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Properties

Label 912.3.z.d
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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Show commands: Magma / PariGP / SageMath

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: \( x^{12} + 79x^{10} + 2442x^{8} + 36927x^{6} + 276170x^{4} + 885347x^{2} + 737881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + 1) q^{3} + \beta_{11} q^{5} - \beta_{5} q^{7} + 3 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + 1) q^{3} + \beta_{11} q^{5} - \beta_{5} q^{7} + 3 \beta_{6} q^{9} + (2 \beta_{11} + 2 \beta_{6} + \beta_{4} + \cdots - 1) q^{11}+ \cdots + (3 \beta_{11} + 3 \beta_{6} - 3 \beta_{3} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{3} + 2 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{3} + 2 q^{5} + 18 q^{9} + 6 q^{13} + 6 q^{15} - 16 q^{17} - 12 q^{19} - 30 q^{23} + 34 q^{25} - 48 q^{29} - 18 q^{33} + 18 q^{35} + 20 q^{37} - 8 q^{41} - 48 q^{43} + 12 q^{45} - 48 q^{47} - 136 q^{49} - 48 q^{51} - 86 q^{53} - 228 q^{55} + 126 q^{59} - 114 q^{61} - 124 q^{65} + 24 q^{67} - 60 q^{69} + 240 q^{71} - 146 q^{73} - 268 q^{77} - 84 q^{79} - 54 q^{81} - 40 q^{85} + 42 q^{89} + 240 q^{91} + 108 q^{93} + 486 q^{95} - 52 q^{97} - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 79x^{10} + 2442x^{8} + 36927x^{6} + 276170x^{4} + 885347x^{2} + 737881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1146\nu^{10} - 73849\nu^{8} - 1714322\nu^{6} - 16951358\nu^{4} - 62389693\nu^{2} - 40172808 ) / 2066819 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 548682 \nu^{11} + 7174368 \nu^{10} - 29427501 \nu^{9} + 369368282 \nu^{8} + \cdots + 3368010150 ) / 7101590084 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 869370 \nu^{11} + 11579320 \nu^{10} + 43015887 \nu^{9} + 715194810 \nu^{8} + \cdots + 355158618100 ) / 7101590084 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 869370 \nu^{11} + 11579320 \nu^{10} - 43015887 \nu^{9} + 715194810 \nu^{8} + \cdots + 355158618100 ) / 7101590084 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 548682 \nu^{11} - 29427501 \nu^{9} - 568458212 \nu^{7} - 4757327157 \nu^{5} + \cdots + 16725820161 \nu ) / 3550795042 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 508\nu^{11} + 30683\nu^{9} + 677032\nu^{7} + 6529333\nu^{5} + 25116204\nu^{3} + 24598521\nu + 1465454 ) / 2930908 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1230884 \nu^{11} + 11812968 \nu^{10} + 74344909 \nu^{9} + 761235492 \nu^{8} + \cdots + 595191852006 ) / 7101590084 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 764768 \nu^{11} - 19941685 \nu^{10} - 43831959 \nu^{9} - 1175061319 \nu^{8} + \cdots - 762111050084 ) / 3550795042 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 764768 \nu^{11} - 19941685 \nu^{10} + 43831959 \nu^{9} - 1175061319 \nu^{8} + \cdots - 762111050084 ) / 3550795042 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2760420 \nu^{11} - 51696338 \nu^{10} + 162008827 \nu^{9} - 3111358130 \nu^{8} + \cdots - 2119413952174 ) / 7101590084 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2042889 \nu^{11} + 984414 \nu^{10} - 118351472 \nu^{9} + 63436291 \nu^{8} + \cdots + 34508442072 ) / 3550795042 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{6} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{7} + 6\beta _1 - 51 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} - 10 \beta_{10} + 9 \beta_{9} + \beta_{8} - 10 \beta_{7} + 18 \beta_{6} + 4 \beta_{5} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{10} - 15\beta_{9} - 15\beta_{7} - 4\beta_{5} - 12\beta_{4} - 12\beta_{3} + 8\beta_{2} - 146\beta _1 + 873 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 168 \beta_{11} + 419 \beta_{10} - 341 \beta_{9} - 78 \beta_{8} + 419 \beta_{7} - 622 \beta_{6} + \cdots + 311 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 81 \beta_{10} + 71 \beta_{9} - 10 \beta_{8} + 81 \beta_{7} + 95 \beta_{5} + 214 \beta_{4} + \cdots - 8066 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4636 \beta_{11} - 8897 \beta_{10} + 6561 \beta_{9} + 2336 \beta_{8} - 8897 \beta_{7} + 11814 \beta_{6} + \cdots - 5907 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 673 \beta_{10} + 231 \beta_{9} + 904 \beta_{8} - 673 \beta_{7} - 5808 \beta_{5} - 11464 \beta_{4} + \cdots + 308435 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 56668 \beta_{11} + 94486 \beta_{10} - 63903 \beta_{9} - 30583 \beta_{8} + 94486 \beta_{7} - 115968 \beta_{6} + \cdots + 57984 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 31535 \beta_{10} - 59871 \beta_{9} - 28336 \beta_{8} - 31535 \beta_{7} + 149214 \beta_{5} + \cdots - 6020529 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2628408 \beta_{11} - 4001455 \beta_{10} + 2516681 \beta_{9} + 1484774 \beta_{8} - 4001455 \beta_{7} + \cdots - 2299231 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\) \(-1\)

