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Properties

Label 1456.2.r.p
Level $1456$
Weight $2$
Character orbit 1456.r
Analytic conductor $11.626$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(417,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), degree \(2\), 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} + ( - \beta_{7} - \beta_{3}) q^{5} + (\beta_{9} + \beta_{5} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} + ( - \beta_{7} - \beta_{3}) q^{5} + (\beta_{9} + \beta_{5} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{9} + ( - 2 \beta_{6} + \beta_{5} + 2) q^{11} + q^{13} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{15} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - q^{7} - 3 q^{9} + 11 q^{11} + 10 q^{13} + 5 q^{17} + 9 q^{19} + 2 q^{21} + 10 q^{23} - 9 q^{25} - 6 q^{29} - 6 q^{31} - 8 q^{33} + 4 q^{35} - 4 q^{37} + 28 q^{41} - 4 q^{43} + 32 q^{45} + q^{47} - 11 q^{49} - 8 q^{51} - 17 q^{53} - 32 q^{57} + 11 q^{59} + 11 q^{61} - 5 q^{63} - 2 q^{65} + 13 q^{67} + 36 q^{69} - 30 q^{71} - 20 q^{75} - 46 q^{77} + 2 q^{79} + 19 q^{81} - 12 q^{83} - 44 q^{85} - 8 q^{87} + 4 q^{89} - q^{91} - 18 q^{93} - 12 q^{95} - 24 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 983 \nu^{9} + 7328 \nu^{8} - 9160 \nu^{7} + 87336 \nu^{6} + 36640 \nu^{5} + 287624 \nu^{4} + \cdots + 22604 ) / 118350 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + \cdots - 348592 ) / 118350 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} + \cdots - 180842 ) / 59175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 629 \nu^{9} - 3164 \nu^{8} + 3955 \nu^{7} - 40293 \nu^{6} - 15820 \nu^{5} - 124187 \nu^{4} + \cdots - 95852 ) / 39450 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8713 \nu^{9} + 13967 \nu^{8} - 61840 \nu^{7} - 70821 \nu^{6} - 166865 \nu^{5} - 117514 \nu^{4} + \cdots - 33244 ) / 236700 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} + \cdots - 2692 ) / 236700 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16307 \nu^{9} - 15688 \nu^{8} + 137960 \nu^{7} + 104769 \nu^{6} + 750010 \nu^{5} + 331046 \nu^{4} + \cdots + 362216 ) / 118350 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1058 \nu^{9} + 1552 \nu^{8} - 9041 \nu^{7} - 3726 \nu^{6} - 39580 \nu^{5} - 2993 \nu^{4} + \cdots + 136 ) / 4734 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9203 \nu^{9} - 9652 \nu^{8} + 71240 \nu^{7} + 67401 \nu^{6} + 346240 \nu^{5} + 153734 \nu^{4} + \cdots + 115064 ) / 39450 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{9} + \beta_{8} + 6\beta_{6} + 2\beta_{5} + \beta_{4} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} + \beta_{3} - 5\beta_{2} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -29\beta_{9} - 13\beta_{8} + 4\beta_{7} - 38\beta_{6} - 18\beta_{5} + 4\beta_{3} - 29\beta_{2} - 18\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -91\beta_{9} - 47\beta_{8} + 18\beta_{7} - 90\beta_{6} - 46\beta_{5} - 47\beta_{4} + 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -64\beta_{4} - 23\beta_{3} + 134\beta_{2} + 78\beta _1 + 154 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 819 \beta_{9} + 407 \beta_{8} - 156 \beta_{7} + 862 \beta_{6} + 442 \beta_{5} - 156 \beta_{3} + \cdots + 442 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2443\beta_{9} + 1187\beta_{8} - 442\beta_{7} + 2698\beta_{6} + 1382\beta_{5} + 1187\beta_{4} - 2698 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1817\beta_{4} + 691\beta_{3} - 3695\beta_{2} - 2036\beta _1 - 3968 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
−0.132804 − 0.230024i
−0.862625 − 1.49411i
0.597828 + 1.03547i
1.50426 + 2.60546i
−0.606661 − 1.05077i
−0.132804 + 0.230024i
−0.862625 + 1.49411i
0.597828 − 1.03547i
1.50426 − 2.60546i
−0.606661 + 1.05077i
0 −1.31364 + 2.27529i 0 1.45130 + 2.51373i 0 1.29536 − 2.30696i 0 −1.95130 − 3.37975i 0
