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Properties

Label 1232.2.q.p
Level $1232$
Weight $2$
Character orbit 1232.q
Analytic conductor $9.838$
Analytic rank $0$
Dimension $10$
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] = [1232,2,Mod(177,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), 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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 9x^{8} + 2x^{7} + 59x^{6} - 12x^{5} + 69x^{4} - 40x^{3} + 76x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 616)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{4}) q^{3} + (\beta_{8} - \beta_{6} - \beta_{4} + \cdots - 1) q^{5}+ \cdots + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{4}) q^{3} + (\beta_{8} - \beta_{6} - \beta_{4} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{6} - \beta_{3} - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{3} - 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{3} - 4 q^{5} - 4 q^{9} + 5 q^{11} + 14 q^{13} - 14 q^{15} - 5 q^{17} + 13 q^{19} - 10 q^{21} + 8 q^{23} - q^{25} + 14 q^{29} + 11 q^{31} - 3 q^{33} + 3 q^{35} + 14 q^{37} - 7 q^{39} - 14 q^{41} - 24 q^{43} - 11 q^{45} + 11 q^{47} - 14 q^{49} - 6 q^{51} - 7 q^{53} - 8 q^{55} + 32 q^{57} - 18 q^{61} - 2 q^{63} + 9 q^{65} - 12 q^{67} - 58 q^{69} - 38 q^{71} + 12 q^{73} + 31 q^{75} + 3 q^{79} + 7 q^{81} - 46 q^{83} + 54 q^{85} + 36 q^{87} - 5 q^{89} + 47 q^{91} + 42 q^{93} + 3 q^{95} - 26 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 9x^{8} + 2x^{7} + 59x^{6} - 12x^{5} + 69x^{4} - 40x^{3} + 76x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1801 \nu^{9} + 10923 \nu^{8} - 30783 \nu^{7} + 136034 \nu^{6} - 124125 \nu^{5} + 512388 \nu^{4} + \cdots + 1104352 ) / 2258756 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 83639 \nu^{9} + 334279 \nu^{8} - 942059 \nu^{7} + 2008958 \nu^{6} - 3798625 \nu^{5} + \cdots + 14339128 ) / 4517512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69022 \nu^{9} + 70823 \nu^{8} - 610275 \nu^{7} - 168827 \nu^{6} - 3936264 \nu^{5} + \cdots - 320148 ) / 2258756 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26342 \nu^{9} + 81664 \nu^{8} - 230144 \nu^{7} + 389793 \nu^{6} - 928000 \nu^{5} + \cdots + 3340924 ) / 564689 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 268927 \nu^{9} + 893607 \nu^{8} - 2518347 \nu^{7} + 4666534 \nu^{6} - 10154625 \nu^{5} + \cdots + 27456352 ) / 4517512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 173304 \nu^{9} + 33459 \nu^{8} - 1429013 \nu^{7} - 1536375 \nu^{6} - 10698624 \nu^{5} + \cdots - 1177500 ) / 2258756 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 208311 \nu^{9} - 5819 \nu^{8} + 1710466 \nu^{7} + 2111791 \nu^{6} + 13053972 \nu^{5} + \cdots + 1504384 ) / 1129378 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 109415 \nu^{9} - 35914 \nu^{8} + 922578 \nu^{7} + 856018 \nu^{6} + 6671028 \nu^{5} + 2986825 \nu^{4} + \cdots + 705132 ) / 564689 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 3\beta_{3} - 5\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{9} - 9\beta_{8} - 9\beta_{7} + 19\beta_{4} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{9} - 14 \beta_{8} - 30 \beta_{7} + 14 \beta_{6} - 5 \beta_{5} + 22 \beta_{4} - 30 \beta_{3} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 75\beta_{6} - 61\beta_{5} - 85\beta_{3} + 57\beta_{2} + 147 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -81\beta_{9} + 156\beta_{8} + 278\beta_{7} - 270\beta_{4} + 293\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 490 \beta_{9} + 646 \beta_{8} + 808 \beta_{7} - 646 \beta_{6} + 490 \beta_{5} - 1232 \beta_{4} + \cdots - 1232 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1593\beta_{6} + 947\beta_{5} + 2573\beta_{3} - 2530\beta_{2} - 2851 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
