# Properties

 Label 1950.2.bc.h Level $1950$ Weight $2$ Character orbit 1950.bc 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.bc (of order $$6$$, degree $$2$$, 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.

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

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 $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 0.5 + 1.72434i 0.5 + 0.822735i 0.5 − 4.41310i 0.5 + 4.71596i 0.5 + 0.414256i 0.5 − 4.99624i 0.5 − 1.72434i 0.5 − 0.822735i 0.5 + 4.41310i 0.5 − 4.71596i 0.5 − 0.414256i 0.5 + 4.99624i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i −3.10934 1.79518i 1.00000i −0.500000 + 0.866025i 0
751.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i −2.32854 1.34438i 1.00000i −0.500000 + 0.866025i 0
751.3 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 2.20583 + 1.27354i 1.00000i −0.500000 + 0.866025i 0
751.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i −3.96812 2.29099i 1.00000i −0.500000 + 0.866025i 0
751.5 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i −0.242731 0.140141i 1.00000i −0.500000 + 0.866025i 0
751.6 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 4.44290 + 2.56511i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i −3.10934 + 1.79518i 1.00000i −0.500000 0.866025i 0
901.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i −2.32854 + 1.34438i 1.00000i −0.500000 0.866025i 0
901.3 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 2.20583 1.27354i 1.00000i −0.500000 0.866025i 0
901.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −3.96812 + 2.29099i 1.00000i −0.500000 0.866025i 0
901.5 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −0.242731 + 0.140141i 1.00000i −0.500000 0.866025i 0
901.6 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 4.44290 2.56511i 1.00000i −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

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

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$( 1 + T + T^{2} )^{6}$$
$5$ $$T^{12}$$
$7$ $$26244 + 174960 T + 410184 T^{2} + 142560 T^{3} - 50994 T^{4} - 27900 T^{5} + 7296 T^{6} + 5244 T^{7} + 421 T^{8} - 186 T^{9} - 19 T^{10} + 6 T^{11} + T^{12}$$
$11$ $$11664 + 23328 T - 4212 T^{2} - 39528 T^{3} + 10809 T^{4} + 62388 T^{5} + 43314 T^{6} + 7440 T^{7} - 725 T^{8} - 264 T^{9} + 26 T^{10} + 12 T^{11} + T^{12}$$
$13$ $$4826809 + 285610 T^{2} - 131820 T^{3} + 11492 T^{4} - 1560 T^{5} + 3058 T^{6} - 120 T^{7} + 68 T^{8} - 60 T^{9} + 10 T^{10} + T^{12}$$
$17$ $$876096 - 449280 T + 587952 T^{2} + 26112 T^{3} + 139444 T^{4} + 15912 T^{5} + 24284 T^{6} + 4680 T^{7} + 2118 T^{8} + 168 T^{9} + 50 T^{10} + T^{12}$$
$19$ $$73513476 - 28602864 T - 7951008 T^{2} + 4536960 T^{3} + 867214 T^{4} - 405516 T^{5} - 48452 T^{6} + 20520 T^{7} + 3009 T^{8} - 426 T^{9} - 59 T^{10} + 6 T^{11} + T^{12}$$
$23$ $$46656 - 85536 T + 126360 T^{2} - 86940 T^{3} + 55305 T^{4} - 16056 T^{5} + 8544 T^{6} - 1572 T^{7} + 1171 T^{8} - 16 T^{9} + 48 T^{10} - 4 T^{11} + T^{12}$$
$29$ $$56070144 + 52475904 T + 37670400 T^{2} + 14302464 T^{3} + 4705600 T^{4} + 922752 T^{5} + 213152 T^{6} + 29088 T^{7} + 6936 T^{8} + 480 T^{9} + 92 T^{10} + T^{12}$$
$31$ $$248629824 + 95507424 T^{2} + 11994948 T^{4} + 649176 T^{6} + 17008 T^{8} + 212 T^{10} + T^{12}$$
$37$ $$56070144 - 38817792 T + 1132992 T^{2} + 5417280 T^{3} - 218951 T^{4} - 449364 T^{5} + 24472 T^{6} + 23712 T^{7} + 27 T^{8} - 672 T^{9} - 8 T^{10} + 12 T^{11} + T^{12}$$
$41$ $$876096 - 2089152 T + 2018160 T^{2} - 852624 T^{3} + 58804 T^{4} + 68760 T^{5} - 10172 T^{6} - 6768 T^{7} + 2118 T^{8} - 50 T^{10} + T^{12}$$
$43$ $$45077796 - 35449920 T + 32981040 T^{2} - 2540064 T^{3} + 2852110 T^{4} + 37180 T^{5} + 206116 T^{6} + 12040 T^{7} + 6145 T^{8} + 526 T^{9} + 145 T^{10} + 10 T^{11} + T^{12}$$
$47$ $$2985984 + 4022784 T^{2} + 1424016 T^{4} + 190752 T^{6} + 9976 T^{8} + 200 T^{10} + T^{12}$$
$53$ $$( 15768 + 11088 T + 870 T^{2} - 612 T^{3} - 74 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$59$ $$746496 + 3856896 T + 4814208 T^{2} - 9445824 T^{3} + 4066192 T^{4} + 457056 T^{5} - 202784 T^{6} - 18000 T^{7} + 8700 T^{8} - 104 T^{10} + T^{12}$$
$61$ $$82955664 + 30712176 T + 32673996 T^{2} - 5482596 T^{3} + 4349449 T^{4} - 1069812 T^{5} + 518876 T^{6} - 131604 T^{7} + 28755 T^{8} - 3864 T^{9} + 404 T^{10} - 24 T^{11} + T^{12}$$
$67$ $$753831936 + 848719872 T + 253501440 T^{2} - 73199616 T^{3} - 23294528 T^{4} + 5234304 T^{5} + 1259392 T^{6} - 418248 T^{7} + 31881 T^{8} + 1182 T^{9} - 185 T^{10} - 6 T^{11} + T^{12}$$
$71$ $$438439973904 + 233057555856 T + 17030349468 T^{2} - 12898013940 T^{3} + 623152233 T^{4} + 141565824 T^{5} - 7873572 T^{6} - 1133148 T^{7} + 79531 T^{8} + 4368 T^{9} - 316 T^{10} - 12 T^{11} + T^{12}$$
$73$ $$2653641 + 11127402 T^{2} + 9572175 T^{4} + 1562700 T^{6} + 59239 T^{8} + 458 T^{10} + T^{12}$$
$79$ $$( 320086 - 80600 T - 30184 T^{2} + 5824 T^{3} - 73 T^{4} - 26 T^{5} + T^{6} )^{2}$$
$83$ $$900388436544 + 85687587936 T^{2} + 2655600993 T^{4} + 36660300 T^{6} + 249286 T^{8} + 812 T^{10} + T^{12}$$
$89$ $$77158656 + 471173760 T + 945819360 T^{2} - 80996400 T^{3} - 42031004 T^{4} + 3467544 T^{5} + 1436620 T^{6} - 155856 T^{7} - 13602 T^{8} + 2064 T^{9} + 106 T^{10} - 24 T^{11} + T^{12}$$
$97$ $$21233664 - 19906560 T - 5054976 T^{2} + 10571040 T^{3} + 2661985 T^{4} - 7021428 T^{5} + 3206746 T^{6} - 522024 T^{7} + 24003 T^{8} + 2184 T^{9} - 134 T^{10} - 12 T^{11} + T^{12}$$