Properties

 Label 429.2.i.c 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: 10.0.7965937851507.1 Defining polynomial: $$x^{10} - 2 x^{9} + 8 x^{8} - 6 x^{7} + 28 x^{6} - 23 x^{5} + 51 x^{4} - 10 x^{3} + 25 x^{2} - 6 x + 9$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 8 x^{8} - 6 x^{7} + 28 x^{6} - 23 x^{5} + 51 x^{4} - 10 x^{3} + 25 x^{2} - 6 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$14222 \nu^{9} - 26977 \nu^{8} + 106429 \nu^{7} - 28188 \nu^{6} + 227648 \nu^{5} + 5147 \nu^{4} + 169389 \nu^{3} + 497359 \nu^{2} - 1147351 \nu + 243657$$$$)/495525$$ $$\beta_{3}$$ $$=$$ $$($$$$18542 \nu^{9} - 15172 \nu^{8} + 95344 \nu^{7} + 77532 \nu^{6} + 339653 \nu^{5} + 182942 \nu^{4} + 286554 \nu^{3} + 1099849 \nu^{2} + 122414 \nu + 27852$$$$)/495525$$ $$\beta_{4}$$ $$=$$ $$($$$$7304 \nu^{9} - 17664 \nu^{8} + 62928 \nu^{7} - 59841 \nu^{6} + 203136 \nu^{5} - 219696 \nu^{4} + 428423 \nu^{3} - 113712 \nu^{2} + 211543 \nu - 55626$$$$)/165175$$ $$\beta_{5}$$ $$=$$ $$($$$$-4696 \nu^{9} - 6409 \nu^{8} - 16397 \nu^{7} - 78216 \nu^{6} - 107989 \nu^{5} - 284116 \nu^{4} - 106257 \nu^{3} - 509942 \nu^{2} - 210292 \nu - 237996$$$$)/99105$$ $$\beta_{6}$$ $$=$$ $$($$$$27916 \nu^{9} - 16856 \nu^{8} + 142637 \nu^{7} + 150936 \nu^{6} + 524194 \nu^{5} + 533641 \nu^{4} + 451092 \nu^{3} + 1791002 \nu^{2} + 704947 \nu + 849546$$$$)/495525$$ $$\beta_{7}$$ $$=$$ $$($$$$-2072 \nu^{9} + 4432 \nu^{8} - 15789 \nu^{7} + 11693 \nu^{6} - 50968 \nu^{5} + 55123 \nu^{4} - 93819 \nu^{3} + 28531 \nu^{2} - 11634 \nu + 64048$$$$)/33035$$ $$\beta_{8}$$ $$=$$ $$($$$$-42346 \nu^{9} + 118511 \nu^{8} - 411872 \nu^{7} + 514509 \nu^{6} - 1445464 \nu^{5} + 1814654 \nu^{4} - 3068877 \nu^{3} + 1433938 \nu^{2} - 1529257 \nu + 445974$$$$)/495525$$ $$\beta_{9}$$ $$=$$ $$($$$$-24689 \nu^{9} + 51024 \nu^{8} - 181773 \nu^{7} + 143681 \nu^{6} - 586776 \nu^{5} + 634611 \nu^{4} - 846593 \nu^{3} + 328467 \nu^{2} - 133938 \nu + 457816$$$$)/165175$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + \beta_{7} + 4 \beta_{4} - \beta_{2} - 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{6} + \beta_{5} - 7 \beta_{3} - \beta_{2} - 7 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} - 6 \beta_{8} - 9 \beta_{7} + 9 \beta_{6} + 2 \beta_{5} - 21 \beta_{4} - 7 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{9} - 9 \beta_{8} - 33 \beta_{7} - 43 \beta_{4} + 9 \beta_{2} + 43 \beta_{1} + 31$$ $$\nu^{7}$$ $$=$$ $$-61 \beta_{6} - 17 \beta_{5} + 45 \beta_{3} + 33 \beta_{2} + 116 \beta_{1} + 45$$ $$\nu^{8}$$ $$=$$ $$-50 \beta_{9} + 61 \beta_{8} + 182 \beta_{7} - 182 \beta_{6} - 50 \beta_{5} + 255 \beta_{4} + 154 \beta_{3}$$ $$\nu^{9}$$ $$=$$ $$-111 \beta_{9} + 182 \beta_{8} + 377 \beta_{7} + 652 \beta_{4} - 182 \beta_{2} - 652 \beta_{1} - 278$$

