Properties

Label 650.2.o.f
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: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{7} + \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_{5} q^{4} + ( - \beta_{5} - \beta_{4}) q^{6} + \beta_{7} q^{7} - \beta_{3} q^{8} + (3 \beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{7} + \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_{5} q^{4} + ( - \beta_{5} - \beta_{4}) q^{6} + \beta_{7} q^{7} - \beta_{3} q^{8} + (3 \beta_{5} + \beta_{4}) q^{9} + (\beta_{6} - 2 \beta_{5} - \beta_{4} + 2) q^{11} + (\beta_{3} - \beta_{2}) q^{12} + ( - 3 \beta_{3} + \beta_1) q^{13} - \beta_{6} q^{14} + (\beta_{5} - 1) q^{16} + (\beta_{7} - \beta_1) q^{17} + ( - 3 \beta_{3} + \beta_{2}) q^{18} + 5 \beta_{5} q^{19} + 5 q^{21} + ( - \beta_{7} - 2 \beta_1) q^{22} + ( - 2 \beta_{7} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{23} + (\beta_{6} - \beta_{5} - \beta_{4} + 1) q^{24} + (3 \beta_{5} - 4) q^{26} + 5 \beta_{3} q^{27} + (\beta_{7} - \beta_{2}) q^{28} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - 2) q^{29} + (2 \beta_{6} + 6) q^{31} + \beta_1 q^{32} + (2 \beta_{7} + 7 \beta_1) q^{33} + ( - \beta_{6} + 1) q^{34} + ( - \beta_{6} + 3 \beta_{5} + \beta_{4} - 3) q^{36} + ( - 3 \beta_{7} - \beta_{3} + 3 \beta_{2} - \beta_1) q^{37} - 5 \beta_{3} q^{38} + (4 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 4) q^{39} + ( - 4 \beta_{6} + 5 \beta_{5} + 4 \beta_{4} - 5) q^{41} + ( - 5 \beta_{3} - 5 \beta_1) q^{42} + ( - 5 \beta_{7} - 2 \beta_1) q^{43} + (\beta_{6} + 2) q^{44} + (\beta_{5} + 2 \beta_{4}) q^{46} + (\beta_{3} + \beta_{2}) q^{47} + ( - \beta_{7} - \beta_1) q^{48} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - 2) q^{49} + ( - \beta_{6} + 4) q^{51} + (\beta_{3} + 4 \beta_1) q^{52} + (\beta_{3} - 2 \beta_{2}) q^{53} + ( - 5 \beta_{5} + 5) q^{54} - \beta_{4} q^{56} + (5 \beta_{3} - 5 \beta_{2}) q^{57} + (\beta_{7} + 2 \beta_1) q^{58} + ( - \beta_{5} - 2 \beta_{4}) q^{59} + (5 \beta_{5} + 2 \beta_{4}) q^{61} + ( - 2 \beta_{7} - 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{62} + (2 \beta_{7} + 5 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{63} - q^{64} + ( - 2 \beta_{6} - 7) q^{66} + (11 \beta_{3} + 11 \beta_1) q^{67} + (\beta_{7} - \beta_{3} - \beta_{2} - \beta_1) q^{68} + ( - 11 \beta_{5} - \beta_{4}) q^{69} + ( - 4 \beta_{5} - 2 \beta_{4}) q^{71} + (\beta_{7} + 3 \beta_1) q^{72} + 4 \beta_{2} q^{73} + (\beta_{5} + 3 \beta_{4}) q^{74} + (5 \beta_{5} - 5) q^{76} + ( - 5 \beta_{3} + \beta_{2}) q^{77} + ( - 4 \beta_{7} - \beta_{3} + \beta_{2} - 4 \beta_1) q^{78} + ( - 5 \beta_{6} - 6) q^{79} + ( - 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 4) q^{81} + (4 \beta_{7} + 5 \beta_1) q^{82} + (\beta_{3} + 4 \beta_{2}) q^{83} + 5 \beta_{5} q^{84} + (5 \beta_{6} + 2) q^{86} + ( - 2 \beta_{7} - 7 \beta_1) q^{87} + ( - \beta_{7} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{88} + ( - 3 \beta_{6} - 6 \beta_{5} + 3 \beta_{4} + 6) q^{89} + (\beta_{6} - 4 \beta_{4}) q^{91} + ( - \beta_{3} + 2 \beta_{2}) q^{92} + (6 \beta_{7} + 16 \beta_{3} - 6 \beta_{2} + 16 \beta_1) q^{93} + ( - \beta_{6} - \beta_{5} + \beta_{4} + 1) q^{94} + (\beta_{6} + 1) q^{96} + ( - \beta_{7} + 11 \beta_1) q^{97} + (\beta_{7} + 2 \beta_1) q^{98} + (4 \beta_{6} + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 2 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 2 q^{6} + 10 q^{9} + 6 q^{11} + 4 q^{14} - 4 q^{16} + 20 q^{19} + 40 q^{21} + 2 q^{24} - 20 q^{26} - 6 q^{29} + 40 q^{31} + 12 q^{34} - 10 q^{36} + 10 q^{39} - 12 q^{41} + 12 q^{44} - 6 q^{49} + 36 q^{51} + 20 q^{54} + 2 q^{56} + 16 q^{61} - 8 q^{64} - 48 q^{66} - 42 q^{69} - 12 q^{71} - 2 q^{74} - 20 q^{76} - 28 q^{79} + 20 q^{81} + 20 q^{84} - 4 q^{86} + 30 q^{89} + 4 q^{91} + 6 q^{94} + 4 q^{96} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} + 5\nu^{2} - 2 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + \nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{7} - 15\nu^{5} - 25\nu^{3} - 8\nu ) / 40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{3} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 4\beta_{5} + \beta_{4} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 7\beta_{3} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{3} + 2\beta_{2} + 16\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{6} - 5\beta_{5} - 10\beta_{4} - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} - 16\beta_{3} - 7\beta_{2} - 23\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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
