Properties

Label 312.2.bt.a
Level $312$
Weight $2$
Character orbit 312.bt
Analytic conductor $2.491$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,2,Mod(19,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.bt (of order \(12\), degree \(4\), 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: \(2.49133254306\)
Analytic rank: \(1\)
Dimension: \(4\)
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: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(-1\) \(\zeta_{12}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−1.00000 + 1.00000i −0.500000 + 0.866025i 2.00000i 0.366025 + 0.366025i −0.366025 1.36603i −2.36603 0.633975i 2.00000 + 2.00000i −0.500000 0.866025i −0.732051
67.1 −1.00000 + 1.00000i −0.500000 0.866025i 2.00000i −1.36603 1.36603i 1.36603 + 0.366025i −0.633975 2.36603i 2.00000 + 2.00000i −0.500000 + 0.866025i 2.73205
115.1 −1.00000 1.00000i −0.500000 0.866025i 2.00000i 0.366025 0.366025i −0.366025 + 1.36603i −2.36603 + 0.633975i 2.00000 2.00000i −0.500000 + 0.866025i −0.732051
163.1 −1.00000 1.00000i −0.500000 + 0.866025i 2.00000i −1.36603 + 1.36603i 1.36603 0.366025i −0.633975 + 2.36603i 2.00000 2.00000i −0.500000 0.866025i 2.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.bt.a 4
3.b odd 2 1 936.2.ed.b 4
8.d odd 2 1 312.2.bt.b yes 4
13.f odd 12 1 312.2.bt.b yes 4
24.f even 2 1 936.2.ed.a 4
39.k even 12 1 936.2.ed.a 4
104.u even 12 1 inner 312.2.bt.a 4
312.bq odd 12 1 936.2.ed.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bt.a 4 1.a even 1 1 trivial
312.2.bt.a 4 104.u even 12 1 inner
312.2.bt.b yes 4 8.d odd 2 1
312.2.bt.b yes 4 13.f odd 12 1
936.2.ed.a 4 24.f even 2 1
936.2.ed.a 4 39.k even 12 1
936.2.ed.b 4 3.b odd 2 1
936.2.ed.b 4 312.bq odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{3} + 2T_{5}^{2} - 2T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 18 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$11$ \( T^{4} + 10 T^{3} + 74 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$13$ \( (T^{2} + 7 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 53 T^{2} - 14 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} + 125 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$53$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 32 T^{3} + 452 T^{2} + \cdots + 21904 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + 215 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 90 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + 98 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$79$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + 8 T^{2} - 376 T + 8836 \) Copy content Toggle raw display
$89$ \( T^{4} + 26 T^{3} + 194 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$97$ \( T^{4} - 30 T^{3} + 234 T^{2} + \cdots + 324 \) Copy content Toggle raw display
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