# Properties

 Label 1040.2.k.e Level $1040$ Weight $2$ Character orbit 1040.k Analytic conductor $8.304$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(961,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

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("1040.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.k (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: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196$$ x^8 - 2*x^7 + 2*x^6 + 6*x^5 + 36*x^4 - 52*x^3 + 50*x^2 + 140*x + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 520) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} + 2) q^{9}+O(q^{10})$$ q + b3 * q^3 - b2 * q^5 + (b5 - b4 - b2) * q^7 + (b7 + 2) * q^9 $$q + \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} + 2) q^{9} - \beta_{4} q^{11} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} + \beta_{2}) q^{15} + ( - \beta_{7} - \beta_{3} + \beta_1 - 1) q^{17} + (\beta_{6} - \beta_{5}) q^{19} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{21} + (\beta_{3} + 2 \beta_1 + 2) q^{23} - q^{25} + ( - \beta_{7} + 3 \beta_{3} - \beta_1 + 1) q^{27} + ( - \beta_{7} - 2 \beta_{3} + 1) q^{29} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{2}) q^{31} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{33} + ( - \beta_{3} + \beta_1 - 2) q^{35} + (3 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{37} + (\beta_{7} + 2 \beta_{5} - \beta_{4} + 5) q^{39} + ( - \beta_{6} + \beta_{5} + \beta_{4}) q^{41} + (\beta_{7} + \beta_1 + 3) q^{43} + (\beta_{6} - 2 \beta_{2}) q^{45} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2}) q^{47} + ( - 2 \beta_{3} + 4 \beta_1 - 5) q^{49} + ( - 4 \beta_{3} - 4) q^{51} + (\beta_{7} - \beta_{3} - \beta_1 - 1) q^{53} + \beta_1 q^{55} + ( - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 7 \beta_{2}) q^{57} + ( - \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{59} + ( - \beta_{7} + 2 \beta_{3} + 2 \beta_1 + 5) q^{61} + (5 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{63} + (\beta_{5} + \beta_{2} - \beta_1 - 1) q^{65} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2}) q^{67} + (\beta_{7} + 4 \beta_{3} - 2 \beta_1 + 9) q^{69} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{71} + (\beta_{5} - 3 \beta_{4} + \beta_{2}) q^{73} - \beta_{3} q^{75} + (\beta_{7} + \beta_{3} + 3 \beta_1 - 5) q^{77} + (2 \beta_{3} + 2 \beta_1 + 4) q^{79} + (\beta_{7} - 4 \beta_{3} + 2 \beta_1 + 6) q^{81} + (3 \beta_{5} - \beta_{4} + 9 \beta_{2}) q^{83} + ( - \beta_{6} - \beta_{5} + \beta_{4}) q^{85} + ( - \beta_{7} - 3 \beta_{3} + \beta_1 - 11) q^{87} + ( - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}) q^{89} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{91} + ( - 7 \beta_{5} - \beta_{4} - 5 \beta_{2}) q^{93} + ( - \beta_{7} + \beta_{3} + 1) q^{95} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2}) q^{97} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{99}+O(q^{100})$$ q + b3 * q^3 - b2 * q^5 + (b5 - b4 - b2) * q^7 + (b7 + 2) * q^9 - b4 * q^11 + (b4 + b3 - b2) * q^13 + (b5 + b2) * q^15 + (-b7 - b3 + b1 - 1) * q^17 + (b6 - b5) * q^19 + (b6 + b5 + b4 - 2*b2) * q^21 + (b3 + 2*b1 + 2) * q^23 - q^25 + (-b7 + 3*b3 - b1 + 1) * q^27 + (-b7 - 2*b3 + 1) * q^29 + (-b6 - b5 + 2*b4 + 4*b2) * q^31 + (-b5 + b4 + b2) * q^33 + (-b3 + b1 - 2) * q^35 + (3*b5 - b4 + 3*b2) * q^37 + (b7 + 2*b5 - b4 + 5) * q^39 + (-b6 + b5 + b4) * q^41 + (b7 + b1 + 3) * q^43 + (b6 - 2*b2) * q^45 + (2*b6 + b5 - b4 + b2) * q^47 + (-2*b3 + 4*b1 - 5) * q^49 + (-4*b3 - 4) * q^51 + (b7 - b3 - b1 - 1) * q^53 + b1 * q^55 + (-2*b6 + 3*b5 - b4 + 7*b2) * q^57 + (-b6 - b5 - 2*b2) * q^59 + (-b7 + 2*b3 + 2*b1 + 5) * q^61 + (5*b5 + b4 + 3*b2) * q^63 + (b5 + b2 - b1 - 1) * q^65 + (-2*b6 - b5 - b4 - b2) * q^67 + (b7 + 4*b3 - 2*b1 + 9) * q^69 + (-2*b6 - 2*b5 + b4 + 2*b2) * q^71 + (b5 - 3*b4 + b2) * q^73 - b3 * q^75 + (b7 + b3 + 3*b1 - 5) * q^77 + (2*b3 + 2*b1 + 4) * q^79 + (b7 - 4*b3 + 2*b1 + 6) * q^81 + (3*b5 - b4 + 9*b2) * q^83 + (-b6 - b5 + b4) * q^85 + (-b7 - 3*b3 + b1 - 11) * q^87 + (-2*b5 + 2*b4 + 6*b2) * q^89 + (-b7 + b6 + b5 + b4 - 2*b3 - 2*b2 - 2*b1 + 3) * q^91 + (-7*b5 - b4 - 5*b2) * q^93 + (-b7 + b3 + 1) * q^95 + (-2*b5 - 2*b4 + 4*b2) * q^97 + (-b6 - b5 + 2*b4 + 2*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{3} + 16 q^{9}+O(q^{10})$$ 8 * q - 4 * q^3 + 16 * q^9 $$8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100})$$ 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196$$ :

 $$\beta_{1}$$ $$=$$ $$( -295\nu^{7} + 581\nu^{6} - 1975\nu^{5} + 12197\nu^{4} - 28834\nu^{3} + 14811\nu^{2} + 52682\nu + 217006 ) / 103451$$ (-295*v^7 + 581*v^6 - 1975*v^5 + 12197*v^4 - 28834*v^3 + 14811*v^2 + 52682*v + 217006) / 103451 $$\beta_{2}$$ $$=$$ $$( - 4733 \nu^{7} + 18790 \nu^{6} - 22920 \nu^{5} - 15070 \nu^{4} - 128766 \nu^{3} + 794160 \nu^{2} - 577130 \nu - 329546 ) / 1448314$$ (-4733*v^7 + 18790*v^6 - 22920*v^5 - 15070*v^4 - 128766*v^3 + 794160*v^2 - 577130*v - 329546) / 1448314 $$\beta_{3}$$ $$=$$ $$( 371\nu^{7} - 380\nu^{6} - 1023\nu^{5} + 15170\nu^{4} + 10312\nu^{3} - 9509\nu^{2} - 26978\nu + 283268 ) / 103451$$ (371*v^7 - 380*v^6 - 1023*v^5 + 15170*v^4 + 10312*v^3 - 9509*v^2 - 26978*v + 283268) / 103451 $$\beta_{4}$$ $$=$$ $$( 4804 \nu^{7} - 30853 \nu^{6} + 39176 \nu^{5} + 9329 \nu^{4} - 16490 \nu^{3} - 693923 \nu^{2} + 263566 \nu + 195258 ) / 724157$$ (4804*v^7 - 30853*v^6 + 39176*v^5 + 9329*v^4 - 16490*v^3 - 693923*v^2 + 263566*v + 195258) / 724157 $$\beta_{5}$$ $$=$$ $$( 23665 \nu^{7} - 93950 \nu^{6} + 114600 \nu^{5} + 75350 \nu^{4} + 643830 \nu^{3} - 2522486 \nu^{2} + 2885650 \nu + 1647730 ) / 1448314$$ (23665*v^7 - 93950*v^6 + 114600*v^5 + 75350*v^4 + 643830*v^3 - 2522486*v^2 + 2885650*v + 1647730) / 1448314 $$\beta_{6}$$ $$=$$ $$( 37489 \nu^{7} - 35610 \nu^{6} + 179096 \nu^{5} + 120284 \nu^{4} + 1042570 \nu^{3} + 42688 \nu^{2} + 4620886 \nu + 2631678 ) / 1448314$$ (37489*v^7 - 35610*v^6 + 179096*v^5 + 120284*v^4 + 1042570*v^3 + 42688*v^2 + 4620886*v + 2631678) / 1448314 $$\beta_{7}$$ $$=$$ $$( 2968\nu^{7} - 3040\nu^{6} - 8184\nu^{5} + 17909\nu^{4} + 82496\nu^{3} - 76072\nu^{2} - 215824\nu + 93673 ) / 103451$$ (2968*v^7 - 3040*v^6 - 8184*v^5 + 17909*v^4 + 82496*v^3 - 76072*v^2 - 215824*v + 93673) / 103451
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b5 - b4 - b3 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 5\beta_{2}$$ b5 + 5*b2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + 5\beta_{5} - 5\beta_{4} + 5\beta_{3} + 8\beta_{2} - 5\beta _1 - 3 ) / 2$$ (-b7 - b6 + 5*b5 - 5*b4 + 5*b3 + 8*b2 - 5*b1 - 3) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 8\beta_{3} - 21$$ -b7 + 8*b3 - 21 $$\nu^{5}$$ $$=$$ $$-5\beta_{7} + 5\beta_{6} - 16\beta_{5} + 13\beta_{4} + 16\beta_{3} - 29\beta_{2} - 13\beta _1 - 13$$ -5*b7 + 5*b6 - 16*b5 + 13*b4 + 16*b3 - 29*b2 - 13*b1 - 13 $$\nu^{6}$$ $$=$$ $$13\beta_{6} - 57\beta_{5} + \beta_{4} - 180\beta_{2}$$ 13*b6 - 57*b5 + b4 - 180*b2 $$\nu^{7}$$ $$=$$ $$41\beta_{7} + 41\beta_{6} - 110\beta_{5} + 70\beta_{4} - 110\beta_{3} - 211\beta_{2} + 70\beta _1 + 101$$ 41*b7 + 41*b6 - 110*b5 + 70*b4 - 110*b3 - 211*b2 + 70*b1 + 101

