↑

Properties

Label 672.4.q.i
Level $672$
Weight $4$
Character orbit 672.q
Analytic conductor $39.649$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,4,Mod(193,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.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: \(39.6492835239\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 218 x^{10} + 953 x^{9} + 39809 x^{8} + 91760 x^{7} + 1571968 x^{6} - 3346176 x^{5} + \cdots + 943718400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + (\beta_{5} + \beta_1 + 1) q^{5} + (\beta_{6} + \beta_{3} - 2 \beta_1 + 1) q^{7} + ( - 9 \beta_1 - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + (\beta_{5} + \beta_1 + 1) q^{5} + (\beta_{6} + \beta_{3} - 2 \beta_1 + 1) q^{7} + ( - 9 \beta_1 - 9) q^{9} + (\beta_{9} + \beta_{8} + \cdots - 4 \beta_1) q^{11}+ \cdots + ( - 9 \beta_{11} + 9 \beta_{10} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{3} + 7 q^{5} + 22 q^{7} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{3} + 7 q^{5} + 22 q^{7} - 54 q^{9} + 23 q^{11} - 146 q^{13} - 42 q^{15} + 66 q^{17} - 51 q^{19} + 21 q^{21} + 134 q^{23} - 125 q^{25} + 324 q^{27} - 170 q^{29} - 134 q^{31} + 69 q^{33} - 348 q^{35} - 9 q^{37} + 219 q^{39} - 108 q^{41} + 74 q^{43} + 63 q^{45} + 130 q^{47} + 10 q^{49} + 198 q^{51} - 657 q^{53} + 1866 q^{55} + 306 q^{57} - 257 q^{59} + 904 q^{61} - 261 q^{63} - 36 q^{65} + 305 q^{67} - 804 q^{69} - 4140 q^{71} - 573 q^{73} - 375 q^{75} + 463 q^{77} + 544 q^{79} - 486 q^{81} + 1142 q^{83} - 4312 q^{85} + 255 q^{87} + 1666 q^{89} - 2071 q^{91} - 402 q^{93} + 1200 q^{95} + 2582 q^{97} - 414 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 218 x^{10} + 953 x^{9} + 39809 x^{8} + 91760 x^{7} + 1571968 x^{6} - 3346176 x^{5} + \cdots + 943718400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6760213949809 \nu^{11} - 120778423488883 \nu^{10} + \cdots - 23\!\cdots\!60 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!87 \nu^{11} + \cdots + 21\!\cdots\!40 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\!\cdots\!03 \nu^{11} + \cdots + 24\!\cdots\!60 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 87\!\cdots\!43 \nu^{11} + \cdots + 64\!\cdots\!20 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 95\!\cdots\!07 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 681654482927241 \nu^{11} + \cdots - 13\!\cdots\!60 ) / 93\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!86 \nu^{11} + \cdots + 20\!\cdots\!40 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 98\!\cdots\!68 \nu^{11} + \cdots - 35\!\cdots\!20 ) / 82\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32\!\cdots\!71 \nu^{11} + \cdots - 20\!\cdots\!00 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35\!\cdots\!73 \nu^{11} + \cdots - 88\!\cdots\!00 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!67 \nu^{11} + \cdots + 22\!\cdots\!00 ) / 65\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} + \beta_{10} + 3\beta_{7} - 4\beta_{6} + 6\beta_{5} - 3\beta_{3} + 4\beta_{2} - 147\beta _1 - 147 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 42 \beta_{11} + 147 \beta_{10} - 147 \beta_{9} + 42 \beta_{8} - 111 \beta_{7} - 111 \beta_{6} + \cdots - 1235 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1013 \beta_{9} + 1088 \beta_{8} - 2397 \beta_{7} + 1420 \beta_{6} + 1559 \beta_{4} + \cdots + 46905 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6333 \beta_{11} - 13560 \beta_{10} - 10505 \beta_{7} + 20397 \beta_{6} + 1107 \beta_{5} + \cdots + 211354 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 222746 \beta_{11} - 289133 \beta_{10} + 289133 \beta_{9} - 222746 \beta_{8} + 260737 \beta_{7} + \cdots + 9148245 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5565453 \beta_{9} - 3092868 \beta_{8} + 8998873 \beta_{7} - 4818144 \beta_{6} - 788355 \beta_{4} + \cdots - 110195801 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 24417271 \beta_{11} + 35814649 \beta_{10} + 36188983 \beta_{7} - 66970336 \beta_{6} + 19219858 \beta_{5} + \cdots - 969058515 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 717789234 \beta_{11} + 1204740903 \beta_{10} - 1204740903 \beta_{9} + 717789234 \beta_{8} + \cdots - 26423399111 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16881087737 \beta_{9} + 10971182000 \beta_{8} - 30548121201 \beta_{7} + 16451450284 \beta_{6} + \cdots + 426566700909 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 82070712729 \beta_{11} - 133794647988 \beta_{10} - 124390852229 \beta_{7} + 230132204997 \beta_{6} + \cdots + 3071274109342 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−5.20519 + 9.01566i
1.78742 − 3.09591i
7.53696 − 13.0544i
−4.24707 + 7.35614i
−0.993837 + 1.72138i
1.62172 − 2.80890i
−5.20519 − 9.01566i
1.78742 + 3.09591i
7.53696 + 13.0544i
−4.24707 − 7.35614i
−0.993837 − 1.72138i
1.62172 + 2.80890i
0 −1.50000 + 2.59808i 0 −7.79155 − 13.4954i 0 17.7068 − 5.42847i 0 −4.50000 − 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 −4.74910 − 8.22568i 0 −7.19776 − 17.0644i 0 −4.50000 − 7.79423i 0
193.3 0 −1.50000 + 2.59808i 0 −1.03577 − 1.79400i 0 16.6687 + 8.07178i 0 −4.50000 − 7.79423i 0
193.4 0 −1.50000 + 2.59808i 0 1.60529 + 2.78044i 0 −18.1998 + 3.43050i 0 −4.50000 − 7.79423i 0
193.5 0 −1.50000 + 2.59808i 0 5.26979 + 9.12754i 0 −4.23488 + 18.0296i 0 −4.50000 − 7.79423i 0
193.6 0 −1.50000 + 2.59808i 0 10.2013 + 17.6692i 0 6.25687 − 17.4313i 0 −4.50000 − 7.79423i 0
289.1 0 −1.50000 − 2.59808i 0 −7.79155 + 13.4954i 0 17.7068 + 5.42847i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 − 2.59808i 0 −4.74910 + 8.22568i 0 −7.19776 + 17.0644i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 − 2.59808i 0 −1.03577 + 1.79400i 0 16.6687 − 8.07178i 0 −4.50000 + 7.79423i 0
289.4 0 −1.50000 − 2.59808i 0 1.60529 − 2.78044i 0 −18.1998 − 3.43050i 0 −4.50000 + 7.79423i 0
289.5 0 −1.50000 − 2.59808i 0 5.26979 − 9.12754i 0 −4.23488 − 18.0296i 0 −4.50000 + 7.79423i 0
289.6 0 −1.50000 − 2.59808i 0 10.2013 − 17.6692i 0 6.25687 + 17.4313i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.6
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 672.4.q.i ✓ 12
4.b odd 2 1 672.4.q.j yes 12
7.c even 3 1 inner 672.4.q.i ✓ 12
28.g odd 6 1 672.4.q.j yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.q.i ✓ 12 1.a even 1 1 trivial
672.4.q.i ✓ 12 7.c even 3 1 inner
672.4.q.j yes 12 4.b odd 2 1
672.4.q.j yes 12 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{12} - 7 T_{5}^{11} + 462 T_{5}^{10} + 245 T_{5}^{9} + 146182 T_{5}^{8} - 116991 T_{5}^{7} + \cdots + 44808422400 \) Copy content Toggle raw display
\( T_{11}^{12} - 23 T_{11}^{11} + 5308 T_{11}^{10} - 142417 T_{11}^{9} + 20845364 T_{11}^{8} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 44808422400 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{6} + 73 T^{5} + \cdots + 195190272)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 90\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + 85 T^{5} + \cdots - 441220586240)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 17\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 43\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 3488384344064)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 72649107924792)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 30854143994112)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 58\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 73\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 80\!\cdots\!12)^{2} \) Copy content Toggle raw display
show more
show less