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Properties

Label 6300.2.ch.e
Level $6300$
Weight $2$
Character orbit 6300.ch
Analytic conductor $50.306$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 32 x^{18} - 124 x^{16} + 5094 x^{14} + 61094 x^{12} + 245850 x^{10} + 420152 x^{8} + \cdots + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{7} + (\beta_{12} - \beta_{10}) q^{11} + (\beta_{8} + \beta_{7} - \beta_1) q^{13} + ( - \beta_{18} - \beta_{16} + \beta_{14}) q^{17} + (\beta_{7} + \beta_1) q^{19} + ( - 2 \beta_{18} + \beta_{17} + \cdots + \beta_{14}) q^{23}+ \cdots + (3 \beta_{9} + 3 \beta_{8} + 3 \beta_{6} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{31} + 2 q^{37} - 20 q^{43} + 4 q^{49} + 6 q^{61} + 28 q^{67} - 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 32 x^{18} - 124 x^{16} + 5094 x^{14} + 61094 x^{12} + 245850 x^{10} + 420152 x^{8} + \cdots + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 32\!\cdots\!70 \nu^{18} + \cdots + 33\!\cdots\!44 ) / 13\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!75 \nu^{18} + \cdots - 11\!\cdots\!24 ) / 45\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65\!\cdots\!57 \nu^{18} + \cdots + 35\!\cdots\!44 ) / 13\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 283399217654 \nu^{18} - 9068037900549 \nu^{16} - 35102450622962 \nu^{14} + \cdots + 23\!\cdots\!96 ) / 52\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69\!\cdots\!56 \nu^{18} + \cdots - 12\!\cdots\!48 ) / 68\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 75\!\cdots\!00 \nu^{18} + \cdots - 13\!\cdots\!30 ) / 68\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51\!\cdots\!93 \nu^{18} + \cdots + 74\!\cdots\!64 ) / 45\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!15 \nu^{18} + \cdots + 12\!\cdots\!48 ) / 13\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67\!\cdots\!97 \nu^{18} + \cdots - 28\!\cdots\!40 ) / 45\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!27 \nu^{19} + \cdots + 12\!\cdots\!56 \nu ) / 95\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!62 \nu^{19} + \cdots - 16\!\cdots\!40 \nu ) / 15\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 93\!\cdots\!75 \nu^{19} + \cdots + 13\!\cdots\!52 \nu ) / 15\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 57\!\cdots\!32 \nu^{19} + \cdots - 40\!\cdots\!74 \nu ) / 29\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!29 \nu^{19} + \cdots - 86\!\cdots\!68 \nu ) / 95\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 56\!\cdots\!27 \nu^{19} + \cdots - 37\!\cdots\!56 \nu ) / 95\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 43\!\cdots\!88 \nu^{19} + \cdots - 44\!\cdots\!52 \nu ) / 68\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 30\!\cdots\!27 \nu^{19} + \cdots + 26\!\cdots\!32 \nu ) / 47\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 31\!\cdots\!66 \nu^{19} + \cdots + 59\!\cdots\!44 \nu ) / 47\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 70\!\cdots\!01 \nu^{19} + \cdots + 43\!\cdots\!62 \nu ) / 88\!