# Properties

 Label 735.2.q.f Level $735$ Weight $2$ Character orbit 735.q Analytic conductor $5.869$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.89539436150784.1 Defining polynomial: $$x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} - \beta_{4} ) q^{2} + ( \beta_{5} + \beta_{7} ) q^{3} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{2} q^{6} + ( -\beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{8} + \beta_{8} q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} - \beta_{4} ) q^{2} + ( \beta_{5} + \beta_{7} ) q^{3} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{2} q^{6} + ( -\beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{8} + \beta_{8} q^{9} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{10} + ( -2 + 2 \beta_{8} ) q^{11} + ( \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{12} + ( 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{13} + ( \beta_{2} - \beta_{6} ) q^{15} + ( -4 \beta_{1} + 4 \beta_{2} - 3 \beta_{8} ) q^{16} + ( 2 \beta_{3} - \beta_{6} - \beta_{9} + \beta_{11} ) q^{17} -\beta_{3} q^{18} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{19} + ( 4 - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{20} + 2 \beta_{4} q^{22} + ( -2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{23} + ( 2 - \beta_{1} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{24} + ( 1 + 2 \beta_{1} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{25} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{26} + \beta_{5} q^{27} + ( -2 - 2 \beta_{6} - 2 \beta_{10} ) q^{29} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{30} + ( 2 + 2 \beta_{1} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{31} + ( \beta_{3} - 8 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} ) q^{32} -2 \beta_{7} q^{33} + ( -4 - 2 \beta_{2} + \beta_{6} + \beta_{10} ) q^{34} + ( 1 - \beta_{6} - \beta_{10} ) q^{36} + ( 4 \beta_{3} + 4 \beta_{4} ) q^{37} + ( -4 \beta_{5} + \beta_{6} - 4 \beta_{7} + \beta_{9} - \beta_{11} ) q^{38} + ( 2 + 2 \beta_{1} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{39} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{40} + ( 4 \beta_{2} - \beta_{6} - \beta_{10} ) q^{41} + ( 4 \beta_{4} + 4 \beta_{6} - 4 \beta_{10} ) q^{43} + ( 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{44} + ( \beta_{3} - \beta_{9} ) q^{45} + ( 2 - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{46} + ( -4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( -4 \beta_{4} - 3 \beta_{5} ) q^{48} + ( -2 - \beta_{4} + 6 \beta_{5} - \beta_{6} + 3 \beta_{10} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{51} + ( -2 \beta_{3} - 6 \beta_{5} + \beta_{6} - 6 \beta_{7} + \beta_{9} - \beta_{11} ) q^{52} + ( 2 \beta_{3} + 6 \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{9} - \beta_{11} ) q^{53} + ( \beta_{1} - \beta_{2} ) q^{54} + ( -2 \beta_{4} + 2 \beta_{10} ) q^{55} + ( -2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{10} ) q^{57} + ( -2 \beta_{3} - 2 \beta_{4} + 