Properties

 Label 1850.2.a.u Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 - \beta ) q^{3} + q^{4} + ( -1 - \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + q^{8} + ( 1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 - \beta ) q^{3} + q^{4} + ( -1 - \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + q^{8} + ( 1 + 3 \beta ) q^{9} -\beta q^{11} + ( -1 - \beta ) q^{12} + ( 1 - \beta ) q^{13} + ( -2 + 2 \beta ) q^{14} + q^{16} + 6 q^{17} + ( 1 + 3 \beta ) q^{18} + 2 q^{19} + ( -4 - 2 \beta ) q^{21} -\beta q^{22} + ( 3 - 3 \beta ) q^{23} + ( -1 - \beta ) q^{24} + ( 1 - \beta ) q^{26} + ( -7 - 4 \beta ) q^{27} + ( -2 + 2 \beta ) q^{28} + ( 3 - 3 \beta ) q^{29} + ( 2 - \beta ) q^{31} + q^{32} + ( 3 + 2 \beta ) q^{33} + 6 q^{34} + ( 1 + 3 \beta ) q^{36} - q^{37} + 2 q^{38} + ( 2 + \beta ) q^{39} + ( 3 + 3 \beta ) q^{41} + ( -4 - 2 \beta ) q^{42} + ( 4 - 2 \beta ) q^{43} -\beta q^{44} + ( 3 - 3 \beta ) q^{46} -2 \beta q^{47} + ( -1 - \beta ) q^{48} + ( 9 - 4 \beta ) q^{49} + ( -6 - 6 \beta ) q^{51} + ( 1 - \beta ) q^{52} + 6 q^{53} + ( -7 - 4 \beta ) q^{54} + ( -2 + 2 \beta ) q^{56} + ( -2 - 2 \beta ) q^{57} + ( 3 - 3 \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( -4 + 5 \beta ) q^{61} + ( 2 - \beta ) q^{62} + ( 16 + 2 \beta ) q^{63} + q^{64} + ( 3 + 2 \beta ) q^{66} + ( -8 + 5 \beta ) q^{67} + 6 q^{68} + ( 6 + 3 \beta ) q^{69} + 6 q^{71} + ( 1 + 3 \beta ) q^{72} + ( 10 + \beta ) q^{73} - q^{74} + 2 q^{76} -6 q^{77} + ( 2 + \beta ) q^{78} + ( -7 + 7 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 3 + 3 \beta ) q^{82} + ( -12 + 4 \beta ) q^{83} + ( -4 - 2 \beta ) q^{84} + ( 4 - 2 \beta ) q^{86} + ( 6 + 3 \beta ) q^{87} -\beta q^{88} -4 \beta q^{89} + ( -8 + 2 \beta ) q^{91} + ( 3 - 3 \beta ) q^{92} + q^{93} -2 \beta q^{94} + ( -1 - \beta ) q^{96} + ( -2 + 8 \beta ) q^{97} + ( 9 - 4 \beta ) q^{98} + ( -9 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24} + q^{26} - 18 q^{27} - 2 q^{28} + 3 q^{29} + 3 q^{31} + 2 q^{32} + 8 q^{33} + 12 q^{34} + 5 q^{36} - 2 q^{37} + 4 q^{38} + 5 q^{39} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 14 q^{49} - 18 q^{51} + q^{52} + 12 q^{53} - 18 q^{54} - 2 q^{56} - 6 q^{57} + 3 q^{58} + 14 q^{59} - 3 q^{61} + 3 q^{62} + 34 q^{63} + 2 q^{64} + 8 q^{66} - 11 q^{67} + 12 q^{68} + 15 q^{69} + 12 q^{71} + 5 q^{72} + 21 q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{77} + 5 q^{78} - 7 q^{79} + 38 q^{81} + 9 q^{82} - 20 q^{83} - 10 q^{84} + 6 q^{86} + 15 q^{87} - q^{88} - 4 q^{89} - 14 q^{91} + 3 q^{92} + 2 q^{93} - 2 q^{94} - 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99} + O(q^{100})$$

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
 2.30278 −1.30278
1.00000 −3.30278 1.00000 0 −3.30278 2.60555 1.00000 7.90833 0
1.2 1.00000 0.302776 1.00000 0 0.302776 −4.60555 1.00000 −2.90833 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$37$$ $$1$$

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 1850.2.a.u 2
5.b even 2 1 74.2.a.a 2
5.c odd 4 2 1850.2.b.i 4
15.d odd 2 1 666.2.a.j 2
20.d odd 2 1 592.2.a.f 2
35.c odd 2 1 3626.2.a.a 2
40.e odd 2 1 2368.2.a.ba 2
40.f even 2 1 2368.2.a.s 2
55.d odd 2 1 8954.2.a.p 2
60.h even 2 1 5328.2.a.bf 2
185.d even 2 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 5.b even 2 1
592.2.a.f 2 20.d odd 2 1
666.2.a.j 2 15.d odd 2 1
1850.2.a.u 2 1.a even 1 1 trivial
1850.2.b.i 4 5.c odd 4 2
2368.2.a.s 2 40.f even 2 1
2368.2.a.ba 2 40.e odd 2 1
2738.2.a.l 2 185.d even 2 1
3626.2.a.a 2 35.c odd 2 1
5328.2.a.bf 2 60.h even 2 1
8954.2.a.p 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3}^{2} + 3 T_{3} - 1$$ $$T_{7}^{2} + 2 T_{7} - 12$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-12 + 2 T + T^{2}$$
$11$ $$-3 + T + T^{2}$$
$13$ $$-3 - T + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$-27 - 3 T + T^{2}$$
$29$ $$-27 - 3 T + T^{2}$$
$31$ $$-1 - 3 T + T^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$-9 - 9 T + T^{2}$$
$43$ $$-4 - 6 T + T^{2}$$
$47$ $$-12 + 2 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$36 - 14 T + T^{2}$$
$61$ $$-79 + 3 T + T^{2}$$
$67$ $$-51 + 11 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$107 - 21 T + T^{2}$$
$79$ $$-147 + 7 T + T^{2}$$
$83$ $$48 + 20 T + T^{2}$$
$89$ $$-48 + 4 T + T^{2}$$
$97$ $$-204 - 4 T + T^{2}$$