# Properties

 Label 1500.2.o.a Level $1500$ Weight $2$ Character orbit 1500.o Analytic conductor $11.978$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1500 = 2^{2} \cdot 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1500.o (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.9775603032$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{60})$$ Defining polynomial: $$x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: no (minimal twist has level 300) 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 $$\zeta_{60}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{60}^{9} q^{3} + ( -3 \zeta_{60} - \zeta_{60}^{3} + \zeta_{60}^{5} + 2 \zeta_{60}^{7} + 2 \zeta_{60}^{9} + \zeta_{60}^{11} - 3 \zeta_{60}^{15} ) q^{7} + ( 1 - \zeta_{60}^{4} - \zeta_{60}^{6} + \zeta_{60}^{12} + \zeta_{60}^{14} ) q^{9} +O(q^{10})$$ $$q + \zeta_{60}^{9} q^{3} + ( -3 \zeta_{60} - \zeta_{60}^{3} + \zeta_{60}^{5} + 2 \zeta_{60}^{7} + 2 \zeta_{60}^{9} + \zeta_{60}^{11} - 3 \zeta_{60}^{15} ) q^{7} + ( 1 - \zeta_{60}^{4} - \zeta_{60}^{6} + \zeta_{60}^{12} + \zeta_{60}^{14} ) q^{9} + ( -2 - \zeta_{60}^{2} + \zeta_{60}^{4} + \zeta_{60}^{6} + 2 \zeta_{60}^{8} + \zeta_{60}^{10} - \zeta_{60}^{12} - 2 \zeta_{60}^{14} ) q^{11} + ( -\zeta_{60} - 2 \zeta_{60}^{3} + 3 \zeta_{60}^{5} + 2 \zeta_{60}^{7} + \zeta_{60}^{9} + 2 \zeta_{60}^{11} - \zeta_{60}^{13} - 3 \zeta_{60}^{15} ) q^{13} + ( -2 \zeta_{60} - 3 \zeta_{60}^{3} - 4 \zeta_{60}^{5} + 2 \zeta_{60}^{7} + 2 \zeta_{60}^{9} + 3 \zeta_{60}^{11} + 2 \zeta_{60}^{13} + \zeta_{60}^{15} ) q^{17} + ( -2 + 3 \zeta_{60}^{2} - \zeta_{60}^{6} + 3 \zeta_{60}^{10} - 2 \zeta_{60}^{12} ) q^{19} + ( -1 - 2 \zeta_{60}^{2} + \zeta_{60}^{4} + 2 \zeta_{60}^{8} + \zeta_{60}^{12} - 2 \zeta_{60}^{14} ) q^{21} + ( -5 \zeta_{60} - 2 \zeta_{60}^{3} + 6 \zeta_{60}^{5} + 3 \zeta_{60}^{7} + 4 \zeta_{60}^{9} + 2 \zeta_{60}^{11} - 3 \zeta_{60}^{13} - 7 \zeta_{60}^{15} ) q^{23} + ( -\zeta_{60} - \zeta_{60}^{3} + \zeta_{60}^{9} + \zeta_{60}^{11} - \zeta_{60}^{15} ) q^{27} + ( -1 - 2 \zeta_{60}^{2} - 2 \zeta_{60}^{4} - 3 \zeta_{60}^{6} + 4 \zeta_{60}^{8} - 2 \zeta_{60}^{10} + 4 \zeta_{60}^{12} + \zeta_{60}^{14} ) q^{29} + ( -\zeta_{60}^{2} - 2 \zeta_{60}^{4} + 2 \zeta_{60}^{6} - 2 \zeta_{60}^{8} - \zeta_{60}^{10} ) q^{31} + ( -\zeta_{60}^{5} + \zeta_{60}^{7} ) q^{33} + ( -6 \zeta_{60} - 3 \zeta_{60}^{3} + 4 \zeta_{60}^{5} + 6 \zeta_{60}^{7} + 3 \zeta_{60}^{9} + 4 \zeta_{60}^{11} - 3 \zeta_{60}^{13} - 7 \zeta_{60}^{15} ) q^{37} + ( -3 - \zeta_{60}^{2} + 2 \zeta_{60}^{4} + \zeta_{60}^{6} + 2 \zeta_{60}^{8} + 3 \zeta_{60}^{10} - 2 \zeta_{60}^{12} - \zeta_{60}^{14} ) q^{39} + ( -1 + \zeta_{60}^{4} - \zeta_{60}^{6} + 2 \zeta_{60}^{8} + 4 \zeta_{60}^{10} - 3 \zeta_{60}^{12} - 5 \zeta_{60}^{14} ) q^{41} + ( 3 \zeta_{60} - 2 \zeta_{60}^{3} - 