# Properties

 Label 300.2.e.e Level $300$ Weight $2$ Character orbit 300.e 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4521217600.1 Defining polynomial: $$x^{8} + x^{6} - 2 x^{4} + 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + \beta_{3} q^{3} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( 1 + \beta_{1} - \beta_{4} ) q^{6} + ( 1 + \beta_{1} - \beta_{4} + \beta_{7} ) q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{8} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{2} + \beta_{3} q^{3} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( 1 + \beta_{1} - \beta_{4} ) q^{6} + ( 1 + \beta_{1} - \beta_{4} + \beta_{7} ) q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{8} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{9} + ( -\beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 + \beta_{5} - \beta_{7} ) q^{12} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( \beta_{2} - \beta_{3} + \beta_{5} ) q^{14} + ( 2 - \beta_{4} + \beta_{7} ) q^{16} + ( -\beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{18} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{19} + ( 1 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{21} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{22} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{24} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{26} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{27} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{28} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{29} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{31} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{32} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{33} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{34} + ( -3 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{36} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{39} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{41} + ( -4 + 2 \beta_{1} - 4 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{42} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{43} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{44} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 3 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{48} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{49} + ( -1 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{51} + ( -6 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{52} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{53} + ( -4 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{54} + ( \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{56} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{57} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{62} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{63} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{64} + ( 3 - \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{66} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{67} + ( -2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{68} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{69} + ( -5 \beta_{2} + 5 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{71} + ( 7 + \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{6} ) q^{72} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + ( 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{74} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{76} + ( -3 \beta_{2} + 3 \beta_{3} - \beta_{4} + 5 \beta_{5} - 7 \beta_{6} - \beta_{7} ) q^{77} + ( -2 + 3 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{78} + ( -2 - 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} ) q^{79} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{81} + ( 6 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{82} + ( -\beta_{2} + \beta_{3} ) q^{83} + ( 6 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{84} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{86} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( 2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{88} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 1 + \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{7} ) q^{91} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{92} + ( 5 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{93} + ( 4 + 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{7} ) q^{94} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{96} + ( 3 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{97} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{98} + ( -3 