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Properties

Label 232.3.j.a
Level $232$
Weight $3$
Character orbit 232.j
Analytic conductor $6.322$
Analytic rank $0$
Dimension $14$
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] = [232,3,Mod(17,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 232.j (of order \(4\), 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: \(6.32154213316\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 8 x^{12} + 26 x^{11} + 743 x^{10} - 2298 x^{9} + 3586 x^{8} + 2776 x^{7} + \cdots + 1623602 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{7} - \beta_{4}) q^{5} + (\beta_{6} + 1) q^{7} + (\beta_{8} + 3 \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{7} - \beta_{4}) q^{5} + (\beta_{6} + 1) q^{7} + (\beta_{8} + 3 \beta_{4}) q^{9} + (\beta_{13} - \beta_{12} + \beta_{7} + \cdots + 1) q^{11}+ \cdots + (\beta_{12} + \beta_{11} - 2 \beta_{9} + \cdots - 30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{3} + 8 q^{7} - 6 q^{15} - 34 q^{17} + 50 q^{19} - 28 q^{21} - 12 q^{23} - 102 q^{25} - 38 q^{27} + 6 q^{29} + 60 q^{31} + 38 q^{37} + 82 q^{39} + 18 q^{41} - 48 q^{43} + 260 q^{45} - 136 q^{47} + 66 q^{49} - 60 q^{53} - 86 q^{55} + 60 q^{59} + 106 q^{61} - 272 q^{65} - 180 q^{69} + 182 q^{73} - 42 q^{75} + 260 q^{77} - 72 q^{79} - 6 q^{81} + 332 q^{83} - 144 q^{85} - 312 q^{87} + 30 q^{89} + 340 q^{95} + 58 q^{97} - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 8 x^{12} + 26 x^{11} + 743 x^{10} - 2298 x^{9} + 3586 x^{8} + 2776 x^{7} + \cdots + 1623602 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 70\!\cdots\!25 \nu^{13} + \cdots + 25\!\cdots\!54 ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\!\cdots\!99 \nu^{13} + \cdots - 59\!\cdots\!20 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!87 \nu^{13} + \cdots + 53\!\cdots\!78 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!45 \nu^{13} + \cdots - 19\!\cdots\!74 ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53\!\cdots\!18 \nu^{13} + \cdots + 21\!\cdots\!06 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!51 \nu^{13} + \cdots + 63\!\cdots\!56 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!61 \nu^{13} + \cdots - 16\!\cdots\!34 ) / 53\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!58 \nu^{13} + \cdots + 83\!\cdots\!70 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29\!\cdots\!06 \nu^{13} + \cdots + 38\!\cdots\!74 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 77\!\cdots\!85 \nu^{13} + \cdots - 11\!\cdots\!60 ) / 44\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!55 \nu^{13} + \cdots - 19\!\cdots\!52 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50\!\cdots\!05 \nu^{13} + \cdots - 58\!\cdots\!34 ) / 21\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - 12\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{13} + 2\beta_{12} - 2\beta_{6} + 21\beta_{5} - 4\beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{13} + 5 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{6} + 10 \beta_{5} - \beta_{3} + \cdots - 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 68 \beta_{12} + 94 \beta_{11} - 8 \beta_{9} + 14 \beta_{8} - 19 \beta_{7} + 26 \beta_{6} - 94 \beta_{5} + \cdots - 150 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 231 \beta_{13} - 318 \beta_{12} + 231 \beta_{11} + 87 \beta_{10} - 87 \beta_{9} + 579 \beta_{8} + \cdots + 231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2773 \beta_{13} - 2055 \beta_{12} + 400 \beta_{10} + 752 \beta_{8} - 840 \beta_{7} + 2055 \beta_{6} + \cdots + 10229 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8324 \beta_{13} - 8324 \beta_{11} + 2895 \beta_{10} + 2895 \beta_{9} + 3167 \beta_{6} - 21927 \beta_{5} + \cdots + 144392 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 61347 \beta_{12} - 82556 \beta_{11} + 15190 \beta_{9} - 29890 \beta_{8} + 28045 \beta_{7} + \cdots + 288257 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 276808 \beta_{13} + 385570 \beta_{12} - 276808 \beta_{11} - 89392 \beta_{10} + 89392 \beta_{9} + \cdots - 276808 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2489027 \beta_{13} + 1841324 \beta_{12} - 524560 \beta_{10} - 1059130 \beta_{8} + 863812 \beta_{7} + \cdots - 12544812 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8896113 \beta_{13} + 8896113 \beta_{11} - 2705136 \beta_{10} - 2705136 \beta_{9} - 3610322 \beta_{6} + \cdots - 115480053 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 55714348 \beta_{12} + 75768642 \beta_{11} - 17329260 \beta_{9} + 35467396 \beta_{8} + \cdots - 333396724 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/232\mathbb{Z}\right)^\times\).

