# Properties

 Label 930.2.n.c Level $930$ Weight $2$ Character orbit 930.n Analytic conductor $7.426$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 25 x^{10} + 205 x^{8} + 675 x^{6} + 795 x^{4} + 230 x^{2} + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{4} q^{3} + \beta_{8} q^{4} - q^{5} + q^{6} + ( -\beta_{8} + \beta_{10} ) q^{7} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{8} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{4} q^{3} + \beta_{8} q^{4} - q^{5} + q^{6} + ( -\beta_{8} + \beta_{10} ) q^{7} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{8} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{9} + \beta_{1} q^{10} + ( -1 - \beta_{4} - \beta_{10} ) q^{11} -\beta_{1} q^{12} + ( 1 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{13} + ( 1 - \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{14} -\beta_{4} q^{15} + \beta_{4} q^{16} + ( \beta_{5} + \beta_{7} ) q^{17} + \beta_{4} q^{18} + ( \beta_{1} + \beta_{6} + \beta_{7} + \beta_{11} ) q^{19} -\beta_{8} q^{20} -\beta_{11} q^{21} + ( -1 + \beta_{1} - \beta_{9} ) q^{22} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{23} + \beta_{8} q^{24} + q^{25} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{26} + \beta_{8} q^{27} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{28} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} - q^{30} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{31} + q^{32} + ( 1 + \beta_{8} + \beta_{11} ) q^{33} + ( 1 + \beta_{3} + \beta_{6} ) q^{34} + ( \beta_{8} - \beta_{10} ) q^{35} + q^{36} + ( 3 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{37} + ( 1 + \beta_{3} + \beta_{7} - \beta_{10} ) q^{38} + ( 1 - \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( 1 - \beta_{1} + \beta_{4} + \beta_{8} ) q^{40} + ( -4 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{41} + ( -\beta_{8} + \beta_{10} ) q^{42} + ( 3 + \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{43} + ( 1 + \beta_{2} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{44} + ( 1 - \beta_{1} + \beta_{4} + \beta_{8} ) q^{45} + ( -1 - \beta_{2} - \beta_{4} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{46} + ( 1 + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{47} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{48} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{49} -\beta_{1} q^{50} + ( -1 - \beta_{3} - \beta_{5} - \beta_{7} ) q^{51} + ( -1 + \beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{52} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{53} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{54} + ( 1 + \beta_{4} + \beta_{10} ) q^{55} + \beta_{2} q^{56} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{57} + ( 2 + 2 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{58} + ( 5 + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{59} + \beta_{1} q^{60} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{9} + 3 \beta_{10} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{62} + \beta_{2} q^{63} -\beta_{1} q^{64} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{65} + ( -1 - \beta_{4} - \beta_{10} ) q^{66} + ( -6 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{67} + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} ) q^{68} + ( 