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Properties

Label 6300.2.ch.c
Level $6300$
Weight $2$
Character orbit 6300.ch
Analytic conductor $50.306$
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] = [6300,2,Mod(1601,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6300.1601");
 
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.ch (of order \(6\), 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 \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{7} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{7} - \beta_{6}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{7} + 12 q^{11} - 6 q^{19} + 12 q^{23} - 6 q^{31} + 2 q^{37} - 8 q^{41} - 36 q^{43} + 16 q^{47} + 22 q^{49} - 12 q^{53} - 2 q^{67} - 6 q^{73} - 28 q^{77} + 14 q^{79} - 40 q^{83} - 20 q^{89} + 10 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} - 9 \nu^{9} + 58 \nu^{8} - 78 \nu^{7} - 298 \nu^{6} + 1341 \nu^{5} + \cdots - 33614 ) / 16807 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 676 \nu^{11} + 4677 \nu^{10} - 6789 \nu^{9} - 33706 \nu^{8} + 181192 \nu^{7} - 308733 \nu^{6} + \cdots - 30370249 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1012 \nu^{11} - 4908 \nu^{10} + 3324 \nu^{9} + 52459 \nu^{8} - 215632 \nu^{7} + 256380 \nu^{6} + \cdots + 43278025 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2014 \nu^{11} - 9873 \nu^{10} + 4491 \nu^{9} + 112234 \nu^{8} - 442153 \nu^{7} + 476127 \nu^{6} + \cdots + 89211556 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2575 \nu^{11} - 12234 \nu^{10} + 11181 \nu^{9} + 126082 \nu^{8} - 568063 \nu^{7} + \cdots + 134892982 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 412 \nu^{11} - 1776 \nu^{10} + 891 \nu^{9} + 19528 \nu^{8} - 79708 \nu^{7} + 89268 \nu^{6} + \cdots + 17008684 ) / 180075 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2917 \nu^{11} + 14808 \nu^{10} - 15264 \nu^{9} - 153079 \nu^{8} + 699802 \nu^{7} + \cdots - 166305265 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 593 \nu^{11} - 2880 \nu^{10} + 2412 \nu^{9} + 28913 \nu^{8} - 126131 \nu^{7} + 165138 \nu^{6} + \cdots + 25244114 ) / 252105 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4898 \nu^{11} - 23334 \nu^{10} + 21459 \nu^{9} + 233642 \nu^{8} - 1051412 \nu^{7} + \cdots + 232575266 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5624 \nu^{11} - 26256 \nu^{10} + 19923 \nu^{9} + 275603 \nu^{8} - 1178894 \nu^{7} + \cdots + 249415880 ) / 1260525 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + 4\beta_{10} - 2\beta_{9} - 3\beta_{6} + \beta_{5} + 4\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + 12 \beta_{10} - 9 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 10 \beta_{6} + 7 \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{11} - \beta_{10} - 31 \beta_{9} + 10 \beta_{8} - 12 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29 \beta_{11} - 22 \beta_{10} - \beta_{9} - 40 \beta_{8} + 30 \beta_{7} - 69 \beta_{6} - 20 \beta_{5} + \cdots + 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 40 \beta_{11} - 104 \beta_{10} + 64 \beta_{9} - 40 \beta_{8} + 64 \beta_{7} + 5 \beta_{6} + \cdots + 210 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 230 \beta_{11} + 194 \beta_{10} + 374 \beta_{9} - 62 \beta_{8} - 196 \beta_{7} - 276 \beta_{6} + \cdots + 1959 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 32 \beta_{11} + 262 \beta_{10} - 350 \beta_{9} + 440 \beta_{8} - 867 \beta_{7} + 678 \beta_{6} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 352 \beta_{11} + 2362 \beta_{10} - 64 \beta_{9} - 1601 \beta_{8} - 320 \beta_{7} - 3699 \beta_{6} + \cdots - 2656 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2 \beta_{11} + 2929 \beta_{10} - 5123 \beta_{9} - 30 \beta_{8} + 2625 \beta_{7} - 3362 \beta_{6} + \cdots - 23632 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
2.61827 + 0.380350i
1.75207 − 1.98249i
1.63107 + 2.08318i
0.260926 − 2.63285i
−2.61674 + 0.390758i
−2.64559 + 0.0290059i
2.61827 − 0.380350i
1.75207 + 1.98249i
1.63107 − 2.08318i
0.260926 + 2.63285i
−2.61674 − 0.390758i
−2.64559 − 0.0290059i
0 0 0 0 0 −2.61827 − 0.380350i 0 0 0
1601.2 0 0 0 0 0 −1.75207 + 1.98249i 0 0 0
1601.3 0 0 0 0 0 −1.63107 − 2.08318i 0 0 0
1601.4 0 0 0 0 0 −0.260926 + 2.63285i 0 0 0
1601.5 0 0 0 0 0 2.61674 − 0.390758i 0 0 0
1601.6 0 0 0 0 0 2.64559 − 0.0290059i 0 0 0
4301.1 0 0 0 0 0 −2.61827 + 0.380350i 0 0 0
4301.2 0 0 0 0 0 −1.75207 − 1.98249i 0 0 0
4301.3 0 0 0 0 0 −1.63107 + 2.08318i 0 0 0
4301.4 0 0 0 0 0 −0.260926 − 2.63285i 0 0 0
4301.5 0 0 0 0 0 2.61674 + 0.390758i 0 0 0
4301.6 0 0 0 0 0 2.64559 + 0.0290059i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1601.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.ch.c 12
3.b odd 2 1 6300.2.ch.b 12
5.b even 2 1 1260.2.cg.a ✓ 12
5.c odd 4 2 6300.2.dd.c 24
7.d odd 6 1 6300.2.ch.b 12
15.d odd 2 1 1260.2.cg.b yes 12
15.e even 4 2 6300.2.dd.b 24
21.g even 6 1 inner 6300.2.ch.c 12
35.i odd 6 1 1260.2.cg.b yes 12
35.i odd 6 1 8820.2.d.a 12
35.j even 6 1 8820.2.d.b 12
35.k even 12 2 6300.2.dd.b 24
105.o odd 6 1 8820.2.d.a 12
105.p even 6 1 1260.2.cg.a ✓ 12
105.p even 6 1 8820.2.d.b 12
105.w odd 12 2 6300.2.dd.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.cg.a ✓ 12 5.b even 2 1
1260.2.cg.a ✓ 12 105.p even 6 1
1260.2.cg.b yes 12 15.d odd 2 1
1260.2.cg.b yes 12 35.i odd 6 1
6300.2.ch.b 12 3.b odd 2 1
6300.2.ch.b 12 7.d odd 6 1
6300.2.ch.c 12 1.a even 1 1 trivial
6300.2.ch.c 12 21.g even 6 1 inner
6300.2.dd.b 24 15.e even 4 2
6300.2.dd.b 24 35.k even 12 2
6300.2.dd.c 24 5.c odd 4 2
6300.2.dd.c 24 105.w odd 12 2
8820.2.d.a 12 35.i odd 6 1
8820.2.d.a 12 105.o odd 6 1
8820.2.d.b 12 35.j even 6 1
8820.2.d.b 12 105.p even 6 1

