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Properties

Label 912.2.bn.n
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
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] = [912,2,Mod(65,912)]
 
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([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), not 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: \(7.28235666434\)
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: \( x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{14} - \beta_{11}) q^{5} + (\beta_{13} - \beta_{5}) q^{7} + (\beta_{12} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{14} - \beta_{11}) q^{5} + (\beta_{13} - \beta_{5}) q^{7} + (\beta_{12} - \beta_{3} - 1) q^{9} + (\beta_{14} - \beta_{12} + \beta_{11} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} + 3 q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} + 3 q^{5} - 5 q^{9} - 3 q^{13} - 12 q^{15} - 3 q^{17} - 11 q^{19} - 12 q^{21} + 3 q^{23} + 11 q^{25} - 4 q^{27} + 5 q^{29} + 14 q^{33} - 24 q^{35} + 9 q^{39} + 6 q^{41} - 13 q^{43} + 33 q^{45} - 27 q^{47} + 8 q^{49} + 18 q^{51} - 7 q^{53} + 12 q^{55} - 36 q^{57} + 10 q^{59} - q^{61} + 26 q^{63} - 30 q^{65} + 24 q^{67} - 41 q^{69} - 27 q^{71} + 2 q^{73} - 21 q^{75} + 21 q^{79} - 13 q^{81} - 5 q^{85} + 23 q^{87} + 25 q^{89} + 78 q^{91} + 22 q^{93} - 13 q^{95} - 60 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{15} - 4 \nu^{14} + 16 \nu^{13} + 33 \nu^{12} - 50 \nu^{11} - 30 \nu^{10} + 316 \nu^{9} + \cdots + 2187 ) / 23328 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 34 \nu^{14} + 18 \nu^{13} - 113 \nu^{12} - 18 \nu^{11} + 598 \nu^{10} + 124 \nu^{9} + \cdots - 19683 ) / 23328 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4 \nu^{15} - 53 \nu^{14} + 153 \nu^{13} + 385 \nu^{12} - 612 \nu^{11} - 1226 \nu^{10} + \cdots + 203391 ) / 34992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25 \nu^{15} - 76 \nu^{14} + 467 \nu^{12} - 54 \nu^{11} - 1162 \nu^{10} - 364 \nu^{9} + 4734 \nu^{8} + \cdots + 111537 ) / 69984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + \nu^{14} + 6 \nu^{13} - 5 \nu^{12} - 21 \nu^{11} + 4 \nu^{10} + 94 \nu^{9} + \cdots + 2187 ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} + 9 \nu^{13} + 13 \nu^{12} - 36 \nu^{11} - 59 \nu^{10} + 106 \nu^{9} + \cdots + 13122 ) / 2187 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35 \nu^{15} + 62 \nu^{14} + 174 \nu^{13} - 31 \nu^{12} - 438 \nu^{11} - 310 \nu^{10} + \cdots + 247131 ) / 69984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{15} + 13 \nu^{14} + 31 \nu^{13} - 25 \nu^{12} - 86 \nu^{11} - 16 \nu^{10} + 450 \nu^{9} + \cdots + 22599 ) / 5832 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{15} - 2 \nu^{14} - 24 \nu^{13} - 2 \nu^{12} + 66 \nu^{11} + 88 \nu^{10} - 293 \nu^{9} + \cdots - 13122 ) / 4374 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15 \nu^{15} - 8 \nu^{14} + 62 \nu^{13} + 15 \nu^{12} - 322 \nu^{11} - 30 \nu^{10} + 1232 \nu^{9} + \cdots + 9477 ) / 11664 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 52 \nu^{15} + 23 \nu^{14} - 171 \nu^{13} - \nu^{12} + 252 \nu^{11} + 746 \nu^{10} - 1300 \nu^{9} + \cdots - 107163 ) / 34992 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 53 \nu^{15} - 73 \nu^{14} + 327 \nu^{13} + 176 \nu^{12} - 1014 \nu^{11} - 508 \nu^{10} + \cdots + 183708 ) / 34992 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 289 \nu^{15} + 86 \nu^{14} - 1398 \nu^{13} - 679 \nu^{12} + 3570 \nu^{11} + 5738 \nu^{10} + \cdots - 925101 ) / 69984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - 2\beta_{13} + \beta_{12} + 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 2\beta_{7} - 2\beta_{6} + 2\beta_{3} + 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} - 2 \beta_{14} + 6 \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{4} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6 \beta_{15} + 4 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{15} + 24 \beta_{14} + 26 \beta_{13} - 12 \beta_{12} - 20 \beta_{11} - 4 \beta_{10} + 34 \beta_{9} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8 \beta_{15} + \beta_{14} - 2 \beta_{13} + 23 \beta_{12} + 32 \beta_{11} + 31 \beta_{10} + 5 \beta_{9} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20 \beta_{15} + 60 \beta_{14} + 24 \beta_{13} - 24 \beta_{11} + 100 \beta_{9} - 7 \beta_{8} - 120 \beta_{7} + \cdots - 83 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 56 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} - 24 \beta_{12} + 112 \beta_{11} + 8 \beta_{10} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16 \beta_{15} - 64 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} - 60 \beta_{11} - 144 \beta_{10} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 104 \beta_{15} + 176 \beta_{14} + 220 \beta_{13} - 256 \beta_{12} - 160 \beta_{11} + 16 \beta_{10} + \cdots - 164 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 68 \beta_{15} - 16 \beta_{14} + 596 \beta_{13} - 252 \beta_{12} - 524 \beta_{11} - 292 \beta_{10} + \cdots - 1127 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 408 \beta_{15} + 595 \beta_{14} + 130 \beta_{13} - 283 \beta_{12} - 598 \beta_{11} + 781 \beta_{10} + \cdots + 416 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.809132 + 1.53144i
