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Properties

Label 1456.2.r.r
Level $1456$
Weight $2$
Character orbit 1456.r
Analytic conductor $11.626$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 20 x^{14} - 26 x^{13} + 249 x^{12} - 317 x^{11} + 1623 x^{10} - 1655 x^{9} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 728)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{10} + \beta_{4}) q^{5} + ( - \beta_{12} + \beta_{5}) q^{7} + (\beta_{11} + \beta_{9} + \cdots + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{10} + \beta_{4}) q^{5} + ( - \beta_{12} + \beta_{5}) q^{7} + (\beta_{11} + \beta_{9} + \cdots + \beta_{5}) q^{9}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} - q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} - q^{7} - 12 q^{9} + 3 q^{11} + 16 q^{13} + 14 q^{15} - 3 q^{17} - q^{19} + 17 q^{21} - 6 q^{23} - 28 q^{25} - 10 q^{27} + 22 q^{29} - 15 q^{31} - 22 q^{33} + 6 q^{35} - 12 q^{37} + 2 q^{39} - 12 q^{41} - 12 q^{43} + 11 q^{45} + 9 q^{47} + 21 q^{49} + 7 q^{51} - 29 q^{53} - 16 q^{55} + 40 q^{57} - 21 q^{59} - 11 q^{61} - 46 q^{63} + 37 q^{67} - 84 q^{69} + 22 q^{71} + 4 q^{73} + 72 q^{75} + 30 q^{77} + 4 q^{79} - 40 q^{81} + 36 q^{83} + 94 q^{85} - 18 q^{87} - 14 q^{89} - q^{91} + 22 q^{93} - 2 q^{95} - 52 q^{97} - 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 20 x^{14} - 26 x^{13} + 249 x^{12} - 317 x^{11} + 1623 x^{10} - 1655 x^{9} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51698621240423 \nu^{15} - 160041918668022 \nu^{14} + 996911979148320 \nu^{13} + \cdots - 61\!\cdots\!44 ) / 71\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 703668782146181 \nu^{15} + \cdots + 31\!\cdots\!20 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 306573688375884 \nu^{15} - 912389654546025 \nu^{14} + \cdots - 13\!\cdots\!32 ) / 53\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\!\cdots\!43 \nu^{15} + \cdots - 54\!\cdots\!64 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!86 \nu^{15} + \cdots + 30\!\cdots\!84 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 75\!\cdots\!76 \nu^{15} + \cdots - 82\!\cdots\!00 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38\!\cdots\!59 \nu^{15} + \cdots + 69\!\cdots\!12 ) / 71\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!79 \nu^{15} + \cdots - 67\!\cdots\!36 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68\!\cdots\!85 \nu^{15} + \cdots + 46\!\cdots\!04 ) / 53\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!12 \nu^{15} + \cdots + 98\!\cdots\!48 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!78 \nu^{15} + \cdots + 44\!\cdots\!12 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33\!\cdots\!56 \nu^{15} + \cdots + 52\!\cdots\!72 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 41\!\cdots\!09 \nu^{15} + \cdots + 12\!\cdots\!96 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!79 \nu^{15} + \cdots - 63\!\cdots\!28 ) / 53\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} - 5\beta_{8} + \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + \beta_{9} + \beta_{6} + 2 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} + \beta_{14} - 8 \beta_{13} + \beta_{12} - \beta_{11} + 3 \beta_{10} - 10 \beta_{9} + \cdots - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 13 \beta_{14} + 14 \beta_{13} + 21 \beta_{12} - 25 \beta_{11} + 18 \beta_{10} + \cdots - 71 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{15} + 18 \beta_{14} + 103 \beta_{13} + 13 \beta_{12} - 98 \beta_{11} - 18 \beta_{9} + \cdots + 370 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 153 \beta_{15} - 35 \beta_{14} + 45 \beta_{13} - 55 \beta_{12} + 35 \beta_{11} - 248 \beta_{10} + \cdots + 395 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 187 \beta_{15} - 440 \beta_{14} - 467 \beta_{13} - 371 \beta_{12} + 1190 \beta_{11} - 804 \beta_{10} + \cdots + 900 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1286 \beta_{15} - 1286 \beta_{14} - 2761 \beta_{13} - 1392 \beta_{12} + 2867 \beta_{11} + 1286 \beta_{9} + \cdots - 5535 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5451 \beta_{15} + 2208 \beta_{14} - 6098 \beta_{13} + 2668 \beta_{12} - 2208 \beta_{11} + \cdots - 35477 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6113 \beta_{15} + 20330 \beta_{14} + 23776 \beta_{13} + 22744 \beta_{12} - 38685 \beta_{11} + \cdots - 67649 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 39717 \beta_{15} + 39717 \beta_{14} + 133386 \beta_{13} + 23412 \beta_{12} - 117081 \beta_{11} + \cdots + 369728 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 233427 \beta_{15} - 74517 \beta_{14} + 126225 \beta_{13} - 114654 \beta_{12} + 74517 \beta_{11} + \cdots + 886554 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 292479 \beta_{15} - 766785 \beta_{14} - 854157 \beta_{13} - 662862 \beta_{12} + 1599410 \beta_{11} + \cdots + 1849347 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1790705 \beta_{15} - 1790705 \beta_{14} - 4692587 \beta_{13} - 1361671 \beta_{12} + 4263553 \beta_{11} + \cdots - 10685240 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{8}\)

