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Properties

Label 2900.2.f.e
Level $2900$
Weight $2$
Character orbit 2900.f
Analytic conductor $23.157$
Analytic rank $0$
Dimension $20$
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(1449,2900)] 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("2900.1449"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,0,0,0,0,4] 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: no
Analytic conductor: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 42 x^{18} + 679 x^{16} + 5502 x^{14} + 24715 x^{12} + 63108 x^{10} + 87988 x^{8} + 57930 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{14} q^{7} - \beta_{4} q^{9} + ( - \beta_{13} + \beta_{10}) q^{11} - \beta_{18} q^{13} + \beta_{5} q^{17} + (\beta_{16} + \beta_{12} + \beta_{10}) q^{19} + ( - \beta_{17} - \beta_{16} + \cdots - \beta_{10}) q^{21}+ \cdots + ( - 2 \beta_{17} - \beta_{16} + \cdots - 3 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{9} - 4 q^{29} - 68 q^{49} + 20 q^{51} + 12 q^{59} + 28 q^{71} + 12 q^{81} - 52 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 42 x^{18} + 679 x^{16} + 5502 x^{14} + 24715 x^{12} + 63108 x^{10} + 87988 x^{8} + 57930 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1060 \nu^{18} + 50601 \nu^{16} + 953350 \nu^{14} + 9126253 \nu^{12} + 47943461 \nu^{10} + \cdots + 1759031 ) / 460278 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2271 \nu^{18} - 94246 \nu^{16} - 1494401 \nu^{14} - 11738030 \nu^{12} - 50309279 \nu^{10} + \cdots + 703135 ) / 460278 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2346 \nu^{18} + 90283 \nu^{16} + 1278440 \nu^{14} + 8531321 \nu^{12} + 29775650 \nu^{10} + \cdots + 514652 ) / 460278 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2346 \nu^{18} - 90283 \nu^{16} - 1278440 \nu^{14} - 8531321 \nu^{12} - 29775650 \nu^{10} + \cdots + 405904 ) / 230139 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8215 \nu^{18} - 364128 \nu^{16} - 6321365 \nu^{14} - 55881053 \nu^{12} - 275274714 \nu^{10} + \cdots - 916327 ) / 460278 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6934 \nu^{18} - 287989 \nu^{16} - 4583734 \nu^{14} - 36397078 \nu^{12} - 159833780 \nu^{10} + \cdots - 205260 ) / 230139 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9356 \nu^{18} + 375279 \nu^{16} + 5665836 \nu^{14} + 41620632 \nu^{12} + 164565416 \nu^{10} + \cdots - 576570 ) / 230139 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21684 \nu^{18} + 918938 \nu^{16} + 15037473 \nu^{14} + 123691440 \nu^{12} + 564227478 \nu^{10} + \cdots + 2569718 ) / 460278 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14642 \nu^{18} - 610770 \nu^{16} - 9780200 \nu^{14} - 78269147 \nu^{12} - 346687750 \nu^{10} + \cdots - 2083813 ) / 230139 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45529 \nu^{19} + 1917871 \nu^{17} + 31126041 \nu^{15} + 253346661 \nu^{13} + 1142162338 \nu^{11} + \cdots - 3782270 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54374 \nu^{19} + 2281362 \nu^{17} + 36829663 \nu^{15} + 297887308 \nu^{13} + 1335322089 \nu^{11} + \cdots + 9471305 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 54374 \nu^{19} + 2281362 \nu^{17} + 36829663 \nu^{15} + 297887308 \nu^{13} + 1335322089 \nu^{11} + \cdots + 9931583 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 79777 \nu^{19} + 3343843 \nu^{17} + 53903251 \nu^{15} + 435092026 \nu^{13} + 1945307696 \nu^{11} + \cdots + 22189605 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 91688 \nu^{19} + 3835105 \nu^{17} + 61634339 \nu^{15} + 495352916 \nu^{13} + 2202209713 \nu^{11} + \cdots + 4772708 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 106402 \nu^{19} + 4472441 \nu^{17} + 72380886 \nu^{15} + 