# Properties

 Label 432.2.u.b Level 432 Weight 2 Character orbit 432.u Analytic conductor 3.450 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.u (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{4} - \beta_{11} ) q^{3} + ( -\beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{4} - \beta_{11} ) q^{3} + ( -\beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{10} ) q^{9} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{11} + ( -\beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{13} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{15} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{8} + \beta_{11} ) q^{19} + ( 2 + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{21} + ( -3 + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} ) q^{23} + ( -2 + 3 \beta_{1} - 3 \beta_{4} + 4 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{25} + ( -1 + \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + 4 \beta_{9} - \beta_{10} ) q^{27} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{11} ) q^{31} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{33} + ( -\beta_{1} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{35} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{11} ) q^{37} + ( 2 + 4 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{39} + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{41} + ( -5 + 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{9} ) q^{43} + ( 3 - 4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{8} + 4 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{45} + ( 2 - 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{47} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{49} + ( 4 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 4 \beta_{9} + 2 \beta_{10} ) q^{51} + ( 2 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{53} + ( -1 + \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{57} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{59} + ( -2 + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{61} + ( 6 - 5 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{63} + ( 3 - 7 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} - 6 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{65} + ( -3 - 3 \beta_{1} + 5 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - 5 \beta_{9} + 3 \beta_{11} ) q^{67} + ( 3 - 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{69} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{71} + ( 2 - 5 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{73} + ( 2 - 8 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{75} + ( 1 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{77} + ( -2 - 3 \beta_{1} - 5 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{79} + ( -3 + 3 \beta_{2} - \beta_{3} - 7 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{81} + ( -4 - 7 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{83} + ( 8 - 3 \beta_{1} - 4 \beta_{2} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 3 \beta_{10} - 2 \beta_{11} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} + 5 \beta_{3} + 4 \beta_{4} + \beta_{6} + \beta_{8} - 3 \beta_{9} - 2 \beta_{11} ) q^{87} + ( 1 + \beta_{1} - \beta_{2} - 9 \beta_{4} + \beta_{5} + 5 \beta_{6} - 5 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{89} + ( 1 + 2 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} - 7 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{91} + ( -6 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{93} + ( 3 - 5 \beta_{1} - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - 5 \beta_{6} + \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{95} + ( -2 + 4 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{7} + 7 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{97} + ( -2 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{5} + 3q^{7} - 12q^{9} + O(q^{10})$$ $$12q - 3q^{5} + 3q^{7} - 12q^{9} + 12q^{11} + 12q^{13} + 18q^{15} - 6q^{17} + 9q^{19} + 24q^{21} - 30q^{23} - 9q^{25} + 15q^{29} + 36q^{33} - 3q^{35} - 15q^{37} + 42q^{39} - 12q^{41} - 9q^{43} + 18q^{45} + 9q^{47} - 39q^{49} + 27q^{51} - 12q^{53} - 18q^{55} + 18q^{57} - 12q^{59} - 36q^{61} - 3q^{63} - 15q^{65} - 36q^{67} + 18q^{69} - 12q^{71} - 21q^{73} - 30q^{75} + 3q^{77} - 39q^{79} - 18q^{83} + 45q^{85} - 27q^{87} + 12q^{89} + 6q^{91} - 33q^{93} + 15q^{95} + 39q^{97} - 9q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060$$$$)/218$$ $$\beta_{2}$$ $$=$$ $$($$$$36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706$$$$)/218$$ $$\beta_{3}$$ $$=$$ $$($$$$-27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263$$$$)/218$$ $$\beta_{4}$$ $$=$$ $$($$$$26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524$$$$)/218$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555$$$$)/218$$ $$\beta_{6}$$ $$=$$ $$($$$$-27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83$$$$)/218$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350$$$$)/218$$ $$\beta_{9}$$ $$=$$ $$($$$$26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201$$$$)/218$$ $$\beta_{10}$$ $$=$$ $$($$$$36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194$$$$)/218$$ $$\beta_{11}$$ $$=$$ $$($$$$91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469$$$$)/218$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_{1} - 7$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta_{1} + 32$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta_{1} + 77$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta_{1} - 121$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta_{1} - 535$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta_{1} + 257$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta_{1} + 3359$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta_{1} + 1451$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta_{1} - 18736$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 − 1.80139i 0.5 − 0.168222i 0.5 + 1.80139i 0.5 + 0.168222i 0.5 + 1.96356i 0.5 − 0.677980i 0.5 − 2.42499i 0.5 + 1.74095i 0.5 + 2.42499i 0.5 − 1.74095i 0.5 − 1.96356i 0.5 + 0.677980i
0 −0.247510 1.71428i 0 −1.96209 + 0.714144i 0 0.696712 + 3.95125i 0 −2.87748 + 0.848600i 0
49.2 0 1.36085 + 1.07149i 0 0.696050 0.253341i 0 −0.717657 4.07003i 0 0.703829 + 2.91627i 0
97.1 0 −0.247510 + 1.71428i 0 −1.96209 0.714144i 0 0.696712 3.95125i 0 −2.87748 0.848600i 0
97.2 0 1.36085 1.07149i 0 0.696050 + 0.253341i 0 −0.717657 + 4.07003i 0 0.703829 2.91627i 0
193.1 0 −0.552775 1.64147i 0 −0.177398 + 1.00607i 0 −2.04289 + 1.71418i 0 −2.38888 + 1.81473i 0
