# Properties

 Label 196.2.f.d Level $196$ Weight $2$ Character orbit 196.f Analytic conductor $1.565$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 196.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.56506787962$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 4 x^{14} + 6 x^{12} + 8 x^{10} + 20 x^{8} + 32 x^{6} + 96 x^{4} + 256 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{6} - \beta_{11} ) q^{2} + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} + \beta_{14} ) q^{4} + ( \beta_{3} - \beta_{13} + \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{6} + ( 2 \beta_{4} - \beta_{6} ) q^{8} + ( -2 - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{6} - \beta_{11} ) q^{2} + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} + \beta_{14} ) q^{4} + ( \beta_{3} - \beta_{13} + \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{6} + ( 2 \beta_{4} - \beta_{6} ) q^{8} + ( -2 - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{10} - \beta_{12} - \beta_{15} ) q^{10} + ( -2 \beta_{4} + 2 \beta_{8} - \beta_{11} - \beta_{14} ) q^{11} + ( \beta_{1} + 3 \beta_{3} + \beta_{9} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{12} + ( \beta_{10} + \beta_{13} + \beta_{15} ) q^{13} + ( -2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{15} + ( -2 - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{11} ) q^{16} + ( -\beta_{3} - 2 \beta_{10} ) q^{17} + ( -\beta_{4} - \beta_{5} + \beta_{8} + 3 \beta_{11} - \beta_{14} ) q^{18} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{12} - \beta_{15} ) q^{19} + ( \beta_{2} + \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{20} + ( 4 - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{14} ) q^{22} + ( -2 - 2 \beta_{5} - 4 \beta_{8} ) q^{23} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{10} - \beta_{12} - \beta_{15} ) q^{24} + ( \beta_{5} - 2 \beta_{11} + 2 \beta_{14} ) q^{25} + ( \beta_{12} + 2 \beta_{15} ) q^{26} + ( 2 \beta_{2} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{27} + ( -6 - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{29} + ( 8 + 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{30} + ( 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{10} + 4 \beta_{12} + 4 \beta_{15} ) q^{31} + ( 6 \beta_{5} + \beta_{11} + 2 \beta_{14} ) q^{32} + ( -5 \beta_{3} + 5 \beta_{13} + \beta_{15} ) q^{33} + ( \beta_{2} + \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{34} + ( -4 - 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{36} + ( 4 + 4 \beta_{5} ) q^{37} + ( \beta_{1} - 3 \beta_{3} + \beta_{10} ) q^{38} + ( 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} ) q^{39} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{9} - 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{40} + ( 3 \beta_{10} + 3 \beta_{15} ) q^{41} + ( -1 - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{14} ) q^{43} + ( -2 - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{11} ) q^{44} + ( -\beta_{3} + 3 \beta_{10} ) q^{45} + ( 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{8} + 2 \beta_{11} - 4 \beta_{14} ) q^{46} + ( 4 \beta_{1} + 2 \beta_{3} + 4 \beta_{9} + 2 \beta_{13} ) q^{47} + ( -4 \beta_{2} + \beta_{9} + 3 \beta_{10} - 2 \beta_{13} + 3 \beta_{15} ) q^{48} + ( 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{14} ) q^{50} + ( 3 + 3 \beta_{5} + \beta_{6} + \beta_{7} + 6 \beta_{8} + \beta_{11} ) q^{51} + ( -2 \beta_{1} + \beta_{2} - \beta_{10} + \beta_{12} + \beta_{15} ) q^{52} + ( 