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Properties

Label 1600.3.g.j
Level $1600$
Weight $3$
Character orbit 1600.g
Analytic conductor $43.597$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,3,Mod(351,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.351");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2342560000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} + \beta_{5}) q^{7} + ( - \beta_{2} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} + \beta_{5}) q^{7} + ( - \beta_{2} + 7) q^{9} + ( - \beta_{3} + 2 \beta_1) q^{11} - \beta_{7} q^{13} + ( - \beta_{2} + 4) q^{17} + ( - \beta_{3} - 4 \beta_1) q^{19} + (\beta_{7} - 5 \beta_{4}) q^{21} - 3 \beta_{6} q^{23} + (2 \beta_{3} + 12 \beta_1) q^{27} + 2 \beta_{7} q^{29} + (2 \beta_{6} + 7 \beta_{5}) q^{31} + ( - \beta_{2} + 34) q^{33} + ( - 3 \beta_{7} - 5 \beta_{4}) q^{37} - 2 \beta_{6} q^{39} + (\beta_{2} + 26) q^{41} + (4 \beta_{3} + 5 \beta_1) q^{43} + ( - 5 \beta_{6} - 12 \beta_{5}) q^{47} + (3 \beta_{2} - 15) q^{49} + (2 \beta_{3} + 18 \beta_1) q^{51} + (\beta_{7} + 4 \beta_{4}) q^{53} + (5 \beta_{2} - 62) q^{57} + ( - \beta_{3} - 12 \beta_1) q^{59} + (\beta_{7} - 7 \beta_{4}) q^{61} + (13 \beta_{6} + 36 \beta_{5}) q^{63} + ( - 4 \beta_{3} + 9 \beta_1) q^{67} + ( - 3 \beta_{7} + 9 \beta_{4}) q^{69} + ( - 10 \beta_{6} + 9 \beta_{5}) q^{71} + (3 \beta_{2} + 80) q^{73} + (6 \beta_{7} - 7 \beta_{4}) q^{77} + (4 \beta_{6} + \beta_{5}) q^{79} + ( - 5 \beta_{2} + 125) q^{81} + ( - 10 \beta_{3} - 13 \beta_1) q^{83} + 4 \beta_{6} q^{87} + (2 \beta_{2} + 10) q^{89} + ( - 4 \beta_{3} + 6 \beta_1) q^{91} + (2 \beta_{7} - 20 \beta_{4}) q^{93} + ( - 5 \beta_{2} - 24) q^{97} + (11 \beta_{3} + 30 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 56 q^{9} + 32 q^{17} + 272 q^{33} + 208 q^{41} - 120 q^{49} - 496 q^{57} + 640 q^{73} + 1000 q^{81} + 80 q^{89} - 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 17\nu^{4} - 29\nu^{3} + 54\nu^{2} - 40\nu + 20 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} - 17\nu^{4} + 29\nu^{3} - 34\nu^{2} + 20\nu + 70 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} + 9\nu^{5} - 61\nu^{4} + 107\nu^{3} - 272\nu^{2} + 220\nu - 130 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\nu^{7} - 77\nu^{6} + 487\nu^{5} - 1025\nu^{4} + 2547\nu^{3} - 2834\nu^{2} + 2640\nu - 880 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -44\nu^{7} + 154\nu^{6} - 974\nu^{5} + 2050\nu^{4} - 5094\nu^{3} + 5668\nu^{2} - 5080\nu + 1660 ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 182\nu^{7} - 637\nu^{6} + 4047\nu^{5} - 8525\nu^{4} + 21307\nu^{3} - 23754\nu^{2} + 21340\nu - 6980 ) / 50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 152\nu^{7} - 532\nu^{6} + 3392\nu^{5} - 7150\nu^{4} + 18052\nu^{3} - 20194\nu^{2} + 18940\nu - 6330 ) / 25 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} + 2\beta_{2} + 4\beta _1 - 32 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 6\beta_{6} - 13\beta_{5} - 15\beta_{4} + 3\beta_{2} + 6\beta _1 - 50 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - 12\beta_{6} - 27\beta_{5} - 32\beta_{4} - 4\beta_{3} - 16\beta_{2} - 56\beta _1 + 236 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{7} + 70\beta_{6} + 146\beta_{5} + 113\beta_{4} - 10\beta_{3} - 45\beta_{2} - 150\beta _1 + 674 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -29\beta_{7} + 120\beta_{6} + 253\beta_{5} + 210\beta_{4} + 19\beta_{3} + 58\beta_{2} + 270\beta _1 - 856 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 106\beta_{7} - 574\beta_{6} - 1205\beta_{5} - 759\beta_{4} + 168\beta_{3} + 567\beta_{2} + 2422\beta _1 - 8410 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0.500000 − 3.27635i
