# Properties

 Label 637.2.r.f Level $637$ Weight $2$ Character orbit 637.r 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.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 11 x^{14} + 85 x^{12} - 334 x^{10} + 952 x^{8} - 1050 x^{6} + 853 x^{4} - 93 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 91) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{8} ) q^{2} + ( 1 - \beta_{4} - \beta_{7} ) q^{3} + ( 1 - \beta_{7} - \beta_{11} ) q^{4} -\beta_{14} q^{5} + ( \beta_{2} + \beta_{6} - \beta_{8} - \beta_{13} + \beta_{14} ) q^{6} + ( \beta_{2} - \beta_{13} ) q^{8} + ( 1 - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{9} - \beta_{12} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{8} ) q^{2} + ( 1 - \beta_{4} - \beta_{7} ) q^{3} + ( 1 - \beta_{7} - \beta_{11} ) q^{4} -\beta_{14} q^{5} + ( \beta_{2} + \beta_{6} - \beta_{8} - \beta_{13} + \beta_{14} ) q^{6} + ( \beta_{2} - \beta_{13} ) q^{8} + ( 1 - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{9} - \beta_{12} ) q^{9} + ( 2 \beta_{4} - \beta_{11} + \beta_{12} ) q^{10} + ( \beta_{1} + \beta_{10} ) q^{11} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{12} + ( 1 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{8} - \beta_{13} - \beta_{14} ) q^{13} + ( -\beta_{6} + 2 \beta_{8} - \beta_{14} ) q^{15} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{11} ) q^{16} + ( -1 + \beta_{7} - \beta_{12} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{6} - \beta_{10} ) q^{18} + ( 2 \beta_{1} + 2 \beta_{8} - \beta_{13} ) q^{19} + ( -2 \beta_{6} - 2 \beta_{14} + \beta_{15} ) q^{20} + ( -3 - \beta_{3} - \beta_{5} - \beta_{9} ) q^{22} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{23} + ( 3 \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} ) q^{24} + ( \beta_{4} - 2 \beta_{11} + \beta_{12} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{26} + ( -2 - 2 \beta_{3} + \beta_{5} + \beta_{9} ) q^{27} + ( -3 - 2 \beta_{3} - 3 \beta_{5} ) q^{29} + ( -2 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 6 \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{30} + ( \beta_{1} + \beta_{2} + \beta_{6} + \beta_{10} ) q^{31} + ( -\beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{32} + ( \beta_{1} + \beta_{8} + \beta_{10} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{33} + ( -\beta_{2} + 2 \beta_{6} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{34} + ( -1 + 5 \beta_{5} + 2 \beta_{9} ) q^{36} + ( -\beta_{1} - \beta_{8} + \beta_{10} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( -4 - \beta_{4} + 4 \beta_{7} + \beta_{11} ) q^{38} + ( 3 - 2 \beta_{2} - \beta_{4} - \beta_{6} - 3 \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{39} + ( -1 + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{12} ) q^{40} + ( -\beta_{2} - 2 \beta_{6} + \beta_{8} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{41} + ( 2 + 2 \beta_{5} - \beta_{9} ) q^{43} + ( -\beta_{10} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{44} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{45} + ( \beta_{1} - \beta_{2} - \beta_{6} ) q^{46} + ( -\beta_{1} - \beta_{8} - \beta_{10} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{47} + ( -2 + \beta_{3} ) q^{48} + ( 2 \beta_{2} - 3 \beta_{6} - \beta_{8} - 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{50} + ( 1 - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{11} ) q^{51} + ( 6 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - 6 \beta_{7} + \beta_{10} - \beta_{11} ) q^{52} + ( -3 + 3 \beta_{7} + 2 \beta_{11} ) q^{53} + ( 6 \beta_{1} + 6 \beta_{8} - \beta_{10} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{54} + ( -4 - 4 \beta_{3} - 3 \beta_{5} - \beta_{9} ) q^{55} + ( -3 \beta_{2} - \beta_{6} - \beta_{8} + 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{8} - \beta_{13} + 3 \beta_{14} ) q^{58} + ( -3 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} + \beta_{10} ) q^{59} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} + \beta_{10} ) q^{60} + ( -2 \beta_{3} + \beta_{7} - 2 \beta_{11} ) q^{61} + ( -2 - \beta_{3} + 2 \beta_{5} ) q^{62} + ( 5 + 2 \beta_{3} - 2 \beta_{5} + \beta_{9} ) q^{64} + ( -\beta_{3} - 4 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{65} -\beta_{11} q^{66} + ( 3 \beta_{2} + 2 \beta_{6} ) q^{67} + ( 4 + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - \beta_{7} + \beta_{9} + 3 \beta_{11} - \beta_{12} ) q^{68} + ( 4 + \beta_{5} - \beta_{9} ) q^{69} + ( 3 \beta_{2} + \beta_{6} + 4 \beta_{8} - 3 \beta_{13} + \beta_{14} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{8} + 3 \beta_{13} - 3 \beta_{14} ) q^{72} + ( 2 \beta_{1} + \beta_{6} - 2 \beta_{10} ) q^{73} + ( 6 - 6 \beta_{7} - 3 \beta_{11} ) q^{74} + ( -2 + 2 \beta_{4} - 2 \beta_{5} + \beta_{9} - \beta_{12} ) q^{75} + ( -2 \beta_{2} + \beta_{6} + \beta_{8} + 2 \beta_{13} + \beta_{14} ) q^{76} + ( \beta_{2} + \beta_{3} - 3 \beta_{5} + 3 \beta_{6} - 5 \beta_{8} - \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{78} + ( -1 + \beta_{4} - \beta_{5} + 3 \beta_{7} - \beta_{9} + \beta_{12} ) q^{79} + ( -2 \beta_{1} + \beta_{6} - \beta_{10} ) q^{80} + ( -5 + 2 \beta_{4} + 5 \beta_{7} + 4 \beta_{11} ) q^{81} + ( -4 + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} - \beta_{9} + \beta_{12} ) q^{82} + ( \beta_{2} + 5 \beta_{8} - \beta_{13} + \beta_{15} ) q^{83} + ( -2 \beta_{2} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{13} + 3 \beta_{14} ) q^{85} + ( -\beta_{1} - \beta_{8} + \beta_{10} + 3 \beta_{13} - \beta_{15} ) q^{86} + ( 8 - 8 \beta_{7} + 2 \beta_{11} - \beta_{12} ) q^{87} + ( 1 - 2 \beta_{4} - \beta_{7} + 2 \beta_{11} ) q^{88} + ( -5 \beta_{1} - 5 \beta_{8} - \beta_{10} + 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{89} + ( 14 + 6 \beta_{3} - 2 \beta_{5} ) q^{90} + ( -\beta_{3} - \beta_{5} - \beta_{9} ) q^{92} + ( 6 \beta_{1} + 6 \beta_{8} - 2 \beta_{13} - 3 \beta_{14} ) q^{93} + ( 6 - 5 \beta_{4} - 6 \beta_{7} - \beta_{11} - \beta_{12} ) q^{94} + ( -1 - 3 \beta_{4} + \beta_{7} + 2 \beta_{11} - \beta_{12} ) q^{95} + ( -5 \beta_{1} - 5 \beta_{8} + 2 \beta_{10} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{96} + ( -\beta_{2} + \beta_{8} + \beta_{13} + \beta_{15} ) q^{97} + ( -\beta_{6} - 3 \beta_{8} - \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{3} + 6q^{4} - 12q^{9} + O(q^{10})$$ $$16q + 4q^{3} + 6q^{4} - 12q^{9} + 6q^{10} - 18q^{12} + 12q^{13} + 2q^{16} - 8q^{17} - 36q^{22} - 12q^{23} + 6q^{26} - 32q^{27} - 16q^{29} + 38q^{30} - 56q^{36} - 34q^{38} + 18q^{39} + 4q^{40} + 16q^{43} - 36q^{48} + 16q^{51} + 42q^{52} - 20q^{53} - 24q^{55} + 12q^{61} - 44q^{62} + 88q^{64} - 30q^{65} - 2q^{66} + 2q^{68} + 56q^{69} + 42q^{74} - 8q^{75} + 20q^{78} + 20q^{79} - 24q^{81} + 16q^{82} + 68q^{87} + 4q^{88} + 216q^{90} + 12q^{92} + 26q^{94} - 16q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 11 x^{14} + 85 x^{12} - 334 x^{10} + 952 x^{8} - 1050 x^{6} + 853 x^{4} - 93 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-24498 \nu^{15} + 246060 \nu^{13} - 1852321 \nu^{11} + 6411671 \nu^{9} - 17193085 \nu^{7} + 6321845 \nu^{5} - 690027 \nu^{3} - 55035781 \nu$$$$)/14163622$$ $$\beta_{3}$$ $$=$$ $$($$$$-24498 \nu^{14} + 246060 \nu^{12} - 1852321 \nu^{10} + 6411671 \nu^{8} - 17193085 \nu^{6} + 6321845 \nu^{4} - 690027 \nu^{2} - 40872159$$$$)/14163622$$ $$\beta_{4}$$ $$=$$ $$($$$$172099 \nu^{14} - 2170865 \nu^{12} + 17340370 \nu^{10} - 78484018 \nu^{8} + 236538400 \nu^{6} - 377649654 \nu^{4} + 218482087 \nu^{2} - 23829231$$$$)/42490866$$ $$\beta_{5}$$ $$=$$ $$($$$$-99072 \nu^{14} + 1000291 \nu^{12} - 7490944 \nu^{10} + 25929344 \nu^{8} - 66564370 \nu^{6} + 25566080 \nu^{4} - 2790528 \nu^{2} - 20454191$$$$)/14163622$$ $$\beta_{6}$$ $$=$$ $$($$$$123570 \nu^{15} - 1246351 \nu^{13} + 9343265 \nu^{11} - 32341015 \nu^{9} + 83757455 \nu^{7} - 31887925 \nu^{5} + 3480555 \nu^{3} + 61326350 \nu$$$$)/14163622$$ $$\beta_{7}$$ $$=$$ $$($$$$539569 \nu^{14} - 5861765 \nu^{12} + 45125185 \nu^{10} - 174659083 \nu^{8} + 494434675 \nu^{6} - 514968195 \nu^{4} + 441286822 \nu^{2} - 5618970$$$$)/42490866$$ $$\beta_{8}$$ $$=$$ $$($$$$-539569 \nu^{15} + 5861765 \nu^{13} - 45125185 \nu^{11} + 174659083 \nu^{9} - 494434675 \nu^{7} + 514968195 \nu^{5} - 441286822 \nu^{3} + 5618970 \nu$$$$)/42490866$$ $$\beta_{9}$$ $$=$$ $$($$$$102312 \nu^{14} - 1048444 \nu^{12} + 7735924 \nu^{10} - 26777324 \nu^{8} + 67022271 \nu^{6} - 26402180 \nu^{4} + 2881788 \nu^{2} + 23341327$$$$)/7081811$$ $$\beta_{10}$$ $$=$$ $$($$$$476262 \nu^{15} - 4835650 \nu^{13} + 36010699 \nu^{11} - 124648349 \nu^{9} + 318752537 \nu^{7} - 122902055 \nu^{5} + 13414713 \nu^{3} + 238534757 \nu$$$$)/14163622$$ $$\beta_{11}$$ $$=$$ $$($$$$-515071 \nu^{14} + 5615705 \nu^{12} - 43272864 \nu^{10} + 168247412 \nu^{8} - 477241590 \nu^{6} + 508646350 \nu^{4} - 426433173 \nu^{2} + 46491129$$$$)/14163622$$ $$\beta_{12}$$ $$=$$ $$($$$$-3598 \nu^{14} + 40712 \nu^{12} - 317920 \nu^{10} + 1287475 \nu^{8} - 3722089 \nu^{6} + 4460568 \nu^{4} - 3361729 \nu^{2} + 366555$$$$)/95271$$ $$\beta_{13}$$ $$=$$ $$($$$$2084782 \nu^{15} - 22708880 \nu^{13} + 