# Properties

 Label 429.2.i.e 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: $$x^{10} - 2 x^{9} + 10 x^{8} - 6 x^{7} + 46 x^{6} - 31 x^{5} + 111 x^{4} - 36 x^{3} + 145 x^{2} - 72 x + 81$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{5} ) q^{3} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} + ( -\beta_{4} - 2 \beta_{5} - \beta_{9} ) q^{7} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{8} + \beta_{5} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{5} ) q^{3} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} + ( -\beta_{4} - 2 \beta_{5} - \beta_{9} ) q^{7} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{8} + \beta_{5} q^{9} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{10} + ( 1 + \beta_{5} ) q^{11} + ( -1 - \beta_{3} - \beta_{6} ) q^{12} + ( -1 + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{9} ) q^{14} + ( \beta_{1} + \beta_{7} - \beta_{8} ) q^{15} + ( -4 \beta_{1} - \beta_{4} + \beta_{7} + 2 \beta_{8} ) q^{16} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{17} -\beta_{3} q^{18} + ( -3 \beta_{1} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{9} ) q^{19} + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{20} + ( 2 - \beta_{9} ) q^{21} + ( \beta_{1} - \beta_{3} ) q^{22} + ( 2 \beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{23} + ( -2 - \beta_{1} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{24} + ( 5 - \beta_{3} + \beta_{6} - \beta_{9} ) q^{25} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{26} - q^{27} + ( 2 + 2 \beta_{1} + 2 \beta_{5} - 3 \beta_{8} ) q^{28} + ( -2 - 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{29} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{30} + ( -5 + \beta_{2} + 2 \beta_{3} - \beta_{6} + \beta_{9} ) q^{31} + ( -5 \beta_{1} + 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{32} + \beta_{5} q^{33} + ( -3 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} ) q^{34} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} ) q^{35} + ( -1 - \beta_{1} - \beta_{5} + \beta_{8} ) q^{36} + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} + 3 \beta_{7} + 2 \beta_{8} ) q^{37} + ( 8 - \beta_{2} + \beta_{3} + 3 \beta_{6} - \beta_{9} ) q^{38} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{39} + ( -4 - \beta_{2} - 8 \beta_{3} - 3 \beta_{6} + 3 \beta_{9} ) q^{40} + ( -5 - 2 \beta_{1} - 5 \beta_{5} - \beta_{7} ) q^{41} + ( -1 + 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{42} + ( 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + 3 \beta_{9} ) q^{43} + ( -1 - \beta_{3} - \beta_{6} ) q^{44} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{45} + ( \beta_{2} + 5 \beta_{5} - \beta_{7} ) q^{46} + ( -4 + \beta_{6} - \beta_{9} ) q^{47} + ( -4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{48} + ( -2 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{49} + ( -5 + 5 \beta_{1} + \beta_{4} - 5 \beta_{5} - 2 \beta_{7} ) q^{50} + ( 2 \beta_{2} + \beta_{3} - \beta_{9} ) q^{51} + ( 1 - 2 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{52} + ( 