LMFDB - Newform orbit 162.9.d.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,9,Mod(53,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	162.d (of order $6$, degree $2$, not 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:	$65.9953348299$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.206763233181696.28
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441 $ x^8 - 80x^6 + 4879x^4 - 121680*x^2 + 2313441
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{3} q^{2} - 128 \beta_1 q^{4} + ( - \beta_{4} + 21 \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - 41 \beta_1 - 41) q^{7} + (128 \beta_{3} - 128 \beta_{2}) q^{8} + ( - 2 \beta_{5} + 2688) q^{10}+ \cdots + ( - 5248 \beta_{7} + \cdots + 8980176 \beta_{2}) q^{98}+O(q^{100}) $ q + b3 * q^2 - 128b1 q^4 + (-b4 + 21b2) q^5 + (b6 - b5 - 41b1 - 41) q^7 + (128b3 - 128b2) * q^8 + (-2b5 + 2688) q^10 + (-29b7 + 21b3) * q^11 + (b6 + 24911b1) q^13 + (-64b4 - 41b2) * q^14 + (-16384b1 - 16384) q^16 + (-53b7 + 53b4 - 8379b3 + 8379b2) * q^17 + (-22b5 + 128009) q^19 + (-128b7 + 2688b3) * q^20 + (-58b6 - 2688b1) * q^22 + (259b4 - 17583b2) * q^23 + (84b6 - 84b5 + 126551b1 + 126551) q^25 + (64b7 - 64b4 - 24911b3 + 24911b2) * q^26 + (-128b5 - 5248) q^28 + (1306b7 + 38838b3) * q^29 + (204b6 + 148886b1) * q^31 - 16384b2 q^32 + (-106b6 + 106b5 + 1072512b1 + 1072512) q^34 + (1303b7 - 1303b4 - 229503b3 + 229503b2) * q^35 + (505b5 - 252745) q^37 + (-1408b7 + 128009b3) * q^38 + (-256b6 - 344064b1) * q^40 + (338b4 - 204162b2) * q^41 + (-588b6 + 588b5 + 2023546b1 + 2023546) q^43 + (-3712b7 + 3712b4 + 2688b3 - 2688b2) * q^44 + (518b5 - 2250624) q^46 + (-2253b7 + 296697b3) * q^47 + (-82b6 + 8980176b1) * q^49 + (-5376b4 + 126551b2) * q^50 + (128b6 - 128b5 + 3188608b1 + 3188608) q^52 + (-1990b7 + 1990b4 + 229086b3 - 229086b2) * q^53 + (-1260b5 + 13417560) q^55 + (-8192b7 - 5248b3) * q^56 + (2612b6 - 4971264b1) * q^58 + (5527b4 + 1402905b2) * q^59 + (1761b6 - 1761b5 - 1196159b1 - 1196159) q^61 + (13056b7 - 13056b4 - 148886b3 + 148886b2) * q^62 - 2097152 * q^64 + (26255b7 - 753495b3) * q^65 + (-5748b6 + 14289425b1) * q^67 + (6784b4 + 1072512b2) * q^68 + (2606b6 - 2606b5 + 29376384b1 + 29376384) q^70 + (1300b7 - 1300b4 - 387324b3 + 387324b2) * q^71 + (-6066b5 + 3309023) q^73 + (32320b7 - 252745b3) * q^74 + (-2816b6 - 16385152b1) * q^76 + (-155b4 + 6679695b2) * q^77 + (-4537b6 + 4537b5 - 12395921b1 - 12395921) q^79 + (-16384b7 + 16384b4 + 344064b3 - 344064b2) * q^80 + (676b5 - 26132736) q^82 + (-101538b7 - 7206b3) * q^83 + (14532b6 - 1895832b1) * q^85 + (37632b4 + 2023546b2) * q^86 + (-7424b6 + 7424b5 - 344064b1 - 344064) q^88 + (-14585b7 + 14585b4 - 6311223b3 + 6311223b2) * q^89 + (24870b5 - 13721945) q^91 + (33152b7 - 2250624b3) * q^92 + (-4506b6 - 37977216b1) * q^94 + (-157577b4 + 7756197b2) * q^95 + (14114b6 - 14114b5 - 102086231b1 - 102086231) q^97 + (-5248b7 + 5248b4 - 8980176b3 + 8980176b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 512 q^{4} - 164 q^{7} + 21504 q^{10} - 99644 q^{13} - 65536 q^{16} + 1024072 q^{19} + 10752 q^{22} + 506204 q^{25} - 41984 q^{28} - 595544 q^{31} + 4290048 q^{34} - 2021960 q^{37} + 1376256 q^{40}+ \cdots - 408344924 q^{97}+O(q^{100}) $ 8 * q + 512 * q^4 - 164 * q^7 + 21504 * q^10 - 99644 * q^13 - 65536 * q^16 + 1024072 * q^19 + 10752 * q^22 + 506204 * q^25 - 41984 * q^28 - 595544 * q^31 + 4290048 * q^34 - 2021960 * q^37 + 1376256 * q^40 + 8094184 * q^43 - 18004992 * q^46 - 35920704 * q^49 + 12754432 * q^52 + 107340480 * q^55 + 19885056 * q^58 - 4784636 * q^61 - 16777216 * q^64 - 57157700 * q^67 + 117505536 * q^70 + 26472184 * q^73 + 65540608 * q^76 - 49583684 * q^79 - 209061888 * q^82 + 7583328 * q^85 - 1376256 * q^88 - 109775560 * q^91 + 151908864 * q^94 - 408344924 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441 $ x^8 - 80*x^6 + 4879*x^4 - 121680*x^2 + 2313441 :

