# Properties

 Label 147.4.c.a Level $147$ Weight $4$ Character orbit 147.c Analytic conductor $8.673$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{9} q^{3} + ( -2 + \beta_{11} ) q^{4} -\beta_{10} q^{5} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{10} ) q^{6} + ( \beta_{3} + \beta_{7} - \beta_{8} ) q^{8} + ( \beta_{1} - 2 \beta_{3} - \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{9} q^{3} + ( -2 + \beta_{11} ) q^{4} -\beta_{10} q^{5} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{10} ) q^{6} + ( \beta_{3} + \beta_{7} - \beta_{8} ) q^{8} + ( \beta_{1} - 2 \beta_{3} - \beta_{7} - \beta_{8} ) q^{9} + ( \beta_{5} - 5 \beta_{6} - 5 \beta_{9} ) q^{10} + ( 2 \beta_{3} - \beta_{7} - 2 \beta_{8} ) q^{11} + ( -10 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{9} - \beta_{10} ) q^{12} + ( -9 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} ) q^{13} + ( 3 - 15 \beta_{1} + 4 \beta_{3} - \beta_{7} + \beta_{8} + 5 \beta_{11} ) q^{15} + ( -24 - 2 \beta_{3} - 2 \beta_{8} - 3 \beta_{11} ) q^{16} + ( -4 \beta_{4} - 9 \beta_{6} + 9 \beta_{9} ) q^{17} + ( -6 - 9 \beta_{1} + 6 \beta_{3} + 12 \beta_{11} ) q^{18} + ( 17 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{19} + ( 12 \beta_{4} - 3 \beta_{6} + 3 \beta_{9} + 3 \beta_{10} ) q^{20} + ( -16 + 8 \beta_{3} + 8 \beta_{8} + 11 \beta_{11} ) q^{22} + ( 14 \beta_{1} - 5 \beta_{3} + 5 \beta_{8} ) q^{23} + ( -20 \beta_{2} + 10 \beta_{4} - 5 \beta_{5} - 15 \beta_{6} - 3 \beta_{9} - 2 \beta_{10} ) q^{24} + ( 13 + 11 \beta_{3} + 11 \beta_{8} + 7 \beta_{11} ) q^{25} + ( 19 \beta_{4} - 9 \beta_{6} + 9 \beta_{9} + 7 \beta_{10} ) q^{26} + ( 21 \beta_{2} - 24 \beta_{4} + 3 \beta_{5} + 6 \beta_{9} + 3 \beta_{10} ) q^{27} + ( 38 \beta_{1} + 7 \beta_{3} - 5 \beta_{7} - 7 \beta_{8} ) q^{29} + ( 138 - 12 \beta_{1} + 7 \beta_{3} + 5 \beta_{7} + 7 \beta_{8} - 4 \beta_{11} ) q^{30} + ( 55 \beta_{2} + 2 \beta_{5} - 10 \beta_{6} - 10 \beta_{9} ) q^{31} + ( -18 \beta_{1} + 9 \beta_{3} + 5 \beta_{7} - 9 \beta_{8} ) q^{32} + ( -49 \beta_{2} - 16 \beta_{4} - 10 \beta_{5} + 6 \beta_{6} + 3 \beta_{9} + 5 \beta_{10} ) q^{33} + ( -4 \beta_{2} - 14 \beta_{5} - 4 \beta_{6} - 4 \beta_{9} ) q^{34} + ( 66 - 52 \beta_{1} - 4 \beta_{3} + 4 \beta_{7} - 8 \beta_{8} - 9 \beta_{11} ) q^{36} + ( -135 - 4 \beta_{3} - 4 \beta_{8} - 19 \beta_{11} ) q^{37} + ( -13 \beta_{4} - 3 \beta_{6} + 3 \beta_{9} + 3 \beta_{10} ) q^{38} + ( 63 - 32 \beta_{1} - 5 \beta_{3} + 2 \beta_{7} - \beta_{8} - 21 \beta_{11} ) q^{39} + ( 132 \beta_{2} - 13 \beta_{5} - 13 \beta_{6} - 13 \beta_{9} ) q^{40} + ( 32 \beta_{4} + 6 \beta_{6} - 6 \beta_{9} - 16 \beta_{10} ) q^{41} + ( -87 - 13 \beta_{3} - 