# Properties

 Label 1176.2.k.a Level 1176 Weight 2 Character orbit 1176.k Analytic conductor 9.390 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1176.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.39040727770$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{8} q^{5} + \beta_{5} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{8} q^{5} + \beta_{5} q^{9} -\beta_{6} q^{11} + ( \beta_{1} - \beta_{7} - \beta_{12} ) q^{13} + ( 1 + \beta_{5} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{15} + ( -\beta_{2} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{17} -\beta_{12} q^{19} + ( \beta_{11} - \beta_{13} + \beta_{14} ) q^{23} + ( 3 + \beta_{3} + \beta_{5} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{25} + ( -\beta_{4} + \beta_{7} + \beta_{8} - \beta_{15} ) q^{27} + ( -\beta_{6} - \beta_{11} + \beta_{13} ) q^{29} + ( -\beta_{1} + 2 \beta_{4} - \beta_{12} - \beta_{15} ) q^{31} + ( -\beta_{1} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{33} + ( -1 - \beta_{3} - 2 \beta_{5} + 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{37} + ( 3 + \beta_{3} - \beta_{5} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{7} ) q^{41} + ( 1 + \beta_{3} - \beta_{5} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{4} - 3 \beta_{7} - \beta_{9} - \beta_{12} - 2 \beta_{15} ) q^{45} + ( \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{8} + \beta_{9} ) q^{47} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{13} - 2 \beta_{14} ) q^{51} + ( -\beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{53} + ( -2 \beta_{1} + 2 \beta_{4} + 3 \beta_{7} + \beta_{15} ) q^{55} + ( \beta_{3} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{57} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{8} - \beta_{12} - \beta_{15} ) q^{59} + ( \beta_{4} - \beta_{7} + \beta_{12} - \beta_{15} ) q^{61} + ( 2 \beta_{6} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{65} + ( -1 + \beta_{3} + \beta_{5} - \beta_{10} ) q^{67} + ( -\beta_{2} + \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{12} ) q^{69} + ( -2 \beta_{5} - 2 \beta_{10} - \beta_{14} ) q^{71} + ( -\beta_{1} - \beta_{4} - 2 \beta_{12} - \beta_{15} ) q^{73} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{4} + 2 \beta_{8} - \beta_{9} + \beta_{12} ) q^{75} + ( -3 + \beta_{5} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{79} + ( -4 - 2 \beta_{6} + \beta_{10} + 2 \beta_{11} ) q^{81} + ( 2 \beta_{1} + \beta_{4} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{12} - \beta_{15} ) q^{83} + ( 2 + 2 \beta_{3} + \beta_{5} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} ) q^{85} + ( -\beta_{1} - 2 \beta_{4} - 2 \beta_{9} - 2 \beta_{12} - \beta_{15} ) q^{87} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{7} - \beta_{12} - \beta_{15} ) q^{89} + ( -2 + \beta_{3} - \beta_{6} + \beta_{10} - 2 \beta_{13} - \beta_{14} ) q^{93} + ( -\beta_{11} + \beta_{13} + \beta_{14} ) q^{95} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{12} - \beta_{15} ) q^{97} + ( 1 + \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{10} + \beta_{13} - \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{9} + O(q^{10})$$ $$16q - 4q^{9} + 8q^{15} + 36q^{25} + 4q^{37} + 44q^{39} + 20q^{43} - 12q^{51} - 8q^{57} - 28q^{67} - 56q^{79} - 60q^{81} + 16q^{85} - 32q^{93} + 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} - 13122 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$11 \nu^{15} + 30 \nu^{14} - 142 \nu^{13} + 363 \nu^{12} - 662 \nu^{11} + 258 \nu^{10} + 1288 \nu^{9} - 3546 \nu^{8} + 8138 \nu^{7} - 7392 \nu^{6} - 846 \nu^{5} + 28890 \nu^{4} - 41067 \nu^{3} + 38880 \nu^{2} - 34992 \nu - 19683$$$$)/69984$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} + 246 \nu^{14} - 962 \nu^{13} + 1965 \nu^{12} - 2374 \nu^{11} - 462 \nu^{10} + 7988 \nu^{9} - 24594 \nu^{8} + 38170 \nu^{7} - 30276 \nu^{6} - 19998 \nu^{5} + 114858 \nu^{4} - 199989 \nu^{3} + 244944 \nu^{2} - 195372 \nu + 150903$$$$)/69984$$ $$\beta_{3}$$ $$=$$ $$($$$$35 \nu^{15} - 162 \nu^{14} + 512 \nu^{13} - 963 \nu^{12} + 1042 \nu^{11} + 306 \nu^{10} - 5162 \nu^{9} + 13134 \nu^{8} - 21322 \nu^{7} + 14538 \nu^{6} + 16074 \nu^{5} - 72198 \nu^{4} + 144099 \nu^{3} - 184680 \nu^{2} + 173502 \nu - 54675$$$$)/34992$$ $$\beta_{4}$$ $$=$$ $$($$$$-22 \nu^{15} + 75 \nu^{14} - 238 \nu^{13} + 489 \nu^{12} - 737 \nu^{11} + 429 \nu^{10} + 1483 \nu^{9} - 5787 \nu^{8} + 11651 \nu^{7} - 14727 \nu^{6} + 5787 \nu^{5} + 23193 \nu^{4} - 70227 \nu^{3} + 126846 \nu^{2} - 147987 \nu + 113724$$$$)/17496$$ $$\beta_{5}$$ $$=$$ $$($$$$61 \nu^{15} - 234 \nu^{14} + 565 \nu^{13} - 729 \nu^{12} + 563 \nu^{11} + 1251 \nu^{10} - 5437 \nu^{9} + 10815 \nu^{8} - 13421 \nu^{7} + 6351 \nu^{6} + 19359 \nu^{5} - 58185 \nu^{4} + 98010 \nu^{3} - 136323 \nu^{2} + 129762 \nu - 113724$$$$)/34992$$ $$\beta_{6}$$ $$=$$ $$($$$$-131 \nu^{15} + 660 \nu^{14} - 1742 \nu^{13} + 3567 \nu^{12} - 5176 \nu^{11} + 2040 \nu^{10} + 12350 \nu^{9} - 42648 \nu^{8} + 81940 \nu^{7} - 96318 \nu^{6} + 29736 \nu^{5} + 167832 \nu^{4} - 496287 \nu^{3} + 862164 \nu^{2} - 1106622 \nu + 754515$$$$)/69984$$ $$\beta_{7}$$ $$=$$ $$($$$$-19 \nu^{15} + 82 \nu^{14} - 226 \nu^{13} + 433 \nu^{12} - 542 \nu^{11} + 74 \nu^{10} + 1912 \nu^{9} - 5482 \nu^{8} + 9482 \nu^{7} - 10040 \nu^{6} + 42 \nu^{5} + 24354 \nu^{4} - 60237 \nu^{3} + 95580 \nu^{2} - 105948 \nu + 75087$$$$)/7776$$ $$\beta_{8}$$ $$=$$ $$($$$$61 \nu^{15} - 302 \nu^{14} + 784 \nu^{13} - 1463 \nu^{12} + 1772 \nu^{11} - 100 \nu^{10} - 6358 \nu^{9} + 18728 \nu^{8} - 32084 \nu^{7} + 33490 \nu^{6} + 1116 \nu^{5} - 85716 \nu^{4} + 197181 \nu^{3} - 331290 \nu^{2} + 374220 \nu - 255879$$$$)/23328$$ $$\beta_{9}$$ $$=$$ $$($$$$-37 \nu^{15} + 178 \nu^{14} - 562 \nu^{13} + 1105 \nu^{12} - 1544 \nu^{11} + 572 \nu^{10} + 4048 \nu^{9} - 13408 \nu^{8} + 25460 \nu^{7} - 30032 \nu^{6} + 8184 \nu^{5} + 55692 \nu^{4} - 157923 \nu^{3} + 266814 \nu^{2} - 310554 \nu + 244215$$$$)/11664$$ $$\beta_{10}$$ $$=$$ $$($$$$2 \nu^{15} - 8 \nu^{14} + 21 \nu^{13} - 38 \nu^{12} + 41 \nu^{11} + 17 \nu^{10} - 195 \nu^{9} + 497 \nu^{8} - 779 \nu^{7} + 661 \nu^{6} + 349 \nu^{5} - 2499 \nu^{4} + 5247 \nu^{3} - 7803 \nu^{2} + 7452 \nu - 4617$$$$)/432$$ $$\beta_{11}$$ $$=$$ $$($$$$107 \nu^{15} - 468 \nu^{14} + 1268 \nu^{13} - 2511 \nu^{12} + 3166 \nu^{11} - 90 \nu^{10} - 11048 \nu^{9} + 32070 \nu^{8} - 54982 \nu^{7} + 57756 \nu^{6} + 5526 \nu^{5} - 151794 \nu^{4} + 356805 \nu^{3} - 554526 \nu^{2} + 581742 \nu - 365229$$$$)/23328$$ $$\beta_{12}$$ $$=$$ $$($$$$-30 \nu^{15} + 137 \nu^{14} - 378 \nu^{13} + 731 \nu^{12} - 939 \nu^{11} + 175 \nu^{10} + 2997 \nu^{9} - 9161 \nu^{8} + 16161 \nu^{7} - 17341 \nu^{6} + 681 \nu^{5} + 41427 \nu^{4} - 104409 \nu^{3} + 162810 \nu^{2} - 181521 \nu + 125388$$$$)/5832$$ $$\beta_{13}$$ $$=$$ $$($$$$-125 \nu^{15} + 512 \nu^{14} - 1364 \nu^{13} + 2537 \nu^{12} - 3082 \nu^{11} + 142 \nu^{10} + 11108 \nu^{9} - 31994 \nu^{8} + 56962 \nu^{7} - 59656 \nu^{6} - 786 \nu^{5} + 141174 \nu^{4} - 357399 \nu^{3} + 579150 \nu^{2} - 650754 \nu + 465831$$$$)/23328$$ $$\beta_{14}$$ $$=$$ $$($$$$-45 \nu^{15} + 188 \nu^{14} - 498 \nu^{13} + 917 \nu^{12} - 1020 \nu^{11} - 236 \nu^{10} + 4386 \nu^{9} - 