Character values

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

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$ $$1$$

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
 1.19780 + 2.07464i 0.724205 + 1.25436i 0.333779 + 0.578123i −0.362459 − 0.627798i −0.893322 − 1.54728i 1.19780 − 2.07464i 0.724205 − 1.25436i 0.333779 − 0.578123i −0.362459 + 0.627798i −0.893322 + 1.54728i
−1.19780 + 2.07464i −0.500000 + 0.866025i −1.86943 3.23795i 3.82235 −1.19780 2.07464i −2.02991 3.51590i 4.16562 −0.500000 0.866025i −4.57839 + 7.93001i
100.2 −0.724205 + 1.25436i −0.500000 + 0.866025i −0.0489465 0.0847779i −0.404514 −0.724205 1.25436i −1.43979 2.49379i −2.75503 −0.500000 0.866025i 0.292951 0.507407i
100.3 −0.333779 + 0.578123i −0.500000 + 0.866025i 0.777183 + 1.34612i 0.849177 −0.333779 0.578123i 1.61499 + 2.79724i −2.37275 −0.500000 0.866025i −0.283438 + 0.490929i
100.4 0.362459 0.627798i −0.500000 + 0.866025i 0.737247 + 1.27695i 4.26830 0.362459 + 0.627798i −0.335344 0.580832i 2.51872 −0.500000 0.866025i 1.54708 2.67963i
100.5 0.893322 1.54728i −0.500000 + 0.866025i −0.596049 1.03239i −0.535309 0.893322 + 1.54728i −2.30994 4.00094i 1.44343 −0.500000 0.866025i −0.478203 + 0.828272i
133.1 −1.19780 2.07464i −0.500000 0.866025i −1.86943 + 3.23795i 3.82235 −1.19780 + 2.07464i −2.02991 + 3.51590i 4.16562 −0.500000 + 0.866025i −4.57839 7.93001i
133.2 −0.724205 1.25436i −0.500000 0.866025i −0.0489465 + 0.0847779i −0.404514 −0.724205 + 1.25436i −1.43979 + 2.49379i −2.75503 −0.500000 + 0.866025i 0.292951 + 0.507407i
133.3 −0.333779 0.578123i −0.500000 0.866025i 0.777183 1.34612i 0.849177 −0.333779 + 0.578123i 1.61499 2.79724i −2.37275 −0.500000 + 0.866025i −0.283438 0.490929i
133.4 0.362459 + 0.627798i −0.500000 0.866025i 0.737247 1.27695i 4.26830 0.362459 0.627798i −0.335344 + 0.580832i 2.51872 −0.500000 + 0.866025i 1.54708 + 2.67963i
133.5 0.893322 + 1.54728i −0.500000 0.866025i −0.596049 + 1.03239i −0.535309 0.893322 1.54728i −2.30994 + 4.00094i 1.44343 −0.500000 + 0.866025i −0.478203 0.828272i
 $$n$$: 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.c 10
13.c even 3 1 inner 429.2.i.c 10
13.c even 3 1 5577.2.a.v 5
13.e even 6 1 5577.2.a.p 5

