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Properties

Label 177.3.c.a
Level $177$
Weight $3$
Character orbit 177.c
Analytic conductor $4.823$
Analytic rank $0$
Dimension $20$
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] = [177,3,Mod(58,177)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.58");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 177.c (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: \(4.82290067918\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{5} + \beta_{9} q^{6} + \beta_{14} q^{7} + (\beta_{3} - \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{5} + \beta_{9} q^{6} + \beta_{14} q^{7} + (\beta_{3} - \beta_1) q^{8} + 3 q^{9} + (\beta_{7} - \beta_{3} + \beta_1) q^{10} + (\beta_{10} + \beta_{7}) q^{11} + (\beta_{15} + 2 \beta_{8} - 1) q^{12} + (\beta_{11} + \beta_{9} + \beta_1) q^{13} + (\beta_{19} + \beta_{13} + \cdots - 2 \beta_1) q^{14}+ \cdots + (3 \beta_{10} + 3 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 40 q^{4} - 8 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 40 q^{4} - 8 q^{7} + 60 q^{9} - 24 q^{12} + 24 q^{15} - 8 q^{16} + 16 q^{17} - 60 q^{19} - 164 q^{20} + 40 q^{22} + 100 q^{25} - 156 q^{26} + 200 q^{28} - 60 q^{29} - 32 q^{35} - 120 q^{36} + 28 q^{41} + 180 q^{46} - 96 q^{48} + 284 q^{49} - 24 q^{51} - 8 q^{53} + 24 q^{57} - 152 q^{59} - 72 q^{60} - 8 q^{62} - 24 q^{63} + 204 q^{64} - 120 q^{66} + 384 q^{68} + 92 q^{71} + 104 q^{74} + 240 q^{75} + 120 q^{76} - 468 q^{78} - 420 q^{79} + 376 q^{80} + 180 q^{81} + 168 q^{84} - 348 q^{85} + 232 q^{86} + 144 q^{87} - 212 q^{88} + 152 q^{94} + 788 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 9\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24659 \nu^{19} + 525507 \nu^{17} - 8932411 \nu^{15} - 381955669 \nu^{13} + \cdots + 66292360677 \nu ) / 1272772800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30094 \nu^{18} - 1782837 \nu^{16} - 43396174 \nu^{14} - 556908571 \nu^{12} + \cdots + 3502798593 ) / 424257600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 224699 \nu^{18} - 12416727 \nu^{16} - 286167029 \nu^{14} - 3569199791 \nu^{12} + \cdots - 27178002297 ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 224699 \nu^{19} + 12416727 \nu^{17} + 286167029 \nu^{15} + 3569199791 \nu^{13} + \cdots + 47542367097 \nu ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 101093 \nu^{18} + 5880439 \nu^{16} + 142869803 \nu^{14} + 1879285587 \nu^{12} + \cdots + 31714629129 ) / 848515200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101093 \nu^{19} - 5880439 \nu^{17} - 142869803 \nu^{15} - 1879285587 \nu^{13} + \cdots - 31714629129 \nu ) / 848515200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 533161 \nu^{19} - 29389353 \nu^{17} - 671121631 \nu^{15} - 8215396849 \nu^{13} + \cdots - 204166271583 \nu ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 288077 \nu^{19} + 17226546 \nu^{17} + 434018867 \nu^{15} + 5992261568 \nu^{13} + \cdots + 252139008006 \nu ) / 1272772800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 650693 \nu^{18} + 37002339 \nu^{16} + 883008803 \nu^{14} + 11503461887 \nu^{12} + \cdots + 182480565429 ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 243091 \nu^{19} + 14075643 \nu^{17} + 341817061 \nu^{15} + 4524039619 \nu^{13} + \cdots + 69057391773 \nu ) / 848515200 