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Properties

Label 650.2.o.g
Level $650$
Weight $2$
Character orbit 650.o
Analytic conductor $5.190$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(399,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.399");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.o (of order \(6\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{5} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{4}) q^{6} + \beta_1 q^{7} - \beta_{3} q^{8} + ( - 7 \beta_{2} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{5} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{4}) q^{6} + \beta_1 q^{7} - \beta_{3} q^{8} + ( - 7 \beta_{2} + 7) q^{9} + (\beta_{4} + \beta_{2}) q^{11} + ( - \beta_{6} + \beta_{5}) q^{12} + ( - \beta_{6} + 2 \beta_{3} - \beta_1) q^{13} + q^{14} - \beta_{2} q^{16} + ( - \beta_{6} + 2 \beta_1) q^{17} - 7 \beta_{3} q^{18} + ( - \beta_{7} + \beta_{4} + 5 \beta_{2} - 5) q^{19} + \beta_{7} q^{21} + (\beta_{6} + \beta_1) q^{22} + (6 \beta_{3} - 6 \beta_1) q^{23} - \beta_{4} q^{24} + ( - \beta_{7} + 2 \beta_{2} - 1) q^{26} + ( - 4 \beta_{6} + 4 \beta_{5}) q^{27} + ( - \beta_{3} + \beta_1) q^{28} + ( - 2 \beta_{4} + 4 \beta_{2}) q^{29} + ( - \beta_{7} + 4) q^{31} - \beta_1 q^{32} + (\beta_{6} + 10 \beta_1) q^{33} + ( - \beta_{7} + 2) q^{34} - 7 \beta_{2} q^{36} + (\beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{37} + (\beta_{6} - \beta_{5} + 5 \beta_{3}) q^{38} + ( - \beta_{7} + 2 \beta_{4} - 10) q^{39} + ( - 2 \beta_{4} + 4 \beta_{2}) q^{41} + \beta_{5} q^{42} + 2 \beta_1 q^{43} + (\beta_{7} + 1) q^{44} + (6 \beta_{2} - 6) q^{46} - 3 \beta_{3} q^{47} - \beta_{6} q^{48} - 6 \beta_{2} q^{49} + (2 \beta_{7} - 10) q^{51} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{52} + ( - \beta_{6} + \beta_{5} - \beta_{3}) q^{53} - 4 \beta_{4} q^{54} + ( - \beta_{2} + 1) q^{56} + (5 \beta_{6} - 5 \beta_{5} + 10 \beta_{3}) q^{57} + ( - 2 \beta_{6} + 4 \beta_1) q^{58} + (2 \beta_{7} - 2 \beta_{4} + 4 \beta_{2} - 4) q^{59} + ( - 3 \beta_{7} + 3 \beta_{4} + \cdots - 2) q^{61}+ \cdots + (7 \beta_{7} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 28 q^{9} + 4 q^{11} + 8 q^{14} - 4 q^{16} - 20 q^{19} + 16 q^{29} + 32 q^{31} + 16 q^{34} - 28 q^{36} - 80 q^{39} + 16 q^{41} + 8 q^{44} - 24 q^{46} - 24 q^{49} - 80 q^{51} + 4 q^{56} - 16 q^{59} - 8 q^{61} - 8 q^{64} + 80 q^{66} - 8 q^{71} + 12 q^{74} + 20 q^{76} + 32 q^{79} - 76 q^{81} + 16 q^{86} - 4 q^{89} - 12 q^{91} - 12 q^{94} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 125\nu ) / 125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 125\nu ) / 125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} + 25\nu^{3} ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{5} + 5\nu^{3} + 25\nu ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 5\beta_{6} - 5\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{7} + 25\beta_{6} + 25\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -125\beta_{5} + 125\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
−0.578737 − 2.15988i
0.578737 + 2.15988i
−2.15988 + 0.578737i
2.15988 − 0.578737i
−0.578737 + 2.15988i
0.578737 − 2.15988i
−2.15988 − 0.578737i
2.15988 + 0.578737i
−0.866025 − 0.500000i −2.73861 − 1.58114i 0.500000 + 0.866025i 0 1.58114 + 2.73861i −0.866025 + 0.500000i − 1.00000i 3.50000 + 6.06218i 0
