Properties

Label 585.4.a.h
Level $585$
Weight $4$
Character orbit 585.a
Self dual yes
Analytic conductor $34.516$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,4,Mod(1,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

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: yes
Analytic conductor: \(34.5161173534\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + ( - 2 \beta + 30) q^{7} + (11 \beta - 4) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + ( - 2 \beta + 30) q^{7} + (11 \beta - 4) q^{8} - 5 \beta q^{10} + ( - 2 \beta + 44) q^{11} - 13 q^{13} + ( - 28 \beta + 8) q^{14} + ( - 15 \beta - 12) q^{16} + (32 \beta - 2) q^{17} + (22 \beta + 72) q^{19} + (5 \beta - 20) q^{20} + ( - 42 \beta + 8) q^{22} + (64 \beta - 62) q^{23} + 25 q^{25} + 13 \beta q^{26} + (36 \beta - 128) q^{28} + ( - 28 \beta - 46) q^{29} + ( - 126 \beta + 24) q^{31} + ( - 61 \beta + 92) q^{32} + ( - 30 \beta - 128) q^{34} + ( - 10 \beta + 150) q^{35} + (36 \beta + 162) q^{37} + ( - 94 \beta - 88) q^{38} + (55 \beta - 20) q^{40} + ( - 124 \beta + 98) q^{41} + ( - 40 \beta - 2) q^{43} + (50 \beta - 184) q^{44} + ( - 2 \beta - 256) q^{46} + ( - 62 \beta - 150) q^{47} + ( - 116 \beta + 573) q^{49} - 25 \beta q^{50} + ( - 13 \beta + 52) q^{52} + ( - 96 \beta + 246) q^{53} + ( - 10 \beta + 220) q^{55} + (316 \beta - 208) q^{56} + (74 \beta + 112) q^{58} + (174 \beta - 96) q^{59} + 442 q^{61} + (102 \beta + 504) q^{62} + (89 \beta + 340) q^{64} - 65 q^{65} + ( - 210 \beta + 766) q^{67} + ( - 98 \beta + 136) q^{68} + ( - 140 \beta + 40) q^{70} + ( - 314 \beta - 160) q^{71} + (336 \beta - 198) q^{73} + ( - 198 \beta - 144) q^{74} + (6 \beta - 200) q^{76} + ( - 144 \beta + 1336) q^{77} + (12 \beta - 96) q^{79} + ( - 75 \beta - 60) q^{80} + (26 \beta + 496) q^{82} + (218 \beta - 466) q^{83} + (160 \beta - 10) q^{85} + (42 \beta + 160) q^{86} + (470 \beta - 264) q^{88} + ( - 120 \beta + 486) q^{89} + (26 \beta - 390) q^{91} + ( - 254 \beta + 504) q^{92} + (212 \beta + 248) q^{94} + (110 \beta + 360) q^{95} + ( - 836 \beta + 434) q^{97} + ( - 457 \beta + 464) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 58 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 58 q^{7} + 3 q^{8} - 5 q^{10} + 86 q^{11} - 26 q^{13} - 12 q^{14} - 39 q^{16} + 28 q^{17} + 166 q^{19} - 35 q^{20} - 26 q^{22} - 60 q^{23} + 50 q^{25} + 13 q^{26} - 220 q^{28} - 120 q^{29} - 78 q^{31} + 123 q^{32} - 286 q^{34} + 290 q^{35} + 360 q^{37} - 270 q^{38} + 15 q^{40} + 72 q^{41} - 44 q^{43} - 318 q^{44} - 514 q^{46} - 362 q^{47} + 1030 q^{49} - 25 q^{50} + 91 q^{52} + 396 q^{53} + 430 q^{55} - 100 q^{56} + 298 q^{58} - 18 q^{59} + 884 q^{61} + 1110 q^{62} + 769 q^{64} - 130 q^{65} + 1322 q^{67} + 174 q^{68} - 60 q^{70} - 634 q^{71} - 60 q^{73} - 486 q^{74} - 394 q^{76} + 2528 q^{77} - 180 q^{79} - 195 q^{80} + 1018 q^{82} - 714 q^{83} + 140 q^{85} + 362 q^{86} - 58 q^{88} + 852 q^{89} - 754 q^{91} + 754 q^{92} + 708 q^{94} + 830 q^{95} + 32 q^{97} + 471 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
−2.56155 0 −1.43845 5.00000 0 24.8769 24.1771 0 −12.8078
1.2 1.56155 0 −5.56155 5.00000 0 33.1231 −21.1771 0 7.80776
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.4.a.h 2
3.b odd 2 1 65.4.a.c 2
12.b even 2 1 1040.4.a.k 2
15.d odd 2 1 325.4.a.g 2
15.e even 4 2 325.4.b.f 4
39.d odd 2 1 845.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.a.c 2 3.b odd 2 1
325.4.a.g 2 15.d odd 2 1
325.4.b.f 4 15.e even 4 2
585.4.a.h 2 1.a even 1 1 trivial
845.4.a.d 2 39.d odd 2 1
1040.4.a.k 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 58T_{7} + 824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 58T + 824 \) Copy content Toggle raw display
$11$ \( T^{2} - 86T + 1832 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 28T - 4156 \) Copy content Toggle raw display
$19$ \( T^{2} - 166T + 4832 \) Copy content Toggle raw display
$23$ \( T^{2} + 60T - 16508 \) Copy content Toggle raw display
$29$ \( T^{2} + 120T + 268 \) Copy content Toggle raw display
$31$ \( T^{2} + 78T - 65952 \) Copy content Toggle raw display
$37$ \( T^{2} - 360T + 26892 \) Copy content Toggle raw display
$41$ \( T^{2} - 72T - 64052 \) Copy content Toggle raw display
$43$ \( T^{2} + 44T - 6316 \) Copy content Toggle raw display
$47$ \( T^{2} + 362T + 16424 \) Copy content Toggle raw display
$53$ \( T^{2} - 396T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T - 128592 \) Copy content Toggle raw display
$61$ \( (T - 442)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 1322 T + 249496 \) Copy content Toggle raw display
$71$ \( T^{2} + 634T - 318544 \) Copy content Toggle raw display
$73$ \( T^{2} + 60T - 478908 \) Copy content Toggle raw display
$79$ \( T^{2} + 180T + 7488 \) Copy content Toggle raw display
$83$ \( T^{2} + 714T - 74528 \) Copy content Toggle raw display
$89$ \( T^{2} - 852T + 120276 \) Copy content Toggle raw display
$97$ \( T^{2} - 32T - 2970052 \) Copy content Toggle raw display
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