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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(49,650)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.n (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: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{6} + \beta_{4}) q^{3} - \beta_{2} q^{4} + (\beta_{4} + \beta_{2} - 1) q^{6} + (\beta_{7} - 2 \beta_{6} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{6} + \beta_{4}) q^{3} - \beta_{2} q^{4} + (\beta_{4} + \beta_{2} - 1) q^{6} + (\beta_{7} - 2 \beta_{6} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - 4 \beta_{7} - 3 \beta_{6} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 6 q^{3} - 4 q^{4} - 6 q^{6} - 8 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 6 q^{3} - 4 q^{4} - 6 q^{6} - 8 q^{8} + 4 q^{9} - 6 q^{11} - 6 q^{13} - 4 q^{16} - 18 q^{17} + 8 q^{18} + 6 q^{19} - 6 q^{22} + 6 q^{24} + 4 q^{32} - 6 q^{33} + 4 q^{36} + 6 q^{37} + 40 q^{39} + 12 q^{41} - 30 q^{42} + 36 q^{43} + 6 q^{48} - 8 q^{49} + 6 q^{52} - 36 q^{54} + 12 q^{57} + 24 q^{59} + 26 q^{61} + 6 q^{62} - 24 q^{63} + 8 q^{64} - 12 q^{66} - 24 q^{67} + 18 q^{68} - 12 q^{69} - 4 q^{72} + 12 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{78} - 20 q^{79} - 28 q^{81} + 12 q^{82} - 36 q^{83} - 30 q^{84} + 24 q^{87} + 6 q^{88} + 24 q^{89} - 66 q^{91} + 12 q^{93} + 18 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} + 26\nu^{6} - 13\nu^{5} - 36\nu^{4} + 55\nu^{3} + 34\nu^{2} - 140\nu + 104 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{7} - 26\nu^{6} + 17\nu^{5} + 36\nu^{4} - 59\nu^{3} - 34\nu^{2} + 144\nu - 120 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 48\nu^{6} - 23\nu^{5} - 74\nu^{4} + 97\nu^{3} + 80\nu^{2} - 268\nu + 192 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 14\nu^{6} - 8\nu^{5} - 20\nu^{4} + 30\nu^{3} + 18\nu^{2} - 79\nu + 62 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 3\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 4\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{4} + 4\beta_{3} - 3\beta_{2} + \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{6} + 3\beta_{5} + 2\beta_{4} + 9\beta_{3} + 7\beta_{2} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} + 2\beta_{6} - 8\beta_{5} + 5\beta_{4} + 5\beta_{3} + 18\beta_{2} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.40994 − 0.109843i
1.20036 + 0.747754i
−1.27597 + 0.609843i
0.665665 − 1.24775i
1.40994 + 0.109843i
1.20036 − 0.747754i
−1.27597 − 0.609843i
0.665665 + 1.24775i
0.500000 + 0.866025i −2.88581 + 1.66612i −0.500000 + 0.866025i 0 −2.88581 − 1.66612i 0.719687 − 1.24653i −1.00000 4.05193 − 7.01815i 0
49.2 0.500000 + 0.866025i −1.58209 + 0.913419i −0.500000 + 0.866025i 0 −1.58209 − 0.913419i 1.99551 − 3.45632i −1.00000 0.168669 − 0.292144i 0
49.3 0.500000 + 0.866025i 0.519785 − 0.300098i −0.500000 + 0.866025i 0 0.519785 + 0.300098i −0.719687 + 1.24653i −1.00000 −1.31988 + 2.28610i 0
49.4 0.500000 + 0.866025i 0.948114 − 0.547394i −0.500000 + 0.866025i 0 0.948114 + 0.547394i −1.99551 + 3.45632i −1.00000 −0.900720 + 1.56009i 0
199.1 0.500000 − 0.866025i −2.88581 − 1.66612i −0.500000 − 0.866025i 0 −2.88581 + 1.66612i 0.719687 + 1.24653i −1.00000 4.05193 + 7.01815i 0
