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Properties

Label 912.3.z.c
Level $912$
Weight $3$
Character orbit 912.z
Analytic conductor $24.850$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,3,Mod(463,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.463");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.z (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: \(24.8502001097\)
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: \( x^{12} + 79x^{10} + 2118x^{8} + 24582x^{6} + 122933x^{4} + 219143x^{2} + 42436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 2) q^{3} + (\beta_{11} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{8} - \beta_{7} + 2 \beta_{2} + 1) q^{7} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 2) q^{3} + (\beta_{11} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{8} - \beta_{7} + 2 \beta_{2} + 1) q^{7} + (3 \beta_{2} + 3) q^{9} + (\beta_{6} - \beta_{5} - \beta_{4} + \cdots - 2) q^{11}+ \cdots + (3 \beta_{10} + 3 \beta_{9} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{3} - 4 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{3} - 4 q^{5} + 18 q^{9} + 6 q^{13} - 12 q^{15} + 8 q^{17} - 18 q^{21} + 36 q^{23} - 14 q^{25} + 24 q^{29} + 24 q^{33} - 196 q^{37} + 52 q^{41} + 126 q^{43} - 24 q^{45} - 12 q^{47} - 16 q^{49} + 24 q^{51} + 16 q^{53} - 24 q^{55} - 18 q^{57} - 12 q^{59} - 30 q^{61} - 54 q^{63} + 56 q^{65} - 6 q^{67} + 72 q^{69} - 108 q^{71} + 22 q^{73} + 56 q^{77} + 30 q^{79} - 54 q^{81} - 4 q^{85} + 36 q^{89} + 18 q^{91} + 66 q^{93} - 180 q^{95} - 172 q^{97} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 79x^{10} + 2118x^{8} + 24582x^{6} + 122933x^{4} + 219143x^{2} + 42436 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23747 \nu^{10} + 1921270 \nu^{8} + 51672008 \nu^{6} + 548477608 \nu^{4} + 2008429407 \nu^{2} + 1412407522 ) / 147368978 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3718 \nu^{11} - 343574 \nu^{9} - 11032807 \nu^{7} - 148061635 \nu^{5} - 838656513 \nu^{3} + \cdots - 489645314 ) / 979290628 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 220608 \nu^{11} - 8626044 \nu^{10} - 12151548 \nu^{9} - 590646496 \nu^{8} + \cdots - 413363737922 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 220608 \nu^{11} - 8626044 \nu^{10} + 12151548 \nu^{9} - 590646496 \nu^{8} + \cdots - 413363737922 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1259260 \nu^{11} - 2719097 \nu^{10} - 88917860 \nu^{9} - 153749748 \nu^{8} + \cdots + 477615621610 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1259260 \nu^{11} - 2719097 \nu^{10} + 88917860 \nu^{9} - 153749748 \nu^{8} + \cdots + 477615621610 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1374518 \nu^{11} - 7614069 \nu^{10} - 99568654 \nu^{9} - 547224580 \nu^{8} + \cdots - 12569335140 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1374518 \nu^{11} + 7614069 \nu^{10} - 99568654 \nu^{9} + 547224580 \nu^{8} + \cdots + 12569335140 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3935932 \nu^{11} + 13149289 \nu^{10} + 326047698 \nu^{9} + 942069930 \nu^{8} + \cdots + 56240131688 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3935932 \nu^{11} + 13149289 \nu^{10} - 326047698 \nu^{9} + 942069930 \nu^{8} + \cdots + 56240131688 ) / 30358009468 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5013062 \nu^{11} + 2445941 \nu^{10} - 380293910 \nu^{9} + 197890810 \nu^{8} + \cdots + 145477974766 ) / 30358009468 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{10} - 4\beta_{9} + 3\beta_{8} - 3\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{3} + 4\beta _1 - 49 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 16 \beta_{11} + 8 \beta_{10} - 8 \beta_{9} + 39 \beta_{8} + 39 \beta_{7} + 27 \beta_{6} - 27 \beta_{5} + \cdots - 103 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 138 \beta_{10} + 138 \beta_{9} - 143 \beta_{8} + 143 \beta_{7} + 21 \beta_{6} + 21 \beta_{5} + \cdots + 1225 