Properties

Label 177.6.a.c
Level $177$
Weight $6$
Character orbit 177.a
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + \cdots + 49172480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + 9 q^{3} + (\beta_{2} - 2 \beta_1 + 17) q^{4} + (\beta_{3} - \beta_1 + 13) q^{5} + ( - 9 \beta_1 + 18) q^{6} + ( - \beta_{11} - \beta_1 + 35) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} - 22 \beta_1 + 64) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + 9 q^{3} + (\beta_{2} - 2 \beta_1 + 17) q^{4} + (\beta_{3} - \beta_1 + 13) q^{5} + ( - 9 \beta_1 + 18) q^{6} + ( - \beta_{11} - \beta_1 + 35) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} - 22 \beta_1 + 64) q^{8} + 81 q^{9} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{3} - 2 \beta_{2} - 25 \beta_1 + 54) q^{10} + (\beta_{11} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 10 \beta_1 + 122) q^{11} + (9 \beta_{2} - 18 \beta_1 + 153) q^{12} + (2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \cdots + 40) q^{13}+ \cdots + (81 \beta_{11} + 162 \beta_{7} + 81 \beta_{6} + 81 \beta_{5} - 81 \beta_{4} + \cdots + 9882) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 22 q^{2} + 108 q^{3} + 198 q^{4} + 158 q^{5} + 198 q^{6} + 413 q^{7} + 723 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 22 q^{2} + 108 q^{3} + 198 q^{4} + 158 q^{5} + 198 q^{6} + 413 q^{7} + 723 q^{8} + 972 q^{9} + 601 q^{10} + 1480 q^{11} + 1782 q^{12} + 472 q^{13} + 1065 q^{14} + 1422 q^{15} + 6370 q^{16} + 1565 q^{17} + 1782 q^{18} + 3939 q^{19} + 8033 q^{20} + 3717 q^{21} - 1738 q^{22} + 7245 q^{23} + 6507 q^{24} + 9690 q^{25} + 3764 q^{26} + 8748 q^{27} + 12154 q^{28} + 10003 q^{29} + 5409 q^{30} + 7295 q^{31} + 11628 q^{32} + 13320 q^{33} - 16344 q^{34} + 11015 q^{35} + 16038 q^{36} + 6741 q^{37} + 3035 q^{38} + 4248 q^{39} + 5572 q^{40} + 34025 q^{41} + 9585 q^{42} - 6336 q^{43} + 41168 q^{44} + 12798 q^{45} + 2345 q^{46} + 66167 q^{47} + 57330 q^{48} + 28319 q^{49} + 31173 q^{50} + 14085 q^{51} + 16440 q^{52} + 62290 q^{53} + 16038 q^{54} + 55764 q^{55} + 107306 q^{56} + 35451 q^{57} + 37952 q^{58} - 41772 q^{59} + 72297 q^{60} + 68469 q^{61} + 99190 q^{62} + 33453 q^{63} + 68525 q^{64} + 80156 q^{65} - 15642 q^{66} + 113310 q^{67} + 33887 q^{68} + 65205 q^{69} + 32034 q^{70} + 84520 q^{71} + 58563 q^{72} + 135895 q^{73} - 31962 q^{74} + 87210 q^{75} - 61848 q^{76} - 3799 q^{77} + 33876 q^{78} + 14122 q^{79} + 77609 q^{80} + 78732 q^{81} - 1501 q^{82} + 114463 q^{83} + 109386 q^{84} - 101097 q^{85} - 203536 q^{86} + 90027 q^{87} - 244967 q^{88} + 189109 q^{89} + 48681 q^{90} - 168249 q^{91} - 71946 q^{92} + 65655 q^{93} - 472284 q^{94} + 21923 q^{95} + 104652 q^{96} - 76192 q^{97} - 17544 q^{98} + 119880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + \cdots + 49172480 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 350941 \nu^{11} - 1428132 \nu^{10} - 70855647 \nu^{9} + 91864075 \nu^{8} + 4536395590 \nu^{7} + 5505304837 \nu^{6} + \cdots - 17699688792576 ) / 277202304512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8192145 \nu^{11} - 67427634 \nu^{10} + 2640430613 \nu^{9} + 18376727045 \nu^{8} - 268943328324 \nu^{7} - 1674621922747 \nu^{6} + \cdots + 10\!