−1.09445 + 0.895644i
0.228425 1.39564i
−0.228425 + 1.39564i
1.09445 0.895644i
−1.09445 0.895644i
0.228425 + 1.39564i
−0.228425 1.39564i
1.09445 + 0.895644i
−0.866025 0.500000i −1.55130 0.895644i 0.500000 + 0.866025i 0 0.895644 + 1.55130i −2.41733 + 1.39564i 1.00000i 0.104356 + 0.180750i 0
399.2 −0.866025 0.500000i 2.41733 + 1.39564i 0.500000 + 0.866025i 0 −1.39564 2.41733i 1.55130 0.895644i 1.00000i 2.39564 + 4.14938i 0
399.3 0.866025 + 0.500000i −2.41733 1.39564i 0.500000 + 0.866025i 0 −1.39564 2.41733i −1.55130 + 0.895644i 1.00000i 2.39564 + 4.14938i 0
399.4 0.866025 + 0.500000i 1.55130 + 0.895644i 0.500000 + 0.866025i 0 0.895644 + 1.55130i 2.41733 1.39564i 1.00000i 0.104356 + 0.180750i 0
549.1 −0.866025 + 0.500000i −1.55130 + 0.895644i 0.500000 0.866025i 0 0.895644 1.55130i −2.41733 1.39564i 1.00000i 0.104356 0.180750i 0
549.2 −0.866025 + 0.500000i 2.41733 1.39564i 0.500000 0.866025i 0 −1.39564 + 2.41733i 1.55130 + 0.895644i 1.00000i 2.39564 4.14938i 0
549.3 0.866025 0.500000i −2.41733 + 1.39564i 0.500000 0.866025i 0 −1.39564 + 2.41733i −1.55130 0.895644i 1.00000i 2.39564 4.14938i 0
549.4 0.866025 0.500000i 1.55130 0.895644i 0.500000 0.866025i 0 0.895644 1.55130i 2.41733 + 1.39564i 1.00000i 0.104356 0.180750i 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.f 8
5.b even 2 1 inner 650.2.o.f 8
5.c odd 4 1 650.2.e.d 4
5.c odd 4 1 650.2.e.i yes 4
13.c even 3 1 inner 650.2.o.f 8
65.n even 6 1 inner 650.2.o.f 8
65.q odd 12 1 650.2.e.d 4
65.q odd 12 1 650.2.e.i yes 4
65.q odd 12 1 8450.2.a.bb 2
65.q odd 12 1 8450.2.a.bk 2
65.r odd 12 1 8450.2.a.be 2
65.r odd 12 1 8450.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.e.d 4 5.c odd 4 1
650.2.e.d 4 65.q odd 12 1
650.2.e.i yes 4 5.c odd 4 1
650.2.e.i yes 4 65.q odd 12 1
650.2.o.f 8 1.a even 1 1 trivial
650.2.o.f 8 5.b even 2 1 inner
650.2.o.f 8 13.c even 3 1 inner
650.2.o.f 8 65.n even 6 1 inner
8450.2.a.bb 2 65.q odd 12 1
8450.2.a.be 2 65.r odd 12 1
8450.2.a.bh 2 65.r odd 12 1
8450.2.a.bk 2 65.q odd 12 1

Hecke kernels

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

\( T_{3}^{8} - 11T_{3}^{6} + 96T_{3}^{4} - 275T_{3}^{2} + 625 \) Copy content Toggle raw display
\( T_{7}^{8} - 11T_{7}^{6} + 96T_{7}^{4} - 275T_{7}^{2} + 625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 11 T^{6} + 96 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 11 T^{6} + 96 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$11$ \( (T^{4} - 3 T^{3} + 12 T^{2} + 9 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 23 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 15 T^{6} + 216 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 3 T^{3} + 12 T^{2} - 9 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 95 T^{6} + 6816 T^{4} + \cdots + 4879681 \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + 111 T^{2} - 450 T + 5625)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 263 T^{6} + \cdots + 294499921 \) Copy content Toggle raw display
$47$ \( (T^{4} + 15 T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + 69 T^{2} + 40 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 121 T^{2} + 14641)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + 48 T^{2} - 72 T + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 176 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 7 T - 119)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 186 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 15 T^{3} + 216 T^{2} - 135 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 275 T^{6} + \cdots + 260144641 \) Copy content Toggle raw display
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