# Properties

 Label 1170.2.bj.c Level $1170$ Weight $2$ Character orbit 1170.bj Analytic conductor $9.342$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + 190 x^{3} - 1196 x^{2} - 338 x + 2197$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 390) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{4} ) q^{2} + \beta_{4} q^{4} + ( -\beta_{1} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{5} + ( \beta_{3} + \beta_{7} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{4} ) q^{2} + \beta_{4} q^{4} + ( -\beta_{1} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{5} + ( \beta_{3} + \beta_{7} ) q^{7} + q^{8} -\beta_{11} q^{10} + ( -\beta_{1} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} + ( 1 - \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{13} + \beta_{9} q^{14} + ( -1 - \beta_{4} ) q^{16} + ( 3 + \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( \beta_{1} + \beta_{3} + \beta_{9} ) q^{20} + ( -\beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{22} + ( -1 - 2 \beta_{2} - \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{23} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{25} + ( 2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{26} + ( -\beta_{3} - \beta_{7} - \beta_{9} ) q^{28} + ( -2 - \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{31} + \beta_{4} q^{32} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{34} + ( 3 + \beta_{2} + 2 \beta_{4} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{35} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{37} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{38} + ( -\beta_{1} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{40} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{41} + ( 5 + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{43} + ( \beta_{1} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{44} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{46} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{47} + ( 1 + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{49} + ( 2 - \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{50} + ( -1 - \beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{52} + ( 2 - \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} + 5 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{53} + ( 2 \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{55} + ( \beta_{3} + \beta_{7} ) q^{56} + ( \beta_{1} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{58} + ( 4 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{59} + ( -\beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{61} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{62} + q^{64} + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{67} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{68} + ( -1 + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{70} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{5} + 4 \beta_{6} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{71} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} ) q^{73} + ( -1 - \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{74} + ( 1 + \beta_{2} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{76} + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{7} - \beta_{9} + 2 \beta_{10} ) q^{77} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{79} -\beta_{11} q^{80} + ( -3 - \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{82} + ( 