Properties

Label 912.3.z.e
Level $912$
Weight $3$
Character orbit 912.z
Analytic conductor $24.850$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 28 x^{14} - 348 x^{13} - 348 x^{12} + 2078 x^{11} + 15746 x^{10} - 116662 x^{9} + 1437888 x^{8} + 6662192 x^{7} + 22843584 x^{6} + 245977140 x^{5} + \cdots + 14339668675 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} - \beta_{6} q^{5} + ( - \beta_{2} - \beta_1) q^{7} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} - \beta_{6} q^{5} + ( - \beta_{2} - \beta_1) q^{7} - 3 \beta_{2} q^{9} - \beta_{7} q^{11} + ( - \beta_{12} - \beta_{3} + \beta_{2}) q^{13} + (2 \beta_{6} - \beta_{3}) q^{15} + ( - \beta_{13} - \beta_{10} - \beta_{8} + \beta_{6} + \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{13} + \beta_{10} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{19} + (\beta_{5} + \beta_{2} + \beta_1) q^{21} + ( - \beta_{15} + \beta_{10} + \beta_{8}) q^{23} + (\beta_{12} - 3 \beta_{8} + 14 \beta_{2}) q^{25} + (6 \beta_{2} + 3) q^{27} + (\beta_{12} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{11} + \beta_{10} + 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_{2} - 1) q^{31} + (2 \beta_{7} - \beta_{4}) q^{33} + (\beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 4 \beta_{5} + \cdots + 1) q^{35}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{3} + 2 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{3} + 2 q^{5} + 24 q^{9} - 8 q^{13} - 6 q^{15} + 8 q^{17} - 12 q^{19} + 6 q^{21} + 6 q^{23} - 120 q^{25} + 24 q^{29} - 6 q^{33} + 18 q^{35} + 136 q^{37} - 68 q^{41} + 114 q^{43} + 12 q^{45} - 132 q^{47} - 264 q^{49} - 24 q^{51} + 94 q^{53} + 180 q^{55} + 6 q^{57} + 114 q^{59} + 44 q^{61} - 18 q^{63} - 412 q^{65} + 30 q^{67} - 12 q^{69} - 12 q^{71} + 104 q^{73} + 20 q^{77} - 66 q^{79} - 72 q^{81} + 140 q^{85} + 66 q^{89} + 102 q^{91} + 6 q^{93} + 90 q^{95} + 240 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 28 x^{14} - 348 x^{13} - 348 x^{12} + 2078 x^{11} + 15746 x^{10} - 116662 x^{9} + 1437888 x^{8} + 6662192 x^{7} + 22843584 x^{6} + 245977140 x^{5} + \cdots + 14339668675 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 70\!\cdots\!21 \nu^{15} + \cdots - 16\!\cdots\!85 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!63 \nu^{15} + \cdots + 29\!\cdots\!35 ) / 38\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!67 \nu^{15} + \cdots - 11\!\cdots\!80 ) / 59\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\!\cdots\!61 \nu^{15} + \cdots - 47\!\cdots\!95 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!67 \nu^{15} + \cdots + 83\!\cdots\!25 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!26 \nu^{15} + \cdots - 28\!\cdots\!50 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20\!\cdots\!82 \nu^{15} + \cdots - 34\!\cdots\!05 ) / 59\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28\!\cdots\!55 \nu^{15} + \cdots + 76\!\cdots\!35 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 60\!\cdots\!23 \nu^{15} + \cdots - 56\!\cdots\!01 ) / 14\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!91 \nu^{15} + \cdots - 27\!\cdots\!30 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40\!\cdots\!63 \nu^{15} + \cdots - 39\!\cdots\!35 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 49\!\cdots\!09 \nu^{15} + \cdots + 85\!\cdots\!75 ) / 59\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 74\!\cdots\!59 \nu^{15} + \cdots - 13\!\cdots\!15 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!73 \nu^{15} + \cdots - 25\!\cdots\!25 ) / 59\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!37 \nu^{15} + \cdots + 22\!