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 $8$

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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:

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

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