13 \beta_{8} + 5 \beta_{11} ) q^{43} + ( -34 \beta_{1} + 11 \beta_{3} + 3 \beta_{7} - 11 \beta_{8} ) q^{44} + ( -135 \beta_{2} + 15 \beta_{5} + 15 \beta_{6} + 6 \beta_{9} + 12 \beta_{10} ) q^{45} + ( -100 - 10 \beta_{3} - 10 \beta_{8} + 14 \beta_{11} ) q^{46} + ( 20 \beta_{4} + 27 \beta_{6} - 27 \beta_{9} - 14 \beta_{10} ) q^{47} + ( 44 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} - 9 \beta_{6} - 23 \beta_{9} + 5 \beta_{10} ) q^{48} + ( 37 \beta_{1} - 15 \beta_{3} + 7 \beta_{7} + 15 \beta_{8} ) q^{50} + ( 219 - 3 \beta_{1} - 22 \beta_{3} - 5 \beta_{7} - 13 \beta_{8} + 4 \beta_{11} ) q^{51} + ( 154 \beta_{2} - 20 \beta_{5} + 38 \beta_{6} + 38 \beta_{9} ) q^{52} + ( 96 \beta_{1} - 28 \beta_{3} - 3 \beta_{7} + 28 \beta_{8} ) q^{53} + ( -228 \beta_{2} - 21 \beta_{4} + 15 \beta_{5} - 18 \beta_{6} - 3 \beta_{10} ) q^{54} + ( -150 \beta_{2} + 19 \beta_{5} + 28 \beta_{6} + 28 \beta_{9} ) q^{55} + ( 30 + 9 \beta_{1} - \beta_{3} + \beta_{7} + 20 \beta_{8} + 13 \beta_{11} ) q^{57} + ( -436 + 34 \beta_{3} + 34 \beta_{8} + 93 \beta_{11} ) q^{58} + ( 16 \beta_{4} + 54 \beta_{6} - 54 \beta_{9} - 19 \beta_{10} ) q^{59} + ( 144 + 84 \beta_{1} - 6 \beta_{3} - 12 \beta_{7} + 6 \beta_{8} - 27 \beta_{11} ) q^{60} + ( 114 \beta_{2} + 38 \beta_{5} + 51 \beta_{6} + 51 \beta_{9} ) q^{61} + ( -31 \beta_{4} - 6 \beta_{6} + 6 \beta_{9} + 22 \beta_{10} ) q^{62} + ( -84 - 18 \beta_{3} - 18 \beta_{8} - 97 \beta_{11} ) q^{64} + ( 96 \beta_{1} - 6 \beta_{3} - 14 \beta_{7} + 6 \beta_{8} ) q^{65} + ( -166 \beta_{2} + 20 \beta_{4} + 14 \beta_{5} + 39 \beta_{6} - 21 \beta_{9} - 19 \beta_{10} ) q^{66} + ( -181 - 62 \beta_{3} - 62 \beta_{8} - 85 \beta_{11} ) q^{67} + ( -48 \beta_{4} - 30 \beta_{6} + 30 \beta_{9} - 6 \beta_{10} ) q^{68} + ( 143 \beta_{2} - 4 \beta_{4} + 11 \beta_{5} + 15 \beta_{6} - 19 \beta_{10} ) q^{69} + ( -32 \beta_{1} - 18 \beta_{3} - 4 \beta_{7} + 18 \beta_{8} ) q^{71} + ( 456 + 12 \beta_{1} + 39 \beta_{3} - 9 \beta_{7} - 15 \beta_{8} ) q^{72} + ( 145 \beta_{2} + 11 \beta_{5} + 49 \beta_{6} + 49 \beta_{9} ) q^{73} + ( -45 \beta_{1} - 11 \beta_{3} - 19 \beta_{7} + 11 \beta_{8} ) q^{74} + ( -147 \beta_{2} + 60 \beta_{4} + 3 \beta_{5} + 78 \beta_{6} - 2 \beta_{9} - 18 \beta_{10} ) q^{75} + ( 18 \beta_{2} + 12 \beta_{5} - 6 \beta_{6} - 6 \beta_{9} ) q^{76} + ( 336 + 171 \beta_{1} - 27 \beta_{3} - 21 \beta_{7} + 3 \beta_{8} - 54 \beta_{11} ) q^{78} + ( -325 - 50 \beta_{3} - 50 \beta_{8} - 88 \beta_{11} ) q^{79} + ( -36 \beta_{4} + 15 \beta_{6} - 15 \beta_{9} + 37 \beta_{10} ) q^{80} + ( -144 - 21 \beta_{1} - 57 \beta_{3} + 21 \beta_{7} - 6 \beta_{8} + 18 \beta_{11} ) q^{81} + ( 296 \beta_{2} - 4 \beta_{5} - 48 \beta_{6} - 48 \beta_{9} ) q^{82} + ( -56 \beta_{4} + 27 \beta_{6} - 27 \beta_{9} + 13 \beta_{10} ) q^{83} + ( 54 + 29 \beta_{3} + 29 \beta_{8} + 46 \beta_{11} ) q^{85} + ( -195 \beta_{1} + 31 \beta_{3} + 5 \beta_{7} - 31 \beta_{8} ) q^{86} + ( -100 \beta_{2} - 112 \beta_{4} - 73 \beta_{5} + 39 \beta_{6} + 15 \beta_{9} - 16 \beta_{10} ) q^{87} + ( 124 + 74 \beta_{3} + 74 \beta_{8} + 21 \beta_{11} ) q^{88} + ( -104 \beta_{4} + 51 \beta_{6} - 51 \beta_{9} + 48 \beta_{10} ) q^{89} + ( -18 \beta_{2} + 144 \beta_{4} - 3 \beta_{5} + 15 \beta_{6} + 105 \beta_{9} - 6 \beta_{10} ) q^{90} + ( -132 \beta_{1} - 6 \beta_{3} + 14 \beta_{7} + 6 \beta_{8} ) q^{92} + ( 276 + 31 \beta_{1} + 14 \beta_{3} + 10 \beta_{7} + 69 \beta_{8} + 47 \beta_{11} ) q^{93} + ( 92 \beta_{2} + 48 \beta_{5} - 50 \beta_{6} - 50 \beta_{9} ) q^{94} + ( 42 \beta_{1} - 3 \beta_{3} + 16 \beta_{7} + 3 \beta_{8} ) q^{95} + ( -228 \beta_{2} + 60 \beta_{4} - 27 \beta_{5} - 87 \beta_{6} - 15 \beta_{9} + 12 \beta_{10} ) q^{96} + ( -156 \beta_{2} - 81 \beta_{5} - 37 \beta_{6} - 37 \beta_{9} ) q^{97} + ( -303 + 51 \beta_{1} - 12 \beta_{3} + 18 \beta_{7} - 93 \beta_{8} - 18 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 28q^{4} + 6q^{9} + O(q^{10})$$ $$12q - 28q^{4} + 6q^{9} + 6q^{15} - 268q^{16} - 132q^{18} - 268q^{22} + 84q^{25} + 1644q^{30} + 852q^{36} - 1528q^{37} + 852q^{39} - 1012q^{43} - 1216q^{46} + 2682q^{51} + 270q^{57} - 5740q^{58} + 1836q^{60} - 548q^{64} - 1584q^{67} + 5424q^{72} + 4296q^{78} - 3348q^{79} - 1674q^{81} + 348q^{85} + 1108q^{88} + 2958q^{93} - 3354q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-661 \nu^{11} - 17114 \nu^{10} - 13815 \nu^{9} + 226448 \nu^{8} + 617943 \nu^{7} + 441628 \nu^{6} - 2790145 \nu^{5} - 21396420 \nu^{4} - 82640331 \nu^{3} + 112284954 \nu^{2} + 410108427 \nu + 575491554$$$$)/ 489398112$$ $$\beta_{2}$$ $$=$$ $$($$$$-857 \nu^{11} + 5426 \nu^{10} - 627 \nu^{9} + 6088 \nu^{8} - 24405 \nu^{7} + 289700 \nu^{6} - 658001 \nu^{5} + 2301324 \nu^{4} - 3432699 \nu^{3} - 5390226 \nu^{2} + 4113747 \nu + 255288510$$$$)/81566352$$ $$\beta_{3}$$ $$=$$ $$($$$$2476 \nu^{11} + 15515 \nu^{10} + 15948 \nu^{9} - 216551 \nu^{8} - 468246 \nu^{7} - 3622009 \nu^{6} + 3357106 \nu^{5} + 18485199 \nu^{4} + 46012374 \nu^{3} - 101793915 \nu^{2} - 260970336 \nu - 1990167813$$$$)/ 163132704$$ $$\beta_{4}$$ $$=$$ $$($$$$-31 \nu^{11} + 220 \nu^{10} + 1107 \nu^{9} + 1790 \nu^{8} - 14415 \nu^{7} - 28370 \nu^{6} - 54343 \nu^{5} + 212958 \nu^{4} + 1014363 \nu^{3} + 1909980 \nu^{2} - 6029559 \nu - 11337408$$$$)/1889568$$ $$\beta_{5}$$ $$=$$ $$($$$$3517 \nu^{11} - 9850 \nu^{10} + 7683 \nu^{9} + 231268 \nu^{8} + 322857 \nu^{7} - 