\cdots\!78 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{18} - \beta_{17} + \beta_{12} + \beta_{11} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} - 2\beta_{6} - 3\beta_{5} - 2\beta_{3} + \beta_{2} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{19} + 25 \beta_{18} - 23 \beta_{17} + 4 \beta_{16} + 8 \beta_{15} - 2 \beta_{14} + \cdots + 6 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 29 \beta_{9} - 50 \beta_{8} - 37 \beta_{7} - 47 \beta_{6} - 35 \beta_{5} + 46 \beta_{4} - 33 \beta_{3} + \cdots + 118 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26 \beta_{19} + 619 \beta_{18} - 613 \beta_{17} + 200 \beta_{16} + 278 \beta_{15} + \cdots + 102 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 866 \beta_{9} - 996 \beta_{8} - 911 \beta_{7} - 1093 \beta_{6} - 988 \beta_{5} + 1452 \beta_{4} + \cdots + 2560 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 412 \beta_{19} + 16161 \beta_{18} - 15733 \beta_{17} + 6906 \beta_{16} + 8622 \beta_{15} + \cdots + 2852 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 23359 \beta_{9} - 24323 \beta_{8} - 25596 \beta_{7} - 27480 \beta_{6} - 22024 \beta_{5} + 49636 \beta_{4} + \cdots + 52493 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4698 \beta_{19} + 404229 \beta_{18} - 404055 \beta_{17} + 217006 \beta_{16} + 258852 \beta_{15} + \cdots + 63772 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 638447 \beta_{9} - 553021 \beta_{8} - 672613 \beta_{7} - 648526 \beta_{6} - 490565 \beta_{5} + \cdots + 1044154 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 481924 \beta_{19} + 10016627 \beta_{18} - 10147217 \beta_{17} + 6486612 \beta_{16} + \cdots + 1470934 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16893157 \beta_{9} - 12317968 \beta_{8} - 17558375 \beta_{7} - 15216671 \beta_{6} - 10582171 \beta_{5} + \cdots + 18509940 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20730430 \beta_{19} + 243867569 \beta_{18} - 251233667 \beta_{17} + 185971900 \beta_{16} + \cdots + 32779922 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 440363212 \beta_{9} - 265291014 \beta_{8} - 449741915 \beta_{7} - 348073469 \beta_{6} - 217105862 \beta_{5} + \cdots + 267059692 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 732366408 \beta_{19} + 5835958103 \beta_{18} - 6117297667 \beta_{17} + 5177125402 \beta_{16} + \cdots + 704862580 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11281503365 \beta_{9} - 5447207631 \beta_{8} - 11338793162 \beta_{7} - 7753465974 \beta_{6} + \cdots + 1633009107 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 23378425050 \beta_{19} + 137023031699 \beta_{18} - 146393309897 \beta_{17} + \cdots + 14484548288 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 284432194833 \beta_{9} - 104249427013 \beta_{8} - 281379879439 \beta_{7} - 166896676164 \beta_{6} + \cdots - 86752416644 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 702916717104 \beta_{19} + 3147676955901 \beta_{18} - 3437139680051 \beta_{17} + \cdots + 277127759150 \beta_{10} ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
4.99853 + 0.314967i
−4.99853 − 0.314967i
−1.08776 + 0.808980i
1.08776 − 0.808980i
0.227585 + 0.209112i
−0.227585 − 0.209112i
−0.713501 + 2.49445i
0.713501 − 2.49445i
0.520188 + 1.99507i
−0.520188 − 1.99507i
4.99853 − 0.314967i
−4.99853 + 0.314967i
−1.08776 − 0.808980i
1.08776 + 0.808980i
0.227585 − 0.209112i
−0.227585 + 0.209112i
−0.713501 − 2.49445i
0.713501 + 2.49445i
0.520188 − 1.99507i
−0.520188 + 1.99507i
0 0 0 0 0 −2.37152 − 1.17299i 0 0 0
1601.2 0 0 0 0 0 −2.37152 − 1.17299i 0 0 0
1601.3 0 0 0 0 0 −1.54876 + 2.14507i 0 0 0
1601.4 0 0 0 0 0 −1.54876 + 2.14507i 0 0 0
1601.5 0 0 0 0 0 0.651797 − 2.56421i 0 0 0
1601.6 0 0 0 0 0 0.651797 − 2.56421i 0 0 0
1601.7 0 0 0 0 0 1.66390 + 2.05705i 0 0 0
1601.8 0 0 0 0 0 1.66390 + 2.05705i 0 0 0
1601.9 0 0 0 0 0 2.60458 − 0.464924i 0 0 0
1601.10 0 0 0 0 0 2.60458 − 0.464924i 0 0 0
