Properties

Label 1400.2.x.a
Level $1400$
Weight $2$
Character orbit 1400.x
Analytic conductor $11.179$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.29960650073923649536.7
Defining polynomial: \(x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} -\beta_{1} q^{7} -\beta_{4} q^{9} +O(q^{10})\) \( q + \beta_{6} q^{3} -\beta_{1} q^{7} -\beta_{4} q^{9} + ( -1 - \beta_{9} ) q^{11} + ( \beta_{5} + \beta_{6} - \beta_{14} ) q^{13} + ( -\beta_{1} - \beta_{3} + \beta_{12} ) q^{17} + ( \beta_{11} + \beta_{13} ) q^{19} + ( -1 + \beta_{7} - \beta_{9} ) q^{21} + ( -\beta_{5} - \beta_{10} - \beta_{14} ) q^{23} + ( \beta_{1} - \beta_{3} - \beta_{12} ) q^{27} + ( \beta_{2} - 3 \beta_{4} ) q^{29} + ( -\beta_{7} - \beta_{8} ) q^{31} + ( \beta_{5} + \beta_{6} - \beta_{14} ) q^{33} -\beta_{15} q^{37} + ( -\beta_{2} - 3 \beta_{4} ) q^{39} + ( \beta_{7} - \beta_{8} ) q^{41} + ( -\beta_{5} + \beta_{10} - \beta_{14} ) q^{43} -\beta_{3} q^{47} + ( -\beta_{2} - 2 \beta_{4} + \beta_{13} ) q^{49} + ( 1 - 3 \beta_{9} ) q^{51} + ( 2 \beta_{5} + 2 \beta_{14} ) q^{53} + ( 2 \beta_{1} + 2 \beta_{12} + \beta_{15} ) q^{57} + 2 \beta_{11} q^{59} + ( \beta_{7} + \beta_{8} ) q^{61} + ( -\beta_{5} + \beta_{6} + \beta_{10} ) q^{63} + ( -\beta_{1} - \beta_{12} ) q^{67} + ( \beta_{11} + \beta_{13} ) q^{69} + ( -4 - 4 \beta_{9} ) q^{71} -4 \beta_{6} q^{73} + ( \beta_{1} - \beta_{3} - \beta_{12} + \beta_{15} ) q^{77} + ( \beta_{2} - 5 \beta_{4} ) q^{79} + ( 5 - 2 \beta_{9} ) q^{81} + ( -3 \beta_{5} - 4 \beta_{6} + 3 \beta_{14} ) q^{83} + ( 3 \beta_{1} + 5 \beta_{3} - 3 \beta_{12} ) q^{87} + ( -\beta_{11} + \beta_{13} ) q^{89} + ( 5 + \beta_{7} + \beta_{8} + \beta_{9} ) q^{91} + ( 2 \beta_{5} - 3 \beta_{10} + 2 \beta_{14} ) q^{93} + ( -3 \beta_{1} + 5 \beta_{3} + 3 \beta_{12} ) q^{97} -4 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{11} - 8q^{21} + 40q^{51} - 32q^{71} + 96q^{81} + 72q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{13} - 5 \nu^{9} - 36 \nu^{5} + 128 \nu \)\()/128\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{10} - 3 \nu^{6} + 12 \nu^{2} \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{9} + 3 \nu^{5} - 20 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{10} + 28 \nu^{6} \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} - 13 \nu^{11} + 52 \nu^{7} - 416 \nu^{3} \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} - \nu^{11} - 16 \nu^{7} + 16 \nu^{3} \)\()/128\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{12} - \nu^{8} + 28 \nu^{4} + 64 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{12} - 11 \nu^{8} + 84 \nu^{4} - 192 \)\()/32\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{12} - 7 \nu^{8} + 24 \nu^{4} - 64 \)\()/16\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{11} + 4 \nu^{7} + 32 \nu^{3} \)\()/64\)
\(\beta_{11}\)\(=\)\((\)\( -3 \nu^{14} + 17 \nu^{10} - 76 \nu^{6} + 256 \nu^{2} \)\()/64\)
\(\beta_{12}\)\(=\)\((\)\( -5 \nu^{13} + 7 \nu^{9} - 20 \nu^{5} \)\()/128\)
\(\beta_{13}\)\(=\)\((\)\( 5 \nu^{14} - 23 \nu^{10} + 68 \nu^{6} - 64 \nu^{2} \)\()/64\)
\(\beta_{14}\)\(=\)\((\)\( 7 \nu^{15} - 53 \nu^{11} + 196 \nu^{7} - 480 \nu^{3} \)\()/256\)
\(\beta_{15}\)\(=\)\((\)\( 3 \nu^{13} - 17 \nu^{9} + 76 \nu^{5} - 128 \nu \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + 2 \beta_{12} - 4 \beta_{3} + 2 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{13} + 3 \beta_{11} + 4 \beta_{4} - 4 \beta_{2}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{14} + 3 \beta_{10} + 4 \beta_{6} - 6 \beta_{5}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{9} + 5 \beta_{8} + \beta_{7} + 12\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{15} + 14 \beta_{12} - 12 \beta_{3} - 10 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{13} + \beta_{11} + 28 \beta_{4} - 12 \beta_{2}\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(6 \beta_{14} + 21 \beta_{10} - 20 \beta_{6} - 2 \beta_{5}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-28 \beta_{9} + 11 \beta_{8} + 15 \beta_{7} - 76\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(-5 \beta_{15} + 2 \beta_{12} - 20 \beta_{3} - 70 \beta_{1}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-27 \beta_{13} - 33 \beta_{11} + 36 \beta_{4} + 140 \beta_{2}\)\()/8\)
