Properties

Label 1340.1.bl.b
Level $1340$
Weight $1$
Character orbit 1340.bl
Analytic conductor $0.669$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -20
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1340.bl (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.668747116928\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{66}^{31} q^{2} + ( -\zeta_{66}^{17} - \zeta_{66}^{25} ) q^{3} -\zeta_{66}^{29} q^{4} -\zeta_{66}^{3} q^{5} + ( -\zeta_{66}^{15} - \zeta_{66}^{23} ) q^{6} + ( \zeta_{66}^{10} + \zeta_{66}^{30} ) q^{7} -\zeta_{66}^{27} q^{8} + ( -\zeta_{66} - \zeta_{66}^{9} - \zeta_{66}^{17} ) q^{9} +O(q^{10})\) \( q -\zeta_{66}^{31} q^{2} + ( -\zeta_{66}^{17} - \zeta_{66}^{25} ) q^{3} -\zeta_{66}^{29} q^{4} -\zeta_{66}^{3} q^{5} + ( -\zeta_{66}^{15} - \zeta_{66}^{23} ) q^{6} + ( \zeta_{66}^{10} + \zeta_{66}^{30} ) q^{7} -\zeta_{66}^{27} q^{8} + ( -\zeta_{66} - \zeta_{66}^{9} - \zeta_{66}^{17} ) q^{9} -\zeta_{66} q^{10} + ( -\zeta_{66}^{13} - \zeta_{66}^{21} ) q^{12} + ( \zeta_{66}^{8} + \zeta_{66}^{28} ) q^{14} + ( \zeta_{66}^{20} + \zeta_{66}^{28} ) q^{15} -\zeta_{66}^{25} q^{16} + ( -\zeta_{66}^{7} - \zeta_{66}^{15} + \zeta_{66}^{32} ) q^{18} + \zeta_{66}^{32} q^{20} + ( \zeta_{66}^{2} + \zeta_{66}^{14} + \zeta_{66}^{22} - \zeta_{66}^{27} ) q^{21} -\zeta_{66}^{10} q^{23} + ( -\zeta_{66}^{11} - \zeta_{66}^{19} ) q^{24} + \zeta_{66}^{6} q^{25} + ( -\zeta_{66} - \zeta_{66}^{9} + \zeta_{66}^{18} + \zeta_{66}^{26} ) q^{27} + ( \zeta_{66}^{6} + \zeta_{66}^{26} ) q^{28} + ( \zeta_{66}^{10} + \zeta_{66}^{12} ) q^{29} + ( \zeta_{66}^{18} + \zeta_{66}^{26} ) q^{30} -\zeta_{66}^{23} q^{32} + ( 1 - \zeta_{66}^{13} ) q^{35} + ( -\zeta_{66}^{5} - \zeta_{66}^{13} + \zeta_{66}^{30} ) q^{36} + \zeta_{66}^{30} q^{40} + ( -\zeta_{66}^{21} - \zeta_{66}^{31} ) q^{41} + ( 1 + \zeta_{66}^{12} + \zeta_{66}^{20} - \zeta_{66}^{25} ) q^{42} + ( \zeta_{66}^{8} + \zeta_{66}^{22} ) q^{43} + ( \zeta_{66}^{4} + \zeta_{66}^{12} + \zeta_{66}^{20} ) q^{45} -\zeta_{66}^{8} q^{46} + ( -\zeta_{66}^{15} + \zeta_{66}^{16} ) q^{47} + ( -\zeta_{66}^{9} - \zeta_{66}^{17} ) q^{48} + ( -\zeta_{66}^{7} + \zeta_{66}^{20} - \zeta_{66}^{27} ) q^{49} + \zeta_{66}^{4} q^{50} + ( -\zeta_{66}^{7} + \zeta_{66}^{16} + \zeta_{66}^{24} + \zeta_{66}^{32} ) q^{54} + ( \zeta_{66}^{4} + \zeta_{66}^{24} ) q^{56} + ( \zeta_{66}^{8} + \zeta_{66}^{10} ) q^{58} + ( \zeta_{66}^{16} + \zeta_{66}^{24} ) q^{60} + ( \zeta_{66}^{16} + \zeta_{66}^{22} ) q^{61} + ( \zeta_{66}^{6} - \zeta_{66}^{11} + \zeta_{66}^{14} - \zeta_{66}^{19} - \zeta_{66}^{27} - \zeta_{66}^{31} ) q^{63} -\zeta_{66}^{21} q^{64} -\zeta_{66}^{7} q^{67} + ( -\zeta_{66}^{2} + \zeta_{66}^{27} ) q^{69} + ( -\zeta_{66}^{11} - \zeta_{66}^{31} ) q^{70} + ( -\zeta_{66}^{3} - \zeta_{66}^{11} + \zeta_{66}^{28} ) q^{72} + ( -\zeta_{66}^{23} - \zeta_{66}^{31} ) q^