# Properties

 Label 150.16.c.a Level $150$ Weight $16$ Character orbit 150.c Analytic conductor $214.040$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 150.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$214.040257650$$ 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.

 $$f(q)$$ $$=$$ $$q -128 i q^{2} -2187 i q^{3} -16384 q^{4} -279936 q^{6} + 3034528 i q^{7} + 2097152 i q^{8} -4782969 q^{9} +O(q^{10})$$ $$q -128 i q^{2} -2187 i q^{3} -16384 q^{4} -279936 q^{6} + 3034528 i q^{7} + 2097152 i q^{8} -4782969 q^{9} -103451700 q^{11} + 35831808 i q^{12} -104365834 i q^{13} + 388419584 q^{14} + 268435456 q^{16} -997689762 i q^{17} + 612220032 i q^{18} -4934015444 q^{19} + 6636512736 q^{21} + 13241817600 i q^{22} + 8324920200 i q^{23} + 4586471424 q^{24} -13358826752 q^{26} + 10460353203 i q^{27} -49717706752 i q^{28} -104128242846 q^{29} -296696681512 q^{31} -34359738368 i q^{32} + 226248867900 i q^{33} -127704289536 q^{34} + 78364164096 q^{36} + 178337455666 i q^{37} + 631553976832 i q^{38} -228248078958 q^{39} -1790882416086 q^{41} -849473630208 i q^{42} -2863459422772 i q^{43} + 1694952652800 q^{44} + 1065589785600 q^{46} -4332907521600 i q^{47} -587068342272 i q^{48} -4460798672841 q^{49} -2181947509494 q^{51} + 1709929824256 i q^{52} + 9732317104422 i q^{53} + 1338925209984 q^{54} -6363866464256 q^{56} + 10790691776028 i q^{57} + 13328415084288 i q^{58} + 13514837176500 q^{59} + 5352663511190 q^{61} + 37977175233536 i q^{62} -14514053353632 i q^{63} -4398046511104 q^{64} + 28959855091200 q^{66} + 53233909720108 i q^{67} + 16346149060608 i q^{68} + 18206600477400 q^{69} -20229661643400 q^{71} -10030613004288 i q^{72} + 26264166466106 i q^{73} + 22827194325248 q^{74} + 80838909034496 q^{76} -313927080297600 i q^{77} + 29215754106624 i q^{78} + 339031361615128 q^{79} + 22876792454961 q^{81} + 229232949259008 i q^{82} + 131684771045076 i q^{83} -108732624666624 q^{84} -366522806114816 q^{86} + 227728467104202 i q^{87} -216953939558400 i q^{88} + 39352148322678 q^{89} + 316701045516352 q^{91} -136395492556800 i q^{92} + 648875642466744 i q^{93} -554612162764800 q^{94} -75144747810816 q^{96} -1128750908801474 i q^{97} + 570982230123648 i q^{98} + 494806274097300 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32768 q^{4} - 559872 q^{6} - 9565938 q^{9} + O(q^{10})$$ $$2 q - 32768 q^{4} - 559872 q^{6} - 9565938 q^{9} - 206903400 q^{11} + 776839168 q^{14} + 536870912 q^{16} - 9868030888 q^{19} + 13273025472 q^{21} + 9172942848 q^{24} - 26717653504 q^{26} - 208256485692 q^{29} - 593393363024 q^{31} - 255408579072 q^{34} + 156728328192 q^{36} - 456496157916 q^{39} - 3581764832172 q^{41} + 3389905305600 q^{44} + 2131179571200 q^{46} - 8921597345682 q^{49} - 4363895018988 q^{51} + 2677850419968 q^{54} - 12727732928512 q^{56} + 27029674353000 q^{59} + 10705327022380 q^{61} - 8796093022208 q^{64} + 57919710182400 q^{66} + 36413200954800 q^{69} - 40459323286800 q^{71} + 45654388650496 q^{74} + 161677818068992 q^{76} + 678062723230256 q^{79} + 45753584909922 q^{81} - 217465249333248 q^{84} - 733045612229632 q^{86} + 78704296645356 q^{89} + 633402091032704 q^{91} - 1109224325529600 q^{94} - 150289495621632 q^{96} + 989612548194600 q^{99} + O(q^{100})$$

## 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
128.000i 2187.00i −16384.0 0 −279936. 3.03453e6i 2.09715e6i −4.78297e6 0
49.2 128.000i 2187.00i −16384.0 0 −279936. 3.03453e6i 2.09715e6i −4.78297e6 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 150.16.c.a 2
5.b even 2 1 inner 150.16.c.a 2
5.c odd 4 1 6.16.a.b 1
5.c odd 4 1 150.16.a.f 1
15.e even 4 1 18.16.a.b 1
20.e even 4 1 48.16.a.d 1
60.l odd 4 1 144.16.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b 1 5.c odd 4 1
18.16.a.b 1 15.e even 4 1
48.16.a.d 1 20.e even 4 1
144.16.a.j 1 60.l odd 4 1
150.16.a.f 1 5.c odd 4 1
150.16.c.a 2 1.a even 1 1 trivial
150.16.c.a 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} +$$$$92\!\cdots\!84$$ acting on $$S_{16}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16384 + T^{2}$$
$3$ $$4782969 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$9208360182784 + T^{2}$$
$11$ $$( 103451700 + T )^{2}$$
$13$ $$10892227306515556 + T^{2}$$
$17$ $$995384861199616644 + T^{2}$$
$19$ $$( 4934015444 + T )^{2}$$
$23$ $$69304296336368040000 + T^{2}$$
$29$ $$( 104128242846 + T )^{2}$$
$31$ $$( 296696681512 + T )^{2}$$
$37$ $$31\!\cdots\!56$$$$+ T^{2}$$
$41$ $$( 1790882416086 + T )^{2}$$
$43$ $$81\!\cdots\!84$$$$+ T^{2}$$
$47$ $$18\!\cdots\!00$$$$+ T^{2}$$
$53$ $$94\!\cdots\!84$$$$+ T^{2}$$
$59$ $$( -13514837176500 + T )^{2}$$
$61$ $$( -5352663511190 + T )^{2}$$
$67$ $$28\!\cdots\!64$$$$+ T^{2}$$
$71$ $$( 20229661643400 + T )^{2}$$
$73$ $$68\!\cdots\!36$$$$+ T^{2}$$
$79$ $$( -339031361615128 + T )^{2}$$
$83$ $$17\!\cdots\!76$$$$+ T^{2}$$
$89$ $$( -39352148322678 + T )^{2}$$
$97$ $$12\!\cdots\!76$$$$+ T^{2}$$