# Properties

 Label 950.2.j.i Level $950$ Weight $2$ Character orbit 950.j 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.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 8 x^{14} - 36 x^{13} + 67 x^{12} + 34 x^{11} - 24 x^{10} + 182 x^{9} - 495 x^{8} - 166 x^{7} + 258 x^{6} - 1292 x^{5} + 2920 x^{4} + 1176 x^{3} + 200 x^{2} + 80 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{4} ) q^{2} -\beta_{14} q^{3} + ( 1 + \beta_{5} ) q^{4} -\beta_{15} q^{6} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{7} -\beta_{4} q^{8} + ( 3 + 3 \beta_{5} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{4} ) q^{2} -\beta_{14} q^{3} + ( 1 + \beta_{5} ) q^{4} -\beta_{15} q^{6} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{7} -\beta_{4} q^{8} + ( 3 + 3 \beta_{5} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{15} ) q^{9} + ( 2 - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{11} + ( -\beta_{3} - \beta_{14} ) q^{12} + ( 3 \beta_{2} - \beta_{7} - \beta_{11} ) q^{13} + ( \beta_{1} - 2 \beta_{5} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{15} ) q^{14} + \beta_{5} q^{16} + ( \beta_{2} + \beta_{4} + \beta_{7} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{17} + ( -\beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{14} ) q^{18} + ( \beta_{1} + \beta_{12} - 2 \beta_{15} ) q^{19} + ( -\beta_{1} + 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{12} + \beta_{15} ) q^{21} + ( -\beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{22} + ( \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{11} + \beta_{13} ) q^{23} + ( \beta_{1} - \beta_{15} ) q^{24} + ( -3 + \beta_{9} + \beta_{10} ) q^{26} + ( -\beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{27} + ( -\beta_{2} - \beta_{3} - \beta_{7} - \beta_{11} ) q^{28} + ( -5 - 4 \beta_{5} - \beta_{8} + \beta_{10} ) q^{29} + ( 3 - 2 \beta_{1} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} ) q^{31} + \beta_{2} q^{32} + ( 2 \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{11} - \beta_{13} - 4 \beta_{14} ) q^{33} + ( -1 - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{15} ) q^{34} + ( -\beta_{1} + 4 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{15} ) q^{36} + ( -3 \beta_{3} - \beta_{4} - \beta_{11} - \beta_{13} - 3 \beta_{14} ) q^{37} + ( -2 \beta_{3} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{38} + ( 4 + 4 \beta_{1} - \beta_{5} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{39} + ( -\beta_{1} - 3 \beta_{5} - 2 \beta_{9} - 2 \beta_{12} + \beta_{15} ) q^{41} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{7} - 2 \beta_{11} ) q^{42} + ( 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{43} + ( 2 + \beta_{5} + \beta_{8} - \beta_{10} ) q^{44} + ( -\beta_{5} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{46} + ( -3 \beta_{2} - \beta_{3} - \beta_{7} - \beta_{11} ) q^{47} -\beta_{3} q^{48} + ( -5 + 2 \beta_{1} + \beta_{5} - \beta_{8} + \beta_{12} ) q^{49} + ( -4 - 2 \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{51} + ( 2 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{11} + \beta_{13} ) q^{52} + ( -3 \beta_{2} - \beta_{3} - \beta_{7} - \beta_{11} ) q^{53} + ( 2 \beta_{1} - 3 \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - 2 \beta_{15} ) q^{54} + ( 1 + \beta_{1} + \beta_{9} + \beta_{10} ) q^{56} + ( 6 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{13} + \beta_{14} ) q^{57} + ( 5 \beta_{4} + \beta_{11} + \beta_{13} ) q^{58} + ( \beta_{1} + 4 \beta_{5} + \beta_{9} + \beta_{12} - \beta_{15} ) q^{59} + ( -2 - 2 \beta_{5} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{61} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{62} + ( -8 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{11} + 3 \beta_{13} ) q^{63} - q^{64} + ( -2 - 2 \beta_{5} - \beta_{9} + \beta_{10} + \beta_{12} - 4 \beta_{15} ) q^{66} + ( -2 \beta_{2} + \beta_{3} + \beta_{7} + \beta_{11} ) q^{67} + ( -\beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{14} ) q^{68} + ( -6 - 3 \beta_{1} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} ) q^{69} + ( 2 \beta_{1} - 3 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{15} ) q^{71} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{13} ) q^{72} + ( 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} ) q^{73} + ( 3 \beta_{1} + \beta_{5} + \beta_{9} + \beta_{12} - 3 \beta_{15} ) q^{74} + ( 2 \beta_{1} + \beta_{10} + \beta_{12} - \beta_{15} ) q^{76} + ( 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{11} - \beta_{13} - 4 \beta_{14} ) q^{77} + ( -6 \beta_{2} - 6 \beta_{4} - 2 \beta_{6} + \beta_{7} - 3 \beta_{11} + 3 \beta_{13} + 4 \beta_{14} ) q^{78} + ( -2 \beta_{1} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{15} ) q^{79} + ( -6 \beta_{1} + 12 \beta_{5} - 3 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{12} + 6 \beta_{15} ) q^{81} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} + 2 \beta_{13} ) q^{82} + ( -\beta_{2} - \beta_{3} + 5 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{14} ) q^{83} + ( -2 - \beta_{1} + 2 \beta_{9} + 2 \beta_{10} ) q^{84} + ( -2 - 4 \beta_{5} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{86} + ( -\beta_{2} + 7 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} + 7 \beta_{14} ) q^{87} + ( -2 \beta_{4} - \beta_{11} - \beta_{13} ) q^{88} + ( -5 - 3 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} ) q^{89} + ( -6 - 4 \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + 7 \beta_{15} ) q^{91} + ( 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} ) q^{92} + ( -2 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} + \beta_{7} + 3 \beta_{11} - 3 \beta_{13} - \beta_{14} ) q^{93} + ( 