# Properties

 Label 750.2.h.b 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: $$\Q(\zeta_{20})$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{20} q^{2} -\zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{6} + ( -\zeta_{20}^{3} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} +O(q^{10})$$ $$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} + ( -\zeta_{20}^{3} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} + ( -1 + \zeta_{20}^{2} - 2 \zeta_{20}^{6} ) q^{11} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{12} + ( 4 \zeta_{20}^{3} - 4 \zeta_{20}^{5} ) q^{13} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{14} + \zeta_{20}^{4} q^{16} + ( 2 \zeta_{20} + 2 \zeta_{20}^{5} ) q^{17} -\zeta_{20}^{5} q^{18} + ( 4 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{19} + ( -1 + 2 \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{21} + ( -\zeta_{20} + \zeta_{20}^{3} - 2 \zeta_{20}^{7} ) q^{22} + ( 2 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{23} + q^{24} + ( 4 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{26} -\zeta_{20} q^{27} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{28} + ( -4 + 6 \zeta_{20}^{2} - 4 \zeta_{20}^{4} ) q^{29} + ( -5 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 5 \zeta_{20}^{6} ) q^{31} + \zeta_{20}^{5} q^{32} + ( \zeta_{20} - 3 \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{33} + ( 2 \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{34} -\zeta_{20}^{6} q^{36} + ( -8 \zeta_{20} + 8 \zeta_{20}^{3} - 8 \zeta_{20}^{5} + 8 \zeta_{20}^{7} ) q^{37} + ( 4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{38} + ( 4 - 4 \zeta_{20}^{2} ) q^{39} + ( -6 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{41} + ( -\zeta_{20} + 2 \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{42} + ( -6 \zeta_{20}^{3} + 4 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{43} + ( 2 - 3 \zeta_{20}^{2} + 3 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{44} + ( 4 - 2 \zeta_{20}^{2} + 4 \zeta_{20}^{4} ) q^{46} + ( -6 \zeta_{20} + 6 \zeta_{20}^{3} + 8 \zeta_{20}^{7} ) q^{47} + \zeta_{20} q^{48} + ( 5 + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{49} + ( 2 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{51} + ( 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{52} + ( -5 \zeta_{20} + 5 \zeta_{20}^{3} + 6 \zeta_{20}^{7} ) q^{53} -\zeta_{20}^{2} q^{54} + ( -1 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{56} + ( -6 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{57} + ( -4 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} ) q^{58} + ( -\zeta_{20}^{2} + 3 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{59} + ( 2 - 2 \zeta_{20}^{2} - 6 \zeta_{20}^{6} ) q^{61} + ( -5 \zeta_{20} + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{62} + ( \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{63} + \zeta_{20}^{6} q^{64} + ( \zeta_{20}^{2} - 3 \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{66} + ( 4 \zeta_{20} + 4 \zeta_{20}^{3} + 4 \zeta_{20}^{5} ) q^{67} + ( 2 \zeta_{20}^{3} + 2 \zeta_{20}^{7} ) q^{68} + ( 2 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{69} + ( -4 - 8 \zeta_{20}^{2} - 4 \zeta_{20}^{4} ) q^{71} -\zeta_{20}^{7} q^{72} + ( -10 \zeta_{20} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{73} -8 q^{74} + ( 4 + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{76} + ( 3 \zeta_{20} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{77} + ( 4 \zeta_{20} - 4 \zeta_{20}^{3} ) q^{78} + ( 1 - 5 \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{79} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{81} + ( -6 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{82} + ( 3 \zeta_{20} + 4 \zeta_{20}^{3} + 3 \zeta_{20}^{5} ) q^{83} + ( -\zeta_{20}^{2} + 2 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{84} + ( 6 - 6 \zeta_{20}^{2} - 2 \zeta_{20}^{6} ) q^{86} + ( 2 \zeta_{20} - 6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{87} + ( 2 \zeta_{20} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{88} + ( -4 + 4 \zeta_{20}^{2} + 10 \zeta_{20}^{6} ) q^{89} + ( 8 \zeta_{20}^{2} - 12 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{91} + ( 4 \zeta_{20} - 2 \zeta_{20}^{3} + 4 \zeta_{20}^{5} ) q^{92} + ( -\zeta_{20}^{3} + 6 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{93} + ( -8 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{94} + \zeta_{20}^{2} q^{96} + ( -\zeta_{20} + \zeta_{20}^{3} + 5 \zeta_{20}^{7} ) q^{97} + ( 5 \zeta_{20} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{98} + ( -2 + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} + 2q^{6} + 2q^{9} + O(q^{10})$$ $$8q + 2q^{4} + 2q^{6} + 2q^{9} - 10q^{11} + 8q^{14} - 2q^{16} + 32q^{19} - 2q^{21} + 8q^{24} - 16q^{26} - 12q^{29} - 6q^{31} + 8q^{34} - 2q^{36} + 24q^{39} - 28q^{41} + 20q^{46} + 28q^{49} + 8q^{51} - 2q^{54} + 2q^{56} - 10q^{59} + 2q^{64} + 10q^{66} + 20q^{69} - 40q^{71} - 64q^{74} + 8q^{76} - 4q^{79} - 2q^{81} - 8q^{84} + 32q^{86} - 4q^{89} + 56q^{91} - 40q^{94} + 2q^{96} - 20q^{99} + O(q^{100})$$

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 7 T_{7}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(750, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$3$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$5$ 
$7$ $$( 1 - 21 T^{2} + 197 T^{4} - 1029 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 + 5 T - T^{2} - 55 T^{3} - 184 T^{4} - 605 T^{5} - 121 T^{6} + 6655 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$1 + 42 T^{2} + 795 T^{4} + 11332 T^{6} + 153429 T^{8} + 1915108 T^{10} + 22705995 T^{12} + 202725978 T^{14} + 815730721 T^{16}$$
$17$ $$1 + 38 T^{2} + 555 T^{4} - 2432 T^{6} - 168571 T^{8} - 702848 T^{10} + 46354155 T^{12} + 917227622 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 - 16 T + 117 T^{2} - 578 T^{3} + 2525 T^{4} - 10982 T^{5} + 42237 T^{6} - 109744 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 + 26 T^{2} + 147 T^{4} - 9932 T^{6} - 335995 T^{8} - 5254028 T^{10} + 41136627 T^{12} + 3848933114 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 + 6 T + 47 T^{2} + 288 T^{3} + 2365 T^{4} + 8352 T^{5} + 39527 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )^{2}( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 10 T^{2} - 1269 T^{4} - 26380 T^{6} + 1473461 T^{8} - 36114220 T^{10} - 2378310309 T^{12} + 25657264090 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 14 T + 95 T^{2} + 786 T^{3} + 6569 T^{4} + 32226 T^{5} + 159695 T^{6} + 964894 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 80 T^{2} + 5118 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 174 T^{2} + 11867 T^{4} + 388392 T^{6} + 10496005 T^{8} + 857957928 T^{10} + 57907174427 T^{12} + 1875583467246 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 150 T^{2} + 5691 T^{4} - 410500 T^{6} - 45809019 T^{8} - 1153094500 T^{10} + 44904727371 T^{12} + 3324654169350 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 5 T - 49 T^{2} - 295 T^{3} + 1736 T^{4} - 17405 T^{5} - 170569 T^{6} + 1026895 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 25010 T^{5} - 78141 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 198 T^{2} + 10235 T^{4} - 834852 T^{6} - 122622251 T^{8} - 3747650628 T^{10} + 206246723435 T^{12} + 17910759669462 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 + 20 T + 89 T^{2} - 1420 T^{3} - 22639 T^{4} - 100820 T^{5} + 448649 T^{6} + 7158220 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 98 T^{2} + 13875 T^{4} - 944068 T^{6} + 101584229 T^{8} - 5030938372 T^{10} + 394025593875 T^{12} - 14830754176322 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 2 T - 55 T^{2} - 578 T^{3} + 3679 T^{4} - 45662 T^{5} - 343255 T^{6} + 986078 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 207 T^{2} + 12695 T^{4} - 1121523 T^{6} - 205113656 T^{8} - 7726171947 T^{10} + 602483385095 T^{12} + 67676657287383 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 2 T + 35 T^{2} + 632 T^{3} + 8789 T^{4} + 56248 T^{5} + 277235 T^{6} + 1409938 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 195 T^{2} + 20651 T^{4} + 1202145 T^{6} + 55933936 T^{8} + 11310982305 T^{10} + 1828218181931 T^{12} + 162429540961155 T^{14} + 7837433594376961 T^{16}$$