174943777 \nu^{11} - 679401319 \nu^{9} + 1926159445 \nu^{7} - 2040907245 \nu^{5} + 1720586341 \nu^{3} - 187583223 \nu$$$$)/42490866$$ $$\beta_{14}$$ $$=$$ $$($$$$205171 \nu^{15} - 2261795 \nu^{13} + 17480377 \nu^{11} - 68898667 \nu^{9} + 196608895 \nu^{7} - 219868809 \nu^{5} + 176278948 \nu^{3} - 19219314 \nu$$$$)/3862806$$ $$\beta_{15}$$ $$=$$ $$($$$$10171607 \nu^{15} - 110994637 \nu^{13} + 854461673 \nu^{11} - 3317989973 \nu^{9} + 9362299115 \nu^{7} - 9751108731 \nu^{5} + 7579585292 \nu^{3} - 106397226 \nu$$$$)/42490866$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + 3 \beta_{7} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} - 4 \beta_{8} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{11} + 14 \beta_{7} + \beta_{4} - 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{14} - 6 \beta_{13} - 19 \beta_{8} - 19 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{9} + 8 \beta_{5} - 24 \beta_{3} - 61$$ $$\nu^{7}$$ $$=$$ $$\beta_{10} - 10 \beta_{6} - 31 \beta_{2} - 94 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-11 \beta_{12} - 115 \beta_{11} + 11 \beta_{9} - 345 \beta_{7} + 52 \beta_{5} - 52 \beta_{4} - 115 \beta_{3} + 52$$ $$\nu^{9}$$ $$=$$ $$11 \beta_{15} - 74 \beta_{14} + 156 \beta_{13} + 471 \beta_{8} - 74 \beta_{6} - 156 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$-85 \beta_{12} - 553 \beta_{11} - 1736 \beta_{7} - 315 \beta_{4} + 1736$$ $$\nu^{11}$$ $$=$$ $$85 \beta_{15} - 485 \beta_{14} + 783 \beta_{13} - 85 \beta_{10} + 2374 \beta_{8} + 2374 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-570 \beta_{9} - 1838 \beta_{5} + 2672 \beta_{3} + 6935$$ $$\nu^{13}$$ $$=$$ $$-570 \beta_{10} + 2978 \beta_{6} + 3940 \beta_{2} + 12015 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$3548 \beta_{12} + 12977 \beta_{11} - 3548 \beta_{9} + 44495 \beta_{7} - 10466 \beta_{5} + 10466 \beta_{4} + 12977 \beta_{3} - 10466$$ $$\nu^{15}$$ $$=$$ $$-3548 \beta_{15} + 17562 \beta_{14} - 19895 \beta_{13} - 61020 \beta_{8} + 17562 \beta_{6} + 19895 \beta_{2}$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 1.97871 + 1.14241i 1.84073 + 1.06275i 0.929293 + 0.536527i 0.287846 + 0.166188i −0.287846 − 0.166188i −0.929293 − 0.536527i −1.84073 − 1.06275i −1.97871 − 1.14241i 1.97871 − 1.14241i 1.84073 − 1.06275i 0.929293 − 0.536527i 0.287846 − 0.166188i −0.287846 + 0.166188i −0.929293 + 0.536527i −1.84073 + 1.06275i −1.97871 + 1.14241i
−1.97871 + 1.14241i 1.57521 2.72835i 1.61019 2.78892i 1.84030 1.06250i 7.19813i 0 2.78832i −3.46258 5.99736i −2.42760 + 4.20473i
116.2 −1.84073 + 1.06275i −0.0894272 + 0.154892i 1.25885 2.18040i −3.12291 + 1.80301i 0.380153i 0 1.10038i 1.48401 + 2.57037i 3.83229 6.63772i
116.3 −0.929293 + 0.536527i −1.21570 + 2.10566i −0.424277 + 0.734868i −0.541640 + 0.312716i 2.60903i 0 3.05665i −1.45586 2.52163i 0.335561 0.581209i
116.4 −0.287846 + 0.166188i 0.729919 1.26426i −0.944763 + 1.63638i 1.25195 0.722811i 0.485214i 0 1.29278i 0.434437 + 0.752468i −0.240245 + 0.416116i
