Properties

Label 2205.4.a.bi
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $3$
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] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, 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("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.4892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{4} - 5 q^{5} + ( - 3 \beta_{2} - 5 \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{4} - 5 q^{5} + ( - 3 \beta_{2} - 5 \beta_1 - 2) q^{8} + ( - 5 \beta_1 + 5) q^{10} + (\beta_{2} + 7 \beta_1) q^{11} + ( - 4 \beta_{2} + 10 \beta_1 - 25) q^{13} + ( - 7 \beta_{2} - 3 \beta_1 - 24) q^{16} + (6 \beta_{2} - 4 \beta_1 + 22) q^{17} + ( - 8 \beta_{2} - 4 \beta_1 - 7) q^{19} + ( - 5 \beta_{2} + 5 \beta_1) q^{20} + (5 \beta_{2} + 3 \beta_1 + 46) q^{22} + (9 \beta_{2} + 19 \beta_1 + 18) q^{23} + 25 q^{25} + (18 \beta_{2} - 37 \beta_1 + 107) q^{26} + ( - 10 \beta_{2} - 36 \beta_1 - 70) q^{29} + (37 \beta_{2} + 57 \beta_1 + 33) q^{31} + (35 \beta_{2} - 5 \beta_1 + 40) q^{32} + ( - 16 \beta_{2} + 40 \beta_1 - 68) q^{34} + (52 \beta_{2} - 18 \beta_1 - 3) q^{37} + (12 \beta_{2} - 31 \beta_1 + 3) q^{38} + (15 \beta_{2} + 25 \beta_1 + 10) q^{40} + (35 \beta_{2} - 11 \beta_1 + 188) q^{41} + ( - 67 \beta_{2} + 11 \beta_1 + 79) q^{43} + ( - 15 \beta_{2} + 5 \beta_1 - 40) q^{44} + (\beta_{2} + 45 \beta_1 + 88) q^{46} + ( - 75 \beta_{2} - 87 \beta_1 + 26) q^{47} + (25 \beta_1 - 25) q^{50} + ( - 41 \beta_{2} + 81 \beta_1 - 220) q^{52} + ( - 19 \beta_{2} + 41 \beta_1 - 104) q^{53} + ( - 5 \beta_{2} - 35 \beta_1) q^{55} + ( - 16 \beta_{2} - 100 \beta_1 - 152) q^{58} + ( - 34 \beta_{2} - 144 \beta_1 + 518) q^{59} + ( - 26 \beta_{2} + 18 \beta_1 - 378) q^{61} + ( - 17 \beta_{2} + 144 \beta_1 + 255) q^{62} + ( - 19 \beta_{2} + 169 \beta_1 + 12) q^{64} + (20 \beta_{2} - 50 \beta_1 + 125) q^{65} + ( - 25 \beta_{2} + 101 \beta_1 + 277) q^{67} + (24 \beta_{2} - 84 \beta_1 + 220) q^{68} + ( - 14 \beta_{2} - 264 \beta_1 + 140) q^{71} + (23 \beta_{2} - 83 \beta_1 - 405) q^{73} + ( - 122 \beta_{2} + 153 \beta_1 - 279) q^{74} + (9 \beta_{2} + 71 \beta_1 - 200) q^{76} + ( - 103 \beta_{2} - 79 \beta_1 - 829) q^{79} + (35 \beta_{2} + 15 \beta_1 + 120) q^{80} + ( - 81 \beta_{2} + 293 \beta_1 - 370) q^{82} + ( - 44 \beta_{2} - 102 \beta_1 + 184) q^{83} + ( - 30 \beta_{2} + 20 \beta_1 - 110) q^{85} + (145 \beta_{2} - 122 \beta_1 + 199) q^{86} + ( - 5 \beta_{2} - 109 \beta_1 - 248) q^{88} + (60 \beta_{2} + 386 \beta_1 - 34) q^{89} + ( - 29 \beta_{2} - 61 \beta_1 + 80) q^{92} + (63 \beta_{2} - 199 \beta_1 - 410) q^{94} + (40 \beta_{2} + 20 \beta_1 + 35) q^{95} + (174 \beta_{2} - 242 \beta_1 - 398) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + q^{4} - 15 q^{5} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + q^{4} - 15 q^{5} - 9 q^{8} + 15 q^{10} + q^{11} - 79 q^{13} - 79 q^{16} + 72 q^{17} - 29 q^{19} - 5 q^{20} + 143 q^{22} + 63 q^{23} + 75 q^{25} + 339 q^{26} - 220 q^{29} + 136 q^{31} + 155 q^{32} - 220 q^{34} + 43 q^{37} + 21 q^{38} + 45 q^{40} + 599 q^{41} + 170 q^{43} - 135 q^{44} + 265 q^{46} + 3 q^{47} - 75 q^{50} - 701 q^{52} - 331 q^{53} - 5 q^{55} - 472 q^{58} + 1520 q^{59} - 1160 q^{61} + 748 q^{62} + 17 q^{64} + 395 q^{65} + 806 q^{67} + 684 q^{68} + 406 q^{71} - 1192 q^{73} - 959 q^{74} - 591 q^{76} - 2590 q^{79} + 395 q^{80} - 1191 q^{82} + 508 q^{83} - 360 q^{85} + 742 q^{86} - 749 q^{88} - 42 q^{89} + 211 q^{92} - 1167 q^{94} + 145 q^{95} - 1020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 11x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 7 \) 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
