# Properties

 Label 1375.1.ch.a Level $1375$ Weight $1$ Character orbit 1375.ch Analytic conductor $0.686$ Analytic rank $0$ Dimension $20$ Projective image $D_{25}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1375 = 5^{3} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1375.ch (of order $$50$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.686214392370$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{50})$$ Defining polynomial: $$x^{20} - x^{15} + x^{10} - x^{5} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{50}^{4} + \zeta_{50}^{18} ) q^{3} -\zeta_{50}^{21} q^{4} -\zeta_{50}^{5} q^{5} + ( \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{22} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{50}^{4} + \zeta_{50}^{18} ) q^{3} -\zeta_{50}^{21} q^{4} -\zeta_{50}^{5} q^{5} + ( \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{22} ) q^{9} -\zeta_{50}^{23} q^{11} + ( 1 + \zeta_{50}^{14} ) q^{12} + ( -\zeta_{50}^{9} - \zeta_{50}^{23} ) q^{15} -\zeta_{50}^{17} q^{16} -\zeta_{50} q^{20} + ( \zeta_{50}^{2} + \zeta_{50}^{24} ) q^{23} + \zeta_{50}^{10} q^{25} + ( -\zeta_{50} + \zeta_{50}^{4} + \zeta_{50}^{12} - \zeta_{50}^{15} ) q^{27} + ( \zeta_{50}^{16} - \zeta_{50}^{17} ) q^{31} + ( \zeta_{50}^{2} + \zeta_{50}^{16} ) q^{33} + ( \zeta_{50}^{4} - \zeta_{50}^{7} + \zeta_{50}^{18} ) q^{36} + ( \zeta_{50}^{2} - \zeta_{50}^{7} ) q^{37} -\zeta_{50}^{19} q^{44} + ( \zeta_{50}^{2} - \zeta_{50}^{13} + \zeta_{50}^{16} ) q^{45} + ( -\zeta_{50} - \zeta_{50}^{11} ) q^{47} + ( \zeta_{50}^{10} - \zeta_{50}^{21} ) q^{48} + \zeta_{50}^{10} q^{49} + ( \zeta_{50}^{14} + \zeta_{50}^{18} ) q^{53} -\zeta_{50}^{3} q^{55} + ( -\zeta_{50}^{15} + \zeta_{50}^{24} ) q^{59} + ( -\zeta_{50}^{5} - \zeta_{50}^{19} ) q^{60} -\zeta_{50}^{13} q^{64} + ( \zeta_{50}^{8} - \zeta_{50}^{15} ) q^{67} + ( -\zeta_{50}^{3} + \zeta_{50}^{6} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{69} + ( -\zeta_{50}^{13} + \zeta_{50}^{24} ) q^{71} + ( -\zeta_{50}^{3} + \zeta_{50}^{14} ) q^{75} + \zeta_{50}^{22} q^{80} + ( -\zeta_{50}^{5} + \zeta_{50}^{8} + \zeta_{50}^{16} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{81} + ( \zeta_{50}^{12} + \zeta_{50}^{24} ) q^{89} + ( \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{92} + ( -\zeta_{50}^{9} + \zeta_{50}^{10} + \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{93} + ( -\zeta_{50}^{9} + \zeta_{50}^{18} ) q^{97} + ( \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{20} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 5q^{5} + O(q^{10})$$ $$20q - 5q^{5} + 20q^{12} - 5q^{25} - 5q^{27} - 5q^{48} - 5q^{49} - 5q^{59} - 5q^{60} - 5q^{67} - 5q^{69} - 5q^{81} - 5q^{92} - 10q^{93} - 5q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$1002$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{50}^{21}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1
 0.992115 + 0.125333i 0.992115 − 0.125333i 0.637424 − 0.770513i −0.535827 + 0.844328i −0.876307 + 0.481754i −0.728969 + 0.684547i −0.0627905 + 0.998027i −0.0627905 − 0.998027i 0.425779 − 0.904827i 0.187381 − 0.982287i −0.535827 − 0.844328i 0.637424 + 0.770513i 0.187381 + 0.982287i 0.425779 + 0.904827i 0.929776 + 0.368125i −0.968583 − 0.248690i −0.728969 − 0.684547i −0.876307 − 0.481754i −0.968583 + 0.248690i 0.929776 − 0.368125i
0 0.238883 + 1.25227i 0.876307 0.481754i −0.809017 0.587785i 0 0 0 −0.581331 + 0.230165i 0