{75} + \zeta_{66}^{28} q^{80} + ( -\zeta_{66} + \zeta_{66}^{2} + \zeta_{66}^{10} + \zeta_{66}^{18} + \zeta_{66}^{26} ) q^{81} + ( -\zeta_{66}^{19} - \zeta_{66}^{29} ) q^{82} + ( -\zeta_{66}^{5} + \zeta_{66}^{12} ) q^{83} + ( \zeta_{66}^{10} + \zeta_{66}^{18} - \zeta_{66}^{23} - \zeta_{66}^{31} ) q^{84} + ( \zeta_{66}^{6} + \zeta_{66}^{20} ) q^{86} + ( \zeta_{66}^{2} + \zeta_{66}^{4} - \zeta_{66}^{27} - \zeta_{66}^{29} ) q^{87} + ( -\zeta_{66}^{3} - \zeta_{66}^{21} ) q^{89} + ( \zeta_{66}^{2} + \zeta_{66}^{10} + \zeta_{66}^{18} ) q^{90} -\zeta_{66}^{6} q^{92} + ( -\zeta_{66}^{13} + \zeta_{66}^{14} ) q^{94} + ( -\zeta_{66}^{7} - \zeta_{66}^{15} ) q^{96} + ( -\zeta_{66}^{5} + \zeta_{66}^{18} - \zeta_{66}^{25} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + q^{2} + 2q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} - 2q^{8} + O(q^{10}) \) \( 20q + q^{2} + 2q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} - 2q^{8} + q^{10} - q^{12} + 2q^{14} + 2q^{15} + q^{16} + q^{20} - 10q^{21} - q^{23} - 9q^{24} - 2q^{25} - 2q^{27} - q^{28} - q^{29} - q^{30} + q^{32} + 21q^{35} - 2q^{40} - q^{41} + 20q^{42} - 9q^{43} - q^{46} - q^{47} - q^{48} + q^{50} + q^{54} - q^{56} + 2q^{58} - q^{60} - 9q^{61} - 11q^{63} - 2q^{64} + q^{67} + q^{69} - 9q^{70} - 11q^{72} + 2q^{75} + q^{80} + 2q^{81} + 2q^{82} - q^{83} + q^{84} - q^{86} + q^{87} - 4q^{89} + 2q^{92} + 2q^{94} - q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times\).

\(n\) \(537\) \(671\) \(1141\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{66}^{29}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.235759 + 0.971812i
0.981929 + 0.189251i
0.723734 + 0.690079i
0.0475819 + 0.998867i
0.0475819 0.998867i
0.928368 0.371662i
−0.786053 + 0.618159i
−0.995472 + 0.0950560i
−0.327068 0.945001i
0.580057 + 0.814576i
−0.888835 0.458227i
−0.786053 0.618159i
0.981929 0.189251i
−0.995472 0.0950560i
−0.327068 + 0.945001i
0.928368 + 0.371662i
0.235759 0.971812i
0.723734 0.690079i
0.580057 0.814576i
−0.888835 + 0.458227i
−0.888835 0.458227i −1.11312 + 0.326842i 0.580057 + 0.814576i −0.654861 0.755750i 1.13915 + 0.219553i 0.0688733 + 1.44583i −0.142315 0.989821i 0.290959 0.186988i 0.235759 + 0.971812i
39.1 0.928368 0.371662i −0.947890 1.09392i 0.723734 0.690079i 0.841254 + 0.540641i −1.28656 0.663268i 0.514186 + 0.404360i 0.415415 0.909632i −0.155858 + 1.08402i 0.981929 + 0.189251i
199.1 0.0475819 0.998867i 1.91030 + 0.560914i −0.995472 0.0950560i −0.654861 + 0.755750i 0.651174 1.88144i −0.419102 + 0.216062i −0.142315 + 0.989821i 2.49336 + 1.60238i 0.723734 + 0.690079i
339.1 −0.995472 0.0950560i 1.65210 + 1.06174i 0.981929 + 0.189251i −0.142315 0.989821i −1.54370 1.21398i −1.03115 + 1.44805i −0.959493 0.281733i 1.18673 + 2.59858i 0.0475819 + 0.998867i
