Properties

Label 177.6.a.d
Level $177$
Weight $6$
Character orbit 177.a
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $13$
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] = [177,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + \cdots - 6400833792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 19) q^{4} + (\beta_{7} - 1) q^{5} - 9 \beta_1 q^{6} + ( - \beta_{5} + \beta_{2} - \beta_1 + 29) q^{7} + (\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 13 \beta_1 + 10) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 19) q^{4} + (\beta_{7} - 1) q^{5} - 9 \beta_1 q^{6} + ( - \beta_{5} + \beta_{2} - \beta_1 + 29) q^{7} + (\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 13 \beta_1 + 10) q^{8} + 81 q^{9} + (2 \beta_{11} - \beta_{10} - 3 \beta_{9} + \beta_{8} + 3 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots + 10) q^{10}+ \cdots + ( - 81 \beta_{11} + 162 \beta_{10} - 162 \beta_{8} + 162 \beta_{7} + \cdots + 1539) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 117 q^{3} + 246 q^{4} - 14 q^{5} + 373 q^{7} + 123 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 117 q^{3} + 246 q^{4} - 14 q^{5} + 373 q^{7} + 123 q^{8} + 1053 q^{9} + 137 q^{10} + 250 q^{11} - 2214 q^{12} + 1054 q^{13} - 575 q^{14} + 126 q^{15} + 922 q^{16} + 271 q^{17} + 671 q^{19} - 5491 q^{20} - 3357 q^{21} + 1094 q^{22} + 3975 q^{23} - 1107 q^{24} + 15569 q^{25} + 4622 q^{26} - 9477 q^{27} + 21214 q^{28} - 10613 q^{29} - 1233 q^{30} + 25597 q^{31} + 15966 q^{32} - 2250 q^{33} + 31796 q^{34} + 6729 q^{35} + 19926 q^{36} + 17585 q^{37} + 34903 q^{38} - 9486 q^{39} + 31382 q^{40} + 12537 q^{41} + 5175 q^{42} + 26644 q^{43} + 6654 q^{44} - 1134 q^{45} + 149005 q^{46} + 52087 q^{47} - 8298 q^{48} + 95384 q^{49} + 121821 q^{50} - 2439 q^{51} + 263630 q^{52} + 20014 q^{53} + 120932 q^{55} + 126688 q^{56} - 6039 q^{57} + 86066 q^{58} + 45253 q^{59} + 49419 q^{60} - 11667 q^{61} + 164794 q^{62} + 30213 q^{63} + 151893 q^{64} - 28674 q^{65} - 9846 q^{66} + 1106 q^{67} - 4043 q^{68} - 35775 q^{69} + 56066 q^{70} + 21230 q^{71} + 9963 q^{72} + 81131 q^{73} + 102042 q^{74} - 140121 q^{75} + 73900 q^{76} - 104655 q^{77} - 41598 q^{78} - 13470 q^{79} - 191969 q^{80} + 85293 q^{81} + 79909 q^{82} - 76149 q^{83} - 190926 q^{84} + 10035 q^{85} - 321496 q^{86} + 95517 q^{87} - 276779 q^{88} - 190205 q^{89} + 11097 q^{90} + 80601 q^{91} + 45672 q^{92} - 230373 q^{93} + 36768 q^{94} + 9875 q^{95} - 143694 q^{96} + 160850 q^{97} - 116644 q^{98} + 20250 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + \cdots - 6400833792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39783793826895 \nu^{12} - 489453600148070 \nu^{11} + \cdots - 11\!\cdots\!32 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 826874692028041 \nu^{12} + \cdots + 13\!\cdots\!08 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 54340442593255 \nu^{12} - 52352870381654 \nu^{11} + \cdots - 18\!\cdots\!52 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{12} + \cdots + 10\!\cdots\!96 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!71 \nu^{12} + \cdots + 17\!\cdots\!20 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!75 \nu^{12} + \cdots - 70\!\cdots\!36 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!33 \nu^{12} + \cdots + 23\!\cdots\!20 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 22\!\cdots\!49 \nu^{12} + \cdots + 29\!\cdots\!28 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!69 \nu^{12} + \cdots - 20\!\cdots\!84 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 33\!\cdots\!37 \nu^{12} + \cdots + 40\!\cdots\!96 ) / 33\!\cdots\!28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 77\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + 5 \beta_{8} + \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 121 \beta_{2} + 24 \beta _1 + 3946 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{12} + 28 \beta_{10} + 146 \beta_{9} - 2 \beta_{8} - 30 \beta_{7} - 124 \beta_{6} + 14 \beta_{5} - 152 \beta_{4} - 140 \beta_{3} + 137 \beta_{2} + 7054 \beta _1 + 2511 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 430 \beta_{12} - 176 \beta_{11} - 384 \beta_{10} + 201 \beta_{9} + 838 \beta_{8} - 70 \beta_{7} + 193 \beta_{6} - 318 \beta_{5} - 647 \beta_{4} + 269 \beta_{3} + 13259 \beta_{2} + 4555 \beta _1 + 362506 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1613 \beta_{12} + 413 \beta_{11} + 5482 \beta_{10} + 17480 \beta_{9} - 295 \beta_{8} - 6826 \beta_{7} - 12847 \beta_{6} + 1744 \beta_{5} - 18150 \beta_{4} - 15200 \beta_{3} + 18219 \beta_{2} + \cdots + 379332 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 44114 \beta_{12} - 26902 \beta_{11} - 49184 \beta_{10} + 49688 \beta_{9} + 111710 \beta_{8} - 