2 - 3 \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{53} -\beta_{1} q^{54} + ( \beta_{1} + \beta_{7} - \beta_{8} ) q^{55} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} - \beta_{7} - 5 \beta_{8} - 2 \beta_{9} ) q^{56} + ( -2 + 3 \beta_{3} - \beta_{9} ) q^{57} + ( \beta_{4} - 2 \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{58} + ( 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{59} + ( -5 - 3 \beta_{3} - 2 \beta_{6} + \beta_{9} ) q^{60} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 3 \beta_{7} + \beta_{9} ) q^{61} + ( 7 - 5 \beta_{1} - 2 \beta_{4} + 7 \beta_{5} + \beta_{7} ) q^{62} + ( 2 + \beta_{4} + 2 \beta_{5} ) q^{63} + ( 10 - 3 \beta_{2} + 6 \beta_{3} + 4 \beta_{6} ) q^{64} + ( 1 - 5 \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{65} -\beta_{3} q^{66} + ( 7 + \beta_{1} - 2 \beta_{4} + 7 \beta_{5} - 2 \beta_{8} ) q^{67} + ( 2 - \beta_{1} + 2 \beta_{5} + \beta_{7} - \beta_{8} ) q^{68} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{69} + ( -3 - 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{6} + 2 \beta_{9} ) q^{70} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{71} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{72} + ( 2 - 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{6} ) q^{73} + ( -5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{74} + ( 5 - \beta_{1} + \beta_{4} + 5 \beta_{5} - \beta_{8} ) q^{75} + ( 4 + 7 \beta_{1} + 4 \beta_{5} - 3 \beta_{7} - 5 \beta_{8} ) q^{76} + ( 2 - \beta_{9} ) q^{77} + ( 2 - \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{78} + ( -2 - \beta_{2} + 4 \beta_{3} - 3 \beta_{9} ) q^{79} + ( -7 - 8 \beta_{1} - 7 \beta_{5} + 7 \beta_{7} + 6 \beta_{8} ) q^{80} + ( -1 - \beta_{5} ) q^{81} + ( -6 \beta_{1} - \beta_{2} + 6 \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{82} + ( -1 + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} - \beta_{9} ) q^{83} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} ) q^{84} + ( 6 \beta_{1} - 6 \beta_{3} + \beta_{4} - 11 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} + \beta_{9} ) q^{85} + ( -4 + 2 \beta_{2} - \beta_{6} + 3 \beta_{9} ) q^{86} + ( \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{87} + ( -2 - \beta_{1} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{88} + ( -3 - \beta_{1} - \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{89} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{6} - \beta_{9} ) q^{90} + ( 3 + 3 \beta_{2} - 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{91} + ( -1 + \beta_{2} - 2 \beta_{3} - 3 \beta_{6} - \beta_{9} ) q^{92} + ( -5 + 2 \beta_{1} - \beta_{4} - 5 \beta_{5} + \beta_{7} + \beta_{8} ) q^{93} + ( -2 - 3 \beta_{1} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{94} + ( -8 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 8 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{95} + ( 6 + 5 \beta_{3} + 3 \beta_{6} - 2 \beta_{9} ) q^{96} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{97} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 