587243295 \nu^{13} + 2640868528 \nu^{11} + \cdots + 17967680 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 108673 \nu^{19} + 4566687 \nu^{17} + 73875287 \nu^{15} + 598981325 \nu^{13} + 2691177807 \nu^{11} + \cdots + 17264545 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 52053 \nu^{19} - 2189758 \nu^{17} - 35479236 \nu^{15} - 288287084 \nu^{13} - 1298701896 \nu^{11} + \cdots - 10436564 \nu ) / 153426 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 161123 \nu^{19} + 6773706 \nu^{17} + 109654518 \nu^{15} + 890068323 \nu^{13} + 4005875546 \nu^{11} + \cdots + 53113941 \nu ) / 460278 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 196879 \nu^{19} + 8263901 \nu^{17} + 133474174 \nu^{15} + 1079984434 \nu^{13} + 4840712423 \nu^{11} + \cdots + 29345056 \nu ) / 460278 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{12} - \beta_{11} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{16} - 3\beta_{15} - 9\beta_{12} + 10\beta_{11} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - 14\beta_{4} - 28\beta_{3} + 4\beta_{2} + 4\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{19} + 21 \beta_{17} - 17 \beta_{16} + 50 \beta_{15} - 5 \beta_{14} + 2 \beta_{13} + \cdots + \beta_{10} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{9} + 12 \beta_{8} - 28 \beta_{7} - 31 \beta_{6} + 18 \beta_{5} + 197 \beta_{4} + 390 \beta_{3} + \cdots - 471 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 126 \beta_{19} + 14 \beta_{18} - 368 \beta_{17} + 228 \beta_{16} - 735 \beta_{15} + \cdots - 35 \beta_{10} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 87 \beta_{9} - 360 \beta_{8} + 567 \beta_{7} + 696 \beta_{6} - 528 \beta_{5} - 2811 \beta_{4} + \cdots + 6409 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2415 \beta_{19} - 447 \beta_{18} + 6042 \beta_{17} - 2931 \beta_{16} + 10635 \beta_{15} + \cdots + 777 \beta_{10} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2157 \beta_{9} + 7644 \beta_{8} - 10167 \beta_{7} - 13311 \beta_{6} + 11046 \beta_{5} + 40537 \beta_{4} + \cdots - 90271 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 41943 \beta_{19} + 9801 \beta_{18} - 96157 \beta_{17} + 37720 \beta_{16} - 154341 \beta_{15} + \cdots - 14556 \beta_{10} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 43080 \beta_{9} - 141156 \beta_{8} + 171379 \beta_{7} + 233404 \beta_{6} - 202044 \beta_{5} + \cdots + 1292894 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 693251 \beta_{19} - 184236 \beta_{18} + 1503243 \beta_{17} - 491969 \beta_{16} + \cdots + 251212 \beta_{10} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 773255 \beta_{9} + 2423826 \beta_{8} - 2785513 \beta_{7} - 3889813 \beta_{6} + 3448122 \beta_{5} + \cdots - 18714906 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11125341 \beta_{19} + 3197081 \beta_{18} - 23231663 \beta_{17} + 6526086 \beta_{16} + \cdots - 4140854 \beta_{10} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 13069488 \beta_{9} - 39872592 \beta_{8} + 44229411 \beta_{7} + 62795094 \beta_{6} - 56494128 \beta_{5} + \cdots + 273026818 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 175142211 \beta_{19} - 52942080 \beta_{18} + 356189496 \beta_{17} - 88095270 \beta_{16} + \cdots + 66347142 \beta_{10} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 212908665 \beta_{9} + 638252898 \beta_{8} - 691298292 \beta_{7} - 992671830 \beta_{6} + \cdots - 4007333734 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2721121062 \beta_{19} + 851161563 \beta_{18} - 5430106648 \beta_{17} + 1209234457 \beta_{16} + \cdots - 1043370249 \beta_{10} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1449.1
− 1.87199i
1.87199i
1.38054i
− 1.38054i
− 0.0651824i
0.0651824i
0.156418i
− 0.156418i