3 + \beta_{1} + \beta_{9} + \beta_{10} ) q^{94} + \beta_{1} q^{96} + ( -5 \beta_{2} - 5 \beta_{4} + \beta_{7} - \beta_{11} + \beta_{13} + 3 \beta_{14} ) q^{97} + ( 5 \beta_{2} + 5 \beta_{4} - \beta_{7} + \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{98} + ( 16 + 17 \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + 7 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{4} + 2q^{6} + 22q^{9} + O(q^{10})$$ $$16q + 8q^{4} + 2q^{6} + 22q^{9} + 20q^{11} + 12q^{14} - 8q^{16} - 2q^{21} - 2q^{24} - 36q^{26} - 34q^{29} + 44q^{31} - 10q^{34} - 22q^{36} + 72q^{39} + 14q^{41} + 10q^{44} - 24q^{46} - 88q^{49} - 18q^{51} + 16q^{54} + 24q^{56} - 28q^{59} - 18q^{61} - 16q^{64} - 8q^{66} - 108q^{69} + 28q^{71} - 8q^{74} + 34q^{79} - 72q^{81} - 4q^{84} - 26q^{86} - 28q^{89} - 50q^{91} + 56q^{94} - 4q^{96} + 120q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 8 x^{14} - 36 x^{13} + 67 x^{12} + 34 x^{11} - 24 x^{10} + 182 x^{9} - 495 x^{8} - 166 x^{7} + 258 x^{6} - 1292 x^{5} + 2920 x^{4} + 1176 x^{3} + 200 x^{2} + 80 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$279227424185 \nu^{15} - 1253191152168 \nu^{14} + 2149480076638 \nu^{13} - 10769326233704 \nu^{12} + 28177085791575 \nu^{11} + 2405387591342 \nu^{10} + 30961320250634 \nu^{9} - 150224876901330 \nu^{8} - 158301324166899 \nu^{7} - 20272123106222 \nu^{6} + 150963883461760 \nu^{5} + 948140298810660 \nu^{4} + 341175880277816 \nu^{3} + 65125552505168 \nu^{2} + 23338264882344 \nu - 5000793621270704$$$$)/ 1648849543827840$$ $$\beta_{2}$$ $$=$$ $$($$$$-2554774305209 \nu^{15} + 4335535617510 \nu^{14} + 5455089919862 \nu^{13} + 35039031777188 \nu^{12} + 61311937532721 \nu^{11} - 569910800650412 \nu^{10} + 35829458499862 \nu^{9} - 276472542171762 \nu^{8} + 132028748158047 \nu^{7} + 3776590780457084 \nu^{6} - 923077071265360 \nu^{5} + 1621043757545628 \nu^{4} + 756816090478912 \nu^{3} - 23978017695078704 \nu^{2} + 45304332735480 \nu + 10377464251808$$$$)/ 1648849543827840$$ $$\beta_{3}$$ $$=$$ $$($$$$997381935613 \nu^{15} - 1649145399090 \nu^{14} - 2330828468434 \nu^{13} - 13267293959116 \nu^{12} - 25635498319497 \nu^{11} + 226486239183484 \nu^{10} - 13731014168234 \nu^{9} + 106263434009934 \nu^{8} - 53029560960579 \nu^{7} - 1500225464318788 \nu^{6} + 386794141646720 \nu^{5} - 616309906049196 \nu^{4} - 291791019656384 \nu^{3} + 9097892662933528 \nu^{2} - 17391304628760 \nu - 4027830549856$$$$)/ 206106192978480$$ $$\beta_{4}$$ $$=$$ $$($$$$-2070672794 \nu^{15} + 9113657058 \nu^{14} - 20042279554 \nu^{13} + 81839364227 \nu^{12} - 169989772743 \nu^{11} - 8962973576 \nu^{10} + 66320346205 \nu^{9} - 399864503514 \nu^{8} + 1180002213903 \nu^{7} - 89988470689 \nu^{6} - 590795011120 \nu^{5} + 2900116876956 \nu^{4} - 7162603244888 \nu^{3} + 196338149584 \nu^{2} + 75791010936 \nu - 90197230528$$$$)/ 104622432984$$ $$\beta_{5}$$ $$=$$ $$($$$$27445864585 \nu^{15} - 114495167832 \nu^{14} + 240112435262 \nu^{13} - 1033250237896 \nu^{12} + 2024996088615 \nu^{11} + 550070722558 \nu^{10} - 678874701974 \nu^{9} + 5115617357070 \nu^{8} - 14492719927011 \nu^{7} - 1895749127518 \nu^{6} + 6891426699680 \nu^{5} - 36790030621500 \nu^{4} + 86716448660344 \nu^{3} + 16138554024592 \nu^{2} + 5934294177576 \nu + 1092234854864$$$$)/ 1275212330880$$ $$\beta_{6}$$ $$=$$ $$($$$$-8686137725825 \nu^{15} + 42313430612454 \nu^{14} - 101803992713434 \nu^{13} + 381862990208372 \nu^{12} - 872518973835831 \nu^{11} + 288288968521732 \nu^{10} + 314330560086838 \nu^{9} - 1805036844483906 \nu^{8} + 5753269990031463 \nu^{7} - 2700285992025508 \nu^{6} - 2426868339941968 \nu^{5} + 13311842117065020 \nu^{4} - 35699844475117376 \nu^{3} + 14601702815636752 \nu^{2} + 346449370868280 \nu - 450005598920032$$$$)/ 329769908765568$$ $$\beta_{7}$$ $$=$$ $$($$$$453854154643 \nu^{15} - 2148242385810 \nu^{14} + 5058579031646 \nu^{13} - 19397985604396 \nu^{12} + 43227903066213 \nu^{11} - 10504651056956 \nu^{10} - 15217534809074 \nu^{9} + 92416778982534 \nu^{8} - 286864003800549 \nu^{7} + 108414463327772 \nu^{6} + 122851299785840 \nu^{5} - 678946120862436 \nu^{4} + 1780650592054816 \nu^{3} - 568865130413072 \nu^{2} - 17667737170920 \nu + 22444040275424$$$$)/ 13973301218880$$ $$\beta_{8}$$ $$=$$ $$($$$$-312593461575 \nu^{15} + 1305052923848 \nu^{14} - 2734307093218 \nu^{13} + 11768193355464 \nu^{12} - 23099923074905 \nu^{11} - 6290866981922 \nu^{10} + 7728403255866 \nu^{9} - 57640587207250 \nu^{8} + 165785687172509 \nu^{7} + 21635639719922 \nu^{6} - 77274021793920 \nu^{5} + 411296583843940 \nu^{4} - 992392514808776 \nu^{3} - 184670937394608 \nu^{2} - 67912631446744 \nu + 4675736035664$$$$)/ 9315534145920$$ $$\beta_{9}$$ $$=$$ $$($$$$10900402862375 \nu^{15} - 45452419955144 \nu^{14} + 95432288434874 \nu^{13} - 410270868691352 \nu^{12} + 802746853240385 \nu^{11} + 218299036166986 \nu^{10} - 276657009730258 \nu^{9} + 2065965717654250 \nu^{8} - 5735126785204397 \nu^{7} - 751081282393786 \nu^{6} + 2767834595485840 \nu^{5} - 14724969351287620 \nu^{4} + 34415942530333288 \nu^{3} + 6405462596584784 \nu^{2} + 2355200038235992 \nu + 1644107573550128$$$$)/ 274808257304640$$ $$\beta_{10}$$ $$=$$ $$($$$$-21911472450815 \nu^{15} + 91435548636808 \nu^{14} - 191728937133858 \nu^{13} + 824801347929704 \nu^{12} - 1617789871722865 \nu^{11} - 437705517794562 \nu^{10} + 542276882972026 \nu^{9} - 4052777591009810 \nu^{8} + 11536992356516389 \nu^{7} + 1509729505360722 \nu^{6} - 5604878957320480 \nu^{5} + 28946443100502980 \nu^{4} - 69013732889708616 \nu^{3} - 12845479319063408 \nu^{2} - 4722839369813144 \nu + 538715661376464$$$$)/ 549616514609280$$ $$\beta_{11}$$ $$=$$ $$($$$$-35876471476807 \nu^{15} + 142013953594170 \nu^{14} - 278222669153654 \nu^{13} + 1267628390050204 \nu^{12} - 