\cdots\!15 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2\beta_{12} + \beta_{11} + 2\beta_{9} - 4\beta_{6} + 3\beta_{5} + \beta_{4} + 12\beta_{2} - \beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 4 \beta_{14} - 4 \beta_{12} + 2 \beta_{11} - 25 \beta_{10} - \beta_{9} - 12 \beta_{8} - 9 \beta_{7} - 29 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 38 \beta_{3} + 44 \beta_{2} - 9 \beta _1 + 145 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13 \beta_{15} - 22 \beta_{14} + 15 \beta_{13} - 20 \beta_{12} + 2 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} + 50 \beta_{8} + 20 \beta_{7} - 79 \beta_{6} + 82 \beta_{5} - 40 \beta_{4} + 99 \beta_{3} - 524 \beta_{2} - 14 \beta _1 - 102 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 282 \beta_{15} - 568 \beta_{14} - 309 \beta_{13} - 594 \beta_{12} + 204 \beta_{11} - 724 \beta_{10} + 584 \beta_{9} - 672 \beta_{8} + 118 \beta_{7} + 115 \beta_{6} + 1766 \beta_{5} + 20 \beta_{4} + 125 \beta_{3} - 555 \beta_{2} + \cdots + 4539 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 697 \beta_{15} + 314 \beta_{14} - 1466 \beta_{13} - 248 \beta_{12} + 779 \beta_{11} - 4480 \beta_{10} + 1464 \beta_{9} - 2324 \beta_{8} + 1922 \beta_{7} - 14418 \beta_{6} + 7727 \beta_{5} - 2189 \beta_{4} + \cdots + 45061 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9861 \beta_{15} - 13062 \beta_{14} - 273 \beta_{13} - 12224 \beta_{12} + 5182 \beta_{11} - 14935 \beta_{10} + 17887 \beta_{9} + 21392 \beta_{8} - 1509 \beta_{7} - 40088 \beta_{6} + 61670 \beta_{5} + \cdots + 181934 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45043 \beta_{15} - 61112 \beta_{14} - 18837 \beta_{13} - 77424 \beta_{12} + 33847 \beta_{11} - 54942 \beta_{10} + 88700 \beta_{9} - 61212 \beta_{8} - 31014 \beta_{7} - 79613 \beta_{6} + 367548 \beta_{5} + \cdots + 550223 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 398758 \beta_{15} + 73040 \beta_{14} - 723218 \beta_{13} - 859160 \beta_{12} + 553306 \beta_{11} - 1487560 \beta_{10} + 917700 \beta_{9} - 260432 \beta_{8} + 466228 \beta_{7} + \cdots + 16119092 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3662263 \beta_{15} - 5350236 \beta_{14} - 386280 \beta_{13} - 5500742 \beta_{12} + 2631551 \beta_{11} - 10286128 \beta_{10} + 5108984 \beta_{9} + 4079860 \beta_{8} - 2253252 \beta_{7} + \cdots + 94688885 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 34467533 \beta_{15} - 37447388 \beta_{14} - 24498028 \beta_{13} - 60682872 \beta_{12} + 30192860 \beta_{11} - 20734165 \beta_{10} + 65353369 \beta_{9} - 5300092 \beta_{8} + \cdots + 121451409 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 98164234 \beta_{15} - 56898970 \beta_{14} - 167947230 \beta_{13} - 221654596 \beta_{12} + 122693629 \beta_{11} - 330460997 \beta_{10} + 219559520 \beta_{9} + \cdots + 2885781726 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1240004016 \beta_{15} - 1150912016 \beta_{14} - 984580883 \beta_{13} - 1763176746 \beta_{12} + 1196240006 \beta_{11} - 3222868068 \beta_{10} + 2166193596 \beta_{9} + \cdots + 29493117491 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 12507652451 \beta_{15} - 12514619762 \beta_{14} - 12537296846 \beta_{13} - 24553856836 \beta_{12} + 12648917881 \beta_{11} - 9046529292 \beta_{10} + \cdots + 55876691553 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 76330091047 \beta_{15} - 44633586462 \beta_{14} - 135601544781 \beta_{13} - 169989014428 \beta_{12} + 102001220272 \beta_{11} - 206822541739 \beta_{10} + \cdots + 1789846977614 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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
−5.13124 + 3.34935i
0.815431 + 4.35480i
3.46265 + 3.84714i
0.278780 + 1.87356i
−4.25015 1.54665i
0.151357 1.99558i
−2.75909 6.08447i
7.43226 0.334054i
−5.13124 3.34935i
0.815431 4.35480i
3.46265 3.84714i
0.278780 1.87356i
−4.25015 + 1.54665i
0.151357 + 1.99558i
−2.75909 + 6.08447i
7.43226 + 0.334054i
0 −1.50000 + 0.866025i 0 −4.96624 8.60179i 0 5.53823i 0 1.50000 2.59808i 0
463.2 0 −1.50000 + 0.866025i 0 −2.86366 4.96000i 0 5.76717i 0 1.50000 2.59808i 0