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{58} + ( -4 - 4 \beta_{1} + 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} ) q^{59} + ( 2 + \beta_{1} - 2 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{60} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{61} + ( 8 \beta_{5} - 3 \beta_{6} + 3 \beta_{10} ) q^{62} + ( -1 + 4 \beta_{2} + 3 \beta_{6} + 3 \beta_{10} ) q^{64} + ( 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{7} - 6 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} + 2 \beta_{1} q^{66} + ( -4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{67} + ( 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{68} + ( -2 - \beta_{6} - \beta_{10} ) q^{69} + 2 q^{71} + ( -\beta_{3} - \beta_{4} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{72} + ( -2 \beta_{3} + 6 \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{9} - \beta_{11} ) q^{73} + ( -12 + 4 \beta_{6} + 12 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} ) q^{74} + ( 2 \beta_{3} + 2 \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{75} + ( 2 + 2 \beta_{2} + 3 \beta_{6} + 3 \beta_{10} ) q^{76} + ( -4 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{10} ) q^{78} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{79} + ( 4 - 4 \beta_{1} + \beta_{3} + 8 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 4 \beta_{11} ) q^{80} + ( -1 + \beta_{8} ) q^{81} + ( -2 \beta_{3} - 2 \beta_{4} - 10 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{82} + ( -4 \beta_{4} + 4 \beta_{5} ) q^{83} + ( 6 + 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{85} + ( -8 \beta_{1} + 8 \beta_{2} + 4 \beta_{8} ) q^{86} + ( -2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{87} + ( -2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{88} + ( -4 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{10} ) q^{90} + ( -4 \beta_{4} - 6 \beta_{5} - \beta_{6} + \beta_{10} ) q^{92} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{93} + ( 4 - 2 \beta_{6} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{94} + ( -2 + 6 \beta_{5} - \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{95} + ( -\beta_{1} + \beta_{2} - 8 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{96} + ( -2 \beta_{4} + 10 \beta_{5} + 3 \beta_{6} - 3 \beta_{10} ) q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 10 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{9} + O(q^{10})$$ $$12 q + 10 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{9} - 12 q^{10} - 12 q^{11} - 26 q^{16} - 12 q^{19} + 60 q^{20} + 18 q^{24} + 2 q^{25} + 20 q^{26} - 8 q^{29} + 10 q^{30} + 4 q^{31} - 48 q^{34} + 20 q^{36} + 12 q^{39} + 4 q^{40} - 8 q^{41} + 20 q^{44} - 2 q^{45} + 16 q^{46} - 32 q^{50} + 2 q^{54} - 8 q^{55} - 32 q^{59} + 8 q^{60} - 12 q^{61} - 52 q^{64} - 32 q^{65} - 4 q^{66} - 16 q^{69} + 24 q^{71} - 88 q^{74} + 8 q^{75} - 8 q^{76} + 24 q^{79} + 46 q^{80} - 6 q^{81} + 64 q^{85} + 8 q^{86} - 28 q^{89} - 24 q^{90} + 32 q^{94} - 4 q^{95} - 58 q^{96} - 24 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} + 