419.1 −0.995472 + 0.0950560i 1.65210 1.06174i 0.981929 0.189251i −0.142315 + 0.989821i −1.54370 + 1.21398i −1.03115 1.44805i −0.959493 + 0.281733i 1.18673 2.59858i 0.0475819 0.998867i
479.1 0.723734 + 0.690079i −0.0135432 0.0941952i 0.0475819 + 0.998867i 0.415415 0.909632i 0.0552004 0.0775182i −0.370638 + 1.52779i −0.654861 + 0.755750i 0.950804 0.279181i 0.928368 0.371662i
559.1 0.235759 + 0.971812i 0.252989 1.75958i −0.888835 + 0.458227i 0.415415 + 0.909632i 1.76962 0.168978i 1.34378 1.28129i −0.654861 0.755750i −2.07261 0.608574i −0.786053 + 0.618159i
619.1 0.981929 + 0.189251i 0.771316 + 1.68895i 0.928368 + 0.371662i −0.959493 + 0.281733i 0.437742 + 1.80440i −0.379436 1.09631i 0.841254 + 0.540641i −1.60275 + 1.84967i −0.995472 + 0.0950560i
639.1 −0.786053 0.618159i −0.308779 0.356349i 0.235759 + 0.971812i 0.841254 + 0.540641i 0.0224357 + 0.470984i 1.82318 0.729892i 0.415415 0.909632i 0.110674 0.769755i −0.327068 0.945001i
659.1 −0.327068 0.945001i −0.653077 1.43004i −0.786053 + 0.618159i −0.959493 + 0.281733i −1.13779 + 1.08488i −1.95496 0.376789i 0.841254 + 0.540641i −0.963639 + 1.11210i 0.580057 + 0.814576i
719.1 0.580057 0.814576i −0.550294 0.353653i −0.327068 0.945001i −0.142315 0.989821i −0.607279 + 0.243118i −0.0947329 0.00904590i −0.959493 0.281733i −0.237662 0.520406i −0.888835 0.458227i
839.1 0.235759 0.971812i 0.252989 + 1.75958i −0.888835 0.458227i 0.415415 0.909632i 1.76962 + 0.168978i 1.34378 + 1.28129i −0.654861 + 0.755750i −2.07261 + 0.608574i −0.786053 0.618159i
859.1 0.928368 + 0.371662i −0.947890 + 1.09392i 0.723734 + 0.690079i 0.841254 0.540641i −1.28656 + 0.663268i 0.514186 0.404360i 0.415415 + 0.909632i −0.155858 1.08402i 0.981929 0.189251i
959.1 0.981929 0.189251i 0.771316 1.68895i 0.928368 0.371662i −0.959493 0.281733i 0.437742 1.80440i −0.379436 + 1.09631i 0.841254 0.540641i −1.60275 1.84967i −0.995472 0.0950560i
1059.1 −0.786053 + 0.618159i −0.308779 + 0.356349i 0.235759 0.971812i 0.841254 0.540641i 0.0224357 0.470984i 1.82318 + 0.729892i 0.415415 + 0.909632i 0.110674 + 0.769755i −0.327068 + 0.945001i
1119.1 0.723734 0.690079i −0.0135432 + 0.0941952i 0.0475819 0.998867i 0.415415 + 0.909632i 0.0552004 + 0.0775182i −0.370638 1.52779i −0.654861 0.755750i 0.950804 + 0.279181i 0.928368 + 0.371662i
1199.1 −0.888835 + 0.458227i −1.11312 0.326842i 0.580057 0.814576i −0.654861 + 0.755750i 1.13915 0.219553i 0.0688733 1.44583i −0.142315 + 0.989821i 0.290959 + 0.186988i 0.235759 0.971812i
1239.1 0.0475819 + 0.998867i 1.91030 0.560914i −0.995472 + 0.0950560i −0.654861 0.755750i 0.651174 + 1.88144i −0.419102 0.216062i −0.142315 0.989821i 2.49336 1.60238i 0.723734 0.690079i
1279.1 −0.327068 + 0.945001i −0.653077 + 1.43004i −0.786053 0.618159i −0.959493 0.281733i −1.13779 1.08488i −1.95496 + 0.376789i 0.841254 0.540641i −0.963639 1.11210i 0.580057 0.814576i