\(n\) \(89\) \(117\) \(175\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−3.14226 − 3.14226i
−2.75393 − 2.75393i
−0.936730 − 0.936730i
1.04254 + 1.04254i
1.35465 + 1.35465i
2.49907 + 2.49907i
3.93666 + 3.93666i
−3.14226 + 3.14226i
−2.75393 + 2.75393i
−0.936730 + 0.936730i
1.04254 − 1.04254i
1.35465 − 1.35465i
2.49907 − 2.49907i
3.93666 − 3.93666i
0 −3.14226 + 3.14226i 0 − 2.76807i 0 −4.43633 0 − 10.7476i 0
17.2 0 −2.75393 + 2.75393i 0 9.57309i 0 11.1232 0 − 6.16827i 0
17.3 0 −0.936730 + 0.936730i 0 − 0.683746i 0 −2.03679 0 7.24508i 0
17.4 0 1.04254 − 1.04254i 0 − 0.481788i 0 9.84475 0 6.82624i 0
17.5 0 1.35465 − 1.35465i 0 5.16278i 0 −10.3945 0 5.32984i 0
17.6 0 2.49907 − 2.49907i 0 − 7.43739i 0 −3.47606 0 − 3.49073i 0
17.7 0 3.93666 − 3.93666i 0 6.63513i 0 3.37572 0 − 21.9946i 0
41.1 0 −3.14226 − 3.14226i 0 2.76807i 0 −4.43633 0 10.7476i 0
41.2 0 −2.75393 − 2.75393i 0 − 9.57309i 0 11.1232 0 6.16827i 0
41.3 0 −0.936730 − 0.936730i 0 0.683746i 0 −2.03679 0 − 7.24508i 0
41.4 0 1.04254 + 1.04254i 0 0.481788i 0 9.84475 0 − 6.82624i 0
41.5 0 1.35465 + 1.35465i 0 − 5.16278i 0 −10.3945 0 − 5.32984i 0
41.6 0 2.49907 + 2.49907i 0 7.43739i 0 −3.47606 0 3.49073i 0
41.7 0 3.93666 + 3.93666i 0 − 6.63513i 0 3.37572 0 21.9946i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.3.j.a ✓ 14
4.b odd 2 1 464.3.l.e 14
29.c odd 4 1 inner 232.3.j.a ✓ 14
116.e even 4 1 464.3.l.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.3.j.a ✓ 14 1.a even 1 1 trivial
232.3.j.a ✓ 14 29.c odd 4 1 inner
464.3.l.e 14 4.b odd 2 1
464.3.l.e 14 116.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 4 T_{3}^{13} + 8 T_{3}^{12} + 26 T_{3}^{11} + 743 T_{3}^{10} - 2298 T_{3}^{9} + \cdots + 1623602 \) acting on \(S_{3}^{\mathrm{new}}(232, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots + 1623602 \) Copy content Toggle raw display
$5$ \( T^{14} + 226 T^{12} + \cdots + 4946176 \) Copy content Toggle raw display
$7$ \( (T^{7} - 4 T^{6} + \cdots - 120688)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 344636258 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 1430875300864 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 47\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 403962797568512 \) Copy content Toggle raw display
$23$ \( (T^{7} + 6 T^{6} + \cdots + 471424)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 12\!\cdots\!98 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 93\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 55\!\cdots\!78 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 99\!\cdots\!82 \) Copy content Toggle raw display
$53$ \( (T^{7} + 30 T^{6} + \cdots + 284391801074)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} - 30 T^{6} + \cdots - 139548737536)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 36\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 72\!\cdots\!02 \) Copy content Toggle raw display
$83$ \( (T^{7} - 166 T^{6} + \cdots + 67013749376)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 25\!\cdots\!32 \) Copy content Toggle raw display
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