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 856 \beta_{11} - 392 \beta_{10} + 392 \beta_{9} - 1593 \beta_{8} - 1593 \beta_{7} - 925 \beta_{6} + \cdots + 4221 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4860 \beta_{10} - 4860 \beta_{9} + 5847 \beta_{8} - 5847 \beta_{7} - 301 \beta_{6} - 301 \beta_{5} + \cdots - 39729 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 36920 \beta_{11} + 16192 \beta_{10} - 16192 \beta_{9} + 63457 \beta_{8} + 63457 \beta_{7} + \cdots - 162189 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 179098 \beta_{10} + 179098 \beta_{9} - 230207 \beta_{8} + 230207 \beta_{7} + 2265 \beta_{6} + \cdots + 1430973 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1489616 \beta_{11} - 641100 \beta_{10} + 641100 \beta_{9} - 2483963 \beta_{8} - 2483963 \beta_{7} + \cdots + 6205523 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6764048 \beta_{10} - 6764048 \beta_{9} + 8951471 \beta_{8} - 8951471 \beta_{7} + 71251 \beta_{6} + \cdots - 53713309 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 58575672 \beta_{11} + 25000804 \beta_{10} - 25000804 \beta_{9} + 96411531 \beta_{8} + \cdots - 238081971 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
463.1
0.468521i
2.55227i
3.69861i
− 6.20960i
1.83835i
− 4.08020i
− 0.468521i
− 2.55227i
− 3.69861i
6.20960i
− 1.83835i
4.08020i
0 1.50000 − 0.866025i 0 −3.88409 − 6.72744i 0 − 3.08354i 0 1.50000 − 2.59808i 0
463.2 0 1.50000 − 0.866025i 0 −2.20001 − 3.81053i 0 9.74394i 0 1.50000 − 2.59808i 0
463.3 0 1.50000 − 0.866025i 0 −1.19448 − 2.06889i 0 − 1.64291i 0 1.50000 − 2.59808i 0
463.4 0 1.50000 − 0.866025i 0 −0.777719 − 1.34705i 0 − 13.5651i 0 1.50000 − 2.59808i 0
463.5 0 1.50000 − 0.866025i 0 2.43536 + 4.21816i 0 − 3.05961i 0 1.50000 − 2.59808i 0
463.6 0 1.50000 − 0.866025i 0 3.62094 + 6.27165i 0 1.21489i 0 1.50000 − 2.59808i 0
847.1 0 1.50000 + 0.866025i 0 −3.88409 + 6.72744i 0 3.08354i 0 1.50000 + 2.59808i 0
847.2 0 1.50000 + 0.866025i 0 −2.20001 + 3.81053i 0 − 9.74394i 0 1.50000 + 2.59808i 0
847.3 0 1.50000 + 0.866025i 0 −1.19448 + 2.06889i 0 1.64291i 0 1.50000 + 2.59808i 0
847.4 0 1.50000 + 0.866025i 0 −0.777719 + 1.34705i 0 13.5651i 0 1.50000 + 2.59808i 0
847.5 0 1.50000 + 0.866025i 0 2.43536 − 4.21816i 0 3.05961i 0 1.50000 + 2.59808i 0
847.6 0 1.50000 + 0.866025i 0 3.62094 − 6.27165i 0 − 1.21489i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 463.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.z.c yes 12
4.b odd 2 1 912.3.z.a ✓ 12
19.c even 3 1 912.3.z.a ✓ 12
76.g odd 6 1 inner 912.3.z.c yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.z.a ✓ 12 4.b odd 2 1
912.3.z.a ✓ 12 19.c even 3 1
912.3.z.c yes 12 1.a even 1 1 trivial
912.3.z.c yes 12 76.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{12} + 4 T_{5}^{11} + 90 T_{5}^{10} + 288 T_{5}^{9} + 5668 T_{5}^{8} + 18612 T_{5}^{7} + \cdots + 20070400 \) Copy content Toggle raw display
\( T_{23}^{12} - 36 T_{23}^{11} - 398 T_{23}^{10} + 29880 T_{23}^{9} + 183456 T_{23}^{8} + \cdots + 66557140961536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots + 20070400 \) Copy content Toggle raw display
$7$ \( T^{12} + 302 T^{10} + \cdots + 6195121 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 281740147264 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 2163296764225 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 1579304943616 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 66557140961536 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 9330532286464 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 359940616354201 \) Copy content Toggle raw display
$37$ \( (T^{6} + 98 T^{5} + \cdots + 218234605)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 9897083197369 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 96\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 85415395782025 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
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