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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [45,11,Mod(44,45)] 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("45.44"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 45.d (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: \(28.5910763703\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 15950 x^{18} + 106482805 x^{16} - 387295108300 x^{14} + 837563369590460 x^{12} + \cdots + 93\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{68}\cdot 5^{12} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( - \beta_1 + 573) q^{4} + (\beta_{12} - 11 \beta_{10}) q^{5} + \beta_{2} q^{7} + ( - \beta_{14} - 4 \beta_{12} + 560 \beta_{10}) q^{8} + (\beta_{3} + \beta_{2} + 35 \beta_1 - 16920) q^{10}+ \cdots + ( - 9017 \beta_{19} + \cdots - 256271143 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 11460 q^{4} - 338408 q^{10} + 6108820 q^{16} - 1898000 q^{19} + 2383676 q^{25} - 19349480 q^{31} - 195513640 q^{34} - 982288972 q^{40} + 1436848520 q^{46} - 2337029140 q^{49} - 1365616152 q^{55}+ \cdots + 65993506880 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 15950 x^{18} + 106482805 x^{16} - 387295108300 x^{14} + 837563369590460 x^{12} + \cdots + 93\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18\!\cdots\!65 \nu^{18} + \cdots - 50\!\cdots\!56 ) / 53\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!75 \nu^{18} + \cdots + 12\!\cdots\!04 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!09 \nu^{18} + \cdots + 32\!\cdots\!52 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!71 \nu^{18} + \cdots + 11\!\cdots\!88 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!09 \nu^{18} + \cdots - 55\!\cdots\!52 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!07 \nu^{18} + \cdots + 21\!\cdots\!96 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!33 \nu^{18} + \cdots + 40\!\cdots\!24 ) / 49\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!99 \nu^{18} + \cdots + 14\!\cdots\!88 ) / 55\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 54\!\cdots\!71 \nu^{18} + \cdots + 16\!\cdots\!88 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!55 \nu^{19} + \cdots - 51\!\cdots\!64 \nu ) / 24\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!55 \nu^{19} + \cdots - 49\!\cdots\!40 \nu ) / 36\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 28\!\cdots\!67 \nu^{19} + \cdots - 81\!\cdots\!76 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!33 \nu^{19} + \cdots - 37\!\cdots\!24 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!67 \nu^{19} + \cdots + 79\!\cdots\!76 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!69 \nu^{19} + \cdots - 42\!\cdots\!52 \nu ) / 42\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 32\!\cdots\!13 \nu^{19} + \cdots - 55\!\cdots\!36 \nu ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 84\!\cdots\!43 \nu^{19} + \cdots + 30\!\cdots\!04 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!19 \nu^{19} + \cdots - 36\!\cdots\!32 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 54\!\cdots\!43 \nu^{19} + \cdots - 19\!\cdots\!04 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 6561\beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 10\beta_{2} - 6561\beta _1 + 10464795 ) / 6561 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6561\beta_{14} - 243\beta_{13} + 26001\beta_{12} + 4762\beta_{11} - 17071722\beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6885 \beta_{9} - 324 \beta_{8} - 7857 \beta_{7} + 648 \beta_{6} + 7640 \beta_{5} + 674 \beta_{4} + \cdots + 27186726762 ) / 6561 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 179334 \beta_{19} - 227448 \beta_{18} - 1386558 \beta_{17} - 196830 \beta_{16} + \cdots - 51332559786 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 38475081 \beta_{9} - 3498390 \beta_{8} - 40126023 \beta_{7} + 8440200 \beta_{6} + \cdots + 81620694210042 ) / 6561 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 914717124 \beta_{19} - 1667810574 \beta_{18} - 8327964780 \beta_{17} + \cdots - 164891305285218 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 163108367307 \beta_{9} - 20986590024 \beta_{8} - 157796645859 \beta_{7} + 52003916784 \beta_{6} + \cdots + 26\!\cdots\!44 ) / 6561 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3340665774054 \beta_{19} - 8640075845268 \beta_{18} - 36389373069294 \beta_{17} + \cdots - 55\!\cdots\!74 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 631553768241993 \beta_{9} - 100314026556138 \beta_{8} - 573506059247667 \beta_{7} + \cdots + 87\!