# Properties

 Label 105.3.e.a Level $105$ Weight $3$ Character orbit 105.e Analytic conductor $2.861$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 105.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.86104277578$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + 9786 x^{8} - 5744 x^{7} + 8608 x^{6} + 4740 x^{5} + 22841 x^{4} + 22790 x^{3} + 18930 x^{2} + 8170 x + 1849$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{2} q^{3} + ( -2 + \beta_{4} ) q^{4} + ( -\beta_{8} + \beta_{12} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{3} + \beta_{14} ) q^{7} + ( 3 \beta_{3} + \beta_{5} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{2} q^{3} + ( -2 + \beta_{4} ) q^{4} + ( -\beta_{8} + \beta_{12} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{3} + \beta_{14} ) q^{7} + ( 3 \beta_{3} + \beta_{5} ) q^{8} + 3 q^{9} + ( -2 \beta_{1} - \beta_{2} + \beta_{8} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{10} + ( -4 + \beta_{4} + \beta_{15} ) q^{11} + ( -\beta_{2} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{12} + \beta_{14} ) q^{12} + ( 2 \beta_{2} - \beta_{6} - \beta_{8} - \beta_{10} - 2 \beta_{12} - \beta_{14} ) q^{13} + ( -5 + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{15} ) q^{14} + ( 2 - \beta_{3} + \beta_{9} - \beta_{15} ) q^{15} + ( 7 - 4 \beta_{4} + 2 \beta_{9} + 2 \beta_{11} ) q^{16} + ( -2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} + 3 \beta_{12} - 2 \beta_{14} ) q^{17} -3 \beta_{3} q^{18} + ( \beta_{1} - \beta_{13} ) q^{19} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{4} + 4 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{20} + ( \beta_{4} + \beta_{7} + \beta_{13} - \beta_{15} ) q^{21} + ( 8 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{22} + ( -2 \beta_{4} - \beta_{5} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{23} + ( -1 + 2 \beta_{1} - \beta_{4} - 2 \beta_{7} - \beta_{9} - \beta_{11} + \beta_{13} + \beta_{15} ) q^{24} + ( -2 \beta_{3} - \beta_{5} - \beta_{9} - 3 \beta_{11} + 2 \beta_{15} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{26} + 3 \beta_{2} q^{27} + ( 2 \beta_{2} + 9 \beta_{3} - \beta_{4} + \beta_{5} - 5 \beta_{6} - 5 \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} + 2 \beta_{12} - 3 \beta_{14} ) q^{28} + ( -3 - 2 \beta_{4} - 3 \beta_{9} - 3 \beta_{11} + 2 \beta_{15} ) q^{29} + ( -4 - 4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} - \beta_{14} - \beta_{15} ) q^{30} + ( 2 + 4 \beta_{1} + 2 \beta_{4} - \beta_{6} + 4 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{31} + ( -17 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} - 2 \beta_{11} ) q^{32} + ( -2 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{12} ) q^{33} + ( 2 - 5 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} - 3 \beta_{13} - 2 \beta_{15} ) q^{34} + ( 1 + 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + \beta_{7} + 4 \beta_{8} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{35} + ( -6 + 3 \beta_{4} ) q^{36} + ( -3 \beta_{4} - 3 \beta_{5} - 3 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - 2 \beta_{14} ) q^{37} + ( 10 \beta_{2} - 3 \beta_{12} ) q^{38} + ( 3 - 2 \beta_{4} - 2 \beta_{9} - 2 \beta_{11} + 3 \beta_{15} ) q^{39} + ( -2 + 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{4} - 12 \beta_{6} - 4 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} + 6 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{40} + ( 1 + 10 \beta_{1} + \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + 6 \beta_{13} - \beta_{15} ) q^{41} + ( -5 