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Properties

Label 160.5.g.a
Level $160$
Weight $5$
Character orbit 160.g
Analytic conductor $16.539$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,5,Mod(111,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.111");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 160.g (of order \(2\), degree \(1\), 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: \(16.5391940934\)
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: \( x^{16} - 6 x^{15} + 14 x^{14} - 84 x^{13} + 628 x^{12} - 1392 x^{11} + 2016 x^{10} - 18048 x^{9} + \cdots + 4294967296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{56}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{7} q^{7} + ( - \beta_{9} + 2 \beta_1 + 27) q^{9} + (\beta_{5} + \beta_1 - 12) q^{11} + ( - \beta_{7} + \beta_{4}) q^{13} - \beta_{10} q^{15} + ( - \beta_{9} + \beta_{3} - 10 \beta_1) q^{17}+ \cdots + ( - \beta_{14} + 40 \beta_{13} + \cdots - 164) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 432 q^{9} - 192 q^{11} + 704 q^{19} - 2000 q^{25} - 3648 q^{27} - 992 q^{33} - 2208 q^{41} - 5568 q^{43} - 2480 q^{49} + 17792 q^{51} + 8608 q^{57} - 14016 q^{59} + 18880 q^{67} - 7360 q^{73} + 10384 q^{81}+ \cdots - 2624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 14 x^{14} - 84 x^{13} + 628 x^{12} - 1392 x^{11} + 2016 x^{10} - 18048 x^{9} + \cdots + 4294967296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2289 \nu^{15} + 7142 \nu^{14} + 74706 \nu^{13} + 421780 \nu^{12} - 1941812 \nu^{11} + \cdots + 8046621229056 ) / 240249733120 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20271 \nu^{15} + 62410 \nu^{14} + 117966 \nu^{13} + 764940 \nu^{12} + \cdots + 14240769376256 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2519 \nu^{15} + 50682 \nu^{14} + 20766 \nu^{13} - 325460 \nu^{12} - 1116972 \nu^{11} + \cdots + 2310155534336 ) / 60062433280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18049 \nu^{15} - 129530 \nu^{14} + 122994 \nu^{13} - 437420 \nu^{12} + \cdots - 69145215369216 ) / 240249733120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7065 \nu^{15} - 8458 \nu^{14} - 141342 \nu^{13} + 479860 \nu^{12} - 1229780 \nu^{11} + \cdots - 352724189184 ) / 48049946624 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22459 \nu^{15} + 85234 \nu^{14} + 117222 \nu^{13} + 777276 \nu^{12} + \cdots + 33014071427072 ) / 144149839872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41431 \nu^{15} - 103210 \nu^{14} + 228354 \nu^{13} - 3335500 \nu^{12} + 11920108 \nu^{11} + \cdots - 32565515780096 ) / 240249733120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3671 \nu^{15} + 3100 \nu^{14} - 184002 \nu^{13} + 662088 \nu^{12} + 386612 \nu^{11} + \cdots + 9531841052672 ) / 18018729984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7669 \nu^{15} + 23822 \nu^{14} - 30214 \nu^{13} + 431940 \nu^{12} - 2314692 \nu^{11} + \cdots + 5439106318336 ) / 30031216640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 252521 \nu^{15} - 136790 \nu^{14} - 582786 \nu^{13} - 17176500 \nu^{12} + \cdots - 138669998473216 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 157393 \nu^{15} - 283350 \nu^{14} + 473742 \nu^{13} - 15252660 \nu^{12} + \cdots - 167488423723008 ) / 360374599680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27859 \nu^{15} - 23682 \nu^{14} - 40086 \nu^{13} - 2207260 \nu^{12} + 11342812 \nu^{11} + \cdots - 30685125410816 ) / 60062433280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 66183 \nu^{15} - 288554 \nu^{14} + 297698 \nu^{13} - 5398860 \nu^{12} + 33070124 \nu^{11} + \cdots - 86258244124672 ) / 120124866560 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37421 \nu^{15} + 263342 \nu^{14} - 364470 \nu^{13} + 3303044 \nu^{12} + \cdots + 52229486673920 ) / 48049946624 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 661709 \nu^{15} + 1760190 \nu^{14} - 3220566 \nu^{13} + 61487460 \nu^{12} + \cdots + 773217325154304 ) / 720749199360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{10} - \beta_{9} + 3\beta_{7} - \beta_{6} + 3\beta_{2} - 6\beta _1 + 48 ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} - 2 \beta_{13} + \beta_{12} - 3 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 32 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 3 \beta_{12} - \beta_{11} + 5 \beta_{10} - 2 \beta_{9} + \cdots + 432 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 