Properties

Label 320.3.e.b
Level $320$
Weight $3$
Character orbit 320.e
Analytic conductor $8.719$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(159,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.159");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.e (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: \(8.71936845953\)
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: \( x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{34} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} - \beta_{11} q^{7} + ( - \beta_{5} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} - \beta_{11} q^{7} + ( - \beta_{5} - 6) q^{9} + ( - \beta_{13} - 2 \beta_{10}) q^{11} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{13} + (\beta_{15} - 3 \beta_{11} - \beta_{9}) q^{15} + (\beta_{12} - \beta_{8}) q^{17} + 3 \beta_{10} q^{19} + ( - \beta_{7} - 3 \beta_{4} - 3 \beta_{2}) q^{21} + ( - 3 \beta_{15} + \beta_{11}) q^{23} + (2 \beta_{12} + \beta_{8} - \beta_{5} + 2) q^{25} + (\beta_{14} - 6 \beta_1) q^{27} + (3 \beta_{7} - 2 \beta_{4} - 2 \beta_{2}) q^{29} + (\beta_{15} - 2 \beta_{9} + \beta_{6}) q^{31} + ( - 3 \beta_{12} - 7 \beta_{8}) q^{33} + ( - 5 \beta_{10} - 5 \beta_1) q^{35} + (\beta_{4} + 4 \beta_{3} - \beta_{2}) q^{37} + (2 \beta_{15} - 4 \beta_{9} - 3 \beta_{6}) q^{39} + ( - 3 \beta_{5} - 3) q^{41} + (\beta_{14} + 7 \beta_1) q^{43} + ( - 5 \beta_{7} - 4 \beta_{4} + 5 \beta_{3} - 5 \beta_{2}) q^{45} + (\beta_{15} - 7 \beta_{11}) q^{47} + (\beta_{5} - 22) q^{49} + (3 \beta_{13} - \beta_{10}) q^{51} + (\beta_{4} - 7 \beta_{3} - \beta_{2}) q^{53} + (2 \beta_{15} - 6 \beta_{11} + 3 \beta_{9} + 5 \beta_{6}) q^{55} + (3 \beta_{12} + 3 \beta_{8}) q^{57} + ( - 4 \beta_{13} + 15 \beta_{10}) q^{59} + ( - 2 \beta_{7} + 9 \beta_{4} + 9 \beta_{2}) q^{61} + ( - 5 \beta_{15} + 13 \beta_{11}) q^{63} + (3 \beta_{12} + 4 \beta_{8} + \beta_{5} + 33) q^{65} + ( - 3 \beta_{14} + 9 \beta_1) q^{67} + (13 \beta_{7} + 3 \beta_{4} + 3 \beta_{2}) q^{69} + ( - \beta_{15} + 2 \beta_{9}) q^{71} + ( - 5 \beta_{12} + 3 \beta_{8}) q^{73} + (\beta_{14} - 3 \beta_{13} - 23 \beta_{10} - 7 \beta_1) q^{75} + (2 \beta_{4} - 11 \beta_{3} - 2 \beta_{2}) q^{77} + ( - 5 \beta_{15} + 10 \beta_{9} - 7 \beta_{6}) q^{79} + (3 \beta_{5} + 30) q^{81} + ( - 3 \beta_{14} + 13 \beta_1) q^{83} + ( - 10 \beta_{7} + 2 \beta_{4} - 5 \beta_{3} + 10 \beta_{2}) q^{85} + 4 \beta_{15} q^{87} + (2 \beta_{5} - 72) q^{89} + ( - 3 \beta_{13} - 17 \beta_{10}) q^{91} + ( - 8 \beta_{4} + 6 \beta_{3} + 8 \beta_{2}) q^{93} + (3 \beta_{15} + 6 \beta_{11} - 3 \beta_{9}) q^{95} + ( - 5 \beta_{12} + 9 \beta_{8}) q^{97} + (12 \beta_{13} + 55 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 96 q^{9} + 32 q^{25} - 48 q^{41} - 352 q^{49} + 528 q^{65} + 480 q^{81} - 1152 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 89 \nu^{15} + 1682 \nu^{13} + 26620 \nu^{11} + 95408 \nu^{9} + 278884 \nu^{7} + 71420 \nu^{5} + 24119 \nu^{3} + 2802 \nu ) / 300 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5491 \nu^{14} - 104672 \nu^{12} - 1659105 \nu^{10} - 6150577 \nu^{8} - 18098609 \nu^{6} - 6967305 \nu^{4} - 1444486 \nu^{2} - 160337 ) / 23400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 484 \nu^{14} - 9043 \nu^{12} - 142780 \nu^{10} - 487388 \nu^{8} - 1399796 \nu^{6} - 45980 \nu^{4} - 2904 \nu^{2} + 7647 ) / 1950 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6793 \nu^{14} - 128996 \nu^{12} - 2043195 \nu^{10} - 7461691 \nu^{8} - 21872627 \nu^{6} - 7090995 \nu^{4} - 1452298 \nu^{2} + 45169 ) / 23400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60\nu^{14} + 1121\nu^{12} + 17700\nu^{10} + 60420\nu^{8} + 173642\nu^{6} + 5700\nu^{4} + 360\nu^{2} - 1729 ) / 150 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2182 \nu^{15} + 42080 \nu^{13} + 668520 \nu^{11} + 2590264 \nu^{9} + 7726640 \nu^{7} + 4329720 \nu^{5} + 1079482 \nu^{3} + 121760 \nu ) / 2925 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1007 \nu^{14} + 19074 \nu^{12} + 302005 \nu^{10} + 1092309 \nu^{8} + 3201253 \nu^{6} + 937805 \nu^{4} + 297502 \nu^{2} + 10979 ) / 1950 