Properties

Label 73.2.a.c
Level $73$
Weight $2$
Character orbit 73.a
Self dual yes
Analytic conductor $0.583$
Analytic rank $0$
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] = [73,2,Mod(1,73)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(73, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("73.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 73.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: \(0.582907934755\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 1) q^{4} - \beta q^{5} - 3 q^{6} - q^{7} + 3 q^{8} + ( - \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 1) q^{4} - \beta q^{5} - 3 q^{6} - q^{7} + 3 q^{8} + ( - \beta + 1) q^{9} + ( - \beta - 3) q^{10} + (\beta + 3) q^{11} + ( - \beta - 2) q^{12} + (\beta - 1) q^{13} - \beta q^{14} + 3 q^{15} + (\beta - 2) q^{16} + (2 \beta - 3) q^{17} - 3 q^{18} - 7 q^{19} + ( - 2 \beta - 3) q^{20} + (\beta - 1) q^{21} + (4 \beta + 3) q^{22} + (\beta + 6) q^{23} + ( - 3 \beta + 3) q^{24} + (\beta - 2) q^{25} + 3 q^{26} + (2 \beta + 1) q^{27} + ( - \beta - 1) q^{28} + ( - 4 \beta + 3) q^{29} + 3 \beta q^{30} + (2 \beta + 2) q^{31} + ( - \beta - 3) q^{32} - 3 \beta q^{33} + ( - \beta + 6) q^{34} + \beta q^{35} + ( - \beta - 2) q^{36} + ( - 2 \beta + 5) q^{37} - 7 \beta q^{38} + (\beta - 4) q^{39} - 3 \beta q^{40} - 6 q^{41} + 3 q^{42} + ( - 4 \beta + 5) q^{43} + (5 \beta + 6) q^{44} + 3 q^{45} + (7 \beta + 3) q^{46} + 9 q^{47} + (2 \beta - 5) q^{48} - 6 q^{49} + ( - \beta + 3) q^{50} + (3 \beta - 9) q^{51} + (\beta + 2) q^{52} + (4 \beta - 3) q^{53} + (3 \beta + 6) q^{54} + ( - 4 \beta - 3) q^{55} - 3 q^{56} + (7 \beta - 7) q^{57} + ( - \beta - 12) q^{58} + (3 \beta + 3) q^{60} + ( - \beta - 4) q^{61} + (4 \beta + 6) q^{62} + (\beta - 1) q^{63} + ( - 6 \beta + 1) q^{64} - 3 q^{65} + ( - 3 \beta - 9) q^{66} + ( - 6 \beta + 5) q^{67} + (\beta + 3) q^{68} + ( - 6 \beta + 3) q^{69} + (\beta + 3) q^{70} + ( - 3 \beta + 3) q^{71} + ( - 3 \beta + 3) q^{72} + q^{73} + (3 \beta - 6) q^{74} + (2 \beta - 5) q^{75} + ( - 7 \beta - 7) q^{76} + ( - \beta - 3) q^{77} + ( - 3 \beta + 3) q^{78} + (3 \beta - 1) q^{79} + (\beta - 3) q^{80} + (2 \beta - 8) q^{81} - 6 \beta q^{82} + ( - 5 \beta + 6) q^{83} + (\beta + 2) q^{84} + (\beta - 6) q^{85} + (\beta - 12) q^{86} + ( - 3 \beta + 15) q^{87} + (3 \beta + 9) q^{88} + (6 \beta + 3) q^{89} + 3 \beta q^{90} + ( - \beta + 1) q^{91} + (8 \beta + 9) q^{92} + ( - 2 \beta - 4) q^{93} + 9 \beta q^{94} + 7 \beta q^{95} + 3 \beta q^{96} + ( - 3 \beta - 1) q^{97} - 6 \beta q^{98} - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + 3 q^{4} - q^{5} - 6 q^{6} - 2 q^{7} + 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + 3 q^{4} - q^{5} - 6 q^{6} - 2 q^{7} + 6 q^{8} + q^{9} - 7 q^{10} + 7 q^{11} - 5 q^{12} - q^{13} - q^{14} + 6 q^{15} - 3 q^{16} - 4 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} - q^{21} + 10 q^{22} + 13 q^{23} + 3 q^{24} - 3 q^{25} + 6 q^{26} + 4 q^{27} - 3 q^{28} + 2 q^{29} + 3 q^{30} + 6 q^{31} - 7 q^{32} - 3 q^{33} + 11 q^{34} + q^{35} - 5 q^{36} + 8 q^{37} - 7 q^{38} - 7 q^{39} - 3 q^{40} - 12 q^{41} + 6 q^{42} + 6 q^{43} + 17 q^{44} + 6 q^{45} + 13 q^{46} + 18 q^{47} - 8 q^{48} - 12 q^{49} + 5 q^{50} - 15 q^{51} + 5 q^{52} - 2 q^{53} + 15 q^{54} - 10 q^{55} - 6 q^{56} - 7 q^{57} - 25 q^{58} + 9 q^{60} - 9 q^{61} + 16 q^{62} - q^{63} - 4 q^{64} - 6 q^{65} - 21 q^{66} + 4 q^{67} + 7 q^{68} + 7 q^{70} + 3 q^{71} + 3 q^{72} + 2 q^{73} - 9 q^{74} - 8 q^{75} - 21 q^{76} - 7 q^{77} + 3 q^{78} + q^{79} - 5 q^{80} - 14 q^{81} - 6 q^{82} + 7 q^{83} + 5 q^{84} - 11 q^{85} - 23 q^{86} + 27 q^{87} + 21 q^{88} + 12 q^{89} + 3 q^{90} + q^{91} + 26 q^{92} - 10 q^{93} + 9 q^{94} + 7 q^{95} + 3 q^{96} - 5 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
−1.30278 2.30278 −0.302776 1.30278 −3.00000 −1.00000 3.00000 2.30278 −1.69722
1.2 2.30278 −1.30278 3.30278 −2.30278 −3.00000 −1.00000 3.00000 −1.30278 −5.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(73\) \(-1\)

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 73.2.a.c 2
3.b odd 2 1 657.2.a.e 2
4.b odd 2 1 1168.2.a.c 2
5.b even 2 1 1825.2.a.b 2
7.b odd 2 1 3577.2.a.c 2
8.b even 2 1 4672.2.a.f 2
8.d odd 2 1 4672.2.a.i 2
11.b odd 2 1 8833.2.a.d 2
73.b even 2 1 5329.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
73.2.a.c 2 1.a even 1 1 trivial
657.2.a.e 2 3.b odd 2 1
1168.2.a.c 2 4.b odd 2 1
1825.2.a.b 2 5.b even 2 1
3577.2.a.c 2 7.b odd 2 1
4672.2.a.f 2 8.b even 2 1
4672.2.a.i 2 8.d odd 2 1
5329.2.a.c 2 73.b even 2 1
8833.2.a.d 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(73))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$3$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 13T + 39 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 51 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 43 \) Copy content Toggle raw display
$47$ \( (T - 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 17 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 113 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - T - 29 \) Copy content Toggle raw display
$83$ \( T^{2} - 7T - 69 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 81 \) Copy content Toggle raw display
$97$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
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