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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,9,Mod(19,36)] 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("36.19"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.6656299622\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 12)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 - 6) q^{4} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 38) q^{5} + (3 \beta_{7} + \beta_{6} - 3 \beta_{5} + \cdots - 4) q^{7} + ( - \beta_{7} - \beta_{6} + \cdots + 1616) q^{8}+ \cdots + ( - 12928 \beta_{7} + 31248 \beta_{6} + \cdots - 85812961) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} - 52 q^{4} + 336 q^{5} + 12960 q^{8} + 36628 q^{10} - 2864 q^{13} - 52728 q^{14} + 99440 q^{16} + 193200 q^{17} - 335592 q^{20} - 556968 q^{22} - 579048 q^{25} - 21564 q^{26} - 594672 q^{28}+ \cdots - 691081830 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13071742 \nu^{7} - 40510221 \nu^{6} + 365720344 \nu^{5} + 4477637544 \nu^{4} + \cdots - 864197730939 ) / 48999153875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19441601 \nu^{7} + 1819057563 \nu^{6} + 3699690118 \nu^{5} - 41753095607 \nu^{4} + \cdots - 22285033186408 ) / 27999516500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50159892 \nu^{7} + 1246942629 \nu^{6} + 703921944 \nu^{5} - 9257586106 \nu^{4} + \cdots + 11422432378061 ) / 48999153875 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 111239673 \nu^{7} - 1585068399 \nu^{6} - 10620483414 \nu^{5} + 1830296511 \nu^{4} + \cdots + 59875308578034 ) / 97998307750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 464456099 \nu^{7} + 4750820763 \nu^{6} + 35087786318 \nu^{5} + 131999004093 \nu^{4} + \cdots - 81416867085508 ) / 195996615500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 198793919 \nu^{7} + 666906657 \nu^{6} + 312336202 \nu^{5} - 76174483873 \nu^{4} + \cdots - 17027383415592 ) / 39199323100 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2928224053 \nu^{7} - 11493506061 \nu^{6} - 64637168146 \nu^{5} - 939040364571 \nu^{4} + \cdots + 213507284304526 ) / 97998307750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 8\beta_{6} - 6\beta_{5} - 8\beta_{4} + 2\beta_{3} + 4\beta_{2} + 50\beta _1 + 341 ) / 864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 47\beta_{6} + 91\beta_{5} + 88\beta_{4} - 175\beta_{3} + 46\beta_{2} - 1509\beta _1 + 18769 ) / 1728 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 32 \beta_{7} - 209 \beta_{6} - 427 \beta_{5} - 128 \beta_{4} - 649 \beta_{3} + 382 \beta_{2} + \cdots + 339765 ) / 1728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 55 \beta_{7} - 2696 \beta_{6} - 1148 \beta_{5} - 1940 \beta_{4} - 2248 \beta_{3} + 1450 \beta_{2} + \cdots + 844507 ) / 864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1758 \beta_{7} - 27813 \beta_{6} + 20977 \beta_{5} + 11544 \beta_{4} - 61293 \beta_{3} + \cdots + 16197455 ) / 1728 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 5420 \beta_{7} - 229337 \beta_{6} - 136051 \beta_{5} - 83312 \beta_{4} - 348241 \beta_{3} + \cdots + 140213985 ) / 1728 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 23505 \beta_{7} - 1134627 \beta_{6} - 192413 \beta_{5} - 431916 \beta_{4} - 1689099 \beta_{3} + \cdots + 525412100 ) / 864 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−3.64622 + 4.31154i
−3.64622 − 4.31154i
−1.11858 + 4.64627i
−1.11858 − 4.64627i
8.23534 − 0.0522875i
8.23534 + 0.0522875i
−1.97054 + 1.25304i
−1.97054 − 1.25304i
−13.6476 − 8.35123i 0 116.514 + 227.948i −904.196 0 888.085i 313.513 − 4083.98i 0 12340.1 + 7551.15i
19.2 −13.6476 + 8.35123i 0 116.514 − 227.948i −904.196 0 − 888.085i 313.513 + 4083.98i 0 12340.1 − 7551.15i
19.3 −7.47945 − 14.1442i 0 −144.116 + 211.581i 159.249 0 707.133i 4070.55 + 455.883i 0 −1191.10 − 2252.45i
19.4 −7.47945 + 14.1442i 0 −144.116 − 211.581i 159.249 0 − 707.133i 4070.55 − 455.883i 0 −1191.10 + 2252.45i
19.5 2.26258 − 15.8392i 0 −245.761 − 71.6749i 538.046 0 − 3350.97i −1691.33 + 3730.50i 0 1217.37 − 8522.23i
19.6 2.26258 + 15.8392i 0 −245.761 + 71.6749i 538.046 0 3350.97i −1691.33 − 3730.50i 0 1217.37 + 8522.23i
19.7 15.8645 − 2.07809i 0 247.363 − 65.9356i 374.901 0 4472.52i 3787.26 − 1560.08i 0 5947.61 − 779.079i
19.8 15.8645 + 2.07809i 0 247.363 + 65.9356i 374.901 0 − 4472.52i 3787.26 + 1560.08i 0 5947.61 + 779.079i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.9.d.c 8
3.b odd 2 1 12.9.d.a ✓ 8
4.b odd 2 1 inner 36.9.d.c 8
12.b even 2 1 12.9.d.a ✓ 8
24.f even 2 1 192.9.g.e 8
24.h odd 2 1 192.9.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.d.a ✓ 8 3.b odd 2 1
12.9.d.a ✓ 8 12.b even 2 1
36.9.d.c 8 1.a even 1 1 trivial
36.9.d.c 8 4.b odd 2 1 inner
192.9.g.e 8 24.f even 2 1
192.9.g.e 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 168T_{5}^{3} - 622376T_{5}^{2} + 281724000T_{5} - 29045327600 \) acting on \(S_{9}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 168 T^{3} + \cdots - 29045327600)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 83\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 48\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 44\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 39\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 42\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 16\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 30\!\cdots\!48)^{2} \) Copy content Toggle raw display
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