3 \nu^{8} - 62 \nu^{7} + 112 \nu^{6} - 276 \nu^{5} + 338 \nu^{4} + 482 \nu^{3} - 170 \nu^{2} + 164 \nu - 164$$$$)/460$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{11} + 2 \nu^{10} - 8 \nu^{9} + 33 \nu^{8} - 13 \nu^{7} + 21 \nu^{6} - 96 \nu^{5} - 154 \nu^{4} + 150 \nu^{3} - 50 \nu^{2} + 50 \nu - 68$$$$)/230$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{11} + 21 \nu^{10} - 21 \nu^{9} + 61 \nu^{8} - 80 \nu^{7} - 130 \nu^{6} + 230 \nu^{5} - 58 \nu^{4} + 18 \nu^{3} - 298 \nu^{2} - 8 \nu + 8$$$$)/460$$ $$\beta_{4}$$ $$=$$ $$($$$$10 \nu^{11} - 10 \nu^{10} - 6 \nu^{9} - 50 \nu^{8} - 50 \nu^{7} + 263 \nu^{6} + 20 \nu^{5} + 80 \nu^{4} + 216 \nu^{3} + 20 \nu^{2} - 20 \nu - 28$$$$)/230$$ $$\beta_{5}$$ $$=$$ $$($$$$31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142$$$$)/230$$ $$\beta_{6}$$ $$=$$ $$($$$$19 \nu^{11} - 19 \nu^{10} + 7 \nu^{9} - 118 \nu^{8} - 72 \nu^{7} + 318 \nu^{6} + 130 \nu^{5} + 290 \nu^{4} + 967 \nu^{3} + 84 \nu^{2} - 84 \nu + 48$$$$)/115$$ $$\beta_{7}$$ $$=$$ $$($$$$-74 \nu^{11} + 105 \nu^{10} - 61 \nu^{9} + 505 \nu^{8} + 52 \nu^{7} - 1384 \nu^{6} - 78 \nu^{5} - 1158 \nu^{4} - 2246 \nu^{3} + 1062 \nu^{2} + 296 \nu + 296$$$$)/460$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12$$$$)/20$$ $$\beta_{9}$$ $$=$$ $$($$$$85 \nu^{11} - 88 \nu^{10} + 88 \nu^{9} - 674 \nu^{8} - 162 \nu^{7} + 1042 \nu^{6} + 966 \nu^{5} + 2232 \nu^{4} + 2908 \nu^{3} + 688 \nu^{2} + 1028 \nu - 1028$$$$)/460$$ $$\beta_{10}$$ $$=$$ $$($$$$-43 \nu^{11} + 43 \nu^{10} - 34 \nu^{9} + 330 \nu^{8} + 100 \nu^{7} - 572 \nu^{6} - 546 \nu^{5} - 1034 \nu^{4} - 1490 \nu^{3} - 316 \nu^{2} + 316 \nu + 516$$$$)/230$$ $$\beta_{11}$$ $$=$$ $$($$$$225 \nu^{11} - 378 \nu^{10} + 330 \nu^{9} - 1680 \nu^{8} + 374 \nu^{7} + 3630 \nu^{6} - 630 \nu^{5} + 3948 \nu^{4} + 6540 \nu^{3} - 2940 \nu^{2} + 756 \nu - 900$$$$)/460$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{9} + 4 \beta_{7} - \beta_{6} + 4 \beta_{5} + 2 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} + 5 \beta_{2} - 5 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-5 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + 9 \beta_{5} + 8 \beta_{3} - 8 \beta_{1} - 9$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{10} - 8 \beta_{6} + 28 \beta_{5} - 22 \beta_{4}$$ $$\nu^{7}$$ $$=$$ $$-22 \beta_{11} + 39 \beta_{8} - 39 \beta_{7} - 33 \beta_{4} - 33 \beta_{3} + 33 \beta_{2} - 33 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-33 \beta_{11} - 33 \beta_{9} + 116 \beta_{8} + 33 \beta_{6} - 94 \beta_{1} - 116$$ $$\nu^{9}$$ $$=$$ $$94 \beta_{10} + 166 \beta_{5} - 138 \beta_{4} - 138 \beta_{2} - 166$$ $$\nu^{10}$$ $$=$$ $$-138 \beta_{11} + 138 \beta_{10} + 138 \beta_{9} - 486 \beta_{7} - 398 \beta_{4} - 398 \beta_{3}$$ $$\nu^{11}$$ $$=$$ $$-398 \beta_{11} + 702 \beta_{8} - 702 \beta_{7} + 398 \beta_{6} - 702 \beta_{5} - 580 \beta_{3} - 580 \beta_{1} - 702$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-\beta_{8}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.312819 + 1.16746i 1.98293 − 0.531325i −0.147520 − 0.550552i 0.550552 − 0.147520i −0.531325 − 1.98293i −1.16746 + 0.312819i 0.312819 − 1.16746i 1.98293 + 0.531325i −0.147520 + 0.550552i 0.550552 + 0.147520i −0.531325 + 1.98293i −1.16746 − 0.312819i
