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Properties

Label 3420.2.f.e
Level $3420$
Weight $2$
Character orbit 3420.f
Analytic conductor $27.309$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 2x^{8} + 18x^{7} - 7x^{6} - 48x^{5} - 35x^{4} + 450x^{3} - 250x^{2} - 1250x + 3125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1140)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} + \beta_{4} q^{7} + (\beta_{5} - 1) q^{11} + (\beta_{9} + \beta_{7} + \cdots - \beta_{3}) q^{13}+ \cdots + ( - \beta_{9} + 2 \beta_{8} + \cdots - \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{5} - 12 q^{11} + 10 q^{19} - 8 q^{25} + 16 q^{31} + 22 q^{35} - 24 q^{41} - 6 q^{49} - 10 q^{55} - 12 q^{59} + 32 q^{65} - 4 q^{71} - 24 q^{79} - 10 q^{85} - 12 q^{89} + 32 q^{91} + 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 2x^{8} + 18x^{7} - 7x^{6} - 48x^{5} - 35x^{4} + 450x^{3} - 250x^{2} - 1250x + 3125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8 \nu^{9} - 49 \nu^{8} - 54 \nu^{7} + 11 \nu^{6} + 36 \nu^{5} - 136 \nu^{4} - 700 \nu^{3} + \cdots - 10625 ) / 12500 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23 \nu^{9} - 169 \nu^{8} - 99 \nu^{7} + 791 \nu^{6} - 1184 \nu^{5} - 2616 \nu^{4} + 1525 \nu^{3} + \cdots - 98125 ) / 25000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 43 \nu^{9} + 321 \nu^{8} + 41 \nu^{7} - 1219 \nu^{6} + 56 \nu^{5} + 6944 \nu^{4} + 3625 \nu^{3} + \cdots + 95625 ) / 25000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14 \nu^{9} - 67 \nu^{8} + 243 \nu^{7} + 13 \nu^{6} - 662 \nu^{5} - 338 \nu^{4} + 4000 \nu^{3} + \cdots + 18125 ) / 6250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14 \nu^{9} - 33 \nu^{8} - 43 \nu^{7} + 187 \nu^{6} + 112 \nu^{5} - 212 \nu^{4} - 2950 \nu^{3} + \cdots - 18125 ) / 6250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73 \nu^{9} + 281 \nu^{8} - 449 \nu^{7} - 59 \nu^{6} + 1216 \nu^{5} + 584 \nu^{4} - 1525 \nu^{3} + \cdots - 104375 ) / 25000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17 \nu^{9} - \nu^{8} + 179 \nu^{7} - 261 \nu^{6} - 536 \nu^{5} + 536 \nu^{4} + 2875 \nu^{3} + \cdots + 29375 ) / 5000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 87 \nu^{9} - 89 \nu^{8} - 169 \nu^{7} + 671 \nu^{6} + 696 \nu^{5} - 1496 \nu^{4} - 5725 \nu^{3} + \cdots - 23125 ) / 25000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47 \nu^{9} - 9 \nu^{8} - 89 \nu^{7} - 49 \nu^{6} + 976 \nu^{5} + 424 \nu^{4} - 4325 \nu^{3} + \cdots + 26875 ) / 12500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} + 2\beta_{8} - 2\beta_{7} - 6\beta_{5} + 2\beta_{4} + 2\beta_{3} + 4\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} - 3\beta_{8} - 4\beta_{7} - \beta_{6} + 2\beta_{5} + 4\beta_{4} + 7\beta_{3} + 5\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20\beta_{9} - 23\beta_{8} - 24\beta_{7} + 8\beta_{6} - 4\beta_{5} + 24\beta_{4} - 15\beta_{2} + 7\beta _1 - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{9} - 6 \beta_{8} - 32 \beta_{7} - 2 \beta_{6} + 14 \beta_{5} + 24 \beta_{4} + 16 \beta_{3} + \cdots + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 111 \beta_{9} + 83 \beta_{8} + 40 \beta_{6} + 84 \beta_{5} + 64 \beta_{4} - 24 \beta_{3} - 203 \beta_{2} + \cdots - 128 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 37 \beta_{9} + 115 \beta_{8} + 111 \beta_{7} + 117 \beta_{6} + 18 \beta_{5} - 28 \beta_{4} - 20 \beta_{3} + \cdots + 171 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 19 \beta_{9} + 438 \beta_{8} + 74 \beta_{7} - 320 \beta_{6} - 310 \beta_{5} - 226 \beta_{4} + \cdots - 1600 ) / 2 \) 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