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{14} ) q^{42} + ( -6 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{43} + ( 36 - 7 \beta_{4} + 2 \beta_{9} + 2 \beta_{11} - \beta_{15} ) q^{44} + ( -3 \beta_{8} + 3 \beta_{12} ) q^{45} + ( -11 + \beta_{4} - 2 \beta_{9} - 2 \beta_{11} + 10 \beta_{15} ) q^{46} + ( 6 \beta_{2} - 3 \beta_{6} - 3 \beta_{8} - 4 \beta_{10} - 5 \beta_{12} - 4 \beta_{14} ) q^{47} + ( \beta_{2} - 6 \beta_{6} - 6 \beta_{8} - 6 \beta_{10} + 6 \beta_{12} - 6 \beta_{14} ) q^{48} + ( -8 + 7 \beta_{1} - 3 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + \beta_{13} - 4 \beta_{15} ) q^{49} + ( -5 - 5 \beta_{3} + 8 \beta_{4} + \beta_{5} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{50} + ( 2 - 4 \beta_{4} + 3 \beta_{9} + 3 \beta_{11} - \beta_{15} ) q^{51} + ( -14 \beta_{2} + 5 \beta_{6} + 5 \beta_{8} + 5 \beta_{10} - 6 \beta_{12} + 5 \beta_{14} ) q^{52} + ( 12 \beta_{3} + 4 \beta_{4} + \beta_{5} + 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{14} ) q^{53} -3 \beta_{1} q^{54} + ( -2 - 3 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{55} + ( 32 - 14 \beta_{1} - 7 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} + 7 \beta_{11} - 2 \beta_{13} - 7 \beta_{15} ) q^{56} + ( 4 \beta_{3} - \beta_{5} + \beta_{10} - \beta_{14} ) q^{57} + ( -6 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{14} ) q^{58} + ( -3 - 8 \beta_{1} - 3 \beta_{4} - 2 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{11} - 4 \beta_{13} + 3 \beta_{15} ) q^{59} + ( -18 + 9 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + \beta_{10} - \beta_{11} - \beta_{14} ) q^{60} + ( -9 \beta_{1} - 5 \beta_{6} + 5 \beta_{8} + 3 \beta_{13} ) q^{61} + ( 12 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} + 6 \beta_{10} + 4 \beta_{12} + 6 \beta_{14} ) q^{62} + ( -3 \beta_{3} + 3 \beta_{14} ) q^{63} + ( -60 + 13 \beta_{4} + 2 \beta_{9} + 2 \beta_{11} - 4 \beta_{15} ) q^{64} + ( -16 - 6 \beta_{3} - 7 \beta_{4} - \beta_{5} + 2 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} + 2 \beta_{14} - 5 \beta_{15} ) q^{65} + ( 11 \beta_{1} + 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{13} ) q^{66} + ( 10 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{14} ) q^{67} + ( -22 \beta_{2} + 26 \beta_{6} + 26 \beta_{8} + 18 \beta_{10} - 13 \beta_{12} + 18 \beta_{14} ) q^{68} + ( -1 - 7 \beta_{1} - \beta_{4} - 6 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} - \beta_{9} - \beta_{11} - 5 \beta_{13} + \beta_{15} ) q^{69} + ( 32 + 7 \beta_{1} + 11 \beta_{2} + 9 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 10 \beta_{12} + 3 \beta_{14} - 2 \beta_{15} ) q^{70} + ( -7 - \beta_{4} + 3 \beta_{9} + 3 \beta_{11} + 7 \beta_{15} ) q^{71} + ( 9 \beta_{3} + 3 \beta_{5} ) q^{72} + ( -10 \beta_{2} + 9 \beta_{6} + 9 \beta_{8} + 4 \beta_{10} + 4 \beta_{12} + 4 \beta_{14} ) q^{73} + ( -9 + 10 \beta_{4} - 5 \beta_{9} - 5 \beta_{11} + 12 \beta_{15} ) q^{74} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} - 5 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 5 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{75} + ( -3 \beta_{1} - \beta_{13} ) q^{76} + ( -10 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 6 \beta_{14} ) q^{77} + ( -14 \beta_{3} + \beta_{4} + \beta_{9} - 3 \beta_{10} - \beta_{11} + 3 \beta_{14} ) q^{78} + ( 36 + 8 \beta_{4} + 4 \beta_{9} + 4 \beta_{11} - 14 \beta_{15} ) q^{79} + ( -22 \beta_{1} + 24 \beta_{2} - 10 \beta_{6} - 11 \beta_{8} - 14 \beta_{10} - 3 \beta_{12} - 6 \beta_{13} - 14 \beta_{14} ) q^{80} + 9 q^{81} + ( 18 \beta_{2} - 23 \beta_{6} - 23 \beta_{8} - 7 \beta_{10} + 