2 - \beta_{1} + 2 \beta_{4} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{69} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} - \beta_{9} ) q^{70} + ( 6 - 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{71} -\beta_{1} q^{72} + ( -3 + \beta_{1} - 3 \beta_{4} - 3 \beta_{5} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{73} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{74} + \beta_{4} q^{75} + ( -\beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{76} + ( -2 - \beta_{2} - \beta_{3} - 5 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{77} + ( 1 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{78} + ( 3 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{79} -\beta_{4} q^{80} -\beta_{1} q^{81} + ( 1 + \beta_{1} + \beta_{3} + 3 \beta_{5} + \beta_{7} - 5 \beta_{8} - \beta_{9} + \beta_{11} ) q^{82} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{83} + ( 1 - \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{84} + ( -\beta_{5} - \beta_{7} ) q^{85} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{86} + ( 1 + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{87} + ( -\beta_{1} - \beta_{2} + \beta_{4} ) q^{88} + ( -1 - \beta_{1} - \beta_{4} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{89} -\beta_{4} q^{90} + ( \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{11} ) q^{91} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{9} - \beta_{10} ) q^{92} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{10} + \beta_{11} ) q^{93} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{94} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{11} ) q^{95} + \beta_{4} q^{96} + ( 5 - \beta_{1} - \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - \beta_{7} + 5 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{97} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{98} + ( -\beta_{1} - \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{2} - 3q^{3} - 3q^{4} - 12q^{5} + 12q^{6} - q^{7} - 3q^{8} - 3q^{9} + O(q^{10})$$ $$12q - 3q^{2} - 3q^{3} - 3q^{4} - 12q^{5} + 12q^{6} - q^{7} - 3q^{8} - 3q^{9} + 3q^{10} - 5q^{11} - 3q^{12} - 3q^{13} - q^{14} + 3q^{15} - 3q^{16} - 3q^{17} - 3q^{18} - 4q^{19} + 3q^{20} + 4q^{21} - 5q^{22} + 10q^{23} - 3q^{24} + 12q^{25} + 2q^{26} - 3q^{27} + 4q^{28} - 5q^{29} - 12q^{30} - 8q^{31} + 12q^{32} + 5q^{33} + 2q^{34} + q^{35} + 12q^{36} + 2q^{37} + 11q^{38} + 2q^{39} + 3q^{40} + 13q^{41} - q^{42} + 16q^{43} + 5q^{44} + 3q^{45} - 5q^{46} + q^{47} - 3q^{48} + 4q^{49} - 3q^{50} - 3q^{51} - 3q^{52} + 3q^{53} - 3q^{54} + 5q^{55} - 6q^{56} - 14q^{57} + 10q^{58} + 23q^{59} + 3q^{60} - 46q^{61} + 7q^{62} - 6q^{63} - 3q^{64} + 3q^{65} - 5q^{66} - 66q^{67} + 2q^{68} + 10q^{69} + q^{70} + 23q^{71} - 3q^{72} - 4q^{73} + 7q^{74} - 3q^{75} + 11q^{76} + 10q^{77} - 3q^{78} + 21q^{79} + 3q^{80} - 3q^{81} + 13q^{82} + 4q^{83} - q^{84} + 3q^{85} - 4q^{86} - 10q^{87} - 5q^{89} + 3q^{90} - 6q^{91} - 10q^{92} - 3q^{93} + 6q^{94} + 4q^{95} - 3q^{96} + 25q^{97} - 6q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 25 x^{10} + 205 x^{8} + 675 x^{6} + 795 x^{4} + 230 x^{2} + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$13 \nu^{11} + 21 \nu^{10} + 294 \nu^{9} + 458 \nu^{8} + 2079 \nu^{7} + 2959 \nu^{6} + 6146 \nu^{5} + 6970 \nu^{4} + 8341 \nu^{3} + 5405 \nu^{2} + 3895 \nu + 907$$$$)/968$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{10} - 166 \nu^{8} - 2409 \nu^{6} - 11122 \nu^{4} - 14167 \nu^{2} - 2317$$$$)/484$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{10} - 160 \nu^{8} - 1111 \nu^{6} - 2734 \nu^{4} - 1765 \nu^{2} - 361$$$$)/242$$ $$\beta_{4}$$ $$=$$ $$($$$$13 \nu^{11} - 21 \nu^{10} + 294 \nu^{9} - 458 \nu^{8} + 2079 \nu^{7} - 2959 \nu^{6} + 6146 \nu^{5} - 6970 \nu^{4} + 8341 \nu^{3} - 5405 \nu^{2} + 3895 \nu - 907$$$$)/968$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{11} + 13 \nu^{10} + 41 \nu^{9} + 261 \nu^{8} + 220 \nu^{7} + 1397 \nu^{6} + 206 \nu^{5} + 2241 \nu^{4} - 734 \nu^{3} + 1125 \nu^{2} - 505 \nu + 1$$$$)/242$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{11} + 13 \nu^{10} - 41 \nu^{9} + 261 \nu^{8} - 220 \nu^{7} + 1397 \nu^{6} - 206 \nu^{5} + 2241 \nu^{4} + 734 \nu^{3} + 1125 \nu^{2} + 505 \nu + 1$$$$)/242$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{11} - 19 \nu^{10} + 146 \nu^{9} - 362 \nu^{8} + 275 \nu^{7} - 1683 \nu^{6} - 2076 \nu^{5} - 1748 \nu^{4} - 4381 \nu^{3} - 485 \nu^{2} - 705 \nu + 117$$$$)/484$$ $$\beta_{8}$$ $$=$$ $$($$$$37 \nu^{11} + 21 \nu^{10} + 918 \nu^{9} + 458 \nu^{8} + 7447 \nu^{7} + 2959 \nu^{6} + 24238 \nu^{5} + 6970 \nu^{4} + 27913 \nu^{3} + 5405 \nu^{2} + 6635 \nu + 423$$$$)/968$$ $$\beta_{9}$$ $$=$$ $$($$$$59 \nu^{11} + 11 \nu^{10} + 1534 \nu^{9} + 330 \nu^{8} + 13277 \nu^{7} + 3289 \nu^{6} + 46150 \nu^{5} + 11946 \nu^{4} + 57371 \nu^{3} + 11231 \nu^{2} + 19945 \nu - 671$$$$)/968$$ $$\beta_{10}$$ $$=$$ $$($$$$-59 \nu^{11} + 11 \nu^{10} - 1534 \nu^{9} + 330 \nu^{8} - 13277 \nu^{7} + 3289 \nu^{6} - 46150 \nu^{5} + 11946 \nu^{4} - 57371 \nu^{3} + 11231 \nu^{2} - 19945 \nu - 671$$$$)/968$$ $$\beta_{11}$$ $$=$$ $$($$$$-133 \nu^{11} - 13 \nu^{10} - 3238 \nu^{9} - 294 \nu^{8} - 25443 \nu^{7} - 2079 \nu^{6} - 79006 \nu^{5} - 6146 \nu^{4} - 84333 \nu^{3} - 8341 \nu^{2} - 17155 \nu - 3411$$$$)/968$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{1} - 2$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{1} - 7$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} + 8 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} - 5 \beta_{4} + 3 \beta_{3} - \beta_{2} - 11 \beta_{1} + 6$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{10} + 9 \beta_{9} + 13 \beta_{6} + 13 \beta_{5} + 40 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 40 \beta_{1} + 67$$ $$\nu^{5}$$ $$=$$ $$-34 \beta_{11} + 30 \beta_{10} + 4 \beta_{9} - 128 \beta_{8} - 76 \beta_{7} - 67 \beta_{6} - 9 \beta_{5} + 55 \beta_{4} - 38 \beta_{3} + 17 \beta_{2} + 149 \beta_{1} - 85$$ $$\nu^{6}$$ $$=$$ $$-96 \beta_{10} - 96 \beta_{9} - 161 \beta_{6} - 161 \beta_{5} - 507 \beta_{4} + 77 \beta_{3} + 47 \beta_{2} + 507 \beta_{1} - 742$$ $$\nu^{7}$$ $$=$$ $$476 \beta_{11} - 399 \beta_{10} - 77 \beta_{9} + 1774 \beta_{8} + 954 \beta_{7} + 800 \beta_{6} + 154 \beta_{5} - 633 \beta_{4} + 477 \beta_{3} - 238 \beta_{2} - 1931 \beta_{1} + 1126$$ $$\nu^{8}$$ $$=$$ $$1123 \beta_{10} + 1123 \beta_{9} + 2004 \beta_{6} + 2004 \beta_{5} + 6386 \beta_{4} - 1118 \beta_{3} - 631 \beta_{2} - 6386 \beta_{1} + 8819$$ $$\nu^{9}$$ $$=$$ $$-6244 \beta_{11} + 