# Properties

 Label 1950.2.y.n Level $1950$ Weight $2$ Character orbit 1950.y Analytic conductor $15.571$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.y (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 87 x^{10} - 380 x^{9} + 2556 x^{8} - 8010 x^{7} + 29687 x^{6} - 62556 x^{5} + 115386 x^{4} - 135130 x^{3} + 113253 x^{2} - 54888 x + 14089$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 87 x^{10} - 380 x^{9} + 2556 x^{8} - 8010 x^{7} + 29687 x^{6} - 62556 x^{5} + 115386 x^{4} - 135130 x^{3} + 113253 x^{2} - 54888 x + 14089$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-176 \nu^{10} + 880 \nu^{9} - 13279 \nu^{8} + 47836 \nu^{7} - 335904 \nu^{6} + 843982 \nu^{5} - 3291723 \nu^{4} + 5230858 \nu^{3} - 9800366 \nu^{2} + 7317892 \nu - 3607897$$$$)/737586$$ $$\beta_{2}$$ $$=$$ $$($$$$1070 \nu^{11} - 9796 \nu^{10} + 107747 \nu^{9} - 581520 \nu^{8} + 3242004 \nu^{7} - 10618216 \nu^{6} + 34751213 \nu^{5} - 61642348 \nu^{4} + 103874238 \nu^{3} - 76354844 \nu^{2} + 78924115 \nu - 30150460$$$$)/33157458$$ $$\beta_{3}$$ $$=$$ $$($$$$-1070 \nu^{11} + 1974 \nu^{10} - 68637 \nu^{9} + 123933 \nu^{8} - 1646316 \nu^{7} + 3281180 \nu^{6} - 18160751 \nu^{5} + 37476279 \nu^{4} - 81409454 \nu^{3} + 108949118 \nu^{2} - 120239919 \nu + 8385745$$$$)/33157458$$ $$\beta_{4}$$ $$=$$ $$($$$$2140 \nu^{11} - 11770 \nu^{10} + 176384 \nu^{9} - 705453 \nu^{8} + 4888320 \nu^{7} - 13899396 \nu^{6} + 52911964 \nu^{5} - 99118627 \nu^{4} + 185283692 \nu^{3} - 185303962 \nu^{2} + 166006576 \nu - 38536205$$$$)/33157458$$ $$\beta_{5}$$ $$=$$ $$($$$$196298 \nu^{11} - 1423807 \nu^{10} + 17298796 \nu^{9} - 87970594 \nu^{8} + 505040292 \nu^{7} - 1815307513 \nu^{6} + 5860211676 \nu^{5} - 14204635072 \nu^{4} + 24048219416 \nu^{3} - 32665367269 \nu^{2} + 22026982222 \nu - 8896864806$$$$)/ 2884698846$$ $$\beta_{6}$$ $$=$$ $$($$$$-199579 \nu^{11} + 43670 \nu^{10} - 10853189 \nu^{9} - 20210311 \nu^{8} - 114973215 \nu^{7} - 1069324210 \nu^{6} + 1898810217 \nu^{5} - 15838418089 \nu^{4} + 31357497929 \nu^{3} - 61853002840 \nu^{2} + 58367776705 \nu - 34891536081$$$$)/ 2884698846$$ $$\beta_{7}$$ $$=$$ $$($$$$292669 \nu^{11} - 3011773 \nu^{10} + 30806433 \nu^{9} - 211570737 \nu^{8} + 1058307265 \nu^{7} - 5033911301 \nu^{6} + 14392851385 \nu^{5} - 45381571559 \nu^{4} + 68030262545 \nu^{3} - 111050635981 \nu^{2} + 71688382647 \nu - 39899132821$$$$)/ 2884698846$$ $$\beta_{8}$$ $$=$$ $$($$$$370049 \nu^{11} - 2416592 \nu^{10} + 32929734 \nu^{9} - 154475736 \nu^{8} + 981881669 \nu^{7} - 3282046186 \nu^{6} + 11487231632 \nu^{5} - 25708183318 \nu^{4} + 44029180333 \nu^{3} - 53595055148 \nu^{2} + 36245295240 \nu - 13717956122$$$$)/ 2884698846$$ $$\beta_{9}$$ $$=$$ $$($$$$-370049 \nu^{11} + 1653947 \nu^{10} - 29116509 \nu^{9} + 94203315 \nu^{8} - 763671335 \nu^{7} + 1695380863 \nu^{6} - 7474956287 \nu^{5} + 