# Properties

 Label 3850.2.c.b Level $3850$ Weight $2$ Character orbit 3850.c Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 − 1.00000i 1.00000i
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
 $$n$$: 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.b 2
5.b even 2 1 inner 3850.2.c.b 2
5.c odd 4 1 770.2.a.f 1
5.c odd 4 1 3850.2.a.k 1
15.e even 4 1 6930.2.a.o 1
20.e even 4 1 6160.2.a.j 1
35.f even 4 1 5390.2.a.bj 1
55.e even 4 1 8470.2.a.c 1

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

 $$T_{3}^{2} + 4$$ $$T_{13}^{2} + 16$$ $$T_{17}$$ $$T_{19} - 4$$ $$T_{37}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 12 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -6 + T )^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$100 + T^{2}$$