4 - \beta_{2} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{83} + ( 4 - \beta_{1} - 3 \beta_{3} + 4 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{85} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{86} + ( -\beta_{1} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( -3 - 4 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} + 3 \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{89} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} + 3 \beta_{7} ) q^{91} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} ) q^{92} + ( -1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{94} + ( -4 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 3 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{95} + ( -2 - 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 12 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{97} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 6 q^{2} - 6 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8} + O(q^{10})$$ $$12 q - 6 q^{2} - 6 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8} + 4 q^{10} - 6 q^{11} + 8 q^{13} - 4 q^{14} - 6 q^{16} + 18 q^{17} - 6 q^{19} - 2 q^{20} + 6 q^{22} + 6 q^{23} - 10 q^{25} + 2 q^{26} + 2 q^{28} - 14 q^{29} - 6 q^{32} + 22 q^{35} + 12 q^{37} - 2 q^{40} + 18 q^{41} + 36 q^{43} - 6 q^{46} + 16 q^{47} + 8 q^{49} + 20 q^{50} - 10 q^{52} + 8 q^{55} + 2 q^{56} - 14 q^{58} + 36 q^{59} + 10 q^{61} + 6 q^{62} + 12 q^{64} + 44 q^{65} - 4 q^{67} - 18 q^{68} + 4 q^{70} + 12 q^{71} - 28 q^{73} + 12 q^{74} + 6 q^{76} + 4 q^{79} + 4 q^{80} - 18 q^{82} + 72 q^{83} + 48 q^{85} - 6 q^{88} - 18 q^{89} + 2 q^{91} - 8 q^{94} - 18 q^{95} + 48 q^{97} + 8 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + 190 x^{3} - 1196 x^{2} - 338 x + 2197$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$203419 \nu^{11} + 163110633 \nu^{10} - 591783880 \nu^{9} - 97338749 \nu^{8} + 4513282461 \nu^{7} - 6489146722 \nu^{6} - 4655211661 \nu^{5} + 5740233327 \nu^{4} + 8165110694 \nu^{3} + 33307875431 \nu^{2} - 128185173545 \nu - 81183629852$$$$)/ 63907274600$$ $$\beta_{2}$$ $$=$$ $$($$$$99774878 \nu^{11} - 1446867854 \nu^{10} + 1099642740 \nu^{9} + 2834573637 \nu^{8} - 11046550568 \nu^{7} + 1368363861 \nu^{6} - 68535056482 \nu^{5} + 38321129974 \nu^{4} + 51767136278 \nu^{3} - 345701461353 \nu^{2} - 110551992590 \nu - 399707750949$$$$)/ 415397284900$$ $$\beta_{3}$$ $$=$$ $$($$$$120243408 \nu^{11} - 454030419 \nu^{10} - 2051209685 \nu^{9} + 7036618932 \nu^{8} - 2513656023 \nu^{7} - 29967292429 \nu^{6} + 18436259198 \nu^{5} - 35262141211 \nu^{4} - 8309942667 \nu^{3} - 28687118858 \nu^{2} - 98318215515 \nu - 618192439839$$$$)/ 415397284900$$ $$\beta_{4}$$ $$=$$ $$($$$$4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + 35557 \nu^{6} + 1844066 \nu^{5} - 940887 \nu^{4} + 77961 \nu^{3} - 7481036 \nu^{2} - 4439955 \nu + 11867687$$$$)/11796200$$ $$\beta_{5}$$ $$=$$ $$($$$$164152294 \nu^{11} + 1358027358 \nu^{10} - 5882189980 \nu^{9} + 1843354751 \nu^{8} + 45609137136 \nu^{7} - 49422223697 \nu^{6} - 70650734586 \nu^{5} + 94378041302 \nu^{4} + 204958669994 \nu^{3} + 189862006481 \nu^{2} - 68215142970 \nu - 1308690012227$$$$)/ 415397284900$$ $$\beta_{6}$$ $$=$$ $$($$$$480376508 \nu^{11} - 958108569 \nu^{10} - 1722573835 \nu^{9} + 8639610832 \nu^{8} + 2577608327 \nu^{7} - 5697780079 \nu^{6} - 37282009602 \nu^{5} + 13460230639 \nu^{4} + 124582190083 \nu^{3} + 197417975542 \nu^{2} - 141527922965 \nu - 997979945989$$$$)/ 415397284900$$ $$\beta_{7}$$ $$=$$ $$($$$$-23159569 \nu^{11} + 44665642 \nu^{10} + 118225633 \nu^{9} - 603932751 \nu^{8} - 6720278 \nu^{7} + 1195412299 \nu^{6} - 1392260235 \nu^{5} - 486856544 \nu^{4} - 1535131351 \nu^{3} - 7949637689 \nu^{2} - 2866001996 \nu + 21026586399$$$$)/ 16615891396$$ $$\beta_{8}$$ $$=$$ $$($$$$-236542471 \nu^{11} + 346656978 \nu^{10} + 1242595595 \nu^{9} - 6207967149 \nu^{8} + 2310806556 \nu^{7} + 6108999933 \nu^{6} - 30747713501 \nu^{5} + 45266458632 \nu^{4} - 73287704061 \nu^{3} - 120820074499 \nu^{2} - 57773591330 \nu + 460186493$$$$)/ 166158913960$$ $$\beta_{9}$$ $$=$$ $$($$$$-632187142 \nu^{11} + 553303131 \nu^{10} + 5382274615 \nu^{9} - 11841029268 \nu^{8} - 18449268373 \nu^{7} + 45918505121 \nu^{6} + 41904457048 \nu^{5} - 52841609661 \nu^{4} - 129070461767 \nu^{3} - 49363987858 \nu^{2} + 441927290935 \nu + 880210105411$$$$)/ 415397284900$$ $$\beta_{10}$$ $$=$$ $$($$$$320232543 \nu^{11} - 1004615314 \nu^{10} - 1143072025 \nu^{9} + 9975044447 \nu^{8} - 4093093078 \nu^{7} - 25465804119 \nu^{6} + 21841076813 \nu^{5} + 35457807644 \nu^{4} + 27228013423 \nu^{3} + 100602839757 \nu^{2} - 286784071080 \nu - 144016360059$$$$)/ 166158913960$$ $$\beta_{11}$$ $$=$$ $$($$$$-1027654074 \nu^{11} + 2460612632 \nu^{10} + 3624161555 \nu^{9} - 25210679196 \nu^{8} + 10960029469 \nu^{7} + 30148216612 \nu^{6} - 9821523994 \nu^{5} - 23732163042 \nu^{4} - 402415418899 \nu^{3} + 37700985074 \nu^{2} + 390356046795 \nu + 27977691092$$$$)/ 415397284900$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{5} - 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + 2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{1} - 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{11} + 2 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - \beta_{1} - 11$$ $$\nu^{5}$$ $$=$$ $$($$$$-6 \beta_{11} + 2 \beta_{10} - \beta_{9} - 10 \beta_{8} - 12 \beta_{7} - 34 \beta_{6} - 9 \beta_{5} - 16 \beta_{4} - 19 \beta_{3} + 11 \beta_{2} + 8 \beta_{1} - 82$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-46 \beta_{11} - 40 \beta_{10} + 46 \beta_{9} + 42 \beta_{8} - 90 \beta_{7} - 13 \beta_{6} - 30 \beta_{5} - 26 \beta_{4} + 13 \beta_{3} - 2 \beta_{2} + 46 \beta_{1} - 39$$ $$\nu^{7}$$ $$=$$ $$($$$$33 \beta_{11} - 67 \beta_{10} + 79 \beta_{9} - 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + 5 \beta_{5} + 52 \beta_{4} - 156 \beta_{3} + 152 \beta_{2} + 146 \beta_{1} - 51$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-93 \beta_{11} - 174 \beta_{10} + 231 \beta_{9} + 181 \beta_{8} - 364 \beta_{7} + 130 \beta_{6} - 3 \beta_{5} + 107 \beta_{4} + 91 \beta_{3} - 23 \beta_{2} + 179 \beta_{1} + 362$$ $$\nu^{9}$$ $$=$$ $$($$$$1124 \beta_{11} + 316 \beta_{10} - 358 \beta_{9} - 776 \beta_{8} + 646 \beta_{7} + 682 \beta_{6} + 798 \beta_{5} + 1378 \beta_{4} - 861 \beta_{3} + 404 \beta_{2} - 248 \beta_{1} + 1700$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$246 \beta_{11} - 81 \beta_{10} - 196 \beta_{9} + 482 \beta_{8} + 630 \beta_{7} + 1314 \beta_{6} + 501 \beta_{5} + 1079 \beta_{4} + 580 \beta_{3} - 926 \beta_{2} - 469 \beta_{1} + 2556$$ $$\nu^{11}$$ $$=$$ $$($$$$7310 \beta_{11} + 5070 \beta_{10} - 9673 \beta_{9} - 6494 \beta_{8} + 12576 \beta_{7} + 2066 \beta_{6} + 3475 \beta_{5} + 4992 \beta_{4} - 5163 \beta_{3} - 325 \beta_{2} - 6944 \beta_{1} + 2574$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$$.

