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Properties

Label 2900.2.d.e
Level $2900$
Weight $2$
Character orbit 2900.d
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(1101,2900)] 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("2900.1101"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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: \( x^{10} + 16x^{8} + 77x^{6} + 115x^{4} + 61x^{2} + 9 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{5} q^{7} + \beta_{2} q^{9} - \beta_{8} q^{11} + \beta_{7} q^{13} + \beta_{9} q^{17} + (\beta_{9} + \beta_{4} + \beta_1) q^{19} + (2 \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{21}+ \cdots + ( - 3 \beta_{9} - \beta_{4} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{9} - 4 q^{13} + 14 q^{23} + 2 q^{29} + 6 q^{33} + 34 q^{49} + 10 q^{51} + 4 q^{53} - 20 q^{57} - 6 q^{59} + 18 q^{63} - 22 q^{67} + 14 q^{71} + 6 q^{81} + 6 q^{83} + 22 q^{87} - 26 q^{91} - 30 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16x^{8} + 77x^{6} + 115x^{4} + 61x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 16\nu^{7} - 77\nu^{5} - 112\nu^{3} - 40\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + 16\nu^{7} + 77\nu^{5} + 115\nu^{3} + 58\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{8} + 31\nu^{6} + 137\nu^{4} + 148\nu^{2} + 33 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{8} + 31\nu^{6} + 140\nu^{4} + 172\nu^{2} + 48 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} + 15\nu^{6} + 62\nu^{4} + 53\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{9} - 61\nu^{7} - 263\nu^{5} - 271\nu^{3} - 61\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{9} - 62\nu^{7} - 277\nu^{5} - 323\nu^{3} - 96\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} - 8\beta_{2} + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{9} - 2\beta_{8} - 7\beta_{4} - 11\beta_{3} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} - 13\beta_{6} + 16\beta_{5} + 62\beta_{2} - 140 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -45\beta_{9} + 31\beta_{8} + 46\beta_{4} + 102\beta_{3} - 339\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 31\beta_{7} + 133\beta_{6} - 178\beta_{5} - 487\beta_{2} + 1074 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 489\beta_{9} - 342\beta_{8} - 309\beta_{4} - 900\beta_{3} + 2668\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
− 2.87199i
− 2.38054i
− 1.06518i
− 0.843582i
− 0.488328i
0.488328i
0.843582i
1.06518i
2.38054i
2.87199i
0 − 2.87199i 0 0 0 1.95804 0 −5.24832 0
1101.2 0 − 2.38054i 0 0 0 −4.97091 0 −2.66697 0
1101.3 0 − 1.06518i 0 0 0 0.173844 0 1.86539 0
1101.4 0 − 0.843582i 0 0 0 4.53366 0 2.28837 0
1101.5 0 − 0.488328i 0 0 0 −1.69464 0 2.76154 0
1101.6 0 0.488328i 0 0 0 −1.69464 0 2.76154 0
1101.7 0 0.843582i 0 0 0 4.53366 0 2.28837 0
1101.8 0 1.06518i 0 0 0 0.173844 0 1.86539 0
1101.9 0 2.38054i 0 0 0 −4.97091 0 −2.66697 0
1101.10 0 2.87199i 0 0 0 1.95804 0 −5.24832 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1101.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.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.d.e ✓ 10
5.b even 2 1 2900.2.d.f yes 10
5.c odd 4 2 2900.2.f.e 20
29.b even 2 1 inner 2900.2.d.e ✓ 10
145.d even 2 1 2900.2.d.f yes 10
145.h odd 4 2 2900.2.f.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2900.2.d.e ✓ 10 1.a even 1 1 trivial
2900.2.d.e ✓ 10 29.b even 2 1 inner
2900.2.d.f yes 10 5.b even 2 1
2900.2.d.f yes 10 145.d even 2 1
2900.2.f.e 20 5.c odd 4 2
2900.2.f.e 20 145.h odd 4 2

Hecke kernels

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

\( T_{3}^{10} + 16T_{3}^{8} + 77T_{3}^{6} + 115T_{3}^{4} + 61T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{5} - 26T_{7}^{3} + 9T_{7}^{2} + 74T_{7} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 16 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 26 T^{3} + \cdots - 13)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 64 T^{8} + \cdots + 69169 \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} - 32 T^{3} + \cdots - 45)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 55 T^{8} + \cdots + 225 \) Copy content Toggle raw display
$19$ \( T^{10} + 68 T^{8} + \cdots + 289 \) Copy content Toggle raw display
$23$ \( (T^{5} - 7 T^{4} + \cdots + 381)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 100260169 \) Copy content Toggle raw display
$37$ \( T^{10} + 152 T^{8} + \cdots + 576081 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 241958025 \) Copy content Toggle raw display
$43$ \( T^{10} + 202 T^{8} + \cdots + 130321 \) Copy content Toggle raw display
$47$ \( T^{10} + 217 T^{8} + \cdots + 5688225 \) Copy content Toggle raw display
$53$ \( (T^{5} - 2 T^{4} - 111 T^{3} + \cdots - 55)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + 3 T^{4} + \cdots + 14415)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 473 T^{8} + \cdots + 491401 \) Copy content Toggle raw display
$67$ \( (T^{5} + 11 T^{4} + 23 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 7 T^{4} + \cdots + 3357)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 244640881 \) Copy content Toggle raw display
$79$ \( T^{10} + 466 T^{8} + \cdots + 14707225 \) Copy content Toggle raw display
$83$ \( (T^{5} - 3 T^{4} + \cdots + 17575)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1356522561 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1587464649 \) Copy content Toggle raw display
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