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Properties

Label 650.6.b.k
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: \( x^{8} + 1094x^{6} + 343849x^{4} + 27023076x^{2} + 498182400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta_{2} q^{2} + (5 \beta_{2} + \beta_1) q^{3} - 16 q^{4} + (4 \beta_{3} - 20) q^{6} + ( - 3 \beta_{7} - \beta_{5} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - 4 \beta_{6} + 5 \beta_{4} + \cdots - 58) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_{2} q^{2} + (5 \beta_{2} + \beta_1) q^{3} - 16 q^{4} + (4 \beta_{3} - 20) q^{6} + ( - 3 \beta_{7} - \beta_{5} + \cdots + 2 \beta_1) q^{7}+ \cdots + (1247 \beta_{6} - 537 \beta_{4} + \cdots - 10864) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{4} - 160 q^{6} - 444 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{4} - 160 q^{6} - 444 q^{9} - 588 q^{11} + 1424 q^{14} + 2048 q^{16} - 928 q^{19} - 2016 q^{21} + 2560 q^{24} + 5408 q^{26} + 9088 q^{29} + 11236 q^{31} + 14448 q^{34} + 7104 q^{36} + 6760 q^{39} + 26464 q^{41} + 9408 q^{44} + 13344 q^{46} - 2932 q^{49} + 3840 q^{51} + 18784 q^{54} - 22784 q^{56} + 89180 q^{59} + 19336 q^{61} - 32768 q^{64} - 25120 q^{66} + 22636 q^{69} + 2320 q^{71} + 89184 q^{74} + 14848 q^{76} + 211480 q^{79} - 110664 q^{81} + 32256 q^{84} - 165184 q^{86} + 145544 q^{89} - 60164 q^{91} + 70224 q^{94} - 40960 q^{96} - 89060 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1094x^{6} + 343849x^{4} + 27023076x^{2} + 498182400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 1094\nu^{5} - 321529\nu^{3} - 14814036\nu ) / 36024480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 547\nu^{2} + 22320 ) / 1614 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 941\nu^{4} - 213628\nu^{2} - 4068288 ) / 116208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 109\nu^{7} + 120641\nu^{5} + 39562276\nu^{3} + 3394554624\nu ) / 216146880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} - 5065\nu^{4} - 1381268\nu^{2} - 60450048 ) / 464832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\nu^{7} + 59759\nu^{5} + 15047434\nu^{3} + 452273256\nu ) / 54036720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -4\beta_{6} + 5\beta_{4} - 5\beta_{3} - 276 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + 60\beta_{5} + 1212\beta_{2} - 469\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2188\beta_{6} - 2735\beta_{4} + 4349\beta_{3} + 128652 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9711\beta_{7} - 39276\beta_{5} - 1108428\beta_{2} + 242293\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1204396\beta_{6} + 1389287\beta_{4} - 3024269\beta_{3} - 66168492 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9659247\beta_{7} + 23676204\beta_{5} + 786902604\beta_{2} - 129085477\beta_1 \) 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
− 20.3395i
− 8.83034i
5.18032i
23.9895i
− 23.9895i
− 5.18032i
8.83034i
20.3395i
− 4.00000i − 25.3395i −16.0000 0 −101.358 154.564i 64.0000i −399.089 0
599.2 − 4.00000i − 13.8303i −16.0000 0 −55.3213 − 120.695i 64.0000i 51.7218 0
599.3 − 4.00000i 0.180317i −16.0000 0 0.721270 − 27.5588i 64.0000i 242.967 0
599.4 − 4.00000i 18.9895i −16.0000 0 75.9579 171.689i 64.0000i −117.601 0
599.5 4.00000i − 18.9895i −16.0000 0 75.9579 − 171.689i − 64.0000i −117.601 0
599.6 4.00000i − 0.180317i −16.0000 0 0.721270 27.5588i − 64.0000i 242.967 0
599.7 4.00000i 13.8303i −16.0000 0 −55.3213 120.695i − 64.0000i 51.7218 0
599.8 4.00000i 25.3395i −16.0000 0 −101.358 − 154.564i − 64.0000i −399.089 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.k 8
5.b even 2 1 inner 650.6.b.k 8
5.c odd 4 1 650.6.a.l ✓ 4
5.c odd 4 1 650.6.a.m yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.6.a.l ✓ 4 5.c odd 4 1
650.6.a.m yes 4 5.c odd 4 1
650.6.b.k 8 1.a even 1 1 trivial
650.6.b.k 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1194T_{3}^{6} + 423369T_{3}^{4} + 44301856T_{3}^{2} + 1440000 \) 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
$3$ \( T^{8} + 1194 T^{6} + \cdots + 1440000 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + 294 T^{3} + \cdots - 443328300)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + 464 T^{3} + \cdots - 28772012160)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 38291708851565)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 9017382657420)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 352330273075200)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 69\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 27\!\cdots\!41)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 80\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
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