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Properties

Label 84.6.i.c
Level $84$
Weight $6$
Character orbit 84.i
Analytic conductor $13.472$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,6,Mod(25,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 84.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: \(13.4722408643\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 \beta_1 q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} + 30 \beta_1 + 10) q^{7} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 \beta_1 q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} + 30 \beta_1 + 10) q^{7} + ( - 81 \beta_1 - 81) q^{9} + (\beta_{7} + \beta_{5} + 3 \beta_{4} + \cdots + 2) q^{11}+ \cdots + ( - 162 \beta_{7} + 243 \beta_{6} + \cdots + 9153) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 202509581 \nu^{7} - 15579673694 \nu^{6} - 97000119635 \nu^{5} - 10465172425490 \nu^{4} + \cdots - 66\!\cdots\!28 ) / 59\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4074390615 \nu^{7} - 75239346984 \nu^{6} - 2524519237925 \nu^{5} - 9170584770500 \nu^{4} + \cdots + 91\!\cdots\!92 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1106077550275 \nu^{7} - 64933012777282 \nu^{6} - 404277402169405 \nu^{5} + \cdots - 27\!\cdots\!84 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2298609090161 \nu^{7} - 10640696920522 \nu^{6} - 380580964066225 \nu^{5} + \cdots - 12\!\cdots\!04 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3837039574679 \nu^{7} + 46767932099422 \nu^{6} + \cdots - 18\!\cdots\!96 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 608647346745 \nu^{7} + 5497197755738 \nu^{6} - 365756430948295 \nu^{5} + \cdots + 77\!\cdots\!76 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15200991659713 \nu^{7} + 29669373236422 \nu^{6} + \cdots + 12\!\cdots\!44 ) / 36\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -16\beta_{7} - 15\beta_{6} - \beta_{5} - 3\beta_{4} - 28\beta_{3} - 17\beta_{2} - 118\beta _1 - 2 ) / 252 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 6\beta_{6} - 29\beta_{5} + 39\beta_{4} + 217\beta_{3} + 221\beta_{2} - 22070\beta _1 - 22066 ) / 63 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1033 \beta_{7} + 1968 \beta_{6} - 1220 \beta_{5} + 1407 \beta_{4} + 1594 \beta_{3} + 1289 \beta_{2} + \cdots - 56104 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3187 \beta_{7} + 3396 \beta_{6} - 209 \beta_{5} - 627 \beta_{4} - 12936 \beta_{3} + 2978 \beta_{2} + \cdots - 418 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 995191 \beta_{7} - 2985573 \beta_{6} + 4334173 \beta_{5} - 9310128 \beta_{4} - 2044322 \beta_{3} + \cdots + 421240394 ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2915935 \beta_{7} - 4285635 \beta_{6} + 3189875 \beta_{5} - 3463815 \beta_{4} - 3737755 \beta_{3} + \cdots + 1038478372 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3473034028 \beta_{7} - 4111074465 \beta_{6} + 638040437 \beta_{5} + 1914121311 \beta_{4} + \cdots + 1276080874 ) / 252 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
4.59067 − 7.95128i
−5.49618 + 9.51967i
13.1471 − 22.7714i
−11.2416 + 19.4709i
4.59067 + 7.95128i
−5.49618 − 9.51967i
13.1471 + 22.7714i
−11.2416 − 19.4709i
0 4.50000 − 7.79423i 0 −39.3359 − 68.1317i 0 −100.606 + 81.7641i 0 −40.5000 − 70.1481i 0
25.2 0 4.50000 − 7.79423i 0 −23.0577 − 39.9371i 0 112.271 + 64.8240i 0 −40.5000 − 70.1481i 0
25.3 0 4.50000 − 7.79423i 0 15.9808 + 27.6796i 0 85.7043 − 97.2716i 0 −40.5000 − 70.1481i 0
25.4 0 4.50000 − 7.79423i 0 46.4128 + 80.3893i 0 −118.369 + 52.8745i 0 −40.5000 − 70.1481i 0
37.1 0 4.50000 + 7.79423i 0 −39.3359 + 68.1317i 0 −100.606 − 81.7641i 0 −40.5000 + 70.1481i 0
37.2 0 4.50000 + 7.79423i 0 −23.0577 + 39.9371i 0 112.271 − 64.8240i 0 −40.5000 + 70.1481i 0
37.3 0 4.50000 + 7.79423i 0 15.9808 − 27.6796i 0 85.7043 + 97.2716i 0 −40.5000 + 70.1481i 0
37.4 0 4.50000 + 7.79423i 0 46.4128 − 80.3893i 0 −118.369 − 52.8745i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.6.i.c ✓ 8
3.b odd 2 1 252.6.k.f 8
4.b odd 2 1 336.6.q.i 8
7.b odd 2 1 588.6.i.o 8
7.c even 3 1 inner 84.6.i.c ✓ 8
7.c even 3 1 588.6.a.n 4
7.d odd 6 1 588.6.a.p 4
7.d odd 6 1 588.6.i.o 8
21.h odd 6 1 252.6.k.f 8
28.g odd 6 1 336.6.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.i.c ✓ 8 1.a even 1 1 trivial
84.6.i.c ✓ 8 7.c even 3 1 inner
252.6.k.f 8 3.b odd 2 1
252.6.k.f 8 21.h odd 6 1
336.6.q.i 8 4.b odd 2 1
336.6.q.i 8 28.g odd 6 1
588.6.a.n 4 7.c even 3 1
588.6.a.p 4 7.d odd 6 1
588.6.i.o 8 7.b odd 2 1
588.6.i.o 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 8977 T_{5}^{6} + 165000 T_{5}^{5} + 69822853 T_{5}^{4} + 740602500 T_{5}^{3} + \cdots + 115856721032976 \) acting on \(S_{6}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 115856721032976 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{4} + 602 T^{3} + \cdots + 755795447424)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 333378202056336)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 15\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 359515701753932)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 53\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 37\!\cdots\!72)^{2} \) Copy content Toggle raw display
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