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Properties

Label 650.6.b.n
Level $650$
Weight $6$
Character orbit 650.b
Analytic conductor $104.249$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), 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: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 1622x^{8} + 821977x^{6} + 134814852x^{4} + 6402281472x^{2} + 3251736576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta_{6} q^{2} + \beta_1 q^{3} - 16 q^{4} - 4 \beta_{2} q^{6} + ( - \beta_{8} + 31 \beta_{6} + \beta_1) q^{7} - 64 \beta_{6} q^{8} + (\beta_{5} - \beta_{4} + 3 \beta_{2} - 83) q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 67) q^{11}+ \cdots + (73 \beta_{5} - 511 \beta_{4} + \cdots - 60353) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 160 q^{4} - 16 q^{6} - 814 q^{9} - 642 q^{11} - 1256 q^{14} + 2560 q^{16} - 2304 q^{19} - 2644 q^{21} + 256 q^{24} - 6760 q^{26} - 34642 q^{29} - 3010 q^{31} - 15640 q^{34} + 13024 q^{36} - 676 q^{39}+ \cdots - 610302 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 1622x^{8} + 821977x^{6} + 134814852x^{4} + 6402281472x^{2} + 3251736576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 163\nu^{8} + 276734\nu^{6} + 131322823\nu^{4} + 12717384588\nu^{2} + 28567997568 ) / 30804785568 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 45491\nu^{8} + 64108498\nu^{6} + 24269111915\nu^{4} + 1270990365132\nu^{2} - 54640541955456 ) / 308047855680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 685541 \nu^{8} - 1045763878 \nu^{6} - 471687654965 \nu^{4} - 56362995024372 \nu^{2} - 10\!\cdots\!44 ) / 4620717835200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 758891 \nu^{8} - 1170294178 \nu^{6} - 530782925315 \nu^{4} - 57465100253772 \nu^{2} + 452324878995456 ) / 4620717835200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 83497 \nu^{9} + 133882982 \nu^{7} + 66002533633 \nu^{5} + 10008543587652 \nu^{3} + 413705272943232 \nu ) / 292768682038272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6240013 \nu^{9} + 10397149934 \nu^{7} + 5503922541445 \nu^{5} + 975233780372196 \nu^{3} + 46\!\cdots\!52 \nu ) / 182980426273920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38170079 \nu^{9} + 62165491882 \nu^{7} + 31852292042135 \nu^{5} + \cdots + 28\!\cdots\!36 \nu ) / 914902131369600 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 168205559 \nu^{9} - 264035184922 \nu^{7} - 125950005013535 \nu^{5} + \cdots - 54\!\cdots\!56 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 3\beta_{2} - 326 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{9} + 8\beta_{8} - 9\beta_{7} + 1034\beta_{6} - 617\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -759\beta_{5} + 609\beta_{4} - 90\beta_{3} - 3971\beta_{2} + 199242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7067\beta_{9} - 4768\beta_{8} + 7461\beta_{7} - 1333570\beta_{6} + 422073\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 542447\beta_{5} - 400937\beta_{4} + 61434\beta_{3} + 3880531\beta_{2} - 136144282 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6247755\beta_{9} + 2313480\beta_{8} - 5194701\beta_{7} + 1289612130\beta_{6} - 297061417\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -387465583\beta_{5} + 268065553\beta_{4} - 31790322\beta_{3} - 3433983459\beta_{2} + 95877479834 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -5270703979\beta_{9} - 899479856\beta_{8} + 3510474021\beta_{7} - 1134102832418\beta_{6} + 211686013721\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
− 24.8846i
− 12.8951i
− 0.716545i
9.02921i
27.4670i
− 27.4670i
− 9.02921i
0.716545i
12.8951i
24.8846i
− 4.00000i − 24.8846i −16.0000 0 −99.5383 − 165.378i 64.0000i −376.242 0
599.2 − 4.00000i − 12.8951i −16.0000 0 −51.5803 48.5387i 64.0000i 76.7173 0
599.3 − 4.00000i − 0.716545i −16.0000 0 −2.86618 187.375i 64.0000i 242.487 0
599.4 − 4.00000i 9.02921i −16.0000 0 36.1168 − 228.688i 64.0000i 161.473 0
599.5 − 4.00000i 27.4670i −16.0000 0 109.868 1.15327i 64.0000i −511.435 0
599.6 4.00000i − 27.4670i −16.0000 0 109.868 − 1.15327i − 64.0000i −511.435 0
599.7 4.00000i − 9.02921i −16.0000 0 36.1168 228.688i − 64.0000i 161.473 0
599.8 4.00000i 0.716545i −16.0000 0 −2.86618 − 187.375i − 64.0000i 242.487 0
599.9 4.00000i 12.8951i −16.0000 0 −51.5803 − 48.5387i − 64.0000i 76.7173 0
599.10 4.00000i 24.8846i −16.0000 0 −99.5383 165.378i − 64.0000i −376.242 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.6.b.n 10
5.b even 2 1 inner 650.6.b.n 10
5.c odd 4 1 650.6.a.n ✓ 5
5.c odd 4 1 650.6.a.q yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.6.a.n ✓ 5 5.c odd 4 1
650.6.a.q yes 5 5.c odd 4 1
650.6.b.n 10 1.a even 1 1 trivial
650.6.b.n 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 1622T_{3}^{8} + 821977T_{3}^{6} + 134814852T_{3}^{4} + 6402281472T_{3}^{2} + 3251736576 \) acting on \(S_{6}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 3251736576 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 2112867422028)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{5} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 103203582703616)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 13\!\cdots\!43)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 80\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 68\!\cdots\!83)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 52\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
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