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Properties

Label 6300.2.d.d
Level $6300$
Weight $2$
Character orbit 6300.d
Analytic conductor $50.306$
Analytic rank $0$
Dimension $12$
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(3401,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6300.3401");
 
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.d (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: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 46x^{10} + 776x^{8} + 6250x^{6} + 25160x^{4} + 46174x^{2} + 26569 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{7} - \beta_{9} q^{11} + \beta_{8} q^{13} + \beta_1 q^{17} - \beta_{8} q^{19} - \beta_{7} q^{23} + \beta_{7} q^{29} - \beta_{11} q^{31} + (\beta_{2} + 3) q^{37} + (\beta_{4} - \beta_{3}) q^{41} + ( - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{43} + (\beta_{4} - \beta_{3} - \beta_1) q^{47} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{49} + (\beta_{9} - \beta_{7}) q^{53} + \beta_1 q^{59} + (\beta_{11} + 2 \beta_{6} + 2 \beta_{5}) q^{61} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} - 3) q^{67} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7}) q^{71} + (\beta_{11} - \beta_{8}) q^{73} + (\beta_{10} - \beta_{9} + \beta_{3} - \beta_1) q^{77} + ( - \beta_{6} + \beta_{5} + 3) q^{79} + ( - \beta_{10} - \beta_{9} + \cdots - 2 \beta_{3}) q^{83}+ \cdots + (\beta_{6} + \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{7} + 32 q^{37} - 12 q^{43} - 2 q^{49} - 20 q^{67} + 32 q^{79} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 46x^{10} + 776x^{8} + 6250x^{6} + 25160x^{4} + 46174x^{2} + 26569 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -115\nu^{10} - 4649\nu^{8} - 63541\nu^{6} - 371351\nu^{4} - 884107\nu^{2} - 560945 ) / 5832 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{10} + 664\nu^{8} + 8564\nu^{6} + 46072\nu^{4} + 98735\nu^{2} + 55657 ) / 729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 731 \nu^{11} - 4727 \nu^{10} + 38353 \nu^{9} - 134149 \nu^{8} + 701405 \nu^{7} - 721601 \nu^{6} + \cdots + 409619 ) / 1901232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 731 \nu^{11} - 112959 \nu^{10} + 38353 \nu^{9} - 4516893 \nu^{8} + 701405 \nu^{7} + \cdots - 547600293 ) / 1901232 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 627 \nu^{11} + 1304 \nu^{10} + 25419 \nu^{9} + 52486 \nu^{8} + 352077 \nu^{7} + 703508 \nu^{6} + \cdots + 5165470 ) / 475308 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 627 \nu^{11} - 1304 \nu^{10} + 25419 \nu^{9} - 52486 \nu^{8} + 352077 \nu^{7} - 703508 \nu^{6} + \cdots - 5165470 ) / 475308 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5039\nu^{11} + 199357\nu^{9} + 2614577\nu^{7} + 14198635\nu^{5} + 29824439\nu^{3} + 13319893\nu ) / 950616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -1351\nu^{11} - 53507\nu^{9} - 703957\nu^{7} - 3868829\nu^{5} - 8597227\nu^{3} - 5889023\nu ) / 237654 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2155\nu^{11} - 89513\nu^{9} - 1273093\nu^{7} - 7807271\nu^{5} - 19804795\nu^{3} - 14252873\nu ) / 316872 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 133\nu^{11} + 5303\nu^{9} + 70771\nu^{7} + 399137\nu^{5} + 905029\nu^{3} + 522695\nu ) / 11736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3217\nu^{11} - 126629\nu^{9} - 1660039\nu^{7} - 9239855\nu^{5} - 21441949\nu^{3} - 15694313\nu ) / 237654 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{10} - 3\beta_{8} - 6\beta_{7} - 3\beta_{6} - 3\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{7} - 4\beta_{6} + 4\beta_{5} - 2\beta_{3} - 2\beta_{2} - \beta _1 - 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -21\beta_{11} - 52\beta_{10} - 18\beta_{9} + 42\beta_{8} + 78\beta_{7} + 3\beta_{6} + 3\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17 \beta_{10} + 17 \beta_{9} - 17 \beta_{7} + 90 \beta_{6} - 90 \beta_{5} - 11 \beta_{4} + 45 \beta_{3} + \cdots + 358 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 531\beta_{11} + 998\beta_{10} + 450\beta_{9} - 849\beta_{8} - 1284\beta_{7} + 348\beta_{6} + 348\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 157 \beta_{10} - 157 \beta_{9} + 157 \beta_{7} - 878 \beta_{6} + 878 \beta_{5} + 152 \beta_{4} + \cdots - 2820 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11361 \beta_{11} - 19126 \beta_{10} - 9450 \beta_{9} + 17583 \beta_{8} + 23388 \beta_{7} + \cdots - 9600 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6121 \beta_{10} + 6121 \beta_{9} - 6121 \beta_{7} + 33966 \beta_{6} - 33966 \beta_{5} - 6667 \beta_{4} + \cdots + 101818 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 231939 \beta_{11} + 371588 \beta_{10} + 189522 \beta_{9} - 357582 \beta_{8} - 446334 \beta_{7} + \cdots + 208707 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 121161 \beta_{10} - 121161 \beta_{9} + 121161 \beta_{7} - 662740 \beta_{6} + 662740 \beta_{5} + \cdots - 1944738 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4653228 \beta_{11} - 7286818 \beta_{10} - 3756168 \beta_{9} + 7175301 \beta_{8} + \cdots - 4266303 \beta_{5} ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3401.1
− 4.45015i
2.96015i
− 2.96015i
4.45015i
− 1.00154i
2.59108i
− 2.59108i
1.00154i
1.98144i
2.40641i
− 2.40641i
− 1.98144i
0 0 0 0 0 −2.54327 − 0.729223i 0 0 0
3401.2 0 0 0 0 0 −2.54327 − 0.729223i 0 0 0
3401.3 0 0 0 0 0 −2.54327 + 0.729223i 0 0 0
3401.4 0 0 0 0 0 −2.54327 + 0.729223i 0 0 0
3401.5 0 0 0 0 0 0.101232 − 2.64381i 0 0 0
3401.6 0 0 0 0 0 0.101232 − 2.64381i 0 0 0
3401.7 0 0 0 0 0 0.101232 + 2.64381i 0 0 0
3401.8 0 0 0 0 0 0.101232 + 2.64381i 0 0 0
3401.9 0 0 0 0 0 1.94204 − 1.79680i 0 0 0
3401.10 0 0 0 0 0 1.94204 − 1.79680i 0 0 0
3401.11 0 0 0 0 0 1.94204 + 1.79680i 0 0 0
3401.12 0 0 0 0 0 1.94204 + 1.79680i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3401.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.d.d ✓ 12
3.b odd 2 1 inner 6300.2.d.d ✓ 12
5.b even 2 1 6300.2.d.e yes 12
5.c odd 4 2 6300.2.f.d 24
7.b odd 2 1 inner 6300.2.d.d ✓ 12
15.d odd 2 1 6300.2.d.e yes 12
15.e even 4 2 6300.2.f.d 24
21.c even 2 1 inner 6300.2.d.d ✓ 12
35.c odd 2 1 6300.2.d.e yes 12
35.f even 4 2 6300.2.f.d 24
105.g even 2 1 6300.2.d.e yes 12
105.k odd 4 2 6300.2.f.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6300.2.d.d ✓ 12 1.a even 1 1 trivial
6300.2.d.d ✓ 12 3.b odd 2 1 inner
6300.2.d.d ✓ 12 7.b odd 2 1 inner
6300.2.d.d ✓ 12 21.c even 2 1 inner
6300.2.d.e yes 12 5.b even 2 1
6300.2.d.e yes 12 15.d odd 2 1
6300.2.d.e yes 12 35.c odd 2 1
6300.2.d.e yes 12 105.g even 2 1
6300.2.f.d 24 5.c odd 4 2
6300.2.f.d 24 15.e even 4 2
6300.2.f.d 24 35.f even 4 2
6300.2.f.d 24 105.k odd 4 2

