# Properties

 Label 3750.2.c.c Level $3750$ Weight $2$ Character orbit 3750.c Analytic conductor $29.944$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3750 = 2 \cdot 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3750.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9439007580$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$$.

 $$n$$ $$2501$$ $$3127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 1.61803i − 0.618034i − 1.61803i 0.618034i
1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.3 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.4 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 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 3750.2.c.c 4
5.b even 2 1 inner 3750.2.c.c 4
5.c odd 4 1 3750.2.a.b 2
5.c odd 4 1 3750.2.a.g 2
25.d even 5 2 750.2.h.a 8
25.e even 10 2 750.2.h.a 8
25.f odd 20 2 150.2.g.b 4
25.f odd 20 2 750.2.g.a 4
75.l even 20 2 450.2.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 25.f odd 20 2
450.2.h.b 4 75.l even 20 2
750.2.g.a 4 25.f odd 20 2
750.2.h.a 8 25.d even 5 2
750.2.h.a 8 25.e even 10 2
3750.2.a.b 2 5.c odd 4 1
3750.2.a.g 2 5.c odd 4 1
3750.2.c.c 4 1.a even 1 1 trivial
3750.2.c.c 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(3750, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 4 + 6 T + T^{2} )^{2}$$
$13$ $$81 + 27 T^{2} + T^{4}$$
$17$ $$81 + 63 T^{2} + T^{4}$$
$19$ $$( 20 - 10 T + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$( 5 - 5 T + T^{2} )^{2}$$
$31$ $$( -36 + 6 T + T^{2} )^{2}$$
$37$ $$361 + 83 T^{2} + T^{4}$$
$41$ $$( -31 + T + T^{2} )^{2}$$
$43$ $$16 + 12 T^{2} + T^{4}$$
$47$ $$1936 + 108 T^{2} + T^{4}$$
$53$ $$9801 + 207 T^{2} + T^{4}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$( 19 + 11 T + T^{2} )^{2}$$
$67$ $$1296 + 108 T^{2} + T^{4}$$
$71$ $$( -116 + 6 T + T^{2} )^{2}$$
$73$ $$961 + 107 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 5 + 5 T + T^{2} )^{2}$$
$97$ $$9801 + 243 T^{2} + T^{4}$$