# Properties

 Label 29.4.b.a Level $29$ Weight $4$ Character orbit 29.b Analytic conductor $1.711$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 29.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.71105539017$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 38 x^{4} + 301 x^{2} + 560$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -5 + \beta_{4} + \beta_{5} ) q^{4} + ( 4 - \beta_{4} ) q^{5} + ( -2 + \beta_{5} ) q^{6} + ( -5 + \beta_{4} ) q^{7} + ( -6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( 7 - \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -5 + \beta_{4} + \beta_{5} ) q^{4} + ( 4 - \beta_{4} ) q^{5} + ( -2 + \beta_{5} ) q^{6} + ( -5 + \beta_{4} ) q^{7} + ( -6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( 7 - \beta_{4} - 2 \beta_{5} ) q^{9} + ( 10 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{10} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{11} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{12} + ( 5 - 6 \beta_{5} ) q^{13} + ( -11 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{14} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{15} + ( 44 - 10 \beta_{4} - 5 \beta_{5} ) q^{16} + ( -5 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{17} + ( 19 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{18} + ( -11 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{19} + ( -96 + 14 \beta_{4} + 13 \beta_{5} ) q^{20} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{21} + ( -39 + 15 \beta_{4} + 7 \beta_{5} ) q^{22} + ( -11 - 5 \beta_{4} + 4 \beta_{5} ) q^{23} + ( 41 - 5 \beta_{4} + 7 \beta_{5} ) q^{24} + ( 6 - 6 \beta_{4} - 8 \beta_{5} ) q^{25} + ( 23 \beta_{1} - 24 \beta_{2} ) q^{26} + ( 13 \beta_{1} - 18 \beta_{2} - \beta_{3} ) q^{27} + ( 101 - 15 \beta_{4} - 14 \beta_{5} ) q^{28} + ( -11 - 19 \beta_{1} + 13 \beta_{2} + \beta_{3} + 6 \beta_{4} + 14 \beta_{5} ) q^{29} + ( 31 + 9 \beta_{4} + 5 \beta_{5} ) q^{30} + ( -46 \beta_{1} - 35 \beta_{2} ) q^{31} + ( 71 \beta_{1} + 18 \beta_{2} + 2 \beta_{3} ) q^{32} + ( 25 + 8 \beta_{4} - 4 \beta_{5} ) q^{33} + ( 91 - 17 \beta_{4} - 22 \beta_{5} ) q^{34} + ( -135 + 7 \beta_{4} + 8 \beta_{5} ) q^{35} + ( -205 + 23 \beta_{4} + 14 \beta_{5} ) q^{36} + ( 50 \beta_{1} + 46 \beta_{2} + 6 \beta_{3} ) q^{37} + ( 161 - 23 \beta_{4} - 24 \beta_{5} ) q^{38} + ( 48 \beta_{1} + 37 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -139 \beta_{1} + 34 \beta_{2} - 6 \beta_{3} ) q^{40} + ( -63 \beta_{1} - 43 \beta_{2} + \beta_{3} ) q^{41} + ( -29 - 9 \beta_{4} - 6 \beta_{5} ) q^{42} + ( 9 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} ) q^{43} + ( -126 \beta_{1} - \beta_{2} - 7 \beta_{3} ) q^{44} + ( 65 - 23 \beta_{4} - 18 \beta_{5} ) q^{45} + ( 7 \beta_{1} + 31 \beta_{2} + 5 \beta_{3} ) q^{46} + ( 45 \beta_{1} - 36 \beta_{2} - 7 \beta_{3} ) q^{47} + ( 10 \beta_{1} + 11 \beta_{2} + 5 \beta_{3} ) q^{48} + ( -203 - 8 \beta_{4} - 8 \beta_{5} ) q^{49} + ( 66 \beta_{1} - 14 \beta_{2} + 6 \beta_{3} ) q^{50} + ( 239 + 5 \beta_{4} + 28 \beta_{5} ) q^{51} + ( -307 + 23 \beta_{4} - \beta_{5} ) q^{52} + ( 155 + 30 \beta_{4} - 14 \beta_{5} ) q^{53} + ( -201 + \beta_{4} + 25 \beta_{5} ) q^{54} + ( 118 \beta_{1} - 37 \beta_{2} ) q^{55} + ( 145 \beta_{1} - 35 \beta_{2} + 7 \beta_{3} ) q^{56} + ( 