# Properties

 Label 1470.2.n.j Level $1470$ Weight $2$ Character orbit 1470.n Analytic conductor $11.738$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.7652750400000000.1 Defining polynomial: $$x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} - 112 x^{3} + 288 x^{2} - 192 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 210) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{1} q^{3} + \beta_{8} q^{4} + \beta_{5} q^{5} - q^{6} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{1} q^{3} + \beta_{8} q^{4} + \beta_{5} q^{5} - q^{6} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{8} ) q^{9} + ( -\beta_{4} - \beta_{11} ) q^{10} + ( \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{11} -\beta_{3} q^{12} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{13} + \beta_{11} q^{15} + ( -1 + \beta_{8} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{11} ) q^{17} + \beta_{1} q^{18} + ( -1 - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{19} + ( \beta_{5} + \beta_{6} ) q^{20} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{22} + ( -3 \beta_{3} - \beta_{5} + \beta_{10} ) q^{23} -\beta_{8} q^{24} + ( \beta_{2} + 2 \beta_{7} ) q^{25} + ( 1 + \beta_{2} - \beta_{4} - \beta_{8} - \beta_{9} ) q^{26} + ( -\beta_{1} + \beta_{3} ) q^{27} + ( -4 - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{29} -\beta_{5} q^{30} + ( \beta_{2} - \beta_{4} - \beta_{11} ) q^{31} -\beta_{1} q^{32} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} ) q^{33} + ( 2 - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} ) q^{34} + q^{36} + ( 3 \beta_{3} + \beta_{5} - \beta_{10} ) q^{37} + ( -\beta_{1} - \beta_{6} - \beta_{7} ) q^{38} + ( -\beta_{2} + \beta_{4} + \beta_{8} + \beta_{11} ) q^{39} -\beta_{4} q^{40} + ( 3 - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{44} -\beta_{6} q^{45} + ( -\beta_{2} + \beta_{4} - 3 \beta_{8} + \beta_{11} ) q^{46} + ( \beta_{3} - 3 \beta_{5} + 3 \beta_{10} ) q^{47} + ( \beta_{1} - \beta_{3} ) q^{48} + ( \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{50} + ( -2 + 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} ) q^{51} + ( \beta_{1} + \beta_{6} + \beta_{7} ) q^{52} + ( -3 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} ) q^{53} + ( -1 + \beta_{8} ) q^{54} + ( -5 - 5 \beta_{1} + 5 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{9} - 3 \beta_{10} ) q^{55} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{57} + ( \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{9} ) q^{58} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} + 4 \beta_{8} - 2 \beta_{11} ) q^{59} + ( \beta_{4} + \beta_{11} ) q^{60} + ( 2 - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{61} + ( \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{62} - q^{64} + ( -5 + \beta_{2} + \beta_{4} + 5 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{65} + ( -\beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{66} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{11} ) q^{67} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{9} ) q^{68} + ( 3 + \beta_{9} - \beta_{11} ) q^{69} + 6 q^{71} + \beta_{3} q^{72} -4 \beta_{1} q^{73} + ( \beta_{2} - \beta_{4} + 3 \beta_{8} - \beta_{11} ) q^{74} + ( -2 \beta_{2} + 2 \beta_{9} + \beta_{10} ) q^{75} + ( -1 + \beta_{9} - \beta_{11} ) q^{76} + ( -\beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{78} + ( 8 - \beta_{2} + \beta_{4} - 8 \beta_{8} + \beta_{9} ) q^{79} + \beta_{6} q^{80} -\beta_{8} q^{81} + ( 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{10} ) q^{82} + ( 4 \beta_{1} - 4 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{83} + ( -10 \beta_{1} + 10 \beta_{3} - 2 \beta_{7} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{85} + ( 2 - 2 \beta_{8} ) q^{86} + ( 4 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{11} ) q^{87} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{88} + ( 2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{89} -\beta_{11} q^{90} + ( 3 \beta_{1} - 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{92} + ( -\beta_{5} + \beta_{10} ) q^{93} + ( -3 \beta_{2} + 3 \beta_{4} + \beta_{8} + 3 \beta_{11} ) q^{94} + ( 5 \beta_{1} + 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{95} + ( 1 - \beta_{8} ) q^{96} + ( 8 \beta_{1} - 8 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{97} + ( -1 - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 6q^{4} - 12q^{6} + 6q^{9} + O(q^{10})$$ $$12q + 6q^{4} - 12q^{6} + 6q^{9} - 6q^{11} - 6q^{16} - 6q^{19} - 6q^{24} + 6q^{26} - 48q^{29} + 24q^{34} + 12q^{36} + 6q^{39} + 36q^{41} + 6q^{44} - 18q^{46} - 12q^{51} - 6q^{54} - 60q^{55} + 24q^{59} + 12q^{61} - 12q^{64} - 30q^{65} + 6q^{66} + 36q^{69} + 72q^{71} + 18q^{74} - 12q^{76} + 48q^{79} - 6q^{81} + 12q^{86} + 12q^{89} + 6q^{94} + 6q^{96} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} - 112 x^{3} + 288 x^{2} - 192 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$10967 \nu^{11} + 32246 \nu^{10} - 320538 \nu^{9} + 1200782 \nu^{8} + 26475 \nu^{7} + 192316 \nu^{6} - 5356872 \nu^{5} + 25802216 \nu^{4} + 4101028 \nu^{3} - 2644000 \nu^{2} - 7541936 \nu + 11668096$$$$)/6321728$$ $$\beta_{2}$$ $$=$$ $$($$$$71537 \nu^{11} - 395262 \nu^{10} + 1115946 \nu^{9} - 559878 \nu^{8} + 1522973 \nu^{7} - 7374100 \nu^{6} + 23734728 \nu^{5} - 5540672 \nu^{4} + 7241340 \nu^{3} - 10074032 \nu^{2} + 29588496 \nu + 376704$$$$)/6321728$$ $$\beta_{3}$$ $$=$$ $$($$$$38425 \nu^{11} - 170111 \nu^{10} + 367550 \nu^{9} + 334360 \nu^{8} + 552915 \nu^{7} - 3125365 \nu^{6} + 8245842 \nu^{5} + 9778420 \nu^{4} + 2618720 \nu^{3} - 1196060 \nu^{2} + 4633520 \nu + 2629264$$$$)/3160864$$ $$\beta_{4}$$ $$=$$ $$($$$$-87643 \nu^{11} + 504878 \nu^{10} - 1490814 \nu^{9} + 1056954 \nu^{8} - 2126039 \nu^{7} + 9354400 \nu^{6} - 31186560 \nu^{5} + 15651488 \nu^{4} - 14480484 \nu^{3} + 11904288 \nu^{2} - 29339888 \nu + 21343872$$$$)/6321728$$ $$\beta_{5}$$ $$=$$ $$($$$$44272 \nu^{11} - 237241 \nu^{10} + 629192 \nu^{9} - 106362 \nu^{8} + 510126 \nu^{7} - 4079825 \nu^{6} + 13520382 \nu^{5} + 801716 \nu^{4} - 1987220 \nu^{3} - 690028 \nu^{2} + 15880448 \nu - 1612208$$$$)/3160864$$ $$\beta_{6}$$ $$=$$ $$($$$$112417 \nu^{11} - 805130 \nu^{10} + 2628966 \nu^{9} - 2969054 \nu^{8} + 1600117 \nu^{7} - 14110440 \nu^{6} + 52377004 \nu^{5} - 54894792 \nu^{4} - 17165628 \nu^{3} - 21185216 \nu^{2} + 32776656 \nu - 33288832$$$$)/6321728$$ $$\beta_{7}$$ $$=$$ $$($$$$131935 \nu^{11} - 773178 \nu^{10} + 2298830 \nu^{9} - 1687650 \nu^{8} + 2890131 \nu^{7} - 13565876 \nu^{6} + 46490256 \nu^{5} - 24737320 \nu^{4} + 15014452 \nu^{3} - 6351360 \nu^{2} + 31227728 \nu - 20220800$$$$)/6321728$$ $$\beta_{8}$$ $$=$$ $$($$$$227 \nu^{11} - 1290 \nu^{10} + 3622 \nu^{9} - 1778 \nu^{8} + 3655 \nu^{7} - 23804 \nu^{6} + 75424 \nu^{5} - 20464 \nu^{4} + 228 \nu^{3} - 36720 \nu^{2} + 54704 \nu - 16768$$$$)/8768$$ $$\beta_{9}$$ $$=$$ $$($$$$-43824 \nu^{11} + 276857 \nu^{10} - 829457 \nu^{9} + 642506 \nu^{8} - 534986 \nu^{7} + 4941719 \nu^{6} - 16904415 \nu^{5} + 10669054 \nu^{4} + 5302254 \nu^{3} + 7880932 \nu^{2} - 12722696 \nu + 9604288$$$$)/1580432$$ $$\beta_{10}$$ $$=$$ $$($$$$-128463 \nu^{11} + 645645 \nu^{10} - 1597836 \nu^{9} - 220844 \nu^{8} - 1660469 \nu^{7} + 11871603 \nu^{6} - 34275032 \nu^{5} - 14186852 \nu^{4} + 1994316 \nu^{3} + 14559756 \nu^{2} - 17921792 \nu - 2470736$$$$)/3160864$$ $$\beta_{11}$$ $$=$$ $$($$$$-80946 \nu^{11} + 432381 \nu^{10} - 1148873 \nu^{9} + 259810 \nu^{8} - 1234124 \nu^{7} + 7919495 \nu^{6} - 24050043 \nu^{5} - 800286 \nu^{4} - 1154946 \nu^{3} + 10439620 \nu^{2} - 13207560 \nu + 782560$$$$)/1580432$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - \beta_{4} + 12 \beta_{3} + \beta_{2} - 12 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$-5 \beta_{11} + 5 \beta_{10} - \beta_{9} - 5 \beta_{8} - 4 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 8 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} - 3$$ $$\nu^{4}$$ $$=$$ $$($$$$-20 \beta_{11} + 32 \beta_{10} + 20 \beta_{9} - 10 \beta_{7} + 32 \beta_{6} + 10 \beta_{5} + 23 \beta_{4} + 28 \beta_{3} + 23 \beta_{2} + 28 \beta_{1} - 99$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$45 \beta_{11} + 45 \beta_{10} + 145 \beta_{9} + 191 \beta_{8} + 60 \beta_{7} + 145 \beta_{6} + 125 \beta_{5} + 105 \beta_{4} - 98 \beta_{3} - 20 \beta_{2} + 289 \beta_{1} - 289$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$138 \beta_{11} - 68 \beta_{10} + 138 \beta_{9} + 290 \beta_{8} + 160 \beta_{7} + 68 \beta_{6} + 160 \beta_{5} + 105 \beta_{4} - 338 \beta_{3} - 105 \beta_{2} + 338 \beta_{1} - 145$$ $$\nu^{7}$$ $$=$$ $$($$$$1579 \beta_{11} - 1579 \beta_{10} + 591 \beta_{9} + 2393 \beta_{8} + 1388 \beta_{7} - 591 \beta_{6} + 849 \beta_{5} + 191 \beta_{4} - 3466 \beta_{3} - 1440 \beta_{2} + 1073 \beta_{1} + 1073$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$2224 \beta_{11} - 5104 \beta_{10} - 2224 \beta_{9} + 1034 \beta_{7} - 5104 \beta_{6} - 1034 \beta_{5} - 3373 \beta_{4} - 5768 \beta_{3} - 3373 \beta_{2} - 5768 \beta_{1} + 11361$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$-2481 \beta_{11} - 2481 \beta_{10} - 6069 \beta_{9} - 9855 \beta_{8} - 3624 \beta_{7} - 6069 \beta_{6} - 5361 \beta_{5} - 6105 \beta_{4} + 4034 \beta_{3} + 708 \beta_{2} - 13889 \beta_{1} + 13889$$ $$\nu^{10}$$ $$=$$ $$($$$$-62050 \beta_{11} + 25420 \beta_{10} - 62050 \beta_{9} - 144778 \beta_{8} - 66620 \beta_{7} - 25420 \beta_{6} - 66620 \beta_{5} - 50645 \beta_{4} + 