# Properties

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

# Learn more about

## 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: yes 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.

 $$f(q)$$ $$=$$ $$q -4 i q^{2} + 9 i q^{3} -16 q^{4} + 36 q^{6} + 233 i q^{7} + 64 i q^{8} -81 q^{9} +O(q^{10})$$ $$q -4 i q^{2} + 9 i q^{3} -16 q^{4} + 36 q^{6} + 233 i q^{7} + 64 i q^{8} -81 q^{9} -498 q^{11} -144 i q^{12} -809 i q^{13} + 932 q^{14} + 256 q^{16} -1002 i q^{17} + 324 i q^{18} + 1705 q^{19} -2097 q^{21} + 1992 i q^{22} -1554 i q^{23} -576 q^{24} -3236 q^{26} -729 i q^{27} -3728 i q^{28} -7830 q^{29} + 977 q^{31} -1024 i q^{32} -4482 i q^{33} -4008 q^{34} + 1296 q^{36} -4822 i q^{37} -6820 i q^{38} + 7281 q^{39} -8148 q^{41} + 8388 i q^{42} -19469 i q^{43} + 7968 q^{44} -6216 q^{46} + 8418 i q^{47} + 2304 i q^{48} -37482 q^{49} + 9018 q^{51} + 12944 i q^{52} -17664 i q^{53} -2916 q^{54} -14912 q^{56} + 15345 i q^{57} + 31320 i q^{58} -35910 q^{59} + 3527 q^{61} -3908 i q^{62} -18873 i q^{63} -4096 q^{64} -17928 q^{66} + 57473 i q^{67} + 16032 i q^{68} + 13986 q^{69} -7548 q^{71} -5184 i q^{72} + 646 i q^{73} -19288 q^{74} -27280 q^{76} -116034 i q^{77} -29124 i q^{78} + 22720 q^{79} + 6561 q^{81} + 32592 i q^{82} -11574 i q^{83} + 33552 q^{84} -77876 q^{86} -70470 i q^{87} -31872 i q^{88} + 78960 q^{89} + 188497 q^{91} + 24864 i q^{92} + 8793 i q^{93} + 33672 q^{94} + 9216 q^{96} + 54593 i q^{97} + 149928 i q^{98} + 40338 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + 72q^{6} - 162q^{9} + O(q^{10})$$ $$2q - 32q^{4} + 72q^{6} - 162q^{9} - 996q^{11} + 1864q^{14} + 512q^{16} + 3410q^{19} - 4194q^{21} - 1152q^{24} - 6472q^{26} - 15660q^{29} + 1954q^{31} - 8016q^{34} + 2592q^{36} + 14562q^{39} - 16296q^{41} + 15936q^{44} - 12432q^{46} - 74964q^{49} + 18036q^{51} - 5832q^{54} - 29824q^{56} - 71820q^{59} + 7054q^{61} - 8192q^{64} - 35856q^{66} + 27972q^{69} - 15096q^{71} - 38576q^{74} - 54560q^{76} + 45440q^{79} + 13122q^{81} + 67104q^{84} - 155752q^{86} + 157920q^{89} + 376994q^{91} + 67344q^{94} + 18432q^{96} + 80676q^{99} + O(q^{100})$$

## 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 233.000i 64.0000i −81.0000 0
49.2 4.00000i 9.00000i −16.0000 0 36.0000 233.000i 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.e 2
3.b odd 2 1 450.6.c.n 2
5.b even 2 1 inner 150.6.c.e 2
5.c odd 4 1 150.6.a.c 1
5.c odd 4 1 150.6.a.l yes 1
15.d odd 2 1 450.6.c.n 2
15.e even 4 1 450.6.a.a 1
15.e even 4 1 450.6.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.c 1 5.c odd 4 1
150.6.a.l yes 1 5.c odd 4 1
150.6.c.e 2 1.a even 1 1 trivial
150.6.c.e 2 5.b even 2 1 inner
450.6.a.a 1 15.e even 4 1
450.6.a.x 1 15.e even 4 1
450.6.c.n 2 3.b odd 2 1
450.6.c.n 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 54289$$ acting on $$S_{6}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$81 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$54289 + T^{2}$$
$11$ $$( 498 + T )^{2}$$
$13$ $$654481 + T^{2}$$
$17$ $$1004004 + T^{2}$$
$19$ $$( -1705 + T )^{2}$$
$23$ $$2414916 + T^{2}$$
$29$ $$( 7830 + T )^{2}$$
$31$ $$( -977 + T )^{2}$$
$37$ $$23251684 + T^{2}$$
$41$ $$( 8148 + T )^{2}$$
$43$ $$379041961 + T^{2}$$
$47$ $$70862724 + T^{2}$$
$53$ $$312016896 + T^{2}$$
$59$ $$( 35910 + T )^{2}$$
$61$ $$( -3527 + T )^{2}$$
$67$ $$3303145729 + T^{2}$$
$71$ $$( 7548 + T )^{2}$$
$73$ $$417316 + T^{2}$$
$79$ $$( -22720 + T )^{2}$$
$83$ $$133957476 + T^{2}$$
$89$ $$( -78960 + T )^{2}$$
$97$ $$2980395649 + T^{2}$$
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