9813617281 \nu^{4} - 18680221561 \nu^{3} + 7276280393 \nu^{2} - 1967511735 \nu - 3683244445$$$$)/ 2884698846$$ $$\beta_{10}$$ $$=$$ $$($$$$-1663873 \nu^{11} + 8187240 \nu^{10} - 136318210 \nu^{9} + 487832491 \nu^{8} - 3738605045 \nu^{7} + 9381467744 \nu^{6} - 39075448160 \nu^{5} + 62438213831 \nu^{4} - 118312226931 \nu^{3} + 95735578536 \nu^{2} - 71180117206 \nu + 14213570375$$$$)/ 2884698846$$ $$\beta_{11}$$ $$=$$ $$($$$$2230220 \nu^{11} - 13193117 \nu^{10} + 192374130 \nu^{9} - 827318553 \nu^{8} + 5578721594 \nu^{7} - 17140264765 \nu^{6} + 63195515414 \nu^{5} - 128944017517 \nu^{4} + 228688296268 \nu^{3} - 244027871345 \nu^{2} + 168521884530 \nu - 52508450723$$$$)/ 2884698846$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$-\beta_{4} - \beta_{3} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_{1} - 12$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{11} + \beta_{10} + 12 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} + 32 \beta_{4} + 15 \beta_{3} - 17 \beta_{2} + 4 \beta_{1} - 15$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{11} - 18 \beta_{10} - 89 \beta_{9} - 95 \beta_{8} + 45 \beta_{7} + 41 \beta_{6} + 8 \beta_{5} + 85 \beta_{4} + 100 \beta_{3} - 55 \beta_{2} - 32 \beta_{1} + 242$$ $$\nu^{5}$$ $$=$$ $$-44 \beta_{11} - 46 \beta_{10} - 321 \beta_{9} - 158 \beta_{8} + 92 \beta_{7} + 133 \beta_{6} + 183 \beta_{5} - 684 \beta_{4} - 235 \beta_{3} + 327 \beta_{2} - 139 \beta_{1} + 592$$ $$\nu^{6}$$ $$=$$ $$329 \beta_{11} + 328 \beta_{10} + 2130 \beta_{9} + 2213 \beta_{8} - 971 \beta_{7} - 838 \beta_{6} + 108 \beta_{5} - 2686 \beta_{4} - 2484 \beta_{3} + 1567 \beta_{2} + 352 \beta_{1} - 4964$$ $$\nu^{7}$$ $$=$$ $$1001 \beta_{11} + 1617 \beta_{10} + 8482 \beta_{9} + 8101 \beta_{8} - 3297 \beta_{7} - 3829 \beta_{6} - 4846 \beta_{5} + 13692 \beta_{4} + 2733 \beta_{3} - 5841 \beta_{2} + 3739 \beta_{1} - 19054$$ $$\nu^{8}$$ $$=$$ $$-7402 \beta_{11} - 4938 \beta_{10} - 51821 \beta_{9} - 43913 \beta_{8} + 19371 \beta_{7} + 16613 \beta_{6} - 10036 \beta_{5} + 77335 \beta_{4} + 59344 \beta_{3} - 41809 \beta_{2} + 1372 \beta_{1} + 96085$$ $$\nu^{9}$$ $$=$$ $$-23870 \beta_{11} - 47752 \beta_{10} - 233457 \beta_{9} - 283244 \beta_{8} + 100886 \beta_{7} + 104755 \beta_{6} + 115911 \beta_{5} - 256271 \beta_{4} - 156 \beta_{3} + 91646 \beta_{2} - 83821 \beta_{1} + 551212$$ $$\nu^{10}$$ $$=$$ $$175288 \beta_{11} + 37401 \beta_{10} + 1241888 \beta_{9} + 697514 \beta_{8} - 348537 \beta_{7} - 307556 \beta_{6} + 418801 \beta_{5} - 2117336 \beta_{4} - 1364810 \beta_{3} + 1056125 \beta_{2} - 242920 \beta_{1} - 1683590$$ $$\nu^{11}$$ $$=$$ $$602019 \beta_{11} + 1254110 \beta_{10} + 6570934 \beta_{9} + 8427566 \beta_{8} - 2812501 \beta_{7} - 2761947 \beta_{6} - 2508922 \beta_{5} + 4247176 \beta_{4} - 1432740 \beta_{3} - 1072366 \beta_{2} + 1551867 \beta_{1} - 14885497$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 + 4.99624i 0.5 − 0.414256i 0.5 − 4.71596i 0.5 − 1.72434i 0.5 − 0.822735i 0.5 + 4.41310i 0.5 − 4.99624i 0.5 + 0.414256i 0.5 + 4.71596i 0.5 + 1.72434i 0.5 + 0.822735i 0.5 − 4.41310i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −2.56511 + 4.44290i −1.00000 0.500000 0.866025i 0