199.2 0.500000 − 0.866025i −1.58209 − 0.913419i −0.500000 − 0.866025i 0 −1.58209 + 0.913419i 1.99551 + 3.45632i −1.00000 0.168669 + 0.292144i 0
199.3 0.500000 − 0.866025i 0.519785 + 0.300098i −0.500000 − 0.866025i 0 0.519785 − 0.300098i −0.719687 − 1.24653i −1.00000 −1.31988 − 2.28610i 0
199.4 0.500000 − 0.866025i 0.948114 + 0.547394i −0.500000 − 0.866025i 0 0.948114 − 0.547394i −1.99551 − 3.45632i −1.00000 −0.900720 − 1.56009i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l 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.n.e 8
5.b even 2 1 650.2.n.d 8
5.c odd 4 1 130.2.l.b ✓ 8
5.c odd 4 1 650.2.m.c 8
13.e even 6 1 650.2.n.d 8
15.e even 4 1 1170.2.bs.g 8
20.e even 4 1 1040.2.da.d 8
65.f even 4 1 1690.2.e.s 8
65.h odd 4 1 1690.2.l.j 8
65.k even 4 1 1690.2.e.t 8
65.l even 6 1 inner 650.2.n.e 8
65.o even 12 1 1690.2.a.t 4
65.o even 12 1 1690.2.e.t 8
65.o even 12 1 8450.2.a.ci 4
65.q odd 12 1 1690.2.d.k 8
65.q odd 12 1 1690.2.l.j 8
65.r odd 12 1 130.2.l.b ✓ 8
65.r odd 12 1 650.2.m.c 8
65.r odd 12 1 1690.2.d.k 8
65.t even 12 1 1690.2.a.u 4
65.t even 12 1 1690.2.e.s 8
65.t even 12 1 8450.2.a.cm 4
195.bf even 12 1 1170.2.bs.g 8
260.bg even 12 1 1040.2.da.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.b ✓ 8 5.c odd 4 1
130.2.l.b ✓ 8 65.r odd 12 1
650.2.m.c 8 5.c odd 4 1
650.2.m.c 8 65.r odd 12 1
650.2.n.d 8 5.b even 2 1
650.2.n.d 8 13.e even 6 1
650.2.n.e 8 1.a even 1 1 trivial
650.2.n.e 8 65.l even 6 1 inner
1040.2.da.d 8 20.e even 4 1
1040.2.da.d 8 260.bg even 12 1
1170.2.bs.g 8 15.e even 4 1
1170.2.bs.g 8 195.bf even 12 1
1690.2.a.t 4 65.o even 12 1
1690.2.a.u 4 65.t even 12 1
1690.2.d.k 8 65.q odd 12 1
1690.2.d.k 8 65.r odd 12 1
1690.2.e.s 8 65.f even 4 1
1690.2.e.s 8 65.t even 12 1
1690.2.e.t 8 65.k even 4 1
1690.2.e.t 8 65.o even 12 1
1690.2.l.j 8 65.h odd 4 1
1690.2.l.j 8 65.q odd 12 1
8450.2.a.ci 4 65.o even 12 1
8450.2.a.cm 4 65.t even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6T_{3}^{7} + 10T_{3}^{6} - 12T_{3}^{5} - 24T_{3}^{4} + 24T_{3}^{3} + 40T_{3}^{2} - 48T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 18 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 18 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{8} - 36 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{8} + 96 T^{6} + \cdots + 4460544 \) Copy content Toggle raw display
$31$ \( T^{8} + 144 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 20736 \) Copy content Toggle raw display
$43$ \( T^{8} - 36 T^{7} + \cdots + 135424 \) Copy content Toggle raw display
$47$ \( (T^{4} - 114 T^{2} + \cdots - 639)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 312 T^{6} + \cdots + 335241 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 26 T^{7} + \cdots + 394384 \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + \cdots + 9437184 \) Copy content Toggle raw display
$71$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} - 114 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots - 188)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 18 T^{3} + \cdots - 2124)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 24 T^{7} + \cdots + 42849 \) Copy content Toggle raw display
$97$ \( T^{8} - 18 T^{7} + \cdots + 18558864 \) Copy content Toggle raw display
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