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Properties

Label 560.5.d.a
Level $560$
Weight $5$
Character orbit 560.d
Analytic conductor $57.887$
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] = [560,5,Mod(351,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.351");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 560.d (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: \(57.8871793270\)
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: \( x^{16} + 1010 x^{14} + 402663 x^{12} + 80209516 x^{10} + 8304007999 x^{8} + 421985349234 x^{6} + \cdots + 20681247856896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 5^{4}\cdot 7^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{12} q^{7} + (\beta_{4} + 3 \beta_{2} - 45) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{12} q^{7} + (\beta_{4} + 3 \beta_{2} - 45) q^{9} + (\beta_{13} + \beta_{11} - \beta_{10}) q^{11} + ( - \beta_{5} - \beta_{4} + 5 \beta_{2} + 25) q^{13} + (\beta_{11} + \beta_{10} + 3 \beta_1) q^{15} + (\beta_{8} + \beta_{7} + 4 \beta_{2} - 45) q^{17} + ( - \beta_{14} + \beta_{13} + \cdots + 10 \beta_1) q^{19}+ \cdots + ( - 16 \beta_{15} + 30 \beta_{14} + \cdots - 172 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 724 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 724 q^{9} + 408 q^{13} - 720 q^{17} - 196 q^{21} + 2000 q^{25} + 4860 q^{29} - 456 q^{33} - 3416 q^{37} + 1632 q^{41} - 5200 q^{45} - 5488 q^{49} + 14400 q^{53} - 19272 q^{57} - 64 q^{61} - 9900 q^{65} + 18856 q^{69} - 21192 q^{73} + 43664 q^{81} - 8700 q^{85} + 55848 q^{89} - 5048 q^{93} - 5984 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1010 x^{14} + 402663 x^{12} + 80209516 x^{10} + 8304007999 x^{8} + 421985349234 x^{6} + \cdots + 20681247856896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7916051581 \nu^{14} - 7091457376733 \nu^{12} + \cdots + 19\!\cdots\!72 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10158383071 \nu^{14} + 17990847719573 \nu^{12} + \cdots - 19\!\cdots\!72 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7916051581 \nu^{14} + 7091457376733 \nu^{12} + \cdots + 55\!\cdots\!28 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!53 \nu^{14} + \cdots - 77\!\cdots\!76 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 82\!\cdots\!83 \nu^{14} + \cdots - 16\!\cdots\!64 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 142331351 \nu^{14} - 128988946423 \nu^{12} - 44713422907422 \nu^{10} + \cdots - 14\!\cdots\!48 ) / 34\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!39 \nu^{14} + \cdots - 12\!\cdots\!88 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32\!\cdots\!64 \nu^{15} + \cdots - 64\!\cdots\!12 \nu ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23\!\cdots\!07 \nu^{15} + \cdots - 19\!\cdots\!76 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!47 \nu^{15} + \cdots + 15\!\cdots\!52 \nu ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 161754014339 \nu^{15} + 158876588086459 \nu^{13} + \cdots + 33\!\cdots\!40 \nu ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 99\!\cdots\!19 \nu^{15} + \cdots - 70\!\cdots\!52 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 44\!\cdots\!99 \nu^{15} + \cdots - 12\!\cdots\!92 \nu ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 83\!\cdots\!73 \nu^{15} + \cdots + 22\!