2325280412563137 \nu^{11} - 1425992145674356 \nu^{10} + 1024417527408026 \nu^{9} - 6430588780352766 \nu^{8} + 17401151964390801 \nu^{7} + 7157630081850772 \nu^{6} - 10641557396482160 \nu^{5} + 45765148270171764 \nu^{4} - 102377588671650784 \nu^{3} - 49960138575606352 \nu^{2} + 1202570377070280 \nu - 1287332125546976$$$$)/ 824424771913920$$ $$\beta_{12}$$ $$=$$ $$($$$$-15156149970385 \nu^{15} + 63147564290984 \nu^{14} - 132518533985934 \nu^{13} + 570659988276592 \nu^{12} - 1116036517703975 \nu^{11} - 303771743125326 \nu^{10} + 367883152197278 \nu^{9} - 2858442575885950 \nu^{8} + 8011689454391267 \nu^{7} + 1049824853870886 \nu^{6} - 3743774633723120 \nu^{5} + 20425874452495180 \nu^{4} - 47829584824712088 \nu^{3} - 8900724034061104 \nu^{2} - 3273138752262952 \nu - 1813136794190928$$$$)/ 274808257304640$$ $$\beta_{13}$$ $$=$$ $$($$$$95383263803887 \nu^{15} - 388596494728170 \nu^{14} + 786381536962454 \nu^{13} - 3469684848879484 \nu^{12} + 6601197866959737 \nu^{11} + 2954873449622116 \nu^{10} - 2891302559786186 \nu^{9} + 17440498236979086 \nu^{8} - 48341286400946361 \nu^{7} - 13285502578596532 \nu^{6} + 28064740508800880 \nu^{5} - 124748926095394884 \nu^{4} + 285608850184508224 \nu^{3} + 97671412723360912 \nu^{2} - 3273329814085320 \nu + 3592742641175456$$$$)/ 1648849543827840$$ $$\beta_{14}$$ $$=$$ $$($$$$-107197556804581 \nu^{15} + 449995772335710 \nu^{14} - 941949725565842 \nu^{13} + 4026927893654452 \nu^{12} - 7942224171410931 \nu^{11} - 2241465050143708 \nu^{10} + 3332661249014798 \nu^{9} - 20015911543649178 \nu^{8} + 56979822291703443 \nu^{7} + 7172073154641196 \nu^{6} - 31409862725090000 \nu^{5} + 143934587182818252 \nu^{4} - 340225833225284992 \nu^{3} - 63372547476611056 \nu^{2} + 3771826243988760 \nu - 4281518066231648$$$$)/ 1648849543827840$$ $$\beta_{15}$$ $$=$$ $$($$$$-13452207709045 \nu^{15} + 56083850298582 \nu^{14} - 117674794077602 \nu^{13} + 506429630647996 \nu^{12} - 991211770226175 \nu^{11} - 270048956304808 \nu^{10} + 332679951349574 \nu^{9} - 2532281570625150 \nu^{8} + 7111279378882731 \nu^{7} + 930691079013928 \nu^{6} - 3335896743811940 \nu^{5} + 18028410763163820 \nu^{4} - 42554153768011024 \nu^{3} - 7919054149479832 \nu^{2} - 2912122790817576 \nu - 1161652513451264$$$$)/ 206106192978480$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{7} + \beta_{3} - 4 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$($$$$21 \beta_{14} + 13 \beta_{13} + \beta_{12} - \beta_{11} - 13 \beta_{10} - 13 \beta_{9} - \beta_{8} + 14 \beta_{7} - 14 \beta_{6} + \beta_{5} + 19 \beta_{4} + 21 \beta_{3} - 14 \beta_{2} - 21 \beta_{1} + 33$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{15} - 10 \beta_{12} - 9 \beta_{10} - \beta_{9} - 9 \beta_{8} - 18 \beta_{5} - 12 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$171 \beta_{15} + 11 \beta_{13} - 116 \beta_{12} + 127 \beta_{11} + 11 \beta_{10} + 116 \beta_{9} - 127 \beta_{8} + 127 \beta_{7} + 11 \beta_{6} - 170 \beta_{5} + 11 \beta_{4} + 171 \beta_{3} - 286 \beta_{2} - 297$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$113 \beta_{14} + 91 \beta_{13} + 12 \beta_{11} + 79 \beta_{7} - 79 \beta_{6} + 137 \beta_{4} + 113 \beta_{3} - 79 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$($$$$1467 \beta_{15} + 1467 \beta_{14} + 998 \beta_{13} - 1147 \beta_{12} - 998 \beta_{11} - 998 \beta_{10} - 149 \beta_{9} - 998 \beta_{8} - 149 \beta_{7} - 1147 \beta_{6} - 1609 \beta_{5} + 1460 \beta_{4} + 1460 \beta_{2} - 1467 \beta_{1}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$1008 \beta_{15} - 691 \beta_{12} + 115 \beta_{10} + 691 \beta_{9} - 806 \beta_{8} - 1031 \beta_{5} - 1837$$ $$\nu^{9}$$ $$=$$ $$($$$$12765 \beta_{14} + 10129 \beta_{13} + 1445 \beta_{12} + 1445 \beta_{11} + 10129 \beta_{10} + 10129 \beta_{9} - 1445 \beta_{8} + 8684 \beta_{7} - 8684 \beta_{6} + 1445 \beta_{5} + 14107 \beta_{4} + 12765 \beta_{3} - 8684 \beta_{2} + 12765 \beta_{1} - 22791$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$8863 \beta_{14} + 6041 \beta_{13} - 6041 \beta_{11} - 1036 \beta_{7} - 7077 \beta_{6} + 8851 \beta_{4} + 8851 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$($$$$111477 \beta_{15} - 13019 \beta_{13} - 75818 \beta_{12} - 88837 \beta_{11} + 13019 \beta_{10} + 75818 \beta_{9} - 88837 \beta_{8} - 88837 \beta_{7} - 13019 \beta_{6} - 110348 \beta_{5} - 13019 \beta_{4} - 111477 \beta_{3} + 186166 \beta_{2} - 199185$$$$)/6$$ $$\nu^{12}$$ $$=$$ $$9149 \beta_{12} + 61962 \beta_{10} + 61962 \beta_{9} - 9149 \beta_{8} + 9149 \beta_{5} + 77616 \beta_{1} - 138915$$ $$\nu^{13}$$ $$=$$ $$($$$$-974403 \beta_{15} + 974403 \beta_{14} + 662612 \beta_{13} + 777487 \beta_{12} - 662612 \beta_{11} + 662612 \beta_{10} + 114875 \beta_{9} + 662612 \beta_{8} - 114875 \beta_{7} - 777487 \beta_{6} + 1078501 \beta_{5} + 963626 \beta_{4} + 963626 \beta_{2} + 974403 \beta_{1}$$$$)/6$$ $$\nu^{14}$$ $$=$$ $$-80260 \beta_{13} - 541995 \beta_{11} - 541995 \beta_{7} - 80260 \beta_{6} - 80260 \beta_{4} - 678929 \beta_{3} + 1133396 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$($$$$-8518923 \beta_{14} - 6799867 \beta_{13} + 1007237 \beta_{12} - 1007237 \beta_{11} + 6799867 \beta_{10} + 6799867 \beta_{9} - 1007237 \beta_{8} - 5792630 \beta_{7} + 5792630 \beta_{6} + 1007237 \beta_{5} - 9428761 \beta_{4} - 8518923 \beta_{3} + 5792630 \beta_{2} + 8518923 \beta_{1} - 15221391$$$$)/6$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{5}$$

## 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
 0.0677143 + 0.252713i −0.765301 − 2.85614i −0.433740 − 1.61874i 0.399276 + 1.49012i −1.49012 + 0.399276i 1.61874 − 0.433740i 2.85614 − 0.765301i −0.252713 + 0.0677143i 0.0677143 − 0.252713i −0.765301 + 2.85614i −0.433740 + 1.61874i 0.399276 − 1.49012i −1.49012 − 0.399276i 1.61874 + 0.433740i 2.85614 + 0.765301i −0.252713 − 0.0677143i
