Properties

Label 676.6.d.d
Level $676$
Weight $6$
Character orbit 676.d
Analytic conductor $108.419$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 125x^{4} + 4036x^{2} + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 2 \beta_{3} + 5) q^{3} + (5 \beta_{4} + 2 \beta_{2} + 5 \beta_1) q^{5} + (9 \beta_{4} + 26 \beta_{2}) q^{7} + ( - 9 \beta_{5} + 15 \beta_{3} + 345) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 2 \beta_{3} + 5) q^{3} + (5 \beta_{4} + 2 \beta_{2} + 5 \beta_1) q^{5} + (9 \beta_{4} + 26 \beta_{2}) q^{7} + ( - 9 \beta_{5} + 15 \beta_{3} + 345) q^{9} + (32 \beta_{4} - 37 \beta_{2} + 2 \beta_1) q^{11} + (71 \beta_{4} - 791 \beta_{2} - 38 \beta_1) q^{15} + ( - 57 \beta_{5} - 69 \beta_{3} - 822) q^{17} + (60 \beta_{4} + 3 \beta_{2} + 138 \beta_1) q^{19} + (155 \beta_{4} - 155 \beta_{2} - 41 \beta_1) q^{21} + ( - 124 \beta_{5} + 32 \beta_{3} - 1236) q^{23} + (285 \beta_{5} - 15 \beta_{3} - 2711) q^{25} + ( - 435 \beta_{5} - 474 \beta_{3} - 2541) q^{27} + (64 \beta_{5} + 400 \beta_{3} + 1602) q^{29} + (18 \beta_{4} + 2749 \beta_{2} + 18 \beta_1) q^{31} + (794 \beta_{4} - 1082 \beta_{2} + 346 \beta_1) q^{33} + (289 \beta_{5} + 190 \beta_{3} - 6357) q^{35} + ( - 177 \beta_{4} - 378 \beta_{2} + 339 \beta_1) q^{37} + ( - 378 \beta_{4} + 4269 \beta_{2} - 186 \beta_1) q^{41} + ( - 141 \beta_{5} - 642 \beta_{3} + 2645) q^{43} + (2304 \beta_{4} + 1467 \beta_{2} + 2940 \beta_1) q^{45} + ( - 259 \beta_{4} + 182 \beta_{2} + 860 \beta_1) q^{47} + ( - 531 \beta_{5} + 1053 \beta_{3} - 3105) q^{49} + (189 \beta_{5} + 2394 \beta_{3} + 16911) q^{51} + ( - 1014 \beta_{5} - 534 \beta_{3} + 18030) q^{53} + (2322 \beta_{5} + 1872 \beta_{3} - 25018) q^{55} + (270 \beta_{4} - 20142 \beta_{2} - 1386 \beta_1) q^{57} + ( - 896 \beta_{4} + 8359 \beta_{2} + 1786 \beta_1) q^{59} + (246 \beta_{5} - 210 \beta_{3} + 25726) q^{61} + (2016 \beta_{4} - 4617 \beta_{2} + 2238 \beta_1) q^{63} + ( - 732 \beta_{4} + 2515 \beta_{2} - 3246 \beta_1) q^{67} + ( - 1780 \beta_{5} + 1372 \beta_{3} - 5248) q^{69} + (493 \beta_{4} - 4958 \beta_{2} + 3688 \beta_1) q^{71} + (300 \beta_{4} + 5797 \beta_{2} + 3444 \beta_1) q^{73} + (9116 \beta_{5} + 7072 \beta_{3} - 30100) q^{75} + (586 \beta_{5} + 3586 \beta_{3} - 58212) q^{77} + (1632 \beta_{5} - 516 \beta_{3} + 13424) q^{79} + ( - 576 \beta_{5} + 6372 \beta_{3} + 50949) q^{81} + ( - 5250 \beta_{4} + 2625 \beta_{2} - 138 \beta_1) q^{83} + ( - 1263 \beta_{4} - 36411 \beta_{2} - 5271 \beta_1) q^{85} + ( - 3794 \beta_{5} - 8884 \beta_{3} - 94934) q^{87} + (1044 \beta_{4} - 9735 \beta_{2} + 2892 \beta_1) q^{89} + ( - 5228 \beta_{4} + \cdots - 11104 \beta_1) q^{93}+ \cdots + (9882 \beta_{4} - 58491 \beta_{2} + 4902 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24 q^{3} + 2082 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 24 q^{3} + 2082 q^{9} - 5184 q^{17} - 7600 q^{23} - 15726 q^{25} - 17064 q^{27} + 10540 q^{29} - 37184 q^{35} + 14304 q^{43} - 17586 q^{49} + 106632 q^{51} + 105084 q^{53} - 141720 q^{55} + 154428 q^{61} - 32304 q^{69} - 148224 q^{75} - 340928 q^{77} + 82776 q^{79} + 317286 q^{81} - 594960 q^{87} - 623736 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 125x^{4} + 4036x^{2} + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 29\nu^{3} - 2012\nu ) / 1632 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 63\nu^{2} + 113 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{5} + 1193\nu^{3} + 21988\nu ) / 1632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{4} - 223\nu^{2} - 1716 ) / 68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} - 3\beta_{3} - 81 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{4} - 52\beta_{2} - 59\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 252\beta_{5} + 223\beta_{3} + 4877 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -116\beta_{4} + 4772\beta_{2} + 3723\beta_1 ) / 2 \) Copy content Toggle raw display

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
8.10760i
8.10760i
7.53647i
7.53647i
1.57113i
1.57113i
0 −29.1356 0 78.6019i 0 40.3466i 0 605.881 0
337.2 0 −29.1356 0 78.6019i 0 40.3466i 0 605.881 0
337.3 0 14.4420 0 17.1549i 0 211.335i 0 −34.4294 0
337.4 0 14.4420 0 17.1549i 0 211.335i 0 −34.4294 0
337.5 0 26.6936 0 103.757i 0 113.682i 0 469.548 0
337.6 0 26.6936 0 103.757i 0 113.682i 0 469.548 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
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.d 6
13.b even 2 1 inner 676.6.d.d 6
13.d odd 4 1 52.6.a.c 3
13.d odd 4 1 676.6.a.d 3
39.f even 4 1 468.6.a.e 3
52.f even 4 1 208.6.a.i 3
104.j odd 4 1 832.6.a.r 3
104.m even 4 1 832.6.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.6.a.c 3 13.d odd 4 1
208.6.a.i 3 52.f even 4 1
468.6.a.e 3 39.f even 4 1
676.6.a.d 3 13.d odd 4 1
676.6.d.d 6 1.a even 1 1 trivial
676.6.d.d 6 13.b even 2 1 inner
832.6.a.r 3 104.j odd 4 1
832.6.a.u 3 104.m even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 12T_{3}^{2} - 813T_{3} + 11232 \) acting on \(S_{6}^{\mathrm{new}}(676, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 12 T^{2} + \cdots + 11232)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 19573688836 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 939592894276 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 659985553953856 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} + 2592 T^{2} + \cdots - 581484906)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{3} + 3800 T^{2} + \cdots - 2219034752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 5270 T^{2} + \cdots - 8846516296)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{3} - 7152 T^{2} + \cdots + 489811424572)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots - 2213570693088)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 16058441694592)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 2886179269312)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 73\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
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