4301.1 0 0 0 0 0 −2.37152 + 1.17299i 0 0 0
4301.2 0 0 0 0 0 −2.37152 + 1.17299i 0 0 0
4301.3 0 0 0 0 0 −1.54876 − 2.14507i 0 0 0
4301.4 0 0 0 0 0 −1.54876 − 2.14507i 0 0 0
4301.5 0 0 0 0 0 0.651797 + 2.56421i 0 0 0
4301.6 0 0 0 0 0 0.651797 + 2.56421i 0 0 0
4301.7 0 0 0 0 0 1.66390 − 2.05705i 0 0 0
4301.8 0 0 0 0 0 1.66390 − 2.05705i 0 0 0
4301.9 0 0 0 0 0 2.60458 + 0.464924i 0 0 0
4301.10 0 0 0 0 0 2.60458 + 0.464924i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1601.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
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.e yes 20
3.b odd 2 1 inner 6300.2.ch.e yes 20
5.b even 2 1 6300.2.ch.d ✓ 20
5.c odd 4 2 6300.2.dd.d 40
7.d odd 6 1 inner 6300.2.ch.e yes 20
15.d odd 2 1 6300.2.ch.d ✓ 20
15.e even 4 2 6300.2.dd.d 40
21.g even 6 1 inner 6300.2.ch.e yes 20
35.i odd 6 1 6300.2.ch.d ✓ 20
35.k even 12 2 6300.2.dd.d 40
105.p even 6 1 6300.2.ch.d ✓ 20
105.w odd 12 2 6300.2.dd.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6300.2.ch.d ✓ 20 5.b even 2 1
6300.2.ch.d ✓ 20 15.d odd 2 1
6300.2.ch.d ✓ 20 35.i odd 6 1
6300.2.ch.d ✓ 20 105.p even 6 1
6300.2.ch.e yes 20 1.a even 1 1 trivial
6300.2.ch.e yes 20 3.b odd 2 1 inner
6300.2.ch.e yes 20 7.d odd 6 1 inner
6300.2.ch.e yes 20 21.g even 6 1 inner
6300.2.dd.d 40 5.c odd 4 2
6300.2.dd.d 40 15.e even 4 2
6300.2.dd.d 40 35.k even 12 2
6300.2.dd.d 40 105.w odd 12 2

Hecke kernels

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

\( T_{11}^{20} - 44 T_{11}^{18} + 1408 T_{11}^{16} - 20576 T_{11}^{14} + 219724 T_{11}^{12} - 645992 T_{11}^{10} + \cdots + 5184 \) Copy content Toggle raw display
\( T_{37}^{10} - T_{37}^{9} + 59 T_{37}^{8} - 418 T_{37}^{7} + 3781 T_{37}^{6} - 14307 T_{37}^{5} + \cdots + 21025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} - 2 T^{9} + \cdots + 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} - 44 T^{18} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( (T^{10} + 72 T^{8} + \cdots + 47628)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 68 T^{18} + \cdots + 46656 \) Copy content Toggle raw display
$19$ \( (T^{10} - 30 T^{8} + \cdots + 21168)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 1991485440000 \) Copy content Toggle raw display
$29$ \( (T^{10} + 112 T^{8} + \cdots + 225792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 3 T^{9} + \cdots + 768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - T^{9} + \cdots + 21025)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 240 T^{8} + \cdots - 48600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 5 T^{4} + \cdots + 328)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 18440978135616 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 59\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 157389469886016 \) Copy content Toggle raw display
$61$ \( (T^{10} - 3 T^{9} + \cdots + 69437163)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 14 T^{9} + \cdots + 777924)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 570 T^{8} + \cdots + 1805764608)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 3 T^{9} + \cdots + 15417867)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - T^{9} + \cdots + 3094081)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 240 T^{8} + \cdots - 48600)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{10} + 863 T^{8} + \cdots + 10250259627)^{2} \) Copy content Toggle raw display
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