# Properties

 Label 62.2.d.a Level 62 Weight 2 Character orbit 62.d Analytic conductor 0.495 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$62 = 2 \cdot 31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 62.d (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.495072492532$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.511890625.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{3} + \beta_{4} q^{4} + ( -1 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{5} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{6} + ( 1 - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{7} + ( -1 + \beta_{2} - \beta_{4} + \beta_{7} ) q^{8} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{3} + \beta_{4} q^{4} + ( -1 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{5} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{6} + ( 1 - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{7} + ( -1 + \beta_{2} - \beta_{4} + \beta_{7} ) q^{8} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{9} + ( \beta_{2} + \beta_{5} - \beta_{6} ) q^{10} + ( 1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( -1 - \beta_{3} - \beta_{4} ) q^{12} + ( -2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{13} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{14} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{15} -\beta_{7} q^{16} + ( -3 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{17} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{18} + ( -3 + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{19} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{20} + ( 1 - 7 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{21} + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{22} + ( 2 + \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{7} ) q^{23} + ( 1 + \beta_{4} + \beta_{5} - \beta_{7} ) q^{24} + ( -1 - \beta_{1} + 3 \beta_{2} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{25} + ( \beta_{1} + 2 \beta_{2} - \beta_{5} + 2 \beta_{7} ) q^{26} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 6 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{27} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{28} + ( 3 - 2 \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{29} + ( \beta_{1} - 4 \beta_{2} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{30} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{31} + q^{32} + ( 1 - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( -7 + \beta_{1} + \beta_{3} - 5 \beta_{4} - \beta_{5} + 7 \beta_{7} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{36} + ( 5 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{37} + ( 3 - \beta_{5} - 3 \beta_{7} ) q^{38} + ( -5 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{39} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{40} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{41} + ( -1 + \beta_{1} + \beta_{3} + 6 \beta_{4} + \beta_{7} ) q^{42} + ( -3 - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{43} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{44} + ( 4 + 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 11 \beta_{4} + 2 \beta_{6} - 11 \beta_{7} ) q^{45} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{46} + ( -2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -\beta_{1} - \beta_{4} + \beta_{7} ) q^{48} + ( -4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} ) q^{49} + ( -3 + \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{50} + ( 4 - \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} - 4 \beta_{7} ) q^{51} + ( -2 - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{52} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( -4 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{6} + 