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Properties

Label 126.9.n.c
Level $126$
Weight $9$
Character orbit 126.n
Analytic conductor $51.330$
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] = [126,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(51.3297048677\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 6683 x^{10} + 105006 x^{9} + 34690411 x^{8} + 548172728 x^{7} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{6}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 42)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - 128 \beta_1 - 128) q^{4} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 126) q^{5}+ \cdots - 128 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - 128 \beta_1 - 128) q^{4} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 126) q^{5}+ \cdots + (5116 \beta_{11} + 11660 \beta_{10} + \cdots + 130616) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 768 q^{4} + 1122 q^{5} - 4434 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 768 q^{4} + 1122 q^{5} - 4434 q^{7} + 4224 q^{10} - 9606 q^{11} + 19968 q^{14} - 98304 q^{16} + 227388 q^{17} + 315408 q^{19} + 557568 q^{22} + 663480 q^{23} + 601008 q^{25} + 585984 q^{26} + 398592 q^{28} - 5944884 q^{29} - 3214278 q^{31} + 1019784 q^{35} + 1607760 q^{37} + 986880 q^{38} - 540672 q^{40} + 12627648 q^{43} - 1229568 q^{44} - 4608768 q^{46} + 7830240 q^{47} + 16477242 q^{49} + 16565760 q^{50} - 4283904 q^{52} + 33200358 q^{53} + 8110080 q^{56} + 5576064 q^{58} + 29657382 q^{59} + 58344096 q^{61} + 25165824 q^{64} - 30909216 q^{65} + 7977444 q^{67} - 29105664 q^{68} - 59774976 q^{70} - 117213264 q^{71} - 170286636 q^{73} + 80311296 q^{74} + 3340686 q^{77} + 35420490 q^{79} - 18382848 q^{80} + 154232832 q^{82} + 416634600 q^{85} + 71958528 q^{86} - 35684352 q^{88} - 78064092 q^{89} + 335513292 q^{91} - 169850880 q^{92} - 18456576 q^{94} + 250727796 q^{95} + 150533376 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 6683 x^{10} + 105006 x^{9} + 34690411 x^{8} + 548172728 x^{7} + \cdots + 30\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!86 \nu^{11} + \cdots + 48\!\cdots\!92 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{11} + \cdots + 43\!\cdots\!24 ) / 38\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\!\cdots\!37 \nu^{11} + \cdots + 68\!\cdots\!04 ) / 49\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!64 \nu^{11} + \cdots - 86\!\cdots\!68 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!12 \nu^{11} + \cdots + 22\!\cdots\!16 ) / 43\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!62 \nu^{11} + \cdots + 29\!\cdots\!16 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!52 \nu^{11} + \cdots - 24\!\cdots\!64 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!04 \nu^{11} + \cdots - 75\!\cdots\!52 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{11} + \cdots - 53\!\cdots\!44 ) / 98\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!86 \nu^{11} + \cdots - 22\!\cdots\!52 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!94 \nu^{11} + \cdots + 72\!\cdots\!28 ) / 19\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5 \beta_{11} + 5 \beta_{10} + 10 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 15 \beta_{6} - 6 \beta_{5} + \cdots + 2 ) / 672 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 74 \beta_{11} - 31 \beta_{10} + 31 \beta_{9} - 37 \beta_{8} - 77 \beta_{7} - 3 \beta_{6} + \cdots - 499319 ) / 224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 23491 \beta_{11} - 46882 \beta_{10} - 23441 \beta_{9} - 23681 \beta_{8} - 46817 \beta_{7} + \cdots - 19793895 ) / 672 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 543303 \beta_{11} - 169655 \beta_{10} - 339310 \beta_{9} - 380554 \beta_{8} + 380554 \beta_{7} + \cdots + 380554 ) / 224 