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Properties

Label 1300.2.r.f
Level $1300$
Weight $2$
Character orbit 1300.r
Analytic conductor $10.381$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(957,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.957");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.r (of order \(4\), degree \(2\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 128x^{8} + 652x^{4} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{8} q^{7} + (\beta_{5} + 2 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{8} q^{7} + (\beta_{5} + 2 \beta_{3}) q^{9} - \beta_{7} q^{11} + (\beta_{11} - \beta_{4}) q^{13} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{8}) q^{17}+ \cdots + ( - \beta_{6} + 2 \beta_{5} + 5 \beta_{3} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{11} + 8 q^{19} + 20 q^{21} + 28 q^{31} - 48 q^{39} - 8 q^{41} + 20 q^{49} - 24 q^{59} + 8 q^{61} + 72 q^{69} - 40 q^{71} - 116 q^{81} + 44 q^{89} - 20 q^{91} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 128x^{8} + 652x^{4} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 162\nu^{4} + 2038 ) / 458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} + 905\nu^{6} + 6022\nu^{2} ) / 4122 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{11} - 905\nu^{7} - 6022\nu^{3} ) / 4122 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -35\nu^{10} - 4525\nu^{6} - 25988\nu^{2} ) / 4122 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 89\nu^{10} - 108\nu^{8} + 11212\nu^{6} - 13374\nu^{4} + 37112\nu^{2} - 30492 ) / 8244 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -89\nu^{10} - 108\nu^{8} - 11212\nu^{6} - 13374\nu^{4} - 37112\nu^{2} - 30492 ) / 8244 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -187\nu^{11} + 126\nu^{9} - 23882\nu^{7} + 16290\nu^{5} - 113176\nu^{3} + 83664\nu ) / 24732 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -187\nu^{11} - 126\nu^{9} - 23882\nu^{7} - 16290\nu^{5} - 113176\nu^{3} - 83664\nu ) / 24732 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -206\nu^{11} + 225\nu^{9} - 26044\nu^{7} + 28206\nu^{5} - 94190\nu^{3} + 75204\nu ) / 24732 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 206\nu^{11} + 225\nu^{9} + 26044\nu^{7} + 28206\nu^{5} + 94190\nu^{3} + 75204\nu ) / 24732 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 12\beta_{2} - 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{11} - 14\beta_{10} - 25\beta_{9} + 25\beta_{8} - 84\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{7} - 14\beta_{6} - 134\beta_{5} - 492\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -162\beta_{11} + 162\beta_{10} - 282\beta_{9} - 282\beta_{8} + 922\beta_{4} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -162\beta_{7} - 162\beta_{6} - 1486\beta_{2} + 5414 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1810\beta_{11} + 1810\beta_{10} + 3134\beta_{9} - 3134\beta_{8} + 10196\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1810\beta_{7} + 1810\beta_{6} + 16464\beta_{5} + 59896\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20084\beta_{11} - 20084\beta_{10} + 34738\beta_{9} + 34738\beta_{8} - 112908\beta_{4} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
957.1
−2.35344 + 2.35344i
−1.04303 + 1.04303i
−0.611071 + 0.611071i
0.611071 − 0.611071i
1.04303 − 1.04303i
2.35344 − 2.35344i
−2.35344 − 2.35344i
−1.04303 − 1.04303i
−0.611071 − 0.611071i
0.611071 + 0.611071i
1.04303 + 1.04303i
2.35344 + 2.35344i
0 −2.35344 + 2.35344i 0 0 0 −3.43215 0 − 8.07737i 0
957.2 0 −1.04303 + 1.04303i 0 0 0 0.790185 0 0.824185i 0
957.3 0 −0.611071 + 0.611071i 0 0 0 3.68727 0 2.25318i 0
957.4 0 0.611071 − 0.611071i 0 0 0 −3.68727 0 2.25318i 0
957.5 0 1.04303 − 1.04303i 0 0 0 −0.790185 0 0.824185i 0
957.6 0 2.35344 − 2.35344i 0 0 0 3.43215 0 − 8.07737i 0
993.1 0 −2.35344 − 2.35344i 0 0 0 −3.43215 0 8.07737i 0
993.2 0 −1.04303 − 1.04303i 0 0 0 0.790185 0 − 0.824185i 0
993.3 0 −0.611071 − 0.611071i 0 0 0 3.68727 0 − 2.25318i 0
993.4 0 0.611071 + 0.611071i 0 0 0 −3.68727 0 − 2.25318i 0
993.5 0 1.04303 + 1.04303i 0 0 0 −0.790185 0 − 0.824185i 0
993.6 0 2.35344 + 2.35344i 0 0 0 3.43215 0 8.07737i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 957.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.f even 4 1 inner
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.r.f yes 12
5.b even 2 1 inner 1300.2.r.f yes 12
5.c odd 4 2 1300.2.m.f ✓ 12
13.d odd 4 1 1300.2.m.f ✓ 12
65.f even 4 1 inner 1300.2.r.f yes 12
65.g odd 4 1 1300.2.m.f ✓ 12
65.k even 4 1 inner 1300.2.r.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1300.2.m.f ✓ 12 5.c odd 4 2
1300.2.m.f ✓ 12 13.d odd 4 1
1300.2.m.f ✓ 12 65.g odd 4 1
1300.2.r.f yes 12 1.a even 1 1 trivial
1300.2.r.f yes 12 5.b even 2 1 inner
1300.2.r.f yes 12 65.f even 4 1 inner
1300.2.r.f yes 12 65.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 128T_{3}^{8} + 652T_{3}^{4} + 324 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 128 T^{8} + \cdots + 324 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 26 T^{4} + \cdots - 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 2 T^{5} + \cdots + 3042)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 50 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 3964 T^{8} + \cdots + 3240000 \) Copy content Toggle raw display
$19$ \( (T^{6} - 4 T^{5} + \cdots + 450)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 3940 T^{8} + \cdots + 27060804 \) Copy content Toggle raw display
$29$ \( (T^{6} + 120 T^{4} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 14 T^{5} + \cdots + 101250)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 98 T^{4} + \cdots - 33124)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} + \cdots + 14112)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 36466485444 \) Copy content Toggle raw display
$47$ \( (T^{6} - 166 T^{4} + \cdots - 44100)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27518828544 \) Copy content Toggle raw display
$59$ \( (T^{6} + 12 T^{5} + \cdots + 246402)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 2 T^{2} - 30 T - 42)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 226 T^{4} + \cdots + 6084)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 20 T^{5} + \cdots + 118098)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 118 T^{4} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 232 T^{4} + \cdots + 24336)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 34 T^{4} + \cdots - 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 22 T^{5} + \cdots + 520200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 368 T^{4} + \cdots + 295936)^{2} \) Copy content Toggle raw display
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