132702 \beta_{3} + 50645 \beta_{2} - 132702 \beta_{1} + 72389$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-215513 \beta_{11} + 215513 \beta_{10} - 91845 \beta_{9} - 361723 \beta_{8} - 190504 \beta_{7} + 91845 \beta_{6} - 134439 \beta_{5} - 25009 \beta_{4} + 502082 \beta_{3} + 226284 \beta_{2} - 140359 \beta_{1} - 140359$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{8}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.685661 + 0.685661i 2.45845 − 2.45845i 0.593239 − 0.593239i 1.68566 + 1.68566i −1.45845 − 1.45845i 0.406761 + 0.406761i −0.685661 − 0.685661i 2.45845 + 2.45845i 0.593239 + 0.593239i 1.68566 − 1.68566i −1.45845 + 1.45845i 0.406761 − 0.406761i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −2.20942 0.344208i −1.00000 0 1.00000i 0.500000 0.866025i 1.74131 + 1.40280i
79.2 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0.806615 + 2.08551i −1.00000 0 1.00000i 0.500000 0.866025i 0.344208 2.20942i
79.3 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.40280 1.74131i −1.00000 0 1.00000i 0.500000 0.866025i −2.08551 + 0.806615i
79.4 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.20942 + 0.344208i −1.00000 0 1.00000i 0.500000 0.866025i −2.08551 0.806615i
79.5 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.806615 2.08551i −1.00000 0 1.00000i 0.500000 0.866025i 1.74131 1.40280i
79.6 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.40280 + 1.74131i −1.00000 0 1.00000i 0.500000 0.866025i 0.344208 + 2.20942i
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −2.20942 + 0.344208i −1.00000 0 1.00000i 0.500000 + 0.866025i 1.74131 1.40280i
949.2 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0.806615 2.08551i −1.00000 0 1.00000i 0.500000 + 0.866025i 0.344208 + 2.20942i
949.3 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 1.40280 + 1.74131i −1.00000 0 1.00000i 0.500000 + 0.866025i −2.08551 0.806615i
949.4 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i −2.20942 0.344208i −1.00000 0 1.00000i 0.500000 + 0.866025i −2.08551 + 0.806615i
949.5 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.806615 + 2.08551i −1.00000 0 1.00000i 0.500000 + 0.866025i 1.74131 + 1.40280i
949.6 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 1.40280 1.74131i −1.00000 0 1.00000i 0.500000 + 0.866025i 0.344208 2.20942i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 949.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.j 12
5.b even 2 1 inner 1470.2.n.j 12
7.b odd 2 1 210.2.n.b 12
7.c even 3 1 1470.2.g.h 6
7.c even 3 1 inner 1470.2.n.j 12
7.d odd 6 1 210.2.n.b 12
7.d odd 6 1 1470.2.g.i 6
21.c even 2 1 630.2.u.f 12
21.g even 6 1 630.2.u.f 12
28.d even 2 1 1680.2.di.c 12
28.f even 6 1 1680.2.di.c 12
35.c odd 2 1 210.2.n.b 12
35.f even 4 1 1050.2.i.u 6
35.f even 4 1 1050.2.i.v 6
35.i odd 6 1 210.2.n.b 12
35.i odd 6 1 1470.2.g.i 6
35.j even 6 1 1470.2.g.h 6
35.j even 6 1 inner 1470.2.n.j 12
35.k even 12 1 1050.2.i.u 6
35.k even 12 1 1050.2.i.v 6
35.k even 12 1 7350.2.a.dn 3
35.k even 12 1 7350.2.a.dq 3
35.l odd 12 1 7350.2.a.do 3
35.l odd 12 1 7350.2.a.dp 3
105.g even 2 1 630.2.u.f 12
105.p even 6 1 630.2.u.f 12
