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Properties

Label 3420.2.f.f
Level $3420$
Weight $2$
Character orbit 3420.f
Analytic conductor $27.309$
Analytic rank $0$
Dimension $12$
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] = [3420,2,Mod(1369,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.f (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: \(27.3088374913\)
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: \( x^{12} - x^{10} - 21x^{8} + 26x^{6} - 525x^{4} - 625x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{5} q^{7} + \beta_{3} q^{11} + (\beta_{6} - \beta_{2}) q^{13} + ( - \beta_{10} + \beta_{7} + \beta_1) q^{17} - q^{19} + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{23}+ \cdots + ( - 2 \beta_{8} + \beta_{6} + \cdots - \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{19} + 2 q^{25} + 16 q^{31} - 48 q^{49} + 26 q^{55} - 12 q^{61} - 48 q^{79} + 42 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{10} - 21x^{8} + 26x^{6} - 525x^{4} - 625x^{2} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 74\nu^{9} - 529\nu^{7} + 5299\nu^{5} + 17950\nu^{3} - 48125\nu ) / 100000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} - 27\nu^{8} - 17\nu^{6} + 1202\nu^{4} + 1425\nu^{2} + 1875 ) / 5000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{10} - 34\nu^{8} + 461\nu^{6} - 491\nu^{4} - 12250\nu^{2} + 15625 ) / 20000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -41\nu^{10} + 66\nu^{8} + 211\nu^{6} - 341\nu^{4} + 29050\nu^{2} + 4375 ) / 20000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - \nu^{9} - 21\nu^{7} + 26\nu^{5} - 525\nu^{3} - 625\nu ) / 3125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 6\nu^{8} + 3\nu^{6} - 29\nu^{4} + 418\nu^{2} + 3975 ) / 400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 14\nu^{9} - 11\nu^{7} + 61\nu^{5} - 410\nu^{3} - 8975\nu ) / 2000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 37\nu^{11} + 188\nu^{9} - 377\nu^{7} - 1263\nu^{5} - 1700\nu^{3} - 103125\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 59\nu^{11} + 66\nu^{9} + 511\nu^{7} + 159\nu^{5} - 23350\nu^{3} - 88125\nu ) / 50000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} - \beta_{9} - 3\beta_{7} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + 2\beta_{6} + 4\beta_{4} - 3\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{11} - 8\beta_{10} + 5\beta_{9} + 12\beta_{3} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{8} + 13\beta_{6} + 39\beta_{5} + 4\beta_{4} + 7\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33\beta_{11} - 16\beta_{10} - \beta_{9} - 83\beta_{7} + 4\beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -49\beta_{8} + 55\beta_{6} - 11\beta_{5} - 12\beta_{4} - 32\beta_{2} + 488 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -29\beta_{11} + 14\beta_{10} + 130\beta_{9} - 130\beta_{7} - 46\beta_{3} + 513\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -86\beta_{8} - 349\beta_{6} + 183\beta_{5} - 32\beta_{4} + 718\beta_{2} + 800 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 586\beta_{11} + 936\beta_{10} - 546\beta_{9} - 323\beta_{7} + 776\beta_{3} + 822\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
2.23113 + 0.148474i
2.23113 − 0.148474i
1.64243 + 1.51737i
1.64243 − 1.51737i
0.272890 + 2.21935i
0.272890 − 2.21935i
−0.272890 + 2.21935i
−0.272890 − 2.21935i
−1.64243 + 1.51737i
−1.64243 − 1.51737i
−2.23113 + 0.148474i
−2.23113 − 0.148474i
0 0 0 −2.23113 − 0.148474i 0 0.880105i 0 0 0
1369.2 0 0 0 −2.23113 + 0.148474i 0 − 0.880105i 0 0 0
1369.3 0 0 0 −1.64243 − 1.51737i 0 − 4.32180i 0 0 0
1369.4 0 0 0 −1.64243 + 1.51737i 0 4.32180i 0 0 0
1369.5 0 0 0 −0.272890 − 2.21935i 0 3.68068i 0 0 0
1369.6 0 0 0 −0.272890 + 2.21935i 0 − 3.68068i 0 0 0
1369.7 0 0 0 0.272890 − 2.21935i 0 − 3.68068i 0 0 0
1369.8 0 0 0 0.272890 + 2.21935i 0 3.68068i 0 0 0
1369.9 0 0 0 1.64243 − 1.51737i 0 4.32180i 0 0 0
1369.10 0 0 0 1.64243 + 1.51737i 0 − 4.32180i 0 0 0
1369.11 0 0 0 2.23113 − 0.148474i 0 − 0.880105i 0 0 0
1369.12 0 0 0 2.23113 + 0.148474i 0 0.880105i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1369.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.f.f ✓ 12
3.b odd 2 1 inner 3420.2.f.f ✓ 12
5.b even 2 1 inner 3420.2.f.f ✓ 12
15.d odd 2 1 inner 3420.2.f.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3420.2.f.f ✓ 12 1.a even 1 1 trivial
3420.2.f.f ✓ 12 3.b odd 2 1 inner
3420.2.f.f ✓ 12 5.b even 2 1 inner
3420.2.f.f ✓ 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 33T_{7}^{4} + 278T_{7}^{2} + 196 \) acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 33 T^{4} + \cdots + 196)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 41 T^{4} + \cdots - 100)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 74 T^{4} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 81 T^{4} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + 116 T^{4} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 46 T^{4} + \cdots - 784)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 56 T + 40)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 134 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 98 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 161 T^{4} + \cdots + 114244)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 81 T^{4} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 216 T^{4} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 296 T^{4} + \cdots - 861184)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 3 T^{2} + \cdots - 284)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 192 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 296 T^{4} + \cdots - 861184)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 203 T^{4} + \cdots + 141376)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{12} \) Copy content Toggle raw display
$83$ \( (T^{6} + 116 T^{4} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 326 T^{4} + \cdots - 2704)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 522 T^{4} + \cdots + 1507984)^{2} \) Copy content Toggle raw display
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