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Properties

Label 160.6.n.b
Level $160$
Weight $6$
Character orbit 160.n
Analytic conductor $25.661$
Analytic rank $0$
Dimension $14$
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] = [160,6,Mod(63,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.63");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.n (of order \(4\), degree \(2\), 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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} + \cdots + 69451154208 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{31}\cdot 5^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 + 1) q^{3} + ( - \beta_{5} + \beta_{2} - 4 \beta_1 + 3) q^{5} + (\beta_{8} + \beta_{6} - 5 \beta_1 + 5) q^{7} + (\beta_{11} + \beta_{7} + \cdots + 58 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 + 1) q^{3} + ( - \beta_{5} + \beta_{2} - 4 \beta_1 + 3) q^{5} + (\beta_{8} + \beta_{6} - 5 \beta_1 + 5) q^{7} + (\beta_{11} + \beta_{7} + \cdots + 58 \beta_1) q^{9}+ \cdots + ( - 10 \beta_{13} + 57 \beta_{10} + \cdots - 28504) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 10 q^{3} + 42 q^{5} + 66 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 10 q^{3} + 42 q^{5} + 66 q^{7} - 414 q^{13} + 278 q^{15} + 1222 q^{17} + 5672 q^{19} + 5924 q^{21} + 2902 q^{23} - 4466 q^{25} - 2168 q^{27} - 2444 q^{33} - 2618 q^{35} - 1790 q^{37} - 11076 q^{39} + 11644 q^{41} - 3982 q^{43} + 14704 q^{45} - 1278 q^{47} + 5882 q^{53} + 65608 q^{55} - 14552 q^{57} - 8504 q^{59} + 20564 q^{61} + 19422 q^{63} + 40798 q^{65} + 107926 q^{67} - 16418 q^{73} + 66586 q^{75} - 13348 q^{77} - 146544 q^{79} + 173806 q^{81} - 36398 q^{83} - 66262 q^{85} + 124384 q^{87} - 306620 q^{93} + 173768 q^{95} - 60314 q^{97} - 388628 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} + \cdots + 69451154208 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!28 \nu^{13} + \cdots + 51\!\cdots\!44 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53\!\cdots\!57 \nu^{13} + \cdots + 29\!\cdots\!84 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!89 \nu^{13} + \cdots + 60\!\cdots\!32 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!53 \nu^{13} + \cdots - 24\!\cdots\!64 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!41 \nu^{13} + \cdots + 32\!\cdots\!92 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69\!\cdots\!36 \nu^{13} + \cdots + 15\!\cdots\!68 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!33 \nu^{13} + \cdots + 46\!\cdots\!04 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 95\!\cdots\!18 \nu^{13} + \cdots + 21\!\cdots\!84 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 96\!\cdots\!91 \nu^{13} + \cdots + 16\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67\!\cdots\!51 \nu^{13} + \cdots + 24\!\cdots\!12 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 96\!\cdots\!11 \nu^{13} + \cdots - 47\!\cdots\!68 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!43 \nu^{13} + \cdots + 35\!\cdots\!84 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!67 \nu^{13} + \cdots - 18\!\cdots\!96 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 6 \beta_{7} + 47 \beta_{6} + \cdots + 62 ) / 160 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 17 \beta_{12} - 36 \beta_{11} - 8 \beta_{9} + 8 \beta_{8} + 34 \beta_{7} - 343 \beta_{6} + 78 \beta_{5} + \cdots + 9 ) / 80 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1347 \beta_{13} + 589 \beta_{12} + 1347 \beta_{11} + 589 \beta_{10} - 98 \beta_{9} + 2788 \beta_{7} + \cdots + 151446 ) / 160 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3975 \beta_{13} - 1802 \beta_{10} - 351 \beta_{9} - 351 \beta_{8} - 8407 \beta_{7} + 38099 \beta_{6} + \cdots - 577119 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 649107 \beta_{13} - 290279 \beta_{12} - 649107 \beta_{11} + 290279 \beta_{10} + 36622 \beta_{8} + \cdots + 85262162 ) / 160 