417.2 0 −0.673208 + 1.16603i 0 −1.09358 − 1.89414i 0 2.19729 + 1.47375i 0 0.593582 + 1.02811i 0
417.3 0 −0.129894 + 0.224983i 0 −1.96625 − 3.40565i 0 −1.12324 − 2.39548i 0 1.46625 + 2.53963i 0
417.4 0 0.879528 − 1.52339i 0 −0.452861 − 0.784378i 0 −0.237709 + 2.63505i 0 −0.0471392 − 0.0816475i 0
417.5 0 1.23721 − 2.14292i 0 1.06140 + 1.83839i 0 −2.63169 − 0.272389i 0 −1.56140 − 2.70442i 0
625.1 0 −1.31364 − 2.27529i 0 1.45130 − 2.51373i 0 1.29536 + 2.30696i 0 −1.95130 + 3.37975i 0
625.2 0 −0.673208 − 1.16603i 0 −1.09358 + 1.89414i 0 2.19729 − 1.47375i 0 0.593582 − 1.02811i 0
625.3 0 −0.129894 − 0.224983i 0 −1.96625 + 3.40565i 0 −1.12324 + 2.39548i 0 1.46625 − 2.53963i 0
625.4 0 0.879528 + 1.52339i 0 −0.452861 + 0.784378i 0 −0.237709 − 2.63505i 0 −0.0471392 + 0.0816475i 0
625.5 0 1.23721 + 2.14292i 0 1.06140 − 1.83839i 0 −2.63169 + 0.272389i 0 −1.56140 + 2.70442i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.r.p 10
4.b odd 2 1 91.2.e.c ✓ 10
7.c even 3 1 inner 1456.2.r.p 10
12.b even 2 1 819.2.j.h 10
28.d even 2 1 637.2.e.m 10
28.f even 6 1 637.2.a.k 5
28.f even 6 1 637.2.e.m 10
28.g odd 6 1 91.2.e.c ✓ 10
28.g odd 6 1 637.2.a.l 5
52.b odd 2 1 1183.2.e.f 10
84.j odd 6 1 5733.2.a.bm 5
84.n even 6 1 819.2.j.h 10
84.n even 6 1 5733.2.a.bl 5
364.x even 6 1 8281.2.a.bx 5
364.bl odd 6 1 1183.2.e.f 10
364.bl odd 6 1 8281.2.a.bw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c ✓ 10 4.b odd 2 1
91.2.e.c ✓ 10 28.g odd 6 1
637.2.a.k 5 28.f even 6 1
637.2.a.l 5 28.g odd 6 1
637.2.e.m 10 28.d even 2 1
637.2.e.m 10 28.f even 6 1
819.2.j.h 10 12.b even 2 1
819.2.j.h 10 84.n even 6 1
1183.2.e.f 10 52.b odd 2 1
1183.2.e.f 10 364.bl odd 6 1
1456.2.r.p 10 1.a even 1 1 trivial
1456.2.r.p 10 7.c even 3 1 inner
5733.2.a.bl 5 84.n even 6 1
5733.2.a.bm 5 84.j odd 6 1
8281.2.a.bw 5 364.bl odd 6 1
8281.2.a.bx 5 364.x even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3}^{10} + 9T_{3}^{8} + 65T_{3}^{6} + 4T_{3}^{5} + 144T_{3}^{4} + 72T_{3}^{3} + 256T_{3}^{2} + 64T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{10} + 2 T_{5}^{9} + 19 T_{5}^{8} + 10 T_{5}^{7} + 217 T_{5}^{6} + 156 T_{5}^{5} + 1024 T_{5}^{4} + \cdots + 2304 \) Copy content Toggle raw display
\( T_{11}^{10} - 11 T_{11}^{9} + 85 T_{11}^{8} - 352 T_{11}^{7} + 1099 T_{11}^{6} - 1749 T_{11}^{5} + \cdots + 1089 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} - 11 T^{9} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( (T - 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} + \cdots + 184041 \) Copy content Toggle raw display
$19$ \( T^{10} - 9 T^{9} + \cdots + 49729 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{5} + 3 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 6 T^{9} + \cdots + 126736 \) Copy content Toggle raw display
$37$ \( T^{10} + 4 T^{9} + \cdots + 49505296 \) Copy content Toggle raw display
$41$ \( (T^{5} - 14 T^{4} + \cdots - 1584)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 2 T^{4} - 72 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - T^{9} + \cdots + 26718561 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 398361681 \) Copy content Toggle raw display
$59$ \( T^{10} - 11 T^{9} + \cdots + 1089 \) Copy content Toggle raw display
$61$ \( T^{10} - 11 T^{9} + \cdots + 71588521 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 515244601 \) Copy content Toggle raw display
$71$ \( (T^{5} + 15 T^{4} + \cdots + 6336)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 75 T^{8} + \cdots + 506944 \) Copy content Toggle raw display
$79$ \( T^{10} - 2 T^{9} + \cdots + 1000000 \) Copy content Toggle raw display
$83$ \( (T^{5} + 6 T^{4} + \cdots + 7488)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 4 T^{9} + \cdots + 59166864 \) Copy content Toggle raw display
$97$ \( (T^{5} + 12 T^{4} + \cdots - 2384)^{2} \) Copy content Toggle raw display
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