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −2.39378 − 0.0429626i 1.75374 − 1.62986i −0.330925 − 1.46916i −1.44229 − 0.433312i 2.00607 + 1.30680i 1.40719 + 0.536449i −2.39378 + 0.0429626i 1.75374 + 1.62986i −0.330925 + 1.46916i −1.44229 + 0.433312i 2.00607 − 1.30680i 1.40719 − 0.536449i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −2.10012 0.767774i 0 0.823063 + 1.42559i 1.00000 0 1.71497 1.43487i
199.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.40066 + 1.74303i 0 −0.763837 1.32301i 1.00000 0 −0.809179 2.08452i
199.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.571769 2.16173i 0 0.603137 + 1.04466i 1.00000 0 2.15800 + 0.585699i
199.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.230377 + 2.22417i 0 −0.432713 0.749482i 1.00000 0 −1.81100 1.31160i
199.5 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.26873 1.84128i 0 2.17283 + 3.76344i 1.00000 0 0.960230 + 2.01940i
199.6 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.03420 0.928463i 0 −1.40247 2.42916i 1.00000 0 −0.213026 + 2.22590i
829.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −2.10012 + 0.767774i 0 0.823063 1.42559i 1.00000 0 1.71497 + 1.43487i
829.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.40066 1.74303i 0 −0.763837 + 1.32301i 1.00000 0 −0.809179 + 2.08452i
829.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.571769 + 2.16173i 0 0.603137 1.04466i 1.00000 0 2.15800 0.585699i
829.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.230377 2.22417i 0 −0.432713 + 0.749482i 1.00000 0 −1.81100 + 1.31160i
829.5 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.26873 + 1.84128i 0 2.17283 3.76344i 1.00000 0 0.960230 2.01940i
829.6 −0.500000 0.866025i 0 −0.500000 + 0.866025i 2.03420 + 0.928463i 0 −1.40247 + 2.42916i 1.00000 0 −0.213026 2.22590i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bj.c 12
3.b odd 2 1 390.2.x.b yes 12
5.b even 2 1 1170.2.bj.d 12
13.e even 6 1 1170.2.bj.d 12
15.d odd 2 1 390.2.x.a 12
15.e even 4 1 1950.2.bc.i 12
15.e even 4 1 1950.2.bc.j 12
39.h odd 6 1 390.2.x.a 12
65.l even 6 1 inner 1170.2.bj.c 12
195.y odd 6 1 390.2.x.b yes 12
195.bf even 12 1 1950.2.bc.i 12
195.bf even 12 1 1950.2.bc.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 15.d odd 2 1
390.2.x.a 12 39.h odd 6 1
390.2.x.b yes 12 3.b odd 2 1
390.2.x.b yes 12 195.y odd 6 1
1170.2.bj.c 12 1.a even 1 1 trivial
1170.2.bj.c 12 65.l even 6 1 inner
1170.2.bj.d 12 5.b even 2 1
1170.2.bj.d 12 13.e even 6 1
1950.2.bc.i 12 15.e even 4 1
1950.2.bc.i 12 195.bf even 12 1
1950.2.bc.j 12 15.e even 4 1
1950.2.bc.j 12 195.bf even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1170, [\chi])$$:

 $$T_{7}^{12} - \cdots$$ $$T_{17}^{12} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{6}$$
$3$ $$T^{12}$$
$5$ $$15625 + 6250 T + 4375 T^{2} + 750 T^{3} + 375 T^{4} + 160 T^{5} + 114 T^{6} + 32 T^{7} + 15 T^{8} + 6 T^{9} + 7 T^{10} + 2 T^{11} + T^{12}$$
$7$ $$1024 + 512 T + 1664 T^{2} + 64 T^{3} + 1648 T^{4} + 112 T^{5} + 708 T^{6} - 20 T^{7} + 205 T^{8} + 6 T^{9} + 19 T^{10} - 2 T^{11} + T^{12}$$
$11$ $$16 + 96 T - 4 T^{2} - 1176 T^{3} + 2585 T^{4} - 1446 T^{5} - 330 T^{6} + 444 T^{7} + 87 T^{8} - 84 T^{9} - 2 T^{10} + 6 T^{11} + T^{12}$$
