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Properties

Label 2010.2.i.d
Level $2010$
Weight $2$
Character orbit 2010.i
Analytic conductor $16.050$
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] = [2010,2,Mod(841,2010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2010, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2010.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2010.i (of order \(3\), degree \(2\), 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: \(16.0499308063\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.2480070301875.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 6x^{7} + 39x^{6} - 33x^{5} + 72x^{4} + 6x^{3} + 12x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + q^{3} + (\beta_{5} - 1) q^{4} + q^{5} - \beta_{5} q^{6} + ( - \beta_{6} - \beta_{5} - \beta_1 + 1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + q^{3} + (\beta_{5} - 1) q^{4} + q^{5} - \beta_{5} q^{6} + ( - \beta_{6} - \beta_{5} - \beta_1 + 1) q^{7} + q^{8} + q^{9} - \beta_{5} q^{10} + (\beta_{8} + \beta_{6} + \beta_{5} + \cdots - 1) q^{11}+ \cdots + (\beta_{8} + \beta_{6} + \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} + 10 q^{3} - 5 q^{4} + 10 q^{5} - 5 q^{6} + q^{7} + 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{2} + 10 q^{3} - 5 q^{4} + 10 q^{5} - 5 q^{6} + q^{7} + 10 q^{8} + 10 q^{9} - 5 q^{10} - q^{11} - 5 q^{12} - 2 q^{14} + 10 q^{15} - 5 q^{16} + 2 q^{17} - 5 q^{18} - q^{19} - 5 q^{20} + q^{21} + 2 q^{22} + q^{23} + 10 q^{24} + 10 q^{25} + 10 q^{27} + q^{28} - 4 q^{29} - 5 q^{30} - 6 q^{31} - 5 q^{32} - q^{33} + 2 q^{34} + q^{35} - 5 q^{36} - 2 q^{37} - q^{38} + 10 q^{40} - 2 q^{42} + 14 q^{43} - q^{44} + 10 q^{45} + q^{46} - 13 q^{47} - 5 q^{48} - 6 q^{49} - 5 q^{50} + 2 q^{51} + 36 q^{53} - 5 q^{54} - q^{55} + q^{56} - q^{57} + 8 q^{58} + 48 q^{59} - 5 q^{60} - q^{61} + 12 q^{62} + q^{63} + 10 q^{64} + 2 q^{66} + 20 q^{67} - 4 q^{68} + q^{69} - 2 q^{70} - 12 q^{71} + 10 q^{72} + 6 q^{73} - 2 q^{74} + 10 q^{75} + 2 q^{76} + 26 q^{77} + 17 q^{79} - 5 q^{80} + 10 q^{81} + q^{84} + 2 q^{85} - 7 q^{86} - 4 q^{87} - q^{88} + 10 q^{89} - 5 q^{90} - 30 q^{91} - 2 q^{92} - 6 q^{93} + 26 q^{94} - q^{95} - 5 q^{96} - 12 q^{97} - 6 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 9x^{8} - 6x^{7} + 39x^{6} - 33x^{5} + 72x^{4} + 6x^{3} + 12x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 145 \nu^{9} + 2385 \nu^{8} - 5889 \nu^{7} + 19970 \nu^{6} - 21495 \nu^{5} + 85005 \nu^{4} + \cdots + 3346 ) / 8097 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 651 \nu^{9} - 1494 \nu^{8} + 6225 \nu^{7} - 5431 \nu^{6} + 26145 \nu^{5} - 27888 \nu^{4} + \cdots - 1546 ) / 8097 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 843 \nu^{9} + 1860 \nu^{8} - 7750 \nu^{7} + 6187 \nu^{6} - 32550 \nu^{5} + 34720 \nu^{4} + \cdots + 17089 ) / 8097 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1546 \nu^{9} + 2441 \nu^{8} - 12420 \nu^{7} + 3051 \nu^{6} - 54863 \nu^{5} + 24873 \nu^{4} + \cdots + 2096 ) / 8097 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2096 \nu^{9} - 2646 \nu^{8} + 16423 \nu^{7} - 156 \nu^{6} + 78693 \nu^{5} - 14305 \nu^{4} + \cdots + 9131 ) / 8097 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2900 \nu^{9} + 4516 \nu^{8} - 23315 \nu^{7} + 5346 \nu^{6} - 103321 \nu^{5} + 42914 \nu^{4} + \cdots - 11351 ) / 8097 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2918 \nu^{9} - 4719 \nu^{8} + 23711 \nu^{7} - 6429 \nu^{6} + 102825 \nu^{5} - 47603 \nu^{4} + \cdots + 11159 ) / 8097 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6001 \nu^{9} + 13548 \nu^{8} - 56450 \nu^{7} + 48426 \nu^{6} - 237090 \nu^{5} + 252896 \nu^{4} + \cdots + 25325 ) / 8097 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{4} + 6\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} - 7\beta_{7} - \beta_{6} + 8\beta_{5} - 9\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 11\beta_{7} - 2\beta_{6} + \beta_{5} - 11\beta_{4} - 38\beta_{3} - 9\beta_{2} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} + 20\beta_{8} - 47\beta_{4} - 72\beta_{3} - 20\beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 67\beta_{8} + 92\beta_{7} + 20\beta_{6} - 19\beta_{5} + 251\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -67\beta_{9} + 318\beta_{7} + 67\beta_{6} - 204\beta_{5} + 318\beta_{4} + 546\beta_{3} + 159\beta_{2} + 546\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -159\beta_{9} - 477\beta_{8} + 705\beta_{4} + 1704\beta_{3} + 477\beta_{2} - 205 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times\).

