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Properties

Label 6348.2.a.t
Level $6348$
Weight $2$
Character orbit 6348.a
Self dual yes
Analytic conductor $50.689$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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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: \(0\)
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: \( x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 \) 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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{4} q^{5} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 2) q^{7}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{4} q^{5} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 2) q^{7}+ \cdots + ( - \beta_{6} + \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9} + 11 q^{11} + 2 q^{15} + 13 q^{17} + 18 q^{19} + 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} + 11 q^{33} - 13 q^{35} + 5 q^{37} + 24 q^{41} + 40 q^{43} + 2 q^{45} - 9 q^{47} + 5 q^{49} + 13 q^{51} + 6 q^{53} - 14 q^{55} + 18 q^{57} + 28 q^{59} + 39 q^{61} + 11 q^{63} + 14 q^{65} + 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} + 33 q^{79} + 10 q^{81} + 29 q^{83} - 21 q^{85} + 5 q^{87} + 17 q^{89} + 36 q^{91} + 15 q^{93} - 42 q^{95} + 46 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9 \nu^{9} - 4 \nu^{8} + 167 \nu^{7} + 216 \nu^{6} - 1140 \nu^{5} - 2057 \nu^{4} + 3380 \nu^{3} + \cdots - 2070 ) / 529 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16 \nu^{9} - 44 \nu^{8} - 279 \nu^{7} + 559 \nu^{6} + 1743 \nu^{5} - 1766 \nu^{4} - 4266 \nu^{3} + \cdots + 1035 ) / 529 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20 \nu^{9} - 55 \nu^{8} - 481 \nu^{7} + 831 \nu^{6} + 3898 \nu^{5} - 3530 \nu^{4} - 11416 \nu^{3} + \cdots - 161 ) / 529 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 50 \nu^{9} + 12 \nu^{8} - 1030 \nu^{7} - 165 \nu^{6} + 7123 \nu^{5} + 927 \nu^{4} - 18006 \nu^{3} + \cdots + 2507 ) / 529 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76 \nu^{9} + 117 \nu^{8} + 1331 \nu^{7} - 1672 \nu^{6} - 7917 \nu^{5} + 7250 \nu^{4} + 17423 \nu^{3} + \cdots + 506 ) / 529 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 95 \nu^{9} + 152 \nu^{8} + 1589 \nu^{7} - 2044 \nu^{6} - 9051 \nu^{5} + 7993 \nu^{4} + 19427 \nu^{3} + \cdots - 161 ) / 529 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 99 \nu^{9} + 163 \nu^{8} + 1791 \nu^{7} - 2316 \nu^{6} - 11206 \nu^{5} + 9757 \nu^{4} + 26577 \nu^{3} + \cdots - 1081 ) / 529 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 233 \nu^{9} + 405 \nu^{8} + 4119 \nu^{7} - 5563 \nu^{6} - 25289 \nu^{5} + 22023 \nu^{4} + \cdots - 2277 ) / 529 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 9\beta_{8} - 6\beta_{7} + \beta_{5} + 6\beta_{4} - 4\beta_{3} + 2\beta_{2} + 8\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} - 12\beta_{8} - 22\beta_{7} + 17\beta_{6} + 10\beta_{4} - 5\beta_{3} + \beta_{2} + 40\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{9} + 69 \beta_{8} - 38 \beta_{7} + 5 \beta_{6} + 12 \beta_{5} + 36 \beta_{4} - 17 \beta_{3} + \cdots + 166 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 69 \beta_{9} - 107 \beta_{8} - 196 \beta_{7} + 180 \beta_{6} + 2 \beta_{5} + 80 \beta_{4} - 61 \beta_{3} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 107 \beta_{9} + 504 \beta_{8} - 270 \beta_{7} + 95 \beta_{6} + 116 \beta_{5} + 225 \beta_{4} + \cdots + 1132 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 504 \beta_{9} - 857 \beta_{8} - 1631 \beta_{7} + 1640 \beta_{6} + 45 \beta_{5} + 595 \beta_{4} + \cdots + 430 \) 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