−0.866025 0.500000i −2.62877 1.51772i 0.500000 + 0.866025i 0 1.51772 + 2.62877i 4.93155i 1.00000i 3.10694 + 5.38138i 0
49.2 −0.866025 0.500000i −1.47519 0.851703i 0.500000 + 0.866025i 0 0.851703 + 1.47519i 3.74324i 1.00000i −0.0492032 0.0852224i 0
49.3 −0.866025 0.500000i 0.410396 + 0.236942i 0.500000 + 0.866025i 0 −0.236942 0.410396i 2.19155i 1.00000i −1.38772 2.40360i 0
49.4 −0.866025 0.500000i 2.82754 + 1.63248i 0.500000 + 0.866025i 0 −1.63248 2.82754i 2.62013i 1.00000i 3.82998 + 6.63372i 0
49.5 0.866025 + 0.500000i −2.82754 1.63248i 0.500000 + 0.866025i 0 −1.63248 2.82754i 2.62013i 1.00000i 3.82998 + 6.63372i 0
49.6 0.866025 + 0.500000i −0.410396 0.236942i 0.500000 + 0.866025i 0 −0.236942 0.410396i 2.19155i 1.00000i −1.38772 2.40360i 0
49.7 0.866025 + 0.500000i 1.47519 + 0.851703i 0.500000 + 0.866025i 0 0.851703 + 1.47519i 3.74324i 1.00000i −0.0492032 0.0852224i 0
49.8 0.866025 + 0.500000i 2.62877 + 1.51772i 0.500000 + 0.866025i 0 1.51772 + 2.62877i 4.93155i 1.00000i 3.10694 + 5.38138i 0
349.1 −0.866025 + 0.500000i −2.62877 + 1.51772i 0.500000 0.866025i 0 1.51772 2.62877i 4.93155i 1.00000i 3.10694 5.38138i 0
349.2 −0.866025 + 0.500000i −1.47519 + 0.851703i 0.500000 0.866025i 0 0.851703 1.47519i 3.74324i 1.00000i −0.0492032 + 0.0852224i 0
349.3 −0.866025 + 0.500000i 0.410396 0.236942i 0.500000 0.866025i 0 −0.236942 + 0.410396i 2.19155i 1.00000i −1.38772 + 2.40360i 0
349.4 −0.866025 + 0.500000i 2.82754 1.63248i 0.500000 0.866025i 0 −1.63248 + 2.82754i 2.62013i 1.00000i 3.82998 6.63372i 0
349.5 0.866025 0.500000i −2.82754 + 1.63248i 0.500000 0.866025i 0 −1.63248 + 2.82754i 2.62013i 1.00000i 3.82998 6.63372i 0
349.6 0.866025 0.500000i −0.410396 + 0.236942i 0.500000 0.866025i 0 −0.236942 + 0.410396i 2.19155i 1.00000i −1.38772 + 2.40360i 0
349.7 0.866025 0.500000i 1.47519 0.851703i 0.500000 0.866025i 0 0.851703 1.47519i 3.74324i 1.00000i −0.0492032 + 0.0852224i 0
349.8 0.866025 0.500000i 2.62877 1.51772i 0.500000 0.866025i 0 1.51772 2.62877i 4.93155i 1.00000i 3.10694 5.38138i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.8 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.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.i 16
5.b even 2 1 inner 950.2.j.i 16
5.c odd 4 1 950.2.e.l 8
5.c odd 4 1 950.2.e.m yes 8
19.c even 3 1 inner 950.2.j.i 16
95.i even 6 1 inner 950.2.j.i 16
95.m odd 12 1 950.2.e.l 8
95.m odd 12 1 950.2.e.m yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.l 8 5.c odd 4 1
950.2.e.l 8 95.m odd 12 1
950.2.e.m yes 8 5.c odd 4 1
950.2.e.m yes 8 95.m odd 12 1
950.2.j.i 16 1.a even 1 1 trivial
950.2.j.i 16 5.b even 2 1 inner
950.2.j.i 16 19.c even 3 1 inner
