# Properties

 Label 1849.2.a.f Level $1849$ Weight $2$ Character orbit 1849.a Self dual yes Analytic conductor $14.764$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7643393337$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -\beta q^{3} + ( -2 - \beta ) q^{5} -2 q^{6} + ( 2 + \beta ) q^{7} -2 \beta q^{8} - q^{9} +O(q^{10})$$ $$q + \beta q^{2} -\beta q^{3} + ( -2 - \beta ) q^{5} -2 q^{6} + ( 2 + \beta ) q^{7} -2 \beta q^{8} - q^{9} + ( -2 - 2 \beta ) q^{10} + ( -1 - 2 \beta ) q^{11} + ( 1 - 2 \beta ) q^{13} + ( 2 + 2 \beta ) q^{14} + ( 2 + 2 \beta ) q^{15} -4 q^{16} + ( 5 - 2 \beta ) q^{17} -\beta q^{18} + ( 2 - 2 \beta ) q^{19} + ( -2 - 2 \beta ) q^{21} + ( -4 - \beta ) q^{22} + ( 1 + 4 \beta ) q^{23} + 4 q^{24} + ( 1 + 4 \beta ) q^{25} + ( -4 + \beta ) q^{26} + 4 \beta q^{27} + 3 \beta q^{29} + ( 4 + 2 \beta ) q^{30} -3 q^{31} + ( 4 + \beta ) q^{33} + ( -4 + 5 \beta ) q^{34} + ( -6 - 4 \beta ) q^{35} -6 \beta q^{37} + ( -4 + 2 \beta ) q^{38} + ( 4 - \beta ) q^{39} + ( 4 + 4 \beta ) q^{40} + ( -1 + 2 \beta ) q^{41} + ( -4 - 2 \beta ) q^{42} + ( 2 + \beta ) q^{45} + ( 8 + \beta ) q^{46} + 6 q^{47} + 4 \beta q^{48} + ( -1 + 4 \beta ) q^{49} + ( 8 + \beta ) q^{50} + ( 4 - 5 \beta ) q^{51} + ( 11 + 2 \beta ) q^{53} + 8 q^{54} + ( 6 + 5 \beta ) q^{55} + ( -4 - 4 \beta ) q^{56} + ( 4 - 2 \beta ) q^{57} + 6 q^{58} + ( -2 - 2 \beta ) q^{59} + ( -4 + 3 \beta ) q^{61} -3 \beta q^{62} + ( -2 - \beta ) q^{63} + 8 q^{64} + ( 2 + 3 \beta ) q^{65} + ( 2 + 4 \beta ) q^{66} + ( 1 - 6 \beta ) q^{67} + ( -8 - \beta ) q^{69} + ( -8 - 6 \beta ) q^{70} + ( 6 - 2 \beta ) q^{71} + 2 \beta q^{72} + ( 12 + 3 \beta ) q^{73} -12 q^{74} + ( -8 - \beta ) q^{75} + ( -6 - 5 \beta ) q^{77} + ( -2 + 4 \beta ) q^{78} + ( 2 + 2 \beta ) q^{79} + ( 8 + 4 \beta ) q^{80} -5 q^{81} + ( 4 - \beta ) q^{82} + ( 9 - 4 \beta ) q^{83} + ( -6 - \beta ) q^{85} -6 q^{87} + ( 8 + 2 \beta ) q^{88} + ( 6 - 3 \beta ) q^{89} + ( 2 + 2 \beta ) q^{90} + ( -2 - 3 \beta ) q^{91} + 3 \beta q^{93} + 6 \beta q^{94} + 2 \beta q^{95} + ( -1 + 2 \beta ) q^{97} + ( 8 - \beta ) q^{98} + ( 1 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 4q^{6} + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{5} - 4q^{6} + 4q^{7} - 2q^{9} - 4q^{10} - 2q^{11} + 2q^{13} + 4q^{14} + 4q^{15} - 8q^{16} + 10q^{17} + 4q^{19} - 4q^{21} - 8q^{22} + 2q^{23} + 8q^{24} + 2q^{25} - 8q^{26} + 8q^{30} - 6q^{31} + 8q^{33} - 8q^{34} - 12q^{35} - 8q^{38} + 8q^{39} + 8q^{40} - 2q^{41} - 8q^{42} + 4q^{45} + 16q^{46} + 12q^{47} - 2q^{49} + 16q^{50} + 8q^{51} + 22q^{53} + 16q^{54} + 12q^{55} - 8q^{56} + 8q^{57} + 12q^{58} - 4q^{59} - 8q^{61} - 4q^{63} + 16q^{64} + 4q^{65} + 4q^{66} + 2q^{67} - 16q^{69} - 16q^{70} + 12q^{71} + 24q^{73} - 24q^{74} - 16q^{75} - 12q^{77} - 4q^{78} + 4q^{79} + 16q^{80} - 10q^{81} + 8q^{82} + 18q^{83} - 12q^{85} - 12q^{87} + 16q^{88} + 12q^{89} + 4q^{90} - 4q^{91} - 2q^{97} + 16q^{98} + 2q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−1.41421 1.41421 0 −0.585786 −2.00000 0.585786 2.82843 −1.00000 0.828427