2q^{2} + 5q^{3} - 6q^{4} + 4q^{5} - 2q^{6} + 9q^{7} - 18q^{8} - 5q^{9} + O(q^{10})$$ $$10q + 2q^{2} + 5q^{3} - 6q^{4} + 4q^{5} - 2q^{6} + 9q^{7} - 18q^{8} - 5q^{9} + 5q^{10} + 5q^{11} - 12q^{12} - 3q^{13} - 6q^{14} + 2q^{15} - 4q^{16} + 3q^{17} - 4q^{18} - 5q^{19} - 28q^{20} + 18q^{21} - 2q^{22} + 5q^{23} - 9q^{24} + 42q^{25} + 20q^{26} - 10q^{27} + 11q^{28} - 12q^{29} - 5q^{30} - 36q^{31} + 35q^{32} - 5q^{33} - 6q^{34} - 6q^{36} + q^{37} + 74q^{38} + 6q^{39} - 62q^{40} - 30q^{41} - 3q^{42} + 3q^{43} - 12q^{44} - 2q^{45} - 24q^{46} - 44q^{47} + 4q^{48} - 14q^{49} - 18q^{50} + 6q^{51} + 35q^{52} + 14q^{53} - 2q^{54} + 2q^{55} - 27q^{56} - 10q^{57} + 3q^{58} + 12q^{59} - 56q^{60} - 18q^{61} + 28q^{62} + 9q^{63} + 110q^{64} - 28q^{65} - 4q^{66} + 37q^{67} + 8q^{68} - 5q^{69} - 32q^{70} + 17q^{71} + 9q^{72} + 4q^{73} + q^{74} + 21q^{75} + 26q^{76} + 18q^{77} + 25q^{78} - 12q^{79} - 38q^{80} - 5q^{81} + 36q^{82} + 8q^{83} - 11q^{84} + 41q^{85} - 28q^{86} + 12q^{87} - 9q^{88} - 14q^{89} - 10q^{90} + 35q^{91} - 12q^{92} - 18q^{93} - 20q^{94} + 7q^{95} + 70q^{96} + 15q^{97} + 4q^{98} - 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 10 x^{8} - 6 x^{7} + 46 x^{6} - 31 x^{5} + 111 x^{4} - 36 x^{3} + 145 x^{2} - 72 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$6866 \nu^{9} + 75296 \nu^{8} - 280912 \nu^{7} + 955837 \nu^{6} - 1077312 \nu^{5} + 1717328 \nu^{4} - 4139410 \nu^{3} + 2701968 \nu^{2} - 1720224 \nu - 2950718$$$$)/3992813$$ $$\beta_{3}$$ $$=$$ $$($$$$-20666 \nu^{9} + 75764 \nu^{8} - 282658 \nu^{7} + 407527 \nu^{6} - 1084008 \nu^{5} + 1728002 \nu^{4} - 4027265 \nu^{3} + 2718762 \nu^{2} - 1730916 \nu + 3224214$$$$)/3992813$$ $$\beta_{4}$$ $$=$$ $$($$$$10738 \nu^{9} + 44485 \nu^{8} + 43450 \nu^{7} + 425376 \nu^{6} + 731689 \nu^{5} + 2005823 \nu^{4} + 1576659 \nu^{3} + 3062220 \nu^{2} + 2110927 \nu + 2697786$$$$)/1088949$$ $$\beta_{5}$$ $$=$$ $$($$$$358246 \nu^{9} - 530498 \nu^{8} + 2900584 \nu^{7} + 394446 \nu^{6} + 12811573 \nu^{5} - 1349554 \nu^{4} + 24213288 \nu^{3} + 23348529 \nu^{2} + 27476812 \nu - 10215468$$$$)/35935317$$ $$\beta_{6}$$ $$=$$ $$($$$$55098 \nu^{9} - 151762 \nu^{8} + 566189 \nu^{7} - 540899 \nu^{6} + 2171364 \nu^{5} - 3461341 \nu^{4} + 6002051 \nu^{3} - 5445921 \nu^{2} + 3467178 \nu - 13528707$$$$)/3992813$$ $$\beta_{7}$$ $$=$$ $$($$$$-749002 \nu^{9} + 1301129 \nu^{8} - 7157758 \nu^{7} + 490164 \nu^{6} - 25680361 \nu^{5} - 9177392 \nu^{4} - 47918541 \nu^{3} - 41304942 \nu^{2} - 8226031 \nu - 47647440$$$$)/35935317$$ $$\beta_{8}$$ $$=$$ $$($$$$254950 \nu^{9} - 302504 \nu^{8} + 2049991 \nu^{7} + 794562 \nu^{6} + 9549505 \nu^{5} + 3850463 \nu^{4} + 18288930 \nu^{3} + 19551567 \nu^{2} + 34246465 \nu + 20698011$$$$)/11978439$$ $$\beta_{9}$$ $$=$$ $$($$$$152201 \nu^{9} - 387192 \nu^{8} + 1444524 \nu^{7} - 1545542 \nu^{6} + 5539824 \nu^{5} - 8830956 \nu^{4} + 8369966 \nu^{3} - 13894236 \nu^{2} + 8845848 \nu - 16559562$$$$)/3992813$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{8} - \beta_{6} + 