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Properties

Label 126.10.g.i
Level $126$
Weight $10$
Character orbit 126.g
Analytic conductor $64.895$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 789419 x^{10} + 45865502 x^{9} + 486536495501 x^{8} + 18032405954777 x^{7} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{9}\cdot 7^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 16 \beta_1 q^{2} + ( - 256 \beta_1 - 256) q^{4} + (\beta_{6} - \beta_{3} - 342 \beta_1) q^{5} + (\beta_{6} - \beta_{4} - \beta_{3} + \cdots - 331) q^{7}+ \cdots - 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 \beta_1 q^{2} + ( - 256 \beta_1 - 256) q^{4} + (\beta_{6} - \beta_{3} - 342 \beta_1) q^{5} + (\beta_{6} - \beta_{4} - \beta_{3} + \cdots - 331) q^{7}+ \cdots + (1568 \beta_{11} + 8176 \beta_{10} + \cdots - 73381328) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 96 q^{2} - 1536 q^{4} + 2051 q^{5} - 5510 q^{7} - 49152 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 96 q^{2} - 1536 q^{4} + 2051 q^{5} - 5510 q^{7} - 49152 q^{8} - 32816 q^{10} - 54707 q^{11} - 10094 q^{13} - 7024 q^{14} - 393216 q^{16} + 337414 q^{17} - 129045 q^{19} - 1050112 q^{20} - 1750624 q^{22} + 1339210 q^{23} - 2884061 q^{25} - 80752 q^{26} + 1298176 q^{28} + 9030830 q^{29} + 2590042 q^{31} + 6291456 q^{32} + 10797248 q^{34} - 16047908 q^{35} + 9981273 q^{37} + 2064720 q^{38} - 8400896 q^{40} - 35820288 q^{41} + 104276670 q^{43} - 14004992 q^{44} - 21427360 q^{46} + 47268060 q^{47} - 82296318 q^{49} - 92289952 q^{50} + 1292032 q^{52} + 55038429 q^{53} - 149798446 q^{55} + 22568960 q^{56} + 72246640 q^{58} - 61175107 q^{59} - 90035918 q^{61} + 82881344 q^{62} + 201326592 q^{64} + 76007430 q^{65} + 169817453 q^{67} + 86377984 q^{68} - 349922320 q^{70} + 5480608 q^{71} - 299018881 q^{73} - 159700368 q^{74} + 66071040 q^{76} - 944159761 q^{77} + 368529472 q^{79} + 134414336 q^{80} - 286562304 q^{82} - 490229138 q^{83} + 2437673804 q^{85} + 834213360 q^{86} + 224079872 q^{88} + 501970266 q^{89} - 2166173333 q^{91} - 685675520 q^{92} - 756288960 q^{94} - 857622986 q^{95} + 1463869190 q^{97} - 1318724208 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 789419 x^{10} + 45865502 x^{9} + 486536495501 x^{8} + 18032405954777 x^{7} + \cdots + 41\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 14\!\cdots\!67 \nu^{11} + \cdots + 76\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 74\!\cdots\!74 \nu^{11} + \cdots + 78\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\!\cdots\!01 \nu^{11} + \cdots + 55\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 98\!\cdots\!73 \nu^{11} + \cdots - 10\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!99 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!08 \nu^{11} + \cdots + 30\!\cdots\!24 ) / 51\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!99 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!96 \nu^{11} + \cdots - 29\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!93 \nu^{11} + \cdots - 26\!\cdots\!00 ) / 91\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!27 \nu^{11} + \cdots + 48\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!92 \nu^{11} + \cdots + 36\!\cdots\!40 ) / 18\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{9} - 16\beta_{8} + \beta_{7} + 5\beta_{6} + 5\beta_{5} + 8\beta_{4} - 10\beta_{3} + 46\beta _1 + 42 ) / 252 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 286 \beta_{11} - 411 \beta_{10} - 431 \beta_{9} - 1513 \beta_{8} + 11313 \beta_{6} - 785 \beta_{5} + \cdots + 145 ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 621493 \beta_{11} - 213768 \beta_{10} - 530701 \beta_{9} + 3019448 \beta_{8} - 621493 \beta_{7} + \cdots - 2939835454 ) / 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 289523838 \beta_{9} + 457091915 \beta_{8} + 68508322 \beta_{7} + 190464083 \beta_{6} + \cdots - 14698400052126 ) / 126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 336624184681 \beta_{11} + 177121097640 \beta_{10} - 898221436325 \beta_{9} + 1547950659248 \beta_{8} + \cdots + 561597251644 ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1117080980938 \beta_{11} + 12776183332773 \beta_{10} + 39685387220135 \beta_{9} + 45403573201850 \beta_{8} + \cdots + 10\!\cdots\!55 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 60\!\cdots\!78 \beta_{9} + \cdots + 33\!\cdots\!50 ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 68\!\cdots\!98 \beta_{11} + \cdots + 24\!\cdots\!01 ) / 126 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 94\!\cdots\!61 \beta_{11} + \cdots - 24\!\cdots\!26 ) / 252 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 88\!\cdots\!54 \beta_{9} + \cdots - 20\!\cdots\!70 ) / 126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 51\!\cdots\!93 \beta_{11} + \cdots + 61\!\cdots\!84 ) / 252 \) Copy content Toggle raw display