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 80\nu^{14} + 1513\nu^{12} + 23950\nu^{10} + 86110\nu^{8} + 252226\nu^{6} + 68750\nu^{4} + 25230\nu^{2} + 913 ) / 150 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2177 \nu^{15} + 41200 \nu^{13} + 652160 \nu^{11} + 2349604 \nu^{9} + 6863340 \nu^{7} + 1846960 \nu^{5} + 400527 \nu^{3} - 67060 \nu ) / 1950 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 379 \nu^{15} + 7182 \nu^{13} + 113715 \nu^{11} + 411883 \nu^{9} + 1205379 \nu^{7} + 353115 \nu^{5} + 84304 \nu^{3} + 1197 \nu ) / 300 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5949 \nu^{15} + 112740 \nu^{13} + 1785050 \nu^{11} + 6466898 \nu^{9} + 18921530 \nu^{7} + 5543050 \nu^{5} + 1271149 \nu^{3} + 18790 \nu ) / 3900 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 100 \nu^{14} - 1899 \nu^{12} - 30080 \nu^{10} - 109880 \nu^{8} - 322398 \nu^{6} - 105880 \nu^{4} - 25980 \nu^{2} - 999 ) / 75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 381 \nu^{15} - 7218 \nu^{13} - 114285 \nu^{11} - 413597 \nu^{9} - 1211421 \nu^{7} - 354885 \nu^{5} - 100416 \nu^{3} - 1203 \nu ) / 100 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 611 \nu^{15} + 11718 \nu^{13} + 185970 \nu^{11} + 705902 \nu^{9} + 2094906 \nu^{7} + 1012770 \nu^{5} + 264491 \nu^{3} + 29988 \nu ) / 150 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4443 \nu^{15} + 84180 \nu^{13} + 1332850 \nu^{11} + 4825036 \nu^{9} + 14128210 \nu^{7} + 4138850 \nu^{5} + 1104293 \nu^{3} + 14030 \nu ) / 975 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{15} - \beta_{14} + 2\beta_{13} - 2\beta_{10} - 4\beta_{9} + 3\beta_{6} - 6\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{12} - 4\beta_{8} + 11\beta_{7} + 2\beta_{5} + 2\beta_{3} - 12\beta_{2} - 38 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{15} - 5\beta_{13} - 4\beta_{11} + 15\beta_{10} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -85\beta_{12} + 44\beta_{8} - 149\beta_{7} + 38\beta_{5} + 200\beta_{4} + 42\beta_{3} + 28\beta_{2} - 482 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 124 \beta_{15} + 281 \beta_{14} + 278 \beta_{13} + 320 \beta_{11} - 998 \beta_{10} + 556 \beta_{9} - 957 \beta_{6} + 726 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -75\beta_{5} - 160\beta_{4} - 85\beta_{3} + 160\beta_{2} + 874 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10116 \beta_{15} - 4279 \beta_{14} + 4118 \beta_{13} + 5120 \beta_{11} - 15398 \beta_{10} - 8236 \beta_{9} + 14637 \beta_{6} - 10314 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 20075 \beta_{12} - 8836 \beta_{8} + 33029 \beta_{7} + 9158 \beta_{5} - 8200 \beta_{4} + 10438 \beta_{3} - 46748 \beta_{2} - 104642 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -22593\beta_{15} - 15485\beta_{13} - 19596\beta_{11} + 58455\beta_{10} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 303835 \beta_{12} + 132716 \beta_{8} - 498491 \beta_{7} + 138722 \beta_{5} + 707200 \beta_{4} + 158318 \beta_{3} + 125132 \beta_{2} - 1577798 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 429756 \beta_{15} + 979439 \beta_{14} + 935042 \beta_{13} + 1188160 \beta_{11} - 3537602 \beta_{10} + 1870084 \beta_{9} - 3354483 \beta_{6} + 2312154 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -262200\beta_{5} - 549680\beta_{4} - 299330\beta_{3} + 549680\beta_{2} + 2979001 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 34749764 \beta_{15} - 14803441 \beta_{14} + 14127362 \beta_{13} + 17968960 \beta_{11} - 53476802 \beta_{10} - 28254724 \beta_{9} + 50703123 \beta_{6} - 34913766 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 69423845 \beta_{12} - 30257524 \beta_{8} + 113808731 \beta_{7} + 31704482 \beta_{5} - 28659200 \beta_{4} + 36196722 \beta_{3} - 161567692 \beta_{2} - 360121478 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -77912207\beta_{15} - 53372525\beta_{13} - 67901204\beta_{11} + 202057575\beta_{10} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
−0.940543 + 1.62907i
0.940543 + 1.62907i
0.265804 0.460386i
−0.265804 0.460386i
1.94376 + 3.36668i
−1.94376 + 3.36668i
