# Properties

 Label 1397.1.l.a Level $1397$ Weight $1$ Character orbit 1397.l Analytic conductor $0.697$ Analytic rank $0$ Dimension $20$ Projective image $D_{25}$ CM discriminant -127 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1397 = 11 \cdot 127$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1397.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.697193822648$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{50})$$ Defining polynomial: $$x^{20} - x^{15} + x^{10} - x^{5} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{25}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{25} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{2} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{4} + ( -\zeta_{50} + \zeta_{50}^{4} - \zeta_{50}^{7} - \zeta_{50}^{23} ) q^{8} -\zeta_{50}^{5} q^{9} +O(q^{10})$$ $$q + ( \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{2} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{4} + ( -\zeta_{50} + \zeta_{50}^{4} - \zeta_{50}^{7} - \zeta_{50}^{23} ) q^{8} -\zeta_{50}^{5} q^{9} -\zeta_{50}^{13} q^{11} + ( -\zeta_{50}^{17} + \zeta_{50}^{18} ) q^{13} + ( -\zeta_{50} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{16} + ( \zeta_{50}^{16} + \zeta_{50}^{24} ) q^{17} + ( -\zeta_{50}^{21} + \zeta_{50}^{24} ) q^{18} + ( -\zeta_{50}^{9} - \zeta_{50}^{21} ) q^{19} + ( \zeta_{50}^{4} - \zeta_{50}^{7} ) q^{22} -\zeta_{50}^{15} q^{25} + ( \zeta_{50}^{8} - \zeta_{50}^{9} - \zeta_{50}^{11} + \zeta_{50}^{12} ) q^{26} + ( \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{31} + ( -\zeta_{50}^{5} + \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{32} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{15} + \zeta_{50}^{18} ) q^{34} + ( \zeta_{50}^{12} - \zeta_{50}^{15} + \zeta_{50}^{18} ) q^{36} + ( \zeta_{50}^{6} + \zeta_{50}^{14} ) q^{37} + ( 1 - \zeta_{50}^{3} + \zeta_{50}^{12} - \zeta_{50}^{15} ) q^{38} + ( -\zeta_{50}^{13} - \zeta_{50}^{17} ) q^{41} + ( -\zeta_{50} + \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{44} + ( \zeta_{50}^{2} - \zeta_{50}^{3} ) q^{47} + \zeta_{50}^{20} q^{49} + ( \zeta_{50}^{6} - \zeta_{50}^{9} ) q^{50} + ( 1 + \zeta_{50}^{2} - \zeta_{50}^{3} - \zeta_{50}^{5} + \zeta_{50}^{6} + \zeta_{50}^{24} ) q^{52} + ( -\zeta_{50}^{19} - \zeta_{50}^{21} ) q^{61} + ( -\zeta_{50}^{7} + 2 \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{62} + ( \zeta_{50}^{2} - \zeta_{50}^{5} + \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{21} + \zeta_{50}^{24} ) q^{64} + ( -\zeta_{50} + \zeta_{50}^{4} + \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{12} - \zeta_{50}^{23} ) q^{68} + ( \zeta_{50}^{4} - \zeta_{50}^{11} ) q^{71} + ( -\zeta_{50}^{3} + \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{12} ) q^{72} + ( -\zeta_{50}^{3} - \zeta_{50}^{17} ) q^{73} + ( 1 - \zeta_{50}^{5} + \zeta_{50}^{8} + \zeta_{50}^{22} ) q^{74} + ( -\zeta_{50}^{3} + \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{16} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{76} + ( \zeta_{50}^{2} + \zeta_{50}^{8} ) q^{79} + \zeta_{50}^{10} q^{81} + ( \zeta_{50}^{4} - \zeta_{50}^{7} + \zeta_{50}^{8} - \zeta_{50}^{11} ) q^{82} + ( -\zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{88} + ( \zeta_{50}^{18} - \zeta_{50}^{19} - \zeta_{50}^{21} + \zeta_{50}^{22} ) q^{94} + ( -\zeta_{50}^{11} + \zeta_{50}^{14} ) q^{98} + \zeta_{50}^{18} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 5q^{4} - 5q^{9} + O(q^{10})$$ $$20q - 5q^{4} - 5q^{9} - 5q^{16} - 5q^{25} - 10q^{32} - 10q^{34} - 5q^{36} + 15q^{38} - 5q^{44} - 5q^{49} + 15q^{52} - 10q^{62} - 5q^{64} + 15q^{74} - 5q^{81} - 5q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1397\mathbb{Z}\right)^\times$$.