−2.34630 1.35464i −0.866025 + 0.500000i 2.67009 + 4.62473i 1.55199 1.60976i 2.70928 0 9.04945i 0.500000 0.866025i −5.82208 + 1.67458i
79.2 −1.64823 0.951606i 0.866025 0.500000i 0.811108 + 1.40488i 2.07321 + 0.837733i −1.90321 0 0.719004i 0.500000 0.866025i −2.61994 3.35366i
79.3 −0.167954 0.0969683i −0.866025 + 0.500000i −0.981194 1.69948i 0.710109 + 2.12032i 0.193937 0 0.768452i 0.500000 0.866025i 0.0863379 0.424974i
79.4 0.167954 + 0.0969683i 0.866025 0.500000i −0.981194 1.69948i −2.19130 + 0.445186i 0.193937 0 0.768452i 0.500000 0.866025i −0.411207 0.137716i
79.5 1.64823 + 0.951606i −0.866025 + 0.500000i 0.811108 + 1.40488i −1.76210 1.37659i −1.90321 0 0.719004i 0.500000 0.866025i −1.59438 3.94576i
79.6 2.34630 + 1.35464i 0.866025 0.500000i 2.67009 + 4.62473i 0.618092 2.14894i 2.70928 0 9.04945i 0.500000 0.866025i 4.36127 4.20478i
214.1 −2.34630 + 1.35464i −0.866025 0.500000i 2.67009 4.62473i 1.55199 + 1.60976i 2.70928 0 9.04945i 0.500000 + 0.866025i −5.82208 1.67458i
214.2 −1.64823 + 0.951606i 0.866025 + 0.500000i 0.811108 1.40488i 2.07321 0.837733i −1.90321 0 0.719004i 0.500000 + 0.866025i −2.61994 + 3.35366i
214.3 −0.167954 + 0.0969683i −0.866025 0.500000i −0.981194 + 1.69948i 0.710109 2.12032i 0.193937 0 0.768452i 0.500000 + 0.866025i 0.0863379 + 0.424974i
214.4 0.167954 0.0969683i 0.866025 + 0.500000i −0.981194 + 1.69948i −2.19130 0.445186i 0.193937 0 0.768452i 0.500000 + 0.866025i −0.411207 + 0.137716i
214.5 1.64823 0.951606i −0.866025 0.500000i 0.811108 1.40488i −1.76210 + 1.37659i −1.90321 0 0.719004i 0.500000 + 0.866025i −1.59438 + 3.94576i
214.6 2.34630 1.35464i 0.866025 + 0.500000i 2.67009 4.62473i 0.618092 + 2.14894i 2.70928 0 9.04945i 0.500000 + 0.866025i 4.36127 + 4.20478i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.6 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 735.2.q.f 12
5.b even 2 1 inner 735.2.q.f 12
7.b odd 2 1 735.2.q.e 12
7.c even 3 1 735.2.d.b 6
7.c even 3 1 inner 735.2.q.f 12
7.d odd 6 1 105.2.d.b 6
7.d odd 6 1 735.2.q.e 12
21.g even 6 1 315.2.d.e 6
21.h odd 6 1 2205.2.d.l 6
28.f even 6 1 1680.2.t.k 6
35.c odd 2 1 735.2.q.e 12
35.i odd 6 1 105.2.d.b 6
35.i odd 6 1 735.2.q.e 12
35.j even 6 1 735.2.d.b 6
35.j even 6 1 inner 735.2.q.f 12
35.k even 12 1 525.2.a.j 3
35.k even 12 1 525.2.a.k 3
35.l odd 12 1 3675.2.a.bi 3
35.l odd 12 1 3675.2.a.bj 3
84.j odd 6 1 5040.2.t.v 6
105.o odd 6 1 2205.2.d.l 6
105.p even 6 1 315.2.d.e 6
105.w odd 12 1 1575.2.a.w 3
105.w odd 12 1 1575.2.a.x 3
140.s even 6 1 1680.2.t.k 6
140.x odd 12 1 8400.2.a.dg 3
140.x odd 12 1 8400.2.a.dj 3
420.be odd 6 1 5040.2.t.v 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 7.d odd 6 1
105.2.d.b 6 35.i odd 6 1
315.2.d.e 6 21.g even 6 1
315.2.d.e 6 105.p even 6 1
525.2.a.j 3 35.k even 12 1
525.2.a.k 3 35.k even 12 1
735.2.d.b 6 7.c even 3 1
