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Properties

Label 961.2.g.q
Level $961$
Weight $2$
Character orbit 961.g
Analytic conductor $7.674$
Analytic rank $0$
Dimension $16$
Inner twists $16$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [961,2,Mod(235,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([26])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.235"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.g (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,4,0,4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: 16.0.26873856000000000000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} + 8x^{10} - 16x^{8} + 32x^{6} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{12} + \beta_{6} + \cdots + 1) q^{2} + (2 \beta_{14} - 2 \beta_{11} + \cdots - 2 \beta_1) q^{3} - \beta_{6} q^{4} - 2 \beta_{5} q^{6} + ( - 4 \beta_{13} + 4 \beta_{10} + \cdots + 4) q^{7}+ \cdots + 10 \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} + 10 q^{9} - 8 q^{14} + 4 q^{16} - 10 q^{18} - 8 q^{19} + 40 q^{25} - 8 q^{28} - 80 q^{32} - 32 q^{33} + 40 q^{36} + 8 q^{38} - 16 q^{39} - 4 q^{41} - 48 q^{47}+ \cdots + 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 8x^{10} - 16x^{8} + 32x^{6} - 128x^{2} + 256 \) : 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
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{12} ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{14} ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{13} + 32\nu^{3} ) / 64 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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
\(\nu^{8}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{11} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 64\beta_{12} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 64\beta_{15} - 64\beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 128\beta_{13} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 128\beta_{14} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{4}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
1.38331 − 0.294032i
−1.38331 + 0.294032i
0.946294 + 1.05097i
−0.946294 − 1.05097i
−1.29195 + 0.575212i
1.29195 − 0.575212i
−1.29195 − 0.575212i
1.29195 + 0.575212i
1.38331 + 0.294032i
−1.38331 − 0.294032i
0.946294 − 1.05097i
−0.946294 + 1.05097i
−0.147826 − 1.40647i
0.147826 + 1.40647i
−0.147826 + 1.40647i
0.147826 − 1.40647i
0.809017 − 0.587785i −2.58390 + 1.15042i −0.309017 + 0.951057i 0 −1.41421 + 2.44949i 2.67652 + 2.97258i 0.927051 + 2.85317i 3.34565 − 3.71572i 0
235.2 0.809017 − 0.587785i 2.58390 − 1.15042i −0.309017 + 0.951057i 0 1.41421 − 2.44949i 2.67652 + 2.97258i 0.927051 + 2.85317i 3.34565 − 3.71572i 0
338.1 0.809017 − 0.587785i −0.295651 + 2.81293i −0.309017 + 0.951057i 0 1.41421 + 2.44949i −3.91259 + 0.831647i 0.927051 + 2.85317i −4.89074 − 1.03956i 0
338.2 0.809017 − 0.587785i 0.295651 − 2.81293i −0.309017 + 0.951057i 0 −1.41421 − 2.44949i −3.91259 + 0.831647i 0.927051 + 2.85317i −4.89074 − 1.03956i 0
448.1 −0.309017 + 0.951057i −1.89259 + 2.10193i 0.809017 + 0.587785i 0 −1.41421 − 2.44949i −0.418114 + 3.97809i −2.42705 + 1.76336i −0.522642 − 4.97261i 0
448.2 −0.309017 + 0.951057i 1.89259 − 2.10193i 0.809017 + 0.587785i 0 1.41421 + 2.44949i −0.418114 + 3.97809i −2.42705 + 1.76336i −0.522642 − 4.97261i 0
547.1 −0.309017 − 0.951057i −1.89259 − 2.10193i 0.809017 − 0.587785i 0 −1.41421 + 2.44949i −0.418114 − 3.97809i −2.42705 − 1.76336i −0.522642 + 4.97261i 0
547.2 −0.309017 − 0.951057i 1.89259 + 2.10193i 0.809017 − 0.587785i 0 1.41421 − 2.44949i −0.418114 − 3.97809i −2.42705 − 1.76336i −0.522642 + 4.97261i 0
