# Properties

 Label 150.6.a.n 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 −233.706 −64.0000 81.0000 0
1.2 −4.00000 −9.00000 16.0000 0 36.0000 119.706 −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.n 2
3.b odd 2 1 450.6.a.bc 2
5.b even 2 1 150.6.a.o 2
5.c odd 4 2 30.6.c.b 4
15.d odd 2 1 450.6.a.bb 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 1.a even 1 1 trivial
150.6.a.o 2 5.b even 2 1
240.6.f.b 4 20.e even 4 2
450.6.a.bb 2 15.d odd 2 1
450.6.a.bc 2 3.b 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

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