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Properties

Label 961.2.d.c
Level $961$
Weight $2$
Character orbit 961.d
Analytic conductor $7.674$
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] = [961,2,Mod(374,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.374");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.d (of order \(5\), degree \(4\), not 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: \(7.67362363425\)
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: no (minimal twist has level 31)
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.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
0.809017 + 0.587785i
0.809017 − 0.587785i
−0.309017 − 0.951057i
−0.309017 + 0.951057i
−1.30902 − 0.951057i 2.61803 − 1.90211i 0.190983 + 0.587785i 1.00000 −5.23607 0.0729490 + 0.224514i −0.690983 + 2.12663i 2.30902 − 7.10642i −1.30902 − 0.951057i
388.1 −1.30902 + 0.951057i 2.61803 + 1.90211i 0.190983 − 0.587785i 1.00000 −5.23607 0.0729490 − 0.224514i −0.690983 − 2.12663i 2.30902 + 7.10642i −1.30902 + 0.951057i
531.1 −0.190983 − 0.587785i 0.381966 − 1.17557i 1.30902 − 0.951057i 1.00000 −0.763932 3.42705 − 2.48990i −1.80902 − 1.31433i 1.19098 + 0.865300i −0.190983 − 0.587785i
628.1 −0.190983 + 0.587785i 0.381966 + 1.17557i 1.30902 + 0.951057i 1.00000 −0.763932 3.42705 + 2.48990i −1.80902 + 1.31433i 1.19098 − 0.865300i −0.190983 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.d.c 4
31.b odd 2 1 961.2.d.a 4
31.c even 3 2 961.2.g.a 8
31.d even 5 1 31.2.a.a ✓ 2
31.d even 5 1 inner 961.2.d.c 4
31.d even 5 2 961.2.d.d 4
31.e odd 6 2 961.2.g.d 8
31.f odd 10 1 961.2.a.f 2
31.f odd 10 1 961.2.d.a 4
31.f odd 10 2 961.2.d.g 4
31.g even 15 2 961.2.c.e 4
31.g even 15 2 961.2.g.a 8
31.g even 15 4 961.2.g.h 8
31.h odd 30 2 961.2.c.c 4
31.h odd 30 2 961.2.g.d 8
31.h odd 30 4 961.2.g.e 8
93.k even 10 1 8649.2.a.c 2
93.l odd 10 1 279.2.a.a 2
124.l odd 10 1 496.2.a.i 2
155.n even 10 1 775.2.a.d 2
155.s odd 20 2 775.2.b.d 4
217.w odd 10 1 1519.2.a.a 2
248.s odd 10 1 1984.2.a.n 2
248.u even 10 1 1984.2.a.r 2
341.z odd 10 1 3751.2.a.b 2
372.t even 10 1 4464.2.a.bf 2
403.y even 10 1 5239.2.a.f 2
465.x odd 10 1 6975.2.a.y 2
527.o even 10 1 8959.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.a.a ✓ 2 31.d even 5 1
279.2.a.a 2 93.l odd 10 1
496.2.a.i 2 124.l odd 10 1
775.2.a.d 2 155.n even 10 1
775.2.b.d 4 155.s odd 20 2
961.2.a.f 2 31.f odd 10 1
961.2.c.c 4 31.h odd 30 2
961.2.c.e 4 31.g even 15 2
961.2.d.a 4 31.b odd 2 1
961.2.d.a 4 31.f odd 10 1
961.2.d.c 4 1.a even 1 1 trivial
961.2.d.c 4 31.d even 5 1 inner
961.2.d.d 4 31.d even 5 2
961.2.d.g 4 31.f odd 10 2
961.2.g.a 8 31.c even 3 2
961.2.g.a 8 31.g even 15 2
961.2.g.d 8 31.e odd 6 2
961.2.g.d 8 31.h odd 30 2
961.2.g.e 8 31.h odd 30 4
961.2.g.h 8 31.g even 15 4
1519.2.a.a 2 217.w odd 10 1
1984.2.a.n 2 248.s odd 10 1
1984.2.a.r 2 248.u even 10 1
3751.2.a.b 2 341.z odd 10 1
4464.2.a.bf 2 372.t even 10 1
5239.2.a.f 2 403.y even 10 1
6975.2.a.y 2 465.x odd 10 1
8649.2.a.c 2 93.k even 10 1
8959.2.a.b 2 527.o even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 6T_{3}^{3} + 16T_{3}^{2} - 16T_{3} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{4} - 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$29$ \( T^{4} + 40 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T + 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 7 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 116)^{2} \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 27 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$97$ \( T^{4} + 13 T^{3} + \cdots + 961 \) Copy content Toggle raw display
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