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Properties

Label 168.5.z.a
Level $168$
Weight $5$
Character orbit 168.z
Analytic conductor $17.366$
Analytic rank $0$
Dimension $16$
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] = [168,5,Mod(73,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 168.z (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: \(17.3661537981\)
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: \( x^{16} - 6 x^{15} + 99 x^{14} - 1810 x^{13} + 14212 x^{12} - 199882 x^{11} + 1800935 x^{10} + \cdots + 41390114348800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{36}\cdot 7^{5} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (3 \beta_1 - 6) q^{3} + (\beta_{7} + \beta_{5} + \beta_1) q^{5} + (\beta_{10} + 2 \beta_1 + 2) q^{7} + ( - 27 \beta_1 + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_1 - 6) q^{3} + (\beta_{7} + \beta_{5} + \beta_1) q^{5} + (\beta_{10} + 2 \beta_1 + 2) q^{7} + ( - 27 \beta_1 + 27) q^{9} + ( - \beta_{12} - \beta_{9} + \cdots - 14 \beta_1) q^{11}+ \cdots + ( - 27 \beta_{10} - 27 \beta_{9} + \cdots - 378) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 99 x^{14} - 1810 x^{13} + 14212 x^{12} - 199882 x^{11} + 1800935 x^{10} + \cdots + 41390114348800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 43\!\cdots\!43 \nu^{15} + \cdots + 10\!\cdots\!20 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51\!\cdots\!23 \nu^{15} + \cdots - 48\!\cdots\!60 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!17 \nu^{15} + \cdots - 26\!\cdots\!20 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!53 \nu^{15} + \cdots + 28\!\cdots\!80 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!03 \nu^{15} + \cdots + 16\!\cdots\!40 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!75 \nu^{15} + \cdots - 52\!\cdots\!20 ) / 74\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!57 \nu^{15} + \cdots - 38\!\cdots\!20 ) / 42\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 95\!\cdots\!63 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50\!\cdots\!61 \nu^{15} + \cdots + 28\!\cdots\!92 ) / 59\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29\!\cdots\!99 \nu^{15} + \cdots - 24\!\cdots\!60 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!29 \nu^{15} + \cdots - 28\!\cdots\!20 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 62\!\cdots\!61 \nu^{15} + \cdots + 10\!\cdots\!40 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 80\!\cdots\!17 \nu^{15} + \cdots + 40\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{15} + \cdots + 10\!\cdots\!80 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{15} + \cdots - 22\!\cdots\!40 ) / 29\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} + \beta_{13} + \beta_{11} - 11 \beta_{10} + 19 \beta_{7} - 2 \beta_{6} + \cdots + 13 ) / 112 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 39 \beta_{15} + 20 \beta_{14} - 56 \beta_{13} + 27 \beta_{12} - 24 \beta_{11} - 112 \beta_{10} + \cdots + 347 ) / 224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 140 \beta_{15} - 625 \beta_{14} - 506 \beta_{13} + 420 \beta_{12} - 196 \beta_{11} + 473 \beta_{10} + \cdots + 36974 ) / 224 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5389 \beta_{15} - 1006 \beta_{14} - 1104 \beta_{13} + 6175 \beta_{12} - 1508 \beta_{11} - 6834 \beta_{10} + \cdots - 59757 ) / 224 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 49432 \beta_{15} + 38657 \beta_{14} - 37742 \beta_{13} + 15092 \beta_{12} - 9580 \beta_{11} + \cdots + 7493826 ) / 224 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 462603 \beta_{15} - 588326 \beta_{14} - 175228 \beta_{13} + 567053 \beta_{12} + 71032 \beta_{11} + \cdots - 6687639 ) / 224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3770334 \beta_{15} + 5147733 \beta_{14} - 1600622 \beta_{13} + 7044102 \beta_{12} - 4888408 \beta_{11} + \cdots - 590466396 ) / 224 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 84771815 \beta_{15} + 28708282 \beta_{14} - 42187436 \beta_{13} - 12984965 \beta_{12} + \cdots + 13042009815 ) / 224 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1109343926 \beta_{15} - 762422693 \beta_{14} + 258316354 \beta_{13} + 