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Properties

Label 1380.2.f.b
Level $1380$
Weight $2$
Character orbit 1380.f
Analytic conductor $11.019$
Analytic rank $0$
Dimension $14$
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] = [1380,2,Mod(829,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1380.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.f (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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{10} q^{5} + \beta_{13} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{10} q^{5} + \beta_{13} q^{7} - q^{9} + ( - \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 1) q^{11}+ \cdots + (\beta_{10} - \beta_{9} + \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{9} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 6 q^{25} - 30 q^{29} + 6 q^{31} - 14 q^{35} + 4 q^{39} + 46 q^{41} - 20 q^{49} + 2 q^{51} - 16 q^{55} - 10 q^{59} + 64 q^{61} - 36 q^{65} + 14 q^{69} + 42 q^{71} - 16 q^{75} - 32 q^{79} + 14 q^{81} - 42 q^{85} - 52 q^{89} + 28 q^{91} - 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} + 6\nu^{10} - 153\nu^{8} - 1732\nu^{6} - 5726\nu^{4} - 5036\nu^{2} + 136 ) / 372 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{12} + 338\nu^{10} + 2851\nu^{8} + 11282\nu^{6} + 21122\nu^{4} + 15228\nu^{2} + 800 ) / 372 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{12} - 86\nu^{10} - 659\nu^{8} - 2124\nu^{6} - 2423\nu^{4} - 6\nu^{2} + 200 ) / 62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{13} + 160\nu^{11} + 1655\nu^{9} + 8332\nu^{7} + 20731\nu^{5} + 22050\nu^{3} + 5218\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 12 \nu^{10} - 6459 \nu^{9} - 306 \nu^{8} - 34215 \nu^{7} + \cdots - 1216 ) / 744 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 74 \nu^{10} - 6459 \nu^{9} - 934 \nu^{8} - 34215 \nu^{7} + \cdots - 1264 ) / 744 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 74 \nu^{10} - 6459 \nu^{9} + 934 \nu^{8} - 34215 \nu^{7} + \cdots + 1264 ) / 744 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 12 \nu^{10} - 6459 \nu^{9} + 306 \nu^{8} - 34215 \nu^{7} + \cdots + 1216 ) / 744 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\nu^{13} + 468\nu^{11} + 5271\nu^{9} + 28460\nu^{7} + 74017\nu^{5} + 78826\nu^{3} + 16870\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -27\nu^{13} - 689\nu^{11} - 6781\nu^{9} - 32255\nu^{7} - 74798\nu^{5} - 71046\nu^{3} - 10244\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -22\nu^{13} - 597\nu^{11} - 6306\nu^{9} - 32483\nu^{7} - 82534\nu^{5} - 88786\nu^{3} - 20848\nu ) / 372 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{10} - 3\beta_{9} - 3\beta_{8} + 3\beta_{7} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{7} + 2\beta_{3} - 7\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{13} + \beta_{11} - 19\beta_{10} + 21\beta_{9} + 21\beta_{8} - 19\beta_{7} - 3\beta_{6} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{10} + 4\beta_{9} - 4\beta_{8} + 17\beta_{7} + 2\beta_{5} + 2\beta_{4} - 24\beta_{3} + 49\beta_{2} - 156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 30 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} + 128 \beta_{10} - 156 \beta_{9} - 156 \beta_{8} + \cdots + 99 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 197 \beta_{10} - 68 \beta_{9} + 68 \beta_{8} - 197 \beta_{7} - 30 \beta_{5} - 28 \beta_{4} + 230 \beta_{3} + \cdots + 1072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 338 \beta_{13} - 32 \beta_{12} + 157 \beta_{11} - 904 \beta_{10} + 1194 \beta_{9} + 1194 \beta_{8} + \cdots - 841 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1965 \beta_{10} + 810 \beta_{9} - 810 \beta_{8} + 1965 \beta_{7} + 322 \beta_{5} + 282 \beta_{4} + \cdots - 7692 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3390 \beta_{13} + 362 \beta_{12} - 1437 \beta_{11} + 6624 \beta_{10} - 9320 \beta_{9} + \cdots + 7003 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18213 \beta_{10} - 8336 \beta_{9} + 8336 \beta_{8} - 18213 \beta_{7} - 3058 \beta_{5} - 2512 \beta_{4} + \cdots + 57120 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 31918 \beta_{13} - 3604 \beta_{12} + 12389 \beta_{11} - 49978 \beta_{10} + 73852 \beta_{9} + \cdots - 57917 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
