# Properties

 Label 700.2.c.k Level $700$ Weight $2$ Character orbit 700.c Analytic conductor $5.590$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.29960650073923649536.7 Defining polynomial: $$x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{3} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{6} + ( -\beta_{1} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{14} ) q^{7} + ( -\beta_{7} + 2 \beta_{14} ) q^{8} + ( -2 + 2 \beta_{2} + 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{7} q^{2} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{3} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{6} + ( -\beta_{1} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{14} ) q^{7} + ( -\beta_{7} + 2 \beta_{14} ) q^{8} + ( -2 + 2 \beta_{2} + 2 \beta_{4} ) q^{9} + ( 1 + \beta_{2} - \beta_{4} ) q^{11} + ( \beta_{1} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{12} - \beta_{14} ) q^{12} + ( -2 \beta_{1} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{14} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{14} + ( \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{16} + ( \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{12} ) q^{17} + ( 2 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{14} ) q^{18} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{10} + \beta_{13} - 3 \beta_{15} ) q^{19} + ( -3 + \beta_{2} - \beta_{3} - 2 \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{21} + ( -3 \beta_{1} - \beta_{6} + \beta_{7} + \beta_{14} ) q^{22} + ( \beta_{6} + \beta_{7} - 2 \beta_{14} ) q^{23} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{10} - \beta_{11} + 3 \beta_{13} ) q^{24} + ( -4 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{13} + 2 \beta_{15} ) q^{26} + ( 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{27} + ( 3 \beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{14} ) q^{28} + ( -4 + 3 \beta_{2} + 3 \beta_{4} ) q^{29} + ( -\beta_{2} - \beta_{4} + 2 \beta_{15} ) q^{31} + ( 6 \beta_{1} - 2 \beta_{6} - \beta_{7} ) q^{32} + ( -\beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{12} ) q^{33} + ( -2 \beta_{3} - 2 \beta_{4} + 3 \beta_{10} + \beta_{11} + \beta_{13} ) q^{34} + ( 4 - 2 \beta_{2} - 4 \beta_{3} ) q^{36} + ( 2 \beta_{1} + 3 \beta_{6} - 3 \beta_{7} ) q^{37} + ( 2 \beta_{1} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{9} - \beta_{12} - 2 \beta_{14} ) q^{38} + ( -6 - \beta_{2} + 10 \beta_{3} + \beta_{4} ) q^{39} + ( \beta_{2} - \beta_{3} + \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{15} ) q^{41} + ( -2 \beta_{1} + 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{14} ) q^{42} + ( 2 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 + 3 \beta_{2} - 2 \beta_{3} ) q^{44} + ( -2 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{46} + ( -6 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{47} + ( -\beta_{1} + 5 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{12} + \beta_{14} ) q^{48} + ( -3 \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{15} ) q^{49} + ( 1 + 3 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{51} + ( -\beta_{1} - 6 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 6 \beta_{9} + 4 \beta_{12} + \beta_{14} ) q^{52} + ( -8 \beta_{1} - 3 \beta_{6} + 3 \beta_{7} ) q^{53} + ( -\beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} + 3 \beta_{13} + 2 \beta_{15} ) q^{54} + ( 3 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{56} + ( 3 \beta_{1} - 8 \beta_{6} + 8 \beta_{7} ) q^{57} + ( 3 \beta_{1} - 3 \beta_{6} + 4 \beta_{7} + 3 \beta_{14} ) q^{58} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{15} ) q^{59} + ( -2 \beta_{2} - 2 \beta_{4} + 4 \beta_{10} - 4 \beta_{13} ) q^{61} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{14} ) q^{62} + ( 4 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{14} ) q^{63} + ( 4 - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{64} + ( 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{10} - \beta_{11} - \beta_{13} ) q^{66} + ( -3 \beta_{1} - \beta_{6} - \beta_{7} + 8 \beta_{14} ) q^{67} + ( 2 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{12} - 2 \beta_{14} ) q^{68} + ( -\beta_{2} - \beta_{4} + 2 \beta_{10} - 2 \beta_{13} ) q^{69} + ( -2 - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 6 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} - 6 \beta_{14} ) q^{72} + ( -3 \beta_{5} + 3 \beta_{8} - 3 \beta_{9} + 6 \beta_{12} ) q^{73} + ( -6 - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{74} + ( 4 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 5 \beta_{10} - 3 \beta_{11} + 7 \beta_{13} + 2 \beta_{15} ) q^{76} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{9} + 2 \beta_{12} + \beta_{14} ) q^{77} + ( -7 \beta_{1} + 11 \beta_{6} - 6 \beta_{7} + 9 \beta_{14} ) q^{78} + ( 4 + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{79} + ( -3 - 2 \beta_{2} - 2 \beta_{4} ) q^{81} + ( -\beta_{1} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{12} + \beta_{14} ) q^{82} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{83} + ( -1 - 6 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + 3 \beta_{10} + \beta_{11} + 5 \beta_{15} ) q^{84} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{86} + ( 10 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} ) q^{87} + ( -\beta_{1} - 5 \beta_{6} + 3 \beta_{7} + \beta_{14} ) q^{88} + ( 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{10} - 6 \beta_{11} + 3 \beta_{13} + 3 \beta_{15} ) q^{89} + ( -7 - \beta_{2} + 7 \beta_{3} + 8 \beta_{4} - \beta_{10} - \beta_{13} - 7 \beta_{15} ) q^{91} + ( -5 \beta_{1} + 3 \beta_{6} + 2 \beta_{7} - \beta_{14} ) q^{92} + ( -4 \beta_{1} + 5 \beta_{6} - 5 \beta_{7} ) q^{93} + ( 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{13} - 6 \beta_{15} ) q^{94} + ( -2 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{13} + 4 \beta_{15} ) q^{96} + ( 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{12} ) q^{97} + ( -5 \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{12} - 3 \beta_{14} ) q^{98} + ( 2 - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} - 32 q^{9} + O(q^{10})$$ $$16 q + 8 q^{4} - 32 q^{9} - 24 q^{14} + 8 q^{16} - 48 q^{21} - 64 q^{29} + 48 q^{36} - 8 q^{44} - 32 q^{46} + 40 q^{56} + 104 q^{64} - 32 q^{74} - 48 q^{81} - 40 q^{84} + 80 q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{10} - 3 \nu^{6} + 12 \nu^{2}$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{13} + 2 \nu^{12} - 3 \nu^{11} + 28 \nu^{9} + 10 \nu^{8} + 28 \nu^{7} - 96 \nu^{5} - 56 \nu^{4} + 256 \nu + 128$$$$)/256$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} + 2 \nu^{13} - 8 \nu^{12} + 11 \nu^{11} + 10 \nu^{9} + 24 \nu^{8} - 20 \nu^{7} - 56 \nu^{5} - 224 \nu^{4} + 384 \nu + 512$$$$)/512$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{13} - \nu^{12} + 3 \nu^{11} + 3 \nu^{9} - 5 \nu^{8} - 12 \nu^{7} - 28 \nu^{5} + 28 \nu^{4} + 16 \nu^{3} - 64$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{15} + 2 \nu^{13} - 12 \nu^{12} - 17 \nu^{11} - 22 \nu^{9} + 4 \nu^{8} + 76 \nu^{7} + 168 \nu^{5} - 112 \nu^{4} - 384 \nu - 256$$$$)/512$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{15} + \nu^{14} + 2 \nu^{13} + 6 \nu^{11} + 5 \nu^{10} - 6 \nu^{9} - 24 \nu^{7} - 28 \nu^{6} + 56 \nu^{5} + 32 \nu^{3} + 192 \nu^{2}$$$$)/256$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} - 4 \nu^{13} + 3 \nu^{11} + 5 \nu^{10} + 28 \nu^{9} - 28 \nu^{7} - 28 \nu^{6} - 96 \nu^{5} + 192 \nu^{2} + 256 \nu$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$4 \nu^{15} + 3 \nu^{14} + 2 \nu^{13} - 2 \nu^{12} - 44 \nu^{11} - 33 \nu^{10} + 10 \nu^{9} + 22 \nu^{8} + 144 \nu^{7} + 124 \nu^{6} - 56 \nu^{5} - 168 \nu^{4} - 448 \nu^{3} - 448 \nu^{2} - 128 \nu + 384$$$$)/512$$ $$\beta_{9}$$ $$=$$ $$($$$$5 \nu^{15} + 3 \nu^{14} + 8 \nu^{13} + 2 \nu^{12} - 39 \nu^{11} - 33 \nu^{10} + 8 \nu^{9} - 22 \nu^{8} + 180 \nu^{7} + 124 \nu^{6} + 168 \nu^{4} - 448 \nu^{3} - 448 \nu^{2} + 256 \nu - 384$$$$)/512$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{15} - 5 \nu^{14} - 4 \nu^{13} - 6 \nu^{12} - 21 \nu^{11} + 23 \nu^{10} + 12 \nu^{9} + 2 \nu^{8} + 76 \nu^{7} - 68 \nu^{6} - 112 \nu^{5} - 56 \nu^{4} - 384 \nu^{3} + 64 \nu^{2} + 512 \nu + 128$$$$)/512$$ $$\beta_{11}$$ $$=$$ $$($$$$-2 \nu^{15} + 3 \nu^{14} - 4 \nu^{13} + 6 \nu^{11} - 17 \nu^{10} + 28 \nu^{9} - 24 \nu^{7} + 140 \nu^{6} - 96 \nu^{5} + 32 \nu^{3} - 192 \nu^{2} + 256 \nu$$$$)/256$$ $$\beta_{12}$$ $$=$$ $$($$$$\nu^{15} + 2 \nu^{13} - 18 \nu^{12} - 3 \nu^{11} - 6 \nu^{9} + 102 \nu^{8} + 28 \nu^{7} + 56 \nu^{5} - 328 \nu^{4} + 640$$$$)/256$$ $$\beta_{13}$$ $$=$$ $$($$$$-5 \nu^{15} - 5 \nu^{14} + 8 \nu^{13} - 6 \nu^{12} + 39 \nu^{11} + 23 \nu^{10} + 8 \nu^{9} + 2 \nu^{8} - 180 \nu^{7} - 68 \nu^{6} - 56 \nu^{4} + 448 \nu^{3} + 64 \nu^{2} + 256 \nu + 128$$$$)/512$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{15} + 8 \nu^{14} + 2 \nu^{13} - 11 \nu^{11} - 24 \nu^{10} + 10 \nu^{9} + 20 \nu^{7} + 96 \nu^{6} - 56 \nu^{5} - 128 \nu^{2} + 384 \nu$$$$)/512$$ $$\beta_{15}$$ $$=$$ $$($$$$-5 \nu^{15} + 6 \nu^{14} - 10 \nu^{13} + 23 \nu^{11} - 34 \nu^{10} + 46 \nu^{9} - 68 \nu^{7} + 152 \nu^{6} - 136 \nu^{5} + 64 \nu^{3} - 512 \nu^{2} + 128 \nu$$$$)/512$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} + \beta_{10} - \beta_{8} - \beta_{4} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{12} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$3 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_{1} + 3$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-4 \beta_{15} - 3 \beta_{14} - \beta_{13} + 7 \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{15} - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 10 \beta_{7} + 7 \beta_{6} + \beta_{5} + 