# Properties

 Label 900.3.p.c Level $900$ Weight $3$ Character orbit 900.p Analytic conductor $24.523$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} - 9720 x^{4} + 43740 x^{3} - 72171 x^{2} - 118098 x + 531441$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{7} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{7} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( 5 + \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{11} + ( 5 + \beta_{1} - \beta_{2} + 6 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{13} + ( 1 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{17} + ( 6 - \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{19} + ( -13 - \beta_{1} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} ) q^{21} + ( 4 - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{23} + ( 13 - \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{27} + ( 18 - 2 \beta_{1} + 7 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{29} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} ) q^{31} + ( -12 - 3 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{33} + ( 1 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 4 \beta_{11} ) q^{37} + ( 4 - \beta_{1} - \beta_{2} + 9 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{39} + ( 7 + \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{41} + ( 4 \beta_{1} - 6 \beta_{2} + 25 \beta_{3} - 6 \beta_{4} - \beta_{6} - 6 \beta_{7} - \beta_{8} - 5 \beta_{9} + \beta_{10} + 7 \beta_{11} ) q^{43} + ( -1 - \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{47} + ( 9 - 11 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 6 \beta_{5} - 7 \beta_{6} + 9 \beta_{7} + 6 \beta_{9} - 3 \beta_{11} ) q^{49} + ( -11 + 5 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 5 \beta_{7} + 2 \beta_{8} - 4 \beta_{10} - 7 \beta_{11} ) q^{51} + ( -8 + 6 \beta_{1} + \beta_{2} - 13 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{53} + ( 9 - 11 \beta_{1} + 10 \beta_{2} - 21 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + 9 \beta_{7} + \beta_{8} + 5 \beta_{9} + \beta_{10} - 5 \beta_{11} ) q^{57} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} ) q^{59} + ( -9 - 13 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} + 5 \beta_{4} - 14 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{61} + ( 24 + 13 \beta_{1} + 13 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{63} + ( 24 + 9 \beta_{1} + 3 \beta_{2} + 30 \beta_{3} + 4 \beta_{4} + 9 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 9 \beta_{9} - 3 \beta_{10} - 6 \beta_{11} ) q^{67} + ( -16 - 14 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} + 4 \beta_{4} + 5 \beta_{5} - 13 \beta_{6} + 13 \beta_{7} + 4 \beta_{9} - 9 \beta_{11} ) q^{69} + ( 8 - 8 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 9 \beta_{6} + 5 \beta_{7} - 6 \beta_{9} - \beta_{11} ) q^{71} + ( 2 + 2 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} - 4 \beta_{11} ) q^{73} + ( 16 + 5 \beta_{1} + 2 \beta_{2} - 11 \beta_{3} + 9 \beta_{4} + 14 \beta_{6} - 2 \beta_{11} ) q^{77} + ( 1 - 2 \beta_{1} + 9 \beta_{2} - 29 \beta_{3} + 5 \beta_{4} - 4 \beta_{6} + 10 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} - 13 \beta_{11} ) q^{79} + ( 