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
−1.38669 + 2.40181i
−1.35948 + 2.35468i
−0.289716 + 0.501804i
−0.0449643 + 0.0778805i
0.436176 − 0.755479i
0.675133 − 1.16937i
1.27520 − 2.20870i
1.69434 − 2.93468i
−1.38669 − 2.40181i
−1.35948 − 2.35468i
−0.289716 − 0.501804i
−0.0449643 − 0.0778805i
0.436176 + 0.755479i
0.675133 + 1.16937i
1.27520 + 2.20870i
1.69434 + 2.93468i
0 −1.38669 + 2.40181i 0 −1.35960 − 2.35490i 0 2.46190 + 0.969047i 0 −2.34580 − 4.06304i 0
417.2 0 −1.35948 + 2.35468i 0 0.838875 + 1.45297i 0 −0.839489 − 2.50904i 0 −2.19636 − 3.80421i 0
417.3 0 −0.289716 + 0.501804i 0 0.570542 + 0.988207i 0 −2.50816 + 0.842101i 0 1.33213 + 2.30731i 0
417.4 0 −0.0449643 + 0.0778805i 0 −1.23909 − 2.14617i 0 −1.58068 − 2.12166i 0 1.49596 + 2.59107i 0
417.5 0 0.436176 − 0.755479i 0 2.00964 + 3.48079i 0 −2.27149 + 1.35659i 0 1.11950 + 1.93903i 0
417.6 0 0.675133 − 1.16937i 0 −0.765707 − 1.32624i 0 2.45141 − 0.995283i 0 0.588390 + 1.01912i 0
417.7 0 1.27520 − 2.20870i 0 −2.02238 − 3.50286i 0 −0.676773 + 2.55773i 0 −1.75225 − 3.03498i 0
417.8 0 1.69434 − 2.93468i 0 1.96773 + 3.40820i 0 2.46329 − 0.965515i 0 −4.24157 − 7.34662i 0
625.1 0 −1.38669 − 2.40181i 0 −1.35960 + 2.35490i 0 2.46190 − 0.969047i 0 −2.34580 + 4.06304i 0
625.2 0 −1.35948 − 2.35468i 0 0.838875 − 1.45297i 0 −0.839489 + 2.50904i 0 −2.19636 + 3.80421i 0
625.3 0 −0.289716 − 0.501804i 0 0.570542 − 0.988207i 0 −2.50816 − 0.842101i 0 1.33213 − 2.30731i 0
625.4 0 −0.0449643 − 0.0778805i 0 −1.23909 + 2.14617i 0 −1.58068 + 2.12166i 0 1.49596 − 2.59107i 0
625.5 0 0.436176 + 0.755479i 0 2.00964 − 3.48079i 0 −2.27149 − 1.35659i 0 1.11950 − 1.93903i 0
625.6 0 0.675133 + 1.16937i 0 −0.765707 + 1.32624i 0 2.45141 + 0.995283i 0 0.588390 − 1.01912i 0
625.7 0 1.27520 + 2.20870i 0 −2.02238 + 3.50286i 0 −0.676773 − 2.55773i 0 −1.75225 + 3.03498i 0
625.8 0 1.69434 + 2.93468i 0 1.96773 − 3.40820i 0 2.46329 + 0.965515i 0 −4.24157 + 7.34662i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.8
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.r 16
4.b odd 2 1 728.2.r.g ✓ 16
7.c even 3 1 inner 1456.2.r.r 16
28.f even 6 1 5096.2.a.bb 8
28.g odd 6 1 728.2.r.g ✓ 16
28.g odd 6 1 5096.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.r.g ✓ 16 4.b odd 2 1
728.2.r.g ✓ 16 28.g odd 6 1
1456.2.r.r 16 1.a even 1 1 trivial
1456.2.r.r 16 7.c even 3 1 inner
5096.2.a.bb 8 28.f even 6 1
5096.2.a.be 8 28.g odd 6 1

Hecke kernels

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

\( T_{3}^{16} - 2 T_{3}^{15} + 20 T_{3}^{14} - 26 T_{3}^{13} + 249 T_{3}^{12} - 317 T_{3}^{11} + 1623 T_{3}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{16} + 34 T_{5}^{14} + 18 T_{5}^{13} + 797 T_{5}^{12} + 477 T_{5}^{11} + 9831 T_{5}^{10} + \cdots + 1597696 \) Copy content Toggle raw display
\( T_{11}^{16} - 3 T_{11}^{15} + 61 T_{11}^{14} - 116 T_{11}^{13} + 2335 T_{11}^{12} - 4195 T_{11}^{11} + \cdots + 4910656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} + 34 T^{14} + \cdots + 1597696 \) Copy content Toggle raw display
$7$ \( T^{16} + T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 3 T^{15} + \cdots + 4910656 \) Copy content Toggle raw display
$13$ \( (T - 1)^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 5335572025 \) Copy content Toggle raw display
$19$ \( T^{16} + T^{15} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 4067378176 \) Copy content Toggle raw display
$29$ \( (T^{8} - 11 T^{7} + \cdots - 169504)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 63191904400 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1173199504 \) Copy content Toggle raw display
$41$ \( (T^{8} + 6 T^{7} + \cdots + 77696)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 6 T^{7} + \cdots - 976896)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3523785980625 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 7468064666176 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 236157121600 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 688568040000 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 562992107584 \) Copy content Toggle raw display
$71$ \( (T^{8} - 11 T^{7} + \cdots - 3497760)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 13271040000 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 300051129696256 \) Copy content Toggle raw display
$83$ \( (T^{8} - 18 T^{7} + \cdots + 151552)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 78198224536576 \) Copy content Toggle raw display
$97$ \( (T^{8} + 26 T^{7} + \cdots - 15488)^{2} \) Copy content Toggle raw display
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