# Properties

 Label 1265.1.bi.b Level $1265$ Weight $1$ Character orbit 1265.bi Analytic conductor $0.631$ Analytic rank $0$ Dimension $20$ Projective image $D_{44}$ CM discriminant -11 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1265 = 5 \cdot 11 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1265.bi (of order $$44$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.631317240981$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{44})$$ Defining polynomial: $$x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{44}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{44} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1265\mathbb{Z}\right)^\times$$.

 $$n$$ $$166$$ $$507$$ $$1036$$ $$\chi(n)$$ $$\zeta_{44}^{2}$$ $$-\zeta_{44}^{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.755750 + 0.654861i −0.755750 + 0.654861i −0.909632 − 0.415415i 0.540641 − 0.841254i 0.540641 + 0.841254i 0.989821 + 0.142315i −0.989821 + 0.142315i 0.281733 − 0.959493i −0.909632 + 0.415415i −0.281733 − 0.959493i −0.755750 − 0.654861i −0.540641 − 0.841254i 0.755750 − 0.654861i 0.909632 + 0.415415i −0.281733 + 0.959493i −0.540641 + 0.841254i 0.909632 − 0.415415i −0.989821 − 0.142315i 0.989821 − 0.142315i 0.281733 + 0.959493i
0 1.83107 0.682956i 0.281733 + 0.959493i −0.540641 + 0.841254i 0 0 0 2.13066 1.84623i 0
153.1 0 −0.148568 + 0.398326i −0.281733 + 0.959493i 0.540641 + 0.841254i 0 0 0 0.619158 + 0.536504i 0
263.1 0 −0.203743 0.936593i 0.989821 + 0.142315i −0.281733 0.959493i 0 0 0 0.0739364 0.0337656i 0
318.1 0 −1.05195 0.574406i 0.755750 + 0.654861i −0.989821 0.142315i 0 0 0 0.236009 + 0.367237i 0
362.1 0 −1.05195 + 0.574406i 0.755750 0.654861i −0.989821 + 0.142315i 0 0 0 0.236009 0.367237i 0
373.1 0 0.697148 0.0498610i 0.540641 0.841254i 0.909632 + 0.415415i 0 0 0 −0.506293 + 0.0727939i 0
428.1 0 0.133682 1.86912i −0.540641 0.841254i −0.909632 + 0.415415i 0 0 0 −2.48594 0.357424i 0
527.1 0 −0.114220 0.0855040i −0.909632 + 0.415415i −0.755750 + 0.654861i 0 0 0 −0.275997 0.939960i 0
582.1 0 −0.203743 + 0.936593i 0.989821 0.142315i −0.281733 + 0.959493i 0 0 0 0.0739364 + 0.0337656i 0
747.1 0 −1.19550 1.59700i 0.909632 + 0.415415i 0.755750 + 0.654861i 0 0 0 −0.839462 + 2.85895i 0
802.1 0 −0.148568 0.398326i −0.281733 0.959493i 0.540641 0.841254i 0 0 0 0.619158 0.536504i 0
868.1 0 0.767317 + 1.40524i −0.755750 + 0.654861i 0.989821 0.142315i 0 0 0 −0.845273 + 1.31527i 0
912.1 0 1.83107 + 0.682956i 0.281733 0.959493i −0.540641 0.841254i 0 0 0 2.13066 + 1.84623i 0
1022.1 0 −1.71524 + 0.373128i −0.989821 0.142315i 0.281733 + 0.959493i 0 0 0 1.89320 0.864596i 0
1033.1 0 −1.19550 + 1.59700i 0.909632 0.415415i 0.755750 0.654861i 0 0 0 −0.839462 2.85895i 0
1077.1 0 0.767317 1.40524i −0.755750 0.654861i 0.989821 + 0.142315i 0 0 0 −0.845273 1.31527i 0
1088.1 0 −1.71524 0.373128i −0.989821 + 0.142315i 0.281733 0.959493i 0 0 0 1.89320 + 0.864596i 0
1132.1 0 0.133682 + 1.86912i −0.540641 + 0.841254i −0.909632 0.415415i 0 0 0 −2.48594 + 0.357424i 0
