# Properties

 Label 3850.2.c.x 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
1.00000i 0.732051i −1.00000 0 −0.732051 1.00000i 1.00000i 2.46410 0
1849.2 1.00000i 2.73205i −1.00000 0 2.73205 1.00000i 1.00000i −4.46410 0
1849.3 1.00000i 2.73205i −1.00000 0 2.73205 1.00000i 1.00000i −4.46410 0
1849.4 1.00000i 0.732051i −1.00000 0 −0.732051 1.00000i 1.00000i 2.46410 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.x 4
5.b even 2 1 inner 3850.2.c.x 4
5.c odd 4 1 770.2.a.j 2
5.c odd 4 1 3850.2.a.bd 2
15.e even 4 1 6930.2.a.bv 2
20.e even 4 1 6160.2.a.t 2
35.f even 4 1 5390.2.a.bs 2
55.e even 4 1 8470.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.j 2 5.c odd 4 1
3850.2.a.bd 2 5.c odd 4 1
3850.2.c.x 4 1.a even 1 1 trivial
3850.2.c.x 4 5.b even 2 1 inner
5390.2.a.bs 2 35.f even 4 1
6160.2.a.t 2 20.e even 4 1
6930.2.a.bv 2 15.e even 4 1
8470.2.a.br 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 $$S_{2}^{\mathrm{new}}(3850, [\chi])$$:

 $$T_{3}^{4} + 8 T_{3}^{2} + 4$$ $$T_{13}^{4} + 32 T_{13}^{2} + 64$$ $$T_{17}^{2} + 12$$ $$T_{19}^{2} + 10 T_{19} + 22$$ $$T_{37}^{4} + 8 T_{37}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$4 + 8 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$64 + 32 T^{2} + T^{4}$$
$17$ $$( 12 + T^{2} )^{2}$$
$19$ $$( 22 + 10 T + T^{2} )^{2}$$
$23$ $$324 + 72 T^{2} + T^{4}$$
$29$ $$( 6 - 6 T + T^{2} )^{2}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$4 + 8 T^{2} + T^{4}$$
$41$ $$( -18 - 6 T + T^{2} )^{2}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$( 48 + T^{2} )^{2}$$
$53$ $$6084 + 168 T^{2} + T^{4}$$
$59$ $$( -48 + T^{2} )^{2}$$
$61$ $$( -44 - 4 T + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( 24 - 12 T + T^{2} )^{2}$$
$73$ $$8464 + 248 T^{2} + T^{4}$$
$79$ $$( 22 - 14 T + T^{2} )^{2}$$
$83$ $$5184 + 288 T^{2} + T^{4}$$
$89$ $$( -12 + T^{2} )^{2}$$
$97$ $$58564 + 488 T^{2} + T^{4}$$