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Properties

Label 1600.3.h.p
Level $1600$
Weight $3$
Character orbit 1600.h
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(1599,1600)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1599");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.h (of order \(2\), degree \(1\), not 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: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{5} q^{7} + ( - \beta_{3} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{5} q^{7} + ( - \beta_{3} + 9) q^{9} - \beta_{2} q^{11} + (\beta_{7} - 2 \beta_{4}) q^{13} - 9 \beta_{4} q^{17} - 2 \beta_{6} q^{19} + (\beta_{3} + 6) q^{21} + ( - 3 \beta_{5} - 2 \beta_1) q^{23} + (6 \beta_{5} + 12 \beta_1) q^{27} + ( - 2 \beta_{3} - 18) q^{29} + ( - \beta_{6} - 4 \beta_{2}) q^{31} + (2 \beta_{7} - 18 \beta_{4}) q^{33} + ( - \beta_{7} + 10 \beta_{4}) q^{37} + (3 \beta_{6} - 8 \beta_{2}) q^{39} + (3 \beta_{3} + 12) q^{41} + (2 \beta_{5} - 5 \beta_1) q^{43} + ( - 9 \beta_{5} + 4 \beta_1) q^{47} + (3 \beta_{3} - 7) q^{49} - 9 \beta_{2} q^{51} + ( - 3 \beta_{7} - 6 \beta_{4}) q^{53} + ( - 4 \beta_{7} - 12 \beta_{4}) q^{57} + 4 \beta_{2} q^{59} + ( - \beta_{3} + 64) q^{61} + (3 \beta_{5} - 6 \beta_1) q^{63} + ( - 2 \beta_{5} + 7 \beta_1) q^{67} + (5 \beta_{3} - 18) q^{69} + (3 \beta_{6} - 8 \beta_{2}) q^{71} + ( - 4 \beta_{7} - 19 \beta_{4}) q^{73} + ( - 2 \beta_{7} - 6 \beta_{4}) q^{77} + 10 \beta_{2} q^{79} + ( - 9 \beta_{3} + 99) q^{81} + (6 \beta_{5} + 15 \beta_1) q^{83} + (12 \beta_{5} + 6 \beta_1) q^{87} + ( - 4 \beta_{3} + 6) q^{89} + (4 \beta_{6} + 3 \beta_{2}) q^{91} + (6 \beta_{7} - 78 \beta_{4}) q^{93} + (10 \beta_{7} + 13 \beta_{4}) q^{97} + (6 \beta_{6} - 21 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 72 q^{9} + 48 q^{21} - 144 q^{29} + 96 q^{41} - 56 q^{49} + 512 q^{61} - 144 q^{69} + 792 q^{81} + 48 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 24\nu^{3} + 17\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{6} - 16\nu^{4} + 32\nu^{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} + 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 20\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 40\nu^{5} + 104\nu^{3} - 73\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\nu^{6} - 48\nu^{4} + 128\nu^{2} - 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 9\nu^{7} - 24\nu^{5} + 66\nu^{3} - 3\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 3\beta_{4} + 6\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} - 2\beta_{3} - 9\beta_{2} + 36 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 6\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + \beta_{3} - 4\beta_{2} - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 9\beta_{5} - 33\beta_{4} - 39\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{3} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 24\beta_{5} + 87\beta_{4} - 102\beta_1 ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1599.1
−1.40126 + 0.809017i
−1.40126 − 0.809017i
−0.535233 − 0.309017i
−0.535233 + 0.309017i
0.535233 + 0.309017i
0.535233 − 0.309017i
1.40126 − 0.809017i
1.40126 + 0.809017i
0 −5.60503 0 0 0 1.32317 0 22.4164 0
1599.2 0 −5.60503 0 0 0 1.32317 0 22.4164 0
1599.3 0 −2.14093 0 0 0 −9.06914 0 −4.41641 0
1599.4 0 −2.14093 0 0 0 −9.06914 0 −4.41641 0
1599.5 0 2.14093 0 0 0 9.06914 0 −4.41641 0
1599.6 0 2.14093 0 0 0 9.06914 0 −4.41641 0
1599.7 0 5.60503 0 0 0 −1.32317 0 22.4164 0
1599.8 0 5.60503 0 0 0 −1.32317 0 22.4164 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1599.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.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.h.p 8
