↑

Properties

Label 102.3.d.a
Level $102$
Weight $3$
Character orbit 102.d
Analytic conductor $2.779$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [102,3,Mod(101,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 102.d (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: \(2.77929869648\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 6x^{10} + 65x^{8} + 1584x^{6} + 5265x^{4} + 39366x^{2} + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{8} q^{5} + \beta_{10} q^{6} + (\beta_{10} - \beta_{4} + \beta_1) q^{7} + 2 \beta_{3} q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{8} q^{5} + \beta_{10} q^{6} + (\beta_{10} - \beta_{4} + \beta_1) q^{7} + 2 \beta_{3} q^{8} + (\beta_{2} - 1) q^{9} - \beta_{5} q^{10} + (\beta_{10} - \beta_{8} + \beta_{4} + \beta_1) q^{11} - 2 \beta_1 q^{12} + ( - \beta_{11} + \beta_{6} + \beta_{2} + 2) q^{13} + (\beta_{10} + \beta_{9} - 2 \beta_1) q^{14} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} - 2) q^{15} + 4 q^{16} + (\beta_{8} + \beta_{6} + \cdots - \beta_{2}) q^{17}+ \cdots + ( - 10 \beta_{10} + 6 \beta_{9} + \cdots + 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{4} - 12 q^{9} + 28 q^{13} - 28 q^{15} + 48 q^{16} + 8 q^{18} - 44 q^{19} - 112 q^{21} + 184 q^{25} - 48 q^{30} + 76 q^{33} + 80 q^{34} + 24 q^{36} - 184 q^{42} - 172 q^{43} - 4 q^{49} + 28 q^{51} - 56 q^{52} + 332 q^{55} + 56 q^{60} - 96 q^{64} + 200 q^{66} - 360 q^{67} + 52 q^{69} + 16 q^{70} - 16 q^{72} + 88 q^{76} - 188 q^{81} + 224 q^{84} - 484 q^{85} + 424 q^{87} - 632 q^{93} - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6x^{10} + 65x^{8} + 1584x^{6} + 5265x^{4} + 39366x^{2} + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{10} + 3\nu^{8} - 80\nu^{6} + 2520\nu^{4} - 12555\nu^{2} - 137781 ) / 52488 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{11} + 483\nu^{9} + 1088\nu^{7} + 14112\nu^{5} + 272565\nu^{3} - 505197\nu ) / 472392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{11} - 57\nu^{9} - 280\nu^{7} - 20232\nu^{5} + 153009\nu^{3} + 98415\nu ) / 472392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{10} - 57\nu^{8} + 728\nu^{6} - 4572\nu^{4} - 9801\nu^{2} + 649539 ) / 26244 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} - 27\nu^{8} - 56\nu^{6} - 924\nu^{4} - 14445\nu^{2} + 35721 ) / 2916 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{11} - 57\nu^{9} - 1216\nu^{7} - 7488\nu^{5} - 57429\nu^{3} - 653913\nu ) / 78732 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\nu^{11} + 627\nu^{9} + 2648\nu^{7} + 52128\nu^{5} + 398925\nu^{3} + 439587\nu ) / 472392 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -5\nu^{11} - 3\nu^{9} + 80\nu^{7} - 2520\nu^{5} + 12555\nu^{3} + 137781\nu ) / 52488 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\nu^{10} + 201\nu^{8} + 2740\nu^{6} + 18180\nu^{4} + 173583\nu^{2} + 1371249 ) / 26244 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + 3\beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 3\beta_{7} - 6\beta_{6} - 10\beta_{3} - 3\beta_{2} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{10} + 23\beta_{9} + 6\beta_{8} - 9\beta_{5} - 26\beta_{4} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{11} - 3\beta_{7} + 24\beta_{6} + 24\beta_{3} - 20\beta_{2} - 631 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20\beta_{10} - 113\beta_{9} - 90\beta_{8} - 60\beta_{5} + 77\beta_{4} - 600\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -40\beta_{11} - 150\beta_{7} + 30\beta_{6} - 304\beta_{3} - 540\beta_{2} + 2927 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 684\beta_{10} - 700\beta_{9} + 150\beta_{8} - 1620\beta_{5} + 1840\beta_{4} + 2407\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -936\beta_{11} - 1470\beta_{7} + 3390\beta_{6} + 16104\beta_{3} + 4027\beta_{2} + 21257 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -19147\beta_{10} - 10469\beta_{9} - 4554\beta_{8} + 12081\beta_{5} + 10721\beta_{4} + 31128\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−2.49708 + 1.66270i
−2.39439 − 1.80745i
−0.177398 − 2.99475i
0.177398 + 2.99475i
2.39439 + 1.80745i
2.49708 − 1.66270i
−2.49708 − 1.66270i
−2.39439 + 1.80745i
