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Properties

Label 6354.2.a.bl
Level $6354$
Weight $2$
Character orbit 6354.a
Self dual yes
Analytic conductor $50.737$
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] = [6354,2,Mod(1,6354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6354.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6354 = 2 \cdot 3^{2} \cdot 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6354.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: \(50.7369454443\)
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: \( x^{10} - 2x^{9} - 26x^{8} + 44x^{7} + 239x^{6} - 340x^{5} - 946x^{4} + 1056x^{3} + 1584x^{2} - 1024x - 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2118)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( - \beta_{9} + 1) q^{5} + (\beta_{8} + \beta_{3} + \beta_1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + ( - \beta_{9} + 1) q^{5} + (\beta_{8} + \beta_{3} + \beta_1) q^{7} + q^{8} + ( - \beta_{9} + 1) q^{10} + (\beta_{8} + \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \beta_{7} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8} + 10 q^{10} + 2 q^{11} - 4 q^{13} + 10 q^{16} + 10 q^{17} + 14 q^{19} + 10 q^{20} + 2 q^{22} + 8 q^{23} + 20 q^{25} - 4 q^{26} + 14 q^{29} + 12 q^{31} + 10 q^{32} + 10 q^{34} + 6 q^{35} - 4 q^{37} + 14 q^{38} + 10 q^{40} + 2 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 8 q^{47} + 32 q^{49} + 20 q^{50} - 4 q^{52} + 36 q^{53} + 10 q^{55} + 14 q^{58} + 10 q^{59} - 6 q^{61} + 12 q^{62} + 10 q^{64} + 26 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} + 12 q^{71} - 8 q^{73} - 4 q^{74} + 14 q^{76} + 56 q^{77} + 28 q^{79} + 10 q^{80} + 2 q^{82} + 22 q^{83} - 4 q^{85} + 10 q^{86} + 2 q^{88} + 28 q^{89} + 8 q^{92} + 8 q^{94} + 16 q^{95} - 8 q^{97} + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 26x^{8} + 44x^{7} + 239x^{6} - 340x^{5} - 946x^{4} + 1056x^{3} + 1584x^{2} - 1024x - 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 313 \nu^{9} + 498 \nu^{8} + 7594 \nu^{7} - 8076 \nu^{6} - 62967 \nu^{5} + 32820 \nu^{4} + \cdots - 158336 ) / 13184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 84 \nu^{9} + 55 \nu^{8} + 2090 \nu^{7} - 470 \nu^{6} - 17400 \nu^{5} - 1807 \nu^{4} + 57484 \nu^{3} + \cdots - 38944 ) / 1648 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 689 \nu^{9} + 342 \nu^{8} + 18146 \nu^{7} - 2548 \nu^{6} - 162159 \nu^{5} - 18656 \nu^{4} + \cdots - 401088 ) / 6592 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1575 \nu^{9} + 542 \nu^{8} + 41814 \nu^{7} - 212 \nu^{6} - 376617 \nu^{5} - 80772 \nu^{4} + \cdots - 943616 ) / 13184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1843 \nu^{9} - 742 \nu^{8} - 48590 \nu^{7} + 2820 \nu^{6} + 431837 \nu^{5} + 77380 \nu^{4} + \cdots + 1070848 ) / 13184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 118 \nu^{9} - 65 \nu^{8} - 3088 \nu^{7} + 518 \nu^{6} + 27474 \nu^{5} + 2997 \nu^{4} - 97506 \nu^{3} + \cdots + 64752 ) / 824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 973 \nu^{9} - 474 \nu^{8} - 25634 \nu^{7} + 3676 \nu^{6} + 228227 \nu^{5} + 21180 \nu^{4} + \cdots + 489280 ) / 6592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2075 \nu^{9} + 374 \nu^{8} + 56030 \nu^{7} + 7516 \nu^{6} - 512501 \nu^{5} - 173668 \nu^{4} + \cdots - 1500160 ) / 13184 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{8} + 2\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 4\beta_{8} + 14\beta_{7} + 3\beta_{6} + 15\beta_{5} - 5\beta_{4} + \beta_{3} + 15\beta_{2} + 16\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{9} - 35 \beta_{8} + 38 \beta_{7} + 28 \beta_{6} + 20 \beta_{5} - 25 \beta_{4} + 15 \beta_{3} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{9} - 86 \beta_{8} + 183 \beta_{7} + 64 \beta_{6} + 197 \beta_{5} - 116 \beta_{4} + 24 \beta_{3} + \cdots + 409 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 251 \beta_{9} - 528 \beta_{8} + 588 \beta_{7} + 374 \beta_{6} + 343 \beta_{5} - 455 \beta_{4} + \cdots + 593 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 587 \beta_{9} - 1476 \beta_{8} + 2462 \beta_{7} + 1079 \beta_{6} + 2565 \beta_{5} - 1987 \beta_{4} + \cdots + 4806 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3603 \beta_{9} - 7655 \beta_{8} + 8674 \beta_{7} + 5158 \beta_{6} + 5590 \beta_{5} - 7371 \beta_{4} + \cdots + 10265 \) 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
1.66086
−2.69491
−1.20839
2.20729
2.70652
3.81477
−2.25056
−0.743823
1.60338
−3.09513
1.00000 0 1.00000 −2.87858 0 −1.03296 1.00000 0 −2.87858
1.2 1.00000 0 1.00000 −1.42874 0 −4.89688 1.00000 0 −1.42874
1.3 1.00000 0 1.00000 −1.25462 0 3.30920 1.00000 0 −1.25462
1.4 1.00000 0 1.00000 −0.107389 0 −0.389547 1.00000 0 −0.107389
1.5 1.00000 0 1.00000 0.215336 0 2.46762 1.00000 0 0.215336
1.6 1.00000 0 1.00000 0.586807 0 −1.49919 1.00000 0 0.586807
1.7 1.00000 0 1.00000 2.38136 0 4.79479 1.00000 0 2.38136
1.8 1.00000 0 1.00000 4.00514 0 −4.94978 1.00000 0 4.00514
1.9 1.00000 0 1.00000 4.13269 0 3.04915 1.00000 0 4.13269
1.10 1.00000 0 1.00000 4.34799 0 −0.852406 1.00000 0 4.34799
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(353\) \(-1\)

