# Properties

 Label 750.2.g.a Level $750$ Weight $2$ Character orbit 750.g Analytic conductor $5.989$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$750 = 2 \cdot 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 750.g (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.98878015160$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0 −0.809017 0.587785i −2.00000 −0.809017 0.587785i 0.309017 0.951057i 0
301.1 −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 0 0.309017 + 0.951057i −2.00000 0.309017 + 0.951057i −0.809017 0.587785i 0
451.1 −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 0 0.309017 0.951057i −2.00000 0.309017 0.951057i −0.809017 + 0.587785i 0
601.1 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0 −0.809017 + 0.587785i −2.00000 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.g.a 4
5.b even 2 1 150.2.g.b 4
5.c odd 4 2 750.2.h.a 8
15.d odd 2 1 450.2.h.b 4
25.d even 5 1 inner 750.2.g.a 4
25.d even 5 1 3750.2.a.g 2
25.e even 10 1 150.2.g.b 4
25.e even 10 1 3750.2.a.b 2
25.f odd 20 2 750.2.h.a 8
25.f odd 20 2 3750.2.c.c 4
75.h odd 10 1 450.2.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 5.b even 2 1
150.2.g.b 4 25.e even 10 1
450.2.h.b 4 15.d odd 2 1
450.2.h.b 4 75.h odd 10 1
750.2.g.a 4 1.a even 1 1 trivial
750.2.g.a 4 25.d even 5 1 inner
750.2.h.a 8 5.c odd 4 2
750.2.h.a 8 25.f odd 20 2
3750.2.a.b 2 25.e even 10 1
3750.2.a.g 2 25.d even 5 1
3750.2.c.c 4 25.f odd 20 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 2$$ acting on $$S_{2}^{\mathrm{new}}(750, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 + T )^{4}$$
$11$ $$16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$81 - 81 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$81 + 27 T + 54 T^{2} - 12 T^{3} + T^{4}$$
$19$ $$400 - 200 T + 40 T^{2} + T^{4}$$
$23$ $$1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$25 - 25 T + 10 T^{2} + T^{4}$$
$31$ $$1296 + 648 T + 144 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$361 - 38 T + 64 T^{2} + 13 T^{3} + T^{4}$$
$41$ $$961 + 403 T + 94 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$( -4 + 2 T + T^{2} )^{2}$$
$47$ $$1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$9801 - 2376 T + 306 T^{2} - 21 T^{3} + T^{4}$$
$59$ $$6400 - 1600 T + 240 T^{2} - 20 T^{3} + T^{4}$$
$61$ $$361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$1296 + 432 T + 144 T^{2} + 18 T^{3} + T^{4}$$
$71$ $$13456 - 2552 T + 384 T^{2} - 28 T^{3} + T^{4}$$
$73$ $$961 - 31 T + 76 T^{2} + 14 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$89$ $$25 + 10 T^{2} + 5 T^{3} + T^{4}$$
$97$ $$9801 + 297 T + 144 T^{2} + 18 T^{3} + T^{4}$$