Properties

Label 177.8.a.d
Level $177$
Weight $8$
Character orbit 177.a
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + 27 q^{3} + (\beta_{2} + \beta_1 + 75) q^{4} + (\beta_{4} + \beta_1 + 37) q^{5} + (27 \beta_1 + 27) q^{6} + (\beta_{7} + 5 \beta_1 + 170) q^{7} + (\beta_{3} + 4 \beta_{2} + 61 \beta_1 + 207) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + 27 q^{3} + (\beta_{2} + \beta_1 + 75) q^{4} + (\beta_{4} + \beta_1 + 37) q^{5} + (27 \beta_1 + 27) q^{6} + (\beta_{7} + 5 \beta_1 + 170) q^{7} + (\beta_{3} + 4 \beta_{2} + 61 \beta_1 + 207) q^{8} + 729 q^{9} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 62 \beta_1 + 179) q^{10} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{4} + 2 \beta_{2} + 69 \beta_1 + 815) q^{11} + (27 \beta_{2} + 27 \beta_1 + 2025) q^{12} + (\beta_{15} + 2 \beta_{7} + \beta_{4} - 13 \beta_{2} + 53 \beta_1 + 743) q^{13} + ( - \beta_{15} + \beta_{14} - \beta_{11} + \beta_{9} - \beta_{8} + 3 \beta_{7} - \beta_{5} + 2 \beta_{4} + \cdots + 1085) q^{14}+ \cdots + (729 \beta_{11} + 729 \beta_{8} + 729 \beta_{7} - 729 \beta_{4} + \cdots + 594135) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} + \cdots + 51\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 202 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 318\nu + 346 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!09 \nu^{17} + \cdots + 78\!\cdots\!20 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63\!\cdots\!95 \nu^{17} + \cdots + 34\!\cdots\!64 ) / 51\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59\!\cdots\!49 \nu^{17} + \cdots + 22\!\cdots\!04 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 69\!\cdots\!03 \nu^{17} + \cdots + 11\!\cdots\!40 ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 87\!\cdots\!53 \nu^{17} + \cdots - 12\!\cdots\!04 ) / 90\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!75 \nu^{17} + \cdots - 41\!\cdots\!84 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!43 \nu^{17} + \cdots + 29\!\cdots\!28 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!61 \nu^{17} + \cdots - 26\!\cdots\!52 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!75 \nu^{17} + \cdots - 56\!\cdots\!64 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!37 \nu^{17} + \cdots + 63\!\cdots\!16 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!43 \nu^{17} + \cdots + 93\!\cdots\!96 ) / 18\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!51 \nu^{17} + \cdots + 26\!\cdots\!16 ) / 72\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!57 \nu^{17} + \cdots - 48\!\cdots\!00 ) / 72\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 67\!\cdots\!09 \nu^{17} + \cdots - 83\!\cdots\!44 ) / 14\!\cdots\!52 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 202 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 317\beta _1 - 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 14 \beta_{4} - \beta_{3} + 441 \beta_{2} - 458 \beta _1 + 64138 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{17} - 5 \beta_{16} - 3 \beta_{15} - 11 \beta_{14} + 9 \beta_{13} - 15 \beta_{12} + 5 \beta_{11} - 9 \beta_{10} - 19 \beta_{9} + 14 \beta_{8} - 3 \beta_{7} - 8 \beta_{6} - 4 \beta_{5} - 207 \beta_{4} + 536 \beta_{3} + 357 \beta_{2} + 115053 \beta _1 - 67329 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 124 \beta_{17} - 64 \beta_{16} - 791 \beta_{15} - 647 \beta_{14} + 659 \beta_{13} + 657 \beta_{12} - 575 \beta_{11} + 12 \beta_{10} - 635 \beta_{9} - 961 \beta_{8} - 1609 \beta_{7} + 1378 \beta_{6} + 1106 \beta_{5} + \cdots + 23297602 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2474 \beta_{17} - 3415 \beta_{16} - 2716 \beta_{15} - 7840 \beta_{14} + 6558 \beta_{13} - 11534 \beta_{12} + 6480 \beta_{11} - 6591 \beta_{10} - 13976 \beta_{9} + 13659 \beta_{8} - 2036 \beta_{7} + \cdots - 31157507 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 116554 \beta_{17} - 52891 \beta_{16} - 447118 \beta_{15} - 328802 \beta_{14} + 351956 \beta_{13} + 359890 \beta_{12} - 257626 \beta_{11} + 7433 \beta_{10} - 318758 \beta_{9} + \cdots + 8984566067 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 388322 \beta_{17} - 1825573 \beta_{16} - 1726918 \beta_{15} - 4078574 \beta_{14} + 3596304 \beta_{13} - 6216366 \beta_{12} + 4630522 \beta_{11} - 3534565 \beta_{10} + \cdots - 14615002473 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 75945018 \beta_{17} - 31120411 \beta_{16} - 221976003 \beta_{15} - 155030463 \beta_{14} + 173120393 \beta_{13} + 184817587 \beta_{12} - 105545031 \beta_{11} + \cdots + 3585271738667 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 228023992 \beta_{17} - 908704302 \beta_{16} - 934849297 \beta_{15} - 1888558801 \beta_{14} + 1763596785 \beta_{13} - 2924352141 \beta_{12} + 2662864647 \beta_{11} + \cdots - 6867264837542 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 42522848694 \beta_{17} - 16097755799 \beta_{16} - 103292091881 \beta_{15} - 70929671893 \beta_{14} + 81470489051 \beta_{13} + 91236803507 \beta_{12} + \cdots + 14\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 278185379156 \beta_{17} - 438056649436 \beta_{16} - 461373418794 \beta_{15} - 829096386854 \beta_{14} + 815874882166 \beta_{13} - 1286465288348 \beta_{12} + \cdots - 32\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 21971913318404 \beta_{17} - 7808170309630 \beta_{16} - 46346257405376 \beta_{15} - 31997668747184 \beta_{14} + 37340874144940 \beta_{13} + \cdots + 60\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 195154928700636 \beta_{17} - 207010529679718 \beta_{16} - 214807477197556 \beta_{15} - 354539077892948 \beta_{14} + 364571231003256 \beta_{13} + \cdots - 14\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 10\!