1) * q^19 + (b7 - b5 - b3 - 2*b2 - b1 - 1) * q^21 - b4 * q^23 + (b6 - 2*b5 - b4 + b2 - 1) * q^25 + (-3*b7 + 2*b6 - b5 - 5*b4) * q^27 + (b7 - b5 - 2*b3 - b2 + b1 - 1) * q^29 + (-b7 + b5 + 2*b3 + 3*b2 + b1 + 1) * q^31 + (-b7 + b6 - 5*b4 - 2*b2 + 2*b1) * q^33 + (-2*b7 + b6 + 2*b3 + 2*b2 - b1 - 3) * q^35 + (-b7 - b6 - 2*b5 + b4 + 2*b2 - 2*b1) * q^37 + (3*b7 - 3*b5 - b2 - b1 - 1) * q^39 + (-4*b7 + 4*b5 + 2*b3 + 2*b2 - 3) * q^41 + (-b6 - b5 - b4 - 4*b2 + 4*b1) * q^43 + (2*b7 - b6 - 3*b5 - b4 + b3 + 3*b1 + 5) * q^45 + (3*b7 + 3*b5 + b4) * q^47 + (-2*b7 + 2*b5 + 4*b3 + 4*b2 + 1) * q^49 + (-5*b7 + 5*b5 - 2*b3 - 2*b2 - 6) * q^51 + (-2*b7 + 4*b6 + 2*b5 - 2*b4) * q^53 + (b7 - b6 - 3*b5 + 2*b4 - b2 + 2*b1 + 3) * q^55 + (b6 + b5 - b4 - 2*b2 + 2*b1) * q^57 + (b7 - b5 - b2 - b1) * q^59 + (-2*b7 + 2*b5 + 3*b3 - 3*b1 - 3) * q^61 + (-4*b4 - 2*b2 + 2*b1) * q^63 + (-b5 + b4 + 4*b3 + 2*b2 - 4*b1 - 1) * q^65 + (b7 - b6 + b4) * q^67 + (b7 - b5 + 1) * q^69 + (-2*b7 + 2*b5 + 2*b3 + 4*b2 + 2*b1 + 7) * q^71 + (b7 - 4*b6 - 3*b5 - 5*b4 - 2*b2 + 2*b1) * q^73 + (4*b7 - b6 + 6*b4 + 2*b3 + 2*b2 + b1 + 2) * q^75 + (-2*b7 + 2*b6) * q^77 + (-b3 - 2*b2 - b1 + 5) * q^79 + (3*b7 - 3*b5 + 4*b3 + 3*b2 - b1 + 5) * q^81 + (-3*b7 + 4*b6 + b5 + 6*b2 - 6*b1) * q^83 + (-4*b7 + b6 + 2*b5 - 5*b4 - 2*b3 - 3*b2 + 4*b1) * q^85 + (-b7 - b5 + b4) * q^87 + (b3 - 2*b2 - 3*b1 + 1) * q^89 + (b7 - b5 + 3*b3 + 2*b2 - b1 - 5) * q^91 + (5*b7 - 4*b6 + b5 + 3*b4 + 2*b2 - 2*b1) * q^93 + (b7 - b5 + 3*b4 - 2*b2 + 2*b1 + 1) * q^95 + (3*b7 - 3*b6 + 9*b4 - 2*b2 + 2*b1) * q^97 + (2*b7 - 2*b5 + 4*b3 + 4*b2 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{5} - 8 q^{9}+O(q^{10})$$ 8 * q - 6 * q^5 - 8 * q^9 $$8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100})$$ 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 25\nu^{7} - 46\nu^{6} + 80\nu^{5} - 124\nu^{4} + 398\nu^{3} - 205\nu^{2} + 354\nu - 740 ) / 467$$ (25*v^7 - 46*v^6 + 80*v^5 - 124*v^4 + 398*v^3 - 205*v^2 + 354*v - 740) / 467 $$\beta_{2}$$ $$=$$ $$( -27\nu^{7} + 31\nu^{6} + 7\nu^{5} - 221\nu^{4} - 187\nu^{3} + 128\nu^{2} - 401\nu - 882 ) / 467$$ (-27*v^7 + 31*v^6 + 7*v^5 - 221*v^4 - 187*v^3 + 128*v^2 - 401*v - 882) / 467 $$\beta_{3}$$ $$=$$ $$( -108\nu^{7} + 124\nu^{6} + 28\nu^{5} - 417\nu^{4} - 748\nu^{3} + 512\nu^{2} + 731\nu - 726 ) / 467$$ (-108*v^7 + 124*v^6 + 28*v^5 - 417*v^4 - 748*v^3 + 512*v^2 + 731*v - 726) / 467 $$\beta_{4}$$ $$=$$ $$( -142\nu^{7} + 336\nu^{6} - 361\nu^{5} - 211\nu^{4} - 897\nu^{3} + 2005\nu^{2} - 1469\nu - 280 ) / 467$$ (-142*v^7 + 336*v^6 - 361*v^5 - 211*v^4 - 897*v^3 + 2005*v^2 - 1469*v - 280) / 467 $$\beta_{5}$$ $$=$$ $$( 232\nu^{7} - 595\nu^{6} + 649\nu^{5} + 325\nu^{4} + 1209\nu^{3} - 3210\nu^{2} + 2650\nu + 418 ) / 467$$ (232*v^7 - 595*v^6 + 649*v^5 + 325*v^4 + 1209*v^3 - 3210*v^2 + 2650*v + 418) / 467 $$\beta_{6}$$ $$=$$ $$( -269\nu^{7} + 551\nu^{6} - 674\nu^{5} - 403\nu^{4} - 1742\nu^{3} + 2486\nu^{2} - 3286\nu - 537 ) / 467$$ (-269*v^7 + 551*v^6 - 674*v^5 - 403*v^4 - 1742*v^3 + 2486*v^2 - 3286*v - 537) / 467 $$\beta_{7}$$ $$=$$ $$( 284\nu^{7} - 672\nu^{6} + 722\nu^{5} + 422\nu^{4} + 1794\nu^{3} - 3543\nu^{2} + 2471\nu + 560 ) / 467$$ (284*v^7 - 672*v^6 + 722*v^5 + 422*v^4 + 1794*v^3 - 3543*v^2 + 2471*v + 560) / 467
 $$\nu$$ $$=$$ $$( -\beta_{7} + \beta_{5} - \beta_{2} + \beta_1 ) / 2$$ (-b7 + b5 - b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{5} + 4\beta_{4} - \beta_{2} + \beta_1 ) / 2$$ (b7 + b5 + 4*b4 - b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 3\beta_{2} + 2\beta _1 - 1 ) / 2$$ (3*b7 + b6 - 2*b5 + b4 + b3 - 3*b2 + 2*b1 - 1) / 2 $$\nu^{4}$$ $$=$$ $$( 5\beta_{7} - 5\beta_{5} + 2\beta_{3} - 3\beta_{2} - 5\beta _1 - 12 ) / 2$$ (5*b7 - 5*b5 + 2*b3 - 3*b2 - 5*b1 - 12) / 2 $$\nu^{5}$$ $$=$$ $$( 10\beta_{7} - 5\beta_{6} - 17\beta_{5} - 7\beta_{4} + 5\beta_{3} + 10\beta_{2} - 17\beta _1 - 7 ) / 2$$ (10*b7 - 5*b6 - 17*b5 - 7*b4 + 5*b3 + 10*b2 - 17*b1 - 7) / 2 $$\nu^{6}$$ $$=$$ $$-5\beta_{7} - 6\beta_{6} - 11\beta_{5} - 21\beta_{4} + 12\beta_{2} - 12\beta_1$$ -5*b7 - 6*b6 - 11*b5 - 21*b4 + 12*b2 - 12*b1 $$\nu^{7}$$ $$=$$ $$( -51\beta_{7} - 22\beta_{6} + 15\beta_{5} - 38\beta_{4} - 22\beta_{3} + 51\beta_{2} - 15\beta _1 + 38 ) / 2$$ (-51*b7 - 22*b6 + 15*b5 - 38*b4 - 22*b3 + 51*b2 - 15*b1 + 38) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 −1.07037 + 1.07037i 1.47984 + 1.47984i −0.199724 + 0.199724i 0.790245 + 0.790245i 0.790245 − 0.790245i −0.199724 − 0.199724i 1.47984 − 1.47984i −1.07037 − 1.07037i
0 3.14073i 0 −2.07037 0.844739i 0 1.20647i 0 −6.86420 0
369.2 0 1.95969i 0 0.479844 2.18398i 0 2.28394i 0 −0.840379 0
369.3 0 1.39945i 0 −1.19972 + 1.88697i 0 4.60747i 0 1.04155 0
369.4 0 0.580491i 0 −0.209755 + 2.22621i 0 0.315061i 0 2.66303 0