5126 \beta_{10} + 1118 \beta_{9} - 23202 \beta_{8} - 11986 \beta_{7} - 9844 \beta_{6} - 2142 \beta_{5} + 7622 \beta_{4} - 5993 \beta_{3} + 3122 \beta_{2} + 24580 \beta_{1} - 14472$$ $$\nu^{10}$$ $$=$$ $$-13695 \beta_{10} - 13695 \beta_{9} - 25078 \beta_{6} - 25078 \beta_{5} - 80364 \beta_{4} + 14861 \beta_{3} + 8135 \beta_{2} + 80364 \beta_{1} - 108266$$ $$\nu^{11}$$ $$=$$ $$79878 \beta_{11} - 65017 \beta_{10} - 14861 \beta_{9} + 296520 \beta_{8} + 150702 \beta_{7} + 122693 \beta_{6} + 28009 \beta_{5} - 94020 \beta_{4} + 75351 \beta_{3} - 39939 \beta_{2} - 310662 \beta_{1} + 183672$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{1} - \beta_{4} - \beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
481.1
 − 0.153723i − 2.29476i 1.27291i 0.620070i 3.54652i − 2.26448i − 0.620070i − 3.54652i 2.26448i 0.153723i 2.29476i − 1.27291i
−0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 1.00000 −1.26952 + 3.90719i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
481.2 −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 1.00000 0.0931451 0.286671i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
481.3 −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 1.00000 0.367362 1.13062i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
721.1 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 1.00000 −2.70650 1.96638i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.2 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 1.00000 0.606949 + 0.440975i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.3 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 1.00000 2.40856 + 1.74992i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
841.1 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 −2.70650 + 1.96638i −0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i
841.2 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 0.606949 0.440975i −0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i
841.3 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 2.40856 1.74992i −0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i
901.1 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 −1.26952 3.90719i 0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
901.2 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 0.0931451 + 0.286671i 0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
901.3 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 0.367362 + 1.13062i 0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.c 12
31.d even 5 1 inner 930.2.n.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.c 12 1.a even 1 1 trivial
930.2.n.c 12 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$3$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$5$ $$( 1 + T )^{12}$$
$7$ $$121 - 561 T + 2379 T^{2} - 4530 T^{3} + 5110 T^{4} - 3041 T^{5} + 1341 T^{6} + 121 T^{7} + 40 T^{8} - 30 T^{9} + 9 T^{10} + T^{11} + T^{12}$$
$11$ $$32400 + 43200 T + 36000 T^{2} + 17850 T^{3} + 8875 T^{4} + 2425 T^{5} + 1340 T^{6} + 285 T^{7} + 155 T^{8} + 45 T^{9} + 20 T^{10} + 5 T^{11} + T^{12}$$
$13$ $$361 + 1843 T + 13571 T^{2} + 19450 T^{3} + 8530 T^{4} - 5637 T^{5} + 1179 T^{6} + 2203 T^{7} + 950 T^{8} + 120 T^{9} + 51 T^{10} + 3 T^{11} + T^{12}$$