Hecke kernels

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

\( T_{11}^{6} + 46T_{11}^{4} + 385T_{11}^{2} + 18 \) Copy content Toggle raw display
\( T_{37}^{3} - 8T_{37}^{2} - 11T_{37} + 136 \) Copy content Toggle raw display
\( T_{41}^{6} - 192T_{41}^{4} + 6208T_{41}^{2} - 55296 \) 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
$7$ \( (T^{6} + T^{5} + T^{4} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 46 T^{4} + \cdots + 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 51 T^{4} + \cdots + 2352)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 94 T^{4} + \cdots - 7776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 51 T^{4} + \cdots + 2352)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 62 T^{4} + \cdots + 882)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 62 T^{4} + \cdots + 882)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 171 T^{4} + \cdots + 120000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 8 T^{2} + \cdots + 136)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 192 T^{4} + \cdots - 55296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 3 T^{2} - 59 T + 23)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 222 T^{4} + \cdots - 249696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 84 T^{4} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 94 T^{4} + \cdots - 7776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 291 T^{4} + \cdots + 120000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 5 T^{2} - 165 T + 63)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 276 T^{4} + \cdots + 20808)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 220 T^{4} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} + T + 46)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 486 T^{4} + \cdots - 3538944)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 528 T^{4} + \cdots - 2906496)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 43 T^{4} + \cdots + 768)^{2} \) Copy content Toggle raw display
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