29 \beta_{10} + \cdots - 1648 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{15} - 22 \beta_{14} + 9 \beta_{13} - 6 \beta_{12} - 46 \beta_{11} + 75 \beta_{10} + \cdots - 8208 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 23 \beta_{15} + 14 \beta_{14} + 66 \beta_{13} - 46 \beta_{12} - 27 \beta_{11} + 121 \beta_{10} + \cdots - 1856 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 278 \beta_{15} + 74 \beta_{14} + 179 \beta_{13} + 318 \beta_{12} - 190 \beta_{11} - 185 \beta_{10} + \cdots - 44304 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 437 \beta_{15} - 307 \beta_{14} + 345 \beta_{13} - 609 \beta_{12} - 453 \beta_{11} + 83 \beta_{10} + \cdots - 292112 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 277 \beta_{15} - 875 \beta_{14} + 256 \beta_{13} + 1041 \beta_{12} + 837 \beta_{11} + 2609 \beta_{10} + \cdots - 40176 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4079 \beta_{15} + 4417 \beta_{14} + 3550 \beta_{13} - 369 \beta_{12} + 2865 \beta_{11} - 12928 \beta_{10} + \cdots + 326752 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1784 \beta_{15} - 4568 \beta_{14} + 1003 \beta_{13} + 11844 \beta_{12} + 900 \beta_{11} - 62631 \beta_{10} + \cdots + 3434064 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10214 \beta_{15} + 515 \beta_{14} - 11626 \beta_{13} + 9743 \beta_{12} - 11586 \beta_{11} + \cdots - 985424 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 113832 \beta_{15} + 178536 \beta_{14} + 102271 \beta_{13} - 5712 \beta_{12} - 106896 \beta_{11} + \cdots + 25914672 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19177 \beta_{15} + 227081 \beta_{14} - 6010 \beta_{13} + 125735 \beta_{12} + 497691 \beta_{11} + \cdots - 14433632 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1318742 \beta_{15} + 349894 \beta_{14} - 1000148 \beta_{13} - 36978 \beta_{12} + 1021578 \beta_{11} + \cdots - 408978144 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
−1.60069 + 3.66576i
−1.60069 − 3.66576i
3.55671 − 1.83026i
3.55671 + 1.83026i
−3.73120 − 1.44159i
−3.73120 + 1.44159i
3.79794 + 1.25525i
3.79794 − 1.25525i
−2.66926 + 2.97910i
−2.66926 − 2.97910i
1.64589 + 3.64569i
1.64589 − 3.64569i
−1.30255 − 3.78198i
−1.30255 + 3.78198i
3.30316 − 2.25591i
3.30316 + 2.25591i
0 −15.9375 0 − 11.1803i 0 − 56.7751i 0 173.005 0
111.2 0 −15.9375 0 11.1803i 0 56.7751i 0 173.005 0
111.3 0 −13.1346 0 − 11.1803i 0 61.6422i 0 91.5179 0
111.4 0 −13.1346 0 11.1803i 0 − 61.6422i 0 91.5179 0
111.5 0 −7.09911 0 − 11.1803i 0 2.59084i 0 −30.6027 0
111.6 0 −7.09911 0 11.1803i 0 − 2.59084i 0 −30.6027 0
111.7 0 0.715641 0 − 11.1803i 0 25.0983i 0 −80.4879 0
111.8 0 0.715641 0 11.1803i 0 − 25.0983i 0 −80.4879 0
111.9 0 4.51805 0 − 11.1803i 0 50.0881i 0 −60.5872 0
111.10 0 4.51805 0 11.1803i 0 − 50.0881i 0 −60.5872 0
111.11 0 5.51460 0 − 11.1803i 0 − 78.3513i 0 −50.5892 0
111.12 0 5.51460 0 11.1803i 0 78.3513i 0 −50.5892 0
111.13 0 10.2034 0 − 11.1803i 0 43.3025i 0 23.1094 0
111.14 0 10.2034 0 11.1803i 0 − 43.3025i 0 23.1094 0
111.15 0 15.2196 0 − 11.1803i 0 − 47.5956i 0 150.635 0
111.16 0 15.2196 0 11.1803i 0 47.5956i 0 150.635 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.5.g.a 16
3.b odd 2 1 1440.5.g.a 16
4.b odd 2 1 40.5.g.a ✓ 16
5.b even 2 1 800.5.g.h 16
5.c odd 4 2 800.5.e.e 32
8.b even 2 1 40.5.g.a ✓ 16
8.d odd 2 1 inner 160.5.g.a 16
12.b even 2 1 360.5.g.a 16
20.d odd 2 1 200.5.g.h 16
20.e even 4 2 200.5.e.e 32
24.f even 2 1 1440.5.g.a 16
24.h odd 2 1 360.5.g.a 16
40.e odd 2 1 800.5.g.h 16
40.f even 2 1 200.5.g.h 16
40.i odd 4 2 200.5.e.e 32
40.k even 4 2 800.5.e.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.5.g.a ✓ 16 4.b odd 2 1
40.5.g.a ✓ 16 8.b even 2 1
160.5.g.a 16 1.a even 1 1 trivial
160.5.g.a 16 8.d odd 2 1 inner
200.5.e.e 32 20.e even 4 2
200.5.e.e 32 40.i odd 4 2
200.5.g.h 16 20.d odd 2 1
200.5.g.h 16 40.f even 2 1
360.5.g.a 16 12.b even 2 1
360.5.g.a 16 24.h odd 2 1
800.5.e.e 32 5.c odd 4 2
800.5.e.e 32 40.k even 4 2
800.5.g.h 16 5.b even 2 1
800.5.g.h 16 40.e odd 2 1
1440.5.g.a 16 3.b odd 2 1
1440.5.g.a 16 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 432 T^{6} + \cdots - 4114800)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 75\!\cdots\!72)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 14\!\cdots\!52)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 47\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
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