$\beta_{1}$	$=$	$ ( 80\nu^{6} - 4879\nu^{4} + 390320\nu^{2} - 9734400 ) / 7420959 $ (80v^6 - 4879v^4 + 390320*v^2 - 9734400) / 7420959
$\beta_{2}$	$=$	$ ( -8\nu^{7} - 626872\nu ) / 190281 $ (-8v^7 - 626872v) / 190281
$\beta_{3}$	$=$	$ ( 14072\nu^{7} - 1600312\nu^{5} + 68657288\nu^{3} - 1712280960\nu ) / 289417401 $ (14072v^7 - 1600312v^5 + 68657288v^3 - 1712280960v) / 289417401
$\beta_{4}$	$=$	$ ( -18\nu^{7} - 8260578\nu ) / 63427 $ (-18v^7 - 8260578v) / 63427
$\beta_{5}$	$=$	$ ( 432\nu^{6} + 31743360 ) / 4879 $ (432*v^6 + 31743360) / 4879
$\beta_{6}$	$=$	$ ( -80592\nu^{6} + 9367680\nu^{4} - 393208368\nu^{2} + 9806434560 ) / 824551 $ (-80592v^6 + 9367680v^4 - 393208368*v^2 + 9806434560) / 824551
$\beta_{7}$	$=$	$ ( 15998\nu^{7} - 1161202\nu^{5} + 78054242\nu^{3} - 1946636640\nu ) / 10719163 $ (15998v^7 - 1161202v^5 + 78054242v^3 - 1946636640v) / 10719163

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -4\beta_{4} + 27\beta_{2} ) / 432 $ (-4b4 + 27b2) / 432
$\nu^{2}$	$=$	$ ( \beta_{6} - \beta_{5} + 17280\beta _1 + 17280 ) / 432 $ (b6 - b5 + 17280*b1 + 17280) / 432
$\nu^{3}$	$=$	$ ( 164\beta_{7} - 164\beta_{4} - 3213\beta_{3} + 3213\beta_{2} ) / 432 $ (164b7 - 164b4 - 3213b3 + 3213b2) / 432
$\nu^{4}$	$=$	$ ( 5\beta_{6} + 45333\beta_1 ) / 27 $ (5b6 + 45333b1) / 27
$\nu^{5}$	$=$	$ ( 7036\beta_{7} - 215973\beta_{3} ) / 432 $ (7036b7 - 215973b3) / 432
$\nu^{6}$	$=$	$ ( 4879\beta_{5} - 31743360 ) / 432 $ (4879*b5 - 31743360) / 432
$\nu^{7}$	$=$	$ ( 313436\beta_{4} - 12390867\beta_{2} ) / 432 $ (313436b4 - 12390867b2) / 432

