Properties

Label 90.5.g.a
Level $90$
Weight $5$
Character orbit 90.g
Analytic conductor $9.303$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 i - 2) q^{2} + 8 i q^{4} + (20 i + 15) q^{5} + ( - 19 i - 19) q^{7} + ( - 16 i + 16) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 i - 2) q^{2} + 8 i q^{4} + (20 i + 15) q^{5} + ( - 19 i - 19) q^{7} + ( - 16 i + 16) q^{8} + ( - 70 i + 10) q^{10} - 202 q^{11} + (99 i - 99) q^{13} + 76 i q^{14} - 64 q^{16} + (239 i + 239) q^{17} + 40 i q^{19} + (120 i - 160) q^{20} + (404 i + 404) q^{22} + (541 i - 541) q^{23} + (600 i - 175) q^{25} + 396 q^{26} + ( - 152 i + 152) q^{28} + 200 i q^{29} - 758 q^{31} + (128 i + 128) q^{32} - 956 i q^{34} + ( - 665 i + 95) q^{35} + (141 i + 141) q^{37} + ( - 80 i + 80) q^{38} + (80 i + 560) q^{40} - 1042 q^{41} + (759 i - 759) q^{43} - 1616 i q^{44} + 2164 q^{46} + (459 i + 459) q^{47} - 1679 i q^{49} + ( - 850 i + 1550) q^{50} + ( - 792 i - 792) q^{52} + ( - 1819 i + 1819) q^{53} + ( - 4040 i - 3030) q^{55} - 608 q^{56} + ( - 400 i + 400) q^{58} - 4600 i q^{59} + 2082 q^{61} + (1516 i + 1516) q^{62} - 512 i q^{64} + ( - 495 i - 3465) q^{65} + (5081 i + 5081) q^{67} + (1912 i - 1912) q^{68} + (1140 i - 1520) q^{70} + 3478 q^{71} + (3479 i - 3479) q^{73} - 564 i q^{74} - 320 q^{76} + (3838 i + 3838) q^{77} - 7680 i q^{79} + ( - 1280 i - 960) q^{80} + (2084 i + 2084) q^{82} + (6081 i - 6081) q^{83} + (8365 i - 1195) q^{85} + 3036 q^{86} + (3232 i - 3232) q^{88} + 5680 i q^{89} + 3762 q^{91} + ( - 4328 i - 4328) q^{92} - 1836 i q^{94} + (600 i - 800) q^{95} + (561 i + 561) q^{97} + (3358 i - 3358) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 30 q^{5} - 38 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 30 q^{5} - 38 q^{7} + 32 q^{8} + 20 q^{10} - 404 q^{11} - 198 q^{13} - 128 q^{16} + 478 q^{17} - 320 q^{20} + 808 q^{22} - 1082 q^{23} - 350 q^{25} + 792 q^{26} + 304 q^{28} - 1516 q^{31} + 256 q^{32} + 190 q^{35} + 282 q^{37} + 160 q^{38} + 1120 q^{40} - 2084 q^{41} - 1518 q^{43} + 4328 q^{46} + 918 q^{47} + 3100 q^{50} - 1584 q^{52} + 3638 q^{53} - 6060 q^{55} - 1216 q^{56} + 800 q^{58} + 4164 q^{61} + 3032 q^{62} - 6930 q^{65} + 10162 q^{67} - 3824 q^{68} - 3040 q^{70} + 6956 q^{71} - 6958 q^{73} - 640 q^{76} + 7676 q^{77} - 1920 q^{80} + 4168 q^{82} - 12162 q^{83} - 2390 q^{85} + 6072 q^{86} - 6464 q^{88} + 7524 q^{91} - 8656 q^{92} - 1600 q^{95} + 1122 q^{97} - 6716 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
1.00000i
1.00000i
−2.00000 2.00000i 0 8.00000i 15.0000 + 20.0000i 0 −19.0000 19.0000i 16.0000 16.0000i 0 10.0000 70.0000i
73.1 −2.00000 + 2.00000i 0 8.00000i 15.0000 20.0000i 0 −19.0000 + 19.0000i 16.0000 + 16.0000i 0 10.0000 + 70.0000i
\(n\): 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.a 2
3.b odd 2 1 10.5.c.b 2
5.b even 2 1 450.5.g.b 2
5.c odd 4 1 inner 90.5.g.a 2
5.c odd 4 1 450.5.g.b 2
12.b even 2 1 80.5.p.c 2
15.d odd 2 1 50.5.c.a 2
15.e even 4 1 10.5.c.b 2
15.e even 4 1 50.5.c.a 2
24.f even 2 1 320.5.p.g 2
24.h odd 2 1 320.5.p.d 2
60.h even 2 1 400.5.p.b 2
60.l odd 4 1 80.5.p.c 2
60.l odd 4 1 400.5.p.b 2
120.q odd 4 1 320.5.p.g 2
120.w even 4 1 320.5.p.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.5.c.b 2 3.b odd 2 1
10.5.c.b 2 15.e even 4 1
50.5.c.a 2 15.d odd 2 1
50.5.c.a 2 15.e even 4 1
80.5.p.c 2 12.b even 2 1
80.5.p.c 2 60.l odd 4 1
90.5.g.a 2 1.a even 1 1 trivial
90.5.g.a 2 5.c odd 4 1 inner
320.5.p.d 2 24.h odd 2 1
320.5.p.d 2 120.w even 4 1
320.5.p.g 2 24.f even 2 1
320.5.p.g 2 120.q odd 4 1
400.5.p.b 2 60.h even 2 1
400.5.p.b 2 60.l odd 4 1
450.5.g.b 2 5.b even 2 1
450.5.g.b 2 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 \(S_{5}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{2} + 38T_{7} + 722 \) Copy content Toggle raw display
\( T_{11} + 202 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 30T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} + 38T + 722 \) Copy content Toggle raw display
$11$ \( (T + 202)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 198T + 19602 \) Copy content Toggle raw display
$17$ \( T^{2} - 478T + 114242 \) Copy content Toggle raw display
$19$ \( T^{2} + 1600 \) Copy content Toggle raw display
$23$ \( T^{2} + 1082 T + 585362 \) Copy content Toggle raw display
$29$ \( T^{2} + 40000 \) Copy content Toggle raw display
$31$ \( (T + 758)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 282T + 39762 \) Copy content Toggle raw display
$41$ \( (T + 1042)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1518 T + 1152162 \) Copy content Toggle raw display
$47$ \( T^{2} - 918T + 421362 \) Copy content Toggle raw display
$53$ \( T^{2} - 3638 T + 6617522 \) Copy content Toggle raw display
$59$ \( T^{2} + 21160000 \) Copy content Toggle raw display
$61$ \( (T - 2082)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 10162 T + 51633122 \) Copy content Toggle raw display
$71$ \( (T - 3478)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6958 T + 24206882 \) Copy content Toggle raw display
$79$ \( T^{2} + 58982400 \) Copy content Toggle raw display
$83$ \( T^{2} + 12162 T + 73957122 \) Copy content Toggle raw display
$89$ \( T^{2} + 32262400 \) Copy content Toggle raw display
$97$ \( T^{2} - 1122 T + 629442 \) Copy content Toggle raw display
show more
show less