# Properties

 Label 750.2.h.a Level $750$ Weight $2$ Character orbit 750.h Analytic conductor $5.989$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$750 = 2 \cdot 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 750.h (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.98878015160$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{6} -2 \zeta_{20}^{5} q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{6} -2 \zeta_{20}^{5} q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} + ( -2 + 2 \zeta_{20}^{2} + 4 \zeta_{20}^{6} ) q^{11} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{12} + ( 3 \zeta_{20} - 3 \zeta_{20}^{7} ) q^{13} -2 \zeta_{20}^{6} q^{14} + \zeta_{20}^{4} q^{16} + ( 3 \zeta_{20} + 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} ) q^{17} -\zeta_{20}^{5} q^{18} + ( -4 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{19} + 2 \zeta_{20}^{2} q^{21} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} + 4 \zeta_{20}^{7} ) q^{22} + 6 \zeta_{20} q^{23} - q^{24} + ( 3 + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{26} + \zeta_{20} q^{27} -2 \zeta_{20}^{7} q^{28} + ( 1 - 3 \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{29} + ( -6 \zeta_{20}^{2} + 6 \zeta_{20}^{4} ) q^{31} + \zeta_{20}^{5} q^{32} + ( -2 \zeta_{20} - 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} ) q^{33} + ( 3 \zeta_{20}^{2} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{6} ) q^{34} -\zeta_{20}^{6} q^{36} + ( -4 \zeta_{20} + 7 \zeta_{20}^{3} - 7 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{37} + ( -4 \zeta_{20} + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{38} + ( -3 + 3 \zeta_{20}^{2} + 3 \zeta_{20}^{6} ) q^{39} + ( -5 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 5 \zeta_{20}^{6} ) q^{41} + 2 \zeta_{20}^{3} q^{42} + ( -2 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{43} + ( -4 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{44} + 6 \zeta_{20}^{2} q^{46} + ( 2 \zeta_{20} - 2 \zeta_{20}^{3} + 6 \zeta_{20}^{7} ) q^{47} -\zeta_{20} q^{48} + 3 q^{49} + ( -6 - 3 \zeta_{20}^{4} + 3 \zeta_{20}^{6} ) q^{51} + ( 3 \zeta_{20} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{52} + ( 9 \zeta_{20} - 9 \zeta_{20}^{3} - 3 \zeta_{20}^{7} ) q^{53} + \zeta_{20}^{2} q^{54} + ( 2 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{56} + ( 2 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{57} + ( \zeta_{20} - 3 \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{58} + ( -8 \zeta_{20}^{2} + 4 \zeta_{20}^{4} - 8 \zeta_{20}^{6} ) q^{59} + ( -3 + 3 \zeta_{20}^{2} + 7 \zeta_{20}^{6} ) q^{61} + ( -6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} ) q^{62} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{63} + \zeta_{20}^{6} q^{64} + ( -2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{66} + ( -6 \zeta_{20} - 6 \zeta_{20}^{5} ) q^{67} + ( 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{68} + ( -6 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{69} + ( 10 - 2 \zeta_{20}^{2} + 10 \zeta_{20}^{4} ) q^{71} -\zeta_{20}^{7} q^{72} + ( -5 \zeta_{20} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{73} + ( -4 + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{74} + ( -4 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{76} + ( 8 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{77} + ( -3 \zeta_{20} + 3 \zeta_{20}^{3} + 3 \zeta_{20}^{7} ) q^{78} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{81} + ( -5 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 5 \zeta_{20}^{7} ) q^{82} -6 \zeta_{20}^{3} q^{83} + 2 \zeta_{20}^{4} q^{84} + ( 2 - 2 \zeta_{20}^{2} ) q^{86} + ( 2 \zeta_{20} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{87} + ( -4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{88} + ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{89} + ( -6 \zeta_{20}^{2} - 6 \zeta_{20}^{6} ) q^{91} + 6 \zeta_{20}^{3} q^{92} + ( -6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{93} + ( -6 + 8 \zeta_{20}^{2} - 8 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{94} -\zeta_{20}^{2} q^{96} + ( 3 \zeta_{20} - 3 \zeta_{20}^{3} - 12 \zeta_{20}^{7} ) q^{97} + 3 \zeta_{20} q^{98} + ( 4 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} - 2q^{6} + 2q^{9} + O(q^{10})$$ $$8q + 2q^{4} - 2q^{6} + 2q^{9} - 4q^{11} - 4q^{14} - 2q^{16} + 4q^{21} - 8q^{24} + 12q^{26} - 24q^{31} + 6q^{34} - 2q^{36} - 12q^{39} - 24q^{41} - 16q^{44} + 12q^{46} + 24q^{49} - 36q^{51} + 2q^{54} + 4q^{56} - 40q^{59} - 4q^{61} + 2q^{64} - 4q^{66} - 12q^{69} + 56q^{71} - 44q^{74} - 40q^{76} - 2q^{81} - 4q^{84} + 12q^{86} + 10q^{89} - 24q^{91} - 4q^{94} - 2q^{96} + 24q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$\zeta_{20}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 + 0.309017i 0.951057 − 0.309017i −0.951057 − 0.309017i 0.951057 + 0.309017i
−0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.00000i 0.951057 + 0.309017i 0.809017 0.587785i 0
49.2 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.00000i −0.951057 0.309017i 0.809017 0.587785i 0
199.1 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.00000i 0.951057 0.309017i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.00000i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
349.1 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.00000i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.2 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.00000i 0.587785 0.809017i −0.309017 + 0.951057i 0
649.1 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.00000i −0.587785 0.809017i −0.309017 0.951057i 0
649.2 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.00000i 0.587785 + 0.809017i −0.309017 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.2 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
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(750, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$3$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 4 + T^{2} )^{4}$$
$11$ $$( 16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$6561 + 729 T^{2} + 486 T^{4} - 36 T^{6} + T^{8}$$
$17$ $$6561 - 8019 T^{2} + 3726 T^{4} + 36 T^{6} + T^{8}$$
$19$ $$( 400 + 200 T + 40 T^{2} + T^{4} )^{2}$$
$23$ $$1679616 - 46656 T^{2} + 1296 T^{4} - 36 T^{6} + T^{8}$$
$29$ $$( 25 + 25 T + 10 T^{2} + T^{4} )^{2}$$
$31$ $$( 1296 + 648 T + 144 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$37$ $$130321 - 44764 T^{2} + 5806 T^{4} + 41 T^{6} + T^{8}$$
$41$ $$( 961 + 403 T + 94 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$43$ $$( 16 + 12 T^{2} + T^{4} )^{2}$$
$47$ $$3748096 + 30976 T^{2} + 5856 T^{4} - 124 T^{6} + T^{8}$$
$53$ $$96059601 - 352836 T^{2} + 13446 T^{4} - 171 T^{6} + T^{8}$$
$59$ $$( 6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$61$ $$( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$67$ $$1679616 - 186624 T^{2} + 7776 T^{4} + 36 T^{6} + T^{8}$$
$71$ $$( 13456 - 2552 T + 384 T^{2} - 28 T^{3} + T^{4} )^{2}$$
$73$ $$923521 - 145111 T^{2} + 8566 T^{4} + 44 T^{6} + T^{8}$$
$79$ $$T^{8}$$
$83$ $$1679616 - 46656 T^{2} + 1296 T^{4} - 36 T^{6} + T^{8}$$
$89$ $$( 25 + 10 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$97$ $$96059601 - 2734479 T^{2} + 29646 T^{4} + 36 T^{6} + T^{8}$$