1187.1 0 0.697148 + 0.0498610i 0.540641 + 0.841254i 0.909632 0.415415i 0 0 0 −0.506293 0.0727939i 0
1253.1 0 −0.114220 + 0.0855040i −0.909632 0.415415i −0.755750 0.654861i 0 0 0 −0.275997 + 0.939960i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1253.1 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 $$\Q(\sqrt{-11})$$
115.l even 44 1 inner
1265.bi odd 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1265.1.bi.b yes 20
5.c odd 4 1 1265.1.bi.a 20
11.b odd 2 1 CM 1265.1.bi.b yes 20
23.d odd 22 1 1265.1.bi.a 20
55.e even 4 1 1265.1.bi.a 20
115.l even 44 1 inner 1265.1.bi.b yes 20
253.l even 22 1 1265.1.bi.a 20
1265.bi odd 44 1 inner 1265.1.bi.b yes 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1265.1.bi.a 20 5.c odd 4 1
1265.1.bi.a 20 23.d odd 22 1
1265.1.bi.a 20 55.e even 4 1
1265.1.bi.a 20 253.l even 22 1
1265.1.bi.b yes 20 1.a even 1 1 trivial
1265.1.bi.b yes 20 11.b odd 2 1 CM
1265.1.bi.b yes 20 115.l even 44 1 inner
1265.1.bi.b yes 20 1265.bi odd 44 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1265, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 + 12 T + 61 T^{2} + 66 T^{3} + 63 T^{4} - 454 T^{5} - 403 T^{6} - 176 T^{7} + 328 T^{8} + 658 T^{9} + 494 T^{10} + 164 T^{11} + 148 T^{12} + 66 T^{13} - 8 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$5$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$7$ $$T^{20}$$
$11$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$29$ $$T^{20}$$
$31$ $$121 + 605 T^{2} + 484 T^{4} - 968 T^{6} + 484 T^{8} + 99 T^{10} + 165 T^{12} + 22 T^{16} + T^{20}$$
$37$ $$1 + 10 T + 94 T^{2} + 462 T^{3} + 1361 T^{4} + 2412 T^{5} + 2413 T^{6} + 1100 T^{7} + 53 T^{8} + 2 T^{9} + 32 T^{10} - 32 T^{11} + 16 T^{12} - 8 T^{14} + 8 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$1 - 10 T + 50 T^{2} - 25 T^{4} + 52 T^{5} + 730 T^{6} + 748 T^{7} + 383 T^{8} - 2 T^{9} + 472 T^{10} + 472 T^{11} + 236 T^{12} + 58 T^{14} + 58 T^{15} + 29 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$53$ $$1 + 12 T + 61 T^{2} + 66 T^{3} + 63 T^{4} - 454 T^{5} - 403 T^{6} - 176 T^{7} + 328 T^{8} + 658 T^{9} + 494 T^{10} + 164 T^{11} + 148 T^{12} + 66 T^{13} - 8 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$59$ $$( 11 - 11 T + 11 T^{2} + 33 T^{3} - 22 T^{5} + T^{10} )^{2}$$
$61$ $$T^{20}$$
$67$ $$1 + 10 T + 94 T^{2} + 462 T^{3} + 1361 T^{4} + 2412 T^{5} + 2413 T^{6} + 1100 T^{7} + 53 T^{8} + 2 T^{9} + 32 T^{10} - 32 T^{11} + 16 T^{12} - 8 T^{14} + 8 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$71$ $$( 1 + 5 T + 25 T^{2} + 70 T^{3} + 130 T^{4} + 166 T^{5} + 148 T^{6} + 91 T^{7} + 37 T^{8} + 9 T^{9} + T^{10} )^{2}$$
$73$ $$T^{20}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20}$$
$97$ $$1 - 12 T + 17 T^{2} + 154 T^{3} - 135 T^{4} - 404 T^{5} + 554 T^{6} - 22 T^{7} - 90 T^{8} + 46 T^{9} + 230 T^{10} - 252 T^{11} + 368 T^{12} - 242 T^{13} + 223 T^{14} - 102 T^{15} + 73 T^{16} - 22 T^{17} + 13 T^{18} - 2 T^{19} + T^{20}$$
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