Properties

Label 310.2.a.c
Level $310$
Weight $2$
Character orbit 310.a
Self dual yes
Analytic conductor $2.475$
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] = [310,2,Mod(1,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.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: \(2.47536246266\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta - 1) q^{3} + q^{4} - q^{5} + ( - \beta + 1) q^{6} - 2 \beta q^{7} - q^{8} + ( - 2 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta - 1) q^{3} + q^{4} - q^{5} + ( - \beta + 1) q^{6} - 2 \beta q^{7} - q^{8} + ( - 2 \beta + 1) q^{9} + q^{10} + ( - \beta - 1) q^{11} + (\beta - 1) q^{12} + (\beta - 3) q^{13} + 2 \beta q^{14} + ( - \beta + 1) q^{15} + q^{16} - 4 q^{17} + (2 \beta - 1) q^{18} + (2 \beta + 2) q^{19} - q^{20} + (2 \beta - 6) q^{21} + (\beta + 1) q^{22} + (2 \beta - 4) q^{23} + ( - \beta + 1) q^{24} + q^{25} + ( - \beta + 3) q^{26} - 4 q^{27} - 2 \beta q^{28} + (\beta - 1) q^{29} + (\beta - 1) q^{30} - q^{31} - q^{32} - 2 q^{33} + 4 q^{34} + 2 \beta q^{35} + ( - 2 \beta + 1) q^{36} + ( - \beta - 5) q^{37} + ( - 2 \beta - 2) q^{38} + ( - 4 \beta + 6) q^{39} + q^{40} + (2 \beta - 6) q^{41} + ( - 2 \beta + 6) q^{42} + ( - \beta + 5) q^{43} + ( - \beta - 1) q^{44} + (2 \beta - 1) q^{45} + ( - 2 \beta + 4) q^{46} - 2 \beta q^{47} + (\beta - 1) q^{48} + 5 q^{49} - q^{50} + ( - 4 \beta + 4) q^{51} + (\beta - 3) q^{52} + (\beta - 7) q^{53} + 4 q^{54} + (\beta + 1) q^{55} + 2 \beta q^{56} + 4 q^{57} + ( - \beta + 1) q^{58} + (2 \beta + 2) q^{59} + ( - \beta + 1) q^{60} + (7 \beta - 3) q^{61} + q^{62} + ( - 2 \beta + 12) q^{63} + q^{64} + ( - \beta + 3) q^{65} + 2 q^{66} + (6 \beta - 2) q^{67} - 4 q^{68} + ( - 6 \beta + 10) q^{69} - 2 \beta q^{70} - 8 \beta q^{71} + (2 \beta - 1) q^{72} + ( - 6 \beta + 6) q^{73} + (\beta + 5) q^{74} + (\beta - 1) q^{75} + (2 \beta + 2) q^{76} + (2 \beta + 6) q^{77} + (4 \beta - 6) q^{78} + (2 \beta + 14) q^{79} - q^{80} + (2 \beta + 1) q^{81} + ( - 2 \beta + 6) q^{82} + ( - \beta - 3) q^{83} + (2 \beta - 6) q^{84} + 4 q^{85} + (\beta - 5) q^{86} + ( - 2 \beta + 4) q^{87} + (\beta + 1) q^{88} + (4 \beta - 6) q^{89} + ( - 2 \beta + 1) q^{90} + (6 \beta - 6) q^{91} + (2 \beta - 4) q^{92} + ( - \beta + 1) q^{93} + 2 \beta q^{94} + ( - 2 \beta - 2) q^{95} + ( - \beta + 1) q^{96} + 4 q^{97} - 5 q^{98} + (\beta + 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} - 6 q^{13} + 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 4 q^{19} - 2 q^{20} - 12 q^{21} + 2 q^{22} - 8 q^{23} + 2 q^{24} + 2 q^{25} + 6 q^{26} - 8 q^{27} - 2 q^{29} - 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{34} + 2 q^{36} - 10 q^{37} - 4 q^{38} + 12 q^{39} + 2 q^{40} - 12 q^{41} + 12 q^{42} + 10 q^{43} - 2 q^{44} - 2 q^{45} + 8 q^{46} - 2 q^{48} + 10 q^{49} - 2 q^{50} + 8 q^{51} - 6 q^{52} - 14 q^{53} + 8 q^{54} + 2 q^{55} + 8 q^{57} + 2 q^{58} + 4 q^{59} + 2 q^{60} - 6 q^{61} + 2 q^{62} + 24 q^{63} + 2 q^{64} + 6 q^{65} + 4 q^{66} - 4 q^{67} - 8 q^{68} + 20 q^{69} - 2 q^{72} + 12 q^{73} + 10 q^{74} - 2 q^{75} + 4 q^{76} + 12 q^{77} - 12 q^{78} + 28 q^{79} - 2 q^{80} + 2 q^{81} + 12 q^{82} - 6 q^{83} - 12 q^{84} + 8 q^{85} - 10 q^{86} + 8 q^{87} + 2 q^{88} - 12 q^{89} + 2 q^{90} - 12 q^{91} - 8 q^{92} + 2 q^{93} - 4 q^{95} + 2 q^{96} + 8 q^{97} - 10 q^{98} + 10 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
−1.00000 −2.73205 1.00000 −1.00000 2.73205 3.46410 −1.00000 4.46410 1.00000
1.2 −1.00000 0.732051 1.00000 −1.00000 −0.732051 −3.46410 −1.00000 −2.46410 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(31\) \( +1 \)

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 310.2.a.c 2
3.b odd 2 1 2790.2.a.bh 2
4.b odd 2 1 2480.2.a.s 2
5.b even 2 1 1550.2.a.j 2
5.c odd 4 2 1550.2.b.h 4
8.b even 2 1 9920.2.a.bt 2
8.d odd 2 1 9920.2.a.bl 2
31.b odd 2 1 9610.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.c 2 1.a even 1 1 trivial
1550.2.a.j 2 5.b even 2 1
1550.2.b.h 4 5.c odd 4 2
2480.2.a.s 2 4.b odd 2 1
2790.2.a.bh 2 3.b odd 2 1
9610.2.a.j 2 31.b odd 2 1
9920.2.a.bl 2 8.d odd 2 1
9920.2.a.bt 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(310))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$47$ \( T^{2} - 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 46 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 138 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 104 \) Copy content Toggle raw display
$71$ \( T^{2} - 192 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 72 \) Copy content Toggle raw display
$79$ \( T^{2} - 28T + 184 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$97$ \( (T - 4)^{2} \) Copy content Toggle raw display
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