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Properties

Label 168.10.q.a
Level $168$
Weight $10$
Character orbit 168.q
Analytic conductor $86.526$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,10,Mod(25,168)]
 
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, 4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.25");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), 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: \(86.5260204755\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18660372 x^{14} - 3458782984 x^{13} + 143123973101310 x^{12} + \cdots + 50\!\cdots\!97 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{5}\cdot 5^{2}\cdot 7^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (81 \beta_1 - 81) q^{3} + ( - \beta_{7} - 24 \beta_1) q^{5} + (\beta_{6} - \beta_{5} + 448 \beta_1 - 235) q^{7} - 6561 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (81 \beta_1 - 81) q^{3} + ( - \beta_{7} - 24 \beta_1) q^{5} + (\beta_{6} - \beta_{5} + 448 \beta_1 - 235) q^{7} - 6561 \beta_1 q^{9} + ( - \beta_{8} + 4 \beta_{7} + \cdots + 4057) q^{11}+ \cdots + (6561 \beta_{10} - 13122 \beta_{7} + \cdots - 26617977) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 18660372 x^{14} - 3458782984 x^{13} + 143123973101310 x^{12} + \cdots + 50\!\cdots\!97 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 75\!\cdots\!68 \nu^{15} + \cdots - 24\!\cdots\!93 ) / 95\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75\!\cdots\!68 \nu^{15} + \cdots - 24\!\cdots\!93 ) / 95\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50\!\cdots\!67 \nu^{15} + \cdots + 52\!\cdots\!45 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!21 \nu^{15} + \cdots + 27\!\cdots\!91 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!09 \nu^{15} + \cdots + 14\!\cdots\!68 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!01 \nu^{15} + \cdots - 64\!\cdots\!56 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!02 \nu^{15} + \cdots - 37\!\cdots\!64 ) / 95\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 76\!\cdots\!60 \nu^{15} + \cdots - 26\!\cdots\!63 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!60 \nu^{15} + \cdots + 38\!\cdots\!21 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!08 \nu^{15} + \cdots + 34\!\cdots\!69 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52\!\cdots\!60 \nu^{15} + \cdots + 15\!\cdots\!93 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43\!\cdots\!40 \nu^{15} + \cdots + 12\!\cdots\!19 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 61\!\cdots\!85 \nu^{15} + \cdots - 11\!\cdots\!54 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 30\!\cdots\!95 \nu^{15} + \cdots + 71\!\cdots\!24 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 71\!\cdots\!80 \nu^{15} + \cdots + 13\!\cdots\!03 ) / 25\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} - 6 \beta_{9} + 96 \beta_{7} - 42 \beta_{6} - 52 \beta_{5} + \cdots + 2332423 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 18 \beta_{15} + 9 \beta_{14} - 21 \beta_{13} + 7015 \beta_{12} - 1520 \beta_{11} + 4444 \beta_{10} + \cdots + 650578596 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9476 \beta_{15} - 6098 \beta_{14} - 28018 \beta_{13} + 43883745 \beta_{12} - 1168920 \beta_{11} + \cdots + 7745566567665 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 141806730 \beta_{15} - 5829310 \beta_{14} - 219348785 \beta_{13} + 60564225350 \beta_{12} + \cdots + 49\!