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Properties

Label 2900.2.s.c
Level $2900$
Weight $2$
Character orbit 2900.s
Analytic conductor $23.157$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2900,2,Mod(157,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 468x^{12} + 2844x^{10} + 8574x^{8} + 12524x^{6} + 8404x^{4} + 2324x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{14} q^{7} + ( - \beta_{3} + 1) q^{9} + \beta_{10} q^{11} + (\beta_{14} - \beta_{13}) q^{13} + (\beta_{15} + \beta_{14} + \cdots - 2 \beta_{6}) q^{17} + ( - \beta_{11} + \beta_{7} - 1) q^{19}+ \cdots + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9} - 4 q^{11} - 12 q^{19} + 16 q^{21} - 28 q^{29} + 28 q^{31} + 32 q^{39} - 16 q^{41} - 24 q^{61} - 72 q^{69} - 4 q^{79} + 8 q^{81} - 24 q^{89} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 468x^{12} + 2844x^{10} + 8574x^{8} + 12524x^{6} + 8404x^{4} + 2324x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1026 \nu^{14} - 36129 \nu^{12} - 452236 \nu^{10} - 2577619 \nu^{8} - 6929706 \nu^{6} + \cdots - 386705 ) / 80464 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 607 \nu^{14} - 21801 \nu^{12} - 282236 \nu^{10} - 1701211 \nu^{8} - 5027133 \nu^{6} + \cdots - 538967 ) / 40232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1026 \nu^{14} - 36129 \nu^{12} - 452236 \nu^{10} - 2577619 \nu^{8} - 6929706 \nu^{6} + \cdots - 185545 ) / 40232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3297 \nu^{14} + 116584 \nu^{12} + 1468619 \nu^{10} + 8443258 \nu^{8} + 22937211 \nu^{6} + \cdots + 991222 ) / 40232 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13905 \nu^{15} + 494275 \nu^{13} + 6286475 \nu^{11} + 36794383 \nu^{9} + 103611135 \nu^{7} + \cdots + 1463165 \nu ) / 1046032 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23557 \nu^{15} + 834714 \nu^{13} + 10554999 \nu^{11} + 61117040 \nu^{9} + 168468671 \nu^{7} + \cdots + 12157116 \nu ) / 1046032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23557 \nu^{15} + 834714 \nu^{13} + 10554999 \nu^{11} + 61117040 \nu^{9} + 168468671 \nu^{7} + \cdots + 11111084 \nu ) / 1046032 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1271 \nu^{15} + 44771 \nu^{13} + 559981 \nu^{11} + 3174874 \nu^{9} + 8353891 \nu^{7} + \cdots - 464322 \nu ) / 40232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33776 \nu^{15} + 1199751 \nu^{13} + 15230930 \nu^{11} + 88725033 \nu^{9} + 246851164 \nu^{7} + \cdots + 16148971 \nu ) / 1046032 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 74597 \nu^{15} + 16848 \nu^{14} - 2631685 \nu^{13} + 600158 \nu^{12} - 33007299 \nu^{11} + \cdots + 13252538 ) / 2092064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 74597 \nu^{15} + 16848 \nu^{14} + 2631685 \nu^{13} + 600158 \nu^{12} + 33007299 \nu^{11} + \cdots + 13252538 ) / 2092064 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 77771 \nu^{15} - 29523 \nu^{14} + 2764539 \nu^{13} - 1045915 \nu^{12} + 35137791 \nu^{11} + \cdots - 14919437 ) / 2092064 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 77771 \nu^{15} + 29523 \nu^{14} + 2764539 \nu^{13} + 1045915 \nu^{12} + 35137791 \nu^{11} + \cdots + 14919437 ) / 2092064 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43057 \nu^{15} - 19500 \nu^{14} + 1530552 \nu^{13} - 695838 \nu^{12} + 19454838 \nu^{11} + \cdots - 6230601 ) / 1046032 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 43057 \nu^{15} + 19500 \nu^{14} + 1530552 \nu^{13} + 695838 \nu^{12} + 19454838 \nu^{11} + \cdots + 6230601 ) / 1046032 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - 3\beta_{9} + 10\beta_{7} - 9\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{13} - 4\beta_{12} - 2\beta_{4} - 16\beta_{3} + 28\beta _1 + 53 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 2 \beta_{14} - 22 \beta_{13} - 22 \beta_{12} + 60 \beta_{9} - 2 \beta_{8} + \cdots - 10 \beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{15} + 12 \beta_{14} - 92 \beta_{13} + 92 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 709 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 68 \beta_{15} - 68 \beta_{14} + 403 \beta_{13} + 403 \beta_{12} - 14 \beta_{11} + \cdots + 238 \beta_{5} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 320 \beta_{15} - 320 \beta_{14} + 1656 \beta_{13} - 1656 \beta_{12} + 84 \beta_{11} + 84 \beta_{10} + \cdots + 10413 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1448 \beta_{15} + 1448 \beta_{14} - 6828 \beta_{13} - 6828 \beta_{12} + 420 \beta_{11} + \cdots - 4356 \beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6224 \beta_{15} + 6224 \beta_{14} - 27624 \beta_{13} + 27624 \beta_{12} - 1968 \beta_{11} + \cdots - 159041 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 26204 \beta_{15} - 26204 \beta_{14} + 111489 \beta_{13} + 111489 \beta_{12} - 8712 \beta_{11} + \cdots + 73260 \beta_{5} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 108176 \beta_{15} - 108176 \beta_{14} + 446956 \beta_{13} - 446956 \beta_{12} + 37404 \beta_{11} + \cdots + 2470361 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 441818 \beta_{15} + 441818 \beta_{14} - 1788002 \beta_{13} - 1788002 \beta_{12} + \cdots - 1190150 \beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1788748 \beta_{15} + 1788748 \beta_{14} - 7133316 \beta_{13} + 7133316 \beta_{12} - 647202 \beta_{11} + \cdots - 38661209 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7202104 \beta_{15} - 7202104 \beta_{14} + 28421867 \beta_{13} + 28421867 \beta_{12} + \cdots + 19032786 \beta_{5} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(\beta_{7}\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
− 3.96928i
− 2.82623i
− 2.33096i
− 1.27711i
− 0.722887i
0.330965i
0.826227i
1.96928i
3.96928i
2.82623i
2.33096i
1.27711i
0.722887i
− 0.330965i
− 0.826227i
− 1.96928i
0 −2.96928 0 0 0 −0.336782 − 0.336782i 0 5.81664 0
157.2 0 −1.82623 0 0 0 −0.547577 − 0.547577i 0 0.335104 0
157.3 0 −1.33096 0 0 0 −0.751335 − 0.751335i 0 −1.22853 0
157.4 0 −0.277113 0 0 0 −3.60863 − 3.60863i 0 −2.92321 0
157.5 0 0.277113 0 0 0 3.60863 + 3.60863i 0 −2.92321 0
157.6 0 1.33096 0 0 0 0.751335 + 0.751335i 0 −1.22853 0
157.7 0 1.82623 0 0 0 0.547577 + 0.547577i 0 0.335104 0
157.8 0 2.96928 0 0 0 0.336782 + 0.336782i 0 5.81664 0
1293.1 0 −2.96928 0 0 0 −0.336782 + 0.336782i 0 5.81664 0
1293.2 0 −1.82623 0 0 0 −0.547577 + 0.547577i 0 0.335104 0
1293.3 0 −1.33096 0 0 0 −0.751335 + 0.751335i 0 −1.22853 0
1293.4 0 −0.277113 0 0 0 −3.60863 + 3.60863i 0 −2.92321 0
1293.5 0 0.277113 0 0 0 3.60863 − 3.60863i 0 −2.92321 0
1293.6 0 1.33096 0 0 0 0.751335 − 0.751335i 0 −1.22853 0
1293.7 0 1.82623 0 0 0 0.547577 − 0.547577i 0 0.335104 0
1293.8 0 2.96928 0 0 0 0.336782 − 0.336782i 0 5.81664 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
145.e even 4 1 inner
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2900.2.s.c yes 16
5.b even 2 1 inner 2900.2.s.c yes 16
5.c odd 4 2 2900.2.j.c ✓ 16
29.c odd 4 1 2900.2.j.c ✓ 16
145.e even 4 1 inner 2900.2.s.c yes 16
145.f odd 4 1 2900.2.j.c ✓ 16
145.j even 4 1 inner 2900.2.s.c yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2900.2.j.c ✓ 16 5.c odd 4 2
2900.2.j.c ✓ 16 29.c odd 4 1
2900.2.j.c ✓ 16 145.f odd 4 1
2900.2.s.c yes 16 1.a even 1 1 trivial
2900.2.s.c yes 16 5.b even 2 1 inner
2900.2.s.c yes 16 145.e even 4 1 inner
2900.2.s.c yes 16 145.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 14T_{3}^{6} + 52T_{3}^{4} - 56T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 14 T^{6} + 52 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 680 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + \cdots + 3364)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 1475789056 \) Copy content Toggle raw display
$17$ \( (T^{8} + 82 T^{6} + \cdots + 114244)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 6 T^{7} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 735265090576 \) Copy content Toggle raw display
$29$ \( (T^{8} + 14 T^{7} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 996004)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 106 T^{6} + \cdots + 8836)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 142 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 182 T^{6} + \cdots + 47524)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 75391979776 \) Copy content Toggle raw display
$59$ \( (T^{8} + 244 T^{6} + \cdots + 3794704)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 1401856)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 10750371856 \) Copy content Toggle raw display
$71$ \( (T^{8} + 68 T^{6} + \cdots + 35344)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 262 T^{6} + \cdots + 15697444)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + \cdots + 454276)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 40199708424976 \) Copy content Toggle raw display
$89$ \( (T^{8} + 12 T^{7} + \cdots + 4892944)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 302 T^{6} + \cdots + 103684)^{2} \) Copy content Toggle raw display
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