# Properties

 Label 448.3.s.g Level 448 Weight 3 Character orbit 448.s Analytic conductor 12.207 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 448.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2071158433$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{20}$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -\beta_{10} q^{5} + \beta_{13} q^{7} + ( 1 + \beta_{1} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -\beta_{10} q^{5} + \beta_{13} q^{7} + ( 1 + \beta_{1} + \beta_{8} ) q^{9} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{12} ) q^{11} + ( 1 + 2 \beta_{1} - \beta_{8} - \beta_{15} ) q^{13} + ( \beta_{4} - \beta_{5} ) q^{15} + ( 4 + 2 \beta_{1} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{14} ) q^{17} + ( 5 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{19} + ( -5 - 3 \beta_{1} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{14} - \beta_{15} ) q^{21} + ( -3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{11} - \beta_{12} - 4 \beta_{13} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{14} ) q^{25} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{27} + ( -7 + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{14} + \beta_{15} ) q^{29} + ( -2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{11} - \beta_{12} - \beta_{13} ) q^{31} + ( 5 - 5 \beta_{1} ) q^{33} + ( 8 \beta_{2} - 3 \beta_{3} - \beta_{4} - 5 \beta_{5} - \beta_{6} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{35} + ( -1 - \beta_{1} - \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{14} + \beta_{15} ) q^{37} + ( 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} + 6 \beta_{6} + 3 \beta_{12} + 4 \beta_{13} ) q^{39} + ( 7 + 14 \beta_{1} - \beta_{8} - 3 \beta_{9} + 2 \beta_{14} + 2 \beta_{15} ) q^{41} + ( -4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{12} + 2 \beta_{13} ) q^{43} + ( 6 + 3 \beta_{1} - \beta_{7} + \beta_{8} + 6 \beta_{10} - 6 \beta_{14} ) q^{45} + ( 10 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} ) q^{47} + ( -16 - 16 \beta_{1} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 6 \beta_{14} + 2 \beta_{15} ) q^{49} + ( -18 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{51} + ( -3 \beta_{1} + \beta_{7} + \beta_{8} + 5 \beta_{10} + 5 \beta_{14} + 2 \beta_{15} ) q^{53} + ( 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 8 \beta_{6} + 4 \beta_{11} + \beta_{12} + \beta_{13} ) q^{55} + ( -33 - 4 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} - 8 \beta_{10} + 4 \beta_{14} - 2 \beta_{15} ) q^{57} + ( \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 12 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{59} + ( 15 - 15 \beta_{1} + \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} + \beta_{15} ) q^{61} + ( 16 \beta_{2} - 6 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 10 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{63} + ( -1 - \beta_{1} + 2 \beta_{7} - 5 \beta_{8} - 4 \beta_{10} + 8 \beta_{14} - 2 \beta_{15} ) q^{65} + ( 7 \beta_{2} - 14 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 12 \beta_{6} + 2 \beta_{12} + 4 \beta_{13} ) q^{67} + ( 18 + 36 \beta_{1} + 4 \beta_{8} + 4 \beta_{9} + 3 \beta_{14} ) q^{69} + ( -8 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{11} ) q^{71} + ( 6 + 3 \beta_{1} + 2 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} - 6 \beta_{14} ) q^{73} + ( 12 \beta_{3} - 6 \beta_{4} + 2 \beta_{6} + 2 \beta_{11} - 6 \beta_{12} ) q^{75} + ( -17 - 34 \beta_{1} + 3 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} - 9 \beta_{14} + \beta_{15} ) q^{77} + ( -36 \beta_{2} + 18 \beta_{3} + 7 \beta_{4} + 8 \beta_{5} + \beta_{12} + 8 \beta_{13} ) q^{79} + ( -4 \beta_{1} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 4 \beta_{15} ) q^{81} + ( 10 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 16 \beta_{6} - 8 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} ) q^{83} + ( -45 - 2 \beta_{7} + 5 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - \beta_{14} - \beta_{15} ) q^{85} + ( 2 \beta_{2} - 5 \beta_{4} + \beta_{5} + 8 \beta_{6} - 16 \beta_{11} + 5 \beta_{12} - 4 \beta_{13} ) q^{87} + ( 17 - 17 \beta_{1} - 2 \beta_{7} - 10 \beta_{10} - 2 \beta_{15} ) q^{89} + ( 6 \beta_{2} + 10 \beta_{3} + 4 \beta_{4} + 12 \beta_{5} + 8 \beta_{6} - 19 \beta_{11} + \beta_{12} + 7 \beta_{13} ) q^{91} + ( 29 + 29 \beta_{1} + \beta_{7} + 6 \beta_{8} - 3 \beta_{10} + 6 \beta_{14} - \beta_{15} ) q^{93} + ( 14 \beta_{2} - 28 \beta_{3} - 14 \beta_{4} - 3 \beta_{5} + 14 \beta_{6} - 14 \beta_{12} - 11 \beta_{13} ) q^{95} + ( 31 + 62 \beta_{1} + 3 \beta_{8} + 3 \beta_{9} - 6 \beta_{14} ) q^{97} + ( 4 \beta_{2} + 4 \beta_{3} + 9 \beta_{11} + 9 \beta_{12} - 9 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{9} + O(q^{10})$$ $$16q + 8q^{9} + 48q^{17} - 56q^{21} + 16q^{25} - 112q^{29} + 120q^{33} - 8q^{37} + 72q^{45} - 128q^{49} + 24q^{53} - 528q^{57} + 360q^{61} - 8q^{65} + 72q^{73} + 32q^{81} - 720q^{85} + 408q^{89} + 232q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 36 x^{14} + 522 x^{12} + 3644 x^{10} + 12219 x^{8} + 15156 x^{6} + 15478 x^{4} - 10992 x^{2} + 11025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{15} - 158 \nu^{13} - 2599 \nu^{11} - 22087 \nu^{9} - 102335 \nu^{7} - 245347 \nu^{5} - 300255 \nu^{3} - 71343 \nu - 128520$$$$)/257040$$ $$\beta_{2}$$ $$=$$ $$($$$$1349 \nu^{15} + 1130 \nu^{14} + 51424 \nu^{13} + 38180 \nu^{12} + 794798 \nu^{11} + 510750 \nu^{10} + 6024806 \nu^{9} + 3135050 \nu^{8} + 22337746 \nu^{7} + 8439230 \nu^{6} + 30903344 \nu^{5} + 4733190 \nu^{4} + 1517787 \nu^{3} + 22925640 \nu^{2} - 17661258 \nu - 5574870$$$$)/54235440$$ $$\beta_{3}$$ $$=$$ $$($$$$1349 \nu^{15} - 1130 \nu^{14} + 51424 \nu^{13} - 38180 \nu^{12} + 794798 \nu^{11} - 510750 \nu^{10} + 6024806 \nu^{9} - 3135050 \nu^{8} + 22337746 \nu^{7} - 8439230 \nu^{6} + 30903344 \nu^{5} - 4733190 \nu^{4} + 1517787 \nu^{3} - 22925640 \nu^{2} - 17661258 \nu + 5574870$$$$)/54235440$$ $$\beta_{4}$$ $$=$$ $$($$$$-3289 \nu^{15} - 17168 \nu^{14} - 108644 \nu^{13} - 624062 \nu^{12} - 1373628 \nu^{11} - 9211662 \nu^{10} - 7277106 \nu^{9} - 66703628 \nu^{8} - 11587916 \nu^{7} - 242434946 \nu^{6} + 12082056 \nu^{5} - 379277136 \nu^{4} - 90706517 \nu^{3} - 428795322 \nu^{2} + 113485638 \nu + 48849570$$$$)/54235440$$ $$\beta_{5}$$ $$=$$ $$($$$$3289 \nu^{15} - 17168 \nu^{14} + 108644 \nu^{13} - 624062 \nu^{12} + 1373628 \nu^{11} - 9211662 \nu^{10} + 7277106 \nu^{9} - 66703628 \nu^{8} + 11587916 \nu^{7} - 242434946 \nu^{6} - 12082056 \nu^{5} - 379277136 \nu^{4} + 90706517 \nu^{3} - 428795322 \nu^{2} - 113485638 \nu + 48849570$$$$)/54235440$$ $$\beta_{6}$$ $$=$$ $$($$$$1954 \nu^{15} - 2193 \nu^{14} + 69214 \nu^{13} - 84762 \nu^{12} + 981808 \nu^{11} - 1343187 \nu^{10} + 6609626 \nu^{9} - 10685163 \nu^{8} + 20740876 \nu^{7} - 43930431 \nu^{6} + 21175594 \nu^{5} - 82671561 \nu^{4} + 25510822 \nu^{3} - 64871592 \nu^{2} - 44404008 \nu + 2607885$$$$)/13558860$$ $$\beta_{7}$$ $$=$$ $$($$$$-1954 \nu^{15} - 4845 \nu^{14} - 69214 \nu^{13} - 169060 \nu^{12} - 981808 \nu^{11} - 2337205 \nu^{10} - 6609626 \nu^{9} - 15009585 \nu^{8} - 20740876 \nu^{7} - 42461905 \nu^{6} - 21175594 \nu^{5} - 22642375 \nu^{4} - 25510822 \nu^{3} + 20402430 \nu^{2} - 9831432 \nu + 143606925$$$$)/13558860$$ $$\beta_{8}$$ $$=$$ $$($$$$10806 \nu^{15} + 2260 \nu^{14} + 412836 \nu^{13} + 76360 \nu^{12} + 6515547 \nu^{11} + 1021500 \nu^{10} + 52453299 \nu^{9} + 6270100 \nu^{8} + 227611779 \nu^{7} + 16878460 \nu^{6} + 504688011 \nu^{5} + 9466380 \nu^{4} + 615809913 \nu^{3} - 8384160 \nu^{2} + 152192673 \nu - 255209220$$$$)/54235440$$ $$\beta_{9}$$ $$=$$ $$($$$$10806 \nu^{15} - 2260 \nu^{14} + 412836 \nu^{13} - 76360 \nu^{12} + 6515547 \nu^{11} - 1021500 \nu^{10} + 52453299 \nu^{9} - 6270100 \nu^{8} + 227611779 \nu^{7} - 16878460 \nu^{6} + 504688011 \nu^{5} - 9466380 \nu^{4} + 615809913 \nu^{3} + 8384160 \nu^{2} + 152192673 \nu + 255209220$$$$)/54235440$$ $$\beta_{10}$$ $$=$$ $$($$$$-9238 \nu^{15} + 14310 \nu^{14} - 334664 \nu^{13} + 512910 \nu^{12} - 4891055 \nu^{11} + 7349880 \nu^{10} - 34544651 \nu^{9} + 49590750 \nu^{8} - 118066943 \nu^{7} + 147716820 \nu^{6} - 156430715 \nu^{5} + 74664030 \nu^{4} - 191515861 \nu^{3} - 68545230 \nu^{2} - 61542453 \nu - 173692260$$$$)/54235440$$ $$\beta_{11}$$ $$=$$ $$($$$$977 \nu^{15} + 34607 \nu^{13} + 490904 \nu^{11} + 3304813 \nu^{9} + 10370438 \nu^{7} + 10587797 \nu^{5} + 12755411 \nu^{3} - 22202004 \nu$$$$)/3389715$$ $$\beta_{12}$$ $$=$$ $$($$$$7670 \nu^{15} + 3982 \nu^{14} + 279280 \nu^{13} + 151848 \nu^{12} + 4108875 \nu^{11} + 2400458 \nu^{10} + 29286930 \nu^{9} + 19351042 \nu^{8} + 100640065 \nu^{7} + 82574874 \nu^{6} + 122637480 \nu^{5} + 161166614 \nu^{4} + 60034105 \nu^{3} + 117185088 \nu^{2} - 141415260 \nu - 2544570$$$$)/27117720$$ $$\beta_{13}$$ $$=$$ $$($$$$-7670 \nu^{15} + 3982 \nu^{14} - 279280 \nu^{13} + 151848 \nu^{12} - 4108875 \nu^{11} + 2400458 \nu^{10} - 29286930 \nu^{9} + 19351042 \nu^{8} - 100640065 \nu^{7} + 82574874 \nu^{6} - 122637480 \nu^{5} + 161166614 \nu^{4} - 60034105 \nu^{3} + 117185088 \nu^{2} + 141415260 \nu - 2544570$$$$)/27117720$$ $$\beta_{14}$$ $$=$$ $$($$$$-9238 \nu^{15} - 334664 \nu^{13} - 4891055 \nu^{11} - 34544651 \nu^{9} - 118066943 \nu^{7} - 156430715 \nu^{5} - 191515861 \nu^{3} - 61542453 \nu$$$$)/27117720$$ $$\beta_{15}$$ $$=$$ $$($$$$26438 \nu^{15} - 2260 \nu^{14} + 966548 \nu^{13} - 76360 \nu^{12} + 14370011 \nu^{11} - 1021500 \nu^{10} + 105330307 \nu^{9} - 6270100 \nu^{8} + 393538787 \nu^{7} - 16878460 \nu^{6} + 674092763 \nu^{5} - 9466380 \nu^{4} + 819896489 \nu^{3} + 8384160 \nu^{2} + 230844129 \nu + 255209220$$$$)/54235440$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{11} - \beta_{9}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} - \beta_{8} - 2 \beta_{3} + 2 \beta_{2} - 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{15} - 6 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} + 13 \beta_{11} + 9 \beta_{9} + 2 \beta_{8} - 6 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 48 \beta_{1} + 24$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 10 \beta_{9} + 9 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 32 \beta_{3} - 32 \beta_{2} + 63$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$20 \beta_{15} + 15 \beta_{14} - 45 \beta_{13} + 45 \beta_{12} - 92 \beta_{11} - 33 \beta_{9} - 13 \beta_{8} + 45 \beta_{5} - 45 \beta_{4} - 85 \beta_{3} - 85 \beta_{2} - 280 \beta_{1} - 140$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-11 \beta_{15} + 6 \beta_{14} + 13 \beta_{13} + 13 \beta_{12} + 52 \beta_{11} - 12 \beta_{10} + 64 \beta_{9} - 53 \beta_{8} - 22 \beta_{7} - 104 \beta_{6} + 62 \beta_{5} + 62 \beta_{4} - 440 \beta_{3} + 440 \beta_{2} - 303$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-49 \beta_{15} + 1176 \beta_{13} - 1176 \beta_{12} + 2239 \beta_{11} + 201 \beta_{9} + 152 \beta_{8} - 1008 \beta_{5} + 1008 \beta_{4} + 2800 \beta_{3} + 2800 \beta_{2} + 2800 \beta_{1} + 1400$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$21 \beta_{15} - 36 \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 816 \beta_{11} + 72 \beta_{10} + 104 \beta_{9} - 125 \beta_{8} + 42 \beta_{7} + 1632 \beta_{6} - 768 \beta_{5} - 768 \beta_{4} + 5264 \beta_{3} - 5264 \beta_{2} - 1401$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-3575 \beta_{15} - 2730 \beta_{14} - 13362 \beta_{13} + 13362 \beta_{12} - 24133 \beta_{11} + 5061 \beta_{9} + 1486 \beta_{8} + 10050 \beta_{5} - 10050 \beta_{4} - 35922 \beta_{3} - 35922 \beta_{2} + 33840 \beta_{1} + 16920$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$1577 \beta_{15} - 630 \beta_{14} - 1291 \beta_{13} - 1291 \beta_{12} + 9806 \beta_{11} + 1260 \beta_{10} - 12131 \beta_{9} + 10554 \beta_{8} + 3154 \beta_{7} - 19612 \beta_{6} + 8146 \beta_{5} + 8146 \beta_{4} - 54698 \beta_{3} + 54698 \beta_{2} + 70056$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$40984 \beta_{15} + 27165 \beta_{14} + 64251 \beta_{13} - 64251 \beta_{12} + 112052 \beta_{11} - 71067 \beta_{9} - 30083 \beta_{8} - 44055 \beta_{5} + 44055 \beta_{4} + 185383 \beta_{3} + 185383 \beta_{2} - 616880 \beta_{1} - 308440$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$-19787 \beta_{15} + 10410 \beta_{14} + 8812 \beta_{13} + 8812 \beta_{12} - 45437 \beta_{11} - 20820 \beta_{10} + 128560 \beta_{9} - 108773 \beta_{8} - 39574 \beta_{7} + 90874 \beta_{6} - 35062 \beta_{5} - 35062 \beta_{4} + 232060 \beta_{3} - 232060 \beta_{2} - 671586$$ $$\nu^{13}$$ $$=$$ $$($$$$-1304123 \beta_{15} - 810468 \beta_{14} - 925548 \beta_{13} + 925548 \beta_{12} - 1569697 \beta_{11} + 2431239 \beta_{9} + 1127116 \beta_{8} + 587340 \beta_{5} - 587340 \beta_{4} - 2812108 \beta_{3} - 2812108 \beta_{2} + 22609184 \beta_{1} + 11304592$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$644022 \beta_{15} - 363636 \beta_{14} - 133956 \beta_{13} - 133956 \beta_{12} + 522108 \beta_{11} + 727272 \beta_{10} - 3949277 \beta_{9} + 3305255 \beta_{8} + 1288044 \beta_{7} - 1044216 \beta_{6} + 371844 \beta_{5} + 371844 \beta_{4} - 2418218 \beta_{3} + 2418218 \beta_{2} + 19796271$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$17302151 \beta_{15} + 10397226 \beta_{14} + 1830030 \beta_{13} - 1830030 \beta_{12} + 2693803 \beta_{11} - 33379797 \beta_{9} - 16077646 \beta_{8} - 725550 \beta_{5} + 725550 \beta_{4} + 6867726 \beta_{3} + 6867726 \beta_{2} - 319686288 \beta_{1} - 159843144$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 −0.707107 − 3.42121i 0.707107 + 2.60548i 0.707107 + 1.17406i −0.707107 − 0.358323i 0.707107 − 0.358323i −0.707107 + 1.17406i −0.707107 + 2.60548i 0.707107 − 3.42121i −0.707107 + 3.42121i 0.707107 − 2.60548i 0.707107 − 1.17406i −0.707107 + 0.358323i 0.707107 + 0.358323i −0.707107 − 1.17406i −0.707107 − 2.60548i 0.707107 + 3.42121i
0 −4.19011 2.41916i 0 0.0446470 0.0257769i 0 6.12357 3.39144i 0 7.20469 + 12.4789i 0
129.2 0 −3.19104 1.84235i 0 2.63938 1.52385i 0 −0.812549 + 6.95268i 0 2.28850 + 3.96380i 0
129.3 0 −1.43792 0.830185i 0 −7.27622 + 4.20093i 0 −3.99843 5.74565i 0 −3.12159 5.40674i 0
129.4 0 −0.438854 0.253372i 0 4.59219 2.65130i 0 −5.27770 + 4.59846i 0 −4.37160 7.57184i 0
129.5 0 0.438854 + 0.253372i 0 4.59219 2.65130i 0 5.27770 4.59846i 0 −4.37160 7.57184i 0
129.6 0 1.43792 + 0.830185i 0 −7.27622 + 4.20093i 0 3.99843 + 5.74565i 0 −3.12159 5.40674i 0
129.7 0 3.19104 + 1.84235i 0 2.63938 1.52385i 0 0.812549 6.95268i 0 2.28850 + 3.96380i 0
129.8 0 4.19011 + 2.41916i 0 0.0446470 0.0257769i 0 −6.12357 + 3.39144i 0 7.20469 + 12.4789i 0
257.1 0 −4.19011 + 2.41916i 0 0.0446470 + 0.0257769i 0 6.12357 + 3.39144i 0 7.20469 12.4789i 0
257.2 0 −3.19104 + 1.84235i 0 2.63938 + 1.52385i 0 −0.812549 6.95268i 0 2.28850 3.96380i 0
257.3 0 −1.43792 + 0.830185i 0 −7.27622 4.20093i 0 −3.99843 + 5.74565i 0 −3.12159 + 5.40674i 0
257.4 0 −0.438854 + 0.253372i 0 4.59219 + 2.65130i 0 −5.27770 4.59846i 0 −4.37160 + 7.57184i 0
257.5 0 0.438854 0.253372i 0 4.59219 + 2.65130i 0 5.27770 + 4.59846i 0 −4.37160 + 7.57184i 0
