Properties

Label 1503.1.f.b
Level $1503$
Weight $1$
Character orbit 1503.f
Analytic conductor $0.750$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -167
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1503 = 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1503.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.750094713987\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{2} + \zeta_{66}^{14} q^{3} + ( \zeta_{66}^{18} + \zeta_{66}^{22} + \zeta_{66}^{26} ) q^{4} + ( -\zeta_{66}^{23} - \zeta_{66}^{27} ) q^{6} + ( -\zeta_{66} - \zeta_{66}^{21} ) q^{7} + ( \zeta_{66}^{2} + \zeta_{66}^{6} - \zeta_{66}^{27} - \zeta_{66}^{31} ) q^{8} + \zeta_{66}^{28} q^{9} +O(q^{10})\) \( q + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{2} + \zeta_{66}^{14} q^{3} + ( \zeta_{66}^{18} + \zeta_{66}^{22} + \zeta_{66}^{26} ) q^{4} + ( -\zeta_{66}^{23} - \zeta_{66}^{27} ) q^{6} + ( -\zeta_{66} - \zeta_{66}^{21} ) q^{7} + ( \zeta_{66}^{2} + \zeta_{66}^{6} - \zeta_{66}^{27} - \zeta_{66}^{31} ) q^{8} + \zeta_{66}^{28} q^{9} + ( \zeta_{66}^{4} + \zeta_{66}^{18} ) q^{11} + ( -\zeta_{66}^{3} - \zeta_{66}^{7} + \zeta_{66}^{32} ) q^{12} + ( -\zeta_{66} + \zeta_{66}^{10} + \zeta_{66}^{14} + \zeta_{66}^{30} ) q^{14} + ( -\zeta_{66}^{3} - \zeta_{66}^{7} - \zeta_{66}^{11} - \zeta_{66}^{15} - \zeta_{66}^{19} ) q^{16} + ( \zeta_{66}^{4} + \zeta_{66}^{8} ) q^{18} + ( \zeta_{66}^{16} - \zeta_{66}^{17} ) q^{19} + ( \zeta_{66}^{2} - \zeta_{66}^{15} ) q^{21} + ( -\zeta_{66}^{13} - \zeta_{66}^{17} - \zeta_{66}^{27} - \zeta_{66}^{31} ) q^{22} + ( \zeta_{66}^{8} + \zeta_{66}^{12} + \zeta_{66}^{16} + \zeta_{66}^{20} ) q^{24} -\zeta_{66}^{11} q^{25} -\zeta_{66}^{9} q^{27} + ( \zeta_{66}^{6} + \zeta_{66}^{10} + \zeta_{66}^{14} - \zeta_{66}^{19} - \zeta_{66}^{23} - \zeta_{66}^{27} ) q^{28} + ( \zeta_{66}^{10} + \zeta_{66}^{12} ) q^{29} + ( \zeta_{66}^{20} + \zeta_{66}^{24} ) q^{31} + ( \zeta_{66}^{12} + \zeta_{66}^{16} + \zeta_{66}^{20} + \zeta_{66}^{24} + \zeta_{66}^{28} + \zeta_{66}^{32} ) q^{32} + ( \zeta_{66}^{18} + \zeta_{66}^{32} ) q^{33} + ( -\zeta_{66}^{13} - \zeta_{66}^{17} - \zeta_{66}^{21} ) q^{36} + ( -\zeta_{66}^{25} + \zeta_{66}^{26} - \zeta_{66}^{29} + \zeta_{66}^{30} ) q^{38} + ( -\zeta_{66}^{11} - \zeta_{66}^{15} + \zeta_{66}^{24} + \zeta_{66}^{28} ) q^{42} + ( -\zeta_{66}^{3} - \zeta_{66}^{7} - \zeta_{66}^{11} + \zeta_{66}^{22} + \zeta_{66}^{26} + \zeta_{66}^{30} ) q^{44} + ( \zeta_{66}^{6} + \zeta_{66}^{16} ) q^{47} + ( 1 - \zeta_{66}^{17} - \zeta_{66}^{21} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{48} + ( \zeta_{66}^{2} - \zeta_{66}^{9} + \zeta_{66}^{22} ) q^{49} + ( \zeta_{66}^{20} + \zeta_{66}^{24} ) q^{50} + ( \zeta_{66}^{18} + \zeta_{66}^{22} ) q^{54} + ( -\zeta_{66}^{3} - \zeta_{66}^{7} - \zeta_{66}^{15} - \zeta_{66}^{19} - \zeta_{66}^{23} - \zeta_{66}^{27} + \zeta_{66}^{28} + \zeta_{66}^{32} ) q^{56} + ( \zeta_{66}^{30} - \zeta_{66}^{31} ) q^{57} + ( -\zeta_{66}^{19} - \zeta_{66}^{21} - \zeta_{66}^{23} - \zeta_{66}^{25} ) q^{58} + ( \zeta_{66}^{8} + \zeta_{66}^{14} ) q^{61} + ( 2 + \zeta_{66}^{4} - \zeta_{66}^{29} ) q^{62} + ( \zeta_{66}^{16} - \zeta_{66}^{29} ) q^{63} + ( 1 + \zeta_{66}^{4} + \zeta_{66}^{8} + \zeta_{66}^{12} - \zeta_{66}^{21} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{64} + ( \zeta_{66}^{8} + \zeta_{66}^{12} - \zeta_{66}^{27} - \zeta_{66}^{31} ) q^{66} + ( -\zeta_{66} + \zeta_{66}^{22} + \zeta_{66}^{26} + \zeta_{66}^{30} ) q^{72} -\zeta_{66}^{25} q^{75} + ( -\zeta_{66} + \zeta_{66}^{2} - \zeta_{66}^{5} + \zeta_{66}^{6} - \zeta_{66}^{9} + \zeta_{66}^{10} ) q^{76} + ( -\zeta_{66}^{5} + \zeta_{66}^{6} - \zeta_{66}^{19} - \zeta_{66}^{25} ) q^{77} -\zeta_{66}^{23} q^{81} + ( 1 + \zeta_{66}^{4} + \zeta_{66}^{8} + \zeta_{66}^{20} + \zeta_{66}^{24} + \zeta_{66}^{28} ) q^{84} + ( \zeta_{66}^{24} + \zeta_{66}^{26} ) q^{87} + ( \zeta_{66}^{2} + \zeta_{66}^{6} + \zeta_{66}^{10} + \zeta_{66}^{12} + \zeta_{66}^{16} + \zeta_{66}^{20} + \zeta_{66}^{24} - \zeta_{66}^{31} ) q^{88} + ( \zeta_{66}^{14} - \zeta_{66}^{19} ) q^{89} + ( -\zeta_{66} - \zeta_{66}^{5} ) q^{93} + ( -\zeta_{66}^{15} - \zeta_{66}^{19} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{94} + ( -\zeta_{66} - \zeta_{66}^{5} - \zeta_{66}^{9} - \zeta_{66}^{13} + \zeta_{66}^{26} + \zeta_{66}^{30} ) q^{96} + ( \zeta_{66}^{26} - \zeta_{66}^{29} ) q^{97} + ( \zeta_{66}^{2} - \zeta_{66}^{11} - \zeta_{66}^{15} + \zeta_{66}^{18} + \zeta_{66}^{22} - \zeta_{66}^{31} ) q^{98} + ( -\zeta_{66}^{13} + \zeta_{66}^{32} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - q^{2} + q^{3} - 11q^{4} - q^{6} - q^{7} - 2q^{8} + q^{9} + O(q^{10}) \) \( 20q - q^{2} + q^{3} - 11q^{4} - q^{6} - q^{7} - 2q^{8} + q^{9} - q^{11} + q^{14} - 12q^{16} + 2q^{18} + 2q^{19} - q^{21} + q^{22} + q^{24} - 10q^{25} - 2q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13q^{42} - 22q^{44} - q^{47} + 21q^{48} - 11q^{49} - q^{50} - 12q^{54} - q^{56} - q^{57} + q^{58} + 2q^{61} + 42q^{62} + 2q^{63} + 20q^{64} - 2q^{66} - 10q^{72} + q^{75} + q^{77} + q^{81} + 22q^{84} - q^{87} - q^{88} + 2q^{89} + 2q^{93} + q^{94} + 2q^{97} - 22q^{98} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(335\)
\(\chi(n)\) \(-1\) \(\zeta_{66}^{22}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1
−0.995472 0.0950560i
0.981929 0.189251i
0.928368 + 0.371662i
−0.888835 + 0.458227i
−0.786053 0.618159i
0.723734 0.690079i
0.580057 + 0.814576i
−0.327068 0.945001i
0.235759 0.971812i
0.0475819 + 0.998867i
−0.995472 + 0.0950560i
0.981929 + 0.189251i
0.928368 0.371662i
−0.888835 0.458227i
−0.786053 + 0.618159i
0.723734 + 0.690079i
0.580057 0.814576i
−0.327068 + 0.945001i
0.235759 + 0.971812i
0.0475819 0.998867i
−0.981929 1.70075i 0.235759 + 0.971812i −1.42837 + 2.47401i 0 1.42131 1.35522i −0.580057 1.00469i 3.64636 −0.888835 + 0.458227i 0
166.2 −0.928368 1.60798i −0.888835 0.458227i −1.22373 + 2.11957i 0 0.0883470 + 1.85463i 0.327068 + 0.566498i 2.68757 0.580057 + 0.814576i 0
166.3 −0.723734 1.25354i 0.580057 0.814576i −0.547582 + 0.948440i 0 −1.44091 0.137591i 0.786053 + 1.36148i 0.137747 −0.327068 0.945001i 0
166.4 −0.580057 1.00469i 0.928368 0.371662i −0.172932 + 0.299527i 0 −0.911911 0.717135i −0.0475819 0.0824143i −0.758872 0.723734 0.690079i 0
166.5 −0.235759 0.408346i −0.995472 + 0.0950560i 0.388835 0.673483i 0 0.273507 + 0.384087i −0.928368 1.60798i −0.838204 0.981929 0.189251i 0
166.6 −0.0475819 0.0824143i −0.327068 + 0.945001i 0.495472 0.858183i 0 0.0934441 0.0180099i −0.235759 0.408346i −0.189466 −0.786053 0.618159i 0
166.7 0.327068 + 0.566498i 0.723734 + 0.690079i 0.286053 0.495458i 0 −0.154218 + 0.635697i 0.995472 + 1.72421i 1.02837 0.0475819 + 0.998867i 0
