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Properties

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

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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 = [16,0,0,0,0,0,0,0,-28,0,0,0,0] 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: \(16\)
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: \( x^{16} + 14x^{14} + 183x^{12} + 1168x^{10} + 7863x^{8} + 20838x^{6} + 21041x^{4} - 62692x^{2} + 29584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 580)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{8} q^{7} + (\beta_{2} - 2) q^{9} + \beta_{14} q^{11} - \beta_{12} q^{13} - \beta_{3} q^{17} + (\beta_{14} - \beta_{11}) q^{19} + (\beta_{15} - \beta_{14} + \cdots - \beta_{9}) q^{21}+ \cdots + (\beta_{15} - 4 \beta_{14} + \cdots - 2 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 28 q^{9} + 8 q^{29} + 24 q^{51} + 24 q^{59} + 72 q^{81} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 14x^{14} + 183x^{12} + 1168x^{10} + 7863x^{8} + 20838x^{6} + 21041x^{4} - 62692x^{2} + 29584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6882048 \nu^{14} - 98179425 \nu^{12} - 1276619884 \nu^{10} - 7672890188 \nu^{8} + \cdots + 1813870380862 ) / 439684249550 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11717354 \nu^{14} + 140938855 \nu^{12} + 1737835562 \nu^{10} + 8891962644 \nu^{8} + \cdots + 82052458844 ) / 439684249550 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 87708007 \nu^{14} - 1423445610 \nu^{12} - 18412115616 \nu^{10} - 132716840482 \nu^{8} + \cdots + 3692603515768 ) / 879368499100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 285778781 \nu^{14} - 4112155050 \nu^{12} - 54465034998 \nu^{10} - 362633046686 \nu^{8} + \cdots + 10655842041764 ) / 2638105497300 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29730017 \nu^{14} - 451115120 \nu^{12} - 5867670106 \nu^{10} - 40703690582 \nu^{8} + \cdots + 1155825750468 ) / 125624071300 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 216833651 \nu^{14} + 3164809530 \nu^{12} + 41597602788 \nu^{10} + 278295975026 \nu^{8} + \cdots - 8025395679224 ) / 376872213900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5011111159 \nu^{15} + 78150020250 \nu^{13} + 976154158197 \nu^{11} + \cdots - 394853800874296 \nu ) / 226877072767800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1117880427 \nu^{15} - 15058469850 \nu^{13} - 196128687591 \nu^{11} + \cdots + 93267036186888 \nu ) / 37812845461300 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1117880427 \nu^{15} + 1748598612 \nu^{14} - 15058469850 \nu^{13} + 21258327315 \nu^{12} + \cdots - 26090718836568 ) / 37812845461300 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1117880427 \nu^{15} - 1748598612 \nu^{14} - 15058469850 \nu^{13} - 21258327315 \nu^{12} + \cdots + 26090718836568 ) / 37812845461300 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5633758289 \nu^{15} - 76401686320 \nu^{13} - 1003868334757 \nu^{11} + \cdots + 121062986492736 \nu ) / 75625690922600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1363494109 \nu^{15} + 18657709484 \nu^{13} + 244689390961 \nu^{11} + \cdots - 116860014769240 \nu ) / 15125138184520 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9221664351 \nu^{15} - 131590946030 \nu^{13} - 1698162315813 \nu^{11} + \cdots + 771014642004024 \nu ) / 75625690922600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17177812311 \nu^{15} - 253089298230 \nu^{13} - 3329846163993 \nu^{11} + \cdots + 596811658776864 \nu ) / 75625690922600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14744443268 \nu^{15} + 1748598612 \nu^{14} - 217781723270 \nu^{13} + \cdots - 26090718836568 ) / 37812845461300 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + 2\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 4\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} - 8 \beta_{14} - 2 \beta_{13} - 8 \beta_{12} + 4 \beta_{11} + \beta_{10} + \cdots - 6 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 3\beta_{5} - 7\beta_{4} - 5\beta_{3} + 4\beta_{2} + 5\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{14} + 42\beta_{13} + 36\beta_{12} - 48\beta_{11} + 45\beta_{10} + 45\beta_{9} + 4\beta_{8} + 90\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 44 \beta_{10} - 44 \beta_{9} - 33 \beta_{6} - 40 \beta_{5} - 120 \beta_{4} + 35 \beta_{3} + 95 \beta_{2} + \cdots + 333 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 520 \beta_{15} + 912 \beta_{14} - 12 \beta_{13} + 204 \beta_{12} + 124 \beta_{11} + \cdots - 132 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 213 \beta_{10} + 213 \beta_{9} + 22 \beta_{6} - 214 \beta_{5} + 382 \beta_{4} + 214 \beta_{3} + \cdots - 411 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2436 \beta_{15} - 4332 \beta_{14} - 4280 \beta_{13} - 5024 \beta_{12} + 32 \beta_{11} + \cdots - 6996 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 