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Properties

Label 1216.2.n.f
Level $1216$
Weight $2$
Character orbit 1216.n
Analytic conductor $9.710$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(255,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.n (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: \(9.70980888579\)
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: \( x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 76)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + (\beta_{10} - \beta_{5}) q^{5} - \beta_{9} q^{7} + ( - \beta_{12} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + (\beta_{10} - \beta_{5}) q^{5} - \beta_{9} q^{7} + ( - \beta_{12} - \beta_{5}) q^{9} + (\beta_{15} - \beta_{9}) q^{11} + ( - \beta_{12} + \beta_{5} - \beta_{4} + \cdots + 2) q^{13}+ \cdots + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{5} - 4 q^{9} + 18 q^{13} + 2 q^{17} - 2 q^{25} + 6 q^{29} + 18 q^{33} - 48 q^{41} - 24 q^{45} + 16 q^{49} - 6 q^{53} - 26 q^{57} + 26 q^{61} + 16 q^{73} - 80 q^{77} + 12 q^{81} - 14 q^{85} + 18 q^{89} + 20 q^{93} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 26 \nu^{15} - 397 \nu^{14} + 335 \nu^{13} - 666 \nu^{12} + 419 \nu^{11} - 428 \nu^{10} + \cdots - 5056 ) / 40768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79 \nu^{15} + 341 \nu^{14} + 1114 \nu^{13} - 629 \nu^{12} + 1716 \nu^{11} - 965 \nu^{10} + \cdots - 32128 ) / 81536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79 \nu^{15} - 341 \nu^{14} - 1114 \nu^{13} + 629 \nu^{12} - 1716 \nu^{11} + 965 \nu^{10} + \cdots + 32128 ) / 81536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{15} + 34 \nu^{14} - 17 \nu^{13} + 165 \nu^{12} - 269 \nu^{11} + 179 \nu^{10} + 62 \nu^{9} + \cdots + 13504 ) / 5824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 230 \nu^{15} + 97 \nu^{14} + 949 \nu^{13} - 290 \nu^{12} - 367 \nu^{11} + 1404 \nu^{10} + \cdots + 52928 ) / 81536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 334 \nu^{15} - 1881 \nu^{14} - 785 \nu^{13} - 1002 \nu^{12} + 671 \nu^{11} - 5272 \nu^{10} + \cdots - 179776 ) / 81536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 885 \nu^{15} - 431 \nu^{14} + 114 \nu^{13} + 4223 \nu^{12} - 4660 \nu^{11} + 5903 \nu^{10} + \cdots + 273536 ) / 163072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1077 \nu^{15} + 13 \nu^{14} + 1334 \nu^{13} - 4609 \nu^{12} + 2272 \nu^{11} - 2425 \nu^{10} + \cdots - 183168 ) / 163072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1261 \nu^{15} + 7769 \nu^{14} - 6660 \nu^{13} + 15817 \nu^{12} - 17286 \nu^{11} + 13101 \nu^{10} + \cdots + 869632 ) / 163072 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1277 \nu^{15} - 6409 \nu^{14} + 12364 \nu^{13} - 20081 \nu^{12} + 17782 \nu^{11} - 13893 \nu^{10} + \cdots - 548096 ) / 163072 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1346 \nu^{15} - 2357 \nu^{14} + 3471 \nu^{13} - 6350 \nu^{12} + 7867 \nu^{11} - 3172 \nu^{10} + \cdots - 249664 ) / 81536 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1481 \nu^{15} - 3622 \nu^{14} + 6053 \nu^{13} - 8885 \nu^{12} + 8867 \nu^{11} - 3803 \nu^{10} + \cdots - 227392 ) / 81536 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 415 \nu^{15} + 887 \nu^{14} - 1364 \nu^{13} + 2381 \nu^{12} - 2428 \nu^{11} + 771 \nu^{10} + \cdots + 80320 ) / 20384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17 \nu^{15} + 33 \nu^{14} - 46 \nu^{13} + 67 \nu^{12} - 74 \nu^{11} + 63 \nu^{10} - 5 \nu^{9} + \cdots + 2304 ) / 832 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2119 \nu^{15} + 4811 \nu^{14} - 9276 \nu^{13} + 10107 \nu^{12} - 8898 \nu^{11} + 1623 \nu^{10} + \cdots + 166528 ) / 81536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + \beta_{12} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} + 2\beta_{12} - 2\beta_{11} - 2\beta_{10} - 2\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} - \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{14} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{9} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} + 4 \beta_{12} - 4 \beta_{11} - \beta_{10} + 3 \beta_{8} + 4 \beta_{7} - \beta_{5} - 4 \beta_{4} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4 \beta_{15} - 4 \beta_{13} - 2 \beta_{12} + 4 \beta_{10} - 2 \beta_{8} + 6 \beta_{7} + 2 \beta_{5} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} + 6 \beta_{10} - 6 \beta_{8} - 6 \beta_{6} - 9 \beta_{5} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4 \beta_{15} - 5 \beta_{14} + 8 \beta_{13} - 4 \beta_{12} + 6 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5 \beta_{15} + 24 \beta_{14} - 5 \beta_{13} + 3 \beta_{12} + 8 \beta_{11} - 11 \beta_{10} - 12 \beta_{9} + \cdots - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5 \beta_{14} + 4 \beta_{12} + 22 \beta_{11} - 22 \beta_{10} - 8 \beta_{9} - 22 \beta_{8} - 4 \beta_{6} + \cdots + 66 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 7 \beta_{15} - 24 \beta_{14} - 6 \beta_{12} - 33 \beta_{10} - 10 \beta_{9} - 39 \beta_{8} - 33 \beta_{5} + \cdots - 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 24 \beta_{15} + 24 \beta_{13} - 18 \beta_{12} + 26 \beta_{10} - 2 \beta_{9} - 18 \beta_{8} + \cdots - 64 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 24 \beta_{14} - 7 \beta_{13} + 29 \beta_{12} - 52 \beta_{11} - 20 \beta_{10} + 20 \beta_{8} + \cdots + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 20 \beta_{15} - 41 \beta_{14} - 40 \beta_{13} - 74 \beta_{12} + 42 \beta_{11} - 26 \beta_{10} + \cdots + 124 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
−0.327894 − 1.37568i
1.34543 + 0.435684i
−0.835469 + 1.14105i
1.16486 − 0.801943i
−0.112075 + 1.40977i
0.570443 − 1.29406i
1.05003 + 0.947334i
−1.35532 + 0.403874i
−0.327894 + 1.37568i
1.34543 − 0.435684i
−0.835469 − 1.14105i
1.16486 + 0.801943i
−0.112075 − 1.40977i
0.570443 + 1.29406i
1.05003 − 0.947334i
−1.35532 − 0.403874i
0 −1.42689 + 2.47144i 0 0.139977 − 0.242447i 0 1.55280i 0 −2.57201 − 4.45486i 0
255.2 0 −0.982349 + 1.70148i 0 0.349646 − 0.605604i 0 − 3.80025i 0 −0.430019 − 0.744815i 0
255.3 0 −0.637123 + 1.10353i 0 1.60333 − 2.77705i 0 1.25044i 0 0.688149 + 1.19191i 0
255.4 0 −0.305055 + 0.528371i 0 −1.59295 + 2.75907i 0 2.36291i 0 1.31388 + 2.27571i 0
255.5 0 0.305055 − 0.528371i 0 −1.59295 + 2.75907i 0 − 2.36291i 0 1.31388 + 2.27571i 0
255.6 0 0.637123 − 1.10353i 0 1.60333 − 2.77705i 0 − 1.25044i 0 0.688149 + 1.19191i 0
255.7 0 0.982349 − 1.70148i 0 0.349646 − 0.605604i 0 3.80025i 0 −0.430019 − 0.744815i 0
255.8 0 1.42689 − 2.47144i 0 0.139977 − 0.242447i 0 − 1.55280i 0 −2.57201 − 4.45486i 0
