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Properties

Label 650.6.b.l
Level $650$
Weight $6$
Character orbit 650.b
Analytic conductor $104.249$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 2910x^{5} + 109404x^{4} + 595404x^{3} + 977202x^{2} - 23402520x + 280227600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 130)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta_1 q^{2} + ( - \beta_{2} - 5 \beta_1) q^{3} - 16 q^{4} + ( - 4 \beta_{3} + 20) q^{6} + ( - \beta_{4} + 15 \beta_1) q^{7} - 64 \beta_1 q^{8} + (\beta_{7} + \beta_{5} + 2 \beta_{3} - 163) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_1 q^{2} + ( - \beta_{2} - 5 \beta_1) q^{3} - 16 q^{4} + ( - 4 \beta_{3} + 20) q^{6} + ( - \beta_{4} + 15 \beta_1) q^{7} - 64 \beta_1 q^{8} + (\beta_{7} + \beta_{5} + 2 \beta_{3} - 163) q^{9} + (2 \beta_{7} - \beta_{5} - 11 \beta_{3} - 2) q^{11} + (16 \beta_{2} + 80 \beta_1) q^{12} - 169 \beta_1 q^{13} + ( - 4 \beta_{5} - 60) q^{14} + 256 q^{16} + ( - 2 \beta_{6} + 8 \beta_{4} + \cdots + 426 \beta_1) q^{17}+ \cdots + ( - 92 \beta_{7} + 649 \beta_{5} + \cdots + 62132) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{4} + 160 q^{6} - 1304 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{4} + 160 q^{6} - 1304 q^{9} - 28 q^{11} - 496 q^{14} + 2048 q^{16} - 5908 q^{19} + 2064 q^{21} - 2560 q^{24} + 5408 q^{26} - 10312 q^{29} - 11284 q^{31} - 13472 q^{34} + 20864 q^{36} - 6760 q^{39} + 14944 q^{41} + 448 q^{44} + 28224 q^{46} + 288 q^{49} + 11520 q^{51} - 5536 q^{54} + 7936 q^{56} + 27860 q^{59} + 8896 q^{61} - 32768 q^{64} + 128320 q^{66} + 182176 q^{69} + 29300 q^{71} - 124736 q^{74} + 94528 q^{76} + 195000 q^{79} - 135784 q^{81} - 33024 q^{84} - 64224 q^{86} + 173904 q^{89} + 20956 q^{91} + 48784 q^{94} + 40960 q^{96} + 500020 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 2910x^{5} + 109404x^{4} + 595404x^{3} + 977202x^{2} - 23402520x + 280227600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 269490001 \nu^{7} + 6578603516 \nu^{6} - 42790834912 \nu^{5} + 933356122950 \nu^{4} + \cdots - 41\!\cdots\!60 ) / 51\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7387073519 \nu^{7} + 51525358436 \nu^{6} - 191931054952 \nu^{5} - 20330793012750 \nu^{4} + \cdots + 71\!\cdots\!40 ) / 51\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1372144 \nu^{7} + 8054974 \nu^{6} - 26727638 \nu^{5} - 3810779415 \nu^{4} + \cdots + 4372047291483 ) / 9161740558737 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 316920964423 \nu^{7} - 8115545516272 \nu^{6} + 88041658790624 \nu^{5} + \cdots - 59\!\cdots\!80 ) / 51\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69912954 \nu^{7} - 362248573 \nu^{6} - 7934099690 \nu^{5} + 407688266416 \nu^{4} + \cdots + 27\!\cdots\!49 ) / 9161740558737 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 98747375999 \nu^{7} + 1223554511504 \nu^{6} - 14722974080752 \nu^{5} + \cdots + 57\!\cdots\!40 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 88767190 \nu^{7} + 474299285 \nu^{6} + 7302504060 \nu^{5} - 410558777638 \nu^{4} + \cdots - 25008270063885 ) / 9161740558737 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} - 10\beta_{2} + 382\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{7} + 11\beta_{6} + 8\beta_{5} + 8\beta_{4} - 308\beta_{3} - 308\beta_{2} + 2189\beta _1 - 2189 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 211\beta_{7} + 205\beta_{5} - 3205\beta_{3} - 60535 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5701 \beta_{7} - 5701 \beta_{6} + 4543 \beta_{5} - 4543 \beta_{4} - 119494 \beta_{3} + 119494 \beta_{2} + \cdots - 1319860 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -90479\beta_{6} - 83798\beta_{4} + 1582070\beta_{2} - 23854721\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1313866 \beta_{7} - 1313866 \beta_{6} - 1102558 \beta_{5} - 1102558 \beta_{4} + 25410841 \beta_{3} + \cdots + 321155017 \) 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
14.9519 − 14.9519i
3.93565 − 3.93565i
−8.02106 + 8.02106i
−8.86647 + 8.86647i
−8.86647 − 8.86647i
−8.02106 − 8.02106i
3.93565 + 3.93565i
14.9519 + 14.9519i
− 4.00000i − 23.9038i −16.0000 0 −95.6151 43.6771i 64.0000i −328.390 0
599.2 − 4.00000i − 1.87130i −16.0000 0 −7.48522 − 86.5977i 64.0000i 239.498 0
599.3 − 4.00000i 22.0421i −16.0000 0 88.1685 − 179.091i 64.0000i −242.855 0
599.4 − 4.00000i 23.7329i −16.0000 0 94.9318 160.011i 64.0000i −320.253 0
599.5 4.00000i − 23.7329i −16.0000 0 94.9318 − 160.011i − 64.0000i −320.253 0
599.6 4.00000i − 22.0421i −16.0000 0 88.1685 179.091i − 64.0000i −242.855 0
599.7 4.00000i 1.87130i −16.0000 0 −7.48522 86.5977i − 64.0000i 239.498 0
599.8 4.00000i 23.9038i −16.0000 0 −95.6151 − 43.6771i − 64.0000i −328.390 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.8
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.l 8
5.b even 2 1 inner 650.6.b.l 8
5.c odd 4 1 130.6.a.h ✓ 4
5.c odd 4 1 650.6.a.k 4
20.e even 4 1 1040.6.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.6.a.h ✓ 4 5.c odd 4 1
650.6.a.k 4 5.c odd 4 1
650.6.b.l 8 1.a even 1 1 trivial
650.6.b.l 8 5.b even 2 1 inner
1040.6.a.m 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1624T_{3}^{6} + 878784T_{3}^{4} + 159423696T_{3}^{2} + 547560000 \) acting on \(S_{6}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 1624 T^{6} + \cdots + 547560000 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + 14 T^{3} + \cdots + 55844170200)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 1287459140200)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 149094276505200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 14\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 717263645799600)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 67\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 20\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 47\!\cdots\!64 \) Copy content Toggle raw display
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