399.2 −0.866025 − 0.500000i 2.73861 + 1.58114i 0.500000 + 0.866025i 0 −1.58114 − 2.73861i −0.866025 + 0.500000i − 1.00000i 3.50000 + 6.06218i 0
399.3 0.866025 + 0.500000i −2.73861 − 1.58114i 0.500000 + 0.866025i 0 −1.58114 − 2.73861i 0.866025 − 0.500000i 1.00000i 3.50000 + 6.06218i 0
399.4 0.866025 + 0.500000i 2.73861 + 1.58114i 0.500000 + 0.866025i 0 1.58114 + 2.73861i 0.866025 − 0.500000i 1.00000i 3.50000 + 6.06218i 0
549.1 −0.866025 + 0.500000i −2.73861 + 1.58114i 0.500000 − 0.866025i 0 1.58114 − 2.73861i −0.866025 − 0.500000i 1.00000i 3.50000 − 6.06218i 0
549.2 −0.866025 + 0.500000i 2.73861 − 1.58114i 0.500000 − 0.866025i 0 −1.58114 + 2.73861i −0.866025 − 0.500000i 1.00000i 3.50000 − 6.06218i 0
549.3 0.866025 − 0.500000i −2.73861 + 1.58114i 0.500000 − 0.866025i 0 −1.58114 + 2.73861i 0.866025 + 0.500000i − 1.00000i 3.50000 − 6.06218i 0
549.4 0.866025 − 0.500000i 2.73861 − 1.58114i 0.500000 − 0.866025i 0 1.58114 − 2.73861i 0.866025 + 0.500000i − 1.00000i 3.50000 − 6.06218i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 399.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.o.g 8
5.b even 2 1 inner 650.2.o.g 8
5.c odd 4 1 130.2.e.c ✓ 4
5.c odd 4 1 650.2.e.h 4
13.c even 3 1 inner 650.2.o.g 8
15.e even 4 1 1170.2.i.q 4
20.e even 4 1 1040.2.q.m 4
65.f even 4 1 1690.2.l.k 8
65.h odd 4 1 1690.2.e.m 4
65.k even 4 1 1690.2.l.k 8
65.n even 6 1 inner 650.2.o.g 8
65.o even 12 1 1690.2.d.g 4
65.o even 12 1 1690.2.l.k 8
65.q odd 12 1 130.2.e.c ✓ 4
65.q odd 12 1 650.2.e.h 4
65.q odd 12 1 1690.2.a.n 2
65.q odd 12 1 8450.2.a.bc 2
65.r odd 12 1 1690.2.a.k 2
65.r odd 12 1 1690.2.e.m 4
65.r odd 12 1 8450.2.a.bj 2
65.t even 12 1 1690.2.d.g 4
65.t even 12 1 1690.2.l.k 8
195.bl even 12 1 1170.2.i.q 4
260.bj even 12 1 1040.2.q.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.c ✓ 4 5.c odd 4 1
130.2.e.c ✓ 4 65.q odd 12 1
650.2.e.h 4 5.c odd 4 1
650.2.e.h 4 65.q odd 12 1
650.2.o.g 8 1.a even 1 1 trivial
650.2.o.g 8 5.b even 2 1 inner
650.2.o.g 8 13.c even 3 1 inner
650.2.o.g 8 65.n even 6 1 inner
1040.2.q.m 4 20.e even 4 1
1040.2.q.m 4 260.bj even 12 1
1170.2.i.q 4 15.e even 4 1
1170.2.i.q 4 195.bl even 12 1
1690.2.a.k 2 65.r odd 12 1
1690.2.a.n 2 65.q odd 12 1
1690.2.d.g 4 65.o even 12 1
1690.2.d.g 4 65.t even 12 1
1690.2.e.m 4 65.h odd 4 1
1690.2.e.m 4 65.r odd 12 1
1690.2.l.k 8 65.f even 4 1
1690.2.l.k 8 65.k even 4 1
1690.2.l.k 8 65.o even 12 1
1690.2.l.k 8 65.t even 12 1
8450.2.a.bc 2 65.q odd 12 1
8450.2.a.bj 2 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + 13 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 28 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{3} + \cdots + 225)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 38 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 22 T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 7396)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 112 T^{6} + \cdots + 331776 \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 92 T^{2} + 676)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 74)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 188 T^{6} + \cdots + 54700816 \) Copy content Toggle raw display
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