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Properties

Label 2900.2.c.i
Level $2900$
Weight $2$
Character orbit 2900.c
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(349,2900)] 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("2900.349"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-6,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
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: \( x^{10} + 18x^{8} + 99x^{6} + 163x^{4} + 75x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{9} + \beta_{3}) q^{7} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{9} + (\beta_{4} + \beta_{2} + 1) q^{11} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{13} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{6} - \beta_{5} - 8 \beta_{2} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} + 8 q^{11} + 8 q^{19} + 6 q^{21} + 10 q^{29} + 6 q^{31} + 36 q^{39} + 10 q^{41} + 6 q^{49} + 18 q^{51} + 46 q^{59} + 10 q^{61} + 60 q^{69} + 46 q^{71} - 8 q^{79} + 18 q^{81} + 12 q^{89} + 54 q^{91}+ \cdots + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 99x^{6} + 163x^{4} + 75x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 21\nu^{6} + 117\nu^{4} + 109\nu^{2} - 3 ) / 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 21\nu^{7} + 162\nu^{5} + 514\nu^{3} + 402\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{8} + 42\nu^{6} + 279\nu^{4} + 623\nu^{2} + 309 ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{8} - 69\nu^{6} - 348\nu^{4} - 436\nu^{2} - 93 ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\nu^{8} + 249\nu^{6} + 1323\nu^{4} + 1886\nu^{2} + 408 ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43\nu^{9} + 768\nu^{7} + 4131\nu^{5} + 6307\nu^{3} + 2436\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -43\nu^{9} - 768\nu^{7} - 4131\nu^{5} - 6172\nu^{3} - 1491\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\nu^{9} + 1389\nu^{7} + 7263\nu^{5} + 10096\nu^{3} + 2328\nu ) / 135 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{5} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{6} + 9\beta_{5} - 8\beta_{4} - 2\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - 8\beta_{8} - 10\beta_{7} + 7\beta_{3} + 54\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -72\beta_{6} - 71\beta_{5} + 64\beta_{4} + 28\beta_{2} - 225 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{9} + 51\beta_{8} + 92\beta_{7} - 104\beta_{3} - 425\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 568\beta_{6} + 547\beta_{5} - 517\beta_{4} - 309\beta_{2} + 1771 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 279\beta_{9} - 289\beta_{8} - 826\beta_{7} + 1185\beta_{3} + 3373\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
− 2.89768i
− 2.68862i
− 1.31654i
− 0.672645i
− 0.434833i
0.434833i
0.672645i
1.31654i
2.68862i
2.89768i
0 − 2.89768i 0 0 0 − 1.71851i 0 −5.39654 0
349.2 0 − 2.68862i 0 0 0 2.75949i 0 −4.22866 0
349.3 0 − 1.31654i 0 0 0 − 0.257197i 0 1.26672 0
349.4 0 − 0.672645i 0 0 0 3.37701i 0 2.54755 0
349.5 0 − 0.434833i 0 0 0 − 3.15620i 0 2.81092 0
349.6 0 0.434833i 0 0 0 3.15620i 0 2.81092 0
349.7 0 0.672645i 0 0 0 − 3.37701i 0 2.54755 0
349.8 0 1.31654i 0 0 0 0.257197i 0 1.26672 0
349.9 0 2.68862i 0 0 0 − 2.75949i 0 −4.22866 0
349.10 0 2.89768i 0 0 0 1.71851i 0 −5.39654 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.10
Significant digits:
Format:

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.c.i 10
5.b even 2 1 inner 2900.2.c.i 10
5.c odd 4 1 2900.2.a.i ✓ 5
5.c odd 4 1 2900.2.a.k yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2900.2.a.i ✓ 5 5.c odd 4 1
2900.2.a.k yes 5 5.c odd 4 1
2900.2.c.i 10 1.a even 1 1 trivial
2900.2.c.i 10 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{10} + 18T_{3}^{8} + 99T_{3}^{6} + 163T_{3}^{4} + 75T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{10} + 32T_{7}^{8} + 364T_{7}^{6} + 1705T_{7}^{4} + 2666T_{7}^{2} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 18 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 32 T^{8} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( (T^{5} - 4 T^{4} + \cdots - 129)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 84 T^{8} + \cdots + 308025 \) Copy content Toggle raw display
$17$ \( T^{10} + 107 T^{8} + \cdots + 149769 \) Copy content Toggle raw display
$19$ \( (T^{5} - 4 T^{4} - 29 T^{3} + \cdots - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 165 T^{8} + \cdots + 5861241 \) Copy content Toggle raw display
$29$ \( (T - 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 3 T^{4} + \cdots - 1665)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 170 T^{8} + \cdots + 434281 \) Copy content Toggle raw display
$41$ \( (T^{5} - 5 T^{4} + \cdots + 2715)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 110 T^{8} + \cdots + 172225 \) Copy content Toggle raw display
$47$ \( T^{10} + 23 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{10} + 230 T^{8} + \cdots + 18225 \) Copy content Toggle raw display
$59$ \( (T^{5} - 23 T^{4} + \cdots + 3405)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 5 T^{4} + \cdots - 9575)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 209641441 \) Copy content Toggle raw display
$71$ \( (T^{5} - 23 T^{4} + \cdots + 48951)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 120 T^{8} + \cdots + 1946025 \) Copy content Toggle raw display
$79$ \( (T^{5} + 4 T^{4} + \cdots + 32525)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 531 T^{8} + \cdots + 43046721 \) Copy content Toggle raw display
$89$ \( (T^{5} - 6 T^{4} + \cdots - 30717)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 447 T^{8} + \cdots + 1946025 \) Copy content Toggle raw display
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