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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [207,3,Mod(116,207)] 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("207.116"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.64034147226\)
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: \( x^{16} + 52x^{14} + 1096x^{12} + 12028x^{10} + 73262x^{8} + 243356x^{6} + 401864x^{4} + 268244x^{2} + 31329 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 3) q^{4} + \beta_{10} q^{5} - \beta_{12} q^{7} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{8} + ( - \beta_{15} - \beta_{7} - \beta_{6} + \cdots + 4) q^{10} + (\beta_{10} + \beta_{4}) q^{11}+ \cdots + ( - 4 \beta_{14} - 3 \beta_{11} + \cdots + 33 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 40 q^{4} + 40 q^{10} - 8 q^{13} + 32 q^{16} - 24 q^{19} + 72 q^{22} - 200 q^{25} - 88 q^{28} + 160 q^{31} + 208 q^{34} + 64 q^{37} - 384 q^{40} + 112 q^{43} + 264 q^{49} - 56 q^{52} - 368 q^{55}+ \cdots - 344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 52x^{14} + 1096x^{12} + 12028x^{10} + 73262x^{8} + 243356x^{6} + 401864x^{4} + 268244x^{2} + 31329 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{15} + 201 \nu^{13} + 2825 \nu^{11} + 14061 \nu^{9} - 20789 \nu^{7} - 391737 \nu^{5} + \cdots - 561117 \nu ) / 33984 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7 \nu^{15} - 423 \nu^{13} - 10327 \nu^{11} - 128151 \nu^{9} - 834089 \nu^{7} - 2649321 \nu^{5} + \cdots - 1658169 \nu ) / 67968 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 111 \nu^{13} + 3397 \nu^{11} + 45363 \nu^{9} + 295751 \nu^{7} + 921561 \nu^{5} + \cdots + 403413 \nu ) / 22656 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} + 39\nu^{12} + 565\nu^{10} + 3771\nu^{8} + 11591\nu^{6} + 15009\nu^{4} + 12755\nu^{2} + 6477 ) / 768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} + 36\nu^{12} + 439\nu^{10} + 1746\nu^{8} - 4165\nu^{6} - 46584\nu^{4} - 96763\nu^{2} - 43110 ) / 1152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} - 125 \nu^{13} - 6515 \nu^{11} - 112049 \nu^{9} - 873865 \nu^{7} - 3068491 \nu^{5} + \cdots - 688087 \nu ) / 45312 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{14} - 39\nu^{12} - 565\nu^{10} - 3771\nu^{8} - 11495\nu^{6} - 12993\nu^{4} - 2675\nu^{2} - 1389 ) / 384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 65 \nu^{15} + 2790 \nu^{13} + 46283 \nu^{11} + 375192 \nu^{9} + 1528771 \nu^{7} + 2780910 \nu^{5} + \cdots - 870660 \nu ) / 67968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 68 \nu^{15} - 3123 \nu^{13} - 56474 \nu^{11} - 511281 \nu^{9} - 2416024 \nu^{7} + \cdots + 272133 \nu ) / 67968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7 \nu^{14} - 297 \nu^{12} - 4819 \nu^{10} - 37557 \nu^{8} - 143825 \nu^{6} - 247599 \nu^{4} + \cdots - 30051 ) / 2304 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4 \nu^{14} - 171 \nu^{12} - 2818 \nu^{10} - 22689 \nu^{8} - 93176 \nu^{6} - 185697 \nu^{4} + \cdots - 22635 ) / 1152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 359 \nu^{15} + 15777 \nu^{13} + 270803 \nu^{11} + 2319309 \nu^{9} + 10509649 \nu^{7} + \cdots + 12150315 \nu ) / 135936 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5\nu^{14} + 219\nu^{12} + 3737\nu^{10} + 31599\nu^{8} + 138499\nu^{6} + 297261\nu^{4} + 257071\nu^{2} + 37593 ) / 768 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{5} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{9} - \beta_{7} - \beta_{6} - 14\beta_{2} + 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 17\beta_{11} - 21\beta_{10} - \beta_{8} - 20\beta_{5} - 3\beta_{4} + 7\beta_{3} + 93\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -21\beta_{13} + 21\beta_{12} + 25\beta_{9} + 21\beta_{7} + 29\beta_{6} + 189\beta_{2} - 830 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 25 \beta_{14} + 252 \beta_{11} + 352 \beta_{10} + 33 \beta_{8} + 331 \beta_{5} + 67 \beta_{4} + \cdots - 1045 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -8\beta_{15} + 328\beta_{13} - 344\beta_{12} - 464\beta_{9} - 360\beta_{7} - 