11588 \nu^{8} + 19032 \nu^{7} - 18226 \nu^{6} - 5868 \nu^{5} + 55620 \nu^{4} - 124497 \nu^{3} + 190512 \nu^{2} - 202662 \nu + 140697$$$$)/5832$$ $$\beta_{15}$$ $$=$$ $$($$$$243 \nu^{15} - 1090 \nu^{14} + 3042 \nu^{13} - 5725 \nu^{12} + 7074 \nu^{11} - 326 \nu^{10} - 24984 \nu^{9} + 72382 \nu^{8} - 125070 \nu^{7} + 129248 \nu^{6} + 3594 \nu^{5} - 332046 \nu^{4} + 796365 \nu^{3} - 1239624 \nu^{2} + 1397736 \nu - 993627$$$$)/23328$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{10} + \beta_{9} + \beta_{7} - 2 \beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - \beta_{10} + \beta_{9} - \beta_{8}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{12} - \beta_{9} - 4 \beta_{8} + 3 \beta_{7} + 5 \beta_{4} + \beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{13} - \beta_{12} - \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{2} + 4 \beta_{1} + 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-16 \beta_{15} - 15 \beta_{14} - 8 \beta_{12} + 10 \beta_{11} - 24 \beta_{10} + \beta_{9} + \beta_{7} - 6 \beta_{6} + 18 \beta_{5} - 15 \beta_{4} + 18 \beta_{3} - 7 \beta_{2} + 8 \beta_{1} - 12$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{14} + 4 \beta_{11} - 6 \beta_{10} + 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 7$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{15} - 35 \beta_{14} + 94 \beta_{13} - 26 \beta_{12} + 16 \beta_{11} - 22 \beta_{10} - 13 \beta_{9} - 29 \beta_{7} + 2 \beta_{6} + 24 \beta_{5} - 11 \beta_{4} - 26 \beta_{3} + 11 \beta_{2} + 70 \beta_{1} + 92$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$11 \beta_{15} + 16 \beta_{14} + 12 \beta_{13} + 2 \beta_{12} + 14 \beta_{11} - 5 \beta_{10} + 15 \beta_{9} + 5 \beta_{8} - 12 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} - 12 \beta_{4} + 4 \beta_{3} - 14 \beta_{2} + 70 \beta_{1} + 4$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$16 \beta_{15} - 68 \beta_{12} + 37 \beta_{9} + 20 \beta_{8} + 241 \beta_{7} - 97 \beta_{4} - 37 \beta_{2} + 102 \beta_{1}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$40 \beta_{15} + 16 \beta_{14} - 32 \beta_{13} + 57 \beta_{12} + 60 \beta_{11} + 24 \beta_{10} - 23 \beta_{9} - 49 \beta_{8} + 183 \beta_{7} - 28 \beta_{6} - 3 \beta_{5} + 51 \beta_{4} - 44 \beta_{3} - 12 \beta_{2} + 36 \beta_{1} + 32$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-176 \beta_{15} + 163 \beta_{14} + 112 \beta_{13} + 80 \beta_{12} + 422 \beta_{11} + 80 \beta_{10} + 3 \beta_{9} - 512 \beta_{8} - 93 \beta_{7} - 682 \beta_{6} + 774 \beta_{5} - 349 \beta_{4} - 74 \beta_{3} - 173 \beta_{2} - 448 \beta_{1} - 436$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$148 \beta_{14} - 76 \beta_{13} + 140 \beta_{11} - 76 \beta_{10} + 24 \beta_{6} + 244 \beta_{5} + 64 \beta_{3} - 39$$ $$\nu^{13}$$ $$=$$ $$($$$$-174 \beta_{15} - 201 \beta_{14} + 114 \beta_{13} - 942 \beta_{12} - 848 \beta_{11} - 594 \beta_{10} - 855 \beta_{9} + 1001 \beta_{7} + 1182 \beta_{6} + 1776 \beta_{5} - 1209 \beta_{4} + 66 \beta_{3} - 327 \beta_{2} - 414 \beta_{1} + 5604$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$405 \beta_{15} - 176 \beta_{14} + 420 \beta_{13} + 764 \beta_{12} + 32 \beta_{11} - 141 \beta_{10} - 191 \beta_{9} - 377 \beta_{8} - 48 \beta_{7} + 196 \beta_{6} - 40 \beta_{5} - 984 \beta_{4} + 460 \beta_{3} - 20 \beta_{2} + 712 \beta_{1} + 1120$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$1456 \beta_{15} + 1548 \beta_{12} + 2919 \beta_{9} - 2276 \beta_{8} - 1301 \beta_{7} - 6899 \beta_{4} + 153 \beta_{2} + 962 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$$.