166.8 0.786053 + 1.36148i 0.0475819 0.998867i −0.735759 + 1.27437i 0 1.39734 0.720381i −0.981929 1.70075i −0.741276 −0.995472 0.0950560i 0
166.9 0.888835 + 1.53951i 0.981929 + 0.189251i −1.08006 + 1.87071i 0 0.581419 + 1.67990i −0.723734 1.25354i −2.06230 0.928368 + 0.371662i 0
166.10 0.995472 + 1.72421i −0.786053 + 0.618159i −1.48193 + 2.56678i 0 −1.84833 0.739959i 0.888835 + 1.53951i −3.90993 0.235759 0.971812i 0
1168.1 −0.981929 + 1.70075i 0.235759 0.971812i −1.42837 2.47401i 0 1.42131 + 1.35522i −0.580057 + 1.00469i 3.64636 −0.888835 0.458227i 0
1168.2 −0.928368 + 1.60798i −0.888835 + 0.458227i −1.22373 2.11957i 0 0.0883470 1.85463i 0.327068 0.566498i 2.68757 0.580057 0.814576i 0
1168.3 −0.723734 + 1.25354i 0.580057 + 0.814576i −0.547582 0.948440i 0 −1.44091 + 0.137591i 0.786053 1.36148i 0.137747 −0.327068 + 0.945001i 0
1168.4 −0.580057 + 1.00469i 0.928368 + 0.371662i −0.172932 0.299527i 0 −0.911911 + 0.717135i −0.0475819 + 0.0824143i −0.758872 0.723734 + 0.690079i 0
1168.5 −0.235759 + 0.408346i −0.995472 0.0950560i 0.388835 + 0.673483i 0 0.273507 0.384087i −0.928368 + 1.60798i −0.838204 0.981929 + 0.189251i 0
1168.6 −0.0475819 + 0.0824143i −0.327068 0.945001i 0.495472 + 0.858183i 0 0.0934441 + 0.0180099i −0.235759 + 0.408346i −0.189466 −0.786053 + 0.618159i 0
1168.7 0.327068 0.566498i 0.723734 0.690079i 0.286053 + 0.495458i 0 −0.154218 0.635697i 0.995472 1.72421i 1.02837 0.0475819 0.998867i 0
1168.8 0.786053 1.36148i 0.0475819 + 0.998867i −0.735759 1.27437i 0 1.39734 + 0.720381i −0.981929 + 1.70075i −0.741276 −0.995472 + 0.0950560i 0
1168.9 0.888835 1.53951i 0.981929 0.189251i −1.08006 1.87071i 0 0.581419 1.67990i −0.723734 + 1.25354i −2.06230 0.928368 0.371662i 0
1168.10 0.995472 1.72421i −0.786053 0.618159i −1.48193 2.56678i 0 −1.84833 + 0.739959i 0.888835 1.53951i −3.90993 0.235759 + 0.971812i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1168.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 CM by \(\Q(\sqrt{-167}) \)
9.c even 3 1 inner
1503.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1503.1.f.b 20
9.c even 3 1 inner 1503.1.f.b 20
167.b odd 2 1 CM 1503.1.f.b 20
1503.f odd 6 1 inner 1503.1.f.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1503.1.f.b 20 1.a even 1 1 trivial
1503.1.f.b 20 9.c even 3 1 inner
1503.1.f.b 20 167.b odd 2 1 CM
1503.1.f.b 20 1503.f odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(1503, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$3$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
$5$ \( T^{20} \)
$7$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$11$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( ( 1 - 12 T + 12 T^{2} + 43 T^{3} - 43 T^{4} - 34 T^{5} + 34 T^{6} + 10 T^{7} - 10 T^{8} - T^{9} + T^{10} )^{2} \)
$23$ \( T^{20} \)
$29$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$31$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$37$ \( T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$53$ \( T^{20} \)
$59$ \( T^{20} \)
$61$ \( ( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2} \)
$67$ \( T^{20} \)
$71$ \( T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( ( 1 - 12 T + 12 T^{2} + 43 T^{3} - 43 T^{4} - 34 T^{5} + 34 T^{6} + 10 T^{7} - 10 T^{8} - T^{9} + T^{10} )^{2} \)
$97$ \( ( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2} \)
show more
show less