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Properties

Label 2900.2.a.m
Level $2900$
Weight $2$
Character orbit 2900.a
Self dual yes
Analytic conductor $23.157$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2900,2,Mod(1,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 22x^{8} + 149x^{6} - 324x^{4} + 252x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 580)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{9} + \beta_1) q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{7} - \beta_{4}) q^{11} + (\beta_{8} + \beta_{3}) q^{13} + ( - \beta_{8} - \beta_{6} + \beta_1) q^{17} + (\beta_{7} + \beta_{5}) q^{19}+ \cdots + (2 \beta_{7} + \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 14 q^{9} + 4 q^{19} + 24 q^{21} + 10 q^{29} + 16 q^{31} + 4 q^{39} + 28 q^{41} + 26 q^{49} + 32 q^{51} - 24 q^{59} + 52 q^{61} + 32 q^{69} + 24 q^{71} + 8 q^{79} + 66 q^{81} + 20 q^{89} + 16 q^{91}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 22x^{8} + 149x^{6} - 324x^{4} + 252x^{2} - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 22\nu^{7} + 149\nu^{5} - 316\nu^{3} + 172\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 22\nu^{6} + 147\nu^{4} - 294\nu^{2} + 144 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{8} - 64\nu^{6} + 403\nu^{4} - 690\nu^{2} + 288 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{9} - 106\nu^{7} + 661\nu^{5} - 1100\nu^{3} + 396\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{8} - 64\nu^{6} + 405\nu^{4} - 710\nu^{2} + 300 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -13\nu^{9} + 278\nu^{7} - 1765\nu^{5} + 3112\nu^{3} - 1316\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\nu^{9} - 406\nu^{7} + 2575\nu^{5} - 4536\nu^{3} + 1948\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{6} + \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 2\beta_{5} + 10\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{9} - 7\beta_{8} + 13\beta_{6} + 15\beta_{3} + 74\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{7} - 36\beta_{5} - 6\beta_{4} + 94\beta_{2} + 334 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -69\beta_{9} - 27\beta_{8} + 155\beta_{6} + 185\beta_{3} + 710\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 271\beta_{7} - 498\beta_{5} - 128\beta_{4} + 892\beta_{2} + 3382 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -493\beta_{9} + 133\beta_{8} + 1789\beta_{6} + 2159\beta_{3} + 6950\beta_1 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−3.27452
−2.85932
−1.35708
−0.809011
−0.778250
0.778250
0.809011
1.35708
2.85932
3.27452
0 −3.27452 0 0 0 −1.44995 0 7.72245 0
1.2 0 −2.85932 0 0 0 −3.01459 0 5.17570 0
1.3 0 −1.35708 0 0 0 0.415357 0 −1.15832 0
1.4 0 −0.809011 0 0 0 −3.65075 0 −2.34550 0
1.5 0 −0.778250 0 0 0 4.82798 0 −2.39433 0
1.6 0 0.778250 0 0 0 −4.82798 0 −2.39433 0
1.7 0 0.809011 0 0 0 3.65075 0 −2.34550 0
1.8 0 1.35708 0 0 0 −0.415357 0 −1.15832 0
1.9 0 2.85932 0 0 0 3.01459 0 5.17570 0
1.10 0 3.27452 0 0 0 1.44995 0 7.72245 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(29\) \( -1 \)

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 2900.2.a.m 10
5.b even 2 1 inner 2900.2.a.m 10
5.c odd 4 2 580.2.c.b ✓ 10
15.e even 4 2 5220.2.g.d 10
20.e even 4 2 2320.2.d.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.c.b ✓ 10 5.c odd 4 2
2320.2.d.i 10 20.e even 4 2
2900.2.a.m 10 1.a even 1 1 trivial
2900.2.a.m 10 5.b even 2 1 inner
5220.2.g.d 10 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2900))\):

\( T_{3}^{10} - 22T_{3}^{8} + 149T_{3}^{6} - 324T_{3}^{4} + 252T_{3}^{2} - 64 \) Copy content Toggle raw display
\( T_{7}^{10} - 48T_{7}^{8} + 748T_{7}^{6} - 4304T_{7}^{4} + 6656T_{7}^{2} - 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 22 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 48 T^{8} + \cdots - 1024 \) Copy content Toggle raw display
$11$ \( (T^{5} - 39 T^{3} + \cdots + 44)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 110 T^{8} + \cdots - 313600 \) Copy content Toggle raw display
$17$ \( T^{10} - 140 T^{8} + \cdots - 2768896 \) Copy content Toggle raw display
$19$ \( (T^{5} - 2 T^{4} - 38 T^{3} + \cdots - 80)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 80 T^{8} + \cdots - 30976 \) Copy content Toggle raw display
$29$ \( (T - 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots - 692)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 164 T^{8} + \cdots - 1048576 \) Copy content Toggle raw display
$41$ \( (T^{5} - 14 T^{4} + \cdots - 16000)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} - 166 T^{8} + \cdots - 7573504 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 128686336 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 1436713216 \) Copy content Toggle raw display
$59$ \( (T^{5} + 12 T^{4} + \cdots + 10048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 26 T^{4} + \cdots + 27584)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 408 T^{8} + \cdots - 57032704 \) Copy content Toggle raw display
$71$ \( (T^{5} - 12 T^{4} + \cdots - 8384)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 300 T^{8} + \cdots - 262144 \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} + \cdots - 10892)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 312 T^{8} + \cdots - 4000000 \) Copy content Toggle raw display
$89$ \( (T^{5} - 10 T^{4} + \cdots - 112448)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 476 T^{8} + \cdots - 76877824 \) Copy content Toggle raw display
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