\cdots\!12 \beta_{17} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 11\!\cdots\!06 \beta_{17} + \cdots - 68\!\cdots\!27 \) 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
−20.9501
−20.5204
−17.7235
−16.0158
−13.7979
−10.5486
−6.66616
−6.28523
2.80127
2.93322
6.43791
8.74396
9.20800
13.6971
15.9283
17.9457
19.9328
20.8794
−19.9501 27.0000 270.005 505.865 −538.652 222.970 −2833.01 729.000 −10092.0
1.2 −19.5204 27.0000 253.046 −49.6011 −527.051 −119.919 −2440.94 729.000 968.233
1.3 −16.7235 27.0000 151.676 −298.819 −451.535 200.095 −395.941 729.000 4997.30
1.4 −15.0158 27.0000 97.4757 6.55078 −405.428 1466.16 458.348 729.000 −98.3656
1.5 −12.7979 27.0000 35.7870 −405.222 −345.544 −375.715 1180.14 729.000 5186.00
1.6 −9.54862 27.0000 −36.8240 248.981 −257.813 −1453.31 1573.84 729.000 −2377.42
1.7 −5.66616 27.0000 −95.8947 −15.0142 −152.986 499.350 1268.62 729.000 85.0729
1.8 −5.28523 27.0000 −100.066 498.961 −142.701 341.176 1205.38 729.000 −2637.13
1.9 3.80127 27.0000 −113.550 −87.8737 102.634 1306.06 −918.197 729.000 −334.031
1.10 3.93322 27.0000 −112.530 −305.648 106.197 −1406.85 −946.057 729.000 −1202.18
1.11 7.43791 27.0000 −72.6775 171.827 200.824 1529.87 −1492.62 729.000 1278.03
1.12 9.74396 27.0000 −33.0553 321.882 263.087 −939.921 −1569.32 729.000 3136.41
1.13 10.2080 27.0000 −23.7967 −397.678 275.616 −505.577 −1549.54 729.000 −4059.50
1.14 14.6971 27.0000 88.0043 457.882 396.821 1576.42 −587.821 729.000 6729.53
1.15 16.9283 27.0000 158.568 −421.077 457.065 −62.1753 517.469 729.000 −7128.13
1.16 18.9457 27.0000 230.941 260.561 511.535 254.948 1950.29 729.000 4936.51
1.17 20.9328 27.0000 310.183 −149.882 565.186 1201.09 3813.60 729.000 −3137.45
1.18 21.8794 27.0000 350.709 336.305 590.745 −653.674 4872.75 729.000 7358.16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.d 18
3.b odd 2 1 531.8.a.e 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.d 18 1.a even 1 1 trivial
531.8.a.e 18 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 24 T_{2}^{17} - 1543 T_{2}^{16} + 38223 T_{2}^{15} + 951840 T_{2}^{14} - 24830263 T_{2}^{13} - 296611506 T_{2}^{12} + 8480054376 T_{2}^{11} + 47652762640 T_{2}^{10} + \cdots + 85\!\cdots\!24 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 24 T^{17} + \cdots + 85\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( (T - 27)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 678 T^{17} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} - 3081 T^{17} + \cdots + 19\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{18} - 15070 T^{17} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{18} - 13662 T^{17} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{18} - 71919 T^{17} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{18} - 56231 T^{17} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} - 150029 T^{17} + \cdots - 34\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{18} - 591285 T^{17} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} - 426733 T^{17} + \cdots - 14\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{18} - 7703 T^{17} + \cdots + 76\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{18} - 770959 T^{17} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{18} - 793050 T^{17} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{18} - 1410373 T^{17} + \cdots - 27\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{18} - 1037934 T^{17} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T - 205379)^{18} \) Copy content Toggle raw display
$61$ \( T^{18} + 1374623 T^{17} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + 2436904 T^{17} + \cdots - 90\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{18} - 14289172 T^{17} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{18} - 5482515 T^{17} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{18} - 19786414 T^{17} + \cdots + 10\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{18} - 30227337 T^{17} + \cdots - 20\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{18} - 31061677 T^{17} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} - 12084118 T^{17} + \cdots - 17\!\cdots\!84 \) Copy content Toggle raw display
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