# Properties

 Label 429.2.e.c Level $429$ Weight $2$ Character orbit 429.e Analytic conductor $3.426$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 20 x^{14} + 164 x^{12} - 666 x^{10} + 1300 x^{8} - 924 x^{6} + 273 x^{4} + 404 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 13^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{5} q^{3} + ( -1 + \beta_{4} + \beta_{5} ) q^{4} + \beta_{10} q^{5} + ( -\beta_{6} + \beta_{8} ) q^{6} -\beta_{8} q^{8} + ( -1 + \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{5} q^{3} + ( -1 + \beta_{4} + \beta_{5} ) q^{4} + \beta_{10} q^{5} + ( -\beta_{6} + \beta_{8} ) q^{6} -\beta_{8} q^{8} + ( -1 + \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} + ( -\beta_{3} + \beta_{7} + \beta_{13} + \beta_{14} ) q^{10} + ( -\beta_{8} + \beta_{10} ) q^{11} + ( -2 - \beta_{4} - \beta_{9} ) q^{12} + ( \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{13} -\beta_{11} q^{15} + ( -1 + \beta_{4} + \beta_{5} ) q^{16} -\beta_{15} q^{17} + ( -\beta_{1} - 2 \beta_{3} - \beta_{6} ) q^{18} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{8} ) q^{19} + ( -\beta_{10} + \beta_{11} - \beta_{12} ) q^{20} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{13} + \beta_{14} ) q^{22} + ( -\beta_{4} + \beta_{5} ) q^{23} + ( \beta_{1} - \beta_{3} + \beta_{8} ) q^{24} + ( 3 - \beta_{4} - \beta_{5} ) q^{25} + ( 2 \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} ) q^{26} + ( -2 + 2 \beta_{4} + \beta_{5} - \beta_{9} ) q^{27} + \beta_{15} q^{29} + ( -\beta_{1} + \beta_{6} - \beta_{8} + 2 \beta_{13} - \beta_{15} ) q^{30} + \beta_{2} q^{31} + ( -2 \beta_{3} - 3 \beta_{8} ) q^{32} + ( \beta_{1} - \beta_{3} + \beta_{8} - \beta_{11} ) q^{33} + ( \beta_{2} + \beta_{11} + \beta_{12} ) q^{34} + ( 3 - 3 \beta_{4} + \beta_{5} ) q^{36} + ( \beta_{11} + \beta_{12} ) q^{37} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{38} + ( \beta_{1} + 2 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{13} ) q^{39} + ( \beta_{7} + \beta_{8} - \beta_{13} - \beta_{14} ) q^{40} + ( 3 \beta_{3} - \beta_{8} ) q^{41} + ( \beta_{3} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{43} + ( 2 \beta_{3} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{44} + ( -\beta_{2} - \beta_{10} - \beta_{12} ) q^{45} + ( \beta_{3} + 2 \beta_{6} - \beta_{8} ) q^{46} + ( -2 - \beta_{4} - \beta_{9} ) q^{48} -7 q^{49} + ( 4 \beta_{3} + \beta_{8} ) q^{50} + ( -\beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{13} - 3 \beta_{14} ) q^{51} + ( -\beta_{1} - \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{13} + \beta_{14} ) q^{52} + ( 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( \beta_{1} - 4 \beta_{3} - 2 \beta_{8} ) q^{54} + ( 8 - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{13} - \beta_{14} ) q^{55} + ( -3 \beta_{1} - 3 \beta_{3} + \beta_{6} + 2 \beta_{8} ) q^{57} + ( -\beta_{2} - \beta_{11} - \beta_{12} ) q^{58} + ( -2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{59} + ( \beta_{2} - 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{60} + ( \beta_{3} - 2 \beta_{7} - \beta_{8} ) q^{61} + \beta_{15} q^{62} + ( 7 - 3 \beta_{4} - 3 \beta_{5} ) q^{64} + ( \beta_{1} - 4 \beta_{3} - \beta_{6} + 