Properties

Label 2205.4.a.bg
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{2} + (6 \beta + 3) q^{4} + 5 q^{5} + (13 \beta - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 3) q^{2} + (6 \beta + 3) q^{4} + 5 q^{5} + (13 \beta - 3) q^{8} + (5 \beta + 15) q^{10} + (10 \beta - 14) q^{11} + (10 \beta + 18) q^{13} + ( - 12 \beta - 7) q^{16} + ( - 54 \beta - 38) q^{17} + (26 \beta - 80) q^{19} + (30 \beta + 15) q^{20} + (16 \beta - 22) q^{22} + ( - 117 \beta + 11) q^{23} + 25 q^{25} + (48 \beta + 74) q^{26} + ( - 60 \beta + 125) q^{29} + ( - 100 \beta + 66) q^{31} + ( - 147 \beta - 21) q^{32} + ( - 200 \beta - 222) q^{34} + ( - 136 \beta - 208) q^{37} + ( - 2 \beta - 188) q^{38} + (65 \beta - 15) q^{40} + ( - 30 \beta - 53) q^{41} + ( - 5 \beta - 333) q^{43} + ( - 54 \beta + 78) q^{44} + ( - 340 \beta - 201) q^{46} + (64 \beta - 98) q^{47} + (25 \beta + 75) q^{50} + (138 \beta + 174) q^{52} + (142 \beta + 476) q^{53} + (50 \beta - 70) q^{55} + ( - 55 \beta + 255) q^{58} + (266 \beta + 420) q^{59} + (570 \beta - 49) q^{61} + ( - 234 \beta - 2) q^{62} + ( - 366 \beta - 301) q^{64} + (50 \beta + 90) q^{65} + ( - 69 \beta - 643) q^{67} + ( - 390 \beta - 762) q^{68} + (350 \beta - 532) q^{71} + ( - 118 \beta + 86) q^{73} + ( - 616 \beta - 896) q^{74} + ( - 402 \beta + 72) q^{76} + (214 \beta - 620) q^{79} + ( - 60 \beta - 35) q^{80} + ( - 143 \beta - 219) q^{82} + (5 \beta - 953) q^{83} + ( - 270 \beta - 190) q^{85} + ( - 348 \beta - 1009) q^{86} + ( - 212 \beta + 302) q^{88} + (324 \beta + 325) q^{89} + ( - 285 \beta - 1371) q^{92} + (94 \beta - 166) q^{94} + (130 \beta - 400) q^{95} + ( - 500 \beta + 314) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 6 q^{4} + 10 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 6 q^{4} + 10 q^{5} - 6 q^{8} + 30 q^{10} - 28 q^{11} + 36 q^{13} - 14 q^{16} - 76 q^{17} - 160 q^{19} + 30 q^{20} - 44 q^{22} + 22 q^{23} + 50 q^{25} + 148 q^{26} + 250 q^{29} + 132 q^{31} - 42 q^{32} - 444 q^{34} - 416 q^{37} - 376 q^{38} - 30 q^{40} - 106 q^{41} - 666 q^{43} + 156 q^{44} - 402 q^{46} - 196 q^{47} + 150 q^{50} + 348 q^{52} + 952 q^{53} - 140 q^{55} + 510 q^{58} + 840 q^{59} - 98 q^{61} - 4 q^{62} - 602 q^{64} + 180 q^{65} - 1286 q^{67} - 1524 q^{68} - 1064 q^{71} + 172 q^{73} - 1792 q^{74} + 144 q^{76} - 1240 q^{79} - 70 q^{80} - 438 q^{82} - 1906 q^{83} - 380 q^{85} - 2018 q^{86} + 604 q^{88} + 650 q^{89} - 2742 q^{92} - 332 q^{94} - 800 q^{95} + 628 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
1.58579 0 −5.48528 5.00000 0 0 −21.3848 0 7.92893
1.2 4.41421 0 11.4853 5.00000 0 0 15.3848 0 22.0711
\(n\): 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.bg 2
3.b odd 2 1 245.4.a.h 2
7.b odd 2 1 2205.4.a.bf 2
7.d odd 6 2 315.4.j.c 4
15.d odd 2 1 1225.4.a.v 2
21.c even 2 1 245.4.a.g 2
21.g even 6 2 35.4.e.b 4
21.h odd 6 2 245.4.e.l 4
84.j odd 6 2 560.4.q.i 4
105.g even 2 1 1225.4.a.x 2
105.p even 6 2 175.4.e.c 4
105.w odd 12 4 175.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 21.g even 6 2
175.4.e.c 4 105.p even 6 2
175.4.k.c 8 105.w odd 12 4
245.4.a.g 2 21.c even 2 1
245.4.a.h 2 3.b odd 2 1
245.4.e.l 4 21.h odd 6 2
315.4.j.c 4 7.d odd 6 2
560.4.q.i 4 84.j odd 6 2
1225.4.a.v 2 15.d odd 2 1
1225.4.a.x 2 105.g even 2 1
2205.4.a.bf 2 7.b odd 2 1
2205.4.a.bg 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} - 6T_{2} + 7 \) Copy content Toggle raw display
\( T_{11}^{2} + 28T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 36T_{13} + 124 \) Copy content Toggle raw display
\( T_{17}^{2} + 76T_{17} - 4388 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 28T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 36T + 124 \) Copy content Toggle raw display
$17$ \( T^{2} + 76T - 4388 \) Copy content Toggle raw display
$19$ \( T^{2} + 160T + 5048 \) Copy content Toggle raw display
$23$ \( T^{2} - 22T - 27257 \) Copy content Toggle raw display
$29$ \( T^{2} - 250T + 8425 \) Copy content Toggle raw display
$31$ \( T^{2} - 132T - 15644 \) Copy content Toggle raw display
$37$ \( T^{2} + 416T + 6272 \) Copy content Toggle raw display
$41$ \( T^{2} + 106T + 1009 \) Copy content Toggle raw display
$43$ \( T^{2} + 666T + 110839 \) Copy content Toggle raw display
$47$ \( T^{2} + 196T + 1412 \) Copy content Toggle raw display
$53$ \( T^{2} - 952T + 186248 \) Copy content Toggle raw display
$59$ \( T^{2} - 840T + 34888 \) Copy content Toggle raw display
$61$ \( T^{2} + 98T - 647399 \) Copy content Toggle raw display
$67$ \( T^{2} + 1286 T + 403927 \) Copy content Toggle raw display
$71$ \( T^{2} + 1064T + 38024 \) Copy content Toggle raw display
$73$ \( T^{2} - 172T - 20452 \) Copy content Toggle raw display
$79$ \( T^{2} + 1240 T + 292808 \) Copy content Toggle raw display
$83$ \( T^{2} + 1906 T + 908159 \) Copy content Toggle raw display
$89$ \( T^{2} - 650T - 104327 \) Copy content Toggle raw display
$97$ \( T^{2} - 628T - 401404 \) Copy content Toggle raw display
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