Properties

Label 161.4.a.c
Level $161$
Weight $4$
Character orbit 161.a
Self dual yes
Analytic conductor $9.499$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,4,Mod(1,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.a (trivial)

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: yes
Analytic conductor: \(9.49930751092\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 60x^{7} - 22x^{6} + 1179x^{5} + 694x^{4} - 7936x^{3} - 4352x^{2} + 11008x + 3072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + \beta_{6} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 4) q^{6} - 7 q^{7} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 8) q^{8} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + \beta_{6} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 4) q^{6} - 7 q^{7} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 8) q^{8} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 14) q^{9} + (\beta_{8} + 4 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{10} + ( - \beta_{8} + \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} + 7 \beta_1 + 24) q^{12} + ( - 2 \beta_{8} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{13} - 7 \beta_1 q^{14} + (2 \beta_{8} + 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \cdots + 11) q^{15}+ \cdots + (3 \beta_{8} + 50 \beta_{7} + 9 \beta_{6} + 24 \beta_{5} + 96 \beta_{4} + \cdots - 102) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} + 48 q^{4} - 4 q^{5} + 46 q^{6} - 63 q^{7} + 66 q^{8} + 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{3} + 48 q^{4} - 4 q^{5} + 46 q^{6} - 63 q^{7} + 66 q^{8} + 122 q^{9} + 50 q^{10} - 8 q^{11} + 220 q^{12} + 25 q^{13} + 88 q^{15} + 180 q^{16} - 28 q^{17} - 54 q^{18} + 254 q^{19} + 302 q^{20} - 63 q^{21} - 122 q^{22} - 207 q^{23} + 624 q^{24} + 295 q^{25} - 6 q^{26} + 633 q^{27} - 336 q^{28} + 25 q^{29} + 80 q^{30} + 1171 q^{31} + 1018 q^{32} + 272 q^{33} + 8 q^{34} + 28 q^{35} + 72 q^{36} + 70 q^{37} + 282 q^{38} + 1185 q^{39} - 54 q^{40} + 221 q^{41} - 322 q^{42} - 42 q^{43} - 1214 q^{44} + 698 q^{45} + 159 q^{47} - 1104 q^{48} + 441 q^{49} - 1680 q^{50} + 308 q^{51} - 664 q^{52} - 774 q^{53} - 2674 q^{54} + 1498 q^{55} - 462 q^{56} - 2524 q^{57} + 842 q^{58} + 1080 q^{59} - 2160 q^{60} + 686 q^{61} - 2078 q^{62} - 854 q^{63} - 1260 q^{64} - 1656 q^{65} + 532 q^{66} - 370 q^{67} - 1936 q^{68} - 207 q^{69} - 350 q^{70} + 1035 q^{71} - 722 q^{72} + 1979 q^{73} - 5494 q^{74} - 1459 q^{75} + 206 q^{76} + 56 q^{77} - 2066 q^{78} + 2336 q^{79} - 242 q^{80} + 997 q^{81} + 1642 q^{82} + 130 q^{83} - 1540 q^{84} - 272 q^{85} - 1906 q^{86} - 581 q^{87} - 3742 q^{88} - 1328 q^{89} - 7650 q^{90} - 175 q^{91} - 1104 q^{92} + 1305 q^{93} + 6078 q^{94} - 484 q^{95} - 136 q^{96} - 104 q^{97} - 618 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 60x^{7} - 22x^{6} + 1179x^{5} + 694x^{4} - 7936x^{3} - 4352x^{2} + 11008x + 3072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 25\nu^{2} - 50\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 25\nu^{2} + 26\nu + 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 35\nu^{4} + 83\nu^{3} + 306\nu^{2} - 480\nu - 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 40\nu^{6} + 142\nu^{5} + 455\nu^{4} - 1354\nu^{3} - 1024\nu^{2} + 2592\nu - 768 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 4\nu^{7} + 44\nu^{6} - 138\nu^{5} - 611\nu^{4} + 1190\nu^{3} + 2520\nu^{2} - 1472\nu - 1536 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} - 56\nu^{6} - 18\nu^{5} + 1007\nu^{4} + 498\nu^{3} - 5912\nu^{2} - 2304\nu + 5120 ) / 128 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{4} + 2\beta_{3} + 19\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 4\beta_{3} + 25\beta_{2} + 37\beta _1 + 269 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} + 8\beta_{6} - 