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Properties

Label 684.2.i.c
Level $684$
Weight $2$
Character orbit 684.i
Analytic conductor $5.462$
Analytic rank $0$
Dimension $16$
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] = [684,2,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: \( x^{16} - x^{15} - 3 x^{14} - 4 x^{13} + 22 x^{12} - 15 x^{11} - 42 x^{10} - 27 x^{9} + 324 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{13} q^{5} + (\beta_{14} + \beta_{11} + \cdots - \beta_{3}) q^{7}+ \cdots + (\beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{13} q^{5} + (\beta_{14} + \beta_{11} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{14} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9} - 7 q^{11} - 7 q^{13} + 4 q^{15} - 12 q^{17} - 16 q^{19} - 15 q^{21} - 16 q^{23} + 2 q^{25} - 22 q^{27} - 5 q^{29} + 10 q^{31} + 37 q^{33} + 24 q^{35} - 20 q^{37} - 8 q^{39} - 17 q^{41} - 11 q^{43} - 43 q^{45} - 10 q^{47} - 13 q^{49} + 30 q^{51} + 14 q^{53} - 20 q^{55} + q^{57} - 24 q^{59} + 5 q^{61} - 27 q^{63} - 5 q^{65} + 6 q^{67} + 10 q^{69} + 68 q^{71} + 36 q^{73} + 31 q^{75} - 6 q^{77} + 23 q^{79} - 41 q^{81} - 51 q^{83} + 12 q^{85} + 17 q^{87} + 24 q^{89} + 12 q^{91} + 26 q^{93} - 2 q^{95} + q^{97} - 61 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 3 x^{14} - 4 x^{13} + 22 x^{12} - 15 x^{11} - 42 x^{10} - 27 x^{9} + 324 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 70 \nu^{14} - 267 \nu^{13} + 509 \nu^{12} + 811 \nu^{11} - 1470 \nu^{10} - 5172 \nu^{9} + \cdots - 137781 ) / 135594 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44 \nu^{15} + 34 \nu^{14} - 66 \nu^{13} - 257 \nu^{12} + 224 \nu^{11} - 870 \nu^{10} - 1686 \nu^{9} + \cdots - 65610 ) / 67797 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 44 \nu^{15} - 127 \nu^{14} + 159 \nu^{13} + 536 \nu^{12} + 148 \nu^{11} - 1176 \nu^{10} + \cdots + 201204 ) / 67797 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44 \nu^{15} + 127 \nu^{14} - 159 \nu^{13} - 536 \nu^{12} - 148 \nu^{11} + 1176 \nu^{10} + \cdots - 269001 ) / 67797 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 163 \nu^{15} + 176 \nu^{14} - 747 \nu^{13} - 1021 \nu^{12} + 3337 \nu^{11} + 828 \nu^{10} + \cdots - 509571 ) / 135594 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 191 \nu^{15} - 190 \nu^{14} + 279 \nu^{13} + 1637 \nu^{12} - 2057 \nu^{11} - 792 \nu^{10} + \cdots + 999459 ) / 135594 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73 \nu^{15} - 80 \nu^{14} + 207 \nu^{13} + 691 \nu^{12} - 661 \nu^{11} - 900 \nu^{10} + \cdots + 244215 ) / 45198 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55 \nu^{15} + 55 \nu^{14} + 16 \nu^{13} - 499 \nu^{12} + 257 \nu^{11} + 161 \nu^{10} - 789 \nu^{9} + \cdots - 371061 ) / 22599 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 131 \nu^{15} - 64 \nu^{14} + 105 \nu^{13} + 611 \nu^{12} - 1265 \nu^{11} + 1578 \nu^{10} + \cdots + 340443 ) / 45198 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 127 \nu^{15} + 98 \nu^{14} - 291 \nu^{13} - 715 \nu^{12} + 1219 \nu^{11} - 240 \nu^{10} + \cdots - 417717 ) / 45198 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 215 \nu^{15} + 2 \nu^{14} + 363 \nu^{13} + 860 \nu^{12} - 2177 \nu^{11} + 