Properties

Label 1280.3.g.g
Level $1280$
Weight $3$
Character orbit 1280.g
Analytic conductor $34.877$
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] = [1280,3,Mod(1151,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.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: \(34.8774738381\)
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: \( x^{16} + 24x^{14} + 228x^{12} + 1110x^{10} + 2970x^{8} + 4308x^{6} + 3085x^{4} + 882x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{36} \)
Twist minimal: no (minimal twist has level 640)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - 1) q^{3} - \beta_{7} q^{5} + (\beta_{9} + \beta_{7}) q^{7} + (\beta_{10} + \beta_{6} - \beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - 1) q^{3} - \beta_{7} q^{5} + (\beta_{9} + \beta_{7}) q^{7} + (\beta_{10} + \beta_{6} - \beta_{4} + 3) q^{9} + ( - \beta_{6} - \beta_1 - 4) q^{11} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{11}) q^{13} + (\beta_{12} + \beta_{7}) q^{15} + (\beta_{10} + \beta_{2} - \beta_1) q^{17} + (\beta_{8} - \beta_{6} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 6) q^{19} + (\beta_{14} - \beta_{13} - 3 \beta_{12} - 2 \beta_{9} - 2 \beta_{7} - \beta_{3}) q^{21} + ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{12} + 2 \beta_{11} - \beta_{9} + 5 \beta_{7} + \beta_{3}) q^{23} - 5 q^{25} + ( - 2 \beta_{10} - 2 \beta_{8} - 3 \beta_{6} - \beta_{5} - 2 \beta_{2} - \beta_1 - 4) q^{27} + (3 \beta_{15} + \beta_{13} - \beta_{12} + 2 \beta_{9} - 3 \beta_{3}) q^{29} + (2 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{9} + 2 \beta_{7} + 4 \beta_{3}) q^{31} + ( - \beta_{10} + \beta_{8} + 2 \beta_{5} - 5 \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 2) q^{33} + (2 \beta_{10} - \beta_{5} - \beta_{4} + 5) q^{35} + ( - \beta_{15} - \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{7}) q^{37} + (2 \beta_{15} + \beta_{14} - 3 \beta_{13} - \beta_{12} - 4 \beta_{11} + 6 \beta_{7} + 6 \beta_{3}) q^{39} + ( - \beta_{10} + 3 \beta_{6} + 4 \beta_{5} + \beta_{4} + 6) q^{41} + (2 \beta_{10} - \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - \beta_1 + 11) q^{43} + (\beta_{15} - \beta_{14} - \beta_{11} - 2 \beta_{9} - 3 \beta_{7} + 5 \beta_{3}) q^{45} + ( - 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{11} + \beta_{9} + 7 \beta_{7} + 4 \beta_{3}) q^{47} + ( - 3 \beta_{10} - 2 \beta_{8} + 3 \beta_{6} + 7 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 11) q^{49} + ( - 4 \beta_{10} - \beta_{8} - 8 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + 3 \beta_{2} + \cdots - 6) q^{51}+ \cdots + ( - 8 \beta_{10} - 5 \beta_{8} - 17 \beta_{6} - 4 \beta_{5} - \beta_{2} - 4 \beta_1 - 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} + 48 q^{9} - 64 q^{11} - 96 q^{19} - 80 q^{25} - 64 q^{27} + 32 q^{33} + 80 q^{35} + 96 q^{41} + 176 q^{43} - 176 q^{49} - 96 q^{51} - 352 q^{57} + 32 q^{59} + 560 q^{67} + 320 q^{73} + 80 q^{75} - 48 q^{81} - 48 q^{83} + 96 q^{89} + 32 q^{91} + 448 q^{97} - 704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 228x^{12} + 1110x^{10} + 2970x^{8} + 4308x^{6} + 3085x^{4} + 882x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 32\nu^{14} + 687\nu^{12} + 5685\nu^{10} + 23766\nu^{8} + 54999\nu^{6} + 70284\nu^{4} + 42011\nu^{2} + 6120 ) / 117 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -43\nu^{14} - 744\nu^{12} - 3855\nu^{10} - 2298\nu^{8} + 33417\nu^{6} + 84873\nu^{4} + 60566\nu^{2} + 10818 ) / 351 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{15} - 291 \nu^{13} - 3084 \nu^{11} - 16791 \nu^{9} - 49644 \nu^{7} - 76503 \nu^{5} - 51980 \nu^{3} - 8802 \nu ) / 351 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 107\nu^{14} + 2235\nu^{12} + 17448\nu^{10} + 64572\nu^{8} + 116946\nu^{6} + 93603\nu^{4} + 22637\nu^{2} + 18 ) / 351 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64 \nu^{14} - 1023 \nu^{12} - 4233 \nu^{10} + 6054 \nu^{8} + 77553 \nu^{6} + 171588 \nu^{4} + 129737 \nu^{2} + 21456 ) / 351 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 