Properties

Label 231.2.i.f
Level $231$
Weight $2$
Character orbit 231.i
Analytic conductor $1.845$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(67,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.i (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: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{4}+ \cdots + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{4}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 5 q^{3} - 10 q^{4} + 4 q^{5} - 4 q^{6} - q^{7} + 12 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} + 5 q^{3} - 10 q^{4} + 4 q^{5} - 4 q^{6} - q^{7} + 12 q^{8} - 5 q^{9} - 2 q^{10} - 5 q^{11} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 8 q^{15} - 16 q^{16} - 2 q^{17} - 2 q^{18} + 3 q^{19} - 16 q^{20} - 2 q^{21} + 4 q^{22} - 16 q^{23} + 6 q^{24} - 7 q^{25} + 10 q^{26} - 10 q^{27} + 4 q^{28} + 2 q^{30} - 5 q^{31} - 4 q^{32} + 5 q^{33} + 40 q^{34} + 26 q^{35} + 20 q^{36} - 15 q^{37} - 6 q^{38} + 5 q^{39} + 6 q^{40} - 44 q^{41} - 14 q^{42} + 6 q^{43} - 10 q^{44} + 4 q^{45} - 16 q^{46} + 2 q^{47} - 32 q^{48} + 31 q^{49} + 68 q^{50} + 2 q^{51} - 40 q^{52} - 6 q^{53} + 2 q^{54} - 8 q^{55} - 12 q^{56} + 6 q^{57} - 12 q^{58} - 16 q^{59} - 8 q^{60} - 12 q^{61} - 8 q^{62} - q^{63} - 8 q^{64} + 28 q^{65} + 2 q^{66} - 7 q^{67} - 10 q^{68} - 32 q^{69} + 32 q^{70} + 48 q^{71} - 6 q^{72} - 17 q^{73} + 36 q^{74} + 7 q^{75} + 60 q^{76} + 2 q^{77} + 20 q^{78} - 7 q^{79} - 16 q^{80} - 5 q^{81} - 8 q^{82} - 24 q^{83} - 28 q^{84} + 4 q^{85} + 18 q^{86} - 6 q^{88} + 6 q^{89} + 4 q^{90} + 11 q^{91} + 136 q^{92} + 5 q^{93} - 82 q^{94} + 18 q^{95} + 4 q^{96} - 28 q^{97} - 38 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} + 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 15\nu^{7} + 70\nu^{5} + 102\nu^{3} + 36\nu - 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 2\nu^{8} + 13\nu^{7} + 28\nu^{6} + 44\nu^{5} + 118\nu^{4} + 6\nu^{3} + 140\nu^{2} - 36\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + \nu^{8} + 14\nu^{7} + 15\nu^{6} + 59\nu^{5} + 70\nu^{4} + 70\nu^{3} + 100\nu^{2} + 18\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 14\nu^{7} - 13\nu^{6} - 59\nu^{5} - 48\nu^{4} - 70\nu^{3} - 36\nu^{2} - 18\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} - 14\nu^{6} - 59\nu^{4} - 70\nu^{2} - 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 2\nu^{8} + 15\nu^{7} - 30\nu^{6} + 74\nu^{5} - 140\nu^{4} + 130\nu^{3} - 204\nu^{2} + 60\nu - 76 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{9} - 43\nu^{7} - 188\nu^{5} - 4\nu^{4} - 246\nu^{3} - 28\nu^{2} - 96\nu - 20 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} + \nu^{8} - 14\nu^{7} + 15\nu^{6} - 59\nu^{5} + 70\nu^{4} - 70\nu^{3} + 100\nu^{2} - 18\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{9} - 2\beta_{8} + 2\beta_{7} + \beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{6} + \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{9} + 2\beta_{8} - 4\beta_{7} - 2\beta_{5} + \beta_{4} - 4\beta_{3} + 2\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} + 7\beta_{6} - 7\beta_{5} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{9} - 10 \beta_{8} + 26 \beta_{7} - \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 24 \beta_{3} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -45\beta_{9} - 45\beta_{6} + 47\beta_{5} + 2\beta_{4} - 11\beta _1 - 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 101 \beta_{9} + 58 \beta_{8} - 170 \beta_{7} + 11 \beta_{6} - 85 \beta_{5} + 16 \beta_{4} - 148 \beta_{3} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 287\beta_{9} + 285\beta_{6} - 315\beta_{5} - 28\beta_{4} + 95\beta _1 + 618 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 607 \beta_{9} - 350 \beta_{8} + 1114 \beta_{7} - 95 \beta_{6} + 557 \beta_{5} - 50 \beta_{4} + \cdots - 326 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.518255i
1.21103i
2.57330i
2.42024i
0.886226i
0.518255i
1.21103i
2.57330i
2.42024i
