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Properties

Label 567.3.r.c
Level $567$
Weight $3$
Character orbit 567.r
Analytic conductor $15.450$
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] = [567,3,Mod(134,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 567.r (of order \(6\), degree \(2\), not 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: \(15.4496309892\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.39033114624.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 6x^{6} - 30x^{5} + 34x^{4} - 102x^{3} + 486x^{2} - 730x + 373 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (3 \beta_{3} - 2 \beta_1 + 3) q^{4} + ( - \beta_{7} - \beta_{5}) q^{5} + ( - \beta_{6} - \beta_1) q^{7} + (3 \beta_{7} + 3 \beta_{4} - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (3 \beta_{3} - 2 \beta_1 + 3) q^{4} + ( - \beta_{7} - \beta_{5}) q^{5} + ( - \beta_{6} - \beta_1) q^{7} + (3 \beta_{7} + 3 \beta_{4} - 2 \beta_{2}) q^{8} + ( - \beta_{6} + 7) q^{10} + 2 \beta_{4} q^{11} + (9 \beta_{3} - \beta_1 + 9) q^{13} + (2 \beta_{7} - \beta_{5}) q^{14} + ( - 4 \beta_{6} + 9 \beta_{3} - 4 \beta_1) q^{16} - 2 \beta_{2} q^{17} + (5 \beta_{6} + 3) q^{19} + (3 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{20} + (14 \beta_{3} - 4 \beta_1 + 14) q^{22} + (6 \beta_{7} - 2 \beta_{5}) q^{23} + ( - 10 \beta_{6} - 3 \beta_{3} - 10 \beta_1) q^{25} + (11 \beta_{7} + 11 \beta_{4} - \beta_{2}) q^{26} + ( - 3 \beta_{6} - 14) q^{28} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{2}) q^{29} + ( - 34 \beta_{3} - 2 \beta_1 - 34) q^{31} + (5 \beta_{7} + 4 \beta_{5}) q^{32} + ( - 6 \beta_{6} - 6 \beta_1) q^{34} + ( - \beta_{7} - \beta_{4} - 3 \beta_{2}) q^{35} + ( - 14 \beta_{6} + 4) q^{37} + (5 \beta_{5} + 13 \beta_{4} - 5 \beta_{2}) q^{38} + ( - 21 \beta_{3} - 15 \beta_1 - 21) q^{40} + ( - 14 \beta_{7} + 8 \beta_{5}) q^{41} + ( - 6 \beta_{6} - 40 \beta_{3} - 6 \beta_1) q^{43} + (14 \beta_{7} + 14 \beta_{4} - 4 \beta_{2}) q^{44} + ( - 18 \beta_{6} - 42) q^{46} + ( - 4 \beta_{5} + 12 \beta_{4} + 4 \beta_{2}) q^{47} + ( - 7 \beta_{3} - 7) q^{49} + (17 \beta_{7} - 10 \beta_{5}) q^{50} + ( - 21 \beta_{6} + 41 \beta_{3} - 21 \beta_1) q^{52} + (6 \beta_{7} + 6 \beta_{4} + 16 \beta_{2}) q^{53} + ( - 2 \beta_{6} + 14) q^{55} + (\beta_{5} - 12 \beta_{4} - \beta_{2}) q^{56} + ( - 28 \beta_{3} + 2 \beta_1 - 28) q^{58} + ( - 27 \beta_{7} - \beta_{5}) q^{59} + (7 \beta_{6} - 39 \beta_{3} + 7 \beta_1) q^{61} + ( - 30 \beta_{7} - 30 \beta_{4} - 2 \beta_{2}) q^{62} + (18 \beta_{6} + 1) q^{64} + ( - 6 \beta_{5} + 8 \beta_{4} + 6 \beta_{2}) q^{65} + (6 \beta_{3} + 8 \beta_1 + 6) q^{67} + (12 \beta_{7} + 2 \beta_{5}) q^{68} + ( - 7 \beta_{6} - 7 \beta_{3} - 7 \beta_1) q^{70} + ( - 24 \beta_{7} - 24 \beta_{4} - 6 \beta_{2}) q^{71} + ( - 26 \beta_{6} - 8) q^{73} + ( - 14 \beta_{5} - 24 \beta_{4} + 14 \beta_{2}) q^{74} + (79 \beta_{3} - 21 \beta_1 + 79) q^{76} + (4 \beta_{7} - 2 \beta_{5}) q^{77} + ( - 36 \beta_{6} + 32 \beta_{3} - 36 \beta_1) q^{79} + (5 \beta_{7} + 5 \beta_{4} - 3 \beta_{2}) q^{80} + (52 \beta_{6} + 98) q^{82} + (9 \beta_{5} - 27 \beta_{4} - 9 \beta_{2}) q^{83} + ( - 42 \beta_{3} - 18 \beta_1 - 42) q^{85} + ( - 28 \beta_{7} - 6 \beta_{5}) q^{86} + ( - 24 \beta_{6} + 42 \beta_{3} - 24 \beta_1) q^{88} + ( - 26 \beta_{7} - 26 \beta_{4} + 16 \beta_{2}) q^{89} + ( - 9 \beta_{6} - 7) q^{91} + ( - 10 \beta_{5} - 54 \beta_{4} + 10 \beta_{2}) q^{92} + (84 \beta_{3} - 12 \beta_1 + 84) q^{94} + (2 \beta_{7} + 12 \beta_{5}) q^{95} + (8 \beta_{6} - 2 \beta_{3} + 8 \beta_1) q^{97} + ( - 7 \beta_{7} - 7 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} + 56 q^{10} + 36 q^{13} - 36 q^{16} + 24 q^{19} + 56 q^{22} + 12 q^{25} - 112 q^{28} - 136 q^{31} + 32 q^{37} - 84 q^{40} + 160 q^{43} - 336 q^{46} - 28 q^{49} - 164 q^{52} + 112 q^{55} - 112 q^{58} + 156 q^{61} + 8 q^{64} + 24 q^{67} + 28 q^{70} - 64 q^{73} + 316 q^{76} - 128 q^{79} + 784 q^{82} - 168 q^{85} - 168 q^{88} - 56 q^{91} + 336 q^{94} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 6x^{6} - 30x^{5} + 34x^{4} - 102x^{3} + 486x^{2} - 730x + 373 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -62\nu^{7} - 1809\nu^{6} - 3078\nu^{5} - 13500\nu^{4} + 11602\nu^{3} + 37037\nu^{2} + 174898\nu - 164120 ) / 46998 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56\nu^{7} - 207\nu^{6} + 121\nu^{5} - 2558\nu^{4} + 2287\nu^{3} - 4070\nu^{2} + 43592\nu - 43268 ) / 2611 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 163\nu^{7} - 63\nu^{6} + 945\nu^{5} - 3276\nu^{4} + 469\nu^{3} - 15883\nu^{2} + 51751\nu - 43268 ) / 6714 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1223\nu^{7} + 255\nu^{6} + 7365\nu^{5} - 20310\nu^{4} - 655\nu^{3} - 114503\nu^{2} + 346799\nu - 202912 ) / 46998 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -448\nu^{7} - 582\nu^{6} - 2460\nu^{5} + 4425\nu^{4} + 14528\nu^{3} + 50464\nu^{2} - 53320\nu - 78703 ) / 15666 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -152\nu^{7} + 29\nu^{6} - 808\nu^{5} + 3373\nu^{4} + 986\nu^{3} + 16429\nu^{2} - 45746\nu + 19023 ) / 5222 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2323\nu^{7} - 534\nu^{6} + 13605\nu^{5} - 44553\nu^{4} + 4597\nu^{3} - 247522\nu^{2} + 669787\nu - 532271 ) / 46998 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} - \beta_{5} + 3\beta_{3} + 2\beta_{2} - 3\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + \beta_{5} + 3\beta_{4} + 6\beta_{3} - 2\beta_{2} + 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 27\beta_{7} + 6\beta_{6} + 7\beta_{5} + 24\beta_{4} - 63\beta_{3} - 2\beta_{2} + 9\beta _1 + 15 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{6} - 6\beta_{5} - 6\beta_{4} + 10\beta_{3} + 4\beta_{2} - 4\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -243\beta_{7} - 138\beta_{6} + 73\beta_{5} - 138\beta_{4} + 597\beta_{3} - 44\beta_{2} - 93\beta _1 + 315 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 57\beta_{7} - 168\beta_{6} + 77\beta_{5} + 375\beta_{4} - 630\beta_{3} + 2\beta_{2} + 9\beta _1 - 732 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1743\beta_{7} + 2100\beta_{6} - 595\beta_{5} + 882\beta_{4} - 2097\beta_{3} - 280\beta_{2} + 1575\beta _1 + 255 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
−0.279898 − 3.02113i
−1.85391 + 1.90397i
1.03103 − 0.478705i
2.10277 − 0.136187i
−0.279898 + 3.02113i
−1.85391 − 1.90397i
1.03103 + 0.478705i
2.10277 + 0.136187i
−3.03622 + 1.75296i 0 4.14575 − 7.18065i −1.07558 − 0.620984i 0 −1.32288 − 2.29129i 15.0457i 0 4.35425
134.2 −1.13198 + 0.653548i 0 −1.14575 + 1.98450i −6.39086 − 3.68977i 0 1.32288 + 2.29129i − 8.22359i 0 9.64575
134.3 1.13198 − 0.653548i 0 −1.14575 + 1.98450i 6.39086 + 3.68977i 0 1.32288 + 2.29129i 8.22359i 0 9.64575
134.4 3.03622 − 1.75296i 0 4.14575 − 7.18065i 1.07558 + 0.620984i 0 −1.32288 − 2.29129i − 15.0457i 0 4.35425
512.1 −3.03622 − 1.75296i 0 4.14575 + 7.18065i −1.07558 + 0.620984i 0 −1.32288 + 2.29129i − 15.0457i 0 4.35425
512.2 −1.13198 − 0.653548i 0 −1.14575 − 1.98450i −6.39086 + 3.68977i 0 1.32288 − 2.29129i 8.22359i 0 9.64575
