# Properties

 Label 1530.2.f.e Level 1530 Weight 2 Character orbit 1530.f Analytic conductor 12.217 Analytic rank 0 Dimension 2 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1530.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2171115093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 170) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + ( 2 + i ) q^{5} -2 q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + ( 2 + i ) q^{5} -2 q^{7} -i q^{8} + ( -1 + 2 i ) q^{10} -i q^{13} -2 i q^{14} + q^{16} + ( 1 + 4 i ) q^{17} -5 q^{19} + ( -2 - i ) q^{20} -4 q^{23} + ( 3 + 4 i ) q^{25} + q^{26} + 2 q^{28} + 9 i q^{29} + 5 i q^{31} + i q^{32} + ( -4 + i ) q^{34} + ( -4 - 2 i ) q^{35} -2 q^{37} -5 i q^{38} + ( 1 - 2 i ) q^{40} + 10 i q^{41} -6 i q^{43} -4 i q^{46} -7 i q^{47} -3 q^{49} + ( -4 + 3 i ) q^{50} + i q^{52} + i q^{53} + 2 i q^{56} -9 q^{58} + 5 q^{59} + 5 i q^{61} -5 q^{62} - q^{64} + ( 1 - 2 i ) q^{65} + 2 i q^{67} + ( -1 - 4 i ) q^{68} + ( 2 - 4 i ) q^{70} -5 i q^{71} -11 q^{73} -2 i q^{74} + 5 q^{76} + 16 i q^{79} + ( 2 + i ) q^{80} -10 q^{82} + 6 i q^{83} + ( -2 + 9 i ) q^{85} + 6 q^{86} -5 q^{89} + 2 i q^{91} + 4 q^{92} + 7 q^{94} + ( -10 - 5 i ) q^{95} -7 q^{97} -3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 4q^{5} - 4q^{7} + O(q^{10})$$ $$2q - 2q^{4} + 4q^{5} - 4q^{7} - 2q^{10} + 2q^{16} + 2q^{17} - 10q^{19} - 4q^{20} - 8q^{23} + 6q^{25} + 2q^{26} + 4q^{28} - 8q^{34} - 8q^{35} - 4q^{37} + 2q^{40} - 6q^{49} - 8q^{50} - 18q^{58} + 10q^{59} - 10q^{62} - 2q^{64} + 2q^{65} - 2q^{68} + 4q^{70} - 22q^{73} + 10q^{76} + 4q^{80} - 20q^{82} - 4q^{85} + 12q^{86} - 10q^{89} + 8q^{92} + 14q^{94} - 20q^{95} - 14q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times$$.

 $$n$$ $$307$$ $$1261$$ $$1361$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1189.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 2.00000 1.00000i 0 −2.00000 1.00000i 0 −1.00000 2.00000i
1189.2 1.00000i 0 −1.00000 2.00000 + 1.00000i 0 −2.00000 1.00000i 0 −1.00000 + 2.00000i
 $$n$$: 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
85.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.f.e 2
3.b odd 2 1 170.2.d.a 2
5.b even 2 1 1530.2.f.b 2
12.b even 2 1 1360.2.o.b 2
15.d odd 2 1 170.2.d.b yes 2
15.e even 4 1 850.2.b.c 2
15.e even 4 1 850.2.b.i 2
17.b even 2 1 1530.2.f.b 2
51.c odd 2 1 170.2.d.b yes 2
60.h even 2 1 1360.2.o.a 2
85.c even 2 1 inner 1530.2.f.e 2
204.h even 2 1 1360.2.o.a 2
255.h odd 2 1 170.2.d.a 2
255.o even 4 1 850.2.b.c 2
255.o even 4 1 850.2.b.i 2
1020.b even 2 1 1360.2.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 3.b odd 2 1
170.2.d.a 2 255.h odd 2 1
170.2.d.b yes 2 15.d odd 2 1
170.2.d.b yes 2 51.c odd 2 1
850.2.b.c 2 15.e even 4 1
850.2.b.c 2 255.o even 4 1
850.2.b.i 2 15.e even 4 1
850.2.b.i 2 255.o even 4 1
1360.2.o.a 2 60.h even 2 1
1360.2.o.a 2 204.h even 2 1
1360.2.o.b 2 12.b even 2 1
1360.2.o.b 2 1020.b even 2 1
1530.2.f.b 2 5.b even 2 1
1530.2.f.b 2 17.b even 2 1
1530.2.f.e 2 1.a even 1 1 trivial
1530.2.f.e 2 85.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1530, [\chi])$$:

 $$T_{7} + 2$$ $$T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$( 1 + 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$( 1 + 5 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 23 T^{2} + 841 T^{4}$$
$31$ $$1 - 37 T^{2} + 961 T^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$
$43$ $$1 - 50 T^{2} + 1849 T^{4}$$
$47$ $$1 - 45 T^{2} + 2209 T^{4}$$
$53$ $$1 - 105 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 5 T + 59 T^{2} )^{2}$$
$61$ $$1 - 97 T^{2} + 3721 T^{4}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$1 - 117 T^{2} + 5041 T^{4}$$
$73$ $$( 1 + 11 T + 73 T^{2} )^{2}$$
$79$ $$1 + 98 T^{2} + 6241 T^{4}$$
$83$ $$1 - 130 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 5 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 7 T + 97 T^{2} )^{2}$$
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