Properties

 Label 2300.2.i.b Level $2300$ Weight $2$ Character orbit 2300.i Analytic conductor $18.366$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.i (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$18.3655924649$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} -\beta_{5} q^{7} + 4 \beta_{3} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} -\beta_{5} q^{7} + 4 \beta_{3} q^{9} -\beta_{1} q^{11} + \beta_{2} q^{13} + \beta_{6} q^{19} -\beta_{1} q^{21} + ( -\beta_{2} + 2 \beta_{4} ) q^{23} + \beta_{7} q^{27} -\beta_{3} q^{29} + 3 q^{31} + 7 \beta_{4} q^{33} + 5 \beta_{5} q^{37} + 7 \beta_{3} q^{39} + 9 q^{41} -3 \beta_{4} q^{43} + \beta_{7} q^{47} -3 \beta_{3} q^{49} -2 \beta_{4} q^{53} + 7 \beta_{5} q^{57} -\beta_{1} q^{61} + 4 \beta_{4} q^{63} + \beta_{5} q^{67} + ( -7 \beta_{3} + 2 \beta_{6} ) q^{69} -9 q^{71} + 5 \beta_{2} q^{73} + 4 \beta_{7} q^{77} + \beta_{6} q^{79} + 5 q^{81} + 6 \beta_{4} q^{83} -\beta_{7} q^{87} -3 \beta_{6} q^{89} -\beta_{1} q^{91} + 3 \beta_{2} q^{93} + \beta_{5} q^{97} + 4 \beta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 24q^{31} + 72q^{41} - 72q^{71} + 40q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{4} + 2$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 11 \nu$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{3}$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{2}$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 13 \nu^{3}$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 2 \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 6 \beta_{3}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 5 \beta_{5}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{4} + 2 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} + 18 \beta_{3}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$14 \beta_{7} - 13 \beta_{5}$$$$)/4$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1057.1
 −0.581861 + 1.28897i −1.28897 + 0.581861i 1.28897 − 0.581861i 0.581861 − 1.28897i −0.581861 − 1.28897i −1.28897 − 0.581861i 1.28897 + 0.581861i 0.581861 + 1.28897i
0 −1.87083 + 1.87083i 0 0 0 −1.41421 + 1.41421i 0 4.00000i 0
1057.2 0 −1.87083 + 1.87083i 0 0 0 1.41421 1.41421i 0 4.00000i 0
1057.3 0 1.87083 1.87083i 0 0 0 −1.41421 + 1.41421i 0 4.00000i 0
1057.4 0 1.87083 1.87083i 0 0 0 1.41421 1.41421i 0 4.00000i 0
1793.1 0 −1.87083 1.87083i 0 0 0 −1.41421 1.41421i 0 4.00000i 0
1793.2 0 −1.87083 1.87083i 0 0 0 1.41421 + 1.41421i 0 4.00000i 0
1793.3 0 1.87083 + 1.87083i 0 0 0 −1.41421 1.41421i 0 4.00000i 0
1793.4 0 1.87083 + 1.87083i 0 0 0 1.41421 + 1.41421i 0 4.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1793.4 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
5.c odd 4 2 inner
23.b odd 2 1 inner
115.c odd 2 1 inner
115.e even 4 2 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.i.b 8
5.b even 2 1 inner 2300.2.i.b 8
5.c odd 4 2 inner 2300.2.i.b 8
23.b odd 2 1 inner 2300.2.i.b 8
115.c odd 2 1 inner 2300.2.i.b 8
115.e even 4 2 inner 2300.2.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.b 8 1.a even 1 1 trivial
2300.2.i.b 8 5.b even 2 1 inner
2300.2.i.b 8 5.c odd 4 2 inner
2300.2.i.b 8 23.b odd 2 1 inner
2300.2.i.b 8 115.c odd 2 1 inner
2300.2.i.b 8 115.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2300, [\chi])$$:

 $$T_{3}^{4} + 49$$ $$T_{19}^{2} - 28$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 49 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 16 + T^{4} )^{2}$$
$11$ $$( 28 + T^{2} )^{4}$$
$13$ $$( 49 + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( -28 + T^{2} )^{4}$$
$23$ $$279841 - 734 T^{4} + T^{8}$$
$29$ $$( 1 + T^{2} )^{4}$$
$31$ $$( -3 + T )^{8}$$
$37$ $$( 10000 + T^{4} )^{2}$$
$41$ $$( -9 + T )^{8}$$
$43$ $$( 1296 + T^{4} )^{2}$$
$47$ $$( 49 + T^{4} )^{2}$$
$53$ $$( 256 + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$( 28 + T^{2} )^{4}$$
$67$ $$( 16 + T^{4} )^{2}$$
$71$ $$( 9 + T )^{8}$$
$73$ $$( 30625 + T^{4} )^{2}$$
$79$ $$( -28 + T^{2} )^{4}$$
$83$ $$( 20736 + T^{4} )^{2}$$
$89$ $$( -252 + T^{2} )^{4}$$
$97$ $$( 16 + T^{4} )^{2}$$