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Properties

Label 126.9.c.c
Level $126$
Weight $9$
Character orbit 126.c
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(55,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.55");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 126.c (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 7731 x^{10} + 218714 x^{9} + 46944238 x^{8} + 954612102 x^{7} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{33}\cdot 3^{18}\cdot 7^{5} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 128 q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} - 2 \beta_{2} + \cdots + 535) q^{7}+ \cdots - 128 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 128 q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} - 2 \beta_{2} + \cdots + 535) q^{7}+ \cdots + (42 \beta_{11} + 2170 \beta_{10} + \cdots - 32312320) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 1536 q^{4} + 6420 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 1536 q^{4} + 6420 q^{7} - 4344 q^{11} + 12288 q^{14} + 196608 q^{16} - 508416 q^{22} - 499800 q^{23} - 3001476 q^{25} + 821760 q^{28} + 1278408 q^{29} - 2028912 q^{35} + 7068648 q^{37} - 11388024 q^{43} - 556032 q^{44} + 8171520 q^{46} - 12346788 q^{49} - 30019584 q^{50} - 19714968 q^{53} + 1572864 q^{56} - 17696256 q^{58} + 25165824 q^{64} + 93770592 q^{65} - 9394008 q^{67} + 11218944 q^{70} - 5393208 q^{71} - 58512384 q^{74} - 24982968 q^{77} + 134560968 q^{79} - 102074640 q^{85} + 282934272 q^{86} - 65077248 q^{88} - 96105408 q^{91} - 63974400 q^{92} + 378351840 q^{95} - 387747840 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 7731 x^{10} + 218714 x^{9} + 46944238 x^{8} + 954612102 x^{7} + \cdots + 37\!\cdots\!84 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 63\!\cdots\!07 \nu^{11} + \cdots - 16\!\cdots\!08 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!69 \nu^{11} + \cdots - 27\!\cdots\!64 ) / 23\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 81\!\cdots\!24 \nu^{11} + \cdots + 75\!\cdots\!64 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!53 \nu^{11} + \cdots - 38\!\cdots\!52 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19\!\cdots\!64 \nu^{11} + \cdots - 27\!\cdots\!60 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 81\!\cdots\!17 \nu^{11} + \cdots + 15\!\cdots\!16 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!56 \nu^{11} + \cdots + 28\!\cdots\!64 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!11 \nu^{11} + \cdots - 45\!\cdots\!12 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!33 \nu^{11} + \cdots + 48\!\cdots\!88 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 63\!\cdots\!07 \nu^{11} + \cdots - 56\!\cdots\!48 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 54\!\cdots\!09 \nu^{11} + \cdots + 67\!\cdots\!04 ) / 23\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 23 \beta_{10} + 12 \beta_{9} - 56 \beta_{8} - 240 \beta_{7} + 535 \beta_{6} + \cdots + 6048 ) / 36288 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 91 \beta_{11} + 35 \beta_{10} + 1890 \beta_{9} - 4794 \beta_{8} - 5888 \beta_{7} - 2193 \beta_{6} + \cdots - 46744992 ) / 36288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 509 \beta_{11} + 55885 \beta_{10} + 30321 \beta_{9} - 106626 \beta_{8} + 107135 \beta_{7} + \cdots - 531105120 ) / 9072 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 441819 \beta_{11} + 3470467 \beta_{10} + 13069770 \beta_{9} + 29205266 \beta_{8} + \cdots - 212023592928 ) / 36288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17956909 \beta_{11} - 624573093 \beta_{10} - 747761802 \beta_{9} + 5167341226 \beta_{8} + \cdots + 11505977991648 ) / 36288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 180679837 \beta_{11} - 2589014389 \beta_{10} - 6167220420 \beta_{9} + 5896286276 \beta_{8} + \cdots + 92345788388736 ) / 1296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 158954978531 \beta_{11} - 3984548534619 \beta_{10} - 6218598200628 \beta_{9} + \cdots + 94\!\cdots\!04 ) / 36288 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5525647141819 \beta_{11} + 98279432005251 \beta_{10} + 198608363285010 \beta_{9} + \cdots - 29\!\cdots\!64 ) / 12096 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 623294225622808 \beta_{11} + \cdots - 35\!\cdots\!00 ) / 9072 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11\!\cdots\!99 \beta_{11} + \cdots - 63\!\cdots\!32 ) / 36288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 93\!\cdots\!03 \beta_{11} + \cdots + 52\!\cdots\!84 ) / 36288 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
42.6889 − 73.9393i
−18.3977 − 31.8658i
−24.4982 + 42.4322i
−24.4982 − 42.4322i
−18.3977 + 31.8658i
42.6889 + 73.9393i
−26.5796 + 46.0372i
16.2382 + 28.1253i
11.5485 + 20.0026i
11.5485 − 20.0026i
16.2382 − 28.1253i
−26.5796 − 46.0372i
−11.3137 0 128.000 − 730.548i 0 −2148.48 + 1071.83i −1448.15 0 8265.21i
55.2 −11.3137 0 128.000 − 640.234i 0 2336.76 − 551.670i −1448.15 0 7243.42i
55.3 −11.3137 0 128.000 − 561.514i 0 1145.19 − 2110.29i −1448.15 0 6352.81i
55.4 −11.3137 0 128.000 561.514i 0 1145.19 + 2110.29i −1448.15 0 − 6352.81i
55.5 −11.3137 0 128.000 640.234i 0 2336.76 + 551.670i −1448.15 0 − 7243.42i
55.6 −11.3137 0 128.000 730.548i 0 −2148.48 − 1071.83i −1448.15 0 − 8265.21i
55.7 11.3137 0 128.000 − 1217.44i 0 1530.45 − 1850.01i 1448.15 0 − 13773.8i
55.8 11.3137 0 128.000 − 1031.06i 0 −283.934 + 2384.15i 1448.15 0 − 11665.2i
55.9 11.3137 0 128.000 − 200.812i 0 630.018 − 2316.87i 1448.15 0 − 2271.93i
55.10 11.3137 0 128.000 200.812i 0 630.018 + 2316.87i 1448.15 0 2271.93i
55.11 11.3137 0 128.000 1031.06i 0 −283.934 − 2384.15i 1448.15 0 11665.2i
55.12 11.3137 0 128.000 1217.44i 0 1530.45 + 1850.01i 1448.15 0 13773.8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.9.c.c 12
3.b odd 2 1 42.9.c.a ✓ 12
7.b odd 2 1 inner 126.9.c.c 12
12.b even 2 1 336.9.f.c 12
21.c even 2 1 42.9.c.a ✓ 12
84.h odd 2 1 336.9.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.9.c.a ✓ 12 3.b odd 2 1
42.9.c.a ✓ 12 21.c even 2 1
126.9.c.c 12 1.a even 1 1 trivial
126.9.c.c 12 7.b odd 2 1 inner
336.9.f.c 12 12.b even 2 1
336.9.f.c 12 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 3844488 T_{5}^{10} + 5449599517464 T_{5}^{8} + \cdots + 43\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 58\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 92\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 45\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 40\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
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