369.5 0 0.580491i 0 −0.209755 2.22621i 0 0.315061i 0 2.66303 0
369.6 0 1.39945i 0 −1.19972 1.88697i 0 4.60747i 0 1.04155 0
369.7 0 1.95969i 0 0.479844 + 2.18398i 0 2.28394i 0 −0.840379 0
369.8 0 3.14073i 0 −2.07037 + 0.844739i 0 1.20647i 0 −6.86420 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1840.2.e.d 8
4.b odd 2 1 115.2.b.b 8
5.b even 2 1 inner 1840.2.e.d 8
5.c odd 4 1 9200.2.a.ck 4
5.c odd 4 1 9200.2.a.cq 4
12.b even 2 1 1035.2.b.e 8
20.d odd 2 1 115.2.b.b 8
20.e even 4 1 575.2.a.i 4
20.e even 4 1 575.2.a.j 4
60.h even 2 1 1035.2.b.e 8
60.l odd 4 1 5175.2.a.bv 4
60.l odd 4 1 5175.2.a.bw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 4.b odd 2 1
115.2.b.b 8 20.d odd 2 1
575.2.a.i 4 20.e even 4 1
575.2.a.j 4 20.e even 4 1
1035.2.b.e 8 12.b even 2 1
1035.2.b.e 8 60.h even 2 1
1840.2.e.d 8 1.a even 1 1 trivial
1840.2.e.d 8 5.b even 2 1 inner
5175.2.a.bv 4 60.l odd 4 1
5175.2.a.bw 4 60.l odd 4 1
9200.2.a.ck 4 5.c odd 4 1
9200.2.a.cq 4 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}^{8} + 16T_{3}^{6} + 70T_{3}^{4} + 96T_{3}^{2} + 25$$ T3^8 + 16*T3^6 + 70*T3^4 + 96*T3^2 + 25 $$T_{7}^{8} + 28T_{7}^{6} + 152T_{7}^{4} + 176T_{7}^{2} + 16$$ T7^8 + 28*T7^6 + 152*T7^4 + 176*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 16 T^{6} + \cdots + 25$$
$5$ $$T^{8} + 6 T^{7} + \cdots + 625$$
$7$ $$T^{8} + 28 T^{6} + \cdots + 16$$
$11$ $$(T^{4} + 2 T^{3} - 16 T^{2} + \cdots - 28)^{2}$$
$13$ $$T^{8} + 64 T^{6} + \cdots + 3721$$
$17$ $$T^{8} + 80 T^{6} + \cdots + 400$$
$19$ $$(T^{4} + 4 T^{3} - 6 T^{2} + \cdots - 20)^{2}$$
$23$ $$(T^{2} + 1)^{4}$$
$29$ $$(T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 5)^{2}$$
$31$ $$(T^{4} - 74 T^{2} + \cdots - 167)^{2}$$
$37$ $$T^{8} + 148 T^{6} + \cdots + 226576$$
$41$ $$(T^{4} + 8 T^{3} + \cdots + 2485)^{2}$$
$43$ $$T^{8} + 252 T^{6} + \cdots + 3857296$$
$47$ $$T^{8} + 280 T^{6} + \cdots + 20367169$$
$53$ $$T^{8} + 224 T^{6} + \cdots + 4804864$$
$59$ $$(T^{4} - 20 T^{2} + \cdots + 16)^{2}$$
$61$ $$(T^{4} + 8 T^{3} + \cdots - 2756)^{2}$$
$67$ $$T^{8} + 20 T^{6} + \cdots + 16$$
$71$ $$(T^{4} - 24 T^{3} + \cdots - 7435)^{2}$$
$73$ $$T^{8} + 368 T^{6} + \cdots + 69538921$$
$79$ $$(T^{4} - 24 T^{3} + \cdots + 28)^{2}$$
$83$ $$T^{8} + 576 T^{6} + \cdots + 223442704$$
$89$ $$(T^{4} - 8 T^{3} + \cdots + 2380)^{2}$$
$97$ $$T^{8} + 460 T^{6} + \cdots + 21864976$$