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Properties

Label 3450.2.d.x
Level $3450$
Weight $2$
Character orbit 3450.d
Analytic conductor $27.548$
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] = [3450,2,Mod(2899,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3450.2899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (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: \(27.5483886973\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 138)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_1 q^{3} - q^{4} + q^{6} + 2 \beta_{2} q^{7} - \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_1 q^{3} - q^{4} + q^{6} + 2 \beta_{2} q^{7} - \beta_1 q^{8} - q^{9} + (\beta_{3} - 3) q^{11} + \beta_1 q^{12} + 2 \beta_{2} q^{13} - 2 \beta_{3} q^{14} + q^{16} - 4 \beta_1 q^{17} - \beta_1 q^{18} + (3 \beta_{3} + 1) q^{19} + 2 \beta_{3} q^{21} + (\beta_{2} - 3 \beta_1) q^{22} - \beta_1 q^{23} - q^{24} - 2 \beta_{3} q^{26} + \beta_1 q^{27} - 2 \beta_{2} q^{28} - 2 \beta_{3} q^{29} + (2 \beta_{3} + 2) q^{31} + \beta_1 q^{32} + ( - \beta_{2} + 3 \beta_1) q^{33} + 4 q^{34} + q^{36} + ( - \beta_{2} + 9 \beta_1) q^{37} + (3 \beta_{2} + \beta_1) q^{38} + 2 \beta_{3} q^{39} - 2 q^{41} + 2 \beta_{2} q^{42} + (\beta_{2} + 7 \beta_1) q^{43} + ( - \beta_{3} + 3) q^{44} + q^{46} + 4 \beta_1 q^{47} - \beta_1 q^{48} - 13 q^{49} - 4 q^{51} - 2 \beta_{2} q^{52} + (\beta_{2} - 3 \beta_1) q^{53} - q^{54} + 2 \beta_{3} q^{56} + ( - 3 \beta_{2} - \beta_1) q^{57} - 2 \beta_{2} q^{58} - 4 \beta_{3} q^{59} + (\beta_{3} + 3) q^{61} + (2 \beta_{2} + 2 \beta_1) q^{62} - 2 \beta_{2} q^{63} - q^{64} + (\beta_{3} - 3) q^{66} + ( - 3 \beta_{2} + 3 \beta_1) q^{67} + 4 \beta_1 q^{68} - q^{69} - 4 \beta_{3} q^{71} + \beta_1 q^{72} - 2 \beta_{2} q^{73} + (\beta_{3} - 9) q^{74} + ( - 3 \beta_{3} - 1) q^{76} + ( - 6 \beta_{2} + 10 \beta_1) q^{77} + 2 \beta_{2} q^{78} + 2 \beta_{3} q^{79} + q^{81} - 2 \beta_1 q^{82} + (\beta_{2} - 11 \beta_1) q^{83} - 2 \beta_{3} q^{84} + ( - \beta_{3} - 7) q^{86} + 2 \beta_{2} q^{87} + ( - \beta_{2} + 3 \beta_1) q^{88} + ( - 2 \beta_{3} + 6) q^{89} - 20 q^{91} + \beta_1 q^{92} + ( - 2 \beta_{2} - 2 \beta_1) q^{93} - 4 q^{94} + q^{96} + ( - 2 \beta_{2} - 4 \beta_1) q^{97} - 13 \beta_1 q^{98} + ( - \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 12 q^{11} + 4 q^{16} + 4 q^{19} - 4 q^{24} + 8 q^{31} + 16 q^{34} + 4 q^{36} - 8 q^{41} + 12 q^{44} + 4 q^{46} - 52 q^{49} - 16 q^{51} - 4 q^{54} + 12 q^{61} - 4 q^{64} - 12 q^{66} - 4 q^{69} - 36 q^{74} - 4 q^{76} + 4 q^{81} - 28 q^{86} + 24 q^{89} - 80 q^{91} - 16 q^{94} + 4 q^{96} + 12 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2899.1
− 0.618034i
1.61803i
− 1.61803i
0.618034i
− 1.00000i 1.00000i −1.00000 0 1.00000 − 4.47214i 1.00000i −1.00000 0
2899.2 − 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.3 1.00000i − 1.00000i −1.00000 0 1.00000 − 4.47214i − 1.00000i −1.00000 0
2899.4 1.00000i − 1.00000i −1.00000 0 1.00000 4.47214i − 1.00000i −1.00000 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 3450.2.d.x 4
5.b even 2 1 inner 3450.2.d.x 4
5.c odd 4 1 138.2.a.d ✓ 2
5.c odd 4 1 3450.2.a.be 2
15.e even 4 1 414.2.a.f 2
20.e even 4 1 1104.2.a.j 2
35.f even 4 1 6762.2.a.cb 2
40.i odd 4 1 4416.2.a.bh 2
40.k even 4 1 4416.2.a.bl 2
60.l odd 4 1 3312.2.a.bc 2
115.e even 4 1 3174.2.a.s 2
345.l odd 4 1 9522.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d ✓ 2 5.c odd 4 1
414.2.a.f 2 15.e even 4 1
1104.2.a.j 2 20.e even 4 1
3174.2.a.s 2 115.e even 4 1
3312.2.a.bc 2 60.l odd 4 1
3450.2.a.be 2 5.c odd 4 1
3450.2.d.x 4 1.a even 1 1 trivial
3450.2.d.x 4 5.b even 2 1 inner
4416.2.a.bh 2 40.i odd 4 1
4416.2.a.bl 2 40.k even 4 1
6762.2.a.cb 2 35.f even 4 1
9522.2.a.q 2 345.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7}^{2} + 20 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 20 \) Copy content Toggle raw display
\( T_{17}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 252 T^{2} + 13456 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
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