257.6 0 1.43792 0.830185i 0 −7.27622 4.20093i 0 3.99843 5.74565i 0 −3.12159 + 5.40674i 0
257.7 0 3.19104 1.84235i 0 2.63938 + 1.52385i 0 0.812549 + 6.95268i 0 2.28850 3.96380i 0
257.8 0 4.19011 2.41916i 0 0.0446470 + 0.0257769i 0 −6.12357 3.39144i 0 7.20469 12.4789i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.g 16
4.b odd 2 1 inner 448.3.s.g 16
7.d odd 6 1 inner 448.3.s.g 16
8.b even 2 1 224.3.s.a 16
8.d odd 2 1 224.3.s.a 16
28.f even 6 1 inner 448.3.s.g 16
56.j odd 6 1 224.3.s.a 16
56.j odd 6 1 1568.3.c.h 16
56.k odd 6 1 1568.3.c.h 16
56.m even 6 1 224.3.s.a 16
56.m even 6 1 1568.3.c.h 16
56.p even 6 1 1568.3.c.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.a 16 8.b even 2 1
224.3.s.a 16 8.d odd 2 1
224.3.s.a 16 56.j odd 6 1
224.3.s.a 16 56.m even 6 1
448.3.s.g 16 1.a even 1 1 trivial
448.3.s.g 16 4.b odd 2 1 inner
448.3.s.g 16 7.d odd 6 1 inner
448.3.s.g 16 28.f even 6 1 inner
1568.3.c.h 16 56.j odd 6 1
1568.3.c.h 16 56.k odd 6 1
1568.3.c.h 16 56.m even 6 1
1568.3.c.h 16 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 32 T^{2} + 486 T^{4} + 3776 T^{6} + 6841 T^{8} - 164352 T^{10} - 1549098 T^{12} - 1704672 T^{14} + 53230932 T^{16} - 138078432 T^{18} - 10163631978 T^{20} - 87343391232 T^{22} + 294482618361 T^{24} + 13166097898176 T^{26} + 137260754729766 T^{28} + 732057358558752 T^{30} + 1853020188851841 T^{32}$$
$5$ $$( 1 + 46 T^{2} + 1073 T^{4} + 5040 T^{5} + 9678 T^{6} + 250320 T^{7} - 145276 T^{8} + 6258000 T^{9} + 6048750 T^{10} + 78750000 T^{11} + 419140625 T^{12} + 11230468750 T^{14} + 152587890625 T^{16} )^{2}$$
$7$ $$1 + 64 T^{2} + 5084 T^{4} + 338880 T^{6} + 15148742 T^{8} + 813650880 T^{10} + 29308248284 T^{12} + 885842380864 T^{14} + 33232930569601 T^{16}$$
$11$ $$1 - 640 T^{2} + 227030 T^{4} - 52241408 T^{6} + 8348702825 T^{8} - 853428477056 T^{10} + 32116185305766 T^{12} + 6409181104515840 T^{14} - 1320087233250529292 T^{16} + 93836820551216413440 T^{18} +$$$$68\!\cdots\!46$$$$T^{20} -$$$$26\!\cdots\!76$$$$T^{22} +$$$$38\!\cdots\!25$$$$T^{24} -$$$$35\!\cdots\!08$$$$T^{26} +$$$$22\!\cdots\!30$$$$T^{28} -$$$$92\!\cdots\!40$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$( 1 - 192 T^{2} - 16036 T^{4} + 1118400 T^{6} + 1003382406 T^{8} + 31942622400 T^{10} - 13081057841956 T^{12} - 4473232343516352 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 - 24 T + 766 T^{2} - 13776 T^{3} + 249905 T^{4} - 2407680 T^{5} - 7157346 T^{6} + 486721512 T^{7} - 15232969756 T^{8} + 140662516968 T^{9} - 597788695266 T^{10} - 58115542129920 T^{11} + 1743276663293105 T^{12} - 27772331972585424 T^{13} + 446288633717996926 T^{14} - 4041067837425622296 T^{15} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$1 + 528 T^{2} + 24262 T^{4} + 11189472 T^{6} + 3728656729 T^{8} - 6420105050688 T^{10} + 940099701135094 T^{12} + 1121910404491838544 T^{14} +$$$$16\!\cdots\!80$$$$T^{16} +$$$$14\!\cdots\!24$$$$T^{18} +$$$$15\!\cdots\!54$$$$T^{20} -$$$$14\!\cdots\!68$$$$T^{22} +$$$$10\!