# Properties

 Label 3850.2.c.ba Level $3850$ Weight $2$ Character orbit 3850.c Analytic conductor $30.742$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{3} + 4 \nu^{2} + 2 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu^{3} - 4 \nu^{2} + 2 \nu - 8$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{3} + 4 \nu + 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - 7 \nu^{3} + 8 \nu^{2} - 6 \nu + 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} + 2 \nu^{3} - 3 \nu^{2} + 3 \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 3 \beta_{1} + 6$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + 3 \beta_{4} + \beta_{3} + 10 \beta_{2} + \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + \beta_{1} + 10$$$$)/4$$

## 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
 0.264658 − 1.38923i −0.671462 + 1.24464i 1.40680 + 0.144584i 1.40680 − 0.144584i −0.671462 − 1.24464i 0.264658 + 1.38923i
1.00000i 2.24914i −1.00000 0 −2.24914 1.00000i 1.00000i −2.05863 0
1849.2 1.00000i 1.14637i −1.00000 0 1.14637 1.00000i 1.00000i 1.68585 0
1849.3 1.00000i 3.10278i −1.00000 0 3.10278 1.00000i 1.00000i −6.62721 0
1849.4 1.00000i 3.10278i −1.00000 0 3.10278 1.00000i 1.00000i −6.62721 0
1849.5 1.00000i 1.14637i −1.00000 0 1.14637 1.00000i 1.00000i 1.68585 0
1849.6 1.00000i 2.24914i −1.00000 0 −2.24914 1.00000i 1.00000i −2.05863 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1849.6 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.ba 6
5.b even 2 1 inner 3850.2.c.ba 6
5.c odd 4 1 770.2.a.m 3
5.c odd 4 1 3850.2.a.bt 3
15.e even 4 1 6930.2.a.ce 3
20.e even 4 1 6160.2.a.bf 3
35.f even 4 1 5390.2.a.ca 3
55.e even 4 1 8470.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.m 3 5.c odd 4 1
3850.2.a.bt 3 5.c odd 4 1
3850.2.c.ba 6 1.a even 1 1 trivial
3850.2.c.ba 6 5.b even 2 1 inner
5390.2.a.ca 3 35.f even 4 1
6160.2.a.bf 3 20.e even 4 1
6930.2.a.ce 3 15.e even 4 1
8470.2.a.ci 3 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 $$S_{2}^{\mathrm{new}}(3850, [\chi])$$:

 $$T_{3}^{6} + 16 T_{3}^{4} + 68 T_{3}^{2} + 64$$ $$T_{13}^{6} + 36 T_{13}^{4} + 320 T_{13}^{2} + 256$$ $$T_{17}^{6} + 60 T_{17}^{4} + 752 T_{17}^{2} + 64$$ $$T_{19}^{3} - 6 T_{19}^{2} - 46 T_{19} + 232$$ $$T_{37}^{6} + 20 T_{37}^{4} + 36 T_{37}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$64 + 68 T^{2} + 16 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( -1 + T )^{6}$$
$13$ $$256 + 320 T^{2} + 36 T^{4} + T^{6}$$
$17$ $$64 + 752 T^{2} + 60 T^{4} + T^{6}$$
$19$ $$( 232 - 46 T - 6 T^{2} + T^{3} )^{2}$$
$23$ $$256 + 356 T^{2} + 48 T^{4} + T^{6}$$
$29$ $$( 92 - 66 T + T^{3} )^{2}$$
$31$ $$( -32 + 20 T + 12 T^{2} + T^{3} )^{2}$$
$37$ $$16 + 36 T^{2} + 20 T^{4} + T^{6}$$
$41$ $$( -164 - 90 T - 4 T^{2} + T^{3} )^{2}$$
$43$ $$719104 + 25232 T^{2} + 280 T^{4} + T^{6}$$
$47$ $$16384 + 12544 T^{2} + 224 T^{4} + T^{6}$$
$53$ $$85264 + 7332 T^{2} + 180 T^{4} + T^{6}$$
$59$ $$( 128 - 64 T - 4 T^{2} + T^{3} )^{2}$$
$61$ $$( -344 - 100 T + 6 T^{2} + T^{3} )^{2}$$
$67$ $$4096 + 12032 T^{2} + 240 T^{4} + T^{6}$$
$71$ $$( 64 + 200 T + 28 T^{2} + T^{3} )^{2}$$
$73$ $$64 + 1520 T^{2} + 124 T^{4} + T^{6}$$
$79$ $$( -256 + 50 T + 18 T^{2} + T^{3} )^{2}$$
$83$ $$1024 + 576 T^{2} + 80 T^{4} + T^{6}$$
$89$ $$( -1208 - 28 T + 18 T^{2} + T^{3} )^{2}$$
$97$ $$99856 + 21188 T^{2} + 324 T^{4} + T^{6}$$