464 \beta_{10} + 464 \beta_{9} + 1121 \beta_{6} + 7220 \beta_{5} - 3604 \beta_{4} - 6683 \beta_{3} + \cdots - 115 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26964 \beta_{15} - 47232 \beta_{14} + 33426 \beta_{13} + 43832 \beta_{12} - 3892 \beta_{11} + \cdots + 53406 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18604 \beta_{10} - 18604 \beta_{9} + 4325 \beta_{6} - 12431 \beta_{5} + 36395 \beta_{4} + 14225 \beta_{3} + \cdots + 65163 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 304272 \beta_{15} + 517532 \beta_{14} + 41998 \beta_{13} - 62812 \beta_{12} + 207112 \beta_{11} + \cdots + 105294 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 288008 \beta_{10} + 288008 \beta_{9} - 95229 \beta_{6} - 283524 \beta_{5} - 102396 \beta_{4} + \cdots - 2419219 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 482680 \beta_{15} - 628984 \beta_{14} - 2448088 \beta_{13} - 3443236 \beta_{12} + \cdots - 3783216 \beta_{7} ) / 4 \) 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
1.98716 + 1.93117i
−1.98716 − 1.93117i
0.826047 − 0.146119i
−0.826047 + 0.146119i
1.39210 + 2.68223i
−1.39210 − 2.68223i
0.656421 − 1.74782i
−0.656421 + 1.74782i
0.656421 + 1.74782i
−0.656421 − 1.74782i
1.39210 − 2.68223i
−1.39210 + 2.68223i
0.826047 + 0.146119i
−0.826047 − 0.146119i
1.98716 − 1.93117i
−1.98716 + 1.93117i
0 − 3.33421i 0 0 0 −3.97432 0 −8.11698 0
1101.2 0 − 3.33421i 0 0 0 3.97432 0 −8.11698 0
1101.3 0 − 2.24181i 0 0 0 −1.65209 0 −2.02571 0
1101.4 0 − 2.24181i 0 0 0 1.65209 0 −2.02571 0
1101.5 0 − 1.27260i 0 0 0 −2.78421 0 1.38049 0
1101.6 0 − 1.27260i 0 0 0 2.78421 0 1.38049 0
1101.7 0 − 1.11256i 0 0 0 −1.31284 0 1.76220 0
1101.8 0 − 1.11256i 0 0 0 1.31284 0 1.76220 0
1101.9 0 1.11256i 0 0 0 −1.31284 0 1.76220 0
1101.10 0 1.11256i 0 0 0 1.31284 0 1.76220 0
1101.11 0 1.27260i 0 0 0 −2.78421 0 1.38049 0
1101.12 0 1.27260i 0 0 0 2.78421 0 1.38049 0
1101.13 0 2.24181i 0 0 0 −1.65209 0 −2.02571 0
1101.14 0 2.24181i 0 0 0 1.65209 0 −2.02571 0
1101.15 0 3.33421i 0 0 0 −3.97432 0 −8.11698 0
1101.16 0 3.33421i 0 0 0 3.97432 0 −8.11698 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1101.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.b even 2 1 inner
145.d 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.g 16
5.b even 2 1 inner 2900.2.d.g 16
5.c odd 4 2 580.2.f.a ✓ 16
15.e even 4 2 5220.2.b.e 16
20.e even 4 2 2320.2.j.f 16
29.b even 2 1 inner 2900.2.d.g 16
145.d even 2 1 inner 2900.2.d.g 16
145.h odd 4 2 580.2.f.a ✓ 16
435.p even 4 2 5220.2.b.e 16
580.o even 4 2 2320.2.j.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.f.a ✓ 16 5.c odd 4 2
580.2.f.a ✓ 16 145.h odd 4 2
2320.2.j.f 16 20.e even 4 2
2320.2.j.f 16 580.o even 4 2
2900.2.d.g 16 1.a even 1 1 trivial
2900.2.d.g 16 5.b even 2 1 inner
2900.2.d.g 16 29.b even 2 1 inner
2900.2.d.g 16 145.d even 2 1 inner
5220.2.b.e 16 15.e even 4 2
5220.2.b.e 16 435.p even 4 2

Hecke kernels

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

\( T_{3}^{8} + 19T_{3}^{6} + 104T_{3}^{4} + 192T_{3}^{2} + 112 \) Copy content Toggle raw display
\( T_{7}^{8} - 28T_{7}^{6} + 232T_{7}^{4} - 656T_{7}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 19 T^{6} + \cdots + 112)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 28 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 57 T^{6} + \cdots + 112)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 43 T^{6} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 92 T^{6} + \cdots + 100800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 16128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 108 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 153 T^{6} + \cdots + 49392)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 240 T^{6} + \cdots + 4845568)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 252 T^{6} + \cdots + 4480000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 19 T^{6} + \cdots + 112)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 219 T^{6} + \cdots + 2041200)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 331 T^{6} + \cdots + 4804864)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} + \cdots + 3360)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 220 T^{6} + \cdots + 64512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 304 T^{6} + \cdots + 1806336)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 112 T^{2} + \cdots - 576)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 248 T^{6} + \cdots + 114688)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 449 T^{6} + \cdots + 4258800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 252 T^{6} + \cdots + 270400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 500 T^{6} + \cdots + 83607552)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 532 T^{6} + \cdots + 64351168)^{2} \) Copy content Toggle raw display
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