# Properties

 Label 1850.2.b.i Level $1850$ Weight $2$ Character orbit 1850.b Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} - q^{4} + ( -2 + \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{7} + \beta_{2} q^{8} + ( -4 + 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} - q^{4} + ( -2 + \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{7} + \beta_{2} q^{8} + ( -4 + 3 \beta_{3} ) q^{9} + ( -1 + \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{12} + ( \beta_{1} + \beta_{2} ) q^{13} + 2 \beta_{3} q^{14} + q^{16} -6 \beta_{2} q^{17} + ( -3 \beta_{1} + \beta_{2} ) q^{18} -2 q^{19} + ( -6 + 2 \beta_{3} ) q^{21} -\beta_{1} q^{22} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{23} + ( 2 - \beta_{3} ) q^{24} + \beta_{3} q^{26} + ( -4 \beta_{1} + 7 \beta_{2} ) q^{27} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{28} -3 \beta_{3} q^{29} + ( 1 + \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{33} -6 q^{34} + ( 4 - 3 \beta_{3} ) q^{36} + \beta_{2} q^{37} + 2 \beta_{2} q^{38} + ( -3 + \beta_{3} ) q^{39} + ( 6 - 3 \beta_{3} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{42} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 1 - \beta_{3} ) q^{44} + 3 \beta_{3} q^{46} -2 \beta_{1} q^{47} + ( \beta_{1} - \beta_{2} ) q^{48} + ( -5 - 4 \beta_{3} ) q^{49} + ( -12 + 6 \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{2} ) q^{52} + 6 \beta_{2} q^{53} + ( 11 - 4 \beta_{3} ) q^{54} -2 \beta_{3} q^{56} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{58} + ( -8 + 2 \beta_{3} ) q^{59} + ( 1 - 5 \beta_{3} ) q^{61} + ( -\beta_{1} - 2 \beta_{2} ) q^{62} + ( -2 \beta_{1} + 16 \beta_{2} ) q^{63} - q^{64} + ( 5 - 2 \beta_{3} ) q^{66} + ( 5 \beta_{1} + 8 \beta_{2} ) q^{67} + 6 \beta_{2} q^{68} + ( -9 + 3 \beta_{3} ) q^{69} + 6 q^{71} + ( 3 \beta_{1} - \beta_{2} ) q^{72} + ( -\beta_{1} + 10 \beta_{2} ) q^{73} + q^{74} + 2 q^{76} + 6 \beta_{2} q^{77} + ( -\beta_{1} + 2 \beta_{2} ) q^{78} + 7 \beta_{3} q^{79} + ( 22 - 6 \beta_{3} ) q^{81} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -4 \beta_{1} - 12 \beta_{2} ) q^{83} + ( 6 - 2 \beta_{3} ) q^{84} + ( 2 + 2 \beta_{3} ) q^{86} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{87} + \beta_{1} q^{88} + ( 4 - 4 \beta_{3} ) q^{89} + ( -6 - 2 \beta_{3} ) q^{91} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{92} + \beta_{2} q^{93} + ( 2 - 2 \beta_{3} ) q^{94} + ( -2 + \beta_{3} ) q^{96} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 4 \beta_{1} + 9 \beta_{2} ) q^{98} + ( 13 - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} - 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 2.30278i 1.30278i − 1.30278i 2.30278i
1.00000i 3.30278i −1.00000 0 −3.30278 2.60555i 1.00000i −7.90833 0
149.2 1.00000i 0.302776i −1.00000 0 0.302776 4.60555i 1.00000i 2.90833 0
149.3 1.00000i 0.302776i −1.00000 0 0.302776 4.60555i 1.00000i 2.90833 0
149.4 1.00000i 3.30278i −1.00000 0 −3.30278 2.60555i 1.00000i −7.90833 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 1850.2.b.i 4
5.b even 2 1 inner 1850.2.b.i 4
5.c odd 4 1 74.2.a.a 2
5.c odd 4 1 1850.2.a.u 2
15.e even 4 1 666.2.a.j 2
20.e even 4 1 592.2.a.f 2
35.f even 4 1 3626.2.a.a 2
40.i odd 4 1 2368.2.a.s 2
40.k even 4 1 2368.2.a.ba 2
55.e even 4 1 8954.2.a.p 2
60.l odd 4 1 5328.2.a.bf 2
185.h odd 4 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 5.c odd 4 1
592.2.a.f 2 20.e even 4 1
666.2.a.j 2 15.e even 4 1
1850.2.a.u 2 5.c odd 4 1
1850.2.b.i 4 1.a even 1 1 trivial
1850.2.b.i 4 5.b even 2 1 inner
2368.2.a.s 2 40.i odd 4 1
2368.2.a.ba 2 40.k even 4 1
2738.2.a.l 2 185.h odd 4 1
3626.2.a.a 2 35.f even 4 1
5328.2.a.bf 2 60.l odd 4 1
8954.2.a.p 2 55.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 $$S_{2}^{\mathrm{new}}(1850, [\chi])$$:

 $$T_{3}^{4} + 11 T_{3}^{2} + 1$$ $$T_{7}^{4} + 28 T_{7}^{2} + 144$$ $$T_{13}^{4} + 7 T_{13}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + 11 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$144 + 28 T^{2} + T^{4}$$
$11$ $$( -3 + T + T^{2} )^{2}$$
$13$ $$9 + 7 T^{2} + T^{4}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$729 + 63 T^{2} + T^{4}$$
$29$ $$( -27 + 3 T + T^{2} )^{2}$$
$31$ $$( -1 - 3 T + T^{2} )^{2}$$
$37$ $$( 1 + T^{2} )^{2}$$
$41$ $$( -9 - 9 T + T^{2} )^{2}$$
$43$ $$16 + 44 T^{2} + T^{4}$$
$47$ $$144 + 28 T^{2} + T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 36 + 14 T + T^{2} )^{2}$$
$61$ $$( -79 + 3 T + T^{2} )^{2}$$
$67$ $$2601 + 223 T^{2} + T^{4}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$11449 + 227 T^{2} + T^{4}$$
$79$ $$( -147 - 7 T + T^{2} )^{2}$$
$83$ $$2304 + 304 T^{2} + T^{4}$$
$89$ $$( -48 - 4 T + T^{2} )^{2}$$
$97$ $$41616 + 424 T^{2} + T^{4}$$