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Properties

Label 1568.3.c.h
Level $1568$
Weight $3$
Character orbit 1568.c
Analytic conductor $42.725$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,3,Mod(97,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.c (of order \(2\), degree \(1\), 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: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{12} q^{5} + (\beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{12} q^{5} + (\beta_{3} - 1) q^{9} + (\beta_{14} - \beta_{10}) q^{11} + (\beta_{15} - \beta_{2}) q^{13} + \beta_{5} q^{15} + (\beta_{13} + 2 \beta_{12} - 2 \beta_{2}) q^{17} + ( - \beta_{9} - 3 \beta_{7} + \cdots + 3 \beta_1) q^{19}+ \cdots + (9 \beta_{14} - 9 \beta_{10} + 5 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 113 \nu^{14} + 3818 \nu^{12} + 51075 \nu^{10} + 313505 \nu^{8} + 843923 \nu^{6} + 473319 \nu^{4} + \cdots - 557487 ) / 2711772 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4 \nu^{15} - 158 \nu^{13} - 2599 \nu^{11} - 22087 \nu^{9} - 102335 \nu^{7} - 245347 \nu^{5} + \cdots - 71343 \nu ) / 128520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 113 \nu^{14} - 3818 \nu^{12} - 51075 \nu^{10} - 313505 \nu^{8} - 843923 \nu^{6} + \cdots + 12760461 ) / 1355886 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1349 \nu^{15} - 51424 \nu^{13} - 794798 \nu^{11} - 6024806 \nu^{9} - 22337746 \nu^{7} + \cdots + 17661258 \nu ) / 27117720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3289 \nu^{15} + 108644 \nu^{13} + 1373628 \nu^{11} + 7277106 \nu^{9} + 11587916 \nu^{7} + \cdots - 113485638 \nu ) / 27117720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43 \nu^{14} + 1662 \nu^{12} + 26337 \nu^{10} + 209513 \nu^{8} + 861381 \nu^{6} + 1621011 \nu^{4} + \cdots - 51135 ) / 132930 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5167 \nu^{14} - 179273 \nu^{12} - 2460748 \nu^{10} - 15568297 \nu^{8} - 42862214 \nu^{6} + \cdots + 24667650 ) / 13558860 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 477 \nu^{14} + 17097 \nu^{12} + 244996 \nu^{10} + 1653025 \nu^{8} + 4923894 \nu^{6} + \cdots - 5789742 ) / 903924 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1307 \nu^{14} + 47303 \nu^{12} + 694458 \nu^{10} + 4988477 \nu^{8} + 17919584 \nu^{6} + \cdots - 3887460 ) / 1936980 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 977 \nu^{15} - 34607 \nu^{13} - 490904 \nu^{11} - 3304813 \nu^{9} - 10370438 \nu^{7} + \cdots + 22202004 \nu ) / 3389715 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 969 \nu^{14} - 33812 \nu^{12} - 467441 \nu^{10} - 3001917 \nu^{8} - 8492381 \nu^{6} + \cdots + 28721385 ) / 1355886 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9238 \nu^{15} - 334664 \nu^{13} - 4891055 \nu^{11} - 34544651 \nu^{9} - 118066943 \nu^{7} + \cdots - 61542453 \nu ) / 27117720 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3602 \nu^{15} - 137612 \nu^{13} - 2171849 \nu^{11} - 17484433 \nu^{9} - 75870593 \nu^{7} + \cdots - 50730891 \nu ) / 9039240 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16689 \nu^{15} + 609984 \nu^{13} + 9012548 \nu^{11} + 64598666 \nu^{9} + 223617876 \nu^{7} + \cdots - 300491778 \nu ) / 27117720 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18622 \nu^{15} + 689692 \nu^{13} + 10442779 \nu^{11} + 78891803 \nu^{9} + 310575283 \nu^{7} + \cdots + 191518401 \nu ) / 27117720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{13} + \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{15} - 6\beta_{14} - 9\beta_{13} - 6\beta_{12} - 13\beta_{10} - 6\beta_{5} - 12\beta_{4} + 24\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} + 2\beta_{9} - 2\beta_{7} - 2\beta_{6} - 9\beta_{3} - 36\beta _1 + 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20 \beta_{15} + 45 \beta_{14} + 33 \beta_{13} + 15 \beta_{12} + 92 \beta_{10} + 45 \beta_{5} + \cdots - 140 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -11\beta_{11} - 49\beta_{9} - 6\beta_{8} + 13\beta_{7} + 52\beta_{6} + 53\beta_{3} + 502\beta _1 - 303 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -49\beta_{15} - 1176\beta_{14} - 201\beta_{13} - 2239\beta_{10} - 1008\beta_{5} - 3976\beta_{4} + 1400\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\beta_{11} + 780\beta_{9} + 36\beta_{8} + 12\beta_{7} - 816\beta_{6} + 125\beta_{3} - 6032\beta _1 - 1401 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3575 \beta_{15} + 13362 \beta_{14} - 5061 \beta_{13} - 2730 \beta_{12} + 24133 \beta_{10} + \cdots + 16920 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1577 \beta_{11} - 9437 \beta_{9} + 630 \beta_{8} - 1291 \beta_{7} + 9806 \beta_{6} - 10554 \beta_{3} + \cdots + 70056 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 