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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,6,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.1905401061\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 290 x^{6} + 896 x^{5} + 416 x^{4} + 13952 x^{3} + 60997 x^{2} + \cdots + 750564 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 24)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_{5} + 20) q^{7} + (\beta_{7} - 3 \beta_{3} + \beta_1) q^{11} + (\beta_{7} - \beta_{4} + \cdots + 4 \beta_1) q^{13} + (\beta_{8} - 2 \beta_{5} + \beta_{2} + 41) q^{17} + ( - 2 \beta_{7} - 3 \beta_{6} + \cdots - 2 \beta_1) q^{19}+ \cdots + (30 \beta_{9} + 68 \beta_{8} + \cdots + 6170) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 196 q^{7} + 404 q^{17} + 1672 q^{23} - 9366 q^{25} - 18692 q^{31} + 2476 q^{41} + 65256 q^{47} + 9378 q^{49} + 103424 q^{55} - 51536 q^{65} - 272232 q^{71} - 27388 q^{73} - 252308 q^{79} - 175580 q^{89}+ \cdots + 59372 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 290 x^{6} + 896 x^{5} + 416 x^{4} + 13952 x^{3} + 60997 x^{2} + \cdots + 750564 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 23 \nu^{9} + 469 \nu^{8} - 1703 \nu^{7} + 1369 \nu^{6} + 21211 \nu^{5} - 44789 \nu^{4} + \cdots + 17544636 ) / 327680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{9} + 147 \nu^{8} - 817 \nu^{7} + 4719 \nu^{6} - 5155 \nu^{5} + 23565 \nu^{4} + 196637 \nu^{3} + \cdots + 5475364 ) / 8192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1321 \nu^{9} - 15083 \nu^{8} + 84121 \nu^{7} - 210023 \nu^{6} + 393883 \nu^{5} - 1181237 \nu^{4} + \cdots - 254458692 ) / 5898240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 161 \nu^{9} + 6643 \nu^{8} - 30161 \nu^{7} + 102223 \nu^{6} + 40957 \nu^{5} + 1293997 \nu^{4} + \cdots + 345177252 ) / 737280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15 \nu^{9} - 53 \nu^{8} + 271 \nu^{7} - 617 \nu^{6} + 4237 \nu^{5} + 16373 \nu^{4} + 5949 \nu^{3} + \cdots + 3313284 ) / 40960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8417 \nu^{9} - 74131 \nu^{8} + 358097 \nu^{7} - 68911 \nu^{6} - 4363069 \nu^{5} + \cdots - 3512348964 ) / 5898240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 695 \nu^{9} - 4789 \nu^{8} + 20807 \nu^{7} - 38329 \nu^{6} + 180485 \nu^{5} + 227285 \nu^{4} + \cdots + 20078532 ) / 393216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 145 \nu^{9} + 443 \nu^{8} + 79 \nu^{7} - 12313 \nu^{6} - 6227 \nu^{5} - 73403 \nu^{4} + \cdots - 20049084 ) / 20480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 215 \nu^{9} + 857 \nu^{8} - 2799 \nu^{7} - 12907 \nu^{6} - 41253 \nu^{5} - 122537 \nu^{4} + \cdots - 44632756 ) / 20480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + 8 \beta_{7} + 2 \beta_{6} - 39 \beta_{5} + 10 \beta_{4} + 18 \beta_{3} + \cdots + 937 ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{9} - 7 \beta_{8} - 60 \beta_{7} - 6 \beta_{6} + 23 \beta_{5} + 42 \beta_{4} + 198 \beta_{3} + \cdots - 5537 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 23 \beta_{9} - 27 \beta_{8} - 188 \beta_{7} - 110 \beta_{6} + 535 \beta_{5} + 98 \beta_{4} + \cdots - 29709 ) / 2304 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{9} + 173 \beta_{8} + 324 \beta_{7} - 414 \beta_{6} + 3115 \beta_{5} + 306 \beta_{4} + \cdots - 261173 ) / 2304 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 729 \beta_{9} + 1085 \beta_{8} + 2004 \beta_{7} - 2622 \beta_{6} + 4663 \beta_{5} - 1806 \beta_{4} + \cdots - 1839461 ) / 2304 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4613 \beta_{9} + 4861 \beta_{8} + 1476 \beta_{7} - 846 \beta_{6} - 15437 \beta_{5} + \cdots - 2414917 ) / 2304 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15585 \beta_{9} + 28733 \beta_{8} + 44244 \beta_{7} + 63954 \beta_{6} - 159329 \beta_{5} + \cdots + 13593307 ) / 2304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 13437 \beta_{9} + 39805 \beta_{8} + 137604 \beta_{7} + 463458 \beta_{6} - 716341 \beta_{5} + \cdots + 52241915 ) / 2304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 274423 \beta_{9} - 600739 \beta_{8} - 162028 \beta_{7} + 1806050 \beta_{6} - 153369 \beta_{5} + \cdots + 238340347 ) / 2304 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−2.89538 − 0.908872i
−0.700222 + 3.62067i
2.80405 − 3.57007i
3.93225 − 2.72065i
−1.14070 − 3.37896i
−1.14070 + 3.37896i
3.93225 + 2.72065i
2.80405 + 3.57007i
−0.700222 − 3.62067i
−2.89538 + 0.908872i
0 0 0 − 100.204i 0 −154.395 0 0 0
145.2 0 0 0 − 67.1492i 0 −37.4101 0 0 0
145.3 0 0 0 − 66.8717i 0 230.960 0 0 0
145.4 0 0 0 − 29.4414i 0 94.2792 0 0 0
145.5 0 0 0 − 20.4846i 0 −35.4345 0 0 0
145.6 0 0 0 20.4846i 0 −35.4345 0 0 0
145.7 0 0 0 29.4414i 0 94.2792 0 0 0
145.8 0 0 0 66.8717i 0 230.960 0 0 0
145.9 0 0 0 67.1492i 0 −37.4101 0 0 0
145.10 0 0 0 100.204i 0 −154.395 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 145.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.6.d.d 10
3.b odd 2 1 96.6.d.a 10
4.b odd 2 1 72.6.d.d 10
8.b even 2 1 inner 288.6.d.d 10
8.d odd 2 1 72.6.d.d 10
12.b even 2 1 24.6.d.a ✓ 10
24.f even 2 1 24.6.d.a ✓ 10
24.h odd 2 1 96.6.d.a 10
48.i odd 4 1 768.6.a.bb 5
48.i odd 4 1 768.6.a.bc 5
48.k even 4 1 768.6.a.ba 5
48.k even 4 1 768.6.a.bd 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.d.a ✓ 10 12.b even 2 1
24.6.d.a ✓ 10 24.f even 2 1
72.6.d.d 10 4.b odd 2 1
72.6.d.d 10 8.d odd 2 1
96.6.d.a 10 3.b odd 2 1
96.6.d.a 10 24.h odd 2 1
288.6.d.d 10 1.a even 1 1 trivial
288.6.d.d 10 8.b even 2 1 inner
768.6.a.ba 5 48.k even 4 1
768.6.a.bb 5 48.i odd 4 1
768.6.a.bc 5 48.i odd 4 1
768.6.a.bd 5 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 20308T_{5}^{8} + 135171232T_{5}^{6} + 351315745408T_{5}^{4} + 300576113673472T_{5}^{2} + 73638316292260864 \) acting on \(S_{6}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( (T^{5} - 98 T^{4} + \cdots + 4456567264)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 25\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 28\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots - 65\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 17\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 85\!\cdots\!20)^{2} \) Copy content Toggle raw display
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