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Properties

Label 84.7.m.b
Level $84$
Weight $7$
Character orbit 84.m
Analytic conductor $19.325$
Analytic rank $0$
Dimension $8$
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] = [84,7,Mod(61,84)]
 
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, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.61");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 84.m (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: \(19.3245430241\)
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: \( x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + \cdots + 5192355364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + (9 \beta_1 + 9) q^{3} + ( - \beta_{2} + 25 \beta_1 - 49) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{2} + \cdots + 37) q^{7}+ \cdots + 243 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_1 + 9) q^{3} + ( - \beta_{2} + 25 \beta_1 - 49) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{2} + \cdots + 37) q^{7}+ \cdots + ( - 729 \beta_{7} + 486 \beta_{6} + \cdots + 24057) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} - 294 q^{5} + 232 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 108 q^{3} - 294 q^{5} + 232 q^{7} + 972 q^{9} + 378 q^{11} - 5292 q^{15} + 852 q^{17} + 3690 q^{19} + 3942 q^{21} + 15600 q^{23} + 3386 q^{25} - 68604 q^{29} + 23028 q^{31} + 10206 q^{33} + 93828 q^{35} + 15914 q^{37} + 25326 q^{39} - 170044 q^{43} - 71442 q^{45} + 102180 q^{47} + 157340 q^{49} + 7668 q^{51} + 196410 q^{53} + 66420 q^{57} - 662550 q^{59} - 23928 q^{61} + 50058 q^{63} + 14892 q^{65} + 774838 q^{67} - 721896 q^{71} - 1219050 q^{73} + 91422 q^{75} + 1584738 q^{77} - 493868 q^{79} - 236196 q^{81} - 1329816 q^{85} - 926154 q^{87} + 604260 q^{89} + 3831690 q^{91} + 207252 q^{93} + 448944 q^{95} + 183708 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + \cdots + 5192355364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 896877863432 \nu^{7} + 6041228263913 \nu^{6} + \cdots - 31\!\cdots\!34 ) / 73\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 183863909546285 \nu^{7} + \cdots + 89\!\cdots\!72 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39954530257266 \nu^{7} + 880422014972009 \nu^{6} + \cdots + 51\!\cdots\!26 ) / 43\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 718139017476094 \nu^{7} + \cdots + 10\!\cdots\!02 ) / 73\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!01 \nu^{7} + \cdots + 97\!\cdots\!32 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 85566866050425 \nu^{7} + \cdots + 39\!\cdots\!66 ) / 56\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!15 \nu^{7} + \cdots - 15\!\cdots\!56 ) / 73\!\cdots\!34 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} + 2\beta_{6} + 12\beta_{5} - 2\beta_{4} - 3\beta_{3} - 6\beta_{2} + 41\beta _1 - 4 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 44\beta_{5} - 15\beta_{4} - 17\beta_{3} + 44\beta_{2} + 11089\beta _1 - 11133 ) / 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1324 \beta_{7} - 1727 \beta_{6} - 3930 \beta_{5} - 403 \beta_{4} + 403 \beta_{3} + 7860 \beta_{2} + \cdots - 590288 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30080 \beta_{7} - 27625 \beta_{6} - 150868 \beta_{5} + 27625 \beta_{4} + 30080 \beta_{3} + 75434 \beta_{2} + \cdots + 47809 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 202582 \beta_{7} + 202582 \beta_{6} - 3069176 \beta_{5} + 1276883 \beta_{4} + 1074301 \beta_{3} + \cdots + 527771423 ) / 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 43604781 \beta_{7} + 49818560 \beta_{6} + 121787402 \beta_{5} + 6213779 \beta_{4} + \cdots + 21577820576 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2012777654 \beta_{7} + 1730791115 \beta_{6} + 9772639964 \beta_{5} - 1730791115 \beta_{4} + \cdots - 3155528867 ) / 21 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−4.36471 + 7.55990i
−8.68580 + 15.0442i
−5.94197 + 10.2918i
19.9925 − 34.6280i
−4.36471 − 7.55990i
−8.68580 − 15.0442i
−5.94197 − 10.2918i
19.9925 + 34.6280i
0 13.5000 − 7.79423i 0 −167.007 − 96.4218i 0 −243.806 − 241.263i 0 121.500 − 210.444i 0
61.2 0 13.5000 − 7.79423i 0 −93.2075 − 53.8134i 0 280.014 + 198.094i 0 121.500 − 210.444i 0
61.3 0 13.5000 − 7.79423i 0 −0.0783677 − 0.0452456i 0 −218.562 + 264.348i 0 121.500 − 210.444i 0
61.4 0 13.5000 − 7.79423i 0 113.293 + 65.4099i 0 298.353 − 169.217i 0 121.500 − 210.444i 0
73.1 0 13.5000 + 7.79423i 0 −167.007 + 96.4218i 0 −243.806 + 241.263i 0 121.500 + 210.444i 0
73.2 0 13.5000 + 7.79423i 0 −93.2075 + 53.8134i 0 280.014 − 198.094i 0 121.500 + 210.444i 0
73.3 0 13.5000 + 7.79423i 0 −0.0783677 + 0.0452456i 0 −218.562 − 264.348i 0 121.500 + 210.444i 0
73.4 0 13.5000 + 7.79423i 0 113.293 − 65.4099i 0 298.353 + 169.217i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.7.m.b ✓ 8
3.b odd 2 1 252.7.z.e 8
4.b odd 2 1 336.7.bh.a 8
7.b odd 2 1 588.7.m.b 8
7.c even 3 1 588.7.d.a 8
7.c even 3 1 588.7.m.b 8
7.d odd 6 1 inner 84.7.m.b ✓ 8
7.d odd 6 1 588.7.d.a 8
21.g even 6 1 252.7.z.e 8
28.f even 6 1 336.7.bh.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.m.b ✓ 8 1.a even 1 1 trivial
84.7.m.b ✓ 8 7.d odd 6 1 inner
252.7.z.e 8 3.b odd 2 1
252.7.z.e 8 21.g even 6 1
336.7.bh.a 8 4.b odd 2 1
336.7.bh.a 8 28.f even 6 1
588.7.d.a 8 7.c even 3 1
588.7.d.a 8 7.d odd 6 1
588.7.m.b 8 7.b odd 2 1
588.7.m.b 8 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 294 T_{5}^{7} + 10275 T_{5}^{6} - 5449878 T_{5}^{5} - 117296451 T_{5}^{4} + \cdots + 60368490000 \) acting on \(S_{7}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 60368490000 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 90\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 86\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 33\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 27\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 59\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
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