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Properties

Label 104.2.s.c
Level $104$
Weight $2$
Character orbit 104.s
Analytic conductor $0.830$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [104,2,Mod(69,104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(104, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("104.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.s (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: \(0.830444181021\)
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: \( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_1) q^{2} + ( - \beta_{11} + \beta_{3}) q^{3} + ( - \beta_{9} + \beta_{3} - \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{11} - \beta_{3}) q^{5} + (\beta_{15} + \beta_{12} - \beta_{11} + \cdots + 2) q^{6}+ \cdots + (\beta_{11} + 2 \beta_{9} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} + \beta_1) q^{2} + ( - \beta_{11} + \beta_{3}) q^{3} + ( - \beta_{9} + \beta_{3} - \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{11} - \beta_{3}) q^{5} + (\beta_{15} + \beta_{12} - \beta_{11} + \cdots + 2) q^{6}+ \cdots + (\beta_{15} - 4 \beta_{13} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} - q^{4} + 6 q^{6} + 18 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} - q^{4} + 6 q^{6} + 18 q^{7} + 2 q^{9} - 9 q^{10} - 16 q^{12} - 24 q^{14} - 36 q^{15} - q^{16} + 8 q^{17} - 15 q^{20} + 22 q^{22} - 2 q^{23} - 12 q^{24} - 12 q^{25} + 23 q^{26} - 8 q^{30} - 27 q^{32} - 30 q^{33} + 33 q^{36} - 14 q^{39} + 26 q^{40} + 24 q^{41} + 14 q^{42} + 42 q^{46} + 6 q^{48} - 14 q^{49} + 6 q^{50} + 28 q^{52} + 6 q^{54} + 4 q^{55} - 8 q^{56} - 33 q^{58} - 4 q^{62} + 50 q^{64} + 6 q^{65} + 40 q^{66} + 5 q^{68} + 6 q^{71} + 39 q^{72} - 29 q^{74} + 24 q^{76} - 20 q^{78} + 32 q^{79} - 33 q^{80} + 12 q^{81} + 9 q^{82} - 24 q^{84} + 34 q^{87} - 8 q^{88} - 30 q^{89} - 62 q^{90} - 4 q^{92} - 24 q^{94} + 28 q^{95} - 30 q^{97} - 93 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} - 3 \nu^{14} + 9 \nu^{13} - 14 \nu^{12} + 12 \nu^{11} - 10 \nu^{10} - 4 \nu^{9} + \cdots + 32 \nu ) / 320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7 \nu^{15} - \nu^{14} + 11 \nu^{13} - 34 \nu^{12} - 2 \nu^{11} + 24 \nu^{10} - 12 \nu^{9} + \cdots + 1664 ) / 640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{15} + 11 \nu^{14} + 3 \nu^{13} + 6 \nu^{12} - 26 \nu^{11} - 12 \nu^{10} + 60 \nu^{9} + \cdots + 128 ) / 640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} + 9 \nu^{14} - 15 \nu^{13} + 8 \nu^{12} - 10 \nu^{11} - 4 \nu^{10} + 56 \nu^{9} + \cdots + 896 ) / 320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + 3 \nu^{14} - 5 \nu^{13} + 6 \nu^{12} - 6 \nu^{11} + 20 \nu^{9} - 48 \nu^{8} + 76 \nu^{7} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + \nu^{14} - \nu^{13} + 2 \nu^{12} - 4 \nu^{10} + 4 \nu^{9} - 8 \nu^{8} + 12 \nu^{7} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{15} + 21 \nu^{14} - 7 \nu^{13} - 4 \nu^{12} + 14 \nu^{11} - 72 \nu^{10} + 180 \nu^{9} + \cdots + 2048 ) / 640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{15} - 15 \nu^{14} + 13 \nu^{13} + 14 \nu^{12} - 26 \nu^{11} + 84 \nu^{10} - 164 \nu^{9} + \cdots - 896 ) / 640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7 \nu^{15} + 21 \nu^{14} - 31 \nu^{13} + 14 \nu^{12} + 2 \nu^{11} - 64 \nu^{10} + 172 \nu^{9} + \cdots + 896 ) / 640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{15} - 10 \nu^{14} + 16 \nu^{13} - 27 \nu^{12} + 28 \nu^{11} - 12 \nu^{10} - 68 \nu^{9} + \cdots - 832 ) / 320 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 2 \nu^{13} - 3 \nu^{12} + 2 \nu^{11} + 8 \nu^{10} - 20 \nu^{9} + 32 \nu^{8} + \cdots - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7 \nu^{15} + 29 \nu^{14} - 27 \nu^{13} + 22 \nu^{12} - 46 \nu^{11} - 20 \nu^{10} + 172 \nu^{9} + \cdots + 1920 ) / 