# Properties

 Label 950.2.l.h Level $950$ Weight $2$ Character orbit 950.l Analytic conductor $7.586$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.l (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 12 x^{10} + 105 x^{8} + 394 x^{6} + 1077 x^{4} + 1443 x^{2} + 1369$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{3} q^{3} -\beta_{7} q^{4} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{6} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{7} + \beta_{5} q^{8} + ( 1 + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{3} q^{3} -\beta_{7} q^{4} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{6} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{7} + \beta_{5} q^{8} + ( 1 + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{11} + ( \beta_{1} + \beta_{6} ) q^{12} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{13} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{10} - \beta_{11} ) q^{14} + ( -\beta_{2} + \beta_{10} ) q^{16} + ( 2 + 2 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{17} + ( 1 + 2 \beta_{2} + \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{18} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{19} + ( 3 + 4 \beta_{2} + 4 \beta_{5} + \beta_{7} - 4 \beta_{8} + \beta_{11} ) q^{21} + ( \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{22} + ( 3 - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{23} + ( -\beta_{3} - \beta_{11} ) q^{24} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{26} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{27} + ( \beta_{1} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{31} + \beta_{8} q^{32} + ( -\beta_{1} + \beta_{3} - 4 \beta_{5} - \beta_{6} + 4 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{33} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{34} + ( 2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{36} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{37} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{38} + ( -3 - \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} + 5 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{39} + ( 1 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{9} + 4 \beta_{10} ) q^{42} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + 4 \beta_{5} - 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{44} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{46} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 5 \beta_{7} - 5 \beta_{8} + \beta_{9} ) q^{47} -\beta_{9} q^{48} + ( 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{49} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - \beta_{8} + 4 \beta_{10} - \beta_{11} ) q^{51} + ( 1 - \beta_{1} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{52} + ( 2 + \beta_{1} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{53} + ( -2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{9} ) q^{54} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{9} - \beta_{11} ) q^{56} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{57} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{10} ) q^{58} + ( -1 - 4 \beta_{1} - \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{59} + ( -1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{62} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{63} + ( -1 - \beta_{5} ) q^{64} + ( 4 \beta_{2} + \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{66} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{11} ) q^{67} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{68} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{69} + ( -4 + \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 4 \beta_{8} + \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{71} + ( 1 + \beta_{2} + 2 \beta_{5} - \beta_{7} + \beta_{10} ) q^{72} + ( -1 + 3 \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} - 3 \beta_{10} ) q^{73} + ( -1 - 3 \beta_{2} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{74} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{76} + ( 2 + 3 \beta_{2} + \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{77} + ( -5 - \beta_{1} - 3 \beta_{2} - 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{78} + ( 4 - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} ) q^{79} + ( 1 - \beta_{5} + \beta_{7} + \beta_{8} + 5 \beta_{10} ) q^{81} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{8} + 3 \beta_{10} ) q^{82} + ( -4 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{83} + ( -\beta_{1} + \beta_{2} + 4 \beta_{5} - 3 \beta_{7} + 4 \beta_{8} + 3 \beta_{10} ) q^{84} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 9 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{11} ) q^{87} + ( 1 - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{88} + ( 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{89} + ( -2 - 6 \beta_{2} + 2 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{91} + ( -3 - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{92} + ( -2 + 4 \beta_{1} + 5 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{93} + ( 5 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{10} ) q^{94} -\beta_{6} q^{96} + ( -3 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{97} + ( -4 - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{98} + ( -3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 6q^{7} - 6q^{8} + 18q^{9} + O(q^{10})$$ $$12q - 6q^{7} - 6q^{8} + 18q^{9} + 6q^{11} - 6q^{13} + 18q^{17} + 12q^{18} + 12q^{21} + 6q^{22} + 30q^{23} - 6q^{29} + 6q^{31} + 24q^{33} + 18q^{36} - 36q^{37} - 18q^{38} - 36q^{39} - 6q^{41} + 30q^{42} + 6q^{44} - 12q^{46} + 6q^{47} - 18q^{49} + 12q^{52} + 12q^{53} + 12q^{56} + 18q^{57} + 36q^{58} - 24q^{59} - 30q^{61} + 6q^{62} - 18q^{63} - 6q^{64} + 24q^{66} - 12q^{67} - 12q^{68} + 6q^{69} - 42q^{71} - 6q^{73} + 6q^{74} + 18q^{76} + 24q^{77} - 48q^{78} + 60q^{79} + 18q^{81} - 6q^{82} - 24q^{83} - 24q^{84} - 36q^{86} - 54q^{87} + 6q^{88} - 12q^{89} + 24q^{91} - 24q^{92} - 6q^{93} + 60q^{94} - 30q^{97} - 36q^{98} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 12 x^{10} + 105 x^{8} + 394 x^{6} + 1077 x^{4} + 1443 x^{2} + 1369$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-450 \nu^{10} - 16352 \nu^{8} - 143080 \nu^{6} - 950193 \nu^{4} - 1861309 \nu^{2} - 3541196$$$$)/1181151$$ $$\beta_{3}$$ $$=$$ $$($$$$450 \nu^{11} + 16352 \nu^{9} + 143080 \nu^{7} + 950193 \nu^{5} + 1861309 \nu^{3} + 3541196 \nu$$$$)/1181151$$ $$\beta_{4}$$ $$=$$ $$($$$$-811 \nu^{11} - 889 \nu^{9} + 25031 \nu^{7} + 578974 \nu^{5} + 2184085 \nu^{3} + 4191175 \nu$$$$)/1181151$$ $$\beta_{5}$$ $$=$$ $$($$$$-455 \nu^{10} - 4868 \nu^{8} - 42595 \nu^{6} - 133945 \nu^{4} - 436903 \nu^{2} - 585377$$$$)/393717$$ $$\beta_{6}$$ $$=$$ $$($$$$455 \nu^{11} + 4868 \nu^{9} + 42595 \nu^{7} + 133945 \nu^{5} + 436903 \nu^{3} + 191660 \nu$$$$)/393717$$ $$\beta_{7}$$ $$=$$ $$($$$$2176 \nu^{10} + 15493 \nu^{8} + 102754 \nu^{6} - 177139 \nu^{4} - 873376 \nu^{2} - 2435044$$$$)/1181151$$ $$\beta_{8}$$ $$=$$ $$($$$$-954 \nu^{10} - 13668 \nu^{8} - 119595 \nu^{6} - 513035 \nu^{4} - 1095464 \nu^{2} - 1118621$$$$)/393717$$ $$\beta_{9}$$ $$=$$ $$($$$$-954 \nu^{11} - 13668 \nu^{9} - 119595 \nu^{7} - 513035 \nu^{5} - 1095464 \nu^{3} - 1118621 \nu$$$$)/393717$$ $$\beta_{10}$$ $$=$$ $$($$$$5224 \nu^{10} + 49257 \nu^{8} + 398189 \nu^{6} + 841289 \nu^{4} + 1763226 \nu^{2} + 50209$$$$)/1181151$$ $$\beta_{11}$$ $$=$$ $$($$$$5224 \nu^{11} + 49257 \nu^{9} + 398189 \nu^{7} + 841289 \nu^{5} + 1763226 \nu^{3} + 50209 \nu$$$$)/1181151$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{8} + \beta_{7} - 4 \beta_{5} - \beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{9} + 5 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$19 \beta_{10} + 11 \beta_{8} - 19 \beta_{7} + 22 \beta_{5} - 8 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$19 \beta_{11} + 11 \beta_{9} - 41 \beta_{6} + 19 \beta_{4} + 8 \beta_{3} - 41 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-93 \beta_{10} - 150 \beta_{8} + 57 \beta_{7} + 150 \beta_{2} + 145$$ $$\nu^{7}$$ $$=$$ $$-93 \beta_{11} - 150 \beta_{9} + 57 \beta_{6} - 57 \beta_{4} - 150 \beta_{3} + 202 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-409 \beta_{10} + 409 \beta_{8} + 724 \beta_{7} - 1030 \beta_{5} - 724 \beta_{2} - 1030$$ $$\nu^{9}$$ $$=$$ $$-409 \beta_{11} + 409 \beta_{9} + 1754 \beta_{6} - 724 \beta_{4} + 724 \beta_{3} + 724 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$8449 \beta_{10} + 5468 \beta_{8} - 8449 \beta_{7} + 7519 \beta_{5} - 2981 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$8449 \beta_{11} + 5468 \beta_{9} - 15968 \beta_{6} + 8449 \beta_{4} + 2981 \beta_{3} - 15968 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.36120 + 2.35767i −1.36120 − 2.35767i −0.665830 − 1.15325i 0.665830 + 1.15325i 1.36120 − 2.35767i −1.36120 + 2.35767i −0.838929 − 1.45307i 0.838929 + 1.45307i −0.665830 + 1.15325i 0.665830 − 1.15325i −0.838929 + 1.45307i 0.838929 − 1.45307i
0.766044 0.642788i −2.55822 0.931116i 0.173648 0.984808i 0 −2.55822 + 0.931116i −2.33394 + 4.04250i −0.500000 0.866025i 3.37939 + 2.83564i 0
101.2 0.766044 0.642788i 2.55822 + 0.931116i 0.173648 0.984808i 0 2.55822 0.931116i 1.33394 2.31045i −0.500000 0.866025i 3.37939 + 2.83564i 0
