# Properties

 Label 693.2.m.a Level 693 Weight 2 Character orbit 693.m Analytic conductor 5.534 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 693.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -3 \zeta_{10}^{3} q^{4} + ( 1 + \zeta_{10}^{2} ) q^{5} -\zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} -3 \zeta_{10}^{3} q^{4} + ( 1 + \zeta_{10}^{2} ) q^{5} -\zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + ( -3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{10} + ( -1 + \zeta_{10} - 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{11} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{13} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{14} + \zeta_{10} q^{16} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{17} + ( -3 \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( 3 - 3 \zeta_{10}^{3} ) q^{20} + ( 6 + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{22} + ( -6 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{25} + ( 2 - 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{26} -3 \zeta_{10} q^{28} -6 \zeta_{10}^{3} q^{29} + ( -5 + 5 \zeta_{10}^{3} ) q^{31} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{32} + ( 6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{34} + ( 1 - \zeta_{10}^{3} ) q^{35} + ( -1 + \zeta_{10} - 3 \zeta_{10}^{3} ) q^{37} + ( 1 + 7 \zeta_{10} + \zeta_{10}^{2} ) q^{38} + ( -1 + \zeta_{10} + 3 \zeta_{10}^{3} ) q^{40} + ( -\zeta_{10} - 8 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{41} + ( -6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{43} + ( -6 - 6 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{44} + ( 8 + 3 \zeta_{10} - 3 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{46} + ( 4 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{47} -\zeta_{10} q^{49} + ( 7 - 6 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{50} + ( -6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{52} + ( 10 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{53} + ( 3 - 2 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{55} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{56} + ( 12 \zeta_{10} - 6 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{58} + ( 2 - 2 \zeta_{10} ) q^{59} + ( 5 - 5 \zeta_{10} - 15 \zeta_{10}^{3} ) q^{62} + ( 13 - 13 \zeta_{10} + 13 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{64} + 2 q^{65} + ( -9 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{68} + ( -1 + \zeta_{10} + 3 \zeta_{10}^{3} ) q^{70} + ( 8 + 8 \zeta_{10}^{2} ) q^{71} + ( -6 + 6 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{73} + ( 7 \zeta_{10} - 6 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{74} + ( -6 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{76} + ( -2 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{77} + ( 10 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{79} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{80} + ( 17 - 6 \zeta_{10} + 17 \zeta_{10}^{2} ) q^{82} + ( -4 + 14 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{83} -3 \zeta_{10}^{2} q^{85} + ( 12 - 18 \zeta_{10} + 18 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{86} + ( 2 + 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{88} + ( -4 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{89} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} + 18 \zeta_{10}^{3} ) q^{92} -10 \zeta_{10} q^{94} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{95} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{97} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 5q^{2} - 3q^{4} + 3q^{5} - q^{7} + 5q^{8} + O(q^{10})$$ $$4q - 5q^{2} - 3q^{4} + 3q^{5} - q^{7} + 5q^{8} - 10q^{10} - q^{11} + 4q^{13} + 5q^{14} + q^{16} - 6q^{17} - 7q^{19} + 9q^{20} + 25q^{22} - 22q^{23} + 6q^{25} + 10q^{26} - 3q^{28} - 6q^{29} - 15q^{31} + 30q^{34} + 3q^{35} - 6q^{37} + 10q^{38} + 6q^{41} + 12q^{43} - 33q^{44} + 30q^{46} + 10q^{47} - q^{49} + 15q^{50} - 18q^{52} + 14q^{53} + 8q^{55} + 30q^{58} + 6q^{59} + 13q^{64} + 8q^{65} - 18q^{68} + 24q^{71} - 22q^{73} + 20q^{74} - 6q^{76} - 11q^{77} + 10q^{79} + 2q^{80} + 45q^{82} + 2q^{83} + 3q^{85} + 5q^{88} - 18q^{89} - 6q^{91} + 9q^{92} - 10q^{94} + q^{95} + 6q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.809017 − 0.587785i
−0.690983 + 2.12663i 0 −2.42705 1.76336i 0.190983 + 0.587785i 0 −0.809017 0.587785i 1.80902 1.31433i 0 −1.38197
190.1 −1.80902 + 1.31433i 0 0.927051 2.85317i 1.30902 + 0.951057i 0 0.309017 0.951057i 0.690983 + 2.12663i 0 −3.61803
379.1 −0.690983 2.12663i 0 −2.42705 + 1.76336i 0.190983 0.587785i 0 −0.809017 + 0.587785i 1.80902 + 1.31433i 0 −1.38197
631.1 −1.80902 1.31433i 0 0.927051 + 2.85317i 1.30902 0.951057i 0 0.309017 + 0.951057i 0.690983 2.12663i 0 −3.61803
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.m.a 4
3.b odd 2 1 231.2.j.e 4
11.c even 5 1 inner 693.2.m.a 4
11.c even 5 1 7623.2.a.bk 2
11.d odd 10 1 7623.2.a.bj 2
33.f even 10 1 2541.2.a.v 2
33.h odd 10 1 231.2.j.e 4
33.h odd 10 1 2541.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.e 4 3.b odd 2 1
231.2.j.e 4 33.h odd 10 1
693.2.m.a 4 1.a even 1 1 trivial
693.2.m.a 4 11.c even 5 1 inner
2541.2.a.v 2 33.f even 10 1
2541.2.a.w 2 33.h odd 10 1
7623.2.a.bj 2 11.d odd 10 1
7623.2.a.bk 2 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5 T_{2}^{3} + 15 T_{2}^{2} + 25 T_{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(693, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8}$$
$3$ 
$5$ $$1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 + 6 T + 19 T^{2} + 132 T^{3} + 829 T^{4} + 2244 T^{5} + 5491 T^{6} + 29478 T^{7} + 83521 T^{8}$$
$19$ $$1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 2033 T^{5} + 5415 T^{6} + 48013 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 11 T + 75 T^{2} + 253 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 15 T + 69 T^{2} + 95 T^{3} + 36 T^{4} + 2945 T^{5} + 66309 T^{6} + 446865 T^{7} + 923521 T^{8}$$
$37$ $$1 + 6 T - 21 T^{2} - 248 T^{3} - 471 T^{4} - 9176 T^{5} - 28749 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 6 T + 35 T^{2} - 84 T^{3} - 371 T^{4} - 3444 T^{5} + 58835 T^{6} - 413526 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - 6 T + 50 T^{2} - 258 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 10 T + 13 T^{2} - 200 T^{3} + 3549 T^{4} - 9400 T^{5} + 28717 T^{6} - 1038230 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 14 T + 43 T^{2} + 650 T^{3} - 8799 T^{4} + 34450 T^{5} + 120787 T^{6} - 2084278 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 6 T - 43 T^{2} + 102 T^{3} + 3025 T^{4} + 6018 T^{5} - 149683 T^{6} - 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 67 T^{2} )^{4}$$
$71$ $$1 - 24 T + 185 T^{2} - 456 T^{3} + 49 T^{4} - 32376 T^{5} + 932585 T^{6} - 8589864 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 22 T + 111 T^{2} - 1054 T^{3} - 17431 T^{4} - 76942 T^{5} + 591519 T^{6} + 8558374 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 2 T + 121 T^{2} - 736 T^{3} + 7989 T^{4} - 61088 T^{5} + 833569 T^{6} - 1143574 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 9 T + 197 T^{2} + 801 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 6 T - 61 T^{2} + 948 T^{3} + 229 T^{4} + 91956 T^{5} - 573949 T^{6} - 5476038 T^{7} + 88529281 T^{8}$$