44 - 12 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 7 \beta_{11} ) q^{81} + ( -15 - 11 \beta_{1} - 2 \beta_{2} - 13 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 19 \beta_{6} + 9 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{83} + ( 46 - 7 \beta_{1} + 24 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} ) q^{87} + ( 9 + 10 \beta_{1} + 4 \beta_{2} + 18 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} - 2 \beta_{8} + 8 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{89} + ( 10 + 12 \beta_{1} - 8 \beta_{2} + 18 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + 33 \beta_{6} - 23 \beta_{7} + 4 \beta_{8} + 5 \beta_{11} ) q^{91} + ( -17 - 4 \beta_{1} + 5 \beta_{2} - 18 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} + 5 \beta_{11} ) q^{93} + ( -13 - 30 \beta_{1} + 16 \beta_{3} + 15 \beta_{4} - 12 \beta_{6} - 13 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{97} + ( -2 - \beta_{1} + 9 \beta_{2} + 21 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} - 17 \beta_{6} + 5 \beta_{7} + 5 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{3} + 6q^{7} + 26q^{9} + O(q^{10})$$ $$12q - 2q^{3} + 6q^{7} + 26q^{9} + 48q^{11} + 30q^{13} + 72q^{19} - 128q^{21} + 78q^{23} + 106q^{27} + 150q^{29} - 12q^{31} - 96q^{33} + 12q^{37} + 40q^{39} + 90q^{41} - 114q^{43} - 12q^{47} + 48q^{49} - 144q^{51} + 158q^{57} + 48q^{59} - 78q^{61} + 212q^{63} + 168q^{67} - 150q^{69} + 24q^{73} + 258q^{77} + 120q^{79} + 434q^{81} - 114q^{83} + 330q^{87} + 120q^{91} - 82q^{93} - 96q^{97} - 120q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} - 9720 x^{4} + 43740 x^{3} - 72171 x^{2} - 118098 x + 531441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$412 \nu^{11} - 17123 \nu^{10} + 22072 \nu^{9} + 48327 \nu^{8} - 317739 \nu^{7} + 1679139 \nu^{6} + 2047122 \nu^{5} - 11186100 \nu^{4} + 15780420 \nu^{3} + 1911438 \nu^{2} - 143915535 \nu + 368052417$$$$)/95482233$$ $$\beta_{3}$$ $$=$$ $$($$$$-1264 \nu^{11} + 7514 \nu^{10} - 7570 \nu^{9} - 69045 \nu^{8} + 447600 \nu^{7} - 644742 \nu^{6} - 2388834 \nu^{5} + 11774160 \nu^{4} - 18696420 \nu^{3} - 22788540 \nu^{2} + 174876894 \nu - 255564072$$$$)/95482233$$ $$\beta_{4}$$ $$=$$ $$($$$$262 \nu^{11} - 2342 \nu^{10} + 7018 \nu^{9} + 7395 \nu^{8} - 133365 \nu^{7} + 477666 \nu^{6} - 50022 \nu^{5} - 4071222 \nu^{4} + 12670020 \nu^{3} - 10847520 \nu^{2} - 50992092 \nu + 188917434$$$$)/10609137$$ $$\beta_{5}$$ $$=$$ $$($$$$3622 \nu^{11} - 28592 \nu^{10} + 70732 \nu^{9} + 135600 \nu^{8} - 1647885 \nu^{7} + 4943736 \nu^{6} + 1938636 \nu^{5} - 48415158 \nu^{4} + 132726600 \nu^{3} + 20643093 \nu^{2} - 729287955 \nu + 1669374279$$$$)/95482233$$ $$\beta_{6}$$ $$=$$ $$($$$$-554 \nu^{11} + 2386 \nu^{10} - 755 \nu^{9} - 32880 \nu^{8} + 119670 \nu^{7} - 115038 \nu^{6} - 875952 \nu^{5} + 3442500 \nu^{4} - 3610980 \nu^{3} - 9294750 \nu^{2} + 34373079 \nu - 74637936$$$$)/10609137$$ $$\beta_{7}$$ $$=$$ $$($$$$-5945 \nu^{11} + 23266 \nu^{10} - 2231 \nu^{9} - 288570 \nu^{8} + 1334805 \nu^{7} - 1995210 \nu^{6} - 10302327 \nu^{5} + 39798216 \nu^{4} - 48182040 \nu^{3} - 91766520 \nu^{2} + 634153455 \nu - 967281669$$$$)/95482233$$ $$\beta_{8}$$ $$=$$ $$($$$$-956 \nu^{11} + 6430 \nu^{10} + 589 \nu^{9} - 82677 \nu^{8} + 330720 \nu^{7} - 477891 \nu^{6} - 2278440 \nu^{5} + 9408879 \nu^{4} - 10032741 \nu^{3} - 15212772 \nu^{2} + 116254359 \nu - 205903863$$$$)/13640319$$ $$\beta_{9}$$ $$=$$ $$($$$$-1031 \nu^{11} + 5815 \nu^{10} - 26 \nu^{9} - 46335 \nu^{8} + 226131 \nu^{7} - 446544 \nu^{6} - 1203111 \nu^{5} + 6708096 \nu^{4} - 11117250 \nu^{3} - 1565892 \nu^{2} + 95154183 \nu - 222575364$$$$)/10609137$$ $$\beta_{10}$$ $$=$$ $$($$$$599 \nu^{11} - 3664 \nu^{10} + 7577 \nu^{9} + 21270 \nu^{8} - 191652 \nu^{7} + 479619 \nu^{6} + 441630 \nu^{5} - 5156136 \nu^{4} + 11030013 \nu^{3} - 251505 \nu^{2} - 66672882 \nu + 145614834$$$$)/4546773$$ $$\beta_{11}$$ $$=$$ $$($$$$1417 \nu^{11} - 5975 \nu^{10} + 5086 \nu^{9} + 56283 \nu^{8} - 326775 \nu^{7} + 552132 \nu^{6} + 1555911 \nu^{5} - 7611003 \nu^{4} + 15789654 \nu^{3} + 8793927 \nu^{2} - 107114886 \nu + 189606339$$$$)/10609137$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - 2 \beta_{10} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + \beta_{2} - 13$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} + 4 \beta_{2} - 12 \beta_{1} + 44$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{11} - 11 \beta_{10} + 9 \beta_{9} + 7 \beta_{8} - 17 \beta_{7} + \beta_{6} - 14 \beta_{5} + 23 \beta_{4} - 56 \beta_{3} + 22 \beta_{2} + 18 \beta_{1} - 43$$ $$\nu^{6}$$ $$=$$ $$16 \beta_{11} + 16 \beta_{10} + 21 \beta_{9} - 38 \beta_{8} - 59 \beta_{7} + 79 \beta_{6} + 28 \beta_{5} - 37 \beta_{4} + 349 \beta_{3} + 13 \beta_{2} + 84 \beta_{1} + 80$$ $$\nu^{7}$$ $$=$$ $$-56 \beta_{11} - 74 \beta_{10} - 12 \beta_{9} + 106 \beta_{8} - 128 \beta_{7} - 284 \beta_{6} + 34 \beta_{5} + 137 \beta_{4} + 349 \beta_{3} + 148 \beta_{2} - 174 \beta_{1} - 1246$$ $$\nu^{8}$$ $$=$$ $$178 \beta_{11} - 74 \beta_{10} + 270 \beta_{9} - 362 \beta_{8} + 370 \beta_{7} - 692 \beta_{6} + 184 \beta_{5} + 140 \beta_{4} + 2779 \beta_{3} + 22 \beta_{2} - 1818 \beta_{1} - 88$$ $$\nu^{9}$$ $$=$$ $$34 \beta_{11} + 34 \beta_{10} + 1254 \beta_{9} + 898 \beta_{8} + 220 \beta_{7} - 3071 \beta_{6} - 1178 \beta_{5} + 1574 \beta_{4} - 2162 \beta_{3} + 1462 \beta_{2} - 4839 \beta_{1} - 64$$ $$\nu^{10}$$ $$=$$ $$376 \beta_{11} + 1114 \beta_{10} + 3669 \beta_{9} - 2216 \beta_{8} - 6275 \beta_{7} + 2677 \beta_{6} - 272 \beta_{5} + 1226 \beta_{4} + 18709 \beta_{3} - 4388 \beta_{2} + 2049 \beta_{1} + 6305$$ $$\nu^{11}$$ $$=$$ $$1501 \beta_{11} + 7837 \beta_{10} - 4473 \beta_{9} + 11491 \beta_{8} - 19466 \beta_{7} - 3806 \beta_{6} - 3902 \beta_{5} - 6736 \beta_{4} + 69361 \beta_{3} - 9374 \beta_{2} + 10530 \beta_{1} - 12220$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1 + \beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 2.89597 + 0.783177i 2.65605 − 1.39478i 0.841761 + 2.87949i 0.459278 − 2.96464i −2.85525 − 0.920635i −2.99781 − 0.114662i 2.89597 − 0.783177i 2.65605 + 1.39478i 0.841761 − 2.87949i 0.459278 + 2.96464i −2.85525 + 0.920635i −2.99781 + 0.114662i
0 −2.89597 0.783177i 0 0 0 2.35650 4.08158i 0 7.77327 + 4.53611i 0
101.2 0 −2.65605 + 1.39478i 0 0 0 −1.41583 + 2.45229i 0 5.10917 7.40921i 0
101.3 0 −0.841761 2.87949i 0 0 0 5.98742 10.3705i 0 −7.58288 + 4.84768i 0
101.4 0 −0.459278 + 2.96464i 0 0 0 −4.13490 + 7.16186i 0 −8.57813 2.72318i 0