−0.128617 0.222771i
0.128617 0.222771i
−1.94376 3.36668i
1.94376 3.36668i
0.128617 + 0.222771i
−0.128617 + 0.222771i
0.940543 1.62907i
−0.940543 1.62907i
−0.265804 + 0.460386i
0.265804 + 0.460386i
0 5.13399i 0 −2.79662 4.14474i 0 −6.19337 0 −17.3578 0
159.2 0 5.13399i 0 −2.79662 + 4.14474i 0 6.19337 0 −17.3578 0
159.3 0 5.13399i 0 2.79662 4.14474i 0 −6.19337 0 −17.3578 0
159.4 0 5.13399i 0 2.79662 + 4.14474i 0 6.19337 0 −17.3578 0
159.5 0 1.90845i 0 −4.37937 2.41269i 0 −3.95502 0 5.35782 0
159.6 0 1.90845i 0 −4.37937 + 2.41269i 0 3.95502 0 5.35782 0
159.7 0 1.90845i 0 4.37937 2.41269i 0 −3.95502 0 5.35782 0
159.8 0 1.90845i 0 4.37937 + 2.41269i 0 3.95502 0 5.35782 0
159.9 0 1.90845i 0 −4.37937 2.41269i 0 3.95502 0 5.35782 0
159.10 0 1.90845i 0 −4.37937 + 2.41269i 0 −3.95502 0 5.35782 0
159.11 0 1.90845i 0 4.37937 2.41269i 0 3.95502 0 5.35782 0
159.12 0 1.90845i 0 4.37937 + 2.41269i 0 −3.95502 0 5.35782 0
159.13 0 5.13399i 0 −2.79662 4.14474i 0 6.19337 0 −17.3578 0
159.14 0 5.13399i 0 −2.79662 + 4.14474i 0 −6.19337 0 −17.3578 0
159.15 0 5.13399i 0 2.79662 4.14474i 0 6.19337 0 −17.3578 0
159.16 0 5.13399i 0 2.79662 + 4.14474i 0 −6.19337 0 −17.3578 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.e.b 16
4.b odd 2 1 inner 320.3.e.b 16
5.b even 2 1 inner 320.3.e.b 16
5.c odd 4 2 1600.3.g.k 16
8.b even 2 1 inner 320.3.e.b 16
8.d odd 2 1 inner 320.3.e.b 16
16.e even 4 2 1280.3.h.n 16
16.f odd 4 2 1280.3.h.n 16
20.d odd 2 1 inner 320.3.e.b 16
20.e even 4 2 1600.3.g.k 16
40.e odd 2 1 inner 320.3.e.b 16
40.f even 2 1 inner 320.3.e.b 16
40.i odd 4 2 1600.3.g.k 16
40.k even 4 2 1600.3.g.k 16
80.k odd 4 2 1280.3.h.n 16
80.q even 4 2 1280.3.h.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.e.b 16 1.a even 1 1 trivial
320.3.e.b 16 4.b odd 2 1 inner
320.3.e.b 16 5.b even 2 1 inner
320.3.e.b 16 8.b even 2 1 inner
320.3.e.b 16 8.d odd 2 1 inner
320.3.e.b 16 20.d odd 2 1 inner
320.3.e.b 16 40.e odd 2 1 inner
320.3.e.b 16 40.f even 2 1 inner
1280.3.h.n 16 16.e even 4 2
1280.3.h.n 16 16.f odd 4 2
1280.3.h.n 16 80.k odd 4 2
1280.3.h.n 16 80.q even 4 2
1600.3.g.k 16 5.c odd 4 2
1600.3.g.k 16 20.e even 4 2
1600.3.g.k 16 40.i odd 4 2
1600.3.g.k 16 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 30T_{3}^{2} + 96 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 30 T^{2} + 96)^{4} \) Copy content Toggle raw display
$5$ \( (T^{8} - 8 T^{6} + 750 T^{4} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 54 T^{2} + 600)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 440 T^{2} + 15376)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 276 T^{2} + 6144)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 552 T^{2} + 24576)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 108)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2070 T^{2} + 786264)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 944 T^{2} + 16384)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1040 T^{2} + 262144)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 1836 T^{2} + 726624)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 1152)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3294 T^{2} + 1548384)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2550 T^{2} + 411864)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 6324 T^{2} + 1935744)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 10904 T^{2} + 2704)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6972 T^{2} + 12110400)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 16686 T^{2} + 67254624)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1104 T^{2} + 230400)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 9384 T^{2} + 960000)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 27152 T^{2} + \cdots + 149426176)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 19038 T^{2} + 89397600)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144 T + 4668)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 27816 T^{2} + \cdots + 147609600)^{4} \) Copy content Toggle raw display
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