\cdots\!40 ) / 3049225349632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 299377 \nu^{11} - 524416 \nu^{10} + 88399319 \nu^{9} + 187648889 \nu^{8} - 8690291434 \nu^{7} - 19612654933 \nu^{6} + \cdots + 16317679113344 ) / 108900905344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5842853 \nu^{11} + 35029081 \nu^{10} - 1903226618 \nu^{9} - 8347251655 \nu^{8} + 191946287467 \nu^{7} + 677570013916 \nu^{6} + \cdots - 324693088418048 ) / 1524612674816 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1506791 \nu^{11} + 7464662 \nu^{10} - 465055595 \nu^{9} - 2148263267 \nu^{8} + 46570135316 \nu^{7} + 202161478261 \nu^{6} + \cdots - 94789227549184 ) / 277202304512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 100767 \nu^{11} + 53452 \nu^{10} - 28964453 \nu^{9} - 48656271 \nu^{8} + 2821805338 \nu^{7} + 6891575767 \nu^{6} - 108469681492 \nu^{5} + \cdots - 4701282618368 ) / 11773070848 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14438755 \nu^{11} + 3300860 \nu^{10} - 4146949925 \nu^{9} - 4582591287 \nu^{8} + 404519679498 \nu^{7} + 624811840007 \nu^{6} + \cdots - 156500452428544 ) / 1524612674816 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24199755 \nu^{11} + 1545329 \nu^{10} - 6527695920 \nu^{9} - 8306848553 \nu^{8} + 593347027999 \nu^{7} + 1132584010846 \nu^{6} + \cdots - 172136514045440 ) / 1524612674816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25955403 \nu^{11} + 47327234 \nu^{10} + 6681309643 \nu^{9} + 346810155 \nu^{8} - 598161796452 \nu^{7} - 781324808245 \nu^{6} + \cdots + 849314392654336 ) / 1524612674816 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{4} + 5\beta_{2} + 86\beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} + 6 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + 132 \beta_{2} + 437 \beta _1 + 3807 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{11} + 18 \beta_{10} + 4 \beta_{9} + 20 \beta_{8} + 162 \beta_{7} - 42 \beta_{6} + 114 \beta_{5} + 166 \beta_{4} + 44 \beta_{3} + 1063 \beta_{2} + 9386 \beta _1 + 18767 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 606 \beta_{11} + 550 \beta_{10} - 40 \beta_{9} + 420 \beta_{8} + 1413 \beta_{7} - 994 \beta_{6} + 673 \beta_{5} + 1453 \beta_{4} + 200 \beta_{3} + 18249 \beta_{2} + 74944 \beta _1 + 411640 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6783 \beta_{11} + 4559 \beta_{10} + 584 \beta_{9} + 4402 \beta_{8} + 24374 \beta_{7} - 10209 \beta_{6} + 13228 \beta_{5} + 25022 \beta_{4} + 8000 \beta_{3} + 183734 \beta_{2} + 1198009 \beta _1 + 3226133 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 100910 \beta_{11} + 85242 \beta_{10} - 6060 \beta_{9} + 69276 \beta_{8} + 253564 \beta_{7} - 162306 \beta_{6} + 106068 \beta_{5} + 263088 \beta_{4} + 56452 \beta_{3} + 2644521 \beta_{2} + \cdots + 52195939 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1187260 \beta_{11} + 850124 \beta_{10} + 55096 \beta_{9} + 768448 \beta_{8} + 3661489 \beta_{7} - 1839132 \beta_{6} + 1696449 \beta_{5} + 3763833 \beta_{4} + 1216848 \beta_{3} + \cdots + 518319894 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 15917723 \beta_{11} + 12861355 \beta_{10} - 684976 \beta_{9} + 10742610 \beta_{8} + 41567962 \beta_{7} - 25302541 \beta_{6} + 16617000 \beta_{5} + 43157930 \beta_{4} + \cdots + 7274855475 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 191789770 \beta_{11} + 142131150 \beta_{10} + 3961508 \beta_{9} + 124512460 \beta_{8} + 553464350 \beta_{7} - 300500302 \beta_{6} + 235486966 \beta_{5} + \cdots + 81136464075 \) 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
12.3715
8.10567
7.96013
4.10256
1.70991
1.38446
−0.689340
−3.21799
−5.43261
−7.23155
−8.49197
−8.57072