30 \beta_{12} - 7 \beta_{14} ) q^{82} + ( -16 \beta_{2} - 20 \beta_{6} - 20 \beta_{8} + 4 \beta_{10} + 20 \beta_{12} + 4 \beta_{14} ) q^{83} + ( 13 + 7 \beta_{1} - 9 \beta_{4} - 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + 2 \beta_{13} + 2 \beta_{15} ) q^{84} + ( 33 - 20 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 3 \beta_{14} + 10 \beta_{15} ) q^{85} + ( -26 + 13 \beta_{4} - 2 \beta_{9} - 2 \beta_{11} - 5 \beta_{15} ) q^{86} + ( -5 \beta_{6} - 5 \beta_{8} - \beta_{10} - 5 \beta_{12} - \beta_{14} ) q^{87} + ( -40 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 5 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} + 3 \beta_{14} ) q^{88} + ( -1 + 14 \beta_{1} - \beta_{4} + 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{13} + \beta_{15} ) q^{89} + ( -6 \beta_{1} - 3 \beta_{2} + 3 \beta_{8} + 3 \beta_{10} - 3 \beta_{13} + 3 \beta_{14} ) q^{90} + ( -7 - 7 \beta_{1} + 5 \beta_{4} + 7 \beta_{6} - 2 \beta_{7} - 7 \beta_{8} + 5 \beta_{13} + 2 \beta_{15} ) q^{91} + ( 8 \beta_{3} - \beta_{5} - 2 \beta_{10} + 2 \beta_{14} ) q^{92} + ( 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{9} - 4 \beta_{10} + \beta_{11} + 4 \beta_{14} ) q^{93} + ( 4 - 7 \beta_{1} + 4 \beta_{4} + 5 \beta_{6} + 8 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} + 4 \beta_{11} + 5 \beta_{13} - 4 \beta_{15} ) q^{94} + ( -3 - 4 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{9} - 4 \beta_{10} - 3 \beta_{11} + 4 \beta_{14} + 3 \beta_{15} ) q^{95} + ( 2 - 11 \beta_{1} + 2 \beta_{4} + 6 \beta_{6} + 4 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{15} ) q^{96} + ( 50 \beta_{2} + 15 \beta_{6} + 15 \beta_{8} - 2 \beta_{10} - 22 \beta_{12} - 2 \beta_{14} ) q^{97} + ( 22 \beta_{2} + \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 20 \beta_{6} - 20 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 15 \beta_{12} - 12 \beta_{14} ) q^{98} + ( -12 + 3 \beta_{4} + 3 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 32q^{4} + 48q^{9} + O(q^{10})$$ $$16q - 32q^{4} + 48q^{9} - 56q^{11} - 84q^{14} + 24q^{15} + 112q^{16} - 12q^{21} + 16q^{25} - 32q^{29} - 72q^{30} - 4q^{35} - 96q^{36} + 72q^{39} + 568q^{44} - 96q^{46} - 152q^{49} - 96q^{50} + 24q^{51} + 444q^{56} - 288q^{60} - 992q^{64} - 296q^{65} + 504q^{70} - 56q^{71} - 48q^{74} + 464q^{79} + 144q^{81} + 228q^{84} + 608q^{85} - 456q^{86} - 88q^{91} - 24q^{95} - 168q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + 9786 x^{8} - 5744 x^{7} + 8608 x^{6} + 4740 x^{5} + 22841 x^{4} + 22790 x^{3} + 18930 x^{2} + 8170 x + 1849$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5949415435597583 \nu^{15} + 151070270839721768 \nu^{14} - 624949116473778384 \nu^{13} + 9047165453214229720 \nu^{12} - 18515998524725867198 \nu^{11} + 110047952065592100156 \nu^{10} - 11864802943760395776 \nu^{9} + 465972238404344424450 \nu^{8} + 591157737446594753241 \nu^{7} + 1900714298970544923428 \nu^{6} + 2657908517465359193108 \nu^{5} + 3400389223503810186228 \nu^{4} + 2702883348828407814349 \nu^{3} + 1619822353424749896430 \nu^{2} + 46718672472112463295825 \nu + 16996042407543909134631$$$$)/$$$$97\!\cdots\!47$$ $$\beta_{2}$$ $$=$$ $$($$$$-1746443419060029636 \nu^{15} + 16140399267467695117 \nu^{14} - 131158336210169583381 \nu^{13} + 596748427306059971829 \nu^{12} - 1959316725208763683837 \nu^{11} + 4668486467467535394123 \nu^{10} - 5580313192434404344503 \nu^{9} + 12579443135006949310047 \nu^{8} + 3413890492780054808007 \nu^{7} + 48589879865008982901288 \nu^{6} + 71327450410630971555412 \nu^{5} + 109105912523064990680616 \nu^{4} + 92431208210955395625696 \nu^{3} + 59672064035483483788643 \nu^{2} + 22275149134867397848407 \nu + 1203489837053775553761837$$$$)/$$$$69\!