# Properties

 Label 737.1.w.a Level $737$ Weight $1$ Character orbit 737.w Analytic conductor $0.368$ Analytic rank $0$ Dimension $20$ Projective image $D_{33}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$737 = 11 \cdot 67$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 737.w (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.367810914311$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{66}^{12} + \zeta_{66}^{30} ) q^{3} -\zeta_{66}^{7} q^{4} + ( \zeta_{66}^{8} - \zeta_{66}^{31} ) q^{5} + ( -\zeta_{66}^{9} + \zeta_{66}^{24} - \zeta_{66}^{27} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{66}^{12} + \zeta_{66}^{30} ) q^{3} -\zeta_{66}^{7} q^{4} + ( \zeta_{66}^{8} - \zeta_{66}^{31} ) q^{5} + ( -\zeta_{66}^{9} + \zeta_{66}^{24} - \zeta_{66}^{27} ) q^{9} -\zeta_{66}^{25} q^{11} + ( \zeta_{66}^{4} - \zeta_{66}^{19} ) q^{12} + ( -\zeta_{66}^{5} + \zeta_{66}^{10} + \zeta_{66}^{20} + \zeta_{66}^{28} ) q^{15} + \zeta_{66}^{14} q^{16} + ( -\zeta_{66}^{5} - \zeta_{66}^{15} ) q^{20} + ( -\zeta_{66}^{11} + \zeta_{66}^{20} ) q^{23} + ( \zeta_{66}^{6} + \zeta_{66}^{16} - \zeta_{66}^{29} ) q^{25} + ( -\zeta_{66}^{3} + \zeta_{66}^{6} - \zeta_{66}^{21} + \zeta_{66}^{24} ) q^{27} + ( \zeta_{66}^{14} + \zeta_{66}^{18} ) q^{31} + ( \zeta_{66}^{4} + \zeta_{66}^{22} ) q^{33} + ( -\zeta_{66} + \zeta_{66}^{16} - \zeta_{66}^{31} ) q^{36} + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{37} + \zeta_{66}^{32} q^{44} + ( \zeta_{66}^{2} - \zeta_{66}^{7} - \zeta_{66}^{17} + \zeta_{66}^{22} - \zeta_{66}^{25} + \zeta_{66}^{32} ) q^{45} + ( -\zeta_{66}^{23} + \zeta_{66}^{30} ) q^{47} + ( -\zeta_{66}^{11} + \zeta_{66}^{26} ) q^{48} -\zeta_{66}^{29} q^{49} + ( \zeta_{66}^{10} + \zeta_{66}^{26} ) q^{53} + ( 1 - \zeta_{66}^{23} ) q^{55} + ( \zeta_{66}^{2} - \zeta_{66}^{19} ) q^{59} + ( \zeta_{66}^{2} + \zeta_{66}^{12} - \zeta_{66}^{17} - \zeta_{66}^{27} ) q^{60} -\zeta_{66}^{21} q^{64} -\zeta_{66} q^{67} + ( \zeta_{66}^{8} - \zeta_{66}^{17} - \zeta_{66}^{23} + \zeta_{66}^{32} ) q^{69} + ( -\zeta_{66} - \zeta_{66}^{13} ) q^{71} + ( -\zeta_{66}^{3} + \zeta_{66}^{8} - \zeta_{66}^{13} + \zeta_{66}^{18} + \zeta_{66}^{26} + \zeta_{66}^{28} ) q^{75} + ( \zeta_{66}^{12} + \zeta_{66}^{22} ) q^{80} + ( 1 - \zeta_{66}^{3} - \zeta_{66}^{15} + \zeta_{66}^{18} - \zeta_{66}^{21} ) q^{81} + ( \zeta_{66}^{28} - \zeta_{66}^{29} ) q^{89} + ( \zeta_{66}^{18} - \zeta_{66}^{27} ) q^{92} + ( -\zeta_{66}^{11} - \zeta_{66}^{15} + \zeta_{66}^{26} + \zeta_{66}^{30} ) q^{93} + ( -\zeta_{66}^{7} - \zeta_{66}^{15} ) q^{97} + ( -\zeta_{66} + \zeta_{66}^{16} - \zeta_{66}^{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 4q^{3} + q^{4} + 2q^{5} - 6q^{9} + O(q^{10})$$ $$20q - 4q^{3} + q^{4} + 2q^{5} - 6q^{9} + q^{11} + 2q^{12} + 4q^{15} + q^{16} - q^{20} - 9q^{23} - 8q^{27} - q^{31} - 9q^{33} + 3q^{36} - q^{37} + q^{44} - 5q^{45} - q^{47} - 9q^{48} + q^{49} + 2q^{53} + 21q^{55} + 2q^{59} - 2q^{60} - 2q^{64} + q^{67} + 4q^{69} + 2q^{71} - 12q^{80} + 12q^{81} + 2q^{89} - 4q^{92} - 13q^{93} - q^{97} + 3q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/737\mathbb{Z}\right)^\times$$.

