# Properties

 Label 1340.1.bl.a 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 20
4.b odd 2 1 1340.1.bl.b yes 20
5.b even 2 1 1340.1.bl.b yes 20
20.d odd 2 1 CM 1340.1.bl.a 20
67.g even 33 1 inner 1340.1.bl.a 20
268.o odd 66 1 1340.1.bl.b yes 20
335.u even 66 1 1340.1.bl.b yes 20
1340.bl odd 66 1 inner 1340.1.bl.a 20

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

## 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

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