732.1 0.809017 + 0.587785i −2.58390 − 1.15042i −0.309017 − 0.951057i 0 −1.41421 − 2.44949i 2.67652 − 2.97258i 0.927051 − 2.85317i 3.34565 + 3.71572i 0
732.2 0.809017 + 0.587785i 2.58390 + 1.15042i −0.309017 − 0.951057i 0 1.41421 + 2.44949i 2.67652 − 2.97258i 0.927051 − 2.85317i 3.34565 + 3.71572i 0
816.1 0.809017 + 0.587785i −0.295651 − 2.81293i −0.309017 − 0.951057i 0 1.41421 − 2.44949i −3.91259 − 0.831647i 0.927051 − 2.85317i −4.89074 + 1.03956i 0
816.2 0.809017 + 0.587785i 0.295651 + 2.81293i −0.309017 − 0.951057i 0 −1.41421 + 2.44949i −3.91259 − 0.831647i 0.927051 − 2.85317i −4.89074 + 1.03956i 0
844.1 −0.309017 − 0.951057i −2.76662 + 0.588063i 0.809017 − 0.587785i 0 1.41421 + 2.44949i 3.65418 + 1.62695i −2.42705 − 1.76336i 4.56773 − 2.03368i 0
844.2 −0.309017 − 0.951057i 2.76662 − 0.588063i 0.809017 − 0.587785i 0 −1.41421 − 2.44949i 3.65418 + 1.62695i −2.42705 − 1.76336i 4.56773 − 2.03368i 0
846.1 −0.309017 + 0.951057i −2.76662 − 0.588063i 0.809017 + 0.587785i 0 1.41421 − 2.44949i 3.65418 − 1.62695i −2.42705 + 1.76336i 4.56773 + 2.03368i 0
846.2 −0.309017 + 0.951057i 2.76662 + 0.588063i 0.809017 + 0.587785i 0 −1.41421 + 2.44949i 3.65418 − 1.62695i −2.42705 + 1.76336i 4.56773 + 2.03368i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner
31.c even 3 1 inner
31.d even 5 3 inner
31.e odd 6 1 inner
31.f odd 10 3 inner
31.g even 15 3 inner
31.h odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.g.q 16
31.b odd 2 1 inner 961.2.g.q 16
31.c even 3 1 961.2.d.k 8
31.c even 3 1 inner 961.2.g.q 16
31.d even 5 1 961.2.c.b 4
31.d even 5 3 inner 961.2.g.q 16
31.e odd 6 1 961.2.d.k 8
31.e odd 6 1 inner 961.2.g.q 16
31.f odd 10 1 961.2.c.b 4
31.f odd 10 3 inner 961.2.g.q 16
31.g even 15 1 961.2.a.b ✓ 2
31.g even 15 1 961.2.c.b 4
31.g even 15 3 961.2.d.k 8
31.g even 15 3 inner 961.2.g.q 16
31.h odd 30 1 961.2.a.b ✓ 2
31.h odd 30 1 961.2.c.b 4
31.h odd 30 3 961.2.d.k 8
31.h odd 30 3 inner 961.2.g.q 16
93.o odd 30 1 8649.2.a.j 2
93.p even 30 1 8649.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.b ✓ 2 31.g even 15 1
961.2.a.b ✓ 2 31.h odd 30 1
961.2.c.b 4 31.d even 5 1
961.2.c.b 4 31.f odd 10 1
961.2.c.b 4 31.g even 15 1
961.2.c.b 4 31.h odd 30 1
961.2.d.k 8 31.c even 3 1
961.2.d.k 8 31.e odd 6 1
961.2.d.k 8 31.g even 15 3
961.2.d.k 8 31.h odd 30 3
961.2.g.q 16 1.a even 1 1 trivial
961.2.g.q 16 31.b odd 2 1 inner
961.2.g.q 16 31.c even 3 1 inner
961.2.g.q 16 31.d even 5 3 inner
961.2.g.q 16 31.e odd 6 1 inner
961.2.g.q 16 31.f odd 10 3 inner
961.2.g.q 16 31.g even 15 3 inner
961.2.g.q 16 31.h odd 30 3 inner
8649.2.a.j 2 93.o odd 30 1
8649.2.a.j 2 93.p even 30 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{16} - 8T_{3}^{14} + 512T_{3}^{10} - 4096T_{3}^{8} + 32768T_{3}^{6} - 2097152T_{3}^{2} + 16777216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 32 T^{6} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{4} + 18 T^{2} + 324)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 2 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 722204136308736 \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} + \cdots + 20736)^{4} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 11019960576 \) Copy content Toggle raw display
$59$ \( (T^{8} - 8 T^{7} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{8} \) Copy content Toggle raw display
$71$ \( (T^{8} - 8 T^{7} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11019960576 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 72\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + 50 T^{6} + \cdots + 6250000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 8 T^{3} + \cdots + 4096)^{4} \) Copy content Toggle raw display
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