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Properties

Label 1413.2.b.c
Level $1413$
Weight $2$
Character orbit 1413.b
Analytic conductor $11.283$
Analytic rank $0$
Dimension $12$
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] = [1413,2,Mod(784,1413)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1413, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1413.784");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1413 = 3^{2} \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1413.b (of order \(2\), degree \(1\), 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: \(11.2828618056\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 15x^{10} + 77x^{8} + 158x^{6} + 111x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 471)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{7} - \beta_1) q^{5} - \beta_{10} q^{7} + (\beta_{7} + \beta_{6} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{7} - \beta_1) q^{5} - \beta_{10} q^{7} + (\beta_{7} + \beta_{6} - \beta_1) q^{8} + ( - \beta_{2} + 2) q^{10} + (\beta_{4} + \beta_{2}) q^{11} - \beta_{3} q^{13} + \beta_{3} q^{14} + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{16} + ( - \beta_{8} + \beta_{5} + \cdots - \beta_{2}) q^{17}+ \cdots + (\beta_{10} + \beta_{9} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{4} + 18 q^{10} + 2 q^{11} + 4 q^{13} - 4 q^{14} + 10 q^{16} - 2 q^{17} + 4 q^{19} + 12 q^{25} + 2 q^{31} + 4 q^{35} - 2 q^{37} - 40 q^{40} + 36 q^{44} - 34 q^{47} + 10 q^{49} - 6 q^{52} - 2 q^{56} + 24 q^{58} + 28 q^{64} - 38 q^{67} - 32 q^{68} + 26 q^{71} - 28 q^{76} - 50 q^{82} - 6 q^{86} + 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 15x^{10} + 77x^{8} + 158x^{6} + 111x^{4} + 21x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 14\nu^{8} + 65\nu^{6} + 115\nu^{4} + 62\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{10} + 15\nu^{8} + 77\nu^{6} + 157\nu^{4} + 103\nu^{2} + 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{10} + 15\nu^{8} + 77\nu^{6} + 158\nu^{4} + 110\nu^{2} + 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{11} - 15\nu^{9} - 77\nu^{7} - 158\nu^{5} - 110\nu^{3} - 16\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{11} + 15\nu^{9} + 77\nu^{7} + 158\nu^{5} + 111\nu^{3} + 21\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{10} - 44\nu^{8} - 217\nu^{6} - 409\nu^{4} - 220\nu^{2} - 13 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -2\nu^{11} - 30\nu^{9} - 154\nu^{7} - 315\nu^{5} - 213\nu^{3} - 24\nu \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5\nu^{11} + 74\nu^{9} + 371\nu^{7} + 725\nu^{5} + 440\nu^{3} + 43\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -9\nu^{11} - 134\nu^{9} - 677\nu^{7} - 1335\nu^{5} - 818\nu^{3} - 81\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} - 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - 7\beta_{7} - 9\beta_{6} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} - 10\beta_{5} + 11\beta_{4} + \beta_{3} + 46\beta_{2} - 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + \beta_{10} - 11\beta_{9} + 46\beta_{7} + 66\beta_{6} - 155\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -12\beta_{8} + 78\beta_{5} - 89\beta_{4} - 14\beta_{3} - 299\beta_{2} + 485 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -14\beta_{11} - 16\beta_{10} + 89\beta_{9} - 299\beta_{7} - 454\beta_{6} + 928\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 103\beta_{8} - 557\beta_{5} + 646\beta_{4} + 133\beta_{3} + 1939\beta_{2} - 2939 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 133\beta_{11} + 163\beta_{10} - 646\beta_{9} + 1939\beta_{7} + 3039\beta_{6} - 5717\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1413\mathbb{Z}\right)^\times\).

\(n\) \(1100\) \(1261\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
784.1
− 2.55080i
− 2.06731i
− 1.76113i
− 0.927515i
− 0.430002i
− 0.269982i
0.269982i
0.430002i
0.927515i
1.76113i
2.06731i
2.55080i
− 2.55080i 0 −4.50658 2.15877i 0 − 0.830530i 6.39378i 0 5.50658
784.2 − 2.06731i 0 −2.27376 1.58359i 0 2.74875i 0.565947i 0 3.27376
784.3 − 1.76113i 0 −1.10159 1.19332i 0 − 3.36519i − 1.58223i 0 2.10159
784.4 − 0.927515i 0 1.13972 − 0.150634i 0 1.39352i − 2.91213i 0 −0.139715
784.5 − 0.430002i 0 1.81510 − 1.89557i 0 − 3.40343i − 1.64050i 0 −0.815098
784.6 − 0.269982i 0 1.92711 − 3.43396i 0 1.97607i − 1.06025i 0 −0.927110
784.7 0.269982i 0 1.92711 3.43396i 0 − 1.97607i 1.06025i 0 −0.927110
784.8 0.430002i 0 1.81510 1.89557i 0 3.40343i 1.64050i 0 −0.815098
784.9 0.927515i 0 1.13972 0.150634i 0 − 1.39352i 2.91213i 0 −0.139715
784.10 1.76113i 0 −1.10159 − 1.19332i 0 3.36519i 1.58223i 0 2.10159
784.11 2.06731i 0 −2.27376 − 1.58359i 0 − 2.74875i − 0.565947i 0 3.27376
784.12 2.55080i 0 −4.50658 − 2.15877i 0 0.830530i − 6.39378i 0 5.50658
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 784.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1413.2.b.c 12
3.b odd 2 1 471.2.b.a ✓ 12
157.b even 2 1 inner 1413.2.b.c 12
471.d odd 2 1 471.2.b.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.b.a ✓ 12 3.b odd 2 1
471.2.b.a ✓ 12 471.d odd 2 1
1413.2.b.c 12 1.a even 1 1 trivial
1413.2.b.c 12 157.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 15T_{2}^{10} + 77T_{2}^{8} + 158T_{2}^{6} + 111T_{2}^{4} + 21T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1413, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 15 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 24 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} + 37 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} - 45 T^{4} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 2 T^{5} - 36 T^{4} + \cdots - 72)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + T^{5} - 55 T^{4} + \cdots - 2088)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} + \cdots - 544)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 187 T^{10} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 1230045184 \) Copy content Toggle raw display
$31$ \( (T^{6} - T^{5} - 127 T^{4} + \cdots + 7152)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + T^{5} + \cdots + 21072)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 240 T^{10} + \cdots + 3610000 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 23740646400 \) Copy content Toggle raw display
$47$ \( (T^{6} + 17 T^{5} + \cdots + 18432)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 282 T^{10} + \cdots + 1937664 \) Copy content Toggle raw display
$59$ \( T^{12} + 201 T^{10} + \cdots + 238144 \) Copy content Toggle raw display
$61$ \( T^{12} + 362 T^{10} + \cdots + 5308416 \) Copy content Toggle raw display
$67$ \( (T^{6} + 19 T^{5} + \cdots - 115200)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 13 T^{5} + \cdots - 660960)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 215737344 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 324000000 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 132066064 \) Copy content Toggle raw display
$89$ \( (T^{6} - 2 T^{5} + \cdots - 253944)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 6737126400 \) Copy content Toggle raw display
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