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Properties

Label 392.3.j.d
Level $392$
Weight $3$
Character orbit 392.j
Analytic conductor $10.681$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,3,Mod(117,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.117");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.j (of order \(6\), degree \(2\), 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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 6 x^{14} + 176 x^{13} - 93 x^{12} - 1812 x^{11} + 1084 x^{10} + 11240 x^{9} + \cdots + 829 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{2}) q^{2} + (\beta_{11} - \beta_{3}) q^{3} + ( - \beta_{6} - 2 \beta_{5} + \beta_1 + 2) q^{4} + ( - \beta_{11} - \beta_{9}) q^{5} + ( - \beta_{15} + \beta_{13} - \beta_{4}) q^{6} + ( - 2 \beta_{10} + 2 \beta_{8} + \cdots + 2 \beta_1) q^{8}+ \cdots + (2 \beta_{8} - 2 \beta_{7} - \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{2}) q^{2} + (\beta_{11} - \beta_{3}) q^{3} + ( - \beta_{6} - 2 \beta_{5} + \beta_1 + 2) q^{4} + ( - \beta_{11} - \beta_{9}) q^{5} + ( - \beta_{15} + \beta_{13} - \beta_{4}) q^{6} + ( - 2 \beta_{10} + 2 \beta_{8} + \cdots + 2 \beta_1) q^{8}+ \cdots + (10 \beta_{10} + 4 \beta_{8} + \cdots + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 16 q^{4} + 16 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 16 q^{4} + 16 q^{8} - 8 q^{9} + 128 q^{15} + 16 q^{16} - 44 q^{18} - 144 q^{22} + 32 q^{23} + 24 q^{25} + 56 q^{30} - 176 q^{32} - 224 q^{36} + 128 q^{39} + 96 q^{44} - 160 q^{46} - 264 q^{50} + 608 q^{57} - 288 q^{58} + 80 q^{60} - 128 q^{64} - 272 q^{65} + 1088 q^{71} + 280 q^{72} + 240 q^{74} - 400 q^{78} + 32 q^{79} + 328 q^{81} + 264 q^{86} - 336 q^{88} - 448 q^{92} + 256 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 6 x^{14} + 176 x^{13} - 93 x^{12} - 1812 x^{11} + 1084 x^{10} + 11240 x^{9} + \cdots + 829 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19\!\cdots\!93 \nu^{15} + \cdots + 52\!\cdots\!01 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91\!\cdots\!31 \nu^{15} + \cdots - 48\!\cdots\!97 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78\!\cdots\!23 \nu^{15} + \cdots - 34\!\cdots\!98 ) / 81\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 85\!\cdots\!53 \nu^{15} + \cdots + 25\!\cdots\!09 ) / 81\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 511912054384101 \nu^{15} + \cdots - 41\!\cdots\!29 ) / 23\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!99 \nu^{15} + \cdots + 41\!\cdots\!44 ) / 81\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 84\!\cdots\!53 \nu^{15} + \cdots - 10\!\cdots\!49 ) / 29\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44\!\cdots\!08 \nu^{15} + \cdots - 13\!\cdots\!26 ) / 97\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 78\!\cdots\!03 \nu^{15} + \cdots - 10\!\cdots\!15 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!19 \nu^{15} + \cdots + 14\!\cdots\!39 ) / 32\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!49 \nu^{15} + \cdots + 14\!\cdots\!57 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 78\!\cdots\!30 \nu^{15} + \cdots - 31\!\cdots\!32 ) / 11\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!89 \nu^{15} + \cdots + 92\!\cdots\!19 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 81\!\cdots\!66 \nu^{15} + \cdots - 83\!\cdots\!50 ) / 11\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!57 \nu^{15} + \cdots + 17\!