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Properties

Label 231.2.c.a
Level $231$
Weight $2$
Character orbit 231.c
Analytic conductor $1.845$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(76,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.c (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{8} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{4} q^{5} - \beta_{12} q^{6} + (\beta_{13} + \beta_{3}) q^{7} + \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{8} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{4} q^{5} - \beta_{12} q^{6} + (\beta_{13} + \beta_{3}) q^{7} + \beta_{3} q^{8} - q^{9} + (\beta_{15} + 2 \beta_{10}) q^{10} + (\beta_{9} - \beta_{7} - \beta_{6} + 1) q^{11} + (\beta_{8} + \beta_{5}) q^{12} + ( - \beta_{15} - \beta_{12} - \beta_{10}) q^{13} + ( - \beta_{14} + \beta_{13} - \beta_{11} + \cdots + 1) q^{14}+ \cdots + ( - \beta_{9} + \beta_{7} + \beta_{6} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} - 16 q^{9} + 16 q^{11} + 8 q^{14} - 8 q^{15} - 4 q^{16} - 20 q^{22} + 24 q^{23} - 24 q^{25} + 12 q^{36} + 8 q^{37} + 12 q^{42} - 32 q^{44} + 24 q^{53} - 40 q^{56} - 12 q^{58} + 36 q^{60} + 88 q^{64} - 32 q^{67} + 36 q^{70} - 48 q^{71} + 12 q^{78} + 16 q^{81} + 32 q^{86} + 16 q^{91} - 128 q^{92} - 40 q^{93} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 75631 \nu^{14} - 26336 \nu^{12} + 423219 \nu^{10} + 3371514 \nu^{8} - 2776621 \nu^{6} + \cdots - 47672250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 122987 \nu^{14} - 97172 \nu^{12} + 701513 \nu^{10} + 5799603 \nu^{8} - 3932417 \nu^{6} + \cdots - 52412250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1660556 \nu^{14} + 460994 \nu^{12} + 5462774 \nu^{10} + 77606794 \nu^{8} - 106849066 \nu^{6} + \cdots - 854571625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 739864 \nu^{15} + 254886 \nu^{13} + 6858931 \nu^{11} + 26926811 \nu^{9} + \cdots + 1014860875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1157122 \nu^{15} - 1428203 \nu^{13} - 8678488 \nu^{11} - 43919978 \nu^{9} + \cdots - 1705295750 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4995212 \nu^{14} + 2030168 \nu^{12} + 16902678 \nu^{10} + 220892193 \nu^{8} + \cdots - 2203374500 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5423411 \nu^{14} - 2714684 \nu^{12} - 18323989 \nu^{10} - 234916384 \nu^{8} + 402906301 \nu^{6} + \cdots + 2546363000 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2293 \nu^{15} - 486 \nu^{13} + 12284 \nu^{11} + 101834 \nu^{9} - 117086 \nu^{7} + \cdots - 955700 \nu ) / 633875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 691353 \nu^{14} - 217862 \nu^{12} - 2440727 \nu^{10} - 30839287 \nu^{8} + 47497193 \nu^{6} + \cdots + 329532250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 631743 \nu^{15} - 346072 \nu^{13} - 1947962 \nu^{11} - 27635822 \nu^{9} + 50248808 \nu^{7} + \cdots + 255027250 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 276139 \nu^{15} + 1073735 \nu^{14} + 41490 \nu^{13} - 529596 \nu^{12} - 1383000 \nu^{11} + \cdots + 434987125 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1923341 \nu^{15} + 737049 \nu^{13} + 6650629 \nu^{11} + 85453649 \nu^{9} + \cdots - 861041125 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23106644 \nu^{15} + 26843375 \nu^{14} - 6020806 \nu^{13} - 13239900 \nu^{12} + \cdots + 10874678125 ) / 858900625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 23106644 \nu^{15} - 26843375 \nu^{14} - 6020806 \nu^{13} + 13239900 \nu^{12} + \cdots - 10874678125 ) / 858900625 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2876824 \nu^{15} + 1459141 \nu^{13} + 9708086 \nu^{11} + 124820466 \nu^{9} + \cdots - 1203761250 \nu ) / 78081875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{13} - 2\beta_{12} - 2\beta_{8} - 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{7} + \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - 9\beta_{10} + 7\beta_{8} - 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{14} - \beta_{13} - 6\beta_{9} - 2\beta_{7} - 12\beta_{6} - 2\beta_{3} + 2\beta_{2} - 8\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{15} + 5\beta_{14} - 15\beta_{13} - 18\beta_{12} + 20\beta_{11} + 2\beta_{10} + 36\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6\beta_{14} + 6\beta_{13} - \beta_{9} - 29\beta_{7} - 31\beta_{6} + 29\beta_{3} - 4\beta_{2} + 14\beta _1 + 75 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11 \beta_{15} - 4 \beta_{14} - 15 \beta_{13} - 35 \beta_{12} + 11 \beta_{11} + 11 \beta_{10} + \cdots - 76 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 25 \beta_{14} - 25 \beta_{13} + 19 \beta_{9} + 19 \beta_{7} - 34 \beta_{6} + 91 \beta_{3} + 15 \beta_{2} + \cdots - 65 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 113 \beta_{15} + 134 \beta_{14} - 57 \beta_{13} + 191 \beta_{12} + 191 \beta_{11} + \cdots - 114 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 265\beta_{14} - 265\beta_{13} - 248\beta_{9} + 170\beta_{7} - 744\beta_{6} - 56\beta_{3} - 398 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 247 \beta_{15} + 494 \beta_{14} - 363 \beta_{13} + 53 \beta_{12} + 857 \beta_{11} + 857 \beta_{10} + \cdots - 78 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 305 \beta_{14} - 305 \beta_{13} + 77 \beta_{9} - 77 \beta_{7} - 1065 \beta_{6} + 1337 \beta_{3} + \cdots + 1793 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 169 \beta_{15} + 571 \beta_{14} + 740 \beta_{13} + 2039 \beta_{12} - 169 \beta_{11} + \cdots - 3012 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3376 \beta_{14} - 3376 \beta_{13} + 1749 \beta_{9} + 5903 \beta_{7} - 287 \beta_{6} + 5903 \beta_{3} + \cdots - 12655 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -1546\beta_{15} + 1660\beta_{14} + 620\beta_{13} + 4638\beta_{12} + 1040\beta_{11} - 367\beta_{10} + 2334\beta_{8} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.917186 − 1.66637i
0.917186 + 1.66637i
0.0566033 − 1.17421i
−0.0566033 + 1.17421i
−1.86824 + 0.357358i
1.86824 − 0.357358i
0.644389 + 0.983224i
−0.644389 − 0.983224i
−0.644389 + 0.983224i
0.644389 − 0.983224i
1.86824 + 0.357358i
−1.86824 − 0.357358i
−0.0566033 − 1.17421i
0.0566033 + 1.17421i
0.917186 − 1.66637i
−0.917186 + 1.66637i
− 2.30927i − 1.00000i −3.33275 − 3.77447i −2.30927 −0.474903 + 2.60278i 3.07768i −1.00000 −8.71628
76.2 − 2.30927i 1.00000i −3.33275 3.77447i 2.30927 0.474903 + 2.60278i 3.07768i −1.00000 8.71628
76.3 − 2.08529i − 1.00000i −2.34841 0.833366i −2.08529 −2.19849 − 1.47195i 0.726543i −1.00000 1.73781
76.4 − 2.08529i 1.00000i −2.34841 − 0.833366i 2.08529 2.19849 − 1.47195i 0.726543i −1.00000 −1.73781
76.5 − 1.13370i − 1.00000i 0.714715 2.77447i −1.13370 2.60278 − 0.474903i − 3.07768i −1.00000 3.14542
76.6 − 1.13370i 1.00000i 0.714715 − 2.77447i 1.13370 −2.60278 − 0.474903i − 3.07768i −1.00000 −3.14542
76.7 − 0.183172i − 1.00000i 1.96645 − 1.83337i −0.183172 −1.47195 − 2.19849i − 0.726543i −1.00000 −0.335821
76.8 − 0.183172i 1.00000i 1.96645 1.83337i 0.183172 1.47195 − 2.19849i − 0.726543i −1.00000 0.335821
76.9 0.183172i − 1.00000i 1.96645 − 1.83337i 0.183172 1.47195 + 2.19849i 0.726543i −1.00000 0.335821
76.10 0.183172i 1.00000i 1.96645 1.83337i −0.183172 −1.47195 + 2.19849i 0.726543i −1.00000 −0.335821
76.11 1.13370i − 1.00000i 0.714715 2.77447i 1.13370 −2.60278 + 0.474903i 3.07768i −1.00000 −3.14542
76.12 1.13370i 1.00000i 0.714715 − 2.77447i −1.13370 2.60278 + 0.474903i 3.07768i −1.00000 3.14542
76.13 2.08529i − 1.00000i −2.34841 0.833366i 2.08529 2.19849 + 1.47195i − 0.726543i −1.00000 −1.73781
76.14 2.08529i 1.00000i −2.34841 − 0.833366i −2.08529 −2.19849 + 1.47195i − 0.726543i −1.00000 1.73781
76.15 2.30927i − 1.00000i −3.33275 − 3.77447i 2.30927 0.474903 − 2.60278i − 3.07768i −1.00000 8.71628
76.16 2.30927i 1.00000i −3.33275 3.77447i −2.30927 −0.474903 − 2.60278i − 3.07768i −1.00000 −8.71628
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.c.a ✓ 16
3.b odd 2 1 693.2.c.e 16
4.b odd 2 1 3696.2.q.e 16
7.b odd 2 1 inner 231.2.c.a ✓ 16
11.b odd 2 1 inner 231.2.c.a ✓ 16
21.c even 2 1 693.2.c.e 16
28.d even 2 1 3696.2.q.e 16
33.d even 2 1 693.2.c.e 16
44.c even 2 1 3696.2.q.e 16
77.b even 2 1 inner 231.2.c.a ✓ 16
231.h odd 2 1 693.2.c.e 16
308.g odd 2 1 3696.2.q.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.c.a ✓ 16 1.a even 1 1 trivial
231.2.c.a ✓ 16 7.b odd 2 1 inner
231.2.c.a ✓ 16 11.b odd 2 1 inner
231.2.c.a ✓ 16 77.b even 2 1 inner
693.2.c.e 16 3.b odd 2 1
693.2.c.e 16 21.c even 2 1
693.2.c.e 16 33.d even 2 1
693.2.c.e 16 231.h odd 2 1
3696.2.q.e 16 4.b odd 2 1
3696.2.q.e 16 28.d even 2 1
3696.2.q.e 16 44.c even 2 1
3696.2.q.e 16 308.g odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 11 T^{6} + 36 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 26 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 34 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 116 T^{6} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 50 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} - 44 T^{2} + \cdots - 64)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 74 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 140 T^{6} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 136 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 204 T^{6} + \cdots + 4946176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 146 T^{6} + \cdots + 1597696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 6 T^{3} + \cdots + 176)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 194 T^{6} + \cdots + 215296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 176 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + \cdots + 176)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots - 1984)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 594 T^{6} + \cdots + 426670336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 464 T^{6} + \cdots + 38738176)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 484 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 184 T^{6} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 716 T^{6} + \cdots + 126877696)^{2} \) Copy content Toggle raw display
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