0.500000 + 3.27635i
0.500000 + 1.04028i
0.500000 − 1.04028i
0.500000 − 0.0402784i
0.500000 + 0.0402784i
0.500000 + 2.27635i
0.500000 − 2.27635i
0 −5.55269 0 0 0 − 10.4162i 0 21.8324 0
351.2 0 −5.55269 0 0 0 10.4162i 0 21.8324 0
351.3 0 −1.08056 0 0 0 − 4.41620i 0 −7.83240 0
351.4 0 −1.08056 0 0 0 4.41620i 0 −7.83240 0
351.5 0 1.08056 0 0 0 − 4.41620i 0 −7.83240 0
351.6 0 1.08056 0 0 0 4.41620i 0 −7.83240 0
351.7 0 5.55269 0 0 0 − 10.4162i 0 21.8324 0
351.8 0 5.55269 0 0 0 10.4162i 0 21.8324 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.g.j 8
4.b odd 2 1 inner 1600.3.g.j 8
5.b even 2 1 320.3.g.b ✓ 8
5.c odd 4 1 1600.3.e.i 8
5.c odd 4 1 1600.3.e.j 8
8.b even 2 1 inner 1600.3.g.j 8
8.d odd 2 1 inner 1600.3.g.j 8
15.d odd 2 1 2880.3.g.e 8
20.d odd 2 1 320.3.g.b ✓ 8
20.e even 4 1 1600.3.e.i 8
20.e even 4 1 1600.3.e.j 8
40.e odd 2 1 320.3.g.b ✓ 8
40.f even 2 1 320.3.g.b ✓ 8
40.i odd 4 1 1600.3.e.i 8
40.i odd 4 1 1600.3.e.j 8
40.k even 4 1 1600.3.e.i 8
40.k even 4 1 1600.3.e.j 8
60.h even 2 1 2880.3.g.e 8
80.k odd 4 2 1280.3.b.e 8
80.q even 4 2 1280.3.b.e 8
120.i odd 2 1 2880.3.g.e 8
120.m even 2 1 2880.3.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.g.b ✓ 8 5.b even 2 1
320.3.g.b ✓ 8 20.d odd 2 1
320.3.g.b ✓ 8 40.e odd 2 1
320.3.g.b ✓ 8 40.f even 2 1
1280.3.b.e 8 80.k odd 4 2
1280.3.b.e 8 80.q even 4 2
1600.3.e.i 8 5.c odd 4 1
1600.3.e.i 8 20.e even 4 1
1600.3.e.i 8 40.i odd 4 1
1600.3.e.i 8 40.k even 4 1
1600.3.e.j 8 5.c odd 4 1
1600.3.e.j 8 20.e even 4 1
1600.3.e.j 8 40.i odd 4 1
1600.3.e.j 8 40.k even 4 1
1600.3.g.j 8 1.a even 1 1 trivial
1600.3.g.j 8 4.b odd 2 1 inner
1600.3.g.j 8 8.b even 2 1 inner
1600.3.g.j 8 8.d odd 2 1 inner
2880.3.g.e 8 15.d odd 2 1
2880.3.g.e 8 60.h even 2 1
2880.3.g.e 8 120.i odd 2 1
2880.3.g.e 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} - 32T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} - 8T_{17} - 204 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 32 T^{2} + 36)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 128 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 392 T^{2} + 24336)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 248 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8 T - 204)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 728 T^{2} + 76176)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1008 T^{2} + 236196)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 992 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1792 T^{2} + 207936)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 5672 T^{2} + 7817616)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 52 T + 456)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4608 T^{2} + 4435236)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6448 T^{2} + 224676)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2360 T^{2} + 1040400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4760 T^{2} + 32400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6408 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 6848 T^{2} + 8608356)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 15232 T^{2} + 11451456)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 160 T + 4420)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 880)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 29168 T^{2} + 182412036)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 20 T - 780)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 48 T - 4924)^{4} \) Copy content Toggle raw display
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