Properties

Label 1300.2.c.f
Level $1300$
Weight $2$
Character orbit 1300.c
Analytic conductor $10.381$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(1249,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, 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("1300.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.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: \(10.3805522628\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 260)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{2} - 4) q^{9} + ( - \beta_{3} - \beta_{2}) q^{11} - \beta_{4} q^{13} - 2 \beta_1 q^{17} + (\beta_{3} - \beta_{2} - 2) q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{21} + (3 \beta_{4} - \beta_1) q^{23} + (2 \beta_{5} + 4 \beta_{4} + 2 \beta_1) q^{27} + ( - \beta_{2} - 3) q^{29} + ( - \beta_{3} - \beta_{2} - 4) q^{31} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{33} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{37} + ( - \beta_{3} - 1) q^{39} + 2 \beta_{3} q^{41} + (\beta_{4} + \beta_1) q^{43} + (\beta_{5} - 3 \beta_{4}) q^{47} + ( - 4 \beta_{3} - 9) q^{49} + (2 \beta_{3} + 2 \beta_{2} - 12) q^{51} + (2 \beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{53} + ( - 3 \beta_{5} - 7 \beta_{4} + 2 \beta_1) q^{57} + (3 \beta_{3} + \beta_{2} + 6) q^{59} + ( - \beta_{2} + 5) q^{61} + (3 \beta_{5} + 19 \beta_{4} + 4 \beta_1) q^{63} + ( - \beta_{5} + 5 \beta_{4} + 2 \beta_1) q^{67} + (4 \beta_{3} + \beta_{2} - 3) q^{69} + ( - \beta_{3} - \beta_{2}) q^{71} + ( - 3 \beta_{5} - 5 \beta_{4} + 2 \beta_1) q^{73} + (2 \beta_{5} - 12 \beta_{4} - 2 \beta_1) q^{77} + (2 \beta_{3} - 2) q^{79} + (4 \beta_{3} - 3 \beta_{2} + 10) q^{81} + ( - \beta_{5} + 3 \beta_{4} + 4 \beta_1) q^{83} + ( - 2 \beta_{5} + 2 \beta_1) q^{87} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{89} + (\beta_{2} - 1) q^{91} + ( - \beta_{5} + 7 \beta_{4} + 2 \beta_1) q^{93} + ( - 2 \beta_{5} + 8 \beta_{4}) q^{97} + (\beta_{3} + \beta_{2} - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 22 q^{9} - 16 q^{19} - 16 q^{21} - 20 q^{29} - 24 q^{31} - 4 q^{39} - 4 q^{41} - 46 q^{49} - 72 q^{51} + 32 q^{59} + 28 q^{61} - 24 q^{69} - 16 q^{79} + 46 q^{81} + 4 q^{89} - 4 q^{91} - 72 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131 \) 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
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -161\nu^{5} + 196\nu^{4} - 49\nu^{3} - 292\nu^{2} - 2702\nu + 1869 ) / 393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 7\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 5\beta_{4} - 4\beta_{3} + \beta_{2} + 4\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} + 3\beta_{2} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{5} - 31\beta_{4} - 18\beta_{3} + 7\beta_{2} - 18\beta _1 - 31 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
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 5.02827i 0 −8.02827 0
1249.2 0 2.67282i 0 0 0 1.14399i 0 −4.14399 0
1249.3 0 1.35194i 0 0 0 4.17226i 0 1.17226 0
1249.4 0 1.35194i 0 0 0 4.17226i 0 1.17226 0
1249.5 0 2.67282i 0 0 0 1.14399i 0 −4.14399 0
1249.6 0 3.32088i 0 0 0 5.02827i 0 −8.02827 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.6
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 1300.2.c.f 6
5.b even 2 1 inner 1300.2.c.f 6
5.c odd 4 1 260.2.a.b 3
5.c odd 4 1 1300.2.a.i 3
15.e even 4 1 2340.2.a.n 3
20.e even 4 1 1040.2.a.o 3
20.e even 4 1 5200.2.a.ci 3
40.i odd 4 1 4160.2.a.bo 3
40.k even 4 1 4160.2.a.br 3
60.l odd 4 1 9360.2.a.da 3
65.f even 4 1 3380.2.f.h 6
65.h odd 4 1 3380.2.a.o 3
65.k even 4 1 3380.2.f.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.a.b 3 5.c odd 4 1
1040.2.a.o 3 20.e even 4 1
1300.2.a.i 3 5.c odd 4 1
1300.2.c.f 6 1.a even 1 1 trivial
1300.2.c.f 6 5.b even 2 1 inner
2340.2.a.n 3 15.e even 4 1
3380.2.a.o 3 65.h odd 4 1
3380.2.f.h 6 65.f even 4 1
3380.2.f.h 6 65.k even 4 1
4160.2.a.bo 3 40.i odd 4 1
4160.2.a.br 3 40.k even 4 1
5200.2.a.ci 3 20.e even 4 1
9360.2.a.da 3 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\):

\( T_{3}^{6} + 20T_{3}^{4} + 112T_{3}^{2} + 144 \) Copy content Toggle raw display
\( T_{7}^{6} + 44T_{7}^{4} + 496T_{7}^{2} + 576 \) 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
$7$ \( T^{6} + 44 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 76 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( (T^{3} + 8 T^{2} + \cdots - 164)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 52 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{3} + 10 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 12 T^{2} + 24 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 92 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 36 T + 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 20 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$47$ \( T^{6} + 76 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( T^{6} + 300 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
$59$ \( (T^{3} - 16 T^{2} + 564)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + 44 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 156 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$71$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 444 T^{4} + \cdots + 3182656 \) Copy content Toggle raw display
$79$ \( (T^{3} + 8 T^{2} - 16 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 300 T^{4} + \cdots + 876096 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} + \cdots - 216)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 396 T^{4} + \cdots + 64 \) Copy content Toggle raw display
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