Properties

Label 1305.4.a.e
Level $1305$
Weight $4$
Character orbit 1305.a
Self dual yes
Analytic conductor $76.997$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,4,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(76.9974925575\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{34}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 435)
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 \(\beta = 4\sqrt{34}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{4} + 5 q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{4} + 5 q^{5} + \beta q^{7} + (2 \beta + 26) q^{11} + (2 \beta - 2) q^{13} + 64 q^{16} + (2 \beta - 28) q^{17} + (4 \beta - 18) q^{19} - 40 q^{20} + (3 \beta - 24) q^{23} + 25 q^{25} - 8 \beta q^{28} - 29 q^{29} + 74 q^{31} + 5 \beta q^{35} + ( - 4 \beta + 280) q^{37} + ( - 4 \beta - 430) q^{41} + (6 \beta - 40) q^{43} + ( - 16 \beta - 208) q^{44} + (6 \beta + 24) q^{47} + 201 q^{49} + ( - 16 \beta + 16) q^{52} + ( - 4 \beta - 394) q^{53} + (10 \beta + 130) q^{55} + ( - 6 \beta - 252) q^{59} + (20 \beta + 382) q^{61} - 512 q^{64} + (10 \beta - 10) q^{65} + ( - 21 \beta + 188) q^{67} + ( - 16 \beta + 224) q^{68} - 444 q^{71} + (26 \beta - 440) q^{73} + ( - 32 \beta + 144) q^{76} + (26 \beta + 1088) q^{77} + ( - 4 \beta - 86) q^{79} + 320 q^{80} + ( - 21 \beta + 156) q^{83} + (10 \beta - 140) q^{85} + (16 \beta + 658) q^{89} + ( - 2 \beta + 1088) q^{91} + ( - 24 \beta + 192) q^{92} + (20 \beta - 90) q^{95} + (6 \beta + 1244) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} + 10 q^{5} + 52 q^{11} - 4 q^{13} + 128 q^{16} - 56 q^{17} - 36 q^{19} - 80 q^{20} - 48 q^{23} + 50 q^{25} - 58 q^{29} + 148 q^{31} + 560 q^{37} - 860 q^{41} - 80 q^{43} - 416 q^{44} + 48 q^{47} + 402 q^{49} + 32 q^{52} - 788 q^{53} + 260 q^{55} - 504 q^{59} + 764 q^{61} - 1024 q^{64} - 20 q^{65} + 376 q^{67} + 448 q^{68} - 888 q^{71} - 880 q^{73} + 288 q^{76} + 2176 q^{77} - 172 q^{79} + 640 q^{80} + 312 q^{83} - 280 q^{85} + 1316 q^{89} + 2176 q^{91} + 384 q^{92} - 180 q^{95} + 2488 q^{97}+O(q^{100}) \) 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
−5.83095
5.83095
0 0 −8.00000 5.00000 0 −23.3238 0 0 0
1.2 0 0 −8.00000 5.00000 0 23.3238 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(29\) \( +1 \)

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 1305.4.a.e 2
3.b odd 2 1 435.4.a.e 2
15.d odd 2 1 2175.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.e 2 3.b odd 2 1
1305.4.a.e 2 1.a even 1 1 trivial
2175.4.a.d 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1305))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 544 \) Copy content Toggle raw display
$11$ \( T^{2} - 52T - 1500 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 2172 \) Copy content Toggle raw display
$17$ \( T^{2} + 56T - 1392 \) Copy content Toggle raw display
$19$ \( T^{2} + 36T - 8380 \) Copy content Toggle raw display
$23$ \( T^{2} + 48T - 4320 \) Copy content Toggle raw display
$29$ \( (T + 29)^{2} \) Copy content Toggle raw display
$31$ \( (T - 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 560T + 69696 \) Copy content Toggle raw display
$41$ \( T^{2} + 860T + 176196 \) Copy content Toggle raw display
$43$ \( T^{2} + 80T - 17984 \) Copy content Toggle raw display
$47$ \( T^{2} - 48T - 19008 \) Copy content Toggle raw display
$53$ \( T^{2} + 788T + 146532 \) Copy content Toggle raw display
$59$ \( T^{2} + 504T + 43920 \) Copy content Toggle raw display
$61$ \( T^{2} - 764T - 71676 \) Copy content Toggle raw display
$67$ \( T^{2} - 376T - 204560 \) Copy content Toggle raw display
$71$ \( (T + 444)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 880T - 174144 \) Copy content Toggle raw display
$79$ \( T^{2} + 172T - 1308 \) Copy content Toggle raw display
$83$ \( T^{2} - 312T - 215568 \) Copy content Toggle raw display
$89$ \( T^{2} - 1316 T + 293700 \) Copy content Toggle raw display
$97$ \( T^{2} - 2488 T + 1527952 \) Copy content Toggle raw display
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