Properties

Label 308.3.g.e
Level $308$
Weight $3$
Character orbit 308.g
Self dual yes
Analytic conductor $8.392$
Analytic rank $0$
Dimension $2$
CM discriminant -308
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,3,Mod(307,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.307");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 308.g (of order \(2\), degree \(1\), minimal)

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: \(8.39239214230\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} + 2 \beta q^{6} - 7 q^{7} + 8 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} + 2 \beta q^{6} - 7 q^{7} + 8 q^{8} + 13 q^{9} + 11 q^{11} + 4 \beta q^{12} - 5 \beta q^{13} - 14 q^{14} + 16 q^{16} + \beta q^{17} + 26 q^{18} - 7 \beta q^{21} + 22 q^{22} + 8 \beta q^{24} + 25 q^{25} - 10 \beta q^{26} + 4 \beta q^{27} - 28 q^{28} - 13 \beta q^{31} + 32 q^{32} + 11 \beta q^{33} + 2 \beta q^{34} + 52 q^{36} - 52 q^{37} - 110 q^{39} - 5 \beta q^{41} - 14 \beta q^{42} - 68 q^{43} + 44 q^{44} + 11 \beta q^{47} + 16 \beta q^{48} + 49 q^{49} + 50 q^{50} + 22 q^{51} - 20 \beta q^{52} + 92 q^{53} + 8 \beta q^{54} - 56 q^{56} - 13 \beta q^{59} + 25 \beta q^{61} - 26 \beta q^{62} - 91 q^{63} + 64 q^{64} + 22 \beta q^{66} + 4 \beta q^{68} + 104 q^{72} + \beta q^{73} - 104 q^{74} + 25 \beta q^{75} - 77 q^{77} - 220 q^{78} + 4 q^{79} - 29 q^{81} - 10 \beta q^{82} - 28 \beta q^{84} - 136 q^{86} + 88 q^{88} + 35 \beta q^{91} - 286 q^{93} + 22 \beta q^{94} + 32 \beta q^{96} + 98 q^{98} + 143 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} + 26 q^{9} + 22 q^{11} - 28 q^{14} + 32 q^{16} + 52 q^{18} + 44 q^{22} + 50 q^{25} - 56 q^{28} + 64 q^{32} + 104 q^{36} - 104 q^{37} - 220 q^{39} - 136 q^{43} + 88 q^{44} + 98 q^{49} + 100 q^{50} + 44 q^{51} + 184 q^{53} - 112 q^{56} - 182 q^{63} + 128 q^{64} + 208 q^{72} - 208 q^{74} - 154 q^{77} - 440 q^{78} + 8 q^{79} - 58 q^{81} - 272 q^{86} + 176 q^{88} - 572 q^{93} + 196 q^{98} + 286 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−4.69042
4.69042
2.00000 −4.69042 4.00000 0 −9.38083 −7.00000 8.00000 13.0000 0
307.2 2.00000 4.69042 4.00000 0 9.38083 −7.00000 8.00000 13.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
308.g odd 2 1 CM by \(\Q(\sqrt{-77}) \)
7.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.3.g.e yes 2
4.b odd 2 1 308.3.g.b 2
7.b odd 2 1 inner 308.3.g.e yes 2
11.b odd 2 1 308.3.g.b 2
28.d even 2 1 308.3.g.b 2
44.c even 2 1 inner 308.3.g.e yes 2
77.b even 2 1 308.3.g.b 2
308.g odd 2 1 CM 308.3.g.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.3.g.b 2 4.b odd 2 1
308.3.g.b 2 11.b odd 2 1
308.3.g.b 2 28.d even 2 1
308.3.g.b 2 77.b even 2 1
308.3.g.e yes 2 1.a even 1 1 trivial
308.3.g.e yes 2 7.b odd 2 1 inner
308.3.g.e yes 2 44.c even 2 1 inner
308.3.g.e yes 2 308.g odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(308, [\chi])\):

\( T_{3}^{2} - 22 \) Copy content Toggle raw display
\( T_{43} + 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 22 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 550 \) Copy content Toggle raw display
$17$ \( T^{2} - 22 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3718 \) Copy content Toggle raw display
$37$ \( (T + 52)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 550 \) Copy content Toggle raw display
$43$ \( (T + 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2662 \) Copy content Toggle raw display
$53$ \( (T - 92)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3718 \) Copy content Toggle raw display
$61$ \( T^{2} - 13750 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 22 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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