49.2 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 0.140141 0.242731i −1.00000 0.500000 0.866025i 0
49.3 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 2.29099 3.96812i −1.00000 0.500000 0.866025i 0
49.4 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −1.79518 + 3.10934i −1.00000 0.500000 0.866025i 0
49.5 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −1.34438 + 2.32854i −1.00000 0.500000 0.866025i 0
49.6 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 1.27354 2.20583i −1.00000 0.500000 0.866025i 0
199.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i −2.56511 4.44290i −1.00000 0.500000 + 0.866025i 0
199.2 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 0.140141 + 0.242731i −1.00000 0.500000 + 0.866025i 0
199.3 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 2.29099 + 3.96812i −1.00000 0.500000 + 0.866025i 0
199.4 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −1.79518 3.10934i −1.00000 0.500000 + 0.866025i 0
199.5 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −1.34438 2.32854i −1.00000 0.500000 + 0.866025i 0
199.6 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 1.27354 + 2.20583i −1.00000 0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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 1950.2.y.n 12
5.b even 2 1 1950.2.y.m 12
5.c odd 4 1 1950.2.bc.h 12
5.c odd 4 1 1950.2.bc.k yes 12
13.e even 6 1 1950.2.y.m 12
65.l even 6 1 inner 1950.2.y.n 12
65.r odd 12 1 1950.2.bc.h 12
65.r odd 12 1 1950.2.bc.k yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.y.m 12 5.b even 2 1
1950.2.y.m 12 13.e even 6 1
1950.2.y.n 12 1.a even 1 1 trivial
1950.2.y.n 12 65.l even 6 1 inner
1950.2.bc.h 12 5.c odd 4 1
1950.2.bc.h 12 65.r odd 12 1
1950.2.bc.k yes 12 5.c odd 4 1
1950.2.bc.k yes 12 65.r odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{6}$$
$3$ $$( 1 - T^{2} + T^{4} )^{3}$$
$5$ $$T^{12}$$
$7$ $$26244 - 87480 T + 318816 T^{2} + 55728 T^{3} + 81846 T^{4} + 12528 T^{5} + 14052 T^{6} + 2328 T^{7} + 1105 T^{8} + 100 T^{9} + 45 T^{10} + 4 T^{11} + T^{12}$$
$11$ $$11664 + 23328 T - 4212 T^{2} - 39528 T^{3} + 10809 T^{4} + 62388 T^{5} + 43314 T^{6} + 7440 T^{7} - 725 T^{8} - 264 T^{9} + 26 T^{10} + 12 T^{11} + T^{12}$$
$13$ $$4826809 - 1485172 T - 57122 T^{2} - 70304 T^{3} + 41236 T^{4} - 5044 T^{5} + 614 T^{6} - 388 T^{7} + 244 T^{8} - 32 T^{9} - 2 T^{10} - 4 T^{11} + T^{12}$$
$17$ $$876096 - 2089152 T + 2018160 T^{2} - 852624 T^{3} + 58804 T^{4} + 68760 T^{5} - 10172 T^{6} - 6768 T^{7} + 2118 T^{8} - 50 T^{10} + T^{12}$$
