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Properties

Label 1260.2.v.b
Level $1260$
Weight $2$
Character orbit 1260.v
Analytic conductor $10.061$
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] = [1260,2,Mod(197,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 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("1260.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.v (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: \(10.0611506547\)
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: \( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{5} - \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{5} - \beta_{6} q^{7} + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{11}+ \cdots + (2 \beta_{11} - \beta_{10} - 2 \beta_{8} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{5} - 12 q^{13} + 8 q^{17} - 8 q^{23} + 8 q^{25} - 32 q^{29} + 16 q^{31} + 4 q^{35} + 4 q^{37} + 8 q^{43} + 8 q^{47} - 8 q^{53} + 32 q^{55} - 16 q^{59} + 64 q^{65} - 12 q^{73} - 56 q^{83} - 24 q^{85} - 72 q^{89} + 72 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 478873536 \nu^{11} + 47412266 \nu^{10} + 3079467412 \nu^{9} - 10994046217 \nu^{8} + \cdots - 16498970487 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 106354558 \nu^{11} - 141053427 \nu^{10} + 466090971 \nu^{9} - 3528346131 \nu^{8} + \cdots + 4497150269 ) / 2782601185 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65564 \nu^{11} - 54701 \nu^{10} + 348926 \nu^{9} - 1928638 \nu^{8} + 2217584 \nu^{7} + \cdots - 1239927 ) / 1476181 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 241501321 \nu^{11} - 317659261 \nu^{10} - 1608964087 \nu^{9} + 3811780527 \nu^{8} + \cdots - 6703875713 ) / 2782601185 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1855275991 \nu^{11} + 118092454 \nu^{10} - 11400048997 \nu^{9} + 44654056002 \nu^{8} + \cdots + 19101313197 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2007824523 \nu^{11} + 642662513 \nu^{10} + 12279123091 \nu^{9} - 44235399731 \nu^{8} + \cdots - 2642262891 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2024040388 \nu^{11} + 481621272 \nu^{10} - 11262741596 \nu^{9} + 52099333786 \nu^{8} + \cdots - 4694601129 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2281630388 \nu^{11} - 1661658328 \nu^{10} - 14577758021 \nu^{9} + 44298602736 \nu^{8} + \cdots - 27875512029 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 511017007 \nu^{11} + 120344967 \nu^{10} + 3187847414 \nu^{9} - 11403406359 \nu^{8} + \cdots - 1016773599 ) / 2782601185 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3200667466 \nu^{11} - 750652796 \nu^{10} - 19636187197 \nu^{9} + 72271053552 \nu^{8} + \cdots + 43277420197 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 771058311 \nu^{11} + 121340051 \nu^{10} + 4638436737 \nu^{9} - 17846443417 \nu^{8} + \cdots - 7445776057 ) / 2782601185 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} + \beta_{7} + 2\beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta_{2} + 6\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} - 5 \beta_{10} - 7 \beta_{9} + 7 \beta_{8} + \beta_{7} + 12 \beta_{6} - 5 \beta_{5} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 17 \beta_{8} - 23 \beta_{7} - 30 \beta_{6} + \cdots - 30 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 45 \beta_{11} + 71 \beta_{10} + 73 \beta_{9} - 17 \beta_{8} + 45 \beta_{7} - 52 \beta_{6} + 73 \beta_{5} + \cdots - 144 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 136 \beta_{11} - 109 \beta_{10} - 80 \beta_{9} + 136 \beta_{8} + 80 \beta_{7} + 276 \beta_{6} + \cdots + 188 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 109 \beta_{11} - 109 \beta_{10} - 253 \beta_{9} - 253 \beta_{8} - 457 \beta_{7} - 348 \beta_{6} + \cdots + 348 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1330 \beta_{11} + 1631 \beta_{10} + 