# Properties

 Label 1400.1.cc.a Level $1400$ Weight $1$ Character orbit 1400.cc Analytic conductor $0.699$ Analytic rank $0$ Dimension $16$ Projective image $D_{20}$ CM discriminant -56 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1400.cc (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.698691017686$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{20}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{40}^{6} q^{2} + ( \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{3} + \zeta_{40}^{12} q^{4} + \zeta_{40}^{11} q^{5} + ( \zeta_{40} + \zeta_{40}^{15} ) q^{6} + \zeta_{40}^{10} q^{7} + \zeta_{40}^{18} q^{8} + ( \zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{9} +O(q^{10})$$ $$q + \zeta_{40}^{6} q^{2} + ( \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{3} + \zeta_{40}^{12} q^{4} + \zeta_{40}^{11} q^{5} + ( \zeta_{40} + \zeta_{40}^{15} ) q^{6} + \zeta_{40}^{10} q^{7} + \zeta_{40}^{18} q^{8} + ( \zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{9} + \zeta_{40}^{17} q^{10} + ( -\zeta_{40} + \zeta_{40}^{7} ) q^{12} + ( -\zeta_{40}^{11} - \zeta_{40}^{17} ) q^{13} + \zeta_{40}^{16} q^{14} + ( -1 + \zeta_{40}^{6} ) q^{15} -\zeta_{40}^{4} q^{16} + ( -\zeta_{40}^{4} + \zeta_{40}^{10} - \zeta_{40}^{16} ) q^{18} + ( -\zeta_{40}^{3} - \zeta_{40}^{13} ) q^{19} -\zeta_{40}^{3} q^{20} + ( \zeta_{40}^{5} + \zeta_{40}^{19} ) q^{21} + ( \zeta_{40}^{4} + \zeta_{40}^{8} ) q^{23} + ( -\zeta_{40}^{7} + \zeta_{40}^{13} ) q^{24} -\zeta_{40}^{2} q^{25} + ( \zeta_{40}^{3} - \zeta_{40}^{17} ) q^{26} + ( -\zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{27} -\zeta_{40}^{2} q^{28} + ( -\zeta_{40}^{6} + \zeta_{40}^{12} ) q^{30} -\zeta_{40}^{10} q^{32} -\zeta_{40} q^{35} + ( \zeta_{40}^{2} - \zeta_{40}^{10} + \zeta_{40}^{16} ) q^{36} + ( -\zeta_{40}^{9} - \zeta_{40}^{19} ) q^{38} + ( 1 - \zeta_{40}^{12} ) q^{39} -\zeta_{40}^{9} q^{40} + ( -\zeta_{40}^{5} + \zeta_{40}^{11} ) q^{42} + ( \zeta_{40} - \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{45} + ( \zeta_{40}^{10} + \zeta_{40}^{14} ) q^{46} + ( -\zeta_{40}^{13} + \zeta_{40}^{19} ) q^{48} - q^{49} -\zeta_{40}^{8} q^{50} + ( \zeta_{40}^{3} + \zeta_{40}^{9} ) q^{52} + ( \zeta_{40}^{5} - \zeta_{40}^{11} - \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{54} -\zeta_{40}^{8} q^{56} + ( \zeta_{40}^{2} - \zeta_{40}^{8} - \zeta_{40}^{12} + \zeta_{40}^{18} ) q^{57} + ( \zeta_{40}^{3} + \zeta_{40}^{5} ) q^{59} + ( -\zeta_{40}^{12} + \zeta_{40}^{18} ) q^{60} + ( -\zeta_{40}^{5} + \zeta_{40}^{7} ) q^{61} + ( 1 - \zeta_{40}^{8} + \zeta_{40}^{14} ) q^{63} -\zeta_{40}^{16} q^{64} + ( \zeta_{40}^{2} + \zeta_{40}^{8} ) q^{65} + ( \zeta_{40}^{3} + \zeta_{40}^{13} + \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{69} -\zeta_{40}^{7} q^{70} + ( -\zeta_{40}^{6} - \zeta_{40}^{18} ) q^{71} + ( -\zeta_{40}^{2} + \zeta_{40}^{8} - \zeta_{40}^{16} ) q^{72} + ( -\zeta_{40}^{11} + \zeta_{40}^{17} ) q^{75} + ( \zeta_{40}^{5} - \zeta_{40}^{15} ) q^{76} + ( \zeta_{40}^{6} - \zeta_{40}^{18} ) q^{78} + ( \zeta_{40}^{10} + \zeta_{40}^{14} ) q^{79} -\zeta_{40}^{15} q^{80} + ( -1 - \zeta_{40}^{2} + \zeta_{40}^{8} - \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{81} + ( \zeta_{40}^{7} - \zeta_{40}^{9} ) q^{83} + ( -\zeta_{40}^{11} + \zeta_{40}^{17} ) q^{84} + ( -\zeta_{40} + \zeta_{40}^{7} - \zeta_{40}^{15} ) q^{90} + ( \zeta_{40} + \zeta_{40}^{7} ) q^{91} + ( -1 + \zeta_{40}^{16} ) q^{92} + ( \zeta_{40}^{4} - \zeta_{40}^{14} ) q^{95} + ( -\zeta_{40}^{5} - \zeta_{40}^{19} ) q^{96} -\zeta_{40}^{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + 4q^{9} + O(q^{10})$$ $$16q + 4q^{4} + 4q^{9} - 4q^{14} - 16q^{15} - 4q^{16} + 4q^{30} - 4q^{36} + 12q^{39} - 16q^{49} + 4q^{50} + 4q^{56} - 4q^{60} + 20q^{63} + 4q^{64} - 4q^{65} - 16q^{81} - 20q^{92} + 4q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{40}^{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.891007 + 0.453990i −0.891007 − 0.453990i 0.453990 − 0.891007i −0.453990 + 0.891007i 0.891007 − 0.453990i −0.891007 + 0.453990i 0.453990 + 0.891007i −0.453990 − 0.891007i −0.156434 − 0.987688i 0.156434 + 0.987688i −0.987688 + 0.156434i 0.987688 − 0.156434i −0.156434 + 0.987688i 0.156434 − 0.987688i −0.987688 − 0.156434i 0.987688 + 0.156434i
−0.951057 + 0.309017i −1.16110 1.59811i 0.809017 0.587785i 0.453990 0.891007i 1.59811 + 1.16110i 1.00000i −0.587785 + 0.809017i −0.896802 + 2.76007i −0.156434 + 0.987688i
69.2 −0.951057 + 0.309017i 1.16110 + 1.59811i 0.809017 0.587785i −0.453990 + 0.891007i −1.59811 1.16110i 1.00000i −0.587785 + 0.809017i −0.896802 + 2.76007i 0.156434 0.987688i
69.3 0.951057 0.309017i −0.183900 0.253116i 0.809017 0.587785i 0.891007 + 0.453990i −0.253116 0.183900i 1.00000i 0.587785 0.809017i 0.278768 0.857960i 0.987688 + 0.156434i
69.4 0.951057 0.309017i 0.183900 + 0.253116i 0.809017 0.587785i −0.891007 0.453990i 0.253116 + 0.183900i 1.00000i 0.587785 0.809017i 0.278768 0.857960i −0.987688 0.156434i
629.1 −0.951057 0.309017i −1.16110 + 1.59811i 0.809017 + 0.587785i 0.453990 + 0.891007i 1.59811 1.16110i 1.00000i −0.587785 0.809017i −0.896802 2.76007i −0.156434 0.987688i
629.2 −0.951057 0.309017i 1.16110 1.59811i 0.809017 + 0.587785i −0.453990 0.891007i −1.59811 + 1.16110i 1.00000i −0.587785 0.809017i −0.896802 2.76007i 0.156434 + 0.987688i
629.3 0.951057 + 0.309017i −0.183900 + 0.253116i 0.809017 + 0.587785i 0.891007 0.453990i −0.253116 + 0.183900i 1.00000i 0.587785 + 0.809017i 0.278768 + 0.857960i 0.987688 0.156434i
629.4 0.951057 + 0.309017i 0.183900 0.253116i 0.809017 + 0.587785i −0.891007 + 0.453990i 0.253116 0.183900i 1.00000i 0.587785 + 0.809017i 0.278768 + 0.857960i −0.987688 + 0.156434i