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22413082 \beta_{11} + 109373171 \beta_{10} - 109373171 \beta_{9} + 92331913 \beta_{8} + \cdots + 96506987983 ) / 672 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 670559331 \beta_{11} + 1842895410 \beta_{10} + 921447705 \beta_{9} + 2638131273 \beta_{8} + \cdots + 7896264848363 ) / 224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 553888077125 \beta_{11} + 507375414005 \beta_{10} + 1014750828010 \beta_{9} + 168033488614 \beta_{8} + \cdots - 168033488614 ) / 672 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9802335487634 \beta_{11} - 4894242561043 \beta_{10} + 4894242561043 \beta_{9} - 2938413657937 \beta_{8} + \cdots - 34\!\cdots\!83 ) / 224 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16\!\cdots\!57 \beta_{11} + \cdots - 22\!\cdots\!77 ) / 672 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 61\!\cdots\!93 \beta_{11} + \cdots + 47\!\cdots\!38 ) / 224 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 54\!\cdots\!78 \beta_{11} + \cdots + 11\!\cdots\!49 ) / 672 \) 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
−12.6321 + 21.8795i
−13.8894 + 24.0571i
27.0215 − 46.8026i
−1.61905 + 2.80427i
34.6228 − 59.9685i
−32.5038 + 56.2982i
−12.6321 − 21.8795i
−13.8894 − 24.0571i
27.0215 + 46.8026i
−1.61905 − 2.80427i
34.6228 + 59.9685i
−32.5038 − 56.2982i
−5.65685 + 9.79796i 0 −64.0000 − 110.851i −607.249 − 350.595i 0 −2120.49 − 1126.19i 1448.15 0 6870.24 − 3966.53i
19.2 −5.65685 + 9.79796i 0 −64.0000 − 110.851i 127.532 + 73.6307i 0 1148.13 + 2108.70i 1448.15 0 −1442.86 + 833.036i
19.3 −5.65685 + 9.79796i 0 −64.0000 − 110.851i 666.879 + 385.023i 0 1263.93 − 2041.39i 1448.15 0 −7544.87 + 4356.03i
19.4 5.65685 − 9.79796i 0 −64.0000 − 110.851i −537.676 − 310.427i 0 −2285.79 − 734.809i −1448.15 0 −6083.11 + 3512.08i
19.5 5.65685 − 9.79796i 0 −64.0000 − 110.851i −124.890 − 72.1052i 0 1980.48 + 1357.39i −1448.15 0 −1412.97 + 815.777i
19.6 5.65685 − 9.79796i 0 −64.0000 − 110.851i 1036.40 + 598.368i 0 −2203.25 + 954.186i −1448.15 0 11725.6 − 6769.76i
73.1 −5.65685 − 9.79796i 0 −64.0000 + 110.851i −607.249 + 350.595i 0 −2120.49 + 1126.19i 1448.15 0 6870.24 + 3966.53i
73.2 −5.65685 − 9.79796i 0 −64.0000 + 110.851i 127.532 − 73.6307i 0 1148.13 − 2108.70i 1448.15 0 −1442.86 − 833.036i
73.3 −5.65685 − 9.79796i 0 −64.0000 + 110.851i 666.879 − 385.023i 0 1263.93 + 2041.39i 1448.15 0 −7544.87 − 4356.03i
73.4 5.65685 + 9.79796i 0 −64.0000 + 110.851i −537.676 + 310.427i 0 −2285.79 + 734.809i −1448.15 0 −6083.11 − 3512.08i
73.5 5.65685 + 9.79796i 0 −64.0000 + 110.851i −124.890 + 72.1052i 0 1980.48 − 1357.39i −1448.15 0 −1412.97 − 815.777i
73.6 5.65685 + 9.79796i 0 −64.0000 + 110.851i 1036.40 − 598.368i 0 −2203.25 − 954.186i −1448.15 0 11725.6 + 6769.76i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
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 126.9.n.c 12
3.b odd 2 1 42.9.g.b ✓ 12
7.d odd 6 1 inner 126.9.n.c 12
21.g even 6 1 42.9.g.b ✓ 12
21.g even 6 1 294.9.c.b 12
21.h odd 6 1 294.9.c.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.9.g.b ✓ 12 3.b odd 2 1
42.9.g.b ✓ 12 21.g even 6 1
126.9.n.c 12 1.a even 1 1 trivial
126.9.n.c 12 7.d odd 6 1 inner
294.9.c.b 12 21.g even 6 1
294.9.c.b 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 1122 T_{5}^{11} - 842937 T_{5}^{10} + 1416597930 T_{5}^{9} + 921696727761 T_{5}^{8} + \cdots + 72\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 128 T^{2} + 16384)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 30\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 94\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 79\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 45\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 23\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
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