Properties

Label 4600.2.e.u
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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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: \( x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + \beta_{3} - 2) q^{9} + (\beta_{4} + \beta_{3}) q^{11} + ( - \beta_{9} - \beta_{8} + \beta_{2}) q^{13} + ( - \beta_{9} + \beta_{6} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{5} - 2 \beta_{4} - 1) q^{19} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{21} - \beta_{2} q^{23} + (\beta_{8} + 2 \beta_{6} - \beta_{2} - 3 \beta_1) q^{27} + (\beta_{7} - \beta_{3} - 1) q^{29} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3} + 4) q^{31} + (\beta_{8} + 2 \beta_{6} + 3 \beta_{2} - 2 \beta_1) q^{33} + ( - \beta_{6} - 3 \beta_{2}) q^{37} + (2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 3) q^{39} + ( - \beta_{7} - \beta_{4} + 5) q^{41} + ( - \beta_{9} - 2 \beta_{6} + 2 \beta_{2}) q^{47} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 6) q^{49} + (2 \beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3) q^{51} + ( - 2 \beta_{9} - 2 \beta_{8} + \beta_{6} + \beta_{2} - 2 \beta_1) q^{53} + (\beta_{9} - \beta_{8} - 9 \beta_{2} - 3 \beta_1) q^{57} + ( - \beta_{7} - 2 \beta_{5} + 1) q^{59} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + (3 \beta_{9} + \beta_{8} - \beta_{6} - 8 \beta_{2} + \beta_1) q^{63} + ( - 2 \beta_{8} + \beta_{6} - \beta_{2} + 2 \beta_1) q^{67} + \beta_{4} q^{69} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + 1) q^{71} + (\beta_{9} + 2 \beta_{6}) q^{73} + (\beta_{9} - \beta_{8} - 2 \beta_{6} - 3 \beta_{2} - \beta_1) q^{77} - 2 \beta_{3} q^{79} + (\beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 10) q^{81} + ( - \beta_{6} - 9 \beta_{2}) q^{83} + (\beta_{9} - 2 \beta_{6} + 2 \beta_{2} + 2 \beta_1) q^{87} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2) q^{89} + ( - 4 \beta_{5} - 5 \beta_{4} + \beta_{3} + 4) q^{91} + (2 \beta_{9} - \beta_{8} - 2 \beta_{6} - 7 \beta_{2} + 5 \beta_1) q^{93} + ( - \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - 6 \beta_{2} - \beta_1) q^{97} + (3 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - 12\nu^{7} - 68\nu^{5} - 833\nu^{3} - 2064\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{8} + 36\nu^{6} - 52\nu^{4} - 829\nu^{2} - 208 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - 12\nu^{6} - 4\nu^{4} + 63\nu^{2} - 16 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{8} + 36\nu^{6} - 52\nu^{4} - 1085\nu^{2} - 1488 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -15\nu^{9} - 436\nu^{7} - 4092\nu^{5} - 12495\nu^{3} - 4592\nu ) / 2048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\nu^{8} + 556\nu^{6} + 3748\nu^{4} + 7513\nu^{2} + 2192 ) / 512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7\nu^{9} + 212\nu^{7} + 2012\nu^{5} + 6343\nu^{3} + 5872\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\nu^{9} + 556\nu^{7} + 3748\nu^{5} + 7513\nu^{3} + 1680\nu ) / 1024 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{6} - \beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{5} - 3\beta_{4} - 14\beta_{3} + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} - 13\beta_{8} - 28\beta_{6} + 3\beta_{2} + 94\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} - 107\beta_{5} + 49\beta_{4} + 164\beta_{3} - 485 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{9} + 156\beta_{8} + 328\beta_{6} + 24\beta_{2} - 1025\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -24\beta_{7} + 1181\beta_{5} - 640\beta_{4} - 1849\beta_{3} + 5305 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -76\beta_{9} - 1821\beta_{8} - 3698\beta_{6} - 683\beta_{2} + 11341\beta_1 \) 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
