Properties

Label 90.5.g.b
Level $90$
Weight $5$
Character orbit 90.g
Analytic conductor $9.303$
Analytic rank $0$
Dimension $2$
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: \(2\)
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.

\(f(q)\) \(=\) \( q + (2 i + 2) q^{2} + 8 i q^{4} + ( - 20 i + 15) q^{5} + (29 i + 29) q^{7} + (16 i - 16) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 i + 2) q^{2} + 8 i q^{4} + ( - 20 i + 15) q^{5} + (29 i + 29) q^{7} + (16 i - 16) q^{8} + ( - 10 i + 70) q^{10} + 118 q^{11} + ( - 69 i + 69) q^{13} + 116 i q^{14} - 64 q^{16} + (271 i + 271) q^{17} + 280 i q^{19} + (120 i + 160) q^{20} + (236 i + 236) q^{22} + (269 i - 269) q^{23} + ( - 600 i - 175) q^{25} + 276 q^{26} + (232 i - 232) q^{28} - 680 i q^{29} + 202 q^{31} + ( - 128 i - 128) q^{32} + 1084 i q^{34} + ( - 145 i + 1015) q^{35} + ( - 651 i - 651) q^{37} + (560 i - 560) q^{38} + (560 i + 80) q^{40} - 1682 q^{41} + ( - 1089 i + 1089) q^{43} + 944 i q^{44} - 1076 q^{46} + ( - 1269 i - 1269) q^{47} - 719 i q^{49} + ( - 1550 i + 850) q^{50} + (552 i + 552) q^{52} + ( - 611 i + 611) q^{53} + ( - 2360 i + 1770) q^{55} - 928 q^{56} + ( - 1360 i + 1360) q^{58} - 1160 i q^{59} - 5598 q^{61} + (404 i + 404) q^{62} - 512 i q^{64} + ( - 2415 i - 345) q^{65} + ( - 751 i - 751) q^{67} + (2168 i - 2168) q^{68} + (1740 i + 2320) q^{70} - 6442 q^{71} + (2951 i - 2951) q^{73} - 2604 i q^{74} - 2240 q^{76} + (3422 i + 3422) q^{77} + 10560 i q^{79} + (1280 i - 960) q^{80} + ( - 3364 i - 3364) q^{82} + ( - 6231 i + 6231) q^{83} + ( - 1355 i + 9485) q^{85} + 4356 q^{86} + (1888 i - 1888) q^{88} + 14480 i q^{89} + 4002 q^{91} + ( - 2152 i - 2152) q^{92} - 5076 i q^{94} + (4200 i + 5600) q^{95} + ( - 7311 i - 7311) q^{97} + ( - 1438 i + 1438) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 30 q^{5} + 58 q^{7} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 30 q^{5} + 58 q^{7} - 32 q^{8} + 140 q^{10} + 236 q^{11} + 138 q^{13} - 128 q^{16} + 542 q^{17} + 320 q^{20} + 472 q^{22} - 538 q^{23} - 350 q^{25} + 552 q^{26} - 464 q^{28} + 404 q^{31} - 256 q^{32} + 2030 q^{35} - 1302 q^{37} - 1120 q^{38} + 160 q^{40} - 3364 q^{41} + 2178 q^{43} - 2152 q^{46} - 2538 q^{47} + 1700 q^{50} + 1104 q^{52} + 1222 q^{53} + 3540 q^{55} - 1856 q^{56} + 2720 q^{58} - 11196 q^{61} + 808 q^{62} - 690 q^{65} - 1502 q^{67} - 4336 q^{68} + 4640 q^{70} - 12884 q^{71} - 5902 q^{73} - 4480 q^{76} + 6844 q^{77} - 1920 q^{80} - 6728 q^{82} + 12462 q^{83} + 18970 q^{85} + 8712 q^{86} - 3776 q^{88} + 8004 q^{91} - 4304 q^{92} + 11200 q^{95} - 14622 q^{97} + 2876 q^{98}+O(q^{100}) \) 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 29.0000 + 29.0000i −16.0000 + 16.0000i 0 70.0000 10.0000i
73.1 2.00000 2.00000i 0 8.00000i 15.0000 + 20.0000i 0 29.0000 29.0000i −16.0000 16.0000i 0 70.0000 + 10.0000i
\(n\): 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.b 2
3.b odd 2 1 10.5.c.a 2
5.b even 2 1 450.5.g.a 2
5.c odd 4 1 inner 90.5.g.b 2
5.c odd 4 1 450.5.g.a 2
12.b even 2 1 80.5.p.b 2
15.d odd 2 1 50.5.c.b 2
15.e even 4 1 10.5.c.a 2
15.e even 4 1 50.5.c.b 2
24.f even 2 1 320.5.p.i 2
24.h odd 2 1 320.5.p.b 2
60.h even 2 1 400.5.p.c 2
60.l odd 4 1 80.5.p.b 2
60.l odd 4 1 400.5.p.c 2
120.q odd 4 1 320.5.p.i 2
120.w even 4 1 320.5.p.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.5.c.a 2 3.b odd 2 1
10.5.c.a 2 15.e even 4 1
50.5.c.b 2 15.d odd 2 1
50.5.c.b 2 15.e even 4 1
80.5.p.b 2 12.b even 2 1
80.5.p.b 2 60.l odd 4 1
90.5.g.b 2 1.a even 1 1 trivial
90.5.g.b 2 5.c odd 4 1 inner
320.5.p.b 2 24.h odd 2 1
320.5.p.b 2 120.w even 4 1
320.5.p.i 2 24.f even 2 1
320.5.p.i 2 120.q odd 4 1
400.5.p.c 2 60.h even 2 1
400.5.p.c 2 60.l odd 4 1
450.5.g.a 2 5.b even 2 1
450.5.g.a 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 \(S_{5}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{2} - 58T_{7} + 1682 \) Copy content Toggle raw display
\( T_{11} - 118 \) 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
$7$ \( T^{2} - 58T + 1682 \) Copy content Toggle raw display
$11$ \( (T - 118)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 138T + 9522 \) Copy content Toggle raw display
$17$ \( T^{2} - 542T + 146882 \) Copy content Toggle raw display
$19$ \( T^{2} + 78400 \) Copy content Toggle raw display
$23$ \( T^{2} + 538T + 144722 \) Copy content Toggle raw display
$29$ \( T^{2} + 462400 \) Copy content Toggle raw display
$31$ \( (T - 202)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1302 T + 847602 \) Copy content Toggle raw display
$41$ \( (T + 1682)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 2178 T + 2371842 \) Copy content Toggle raw display
$47$ \( T^{2} + 2538 T + 3220722 \) Copy content Toggle raw display
$53$ \( T^{2} - 1222 T + 746642 \) Copy content Toggle raw display
$59$ \( T^{2} + 1345600 \) Copy content Toggle raw display
$61$ \( (T + 5598)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1502 T + 1128002 \) Copy content Toggle raw display
$71$ \( (T + 6442)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5902 T + 17416802 \) Copy content Toggle raw display
$79$ \( T^{2} + 111513600 \) Copy content Toggle raw display
$83$ \( T^{2} - 12462 T + 77650722 \) Copy content Toggle raw display
$89$ \( T^{2} + 209670400 \) Copy content Toggle raw display
$97$ \( T^{2} + 14622 T + 106901442 \) Copy content Toggle raw display
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