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-\beta_{1}$

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} $

53.1

−6.05526	−	3.49600i
4.83051	+	2.78890i
−4.83051	−	2.78890i
6.05526	+	3.49600i
−6.05526	+	3.49600i
4.83051	−	2.78890i
−4.83051	+	2.78890i
6.05526	−	3.49600i

−9.79796

5.65685i

64.0000

−

110.851i

−793.589

−

458.179i

1899.35

3289.77i

1448.15i

10367.4

53.2

−9.79796

5.65685i

64.0000

−

110.851i

382.074

220.591i

−1940.35

−

3360.78i

1448.15i

−4991.40

53.3

9.79796

−

5.65685i

64.0000

−

110.851i

−382.074

−

220.591i

−1940.35

−

3360.78i

−

1448.15i

−4991.40

53.4

9.79796

−

5.65685i

64.0000

−

110.851i

793.589

458.179i

1899.35

3289.77i

−

1448.15i

10367.4

107.1

−9.79796

−

5.65685i

64.0000

110.851i

−793.589

458.179i

1899.35

−

3289.77i

−

1448.15i

10367.4

107.2

−9.79796

−

5.65685i

64.0000

110.851i

382.074

−

220.591i

−1940.35

3360.78i

−

1448.15i

−4991.40

107.3

9.79796

5.65685i

64.0000

110.851i

−382.074

220.591i

−1940.35

3360.78i

1448.15i

−4991.40

107.4

9.79796

5.65685i

64.0000

110.851i

793.589

−

458.179i

1899.35

−

3289.77i

1448.15i

10367.4

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.9.d.f		8
3.b	odd	2	1	inner	162.9.d.f		8
9.c	even	3	1		54.9.b.b	✓	4
9.c	even	3	1	inner	162.9.d.f		8
9.d	odd	6	1		54.9.b.b	✓	4
9.d	odd	6	1	inner	162.9.d.f		8
36.f	odd	6	1		432.9.e.i		4
36.h	even	6	1		432.9.e.i		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.9.b.b	✓	4	9.c	even	3	1
54.9.b.b	✓	4	9.d	odd	6	1
162.9.d.f		8	1.a	even	1	1	trivial
162.9.d.f		8	3.b	odd	2	1	inner
162.9.d.f		8	9.c	even	3	1	inner
162.9.d.f		8	9.d	odd	6	1	inner
432.9.e.i		4	36.f	odd	6	1
432.9.e.i		4	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 1034352T_{5}^{6} + 906441741504T_{5}^{4} - 169056888921676800T_{5}^{2} + 26713391443966978560000 $ T5^8 - 1034352*T5^6 + 906441741504*T5^4 - 169056888921676800*T5^2 + 26713391443966978560000 acting on $S_{9}^{\mathrm{new}}(162, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - 128 T^{2} + 16384)^{2} $ (T^4 - 128*T^2 + 16384)^2
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + \cdots + 26\!\cdots\!00 $ T^8 - 1034352T^6 + 906441741504T^4 - 169056888921676800*T^2 + 26713391443966978560000
$7$	$ (T^{4} + \cdots + 217315212808225)^{2} $ (T^4 + 82T^3 + 14748339T^2 - 1208812430*T + 217315212808225)^2
$11$	$ T^{8} + \cdots + 22\!\cdots\!00 $ T^8 - 775057392T^6 + 450622958804201664T^4 - 116329140642262928618880000*T^2 + 22527308908272318846595329600000000
$13$	$ (T^{4} + \cdots + 36\!\cdots\!25)^{2} $ (T^4 + 49822T^3 + 1876417059T^2 + 30182896246750*T + 367011359863890625)^2
$17$	$ (T^{4} + \cdots + 59\!\cdots\!16)^{2} $ (T^4 + 20561526000*T^2 + 59172911543388465216)^2
$19$	$ (T^{2} - 256018 T + 9250548817)^{4} $ (T^2 - 256018*T + 9250548817)^4