193.2 0 1.14517 + 1.29945i 0 0.617090 3.49969i 0 0.244752 0.205371i 0 −0.377165 + 2.97620i 0
241.1 0 −1.56529 + 0.741539i 0 −3.10057 + 2.60168i 0 −0.144365 + 0.0525446i 0 1.90024 2.32144i 0
241.2 0 −0.140451 1.72635i 0 2.42692 2.03643i 0 3.46344 1.26059i 0 −2.96055 + 0.484935i 0
337.1 0 −1.56529 0.741539i 0 −3.10057 2.60168i 0 −0.144365 0.0525446i 0 1.90024 + 2.32144i 0
337.2 0 −0.140451 + 1.72635i 0 2.42692 + 2.03643i 0 3.46344 + 1.26059i 0 −2.96055 0.484935i 0
385.1 0 −0.552775 + 1.64147i 0 −0.177398 1.00607i 0 −2.04289 1.71418i 0 −2.38888 1.81473i 0
385.2 0 1.14517 1.29945i 0 0.617090 + 3.49969i 0 0.244752 + 0.205371i 0 −0.377165 2.97620i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 385.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.u.b 12
4.b odd 2 1 54.2.e.b 12
12.b even 2 1 162.2.e.b 12
27.e even 9 1 inner 432.2.u.b 12
36.f odd 6 1 486.2.e.f 12
36.f odd 6 1 486.2.e.h 12
36.h even 6 1 486.2.e.e 12
36.h even 6 1 486.2.e.g 12
108.j odd 18 1 54.2.e.b 12
108.j odd 18 1 486.2.e.f 12
108.j odd 18 1 486.2.e.h 12
108.j odd 18 1 1458.2.a.g 6
108.j odd 18 2 1458.2.c.f 12
108.l even 18 1 162.2.e.b 12
108.l even 18 1 486.2.e.e 12
108.l even 18 1 486.2.e.g 12
108.l even 18 1 1458.2.a.f 6
108.l even 18 2 1458.2.c.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.b 12 4.b odd 2 1
54.2.e.b 12 108.j odd 18 1
162.2.e.b 12 12.b even 2 1
162.2.e.b 12 108.l even 18 1
432.2.u.b 12 1.a even 1 1 trivial
432.2.u.b 12 27.e even 9 1 inner
486.2.e.e 12 36.h even 6 1
486.2.e.e 12 108.l even 18 1
486.2.e.f 12 36.f odd 6 1
486.2.e.f 12 108.j odd 18 1
486.2.e.g 12 36.h even 6 1
486.2.e.g 12 108.l even 18 1
486.2.e.h 12 36.f odd 6 1
486.2.e.h 12 108.j odd 18 1
1458.2.a.f 6 108.l even 18 1
1458.2.a.g 6 108.j odd 18 1
1458.2.c.f 12 108.j odd 18 2
1458.2.c.g 12 108.l even 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 6 T^{2} + 18 T^{4} + 18 T^{5} + 57 T^{6} + 54 T^{7} + 162 T^{8} + 486 T^{10} + 729 T^{12}$$
$5$ $$1 + 3 T + 9 T^{2} + 9 T^{3} - 18 T^{4} - 42 T^{5} - 142 T^{6} + 9 T^{7} - 171 T^{8} - 702 T^{9} - 603 T^{10} - 1791 T^{11} + 8049 T^{12} - 8955 T^{13} - 15075 T^{14} - 87750 T^{15} - 106875 T^{16} + 28125 T^{17} - 2218750 T^{18} - 3281250 T^{19} - 7031250 T^{20} + 17578125 T^{21} + 87890625 T^{22} + 146484375 T^{23} + 244140625 T^{24}$$
$7$ $$1 - 3 T + 24 T^{2} - 46 T^{3} + 321 T^{4} - 561 T^{5} + 3304 T^{6} - 4068 T^{7} + 25767 T^{8} - 26386 T^{9} + 193845 T^{10} - 149838 T^{11} + 1294315 T^{12} - 1048866 T^{13} + 9498405 T^{14} - 9050398 T^{15} + 61866567 T^{16} - 68370876 T^{17} + 388712296 T^{18} - 462007623 T^{19} + 1850501121 T^{20} - 1856265922 T^{21} + 6779405976 T^{22} - 5931980229 T^{23} + 13841287201 T^{24}$$
$11$ $$1 - 12 T + 90 T^{2} - 504 T^{3} + 2628 T^{4} - 12720 T^{5} + 57872 T^{6} - 239643 T^{7} + 957330 T^{8} - 3660336 T^{9} + 13502205 T^{10} - 46654983 T^{11} + 157089783 T^{12} - 513204813 T^{13} + 1633766805 T^{14} - 4871907216 T^{15} + 14016268530 T^{16} - 38594744793 T^{17} + 102523778192 T^{18} - 247876815120 T^{19} + 563335139268 T^{20} - 1188405636264 T^{21} + 2334368214090 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 - 12 T + 48 T^{2} - 74 T^{3} + 246 T^{4} - 1974 T^{5} + 8974 T^{6} - 33102 T^{7} + 94338 T^{8} - 225668 T^{9} + 