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Properties

Label 1008.2.r.n
Level $1008$
Weight $2$
Character orbit 1008.r
Analytic conductor $8.049$
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] = [1008,2,Mod(337,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.6095158642368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 6x^{8} - 7x^{7} + 25x^{6} - 66x^{5} + 75x^{4} - 63x^{3} + 162x^{2} - 324x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 504)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{7} - \beta_{2}) q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{7} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + ( - \beta_{7} - \beta_{2}) q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{7} + \beta_{3}) q^{9} + (\beta_{8} - \beta_{7} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{9} - \beta_{7} + \cdots - \beta_1) q^{13}+ \cdots + (2 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{5} - 5 q^{7} - 4 q^{11} - 3 q^{13} - 15 q^{15} - 2 q^{19} - 8 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} + 3 q^{31} + 30 q^{33} + 6 q^{35} - 6 q^{37} + 18 q^{39} - 12 q^{41} + 5 q^{43} - 9 q^{45} - 3 q^{47} - 5 q^{49} - 9 q^{51} + 60 q^{53} - 44 q^{55} - 21 q^{57} - 7 q^{59} - 14 q^{61} - 6 q^{63} - 11 q^{65} + 8 q^{67} + 21 q^{69} + 18 q^{71} + 30 q^{73} + 51 q^{75} - 4 q^{77} + 3 q^{79} - 12 q^{81} - 20 q^{83} - 21 q^{85} - 9 q^{87} - 24 q^{89} + 6 q^{91} - 39 q^{93} + 12 q^{95} - 37 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 6x^{8} - 7x^{7} + 25x^{6} - 66x^{5} + 75x^{4} - 63x^{3} + 162x^{2} - 324x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - \nu^{8} + 3\nu^{7} + 2\nu^{6} + 31\nu^{5} + 27\nu^{4} + 156\nu^{3} - 243\nu^{2} + 81\nu - 81 ) / 648 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{9} + 3\nu^{8} - 4\nu^{7} + 5\nu^{6} - 33\nu^{5} + 64\nu^{4} - 39\nu^{3} + 57\nu^{2} - 180\nu + 405 ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{9} - 59 \nu^{8} + 45 \nu^{7} - 86 \nu^{6} + 461 \nu^{5} - 939 \nu^{4} + 600 \nu^{3} + \cdots - 3807 ) / 648 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} - 11\nu^{8} + 3\nu^{7} - 20\nu^{6} + 101\nu^{5} - 141\nu^{4} + 54\nu^{3} - 201\nu^{2} + 657\nu - 495 ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 55 \nu^{9} - 127 \nu^{8} + 129 \nu^{7} - 214 \nu^{6} + 1129 \nu^{5} - 1935 \nu^{4} + 912 \nu^{3} + \cdots - 8019 ) / 648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29 \nu^{9} - 74 \nu^{8} + 51 \nu^{7} - 77 \nu^{6} + 566 \nu^{5} - 1017 \nu^{4} + 393 \nu^{3} + \cdots - 4050 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71 \nu^{9} - 155 \nu^{8} + 153 \nu^{7} - 290 \nu^{6} + 1277 \nu^{5} - 2271 \nu^{4} + 1428 \nu^{3} + \cdots - 9639 ) / 648 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25 \nu^{9} - 49 \nu^{8} + 27 \nu^{7} - 94 \nu^{6} + 439 \nu^{5} - 645 \nu^{4} + 300 \nu^{3} + \cdots - 2457 ) / 216 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 89 \nu^{9} + 167 \nu^{8} - 111 \nu^{7} + 308 \nu^{6} - 1577 \nu^{5} + 2265 \nu^{4} - 798 \nu^{3} + \cdots + 8667 ) / 648 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{8} + \beta_{7} + 2\beta_{4} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} - 2\beta_{8} + \beta_{7} - 3\beta_{5} - \beta_{4} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + \beta_{7} - 3\beta_{5} - \beta_{4} + 2\beta_{3} - 3\beta_{2} + 6\beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{9} - 8\beta_{8} + 4\beta_{7} - 3\beta_{5} + 2\beta_{4} - \beta_{3} + 12\beta_{2} + 12\beta _1 - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{9} - 2\beta_{8} - 8\beta_{7} - 3\beta_{6} - 6\beta_{5} - 13\beta_{4} + 17\beta_{3} + 18\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{9} - 2 \beta_{8} - 11 \beta_{7} + 15 \beta_{6} + 3 \beta_{5} - \beta_{4} - 7 \beta_{3} + \cdots + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21 \beta_{9} - 59 \beta_{8} + 25 \beta_{7} - 3 \beta_{6} + 21 \beta_{5} + 14 \beta_{4} - 25 \beta_{3} + \cdots - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 42 \beta_{9} - 29 \beta_{8} - 17 \beta_{7} - 42 \beta_{6} - 9 \beta_{5} + 2 \beta_{4} + 32 \beta_{3} + \cdots + 216 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6 \beta_{9} - 50 \beta_{8} + 76 \beta_{7} + 30 \beta_{6} - 72 \beta_{5} + 62 \beta_{4} - 124 \beta_{3} + \cdots + 177 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.11541 + 1.32509i