1194 \nu^{10} + \cdots + 238383 ) / 67797 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 647 \nu^{15} - 256 \nu^{14} + 1089 \nu^{13} + 4109 \nu^{12} - 6275 \nu^{11} + 1368 \nu^{10} + \cdots + 2009853 ) / 135594 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 237 \nu^{15} + 364 \nu^{14} - 421 \nu^{13} - 2049 \nu^{12} + 1937 \nu^{11} + 2926 \nu^{10} + \cdots - 909063 ) / 45198 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 289 \nu^{15} + 102 \nu^{14} - 565 \nu^{13} - 1843 \nu^{12} + 2985 \nu^{11} - 656 \nu^{10} + \cdots - 1092771 ) / 45198 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} + 2\beta_{3} + 2\beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + \beta_{10} + 3 \beta_{9} - 4 \beta_{8} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} - 5 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{15} - 4 \beta_{14} + 3 \beta_{13} - 7 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} + 6 \beta_{9} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} - 2 \beta_{14} + \beta_{13} + 5 \beta_{12} + 15 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11 \beta_{15} + 2 \beta_{14} - 24 \beta_{13} + 11 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 38 \beta_{15} - 17 \beta_{14} - 6 \beta_{13} - 28 \beta_{12} + 42 \beta_{11} + 36 \beta_{10} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14 \beta_{15} + 18 \beta_{14} + 28 \beta_{13} - 40 \beta_{12} + 27 \beta_{11} + 45 \beta_{10} + \cdots - 67 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 112 \beta_{15} - 41 \beta_{14} + 36 \beta_{13} - 94 \beta_{12} - 12 \beta_{11} - 23 \beta_{10} + \cdots + 183 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 109 \beta_{15} - 54 \beta_{14} + 90 \beta_{13} - 112 \beta_{12} + 414 \beta_{11} + 139 \beta_{10} + \cdots + 266 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 82 \beta_{15} + 26 \beta_{14} - 200 \beta_{13} + 113 \beta_{12} + 75 \beta_{11} - 152 \beta_{10} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 339 \beta_{15} - 3 \beta_{14} - 690 \beta_{13} + 728 \beta_{12} + 864 \beta_{11} - 295 \beta_{10} + \cdots + 256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 741 \beta_{15} + 290 \beta_{14} - 987 \beta_{13} - 466 \beta_{12} + 1371 \beta_{11} + 905 \beta_{10} + \cdots - 1914 \) 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
1.73112 + 0.0567097i
1.69976 + 0.332884i
1.00579 − 1.41010i
0.885535 + 1.48857i
−0.894231 + 1.48336i
−1.04260 − 1.38311i
−1.34551 + 1.09069i
−1.53987 − 0.792975i
1.73112 − 0.0567097i
1.69976 − 0.332884i
1.00579 + 1.41010i
0.885535 − 1.48857i
−0.894231 − 1.48336i
−1.04260 + 1.38311i
−1.34551 − 1.09069i
−1.53987 + 0.792975i
0 −1.73112 − 0.0567097i 0 −1.01480 + 1.75769i 0 2.33899 + 4.05126i 0 2.99357 + 0.196343i 0
229.2 0 −1.69976 − 0.332884i 0 0.0704417 − 0.122009i 0 −2.11813 − 3.66871i 0 2.77838 + 1.13165i 0
229.3 0 −1.00579 + 1.41010i 0 1.70349 − 2.95052i 0 0.725004 + 1.25574i 0 −0.976782 − 2.83653i 0
229.4 0 −0.885535 − 1.48857i 0 0.0223771 − 0.0387582i 0 1.14836 + 1.98901i 0 −1.43165 + 2.63635i 0
229.5 0 0.894231 − 1.48336i 0 1.22580 − 2.12315i 0 0.597749 + 1.03533i 0 −1.40070 − 2.65293i 0
229.6 0 1.04260 + 1.38311i 0 1.16403 − 2.01615i 0 1.21545 + 2.10522i 0 −0.825965 + 2.88406i 0