735.2.d.b 6 35.j even 6 1
735.2.q.e 12 7.b odd 2 1
735.2.q.e 12 7.d odd 6 1
735.2.q.e 12 35.c odd 2 1
735.2.q.e 12 35.i odd 6 1
735.2.q.f 12 1.a even 1 1 trivial
735.2.q.f 12 5.b even 2 1 inner
735.2.q.f 12 7.c even 3 1 inner
735.2.q.f 12 35.j even 6 1 inner
1575.2.a.w 3 105.w odd 12 1
1575.2.a.x 3 105.w odd 12 1
1680.2.t.k 6 28.f even 6 1
1680.2.t.k 6 140.s even 6 1
2205.2.d.l 6 21.h odd 6 1
2205.2.d.l 6 105.o odd 6 1
3675.2.a.bi 3 35.l odd 12 1
3675.2.a.bj 3 35.l odd 12 1
5040.2.t.v 6 84.j odd 6 1
5040.2.t.v 6 420.be odd 6 1
8400.2.a.dg 3 140.x odd 12 1
8400.2.a.dj 3 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(735, [\chi])$$:

 $$T_{2}^{12} - 11 T_{2}^{10} + 94 T_{2}^{8} - 295 T_{2}^{6} + 718 T_{2}^{4} - 27 T_{2}^{2} + 1$$ $$T_{19}^{6} + 6 T_{19}^{5} + 40 T_{19}^{4} + 56 T_{19}^{3} + 256 T_{19}^{2} + 160 T_{19} + 1600$$ $$T_{73}^{12} - 140 T_{73}^{10} + 14880 T_{73}^{8} - 639168 T_{73}^{6} + 20764160 T_{73}^{4} - 51051520 T_{73}^{2} + 116985856$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 27 T^{2} + 718 T^{4} - 295 T^{6} + 94 T^{8} - 11 T^{10} + T^{12}$$
$3$ $$( 1 - T^{2} + T^{4} )^{3}$$
$5$ $$15625 - 6250 T + 625 T^{2} + 2250 T^{3} - 750 T^{4} - 130 T^{5} + 249 T^{6} - 26 T^{7} - 30 T^{8} + 18 T^{9} + T^{10} - 2 T^{11} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$( 4 + 2 T + T^{2} )^{6}$$
$13$ $$( 64 + 112 T^{2} + 44 T^{4} + T^{6} )^{2}$$
$17$ $$65536 - 65536 T^{2} + 57344 T^{4} - 7680 T^{6} + 768 T^{8} - 32 T^{10} + T^{12}$$
$19$ $$( 1600 + 160 T + 256 T^{2} + 56 T^{3} + 40 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$23$ $$65536 - 49152 T^{2} + 28672 T^{4} - 5632 T^{6} + 832 T^{8} - 32 T^{10} + T^{12}$$
$29$ $$( -40 - 52 T + 2 T^{2} + T^{3} )^{4}$$
$31$ $$( 33856 - 9568 T + 3072 T^{2} - 264 T^{3} + 56 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$37$ $$16777216 - 28311552 T^{2} + 47054848 T^{4} - 1208320 T^{6} + 24064 T^{8} - 176 T^{10} + T^{12}$$
$41$ $$( -200 - 60 T + 2 T^{2} + T^{3} )^{4}$$
$43$ $$( 692224 + 27392 T^{2} + 304 T^{4} + T^{6} )^{2}$$
$47$ $$268435456 - 50331648 T^{2} + 7340032 T^{4} - 360448 T^{6} + 13312 T^{8} - 128 T^{10} + T^{12}$$
$53$ $$7676563456 - 738778112 T^{2} + 56028672 T^{4} - 1275072 T^{6} + 21152 T^{8} - 172 T^{10} + T^{12}$$
$59$ $$( 1638400 + 81920 T + 24576 T^{2} + 1536 T^{3} + 320 T^{4} + 16 T^{5} + T^{6} )^{2}$$
$61$ $$( 61504 + 12896 T + 4192 T^{2} + 184 T^{3} + 88 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$67$ $$268435456 - 50331648 T^{2} + 7340032 T^{4} - 360448 T^{6} + 13312 T^{8} - 128 T^{10} + T^{12}$$
$71$ $$( -2 + T )^{12}$$
$73$ $$116985856 - 51051520 T^{2} + 20764160 T^{4} - 639168 T^{6} + 14880 T^{8} - 140 T^{10} + T^{12}$$
$79$ $$( 102400 - 5120 T + 4096 T^{2} - 448 T^{3} + 160 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$83$ $$( 65536 + 8192 T^{2} + 192 T^{4} + T^{6} )^{2}$$
$89$ $$( 1600 + 2080 T + 2144 T^{2} + 648 T^{3} + 144 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$97$ $$( 3474496 + 83312 T^{2} + 556 T^{4} + T^{6} )^{2}$$