1299.1 0.580057 + 0.814576i −0.550294 + 0.353653i −0.327068 + 0.945001i −0.142315 + 0.989821i −0.607279 0.243118i −0.0947329 + 0.00904590i −0.959493 + 0.281733i −0.237662 + 0.520406i −0.888835 + 0.458227i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1299.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
67.g even 33 1 inner
1340.bl odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.1.bl.b yes 20
4.b odd 2 1 1340.1.bl.a 20
5.b even 2 1 1340.1.bl.a 20
20.d odd 2 1 CM 1340.1.bl.b yes 20
67.g even 33 1 inner 1340.1.bl.b yes 20
268.o odd 66 1 1340.1.bl.a 20
335.u even 66 1 1340.1.bl.a 20
1340.bl odd 66 1 inner 1340.1.bl.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.1.bl.a 20 4.b odd 2 1
1340.1.bl.a 20 5.b even 2 1
1340.1.bl.a 20 268.o odd 66 1
1340.1.bl.a 20 335.u even 66 1
1340.1.bl.b yes 20 1.a even 1 1 trivial
1340.1.bl.b yes 20 20.d odd 2 1 CM
1340.1.bl.b yes 20 67.g even 33 1 inner
1340.1.bl.b yes 20 1340.bl odd 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} - \cdots\) acting on \(S_{1}^{\mathrm{new}}(1340, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
$3$ \( 1 + 9 T + 146 T^{2} + 744 T^{3} + 2051 T^{4} + 3151 T^{5} + 2658 T^{6} + 971 T^{7} + 119 T^{8} + T^{9} + T^{11} + 9 T^{12} + 25 T^{13} + 18 T^{14} - 6 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$7$ \( 1 + 23 T + 154 T^{2} + 230 T^{3} - 12 T^{4} + 881 T^{6} + 760 T^{7} + 1452 T^{8} + 450 T^{9} + 450 T^{10} - T^{11} + 22 T^{12} + 34 T^{13} + 23 T^{14} + 22 T^{15} - T^{16} - T^{17} + T^{19} + T^{20} \)
$11$ \( T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} \)
$29$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
$31$ \( T^{20} \)
$37$ \( T^{20} \)
$41$ \( 1 - 21 T + 143 T^{2} - 243 T^{3} + 593 T^{4} - 1331 T^{5} + 1464 T^{6} - 472 T^{7} + 242 T^{8} + 109 T^{9} - 12 T^{10} - 122 T^{11} - 99 T^{12} + 67 T^{13} + 12 T^{14} - T^{16} - T^{17} + T^{19} + T^{20} \)
$43$ \( 1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20} \)
$47$ \( 1 + 23 T + 154 T^{2} + 230 T^{3} - 12 T^{4} + 881 T^{6} + 760 T^{7} + 1452 T^{8} + 450 T^{9} + 450 T^{10} - T^{11} + 22 T^{12} + 34 T^{13} + 23 T^{14} + 22 T^{15} - T^{16} - T^{17} + T^{19} + T^{20} \)
$53$ \( T^{20} \)
$59$ \( T^{20} \)
$61$ \( 1 + 5 T - 15 T^{3} + 145 T^{4} + 616 T^{5} + 1116 T^{6} + 1631 T^{7} + 2453 T^{8} + 3302 T^{9} + 3750 T^{10} + 3713 T^{11} + 3223 T^{12} + 2424 T^{13} + 1571 T^{14} + 869 T^{15} + 402 T^{16} + 151 T^{17} + 44 T^{18} + 9 T^{19} + T^{20} \)
$67$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
$71$ \( T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( 1 + 12 T + 154 T^{2} + 626 T^{3} + 934 T^{4} + 253 T^{5} - 560 T^{6} - 87 T^{7} + 605 T^{8} - 155 T^{9} + 87 T^{10} + 241 T^{11} + 11 T^{12} + 89 T^{13} + 78 T^{14} + 10 T^{16} + 10 T^{17} + T^{19} + T^{20} \)
$89$ \( ( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$97$ \( T^{20} \)
show more
show less