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Properties

Label 684.4.k.c
Level $684$
Weight $4$
Character orbit 684.k
Analytic conductor $40.357$
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] = [684,4,Mod(505,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.505");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 684.k (of order \(3\), degree \(2\), 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: \(40.3573064439\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + \cdots + 1016064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{7} - \beta_{3}) q^{5} + (\beta_{5} - \beta_{2} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{7} - \beta_{3}) q^{5} + (\beta_{5} - \beta_{2} - 2) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{4} + 6) q^{11} + (\beta_{9} + \beta_{8} - \beta_{4} + \cdots - 11) q^{13}+ \cdots + (16 \beta_{9} - 22 \beta_{8} + \cdots + 28 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{5} - 20 q^{7} + 50 q^{11} - 56 q^{13} - 32 q^{17} + 77 q^{19} - 184 q^{23} - 121 q^{25} - 352 q^{29} + 264 q^{31} + 132 q^{35} - 640 q^{37} + 57 q^{41} - 528 q^{43} + 434 q^{47} + 2138 q^{49} - 780 q^{53} + 598 q^{55} + 343 q^{59} + 536 q^{61} + 1988 q^{65} + 779 q^{67} - 474 q^{71} + 1453 q^{73} + 2956 q^{77} - 1968 q^{79} + 698 q^{83} - 2334 q^{85} - 380 q^{89} + 1348 q^{91} - 4312 q^{95} + 883 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + \cdots + 1016064 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1165047 \nu^{9} - 9250992 \nu^{8} + 143004918 \nu^{7} - 836736016 \nu^{6} + \cdots - 3174016519296 ) / 2610376990464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 393603239 \nu^{9} - 640410556 \nu^{8} + 34258666518 \nu^{7} - 65425267000 \nu^{6} + \cdots - 235285623828864 ) / 328907500798464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24604755 \nu^{9} + 228981360 \nu^{8} - 3539670190 \nu^{7} + 21170339856 \nu^{6} + \cdots - 127823859374976 ) / 7831130971392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45367449 \nu^{9} - 343433424 \nu^{8} + 5308908346 \nu^{7} - 30833262864 \nu^{6} + \cdots - 150545568650112 ) / 7831130971392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 430238075 \nu^{9} + 257384052 \nu^{8} - 36268548014 \nu^{7} + 37431389032 \nu^{6} + \cdots - 109947804333696 ) / 54817916799744 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65962847 \nu^{9} - 378416976 \nu^{8} + 5849695754 \nu^{7} - 35865870088 \nu^{6} + \cdots - 179178099194880 ) / 7831130971392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1518356977 \nu^{9} + 423607099 \nu^{8} + 133515913632 \nu^{7} - 29537922338 \nu^{6} + \cdots + 410923373422464 ) / 82226875199616 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 403452304 \nu^{9} - 295261329 \nu^{8} + 33030273168 \nu^{7} - 34821843230 \nu^{6} + \cdots - 116960641689024 ) / 13704479199936 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{6} - \beta_{5} + 36\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{5} + 2\beta_{4} + 66\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 82\beta_{9} - 24\beta_{8} - 82\beta_{6} - 82\beta_{4} - 2400\beta_{3} + 36\beta _1 - 2400 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 200 \beta_{9} - 48 \beta_{8} - 48 \beta_{7} - 316 \beta_{6} + 316 \beta_{5} + 4200 \beta_{3} + \cdots - 4836 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2160\beta_{7} + 6148\beta_{5} + 6268\beta_{4} + 5088\beta_{2} + 177264 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17864\beta_{9} + 3840\beta_{8} + 21664\beta_{6} - 17864\beta_{4} - 403536\beta_{3} + 361944\beta _1 - 403536 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 476728 \beta_{9} + 165792 \beta_{8} + 165792 \beta_{7} + 452440 \beta_{6} - 452440 \beta_{5} + \cdots - 548016 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 234432\beta_{7} - 1427536\beta_{5} + 1550048\beta_{4} + 27257712\beta_{2} + 36364896 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
2.06508 + 3.57683i
−1.41615 − 2.45284i
−4.43560 − 7.68268i
4.20983 + 7.29165i
0.576829 + 0.999096i
2.06508 − 3.57683i
−1.41615 + 2.45284i
−4.43560 + 7.68268i
4.20983 − 7.29165i
0.576829 − 0.999096i
0 0 0 −7.98141 + 13.8242i 0 −31.8733 0 0 0
505.2 0 0 0 −2.48769 + 4.30880i 0 19.2102 0 0 0
505.3 0 0 0 −0.0670670 + 0.116164i 0 −9.41300 0 0 0
505.4 0 0 0 1.90769 − 3.30421i 0 30.9131 0 0 0
505.5 0 0 0 10.6285 − 18.4091i 0 −18.8370 0 0 0
577.1 0 0 0 −7.98141 − 13.8242i 0 −31.8733 0 0 0
577.2 0 0 0 −2.48769 − 4.30880i 0 19.2102 0 0 0
577.3 0 0 0 −0.0670670 − 0.116164i 0 −9.41300 0 0 0
577.4 0 0 0 1.90769 + 3.30421i 0 30.9131 0 0 0
577.5 0 0 0 10.6285 + 18.4091i 0 −18.8370 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.4.k.c 10
3.b odd 2 1 76.4.e.a ✓ 10
12.b even 2 1 304.4.i.f 10
19.c even 3 1 inner 684.4.k.c 10
57.f even 6 1 1444.4.a.g 5
57.h odd 6 1 76.4.e.a ✓ 10
57.h odd 6 1 1444.4.a.f 5
228.m even 6 1 304.4.i.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.e.a ✓ 10 3.b odd 2 1
76.4.e.a ✓ 10 57.h odd 6 1
304.4.i.f 10 12.b even 2 1
304.4.i.f 10 228.m even 6 1
684.4.k.c 10 1.a even 1 1 trivial
684.4.k.c 10 19.c even 3 1 inner
1444.4.a.f 5 57.h odd 6 1
1444.4.a.g 5 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 4 T_{5}^{9} + 381 T_{5}^{8} + 2144 T_{5}^{7} + 125455 T_{5}^{6} + 176910 T_{5}^{5} + \cdots + 746496 \) acting on \(S_{4}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots + 746496 \) Copy content Toggle raw display
$7$ \( (T^{5} + 10 T^{4} + \cdots + 3356160)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 25 T^{4} + \cdots + 14618352)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 261619974144 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{5} - 132 T^{4} + \cdots - 11017558272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 320 T^{4} + \cdots - 309775727920)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 98\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 138587802934320)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
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