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4550703 \beta_{12} + 10121334 \beta_{11} + 561642 \beta_{9} - 561642 \beta_{8} - 2468176 \beta_{7} + \cdots + 87259 ) / 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 322070793 \beta_{13} - 144504171 \beta_{12} - 322070793 \beta_{11} - 144504171 \beta_{10} + \cdots - 43346099354 ) / 160 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1015819605 \beta_{13} + 456147646 \beta_{10} + 49378493 \beta_{9} + 49378493 \beta_{8} + \cdots + 137706097181 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 160815790533 \beta_{13} + 72189394841 \beta_{12} + 160815790533 \beta_{11} - 72189394841 \beta_{10} + \cdots - 21729374219838 ) / 160 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1141073051377 \beta_{12} - 2541616650546 \beta_{11} - 120090369758 \beta_{9} + 120090369758 \beta_{8} + \cdots - 37301571341 ) / 80 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 80396898736167 \beta_{13} + 36092621151669 \beta_{12} + 80396898736167 \beta_{11} + \cdots + 10\!\cdots\!46 ) / 160 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 254232041165115 \beta_{13} - 114135193791106 \beta_{10} - 11945785122003 \beta_{9} + \cdots - 34\!\cdots\!19 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 40\!\cdots\!27 \beta_{13} + \cdots + 54\!\cdots\!62 ) / 160 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
−15.8126 − 15.8126i
3.77108 + 3.77108i
2.04998 + 2.04998i
6.86993 + 6.86993i
4.57273 + 4.57273i
−2.20370 − 2.20370i
2.75256 + 2.75256i
−15.8126 + 15.8126i
3.77108 − 3.77108i
2.04998 − 2.04998i
6.86993 − 6.86993i
4.57273 − 4.57273i
−2.20370 + 2.20370i
2.75256 − 2.75256i
0 −16.5519 + 16.5519i 0 13.9288 − 54.1386i 0 −2.15894 − 2.15894i 0 − 304.928i 0
63.2 0 −13.4331 + 13.4331i 0 15.0352 + 53.8418i 0 −76.4413 − 76.4413i 0 − 117.896i 0
63.3 0 −5.65790 + 5.65790i 0 −54.5514 + 12.2124i 0 23.8754 + 23.8754i 0 178.976i 0
63.4 0 1.47740 − 1.47740i 0 55.2474 − 8.52775i 0 156.265 + 156.265i 0 238.635i 0
63.5 0 9.28112 − 9.28112i 0 42.6946 + 36.0856i 0 −105.819 − 105.819i 0 70.7216i 0
63.6 0 10.9598 − 10.9598i 0 −14.9228 − 53.8731i 0 −75.2427 − 75.2427i 0 2.76340i 0
63.7 0 18.9245 − 18.9245i 0 −36.4318 + 42.3996i 0 112.521 + 112.521i 0 − 473.272i 0
127.1 0 −16.5519 − 16.5519i 0 13.9288 + 54.1386i 0 −2.15894 + 2.15894i 0 304.928i 0
127.2 0 −13.4331 − 13.4331i 0 15.0352 − 53.8418i 0 −76.4413 + 76.4413i 0 117.896i 0
127.3 0 −5.65790 − 5.65790i 0 −54.5514 − 12.2124i 0 23.8754 − 23.8754i 0 − 178.976i 0
127.4 0 1.47740 + 1.47740i 0 55.2474 + 8.52775i 0 156.265 − 156.265i 0 − 238.635i 0
127.5 0 9.28112 + 9.28112i 0 42.6946 − 36.0856i 0 −105.819 + 105.819i 0 − 70.7216i 0
127.6 0 10.9598 + 10.9598i 0 −14.9228 + 53.8731i 0 −75.2427 + 75.2427i 0 − 2.76340i 0
127.7 0 18.9245 + 18.9245i 0 −36.4318 − 42.3996i 0 112.521 − 112.521i 0 473.272i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.n.b yes 14
4.b odd 2 1 160.6.n.a ✓ 14
5.c odd 4 1 160.6.n.a ✓ 14
20.e even 4 1 inner 160.6.n.b yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.n.a ✓ 14 4.b odd 2 1
160.6.n.a ✓ 14 5.c odd 4 1
160.6.n.b yes 14 1.a even 1 1 trivial
160.6.n.b yes 14 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 10 T_{3}^{13} + 50 T_{3}^{12} + 2176 T_{3}^{11} + 494504 T_{3}^{10} - 3508816 T_{3}^{9} + \cdots + 16\!\cdots\!68 \) acting on \(S_{6}^{\mathrm{new}}(160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 38\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 87\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 87\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 88\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 20\!\cdots\!88)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 24\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 42\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 29\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 76\!\cdots\!32 \) Copy content Toggle raw display
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