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 6354.2.a.bl 10
3.b odd 2 1 2118.2.a.t ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2118.2.a.t ✓ 10 3.b odd 2 1
6354.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6354))\):

\( T_{5}^{10} - 10 T_{5}^{9} + 15 T_{5}^{8} + 116 T_{5}^{7} - 301 T_{5}^{6} - 332 T_{5}^{5} + 953 T_{5}^{4} + \cdots + 12 \) Copy content Toggle raw display
\( T_{7}^{10} - 51T_{7}^{8} + 18T_{7}^{7} + 812T_{7}^{6} - 416T_{7}^{5} - 4200T_{7}^{4} - 312T_{7}^{3} + 8000T_{7}^{2} + 6752T_{7} + 1488 \) Copy content Toggle raw display
\( T_{11}^{10} - 2 T_{11}^{9} - 60 T_{11}^{8} + 168 T_{11}^{7} + 1005 T_{11}^{6} - 3662 T_{11}^{5} + \cdots - 256 \) Copy content Toggle raw display
\( T_{13}^{10} + 4 T_{13}^{9} - 61 T_{13}^{8} - 190 T_{13}^{7} + 1125 T_{13}^{6} + 1414 T_{13}^{5} + \cdots - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 10 T^{9} + \cdots + 12 \) Copy content Toggle raw display
$7$ \( T^{10} - 51 T^{8} + \cdots + 1488 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{10} - 10 T^{9} + \cdots - 15744 \) Copy content Toggle raw display
$19$ \( T^{10} - 14 T^{9} + \cdots - 24896 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} + \cdots - 374016 \) Copy content Toggle raw display
$29$ \( T^{10} - 14 T^{9} + \cdots - 1848064 \) Copy content Toggle raw display
$31$ \( T^{10} - 12 T^{9} + \cdots - 10083088 \) Copy content Toggle raw display
$37$ \( T^{10} + 4 T^{9} + \cdots - 41092 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 252099696 \) Copy content Toggle raw display
$43$ \( T^{10} - 10 T^{9} + \cdots + 2097796 \) Copy content Toggle raw display
$47$ \( T^{10} - 8 T^{9} + \cdots + 10473984 \) Copy content Toggle raw display
$53$ \( T^{10} - 36 T^{9} + \cdots + 190656 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 317087744 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 141307904 \) Copy content Toggle raw display
$67$ \( T^{10} - 2 T^{9} + \cdots - 1911296 \) Copy content Toggle raw display
$71$ \( T^{10} - 12 T^{9} + \cdots - 141536 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 1819886316 \) Copy content Toggle raw display
$79$ \( T^{10} - 28 T^{9} + \cdots - 5021552 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 273381936 \) Copy content Toggle raw display
$89$ \( T^{10} - 28 T^{9} + \cdots - 755712 \) Copy content Toggle raw display
$97$ \( T^{10} + 8 T^{9} + \cdots + 32256612 \) Copy content Toggle raw display
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