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 1.73018 + 0.0805675i 1.73018 − 0.0805675i −0.601642 + 1.62420i −0.601642 − 1.62420i 0.247636 − 1.71426i 0.247636 + 1.71426i 1.22961 − 1.21986i 1.22961 + 1.21986i −0.441628 + 1.67480i −0.441628 − 1.67480i 1.60841 + 0.642670i 1.60841 − 0.642670i −1.70742 − 0.291063i −1.70742 + 0.291063i 0.934861 + 1.45809i 0.934861 − 1.45809i
0 −1.53866 0.795315i 0 −3.80034 0 0 0 1.73495 + 2.44744i 0
881.2 0 −1.53866 + 0.795315i 0 −3.80034 0 0 0 1.73495 2.44744i 0
881.3 0 −1.33314 1.10578i 0 −0.145339 0 0 0 0.554510 + 2.94831i 0
881.4 0 −1.33314 + 1.10578i 0 −0.145339 0 0 0 0.554510 2.94831i 0
881.5 0 −1.07159 1.36077i 0 2.57910 0 0 0 −0.703402 + 2.91637i 0
881.6 0 −1.07159 + 1.36077i 0 2.57910 0 0 0 −0.703402 2.91637i 0
881.7 0 −0.454941 1.67124i 0 2.80795 0 0 0 −2.58606 + 1.52063i 0
881.8 0 −0.454941 + 1.67124i 0 2.80795 0 0 0 −2.58606 1.52063i 0
881.9 0 0.454941 1.67124i 0 −2.80795 0 0 0 −2.58606 1.52063i 0
881.10 0 0.454941 + 1.67124i 0 −2.80795 0 0 0 −2.58606 + 1.52063i 0
881.11 0 1.07159 1.36077i 0 −2.57910 0 0 0 −0.703402 2.91637i 0
881.12 0 1.07159 + 1.36077i 0 −2.57910 0 0 0 −0.703402 + 2.91637i 0
881.13 0 1.33314 1.10578i 0 0.145339 0 0 0 0.554510 2.94831i 0
881.14 0 1.33314 + 1.10578i 0 0.145339 0 0 0 0.554510 + 2.94831i 0
881.15 0 1.53866 0.795315i 0 3.80034 0 0 0 1.73495 2.44744i 0
881.16 0 1.53866 + 0.795315i 0 3.80034 0 0 0 1.73495 + 2.44744i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.16 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
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 1176.2.k.a 16
3.b odd 2 1 inner 1176.2.k.a 16
4.b odd 2 1 2352.2.k.i 16
7.b odd 2 1 inner 1176.2.k.a 16
7.c even 3 1 168.2.u.a 16
7.c even 3 1 1176.2.u.b 16
7.d odd 6 1 168.2.u.a 16
7.d odd 6 1 1176.2.u.b 16
12.b even 2 1 2352.2.k.i 16
21.c even 2 1 inner 1176.2.k.a 16
21.g even 6 1 168.2.u.a 16
21.g even 6 1 1176.2.u.b 16
21.h odd 6 1 168.2.u.a 16
21.h odd 6 1 1176.2.u.b 16
28.d even 2 1 2352.2.k.i 16
28.f even 6 1 336.2.bc.f 16
28.g odd 6 1 336.2.bc.f 16
84.h odd 2 1 2352.2.k.i 16
84.j odd 6 1 336.2.bc.f 16
84.n even 6 1 336.2.bc.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.u.a 16 7.c even 3 1
168.2.u.a 16 7.d odd 6 1
168.2.u.a 16 21.g even 6 1
168.2.u.a 16 21.h odd 6 1
336.2.bc.f 16 28.f even 6 1
336.2.bc.f 16 28.g odd 6 1
336.2.bc.f 16 84.j odd 6 1
336.2.bc.f 16 84.n even 6 1
1176.2.k.a 16 1.a even 1 1 trivial
1176.2.k.a 16 3.b odd 2 1 inner
1176.2.k.a 16 7.b odd 2 1 inner
1176.2.k.a 16 21.c even 2 1 inner
1176.2.u.b 16 7.c even 3 1
1176.2.u.b 16 7.d odd 6 1
1176.2.u.b 16 21.g even 6 1
1176.2.u.b 16 21.h odd 6 1
2352.2.k.i 16 4.b odd 2 1
2352.2.k.i 16 12.b even 2 1
2352.2.k.i 16 28.d even 2 1
2352.2.k.i 16 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 29 T_{5}^{6} + 263 T_{5}^{4} - 763 T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(1176, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 2 T^{2} + 17 T^{4} + 46 T^{6} + 172 T^{8} + 414 T^{10} + 1377 T^{12} + 1458 T^{14} + 6561 T^{16}$$
