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Properties

Label 1216.2.s.g
Level $1216$
Weight $2$
Character orbit 1216.s
Analytic conductor $9.710$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(31,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.s (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} - 18 x^{9} + 11 x^{8} - 36 x^{7} + 180 x^{6} - 120 x^{5} + 31 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + ( - \beta_{5} - 1) q^{5} + ( - \beta_{8} - \beta_{6}) q^{7} + (2 \beta_{5} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} + ( - \beta_{5} - 1) q^{5} + ( - \beta_{8} - \beta_{6}) q^{7} + (2 \beta_{5} + \beta_1) q^{9} - \beta_{11} q^{13} + ( - 2 \beta_{9} + \beta_{7}) q^{15} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{10} - \beta_{8} + \cdots - \beta_{6}) q^{19}+ \cdots + (\beta_{11} - \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{5} + 12 q^{9} - 2 q^{13} - 6 q^{17} - 4 q^{21} - 12 q^{25} + 6 q^{29} - 32 q^{37} + 6 q^{41} + 36 q^{49} - 6 q^{53} + 54 q^{57} + 30 q^{61} + 132 q^{69} + 30 q^{73} - 30 q^{81} + 18 q^{85} + 78 q^{89} - 84 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} - 18 x^{9} + 11 x^{8} - 36 x^{7} + 180 x^{6} - 120 x^{5} + 31 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 242175 \nu^{11} - 893909 \nu^{10} + 409625 \nu^{9} + 9968412 \nu^{8} - 20916691 \nu^{7} + \cdots + 2624424 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 380120 \nu^{11} + 2635014 \nu^{10} - 9259311 \nu^{9} + 15006210 \nu^{8} - 15001060 \nu^{7} + \cdots + 3615945 ) / 26550741 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2412999 \nu^{11} - 14572043 \nu^{10} + 43590990 \nu^{9} - 42887745 \nu^{8} + 23608350 \nu^{7} + \cdots - 356426955 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 229132 \nu^{11} - 1571034 \nu^{10} + 5437147 \nu^{9} - 8460709 \nu^{8} + 8042240 \nu^{7} + \cdots - 50023417 ) / 8850247 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3023 \nu^{11} - 17672 \nu^{10} + 51530 \nu^{9} - 46380 \nu^{8} + 28152 \nu^{7} - 115112 \nu^{6} + \cdots - 36570 ) / 96636 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1731304 \nu^{11} - 8906435 \nu^{10} + 22643951 \nu^{9} - 6494647 \nu^{8} - 2719650 \nu^{7} + \cdots + 42893811 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3684547 \nu^{11} + 23250277 \nu^{10} - 72274600 \nu^{9} + 81050027 \nu^{8} - 43329016 \nu^{7} + \cdots + 322108929 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1956788 \nu^{11} - 12578603 \nu^{10} + 39909583 \nu^{9} - 47983703 \nu^{8} + 29508456 \nu^{7} + \cdots - 139603275 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4335151 \nu^{11} - 23551921 \nu^{10} + 64739774 \nu^{9} - 41686879 \nu^{8} + 24298878 \nu^{7} + \cdots + 95497173 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5219629 \nu^{11} + 30042221 \nu^{10} - 89043714 \nu^{9} + 86354271 \nu^{8} - 78957164 \nu^{7} + \cdots - 97712469 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19116051 \nu^{11} - 109968826 \nu^{10} + 312962446 \nu^{9} - 243581220 \nu^{8} + \cdots - 205200354 ) / 106202964 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{3} + \beta_{2} - 2\beta _1 + 4 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{9} + 9\beta_{8} + 6\beta_{7} + 9\beta_{6} - 4\beta_{5} + 2\beta_{2} - 4\beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} - 5 \beta_{10} - 17 \beta_{9} + 8 \beta_{8} + \beta_{7} + 16 \beta_{6} - 19 \beta_{5} + \cdots - 19 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{10} - 14\beta_{9} - 7\beta_{8} - 14\beta_{7} + 7\beta_{6} - 12\beta_{4} - 8\beta_{3} - 15\beta_{2} - 69 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 36 \beta_{11} - 19 \beta_{10} + 35 \beta_{9} - 143 \beta_{8} - 175 \beta_{7} - 64 \beta_{6} + \cdots - 349 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 156 \beta_{11} + 96 \beta_{10} + 498 \beta_{9} - 453 \beta_{8} - 498 \beta_{7} - 453 \beta_{6} + \cdots - 416 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 375 \beta_{11} + 497 \beta_{10} + 1691 \beta_{9} - 509 \beta_{8} - 409 \beta_{7} - 1321 \beta_{6} + \cdots + 1240 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 536 \beta_{10} + 