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Properties

Label 177.8.a.c
Level $177$
Weight $8$
Character orbit 177.a
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 27 q^{3} + (\beta_{2} + \beta_1 + 68) q^{4} + ( - \beta_{4} + 2 \beta_1 - 19) q^{5} - 27 \beta_1 q^{6} + (\beta_{10} - 2 \beta_1 + 185) q^{7} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots + 128) q^{8}+ \cdots + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 27 q^{3} + (\beta_{2} + \beta_1 + 68) q^{4} + ( - \beta_{4} + 2 \beta_1 - 19) q^{5} - 27 \beta_1 q^{6} + (\beta_{10} - 2 \beta_1 + 185) q^{7} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots + 128) q^{8}+ \cdots + ( - 729 \beta_{16} + 729 \beta_{15} + \cdots - 69984) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 2 q^{2} - 459 q^{3} + 1166 q^{4} - 318 q^{5} - 54 q^{6} + 3145 q^{7} + 2355 q^{8} + 12393 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 2 q^{2} - 459 q^{3} + 1166 q^{4} - 318 q^{5} - 54 q^{6} + 3145 q^{7} + 2355 q^{8} + 12393 q^{9} + 6521 q^{10} - 1764 q^{11} - 31482 q^{12} + 18192 q^{13} - 7827 q^{14} + 8586 q^{15} + 139226 q^{16} - 15507 q^{17} + 1458 q^{18} + 52083 q^{19} + 721 q^{20} - 84915 q^{21} - 234434 q^{22} + 63823 q^{23} - 63585 q^{24} + 202153 q^{25} - 367956 q^{26} - 334611 q^{27} + 182306 q^{28} - 502955 q^{29} - 176067 q^{30} + 347531 q^{31} - 243908 q^{32} + 47628 q^{33} - 330872 q^{34} + 92641 q^{35} + 850014 q^{36} + 447615 q^{37} + 775669 q^{38} - 491184 q^{39} + 2203270 q^{40} + 940335 q^{41} + 211329 q^{42} + 478562 q^{43} - 596924 q^{44} - 231822 q^{45} - 3078663 q^{46} + 703121 q^{47} - 3759102 q^{48} + 1895082 q^{49} - 876967 q^{50} + 418689 q^{51} + 6278296 q^{52} - 1005974 q^{53} - 39366 q^{54} + 5212846 q^{55} + 3425294 q^{56} - 1406241 q^{57} + 6710166 q^{58} - 3491443 q^{59} - 19467 q^{60} + 11510749 q^{61} + 5996234 q^{62} + 2292705 q^{63} + 29496941 q^{64} + 11094180 q^{65} + 6329718 q^{66} + 14007144 q^{67} + 19688159 q^{68} - 1723221 q^{69} + 30909708 q^{70} + 5229074 q^{71} + 1716795 q^{72} + 5452211 q^{73} + 12819662 q^{74} - 5458131 q^{75} + 41929340 q^{76} + 9930777 q^{77} + 9934812 q^{78} + 15275654 q^{79} + 36576105 q^{80} + 9034497 q^{81} + 32025935 q^{82} + 7826609 q^{83} - 4922262 q^{84} + 11836945 q^{85} + 51649136 q^{86} + 13579785 q^{87} + 30223741 q^{88} - 6436185 q^{89} + 4753809 q^{90} + 11633535 q^{91} + 43357972 q^{92} - 9383337 q^{93} - 4494252 q^{94} + 23741055 q^{95} + 6585516 q^{96} + 26377540 q^{97} + 26517816 q^{98} - 1285956 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + \cdots + 24\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 196 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!23 \nu^{16} + \cdots - 18\!\cdots\!68 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!41 \nu^{16} + \cdots - 85\!\cdots\!48 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!41 \nu^{16} + \cdots - 15\!\cdots\!40 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!95 \nu^{16} + \cdots + 62\!\cdots\!00 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!01 \nu^{16} + \cdots - 16\!\cdots\!76 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!77 \nu^{16} + \cdots + 11\!\cdots\!44 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55\!\cdots\!51 \nu^{16} + \cdots - 31\!\cdots\!96 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!87 \nu^{16} + \cdots + 45\!\cdots\!88 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{16} + \cdots - 74\!\cdots\!20 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!33 \nu^{16} + \cdots - 91\!\cdots\!52 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!47 \nu^{16} + \cdots - 72\!\cdots\!96 ) / 82\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!15 \nu^{16} + \cdots - 26\!\cdots\!04 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 43\!\cdots\!78 \nu^{16} + \cdots + 30\!\cdots\!64 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{16} + \cdots - 65\!