Properties

Label 2900.2.c.f
Level $2900$
Weight $2$
Character orbit 2900.c
Analytic conductor $23.157$
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] = [2900,2,Mod(349,2900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2900.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.c (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: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 580)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{4} + \beta_{2}) q^{7} + (\beta_1 - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{4} + \beta_{2}) q^{7} + (\beta_1 - 4) q^{9} + ( - \beta_{3} + 3) q^{11} + \beta_{5} q^{13} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{17} + ( - \beta_{3} - \beta_1 - 2) q^{19} + (2 \beta_{3} - \beta_1 + 5) q^{21} + (\beta_{5} - \beta_{2}) q^{23} + (2 \beta_{5} + 2 \beta_{2}) q^{27} + q^{29} + ( - \beta_{3} - \beta_1 + 2) q^{31} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{2}) q^{33} + (\beta_{5} + \beta_{4} - 3 \beta_{2}) q^{37} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{39} + (\beta_1 + 3) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{43} + (2 \beta_{4} + \beta_{2}) q^{47} + ( - 4 \beta_{3} + \beta_1) q^{49} + (3 \beta_1 - 9) q^{51} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{53} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2}) q^{57} + ( - 2 \beta_{3} + 3 \beta_1 - 3) q^{59} + ( - 2 \beta_{3} + \beta_1 - 7) q^{61} + ( - 3 \beta_{4} - 5 \beta_{2}) q^{63} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{67} + ( - 2 \beta_{3} - \beta_1 - 3) q^{69} + (2 \beta_{3} - \beta_1 - 3) q^{71} + (2 \beta_{5} - \beta_{2}) q^{73} + (\beta_{5} - \beta_{4}) q^{77} + ( - \beta_{3} - 2 \beta_1 + 1) q^{79} + ( - 4 \beta_{3} - 3 \beta_1 + 10) q^{81} + (3 \beta_{4} + 3 \beta_{2}) q^{83} - \beta_{2} q^{87} + (2 \beta_{3} - 2 \beta_1 - 6) q^{89} + (2 \beta_{3} - 2) q^{91} + ( - 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{2}) q^{93} + (6 \beta_{4} - \beta_{2}) q^{97} + (5 \beta_{3} + 4 \beta_1 - 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 226\nu - 138 ) / 131 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -46\nu^{5} + 56\nu^{4} - 14\nu^{3} - 308\nu^{2} - 772\nu + 534 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -46\nu^{5} + 56\nu^{4} - 14\nu^{3} - 46\nu^{2} - 772\nu + 534 ) / 131 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 4\beta_{3} + 4\beta_{2} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 3\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{5} + 3\beta_{4} + 18\beta_{3} - 18\beta_{2} + 7\beta _1 - 31 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.66044 + 1.66044i
−1.33641 + 1.33641i
0.675970 + 0.675970i
0.675970 0.675970i
−1.33641 1.33641i
1.66044 1.66044i
0 3.32088i 0 0 0 1.32088i 0 −8.02827 0
349.2 0 2.67282i 0 0 0 4.67282i 0 −4.14399 0
349.3 0 1.35194i 0 0 0 0.648061i 0 1.17226 0
349.4 0 1.35194i 0 0 0 0.648061i 0 1.17226 0
349.5 0 2.67282i 0 0 0 4.67282i 0 −4.14399 0
349.6 0 3.32088i 0 0 0 1.32088i 0 −8.02827 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
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 2900.2.c.f 6
5.b even 2 1 inner 2900.2.c.f 6
5.c odd 4 1 580.2.a.c 3
5.c odd 4 1 2900.2.a.g 3
15.e even 4 1 5220.2.a.x 3
20.e even 4 1 2320.2.a.m 3
40.i odd 4 1 9280.2.a.bk 3
40.k even 4 1 9280.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.a.c 3 5.c odd 4 1
2320.2.a.m 3 20.e even 4 1
2900.2.a.g 3 5.c odd 4 1
2900.2.c.f 6 1.a even 1 1 trivial
2900.2.c.f 6 5.b even 2 1 inner
5220.2.a.x 3 15.e even 4 1
9280.2.a.bk 3 40.i odd 4 1
9280.2.a.bw 3 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\):

\( T_{3}^{6} + 20T_{3}^{4} + 112T_{3}^{2} + 144 \) Copy content Toggle raw display
\( T_{7}^{6} + 24T_{7}^{4} + 48T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 20 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} - 8 T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 44 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$17$ \( T^{6} + 76 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( (T^{3} + 8 T^{2} + \cdots - 164)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 48 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T - 1)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 32 T - 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 204 T^{4} + \cdots + 258064 \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 228 T^{4} + \cdots + 300304 \) Copy content Toggle raw display
$47$ \( T^{6} + 52 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{6} + 124 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} + \cdots - 1488)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 22 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 96 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + \cdots - 144)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 164 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$79$ \( (T^{3} - 108 T - 244)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 216 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 500 T^{4} + \cdots + 3640464 \) Copy content Toggle raw display
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