251.1 −0.939693 0.342020i −0.231240 1.31143i 0.766044 + 0.642788i 0 −0.231240 + 1.31143i 1.18594 2.05411i −0.500000 0.866025i 1.15270 0.419550i 0
251.2 −0.939693 0.342020i 0.231240 + 1.31143i 0.766044 + 0.642788i 0 0.231240 1.31143i −2.18594 + 3.78616i −0.500000 0.866025i 1.15270 0.419550i 0
301.1 0.766044 + 0.642788i −2.55822 + 0.931116i 0.173648 + 0.984808i 0 −2.55822 0.931116i −2.33394 4.04250i −0.500000 + 0.866025i 3.37939 2.83564i 0
301.2 0.766044 + 0.642788i 2.55822 0.931116i 0.173648 + 0.984808i 0 2.55822 + 0.931116i 1.33394 + 2.31045i −0.500000 + 0.866025i 3.37939 2.83564i 0
351.1 0.173648 + 0.984808i −1.28531 + 1.07851i −0.939693 + 0.342020i 0 −1.28531 1.07851i −1.23774 + 2.14383i −0.500000 0.866025i −0.0320889 + 0.181985i 0
351.2 0.173648 + 0.984808i 1.28531 1.07851i −0.939693 + 0.342020i 0 1.28531 + 1.07851i 0.237742 0.411781i −0.500000 0.866025i −0.0320889 + 0.181985i 0
651.1 −0.939693 + 0.342020i −0.231240 + 1.31143i 0.766044 0.642788i 0 −0.231240 1.31143i 1.18594 + 2.05411i −0.500000 + 0.866025i 1.15270 + 0.419550i 0
651.2 −0.939693 + 0.342020i 0.231240 1.31143i 0.766044 0.642788i 0 0.231240 + 1.31143i −2.18594 3.78616i −0.500000 + 0.866025i 1.15270 + 0.419550i 0
701.1 0.173648 0.984808i −1.28531 1.07851i −0.939693 0.342020i 0 −1.28531 + 1.07851i −1.23774 2.14383i −0.500000 + 0.866025i −0.0320889 0.181985i 0
701.2 0.173648 0.984808i 1.28531 + 1.07851i −0.939693 0.342020i 0 1.28531 1.07851i 0.237742 + 0.411781i −0.500000 + 0.866025i −0.0320889 0.181985i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.l.h 12
5.b even 2 1 190.2.k.b 12
5.c odd 4 2 950.2.u.e 24
19.e even 9 1 inner 950.2.l.h 12
95.o odd 18 1 3610.2.a.be 6
95.p even 18 1 190.2.k.b 12
95.p even 18 1 3610.2.a.bc 6
95.q odd 36 2 950.2.u.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.b 12 5.b even 2 1
190.2.k.b 12 95.p even 18 1
950.2.l.h 12 1.a even 1 1 trivial
950.2.l.h 12 19.e even 9 1 inner
950.2.u.e 24 5.c odd 4 2
950.2.u.e 24 95.q odd 36 2
3610.2.a.bc 6 95.p even 18 1
3610.2.a.be 6 95.o odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 9 T_{3}^{10} + 36 T_{3}^{8} + 64 T_{3}^{6} + 189 T_{3}^{4} + 999 T_{3}^{2} + 1369$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{3} + T^{6} )^{2}$$
$3$ $$1369 + 999 T^{2} + 189 T^{4} + 64 T^{6} + 36 T^{8} - 9 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$23104 - 47424 T + 106464 T^{2} - 8032 T^{3} + 29232 T^{4} + 1296 T^{5} + 6288 T^{6} + 648 T^{7} + 612 T^{8} + 104 T^{9} + 48 T^{10} + 6 T^{11} + T^{12}$$
$11$ $$729 - 5832 T + 43011 T^{2} - 34344 T^{3} + 39447 T^{4} + 5022 T^{5} + 10404 T^{6} - 324 T^{7} + 765 T^{8} - 84 T^{9} + 54 T^{10} - 6 T^{11} + T^{12}$$
$13$ $$46656 + 93312 T + 132192 T^{2} + 114912 T^{3} + 63072 T^{4} + 13536 T^{5} - 944 T^{6} - 1032 T^{7} - 60 T^{8} - 8 T^{9} + 6 T^{10} + 6 T^{11} + T^{12}$$
$17$ $$12766329 - 4952178 T + 9453348 T^{2} - 5393700 T^{3} + 2135304 T^{4} - 523926 T^{5} + 45091 T^{6} + 14202 T^{7} - 4104 T^{8} + 100 T^{9} + 108 T^{10} - 18 T^{11} + T^{12}$$
$19$ $$47045881 - 781926 T^{2} + 1111158 T^{3} + 112632 T^{4} + 5130 T^{5} + 12913 T^{6} + 270 T^{7} + 312 T^{8} + 162 T^{9} - 6 T^{10} + T^{12}$$