\cdots\!52 ) / 41\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 196 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} + \beta_{3} + 343\beta _1 + 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{16} + \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - \beta_{11} + 6 \beta_{10} + \cdots + 67005 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{16} - 9 \beta_{15} - 30 \beta_{14} - 52 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} + \cdots + 49443 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4428 \beta_{16} + 620 \beta_{15} + 1572 \beta_{14} - 3762 \beta_{13} - 2026 \beta_{12} - 518 \beta_{11} + \cdots + 27441788 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9070 \beta_{16} - 4023 \beta_{15} - 20770 \beta_{14} - 39287 \beta_{13} - 7888 \beta_{12} + \cdots + 16725561 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2479010 \beta_{16} + 290925 \beta_{15} + 963786 \beta_{14} - 2165060 \beta_{13} - 1340093 \beta_{12} + \cdots + 12112167141 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4315240 \beta_{16} - 408182 \beta_{15} - 10904788 \beta_{14} - 22246216 \beta_{13} - 3753658 \beta_{12} + \cdots + 5128297790 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1271346030 \beta_{16} + 122290725 \beta_{15} + 543072090 \beta_{14} - 1138061973 \beta_{13} + \cdots + 5534138272313 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1772327478 \beta_{16} + 788518275 \beta_{15} - 5296131630 \beta_{14} - 11335515344 \beta_{13} + \cdots + 1353066234647 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 630160762572 \beta_{16} + 47674381828 \beta_{15} + 293440025772 \beta_{14} - 574597910894 \beta_{13} + \cdots + 25\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 638016799126 \beta_{16} + 872848948757 \beta_{15} - 2522693475330 \beta_{14} - 5490034897119 \beta_{13} + \cdots + 217406968471469 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 307896248891050 \beta_{16} + 17087373773281 \beta_{15} + 154652656782170 \beta_{14} + \cdots + 12\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 186458253689432 \beta_{16} + 647735166648634 \beta_{15} + \cdots - 77\!\cdots\!26 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 14\!\cdots\!22 \beta_{16} + \cdots + 56\!\cdots\!25 \) 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
−22.1687
−20.9857
−14.3045
−13.8304
−12.3679
−8.14464
−5.66556
−3.88629
−3.62331
3.56813
7.49302
9.14085
11.5967
15.1891
16.5905
21.5848
21.8139
−22.1687 −27.0000 363.451 258.765 598.555 411.003 −5219.64 729.000 −5736.49
1.2 −20.9857 −27.0000 312.401 −465.631 566.615 286.579 −3869.79 729.000 9771.60
1.3 −14.3045 −27.0000 76.6177 348.885 386.221 −1316.66 734.996 729.000 −4990.62
1.4 −13.8304 −27.0000 63.2803 −149.469 373.421 14.2725 895.100 729.000 2067.22
1.5 −12.3679 −27.0000 24.9655 −477.980 333.934 1344.57 1274.32 729.000 5911.62
1.6 −8.14464 −27.0000 −61.6648 −54.3103 219.905 −413.400 1544.75 729.000 442.338
1.7 −5.66556 −27.0000 −95.9014 117.894 152.970 1197.66 1268.53 729.000 −667.937
1.8 −3.88629 −27.0000 −112.897 −26.2639 104.930 −644.460 936.194 729.000 102.069
1.9 −3.62331 −27.0000 −114.872 300.817 97.8295 1353.78 880.000 729.000 −1089.96
1.10 3.56813 −27.0000 −115.268 −210.232 −96.3395 1544.27 −868.013 729.000 −750.136
1.11 7.49302 −27.0000 −71.8546 400.807 −202.312 −272.200 −1497.52 729.000 3003.25
1.12 9.14085 −27.0000 −44.4448 −27.9722 −246.803 −1415.76 −1576.29 729.000 −255.690
1.13 11.5967 −27.0000 6.48306 −401.104 −313.110 213.573 −1409.19 729.000 −4651.48
1.14 15.1891 −27.0000 102.710 −418.807 −410.107 −814.653 −384.131 729.000 −6361.32
1.15 16.5905 −27.0000 147.246 174.998 −447.945 1111.07 319.307 729.000 2903.31
1.16 21.5848 −27.0000 337.903 −113.334 −582.789 −647.523 4530.72 729.000 −2446.30
1.17 21.8139 −27.0000 347.844 424.937 −588.974 1192.89 4795.65 729.000 9269.52
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.c ✓ 17
3.b odd 2 1 531.8.a.c 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.c ✓ 17 1.a even 1 1 trivial
531.8.a.c 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 2 T_{2}^{16} - 1669 T_{2}^{15} + 2385 T_{2}^{14} + 1108684 T_{2}^{13} + \cdots + 24\!\cdots\!16 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( (T + 27)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 47\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 46\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 41\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 98\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 52\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 52\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 44\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T + 205379)^{17} \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 99\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 22\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 45\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 44\!\cdots\!88 \) Copy content Toggle raw display
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