Properties

Label 452.2.a.b
Level $452$
Weight $2$
Character orbit 452.a
Self dual yes
Analytic conductor $3.609$
Analytic rank $0$
Dimension $7$
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] = [452,2,Mod(1,452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("452.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 452 = 2^{2} \cdot 113 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 452.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: \(3.60923817136\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 12x^{5} + 33x^{4} + 40x^{3} - 98x^{2} - 16x + 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{6} q^{7} + (\beta_{6} + \beta_{5} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{6} q^{7} + (\beta_{6} + \beta_{5} + \beta_1 + 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{11} + (\beta_{3} - \beta_1 + 3) q^{13} + ( - \beta_{5} + \beta_{3} + 1) q^{15} + (\beta_{6} + 2) q^{17} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{21} + (\beta_{4} - \beta_{3} - \beta_1) q^{23} + ( - 2 \beta_{4} - \beta_{2} + 3) q^{25} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{27} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + 1) q^{29} + ( - \beta_{6} + 2 \beta_{4} + \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{6} - \beta_{5} - 3 \beta_{3} - \beta_{2} - 1) q^{33} + ( - \beta_{6} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 - 2) q^{35} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{37} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{39} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{41} + (\beta_{6} - \beta_{2} - \beta_1 + 2) q^{43} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2) q^{45} + (\beta_{6} - 3 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{47} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{49} + (\beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 4 \beta_1 - 1) q^{51} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{53} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} - 3) q^{55} + (\beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{57} + (\beta_{6} + 2 \beta_{4} - \beta_1 - 2) q^{59} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} + 3) q^{61} + ( - \beta_{6} - \beta_{5} - 3 \beta_{3} - \beta_{2} - 5) q^{63} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 5) q^{65} + (2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{67} + ( - 2 \beta_{6} - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{69} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_1 - 3) q^{71} + (3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 6) q^{73} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 - 3) q^{75} + (2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{77} + (2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_1 - 5) q^{79} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 5 \beta_1) q^{81} + ( - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{83} + (\beta_{6} - 2 \beta_{4} + 2 \beta_1 + 2) q^{85} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{87} + ( - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 5) q^{89} + ( - 4 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - 1) q^{91} + ( - 3 \beta_{6} - 4) q^{93} + (\beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{2} - 2 \beta_1 - 6) q^{95} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2 \beta_1 + 4) q^{97} + ( - 4 \beta_{6} - 3 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} + 3 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3 q^{3} + 3 q^{5} + 12 q^{9} + 4 q^{11} + 16 q^{13} + 3 q^{15} + 14 q^{17} + 8 q^{19} + 3 q^{21} + 16 q^{25} + 18 q^{27} + 5 q^{29} - q^{31} - 6 q^{33} - 12 q^{35} + 6 q^{37} - 12 q^{39} + q^{41} + 8 q^{43} + 12 q^{45} - 15 q^{47} + 9 q^{49} + 3 q^{51} - q^{53} - 22 q^{55} + 11 q^{57} - 15 q^{59} + 15 q^{61} - 34 q^{63} - 14 q^{65} - q^{67} - 29 q^{69} - 24 q^{71} + 41 q^{73} - 26 q^{75} + q^{77} - 21 q^{79} + 11 q^{81} - 14 q^{83} + 18 q^{85} - 23 q^{87} - 40 q^{89} - 13 q^{91} - 28 q^{93} - 30 q^{95} + 24 q^{97} - 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 12x^{5} + 33x^{4} + 40x^{3} - 98x^{2} - 16x + 58 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 