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Properties

Label 650.2.w.c
Level $650$
Weight $2$
Character orbit 650.w
Analytic conductor $5.190$
Analytic rank $1$
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(193,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.w (of order \(12\), degree \(4\), 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: \(1\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-\beta_{4} + \beta_{5}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 − 1.56488i
0.500000 + 0.564882i
0.500000 − 2.19293i
0.500000 + 1.19293i
0.500000 + 1.56488i
0.500000 − 0.564882i
0.500000 + 2.19293i
0.500000 − 1.19293i
−0.500000 − 0.866025i −0.849429 + 3.17011i −0.500000 + 0.866025i 0 3.17011 − 0.849429i 0.344430 + 0.198857i 1.00000 −6.72999 − 3.88556i 0
193.2 −0.500000 − 0.866025i 0.215454 − 0.804085i −0.500000 + 0.866025i 0 −0.804085 + 0.215454i −3.34443 − 1.93091i 1.00000 1.99794 + 1.15351i 0
293.1 −0.500000 + 0.866025i −2.02948 + 0.543797i −0.500000 − 0.866025i 0 0.543797 − 2.02948i −4.43225 + 2.55896i 1.00000 1.22500 − 0.707252i 0
293.2 −0.500000 + 0.866025i −0.336546 + 0.0901772i −0.500000 − 0.866025i 0 0.0901772 − 0.336546i 1.43225 − 0.826909i 1.00000 −2.49295 + 1.43930i 0
357.1 −0.500000 + 0.866025i −0.849429 − 3.17011i −0.500000 − 0.866025i 0 3.17011 + 0.849429i 0.344430 − 0.198857i 1.00000 −6.72999 + 3.88556i 0
357.2 −0.500000 + 0.866025i 0.215454 + 0.804085i −0.500000 − 0.866025i 0 −0.804085 − 0.215454i −3.34443 + 1.93091i 1.00000 1.99794 − 1.15351i 0
457.1 −0.500000 − 0.866025i −2.02948 − 0.543797i −0.500000 + 0.866025i 0 0.543797 + 2.02948i −4.43225 − 2.55896i 1.00000 1.22500 + 0.707252i 0
457.2 −0.500000 − 0.866025i −0.336546 − 0.0901772i −0.500000 + 0.866025i 0 0.0901772 + 0.336546i 1.43225 + 0.826909i 1.00000 −2.49295 − 1.43930i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.w.c yes 8
5.b even 2 1 650.2.w.d yes 8
5.c odd 4 1 650.2.t.c ✓ 8
5.c odd 4 1 650.2.t.d yes 8
13.f odd 12 1 650.2.t.d yes 8
65.o even 12 1 inner 650.2.w.c yes 8
65.s odd 12 1 650.2.t.c ✓ 8
65.t even 12 1 650.2.w.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.t.c ✓ 8 5.c odd 4 1
650.2.t.c ✓ 8 65.s odd 12 1
650.2.t.d yes 8 5.c odd 4 1
650.2.t.d yes 8 13.f odd 12 1
650.2.w.c yes 8 1.a even 1 1 trivial
650.2.w.c yes 8 65.o even 12 1 inner
650.2.w.d yes 8 5.b even 2 1
650.2.w.d yes 8 65.t even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6T_{3}^{7} + 24T_{3}^{6} + 60T_{3}^{5} + 74T_{3}^{4} + 48T_{3}^{3} + 48T_{3}^{2} + 24T_{3} + 4 \) 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
$3$ \( T^{8} + 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{7} + \cdots + 408321 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + \cdots + 20449 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{8} + 24 T^{7} + \cdots + 876096 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + \cdots + 45796 \) Copy content Toggle raw display
$37$ \( T^{8} + 30 T^{7} + \cdots + 1771561 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 876096 \) Copy content Toggle raw display
$43$ \( T^{8} + 24 T^{7} + \cdots + 937024 \) Copy content Toggle raw display
$47$ \( T^{8} + 252 T^{6} + \cdots + 2313441 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + \cdots + 219024 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + \cdots + 6625476 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + \cdots + 419904 \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots - 242)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 276 T^{6} + \cdots + 5560164 \) Copy content Toggle raw display
$83$ \( T^{8} + 324 T^{6} + \cdots + 26244 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + \cdots + 178929 \) Copy content Toggle raw display
$97$ \( T^{8} + 18 T^{7} + \cdots + 1503076 \) Copy content Toggle raw display
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