Properties

Label 4600.2.e.v
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
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] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (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: \(36.7311849298\)
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: \( x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{2}) q^{3} + \beta_{8} q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{2}) q^{3} + \beta_{8} q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} - 2) q^{9} + (\beta_{9} + \beta_{7} + \beta_{4}) q^{11} + (2 \beta_{5} + \beta_{2} + \beta_1) q^{13} + (\beta_{6} + \beta_{5} + 2 \beta_1) q^{17} + (\beta_{9} - \beta_{7} + \beta_{4} - 2 \beta_{3}) q^{19} + (2 \beta_{3} - 2) q^{21} + \beta_{2} q^{23} + (\beta_{8} + 3 \beta_{6} - 2 \beta_{2}) q^{27} + ( - 2 \beta_{7} + \beta_{4} - \beta_{3} + 2) q^{29} + (\beta_{9} - \beta_{4} - 3 \beta_{3} + 2) q^{31} + ( - 2 \beta_{8} - 3 \beta_{6} + \beta_{5} + 4 \beta_{2}) q^{33} + (2 \beta_{8} - 2 \beta_{6} - 2 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{9} - \beta_{7} - 2 \beta_{4} + 2 \beta_{3} - 8) q^{39} + ( - \beta_{9} + \beta_{7} - 2 \beta_{4} - 2 \beta_{3} - 1) q^{41} + 3 \beta_{8} q^{43} + ( - 3 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_1) q^{47} + (\beta_{9} + 2 \beta_{4} + 1) q^{49} + (2 \beta_{9} + \beta_{7} + \beta_{4}) q^{51} + ( - 2 \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + 6 \beta_{2} + 4 \beta_1) q^{53} + (2 \beta_{8} - \beta_{6} + \beta_{5} - 6 \beta_{2} - 2 \beta_1) q^{57} + (\beta_{9} - 3 \beta_{7} + 3 \beta_{4} - 5) q^{59} + (2 \beta_{3} - 6) q^{61} + (\beta_{8} - 2 \beta_{5}) q^{63} + (\beta_{6} - \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{7} - 1) q^{69} + (\beta_{9} - 3 \beta_{3} + 1) q^{71} + ( - 4 \beta_{6} + \beta_{5} - 5 \beta_1) q^{73} + ( - 3 \beta_{8} - 2 \beta_{6} - 2 \beta_{2}) q^{77} + (\beta_{9} + 2 \beta_{7} - 2 \beta_{4} - 8) q^{79} + (2 \beta_{7} + 3 \beta_{4} + 2 \beta_{3}) q^{81} + ( - \beta_{8} + \beta_{6} - 3 \beta_{5} + 4 \beta_{2} - 2 \beta_1) q^{83} + (2 \beta_{8} + 3 \beta_{5} - 5 \beta_{2} - \beta_1) q^{87} + ( - 4 \beta_{9} + \beta_{7} - 3 \beta_{4} - 4) q^{89} + ( - \beta_{9} - 2 \beta_{7} + 2 \beta_{4} + 4 \beta_{3} - 6) q^{91} + (4 \beta_{8} + 2 \beta_{6} + \beta_{5} - 5 \beta_{2} - 3 \beta_1) q^{93} + ( - 4 \beta_{8} - 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{97} + ( - \beta_{9} - 4 \beta_{7} - 4 \beta_{4} - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 4\nu^{7} - 101\nu^{5} - 493\nu^{3} - 523\nu ) / 74 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{8} - 102\nu^{6} - 366\nu^{4} - 360\nu^{2} - 2 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{8} - 102\nu^{6} - 366\nu^{4} - 323\nu^{2} + 109 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{9} + 102\nu^{7} + 329\nu^{5} - 121\nu^{3} - 1071\nu ) / 74 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{9} + 192\nu^{7} + 1035\nu^{5} + 2199\nu^{3} + 1647\nu ) / 74 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{8} + 249\nu^{6} + 1048\nu^{4} + 1362\nu^{2} + 105 ) / 37 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\nu^{9} + 502\nu^{7} + 1995\nu^{5} + 2231\nu^{3} - 239\nu ) / 74 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -19\nu^{8} - 298\nu^{6} - 1263\nu^{4} - 1585\nu^{2} - 53 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 2\beta_{6} + \beta_{5} + 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 4\beta_{7} - 9\beta_{4} + 10\beta_{3} + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{8} - 25\beta_{6} - 15\beta_{5} - 49\beta_{2} + 35\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{9} - 51\beta_{7} + 86\beta_{4} - 94\beta_{3} - 171 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -137\beta_{8} + 260\beta_{6} + 166\beta_{5} + 499\beta_{2} - 300\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 426\beta_{9} + 534\beta_{7} - 834\beta_{4} + 893\beta_{3} + 1600 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1368\beta_{8} - 2579\beta_{6} - 1686\beta_{5} - 4899\beta_{2} + 2793\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