6 \beta_{7} ) q^{54} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{55} + ( -2 + \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{56} + ( -3 \beta_{1} + 3 \beta_{5} + \beta_{6} ) q^{57} + ( -3 - \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{7} ) q^{58} + ( -\beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 4 + \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{60} + ( -3 + 6 \beta_{2} - 2 \beta_{6} + 6 \beta_{7} ) q^{61} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{62} + ( 5 + 6 \beta_{2} - \beta_{6} + 6 \beta_{7} ) q^{63} -\beta_{2} q^{64} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{65} + ( -1 + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{66} + ( -4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{67} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{68} + ( -3 - \beta_{1} - \beta_{3} - 7 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{69} + ( -2 + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} - \beta_{6} - 5 \beta_{7} ) q^{70} + ( 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{71} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{72} + ( -5 + 4 \beta_{1} + 4 \beta_{3} + 3 \beta_{4} - \beta_{5} + 5 \beta_{7} ) q^{73} + ( -2 - 5 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{74} + ( 3 \beta_{1} - 5 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} + 3 \beta_{6} - 8 \beta_{7} ) q^{75} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{76} + ( 5 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{77} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{78} + ( 7 - 4 \beta_{1} - 7 \beta_{2} + \beta_{3} + 7 \beta_{4} + \beta_{6} - 7 \beta_{7} ) q^{79} + ( -\beta_{1} - \beta_{6} + \beta_{7} ) q^{80} + ( -9 + 4 \beta_{2} - 9 \beta_{4} - \beta_{5} + \beta_{6} ) q^{81} + ( 1 + 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{82} + ( 9 - 7 \beta_{2} + 9 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{83} + ( -7 - \beta_{1} + 7 \beta_{2} - \beta_{3} - 6 \beta_{4} - \beta_{6} + 6 \beta_{7} ) q^{84} + ( 6 - \beta_{1} - 6 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{85} + ( 3 + 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{86} + ( -3 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{87} + ( 1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{88} + ( 5 - \beta_{1} - \beta_{3} + 11 \beta_{4} + 2 \beta_{5} - 5 \beta_{7} ) q^{89} + ( -3 \beta_{1} + 7 \beta_{2} - \beta_{3} - 7 \beta_{4} + \beta_{5} - 3 \beta_{6} + 11 \beta_{7} ) q^{90} + ( 8 - 8 \beta_{2} - \beta_{3} + 8 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{91} + ( -2 - \beta_{2} + \beta_{6} - \beta_{7} ) q^{92} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} - 6 \beta_{4} - 4 \beta_{5} + 5 \beta_{7} ) q^{93} + ( -5 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{94} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{95} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{96} + ( -5 - 2 \beta_{1} - 2 \beta_{3} - 11 \beta_{4} + 5 \beta_{7} ) q^{97} + ( 10 - 3 \beta_{2} + 4 \beta_{6} - 3 \beta_{7} ) q^{98} + ( 2 - 2 \beta_{1} + 2 \beta_{5} - 5 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 4q^{3} - 2q^{4} - 4q^{5} + 6q^{6} + 2q^{7} - 2q^{8} + O(q^{10})$$ $$8q - 2q^{2} - 4q^{3} - 2q^{4} - 4q^{5} + 6q^{6} + 2q^{7} - 2q^{8} + q^{10} - 2q^{11} - 4q^{12} - 11q^{13} + 2q^{14} + 21q^{15} - 2q^{16} - 7q^{17} - 5q^{18} - 14q^{19} + q^{20} - 7q^{21} - 2q^{22} + 3q^{23} + q^{24} + 14q^{26} + 5q^{27} + 2q^{28} + 13q^{29} - 14q^{30} + 15q^{31} + 8q^{32} + 2q^{33} + 3q^{34} - 28q^{35} + 10q^{36} + 52q^{37} + 21q^{38} - 16q^{39} + q^{40} - 11q^{41} - 17q^{42} - 22q^{43} - 7q^{44} - 19q^{45} + 8q^{46} - 10q^{47} + q^{48} - 10q^{49} - 15q^{50} + 28q^{51} - 