# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 29.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.231566165862$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/29\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 − 2.23607i 2.23607i
2.23607i 2.23607i −3.00000 −3.00000 5.00000 2.00000 2.23607i −2.00000 6.70820i
28.2 2.23607i 2.23607i −3.00000 −3.00000 5.00000 2.00000 2.23607i −2.00000 6.70820i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.2.b.a 2
3.b odd 2 1 261.2.c.a 2
4.b odd 2 1 464.2.e.a 2
5.b even 2 1 725.2.c.c 2
5.c odd 4 2 725.2.d.a 4
7.b odd 2 1 1421.2.b.b 2
8.b even 2 1 1856.2.e.g 2
8.d odd 2 1 1856.2.e.f 2
12.b even 2 1 4176.2.o.k 2
29.b even 2 1 inner 29.2.b.a 2
29.c odd 4 2 841.2.a.b 2
29.d even 7 6 841.2.e.g 12
29.e even 14 6 841.2.e.g 12
29.f odd 28 12 841.2.d.h 12
87.d odd 2 1 261.2.c.a 2
87.f even 4 2 7569.2.a.i 2
116.d odd 2 1 464.2.e.a 2
145.d even 2 1 725.2.c.c 2
145.h odd 4 2 725.2.d.a 4
203.c odd 2 1 1421.2.b.b 2
232.b odd 2 1 1856.2.e.f 2
232.g even 2 1 1856.2.e.g 2
348.b even 2 1 4176.2.o.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 1.a even 1 1 trivial
29.2.b.a 2 29.b even 2 1 inner
261.2.c.a 2 3.b odd 2 1
261.2.c.a 2 87.d odd 2 1
464.2.e.a 2 4.b odd 2 1
464.2.e.a 2 116.d odd 2 1
725.2.c.c 2 5.b even 2 1
725.2.c.c 2 145.d even 2 1
725.2.d.a 4 5.c odd 4 2
725.2.d.a 4 145.h odd 4 2
841.2.a.b 2 29.c odd 4 2
841.2.d.h 12 29.f odd 28 12
841.2.e.g 12 29.d even 7 6
841.2.e.g 12 29.e even 14 6
1421.2.b.b 2 7.b odd 2 1
1421.2.b.b 2 203.c odd 2 1
1856.2.e.f 2 8.d odd 2 1
1856.2.e.f 2 232.b odd 2 1
1856.2.e.g 2 8.b even 2 1
1856.2.e.g 2 232.g even 2 1
4176.2.o.k 2 12.b even 2 1
4176.2.o.k 2 348.b even 2 1
7569.2.a.i 2 87.f even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 4 T^{4}$$
$3$ $$1 - T^{2} + 9 T^{4}$$
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$1 - 17 T^{2} + 121 T^{4}$$
$13$ $$( 1 + T + 13 T^{2} )^{2}$$
$17$ $$1 - 14 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 19 T^{2} )^{2}$$
$23$ $$( 1 - 6 T + 23 T^{2} )^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 - 17 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 37 T^{2} )^{2}$$
$41$ $$( 1 - 12 T + 41 T^{2} )( 1 + 12 T + 41 T^{2} )$$
$43$ $$1 - 41 T^{2} + 1849 T^{4}$$
$47$ $$1 - 89 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 9 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 6 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 8 T + 61 T^{2} )( 1 + 8 T + 61 T^{2} )$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{2}$$
$79$ $$1 - 113 T^{2} + 6241 T^{4}$$
$83$ $$( 1 + 6 T + 83 T^{2} )^{2}$$
$89$ $$1 - 158 T^{2} + 7921 T^{4}$$
$97$ $$1 - 14 T^{2} + 9409 T^{4}$$