# Properties

 Label 3703.2.a.b Level 3703 Weight 2 Character orbit 3703.a Self dual yes Analytic conductor 29.569 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3703 = 7 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3703.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.5686038685$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 161) 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + ( 2 - 2 \beta ) q^{5} + \beta q^{6} + q^{7} + ( -1 + 2 \beta ) q^{8} -2 q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + ( 2 - 2 \beta ) q^{5} + \beta q^{6} + q^{7} + ( -1 + 2 \beta ) q^{8} -2 q^{9} + 2 q^{10} + ( -2 + 4 \beta ) q^{11} + ( 1 - \beta ) q^{12} + ( -1 - 2 \beta ) q^{13} -\beta q^{14} + ( -2 + 2 \beta ) q^{15} -3 \beta q^{16} + 2 \beta q^{18} + ( 6 - 2 \beta ) q^{19} + ( -4 + 2 \beta ) q^{20} - q^{21} + ( -4 - 2 \beta ) q^{22} + ( 1 - 2 \beta ) q^{24} + ( 3 - 4 \beta ) q^{25} + ( 2 + 3 \beta ) q^{26} + 5 q^{27} + ( -1 + \beta ) q^{28} + ( 1 + 4 \beta ) q^{29} -2 q^{30} -9 q^{31} + ( 5 - \beta ) q^{32} + ( 2 - 4 \beta ) q^{33} + ( 2 - 2 \beta ) q^{35} + ( 2 - 2 \beta ) q^{36} + ( 2 - 6 \beta ) q^{37} + ( 2 - 4 \beta ) q^{38} + ( 1 + 2 \beta ) q^{39} + ( -6 + 2 \beta ) q^{40} + ( -1 + 2 \beta ) q^{41} + \beta q^{42} + 4 \beta q^{43} + ( 6 - 2 \beta ) q^{44} + ( -4 + 4 \beta ) q^{45} + ( -1 + 4 \beta ) q^{47} + 3 \beta q^{48} + q^{49} + ( 4 + \beta ) q^{50} + ( -1 - \beta ) q^{52} + ( -10 + 2 \beta ) q^{53} -5 \beta q^{54} + ( -12 + 4 \beta ) q^{55} + ( -1 + 2 \beta ) q^{56} + ( -6 + 2 \beta ) q^{57} + ( -4 - 5 \beta ) q^{58} + ( -4 - 4 \beta ) q^{59} + ( 4 - 2 \beta ) q^{60} + ( -6 + 12 \beta ) q^{61} + 9 \beta q^{62} -2 q^{63} + ( 1 + 2 \beta ) q^{64} + ( 2 + 2 \beta ) q^{65} + ( 4 + 2 \beta ) q^{66} + ( 6 - 10 \beta ) q^{67} + 2 q^{70} + ( -9 + 2 \beta ) q^{71} + ( 2 - 4 \beta ) q^{72} + ( -3 + 6 \beta ) q^{73} + ( 6 + 4 \beta ) q^{74} + ( -3 + 4 \beta ) q^{75} + ( -8 + 6 \beta ) q^{76} + ( -2 + 4 \beta ) q^{77} + ( -2 - 3 \beta ) q^{78} + ( 6 - 2 \beta ) q^{79} + 6 q^{80} + q^{81} + ( -2 - \beta ) q^{82} + ( -4 + 4 \beta ) q^{83} + ( 1 - \beta ) q^{84} + ( -4 - 4 \beta ) q^{86} + ( -1 - 4 \beta ) q^{87} + 10 q^{88} + ( -4 + 8 \beta ) q^{89} -4 q^{90} + ( -1 - 2 \beta ) q^{91} + 9 q^{93} + ( -4 - 3 \beta ) q^{94} + ( 16 - 12 \beta ) q^{95} + ( -5 + \beta ) q^{96} + 6 \beta q^{97} -\beta q^{98} + ( 4 - 8 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 2q^{3} - q^{4} + 2q^{5} + q^{6} + 2q^{7} - 4q^{9} + O(q^{10})$$ $$2q - q^{2} - 2q^{3} - q^{4} + 2q^{5} + q^{6} + 2q^{7} - 4q^{9} + 4q^{10} + q^{12} - 4q^{13} - q^{14} - 2q^{15} - 3q^{16} + 2q^{18} + 10q^{19} - 6q^{20} - 2q^{21} - 10q^{22} + 2q^{25} + 7q^{26} + 10q^{27} - q^{28} + 6q^{29} - 4q^{30} - 18q^{31} + 9q^{32} + 2q^{35} + 2q^{36} - 2q^{37} + 4q^{39} - 10q^{40} + q^{42} + 4q^{43} + 10q^{44} - 4q^{45} + 2q^{47} + 3q^{48} + 2q^{49} + 9q^{50} - 3q^{52} - 18q^{53} - 5q^{54} - 20q^{55} - 10q^{57} - 13q^{58} - 12q^{59} + 6q^{60} + 9q^{62} - 4q^{63} + 4q^{64} + 6q^{65} + 10q^{66} + 2q^{67} + 4q^{70} - 16q^{71} + 16q^{74} - 2q^{75} - 10q^{76} - 7q^{78} + 10q^{79} + 12q^{80} + 2q^{81} - 5q^{82} - 4q^{83} + q^{84} - 12q^{86} - 6q^{87} + 20q^{88} - 8q^{90} - 4q^{91} + 18q^{93} - 11q^{94} + 20q^{95} - 9q^{96} + 6q^{97} - q^{98} + O(q^{100})$$

## 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
−1.61803 −1.00000 0.618034 −1.23607 1.61803 1.00000 2.23607 −2.00000 2.00000
1.2 0.618034 −1.00000 −1.61803 3.23607 −0.618034 1.00000 −2.23607 −2.00000 2.00000
 $$n$$: 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 3703.2.a.b 2
23.b odd 2 1 161.2.a.b 2
69.c even 2 1 1449.2.a.i 2
92.b even 2 1 2576.2.a.s 2
115.c odd 2 1 4025.2.a.i 2
161.c even 2 1 1127.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.b 2 23.b odd 2 1
1127.2.a.d 2 161.c even 2 1
1449.2.a.i 2 69.c even 2 1
2576.2.a.s 2 92.b even 2 1
3703.2.a.b 2 1.a even 1 1 trivial
4025.2.a.i 2 115.c odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$23$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3703))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{5}^{2} - 2 T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 + T + 3 T^{2} )^{2}$$
$5$ $$1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 + 2 T^{2} + 121 T^{4}$$
$13$ $$1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4}$$
$23$ 1
$29$ $$1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 9 T + 31 T^{2} )^{2}$$
$37$ $$1 + 2 T + 30 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$1 + 77 T^{2} + 1681 T^{4}$$
$43$ $$1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 2 T + 75 T^{2} - 94 T^{3} + 2209 T^{4}$$
$53$ $$1 + 18 T + 182 T^{2} + 954 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 134 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 58 T^{2} + 3721 T^{4}$$
$67$ $$1 - 2 T + 10 T^{2} - 134 T^{3} + 4489 T^{4}$$
$71$ $$1 + 16 T + 201 T^{2} + 1136 T^{3} + 5041 T^{4}$$
$73$ $$1 + 101 T^{2} + 5329 T^{4}$$
$79$ $$1 - 10 T + 178 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 + 4 T + 150 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 98 T^{2} + 7921 T^{4}$$
$97$ $$1 - 6 T + 158 T^{2} - 582 T^{3} + 9409 T^{4}$$