Properties

Label 1012.1.r.a
Level $1012$
Weight $1$
Character orbit 1012.r
Analytic conductor $0.505$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1012 = 2^{2} \cdot 11 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1012.r (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.505053792785\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
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: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{66}^{2} - \zeta_{66}^{19} ) q^{3} + ( -\zeta_{66}^{25} + \zeta_{66}^{32} ) q^{5} + ( \zeta_{66}^{4} - \zeta_{66}^{5} - \zeta_{66}^{21} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{66}^{2} - \zeta_{66}^{19} ) q^{3} + ( -\zeta_{66}^{25} + \zeta_{66}^{32} ) q^{5} + ( \zeta_{66}^{4} - \zeta_{66}^{5} - \zeta_{66}^{21} ) q^{9} -\zeta_{66}^{9} q^{11} + ( -\zeta_{66} - \zeta_{66}^{11} + \zeta_{66}^{18} - \zeta_{66}^{27} ) q^{15} + \zeta_{66}^{20} q^{23} + ( -\zeta_{66}^{17} + \zeta_{66}^{24} - \zeta_{66}^{31} ) q^{25} + ( \zeta_{66}^{6} - \zeta_{66}^{7} - \zeta_{66}^{23} + \zeta_{66}^{24} ) q^{27} + ( \zeta_{66}^{16} - \zeta_{66}^{29} ) q^{31} + ( -\zeta_{66}^{11} + \zeta_{66}^{28} ) q^{33} + ( \zeta_{66}^{14} + \zeta_{66}^{28} ) q^{37} + ( -\zeta_{66}^{3} + \zeta_{66}^{4} - \zeta_{66}^{13} + \zeta_{66}^{20} - \zeta_{66}^{29} + \zeta_{66}^{30} ) q^{45} + ( \zeta_{66}^{12} - \zeta_{66}^{21} ) q^{47} -\zeta_{66}^{27} q^{49} + ( \zeta_{66}^{12} - \zeta_{66}^{15} ) q^{53} + ( -\zeta_{66} + \zeta_{66}^{8} ) q^{55} + ( -\zeta_{66}^{7} + \zeta_{66}^{32} ) q^{59} + ( -\zeta_{66}^{5} + \zeta_{66}^{10} ) q^{67} + ( \zeta_{66}^{6} + \zeta_{66}^{22} ) q^{69} + ( \zeta_{66}^{22} + \zeta_{66}^{26} ) q^{71} + ( 1 - \zeta_{66}^{3} + \zeta_{66}^{10} - \zeta_{66}^{17} - \zeta_{66}^{19} + \zeta_{66}^{26} ) q^{75} + ( \zeta_{66}^{8} - \zeta_{66}^{9} + \zeta_{66}^{10} - \zeta_{66}^{25} + \zeta_{66}^{26} ) q^{81} + ( \zeta_{66}^{8} - \zeta_{66}^{13} ) q^{89} + ( \zeta_{66}^{2} - \zeta_{66}^{15} + \zeta_{66}^{18} - \zeta_{66}^{31} ) q^{93} + ( \zeta_{66}^{2} + \zeta_{66}^{22} ) q^{97} + ( -\zeta_{66}^{13} + \zeta_{66}^{14} + \zeta_{66}^{30} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{3} + 2q^{5} + O(q^{10}) \) \( 20q + 2q^{3} + 2q^{5} - 2q^{11} - 13q^{15} + q^{23} - 2q^{27} + 2q^{31} - 9q^{33} + 2q^{37} - 4q^{47} - 2q^{49} - 4q^{53} + 2q^{55} + 2q^{59} + 2q^{67} - 12q^{69} - 9q^{71} + 22q^{75} + 2q^{81} + 2q^{89} - 2q^{93} - 9q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1012\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(507\) \(925\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{66}^{24}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0.580057 + 0.814576i
−0.995472 + 0.0950560i
−0.786053 + 0.618159i
0.928368 + 0.371662i
−0.327068 0.945001i
0.981929 + 0.189251i
−0.786053 0.618159i
0.928368 0.371662i
0.580057 0.814576i
−0.995472 0.0950560i
0.723734 + 0.690079i
0.235759 0.971812i
−0.327068 + 0.945001i
0.981929 0.189251i
0.0475819 + 0.998867i
−0.888835 0.458227i
0.0475819 0.998867i
−0.888835 + 0.458227i
0.723734 0.690079i
0.235759 + 0.971812i
