↑

Properties

Label 684.2.i.d
Level $684$
Weight $2$
Character orbit 684.i
Analytic conductor $5.462$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(229,684)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.i (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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 18x^{14} + 123x^{12} + 399x^{10} + 631x^{8} + 465x^{6} + 153x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{13} + \beta_{2}) q^{3} + (\beta_{8} + \beta_{7}) q^{5} + (\beta_{7} + \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{14} + \beta_{11} + \beta_{10} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} + \beta_{2}) q^{3} + (\beta_{8} + \beta_{7}) q^{5} + (\beta_{7} + \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{14} + \beta_{11} + \beta_{10} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} + 6 q^{5} - 5 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} + 6 q^{5} - 5 q^{7} + 15 q^{9} + 13 q^{11} - 3 q^{13} + 6 q^{15} - 20 q^{17} + 16 q^{19} + 3 q^{21} + 16 q^{23} - 14 q^{25} + 20 q^{27} + 3 q^{29} - 6 q^{31} - 17 q^{33} + 16 q^{37} - 16 q^{39} + 7 q^{41} - 3 q^{43} + 21 q^{45} + 10 q^{47} + 3 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} - q^{57} + 28 q^{59} - 7 q^{61} + 49 q^{63} + 21 q^{65} - 6 q^{67} - 56 q^{69} - 40 q^{71} - 20 q^{73} - 7 q^{75} + 22 q^{77} + 7 q^{79} + 39 q^{81} + 17 q^{83} + 24 q^{85} - 27 q^{87} - 32 q^{89} - 24 q^{91} - 22 q^{93} + 6 q^{95} + 11 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 18x^{14} + 123x^{12} + 399x^{10} + 631x^{8} + 465x^{6} + 153x^{4} + 21x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{14} - 18\nu^{12} - 123\nu^{10} - 399\nu^{8} - 631\nu^{6} - 464\nu^{4} - 146\nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10 \nu^{15} - \nu^{14} + 179 \nu^{13} - 18 \nu^{12} + 1212 \nu^{11} - 123 \nu^{10} + 3867 \nu^{9} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15 \nu^{15} + \nu^{14} - 267 \nu^{13} + 18 \nu^{12} - 1792 \nu^{11} + 123 \nu^{10} - 5633 \nu^{9} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -7\nu^{14} - 125\nu^{12} - 843\nu^{10} - 2671\nu^{8} - 4031\nu^{6} - 2680\nu^{4} - 699\nu^{2} - 52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10 \nu^{15} + 28 \nu^{14} + 179 \nu^{13} + 500 \nu^{12} + 1212 \nu^{11} + 3373 \nu^{10} + 3867 \nu^{9} + \cdots + 221 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39 \nu^{15} + 12 \nu^{14} + 697 \nu^{13} + 215 \nu^{12} + 4708 \nu^{11} + 1458 \nu^{10} + 14963 \nu^{9} + \cdots + 112 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\nu^{15} + 947\nu^{13} + 6394\nu^{11} + 20304\nu^{9} + 30772\nu^{7} + 20614\nu^{5} + 5429\nu^{3} + 414\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53 \nu^{15} + 7 \nu^{14} - 947 \nu^{13} + 125 \nu^{12} - 6394 \nu^{11} + 843 \nu^{10} - 20304 \nu^{9} + \cdots + 52 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13 \nu^{15} + 47 \nu^{14} + 233 \nu^{13} + 840 \nu^{12} + 1581 \nu^{11} + 5674 \nu^{10} + 5064 \nu^{9} + \cdots + 377 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50 \nu^{15} - 26 \nu^{14} + 894 \nu^{13} - 465 \nu^{12} + 6043 \nu^{11} - 3144 \nu^{10} + 19229 \nu^{9} + \cdots - 205 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{15} + 53 \nu^{14} + 54 \nu^{13} + 947 \nu^{12} + 369 \nu^{11} + 6394 \nu^{10} + 1197 \nu^{9} + \cdots + 414 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{15} - 53 \nu^{14} + 54 \nu^{13} - 947 \nu^{12} + 369 \nu^{11} - 6394 \nu^{10} + 1197 \nu^{9} + \cdots - 414 ) / 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 83 \nu^{15} + 16 \nu^{14} - 1484 \nu^{13} + 286 \nu^{12} - 10030 \nu^{11} + 1932 \nu^{10} + \cdots + 127 ) / 2 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 80 \nu^{15} - 30 \nu^{14} + 1430 \nu^{13} - 536 \nu^{12} + 9661 \nu^{11} - 3619 \nu^{10} + 30708 \nu^{9} + \cdots - 234 ) / 2 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 73 \nu^{15} + 62 \nu^{14} - 1305 \nu^{13} + 1108 \nu^{12} - 8818 \nu^{11} + 7483 \nu^{10} + \cdots + 478 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{8} + 2\beta_{7} + \beta_{4} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} - \beta_{9} + 2 \beta_{6} + \beta_{5} + \cdots - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} - \beta_{9} - 10 \beta_{8} - 10 \beta_{7} + \beta_{5} + \cdots + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{15} + 7 \beta_{14} - 6 \beta_{13} + 7 \beta_{12} - 2 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} + \cdots + 37 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7 \beta_{15} - 7 \beta_{14} - 3 \beta_{12} - 9 \beta_{11} - 3 \beta_{10} + 10 \beta_{9} + 53 \beta_{8} + \cdots - 25 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 77 \beta_{15} - 39 \beta_{14} + 38 \beta_{13} - 42 \beta_{12} + 20 \beta_{11} - 32 \beta_{10} + \cdots - 188 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39 \beta_{15} + 42 \beta_{14} - 3 \beta_{13} - \beta_{12} + 71 \beta_{11} + 30 \beta_{10} - 78 \beta_{9} + \cdots + 129 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 150 \beta_{15} + 69 \beta_{14} - 81 \beta_{13} + 82 \beta_{12} - 52 \beta_{11} + 56 \beta_{10} + \cdots + 333 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 207 \beta_{15} - 249 \beta_{14} + 42 \beta_{13} + 99 \beta_{12} - 513 \beta_{11} - 228 \beta_{10} + \cdots - 691 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2633 \beta_{15} - 1102 \beta_{14} + 1531 \beta_{13} - 1441 \beta_{12} + 1095 \beta_{11} - 892 \beta_{10} + \cdots - 5477 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1102 \beta_{15} + 1492 \beta_{14} - 390 \beta_{13} - 1025 \beta_{12} + 3502 \beta_{11} + 1581 \beta_{10} + \cdots + 3830 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 15478 \beta_{15} + 5983 \beta_{14} - 9495 \beta_{13} + 8485 \beta_{12} - 7262 \beta_{11} + 4827 \beta_{10} + \cdots + 30748 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5983 \beta_{15} - 9034 \beta_{14} + 3051 \beta_{13} + 8115 \beta_{12} - 23040 \beta_{11} + \cdots - 21817 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 91448 \beta_{15} - 33234 \beta_{14} + 58214 \beta_{13} - 50241 \beta_{12} + 46583 \beta_{11} + \cdots - 175802 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 33234 \beta_{15} + 55047 \beta_{14} - 21813 \beta_{13} - 57622 \beta_{12} + 147893 \beta_{11} + \cdots + 126807 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
− 2.45728i
0.359966i
2.13120i
− 1.42151i
− 0.351047i
0.879951i
− 2.00195i
− 0.603439i
2.45728i
− 0.359966i
− 2.13120i
1.42151i
0.351047i
− 0.879951i
2.00195i
0.603439i
0 −1.70747 − 0.290768i 0 2.12807 − 3.68592i 0 1.01351 + 1.75545i 0 2.83091 + 0.992956i 0
229.2 0 −1.56626 + 0.739484i 0 −0.311740 + 0.539949i 0 −0.303873 − 0.526323i 0 1.90633 − 2.31645i 0
229.3 0 −1.43701 − 0.966955i 0 −1.84568 + 3.19681i 0 −1.69217 − 2.93093i 0 1.13000 + 2.77905i 0
229.4 0 −0.424199 − 1.67930i 0 1.23106 − 2.13226i 0 −1.27825 − 2.21399i 0 −2.64011 + 1.42472i 0
