Properties

Label 1600.2.l.h
Level $1600$
Weight $2$
Character orbit 1600.l
Analytic conductor $12.776$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.534694406811304329216.1
Defining polynomial: \(x^{16} - 2 x^{14} - 2 x^{12} + 4 x^{10} + 4 x^{8} + 16 x^{6} - 32 x^{4} - 128 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q -\beta_{14} q^{3} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{14} ) q^{7} + ( 1 - \beta_{4} - \beta_{10} - \beta_{15} ) q^{9} +O(q^{10})\) \( q -\beta_{14} q^{3} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{14} ) q^{7} + ( 1 - \beta_{4} - \beta_{10} - \beta_{15} ) q^{9} + ( -1 + \beta_{4} + \beta_{9} + \beta_{10} ) q^{11} + ( -\beta_{5} - \beta_{14} ) q^{13} + ( -\beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{17} + ( -\beta_{9} - \beta_{13} ) q^{19} + ( -1 - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{21} + ( -\beta_{2} + \beta_{3} - \beta_{14} ) q^{23} + ( \beta_{1} + \beta_{3} + \beta_{11} + \beta_{14} ) q^{27} + ( 1 + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{29} + ( -1 - \beta_{4} + 2 \beta_{12} + \beta_{15} ) q^{31} + ( -\beta_{3} + \beta_{6} - \beta_{11} - \beta_{14} ) q^{33} + ( -2 \beta_{3} + 2 \beta_{8} ) q^{37} + ( 1 - \beta_{4} + 4 \beta_{9} - \beta_{15} ) q^{39} + ( 1 - \beta_{4} + \beta_{7} - \beta_{9} - 3 \beta_{10} - \beta_{12} - \beta_{13} ) q^{41} + ( \beta_{1} - 2 \beta_{8} + \beta_{11} + \beta_{14} ) q^{43} + ( -2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{11} + \beta_{14} ) q^{47} + ( -1 + \beta_{4} - \beta_{7} - \beta_{13} ) q^{49} + ( 2 - 3 \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{11} + \beta_{14} ) q^{53} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{8} + \beta_{14} ) q^{57} + ( -1 - \beta_{4} - \beta_{7} + \beta_{9} - \beta_{12} ) q^{59} + ( 2 \beta_{10} + 2 \beta_{12} ) q^{61} + ( -2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} + 3 \beta_{14} ) q^{63} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} - \beta_{11} - 2 \beta_{14} ) q^{67} + ( -2 - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} ) q^{69} + ( 2 - 2 \beta_{4} - 2 \beta_{15} ) q^{71} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{14} ) q^{73} + ( -3 \beta_{5} + \beta_{14} ) q^{77} + ( 1 + \beta_{4} - \beta_{7} + 3 \beta_{12} - \beta_{13} ) q^{79} + ( -1 + \beta_{4} + 3 \beta_{12} - \beta_{15} ) q^{81} + ( 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{11} ) q^{83} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{8} ) q^{87} + ( \beta_{7} + 4 \beta_{9} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{89} + ( 2 - 2 \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} - 3 \beta_{12} ) q^{91} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{6} + \beta_{11} + \beta_{14} ) q^{93} + ( \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - \beta_{11} - 3 \beta_{14} ) q^{97} + ( 6 + 5 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{11} - 8q^{19} - 16q^{21} + 