# Properties

 Label 429.2.i.f Level $429$ Weight $2$ Character orbit 429.i Analytic conductor $3.426$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} + 7 x^{8} + 4 x^{7} + 32 x^{6} + 3 x^{5} + 30 x^{4} - 7 x^{3} + 26 x^{2} - 5 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 7 x^{8} + 4 x^{7} + 32 x^{6} + 3 x^{5} + 30 x^{4} - 7 x^{3} + 26 x^{2} - 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$911 \nu^{9} + 318 \nu^{8} - 583 \nu^{7} + 16404 \nu^{6} - 265 \nu^{5} + 8533 \nu^{4} - 159430 \nu^{3} + 8480 \nu^{2} - 1643 \nu + 31405$$$$)/160271$$ $$\beta_{3}$$ $$=$$ $$($$$$-4571 \nu^{9} + 29016 \nu^{8} - 53196 \nu^{7} + 154140 \nu^{6} - 24180 \nu^{5} + 778596 \nu^{4} + 122097 \nu^{3} + 773760 \nu^{2} - 149916 \nu + 319366$$$$)/160271$$ $$\beta_{4}$$ $$=$$ $$($$$$4889 \nu^{9} - 36294 \nu^{8} + 66539 \nu^{7} - 199961 \nu^{6} + 30245 \nu^{5} - 973889 \nu^{4} + 52190 \nu^{3} - 967840 \nu^{2} + 187519 \nu - 672224$$$$)/160271$$ $$\beta_{5}$$ $$=$$ $$($$$$-5191 \nu^{9} + 12966 \nu^{8} - 23771 \nu^{7} + 9622 \nu^{6} - 10805 \nu^{5} + 347921 \nu^{4} + 288305 \nu^{3} + 345760 \nu^{2} - 66991 \nu + 388420$$$$)/160271$$ $$\beta_{6}$$ $$=$$ $$($$$$-31405 \nu^{9} + 32316 \nu^{8} - 219517 \nu^{7} - 126203 \nu^{6} - 988556 \nu^{5} - 94480 \nu^{4} - 933617 \nu^{3} + 60405 \nu^{2} - 808050 \nu + 155382$$$$)/160271$$ $$\beta_{7}$$ $$=$$ $$($$$$-62396 \nu^{9} + 33989 \nu^{8} - 409567 \nu^{7} - 433019 \nu^{6} - 2138559 \nu^{5} - 984502 \nu^{4} - 1969946 \nu^{3} - 1829 \nu^{2} - 1431066 \nu - 15562$$$$)/160271$$ $$\beta_{8}$$ $$=$$ $$($$$$-91217 \nu^{9} + 91837 \nu^{8} - 622469 \nu^{7} - 394293 \nu^{6} - 2774426 \nu^{5} - 287026 \nu^{4} - 2305835 \nu^{3} + 472311 \nu^{2} - 1943642 \nu + 373160$$$$)/160271$$ $$\beta_{9}$$ $$=$$ $$($$$$-128866 \nu^{9} + 127955 \nu^{8} - 902380 \nu^{7} - 514881 \nu^{6} - 4140116 \nu^{5} - 386333 \nu^{4} - 3874513 \nu^{3} + 1061492 \nu^{2} - 3358996 \nu + 645973$$$$)/160271$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{9} + \beta_{8} + 8 \beta_{7} + 9 \beta_{6} - \beta_{5} - 7 \beta_{4} - 10 \beta_{1} - 9$$ $$\nu^{5}$$ $$=$$ $$-18 \beta_{9} + 7 \beta_{8} + 19 \beta_{7} + 14 \beta_{6} + 19 \beta_{3} + 34 \beta_{2} - 34 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$12 \beta_{5} + 53 \beta_{4} + 60 \beta_{3} + 85 \beta_{2} + 57$$ $$\nu^{7}$$ $$=$$ $$145 \beta_{9} - 48 \beta_{8} - 157 \beta_{7} - 129 \beta_{6} + 48 \beta_{5} + 145 \beta_{4} + 255 \beta_{1} + 129$$ $$\nu^{8}$$ $$=$$ $$412 \beta_{9} - 109 \beta_{8} - 460 \beta_{7} - 413 \beta_{6} - 460 \beta_{3} - 686 \beta_{2} + 686 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$-351 \beta_{5} - 1146 \beta_{4} - 1255 \beta_{3} - 1971 \beta_{2} - 1069$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-\beta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 1.40131 − 2.42714i 0.440981 − 0.763802i 0.100998 − 0.174933i −0.580000 + 1.00459i −0.863288 + 1.49526i 1.40131 + 2.42714i 0.440981 + 0.763802i 0.100998 + 0.174933i −0.580000 − 1.00459i −0.863288 − 1.49526i
