# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3655924649$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 14 x^{6} + 53 x^{4} + 29 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{5} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{5} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -\beta_{3} + \beta_{6} ) q^{11} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{13} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{17} + ( -2 - \beta_{3} + \beta_{6} ) q^{19} + 2 q^{21} -\beta_{4} q^{23} + ( -\beta_{4} + \beta_{5} ) q^{27} + ( 2 - \beta_{2} - \beta_{6} ) q^{29} + ( 4 - \beta_{2} - \beta_{3} ) q^{31} + ( -6 \beta_{4} - 2 \beta_{7} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{37} + ( -3 - 2 \beta_{3} + \beta_{6} ) q^{39} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{41} + ( -\beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( -4 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{47} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{6} ) q^{49} + ( 8 - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -2 \beta_{1} - 6 \beta_{4} - 2 \beta_{7} ) q^{57} + ( -3 - \beta_{3} + \beta_{6} ) q^{59} + ( 6 - 2 \beta_{6} ) q^{61} + ( -\beta_{1} - 3 \beta_{5} ) q^{63} + ( -4 \beta_{4} - 2 \beta_{7} ) q^{67} + \beta_{3} q^{69} + ( 4 + \beta_{2} - 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{73} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{77} + ( -6 - 3 \beta_{3} - \beta_{6} ) q^{79} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{81} + ( -3 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{83} + ( 4 \beta_{1} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{87} + ( 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} ) q^{89} + ( 6 + 2 \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{91} + ( 6 \beta_{1} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{93} + ( -2 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} ) q^{97} + ( 2 + 7 \beta_{3} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{9} + O(q^{10})$$ $$8q - 4q^{9} + 2q^{11} - 14q^{19} + 16q^{21} + 10q^{29} + 28q^{31} - 22q^{39} + 6q^{41} - 2q^{49} + 56q^{51} - 22q^{59} + 44q^{61} + 36q^{71} - 50q^{79} + 50q^{91} + 18q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 53 x^{4} + 29 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} + 7 \nu^{2} + 2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} + 51 \nu^{3} + 15 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} + 53 \nu^{3} + 27 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} + 13 \nu^{4} + 44 \nu^{2} + 13$$ $$\beta_{7}$$ $$=$$ $$-2 \nu^{7} - 27 \nu^{5} - 95 \nu^{3} - 28 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - \beta_{4} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} - 7 \beta_{2} + 26$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} - 7 \beta_{5} + 11 \beta_{4} + 40 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} - 13 \beta_{3} + 47 \beta_{2} - 175$$ $$\nu^{7}$$ $$=$$ $$-14 \beta_{7} + 47 \beta_{5} - 101 \beta_{4} - 269 \beta_{1}$$

## Character values

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

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1749.1
 − 2.66337i − 2.50653i − 0.631352i − 0.474520i 0.474520i 0.631352i 2.50653i 2.66337i
0 2.66337i 0 0 0 0.750930i 0 −4.09352 0
1749.2 0 2.50653i 0 0 0 0.797915i 0 −3.28271 0
1749.3 0 0.631352i 0 0 0 3.16780i 0 2.60139 0
1749.4 0 0.474520i 0 0 0 4.21479i 0 2.77483 0
1749.5 0 0.474520i 0 0 0 4.21479i 0 2.77483 0
1749.6 0 0.631352i 0 0 0 3.16780i 0 2.60139 0
1749.7 0 2.50653i 0 0 0 0.797915i 0 −3.28271 0
1749.8 0 2.66337i 0 0 0 0.750930i 0 −4.09352 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1749.8 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 2300.2.c.j 8
5.b even 2 1 inner 2300.2.c.j 8
5.c odd 4 1 2300.2.a.l 4
5.c odd 4 1 2300.2.a.m yes 4
20.e even 4 1 9200.2.a.cm 4
20.e even 4 1 9200.2.a.co 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.l 4 5.c odd 4 1
2300.2.a.m yes 4 5.c odd 4 1
2300.2.c.j 8 1.a even 1 1 trivial
2300.2.c.j 8 5.b even 2 1 inner
9200.2.a.cm 4 20.e even 4 1
9200.2.a.co 4 20.e even 4 1

## Hecke kernels

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

 $$T_{3}^{8} + 14 T_{3}^{6} + 53 T_{3}^{4} + 29 T_{3}^{2} + 4$$ $$T_{7}^{8} + 29 T_{7}^{6} + 212 T_{7}^{4} + 224 T_{7}^{2} + 64$$ $$T_{11}^{4} - T_{11}^{3} - 26 T_{11}^{2} + 24 T_{11} + 120$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 + 29 T^{2} + 53 T^{4} + 14 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$64 + 224 T^{2} + 212 T^{4} + 29 T^{6} + T^{8}$$
$11$ $$( 120 + 24 T - 26 T^{2} - T^{3} + T^{4} )^{2}$$
$13$ $$289 + 1650 T^{2} + 823 T^{4} + 63 T^{6} + T^{8}$$
$17$ $$2304 + 4800 T^{2} + 1072 T^{4} + 72 T^{6} + T^{8}$$
$19$ $$( 72 - 60 T - 8 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$23$ $$( 1 + T^{2} )^{4}$$
$29$ $$( -93 + 132 T - 31 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$31$ $$( -10 - 27 T + 49 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$37$ $$173056 + 243008 T^{2} + 12560 T^{4} + 200 T^{6} + T^{8}$$
$41$ $$( 381 + 174 T - 107 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$43$ $$5184 + 4752 T^{2} + 1048 T^{4} + 65 T^{6} + T^{8}$$
$47$ $$1542564 + 281637 T^{2} + 13077 T^{4} + 210 T^{6} + T^{8}$$
$53$ $$82944 + 37440 T^{2} + 4192 T^{4} + 132 T^{6} + T^{8}$$
$59$ $$( 12 - 51 T + 19 T^{2} + 11 T^{3} + T^{4} )^{2}$$
$61$ $$( -416 - 40 T + 128 T^{2} - 22 T^{3} + T^{4} )^{2}$$
$67$ $$984064 + 196928 T^{2} + 11792 T^{4} + 200 T^{6} + T^{8}$$
$71$ $$( -1800 + 327 T + 67 T^{2} - 18 T^{3} + T^{4} )^{2}$$
$73$ $$19321 + 19682 T^{2} + 2927 T^{4} + 119 T^{6} + T^{8}$$
$79$ $$( 40 + 244 T + 176 T^{2} + 25 T^{3} + T^{4} )^{2}$$
$83$ $$5184 + 52560 T^{2} + 8776 T^{4} + 201 T^{6} + T^{8}$$
$89$ $$( -3840 - 1848 T - 236 T^{2} + T^{4} )^{2}$$
$97$ $$1081600 + 3769536 T^{2} + 79216 T^{4} + 504 T^{6} + T^{8}$$