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Properties

Label 1600.3.b.w
Level $1600$
Weight $3$
Character orbit 1600.b
Analytic conductor $43.597$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,3,Mod(1151,1600)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.b (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: \(6\)
Coefficient field: 6.0.1827904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 160)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{7} + (\beta_{4} - \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{7} + (\beta_{4} - \beta_{2} - 3) q^{9} + (2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{11} + (\beta_{4} + 2 \beta_{2} - 1) q^{13} + (2 \beta_{4} + 2 \beta_{2} + 12) q^{17} + ( - 2 \beta_{5} - 3 \beta_{3} - 6 \beta_1) q^{19} + (\beta_{4} + \beta_{2} - 2) q^{21} + ( - 3 \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{23} + (2 \beta_{5} + 8 \beta_{3} - 4 \beta_1) q^{27} + (4 \beta_{2} + 6) q^{29} + (4 \beta_{5} + 4 \beta_1) q^{31} + (6 \beta_{2} + 22) q^{33} + (3 \beta_{4} + 4 \beta_{2} - 37) q^{37} + ( - 4 \beta_{5} - 10 \beta_{3} + 4 \beta_1) q^{39} + ( - \beta_{4} - 3 \beta_{2} - 10) q^{41} + ( - 6 \beta_{5} - \beta_1) q^{43} + (\beta_{5} - 12 \beta_{3} - 8 \beta_1) q^{47} + ( - 7 \beta_{4} - \beta_{2} + 13) q^{49} + ( - 4 \beta_{5} - 8 \beta_{3} + 12 \beta_1) q^{51} + ( - 5 \beta_{4} + 6 \beta_{2} - 11) q^{53} + ( - 8 \beta_{4} - 2 \beta_{2} + 70) q^{57} + (2 \beta_{5} - 13 \beta_{3} + 6 \beta_1) q^{59} + ( - 7 \beta_{4} - 3 \beta_{2} + 20) q^{61} + (7 \beta_{5} - 4 \beta_{3} - 2 \beta_1) q^{63} + (2 \beta_{5} + 16 \beta_{3} + 13 \beta_1) q^{67} + ( - \beta_{4} - 13 \beta_{2} - 26) q^{69} + ( - 8 \beta_{5} - 10 \beta_{3}) q^{71} + (4 \beta_{4} + 6 \beta_{2} + 10) q^{73} + ( - 12 \beta_{4} - 2 \beta_{2} - 62) q^{77} + (4 \beta_{5} - 20 \beta_{3} - 12 \beta_1) q^{79} + (7 \beta_{4} + 13 \beta_{2} + 33) q^{81} + (14 \beta_{5} + 16 \beta_{3} + 3 \beta_1) q^{83} + ( - 8 \beta_{5} - 24 \beta_{3} + 26 \beta_1) q^{87} + (16 \beta_{4} + 12 \beta_{2} - 22) q^{89} + ( - 10 \beta_{3} - 8 \beta_1) q^{91} + (8 \beta_{4} - 56) q^{93} + (6 \beta_{4} + 2 \beta_{2} + 32) q^{97} + (6 \beta_{5} - 27 \beta_{3} + 34 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 18 q^{9} + 80 q^{17} - 8 q^{21} + 44 q^{29} + 144 q^{33} - 208 q^{37} - 68 q^{41} + 62 q^{49} - 64 q^{53} + 400 q^{57} + 100 q^{61} - 184 q^{69} + 80 q^{73} - 400 q^{77} + 238 q^{81} - 76 q^{89} - 320 q^{93} + 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 14x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{4} + 20\nu^{2} - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} + 40\nu^{3} + 76\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{4} + 40\nu^{2} + 51 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} - 140\nu^{3} - 194\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{2} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 4\beta_{3} - 11\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{4} + 10\beta_{2} + 69 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -10\beta_{5} - 35\beta_{3} + 72\beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
− 2.65109i
− 1.37720i
− 0.273891i
0.273891i
1.37720i
2.65109i
0 − 5.30219i 0 0 0 0.206625i 0 −19.1132 0
1151.2 0 − 2.75441i 0 0 0 − 3.84997i 0 1.41325 0
1151.3 0 − 0.547781i 0 0 0 10.0566i 0 8.69994 0
1151.4 0 0.547781i 0 0 0 − 10.0566i 0 8.69994 0
1151.5 0 2.75441i 0 0 0 3.84997i 0 1.41325 0
1151.6 0 5.30219i 0 0 0 − 0.206625i 0 −19.1132 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.b.w 6
