# Properties

 Label 90.6.c.b Level $90$ Weight $6$ Character orbit 90.c Analytic conductor $14.435$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 90.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4345437832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -16 q^{4} + ( 55 + 10 i ) q^{5} + 4 i q^{7} -64 i q^{8} +O(q^{10})$$ $$q + 4 i q^{2} -16 q^{4} + ( 55 + 10 i ) q^{5} + 4 i q^{7} -64 i q^{8} + ( -40 + 220 i ) q^{10} + 500 q^{11} -288 i q^{13} -16 q^{14} + 256 q^{16} + 1516 i q^{17} + 1344 q^{19} + ( -880 - 160 i ) q^{20} + 2000 i q^{22} + 4100 i q^{23} + ( 2925 + 1100 i ) q^{25} + 1152 q^{26} -64 i q^{28} -2646 q^{29} -5612 q^{31} + 1024 i q^{32} -6064 q^{34} + ( -40 + 220 i ) q^{35} + 7288 i q^{37} + 5376 i q^{38} + ( 640 - 3520 i ) q^{40} + 18986 q^{41} -2404 i q^{43} -8000 q^{44} -16400 q^{46} + 8900 i q^{47} + 16791 q^{49} + ( -4400 + 11700 i ) q^{50} + 4608 i q^{52} -39804 i q^{53} + ( 27500 + 5000 i ) q^{55} + 256 q^{56} -10584 i q^{58} -28300 q^{59} + 18290 q^{61} -22448 i q^{62} -4096 q^{64} + ( 2880 - 15840 i ) q^{65} -65956 i q^{67} -24256 i q^{68} + ( -880 - 160 i ) q^{70} + 28800 q^{71} -30808 i q^{73} -29152 q^{74} -21504 q^{76} + 2000 i q^{77} -60228 q^{79} + ( 14080 + 2560 i ) q^{80} + 75944 i q^{82} + 2468 i q^{83} + ( -15160 + 83380 i ) q^{85} + 9616 q^{86} -32000 i q^{88} + 22678 q^{89} + 1152 q^{91} -65600 i q^{92} -35600 q^{94} + ( 73920 + 13440 i ) q^{95} + 36968 i q^{97} + 67164 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} + 110 q^{5} + O(q^{10})$$ $$2 q - 32 q^{4} + 110 q^{5} - 80 q^{10} + 1000 q^{11} - 32 q^{14} + 512 q^{16} + 2688 q^{19} - 1760 q^{20} + 5850 q^{25} + 2304 q^{26} - 5292 q^{29} - 11224 q^{31} - 12128 q^{34} - 80 q^{35} + 1280 q^{40} + 37972 q^{41} - 16000 q^{44} - 32800 q^{46} + 33582 q^{49} - 8800 q^{50} + 55000 q^{55} + 512 q^{56} - 56600 q^{59} + 36580 q^{61} - 8192 q^{64} + 5760 q^{65} - 1760 q^{70} + 57600 q^{71} - 58304 q^{74} - 43008 q^{76} - 120456 q^{79} + 28160 q^{80} - 30320 q^{85} + 19232 q^{86} + 45356 q^{89} + 2304 q^{91} - 71200 q^{94} + 147840 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
4.00000i 0 −16.0000 55.0000 10.0000i 0 4.00000i 64.0000i 0 −40.0000 220.000i
19.2 4.00000i 0 −16.0000 55.0000 + 10.0000i 0 4.00000i 64.0000i 0 −40.0000 + 220.000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.6.c.b 2
3.b odd 2 1 30.6.c.a 2
4.b odd 2 1 720.6.f.g 2
5.b even 2 1 inner 90.6.c.b 2
5.c odd 4 1 450.6.a.f 1
5.c odd 4 1 450.6.a.s 1
12.b even 2 1 240.6.f.a 2
15.d odd 2 1 30.6.c.a 2
15.e even 4 1 150.6.a.f 1
15.e even 4 1 150.6.a.j 1
20.d odd 2 1 720.6.f.g 2
60.h even 2 1 240.6.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 3.b odd 2 1
30.6.c.a 2 15.d odd 2 1
90.6.c.b 2 1.a even 1 1 trivial
90.6.c.b 2 5.b even 2 1 inner
150.6.a.f 1 15.e even 4 1
150.6.a.j 1 15.e even 4 1
240.6.f.a 2 12.b even 2 1
240.6.f.a 2 60.h even 2 1
450.6.a.f 1 5.c odd 4 1
450.6.a.s 1 5.c odd 4 1
720.6.f.g 2 4.b odd 2 1
720.6.f.g 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 16$$ acting on $$S_{6}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$3125 - 110 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( -500 + T )^{2}$$
$13$ $$82944 + T^{2}$$
$17$ $$2298256 + T^{2}$$
$19$ $$( -1344 + T )^{2}$$
$23$ $$16810000 + T^{2}$$
$29$ $$( 2646 + T )^{2}$$
$31$ $$( 5612 + T )^{2}$$
$37$ $$53114944 + T^{2}$$
$41$ $$( -18986 + T )^{2}$$
$43$ $$5779216 + T^{2}$$
$47$ $$79210000 + T^{2}$$
$53$ $$1584358416 + T^{2}$$
$59$ $$( 28300 + T )^{2}$$
$61$ $$( -18290 + T )^{2}$$
$67$ $$4350193936 + T^{2}$$
$71$ $$( -28800 + T )^{2}$$
$73$ $$949132864 + T^{2}$$
$79$ $$( 60228 + T )^{2}$$
$83$ $$6091024 + T^{2}$$
$89$ $$( -22678 + T )^{2}$$
$97$ $$1366633024 + T^{2}$$