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 455827 \nu^{18} - 27087171 \nu^{16} - 671585917 \nu^{14} - 8996141443 \nu^{12} + \cdots - 127003783581 ) / 1272772800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 421417 \nu^{18} - 24288891 \nu^{16} - 584638807 \nu^{14} - 7622076103 \nu^{12} + \cdots - 131729208501 ) / 848515200 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 447991 \nu^{18} - 24915743 \nu^{16} - 575077561 \nu^{14} - 7131169119 \nu^{12} + \cdots - 64472580273 ) / 848515200 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 596207 \nu^{19} + 34251811 \nu^{17} + 822056297 \nu^{15} + 10690895763 \nu^{13} + \cdots + 86604679221 \nu ) / 848515200 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2097083 \nu^{18} - 119728059 \nu^{16} - 2851123493 \nu^{14} - 36718884347 \nu^{12} + \cdots - 421529033349 ) / 2545545600 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 90587 \nu^{19} - 5241276 \nu^{17} - 126656027 \nu^{15} - 1657491608 \nu^{13} + \cdots - 27510621786 \nu ) / 106064400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + 2\beta_{8} - \beta_{6} + \beta_{5} - 13\beta_{2} + 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} + \beta_{10} - 4\beta_{9} + \beta_{4} - 15\beta_{3} + 95\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{16} - 6 \beta_{15} + 2 \beta_{14} + 18 \beta_{12} - 50 \beta_{8} + 16 \beta_{6} - 22 \beta_{5} + \cdots - 597 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{19} + 20 \beta_{13} + 2 \beta_{11} - 24 \beta_{10} + 106 \beta_{9} + 2 \beta_{7} + \cdots - 1075 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{18} - 48 \beta_{16} + 158 \beta_{15} - 38 \beta_{14} - 266 \beta_{12} + 950 \beta_{8} + \cdots + 6756 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 72 \beta_{19} + 2 \beta_{17} - 304 \beta_{13} - 48 \beta_{11} + 398 \beta_{10} - 2004 \beta_{9} + \cdots + 12663 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 104 \beta_{18} + 794 \beta_{16} - 2944 \beta_{15} + 470 \beta_{14} + 3739 \beta_{12} - 15914 \beta_{8} + \cdots - 79374 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1680 \beta_{19} - 104 \beta_{17} + 4209 \beta_{13} + 794 \beta_{11} - 5721 \beta_{10} + \cdots - 153405 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2984 \beta_{18} - 11276 \beta_{16} + 47820 \beta_{15} - 4256 \beta_{14} - 51676 \beta_{12} + \cdots + 958629 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 32288 \beta_{19} + 2984 \beta_{17} - 55932 \beta_{13} - 11276 \beta_{11} + 76652 \beta_{10} + \cdots + 1897953 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 65740 \beta_{18} + 147808 \beta_{16} - 724052 \beta_{15} + 19600 \beta_{14} + 709288 \beta_{12} + \cdots - 11827082 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 556644 \beta_{19} - 65740 \beta_{17} + 728888 \beta_{13} + 147808 \beta_{11} - 987900 \beta_{10} + \cdots - 23863777 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1250416 \beta_{18} - 1846204 \beta_{16} + 10525640 \beta_{15} + 286768 \beta_{14} - 9701529 \beta_{12} + \cdots + 148356592 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 8966204 \beta_{19} + 1250416 \beta_{17} - 9414761 \beta_{13} - 1846204 \beta_{11} + \cdots + 303804087 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 21697080 \beta_{18} + 22328214 \beta_{16} - 149178658 \beta_{15} - 11158094 \beta_{14} + \cdots - 1885103977 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 138008538 \beta_{19} - 21697080 \beta_{17} + 121233592 \beta_{13} + 22328214 \beta_{11} + \cdots - 3905050051 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