\cdots\!37$$ $$\beta_{3}$$ $$=$$ $$($$$$50438416627615303 \nu^{15} - 125916512508080223 \nu^{14} + 2018301138151891119 \nu^{13} + 1060472555166286937 \nu^{12} + 14992577440772723553 \nu^{11} + 73137630911383807371 \nu^{10} + 110184004265753866149 \nu^{9} + 476850832365151275021 \nu^{8} + 896704857347166806328 \nu^{7} + 2008391520044179224700 \nu^{6} + 2729861728590919329192 \nu^{5} + 3077249356876872849804 \nu^{4} + 2313337431531664167281 \nu^{3} + 1286867978085436252509 \nu^{2} + 44420675627597747968458 \nu + 16175812699559282826807$$$$)/$$$$16\!\cdots\!59$$ $$\beta_{4}$$ $$=$$ $$($$$$9240260201139648 \nu^{15} - 48991024550549040 \nu^{14} + 501599756298080184 \nu^{13} - 1178880346269390348 \nu^{12} + 5756065359498089832 \nu^{11} - 1934329416744461416 \nu^{10} + 22915586684194305680 \nu^{9} + 24787756185759851442 \nu^{8} + 89994778661982206520 \nu^{7} + 121241672064917916216 \nu^{6} + 156845999736701285712 \nu^{5} + 119395011668089461444 \nu^{4} + 69034194216161906040 \nu^{3} + 1714181972153862763032 \nu^{2} + 1241794107909005977952 \nu + 1586882849255326084865$$$$)/$$$$22\!\cdots\!29$$ $$\beta_{5}$$ $$=$$ $$($$$$-6016452986261628914 \nu^{15} + 36630713528299685658 \nu^{14} - 351861925648827878550 \nu^{13} + 1018576848078050136218 \nu^{12} - 4332560092633932383430 \nu^{11} + 3928057616029593726870 \nu^{10} - 15659475775851327640992 \nu^{9} - 6584234820175225034190 \nu^{8} - 49683543781496578860456 \nu^{7} - 44583625184705533835864 \nu^{6} - 54320050181301314482500 \nu^{5} - 12830679531167566790592 \nu^{4} - 320451334958550504665872 \nu^{3} - 341865029718639426849336 \nu^{2} - 605700484189793968186938 \nu - 190620790171914320921004$$$$)/$$$$16\!\cdots\!59$$ $$\beta_{6}$$ $$=$$ $$($$$$-2633593782927862323156 \nu^{15} + 20054201752890775950261 \nu^{14} - 179451679553128712414993 \nu^{13} + 680674554894217548453757 \nu^{12} - 2592024903243541775506146 \nu^{11} + 4415754689101438441354204 \nu^{10} - 9067504097822928242519180 \nu^{9} + 4505001486321326588676936 \nu^{8} - 18180151232499897388362939 \nu^{7} + 11662532775646405153270734 \nu^{6} - 22129538411938522556418909 \nu^{5} - 2465926997316334277367257 \nu^{4} - 84411416553156037747263764 \nu^{3} - 44016615044968868964486621 \nu^{2} - 44068198864538500764791564 \nu + 164640522615079640742946$$$$)/$$$$20\!\cdots\!11$$ $$\beta_{7}$$ $$=$$ $$($$$$-15360048755022399771 \nu^{15} + 89122278553051220092 \nu^{14} - 793380149942379006891 \nu^{13} + 1705309885230511409994 \nu^{12} - 4664528024255898202544 \nu^{11} - 16244462874664660731616 \nu^{10} + 50782003486035890500485 \nu^{9} - 200371504470695927278214 \nu^{8} + 204465360246444871692776 \nu^{7} - 409827072816334488445310 \nu^{6} + 385477149182092498342221 \nu^{5} - 560478702880629389677034 \nu^{4} - 267785635096807625434690 \nu^{3} - 831282110023405371102088 \nu^{2} - 286830159331788706929236 \nu - 144015204945851594836046$$$$)/$$$$97\!\cdots\!47$$ $$\beta_{8}$$ $$=$$ $$($$$$4114759305682869183566 \nu^{15} - 29952149581024003840627 \nu^{14} + 271310662894403293708127 \nu^{13} - 980616214878742280612459 \nu^{12} + 3792182787571991120463632 \nu^{11} - 5917983604838226406141396 \nu^{10} + 13243994737858977874458920 \nu^{9} - 4456937405661984103352400 \nu^{8} + 29520039848668763089898331 \nu^{7} - 6497173910582563226050546 \nu^{6} + 27017176177112840905458785 \nu^{5} + 26594540461815545684521025 \nu^{4} + 120959192524162995997644700 \nu^{3} + 140209427531179606517212763 \nu^{2} + 104653177187372851630370522 \nu + 37101555119267401851106892$$$$)/$$$$20\!