Properties

 Label 1728.3.o.g Level 1728 Weight 3 Character orbit 1728.o Analytic conductor 47.085 Analytic rank 0 Dimension 16 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.o (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{16}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} + \beta_{6} ) q^{5} + ( \beta_{3} - \beta_{8} - \beta_{9} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{1} + \beta_{6} ) q^{5} + ( \beta_{3} - \beta_{8} - \beta_{9} ) q^{7} + ( \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{11} + ( 6 - 6 \beta_{1} + \beta_{15} ) q^{13} + ( -1 - \beta_{4} + \beta_{11} - 3 \beta_{13} - 3 \beta_{15} ) q^{17} + ( \beta_{2} + 2 \beta_{5} - 2 \beta_{9} - \beta_{10} + \beta_{12} ) q^{19} + ( 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - 2 \beta_{8} + 3 \beta_{12} ) q^{23} + ( -4 \beta_{1} + 3 \beta_{4} + 2 \beta_{13} - 3 \beta_{14} ) q^{25} + ( 4 \beta_{1} - 3 \beta_{13} + 2 \beta_{14} ) q^{29} + ( 3 \beta_{3} + \beta_{5} + \beta_{7} + 5 \beta_{12} ) q^{31} + ( -\beta_{2} - 4 \beta_{5} - 5 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} ) q^{35} + ( -7 + 6 \beta_{4} - 3 \beta_{6} - 6 \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{37} + ( -8 + 8 \beta_{1} + 3 \beta_{6} + 2 \beta_{11} + 3 \beta_{15} ) q^{41} + ( 5 \beta_{3} - 2 \beta_{8} - 5 \beta_{9} + 6 \beta_{10} ) q^{43} + ( 3 \beta_{3} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} ) q^{47} + ( 3 - 3 \beta_{1} - 9 \beta_{11} + \beta_{15} ) q^{49} + ( -3 - 4 \beta_{4} + \beta_{6} + 4 \beta_{11} - 3 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{53} + ( 2 \beta_{2} + 9 \beta_{5} - 7 \beta_{9} + 9 \beta_{10} - 9 \beta_{12} ) q^{55} + ( 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{5} + 6 \beta_{7} - 6 \beta_{8} + 2 \beta_{12} ) q^{59} + ( 12 \beta_{1} - 6 \beta_{4} + 7 \beta_{13} - 12 \beta_{14} ) q^{61} + ( 2 \beta_{1} - \beta_{4} - 3 \beta_{13} + 6 \beta_{14} ) q^{65} + ( -\beta_{2} + 11 \beta_{3} + 9 \beta_{5} + 9 \beta_{7} + \beta_{8} - 3 \beta_{12} ) q^{67} + ( -5 \beta_{2} - 5 \beta_{5} - 22 \beta_{9} - \beta_{10} + \beta_{12} ) q^{71} + ( 7 - 3 \beta_{4} - 12 \beta_{6} + 3 \beta_{11} + 5 \beta_{13} + 12 \beta_{14} + 5 \beta_{15} ) q^{73} + ( -37 + 37 \beta_{1} + \beta_{6} + 8 \beta_{11} ) q^{77} + ( 11 \beta_{3} + 5 \beta_{7} + 4 \beta_{8} - 11 \beta_{9} - 11 \beta_{10} ) q^{79} + ( \beta_{3} + 26 \beta_{7} + \beta_{8} - \beta_{9} - 5 \beta_{10} ) q^{83} + ( 19 - 19 \beta_{1} + 3 \beta_{6} + 6 \beta_{11} - 9 \beta_{15} ) q^{85} + ( 29 - 4 \beta_{4} + 3 \beta_{6} + 4 \beta_{11} + 9 \beta_{13} - 3 \beta_{14} + 9 \beta_{15} ) q^{89} + ( -4 \beta_{2} + 11 \beta_{5} - \beta_{9} - 4 \beta_{10} + 4 \beta_{12} ) q^{91} + ( -2 \beta_{2} - 4 \beta_{3} + 22 \beta_{5} + 22 \beta_{7} + 2 \beta_{8} - 14 \beta_{12} ) q^{95} + ( -16 \beta_{1} + \beta_{13} - 9 \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 6q^{5} + O(q^{10})$$ $$16q + 6q^{5} + 46q^{13} - 12q^{17} - 30q^{25} + 42q^{29} - 56q^{37} - 84q^{41} + 58q^{49} - 72q^{53} + 34q^{61} + 30q^{65} + 116q^{73} - 330q^{77} + 140q^{85} + 384q^{89} - 148q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + 1552 x^{8} - 3648 x^{7} + 6784 x^{6} - 9216 x^{5} + 19456 x^{4} - 30720 x^{3} + 28672 x^{2} - 49152 x + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} - 7664 \nu^{8} + 26384 \nu^{7} - 63104 \nu^{6} + 58368 \nu^{5} - 187904 \nu^{4} + 372736 \nu^{3} - 258048 \nu^{2} + 634880 \nu - 1228800$$$$)/540672$$ $$\beta_{2}$$ $$=$$ $$($$$$-17 \nu^{15} - 179 \nu^{14} + 371 \nu^{13} - 628 \nu^{12} + 3440 \nu^{11} - 6984 \nu^{10} + 9048 \nu^{9} - 37088 \nu^{8} + 62224 \nu^{7} - 105376 \nu^{6} + 234112 \nu^{5} - 334080 \nu^{4} + 241664 \nu^{3} - 929792 \nu^{2} + 786432 \nu - 286720$$$$)/270336$$ $$\beta_{3}$$ $$=$$ $$($$$$41 \nu^{15} - 7 \nu^{14} + 59 \nu^{13} - 842 \nu^{12} + 796 \nu^{11} - 368 \nu^{10} + 9672 \nu^{9} - 8688 \nu^{8} + 24336 \nu^{7} - 69760 \nu^{6} + 47360 \nu^{5} - 151552 \nu^{4} + 391168 \nu^{3} - 81920 \nu^{2} + 348160 \nu - 1179648$$$$)/540672$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{15} - 79 \nu^{14} + 339 \nu^{13} - 756 \nu^{12} + 2396 \nu^{11} - 6416 \nu^{10} + 13512 \nu^{9} - 31296 \nu^{8} + 68528 \nu^{7} - 117536 \nu^{6} + 230464 \nu^{5} - 422656 \nu^{4} + 621056 \nu^{3} - 939008 \nu^{2} + 1650688 \nu - 1392640$$$$)/135168$$ $$\beta_{5}$$ $$=$$ $$($$$$25 \nu^{15} - 37 \nu^{14} + 293 \nu^{13} - 1116 \nu^{12} + 1360 \nu^{11} - 4472 \nu^{10} + 13736 \nu^{9} - 17184 \nu^{8} + 47472 \nu^{7} - 123488 \nu^{6} + 131456 \nu^{5} - 290560 \nu^{4} + 746496 \nu^{3} - 413696 \nu^{2} + 1048576 \nu - 2695168$$$$)/270336$$ $$\beta_{6}$$ $$=$$ $$($$$$49 \nu^{15} - 135 \nu^{14} + 723 \nu^{13} - 2762 \nu^{12} + 4100 \nu^{11} - 12352 \nu^{10} + 39144 \nu^{9} - 63664 \nu^{8} + 120656 \nu^{7} - 352128 \nu^{6} + 428416 \nu^{5} - 894464 \nu^{4} + 1829888 \nu^{3} - 1673216 \nu^{2} + 2363392 \nu - 7045120$$$$)/540672$$ $$\beta_{7}$$ $$=$$ $$($$$$63 \nu^{15} - 29 \nu^{14} + 169 \nu^{13} - 50 \nu^{12} + 356 \nu^{11} - 16 \nu^{10} - 1768 \nu^{9} + 3984 \nu^{8} - 14736 \nu^{7} + 68928 \nu^{6} - 97664 \nu^{5} + 265216 \nu^{4} - 588800 \nu^{3} + 481280 \nu^{2} - 913408 \nu + 3325952$$$$)/540672$$ $$\beta_{8}$$ $$=$$ $$($$$$71 \nu^{15} + 283 \nu^{14} - 223 \nu^{13} - 1090 \nu^{12} - 652 \nu^{11} + 2256 \nu^{10} + 8344 \nu^{9} + 17296 \nu^{8} - 19088 \nu^{7} - 81088 \nu^{6} - 146048 \nu^{5} + 113664 \nu^{4} + 500736 \nu^{3} + 894976 \nu^{2} - 1241088 \nu - 2179072$$$$)/540672$$ $$\beta_{9}$$ $$=$$ $$($$$$-47 \nu^{15} + 125 \nu^{14} - 73 \nu^{13} + 786 \nu^{12} - 1844 \nu^{11} + 2272 \nu^{10} - 8632 \nu^{9} + 17712 \nu^{8} - 13680 \nu^{7} + 74560 \nu^{6} - 92032 \nu^{5} + 42752 \nu^{4} - 374784 \nu^{3} + 458752 \nu^{2} + 212992 \nu + 892928$$$$)/270336$$ $$\beta_{10}$$ $$=$$ $$($$$$-53 \nu^{15} + 23 \nu^{14} + 133 \nu^{13} + 862 \nu^{12} - 428 \nu^{11} - 1104 \nu^{10} - 7240 \nu^{9} + 5264 \nu^{8} + 8240 \nu^{7} + 68800 \nu^{6} - 45184 \nu^{5} - 49152 \nu^{4} - 414720 \nu^{3} + 407552 \nu^{2} + 774144 \nu + 1687552$$$$)/270336$$ $$\beta_{11}$$ $$=$$ $$($$$$-37 \nu^{15} + 119 \nu^{14} - 255 \nu^{13} + 762 \nu^{12} - 1960 \nu^{11} + 3880 \nu^{10} - 8312 \nu^{9} + 17808 \nu^{8} - 29680 \nu^{7} + 52480 \nu^{6} - 99008 \nu^{5} + 134912 \nu^{4} - 167424 \nu^{3} + 277504 \nu^{2} + 28672 \nu - 311296$$$$)/135168$$ $$\beta_{12}$$ $$=$$ $$($$$$75 \nu^{15} - 221 \nu^{14} + 593 \nu^{13} - 1918 \nu^{12} + 4124 \nu^{11} - 9984 \nu^{10} + 22904 \nu^{9} - 43280 \nu^{8} + 85744 \nu^{7} - 168064 \nu^{6} + 256384 \nu^{5} - 457728 \nu^{4} + 707584 \nu^{3} - 813056 \nu^{2} + 1658880 \nu - 1146880$$$$)/270336$$ $$\beta_{13}$$ $$=$$ $$($$$$85 \nu^{15} - 29 \nu^{14} - 7 \nu^{13} - 1084 \nu^{12} + 708 \nu^{11} - 16 \nu^{10} + 9496 \nu^{9} - 7104 \nu^{8} - 2416 \nu^{7} - 65888 \nu^{6} + 25536 \nu^{5} + 87808 \nu^{4} + 306688 \nu^{3} + 244736 \nu^{2} - 913408 \nu - 638976$$$$)/270336$$ $$\beta_{14}$$ $$=$$ $$($$$$-193 \nu^{15} + 459 \nu^{14} - 1455 \nu^{13} + 4102 \nu^{12} - 10156 \nu^{11} + 24080 \nu^{10} - 59240 \nu^{9} + 101072 \nu^{8} - 237328 \nu^{7} + 435648 \nu^{6} - 712064 \nu^{5} + 1183744 \nu^{4} - 1999872 \nu^{3} + 2381824 \nu^{2} - 4124672 \nu + 2146304$$$$)/540672$$ $$\beta_{15}$$ $$=$$ $$($$$$-59 \nu^{15} + 163 \nu^{14} - 277 \nu^{13} + 1510 \nu^{12} - 3126 \nu^{11} + 5420 \nu^{10} - 18080 \nu^{9} + 33120 \nu^{8} - 46048 \nu^{7} + 133440 \nu^{6} - 196512 \nu^{5} + 205312 \nu^{4} - 575744 \nu^{3} + 621056 \nu^{2} - 83968 \nu + 1040384$$$$)/135168$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} + \beta_{11} - 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{13} + \beta_{10} - \beta_{7} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} + 6$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} - \beta_{12} + 3 \beta_{11} - \beta_{8} + 4 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} + \beta_{2} + 4 \beta_{1} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{14} + 6 \beta_{13} - 3 \beta_{10} + 10 \beta_{9} - \beta_{8} - 8 \beta_{7} + \beta_{4} - 10 \beta_{3} - 20 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-6 \beta_{15} + 9 \beta_{14} - 6 \beta_{13} - \beta_{12} - 9 \beta_{11} + \beta_{10} + 2 \beta_{9} - 9 \beta_{6} + 4 \beta_{5} + 9 \beta_{4} - 13 \beta_{2} - 40$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-6 \beta_{15} - 17 \beta_{12} - \beta_{11} - 11 \beta_{8} + 20 \beta_{7} - 33 \beta_{6} + 20 \beta_{5} + 