−3.11718
−0.368173
3.48535
−4.11718 0 8.95114 −5.00000 0 0 −3.91601 0 20.5859
1.2 −1.36817 0 −6.12810 −5.00000 0 0 19.3297 0 6.84087
1.3 2.48535 0 −1.82304 −5.00000 0 0 −24.4137 0 −12.4267
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 2205.4.a.bi 3
3.b odd 2 1 735.4.a.r 3
7.b odd 2 1 2205.4.a.bj 3
7.c even 3 2 315.4.j.e 6
21.c even 2 1 735.4.a.s 3
21.h odd 6 2 105.4.i.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.c 6 21.h odd 6 2
315.4.j.e 6 7.c even 3 2
735.4.a.r 3 3.b odd 2 1
735.4.a.s 3 21.c even 2 1
2205.4.a.bi 3 1.a even 1 1 trivial
2205.4.a.bj 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{3} + 3T_{2}^{2} - 8T_{2} - 14 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 508T_{11} - 3780 \) Copy content Toggle raw display
\( T_{13}^{3} + 79T_{13}^{2} - 49T_{13} - 687 \) Copy content Toggle raw display
\( T_{17}^{3} - 72T_{17}^{2} - 104T_{17} + 19424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3 T^{2} - 8 T - 14 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 508 T - 3780 \) Copy content Toggle raw display
$13$ \( T^{3} + 79 T^{2} - 49 T - 687 \) Copy content Toggle raw display
$17$ \( T^{3} - 72 T^{2} - 104 T + 19424 \) Copy content Toggle raw display
$19$ \( T^{3} + 29 T^{2} - 2093 T - 65521 \) Copy content Toggle raw display
$23$ \( T^{3} - 63 T^{2} - 4124 T + 53096 \) Copy content Toggle raw display
$29$ \( T^{3} + 220 T^{2} + 1544 T - 28096 \) Copy content Toggle raw display
$31$ \( T^{3} - 136 T^{2} - 62331 T + 4764978 \) Copy content Toggle raw display
$37$ \( T^{3} - 43 T^{2} - 118665 T + 2466387 \) Copy content Toggle raw display
$41$ \( T^{3} - 599 T^{2} + 66236 T + 3130740 \) Copy content Toggle raw display
$43$ \( T^{3} - 170 T^{2} - 175635 T + 1053316 \) Copy content Toggle raw display
$47$ \( T^{3} - 3 T^{2} - 239256 T - 30756380 \) Copy content Toggle raw display
$53$ \( T^{3} + 331 T^{2} - 3960 T + 10864 \) Copy content Toggle raw display
$59$ \( T^{3} - 1520 T^{2} + \cdots + 24592160 \) Copy content Toggle raw display
$61$ \( T^{3} + 1160 T^{2} + \cdots + 45301392 \) Copy content Toggle raw display
$67$ \( T^{3} - 806 T^{2} + \cdots + 43402660 \) Copy content Toggle raw display
$71$ \( T^{3} - 406 T^{2} + \cdots + 232329576 \) Copy content Toggle raw display
$73$ \( T^{3} + 1192 T^{2} + \cdots + 4156150 \) Copy content Toggle raw display
$79$ \( T^{3} + 2590 T^{2} + \cdots + 197479612 \) Copy content Toggle raw display
$83$ \( T^{3} - 508 T^{2} + \cdots + 30444208 \) Copy content Toggle raw display
$89$ \( T^{3} + 42 T^{2} + \cdots - 708440784 \) Copy content Toggle raw display
$97$ \( T^{3} + 1020 T^{2} + \cdots - 1879560768 \) Copy content Toggle raw display
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