176 \nu^{14} + 3720 \nu^{12} + 29688 \nu^{10} + 114684 \nu^{8} + 227088 \nu^{6} + 223932 \nu^{4} + 98792 \nu^{2} + 13302 ) / 351 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{15} - 219\nu^{13} - 1590\nu^{11} - 5271\nu^{9} - 7992\nu^{7} - 4377\nu^{5} + 598\nu^{3} + 1062\nu ) / 162 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -71\nu^{14} - 1467\nu^{12} - 11340\nu^{10} - 41940\nu^{8} - 78126\nu^{6} - 69543\nu^{4} - 23837\nu^{2} - 1674 ) / 117 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 227 \nu^{15} + 4431 \nu^{13} + 31074 \nu^{11} + 96423 \nu^{9} + 126252 \nu^{7} + 46065 \nu^{5} + 12254 \nu^{3} + 40518 \nu ) / 2106 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 283 \nu^{14} + 5955 \nu^{12} + 47136 \nu^{10} + 179256 \nu^{8} + 344034 \nu^{6} + 317535 \nu^{4} + 122833 \nu^{2} + 17532 ) / 351 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 184 \nu^{15} + 4974 \nu^{13} + 53076 \nu^{11} + 285186 \nu^{9} + 815454 \nu^{7} + 1198914 \nu^{5} + 782074 \nu^{3} + 136278 \nu ) / 1053 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 763 \nu^{15} + 16143 \nu^{13} + 129252 \nu^{11} + 504273 \nu^{9} + 1028268 \nu^{7} + 1103163 \nu^{5} + 604174 \nu^{3} + 130392 \nu ) / 2106 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 997 \nu^{15} - 21057 \nu^{13} - 167160 \nu^{11} - 634143 \nu^{9} - 1191132 \nu^{7} - 1011201 \nu^{5} - 284998 \nu^{3} + 4392 \nu ) / 2106 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 565 \nu^{15} - 12309 \nu^{13} - 102522 \nu^{11} - 419655 \nu^{9} - 895644 \nu^{7} - 963735 \nu^{5} - 455476 \nu^{3} - 60408 \nu ) / 1053 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 631 \nu^{15} - 14055 \nu^{13} - 121026 \nu^{11} - 520401 \nu^{9} - 1193508 \nu^{7} - 1422753 \nu^{5} - 767356 \nu^{3} - 121644 \nu ) / 1053 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{14} + 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{6} - \beta_{4} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{15} - 7\beta_{14} + 2\beta_{13} - 2\beta_{11} + 6\beta_{7} - 11\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -14\beta_{10} + 3\beta_{8} + 18\beta_{6} - 2\beta_{5} + 13\beta_{4} + 2\beta_{2} + 120 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -23\beta_{15} + 52\beta_{14} - 19\beta_{13} - 3\beta_{12} + 22\beta_{11} + 8\beta_{9} - 60\beta_{7} + 82\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 200\beta_{10} - 77\beta_{8} - 278\beta_{6} + 50\beta_{5} - 207\beta_{4} - 52\beta_{2} - 10\beta _1 - 1476 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 319 \beta_{15} - 797 \beta_{14} + 306 \beta_{13} + 90 \beta_{12} - 400 \beta_{11} - 204 \beta_{9} + 1092 \beta_{7} - 1334 \beta_{3} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -373\beta_{10} + 191\beta_{8} + 531\beta_{6} - 120\beta_{5} + 435\beta_{4} + 125\beta_{2} + 41\beta _1 + 2556 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1198 \beta_{15} + 3111 \beta_{14} - 1192 \beta_{13} - 462 \beta_{12} + 1714 \beta_{11} + 980 \beta_{9} - 4746 \beta_{7} + 5519 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5732 \beta_{10} - 3445 \beta_{8} - 8210 \beta_{6} + 2104 \beta_{5} - 7309 \beta_{4} - 2174 \beta_{2} - 914 \beta _1 - 37884 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9316 \beta_{15} - 24570 \beta_{14} + 9283 \beta_{13} + 4161 \beta_{12} - 14304 \beta_{11} - 8556 \beta_{9} + 40062 \beta_{7} - 45575 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 89724 \beta_{10} + 59267 \beta_{8} + 128574 \beta_{6} - 35426 \beta_{5} + 121513 \beta_{4} + 36304 \beta_{2} + 17486 \beta _1 + 582684 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 147283 \beta_{15} + 390967 \beta_{14} - 145680 \beta_{13} - 70896 \beta_{12} + 235380 \beta_{11} + 143460 \beta_{9} - 664092 \beta_{7} + 748154 \beta_{3} ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 710204 \beta_{10} - 496587 \beta_{8} - 1016690 \beta_{6} + 292314 \beta_{5} - 1000109 \beta_{4} - 297708 \beta_{2} - 155202 \beta _1 - 4572420 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1173997 \beta_{15} - 3125197 \beta_{14} + 1152110 \beta_{13} + 588330 \beta_{12} - 1920788 \beta_{11} - 1180044 \beta_{9} + 5443080 \beta_{7} - 6105812 \beta_{3} ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.424128i