$23$	$ T^{8} + \cdots + 56\!\cdots\!76 $ T^8 - 140957633520T^6 + 19793944027052151263424T^4 - 10587387127284982884438697835520*T^2 + 5641575268659699534967649627100065304576
$29$	$ T^{8} + \cdots + 12\!\cdots\!76 $ T^8 - 1957816428480T^6 + 3481680380409402230836224T^4 - 687907752802392797121774896614932480*T^2 + 123457213695893595067597042097860319642020478976
$31$	$ (T^{4} + \cdots + 34\!\cdots\!00)^{2} $ (T^4 + 297772T^3 + 680058129324T^2 - 176099372759222480*T + 349742091104846401315600)^2
$37$	$ (T^{2} + \cdots - 3696029027375)^{4} $ (T^2 + 505490*T - 3696029027375)^4
$41$	$ T^{8} + \cdots + 77\!\cdots\!00 $ T^8 - 10775894113728T^6 + 88213225718974066944297984T^4 - 300719301927465627679308705838817280000*T^2 + 778782131772054713813100316017683483722137600000000
$43$	$ (T^{4} + \cdots + 10\!\cdots\!64)^{2} $ (T^4 - 4047092T^3 + 17381621374572T^2 + 4057888500613141936*T + 1005342552935903751099664)^2
$47$	$ T^{8} + \cdots + 63\!\cdots\!00 $ T^8 - 27212771060208T^6 + 660806678767363006800003264T^4 - 2169626070241432563555234958808688000000*T^2 + 6356590660197553898015315289897455852721000000000000
$53$	$ (T^{4} + \cdots + 23\!\cdots\!44)^{2} $ (T^4 + 17084039126976*T^2 + 23941073785562085552620544)^2
$59$	$ T^{8} + \cdots + 32\!\cdots\!36 $ T^8 - 531992852563824T^6 + 226444705946937819985135856832T^4 - 30095734328916041637239638422749820167494656*T^2 + 3200356022568406757220892511401581691858334377260318068736
$61$	$ (T^{4} + \cdots + 19\!\cdots\!25)^{2} $ (T^4 + 2392318T^3 + 50013133894659T^2 - 105955640971448848130*T + 1961599536497024452235956225)^2
$67$	$ (T^{4} + \cdots + 80\!\cdots\!81)^{2} $ (T^4 + 28578850T^3 + 1099674187657059T^2 - 8085628849113311477150*T + 80045718358499619967289724481)^2
$71$	$ (T^{4} + \cdots + 33\!\cdots\!84)^{2} $ (T^4 + 39962350169856*T^2 + 339440623474026614344925184)^2
$73$	$ (T^{2} + \cdots - 531549935014847)^{4} $ (T^2 - 6618046*T - 531549935014847)^4
$79$	$ (T^{4} + \cdots + 22\!\cdots\!89)^{2} $ (T^4 + 24791842T^3 + 764458017454947T^2 - 3714377922338705622686*T + 22446807785718387672822132289)^2
$83$	$ T^{8} + \cdots + 50\!\cdots\!76 $ T^8 - 9500192811334080T^6 + 67690373876871514989204969114624T^4 - 214355601426660362685560131740257680125175726080*T^2 + 509102036474735937739388820632539993160361358963815738875314176
$89$	$ (T^{4} + \cdots + 25\!\cdots\!44)^{2} $ (T^4 + 10392887324026224*T^2 + 25004295098998028639850457134144)^2
$97$	$ (T^{4} + \cdots + 56\!\cdots\!25)^{2} $ (T^4 + 204172462T^3 + 34201733900062899T^2 + 1528161528663421344027790*T + 56020140391375360213659679317025)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{2} - 128 \beta_1 q^{4} + ( - \beta_{4} + 21 \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - 41 \beta_1 - 41) q^{7} + (128 \beta_{3} - 128 \beta_{2}) q^{8} + ( - 2 \beta_{5} + 2688) q^{10}+ \cdots + ( - 5248 \beta_{7} + \cdots + 8980176 \beta_{2}) q^{98}+O(q^{100}) \) q + b3 * q^2 - 128b1 q^4 + (-b4 + 21b2) q^5 + (b6 - b5 - 41b1 - 41) q^7 + (128b3 - 128b2) * q^8 + (-2b5 + 2688) q^10 + (-29b7 + 21b3) * q^11 + (b6 + 24911b1) q^13 + (-64b4 - 41b2) * q^14 + (-16384b1 - 16384) q^16 + (-53b7 + 53b4 - 8379b3 + 8379b2) * q^17 + (-22b5 + 128009) q^19 + (-128b7 + 2688b3) * q^20 + (-58b6 - 2688b1) * q^22 + (259b4 - 17583b2) * q^23 + (84b6 - 84b5 + 126551b1 + 126551) q^25 + (64b7 - 64b4 - 24911b3 + 24911b2) * q^26 + (-128b5 - 5248) q^28 + (1306b7 + 38838b3) * q^29 + (204b6 + 148886b1) * q^31 - 16384b2 q^32 + (-106b6 + 106b5 + 1072512b1 + 1072512) q^34 + (1303b7 - 1303b4 - 229503b3 + 229503b2) * q^35 + (505b5 - 252745) q^37 + (-1408b7 + 128009b3) * q^38 + (-256b6 - 344064b1) * q^40 + (338b4 - 204162b2) * q^41 + (-588b6 + 588b5 + 2023546b1 + 2023546) q^43 + (-3712b7 + 3712b4 + 2688b3 - 2688b2) * q^44 + (518b5 - 2250624) q^46 + (-2253b7 + 296697b3) * q^47 + (-82b6 + 8980176b1) * q^49 + (-5376b4 + 126551b2) * q^50 + (128b6 - 128b5 + 3188608b1 + 3188608) q^52 + (-1990b7 + 1990b4 + 229086b3 - 229086b2) * q^53 + (-1260b5 + 13417560) q^55 + (-8192b7 - 5248b3) * q^56 + (2612b6 - 4971264b1) * q^58 + (5527b4 + 1402905b2) * q^59 + (1761b6 - 1761b5 - 1196159b1 - 1196159) q^61 + (13056b7 - 13056b4 - 148886b3 + 148886b2) * q^62 - 2097152 * q^64 + (26255b7 - 753495b3) * q^65 + (-5748b6 + 14289425b1) * q^67 + (6784b4 + 1072512b2) * q^68 + (2606b6 - 2606b5 + 29376384b1 + 29376384) q^70 + (1300b7 - 1300b4 - 387324b3 + 387324b2) * q^71 + (-6066b5 + 3309023) q^73 + (32320b7 - 252745b3) * q^74 + (-2816b6 - 16385152b1) * q^76 + (-155b4 + 6679695b2) * q^77 + (-4537b6 + 4537b5 - 12395921b1 - 12395921) q^79 + (-16384b7 + 16384b4 + 344064b3 - 344064b2) * q^80 + (676b5 - 26132736) q^82 + (-101538b7 - 7206b3) * q^83 + (14532b6 - 1895832b1) * q^85 + (37632b4 + 2023546b2) * q^86 + (-7424b6 + 7424b5 - 344064b1 - 344064) q^88 + (-14585b7 + 14585b4 - 6311223b3 + 6311223b2) * q^89 + (24870b5 - 13721945) q^91 + (33152b7 - 2250624b3) * q^92 + (-4506b6 - 37977216b1) * q^94 + (-157577b4 + 7756197b2) * q^95 + (14114b6 - 14114b5 - 102086231b1 - 102086231) q^97 + (-5248b7 + 5248b4 - 8980176b3 + 8980176b2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 512 q^{4} - 164 q^{7} + 21504 q^{10} - 99644 q^{13} - 65536 q^{16} + 1024072 q^{19} + 10752 q^{22} + 506204 q^{25} - 41984 q^{28} - 595544 q^{31} + 4290048 q^{34} - 2021960 q^{37} + 1376256 q^{40}+ \cdots - 408344924 q^{97}+O(q^{100}) \) 8 * q + 512 * q^4 - 164 * q^7 + 21504 * q^10 - 99644 * q^13 - 65536 * q^16 + 1024072 * q^19 + 10752 * q^22 + 506204 * q^25 - 41984 * q^28 - 595544 * q^31 + 4290048 * q^34 - 2021960 * q^37 + 1376256 * q^40 + 8094184 * q^43 - 18004992 * q^46 - 35920704 * q^49 + 12754432 * q^52 + 107340480 * q^55 + 19885056 * q^58 - 4784636 * q^61 - 16777216 * q^64 - 57157700 * q^67 + 117505536 * q^70 + 26472184 * q^73 + 65540608 * q^76 - 49583684 * q^79 - 209061888 * q^82 + 7583328 * q^85 - 1376256 * q^88 - 109775560 * q^91 + 151908864 * q^94 - 408344924 * q^97