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{4} + 3q^{6} + 2q^{9} + O(q^{10})$$ $$8q - 2q^{4} + 3q^{6} + 2q^{9} + 11q^{12} - 4q^{13} + 10q^{16} - 7q^{18} + 4q^{21} + 22q^{22} + 13q^{24} - 4q^{28} - 14q^{33} + 22q^{34} - 21q^{36} - 8q^{37} - 36q^{42} - 28q^{46} + 15q^{48} - 20q^{49} - 40q^{52} - 28q^{54} - 18q^{57} + 36q^{58} - 20q^{61} - 38q^{64} + 29q^{66} - 12q^{69} + 51q^{72} + 36q^{73} + 18q^{76} - 22q^{78} - 30q^{81} + 50q^{82} + 40q^{84} - 14q^{88} + 40q^{93} + 12q^{94} - 39q^{96} + 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} - 2 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - 3 \nu^{5} + 2 \nu^{4} + 2 \nu^{3} + 4 \nu^{2} + 12 \nu + 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} + 3 \nu^{5} + 2 \nu^{4} - 2 \nu^{3} + 4 \nu^{2} - 12 \nu + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} + 3 \nu^{5} + 6 \nu^{4} + 6 \nu^{3} + 12 \nu^{2} + 4 \nu - 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 6 \nu^{3} + 4 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} - 2 \nu^{3} + 4 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} + 3 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} - 12 \nu^{2} + 4 \nu + 8$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} + \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$\beta_{7} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 4$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2}$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.29437 − 0.569745i 1.29437 + 0.569745i 0.273147 − 1.38758i 0.273147 + 1.38758i −0.273147 − 1.38758i −0.273147 + 1.38758i −1.29437 − 0.569745i −1.29437 + 0.569745i
−1.29437 0.569745i −0.908080 1.47492i 1.35078 + 1.47492i 0 0.335062 + 2.42647i 2.50967i −0.908080 2.67869i −1.35078 + 2.67869i 0
251.2 −1.29437 + 0.569745i −0.908080 + 1.47492i 1.35078 1.47492i 0 0.335062 2.42647i 2.50967i −0.908080 + 2.67869i −1.35078 2.67869i 0
251.3 −0.273147 1.38758i 1.55737 0.758030i −1.85078 + 0.758030i 0 −1.47722 1.95392i 3.56393i 1.55737 + 2.36106i 1.85078 2.36106i 0
251.4 −0.273147 + 1.38758i 1.55737 + 0.758030i −1.85078 0.758030i 0 −1.47722 + 1.95392i 3.56393i 1.55737 2.36106i 1.85078 + 2.36106i 0
251.5 0.273147 1.38758i −1.55737 + 0.758030i −1.85078 0.758030i 0 0.626440 + 2.36803i 3.56393i −1.55737 + 2.36106i 1.85078 2.36106i 0
251.6 0.273147 + 1.38758i −1.55737 0.758030i −1.85078 + 0.758030i 0 0.626440 2.36803i 3.56393i −1.55737 2.36106i 1.85078 + 2.36106i 0
251.7 1.29437 0.569745i 0.908080 + 1.47492i 1.35078 1.47492i 0 2.01572 + 1.39172i 2.50967i 0.908080 2.67869i −1.35078 + 2.67869i 0
251.8 1.29437 + 0.569745i 0.908080 1.47492i 1.35078 + 1.47492i 0 2.01572 1.39172i 2.50967i 0.908080 + 2.67869i −1.35078 2.67869i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.e.e yes 8
3.b odd 2 1 inner 300.2.e.e yes 8
4.b odd 2 1 inner 300.2.e.e yes 8
5.b even 2 1 300.2.e.d 8
5.c odd 4 2 300.2.h.c 16
12.b even 2 1 inner 300.2.e.e yes 8
15.d odd 2 1 300.2.e.d 8
15.e even 4 2 300.2.h.c 16
20.d odd 2 1 300.2.e.d 8
20.e even 4 2 300.2.h.c 16
60.h even 2 1 300.2.e.d 8
60.l odd 4 2 300.2.h.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.e.d 8 5.b even 2 1
300.2.e.d 8 15.d odd 2 1
300.2.e.d 8 20.d odd 2 1
300.2.e.d 8 60.h even 2 1
300.2.e.e yes 8 1.a even 1 1 trivial
300.2.e.e yes 8 3.b odd 2 1 inner
300.2.e.e yes 8 4.b odd 2 1 inner
300.2.e.e yes 8 12.b even 2 1 inner
300.2.h.c 16 5.c odd 4 2
300.2.h.c 16 15.e even 4 2
300.2.h.c 16 20.e even 4 2
300.2.h.c 16 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 $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{4} + 19 T_{7}^{2} + 80$$ $$T_{13}^{2} + T_{13} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T^{2} - 2 T^{4} + T^{6} + T^{8}$$
$3$ $$81 - 9 T^{2} + 8 T^{4} - T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 80 + 19 T^{2} + T^{4} )^{2}$$
$11$ $$( 200 - 29 T^{2} + T^{4} )^{2}$$
$13$ $$( -10 + T + T^{2} )^{4}$$
$17$ $$( 40 + 59 T^{2} + T^{4} )^{2}$$
$19$ $$( 5 + 26 T^{2} + T^{4} )^{2}$$
$23$ $$( 32 - 28 T^{2} + T^{4} )^{2}$$
$29$ $$( 160 + 36 T^{2} + T^{4} )^{2}$$
$31$ $$( 500 + 55 T^{2} + T^{4} )^{2}$$
$37$ $$( -40 + 2 T + T^{2} )^{4}$$
$41$ $$( 1000 + 115 T^{2} + T^{4} )^{2}$$
$43$ $$( 20 + 71 T^{2} + T^{4} )^{2}$$
$47$ $$( 512 - 52 T^{2} + T^{4} )^{2}$$
$53$ $$( 160 + 36 T^{2} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$( -4 + 5 T + T^{2} )^{4}$$
$67$ $$( 125 + 34 T^{2} + T^{4} )^{2}$$
$71$ $$( 20000 - 284 T^{2} + T^{4} )^{2}$$
$73$ $$( 10 - 9 T + T^{2} )^{4}$$
$79$ $$( 1280 + 76 T^{2} + T^{4} )^{2}$$
$83$ $$( 32 - 13 T^{2} + T^{4} )^{2}$$
$89$ $$( 640 + 51 T^{2} + T^{4} )^{2}$$
$97$ $$( -90 - 3 T + T^{2} )^{4}$$