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Properties

Label 1650.2.d.h
Level $1650$
Weight $2$
Character orbit 1650.d
Analytic conductor $13.175$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: 8.0.1350020905216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + x^{5} + 2x^{4} + 3x^{3} - 18x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{4} q^{3} + q^{4} - \beta_{4} q^{6} - \beta_{7} q^{7} + q^{8} + (\beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{4} q^{3} + q^{4} - \beta_{4} q^{6} - \beta_{7} q^{7} + q^{8} + (\beta_{3} + 1) q^{9} + ( - \beta_{5} - \beta_{2}) q^{11} - \beta_{4} q^{12} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{13}+ \cdots + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + q^{3} + 8 q^{4} + q^{6} + 8 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + q^{3} + 8 q^{4} + q^{6} + 8 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} + 8 q^{16} - 8 q^{17} + 5 q^{18} + 8 q^{21} + 3 q^{22} + q^{24} + 4 q^{27} - 8 q^{29} + 8 q^{31} + 8 q^{32} - 9 q^{33} - 8 q^{34} + 5 q^{36} + 6 q^{37} - 4 q^{39} + 4 q^{41} + 8 q^{42} + 3 q^{44} + q^{48} + 2 q^{49} + q^{51} + 4 q^{54} - 13 q^{57} - 8 q^{58} + 8 q^{62} + 2 q^{63} + 8 q^{64} - 9 q^{66} - 6 q^{67} - 8 q^{68} - 16 q^{69} + 5 q^{72} + 6 q^{74} - 12 q^{77} - 4 q^{78} + 5 q^{81} + 4 q^{82} + 30 q^{83} + 8 q^{84} + 20 q^{87} + 3 q^{88} - 24 q^{91} + 22 q^{93} + q^{96} - 46 q^{97} + 2 q^{98} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + x^{5} + 2x^{4} + 3x^{3} - 18x^{2} - 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 5\nu^{5} + 5\nu^{4} - 5\nu^{3} - 9\nu^{2} + 9\nu + 54 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} + \nu^{4} + 2\nu^{3} + 3\nu^{2} - 18\nu - 27 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 2\nu^{5} - \nu^{4} + \nu^{3} - 12\nu^{2} + 12\nu + 63 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} + \nu^{5} + 6\nu^{4} + 11\nu^{3} - 19\nu^{2} - 3\nu + 54 ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - \nu^{6} + \nu^{5} + \nu^{4} - 4\nu^{3} - 9\nu^{2} + 15\nu + 81 ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - 3\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} + 2\beta_{5} - 6\beta_{4} + 3\beta_{3} + \beta_{2} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} - 5\beta_{5} - 10\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{7} - 2\beta_{6} + 5\beta_{4} + 5\beta_{3} - 5\beta_{2} + 8\beta _1 + 32 \) 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
−1.64269 + 0.549154i
−1.64269 − 0.549154i
−0.656040 + 1.60300i
−0.656040 − 1.60300i
1.07942 + 1.35457i
1.07942 − 1.35457i
1.71931 + 0.209658i
1.71931 − 0.209658i
1.00000 −1.64269 − 0.549154i 1.00000 0 −1.64269 − 0.549154i 1.82098i 1.00000 2.39686 + 1.80418i 0
1451.2 1.00000 −1.64269 + 0.549154i 1.00000 0 −1.64269 + 0.549154i − 1.82098i 1.00000 2.39686 − 1.80418i 0
1451.3 1.00000 −0.656040 − 1.60300i 1.00000 0 −0.656040 − 1.60300i 0.623830i 1.00000 −2.13922 + 2.10327i 0
1451.4 1.00000 −0.656040 + 1.60300i 1.00000 0 −0.656040 + 1.60300i − 0.623830i 1.00000 −2.13922 − 2.10327i 0
1451.5 1.00000 1.07942 − 1.35457i 1.00000 0 1.07942 − 1.35457i 0.738241i 1.00000 −0.669725 − 2.92429i 0
1451.6 1.00000 1.07942 + 1.35457i 1.00000 0 1.07942 + 1.35457i − 0.738241i 1.00000 −0.669725 + 2.92429i 0
1451.7 1.00000 1.71931 − 0.209658i 1.00000 0 1.71931 − 0.209658i 4.76968i 1.00000 2.91209 − 0.720935i 0
1451.8 1.00000 1.71931 + 0.209658i 1.00000 0 1.71931 + 0.209658i − 4.76968i 1.00000 2.91209 + 0.720935i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1451.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 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.h yes 8
3.b odd 2 1 1650.2.d.e yes 8
5.b even 2 1 1650.2.d.d ✓ 8
5.c odd 4 2 1650.2.f.h 16
11.b odd 2 1 1650.2.d.e yes 8
15.d odd 2 1 1650.2.d.g yes 8
15.e even 4 2 1650.2.f.g 16
33.d even 2 1 inner 1650.2.d.h yes 8
55.d odd 2 1 1650.2.d.g yes 8
55.e even 4 2 1650.2.f.g 16
165.d even 2 1 1650.2.d.d ✓ 8
165.l odd 4 2 1650.2.f.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1650.2.d.d ✓ 8 5.b even 2 1
1650.2.d.d ✓ 8 165.d even 2 1
1650.2.d.e yes 8 3.b odd 2 1
1650.2.d.e yes 8 11.b odd 2 1
1650.2.d.g yes 8 15.d odd 2 1
1650.2.d.g yes 8 55.d odd 2 1
1650.2.d.h yes 8 1.a even 1 1 trivial
1650.2.d.h yes 8 33.d even 2 1 inner
1650.2.f.g 16 15.e even 4 2
1650.2.f.g 16 55.e even 4 2
1650.2.f.h 16 5.c odd 4 2
1650.2.f.h 16 165.l odd 4 2

Hecke kernels

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

\( T_{7}^{8} + 27T_{7}^{6} + 100T_{7}^{4} + 76T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} + 4T_{17}^{3} - 37T_{17}^{2} - 172T_{17} - 140 \) Copy content Toggle raw display
\( T_{29}^{4} + 4T_{29}^{3} - 49T_{29}^{2} - 82T_{29} + 440 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 27 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 38 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} + \cdots - 140)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 56 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{8} + 125 T^{6} + \cdots + 75076 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 440)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + \cdots + 392)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 3 T^{3} + \cdots - 452)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 2 T^{3} - 69 T^{2} + \cdots - 86)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 178 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$47$ \( T^{8} + 163 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$53$ \( T^{8} + 400 T^{6} + \cdots + 20070400 \) Copy content Toggle raw display
$59$ \( T^{8} + 299 T^{6} + \cdots + 14961424 \) Copy content Toggle raw display
$61$ \( T^{8} + 196 T^{6} + \cdots + 25600 \) Copy content Toggle raw display
$67$ \( (T^{4} + 3 T^{3} + \cdots + 224)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 401 T^{6} + \cdots + 1000000 \) Copy content Toggle raw display
$73$ \( T^{8} + 455 T^{6} + \cdots + 58369600 \) Copy content Toggle raw display
$79$ \( T^{8} + 319 T^{6} + \cdots + 7795264 \) Copy content Toggle raw display
$83$ \( (T^{4} - 15 T^{3} + \cdots - 112)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 157 T^{6} + \cdots + 71824 \) Copy content Toggle raw display
$97$ \( (T^{4} + 23 T^{3} + \cdots - 142)^{2} \) Copy content Toggle raw display
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