Properties

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

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.m (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{6} q^{5} + 2 \beta_{5} q^{7} + 2 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{6} q^{5} + 2 \beta_{5} q^{7} + 2 q^{9} + \beta_{4} q^{13} + ( -3 \beta_{5} - \beta_{7} ) q^{15} + ( \beta_{1} - \beta_{6} ) q^{17} + ( -\beta_{5} - 2 \beta_{7} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{6} ) q^{21} + ( -\beta_{2} + 3 \beta_{5} ) q^{23} + \beta_{4} q^{25} + \beta_{2} q^{27} + 9 q^{29} -3 \beta_{3} q^{31} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{35} -5 \beta_{3} q^{39} -3 q^{41} -\beta_{5} q^{43} + 2 \beta_{6} q^{45} -3 \beta_{2} q^{47} - q^{49} + ( \beta_{5} + 2 \beta_{7} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{53} + ( -5 \beta_{1} + 5 \beta_{6} ) q^{57} + 2 \beta_{3} q^{59} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{61} + 4 \beta_{5} q^{63} + 5 \beta_{1} q^{65} + 2 \beta_{5} q^{67} + ( 5 - 3 \beta_{1} - 3 \beta_{6} ) q^{69} -3 \beta_{3} q^{71} -3 \beta_{4} q^{73} -5 \beta_{3} q^{75} + ( \beta_{5} + 2 \beta_{7} ) q^{79} -11 q^{81} + 9 \beta_{5} q^{83} + ( -5 - \beta_{4} ) q^{85} -9 \beta_{2} q^{87} + ( \beta_{1} + \beta_{6} ) q^{89} + ( -2 \beta_{5} - 4 \beta_{7} ) q^{91} + 3 \beta_{4} q^{93} + ( -5 \beta_{2} - 5 \beta_{3} ) q^{95} + ( 6 \beta_{1} - 6 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} + 72q^{29} - 24q^{41} - 8q^{49} + 40q^{69} - 88q^{81} - 40q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 11 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} + 7$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} - 6 \nu^{2}$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{6} - 40 \nu^{2}$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{7} + 29 \nu^{3}$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} - 3 \nu^{5} + 26 \nu^{3} - 15 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{5} + 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{4} + 5 \beta_{3}$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 5 \beta_{6} - 6 \beta_{5}$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{2} - 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{7} + 22 \beta_{5} - 25 \beta_{1}$$$$)/10$$ $$\nu^{6}$$ $$=$$ $$($$$$9 \beta_{4} - 20 \beta_{3}$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{7} - 65 \beta_{6} + 87 \beta_{5}$$$$)/10$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1839.1
 0.437016 − 0.437016i 0.437016 + 0.437016i −0.437016 − 0.437016i −0.437016 + 0.437016i 1.14412 − 1.14412i 1.14412 + 1.14412i −1.14412 − 1.14412i −1.14412 + 1.14412i
0 −2.23607 0 −1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.2 0 −2.23607 0 −1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
1839.3 0 −2.23607 0 1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.4 0 −2.23607 0 1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
1839.5 0 2.23607 0 −1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.6 0 2.23607 0 −1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
1839.7 0 2.23607 0 1.58114 1.58114i 0 2.82843i 0 2.00000 0
1839.8 0 2.23607 0 1.58114 + 1.58114i 0 2.82843i 0 2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1839.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c odd 2 1 inner
460.g 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.m.f 8
4.b odd 2 1 inner 1840.2.m.f 8
5.b even 2 1 inner 1840.2.m.f 8
20.d odd 2 1 inner 1840.2.m.f 8
23.b odd 2 1 inner 1840.2.m.f 8
92.b even 2 1 inner 1840.2.m.f 8
115.c odd 2 1 inner 1840.2.m.f 8
460.g even 2 1 inner 1840.2.m.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.f 8 1.a even 1 1 trivial
1840.2.m.f 8 4.b odd 2 1 inner
1840.2.m.f 8 5.b even 2 1 inner
1840.2.m.f 8 20.d odd 2 1 inner
1840.2.m.f 8 23.b odd 2 1 inner
1840.2.m.f 8 92.b even 2 1 inner
1840.2.m.f 8 115.c odd 2 1 inner
1840.2.m.f 8 460.g even 2 1 inner

Hecke kernels

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

 $$T_{3}^{2} - 5$$ $$T_{7}^{2} + 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -5 + T^{2} )^{4}$$
$5$ $$( 25 + T^{4} )^{2}$$
$7$ $$( 8 + T^{2} )^{4}$$
$11$ $$T^{8}$$
$13$ $$( 25 + T^{2} )^{4}$$
$17$ $$( -10 + T^{2} )^{4}$$
$19$ $$( -50 + T^{2} )^{4}$$
$23$ $$( 529 + 26 T^{2} + T^{4} )^{2}$$
$29$ $$( -9 + T )^{8}$$
$31$ $$( 45 + T^{2} )^{4}$$
$37$ $$T^{8}$$
$41$ $$( 3 + T )^{8}$$
$43$ $$( 2 + T^{2} )^{4}$$
$47$ $$( -45 + T^{2} )^{4}$$
$53$ $$( -40 + T^{2} )^{4}$$
$59$ $$( 20 + T^{2} )^{4}$$
$61$ $$( 90 + T^{2} )^{4}$$
$67$ $$( 8 + T^{2} )^{4}$$
$71$ $$( 45 + T^{2} )^{4}$$
$73$ $$( 225 + T^{2} )^{4}$$
$79$ $$( -50 + T^{2} )^{4}$$
$83$ $$( 162 + T^{2} )^{4}$$
$89$ $$( 10 + T^{2} )^{4}$$
$97$ $$( -360 + T^{2} )^{4}$$