2 \beta_{5} + 6 \beta_{11} - 6 \beta_{14} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{9} - \beta_{12} + 4 \beta_{13} ) q^{54} + ( -4 \beta_{2} + 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{55} + ( 9 + 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{14} ) q^{57} + ( 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{11} ) q^{58} + ( -2 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 3 \beta_{10} - 6 \beta_{12} - 6 \beta_{15} ) q^{59} + ( -4 \beta_{4} + 4 \beta_{8} - 6 \beta_{11} ) q^{60} + ( 5 \beta_{3} - 5 \beta_{13} - 5 \beta_{15} ) q^{61} + ( 2 \beta_{2} + 4 \beta_{9} + 2 \beta_{10} + 8 \beta_{13} + 2 \beta_{15} ) q^{62} + ( -6 - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{14} ) q^{64} + ( 2 + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} ) q^{65} + ( 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - \beta_{10} + 5 \beta_{12} + 5 \beta_{15} ) q^{66} + ( -4 \beta_{4} + 4 \beta_{5} + 4 \beta_{8} + 2 \beta_{11} + 2 \beta_{14} ) q^{67} + ( 3 \beta_{1} + \beta_{3} + 3 \beta_{9} - 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{15} ) q^{68} + ( -8 \beta_{10} - 6 \beta_{13} - 8 \beta_{15} ) q^{69} + ( -2 - 2 \beta_{5} + \beta_{6} - \beta_{7} - 7 \beta_{8} + \beta_{11} ) q^{72} + ( 8 \beta_{3} + \beta_{10} ) q^{73} -4 \beta_{11} q^{74} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{9} + 4 \beta_{12} - 3 \beta_{13} + 2 \beta_{15} ) q^{75} + ( 3 \beta_{2} - 5 \beta_{9} ) q^{76} + ( 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{14} ) q^{78} + ( 2 + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{11} ) q^{79} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{80} + ( -7 \beta_{5} - 3 \beta_{11} + 3 \beta_{14} ) q^{81} + ( -3 \beta_{1} - 3 \beta_{3} - 3 \beta_{9} + 3 \beta_{15} ) q^{82} + ( 2 \beta_{2} - 6 \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{83} + ( -2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{85} + ( -12 - 12 \beta_{5} - 2 \beta_{6} + \beta_{7} - 5 \beta_{8} - 2 \beta_{11} ) q^{86} + ( -8 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} - 2 \beta_{10} - 4 \beta_{12} - 4 \beta_{15} ) q^{87} + ( \beta_{4} - 7 \beta_{5} - \beta_{8} + 3 \beta_{14} ) q^{88} + ( 2 \beta_{3} - 2 \beta_{13} + 3 \beta_{15} ) q^{89} + ( \beta_{2} - 4 \beta_{9} + 3 \beta_{10} + 3 \beta_{15} ) q^{90} + ( 4 + 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} - 6 \beta_{14} ) q^{92} + ( -8 - 8 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} - 12 \beta_{11} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{10} - 2 \beta_{12} - 2 \beta_{15} ) q^{94} + ( 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{8} + 4 \beta_{11} + 4 \beta_{14} ) q^{95} + ( -9 \beta_{1} - \beta_{3} - 9 \beta_{9} + \beta_{12} - 8 \beta_{13} + 5 \beta_{15} ) q^{96} + 5 \beta_{13} q^{97} + ( 3 + 6 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{2} + 4q^{4} - 8q^{8} - 24q^{9} + O(q^{10})$$ $$16q + 4q^{2} + 4q^{4} - 8q^{8} - 24q^{9} - 4q^{16} + 20q^{18} + 32q^{22} - 8q^{25} - 64q^{29} + 40q^{30} - 36q^{32} + 8q^{36} + 32q^{37} - 24q^{44} + 8q^{46} - 40q^{50} - 16q^{53} + 64q^{57} - 8q^{60} - 104q^{64} + 4q^{72} - 16q^{74} + 16q^{78} + 56q^{81} + 32q^{85} - 64q^{86} + 64q^{88} + 112q^{92} + 32q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 4 x^{14} + 6 x^{12} + 8 x^{10} + 20 x^{8} + 32 x^{6} + 96 x^{4} + 256 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} + 6 \nu^{13} + 4 \nu^{11} - 12 \nu^{9} + 