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Properties

Label 961.2.g.r
Level $961$
Weight $2$
Character orbit 961.g
Analytic conductor $7.674$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,2,Mod(235,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: 16.0.26873856000000000000.2
comment: defining polynomial
 
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

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 \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
−0.946294 − 1.05097i
0.946294 + 1.05097i
1.38331 − 0.294032i
−1.38331 + 0.294032i
−0.147826 + 1.40647i
0.147826 − 1.40647i
−0.147826 − 1.40647i
0.147826 + 1.40647i
−0.946294 + 1.05097i
0.946294 − 1.05097i
1.38331 + 0.294032i
−1.38331 − 0.294032i
1.29195 + 0.575212i
−1.29195 − 0.575212i
1.29195 − 0.575212i
−1.29195 + 0.575212i
−0.335106 + 0.243469i −0.378403 + 0.168476i −0.565015 + 1.73894i −0.500000 − 0.866025i 0.0857864 − 0.148586i −0.277163 − 0.307821i −0.490035 − 1.50817i −1.89259 + 2.10193i 0.378403 + 0.168476i
235.2 1.95314 − 1.41904i 2.20549 − 0.981949i 1.18305 − 3.64105i −0.500000 − 0.866025i 2.91421 − 5.04757i 1.61542 + 1.79411i −1.36407 − 4.19817i 1.89259 − 2.10193i −2.20549 − 0.981949i
338.1 −0.335106 + 0.243469i 0.0432971 − 0.411944i −0.565015 + 1.73894i −0.500000 + 0.866025i 0.0857864 + 0.148586i 0.405162 − 0.0861198i −0.490035 − 1.50817i 2.76662 + 0.588063i −0.0432971 − 0.411944i
338.2 1.95314 − 1.41904i −0.252354 + 2.40099i 1.18305 − 3.64105i −0.500000 + 0.866025i 2.91421 + 5.04757i −2.36146 + 0.501943i −1.36407 − 4.19817i −2.76662 − 0.588063i 0.252354 + 2.40099i
448.1 −0.746033 + 2.29605i 1.61542 − 1.79411i −3.09726 − 2.25029i −0.500000 + 0.866025i 2.91421 + 5.04757i −0.252354 + 2.40099i 3.57117 − 2.59461i −0.295651 − 2.81293i −1.61542 − 1.79411i
448.2 0.127999 − 0.393941i −0.277163 + 0.307821i 1.47923 + 1.07472i −0.500000 + 0.866025i 0.0857864 + 0.148586i 0.0432971 − 0.411944i 1.28293 − 0.932102i 0.295651 + 2.81293i 0.277163 + 0.307821i
547.1 −0.746033 − 2.29605i 1.61542 + 1.79411i −3.09726 + 2.25029i −0.500000 − 0.866025i 2.91421 − 5.04757i −0.252354 − 2.40099i 3.57117 + 2.59461i −0.295651 + 2.81293i −1.61542 + 1.79411i
547.2 0.127999 + 0.393941i −0.277163 − 0.307821i 1.47923 − 1.07472i −0.500000 − 0.866025i 0.0857864 − 0.148586i 0.0432971 + 0.411944i 1.28293 + 0.932102i 0.295651 − 2.81293i 0.277163 − 0.307821i
732.1 −0.335106 − 0.243469i −0.378403 − 0.168476i −0.565015 − 1.73894i −0.500000 + 0.866025i 0.0857864 + 0.148586i −0.277163 + 0.307821i −0.490035 + 1.50817i −1.89259 − 2.10193i 0.378403 − 0.168476i
732.2 1.95314 + 1.41904i 2.20549 + 0.981949i 1.18305 + 3.64105i −0.500000 + 0.866025i 2.91421 + 5.04757i 1.61542 − 1.79411i −1.36407 + 4.19817i 1.89259 + 2.10193i −2.20549 + 0.981949i
816.1 −0.335106 − 0.243469i 0.0432971 + 0.411944i −0.565015 − 1.73894i −0.500000 − 0.866025i 0.0857864 − 0.148586i 0.405162 + 0.0861198i −0.490035 + 1.50817i 2.76662 − 0.588063i −0.0432971 + 0.411944i
816.2 1.95314 + 1.41904i −0.252354 − 2.40099i 1.18305 + 3.64105i −0.500000 − 0.866025i 2.91421 − 5.04757i −2.36146 − 0.501943i −1.36407 + 4.19817i −2.76662 + 0.588063i 0.252354 − 2.40099i
844.1 −0.746033 − 2.29605i −2.36146 + 0.501943i −3.09726 + 2.25029i −0.500000 + 0.866025i 2.91421 + 5.04757i 2.20549 + 0.981949i 3.57117 + 2.59461i 2.58390 − 1.15042i 2.36146 + 0.501943i
844.2 0.127999 + 0.393941i 0.405162 − 0.0861198i 1.47923 − 1.07472i −0.500000 + 0.866025i 0.0857864 + 0.148586i −0.378403 − 0.168476i 1.28293 + 0.932102i −2.58390 + 1.15042i −0.405162 − 0.0861198i