229.5 0 −0.276297 + 1.70987i 0 0.304016 − 0.526570i 0 −0.481049 − 0.833201i 0 −2.84732 − 0.944864i 0
229.6 0 1.53086 − 0.810232i 0 −0.762060 + 1.31993i 0 0.943950 + 1.63497i 0 1.68705 − 2.48070i 0
229.7 0 1.65079 + 0.524293i 0 1.73374 − 3.00292i 0 −2.12083 − 3.67338i 0 2.45023 + 1.73100i 0
229.8 0 1.72958 − 0.0924174i 0 0.522594 − 0.905159i 0 1.41871 + 2.45728i 0 2.98292 − 0.319687i 0
457.1 0 −1.70747 + 0.290768i 0 2.12807 + 3.68592i 0 1.01351 − 1.75545i 0 2.83091 − 0.992956i 0
457.2 0 −1.56626 − 0.739484i 0 −0.311740 − 0.539949i 0 −0.303873 + 0.526323i 0 1.90633 + 2.31645i 0
457.3 0 −1.43701 + 0.966955i 0 −1.84568 − 3.19681i 0 −1.69217 + 2.93093i 0 1.13000 − 2.77905i 0
457.4 0 −0.424199 + 1.67930i 0 1.23106 + 2.13226i 0 −1.27825 + 2.21399i 0 −2.64011 − 1.42472i 0
457.5 0 −0.276297 − 1.70987i 0 0.304016 + 0.526570i 0 −0.481049 + 0.833201i 0 −2.84732 + 0.944864i 0
457.6 0 1.53086 + 0.810232i 0 −0.762060 − 1.31993i 0 0.943950 − 1.63497i 0 1.68705 + 2.48070i 0
457.7 0 1.65079 − 0.524293i 0 1.73374 + 3.00292i 0 −2.12083 + 3.67338i 0 2.45023 − 1.73100i 0
457.8 0 1.72958 + 0.0924174i 0 0.522594 + 0.905159i 0 1.41871 − 2.45728i 0 2.98292 + 0.319687i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.i.d ✓ 16
3.b odd 2 1 2052.2.i.c 16
9.c even 3 1 inner 684.2.i.d ✓ 16
9.c even 3 1 6156.2.a.o 8
9.d odd 6 1 2052.2.i.c 16
9.d odd 6 1 6156.2.a.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.i.d ✓ 16 1.a even 1 1 trivial
684.2.i.d ✓ 16 9.c even 3 1 inner
2052.2.i.c 16 3.b odd 2 1
2052.2.i.c 16 9.d odd 6 1
6156.2.a.o 8 9.c even 3 1
6156.2.a.t 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 6 T_{5}^{15} + 45 T_{5}^{14} - 144 T_{5}^{13} + 756 T_{5}^{12} - 2106 T_{5}^{11} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} - 6 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{16} + 5 T^{15} + \cdots + 54289 \) Copy content Toggle raw display
$11$ \( T^{16} - 13 T^{15} + \cdots + 1750329 \) Copy content Toggle raw display
$13$ \( T^{16} + 3 T^{15} + \cdots + 3560769 \) Copy content Toggle raw display
$17$ \( (T^{8} + 10 T^{7} + \cdots + 54)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{16} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 218123361 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 474760521 \) Copy content Toggle raw display
$31$ \( T^{16} + 6 T^{15} + \cdots + 1687401 \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots - 2798)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 376673105169 \) Copy content Toggle raw display
$43$ \( T^{16} + 3 T^{15} + \cdots + 18224361 \) Copy content Toggle raw display
$47$ \( T^{16} - 10 T^{15} + \cdots + 6859161 \) Copy content Toggle raw display
$53$ \( (T^{8} - T^{7} - 158 T^{6} + \cdots + 54)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60355222929 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 217666103209 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1084936809609 \) Copy content Toggle raw display
$71$ \( (T^{8} + 20 T^{7} + \cdots + 168156)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 10 T^{7} + \cdots - 11183342)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 864105962329 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 319071728531241 \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} + \cdots - 62975502)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 311557801 \) Copy content Toggle raw display
show more
show less