↑

Properties

Label 6354.2.a.bk
Level $6354$
Weight $2$
Character orbit 6354.a
Self dual yes
Analytic conductor $50.737$
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] = [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: \(1\)
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: \( x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 164x^{6} - 186x^{5} - 510x^{4} + 200x^{3} + 579x^{2} + 108x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 706)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1) q^{5}+ \cdots - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1) q^{5}+ \cdots + (\beta_{8} - \beta_{5} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 12 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 12 q^{7} - 10 q^{8} + 3 q^{10} - 15 q^{11} + 11 q^{13} + 12 q^{14} + 10 q^{16} - 7 q^{17} + 4 q^{19} - 3 q^{20} + 15 q^{22} - 10 q^{23} + 27 q^{25} - 11 q^{26} - 12 q^{28} - 14 q^{29} - q^{31} - 10 q^{32} + 7 q^{34} + 20 q^{35} + 9 q^{37} - 4 q^{38} + 3 q^{40} - 7 q^{41} + 19 q^{43} - 15 q^{44} + 10 q^{46} + 22 q^{47} - 6 q^{49} - 27 q^{50} + 11 q^{52} + 2 q^{53} - 33 q^{55} + 12 q^{56} + 14 q^{58} + 6 q^{59} + 6 q^{61} + q^{62} + 10 q^{64} + 9 q^{65} - 20 q^{67} - 7 q^{68} - 20 q^{70} + 11 q^{71} - 9 q^{73} - 9 q^{74} + 4 q^{76} + 20 q^{77} - 10 q^{79} - 3 q^{80} + 7 q^{82} + 7 q^{83} - 21 q^{85} - 19 q^{86} + 15 q^{88} - 18 q^{89} - 8 q^{91} - 10 q^{92} - 22 q^{94} - 2 q^{95} + 19 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 164x^{6} - 186x^{5} - 510x^{4} + 200x^{3} + 579x^{2} + 108x - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1373 \nu^{9} - 7954 \nu^{8} - 13810 \nu^{7} + 149396 \nu^{6} - 112496 \nu^{5} - 631680 \nu^{4} + \cdots - 518760 ) / 80604 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1537 \nu^{9} + 5186 \nu^{8} + 28238 \nu^{7} - 105736 \nu^{6} - 113462 \nu^{5} + 541746 \nu^{4} + \cdots + 178614 ) / 80604 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1183 \nu^{9} - 4280 \nu^{8} - 22652 \nu^{7} + 87427 \nu^{6} + 104924 \nu^{5} - 446877 \nu^{4} + \cdots - 130383 ) / 40302 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1002 \nu^{9} - 2147 \nu^{8} - 21200 \nu^{7} + 39718 \nu^{6} + 130804 \nu^{5} - 145762 \nu^{4} + \cdots + 107343 ) / 26868 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1862 \nu^{9} + 5107 \nu^{8} + 39628 \nu^{7} - 106208 \nu^{6} - 233668 \nu^{5} + 560364 \nu^{4} + \cdots + 146673 ) / 40302 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4925 \nu^{9} - 15337 \nu^{8} - 99232 \nu^{7} + 303734 \nu^{6} + 533986 \nu^{5} - 1431258 \nu^{4} + \cdots + 40707 ) / 80604 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1736 \nu^{9} - 5923 \nu^{8} - 31492 \nu^{7} + 112556 \nu^{6} + 125434 \nu^{5} - 473658 \nu^{4} + \cdots + 68247 ) / 26868 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2327 \nu^{9} - 7853 \nu^{8} - 42146 \nu^{7} + 151870 \nu^{6} + 160674 \nu^{5} - 670844 \nu^{4} + \cdots - 78135 ) / 26868 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} - 4\beta_{8} + 4\beta_{7} + 3\beta_{6} - 2\beta_{4} - 2\beta_{3} + 10\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{9} - 13 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 12 \beta_{5} + 17 \beta_{4} + 16 \beta_{3} + \cdots + 65 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29\beta_{9} - 59\beta_{8} + 62\beta_{7} + 50\beta_{6} - 29\beta_{4} - 38\beta_{3} - 3\beta_{2} + 118\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 47 \beta_{9} - 156 \beta_{8} - 46 \beta_{7} + 30 \beta_{6} + 147 \beta_{5} + 242 \beta_{4} + 207 \beta_{3} + \cdots + 776 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 379 \beta_{9} - 741 \beta_{8} + 812 \beta_{7} + 687 \beta_{6} - 24 \beta_{5} - 401 \beta_{4} + \cdots - 371 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 592 \beta_{9} - 1814 \beta_{8} - 848 \beta_{7} + 318 \beta_{6} + 1854 \beta_{5} + 3262 \beta_{4} + \cdots + 9565 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4834 \beta_{9} - 9004 \beta_{8} + 10320 \beta_{7} + 8930 \beta_{6} - 674 \beta_{5} - 5672 \beta_{4} + \cdots - 7848 \) 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
1.56329
0.140678
3.10004
2.37985
−1.51097
−1.20885
−1.79126
−3.67678
−0.405705
3.40971
−1.00000 0 1.00000 −3.97309 0 −2.56329 −1.00000 0 3.97309
1.2 −1.00000 0 1.00000 −3.35597 0 −1.14068 −1.00000 0 3.35597
1.3 −1.00000 0 1.00000 −3.34822 0 −4.10004 −1.00000 0 3.34822
1.4 −1.00000 0 1.00000 −1.40919 0 −3.37985 −1.00000 0 1.40919
1.5 −1.00000 0 1.00000 −0.988722 0 0.510966 −1.00000 0 0.988722
1.6 −1.00000 0 1.00000 −0.825583 0 0.208846 −1.00000 0 0.825583
1.7 −1.00000 0 1.00000 0.924826 0 0.791263 −1.00000 0 −0.924826
1.8 −1.00000 0 1.00000 2.49680 0 2.67678 −1.00000 0 −2.49680
1.9 −1.00000 0 1.00000 3.60127 0 −0.594295 −1.00000 0 −3.60127
1.10 −1.00000 0 1.00000 3.87788 0 −4.40971 −1.00000 0 −3.87788
\(n\): 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.bk 10
3.b odd 2 1 706.2.a.h ✓ 10
12.b even 2 1 5648.2.a.m 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
706.2.a.h ✓ 10 3.b odd 2 1
5648.2.a.m 10 12.b even 2 1
6354.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6354))\):