34 \beta_{3} + 11 \beta_{2} + 104 \beta_{1} - 104$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-57 \beta_{14} - 26 \beta_{13} - 33 \beta_{10} + 46 \beta_{9} + 13 \beta_{8} - 60 \beta_{7} - 33 \beta_{4} - 46 \beta_{3} - 88 \beta_{1}$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-90 \beta_{15} - 65 \beta_{14} - 90 \beta_{13} - 39 \beta_{12} + 41 \beta_{11} + 39 \beta_{10} + 30 \beta_{9} + 65 \beta_{6} - 84 \beta_{5} - 41 \beta_{4} - 75 \beta_{2} - 168$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$22 \beta_{15} - 111 \beta_{12} + 33 \beta_{11} - 77 \beta_{8} + 108 \beta_{7} + 57 \beta_{6} + 108 \beta_{5} + 494 \beta_{3} + 77 \beta_{2} - 440 \beta_{1} + 440$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-111 \beta_{14} - 294 \beta_{13} + 73 \beta_{10} - 190 \beta_{9} + 379 \beta_{8} - 244 \beta_{7} + 25 \beta_{4} + 190 \beta_{3} - 1592 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$138 \beta_{15} - 855 \beta_{14} + 138 \beta_{13} - 929 \beta_{12} + 207 \beta_{11} + 929 \beta_{10} - 1166 \beta_{9} + 855 \beta_{6} - 748 \beta_{5} - 207 \beta_{4} - 125 \beta_{2} - 1336$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$762 \beta_{15} - 1401 \beta_{12} - 265 \beta_{11} - 523 \beta_{8} + 2548 \beta_{7} + 1983 \beta_{6} + 2548 \beta_{5} + 674 \beta_{3} + 523 \beta_{2} - 7048 \beta_{1} + 7048$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$1767 \beta_{14} + 22 \beta_{13} - 1297 \beta_{10} + 2222 \beta_{9} + 3245 \beta_{8} + 3028 \beta_{7} + 3039 \beta_{4} - 2222 \beta_{3} - 6472 \beta_{1}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$9126 \beta_{15} - 2609 \beta_{14} + 9126 \beta_{13} + 1129 \beta_{12} - 3079 \beta_{11} - 1129 \beta_{10} - 6274 \beta_{9} + 2609 \beta_{6} - 7604 \beta_{5} + 3079 \beta_{4} - 347 \beta_{2} - 23304$$$$)/2$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.523926 + 1.93016i 1.93353 + 0.511345i 1.84233 − 0.778342i −1.59523 + 1.20633i −1.26364 − 1.55023i −0.710719 − 1.86946i 0.186266 + 1.99131i 1.63139 + 1.15696i −0.523926 − 1.93016i 1.93353 − 0.511345i 1.84233 + 0.778342i −1.59523 − 1.20633i −1.26364 + 1.55023i −0.710719 + 1.86946i 0.186266 − 1.99131i 1.63139 − 1.15696i
0 0 0 −4.03104 6.98197i 0 −3.90254 2.25313i 0 0 0
127.2 0 0 0 −4.03104 6.98197i 0 3.90254 + 2.25313i 0 0 0
127.3 0 0 0 1.10093 + 1.90686i 0 −7.23844 4.17912i 0 0 0
127.4 0 0 0 1.10093 + 1.90686i 0 7.23844 + 4.17912i 0 0 0
127.5 0 0 0 1.35609 + 2.34881i 0 −10.0431 5.79837i 0 0 0
127.6 0 0 0 1.35609 + 2.34881i 0 10.0431 + 5.79837i 0 0 0
127.7 0 0 0 3.07403 + 5.32438i 0 −0.511543 0.295340i 0 0 0
127.8 0 0 0 3.07403 + 5.32438i 0 0.511543 + 0.295340i 0 0 0
1279.1 0 0 0 −4.03104 + 6.98197i 0 −3.90254 + 2.25313i 0 0 0
1279.2 0 0 0 −4.03104 + 6.98197i 0 3.90254 2.25313i 0 0 0
