# 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 $4$

# Related objects

## 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: 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 $$\Q(\sqrt{-11})$$
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

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