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Properties

Label 1005.2.i.b
Level $1005$
Weight $2$
Character orbit 1005.i
Analytic conductor $8.025$
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] = [1005,2,Mod(766,1005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1005, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1005.766");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1005 = 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1005.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: \(8.02496540314\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.142510530627.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 7x^{8} - 2x^{7} + 15x^{6} - 5x^{5} + 20x^{4} + 2x^{3} + 8x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + ( - \beta_{9} + \beta_{7} + \cdots + 2 \beta_1) q^{4}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + ( - \beta_{9} + \beta_{7} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - 2 \beta_{9} - \beta_{8} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 10 q^{3} - 10 q^{5} - 2 q^{6} - 5 q^{7} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} - 10 q^{3} - 10 q^{5} - 2 q^{6} - 5 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{10} - 8 q^{11} + 5 q^{13} - 4 q^{14} + 10 q^{15} - 2 q^{16} + 2 q^{17} + 2 q^{18} - 8 q^{19} + 5 q^{21} - 14 q^{22} - 8 q^{23} + 12 q^{24} + 10 q^{25} - 8 q^{26} - 10 q^{27} + 2 q^{28} + 13 q^{29} + 2 q^{30} + 21 q^{31} - q^{32} + 8 q^{33} + 28 q^{34} + 5 q^{35} - 6 q^{37} + 16 q^{38} - 5 q^{39} + 12 q^{40} + 14 q^{41} + 4 q^{42} + 6 q^{43} - 19 q^{44} - 10 q^{45} + 16 q^{46} - 19 q^{47} + 2 q^{48} + 8 q^{49} + 2 q^{50} - 2 q^{51} - 28 q^{52} + 4 q^{53} - 2 q^{54} + 8 q^{55} + q^{56} + 8 q^{57} + 10 q^{58} - 2 q^{59} + 10 q^{61} - 30 q^{62} - 5 q^{63} + 12 q^{64} - 5 q^{65} + 14 q^{66} + q^{67} + 46 q^{68} + 8 q^{69} + 4 q^{70} - 30 q^{71} - 12 q^{72} - 5 q^{73} + 31 q^{74} - 10 q^{75} + 70 q^{76} + 7 q^{77} + 8 q^{78} + 11 q^{79} + 2 q^{80} + 10 q^{81} + 22 q^{82} - 3 q^{83} - 2 q^{84} - 2 q^{85} + 2 q^{86} - 13 q^{87} + 45 q^{88} + 68 q^{89} - 2 q^{90} + 42 q^{91} - 24 q^{92} - 21 q^{93} + 14 q^{94} + 8 q^{95} + q^{96} + 12 q^{97} - 13 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 7x^{8} - 2x^{7} + 15x^{6} - 5x^{5} + 20x^{4} + 2x^{3} + 8x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 742 \nu^{9} + 3360 \nu^{8} - 10416 \nu^{7} + 15194 \nu^{6} - 19152 \nu^{5} + 16128 \nu^{4} + \cdots - 14164 ) / 12391 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1287 \nu^{9} - 3490 \nu^{8} + 10819 \nu^{7} - 8185 \nu^{6} + 19893 \nu^{5} - 16752 \nu^{4} + \cdots - 4022 ) / 12391 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1876 \nu^{9} + 5222 \nu^{8} - 13710 \nu^{7} + 8022 \nu^{6} - 12418 \nu^{5} + 17631 \nu^{4} + \cdots - 742 ) / 12391 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3649 \nu^{9} + 9510 \nu^{8} - 29481 \nu^{7} + 21984 \nu^{6} - 54207 \nu^{5} + 45648 \nu^{4} + \cdots + 5492 ) / 12391 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4022 \nu^{9} + 6757 \nu^{8} - 24664 \nu^{7} - 2775 \nu^{6} - 52145 \nu^{5} + 217 \nu^{4} + \cdots + 5252 ) / 12391 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5252 \nu^{9} + 6482 \nu^{8} - 30007 \nu^{7} - 14160 \nu^{6} - 81555 \nu^{5} - 25885 \nu^{4} + \cdots - 12249 ) / 12391 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6234 \nu^{9} + 10695 \nu^{8} - 39350 \nu^{7} - 3303 \nu^{6} - 79548 \nu^{5} - 10619 \nu^{4} + \cdots - 10788 ) / 12391 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7139 \nu^{9} - 18300 \nu^{8} + 56730 \nu^{7} - 38942 \nu^{6} + 104310 \nu^{5} - 87840 \nu^{4} + \cdots - 37031 ) / 12391 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} - 2\beta_{5} + 6\beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} - 6\beta_{7} + 8\beta_{6} - 7\beta_{4} - 14\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{9} + 7 \beta_{8} - 14 \beta_{7} + 15 \beta_{6} + 16 \beta_{5} - 16 \beta_{4} - 36 \beta_{3} + \cdots - 36 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{9} + 43\beta_{5} - 87\beta_{3} - 16\beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -43\beta_{8} + 87\beta_{7} - 95\beta_{6} + 103\beta_{4} + 216\beta _1 + 95 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 216 \beta_{9} - 103 \beta_{8} + 216 \beta_{7} - 242 \beta_{6} - 259 \beta_{5} + 259 \beta_{4} + \cdots + 527 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -527\beta_{9} - 630\beta_{5} + 1296\beta_{3} + 259\beta_{2} - 579 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1005\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(202\) \(671\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
