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Properties

Label 90.17.g.b
Level $90$
Weight $17$
Character orbit 90.g
Analytic conductor $146.092$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,17,Mod(37,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.37");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(146.092089471\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7213550 x^{5} + 3043721913 x^{4} - 386278388950 x^{3} + 26017651801250 x^{2} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{12}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - 128 \beta_1 + 128) q^{2} - 32768 \beta_1 q^{4} + ( - \beta_{6} + \beta_{3} + \cdots - 23104) q^{5}+ \cdots + ( - 4194304 \beta_1 - 4194304) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 128 \beta_1 + 128) q^{2} - 32768 \beta_1 q^{4} + ( - \beta_{6} + \beta_{3} + \cdots - 23104) q^{5}+ \cdots + ( - 7224141440 \beta_{7} + \cdots - 22\!\cdots\!40) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1024 q^{2} - 184830 q^{5} - 1586702 q^{7} - 33554432 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1024 q^{2} - 184830 q^{5} - 1586702 q^{7} - 33554432 q^{8} - 179537920 q^{10} - 307957276 q^{11} + 202095228 q^{13} - 8589934592 q^{16} + 10825054172 q^{17} - 39905198080 q^{20} - 39418531328 q^{22} + 58166716742 q^{23} + 48765928900 q^{25} + 51736378368 q^{26} + 51993051136 q^{28} - 1503757815484 q^{31} - 1099511627776 q^{32} - 1361150225890 q^{35} + 5719558248048 q^{37} - 10471905756160 q^{38} - 4332632145920 q^{40} + 21624661426724 q^{41} + 694778360778 q^{43} + 14890679485952 q^{46} - 17454156046938 q^{47} + 38811694553600 q^{50} + 6622256431104 q^{52} - 315933715243808 q^{53} - 170380752243540 q^{55} + 13310221090816 q^{56} + 231112067379200 q^{58} - 260671832048484 q^{61} - 192481000381952 q^{62} - 442440687522120 q^{65} + 635885550643738 q^{67} - 354715375108096 q^{68} - 971840253967360 q^{70} + 11\!\cdots\!04 q^{71}+ \cdots - 18\!\cdots\!04 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7213550 x^{5} + 3043721913 x^{4} - 386278388950 x^{3} + 26017651801250 x^{2} + \cdots + 77\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12\!\cdots\!75 \nu^{7} + \cdots - 38\!\cdots\!00 ) / 24\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!81 \nu^{7} + \cdots + 18\!\cdots\!80 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{7} + \cdots - 12\!\cdots\!72 ) / 75\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!75 \nu^{7} + \cdots - 17\!\cdots\!28 ) / 75\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50\!\cdots\!25 \nu^{7} + \cdots + 30\!\cdots\!68 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{7} + \cdots + 55\!\cdots\!52 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 77\!\cdots\!59 \nu^{7} + \cdots + 16\!\cdots\!56 ) / 30\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 15\beta_{7} - 6\beta_{6} - 18\beta_{5} + 2\beta_{4} + 6\beta_{3} + 15\beta_{2} + 2\beta _1 - 1 ) / 54000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{7} - 4\beta_{6} + 26\beta_{5} + 26\beta_{4} + 22\beta_{3} - 19\beta_{2} + 9638846\beta _1 - 13 ) / 360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 937605 \beta_{7} + 1084266 \beta_{6} - 497622 \beta_{5} - 293322 \beta_{4} + 121174 \beta_{3} + \cdots + 146075119331 ) / 54000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5216358 \beta_{7} - 5908074 \beta_{6} - 1508214 \beta_{5} - 2957150 \beta_{4} + 4465364 \beta_{3} + \cdots - 1643613573916 ) / 1080 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 45999610785 \beta_{7} + 39078197514 \beta_{6} + 75877886142 \beta_{5} + 19303806962 \beta_{4} + \cdots + 13\!\cdots\!19 ) / 54000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 94538716389 \beta_{7} + 124138568820 \beta_{6} - 437354570418 \beta_{5} - 258406592858 \beta_{4} + \cdots + 218677285209 ) / 1080 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 48\!\cdots\!95 \beta_{7} + \cdots - 10\!\cdots\!89 ) / 54000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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.

comment: embeddings in the coefficient field
 
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
−191.774 + 191.774i
110.950 − 110.950i
64.8621 − 64.8621i
15.9615 − 15.9615i
−191.774 − 191.774i
110.950 + 110.950i
64.8621 + 64.8621i
15.9615 + 15.9615i
128.000 + 128.000i 0 32768.0i −390295. − 16042.9i 0 −6.21755e6 − 6.21755e6i −4.19430e6 + 4.19430e6i 0 −4.79043e7 − 5.20113e7i
37.2 128.000 + 128.000i 0 32768.0i −147024. + 361900.i 0 7.60621e6 + 7.60621e6i −4.19430e6 + 4.19430e6i 0 −6.51423e7 + 2.75042e7i
37.3 128.000 + 128.000i 0 32768.0i 73370.4 + 383673.i 0 −2.41428e6 − 2.41428e6i −4.19430e6 + 4.19430e6i 0 −3.97187e7 + 5.85015e7i
37.4 128.000 + 128.000i 0 32768.0i 371534. − 120625.i 0 232276. + 232276.i −4.19430e6 + 4.19430e6i 0 6.29964e7 + 3.21163e7i
73.1 128.000 − 128.000i 0 − 32768.0i −390295. + 16042.9i 0 −6.21755e6 + 6.21755e6i −4.19430e6 − 4.19430e6i 0 −4.79043e7 + 5.20113e7i
73.2 128.000 − 128.000i 0 − 32768.0i −147024. − 361900.i 0 7.60621e6 − 7.60621e6i −4.19430e6 − 4.19430e6i 0 −6.51423e7 − 2.75042e7i
73.3 128.000 − 128.000i 0 − 32768.0i 73370.4 − 383673.i 0 −2.41428e6 + 2.41428e6i −4.19430e6 − 4.19430e6i 0 −3.97187e7 − 5.85015e7i
73.4 128.000 − 128.000i 0 − 32768.0i 371534. + 120625.i 0 232276. − 232276.i −4.19430e6 − 4.19430e6i 0 6.29964e7 − 3.21163e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
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 90.17.g.b 8
3.b odd 2 1 10.17.c.a ✓ 8
5.c odd 4 1 inner 90.17.g.b 8
12.b even 2 1 80.17.p.a 8
15.d odd 2 1 50.17.c.d 8
15.e even 4 1 10.17.c.a ✓ 8
15.e even 4 1 50.17.c.d 8
60.l odd 4 1 80.17.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.17.c.a ✓ 8 3.b odd 2 1
10.17.c.a ✓ 8 15.e even 4 1
50.17.c.d 8 15.d odd 2 1
50.17.c.d 8 15.e even 4 1
80.17.p.a 8 12.b even 2 1
80.17.p.a 8 60.l odd 4 1
90.17.g.b 8 1.a even 1 1 trivial
90.17.g.b 8 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{17}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{8} + 1586702 T_{7}^{7} + 1258811618402 T_{7}^{6} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
\( T_{11}^{4} + 153978638 T_{11}^{3} + \cdots - 43\!\cdots\!44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 256 T + 32768)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 43\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 98\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 46\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
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