# Properties

 Label 597.1.s.a Level $597$ Weight $1$ Character orbit 597.s Analytic conductor $0.298$ Analytic rank $0$ Dimension $20$ Projective image $D_{33}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$597 = 3 \cdot 199$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 597.s (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.297941812542$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{33})$$ Defining polynomial: $$x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{33}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{33} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{66}^{7} q^{3} + \zeta_{66}^{32} q^{4} + ( \zeta_{66}^{12} - \zeta_{66}^{29} ) q^{7} + \zeta_{66}^{14} q^{9} +O(q^{10})$$ $$q -\zeta_{66}^{7} q^{3} + \zeta_{66}^{32} q^{4} + ( \zeta_{66}^{12} - \zeta_{66}^{29} ) q^{7} + \zeta_{66}^{14} q^{9} + \zeta_{66}^{6} q^{12} + ( -\zeta_{66}^{9} - \zeta_{66}^{23} ) q^{13} -\zeta_{66}^{31} q^{16} + ( -\zeta_{66}^{17} - \zeta_{66}^{27} ) q^{19} + ( -\zeta_{66}^{3} - \zeta_{66}^{19} ) q^{21} + \zeta_{66}^{18} q^{25} -\zeta_{66}^{21} q^{27} + ( -\zeta_{66}^{11} + \zeta_{66}^{28} ) q^{28} -\zeta_{66}^{26} q^{31} -\zeta_{66}^{13} q^{36} + ( \zeta_{66}^{2} + \zeta_{66}^{20} ) q^{37} + ( \zeta_{66}^{16} + \zeta_{66}^{30} ) q^{39} + ( -\zeta_{66} + \zeta_{66}^{10} ) q^{43} -\zeta_{66}^{5} q^{48} + ( \zeta_{66}^{8} + \zeta_{66}^{24} - \zeta_{66}^{25} ) q^{49} + ( \zeta_{66}^{8} + \zeta_{66}^{22} ) q^{52} + ( -\zeta_{66} + \zeta_{66}^{24} ) q^{57} + ( \zeta_{66}^{22} + \zeta_{66}^{26} ) q^{61} + ( \zeta_{66}^{10} + \zeta_{66}^{26} ) q^{63} + \zeta_{66}^{30} q^{64} + ( -\zeta_{66}^{5} - \zeta_{66}^{13} ) q^{67} + ( -\zeta_{66}^{15} + \zeta_{66}^{20} ) q^{73} -\zeta_{66}^{25} q^{75} + ( \zeta_{66}^{16} + \zeta_{66}^{26} ) q^{76} + ( \zeta_{66}^{4} - \zeta_{66}^{25} ) q^{79} + \zeta_{66}^{28} q^{81} + ( \zeta_{66}^{2} + \zeta_{66}^{18} ) q^{84} + ( \zeta_{66}^{2} - \zeta_{66}^{5} - \zeta_{66}^{19} - \zeta_{66}^{21} ) q^{91} - q^{93} + ( -\zeta_{66}^{3} + \zeta_{66}^{16} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + q^{3} + q^{4} - q^{7} + q^{9} + O(q^{10})$$ $$20q + q^{3} + q^{4} - q^{7} + q^{9} - 2q^{12} - q^{13} + q^{16} - q^{19} - q^{21} - 2q^{25} - 2q^{27} - 9q^{28} - q^{31} + q^{36} + 2q^{37} - q^{39} + 2q^{43} + q^{48} - 9q^{52} - q^{57} - 9q^{61} + 2q^{63} - 2q^{64} + 2q^{67} - q^{73} + q^{75} + 2q^{76} + 2q^{79} + q^{81} - q^{84} + q^{91} - 20q^{93} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/597\mathbb{Z}\right)^\times$$.