1.35419 − 1.77937i
1.35419 + 1.77937i
2.09727 − 0.775529i
2.09727 + 0.775529i
−2.23345 + 0.108197i
−2.23345 − 0.108197i
−1.38485 − 1.75562i
−1.38485 + 1.75562i
1.16684 + 1.90748i
1.16684 − 1.90748i
0 0 0 −1.77937 − 1.35419i 0 4.11171i 0 0 0
1369.2 0 0 0 −1.77937 + 1.35419i 0 − 4.11171i 0 0 0
1369.3 0 0 0 −0.775529 − 2.09727i 0 − 2.02158i 0 0 0
1369.4 0 0 0 −0.775529 + 2.09727i 0 2.02158i 0 0 0
1369.5 0 0 0 −0.108197 − 2.23345i 0 3.86839i 0 0 0
1369.6 0 0 0 −0.108197 + 2.23345i 0 − 3.86839i 0 0 0
1369.7 0 0 0 1.75562 − 1.38485i 0 − 0.408758i 0 0 0
1369.8 0 0 0 1.75562 + 1.38485i 0 0.408758i 0 0 0
1369.9 0 0 0 1.90748 − 1.16684i 0 1.36950i 0 0 0
1369.10 0 0 0 1.90748 + 1.16684i 0 − 1.36950i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1369.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 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.e 10
3.b odd 2 1 1140.2.f.b ✓ 10
5.b even 2 1 inner 3420.2.f.e 10
15.d odd 2 1 1140.2.f.b ✓ 10
15.e even 4 1 5700.2.a.bc 5
15.e even 4 1 5700.2.a.bd 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.f.b ✓ 10 3.b odd 2 1
1140.2.f.b ✓ 10 15.d odd 2 1
3420.2.f.e 10 1.a even 1 1 trivial
3420.2.f.e 10 5.b even 2 1 inner
5700.2.a.bc 5 15.e even 4 1
5700.2.a.bd 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 38T_{7}^{8} + 457T_{7}^{6} + 1828T_{7}^{4} + 2232T_{7}^{2} + 324 \) acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 38 T^{8} + \cdots + 324 \) Copy content Toggle raw display
$11$ \( (T^{5} + 6 T^{4} - 13 T^{3} + \cdots + 50)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 64 T^{8} + \cdots + 40000 \) Copy content Toggle raw display
$17$ \( T^{10} + 78 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( (T - 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} + 128 T^{8} + \cdots + 3873024 \) Copy content Toggle raw display
$29$ \( (T^{5} - 126 T^{3} + \cdots + 8920)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} - 32 T^{3} + \cdots + 48)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 252 T^{8} + \cdots + 1893376 \) Copy content Toggle raw display
$41$ \( (T^{5} + 12 T^{4} + \cdots - 480)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 258 T^{8} + \cdots + 10771524 \) Copy content Toggle raw display
$47$ \( T^{10} + 294 T^{8} + \cdots + 38192400 \) Copy content Toggle raw display
$53$ \( T^{10} + 180 T^{8} + \cdots + 2715904 \) Copy content Toggle raw display
$59$ \( (T^{5} + 6 T^{4} + \cdots + 3200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 99 T^{3} + \cdots - 180)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 172 T^{8} + \cdots + 10240000 \) Copy content Toggle raw display
$71$ \( (T^{5} + 2 T^{4} + \cdots + 9600)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 2171560000 \) Copy content Toggle raw display
$79$ \( (T^{5} + 12 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 274233600 \) Copy content Toggle raw display
$89$ \( (T^{5} + 6 T^{4} + \cdots - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1737889344 \) Copy content Toggle raw display
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