↑

Properties

Label 6348.2.a.q
Level $6348$
Weight $2$
Character orbit 6348.a
Self dual yes
Analytic conductor $50.689$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6348,2,Mod(1,6348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6348, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6348.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6348.a (trivial)

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: yes
Analytic conductor: \(50.6890352031\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 15x^{8} + 29x^{7} + 74x^{6} - 143x^{5} - 126x^{4} + 259x^{3} + 21x^{2} - 98x + 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 276)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{7} + \beta_{5}) q^{5} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{7}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_{7} + \beta_{5}) q^{5} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{7}+ \cdots + (\beta_{8} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{3} - 9 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{3} - 9 q^{7} + 10 q^{9} - 11 q^{11} - 4 q^{13} - 7 q^{17} + 2 q^{19} + 9 q^{21} + 6 q^{25} - 10 q^{27} + 9 q^{29} + 11 q^{31} + 11 q^{33} + 43 q^{35} + 9 q^{37} + 4 q^{39} + 2 q^{41} - 24 q^{43} - 27 q^{47} + 37 q^{49} + 7 q^{51} - 4 q^{53} - 22 q^{55} - 2 q^{57} + 2 q^{59} + 27 q^{61} - 9 q^{63} - 22 q^{65} - 24 q^{67} + 11 q^{71} + 2 q^{73} - 6 q^{75} - 11 q^{77} - 23 q^{79} + 10 q^{81} - 37 q^{83} - q^{85} - 9 q^{87} - 11 q^{89} + 8 q^{91} - 11 q^{93} - 10 q^{95} - 10 q^{97} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 15x^{8} + 29x^{7} + 74x^{6} - 143x^{5} - 126x^{4} + 259x^{3} + 21x^{2} - 98x + 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33 \nu^{9} + 30 \nu^{8} - 493 \nu^{7} - 465 \nu^{6} + 2277 \nu^{5} + 1972 \nu^{4} - 3562 \nu^{3} + \cdots + 206 ) / 67 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 55 \nu^{9} + 17 \nu^{8} + 844 \nu^{7} - 163 \nu^{6} - 4197 \nu^{5} + 711 \nu^{4} + 7433 \nu^{3} + \cdots + 885 ) / 67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 57 \nu^{9} - 3 \nu^{8} - 882 \nu^{7} - 54 \nu^{6} + 4402 \nu^{5} + 379 \nu^{4} - 7724 \nu^{3} + \cdots - 436 ) / 67 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 104 \nu^{9} - 9 \nu^{8} - 1574 \nu^{7} - 28 \nu^{6} + 7645 \nu^{5} + 132 \nu^{4} - 13122 \nu^{3} + \cdots - 1040 ) / 67 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 166 \nu^{9} + 44 \nu^{8} + 2551 \nu^{7} - 414 \nu^{6} - 12727 \nu^{5} + 1990 \nu^{4} + 22813 \nu^{3} + \cdots + 2665 ) / 67 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 199 \nu^{9} - 14 \nu^{8} - 3044 \nu^{7} - 118 \nu^{6} + 15004 \nu^{5} + 719 \nu^{4} - 26174 \nu^{3} + \cdots - 1789 ) / 67 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 233 \nu^{9} - 44 \nu^{8} - 3556 \nu^{7} + 280 \nu^{6} + 17551 \nu^{5} - 1186 \nu^{4} - 30987 \nu^{3} + \cdots - 2866 ) / 67 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 399 \nu^{9} - 88 \nu^{8} - 6107 \nu^{7} + 694 \nu^{6} + 30278 \nu^{5} - 3176 \nu^{4} - 53800 \nu^{3} + \cdots - 5732 ) / 67 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{8} + 2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} - 7\beta_{8} - 2\beta_{7} + 6\beta_{6} + 3\beta_{5} + 4\beta_{4} + 5\beta_{3} - \beta_{2} - \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{9} - 20 \beta_{8} - \beta_{7} + 2 \beta_{6} + 19 \beta_{5} + 11 \beta_{4} + 34 \beta_{3} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51 \beta_{9} - 54 \beta_{8} - 23 \beta_{7} + 36 \beta_{6} + 39 \beta_{5} + 47 \beta_{4} + 64 \beta_{3} + \cdots + 113 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 115 \beta_{9} - 174 \beta_{8} - 17 \beta_{7} + 31 \beta_{6} + 163 \beta_{5} + 102 \beta_{4} + 305 \beta_{3} + \cdots + 135 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 393 \beta_{9} - 441 \beta_{8} - 203 \beta_{7} + 233 \beta_{6} + 382 \beta_{5} + 429 \beta_{4} + \cdots + 816 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1018 \beta_{9} - 1454 \beta_{8} - 205 \beta_{7} + 339 \beta_{6} + 1363 \beta_{5} + 906 \beta_{4} + \cdots + 1287 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.87071
0.343956
1.92335
−2.18686
1.84283
0.472836
−0.754674
−2.46955
1.81408
2.88475
0 −1.00000 0 −3.94607 0 −4.45541 0 1.00000 0
1.2 0 −1.00000 0 −2.59067 0 −3.72003 0 1.00000 0
1.3 0 −1.00000 0 −2.53105 0 −3.36448 0 1.00000 0
1.4 0 −1.00000 0 −0.360801 0 −3.33122 0 1.00000 0
1.5 0 −1.00000 0 0.0235872 0 3.81556 0 1.00000 0
1.6 0 −1.00000 0 0.344574 0 1.62458 0 1.00000 0
1.7 0 −1.00000 0 0.828090 0 −4.50584 0 1.00000 0
1.8 0 −1.00000 0 1.50596 0 3.28346 0 1.00000 0
1.9 0 −1.00000 0 2.50135 0 1.61585 0 1.00000 0
1.10 0 −1.00000 0 4.22503 0 0.0375262 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6348.2.a.q 10
23.b odd 2 1 6348.2.a.r 10
23.d odd 22 2 276.2.i.b ✓ 20
69.g even 22 2 828.2.q.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.b ✓ 20 23.d odd 22 2
828.2.q.b 20 69.g even 22 2
6348.2.a.q 10 1.a even 1 1 trivial
6348.2.a.r 10 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6348))\):

