# Properties

 Label 637.2.f.l Level $637$ Weight $2$ Character orbit 637.f Analytic conductor $5.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 8 x^{14} + 45 x^{12} + 124 x^{10} + 248 x^{8} + 250 x^{6} + 177 x^{4} + 14 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{4} + \beta_{10} ) q^{2} + ( \beta_{8} - \beta_{15} ) q^{3} + ( -2 + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{4} + ( \beta_{3} + \beta_{13} - \beta_{14} ) q^{5} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{6} + ( 1 - \beta_{4} - \beta_{6} ) q^{8} + \beta_{10} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{4} + \beta_{10} ) q^{2} + ( \beta_{8} - \beta_{15} ) q^{3} + ( -2 + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{4} + ( \beta_{3} + \beta_{13} - \beta_{14} ) q^{5} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{6} + ( 1 - \beta_{4} - \beta_{6} ) q^{8} + \beta_{10} q^{9} + ( 2 \beta_{7} - \beta_{8} + 2 \beta_{13} - 2 \beta_{14} ) q^{10} + ( -1 - \beta_{4} - \beta_{10} ) q^{11} + ( 2 \beta_{3} + \beta_{5} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{12} + ( \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{13} + ( \beta_{2} - 2 \beta_{6} - \beta_{9} + 2 \beta_{11} - \beta_{12} ) q^{15} + ( 1 - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{16} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{17} + ( -3 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{18} + ( \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} + 3 \beta_{7} ) q^{20} + ( 4 - 4 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{22} + ( -1 - \beta_{2} - \beta_{4} + 2 \beta_{6} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{23} + ( 3 \beta_{7} + 3 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{24} + ( -1 + \beta_{4} + 2 \beta_{6} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} - \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{14} - 3 \beta_{15} ) q^{26} + ( \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{8} + \beta_{13} - 2 \beta_{15} ) q^{27} + ( 2 + 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} ) q^{29} + ( 3 - 3 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{12} ) q^{30} + ( \beta_{5} + 3 \beta_{14} + \beta_{15} ) q^{31} + ( -\beta_{11} + 2 \beta_{12} ) q^{32} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{33} + ( -4 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} - 4 \beta_{8} + 3 \beta_{13} + 2 \beta_{15} ) q^{34} + ( -3 - 3 \beta_{4} - \beta_{6} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{36} + ( -4 - 3 \beta_{2} - 4 \beta_{4} + \beta_{6} + \beta_{9} - 4 \beta_{10} - \beta_{11} + 3 \beta_{12} ) q^{37} + ( -6 \beta_{1} + 5 \beta_{3} - \beta_{5} - 6 \beta_{8} + 5 \beta_{13} - \beta_{14} - \beta_{15} ) q^{38} + ( -2 \beta_{2} + \beta_{4} + \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{39} + ( 3 \beta_{1} - 2 \beta_{3} - \beta_{5} + 3 \beta_{8} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{40} + ( -2 \beta_{7} - 2 \beta_{8} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{41} + ( 