↑

Properties

Label 6300.2.v.e
Level $6300$
Weight $2$
Character orbit 6300.v
Analytic conductor $50.306$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6300,2,Mod(1457,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, 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("6300.1457");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.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: \(50.3057532734\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1260)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{7} + ( - \beta_{11} + \beta_{10} + 2 \beta_{3}) q^{11} + ( - \beta_{10} - \beta_{8} + \beta_{6} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{11} + \beta_{10} - \beta_{8} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{13} - 8 q^{17} + 8 q^{23} - 32 q^{29} + 16 q^{31} - 4 q^{37} - 8 q^{43} - 8 q^{47} + 8 q^{53} - 16 q^{59} + 12 q^{73} + 56 q^{83} - 72 q^{89} - 20 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 52519139 \nu^{11} + 1732338516 \nu^{10} + 98241612 \nu^{9} + 11349499483 \nu^{8} + \cdots + 63475795713 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 478873536 \nu^{11} - 47412266 \nu^{10} - 3079467412 \nu^{9} + 10994046217 \nu^{8} + \cdots + 16498970487 ) / 13913005925 \) 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
\(\beta_{4}\)\(=\) \( ( 2007824523 \nu^{11} + 642662513 \nu^{10} + 12279123091 \nu^{9} - 44235399731 \nu^{8} + \cdots - 2642262891 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2052534703 \nu^{11} - 1007545807 \nu^{10} + 11693893076 \nu^{9} - 55927660566 \nu^{8} + \cdots - 81603362801 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 375723 \nu^{11} + 246804 \nu^{10} - 2079388 \nu^{9} + 10642382 \nu^{8} - 11520532 \nu^{7} + \cdots + 15136061 ) / 2451631 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 127569664 \nu^{11} + 41524635 \nu^{10} - 728459325 \nu^{9} + 3322335160 \nu^{8} + \cdots - 179187599 ) / 556520237 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3231546993 \nu^{11} + 1106675033 \nu^{10} + 19307562031 \nu^{9} - 71158236421 \nu^{8} + \cdots + 52126985619 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 868470833 \nu^{11} + 310328288 \nu^{10} + 5221573711 \nu^{9} - 19036337381 \nu^{8} + \cdots + 10060993009 ) / 2782601185 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6373693201 \nu^{11} - 836820706 \nu^{10} - 38864051817 \nu^{9} + 147465498497 \nu^{8} + \cdots + 60047600167 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1515864981 \nu^{11} - 338847901 \nu^{10} - 9332439127 \nu^{9} + 34187661177 \nu^{8} + \cdots + 12757022977 ) / 2782601185 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} - \beta_{10} + 2\beta_{9} - \beta_{8} + 2\beta_{6} + 2\beta_{5} + \beta_{3} - 6\beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} - 2 \beta_{10} - \beta_{9} + 8 \beta_{8} + 9 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \cdots + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 30 \beta_{11} + 26 \beta_{10} - 17 \beta_{9} - 23 \beta_{7} - 23 \beta_{6} - 17 \beta_{5} - 20 \beta_{4} + \cdots - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 41 \beta_{11} - 32 \beta_{10} + 71 \beta_{9} - 86 \beta_{8} - 41 \beta_{7} + 103 \beta_{6} + 86 \beta_{5} + \cdots - 131 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 136 \beta_{11} - 109 \beta_{10} + 21 \beta_{9} + 109 \beta_{8} + 189 \beta_{7} + 202 \beta_{4} + \cdots + 289 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1229 \beta_{11} + 1011 \beta_{10} - 914 \beta_{9} + 409 \beta_{8} - 505 \beta_{7} - 1229 \beta_{6} + \cdots + 97 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 968 \beta_{9} - 1889 \beta_{8} - 1631 \beta_{7} + 1631 \beta_{6} + 1330 \beta_{5} - 1624 \beta_{4} + \cdots - 3667 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7390 \beta_{11} - 6060 \beta_{10} + 3004 \beta_{9} + 2497 \beta_{8} + 7390 \beta_{7} + 3056 \beta_{6} + \cdots + 9125 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 19630 \beta_{11} + 16067 \beta_{10} - 19630 \beta_{9} + 16067 \beta_{8} - 27739 \beta_{6} + \cdots + 19663 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 36855 \beta_{11} + 30177 \beta_{10} + 16067 \beta_{9} - 72939 \beta_{8} - 89006 \beta_{7} + \cdots - 162235 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1
1.52752 − 0.632721i
0.649118 − 0.268874i
−1.46953 + 0.608701i
−1.32833 − 3.20687i
0.0572576 + 0.138232i
0.563963 + 1.36153i
−1.46953 − 0.608701i
0.649118 + 0.268874i
1.52752 + 0.632721i
0.563963 − 1.36153i
0.0572576 − 0.138232i
−1.32833 + 3.20687i
0 0 0 0 0 −0.707107 − 0.707107i 0 0 0
1457.2 0 0 0 0 0 −0.707107 − 0.707107i 0 0 0
1457.3 0 0 0 0 0 −0.707107 − 0.707107i 0 0 0
1457.4 0 0 0 0 0 0.707107 + 0.707107i 0 0 0
1457.5 0 0 0 0 0 0.707107 + 0.707107i 0 0 0
1457.6 0 0 0 0 0 0.707107 + 0.707107i 0 0 0
5993.1 0 0 0 0 0 −0.707107 + 0.707107i 0 0 0
5993.2 0 0 0 0 0 −0.707107 + 0.707107i 0 0 0
5993.3 0 0 0 0 0 −0.707107 + 0.707107i 0 0 0
5993.4 0 0 0 0 0 0.707107 − 0.707107i 0 0 0
5993.5 0 0 0 0 0 0.707107 − 0.707107i 0 0 0
5993.6 0 0 0 0 0 0.707107 − 0.707107i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1457.6
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 6300.2.v.e 12
3.b odd 2 1 6300.2.v.f 12
5.b even 2 1 1260.2.v.b yes 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 1260.2.v.a ✓ 12
15.e even 4 1 1260.2.v.b yes 12
15.e even 4 1 inner 6300.2.v.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.v.a ✓ 12 5.c odd 4 1
1260.2.v.a ✓ 12 15.d odd 2 1
1260.2.v.b yes 12 5.b even 2 1
1260.2.v.b yes 12 15.e even 4 1
6300.2.v.e 12 1.a even 1 1 trivial
6300.2.v.e 12 15.e even 4 1 inner
6300.2.v.f 12 3.b odd 2 1
6300.2.v.f 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\):

\( T_{11}^{12} + 64T_{11}^{10} + 1216T_{11}^{8} + 9408T_{11}^{6} + 31744T_{11}^{4} + 38912T_{11}^{2} + 1024 \) Copy content Toggle raw display
\( T_{13}^{12} - 12 T_{13}^{11} + 72 T_{13}^{10} - 136 T_{13}^{9} + 60 T_{13}^{8} - 320 T_{13}^{7} + \cdots + 18496 \) Copy content Toggle raw display
\( T_{17}^{12} + 8 T_{17}^{11} + 32 T_{17}^{10} + 16 T_{17}^{9} + 1456 T_{17}^{8} + 10208 T_{17}^{7} + \cdots + 73984 \) Copy content Toggle raw display
\( T_{23}^{12} - 8 T_{23}^{11} + 32 T_{23}^{10} + 64 T_{23}^{9} - 48 T_{23}^{8} - 320 T_{23}^{7} + \cdots + 256 \) 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
$5$ \( T^{12} \) 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
show more
show less