Properties

Label 315.2.bh.b
Level $315$
Weight $2$
Character orbit 315.bh
Analytic conductor $2.515$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(169,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bh (of order \(6\), degree \(2\), 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: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−1.73205 1.00000i −0.866025 + 1.50000i 1.00000 + 1.73205i 0.133975 + 2.23205i 3.00000 1.73205i 0.866025 + 0.500000i 0 −1.50000 2.59808i 2.00000 4.00000i
169.2 1.73205 + 1.00000i 0.866025 1.50000i 1.00000 + 1.73205i 1.86603 + 1.23205i 3.00000 1.73205i −0.866025 0.500000i 0 −1.50000 2.59808i 2.00000 + 4.00000i
274.1 −1.73205 + 1.00000i −0.866025 1.50000i 1.00000 1.73205i 0.133975 2.23205i 3.00000 + 1.73205i 0.866025 0.500000i 0 −1.50000 + 2.59808i 2.00000 + 4.00000i
274.2 1.73205 1.00000i 0.866025 + 1.50000i 1.00000 1.73205i 1.86603 1.23205i 3.00000 + 1.73205i −0.866025 + 0.500000i 0 −1.50000 + 2.59808i 2.00000 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bh.b 4
3.b odd 2 1 945.2.bh.b 4
5.b even 2 1 inner 315.2.bh.b 4
9.c even 3 1 inner 315.2.bh.b 4
9.d odd 6 1 945.2.bh.b 4
15.d odd 2 1 945.2.bh.b 4
45.h odd 6 1 945.2.bh.b 4
45.j even 6 1 inner 315.2.bh.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bh.b 4 1.a even 1 1 trivial
315.2.bh.b 4 5.b even 2 1 inner
315.2.bh.b 4 9.c even 3 1 inner
315.2.bh.b 4 45.j even 6 1 inner
945.2.bh.b 4 3.b odd 2 1
945.2.bh.b 4 9.d odd 6 1
945.2.bh.b 4 15.d odd 2 1
945.2.bh.b 4 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T + 6)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$71$ \( (T - 7)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 289 T^{2} + 83521 \) Copy content Toggle raw display
$89$ \( (T - 12)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
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