16q^{29} - 16q^{31} - 16q^{49} + 16q^{51} - 24q^{59} - 32q^{69} + 16q^{79} - 16q^{81} + 16q^{91} + 88q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{14} - 2 x^{12} + 4 x^{10} + 4 x^{8} + 16 x^{6} - 32 x^{4} - 128 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{15} + 2 \nu^{13} + 6 \nu^{11} - 20 \nu^{9} - 76 \nu^{7} + 192 \nu^{5} - 224 \nu^{3} - 448 \nu \)\()/576\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{13} + \nu^{11} - 2 \nu^{7} + 8 \nu^{5} - 36 \nu^{3} + 48 \nu \)\()/48\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} - \nu^{13} + 6 \nu^{11} - 2 \nu^{9} + 20 \nu^{7} + 12 \nu^{5} - 8 \nu^{3} + 224 \nu \)\()/288\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{14} + 2 \nu^{12} + 6 \nu^{10} + 4 \nu^{8} - 28 \nu^{6} + 48 \nu^{4} - 32 \nu^{2} + 80 \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{15} - \nu^{13} - 2 \nu^{9} + 32 \nu^{7} + 36 \nu^{5} - 104 \nu^{3} + 32 \nu \)\()/288\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{13} + 2 \nu^{11} + 12 \nu^{9} - 4 \nu^{7} + 16 \nu^{5} - 32 \nu^{3} + 320 \nu \)\()/192\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{14} + 6 \nu^{12} + 6 \nu^{10} - 28 \nu^{8} + 20 \nu^{6} + 144 \nu^{4} + 192 \nu^{2} - 128 \)\()/192\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} + 3 \nu^{11} + 8 \nu^{9} + 10 \nu^{7} - 48 \nu^{5} - 52 \nu^{3} + 160 \nu \)\()/144\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{14} + 2 \nu^{12} + 18 \nu^{10} - 20 \nu^{8} - 4 \nu^{6} - 144 \nu^{4} - 128 \nu^{2} + 704 \)\()/576\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{14} - 2 \nu^{10} + 4 \nu^{6} + 8 \nu^{4} - 32 \nu^{2} + 32 \)\()/96\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{15} - 2 \nu^{13} + 2 \nu^{11} + 4 \nu^{9} - 4 \nu^{7} + 16 \nu^{5} + 16 \nu^{3} - 96 \nu \)\()/96\)
\(\beta_{12}\)\(=\)\((\)\( -3 \nu^{14} + 2 \nu^{12} + 6 \nu^{10} + 12 \nu^{8} + 20 \nu^{6} - 96 \nu^{4} + 320 \)\()/192\)
\(\beta_{13}\)\(=\)\((\)\( -2 \nu^{14} + 5 \nu^{12} + 6 \nu^{10} - 2 \nu^{8} - 76 \nu^{6} - 60 \nu^{4} + 256 \nu^{2} + 368 \)\()/144\)
\(\beta_{14}\)\(=\)\((\)\( -5 \nu^{15} + 2 \nu^{13} + 12 \nu^{11} - 8 \nu^{9} + 8 \nu^{7} - 120 \nu^{5} + 40 \nu^{3} + 512 \nu \)\()/288\)
\(\beta_{15}\)\(=\)\((\)\( 5 \nu^{14} + 10 \nu^{12} - 18 \nu^{10} - 4 \nu^{8} + 4 \nu^{6} + 224 \nu^{2} - 320 \)\()/288\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{11} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} + \beta_{13} + \beta_{12} - 2 \beta_{10} - 4 \beta_{9} + \beta_{7} - 2 \beta_{4} + 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{14} + \beta_{11} - 2 \beta_{8} + \beta_{6} - \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{15} + 2 \beta_{10} - 3 \beta_{9} + \beta_{7} + \beta_{4} + 2\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{11} - 3 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} + \beta_{3} - \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-\beta_{13} + 2 \beta_{12} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{4} + 2\)