$23$ $$1871424 - 6303744 T + 10051776 T^{2} - 9151776 T^{3} + 5436864 T^{4} - 2318256 T^{5} + 746704 T^{6} - 181896 T^{7} + 33360 T^{8} - 4600 T^{9} + 462 T^{10} - 30 T^{11} + T^{12}$$
$29$ $$5184 + 103680 T + 313632 T^{2} - 3327552 T^{3} + 10647648 T^{4} - 246096 T^{5} + 221392 T^{6} + 6384 T^{7} + 1776 T^{8} + 748 T^{9} + 60 T^{10} + 6 T^{11} + T^{12}$$
$31$ $$23104 - 1174656 T + 59526816 T^{2} - 10054080 T^{3} + 4998096 T^{4} - 559056 T^{5} + 233008 T^{6} - 23424 T^{7} + 6492 T^{8} - 432 T^{9} + 108 T^{10} - 6 T^{11} + T^{12}$$
$37$ $$( -2168 - 4368 T - 2844 T^{2} - 560 T^{3} + 48 T^{4} + 18 T^{5} + T^{6} )^{2}$$
$41$ $$114896961 + 41675472 T - 17953812 T^{2} + 1481112 T^{3} + 5461668 T^{4} - 321570 T^{5} + 248374 T^{6} + 27084 T^{7} - 1668 T^{8} + 332 T^{9} + 93 T^{10} + 6 T^{11} + T^{12}$$
$43$ $$12341169 - 51598944 T + 64800306 T^{2} - 12206748 T^{3} + 7564356 T^{4} - 1412700 T^{5} + 185821 T^{6} - 1428 T^{7} + 816 T^{8} + 340 T^{9} - 30 T^{10} + T^{12}$$
$47$ $$16842816 + 7091712 T - 1796256 T^{2} + 876096 T^{3} + 899424 T^{4} - 151632 T^{5} + 134928 T^{6} + 23760 T^{7} - 2880 T^{8} + 60 T^{9} + 108 T^{10} - 6 T^{11} + T^{12}$$
$53$ $$46656 - 933120 T + 8281440 T^{2} - 19048608 T^{3} + 19176912 T^{4} + 1881792 T^{5} + 1397232 T^{6} + 80136 T^{7} - 15624 T^{8} - 168 T^{9} + 90 T^{10} - 12 T^{11} + T^{12}$$
$59$ $$3234310641 - 1032891102 T + 617009724 T^{2} - 338533596 T^{3} + 34271730 T^{4} + 9116352 T^{5} + 1162072 T^{6} + 106320 T^{7} + 3138 T^{8} + 440 T^{9} + 219 T^{10} + 24 T^{11} + T^{12}$$
$61$ $$224041024 - 97711104 T - 80018880 T^{2} + 11464288 T^{3} + 20139264 T^{4} + 8342064 T^{5} + 2327664 T^{6} + 426744 T^{7} + 57744 T^{8} + 5992 T^{9} + 510 T^{10} + 30 T^{11} + T^{12}$$
$67$ $$2859481 + 6716652 T + 125542143 T^{2} + 6725088 T^{3} + 5002659 T^{4} - 1601004 T^{5} - 215000 T^{6} - 18060 T^{7} + 23886 T^{8} - 792 T^{9} - 135 T^{10} + 12 T^{11} + T^{12}$$
$71$ $$6701714496 + 1520705664 T + 518114880 T^{2} + 176823648 T^{3} + 26754624 T^{4} + 4521744 T^{5} + 1221840 T^{6} + 147960 T^{7} + 27792 T^{8} + 5736 T^{9} + 702 T^{10} + 42 T^{11} + T^{12}$$
$73$ $$136258929 + 420648228 T + 573942753 T^{2} + 178257600 T^{3} + 31256532 T^{4} + 5265990 T^{5} + 586819 T^{6} - 9636 T^{7} - 4299 T^{8} - 746 T^{9} + 6 T^{10} + 6 T^{11} + T^{12}$$
$79$ $$1036324864 + 1897525248 T + 503758848 T^{2} - 366187520 T^{3} + 385198848 T^{4} - 165814272 T^{5} + 39627840 T^{6} - 5787072 T^{7} + 563904 T^{8} - 38432 T^{9} + 1860 T^{10} - 60 T^{11} + T^{12}$$
$83$ $$51075548001 - 602061336 T + 5382935109 T^{2} + 1605585744 T^{3} + 554019813 T^{4} + 86649156 T^{5} + 12443122 T^{6} + 1098168 T^{7} + 105819 T^{8} + 7112 T^{9} + 564 T^{10} + 24 T^{11} + T^{12}$$
$89$ $$2653641 + 3225420 T + 7649640 T^{2} + 7028442 T^{3} + 6984288 T^{4} + 4663656 T^{5} + 2151703 T^{6} + 307320 T^{7} - 240 T^{8} - 3070 T^{9} - 120 T^{10} + 12 T^{11} + T^{12}$$
$97$ $$2521747089 + 703238868 T + 440892243 T^{2} + 133565982 T^{3} + 26697609 T^{4} + 2427096 T^{5} - 182324 T^{6} - 99000 T^{7} - 9006 T^{8} + 1078 T^{9} + 333 T^{10} + 30 T^{11} + T^{12}$$