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Properties

Label 6162.2.a.bi
Level $6162$
Weight $2$
Character orbit 6162.a
Self dual yes
Analytic conductor $49.204$
Analytic rank $0$
Dimension $9$
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 21x^{7} + 41x^{6} + 150x^{5} - 116x^{4} - 354x^{3} - 38x^{2} + 92x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 21x^{7} + 41x^{6} + 150x^{5} - 116x^{4} - 354x^{3} - 38x^{2} + 92x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2201 \nu^{8} + 11201 \nu^{7} + 25189 \nu^{6} - 150921 \nu^{5} - 58200 \nu^{4} + 477384 \nu^{3} + \cdots + 95656 ) / 105268 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2204 \nu^{8} + 7689 \nu^{7} + 43876 \nu^{6} - 114479 \nu^{5} - 307412 \nu^{4} + 455508 \nu^{3} + \cdots - 191656 ) / 105268 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2302 \nu^{8} + 7004 \nu^{7} + 49027 \nu^{6} - 99533 \nu^{5} - 360246 \nu^{4} + 319866 \nu^{3} + \cdots - 173240 ) / 52634 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2458 \nu^{8} + 8599 \nu^{7} + 47022 \nu^{6} - 125690 \nu^{5} - 292355 \nu^{4} + 445530 \nu^{3} + \cdots + 45128 ) / 52634 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2498 \nu^{8} - 5634 \nu^{7} - 59329 \nu^{6} + 69641 \nu^{5} + 465914 \nu^{4} - 101216 \nu^{3} + \cdots + 83774 ) / 52634 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6269 \nu^{8} + 12306 \nu^{7} + 153075 \nu^{6} - 132578 \nu^{5} - 1237340 \nu^{4} - 27236 \nu^{3} + \cdots - 639424 ) / 105268 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12113 \nu^{8} - 39667 \nu^{7} - 237891 \nu^{6} + 541583 \nu^{5} + 1574738 \nu^{4} - 1570876 \nu^{3} + \cdots + 613244 ) / 105268 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6620 \nu^{8} + 22474 \nu^{7} + 128826 \nu^{6} - 316437 \nu^{5} - 841153 \nu^{4} + 992384 \nu^{3} + \cdots - 200730 ) / 52634 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{8} + 3\beta_{7} + 5\beta_{6} + 6\beta_{5} - 2\beta_{4} - 8\beta_{3} + 9\beta_{2} - 2\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 20\beta_{8} + 2\beta_{7} + 15\beta_{6} + 27\beta_{5} - 18\beta_{4} - 28\beta_{3} + 25\beta_{2} - 17\beta _1 + 97 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 89\beta_{8} + 31\beta_{7} + 68\beta_{6} + 90\beta_{5} - 68\beta_{4} - 147\beta_{3} + 154\beta_{2} - 49\beta _1 + 328 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 406 \beta_{8} + 68 \beta_{7} + 243 \beta_{6} + 407 \beta_{5} - 380 \beta_{4} - 592 \beta_{3} + \cdots + 1629 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1798 \beta_{8} + 480 \beta_{7} + 1071 \beta_{6} + 1525 \beta_{5} - 1644 \beta_{4} - 2800 \beta_{3} + \cdots + 6431 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8220 \beta_{8} + 1668 \beta_{7} + 4227 \beta_{6} + 6801 \beta_{5} - 8006 \beta_{4} - 11910 \beta_{3} + \cdots + 29773 \) 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
−0.875706
3.27353
0.388928
2.09848
0.0938633
−1.24389
−2.93394
4.44458
−2.24586
−1.00000 −1.00000 1.00000 −3.99591 1.00000 4.63074 −1.00000 1.00000 3.99591
1.2 −1.00000 −1.00000 1.00000 −3.69206 1.00000 −1.23956 −1.00000 1.00000 3.69206