− 0.511672i
0.511672i
− 1.48833i
1.48833i
1.84358i
− 1.84358i
2.06518i
− 2.06518i
− 3.38054i
3.38054i
3.87199i
− 3.87199i
0 −2.87199 0 0 0 − 1.95804i 0 5.24832 0
1449.2 0 −2.87199 0 0 0 1.95804i 0 5.24832 0
1449.3 0 −2.38054 0 0 0 − 4.97091i 0 2.66697 0
1449.4 0 −2.38054 0 0 0 4.97091i 0 2.66697 0
1449.5 0 −1.06518 0 0 0 − 0.173844i 0 −1.86539 0
1449.6 0 −1.06518 0 0 0 0.173844i 0 −1.86539 0
1449.7 0 −0.843582 0 0 0 − 4.53366i 0 −2.28837 0
1449.8 0 −0.843582 0 0 0 4.53366i 0 −2.28837 0
1449.9 0 −0.488328 0 0 0 − 1.69464i 0 −2.76154 0
1449.10 0 −0.488328 0 0 0 1.69464i 0 −2.76154 0
1449.11 0 0.488328 0 0 0 − 1.69464i 0 −2.76154 0
1449.12 0 0.488328 0 0 0 1.69464i 0 −2.76154 0
1449.13 0 0.843582 0 0 0 − 4.53366i 0 −2.28837 0
1449.14 0 0.843582 0 0 0 4.53366i 0 −2.28837 0
1449.15 0 1.06518 0 0 0 − 0.173844i 0 −1.86539 0
1449.16 0 1.06518 0 0 0 0.173844i 0 −1.86539 0
1449.17 0 2.38054 0 0 0 − 4.97091i 0 2.66697 0
1449.18 0 2.38054 0 0 0 4.97091i 0 2.66697 0
1449.19 0 2.87199 0 0 0 − 1.95804i 0 5.24832 0
1449.20 0 2.87199 0 0 0 1.95804i 0 5.24832 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1449.20
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.f.e 20
5.b even 2 1 inner 2900.2.f.e 20
5.c odd 4 1 2900.2.d.e ✓ 10
5.c odd 4 1 2900.2.d.f yes 10
29.b even 2 1 inner 2900.2.f.e 20
145.d even 2 1 inner 2900.2.f.e 20
145.h odd 4 1 2900.2.d.e ✓ 10
145.h odd 4 1 2900.2.d.f yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2900.2.d.e ✓ 10 5.c odd 4 1
2900.2.d.e ✓ 10 145.h odd 4 1
2900.2.d.f yes 10 5.c odd 4 1
2900.2.d.f yes 10 145.h odd 4 1
2900.2.f.e 20 1.a even 1 1 trivial
2900.2.f.e 20 5.b even 2 1 inner
2900.2.f.e 20 29.b even 2 1 inner
2900.2.f.e 20 145.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 16T_{3}^{8} + 77T_{3}^{6} - 115T_{3}^{4} + 61T_{3}^{2} - 9 \) acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - 16 T^{8} + \cdots - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 52 T^{8} + \cdots + 169)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 64 T^{8} + \cdots + 69169)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 68 T^{8} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 55 T^{8} + \cdots - 225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 68 T^{8} + \cdots + 289)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + 109 T^{8} + \cdots + 145161)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 2 T^{9} + \cdots + 20511149)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 225 T^{8} + \cdots + 100260169)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - 152 T^{8} + \cdots - 576081)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 325 T^{8} + \cdots + 241958025)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 202 T^{8} + \cdots - 130321)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 217 T^{8} + \cdots - 5688225)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 226 T^{8} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 3 T^{4} + \cdots - 14415)^{4} \) Copy content Toggle raw display
$61$ \( (T^{10} + 473 T^{8} + \cdots + 491401)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 75 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 7 T^{4} + \cdots + 3357)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} - 320 T^{8} + \cdots - 244640881)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 466 T^{8} + \cdots + 14707225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 331 T^{8} + \cdots + 308880625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 538 T^{8} + \cdots + 1356522561)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 585 T^{8} + \cdots - 1587464649)^{2} \) Copy content Toggle raw display
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