# Properties

 Label 1840.2.e.d Level $1840$ Weight $2$ Character orbit 1840.e Analytic conductor $14.692$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.527896576.2 Defining polynomial: $$x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} + 4 x + 1$$ 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

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$25 \nu^{7} - 46 \nu^{6} + 80 \nu^{5} - 124 \nu^{4} + 398 \nu^{3} - 205 \nu^{2} + 354 \nu - 740$$$$)/467$$ $$\beta_{2}$$ $$=$$ $$($$$$-27 \nu^{7} + 31 \nu^{6} + 7 \nu^{5} - 221 \nu^{4} - 187 \nu^{3} + 128 \nu^{2} - 401 \nu - 882$$$$)/467$$ $$\beta_{3}$$ $$=$$ $$($$$$-108 \nu^{7} + 124 \nu^{6} + 28 \nu^{5} - 417 \nu^{4} - 748 \nu^{3} + 512 \nu^{2} + 731 \nu - 726$$$$)/467$$ $$\beta_{4}$$ $$=$$ $$($$$$-142 \nu^{7} + 336 \nu^{6} - 361 \nu^{5} - 211 \nu^{4} - 897 \nu^{3} + 2005 \nu^{2} - 1469 \nu - 280$$$$)/467$$ $$\beta_{5}$$ $$=$$ $$($$$$232 \nu^{7} - 595 \nu^{6} + 649 \nu^{5} + 325 \nu^{4} + 1209 \nu^{3} - 3210 \nu^{2} + 2650 \nu + 418$$$$)/467$$ $$\beta_{6}$$ $$=$$ $$($$$$-269 \nu^{7} + 551 \nu^{6} - 674 \nu^{5} - 403 \nu^{4} - 1742 \nu^{3} + 2486 \nu^{2} - 3286 \nu - 537$$$$)/467$$ $$\beta_{7}$$ $$=$$ $$($$$$284 \nu^{7} - 672 \nu^{6} + 722 \nu^{5} + 422 \nu^{4} + 1794 \nu^{3} - 3543 \nu^{2} + 2471 \nu + 560$$$$)/467$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 4 \beta_{4} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} - 5 \beta_{1} - 12$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$10 \beta_{7} - 5 \beta_{6} - 17 \beta_{5} - 7 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} - 7$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{7} - 6 \beta_{6} - 11 \beta_{5} - 21 \beta_{4} + 12 \beta_{2} - 12 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-51 \beta_{7} - 22 \beta_{6} + 15 \beta_{5} - 38 \beta_{4} - 22 \beta_{3} + 51 \beta_{2} - 15 \beta_{1} + 38$$$$)/2$$

## 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.

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 $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$25 + 96 T^{2} + 70 T^{4} + 16 T^{6} + T^{8}$$
$5$ $$625 + 750 T + 650 T^{2} + 410 T^{3} + 206 T^{4} + 82 T^{5} + 26 T^{6} + 6 T^{7} + T^{8}$$
$7$ $$16 + 176 T^{2} + 152 T^{4} + 28 T^{6} + T^{8}$$
$11$ $$( -28 - 44 T - 16 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$3721 + 5872 T^{2} + 1174 T^{4} + 64 T^{6} + T^{8}$$
$17$ $$400 + 1776 T^{2} + 1612 T^{4} + 80 T^{6} + T^{8}$$
$19$ $$( -20 - 28 T - 6 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$23$ $$( 1 + T^{2} )^{4}$$
$29$ $$( 5 - 4 T - 22 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$( -167 + 256 T - 74 T^{2} + T^{4} )^{2}$$
$37$ $$226576 + 75856 T^{2} + 5752 T^{4} + 148 T^{6} + T^{8}$$
$41$ $$( 2485 - 368 T - 94 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$43$ $$3857296 + 653600 T^{2} + 21340 T^{4} + 252 T^{6} + T^{8}$$
$47$ $$20367169 + 1273144 T^{2} + 28734 T^{4} + 280 T^{6} + T^{8}$$
$53$ $$4804864 + 468480 T^{2} + 16096 T^{4} + 224 T^{6} + T^{8}$$
$59$ $$( 16 + 16 T - 20 T^{2} + T^{4} )^{2}$$
$61$ $$( -2756 - 1148 T - 102 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$67$ $$16 + 208 T^{2} + 120 T^{4} + 20 T^{6} + T^{8}$$
$71$ $$( -7435 + 1568 T + 66 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$73$ $$69538921 + 3070400 T^{2} + 50550 T^{4} + 368 T^{6} + T^{8}$$
$79$ $$( 28 - 412 T + 178 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$83$ $$223442704 + 9024496 T^{2} + 114508 T^{4} + 576 T^{6} + T^{8}$$
$89$ $$( 2380 + 372 T - 86 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$97$ $$21864976 + 3638896 T^{2} + 69592 T^{4} + 460 T^{6} + T^{8}$$