2788588 \nu^{6} + 2276065 \nu^{5} + 386256 \nu^{4} + 27320031 \nu^{3} + 249912378 \nu^{2} + 284920173 \nu - 1904802642$$$$)/ 163132704$$ $$\beta_{6}$$ $$=$$ $$($$$$1523 \nu^{11} - 209 \nu^{10} - 6255 \nu^{9} - 6421 \nu^{8} + 82569 \nu^{7} + 227449 \nu^{6} + 641129 \nu^{5} - 1005399 \nu^{4} - 7836021 \nu^{3} + 1371249 \nu^{2} - 13338513 \nu + 151814979$$$$)/54377568$$ $$\beta_{7}$$ $$=$$ $$($$$$6025 \nu^{11} + 87566 \nu^{10} - 97641 \nu^{9} - 1174508 \nu^{8} - 3032283 \nu^{7} + 8001092 \nu^{6} + 15441661 \nu^{5} + 108193104 \nu^{4} - 106799229 \nu^{3} - 574520526 \nu^{2} - 1942311879 \nu + 4735611702$$$$)/ 163132704$$ $$\beta_{8}$$ $$=$$ $$($$$$23393 \nu^{11} + 32191 \nu^{10} + 346095 \nu^{9} - 558841 \nu^{8} - 77001 \nu^{7} - 6287543 \nu^{6} + 15016199 \nu^{5} + 29481381 \nu^{4} + 265284153 \nu^{3} - 211205151 \nu^{2} + 536512653 \nu - 1125769185$$$$)/ 489398112$$ $$\beta_{9}$$ $$=$$ $$($$$$-25402 \nu^{11} + 2263 \nu^{10} + 146502 \nu^{9} + 719729 \nu^{8} + 11964 \nu^{7} + 585763 \nu^{6} - 31906828 \nu^{5} - 6563745 \nu^{4} + 49790376 \nu^{3} + 444340809 \nu^{2} - 145536102 \nu + 1611033867$$$$)/ 489398112$$ $$\beta_{10}$$ $$=$$ $$($$$$103 \nu^{11} - 112 \nu^{10} + 657 \nu^{9} - 2906 \nu^{8} + 6711 \nu^{7} - 30454 \nu^{6} + 85231 \nu^{5} - 277146 \nu^{4} + 657801 \nu^{3} + 594864 \nu^{2} + 5662143 \nu - 12282192$$$$)/1889568$$ $$\beta_{11}$$ $$=$$ $$($$$$-95 \nu^{11} - 112 \nu^{10} + 207 \nu^{9} + 2026 \nu^{8} - 399 \nu^{7} + 9974 \nu^{6} - 58247 \nu^{5} - 95958 \nu^{4} - 17577 \nu^{3} + 734832 \nu^{2} - 2670327 \nu + 4723920$$$$)/944784$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} - 3 \beta_{6} - \beta_{3} + \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + \beta_{7} + 3 \beta_{5} - \beta_{3} - \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{8} - \beta_{7} - 3 \beta_{3} - 22 \beta_{1} + 21$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{11} - 21 \beta_{10} + 30 \beta_{9} + 19 \beta_{8} + 7 \beta_{7} - 21 \beta_{6} + 27 \beta_{5} - 24 \beta_{4} + 2 \beta_{3} - 48 \beta_{2} - 7 \beta_{1} + 48$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$95 \beta_{11} + 18 \beta_{10} - 171 \beta_{9} + 33 \beta_{8} + 6 \beta_{7} - 42 \beta_{6} + 72 \beta_{5} - 10 \beta_{3} + 216 \beta_{2} + 43 \beta_{1} + 216$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$-6 \beta_{11} + 20 \beta_{8} + 8 \beta_{7} - 140 \beta_{3} - 200 \beta_{1} - 1629$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$535 \beta_{11} - 30 \beta_{10} + 666 \beta_{9} + 406 \beta_{8} + 10 \beta_{7} + 1491 \beta_{6} + 552 \beta_{5} - 912 \beta_{4} + 681 \beta_{3} + 984 \beta_{2} - 1187 \beta_{1} - 