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Properties

Label 6162.2.a.bl
Level $6162$
Weight $2$
Character orbit 6162.a
Self dual yes
Analytic conductor $49.204$
Analytic rank $1$
Dimension $12$
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] = [6162,2,Mod(1,6162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6162, 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("6162.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6162 = 2 \cdot 3 \cdot 13 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6162.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: \(49.2038177255\)
Analytic rank: \(1\)
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: \( x^{12} - 6 x^{11} - 25 x^{10} + 175 x^{9} + 198 x^{8} - 1833 x^{7} - 254 x^{6} + 8244 x^{5} - 3124 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + (\beta_1 - 1) q^{5} - q^{6} + \beta_{9} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + (\beta_1 - 1) q^{5} - q^{6} + \beta_{9} q^{7} - q^{8} + q^{9} + ( - \beta_1 + 1) q^{10} + (\beta_{8} - 1) q^{11} + q^{12} - q^{13} - \beta_{9} q^{14} + (\beta_1 - 1) q^{15} + q^{16} + ( - \beta_{10} - \beta_{9} - 1) q^{17} - q^{18} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{19}+ \cdots + (\beta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 6 q^{5} - 12 q^{6} - 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 6 q^{5} - 12 q^{6} - 5 q^{7} - 12 q^{8} + 12 q^{9} + 6 q^{10} - 9 q^{11} + 12 q^{12} - 12 q^{13} + 5 q^{14} - 6 q^{15} + 12 q^{16} - 2 q^{17} - 12 q^{18} - 6 q^{19} - 6 q^{20} - 5 q^{21} + 9 q^{22} - 8 q^{23} - 12 q^{24} + 26 q^{25} + 12 q^{26} + 12 q^{27} - 5 q^{28} - 3 q^{29} + 6 q^{30} - 14 q^{31} - 12 q^{32} - 9 q^{33} + 2 q^{34} + q^{35} + 12 q^{36} - 17 q^{37} + 6 q^{38} - 12 q^{39} + 6 q^{40} - 18 q^{41} + 5 q^{42} + 6 q^{43} - 9 q^{44} - 6 q^{45} + 8 q^{46} - 44 q^{47} + 12 q^{48} + 33 q^{49} - 26 q^{50} - 2 q^{51} - 12 q^{52} + 3 q^{53} - 12 q^{54} - 15 q^{55} + 5 q^{56} - 6 q^{57} + 3 q^{58} - 10 q^{59} - 6 q^{60} + 12 q^{61} + 14 q^{62} - 5 q^{63} + 12 q^{64} + 6 q^{65} + 9 q^{66} - q^{67} - 2 q^{68} - 8 q^{69} - q^{70} - 55 q^{71} - 12 q^{72} - q^{73} + 17 q^{74} + 26 q^{75} - 6 q^{76} - 37 q^{77} + 12 q^{78} + 12 q^{79} - 6 q^{80} + 12 q^{81} + 18 q^{82} - 29 q^{83} - 5 q^{84} - 11 q^{85} - 6 q^{86} - 3 q^{87} + 9 q^{88} - 37 q^{89} + 6 q^{90} + 5 q^{91} - 8 q^{92} - 14 q^{93} + 44 q^{94} - 34 q^{95} - 12 q^{96} - 5 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 25 x^{10} + 175 x^{9} + 198 x^{8} - 1833 x^{7} - 254 x^{6} + 8244 x^{5} - 3124 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1118570 \nu^{11} + 5513169 \nu^{10} + 36906906 \nu^{9} - 182510233 \nu^{8} - 428679287 \nu^{7} + \cdots + 813464392 ) / 91731124 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2757219 \nu^{11} - 8930614 \nu^{10} - 97987693 \nu^{9} + 225151451 \nu^{8} + 1341478986 \nu^{7} + \cdots + 427921736 ) / 183462248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2851751 \nu^{11} + 25720378 \nu^{10} + 48446211 \nu^{9} - 846905949 \nu^{8} + \cdots + 3992241776 ) / 183462248 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3830073 \nu^{11} - 3643060 \nu^{10} - 172760389 \nu^{9} + 43764245 \nu^{8} + 2800689864 \nu^{7} + \cdots + 1752948584 ) / 183462248 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4925175 \nu^{11} - 30816462 \nu^{10} - 124909673 \nu^{9} + 946624303 \nu^{8} + 962319958 \nu^{7} + \cdots - 2421820296 ) / 183462248 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5424545 \nu^{11} - 36466806 \nu^{10} - 119282357 \nu^{9} + 1070391235 \nu^{8} + \cdots - 3853870760 ) / 183462248 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3180245 \nu^{11} + 20131071 \nu^{10} + 78780339 \nu^{9} - 610736728 \nu^{8} + \cdots + 1747694740 ) / 91731124 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4466642 \nu^{11} - 25504285 \nu^{10} - 114198412 \nu^{9} + 722479371 \nu^{8} + 976890791 \nu^{7} + \cdots - 1128122488 ) / 91731124 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5408215 \nu^{11} + 33998261 \nu^{10} + 128829245 \nu^{9} - 995004162 \nu^{8} + \cdots + 2591853876 ) / 91731124 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9163835 \nu^{11} - 49888076 \nu^{10} - 249405537 \nu^{9} + 1437710593 \nu^{8} + 2361295704 \nu^{7} + \cdots - 2313764224 ) / 91731124 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} - 4\beta_{8} + \beta_{7} - \beta_{5} + 4\beta_{4} - \beta_{2} + 12\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{11} + 4 \beta_{10} - 19 \beta_{9} - 28 \beta_{8} + 16 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 8 \beta_{10} - 49 \beta_{9} - 109 \beta_{8} + 35 \beta_{7} - 4 \beta_{6} - 24 \beta_{5} + \cdots + 146 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46 \beta_{11} + 114 \beta_{10} - 305 \beta_{9} - 591 \beta_{8} + 292 \beta_{7} - 49 \beta_{6} + \cdots + 1175 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + 392 \beta_{10} - 889 \beta_{9} - 2408 \beta_{8} + 952 \beta_{7} - 49 \beta_{6} + \cdots + 3753 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 399 \beta_{11} + 2950 \beta_{10} - 4529 \beta_{9} - 11891 \beta_{8} + 5895 \beta_{7} - 364 \beta_{6} + \cdots + 22140 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1054 \beta_{11} + 12636 \beta_{10} - 13785 \beta_{9} - 50497 \beta_{8} + 23422 \beta_{7} + \cdots + 86851 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2533 \beta_{11} + 74786 \beta_{10} - 60885 \beta_{9} - 239040 \beta_{8} + 126030 \beta_{7} + \cdots + 449805 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 51747 \beta_{11} + 348736 \beta_{10} - 175849 \beta_{9} - 1044981 \beta_{8} + 550359 \beta_{7} + \cdots + 1935022 \) 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
−3.25657
−2.72012
−2.38640
−2.30799
0.0345069
0.0542661
0.742544
2.12231
2.18544
2.49715
4.30764
4.72722
−1.00000 1.00000 1.00000 −4.25657 −1.00000 1.88312 −1.00000 1.00000 4.25657
1.2 −1.00000 1.00000 1.00000 −3.72012 −1.00000 −4.44725 −1.00000 1.00000 3.72012
1.3 −1.00000 1.00000 1.00000 −3.38640 −1.00000 −4.09451 −1.00000 1.00000 3.38640
1.4 −1.00000 1.00000 1.00000 −3.30799 −1.00000 3.08293 −1.00000 1.00000 3.30799
1.5 −1.00000 1.00000 1.00000 −0.965493 −1.00000 −1.33935 −1.00000 1.00000 0.965493
1.6 −1.00000 1.00000 1.00000 −0.945734 −1.00000 4.37051 −1.00000 1.00000 0.945734
1.7 −1.00000 1.00000 1.00000 −0.257456 −1.00000 0.527372 −1.00000 1.00000 0.257456
1.8 −1.00000 1.00000 1.00000 1.12231 −1.00000 1.70005 −1.00000 1.00000 −1.12231
1.9 −1.00000 1.00000 1.00000 1.18544 −1.00000 −2.28547 −1.00000 1.00000 −1.18544
1.10 −1.00000 1.00000 1.00000 1.49715 −1.00000 −3.22012 −1.00000 1.00000 −1.49715
1.11 −1.00000 1.00000 1.00000 3.30764 −1.00000 −4.26678 −1.00000 1.00000 −3.30764
1.12 −1.00000 1.00000 1.00000 3.72722 −1.00000 3.08949 −1.00000 1.00000 −3.72722
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(13\) \( +1 \)
\(79\) \( -1 \)

