# Properties

 Label 950.2.f.c Level $950$ Weight $2$ Character orbit 950.f Analytic conductor $7.586$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 24 x^{14} + 212 x^{12} + 880 x^{10} + 1858 x^{8} + 1960 x^{6} + 892 x^{4} + 96 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{2} + \beta_{13} q^{3} + \beta_{1} q^{4} + \beta_{2} q^{6} -\beta_{14} q^{7} -\beta_{10} q^{8} + ( -2 \beta_{1} - \beta_{8} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{9} q^{2} + \beta_{13} q^{3} + \beta_{1} q^{4} + \beta_{2} q^{6} -\beta_{14} q^{7} -\beta_{10} q^{8} + ( -2 \beta_{1} - \beta_{8} - \beta_{11} ) q^{9} + ( \beta_{2} + \beta_{14} + \beta_{15} ) q^{11} + \beta_{12} q^{12} + ( -\beta_{4} - \beta_{5} - \beta_{9} - 2 \beta_{10} ) q^{13} -\beta_{4} q^{14} - q^{16} + ( 2 - 2 \beta_{1} - \beta_{8} ) q^{17} + ( -\beta_{3} - \beta_{6} + 2 \beta_{10} ) q^{18} + ( \beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{19} + ( -2 \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{21} + ( \beta_{4} - \beta_{5} - \beta_{10} + \beta_{12} ) q^{22} + ( 1 + \beta_{2} + \beta_{7} - 2 \beta_{11} + \beta_{15} ) q^{23} + \beta_{7} q^{24} + ( -2 - \beta_{14} - \beta_{15} ) q^{26} + ( -2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{9} - \beta_{10} + \beta_{12} ) q^{27} + ( -1 - \beta_{15} ) q^{28} + ( -3 \beta_{3} + \beta_{12} + \beta_{13} ) q^{29} + ( \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{31} + \beta_{9} q^{32} + ( 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{9} + \beta_{10} + \beta_{13} ) q^{33} + ( -\beta_{3} - 2 \beta_{9} + 2 \beta_{10} ) q^{34} + ( 2 - \beta_{8} + \beta_{11} ) q^{36} + ( -\beta_{3} + \beta_{6} + 4 \beta_{9} ) q^{37} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{38} + ( 2 - 6 \beta_{1} + 3 \beta_{7} - \beta_{8} - \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{39} + ( \beta_{5} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{41} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - 2 \beta_{14} ) q^{42} + ( -2 - \beta_{1} - \beta_{2} - \beta_{7} - \beta_{11} - \beta_{15} ) q^{43} + ( 1 - \beta_{1} + \beta_{7} - \beta_{14} + \beta_{15} ) q^{44} + ( -\beta_{5} - 2 \beta_{6} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{8} ) q^{47} -\beta_{13} q^{48} + ( -2 \beta_{7} - \beta_{8} - \beta_{11} ) q^{49} + ( -\beta_{5} + 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{51} + ( -\beta_{4} + \beta_{5} + 2 \beta_{9} + \beta_{10} ) q^{52} + ( -\beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + 4 \beta_{10} + 2 \beta_{13} ) q^{53} + ( 1 + 2 \beta_{1} + \beta_{7} + 2 \beta_{8} + 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{54} + ( \beta_{5} + \beta_{9} + \beta_{10} ) q^{56} + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{14} ) q^{57} + ( \beta_{2} + \beta_{7} + 3 \beta_{11} ) q^{58} + ( -2 \beta_{3} + \beta_{4} - \beta_{9} + \beta_{10} + 3 \beta_{12} + 3 \beta_{13} ) q^{59} + ( -4 - \beta_{2} - \beta_{8} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{61} + ( 2 \beta_{2} - 2 \beta_{7} - \beta_{8} + \beta_{14} ) q^{62} + ( 2 - \beta_{2} - \beta_{7} + 2 \beta_{11} + 2 \beta_{15} ) q^{63} -\beta_{1} q^{64} + ( 1 + \beta_{2} + 2 \beta_{14} + 2 \beta_{15} ) q^{66} + ( -2 \beta_{4} + 2 \beta_{5} - 2 \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{67} + ( 2 + 2 \beta_{1} + \beta_{11} ) q^{68} + ( -5 \beta_{3} + 4 \beta_{4} - 4 \beta_{9} + 4 \beta_{10} + \beta_{12} + \beta_{13} ) q^{69} + ( 3 \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{71} + ( -\beta_{3} + \beta_{6} - 2 \beta_{9} ) q^{72} + ( -2 \beta_{1} - \beta_{11} + 2 \beta_{15} ) q^{73} + ( -4 \beta_{1} + \beta_{8} + \beta_{11} ) q^{74} + ( -1 - \beta_{2} - \beta_{5} + \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{76} + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{7} + \beta_{8} - \beta_{14} ) q^{77} + ( -\beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{13} ) q^{78} + ( 4 \beta_{3} - 6 \beta_{4} - 5 \beta_{9} + 5 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} ) q^{79} + ( 1 + 4 \beta_{2} + \beta_{8} - \beta_{11} ) q^{81} + ( -3 + 3 \beta_{1} - \beta_{8} + \beta_{14} ) q^{82} + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} ) q^{83} + ( -\beta_{3} - 2 \beta_{4} - 3 \beta_{9} + 3 \beta_{10} - \beta_{12} - \beta_{13} ) q^{84} + ( \beta_{5} - \beta_{6} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{86} + ( 2 - 2 \beta_{1} + 4 \beta_{8} - 3 \beta_{14} ) q^{87} + ( -\beta_{4} - \beta_{5} - \beta_{9} - \beta_{13} ) q^{88} + ( \beta_{3} + \beta_{4} + 8 \beta_{9} - 8 \beta_{10} + \beta_{12} + \beta_{13} ) q^{89} + ( \beta_{5} - 2 \beta_{6} + 8 \beta_{9} + 8 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{91} + ( -\beta_{2} + \beta_{7} - 2 \beta_{8} - \beta_{14} ) q^{92} + ( -7 - 6 \beta_{1} - \beta_{2} - \beta_{7} - \beta_{11} - \beta_{15} ) q^{93} + ( 2 \beta_{3} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} ) q^{94} -\beta_{2} q^{96} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{97} + ( -\beta_{3} - \beta_{6} + 2 \beta_{13} ) q^{98} + ( -1 + 4 \beta_{1} + \beta_{8} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{7} + O(q^{10})$$ $$16q - 8q^{7} - 16q^{16} + 32q^{17} + 8q^{23} - 32q^{26} - 8q^{28} + 32q^{36} - 8q^{38} + 32q^{42} - 24q^{43} - 32q^{47} + 48q^{57} - 64q^{61} + 8q^{62} + 16q^{63} + 16q^{66} + 32q^{68} - 16q^{73} - 16q^{76} - 72q^{77} + 16q^{81} - 40q^{82} + 16q^{83} + 8q^{87} - 8q^{92} - 104q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 24 x^{14} + 212 x^{12} + 880 x^{10} + 1858 x^{8} + 1960 x^{6} + 892 x^{4} + 96 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} - 26 \nu^{13} - 259 \nu^{11} - 1278 \nu^{9} - 3369 \nu^{7} - 4598 \nu^{5} - 2743 \nu^{3} - 402 \nu$$$$)/40$$ $$\beta_{2}$$ $$=$$ $$($$$$15 \nu^{14} + 334 \nu^{12} + 2603 \nu^{10} + 8721 \nu^{8} + 12887 \nu^{6} + 6904 \nu^{4} + 255 \nu^{2} + 215$$$$)/116$$ $$\beta_{3}$$ $$=$$ $$($$$$15 \nu^{14} + 334 \nu^{12} + 2603 \nu^{10} + 8721 \nu^{8} + 12887 \nu^{6} + 6904 \nu^{4} + 139 \nu^{2} - 133$$$$)/116$$ $$\beta_{4}$$ $$=$$ $$($$$$113 \nu^{14} + 2578 \nu^{12} + 20922 \nu^{10} + 75129 \nu^{8} + 124547 \nu^{6} + 84784 \nu^{4} + 12564 \nu^{2} - 1609$$$$)/580$$ $$\beta_{5}$$ $$=$$ $$($$$$-20 \nu^{15} - 455 \nu^{13} - 3664 \nu^{11} - 12875 \nu^{9} - 20160 \nu^{7} - 12347 \nu^{5} - 3820 \nu^{3} - 3467 \nu$$$$)/232$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{15} - 332 \nu^{13} - 3211 \nu^{11} - 15104 \nu^{9} - 37029 \nu^{7} - 45640 \nu^{5} - 23711 \nu^{3} - 2748 \nu$$$$)/232$$ $$\beta_{7}$$ $$=$$ $$($$$$-16 \nu^{15} - 393 \nu^{13} - 3604 \nu^{11} - 15897 \nu^{9} - 36892 \nu^{7} - 45101 \nu^{5} - 25792 \nu^{3} - 4241 \nu$$$$)/232$$ $$\beta_{8}$$ $$=$$ $$($$$$17 \nu^{15} - 8 \nu^{14} + 423 \nu^{13} - 182 \nu^{12} + 3938 \nu^{11} - 1454 \nu^{10} + 17563 \nu^{9} - 4918 \nu^{8} + 40307 \nu^{7} - 6556 \nu^{6} + 46207 \nu^{5} - 1250 \nu^{4} + 22068 \nu^{3} + 2126 \nu^{2} + 2003 \nu + 214$$$$)/232$$ $$\beta_{9}$$ $$=$$ $$($$$$-85 \nu^{15} + 18 \nu^{14} - 2115 \nu^{13} + 453 \nu^{12} - 19690 \nu^{11} + 4272 \nu^{10} - 87815 \nu^{9} + 19229 \nu^{8} - 201535 \nu^{7} + 43722 \nu^{6} - 231035 \nu^{5} + 47429 \nu^{4} - 110340 \nu^{3} + 19504 \nu^{2} - 8855 \nu + 1041$$$$)/1160$$ $$\beta_{10}$$ $$=$$ $$($$$$-85 \nu^{15} - 18 \nu^{14} - 2115 \nu^{13} - 453 \nu^{12} - 19690 \nu^{11} - 4272 \nu^{10} - 87815 \nu^{9} - 19229 \nu^{8} - 201535 \nu^{7} - 43722 \nu^{6} - 231035 \nu^{5} - 47429 \nu^{4} - 110340 \nu^{3} - 19504 \nu^{2} - 8855 \nu - 1041$$$$)/1160$$ $$\beta_{11}$$ $$=$$ $$($$$$17 \nu^{15} + 8 \nu^{14} + 423 \nu^{13} + 182 \nu^{12} + 3938 \nu^{11} + 1454 \nu^{10} + 17563 \nu^{9} + 4918 \nu^{8} + 40307 \nu^{7} + 6556 \nu^{6} + 46207 \nu^{5} + 1250 \nu^{4} + 22068 \nu^{3} - 2126 \nu^{2} + 2003 \nu - 214$$$$)/232$$ $$\beta_{12}$$ $$=$$ $$($$$$49 \nu^{15} - 23 \nu^{14} + 1180 \nu^{13} - 545 \nu^{12} + 10479 \nu^{11} - 4695 \nu^{10} + 43847 \nu^{9} - 18511 \nu^{8} + 93501 \nu^{7} - 35219 \nu^{6} + 99318 \nu^{5} - 30397 \nu^{4} + 44275 \nu^{3} - 9439 \nu^{2} + 3293 \nu - 407$$$$)/232$$ $$\beta_{13}$$ $$=$$ $$($$$$-49 \nu^{15} - 23 \nu^{14} - 1180 \nu^{13} - 545 \nu^{12} - 10479 \nu^{11} - 4695 \nu^{10} - 43847 \nu^{9} - 18511 \nu^{8} - 93501 \nu^{7} - 35219 \nu^{6} - 99318 \nu^{5} - 30397 \nu^{4} - 44275 \nu^{3} - 9439 \nu^{2} - 3293 \nu - 407$$$$)/232$$ $$\beta_{14}$$ $$=$$ $$($$$$536 \nu^{15} - 10 \nu^{14} + 12716 \nu^{13} - 155 \nu^{12} + 110294 \nu^{11} - 150 \nu^{10} + 445013 \nu^{9} + 7555 \nu^{8} + 901744 \nu^{7} + 41830 \nu^{6} + 894358 \nu^{5} + 81885 \nu^{4} + 