Properties

Label 5225.2.a.p
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $9$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1045)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{8} q^{6} + ( - \beta_{4} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{8} q^{6} + ( - \beta_{4} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{2} + 1) q^{9} + q^{11} + (\beta_{8} + \beta_{6} + 1) q^{12} + ( - \beta_{6} - 1) q^{13} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 1) q^{14} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{16}+ \cdots + (\beta_{7} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26} - 6 q^{27} - 10 q^{28} - 3 q^{29} - 9 q^{31} - 6 q^{32} - 3 q^{33} - 2 q^{34} - 15 q^{36} + 7 q^{37} - 3 q^{38} - 2 q^{39} + 9 q^{41} + 9 q^{42} - 23 q^{43} + 9 q^{44} - 32 q^{46} - 20 q^{47} + 18 q^{48} - 4 q^{49} + 8 q^{51} - 9 q^{52} + 5 q^{53} + 9 q^{54} + 4 q^{56} - 3 q^{57} - 3 q^{58} + 19 q^{59} + q^{61} - 18 q^{62} - 24 q^{63} + 23 q^{64} + 10 q^{67} - 9 q^{68} - 28 q^{69} - 24 q^{72} - 12 q^{73} + 5 q^{74} + 9 q^{76} - 13 q^{77} - 19 q^{78} - 21 q^{79} - 3 q^{81} + 14 q^{82} - 47 q^{83} - 11 q^{84} + 12 q^{86} + 2 q^{87} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 19 q^{92} + 2 q^{93} + 26 q^{94} - 55 q^{96} + 32 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 21\nu^{4} - 31\nu^{3} - 12\nu^{2} + 15\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 5\nu^{6} + 32\nu^{5} - 7\nu^{4} - 59\nu^{3} + 24\nu^{2} + 25\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 8\nu^{3} - 45\nu^{2} + 19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 2\nu^{6} + 9\nu^{5} - 16\nu^{4} - 21\nu^{3} + 31\nu^{2} + 13\nu - 15 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 3\nu^{6} + 30\nu^{5} - 25\nu^{4} - 47\nu^{3} + 64\nu^{2} + 15\nu - 27 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + 7\beta_{3} + \beta_{2} + 27\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{8} + \beta_{7} + 11\beta_{6} - 2\beta_{5} - 10\beta_{4} + \beta_{3} + 44\beta_{2} + 2\beta _1 + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26\beta_{8} + 10\beta_{7} + 24\beta_{6} - 13\beta_{5} - 13\beta_{4} + 44\beta_{3} + 16\beta_{2} + 155\beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 107\beta_{8} + 13\beta_{7} + 94\beta_{6} - 28\beta_{5} - 77\beta_{4} + 16\beta_{3} + 277\beta_{2} + 36\beta _1 + 564 \) 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
2.69878
2.14345
1.92391
1.22961
0.682920
−0.730132
−1.24377
−1.35249
−2.35228
−2.69878 1.59120 5.28341 0 −4.29429 −0.342738 −8.86121 −0.468091 0
1.2 −2.14345 −2.53942 2.59437 0 5.44311 1.53582 −1.27399 3.44866 0
1.3 −1.92391 −0.621031 1.70143 0 1.19481 −5.15278 0.574421 −2.61432 0
1.4 −1.22961 2.38474 −0.488068 0 −2.93229 −1.13367 3.05934 2.68698 0
1.5 −0.682920 −2.48419 −1.53362 0 1.69650 −0.856948 2.41318 3.17118 0
1.6 0.730132 −0.820975 −1.46691 0 −0.599420 1.34825 −2.53130 −2.32600 0
1.7 1.24377 −3.09454 −0.453038 0 −3.84889 −3.83759 −3.05101 6.57616 0
1.8 1.35249 2.73889 −0.170779 0 3.70431 −2.99075 −2.93595 4.50151 0
1.9 2.35228 −0.154676 3.53320 0 −0.363841 −1.56958 3.60652 −2.97608 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)
\(19\) \(-1\)

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 5225.2.a.p 9
5.b even 2 1 1045.2.a.k 9
15.d odd 2 1 9405.2.a.bh 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.k 9 5.b even 2 1
5225.2.a.p 9 1.a even 1 1 trivial
9405.2.a.bh 9 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{9} + 3T_{2}^{8} - 9T_{2}^{7} - 29T_{2}^{6} + 23T_{2}^{5} + 84T_{2}^{4} - 23T_{2}^{3} - 89T_{2}^{2} + 8T_{2} + 27 \) Copy content Toggle raw display
\( T_{7}^{9} + 13T_{7}^{8} + 55T_{7}^{7} + 56T_{7}^{6} - 169T_{7}^{5} - 389T_{7}^{4} - 56T_{7}^{3} + 408T_{7}^{2} + 320T_{7} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 3 T^{8} + \cdots + 27 \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 13 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} + 5 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{9} + 13 T^{8} + \cdots - 5904 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 8 T^{8} + \cdots + 192 \) Copy content Toggle raw display
$29$ \( T^{9} + 3 T^{8} + \cdots - 1680 \) Copy content Toggle raw display
$31$ \( T^{9} + 9 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{9} - 7 T^{8} + \cdots + 32192 \) Copy content Toggle raw display
$41$ \( T^{9} - 9 T^{8} + \cdots + 1696176 \) Copy content Toggle raw display
$43$ \( T^{9} + 23 T^{8} + \cdots - 29632 \) Copy content Toggle raw display
$47$ \( T^{9} + 20 T^{8} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( T^{9} - 5 T^{8} + \cdots - 768192 \) Copy content Toggle raw display
$59$ \( T^{9} - 19 T^{8} + \cdots - 3341760 \) Copy content Toggle raw display
$61$ \( T^{9} - T^{8} + \cdots - 23824 \) Copy content Toggle raw display
$67$ \( T^{9} - 10 T^{8} + \cdots - 46441456 \) Copy content Toggle raw display
$71$ \( T^{9} - 298 T^{7} + \cdots - 34636224 \) Copy content Toggle raw display
$73$ \( T^{9} + 12 T^{8} + \cdots + 160279408 \) Copy content Toggle raw display
$79$ \( T^{9} + 21 T^{8} + \cdots - 71054080 \) Copy content Toggle raw display
$83$ \( T^{9} + 47 T^{8} + \cdots - 909504 \) Copy content Toggle raw display
$89$ \( T^{9} + 2 T^{8} + \cdots + 37547280 \) Copy content Toggle raw display
$97$ \( T^{9} - 32 T^{8} + \cdots + 165140288 \) Copy content Toggle raw display
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