# Properties

 Label 2150.2.b.o Level 2150 Weight 2 Character orbit 2150.b Analytic conductor 17.168 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2150 = 2 \cdot 5^{2} \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2150.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.1678364346$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 430) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{2} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} - q^{4} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{6} -\zeta_{8}^{2} q^{7} + \zeta_{8}^{2} q^{8} + q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{2} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} - q^{4} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{6} -\zeta_{8}^{2} q^{7} + \zeta_{8}^{2} q^{8} + q^{9} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{12} + ( -2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} - q^{14} + q^{16} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} -\zeta_{8}^{2} q^{18} + q^{19} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{21} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} + ( -5 \zeta_{8} + 2 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{23} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{24} + ( 1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{26} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{27} + \zeta_{8}^{2} q^{28} + ( 3 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + ( 1 - \zeta_{8} + \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{2} q^{32} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{33} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{34} - q^{36} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{37} -\zeta_{8}^{2} q^{38} + ( 4 - \zeta_{8} + \zeta_{8}^{3} ) q^{39} + ( 1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{41} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{42} + \zeta_{8}^{2} q^{43} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{44} + ( 2 - 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{46} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{47} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{48} + 6 q^{49} + 2 q^{51} + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{52} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{53} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{54} + q^{56} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{57} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{58} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + ( -7 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{61} + ( \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{62} -\zeta_{8}^{2} q^{63} - q^{64} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{66} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{67} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{68} + ( 10 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} + ( 2 + 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{71} + \zeta_{8}^{2} q^{72} + ( -3 \zeta_{8} - 5 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{73} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{74} - q^{76} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{77} + ( \zeta_{8} - 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{78} + ( 7 + \zeta_{8} - \zeta_{8}^{3} ) q^{79} -5 q^{81} + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{82} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{83} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{84} + q^{86} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{87} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{88} + ( 2 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{89} + ( 1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{91} + ( 5 \zeta_{8} - 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{92} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{93} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{94} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{96} + ( -11 \zeta_{8} + 2 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{97} -6 \zeta_{8}^{2} q^{98} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{9} + 8q^{11} - 4q^{14} + 4q^{16} + 4q^{19} + 4q^{26} + 12q^{29} + 4q^{31} - 4q^{36} + 16q^{39} + 4q^{41} - 8q^{44} + 8q^{46} + 24q^{49} + 8q^{51} + 4q^{56} + 16q^{59} - 28q^{61} - 4q^{64} - 8q^{66} + 40q^{69} + 8q^{71} + 8q^{74} - 4q^{76} + 28q^{79} - 20q^{81} + 4q^{86} + 8q^{89} + 4q^{91} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2150\mathbb{Z}\right)^\times$$.

 $$n$$ $$1377$$ $$1551$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1549.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
1.00000i 1.41421i −1.00000 0 −1.41421 1.00000i 1.00000i 1.00000 0
1549.2 1.00000i 1.41421i −1.00000 0 1.41421 1.00000i 1.00000i 1.00000 0
1549.3 1.00000i 1.41421i −1.00000 0 1.41421 1.00000i 1.00000i 1.00000 0
1549.4 1.00000i 1.41421i −1.00000 0 −1.41421 1.00000i 1.00000i 1.00000 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 2150.2.b.o 4
5.b even 2 1 inner 2150.2.b.o 4
5.c odd 4 1 430.2.a.g 2
5.c odd 4 1 2150.2.a.v 2
15.e even 4 1 3870.2.a.bc 2
20.e even 4 1 3440.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.g 2 5.c odd 4 1
2150.2.a.v 2 5.c odd 4 1
2150.2.b.o 4 1.a even 1 1 trivial
2150.2.b.o 4 5.b even 2 1 inner
3440.2.a.j 2 20.e even 4 1
3870.2.a.bc 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2150, [\chi])$$:

 $$T_{3}^{2} + 2$$ $$T_{7}^{2} + 1$$ $$T_{11}^{2} - 4 T_{11} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 - 4 T^{2} + 9 T^{4} )^{2}$$
$5$ 
$7$ $$( 1 - 13 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 - 34 T^{2} + 595 T^{4} - 5746 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 32 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - T + 19 T^{2} )^{4}$$
$23$ $$1 + 16 T^{2} + 322 T^{4} + 8464 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 6 T + 49 T^{2} - 174 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 2 T + 61 T^{2} - 62 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 - 104 T^{2} + 5154 T^{4} - 142376 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 2 T + 75 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + T^{2} )^{2}$$
$47$ $$( 1 - 44 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 14 T + 121 T^{2} + 854 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 214 T^{2} + 19779 T^{4} - 960646 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 4 T + 48 T^{2} - 284 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 206 T^{2} + 19467 T^{4} - 1097774 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 14 T + 205 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 260 T^{2} + 30166 T^{4} - 1791140 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 4 T + 132 T^{2} - 356 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 104 T^{2} + 17650 T^{4} + 978536 T^{6} + 88529281 T^{8}$$