984$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$762 \beta_{11} - 2373 \beta_{10} - 3819 \beta_{9} + 1282 \beta_{8} - 791 \beta_{7} + 1506 \beta_{6} - 27 \beta_{5} + 1176 \beta_{4} - 493 \beta_{3} + 3648 \beta_{2} + 599 \beta_{1} + 3648$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$2311 \beta_{11} + 4533 \beta_{8} + 681 \beta_{7} + 487 \beta_{3} + 12470 \beta_{1} - 29619$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$5664 \beta_{11} + 11715 \beta_{10} - 17760 \beta_{9} - 7409 \beta_{8} - 3905 \beta_{7} + 30285 \beta_{6} - 4467 \beta_{5} + 11088 \beta_{4} + 8606 \beta_{3} + 114192 \beta_{2} - 4927 \beta_{1} - 114192$$$$)/6$$ $$\nu^{11}$$ $$=$$ $$($$$$-67241 \beta_{11} - 30912 \beta_{10} + 34443 \beta_{9} + 3799 \beta_{8} - 10304 \beta_{7} + 121440 \beta_{6} - 45840 \beta_{5} + 40080 \beta_{4} - 25200 \beta_{3} - 144912 \beta_{2} - 11081 \beta_{1} - 144912$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 −0.232749 − 2.99096i 2.70662 + 1.29391i −2.59957 + 1.49740i 0.00299931 − 3.00000i −2.23014 + 2.00661i 2.85284 + 0.928053i −2.23014 − 2.00661i 2.85284 − 0.928053i −2.59957 − 1.49740i 0.00299931 + 3.00000i −0.232749 + 2.99096i 2.70662 − 1.29391i
4.54551i −2.93937 + 4.28487i −12.6617 11.6039 19.4769 + 13.3609i 0 21.1897i −9.72022 25.1896i 52.7455i
146.2 4.54551i 2.93937 4.28487i −12.6617 −11.6039 −19.4769 13.3609i 0 21.1897i −9.72022 25.1896i 52.7455i
146.3 2.58741i −2.60257 4.49740i 1.30532 −16.1181 −11.6366 + 6.73392i 0 24.0767i −13.4532 + 23.4096i 41.7042i
146.4 2.58741i 2.60257 + 4.49740i 1.30532 16.1181 11.6366 6.73392i 0 24.0767i −13.4532 + 23.4096i 41.7042i
146.5 1.90883i −5.08298 1.07856i 4.35636 1.24741 −2.05878 + 9.70256i 0 23.5862i 24.6734 + 10.9646i 2.38110i
146.6 1.90883i 5.08298 + 1.07856i 4.35636 −1.24741 2.05878 9.70256i 0 23.5862i 24.6734 + 10.9646i 2.38110i
146.7 1.90883i −5.08298 + 1.07856i 4.35636 1.24741 −2.05878 9.70256i 0 23.5862i 24.6734 10.9646i 2.38110i
146.8 1.90883i 5.08298 1.07856i 4.35636 −1.24741 2.05878 + 9.70256i 0 23.5862i 24.6734 10.9646i 2.38110i
146.9 2.58741i −2.60257 + 4.49740i 1.30532 −16.1181 −11.6366 6.73392i 0 24.0767i −13.4532 23.4096i 41.7042i
146.10 2.58741i 2.60257 4.49740i 1.30532 16.1181 11.6366 + 6.73392i 0 24.0767i −13.4532 23.4096i 41.7042i
146.11 4.54551i −2.93937 4.28487i −12.6617 11.6039 19.4769 13.3609i 0 21.1897i −9.72022 + 25.1896i 52.7455i
146.12 4.54551i 2.93937 + 4.28487i −12.6617 −11.6039 −19.4769 + 13.3609i 0 21.1897i −9.72022 + 25.1896i 52.7455i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 146.12 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.c.a 12
3.b odd 2 1 inner 147.4.c.a 12
7.b odd 2 1 inner 147.4.c.a 12