3 \beta_{8} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{65} + ( 2 - \beta_{1} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{13} - \beta_{15} ) q^{66} -\beta_{2} q^{67} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{8} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{68} + ( 4 - \beta_{4} - \beta_{5} - \beta_{9} ) q^{69} + ( -\beta_{11} + \beta_{12} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - \beta_{8} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{3} + 4 \beta_{6} - 2 \beta_{8} ) q^{73} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{8} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{74} + ( 2 + \beta_{4} - 2 \beta_{5} + \beta_{9} ) q^{75} + 4 \beta_{1} q^{76} + ( -7 + 4 \beta_{4} + 2 \beta_{5} + \beta_{9} - \beta_{11} ) q^{78} + ( \beta_{3} - 3 \beta_{7} - 2 \beta_{8} + \beta_{13} + \beta_{14} ) q^{79} + ( -\beta_{10} + \beta_{11} - \beta_{12} ) q^{80} + ( -5 - 4 \beta_{4} - \beta_{9} ) q^{81} + ( -8 + 2 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 3 \beta_{3} + 5 \beta_{8} ) q^{83} + ( -6 \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{8} ) q^{85} + ( 3 \beta_{10} - \beta_{11} + \beta_{12} ) q^{86} + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{13} + 3 \beta_{14} ) q^{87} + ( -3 - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{13} - \beta_{14} ) q^{88} + ( -\beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{89} + ( -\beta_{1} + 2 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{90} + ( 2 \beta_{4} + 2 \beta_{9} ) q^{92} + ( -\beta_{2} - \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{93} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{8} + 2 \beta_{13} - 2 \beta_{14} ) q^{95} + ( 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{6} + \beta_{8} ) q^{96} + ( -2 \beta_{2} - \beta_{11} - \beta_{12} ) q^{97} -7 \beta_{3} q^{98} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{3} - 8q^{4} - 12q^{9} + O(q^{10})$$ $$16q - 4q^{3} - 8q^{4} - 12q^{9} - 32q^{12} - 8q^{16} + 8q^{22} + 40q^{25} - 16q^{27} + 40q^{36} - 32q^{48} - 112q^{49} + 120q^{55} + 88q^{64} + 32q^{66} + 60q^{69} + 24q^{75} - 92q^{78} - 92q^{81} - 112q^{82} - 56q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 20 x^{14} + 164 x^{12} - 666 x^{10} + 1300 x^{8} - 924 x^{6} + 273 x^{4} + 404 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1017 \nu^{14} - 18889 \nu^{12} + 141809 \nu^{10} - 504639 \nu^{8} + 779299 \nu^{6} - 250803 \nu^{4} - 305344 \nu^{2} + 1865592$$$$)/695368$$ $$\beta_{2}$$ $$=$$ $$($$$$31233 \nu^{15} - 628770 \nu^{13} + 5186482 \nu^{11} - 21171500 \nu^{9} + 41344590 \nu^{7} - 28056790 \nu^{5} + 3745023 \nu^{3} + 16866948 \nu$$$$)/695368$$ $$\beta_{3}$$ $$=$$ $$($$$$5799 \nu^{14} - 117329 \nu^{12} + 976398 \nu^{10} - 4060171 \nu^{8} + 8332329 \nu^{6} - 7059146 \nu^{4} + 3413416 \nu^{2} + 1437494$$$$)/173842$$ $$\beta_{4}$$ $$=$$ $$($$$$-36955 \nu^{14} + 751451 \nu^{12} - 6309143 \nu^{10} + 26657365 \nu^{8} - 56418201 \nu^{6} + 51028693 \nu^{4} - 23948824 \nu^{2} - 9249288$$$$)/695368$$ $$\beta_{5}$$ $$=$$ $$($$$$40155 \nu^{14} - 814827 \nu^{12} + 6814631 \nu^{10} - 28599597 \nu^{8} + 59716185 \nu^{6} - 52062909 \nu^{4} + 