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Properties

Label 684.2.r.a
Level $684$
Weight $2$
Character orbit 684.r
Analytic conductor $5.462$
Analytic rank $0$
Dimension $16$
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] = [684,2,Mod(487,684)]
 
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([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.487");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.r (of order \(6\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} + (\beta_{14} - \beta_{13} - \beta_{12} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} + (\beta_{14} - \beta_{13} - \beta_{12} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} - 3 \beta_{14} - \beta_{12} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} - 3 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} - 3 q^{4} + 2 q^{5} - 6 q^{10} - 18 q^{13} + 6 q^{14} - 3 q^{16} - 2 q^{17} - 4 q^{20} + 3 q^{22} - 2 q^{25} + 24 q^{26} - 4 q^{28} + 6 q^{29} - 27 q^{32} + 36 q^{34} + 24 q^{38} + 48 q^{40} + 48 q^{41} + 25 q^{44} + 16 q^{49} + 6 q^{52} - 6 q^{53} - 40 q^{58} - 26 q^{61} - 32 q^{62} - 18 q^{64} + 36 q^{68} - 54 q^{70} + 16 q^{73} + 2 q^{74} + 43 q^{76} - 80 q^{77} + 30 q^{80} + 11 q^{82} + 14 q^{85} + 48 q^{86} - 18 q^{89} + 52 q^{92} - 12 q^{97} - 51 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26 \nu^{15} - 397 \nu^{14} + 335 \nu^{13} - 666 \nu^{12} + 419 \nu^{11} - 428 \nu^{10} + \cdots - 5056 ) / 40768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79 \nu^{15} - 341 \nu^{14} - 1114 \nu^{13} + 629 \nu^{12} - 1716 \nu^{11} + 965 \nu^{10} + \cdots + 32128 ) / 81536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 130 \nu^{15} - 269 \nu^{14} + 1363 \nu^{13} - 2 \nu^{12} - 37 \nu^{11} + 1552 \nu^{10} + \cdots + 62912 ) / 81536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 339 \nu^{15} + 2061 \nu^{14} + 1416 \nu^{13} - 2119 \nu^{12} - 1594 \nu^{11} + 3285 \nu^{10} + \cdots + 45056 ) / 163072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 425 \nu^{15} - 845 \nu^{14} - 666 \nu^{13} - 883 \nu^{12} - 1268 \nu^{11} - 5251 \nu^{10} + \cdots + 51840 ) / 163072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 230 \nu^{15} - 97 \nu^{14} - 949 \nu^{13} + 290 \nu^{12} + 367 \nu^{11} - 1404 \nu^{10} + \cdots - 52928 ) / 81536 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 537 \nu^{15} + 2473 \nu^{14} - 5608 \nu^{13} + 1525 \nu^{12} - 1506 \nu^{11} + 97 \nu^{10} + \cdots - 147968 ) / 163072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 361 \nu^{15} - 550 \nu^{14} + 835 \nu^{13} - 1671 \nu^{12} + 2921 \nu^{11} - 3225 \nu^{10} + \cdots - 95168 ) / 81536 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 885 \nu^{15} - 431 \nu^{14} + 114 \nu^{13} + 4223 \nu^{12} - 4660 \nu^{11} + 5903 \nu^{10} + \cdots + 273536 ) / 163072 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 621 \nu^{15} + 2879 \nu^{14} - 5566 \nu^{13} + 9309 \nu^{12} - 9892 \nu^{11} + 5725 \nu^{10} + \cdots + 352896 ) / 81536 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 3 \nu^{14} + 6 \nu^{13} - 9 \nu^{12} + 12 \nu^{11} - 9 \nu^{10} + 3 \nu^{9} + 6 \nu^{8} + \cdots - 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1441 \nu^{15} + 1063 \nu^{14} + 2720 \nu^{13} - 1949 \nu^{12} + 522 \nu^{11} + 2391 \nu^{10} + \cdots + 125952 ) / 163072 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 475 \nu^{15} - 491 \nu^{14} + 900 \nu^{13} - 731 \nu^{12} + 662 \nu^{11} - 703 \nu^{10} + \cdots - 6656 ) / 40768 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1346 \nu^{15} - 2357 \nu^{14} + 3471 \nu^{13} - 6350 \nu^{12} + 7867 \nu^{11} - 3172 \nu^{10} + \cdots - 249664 ) / 81536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{7} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{14} + \beta_{13} + 2\beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2\beta_{2} - 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} + 3 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} - 3 \beta_{9} + 9 \beta_{7} + 6 \beta_{5} + 6 \beta_{4} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{15} + 8 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} + 3 \beta_{10} + 8 \beta_{9} + \cdots - 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4 \beta_{15} - 16 \beta_{14} + 5 \beta_{13} - 8 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - 5 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11 \beta_{15} - 13 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} + 5 \beta_{11} + 9 \beta_{7} - 13 \beta_{6} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 24 \beta_{14} - 19 \beta_{13} + 11 \beta_{11} - 7 \beta_{8} + 31 \beta_{7} - 20 \beta_{6} - 2 \beta_{5} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 21 \beta_{14} - 18 \beta_{13} - 21 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + 24 \beta_{9} - 24 \beta_{8} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 26 \beta_{15} + 18 \beta_{14} + 28 \beta_{13} + 6 \beta_{12} - 8 \beta_{11} - 52 \beta_{10} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 21 \beta_{15} + 2 \beta_{14} + 43 \beta_{13} + 39 \beta_{12} + 32 \beta_{11} + 21 \beta_{10} - 40 \beta_{9} + \cdots - 24 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
487.1
−1.35532 + 0.403874i
−0.835469 + 1.14105i
−0.327894 − 1.37568i
−0.112075 + 1.40977i
0.570443 − 1.29406i
1.05003 + 0.947334i
1.16486 − 0.801943i
1.34543 + 0.435684i
−1.35532 − 0.403874i
−0.835469 − 1.14105i
−0.327894 + 1.37568i
−0.112075 − 1.40977i
0.570443 + 1.29406i
1.05003 − 0.947334i
1.16486 + 0.801943i
1.34543 − 0.435684i
−1.35532 − 0.403874i 0 1.67377 + 1.09475i 0.139977 + 0.242447i 0 1.55280i −1.82635 − 2.15973i 0 −0.0917953 − 0.385126i
487.2 −0.835469 − 1.14105i 0 −0.603985 + 1.90662i 1.60333 + 2.77705i 0 − 1.25044i 2.68016 − 0.903746i 0 1.82921 − 4.14961i
487.3 −0.327894 + 1.37568i 0 −1.78497 − 0.902152i 0.139977 + 0.242447i 0 − 1.55280i 1.82635 − 2.15973i 0 −0.379427 + 0.113066i
487.4 −0.112075 − 1.40977i 0 −1.97488 + 0.316000i −1.59295 − 2.75907i 0 2.36291i 0.666820 + 2.74870i 0 −3.71111 + 2.55491i
487.5 0.570443 + 1.29406i 0 −1.34919 + 1.47638i 1.60333 + 2.77705i 0 1.25044i −2.68016 − 0.903746i 0 −2.67906 + 3.65895i
487.6 1.05003 − 0.947334i 0 0.205118 − 1.98945i 0.349646 + 0.605604i 0 − 3.80025i −1.66930 − 2.28330i 0 0.940847 + 0.304670i
487.7 1.16486 + 0.801943i 0 0.713775 + 1.86829i −1.59295 − 2.75907i 0 − 2.36291i −0.666820 + 2.74870i 0 0.357061 − 4.49138i
487.8 1.34543 − 0.435684i 0 1.62036 − 1.17236i 0.349646 + 0.605604i 0 3.80025i 1.66930 − 2.28330i 0 0.734275 + 0.662462i
559.1 −1.35532 + 0.403874i 0 1.67377 − 1.09475i 0.139977 − 0.242447i 0 − 1.55280i −1.82635 + 2.15973i 0 −0.0917953 + 0.385126i
559.2 −0.835469 + 1.14105i 0 −0.603985 − 1.90662i 1.60333 − 2.77705i 0 1.25044i 2.68016 + 0.903746i 0 1.82921 + 4.14961i
559.3 −0.327894 − 1.37568i 0 −1.78497 + 0.902152i 0.139977 − 0.242447i 0 1.55280i 1.82635 + 2.15973i 0 −0.379427 − 0.113066i