4.b odd 2 1 inner 1600.3.b.w 6
5.b even 2 1 1600.3.b.v 6
5.c odd 4 1 320.3.h.f 6
5.c odd 4 1 320.3.h.g 6
8.b even 2 1 800.3.b.i 6
8.d odd 2 1 800.3.b.i 6
20.d odd 2 1 1600.3.b.v 6
20.e even 4 1 320.3.h.f 6
20.e even 4 1 320.3.h.g 6
40.e odd 2 1 800.3.b.h 6
40.f even 2 1 800.3.b.h 6
40.i odd 4 1 160.3.h.a ✓ 6
40.i odd 4 1 160.3.h.b yes 6
40.k even 4 1 160.3.h.a ✓ 6
40.k even 4 1 160.3.h.b yes 6
80.i odd 4 1 1280.3.e.f 6
80.i odd 4 1 1280.3.e.g 6
80.j even 4 1 1280.3.e.h 6
80.j even 4 1 1280.3.e.i 6
80.s even 4 1 1280.3.e.f 6
80.s even 4 1 1280.3.e.g 6
80.t odd 4 1 1280.3.e.h 6
80.t odd 4 1 1280.3.e.i 6
120.q odd 4 1 1440.3.j.a 6
120.q odd 4 1 1440.3.j.b 6
120.w even 4 1 1440.3.j.a 6
120.w even 4 1 1440.3.j.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a ✓ 6 40.i odd 4 1
160.3.h.a ✓ 6 40.k even 4 1
160.3.h.b yes 6 40.i odd 4 1
160.3.h.b yes 6 40.k even 4 1
320.3.h.f 6 5.c odd 4 1
320.3.h.f 6 20.e even 4 1
320.3.h.g 6 5.c odd 4 1
320.3.h.g 6 20.e even 4 1
800.3.b.h 6 40.e odd 2 1
800.3.b.h 6 40.f even 2 1
800.3.b.i 6 8.b even 2 1
800.3.b.i 6 8.d odd 2 1
1280.3.e.f 6 80.i odd 4 1
1280.3.e.f 6 80.s even 4 1
1280.3.e.g 6 80.i odd 4 1
1280.3.e.g 6 80.s even 4 1
1280.3.e.h 6 80.j even 4 1
1280.3.e.h 6 80.t odd 4 1
1280.3.e.i 6 80.j even 4 1
1280.3.e.i 6 80.t odd 4 1
1440.3.j.a 6 120.q odd 4 1
1440.3.j.a 6 120.w even 4 1
1440.3.j.b 6 120.q odd 4 1
1440.3.j.b 6 120.w even 4 1
1600.3.b.v 6 5.b even 2 1
1600.3.b.v 6 20.d odd 2 1
1600.3.b.w 6 1.a even 1 1 trivial
1600.3.b.w 6 4.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{6} + 36T_{3}^{4} + 224T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{6} + 116T_{7}^{4} + 1504T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{3} - 208T_{13} + 832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 36 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 116 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 560 T^{4} + \cdots + 2560000 \) Copy content Toggle raw display
$13$ \( (T^{3} - 208 T + 832)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 40 T^{2} + \cdots + 2560)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1712 T^{4} + \cdots + 11505664 \) Copy content Toggle raw display
$23$ \( T^{6} + 1460 T^{4} + \cdots + 83905600 \) Copy content Toggle raw display
$29$ \( (T^{3} - 22 T^{2} + \cdots - 2120)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 2560 T^{4} + \cdots + 419430400 \) Copy content Toggle raw display
$37$ \( (T^{3} + 104 T^{2} + \cdots + 20800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 34 T^{2} + \cdots - 5000)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 4260 T^{4} + \cdots + 39438400 \) Copy content Toggle raw display
$47$ \( T^{6} + 8308 T^{4} + \cdots + 493550656 \) Copy content Toggle raw display
$53$ \( (T^{3} + 32 T^{2} + \cdots - 224320)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 25416011776 \) Copy content Toggle raw display
$61$ \( (T^{3} - 50 T^{2} + \cdots + 81544)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 14180 T^{4} + \cdots + 613057600 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 14231535616 \) Copy content Toggle raw display
$73$ \( (T^{3} - 40 T^{2} + \cdots + 55808)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 37060870144 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 233675560000 \) Copy content Toggle raw display
$89$ \( (T^{3} + 38 T^{2} + \cdots - 155000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 104 T^{2} + \cdots - 6656)^{2} \) Copy content Toggle raw display
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