640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} - 3 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} + 8 \nu^{11} - 16 \nu^{9} + 32 \nu^{8} - 52 \nu^{7} + \cdots - 256 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23 \nu^{15} - 57 \nu^{14} + 55 \nu^{13} - 54 \nu^{12} + 50 \nu^{11} + 112 \nu^{10} - 468 \nu^{9} + \cdots - 3968 ) / 640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{9} - \beta_{5} - \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} - \beta_{14} - 3 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + \beta_{14} - 3 \beta_{11} + \beta_{10} - 3 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{15} + 3 \beta_{13} + \beta_{12} + \beta_{10} - 2 \beta_{9} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{15} - \beta_{14} - \beta_{13} + 3 \beta_{12} + 7 \beta_{11} + 4 \beta_{10} + 7 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7 \beta_{15} + 9 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2 \beta_{14} + 7 \beta_{13} + 13 \beta_{12} - 14 \beta_{11} + 3 \beta_{10} + 3 \beta_{8} - 5 \beta_{7} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7 \beta_{15} + 7 \beta_{14} + \beta_{13} + 17 \beta_{12} + 7 \beta_{11} + 6 \beta_{10} + 15 \beta_{9} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9 \beta_{15} + 3 \beta_{14} - 14 \beta_{13} - 5 \beta_{11} - 7 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2 \beta_{15} - 21 \beta_{13} + 5 \beta_{12} - 16 \beta_{11} - 27 \beta_{10} - 22 \beta_{9} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13 \beta_{15} + 7 \beta_{14} + 15 \beta_{13} + 35 \beta_{12} - 17 \beta_{11} + 64 \beta_{10} + 51 \beta_{9} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 17 \beta_{15} + 49 \beta_{14} + 80 \beta_{13} - 50 \beta_{12} + 9 \beta_{11} + 73 \beta_{10} + 103 \beta_{9} + \cdots - 25 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(-\beta_{12}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
−0.741068 + 1.20450i
−1.41366 + 0.0395346i
0.801337 + 1.16527i
1.21425 + 0.724984i
−0.608487 − 1.27661i
1.40104 − 0.192615i
−0.0207305 − 1.41406i
0.867327 − 1.11702i
−0.741068 − 1.20450i
−1.41366 − 0.0395346i
0.801337 − 1.16527i
1.21425 − 0.724984i
−0.608487 + 1.27661i
1.40104 + 0.192615i
−0.0207305 + 1.41406i
0.867327 + 1.11702i
−1.41366 − 0.0395346i −0.779193 − 0.449867i 1.99687 + 0.111777i −0.893415 1.08373 + 0.666765i 3.65473 − 2.11006i −2.81848 − 0.236960i −1.09524 − 1.89701i 1.26299 + 0.0353208i
69.2 −0.741068 − 1.20450i 0.779193 + 0.449867i −0.901635 + 1.78523i 0.893415 −0.0355706 − 1.27192i 3.65473 − 2.11006i 2.81848 − 0.236960i −1.09524 − 1.89701i −0.662082 − 1.07612i
69.3 −0.608487 + 1.27661i −2.28316 − 1.31818i −1.25949 − 1.55361i 3.40672 3.07208 − 2.11261i 1.30715 − 0.754684i 2.74974 − 0.662529i 1.97521 + 3.42116i −2.07295 + 4.34907i
69.4 −0.0207305 + 1.41406i 1.95589 + 1.12924i −1.99914 − 0.0586285i −0.642566 −1.63735 + 2.74234i 0.306483 − 0.176948i 0.124348 − 2.82569i 1.05034 + 1.81925i 0.0133207 − 0.908627i
69.5 0.801337 − 1.16527i 2.28316 + 1.31818i −0.715719 − 1.86755i −3.40672 3.36562 − 1.60419i 1.30715 − 0.754684i −2.74974 − 0.662529i 1.97521 + 3.42116i −2.72993 + 3.96976i
69.6 0.867327 + 1.11702i −0.323315 − 0.186666i −0.495487 + 1.93765i 2.04528 −0.0719095 − 0.523052i −0.768362 + 0.443614i −2.59415 + 1.12711i −1.43031 − 2.47737i 1.77393 + 2.28463i
69.7 1.21425 − 0.724984i −1.95589 − 1.12924i 0.948796 − 1.76062i 0.642566 −3.19362 + 0.0468193i 0.306483 − 0.176948i −0.124348 − 2.82569i 1.05034 + 1.81925i 0.780234 − 0.465850i
69.8 1.40104 + 0.192615i 0.323315 + 0.186666i 1.92580 + 0.539721i −2.04528 0.417022 + 0.323801i −0.768362 + 0.443614i 2.59415 + 1.12711i −1.43031 − 2.47737i −2.86551 − 0.393951i
101.1 −1.41366 + 0.0395346i −0.779193 + 0.449867i 1.99687 − 0.111777i −0.893415 1.08373 − 0.666765i 3.65473 + 2.11006i −2.81848 + 0.236960i −1.09524 + 1.89701i 1.26299 − 0.0353208i