$5$ $$( 1 + 11 T^{2} + 93 T^{4} + 622 T^{6} + 3086 T^{8} + 15550 T^{10} + 58125 T^{12} + 171875 T^{14} + 390625 T^{16} )^{2}$$
$7$ 1
$11$ $$( 1 - 49 T^{2} + 1101 T^{4} - 15842 T^{6} + 183302 T^{8} - 1916882 T^{10} + 16119741 T^{12} - 86806489 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 3970655 T^{10} + 36500958 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 42 T^{2} + 705 T^{4} + 11358 T^{6} + 224396 T^{8} + 3282462 T^{10} + 58882305 T^{12} + 1013777898 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 115 T^{2} + 6345 T^{4} - 216074 T^{6} + 4950494 T^{8} - 78002714 T^{10} + 826886745 T^{12} - 5410276315 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 102 T^{2} + 4081 T^{4} - 83682 T^{6} + 1454124 T^{8} - 44267778 T^{10} + 1142031121 T^{12} - 15099660678 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 247227929 T^{10} + 4672298286 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 124 T^{2} + 8778 T^{4} - 418928 T^{6} + 15011795 T^{8} - 402589808 T^{10} + 8106667338 T^{12} - 110050456444 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - T + 61 T^{2} + 314 T^{3} + 1294 T^{4} + 11618 T^{5} + 83509 T^{6} - 50653 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 3422058768 T^{10} + 79110004956 T^{12} + 1140025017840 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 5 T + 76 T^{2} - 341 T^{3} + 2710 T^{4} - 14663 T^{5} + 140524 T^{6} - 397535 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 158 T^{2} + 11793 T^{4} + 733630 T^{6} + 39908708 T^{8} + 1620588670 T^{10} + 57546078033 T^{12} + 1703116021982 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 265 T^{2} + 37257 T^{4} - 3352538 T^{6} + 211268186 T^{8} - 9417279242 T^{10} + 293975650617 T^{12} - 5873555699185 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 187 T^{2} + 23065 T^{4} + 2022838 T^{6} + 135204754 T^{8} + 7041499078 T^{10} + 279486931465 T^{12} + 7887759790867 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 338 T^{2} + 55633 T^{4} - 5801762 T^{6} + 419624596 T^{8} - 21588356402 T^{10} + 770285672353 T^{12} - 17413886534018 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 7 T + 249 T^{2} + 1304 T^{3} + 24614 T^{4} + 87368 T^{5} + 1117761 T^{6} + 2105341 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 4982766368 T^{10} + 539845751164 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 399 T^{2} + 70689 T^{4} - 7728594 T^{6} + 628610462 T^{8} - 41185677426 T^{10} + 2007443258049 T^{12} - 60382356289311 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 14 T + 348 T^{2} + 3160 T^{3} + 42101 T^{4} + 249640 T^{5} + 2171868 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 12391788531 T^{10} + 1057276475238 T^{12} + 46098592645029 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 378 T^{2} + 73089 T^{4} + 9573054 T^{6} + 954583820 T^{8} + 75828160734 T^{10} + 4585767652449 T^{12} + 187858927983258 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 107733228771 T^{10} + 7724888001498 T^{12} - 357344990114541 T^{14} + 7837433594376961 T^{16} )^{2}$$
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