22684152 \nu^{2} + 10837864$$$$)/695368$$ $$\beta_{6}$$ $$=$$ $$($$$$-53617 \nu^{14} + 1086197 \nu^{12} - 9069441 \nu^{10} + 37939335 \nu^{8} - 78591519 \nu^{6} + 66859611 \nu^{4} - 29371920 \nu^{2} - 13405760$$$$)/695368$$ $$\beta_{7}$$ $$=$$ $$($$$$-11595 \nu^{15} + 51170 \nu^{14} + 231812 \nu^{13} - 1036686 \nu^{12} - 1903314 \nu^{11} + 8656492 \nu^{10} + 7753568 \nu^{9} - 36253462 \nu^{8} - 15174808 \nu^{7} + 75511150 \nu^{6} + 10393614 \nu^{5} - 65914896 \nu^{4} - 1236901 \nu^{3} + 30533344 \nu^{2} - 6116828 \nu + 13045736$$$$)/695368$$ $$\beta_{8}$$ $$=$$ $$($$$$-9893 \nu^{14} + 200507 \nu^{12} - 1675924 \nu^{10} + 7033280 \nu^{8} - 14711623 \nu^{6} + 12949151 \nu^{4} - 5926628 \nu^{2} - 2542687$$$$)/86921$$ $$\beta_{9}$$ $$=$$ $$($$$$25297 \nu^{14} - 511003 \nu^{12} + 4256369 \nu^{10} - 17787087 \nu^{8} + 36994087 \nu^{6} - 32342229 \nu^{4} + 15293776 \nu^{2} + 6255548$$$$)/173842$$ $$\beta_{10}$$ $$=$$ $$($$$$155031 \nu^{15} - 3141792 \nu^{13} + 26258800 \nu^{11} - 110207086 \nu^{9} + 230644536 \nu^{7} - 203744128 \nu^{5} + 94637175 \nu^{3} + 40253716 \nu$$$$)/695368$$ $$\beta_{11}$$ $$=$$ $$($$$$194169 \nu^{15} - 3937730 \nu^{13} + 32931622 \nu^{11} - 138302044 \nu^{9} + 289581422 \nu^{7} - 255556234 \nu^{5} + 117626671 \nu^{3} + 46444516 \nu$$$$)/695368$$ $$\beta_{12}$$ $$=$$ $$($$$$-213619 \nu^{15} + 4316546 \nu^{13} - 35936294 \nu^{11} + 149903292 \nu^{9} - 310245654 \nu^{7} + 267041610 \nu^{5} - 122356757 \nu^{3} - 56266252 \nu$$$$)/695368$$ $$\beta_{13}$$ $$=$$ $$($$$$282598 \nu^{15} - 12255 \nu^{14} - 5719412 \nu^{13} + 249485 \nu^{12} + 47711512 \nu^{11} - 2098071 \nu^{10} - 199597198 \nu^{9} + 8911133 \nu^{8} + 415064900 \nu^{7} - 19161083 \nu^{6} - 360935116 \nu^{5} + 18241397 \nu^{4} + 166286064 \nu^{3} - 9173224 \nu^{2} + 72462192 \nu - 2535072$$$$)/695368$$ $$\beta_{14}$$ $$=$$ $$($$$$-275554 \nu^{15} - 15719 \nu^{14} + 5581644 \nu^{13} + 317885 \nu^{12} - 46621406 \nu^{11} - 2652829 \nu^{10} + 195442680 \nu^{9} + 11101645 \nu^{8} - 407951240 \nu^{7} - 23020751 \nu^{6} + 357404278 \nu^{5} + 19436915 \nu^{4} - 164164102 \nu^{3} - 7706456 \nu^{2} - 68836696 \nu - 4760688$$$$)/695368$$ $$\beta_{15}$$ $$=$$ $$($$$$429155 \nu^{15} - 8692814 \nu^{13} + 72609066 \nu^{11} - 304468532 \nu^{9} + 636294706 \nu^{7} - 560454494 \nu^{5} + 259657357 \nu^{3} + 110595980 \nu$$$$)/695368$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-7 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} + 26 \beta_{10} + \beta_{8} - 2 \beta_{6} - \beta_{3} + \beta_{2} + 2 \beta_{1}$$$$)/52$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{8} - \beta_{5} + \beta_{4} - 2 \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-40 \beta_{15} - 35 \beta_{14} - 43 \beta_{13} - 37 \beta_{12} - 11 \beta_{11} + 91 \beta_{10} + 41 \beta_{8} + 39 \beta_{7} - 4 \beta_{6} - 2 \beta_{3} + 8 \beta_{2} + 4 \beta_{1}$$$$)/52$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{9} - 11 \beta_{8} - 7 \beta_{5} + 9 \beta_{4} - 2 \beta_{3} - 8 \beta_{1} + 17$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-124 \beta_{15} - 297 \beta_{14} - 353 \beta_{13} - 223 \beta_{12} - 93 \beta_{11} + 273 \beta_{10} + 339 \beta_{8} + 325 \beta_{7} - 