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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(32,75)] 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("75.32"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,-24480] 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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 370x^{6} - 20x^{5} + 51573x^{4} - 3720x^{3} - 3172180x^{2} + 516580x + 75229001 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{3} q^{2} + ( - \beta_{5} - 9 \beta_{2}) q^{3} + 208 \beta_1 q^{4} + ( - 2 \beta_{7} - 3060) q^{6} + ( - 114 \beta_{4} - 57 \beta_{3}) q^{7} + 608 \beta_{2} q^{8} + (17 \beta_{6} - 6327 \beta_1) q^{9}+ \cdots + ( - 265734 \beta_{6} - 858256560 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24480 q^{6} + 2603008 q^{16} + 6090336 q^{21} + 79029056 q^{31} + 10528128 q^{36} - 155350080 q^{46} - 245118240 q^{51} - 283170224 q^{61} - 807770880 q^{66} - 129718784 q^{76} + 2458869048 q^{81}+ \cdots - 4154941440 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 370x^{6} - 20x^{5} + 51573x^{4} - 3720x^{3} - 3172180x^{2} + 516580x + 75229001 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 23362 \nu^{7} + 115230 \nu^{6} + 6506688 \nu^{5} - 52942655 \nu^{4} - 615714322 \nu^{3} + \cdots - 279766247600 ) / 39441307653 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7448345780 \nu^{7} + 15747194550 \nu^{6} + 2539809651876 \nu^{5} - 7160605025500 \nu^{4} + \cdots - 40\!\cdots\!50 ) / 18\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31930598660 \nu^{7} - 1164411084 \nu^{6} - 8855530432890 \nu^{5} - 19663144658 \nu^{4} + \cdots + 89\!\cdots\!98 ) / 18\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52001660128 \nu^{7} - 590479742326 \nu^{6} - 20814895781503 \nu^{5} + 162491838114844 \nu^{4} + \cdots + 49\!\cdots\!56 ) / 18\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 55795922692 \nu^{7} + 583506890353 \nu^{6} + 20689516304674 \nu^{5} + \cdots - 47\!\cdots\!30 ) / 18\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12 \nu^{7} - 1146 \nu^{6} + 4410 \nu^{5} + 311982 \nu^{4} - 475518 \nu^{3} - 26220054 \nu^{2} + \cdots + 630316068 ) / 133727 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4 \nu^{7} - 362 \nu^{6} - 1490 \nu^{5} + 101994 \nu^{4} + 184486 \nu^{3} - 10320518 \nu^{2} + \cdots + 358593096 ) / 32771 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - 5\beta_{2} + 12\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 6\beta_{5} + 6\beta_{4} - 9\beta_{3} - 3\beta_{2} - 360\beta _1 + 6660 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - 154\beta_{5} - 214\beta_{4} - 87\beta_{3} - 1187\beta_{2} + 3336\beta _1 + 180 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 388 \beta_{7} + 348 \beta_{6} - 1290 \beta_{5} + 930 \beta_{4} - 6207 \beta_{3} - 765 \beta_{2} + \cdots + 607572 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 875 \beta_{7} + 975 \beta_{6} - 21876 \beta_{5} - 77076 \beta_{4} + 16962 \beta_{3} - 518448 \beta_{2} + \cdots + 500400 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 29638 \beta_{7} + 20438 \beta_{6} - 117228 \beta_{5} + 33978 \beta_{4} - 752619 \beta_{3} + \cdots + 26316990 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 146132 \beta_{7} + 210882 \beta_{6} + 1233918 \beta_{5} - 9303042 \beta_{4} + 13120329 \beta_{3} + \cdots + 161680680 ) / 72 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−8.04954 + 1.08114i
11.2118 + 1.08114i
−11.2118 − 2.08114i
8.04954 − 2.08114i
−8.04954 − 1.08114i
11.2118 − 1.08114i
−11.2118 + 2.08114i
8.04954 + 2.08114i
−18.9737 + 18.9737i 22.8540 + 138.422i − 208.000i 0 −3060.00 − 2192.75i −6587.39 − 6587.39i −5767.99 − 5767.99i −18638.4 + 6327.00i 0
32.2 −18.9737 + 18.9737i 138.422 + 22.8540i − 208.000i 0 −3060.00 + 2192.75i 6587.39 + 6587.39i −5767.99 − 5767.99i 18638.4 + 6327.00i 0
32.3 18.9737 − 18.9737i −138.422 − 22.8540i − 208.000i 0 −3060.00 + 2192.75i −6587.39 − 6587.39i 5767.99 + 5767.99i 18638.4 + 6327.00i 0
32.4 18.9737 − 18.9737i −22.8540 − 138.422i − 208.000i 0 −3060.00 − 2192.75i 6587.39 + 6587.39i 5767.99 + 5767.99i −18638.4 + 6327.00i 0
68.1 −18.9737 − 18.9737i 22.8540 − 138.422i 208.000i 0 −3060.00 + 2192.75i −6587.39 + 6587.39i −5767.99 + 5767.99i −18638.4 − 6327.00i 0
68.2 −18.9737 − 18.9737i 138.422 − 22.8540i 208.000i 0 −3060.00 − 2192.75i 6587.39 − 6587.39i −5767.99 + 5767.99i 18638.4 − 6327.00i 0
68.3 18.9737 + 18.9737i −138.422 + 22.8540i 208.000i 0 −3060.00 − 2192.75i −6587.39 + 6587.39i 5767.99 − 5767.99i 18638.4 − 6327.00i 0
68.4 18.9737 + 18.9737i −22.8540 + 138.422i 208.000i 0 −3060.00 + 2192.75i 6587.39 − 6587.39i 5767.99 − 5767.99i −18638.4 − 6327.00i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 32.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.10.e.e ✓ 8
3.b odd 2 1 inner 75.10.e.e ✓ 8
5.b even 2 1 inner 75.10.e.e ✓ 8
5.c odd 4 2 inner 75.10.e.e ✓ 8
15.d odd 2 1 inner 75.10.e.e ✓ 8
15.e even 4 2 inner 75.10.e.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.10.e.e ✓ 8 1.a even 1 1 trivial
75.10.e.e ✓ 8 3.b odd 2 1 inner
75.10.e.e ✓ 8 5.b even 2 1 inner
75.10.e.e ✓ 8 5.c odd 4 2 inner
75.10.e.e ✓ 8 15.d odd 2 1 inner
75.10.e.e ✓ 8 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{4} + 518400 \) Copy content Toggle raw display
\( T_{7}^{4} + 7532033358394944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 518400)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 75\!\cdots\!44)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2120398560)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6077137936)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 17012044193760)^{4} \) Copy content Toggle raw display
$31$ \( (T - 9878632)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 24\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8263885563360)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 32\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20\!\cdots\!40)^{4} \) Copy content Toggle raw display
$61$ \( (T + 35396278)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 43\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 62\!\cdots\!40)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 74\!\cdots\!36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 41\!\cdots\!40)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 39\!\cdots\!04)^{2} \) Copy content Toggle raw display
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