1.53014 + 2.65028i
−0.600543 − 1.04017i
−1.13549 − 1.96672i
0.422020 + 0.730960i
0.283869 + 0.491676i
1.53014 − 2.65028i
−0.600543 + 1.04017i
−1.13549 + 1.96672i
0.422020 − 0.730960i
0.283869 − 0.491676i
0 −1.20337 − 2.08431i 0 −0.0393108 + 0.0680883i 0 0.659613 − 2.56221i 0 −1.39622 + 2.41832i 0
177.2 0 −0.232036 − 0.401898i 0 −0.540804 + 0.936700i 0 0.391251 + 2.61666i 0 1.39232 − 2.41157i 0
177.3 0 0.695148 + 1.20403i 0 −1.54229 + 2.67132i 0 −1.46021 − 2.20631i 0 0.533539 − 0.924116i 0
177.4 0 0.762757 + 1.32113i 0 1.44320 − 2.49970i 0 −2.16132 + 1.52601i 0 0.336403 − 0.582666i 0
177.5 0 1.47750 + 2.55911i 0 −1.32080 + 2.28769i 0 2.57066 + 0.625845i 0 −2.86604 + 4.96413i 0
529.1 0 −1.20337 + 2.08431i 0 −0.0393108 − 0.0680883i 0 0.659613 + 2.56221i 0 −1.39622 − 2.41832i 0
529.2 0 −0.232036 + 0.401898i 0 −0.540804 − 0.936700i 0 0.391251 − 2.61666i 0 1.39232 + 2.41157i 0
529.3 0 0.695148 − 1.20403i 0 −1.54229 − 2.67132i 0 −1.46021 + 2.20631i 0 0.533539 + 0.924116i 0
529.4 0 0.762757 − 1.32113i 0 1.44320 + 2.49970i 0 −2.16132 − 1.52601i 0 0.336403 + 0.582666i 0
529.5 0 1.47750 − 2.55911i 0 −1.32080 − 2.28769i 0 2.57066 − 0.625845i 0 −2.86604 − 4.96413i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.p 10
4.b odd 2 1 616.2.q.e ✓ 10
7.c even 3 1 inner 1232.2.q.p 10
7.c even 3 1 8624.2.a.da 5
7.d odd 6 1 8624.2.a.dd 5
28.f even 6 1 4312.2.a.be 5
28.g odd 6 1 616.2.q.e ✓ 10
28.g odd 6 1 4312.2.a.bh 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.e ✓ 10 4.b odd 2 1
616.2.q.e ✓ 10 28.g odd 6 1
1232.2.q.p 10 1.a even 1 1 trivial
1232.2.q.p 10 7.c even 3 1 inner
4312.2.a.be 5 28.f even 6 1
4312.2.a.bh 5 28.g odd 6 1
8624.2.a.da 5 7.c even 3 1
8624.2.a.dd 5 7.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 \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{10} - 3 T_{3}^{9} + 14 T_{3}^{8} - 21 T_{3}^{7} + 85 T_{3}^{6} - 133 T_{3}^{5} + 273 T_{3}^{4} + \cdots + 49 \) Copy content Toggle raw display
\( T_{13}^{5} - 7T_{13}^{4} - 29T_{13}^{3} + 272T_{13}^{2} - 410T_{13} + 141 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{10} + 7 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$13$ \( (T^{5} - 7 T^{4} + \cdots + 141)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 5 T^{9} + \cdots + 21904 \) Copy content Toggle raw display
$19$ \( T^{10} - 13 T^{9} + \cdots + 21215236 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} + \cdots + 835396 \) Copy content Toggle raw display
$29$ \( (T^{5} - 7 T^{4} + \cdots + 1057)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 11 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{10} - 14 T^{9} + \cdots + 1937664 \) Copy content Toggle raw display
$41$ \( (T^{5} + 7 T^{4} + \cdots + 134)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 12 T^{4} + \cdots + 14864)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 11 T^{9} + \cdots + 620944 \) Copy content Toggle raw display
$53$ \( T^{10} + 7 T^{9} + \cdots + 1965604 \) Copy content Toggle raw display
$59$ \( T^{10} + 114 T^{8} + \cdots + 40401 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 195328576 \) Copy content Toggle raw display
$67$ \( T^{10} + 12 T^{9} + \cdots + 4359744 \) Copy content Toggle raw display
$71$ \( (T^{5} + 19 T^{4} + \cdots - 954)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 129367876 \) Copy content Toggle raw display
$79$ \( T^{10} - 3 T^{9} + \cdots + 19035769 \) Copy content Toggle raw display
$83$ \( (T^{5} + 23 T^{4} + \cdots - 636)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 7848542464 \) Copy content Toggle raw display
$97$ \( (T^{5} + 13 T^{4} + \cdots - 2693)^{2} \) Copy content Toggle raw display
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