4.b odd 2 1 inner 1600.3.h.p 8
5.b even 2 1 inner 1600.3.h.p 8
5.c odd 4 1 320.3.b.a 4
5.c odd 4 1 1600.3.b.k 4
8.b even 2 1 400.3.h.d 8
8.d odd 2 1 400.3.h.d 8
15.e even 4 1 2880.3.e.b 4
20.d odd 2 1 inner 1600.3.h.p 8
20.e even 4 1 320.3.b.a 4
20.e even 4 1 1600.3.b.k 4
24.f even 2 1 3600.3.j.k 8
24.h odd 2 1 3600.3.j.k 8
40.e odd 2 1 400.3.h.d 8
40.f even 2 1 400.3.h.d 8
40.i odd 4 1 80.3.b.a ✓ 4
40.i odd 4 1 400.3.b.g 4
40.k even 4 1 80.3.b.a ✓ 4
40.k even 4 1 400.3.b.g 4
60.l odd 4 1 2880.3.e.b 4
80.i odd 4 1 1280.3.g.f 8
80.j even 4 1 1280.3.g.f 8
80.s even 4 1 1280.3.g.f 8
80.t odd 4 1 1280.3.g.f 8
120.i odd 2 1 3600.3.j.k 8
120.m even 2 1 3600.3.j.k 8
120.q odd 4 1 720.3.e.c 4
120.q odd 4 1 3600.3.e.bb 4
120.w even 4 1 720.3.e.c 4
120.w even 4 1 3600.3.e.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.b.a ✓ 4 40.i odd 4 1
80.3.b.a ✓ 4 40.k even 4 1
320.3.b.a 4 5.c odd 4 1
320.3.b.a 4 20.e even 4 1
400.3.b.g 4 40.i odd 4 1
400.3.b.g 4 40.k even 4 1
400.3.h.d 8 8.b even 2 1
400.3.h.d 8 8.d odd 2 1
400.3.h.d 8 40.e odd 2 1
400.3.h.d 8 40.f even 2 1
720.3.e.c 4 120.q odd 4 1
720.3.e.c 4 120.w even 4 1
1280.3.g.f 8 80.i odd 4 1
1280.3.g.f 8 80.j even 4 1
1280.3.g.f 8 80.s even 4 1
1280.3.g.f 8 80.t odd 4 1
1600.3.b.k 4 5.c odd 4 1
1600.3.b.k 4 20.e even 4 1
1600.3.h.p 8 1.a even 1 1 trivial
1600.3.h.p 8 4.b odd 2 1 inner
1600.3.h.p 8 5.b even 2 1 inner
1600.3.h.p 8 20.d odd 2 1 inner
2880.3.e.b 4 15.e even 4 1
2880.3.e.b 4 60.l odd 4 1
3600.3.e.bb 4 120.q odd 4 1
3600.3.e.bb 4 120.w even 4 1
3600.3.j.k 8 24.f even 2 1
3600.3.j.k 8 24.h odd 2 1
3600.3.j.k 8 120.i odd 2 1
3600.3.j.k 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 \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} - 36T_{3}^{2} + 144 \) Copy content Toggle raw display
\( T_{7}^{4} - 84T_{7}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 36 T^{2} + 144)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 84 T^{2} + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 144 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 392 T^{2} + 26896)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 324)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1344 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 756 T^{2} + 121104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36 T - 396)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3024 T^{2} + 2214144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1160 T^{2} + 48400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 24 T - 1476)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1476 T^{2} + 535824)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 8244 T^{2} + 898704)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3528 T^{2} + 2178576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2304 T^{2} + 589824)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 128 T + 3916)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2436 T^{2} + 1468944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 9936 T^{2} + 3873024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8648 T^{2} + 2062096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 14400 T^{2} + 23040000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8964 T^{2} + 4210704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T - 2844)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 37352 T^{2} + 300120976)^{2} \) Copy content Toggle raw display
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