559.4 −0.112075 + 1.40977i 0 −1.97488 − 0.316000i −1.59295 + 2.75907i 0 − 2.36291i 0.666820 − 2.74870i 0 −3.71111 − 2.55491i
559.5 0.570443 − 1.29406i 0 −1.34919 − 1.47638i 1.60333 − 2.77705i 0 − 1.25044i −2.68016 + 0.903746i 0 −2.67906 − 3.65895i
559.6 1.05003 + 0.947334i 0 0.205118 + 1.98945i 0.349646 − 0.605604i 0 3.80025i −1.66930 + 2.28330i 0 0.940847 − 0.304670i
559.7 1.16486 − 0.801943i 0 0.713775 − 1.86829i −1.59295 + 2.75907i 0 2.36291i −0.666820 − 2.74870i 0 0.357061 + 4.49138i
559.8 1.34543 + 0.435684i 0 1.62036 + 1.17236i 0.349646 − 0.605604i 0 − 3.80025i 1.66930 + 2.28330i 0 0.734275 − 0.662462i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 487.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.r.a 16
3.b odd 2 1 76.2.f.a ✓ 16
4.b odd 2 1 inner 684.2.r.a 16
12.b even 2 1 76.2.f.a ✓ 16
19.d odd 6 1 inner 684.2.r.a 16
24.f even 2 1 1216.2.n.f 16
24.h odd 2 1 1216.2.n.f 16
57.f even 6 1 76.2.f.a ✓ 16
76.f even 6 1 inner 684.2.r.a 16
228.n odd 6 1 76.2.f.a ✓ 16
456.s odd 6 1 1216.2.n.f 16
456.v even 6 1 1216.2.n.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.f.a ✓ 16 3.b odd 2 1
76.2.f.a ✓ 16 12.b even 2 1
76.2.f.a ✓ 16 57.f even 6 1
76.2.f.a ✓ 16 228.n odd 6 1
684.2.r.a 16 1.a even 1 1 trivial
684.2.r.a 16 4.b odd 2 1 inner
684.2.r.a 16 19.d odd 6 1 inner
684.2.r.a 16 76.f even 6 1 inner
1216.2.n.f 16 24.f even 2 1
1216.2.n.f 16 24.h odd 2 1
1216.2.n.f 16 456.s odd 6 1
1216.2.n.f 16 456.v even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{8} - T_{5}^{7} + 11T_{5}^{6} - 10T_{5}^{5} + 112T_{5}^{4} - 104T_{5}^{3} + 80T_{5}^{2} - 20T_{5} + 4 \) Copy content Toggle raw display
\( T_{23}^{16} - 161 T_{23}^{14} + 17449 T_{23}^{12} - 1031672 T_{23}^{10} + 44152900 T_{23}^{8} + \cdots + 757115775376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - T^{7} + 11 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 24 T^{6} + \cdots + 304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 45 T^{6} + \cdots + 3724)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 9 T^{7} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + T^{7} + 19 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 757115775376 \) Copy content Toggle raw display
$29$ \( (T^{8} - 3 T^{7} + \cdots + 98596)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 124 T^{6} + \cdots + 7600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 124 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 24 T^{7} + \cdots + 841)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 105 T^{14} + \cdots + 1478656 \) Copy content Toggle raw display
$47$ \( T^{16} - 69 T^{14} + \cdots + 37896336 \) Copy content Toggle raw display
$53$ \( (T^{8} + 3 T^{7} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1300683225625 \) Copy content Toggle raw display
$61$ \( (T^{8} + 13 T^{7} + \cdots + 209764)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 106 T^{14} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3550253056 \) Copy content Toggle raw display
$73$ \( (T^{8} - 8 T^{7} + \cdots + 1190281)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 136386521399296 \) Copy content Toggle raw display
$83$ \( (T^{8} + 181 T^{6} + \cdots + 255664)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 9 T^{7} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 6 T^{7} + \cdots + 24649)^{2} \) Copy content Toggle raw display
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