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Properties

Label 126.10.g.e
Level $126$
Weight $10$
Character orbit 126.g
Analytic conductor $64.895$
Analytic rank $0$
Dimension $6$
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] = [126,10,Mod(37,126)]
 
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, 2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.37");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.g (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: \(64.8945153566\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16 \beta_1 q^{2} + ( - 256 \beta_1 - 256) q^{4} + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \cdots - 3) q^{5}+ \cdots + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 \beta_1 q^{2} + ( - 256 \beta_1 - 256) q^{4} + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \cdots - 3) q^{5}+ \cdots + (140304 \beta_{5} + 1987696 \beta_{4} + \cdots + 327036304) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} - 768 q^{4} + 1085 q^{5} - 6796 q^{7} + 24576 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 48 q^{2} - 768 q^{4} + 1085 q^{5} - 6796 q^{7} + 24576 q^{8} + 17360 q^{10} - 2555 q^{11} + 36140 q^{13} - 43504 q^{14} - 196608 q^{16} - 20759 q^{17} + 1220649 q^{19} - 555520 q^{20} + 81760 q^{22} + 1960903 q^{23} - 863572 q^{25} - 289120 q^{26} + 2435840 q^{28} + 5212292 q^{29} - 9377989 q^{31} - 3145728 q^{32} + 664288 q^{34} - 20361719 q^{35} - 25814913 q^{37} + 19530384 q^{38} + 4444160 q^{40} - 418500 q^{41} + 5888616 q^{43} - 654080 q^{44} + 31374448 q^{46} - 48391269 q^{47} + 108466086 q^{49} + 27634304 q^{50} - 4625920 q^{52} - 102186411 q^{53} - 456557402 q^{55} - 27836416 q^{56} - 41698336 q^{58} - 144220135 q^{59} + 280936871 q^{61} + 300095648 q^{62} + 100663296 q^{64} + 186819738 q^{65} + 170710399 q^{67} - 5314304 q^{68} - 347247824 q^{70} - 939517376 q^{71} + 613838539 q^{73} - 413038608 q^{74} - 624972288 q^{76} + 729499715 q^{77} + 197445809 q^{79} + 71106560 q^{80} + 3348000 q^{82} + 2148362872 q^{83} + 822545038 q^{85} - 47108928 q^{86} - 10465280 q^{88} - 805730427 q^{89} + 17178248 q^{91} - 1003982336 q^{92} - 774260304 q^{94} - 1799421743 q^{95} - 4525836188 q^{97} + 597954960 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2716901 \nu^{5} - 2705071 \nu^{4} + 10923076698 \nu^{3} - 181829734103 \nu^{2} + \cdots - 775127978084460 ) / 778518660033660 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 689158117 \nu^{5} - 35991803473 \nu^{4} - 2954366153866 \nu^{3} - 27838776785169 \nu^{2} + \cdots - 18\!\cdots\!20 ) / 778518660033660 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 694639239 \nu^{5} - 35795315171 \nu^{4} - 3747785917342 \nu^{3} - 26700567042283 \nu^{2} + \cdots - 13\!\cdots\!80 ) / 778518660033660 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 702769377 \nu^{5} + 18970464407 \nu^{4} - 2458100986546 \nu^{3} + 194955077224571 \nu^{2} + \cdots + 37\!\cdots\!80 ) / 778518660033660 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 352672159 \nu^{5} + 9199967499 \nu^{4} - 77151616502 \nu^{3} + 96406628067317 \nu^{2} + \cdots + 10\!\cdots\!10 ) / 389259330016830 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + 3\beta_{3} - 3\beta_{2} + 7\beta _1 + 6 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -15\beta_{5} + 113\beta_{4} - 10\beta_{3} + 59\beta_{2} + 37744\beta _1 + 113 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8054\beta_{5} - 7956\beta_{4} - 4027\beta_{3} + 3929\beta_{2} + 1875737 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 107323\beta_{5} - 503047\beta_{4} + 321969\beta_{3} - 1113417\beta_{2} - 302364503\beta _1 - 302867550 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 25289214 \beta_{5} + 31453904 \beta_{4} - 16859476 \beta_{3} + 19941821 \beta_{2} + 6330121189 \beta _1 + 31453904 ) / 14 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
26.1296 + 45.2579i
9.63015 + 16.6799i
−35.2598 − 61.0717i
26.1296 − 45.2579i
9.63015 − 16.6799i
−35.2598 + 61.0717i
−8.00000 + 13.8564i 0 −128.000 − 221.703i −546.130 + 945.925i 0 6111.84 − 1731.76i 4096.00 0 −8738.08 − 15134.8i
37.2 −8.00000 + 13.8564i 0 −128.000 − 221.703i −84.1442 + 145.742i 0 −3162.16 + 5509.48i 4096.00 0 −1346.31 − 2331.87i
37.3 −8.00000 + 13.8564i 0 −128.000 − 221.703i 1172.77 − 2031.30i 0 −6347.68 − 246.066i 4096.00 0 18764.4 + 32500.9i
109.1 −8.00000 − 13.8564i 0 −128.000 + 221.703i −546.130 − 945.925i 0 6111.84 + 1731.76i 4096.00 0 −8738.08 + 15134.8i
109.2 −8.00000 − 13.8564i 0 −128.000 + 221.703i −84.1442 − 145.742i 0 −3162.16 − 5509.48i 4096.00 0 −1346.31 + 2331.87i
109.3 −8.00000 − 13.8564i 0 −128.000 + 221.703i 1172.77 + 2031.30i 0 −6347.68 + 246.066i 4096.00 0 18764.4 − 32500.9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
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 126.10.g.e 6
3.b odd 2 1 14.10.c.b ✓ 6
7.c even 3 1 inner 126.10.g.e 6
12.b even 2 1 112.10.i.a 6
21.c even 2 1 98.10.c.l 6
21.g even 6 1 98.10.a.h 3
21.g even 6 1 98.10.c.l 6
21.h odd 6 1 14.10.c.b ✓ 6
21.h odd 6 1 98.10.a.g 3
84.n even 6 1 112.10.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.c.b ✓ 6 3.b odd 2 1
14.10.c.b ✓ 6 21.h odd 6 1
98.10.a.g 3 21.h odd 6 1
98.10.a.h 3 21.g even 6 1
98.10.c.l 6 21.c even 2 1
98.10.c.l 6 21.g even 6 1
112.10.i.a 6 12.b even 2 1
112.10.i.a 6 84.n even 6 1
126.10.g.e 6 1.a even 1 1 trivial
126.10.g.e 6 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 1085 T_{5}^{5} + 3950086 T_{5}^{4} + 3870845895 T_{5}^{3} + 7220964872646 T_{5}^{2} + \cdots + 18\!\cdots\!25 \) acting on \(S_{10}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16 T + 256)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 65\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 31\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 368974841338200)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 97\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 22\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 85\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 12\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{3} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
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