−0.177398 + 2.99475i
0.177398 − 2.99475i
2.39439 − 1.80745i
2.49708 + 1.66270i
− 1.41421i −2.49708 + 1.66270i −2.00000 5.90424 2.35142 + 3.53141i 10.3882i 2.82843i 3.47083 − 8.30381i − 8.34985i
101.2 − 1.41421i −2.39439 − 1.80745i −2.00000 −3.85932 −2.55612 + 3.38618i 3.15746i 2.82843i 2.46623 + 8.65550i 5.45791i
101.3 − 1.41421i −0.177398 − 2.99475i −2.00000 8.44071 −4.23522 + 0.250878i − 5.48774i 2.82843i −8.93706 + 1.06252i − 11.9370i
101.4 − 1.41421i 0.177398 + 2.99475i −2.00000 −8.44071 4.23522 − 0.250878i 5.48774i 2.82843i −8.93706 + 1.06252i 11.9370i
101.5 − 1.41421i 2.39439 + 1.80745i −2.00000 3.85932 2.55612 − 3.38618i − 3.15746i 2.82843i 2.46623 + 8.65550i − 5.45791i
101.6 − 1.41421i 2.49708 − 1.66270i −2.00000 −5.90424 −2.35142 − 3.53141i − 10.3882i 2.82843i 3.47083 − 8.30381i 8.34985i
101.7 1.41421i −2.49708 − 1.66270i −2.00000 5.90424 2.35142 − 3.53141i − 10.3882i − 2.82843i 3.47083 + 8.30381i 8.34985i
101.8 1.41421i −2.39439 + 1.80745i −2.00000 −3.85932 −2.55612 − 3.38618i − 3.15746i − 2.82843i 2.46623 − 8.65550i − 5.45791i
101.9 1.41421i −0.177398 + 2.99475i −2.00000 8.44071 −4.23522 − 0.250878i 5.48774i − 2.82843i −8.93706 − 1.06252i 11.9370i
101.10 1.41421i 0.177398 − 2.99475i −2.00000 −8.44071 4.23522 + 0.250878i − 5.48774i − 2.82843i −8.93706 − 1.06252i − 11.9370i
101.11 1.41421i 2.39439 − 1.80745i −2.00000 3.85932 2.55612 + 3.38618i 3.15746i − 2.82843i 2.46623 − 8.65550i 5.45791i
101.12 1.41421i 2.49708 + 1.66270i −2.00000 −5.90424 −2.35142 + 3.53141i 10.3882i − 2.82843i 3.47083 + 8.30381i − 8.34985i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.b even 2 1 inner
51.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.3.d.a ✓ 12
3.b odd 2 1 inner 102.3.d.a ✓ 12
4.b odd 2 1 816.3.m.g 12
12.b even 2 1 816.3.m.g 12
17.b even 2 1 inner 102.3.d.a ✓ 12
51.c odd 2 1 inner 102.3.d.a ✓ 12
68.d odd 2 1 816.3.m.g 12
204.h even 2 1 816.3.m.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.3.d.a ✓ 12 1.a even 1 1 trivial
102.3.d.a ✓ 12 3.b odd 2 1 inner
102.3.d.a ✓ 12 17.b even 2 1 inner
102.3.d.a ✓ 12 51.c odd 2 1 inner
816.3.m.g 12 4.b odd 2 1
816.3.m.g 12 12.b even 2 1
816.3.m.g 12 68.d odd 2 1
816.3.m.g 12 204.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(102, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{6} - 121 T^{4} + \cdots - 36992)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 148 T^{4} + \cdots + 32400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 221 T^{4} + \cdots - 882)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 7 T^{2} + \cdots - 786)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( (T^{3} + 11 T^{2} + \cdots - 7248)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 701 T^{4} + \cdots - 112338)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 1700 T^{4} + \cdots - 4608)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 3196 T^{4} + \cdots + 281434176)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 4096 T^{4} + \cdots + 862949376)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 4121 T^{4} + \cdots - 2111980032)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 43 T^{2} + \cdots - 192352)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 6056 T^{4} + \cdots + 1198540800)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 8784 T^{4} + \cdots + 20995200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1152)^{6} \) Copy content Toggle raw display
$61$ \( (T^{6} + 16744 T^{4} + \cdots + 69442790400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 90 T^{2} + \cdots - 102944)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} - 20128 T^{4} + \cdots - 3264643208)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 10696 T^{4} + \cdots + 8707129344)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 14652 T^{4} + \cdots + 27304257600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 20552 T^{4} + \cdots + 89373155328)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 30284 T^{4} + \cdots + 681000517152)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 30600 T^{4} + \cdots + 26873856)^{2} \) Copy content Toggle raw display
show more
show less