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Properties

Label 684.3.e.a
Level $684$
Weight $3$
Character orbit 684.e
Analytic conductor $18.638$
Analytic rank $0$
Dimension $12$
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] = [684,3,Mod(305,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.e (of order \(2\), degree \(1\), 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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 156x^{10} + 8721x^{8} + 208784x^{6} + 2024760x^{4} + 7117056x^{2} + 6533136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{4} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{4} + 1) q^{7} - \beta_{10} q^{11} + (\beta_{3} - \beta_{2} - 1) q^{13} + ( - \beta_{11} - \beta_{9} + \cdots - \beta_1) q^{17}+ \cdots + (2 \beta_{6} + 4 \beta_{4} - 6 \beta_{2} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 156x^{10} + 8721x^{8} + 208784x^{6} + 2024760x^{4} + 7117056x^{2} + 6533136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5283\nu^{10} + 747468\nu^{8} + 35746495\nu^{6} + 640600728\nu^{4} + 2908500948\nu^{2} - 2437936256 ) / 1226974736 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10699 \nu^{10} - 1453458 \nu^{8} - 71124471 \nu^{6} - 1579317638 \nu^{4} + \cdots - 50749430904 ) / 1698888096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 247303 \nu^{10} + 39300927 \nu^{8} + 2207100159 \nu^{6} + 51133155599 \nu^{4} + \cdots + 789616257444 ) / 27606931560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 439759 \nu^{10} + 64571058 \nu^{8} + 3297682539 \nu^{6} + 66047797334 \nu^{4} + \cdots + 322447888440 ) / 22085545248 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1207811 \nu^{10} + 179510034 \nu^{8} + 9239977143 \nu^{6} + 188747409478 \nu^{4} + \cdots + 2685671561448 ) / 55213863120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1217 \nu^{11} + 189923 \nu^{9} + 10410681 \nu^{7} + 240839611 \nu^{5} + 2413128508 \nu^{3} + 11894428236 \nu ) / 2498524080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1402267 \nu^{11} + 254053290 \nu^{9} + 18225308799 \nu^{7} + 639808808750 \nu^{5} + \cdots + 33589159747272 \nu ) / 2352110568912 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29570539 \nu^{11} + 4740331224 \nu^{9} + 270890863839 \nu^{7} + 6494446864916 \nu^{5} + \cdots + 151480250456112 \nu ) / 4704221137824 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87410753 \nu^{11} + 13229974437 \nu^{9} + 707856730569 \nu^{7} + 15696035968309 \nu^{5} + \cdots + 318148928199684 \nu ) / 11760552844560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 237670967 \nu^{11} - 36242682558 \nu^{9} - 1941708477411 \nu^{7} + \cdots - 551649655461096 \nu ) / 23521105689120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} - 3\beta_{2} - 26 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{11} + 3\beta_{10} + \beta_{9} - 2\beta_{8} + 6\beta_{7} - 43\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -51\beta_{6} - 13\beta_{5} + 61\beta_{4} - 5\beta_{3} + 185\beta_{2} + 1144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -203\beta_{11} - 208\beta_{10} - 83\beta_{9} + 151\beta_{8} - 151\beta_{7} + 2101\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2403\beta_{6} + 1212\beta_{5} - 3273\beta_{4} + 192\beta_{3} - 10723\beta_{2} - 56536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12613\beta_{11} + 13371\beta_{10} + 5431\beta_{9} - 8826\beta_{8} - 1660\beta_{7} - 106839\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -114005\beta_{6} - 86295\beta_{5} + 179075\beta_{4} + 753\beta_{3} + 606807\beta_{2} + 2911516 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -758925\beta_{11} - 841572\beta_{10} - 312221\beta_{9} + 485389\beta_{8} + 533019\beta_{7} + 5551775\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5504145\beta_{6} + 5585048\beta_{5} - 10036475\beta_{4} - 799388\beta_{3} - 33847777\beta_{2} - 153338912 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 44764543 \beta_{11} + 52167557 \beta_{10} + 16708381 \beta_{9} - 26164658 \beta_{8} + \cdots - 292750157 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
− 7.53519i
− 7.04185i
− 5.71859i
− 3.29597i
− 2.16068i
− 1.18282i
1.18282i
2.16068i
3.29597i
5.71859i
7.04185i
7.53519i
0 0 0 − 7.53519i 0 5.18161 0 0 0
305.2 0 0 0 − 7.04185i 0 −3.07744 0 0 0
305.3 0 0 0 − 5.71859i 0 5.91816 0 0 0
305.4 0 0 0 − 3.29597i 0 −2.46307 0 0 0
305.5 0 0 0 − 2.16068i 0 −9.11429 0 0 0
305.6 0 0 0 − 1.18282i 0 11.5550 0 0 0
305.7 0 0 0 1.18282i 0 11.5550 0 0 0
305.8 0 0 0 2.16068i 0 −9.11429 0 0 0
305.9 0 0 0 3.29597i 0 −2.46307 0 0 0
305.10 0 0 0 5.71859i 0 5.91816 0 0 0
305.11 0 0 0 7.04185i 0 −3.07744 0 0 0
305.12 0 0 0 7.53519i 0 5.18161 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.e.a ✓ 12
3.b odd 2 1 inner 684.3.e.a ✓ 12
4.b odd 2 1 2736.3.h.c 12
12.b even 2 1 2736.3.h.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.e.a ✓ 12 1.a even 1 1 trivial
684.3.e.a ✓ 12 3.b odd 2 1 inner
2736.3.h.c 12 4.b odd 2 1
2736.3.h.c 12 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 156 T^{10} + \cdots + 6533136 \) Copy content Toggle raw display
$7$ \( (T^{6} - 8 T^{5} + \cdots - 24480)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 4706508816 \) Copy content Toggle raw display
$13$ \( (T^{6} + 8 T^{5} + \cdots + 189408)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 22829551568784 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 5361077160000 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 67\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{6} + 20 T^{5} + \cdots - 14066208)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 16 T^{5} + \cdots + 641267584)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} - 46 T^{5} + \cdots + 86899920)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + 24 T^{5} + \cdots - 12101473952)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 44 T^{5} + \cdots + 7409664)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{6} - 74 T^{5} + \cdots - 22640659056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 28 T^{5} + \cdots - 63064430272)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} - 36 T^{5} + \cdots - 973638592)^{2} \) Copy content Toggle raw display
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