\cdots\!11$$ $$\beta_{9}$$ $$=$$ $$($$$$-36289516015651703993 \nu^{15} + 274050459470938858125 \nu^{14} - 2459366978577148857409 \nu^{13} + 9250740310870536080193 \nu^{12} - 35371574784708671273370 \nu^{11} + 59391118053427904660182 \nu^{10} - 124084030311785834407554 \nu^{9} + 57434047976414727730302 \nu^{8} - 254131940257000295386722 \nu^{7} + 141868236352045972811902 \nu^{6} - 316373252883130110076473 \nu^{5} - 49190601584590184744743 \nu^{4} - 1143425409803416390227663 \nu^{3} - 715118701411432091738931 \nu^{2} - 636560631745825009713221 \nu - 94450752603254934955638$$$$)/$$$$16\!\cdots\!59$$ $$\beta_{10}$$ $$=$$ $$($$$$-6200464855038343188657 \nu^{15} + 61241519576323447409140 \nu^{14} - 548847328865015199242145 \nu^{13} + 2722935542465529357628070 \nu^{12} - 11179640147741399646999104 \nu^{11} + 30301272404394388793528490 \nu^{10} - 67916921244080072344542889 \nu^{9} + 104906059800656347988170458 \nu^{8} - 149483420657785923268010670 \nu^{7} + 168207242051636646470309910 \nu^{6} - 163609325747782376387266793 \nu^{5} + 70736842946247431299511238 \nu^{4} - 89322976058237643037406968 \nu^{3} + 20440009555194445503409654 \nu^{2} - 7174813320990018832301862 \nu - 2734585224654296517295978$$$$)/$$$$20\!\cdots\!11$$ $$\beta_{11}$$ $$=$$ $$($$$$55887104219314688633 \nu^{15} - 406916954174863607709 \nu^{14} + 3685211110613857871425 \nu^{13} - 13324430878338532067813 \nu^{12} + 51508785441493315877442 \nu^{11} - 80479723904094126573590 \nu^{10} + 179807207621961075495246 \nu^{9} - 61438547413365777880704 \nu^{8} + 399016513201961099688954 \nu^{7} - 92896321113284717786854 \nu^{6} + 353436896978794230693057 \nu^{5} + 338412886814232577839037 \nu^{4} + 1607081819735839761515731 \nu^{3} + 1870762579698903902042607 \nu^{2} + 1356575175420994020150745 \nu + 481123898044641750986643$$$$)/$$$$16\!\cdots\!59$$ $$\beta_{12}$$ $$=$$ $$($$$$11018483493469604673274 \nu^{15} - 87988201551485884951734 \nu^{14} + 786118720966943786556438 \nu^{13} - 3153935262986323085771962 \nu^{12} + 12139694969872455071188044 \nu^{11} - 23151592821396571165538880 \nu^{10} + 46596675619220247132263332 \nu^{9} - 30097753787437611033922872 \nu^{8} + 68199148186629326731506504 \nu^{7} - 16268015656147790087728442 \nu^{6} + 24041936689987440396711810 \nu^{5} + 115718370089913421671124638 \nu^{4} + 195094171393999055690478374 \nu^{3} + 235812624704167970167273206 \nu^{2} + 119873667189599828512150362 \nu + 39178479640488652675382002$$$$)/$$$$20\!\cdots\!11$$ $$\beta_{13}$$ $$=$$ $$($$$$53562517124571748037 \nu^{15} - 504798929746713583544 \nu^{14} + 4494764702161373132238 \nu^{13} - 21335078577054283539650 \nu^{12} + 85571608018273199170406 \nu^{11} - 217322452447820213949048 \nu^{10} + 466787123091048279714180 \nu^{9} - 651518174650085495797174 \nu^{8} + 919131959220802651270693 \nu^{7} - 993688824581634565029488 \nu^{6} + 914678802196350278556398 \nu^{5} - 196224703541739330321646 \nu^{4} + 709477967763952161049703 \nu^{3} - 36244212724108806596882 \nu^{2} - 137317148134623324820553 \nu - 94578759179951502891727$$$$)/$$$$97\!\cdots\!47$$ $$\beta_{14}$$ $$=$$ $$($$$$14381431296329404931553 \nu^{15} - 130008229412218330685222 \nu^{14} + 1157554815739851815613267 \nu^{13} - 5288114345893489494014116 \nu^{12} + 20985323168178764928829012 \nu^{11} - 50418677192934030458918778 \nu^{10} + 106451510189544907728705485 \nu^{9} - 134542477169921978975402418 \nu^{8} + 196481416143282180605991252 \nu^{7} - 192228617663628190570983456 \nu^{6} + 181328681712012974538637147 \nu^{5} + 81354129744566949038712 \nu^{4} + 213653084570336463788068220 \nu^{3} + 50975476123563924806145274 \nu^{2} + 25874095972496197303897050 \nu - 10985256794652743573274298$$$$)/$$$$20\!