\cdots\!29 ) / 16\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{10} + 3\beta_{6} - 3\beta_{4} - \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{8} + \cdots + 30 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 6 \beta_{14} + 6 \beta_{13} + 12 \beta_{12} - 6 \beta_{11} + 27 \beta_{10} + 3 \beta_{9} + \cdots + 38 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 44 \beta_{15} - 28 \beta_{14} + 40 \beta_{13} + 56 \beta_{12} + 12 \beta_{11} + 42 \beta_{10} + \cdots + 186 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 70 \beta_{15} - 110 \beta_{14} + 125 \beta_{13} + 220 \beta_{12} - 110 \beta_{11} + 285 \beta_{10} + \cdots + 156 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 438 \beta_{15} - 324 \beta_{14} + 546 \beta_{13} + 690 \beta_{12} - 170 \beta_{11} + 654 \beta_{10} + \cdots + 422 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 966 \beta_{15} - 1344 \beta_{14} + 1428 \beta_{13} + 2457 \beta_{12} - 2107 \beta_{11} + 2920 \beta_{10} + \cdots - 2692 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4464 \beta_{15} - 3984 \beta_{14} + 5448 \beta_{13} + 6600 \beta_{12} - 5928 \beta_{11} + \cdots - 14104 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12465 \beta_{15} - 15876 \beta_{14} + 10782 \beta_{13} + 19116 \beta_{12} - 33060 \beta_{11} + \cdots - 78026 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 50882 \beta_{15} - 54694 \beta_{14} + 34552 \beta_{13} + 38870 \beta_{12} - 99832 \beta_{11} + \cdots - 341746 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 171050 \beta_{15} - 209935 \beta_{14} + 22495 \beta_{13} + 63008 \beta_{12} - 401511 \beta_{11} + \cdots - 1324924 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 106442 \beta_{15} - 129630 \beta_{14} - 4762 \beta_{13} - 17386 \beta_{12} - 190896 \beta_{11} + \cdots - 893963 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2298374 \beta_{15} - 2819518 \beta_{14} - 1048580 \beta_{13} - 1241578 \beta_{12} - 3481088 \beta_{11} + \cdots - 18651702 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 7691488 \beta_{15} - 9948362 \beta_{14} - 5262730 \beta_{13} - 7597940 \beta_{12} - 7888094 \beta_{11} + \cdots - 70729874 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 26046471 \beta_{15} - 32010852 \beta_{14} - 26529732 \beta_{13} - 35555514 \beta_{12} + \cdots - 233145150 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
2.91591 + 0.109843i
−0.0960299 + 0.109843i
−1.19381 − 0.747754i
3.59453 − 0.747754i
0.230004 − 0.609843i
−2.78194 − 0.609843i
−1.72850 + 1.24775i
3.05984 + 1.24775i
2.91591 − 0.109843i
−0.0960299 − 0.109843i
−1.19381 + 0.747754i
3.59453 + 0.747754i
0.230004 + 0.609843i
−2.78194 + 0.609843i
−1.72850 − 1.24775i
3.05984 − 1.24775i
−1.88581 − 0.666123i −2.05719 − 3.56317i 3.11256 + 2.51236i −0.551224 + 0.954747i 1.50597 + 8.08980i 0 −4.19615 − 6.81119i −3.96410 + 6.86603i 1.67548 − 1.43329i
117.2 −1.88581 − 0.666123i 2.05719 + 3.56317i 3.11256 + 2.51236i 0.551224 − 0.954747i −1.50597 − 8.08980i 0 −4.19615 − 6.81119i −3.96410 + 6.86603i −1.67548 + 1.43329i
117.3 −0.582088 − 1.91342i −0.876327 − 1.51784i −3.32235 + 2.22756i −3.27050 + 5.66467i −2.39417 + 2.56030i 0 6.19615 + 5.06040i 2.96410 − 5.13397i 12.7426 + 2.96049i
117.4 −0.582088 − 1.91342i 0.876327 + 1.51784i −3.32235 + 2.22756i 3.27050 − 5.66467i 2.39417 − 2.56030i 0 6.19615 + 5.06040i 2.96410 − 5.13397i −12.7426 − 2.96049i
117.5 1.51978 + 1.30010i −2.05719 − 3.56317i 0.619491 + 3.95174i −0.551224 + 0.954747i 1.50597 − 8.08980i 0 −4.19615 + 6.81119i −3.96410 + 6.86603i −2.07901 + 0.734366i
117.6 1.51978 + 1.30010i 2.05719 + 3.56317i 0.619491 + 3.95174i 0.551224 − 0.954747i −1.50597 + 8.08980i 0 −4.19615 + 6.81119i −3.96410 + 6.86603i 2.07901 − 0.734366i
117.7 1.94811 − 0.452606i −0.876327 − 1.51784i 3.59030 − 1.76346i −3.27050 + 5.66467i −2.39417 − 2.56030i 0 6.19615 − 5.06040i 2.96410 − 5.13397i −3.80744 + 12.5157i
117.8 1.94811 − 0.452606i 0.876327 + 1.51784i 3.59030 − 1.76346i 3.27050 − 5.66467i 2.39417 + 2.56030i 0 6.19615 − 5.06040i 2.96410 − 5.13397i 3.80744 − 12.5157i
325.1 −1.88581 + 0.666123i −2.05719 + 3.56317i 3.11256 − 2.51236i −0.551224 − 0.954747i 1.50597 − 8.08980i 0 −4.19615 + 6.81119i −3.96410 − 6.86603i 1.67548 + 1.43329i
325.2 −1.88581 + 0.666123i 2.05719 − 3.56317i 3.11256 − 2.51236i 0.551224 + 0.954747i −1.50597 + 8.08980i 0 −4.19615 + 6.81119i −3.96410 − 6.86603i −1.67548 − 1.43329i
325.3 −0.582088 + 1.91342i −0.876327 + 1.51784i −3.32235 − 2.22756i −3.27050 − 5.66467i −2.39417 − 2.56030i 0 6.19615 − 5.06040i 2.96410 + 5.13397i 12.7426 − 2.96049i