896478 T^{10} - 3069618 T^{11} + 8300191 T^{12} - 39905034 T^{13} + 151504782 T^{14} - 495792596 T^{15} + 2694387618 T^{16} - 12290540886 T^{17} + 43315783966 T^{18} - 123865572558 T^{19} + 200669757366 T^{20} - 784732953602 T^{21} + 6617207608752 T^{22} - 21505924728444 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 + 6 T - 48 T^{2} - 252 T^{3} + 1722 T^{4} + 5946 T^{5} - 48874 T^{6} - 104832 T^{7} + 1071234 T^{8} + 1189944 T^{9} - 21101022 T^{10} - 6521976 T^{11} + 379698279 T^{12} - 110873592 T^{13} - 6098195358 T^{14} + 5846194872 T^{15} + 89470534914 T^{16} - 148846449024 T^{17} - 1179699547306 T^{18} + 2439873749658 T^{19} + 12012254313402 T^{20} - 29884144877244 T^{21} - 96767707221552 T^{22} + 205631377845798 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 - 9 T - 36 T^{2} + 419 T^{3} + 1611 T^{4} - 13806 T^{5} - 55610 T^{6} + 315819 T^{7} + 1580067 T^{8} - 4822864 T^{9} - 38744127 T^{10} + 34656777 T^{11} + 805313071 T^{12} + 658478763 T^{13} - 13986629847 T^{14} - 33080024176 T^{15} + 205915911507 T^{16} + 781999110081 T^{17} - 2616221442410 T^{18} - 12340793228634 T^{19} + 27360520059051 T^{20} + 135206145369401 T^{21} - 220718385280836 T^{22} - 1048412330083971 T^{23} + 2213314919066161 T^{24}$$
$23$ $$1 + 30 T + 414 T^{2} + 3222 T^{3} + 12348 T^{4} - 19608 T^{5} - 550618 T^{6} - 3222558 T^{7} - 7146234 T^{8} + 22769856 T^{9} + 232883658 T^{10} + 864182466 T^{11} + 2884098855 T^{12} + 19876196718 T^{13} + 123195455082 T^{14} + 277040837952 T^{15} - 1999809268794 T^{16} - 20741488625394 T^{17} - 81511225129402 T^{18} - 66761817364776 T^{19} + 966984046249788 T^{20} + 5803313875233786 T^{21} + 17150575642450686 T^{22} + 28584292737417810 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 - 15 T + 81 T^{2} + 45 T^{3} - 4518 T^{4} + 31692 T^{5} - 51964 T^{6} - 564705 T^{7} + 4930839 T^{8} - 14676930 T^{9} - 17187291 T^{10} + 338065677 T^{11} - 2175351603 T^{12} + 9803904633 T^{13} - 14454511731 T^{14} - 357955645770 T^{15} + 3487488738759 T^{16} - 11582748396045 T^{17} - 30909399052444 T^{18} + 546683079984828 T^{19} - 2260113293757798 T^{20} + 652821568914105 T^{21} + 34077285897316281 T^{22} - 183007646485587435 T^{23} + 353814783205469041 T^{24}$$
$31$ $$1 + 81 T^{2} - 49 T^{3} + 4023 T^{4} + 2565 T^{5} + 163531 T^{6} + 464805 T^{7} + 4701186 T^{8} + 27482744 T^{9} + 112711716 T^{10} + 1250347077 T^{11} + 3229099084 T^{12} + 38760759387 T^{13} + 108315959076 T^{14} + 818738426504 T^{15} + 4341643995906 T^{16} + 13306972530555 T^{17} + 145134364457611 T^{18} + 70569855194715 T^{19} + 3431180643625143 T^{20} - 1295541485872879 T^{21} + 66389891245444881 T^{22} + 787662783788549761 T^{24}$$
$37$ $$1 + 15 T + 3 T^{2} - 578 T^{3} + 2505 T^{4} + 24945 T^{5} - 192167 T^{6} - 791163 T^{7} + 6816672 T^{8} + 1096072 T^{9} - 278194485 T^{10} + 271278831 T^{11} + 11925628483 T^{12} + 10037316747 T^{13} - 380848249965 T^{14} + 55519335016 T^{15} + 12775540812192 T^{16} - 54862373051991 T^{17} - 493047946838303 T^{18} + 2368075675082685 T^{19} + 8798761032072105 T^{20} - 75117885601554506 T^{21} + 14425753117253547 T^{22} + 2668764326691906195 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 + 12 T + 117 T^{2} + 630 T^{3} + 5031 T^{4} + 23943 T^{5} + 171524 T^{6} + 408330 T^{7} + 3551247 T^{8} - 22823100 T^{9} - 180203112 T^{10} - 2500776837 T^{11} - 9745515051 T^{12} - 102531850317 T^{13} - 302921431272 T^{14} - 1572990875100 T^{15} + 10034975273967 T^{16} + 47307562554330 T^{17} + 814756879833284 T^{18} + 4663001579532783 T^{19} + 40172158827707751 T^{20} + 206250618668195430 T^{21} + 1570451139287830917 T^{22} + 6603948380594981292 T^{23} + 22563490300366186081 T^{24}$$
