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Properties

Label 90.7.g.f
Level $90$
Weight $7$
Character orbit 90.g
Analytic conductor $20.705$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,7,Mod(37,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.37"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,32,0,0,-60] 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: \(20.7048675258\)
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: \( x^{8} + 1602x^{6} + 816401x^{4} + 140305200x^{2} + 7638760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{8}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + (4 \beta_1 + 4) q^{2} + 32 \beta_1 q^{4} + ( - \beta_{3} + 23 \beta_1 - 7) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + 49) q^{7} + (128 \beta_1 - 128) q^{8} + ( - 4 \beta_{3} - 4 \beta_{2} + \cdots - 120) q^{10}+ \cdots + (496 \beta_{7} + 272 \beta_{6} + \cdots + 37516) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{2} - 60 q^{5} + 388 q^{7} - 1024 q^{8} - 976 q^{10} + 4192 q^{11} + 7884 q^{13} - 8192 q^{16} + 1372 q^{17} - 5888 q^{20} + 16768 q^{22} - 14888 q^{23} + 17932 q^{25} + 63072 q^{26} - 12416 q^{28}+ \cdots + 301888 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1602x^{6} + 816401x^{4} + 140305200x^{2} + 7638760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -21\nu^{7} - 29042\nu^{5} - 11615221\nu^{3} - 1070524600\nu ) / 664240000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2439 \nu^{7} - 5520 \nu^{6} - 3497878 \nu^{5} + 43182960 \nu^{4} - 1283225239 \nu^{3} + \cdots + 4480431648000 ) / 7306640000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 387 \nu^{7} - 15364 \nu^{6} + 647574 \nu^{5} - 26770528 \nu^{4} + 314249787 \nu^{3} + \cdots - 1251567650400 ) / 730664000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 843 \nu^{7} - 305900 \nu^{6} + 1290686 \nu^{5} - 434115800 \nu^{4} + 599740443 \nu^{3} + \cdots - 16362057860000 ) / 3653320000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 171 \nu^{7} - 207 \nu^{6} + 273942 \nu^{5} - 248814 \nu^{4} + 124659171 \nu^{3} + \cdots + 4737691800 ) / 182666000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 171 \nu^{7} - 207 \nu^{6} - 273942 \nu^{5} - 248814 \nu^{4} - 124659171 \nu^{3} + \cdots + 4737691800 ) / 182666000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2127 \nu^{7} - 303140 \nu^{6} + 3191254 \nu^{5} - 380980280 \nu^{4} + 1244128527 \nu^{3} + \cdots - 12071133884000 ) / 3653320000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} - 3\beta_{6} + 3\beta_{5} + 5\beta_{4} - 8\beta_{3} - 2\beta_{2} + 46\beta _1 + 4 ) / 90 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} + 278\beta_{6} + 278\beta_{5} - 3\beta_{4} + 2\beta_{3} - 8\beta_{2} + 4\beta _1 - 36046 ) / 90 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1203 \beta_{7} + 1038 \beta_{6} - 3018 \beta_{5} - 2665 \beta_{4} + 5848 \beta_{3} + 3442 \beta_{2} + \cdots - 2924 ) / 90 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5325 \beta_{7} - 225758 \beta_{6} - 221798 \beta_{5} + 2403 \beta_{4} - 6882 \beta_{3} + \cdots + 21009486 ) / 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 701403 \beta_{7} - 788358 \beta_{6} + 2378298 \beta_{5} + 1698985 \beta_{4} - 3990328 \beta_{3} + \cdots + 1995164 ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 5083645 \beta_{7} + 158425638 \beta_{6} + 153665718 \beta_{5} - 2098203 \beta_{4} + \cdots - 13698983726 ) / 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 457552803 \beta_{7} + 669069878 \beta_{6} - 1772731778 \beta_{5} - 1130475305 \beta_{4} + \cdots - 1345845004 ) / 90 \) 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.

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
11.9371i
− 25.3784i
− 10.9371i
26.3784i
− 11.9371i
25.3784i
10.9371i
− 26.3784i
4.00000 + 4.00000i 0 32.0000i −124.708 + 8.53435i 0 −288.850 − 288.850i −128.000 + 128.000i 0 −532.971 − 464.696i
37.2 4.00000 + 4.00000i 0 32.0000i −75.4773 − 99.6402i 0 145.451 + 145.451i −128.000 + 128.000i 0 96.6517 − 700.470i
37.3 4.00000 + 4.00000i 0 32.0000i 83.9887 + 92.5792i 0 334.411 + 334.411i −128.000 + 128.000i 0 −34.3620 + 706.271i
37.4 4.00000 + 4.00000i 0 32.0000i 86.1969 + 90.5267i 0 2.98790 + 2.98790i −128.000 + 128.000i 0 −17.3191 + 706.895i
73.1 4.00000 − 4.00000i 0 − 32.0000i −124.708 − 8.53435i 0 −288.850 + 288.850i −128.000 − 128.000i 0 −532.971 + 464.696i
73.2 4.00000 − 4.00000i 0 − 32.0000i −75.4773 + 99.6402i 0 145.451 − 145.451i −128.000 − 128.000i 0 96.6517 + 700.470i
73.3 4.00000 − 4.00000i 0 − 32.0000i 83.9887 − 92.5792i 0 334.411 − 334.411i −128.000 − 128.000i 0 −34.3620 − 706.271i
73.4 4.00000 − 4.00000i 0 − 32.0000i 86.1969 − 90.5267i 0 2.98790 − 2.98790i −128.000 − 128.000i 0 −17.3191 − 706.895i
\(n\): 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 90.7.g.f 8
3.b odd 2 1 30.7.f.b ✓ 8
5.b even 2 1 450.7.g.r 8
5.c odd 4 1 inner 90.7.g.f 8
5.c odd 4 1 450.7.g.r 8
12.b even 2 1 240.7.bg.b 8
15.d odd 2 1 150.7.f.h 8
15.e even 4 1 30.7.f.b ✓ 8
15.e even 4 1 150.7.f.h 8
60.l odd 4 1 240.7.bg.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.7.f.b ✓ 8 3.b odd 2 1
30.7.f.b ✓ 8 15.e even 4 1
90.7.g.f 8 1.a even 1 1 trivial
90.7.g.f 8 5.c odd 4 1 inner
150.7.f.h 8 15.d odd 2 1
150.7.f.h 8 15.e even 4 1
240.7.bg.b 8 12.b even 2 1
240.7.bg.b 8 60.l odd 4 1
450.7.g.r 8 5.b even 2 1
450.7.g.r 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{8} - 388 T_{7}^{7} + 75272 T_{7}^{6} + 12097544 T_{7}^{5} + 32303129764 T_{7}^{4} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
\( T_{11}^{4} - 2096T_{11}^{3} - 4964886T_{11}^{2} + 6193843120T_{11} + 8268865516000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 32)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 8268865516000)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 76\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 98\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 85\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
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