576\beta_{6} - 2592\beta_{2} + 10201 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 472 \beta_{14} - 3616 \beta_{11} - 5488 \beta_{10} - 712 \beta_{8} - 5128 \beta_{5} - 1120 \beta_{4} + \cdots + 12489 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 280 \beta_{15} - 4672 \beta_{13} + 5248 \beta_{12} + 7672 \beta_{9} + 5728 \beta_{7} + 9912 \beta_{6} + \cdots - 130999 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7968 \beta_{14} + 51433 \beta_{11} + 82777 \beta_{10} + 12912 \beta_{8} + 76953 \beta_{5} + \cdots - 156513 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6360 \beta_{15} + 64585 \beta_{13} - 77977 \beta_{12} - 119793 \beta_{9} - 87721 \beta_{7} + \cdots + 1736224 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 126825 \beta_{14} - 730753 \beta_{11} - 1226549 \beta_{10} - 214233 \beta_{8} - 1134796 \beta_{5} + \cdots + 2033941 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 120008 \beta_{15} - 887621 \beta_{13} + 1144805 \beta_{12} + 1812225 \beta_{9} + 1313957 \beta_{7} + \cdots - 23543582 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1949401 \beta_{14} + 10397124 \beta_{11} + 17985592 \beta_{10} + 3378033 \beta_{8} + \cdots - 27164029 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
− 3.79102i
− 3.20575i
− 3.17595i
− 2.71856i
− 2.45134i
− 1.52431i
− 1.18130i
− 0.382152i
0.382152i
1.18130i
1.52431i
2.45134i
2.71856i
3.17595i
3.20575i
3.79102i
− 3.79102i 0 −10.3718 8.40012i 0 10.6628 24.1556i 0 31.8450
116.2 − 3.20575i 0 −6.27681 − 9.85252i 0 2.09296 7.29889i 0 −31.5847
116.3 − 3.17595i 0 −6.08668 6.53360i 0 −11.4829 6.62719i 0 20.7504
116.4 − 2.71856i 0 −3.39058 0.300866i 0 8.61863 − 1.65675i 0 0.817922
116.5 − 2.45134i 0 −2.00908 1.66305i 0 −11.6414 − 4.88044i 0 4.07669
116.6 − 1.52431i 0 1.67647 − 7.07613i 0 −5.85251 − 8.65272i 0 −10.7862
116.7 − 1.18130i 0 2.60453 2.31996i 0 3.05502 − 7.80194i 0 2.74057
116.8 − 0.382152i 0 3.85396 5.60069i 0 4.54739 − 3.00141i 0 2.14032
116.9 0.382152i 0 3.85396 − 5.60069i 0 4.54739 3.00141i 0 2.14032
116.10 1.18130i 0 2.60453 − 2.31996i 0 3.05502 7.80194i 0 2.74057
116.11 1.52431i 0 1.67647 7.07613i 0 −5.85251 8.65272i 0 −10.7862
116.12 2.45134i 0 −2.00908 − 1.66305i 0 −11.6414 4.88044i 0 4.07669
116.13 2.71856i 0 −3.39058 − 0.300866i 0 8.61863 1.65675i 0 0.817922
116.14 3.17595i 0 −6.08668 − 6.53360i 0 −11.4829 − 6.62719i 0 20.7504
116.15 3.20575i 0 −6.27681 9.85252i 0 2.09296 − 7.29889i 0 −31.5847
116.16 3.79102i 0 −10.3718 − 8.40012i 0 10.6628 − 24.1556i 0 31.8450
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 116.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.3.b.a ✓ 16
3.b odd 2 1 inner 207.3.b.a ✓ 16
4.b odd 2 1 3312.3.g.d 16
12.b even 2 1 3312.3.g.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.3.b.a ✓ 16 1.a even 1 1 trivial
207.3.b.a ✓ 16 3.b odd 2 1 inner
3312.3.g.d 16 4.b odd 2 1
3312.3.g.d 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(207, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 52 T^{14} + \cdots + 31329 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 618815376 \) Copy content Toggle raw display
$7$ \( (T^{8} - 262 T^{6} + \cdots - 2090480)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 2146107801600 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots - 86500224)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} + \cdots + 1826156)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} - 80 T^{7} + \cdots - 226525864896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 32 T^{7} + \cdots + 233663876656)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 1103520784428)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 12699104351280)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 7253093553100)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{8} - 88 T^{7} + \cdots - 720222798272)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 30974206135824)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 15543338447488)^{2} \) Copy content Toggle raw display
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