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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(24,575)] 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("575.24"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-66,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.9260982533\)
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: \( x^{10} + 68x^{8} + 1676x^{6} + 17761x^{4} + 68680x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{6} + \cdots + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} + \beta_{3} - 6) q^{4} + (3 \beta_{8} - 3 \beta_{7} + 5 \beta_{4} + \cdots - 7) q^{6}+ \cdots + ( - 113 \beta_{8} + 5 \beta_{7} + \cdots - 40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 66 q^{4} - 72 q^{6} - 158 q^{9} - 306 q^{11} - 138 q^{14} + 290 q^{16} - 6 q^{19} - 424 q^{21} + 486 q^{24} - 532 q^{26} + 1166 q^{29} + 1324 q^{31} - 1598 q^{34} + 4094 q^{36} - 166 q^{39} + 688 q^{41}+ \cdots + 182 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 68x^{8} + 1676x^{6} + 17761x^{4} + 68680x^{2} + 3600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{9} + 3008\nu^{7} + 147716\nu^{5} + 2523931\nu^{3} + 13648940\nu ) / 336000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{8} + 576\nu^{6} + 2252\nu^{4} - 28543\nu^{2} + 114180 ) / 33600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{8} + 184\nu^{6} + 3568\nu^{4} + 21563\nu^{2} + 1620 ) / 2100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{9} + 552\nu^{7} + 11404\nu^{5} + 88489\nu^{3} + 186860\nu ) / 42000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 349\nu^{9} + 19072\nu^{7} + 308444\nu^{5} + 679629\nu^{3} - 11886540\nu ) / 1344000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{8} - 704\nu^{6} - 11868\nu^{4} - 56573\nu^{2} + 59340 ) / 6720 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 431\nu^{8} + 23168\nu^{6} + 403636\nu^{4} + 2311551\nu^{2} + 396540 ) / 134400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 89\nu^{9} + 4992\nu^{7} + 93484\nu^{5} + 645769\nu^{3} + 1037860\nu ) / 67200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{4} + \beta_{3} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} + 5\beta_{5} - \beta_{2} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{8} - 16\beta_{7} - 29\beta_{4} - 30\beta_{3} + 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + 136\beta_{6} - 227\beta_{5} + 43\beta_{2} + 327\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -308\beta_{8} + 293\beta_{7} + 819\beta_{4} + 760\beta_{3} - 4893 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -516\beta_{9} - 3656\beta_{6} + 7816\beta_{5} - 1286\beta_{2} - 6899\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9376\beta_{8} - 6129\beta_{7} - 22229\beta_{4} - 18121\beta_{3} + 105185 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20244\beta_{9} + 91236\beta_{6} - 236241\beta_{5} + 34221\beta_{2} + 155177\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 4.34884i
− 3.67392i
− 4.98640i
− 3.26689i
− 0.230529i
0.230529i
3.26689i
4.98640i
3.67392i
4.34884i
− 5.34884i − 8.28008i −20.6101 0 −44.2888 − 30.9947i 67.4492i −41.5597 0
24.2 − 4.67392i 9.09294i −13.8456 0 42.4997 6.08885i 27.3217i −55.6816 0
24.3 − 3.98640i − 7.39409i −7.89136 0 −29.4758 3.57544i − 0.433099i −27.6726 0
24.4 − 2.26689i − 0.577397i 2.86119 0 −1.30890 10.7178i − 24.6212i 26.6666 0
24.5 − 1.23053i − 2.78435i 6.48580 0 −3.42623 24.1991i − 17.8252i 19.2474 0
24.6 1.23053i 2.78435i 6.48580 0 −3.42623 − 24.1991i 17.8252i 19.2474 0
24.7 2.26689i 0.577397i 2.86119 0 −1.30890 − 10.7178i 24.6212i 26.6666 0
24.8 3.98640i 7.39409i −7.89136 0 −29.4758 − 3.57544i 0.433099i −27.6726 0
24.9 4.67392i − 9.09294i −13.8456 0 42.4997 − 6.08885i − 27.3217i −55.6816 0
24.10 5.34884i 8.28008i −20.6101 0 −44.2888 30.9947i − 67.4492i −41.5597 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 24.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.4.b.h 10
5.b even 2 1 inner 575.4.b.h 10
5.c odd 4 1 115.4.a.d ✓ 5
5.c odd 4 1 575.4.a.k 5
15.e even 4 1 1035.4.a.m 5
20.e even 4 1 1840.4.a.p 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.d ✓ 5 5.c odd 4 1
575.4.a.k 5 5.c odd 4 1
575.4.b.h 10 1.a even 1 1 trivial
575.4.b.h 10 5.b even 2 1 inner
1035.4.a.m 5 15.e even 4 1
1840.4.a.p 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(575, [\chi])\):

\( T_{2}^{10} + 73T_{2}^{8} + 1876T_{2}^{6} + 19941T_{2}^{4} + 77181T_{2}^{2} + 77284 \) Copy content Toggle raw display
\( T_{3}^{10} + 214T_{3}^{8} + 15605T_{3}^{6} + 423149T_{3}^{4} + 2542030T_{3}^{2} + 801025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 73 T^{8} + \cdots + 77284 \) Copy content Toggle raw display
$3$ \( T^{10} + 214 T^{8} + \cdots + 801025 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 30627800064 \) Copy content Toggle raw display
$11$ \( (T^{5} + 153 T^{4} + \cdots - 58105432)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 18\!\cdots\!69 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} + \cdots + 5566328)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} - 583 T^{4} + \cdots + 63213004636)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 662 T^{4} + \cdots + 341100199935)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} - 344 T^{4} + \cdots - 554461833173)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} - 1166 T^{4} + \cdots + 446741827072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 499 T^{4} + \cdots + 53636443112)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 14 T^{4} + \cdots + 256345940645)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 22913376438144)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 279901843479552)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
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