8\beta_{5} - 58\beta_{4} + 66\beta_{3} + 8\beta_{2} + 419\beta _1 + 368 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{7} + 24\beta_{6} + 8\beta_{5} + 132\beta_{4} + 172\beta_{3} + 593\beta_{2} + 1149\beta _1 + 6165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32 \beta_{8} + 416 \beta_{7} + 384 \beta_{6} - 288 \beta_{5} - 1418 \beta_{4} + 1850 \beta_{3} + 464 \beta_{2} + 9899 \beta _1 + 12968 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 128 \beta_{8} + 1488 \beta_{7} + 1488 \beta_{6} + 304 \beta_{5} + 3316 \beta_{4} + 5796 \beta_{3} + 14089 \beta_{2} + 33381 \beta _1 + 148733 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.55260
−4.38565
−3.73922
−1.58693
−0.264899
1.22159
3.04896
4.95572
5.30303
−4.55260 −5.72089 12.7261 17.5795 26.0449 −7.00000 −21.5162 5.72860 −80.0322
1.2 −4.38565 −0.971823 11.2339 −20.4381 4.26207 −7.00000 −14.1827 −26.0556 89.6341
1.3 −3.73922 10.3390 5.98179 7.23273 −38.6599 −7.00000 7.54655 79.8951 −27.0448
1.4 −1.58693 3.23551 −5.48164 −8.51310 −5.13454 −7.00000 21.3945 −16.5315 13.5097
1.5 −0.264899 −6.92505 −7.92983 −11.7130 1.83444 −7.00000 4.21980 20.9564 3.10276
1.6 1.22159 −7.12940 −6.50771 −2.67729 −8.70922 −7.00000 −17.7225 23.8284 −3.27056
1.7 3.04896 7.42499 1.29618 11.7973 22.6385 −7.00000 −20.4397 28.1305 35.9694
1.8 4.95572 7.67438 16.5591 −10.4913 38.0320 −7.00000 42.4166 31.8961 −51.9921
1.9 5.30303 1.07328 20.1221 13.2233 5.69164 −7.00000 64.2837 −25.8481 70.1237
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.a.c 9
3.b odd 2 1 1449.4.a.n 9
7.b odd 2 1 1127.4.a.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.a.c 9 1.a even 1 1 trivial
1127.4.a.f 9 7.b odd 2 1
1449.4.a.n 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 60T_{2}^{7} - 22T_{2}^{6} + 1179T_{2}^{5} + 694T_{2}^{4} - 7936T_{2}^{3} - 4352T_{2}^{2} + 11008T_{2} + 3072 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 60 T^{7} - 22 T^{6} + \cdots + 3072 \) Copy content Toggle raw display
$3$ \( T^{9} - 9 T^{8} - 142 T^{7} + \cdots - 561568 \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots + 1135409856 \) Copy content Toggle raw display
$7$ \( (T + 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 8 T^{8} + \cdots + 43666792951552 \) Copy content Toggle raw display
$13$ \( T^{9} - 25 T^{8} + \cdots - 6636351117312 \) Copy content Toggle raw display
$17$ \( T^{9} + 28 T^{8} + \cdots + 80\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{9} - 254 T^{8} + \cdots + 95\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( (T + 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} - 25 T^{8} + \cdots + 87\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{9} - 1171 T^{8} + \cdots - 42\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{9} - 70 T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{9} - 221 T^{8} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{9} + 42 T^{8} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{9} - 159 T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{9} + 774 T^{8} + \cdots + 27\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{9} - 1080 T^{8} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{9} - 686 T^{8} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{9} + 370 T^{8} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{9} - 1035 T^{8} + \cdots - 36\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{9} - 1979 T^{8} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{9} - 2336 T^{8} + \cdots - 92\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{9} - 130 T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{9} + 1328 T^{8} + \cdots + 79\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{9} + 104 T^{8} + \cdots - 44\!\cdots\!92 \) Copy content Toggle raw display
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