Properties

Label 861.2.n.a
Level $861$
Weight $2$
Character orbit 861.n
Analytic conductor $6.875$
Analytic rank $0$
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] = [861,2,Mod(379,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.n (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: \(6.87511961403\)
Analytic rank: \(0\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\) \(575\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.809017 0.587785i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
1.30902 + 0.951057i −1.00000 0.190983 + 0.587785i −0.309017 0.951057i −1.30902 0.951057i −0.809017 + 0.587785i 0.690983 2.12663i 1.00000 0.500000 1.53884i
631.1 0.190983 0.587785i −1.00000 1.30902 + 0.951057i 0.809017 + 0.587785i −0.190983 + 0.587785i 0.309017 + 0.951057i 1.80902 1.31433i 1.00000 0.500000 0.363271i
652.1 1.30902 0.951057i −1.00000 0.190983 0.587785i −0.309017 + 0.951057i −1.30902 + 0.951057i −0.809017 0.587785i 0.690983 + 2.12663i 1.00000 0.500000 + 1.53884i
715.1 0.190983 + 0.587785i −1.00000 1.30902 0.951057i 0.809017 0.587785i −0.190983 0.587785i 0.309017 0.951057i 1.80902 + 1.31433i 1.00000 0.500000 + 0.363271i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.n.a 4
41.d even 5 1 inner 861.2.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.n.a 4 1.a even 1 1 trivial
861.2.n.a 4 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$23$ \( T^{4} + 17 T^{3} + 184 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$31$ \( T^{4} - 20 T^{3} + 190 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} + 79 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$41$ \( T^{4} - 11 T^{3} + 111 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$53$ \( T^{4} + 19 T^{3} + 151 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$59$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 11 T^{3} + 96 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121 \) Copy content Toggle raw display
$71$ \( T^{4} + 13 T^{3} + 139 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$73$ \( (T^{2} + T - 11)^{2} \) Copy content Toggle raw display
$79$ \( (T - 15)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 17 T + 61)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} + 214 T^{2} + \cdots + 961 \) Copy content Toggle raw display
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