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

$p$ $F_p(T)$
$2$ $$9 + 6 T + 25 T^{2} + 10 T^{3} + 51 T^{4} + 23 T^{5} + 28 T^{6} + 6 T^{7} + 8 T^{8} + 2 T^{9} + T^{10}$$
$3$ $$( 1 + T + T^{2} )^{5}$$
$5$ $$( -3 - 8 T + 6 T^{2} + 15 T^{3} - 8 T^{4} + T^{5} )^{2}$$
$7$ $$13689 + 27261 T + 44578 T^{2} + 22147 T^{3} + 10738 T^{4} + 3081 T^{5} + 1124 T^{6} + 274 T^{7} + 69 T^{8} + 9 T^{9} + T^{10}$$
$11$ $$( 1 - T + T^{2} )^{5}$$
$13$ $$371293 - 28561 T + 19773 T^{2} + 5239 T^{3} + 2951 T^{4} - 279 T^{5} + 227 T^{6} + 31 T^{7} + 9 T^{8} - T^{9} + T^{10}$$
$17$ $$9801 - 1485 T + 11907 T^{2} + 7908 T^{3} + 13162 T^{4} + 3847 T^{5} + 1330 T^{6} + 143 T^{7} + 40 T^{8} + 3 T^{9} + T^{10}$$
$19$ $$2199289 + 1431095 T + 880803 T^{2} + 207804 T^{3} + 62540 T^{4} + 5267 T^{5} + 2618 T^{6} + 109 T^{7} + 68 T^{8} - 3 T^{9} + T^{10}$$
$23$ $$15848361 - 5664963 T + 2622079 T^{2} - 383700 T^{3} + 125244 T^{4} - 12385 T^{5} + 4352 T^{6} - 225 T^{7} + 76 T^{8} - T^{9} + T^{10}$$
$29$ $$328329 + 332340 T + 324367 T^{2} + 67188 T^{3} + 29427 T^{4} + 1885 T^{5} + 1766 T^{6} + 54 T^{7} + 52 T^{8} - 2 T^{9} + T^{10}$$
$31$ $$( 3 - 89 T - 123 T^{2} - 33 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$37$ $$1034289 + 791226 T + 614437 T^{2} + 104868 T^{3} + 43888 T^{4} + 3068 T^{5} + 2238 T^{6} + 73 T^{7} + 56 T^{8} - T^{9} + T^{10}$$
$41$ $$31505769 - 5175186 T + 5340484 T^{2} + 591662 T^{3} + 550952 T^{4} + 28405 T^{5} + 15491 T^{6} + 1834 T^{7} + 311 T^{8} + 18 T^{9} + T^{10}$$
$43$ $$1054729 - 1180023 T + 911455 T^{2} - 405952 T^{3} + 138922 T^{4} - 29605 T^{5} + 5356 T^{6} - 571 T^{7} + 106 T^{8} - 9 T^{9} + T^{10}$$
$47$ $$( 291 - 265 T - 19 T^{2} + 67 T^{3} - 16 T^{4} + T^{5} )^{2}$$
$53$ $$( -711 + 808 T + 103 T^{2} - 61 T^{3} - 3 T^{4} + T^{5} )^{2}$$
$59$ $$123143409 + 76502718 T + 33655986 T^{2} + 8462142 T^{3} + 1691794 T^{4} + 218261 T^{5} + 26943 T^{6} + 2388 T^{7} + 263 T^{8} + 16 T^{9} + T^{10}$$
$61$ $$196812841 - 162736400 T + 124290772 T^{2} - 14243090 T^{3} + 3026056 T^{4} - 49569 T^{5} + 36281 T^{6} - 176 T^{7} + 269 T^{8} + 8 T^{9} + T^{10}$$
$67$ $$1284577281 - 117020865 T + 90191404 T^{2} + 12119411 T^{3} + 4020962 T^{4} + 310803 T^{5} + 50050 T^{6} + 3146 T^{7} + 429 T^{8} + 19 T^{9} + T^{10}$$
$71$ $$20169081 + 7261947 T + 4024863 T^{2} + 1225788 T^{3} + 522952 T^{4} + 136961 T^{5} + 31016 T^{6} + 4197 T^{7} + 432 T^{8} + 25 T^{9} + T^{10}$$
$73$ $$( -3 - 472 T + 319 T^{2} - 28 T^{3} - 8 T^{4} + T^{5} )^{2}$$
$79$ $$( -3091 - 466 T + 491 T^{2} + 34 T^{3} - 18 T^{4} + T^{5} )^{2}$$
$83$ $$( 56421 - 29349 T + 4291 T^{2} - 75 T^{3} - 22 T^{4} + T^{5} )^{2}$$
$89$ $$213364449 + 29549961 T + 19386058 T^{2} - 1270875 T^{3} + 862736 T^{4} - 65164 T^{5} + 23804 T^{6} - 2674 T^{7} + 371 T^{8} - 20 T^{9} + T^{10}$$
$97$ $$13810245289 + 6937028510 T + 2563795205 T^{2} + 414788148 T^{3} + 51871992 T^{4} + 3952248 T^{5} + 264774 T^{6} + 11407 T^{7} + 644 T^{8} + 21 T^{9} + T^{10}$$