Properties

Label 585.4.c.b
Level $585$
Weight $4$
Character orbit 585.c
Analytic conductor $34.516$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,4,Mod(469,585)]
 
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, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.469");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 585.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: \(34.5161173534\)
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 65)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} - q^{4} + ( - 11 i - 2) q^{5} + 28 i q^{7} + 21 i q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} - q^{4} + ( - 11 i - 2) q^{5} + 28 i q^{7} + 21 i q^{8} + ( - 6 i + 33) q^{10} - 2 q^{11} - 13 i q^{13} - 84 q^{14} - 71 q^{16} + 44 i q^{17} + 94 q^{19} + (11 i + 2) q^{20} - 6 i q^{22} + 18 i q^{23} + (44 i - 117) q^{25} + 39 q^{26} - 28 i q^{28} + 118 q^{29} - 100 q^{31} - 45 i q^{32} - 132 q^{34} + ( - 56 i + 308) q^{35} - 126 i q^{37} + 282 i q^{38} + ( - 42 i + 231) q^{40} - 474 q^{41} - 200 i q^{43} + 2 q^{44} - 54 q^{46} + 448 i q^{47} - 441 q^{49} + ( - 351 i - 132) q^{50} + 13 i q^{52} + 754 i q^{53} + (22 i + 4) q^{55} - 588 q^{56} + 354 i q^{58} - 446 q^{59} - 638 q^{61} - 300 i q^{62} - 433 q^{64} + (26 i - 143) q^{65} + 868 i q^{67} - 44 i q^{68} + (924 i + 168) q^{70} - 536 q^{71} - 58 i q^{73} + 378 q^{74} - 94 q^{76} - 56 i q^{77} - 232 q^{79} + (781 i + 142) q^{80} - 1422 i q^{82} + 108 i q^{83} + ( - 88 i + 484) q^{85} + 600 q^{86} - 42 i q^{88} + 1038 q^{89} + 364 q^{91} - 18 i q^{92} - 1344 q^{94} + ( - 1034 i - 188) q^{95} + 774 i q^{97} - 1323 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{5} + 66 q^{10} - 4 q^{11} - 168 q^{14} - 142 q^{16} + 188 q^{19} + 4 q^{20} - 234 q^{25} + 78 q^{26} + 236 q^{29} - 200 q^{31} - 264 q^{34} + 616 q^{35} + 462 q^{40} - 948 q^{41} + 4 q^{44} - 108 q^{46} - 882 q^{49} - 264 q^{50} + 8 q^{55} - 1176 q^{56} - 892 q^{59} - 1276 q^{61} - 866 q^{64} - 286 q^{65} + 336 q^{70} - 1072 q^{71} + 756 q^{74} - 188 q^{76} - 464 q^{79} + 284 q^{80} + 968 q^{85} + 1200 q^{86} + 2076 q^{89} + 728 q^{91} - 2688 q^{94} - 376 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
1.00000i
1.00000i
3.00000i 0 −1.00000 −2.00000 + 11.0000i 0 28.0000i 21.0000i 0 33.0000 + 6.00000i
469.2 3.00000i 0 −1.00000 −2.00000 11.0000i 0 28.0000i 21.0000i 0 33.0000 6.00000i
\(n\): 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 585.4.c.b 2
3.b odd 2 1 65.4.b.a 2
5.b even 2 1 inner 585.4.c.b 2
15.d odd 2 1 65.4.b.a 2
15.e even 4 1 325.4.a.b 1
15.e even 4 1 325.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.b.a 2 3.b odd 2 1
65.4.b.a 2 15.d odd 2 1
325.4.a.b 1 15.e even 4 1
325.4.a.c 1 15.e even 4 1
585.4.c.b 2 1.a even 1 1 trivial
585.4.c.b 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 784 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{2} + 1936 \) Copy content Toggle raw display
$19$ \( (T - 94)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 324 \) Copy content Toggle raw display
$29$ \( (T - 118)^{2} \) Copy content Toggle raw display
$31$ \( (T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 15876 \) Copy content Toggle raw display
$41$ \( (T + 474)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 40000 \) Copy content Toggle raw display
$47$ \( T^{2} + 200704 \) Copy content Toggle raw display
$53$ \( T^{2} + 568516 \) Copy content Toggle raw display
$59$ \( (T + 446)^{2} \) Copy content Toggle raw display
$61$ \( (T + 638)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 753424 \) Copy content Toggle raw display
$71$ \( (T + 536)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3364 \) Copy content Toggle raw display
$79$ \( (T + 232)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11664 \) Copy content Toggle raw display
$89$ \( (T - 1038)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 599076 \) Copy content Toggle raw display
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