Properties

Label 90.5.g.d
Level $90$
Weight $5$
Character orbit 90.g
Analytic conductor $9.303$
Analytic rank $0$
Dimension $4$
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] = [90,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(9.30329667755\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + 8 \beta_{2} q^{4} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 6) q^{5} + ( - 25 \beta_{2} + 4 \beta_1 - 25) q^{7} + ( - 16 \beta_{2} + 16) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + 8 \beta_{2} q^{4} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 6) q^{5} + ( - 25 \beta_{2} + 4 \beta_1 - 25) q^{7} + ( - 16 \beta_{2} + 16) q^{8} + (6 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 16) q^{10} + (7 \beta_{3} - 7 \beta_1 + 80) q^{11} + (16 \beta_{3} + 30 \beta_{2} - 30) q^{13} + ( - 8 \beta_{3} + 100 \beta_{2} - 8 \beta_1) q^{14} - 64 q^{16} + ( - 172 \beta_{2} + 10 \beta_1 - 172) q^{17} + (40 \beta_{3} - 62 \beta_{2} + 40 \beta_1) q^{19} + ( - 16 \beta_{3} - 48 \beta_{2} + 8 \beta_1 - 16) q^{20} + ( - 160 \beta_{2} + 28 \beta_1 - 160) q^{22} + (30 \beta_{3} + 100 \beta_{2} - 100) q^{23} + (4 \beta_{3} + 327 \beta_{2} + 28 \beta_1 - 436) q^{25} + ( - 32 \beta_{3} + 32 \beta_1 + 120) q^{26} + (32 \beta_{3} - 200 \beta_{2} + 200) q^{28} + (41 \beta_{3} + 500 \beta_{2} + 41 \beta_1) q^{29} + (20 \beta_{3} - 20 \beta_1 + 616) q^{31} + (128 \beta_{2} + 128) q^{32} + ( - 20 \beta_{3} + 688 \beta_{2} - 20 \beta_1) q^{34} + (83 \beta_{3} - 836 \beta_{2} + \beta_1 + 668) q^{35} + ( - 750 \beta_{2} - 24 \beta_1 - 750) q^{37} + ( - 160 \beta_{3} + 124 \beta_{2} - 124) q^{38} + (16 \beta_{3} + 128 \beta_{2} - 48 \beta_1 - 64) q^{40} + (50 \beta_{3} - 50 \beta_1 - 1360) q^{41} + ( - 112 \beta_{3} - 810 \beta_{2} + 810) q^{43} + ( - 56 \beta_{3} + 640 \beta_{2} - 56 \beta_1) q^{44} + ( - 60 \beta_{3} + 60 \beta_1 + 400) q^{46} + ( - 900 \beta_{2} - 150 \beta_1 - 900) q^{47} + ( - 200 \beta_{3} + 721 \beta_{2} - 200 \beta_1) q^{49} + ( - 64 \beta_{3} + 218 \beta_{2} - 48 \beta_1 + 1526) q^{50} + ( - 240 \beta_{2} - 128 \beta_1 - 240) q^{52} + (240 \beta_{3} - 1756 \beta_{2} + 1756) q^{53} + ( - 136 \beta_{3} + 2617 \beta_{2} - 132 \beta_1 + 339) q^{55} + ( - 64 \beta_{3} + 64 \beta_1 - 800) q^{56} + ( - 164 \beta_{3} - 1000 \beta_{2} + 1000) q^{58} + ( - 21 \beta_{3} + 3320 \beta_{2} - 21 \beta_1) q^{59} + ( - 40 \beta_{3} + 40 \beta_1 + 3582) q^{61} + ( - 1232 \beta_{2} + 80 \beta_1 - 1232) q^{62} - 512 \beta_{2} q^{64} + ( - 126 \beta_{3} + 1632 \beta_{2} + 58 \beta_1 + 3864) q^{65} + ( - 2650 \beta_{2} - 168 \beta_1 - 2650) q^{67} + (80 \beta_{3} - 1376 \beta_{2} + 1376) q^{68} + ( - 168 \beta_{3} + 336 \beta_{2} + 164 \beta_1 - 3008) q^{70} + (76 \beta_{3} - 76 \beta_1 - 5240) q^{71} + (40 \beta_{3} + 1025 \beta_{2} - 1025) q^{73} + (48 \beta_{3} + 3000 \beta_{2} + 48 \beta_1) q^{74} + (320 \beta_{3} - 320 \beta_1 + 496) q^{76} + ( - 5276 \beta_{2} + 670 \beta_1 - 5276) q^{77} + (220 \beta_{3} - 132 \beta_{2} + 220 \beta_1) q^{79} + (64 \beta_{3} - 128 \beta_{2} + 128 \beta_1 + 384) q^{80} + (2720 \beta_{2} + 200 \beta_1 + 2720) q^{82} + ( - 550 \beta_{3} - 3408 \beta_{2} + 3408) q^{83} + (536 \beta_{3} - 1652 \beta_{2} + 112 \beta_1 + 2546) q^{85} + (224 \beta_{3} - 224 \beta_1 - 3240) q^{86} + (224 \beta_{3} - 1280 \beta_{2} + 1280) q^{88} + (48 \beta_{3} + 13240 \beta_{2} + 48 \beta_1) q^{89} + ( - 280 \beta_{3} + 280 \beta_1 - 5988) q^{91} + ( - 800 \beta_{2} - 240 \beta_1 - 800) q^{92} + (300 \beta_{3} + 3600 \beta_{2} + 300 \beta_1) q^{94} + ( - 36 \beta_{3} - 4308 \beta_{2} - 382 \beta_1 + 14164) q^{95} + ( - 1605 \beta_{2} + 1024 \beta_1 - 1605) q^{97} + (800 \beta_{3} - 1442 \beta_{2} + 1442) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 24 q^{5} - 100 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} - 24 q^{5} - 100 q^{7} + 64 q^{8} + 64 q^{10} + 320 q^{11} - 120 