Properties

Label 220.6.a.a
Level $220$
Weight $6$
Character orbit 220.a
Self dual yes
Analytic conductor $35.284$
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] = [220,6,Mod(1,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 220.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: \(35.2844403589\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1761}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{1761})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 9) q^{3} + 25 q^{5} + ( - 5 \beta - 63) q^{7} + (19 \beta + 278) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 9) q^{3} + 25 q^{5} + ( - 5 \beta - 63) q^{7} + (19 \beta + 278) q^{9} - 121 q^{11} + (18 \beta - 814) q^{13} + ( - 25 \beta - 225) q^{15} + (97 \beta - 209) q^{17} + ( - 7 \beta + 1723) q^{19} + (113 \beta + 2767) q^{21} + (92 \beta + 2222) q^{23} + 625 q^{25} + ( - 225 \beta - 8675) q^{27} + (139 \beta - 2585) q^{29} + ( - 113 \beta - 2175) q^{31} + (121 \beta + 1089) q^{33} + ( - 125 \beta - 1575) q^{35} + ( - 341 \beta - 1041) q^{37} + (634 \beta - 594) q^{39} + (248 \beta + 9710) q^{41} + ( - 300 \beta - 7300) q^{43} + (475 \beta + 6950) q^{45} + ( - 712 \beta - 11170) q^{47} + (655 \beta - 1838) q^{49} + ( - 761 \beta - 40799) q^{51} + ( - 441 \beta - 3381) q^{53} - 3025 q^{55} + ( - 1653 \beta - 12427) q^{57} + (562 \beta + 25138) q^{59} + (581 \beta - 22379) q^{61} + ( - 2682 \beta - 59314) q^{63} + (450 \beta - 20350) q^{65} + (222 \beta - 5416) q^{67} + ( - 3142 \beta - 60478) q^{69} + (921 \beta - 1005) q^{71} + (254 \beta - 46454) q^{73} + ( - 625 \beta - 5625) q^{75} + (605 \beta + 7623) q^{77} + ( - 2818 \beta - 6758) q^{79} + (6308 \beta + 109521) q^{81} + (2518 \beta + 42946) q^{83} + (2425 \beta - 5225) q^{85} + (1195 \beta - 37895) q^{87} + ( - 499 \beta - 40159) q^{89} + (2846 \beta + 11682) q^{91} + (3305 \beta + 69295) q^{93} + ( - 175 \beta + 43075) q^{95} + (118 \beta + 51288) q^{97} + ( - 2299 \beta - 33638) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 19 q^{3} + 50 q^{5} - 131 q^{7} + 575 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 19 q^{3} + 50 q^{5} - 131 q^{7} + 575 q^{9} - 242 q^{11} - 1610 q^{13} - 475 q^{15} - 321 q^{17} + 3439 q^{19} + 5647 q^{21} + 4536 q^{23} + 1250 q^{25} - 17575 q^{27} - 5031 q^{29} - 4463 q^{31} + 2299 q^{33} - 3275 q^{35} - 2423 q^{37} - 554 q^{39} + 19668 q^{41} - 14900 q^{43} + 14375 q^{45} - 23052 q^{47} - 3021 q^{49} - 82359 q^{51} - 7203 q^{53} - 6050 q^{55} - 26507 q^{57} + 50838 q^{59} - 44177 q^{61} - 121310 q^{63} - 40250 q^{65} - 10610 q^{67} - 124098 q^{69} - 1089 q^{71} - 92654 q^{73} - 11875 q^{75} + 15851 q^{77} - 16334 q^{79} + 225350 q^{81} + 88410 q^{83} - 8025 q^{85} - 74595 q^{87} - 80817 q^{89} + 26210 q^{91} + 141895 q^{93} + 85975 q^{95} + 102694 q^{97} - 69575 q^{99}+O(q^{100}) \) 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
21.4821
−20.4821
0 −30.4821 0 25.0000 0 −170.411 0 686.161 0
1.2 0 11.4821 0 25.0000 0 39.4107 0 −111.161 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)

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 220.6.a.a 2
4.b odd 2 1 880.6.a.h 2
5.b even 2 1 1100.6.a.c 2
5.c odd 4 2 1100.6.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.6.a.a 2 1.a even 1 1 trivial
880.6.a.h 2 4.b odd 2 1
1100.6.a.c 2 5.b even 2 1
1100.6.b.b 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 19T_{3} - 350 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(220))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 19T - 350 \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 131T - 6716 \) Copy content Toggle raw display
$11$ \( (T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1610 T + 505384 \) Copy content Toggle raw display
$17$ \( T^{2} + 321 T - 4116552 \) Copy content Toggle raw display
$19$ \( T^{2} - 3439 T + 2935108 \) Copy content Toggle raw display
$23$ \( T^{2} - 4536 T + 1417548 \) Copy content Toggle raw display
$29$ \( T^{2} + 5031 T - 2178330 \) Copy content Toggle raw display
$31$ \( T^{2} + 4463 T - 641960 \) Copy content Toggle raw display
$37$ \( T^{2} + 2423 T - 49724978 \) Copy content Toggle raw display
$41$ \( T^{2} - 19668 T + 69630420 \) Copy content Toggle raw display
$43$ \( T^{2} + 14900 T + 15880000 \) Copy content Toggle raw display
$47$ \( T^{2} + 23052 T - 90333420 \) Copy content Toggle raw display
$53$ \( T^{2} + 7203 T - 72649458 \) Copy content Toggle raw display
$59$ \( T^{2} - 50838 T + 507075240 \) Copy content Toggle raw display
$61$ \( T^{2} + 44177 T + 339290602 \) Copy content Toggle raw display
$67$ \( T^{2} + 10610 T + 6445744 \) Copy content Toggle raw display
$71$ \( T^{2} + 1089 T - 373141620 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 2117787760 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 3429379952 \) Copy content Toggle raw display
$83$ \( T^{2} - 88410 T - 837245616 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1523224182 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 2630384368 \) Copy content Toggle raw display
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