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Properties

Label 2100.3.j.g
Level $2100$
Weight $3$
Character orbit 2100.j
Analytic conductor $57.221$
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] = [2100,3,Mod(601,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.601"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2100.j (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,-48,0,-24] 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: \(57.2208555157\)
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: \( x^{16} - 56 x^{14} + 2052 x^{12} - 43310 x^{10} + 663499 x^{8} - 6680748 x^{6} + 49052709 x^{4} + \cdots + 601475625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 420)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{7} q^{7} - 3 q^{9} + (\beta_{6} - 2) q^{11} + ( - \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{13} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{17} + \beta_{10} q^{19}+ \cdots + ( - 3 \beta_{6} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 56 x^{14} + 2052 x^{12} - 43310 x^{10} + 663499 x^{8} - 6680748 x^{6} + 49052709 x^{4} + \cdots + 601475625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 399609932752 \nu^{14} - 20296411750952 \nu^{12} + 722885544506784 \nu^{10} + \cdots - 46\!\cdots\!25 ) / 17\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 399609932752 \nu^{15} + 20296411750952 \nu^{13} - 722885544506784 \nu^{11} + \cdots + 28\!\cdots\!00 \nu ) / 17\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 399609932752 \nu^{15} + 20296411750952 \nu^{13} - 722885544506784 \nu^{11} + \cdots + 98\!\cdots\!00 \nu ) / 17\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 43654472952294 \nu^{15} + 117068050702150 \nu^{14} + \cdots - 20\!\cdots\!25 ) / 78\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47927470426472 \nu^{14} + \cdots + 10\!\cdots\!75 ) / 15\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97610193536 \nu^{14} + 5049407244982 \nu^{12} - 158649071211744 \nu^{10} + \cdots + 61\!\cdots\!00 ) / 28\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43654472952294 \nu^{15} + 117068050702150 \nu^{14} + \cdots - 20\!\cdots\!25 ) / 78\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 76169013019072 \nu^{14} + \cdots + 49\!\cdots\!25 ) / 15\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43304008251082 \nu^{15} + 471022079352700 \nu^{14} + \cdots - 47\!\cdots\!50 ) / 78\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 118538014513478 \nu^{15} + \cdots - 95\!\cdots\!00 \nu ) / 78\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43304008251082 \nu^{15} - 471022079352700 \nu^{14} + \cdots + 47\!\cdots\!50 ) / 78\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 477691903792 \nu^{14} + 23385047455394 \nu^{12} - 776408427404868 \nu^{10} + \cdots + 52\!\cdots\!55 ) / 28\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 24795765041626 \nu^{15} + \cdots - 82\!\cdots\!00 \nu ) / 52\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 144912124486304 \nu^{15} + \cdots + 96\!\cdots\!00 \nu ) / 26\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89761133871886 \nu^{15} + \cdots - 72\!\cdots\!00 \nu ) / 71\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} + 27\beta _1 + 29 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 2\beta_{10} + \beta_{9} - 18\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 31 \beta_{12} + 34 \beta_{11} - 34 \beta_{9} + 28 \beta_{8} + 74 \beta_{7} + 28 \beta_{6} + \cdots - 515 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{15} + 65 \beta_{14} + 74 \beta_{13} + 25 \beta_{11} - 71 \beta_{10} + 25 \beta_{9} + \cdots - 375 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -811\beta_{12} + 934\beta_{11} - 934\beta_{9} + 661\beta_{6} - 10595 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 123 \beta_{15} + 1772 \beta_{14} + 2141 \beta_{13} - 511 \beta_{11} + 2018 \beta_{10} + \cdots + 8505 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20509 \beta_{12} + 24145 \beta_{11} - 24145 \beta_{9} - 15361 \beta_{8} - 57074 \beta_{7} + \cdots - 238223 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3636\beta_{15} - 10213\beta_{11} + 53438\beta_{10} - 10213\beta_{9} + 201801\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 514462 \beta_{12} - 610837 \beta_{11} + 610837 \beta_{9} - 362683 \beta_{8} - 1469828 \beta_{7} + \cdots + 5623238 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 