325.4 −0.582088 + 1.91342i 0.876327 − 1.51784i −3.32235 − 2.22756i 3.27050 + 5.66467i 2.39417 + 2.56030i 0 6.19615 − 5.06040i 2.96410 + 5.13397i −12.7426 + 2.96049i
325.5 1.51978 − 1.30010i −2.05719 + 3.56317i 0.619491 − 3.95174i −0.551224 − 0.954747i 1.50597 + 8.08980i 0 −4.19615 − 6.81119i −3.96410 − 6.86603i −2.07901 − 0.734366i
325.6 1.51978 − 1.30010i 2.05719 − 3.56317i 0.619491 − 3.95174i 0.551224 + 0.954747i −1.50597 − 8.08980i 0 −4.19615 − 6.81119i −3.96410 − 6.86603i 2.07901 + 0.734366i
325.7 1.94811 + 0.452606i −0.876327 + 1.51784i 3.59030 + 1.76346i −3.27050 − 5.66467i −2.39417 + 2.56030i 0 6.19615 + 5.06040i 2.96410 + 5.13397i −3.80744 − 12.5157i
325.8 1.94811 + 0.452606i 0.876327 − 1.51784i 3.59030 + 1.76346i 3.27050 + 5.66467i 2.39417 − 2.56030i 0 6.19615 + 5.06040i 2.96410 + 5.13397i 3.80744 + 12.5157i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.j.d 16
7.b odd 2 1 inner 392.3.j.d 16
7.c even 3 1 56.3.h.d ✓ 8
7.c even 3 1 inner 392.3.j.d 16
7.d odd 6 1 56.3.h.d ✓ 8
7.d odd 6 1 inner 392.3.j.d 16
8.b even 2 1 inner 392.3.j.d 16
21.g even 6 1 504.3.l.f 8
21.h odd 6 1 504.3.l.f 8
28.f even 6 1 224.3.h.d 8
28.g odd 6 1 224.3.h.d 8
56.h odd 2 1 inner 392.3.j.d 16
56.j odd 6 1 56.3.h.d ✓ 8
56.j odd 6 1 inner 392.3.j.d 16
56.k odd 6 1 224.3.h.d 8
56.m even 6 1 224.3.h.d 8
56.p even 6 1 56.3.h.d ✓ 8
56.p even 6 1 inner 392.3.j.d 16
84.j odd 6 1 2016.3.l.f 8
84.n even 6 1 2016.3.l.f 8
168.s odd 6 1 504.3.l.f 8
168.v even 6 1 2016.3.l.f 8
168.ba even 6 1 504.3.l.f 8
168.be odd 6 1 2016.3.l.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.d ✓ 8 7.c even 3 1
56.3.h.d ✓ 8 7.d odd 6 1
56.3.h.d ✓ 8 56.j odd 6 1
56.3.h.d ✓ 8 56.p even 6 1
224.3.h.d 8 28.f even 6 1
224.3.h.d 8 28.g odd 6 1
224.3.h.d 8 56.k odd 6 1
224.3.h.d 8 56.m even 6 1
392.3.j.d 16 1.a even 1 1 trivial
392.3.j.d 16 7.b odd 2 1 inner
392.3.j.d 16 7.c even 3 1 inner
392.3.j.d 16 7.d odd 6 1 inner
392.3.j.d 16 8.b even 2 1 inner
392.3.j.d 16 56.h odd 2 1 inner
392.3.j.d 16 56.j odd 6 1 inner
392.3.j.d 16 56.p even 6 1 inner
504.3.l.f 8 21.g even 6 1
504.3.l.f 8 21.h odd 6 1
504.3.l.f 8 168.s odd 6 1
504.3.l.f 8 168.ba even 6 1
2016.3.l.f 8 84.j odd 6 1
2016.3.l.f 8 84.n even 6 1
2016.3.l.f 8 168.v even 6 1
2016.3.l.f 8 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 20T_{3}^{6} + 348T_{3}^{4} + 1040T_{3}^{2} + 2704 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 20 T^{6} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 44 T^{6} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 120 T^{6} + \cdots + 278784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 332 T^{2} + 27508)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1008 T^{6} + \cdots + 61060386816)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 788 T^{6} + \cdots + 13194657424)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + \cdots + 173056)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1920 T^{2} + 33792)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 11036853374976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 864 T^{6} + \cdots + 5780865024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1344 T^{2} + 439296)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6072 T^{2} + 7730448)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 11036853374976)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 7492001071104)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 25295789778064)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 44 T^{6} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 59759826280704)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 136 T + 2596)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 893985123373056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 913936)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 20948 T^{2} + 114868)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 893985123373056)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 24048 T^{2} + 102163776)^{4} \) Copy content Toggle raw display
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