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Properties

Label 912.6.k.b
Level $912$
Weight $6$
Character orbit 912.k
Analytic conductor $146.270$
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] = [912,6,Mod(607,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.607");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.k (of order \(2\), degree \(1\), 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: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 32108 x^{14} + 58477 x^{13} + 379886766 x^{12} - 1462364091 x^{11} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{38}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + (\beta_{2} + 1) q^{5} + \beta_1 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + (\beta_{2} + 1) q^{5} + \beta_1 q^{7} + 81 q^{9} + \beta_{3} q^{11} - \beta_{4} q^{13} + (9 \beta_{2} + 9) q^{15} + (\beta_{9} - 4 \beta_{2} + 37) q^{17} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \cdots - 27) q^{19}+ \cdots + 81 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 144 q^{3} + 22 q^{5} + 1296 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 144 q^{3} + 22 q^{5} + 1296 q^{9} + 198 q^{15} + 574 q^{17} - 438 q^{19} + 6878 q^{25} + 11664 q^{27} - 820 q^{31} + 1782 q^{45} - 44706 q^{49} + 5166 q^{51} - 3942 q^{57} - 94808 q^{59} + 15282 q^{61} - 35724 q^{67} + 106120 q^{71} + 86162 q^{73} + 61902 q^{75} + 143398 q^{77} - 45664 q^{79} + 104976 q^{81} - 227674 q^{85} + 124568 q^{91} - 7380 q^{93} - 60460 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 32108 x^{14} + 58477 x^{13} + 379886766 x^{12} - 1462364091 x^{11} + \cdots + 12\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 92\!\cdots\!31 \nu^{15} + \cdots + 20\!\cdots\!08 ) / 58\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 92\!\cdots\!31 \nu^{15} + \cdots + 21\!\cdots\!96 ) / 58\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36\!\cdots\!69 \nu^{15} + \cdots - 18\!\cdots\!72 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!21 \nu^{15} + \cdots + 31\!\cdots\!88 ) / 24\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\!\cdots\!67 \nu^{15} + \cdots + 63\!\cdots\!56 ) / 48\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!07 \nu^{15} + \cdots - 22\!\cdots\!76 ) / 24\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!01 \nu^{15} + \cdots + 74\!\cdots\!32 ) / 96\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 40\!\cdots\!51 \nu^{15} + \cdots + 11\!\cdots\!08 ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51\!\cdots\!84 \nu^{15} + \cdots + 12\!\cdots\!28 ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 49\!\cdots\!09 \nu^{15} + \cdots - 76\!\cdots\!72 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65\!\cdots\!13 \nu^{15} + \cdots + 13\!\cdots\!96 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!27 \nu^{15} + \cdots - 41\!\cdots\!28 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 90\!\cdots\!85 \nu^{15} + \cdots - 11\!\cdots\!72 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 57\!\cdots\!35 \nu^{15} + \cdots + 96\!\cdots\!28 ) / 52\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!51 \nu^{15} + \cdots + 12\!\cdots\!20 ) / 10\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - 4 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} - \beta_{8} + \cdots - 16055 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 75 \beta_{15} + 99 \beta_{14} - 120 \beta_{13} - 476 \beta_{12} + 752 \beta_{11} + 235 \beta_{10} + \cdots - 339026 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 612 \beta_{15} - 5901 \beta_{14} + 13031 \beta_{13} - 37749 \beta_{12} + 20688 \beta_{11} + \cdots + 67392068 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 776487 \beta_{15} - 1526492 \beta_{14} + 419113 \beta_{13} + 4940144 \beta_{12} - 3310121 \beta_{11} + \cdots + 8321277125 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9985706 \beta_{15} + 50392313 \beta_{14} - 65311797 \beta_{13} + 758093975 \beta_{12} + \cdots - 498815138062 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6057679154 \beta_{15} + 14845684760 \beta_{14} - 8175808808 \beta_{13} - 18102842821 \beta_{12} + \cdots - 97823894970539 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17670957984 \beta_{15} - 9215483557 \beta_{14} - 91969607309 \beta_{13} - 12293135407769 \beta_{12} + \cdots - 197406736025614 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 172227473552799 \beta_{15} - 483616762822386 \beta_{14} + 371568034929939 \beta_{13} + \cdots + 35\!\cdots\!91 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 29\!\cdots\!03 \beta_{15} + \cdots + 10\!\cdots\!59 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 21\!\cdots\!73 \beta_{15} + \cdots - 51\!\cdots\!93 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 48\!\cdots\!54 \beta_{15} + \cdots - 15\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!08 \beta_{15} + \cdots + 29\!\cdots\!16 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 21\!\cdots\!09 \beta_{15} + \cdots + 61\!\cdots\!67 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 13\!\cdots\!33 \beta_{15} + \cdots - 37\!\cdots\!85 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
49.0932 − 22.6994i
49.0932 + 22.6994i
27.2339 − 45.2300i
27.2339 + 45.2300i
24.3797 − 123.738i
24.3797 + 123.738i
5.69256 − 59.5526i
5.69256 + 59.5526i
−7.70314 − 79.2805i
−7.70314 + 79.2805i
−23.3430 − 97.2316i
−23.3430 + 97.2316i
−30.5052 − 14.7341i
−30.5052 + 14.7341i
−42.3479 − 42.7442i
−42.3479 + 42.7442i
0 9.00000 0 −96.1864 0 − 45.3988i 0 81.0000 0
607.2 0 9.00000 0 −96.1864 0 45.3988i 0 81.0000 0
607.3 0 9.00000 0 −52.4677 0 − 90.4599i 0 81.0000 0
607.4 0 9.00000 0 −52.4677 0 90.4599i 0 81.0000 0
607.5 0 9.00000 0 −46.7594 0 − 247.477i 0 81.0000 0
607.6 0 9.00000 0 −46.7594 0 247.477i 0 81.0000 0
607.7 0 9.00000 0 −9.38512 0 − 119.105i 0 81.0000 0
607.8 0 9.00000 0 −9.38512 0 119.105i 0 81.0000 0
607.9 0 9.00000 0 17.4063 0 − 158.561i 0 81.0000 0
607.10 0 9.00000 0 17.4063 0 158.561i 0 81.0000 0
607.11 0 9.00000 0 48.6861 0 − 194.463i 0 81.0000 0
607.12 0 9.00000 0 48.6861 0 194.463i 0 81.0000 0
607.13 0 9.00000 0 63.0104 0 − 29.4682i 0 81.0000 0
607.14 0 9.00000 0 63.0104 0 29.4682i 0 81.0000 0
607.15 0 9.00000 0 86.6958 0 − 85.4883i 0 81.0000 0
607.16 0 9.00000 0 86.6958 0 85.4883i 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.6.k.b yes 16
4.b odd 2 1 912.6.k.a ✓ 16
19.b odd 2 1 912.6.k.a ✓ 16
76.d even 2 1 inner 912.6.k.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.6.k.a ✓ 16 4.b odd 2 1
912.6.k.a ✓ 16 19.b odd 2 1
912.6.k.b yes 16 1.a even 1 1 trivial
912.6.k.b yes 16 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{8} - 11 T_{5}^{7} - 14159 T_{5}^{6} + 194503 T_{5}^{5} + 55833706 T_{5}^{4} + \cdots + 10252637548416 \) Copy content Toggle raw display
\( T_{31}^{8} + 410 T_{31}^{7} - 136460320 T_{31}^{6} + 52911996880 T_{31}^{5} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T - 9)^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 10252637548416)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 88\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 74\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 25\!\cdots\!88)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 68\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 49\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 20\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 56\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
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