16\nu^{3} + 52\nu^{2} - 22\nu - 44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{6} + 4\nu^{5} + 27\nu^{4} - 34\nu^{3} - 111\nu^{2} + 53\nu + 93 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{6} + 6\nu^{5} + 40\nu^{4} - 51\nu^{3} - 161\nu^{2} + 81\nu + 131 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{6} + 5\nu^{5} + 44\nu^{4} - 46\nu^{3} - 187\nu^{2} + 81\nu + 152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\nu^{6} - 5\nu^{5} - 44\nu^{4} + 46\nu^{3} + 188\nu^{2} - 82\nu - 156 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{6} + 11\beta_{5} - 2\beta_{4} + 3\beta_{3} + 14\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28\beta_{6} + 27\beta_{5} - 12\beta_{4} + 17\beta_{3} - 5\beta_{2} + 70\beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 115\beta_{6} + 113\beta_{5} - 34\beta_{4} + 57\beta_{3} + 7\beta_{2} + 164\beta _1 + 225 \) 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
−2.36581
−2.27178
−0.830239
1.17972
1.20346
2.72519
3.35945
0 −2.36581 0 3.54480 0 0.672272 0 2.59705 0
1.2 0 −2.27178 0 −1.02046 0 −4.19792 0 2.16100 0
1.3 0 −0.830239 0 −4.10804 0 2.60835 0 −2.31070 0
1.4 0 1.17972 0 1.63218 0 4.12658 0 −1.60825 0
1.5 0 1.20346 0 3.44450 0 −1.97464 0 −1.55168 0
1.6 0 2.72519 0 −1.95923 0 1.77350 0 4.42666 0
1.7 0 3.35945 0 1.46625 0 −3.00813 0 8.28593 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(113\) \(1\)

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 452.2.a.b 7
3.b odd 2 1 4068.2.a.l 7
4.b odd 2 1 1808.2.a.n 7
8.b even 2 1 7232.2.a.bb 7
8.d odd 2 1 7232.2.a.bc 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
452.2.a.b 7 1.a even 1 1 trivial
1808.2.a.n 7 4.b odd 2 1
4068.2.a.l 7 3.b odd 2 1
7232.2.a.bb 7 8.b even 2 1
7232.2.a.bc 7 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 3T_{3}^{6} - 12T_{3}^{5} + 33T_{3}^{4} + 40T_{3}^{3} - 98T_{3}^{2} - 16T_{3} + 58 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(452))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 3 T^{6} - 12 T^{5} + 33 T^{4} + \cdots + 58 \) Copy content Toggle raw display
$5$ \( T^{7} - 3 T^{6} - 21 T^{5} + 67 T^{4} + \cdots + 240 \) Copy content Toggle raw display
$7$ \( T^{7} - 29 T^{5} + 5 T^{4} + 232 T^{3} + \cdots + 320 \) Copy content Toggle raw display
$11$ \( T^{7} - 4 T^{6} - 49 T^{5} + \cdots + 4704 \) Copy content Toggle raw display
$13$ \( T^{7} - 16 T^{6} + 69 T^{5} + \cdots - 892 \) Copy content Toggle raw display
$17$ \( T^{7} - 14 T^{6} + 55 T^{5} + 5 T^{4} + \cdots - 48 \) Copy content Toggle raw display
$19$ \( T^{7} - 8 T^{6} - 31 T^{5} + \cdots + 3398 \) Copy content Toggle raw display
$23$ \( T^{7} - 59 T^{5} + 3 T^{4} + \cdots - 2430 \) Copy content Toggle raw display
$29$ \( T^{7} - 5 T^{6} - 114 T^{5} + 381 T^{4} + \cdots - 48 \) Copy content Toggle raw display
$31$ \( T^{7} + T^{6} - 104 T^{5} - 115 T^{4} + \cdots + 3104 \) Copy content Toggle raw display
$37$ \( T^{7} - 6 T^{6} - 103 T^{5} + \cdots - 18000 \) Copy content Toggle raw display
$41$ \( T^{7} - T^{6} - 152 T^{5} + 221 T^{4} + \cdots + 1380 \) Copy content Toggle raw display
$43$ \( T^{7} - 8 T^{6} - 35 T^{5} + 179 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$47$ \( T^{7} + 15 T^{6} - 194 T^{5} + \cdots - 1796634 \) Copy content Toggle raw display
$53$ \( T^{7} + T^{6} - 140 T^{5} + \cdots + 116868 \) Copy content Toggle raw display
$59$ \( T^{7} + 15 T^{6} - 82 T^{5} + \cdots + 21162 \) Copy content Toggle raw display
$61$ \( T^{7} - 15 T^{6} - 8 T^{5} + \cdots + 69760 \) Copy content Toggle raw display
$67$ \( T^{7} + T^{6} - 224 T^{5} + \cdots + 157754 \) Copy content Toggle raw display
$71$ \( T^{7} + 24 T^{6} - 68 T^{5} + \cdots + 1451376 \) Copy content Toggle raw display
$73$ \( T^{7} - 41 T^{6} + 458 T^{5} + \cdots - 185392 \) Copy content Toggle raw display
$79$ \( T^{7} + 21 T^{6} - 78 T^{5} + \cdots - 2349362 \) Copy content Toggle raw display
$83$ \( T^{7} + 14 T^{6} - 43 T^{5} + \cdots + 13632 \) Copy content Toggle raw display
$89$ \( T^{7} + 40 T^{6} + 463 T^{5} + \cdots - 1465104 \) Copy content Toggle raw display
$97$ \( T^{7} - 24 T^{6} - 73 T^{5} + \cdots - 1300608 \) Copy content Toggle raw display
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