Properties

Label 3450.2.a.bh
Level $3450$
Weight $2$
Character orbit 3450.a
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3450,2,Mod(1,3450)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3450.1");
 
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.a (trivial)

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: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} + q^{9} + ( - \beta + 2) q^{11} + q^{12} + \beta q^{13} + 3 q^{14} + q^{16} + ( - \beta - 3) q^{17} - q^{18} + ( - \beta + 4) q^{19} - 3 q^{21} + (\beta - 2) q^{22} + q^{23} - q^{24} - \beta q^{26} + q^{27} - 3 q^{28} + 3 q^{29} + (2 \beta - 2) q^{31} - q^{32} + ( - \beta + 2) q^{33} + (\beta + 3) q^{34} + q^{36} + ( - \beta + 3) q^{37} + (\beta - 4) q^{38} + \beta q^{39} + (\beta - 2) q^{41} + 3 q^{42} + ( - \beta + 4) q^{43} + ( - \beta + 2) q^{44} - q^{46} + (\beta - 5) q^{47} + q^{48} + 2 q^{49} + ( - \beta - 3) q^{51} + \beta q^{52} + 4 q^{53} - q^{54} + 3 q^{56} + ( - \beta + 4) q^{57} - 3 q^{58} + 6 q^{59} + (2 \beta - 2) q^{61} + ( - 2 \beta + 2) q^{62} - 3 q^{63} + q^{64} + (\beta - 2) q^{66} + (2 \beta - 2) q^{67} + ( - \beta - 3) q^{68} + q^{69} + (\beta - 1) q^{71} - q^{72} + ( - 2 \beta - 1) q^{73} + (\beta - 3) q^{74} + ( - \beta + 4) q^{76} + (3 \beta - 6) q^{77} - \beta q^{78} + (\beta + 6) q^{79} + q^{81} + ( - \beta + 2) q^{82} - q^{83} - 3 q^{84} + (\beta - 4) q^{86} + 3 q^{87} + (\beta - 2) q^{88} + ( - \beta + 13) q^{89} - 3 \beta q^{91} + q^{92} + (2 \beta - 2) q^{93} + ( - \beta + 5) q^{94} - q^{96} + ( - 2 \beta + 10) q^{97} - 2 q^{98} + ( - \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{11} + 2 q^{12} + q^{13} + 6 q^{14} + 2 q^{16} - 7 q^{17} - 2 q^{18} + 7 q^{19} - 6 q^{21} - 3 q^{22} + 2 q^{23} - 2 q^{24} - q^{26} + 2 q^{27} - 6 q^{28} + 6 q^{29} - 2 q^{31} - 2 q^{32} + 3 q^{33} + 7 q^{34} + 2 q^{36} + 5 q^{37} - 7 q^{38} + q^{39} - 3 q^{41} + 6 q^{42} + 7 q^{43} + 3 q^{44} - 2 q^{46} - 9 q^{47} + 2 q^{48} + 4 q^{49} - 7 q^{51} + q^{52} + 8 q^{53} - 2 q^{54} + 6 q^{56} + 7 q^{57} - 6 q^{58} + 12 q^{59} - 2 q^{61} + 2 q^{62} - 6 q^{63} + 2 q^{64} - 3 q^{66} - 2 q^{67} - 7 q^{68} + 2 q^{69} - q^{71} - 2 q^{72} - 4 q^{73} - 5 q^{74} + 7 q^{76} - 9 q^{77} - q^{78} + 13 q^{79} + 2 q^{81} + 3 q^{82} - 2 q^{83} - 6 q^{84} - 7 q^{86} + 6 q^{87} - 3 q^{88} + 25 q^{89} - 3 q^{91} + 2 q^{92} - 2 q^{93} + 9 q^{94} - 2 q^{96} + 18 q^{97} - 4 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.77200
−3.77200
−1.00000 1.00000 1.00000 0 −1.00000 −3.00000 −1.00000 1.00000 0
1.2 −1.00000 1.00000 1.00000 0 −1.00000 −3.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3450.2.a.bh 2
5.b even 2 1 3450.2.a.bj yes 2
5.c odd 4 2 3450.2.d.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bh 2 1.a even 1 1 trivial
3450.2.a.bj yes 2 5.b even 2 1
3450.2.d.w 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3450))\):

\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 16 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 18 \) Copy content Toggle raw display
\( T_{17}^{2} + 7T_{17} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 7T - 6 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T - 6 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 72 \) Copy content Toggle raw display
$37$ \( T^{2} - 5T - 12 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T - 6 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 2 \) Copy content Toggle raw display
$53$ \( (T - 4)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 72 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 72 \) Copy content Toggle raw display
$71$ \( T^{2} + T - 18 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 69 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 24 \) Copy content Toggle raw display
$83$ \( (T + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 25T + 138 \) Copy content Toggle raw display
$97$ \( T^{2} - 18T + 8 \) Copy content Toggle raw display
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