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Properties

Label 1734.2.f.o
Level $1734$
Weight $2$
Character orbit 1734.f
Analytic conductor $13.846$
Analytic rank $0$
Dimension $12$
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] = [1734,2,Mod(829,1734)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1734, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("1734.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.f (of order \(4\), 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: \(13.8460597105\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.722204136308736.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{8} + 69x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{10} q^{3} - q^{4} + ( - \beta_{11} + 2 \beta_{10}) q^{5} - \beta_{3} q^{6} + ( - \beta_{3} + 2 \beta_1) q^{7} + \beta_{7} q^{8} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_{10} q^{3} - q^{4} + ( - \beta_{11} + 2 \beta_{10}) q^{5} - \beta_{3} q^{6} + ( - \beta_{3} + 2 \beta_1) q^{7} + \beta_{7} q^{8} + \beta_{7} q^{9} + (\beta_{5} + 2 \beta_{3} - \beta_1) q^{10} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_1) q^{11} + \beta_{10} q^{12} + ( - 2 \beta_{2} - 2) q^{13} + (\beta_{10} - 2 \beta_{8}) q^{14} + (\beta_{9} - 2 \beta_{7}) q^{15} + q^{16} + q^{18} + ( - 2 \beta_{7} - 2 \beta_{4}) q^{19} + (\beta_{11} - 2 \beta_{10}) q^{20} + ( - 2 \beta_{6} - 2 \beta_{2} + 1) q^{21} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{8}) q^{22} + (4 \beta_{5} + 2 \beta_{3}) q^{23} + \beta_{3} q^{24} + ( - 3 \beta_{9} + \beta_{7} + \beta_{4}) q^{25} + (2 \beta_{9} + 2 \beta_{7}) q^{26} + \beta_{3} q^{27} + (\beta_{3} - 2 \beta_1) q^{28} + (5 \beta_{11} - 3 \beta_{8}) q^{29} + (\beta_{2} - 2) q^{30} + (\beta_{11} + 2 \beta_{10} + 5 \beta_{8}) q^{31} - \beta_{7} q^{32} + (\beta_{6} - \beta_{2} + 2) q^{33} + (6 \beta_{6} + 7 \beta_{2} - 4) q^{35} - \beta_{7} q^{36} + ( - 2 \beta_{11} + 6 \beta_{10} + 2 \beta_{8}) q^{37} + (2 \beta_{6} + 2 \beta_{2} - 2) q^{38} + (2 \beta_{11} + 2 \beta_{10}) q^{39} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{40} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{41} + ( - \beta_{7} - 2 \beta_{4}) q^{42} + 6 \beta_{7} q^{43} + (2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{44} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{45} + (4 \beta_{11} - 2 \beta_{10} - 4 \beta_{8}) q^{46} + (6 \beta_{6} + 2 \beta_{2} - 8) q^{47} - \beta_{10} q^{48} + (4 \beta_{9} - 2 \beta_{7} - 4 \beta_{4}) q^{49} + ( - \beta_{6} - 4 \beta_{2} + 1) q^{50} + (2 \beta_{2} + 2) q^{52} + ( - 6 \beta_{9} + 2 \beta_{7} - 5 \beta_{4}) q^{53} - \beta_{10} q^{54} + ( - \beta_{6} + 3 \beta_{2} - 7) q^{55} + ( - \beta_{10} + 2 \beta_{8}) q^{56} + ( - 2 \beta_{3} + 2 \beta_1) q^{57} + ( - 5 \beta_{5} + 2 \beta_1) q^{58} + (\beta_{9} - 4 \beta_{4}) q^{59} + ( - \beta_{9} + 2 \beta_{7}) q^{60} + (8 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{61} + ( - \beta_{5} + 2 \beta_{3} + 6 \beta_1) q^{62} + ( - \beta_{10} + 2 \beta_{8}) q^{63} - q^{64} - 2 \beta_{8} q^{65} + (2 \beta_{9} - 2 \beta_{7} + \beta_{4}) q^{66} + (4 \beta_{6} + 6 \beta_{2} + 8) q^{67} + ( - 4 \beta_{6} - 2) q^{69} + ( - \beta_{9} + 4 \beta_{7} + 6 \beta_{4}) q^{70} + (2 \beta_{11} + 6 \beta_{10} - 4 \beta_{8}) q^{71} - q^{72} + (3 \beta_{11} + 5 \beta_{8}) q^{73} + (2 \beta_{5} + 6 \beta_{3}) q^{74} + (3 \beta_{5} + \beta_{3} - 4 \beta_1) q^{75} + (2 \beta_{7} + 2 \beta_{4}) q^{76} + ( - 2 \beta_{7} - 7 \beta_{4}) q^{77} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{78} + (4 \beta_{5} + 8 \beta_{3} + \beta_1) q^{79} + ( - \beta_{11} + 2 \beta_{10}) q^{80} - q^{81} + (2 \beta_{11} + 2 \beta_{10} - 4 \beta_{8}) q^{82} + (2 \beta_{9} + 3 \beta_{7}) q^{83} + (2 \beta_{6} + 2 \beta_{2} - 1) q^{84} + 6 q^{86} + ( - 5 \beta_{9} - 3 \beta_{4}) q^{87} + (2 \beta_{11} - 2 \beta_{10} - \beta_{8}) q^{88} + ( - 4 \beta_{6} + 2 \beta_{2} - 4) q^{89} + ( - \beta_{11} + 2 \beta_{10}) q^{90} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{91} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{92} + ( - \beta_{9} - 2 \beta_{7} + 5 \beta_{4}) q^{93} + (4 \beta_{9} + 8 \beta_{7} + 6 \beta_{4}) q^{94} + (2 \beta_{5} + 6 \beta_{3} - 8 \beta_1) q^{95} - \beta_{3} q^{96} + (4 \beta_{11} - 4 \beta_{10} - 7 \beta_{8}) q^{97} + (4 \beta_{6} + 8 \beta_{2} - 2) q^{98} + (2 \beta_{11} - 2 \beta_{10} - \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 24 q^{13} + 12 q^{16} + 12 q^{18} + 12 q^{21} - 24 q^{30} + 24 q^{33} - 48 q^{35} - 24 q^{38} - 96 q^{47} + 12 q^{50} + 24 q^{52} - 84 q^{55} - 12 q^{64} + 96 q^{67} - 24 q^{69} - 12 q^{72} - 12 q^{81} - 12 q^{84} + 72 q^{86} - 48 q^{89} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{8} + 69x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 27\nu^{4} + 100 ) / 53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 27\nu^{5} + 153\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 27\nu^{6} + 153\nu^{2} ) / 53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} - 82\nu^{5} - 235\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{8} - 55\nu^{4} - 82 ) / 53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{10} - 109\nu^{6} - 441\nu^{2} ) / 53 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6\nu^{11} - 109\nu^{7} - 441\nu^{3} ) / 53 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -12\nu^{10} - 218\nu^{6} - 829\nu^{2} ) / 53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -18\nu^{11} - 327\nu^{7} - 1270\nu^{3} ) / 53 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -35\nu^{11} - 627\nu^{7} - 2387\nu^{3} ) / 53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{9} + 19\beta_{7} + 6\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6\beta_{11} - 21\beta_{10} + 28\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -27\beta_{6} - 55\beta_{2} + 62 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -27\beta_{5} - 82\beta_{3} + 117\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 90\beta_{9} - 207\beta_{7} - 109\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -109\beta_{11} + 308\beta_{10} - 297\beta_{8} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1159\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
0.245576 − 0.245576i
1.08335 − 1.08335i
−1.32893 + 1.32893i
1.32893 − 1.32893i
−1.08335 + 1.08335i
−0.245576 + 0.245576i
0.245576 + 0.245576i
1.08335 + 1.08335i
−1.32893 − 1.32893i
1.32893 + 1.32893i
−1.08335 − 1.08335i
−0.245576 − 0.245576i
− 1.00000i −0.707107 − 0.707107i −1.00000 0.0852875 + 0.0852875i −0.707107 + 0.707107i −0.215956 + 0.215956i 1.00000i 1.00000i 0.0852875 − 0.0852875i
829.2 − 1.00000i −0.707107 − 0.707107i −1.00000 1.65979 + 1.65979i −0.707107 + 0.707107i 1.45959 − 1.45959i 1.00000i 1.00000i 1.65979 − 1.65979i
829.3 − 1.00000i −0.707107 − 0.707107i −1.00000 2.49756 + 2.49756i −0.707107 + 0.707107i −3.36496 + 3.36496i 1.00000i 1.00000i 2.49756 − 2.49756i
829.4 − 1.00000i 0.707107 + 0.707107i −1.00000 −2.49756 − 2.49756i 0.707107 − 0.707107i 3.36496 − 3.36496i 1.00000i 1.00000i −2.49756 + 2.49756i
