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Properties

Label 2800.2.k.m
Level $2800$
Weight $2$
Character orbit 2800.k
Analytic conductor $22.358$
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] = [2800,2,Mod(2351,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.k (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: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.116319195136.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 77x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 560)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{4} q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{4} q^{7} + (\beta_1 + 1) q^{9} - \beta_{7} q^{11} - \beta_{3} q^{13} + \beta_{3} q^{17} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{19} + ( - \beta_{3} + \beta_{2} - 1) q^{21} + ( - \beta_{5} + \beta_{4}) q^{23} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4}) q^{27} - \beta_1 q^{29} + 2 \beta_{6} q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{33} + (2 \beta_1 + 2) q^{37} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4}) q^{39} + 2 \beta_{2} q^{41} + (2 \beta_{7} - \beta_{5} + \beta_{4}) q^{43} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{47} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{49} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4}) q^{51} + 6 q^{53} + ( - 2 \beta_1 - 4) q^{57} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{59} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{61} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5}) q^{63} + (2 \beta_{7} + \beta_{5} - \beta_{4}) q^{67} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{69} + 2 \beta_{7} q^{71} + 2 \beta_{2} q^{73} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{77} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{4}) q^{79} + 7 q^{81} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{83} + (5 \beta_{6} - \beta_{5} - \beta_{4}) q^{87} - 4 \beta_{2} q^{89} + ( - \beta_{7} + 4 \beta_{6} + 2 \beta_{5}) q^{91} + ( - 2 \beta_1 - 8) q^{93} + (\beta_{3} - 4 \beta_{2}) q^{97} + ( - 4 \beta_{5} + 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{9} - 8 q^{21} - 4 q^{29} + 24 q^{37} - 20 q^{49} + 48 q^{53} - 40 q^{57} - 8 q^{77} + 56 q^{81} - 72 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 77x^{4} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 43 ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 79\nu^{2} ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{6} + 377\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{7} - 2\nu^{5} - 535\nu^{3} - 158\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} + 2\nu^{5} + 693\nu^{3} + 122\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 2\nu^{5} + 693\nu^{3} - 158\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 2\nu^{5} + 851\nu^{3} + 194\nu ) / 36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{3} - 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{7} - 3\beta_{6} + 8\beta_{5} + 19\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -70\beta_{7} - 27\beta_{6} + 167\beta_{5} + 70\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 237\beta_{3} + 377\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -614\beta_{7} + 237\beta_{6} - 614\beta_{5} - 1465\beta_{4} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
−0.337637 + 0.337637i
−0.337637 − 0.337637i
−2.09428 − 2.09428i
−2.09428 + 2.09428i
2.09428 − 2.09428i
2.09428 + 2.09428i
0.337637 + 0.337637i
0.337637 − 0.337637i
0 −2.96176 0 0 0 0.337637 − 2.62412i 0 5.77200 0
2351.2 0 −2.96176 0 0 0 0.337637 + 2.62412i 0 5.77200 0
2351.3 0 −0.477491 0 0 0 2.09428 − 1.61679i 0 −2.77200 0
2351.4 0 −0.477491 0 0 0 2.09428 + 1.61679i 0 −2.77200 0
2351.5 0 0.477491 0 0 0 −2.09428 − 1.61679i 0 −2.77200 0
2351.6 0 0.477491 0 0 0 −2.09428 + 1.61679i 0 −2.77200 0
2351.7 0 2.96176 0 0 0 −0.337637 − 2.62412i 0 5.77200 0
2351.8 0 2.96176 0 0 0 −0.337637 + 2.62412i 0 5.77200 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2351.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.k.m 8
4.b odd 2 1 inner 2800.2.k.m 8
5.b even 2 1 560.2.k.b ✓ 8
5.c odd 4 1 2800.2.e.g 8
5.c odd 4 1 2800.2.e.h 8
7.b odd 2 1 inner 2800.2.k.m 8
15.d odd 2 1 5040.2.d.d 8
20.d odd 2 1 560.2.k.b ✓ 8
20.e even 4 1 2800.2.e.g 8
20.e even 4 1 2800.2.e.h 8
28.d even 2 1 inner 2800.2.k.m 8
35.c odd 2 1 560.2.k.b ✓ 8
35.f even 4 1 2800.2.e.g 8
35.f even 4 1 2800.2.e.h 8
40.e odd 2 1 2240.2.k.d 8
40.f even 2 1 2240.2.k.d 8
60.h even 2 1 5040.2.d.d 8
105.g even 2 1 5040.2.d.d 8
140.c even 2 1 560.2.k.b ✓ 8
140.j odd 4 1 2800.2.e.g 8
140.j odd 4 1 2800.2.e.h 8
280.c odd 2 1 2240.2.k.d 8
280.n even 2 1 2240.2.k.d 8
420.o odd 2 1 5040.2.d.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.b ✓ 8 5.b even 2 1
560.2.k.b ✓ 8 20.d odd 2 1
560.2.k.b ✓ 8 35.c odd 2 1
560.2.k.b ✓ 8 140.c even 2 1
2240.2.k.d 8 40.e odd 2 1
2240.2.k.d 8 40.f even 2 1
2240.2.k.d 8 280.c odd 2 1
2240.2.k.d 8 280.n even 2 1
2800.2.e.g 8 5.c odd 4 1
2800.2.e.g 8 20.e even 4 1
2800.2.e.g 8 35.f even 4 1
2800.2.e.g 8 140.j odd 4 1
2800.2.e.h 8 5.c odd 4 1
2800.2.e.h 8 20.e even 4 1
2800.2.e.h 8 35.f even 4 1
2800.2.e.h 8 140.j odd 4 1
2800.2.k.m 8 1.a even 1 1 trivial
2800.2.k.m 8 4.b odd 2 1 inner
2800.2.k.m 8 7.b odd 2 1 inner
2800.2.k.m 8 28.d even 2 1 inner
5040.2.d.d 8 15.d odd 2 1
5040.2.d.d 8 60.h even 2 1
5040.2.d.d 8 105.g even 2 1
5040.2.d.d 8 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3}^{4} - 9T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} + 35T_{11}^{2} + 288 \) Copy content Toggle raw display
\( T_{19}^{4} - 76T_{19}^{2} + 1152 \) Copy content Toggle raw display
\( T_{37}^{2} - 6T_{37} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 9 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 35 T^{2} + 288)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 37 T^{2} + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 37 T^{2} + 324)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 76 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 38 T^{2} + 288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 36 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 162 T^{2} + 648)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 65 T^{2} + 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 140 T^{2} + 4608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 196 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 194 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 140 T^{2} + 4608)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 171 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 162 T^{2} + 648)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 349 T^{2} + 19044)^{2} \) Copy content Toggle raw display
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