7.c even 3 1 21.4.g.a 12
7.c even 3 1 147.4.g.d 12
7.d odd 6 1 21.4.g.a 12
7.d odd 6 1 147.4.g.d 12
21.c even 2 1 inner 147.4.c.a 12
21.g even 6 1 21.4.g.a 12
21.g even 6 1 147.4.g.d 12
21.h odd 6 1 21.4.g.a 12
21.h odd 6 1 147.4.g.d 12
28.f even 6 1 336.4.bc.d 12
28.g odd 6 1 336.4.bc.d 12
84.j odd 6 1 336.4.bc.d 12
84.n even 6 1 336.4.bc.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a 12 7.c even 3 1
21.4.g.a 12 7.d odd 6 1
21.4.g.a 12 21.g even 6 1
21.4.g.a 12 21.h odd 6 1
147.4.c.a 12 1.a even 1 1 trivial
147.4.c.a 12 3.b odd 2 1 inner
147.4.c.a 12 7.b odd 2 1 inner
147.4.c.a 12 21.c even 2 1 inner
147.4.g.d 12 7.c even 3 1
147.4.g.d 12 7.d odd 6 1
147.4.g.d 12 21.g even 6 1
147.4.g.d 12 21.h odd 6 1
336.4.bc.d 12 28.f even 6 1
336.4.bc.d 12 28.g odd 6 1
336.4.bc.d 12 84.j odd 6 1
336.4.bc.d 12 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 31 T_{2}^{4} + 238 T_{2}^{2} + 504$$ acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 504 + 238 T^{2} + 31 T^{4} + T^{6} )^{2}$$
$3$ $$387420489 - 1594323 T^{2} + 308367 T^{4} - 30186 T^{6} + 423 T^{8} - 3 T^{10} + T^{12}$$
$5$ $$( -54432 + 35595 T^{2} - 396 T^{4} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$( 673513344 + 3036523 T^{2} + 3244 T^{4} + T^{6} )^{2}$$
$13$ $$( 82121472 + 1731204 T^{2} + 4335 T^{4} + T^{6} )^{2}$$
$17$ $$( -4206722688 + 14585004 T^{2} - 12261 T^{4} + T^{6} )^{2}$$
$19$ $$( 243972972 + 2370609 T^{2} + 2994 T^{4} + T^{6} )^{2}$$
$23$ $$( 94816842624 + 65390668 T^{2} + 14311 T^{4} + T^{6} )^{2}$$
$29$ $$( 14683734245376 + 3697274560 T^{2} + 120001 T^{4} + T^{6} )^{2}$$
$31$ $$( 33414175107 + 323779851 T^{2} + 42033 T^{4} + T^{6} )^{2}$$
$37$ $$( -1849018 + 26747 T + 382 T^{2} + T^{3} )^{4}$$
$41$ $$( -4591113633792 + 4941510336 T^{2} - 172788 T^{4} + T^{6} )^{2}$$
$43$ $$( -6662944 - 23284 T + 253 T^{2} + T^{3} )^{4}$$
$47$ $$( -104940131240448 + 8170201872 T^{2} - 185553 T^{4} + T^{6} )^{2}$$
$53$ $$( 3687131688228576 + 83627725867 T^{2} + 531100 T^{4} + T^{6} )^{2}$$
$59$ $$( -623203616502432 + 71676273123 T^{2} - 570420 T^{4} + T^{6} )^{2}$$
$61$ $$( 6477700166618112 + 150499014480 T^{2} + 867963 T^{4} + T^{6} )^{2}$$
$67$ $$( 98311462 - 536655 T + 396 T^{2} + T^{3} )^{4}$$
$71$ $$( 6720226523136 + 3717765184 T^{2} + 225148 T^{4} + T^{6} )^{2}$$
$73$ $$( 4047431204396592 + 108435730329 T^{2} + 618630 T^{4} + T^{6} )^{2}$$
$79$ $$( -19057853 - 259833 T + 837 T^{2} + T^{3} )^{4}$$
$83$ $$( -388952511994368 + 75760581456 T^{2} - 567987 T^{4} + T^{6} )^{2}$$
$89$ $$( -9409280439379968 + 748631730924 T^{2} - 2594253 T^{4} + T^{6} )^{2}$$
$97$ $$( 9887068459035648 + 312291915984 T^{2} + 2159691 T^{4} + T^{6} )^{2}$$