101.5 0 2.85525 + 0.920635i 0 0 0 −0.594587 + 1.02985i 0 7.30486 + 5.25728i 0
101.6 0 2.99781 + 0.114662i 0 0 0 0.801399 1.38806i 0 8.97371 + 0.687471i 0
401.1 0 −2.89597 + 0.783177i 0 0 0 2.35650 + 4.08158i 0 7.77327 4.53611i 0
401.2 0 −2.65605 1.39478i 0 0 0 −1.41583 2.45229i 0 5.10917 + 7.40921i 0
401.3 0 −0.841761 + 2.87949i 0 0 0 5.98742 + 10.3705i 0 −7.58288 4.84768i 0
401.4 0 −0.459278 2.96464i 0 0 0 −4.13490 7.16186i 0 −8.57813 + 2.72318i 0
401.5 0 2.85525 0.920635i 0 0 0 −0.594587 1.02985i 0 7.30486 5.25728i 0
401.6 0 2.99781 0.114662i 0 0 0 0.801399 + 1.38806i 0 8.97371 0.687471i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.p.c 12
3.b odd 2 1 2700.3.p.c 12
5.b even 2 1 180.3.o.b 12
5.c odd 4 2 900.3.u.c 24
9.c even 3 1 2700.3.p.c 12
9.d odd 6 1 inner 900.3.p.c 12
15.d odd 2 1 540.3.o.b 12
15.e even 4 2 2700.3.u.c 24
20.d odd 2 1 720.3.bs.b 12
45.h odd 6 1 180.3.o.b 12
45.h odd 6 1 1620.3.g.b 12
45.j even 6 1 540.3.o.b 12
45.j even 6 1 1620.3.g.b 12
45.k odd 12 2 2700.3.u.c 24
45.l even 12 2 900.3.u.c 24
60.h even 2 1 2160.3.bs.b 12
180.n even 6 1 720.3.bs.b 12
180.p odd 6 1 2160.3.bs.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b 12 5.b even 2 1
180.3.o.b 12 45.h odd 6 1
540.3.o.b 12 15.d odd 2 1
540.3.o.b 12 45.j even 6 1
720.3.bs.b 12 20.d odd 2 1
720.3.bs.b 12 180.n even 6 1
900.3.p.c 12 1.a even 1 1 trivial
900.3.p.c 12 9.d odd 6 1 inner
900.3.u.c 24 5.c odd 4 2
900.3.u.c 24 45.l even 12 2
1620.3.g.b 12 45.h odd 6 1
1620.3.g.b 12 45.j even 6 1
2160.3.bs.b 12 60.h even 2 1
2160.3.bs.b 12 180.p odd 6 1
2700.3.p.c 12 3.b odd 2 1
2700.3.p.c 12 9.c even 3 1
2700.3.u.c 24 15.e even 4 2
2700.3.u.c 24 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$531441 + 118098 T - 72171 T^{2} - 43740 T^{3} - 9720 T^{4} + 3078 T^{5} + 2709 T^{6} + 342 T^{7} - 120 T^{8} - 60 T^{9} - 11 T^{10} + 2 T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$6345361 + 2508924 T + 4581591 T^{2} + 41720 T^{3} + 2054970 T^{4} + 219204 T^{5} + 222711 T^{6} - 14346 T^{7} + 11340 T^{8} + 50 T^{9} + 141 T^{10} - 6 T^{11} + T^{12}$$
$11$ $$416649744 + 277766496 T - 114633792 T^{2} - 117573120 T^{3} + 81997920 T^{4} - 17682624 T^{5} + 695304 T^{6} + 245592 T^{7} - 11520 T^{8} - 5760 T^{9} + 888 T^{10} - 48 T^{11} + T^{12}$$
$13$ $$15716943091600 + 4050409492800 T + 1162525954800 T^{2} + 56153285600 T^{3} + 11003178600 T^{4} - 105229800 T^{5} + 89188080 T^{6} - 2179080 T^{7} + 371160 T^{8} - 13780 T^{9} + 1170 T^{10} - 30 T^{11} + T^{12}$$
$17$ $$191070131587344 + 8008056158688 T^{2} + 115286042880 T^{4} + 686092680 T^{6} + 1885500 T^{8} + 2328 T^{10} + T^{12}$$
$19$ $$( -19580204 - 2714376 T + 190260 T^{2} + 22040 T^{3} - 660 T^{4} - 36 T^{5} + T^{6} )^{2}$$
$23$ $$13626529936569 + 19289375271936 T + 8883080949483 T^{2} - 309687598080 T^{3} - 71707859010 T^{4} + 2224313136 T^{5} + 448266879 T^{6} - 10670778 T^{7} - 1020780 T^{8} + 29250 T^{9} + 1653 T^{10} - 78 T^{11} + T^{12}$$
$29$ $$3840796442025 + 2262093378750 T + 291968600625 T^{2} - 89598656250 T^{3} - 9542427750 T^{4} + 2515596750 T^{5} + 146349585 T^{6} - 67695750 T^{7} + 6914250 T^{8} - 348750 T^{9} + 9825 T^{10} - 150 T^{11} + T^{12}$$