1279.3 0 0 0 1.10093 1.90686i 0 −7.23844 + 4.17912i 0 0 0
1279.4 0 0 0 1.10093 1.90686i 0 7.23844 4.17912i 0 0 0
1279.5 0 0 0 1.35609 2.34881i 0 −10.0431 + 5.79837i 0 0 0
1279.6 0 0 0 1.35609 2.34881i 0 10.0431 5.79837i 0 0 0
1279.7 0 0 0 3.07403 5.32438i 0 −0.511543 + 0.295340i 0 0 0
1279.8 0 0 0 3.07403 5.32438i 0 0.511543 0.295340i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1279.8 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
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.o.g 16
3.b odd 2 1 576.3.o.g 16
4.b odd 2 1 inner 1728.3.o.g 16
8.b even 2 1 108.3.f.c 16
8.d odd 2 1 108.3.f.c 16
9.c even 3 1 inner 1728.3.o.g 16
9.d odd 6 1 576.3.o.g 16
12.b even 2 1 576.3.o.g 16
24.f even 2 1 36.3.f.c 16
24.h odd 2 1 36.3.f.c 16
36.f odd 6 1 inner 1728.3.o.g 16
36.h even 6 1 576.3.o.g 16
72.j odd 6 1 36.3.f.c 16
72.j odd 6 1 324.3.d.i 8
72.l even 6 1 36.3.f.c 16
72.l even 6 1 324.3.d.i 8
72.n even 6 1 108.3.f.c 16
72.n even 6 1 324.3.d.g 8
72.p odd 6 1 108.3.f.c 16
72.p odd 6 1 324.3.d.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c 16 24.f even 2 1
36.3.f.c 16 24.h odd 2 1
36.3.f.c 16 72.j odd 6 1
36.3.f.c 16 72.l even 6 1
108.3.f.c 16 8.b even 2 1
108.3.f.c 16 8.d odd 2 1
108.3.f.c 16 72.n even 6 1
108.3.f.c 16 72.p odd 6 1
324.3.d.g 8 72.n even 6 1
324.3.d.g 8 72.p odd 6 1
324.3.d.i 8 72.j odd 6 1
324.3.d.i 8 72.l even 6 1
576.3.o.g 16 3.b odd 2 1
576.3.o.g 16 9.d odd 6 1
576.3.o.g 16 12.b even 2 1
576.3.o.g 16 36.h even 6 1
1728.3.o.g 16 1.a even 1 1 trivial
1728.3.o.g 16 4.b odd 2 1 inner
1728.3.o.g 16 9.c even 3 1 inner
1728.3.o.g 16 36.f odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{16} - \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 3 T - 38 T^{2} - 201 T^{3} + 1495 T^{4} + 6984 T^{5} + 24112 T^{6} - 181380 T^{7} - 866084 T^{8} - 4534500 T^{9} + 15070000 T^{10} + 109125000 T^{11} + 583984375 T^{12} - 1962890625 T^{13} - 9277343750 T^{14} - 18310546875 T^{15} + 152587890625 T^{16} )^{2}$$
$7$ $$1 + 167 T^{2} + 13188 T^{4} + 535927 T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 4977965952336286 T^{18} - 278348169589725074 T^{20} - 9902233810610977584 T^{22} +$$$$15\!\cdots\!25$$$$T^{24} +$$$$42\!\cdots\!27$$$$T^{26} +$$$$25\!\cdots\!88$$$$T^{28} +$$$$76\!\cdots\!67$$$$T^{30} +$$$$11\!\cdots\!01$$$$T^{32}$$
$11$ $$1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 48469876840225802764 T^{18} -$$$$51\!\cdots\!42$$$$T^{20} +$$$$67\!\cdots\!68$$$$T^{22} +$$$$13\!\cdots\!45$$$$T^{24} +$$$$16\!\cdots\!04$$$$T^{26} +$$$$13\!\cdots\!54$$$$T^{28} +$$$$75\!\cdots\!44$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$( 1 - 23 T - 276 T^{2} + 5009 T^{3} + 162641 T^{4} - 1572720 T^{5} - 34351730 T^{6} + 33339262 T^{7} + 8566506504 T^{8} + 5634335278 T^{9} - 981119760530 T^{10} - 7591219050480 T^{11} + 132671260194161 T^{12} + 690533185671641 T^{13} - 6430271493804756 T^{14} - 90559656871083647 T^{15} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 3 T + 578 T^{2} + 6549 T^{3} + 169242 T^{4} + 1892661 T^{5} + 48275138 T^{6} + 72412707 T^{7} + 6975757441 T^{8} )^{4}$$
