# Properties

 Label 2300.2.i.a Level $2300$ Weight $2$ Character orbit 2300.i Analytic conductor $18.366$ Analytic rank $0$ Dimension $8$ CM discriminant -115 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: 8.0.928445276160000.122 Defining polynomial: $$x^{8} + 353 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{7} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{7} -3 \beta_{2} q^{9} + ( \beta_{1} - 2 \beta_{5} ) q^{17} + ( \beta_{3} + \beta_{6} ) q^{23} -\beta_{7} q^{29} + ( 2 - \beta_{4} ) q^{31} + ( -3 \beta_{1} + 2 \beta_{5} ) q^{37} + ( -4 + \beta_{4} ) q^{41} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{43} + ( 3 \beta_{2} + \beta_{7} ) q^{49} + ( \beta_{3} + 2 \beta_{6} ) q^{53} + ( 6 \beta_{2} + \beta_{7} ) q^{59} + 3 \beta_{3} q^{63} + ( 3 \beta_{1} - 4 \beta_{5} ) q^{67} + ( 6 + \beta_{4} ) q^{71} -9 q^{81} + ( -\beta_{3} + 4 \beta_{6} ) q^{83} + ( -4 \beta_{1} - 4 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 12q^{31} - 28q^{41} + 52q^{71} - 72q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 353 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 357 \nu^{2}$$$$)/76$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 357 \nu^{3}$$$$)/76$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 186$$$$)/19$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + 357 \nu$$$$)/38$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 353 \nu^{3}$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{6} - 1747 \nu^{2}$$$$)/38$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{6} + 19 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$19 \beta_{4} - 186$$ $$\nu^{5}$$ $$=$$ $$38 \beta_{5} - 357 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-357 \beta_{7} - 3494 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$714 \beta_{6} - 6707 \beta_{3}$$

## Character values

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

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$-1$$

## 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
 3.06489 − 3.06489i 0.326276 − 0.326276i −0.326276 + 0.326276i −3.06489 + 3.06489i 3.06489 + 3.06489i 0.326276 + 0.326276i −0.326276 − 0.326276i −3.06489 − 3.06489i
0 0 0 0 0 −3.06489 + 3.06489i 0 3.00000i 0
1057.2 0 0 0 0 0 −0.326276 + 0.326276i 0 3.00000i 0
1057.3 0 0 0 0 0 0.326276 0.326276i 0 3.00000i 0
1057.4 0 0 0 0 0 3.06489 3.06489i 0 3.00000i 0
1793.1 0 0 0 0 0 −3.06489 3.06489i 0 3.00000i 0
1793.2 0 0 0 0 0 −0.326276 0.326276i 0 3.00000i 0
1793.3 0 0 0 0 0 0.326276 + 0.326276i 0 3.00000i 0
1793.4 0 0 0 0 0 3.06489 + 3.06489i 0 3.00000i 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
115.c odd 2 1 CM by $$\Q(\sqrt{-115})$$
5.b even 2 1 inner
5.c odd 4 2 inner
23.b 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.a 8
5.b even 2 1 inner 2300.2.i.a 8
5.c odd 4 2 inner 2300.2.i.a 8
23.b odd 2 1 inner 2300.2.i.a 8
115.c odd 2 1 CM 2300.2.i.a 8
115.e even 4 2 inner 2300.2.i.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.a 8 1.a even 1 1 trivial
2300.2.i.a 8 5.b even 2 1 inner
2300.2.i.a 8 5.c odd 4 2 inner
2300.2.i.a 8 23.b odd 2 1 inner
2300.2.i.a 8 115.c odd 2 1 CM
2300.2.i.a 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 $$S_{2}^{\mathrm{new}}(2300, [\chi])$$:

 $$T_{3}$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$16 + 353 T^{4} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$614656 + 4673 T^{4} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 529 + T^{4} )^{2}$$
$29$ $$( 7396 + 173 T^{2} + T^{4} )^{2}$$
$31$ $$( -84 - 3 T + T^{2} )^{4}$$
$37$ $$59969536 + 24113 T^{4} + T^{8}$$
$41$ $$( -74 + 7 T + T^{2} )^{4}$$
$43$ $$( 8464 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$5308416 + 7713 T^{4} + T^{8}$$
$59$ $$( 3136 + 233 T^{2} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$1003875856 + 80273 T^{4} + T^{8}$$
$71$ $$( -44 - 13 T + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$3111696 + 81153 T^{4} + T^{8}$$
$89$ $$T^{8}$$
$97$ $$( 135424 + T^{4} )^{2}$$