\(\nu^{11}\)\(=\)\((\)\(-38 \beta_{14} + 11 \beta_{10} - 172 \beta_{6} + 34 \beta_{5}\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(28 \beta_{9} - 43 \beta_{8} + 81 \beta_{7} - 308\)\()/8\)
\(\nu^{13}\)\(=\)\((\)\(-27 \beta_{15} - 258 \beta_{12} + 20 \beta_{3} - 58 \beta_{1}\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(59 \beta_{13} - 127 \beta_{11} - 164 \beta_{4} + 756 \beta_{2}\)\()/8\)
\(\nu^{15}\)\(=\)\((\)\(-26 \beta_{14} - 299 \beta_{10} - 468 \beta_{6} - 98 \beta_{5}\)\()/8\)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
657.1
−0.281691 + 1.38588i
−1.38588 + 0.281691i
0.481610 + 1.32968i
−1.32968 0.481610i
1.32968 + 0.481610i
−0.481610 1.32968i
1.38588 0.281691i
0.281691 1.38588i
−0.281691 1.38588i
−1.38588 0.281691i
0.481610 1.32968i
−1.32968 + 0.481610i
1.32968 0.481610i
−0.481610 + 1.32968i
1.38588 + 0.281691i
0.281691 + 1.38588i
0 −1.66757 1.66757i 0 0 0 −2.27950 1.34308i 0 2.56155i 0
657.2 0 −1.66757 1.66757i 0 0 0 1.34308 + 2.27950i 0 2.56155i 0
657.3 0 −0.848071 0.848071i 0 0 0 0.406039 2.61441i 0 1.56155i 0
657.4 0 −0.848071 0.848071i 0 0 0 2.61441 0.406039i 0 1.56155i 0
657.5 0 0.848071 + 0.848071i 0 0 0 −2.61441 + 0.406039i 0 1.56155i 0
657.6 0 0.848071 + 0.848071i 0 0 0 −0.406039 + 2.61441i 0 1.56155i 0
657.7 0 1.66757 + 1.66757i 0 0 0 −1.34308 2.27950i 0 2.56155i 0
657.8 0 1.66757 + 1.66757i 0 0 0 2.27950 + 1.34308i 0 2.56155i 0
993.1 0 −1.66757 + 1.66757i 0 0 0 −2.27950 + 1.34308i 0 2.56155i 0
993.2 0 −1.66757 + 1.66757i 0 0 0 1.34308 2.27950i 0 2.56155i 0
993.3 0 −0.848071 + 0.848071i 0 0 0 0.406039 + 2.61441i 0 1.56155i 0
993.4 0 −0.848071 + 0.848071i 0 0 0 2.61441 + 0.406039i 0 1.56155i 0
993.5 0 0.848071 0.848071i 0 0 0 −2.61441 0.406039i 0 1.56155i 0
993.6 0 0.848071 0.848071i 0 0 0 −0.406039 2.61441i 0 1.56155i 0
993.7 0 1.66757 1.66757i 0 0 0 −1.34308 + 2.27950i 0 2.56155i 0
993.8 0 1.66757 1.66757i 0 0 0 2.27950 1.34308i 0 2.56155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 993.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.x.a 16
5.b even 2 1 inner 1400.2.x.a 16
5.c odd 4 2 inner 1400.2.x.a 16
7.b odd 2 1 inner 1400.2.x.a 16
35.c odd 2 1 inner 1400.2.x.a 16
35.f even 4 2 inner 1400.2.x.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.x.a 16 1.a even 1 1 trivial
1400.2.x.a 16 5.b even 2 1 inner
1400.2.x.a 16 5.c odd 4 2 inner
1400.2.x.a 16 7.b odd 2 1 inner
1400.2.x.a 16 35.c odd 2 1 inner
1400.2.x.a 16 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 33 T_{3}^{4} + 64 \) acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 64 + 33 T^{4} + T^{8} )^{2} \)
$5$ \( T^{16} \)
$7$ \( 5764801 - 67228 T^{4} + 646 T^{8} - 28 T^{12} + T^{16} \)
$11$ \( ( -4 + T + T^{2} )^{8} \)
$13$ \( ( 16384 + 273 T^{4} + T^{8} )^{2} \)
$17$ \( ( 16384 + 273 T^{4} + T^{8} )^{2} \)
$19$ \( ( 512 - 80 T^{2} + T^{4} )^{4} \)
$23$ \( ( 4096 + 2576 T^{4} + T^{8} )^{2} \)
$29$ \( ( 1444 + 77 T^{2} + T^{4} )^{4} \)
$31$ \( ( 2048 + 112 T^{2} + T^{4} )^{4} \)
$37$ \( ( 256 + T^{4} )^{4} \)
$41$ \( ( 512 + 80 T^{2} + T^{4} )^{4} \)
$43$ \( ( 4096 + 6928 T^{4} + T^{8} )^{2} \)
$47$ \( ( 64 + 33 T^{4} + T^{8} )^{2} \)
$53$ \( ( 16777216 + 12544 T^{4} + T^{8} )^{2} \)
$59$ \( ( 2048 - 160 T^{2} + T^{4} )^{4} \)
$61$ \( ( 2048 + 112 T^{2} + T^{4} )^{4} \)
$67$ \( ( 65536 + 784 T^{4} + T^{8} )^{2} \)
$71$ \( ( -64 + 4 T + T^{2} )^{8} \)
$73$ \( ( 4194304 + 8448 T^{4} + T^{8} )^{2} \)
$79$ \( ( 10816 + 217 T^{2} + T^{4} )^{4} \)
$83$ \( ( 133448704 + 36432 T^{4} + T^{8} )^{2} \)
$89$ \( ( 2048 - 112 T^{2} + T^{4} )^{4} \)
$97$ \( ( 467943424 + 128961 T^{4} + T^{8} )^{2} \)
show more
show less