Properties

Label 90.4.c.a
Level $90$
Weight $4$
Character orbit 90.c
Analytic conductor $5.310$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,4,Mod(19,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\), degree \(1\), 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: \(5.31017190052\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + (11 i - 2) q^{5} + 2 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + (11 i - 2) q^{5} + 2 i q^{7} - 8 i q^{8} + ( - 4 i - 22) q^{10} - 70 q^{11} + 54 i q^{13} - 4 q^{14} + 16 q^{16} - 22 i q^{17} - 24 q^{19} + ( - 44 i + 8) q^{20} - 140 i q^{22} + 100 i q^{23} + ( - 44 i - 117) q^{25} - 108 q^{26} - 8 i q^{28} + 216 q^{29} + 208 q^{31} + 32 i q^{32} + 44 q^{34} + ( - 4 i - 22) q^{35} + 254 i q^{37} - 48 i q^{38} + (16 i + 88) q^{40} + 206 q^{41} + 292 i q^{43} + 280 q^{44} - 200 q^{46} - 320 i q^{47} + 339 q^{49} + ( - 234 i + 88) q^{50} - 216 i q^{52} + 402 i q^{53} + ( - 770 i + 140) q^{55} + 16 q^{56} + 432 i q^{58} - 370 q^{59} - 550 q^{61} + 416 i q^{62} - 64 q^{64} + ( - 108 i - 594) q^{65} - 728 i q^{67} + 88 i q^{68} + ( - 44 i + 8) q^{70} + 540 q^{71} + 604 i q^{73} - 508 q^{74} + 96 q^{76} - 140 i q^{77} - 792 q^{79} + (176 i - 32) q^{80} + 412 i q^{82} - 404 i q^{83} + (44 i + 242) q^{85} - 584 q^{86} + 560 i q^{88} - 938 q^{89} - 108 q^{91} - 400 i q^{92} + 640 q^{94} + ( - 264 i + 48) q^{95} - 56 i q^{97} + 678 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 4 q^{5} - 44 q^{10} - 140 q^{11} - 8 q^{14} + 32 q^{16} - 48 q^{19} + 16 q^{20} - 234 q^{25} - 216 q^{26} + 432 q^{29} + 416 q^{31} + 88 q^{34} - 44 q^{35} + 176 q^{40} + 412 q^{41} + 560 q^{44} - 400 q^{46} + 678 q^{49} + 176 q^{50} + 280 q^{55} + 32 q^{56} - 740 q^{59} - 1100 q^{61} - 128 q^{64} - 1188 q^{65} + 16 q^{70} + 1080 q^{71} - 1016 q^{74} + 192 q^{76} - 1584 q^{79} - 64 q^{80} + 484 q^{85} - 1168 q^{86} - 1876 q^{89} - 216 q^{91} + 1280 q^{94} + 96 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.00000i
1.00000i
2.00000i 0 −4.00000 −2.00000 11.0000i 0 2.00000i 8.00000i 0 −22.0000 + 4.00000i
19.2 2.00000i 0 −4.00000 −2.00000 + 11.0000i 0 2.00000i 8.00000i 0 −22.0000 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.4.c.a 2
3.b odd 2 1 30.4.c.a 2
4.b odd 2 1 720.4.f.c 2
5.b even 2 1 inner 90.4.c.a 2
5.c odd 4 1 450.4.a.e 1
5.c odd 4 1 450.4.a.p 1
12.b even 2 1 240.4.f.d 2
15.d odd 2 1 30.4.c.a 2
15.e even 4 1 150.4.a.d 1
15.e even 4 1 150.4.a.f 1
20.d odd 2 1 720.4.f.c 2
24.f even 2 1 960.4.f.d 2
24.h odd 2 1 960.4.f.c 2
60.h even 2 1 240.4.f.d 2
60.l odd 4 1 1200.4.a.h 1
60.l odd 4 1 1200.4.a.bc 1
120.i odd 2 1 960.4.f.c 2
120.m even 2 1 960.4.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.c.a 2 3.b odd 2 1
30.4.c.a 2 15.d odd 2 1
90.4.c.a 2 1.a even 1 1 trivial
90.4.c.a 2 5.b even 2 1 inner
150.4.a.d 1 15.e even 4 1
150.4.a.f 1 15.e even 4 1
240.4.f.d 2 12.b even 2 1
240.4.f.d 2 60.h even 2 1
450.4.a.e 1 5.c odd 4 1
450.4.a.p 1 5.c odd 4 1
720.4.f.c 2 4.b odd 2 1
720.4.f.c 2 20.d odd 2 1
960.4.f.c 2 24.h odd 2 1
960.4.f.c 2 120.i odd 2 1
960.4.f.d 2 24.f even 2 1
960.4.f.d 2 120.m even 2 1
1200.4.a.h 1 60.l odd 4 1
1200.4.a.bc 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 70)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2916 \) Copy content Toggle raw display
$17$ \( T^{2} + 484 \) Copy content Toggle raw display
$19$ \( (T + 24)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10000 \) Copy content Toggle raw display
$29$ \( (T - 216)^{2} \) Copy content Toggle raw display
$31$ \( (T - 208)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T - 206)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 85264 \) Copy content Toggle raw display
$47$ \( T^{2} + 102400 \) Copy content Toggle raw display
$53$ \( T^{2} + 161604 \) Copy content Toggle raw display
$59$ \( (T + 370)^{2} \) Copy content Toggle raw display
$61$ \( (T + 550)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 529984 \) Copy content Toggle raw display
$71$ \( (T - 540)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 364816 \) Copy content Toggle raw display
$79$ \( (T + 792)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 163216 \) Copy content Toggle raw display
$89$ \( (T + 938)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3136 \) Copy content Toggle raw display
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