5 - 5 \beta_{9} + 2 \beta_{10} - 5 \beta_{12} ) q^{43} + ( 1 + 3 \beta_{4} + \beta_{6} ) q^{44} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{7} ) q^{45} + ( 1 - \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{3} - \beta_{5} - 4 \beta_{8} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{47} + ( -\beta_{1} + 2 \beta_{5} + 4 \beta_{7} ) q^{48} + ( -3 - \beta_{2} - 3 \beta_{4} + 5 \beta_{6} + 6 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} + \beta_{12} ) q^{50} + ( -6 + 3 \beta_{2} - \beta_{4} - \beta_{6} ) q^{51} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} + 6 \beta_{8} - 3 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{52} + ( -1 - 4 \beta_{2} - \beta_{4} - \beta_{6} ) q^{53} + ( 2 \beta_{7} + 4 \beta_{8} - \beta_{13} - 2 \beta_{14} ) q^{54} + ( -2 \beta_{7} + \beta_{8} - 2 \beta_{13} + 2 \beta_{14} ) q^{55} + ( -6 + 2 \beta_{2} - 3 \beta_{6} ) q^{57} + ( -7 + 7 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} - \beta_{12} ) q^{58} + ( 3 \beta_{1} - \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{59} + ( 9 + 2 \beta_{4} + 5 \beta_{6} ) q^{60} + ( -\beta_{1} - 3 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{61} + ( -4 \beta_{7} + \beta_{8} - 2 \beta_{13} + 4 \beta_{14} + 3 \beta_{15} ) q^{62} + ( -3 + 2 \beta_{4} - 4 \beta_{6} ) q^{64} + ( \beta_{4} + \beta_{6} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{8} - 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{66} + ( -1 - \beta_{2} - \beta_{4} - 2 \beta_{6} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{67} + ( 5 \beta_{7} - 2 \beta_{8} + 3 \beta_{13} - 5 \beta_{14} + 3 \beta_{15} ) q^{68} + ( 4 \beta_{3} - 5 \beta_{7} ) q^{69} + ( -1 + \beta_{9} - 4 \beta_{10} + \beta_{11} + 5 \beta_{12} ) q^{71} + ( 5 - 5 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - \beta_{12} ) q^{72} + ( -\beta_{1} - 3 \beta_{3} + 7 \beta_{5} - \beta_{8} - 3 \beta_{13} + 4 \beta_{14} + 7 \beta_{15} ) q^{73} + ( 12 - 12 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{74} + ( -6 \beta_{7} - 5 \beta_{13} + 6 \beta_{14} + 3 \beta_{15} ) q^{75} + ( 9 \beta_{7} + \beta_{8} + 3 \beta_{13} - 9 \beta_{14} + 6 \beta_{15} ) q^{76} + ( -2 - 2 \beta_{2} - \beta_{4} + 6 \beta_{6} + 6 \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} ) q^{78} + ( -1 + 7 \beta_{2} - \beta_{4} ) q^{79} + ( -\beta_{13} - \beta_{15} ) q^{80} + ( 4 + \beta_{2} + 4 \beta_{4} - \beta_{6} + 5 \beta_{9} + 4 \beta_{10} + \beta_{11} - \beta_{12} ) q^{81} + ( -6 \beta_{1} + 2 \beta_{3} - 6 \beta_{5} + 4 \beta_{7} ) q^{82} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{5} - \beta_{8} - 3 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{83} + ( 4 - 4 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{85} + ( 4 - 3 \beta_{2} + 7 \beta_{4} - 2 \beta_{6} ) q^{86} + ( -4 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} - 9 \beta_{7} ) q^{87} + ( -3 - \beta_{2} - 3 \beta_{4} + 3 \beta_{6} + 5 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + \beta_{12} ) q^{88} + ( -10 \beta_{7} - 4 \beta_{8} - 3 \beta_{13} + 10 \beta_{14} + 2 \beta_{15} ) q^{89} + ( 2 \beta_{1} - 4 \beta_{3} - \beta_{5} + 2 \beta_{8} - 4 \beta_{13} + 5 \beta_{14} - \beta_{15} ) q^{90} + ( -9 + \beta_{2} + 2 \beta_{4} - 5 \beta_{6} ) q^{92} + ( 4 \beta_{2} + 3 \beta_{6} + \beta_{9} - 3 \beta_{11} - 4 \beta_{12} ) q^{93} + ( \beta_{7} + 4 \beta_{8} - 5 \beta_{13} - \beta_{14} + 4 \beta_{15} ) q^{94} + ( 6 - 6 \beta_{9} - \beta_{10} + 5 \beta_{11} - \beta_{12} ) q^{95} + ( 4 \beta_{3} + \beta_{5} + 4 \beta_{13} - \beta_{14} + \beta_{15} ) q^{96} + ( -6 \beta_{1} - \beta_{3} - \beta_{5} ) q^{97} + ( 3 - \beta_{2} - \beta_{4} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{2} - 12q^{4} + 24q^{8} - 4q^{9} + O(q^{10})$$ $$16q + 4q^{2} - 12q^{4} + 24q^{8} - 4q^{9} - 4q^{11} - 8q^{15} - 4q^{16} - 56q^{18} + 28q^{22} + 12q^{23} - 24q^{25} + 8q^{29} + 28q^{30} + 4q^{36} - 8q^{37} - 4q^{39} + 32q^{43} - 8q^{44} - 4q^{46} + 36q^{50} - 88q^{51} - 8q^{53} - 96q^{57} - 48q^{58} + 128q^{60} - 64q^{64} + 16q^{65} + 20q^{67} + 8q^{71} + 28q^{72} + 76q^{74} + 28q^{78} - 8q^{79} + 56q^{81} + 36q^{85} + 8q^{86} + 28q^{88} - 160q^{92} + 8q^{93} + 52q^{95} + 56q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 8 x^{14} + 45 x^{12} + 124 x^{10} + 248 x^{8} + 250 x^{6} + 177 x^{4} + 14 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$7068 \nu^{14} + 50635 \nu^{12} + 276768 \nu^{10} + 645048 \nu^{8} + 1213590 \nu^{6} + 521544 \nu^{4} + 41292 \nu^{2} - 1483013$$$$)/773722$$ $$\beta_{3}$$ $$=$$ $$($$$$7068 \nu^{15} + 50635 \nu^{13} + 276768 \nu^{11} + 645048 \nu^{9} + 1213590 \nu^{7} + 521544 \nu^{5} + 41292 \nu^{3} - 2256735 \nu$$$$)/773722$$ $$\beta_{4}$$ $$=$$ $$($$$$14022 \nu^{14} + 106693 \nu^{12} + 549072 \nu^{10} + 1279692 \nu^{8} + 1796116 \nu^{6} + 1034676 \nu^{4} + 81918 \nu^{2} + 408609$$$$)/773722$$ $$\beta_{5}$$ $$=$$ $$($$$$36499 \nu^{15} + 262518 \nu^{13} + 1429224 \nu^{11} + 3331014 \nu^{9} + 6036084 \nu^{7} + 2693242 \nu^{5} + 213231 \nu^{3} - 5036536 \nu$$$$)/773722$$ $$\beta_{6}$$ $$=$$ $$($$$$43453 \nu^{14} + 318576 \nu^{12} + 1701528 \nu^{10} + 3965658 \nu^{8} + 6618610 \nu^{6} + 3206374 \nu^{4} + 253857 \nu^{2} - 1597470$$$$)/773722$$ $$\beta_{7}$$ $$=$$ $$($$$$50521 \nu^{15} + 369211 \nu^{13} + 1978296 \nu^{11} + 4610706 \nu^{9} + 7832200 \nu^{7} + 3727918 \nu^{5} + 295149 \nu^{3} - 3854205 \nu$$$$)/773722$$ $$\beta_{8}$$ $$=$$ $$($$$$-64431 \nu^{15} - 508380 \nu^{13} - 2848760 \nu^{11} - 7712676 \nu^{9} - 15333840 \nu^{7} - 14894160 \nu^{5} - 10882743 \nu^{3} - 860742 \nu$$$$)/773722$$ $$\beta_{9}$$ $$=$$ $$($$$$-64431 \nu^{14} - 508380 \nu^{12} - 2848760 \nu^{10} - 7712676 \nu^{8} - 15333840 \nu^{6} - 14894160 \nu^{4} - 10882743 \nu^{2} - 87020$$$$)/773722$$ $$\beta_{10}$$ $$=$$ $$($$$$-107927 \nu^{14} - 828304 \nu^{12} - 4592694 \nu^{10} - 12008036 \nu^{8} - 23561464 \nu^{6} - 22175928 \nu^{4} - 16512183 \nu^{2} - 1305850$$$$)/773722$$ $$\beta_{11}$$ $$=$$ $$($$$$-114457 \nu^{14} - 952041 \nu^{12} - 5418506 \nu^{10} - 15617428 \nu^{8} - 31705946 \nu^{6} - 