365058 \nu^{3} + 54930 \nu^{2} + 26987 \nu + 3095$$$$)/1160$$ $$\beta_{15}$$ $$=$$ $$($$$$-536 \nu^{15} - 10 \nu^{14} - 12716 \nu^{13} - 155 \nu^{12} - 110294 \nu^{11} - 150 \nu^{10} - 445013 \nu^{9} + 7555 \nu^{8} - 901744 \nu^{7} + 41830 \nu^{6} - 894358 \nu^{5} + 81885 \nu^{4} - 365058 \nu^{3} + 54930 \nu^{2} - 26987 \nu + 1935$$$$)/1160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + 3 \beta_{13} - 3 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} - 8 \beta_{9} - 6 \beta_{8} - 2 \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} - 2 \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{4} + 10 \beta_{3} - 8 \beta_{2} + 19$$ $$\nu^{5}$$ $$=$$ $$($$$$10 \beta_{15} - 10 \beta_{14} - 30 \beta_{13} + 30 \beta_{12} + 45 \beta_{11} + 69 \beta_{10} + 69 \beta_{9} + 45 \beta_{8} - 4 \beta_{7} - 10 \beta_{6} + 30 \beta_{1} + 10$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-\beta_{15} - \beta_{14} - 6 \beta_{13} - 6 \beta_{12} - 17 \beta_{11} + 35 \beta_{10} - 35 \beta_{9} + 17 \beta_{8} + 20 \beta_{4} - 87 \beta_{3} + 65 \beta_{2} - 150$$ $$\nu^{7}$$ $$=$$ $$($$$$-85 \beta_{15} + 85 \beta_{14} + 259 \beta_{13} - 259 \beta_{12} - 368 \beta_{11} - 610 \beta_{10} - 610 \beta_{9} - 368 \beta_{8} + 82 \beta_{7} + 168 \beta_{6} - 14 \beta_{5} - 370 \beta_{1} - 85$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$24 \beta_{15} + 24 \beta_{14} + 108 \beta_{13} + 108 \beta_{12} + 218 \beta_{11} - 432 \beta_{10} + 432 \beta_{9} - 218 \beta_{8} - 172 \beta_{4} + 748 \beta_{3} - 540 \beta_{2} + 1283$$ $$\nu^{9}$$ $$=$$ $$($$$$712 \beta_{15} - 712 \beta_{14} - 2208 \beta_{13} + 2208 \beta_{12} + 3135 \beta_{11} + 5479 \beta_{10} + 5479 \beta_{9} + 3135 \beta_{8} - 1136 \beta_{7} - 2076 \beta_{6} + 264 \beta_{5} + 4124 \beta_{1} + 712$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-350 \beta_{15} - 350 \beta_{14} - 1388 \beta_{13} - 1388 \beta_{12} - 2474 \beta_{11} + 4730 \beta_{10} - 4730 \beta_{9} + 2474 \beta_{8} + 1460 \beta_{4} - 6515 \beta_{3} + 4595 \beta_{2} - 11363$$ $$\nu^{11}$$ $$=$$ $$($$$$-6055 \beta_{15} + 6055 \beta_{14} + 19085 \beta_{13} - 19085 \beta_{12} - 27418 \beta_{11} - 49840 \beta_{10} - 49840 \beta_{9} - 27418 \beta_{8} + 13372 \beta_{7} + 22880 \beta_{6} - 3476 \beta_{5} - 43390 \beta_{1} - 6055$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$4212 \beta_{15} + 4212 \beta_{14} + 15652 \beta_{13} + 15652 \beta_{12} + 26281 \beta_{11} - 48990 \beta_{10} + 48990 \beta_{9} - 26281 \beta_{8} - 12570 \beta_{4} + 57714 \beta_{3} - 39988 \beta_{2} + 102625$$ $$\nu^{13}$$ $$=$$ $$($$$$52558 \beta_{15} - 52558 \beta_{14} - 167986 \beta_{13} + 167986 \beta_{12} + 244527 \beta_{11} + 458015 \beta_{10} + 458015 \beta_{9} + 244527 \beta_{8} - 144852 \beta_{7} - 238290 \beta_{6} + 39728 \beta_{5} + 440766 \beta_{1} + 