40984 \beta_{15} - 64251 \beta_{14} + 71067 \beta_{13} + 27165 \beta_{12} - 112052 \beta_{10} + \cdots - 308440 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 19787 \beta_{11} + 43874 \beta_{9} - 10410 \beta_{8} + 8812 \beta_{7} - 45437 \beta_{6} + \cdots - 671586 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1304123 \beta_{15} + 925548 \beta_{14} - 2431239 \beta_{13} - 810468 \beta_{12} + \cdots + 11304592 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 644022 \beta_{11} - 505800 \beta_{9} + 363636 \beta_{8} - 133956 \beta_{7} + 522108 \beta_{6} + \cdots + 19796271 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17302151 \beta_{15} - 1830030 \beta_{14} + 33379797 \beta_{13} + 10397226 \beta_{12} + \cdots - 159843144 \beta_{2} ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.707107 − 3.42121i
−0.707107 + 3.42121i
−0.707107 + 2.60548i
0.707107 − 2.60548i
0.707107 − 1.17406i
−0.707107 + 1.17406i
0.707107 − 0.358323i
−0.707107 + 0.358323i
−0.707107 − 0.358323i
0.707107 + 0.358323i
−0.707107 − 1.17406i
0.707107 + 1.17406i
0.707107 + 2.60548i
−0.707107 − 2.60548i
−0.707107 − 3.42121i
0.707107 + 3.42121i
0 − 4.83832i 0 − 0.0515539i 0 0 0 −14.4094 0
97.2 0 − 4.83832i 0 0.0515539i 0 0 0 −14.4094 0
97.3 0 − 3.68470i 0 − 3.04770i 0 0 0 −4.57700 0
97.4 0 − 3.68470i 0 3.04770i 0 0 0 −4.57700 0
97.5 0 − 1.66037i 0 − 8.40186i 0 0 0 6.24317 0
97.6 0 − 1.66037i 0 8.40186i 0 0 0 6.24317 0
97.7 0 − 0.506745i 0 − 5.30261i 0 0 0 8.74321 0
97.8 0 − 0.506745i 0 5.30261i 0 0 0 8.74321 0
97.9 0 0.506745i 0 − 5.30261i 0 0 0 8.74321 0
97.10 0 0.506745i 0 5.30261i 0 0 0 8.74321 0
97.11 0 1.66037i 0 − 8.40186i 0 0 0 6.24317 0
97.12 0 1.66037i 0 8.40186i 0 0 0 6.24317 0
97.13 0 3.68470i 0 − 3.04770i 0 0 0 −4.57700 0
97.14 0 3.68470i 0 3.04770i 0 0 0 −4.57700 0
97.15 0 4.83832i 0 − 0.0515539i 0 0 0 −14.4094 0
97.16 0 4.83832i 0 0.0515539i 0 0 0 −14.4094 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.c.h 16
4.b odd 2 1 inner 1568.3.c.h 16
7.b odd 2 1 inner 1568.3.c.h 16
7.c even 3 1 224.3.s.a ✓ 16
7.d odd 6 1 224.3.s.a ✓ 16
28.d even 2 1 inner 1568.3.c.h 16
28.f even 6 1 224.3.s.a ✓ 16
28.g odd 6 1 224.3.s.a ✓ 16
56.j odd 6 1 448.3.s.g 16
56.k odd 6 1 448.3.s.g 16
56.m even 6 1 448.3.s.g 16
56.p even 6 1 448.3.s.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.a ✓ 16 7.c even 3 1
224.3.s.a ✓ 16 7.d odd 6 1
224.3.s.a ✓ 16 28.f even 6 1
224.3.s.a ✓ 16 28.g odd 6 1
448.3.s.g 16 56.j odd 6 1
448.3.s.g 16 56.k odd 6 1
448.3.s.g 16 56.m even 6 1
448.3.s.g 16 56.p even 6 1
1568.3.c.h 16 1.a even 1 1 trivial
1568.3.c.h 16 4.b odd 2 1 inner
1568.3.c.h 16 7.b odd 2 1 inner
1568.3.c.h 16 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{8} + 40T_{3}^{6} + 430T_{3}^{4} + 984T_{3}^{2} + 225 \) Copy content Toggle raw display
\( T_{5}^{8} + 108T_{5}^{6} + 2902T_{5}^{4} + 18444T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{8} - 328T_{11}^{6} + 10750T_{11}^{4} - 75000T_{11}^{2} + 140625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 40 T^{6} + \cdots + 225)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 108 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 328 T^{6} + \cdots + 140625)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1160 T^{6} + \cdots + 243360000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1356 T^{6} + \cdots + 308880625)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2360 T^{6} + \cdots + 34679750625)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 3304 T^{6} + \cdots + 236910960225)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 28 T^{3} + \cdots + 1155216)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3208 T^{6} + \cdots + 48059600625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots - 234375)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10159773753600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 3616 T^{6} + \cdots + 34857216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 16335961317729)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 484425)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 6739086200625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 6716 T^{6} + \cdots + 371063169)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 568716467841729)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 19561159840000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 115300885730625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 62\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 179911788134400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 41805935994001)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 161218887840000)^{2} \) Copy content Toggle raw display
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