34019270 \nu^{4} - 22943719 \nu^{2} - 1814963$$$$)/773722$$ $$\beta_{12}$$ $$=$$ $$($$$$-121794 \nu^{14} - 966125 \nu^{12} - 5420752 \nu^{10} - 14780304 \nu^{8} - 29454090 \nu^{6} - 29266776 \nu^{4} - 20950472 \nu^{2} - 1657053$$$$)/773722$$ $$\beta_{13}$$ $$=$$ $$($$$$186225 \nu^{15} + 1474505 \nu^{13} + 8269512 \nu^{11} + 22492980 \nu^{9} + 44787930 \nu^{7} + 44160936 \nu^{5} + 31833215 \nu^{3} + 2517795 \nu$$$$)/773722$$ $$\beta_{14}$$ $$=$$ $$($$$$1241 \nu^{15} + 9879 \nu^{13} + 55358 \nu^{11} + 150958 \nu^{9} + 297972 \nu^{7} + 289428 \nu^{5} + 194601 \nu^{3} + 1691 \nu$$$$)/2734$$ $$\beta_{15}$$ $$=$$ $$($$$$408609 \nu^{15} + 3254850 \nu^{13} + 18280712 \nu^{11} + 50118444 \nu^{9} + 100055340 \nu^{7} + 100356134 \nu^{5} + 71289117 \nu^{3} + 5638608 \nu$$$$)/773722$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} - 2 \beta_{9} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} - 3 \beta_{8} - \beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{12} + \beta_{11} + \beta_{10} + 6 \beta_{9} - 6$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} + 6 \beta_{13} + 11 \beta_{8}$$ $$\nu^{6}$$ $$=$$ $$-6 \beta_{6} + 7 \beta_{4} + 23 \beta_{2} + 28$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} - 7 \beta_{5} + 29 \beta_{3} + 44 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$103 \beta_{12} - 29 \beta_{11} - 37 \beta_{10} - 81 \beta_{9} + 29 \beta_{6} - 37 \beta_{4} - 103 \beta_{2} - 37$$ $$\nu^{9}$$ $$=$$ $$37 \beta_{15} - 8 \beta_{14} - 132 \beta_{13} - 184 \beta_{8} + 37 \beta_{5} - 132 \beta_{3} - 184 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-456 \beta_{12} + 132 \beta_{11} + 177 \beta_{10} + 331 \beta_{9} - 331$$ $$\nu^{11}$$ $$=$$ $$-177 \beta_{15} + 45 \beta_{14} + 588 \beta_{13} + 787 \beta_{8} - 45 \beta_{7}$$ $$\nu^{12}$$ $$=$$ $$-588 \beta_{6} + 810 \beta_{4} + 2008 \beta_{2} + 2207$$ $$\nu^{13}$$ $$=$$ $$222 \beta_{7} - 810 \beta_{5} + 2596 \beta_{3} + 3405 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$8819 \beta_{12} - 2596 \beta_{11} - 3628 \beta_{10} - 6000 \beta_{9} + 2596 \beta_{6} - 3628 \beta_{4} - 8819 \beta_{2} - 3628$$ $$\nu^{15}$$ $$=$$ $$3628 \beta_{15} - 1032 \beta_{14} - 11415 \beta_{13} - 14819 \beta_{8} + 3628 \beta_{5} - 11415 \beta_{3} - 14819 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1 + \beta_{9}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.756863 + 1.31093i −0.756863 − 1.31093i −1.04641 − 1.81243i 1.04641 + 1.81243i −0.141226 − 0.244611i 0.141226 + 0.244611i −0.558788 − 0.967849i 0.558788 + 0.967849i 0.756863 − 1.31093i −0.756863 + 1.31093i −1.04641 + 1.81243i 1.04641 − 1.81243i −0.141226 + 0.244611i 0.141226 − 0.244611i −0.558788 + 0.967849i 0.558788 − 0.967849i
−1.21605 + 2.10626i −0.376796 + 0.652630i −1.95755 3.39058i 0.341537 −0.916405 1.58726i 0 4.65773 1.21605 + 2.10626i −0.415326 + 0.719366i
295.2 −1.21605 + 2.10626i 0.376796 0.652630i −1.95755 3.39058i −0.341537 0.916405 + 1.58726i 0 4.65773 1.21605 + 2.10626i 0.415326 0.719366i