$19$ $$73513476 + 28602864 T - 7951008 T^{2} - 4536960 T^{3} + 867214 T^{4} + 405516 T^{5} - 48452 T^{6} - 20520 T^{7} + 3009 T^{8} + 426 T^{9} - 59 T^{10} - 6 T^{11} + T^{12}$$
$23$ $$46656 + 69984 T + 4536 T^{2} - 45684 T^{3} - 1719 T^{4} + 21096 T^{5} + 6288 T^{6} - 2364 T^{7} - 557 T^{8} + 192 T^{9} + 32 T^{10} - 12 T^{11} + T^{12}$$
$29$ $$56070144 - 52475904 T + 37670400 T^{2} - 14302464 T^{3} + 4705600 T^{4} - 922752 T^{5} + 213152 T^{6} - 29088 T^{7} + 6936 T^{8} - 480 T^{9} + 92 T^{10} + T^{12}$$
$31$ $$248629824 + 95507424 T^{2} + 11994948 T^{4} + 649176 T^{6} + 17008 T^{8} + 212 T^{10} + T^{12}$$
$37$ $$56070144 + 28394496 T + 19493568 T^{2} + 4059408 T^{3} + 2210041 T^{4} + 453780 T^{5} + 161144 T^{6} + 23736 T^{7} + 6075 T^{8} + 792 T^{9} + 152 T^{10} + 12 T^{11} + T^{12}$$
$41$ $$876096 - 2089152 T + 2018160 T^{2} - 852624 T^{3} + 58804 T^{4} + 68760 T^{5} - 10172 T^{6} - 6768 T^{7} + 2118 T^{8} - 50 T^{10} + T^{12}$$
$43$ $$45077796 - 74203128 T + 42031512 T^{2} - 2166192 T^{3} - 3109946 T^{4} + 273432 T^{5} + 251200 T^{6} - 58812 T^{7} + 909 T^{8} + 852 T^{9} - 23 T^{10} - 12 T^{11} + T^{12}$$
$47$ $$( -1728 + 288 T + 1140 T^{2} + 192 T^{3} - 68 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$53$ $$248629824 + 95507424 T^{2} + 11994948 T^{4} + 649176 T^{6} + 17008 T^{8} + 212 T^{10} + T^{12}$$
$59$ $$746496 - 3856896 T + 4814208 T^{2} + 9445824 T^{3} + 4066192 T^{4} - 457056 T^{5} - 202784 T^{6} + 18000 T^{7} + 8700 T^{8} - 104 T^{10} + T^{12}$$
$61$ $$82955664 + 30712176 T + 32673996 T^{2} - 5482596 T^{3} + 4349449 T^{4} - 1069812 T^{5} + 518876 T^{6} - 131604 T^{7} + 28755 T^{8} - 3864 T^{9} + 404 T^{10} - 24 T^{11} + T^{12}$$
$67$ $$753831936 - 332107776 T + 297431040 T^{2} + 1999872 T^{3} + 42185152 T^{4} - 5075328 T^{5} + 1615424 T^{6} - 176328 T^{7} + 40953 T^{8} - 4392 T^{9} + 491 T^{10} - 24 T^{11} + T^{12}$$
$71$ $$438439973904 + 233057555856 T + 17030349468 T^{2} - 12898013940 T^{3} + 623152233 T^{4} + 141565824 T^{5} - 7873572 T^{6} - 1133148 T^{7} + 79531 T^{8} + 4368 T^{9} - 316 T^{10} - 12 T^{11} + T^{12}$$
$73$ $$( 1629 - 5148 T + 4719 T^{2} - 1224 T^{3} - 29 T^{4} + 20 T^{5} + T^{6} )^{2}$$
$79$ $$( 320086 + 80600 T - 30184 T^{2} - 5824 T^{3} - 73 T^{4} + 26 T^{5} + T^{6} )^{2}$$
$83$ $$( 948888 - 265932 T - 7887 T^{2} + 5868 T^{3} - 278 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$89$ $$77158656 - 471173760 T + 945819360 T^{2} + 80996400 T^{3} - 42031004 T^{4} - 3467544 T^{5} + 1436620 T^{6} + 155856 T^{7} - 13602 T^{8} - 2064 T^{9} + 106 T^{10} + 24 T^{11} + T^{12}$$
$97$ $$21233664 - 42024960 T + 55973376 T^{2} - 42997344 T^{3} + 24498145 T^{4} - 8327016 T^{5} + 2076686 T^{6} - 177792 T^{7} + 40851 T^{8} - 4320 T^{9} + 494 T^{10} - 24 T^{11} + T^{12}$$