3 \beta_{5} - \beta_{3} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{6} - 5 \beta_{3} + \beta_{2} - 2$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{8} + \beta_{7} - 14 \beta_{5} - \beta_{4} - 10 \beta_{1} - 14$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{9} + 19 \beta_{8} + 8 \beta_{7} + 19 \beta_{6} - 22 \beta_{5} - 2 \beta_{4} + 33 \beta_{3} - 8 \beta_{2} - 33 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{9} + 60 \beta_{6} + 82 \beta_{3} - 13 \beta_{2} + 86$$ $$\nu^{7}$$ $$=$$ $$-155 \beta_{8} - 57 \beta_{7} + 189 \beta_{5} + 23 \beta_{4} + 241 \beta_{1} + 189$$ $$\nu^{8}$$ $$=$$ $$80 \beta_{9} - 453 \beta_{8} - 121 \beta_{7} - 453 \beta_{6} + 602 \beta_{5} + 80 \beta_{4} - 642 \beta_{3} + 121 \beta_{2} + 642 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$201 \beta_{9} - 1216 \beta_{6} - 1818 \beta_{3} + 412 \beta_{2} - 1514$$

## Character values

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

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$1$$

## 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
 −0.922622 + 1.59803i −0.665890 + 1.15336i 0.423911 − 0.734236i 0.779885 − 1.35080i 1.38472 − 2.39840i −0.922622 − 1.59803i −0.665890 − 1.15336i 0.423911 + 0.734236i 0.779885 + 1.35080i 1.38472 + 2.39840i
−0.922622 + 1.59803i 0.500000 0.866025i −0.702461 1.21670i 3.96195 0.922622 + 1.59803i 1.80081 + 3.11910i −1.09806 −0.500000 0.866025i −3.65538 + 6.33130i
100.2 −0.665890 + 1.15336i 0.500000 0.866025i 0.113180 + 0.196033i −3.31232 0.665890 + 1.15336i 0.767139 + 1.32872i −2.96502 −0.500000 0.866025i 2.20564 3.82029i
100.3 0.423911 0.734236i 0.500000 0.866025i 0.640598 + 1.10955i −2.90954 −0.423911 0.734236i 2.22113 + 3.84711i 2.78187 −0.500000 0.866025i −1.23339 + 2.13629i
100.4 0.779885 1.35080i 0.500000 0.866025i −0.216440 0.374886i 1.18322 −0.779885 1.35080i −1.45667 2.52303i 2.44434 −0.500000 0.866025i 0.922773 1.59829i
100.5 1.38472 2.39840i 0.500000 0.866025i −2.83488 4.91015i 3.07670 −1.38472 2.39840i 1.16758 + 2.02232i −10.1631 −0.500000 0.866025i 4.26035 7.37914i
133.1 −0.922622 1.59803i 0.500000 + 0.866025i −0.702461 + 1.21670i 3.96195 0.922622 1.59803i 1.80081 3.11910i −1.09806 −0.500000 + 0.866025i −3.65538 6.33130i
133.2 −0.665890 1.15336i 0.500000 + 0.866025i 0.113180 0.196033i −3.31232 0.665890 1.15336i 0.767139 1.32872i −2.96502 −0.500000 + 0.866025i 2.20564 + 3.82029i
133.3 0.423911 + 0.734236i 0.500000 + 0.866025i 0.640598 1.10955i −2.90954 −0.423911 + 0.734236i 2.22113 3.84711i 2.78187 −0.500000 + 0.866025i −1.23339 2.13629i
133.4 0.779885 + 1.35080i 0.500000 + 0.866025i −0.216440 + 0.374886i 1.18322 −0.779885 + 1.35080i −1.45667 + 2.52303i 2.44434 −0.500000 + 0.866025i 0.922773 + 1.59829i
133.5 1.38472 + 2.39840i 0.500000 + 0.866025i −2.83488 + 4.91015i 3.07670 −1.38472 + 2.39840i 1.16758 2.02232i −10.1631 −0.500000 + 0.866025i 4.26035 + 7.37914i
 $$n$$: 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.e 10
13.c even 3 1 inner 429.2.i.e 10
13.c even 3 1 5577.2.a.o 5