Hecke kernels

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

\( T_{11}^{12} - 12 T_{11}^{11} + 38 T_{11}^{10} + 120 T_{11}^{9} - 636 T_{11}^{8} - 1320 T_{11}^{7} + \cdots + 272484 \) Copy content Toggle raw display
\( T_{37}^{12} - 2 T_{37}^{11} + 109 T_{37}^{10} - 286 T_{37}^{9} + 9604 T_{37}^{8} - 21338 T_{37}^{7} + \cdots + 2002225 \) 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
$7$ \( T^{12} + 2 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 12 T^{11} + \cdots + 272484 \) Copy content Toggle raw display
$13$ \( T^{12} + 102 T^{10} + \cdots + 189225 \) Copy content Toggle raw display
$17$ \( T^{12} + 52 T^{10} + \cdots + 202500 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 2064969 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 170302500 \) Copy content Toggle raw display
$29$ \( T^{12} + 148 T^{10} + \cdots + 8100 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + \cdots + 9126441 \) Copy content Toggle raw display
$37$ \( T^{12} - 2 T^{11} + \cdots + 2002225 \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} + \cdots - 450)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 18 T^{5} + \cdots + 142465)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 232989696 \) Copy content Toggle raw display
$53$ \( T^{12} + 12 T^{11} + \cdots + 82944 \) Copy content Toggle raw display
$59$ \( T^{12} + 166 T^{10} + \cdots + 6812100 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1497690000 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 4482436401 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 239073444 \) Copy content Toggle raw display
$73$ \( T^{12} + 6 T^{11} + \cdots + 1172889 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 27773889025 \) Copy content Toggle raw display
$83$ \( (T^{6} + 20 T^{5} + \cdots - 13410)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 610976722500 \) Copy content Toggle raw display
$97$ \( T^{12} + 92 T^{10} + \cdots + 1089936 \) Copy content Toggle raw display
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