 $$n$$ $$255$$ $$892$$ $$\chi(n)$$ $$\zeta_{50}^{10}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 −0.968583 − 0.248690i −0.0627905 − 0.998027i −0.535827 + 0.844328i 0.929776 − 0.368125i 0.637424 + 0.770513i 0.187381 − 0.982287i −0.876307 − 0.481754i 0.992115 − 0.125333i −0.728969 + 0.684547i 0.425779 + 0.904827i −0.968583 + 0.248690i −0.0627905 + 0.998027i −0.535827 − 0.844328i 0.929776 + 0.368125i 0.637424 − 0.770513i 0.187381 + 0.982287i −0.876307 + 0.481754i 0.992115 + 0.125333i −0.728969 − 0.684547i 0.425779 − 0.904827i
−0.574633 1.76854i 0 −1.98851 + 1.44474i 0 0 0 2.19334 + 1.59355i 0.309017 + 0.951057i 0
126.2 −0.393950 1.21245i 0 −0.505828 + 0.367505i 0 0 0 −0.386520 0.280823i 0.309017 + 0.951057i 0
126.3 0.0388067 + 0.119435i 0 0.796258 0.578516i 0 0 0 0.201592 + 0.146465i 0.309017 + 0.951057i 0
126.4 0.331159 + 1.01920i 0 −0.120092 + 0.0872517i 0 0 0 0.738289 + 0.536399i 0.309017 + 0.951057i 0
126.5 0.598617 + 1.84235i 0 −2.22691 + 1.61795i 0 0 0 −2.74670 1.99559i 0.309017 + 0.951057i 0
1015.1 −1.41789 + 1.03016i 0 0.640176 1.97026i 0 0 0 0.580394 + 1.78627i −0.809017 + 0.587785i 0
1015.2 −1.17950 + 0.856954i 0 0.347824 1.07049i 0 0 0 0.0565777 + 0.174128i −0.809017 + 0.587785i 0
1015.3 0.303189 0.220280i 0 −0.265616 + 0.817483i 0 0 0 0.215351 + 0.662783i −0.809017 + 0.587785i 0
1015.4 0.688925 0.500534i 0 −0.0849327 + 0.261396i 0 0 0 0.335471 + 1.03247i −0.809017 + 0.587785i 0
1015.5 1.60528 1.16630i 0 0.907634 2.79341i 0 0 0 −1.18779 3.65565i −0.809017 + 0.587785i 0
1142.1 −0.574633 + 1.76854i 0 −1.98851 1.44474i 0 0 0 2.19334 1.59355i 0.309017 0.951057i 0
1142.2 −0.393950 + 1.21245i 0 −0.505828 0.367505i 0 0 0 −0.386520 + 0.280823i 0.309017 0.951057i 0
1142.3 0.0388067 0.119435i 0 0.796258 + 0.578516i 0 0 0 0.201592 0.146465i 0.309017 0.951057i 0
1142.4 0.331159 1.01920i 0 −0.120092 0.0872517i 0 0 0 0.738289 0.536399i 0.309017 0.951057i 0
1142.5 0.598617 1.84235i 0 −2.22691 1.61795i 0 0 0 −2.74670 + 1.99559i 0.309017 0.951057i 0
1269.1 −1.41789 1.03016i 0 0.640176 + 1.97026i 0 0 0 0.580394 1.78627i −0.809017 0.587785i 0
1269.2 −1.17950 0.856954i 0 0.347824 + 1.07049i 0 0 0 0.0565777 0.174128i −0.809017 0.587785i 0
1269.3 0.303189 + 0.220280i 0 −0.265616 0.817483i 0 0 0 0.215351 0.662783i −0.809017 0.587785i 0
1269.4 0.688925 + 0.500534i 0 −0.0849327 0.261396i 0 0 0 0.335471 1.03247i −0.809017 0.587785i 0
1269.5 1.60528 + 1.16630i 0 0.907634 + 2.79341i 0 0 0 −1.18779 + 3.65565i −0.809017 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1269.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.b odd 2 1 CM by $$\Q(\sqrt{-127})$$
11.c even 5 1 inner
1397.l odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1397.1.l.a 20
11.c even 5 1 inner 1397.1.l.a 20
127.b odd 2 1 CM 1397.1.l.a 20
1397.l odd 10 1 inner 1397.1.l.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1397.1.l.a 20 1.a even 1 1 trivial
1397.1.l.a 20 11.c even 5 1 inner
1397.1.l.a 20 127.b odd 2 1 CM
1397.1.l.a 20 1397.l odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1397, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$3$ $$T^{20}$$
$5$ $$T^{20}$$
$7$ $$T^{20}$$
$11$ $$1 + T^{5} + T^{10} + T^{15} + T^{20}$$
$13$ $$1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$17$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$19$ $$1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$37$ $$1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$41$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$43$ $$T^{20}$$
$47$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$67$ $$T^{20}$$
$71$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$73$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$79$ $$1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$