28 \beta_{6} - 14 \beta_{3} + 96 \beta_{2} + 28 \beta_{1}$$$$)/52$$ $$\nu^{6}$$ $$=$$ $$($$$$-20 \beta_{9} - 81 \beta_{8} + 4 \beta_{6} - 53 \beta_{5} + 51 \beta_{4} - 6 \beta_{3} - 20 \beta_{1} + 35$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-52 \beta_{15} - 1807 \beta_{14} - 2015 \beta_{13} - 1029 \beta_{12} - 743 \beta_{11} + 143 \beta_{10} + 2145 \beta_{8} + 2275 \beta_{7} - 104 \beta_{6} - 234 \beta_{3} + 720 \beta_{2} + 104 \beta_{1}$$$$)/52$$ $$\nu^{8}$$ $$=$$ $$($$$$-142 \beta_{9} - 467 \beta_{8} + 16 \beta_{6} - 349 \beta_{5} + 219 \beta_{4} + 14 \beta_{3} + 48 \beta_{1} - 135$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$2884 \beta_{15} - 9521 \beta_{14} - 8809 \beta_{13} - 3099 \beta_{12} - 5205 \beta_{11} - 6487 \beta_{10} + 11015 \beta_{8} + 13221 \beta_{7} + 356 \beta_{6} - 1850 \beta_{3} + 4348 \beta_{2} - 356 \beta_{1}$$$$)/52$$ $$\nu^{10}$$ $$=$$ $$($$$$-810 \beta_{9} - 2169 \beta_{8} - 96 \beta_{6} - 2033 \beta_{5} + 543 \beta_{4} + 246 \beta_{3} + 1128 \beta_{1} - 2305$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$29188 \beta_{15} - 44407 \beta_{14} - 25663 \beta_{13} + 2255 \beta_{12} - 33027 \beta_{11} - 68081 \beta_{10} + 44077 \beta_{8} + 62491 \beta_{7} + 9372 \beta_{6} - 9042 \beta_{3} + 21240 \beta_{2} - 9372 \beta_{1}$$$$)/52$$ $$\nu^{12}$$ $$=$$ $$($$$$-3584 \beta_{9} - 7259 \beta_{8} - 2024 \beta_{6} - 10415 \beta_{5} - 1567 \beta_{4} + 1278 \beta_{3} + 10120 \beta_{1} - 18635$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$202092 \beta_{15} - 176321 \beta_{14} + 11039 \beta_{13} + 123133 \beta_{12} - 190505 \beta_{11} - 483951 \beta_{10} + 99007 \beta_{8} + 209053 \beta_{7} + 93680 \beta_{6} - 16366 \beta_{3} + 73984 \beta_{2} - 93680 \beta_{1}$$$$)/52$$ $$\nu^{14}$$ $$=$$ $$($$$$-9250 \beta_{9} - 5053 \beta_{8} - 19332 \beta_{6} - 44551 \beta_{5} - 32711 \beta_{4} + 486 \beta_{3} + 68612 \beta_{1} - 116489$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$1142500 \beta_{15} - 501039 \beta_{14} + 869017 \beta_{13} + 1171595 \beta_{12} - 976083 \beta_{11} - 2803593 \beta_{10} - 399935 \beta_{8} + 69147 \beta_{7} + 685028 \beta_{6} + 215946 \beta_{3} + 42468 \beta_{2} - 685028 \beta_{1}$$$$)/52$$

## Character values

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

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
428.1
 −0.946412 + 0.500000i 0.946412 − 0.500000i −2.14576 − 0.500000i 2.14576 + 0.500000i −2.34517 + 0.500000i 2.34517 − 0.500000i 0.0131233 − 0.500000i −0.0131233 + 0.500000i 0.0131233 + 0.500000i −0.0131233 − 0.500000i −2.34517 − 0.500000i 2.34517 + 0.500000i −2.14576 + 0.500000i 2.14576 − 0.500000i −0.946412 − 0.500000i 0.946412 + 0.500000i
2.13578i 0.780776 1.54609i −2.56155 −3.09218 −3.30210 1.66757i 0 1.19935i −1.78078 2.41430i 6.60421i
428.2 2.13578i 0.780776 1.54609i −2.56155 3.09218 −3.30210 1.66757i 0 1.19935i −1.78078 2.41430i 6.60421i
428.3 2.13578i 0.780776 + 1.54609i −2.56155 −3.09218 3.30210 1.66757i 0 1.19935i −1.78078 + 2.41430i 6.60421i
428.4 2.13578i 0.780776 + 1.54609i −2.56155 3.09218 3.30210 1.66757i 0 1.19935i −1.78078 + 2.41430i 6.60421i
428.5 0.662153i −1.28078 1.16602i 1.56155 −2.33205 −0.772087 + 0.848071i 0 2.35829i 0.280776 + 2.98683i 1.54417i