0.424128i
0.555295i
0.555295i
2.22580i
2.22580i
1.56323i
1.56323i
1.85892i
1.85892i
2.83834i
2.83834i
1.14902i
1.14902i
1.81159i
1.81159i
0 −5.67668 0 2.23607i 0 10.1235i 0 23.2247 0
1151.2 0 −5.67668 0 2.23607i 0 10.1235i 0 23.2247 0
1151.3 0 −3.71784 0 2.23607i 0 12.3101i 0 4.82232 0
1151.4 0 −3.71784 0 2.23607i 0 12.3101i 0 4.82232 0
1151.5 0 −3.62318 0 2.23607i 0 5.28267i 0 4.12745 0
1151.6 0 −3.62318 0 2.23607i 0 5.28267i 0 4.12745 0
1151.7 0 −2.29803 0 2.23607i 0 2.72697i 0 −3.71904 0
1151.8 0 −2.29803 0 2.23607i 0 2.72697i 0 −3.71904 0
1151.9 0 −1.11059 0 2.23607i 0 10.6664i 0 −7.76659 0
1151.10 0 −1.11059 0 2.23607i 0 10.6664i 0 −7.76659 0
1151.11 0 0.848256 0 2.23607i 0 2.82289i 0 −8.28046 0
1151.12 0 0.848256 0 2.23607i 0 2.82289i 0 −8.28046 0
1151.13 0 3.12646 0 2.23607i 0 4.57359i 0 0.774764 0
1151.14 0 3.12646 0 2.23607i 0 4.57359i 0 0.774764 0
1151.15 0 4.45161 0 2.23607i 0 6.92638i 0 10.8168 0
1151.16 0 4.45161 0 2.23607i 0 6.92638i 0 10.8168 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.16
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 1280.3.g.g 16
4.b odd 2 1 1280.3.g.h 16
8.b even 2 1 1280.3.g.h 16
8.d odd 2 1 inner 1280.3.g.g 16
16.e even 4 1 640.3.b.a 16
16.e even 4 1 640.3.b.b yes 16
16.f odd 4 1 640.3.b.a 16
16.f odd 4 1 640.3.b.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.3.b.a 16 16.e even 4 1
640.3.b.a 16 16.f odd 4 1
640.3.b.b yes 16 16.e even 4 1
640.3.b.b yes 16 16.f odd 4 1
1280.3.g.g 16 1.a even 1 1 trivial
1280.3.g.g 16 8.d odd 2 1 inner
1280.3.g.h 16 4.b odd 2 1
1280.3.g.h 16 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 8T_{3}^{7} - 16T_{3}^{6} - 240T_{3}^{5} - 224T_{3}^{4} + 1664T_{3}^{3} + 3008T_{3}^{2} - 768T_{3} - 2304 \) acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 8 T^{7} - 16 T^{6} - 240 T^{5} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 480 T^{14} + \cdots + 2932258963456 \) Copy content Toggle raw display
$11$ \( (T^{8} + 32 T^{7} - 16 T^{6} + \cdots - 2569984)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 1760 T^{14} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1392 T^{6} - 1024 T^{5} + \cdots - 482211584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 48 T^{7} - 624 T^{6} + \cdots + 1870799104)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 4576 T^{14} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + 6944 T^{14} + \cdots + 56\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{16} + 4352 T^{14} + \cdots + 53\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + 13856 T^{14} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{8} - 48 T^{7} + \cdots - 966341086976)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 88 T^{7} + \cdots + 200274921216)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 15008 T^{14} + \cdots + 80\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + 15072 T^{14} + \cdots + 94\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{8} - 16 T^{7} + \cdots + 2914995568896)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 20768 T^{14} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{8} - 280 T^{7} + \cdots + 196984704870144)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 29056 T^{14} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} - 160 T^{7} + \cdots - 1312077678336)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 53248 T^{14} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{8} + 24 T^{7} + \cdots - 6286784368896)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 48 T^{7} + \cdots - 255121887096576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 224 T^{7} + \cdots + 24\!\cdots\!96)^{2} \) Copy content Toggle raw display
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