2 \zeta_{60}^{5} - \zeta_{60}^{9} + \zeta_{60}^{11} + 4 \zeta_{60}^{13} + 10 \zeta_{60}^{15} ) q^{43} + ( -4 \zeta_{60}^{3} - \zeta_{60}^{5} + 5 \zeta_{60}^{7} + 5 \zeta_{60}^{9} + 5 \zeta_{60}^{11} - \zeta_{60}^{13} - 4 \zeta_{60}^{15} ) q^{47} + ( -8 - 2 \zeta_{60}^{2} + 4 \zeta_{60}^{4} + 9 \zeta_{60}^{6} + 8 \zeta_{60}^{8} + 5 \zeta_{60}^{10} - 3 \zeta_{60}^{12} - 14 \zeta_{60}^{14} ) q^{49} + ( -3 - 4 \zeta_{60}^{2} - 3 \zeta_{60}^{4} + 2 \zeta_{60}^{8} + 3 \zeta_{60}^{10} + \zeta_{60}^{12} - 3 \zeta_{60}^{14} ) q^{51} + ( \zeta_{60}^{3} + 3 \zeta_{60}^{5} - \zeta_{60}^{7} - \zeta_{60}^{9} - \zeta_{60}^{11} + 3 \zeta_{60}^{13} + \zeta_{60}^{15} ) q^{53} + ( 5 \zeta_{60} - 3 \zeta_{60}^{5} - 3 \zeta_{60}^{7} - 2 \zeta_{60}^{9} + \zeta_{60}^{11} + 3 \zeta_{60}^{13} + 2 \zeta_{60}^{15} ) q^{57} + ( -9 - 7 \zeta_{60}^{2} + 3 \zeta_{60}^{6} + 6 \zeta_{60}^{8} + 12 \zeta_{60}^{10} + \zeta_{60}^{12} - 5 \zeta_{60}^{14} ) q^{59} + ( 2 + 2 \zeta_{60}^{2} - 2 \zeta_{60}^{6} - 4 \zeta_{60}^{8} - 6 \zeta_{60}^{10} - \zeta_{60}^{12} - 2 \zeta_{60}^{14} ) q^{61} + ( -3 \zeta_{60} + 2 \zeta_{60}^{7} + \zeta_{60}^{9} + \zeta_{60}^{11} - \zeta_{60}^{13} - 2 \zeta_{60}^{15} ) q^{63} + ( -5 \zeta_{60} - 3 \zeta_{60}^{3} + 6 \zeta_{60}^{5} + 2 \zeta_{60}^{7} + 7 \zeta_{60}^{9} + \zeta_{60}^{11} - 8 \zeta_{60}^{13} - 4 \zeta_{60}^{15} ) q^{67} + ( -1 + 3 \zeta_{60}^{4} - \zeta_{60}^{6} + 3 \zeta_{60}^{8} - \zeta_{60}^{12} ) q^{69} + ( -3 - 6 \zeta_{60}^{2} + 2 \zeta_{60}^{4} - 4 \zeta_{60}^{6} + 2 \zeta_{60}^{8} + 4 \zeta_{60}^{10} + 7 \zeta_{60}^{12} - 5 \zeta_{60}^{14} ) q^{71} + ( -10 \zeta_{60} - 6 \zeta_{60}^{3} + 8 \zeta_{60}^{5} + 4 \zeta_{60}^{7} + 5 \zeta_{60}^{9} + 2 \zeta_{60}^{13} - 9 \zeta_{60}^{15} ) q^{73} + ( 3 \zeta_{60} + 2 \zeta_{60}^{3} - 2 \zeta_{60}^{5} - \zeta_{60}^{7} - 4 \zeta_{60}^{9} - \zeta_{60}^{13} + 5 \zeta_{60}^{15} ) q^{77} + ( 1 + 2 \zeta_{60}^{2} - 6 \zeta_{60}^{4} - 8 \zeta_{60}^{6} - 7 \zeta_{60}^{8} + 5 \zeta_{60}^{10} + 7 \zeta_{60}^{12} + 7 \zeta_{60}^{14} ) q^{79} -\zeta_{60}^{6} q^{81} + ( \zeta_{60} + 2 \zeta_{60}^{5} + 2 \zeta_{60}^{7} + 2 \zeta_{60}^{9} - 3 \zeta_{60}^{11} - 4 \zeta_{60}^{13} - 2 \zeta_{60}^{15} ) q^{83} + ( -10 \zeta_{60} - 5 \zeta_{60}^{3} + 2 \zeta_{60}^{5} + 6 \zeta_{60}^{7} + 3 \zeta_{60}^{9} + 6 \zeta_{60}^{11} - 3 \zeta_{60}^{13} - 9 \zeta_{60}^{15} ) q^{87} + ( -7 - 3 \zeta_{60}^{2} + 4 \zeta_{60}^{4} + 3 \zeta_{60}^{6} + 6 \zeta_{60}^{8} + 5 \zeta_{60}^{10} - 6 \zeta_{60}^{12} - 5 \zeta_{60}^{14} ) q^{89} + ( 4 - 2 \zeta_{60}^{4} - 3 \zeta_{60}^{6} - \zeta_{60}^{8} - 2 \zeta_{60}^{10} + 3 \zeta_{60}^{12} + 4 \zeta_{60}^{14} ) q^{91} + ( \zeta_{60} + 2 \zeta_{60}^{3} + \zeta_{60}^{5} - \zeta_{60}^{7} - 2 \zeta_{60}^{9} - 3 \zeta_{60}^{11} - 3 \zeta_{60}^{13} + 3 \zeta_{60}^{15} ) q^{93} + ( 2 \zeta_{60}^{3} + 4 \zeta_{60}^{5} + 5 \zeta_{60}^{9} + 4 \zeta_{60}^{13} + 2 \zeta_{60}^{15} ) q^{97} + ( -1 - \zeta_{60}^{2} + \zeta_{60}^{6} + \zeta_{60}^{8} + \zeta_{60}^{10} - 2 \zeta_{60}^{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{9} + O(q^{10})$$ $$16q + 4q^{9} - 4q^{11} - 10q^{19} - 6q^{21} - 54q^{29} - 6q^{31} + 40q^{41} + 16q^{49} - 16q^{51} - 4q^{59} - 28q^{61} - 4q^{69} - 30q^{71} - 48q^{79} - 4q^{81} + 10q^{91} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$751$$ $$877$$ $$1001$$ $$\chi(n)$$ $$1$$ $$\zeta_{60}^{9}$$ $$1$$

## 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
 −0.743145 + 0.669131i −0.207912 − 0.978148i 0.207912 + 0.978148i 0.743145 − 0.669131i −0.994522 − 0.104528i 0.406737 + 0.913545i −0.406737 − 0.913545i 0.994522 + 0.104528i 0.406737 − 0.913545i −0.994522 + 0.104528i 0.994522 − 0.104528i −0.406737 + 0.913545i −0.207912 + 0.978148i −0.743145 − 0.669131i 0.743145 + 0.669131i 0.207912 − 0.978148i
0 −0.951057 + 0.309017i 0 0 0 0.547318i 0 0.809017 0.587785i 0
49.2 0 −0.951057 + 0.309017i 0 0 0 4.78339i 0 0.809017 0.587785i 0
49.3 0 0.951057 0.309017i 0 0 0 4.78339i 0 0.809017 0.587785i 0
49.4 0 0.951057 0.309017i 0 0 0 0.547318i 0 0.809017 0.587785i 0
349.1 0 −0.587785 0.809017i 0 0 0 0.747238i 0 −0.309017 + 0.951057i 0
349.2 0 −0.587785 0.809017i 0 0 0 0.511170i 0 −0.309017 + 0.951057i 0
349.3 0 0.587785 + 0.809017i 0 0 0 0.511170i 0 −0.309017 + 0.951057i 0
349.4 0 0.587785 + 0.809017i 0 0 0 0.747238i 0 −0.309017 + 0.951057i 0
649.1 0 −0.587785 + 0.809017i 0 0 0 0.511170i 0 −0.309017 0.951057i 0
649.2 0 −0.587785 + 0.809017i 0 0 0 0.747238i 0 −0.309017 0.951057i 0
649.3 0 0.587785 0.809017i 0 0 0 0.747238i 0 −0.309017 0.951057i 0
649.4 0 0.587785 0.809017i 0 0 0 0.511170i 0 −0.309017 0.951057i 0
949.1 0 −0.951057 0.309017i 0 0 0 4.78339i 0 0.809017 + 0.587785i 0
949.2 0 −0.951057 0.309017i 0 0 0 0.547318i 0 0.809017 + 0.587785i 0
949.3 0 0.951057 + 0.309017i 0 0 0 0.547318i 0 0.809017 + 0.587785i 0
949.4 0 0.951057 + 0.309017i 0 0 0 4.78339i 0 0.809017 + 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 949.4 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 1500.2.o.a 16
5.b even 2 1 inner 1500.2.o.a 16
5.c odd 4 1 300.2.m.a 8
5.c odd 4 1 1500.2.m.b 8
15.e even 4 1 900.2.n.a 8
25.d even 5 1 inner 1500.2.o.a 16
25.d even 5 1 7500.2.d.d 8
25.e even 10 1 inner 1500.2.o.a 16
25.e even 10 1 7500.2.d.d 8
25.f odd 20 1 300.2.m.a 8
25.f odd 20 1 1500.2.m.b 8
25.f odd 20 1 7500.2.a.d 4
25.f odd 20 1 7500.2.a.g 4
75.l even 20 1 900.2.n.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 24 T_{7}^{6} + 26 T_{7}^{4} + 9 T_{7}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1500, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$( 1 + 9 T^{2} + 26 T^{4} + 24 T^{6} + T^{8} )^{2}$$
