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Properties

Label 552.2.f.c
Level $552$
Weight $2$
Character orbit 552.f
Analytic conductor $4.408$
Analytic rank $0$
Dimension $18$
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] = [552,2,Mod(277,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.f (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: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{16} - 2 x^{15} + 5 x^{14} + 2 x^{13} + 6 x^{12} + 24 x^{11} - 12 x^{10} - 88 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{4} q^{3} - \beta_{2} q^{4} + \beta_{9} q^{5} - \beta_1 q^{6} + ( - \beta_{12} - \beta_{10} + \beta_{8} + \cdots + 1) q^{7}+ \cdots - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{4} q^{3} - \beta_{2} q^{4} + \beta_{9} q^{5} - \beta_1 q^{6} + ( - \beta_{12} - \beta_{10} + \beta_{8} + \cdots + 1) q^{7}+ \cdots + (\beta_{16} + \beta_{14} + \beta_{11} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 8 q^{4} + 8 q^{7} - 6 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 8 q^{4} + 8 q^{7} - 6 q^{8} - 18 q^{9} + 12 q^{10} - 4 q^{12} - 14 q^{14} + 8 q^{15} + 12 q^{16} - 8 q^{17} - 16 q^{20} - 30 q^{22} - 18 q^{23} + 6 q^{24} - 22 q^{25} + 8 q^{26} - 2 q^{28} + 4 q^{30} - 44 q^{31} + 10 q^{32} - 24 q^{33} + 18 q^{34} + 8 q^{36} - 20 q^{38} - 8 q^{39} + 40 q^{40} + 28 q^{41} + 6 q^{42} - 26 q^{44} + 42 q^{49} + 60 q^{50} - 36 q^{52} + 40 q^{55} - 2 q^{56} + 12 q^{57} + 52 q^{58} + 16 q^{60} + 24 q^{62} - 8 q^{63} + 16 q^{64} - 104 q^{65} + 2 q^{66} + 54 q^{68} - 48 q^{70} - 24 q^{71} + 6 q^{72} + 12 q^{73} - 22 q^{74} - 4 q^{78} + 8 q^{79} - 32 q^{80} + 18 q^{81} - 20 q^{82} + 34 q^{84} + 12 q^{87} + 10 q^{88} + 24 q^{89} - 12 q^{90} + 8 q^{92} - 56 q^{94} - 16 q^{95} - 30 q^{96} + 12 q^{97} - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{16} - 2 x^{15} + 5 x^{14} + 2 x^{13} + 6 x^{12} + 24 x^{11} - 12 x^{10} - 88 x^{9} + \cdots + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} - 2 \nu^{14} + \nu^{13} + 4 \nu^{12} - 18 \nu^{11} - 12 \nu^{10} + 28 \nu^{9} + \cdots + 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{17} + 2 \nu^{16} + 8 \nu^{15} + 2 \nu^{14} - 9 \nu^{13} + 2 \nu^{11} - 60 \nu^{10} + \cdots + 768 ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} - 4 \nu^{15} - 10 \nu^{14} + \nu^{13} + 12 \nu^{12} + 10 \nu^{11} + 36 \nu^{10} + \cdots - 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{17} - 2 \nu^{16} - 4 \nu^{15} + 6 \nu^{14} + 9 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 12 \nu^{10} + \cdots + 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{17} + \nu^{16} - 2 \nu^{15} - 6 \nu^{14} - 5 \nu^{13} + 3 \nu^{12} + 18 \nu^{11} + 34 \nu^{10} + \cdots - 320 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} - 2 \nu^{14} + \nu^{13} - 12 \nu^{12} - 18 \nu^{11} + 20 \nu^{10} + 60 \nu^{9} + \cdots - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{17} - 6 \nu^{16} + 8 \nu^{15} + 10 \nu^{14} + 3 \nu^{13} + 48 \nu^{12} + 42 \nu^{11} + \cdots + 2304 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9 \nu^{17} + 18 \nu^{16} - 24 \nu^{15} - 66 \nu^{14} - 7 \nu^{13} + 20 \nu^{12} + 6 \nu^{11} + \cdots - 4352 ) / 512 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{17} - 18 \nu^{16} + 24 \nu^{15} + 62 \nu^{14} + 33 \nu^{13} - 50 \nu^{11} - 372 \nu^{10} + \cdots + 5888 ) / 512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{17} - 4 \nu^{16} + 20 \nu^{15} + 34 \nu^{14} - \nu^{13} - 30 \nu^{12} - 54 \nu^{11} + \cdots + 1792 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13 \nu^{17} - 22 \nu^{16} + 24 \nu^{15} + 74 \nu^{14} + 43 \nu^{13} + 16 \nu^{12} - 54 \nu^{11} + \cdots + 4864 ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2 \nu^{17} + 5 \nu^{16} - 12 \nu^{14} - 20 \nu^{13} - 19 \nu^{12} + 14 \nu^{11} + 98 \nu^{10} + \cdots - 1024 ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15 \nu^{17} - 18 \nu^{16} + 40 \nu^{15} + 78 \nu^{14} + 9 \nu^{13} - 16 \nu^{12} - 82 \nu^{11} + \cdots + 4864 ) / 512 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13 \nu^{17} - 26 \nu^{16} + 24 \nu^{15} + 106 \nu^{14} + 67 \nu^{13} - 4 \nu^{12} - 126 \nu^{11} + \cdots + 6400 ) / 512 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15 \nu^{17} + 18 \nu^{16} - 40 \nu^{15} - 78 \nu^{14} - 9 \nu^{13} + 16 \nu^{12} + 82 \nu^{11} + \cdots - 5376 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{15} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} - \beta_{14} - \beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{8} + \beta_{7} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} - 2 \beta_{16} + \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{11} - 2 \beta_{10} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{17} - 2 \beta_{16} + 3 \beta_{15} + 4 \beta_{13} - \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 5 \beta_{14} - 3 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{17} - 7 \beta_{16} + 2 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{12} - 4 \beta_{11} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - \beta_{17} - 6 \beta_{16} + \beta_{15} - 6 \beta_{14} + 7 \beta_{13} + 4 \beta_{12} - 5 \beta_{11} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - \beta_{17} - 10 \beta_{16} - \beta_{15} - 8 \beta_{14} + 4 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6 \beta_{17} - 19 \beta_{16} - 13 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} - 7 \beta_{11} - 4 \beta_{10} + \cdots - 48 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2 \beta_{17} - 27 \beta_{16} - 14 \beta_{15} - 31 \beta_{14} + 17 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} + \cdots - 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7 \beta_{17} + 22 \beta_{16} - 15 \beta_{15} + 6 \beta_{14} - 9 \beta_{13} + 20 \beta_{12} - \beta_{11} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 21 \beta_{17} - 50 \beta_{16} - 37 \beta_{15} - 16 \beta_{14} + 12 \beta_{13} + 55 \beta_{12} + \cdots - 101 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2 \beta_{17} + 45 \beta_{16} - 28 \beta_{15} - 5 \beta_{14} + 40 \beta_{12} - 51 \beta_{11} + 44 \beta_{10} + \cdots - 128 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 50 \beta_{17} + \beta_{16} - 30 \beta_{15} + 21 \beta_{14} - 67 \beta_{13} + 31 \beta_{12} + 124 \beta_{11} + \cdots - 55 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(185\) \(277\) \(415\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
0.238365 − 1.39398i
0.238365 + 1.39398i
−0.809892 + 1.15934i
−0.809892 − 1.15934i
−1.30337 + 0.548841i
−1.30337 − 0.548841i
1.37219 − 0.342174i
1.37219 + 0.342174i
1.39509 + 0.231756i
1.39509 − 0.231756i
−1.38678 − 0.277192i
−1.38678 + 0.277192i
1.26873 + 0.624769i
1.26873 − 0.624769i
−1.06785 − 0.927201i
−1.06785 + 0.927201i
0.293513 + 1.38342i
0.293513 − 1.38342i
−1.39398 − 0.238365i − 1.00000i 1.88636 + 0.664552i 4.40892i −0.238365 + 1.39398i 4.93691 −2.47115 − 1.37602i −1.00000 1.05093 − 6.14595i
277.2 −1.39398 + 0.238365i 1.00000i 1.88636 − 0.664552i − 4.40892i −0.238365 − 1.39398i 4.93691 −2.47115 + 1.37602i −1.00000 1.05093 + 6.14595i
277.3 −1.15934 − 0.809892i 1.00000i 0.688150 + 1.87788i 3.56420i 0.809892 − 1.15934i −2.96155 0.723081 − 2.73444i −1.00000 2.88661 − 4.13212i
277.4 −1.15934 + 0.809892i − 1.00000i 0.688150 − 1.87788i − 3.56420i 0.809892 + 1.15934i −2.96155 0.723081 + 2.73444i −1.00000 2.88661 + 4.13212i
277.5 −0.548841 − 1.30337i 1.00000i −1.39755 + 1.43069i 0.625627i 1.30337 − 0.548841i 0.327807 2.63174 + 1.03630i −1.00000 0.815424 − 0.343370i
277.6 −0.548841 + 1.30337i − 1.00000i −1.39755 − 1.43069i − 0.625627i 1.30337 + 0.548841i 0.327807 2.63174 − 1.03630i −1.00000 0.815424 + 0.343370i