950.2.j.i 16 95.i even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{16} - \cdots$$ $$T_{7}^{8} + 50 T_{7}^{6} + 821 T_{7}^{4} + 5240 T_{7}^{2} + 11236$$ $$T_{11}^{4} - 5 T_{11}^{3} - 20 T_{11}^{2} + 123 T_{11} - 117$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$4096 - 20480 T^{2} + 92096 T^{4} - 48576 T^{6} + 18497 T^{8} - 3063 T^{10} + 368 T^{12} - 23 T^{14} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 11236 + 5240 T^{2} + 821 T^{4} + 50 T^{6} + T^{8} )^{2}$$
$11$ $$( -117 + 123 T - 20 T^{2} - 5 T^{3} + T^{4} )^{4}$$
$13$ $$4162314256 - 1370577904 T^{2} + 316791676 T^{4} - 34100212 T^{6} + 2604433 T^{8} - 122227 T^{10} + 4156 T^{12} - 79 T^{14} + T^{16}$$
$17$ $$2562890625 - 922640625 T^{2} + 231710625 T^{4} - 27957150 T^{6} + 2409406 T^{8} - 124254 T^{10} + 4577 T^{12} - 81 T^{14} + T^{16}$$
$19$ $$( 130321 + 12635 T^{2} + 684 T^{4} + 35 T^{6} + T^{8} )^{2}$$
$23$ $$( 1679616 - 104976 T^{2} + 5265 T^{4} - 81 T^{6} + T^{8} )^{2}$$
$29$ $$( 202500 + 35550 T^{2} + 15300 T^{3} + 6691 T^{4} + 1343 T^{5} + 210 T^{6} + 17 T^{7} + T^{8} )^{2}$$
$31$ $$( 1118 + 360 T - 41 T^{2} - 11 T^{3} + T^{4} )^{4}$$
$37$ $$( 9585216 + 733104 T^{2} + 20116 T^{4} + 236 T^{6} + T^{8} )^{2}$$
$41$ $$( 5948721 - 2538999 T + 839781 T^{2} - 138246 T^{3} + 19726 T^{4} - 1382 T^{5} + 149 T^{6} - 7 T^{7} + T^{8} )^{2}$$
$43$ $$875213056 - 2672736896 T^{2} + 7941430448 T^{4} - 663814152 T^{6} + 40489817 T^{8} - 1064631 T^{10} + 20432 T^{12} - 167 T^{14} + T^{16}$$
$47$ $$2998219536 - 1360139040 T^{2} + 479642796 T^{4} - 52467480 T^{6} + 4004725 T^{8} - 176130 T^{10} + 5591 T^{12} - 90 T^{14} + T^{16}$$
$53$ $$2998219536 - 1360139040 T^{2} + 479642796 T^{4} - 52467480 T^{6} + 4004725 T^{8} - 176130 T^{10} + 5591 T^{12} - 90 T^{14} + T^{16}$$
$59$ $$( 363609 + 97686 T + 49158 T^{2} + 10728 T^{3} + 4315 T^{4} + 856 T^{5} + 158 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$61$ $$( 256 + 2048 T + 15792 T^{2} + 5024 T^{3} + 2537 T^{4} - 77 T^{5} + 118 T^{6} + 9 T^{7} + T^{8} )^{2}$$
$67$ $$256 - 2240 T^{2} + 16256 T^{4} - 27660 T^{6} + 36665 T^{8} - 10170 T^{10} + 2291 T^{12} - 50 T^{14} + T^{16}$$
$71$ $$( 2862864 - 1664928 T + 895500 T^{2} - 89688 T^{3} + 17317 T^{4} - 1366 T^{5} + 239 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$73$ $$45137758519296 - 4040108015616 T^{2} + 239224347648 T^{4} - 7904500992 T^{6} + 188635537 T^{8} - 2932571 T^{10} + 33312 T^{12} - 227 T^{14} + T^{16}$$
$79$ $$( 7022500 - 2703000 T + 963550 T^{2} - 119680 T^{3} + 20831 T^{4} - 1547 T^{5} + 318 T^{6} - 17 T^{7} + T^{8} )^{2}$$
$83$ $$( 3504384 + 511200 T^{2} + 20041 T^{4} + 267 T^{6} + T^{8} )^{2}$$
$89$ $$( 8555625 + 3246750 T + 1103400 T^{2} + 130740 T^{3} + 20401 T^{4} + 1604 T^{5} + 240 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$97$ $$6561 - 225261 T^{2} + 7586865 T^{4} - 5006718 T^{6} + 2549686 T^{8} - 482942 T^{10} + 70545 T^{12} - 269 T^{14} + T^{16}$$