829.5 − 1.00000i 0.707107 + 0.707107i −1.00000 −1.65979 − 1.65979i 0.707107 − 0.707107i −1.45959 + 1.45959i 1.00000i 1.00000i −1.65979 + 1.65979i
829.6 − 1.00000i 0.707107 + 0.707107i −1.00000 −0.0852875 − 0.0852875i 0.707107 − 0.707107i 0.215956 − 0.215956i 1.00000i 1.00000i −0.0852875 + 0.0852875i
1483.1 1.00000i −0.707107 + 0.707107i −1.00000 0.0852875 − 0.0852875i −0.707107 − 0.707107i −0.215956 − 0.215956i − 1.00000i − 1.00000i 0.0852875 + 0.0852875i
1483.2 1.00000i −0.707107 + 0.707107i −1.00000 1.65979 − 1.65979i −0.707107 − 0.707107i 1.45959 + 1.45959i − 1.00000i − 1.00000i 1.65979 + 1.65979i
1483.3 1.00000i −0.707107 + 0.707107i −1.00000 2.49756 − 2.49756i −0.707107 − 0.707107i −3.36496 − 3.36496i − 1.00000i − 1.00000i 2.49756 + 2.49756i
1483.4 1.00000i 0.707107 − 0.707107i −1.00000 −2.49756 + 2.49756i 0.707107 + 0.707107i 3.36496 + 3.36496i − 1.00000i − 1.00000i −2.49756 − 2.49756i
1483.5 1.00000i 0.707107 − 0.707107i −1.00000 −1.65979 + 1.65979i 0.707107 + 0.707107i −1.45959 − 1.45959i − 1.00000i − 1.00000i −1.65979 − 1.65979i
1483.6 1.00000i 0.707107 − 0.707107i −1.00000 −0.0852875 + 0.0852875i 0.707107 + 0.707107i 0.215956 + 0.215956i − 1.00000i − 1.00000i −0.0852875 − 0.0852875i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1734.2.f.o 12
17.b even 2 1 inner 1734.2.f.o 12
17.c even 4 2 inner 1734.2.f.o 12
17.d even 8 1 1734.2.a.r ✓ 3
17.d even 8 1 1734.2.a.s yes 3
17.d even 8 2 1734.2.b.i 6
51.g odd 8 1 5202.2.a.bf 3
51.g odd 8 1 5202.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1734.2.a.r ✓ 3 17.d even 8 1
1734.2.a.s yes 3 17.d even 8 1
1734.2.b.i 6 17.d even 8 2
1734.2.f.o 12 1.a even 1 1 trivial
1734.2.f.o 12 17.b even 2 1 inner
1734.2.f.o 12 17.c even 4 2 inner
5202.2.a.bf 3 51.g odd 8 1
5202.2.a.bk 3 51.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1734, [\chi])\):

\( T_{5}^{12} + 186T_{5}^{8} + 4725T_{5}^{4} + 1 \) Copy content Toggle raw display
\( T_{7}^{12} + 531T_{7}^{8} + 9315T_{7}^{4} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 186 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + 531 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{12} + 426 T^{8} + \cdots + 130321 \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} - 8)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 36 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 8496 T^{8} + \cdots + 331776 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 131079601 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 47562811921 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 533794816 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 342102016 \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{6} \) Copy content Toggle raw display
$47$ \( (T^{3} + 24 T^{2} + \cdots - 456)^{4} \) Copy content Toggle raw display
$53$ \( (T^{6} + 198 T^{4} + \cdots + 25281)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 126 T^{4} + \cdots + 29241)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 533794816 \) Copy content Toggle raw display
$67$ \( (T^{3} - 24 T^{2} + \cdots + 456)^{4} \) Copy content Toggle raw display
$71$ \( T^{12} + 16848 T^{8} + \cdots + 26873856 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 217611987121 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1026625681 \) Copy content Toggle raw display
$83$ \( (T^{6} + 51 T^{4} + 243 T^{2} + 1)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} + \cdots - 408)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + 67506 T^{8} + \cdots + 7890481 \) Copy content Toggle raw display
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