# Properties

 Label 300.2.j.c Level $300$ Weight $2$ Character orbit 300.j Analytic conductor $2.396$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{4} q^{4} + ( 1 + \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{6} + ( -2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} -\zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{4} q^{4} + ( 1 + \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{6} + ( -2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} -\zeta_{24}^{6} q^{9} -2 \zeta_{24}^{6} q^{11} -2 \zeta_{24} q^{12} + ( \zeta_{24} + 4 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{13} + ( 3 - 3 \zeta_{24}^{2} + \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{14} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{17} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{18} + ( 2 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{19} + ( -1 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{21} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{22} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{23} + ( 2 + 2 \zeta_{24}^{6} ) q^{24} + ( 3 - 3 \zeta_{24}^{2} - 5 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{26} + \zeta_{24}^{3} q^{27} + ( 4 \zeta_{24} - 8 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{28} + ( 2 - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{29} + ( -1 + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{31} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{32} + 2 \zeta_{24}^{3} q^{33} + ( -6 - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} ) q^{34} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{36} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{37} + ( -6 \zeta_{24} - 5 \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{38} + ( -4 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{39} + ( -2 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{41} + ( -4 \zeta_{24} - \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{42} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{43} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{44} + ( 2 + 2 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{46} + ( 4 \zeta_{24} + 6 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{47} -4 \zeta_{24}^{5} q^{48} + ( 4 - 8 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{49} + 6 \zeta_{24}^{6} q^{51} + ( -2 \zeta_{24} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{52} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{53} + ( 1 - \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{54} + ( -2 - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{56} + ( -2 \zeta_{24} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{57} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{58} + ( -8 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{59} + 3 q^{61} + ( 2 \zeta_{24} - 3 \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{62} + ( \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( 2 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} ) q^{66} + ( -2 \zeta_{24} - 9 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{67} + 12 \zeta_{24} q^{68} + ( -2 + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{69} + ( -2 + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{71} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{72} -4 \zeta_{24}^{3} q^{73} + ( -8 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{74} + ( 6 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{76} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{77} + ( 2 \zeta_{24} + 5 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{78} + ( 4 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{79} - q^{81} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} ) q^{82} + ( 6 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{83} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{84} + ( -1 - \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{86} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{87} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{88} + ( -4 + 8 \zeta_{24}^{4} ) q^{89} + ( 7 - 14 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{91} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{92} + ( -\zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{93} + ( 2 - 2 \zeta_{24}^{2} - 10 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{94} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{96} + ( -8 \zeta_{24} + \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{97} + ( -8 \zeta_{24} + 10 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} + 4q^{6} + O(q^{10})$$ $$8q + 8q^{4} + 4q^{6} + 28q^{14} - 16q^{16} + 16q^{19} - 8q^{21} + 16q^{24} + 4q^{26} - 24q^{34} - 32q^{39} - 16q^{41} - 8q^{46} + 4q^{54} + 16q^{56} - 64q^{59} + 24q^{61} - 64q^{64} + 8q^{66} - 32q^{74} + 16q^{76} + 32q^{79} - 8q^{81} - 8q^{84} - 20q^{86} - 24q^{94} + 16q^{96} - 16q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{24}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i
−1.22474 0.707107i 0.707107 + 0.707107i 1.00000 + 1.73205i 0 −0.366025 1.36603i −3.15660 + 3.15660i 2.82843i 1.00000i 0
7.2 −1.22474 + 0.707107i −0.707107 0.707107i 1.00000 1.73205i 0 1.36603 + 0.366025i −1.74238 + 1.74238i 2.82843i 1.00000i 0
7.3 1.22474 0.707107i 0.707107 + 0.707107i 1.00000 1.73205i 0 1.36603 + 0.366025i 1.74238 1.74238i 2.82843i 1.00000i 0
7.4 1.22474 + 0.707107i −0.707107 0.707107i 1.00000 + 1.73205i 0 −0.366025 1.36603i 3.15660 3.15660i 2.82843i 1.00000i 0
43.1 −1.22474 0.707107i −0.707107 + 0.707107i 1.00000 + 1.73205i 0 1.36603 0.366025i −1.74238 1.74238i 2.82843i 1.00000i 0
43.2 −1.22474 + 0.707107i 0.707107 0.707107i 1.00000 1.73205i 0 −0.366025 + 1.36603i −3.15660 3.15660i 2.82843i 1.00000i 0
43.3 1.22474 0.707107i −0.707107 + 0.707107i 1.00000 1.73205i 0 −0.366025 + 1.36603i 3.15660 + 3.15660i 2.82843i 1.00000i 0
43.4 1.22474 + 0.707107i 0.707107 0.707107i 1.00000 + 1.73205i 0 1.36603 0.366025i 1.74238 + 1.74238i 2.82843i 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.4 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
20.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.j.c yes 8
3.b odd 2 1 900.2.k.l 8
4.b odd 2 1 300.2.j.a 8
5.b even 2 1 inner 300.2.j.c yes 8
5.c odd 4 2 300.2.j.a 8
12.b even 2 1 900.2.k.g 8
15.d odd 2 1 900.2.k.l 8
15.e even 4 2 900.2.k.g 8
20.d odd 2 1 300.2.j.a 8
20.e even 4 2 inner 300.2.j.c yes 8
60.h even 2 1 900.2.k.g 8
60.l odd 4 2 900.2.k.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.a 8 4.b odd 2 1
300.2.j.a 8 5.c odd 4 2
300.2.j.a 8 20.d odd 2 1
300.2.j.c yes 8 1.a even 1 1 trivial
300.2.j.c yes 8 5.b even 2 1 inner
300.2.j.c yes 8 20.e even 4 2 inner
900.2.k.g 8 12.b even 2 1
900.2.k.g 8 15.e even 4 2
900.2.k.g 8 60.h even 2 1
900.2.k.l 8 3.b odd 2 1
900.2.k.l 8 15.d odd 2 1
900.2.k.l 8 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{8} + 434 T_{7}^{4} + 14641$$ $$T_{19}^{2} - 4 T_{19} - 23$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$14641 + 434 T^{4} + T^{8}$$
$11$ $$( 4 + T^{2} )^{4}$$
$13$ $$28561 + 1106 T^{4} + T^{8}$$
$17$ $$( 1296 + T^{4} )^{2}$$
$19$ $$( -23 - 4 T + T^{2} )^{4}$$
$23$ $$256 + 3104 T^{4} + T^{8}$$
$29$ $$( 16 + 56 T^{2} + T^{4} )^{2}$$
$31$ $$( 1 + 14 T^{2} + T^{4} )^{2}$$
$37$ $$1048576 + 14336 T^{4} + T^{8}$$
$41$ $$( -8 + 4 T + T^{2} )^{4}$$
$43$ $$14641 + 434 T^{4} + T^{8}$$
$47$ $$20736 + 27936 T^{4} + T^{8}$$
$53$ $$4096 + 896 T^{4} + T^{8}$$
$59$ $$( 52 + 16 T + T^{2} )^{4}$$
$61$ $$( -3 + T )^{8}$$
$67$ $$22667121 + 25074 T^{4} + T^{8}$$
$71$ $$( 576 + 96 T^{2} + T^{4} )^{2}$$
$73$ $$( 256 + T^{4} )^{2}$$
$79$ $$( 4 - 8 T + T^{2} )^{4}$$
$83$ $$20736 + 27936 T^{4} + T^{8}$$
$89$ $$( 48 + T^{2} )^{4}$$
$97$ $$13845841 + 10514 T^{4} + T^{8}$$