\cdots\!11$$ $$\beta_{15}$$ $$=$$ $$($$$$-1810215665166888624 \nu^{15} + 14793222179735906016 \nu^{14} - 131601911148799346016 \nu^{13} + 540049809028096859366 \nu^{12} - 2071747030109850612660 \nu^{11} + 4089523456364511747292 \nu^{10} - 8037074797030273373076 \nu^{9} + 5658529035165969328950 \nu^{8} - 10847332329991229337192 \nu^{7} + 3864304961247207676368 \nu^{6} - 3878318142837971994924 \nu^{5} - 17591612431903574711556 \nu^{4} - 30180137670701941410448 \nu^{3} - 28200556819588429131288 \nu^{2} - 12484813682591646238340 \nu - 1501008412603867941195$$$$)/$$$$22\!\cdots\!29$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{12} - \beta_{10} - 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - 9 \beta_{14} - 3 \beta_{13} + 9 \beta_{12} + 3 \beta_{11} - 9 \beta_{10} + 3 \beta_{9} - 18 \beta_{8} + 6 \beta_{7} - 18 \beta_{6} + 5 \beta_{5} + 18 \beta_{4} + 40 \beta_{3} + 42 \beta_{2} - 21 \beta_{1} - 82$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-8 \beta_{15} + 24 \beta_{14} - 8 \beta_{13} - 24 \beta_{12} + 15 \beta_{11} + 24 \beta_{10} + 15 \beta_{9} + 36 \beta_{8} + 16 \beta_{7} + 36 \beta_{6} + 14 \beta_{5} - 27 \beta_{4} + 140 \beta_{3} - 32 \beta_{2} - 72 \beta_{1} + 106$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$22 \beta_{15} + 220 \beta_{14} + 22 \beta_{13} - 220 \beta_{12} + 16 \beta_{11} + 220 \beta_{10} + 35 \beta_{9} + 374 \beta_{8} - 44 \beta_{7} + 341 \beta_{6} - 38 \beta_{5} - 345 \beta_{4} - 190 \beta_{3} - 484 \beta_{2} + 77 \beta_{1} + 1194$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$304 \beta_{15} - 90 \beta_{14} + 330 \beta_{13} + 45 \beta_{12} - 460 \beta_{11} - 90 \beta_{10} - 304 \beta_{9} + 90 \beta_{8} - 660 \beta_{7} - 180 \beta_{6} - 572 \beta_{5} - 174 \beta_{4} - 4264 \beta_{3} - 240 \beta_{2} + 2040 \beta_{1} - 18$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$159 \beta_{15} - 13847 \beta_{14} + 779 \beta_{13} + 12915 \beta_{12} - 4135 \beta_{11} - 13705 \beta_{10} - 4135 \beta_{9} - 20664 \beta_{8} - 1312 \beta_{7} - 20664 \beta_{6} - 1207 \beta_{5} + 19224 \beta_{4} - 11218 \beta_{3} + 23452 \beta_{2} + 5863 \beta_{1} - 64130$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-9154 \beta_{15} - 17468 \beta_{14} - 8288 \beta_{13} + 16744 \beta_{12} + 7602 \beta_{11} - 16692 \beta_{10} + 2170 \beta_{9} - 31472 \beta_{8} + 17920 \beta_{7} - 22064 \beta_{6} + 15132 \beta_{5} + 31561 \beta_{4} + 106312 \beta_{3} + 33376 \beta_{2} - 49056 \beta_{1} - 78049$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-53655 \beta_{15} + 305481 \beta_{14} - 68391 \beta_{13} - 276777 \beta_{12} + 168135 \beta_{11} + 308661 \beta_{10} + 129975 \beta_{9} + 415854 \beta_{8} + 142290 \beta_{7} + 481950 \beta_{6} + 121635 \beta_{5} - 384390 \beta_{4} + 885630 \beta_{3} - 491436 \beta_{2} - 417231 \beta_{1} + 1419200$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$202407 \beta_{15} + 937566 \beta_{14} + 142329 \beta_{13} - 855228 \beta_{12} + 18051 \beta_{11} + 915846 \beta_{10} + 122307 \beta_{9} + 1471569 \beta_{8} - 322278 \beta_{7} + 1290993 \beta_{6} - 268242 \beta_{5} - 1441533 \beta_{4} - 1871540 \beta_{3} - 1572098 \beta_{2} + 852929 \beta_{1} + 4084397$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$2710959 \beta_{15} - 3619903 \beta_{14} + 2725383 \beta_{13} + 3310087 \beta_{12} - 4805661 \beta_{11} - 3904735 \beta_{10} - 3031395 \beta_{9} - 3896504 \beta_{8} - 5944110 \beta_{7} - 6969626 \beta_{6} - 5005329 \beta_{5} + 3224030 \beta_{4} - 35659384 \beta_{3} + 5905282 \beta_{2} + 16509323 \beta_{1} - 19194622$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-2221755 \beta_{15} - 32096940 \beta_{14} - 399360 \beta_{13} + 29102580 \beta_{12} - 7233894 \beta_{11} - 31934820 \beta_{10} - 7671618 \beta_{9} - 47654100 \beta_{8} + 1079520 \beta_{7} - 46895940 \beta_{6} + 853832 \beta_{5} + 46099533 \beta_{4} + 5679604 \beta_{3} + 53205360 \beta_{2} - 2439840 \beta_{1} - 143703808$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-91678709 \beta_{15} - 72135045 \beta_{14} - 80600813 \beta_{13} + 62581077 \beta_{12} + 99178821 \beta_{11} - 62501535 \beta_{10} + 45120435 \beta_{9} - 147627156 \beta_{8} + 177887356 \beta_{7} - 53995278 \beta_{6} + 149238203 \beta_{5} + 158984094 \beta_{4} + 1061376578 \beta_{3} + 115414960 \beta_{2} - 489850839 \beta_{1} - 222089510$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-61804684 \beta_{15} + 827687867 \beta_{14} - 97795045 \beta_{13} - 755034803 \beta_{12} + 346092812 \beta_{11} + 838003799 \beta_{10} + 282493024 \beta_{9} + 1173290194 \beta_{8} + 213457808 \beta_{7} + 1283448256 \beta_{6} + 179701922 \beta_{5} - 1115114082 \beta_{4} + 1282210852 \beta_{3} - 1382736644 \beta_{2} - 593858555 \beta_{1} + 3854381958$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$1161278471 \beta_{15} + 3307866177 \beta_{14} + 897806217 \beta_{13} - 2960162271 \beta_{12} - 482474021 \beta_{11} + 3199222752 \beta_{10} + 120457729 \beta_{9} + 5331365127 \beta_{8} - 1983788367 \beta_{7} + 4287056697 \beta_{6} - 1663689400 \beta_{5} - 5345462271 \beta_{4} - 11840907350 \beta_{3} - 5427508944 \beta_{2} + 5464991433 \beta_{1} + 13705362729$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 1.36603 − 5.19914i −0.366025 + 1.39311i 1.36603 − 3.14303i −0.366025 + 0.842173i 1.36603 − 2.63709i −0.366025 + 0.706606i 1.36603 − 0.959486i −0.366025 + 0.257094i 1.36603 + 0.959486i −0.366025 − 0.257094i 1.36603 + 2.63709i −0.366025 − 0.706606i 1.36603 + 3.14303i −0.366025 − 0.842173i 1.36603 + 5.19914i −0.366025 − 1.39311i
3.80604i −1.73205 −10.4859 −3.71318 + 3.34847i 6.59225i 5.08005 4.81592i 24.6856i 3.00000 12.7444 + 14.1325i
34.2 3.80604i 1.73205 −10.4859 3.71318 3.34847i 6.59225i −5.08005 4.81592i 24.6856i 3.00000 −12.7444 14.1325i
34.3 2.30086i −1.73205 −1.29396 −3.65761 3.40909i 3.98521i −6.39480 + 2.84720i 6.22623i 3.00000 −7.84383 + 8.41565i
34.4 2.30086i 1.73205 −1.29396 3.65761 + 3.40909i 3.98521i 6.39480 + 2.84720i 6.22623i 3.00000 7.84383 8.41565i
34.5 1.93048i −1.73205 0.273228 4.88618 + 1.06077i 3.34370i −0.433408 6.98657i 8.24940i 3.00000 2.04780 9.43270i
34.6 1.93048i 1.73205 0.273228 −4.88618 1.06077i 3.34370i 0.433408 6.98657i 8.24940i 3.00000 −2.04780 + 9.43270i
34.7 0.702393i −1.73205 3.50664 −0.979490 + 4.90312i 1.21658i 3.48021 + 6.07356i 5.27261i 3.00000 3.44392 + 0.687987i
34.8 0.702393i 1.73205 3.50664 0.979490 4.90312i 1.21658i −3.48021 + 6.07356i 5.27261i 3.00000 −3.44392 0.687987i
34.9 0.702393i −1.73205 3.50664 −0.979490 4.90312i 1.21658i 3.48021 6.07356i 5.27261i 3.00000 3.44392 0.687987i
34.10 0.702393i 1.73205 3.50664 0.979490 + 4.90312i 1.21658i −3.48021 6.07356i 5.27261i 3.00000 −3.44392 + 0.687987i
34.11 1.93048i −1.73205 0.273228 4.88618 1.06077i 3.34370i −0.433408 + 6.98657i 8.24940i 3.00000 2.04780 + 9.43270i
34.12 1.93048i 1.73205 0.273228 −4.88618 + 1.06077i 3.34370i 0.433408 + 6.98657i 8.24940i 3.00000 −2.04780 9.43270i
34.13 2.30086i −1.73205 −1.29396 −3.65761 + 3.40909i 3.98521i −6.39480 2.84720i 6.22623i 3.00000 −7.84383 8.41565i