−0.901309 + 1.56111i −0.500000 + 0.866025i −0.624717 1.08204i −2.90617 −0.901309 1.56111i −0.704431 1.22011i −1.35298 −0.500000 0.866025i 2.61936 4.53686i
100.2 0.0590186 0.102223i −0.500000 + 0.866025i 0.993034 + 1.71998i −2.39753 0.0590186 + 0.102223i 1.48513 + 2.57233i 0.470505 −0.500000 0.866025i −0.141499 + 0.245083i
100.3 0.399002 0.691092i −0.500000 + 0.866025i 0.681594 + 1.18056i 1.11866 0.399002 + 0.691092i −0.394706 0.683651i 2.68384 −0.500000 0.866025i 0.446346 0.773094i
100.4 1.08000 1.87062i −0.500000 + 0.866025i −1.33280 2.30848i 2.38790 1.08000 + 1.87062i 1.17823 + 2.04076i −1.43770 −0.500000 0.866025i 2.57893 4.46684i
100.5 1.36329 2.36128i −0.500000 + 0.866025i −2.71711 4.70617i −2.20286 1.36329 + 2.36128i −0.0642304 0.111250i −9.36366 −0.500000 0.866025i −3.00314 + 5.20159i
133.1 −0.901309 1.56111i −0.500000 0.866025i −0.624717 + 1.08204i −2.90617 −0.901309 + 1.56111i −0.704431 + 1.22011i −1.35298 −0.500000 + 0.866025i 2.61936 + 4.53686i
133.2 0.0590186 + 0.102223i −0.500000 0.866025i 0.993034 1.71998i −2.39753 0.0590186 0.102223i 1.48513 2.57233i 0.470505 −0.500000 + 0.866025i −0.141499 0.245083i
133.3 0.399002 + 0.691092i −0.500000 0.866025i 0.681594 1.18056i 1.11866 0.399002 0.691092i −0.394706 + 0.683651i 2.68384 −0.500000 + 0.866025i 0.446346 + 0.773094i
133.4 1.08000 + 1.87062i −0.500000 0.866025i −1.33280 + 2.30848i 2.38790 1.08000 1.87062i 1.17823 2.04076i −1.43770 −0.500000 + 0.866025i 2.57893 + 4.46684i
133.5 1.36329 + 2.36128i −0.500000 0.866025i −2.71711 + 4.70617i −2.20286 1.36329 2.36128i −0.0642304 + 0.111250i −9.36366 −0.500000 + 0.866025i −3.00314 5.20159i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.i.f 10
13.c even 3 1 inner 429.2.i.f 10
13.c even 3 1 5577.2.a.n 5
13.e even 6 1 5577.2.a.w 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.f 10 1.a even 1 1 trivial
429.2.i.f 10 13.c even 3 1 inner
5577.2.a.n 5 13.c even 3 1
5577.2.a.w 5 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$:

 $$T_{2}^{10} - \cdots$$ $$T_{7}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 87 T^{2} - 130 T^{3} + 173 T^{4} - 79 T^{5} + 62 T^{6} - 26 T^{7} + 16 T^{8} - 4 T^{9} + T^{10}$$
$3$ $$( 1 + T + T^{2} )^{5}$$