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 1.90427 + 1.90427i 1.90427 − 1.90427i −1.61013 − 1.61013i −1.61013 + 1.61013i 1.47812 + 1.47812i 1.47812 − 1.47812i −0.772270 − 0.772270i −0.772270 + 0.772270i
0 −3.25249 0 1.00000i 0 1.80854i 0 7.57872 0
961.2 0 −3.25249 0 1.00000i 0 1.80854i 0 7.57872 0
961.3 0 −1.18501 0 1.00000i 0 5.22025i 0 −1.59576 0
961.4 0 −1.18501 0 1.00000i 0 5.22025i 0 −1.59576 0
961.5 0 −0.369700 0 1.00000i 0 0.956248i 0 −2.86332 0
961.6 0 −0.369700 0 1.00000i 0 0.956248i 0 −2.86332 0
961.7 0 2.80720 0 1.00000i 0 3.54454i 0 4.88037 0
961.8 0 2.80720 0 1.00000i 0 3.54454i 0 4.88037 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.8 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.k.e 8
4.b odd 2 1 520.2.k.b 8
12.b even 2 1 4680.2.g.k 8
13.b even 2 1 inner 1040.2.k.e 8
20.d odd 2 1 2600.2.k.c 8
20.e even 4 1 2600.2.f.e 8
20.e even 4 1 2600.2.f.f 8
52.b odd 2 1 520.2.k.b 8
52.f even 4 1 6760.2.a.bc 4
52.f even 4 1 6760.2.a.bd 4
156.h even 2 1 4680.2.g.k 8
260.g odd 2 1 2600.2.k.c 8
260.p even 4 1 2600.2.f.e 8
260.p even 4 1 2600.2.f.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.k.b 8 4.b odd 2 1
520.2.k.b 8 52.b odd 2 1
1040.2.k.e 8 1.a even 1 1 trivial
1040.2.k.e 8 13.b even 2 1 inner
2600.2.f.e 8 20.e even 4 1
2600.2.f.e 8 260.p even 4 1
2600.2.f.f 8 20.e even 4 1
2600.2.f.f 8 260.p even 4 1
2600.2.k.c 8 20.d odd 2 1
2600.2.k.c 8 260.g odd 2 1
4680.2.g.k 8 12.b even 2 1
4680.2.g.k 8 156.h even 2 1
6760.2.a.bc 4 52.f even 4 1
6760.2.a.bd 4 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 14T_{3} - 4$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 2 T^{3} - 8 T^{2} - 14 T - 4)^{2}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} + 44 T^{6} + 512 T^{4} + \cdots + 1024$$
$11$ $$T^{8} + 28 T^{6} + 180 T^{4} + \cdots + 64$$
$13$ $$T^{8} + 4 T^{7} + 20 T^{6} + \cdots + 28561$$
$17$ $$(T^{4} + 2 T^{3} - 40 T^{2} - 128 T - 64)^{2}$$
$19$ $$T^{8} + 104 T^{6} + 3004 T^{4} + \cdots + 18496$$
$23$ $$(T^{4} - 6 T^{3} - 64 T^{2} + 390 T + 56)^{2}$$
$29$ $$(T^{4} - 8 T^{3} - 44 T^{2} + 452 T - 664)^{2}$$
$31$ $$T^{8} + 216 T^{6} + 16612 T^{4} + \cdots + 6130576$$
$37$ $$T^{8} + 172 T^{6} + 7728 T^{4} + \cdots + 53824$$
$41$ $$T^{8} + 124 T^{6} + 4144 T^{4} + \cdots + 16384$$
$43$ $$(T^{4} - 12 T^{3} - 8 T^{2} + 338 T - 236)^{2}$$
$47$ $$T^{8} + 284 T^{6} + 24384 T^{4} + \cdots + 7311616$$
$53$ $$(T^{4} + 2 T^{3} - 60 T^{2} - 72 T + 128)^{2}$$
$59$ $$T^{8} + 96 T^{6} + 3052 T^{4} + \cdots + 50176$$
$61$ $$(T^{4} - 16 T^{3} - 48 T^{2} + 876 T + 2264)^{2}$$
$67$ $$T^{8} + 388 T^{6} + \cdots + 63744256$$
$71$ $$T^{8} + 348 T^{6} + 35212 T^{4} + \cdots + 1430416$$
$73$ $$T^{8} + 236 T^{6} + 14896 T^{4} + \cdots + 222784$$
$79$ $$(T^{4} - 12 T^{3} - 64 T^{2} + 672 T + 1024)^{2}$$
$83$ $$T^{8} + 388 T^{6} + 35088 T^{4} + \cdots + 891136$$
$89$ $$T^{8} + 336 T^{6} + 27136 T^{4} + \cdots + 16384$$
$97$ $$T^{8} + 336 T^{6} + 30176 T^{4} + \cdots + 215296$$