1.3 −1.00000 −1.00000 1.00000 −1.56948 1.00000 3.11429 −1.00000 1.00000 1.56948
1.4 −1.00000 −1.00000 1.00000 −0.887420 1.00000 4.10112 −1.00000 1.00000 0.887420
1.5 −1.00000 −1.00000 1.00000 −0.666873 1.00000 −2.47001 −1.00000 1.00000 0.666873
1.6 −1.00000 −1.00000 1.00000 −0.328498 1.00000 −1.41359 −1.00000 1.00000 0.328498
1.7 −1.00000 −1.00000 1.00000 0.486642 1.00000 4.21368 −1.00000 1.00000 −0.486642
1.8 −1.00000 −1.00000 1.00000 3.63655 1.00000 −0.145264 −1.00000 1.00000 −3.63655
1.9 −1.00000 −1.00000 1.00000 4.01704 1.00000 −0.791404 −1.00000 1.00000 −4.01704
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.bi ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6162.2.a.bi ✓ 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6162))\):

\( T_{5}^{9} + 3T_{5}^{8} - 27T_{5}^{7} - 88T_{5}^{6} + 143T_{5}^{5} + 639T_{5}^{4} + 531T_{5}^{3} - 6T_{5}^{2} - 136T_{5} - 32 \) Copy content Toggle raw display
\( T_{7}^{9} - 10T_{7}^{8} + 12T_{7}^{7} + 129T_{7}^{6} - 217T_{7}^{5} - 704T_{7}^{4} + 546T_{7}^{3} + 1941T_{7}^{2} + 1122T_{7} + 124 \) Copy content Toggle raw display
\( T_{11}^{9} + 3T_{11}^{8} - 35T_{11}^{7} - 76T_{11}^{6} + 323T_{11}^{5} + 399T_{11}^{4} - 669T_{11}^{3} - 822T_{11}^{2} - 76T_{11} + 8 \) Copy content Toggle raw display
\( T_{17}^{9} + 9 T_{17}^{8} - 61 T_{17}^{7} - 605 T_{17}^{6} + 1315 T_{17}^{5} + 14193 T_{17}^{4} + \cdots + 366724 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 3 T^{8} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( T^{9} - 10 T^{8} + \cdots + 124 \) Copy content Toggle raw display
$11$ \( T^{9} + 3 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 9 T^{8} + \cdots + 366724 \) Copy content Toggle raw display
$19$ \( T^{9} - 10 T^{8} + \cdots - 390848 \) Copy content Toggle raw display
$23$ \( T^{9} + 4 T^{8} + \cdots + 1475848 \) Copy content Toggle raw display
$29$ \( T^{9} + 2 T^{8} + \cdots - 75008 \) Copy content Toggle raw display
$31$ \( T^{9} - 29 T^{8} + \cdots - 175456 \) Copy content Toggle raw display
$37$ \( T^{9} - 3 T^{8} + \cdots - 5248 \) Copy content Toggle raw display
$41$ \( T^{9} + 13 T^{8} + \cdots - 45112 \) Copy content Toggle raw display
$43$ \( T^{9} - 19 T^{8} + \cdots - 315136 \) Copy content Toggle raw display
$47$ \( T^{9} - 5 T^{8} + \cdots + 389504 \) Copy content Toggle raw display
$53$ \( T^{9} - 16 T^{8} + \cdots - 515072 \) Copy content Toggle raw display
$59$ \( T^{9} - 2 T^{8} + \cdots - 10843136 \) Copy content Toggle raw display
$61$ \( T^{9} - 3 T^{8} + \cdots + 8768 \) Copy content Toggle raw display
$67$ \( T^{9} - 34 T^{8} + \cdots - 2772892 \) Copy content Toggle raw display
$71$ \( T^{9} + 3 T^{8} + \cdots + 17638424 \) Copy content Toggle raw display
$73$ \( T^{9} - 23 T^{8} + \cdots - 17391008 \) Copy content Toggle raw display
$79$ \( (T + 1)^{9} \) Copy content Toggle raw display
$83$ \( T^{9} - 13 T^{8} + \cdots + 8437472 \) Copy content Toggle raw display
$89$ \( T^{9} + 20 T^{8} + \cdots + 23079856 \) Copy content Toggle raw display
$97$ \( T^{9} - 35 T^{8} + \cdots - 1988608 \) Copy content Toggle raw display
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