$31$ $$177690926165776 - 3388931881632 T + 3223062117264 T^{2} - 53867720480 T^{3} + 42832194480 T^{4} - 624922992 T^{5} + 244502664 T^{6} - 809928 T^{7} + 980820 T^{8} - 4400 T^{9} + 1224 T^{10} + 12 T^{11} + T^{12}$$
$37$ $$( -2144769884 + 12967464 T + 6681840 T^{2} + 6640 T^{3} - 4890 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$41$ $$2909846419201568025 - 328259372200165350 T + 8722834129709325 T^{2} + 408461646508650 T^{3} - 18006800640750 T^{4} - 792096079950 T^{5} + 64517932485 T^{6} - 1391318370 T^{7} + 1395630 T^{8} + 319950 T^{9} - 855 T^{10} - 90 T^{11} + T^{12}$$
$43$ $$67\!\cdots\!96$$$$+ 26603059037328012384 T + 1274715378500498256 T^{2} + 27629774869547840 T^{3} + 879393801084120 T^{4} + 15769155299544 T^{5} + 399032156256 T^{6} + 5305685904 T^{7} + 99617760 T^{8} + 1024100 T^{9} + 16326 T^{10} + 114 T^{11} + T^{12}$$
$47$ $$47\!\cdots\!49$$$$+ 24495191940975499926 T - 515270171690466507 T^{2} - 48252401152784010 T^{3} + 1516918493034900 T^{4} + 6464648036466 T^{5} - 482806306161 T^{6} - 685586268 T^{7} + 114510870 T^{8} - 150300 T^{9} - 12477 T^{10} + 12 T^{11} + T^{12}$$
$53$ $$143734073215981824 + 2246720501518848 T^{2} + 11548556026560 T^{4} + 22547976480 T^{6} + 19480320 T^{8} + 7368 T^{10} + T^{12}$$
$59$ $$29961364014979190784 - 13714886927293968384 T + 2281313174787836928 T^{2} - 86349932128281600 T^{3} + 571616406950400 T^{4} + 28624243366656 T^{5} - 273988236096 T^{6} - 8209340928 T^{7} + 121890240 T^{8} + 624960 T^{9} - 12252 T^{10} - 48 T^{11} + T^{12}$$
$61$ $$1183261349304105241 - 83635824013613298 T + 24249323069901969 T^{2} - 12353724626270 T^{3} + 319617724773510 T^{4} - 8694965133018 T^{5} + 521233963269 T^{6} + 3428562438 T^{7} + 128958930 T^{8} + 427210 T^{9} + 16029 T^{10} + 78 T^{11} + T^{12}$$
$67$ $$50\!\cdots\!41$$$$-$$$$19\!\cdots\!82$$$$T + 11561778337659779229 T^{2} - 164717601842137630 T^{3} + 9568793447907840 T^{4} - 173185051603842 T^{5} + 4136604374139 T^{6} - 37791483768 T^{7} + 483761070 T^{8} - 3122980 T^{9} + 35919 T^{10} - 168 T^{11} + T^{12}$$
$71$ $$28684371824616021264 + 260491775334992928 T^{2} + 741856665870000 T^{4} + 830026832040 T^{6} + 327262680 T^{8} + 33768 T^{10} + T^{12}$$
$73$ $$( -2761132736 + 121304448 T + 31430640 T^{2} + 24400 T^{3} - 11340 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$79$ $$86\!\cdots\!00$$$$-$$$$38\!\cdots\!00$$$$T + 17178092590185600000 T^{2} - 333973732296704000 T^{3} + 8491310899776000 T^{4} - 115027609392000 T^{5} + 2764546317600 T^{6} - 26217043200 T^{7} + 375048000 T^{8} - 1936640 T^{9} + 27360 T^{10} - 120 T^{11} + T^{12}$$
$83$ $$15\!\cdots\!29$$$$+$$$$18\!\cdots\!88$$$$T + 8460209320379254827 T^{2} + 146272005497773980 T^{3} - 795950419116150 T^{4} - 50280198471972 T^{5} + 68476782051 T^{6} + 13214564694 T^{7} + 63358560 T^{8} - 1397070 T^{9} - 7923 T^{10} + 114 T^{11} + T^{12}$$
$89$ $$13\!\cdots\!29$$$$+ 8790215611160750058 T^{2} + 9136530240432975 T^{4} + 3650550649740 T^{6} + 652077135 T^{8} + 48618 T^{10} + T^{12}$$
$97$ $$60\!\cdots\!16$$$$-$$$$75\!\cdots\!84$$$$T +$$$$29\!\cdots\!16$$$$T^{2} - 856008631323760000 T^{3} + 62132803484109120 T^{4} - 43854635430144 T^{5} + 9564389628576 T^{6} + 24019258896 T^{7} + 822735000 T^{8} + 1436480 T^{9} + 38736 T^{10} + 96 T^{11} + T^{12}$$