$11$ $$( 1 - 3 T + 23 T^{2} - T^{3} - T^{5} + 3 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$13$ $$625 + 4375 T^{2} + 76375 T^{4} + 19625 T^{6} + 45150 T^{8} - 15125 T^{10} + 2005 T^{12} - 70 T^{14} + T^{16}$$
$17$ $$5393580481 - 565642582 T^{2} + 115441573 T^{4} - 19225589 T^{6} + 1805190 T^{8} - 81559 T^{10} + 1783 T^{12} - 17 T^{14} + T^{16}$$
$19$ $$( 21025 - 19575 T + 6125 T^{2} + 2075 T^{3} + 1140 T^{4} + 215 T^{5} + 45 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$23$ $$923521 + 9334193 T^{2} + 253844508 T^{4} - 43797449 T^{6} + 3199175 T^{8} - 75929 T^{10} + 2568 T^{12} - 67 T^{14} + T^{16}$$
$29$ $$( 358801 + 43727 T + 154128 T^{2} + 93029 T^{3} + 26655 T^{4} + 4229 T^{5} + 428 T^{6} + 27 T^{7} + T^{8} )^{2}$$
$31$ $$( 77841 + 32643 T + 12528 T^{2} + 1701 T^{3} + 225 T^{4} - 9 T^{5} + 18 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$37$ $$1073283121 - 321647498 T^{2} + 64615813 T^{4} - 11196061 T^{6} + 3294990 T^{8} - 40331 T^{10} + 2563 T^{12} - 73 T^{14} + T^{16}$$
$41$ $$( 24025 + 34100 T + 16525 T^{2} - 6800 T^{3} + 4965 T^{4} - 1120 T^{5} + 205 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$43$ $$( 5621641 + 1498794 T^{2} + 40151 T^{4} + 354 T^{6} + T^{8} )^{2}$$
$47$ $$6904497224881 - 1172839677427 T^{2} + 77709031188 T^{4} - 38366789 T^{6} + 62210915 T^{8} - 363209 T^{10} + 8988 T^{12} - 127 T^{14} + T^{16}$$
$53$ $$1026625681 + 274975862 T^{2} + 304695493 T^{4} - 25470821 T^{6} + 1570830 T^{8} - 92851 T^{10} + 5443 T^{12} - 113 T^{14} + T^{16}$$
$59$ $$( 5480281 - 323058 T - 87277 T^{2} + 11924 T^{3} + 11925 T^{4} + 44 T^{5} + 183 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$61$ $$( 201601 + 230786 T + 121137 T^{2} + 33298 T^{3} + 7115 T^{4} + 1402 T^{5} + 237 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$67$ $$36562351115761 + 712347395248 T^{2} + 409082935428 T^{4} - 24040052014 T^{6} + 591999590 T^{8} - 6594604 T^{10} + 33813 T^{12} - 62 T^{14} + T^{16}$$
$71$ $$( 15015625 + 968750 T - 9375 T^{2} - 8125 T^{3} + 9750 T^{4} + 1625 T^{5} + 275 T^{6} + 15 T^{7} + T^{8} )^{2}$$
$73$ $$7041810556881 - 2684232173448 T^{2} + 443216612268 T^{4} - 32452285986 T^{6} + 973538190 T^{8} + 5494284 T^{10} + 103113 T^{12} + 222 T^{14} + T^{16}$$
$79$ $$( 1846881 + 2458431 T + 1343817 T^{2} + 174123 T^{3} + 35190 T^{4} + 4887 T^{5} + 477 T^{6} + 24 T^{7} + T^{8} )^{2}$$
$83$ $$62742241 + 30242378 T^{2} + 80074903 T^{4} - 22285364 T^{6} + 2460165 T^{8} - 53524 T^{10} + 2143 T^{12} - 62 T^{14} + T^{16}$$
$89$ $$( 2025 + 2025 T + 2025 T^{2} + 2025 T^{3} + 2790 T^{4} - 855 T^{5} + 105 T^{6} + T^{8} )^{2}$$
$97$ $$329026259524561 - 34127193496858 T^{2} + 1474741769203 T^{4} - 16449229676 T^{6} + 461359725 T^{8} - 6532636 T^{10} + 47683 T^{12} + 22 T^{14} + T^{16}$$