$43$ $$1 + 9 T + 36 T^{2} + 248 T^{3} + 1629 T^{4} + 13671 T^{5} + 99658 T^{6} + 1291248 T^{7} + 7996347 T^{8} + 44034284 T^{9} + 342067257 T^{10} + 2064609882 T^{11} + 16750938331 T^{12} + 88778224926 T^{13} + 632482358193 T^{14} + 3501033817988 T^{15} + 27337919119947 T^{16} + 189824358006864 T^{17} + 629974398737242 T^{18} + 3716032232443797 T^{19} + 19040078252212029 T^{20} + 124642967760337064 T^{21} + 778013363278232964 T^{22} + 8363643655241004363 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 - 9 T + 99 T^{2} - 981 T^{3} + 6354 T^{4} - 60768 T^{5} + 331868 T^{6} - 2211183 T^{7} + 13193361 T^{8} - 82909494 T^{9} + 638454717 T^{10} - 3119341005 T^{11} + 27438934035 T^{12} - 146609027235 T^{13} + 1410346469853 T^{14} - 8607912395562 T^{15} + 64379392997841 T^{16} - 507123780613281 T^{17} + 3577276632804572 T^{18} - 30786473784295584 T^{19} + 151296915448829394 T^{20} - 1097866994113814427 T^{21} + 5207314091347174851 T^{22} - 22249432935756110727 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$( 1 + 6 T + 255 T^{2} + 1593 T^{3} + 29193 T^{4} + 168801 T^{5} + 1959550 T^{6} + 8946453 T^{7} + 82003137 T^{8} + 237161061 T^{9} + 2012072655 T^{10} + 2509172958 T^{11} + 22164361129 T^{12} )^{2}$$
$59$ $$1 + 12 T + 9 T^{2} - 288 T^{3} - 7416 T^{4} - 73437 T^{5} + 128189 T^{6} + 3672837 T^{7} + 36794322 T^{8} + 361771704 T^{9} - 137989746 T^{10} - 18392855166 T^{11} - 121796603070 T^{12} - 1085178454794 T^{13} - 480342305826 T^{14} + 74300310795816 T^{15} + 445850082424242 T^{16} + 2625800417566263 T^{17} + 5407080426906149 T^{18} - 182759099090652903 T^{19} - 1088894525273644536 T^{20} - 2494942795772622432 T^{21} + 4600050779705772609 T^{22} +$$$$36\!\cdots\!08$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 36 T + 531 T^{2} + 3559 T^{3} + 3123 T^{4} - 91953 T^{5} - 447281 T^{6} - 4553469 T^{7} - 84400488 T^{8} - 593953490 T^{9} - 1210754052 T^{10} - 5462757351 T^{11} - 104608387268 T^{12} - 333228198411 T^{13} - 4505215827492 T^{14} - 134816157113690 T^{15} - 1168595737170408 T^{16} - 3845843074118169 T^{17} - 23044084564562441 T^{18} - 288984632000639013 T^{19} + 598701938490508563 T^{20} + 41619465944396707819 T^{21} +$$$$37\!\cdots\!31$$$$T^{22} +$$$$15\!\cdots\!96$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 36 T + 630 T^{2} + 7592 T^{3} + 71910 T^{4} + 581724 T^{5} + 4963294 T^{6} + 52102143 T^{7} + 557562474 T^{8} + 5169740204 T^{9} + 40448438361 T^{10} + 276054423399 T^{11} + 1991438619445 T^{12} + 18495646367733 T^{13} + 181573039802529 T^{14} + 1554866572975652 T^{15} + 11235508878633354 T^{16} + 70344411392804301 T^{17} + 448971545469104686 T^{18} + 3525661397894916852 T^{19} + 29200326693098054310 T^{20} +$$$$20\!\cdots\!24$$$$T^{21} +$$$$11\!\cdots\!70$$$$T^{22} +$$$$43\!