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Properties

Label 1650.2.d.i
Level $1650$
Weight $2$
Character orbit 1650.d
Analytic conductor $13.175$
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] = [1650,2,Mod(1451,1650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1650.1451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.d (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: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - 5x^{8} - 46x^{6} - 45x^{4} + 243x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 330)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} - \beta_{3} q^{7} - q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} - \beta_{3} q^{7} - q^{8} + \beta_{2} q^{9} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{11} - \beta_1 q^{12} + (\beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - \beta_{11} - \beta_{9} + \beta_{7} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} - 6 q^{9} + 12 q^{16} - 4 q^{17} + 6 q^{18} + 12 q^{31} - 12 q^{32} - 18 q^{33} + 4 q^{34} - 6 q^{36} + 8 q^{49} - 14 q^{57} - 12 q^{62} - 22 q^{63} + 12 q^{64} + 18 q^{66} - 4 q^{68} - 4 q^{69} + 6 q^{72} + 40 q^{77} + 38 q^{81} + 8 q^{83} + 14 q^{87} + 40 q^{91} - 8 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - 5x^{8} - 46x^{6} - 45x^{4} + 243x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 6\nu^{8} + 49\nu^{6} + 161\nu^{4} - 198\nu^{2} - 1215 ) / 648 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + 24 \nu^{9} + 54 \nu^{8} + 5 \nu^{7} + 162 \nu^{6} - 89 \nu^{5} + 216 \nu^{4} + \cdots - 2430 ) / 1944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 24 \nu^{9} + 54 \nu^{8} - 5 \nu^{7} + 162 \nu^{6} + 89 \nu^{5} + 216 \nu^{4} + 792 \nu^{3} + \cdots - 2430 ) / 1944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 24\nu^{9} + 5\nu^{7} - 89\nu^{5} + 180\nu^{3} + 1053\nu ) / 972 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} - 3\nu^{8} + 5\nu^{6} + 46\nu^{4} + 45\nu^{2} - 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7\nu^{11} + 12\nu^{9} - 143\nu^{7} - 277\nu^{5} + 504\nu^{3} + 2673\nu ) / 972 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4\nu^{10} + 3\nu^{8} - 47\nu^{6} - 58\nu^{4} + 153\nu^{2} + 729 ) / 162 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -17\nu^{11} - 6\nu^{9} + 139\nu^{7} + 71\nu^{5} - 414\nu^{3} - 1701\nu ) / 1944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\nu^{11} + 12\nu^{9} - 65\nu^{7} - 463\nu^{5} + 252\nu^{3} + 1863\nu ) / 972 \) 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
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 2\beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} - 4\beta_{10} - \beta_{8} + \beta_{6} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{9} - 5\beta_{7} - \beta_{5} - \beta_{4} + 8\beta_{3} + 7\beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5\beta_{11} - 9\beta_{8} + 5\beta_{6} + 3\beta_{5} - 3\beta_{4} + 13\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5\beta_{9} + 7\beta_{7} + 17\beta_{5} + 17\beta_{4} - 24\beta_{3} + 2\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -7\beta_{11} - 20\beta_{10} - 5\beta_{8} + 36\beta_{6} - 8\beta_{5} + 8\beta_{4} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16\beta_{9} - 35\beta_{7} - 10\beta_{5} - 10\beta_{4} + 112\beta_{3} + 28\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 35\beta_{11} - 124\beta_{10} - 76\beta_{8} + 8\beta_{6} + 3\beta_{5} - 3\beta_{4} + 28\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1451.1
1.71007 + 0.275064i
1.71007 − 0.275064i
0.839210 + 1.51517i
0.839210 − 1.51517i
0.348406 + 1.69665i
0.348406 − 1.69665i
−0.348406 + 1.69665i
−0.348406 − 1.69665i
−0.839210 + 1.51517i
−0.839210 − 1.51517i
−1.71007 + 0.275064i
−1.71007 − 0.275064i
−1.00000 −1.71007 − 0.275064i 1.00000 0 1.71007 + 0.275064i − 2.37039i −1.00000 2.84868 + 0.940756i 0