34.14 2.30086i 1.73205 −1.29396 3.65761 3.40909i 3.98521i 6.39480 2.84720i 6.22623i 3.00000 7.84383 + 8.41565i
34.15 3.80604i −1.73205 −10.4859 −3.71318 3.34847i 6.59225i 5.08005 + 4.81592i 24.6856i 3.00000 12.7444 14.1325i
34.16 3.80604i 1.73205 −10.4859 3.71318 + 3.34847i 6.59225i −5.08005 + 4.81592i 24.6856i 3.00000 −12.7444 + 14.1325i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.16 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
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.e.a 16
3.b odd 2 1 315.3.e.e 16
4.b odd 2 1 1680.3.bd.c 16
5.b even 2 1 inner 105.3.e.a 16
5.c odd 4 2 525.3.h.e 16
7.b odd 2 1 inner 105.3.e.a 16
15.d odd 2 1 315.3.e.e 16
20.d odd 2 1 1680.3.bd.c 16
21.c even 2 1 315.3.e.e 16
28.d even 2 1 1680.3.bd.c 16
35.c odd 2 1 inner 105.3.e.a 16
35.f even 4 2 525.3.h.e 16
105.g even 2 1 315.3.e.e 16
140.c even 2 1 1680.3.bd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.e.a 16 1.a even 1 1 trivial
105.3.e.a 16 5.b even 2 1 inner
105.3.e.a 16 7.b odd 2 1 inner
105.3.e.a 16 35.c odd 2 1 inner
315.3.e.e 16 3.b odd 2 1
315.3.e.e 16 15.d odd 2 1
315.3.e.e 16 21.c even 2 1
315.3.e.e 16 105.g even 2 1
525.3.h.e 16 5.c odd 4 2
525.3.h.e 16 35.f even 4 2
1680.3.bd.c 16 4.b odd 2 1
1680.3.bd.c 16 20.d odd 2 1
1680.3.bd.c 16 28.d even 2 1
1680.3.bd.c 16 140.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(105, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 141 + 360 T^{2} + 162 T^{4} + 24 T^{6} + T^{8} )^{2}$$
$3$ $$( -3 + T^{2} )^{8}$$
$5$ $$152587890625 - 1953125000 T^{2} + 160937500 T^{4} + 2005000 T^{6} - 304250 T^{8} + 3208 T^{10} + 412 T^{12} - 8 T^{14} + T^{16}$$
$7$ $$33232930569601 + 1051937827276 T^{2} + 25203709972 T^{4} + 616010164 T^{6} + 11116630 T^{8} + 256564 T^{10} + 4372 T^{12} + 76 T^{14} + T^{16}$$
$11$ $$( 3280 - 784 T - 72 T^{2} + 14 T^{3} + T^{4} )^{4}$$
$13$ $$( 60715264 - 5488384 T^{2} + 95136 T^{4} - 556 T^{6} + T^{8} )^{2}$$
$17$ $$( 4251040000 - 100611328 T^{2} + 643872 T^{4} - 1468 T^{6} + T^{8} )^{2}$$
$19$ $$( 324864 + 155520 T^{2} + 19872 T^{4} + 396 T^{6} + T^{8} )^{2}$$
$23$ $$( 382942464 + 247320384 T^{2} + 2231328 T^{4} + 2904 T^{6} + T^{8} )^{2}$$
$29$ $$( -26384 + 13616 T - 1284 T^{2} + 8 T^{3} + T^{4} )^{4}$$
$31$ $$( 1892910336 + 269527488 T^{2} + 3403008 T^{4} + 3840 T^{6} + T^{8} )^{2}$$
$37$ $$( 3497148290304 + 12867676416 T^{2} + 15493392 T^{4} + 7056 T^{6} + T^{8} )^{2}$$
$41$ $$( 26090305209600 + 49725697152 T^{2} + 33709152 T^{4} + 9684 T^{6} + T^{8} )^{2}$$
$43$ $$( 86666496 + 68744256 T^{2} + 1341552 T^{4} + 2508 T^{6} + T^{8} )^{2}$$
$47$ $$( 1152376486144 - 5230768384 T^{2} + 8083296 T^{4} - 4876 T^{6} + T^{8} )^{2}$$
$53$ $$( 55452408710400 + 109043164224 T^{2} + 65082528 T^{4} + 14472 T^{6} + T^{8} )^{2}$$
$59$ $$( 2149592204544 + 12460079616 T^{2} + 21372672 T^{4} + 9540 T^{6} + T^{8} )^{2}$$
$61$ $$( 9868871097600 + 67103822592 T^{2} + 51638592 T^{4} + 12828 T^{6} + T^{8} )^{2}$$
$67$ $$( 674847686079744 + 1004067039552 T^{2} + 308052144 T^{4} + 31212 T^{6} + T^{8} )^{2}$$
$71$ $$( -3270032 + 348512 T - 8952 T^{2} + 14 T^{3} + T^{4} )^{4}$$
$73$ $$( 36785098365184 - 101126623168 T^{2} + 60395616 T^{4} - 13360 T^{6} + T^{8} )^{2}$$
$79$ $$( 28622512 + 548560 T - 9072 T^{2} - 116 T^{3} + T^{4} )^{4}$$
$83$ $$( 2618504809283584 - 5472870989824 T^{2} + 985411584 T^{4} - 55936 T^{6} + T^{8} )^{2}$$
$89$ $$( 162581970541824 + 241884683136 T^{2} + 104674464 T^{4} + 17604 T^{6} + T^{8} )^{2}$$
$97$ $$( 338426355591424 - 1197256611520 T^{2} + 604538208 T^{4} - 50464 T^{6} + T^{8} )^{2}$$