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 $4$

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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:

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

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