131.1 0 0.238883 1.25227i 0.876307 + 0.481754i −0.809017 + 0.587785i 0 0 0 −0.581331 0.230165i 0
186.1 0 −1.92189 + 0.493458i −0.929776 0.368125i 0.309017 0.951057i 0 0 0 2.57386 1.41499i 0
241.1 0 0.0915446 + 1.45506i −0.637424 0.770513i 0.309017 + 0.951057i 0 0 0 −1.11671 + 0.141073i 0
296.1 0 −1.35556 1.27295i −0.425779 + 0.904827i −0.809017 0.587785i 0 0 0 0.154335 + 2.45309i 0
406.1 0 −0.456288 0.969661i −0.992115 + 0.125333i −0.809017 + 0.587785i 0 0 0 −0.0946201 + 0.114376i 0
461.1 0 0.542804 0.656137i 0.968583 0.248690i 0.309017 0.951057i 0 0 0 0.0515014 + 0.269980i 0
516.1 0 0.542804 + 0.656137i 0.968583 + 0.248690i 0.309017 + 0.951057i 0 0 0 0.0515014 0.269980i 0
571.1 0 −0.124591 0.0157395i −0.187381 0.982287i −0.809017 0.587785i 0 0 0 −0.953308 0.244768i 0
681.1 0 1.69755 + 0.933237i 0.728969 0.684547i −0.809017 + 0.587785i 0 0 0 1.47492 + 2.32411i 0
736.1 0 0.0915446 1.45506i −0.637424 + 0.770513i 0.309017 0.951057i 0 0 0 −1.11671 0.141073i 0
791.1 0 −1.92189 0.493458i −0.929776 + 0.368125i 0.309017 + 0.951057i 0 0 0 2.57386 + 1.41499i 0
846.1 0 1.69755 0.933237i 0.728969 + 0.684547i −0.809017 0.587785i 0 0 0 1.47492 2.32411i 0
956.1 0 −0.124591 + 0.0157395i −0.187381 + 0.982287i −0.809017 + 0.587785i 0 0 0 −0.953308 + 0.244768i 0
1011.1 0 0.939097 + 1.47978i 0.0627905 0.998027i 0.309017 0.951057i 0 0 0 −0.882067 + 1.87449i 0
1066.1 0 0.348445 0.137959i 0.535827 0.844328i 0.309017 + 0.951057i 0 0 0 −0.626587 + 0.588404i 0
1121.1 0 −0.456288 + 0.969661i −0.992115 0.125333i −0.809017 0.587785i 0 0 0 −0.0946201 0.114376i 0
1231.1 0 −1.35556 + 1.27295i −0.425779 0.904827i −0.809017 + 0.587785i 0 0 0 0.154335 2.45309i 0
1286.1 0 0.348445 + 0.137959i 0.535827 + 0.844328i 0.309017 0.951057i 0 0 0 −0.626587 0.588404i 0
1341.1 0 0.939097 1.47978i 0.0627905 + 0.998027i 0.309017 + 0.951057i 0 0 0 −0.882067 1.87449i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1341.1 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 $$\Q(\sqrt{-11})$$
125.g even 25 1 inner
1375.ch odd 50 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.1.ch.a 20
11.b odd 2 1 CM 1375.1.ch.a 20
125.g even 25 1 inner 1375.1.ch.a 20
1375.ch odd 50 1 inner 1375.1.ch.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.1.ch.a 20 1.a even 1 1 trivial
1375.1.ch.a 20 11.b odd 2 1 CM
1375.1.ch.a 20 125.g even 25 1 inner
1375.1.ch.a 20 1375.ch odd 50 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 + 10 T - 15 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 1010 T^{6} - 715 T^{7} + 925 T^{8} - 225 T^{9} + 379 T^{10} + 120 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
$7$ $$T^{20}$$
$11$ $$1 + T^{5} + T^{10} + T^{15} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20}$$
$37$ $$1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20}$$
$53$ $$1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20}$$
$59$ $$1 - 10 T - 5 T^{2} + 130 T^{3} + 660 T^{4} + 998 T^{5} + 1315 T^{6} + 1270 T^{7} + 805 T^{8} + 85 T^{9} - 246 T^{10} - 45 T^{11} + 165 T^{12} + 235 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$61$ $$T^{20}$$
$67$ $$1 + 15 T + 120 T^{2} + 380 T^{3} + 435 T^{4} - 252 T^{5} - 735 T^{6} - 905 T^{7} - 645 T^{8} + 85 T^{9} + 629 T^{10} + 730 T^{11} + 540 T^{12} + 310 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$71$ $$1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20}$$
$73$ $$T^{20}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$97$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$