116.5 0.287846 0.166188i 0.729919 1.26426i −0.944763 + 1.63638i −1.25195 + 0.722811i 0.485214i 0 1.29278i 0.434437 + 0.752468i −0.240245 + 0.416116i
116.6 0.929293 0.536527i −1.21570 + 2.10566i −0.424277 + 0.734868i 0.541640 0.312716i 2.60903i 0 3.05665i −1.45586 2.52163i 0.335561 0.581209i
116.7 1.84073 1.06275i −0.0894272 + 0.154892i 1.25885 2.18040i 3.12291 1.80301i 0.380153i 0 1.10038i 1.48401 + 2.57037i 3.83229 6.63772i
116.8 1.97871 1.14241i 1.57521 2.72835i 1.61019 2.78892i −1.84030 + 1.06250i 7.19813i 0 2.78832i −3.46258 5.99736i −2.42760 + 4.20473i
324.1 −1.97871 1.14241i 1.57521 + 2.72835i 1.61019 + 2.78892i 1.84030 + 1.06250i 7.19813i 0 2.78832i −3.46258 + 5.99736i −2.42760 4.20473i
324.2 −1.84073 1.06275i −0.0894272 0.154892i 1.25885 + 2.18040i −3.12291 1.80301i 0.380153i 0 1.10038i 1.48401 2.57037i 3.83229 + 6.63772i
324.3 −0.929293 0.536527i −1.21570 2.10566i −0.424277 0.734868i −0.541640 0.312716i 2.60903i 0 3.05665i −1.45586 + 2.52163i 0.335561 + 0.581209i
324.4 −0.287846 0.166188i 0.729919 + 1.26426i −0.944763 1.63638i 1.25195 + 0.722811i 0.485214i 0 1.29278i 0.434437 0.752468i −0.240245 0.416116i
324.5 0.287846 + 0.166188i 0.729919 + 1.26426i −0.944763 1.63638i −1.25195 0.722811i 0.485214i 0 1.29278i 0.434437 0.752468i −0.240245 0.416116i
324.6 0.929293 + 0.536527i −1.21570 2.10566i −0.424277 0.734868i 0.541640 + 0.312716i 2.60903i 0 3.05665i −1.45586 + 2.52163i 0.335561 + 0.581209i
324.7 1.84073 + 1.06275i −0.0894272 0.154892i 1.25885 + 2.18040i 3.12291 + 1.80301i 0.380153i 0 1.10038i 1.48401 2.57037i 3.83229 + 6.63772i
324.8 1.97871 + 1.14241i 1.57521 + 2.72835i 1.61019 + 2.78892i −1.84030 1.06250i 7.19813i 0 2.78832i −3.46258 + 5.99736i −2.42760 4.20473i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.8 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.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.r.f 16
7.b odd 2 1 91.2.r.a 16
7.c even 3 1 637.2.c.e 8
7.c even 3 1 inner 637.2.r.f 16
7.d odd 6 1 91.2.r.a 16
7.d odd 6 1 637.2.c.f 8
13.b even 2 1 inner 637.2.r.f 16
21.c even 2 1 819.2.dl.e 16
21.g even 6 1 819.2.dl.e 16
91.b odd 2 1 91.2.r.a 16
91.i even 4 2 1183.2.e.i 16
91.r even 6 1 637.2.c.e 8
91.r even 6 1 inner 637.2.r.f 16
91.s odd 6 1 91.2.r.a 16
91.s odd 6 1 637.2.c.f 8
91.z odd 12 2 8281.2.a.cj 8
91.bb even 12 2 1183.2.e.i 16
91.bb even 12 2 8281.2.a.ck 8
273.g even 2 1 819.2.dl.e 16
273.ba even 6 1 819.2.dl.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a 16 7.b odd 2 1
91.2.r.a 16 7.d odd 6 1
91.2.r.a 16 91.b odd 2 1
91.2.r.a 16 91.s odd 6 1
637.2.c.e 8 7.c even 3 1
637.2.c.e 8 91.r even 6 1
637.2.c.f 8 7.d odd 6 1
637.2.c.f 8 91.s odd 6 1
637.2.r.f 16 1.a even 1 1 trivial
637.2.r.f 16 7.c even 3 1 inner
637.2.r.f 16 13.b even 2 1 inner
637.2.r.f 16 91.r even 6 1 inner
819.2.dl.e 16 21.c even 2 1
819.2.dl.e 16 21.g even 6 1
819.2.dl.e 16 273.g even 2 1
819.2.dl.e 16 273.ba even 6 1
1183.2.e.i 16 91.i even 4 2
1183.2.e.i 16 91.bb even 12 2
8281.2.a.cj 8 91.z odd 12 2
8281.2.a.ck 8 91.bb even 12 2

## Hecke kernels

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