846.1 −0.746033 + 2.29605i −2.36146 − 0.501943i −3.09726 − 2.25029i −0.500000 − 0.866025i 2.91421 − 5.04757i 2.20549 − 0.981949i 3.57117 − 2.59461i 2.58390 + 1.15042i 2.36146 − 0.501943i
846.2 0.127999 − 0.393941i 0.405162 + 0.0861198i 1.47923 + 1.07472i −0.500000 − 0.866025i 0.0857864 − 0.148586i −0.378403 + 0.168476i 1.28293 − 0.932102i −2.58390 − 1.15042i −0.405162 + 0.0861198i
\(n\): 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.c even 3 1 inner
31.d even 5 3 inner
31.g even 15 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.g.r 16
31.b odd 2 1 961.2.g.o 16
31.c even 3 1 961.2.d.i 8
31.c even 3 1 inner 961.2.g.r 16
31.d even 5 1 961.2.c.a 4
31.d even 5 3 inner 961.2.g.r 16
31.e odd 6 1 961.2.d.l 8
31.e odd 6 1 961.2.g.o 16
31.f odd 10 1 31.2.c.a ✓ 4
31.f odd 10 3 961.2.g.o 16
31.g even 15 1 961.2.a.c 2
31.g even 15 1 961.2.c.a 4
31.g even 15 3 961.2.d.i 8
31.g even 15 3 inner 961.2.g.r 16
31.h odd 30 1 31.2.c.a ✓ 4
31.h odd 30 1 961.2.a.a 2
31.h odd 30 3 961.2.d.l 8
31.h odd 30 3 961.2.g.o 16
93.k even 10 1 279.2.h.c 4
93.o odd 30 1 8649.2.a.k 2
93.p even 30 1 279.2.h.c 4
93.p even 30 1 8649.2.a.l 2
124.j even 10 1 496.2.i.h 4
124.p even 30 1 496.2.i.h 4
155.m odd 10 1 775.2.e.e 4
155.r even 20 2 775.2.o.d 8
155.v odd 30 1 775.2.e.e 4
155.x even 60 2 775.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a ✓ 4 31.f odd 10 1
31.2.c.a ✓ 4 31.h odd 30 1
279.2.h.c 4 93.k even 10 1
279.2.h.c 4 93.p even 30 1
496.2.i.h 4 124.j even 10 1
496.2.i.h 4 124.p even 30 1
775.2.e.e 4 155.m odd 10 1
775.2.e.e 4 155.v odd 30 1
775.2.o.d 8 155.r even 20 2
775.2.o.d 8 155.x even 60 2
961.2.a.a 2 31.h odd 30 1
961.2.a.c 2 31.g even 15 1
961.2.c.a 4 31.d even 5 1
961.2.c.a 4 31.g even 15 1
961.2.d.i 8 31.c even 3 1
961.2.d.i 8 31.g even 15 3
961.2.d.l 8 31.e odd 6 1
961.2.d.l 8 31.h odd 30 3
961.2.g.o 16 31.b odd 2 1
961.2.g.o 16 31.e odd 6 1
961.2.g.o 16 31.f odd 10 3
961.2.g.o 16 31.h odd 30 3
961.2.g.r 16 1.a even 1 1 trivial
961.2.g.r 16 31.c even 3 1 inner
961.2.g.r 16 31.d even 5 3 inner
961.2.g.r 16 31.g even 15 3 inner
8649.2.a.k 2 93.o odd 30 1
8649.2.a.l 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 \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 5T_{2}^{6} - 12T_{2}^{5} + 29T_{2}^{4} + 12T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{16} - 2 T_{3}^{15} - T_{3}^{14} + 14 T_{3}^{13} - 28 T_{3}^{12} + 68 T_{3}^{11} + 33 T_{3}^{10} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + 5 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$17$ \( T^{16} - 6 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} - 6 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 8 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 645753531245761 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 78\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 6 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 7984925229121 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} + \cdots + 289)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + 14 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{16} + 2 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 7984925229121 \) Copy content Toggle raw display
$89$ \( (T^{8} + 8 T^{7} + \cdots + 9834496)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 16 T^{7} + \cdots + 9834496)^{2} \) Copy content Toggle raw display
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