\cdots\!84 \nu ) / 76\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 3\beta_{2} - 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 4\beta_{14} + 8\beta_{12} - 6\beta_{11} - 2\beta_{10} - 4\beta_{9} - 215\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} + 12\beta_{7} - 8\beta_{6} + 10\beta_{5} - 253\beta_{4} - 26\beta_{3} - 881\beta_{2} + 26782 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 530 \beta_{15} - 1460 \beta_{14} + 100 \beta_{13} - 4506 \beta_{12} + 2732 \beta_{11} + \cdots + 51045 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1624 \beta_{8} - 6256 \beta_{7} + 4256 \beta_{6} - 6532 \beta_{5} + 61101 \beta_{4} + 10572 \beta_{3} + \cdots - 6341718 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 116402 \beta_{15} + 452988 \beta_{14} - 72344 \beta_{13} + 1762420 \beta_{12} - 1009746 \beta_{11} + \cdots - 12726159 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 518188 \beta_{8} + 2398652 \beta_{7} - 1718360 \beta_{6} + 2831926 \beta_{5} - 15045481 \beta_{4} + \cdots + 1581459478 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24331026 \beta_{15} - 136034380 \beta_{14} + 32828332 \beta_{13} - 608191342 \beta_{12} + \cdots + 3286412269 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 153442704 \beta_{8} - 818575792 \beta_{7} + 615262416 \beta_{6} - 1050353352 \beta_{5} + \cdots - 409016008158 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4929594882 \beta_{15} + 40453975380 \beta_{14} - 12488512560 \beta_{13} + 197639061024 \beta_{12} + \cdots - 873466280007 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 44247239508 \beta_{8} + 263976241932 \beta_{7} - 206364581160 \beta_{6} + 359568659586 \beta_{5} + \cdots + 108937363324430 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 957425860530 \beta_{15} - 11980103699524 \beta_{14} + 4344394330228 \beta_{13} + \cdots + 237715547183541 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 12660417033288 \beta_{8} - 82510712463184 \beta_{7} + 66508832991360 \beta_{6} + \cdots - 29\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 172500269925682 \beta_{15} + \cdots - 65\!\cdots\!55 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
− 17.1570i
− 14.9289i
− 14.1753i
− 13.4371i
− 7.41838i
− 7.00075i
− 2.59834i
− 0.690767i
0.690767i
2.59834i
7.00075i
7.41838i
13.4371i
14.1753i
14.9289i
17.1570i
0 − 17.1570i 0 11.1803 0 − 18.5203i 0 −213.362 0
351.2 0 − 14.9289i 0 11.1803 0 18.5203i 0 −141.871 0
351.3 0 − 14.1753i 0 −11.1803 0 − 18.5203i 0 −119.939 0
351.4 0 − 13.4371i 0 −11.1803 0 18.5203i 0 −99.5565 0
351.5 0 − 7.41838i 0 11.1803 0 − 18.5203i 0 25.9676 0
351.6 0 − 7.00075i 0 11.1803 0 18.5203i 0 31.9895 0
351.7 0 − 2.59834i 0 −11.1803 0 − 18.5203i 0 74.2486 0
351.8 0 − 0.690767i 0 −11.1803 0 18.5203i 0 80.5228 0
351.9 0 0.690767i 0 −11.1803 0 − 18.5203i 0 80.5228 0
351.10 0 2.59834i 0 −11.1803 0 18.5203i 0 74.2486 0
351.11 0 7.00075i 0 11.1803 0 − 18.5203i 0 31.9895 0
351.12 0 7.41838i 0 11.1803 0 18.5203i 0 25.9676 0
351.13 0 13.4371i 0 −11.1803 0 − 18.5203i 0 −99.5565 0
351.14 0 14.1753i 0 −11.1803 0 18.5203i 0 −119.939 0
351.15 0 14.9289i 0 11.1803 0 − 18.5203i 0 −141.871 0
351.16 0 17.1570i 0 11.1803 0 18.5203i 0 −213.362 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.d.a ✓ 16
4.b odd 2 1 inner 560.5.d.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.5.d.a ✓ 16 1.a even 1 1 trivial
560.5.d.a ✓ 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 1010 T_{3}^{14} + 402663 T_{3}^{12} + 80209516 T_{3}^{10} + 8304007999 T_{3}^{8} + \cdots + 20681247856896 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 20681247856896 \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 343)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 66\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 31\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 68\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 95\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 55\!\cdots\!56)^{2} \) Copy content Toggle raw display
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