24 \nu^{7} + 24 \nu^{5} + 88 \nu^{3} + 320 \nu$$$$)/224$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{15} - 44 \nu^{13} - 6 \nu^{11} - 24 \nu^{9} - 148 \nu^{7} - 64 \nu^{5} - 384 \nu^{3} - 2944 \nu$$$$)/896$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{14} - 12 \nu^{12} + 6 \nu^{10} + 24 \nu^{8} - 76 \nu^{6} + 64 \nu^{4} + 384 \nu^{2} - 1088$$$$)/448$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{14} - 3 \nu^{12} - 2 \nu^{10} + 6 \nu^{8} - 12 \nu^{6} - 12 \nu^{4} + 96 \nu^{2} - 384$$$$)/224$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{14} + 16 \nu^{12} + 6 \nu^{10} + 24 \nu^{8} + 36 \nu^{6} + 64 \nu^{4} + 384 \nu^{2} + 704$$$$)/224$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{14} + 9 \nu^{12} + 6 \nu^{10} + 38 \nu^{8} + 36 \nu^{6} + 36 \nu^{4} + 272 \nu^{2} + 480$$$$)/224$$ $$\beta_{8}$$ $$=$$ $$($$$$5 \nu^{14} + 9 \nu^{12} + 6 \nu^{10} + 38 \nu^{8} + 36 \nu^{6} + 36 \nu^{4} + 496 \nu^{2} + 480$$$$)/224$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{15} + 18 \nu^{13} + 12 \nu^{11} + 20 \nu^{9} + 72 \nu^{7} + 72 \nu^{5} + 264 \nu^{3} + 960 \nu$$$$)/224$$ $$\beta_{10}$$ $$=$$ $$($$$$15 \nu^{15} + 20 \nu^{13} + 18 \nu^{11} + 72 \nu^{9} - 4 \nu^{7} + 192 \nu^{5} + 1152 \nu^{3} + 768 \nu$$$$)/896$$ $$\beta_{11}$$ $$=$$ $$($$$$9 \nu^{14} + 26 \nu^{12} + 22 \nu^{10} + 60 \nu^{8} + 132 \nu^{6} + 104 \nu^{4} + 736 \nu^{2} + 1536$$$$)/224$$ $$\beta_{12}$$ $$=$$ $$($$$$11 \nu^{15} + 24 \nu^{13} + 2 \nu^{11} + 64 \nu^{9} + 124 \nu^{7} + 208 \nu^{5} + 800 \nu^{3} + 1280 \nu$$$$)/448$$ $$\beta_{13}$$ $$=$$ $$($$$$-6 \nu^{15} - 15 \nu^{13} - 10 \nu^{11} - 26 \nu^{9} - 60 \nu^{7} - 60 \nu^{5} - 360 \nu^{3} - 800 \nu$$$$)/224$$ $$\beta_{14}$$ $$=$$ $$($$$$29 \nu^{14} + 76 \nu^{12} + 46 \nu^{10} + 184 \nu^{8} + 388 \nu^{6} + 416 \nu^{4} + 2272 \nu^{2} + 4352$$$$)/448$$ $$\beta_{15}$$ $$=$$ $$($$$$29 \nu^{15} + 76 \nu^{13} + 46 \nu^{11} + 184 \nu^{9} + 388 \nu^{7} + 192 \nu^{5} + 2272 \nu^{3} + 4352 \nu$$$$)/896$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7}$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + 2 \beta_{13} + \beta_{10} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{14} - 2 \beta_{11} - 2 \beta_{8} - 2 \beta_{5} + 2 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{15} + 2 \beta_{12} + 2 \beta_{9} + 2 \beta_{3} + 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{14} - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 4$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{15} + 2 \beta_{12} - 6 \beta_{10} + 4 \beta_{3} + 2 \beta_{2}$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{11} + 4 \beta_{8} + 12 \beta_{7} - 4 \beta_{6} - 8 \beta_{5} - 8$$ $$\nu^{9}$$ $$=$$ $$4 \beta_{9} - 12 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$-12 \beta_{14} + 24 \beta_{11} - 12 \beta_{8} - 4 \beta_{5} + 12 \beta_{4}$$ $$\nu^{11}$$ $$=$$ $$-8 \beta_{13} - 12 \beta_{12} + 16 \beta_{9} + 24 \beta_{3} + 16 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-8 \beta_{14} + 8 \beta_{7} + 24 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 48$$ $$\nu^{13}$$ $$=$$ $$-8 \beta_{15} - 8 \beta_{12} + 16 \beta_{10} - 40 \beta_{3} - 8 \beta_{2} - 72 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$16 \beta_{11} - 56 \beta_{8} + 8 \beta_{7} + 16 \beta_{6} + 80 \beta_{5} + 80$$ $$\nu^{15}$$ $$=$$ $$-40 \beta_{15} - 144 \beta_{13} - 40 \beta_{10} - 64 \beta_{9} - 8 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$$.