101.2 −0.741068 + 1.20450i 0.779193 − 0.449867i −0.901635 − 1.78523i 0.893415 −0.0355706 + 1.27192i 3.65473 + 2.11006i 2.81848 + 0.236960i −1.09524 + 1.89701i −0.662082 + 1.07612i
101.3 −0.608487 − 1.27661i −2.28316 + 1.31818i −1.25949 + 1.55361i 3.40672 3.07208 + 2.11261i 1.30715 + 0.754684i 2.74974 + 0.662529i 1.97521 − 3.42116i −2.07295 − 4.34907i
101.4 −0.0207305 − 1.41406i 1.95589 − 1.12924i −1.99914 + 0.0586285i −0.642566 −1.63735 − 2.74234i 0.306483 + 0.176948i 0.124348 + 2.82569i 1.05034 − 1.81925i 0.0133207 + 0.908627i
101.5 0.801337 + 1.16527i 2.28316 − 1.31818i −0.715719 + 1.86755i −3.40672 3.36562 + 1.60419i 1.30715 + 0.754684i −2.74974 + 0.662529i 1.97521 − 3.42116i −2.72993 − 3.96976i
101.6 0.867327 − 1.11702i −0.323315 + 0.186666i −0.495487 − 1.93765i 2.04528 −0.0719095 + 0.523052i −0.768362 − 0.443614i −2.59415 − 1.12711i −1.43031 + 2.47737i 1.77393 − 2.28463i
101.7 1.21425 + 0.724984i −1.95589 + 1.12924i 0.948796 + 1.76062i 0.642566 −3.19362 − 0.0468193i 0.306483 + 0.176948i −0.124348 + 2.82569i 1.05034 − 1.81925i 0.780234 + 0.465850i
101.8 1.40104 − 0.192615i 0.323315 − 0.186666i 1.92580 − 0.539721i −2.04528 0.417022 − 0.323801i −0.768362 − 0.443614i 2.59415 − 1.12711i −1.43031 + 2.47737i −2.86551 + 0.393951i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.e even 6 1 inner
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.2.s.c ✓ 16
3.b odd 2 1 936.2.dg.d 16
4.b odd 2 1 416.2.ba.c 16
8.b even 2 1 inner 104.2.s.c ✓ 16
8.d odd 2 1 416.2.ba.c 16
13.e even 6 1 inner 104.2.s.c ✓ 16
24.h odd 2 1 936.2.dg.d 16
39.h odd 6 1 936.2.dg.d 16
52.i odd 6 1 416.2.ba.c 16
104.p odd 6 1 416.2.ba.c 16
104.s even 6 1 inner 104.2.s.c ✓ 16
312.bg odd 6 1 936.2.dg.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.s.c ✓ 16 1.a even 1 1 trivial
104.2.s.c ✓ 16 8.b even 2 1 inner
104.2.s.c ✓ 16 13.e even 6 1 inner
104.2.s.c ✓ 16 104.s even 6 1 inner
416.2.ba.c 16 4.b odd 2 1
416.2.ba.c 16 8.d odd 2 1
416.2.ba.c 16 52.i odd 6 1
416.2.ba.c 16 104.p odd 6 1
936.2.dg.d 16 3.b odd 2 1
936.2.dg.d 16 24.h odd 2 1
936.2.dg.d 16 39.h odd 6 1
936.2.dg.d 16 312.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 13T_{3}^{14} + 122T_{3}^{12} - 541T_{3}^{10} + 1750T_{3}^{8} - 1541T_{3}^{6} + 1037T_{3}^{4} - 140T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - 13 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{8} - 17 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 9 T^{7} + 30 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 55 T^{14} + \cdots + 14776336 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 31713911056 \) Copy content Toggle raw display
$23$ \( (T^{8} + T^{7} + 22 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12594450625 \) Copy content Toggle raw display
$31$ \( (T^{8} + 132 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2947295521 \) Copy content Toggle raw display
$41$ \( (T^{8} - 12 T^{7} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 145 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( (T^{8} + 204 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 31 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 121173610000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2897022976 \) Copy content Toggle raw display
$71$ \( (T^{8} - 3 T^{7} + \cdots + 1295044)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 307 T^{6} + \cdots + 9610000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} - 4 T^{2} + \cdots + 40)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 560 T^{6} + \cdots + 189337600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 15 T^{7} + \cdots + 67600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 15 T^{7} + \cdots + 473344)^{2} \) Copy content Toggle raw display
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