909.1 −0.587785 + 0.809017i −1.69480 + 0.550672i −0.309017 0.951057i 0.987688 0.156434i 0.550672 1.69480i 1.00000i 0.951057 + 0.309017i 1.76007 1.27877i −0.453990 + 0.891007i
909.2 −0.587785 + 0.809017i 1.69480 0.550672i −0.309017 0.951057i −0.987688 + 0.156434i −0.550672 + 1.69480i 1.00000i 0.951057 + 0.309017i 1.76007 1.27877i 0.453990 0.891007i
909.3 0.587785 0.809017i −0.863541 + 0.280582i −0.309017 0.951057i 0.156434 + 0.987688i −0.280582 + 0.863541i 1.00000i −0.951057 0.309017i −0.142040 + 0.103198i 0.891007 + 0.453990i
909.4 0.587785 0.809017i 0.863541 0.280582i −0.309017 0.951057i −0.156434 0.987688i 0.280582 0.863541i 1.00000i −0.951057 0.309017i −0.142040 + 0.103198i −0.891007 0.453990i
1189.1 −0.587785 0.809017i −1.69480 0.550672i −0.309017 + 0.951057i 0.987688 + 0.156434i 0.550672 + 1.69480i 1.00000i 0.951057 0.309017i 1.76007 + 1.27877i −0.453990 0.891007i
1189.2 −0.587785 0.809017i 1.69480 + 0.550672i −0.309017 + 0.951057i −0.987688 0.156434i −0.550672 1.69480i 1.00000i 0.951057 0.309017i 1.76007 + 1.27877i 0.453990 + 0.891007i
1189.3 0.587785 + 0.809017i −0.863541 0.280582i −0.309017 + 0.951057i 0.156434 0.987688i −0.280582 0.863541i 1.00000i −0.951057 + 0.309017i −0.142040 0.103198i 0.891007 0.453990i
1189.4 0.587785 + 0.809017i 0.863541 + 0.280582i −0.309017 + 0.951057i −0.156434 + 0.987688i 0.280582 + 0.863541i 1.00000i −0.951057 + 0.309017i −0.142040 0.103198i −0.891007 + 0.453990i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1189.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner
25.e even 10 1 inner
175.m odd 10 1 inner
200.o even 10 1 inner
1400.cc odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.cc.a 16
7.b odd 2 1 inner 1400.1.cc.a 16
8.b even 2 1 inner 1400.1.cc.a 16
25.e even 10 1 inner 1400.1.cc.a 16
56.h odd 2 1 CM 1400.1.cc.a 16
175.m odd 10 1 inner 1400.1.cc.a 16
200.o even 10 1 inner 1400.1.cc.a 16
1400.cc odd 10 1 inner 1400.1.cc.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.cc.a 16 1.a even 1 1 trivial
1400.1.cc.a 16 7.b odd 2 1 inner
1400.1.cc.a 16 8.b even 2 1 inner
1400.1.cc.a 16 25.e even 10 1 inner
1400.1.cc.a 16 56.h odd 2 1 CM
1400.1.cc.a 16 175.m odd 10 1 inner
1400.1.cc.a 16 200.o even 10 1 inner
1400.1.cc.a 16 1400.cc odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$3$ $$1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16}$$
$5$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$7$ $$( 1 + T^{2} )^{8}$$
$11$ $$T^{16}$$
$13$ $$1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16}$$
$17$ $$T^{16}$$
$19$ $$( 16 + 8 T^{2} + 4 T^{4} + 2 T^{6} + T^{8} )^{2}$$
$23$ $$( 5 + 5 T + T^{4} )^{4}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16}$$
$61$ $$1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$( 25 + 25 T^{2} + 10 T^{4} + T^{8} )^{2}$$
$73$ $$T^{16}$$
$79$ $$( 25 + 10 T^{4} + 5 T^{6} + T^{8} )^{2}$$
$83$ $$1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$