Properties

Label 684.3.y.g
Level $684$
Weight $3$
Character orbit 684.y
Analytic conductor $18.638$
Analytic rank $0$
Dimension $8$
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] = [684,3,Mod(145,684)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 34x^{6} + 921x^{4} - 7990x^{2} + 55225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{5} + (\beta_{3} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{5} + (\beta_{3} - 1) q^{7} - \beta_{4} q^{11} + (2 \beta_{6} - 4 \beta_{3} + 5 \beta_{2} - 5) q^{13} + (\beta_{7} + \beta_{5} - \beta_1) q^{17} + (8 \beta_{6} - 7 \beta_{3} + 2 \beta_{2} + 5) q^{19} + \beta_{5} q^{23} + (6 \beta_{6} - 9 \beta_{2} - 9) q^{25} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_1) q^{29} + (10 \beta_{6} - 5 \beta_{3} - 18 \beta_{2} - 9) q^{31} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{35} + (12 \beta_{6} - 6 \beta_{3} - 2 \beta_{2} - 1) q^{37} + (2 \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_1) q^{41} + (11 \beta_{6} - 11 \beta_{3} + 5 \beta_{2}) q^{43} + ( - 4 \beta_{7} - 4 \beta_{5} - 4 \beta_{4}) q^{47} + ( - 2 \beta_{3} - 42) q^{49} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} - 4 \beta_1) q^{53} + (28 \beta_{6} - 28 \beta_{3} - 2 \beta_{2}) q^{55} + (2 \beta_{7} + 5 \beta_{5} + 4 \beta_{4} + 5 \beta_1) q^{59} + (34 \beta_{6} + 31 \beta_{2} + 31) q^{61} + (4 \beta_{7} + 6 \beta_{5} + 2 \beta_{4} - 3 \beta_1) q^{65} + (19 \beta_{6} - 38 \beta_{3} + 19 \beta_{2} - 19) q^{67} + ( - 3 \beta_{7} - 3 \beta_{5} - 6 \beta_{4} - 3 \beta_1) q^{71} + (2 \beta_{6} - 2 \beta_{3} - 25 \beta_{2}) q^{73} - 5 \beta_1 q^{77} + (\beta_{6} + \beta_{3} - 33 \beta_{2} - 66) q^{79} + ( - 3 \beta_{4} + 17 \beta_1) q^{83} + (22 \beta_{6} + 32 \beta_{2} + 32) q^{85} + (5 \beta_{7} + 2 \beta_{5} - 5 \beta_{4} - 4 \beta_1) q^{89} + (3 \beta_{6} - 6 \beta_{3} + 7 \beta_{2} - 7) q^{91} + (7 \beta_{7} - 10 \beta_{5} + 8 \beta_{4} + 4 \beta_1) q^{95} + (18 \beta_{2} + 36) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 34x^{6} + 921x^{4} - 7990x^{2} + 55225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 36\nu^{7} - 5219\nu^{5} + 105301\nu^{3} - 1514105\nu ) / 649305 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -34\nu^{6} + 921\nu^{4} - 31314\nu^{2} + 55225 ) / 216435 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 7667 ) / 2763 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\nu^{7} - 2149\nu^{5} + 58637\nu^{3} - 888535\nu ) / 129861 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 34\nu^{5} + 686\nu^{3} - 3995\nu ) / 2115 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -343\nu^{6} + 15657\nu^{4} - 315903\nu^{2} + 2740570 ) / 649305 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95\nu^{7} - 2149\nu^{5} + 58637\nu^{3} + 250980\nu ) / 129861 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{4} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{6} - 3\beta_{3} - 17\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 34\beta_{7} - 70\beta_{5} + 17\beta_{4} + 35\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 102\beta_{6} - 343\beta_{2} - 343 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 343\beta_{7} - 955\beta_{5} + 686\beta_{4} - 955\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2763\beta_{3} - 7667 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7667\beta_{7} + 24245\beta_{5} + 7667\beta_{4} - 48490\beta_1 ) / 6 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−4.27333 2.46721i
2.69047 + 1.55335i
−2.69047 1.55335i
4.27333 + 2.46721i
−4.27333 + 2.46721i
2.69047 1.55335i
−2.69047 + 1.55335i
4.27333 2.46721i
0 0 0 −3.48916 6.04340i 0 −3.44949 0 0 0
145.2 0 0 0 −2.19676 3.80490i 0 1.44949 0 0 0
145.3 0 0 0 2.19676 + 3.80490i 0 1.44949 0 0 0
145.4 0 0 0 3.48916 + 6.04340i 0 −3.44949 0 0 0
217.1 0 0 0 −3.48916 + 6.04340i 0 −3.44949 0 0 0
217.2 0 0 0 −2.19676 + 3.80490i 0 1.44949 0 0 0
217.3 0 0 0 2.19676 3.80490i 0 1.44949 0 0 0
217.4 0 0 0 3.48916 6.04340i 0 −3.44949 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.y.g 8
3.b odd 2 1 inner 684.3.y.g 8
19.d odd 6 1 inner 684.3.y.g 8
57.f even 6 1 inner 684.3.y.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.y.g 8 1.a even 1 1 trivial
684.3.y.g 8 3.b odd 2 1 inner
684.3.y.g 8 19.d odd 6 1 inner
684.3.y.g 8 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{8} + 68T_{5}^{6} + 3684T_{5}^{4} + 63920T_{5}^{2} + 883600 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 68 T^{6} + 3684 T^{4} + \cdots + 883600 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 5)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 332 T^{2} + 23500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 30 T^{3} + 303 T^{2} + 90 T + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 392 T^{6} + \cdots + 14137600 \) Copy content Toggle raw display
$19$ \( (T^{4} - 16 T^{3} + 570 T^{2} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 68 T^{6} + 3684 T^{4} + \cdots + 883600 \) Copy content Toggle raw display
$29$ \( T^{8} - 1176 T^{6} + \cdots + 1145145600 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1386 T^{2} + 42849)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1302 T^{2} + 416025)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 4896 T^{6} + \cdots + 23745739161600 \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + 801 T^{2} + \cdots + 491401)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 6272 T^{6} + \cdots + 926521753600 \) Copy content Toggle raw display
$53$ \( T^{8} - 1860 T^{6} + \cdots + 44732250000 \) Copy content Toggle raw display
$59$ \( T^{8} - 8844 T^{6} + \cdots + 27957656250000 \) Copy content Toggle raw display
$61$ \( (T^{4} - 62 T^{3} + 9819 T^{2} + \cdots + 35700625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 114 T^{3} - 1083 T^{2} + \cdots + 29322225)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 10584 T^{6} + \cdots + 7513300281600 \) Copy content Toggle raw display
$73$ \( (T^{4} - 50 T^{3} + 1899 T^{2} + \cdots + 361201)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 198 T^{3} + 16317 T^{2} + \cdots + 10556001)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 23048 T^{2} + 18954160)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 25476 T^{6} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} - 54 T + 972)^{4} \) Copy content Toggle raw display
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