$19$ $$( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 122204089833383 T^{10} + 26492490152025490 T^{12} - 3702875859597687353 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$1 + 2687 T^{2} + 3719652 T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} +$$$$39\!\cdots\!18$$$$T^{14} +$$$$20\!\cdots\!24$$$$T^{16} +$$$$11\!\cdots\!38$$$$T^{18} +$$$$59\!\cdots\!94$$$$T^{20} +$$$$31\!\cdots\!76$$$$T^{22} +$$$$14\!\cdots\!85$$$$T^{24} +$$$$59\!\cdots\!55$$$$T^{26} +$$$$17\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!47$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 - 21 T - 2432 T^{2} + 19167 T^{3} + 4062037 T^{4} - 6623136 T^{5} - 4703569190 T^{6} + 3469324686 T^{7} + 4129311885376 T^{8} + 2917702060926 T^{9} - 3326745120272390 T^{10} - 3939595750954656 T^{11} + 2032019438564861557 T^{12} + 8063695540664952567 T^{13} -$$$$86\!\cdots\!12$$$$T^{14} -$$$$62\!\cdots\!01$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} +$$$$40\!\cdots\!98$$$$T^{14} +$$$$41\!\cdots\!28$$$$T^{16} +$$$$37\!\cdots\!58$$$$T^{18} +$$$$31\!\cdots\!94$$$$T^{20} +$$$$25\!\cdots\!88$$$$T^{22} +$$$$18\!\cdots\!37$$$$T^{24} +$$$$11\!\cdots\!27$$$$T^{26} +$$$$66\!\cdots\!64$$$$T^{28} +$$$$27\!\cdots\!15$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$( 1 + 14 T + 3040 T^{2} + 66050 T^{3} + 4581118 T^{4} + 90422450 T^{5} + 5697449440 T^{6} + 35920169726 T^{7} + 3512479453921 T^{8} )^{4}$$
$41$ $$( 1 + 42 T - 4424 T^{2} - 125040 T^{3} + 14943835 T^{4} + 232367040 T^{5} - 36025798940 T^{6} - 132403432026 T^{7} + 71285616374608 T^{8} - 222570169235706 T^{9} - 101800297638493340 T^{10} + 1103767662172616640 T^{11} +$$$$11\!\cdots\!35$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{13} -$$$$99\!\cdots\!44$$$$T^{14} +$$$$15\!\cdots\!62$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} +$$$$45\!\cdots\!90$$$$T^{12} +$$$$10\!\cdots\!64$$$$T^{14} +$$$$19\!\cdots\!28$$$$T^{16} +$$$$34\!\cdots\!64$$$$T^{18} +$$$$53\!\cdots\!90$$$$T^{20} +$$$$75\!\cdots\!08$$$$T^{22} +$$$$90\!\cdots\!65$$$$T^{24} +$$$$94\!\cdots\!36$$$$T^{26} +$$$$84\!\cdots\!02$$$$T^{28} +$$$$56\!\cdots\!92$$$$T^{30} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 1551988218895697 T^{8} + 4556649670813575888 T^{10} +$$$$11\!\cdots\!54$$$$T^{12} +$$$$26\!\cdots\!06$$$$T^{14} +$$$$58\!\cdots\!04$$$$T^{16} +$$$$12\!\cdots\!86$$$$T^{18} +$$$$27\!\cdots\!94$$$$T^{20} +$$$$52\!\cdots\!08$$$$T^{22} +$$$$87\!\cdots\!37$$$$T^{24} +$$$$11\!\cdots\!79$$$$T^{26} +$$$$12\!\cdots\!40$$$$T^{28} +$$$$85\!\cdots\!63$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$( 1 + 18 T + 10016 T^{2} + 126558 T^{3} + 40472766 T^{4} + 355501422 T^{5} + 79031057696 T^{6} + 398958500322 T^{7} + 62259690411361 T^{8} )^{4}$$
