# Properties

 Label 693.2.m.c 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{3} q^{4} + ( -3 + 2 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{5} -\zeta_{10}^{3} q^{7} -3 \zeta_{10}^{2} q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{3} q^{4} + ( -3 + 2 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{5} -\zeta_{10}^{3} q^{7} -3 \zeta_{10}^{2} q^{8} + ( 1 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{10} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{11} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{13} + \zeta_{10}^{2} q^{14} + \zeta_{10} q^{16} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{17} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{19} + ( 1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{20} + ( 2 - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{22} + ( 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{23} + ( -3 \zeta_{10} + 8 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{25} + ( -2 + 2 \zeta_{10} ) q^{26} + \zeta_{10} q^{28} -2 \zeta_{10}^{3} q^{29} + ( 3 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{31} + 5 q^{32} + ( -6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{34} + ( -1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{35} + ( -5 + 5 \zeta_{10} + 9 \zeta_{10}^{3} ) q^{37} + ( -1 - \zeta_{10} - \zeta_{10}^{2} ) q^{38} + ( -9 + 9 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{40} + ( 3 \zeta_{10} + 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{41} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{43} + ( 2 - 4 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{44} + ( -5 \zeta_{10} + 5 \zeta_{10}^{2} ) q^{46} -2 \zeta_{10}^{2} q^{47} -\zeta_{10} q^{49} + ( 3 - 8 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{50} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{52} + ( 6 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} + ( 3 + 10 \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{55} -3 q^{56} + 2 \zeta_{10}^{2} q^{58} + ( 2 - 2 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{59} + ( -8 + 4 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{61} + ( -7 + 7 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{62} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} + ( -6 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{65} + 8 q^{67} + ( -6 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{68} + ( 3 - 3 \zeta_{10} - \zeta_{10}^{3} ) q^{70} + ( 8 - 4 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{71} + ( -2 + 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{73} + ( -5 \zeta_{10} - 4 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{74} + ( -2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{76} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{77} + ( 14 - 14 \zeta_{10} + 14 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{79} + ( -3 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{80} + ( -3 - 6 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{82} + ( 8 - 10 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{83} + ( -12 \zeta_{10} - 3 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{85} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{86} + ( 6 + 3 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{88} + ( 2 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{89} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{91} + ( -5 + 5 \zeta_{10} ) q^{92} + 2 \zeta_{10} q^{94} + ( 4 - 4 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{95} + ( 6 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} + q^{4} - 7q^{5} - q^{7} + 3q^{8} + O(q^{10})$$ $$4q - q^{2} + q^{4} - 7q^{5} - q^{7} + 3q^{8} - 2q^{10} - q^{11} + 4q^{13} - q^{14} + q^{16} + 12q^{17} + q^{19} + 7q^{20} + 9q^{22} - 10q^{23} - 14q^{25} - 6q^{26} + q^{28} - 2q^{29} + 17q^{31} + 20q^{32} - 18q^{34} - 7q^{35} - 6q^{37} - 4q^{38} - 24q^{40} + 4q^{43} + q^{44} - 10q^{46} + 2q^{47} - q^{49} + q^{50} + 6q^{52} + 26q^{53} + 28q^{55} - 12q^{56} - 2q^{58} + 14q^{59} - 20q^{61} - 18q^{62} - 7q^{64} - 32q^{65} + 32q^{67} - 12q^{68} + 8q^{70} + 20q^{71} - 2q^{73} - 6q^{74} - 6q^{76} - q^{77} + 14q^{79} - 8q^{80} - 15q^{82} + 14q^{83} - 21q^{85} + 4q^{86} + 33q^{88} + 18q^{89} - 6q^{91} - 15q^{92} + 2q^{94} + 7q^{95} - 2q^{97} + 4q^{98} + O(q^{100})$$

## 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
0.309017 0.951057i 0 0.809017 + 0.587785i −1.19098 3.66547i 0 −0.809017 0.587785i 2.42705 1.76336i 0 −3.85410
190.1 −0.809017 + 0.587785i 0 −0.309017 + 0.951057i −2.30902 1.67760i 0 0.309017 0.951057i −0.927051 2.85317i 0 2.85410
379.1 0.309017 + 0.951057i 0 0.809017 0.587785i −1.19098 + 3.66547i 0 −0.809017 + 0.587785i 2.42705 + 1.76336i 0 −3.85410
631.1 −0.809017 0.587785i 0 −0.309017 0.951057i −2.30902 + 1.67760i 0 0.309017 + 0.951057i −0.927051 + 2.85317i 0 2.85410
 $$n$$: 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.c 4
3.b odd 2 1 231.2.j.c 4
11.c even 5 1 inner 693.2.m.c 4
11.c even 5 1 7623.2.a.bu 2
11.d odd 10 1 7623.2.a.w 2
33.f even 10 1 2541.2.a.bd 2
33.h odd 10 1 231.2.j.c 4
33.h odd 10 1 2541.2.a.n 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(693, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8}$$
$3$ 
$5$ $$1 + 7 T + 29 T^{2} + 93 T^{3} + 236 T^{4} + 465 T^{5} + 725 T^{6} + 875 T^{7} + 625 T^{8}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$1 + T - 9 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 3060 T^{5} + 10693 T^{6} - 58956 T^{7} + 83521 T^{8}$$
$19$ $$1 - T - 13 T^{2} - 53 T^{3} + 400 T^{4} - 1007 T^{5} - 4693 T^{6} - 6859 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 5 T + 21 T^{2} + 115 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8}$$
$31$ $$1 - 17 T + 153 T^{2} - 1129 T^{3} + 7100 T^{4} - 34999 T^{5} + 147033 T^{6} - 506447 T^{7} + 923521 T^{8}$$
$37$ $$1 + 6 T + 99 T^{2} + 272 T^{3} + 3969 T^{4} + 10064 T^{5} + 135531 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 49 T^{2} + 60 T^{3} + 1861 T^{4} + 2460 T^{5} + 82369 T^{6} + 2825761 T^{8}$$
$43$ $$( 1 - 2 T + 82 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 2 T - 43 T^{2} + 180 T^{3} + 1661 T^{4} + 8460 T^{5} - 94987 T^{6} - 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 26 T + 323 T^{2} - 2870 T^{3} + 22001 T^{4} - 152110 T^{5} + 907307 T^{6} - 3870802 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 14 T + 37 T^{2} + 758 T^{3} - 10095 T^{4} + 44722 T^{5} + 128797 T^{6} - 2875306 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 97600 T^{5} + 666059 T^{6} + 4539620 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{4}$$
$71$ $$1 - 20 T + 169 T^{2} - 1600 T^{3} + 17121 T^{4} - 113600 T^{5} + 851929 T^{6} - 7158220 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 2 T - 49 T^{2} + 406 T^{3} + 5929 T^{4} + 29638 T^{5} - 261121 T^{6} + 778034 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 14 T + 117 T^{2} - 532 T^{3} - 1795 T^{4} - 42028 T^{5} + 730197 T^{6} - 6902546 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 14 T + 193 T^{2} - 2140 T^{3} + 26421 T^{4} - 177620 T^{5} + 1329577 T^{6} - 8005018 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 9 T + 167 T^{2} - 801 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 2 T + 27 T^{2} + 760 T^{3} + 10181 T^{4} + 73720 T^{5} + 254043 T^{6} + 1825346 T^{7} + 88529281 T^{8}$$