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Properties

Label 177.8.a.a
Level $177$
Weight $8$
Character orbit 177.a
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
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] = [177,8,Mod(1,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.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: \(55.2921495107\)
Analytic rank: \(1\)
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: \( x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 27 q^{3} + (\beta_{2} + 61) q^{4} + (\beta_{4} + \beta_1 - 5) q^{5} + 27 \beta_1 q^{6} + ( - \beta_{9} + \beta_{2} + 5 \beta_1 - 148) q^{7} + ( - \beta_{3} - 42 \beta_1 + 67) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 27 q^{3} + (\beta_{2} + 61) q^{4} + (\beta_{4} + \beta_1 - 5) q^{5} + 27 \beta_1 q^{6} + ( - \beta_{9} + \beta_{2} + 5 \beta_1 - 148) q^{7} + ( - \beta_{3} - 42 \beta_1 + 67) q^{8} + 729 q^{9} + (\beta_{14} - \beta_{13} - \beta_{10} + \cdots - 204) q^{10}+ \cdots + ( - 1458 \beta_{14} + 1458 \beta_{12} + \cdots + 53217) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 298\nu + 67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 72\!\cdots\!81 \nu^{15} + \cdots + 98\!\cdots\!76 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!49 \nu^{15} + \cdots + 18\!\cdots\!64 ) / 80\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!57 \nu^{15} + \cdots + 12\!\cdots\!48 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50\!\cdots\!83 \nu^{15} + \cdots + 10\!\cdots\!52 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!91 \nu^{15} + \cdots - 48\!\cdots\!64 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!16 \nu^{15} + \cdots + 10\!\cdots\!64 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!59 \nu^{15} + \cdots + 79\!\cdots\!16 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 77\!\cdots\!87 \nu^{15} + \cdots - 87\!\cdots\!08 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42\!\cdots\!39 \nu^{15} + \cdots - 20\!\cdots\!96 ) / 80\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 63\!\cdots\!26 \nu^{15} + \cdots - 41\!\cdots\!04 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!11 \nu^{15} + \cdots - 22\!\cdots\!24 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 69\!\cdots\!33 \nu^{15} + \cdots + 11\!\cdots\!92 ) / 10\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 189 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 298\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 4 \beta_{14} + 4 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \cdots + 56410 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{15} + 2 \beta_{14} + 20 \beta_{13} + 16 \beta_{12} + 20 \beta_{11} - 6 \beta_{10} + \cdots - 29763 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 322 \beta_{15} - 2056 \beta_{14} + 2752 \beta_{13} + 1738 \beta_{12} - 1718 \beta_{11} + \cdots + 18428719 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3715 \beta_{15} + 1540 \beta_{14} + 14628 \beta_{13} + 11923 \beta_{12} + 7041 \beta_{11} + \cdots - 12727306 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 54326 \beta_{15} - 837682 \beta_{14} + 1420036 \beta_{13} + 734014 \beta_{12} - 710438 \beta_{11} + \cdots + 6300524291 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1081644 \beta_{15} + 1020800 \beta_{14} + 7884360 \beta_{13} + 6213940 \beta_{12} + 1058156 \beta_{11} + \cdots - 4947659831 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9178535 \beta_{15} - 317171524 \beta_{14} + 657958388 \beta_{13} + 275901219 \beta_{12} + \cdots + 2218326673922 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 192171720 \beta_{15} + 611574026 \beta_{14} + 3800835556 \beta_{13} + 2776451832 \beta_{12} + \cdots - 1752912931667 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 14484661278 \beta_{15} - 116815310624 \beta_{14} + 289230246224 \beta_{13} + 97765146802 \beta_{12} + \cdots + 796832833506959 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 21120806365 \beta_{15} + 332609278476 \beta_{14} + 1734027120372 \beta_{13} + 1141712395035 \beta_{12} + \cdots - 570094464406250 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9321682989434 \beta_{15} - 42615689960586 \beta_{14} + 123413715766356 \beta_{13} + \cdots + 29\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 43827714804276 \beta_{15} + 167484351779752 \beta_{14} + 765851434829912 \beta_{13} + \cdots - 16\!\cdots\!91 \) Copy content Toggle raw display

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

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
19.7363
19.0314
16.2952
15.0467
14.7989
7.00808
4.55626
1.97136
1.05882
−3.09726
−7.02227
−13.0039
−14.0604
−17.9442
−18.7189
−19.6562
−19.7363 −27.0000 261.522 −90.5952 532.880 −807.818 −2635.23 729.000 1788.01
1.2 −19.0314 −27.0000 234.195 37.6075 513.848 1158.54 −2021.05 729.000 −715.724
1.3 −16.2952 −27.0000 137.534 495.569 439.971 565.301 −155.363 729.000 −8075.41
1.4 −15.0467 −27.0000 98.4026 159.890 406.260 980.332 445.343 729.000 −2405.81
1.5 −14.7989 −27.0000 91.0084 −296.536 399.571 −1410.76 547.437 729.000 4388.42
1.6 −7.00808 −27.0000 −78.8868 449.079 189.218 −271.717 1449.88 729.000 −3147.18
1.7 −4.55626 −27.0000 −107.241 −540.445 123.019 −1238.10 1071.82 729.000 2462.40
1.8 −1.97136 −27.0000 −124.114 −339.775 53.2268 364.700 497.007 729.000 669.820
1.9 −1.05882 −27.0000 −126.879 151.597 28.5880 −1574.54 269.870 729.000 −160.513
1.10 3.09726 −27.0000 −118.407 −156.435 −83.6261 11.3597 −763.187 729.000 −484.520
1.11 7.02227 −27.0000 −78.6878 −266.773 −189.601 665.758 −1451.42 729.000 −1873.35
1.12 13.0039 −27.0000 41.1009 167.303 −351.105 887.373 −1130.03 729.000 2175.59
1.13 14.0604 −27.0000 69.6949 153.219 −379.631 −215.221 −819.793 729.000 2154.32
1.14 17.9442 −27.0000 193.993 1.22320 −484.493 −719.259 1184.20 729.000 21.9494
1.15 18.7189 −27.0000 222.398 443.832 −505.411 −1695.79 1767.03 729.000 8308.06
1.16 19.6562 −27.0000 258.365 −436.761 −530.717 956.841 2562.48 729.000 −8585.06
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(59\) \(-1\)

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 177.8.a.a ✓ 16
3.b odd 2 1 531.8.a.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.a ✓ 16 1.a even 1 1 trivial
531.8.a.b 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 6 T_{2}^{15} - 1493 T_{2}^{14} - 8791 T_{2}^{13} + 890490 T_{2}^{12} + \cdots - 23\!\cdots\!32 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots - 23\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( (T + 27)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 51\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 41\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 67\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 50\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 82\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( (T - 205379)^{16} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 35\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 74\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 49\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 91\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 38\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
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