0.144312i
1.45894i
1.64975i
3.11721i
1.84717i
1.84717i
3.11721i
1.64975i
1.45894i
0.144312i
0 3.08344i 0 0 0 0.555022i 0 −6.50759 0
4049.2 0 2.21042i 0 0 0 2.22487i 0 −1.88594 0
4049.3 0 1.51466i 0 0 0 3.49880i 0 0.705809 0
4049.4 0 1.33689i 0 0 0 3.16736i 0 1.21273 0
4049.5 0 0.724570i 0 0 0 2.33840i 0 2.47500 0
4049.6 0 0.724570i 0 0 0 2.33840i 0 2.47500 0
4049.7 0 1.33689i 0 0 0 3.16736i 0 1.21273 0
4049.8 0 1.51466i 0 0 0 3.49880i 0 0.705809 0
4049.9 0 2.21042i 0 0 0 2.22487i 0 −1.88594 0
4049.10 0 3.08344i 0 0 0 0.555022i 0 −6.50759 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4600.2.e.v 10
5.b even 2 1 inner 4600.2.e.v 10
5.c odd 4 1 4600.2.a.bc 5
5.c odd 4 1 4600.2.a.bg yes 5
20.e even 4 1 9200.2.a.cs 5
20.e even 4 1 9200.2.a.cw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bc 5 5.c odd 4 1
4600.2.a.bg yes 5 5.c odd 4 1
4600.2.e.v 10 1.a even 1 1 trivial
4600.2.e.v 10 5.b even 2 1 inner
9200.2.a.cs 5 20.e even 4 1
9200.2.a.cw 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{10} + 19T_{3}^{8} + 119T_{3}^{6} + 306T_{3}^{4} + 321T_{3}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{10} + 33T_{7}^{8} + 392T_{7}^{6} + 2000T_{7}^{4} + 3904T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{5} + 4T_{11}^{4} - 17T_{11}^{3} - 88T_{11}^{2} - 92T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{10} + 75T_{13}^{8} + 2107T_{13}^{6} + 26966T_{13}^{4} + 146625T_{13}^{2} + 206116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 19 T^{8} + 119 T^{6} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 33 T^{8} + 392 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{5} + 4 T^{4} - 17 T^{3} - 88 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 75 T^{8} + 2107 T^{6} + \cdots + 206116 \) Copy content Toggle raw display
$17$ \( T^{10} + 81 T^{8} + 1800 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{5} + 4 T^{4} - 53 T^{3} - 264 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} - 11 T^{4} - 11 T^{3} + 306 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 4 T^{4} - 75 T^{3} + 159 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 316 T^{8} + \cdots + 28558336 \) Copy content Toggle raw display
$41$ \( (T^{5} + 8 T^{4} - 52 T^{3} - 671 T^{2} + \cdots - 2069)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 297 T^{8} + \cdots + 60466176 \) Copy content Toggle raw display
$47$ \( T^{10} + 338 T^{8} + \cdots + 34951744 \) Copy content Toggle raw display
$53$ \( T^{10} + 428 T^{8} + \cdots + 490356736 \) Copy content Toggle raw display
$59$ \( (T^{5} + 23 T^{4} + 53 T^{3} - 1087 T^{2} + \cdots + 5128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 26 T^{4} + 236 T^{3} + 848 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 129 T^{8} + 5392 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$71$ \( (T^{5} + 2 T^{4} - 67 T^{3} - 133 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 752 T^{8} + \cdots + 2756355001 \) Copy content Toggle raw display
$79$ \( (T^{5} + 43 T^{4} + 634 T^{3} + \cdots - 45872)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 562 T^{8} + \cdots + 1086361600 \) Copy content Toggle raw display
$89$ \( (T^{5} + 15 T^{4} - 154 T^{3} - 3188 T^{2} + \cdots - 7120)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 520 T^{8} + 85872 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
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