# Properties

 Label 3750.2.a.d Level 3750 Weight 2 Character orbit 3750.a Self dual yes Analytic conductor 29.944 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3750 = 2 \cdot 3 \cdot 5^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3750.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9439007580$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−1.00000 1.00000 1.00000 0 −1.00000 −2.61803 −1.00000 1.00000 0
1.2 −1.00000 1.00000 1.00000 0 −1.00000 −0.381966 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 3 T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3750))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 - T )^{2}$$
$5$ 
$7$ $$1 + 3 T + 15 T^{2} + 21 T^{3} + 49 T^{4}$$
$11$ $$1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4}$$
$13$ $$1 - 4 T + 10 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$1 - 2 T + 30 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4}$$
$23$ $$1 + 26 T^{2} + 529 T^{4}$$
$29$ $$1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 11 T + 91 T^{2} - 341 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{2}$$
$41$ $$1 + 2 T + 38 T^{2} + 82 T^{3} + 1681 T^{4}$$
$43$ $$1 - 2 T + 42 T^{2} - 86 T^{3} + 1849 T^{4}$$
$47$ $$1 - 10 T + 74 T^{2} - 470 T^{3} + 2209 T^{4}$$
$53$ $$1 + 7 T + 87 T^{2} + 371 T^{3} + 2809 T^{4}$$
$59$ $$1 + 5 T + 123 T^{2} + 295 T^{3} + 3481 T^{4}$$
$61$ $$1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4}$$
$67$ $$1 - 12 T + 150 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4}$$
$73$ $$1 + 16 T + 190 T^{2} + 1168 T^{3} + 5329 T^{4}$$
$79$ $$1 + 9 T + 177 T^{2} + 711 T^{3} + 6241 T^{4}$$
$83$ $$1 + 11 T + 185 T^{2} + 913 T^{3} + 6889 T^{4}$$
$89$ $$1 - 16 T + 222 T^{2} - 1424 T^{3} + 7921 T^{4}$$
$97$ $$1 - 9 T + 213 T^{2} - 873 T^{3} + 9409 T^{4}$$