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Properties

Label 1300.2.ba.c
Level $1300$
Weight $2$
Character orbit 1300.ba
Analytic conductor $10.381$
Analytic rank $0$
Dimension $8$
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] = [1300,2,Mod(49,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.ba (of order \(6\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{2}) q^{3} + (\beta_{6} + \beta_{5} + \beta_{4} - 1) q^{7} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{2}) q^{3} + (\beta_{6} + \beta_{5} + \beta_{4} - 1) q^{7} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{9} + ( - \beta_{6} + 1) q^{11} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{13} + (\beta_{6} - \beta_{5} + \beta_{4} + 3) q^{17} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{21} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{23} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_1) q^{27} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{7} - 4 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{31} + ( - 3 \beta_{6} - \beta_{3} - 2 \beta_{2} - 1) q^{33} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 2 \beta_1 - 3) q^{37} + ( - \beta_{7} - 5 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{39} + ( - 3 \beta_{7} - \beta_{6} + 1) q^{41} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_1 - 3) q^{43} + (\beta_{7} - \beta_1) q^{47} + (4 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{49} + (\beta_{7} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{51} + ( - 4 \beta_{7} + 3 \beta_{5} + 6 \beta_{4} - \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{53} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_1 + 4) q^{57} + (3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 4) q^{59} + (4 \beta_{7} + \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{61} + ( - \beta_{7} + 4 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{63} + ( - 2 \beta_{7} + 5 \beta_{6} - \beta_{4} - 4 \beta_1 + 5) q^{67} + (4 \beta_{7} - 7 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + \cdots + 2) q^{69}+ \cdots + ( - 4 \beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} - 6 q^{7} + 4 q^{9} + 12 q^{11} + 18 q^{17} + 6 q^{23} + 6 q^{33} - 18 q^{37} + 4 q^{39} + 12 q^{41} - 18 q^{43} + 4 q^{49} + 36 q^{57} + 24 q^{59} - 4 q^{61} + 12 q^{63} + 18 q^{67} + 24 q^{69} - 36 q^{71} - 48 q^{73} + 16 q^{79} + 8 q^{81} - 72 q^{83} - 18 q^{87} + 24 q^{89} - 48 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 14\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 6\nu^{6} - \nu^{5} + 12\nu^{4} - 5\nu^{3} - 22\nu^{2} + 36\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 3\nu^{5} - 3\nu^{4} + 7\nu^{3} + 3\nu^{2} - 22\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} - 3\nu^{5} - 10\nu^{4} + 13\nu^{3} + 8\nu^{2} - 32\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} + 4\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{6} + 3\beta_{5} - \beta_{4} + \beta_{3} + 4\beta_{2} + 4\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{7} + 6\beta_{6} + 9\beta_{5} - \beta_{3} + \beta_{2} + 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6\beta_{7} + 16\beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 3\beta_{2} + 2\beta _1 + 17 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.27597 − 0.609843i
0.665665 + 1.24775i
1.40994 + 0.109843i
1.20036 − 0.747754i
−1.27597 + 0.609843i
0.665665 − 1.24775i
1.40994 − 0.109843i
1.20036 + 0.747754i
0 −1.38581 + 0.800098i 0 0 0 −2.16612 + 3.75184i 0 −0.219687 + 0.380509i 0
49.2 0 −0.0820885 + 0.0473938i 0 0 0 0.413419 − 0.716063i 0 −1.49551 + 2.59030i 0
49.3 0 2.01978 − 1.16612i 0 0 0 −0.199902 + 0.346241i 0 1.21969 − 2.11256i 0
49.4 0 2.44811 − 1.41342i 0 0 0 −1.04739 + 1.81414i 0 2.49551 − 4.32235i 0
849.1 0 −1.38581 − 0.800098i 0 0 0 −2.16612 − 3.75184i 0 −0.219687 − 0.380509i 0
849.2 0 −0.0820885 − 0.0473938i 0 0 0 0.413419 + 0.716063i 0 −1.49551 − 2.59030i 0
849.3 0 2.01978 + 1.16612i 0 0 0 −0.199902 − 0.346241i 0 1.21969 + 2.11256i 0
849.4 0 2.44811 + 1.41342i 0 0 0 −1.04739 − 1.81414i 0 2.49551 + 4.32235i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 849.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.ba.c 8
5.b even 2 1 1300.2.ba.b 8
5.c odd 4 1 260.2.x.a ✓ 8
5.c odd 4 1 1300.2.y.b 8
13.e even 6 1 1300.2.ba.b 8
15.e even 4 1 2340.2.dj.d 8
20.e even 4 1 1040.2.da.c 8
65.l even 6 1 inner 1300.2.ba.c 8
65.o even 12 1 3380.2.a.p 4
65.q odd 12 1 3380.2.f.i 8
65.r odd 12 1 260.2.x.a ✓ 8
65.r odd 12 1 1300.2.y.b 8
65.r odd 12 1 3380.2.f.i 8
65.t even 12 1 3380.2.a.q 4
195.bf even 12 1 2340.2.dj.d 8
260.bg even 12 1 1040.2.da.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.x.a ✓ 8 5.c odd 4 1
260.2.x.a ✓ 8 65.r odd 12 1
1040.2.da.c 8 20.e even 4 1
1040.2.da.c 8 260.bg even 12 1
1300.2.y.b 8 5.c odd 4 1
1300.2.y.b 8 65.r odd 12 1
1300.2.ba.b 8 5.b even 2 1
1300.2.ba.b 8 13.e even 6 1
1300.2.ba.c 8 1.a even 1 1 trivial
1300.2.ba.c 8 65.l even 6 1 inner
2340.2.dj.d 8 15.e even 4 1
2340.2.dj.d 8 195.bf even 12 1
3380.2.a.p 4 65.o even 12 1
3380.2.a.q 4 65.t even 12 1
3380.2.f.i 8 65.q odd 12 1
3380.2.f.i 8 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 6T_{3}^{7} + 10T_{3}^{6} + 12T_{3}^{5} - 33T_{3}^{4} - 36T_{3}^{3} + 106T_{3}^{2} + 18T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + 10 T^{6} + 12 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + 30 T^{6} + 48 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 16 T^{6} - 96 T^{5} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 18 T^{7} + 126 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{8} - 30 T^{6} + 867 T^{4} + \cdots + 1089 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} - 18 T^{6} + 180 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{8} + 42 T^{6} + 192 T^{5} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( (T^{4} + 96 T^{2} + 2112)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 18 T^{7} + 258 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} - 21 T^{2} + 198 T + 1089)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + 46 T^{6} + \cdots + 5031049 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} + 456 T^{6} + 69168 T^{4} + \cdots + 389376 \) Copy content Toggle raw display
$59$ \( T^{8} - 24 T^{7} + 174 T^{6} + \cdots + 558009 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + 118 T^{6} + \cdots + 942841 \) Copy content Toggle raw display
$67$ \( T^{8} - 18 T^{7} + 330 T^{6} + \cdots + 1083681 \) Copy content Toggle raw display
$71$ \( T^{8} + 36 T^{7} + 414 T^{6} + \cdots + 45198729 \) Copy content Toggle raw display
$73$ \( (T^{4} + 24 T^{3} + 156 T^{2} - 1584)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} - 180 T^{2} + 1504 T - 368)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + 408 T^{2} + 1440 T + 576)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 24 T^{7} + 174 T^{6} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{8} - 6 T^{7} + 222 T^{6} + \cdots + 12981609 \) Copy content Toggle raw display
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