Properties

Label 961.2.a.c
Level $961$
Weight $2$
Character orbit 961.a
Self dual yes
Analytic conductor $7.674$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,2,Mod(1,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

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: yes
Analytic conductor: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−2.41421 2.41421 3.82843 1.00000 −5.82843 2.41421 −4.41421 2.82843 −2.41421
1.2 0.414214 −0.414214 −1.82843 1.00000 −0.171573 −0.414214 −1.58579 −2.82843 0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(31\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.a.c 2
3.b odd 2 1 8649.2.a.k 2
31.b odd 2 1 961.2.a.a 2
31.c even 3 2 961.2.c.a 4
31.d even 5 4 961.2.d.i 8
31.e odd 6 2 31.2.c.a 4
31.f odd 10 4 961.2.d.l 8
31.g even 15 8 961.2.g.r 16
31.h odd 30 8 961.2.g.o 16
93.c even 2 1 8649.2.a.l 2
93.g even 6 2 279.2.h.c 4
124.g even 6 2 496.2.i.h 4
155.i odd 6 2 775.2.e.e 4
155.p even 12 4 775.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 31.e odd 6 2
279.2.h.c 4 93.g even 6 2
496.2.i.h 4 124.g even 6 2
775.2.e.e 4 155.i odd 6 2
775.2.o.d 8 155.p even 12 4
961.2.a.a 2 31.b odd 2 1
961.2.a.c 2 1.a even 1 1 trivial
961.2.c.a 4 31.c even 3 2
961.2.d.i 8 31.d even 5 4
961.2.d.l 8 31.f odd 10 4
961.2.g.o 16 31.h odd 30 8
961.2.g.r 16 31.g even 15 8
8649.2.a.k 2 3.b odd 2 1
8649.2.a.l 2 93.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(961))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 97 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 41 \) Copy content Toggle raw display
$61$ \( T^{2} - 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$71$ \( T^{2} + 14T - 1 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$79$ \( T^{2} - 22T + 103 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 41 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
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