 $$n$$ $$99$$ $$101$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.264742 − 1.38921i −0.264742 + 1.38921i −1.01214 − 0.987711i 1.01214 + 0.987711i −1.33546 − 0.465333i 1.33546 + 0.465333i −0.349313 − 1.37039i 0.349313 + 1.37039i 0.264742 + 1.38921i −0.264742 − 1.38921i −1.01214 + 0.987711i 1.01214 − 0.987711i −1.33546 + 0.465333i 1.33546 − 0.465333i −0.349313 + 1.37039i 0.349313 − 1.37039i
−1.31509 0.520123i −1.07072 1.85455i 1.45894 + 1.36802i −2.26303 1.30656i 0.443508 + 2.99581i 0 −1.20711 2.55791i −0.792893 + 1.37333i 2.29653 + 2.89531i
19.2 −1.31509 0.520123i 1.07072 + 1.85455i 1.45894 + 1.36802i 2.26303 + 1.30656i −0.443508 2.99581i 0 −1.20711 2.55791i −0.792893 + 1.37333i −2.29653 2.89531i
19.3 −0.0345453 1.41379i −1.36145 2.35811i −1.99761 + 0.0976797i −0.937379 0.541196i −3.28684 + 2.00627i 0 0.207107 + 2.82083i −2.20711 + 3.82282i −0.732756 + 1.34395i
19.4 −0.0345453 1.41379i 1.36145 + 2.35811i −1.99761 + 0.0976797i 0.937379 + 0.541196i 3.28684 2.00627i 0 0.207107 + 2.82083i −2.20711 + 3.82282i 0.732756 1.34395i
19.5 1.10799 + 0.878843i −1.07072 1.85455i 0.455270 + 1.94749i 2.26303 + 1.30656i 0.443508 2.99581i 0 −1.20711 + 2.55791i −0.792893 + 1.37333i 1.35915 + 3.43651i
19.6 1.10799 + 0.878843i 1.07072 + 1.85455i 0.455270 + 1.94749i −2.26303 1.30656i −0.443508 + 2.99581i 0 −1.20711 + 2.55791i −0.792893 + 1.37333i −1.35915 3.43651i
19.7 1.24165 0.676979i −1.36145 2.35811i 1.08340 1.68114i 0.937379 + 0.541196i −3.28684 2.00627i 0 0.207107 2.82083i −2.20711 + 3.82282i 1.53028 + 0.0373915i
19.8 1.24165 0.676979i 1.36145 + 2.35811i 1.08340 1.68114i −0.937379 0.541196i 3.28684 + 2.00627i 0 0.207107 2.82083i −2.20711 + 3.82282i −1.53028 0.0373915i
31.1 −1.31509 + 0.520123i −1.07072 + 1.85455i 1.45894 1.36802i −2.26303 + 1.30656i 0.443508 2.99581i 0 −1.20711 + 2.55791i −0.792893 1.37333i 2.29653 2.89531i
31.2 −1.31509 + 0.520123i 1.07072 1.85455i 1.45894 1.36802i 2.26303 1.30656i −0.443508 + 2.99581i 0 −1.20711 + 2.55791i −0.792893 1.37333i −2.29653 + 2.89531i
31.3 −0.0345453 + 1.41379i −1.36145 + 2.35811i −1.99761 0.0976797i −0.937379 + 0.541196i −3.28684 2.00627i 0 0.207107 2.82083i −2.20711 3.82282i −0.732756 1.34395i
31.4 −0.0345453 + 1.41379i 1.36145 2.35811i −1.99761 0.0976797i 0.937379 0.541196i 3.28684 + 2.00627i 0 0.207107 2.82083i −2.20711 3.82282i 0.732756 + 1.34395i
31.5 1.10799 0.878843i −1.07072 + 1.85455i 0.455270 1.94749i 2.26303 1.30656i 0.443508 + 2.99581i 0 −1.20711 2.55791i −0.792893 1.37333i 1.35915 3.43651i
31.6 1.10799 0.878843i 1.07072 1.85455i 0.455270 1.94749i −2.26303 + 1.30656i −0.443508 2.99581i 0 −1.20711 2.55791i −0.792893 1.37333i −1.35915 + 3.43651i
31.7 1.24165 + 0.676979i −1.36145 + 2.35811i 1.08340 + 1.68114i 0.937379 0.541196i −3.28684 + 2.00627i 0 0.207107 + 2.82083i −2.20711 3.82282i 1.53028 0.0373915i