52558$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$-46145 \beta_{15} - 46145 \beta_{14} - 165290 \beta_{13} - 165290 \beta_{12} - 268469 \beta_{11} + 492047 \beta_{10} - 492047 \beta_{9} + 268469 \beta_{8} + 110272 \beta_{4} - 519257 \beta_{3} + 354799 \beta_{2} - 939032$$ $$\nu^{15}$$ $$=$$ $$($$$$-465071 \beta_{15} + 465071 \beta_{14} + 1503585 \beta_{13} - 1503585 \beta_{12} - 2214032 \beta_{11} - 4242622 \beta_{10} - 4242622 \beta_{9} - 2214032 \beta_{8} + 1496746 \beta_{7} + 2402736 \beta_{6} - 422870 \beta_{5} - 4378382 \beta_{1} - 465071$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1
 1.04813i − 0.107911i − 1.18980i 3.07801i 1.66380i − 2.60402i − 1.52212i − 0.366085i − 1.04813i 0.107911i 1.18980i − 3.07801i − 1.66380i 2.60402i 1.52212i 0.366085i
−0.707107 0.707107i −1.68553 + 1.68553i 1.00000i 0 2.38370 2.49494 2.49494i 0.707107 0.707107i 2.68204i 0
493.2 −0.707107 0.707107i −1.29807 + 1.29807i 1.00000i 0 1.83575 −2.12620 + 2.12620i 0.707107 0.707107i 0.369965i 0
493.3 −0.707107 0.707107i 0.776589 0.776589i 1.00000i 0 −1.09826 −0.710512 + 0.710512i 0.707107 0.707107i 1.79382i 0
493.4 −0.707107 0.707107i 2.20701 2.20701i 1.00000i 0 −3.12119 −1.65823 + 1.65823i 0.707107 0.707107i 6.74181i 0
493.5 0.707107 + 0.707107i −2.20701 + 2.20701i 1.00000i 0 −3.12119 −1.65823 + 1.65823i −0.707107 + 0.707107i 6.74181i 0
493.6 0.707107 + 0.707107i −0.776589 + 0.776589i 1.00000i 0 −1.09826 −0.710512 + 0.710512i −0.707107 + 0.707107i 1.79382i 0
493.7 0.707107 + 0.707107i 1.29807 1.29807i 1.00000i 0 1.83575 −2.12620 + 2.12620i −0.707107 + 0.707107i 0.369965i 0
493.8 0.707107 + 0.707107i 1.68553 1.68553i 1.00000i 0 2.38370 2.49494 2.49494i −0.707107 + 0.707107i 2.68204i 0
607.1 −0.707107 + 0.707107i −1.68553 1.68553i 1.00000i 0 2.38370 2.49494 + 2.49494i 0.707107 + 0.707107i 2.68204i 0
607.2 −0.707107 + 0.707107i −1.29807 1.29807i 1.00000i 0 1.83575 −2.12620 2.12620i 0.707107 + 0.707107i 0.369965i 0
607.3 −0.707107 + 0.707107i 0.776589 + 0.776589i 1.00000i 0 −1.09826 −0.710512 0.710512i 0.707107 + 0.707107i 1.79382i 0
607.4 −0.707107 + 0.707107i 2.20701 + 2.20701i 1.00000i 0 −3.12119 −1.65823 1.65823i 0.707107 + 0.707107i 6.74181i 0
607.5 0.707107 0.707107i −2.20701 2.20701i 1.00000i 0 −3.12119 −1.65823 1.65823i −0.707107 0.707107i 6.74181i 0
607.6 0.707107 0.707107i −0.776589 0.776589i 1.00000i 0 −1.09826 −0.710512 0.710512i −0.707107 0.707107i 1.79382i 0
607.7 0.707107 0.707107i 1.29807 + 1.29807i 1.00000i 0 1.83575 −2.12620 2.12620i −0.707107 0.707107i 0.369965i 0
607.8 0.707107 0.707107i 1.68553 + 1.68553i 1.00000i 0 2.38370 2.49494 + 2.49494i −0.707107 0.707107i 2.68204i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.8 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.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.c 16
5.b even 2 1 190.2.f.b 16
5.c odd 4 1 190.2.f.b 16
5.c odd 4 1 inner 950.2.f.c 16