\cdots\!49$$$$T^{24} +$$$$42\!\cdots\!72$$$$T^{26} +$$$$11\!\cdots\!02$$$$T^{28} +$$$$33\!\cdots\!48$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$1 - 928 T^{2} - 110362 T^{4} + 76027072 T^{6} + 161836431161 T^{8} - 2090789193920 T^{10} - 54941991557559978 T^{12} + 14303404978544854944 T^{14} -$$$$12\!\cdots\!04$$$$T^{16} +$$$$40\!\cdots\!04$$$$T^{18} -$$$$43\!\cdots\!18$$$$T^{20} -$$$$45\!\cdots\!20$$$$T^{22} +$$$$99\!\cdots\!21$$$$T^{24} +$$$$13\!\cdots\!72$$$$T^{26} -$$$$53\!\cdots\!42$$$$T^{28} -$$$$12\!\cdots\!68$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 + 28 T + 1376 T^{2} + 41412 T^{3} + 2055086 T^{4} + 34827492 T^{5} + 973218656 T^{6} + 16655052988 T^{7} + 500246412961 T^{8} )^{4}$$
$31$ $$1 + 4480 T^{2} + 9494774 T^{4} + 14306099840 T^{6} + 19217452630025 T^{8} + 24003699927942080 T^{10} + 26963891194383292806 T^{12} +$$$$27\!\cdots\!40$$$$T^{14} +$$$$26\!\cdots\!04$$$$T^{16} +$$$$25\!\cdots\!40$$$$T^{18} +$$$$22\!\cdots\!46$$$$T^{20} +$$$$18\!\cdots\!80$$$$T^{22} +$$$$13\!\cdots\!25$$$$T^{24} +$$$$96\!\cdots\!40$$$$T^{26} +$$$$58\!\cdots\!54$$$$T^{28} +$$$$25\!\cdots\!80$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$( 1 + 4 T - 2770 T^{2} - 121912 T^{3} + 3850313 T^{4} + 247260872 T^{5} + 4348488942 T^{6} - 233812310196 T^{7} - 10747661313644 T^{8} - 320089052658324 T^{9} + 8149768384027662 T^{10} + 634403749202768648 T^{11} + 13524145303664927273 T^{12} -$$$$58\!\cdots\!88$$$$T^{13} -$$$$18\!\cdots\!70$$$$T^{14} +$$$$36\!\cdots\!56$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$( 1 - 4384 T^{2} + 14370524 T^{4} - 32104130784 T^{6} + 62557592438726 T^{8} - 90718600708326624 T^{10} +$$$$11\!\cdots\!04$$$$T^{12} -$$$$98\!\cdots\!04$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$( 1 + 11176 T^{2} + 58293116 T^{4} + 188186249880 T^{6} + 415246366681670 T^{8} + 643371339275993880 T^{10} +$$$$68\!\cdots\!16$$$$T^{12} +$$$$44\!\cdots\!76$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$1 + 7504 T^{2} + 21182918 T^{4} + 37880838368 T^{6} + 100768600793177 T^{8} + 218689095748226816 T^{10} +$$$$22\!\cdots\!70$$$$T^{12} +$$$$76\!\cdots\!12$$$$T^{14} +$$$$29\!\cdots\!24$$$$T^{16} +$$$$37\!\cdots\!72$$$$T^{18} +$$$$52\!\cdots\!70$$$$T^{20} +$$$$25\!\cdots\!56$$$$T^{22} +$$$$57\!\cdots\!17$$$$T^{24} +$$$$10\!\cdots\!68$$$$T^{26} +$$$$28\!\cdots\!58$$$$T^{28} +$$$$49\!\cdots\!44$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$( 1 - 12 T - 5314 T^{2} + 123432 T^{3} + 13289689 T^{4} - 412137720 T^{5} + 8001350174 T^{6} + 662204794044 T^{7} - 79868347316108 T^{8} + 1860133266469596 T^{9} + 63134501522293694 T^{10} - 9134769260962685880 T^{11} +$$$$82\!\cdots\!29$$$$T^{12} +$$$$21\!\cdots\!68$$$$T^{13} -$$$$26\!\cdots\!74$$$$T^{14} -$$$$16\!\cdots\!28$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$59$ $$1 + 12496 T^{2} + 69591254 T^{4} + 231196127072 T^{6} + 548940902299433 T^{8} + 1224313485336173888 T^{10} +$$$$76\!\cdots\!62$$$$T^{12} -$$$$17\!\cdots\!52$$$$T^{14} -$$$$10\!