1364 \beta_{9} + 898 \beta_{8} + 1364 \beta_{7} - 898 \beta_{6} + 885 \beta_{4} + \cdots + 4818 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3750 \beta_{11} - 419 \beta_{10} - 4139 \beta_{9} + 12350 \beta_{8} + 16129 \beta_{7} + 4195 \beta_{6} + \cdots + 30310 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16548 \beta_{11} - 12324 \beta_{10} - 44514 \beta_{9} + 35565 \beta_{8} + 44514 \beta_{7} + 35565 \beta_{6} + \cdots + 44336 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 36693 \beta_{11} - 47633 \beta_{10} - 153329 \beta_{9} + 36062 \beta_{8} + 40249 \beta_{7} + \cdots - 88279 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.17819 + 1.17819i
0.836844 + 0.836844i
1.64901 − 1.64901i
−0.649007 − 0.649007i
0.163156 − 0.163156i
2.17819 + 2.17819i
−1.17819 − 1.17819i
0.836844 − 0.836844i
1.64901 + 1.64901i
−0.649007 + 0.649007i
0.163156 + 0.163156i
2.17819 − 2.17819i
0 −2.71294 + 1.56632i 0 −1.50000 + 0.866025i 0 2.00000i 0 3.40671 − 5.90059i 0
31.2 0 −1.96854 + 1.13654i 0 −1.50000 + 0.866025i 0 − 2.00000i 0 1.08343 − 1.87656i 0
31.3 0 −0.121621 + 0.0702177i 0 −1.50000 + 0.866025i 0 2.00000i 0 −1.49014 + 2.58100i 0
31.4 0 0.121621 − 0.0702177i 0 −1.50000 + 0.866025i 0 − 2.00000i 0 −1.49014 + 2.58100i 0
31.5 0 1.96854 − 1.13654i 0 −1.50000 + 0.866025i 0 2.00000i 0 1.08343 − 1.87656i 0
31.6 0 2.71294 − 1.56632i 0 −1.50000 + 0.866025i 0 − 2.00000i 0 3.40671 − 5.90059i 0
863.1 0 −2.71294 − 1.56632i 0 −1.50000 − 0.866025i 0 − 2.00000i 0 3.40671 + 5.90059i 0
863.2 0 −1.96854 − 1.13654i 0 −1.50000 − 0.866025i 0 2.00000i 0 1.08343 + 1.87656i 0
863.3 0 −0.121621 − 0.0702177i 0 −1.50000 − 0.866025i 0 − 2.00000i 0 −1.49014 − 2.58100i 0
863.4 0 0.121621 + 0.0702177i 0 −1.50000 − 0.866025i 0 2.00000i 0 −1.49014 − 2.58100i 0
863.5 0 1.96854 + 1.13654i 0 −1.50000 − 0.866025i 0 − 2.00000i 0 1.08343 + 1.87656i 0
863.6 0 2.71294 + 1.56632i 0 −1.50000 − 0.866025i 0 2.00000i 0 3.40671 + 5.90059i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
152.l odd 6 1 inner
152.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.s.g ✓ 12
4.b odd 2 1 inner 1216.2.s.g ✓ 12
8.b even 2 1 1216.2.s.h yes 12
8.d odd 2 1 1216.2.s.h yes 12
19.d odd 6 1 1216.2.s.h yes 12
76.f even 6 1 1216.2.s.h yes 12
152.l odd 6 1 inner 1216.2.s.g ✓ 12
152.o even 6 1 inner 1216.2.s.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.s.g ✓ 12 1.a even 1 1 trivial
1216.2.s.g ✓ 12 4.b odd 2 1 inner
1216.2.s.g ✓ 12 152.l odd 6 1 inner
1216.2.s.g ✓ 12 152.o even 6 1 inner
1216.2.s.h yes 12 8.b even 2 1
1216.2.s.h yes 12 8.d odd 2 1
1216.2.s.h yes 12 19.d odd 6 1
1216.2.s.h yes 12 76.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{12} - 15T_{3}^{10} + 174T_{3}^{8} - 763T_{3}^{6} + 2586T_{3}^{4} - 51T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 15 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + T^{5} + 26 T^{4} + \cdots + 121)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 30 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 3486784401 \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 144 T^{4} + \cdots - 432)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 8 T^{2} - 4 T - 68)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} + \cdots + 3267)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 153 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$47$ \( T^{12} - 135 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$53$ \( (T^{6} + 3 T^{5} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 123 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( (T^{6} - 15 T^{5} + \cdots + 29403)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10884540241 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2059979769 \) Copy content Toggle raw display
$73$ \( (T^{2} - 5 T + 25)^{6} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4931831529 \) Copy content Toggle raw display
$83$ \( (T^{6} - 144 T^{4} + \cdots - 432)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 39 T^{5} + \cdots + 8256843)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 3 T^{5} + \cdots + 3267)^{2} \) Copy content Toggle raw display
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