 $$T_{2}^{16} - \cdots$$ $$T_{3}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 93 T^{2} + 853 T^{4} - 1050 T^{6} + 952 T^{8} - 334 T^{10} + 85 T^{12} - 11 T^{14} + T^{16}$$
$3$ $$( 4 + 20 T + 114 T^{2} - 62 T^{3} + 67 T^{4} - 6 T^{5} + 11 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$5$ $$2304 - 7680 T^{2} + 20656 T^{4} - 14560 T^{6} + 7361 T^{8} - 1740 T^{10} + 297 T^{12} - 20 T^{14} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$729 - 9180 T^{2} + 99508 T^{4} - 199832 T^{6} + 337509 T^{8} - 30312 T^{10} + 2108 T^{12} - 52 T^{14} + T^{16}$$
$13$ $$( 28561 - 13182 T + 4732 T^{2} - 1690 T^{3} + 598 T^{4} - 130 T^{5} + 28 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$17$ $$( 15129 - 6396 T + 5164 T^{2} + 56 T^{3} + 485 T^{4} + 24 T^{5} + 36 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$19$ $$10673289 - 7618644 T^{2} + 3674044 T^{4} - 971784 T^{6} + 185725 T^{8} - 19096 T^{10} + 1396 T^{12} - 44 T^{14} + T^{16}$$
$23$ $$( 36 + 60 T + 130 T^{2} + 22 T^{3} + 91 T^{4} + 50 T^{5} + 31 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$29$ $$( 624 - 208 T - 63 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$31$ $$1136229264 - 657036336 T^{2} + 309454636 T^{4} - 35364492 T^{6} + 2779213 T^{8} - 128296 T^{10} + 4309 T^{12} - 80 T^{14} + T^{16}$$
$37$ $$76527504 - 433655856 T^{2} + 2419355628 T^{4} - 213389964 T^{6} + 12939021 T^{8} - 422496 T^{10} + 10053 T^{12} - 120 T^{14} + T^{16}$$
$41$ $$( 292032 + 88192 T^{2} + 5732 T^{4} + 132 T^{6} + T^{8} )^{2}$$
$43$ $$( -104 - 156 T - 66 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$47$ $$57728231289 - 58599199164 T^{2} + 56549167060 T^{4} - 2884224440 T^{6} + 101089845 T^{8} - 1905768 T^{10} + 26204 T^{12} - 196 T^{14} + T^{16}$$
$53$ $$( 7569 - 11310 T + 16900 T^{2} - 1740 T^{3} + 1213 T^{4} + 260 T^{5} + 100 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$59$ $$12487392009 - 18073959780 T^{2} + 24989166028 T^{4} - 1652371368 T^{6} + 79227709 T^{8} - 1646008 T^{10} + 24868 T^{12} - 188 T^{14} + T^{16}$$
$61$ $$( 49729 + 20962 T + 14188 T^{2} + 420 T^{3} + 917 T^{4} - 44 T^{5} + 60 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$67$ $$66330457209 - 73439011956 T^{2} + 75732974260 T^{4} - 6027737800 T^{6} + 387569525 T^{8} - 5578872 T^{10} + 59004 T^{12} - 284 T^{14} + T^{16}$$
$71$ $$( 397488 + 253084 T^{2} + 19829 T^{4} + 292 T^{6} + T^{8} )^{2}$$
$73$ $$8437677133824 - 1371747640320 T^{2} + 164987876800 T^{4} - 7922514640 T^{6} + 273313457 T^{8} - 4249020 T^{10} + 47625 T^{12} - 260 T^{14} + T^{16}$$
$79$ $$( 64 - 480 T + 3672 T^{2} + 380 T^{3} + 689 T^{4} - 210 T^{5} + 91 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$83$ $$( 5483712 + 959920 T^{2} + 28692 T^{4} + 296 T^{6} + T^{8} )^{2}$$
$89$ $$58102628210064 - 22401057000432 T^{2} + 8157120819724 T^{4} - 178140025756 T^{6} + 2655587933 T^{8} - 21797952 T^{10} + 130701 T^{12} - 440 T^{14} + T^{16}$$
$97$ $$( 192 + 4816 T^{2} + 2740 T^{4} + 104 T^{6} + T^{8} )^{2}$$