3.36002i
3.30649i
1.93283i
1.31091i
0.568386i
0.568386i
1.31091i
1.93283i
3.30649i
3.36002i
0 3.36002i 0 0 0 1.90754i 0 −8.28974 0
4049.2 0 3.30649i 0 0 0 2.55040i 0 −7.93288 0
4049.3 0 1.93283i 0 0 0 2.38236i 0 −0.735829 0
4049.4 0 1.31091i 0 0 0 4.66212i 0 1.28151 0
4049.5 0 0.568386i 0 0 0 4.73770i 0 2.67694 0
4049.6 0 0.568386i 0 0 0 4.73770i 0 2.67694 0
4049.7 0 1.31091i 0 0 0 4.66212i 0 1.28151 0
4049.8 0 1.93283i 0 0 0 2.38236i 0 −0.735829 0
4049.9 0 3.30649i 0 0 0 2.55040i 0 −7.93288 0
4049.10 0 3.36002i 0 0 0 1.90754i 0 −8.28974 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.u 10
5.b even 2 1 inner 4600.2.e.u 10
5.c odd 4 1 920.2.a.j 5
5.c odd 4 1 4600.2.a.be 5
15.e even 4 1 8280.2.a.bs 5
20.e even 4 1 1840.2.a.v 5
20.e even 4 1 9200.2.a.cu 5
40.i odd 4 1 7360.2.a.co 5
40.k even 4 1 7360.2.a.cp 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 5.c odd 4 1
1840.2.a.v 5 20.e even 4 1
4600.2.a.be 5 5.c odd 4 1
4600.2.e.u 10 1.a even 1 1 trivial
4600.2.e.u 10 5.b even 2 1 inner
7360.2.a.co 5 40.i odd 4 1
7360.2.a.cp 5 40.k even 4 1
8280.2.a.bs 5 15.e even 4 1
9200.2.a.cu 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{10} + 28T_{3}^{8} + 260T_{3}^{6} + 897T_{3}^{4} + 1056T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{10} + 60T_{7}^{8} + 1268T_{7}^{6} + 11441T_{7}^{4} + 45568T_{7}^{2} + 65536 \) Copy content Toggle raw display
\( T_{11}^{5} + T_{11}^{4} - 35T_{11}^{3} - 28T_{11}^{2} + 172T_{11} + 64 \) Copy content Toggle raw display
\( T_{13}^{10} + 108T_{13}^{8} + 3916T_{13}^{6} + 56825T_{13}^{4} + 285000T_{13}^{2} + 250000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 28 T^{8} + 260 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 60 T^{8} + 1268 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$11$ \( (T^{5} + T^{4} - 35 T^{3} - 28 T^{2} + 172 T + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 108 T^{8} + 3916 T^{6} + \cdots + 250000 \) Copy content Toggle raw display
$17$ \( T^{10} + 108 T^{8} + 3096 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{5} + 7 T^{4} - 41 T^{3} - 180 T^{2} + \cdots - 512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} + 4 T^{4} - 41 T^{3} + 36 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 19 T^{4} + 72 T^{3} + 183 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 97 T^{8} + 2664 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{5} - 25 T^{4} + 212 T^{3} - 653 T^{2} + \cdots + 2182)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( T^{10} + 341 T^{8} + 35052 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$53$ \( T^{10} + 329 T^{8} + \cdots + 410953984 \) Copy content Toggle raw display
$59$ \( (T^{5} - T^{4} - 142 T^{3} + 236 T^{2} + \cdots - 13568)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 5 T^{4} - 115 T^{3} - 568 T^{2} + \cdots + 7664)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 333 T^{8} + \cdots + 67108864 \) Copy content Toggle raw display
$71$ \( (T^{5} - T^{4} - 214 T^{3} + 407 T^{2} + \cdots - 3968)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 317 T^{8} + 29652 T^{6} + \cdots + 1763584 \) Copy content Toggle raw display
$79$ \( (T^{5} - 2 T^{4} - 128 T^{3} - 128 T^{2} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 457 T^{8} + \cdots + 1698758656 \) Copy content Toggle raw display
$89$ \( (T^{5} + 6 T^{4} - 136 T^{3} - 512 T^{2} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 703 T^{8} + \cdots + 2461747456 \) Copy content Toggle raw display
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