# Properties

 Label 29.2.a.a Level 29 Weight 2 Character orbit 29.a Self dual yes Analytic conductor 0.232 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 29.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.231566165862$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( -3 + 2 \beta ) q^{6} + 2 \beta q^{7} + ( -3 + \beta ) q^{8} -2 \beta q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( -3 + 2 \beta ) q^{6} + 2 \beta q^{7} + ( -3 + \beta ) q^{8} -2 \beta q^{9} + ( 1 - \beta ) q^{10} + ( 1 + \beta ) q^{11} + ( 5 - 3 \beta ) q^{12} + ( -1 + 2 \beta ) q^{13} + ( 4 - 2 \beta ) q^{14} + ( -1 + \beta ) q^{15} + 3 q^{16} + ( -2 - 2 \beta ) q^{17} + ( -4 + 2 \beta ) q^{18} + 6 q^{19} + ( -1 + 2 \beta ) q^{20} + ( -4 + 2 \beta ) q^{21} + q^{22} + ( -2 - 4 \beta ) q^{23} + ( -5 + 4 \beta ) q^{24} -4 q^{25} + ( 5 - 3 \beta ) q^{26} + ( 1 + \beta ) q^{27} + ( -8 + 2 \beta ) q^{28} + q^{29} + ( 3 - 2 \beta ) q^{30} + ( 3 - 5 \beta ) q^{31} + ( 3 + \beta ) q^{32} - q^{33} -2 q^{34} -2 \beta q^{35} + ( 8 - 2 \beta ) q^{36} -4 q^{37} + ( -6 + 6 \beta ) q^{38} + ( -5 + 3 \beta ) q^{39} + ( 3 - \beta ) q^{40} + ( 4 + 6 \beta ) q^{41} + ( 8 - 6 \beta ) q^{42} + ( 5 + \beta ) q^{43} + ( -3 - \beta ) q^{44} + 2 \beta q^{45} + ( -6 + 2 \beta ) q^{46} + ( 1 + 3 \beta ) q^{47} + ( 3 - 3 \beta ) q^{48} + q^{49} + ( 4 - 4 \beta ) q^{50} + 2 q^{51} + ( -9 + 4 \beta ) q^{52} + ( 1 - 6 \beta ) q^{53} + q^{54} + ( -1 - \beta ) q^{55} + ( 4 - 6 \beta ) q^{56} + ( 6 - 6 \beta ) q^{57} + ( -1 + \beta ) q^{58} + ( 2 + 4 \beta ) q^{59} + ( -5 + 3 \beta ) q^{60} + ( -2 + 2 \beta ) q^{61} + ( -13 + 8 \beta ) q^{62} -8 q^{63} + ( -7 + 2 \beta ) q^{64} + ( 1 - 2 \beta ) q^{65} + ( 1 - \beta ) q^{66} -4 \beta q^{67} + ( 6 + 2 \beta ) q^{68} + ( 6 - 2 \beta ) q^{69} + ( -4 + 2 \beta ) q^{70} + ( -6 + 2 \beta ) q^{71} + ( -4 + 6 \beta ) q^{72} + 4 q^{73} + ( 4 - 4 \beta ) q^{74} + ( -4 + 4 \beta ) q^{75} + ( 6 - 12 \beta ) q^{76} + ( 4 + 2 \beta ) q^{77} + ( 11 - 8 \beta ) q^{78} + ( -1 + \beta ) q^{79} -3 q^{80} + ( -1 + 6 \beta ) q^{81} + ( 8 - 2 \beta ) q^{82} + ( 2 - 4 \beta ) q^{83} + ( -12 + 10 \beta ) q^{84} + ( 2 + 2 \beta ) q^{85} + ( -3 + 4 \beta ) q^{86} + ( 1 - \beta ) q^{87} + ( -1 - 2 \beta ) q^{88} + ( -4 + 6 \beta ) q^{89} + ( 4 - 2 \beta ) q^{90} + ( 8 - 2 \beta ) q^{91} + 14 q^{92} + ( 13 - 8 \beta ) q^{93} + ( 5 - 2 \beta ) q^{94} -6 q^{95} + ( 1 - 2 \beta ) q^{96} + ( -4 - 6 \beta ) q^{97} + ( -1 + \beta ) q^{98} + ( -4 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 6q^{6} - 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 6q^{6} - 6q^{8} + 2q^{10} + 2q^{11} + 10q^{12} - 2q^{13} + 8q^{14} - 2q^{15} + 6q^{16} - 4q^{17} - 8q^{18} + 12q^{19} - 2q^{20} - 8q^{21} + 2q^{22} - 4q^{23} - 10q^{24} - 8q^{25} + 10q^{26} + 2q^{27} - 16q^{28} + 2q^{29} + 6q^{30} + 6q^{31} + 6q^{32} - 2q^{33} - 4q^{34} + 16q^{36} - 8q^{37} - 12q^{38} - 10q^{39} + 6q^{40} + 8q^{41} + 16q^{42} + 10q^{43} - 6q^{44} - 12q^{46} + 2q^{47} + 6q^{48} + 2q^{49} + 8q^{50} + 4q^{51} - 18q^{52} + 2q^{53} + 2q^{54} - 2q^{55} + 8q^{56} + 12q^{57} - 2q^{58} + 4q^{59} - 10q^{60} - 4q^{61} - 26q^{62} - 16q^{63} - 14q^{64} + 2q^{65} + 2q^{66} + 12q^{68} + 12q^{69} - 8q^{70} - 12q^{71} - 8q^{72} + 8q^{73} + 8q^{74} - 8q^{75} + 12q^{76} + 8q^{77} + 22q^{78} - 2q^{79} - 6q^{80} - 2q^{81} + 16q^{82} + 4q^{83} - 24q^{84} + 4q^{85} - 6q^{86} + 2q^{87} - 2q^{88} - 8q^{89} + 8q^{90} + 16q^{91} + 28q^{92} + 26q^{93} + 10q^{94} - 12q^{95} + 2q^{96} - 8q^{97} - 2q^{98} - 8q^{99} + O(q^{100})$$

