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Properties

Label 192.9.b.c
Level $192$
Weight $9$
Character orbit 192.b
Analytic conductor $78.217$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,9,Mod(31,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.31");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.b (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: \(78.2166931317\)
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: \( x^{12} + 782x^{10} + 458643x^{8} + 116337998x^{6} + 22115557057x^{4} + 245751926832x^{2} + 2583966230784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 27 \beta_{6} q^{3} + (\beta_{4} - 3 \beta_1) q^{5} + (\beta_{9} + \beta_{8} - 3 \beta_{7}) q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 \beta_{6} q^{3} + (\beta_{4} - 3 \beta_1) q^{5} + (\beta_{9} + \beta_{8} - 3 \beta_{7}) q^{7} + 2187 q^{9} + ( - 2 \beta_{11} + \cdots - 2069 \beta_{6}) q^{11}+ \cdots + ( - 4374 \beta_{11} + \cdots - 4524903 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 26244 q^{9} + 5736 q^{17} - 184836 q^{25} + 2008800 q^{33} + 22717560 q^{41} + 21977052 q^{49} + 32958576 q^{57} + 49401696 q^{65} + 116904888 q^{73} + 57395628 q^{81} + 668411880 q^{89} + 487299816 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 782x^{10} + 458643x^{8} + 116337998x^{6} + 22115557057x^{4} + 245751926832x^{2} + 2583966230784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 611524 \nu^{10} - 469638584 \nu^{8} - 275443029516 \nu^{6} - 68194454766968 \nu^{4} + \cdots - 77\!\cdots\!68 ) / 21\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7420 \nu^{10} - 4351830 \nu^{8} - 2385878658 \nu^{6} - 209843265170 \nu^{4} + \cdots + 23\!\cdots\!28 ) / 3970467312087 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12512 \nu^{10} + 7338288 \nu^{8} + 2972148816 \nu^{6} + 353848912912 \nu^{4} + \cdots + 44\!\cdots\!00 ) / 3970467312087 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3837457 \nu^{10} - 3055271776 \nu^{8} - 1774992483529 \nu^{6} - 451082078103696 \nu^{4} + \cdots - 48\!\cdots\!00 ) / 968794024149228 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54838615 \nu^{10} - 39348787028 \nu^{8} - 23656878519771 \nu^{6} + \cdots - 69\!\cdots\!64 ) / 87\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 152881 \nu^{11} - 117409646 \nu^{9} - 68860757379 \nu^{7} - 17048613691742 \nu^{5} + \cdots - 20\!\cdots\!88 \nu ) / 65\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} - 782\nu^{9} - 458643\nu^{7} - 117945470\nu^{5} - 22744078609\nu^{3} - 491503853664\nu ) / 43132291506 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3973230233 \nu^{11} + 3050585906014 \nu^{9} + \cdots + 18\!\cdots\!52 \nu ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 427673405 \nu^{11} - 344414132966 \nu^{9} - 203650711702631 \nu^{7} + \cdots - 22\!\cdots\!16 \nu ) / 47\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46694173649 \nu^{11} - 36063001989550 \nu^{9} + \cdots - 61\!\cdots\!52 \nu ) / 85\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 215765296573 \nu^{11} + 166920255246614 \nu^{9} + \cdots + 28\!\cdots\!04 \nu ) / 25\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} + 4\beta_{10} + 24\beta_{8} - 3\beta_{7} + 2\beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 40\beta_{5} + 152\beta_{4} + 2\beta_{3} + 16\beta_{2} - 3127\beta _1 - 100096 ) / 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -391\beta_{11} - 782\beta_{10} + 421241\beta_{6} ) / 576 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -21496\beta_{5} - 53576\beta_{4} + 2246\beta_{3} + 6256\beta_{2} + 1221925\beta _1 - 39137536 ) / 768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 275018 \beta_{11} + 655444 \beta_{10} + 421632 \beta_{9} - 3932664 \beta_{8} + \cdots - 549216886 \beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -725305\beta_{3} - 1223048\beta_{2} + 7960022912 ) / 192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 98726078 \beta_{11} + 279881212 \beta_{10} - 329716224 \beta_{9} + 1679287272 \beta_{8} + \cdots - 300684899266 \beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5141171128 \beta_{5} + 6644545928 \beta_{4} + 794221238 \beta_{3} + 982143088 \beta_{2} + \cdots - 6626991710464 ) / 768 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -18207904621\beta_{11} - 60588129914\beta_{10} + 76262515351475\beta_{6} ) / 576 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2419944005416 \beta_{5} - 2429623639448 \beta_{4} + 402920682818 \beta_{3} + 404130637072 \beta_{2} + \cdots - 28\!\cdots\!96 ) / 768 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13825905031022 \beta_{11} + 53024923491292 \beta_{10} + 101492453716992 \beta_{9} + \cdots - 74\!\cdots\!46 \beta_{6} ) / 2304 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
8.94623 − 15.4953i
1.66885 + 2.89054i
−10.6151 + 18.3859i
−10.6151 − 18.3859i
1.66885 − 2.89054i
8.94623 + 15.4953i
−8.94623 + 15.4953i
−1.66885 − 2.89054i
10.6151 − 18.3859i
10.6151 + 18.3859i
−1.66885 + 2.89054i
−8.94623 − 15.4953i
0 −46.7654 0 − 907.917i 0 − 2864.11i 0 2187.00 0
31.2 0 −46.7654 0 − 599.131i 0 − 956.234i 0 2187.00 0
31.3 0 −46.7654 0 − 186.581i 0 1637.88i 0 2187.00 0
31.4 0 −46.7654 0 186.581i 0 − 1637.88i 0 2187.00 0
31.5 0 −46.7654 0 599.131i 0 956.234i 0 2187.00 0
31.6 0 −46.7654 0 907.917i 0 2864.11i 0 2187.00 0
31.7 0 46.7654 0 − 907.917i 0 2864.11i 0 2187.00 0
31.8 0 46.7654 0 − 599.131i 0 956.234i 0 2187.00 0
31.9 0 46.7654 0 − 186.581i 0 − 1637.88i 0 2187.00 0
31.10 0 46.7654 0 186.581i 0 1637.88i 0 2187.00 0
31.11 0 46.7654 0 599.131i 0 − 956.234i 0 2187.00 0
31.12 0 46.7654 0 907.917i 0 − 2864.11i 0 2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.9.b.c ✓ 12
4.b odd 2 1 inner 192.9.b.c ✓ 12
8.b even 2 1 inner 192.9.b.c ✓ 12
8.d odd 2 1 inner 192.9.b.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.9.b.c ✓ 12 1.a even 1 1 trivial
192.9.b.c ✓ 12 4.b odd 2 1 inner
192.9.b.c ✓ 12 8.b even 2 1 inner
192.9.b.c ✓ 12 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 1218084T_{5}^{4} + 337086554544T_{5}^{2} + 10300768973956800 \) acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2187)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 66\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 35\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots + 72780358195080)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 40\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 39\!\cdots\!12)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 54\!\cdots\!64)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 24\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 51\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 76\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 14\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 95\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 36\!\cdots\!28)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 19\!\cdots\!20)^{4} \) Copy content Toggle raw display
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