Properties

Label 1232.2.q.o
Level $1232$
Weight $2$
Character orbit 1232.q
Analytic conductor $9.838$
Analytic rank $0$
Dimension $10$
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] = [1232,2,Mod(177,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), 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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.939795628203.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - 9x^{7} + 10x^{6} - 26x^{5} + 87x^{4} - 48x^{3} - 65x^{2} + 30x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 616)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{2}) q^{3} + (\beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} - 1) q^{5} + ( - \beta_{5} + \beta_1 + 1) q^{7} + (\beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{2}) q^{3} + (\beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} - 1) q^{5} + ( - \beta_{5} + \beta_1 + 1) q^{7} + (\beta_{4} + \beta_{3}) q^{9} - \beta_{7} q^{11} + (\beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{13} + (\beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{15} + (2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{17} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_1 - 2) q^{21} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{23} + (2 \beta_{9} - 2 \beta_{8} - 3 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{25}+ \cdots + (\beta_{6} - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} - 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} - 4 q^{5} + 2 q^{7} - 5 q^{11} + 2 q^{13} - 14 q^{15} - 9 q^{17} + q^{19} - 12 q^{21} - 8 q^{23} - 5 q^{25} + 8 q^{27} + 18 q^{29} + 3 q^{31} - q^{33} + 15 q^{35} - 2 q^{37} - 7 q^{39} + 30 q^{41} - 28 q^{43} - 19 q^{45} + 11 q^{47} - 18 q^{49} - 14 q^{51} + 9 q^{53} + 8 q^{55} + 8 q^{57} + 4 q^{59} + 2 q^{61} + 28 q^{63} - 7 q^{65} - 8 q^{67} + 22 q^{69} - 30 q^{71} - 26 q^{73} + 27 q^{75} + 2 q^{77} - 3 q^{79} + 19 q^{81} + 2 q^{83} + 26 q^{85} + 14 q^{87} - 41 q^{89} + 39 q^{91} - 10 q^{93} - 19 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + x^{8} - 9x^{7} + 10x^{6} - 26x^{5} + 87x^{4} - 48x^{3} - 65x^{2} + 30x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 175931 \nu^{9} + 2122814 \nu^{8} - 2291723 \nu^{7} + 1252650 \nu^{6} - 23214830 \nu^{5} + 19749496 \nu^{4} - 57379965 \nu^{3} + \cdots - 166645188 ) / 41020284 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 155541 \nu^{9} + 15964 \nu^{8} - 240025 \nu^{7} + 996486 \nu^{6} - 372550 \nu^{5} + 4073080 \nu^{4} - 7882773 \nu^{3} - 1132357 \nu^{2} + \cdots - 12840162 ) / 6836714 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1013393 \nu^{9} - 1615910 \nu^{8} + 4712021 \nu^{7} - 8349810 \nu^{6} + 23040746 \nu^{5} - 53645260 \nu^{4} + 108969915 \nu^{3} - 191378187 \nu^{2} + \cdots + 92028048 ) / 41020284 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1065671 \nu^{9} - 737758 \nu^{8} - 1671755 \nu^{7} + 6616590 \nu^{6} + 2989378 \nu^{5} + 33742060 \nu^{4} - 34920825 \nu^{3} - 36613935 \nu^{2} + \cdots - 25274400 ) / 41020284 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 276392 \nu^{9} - 372211 \nu^{8} - 612335 \nu^{7} + 1491771 \nu^{6} + 2225113 \nu^{5} + 8490406 \nu^{4} - 3531393 \nu^{3} - 15167460 \nu^{2} + \cdots - 131250 ) / 10255071 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1145905 \nu^{9} - 1178626 \nu^{8} + 950941 \nu^{7} - 9427146 \nu^{6} + 12029062 \nu^{5} - 21697700 \nu^{4} + 99152043 \nu^{3} - 43882995 \nu^{2} + \cdots + 38723880 ) / 41020284 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4793 \nu^{9} - 3458 \nu^{8} + 4601 \nu^{7} - 47178 \nu^{6} + 31478 \nu^{5} - 137080 \nu^{4} + 395715 \nu^{3} - 129855 \nu^{2} - 142600 \nu + 40428 ) / 144948 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 673853 \nu^{9} + 685446 \nu^{8} - 904397 \nu^{7} + 6820122 \nu^{6} - 6623314 \nu^{5} + 21055420 \nu^{4} - 65582787 \nu^{3} + 40792183 \nu^{2} + \cdots + 9226776 ) / 13673428 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1505423 \nu^{9} + 725326 \nu^{8} + 2850521 \nu^{7} - 9602418 \nu^{6} + 425318 \nu^{5} - 40881364 \nu^{4} + 69828225 \nu^{3} + 24370503 \nu^{2} + \cdots - 9415896 ) / 20510142 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{3} - \beta_{2} + 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} - 6\beta_{8} - 11\beta_{7} + 2\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} - \beta_{2} + 3\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{9} + 2\beta_{8} + 9\beta_{7} - 8\beta_{6} - \beta_{5} + 17\beta_{4} - 7\beta_{3} - 3\beta_{2} - 7\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{9} + 30 \beta_{8} + 35 \beta_{7} + 12 \beta_{6} + 15 \beta_{5} - 13 \beta_{4} + 13 \beta_{3} + 27 \beta_{2} + 5 \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7 \beta_{9} - 50 \beta_{8} - 85 \beta_{7} - 24 \beta_{6} + 33 \beta_{5} - 29 \beta_{4} - 23 \beta_{3} - 51 \beta_{2} - 11 \beta _1 - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 43 \beta_{9} + 36 \beta_{8} + 3 \beta_{7} + 78 \beta_{6} - 81 \beta_{5} + 87 \beta_{4} + 93 \beta_{3} - 35 \beta_{2} + 85 \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 53 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} + 82 \beta_{6} + 35 \beta_{5} - 133 \beta_{4} - \beta_{3} + 331 \beta_{2} - 11 \beta _1 + 435 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 401 \beta_{9} - 104 \beta_{8} + 207 \beta_{7} - 730 \beta_{6} + 275 \beta_{5} + 357 \beta_{4} - 633 \beta_{3} - 467 \beta_{2} - 627 \beta _1 - 895 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−0.574432 + 0.269593i
1.13198 + 0.475002i
1.60172 0.387640i
−0.0561490 + 2.08099i
−1.60312 1.57192i
−0.574432 0.269593i
1.13198 0.475002i
1.60172 + 0.387640i
−0.0561490 2.08099i
−1.60312 + 1.57192i
0 −1.29867 2.24937i 0 −1.07443 + 1.86097i 0 −0.726377 2.54409i 0 −1.87311 + 3.24431i 0
177.2 0 −0.746498 1.29297i 0 0.631979 1.09462i 0 2.25927 + 1.37685i 0 0.385481 0.667672i 0
177.3 0 −0.142113 0.246147i 0 1.10172 1.90824i 0 −1.07405 + 2.41794i 0 1.45961 2.52811i 0
177.4 0 0.666828 + 1.15498i 0 −0.556149 + 0.963278i 0 2.01243 1.71759i 0 0.610680 1.05773i 0
177.5 0 1.02046 + 1.76748i 0 −2.10312 + 3.64271i 0 −1.47127 + 2.19895i 0 −0.582662 + 1.00920i 0
529.1 0 −1.29867 + 2.24937i 0 −1.07443 1.86097i 0 −0.726377 + 2.54409i 0 −1.87311 3.24431i 0
529.2 0 −0.746498 + 1.29297i 0 0.631979 + 1.09462i 0 2.25927 1.37685i 0 0.385481 + 0.667672i 0