295.3 0.289905 0.502131i −0.946019 + 1.63855i 0.831910 + 1.44091i −1.47362 0.548512 + 0.950050i 0 2.12432 −0.289905 0.502131i −0.427209 + 0.739948i
295.4 0.289905 0.502131i 0.946019 1.63855i 0.831910 + 1.44091i 1.47362 −0.548512 0.950050i 0 2.12432 −0.289905 0.502131i 0.427209 0.739948i
295.5 0.760387 1.31703i −1.06311 + 1.84135i −0.156376 0.270851i −0.589391 1.61674 + 2.80028i 0 2.56592 −0.760387 1.31703i −0.448165 + 0.776245i
295.6 0.760387 1.31703i 1.06311 1.84135i −0.156376 0.270851i 0.589391 −1.61674 2.80028i 0 2.56592 −0.760387 1.31703i 0.448165 0.776245i
295.7 1.16576 2.01915i −1.15450 + 1.99966i −1.71798 2.97563i 3.37112 2.69174 + 4.66224i 0 −3.34797 −1.16576 2.01915i 3.92990 6.80679i
295.8 1.16576 2.01915i 1.15450 1.99966i −1.71798 2.97563i −3.37112 −2.69174 4.66224i 0 −3.34797 −1.16576 2.01915i −3.92990 + 6.80679i
393.1 −1.21605 2.10626i −0.376796 0.652630i −1.95755 + 3.39058i 0.341537 −0.916405 + 1.58726i 0 4.65773 1.21605 2.10626i −0.415326 0.719366i
393.2 −1.21605 2.10626i 0.376796 + 0.652630i −1.95755 + 3.39058i −0.341537 0.916405 1.58726i 0 4.65773 1.21605 2.10626i 0.415326 + 0.719366i
393.3 0.289905 + 0.502131i −0.946019 1.63855i 0.831910 1.44091i −1.47362 0.548512 0.950050i 0 2.12432 −0.289905 + 0.502131i −0.427209 0.739948i
393.4 0.289905 + 0.502131i 0.946019 + 1.63855i 0.831910 1.44091i 1.47362 −0.548512 + 0.950050i 0 2.12432 −0.289905 + 0.502131i 0.427209 + 0.739948i
393.5 0.760387 + 1.31703i −1.06311 1.84135i −0.156376 + 0.270851i −0.589391 1.61674 2.80028i 0 2.56592 −0.760387 + 1.31703i −0.448165 0.776245i
393.6 0.760387 + 1.31703i 1.06311 + 1.84135i −0.156376 + 0.270851i 0.589391 −1.61674 + 2.80028i 0 2.56592 −0.760387 + 1.31703i 0.448165 + 0.776245i
393.7 1.16576 + 2.01915i −1.15450 1.99966i −1.71798 + 2.97563i 3.37112 2.69174 4.66224i 0 −3.34797 −1.16576 + 2.01915i 3.92990 + 6.80679i
393.8 1.16576 + 2.01915i 1.15450 + 1.99966i −1.71798 + 2.97563i −3.37112 −2.69174 + 4.66224i 0 −3.34797 −1.16576 + 2.01915i −3.92990 6.80679i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 393.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
91.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.f.l 16
7.b odd 2 1 inner 637.2.f.l 16
7.c even 3 1 637.2.g.m 16
7.c even 3 1 637.2.h.m 16
7.d odd 6 1 637.2.g.m 16
7.d odd 6 1 637.2.h.m 16
13.c even 3 1 inner 637.2.f.l 16
13.c even 3 1 8281.2.a.ci 8
13.e even 6 1 8281.2.a.cl 8
91.g even 3 1 637.2.h.m 16
91.h even 3 1 637.2.g.m 16
91.m odd 6 1 637.2.h.m 16
91.n odd 6 1 inner 637.2.f.l 16
91.n odd 6 1 8281.2.a.ci 8
91.t odd 6 1 8281.2.a.cl 8
91.v odd 6 1 637.2.g.m 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.l 16 1.a even 1 1 trivial
637.2.f.l 16 7.b odd 2 1 inner
637.2.f.l 16 13.c even 3 1 inner
637.2.f.l 16 91.n odd 6 1 inner
637.2.g.m 16 7.c even 3 1
637.2.g.m 16 7.d odd 6 1
637.2.g.m 16 91.h even 3 1
637.2.g.m 16 91.v odd 6 1
637.2.h.m 16 7.c even 3 1
637.2.h.m 16 7.d odd 6 1
637.2.h.m 16 91.g even 3 1
637.2.h.m 16 91.m odd 6 1
8281.2.a.ci 8 13.c even 3 1
8281.2.a.ci 8 91.n odd 6 1
8281.2.a.cl 8 13.e even 6 1
