# Properties

 Label 750.2.g.b 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: $$\Q(\zeta_{10})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{3} q^{3} -\zeta_{10}^{3} q^{4} -\zeta_{10}^{2} q^{6} + ( -2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} -\zeta_{10} q^{9} +O(q^{10})$$ $$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 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} -\zeta_{10} q^{9} + ( -3 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{11} -\zeta_{10} q^{12} + ( -4 - 4 \zeta_{10}^{2} ) q^{13} + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{14} -\zeta_{10} q^{16} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{17} - q^{18} + ( -6 \zeta_{10} + 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{19} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{21} + ( -1 + \zeta_{10} + 3 \zeta_{10}^{3} ) q^{22} + ( 2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} - q^{24} + ( -4 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{26} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{27} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{28} + ( 4 - 4 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{29} + ( \zeta_{10} + 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{31} - q^{32} + ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{34} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{36} + 8 \zeta_{10} q^{37} + ( -6 + 4 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{38} + ( -4 + 4 \zeta_{10}^{3} ) q^{39} + ( -6 + 4 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{41} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{42} + ( 4 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{43} + ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{44} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{46} + ( -6 + 6 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{47} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{48} + ( -2 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{49} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{51} + ( -4 + 4 \zeta_{10}^{3} ) q^{52} + ( 5 - 5 \zeta_{10} + \zeta_{10}^{3} ) q^{53} + \zeta_{10}^{3} q^{54} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{56} + ( -2 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{57} + ( -4 \zeta_{10} - 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{58} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{59} + ( -4 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{61} + ( 1 + 5 \zeta_{10} + \zeta_{10}^{2} ) q^{62} + ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{63} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{66} + ( -4 \zeta_{10} + 8 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{67} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{68} + ( -4 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69} + ( -4 + 4 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{71} + \zeta_{10}^{3} q^{72} + ( 6 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{73} + 8 q^{74} + ( -2 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{76} + ( 7 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{77} + ( -4 + 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{78} + ( -1 + \zeta_{10} + 5 \zeta_{10}^{3} ) q^{79} + \zeta_{10}^{2} q^{81} + ( -2 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{82} + ( 3 \zeta_{10} - 7 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{83} + ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{84} + ( 4 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{86} + ( -4 - 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{87} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{88} + ( -6 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{89} + ( 8 + 4 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{91} + ( -4 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{92} + ( 6 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{93} + ( 6 \zeta_{10} - 8 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{94} + \zeta_{10}^{3} q^{96} + ( -1 + \zeta_{10} - 4 \zeta_{10}^{3} ) q^{97} + ( -2 