512.3 1.13198 + 0.653548i 0 −1.14575 − 1.98450i 6.39086 − 3.68977i 0 1.32288 − 2.29129i − 8.22359i 0 9.64575
512.4 3.03622 + 1.75296i 0 4.14575 + 7.18065i 1.07558 − 0.620984i 0 −1.32288 + 2.29129i 15.0457i 0 4.35425
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.3.r.c 8
3.b odd 2 1 inner 567.3.r.c 8
9.c even 3 1 21.3.b.a ✓ 4
9.c even 3 1 inner 567.3.r.c 8
9.d odd 6 1 21.3.b.a ✓ 4
9.d odd 6 1 inner 567.3.r.c 8
36.f odd 6 1 336.3.d.c 4
36.h even 6 1 336.3.d.c 4
45.h odd 6 1 525.3.c.a 4
45.j even 6 1 525.3.c.a 4
45.k odd 12 2 525.3.f.a 8
45.l even 12 2 525.3.f.a 8
63.g even 3 1 147.3.h.e 8
63.h even 3 1 147.3.h.e 8
63.i even 6 1 147.3.h.c 8
63.j odd 6 1 147.3.h.e 8
63.k odd 6 1 147.3.h.c 8
63.l odd 6 1 147.3.b.f 4
63.n odd 6 1 147.3.h.e 8
63.o even 6 1 147.3.b.f 4
63.s even 6 1 147.3.h.c 8
63.t odd 6 1 147.3.h.c 8
72.j odd 6 1 1344.3.d.f 4
72.l even 6 1 1344.3.d.b 4
72.n even 6 1 1344.3.d.f 4
72.p odd 6 1 1344.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a ✓ 4 9.c even 3 1
21.3.b.a ✓ 4 9.d odd 6 1
147.3.b.f 4 63.l odd 6 1
147.3.b.f 4 63.o even 6 1
147.3.h.c 8 63.i even 6 1
147.3.h.c 8 63.k odd 6 1
147.3.h.c 8 63.s even 6 1
147.3.h.c 8 63.t odd 6 1
147.3.h.e 8 63.g even 3 1
147.3.h.e 8 63.h even 3 1
147.3.h.e 8 63.j odd 6 1
147.3.h.e 8 63.n odd 6 1
336.3.d.c 4 36.f odd 6 1
336.3.d.c 4 36.h even 6 1
525.3.c.a 4 45.h odd 6 1
525.3.c.a 4 45.j even 6 1
525.3.f.a 8 45.k odd 12 2
525.3.f.a 8 45.l even 12 2
567.3.r.c 8 1.a even 1 1 trivial
567.3.r.c 8 3.b odd 2 1 inner
567.3.r.c 8 9.c even 3 1 inner
567.3.r.c 8 9.d odd 6 1 inner
1344.3.d.b 4 72.l even 6 1
1344.3.d.b 4 72.p odd 6 1
1344.3.d.f 4 72.j odd 6 1
1344.3.d.f 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{8} - 14T_{2}^{6} + 175T_{2}^{4} - 294T_{2}^{2} + 441 \) Copy content Toggle raw display
\( T_{5}^{8} - 56T_{5}^{6} + 3052T_{5}^{4} - 4704T_{5}^{2} + 7056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 14 T^{6} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 56 T^{6} + \cdots + 7056 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 56 T^{6} + \cdots + 112896 \) Copy content Toggle raw display
$13$ \( (T^{4} - 18 T^{3} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 168 T^{2} + 3024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 166)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 672 T^{6} + \cdots + 146313216 \) Copy content Toggle raw display
$29$ \( T^{8} - 392 T^{6} + \cdots + 740710656 \) Copy content Toggle raw display
$31$ \( (T^{4} + 68 T^{3} + \cdots + 1272384)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 1356)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 17138407700736 \) Copy content Toggle raw display
$43$ \( (T^{4} - 80 T^{3} + \cdots + 1817104)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 3033950846976 \) Copy content Toggle raw display
$53$ \( (T^{4} + 11256 T^{2} + 2543184)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 213354983768976 \) Copy content Toggle raw display
$61$ \( (T^{4} - 78 T^{3} + \cdots + 1387684)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 12 T^{3} + \cdots + 169744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 9576 T^{2} + 22888656)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T - 4668)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 64 T^{3} + \cdots + 64770304)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + 20216 T^{2} + 64754256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 197136)^{2} \) Copy content Toggle raw display
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