0 0.396666 + 0.254922i 0 0.815816 1.78639i 0 0 0 −0.323056 0.707394i 0
197.2 0 1.21769 + 0.782560i 0 −0.271738 + 0.595023i 0 0 0 0.454947 + 0.996196i 0
285.1 0 −0.759713 0.876756i 0 −0.205996 1.43273i 0 0 0 −0.0492216 + 0.342344i 0
285.2 0 1.30379 + 1.50465i 0 −0.0671040 0.466718i 0 0 0 −0.421801 + 2.93369i 0
417.1 0 −0.738471 + 1.61703i 0 −1.21590 + 1.40323i 0 0 0 −1.41457 1.63251i 0
417.2 0 0.0395325 0.0865641i 0 1.02951 1.18812i 0 0 0 0.648930 + 0.748905i 0
593.1 0 −0.759713 + 0.876756i 0 −0.205996 + 1.43273i 0 0 0 −0.0492216 0.342344i 0
593.2 0 1.30379 1.50465i 0 −0.0671040 + 0.466718i 0 0 0 −0.421801 2.93369i 0
637.1 0 0.396666 0.254922i 0 0.815816 + 1.78639i 0 0 0 −0.323056 + 0.707394i 0
637.2 0 1.21769 0.782560i 0 −0.271738 0.595023i 0 0 0 0.454947 0.996196i 0
725.1 0 −0.279486 + 1.94387i 0 1.70566 0.500828i 0 0 0 −2.74102 0.804835i 0
725.2 0 0.0930932 0.647478i 0 −0.0913090 + 0.0268107i 0 0 0 0.548932 + 0.161181i 0
813.1 0 −0.738471 1.61703i 0 −1.21590 1.40323i 0 0 0 −1.41457 + 1.63251i 0
813.2 0 0.0395325 + 0.0865641i 0 1.02951 + 1.18812i 0 0 0 0.648930 0.748905i 0
857.1 0 −1.78153 0.523103i 0 0.975950 0.627205i 0 0 0 2.05894 + 1.32320i 0
857.2 0 1.50842 + 0.442913i 0 −1.67489 + 1.07639i 0 0 0 1.23792 + 0.795563i 0
901.1 0 −1.78153 + 0.523103i 0 0.975950 + 0.627205i 0 0 0 2.05894 1.32320i 0
901.2 0 1.50842 0.442913i 0 −1.67489 1.07639i 0 0 0 1.23792 0.795563i 0
945.1 0 −0.279486 1.94387i 0 1.70566 + 0.500828i 0 0 0 −2.74102 + 0.804835i 0
945.2 0 0.0930932 + 0.647478i 0 −0.0913090 0.0268107i 0 0 0 0.548932 0.161181i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 945.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
23.c even 11 1 inner
253.k odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1012.1.r.a 20
11.b odd 2 1 CM 1012.1.r.a 20
23.c even 11 1 inner 1012.1.r.a 20
253.k odd 22 1 inner 1012.1.r.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1012.1.r.a 20 1.a even 1 1 trivial
1012.1.r.a 20 11.b odd 2 1 CM
1012.1.r.a 20 23.c even 11 1 inner
1012.1.r.a 20 253.k odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1012, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$5$ \( 1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
$29$ \( T^{20} \)
$31$ \( 1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$37$ \( 1 + 9 T + 146 T^{2} + 744 T^{3} + 2051 T^{4} + 3151 T^{5} + 2658 T^{6} + 971 T^{7} + 119 T^{8} + T^{9} + T^{11} + 9 T^{12} + 25 T^{13} + 18 T^{14} - 6 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( ( 1 + 3 T - 3 T^{2} - 4 T^{3} + T^{4} + T^{5} )^{4} \)
$53$ \( ( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$59$ \( 1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( 1 + 9 T + 146 T^{2} + 744 T^{3} + 2051 T^{4} + 3151 T^{5} + 2658 T^{6} + 971 T^{7} + 119 T^{8} + T^{9} + T^{11} + 9 T^{12} + 25 T^{13} + 18 T^{14} - 6 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$71$ \( 1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( 1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$97$ \( 1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20} \)
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