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
− 1.11334i
2.47654i
4.04392i
− 3.71618i
− 4.53264i
4.57375i
1.11334i
− 2.47654i
− 4.04392i
3.71618i
4.53264i
− 4.57375i
0 1.50000 − 0.866025i 0 −3.43763 − 5.95416i 0 8.79858i 0 1.50000 − 2.59808i 0
463.2 0 1.50000 − 0.866025i 0 −1.40024 − 2.42529i 0 − 13.4461i 0 1.50000 − 2.59808i 0
463.3 0 1.50000 − 0.866025i 0 −0.696910 − 1.20708i 0 − 0.281506i 0 1.50000 − 2.59808i 0
463.4 0 1.50000 − 0.866025i 0 1.63172 + 2.82623i 0 8.68936i 0 1.50000 − 2.59808i 0
463.5 0 1.50000 − 0.866025i 0 2.28752 + 3.96210i 0 − 5.13090i 0 1.50000 − 2.59808i 0
463.6 0 1.50000 − 0.866025i 0 2.61554 + 4.53025i 0 1.37053i 0 1.50000 − 2.59808i 0
847.1 0 1.50000 + 0.866025i 0 −3.43763 + 5.95416i 0 − 8.79858i 0 1.50000 + 2.59808i 0
847.2 0 1.50000 + 0.866025i 0 −1.40024 + 2.42529i 0 13.4461i 0 1.50000 + 2.59808i 0
847.3 0 1.50000 + 0.866025i 0 −0.696910 + 1.20708i 0 0.281506i 0 1.50000 + 2.59808i 0
847.4 0 1.50000 + 0.866025i 0 1.63172 − 2.82623i 0 − 8.68936i 0 1.50000 + 2.59808i 0
847.5 0 1.50000 + 0.866025i 0 2.28752 − 3.96210i 0 5.13090i 0 1.50000 + 2.59808i 0
847.6 0 1.50000 + 0.866025i 0 2.61554 − 4.53025i 0 − 1.37053i 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.d yes 12
4.b odd 2 1 912.3.z.b ✓ 12
19.c even 3 1 912.3.z.b ✓ 12
76.g odd 6 1 inner 912.3.z.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.z.b ✓ 12 4.b odd 2 1
912.3.z.b ✓ 12 19.c even 3 1
912.3.z.d yes 12 1.a even 1 1 trivial
912.3.z.d 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 \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{12} - 2 T_{5}^{11} + 60 T_{5}^{10} - 168 T_{5}^{9} + 2800 T_{5}^{8} - 6432 T_{5}^{7} + \cdots + 4393216 \) Copy content Toggle raw display
\( T_{23}^{12} + 30 T_{23}^{11} - 320 T_{23}^{10} - 18600 T_{23}^{9} + 219456 T_{23}^{8} + \cdots + 47370651904 \) 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
$5$ \( T^{12} - 2 T^{11} + \cdots + 4393216 \) Copy content Toggle raw display
$7$ \( T^{12} + 362 T^{10} + \cdots + 4141225 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 417466624 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 19040964121 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 2812516827136 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 47370651904 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3203011772416 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 42\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( (T^{6} - 10 T^{5} + \cdots + 682976641)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 31730959384576 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 95\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 16\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 31\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 75\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
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