140.c even 2 1 1680.2.di.c 12
140.s even 6 1 1680.2.di.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b 12 7.b odd 2 1
210.2.n.b 12 7.d odd 6 1
210.2.n.b 12 35.c odd 2 1
210.2.n.b 12 35.i odd 6 1
630.2.u.f 12 21.c even 2 1
630.2.u.f 12 21.g even 6 1
630.2.u.f 12 105.g even 2 1
630.2.u.f 12 105.p even 6 1
1050.2.i.u 6 35.f even 4 1
1050.2.i.u 6 35.k even 12 1
1050.2.i.v 6 35.f even 4 1
1050.2.i.v 6 35.k even 12 1
1470.2.g.h 6 7.c even 3 1
1470.2.g.h 6 35.j even 6 1
1470.2.g.i 6 7.d odd 6 1
1470.2.g.i 6 35.i odd 6 1
1470.2.n.j 12 1.a even 1 1 trivial
1470.2.n.j 12 5.b even 2 1 inner
1470.2.n.j 12 7.c even 3 1 inner
1470.2.n.j 12 35.j even 6 1 inner
1680.2.di.c 12 28.d even 2 1
1680.2.di.c 12 28.f even 6 1
1680.2.di.c 12 140.c even 2 1
1680.2.di.c 12 140.s even 6 1
7350.2.a.dn 3 35.k even 12 1
7350.2.a.do 3 35.l odd 12 1
7350.2.a.dp 3 35.l odd 12 1
7350.2.a.dq 3 35.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{6} + 3 T_{11}^{5} + 36 T_{11}^{4} + 17 T_{11}^{3} + 876 T_{11}^{2} + 1323 T_{11} + 2401$$ $$T_{17}^{12} - 132 T_{17}^{10} + 11856 T_{17}^{8} - 587008 T_{17}^{6} + 21236736 T_{17}^{4} - 411942912 T_{17}^{2} + 5473632256$$ $$T_{19}^{6} + 3 T_{19}^{5} + 21 T_{19}^{4} + 12 T_{19}^{3} + 216 T_{19}^{2} + 288 T_{19} + 576$$ $$T_{31}^{6} + 15 T_{31}^{4} + 20 T_{31}^{3} + 225 T_{31}^{2} + 150 T_{31} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$( 1 - T^{2} + T^{4} )^{3}$$
$5$ $$( 125 + 20 T^{3} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$( 2401 + 1323 T + 876 T^{2} + 17 T^{3} + 36 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$13$ $$( 576 + 288 T^{2} + 33 T^{4} + T^{6} )^{2}$$
$17$ $$5473632256 - 411942912 T^{2} + 21236736 T^{4} - 587008 T^{6} + 11856 T^{8} - 132 T^{10} + T^{12}$$
$19$ $$( 576 + 288 T + 216 T^{2} + 12 T^{3} + 21 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$23$ $$4096 - 18432 T^{2} + 79296 T^{4} - 16288 T^{6} + 2961 T^{8} - 57 T^{10} + T^{12}$$
$29$ $$( 24 + 33 T + 12 T^{2} + T^{3} )^{4}$$
$31$ $$( 100 + 150 T + 225 T^{2} + 20 T^{3} + 15 T^{4} + T^{6} )^{2}$$
$37$ $$4096 - 18432 T^{2} + 79296 T^{4} - 16288 T^{6} + 2961 T^{8} - 57 T^{10} + T^{12}$$
$41$ $$( 128 - 48 T - 9 T^{2} + T^{3} )^{4}$$
$43$ $$( 4 + T^{2} )^{6}$$
$47$ $$26639462656 - 3239511168 T^{2} + 349385136 T^{4} - 5092072 T^{6} + 54681 T^{8} - 273 T^{10} + T^{12}$$
$53$ $$436880018961 - 15389341227 T^{2} + 365619366 T^{4} - 4894623 T^{6} + 48006 T^{8} - 267 T^{10} + T^{12}$$
$59$ $$( 190096 - 11772 T + 5961 T^{2} - 548 T^{3} + 171 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$61$ $$( 506944 - 76896 T + 15936 T^{2} - 776 T^{3} + 144 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$67$ $$23612624896 - 2515172352 T^{2} + 229188096 T^{4} - 3817408 T^{6} + 47136 T^{8} - 252 T^{10} + T^{12}$$
$71$ $$( -6 + T )^{12}$$
$73$ $$( 256 - 16 T^{2} + T^{4} )^{3}$$
$79$ $$( 161604 - 71154 T + 21681 T^{2} - 3444 T^{3} + 399 T^{4} - 24 T^{5} + T^{6} )^{2}$$
$83$ $$( 7396 + 2793 T^{2} + 198 T^{4} + T^{6} )^{2}$$
$89$ $$( 153664 - 42336 T + 14016 T^{2} - 136 T^{3} + 144 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$97$ $$( 12544 + 8313 T^{2} + 342 T^{4} + T^{6} )^{2}$$