\(\nu^{7}\)\(=\)\(-\beta_{14} + \beta_{8} - \beta_{6} + 4 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - \beta_{1}\)
\(\nu^{8}\)\(=\)\(-\beta_{15} + 6 \beta_{12} - 11 \beta_{9} - \beta_{7} + 3 \beta_{4}\)
\(\nu^{9}\)\(=\)\(-2 \beta_{14} + 6 \beta_{11} + 4 \beta_{8} + 6 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-5 \beta_{15} - \beta_{13} + 5 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} + 3 \beta_{7} + 4 \beta_{4} - 10\)
\(\nu^{11}\)\(=\)\(8 \beta_{14} + 12 \beta_{11} + 8 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + 14 \beta_{3} - 8 \beta_{2} + 14 \beta_{1}\)
\(\nu^{12}\)\(=\)\(18 \beta_{15} - 2 \beta_{13} + 10 \beta_{12} + 16 \beta_{10} + 16 \beta_{9} + 2 \beta_{7} + 16 \beta_{4} - 24\)
\(\nu^{13}\)\(=\)\(-8 \beta_{14} - 14 \beta_{11} + 18 \beta_{8} + 6 \beta_{6} - 6 \beta_{5} + 22 \beta_{3} - 26 \beta_{2} + 26 \beta_{1}\)
\(\nu^{14}\)\(=\)\(14 \beta_{15} - 10 \beta_{13} - 10 \beta_{12} - 52 \beta_{10} + 12 \beta_{9} - 6 \beta_{7} + 8 \beta_{4} + 52\)
\(\nu^{15}\)\(=\)\(-36 \beta_{14} - 20 \beta_{11} + 24 \beta_{8} - 20 \beta_{6} - 56 \beta_{5} + 72 \beta_{3} + 4 \beta_{2} + 4 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−0.238945 1.39388i
1.40501 0.161069i
0.841995 1.13624i
1.32661 0.490008i
−1.32661 + 0.490008i
−0.841995 + 1.13624i
−1.40501 + 0.161069i
0.238945 + 1.39388i
−0.238945 + 1.39388i
1.40501 + 0.161069i
0.841995 + 1.13624i
1.32661 + 0.490008i
−1.32661 0.490008i
−0.841995 1.13624i
−1.40501 0.161069i
0.238945 1.39388i
0 −1.99154 1.99154i 0 0 0 1.09033i 0 4.93244i 0
401.2 0 −1.86033 1.86033i 0 0 0 3.61392i 0 3.92163i 0
401.3 0 −0.734294 0.734294i 0 0 0 1.71452i 0 1.92163i 0
401.4 0 −0.183790 0.183790i 0 0 0 3.84853i 0 2.93244i 0
401.5 0 0.183790 + 0.183790i 0 0 0 3.84853i 0 2.93244i 0
401.6 0 0.734294 + 0.734294i 0 0 0 1.71452i 0 1.92163i 0
401.7 0 1.86033 + 1.86033i 0 0 0 3.61392i 0 3.92163i 0
401.8 0 1.99154 + 1.99154i 0 0 0 1.09033i 0 4.93244i 0
1201.1 0 −1.99154 + 1.99154i 0 0 0 1.09033i 0 4.93244i 0
1201.2 0 −1.86033 + 1.86033i 0 0 0 3.61392i 0 3.92163i 0
1201.3 0 −0.734294 + 0.734294i 0 0 0 1.71452i 0 1.92163i 0
1201.4 0 −0.183790 + 0.183790i 0 0 0 3.84853i 0 2.93244i 0
1201.5 0 0.183790 0.183790i 0 0 0 3.84853i 0 2.93244i 0
1201.6 0 0.734294 0.734294i 0 0 0 1.71452i 0 1.92163i 0
1201.7 0 1.86033 1.86033i 0 0 0 3.61392i 0 3.92163i 0
1201.8 0 1.99154 1.99154i 0 0 0 1.09033i 0 4.93244i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1201.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.l.h 16
4.b odd 2 1 400.2.l.i 16
5.b even 2 1 inner 1600.2.l.h 16
5.c odd 4 2 320.2.q.c 16
16.e even 4 1 inner 1600.2.l.h 16
16.f odd 4 1 400.2.l.i 16
20.d odd 2 1 400.2.l.i 16
20.e even 4 2 80.2.q.c 16
40.i odd 4 2 640.2.q.f 16
40.k even 4 2 640.2.q.e 16
60.l odd 4 2 720.2.bm.f 16
80.i odd 4 1 320.2.q.c 16
80.i odd 4 1 640.2.q.f 16
80.j even 4 1 80.2.q.c 16
80.j even 4 1 640.2.q.e 16
80.k odd 4 1 400.2.l.i 16