1.34147 − 1.09565i
−0.902451 − 1.47837i
1.72987 + 0.0867982i
−1.28430 + 1.16214i
1.11541 − 1.32509i
1.34147 + 1.09565i
−0.902451 + 1.47837i
1.72987 − 0.0867982i
−1.28430 − 1.16214i
0 −1.62713 + 0.593666i 0 1.50470 − 2.60622i 0 −0.500000 − 0.866025i 0 2.29512 − 1.93195i 0
337.2 0 −0.742385 + 1.56488i 0 −1.76217 + 3.05216i 0 −0.500000 − 0.866025i 0 −1.89773 − 2.32349i 0
337.3 0 −0.468714 − 1.66743i 0 −0.553143 + 0.958072i 0 −0.500000 − 0.866025i 0 −2.56061 + 1.56309i 0
337.4 0 1.25506 − 1.19366i 0 0.846320 − 1.46587i 0 −0.500000 − 0.866025i 0 0.150339 − 2.99623i 0
337.5 0 1.58317 + 0.702538i 0 −1.53571 + 2.65993i 0 −0.500000 − 0.866025i 0 2.01288 + 2.22448i 0
673.1 0 −1.62713 − 0.593666i 0 1.50470 + 2.60622i 0 −0.500000 + 0.866025i 0 2.29512 + 1.93195i 0
673.2 0 −0.742385 − 1.56488i 0 −1.76217 − 3.05216i 0 −0.500000 + 0.866025i 0 −1.89773 + 2.32349i 0
673.3 0 −0.468714 + 1.66743i 0 −0.553143 − 0.958072i 0 −0.500000 + 0.866025i 0 −2.56061 − 1.56309i 0
673.4 0 1.25506 + 1.19366i 0 0.846320 + 1.46587i 0 −0.500000 + 0.866025i 0 0.150339 + 2.99623i 0
673.5 0 1.58317 − 0.702538i 0 −1.53571 − 2.65993i 0 −0.500000 + 0.866025i 0 2.01288 − 2.22448i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.n 10
3.b odd 2 1 3024.2.r.n 10
4.b odd 2 1 504.2.r.f ✓ 10
9.c even 3 1 inner 1008.2.r.n 10
9.c even 3 1 9072.2.a.cn 5
9.d odd 6 1 3024.2.r.n 10
9.d odd 6 1 9072.2.a.cm 5
12.b even 2 1 1512.2.r.f 10
36.f odd 6 1 504.2.r.f ✓ 10
36.f odd 6 1 4536.2.a.bd 5
36.h even 6 1 1512.2.r.f 10
36.h even 6 1 4536.2.a.bc 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.f ✓ 10 4.b odd 2 1
504.2.r.f ✓ 10 36.f odd 6 1
1008.2.r.n 10 1.a even 1 1 trivial
1008.2.r.n 10 9.c even 3 1 inner
1512.2.r.f 10 12.b even 2 1
1512.2.r.f 10 36.h even 6 1
3024.2.r.n 10 3.b odd 2 1
3024.2.r.n 10 9.d odd 6 1
4536.2.a.bc 5 36.h even 6 1
4536.2.a.bd 5 36.f odd 6 1
9072.2.a.cm 5 9.d odd 6 1
9072.2.a.cn 5 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{10} + 3 T_{5}^{9} + 22 T_{5}^{8} + 29 T_{5}^{7} + 235 T_{5}^{6} + 287 T_{5}^{5} + 1441 T_{5}^{4} + \cdots + 3721 \) Copy content Toggle raw display
\( T_{11}^{10} + 4 T_{11}^{9} + 49 T_{11}^{8} + 142 T_{11}^{7} + 1621 T_{11}^{6} + 4501 T_{11}^{5} + \cdots + 11664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{7} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots + 3721 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 11664 \) Copy content Toggle raw display
$13$ \( T^{10} + 3 T^{9} + \cdots + 18496 \) Copy content Toggle raw display
$17$ \( (T^{5} - 39 T^{3} + \cdots + 108)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + T^{4} - 53 T^{3} + \cdots - 577)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 8 T^{9} + \cdots + 1846881 \) Copy content Toggle raw display
$29$ \( T^{10} + 9 T^{9} + \cdots + 104976 \) Copy content Toggle raw display
$31$ \( T^{10} - 3 T^{9} + \cdots + 29584 \) Copy content Toggle raw display
$37$ \( (T^{5} + 3 T^{4} + \cdots + 4948)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 12 T^{9} + \cdots + 22733824 \) Copy content Toggle raw display
$43$ \( T^{10} - 5 T^{9} + \cdots + 36144144 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 197346304 \) Copy content Toggle raw display
$53$ \( (T^{5} - 30 T^{4} + \cdots - 324)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 7 T^{9} + \cdots + 51322896 \) Copy content Toggle raw display
$61$ \( T^{10} + 14 T^{9} + \cdots + 2152089 \) Copy content Toggle raw display
$67$ \( T^{10} - 8 T^{9} + \cdots + 75898944 \) Copy content Toggle raw display
$71$ \( (T^{5} - 9 T^{4} + \cdots - 3079)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 15 T^{4} + \cdots - 5924)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 3 T^{9} + \cdots + 2271049 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 8283184144 \) Copy content Toggle raw display
$89$ \( (T^{5} + 12 T^{4} + \cdots + 43416)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1464439824 \) Copy content Toggle raw display
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