766.1
−0.667680 + 1.15645i
−0.357267 + 0.618804i
0.169664 − 0.293867i
0.630367 − 1.09183i
1.22491 − 2.12161i
−0.667680 − 1.15645i
−0.357267 − 0.618804i
0.169664 + 0.293867i
0.630367 + 1.09183i
1.22491 + 2.12161i
−0.667680 + 1.15645i −1.00000 0.108408 + 0.187768i −1.00000 0.667680 − 1.15645i −0.978089 − 1.69410i −2.96025 1.00000 0.667680 − 1.15645i
766.2 −0.357267 + 0.618804i −1.00000 0.744721 + 1.28990i −1.00000 0.357267 − 0.618804i 0.929704 + 1.61029i −2.49332 1.00000 0.357267 − 0.618804i
766.3 0.169664 − 0.293867i −1.00000 0.942428 + 1.63233i −1.00000 −0.169664 + 0.293867i −2.16524 − 3.75031i 1.31824 1.00000 −0.169664 + 0.293867i
766.4 0.630367 − 1.09183i −1.00000 0.205275 + 0.355546i −1.00000 −0.630367 + 1.09183i 0.172820 + 0.299333i 3.03906 1.00000 −0.630367 + 1.09183i
766.5 1.22491 − 2.12161i −1.00000 −2.00083 − 3.46554i −1.00000 −1.22491 + 2.12161i −0.459194 − 0.795348i −4.90374 1.00000 −1.22491 + 2.12161i
841.1 −0.667680 − 1.15645i −1.00000 0.108408 − 0.187768i −1.00000 0.667680 + 1.15645i −0.978089 + 1.69410i −2.96025 1.00000 0.667680 + 1.15645i
841.2 −0.357267 − 0.618804i −1.00000 0.744721 − 1.28990i −1.00000 0.357267 + 0.618804i 0.929704 − 1.61029i −2.49332 1.00000 0.357267 + 0.618804i
841.3 0.169664 + 0.293867i −1.00000 0.942428 − 1.63233i −1.00000 −0.169664 − 0.293867i −2.16524 + 3.75031i 1.31824 1.00000 −0.169664 − 0.293867i
841.4 0.630367 + 1.09183i −1.00000 0.205275 − 0.355546i −1.00000 −0.630367 − 1.09183i 0.172820 − 0.299333i 3.03906 1.00000 −0.630367 − 1.09183i
841.5 1.22491 + 2.12161i −1.00000 −2.00083 + 3.46554i −1.00000 −1.22491 − 2.12161i −0.459194 + 0.795348i −4.90374 1.00000 −1.22491 − 2.12161i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 766.5
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 1005.2.i.b ✓ 10
67.c even 3 1 inner 1005.2.i.b ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1005.2.i.b ✓ 10 1.a even 1 1 trivial
1005.2.i.b ✓ 10 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 2T_{2}^{9} + 7T_{2}^{8} - 2T_{2}^{7} + 15T_{2}^{6} - 5T_{2}^{5} + 20T_{2}^{4} + 2T_{2}^{3} + 8T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1005, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 5 T^{9} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{10} + 8 T^{9} + \cdots + 1681 \) Copy content Toggle raw display
$13$ \( T^{10} - 5 T^{9} + \cdots + 55225 \) Copy content Toggle raw display
$17$ \( T^{10} - 2 T^{9} + \cdots + 5329 \) Copy content Toggle raw display
$19$ \( T^{10} + 8 T^{9} + \cdots + 1868689 \) Copy content Toggle raw display
$23$ \( T^{10} + 8 T^{9} + \cdots + 1852321 \) Copy content Toggle raw display
$29$ \( T^{10} - 13 T^{9} + \cdots + 4020025 \) Copy content Toggle raw display
$31$ \( T^{10} - 21 T^{9} + \cdots + 1646089 \) Copy content Toggle raw display
$37$ \( T^{10} + 6 T^{9} + \cdots + 3690241 \) Copy content Toggle raw display
$41$ \( T^{10} - 14 T^{9} + \cdots + 6225025 \) Copy content Toggle raw display
$43$ \( (T^{5} - 3 T^{4} + \cdots + 691)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 458002801 \) Copy content Toggle raw display
$53$ \( (T^{5} - 2 T^{4} - 98 T^{3} + \cdots - 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + T^{4} - 100 T^{3} + \cdots - 347)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 106151809 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1350125107 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 403567921 \) Copy content Toggle raw display
$73$ \( T^{10} + 5 T^{9} + \cdots + 16736281 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 182367432025 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 295874401 \) Copy content Toggle raw display
$89$ \( (T^{5} - 34 T^{4} + \cdots - 6823)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 616975921 \) Copy content Toggle raw display
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