# Properties

 Label 3850.2.c.y Level $3850$ Weight $2$ Character orbit 3850.c Analytic conductor $30.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 770) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 25 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 17$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{2} + 25 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 3.37228i − 2.37228i 2.37228i − 3.37228i
1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
1849.2 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
1849.3 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
1849.4 1.00000i 2.00000i −1.00000 0 2.00000 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 3850.2.c.y 4
5.b even 2 1 inner 3850.2.c.y 4
5.c odd 4 1 770.2.a.k 2
5.c odd 4 1 3850.2.a.bc 2
15.e even 4 1 6930.2.a.bo 2
20.e even 4 1 6160.2.a.r 2
35.f even 4 1 5390.2.a.bq 2
55.e even 4 1 8470.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.k 2 5.c odd 4 1
3850.2.a.bc 2 5.c odd 4 1
3850.2.c.y 4 1.a even 1 1 trivial
3850.2.c.y 4 5.b even 2 1 inner
5390.2.a.bq 2 35.f even 4 1
6160.2.a.r 2 20.e even 4 1
6930.2.a.bo 2 15.e even 4 1
8470.2.a.bu 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 $$S_{2}^{\mathrm{new}}(3850, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{13}^{4} + 68 T_{13}^{2} + 1024$$ $$T_{17}^{4} + 68 T_{17}^{2} + 1024$$ $$T_{19}^{2} - 2 T_{19} - 32$$ $$T_{37}^{4} + 116 T_{37}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 4 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$1024 + 68 T^{2} + T^{4}$$
$17$ $$1024 + 68 T^{2} + T^{4}$$
$19$ $$( -32 - 2 T + T^{2} )^{2}$$
$23$ $$1024 + 68 T^{2} + T^{4}$$
$29$ $$( -24 + 6 T + T^{2} )^{2}$$
$31$ $$( -32 + 2 T + T^{2} )^{2}$$
$37$ $$64 + 116 T^{2} + T^{4}$$
$41$ $$( 4 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$1024 + 68 T^{2} + T^{4}$$
$53$ $$256 + 164 T^{2} + T^{4}$$
$59$ $$( -24 - 6 T + T^{2} )^{2}$$
$61$ $$( 16 - 14 T + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( 4 + T )^{4}$$
$73$ $$64 + 116 T^{2} + T^{4}$$
$79$ $$( -32 + 2 T + T^{2} )^{2}$$
$83$ $$( 64 + T^{2} )^{2}$$
$89$ $$( -116 + 8 T + T^{2} )^{2}$$
$97$ $$7744 + 308 T^{2} + T^{4}$$