\( T_{5}^{10} + 3 T_{5}^{9} - 34 T_{5}^{8} - 104 T_{5}^{7} + 357 T_{5}^{6} + 1161 T_{5}^{5} - 1002 T_{5}^{4} + \cdots + 1656 \) Copy content Toggle raw display
\( T_{7}^{10} + 12 T_{7}^{9} + 40 T_{7}^{8} - 32 T_{7}^{7} - 382 T_{7}^{6} - 454 T_{7}^{5} + 332 T_{7}^{4} + \cdots + 24 \) Copy content Toggle raw display
\( T_{11}^{10} + 15 T_{11}^{9} + 42 T_{11}^{8} - 346 T_{11}^{7} - 2011 T_{11}^{6} - 189 T_{11}^{5} + \cdots - 38208 \) Copy content Toggle raw display
\( T_{13}^{10} - 11 T_{13}^{9} - 22 T_{13}^{8} + 614 T_{13}^{7} - 1427 T_{13}^{6} - 6061 T_{13}^{5} + \cdots + 67784 \) 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
$5$ \( T^{10} + 3 T^{9} + \cdots + 1656 \) Copy content Toggle raw display
$7$ \( T^{10} + 12 T^{9} + \cdots + 24 \) Copy content Toggle raw display
$11$ \( T^{10} + 15 T^{9} + \cdots - 38208 \) Copy content Toggle raw display
$13$ \( T^{10} - 11 T^{9} + \cdots + 67784 \) Copy content Toggle raw display
$17$ \( T^{10} + 7 T^{9} + \cdots + 149648 \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + \cdots - 50944 \) Copy content Toggle raw display
$23$ \( T^{10} + 10 T^{9} + \cdots + 6144 \) Copy content Toggle raw display
$29$ \( T^{10} + 14 T^{9} + \cdots - 7006784 \) Copy content Toggle raw display
$31$ \( T^{10} + T^{9} + \cdots - 8354 \) Copy content Toggle raw display
$37$ \( T^{10} - 9 T^{9} + \cdots + 61362776 \) Copy content Toggle raw display
$41$ \( T^{10} + 7 T^{9} + \cdots + 1989004 \) Copy content Toggle raw display
$43$ \( T^{10} - 19 T^{9} + \cdots - 1906832 \) Copy content Toggle raw display
$47$ \( T^{10} - 22 T^{9} + \cdots + 1718784 \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + \cdots - 11193568 \) Copy content Toggle raw display
$59$ \( T^{10} - 6 T^{9} + \cdots - 318112 \) Copy content Toggle raw display
$61$ \( T^{10} - 6 T^{9} + \cdots - 8692928 \) Copy content Toggle raw display
$67$ \( T^{10} + 20 T^{9} + \cdots - 3331264 \) Copy content Toggle raw display
$71$ \( T^{10} - 11 T^{9} + \cdots + 43346 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 412620132 \) Copy content Toggle raw display
$79$ \( T^{10} + 10 T^{9} + \cdots + 599288 \) Copy content Toggle raw display
$83$ \( T^{10} - 7 T^{9} + \cdots + 19263632 \) Copy content Toggle raw display
$89$ \( T^{10} + 18 T^{9} + \cdots + 8676352 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 109331504 \) Copy content Toggle raw display
show more
show less