80.q even 4 1 inner 1600.2.l.h 16
80.s even 4 1 80.2.q.c 16
80.s even 4 1 640.2.q.e 16
80.t odd 4 1 320.2.q.c 16
80.t odd 4 1 640.2.q.f 16
240.z odd 4 1 720.2.bm.f 16
240.bd odd 4 1 720.2.bm.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.c 16 20.e even 4 2
80.2.q.c 16 80.j even 4 1
80.2.q.c 16 80.s even 4 1
320.2.q.c 16 5.c odd 4 2
320.2.q.c 16 80.i odd 4 1
320.2.q.c 16 80.t odd 4 1
400.2.l.i 16 4.b odd 2 1
400.2.l.i 16 16.f odd 4 1
400.2.l.i 16 20.d odd 2 1
400.2.l.i 16 80.k odd 4 1
640.2.q.e 16 40.k even 4 2
640.2.q.e 16 80.j even 4 1
640.2.q.e 16 80.s even 4 1
640.2.q.f 16 40.i odd 4 2
640.2.q.f 16 80.i odd 4 1
640.2.q.f 16 80.t odd 4 1
720.2.bm.f 16 60.l odd 4 2
720.2.bm.f 16 240.z odd 4 1
720.2.bm.f 16 240.bd odd 4 1
1600.2.l.h 16 1.a even 1 1 trivial
1600.2.l.h 16 5.b even 2 1 inner
1600.2.l.h 16 16.e even 4 1 inner
1600.2.l.h 16 80.q even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{16} + 112 T_{3}^{12} + 3144 T_{3}^{8} + 3520 T_{3}^{4} + 16 \)
\( T_{7}^{8} + 32 T_{7}^{6} + 312 T_{7}^{4} + 896 T_{7}^{2} + 676 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 16 + 3520 T^{4} + 3144 T^{8} + 112 T^{12} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 676 + 896 T^{2} + 312 T^{4} + 32 T^{6} + T^{8} )^{2} \)
$11$ \( ( 16 - 160 T + 800 T^{2} + 464 T^{3} + 136 T^{4} - 8 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$13$ \( ( 7744 + 224 T^{4} + T^{8} )^{2} \)
$17$ \( ( 88 - 28 T^{2} + T^{4} )^{4} \)
$19$ \( ( 59536 - 9760 T + 800 T^{2} + 464 T^{3} + 808 T^{4} - 104 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$23$ \( ( 4 + 224 T^{2} + 168 T^{4} + 32 T^{6} + T^{8} )^{2} \)
$29$ \( ( 2704 - 832 T + 128 T^{2} + 992 T^{3} + 1192 T^{4} + 304 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$31$ \( ( 208 - 80 T - 72 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$37$ \( ( 147456 + 1536 T^{4} + T^{8} )^{2} \)
$41$ \( ( 219024 + 110592 T^{2} + 10152 T^{4} + 192 T^{6} + T^{8} )^{2} \)
$43$ \( 9721171216 + 502726720 T^{4} + 5803656 T^{8} + 17296 T^{12} + T^{16} \)
$47$ \( ( 27556 - 12512 T^{2} + 1752 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$53$ \( 4096 + 495616 T^{4} + 2212992 T^{8} + 4672 T^{12} + T^{16} \)
$59$ \( ( 144 + 1440 T + 7200 T^{2} + 4464 T^{3} + 1320 T^{4} - 312 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$61$ \( ( 576 + T^{4} )^{4} \)
$67$ \( 36804120336 + 1578000960 T^{4} + 17125128 T^{8} + 8976 T^{12} + T^{16} \)
$71$ \( ( 65536 + 458752 T^{2} + 19968 T^{4} + 256 T^{6} + T^{8} )^{2} \)
$73$ \( ( 48776256 + 3160512 T^{2} + 62208 T^{4} + 456 T^{6} + T^{8} )^{2} \)
$79$ \( ( -368 + 464 T - 120 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$83$ \( 1475727898361616 + 2351652312384 T^{4} + 660646152 T^{8} + 47568 T^{12} + T^{16} \)
$89$ \( ( 135424 + 120064 T^{2} + 10080 T^{4} + 208 T^{6} + T^{8} )^{2} \)
$97$ \( ( 2383936 - 310208 T^{2} + 12672 T^{4} - 200 T^{6} + T^{8} )^{2} \)
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