# Properties

 Label 90.6.c.a 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 10) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -16 q^{4} + ( -55 + 10 i ) q^{5} + 158 i q^{7} -64 i q^{8} +O(q^{10})$$ $$q + 4 i q^{2} -16 q^{4} + ( -55 + 10 i ) q^{5} + 158 i q^{7} -64 i q^{8} + ( -40 - 220 i ) q^{10} + 148 q^{11} -684 i q^{13} -632 q^{14} + 256 q^{16} -2048 i q^{17} -2220 q^{19} + ( 880 - 160 i ) q^{20} + 592 i q^{22} -1246 i q^{23} + ( 2925 - 1100 i ) q^{25} + 2736 q^{26} -2528 i q^{28} -270 q^{29} -2048 q^{31} + 1024 i q^{32} + 8192 q^{34} + ( -1580 - 8690 i ) q^{35} -4372 i q^{37} -8880 i q^{38} + ( 640 + 3520 i ) q^{40} + 2398 q^{41} -2294 i q^{43} -2368 q^{44} + 4984 q^{46} + 10682 i q^{47} -8157 q^{49} + ( 4400 + 11700 i ) q^{50} + 10944 i q^{52} + 2964 i q^{53} + ( -8140 + 1480 i ) q^{55} + 10112 q^{56} -1080 i q^{58} -39740 q^{59} -42298 q^{61} -8192 i q^{62} -4096 q^{64} + ( 6840 + 37620 i ) q^{65} + 32098 i q^{67} + 32768 i q^{68} + ( 34760 - 6320 i ) q^{70} + 4248 q^{71} -30104 i q^{73} + 17488 q^{74} + 35520 q^{76} + 23384 i q^{77} -35280 q^{79} + ( -14080 + 2560 i ) q^{80} + 9592 i q^{82} -27826 i q^{83} + ( 20480 + 112640 i ) q^{85} + 9176 q^{86} -9472 i q^{88} -85210 q^{89} + 108072 q^{91} + 19936 i q^{92} -42728 q^{94} + ( 122100 - 22200 i ) q^{95} -97232 i q^{97} -32628 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} - 110q^{5} + O(q^{10})$$ $$2q - 32q^{4} - 110q^{5} - 80q^{10} + 296q^{11} - 1264q^{14} + 512q^{16} - 4440q^{19} + 1760q^{20} + 5850q^{25} + 5472q^{26} - 540q^{29} - 4096q^{31} + 16384q^{34} - 3160q^{35} + 1280q^{40} + 4796q^{41} - 4736q^{44} + 9968q^{46} - 16314q^{49} + 8800q^{50} - 16280q^{55} + 20224q^{56} - 79480q^{59} - 84596q^{61} - 8192q^{64} + 13680q^{65} + 69520q^{70} + 8496q^{71} + 34976q^{74} + 71040q^{76} - 70560q^{79} - 28160q^{80} + 40960q^{85} + 18352q^{86} - 170420q^{89} + 216144q^{91} - 85456q^{94} + 244200q^{95} + O(q^{100})$$

## 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 0 158.000i 64.0000i 0 −40.0000 + 220.000i
19.2 4.00000i 0 −16.0000 −55.0000 + 10.0000i 0 158.000i 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.a 2
3.b odd 2 1 10.6.b.a 2
4.b odd 2 1 720.6.f.a 2
5.b even 2 1 inner 90.6.c.a 2
5.c odd 4 1 450.6.a.c 1
5.c odd 4 1 450.6.a.w 1
12.b even 2 1 80.6.c.c 2
15.d odd 2 1 10.6.b.a 2
15.e even 4 1 50.6.a.c 1
15.e even 4 1 50.6.a.e 1
20.d odd 2 1 720.6.f.a 2
24.f even 2 1 320.6.c.a 2
24.h odd 2 1 320.6.c.b 2
60.h even 2 1 80.6.c.c 2
60.l odd 4 1 400.6.a.c 1
60.l odd 4 1 400.6.a.k 1
120.i odd 2 1 320.6.c.b 2
120.m even 2 1 320.6.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 3.b odd 2 1
10.6.b.a 2 15.d odd 2 1
50.6.a.c 1 15.e even 4 1
50.6.a.e 1 15.e even 4 1
80.6.c.c 2 12.b even 2 1
80.6.c.c 2 60.h even 2 1
90.6.c.a 2 1.a even 1 1 trivial
90.6.c.a 2 5.b even 2 1 inner
320.6.c.a 2 24.f even 2 1
320.6.c.a 2 120.m even 2 1
320.6.c.b 2 24.h odd 2 1
320.6.c.b 2 120.i odd 2 1
400.6.a.c 1 60.l odd 4 1
400.6.a.k 1 60.l odd 4 1
450.6.a.c 1 5.c odd 4 1
450.6.a.w 1 5.c odd 4 1
720.6.f.a 2 4.b odd 2 1
720.6.f.a 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 24964$$ acting on $$S_{6}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 16 T^{2}$$
$3$ 1
$5$ $$1 + 110 T + 3125 T^{2}$$
$7$ $$1 - 8650 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 148 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 274730 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 1354590 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 2220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 11320170 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 270 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 2048 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 119573530 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 2398 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 288754450 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 344584890 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 827605690 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 39740 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 42298 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 1669968610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 4248 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3239892370 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 35280 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7103795010 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 85210 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 7720618690 T^{2} + 73742412689492826049 T^{4}$$