\cdots\!49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4046612880 \beta_{15} - 159729344025 \beta_{14} - 362727515880 \beta_{13} + 233244008977912 \beta_{12} + \cdots + 29\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 917603907520216 \beta_{15} - 398483932383887 \beta_{14} + \cdots + 29\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 63\!\cdots\!52 \beta_{15} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 56\!\cdots\!94 \beta_{15} + \cdots + 16\!\cdots\!83 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 69\!\cdots\!60 \beta_{15} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 33\!\cdots\!74 \beta_{15} + \cdots + 90\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 54\!\cdots\!40 \beta_{15} + \cdots + 26\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 19\!\cdots\!84 \beta_{15} + \cdots + 47\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 37\!\cdots\!48 \beta_{15} + \cdots + 12\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 11\!\cdots\!22 \beta_{15} + \cdots + 25\!\cdots\!74 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−1807.20 − 0.866025i
−1797.08 − 0.866025i
−1071.14 − 0.866025i
−1013.60 − 0.866025i
397.617 − 0.866025i
1263.53 − 0.866025i
1765.22 − 0.866025i
2262.65 − 0.866025i
−1807.20 + 0.866025i
−1797.08 + 0.866025i
−1071.14 + 0.866025i
−1013.60 + 0.866025i
397.617 + 0.866025i
1263.53 + 0.866025i
1765.22 + 0.866025i
2262.65 + 0.866025i
0 −40.5000 + 70.1481i 0 −915.848 − 1586.30i 0 5121.10 − 3758.71i 0 −3280.50 − 5681.99i 0
25.2 0 −40.5000 + 70.1481i 0 −910.790 − 1577.53i 0 2936.64 + 5632.92i 0 −3280.50 − 5681.99i 0
25.3 0 −40.5000 + 70.1481i 0 −547.821 − 948.853i 0 −5934.92 − 2265.02i 0 −3280.50 − 5681.99i 0
25.4 0 −40.5000 + 70.1481i 0 −519.049 − 899.019i 0 −5773.43 + 2649.73i 0 −3280.50 − 5681.99i 0
25.5 0 −40.5000 + 70.1481i 0 186.559 + 323.129i 0 6073.24 − 1862.63i 0 −3280.50 − 5681.99i 0
25.6 0 −40.5000 + 70.1481i 0 619.513 + 1073.03i 0 −1334.90 − 6210.61i 0 −3280.50 − 5681.99i 0
25.7 0 −40.5000 + 70.1481i 0 870.360 + 1507.51i 0 3862.22 + 5043.50i 0 −3280.50 − 5681.99i 0
25.8 0 −40.5000 + 70.1481i 0 1119.08 + 1938.30i 0 −5033.95 + 3874.66i 0 −3280.50 − 5681.99i 0
121.1 0 −40.5000 − 70.1481i 0 −915.848 + 1586.30i 0 5121.10 + 3758.71i 0 −3280.50 + 5681.99i 0
121.2 0 −40.5000 − 70.1481i 0 −910.790 + 1577.53i 0 2936.64 − 5632.92i 0 −3280.50 + 5681.99i 0
121.3 0 −40.5000 − 70.1481i 0 −547.821 + 948.853i 0 −5934.92 + 2265.02i 0 −3280.50 + 5681.99i 0
121.4 0 −40.5000 − 70.1481i 0 −519.049 + 899.019i 0 −5773.43 − 2649.73i 0 −3280.50 + 5681.99i 0
121.5 0 −40.5000 − 70.1481i 0 186.559 − 323.129i 0 6073.24 + 1862.63i 0 −3280.50 + 5681.99i 0
121.6 0 −40.5000 − 70.1481i 0 619.513 − 1073.03i 0 −1334.90 + 6210.61i 0 −3280.50 + 5681.99i 0
121.7 0 −40.5000 − 70.1481i 0 870.360 − 1507.51i 0 3862.22 − 5043.50i 0 −3280.50 + 5681.99i 0
121.8 0 −40.5000 − 70.1481i 0 1119.08 − 1938.30i 0 −5033.95 − 3874.66i 0 −3280.50 + 5681.99i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.10.q.a ✓ 16
7.c even 3 1 inner 168.10.q.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.10.q.a ✓ 16 1.a even 1 1 trivial
168.10.q.a ✓ 16 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 196 T_{5}^{15} + 9351798 T_{5}^{14} + 4374788592 T_{5}^{13} + 59606728776375 T_{5}^{12} + \cdots + 46\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81 T + 6561)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 70\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 73\!\cdots\!28)^{2} \) Copy content Toggle raw display
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