Properties

Label 3450.2.d.z
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: \(\Q(i, \sqrt{57})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 29x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} - 3 \beta_{2} q^{7} + \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} - 3 \beta_{2} q^{7} + \beta_{2} q^{8} - q^{9} + ( - \beta_{3} + 3) q^{11} - \beta_{2} q^{12} + \beta_1 q^{13} - 3 q^{14} + q^{16} + ( - \beta_{2} + \beta_1) q^{17} + \beta_{2} q^{18} + ( - \beta_{3} + 1) q^{19} + 3 q^{21} + ( - 2 \beta_{2} + \beta_1) q^{22} - \beta_{2} q^{23} - q^{24} + (\beta_{3} - 1) q^{26} - \beta_{2} q^{27} + 3 \beta_{2} q^{28} + (2 \beta_{3} + 1) q^{29} + 6 q^{31} - \beta_{2} q^{32} + (2 \beta_{2} - \beta_1) q^{33} + (\beta_{3} - 2) q^{34} + q^{36} + ( - \beta_{2} - 3 \beta_1) q^{37} + \beta_1 q^{38} + ( - \beta_{3} + 1) q^{39} + (\beta_{3} - 7) q^{41} - 3 \beta_{2} q^{42} + (4 \beta_{2} + \beta_1) q^{43} + (\beta_{3} - 3) q^{44} - q^{46} + ( - 5 \beta_{2} - \beta_1) q^{47} + \beta_{2} q^{48} - 2 q^{49} + ( - \beta_{3} + 2) q^{51} - \beta_1 q^{52} + ( - 4 \beta_{2} - 2 \beta_1) q^{53} - q^{54} + 3 q^{56} - \beta_1 q^{57} + ( - 3 \beta_{2} - 2 \beta_1) q^{58} + ( - 2 \beta_{3} - 4) q^{59} + (2 \beta_{3} - 8) q^{61} - 6 \beta_{2} q^{62} + 3 \beta_{2} q^{63} - q^{64} + ( - \beta_{3} + 3) q^{66} + ( - 2 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} - \beta_1) q^{68} + q^{69} + ( - 3 \beta_{3} + 4) q^{71} - \beta_{2} q^{72} + ( - 9 \beta_{2} - 2 \beta_1) q^{73} + ( - 3 \beta_{3} + 2) q^{74} + (\beta_{3} - 1) q^{76} + ( - 6 \beta_{2} + 3 \beta_1) q^{77} + \beta_1 q^{78} + (3 \beta_{3} - 1) q^{79} + q^{81} + (6 \beta_{2} - \beta_1) q^{82} + ( - 9 \beta_{2} + 2 \beta_1) q^{83} - 3 q^{84} + (\beta_{3} + 3) q^{86} + (3 \beta_{2} + 2 \beta_1) q^{87} + (2 \beta_{2} - \beta_1) q^{88} + ( - \beta_{3} - 4) q^{89} + (3 \beta_{3} - 3) q^{91} + \beta_{2} q^{92} + 6 \beta_{2} q^{93} + ( - \beta_{3} - 4) q^{94} + q^{96} - 14 \beta_{2} q^{97} + 2 \beta_{2} q^{98} + (\beta_{3} - 3) q^{99}+O(q^{100}) \) 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 \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 10 q^{11} - 12 q^{14} + 4 q^{16} + 2 q^{19} + 12 q^{21} - 4 q^{24} - 2 q^{26} + 8 q^{29} + 24 q^{31} - 6 q^{34} + 4 q^{36} + 2 q^{39} - 26 q^{41} - 10 q^{44} - 4 q^{46} - 8 q^{49} + 6 q^{51} - 4 q^{54} + 12 q^{56} - 20 q^{59} - 28 q^{61} - 4 q^{64} + 10 q^{66} + 4 q^{69} + 10 q^{71} + 2 q^{74} - 2 q^{76} + 2 q^{79} + 4 q^{81} - 12 q^{84} + 14 q^{86} - 18 q^{89} - 6 q^{91} - 18 q^{94} + 4 q^{96} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 29x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 15\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 14\beta_{2} - 15\beta_1 \) Copy content Toggle raw display

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
3.27492i
4.27492i
3.27492i
4.27492i
1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 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.z 4
5.b even 2 1 inner 3450.2.d.z 4
5.c odd 4 1 3450.2.a.bd 2
5.c odd 4 1 3450.2.a.bm yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bd 2 5.c odd 4 1
3450.2.a.bm yes 2 5.c odd 4 1
3450.2.d.z 4 1.a even 1 1 trivial
3450.2.d.z 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{4} + 29T_{13}^{2} + 196 \) Copy content Toggle raw display
\( T_{17}^{4} + 33T_{17}^{2} + 144 \) 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
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 29T^{2} + 196 \) Copy content Toggle raw display
$17$ \( T^{4} + 33T^{2} + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 14)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 53)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 257 T^{2} + 16384 \) Copy content Toggle raw display
$41$ \( (T^{2} + 13 T + 28)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 53T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 69T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{4} + 132T^{2} + 2304 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T - 32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 14 T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 132T^{2} + 2304 \) Copy content Toggle raw display
$71$ \( (T^{2} - 5 T - 122)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 242T^{2} + 49 \) Copy content Toggle raw display
$79$ \( (T^{2} - T - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 314T^{2} + 1849 \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
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