q^{13} - 256 q^{16} - 688 q^{17} - 64 q^{20} - 640 q^{22} - 400 q^{23} - 1744 q^{25} + 480 q^{26} + 800 q^{28} + 2464 q^{31} + 512 q^{32} + 2672 q^{35} - 3000 q^{37} - 496 q^{38} - 256 q^{40} - 5440 q^{41} + 3240 q^{43} + 1600 q^{46} - 3600 q^{47} + 6104 q^{50} - 960 q^{52} + 7024 q^{53} + 1356 q^{55} - 3200 q^{56} + 4000 q^{58} + 14328 q^{61} - 4928 q^{62} + 15456 q^{65} - 10600 q^{67} + 5504 q^{68} - 12032 q^{70} - 20960 q^{71} - 4100 q^{73} + 1984 q^{76} - 21104 q^{77} + 1536 q^{80} + 10880 q^{82} + 13632 q^{83} + 10184 q^{85} - 12960 q^{86} + 5120 q^{88} - 23952 q^{91} - 3200 q^{92} + 56656 q^{95} - 6420 q^{97} + 5768 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
2.54951 + 2.54951i
−2.54951 2.54951i
2.54951 2.54951i
−2.54951 + 2.54951i
−2.00000 2.00000i 0 8.00000i −13.6485 20.9456i 0 5.59412 + 5.59412i 16.0000 16.0000i 0 −14.5941 + 69.1882i
37.2 −2.00000 2.00000i 0 8.00000i 1.64853 + 24.9456i 0 −55.5941 55.5941i 16.0000 16.0000i 0 46.5941 53.1882i
73.1 −2.00000 + 2.00000i 0 8.00000i −13.6485 + 20.9456i 0 5.59412 5.59412i 16.0000 + 16.0000i 0 −14.5941 69.1882i
73.2 −2.00000 + 2.00000i 0 8.00000i 1.64853 24.9456i 0 −55.5941 + 55.5941i 16.0000 + 16.0000i 0 46.5941 + 53.1882i
\(n\): e.g. 2-40 or 990-1000
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 90.5.g.d 4
3.b odd 2 1 90.5.g.f yes 4
5.b even 2 1 450.5.g.o 4
5.c odd 4 1 inner 90.5.g.d 4
5.c odd 4 1 450.5.g.o 4
15.d odd 2 1 450.5.g.h 4
15.e even 4 1 90.5.g.f yes 4
15.e even 4 1 450.5.g.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.5.g.d 4 1.a even 1 1 trivial
90.5.g.d 4 5.c odd 4 1 inner
90.5.g.f yes 4 3.b odd 2 1
90.5.g.f yes 4 15.e even 4 1
450.5.g.h 4 15.d odd 2 1
450.5.g.h 4 15.e even 4 1
450.5.g.o 4 5.b even 2 1
450.5.g.o 4 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 \(S_{5}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{4} + 100T_{7}^{3} + 5000T_{7}^{2} - 62200T_{7} + 386884 \) Copy content Toggle raw display
\( T_{11}^{2} - 160T_{11} - 5066 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 24 T^{3} + 1160 T^{2} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{4} + 100 T^{3} + 5000 T^{2} + \cdots + 386884 \) Copy content Toggle raw display
$11$ \( (T^{2} - 160 T - 5066)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 120 T^{3} + \cdots + 792535104 \) Copy content Toggle raw display
$17$ \( T^{4} + 688 T^{3} + \cdots + 2253211024 \) Copy content Toggle raw display
$19$ \( T^{4} + 756488 T^{2} + \cdots + 137311749136 \) Copy content Toggle raw display
$23$ \( T^{4} + 400 T^{3} + \cdots + 7276090000 \) Copy content Toggle raw display
$29$ \( T^{4} + 1286708 T^{2} + \cdots + 20550369316 \) Copy content Toggle raw display
$31$ \( (T^{2} - 1232 T + 285856)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 3000 T^{3} + \cdots + 1118534681664 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2720 T + 1264600)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 3240 T^{3} + \cdots + 24164080704 \) Copy content Toggle raw display
$47$ \( T^{4} + 3600 T^{3} + \cdots + 1025156250000 \) Copy content Toggle raw display
$53$ \( T^{4} - 7024 T^{3} + \cdots + 327330448384 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 119229059670436 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7164 T + 12456324)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 115407579955264 \) Copy content Toggle raw display
$71$ \( (T^{2} + 10480 T + 26106016)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4100 T^{3} + \cdots + 3663587402500 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 127874844446976 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 147952483799184 \) Copy content Toggle raw display
$89$ \( T^{4} + 351673472 T^{2} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{4} + 6420 T^{3} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
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