2.39625i
0.511427i
2.47928i
− 2.04690i
− 2.88373i
0.0729221i
− 1.52925i
− 2.39625i
− 0.511427i
− 2.47928i
2.04690i
2.88373i
− 0.0729221i
1.52925i
0 − 1.00000i 0 −2.22987 − 0.166381i 0 1.74202i 0 −1.00000 0
829.2 0 − 1.00000i 0 −1.81604 + 1.30461i 0 − 3.73844i 0 −1.00000 0
829.3 0 − 1.00000i 0 −0.258163 − 2.22111i 0 2.14682i 0 −1.00000 0
829.4 0 − 1.00000i 0 −0.181772 + 2.22867i 0 0.189781i 0 −1.00000 0
829.5 0 − 1.00000i 0 0.792997 + 2.09073i 0 4.31589i 0 −1.00000 0
829.6 0 − 1.00000i 0 1.54426 − 1.61718i 0 − 3.99468i 0 −1.00000 0
829.7 0 − 1.00000i 0 2.14859 − 0.619333i 0 − 1.66138i 0 −1.00000 0
829.8 0 1.00000i 0 −2.22987 + 0.166381i 0 − 1.74202i 0 −1.00000 0
829.9 0 1.00000i 0 −1.81604 − 1.30461i 0 3.73844i 0 −1.00000 0
829.10 0 1.00000i 0 −0.258163 + 2.22111i 0 − 2.14682i 0 −1.00000 0
829.11 0 1.00000i 0 −0.181772 − 2.22867i 0 − 0.189781i 0 −1.00000 0
829.12 0 1.00000i 0 0.792997 − 2.09073i 0 − 4.31589i 0 −1.00000 0
829.13 0 1.00000i 0 1.54426 + 1.61718i 0 3.99468i 0 −1.00000 0
829.14 0 1.00000i 0 2.14859 + 0.619333i 0 1.66138i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.14
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 1380.2.f.b ✓ 14
3.b odd 2 1 4140.2.f.c 14
5.b even 2 1 inner 1380.2.f.b ✓ 14
5.c odd 4 1 6900.2.a.bc 7
5.c odd 4 1 6900.2.a.bd 7
15.d odd 2 1 4140.2.f.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.b ✓ 14 1.a even 1 1 trivial
1380.2.f.b ✓ 14 5.b even 2 1 inner
4140.2.f.c 14 3.b odd 2 1
4140.2.f.c 14 15.d odd 2 1
6900.2.a.bc 7 5.c odd 4 1
6900.2.a.bd 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 59T_{7}^{12} + 1323T_{7}^{10} + 14065T_{7}^{8} + 72984T_{7}^{6} + 178488T_{7}^{4} + 166704T_{7}^{2} + 5776 \) acting on \(S_{2}^{\mathrm{new}}(1380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + 3 T^{12} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 59 T^{12} + \cdots + 5776 \) Copy content Toggle raw display
$11$ \( (T^{7} - 64 T^{5} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 104 T^{12} + \cdots + 6071296 \) Copy content Toggle raw display
$17$ \( T^{14} + 167 T^{12} + \cdots + 10000 \) Copy content Toggle raw display
$19$ \( (T^{7} - 2 T^{6} + \cdots - 36680)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} + 15 T^{6} + \cdots - 12272)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 3 T^{6} + \cdots + 54720)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 16126968064 \) Copy content Toggle raw display
$41$ \( (T^{7} - 23 T^{6} + \cdots + 15248)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 389193984 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 1973013529600 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 374345104 \) Copy content Toggle raw display
$59$ \( (T^{7} + 5 T^{6} + \cdots - 1483200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 32 T^{6} + \cdots + 377248)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 315 T^{12} + \cdots + 300304 \) Copy content Toggle raw display
$71$ \( (T^{7} - 21 T^{6} + \cdots - 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 47127199744 \) Copy content Toggle raw display
$79$ \( (T^{7} + 16 T^{6} + \cdots - 4755584)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 415 T^{12} + \cdots + 1638400 \) Copy content Toggle raw display
$89$ \( (T^{7} + 26 T^{6} + \cdots - 526112)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 804 T^{12} + \cdots + 56070144 \) Copy content Toggle raw display
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