Properties

Label 946.2.f.a
Level $946$
Weight $2$
Character orbit 946.f
Analytic conductor $7.554$
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] = [946,2,Mod(345,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.345");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.f (of order \(5\), 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: \(7.55384803121\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/946\mathbb{Z}\right)^\times\).

\(n\) \(89\) \(431\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
345.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.809017 0.587785i −0.381966 + 1.17557i 0.309017 + 0.951057i −1.00000 + 0.726543i 1.00000 0.726543i −1.38197 4.25325i 0.309017 0.951057i 1.19098 + 0.865300i 1.23607
603.1 0.309017 0.951057i −2.61803 + 1.90211i −0.809017 0.587785i −1.00000 3.07768i 1.00000 + 3.07768i −3.61803 2.62866i −0.809017 + 0.587785i 2.30902 7.10642i −3.23607
775.1 0.309017 + 0.951057i −2.61803 1.90211i −0.809017 + 0.587785i −1.00000 + 3.07768i 1.00000 3.07768i −3.61803 + 2.62866i −0.809017 0.587785i 2.30902 + 7.10642i −3.23607
861.1 −0.809017 + 0.587785i −0.381966 1.17557i 0.309017 0.951057i −1.00000 0.726543i 1.00000 + 0.726543i −1.38197 + 4.25325i 0.309017 + 0.951057i 1.19098 0.865300i 1.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.f.a 4
11.c even 5 1 inner 946.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.f.a 4 1.a even 1 1 trivial
946.2.f.a 4 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6T_{3}^{3} + 16T_{3}^{2} + 16T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 6 T^{2} + 44 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 36 T^{2} + 81 T + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T - 29)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16 \) Copy content Toggle raw display
$43$ \( (T + 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \) Copy content Toggle raw display
$53$ \( T^{4} + 26 T^{3} + 456 T^{2} + \cdots + 22201 \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16 \) Copy content Toggle raw display
$67$ \( (T^{2} + 7 T - 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} + 96 T^{2} - 256 T + 256 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 7 T^{3} + 124 T^{2} - 18 T + 1 \) Copy content Toggle raw display
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