10 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} + 3 \beta_{1} + 1$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$7 \beta_{12} + 2 \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 11 \beta_{5} - \beta_{4} - 13 \beta_{3} + \beta_{2} - 8$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$9 \beta_{15} - 15 \beta_{14} + 11 \beta_{13} - 6 \beta_{10} + 11 \beta_{9} + 6 \beta_{8} + 14 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} - 9 \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 9$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$12 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 9 \beta_{11} + 9 \beta_{10} - 15 \beta_{7} - 15 \beta_{6} - 15 \beta_{4} - 9 \beta_{3} - 6 \beta_{2} + 53 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$9 \beta_{15} - 30 \beta_{14} - 21 \beta_{13} - 22 \beta_{10} + 21 \beta_{9} - 22 \beta_{8} - 6 \beta_{7} - 7 \beta_{6} - 9 \beta_{5} + 6 \beta_{4} + 52 \beta_{3} - 3 \beta_{2} + 21 \beta_{1} - 9$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-7 \beta_{12} - 18 \beta_{9} + 18 \beta_{8} - 31 \beta_{7} + 31 \beta_{6} - 37 \beta_{5} - 31 \beta_{4} - 19 \beta_{3} + 31 \beta_{2} - 40$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-25 \beta_{15} - 17 \beta_{14} + 37 \beta_{13} - 42 \beta_{10} + 37 \beta_{9} + 42 \beta_{8} - 14 \beta_{7} + 26 \beta_{6} - 25 \beta_{5} + 41 \beta_{4} + 25 \beta_{3} + 11 \beta_{2} + 42 \beta_{1} - 25$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$84 \beta_{15} + 93 \beta_{14} - 9 \beta_{13} - 41 \beta_{11} - 9 \beta_{10} - 17 \beta_{7} - 17 \beta_{6} - 17 \beta_{4} + 9 \beta_{3} - 26 \beta_{2} + 139 \beta_{1}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-9 \beta_{15} - 34 \beta_{14} - 43 \beta_{13} - 74 \beta_{10} + 43 \beta_{9} - 74 \beta_{8} + 134 \beta_{7} - 217 \beta_{6} + 9 \beta_{5} - 134 \beta_{4} + 108 \beta_{3} - 125 \beta_{2} + 43 \beta_{1} + 9$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
699.1
 −1.32968 − 0.481610i 1.38588 − 0.281691i 1.38588 + 0.281691i −1.32968 + 0.481610i 0.281691 + 1.38588i 1.32968 − 0.481610i 1.32968 + 0.481610i 0.281691 − 1.38588i 0.481610 − 1.32968i −1.38588 − 0.281691i −1.38588 + 0.281691i 0.481610 + 1.32968i −0.281691 + 1.38588i −0.481610 − 1.32968i −0.481610 + 1.32968i −0.281691 − 1.38588i
−1.39897 0.207107i 1.47363i 1.91421 + 0.579471i 0 −0.305198 + 2.06155i 0.819496 2.51564i −2.55791 1.20711i 0.828427 0
699.2 −1.39897 0.207107i 1.47363i 1.91421 + 0.579471i 0 0.305198 2.06155i 0.819496 + 2.51564i −2.55791 1.20711i 0.828427 0
699.3 −1.39897 + 0.207107i 1.47363i 1.91421 0.579471i 0 0.305198 + 2.06155i 0.819496 2.51564i −2.55791 + 1.20711i 0.828427 0
699.4 −1.39897 + 0.207107i 1.47363i 1.91421 0.579471i 0 −0.305198 2.06155i 0.819496 + 2.51564i −2.55791 + 1.20711i 0.828427 0
699.5 −0.736813 1.20711i 2.79793i −0.914214 + 1.77882i 0 −3.37740 + 2.06155i 2.51564 0.819496i 2.82083 0.207107i −4.82843 0
699.6 −0.736813 1.20711i 2.79793i −0.914214 + 1.77882i 0 3.37740 2.06155i 2.51564 + 0.819496i 2.82083 0.207107i −4.82843 0
699.7 −0.736813 + 1.20711i 2.79793i −0.914214 1.77882i 0 3.37740 + 2.06155i 2.51564 0.819496i 2.82083 + 0.207107i −4.82843 0