\cdots\!42 ) / 6561 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10\!\cdots\!12 \beta_{19} + \cdots - 18\!\cdots\!90 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 23\!\cdots\!19 \beta_{9} + \cdots + 29\!\cdots\!24 ) / 6561 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 26\!\cdots\!94 \beta_{19} + \cdots - 65\!\cdots\!34 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 86\!\cdots\!69 \beta_{9} + \cdots + 10\!\cdots\!70 ) / 6561 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 51\!\cdots\!44 \beta_{19} + \cdots - 22\!\cdots\!86 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 31\!\cdots\!71 \beta_{9} + \cdots + 35\!\cdots\!88 ) / 6561 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 78\!\cdots\!34 \beta_{19} + \cdots - 79\!\cdots\!30 \beta_{10} ) / 6561 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 11\!\cdots\!09 \beta_{9} + \cdots + 12\!\cdots\!66 ) / 6561 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 64\!\cdots\!04 \beta_{19} + \cdots - 27\!\cdots\!62 \beta_{10} ) / 6561 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
60.0339 − 1.41421i
60.0339 + 1.41421i
48.1887 + 1.41421i
48.1887 − 1.41421i
38.8685 − 1.41421i
38.8685 + 1.41421i
16.6558 − 1.41421i
16.6558 + 1.41421i
16.4502 + 1.41421i
16.4502 − 1.41421i
−16.4502 + 1.41421i
−16.4502 − 1.41421i
−16.6558 − 1.41421i
−16.6558 + 1.41421i
−38.8685 − 1.41421i
−38.8685 + 1.41421i
−48.1887 + 1.41421i
−48.1887 − 1.41421i
−60.0339 − 1.41421i
−60.0339 + 1.41421i
−60.0339 0 2580.06 2954.78 − 1017.31i 0 30036.9i −93416.5 0 −177387. + 61073.3i
44.2 −60.0339 0 2580.06 2954.78 + 1017.31i 0 − 30036.9i −93416.5 0 −177387. − 61073.3i
44.3 −48.1887 0 1298.15 −1954.96 − 2437.98i 0 11278.9i −13211.1 0 94207.0 + 117483.i
44.4 −48.1887 0 1298.15 −1954.96 + 2437.98i 0 − 11278.9i −13211.1 0 94207.0 − 117483.i
44.5 −38.8685 0 486.757 −37.9548 − 3124.77i 0 − 19763.0i 20881.8 0 1475.24 + 121455.i
44.6 −38.8685 0 486.757 −37.9548 + 3124.77i 0 19763.0i 20881.8 0 1475.24 − 121455.i
44.7 −16.6558 0 −746.583 −2360.59 − 2047.74i 0 23573.7i 29490.5 0 39317.6 + 34106.8i
44.8 −16.6558 0 −746.583 −2360.59 + 2047.74i 0 − 23573.7i 29490.5 0 39317.6 − 34106.8i
44.9 −16.4502 0 −753.390 2566.25 − 1783.26i 0 − 4572.15i 29238.5 0 −42215.3 + 29335.0i
44.10 −16.4502 0 −753.390 2566.25 + 1783.26i 0 4572.15i 29238.5 0 −42215.3 − 29335.0i
44.11 16.4502 0 −753.390 −2566.25 − 1783.26i 0 4572.15i −29238.5 0 −42215.3 − 29335.0i
44.12 16.4502 0 −753.390 −2566.25 + 1783.26i 0 − 4572.15i −29238.5 0 −42215.3 + 29335.0i
44.13 16.6558 0 −746.583 2360.59 − 2047.74i 0 − 23573.7i −29490.5 0 39317.6 − 34106.8i
44.14 16.6558 0 −746.583 2360.59 + 2047.74i 0 23573.7i −29490.5 0 39317.6 + 34106.8i
44.15 38.8685 0 486.757 37.9548 − 3124.77i 0 19763.0i −20881.8 0 1475.24 − 121455.i
44.16 38.8685 0 486.757 37.9548 + 3124.77i 0 − 19763.0i −20881.8 0 1475.24 + 121455.i
44.17 48.1887 0 1298.15 1954.96 − 2437.98i 0 − 11278.9i 13211.1 0 94207.0 − 117483.i
44.18 48.1887 0 1298.15 1954.96 + 2437.98i 0 11278.9i 13211.1 0 94207.0 + 117483.i
44.19 60.0339 0 2580.06 −2954.78 − 1017.31i 0 − 30036.9i 93416.5 0 −177387. − 61073.3i
44.20 60.0339 0 2580.06 −2954.78 + 1017.31i 0 30036.9i 93416.5 0 −177387. + 61073.3i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 44.20
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.11.d.a ✓ 20
3.b odd 2 1 inner 45.11.d.a ✓ 20
5.b even 2 1 inner 45.11.d.a ✓ 20
5.c odd 4 2 225.11.c.e 20
15.d odd 2 1 inner 45.11.d.a ✓ 20
15.e even 4 2 225.11.c.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.11.d.a ✓ 20 1.a even 1 1 trivial
45.11.d.a ✓ 20 3.b odd 2 1 inner
45.11.d.a ✓ 20 5.b even 2 1 inner
45.11.d.a ✓ 20 15.d odd 2 1 inner
225.11.c.e 20 5.c odd 4 2
225.11.c.e 20 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + \cdots - 949192717484032)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 78\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 52\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 26\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 52\!\cdots\!00)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 52\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 18\!\cdots\!24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 69\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 52\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 31\!\cdots\!24)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 82\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 32\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 62\!\cdots\!24)^{2} \) Copy content Toggle raw display
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