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{98} + ( 3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - q^{3} - q^{4} + q^{6} - 6q^{7} + q^{8} - q^{9} + O(q^{10})$$ $$4q + q^{2} - q^{3} - q^{4} + q^{6} - 6q^{7} + q^{8} - q^{9} - 5q^{11} - q^{12} - 12q^{13} - 4q^{14} - q^{16} - 6q^{17} - 4q^{18} - 16q^{19} - q^{21} + 10q^{23} - 4q^{24} - 8q^{26} - q^{27} - q^{28} + 6q^{29} - 3q^{31} - 4q^{32} - 4q^{34} - q^{36} + 8q^{37} - 14q^{38} - 12q^{39} - 14q^{41} + q^{42} + 4q^{43} + 10q^{46} - 20q^{47} - q^{48} - 14q^{49} + 4q^{51} - 12q^{52} + 16q^{53} + q^{54} + q^{56} + 4q^{57} - 6q^{58} + 5q^{59} + 8q^{62} + 4q^{63} - q^{64} + 5q^{66} - 16q^{67} + 4q^{68} - 10q^{69} - 20q^{71} + q^{72} - 2q^{73} + 32q^{74} + 4q^{76} + 15q^{77} - 8q^{78} + 2q^{79} - q^{81} + 4q^{82} + 13q^{83} + 4q^{84} + 16q^{86} - 14q^{87} + 5q^{88} + 2q^{89} + 28q^{91} - 10q^{92} + 22q^{93} + 20q^{94} + q^{96} - 7q^{97} + 4q^{98} + 10q^{99} + O(q^{100})$$

## 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 −0.381966 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 −2.61803 −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 −2.61803 −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 −0.381966 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.b 4
5.b even 2 1 150.2.g.a 4
5.c odd 4 2 750.2.h.b 8
15.d odd 2 1 450.2.h.c 4
25.d even 5 1 inner 750.2.g.b 4
25.d even 5 1 3750.2.a.d 2
25.e even 10 1 150.2.g.a 4
25.e even 10 1 3750.2.a.f 2
25.f odd 20 2 750.2.h.b 8
25.f odd 20 2 3750.2.c.b 4
75.h odd 10 1 450.2.h.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 3 T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(750, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ 
$7$ $$( 1 + 3 T + 15 T^{2} + 21 T^{3} + 49 T^{4} )^{2}$$
$11$ $$1 + 5 T - T^{2} - 55 T^{3} - 184 T^{4} - 605 T^{5} - 121 T^{6} + 6655 T^{7} + 14641 T^{8}$$
$13$ $$1 + 12 T + 51 T^{2} + 76 T^{3} + 9 T^{4} + 988 T^{5} + 8619 T^{6} + 26364 T^{7} + 28561 T^{8}$$
$17$ $$1 + 6 T - T^{2} - 18 T^{3} + 169 T^{4} - 306 T^{5} - 289 T^{6} + 29478 T^{7} + 83521 T^{8}$$
$19$ $$1 + 16 T + 117 T^{2} + 578 T^{3} + 2525 T^{4} + 10982 T^{5} + 42237 T^{6} + 109744 T^{7} + 130321 T^{8}$$
$23$ $$1 - 10 T + 37 T^{2} - 200 T^{3} + 1389 T^{4} - 4600 T^{5} + 19573 T^{6} - 121670 T^{7} + 279841 T^{8}$$
$29$ $$1 - 6 T + 47 T^{2} - 288 T^{3} + 2365 T^{4} - 8352 T^{5} + 39527 T^{6} - 146334 T^{7} + 707281 T^{8}$$
$31$ $$( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} )$$
$37$ $$1 - 8 T + 27 T^{2} + 80 T^{3} - 1639 T^{4} + 2960 T^{5} + 36963 T^{6} - 405224 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 14 T + 95 T^{2} + 786 T^{3} + 6569 T^{4} + 32226 T^{5} + 159695 T^{6} + 964894 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - 2 T + 42 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 + 20 T + 113 T^{2} - 270 T^{3} - 5851 T^{4} - 12690 T^{5} + 249617 T^{6} + 2076460 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 16 T + 53 T^{2} + 200 T^{3} - 1759 T^{4} + 10600 T^{5} + 148877 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 5 T - 49 T^{2} + 295 T^{3} + 1736 T^{4} + 17405 T^{5} - 170569 T^{6} - 1026895 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 25010 T^{5} - 78141 T^{6} + 13845841 T^{8}$$
$67$ $$1 + 16 T + 29 T^{2} - 868 T^{3} - 9191 T^{4} - 58156 T^{5} + 130181 T^{6} + 4812208 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 20 T + 89 T^{2} - 1420 T^{3} - 22639 T^{4} - 100820 T^{5} + 448649 T^{6} + 7158220 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 2 T + 51 T^{2} + 376 T^{3} + 6389 T^{4} + 27448 T^{5} + 271779 T^{6} + 778034 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 2 T - 55 T^{2} + 578 T^{3} + 3679 T^{4} + 45662 T^{5} - 343255 T^{6} - 986078 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 13 T - 19 T^{2} + 951 T^{3} - 6196 T^{4} + 78933 T^{5} - 130891 T^{6} - 7433231 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 2 T + 35 T^{2} - 632 T^{3} + 8789 T^{4} - 56248 T^{5} + 277235 T^{6} - 1409938 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 7 T - 73 T^{2} - 835 T^{3} + 1816 T^{4} - 80995 T^{5} - 686857 T^{6} + 6388711 T^{7} + 88529281 T^{8}$$