639.1 0 −1.42689 − 2.47144i 0 0.139977 + 0.242447i 0 − 1.55280i 0 −2.57201 + 4.45486i 0
639.2 0 −0.982349 − 1.70148i 0 0.349646 + 0.605604i 0 3.80025i 0 −0.430019 + 0.744815i 0
639.3 0 −0.637123 − 1.10353i 0 1.60333 + 2.77705i 0 − 1.25044i 0 0.688149 − 1.19191i 0
639.4 0 −0.305055 − 0.528371i 0 −1.59295 − 2.75907i 0 − 2.36291i 0 1.31388 − 2.27571i 0
639.5 0 0.305055 + 0.528371i 0 −1.59295 − 2.75907i 0 2.36291i 0 1.31388 − 2.27571i 0
639.6 0 0.637123 + 1.10353i 0 1.60333 + 2.77705i 0 1.25044i 0 0.688149 − 1.19191i 0
639.7 0 0.982349 + 1.70148i 0 0.349646 + 0.605604i 0 − 3.80025i 0 −0.430019 + 0.744815i 0
639.8 0 1.42689 + 2.47144i 0 0.139977 + 0.242447i 0 1.55280i 0 −2.57201 + 4.45486i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.n.f 16
4.b odd 2 1 inner 1216.2.n.f 16
8.b even 2 1 76.2.f.a ✓ 16
8.d odd 2 1 76.2.f.a ✓ 16
19.d odd 6 1 inner 1216.2.n.f 16
24.f even 2 1 684.2.r.a 16
24.h odd 2 1 684.2.r.a 16
76.f even 6 1 inner 1216.2.n.f 16
152.l odd 6 1 76.2.f.a ✓ 16
152.o even 6 1 76.2.f.a ✓ 16
456.s odd 6 1 684.2.r.a 16
456.v even 6 1 684.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.f.a ✓ 16 8.b even 2 1
76.2.f.a ✓ 16 8.d odd 2 1
76.2.f.a ✓ 16 152.l odd 6 1
76.2.f.a ✓ 16 152.o even 6 1
684.2.r.a 16 24.f even 2 1
684.2.r.a 16 24.h odd 2 1
684.2.r.a 16 456.s odd 6 1
684.2.r.a 16 456.v even 6 1
1216.2.n.f 16 1.a even 1 1 trivial
1216.2.n.f 16 4.b odd 2 1 inner
1216.2.n.f 16 19.d odd 6 1 inner
1216.2.n.f 16 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 14T_{3}^{14} + 140T_{3}^{12} + 644T_{3}^{10} + 2137T_{3}^{8} + 3388T_{3}^{6} + 3836T_{3}^{4} + 1330T_{3}^{2} + 361 \) acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 14 T^{14} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( (T^{8} - T^{7} + 11 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 24 T^{6} + \cdots + 304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 45 T^{6} + \cdots + 3724)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 9 T^{7} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - T^{7} + 19 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 757115775376 \) Copy content Toggle raw display
$29$ \( (T^{8} - 3 T^{7} + \cdots + 98596)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 124 T^{6} + \cdots + 7600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 124 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 24 T^{7} + \cdots + 841)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 105 T^{14} + \cdots + 1478656 \) Copy content Toggle raw display
$47$ \( T^{16} - 69 T^{14} + \cdots + 37896336 \) Copy content Toggle raw display
$53$ \( (T^{8} + 3 T^{7} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1300683225625 \) Copy content Toggle raw display
$61$ \( (T^{8} - 13 T^{7} + \cdots + 209764)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 106 T^{14} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3550253056 \) Copy content Toggle raw display
$73$ \( (T^{8} - 8 T^{7} + \cdots + 1190281)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 136386521399296 \) Copy content Toggle raw display
$83$ \( (T^{8} + 181 T^{6} + \cdots + 255664)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 9 T^{7} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 6 T^{7} + \cdots + 24649)^{2} \) Copy content Toggle raw display
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