428.6 0.662153i −1.28078 1.16602i 1.56155 2.33205 −0.772087 + 0.848071i 0 2.35829i 0.280776 + 2.98683i 1.54417i
428.7 0.662153i −1.28078 + 1.16602i 1.56155 −2.33205 0.772087 + 0.848071i 0 2.35829i 0.280776 2.98683i 1.54417i
428.8 0.662153i −1.28078 + 1.16602i 1.56155 2.33205 0.772087 + 0.848071i 0 2.35829i 0.280776 2.98683i 1.54417i
428.9 0.662153i −1.28078 1.16602i 1.56155 −2.33205 0.772087 0.848071i 0 2.35829i 0.280776 + 2.98683i 1.54417i
428.10 0.662153i −1.28078 1.16602i 1.56155 2.33205 0.772087 0.848071i 0 2.35829i 0.280776 + 2.98683i 1.54417i
428.11 0.662153i −1.28078 + 1.16602i 1.56155 −2.33205 −0.772087 0.848071i 0 2.35829i 0.280776 2.98683i 1.54417i
428.12 0.662153i −1.28078 + 1.16602i 1.56155 2.33205 −0.772087 0.848071i 0 2.35829i 0.280776 2.98683i 1.54417i
428.13 2.13578i 0.780776 1.54609i −2.56155 −3.09218 3.30210 + 1.66757i 0 1.19935i −1.78078 2.41430i 6.60421i
428.14 2.13578i 0.780776 1.54609i −2.56155 3.09218 3.30210 + 1.66757i 0 1.19935i −1.78078 2.41430i 6.60421i
428.15 2.13578i 0.780776 + 1.54609i −2.56155 −3.09218 −3.30210 + 1.66757i 0 1.19935i −1.78078 + 2.41430i 6.60421i
428.16 2.13578i 0.780776 + 1.54609i −2.56155 3.09218 −3.30210 + 1.66757i 0 1.19935i −1.78078 + 2.41430i 6.60421i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 428.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
39.d odd 2 1 inner
143.d odd 2 1 inner
429.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.e.c 16
3.b odd 2 1 inner 429.2.e.c 16
11.b odd 2 1 inner 429.2.e.c 16
13.b even 2 1 inner 429.2.e.c 16
33.d even 2 1 inner 429.2.e.c 16
39.d odd 2 1 inner 429.2.e.c 16
143.d odd 2 1 inner 429.2.e.c 16
429.e even 2 1 inner 429.2.e.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.c 16 1.a even 1 1 trivial
429.2.e.c 16 3.b odd 2 1 inner
429.2.e.c 16 11.b odd 2 1 inner
429.2.e.c 16 13.b even 2 1 inner
429.2.e.c 16 33.d even 2 1 inner
429.2.e.c 16 39.d odd 2 1 inner
429.2.e.c 16 143.d odd 2 1 inner
429.2.e.c 16 429.e even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5 T_{2}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + 5 T^{2} + T^{4} )^{4}$$
$3$ $$( 9 + 3 T + 2 T^{2} + T^{3} + T^{4} )^{4}$$
$5$ $$( 52 - 15 T^{2} + T^{4} )^{4}$$
$7$ $$T^{16}$$
$11$ $$( 14641 - 1936 T^{2} + 238 T^{4} - 16 T^{6} + T^{8} )^{2}$$
$13$ $$( 28561 + 1014 T^{2} - 78 T^{4} + 6 T^{6} + T^{8} )^{2}$$
$17$ $$( 1352 - 74 T^{2} + T^{4} )^{4}$$
$19$ $$( 416 - 58 T^{2} + T^{4} )^{4}$$
$23$ $$( 52 + 15 T^{2} + T^{4} )^{4}$$
$29$ $$( 1352 - 74 T^{2} + T^{4} )^{4}$$
$31$ $$( 676 + 101 T^{2} + T^{4} )^{4}$$
$37$ $$( 2704 + 121 T^{2} + T^{4} )^{4}$$
$41$ $$( 512 + 46 T^{2} + T^{4} )^{4}$$
$43$ $$( 104 + 46 T^{2} + T^{4} )^{4}$$
$47$ $$T^{16}$$
$53$ $$( 832 + 60 T^{2} + T^{4} )^{4}$$
$59$ $$( 208 - 123 T^{2} + T^{4} )^{4}$$
$61$ $$( 416 + 58 T^{2} + T^{4} )^{4}$$
$67$ $$( 676 + 101 T^{2} + T^{4} )^{4}$$
$71$ $$( 832 - 59 T^{2} + T^{4} )^{4}$$
$73$ $$( 6656 - 164 T^{2} + T^{4} )^{4}$$
$79$ $$( 104 + 158 T^{2} + T^{4} )^{4}$$
$83$ $$( 14792 + 250 T^{2} + T^{4} )^{4}$$
$89$ $$( 8788 - 247 T^{2} + T^{4} )^{4}$$
$97$ $$( 43264 + 417 T^{2} + T^{4} )^{4}$$
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