1451.2 −1.00000 −1.71007 + 0.275064i 1.00000 0 1.71007 − 0.275064i 2.37039i −1.00000 2.84868 − 0.940756i 0
1451.3 −1.00000 −0.839210 − 1.51517i 1.00000 0 0.839210 + 1.51517i 3.51327i −1.00000 −1.59145 + 2.54308i 0
1451.4 −1.00000 −0.839210 + 1.51517i 1.00000 0 0.839210 − 1.51517i − 3.51327i −1.00000 −1.59145 − 2.54308i 0
1451.5 −1.00000 −0.348406 − 1.69665i 1.00000 0 0.348406 + 1.69665i − 1.01891i −1.00000 −2.75723 + 1.18224i 0
1451.6 −1.00000 −0.348406 + 1.69665i 1.00000 0 0.348406 − 1.69665i 1.01891i −1.00000 −2.75723 − 1.18224i 0
1451.7 −1.00000 0.348406 − 1.69665i 1.00000 0 −0.348406 + 1.69665i 1.01891i −1.00000 −2.75723 − 1.18224i 0
1451.8 −1.00000 0.348406 + 1.69665i 1.00000 0 −0.348406 − 1.69665i − 1.01891i −1.00000 −2.75723 + 1.18224i 0
1451.9 −1.00000 0.839210 − 1.51517i 1.00000 0 −0.839210 + 1.51517i − 3.51327i −1.00000 −1.59145 − 2.54308i 0
1451.10 −1.00000 0.839210 + 1.51517i 1.00000 0 −0.839210 − 1.51517i 3.51327i −1.00000 −1.59145 + 2.54308i 0
1451.11 −1.00000 1.71007 − 0.275064i 1.00000 0 −1.71007 + 0.275064i 2.37039i −1.00000 2.84868 − 0.940756i 0
1451.12 −1.00000 1.71007 + 0.275064i 1.00000 0 −1.71007 − 0.275064i − 2.37039i −1.00000 2.84868 + 0.940756i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1451.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1650.2.d.i 12
3.b odd 2 1 1650.2.d.j 12
5.b even 2 1 1650.2.d.j 12
5.c odd 4 2 330.2.f.a ✓ 24
11.b odd 2 1 1650.2.d.j 12
15.d odd 2 1 inner 1650.2.d.i 12
15.e even 4 2 330.2.f.a ✓ 24
33.d even 2 1 inner 1650.2.d.i 12
55.d odd 2 1 inner 1650.2.d.i 12
55.e even 4 2 330.2.f.a ✓ 24
165.d even 2 1 1650.2.d.j 12
165.l odd 4 2 330.2.f.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.f.a ✓ 24 5.c odd 4 2
330.2.f.a ✓ 24 15.e even 4 2
330.2.f.a ✓ 24 55.e even 4 2
330.2.f.a ✓ 24 165.l odd 4 2
1650.2.d.i 12 1.a even 1 1 trivial
1650.2.d.i 12 15.d odd 2 1 inner
1650.2.d.i 12 33.d even 2 1 inner
1650.2.d.i 12 55.d odd 2 1 inner
1650.2.d.j 12 3.b odd 2 1
1650.2.d.j 12 5.b even 2 1
1650.2.d.j 12 11.b odd 2 1
1650.2.d.j 12 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{6} + 19T_{7}^{4} + 88T_{7}^{2} + 72 \) Copy content Toggle raw display
\( T_{17}^{3} + T_{17}^{2} - 32T_{17} + 12 \) Copy content Toggle raw display
\( T_{29}^{6} - 143T_{29}^{4} + 4664T_{29}^{2} - 3600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 19 T^{4} + \cdots + 72)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 14 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{6} + 60 T^{4} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + T^{2} - 32 T + 12)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 61 T^{4} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 28 T^{4} + \cdots + 288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 143 T^{4} + \cdots - 3600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 32 T - 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 127 T^{4} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 176 T^{4} + \cdots - 147456)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 76 T^{4} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 200 T^{4} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 205 T^{4} + \cdots + 121032)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 76 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 293 T^{4} + \cdots + 145800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 52 T^{4} + \cdots - 576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 131 T^{4} + \cdots + 16200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 240 T^{4} + \cdots + 41472)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 316 T^{4} + \cdots + 23328)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 2 T^{2} - 128 T - 96)^{4} \) Copy content Toggle raw display
$89$ \( (T^{6} + 35 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 236 T^{4} + \cdots - 20736)^{2} \) Copy content Toggle raw display
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