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Properties

Label 126.7.n.c
Level $126$
Weight $7$
Character orbit 126.n
Analytic conductor $28.987$
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] = [126,7,Mod(19,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("126.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 126.n (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: \(28.9868145361\)
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: \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (32 \beta_1 - 32) q^{4} + ( - \beta_{7} + 3 \beta_{6} + \cdots + 56) q^{5}+ \cdots - 32 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (32 \beta_1 - 32) q^{4} + ( - \beta_{7} + 3 \beta_{6} + \cdots + 56) q^{5}+ \cdots + ( - 5168 \beta_{7} - 12680 \beta_{6} + \cdots + 349424) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{4} + 336 q^{5} + 652 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{4} + 336 q^{5} + 652 q^{7} - 2016 q^{10} + 1356 q^{11} - 2064 q^{14} - 4096 q^{16} + 17304 q^{17} - 32004 q^{19} + 25248 q^{22} + 4128 q^{23} + 4664 q^{25} + 4704 q^{26} - 7552 q^{28} + 30312 q^{29} - 3108 q^{31} - 98028 q^{35} - 6124 q^{37} - 155568 q^{38} + 64512 q^{40} - 297376 q^{43} + 43392 q^{44} - 194064 q^{46} - 313908 q^{47} + 32432 q^{49} - 38784 q^{50} + 255360 q^{52} - 278484 q^{53} + 125952 q^{56} - 169824 q^{58} + 835464 q^{59} - 995316 q^{61} + 262144 q^{64} - 8316 q^{65} + 648808 q^{67} - 553728 q^{68} - 1572816 q^{70} - 190128 q^{71} - 1617084 q^{73} + 1158144 q^{74} - 1456224 q^{77} + 70096 q^{79} - 344064 q^{80} + 66528 q^{82} + 2190984 q^{85} + 573024 q^{86} - 403968 q^{88} - 739116 q^{89} + 2233752 q^{91} - 264192 q^{92} + 3795120 q^{94} - 725640 q^{95} + 3532320 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10737582 \nu^{7} + 884171859 \nu^{6} - 3763944460 \nu^{5} + 189328804101 \nu^{4} + \cdots + 41\!\cdots\!50 ) / 33\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 512915233 \nu^{7} + 35396730954 \nu^{6} - 150684878760 \nu^{5} + 14474316021931 \nu^{4} + \cdots + 16\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14893 \nu^{7} - 236741 \nu^{6} + 3320715 \nu^{5} + 8132426 \nu^{4} + 331237078 \nu^{3} + \cdots + 846958238000 ) / 106489495000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10421298761 \nu^{7} - 383100933507 \nu^{6} + 14696779653705 \nu^{5} - 107198920144948 \nu^{4} + \cdots - 61\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4954059529 \nu^{7} + 53601861423 \nu^{6} - 2878465568645 \nu^{5} + 14314185841272 \nu^{4} + \cdots + 11\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 98330981563 \nu^{7} + 465055977756 \nu^{6} - 28111572790890 \nu^{5} + \cdots - 20\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112677227274 \nu^{7} + 39166570437 \nu^{6} + 18385233372395 \nu^{5} + 149484497566193 \nu^{4} + \cdots + 11\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 4\beta_{4} + \beta_{2} + 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 54\beta_{3} - 53\beta_{2} + 849\beta _1 - 849 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 286\beta_{7} + 434\beta_{6} - 143\beta_{5} + 217\beta_{4} + 477\beta_{3} - 143\beta_{2} - 1911 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 427\beta_{7} + 1137\beta_{6} + 427\beta_{5} + 2274\beta_{4} + 15064\beta_{2} - 139071\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23747 \beta_{7} - 45681 \beta_{6} + 47494 \beta_{5} + 45681 \beta_{4} - 132588 \beta_{3} + \cdots + 644823 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 260622 \beta_{7} - 658058 \beta_{6} + 130311 \beta_{5} - 329029 \beta_{4} - 3745254 \beta_{3} + \cdots + 25540707 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4421299 \beta_{7} - 9613689 \beta_{6} - 4421299 \beta_{5} - 19227378 \beta_{4} + \cdots + 180033087 \beta_1 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
7.51287 − 13.0127i
−6.30576 + 10.9219i
−4.86132 + 8.42006i
4.65421 − 8.06134i
7.51287 + 13.0127i
−6.30576 − 10.9219i
−4.86132 − 8.42006i
4.65421 + 8.06134i
−2.82843 + 4.89898i 0 −16.0000 − 27.7128i 10.7486 + 6.20573i 0 195.705 − 281.689i 181.019 0 −60.8035 + 35.1049i
19.2 −2.82843 + 4.89898i 0 −16.0000 − 27.7128i 162.347 + 93.7310i 0 141.244 + 312.569i 181.019 0 −918.372 + 530.222i
19.3 2.82843 − 4.89898i 0 −16.0000 − 27.7128i −111.836 − 64.5687i 0 298.743 + 168.528i −181.019 0 −632.642 + 365.256i
19.4 2.82843 − 4.89898i 0 −16.0000 − 27.7128i 106.741 + 61.6269i 0 −309.691 − 147.446i −181.019 0 603.818 − 348.614i
73.1 −2.82843 − 4.89898i 0 −16.0000 + 27.7128i 10.7486 − 6.20573i 0 195.705 + 281.689i 181.019 0 −60.8035 − 35.1049i
73.2 −2.82843 − 4.89898i 0 −16.0000 + 27.7128i 162.347 − 93.7310i 0 141.244 − 312.569i 181.019 0 −918.372 − 530.222i
73.3 2.82843 + 4.89898i 0 −16.0000 + 27.7128i −111.836 + 64.5687i 0 298.743 − 168.528i −181.019 0 −632.642 − 365.256i
73.4 2.82843 + 4.89898i 0 −16.0000 + 27.7128i 106.741 − 61.6269i 0 −309.691 + 147.446i −181.019 0 603.818 + 348.614i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
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 126.7.n.c 8
3.b odd 2 1 14.7.d.a ✓ 8
7.d odd 6 1 inner 126.7.n.c 8
12.b even 2 1 112.7.s.c 8
21.c even 2 1 98.7.d.c 8
21.g even 6 1 14.7.d.a ✓ 8
21.g even 6 1 98.7.b.c 8
21.h odd 6 1 98.7.b.c 8
21.h odd 6 1 98.7.d.c 8
84.j odd 6 1 112.7.s.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.d.a ✓ 8 3.b odd 2 1
14.7.d.a ✓ 8 21.g even 6 1
98.7.b.c 8 21.g even 6 1
98.7.b.c 8 21.h odd 6 1
98.7.d.c 8 21.c even 2 1
98.7.d.c 8 21.h odd 6 1
112.7.s.c 8 12.b even 2 1
112.7.s.c 8 84.j odd 6 1
126.7.n.c 8 1.a even 1 1 trivial
126.7.n.c 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 336 T_{5}^{7} + 22866 T_{5}^{6} + 4961376 T_{5}^{5} - 364731309 T_{5}^{4} + \cdots + 13\!\cdots\!25 \) acting on \(S_{7}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 57\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 50\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 56\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 79\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 83\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
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