\cdots\!32$$$$T^{16} -$$$$21\!\cdots\!72$$$$T^{18} +$$$$11\!\cdots\!02$$$$T^{20} +$$$$21\!\cdots\!28$$$$T^{22} +$$$$11\!\cdots\!53$$$$T^{24} +$$$$60\!\cdots\!72$$$$T^{26} +$$$$22\!\cdots\!94$$$$T^{28} +$$$$47\!\cdots\!16$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 - 180 T + 27726 T^{2} - 3046680 T^{3} + 308297129 T^{4} - 26423205000 T^{5} + 2082233555790 T^{6} - 145145432104140 T^{7} + 9348584465944212 T^{8} - 540086152859504940 T^{9} + 28830274738332969390 T^{10} -$$$$13\!\cdots\!00$$$$T^{11} +$$$$59\!\cdots\!49$$$$T^{12} -$$$$21\!\cdots\!80$$$$T^{13} +$$$$73\!\cdots\!46$$$$T^{14} -$$$$17\!\cdots\!80$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$1 - 5792 T^{2} - 24227898 T^{4} + 351601426496 T^{6} - 205859799182695 T^{8} - 8707473271916577024 T^{10} +$$$$34\!\cdots\!66$$$$T^{12} +$$$$81\!\cdots\!24$$$$T^{14} -$$$$92\!\cdots\!56$$$$T^{16} +$$$$16\!\cdots\!04$$$$T^{18} +$$$$14\!\cdots\!06$$$$T^{20} -$$$$71\!\cdots\!64$$$$T^{22} -$$$$33\!\cdots\!95$$$$T^{24} +$$$$11\!\cdots\!96$$$$T^{26} -$$$$16\!\cdots\!58$$$$T^{28} -$$$$78\!\cdots\!72$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 27624 T^{2} + 370534588 T^{4} + 3151580461272 T^{6} + 18747759294515334 T^{8} + 80086957327676918232 T^{10} +$$$$23\!\cdots\!68$$$$T^{12} +$$$$45\!\cdots\!84$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 36 T + 13374 T^{2} - 465912 T^{3} + 80365289 T^{4} - 3011641896 T^{5} + 480881406846 T^{6} - 17666763587772 T^{7} + 3025613230783092 T^{8} - 94146183159236988 T^{9} + 13656186084031757886 T^{10} -$$$$45\!\cdots\!44$$$$T^{11} +$$$$64\!\cdots\!09$$$$T^{12} -$$$$20\!\cdots\!88$$$$T^{13} +$$$$30\!\cdots\!54$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$1 - 10112 T^{2} - 55382202 T^{4} + 565038672512 T^{6} + 4616239476368153 T^{8} - 23803283016229531968 T^{10} -$$$$25\!\cdots\!02$$$$T^{12} +$$$$18\!\cdots\!72$$$$T^{14} +$$$$13\!\cdots\!08$$$$T^{16} +$$$$71\!\cdots\!32$$$$T^{18} -$$$$37\!\cdots\!22$$$$T^{20} -$$$$14\!\cdots\!88$$$$T^{22} +$$$$10\!\cdots\!13$$$$T^{24} +$$$$50\!\cdots\!12$$$$T^{26} -$$$$19\!\cdots\!62$$$$T^{28} -$$$$13\!\cdots\!32$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$( 1 - 24232 T^{2} + 235912988 T^{4} - 1035732354840 T^{6} + 3398417338650758 T^{8} - 49154118566082623640 T^{10} +$$$$53\!\cdots\!08$$$$T^{12} -$$$$25\!\cdots\!52$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 204 T + 42982 T^{2} - 5938440 T^{3} + 800796785 T^{4} - 88063687656 T^{5} + 9449600595510 T^{6} - 894257389252164 T^{7} + 83094368751630116 T^{8} - 7083412780266391044 T^{9} +$$$$59\!\cdots\!10$$$$T^{10} -$$$$43\!\cdots\!16$$$$T^{11} +$$$$31\!\cdots\!85$$$$T^{12} -$$$$18\!\cdots\!40$$$$T^{13} +$$$$10\!\cdots\!22$$$$T^{14} -$$$$39\!\cdots\!64$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 51968 T^{2} + 1305731356 T^{4} - 20766356196608 T^{6} + 230652504659485894 T^{8} -$$$$18\!\cdots\!48$$$$T^{10} +$$$$10\!\cdots\!16$$$$T^{12} -$$$$36\!\cdots\!88$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$