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Properties

Label 676.6.d.e
Level $676$
Weight $6$
Character orbit 676.d
Analytic conductor $108.419$
Analytic rank $0$
Dimension $10$
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] = [676,6,Mod(337,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.337");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 676.d (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: \(108.419462194\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 311 x^{8} - 7364 x^{7} + 10751 x^{6} + 888384 x^{5} + 18980275 x^{4} + \cdots + 19967455989 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 52)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 2) q^{3} + ( - \beta_{8} - 6 \beta_1) q^{5} + ( - \beta_{8} + \beta_{6} - 6 \beta_1) q^{7} + (\beta_{7} + \beta_{3} + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 2) q^{3} + ( - \beta_{8} - 6 \beta_1) q^{5} + ( - \beta_{8} + \beta_{6} - 6 \beta_1) q^{7} + (\beta_{7} + \beta_{3} + 11) q^{9} + ( - \beta_{8} - \beta_{6} + \cdots + 20 \beta_1) q^{11}+ \cdots + ( - 286 \beta_{8} + 146 \beta_{6} + \cdots + 17408 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{3} + 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{3} + 104 q^{9} + 1654 q^{17} - 3602 q^{23} - 3338 q^{25} + 3114 q^{27} - 5578 q^{29} - 24848 q^{35} + 3850 q^{43} - 29964 q^{49} - 4646 q^{51} + 40632 q^{53} - 31920 q^{55} - 15802 q^{61} + 122422 q^{69} + 101366 q^{75} + 114750 q^{77} + 150792 q^{79} - 286774 q^{81} - 127990 q^{87} + 105600 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 311 x^{8} - 7364 x^{7} + 10751 x^{6} + 888384 x^{5} + 18980275 x^{4} + \cdots + 19967455989 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30773149710427 \nu^{9} - 205402957780069 \nu^{8} + \cdots - 15\!\cdots\!04 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!56 \nu^{9} + \cdots + 76\!\cdots\!57 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!92 \nu^{9} + \cdots + 18\!\cdots\!93 ) / 13\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 67\!\cdots\!46 \nu^{9} + \cdots - 16\!\cdots\!13 ) / 43\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27\!\cdots\!74 \nu^{9} + \cdots + 37\!\cdots\!43 ) / 13\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!13 \nu^{9} + \cdots + 19\!\cdots\!25 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!21 \nu^{9} + \cdots - 27\!\cdots\!37 ) / 13\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83\!\cdots\!21 \nu^{9} + \cdots - 47\!\cdots\!82 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 60\!\cdots\!85 \nu^{9} + \cdots + 86\!\cdots\!57 ) / 13\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{9} + \beta_{8} + 3\beta_{6} + 43\beta_{5} - \beta_{4} + 5\beta_{3} - 21\beta_{2} - 26\beta _1 + 31 ) / 104 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 24 \beta_{9} - 61 \beta_{8} + 78 \beta_{7} - 27 \beta_{6} + 763 \beta_{5} + 35 \beta_{4} + \cdots + 6703 ) / 104 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 100 \beta_{9} + 279 \beta_{8} + 455 \beta_{7} - 8 \beta_{6} + 3229 \beta_{5} - 71 \beta_{4} + \cdots + 63144 ) / 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12572 \beta_{9} + 2561 \beta_{8} + 11674 \beta_{7} - 4849 \beta_{6} + 284351 \beta_{5} + \cdots + 2300931 ) / 104 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 201394 \beta_{9} - 56151 \beta_{8} + 617552 \beta_{7} - 158105 \beta_{6} + 5941009 \beta_{5} + \cdots + 84970569 ) / 104 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1049953 \beta_{9} + 1868688 \beta_{8} + 2510235 \beta_{7} - 405760 \beta_{6} + 28403456 \beta_{5} + \cdots + 362928872 ) / 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 89296498 \beta_{9} + 15459117 \beta_{8} + 176818928 \beta_{7} - 80998417 \beta_{6} + \cdots + 26576740131 ) / 104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1695918588 \beta_{9} + 1778431815 \beta_{8} + 4557616154 \beta_{7} - 1488535335 \beta_{6} + \cdots + 645022157611 ) / 104 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8764865389 \beta_{9} + 11714186043 \beta_{8} + 18860051678 \beta_{7} - 8056148968 \beta_{6} + \cdots + 2788930883646 ) / 26 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
20.3033 + 0.777135i
20.3033 − 0.777135i
−0.672765 − 5.92603i
−0.672765 + 5.92603i
−10.7537 + 10.4807i
−10.7537 − 10.4807i
−0.803404 + 7.00003i
−0.803404 − 7.00003i
−7.07349 − 8.45179i
−7.07349 + 8.45179i
0 −22.6494 0 − 1.41734i 0 − 169.499i 0 269.995 0
337.2 0 −22.6494 0 1.41734i 0 169.499i 0 269.995 0
337.3 0 −10.5914 0 − 98.6239i 0 − 230.964i 0 −130.822 0
337.4 0 −10.5914 0 98.6239i 0 230.964i 0 −130.822 0
337.5 0 −8.39942 0 − 61.3828i 0 73.5030i 0 −172.450 0
337.6 0 −8.39942 0 61.3828i 0 − 73.5030i 0 −172.450 0
337.7 0 11.9278 0 − 50.8897i 0 105.415i 0 −100.727 0
337.8 0 11.9278 0 50.8897i 0 − 105.415i 0 −100.727 0
337.9 0 20.7124 0 − 34.7520i 0 20.6758i 0 186.004 0
337.10 0 20.7124 0 34.7520i 0 − 20.6758i 0 186.004 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.6.d.e 10
13.b even 2 1 inner 676.6.d.e 10
13.c even 3 1 52.6.h.a ✓ 10
13.d odd 4 2 676.6.a.h 10
13.e even 6 1 52.6.h.a ✓ 10
39.h odd 6 1 468.6.t.b 10
39.i odd 6 1 468.6.t.b 10
52.i odd 6 1 208.6.w.a 10
52.j odd 6 1 208.6.w.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.6.h.a ✓ 10 13.c even 3 1
52.6.h.a ✓ 10 13.e even 6 1
208.6.w.a 10 52.i odd 6 1
208.6.w.a 10 52.j odd 6 1
468.6.t.b 10 39.h odd 6 1
468.6.t.b 10 39.i odd 6 1
676.6.a.h 10 13.d odd 4 2
676.6.d.e 10 1.a even 1 1 trivial
676.6.d.e 10 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 9T_{3}^{4} - 593T_{3}^{3} - 4641T_{3}^{2} + 62476T_{3} + 497796 \) acting on \(S_{6}^{\mathrm{new}}(676, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} + 9 T^{4} + \cdots + 497796)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 230263260364800 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 92058155204139)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 19\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 11\!\cdots\!69)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 23\!\cdots\!03 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 46\!\cdots\!07 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 18\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 99\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 11\!\cdots\!65)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 43\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 53\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
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