$13$ $$4826809 - 2970344 T + 1028196 T^{2} - 87880 T^{3} - 39208 T^{4} + 26520 T^{5} - 7838 T^{6} + 2040 T^{7} - 232 T^{8} - 40 T^{9} + 36 T^{10} - 8 T^{11} + T^{12}$$
$17$ $$65536 + 491520 T + 1269760 T^{2} + 307200 T^{3} - 300288 T^{4} - 85248 T^{5} + 75744 T^{6} - 4632 T^{7} - 2695 T^{8} + 234 T^{9} + 95 T^{10} - 18 T^{11} + T^{12}$$
$19$ $$1982464 - 3649536 T + 837120 T^{2} + 2581632 T^{3} + 672400 T^{4} - 254496 T^{5} - 43876 T^{6} + 14568 T^{7} + 2701 T^{8} - 390 T^{9} - 53 T^{10} + 6 T^{11} + T^{12}$$
$23$ $$190660864 - 284334336 T + 171279232 T^{2} - 44643456 T^{3} + 1290352 T^{4} + 1617792 T^{5} - 129352 T^{6} - 53748 T^{7} + 7441 T^{8} + 630 T^{9} - 93 T^{10} - 6 T^{11} + T^{12}$$
$29$ $$21904 + 490768 T + 11075036 T^{2} - 1644412 T^{3} + 1737745 T^{4} + 196610 T^{5} + 148926 T^{6} + 15668 T^{7} + 6703 T^{8} + 960 T^{9} + 190 T^{10} + 14 T^{11} + T^{12}$$
$31$ $$177209344 + 93061120 T^{2} + 11339776 T^{4} + 592640 T^{6} + 15433 T^{8} + 198 T^{10} + T^{12}$$
$37$ $$227195329 - 175510012 T + 129327441 T^{2} - 32084244 T^{3} + 11678146 T^{4} - 1319436 T^{5} + 734609 T^{6} - 60444 T^{7} + 14658 T^{8} - 1028 T^{9} + 209 T^{10} - 12 T^{11} + T^{12}$$
$41$ $$65536 + 491520 T + 1269760 T^{2} + 307200 T^{3} - 300288 T^{4} - 85248 T^{5} + 75744 T^{6} - 4632 T^{7} - 2695 T^{8} + 234 T^{9} + 95 T^{10} - 18 T^{11} + T^{12}$$
$43$ $$349241344 - 627916800 T + 494054400 T^{2} - 211680000 T^{3} + 51560464 T^{4} - 6267168 T^{5} + 54524 T^{6} + 57864 T^{7} + 6253 T^{8} - 3708 T^{9} + 535 T^{10} - 36 T^{11} + T^{12}$$
$47$ $$( 5956 + 2352 T - 4283 T^{2} + 1280 T^{3} - 98 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$53$ $$2473271824 + 989681000 T^{2} + 127872521 T^{4} + 5778420 T^{6} + 87438 T^{8} + 508 T^{10} + T^{12}$$
$59$ $$4983230464 + 752228352 T - 820266240 T^{2} - 129534336 T^{3} + 157498704 T^{4} - 37281744 T^{5} + 3235868 T^{6} + 77796 T^{7} - 26427 T^{8} - 972 T^{9} + 459 T^{10} - 36 T^{11} + T^{12}$$
$61$ $$89718784 - 41828352 T + 26851328 T^{2} - 7106048 T^{3} + 3388992 T^{4} - 835296 T^{5} + 256760 T^{6} - 39396 T^{7} + 7561 T^{8} - 762 T^{9} + 135 T^{10} - 10 T^{11} + T^{12}$$
$67$ $$83759104 - 2928640 T + 25142272 T^{2} + 2632704 T^{3} + 6100480 T^{4} + 320768 T^{5} + 394560 T^{6} - 35776 T^{7} + 18784 T^{8} - 784 T^{9} + 164 T^{10} + 4 T^{11} + T^{12}$$
$71$ $$4194304 - 25165824 T + 56295424 T^{2} - 35782656 T^{3} - 703488 T^{4} + 6873600 T^{5} + 1150080 T^{6} - 505152 T^{7} + 36080 T^{8} + 2640 T^{9} - 172 T^{10} - 12 T^{11} + T^{12}$$
$73$ $$( -230528 + 114240 T + 18280 T^{2} - 3068 T^{3} - 263 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$79$ $$( -29312 - 12576 T + 3892 T^{2} + 788 T^{3} - 179 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$83$ $$( -6912 + 51840 T - 16560 T^{2} + 144 T^{3} + 360 T^{4} - 36 T^{5} + T^{6} )^{2}$$
$89$ $$1341001056256 - 39576354816 T - 61874871296 T^{2} + 1837575168 T^{3} + 2363129536 T^{4} - 191431776 T^{5} - 15114712 T^{6} + 1612140 T^{7} + 96433 T^{8} - 7506 T^{9} - 309 T^{10} + 18 T^{11} + T^{12}$$
$97$ $$415519473664 + 141504348160 T + 88030961664 T^{2} - 14104406016 T^{3} + 3300888832 T^{4} - 348175872 T^{5} + 53039744 T^{6} - 5437824 T^{7} + 522672 T^{8} - 32704 T^{9} + 1640 T^{10} - 48 T^{11} + T^{12}$$