 $$n$$ $$200$$ $$202$$ $$\chi(n)$$ $$-1$$ $$\zeta_{66}^{14}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.723734 + 0.690079i 0.0475819 − 0.998867i 0.235759 + 0.971812i 0.981929 + 0.189251i −0.786053 − 0.618159i 0.580057 + 0.814576i 0.0475819 + 0.998867i −0.995472 + 0.0950560i 0.723734 − 0.690079i −0.888835 + 0.458227i −0.995472 − 0.0950560i −0.327068 − 0.945001i −0.786053 + 0.618159i 0.981929 − 0.189251i −0.327068 + 0.945001i −0.888835 − 0.458227i 0.580057 − 0.814576i 0.928368 + 0.371662i 0.235759 − 0.971812i 0.928368 − 0.371662i
0 0.580057 0.814576i 0.723734 0.690079i 0 0 −1.95496 + 0.186677i 0 −0.327068 0.945001i 0
8.1 0 −0.327068 + 0.945001i 0.0475819 + 0.998867i 0 0 1.82318 + 0.351390i 0 −0.786053 0.618159i 0
98.1 0 −0.995472 + 0.0950560i 0.235759 0.971812i 0 0 −0.379436 + 0.532843i 0 0.981929 0.189251i 0
116.1 0 0.235759 + 0.971812i 0.981929 0.189251i 0 0 0.0688733 + 0.0656706i 0 −0.888835 + 0.458227i 0
140.1 0 0.0475819 + 0.998867i −0.786053 + 0.618159i 0 0 −1.03115 + 0.531595i 0 −0.995472 + 0.0950560i 0
182.1 0 0.928368 + 0.371662i 0.580057 0.814576i 0 0 −0.370638 0.291473i 0 0.723734 + 0.690079i 0
224.1 0 −0.327068 0.945001i 0.0475819 0.998867i 0 0 1.82318 0.351390i 0 −0.786053 + 0.618159i 0
227.1 0 −0.786053 + 0.618159i −0.995472 0.0950560i 0 0 1.34378 0.537970i 0 0.235759 0.971812i 0
239.1 0 0.580057 + 0.814576i 0.723734 + 0.690079i 0 0 −1.95496 0.186677i 0 −0.327068 + 0.945001i 0
251.1 0 0.981929 0.189251i −0.888835 0.458227i 0 0 0.514186 + 1.48564i 0 0.928368 0.371662i 0
263.1 0 −0.786053 0.618159i −0.995472 + 0.0950560i 0 0 1.34378 + 0.537970i 0 0.235759 + 0.971812i 0
356.1 0 0.723734 0.690079i −0.327068 + 0.945001i 0 0 −0.419102 + 1.72756i 0 0.0475819 0.998867i 0
371.1 0 0.0475819 0.998867i −0.786053 0.618159i 0 0 −1.03115 0.531595i 0 −0.995472 0.0950560i 0
386.1 0 0.235759 0.971812i 0.981929 + 0.189251i 0 0 0.0688733 0.0656706i 0 −0.888835 0.458227i 0
488.1 0 0.723734 + 0.690079i −0.327068 0.945001i 0 0 −0.419102 1.72756i 0 0.0475819 + 0.998867i 0
509.1 0 0.981929 + 0.189251i −0.888835 + 0.458227i 0 0 0.514186 1.48564i 0 0.928368 + 0.371662i 0
515.1 0 0.928368 0.371662i 0.580057 + 0.814576i 0 0 −0.370638 + 0.291473i 0 0.723734 0.690079i 0
521.1 0 −0.888835 + 0.458227i 0.928368 0.371662i 0 0 −0.0947329 1.98869i 0 0.580057 0.814576i 0
530.1 0 −0.995472 0.0950560i 0.235759 + 0.971812i 0 0 −0.379436 0.532843i 0 0.981929 + 0.189251i 0
542.1 0 −0.888835 0.458227i 0.928368 + 0.371662i 0 0 −0.0947329 + 1.98869i 0 0.580057 + 0.814576i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 542.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
199.i even 33 1 inner
597.s odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 597.1.s.a 20
3.b odd 2 1 CM 597.1.s.a 20
199.i even 33 1 inner 597.1.s.a 20
597.s odd 66 1 inner 597.1.s.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
597.1.s.a 20 1.a even 1 1 trivial
597.1.s.a 20 3.b odd 2 1 CM
597.1.s.a 20 199.i even 33 1 inner
597.1.s.a 20 597.s odd 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(597, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$1 - 10 T + 44 T^{2} + 395 T^{3} + 1198 T^{4} + 1529 T^{5} + 793 T^{6} - 604 T^{7} - 550 T^{8} - 67 T^{9} - T^{10} + 65 T^{11} + 176 T^{12} + 23 T^{13} + 12 T^{14} - 11 T^{15} - 23 T^{16} - T^{17} + T^{19} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 + 12 T + 154 T^{2} + 626 T^{3} + 934 T^{4} + 253 T^{5} - 560 T^{6} - 87 T^{7} + 605 T^{8} - 155 T^{9} + 87 T^{10} + 241 T^{11} + 11 T^{12} + 89 T^{13} + 78 T^{14} + 10 T^{16} + 10 T^{17} + T^{19} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20}$$
$37$ $$( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2}$$
$41$ $$T^{20}$$
$43$ $$( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20}$$
$67$ $$1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$1 + 12 T + 154 T^{2} + 626 T^{3} + 934 T^{4} + 253 T^{5} - 560 T^{6} - 87 T^{7} + 605 T^{8} - 155 T^{9} + 87 T^{10} + 241 T^{11} + 11 T^{12} + 89 T^{13} + 78 T^{14} + 10 T^{16} + 10 T^{17} + T^{19} + T^{20}$$
$79$ $$1 - 6 T + 22 T^{2} - 70 T^{3} + 145 T^{4} + 22 T^{5} - 17 T^{6} - 360 T^{7} + 209 T^{8} + 46 T^{9} + 318 T^{10} - 170 T^{11} - 22 T^{12} - 139 T^{13} + 75 T^{14} + 28 T^{16} - 14 T^{17} - 2 T^{19} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$1 - 21 T + 121 T^{2} - 100 T^{3} + 274 T^{4} + 220 T^{5} + 793 T^{6} + 1299 T^{7} + 605 T^{8} - 89 T^{9} + 153 T^{10} + 241 T^{11} - 22 T^{12} + 56 T^{13} + 78 T^{14} + 10 T^{16} + 10 T^{17} + T^{19} + T^{20}$$