\(n\) \(671\) \(1141\) \(1207\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
0.133467 − 0.231172i
1.33180 − 2.30675i
0.780042 − 1.35107i
−1.02554 + 1.77629i
−0.219769 + 0.380651i
0.133467 + 0.231172i
1.33180 + 2.30675i
0.780042 + 1.35107i
−1.02554 − 1.77629i
−0.219769 − 0.380651i
−0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −1.50659 + 2.60949i 1.00000 1.00000 −0.500000 − 0.866025i
841.2 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −1.01952 + 1.76585i 1.00000 1.00000 −0.500000 − 0.866025i
841.3 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −0.600537 + 1.04016i 1.00000 1.00000 −0.500000 − 0.866025i
841.4 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.76931 − 3.06454i 1.00000 1.00000 −0.500000 − 0.866025i
841.5 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.85733 − 3.21698i 1.00000 1.00000 −0.500000 − 0.866025i
1771.1 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −1.50659 − 2.60949i 1.00000 1.00000 −0.500000 + 0.866025i
1771.2 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −1.01952 − 1.76585i 1.00000 1.00000 −0.500000 + 0.866025i
1771.3 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −0.600537 − 1.04016i 1.00000 1.00000 −0.500000 + 0.866025i
1771.4 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.76931 + 3.06454i 1.00000 1.00000 −0.500000 + 0.866025i
1771.5 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.85733 + 3.21698i 1.00000 1.00000 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2010.2.i.d ✓ 10
67.c even 3 1 inner 2010.2.i.d ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2010.2.i.d ✓ 10 1.a even 1 1 trivial
2010.2.i.d ✓ 10 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - T_{7}^{9} + 21 T_{7}^{8} + 18 T_{7}^{7} + 294 T_{7}^{6} + 291 T_{7}^{5} + 2238 T_{7}^{4} + \cdots + 9409 \) acting on \(S_{2}^{\mathrm{new}}(2010, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - T^{9} + \cdots + 9409 \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots + 625 \) Copy content Toggle raw display
$13$ \( T^{10} + 18 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{10} - 2 T^{9} + \cdots + 1225 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 3613801 \) Copy content Toggle raw display
$23$ \( T^{10} - T^{9} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{10} + 4 T^{9} + \cdots + 112225 \) Copy content Toggle raw display
$31$ \( T^{10} + 6 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + \cdots + 21353641 \) Copy content Toggle raw display
$41$ \( T^{10} + 153 T^{8} + \cdots + 87890625 \) Copy content Toggle raw display
$43$ \( (T^{5} - 7 T^{4} + \cdots - 1025)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 13 T^{9} + \cdots + 3545689 \) Copy content Toggle raw display
$53$ \( (T^{5} - 18 T^{4} + \cdots + 207)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 24 T^{4} + \cdots + 75)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + T^{9} + \cdots + 606841 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1350125107 \) Copy content Toggle raw display
$71$ \( T^{10} + 12 T^{9} + \cdots + 7273809 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 3313268721 \) Copy content Toggle raw display
$79$ \( T^{10} - 17 T^{9} + \cdots + 4489 \) Copy content Toggle raw display
$83$ \( T^{10} + 75 T^{8} + \cdots + 168921 \) Copy content Toggle raw display
$89$ \( (T^{5} - 5 T^{4} + \cdots + 1835)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 12 T^{9} + \cdots + 37197801 \) Copy content Toggle raw display
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