0.886226i
−1.29725 2.24690i 0.500000 0.866025i −2.36571 + 4.09752i −1.09601 1.89835i −2.59450 −2.61329 + 0.413178i 7.08664 −0.500000 0.866025i −2.84359 + 4.92525i
67.2 −1.17616 2.03717i 0.500000 0.866025i −1.76671 + 3.06003i 2.05761 + 3.56389i −2.35232 2.51585 0.818848i 3.60708 −0.500000 0.866025i 4.84017 8.38342i
67.3 −0.307468 0.532550i 0.500000 0.866025i 0.810927 1.40457i −0.747986 1.29555i −0.614936 2.59895 + 0.495442i −2.22721 −0.500000 0.866025i −0.459963 + 0.796680i
67.4 0.534421 + 0.925645i 0.500000 0.866025i 0.428788 0.742682i 1.34592 + 2.33120i 1.06884 −0.855706 2.50355i 3.05430 −0.500000 0.866025i −1.43858 + 2.49169i
67.5 1.24646 + 2.15892i 0.500000 0.866025i −2.10730 + 3.64995i 0.440463 + 0.762904i 2.49291 −2.14580 + 1.54775i −5.52081 −0.500000 0.866025i −1.09804 + 1.90185i
100.1 −1.29725 + 2.24690i 0.500000 + 0.866025i −2.36571 4.09752i −1.09601 + 1.89835i −2.59450 −2.61329 0.413178i 7.08664 −0.500000 + 0.866025i −2.84359 4.92525i
100.2 −1.17616 + 2.03717i 0.500000 + 0.866025i −1.76671 3.06003i 2.05761 3.56389i −2.35232 2.51585 + 0.818848i 3.60708 −0.500000 + 0.866025i 4.84017 + 8.38342i
100.3 −0.307468 + 0.532550i 0.500000 + 0.866025i 0.810927 + 1.40457i −0.747986 + 1.29555i −0.614936 2.59895 0.495442i −2.22721 −0.500000 + 0.866025i −0.459963 0.796680i
100.4 0.534421 0.925645i 0.500000 + 0.866025i 0.428788 + 0.742682i 1.34592 2.33120i 1.06884 −0.855706 + 2.50355i 3.05430 −0.500000 + 0.866025i −1.43858 2.49169i
100.5 1.24646 2.15892i 0.500000 + 0.866025i −2.10730 3.64995i 0.440463 0.762904i 2.49291 −2.14580 1.54775i −5.52081 −0.500000 + 0.866025i −1.09804 1.90185i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.5
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 231.2.i.f 10
3.b odd 2 1 693.2.i.j 10
7.c even 3 1 inner 231.2.i.f 10
7.c even 3 1 1617.2.a.ba 5
7.d odd 6 1 1617.2.a.bb 5
21.g even 6 1 4851.2.a.bz 5
21.h odd 6 1 693.2.i.j 10
21.h odd 6 1 4851.2.a.ca 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.f 10 1.a even 1 1 trivial
231.2.i.f 10 7.c even 3 1 inner
693.2.i.j 10 3.b odd 2 1
693.2.i.j 10 21.h odd 6 1
1617.2.a.ba 5 7.c even 3 1
1617.2.a.bb 5 7.d odd 6 1
4851.2.a.bz 5 21.g even 6 1
4851.2.a.ca 5 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 2T_{2}^{9} + 12T_{2}^{8} + 12T_{2}^{7} + 81T_{2}^{6} + 78T_{2}^{5} + 264T_{2}^{4} + 6T_{2}^{3} + 261T_{2}^{2} + 110T_{2} + 100 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$13$ \( (T^{5} - 5 T^{4} + \cdots + 476)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots + 100 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} + \cdots + 363609 \) Copy content Toggle raw display
$23$ \( T^{10} + 16 T^{9} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( (T^{5} - 96 T^{3} + \cdots - 1290)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 5 T^{9} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{10} + 15 T^{9} + \cdots + 40401 \) Copy content Toggle raw display
$41$ \( (T^{5} + 22 T^{4} + \cdots - 11536)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 3 T^{4} + \cdots - 2973)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 783664036 \) Copy content Toggle raw display
$53$ \( T^{10} + 6 T^{9} + \cdots + 42302016 \) Copy content Toggle raw display
$59$ \( T^{10} + 16 T^{9} + \cdots + 38416 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 2361960000 \) Copy content Toggle raw display
$67$ \( T^{10} + 7 T^{9} + \cdots + 9265936 \) Copy content Toggle raw display
$71$ \( (T^{5} - 24 T^{4} + \cdots + 5232)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 17 T^{9} + \cdots + 9265936 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 8999937424 \) Copy content Toggle raw display
$83$ \( (T^{5} + 12 T^{4} + \cdots + 2688)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 315417600 \) Copy content Toggle raw display
$97$ \( (T^{5} + 14 T^{4} + \cdots - 38906)^{2} \) Copy content Toggle raw display
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