Properties

Label 961.2.d.e
Level $961$
Weight $2$
Character orbit 961.d
Analytic conductor $7.674$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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.

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

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
0.500000 + 0.363271i −0.809017 + 0.587785i −0.500000 1.53884i −2.61803 −0.618034 0.927051 + 2.85317i 0.690983 2.12663i −0.618034 + 1.90211i −1.30902 0.951057i
388.1 0.500000 0.363271i −0.809017 0.587785i −0.500000 + 1.53884i −2.61803 −0.618034 0.927051 2.85317i 0.690983 + 2.12663i −0.618034 1.90211i −1.30902 + 0.951057i
531.1 0.500000 + 1.53884i 0.309017 0.951057i −0.500000 + 0.363271i −0.381966 1.61803 −2.42705 + 1.76336i 1.80902 + 1.31433i 1.61803 + 1.17557i −0.190983 0.587785i
628.1 0.500000 1.53884i 0.309017 + 0.951057i −0.500000 0.363271i −0.381966 1.61803 −2.42705 1.76336i 1.80902 1.31433i 1.61803 1.17557i −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.e 4
31.b odd 2 1 961.2.d.f 4
31.c even 3 2 961.2.g.g 8
31.d even 5 1 961.2.a.e 2
31.d even 5 2 961.2.d.b 4
31.d even 5 1 inner 961.2.d.e 4
31.e odd 6 2 961.2.g.f 8
31.f odd 10 2 31.2.d.a 4
31.f odd 10 1 961.2.a.d 2
31.f odd 10 1 961.2.d.f 4
31.g even 15 2 961.2.c.d 4
31.g even 15 4 961.2.g.c 8
31.g even 15 2 961.2.g.g 8
31.h odd 30 2 961.2.c.f 4
31.h odd 30 4 961.2.g.b 8
31.h odd 30 2 961.2.g.f 8
93.k even 10 2 279.2.i.a 4
93.k even 10 1 8649.2.a.g 2
93.l odd 10 1 8649.2.a.f 2
124.j even 10 2 496.2.n.b 4
155.m odd 10 2 775.2.k.c 4
155.r even 20 4 775.2.bf.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.d.a 4 31.f odd 10 2
279.2.i.a 4 93.k even 10 2
496.2.n.b 4 124.j even 10 2
775.2.k.c 4 155.m odd 10 2
775.2.bf.a 8 155.r even 20 4
961.2.a.d 2 31.f odd 10 1
961.2.a.e 2 31.d even 5 1
961.2.c.d 4 31.g even 15 2
961.2.c.f 4 31.h odd 30 2
961.2.d.b 4 31.d even 5 2
961.2.d.e 4 1.a even 1 1 trivial
961.2.d.e 4 31.d even 5 1 inner
961.2.d.f 4 31.b odd 2 1
961.2.d.f 4 31.f odd 10 1
961.2.g.b 8 31.h odd 30 4
961.2.g.c 8 31.g even 15 4
961.2.g.f 8 31.e odd 6 2
961.2.g.f 8 31.h odd 30 2
961.2.g.g 8 31.c even 3 2
961.2.g.g 8 31.g even 15 2
8649.2.a.f 2 93.l odd 10 1
8649.2.a.g 2 93.k even 10 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81 \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625 \) Copy content Toggle raw display
$23$ \( T^{4} - 9 T^{3} + 61 T^{2} - 209 T + 361 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + 24 T^{2} - 133 T + 361 \) Copy content Toggle raw display
$53$ \( T^{4} - 9 T^{3} + 171 T^{2} - 189 T + 81 \) Copy content Toggle raw display
$59$ \( T^{4} + 15 T^{3} + 85 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + 124 T^{2} - 7 T + 1 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 11 T^{3} + 96 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$97$ \( T^{4} - 27 T^{3} + 279 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
show more
show less