−10.3715 9.00000 75.5670 33.2592 −93.3431 −56.9512 −451.853 81.0000 −344.946
1.2 −6.10567 9.00000 5.27923 −41.8018 −54.9510 208.248 163.148 81.0000 255.228
1.3 −5.96013 9.00000 3.52316 65.6425 −53.6412 119.026 169.726 81.0000 −391.238
1.4 −2.10256 9.00000 −27.5792 −81.1594 −18.9230 −97.6213 125.269 81.0000 170.642
1.5 0.290087 9.00000 −31.9158 87.1048 2.61079 167.610 −18.5412 81.0000 25.2680
1.6 0.615542 9.00000 −31.6211 61.9372 5.53988 −209.534 −39.1615 81.0000 38.1250
1.7 2.68934 9.00000 −24.7675 −83.5049 24.2041 −48.3401 −152.667 81.0000 −224.573
1.8 5.21799 9.00000 −4.77260 −21.6600 46.9619 150.168 −191.879 81.0000 −113.022
1.9 7.43261 9.00000 23.2437 48.3022 66.8935 120.542 −65.0821 81.0000 359.012
1.10 9.23155 9.00000 53.2215 89.5822 83.0840 −121.386 195.908 81.0000 826.983
1.11 10.4920 9.00000 78.0815 46.0665 94.4278 183.145 483.486 81.0000 483.328
1.12 10.5707 9.00000 79.7401 −45.7686 95.1365 −1.90644 504.647 81.0000 −483.807
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.c 12
3.b odd 2 1 531.6.a.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.c 12 1.a even 1 1 trivial
531.6.a.c 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 22 T_{2}^{11} - 49 T_{2}^{10} + 3917 T_{2}^{9} - 15182 T_{2}^{8} - 191819 T_{2}^{7} + 1174642 T_{2}^{6} + 2586728 T_{2}^{5} - 24399488 T_{2}^{4} + 12144672 T_{2}^{3} + 88035904 T_{2}^{2} + \cdots + 15131776 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(177))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 22 T^{11} - 49 T^{10} + \cdots + 15131776 \) Copy content Toggle raw display
$3$ \( (T - 9)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 158 T^{11} + \cdots - 65\!\cdots\!40 \) Copy content Toggle raw display
$7$ \( T^{12} - 413 T^{11} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{12} - 1480 T^{11} + \cdots - 33\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{12} - 472 T^{11} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{12} - 1565 T^{11} + \cdots + 68\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{12} - 3939 T^{11} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} - 7245 T^{11} + \cdots - 83\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{12} - 10003 T^{11} + \cdots - 34\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} - 7295 T^{11} + \cdots - 26\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{12} - 6741 T^{11} + \cdots + 35\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{12} - 34025 T^{11} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + 6336 T^{11} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{12} - 66167 T^{11} + \cdots - 14\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{12} - 62290 T^{11} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( (T + 3481)^{12} \) Copy content Toggle raw display
$61$ \( T^{12} - 68469 T^{11} + \cdots - 74\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{12} - 113310 T^{11} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{12} - 84520 T^{11} + \cdots - 44\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{12} - 135895 T^{11} + \cdots + 51\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{12} - 14122 T^{11} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{12} - 114463 T^{11} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{12} - 189109 T^{11} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{12} + 76192 T^{11} + \cdots - 10\!\cdots\!92 \) Copy content Toggle raw display
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