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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,11,Mod(37,180)] 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("180.37"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 180.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,-894] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(114.364305481\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
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: \( x^{10} + 75402 x^{8} + 1918432665 x^{6} + 20025190470928 x^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 20)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{2} - 175 \beta_1 - 89) q^{5} + ( - \beta_{8} + \beta_{5} + \cdots + 2229) q^{7} + (5 \beta_{9} - 6 \beta_{8} + \cdots + 20147) q^{11} + ( - 15 \beta_{9} + 14 \beta_{7} + \cdots + 23962) q^{13}+ \cdots + ( - 180600 \beta_{9} + \cdots - 4892543931) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 894 q^{5} + 22286 q^{7} + 201700 q^{11} + 239298 q^{13} - 1045442 q^{17} + 4097986 q^{23} - 4233934 q^{25} + 23221660 q^{31} + 55388242 q^{35} - 87811974 q^{37} - 29776460 q^{41} + 156325470 q^{43}+ \cdots - 48945511254 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 75402 x^{8} + 1918432665 x^{6} + 20025190470928 x^{4} + \cdots + 13\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4541161 \nu^{9} + 296647947642 \nu^{7} + \cdots + 25\!\cdots\!24 \nu ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 202965048310579 \nu^{9} + \cdots + 11\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 202965048310579 \nu^{9} + \cdots + 11\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 118655438285131 \nu^{9} + \cdots + 89\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!97 \nu^{9} + \cdots + 26\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\!\cdots\!81 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!59 \nu^{9} + \cdots - 92\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41\!\cdots\!87 \nu^{9} + \cdots - 14\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30\!\cdots\!15 \nu^{9} + \cdots - 55\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{9} - 6\beta_{6} + 15\beta_{5} + 30\beta_{4} - 11\beta_{3} + 8\beta_{2} + 7\beta _1 - 6 ) / 1200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 168 \beta_{9} + 150 \beta_{8} - 150 \beta_{7} + 339 \beta_{6} + 1710 \beta_{5} + 195 \beta_{4} + \cdots - 18088231 ) / 1200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 95631 \beta_{9} - 39900 \beta_{8} - 39900 \beta_{7} + 140082 \beta_{6} - 233535 \beta_{5} + \cdots + 140082 ) / 1200 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1809264 \beta_{9} - 408630 \beta_{8} + 408630 \beta_{7} - 2689239 \beta_{6} - 15776118 \beta_{5} + \cdots + 88671356507 ) / 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3244105023 \beta_{9} + 1334216100 \beta_{8} + 1334216100 \beta_{7} - 3980489946 \beta_{6} + \cdots - 3980489946 ) / 1200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 389561132328 \beta_{9} - 5485557450 \beta_{8} + 5485557450 \beta_{7} + 496823010819 \beta_{6} + \cdots - 13\!\cdots\!51 ) / 1200 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 109909231475631 \beta_{9} - 39782245723500 \beta_{8} - 39782245723500 \beta_{7} + \cdots + 123589693454082 ) / 1200 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 30\!\cdots\!00 \beta_{9} + 354266529252090 \beta_{8} - 354266529252090 \beta_{7} + \cdots + 83\!\cdots\!79 ) / 240 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 37\!\cdots\!43 \beta_{9} + \cdots - 40\!\cdots\!86 ) / 1200 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
− 149.896i
− 56.3206i
− 75.9602i
95.3750i
186.802i
149.896i
56.3206i
75.9602i
− 95.3750i
− 186.802i
0 0 0 −2756.18 + 1472.78i 0 14849.7 + 14849.7i 0 0 0
37.2 0 0 0 −2223.57 − 2195.76i 0 −14037.3 − 14037.3i 0 0 0
37.3 0 0 0 473.688 + 3088.89i 0 −7250.16 − 7250.16i 0 0 0
37.4 0 0 0 944.998 − 2978.69i 0 21520.5 + 21520.5i 0 0 0
37.5 0 0 0 3114.06 − 261.220i 0 −3939.73 − 3939.73i 0 0 0
73.1 0 0 0 −2756.18 − 1472.78i 0 14849.7 − 14849.7i 0 0 0
73.2 0 0 0 −2223.57 + 2195.76i 0 −14037.3 + 14037.3i 0 0 0
73.3 0 0 0 473.688 − 3088.89i 0 −7250.16 + 7250.16i 0 0 0
73.4 0 0 0 944.998 + 2978.69i 0 21520.5 − 21520.5i 0 0 0
73.5 0 0 0 3114.06 + 261.220i 0 −3939.73 + 3939.73i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.11.l.a 10
3.b odd 2 1 20.11.f.a ✓ 10
5.c odd 4 1 inner 180.11.l.a 10
12.b even 2 1 80.11.p.e 10
15.d odd 2 1 100.11.f.b 10
15.e even 4 1 20.11.f.a ✓ 10
15.e even 4 1 100.11.f.b 10
60.l odd 4 1 80.11.p.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.f.a ✓ 10 3.b odd 2 1
20.11.f.a ✓ 10 15.e even 4 1
80.11.p.e 10 12.b even 2 1
80.11.p.e 10 60.l odd 4 1
100.11.f.b 10 15.d odd 2 1
100.11.f.b 10 15.e even 4 1
180.11.l.a 10 1.a even 1 1 trivial
180.11.l.a 10 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 22286 T_{7}^{9} + 248332898 T_{7}^{8} + 11532754764400 T_{7}^{7} + \cdots + 52\!\cdots\!68 \) acting on \(S_{11}^{\mathrm{new}}(180, [\chi])\). 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
$5$ \( T^{10} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 52\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 71\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 67\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 37\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 29\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 90\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 72\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 23\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 57\!\cdots\!68 \) Copy content Toggle raw display
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