21616 \beta_{7} + 27170 \beta_{6} - 36688 \beta_{5} - 81208 \beta_{4} + 26628 \beta_{3} + \cdots + 36059531 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 192940 \beta_{12} + 99996 \beta_{11} + 801488 \beta_{10} + 1993913 \beta_{9} - 23868 \beta_{8} - 1125152 \beta_{7} - 1269021 \beta_{6} + 152808 \beta_{5} - 2033353 \beta_{4} + \cdots + 50738702 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4003155 \beta_{12} - 3748361 \beta_{11} - 5521578 \beta_{10} + 8462900 \beta_{9} + 13829037 \beta_{8} - 4305344 \beta_{7} + 3479885 \beta_{6} - 3826458 \beta_{5} + \cdots + 3721822926 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21290390 \beta_{12} + 16375980 \beta_{11} + 105394716 \beta_{10} + 225028878 \beta_{9} - 387470 \beta_{8} - 162036366 \beta_{7} - 123324220 \beta_{6} + \cdots + 6455842751 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 338534002 \beta_{12} - 488092428 \beta_{11} - 585026064 \beta_{10} + 1236018841 \beta_{9} + 1655369514 \beta_{8} - 703594110 \beta_{7} + 428704365 \beta_{6} + \cdots + 391375714806 \) 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
−10.3652
−9.47242
−6.40804
−6.29094
−6.06926
−3.96558
1.05732
2.69937
5.29201
6.46075
6.57049
9.82137
10.6702
−10.3652 −9.00000 75.4381 −24.0297 93.2871 −107.618 −450.246 81.0000 249.073
1.2 −9.47242 −9.00000 57.7268 −42.5216 85.2518 243.042 −243.695 81.0000 402.782
1.3 −6.40804 −9.00000 9.06300 −77.9071 57.6724 112.353 146.981 81.0000 499.232
1.4 −6.29094 −9.00000 7.57589 82.0909 56.6184 51.5315 153.651 81.0000 −516.429
1.5 −6.06926 −9.00000 4.83596 26.0687 54.6234 −152.673 164.866 81.0000 −158.218
1.6 −3.96558 −9.00000 −16.2741 45.8225 35.6903 214.983 191.435 81.0000 −181.713
1.7 1.05732 −9.00000 −30.8821 33.7931 −9.51585 −85.6226 −66.4863 81.0000 35.7300
1.8 2.69937 −9.00000 −24.7134 −80.3454 −24.2943 −162.093 −153.090 81.0000 −216.882
1.9 5.29201 −9.00000 −3.99466 108.071 −47.6281 212.139 −190.484 81.0000 571.910
1.10 6.46075 −9.00000 9.74133 −95.9365 −58.1468 79.3965 −143.808 81.0000 −619.822
1.11 6.57049 −9.00000 11.1714 −5.02964 −59.1344 −195.140 −136.854 81.0000 −33.0472
1.12 9.82137 −9.00000 64.4594 77.2018 −88.3924 −23.9833 318.795 81.0000 758.228
1.13 10.6702 −9.00000 81.8525 −61.2778 −96.0315 186.685 531.935 81.0000 −653.844
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 177.6.a.d 13
3.b odd 2 1 531.6.a.e 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.d 13 1.a even 1 1 trivial
531.6.a.e 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 331 T_{2}^{11} - 41 T_{2}^{10} + 41990 T_{2}^{9} + 7229 T_{2}^{8} - 2592364 T_{2}^{7} - 312100 T_{2}^{6} + 81977088 T_{2}^{5} - 3773728 T_{2}^{4} - 1245415104 T_{2}^{3} + \cdots - 6400833792 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 331 T^{11} + \cdots - 6400833792 \) Copy content Toggle raw display
$3$ \( (T + 9)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + 14 T^{12} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{13} - 373 T^{12} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{13} - 250 T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{13} - 1054 T^{12} + \cdots - 63\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{13} - 271 T^{12} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{13} - 671 T^{12} + \cdots + 31\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{13} - 3975 T^{12} + \cdots - 43\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{13} + 10613 T^{12} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{13} - 25597 T^{12} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{13} - 17585 T^{12} + \cdots + 50\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{13} - 12537 T^{12} + \cdots - 34\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{13} - 26644 T^{12} + \cdots - 77\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{13} - 52087 T^{12} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{13} - 20014 T^{12} + \cdots + 71\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( (T - 3481)^{13} \) Copy content Toggle raw display
$61$ \( T^{13} + 11667 T^{12} + \cdots + 58\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{13} - 1106 T^{12} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{13} - 21230 T^{12} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{13} - 81131 T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{13} + 13470 T^{12} + \cdots - 85\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{13} + 76149 T^{12} + \cdots - 87\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{13} + 190205 T^{12} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{13} - 160850 T^{12} + \cdots - 79\!\cdots\!20 \) Copy content Toggle raw display
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