2.75582
−1.92165
−2.66501
2.46393
−2.12473
−0.0733115
−0.633097
2.84010
1.00266
2.35529
0 1.00000 0 −3.63668 0 4.36939 0 1.00000 0
1.2 0 1.00000 0 −2.68761 0 0.168413 0 1.00000 0
1.3 0 1.00000 0 −2.49042 0 4.04500 0 1.00000 0
1.4 0 1.00000 0 −1.04710 0 −1.46307 0 1.00000 0
1.5 0 1.00000 0 0.395239 0 −3.09253 0 1.00000 0
1.6 0 1.00000 0 1.12335 0 −1.05967 0 1.00000 0
1.7 0 1.00000 0 1.52600 0 3.74770 0 1.00000 0
1.8 0 1.00000 0 1.80838 0 3.41002 0 1.00000 0
1.9 0 1.00000 0 2.92409 0 1.00076 0 1.00000 0
1.10 0 1.00000 0 4.08477 0 −0.126013 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.t 10
23.b odd 2 1 6348.2.a.s 10
23.d odd 22 2 276.2.i.a ✓ 20
69.g even 22 2 828.2.q.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.a ✓ 20 23.d odd 22 2
828.2.q.c 20 69.g even 22 2
6348.2.a.s 10 23.b odd 2 1
6348.2.a.t 10 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5}^{10} - 2 T_{5}^{9} - 28 T_{5}^{8} + 52 T_{5}^{7} + 245 T_{5}^{6} - 457 T_{5}^{5} - 709 T_{5}^{4} + \cdots + 373 \) Copy content Toggle raw display
\( T_{7}^{10} - 11 T_{7}^{9} + 23 T_{7}^{8} + 121 T_{7}^{7} - 455 T_{7}^{6} - 154 T_{7}^{5} + 1493 T_{7}^{4} + \cdots + 23 \) 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
$5$ \( T^{10} - 2 T^{9} + \cdots + 373 \) Copy content Toggle raw display
$7$ \( T^{10} - 11 T^{9} + \cdots + 23 \) Copy content Toggle raw display
$11$ \( T^{10} - 11 T^{9} + \cdots - 6049 \) Copy content Toggle raw display
$13$ \( T^{10} - 83 T^{8} + \cdots - 27323 \) Copy content Toggle raw display
$17$ \( T^{10} - 13 T^{9} + \cdots + 13463 \) Copy content Toggle raw display
$19$ \( T^{10} - 18 T^{9} + \cdots + 22639 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 5 T^{9} + \cdots + 307 \) Copy content Toggle raw display
$31$ \( T^{10} - 15 T^{9} + \cdots - 36961 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 204113471 \) Copy content Toggle raw display
$41$ \( T^{10} - 24 T^{9} + \cdots + 19742141 \) Copy content Toggle raw display
$43$ \( T^{10} - 40 T^{9} + \cdots + 24034781 \) Copy content Toggle raw display
$47$ \( T^{10} + 9 T^{9} + \cdots - 337853 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 109070807 \) Copy content Toggle raw display
$59$ \( T^{10} - 28 T^{9} + \cdots + 1346311 \) Copy content Toggle raw display
$61$ \( T^{10} - 39 T^{9} + \cdots - 4112263 \) Copy content Toggle raw display
$67$ \( T^{10} - 32 T^{9} + \cdots + 4947251 \) Copy content Toggle raw display
$71$ \( T^{10} + 33 T^{9} + \cdots + 3825889 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 204421579 \) Copy content Toggle raw display
$79$ \( T^{10} - 33 T^{9} + \cdots - 26171441 \) Copy content Toggle raw display
$83$ \( T^{10} - 29 T^{9} + \cdots - 35374439 \) Copy content Toggle raw display
$89$ \( T^{10} - 17 T^{9} + \cdots + 82902247 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 242728531 \) Copy content Toggle raw display
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