171 + \beta_{4} + 14 \beta_{5} ) q^{57} + ( 269 - 89 \beta_{1} + 38 \beta_{2} - 6 \beta_{3} - 7 \beta_{4} - 26 \beta_{5} ) q^{58} + ( 205 + 15 \beta_{4} + 32 \beta_{5} ) q^{59} + ( -62 \beta_{1} - 23 \beta_{2} - \beta_{3} ) q^{60} + ( 51 \beta_{1} - 13 \beta_{2} - 13 \beta_{3} ) q^{61} + ( 528 - 46 \beta_{4} - 11 \beta_{5} ) q^{62} + ( -72 + 24 \beta_{4} + 20 \beta_{5} ) q^{63} + ( -543 + 15 \beta_{4} + 25 \beta_{5} ) q^{64} + ( -214 - 47 \beta_{4} - 30 \beta_{5} ) q^{65} + ( -11 \beta_{1} - 40 \beta_{2} - 8 \beta_{3} ) q^{66} + ( -124 + 40 \beta_{4} + 44 \beta_{5} ) q^{67} + ( 219 \beta_{1} + 51 \beta_{2} + 9 \beta_{3} ) q^{68} + ( -47 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} ) q^{69} + ( -201 \beta_{1} + 11 \beta_{2} - 7 \beta_{3} ) q^{70} + ( -142 - 22 \beta_{4} + 32 \beta_{5} ) q^{71} + ( -233 \beta_{1} - 53 \beta_{2} - 15 \beta_{3} ) q^{72} + ( -124 \beta_{1} - 28 \beta_{2} + 16 \beta_{3} ) q^{73} + ( -582 + 122 \beta_{4} + 40 \beta_{5} ) q^{74} + ( 46 \beta_{1} + 62 \beta_{2} - 2 \beta_{3} ) q^{75} + ( 283 \beta_{1} + 29 \beta_{2} + 15 \beta_{3} ) q^{76} + ( -121 \beta_{1} + 35 \beta_{2} - \beta_{3} ) q^{77} + ( -526 - 24 \beta_{4} - 25 \beta_{5} ) q^{78} + ( 75 \beta_{1} + 42 \beta_{2} - 9 \beta_{3} ) q^{79} + ( 1131 - 99 \beta_{4} - 105 \beta_{5} ) q^{80} + ( -188 - 51 \beta_{4} - 66 \beta_{5} ) q^{81} + ( 729 - 51 \beta_{4} - 14 \beta_{5} ) q^{82} + ( -375 + 3 \beta_{4} - 60 \beta_{5} ) q^{83} + ( 67 \beta_{1} + 27 \beta_{2} + \beta_{3} ) q^{84} + ( -99 \beta_{1} + 57 \beta_{2} - 15 \beta_{3} ) q^{85} + ( -161 + 93 \beta_{4} + 59 \beta_{5} ) q^{86} + ( 289 - 94 \beta_{1} - 99 \beta_{2} + 8 \beta_{3} + 19 \beta_{4} - 4 \beta_{5} ) q^{87} + ( 1352 - 90 \beta_{4} - 111 \beta_{5} ) q^{88} + ( -105 \beta_{1} + 71 \beta_{2} + 7 \beta_{3} ) q^{89} + ( 257 \beta_{1} - 3 \beta_{2} + 23 \beta_{3} ) q^{90} + ( 209 + 47 \beta_{4} + 36 \beta_{5} ) q^{91} + ( -137 + 27 \beta_{4} + 38 \beta_{5} ) q^{92} + ( -608 - 35 \beta_{4} - 116 \beta_{5} ) q^{93} + ( -629 - 39 \beta_{4} + 39 \beta_{5} ) q^{94} + ( -171 \beta_{1} + 31 \beta_{2} - 17 \beta_{3} ) q^{95} + ( 200 + 30 \beta_{4} + 85 \beta_{5} ) q^{96} + ( 125 \beta_{1} - 215 \beta_{2} + 9 \beta_{3} ) q^{97} + ( -131 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{98} + ( 137 \beta_{1} + 41 \beta_{2} + 15 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 28 q^{4} + 22 q^{5} - 12 q^{6} - 28 q^{7} + 40 q^{9} + O(q^{10})$$ $$6 q - 28 q^{4} + 22 q^{5} - 12 q^{6} - 28 q^{7} + 40 q^{9} + 30 q^{13} + 244 q^{16} - 548 q^{20} - 204 q^{22} - 76 q^{23} + 236 q^{24} + 24 q^{25} + 576 q^{28} - 54 q^{29} + 204 q^{30} + 166 q^{33} + 512 q^{34} - 796 q^{35} - 1184 q^{36} + 920 q^{38} - 192 q^{42} + 344 q^{45} - 1234 q^{49} + 1444 q^{51} - 1796 q^{52} + 990 q^{53} - 1204 q^{54} + 1028 q^{57} + 1600 q^{58} + 1260 q^{59} + 3076 q^{62} - 384 q^{63} - 3228 q^{64} - 1378 q^{65} - 664 q^{67} - 896 q^{71} - 3248 q^{74} - 3204 q^{78} + 6588 q^{80} - 1230 q^{81} + 4272 q^{82} - 2244 q^{83} - 780 q^{86} + 1772 q^{87} + 7932 q^{88} + 1348 q^{91} - 768 q^{92} - 3718 q^{93} - 3852 q^{94} + 1260 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 38 x^{4} + 301 x^{2} + 560$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 34 \nu^{3} + 165 \nu$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 14 \nu^{3} - 275 \nu$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{4} - 29 \nu^{2} - 85$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{4} + 34 \nu^{2} + 150$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - 13$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} - 22 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-29 \beta_{5} - 34 \beta_{4} + 292$$ $$\nu^{5}$$ $$=$$ $$34 \beta_{3} - 14 \beta_{2} + 583 \beta_{1}$$

## 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
 − 5.28644i − 2.70440i − 1.65524i 1.65524i 2.70440i 5.28644i
5.28644i 1.10381i −19.9464 15.1112 −5.83520 −16.1112 63.1540i 25.7816 79.8845i
28.2 2.70440i 4.08054i 0.686224 −10.7216 −11.0354 9.72164 23.4910i 10.3492 28.9956i
28.3 1.65524i 6.56740i 5.26019 6.61042 10.8706 −7.61042 21.9488i −16.1308 10.9418i
28.4 1.65524i 6.56740i 5.26019 6.61042 10.8706 −7.61042 21.9488i −16.1308 10.9418i
28.5 2.70440i 4.08054i 0.686224 −10.7216 −11.0354 9.72164 23.4910i 10.3492 28.9956i
28.6 5.28644i 1.10381i −19.9464 15.1112 −5.83520 −16.1112 63.1540i 25.7816 79.8845i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.6 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 29.4.b.a 6
3.b odd 2 1 261.4.c.c 6
4.b odd 2 1 464.4.e.a 6
29.b even 2 1 inner 29.4.b.a 6
29.c odd 4 2 841.4.a.c 6
87.d odd 2 1 261.4.c.c 6
116.d odd 2 1 464.4.e.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.b.a 6 1.a even 1 1 trivial
29.4.b.a 6 29.b even 2 1 inner
261.4.c.c 6 3.b odd 2 1
261.4.c.c 6 87.d odd 2 1
464.4.e.a 6 4.b odd 2 1
464.4.e.a 6 116.d odd 2 1
841.4.a.c 6 29.c odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(29, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$560 + 301 T^{2} + 38 T^{4} + T^{6}$$
$3$ $$875 + 791 T^{2} + 61 T^{4} + T^{6}$$
$5$ $$( 1071 - 133 T - 11 T^{2} + T^{3} )^{2}$$
$7$ $$( -1192 - 108 T + 14 T^{2} + T^{3} )^{2}$$
$11$ $$193452035 + 3919671 T^{2} + 3909 T^{4} + T^{6}$$
$13$ $$( 119791 - 4641 T - 15 T^{2} + T^{3} )^{2}$$
$17$ $$8077890240 + 40558896 T^{2} + 12932 T^{4} + T^{6}$$
$19$ $$5493277440 + 28619584 T^{2} + 12284 T^{4} + T^{6}$$
$23$ $$( 188856 - 8428 T + 38 T^{2} + T^{3} )^{2}$$
$29$ $$14507145975869 + 32120459334 T - 289277929 T^{2} - 5298300 T^{3} - 11861 T^{4} + 54 T^{5} + T^{6}$$
$31$ $$1328705534835 + 3963474151 T^{2} + 135813 T^{4} + T^{6}$$
$37$ $$844118984540160 + 29438424064 T^{2} + 310128 T^{4} + T^{6}$$
$41$ $$16572819704000 + 11410815024 T^{2} + 236388 T^{4} + T^{6}$$
$43$ $$207644764642875 + 10707373159 T^{2} + 180669 T^{4} + T^{6}$$
$47$ $$328073602115 + 1488882199 T^{2} + 335333 T^{4} + T^{6}$$
$53$ $$( -9329121 - 152081 T - 495 T^{2} + T^{3} )^{2}$$
$59$ $$( 6664392 + 18676 T - 630 T^{2} + T^{3} )^{2}$$
$61$ $$757905028251840 + 70588615216 T^{2} + 697956 T^{4} + T^{6}$$
$67$ $$( 16765696 - 275968 T + 332 T^{2} + T^{3} )^{2}$$
$71$ $$( -5011200 - 238432 T + 448 T^{2} + T^{3} )^{2}$$
$73$ $$10400839666237440 + 564679327744 T^{2} + 1499888 T^{4} + T^{6}$$
$79$ $$424851885738315 + 74821656231 T^{2} + 581229 T^{4} + T^{6}$$
$83$ $$( -43495704 - 75852 T + 1122 T^{2} + T^{3} )^{2}$$
$89$ $$1826762940175040 + 150078203184 T^{2} + 965508 T^{4} + T^{6}$$
$97$ $$1898924295636768960 + 5125383625264 T^{2} + 4099044 T^{4} + T^{6}$$