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Properties

Label 4100.2.b.h
Level $4100$
Weight $2$
Character orbit 4100.b
Analytic conductor $32.739$
Analytic rank $0$
Dimension $10$
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] = [4100,2,Mod(901,4100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4100.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.b (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: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 22x^{8} + 160x^{6} + 467x^{4} + 502x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{9} q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{9} + \beta_{8} - \beta_1) q^{11} + ( - \beta_{9} + \beta_{5} - \beta_1) q^{13} + (\beta_{5} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{5} + \beta_1) q^{19} + ( - \beta_{6} + \beta_{2} - 1) q^{21} + (\beta_{3} - 1) q^{23} + (\beta_{9} + 2 \beta_{8} - 2 \beta_1) q^{27} + ( - \beta_{8} + 2 \beta_1) q^{29} + (\beta_{4} - \beta_{3} - 2) q^{31} + ( - \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{33} + (\beta_{6} - 2 \beta_{3}) q^{37} + ( - \beta_{6} + \beta_{4} - \beta_{3} + 3) q^{39} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 2) q^{41}+ \cdots + ( - \beta_{9} - \beta_{8} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 14 q^{9} - 12 q^{21} - 14 q^{23} - 14 q^{31} + 14 q^{33} + 6 q^{37} + 38 q^{39} + 16 q^{41} + 40 q^{43} - 12 q^{49} - 38 q^{51} - 42 q^{57} - 2 q^{59} - 2 q^{61} - 26 q^{73} - 46 q^{77} + 22 q^{81} - 28 q^{83} - 70 q^{87} - 50 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 22x^{8} + 160x^{6} + 467x^{4} + 502x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} - 104\nu^{6} - 661\nu^{4} - 1343\nu^{2} - 444 ) / 71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{8} - 173\nu^{6} - 991\nu^{4} - 1892\nu^{2} - 714 ) / 71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\nu^{9} + 208\nu^{7} + 1393\nu^{5} + 3680\nu^{3} + 3373\nu ) / 213 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -18\nu^{8} - 346\nu^{6} - 1911\nu^{4} - 3003\nu^{2} - 363 ) / 71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43\nu^{9} + 937\nu^{7} + 6565\nu^{5} + 17102\nu^{3} + 13801\nu ) / 639 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -32\nu^{9} - 623\nu^{7} - 3563\nu^{5} - 6025\nu^{3} + 325\nu ) / 639 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 64\nu^{9} + 1246\nu^{7} + 7126\nu^{5} + 12689\nu^{3} + 4462\nu ) / 639 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{8} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 2\beta_{4} - 11\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{9} - 26\beta_{8} - 2\beta_{7} + 5\beta_{5} + 75\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{6} + 33\beta_{4} - 9\beta_{3} + 117\beta_{2} - 264 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 159\beta_{9} + 291\beta_{8} + 42\beta_{7} - 89\beta_{5} - 756\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 159\beta_{6} - 422\beta_{4} + 173\beta_{3} - 1248\beta_{2} + 2643 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1725\beta_{9} - 3167\beta_{8} - 595\beta_{7} + 1176\beta_{5} + 7884\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
− 3.28424i
− 2.41265i
− 1.71283i
− 1.50498i
− 0.440624i
0.440624i
1.50498i
1.71283i
2.41265i
3.28424i
0 − 3.28424i 0 0 0 − 3.44721i 0 −7.78621 0
901.2 0 − 2.41265i 0 0 0 − 0.837962i 0 −2.82089 0
901.3 0 − 1.71283i 0 0 0 4.95182i 0 0.0662279 0
901.4 0 − 1.50498i 0 0 0 − 1.21231i 0 0.735027 0
901.5 0 − 0.440624i 0 0 0 1.55702i 0 2.80585 0
901.6 0 0.440624i 0 0 0 − 1.55702i 0 2.80585 0
901.7 0 1.50498i 0 0 0 1.21231i 0 0.735027 0
901.8 0 1.71283i 0 0 0 − 4.95182i 0 0.0662279 0
901.9 0 2.41265i 0 0 0 0.837962i 0 −2.82089 0
901.10 0 3.28424i 0 0 0 3.44721i 0 −7.78621 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4100.2.b.h ✓ 10
5.b even 2 1 4100.2.b.i yes 10
5.c odd 4 2 4100.2.g.f 20
41.b even 2 1 inner 4100.2.b.h ✓ 10
205.c even 2 1 4100.2.b.i yes 10
205.g odd 4 2 4100.2.g.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4100.2.b.h ✓ 10 1.a even 1 1 trivial
4100.2.b.h ✓ 10 41.b even 2 1 inner
4100.2.b.i yes 10 5.b even 2 1
4100.2.b.i yes 10 205.c even 2 1
4100.2.g.f 20 5.c odd 4 2
4100.2.g.f 20 205.g odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4100, [\chi])\):

\( T_{3}^{10} + 22T_{3}^{8} + 160T_{3}^{6} + 467T_{3}^{4} + 502T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{23}^{5} + 7T_{23}^{4} + T_{23}^{3} - 64T_{23}^{2} - 83T_{23} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 22 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 41 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( T^{10} + 55 T^{8} + \cdots + 8649 \) Copy content Toggle raw display
$13$ \( T^{10} + 90 T^{8} + \cdots + 225 \) Copy content Toggle raw display
$17$ \( T^{10} + 69 T^{8} + \cdots + 42849 \) Copy content Toggle raw display
$19$ \( T^{10} + 139 T^{8} + \cdots + 859329 \) Copy content Toggle raw display
$23$ \( (T^{5} + 7 T^{4} + T^{3} + \cdots - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 98 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( (T^{5} + 7 T^{4} + \cdots - 1539)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 3 T^{4} + \cdots - 12063)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 115856201 \) Copy content Toggle raw display
$43$ \( (T^{5} - 20 T^{4} + \cdots - 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 180 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 2799362281 \) Copy content Toggle raw display
$59$ \( (T^{5} + T^{4} + \cdots + 70839)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + T^{4} - 57 T^{3} + \cdots + 1011)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 238 T^{8} + \cdots + 110889 \) Copy content Toggle raw display
$71$ \( T^{10} + 341 T^{8} + \cdots + 4100625 \) Copy content Toggle raw display
$73$ \( (T^{5} + 13 T^{4} + \cdots + 645)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 152695449 \) Copy content Toggle raw display
$83$ \( (T^{5} + 14 T^{4} + \cdots - 38475)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 512 T^{8} + \cdots + 53450721 \) Copy content Toggle raw display
$97$ \( T^{10} + 380 T^{8} + \cdots + 18275625 \) Copy content Toggle raw display
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