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Properties

Label 4450.2.a.bk
Level $4450$
Weight $2$
Character orbit 4450.a
Self dual yes
Analytic conductor $35.533$
Analytic rank $0$
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] = [4450,2,Mod(1,4450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4450, 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("4450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4450 = 2 \cdot 5^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4450.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: \(35.5334288995\)
Analytic rank: \(0\)
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: \( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 163x^{5} - 153x^{4} - 480x^{3} + 416x^{2} + 448x - 384 \) 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 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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + \beta_{8} q^{7} - q^{8} + (\beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + \beta_{8} q^{7} - q^{8} + (\beta_{3} + \beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_{2} + 1) q^{11} - \beta_1 q^{12} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{13} - \beta_{8} q^{14} + q^{16} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{17} + ( - \beta_{3} - \beta_{2} - 2) q^{18} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{19} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 1) q^{21}+ \cdots + (\beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - q^{3} + 9 q^{4} + q^{6} - 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - q^{3} + 9 q^{4} + q^{6} - 9 q^{8} + 18 q^{9} + 10 q^{11} - q^{12} - 5 q^{13} + 9 q^{16} - 3 q^{17} - 18 q^{18} + 10 q^{21} - 10 q^{22} - 4 q^{23} + q^{24} + 5 q^{26} + 5 q^{27} + 19 q^{29} - 2 q^{31} - 9 q^{32} + 2 q^{33} + 3 q^{34} + 18 q^{36} - 17 q^{37} + 37 q^{39} + 12 q^{41} - 10 q^{42} + 2 q^{43} + 10 q^{44} + 4 q^{46} - q^{48} + 29 q^{49} - 9 q^{51} - 5 q^{52} - 5 q^{54} - 12 q^{57} - 19 q^{58} + 41 q^{59} - 10 q^{61} + 2 q^{62} + 24 q^{63} + 9 q^{64} - 2 q^{66} + 4 q^{67} - 3 q^{68} + 40 q^{69} + q^{71} - 18 q^{72} - 25 q^{73} + 17 q^{74} + 12 q^{77} - 37 q^{78} + 35 q^{79} - 7 q^{81} - 12 q^{82} - 2 q^{83} + 10 q^{84} - 2 q^{86} + 35 q^{87} - 10 q^{88} - 9 q^{89} - 2 q^{91} - 4 q^{92} - 32 q^{93} + q^{96} - 5 q^{97} - 29 q^{98} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 163x^{5} - 153x^{4} - 480x^{3} + 416x^{2} + 448x - 384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{8} - 13\nu^{7} - 134\nu^{6} + 190\nu^{5} + 1183\nu^{4} - 597\nu^{3} - 3864\nu^{2} - 160\nu + 2944 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} + 13\nu^{7} + 134\nu^{6} - 190\nu^{5} - 1183\nu^{4} + 597\nu^{3} + 4088\nu^{2} + 160\nu - 4064 ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{8} - 15\nu^{7} + 106\nu^{6} + 258\nu^{5} - 791\nu^{4} - 1279\nu^{3} + 2380\nu^{2} + 1672\nu - 2384 ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{8} - 39\nu^{7} - 290\nu^{6} + 682\nu^{5} + 1757\nu^{4} - 3135\nu^{3} - 4088\nu^{2} + 3440\nu + 2336 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{8} - 5\nu^{7} + 89\nu^{6} + 86\nu^{5} - 658\nu^{4} - 431\nu^{3} + 1785\nu^{2} + 576\nu - 1336 ) / 56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\nu^{8} - 31\nu^{7} - 494\nu^{6} + 522\nu^{5} + 2681\nu^{4} - 2031\nu^{3} - 4956\nu^{2} + 1376\nu + 2368 ) / 224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} - 7\nu^{7} + 30\nu^{6} + 122\nu^{5} - 307\nu^{4} - 607\nu^{3} + 1128\nu^{2} + 864\nu - 1184 ) / 32 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{4} + \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 3\beta_{4} + 10\beta_{3} + 11\beta_{2} - \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -27\beta_{8} + 11\beta_{7} + 11\beta_{6} + 4\beta_{5} + 28\beta_{4} + 12\beta_{3} - 6\beta_{2} + 56\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{8} - 15\beta_{7} + 17\beta_{6} + 14\beta_{5} - 52\beta_{4} + 93\beta_{3} + 109\beta_{2} - 23\beta _1 + 269 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 303 \beta_{8} + 110 \beta_{7} + 106 \beta_{6} + 64 \beta_{5} + 320 \beta_{4} + 119 \beta_{3} + \cdots - 70 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 272 \beta_{8} - 178 \beta_{7} + 196 \beta_{6} + 153 \beta_{5} - 677 \beta_{4} + 872 \beta_{3} + \cdots + 2334 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.99254
2.59209
2.29406
1.19351
0.874663
−1.35385
−2.09276
−2.23310
−3.26715
−1.00000 −2.99254 1.00000 0 2.99254 −1.11393 −1.00000 5.95528 0
1.2 −1.00000 −2.59209 1.00000 0 2.59209 1.18289 −1.00000 3.71895 0
1.3 −1.00000 −2.29406 1.00000 0 2.29406 −2.50424 −1.00000 2.26269 0
1.4 −1.00000 −1.19351 1.00000 0 1.19351 4.78088 −1.00000 −1.57553 0
1.5 −1.00000 −0.874663 1.00000 0 0.874663 −2.44941 −1.00000 −2.23496 0
1.6 −1.00000 1.35385 1.00000 0 −1.35385 −4.20045 −1.00000 −1.16708 0
1.7 −1.00000 2.09276 1.00000 0 −2.09276 3.39264 −1.00000 1.37965 0
1.8 −1.00000 2.23310 1.00000 0 −2.23310 −3.05581 −1.00000 1.98673 0
1.9 −1.00000 3.26715 1.00000 0 −3.26715 3.96744 −1.00000 7.67426 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
\(2\) \( +1 \)
\(5\) \( -1 \)
\(89\) \( +1 \)