− 3.65759i
− 3.38526i
− 3.07256i
− 2.96307i
− 2.72775i
− 1.86235i
− 1.61360i
− 1.39995i
− 1.08628i
− 0.537675i
0.537675i
1.08628i
1.39995i
1.61360i
1.86235i
2.72775i
2.96307i
3.07256i
3.38526i
3.65759i
− 3.65759i −1.73205 −9.37798 0.395869 6.33514i −6.20847 19.6705i 3.00000 − 1.44793i
58.2 − 3.38526i 1.73205 −7.45997 9.27487 − 5.86344i −9.96030 11.7129i 3.00000 − 31.3978i
58.3 − 3.07256i 1.73205 −5.44061 −6.03179 − 5.32183i −9.30133 4.42637i 3.00000 18.5330i
58.4 − 2.96307i 1.73205 −4.77980 3.25089 − 5.13219i 11.8224 2.31061i 3.00000 − 9.63262i
58.5 − 2.72775i −1.73205 −3.44061 5.71516 4.72460i 8.69147 − 1.52587i 3.00000 − 15.5895i
58.6 − 1.86235i −1.73205 0.531642 −9.04540 3.22569i 1.29987 − 8.43952i 3.00000 16.8457i
58.7 − 1.61360i 1.73205 1.39629 −6.36659 − 2.79484i 6.66564 − 8.70746i 3.00000 10.2731i
58.8 − 1.39995i −1.73205 2.04015 0.273484 2.42478i −10.8938 − 8.45589i 3.00000 − 0.382863i
58.9 − 1.08628i 1.73205 2.82000 3.33671 − 1.88149i −1.22637 − 7.40842i 3.00000 − 3.62460i
58.10 − 0.537675i −1.73205 3.71091 −0.803210 0.931281i 5.11098 − 4.14596i 3.00000 0.431866i
58.11 0.537675i −1.73205 3.71091 −0.803210 − 0.931281i 5.11098 4.14596i 3.00000 − 0.431866i
58.12 1.08628i 1.73205 2.82000 3.33671 1.88149i −1.22637 7.40842i 3.00000 3.62460i
58.13 1.39995i −1.73205 2.04015 0.273484 − 2.42478i −10.8938 8.45589i 3.00000 0.382863i
58.14 1.61360i 1.73205 1.39629 −6.36659 2.79484i 6.66564 8.70746i 3.00000 − 10.2731i
58.15 1.86235i −1.73205 0.531642 −9.04540 − 3.22569i 1.29987 8.43952i 3.00000 − 16.8457i
58.16 2.72775i −1.73205 −3.44061 5.71516 − 4.72460i 8.69147 1.52587i 3.00000 15.5895i
58.17 2.96307i 1.73205 −4.77980 3.25089 5.13219i 11.8224 − 2.31061i 3.00000 9.63262i
58.18 3.07256i 1.73205 −5.44061 −6.03179 5.32183i −9.30133 − 4.42637i 3.00000 − 18.5330i
58.19 3.38526i 1.73205 −7.45997 9.27487 5.86344i −9.96030 − 11.7129i 3.00000 31.3978i
58.20 3.65759i −1.73205 −9.37798 0.395869 − 6.33514i −6.20847 − 19.6705i 3.00000 1.44793i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.3.c.a ✓ 20
3.b odd 2 1 531.3.c.c 20
59.b odd 2 1 inner 177.3.c.a ✓ 20
177.d even 2 1 531.3.c.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.3.c.a ✓ 20 1.a even 1 1 trivial
177.3.c.a ✓ 20 59.b odd 2 1 inner
531.3.c.c 20 3.b odd 2 1
531.3.c.c 20 177.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 60 T^{18} + \cdots + 570861 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{10} \) Copy content Toggle raw display
$5$ \( (T^{10} - 150 T^{8} + \cdots + 17368)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 4 T^{9} + \cdots - 34966256)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{10} - 8 T^{9} + \cdots - 139536176)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 726592156400)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 14765383626976)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 103416495474416)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 242741808356634)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 37\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
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