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Properties

Label 2205.4.a.bz
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.1163891200.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 245)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + ( - 2 \beta_{5} - 3 \beta_{4} + \cdots + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + (77 \beta_{5} + 84 \beta_{4} + \cdots - 63) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8} - 10 q^{10} + 16 q^{11} - 168 q^{13} + 298 q^{16} - 4 q^{17} - 308 q^{19} - 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} + 56 q^{26} - 176 q^{29} - 392 q^{31} + 770 q^{32} - 812 q^{34} - 140 q^{37} + 20 q^{38} - 330 q^{40} + 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} + 628 q^{47} + 50 q^{50} - 1520 q^{52} + 676 q^{53} - 80 q^{55} - 2012 q^{58} + 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} - 2940 q^{68} + 224 q^{71} - 2640 q^{73} - 928 q^{74} + 1340 q^{76} + 1636 q^{79} - 1490 q^{80} + 1756 q^{82} + 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 3332 q^{94} + 1540 q^{95} - 516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−3.05323
−0.241849
−2.05886
−1.04490
4.29508
4.10376
−4.46745 0 11.9581 −5.00000 0 0 −17.6824 0 22.3372
1.2 −1.65606 0 −5.25746 −5.00000 0 0 21.9552 0 8.28031
1.3 −0.644648 0 −7.58443 −5.00000 0 0 10.0465 0 3.22324
1.4 0.369315 0 −7.86361 −5.00000 0 0 −5.85867 0 −1.84657
1.5 2.88087 0 0.299392 −5.00000 0 0 −22.1844 0 −14.4043
1.6 5.51797 0 22.4480 −5.00000 0 0 79.7239 0 −27.5899
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bz 6
3.b odd 2 1 245.4.a.o ✓ 6
7.b odd 2 1 2205.4.a.ca 6
15.d odd 2 1 1225.4.a.bj 6
21.c even 2 1 245.4.a.p yes 6
21.g even 6 2 245.4.e.p 12
21.h odd 6 2 245.4.e.q 12
105.g even 2 1 1225.4.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o ✓ 6 3.b odd 2 1
245.4.a.p yes 6 21.c even 2 1
245.4.e.p 12 21.g even 6 2
245.4.e.q 12 21.h odd 6 2
1225.4.a.bi 6 105.g even 2 1
1225.4.a.bj 6 15.d odd 2 1
2205.4.a.bz 6 1.a even 1 1 trivial
2205.4.a.ca 6 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{6} - 2T_{2}^{5} - 29T_{2}^{4} + 28T_{2}^{3} + 134T_{2}^{2} + 24T_{2} - 28 \) Copy content Toggle raw display
\( T_{11}^{6} - 16T_{11}^{5} - 7174T_{11}^{4} + 121780T_{11}^{3} + 14188145T_{11}^{2} - 145959100T_{11} - 9225436100 \) Copy content Toggle raw display
\( T_{13}^{6} + 168T_{13}^{5} + 5774T_{13}^{4} - 335452T_{13}^{3} - 18677143T_{13}^{2} + 142488452T_{13} + 12513937372 \) Copy content Toggle raw display
\( T_{17}^{6} + 4T_{17}^{5} - 15392T_{17}^{4} - 258596T_{17}^{3} + 56343501T_{17}^{2} + 1617286496T_{17} - 3807091762 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots - 28 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 9225436100 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 12513937372 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 3807091762 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 1617325344 \) Copy content Toggle raw display
$23$ \( T^{6} - 336 T^{5} + \cdots + 298897696 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 544215793700 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 16539893268192 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 271258136464 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 144691772208184 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 440374360000 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 251564448569400 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 590408333736048 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 842000334839552 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 885207397312 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 85\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 510868966648482 \) Copy content Toggle raw display
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