# Properties

 Label 1500.2.m.b Level $1500$ Weight $2$ Character orbit 1500.m Analytic conductor $11.978$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1500 = 2^{2} \cdot 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1500.m (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.9775603032$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\Q(\zeta_{15})$$ Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 300) 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 $$\zeta_{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{15}^{2} + \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{3} + ( 1 - 2 \zeta_{15} + 2 \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} + \zeta_{15}^{3} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{15}^{2} + \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{3} + ( 1 - 2 \zeta_{15} + 2 \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} + \zeta_{15}^{3} q^{9} + ( -1 + 2 \zeta_{15} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{11} + ( -1 - 2 \zeta_{15} - \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{4} - 2 \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{13} + ( -1 - \zeta_{15} + 2 \zeta_{15}^{2} + \zeta_{15}^{3} - 2 \zeta_{15}^{4} - 2 \zeta_{15}^{5} + \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{17} + ( -1 + \zeta_{15}^{2} - 3 \zeta_{15}^{5} - \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{19} + ( -1 + 2 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{4} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{21} + ( -4 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{23} + ( \zeta_{15}^{2} + \zeta_{15}^{7} ) q^{27} + ( 8 - 4 \zeta_{15} + \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 3 \zeta_{15}^{4} + 6 \zeta_{15}^{5} - 2 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{29} + ( 1 - 2 \zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + \zeta_{15}^{5} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{31} + ( \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{33} + ( 1 + \zeta_{15} + 3 \zeta_{15}^{2} + 3 \zeta_{15}^{4} + \zeta_{15}^{5} + \zeta_{15}^{6} ) q^{37} + ( 1 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + \zeta_{15}^{3} - 3 \zeta_{15}^{4} - \zeta_{15}^{5} - 3 \zeta_{15}^{7} ) q^{39} + ( 6 + 2 \zeta_{15} - 2 \zeta_{15}^{2} + 3 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 6 \zeta_{15}^{6} ) q^{41} + ( 5 + \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{43} + ( -2 + 6 \zeta_{15} + 5 \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} - 2 \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{47} + ( -3 - 8 \zeta_{15} + 9 \zeta_{15}^{2} - 6 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + 3 \zeta_{15}^{5} - 5 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{49} + ( 1 + 2 \zeta_{15} + \zeta_{15}^{3} - 4 \zeta_{15}^{4} + \zeta_{15}^{5} + 3 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{51} + ( -3 + 4 \zeta_{15} + \zeta_{15}^{2} + 3 \zeta_{15}^{4} - 6 \zeta_{15}^{5} + 3 \zeta_{15}^{7} ) q^{53} + ( 3 + 3 \zeta_{15} - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{57} + ( -2 - 6 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} - 6 \zeta_{15}^{5} - 2 \zeta_{15}^{6} ) q^{59} + ( 2 - 4 \zeta_{15} - 3 \zeta_{15}^{2} + 6 \zeta_{15}^{3} - 6 \zeta_{15}^{4} - 2 \zeta_{15}^{5} + 4 \zeta_{15}^{6} - 5 \zeta_{15}^{7} ) q^{61} + ( \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{5} ) q^{63} + ( -11 + 4 \zeta_{15} + 7 \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 8 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 6 \zeta_{15}^{6} + 13 \zeta_{15}^{7} ) q^{67} + ( 4 - 3 \zeta_{15} - \zeta_{15}^{2} + 3 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 3 \zeta_{15}^{5} - \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{69} + ( 4 + 2 \zeta_{15} - 6 \zeta_{15}^{2} + 3 \zeta_{15}^{3} - \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 9 \zeta_{15}^{6} - 10 \zeta_{15}^{7} ) q^{71} + ( -6 + 12 \zeta_{15} - 5 \zeta_{15}^{2} - \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 10 \zeta_{15}^{5} + 5 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{73} + ( -2 + 4 \zeta_{15} - 4 \zeta_{15}^{2} + 2 \zeta_{15}^{3} + \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 4 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{77} + ( -6 + 7 \zeta_{15} + 2 \zeta_{15}^{2} + 6 \zeta_{15}^{4} - 12 \zeta_{15}^{5} - \zeta_{15}^{6} + 7 \zeta_{15}^{7} ) q^{79} + \zeta_{15}^{6} q^{81} + ( 4 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 5 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{83} + ( 3 - \zeta_{15} + 3 \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - \zeta_{15}^{5} + 3 \zeta_{15}^{6} ) q^{87} + ( 3 - 6 \zeta_{15} - 4 \zeta_{15}^{2} + \zeta_{15}^{3} - 5 \zeta_{15}^{4} + \zeta_{15}^{5} - 2 \zeta_{15}^{6} - 7 \zeta_{15}^{7} ) q^{89} + ( -1 - \zeta_{15} + \zeta_{15}^{2} - 3 \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{91} + ( 4 - \zeta_{15} + 2 \zeta_{15}^{2} - 3 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{93} + ( -3 - 4 \zeta_{15} + 5 \zeta_{15}^{2} - 7 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + 8 \zeta_{15}^{5} - 5 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{97} + ( -1 - \zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{3} + 8q^{7} - 2q^{9} + O(q^{10})$$ $$8q + 2q^{3} + 8q^{7} - 2q^{9} - 2q^{11} - 7q^{17} + 5q^{19} - 3q^{21} - 7q^{23} + 2q^{27} + 27q^{29} - 3q^{31} - 3q^{33} + 9q^{37} + 20q^{41} + 68q^{43} + 7q^{47} - 8q^{49} - 8q^{51} + 11q^{53} + 10q^{57} + 2q^{59} - 14q^{61} - 7q^{63} - 28q^{67} + 2q^{69} - 15q^{71} - 6q^{73} - 17q^{77} + 24q^{79} - 2q^{81} - 2q^{83} + 23q^{87} + 5q^{91} + 18q^{93} - 34q^{97} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$877$$ $$1001$$ $$\chi(n)$$ $$1$$ $$-1 - \zeta_{15}^{2} - \zeta_{15}^{4} - \zeta_{15}^{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 0.669131 − 0.743145i −0.978148 − 0.207912i −0.104528 − 0.994522i 0.913545 + 0.406737i −0.104528 + 0.994522i 0.913545 − 0.406737i 0.669131 + 0.743145i −0.978148 + 0.207912i
0 −0.309017 + 0.951057i 0 0 0 −0.547318 0 −0.809017 0.587785i 0
301.2 0 −0.309017 + 0.951057i 0 0 0 4.78339 0 −0.809017 0.587785i 0
601.1 0 0.809017 + 0.587785i 0 0 0 −0.747238 0 0.309017 + 0.951057i 0
601.2 0 0.809017 + 0.587785i 0 0 0 0.511170 0 0.309017 + 0.951057i 0
901.1 0 0.809017 0.587785i 0 0 0 −0.747238 0 0.309017 0.951057i 0
901.2 0 0.809017 0.587785i 0 0 0 0.511170 0 0.309017 0.951057i 0
1201.1 0 −0.309017 0.951057i 0 0 0 −0.547318 0 −0.809017 + 0.587785i 0