## 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
−2.41421 2.41421 3.82843 −1.00000 −5.82843 −2.82843 −4.41421 2.82843 2.41421
1.2 0.414214 −0.414214 −1.82843 −1.00000 −0.171573 2.82843 −1.58579 −2.82843 −0.414214
 $$n$$: 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 29.2.a.a 2
3.b odd 2 1 261.2.a.d 2
4.b odd 2 1 464.2.a.h 2
5.b even 2 1 725.2.a.b 2
5.c odd 4 2 725.2.b.b 4
7.b odd 2 1 1421.2.a.j 2
8.b even 2 1 1856.2.a.r 2
8.d odd 2 1 1856.2.a.w 2
11.b odd 2 1 3509.2.a.j 2
12.b even 2 1 4176.2.a.bq 2
13.b even 2 1 4901.2.a.g 2
15.d odd 2 1 6525.2.a.o 2
17.b even 2 1 8381.2.a.e 2
29.b even 2 1 841.2.a.d 2
29.c odd 4 2 841.2.b.a 4
29.d even 7 6 841.2.d.j 12
29.e even 14 6 841.2.d.f 12
29.f odd 28 12 841.2.e.k 24
87.d odd 2 1 7569.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 1.a even 1 1 trivial
261.2.a.d 2 3.b odd 2 1
464.2.a.h 2 4.b odd 2 1
725.2.a.b 2 5.b even 2 1
725.2.b.b 4 5.c odd 4 2
841.2.a.d 2 29.b even 2 1
841.2.b.a 4 29.c odd 4 2
841.2.d.f 12 29.e even 14 6
841.2.d.j 12 29.d even 7 6
841.2.e.k 24 29.f odd 28 12
1421.2.a.j 2 7.b odd 2 1
1856.2.a.r 2 8.b even 2 1
1856.2.a.w 2 8.d odd 2 1
3509.2.a.j 2 11.b odd 2 1
4176.2.a.bq 2 12.b even 2 1
4901.2.a.g 2 13.b even 2 1
6525.2.a.o 2 15.d odd 2 1
7569.2.a.c 2 87.d odd 2 1
8381.2.a.e 2 17.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$29$$ $$-1$$

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(29))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ $$1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T + 5 T^{2} )^{2}$$
$7$ $$1 + 6 T^{2} + 49 T^{4}$$
$11$ $$1 - 2 T + 21 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 4 T + 30 T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 6 T + 19 T^{2} )^{2}$$
$23$ $$1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 - T )^{2}$$
$31$ $$1 - 6 T + 21 T^{2} - 186 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 4 T + 37 T^{2} )^{2}$$
$41$ $$1 - 8 T + 26 T^{2} - 328 T^{3} + 1681 T^{4}$$
$43$ $$1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4}$$
$47$ $$1 - 2 T + 77 T^{2} - 94 T^{3} + 2209 T^{4}$$
$53$ $$1 - 2 T + 35 T^{2} - 106 T^{3} + 2809 T^{4}$$
$59$ $$1 - 4 T + 90 T^{2} - 236 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T + 118 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 102 T^{2} + 4489 T^{4}$$
$71$ $$1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{2}$$
$79$ $$1 + 2 T + 157 T^{2} + 158 T^{3} + 6241 T^{4}$$
$83$ $$1 - 4 T + 138 T^{2} - 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 8 T + 122 T^{2} + 712 T^{3} + 7921 T^{4}$$
$97$ $$1 + 8 T + 138 T^{2} + 776 T^{3} + 9409 T^{4}$$