$59$ $$1 + 18092 T^{2} + 175903662 T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} +$$$$35\!\cdots\!18$$$$T^{12} +$$$$21\!\cdots\!16$$$$T^{14} -$$$$83\!\cdots\!68$$$$T^{16} +$$$$26\!\cdots\!76$$$$T^{18} +$$$$52\!\cdots\!78$$$$T^{20} +$$$$30\!\cdots\!68$$$$T^{22} +$$$$11\!\cdots\!61$$$$T^{24} +$$$$29\!\cdots\!84$$$$T^{26} +$$$$55\!\cdots\!82$$$$T^{28} +$$$$69\!\cdots\!32$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 - 17 T - 3582 T^{2} + 125549 T^{3} - 18305593 T^{4} + 350178696 T^{5} - 12273736208 T^{6} - 2060229775052 T^{7} + 599123836260396 T^{8} - 7666114992968492 T^{9} - 169940200011910928 T^{10} + 18041337511166813256 T^{11} -$$$$35\!\cdots\!33$$$$T^{12} +$$$$89\!\cdots\!49$$$$T^{13} -$$$$95\!\cdots\!22$$$$T^{14} -$$$$16\!\cdots\!97$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} +$$$$12\!\cdots\!60$$$$T^{10} +$$$$66\!\cdots\!50$$$$T^{12} +$$$$31\!\cdots\!76$$$$T^{14} +$$$$14\!\cdots\!68$$$$T^{16} +$$$$63\!\cdots\!96$$$$T^{18} +$$$$27\!\cdots\!50$$$$T^{20} +$$$$10\!\cdots\!60$$$$T^{22} +$$$$35\!\cdots\!25$$$$T^{24} +$$$$10\!\cdots\!00$$$$T^{26} +$$$$22\!\cdots\!66$$$$T^{28} +$$$$33\!\cdots\!68$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 23176426971828216728 T^{10} +$$$$88\!\cdots\!80$$$$T^{12} -$$$$21\!\cdots\!88$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 3307960763 T^{5} + 220767925534 T^{6} - 4388692562381 T^{7} + 806460091894081 T^{8} )^{4}$$
$79$ $$1 + 23147 T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} +$$$$59\!\cdots\!06$$$$T^{12} -$$$$27\!\cdots\!10$$$$T^{14} -$$$$25\!\cdots\!44$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$90\!\cdots\!66$$$$T^{20} +$$$$53\!\cdots\!92$$$$T^{22} +$$$$49\!\cdots\!89$$$$T^{24} +$$$$28\!\cdots\!59$$$$T^{26} +$$$$11\!\cdots\!16$$$$T^{28} +$$$$31\!\cdots\!67$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} -$$$$32\!\cdots\!46$$$$T^{12} +$$$$11\!\cdots\!22$$$$T^{14} +$$$$52\!\cdots\!80$$$$T^{16} +$$$$55\!\cdots\!62$$$$T^{18} -$$$$73\!\cdots\!86$$$$T^{20} +$$$$96\!\cdots\!28$$$$T^{22} +$$$$63\!\cdots\!17$$$$T^{24} -$$$$57\!\cdots\!09$$$$T^{26} +$$$$32\!\cdots\!12$$$$T^{28} +$$$$96\!\cdots\!95$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 15366866736 T^{5} + 1738963951556 T^{6} - 47710203932256 T^{7} + 3936588805702081 T^{8} )^{4}$$
$97$ $$( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 701815562883455698 T^{9} -$$$$71\!\cdots\!00$$$$T^{10} +$$$$17\!\cdots\!44$$$$T^{11} +$$$$47\!\cdots\!03$$$$T^{12} -$$$$11\!\cdots\!72$$$$T^{13} -$$$$20\!\cdots\!88$$$$T^{14} +$$$$48\!\cdots\!06$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$