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 6162.2.a.bl ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6162.2.a.bl ✓ 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6162))\):

\( T_{5}^{12} + 6 T_{5}^{11} - 25 T_{5}^{10} - 185 T_{5}^{9} + 153 T_{5}^{8} + 1863 T_{5}^{7} + 61 T_{5}^{6} + \cdots - 1024 \) Copy content Toggle raw display
\( T_{7}^{12} + 5 T_{7}^{11} - 46 T_{7}^{10} - 229 T_{7}^{9} + 792 T_{7}^{8} + 3769 T_{7}^{7} + \cdots + 53824 \) Copy content Toggle raw display
\( T_{11}^{12} + 9 T_{11}^{11} - 50 T_{11}^{10} - 618 T_{11}^{9} + 157 T_{11}^{8} + 13597 T_{11}^{7} + \cdots + 32768 \) Copy content Toggle raw display
\( T_{17}^{12} + 2 T_{17}^{11} - 128 T_{17}^{10} - 135 T_{17}^{9} + 6236 T_{17}^{8} + 1448 T_{17}^{7} + \cdots + 120832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots - 1024 \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} + \cdots + 53824 \) Copy content Toggle raw display
$11$ \( T^{12} + 9 T^{11} + \cdots + 32768 \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots + 120832 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots - 17757056 \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} + \cdots + 11552 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 23040 \) Copy content Toggle raw display
$31$ \( T^{12} + 14 T^{11} + \cdots - 425984 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 215966720 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 478622720 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 15641154304 \) Copy content Toggle raw display
$47$ \( T^{12} + 44 T^{11} + \cdots - 18055168 \) Copy content Toggle raw display
$53$ \( T^{12} - 3 T^{11} + \cdots + 24724736 \) Copy content Toggle raw display
$59$ \( T^{12} + 10 T^{11} + \cdots + 19456000 \) Copy content Toggle raw display
$61$ \( T^{12} - 12 T^{11} + \cdots - 7461376 \) Copy content Toggle raw display
$67$ \( T^{12} + T^{11} + \cdots - 766760 \) Copy content Toggle raw display
$71$ \( T^{12} + 55 T^{11} + \cdots + 2200960 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 335191936 \) Copy content Toggle raw display
$79$ \( (T - 1)^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 94722977792 \) Copy content Toggle raw display
$89$ \( T^{12} + 37 T^{11} + \cdots + 17107264 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 154238976 \) Copy content Toggle raw display
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