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Properties

Label 2009.4.a.c
Level $2009$
Weight $4$
Character orbit 2009.a
Self dual yes
Analytic conductor $118.535$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2009,4,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2009.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

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: yes
Analytic conductor: \(118.534837202\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 49x^{5} + 33x^{4} + 720x^{3} - 320x^{2} - 3200x + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 41)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{4} - 1) q^{3} + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots + 6) q^{4}+ \cdots + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{4} - 1) q^{3} + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots + 6) q^{4}+ \cdots + ( - 9 \beta_{6} - 24 \beta_{5} + \cdots + 145) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 4 q^{3} + 43 q^{4} - 10 q^{5} - 18 q^{6} + 33 q^{8} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} - 4 q^{3} + 43 q^{4} - 10 q^{5} - 18 q^{6} + 33 q^{8} + 99 q^{9} - 82 q^{10} + 34 q^{11} + 218 q^{12} + 60 q^{13} + 46 q^{15} + 59 q^{16} + 82 q^{17} - 563 q^{18} - 144 q^{19} + 410 q^{20} + 82 q^{22} + 204 q^{23} + 86 q^{24} + 169 q^{25} + 292 q^{26} - 46 q^{27} + 68 q^{29} - 444 q^{30} - 696 q^{31} + 625 q^{32} - 124 q^{33} - 182 q^{34} + 831 q^{36} + 730 q^{37} - 34 q^{38} + 848 q^{39} - 778 q^{40} + 287 q^{41} + 368 q^{43} + 230 q^{44} + 1510 q^{45} + 432 q^{46} + 26 q^{47} + 1770 q^{48} - 1137 q^{50} + 424 q^{51} + 1308 q^{52} - 892 q^{53} - 1340 q^{54} - 358 q^{55} - 3788 q^{57} + 1416 q^{58} + 916 q^{59} + 220 q^{60} + 450 q^{61} - 208 q^{62} + 939 q^{64} - 1092 q^{65} + 576 q^{66} - 142 q^{67} - 226 q^{68} + 3396 q^{69} + 390 q^{71} - 2875 q^{72} - 882 q^{73} + 1822 q^{74} + 292 q^{75} - 4278 q^{76} + 232 q^{78} + 2890 q^{79} - 38 q^{80} + 951 q^{81} + 41 q^{82} - 1368 q^{83} - 1740 q^{85} - 132 q^{86} + 812 q^{87} + 4506 q^{88} + 2006 q^{89} - 5746 q^{90} + 4112 q^{92} + 2508 q^{93} - 104 q^{94} - 1002 q^{95} + 1782 q^{96} + 1950 q^{97} + 892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 49x^{5} + 33x^{4} + 720x^{3} - 320x^{2} - 3200x + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 33\nu^{2} + 176 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + \nu^{5} + 37\nu^{4} - 17\nu^{3} - 260\nu^{2} - 16\nu - 448 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 37\nu^{3} + 17\nu^{2} + 292\nu - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 3\nu^{5} - 45\nu^{4} - 115\nu^{3} + 460\nu^{2} + 848\nu - 448 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 3\nu^{5} - 41\nu^{4} - 131\nu^{3} + 392\nu^{2} + 1120\nu - 576 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} - 4\beta_{4} + 4\beta_{3} + 4\beta_{2} + 21\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 33\beta_{6} - 33\beta_{4} + 33\beta_{3} + 16\beta_{2} + 33\beta _1 + 286 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{6} + 148\beta_{5} - 148\beta_{4} + 164\beta_{3} + 164\beta_{2} + 501\beta _1 + 228 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 977\beta_{6} + 80\beta_{5} - 1041\beta_{4} + 993\beta_{3} + 688\beta_{2} + 1089\beta _1 + 6654 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.84699
−3.89242
−2.69942
0.158390
3.23284
3.61196
5.43564
−4.84699 9.64203 15.4934 14.2831 −46.7349 0 −36.3203 65.9687 −69.2303
1.2 −3.89242 −8.44968 7.15096 10.0474 32.8897 0 3.30483 44.3971 −39.1088
1.3 −2.69942 3.27713 −0.713139 −15.5245 −8.84635 0 23.5204 −16.2604 41.9070
1.4 0.158390 −7.99110 −7.97491 −14.3619 −1.26571 0 −2.53027 36.8576 −2.27479
1.5 3.23284 0.631321 2.45129 −13.5140 2.04096 0 −17.9381 −26.6014 −43.6887
1.6 3.61196 −5.45499 5.04626 10.3662 −19.7032 0 −10.6688 2.75692 37.4424
1.7 5.43564 4.34529 21.5462 −1.29640 23.6194 0 73.6322 −8.11846 −7.04677
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(41\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2009.4.a.c 7
7.b odd 2 1 41.4.a.b ✓ 7
21.c even 2 1 369.4.a.h 7
28.d even 2 1 656.4.a.j 7
35.c odd 2 1 1025.4.a.d 7
287.d odd 2 1 1681.4.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.4.a.b ✓ 7 7.b odd 2 1
369.4.a.h 7 21.c even 2 1
656.4.a.j 7 28.d even 2 1
1025.4.a.d 7 35.c odd 2 1
1681.4.a.c 7 287.d odd 2 1
2009.4.a.c 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{7} - T_{2}^{6} - 49T_{2}^{5} + 33T_{2}^{4} + 720T_{2}^{3} - 320T_{2}^{2} - 3200T_{2} + 512 \) Copy content Toggle raw display
\( T_{3}^{7} + 4T_{3}^{6} - 136T_{3}^{5} - 478T_{3}^{4} + 4782T_{3}^{3} + 7936T_{3}^{2} - 57348T_{3} + 31928 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( T^{7} + 4 T^{6} + \cdots + 31928 \) Copy content Toggle raw display
$5$ \( T^{7} + 10 T^{6} + \cdots - 5811008 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 1089373224 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 1344939417600 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 359921618304 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 10579181757624 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 5799910821888 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 503094100381696 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 298979937161216 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T - 41)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 43\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 10\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 28\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 94\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 58\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 27\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 32\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 36\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
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