15.d odd 2 1 1710.2.p.b 16
15.e even 4 1 1710.2.p.b 16
19.b odd 2 1 inner 950.2.f.c 16
95.d odd 2 1 190.2.f.b 16
95.g even 4 1 190.2.f.b 16
95.g even 4 1 inner 950.2.f.c 16
285.b even 2 1 1710.2.p.b 16
285.j odd 4 1 1710.2.p.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.f.b 16 5.b even 2 1
190.2.f.b 16 5.c odd 4 1
190.2.f.b 16 95.d odd 2 1
190.2.f.b 16 95.g even 4 1
950.2.f.c 16 1.a even 1 1 trivial
950.2.f.c 16 5.c odd 4 1 inner
950.2.f.c 16 19.b odd 2 1 inner
950.2.f.c 16 95.g even 4 1 inner
1710.2.p.b 16 15.d odd 2 1
1710.2.p.b 16 15.e even 4 1
1710.2.p.b 16 285.b even 2 1
1710.2.p.b 16 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + 140 T_{3}^{12} + 4710 T_{3}^{8} + 41356 T_{3}^{4} + 50625$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{4}$$
$3$ $$50625 + 41356 T^{4} + 4710 T^{8} + 140 T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 625 + 1300 T + 1352 T^{2} + 620 T^{3} + 150 T^{4} + 12 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$11$ $$( -10 - 28 T - 20 T^{2} + T^{4} )^{4}$$
$13$ $$2300257521 + 85679956 T^{4} + 591830 T^{8} + 1396 T^{12} + T^{16}$$
$17$ $$( 25 - 280 T + 1568 T^{2} - 2048 T^{3} + 1434 T^{4} - 552 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$19$ $$16983563041 + 376367048 T^{2} + 32840892 T^{4} - 464968 T^{6} + 136614 T^{8} - 1288 T^{10} + 252 T^{12} + 8 T^{14} + T^{16}$$
$23$ $$( 140625 + 133500 T + 63368 T^{2} - 26268 T^{3} + 5334 T^{4} - 44 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$29$ $$( 50625 - 72500 T^{2} + 5926 T^{4} - 148 T^{6} + T^{8} )^{2}$$
$31$ $$( 152100 + 36368 T^{2} + 2972 T^{4} + 96 T^{6} + T^{8} )^{2}$$
$37$ $$136048896 + 450585856 T^{4} + 4277600 T^{8} + 4816 T^{12} + T^{16}$$
$41$ $$( 22500 + 15632 T^{2} + 2564 T^{4} + 96 T^{6} + T^{8} )^{2}$$
$43$ $$( 2500 + 2000 T + 800 T^{2} + 120 T^{3} + 244 T^{4} + 184 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$47$ $$( 57600 + 122880 T + 131072 T^{2} + 16128 T^{3} + 1056 T^{4} + 128 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$53$ $$3486784401 + 606173428 T^{4} + 5208374 T^{8} + 8148 T^{12} + T^{16}$$
$59$ $$( 3515625 - 596300 T^{2} + 20026 T^{4} - 244 T^{6} + T^{8} )^{2}$$
$61$ $$( -1910 - 612 T + 12 T^{2} + 16 T^{3} + T^{4} )^{4}$$
$67$ $$43046721 + 758993164 T^{4} + 29736166 T^{8} + 19724 T^{12} + T^{16}$$
$71$ $$( 14516100 + 1191248 T^{2} + 33836 T^{4} + 368 T^{6} + T^{8} )^{2}$$
$73$ $$( 15625 - 1000 T^{3} + 2250 T^{4} - 400 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$79$ $$( 729000000 - 23303168 T^{2} + 215088 T^{4} - 784 T^{6} + T^{8} )^{2}$$
$83$ $$( 1600 + 1920 T + 1152 T^{2} - 64 T^{3} - 16 T^{4} + 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$89$ $$( 70056900 - 4943888 T^{2} + 86196 T^{4} - 528 T^{6} + T^{8} )^{2}$$
$97$ $$9475854336 + 10229929984 T^{4} + 28926336 T^{8} + 20064 T^{12} + T^{16}$$