13.e even 6 1 5577.2.a.u 5

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

$p$ $F_p(T)$
$2$ $$81 - 72 T + 145 T^{2} - 36 T^{3} + 111 T^{4} - 31 T^{5} + 46 T^{6} - 6 T^{7} + 10 T^{8} - 2 T^{9} + T^{10}$$
$3$ $$( 1 - T + T^{2} )^{5}$$
$5$ $$( -139 + 108 T + 34 T^{2} - 21 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$7$ $$27889 - 34569 T + 33330 T^{2} - 17143 T^{3} + 8064 T^{4} - 2647 T^{5} + 976 T^{6} - 258 T^{7} + 65 T^{8} - 9 T^{9} + T^{10}$$
$11$ $$( 1 - T + T^{2} )^{5}$$
$13$ $$371293 + 85683 T + 41743 T^{2} - 2873 T^{3} - 1859 T^{4} - 1451 T^{5} - 143 T^{6} - 17 T^{7} + 19 T^{8} + 3 T^{9} + T^{10}$$
$17$ $$1 + 5 T + 111 T^{2} - 508 T^{3} + 7198 T^{4} - 3385 T^{5} + 1784 T^{6} - 55 T^{7} + 48 T^{8} - 3 T^{9} + T^{10}$$
$19$ $$2886601 + 824015 T + 676965 T^{2} + 60790 T^{3} + 85780 T^{4} + 11149 T^{5} + 3840 T^{6} + 245 T^{7} + 80 T^{8} + 5 T^{9} + T^{10}$$
$23$ $$18769 + 7809 T + 21333 T^{2} - 15470 T^{3} + 15086 T^{4} - 4535 T^{5} + 1558 T^{6} - 119 T^{7} + 54 T^{8} - 5 T^{9} + T^{10}$$
$29$ $$625 - 3250 T + 17175 T^{2} - 470 T^{3} + 4761 T^{4} + 2727 T^{5} + 1706 T^{6} + 478 T^{7} + 106 T^{8} + 12 T^{9} + T^{10}$$
$31$ $$( -6225 - 4265 T - 739 T^{2} + 43 T^{3} + 18 T^{4} + T^{5} )^{2}$$
$37$ $$102961609 - 29324830 T + 11223701 T^{2} - 1434764 T^{3} + 390732 T^{4} - 35780 T^{5} + 9714 T^{6} - 455 T^{7} + 112 T^{8} - T^{9} + T^{10}$$
$41$ $$426409 + 1951164 T + 7882038 T^{2} + 4354490 T^{3} + 1557786 T^{4} + 351635 T^{5} + 58513 T^{6} + 6726 T^{7} + 569 T^{8} + 30 T^{9} + T^{10}$$
$43$ $$22137025 - 8699545 T + 5573691 T^{2} - 423508 T^{3} + 445264 T^{4} - 55441 T^{5} + 17750 T^{6} - 511 T^{7} + 144 T^{8} - 3 T^{9} + T^{10}$$
$47$ $$( 59 + 463 T + 461 T^{2} + 159 T^{3} + 22 T^{4} + T^{5} )^{2}$$
$53$ $$( 5197 + 4028 T + 233 T^{2} - 127 T^{3} - 7 T^{4} + T^{5} )^{2}$$
$59$ $$32775625 - 3721250 T + 5689500 T^{2} - 237850 T^{3} + 825150 T^{4} - 57285 T^{5} + 15719 T^{6} - 964 T^{7} + 217 T^{8} - 12 T^{9} + T^{10}$$
$61$ $$862655641 - 43527822 T + 44666790 T^{2} + 3376554 T^{3} + 1531116 T^{4} + 113089 T^{5} + 27951 T^{6} + 2514 T^{7} + 345 T^{8} + 18 T^{9} + T^{10}$$
$67$ $$696590449 - 102114517 T + 59283008 T^{2} - 17363221 T^{3} + 5544370 T^{4} - 1018821 T^{5} + 146050 T^{6} - 13366 T^{7} + 917 T^{8} - 37 T^{9} + T^{10}$$
$71$ $$2654207361 + 1906203 T + 110458105 T^{2} - 9455786 T^{3} + 3717546 T^{4} - 247881 T^{5} + 44766 T^{6} - 2741 T^{7} + 380 T^{8} - 17 T^{9} + T^{10}$$
$73$ $$( 32777 + 12108 T - 137 T^{2} - 222 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$79$ $$( 72185 + 11736 T - 1329 T^{2} - 224 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$83$ $$( -879 + 1103 T + 429 T^{2} - 105 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$89$ $$40056241 - 2512613 T + 3758810 T^{2} + 289183 T^{3} + 233170 T^{4} + 20290 T^{5} + 8388 T^{6} + 1068 T^{7} + 201 T^{8} + 14 T^{9} + T^{10}$$
$97$ $$72361 + 27438 T + 107513 T^{2} - 31980 T^{3} + 127204 T^{4} - 80 T^{5} + 5598 T^{6} - 857 T^{7} + 216 T^{8} - 15 T^{9} + T^{10}$$