−1.63023 + 0.585107i
0.956703 + 1.44386i
−1.72340 − 0.172852i
1.66415 + 0.480229i
−0.909329 − 1.47415i
1.70042 − 0.329508i
1.25083 − 1.19809i
−0.809132 − 1.53144i
−1.63023 − 0.585107i
0.956703 − 1.44386i
−1.72340 + 0.172852i
1.66415 − 0.480229i
−0.909329 + 1.47415i
1.70042 + 0.329508i
1.25083 + 1.19809i
0 −1.73083 + 0.0649909i 0 3.15813 + 1.82335i 0 −1.32651 0 2.99155 − 0.224976i 0
65.2 0 −1.32183 − 1.11927i 0 −2.85312 − 1.64725i 0 4.86833 0 0.494481 + 2.95897i 0
65.3 0 −0.772064 + 1.55046i 0 1.67415 + 0.966572i 0 1.30064 0 −1.80783 − 2.39410i 0
65.4 0 −0.712008 − 1.57894i 0 1.02997 + 0.594652i 0 −1.04533 0 −1.98609 + 2.24843i 0
65.5 0 0.416182 + 1.68131i 0 −2.52758 − 1.45930i 0 −0.106684 0 −2.65359 + 1.39946i 0
65.6 0 0.821988 − 1.52458i 0 −2.16072 − 1.24749i 0 −3.30744 0 −1.64867 − 2.50637i 0
65.7 0 1.13557 + 1.30785i 0 1.87360 + 1.08173i 0 −3.41109 0 −0.420952 + 2.97032i 0
65.8 0 1.66299 + 0.484201i 0 1.30557 + 0.753769i 0 3.02808 0 2.53110 + 1.61045i 0
449.1 0 −1.73083 − 0.0649909i 0 3.15813 − 1.82335i 0 −1.32651 0 2.99155 + 0.224976i 0
449.2 0 −1.32183 + 1.11927i 0 −2.85312 + 1.64725i 0 4.86833 0 0.494481 − 2.95897i 0
449.3 0 −0.772064 − 1.55046i 0 1.67415 − 0.966572i 0 1.30064 0 −1.80783 + 2.39410i 0
449.4 0 −0.712008 + 1.57894i 0 1.02997 − 0.594652i 0 −1.04533 0 −1.98609 − 2.24843i 0
449.5 0 0.416182 − 1.68131i 0 −2.52758 + 1.45930i 0 −0.106684 0 −2.65359 − 1.39946i 0
449.6 0 0.821988 + 1.52458i 0 −2.16072 + 1.24749i 0 −3.30744 0 −1.64867 + 2.50637i 0
449.7 0 1.13557 − 1.30785i 0 1.87360 − 1.08173i 0 −3.41109 0 −0.420952 − 2.97032i 0
449.8 0 1.66299 − 0.484201i 0 1.30557 − 0.753769i 0 3.02808 0 2.53110 − 1.61045i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.n 16
3.b odd 2 1 912.2.bn.o 16
4.b odd 2 1 456.2.bf.d yes 16
12.b even 2 1 456.2.bf.c ✓ 16
19.d odd 6 1 912.2.bn.o 16
57.f even 6 1 inner 912.2.bn.n 16
76.f even 6 1 456.2.bf.c ✓ 16
228.n odd 6 1 456.2.bf.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.c ✓ 16 12.b even 2 1
456.2.bf.c ✓ 16 76.f even 6 1
456.2.bf.d yes 16 4.b odd 2 1
456.2.bf.d yes 16 228.n odd 6 1
912.2.bn.n 16 1.a even 1 1 trivial
912.2.bn.n 16 57.f even 6 1 inner
912.2.bn.o 16 3.b odd 2 1
912.2.bn.o 16 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{16} - 3 T_{5}^{15} - 21 T_{5}^{14} + 72 T_{5}^{13} + 326 T_{5}^{12} - 1188 T_{5}^{11} + \cdots + 430336 \) Copy content Toggle raw display
\( T_{7}^{8} - 30T_{7}^{6} - 22T_{7}^{5} + 223T_{7}^{4} + 234T_{7}^{3} - 278T_{7}^{2} - 332T_{7} - 32 \) Copy content Toggle raw display
\( T_{17}^{16} + 3 T_{17}^{15} - 69 T_{17}^{14} - 216 T_{17}^{13} + 3260 T_{17}^{12} + 13344 T_{17}^{11} + \cdots + 2945449984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} - 3 T^{15} + \cdots + 430336 \) Copy content Toggle raw display
$7$ \( (T^{8} - 30 T^{6} + \cdots - 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 121 T^{14} + \cdots + 2637376 \) Copy content Toggle raw display
$13$ \( T^{16} + 3 T^{15} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 2945449984 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1328456704 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 554696704 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 5554422784 \) Copy content Toggle raw display
$37$ \( T^{16} + 80 T^{14} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( T^{16} - 6 T^{15} + \cdots + 27541504 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 334805733376 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18653553688576 \) Copy content Toggle raw display
$53$ \( T^{16} + 7 T^{15} + \cdots + 50176 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 933720229264 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 215225477776 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5605723904881 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 59895709696 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 24812542725961 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 3046449086464 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 8981961048064 \) Copy content Toggle raw display
$89$ \( T^{16} - 25 T^{15} + \cdots + 6885376 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 14736876810496 \) Copy content Toggle raw display
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