31.8 1.24165 + 0.676979i 1.36145 2.35811i 1.08340 + 1.68114i −0.937379 + 0.541196i 3.28684 2.00627i 0 0.207107 + 2.82083i −2.20711 3.82282i −1.53028 + 0.0373915i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.f.d 16
4.b odd 2 1 inner 196.2.f.d 16
7.b odd 2 1 inner 196.2.f.d 16
7.c even 3 1 196.2.d.c 8
7.c even 3 1 inner 196.2.f.d 16
7.d odd 6 1 196.2.d.c 8
7.d odd 6 1 inner 196.2.f.d 16
21.g even 6 1 1764.2.b.k 8
21.h odd 6 1 1764.2.b.k 8
28.d even 2 1 inner 196.2.f.d 16
28.f even 6 1 196.2.d.c 8
28.f even 6 1 inner 196.2.f.d 16
28.g odd 6 1 196.2.d.c 8
28.g odd 6 1 inner 196.2.f.d 16
56.j odd 6 1 3136.2.f.i 8
56.k odd 6 1 3136.2.f.i 8
56.m even 6 1 3136.2.f.i 8
56.p even 6 1 3136.2.f.i 8
84.j odd 6 1 1764.2.b.k 8
84.n even 6 1 1764.2.b.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 7.c even 3 1
196.2.d.c 8 7.d odd 6 1
196.2.d.c 8 28.f even 6 1
196.2.d.c 8 28.g odd 6 1
196.2.f.d 16 1.a even 1 1 trivial
196.2.f.d 16 4.b odd 2 1 inner
196.2.f.d 16 7.b odd 2 1 inner
196.2.f.d 16 7.c even 3 1 inner
196.2.f.d 16 7.d odd 6 1 inner
196.2.f.d 16 28.d even 2 1 inner
196.2.f.d 16 28.f even 6 1 inner
196.2.f.d 16 28.g odd 6 1 inner
1764.2.b.k 8 21.g even 6 1
1764.2.b.k 8 21.h odd 6 1
1764.2.b.k 8 84.j odd 6 1
1764.2.b.k 8 84.n even 6 1
3136.2.f.i 8 56.j odd 6 1
3136.2.f.i 8 56.k odd 6 1
3136.2.f.i 8 56.m even 6 1
3136.2.f.i 8 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(196, [\chi])$$:

 $$T_{3}^{8} + 12 T_{3}^{6} + 110 T_{3}^{4} + 408 T_{3}^{2} + 1156$$ $$T_{5}^{8} - 8 T_{5}^{6} + 56 T_{5}^{4} - 64 T_{5}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 16 T + 4 T^{2} + 4 T^{3} - 3 T^{4} + 2 T^{5} + T^{6} - 2 T^{7} + T^{8} )^{2}$$
$3$ $$( 1156 + 408 T^{2} + 110 T^{4} + 12 T^{6} + T^{8} )^{2}$$
$5$ $$( 64 - 64 T^{2} + 56 T^{4} - 8 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 4624 - 1360 T^{2} + 332 T^{4} - 20 T^{6} + T^{8} )^{2}$$
$13$ $$( 8 + 8 T^{2} + T^{4} )^{4}$$
$17$ $$( 4 - 40 T^{2} + 398 T^{4} - 20 T^{6} + T^{8} )^{2}$$
$19$ $$( 1156 + 952 T^{2} + 750 T^{4} + 28 T^{6} + T^{8} )^{2}$$
$23$ $$( 73984 - 15232 T^{2} + 2864 T^{4} - 56 T^{6} + T^{8} )^{2}$$
$29$ $$( 8 + 8 T + T^{2} )^{8}$$
$31$ $$( 4734976 + 208896 T^{2} + 7040 T^{4} + 96 T^{6} + T^{8} )^{2}$$
$37$ $$( 16 - 4 T + T^{2} )^{8}$$
$41$ $$( 162 + 36 T^{2} + T^{4} )^{4}$$
$43$ $$( 3332 + 116 T^{2} + T^{4} )^{4}$$
$47$ $$( 295936 + 26112 T^{2} + 1760 T^{4} + 48 T^{6} + T^{8} )^{2}$$
$53$ $$( 4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$59$ $$( 96550276 + 2004504 T^{2} + 31790 T^{4} + 204 T^{6} + T^{8} )^{2}$$
$61$ $$( 25000000 - 1000000 T^{2} + 35000 T^{4} - 200 T^{6} + T^{8} )^{2}$$
$67$ $$( 1183744 - 121856 T^{2} + 11456 T^{4} - 112 T^{6} + T^{8} )^{2}$$
$71$ $$T^{16}$$
$73$ $$( 155800324 - 3245320 T^{2} + 55118 T^{4} - 260 T^{6} + T^{8} )^{2}$$
$79$ $$( 1183744 - 87040 T^{2} + 5312 T^{4} - 80 T^{6} + T^{8} )^{2}$$
$83$ $$( 1666 - 108 T^{2} + T^{4} )^{4}$$
$89$ $$( 334084 - 30056 T^{2} + 2126 T^{4} - 52 T^{6} + T^{8} )^{2}$$
$97$ $$( 1250 + 100 T^{2} + T^{4} )^{4}$$