Properties

Label 961.2.d.i
Level $961$
Weight $2$
Character orbit 961.d
Analytic conductor $7.674$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \cdots - 1) q^{3}+ \cdots - 2 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{5} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 8 q^{5} - 24 q^{6} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 8 q^{5} - 24 q^{6} - 2 q^{7} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 10 q^{12} - 2 q^{13} + 6 q^{14} - 2 q^{15} - 6 q^{16} - 6 q^{17} + 8 q^{18} - 6 q^{19} - 2 q^{20} - 6 q^{21} + 14 q^{22} - 8 q^{23} + 10 q^{24} - 32 q^{25} + 24 q^{26} - 2 q^{27} - 10 q^{28} - 8 q^{29} - 24 q^{30} + 24 q^{32} - 14 q^{33} - 2 q^{34} - 2 q^{35} + 64 q^{36} - 8 q^{37} + 2 q^{38} + 6 q^{39} + 6 q^{40} - 2 q^{41} + 14 q^{42} - 2 q^{43} - 26 q^{44} + 8 q^{46} - 8 q^{47} - 6 q^{48} + 8 q^{49} - 8 q^{50} + 2 q^{51} + 14 q^{52} + 6 q^{53} - 2 q^{54} - 2 q^{55} - 40 q^{56} + 8 q^{57} + 6 q^{59} - 10 q^{60} + 32 q^{63} + 14 q^{64} - 2 q^{65} + 30 q^{66} - 8 q^{67} - 40 q^{68} - 8 q^{69} + 6 q^{70} + 14 q^{71} + 8 q^{72} + 2 q^{73} - 2 q^{74} + 8 q^{75} + 2 q^{76} - 14 q^{77} - 10 q^{78} - 22 q^{79} - 6 q^{80} + 2 q^{81} + 26 q^{82} - 6 q^{83} - 22 q^{84} - 6 q^{85} - 26 q^{86} - 72 q^{88} - 8 q^{89} + 8 q^{90} + 6 q^{91} + 32 q^{92} + 32 q^{94} - 6 q^{95} - 2 q^{96} - 16 q^{97} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{6}\)

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
0.437016 1.34500i
−0.437016 + 1.34500i
0.437016 + 1.34500i
−0.437016 1.34500i
1.14412 + 0.831254i
−1.14412 0.831254i
1.14412 0.831254i
−1.14412 + 0.831254i
−0.335106 0.243469i 0.335106 0.243469i −0.565015 1.73894i 1.00000 −0.171573 −0.127999 0.393941i −0.490035 + 1.50817i −0.874032 + 2.68999i −0.335106 0.243469i
374.2 1.95314 + 1.41904i −1.95314 + 1.41904i 1.18305 + 3.64105i 1.00000 −5.82843 0.746033 + 2.29605i −1.36407 + 4.19817i 0.874032 2.68999i 1.95314 + 1.41904i
388.1 −0.335106 + 0.243469i 0.335106 + 0.243469i −0.565015 + 1.73894i 1.00000 −0.171573 −0.127999 + 0.393941i −0.490035 1.50817i −0.874032 2.68999i −0.335106 + 0.243469i
388.2 1.95314 1.41904i −1.95314 1.41904i 1.18305 3.64105i 1.00000 −5.82843 0.746033 2.29605i −1.36407 4.19817i 0.874032 + 2.68999i 1.95314 1.41904i
531.1 −0.746033 2.29605i 0.746033 2.29605i −3.09726 + 2.25029i 1.00000 −5.82843 −1.95314 + 1.41904i 3.57117 + 2.59461i −2.28825 1.66251i −0.746033 2.29605i
531.2 0.127999 + 0.393941i −0.127999 + 0.393941i 1.47923 1.07472i 1.00000 −0.171573 0.335106 0.243469i 1.28293 + 0.932102i 2.28825 + 1.66251i 0.127999 + 0.393941i
628.1 −0.746033 + 2.29605i 0.746033 + 2.29605i −3.09726 2.25029i 1.00000 −5.82843 −1.95314 1.41904i 3.57117 2.59461i −2.28825 + 1.66251i −0.746033 + 2.29605i
628.2 0.127999 0.393941i −0.127999 0.393941i 1.47923 + 1.07472i 1.00000 −0.171573 0.335106 + 0.243469i 1.28293 0.932102i 2.28825 1.66251i 0.127999 0.393941i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.d.i 8
31.b odd 2 1 961.2.d.l 8
31.c even 3 2 961.2.g.r 16
31.d even 5 1 961.2.a.c 2
31.d even 5 3 inner 961.2.d.i 8
31.e odd 6 2 961.2.g.o 16
31.f odd 10 1 961.2.a.a 2
31.f odd 10 3 961.2.d.l 8
31.g even 15 2 961.2.c.a 4
31.g even 15 6 961.2.g.r 16
31.h odd 30 2 31.2.c.a 4
31.h odd 30 6 961.2.g.o 16
93.k even 10 1 8649.2.a.l 2
93.l odd 10 1 8649.2.a.k 2
93.p even 30 2 279.2.h.c 4
124.p even 30 2 496.2.i.h 4
155.v odd 30 2 775.2.e.e 4
155.x even 60 4 775.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 31.h odd 30 2
279.2.h.c 4 93.p even 30 2
496.2.i.h 4 124.p even 30 2
775.2.e.e 4 155.v odd 30 2
775.2.o.d 8 155.x even 60 4
961.2.a.a 2 31.f odd 10 1
961.2.a.c 2 31.d even 5 1
961.2.c.a 4 31.g even 15 2
961.2.d.i 8 1.a even 1 1 trivial
961.2.d.i 8 31.d even 5 3 inner
961.2.d.l 8 31.b odd 2 1
961.2.d.l 8 31.f odd 10 3
961.2.g.o 16 31.e odd 6 2
961.2.g.o 16 31.h odd 30 6
961.2.g.r 16 31.c even 3 2
961.2.g.r 16 31.g even 15 6
8649.2.a.k 2 93.l odd 10 1
8649.2.a.l 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 \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 5T_{2}^{6} - 12T_{2}^{5} + 29T_{2}^{4} + 12T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 2T_{3}^{7} + 5T_{3}^{6} + 12T_{3}^{5} + 29T_{3}^{4} - 12T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T + 1)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + \cdots + 25411681 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} + \cdots + 88529281 \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} - 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots + 2825761 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T - 17)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 112550881 \) Copy content Toggle raw display
$83$ \( T^{8} + 6 T^{7} + \cdots + 2825761 \) Copy content Toggle raw display
$89$ \( T^{8} + 8 T^{7} + \cdots + 9834496 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots + 9834496 \) Copy content Toggle raw display
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