\cdots\!88$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 + 12 T - 201 T^{2} - 2430 T^{3} + 25608 T^{4} + 247278 T^{5} - 2990308 T^{6} - 17740170 T^{7} + 310819581 T^{8} + 1005967566 T^{9} - 26322276303 T^{10} - 30298327380 T^{11} + 1913975417427 T^{12} - 2151181243980 T^{13} - 132690594843423 T^{14} + 360046857514626 T^{15} + 7898448040925661 T^{16} - 32007335405729670 T^{17} - 383059303811237668 T^{18} + 2249023122526609698 T^{19} + 16536456428141447688 T^{20} -$$$$11\!\cdots\!30$$$$T^{21} -$$$$65\!\cdots\!01$$$$T^{22} +$$$$27\!\cdots\!52$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 + 21 T - 48 T^{2} - 3887 T^{3} - 4845 T^{4} + 448266 T^{5} + 1641574 T^{6} - 22871205 T^{7} - 33688827 T^{8} + 739848004 T^{9} - 9072236607 T^{10} + 8731491045 T^{11} + 1415682555505 T^{12} + 637398846285 T^{13} - 48345948878703 T^{14} + 287813450972068 T^{15} - 956703428153307 T^{16} - 47413645383179565 T^{17} + 248426331186138886 T^{18} + 4952173144561535802 T^{19} - 3907299145226822445 T^{20} -$$$$22\!\cdots\!31$$$$T^{21} -$$$$20\!\cdots\!52$$$$T^{22} +$$$$65\!\cdots\!17$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 39 T + 822 T^{2} + 11936 T^{3} + 137379 T^{4} + 1304607 T^{5} + 9780148 T^{6} + 42815682 T^{7} - 154130391 T^{8} - 6081344458 T^{9} - 88425769017 T^{10} - 1013814962454 T^{11} - 9774915393497 T^{12} - 80091382033866 T^{13} - 551865224435097 T^{14} - 2998339990227862 T^{15} - 6003391214011671 T^{16} + 131746268275649118 T^{17} + 2377431291938797108 T^{18} + 25053554090705934513 T^{19} +$$$$20\!\cdots\!19$$$$T^{20} +$$$$14\!\cdots\!84$$$$T^{21} +$$$$77\!\cdots\!22$$$$T^{22} +$$$$29\!\cdots\!81$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 + 18 T + 126 T^{2} - 1260 T^{3} - 41166 T^{4} - 525114 T^{5} - 2039866 T^{6} + 36031104 T^{7} + 749106684 T^{8} + 6189995052 T^{9} + 9354005928 T^{10} - 495774705924 T^{11} - 6862378086069 T^{12} - 41149300591692 T^{13} + 64439746837992 T^{14} + 3539358700797924 T^{15} + 35551345472517564 T^{16} + 141927983068159872 T^{17} - 666914551662728554 T^{18} - 14249520279366992478 T^{19} - 92717862028235761806 T^{20} -$$$$23\!\cdots\!80$$$$T^{21} +$$$$19\!\cdots\!74$$$$T^{22} +$$$$23\!\cdots\!06$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$1 - 12 T - 120 T^{2} + 3294 T^{3} - 12273 T^{4} - 248142 T^{5} + 2973695 T^{6} - 4697397 T^{7} - 153066366 T^{8} + 1115642430 T^{9} + 2098738827 T^{10} - 25141746861 T^{11} - 47336761416 T^{12} - 2237615470629 T^{13} + 16624110248667 T^{14} + 786493328234670 T^{15} - 9603726824566206 T^{16} - 26230544103554253 T^{17} + 1477870780024270895 T^{18} - 10975651903646357118 T^{19} - 48313754412381640113 T^{20} +$$$$11\!\cdots\!46$$$$T^{21} -$$$$37\!\cdots\!20$$$$T^{22} -$$$$33\!\cdots\!68$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 39 T + 993 T^{2} - 19028 T^{3} + 304221 T^{4} - 4077399 T^{5} + 47593567 T^{6} - 476249436 T^{7} + 4125278700 T^{8} - 29493453512 T^{9} + 169040152338 T^{10} - 681702895626 T^{11} + 3451348757128 T^{12} - 66125180875722 T^{13} + 1590498793348242 T^{14} - 26917878697157576 T^{15} + 365207957235614700 T^{16} - 4089715954136345052 T^{17} + 39644108925712691743 T^{18} -$$$$32\!\cdots\!87$$$$T^{19} +$$$$23\!\cdots\!81$$$$T^{20} -$$$$14\!\cdots\!76$$$$T^{21} +$$$$73\!\cdots\!57$$$$T^{22} -$$$$27\!\cdots\!67$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$