# Properties

 Label 1045.2.a.c Level $1045$ Weight $2$ Character orbit 1045.a Self dual yes Analytic conductor $8.344$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + 2 q^{3} + (2 \beta + 1) q^{4} + q^{5} + (2 \beta + 2) q^{6} - 2 \beta q^{7} + (\beta + 3) q^{8} + q^{9} +O(q^{10})$$ q + (b + 1) * q^2 + 2 * q^3 + (2*b + 1) * q^4 + q^5 + (2*b + 2) * q^6 - 2*b * q^7 + (b + 3) * q^8 + q^9 $$q + (\beta + 1) q^{2} + 2 q^{3} + (2 \beta + 1) q^{4} + q^{5} + (2 \beta + 2) q^{6} - 2 \beta q^{7} + (\beta + 3) q^{8} + q^{9} + (\beta + 1) q^{10} - q^{11} + (4 \beta + 2) q^{12} + (2 \beta + 4) q^{13} + ( - 2 \beta - 4) q^{14} + 2 q^{15} + 3 q^{16} + ( - 2 \beta - 2) q^{17} + (\beta + 1) q^{18} - q^{19} + (2 \beta + 1) q^{20} - 4 \beta q^{21} + ( - \beta - 1) q^{22} + 4 q^{23} + (2 \beta + 6) q^{24} + q^{25} + (6 \beta + 8) q^{26} - 4 q^{27} + ( - 2 \beta - 8) q^{28} + ( - 2 \beta + 6) q^{29} + (2 \beta + 2) q^{30} + ( - 2 \beta + 4) q^{31} + (\beta - 3) q^{32} - 2 q^{33} + ( - 4 \beta - 6) q^{34} - 2 \beta q^{35} + (2 \beta + 1) q^{36} + (2 \beta - 4) q^{37} + ( - \beta - 1) q^{38} + (4 \beta + 8) q^{39} + (\beta + 3) q^{40} + ( - 2 \beta - 6) q^{41} + ( - 4 \beta - 8) q^{42} + ( - 2 \beta - 4) q^{43} + ( - 2 \beta - 1) q^{44} + q^{45} + (4 \beta + 4) q^{46} + 6 q^{48} + q^{49} + (\beta + 1) q^{50} + ( - 4 \beta - 4) q^{51} + (10 \beta + 12) q^{52} - 2 \beta q^{53} + ( - 4 \beta - 4) q^{54} - q^{55} + ( - 6 \beta - 4) q^{56} - 2 q^{57} + (4 \beta + 2) q^{58} + (6 \beta + 4) q^{59} + (4 \beta + 2) q^{60} + ( - 4 \beta + 2) q^{61} + 2 \beta q^{62} - 2 \beta q^{63} + ( - 2 \beta - 7) q^{64} + (2 \beta + 4) q^{65} + ( - 2 \beta - 2) q^{66} + ( - 4 \beta + 6) q^{67} + ( - 6 \beta - 10) q^{68} + 8 q^{69} + ( - 2 \beta - 4) q^{70} + ( - 2 \beta + 12) q^{71} + (\beta + 3) q^{72} + ( - 6 \beta - 2) q^{73} - 2 \beta q^{74} + 2 q^{75} + ( - 2 \beta - 1) q^{76} + 2 \beta q^{77} + (12 \beta + 16) q^{78} - 12 q^{79} + 3 q^{80} - 11 q^{81} + ( - 8 \beta - 10) q^{82} + 2 \beta q^{83} + ( - 4 \beta - 16) q^{84} + ( - 2 \beta - 2) q^{85} + ( - 6 \beta - 8) q^{86} + ( - 4 \beta + 12) q^{87} + ( - \beta - 3) q^{88} + ( - 8 \beta - 2) q^{89} + (\beta + 1) q^{90} + ( - 8 \beta - 8) q^{91} + (8 \beta + 4) q^{92} + ( - 4 \beta + 8) q^{93} - q^{95} + (2 \beta - 6) q^{96} + (10 \beta - 4) q^{97} + (\beta + 1) q^{98} - q^{99} +O(q^{100})$$ q + (b + 1) * q^2 + 2 * q^3 + (2*b + 1) * q^4 + q^5 + (2*b + 2) * q^6 - 2*b * q^7 + (b + 3) * q^8 + q^9 + (b + 1) * q^10 - q^11 + (4*b + 2) * q^12 + (2*b + 4) * q^13 + (-2*b - 4) * q^14 + 2 * q^15 + 3 * q^16 + (-2*b - 2) * q^17 + (b + 1) * q^18 - q^19 + (2*b + 1) * q^20 - 4*b * q^21 + (-b - 1) * q^22 + 4 * q^23 + (2*b + 6) * q^24 + q^25 + (6*b + 8) * q^26 - 4 * q^27 + (-2*b - 8) * q^28 + (-2*b + 6) * q^29 + (2*b + 2) * q^30 + (-2*b + 4) * q^31 + (b - 3) * q^32 - 2 * q^33 + (-4*b - 6) * q^34 - 2*b * q^35 + (2*b + 1) * q^36 + (2*b - 4) * q^37 + (-b - 1) * q^38 + (4*b + 8) * q^39 + (b + 3) * q^40 + (-2*b - 6) * q^41 + (-4*b - 8) * q^42 + (-2*b - 4) * q^43 + (-2*b - 1) * q^44 + q^45 + (4*b + 4) * q^46 + 6 * q^48 + q^49 + (b + 1) * q^50 + (-4*b - 4) * q^51 + (10*b + 12) * q^52 - 2*b * q^53 + (-4*b - 4) * q^54 - q^55 + (-6*b - 4) * q^56 - 2 * q^57 + (4*b + 2) * q^58 + (6*b + 4) * q^59 + (4*b + 2) * q^60 + (-4*b + 2) * q^61 + 2*b * q^62 - 2*b * q^63 + (-2*b - 7) * q^64 + (2*b + 4) * q^65 + (-2*b - 2) * q^66 + (-4*b + 6) * q^67 + (-6*b - 10) * q^68 + 8 * q^69 + (-2*b - 4) * q^70 + (-2*b + 12) * q^71 + (b + 3) * q^72 + (-6*b - 2) * q^73 - 2*b * q^74 + 2 * q^75 + (-2*b - 1) * q^76 + 2*b * q^77 + (12*b + 16) * q^78 - 12 * q^79 + 3 * q^80 - 11 * q^81 + (-8*b - 10) * q^82 + 2*b * q^83 + (-4*b - 16) * q^84 + (-2*b - 2) * q^85 + (-6*b - 8) * q^86 + (-4*b + 12) * q^87 + (-b - 3) * q^88 + (-8*b - 2) * q^89 + (b + 1) * q^90 + (-8*b - 8) * q^91 + (8*b + 4) * q^92 + (-4*b + 8) * q^93 - q^95 + (2*b - 6) * q^96 + (10*b - 4) * q^97 + (b + 1) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + 8 q^{13} - 8 q^{14} + 4 q^{15} + 6 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{22} + 8 q^{23} + 12 q^{24} + 2 q^{25} + 16 q^{26} - 8 q^{27} - 16 q^{28} + 12 q^{29} + 4 q^{30} + 8 q^{31} - 6 q^{32} - 4 q^{33} - 12 q^{34} + 2 q^{36} - 8 q^{37} - 2 q^{38} + 16 q^{39} + 6 q^{40} - 12 q^{41} - 16 q^{42} - 8 q^{43} - 2 q^{44} + 2 q^{45} + 8 q^{46} + 12 q^{48} + 2 q^{49} + 2 q^{50} - 8 q^{51} + 24 q^{52} - 8 q^{54} - 2 q^{55} - 8 q^{56} - 4 q^{57} + 4 q^{58} + 8 q^{59} + 4 q^{60} + 4 q^{61} - 14 q^{64} + 8 q^{65} - 4 q^{66} + 12 q^{67} - 20 q^{68} + 16 q^{69} - 8 q^{70} + 24 q^{71} + 6 q^{72} - 4 q^{73} + 4 q^{75} - 2 q^{76} + 32 q^{78} - 24 q^{79} + 6 q^{80} - 22 q^{81} - 20 q^{82} - 32 q^{84} - 4 q^{85} - 16 q^{86} + 24 q^{87} - 6 q^{88} - 4 q^{89} + 2 q^{90} - 16 q^{91} + 8 q^{92} + 16 q^{93} - 2 q^{95} - 12 q^{96} - 8 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 + 8 * q^13 - 8 * q^14 + 4 * q^15 + 6 * q^16 - 4 * q^17 + 2 * q^18 - 2 * q^19 + 2 * q^20 - 2 * q^22 + 8 * q^23 + 12 * q^24 + 2 * q^25 + 16 * q^26 - 8 * q^27 - 16 * q^28 + 12 * q^29 + 4 * q^30 + 8 * q^31 - 6 * q^32 - 4 * q^33 - 12 * q^34 + 2 * q^36 - 8 * q^37 - 2 * q^38 + 16 * q^39 + 6 * q^40 - 12 * q^41 - 16 * q^42 - 8 * q^43 - 2 * q^44 + 2 * q^45 + 8 * q^46 + 12 * q^48 + 2 * q^49 + 2 * q^50 - 8 * q^51 + 24 * q^52 - 8 * q^54 - 2 * q^55 - 8 * q^56 - 4 * q^57 + 4 * q^58 + 8 * q^59 + 4 * q^60 + 4 * q^61 - 14 * q^64 + 8 * q^65 - 4 * q^66 + 12 * q^67 - 20 * q^68 + 16 * q^69 - 8 * q^70 + 24 * q^71 + 6 * q^72 - 4 * q^73 + 4 * q^75 - 2 * q^76 + 32 * q^78 - 24 * q^79 + 6 * q^80 - 22 * q^81 - 20 * q^82 - 32 * q^84 - 4 * q^85 - 16 * q^86 + 24 * q^87 - 6 * q^88 - 4 * q^89 + 2 * q^90 - 16 * q^91 + 8 * q^92 + 16 * q^93 - 2 * q^95 - 12 * q^96 - 8 * q^97 + 2 * q^98 - 2 * q^99

## 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
−0.414214 2.00000 −1.82843 1.00000 −0.828427 2.82843 1.58579 1.00000 −0.414214
1.2 2.41421 2.00000 3.82843 1.00000 4.82843 −2.82843 4.41421 1.00000 2.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$1$$

## 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 1045.2.a.c 2
3.b odd 2 1 9405.2.a.o 2
5.b even 2 1 5225.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.c 2 1.a even 1 1 trivial
5225.2.a.e 2 5.b even 2 1
9405.2.a.o 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1045))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$(T - 2)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 8T + 8$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 12T + 28$$
$31$ $$T^{2} - 8T + 8$$
$37$ $$T^{2} + 8T + 8$$
$41$ $$T^{2} + 12T + 28$$
$43$ $$T^{2} + 8T + 8$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 8$$
$59$ $$T^{2} - 8T - 56$$
$61$ $$T^{2} - 4T - 28$$
$67$ $$T^{2} - 12T + 4$$
$71$ $$T^{2} - 24T + 136$$
$73$ $$T^{2} + 4T - 68$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} - 8$$
$89$ $$T^{2} + 4T - 124$$
$97$ $$T^{2} + 8T - 184$$