96375 \beta_{15} - 1180703 \beta_{14} - 1469828 \beta_{13} - 210904 \beta_{11} + \cdots + 4901361 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12871108\beta_{12} - 15322840\beta_{11} + 15322840\beta_{9} - 8720941\beta_{6} + 136181936 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2451732 \beta_{15} - 29892383 \beta_{14} - 37247579 \beta_{13} + 4570774 \beta_{11} + \cdots - 120495042 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 321582238 \beta_{12} - 383014447 \beta_{11} + 383014447 \beta_{9} + 212607916 \beta_{8} + \cdots + 3342581480 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 61432209\beta_{15} + 103633594\beta_{11} - 875003216\beta_{10} + 103633594\beta_{9} - 2981125200\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−2.27298 − 1.31230i
3.23362 + 1.86693i
−2.77207 − 1.60046i
4.32353 + 2.49619i
−4.32353 − 2.49619i
2.77207 + 1.60046i
−3.23362 − 1.86693i
2.27298 + 1.31230i
−2.27298 + 1.31230i
3.23362 − 1.86693i
−2.77207 + 1.60046i
4.32353 − 2.49619i
−4.32353 + 2.49619i
2.77207 − 1.60046i
−3.23362 + 1.86693i
2.27298 − 1.31230i
0 − 1.73205i 0 0 0 −6.51200 − 2.56785i 0 −3.00000 0
601.2 0 − 1.73205i 0 0 0 −6.43149 + 2.76332i 0 −3.00000 0
601.3 0 − 1.73205i 0 0 0 −4.90840 + 4.99075i 0 −3.00000 0
601.4 0 − 1.73205i 0 0 0 −1.06651 − 6.91828i 0 −3.00000 0
601.5 0 − 1.73205i 0 0 0 1.06651 − 6.91828i 0 −3.00000 0
601.6 0 − 1.73205i 0 0 0 4.90840 + 4.99075i 0 −3.00000 0
601.7 0 − 1.73205i 0 0 0 6.43149 + 2.76332i 0 −3.00000 0
601.8 0 − 1.73205i 0 0 0 6.51200 − 2.56785i 0 −3.00000 0
601.9 0 1.73205i 0 0 0 −6.51200 + 2.56785i 0 −3.00000 0
601.10 0 1.73205i 0 0 0 −6.43149 − 2.76332i 0 −3.00000 0
601.11 0 1.73205i 0 0 0 −4.90840 − 4.99075i 0 −3.00000 0
601.12 0 1.73205i 0 0 0 −1.06651 + 6.91828i 0 −3.00000 0
601.13 0 1.73205i 0 0 0 1.06651 + 6.91828i 0 −3.00000 0
601.14 0 1.73205i 0 0 0 4.90840 − 4.99075i 0 −3.00000 0
601.15 0 1.73205i 0 0 0 6.43149 − 2.76332i 0 −3.00000 0
601.16 0 1.73205i 0 0 0 6.51200 + 2.56785i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.3.j.g 16
5.b even 2 1 inner 2100.3.j.g 16
5.c odd 4 2 420.3.p.a ✓ 16
7.b odd 2 1 inner 2100.3.j.g 16
15.e even 4 2 1260.3.p.e 16
20.e even 4 2 1680.3.bd.b 16
35.c odd 2 1 inner 2100.3.j.g 16
35.f even 4 2 420.3.p.a ✓ 16
105.k odd 4 2 1260.3.p.e 16
140.j odd 4 2 1680.3.bd.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.3.p.a ✓ 16 5.c odd 4 2
420.3.p.a ✓ 16 35.f even 4 2
1260.3.p.e 16 15.e even 4 2
1260.3.p.e 16 105.k odd 4 2
1680.3.bd.b 16 20.e even 4 2
1680.3.bd.b 16 140.j odd 4 2
2100.3.j.g 16 1.a even 1 1 trivial
2100.3.j.g 16 5.b even 2 1 inner
2100.3.j.g 16 7.b odd 2 1 inner
2100.3.j.g 16 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{4} + 6T_{11}^{3} - 248T_{11}^{2} + 1296T_{11} - 1904 \) Copy content Toggle raw display
\( T_{23}^{8} - 2584T_{23}^{6} + 1920160T_{23}^{4} - 514679616T_{23}^{2} + 44935513344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + \cdots - 1904)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 396 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1500 T^{6} + \cdots + 132342016)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2348 T^{6} + \cdots + 63035387136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2584 T^{6} + \cdots + 44935513344)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots - 19664)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5792 T^{6} + \cdots + 422463239424)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 4504180665600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 31643292960000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 17480484267264)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 4076 T^{6} + \cdots + 462030153984)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 5512 T^{6} + \cdots + 45361440000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 20708 T^{6} + \cdots + 985548324096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 136350926228736)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 4662962908416)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 42 T^{3} + \cdots + 3482224)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 8315148960000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 27896112)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 4449737113600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 95144885842176)^{2} \) Copy content Toggle raw display
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