$17$ $$1296 + 1728 T + 3456 T^{2} + 6810 T^{3} + 11815 T^{4} - 6447 T^{5} + 669 T^{6} + 1693 T^{7} + 840 T^{8} + 85 T^{9} + 51 T^{10} + 3 T^{11} + T^{12}$$
$19$ $$1771561 + 3235661 T + 2521519 T^{2} + 651805 T^{3} + 189920 T^{4} + 43561 T^{5} + 14976 T^{6} + 1249 T^{7} + 1040 T^{8} + 85 T^{9} + 59 T^{10} + 4 T^{11} + T^{12}$$
$23$ $$32400 - 37800 T + 121500 T^{2} - 86250 T^{3} + 150925 T^{4} - 83150 T^{5} + 75665 T^{6} - 23490 T^{7} + 2405 T^{8} + 390 T^{9} + 5 T^{10} - 10 T^{11} + T^{12}$$
$29$ $$20250000 - 40500000 T + 33300000 T^{2} - 3131250 T^{3} + 10384375 T^{4} - 839375 T^{5} + 263750 T^{6} + 4125 T^{7} + 2375 T^{8} - 75 T^{9} + 80 T^{10} + 5 T^{11} + T^{12}$$
$31$ $$887503681 + 229033208 T + 88658016 T^{2} + 18023555 T^{3} + 5016420 T^{4} + 899248 T^{5} + 209059 T^{6} + 29008 T^{7} + 5220 T^{8} + 605 T^{9} + 96 T^{10} + 8 T^{11} + T^{12}$$
$37$ $$( -7229 - 301 T + 1160 T^{2} + 35 T^{3} - 60 T^{4} - T^{5} + T^{6} )^{2}$$
$41$ $$31768071696 - 7777862568 T + 7310348856 T^{2} + 34352280 T^{3} + 186282985 T^{4} - 13382268 T^{5} + 1046369 T^{6} - 104243 T^{7} + 20380 T^{8} - 115 T^{9} + 61 T^{10} - 13 T^{11} + T^{12}$$
$43$ $$114041041 + 18357201 T + 31416219 T^{2} - 4633650 T^{3} + 1498450 T^{4} - 128619 T^{5} + 84531 T^{6} - 5126 T^{7} + 3650 T^{8} - 350 T^{9} + 124 T^{10} - 16 T^{11} + T^{12}$$
$47$ $$13220496 - 34229304 T + 37682424 T^{2} - 12409140 T^{3} + 5794885 T^{4} - 831999 T^{5} + 285721 T^{6} - 19911 T^{7} + 5780 T^{8} - 45 T^{9} - 11 T^{10} - T^{11} + T^{12}$$
$53$ $$102333456 + 87523632 T + 29079576 T^{2} + 358890 T^{3} + 1293535 T^{4} + 432897 T^{5} + 85479 T^{6} + 177 T^{7} + 11890 T^{8} + 135 T^{9} + 191 T^{10} - 3 T^{11} + T^{12}$$
$59$ $$1628606736 + 1563472152 T + 4061619756 T^{2} + 115711110 T^{3} + 321190915 T^{4} - 65515543 T^{5} + 8305804 T^{6} - 474143 T^{7} + 8535 T^{8} + 365 T^{9} + 196 T^{10} - 23 T^{11} + T^{12}$$
$61$ $$( 251579 + 72203 T - 8660 T^{2} - 3225 T^{3} - 40 T^{4} + 23 T^{5} + T^{6} )^{2}$$
$67$ $$( -60731 - 35512 T - 2890 T^{2} + 1555 T^{3} + 385 T^{4} + 33 T^{5} + T^{6} )^{2}$$
$71$ $$22525207056 - 53721367128 T + 52828880976 T^{2} - 14468399280 T^{3} + 2385754585 T^{4} - 281100943 T^{5} + 30092749 T^{6} - 2588023 T^{7} + 193470 T^{8} - 11515 T^{9} + 651 T^{10} - 23 T^{11} + T^{12}$$
$73$ $$68740681 - 182808259 T + 820123279 T^{2} - 505446455 T^{3} + 147057380 T^{4} - 17535659 T^{5} + 1959216 T^{6} - 4271 T^{7} + 6860 T^{8} - 935 T^{9} + 119 T^{10} + 4 T^{11} + T^{12}$$
$79$ $$78092861401 - 35197971254 T + 18850159249 T^{2} - 2662856195 T^{3} + 82119290 T^{4} + 13170161 T^{5} + 7568056 T^{6} - 945061 T^{7} + 111040 T^{8} - 7405 T^{9} + 499 T^{10} - 21 T^{11} + T^{12}$$
$83$ $$3794067216 + 2566335744 T + 1108435464 T^{2} + 315234030 T^{3} + 68449345 T^{4} + 11196124 T^{5} + 1583236 T^{6} + 169066 T^{7} + 14610 T^{8} + 490 T^{9} + 9 T^{10} - 4 T^{11} + T^{12}$$
$89$ $$32400 + 91800 T + 81000 T^{2} - 59700 T^{3} - 5675 T^{4} + 74875 T^{5} + 66590 T^{6} + 12165 T^{7} + 4505 T^{8} + 375 T^{9} + 110 T^{10} + 5 T^{11} + T^{12}$$
$97$ $$11471481025 - 911999075 T + 5965428275 T^{2} - 3778721000 T^{3} + 1067119750 T^{4} - 85822375 T^{5} + 7099915 T^{6} - 78735 T^{7} + 28500 T^{8} - 4410 T^{9} + 515 T^{10} - 25 T^{11} + T^{12}$$
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