277.7 −0.342174 − 1.37219i − 1.00000i −1.76583 + 0.939057i 0.957976i −1.37219 + 0.342174i 2.14538 1.89279 + 2.10175i −1.00000 1.31453 − 0.327794i
277.8 −0.342174 + 1.37219i 1.00000i −1.76583 − 0.939057i − 0.957976i −1.37219 − 0.342174i 2.14538 1.89279 − 2.10175i −1.00000 1.31453 + 0.327794i
277.9 0.231756 − 1.39509i − 1.00000i −1.89258 − 0.646644i − 2.83774i −1.39509 − 0.231756i 0.890209 −1.34075 + 2.49046i −1.00000 −3.95892 − 0.657665i
277.10 0.231756 + 1.39509i 1.00000i −1.89258 + 0.646644i 2.83774i −1.39509 + 0.231756i 0.890209 −1.34075 − 2.49046i −1.00000 −3.95892 + 0.657665i
277.11 0.277192 − 1.38678i 1.00000i −1.84633 − 0.768809i − 0.208104i 1.38678 + 0.277192i 4.69627 −1.57796 + 2.34735i −1.00000 −0.288595 − 0.0576846i
277.12 0.277192 + 1.38678i − 1.00000i −1.84633 + 0.768809i 0.208104i 1.38678 − 0.277192i 4.69627 −1.57796 − 2.34735i −1.00000 −0.288595 + 0.0576846i
277.13 0.624769 − 1.26873i − 1.00000i −1.21933 − 1.58532i 3.50951i −1.26873 − 0.624769i −4.50530 −2.77313 + 0.556532i −1.00000 4.45260 + 2.19263i
277.14 0.624769 + 1.26873i 1.00000i −1.21933 + 1.58532i − 3.50951i −1.26873 + 0.624769i −4.50530 −2.77313 − 0.556532i −1.00000 4.45260 − 2.19263i
277.15 0.927201 − 1.06785i 1.00000i −0.280598 − 1.98022i − 0.619156i 1.06785 + 0.927201i −1.72166 −2.37474 − 1.53642i −1.00000 −0.661164 − 0.574082i
277.16 0.927201 + 1.06785i − 1.00000i −0.280598 + 1.98022i 0.619156i 1.06785 − 0.927201i −1.72166 −2.37474 + 1.53642i −1.00000 −0.661164 + 0.574082i
277.17 1.38342 − 0.293513i − 1.00000i 1.82770 − 0.812103i 1.32391i −0.293513 − 1.38342i 0.191941 2.29011 − 1.65993i −1.00000 0.388583 + 1.83152i
277.18 1.38342 + 0.293513i 1.00000i 1.82770 + 0.812103i − 1.32391i −0.293513 + 1.38342i 0.191941 2.29011 + 1.65993i −1.00000 0.388583 − 1.83152i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.f.c ✓ 18
4.b odd 2 1 2208.2.f.c 18
8.b even 2 1 inner 552.2.f.c ✓ 18
8.d odd 2 1 2208.2.f.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.f.c ✓ 18 1.a even 1 1 trivial
552.2.f.c ✓ 18 8.b even 2 1 inner
2208.2.f.c 18 4.b odd 2 1
2208.2.f.c 18 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 56 T_{5}^{16} + 1188 T_{5}^{14} + 11920 T_{5}^{12} + 57236 T_{5}^{10} + 119952 T_{5}^{8} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 4 T^{16} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{18} + 56 T^{16} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{9} - 4 T^{8} - 34 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + 92 T^{16} + \cdots + 200704 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 750321664 \) Copy content Toggle raw display
$17$ \( (T^{9} + 4 T^{8} + \cdots + 6976)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 3650093056 \) Copy content Toggle raw display
$23$ \( (T + 1)^{18} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 440664064 \) Copy content Toggle raw display
$31$ \( (T^{9} + 22 T^{8} + \cdots + 67072)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 2801579659264 \) Copy content Toggle raw display
$41$ \( (T^{9} - 14 T^{8} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 23406089184256 \) Copy content Toggle raw display
$47$ \( (T^{9} - 172 T^{7} + \cdots - 1608704)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 3540726016 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 20495829581824 \) Copy content Toggle raw display
$71$ \( (T^{9} + 12 T^{8} + \cdots + 2633728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 6 T^{8} + \cdots + 34816)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} - 4 T^{8} + \cdots - 371744)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 28955763675136 \) Copy content Toggle raw display
$89$ \( (T^{9} - 12 T^{8} + \cdots - 203802464)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} - 6 T^{8} + \cdots - 53447168)^{2} \) Copy content Toggle raw display
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