11q^{52} + 7q^{53} + 5q^{54} - 7q^{55} - 8q^{56} - 20q^{57} - 17q^{58} + 22q^{59} + 21q^{60} + 4q^{61} + 5q^{62} + 66q^{63} - 2q^{64} - 18q^{66} - 26q^{67} + 8q^{68} + 4q^{69} - 28q^{70} - 15q^{71} - 5q^{72} - 29q^{73} - 23q^{74} - 34q^{75} + 21q^{76} + 34q^{77} - q^{78} - 2q^{79} + q^{80} - 45q^{81} + 9q^{82} + 38q^{83} - 17q^{84} + 25q^{85} + 18q^{86} - 20q^{87} + 18q^{88} + q^{89} + 46q^{90} + 38q^{91} - 22q^{92} + 21q^{93} - 20q^{94} - 12q^{95} - 4q^{96} - 10q^{97} + 60q^{98} + 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 7 x^{6} - 5 x^{5} + 16 x^{4} + 15 x^{3} + 63 x^{2} + 81 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 6 \nu^{6} + 25 \nu^{5} - 80 \nu^{4} + 256 \nu^{3} - 753 \nu^{2} + 594 \nu + 27$$$$)/1728$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 30 \nu^{6} - 97 \nu^{5} + 176 \nu^{4} - 160 \nu^{3} + 273 \nu^{2} + 54 \nu + 621$$$$)/576$$ $$\beta_{4}$$ $$=$$ $$($$$$-13 \nu^{7} + 66 \nu^{6} - 217 \nu^{5} + 416 \nu^{4} - 496 \nu^{3} - 51 \nu^{2} - 270 \nu - 459$$$$)/1728$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 5 \nu^{5} + 16 \nu^{3} + 31 \nu^{2} + 94 \nu + 111$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 14 \nu^{6} + 39 \nu^{5} - 32 \nu^{4} + 16 \nu^{3} + 61 \nu^{2} + 66 \nu + 117$$$$)/192$$ $$\beta_{7}$$ $$=$$ $$($$$$-37 \nu^{7} + 138 \nu^{6} - 313 \nu^{5} + 320 \nu^{4} - 592 \nu^{3} - 123 \nu^{2} - 1494 \nu - 459$$$$)/1728$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{3} - 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{6} + 5 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{7} + 9 \beta_{5} + 12 \beta_{4} - 8 \beta_{3} - 8 \beta_{1} - 3$$ $$\nu^{5}$$ $$=$$ $$24 \beta_{7} + 20 \beta_{6} + 12 \beta_{5} + 24 \beta_{2} - 12 \beta_{1} - 27$$ $$\nu^{6}$$ $$=$$ $$60 \beta_{7} + 56 \beta_{6} - 60 \beta_{4} + 56 \beta_{3} + 96 \beta_{2} - 15 \beta_{1} - 96$$ $$\nu^{7}$$ $$=$$ $$107 \beta_{6} - 107 \beta_{5} - 168 \beta_{4} + 223 \beta_{3} + 213 \beta_{2} - 168$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/62\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1 + \beta_{2} - \beta_{4} + \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 −0.448193 + 1.37940i 0.639176 − 1.96718i −0.892071 − 0.648127i 2.20109 + 1.59918i −0.892071 + 0.648127i 2.20109 − 1.59918i −0.448193 − 1.37940i 0.639176 + 1.96718i
−0.809017 0.587785i −2.48240 + 1.80357i 0.309017 + 0.951057i −3.34677 3.06842 0.778353 + 2.39552i 0.309017 0.951057i 1.98240 6.10121i 2.70759 + 1.96718i
33.2 −0.809017 0.587785i 0.364368 0.264729i 0.309017 + 0.951057i 2.34677 −0.450384 −1.39639 4.29764i 0.309017 0.951057i −0.864368 + 2.66025i −1.89858 1.37940i
35.1 0.309017 + 0.951057i −0.531724 + 1.63648i −0.809017 + 0.587785i −1.68148 −1.72069 3.90218 2.83510i −0.809017 0.587785i 0.0317236 + 0.0230486i −0.519606 1.59918i
35.2 0.309017 + 0.951057i 0.649758 1.99975i −0.809017 + 0.587785i 0.681481 2.10266 −2.28414 + 1.65953i −0.809017 0.587785i −1.14976 0.835348i 0.210589 + 0.648127i
39.1 0.309017 0.951057i −0.531724 1.63648i −0.809017 0.587785i −1.68148 −1.72069 3.90218 + 2.83510i −0.809017 + 0.587785i 0.0317236 0.0230486i −0.519606 + 1.59918i
39.2 0.309017 0.951057i 0.649758 + 1.99975i −0.809017 0.587785i 0.681481 2.10266 −2.28414 1.65953i −0.809017 + 0.587785i −1.14976 + 0.835348i 0.210589 0.648127i
47.1 −0.809017 + 0.587785i −2.48240 1.80357i 0.309017 0.951057i −3.34677 3.06842 0.778353 2.39552i 0.309017 + 0.951057i 1.98240 + 6.10121i 2.70759 1.96718i
47.2 −0.809017 + 0.587785i 0.364368 + 0.264729i 0.309017 0.951057i 2.34677 −0.450384 −1.39639 + 4.29764i 0.309017 + 0.951057i −0.864368 2.66025i −1.89858 + 1.37940i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.d Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(62, [\chi])$$.