1201.2 0 −0.309017 0.951057i 0 0 0 4.78339 0 −0.809017 + 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1201.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1500.2.m.b 8
5.b even 2 1 300.2.m.a 8
5.c odd 4 2 1500.2.o.a 16
15.d odd 2 1 900.2.n.a 8
25.d even 5 1 inner 1500.2.m.b 8
25.d even 5 1 7500.2.a.d 4
25.e even 10 1 300.2.m.a 8
25.e even 10 1 7500.2.a.g 4
25.f odd 20 2 1500.2.o.a 16
25.f odd 20 2 7500.2.d.d 8
75.h odd 10 1 900.2.n.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.a 8 5.b even 2 1
300.2.m.a 8 25.e even 10 1
900.2.n.a 8 15.d odd 2 1
900.2.n.a 8 75.h odd 10 1
1500.2.m.b 8 1.a even 1 1 trivial
1500.2.m.b 8 25.d even 5 1 inner
1500.2.o.a 16 5.c odd 4 2
1500.2.o.a 16 25.f odd 20 2
7500.2.a.d 4 25.d even 5 1
7500.2.a.g 4 25.e even 10 1
7500.2.d.d 8 25.f odd 20 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4 T_{7}^{3} - 4 T_{7}^{2} + T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1500, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 1 + T - 4 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$11$ $$1 - 3 T + 23 T^{2} - T^{3} - T^{5} + 3 T^{6} + 2 T^{7} + T^{8}$$
$13$ $$25 - 175 T + 525 T^{2} - 625 T^{3} + 390 T^{4} - 115 T^{5} + 35 T^{6} + T^{8}$$
$17$ $$73441 - 29268 T + 9683 T^{2} - 1361 T^{3} + 690 T^{4} + 49 T^{5} + 33 T^{6} + 7 T^{7} + T^{8}$$
$19$ $$21025 + 19575 T + 6125 T^{2} - 2075 T^{3} + 1140 T^{4} - 215 T^{5} + 45 T^{6} - 5 T^{7} + T^{8}$$
$23$ $$961 - 5983 T + 13768 T^{2} + 4999 T^{3} + 2325 T^{4} + 389 T^{5} + 58 T^{6} + 7 T^{7} + T^{8}$$
$29$ $$358801 - 43727 T + 154128 T^{2} - 93029 T^{3} + 26655 T^{4} - 4229 T^{5} + 428 T^{6} - 27 T^{7} + T^{8}$$
$31$ $$77841 + 32643 T + 12528 T^{2} + 1701 T^{3} + 225 T^{4} - 9 T^{5} + 18 T^{6} + 3 T^{7} + T^{8}$$
$37$ $$32761 + 22444 T + 12597 T^{2} + 2227 T^{3} + 90 T^{4} - 197 T^{5} + 77 T^{6} - 9 T^{7} + T^{8}$$
$41$ $$24025 + 34100 T + 16525 T^{2} - 6800 T^{3} + 4965 T^{4} - 1120 T^{5} + 205 T^{6} - 20 T^{7} + T^{8}$$
$43$ $$( 2371 - 1844 T + 401 T^{2} - 34 T^{3} + T^{4} )^{2}$$
$47$ $$2627641 + 507373 T + 272158 T^{2} - 5839 T^{3} - 435 T^{4} + 151 T^{5} + 88 T^{6} - 7 T^{7} + T^{8}$$
$53$ $$32041 - 49404 T + 33797 T^{2} - 10247 T^{3} + 2730 T^{4} - 623 T^{5} + 117 T^{6} - 11 T^{7} + T^{8}$$
$59$ $$5480281 + 323058 T - 87277 T^{2} - 11924 T^{3} + 11925 T^{4} - 44 T^{5} + 183 T^{6} - 2 T^{7} + T^{8}$$
$61$ $$201601 + 230786 T + 121137 T^{2} + 33298 T^{3} + 7115 T^{4} + 1402 T^{5} + 237 T^{6} + 14 T^{7} + T^{8}$$
$67$ $$6046681 - 1691792 T + 177768 T^{2} + 29036 T^{3} + 23090 T^{4} + 3416 T^{5} + 423 T^{6} + 28 T^{7} + T^{8}$$
$71$ $$15015625 + 968750 T - 9375 T^{2} - 8125 T^{3} + 9750 T^{4} + 1625 T^{5} + 275 T^{6} + 15 T^{7} + T^{8}$$
$73$ $$2653641 - 1260846 T + 805302 T^{2} - 163998 T^{3} + 39240 T^{4} - 1332 T^{5} - 93 T^{6} + 6 T^{7} + T^{8}$$
$79$ $$1846881 - 2458431 T + 1343817 T^{2} - 174123 T^{3} + 35190 T^{4} - 4887 T^{5} + 477 T^{6} - 24 T^{7} + T^{8}$$
$83$ $$7921 - 14418 T + 11213 T^{2} - 2086 T^{3} + 915 T^{4} + 194 T^{5} + 33 T^{6} + 2 T^{7} + T^{8}$$
$89$ $$2025 - 2025 T + 2025 T^{2} - 2025 T^{3} + 2790 T^{4} + 855 T^{5} + 105 T^{6} + T^{8}$$
$97$ $$18139081 + 7385106 T + 2444087 T^{2} + 443668 T^{3} + 56625 T^{4} + 5692 T^{5} + 567 T^{6} + 34 T^{7} + T^{8}$$