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 4450.2.a.bk ✓ 9
5.b even 2 1 4450.2.a.bl yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4450.2.a.bk ✓ 9 1.a even 1 1 trivial
4450.2.a.bl yes 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4450))\):

\( T_{3}^{9} + T_{3}^{8} - 22T_{3}^{7} - 22T_{3}^{6} + 163T_{3}^{5} + 153T_{3}^{4} - 480T_{3}^{3} - 416T_{3}^{2} + 448T_{3} + 384 \) Copy content Toggle raw display
\( T_{7}^{9} - 46T_{7}^{7} - 26T_{7}^{6} + 713T_{7}^{5} + 762T_{7}^{4} - 4038T_{7}^{3} - 5864T_{7}^{2} + 4468T_{7} + 6676 \) Copy content Toggle raw display
\( T_{11}^{9} - 10 T_{11}^{8} + 2 T_{11}^{7} + 224 T_{11}^{6} - 391 T_{11}^{5} - 1422 T_{11}^{4} + \cdots + 2468 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + T^{8} + \cdots + 384 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 46 T^{7} + \cdots + 6676 \) Copy content Toggle raw display
$11$ \( T^{9} - 10 T^{8} + \cdots + 2468 \) Copy content Toggle raw display
$13$ \( T^{9} + 5 T^{8} + \cdots + 11268 \) Copy content Toggle raw display
$17$ \( T^{9} + 3 T^{8} + \cdots + 15552 \) Copy content Toggle raw display
$19$ \( T^{9} - 78 T^{7} + \cdots + 80000 \) Copy content Toggle raw display
$23$ \( T^{9} + 4 T^{8} + \cdots - 2628 \) Copy content Toggle raw display
$29$ \( T^{9} - 19 T^{8} + \cdots + 707175 \) Copy content Toggle raw display
$31$ \( T^{9} + 2 T^{8} + \cdots + 2000 \) Copy content Toggle raw display
$37$ \( T^{9} + 17 T^{8} + \cdots - 682365 \) Copy content Toggle raw display
$41$ \( T^{9} - 12 T^{8} + \cdots + 5589440 \) Copy content Toggle raw display
$43$ \( T^{9} - 2 T^{8} + \cdots - 5184 \) Copy content Toggle raw display
$47$ \( T^{9} - 300 T^{7} + \cdots + 22523392 \) Copy content Toggle raw display
$53$ \( T^{9} - 232 T^{7} + \cdots - 1219584 \) Copy content Toggle raw display
$59$ \( T^{9} - 41 T^{8} + \cdots - 84000 \) Copy content Toggle raw display
$61$ \( T^{9} + 10 T^{8} + \cdots + 234787500 \) Copy content Toggle raw display
$67$ \( T^{9} - 4 T^{8} + \cdots + 136592 \) Copy content Toggle raw display
$71$ \( T^{9} - T^{8} + \cdots + 3898800 \) Copy content Toggle raw display
$73$ \( T^{9} + 25 T^{8} + \cdots - 51319419 \) Copy content Toggle raw display
$79$ \( T^{9} - 35 T^{8} + \cdots - 18006000 \) Copy content Toggle raw display
$83$ \( T^{9} + 2 T^{8} + \cdots - 173952 \) Copy content Toggle raw display
$89$ \( (T + 1)^{9} \) Copy content Toggle raw display
$97$ \( T^{9} + 5 T^{8} + \cdots + 1913 \) Copy content Toggle raw display
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