\( T_{5}^{10} - 28T_{5}^{8} + 219T_{5}^{6} - 55T_{5}^{5} - 529T_{5}^{4} + 352T_{5}^{3} + 60T_{5}^{2} - 44T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{10} + 9 T_{7}^{9} - 13 T_{7}^{8} - 291 T_{7}^{7} - 239 T_{7}^{6} + 3186 T_{7}^{5} + 4041 T_{7}^{4} + \cdots - 1033 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 28 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} + 9 T^{9} + \cdots - 1033 \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + \cdots - 64009 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{10} + 7 T^{9} + \cdots + 11527 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 23299 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 9 T^{9} + \cdots + 34583 \) Copy content Toggle raw display
$31$ \( T^{10} - 11 T^{9} + \cdots + 76451 \) Copy content Toggle raw display
$37$ \( T^{10} - 9 T^{9} + \cdots - 1607 \) Copy content Toggle raw display
$41$ \( T^{10} - 2 T^{9} + \cdots + 35077769 \) Copy content Toggle raw display
$43$ \( T^{10} + 24 T^{9} + \cdots + 14059057 \) Copy content Toggle raw display
$47$ \( T^{10} + 27 T^{9} + \cdots - 12485353 \) Copy content Toggle raw display
$53$ \( T^{10} + 4 T^{9} + \cdots - 32867 \) Copy content Toggle raw display
$59$ \( T^{10} - 2 T^{9} + \cdots + 3299 \) Copy content Toggle raw display
$61$ \( T^{10} - 27 T^{9} + \cdots + 64877 \) Copy content Toggle raw display
$67$ \( T^{10} + 24 T^{9} + \cdots + 1850971 \) Copy content Toggle raw display
$71$ \( T^{10} - 11 T^{9} + \cdots + 26353921 \) Copy content Toggle raw display
$73$ \( T^{10} - 2 T^{9} + \cdots + 203501 \) Copy content Toggle raw display
$79$ \( T^{10} + 23 T^{9} + \cdots - 175561 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 163352333 \) Copy content Toggle raw display
$89$ \( T^{10} + 11 T^{9} + \cdots + 8229827 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 220555369 \) Copy content Toggle raw display
show more
show less