529.3 0 −0.142113 + 0.246147i 0 1.10172 + 1.90824i 0 −1.07405 2.41794i 0 1.45961 + 2.52811i 0
529.4 0 0.666828 1.15498i 0 −0.556149 0.963278i 0 2.01243 + 1.71759i 0 0.610680 + 1.05773i 0
529.5 0 1.02046 1.76748i 0 −2.10312 3.64271i 0 −1.47127 2.19895i 0 −0.582662 1.00920i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.5
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 1232.2.q.o 10
4.b odd 2 1 616.2.q.f 10
7.c even 3 1 inner 1232.2.q.o 10
7.c even 3 1 8624.2.a.dc 5
7.d odd 6 1 8624.2.a.db 5
28.f even 6 1 4312.2.a.bg 5
28.g odd 6 1 616.2.q.f 10
28.g odd 6 1 4312.2.a.bf 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.f 10 4.b odd 2 1
616.2.q.f 10 28.g odd 6 1
1232.2.q.o 10 1.a even 1 1 trivial
1232.2.q.o 10 7.c even 3 1 inner
4312.2.a.bf 5 28.g odd 6 1
4312.2.a.bg 5 28.f even 6 1
8624.2.a.db 5 7.d odd 6 1
8624.2.a.dc 5 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{10} + T_{3}^{9} + 8T_{3}^{8} + T_{3}^{7} + 43T_{3}^{6} + 11T_{3}^{5} + 83T_{3}^{4} + 2T_{3}^{3} + 112T_{3}^{2} + 30T_{3} + 9 \) Copy content Toggle raw display
\( T_{13}^{5} - T_{13}^{4} - 39T_{13}^{3} + 10T_{13}^{2} + 12T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + 8 T^{8} + T^{7} + 43 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + 23 T^{8} + 22 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + 11 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$13$ \( (T^{5} - T^{4} - 39 T^{3} + 10 T^{2} + 12 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + 79 T^{8} + \cdots + 71824 \) Copy content Toggle raw display
$19$ \( T^{10} - T^{9} + 27 T^{8} + 24 T^{7} + \cdots + 29584 \) Copy content Toggle raw display
$23$ \( T^{10} + 8 T^{9} + 83 T^{8} + \cdots + 753424 \) Copy content Toggle raw display
$29$ \( (T^{5} - 9 T^{4} - 35 T^{3} + 474 T^{2} + \cdots + 401)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 3 T^{9} + 35 T^{8} + 84 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + 116 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
$41$ \( (T^{5} - 15 T^{4} + 8 T^{3} + 645 T^{2} + \cdots - 516)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 14 T^{4} - 109 T^{3} - 2099 T^{2} + \cdots - 5488)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 11 T^{9} + 103 T^{8} + \cdots + 1296 \) Copy content Toggle raw display
$53$ \( T^{10} - 9 T^{9} + 233 T^{8} + \cdots + 6091024 \) Copy content Toggle raw display
$59$ \( T^{10} - 4 T^{9} + 196 T^{8} + \cdots + 206008609 \) Copy content Toggle raw display
$61$ \( T^{10} - 2 T^{9} + 119 T^{8} + \cdots + 1577536 \) Copy content Toggle raw display
$67$ \( T^{10} + 8 T^{9} + 161 T^{8} + \cdots + 17106496 \) Copy content Toggle raw display
$71$ \( (T^{5} + 15 T^{4} - 88 T^{3} - 703 T^{2} + \cdots - 5348)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 26 T^{9} + \cdots + 116553616 \) Copy content Toggle raw display
$79$ \( T^{10} + 3 T^{9} + 152 T^{8} + \cdots + 46063369 \) Copy content Toggle raw display
$83$ \( (T^{5} - T^{4} - 292 T^{3} - 23 T^{2} + \cdots - 37116)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 41 T^{9} + \cdots + 147573904 \) Copy content Toggle raw display
$97$ \( (T^{5} - 7 T^{4} - 133 T^{3} + 670 T^{2} + \cdots - 709)^{2} \) Copy content Toggle raw display
show more
show less