# Properties

 Label 150.6.a.o Level $150$ Weight $6$ Character orbit 150.a Self dual yes Analytic conductor $24.058$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.0575729719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1249})$$ Defining polynomial: $$x^{2} - x - 312$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 5$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 5\sqrt{1249}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 36 q^{6} + ( 57 - \beta ) q^{7} + 64 q^{8} + 81 q^{9} +O(q^{10})$$ $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 36 q^{6} + ( 57 - \beta ) q^{7} + 64 q^{8} + 81 q^{9} + ( 87 + \beta ) q^{11} + 144 q^{12} + ( 321 + 3 \beta ) q^{13} + ( 228 - 4 \beta ) q^{14} + 256 q^{16} + ( 757 + 3 \beta ) q^{17} + 324 q^{18} + ( -60 + 12 \beta ) q^{19} + ( 513 - 9 \beta ) q^{21} + ( 348 + 4 \beta ) q^{22} + ( 2176 - 12 \beta ) q^{23} + 576 q^{24} + ( 1284 + 12 \beta ) q^{26} + 729 q^{27} + ( 912 - 16 \beta ) q^{28} + ( -1215 - 27 \beta ) q^{29} + ( 5842 - 6 \beta ) q^{31} + 1024 q^{32} + ( 783 + 9 \beta ) q^{33} + ( 3028 + 12 \beta ) q^{34} + 1296 q^{36} + ( -8793 - 19 \beta ) q^{37} + ( -240 + 48 \beta ) q^{38} + ( 2889 + 27 \beta ) q^{39} + ( 12492 + 34 \beta ) q^{41} + ( 2052 - 36 \beta ) q^{42} + ( -12084 + 56 \beta ) q^{43} + ( 1392 + 16 \beta ) q^{44} + ( 8704 - 48 \beta ) q^{46} + ( -6658 + 6 \beta ) q^{47} + 2304 q^{48} + ( 17667 - 114 \beta ) q^{49} + ( 6813 + 27 \beta ) q^{51} + ( 5136 + 48 \beta ) q^{52} + ( -6849 - 111 \beta ) q^{53} + 2916 q^{54} + ( 3648 - 64 \beta ) q^{56} + ( -540 + 108 \beta ) q^{57} + ( -4860 - 108 \beta ) q^{58} + ( -11865 + 185 \beta ) q^{59} + ( 28562 + 96 \beta ) q^{61} + ( 23368 - 24 \beta ) q^{62} + ( 4617 - 81 \beta ) q^{63} + 4096 q^{64} + ( 3132 + 36 \beta ) q^{66} + ( -19158 - 86 \beta ) q^{67} + ( 12112 + 48 \beta ) q^{68} + ( 19584 - 108 \beta ) q^{69} + ( -5538 - 294 \beta ) q^{71} + 5184 q^{72} + ( -44274 + 98 \beta ) q^{73} + ( -35172 - 76 \beta ) q^{74} + ( -960 + 192 \beta ) q^{76} + ( -26266 - 30 \beta ) q^{77} + ( 11556 + 108 \beta ) q^{78} + ( 7110 + 198 \beta ) q^{79} + 6561 q^{81} + ( 49968 + 136 \beta ) q^{82} + ( 25396 + 432 \beta ) q^{83} + ( 8208 - 144 \beta ) q^{84} + ( -48336 + 224 \beta ) q^{86} + ( -10935 - 243 \beta ) q^{87} + ( 5568 + 64 \beta ) q^{88} + ( -30210 - 200 \beta ) q^{89} + ( -75378 - 150 \beta ) q^{91} + ( 34816 - 192 \beta ) q^{92} + ( 52578 - 54 \beta ) q^{93} + ( -26632 + 24 \beta ) q^{94} + 9216 q^{96} + ( -85608 + 16 \beta ) q^{97} + ( 70668 - 456 \beta ) q^{98} + ( 7047 + 81 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{2} + 18q^{3} + 32q^{4} + 72q^{6} + 114q^{7} + 128q^{8} + 162q^{9} + O(q^{10})$$ $$2q + 8q^{2} + 18q^{3} + 32q^{4} + 