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Properties

Label 3400.2.e.f
Level $3400$
Weight $2$
Character orbit 3400.e
Analytic conductor $27.149$
Analytic rank $0$
Dimension $4$
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] = [3400,2,Mod(2449,3400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3400, 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("3400.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3400.e (of order \(2\), degree \(1\), not 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: \(27.1491366872\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 136)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} - \beta_1) q^{7} + (2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} - \beta_1) q^{7} + (2 \beta_{3} - 3) q^{9} + ( - \beta_{3} + 1) q^{11} + 2 \beta_{2} q^{13} - \beta_1 q^{17} + (2 \beta_{3} + 2) q^{19} + (2 \beta_{3} - 6) q^{21} + ( - \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{2} + 10 \beta_1) q^{27} - 2 q^{29} + (\beta_{3} - 1) q^{31} + (2 \beta_{2} - 6 \beta_1) q^{33} + (4 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{3} - 10) q^{39} + 2 q^{41} + (2 \beta_{2} - 6 \beta_1) q^{43} + ( - 4 \beta_{2} - 4 \beta_1) q^{47} + (2 \beta_{3} + 1) q^{49} + (\beta_{3} - 1) q^{51} - 2 \beta_1 q^{53} + 8 \beta_1 q^{57} + ( - 2 \beta_{3} - 10) q^{59} + (4 \beta_{3} - 2) q^{61} + ( - 5 \beta_{2} + 13 \beta_1) q^{63} + 12 \beta_1 q^{67} + ( - 2 \beta_{3} + 6) q^{69} + (\beta_{3} + 7) q^{71} + (4 \beta_{2} + 6 \beta_1) q^{73} + (2 \beta_{2} - 6 \beta_1) q^{77} + ( - 3 \beta_{3} - 5) q^{79} + ( - 6 \beta_{3} + 11) q^{81} + ( - 2 \beta_{2} + 6 \beta_1) q^{83} + ( - 2 \beta_{2} + 2 \beta_1) q^{87} + ( - 2 \beta_{3} + 12) q^{89} + (2 \beta_{3} - 10) q^{91} + ( - 2 \beta_{2} + 6 \beta_1) q^{93} - 2 \beta_1 q^{97} + (5 \beta_{3} - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 4 q^{11} + 8 q^{19} - 24 q^{21} - 8 q^{29} - 4 q^{31} - 40 q^{39} + 8 q^{41} + 4 q^{49} - 4 q^{51} - 40 q^{59} - 8 q^{61} + 24 q^{69} + 28 q^{71} - 20 q^{79} + 44 q^{81} + 48 q^{89} - 40 q^{91} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times\).

\(n\) \(1601\) \(1701\) \(2177\) \(2551\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
− 1.61803i
− 0.618034i
0.618034i
1.61803i
0 − 3.23607i 0 0 0 − 3.23607i 0 −7.47214 0
2449.2 0 − 1.23607i 0 0 0 − 1.23607i 0 1.47214 0
2449.3 0 1.23607i 0 0 0 1.23607i 0 1.47214 0
2449.4 0 3.23607i 0 0 0 3.23607i 0 −7.47214 0
\(n\): e.g. 2-40 or 990-1000
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 3400.2.e.f 4
5.b even 2 1 inner 3400.2.e.f 4
5.c odd 4 1 136.2.a.c ✓ 2
5.c odd 4 1 3400.2.a.i 2
15.e even 4 1 1224.2.a.i 2
20.e even 4 1 272.2.a.f 2
20.e even 4 1 6800.2.a.bd 2
35.f even 4 1 6664.2.a.i 2
40.i odd 4 1 1088.2.a.s 2
40.k even 4 1 1088.2.a.o 2
60.l odd 4 1 2448.2.a.u 2
85.f odd 4 1 2312.2.b.g 4
85.g odd 4 1 2312.2.a.m 2
85.i odd 4 1 2312.2.b.g 4
120.q odd 4 1 9792.2.a.da 2
120.w even 4 1 9792.2.a.db 2
340.r even 4 1 4624.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.c ✓ 2 5.c odd 4 1
272.2.a.f 2 20.e even 4 1
1088.2.a.o 2 40.k even 4 1
1088.2.a.s 2 40.i odd 4 1
1224.2.a.i 2 15.e even 4 1
2312.2.a.m 2 85.g odd 4 1
2312.2.b.g 4 85.f odd 4 1
2312.2.b.g 4 85.i odd 4 1
2448.2.a.u 2 60.l odd 4 1
3400.2.a.i 2 5.c odd 4 1
3400.2.e.f 4 1.a even 1 1 trivial
3400.2.e.f 4 5.b even 2 1 inner
4624.2.a.h 2 340.r even 4 1
6664.2.a.i 2 35.f even 4 1
6800.2.a.bd 2 20.e even 4 1
9792.2.a.da 2 120.q odd 4 1
9792.2.a.db 2 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3400, [\chi])\):

\( T_{3}^{4} + 12T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 12T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 20 T + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 14 T + 44)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 24 T + 124)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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