699.8 −0.736813 + 1.20711i 2.79793i −0.914214 1.77882i 0 −3.37740 2.06155i 2.51564 + 0.819496i 2.82083 + 0.207107i −4.82843 0
699.9 0.736813 1.20711i 2.79793i −0.914214 1.77882i 0 −3.37740 2.06155i −2.51564 0.819496i −2.82083 0.207107i −4.82843 0
699.10 0.736813 1.20711i 2.79793i −0.914214 1.77882i 0 3.37740 + 2.06155i −2.51564 + 0.819496i −2.82083 0.207107i −4.82843 0
699.11 0.736813 + 1.20711i 2.79793i −0.914214 + 1.77882i 0 3.37740 2.06155i −2.51564 0.819496i −2.82083 + 0.207107i −4.82843 0
699.12 0.736813 + 1.20711i 2.79793i −0.914214 + 1.77882i 0 −3.37740 + 2.06155i −2.51564 + 0.819496i −2.82083 + 0.207107i −4.82843 0
699.13 1.39897 0.207107i 1.47363i 1.91421 0.579471i 0 −0.305198 2.06155i −0.819496 2.51564i 2.55791 1.20711i 0.828427 0
699.14 1.39897 0.207107i 1.47363i 1.91421 0.579471i 0 0.305198 + 2.06155i −0.819496 + 2.51564i 2.55791 1.20711i 0.828427 0
699.15 1.39897 + 0.207107i 1.47363i 1.91421 + 0.579471i 0 0.305198 2.06155i −0.819496 2.51564i 2.55791 + 1.20711i 0.828427 0
699.16 1.39897 + 0.207107i 1.47363i 1.91421 + 0.579471i 0 −0.305198 + 2.06155i −0.819496 + 2.51564i 2.55791 + 1.20711i 0.828427 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.c.k 16
4.b odd 2 1 inner 700.2.c.k 16
5.b even 2 1 inner 700.2.c.k 16
5.c odd 4 1 700.2.g.i 8
5.c odd 4 1 700.2.g.k yes 8
7.b odd 2 1 inner 700.2.c.k 16
20.d odd 2 1 inner 700.2.c.k 16
20.e even 4 1 700.2.g.i 8
20.e even 4 1 700.2.g.k yes 8
28.d even 2 1 inner 700.2.c.k 16
35.c odd 2 1 inner 700.2.c.k 16
35.f even 4 1 700.2.g.i 8
35.f even 4 1 700.2.g.k yes 8
140.c even 2 1 inner 700.2.c.k 16
140.j odd 4 1 700.2.g.i 8
140.j odd 4 1 700.2.g.k yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.k 16 1.a even 1 1 trivial
700.2.c.k 16 4.b odd 2 1 inner
700.2.c.k 16 5.b even 2 1 inner
700.2.c.k 16 7.b odd 2 1 inner
700.2.c.k 16 20.d odd 2 1 inner
700.2.c.k 16 28.d even 2 1 inner
700.2.c.k 16 35.c odd 2 1 inner
700.2.c.k 16 140.c even 2 1 inner
700.2.g.i 8 5.c odd 4 1
700.2.g.i 8 20.e even 4 1
700.2.g.i 8 35.f even 4 1
700.2.g.i 8 140.j odd 4 1
700.2.g.k yes 8 5.c odd 4 1
700.2.g.k yes 8 20.e even 4 1
700.2.g.k yes 8 35.f even 4 1
700.2.g.k yes 8 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{4} + 10 T_{3}^{2} + 17$$ $$T_{11}^{4} + 10 T_{11}^{2} + 17$$ $$T_{13}^{2} - 34$$ $$T_{19}^{4} - 58 T_{19}^{2} + 833$$ $$T_{23}^{4} - 20 T_{23}^{2} + 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 8 T^{2} + T^{4} - 2 T^{6} + T^{8} )^{2}$$
$3$ $$( 17 + 10 T^{2} + T^{4} )^{4}$$
$5$ $$T^{16}$$
$7$ $$( 2401 - 30 T^{4} + T^{8} )^{2}$$
$11$ $$( 17 + 10 T^{2} + T^{4} )^{4}$$
$13$ $$( -34 + T^{2} )^{8}$$
$17$ $$( -17 + T^{2} )^{8}$$
$19$ $$( 833 - 58 T^{2} + T^{4} )^{4}$$
$23$ $$( 68 - 20 T^{2} + T^{4} )^{4}$$
$29$ $$( -2 + 8 T + T^{2} )^{8}$$
$31$ $$( 68 - 20 T^{2} + T^{4} )^{4}$$
$37$ $$( 196 + 44 T^{2} + T^{4} )^{4}$$
$41$ $$( 17 + T^{2} )^{8}$$
$43$ $$( 272 - 40 T^{2} + T^{4} )^{4}$$
$47$ $$( 5508 + 180 T^{2} + T^{4} )^{4}$$
$53$ $$( 2116 + 164 T^{2} + T^{4} )^{4}$$
$59$ $$( 272 - 40 T^{2} + T^{4} )^{4}$$
$61$ $$( 136 + T^{2} )^{8}$$
$67$ $$( 8993 - 218 T^{2} + T^{4} )^{4}$$
$71$ $$( 3332 + 116 T^{2} + T^{4} )^{4}$$
$73$ $$( -153 + T^{2} )^{8}$$
$79$ $$( 1088 + 80 T^{2} + T^{4} )^{4}$$
$83$ $$( 17 + 10 T^{2} + T^{4} )^{4}$$
$89$ $$( 153 + T^{2} )^{8}$$
$97$ $$( -68 + T^{2} )^{8}$$