Properties

Label 276.4.a.c
Level $276$
Weight $4$
Character orbit 276.a
Self dual yes
Analytic conductor $16.285$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [276,4,Mod(1,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, 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("276.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 276.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: \(16.2845271616\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta - 1) q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta - 1) q^{5} + 2 q^{7} + 9 q^{9} + (7 \beta + 7) q^{11} + (10 \beta - 12) q^{13} + (3 \beta + 3) q^{15} + ( - 10 \beta - 10) q^{17} + ( - 35 \beta - 9) q^{19} - 6 q^{21} - 23 q^{23} + (2 \beta - 111) q^{25} - 27 q^{27} + (22 \beta - 116) q^{29} + (38 \beta - 146) q^{31} + ( - 21 \beta - 21) q^{33} + ( - 2 \beta - 2) q^{35} + (45 \beta - 205) q^{37} + ( - 30 \beta + 36) q^{39} + ( - 12 \beta - 150) q^{41} + ( - 77 \beta - 99) q^{43} + ( - 9 \beta - 9) q^{45} + (72 \beta - 60) q^{47} - 339 q^{49} + (30 \beta + 30) q^{51} + ( - 125 \beta - 113) q^{53} + ( - 14 \beta - 98) q^{55} + (105 \beta + 27) q^{57} + (144 \beta + 12) q^{59} + ( - 29 \beta - 231) q^{61} + 18 q^{63} + (2 \beta - 118) q^{65} + (9 \beta + 95) q^{67} + 69 q^{69} + ( - 240 \beta + 324) q^{71} + (26 \beta - 320) q^{73} + ( - 6 \beta + 333) q^{75} + (14 \beta + 14) q^{77} + (104 \beta - 314) q^{79} + 81 q^{81} + ( - 119 \beta + 865) q^{83} + (20 \beta + 140) q^{85} + ( - 66 \beta + 348) q^{87} + (96 \beta - 36) q^{89} + (20 \beta - 24) q^{91} + ( - 114 \beta + 438) q^{93} + (44 \beta + 464) q^{95} + ( - 106 \beta - 380) q^{97} + (63 \beta + 63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 2 q^{5} + 4 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 2 q^{5} + 4 q^{7} + 18 q^{9} + 14 q^{11} - 24 q^{13} + 6 q^{15} - 20 q^{17} - 18 q^{19} - 12 q^{21} - 46 q^{23} - 222 q^{25} - 54 q^{27} - 232 q^{29} - 292 q^{31} - 42 q^{33} - 4 q^{35} - 410 q^{37} + 72 q^{39} - 300 q^{41} - 198 q^{43} - 18 q^{45} - 120 q^{47} - 678 q^{49} + 60 q^{51} - 226 q^{53} - 196 q^{55} + 54 q^{57} + 24 q^{59} - 462 q^{61} + 36 q^{63} - 236 q^{65} + 190 q^{67} + 138 q^{69} + 648 q^{71} - 640 q^{73} + 666 q^{75} + 28 q^{77} - 628 q^{79} + 162 q^{81} + 1730 q^{83} + 280 q^{85} + 696 q^{87} - 72 q^{89} - 48 q^{91} + 876 q^{93} + 928 q^{95} - 760 q^{97} + 126 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 −3.00000 0 −4.60555 0 2.00000 0 9.00000 0
1.2 0 −3.00000 0 2.60555 0 2.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)

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 276.4.a.c 2
3.b odd 2 1 828.4.a.d 2
4.b odd 2 1 1104.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.4.a.c 2 1.a even 1 1 trivial
828.4.a.d 2 3.b odd 2 1
1104.4.a.n 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(276))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 14T - 588 \) Copy content Toggle raw display
$13$ \( T^{2} + 24T - 1156 \) Copy content Toggle raw display
$17$ \( T^{2} + 20T - 1200 \) Copy content Toggle raw display
$19$ \( T^{2} + 18T - 15844 \) Copy content Toggle raw display
$23$ \( (T + 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 232T + 7164 \) Copy content Toggle raw display
$31$ \( T^{2} + 292T + 2544 \) Copy content Toggle raw display
$37$ \( T^{2} + 410T + 15700 \) Copy content Toggle raw display
$41$ \( T^{2} + 300T + 20628 \) Copy content Toggle raw display
$43$ \( T^{2} + 198T - 67276 \) Copy content Toggle raw display
$47$ \( T^{2} + 120T - 63792 \) Copy content Toggle raw display
$53$ \( T^{2} + 226T - 190356 \) Copy content Toggle raw display
$59$ \( T^{2} - 24T - 269424 \) Copy content Toggle raw display
$61$ \( T^{2} + 462T + 42428 \) Copy content Toggle raw display
$67$ \( T^{2} - 190T + 7972 \) Copy content Toggle raw display
$71$ \( T^{2} - 648T - 643824 \) Copy content Toggle raw display
$73$ \( T^{2} + 640T + 93612 \) Copy content Toggle raw display
$79$ \( T^{2} + 628T - 42012 \) Copy content Toggle raw display
$83$ \( T^{2} - 1730 T + 564132 \) Copy content Toggle raw display
$89$ \( T^{2} + 72T - 118512 \) Copy content Toggle raw display
$97$ \( T^{2} + 760T - 1668 \) Copy content Toggle raw display
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