$5$ $$( 41 - 4 T - 30 T^{2} - 5 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$7$ $$1 + 9 T + 72 T^{2} + 89 T^{3} + 120 T^{4} + 19 T^{5} + 34 T^{6} - 6 T^{7} + 13 T^{8} - 3 T^{9} + T^{10}$$
$11$ $$( 1 + T + T^{2} )^{5}$$
$13$ $$371293 - 142805 T - 32955 T^{2} + 22139 T^{3} + 1859 T^{4} - 2517 T^{5} + 143 T^{6} + 131 T^{7} - 15 T^{8} - 5 T^{9} + T^{10}$$
$17$ $$2968729 + 2848119 T + 1895031 T^{2} + 717208 T^{3} + 210378 T^{4} + 40181 T^{5} + 6652 T^{6} + 747 T^{7} + 106 T^{8} + 9 T^{9} + T^{10}$$
$19$ $$81 - 189 T + 531 T^{2} - 204 T^{3} + 664 T^{4} - 599 T^{5} + 460 T^{6} - 187 T^{7} + 58 T^{8} - 9 T^{9} + T^{10}$$
$23$ $$130321 + 19133 T + 46851 T^{2} - 5744 T^{3} + 11688 T^{4} - 1193 T^{5} + 1152 T^{6} - 253 T^{7} + 80 T^{8} - 9 T^{9} + T^{10}$$
$29$ $$29241 + 24624 T + 43137 T^{2} - 14076 T^{3} + 17809 T^{4} - 299 T^{5} + 1100 T^{6} + 150 T^{7} + 78 T^{8} + 8 T^{9} + T^{10}$$
$31$ $$( 3087 + 735 T - 679 T^{2} - 109 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$37$ $$2809 + 3922 T + 16129 T^{2} - 25156 T^{3} + 34124 T^{4} - 16928 T^{5} + 5918 T^{6} - 1247 T^{7} + 192 T^{8} - 17 T^{9} + T^{10}$$
$41$ $$6561 - 46656 T + 343602 T^{2} + 75186 T^{3} + 52510 T^{4} - 1199 T^{5} + 3325 T^{6} + 38 T^{7} + 91 T^{8} - 6 T^{9} + T^{10}$$
$43$ $$36638809 - 2390935 T + 4780517 T^{2} + 798126 T^{3} + 488812 T^{4} + 47647 T^{5} + 12008 T^{6} + 995 T^{7} + 210 T^{8} + 13 T^{9} + T^{10}$$
$47$ $$( 36279 - 5397 T - 2671 T^{2} - 117 T^{3} + 16 T^{4} + T^{5} )^{2}$$
$53$ $$( 361 + 128 T - 329 T^{2} - 13 T^{3} + 13 T^{4} + T^{5} )^{2}$$
$59$ $$10374841 - 818134 T + 1488198 T^{2} - 242042 T^{3} + 183566 T^{4} - 23467 T^{5} + 6307 T^{6} - 444 T^{7} + 119 T^{8} - 8 T^{9} + T^{10}$$
$61$ $$124609 - 82602 T + 149360 T^{2} + 95894 T^{3} + 59414 T^{4} + 14821 T^{5} + 3515 T^{6} + 348 T^{7} + 63 T^{8} + 4 T^{9} + T^{10}$$
$67$ $$2107452649 - 295227917 T + 114303984 T^{2} - 10714733 T^{3} + 3761654 T^{4} - 343889 T^{5} + 53498 T^{6} - 2038 T^{7} + 253 T^{8} - 5 T^{9} + T^{10}$$
$71$ $$682097689 - 534379937 T + 300238043 T^{2} - 84464968 T^{3} + 17800080 T^{4} - 1472307 T^{5} + 131888 T^{6} - 6047 T^{7} + 520 T^{8} - 19 T^{9} + T^{10}$$
$73$ $$( 14879 + 5912 T - 213 T^{2} - 178 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$79$ $$( -7457 + 4394 T - 371 T^{2} - 108 T^{3} + 8 T^{4} + T^{5} )^{2}$$
$83$ $$( 198059 + 22219 T - 3099 T^{2} - 337 T^{3} + 10 T^{4} + T^{5} )^{2}$$
$89$ $$49 + 1897 T + 66630 T^{2} + 268177 T^{3} + 1033496 T^{4} - 329684 T^{5} + 72176 T^{6} - 8326 T^{7} + 703 T^{8} - 32 T^{9} + T^{10}$$
$97$ $$32001649 - 23272898 T + 16138673 T^{2} - 2778076 T^{3} + 849836 T^{4} - 19692 T^{5} + 34606 T^{6} - 697 T^{7} + 220 T^{8} + 5 T^{9} + T^{10}$$