8281.2.a.cl 8 91.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{8} - \cdots$$ $$T_{3}^{16} + \cdots$$ $$T_{5}^{8} - 14 T_{5}^{6} + 31 T_{5}^{4} - 12 T_{5}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 25 - 60 T + 119 T^{2} - 80 T^{3} + 54 T^{4} - 14 T^{5} + 9 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$3$ $$2401 + 5880 T^{2} + 11117 T^{4} + 6668 T^{6} + 2760 T^{8} + 698 T^{10} + 129 T^{12} + 14 T^{14} + T^{16}$$
$5$ $$( 1 - 12 T^{2} + 31 T^{4} - 14 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 25 + 60 T + 119 T^{2} + 80 T^{3} + 54 T^{4} + 14 T^{5} + 9 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$13$ $$815730721 + 164111506 T^{2} + 12281230 T^{4} + 842296 T^{6} + 76567 T^{8} + 4984 T^{10} + 430 T^{12} + 34 T^{14} + T^{16}$$
$17$ $$13845841 + 14385386 T^{2} + 11351470 T^{4} + 3302920 T^{6} + 705207 T^{8} + 48296 T^{10} + 2398 T^{12} + 58 T^{14} + T^{16}$$
$19$ $$1500625 + 7658700 T^{2} + 36256529 T^{4} + 14218072 T^{6} + 4751808 T^{8} + 204730 T^{10} + 6525 T^{12} + 94 T^{14} + T^{16}$$
$23$ $$( 10000 + 5600 T + 4536 T^{2} + 416 T^{3} + 432 T^{4} - 28 T^{5} + 50 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$29$ $$( 3171961 + 338390 T + 187485 T^{2} - 1902 T^{3} + 6204 T^{4} - 40 T^{5} + 101 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$31$ $$( 26569 - 11726 T^{2} + 1518 T^{4} - 70 T^{6} + T^{8} )^{2}$$
$37$ $$( 144400 + 150480 T + 124136 T^{2} + 37096 T^{3} + 9360 T^{4} + 448 T^{5} + 102 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$41$ $$16777216 + 29360128 T^{2} + 39059456 T^{4} + 20119552 T^{6} + 7782400 T^{8} + 515072 T^{10} + 27968 T^{12} + 176 T^{14} + T^{16}$$
$43$ $$( 10272025 - 3346020 T + 1041861 T^{2} - 118220 T^{3} + 20134 T^{4} - 1848 T^{5} + 271 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$47$ $$( 27889 - 131954 T^{2} + 13374 T^{4} - 226 T^{6} + T^{8} )^{2}$$
$53$ $$( 271 - 70 T - 80 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$59$ $$28561 + 565474 T^{2} + 9929230 T^{4} + 25009352 T^{6} + 55510743 T^{8} + 1447144 T^{10} + 30142 T^{12} + 194 T^{14} + T^{16}$$
$61$ $$384160000 + 372243200 T^{2} + 298054464 T^{4} + 56464832 T^{6} + 8143680 T^{8} + 307184 T^{10} + 8468 T^{12} + 108 T^{14} + T^{16}$$
$67$ $$( 80089 - 62826 T + 47586 T^{2} - 6992 T^{3} + 2539 T^{4} - 384 T^{5} + 106 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$71$ $$( 35473936 + 1596208 T + 1060520 T^{2} + 3160 T^{3} + 22672 T^{4} + 128 T^{5} + 182 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$73$ $$( 2226064 - 706640 T^{2} + 45484 T^{4} - 428 T^{6} + T^{8} )^{2}$$
$79$ $$( 8164 - 712 T - 278 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$83$ $$( 405769 - 257574 T^{2} + 27838 T^{4} - 350 T^{6} + T^{8} )^{2}$$
$89$ $$2045357156640625 + 362153599117500 T^{2} + 58710468033889 T^{4} + 903605089896 T^{6} + 9427081296 T^{8} + 56514906 T^{10} + 247549 T^{12} + 606 T^{14} + T^{16}$$
$97$ $$96254442001 + 110400865654 T^{2} + 117123138597 T^{4} + 10674057554 T^{6} + 808419968 T^{8} + 10437992 T^{10} + 101865 T^{12} + 364 T^{14} + T^{16}$$