487452842 \beta_{12} + \cdots - 71175816172 ) / 224 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2633682805 \beta_{15} + 13298265202 \beta_{14} - 2055751716 \beta_{13} + 3633202761 \beta_{12} + \cdots - 542810878779 ) / 224 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 98651467372 \beta_{15} - 27165895451 \beta_{14} - 37410796558 \beta_{13} - 47097765740 \beta_{12} + \cdots + 15999082509542 ) / 224 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1906226401741 \beta_{15} - 718682432154 \beta_{14} + 668705421080 \beta_{13} + 674129497827 \beta_{12} + \cdots - 196831800659945 ) / 224 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 15335888529194 \beta_{15} + 21426818703465 \beta_{14} - 4425275576606 \beta_{13} + \cdots + 460533911650868 ) / 224 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 37403894209717 \beta_{15} - 155310789295666 \beta_{14} - 4054547826344 \beta_{13} + \cdots + 15\!\cdots\!25 ) / 224 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 25\!\cdots\!26 \beta_{15} + 65430709294215 \beta_{14} + 957125745471010 \beta_{13} + \cdots - 35\!\cdots\!88 ) / 224 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1 - \beta_{1}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−0.154925 + 7.21452i
−5.43765 − 1.98832i
2.19101 + 3.06642i
−0.530737 + 7.91948i
4.88974 − 3.43978i
9.32119 + 2.12492i
0.268520 − 7.69736i
−7.54715 − 8.93193i
−0.154925 − 7.21452i
−5.43765 + 1.98832i
2.19101 − 3.06642i
−0.530737 − 7.91948i
4.88974 + 3.43978i
9.32119 − 2.12492i
0.268520 + 7.69736i
−7.54715 + 8.93193i
0 −4.50000 − 2.59808i 0 −40.6814 + 23.4874i 0 48.9386 + 2.45242i 0 13.5000 + 23.3827i 0
73.2 0 −4.50000 − 2.59808i 0 −31.0314 + 17.9160i 0 −41.3748 + 26.2512i 0 13.5000 + 23.3827i 0
73.3 0 −4.50000 − 2.59808i 0 −6.72797 + 3.88439i 0 −35.4710 − 33.8055i 0 13.5000 + 23.3827i 0
73.4 0 −4.50000 − 2.59808i 0 −2.25900 + 1.30423i 0 16.7036 − 46.0651i 0 13.5000 + 23.3827i 0
73.5 0 −4.50000 − 2.59808i 0 1.51540 − 0.874918i 0 −2.69843 + 48.9256i 0 13.5000 + 23.3827i 0
73.6 0 −4.50000 − 2.59808i 0 23.7919 − 13.7363i 0 −26.3445 − 41.3155i 0 13.5000 + 23.3827i 0
73.7 0 −4.50000 − 2.59808i 0 25.2264 − 14.5645i 0 14.3466 + 46.8527i 0 13.5000 + 23.3827i 0
73.8 0 −4.50000 − 2.59808i 0 36.1660 − 20.8805i 0 45.8999 − 17.1523i 0 13.5000 + 23.3827i 0
145.1 0 −4.50000 + 2.59808i 0 −40.6814 − 23.4874i 0 48.9386 − 2.45242i 0 13.5000 − 23.3827i 0
145.2 0 −4.50000 + 2.59808i 0 −31.0314 − 17.9160i 0 −41.3748 − 26.2512i 0 13.5000 − 23.3827i 0
145.3 0 −4.50000 + 2.59808i 0 −6.72797 − 3.88439i 0 −35.4710 + 33.8055i 0 13.5000 − 23.3827i 0
145.4 0 −4.50000 + 2.59808i 0 −2.25900 − 1.30423i 0 16.7036 + 46.0651i 0 13.5000 − 23.3827i 0
145.5 0 −4.50000 + 2.59808i 0 1.51540 + 0.874918i 0 −2.69843 − 48.9256i 0 13.5000 − 23.3827i 0
145.6 0 −4.50000 + 2.59808i 0 23.7919 + 13.7363i 0 −26.3445 + 41.3155i 0 13.5000 − 23.3827i 0
145.7 0 −4.50000 + 2.59808i 0 25.2264 + 14.5645i 0 14.3466 − 46.8527i 0 13.5000 − 23.3827i 0
145.8 0 −4.50000 + 2.59808i 0 36.1660 + 20.8805i 0 45.8999 + 17.1523i 0 13.5000 − 23.3827i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
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 168.5.z.a ✓ 16
3.b odd 2 1 504.5.by.a 16
4.b odd 2 1 336.5.bh.i 16
7.c even 3 1 1176.5.f.b 16
7.d odd 6 1 inner 168.5.z.a ✓ 16
7.d odd 6 1 1176.5.f.b 16
21.g even 6 1 504.5.by.a 16
28.f even 6 1 336.5.bh.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.5.z.a ✓ 16 1.a even 1 1 trivial
168.5.z.a ✓ 16 7.d odd 6 1 inner
336.5.bh.i 16 4.b odd 2 1
336.5.bh.i 16 28.f even 6 1
504.5.by.a 16 3.b odd 2 1
504.5.by.a 16 21.g even 6 1
1176.5.f.b 16 7.c even 3 1
1176.5.f.b 16 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 12 T_{5}^{15} - 3382 T_{5}^{14} + 41160 T_{5}^{13} + 8430347 T_{5}^{12} + \cdots + 39\!\cdots\!44 \) acting on \(S_{5}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 27)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 26\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 73\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 81\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 88\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
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