72q^{6} + 114q^{7} + 128q^{8} + 162q^{9} + 174q^{11} + 288q^{12} + 642q^{13} + 456q^{14} + 512q^{16} + 1514q^{17} + 648q^{18} - 120q^{19} + 1026q^{21} + 696q^{22} + 4352q^{23} + 1152q^{24} + 2568q^{26} + 1458q^{27} + 1824q^{28} - 2430q^{29} + 11684q^{31} + 2048q^{32} + 1566q^{33} + 6056q^{34} + 2592q^{36} - 17586q^{37} - 480q^{38} + 5778q^{39} + 24984q^{41} + 4104q^{42} - 24168q^{43} + 2784q^{44} + 17408q^{46} - 13316q^{47} + 4608q^{48} + 35334q^{49} + 13626q^{51} + 10272q^{52} - 13698q^{53} + 5832q^{54} + 7296q^{56} - 1080q^{57} - 9720q^{58} - 23730q^{59} + 57124q^{61} + 46736q^{62} + 9234q^{63} + 8192q^{64} + 6264q^{66} - 38316q^{67} + 24224q^{68} + 39168q^{69} - 11076q^{71} + 10368q^{72} - 88548q^{73} - 70344q^{74} - 1920q^{76} - 52532q^{77} + 23112q^{78} + 14220q^{79} + 13122q^{81} + 99936q^{82} + 50792q^{83} + 16416q^{84} - 96672q^{86} - 21870q^{87} + 11136q^{88} - 60420q^{89} - 150756q^{91} + 69632q^{92} + 105156q^{93} - 53264q^{94} + 18432q^{96} - 171216q^{97} + 141336q^{98} + 14094q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 18.1706 −17.1706
4.00000 9.00000 16.0000 0 36.0000 −119.706 64.0000 81.0000 0
1.2 4.00000 9.00000 16.0000 0 36.0000 233.706 64.0000 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.6.a.o 2
3.b odd 2 1 450.6.a.bb 2
5.b even 2 1 150.6.a.n 2
5.c odd 4 2 30.6.c.b 4
15.d odd 2 1 450.6.a.bc 2
15.e even 4 2 90.6.c.c 4
20.e even 4 2 240.6.f.b 4
60.l odd 4 2 720.6.f.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.b 4 5.c odd 4 2
90.6.c.c 4 15.e even 4 2
150.6.a.n 2 5.b even 2 1
150.6.a.o 2 1.a even 1 1 trivial
240.6.f.b 4 20.e even 4 2
450.6.a.bb 2 3.b odd 2 1
450.6.a.bc 2 15.d odd 2 1
720.6.f.i 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 114 T_{7} - 27976$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(150))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4 + T )^{2}$$
$3$ $$( -9 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-27976 - 114 T + T^{2}$$
$11$ $$-23656 - 174 T + T^{2}$$
$13$ $$-177984 - 642 T + T^{2}$$
$17$ $$292024 - 1514 T + T^{2}$$
$19$ $$-4492800 + 120 T + T^{2}$$
$23$ $$238576 - 4352 T + T^{2}$$
$29$ $$-21286800 + 2430 T + T^{2}$$
$31$ $$33004864 - 11684 T + T^{2}$$
$37$ $$66044624 + 17586 T + T^{2}$$
$41$ $$119953964 - 24984 T + T^{2}$$
$43$ $$48101456 + 24168 T + T^{2}$$
$47$ $$43204864 + 13316 T + T^{2}$$
$53$ $$-337814424 + 13698 T + T^{2}$$
$59$ $$-927897400 + 23730 T + T^{2}$$
$61$ $$528018244 - 57124 T + T^{2}$$
$67$ $$136088864 + 38316 T + T^{2}$$
$71$ $$-2668294656 + 11076 T + T^{2}$$
$73$ $$1660302176 + 88548 T + T^{2}$$
$79$ $$-1173592800 - 14220 T + T^{2}$$
$83$ $$-5182377584 - 50792 T + T^{2}$$
$89$ $$-336355900 + 60420 T + T^{2}$$
$97$ $$7320736064 + 171216 T + T^{2}$$