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Properties

Label 192.9.g.f
Level $192$
Weight $9$
Character orbit 192.g
Analytic conductor $78.217$
Analytic rank $0$
Dimension $8$
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] = [192,9,Mod(127,192)]
 
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, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.g (of order \(2\), degree \(1\), not 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 96)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{6} + 154) q^{5} + ( - \beta_{4} - \beta_{3} - 9 \beta_{2} - 17 \beta_1) q^{7} - 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{6} + 154) q^{5} + ( - \beta_{4} - \beta_{3} - 9 \beta_{2} - 17 \beta_1) q^{7} - 2187 q^{9} + (22 \beta_{4} - 4 \beta_{3} + 47 \beta_{2} + 56 \beta_1) q^{11} + (\beta_{7} + 13 \beta_{6} + \beta_{5} + 6782) q^{13} + (27 \beta_{4} - 54 \beta_{2} + 166 \beta_1) q^{15} + (5 \beta_{7} + 17 \beta_{6} - 22 \beta_{5} - 25542) q^{17} + ( - 18 \beta_{4} + 38 \beta_{3} + 729 \beta_{2} + 42 \beta_1) q^{19} + ( - 5 \beta_{7} - 76 \beta_{6} - 27 \beta_{5} + 37260) q^{21} + (94 \beta_{4} + 102 \beta_{3} - 1606 \beta_{2} + 1518 \beta_1) q^{23} + ( - 12 \beta_{7} - 296 \beta_{6} + 22 \beta_{5} - 97677) q^{25} - 2187 \beta_1 q^{27} + ( - 120 \beta_{7} + 385 \beta_{6} - 182 \beta_{5} + 41714) q^{29} + ( - 345 \beta_{4} - 325 \beta_{3} - 7679 \beta_{2} + 4675 \beta_1) q^{31} + (115 \beta_{7} + 1667 \beta_{6} - 108 \beta_{5} - 96876) q^{33} + ( - 438 \beta_{4} - 264 \beta_{3} - 17495 \beta_{2} - 13336 \beta_1) q^{35} + ( - 317 \beta_{7} - 383 \beta_{6} + 431 \beta_{5} - 493146) q^{37} + ( - 378 \beta_{4} - 81 \beta_{3} - 1917 \beta_{2} + 6653 \beta_1) q^{39} + (235 \beta_{7} - 5489 \beta_{6} - 246 \beta_{5} + 1475386) q^{41} + (466 \beta_{4} - 394 \beta_{3} - 35073 \beta_{2} - 46014 \beta_1) q^{43} + (2187 \beta_{6} - 336798) q^{45} + ( - 1086 \beta_{4} + 958 \beta_{3} - 1994 \beta_{2} + 57598 \beta_1) q^{47} + (1578 \beta_{7} + 5758 \beta_{6} + 1206 \beta_{5} - 497151) q^{49} + ( - 594 \beta_{4} + 1782 \beta_{3} + 945 \beta_{2} - 26664 \beta_1) q^{51} + ( - 3148 \beta_{7} - 2755 \beta_{6} + 642 \beta_{5} + \cdots - 98574) q^{53}+ \cdots + ( - 48114 \beta_{4} + 8748 \beta_{3} - 102789 \beta_{2} - 122472 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1232 q^{5} - 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1232 q^{5} - 17496 q^{9} + 54256 q^{13} - 204336 q^{17} + 298080 q^{21} - 781416 q^{25} + 333712 q^{29} - 775008 q^{33} - 3945168 q^{37} + 11803088 q^{41} - 2694384 q^{45} - 3977208 q^{49} - 788592 q^{53} - 1194912 q^{57} + 40252336 q^{61} - 20747936 q^{65} - 26687232 q^{69} - 56712048 q^{73} + 52014208 q^{77} + 38263752 q^{81} - 52697568 q^{85} + 61660176 q^{89} - 81738720 q^{93} + 27830288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -27\nu^{7} + 9234\nu^{5} - 987903\nu^{3} + 33920532\nu ) / 58784 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 348\nu^{7} - 89624\nu^{5} + 7765724\nu^{3} - 226222192\nu ) / 123079 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 871 \nu^{7} + 235136 \nu^{6} + 297882 \nu^{5} - 123211264 \nu^{4} - 31869019 \nu^{3} + 15898720640 \nu^{2} + 1094251236 \nu - 607942246912 ) / 3938528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27 \nu^{7} - 352704 \nu^{6} - 23930 \nu^{5} + 90292224 \nu^{4} + 3471527 \nu^{3} - 7778886720 \nu^{2} - 139408420 \nu + 225427939968 ) / 984632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1044 \nu^{7} + 44088 \nu^{6} + 268872 \nu^{5} - 11286528 \nu^{4} - 23297172 \nu^{3} + 960545256 \nu^{2} + 725928912 \nu - 27168260064 ) / 123079 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14562 \nu^{7} - 20207 \nu^{6} + 3716348 \nu^{5} + 5172992 \nu^{4} - 315278826 \nu^{3} - 440249909 \nu^{2} + 8887740344 \nu + 12452119196 ) / 492316 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14562 \nu^{7} + 773377 \nu^{6} + 3716348 \nu^{5} - 197984512 \nu^{4} - 315278826 \nu^{3} + 16849564699 \nu^{2} + 8887740344 \nu - 476576561956 ) / 492316 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 9\beta_{5} + 27\beta_{2} ) / 3456 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -32\beta_{7} + 32\beta_{6} - 144\beta_{4} - 9\beta_{2} - 64\beta _1 + 1181952 ) / 13824 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -53\beta_{7} + 197\beta_{6} + 255\beta_{5} + 2313\beta_{2} + 256\beta_1 ) / 1152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5440\beta_{7} + 5440\beta_{6} - 24624\beta_{4} - 864\beta_{3} - 1539\beta_{2} - 10528\beta _1 + 100535040 ) / 13824 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -12515\beta_{7} + 85523\beta_{6} + 64683\beta_{5} + 993357\beta_{2} + 218880\beta_1 ) / 3456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 228960 \beta_{7} + 228960 \beta_{6} - 1055472 \beta_{4} - 73728 \beta_{3} - 65967 \beta_{2} - 433600 \beta _1 + 2834839296 ) / 4608 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -975111\beta_{7} + 10137399\beta_{6} + 5437845\beta_{5} + 119757555\beta_{2} + 39232256\beta_1 ) / 3456 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
9.62695 − 0.500000i
−8.87816 + 0.500000i
−9.62695 − 0.500000i
8.87816 + 0.500000i
9.62695 + 0.500000i
−8.87816 − 0.500000i
−9.62695 + 0.500000i
8.87816 − 0.500000i
0 − 46.7654i 0 −455.790 0 − 1689.76i 0 −2187.00 0
127.2 0 − 46.7654i 0 −261.868 0 1340.75i 0 −2187.00 0
127.3 0 − 46.7654i 0 611.370 0 4427.25i 0 −2187.00 0
127.4 0 − 46.7654i 0 722.288 0 − 891.264i 0 −2187.00 0
127.5 0 46.7654i 0 −455.790 0 1689.76i 0 −2187.00 0
127.6 0 46.7654i 0 −261.868 0 − 1340.75i 0 −2187.00 0
127.7 0 46.7654i 0 611.370 0 − 4427.25i 0 −2187.00 0
127.8 0 46.7654i 0 722.288 0 891.264i 0 −2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.g.f 8
4.b odd 2 1 inner 192.9.g.f 8
8.b even 2 1 96.9.g.a ✓ 8
8.d odd 2 1 96.9.g.a ✓ 8
24.f even 2 1 288.9.g.f 8
24.h odd 2 1 288.9.g.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.g.a ✓ 8 8.b even 2 1
96.9.g.a ✓ 8 8.d odd 2 1
192.9.g.f 8 1.a even 1 1 trivial
192.9.g.f 8 4.b odd 2 1 inner
288.9.g.f 8 24.f even 2 1
288.9.g.f 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 616T_{5}^{3} - 396168T_{5}^{2} + 157725920T_{5} + 52706093200 \) acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 616 T^{3} + \cdots + 52706093200)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 25047808 T^{6} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{8} + 1747174336 T^{6} + \cdots + 65\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{4} - 27128 T^{3} + \cdots - 81\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 102168 T^{3} + \cdots + 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 33336733632 T^{6} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{8} + 277565694208 T^{6} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} - 166856 T^{3} + \cdots - 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3440896309504 T^{6} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} + 1972584 T^{3} + \cdots + 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 5901544 T^{3} + \cdots - 37\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 42117576693184 T^{6} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + 48015145188352 T^{6} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + 394296 T^{3} + \cdots + 34\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 556916566724800 T^{6} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{4} - 20126168 T^{3} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} + 28356024 T^{3} + \cdots + 60\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} - 30830088 T^{3} + \cdots - 54\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 13915144 T^{3} + \cdots - 66\!\cdots\!12)^{2} \) Copy content Toggle raw display
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