463.3 0 −1.50000 + 0.866025i 0 −1.10039 1.90593i 0 9.84463i 0 1.50000 2.59808i 0
463.4 0 −1.50000 + 0.866025i 0 −0.983163 1.70289i 0 2.35642i 0 1.50000 2.59808i 0
463.5 0 −1.50000 + 0.866025i 0 −0.285634 0.494733i 0 8.90812i 0 1.50000 2.59808i 0
463.6 0 −1.50000 + 0.866025i 0 2.30390 + 3.99047i 0 1.73342i 0 1.50000 2.59808i 0
463.7 0 −1.50000 + 0.866025i 0 4.38976 + 7.60329i 0 10.8634i 0 1.50000 2.59808i 0
463.8 0 −1.50000 + 0.866025i 0 4.50543 + 7.80363i 0 12.5390i 0 1.50000 2.59808i 0
847.1 0 −1.50000 0.866025i 0 −4.96624 + 8.60179i 0 5.53823i 0 1.50000 + 2.59808i 0
847.2 0 −1.50000 0.866025i 0 −2.86366 + 4.96000i 0 5.76717i 0 1.50000 + 2.59808i 0
847.3 0 −1.50000 0.866025i 0 −1.10039 + 1.90593i 0 9.84463i 0 1.50000 + 2.59808i 0
847.4 0 −1.50000 0.866025i 0 −0.983163 + 1.70289i 0 2.35642i 0 1.50000 + 2.59808i 0
847.5 0 −1.50000 0.866025i 0 −0.285634 + 0.494733i 0 8.90812i 0 1.50000 + 2.59808i 0
847.6 0 −1.50000 0.866025i 0 2.30390 3.99047i 0 1.73342i 0 1.50000 + 2.59808i 0
847.7 0 −1.50000 0.866025i 0 4.38976 7.60329i 0 10.8634i 0 1.50000 + 2.59808i 0
847.8 0 −1.50000 0.866025i 0 4.50543 7.80363i 0 12.5390i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 463.8
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.e 16
4.b odd 2 1 912.3.z.f yes 16
19.c even 3 1 912.3.z.f yes 16
76.g odd 6 1 inner 912.3.z.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.z.e 16 1.a even 1 1 trivial
912.3.z.e 16 76.g odd 6 1 inner
912.3.z.f yes 16 4.b odd 2 1
912.3.z.f yes 16 19.c even 3 1

Hecke kernels

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

\( T_{5}^{16} - 2 T_{5}^{15} + 162 T_{5}^{14} - 108 T_{5}^{13} + 18652 T_{5}^{12} - 5316 T_{5}^{11} + 965912 T_{5}^{10} + 1833840 T_{5}^{9} + 34167600 T_{5}^{8} + 59964608 T_{5}^{7} + 536433296 T_{5}^{6} + \cdots + 2627997696 \) Copy content Toggle raw display
\( T_{23}^{16} - 6 T_{23}^{15} - 3842 T_{23}^{14} + 23124 T_{23}^{13} + 9772920 T_{23}^{12} - 93850572 T_{23}^{11} - 14004849744 T_{23}^{10} + 162732894624 T_{23}^{9} + 14575398795632 T_{23}^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{15} + \cdots + 2627997696 \) Copy content Toggle raw display
$7$ \( T^{16} + 524 T^{14} + \cdots + 2428875495225 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 135867557938176 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 554709375048361 \) Copy content Toggle raw display
$17$ \( T^{16} - 8 T^{15} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{16} + 12 T^{15} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$23$ \( T^{16} - 6 T^{15} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} - 24 T^{15} + \cdots + 83\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{16} + 9868 T^{14} + \cdots + 69\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( (T^{8} - 68 T^{7} - 776 T^{6} + \cdots + 4961117365)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 68 T^{15} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{16} - 114 T^{15} + \cdots + 10\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{16} + 132 T^{15} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} - 94 T^{15} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} - 114 T^{15} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{16} - 44 T^{15} + \cdots + 63\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{16} - 30 T^{15} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( T^{16} + 12 T^{15} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{16} - 104 T^{15} + \cdots + 81\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{16} + 66 T^{15} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + 44464 T^{14} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} - 66 T^{15} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} - 240 T^{15} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
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