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

Character values

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

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

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
384.004 + 665.114i
−340.699 − 590.108i
35.1592 + 60.8975i
−274.464 − 475.385i
231.107 + 400.289i
−34.6068 − 59.9407i
384.004 − 665.114i
−340.699 + 590.108i
35.1592 − 60.8975i
−274.464 + 475.385i
231.107 − 400.289i
−34.6068 + 59.9407i
8.00000 − 13.8564i 0 −128.000 − 221.703i −874.188 + 1514.14i 0 −5772.81 + 2651.09i −4096.00 0 13987.0 + 24226.2i
37.2 8.00000 − 13.8564i 0 −128.000 − 221.703i −362.830 + 628.440i 0 −35.6254 − 6352.35i −4096.00 0 5805.28 + 10055.0i
37.3 8.00000 − 13.8564i 0 −128.000 − 221.703i −330.093 + 571.739i 0 2080.98 + 6001.93i −4096.00 0 5281.49 + 9147.82i
37.4 8.00000 − 13.8564i 0 −128.000 − 221.703i 367.920 − 637.256i 0 6203.00 + 1369.83i −4096.00 0 −5886.71 − 10196.1i
37.5 8.00000 − 13.8564i 0 −128.000 − 221.703i 978.371 − 1694.59i 0 −4925.38 + 4011.76i −4096.00 0 −15653.9 − 27113.4i
37.6 8.00000 − 13.8564i 0 −128.000 − 221.703i 1246.32 − 2158.69i 0 −305.157 − 6345.12i −4096.00 0 −19941.1 − 34539.1i
109.1 8.00000 + 13.8564i 0 −128.000 + 221.703i −874.188 − 1514.14i 0 −5772.81 − 2651.09i −4096.00 0 13987.0 − 24226.2i
109.2 8.00000 + 13.8564i 0 −128.000 + 221.703i −362.830 − 628.440i 0 −35.6254 + 6352.35i −4096.00 0 5805.28 − 10055.0i
109.3 8.00000 + 13.8564i 0 −128.000 + 221.703i −330.093 − 571.739i 0 2080.98 − 6001.93i −4096.00 0 5281.49 − 9147.82i
109.4 8.00000 + 13.8564i 0 −128.000 + 221.703i 367.920 + 637.256i 0 6203.00 − 1369.83i −4096.00 0 −5886.71 + 10196.1i
109.5 8.00000 + 13.8564i 0 −128.000 + 221.703i 978.371 + 1694.59i 0 −4925.38 − 4011.76i −4096.00 0 −15653.9 + 27113.4i
109.6 8.00000 + 13.8564i 0 −128.000 + 221.703i 1246.32 + 2158.69i 0 −305.157 + 6345.12i −4096.00 0 −19941.1 + 34539.1i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
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.i yes 12
3.b odd 2 1 126.10.g.h ✓ 12
7.c even 3 1 inner 126.10.g.i yes 12
21.h odd 6 1 126.10.g.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.10.g.h ✓ 12 3.b odd 2 1
126.10.g.h ✓ 12 21.h odd 6 1
126.10.g.i yes 12 1.a even 1 1 trivial
126.10.g.i yes 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 2051 T_{5}^{11} + 9404706 T_{5}^{10} - 4837411019 T_{5}^{9} + 34850111917522 T_{5}^{8} + \cdots + 90\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 16 T + 256)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 17\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 83\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 73\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 50\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
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