 $$n$$ $$68$$ $$672$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{66}^{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −0.786053 − 0.618159i 0.580057 − 0.814576i 0.981929 − 0.189251i 0.235759 − 0.971812i 0.723734 − 0.690079i −0.995472 − 0.0950560i −0.888835 + 0.458227i −0.327068 − 0.945001i 0.580057 + 0.814576i 0.0475819 + 0.998867i 0.723734 + 0.690079i −0.888835 − 0.458227i 0.0475819 − 0.998867i 0.981929 + 0.189251i −0.786053 + 0.618159i 0.928368 − 0.371662i 0.928368 + 0.371662i −0.995472 + 0.0950560i 0.235759 + 0.971812i −0.327068 + 0.945001i
0 0.273100 + 1.89945i 0.0475819 + 0.998867i 0.815816 1.78639i 0 0 0 −2.57385 + 0.755750i 0
21.1 0 −0.544078 + 1.19136i 0.928368 0.371662i −0.0913090 0.0268107i 0 0 0 −0.468468 0.540641i 0
54.1 0 0.186393 0.215109i 0.235759 0.971812i 0.975950 0.627205i 0 0 0 0.130785 + 0.909632i 0
65.1 0 −1.61435 0.474017i −0.995472 0.0950560i −1.21590 + 1.40323i 0 0 0 1.54019 + 0.989821i 0
153.1 0 −1.61435 + 0.474017i 0.580057 + 0.814576i 1.02951 + 1.18812i 0 0 0 1.54019 0.989821i 0
274.1 0 −0.544078 + 1.19136i −0.786053 0.618159i 1.70566 + 0.500828i 0 0 0 −0.468468 0.540641i 0
285.1 0 0.698939 0.449181i 0.981929 0.189251i −0.205996 + 1.43273i 0 0 0 −0.128663 + 0.281733i 0
307.1 0 0.186393 + 0.215109i 0.723734 0.690079i −1.67489 1.07639i 0 0 0 0.130785 0.909632i 0
351.1 0 −0.544078 1.19136i 0.928368 + 0.371662i −0.0913090 + 0.0268107i 0 0 0 −0.468468 + 0.540641i 0
384.1 0 0.698939 + 0.449181i −0.327068 0.945001i −0.0671040 0.466718i 0 0 0 −0.128663 0.281733i 0
395.1 0 −1.61435 0.474017i 0.580057 0.814576i 1.02951 1.18812i 0 0 0 1.54019 + 0.989821i 0
406.1 0 0.698939 + 0.449181i 0.981929 + 0.189251i −0.205996 1.43273i 0 0 0 −0.128663 0.281733i 0
428.1 0 0.698939 0.449181i −0.327068 + 0.945001i −0.0671040 + 0.466718i 0 0 0 −0.128663 + 0.281733i 0
505.1 0 0.186393 + 0.215109i 0.235759 + 0.971812i 0.975950 + 0.627205i 0 0 0 0.130785 0.909632i 0
516.1 0 0.273100 1.89945i 0.0475819 0.998867i 0.815816 + 1.78639i 0 0 0 −2.57385 0.755750i 0
571.1 0 0.273100 + 1.89945i −0.888835 0.458227i −0.271738 + 0.595023i 0 0 0 −2.57385 + 0.755750i 0
626.1 0 0.273100 1.89945i −0.888835 + 0.458227i −0.271738 0.595023i 0 0 0 −2.57385 0.755750i 0
659.1 0 −0.544078 1.19136i −0.786053 + 0.618159i 1.70566 0.500828i 0 0 0 −0.468468 + 0.540641i 0
703.1 0 −1.61435 + 0.474017i −0.995472 + 0.0950560i −1.21590 1.40323i 0 0 0 1.54019 0.989821i 0
725.1 0 0.186393 0.215109i 0.723734 + 0.690079i −1.67489 + 1.07639i 0 0 0 0.130785 + 0.909632i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 725.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
67.g even 33 1 inner
737.w odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 737.1.w.a 20
11.b odd 2 1 CM 737.1.w.a 20
67.g even 33 1 inner 737.1.w.a 20
737.w odd 66 1 inner 737.1.w.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
737.1.w.a 20 1.a even 1 1 trivial
737.1.w.a 20 11.b odd 2 1 CM
737.1.w.a 20 67.g even 33 1 inner
737.1.w.a 20 737.w odd 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(737, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$5$ $$1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$7$ $$T^{20}$$
$11$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$1 + 5 T - 15 T^{3} + 145 T^{4} + 616 T^{5} + 1116 T^{6} + 1631 T^{7} + 2453 T^{8} + 3302 T^{9} + 3750 T^{10} + 3713 T^{11} + 3223 T^{12} + 2424 T^{13} + 1571 T^{14} + 869 T^{15} + 402 T^{16} + 151 T^{17} + 44 T^{18} + 9 T^{19} + T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 - 21 T + 143 T^{2} - 243 T^{3} + 593 T^{4} - 1331 T^{5} + 1464 T^{6} - 472 T^{7} + 242 T^{8} + 109 T^{9} - 12 T^{10} - 122 T^{11} - 99 T^{12} + 67 T^{13} + 12 T^{14} - T^{16} - T^{17} + T^{19} + T^{20}$$
$37$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$1 - 10 T + 44 T^{2} + 395 T^{3} + 1198 T^{4} + 1529 T^{5} + 793 T^{6} - 604 T^{7} - 550 T^{8} - 67 T^{9} - T^{10} + 65 T^{11} + 176 T^{12} + 23 T^{13} + 12 T^{14} - 11 T^{15} - 23 T^{16} - T^{17} + T^{19} + T^{20}$$
$53$ $$1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$59$ $$1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$61$ $$T^{20}$$
$67$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$71$ $$1 - 6 T + 11 T^{2} - 48 T^{3} + 266 T^{4} - 671 T^{5} + 1116 T^{6} - 1207 T^{7} + 869 T^{8} - 284 T^{9} - 45 T^{10} + 94 T^{11} - 33 T^{12} - 29 T^{13} + 42 T^{14} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$73$ $$T^{20}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$97$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$