1.2 1.41421 −1.41421 0 −3.41421 −2.00000 3.41421 −2.82843 −1.00000 −4.82843
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$-1$$

## 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 1849.2.a.f 2
43.b odd 2 1 43.2.a.b 2
129.d even 2 1 387.2.a.h 2
172.d even 2 1 688.2.a.f 2
215.d odd 2 1 1075.2.a.i 2
215.g even 4 2 1075.2.b.f 4
301.c even 2 1 2107.2.a.b 2
344.e even 2 1 2752.2.a.m 2
344.h odd 2 1 2752.2.a.l 2
473.d even 2 1 5203.2.a.f 2
516.h odd 2 1 6192.2.a.bd 2
559.d odd 2 1 7267.2.a.b 2
645.d even 2 1 9675.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.a.b 2 43.b odd 2 1
387.2.a.h 2 129.d even 2 1
688.2.a.f 2 172.d even 2 1
1075.2.a.i 2 215.d odd 2 1
1075.2.b.f 4 215.g even 4 2
1849.2.a.f 2 1.a even 1 1 trivial
2107.2.a.b 2 301.c even 2 1
2752.2.a.l 2 344.h odd 2 1
2752.2.a.m 2 344.e even 2 1
5203.2.a.f 2 473.d even 2 1
6192.2.a.bd 2 516.h odd 2 1
7267.2.a.b 2 559.d odd 2 1
9675.2.a.bf 2 645.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1849))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 4 T^{4}$$
$3$ $$1 + 4 T^{2} + 9 T^{4}$$
$5$ $$1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$1 - 4 T + 16 T^{2} - 28 T^{3} + 49 T^{4}$$
$11$ $$1 + 2 T + 15 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4}$$
$17$ $$1 - 10 T + 51 T^{2} - 170 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T + 34 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 2 T + 15 T^{2} - 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 40 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 3 T + 31 T^{2} )^{2}$$
$37$ $$1 + 2 T^{2} + 1369 T^{4}$$
$41$ $$1 + 2 T + 75 T^{2} + 82 T^{3} + 1681 T^{4}$$
$43$ 1
$47$ $$( 1 - 6 T + 47 T^{2} )^{2}$$
$53$ $$1 - 22 T + 219 T^{2} - 1166 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T + 114 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 + 8 T + 120 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 - 2 T + 63 T^{2} - 134 T^{3} + 4489 T^{4}$$
$71$ $$1 - 12 T + 170 T^{2} - 852 T^{3} + 5041 T^{4}$$
$73$ $$1 - 24 T + 272 T^{2} - 1752 T^{3} + 5329 T^{4}$$
$79$ $$1 - 4 T + 154 T^{2} - 316 T^{3} + 6241 T^{4}$$
$83$ $$1 - 18 T + 215 T^{2} - 1494 T^{3} + 6889 T^{4}$$
$89$ $$1 - 12 T + 196 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 + 2 T + 187 T^{2} + 194 T^{3} + 9409 T^{4}$$