229.7 0 1.34551 − 1.09069i 0 −0.668541 + 1.15795i 0 −1.80196 − 3.12109i 0 0.620779 − 2.93507i 0
229.8 0 1.53987 + 0.792975i 0 −1.50279 + 2.60290i 0 −0.605459 − 1.04868i 0 1.74238 + 2.44215i 0
457.1 0 −1.73112 + 0.0567097i 0 −1.01480 − 1.75769i 0 2.33899 − 4.05126i 0 2.99357 − 0.196343i 0
457.2 0 −1.69976 + 0.332884i 0 0.0704417 + 0.122009i 0 −2.11813 + 3.66871i 0 2.77838 − 1.13165i 0
457.3 0 −1.00579 − 1.41010i 0 1.70349 + 2.95052i 0 0.725004 − 1.25574i 0 −0.976782 + 2.83653i 0
457.4 0 −0.885535 + 1.48857i 0 0.0223771 + 0.0387582i 0 1.14836 − 1.98901i 0 −1.43165 − 2.63635i 0
457.5 0 0.894231 + 1.48336i 0 1.22580 + 2.12315i 0 0.597749 − 1.03533i 0 −1.40070 + 2.65293i 0
457.6 0 1.04260 − 1.38311i 0 1.16403 + 2.01615i 0 1.21545 − 2.10522i 0 −0.825965 − 2.88406i 0
457.7 0 1.34551 + 1.09069i 0 −0.668541 − 1.15795i 0 −1.80196 + 3.12109i 0 0.620779 + 2.93507i 0
457.8 0 1.53987 − 0.792975i 0 −1.50279 − 2.60290i 0 −0.605459 + 1.04868i 0 1.74238 − 2.44215i 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.c ✓ 16
3.b odd 2 1 2052.2.i.d 16
9.c even 3 1 inner 684.2.i.c ✓ 16
9.c even 3 1 6156.2.a.p 8
9.d odd 6 1 2052.2.i.d 16
9.d odd 6 1 6156.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.i.c ✓ 16 1.a even 1 1 trivial
684.2.i.c ✓ 16 9.c even 3 1 inner
2052.2.i.d 16 3.b odd 2 1
2052.2.i.d 16 9.d odd 6 1
6156.2.a.p 8 9.c even 3 1
6156.2.a.s 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 2 T_{5}^{15} + 21 T_{5}^{14} - 20 T_{5}^{13} + 256 T_{5}^{12} - 198 T_{5}^{11} + 1744 T_{5}^{10} + \cdots + 1 \) 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
$5$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} - 3 T^{15} + \cdots + 700569 \) Copy content Toggle raw display
$11$ \( T^{16} + 7 T^{15} + \cdots + 32160241 \) Copy content Toggle raw display
$13$ \( T^{16} + 7 T^{15} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( (T^{8} + 6 T^{7} + \cdots + 4374)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{16} \) Copy content Toggle raw display
$23$ \( T^{16} + 16 T^{15} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2884871521 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 325227262369 \) Copy content Toggle raw display
$37$ \( (T^{8} + 10 T^{7} + \cdots + 94606)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 3227716969 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2224724385601 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 243987590401 \) Copy content Toggle raw display
$53$ \( (T^{8} - 7 T^{7} + \cdots - 117934)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11398098475449 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 58816435441 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 231070401 \) Copy content Toggle raw display
$71$ \( (T^{8} - 34 T^{7} + \cdots - 132868)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 18 T^{7} + \cdots + 1327698)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 187443481 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 72017100991521 \) Copy content Toggle raw display
$89$ \( (T^{8} - 12 T^{7} + \cdots + 3506814)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 2542452573049 \) Copy content Toggle raw display
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