1631 \beta_{9} - 968 \beta_{8} + 258 \beta_{7} - 2287 \beta_{6} + \cdots - 3262 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4893 \beta_{11} - 3004 \beta_{10} - 1330 \beta_{9} + 5553 \beta_{8} + 4893 \beta_{7} + 10410 \beta_{6} + \cdots + 4284 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3115 \beta_{11} - 11672 \beta_{10} - 16067 \beta_{9} - 3115 \beta_{8} - 16067 \beta_{7} + \cdots + 27621 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 67032 \beta_{11} + 67032 \beta_{10} + 58829 \beta_{9} - 58829 \beta_{8} - 16067 \beta_{7} + \cdots - 125380 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0.649118 − 0.268874i
0.0572576 + 0.138232i
0.563963 + 1.36153i
−1.46953 + 0.608701i
1.52752 − 0.632721i
−1.32833 − 3.20687i
0.649118 + 0.268874i
0.0572576 − 0.138232i
0.563963 − 1.36153i
−1.46953 − 0.608701i
1.52752 + 0.632721i
−1.32833 + 3.20687i
0 0 0 −1.86440 − 1.23451i 0 0.707107 + 0.707107i 0 0 0
197.2 0 0 0 −1.28634 + 1.82903i 0 −0.707107 − 0.707107i 0 0 0
197.3 0 0 0 −0.656785 − 2.13744i 0 −0.707107 − 0.707107i 0 0 0
197.4 0 0 0 1.61196 − 1.54971i 0 0.707107 + 0.707107i 0 0 0
197.5 0 0 0 1.95955 + 1.07711i 0 0.707107 + 0.707107i 0 0 0
197.6 0 0 0 2.23601 + 0.0155164i 0 −0.707107 − 0.707107i 0 0 0
953.1 0 0 0 −1.86440 + 1.23451i 0 0.707107 − 0.707107i 0 0 0
953.2 0 0 0 −1.28634 − 1.82903i 0 −0.707107 + 0.707107i 0 0 0
953.3 0 0 0 −0.656785 + 2.13744i 0 −0.707107 + 0.707107i 0 0 0
953.4 0 0 0 1.61196 + 1.54971i 0 0.707107 − 0.707107i 0 0 0
953.5 0 0 0 1.95955 − 1.07711i 0 0.707107 − 0.707107i 0 0 0
953.6 0 0 0 2.23601 − 0.0155164i 0 −0.707107 + 0.707107i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.v.b yes 12
3.b odd 2 1 1260.2.v.a ✓ 12
5.b even 2 1 6300.2.v.e 12
5.c odd 4 1 1260.2.v.a ✓ 12
5.c odd 4 1 6300.2.v.f 12
15.d odd 2 1 6300.2.v.f 12
15.e even 4 1 inner 1260.2.v.b yes 12
15.e even 4 1 6300.2.v.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.v.a ✓ 12 3.b odd 2 1
1260.2.v.a ✓ 12 5.c odd 4 1
1260.2.v.b yes 12 1.a even 1 1 trivial
1260.2.v.b yes 12 15.e even 4 1 inner
6300.2.v.e 12 5.b even 2 1
6300.2.v.e 12 15.e even 4 1
6300.2.v.f 12 5.c odd 4 1
6300.2.v.f 12 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{12} - 8 T_{17}^{11} + 32 T_{17}^{10} - 16 T_{17}^{9} + 1456 T_{17}^{8} - 10208 T_{17}^{7} + \cdots + 73984 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} + 64 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 18496 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 73984 \) Copy content Toggle raw display
$19$ \( T^{12} + 160 T^{10} + \cdots + 1679616 \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{6} + 16 T^{5} + \cdots - 392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 8 T^{5} + \cdots + 16272)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 4 T^{11} + \cdots + 12902464 \) Copy content Toggle raw display
$41$ \( T^{12} + 148 T^{10} + \cdots + 6927424 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 232013824 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26927497216 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35942093056 \) Copy content Toggle raw display
$59$ \( (T^{6} + 8 T^{5} + \cdots - 78976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 320 T^{4} + \cdots - 79744)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 32 T^{9} + \cdots + 802816 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 150209536 \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots + 906304 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 205520896 \) Copy content Toggle raw display
$83$ \( T^{12} + 56 T^{11} + \cdots + 18939904 \) Copy content Toggle raw display
$89$ \( (T^{6} + 36 T^{5} + \cdots + 100088)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 20 T^{11} + \cdots + 10291264 \) Copy content Toggle raw display
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