# Properties

 Label 150.6.c.b Level 150 Weight 6 Character orbit 150.c Analytic conductor 24.058 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 150.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0575729719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 9 i q^{3} -16 q^{4} -36 q^{6} + 176 i q^{7} -64 i q^{8} -81 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 9 i q^{3} -16 q^{4} -36 q^{6} + 176 i q^{7} -64 i q^{8} -81 q^{9} -60 q^{11} -144 i q^{12} + 658 i q^{13} -704 q^{14} + 256 q^{16} -414 i q^{17} -324 i q^{18} -956 q^{19} -1584 q^{21} -240 i q^{22} -600 i q^{23} + 576 q^{24} -2632 q^{26} -729 i q^{27} -2816 i q^{28} -5574 q^{29} -3592 q^{31} + 1024 i q^{32} -540 i q^{33} + 1656 q^{34} + 1296 q^{36} -8458 i q^{37} -3824 i q^{38} -5922 q^{39} + 19194 q^{41} -6336 i q^{42} -13316 i q^{43} + 960 q^{44} + 2400 q^{46} -19680 i q^{47} + 2304 i q^{48} -14169 q^{49} + 3726 q^{51} -10528 i q^{52} + 31266 i q^{53} + 2916 q^{54} + 11264 q^{56} -8604 i q^{57} -22296 i q^{58} -26340 q^{59} -31090 q^{61} -14368 i q^{62} -14256 i q^{63} -4096 q^{64} + 2160 q^{66} -16804 i q^{67} + 6624 i q^{68} + 5400 q^{69} + 6120 q^{71} + 5184 i q^{72} + 25558 i q^{73} + 33832 q^{74} + 15296 q^{76} -10560 i q^{77} -23688 i q^{78} -74408 q^{79} + 6561 q^{81} + 76776 i q^{82} + 6468 i q^{83} + 25344 q^{84} + 53264 q^{86} -50166 i q^{87} + 3840 i q^{88} + 32742 q^{89} -115808 q^{91} + 9600 i q^{92} -32328 i q^{93} + 78720 q^{94} -9216 q^{96} + 166082 i q^{97} -56676 i q^{98} + 4860 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} - 72q^{6} - 162q^{9} + O(q^{10})$$ $$2q - 32q^{4} - 72q^{6} - 162q^{9} - 120q^{11} - 1408q^{14} + 512q^{16} - 1912q^{19} - 3168q^{21} + 1152q^{24} - 5264q^{26} - 11148q^{29} - 7184q^{31} + 3312q^{34} + 2592q^{36} - 11844q^{39} + 38388q^{41} + 1920q^{44} + 4800q^{46} - 28338q^{49} + 7452q^{51} + 5832q^{54} + 22528q^{56} - 52680q^{59} - 62180q^{61} - 8192q^{64} + 4320q^{66} + 10800q^{69} + 12240q^{71} + 67664q^{74} + 30592q^{76} - 148816q^{79} + 13122q^{81} + 50688q^{84} + 106528q^{86} + 65484q^{89} - 231616q^{91} + 157440q^{94} - 18432q^{96} + 9720q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
4.00000i 9.00000i −16.0000 0 −36.0000 176.000i 64.0000i −81.0000 0
49.2 4.00000i 9.00000i −16.0000 0 −36.0000 176.000i 64.0000i −81.0000 0
 $$n$$: 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 150.6.c.b 2
3.b odd 2 1 450.6.c.j 2
5.b even 2 1 inner 150.6.c.b 2
5.c odd 4 1 6.6.a.a 1
5.c odd 4 1 150.6.a.d 1
15.d odd 2 1 450.6.c.j 2
15.e even 4 1 18.6.a.b 1
15.e even 4 1 450.6.a.m 1
20.e even 4 1 48.6.a.c 1
35.f even 4 1 294.6.a.m 1
35.k even 12 2 294.6.e.a 2
35.l odd 12 2 294.6.e.g 2
40.i odd 4 1 192.6.a.o 1
40.k even 4 1 192.6.a.g 1
45.k odd 12 2 162.6.c.e 2
45.l even 12 2 162.6.c.h 2
55.e even 4 1 726.6.a.a 1
60.l odd 4 1 144.6.a.j 1
65.h odd 4 1 1014.6.a.c 1
80.i odd 4 1 768.6.d.c 2
80.j even 4 1 768.6.d.p 2
80.s even 4 1 768.6.d.p 2
80.t odd 4 1 768.6.d.c 2
105.k odd 4 1 882.6.a.a 1
120.q odd 4 1 576.6.a.i 1
120.w even 4 1 576.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 5.c odd 4 1
18.6.a.b 1 15.e even 4 1
48.6.a.c 1 20.e even 4 1
144.6.a.j 1 60.l odd 4 1
150.6.a.d 1 5.c odd 4 1
150.6.c.b 2 1.a even 1 1 trivial
150.6.c.b 2 5.b even 2 1 inner
162.6.c.e 2 45.k odd 12 2
162.6.c.h 2 45.l even 12 2
192.6.a.g 1 40.k even 4 1
192.6.a.o 1 40.i odd 4 1
294.6.a.m 1 35.f even 4 1
294.6.e.a 2 35.k even 12 2
294.6.e.g 2 35.l odd 12 2
450.6.a.m 1 15.e even 4 1
450.6.c.j 2 3.b odd 2 1
450.6.c.j 2 15.d odd 2 1
576.6.a.i 1 120.q odd 4 1
576.6.a.j 1 120.w even 4 1
726.6.a.a 1 55.e even 4 1
768.6.d.c 2 80.i odd 4 1
768.6.d.c 2 80.t odd 4 1
768.6.d.p 2 80.j even 4 1
768.6.d.p 2 80.s even 4 1
882.6.a.a 1 105.k odd 4 1
1014.6.a.c 1 65.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 30976$$ acting on $$S_{6}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 16 T^{2}$$
$3$ $$1 + 81 T^{2}$$
$5$ 1
$7$ $$1 - 2638 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 60 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 309622 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2668318 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 956 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 12512686 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 5574 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 3592 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 67150150 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 19194 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 116701030 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 71387614 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 + 141171770 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 26340 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 31090 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 2417875798 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 6120 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3492931822 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 74408 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7836246262 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 32742 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 + 10408550210 T^{2} + 73742412689492826049 T^{4}$$