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	162.9.d.f

Level	$162$
Weight	$9$
Character orbit	162.d
Analytic conductor	$65.995$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(65.9953348299\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.206763233181696.28
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441 \) x^8 - 80x^6 + 4879x^4 - 121680*x^2 + 2313441
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{12} \)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 80\nu^{6} - 4879\nu^{4} + 390320\nu^{2} - 9734400 ) / 7420959 \) (80v^6 - 4879v^4 + 390320*v^2 - 9734400) / 7420959
\(\beta_{2}\)	\(=\)	\( ( -8\nu^{7} - 626872\nu ) / 190281 \) (-8v^7 - 626872v) / 190281
\(\beta_{3}\)	\(=\)	\( ( 14072\nu^{7} - 1600312\nu^{5} + 68657288\nu^{3} - 1712280960\nu ) / 289417401 \) (14072v^7 - 1600312v^5 + 68657288v^3 - 1712280960v) / 289417401
\(\beta_{4}\)	\(=\)	\( ( -18\nu^{7} - 8260578\nu ) / 63427 \) (-18v^7 - 8260578v) / 63427
\(\beta_{5}\)	\(=\)	\( ( 432\nu^{6} + 31743360 ) / 4879 \) (432*v^6 + 31743360) / 4879
\(\beta_{6}\)	\(=\)	\( ( -80592\nu^{6} + 9367680\nu^{4} - 393208368\nu^{2} + 9806434560 ) / 824551 \) (-80592v^6 + 9367680v^4 - 393208368*v^2 + 9806434560) / 824551
\(\beta_{7}\)	\(=\)	\( ( 15998\nu^{7} - 1161202\nu^{5} + 78054242\nu^{3} - 1946636640\nu ) / 10719163 \) (15998v^7 - 1161202v^5 + 78054242v^3 - 1946636640v) / 10719163

\(\nu\)	\(=\)	\( ( -4\beta_{4} + 27\beta_{2} ) / 432 \) (-4b4 + 27b2) / 432
\(\nu^{2}\)	\(=\)	\( ( \beta_{6} - \beta_{5} + 17280\beta _1 + 17280 ) / 432 \) (b6 - b5 + 17280*b1 + 17280) / 432
\(\nu^{3}\)	\(=\)	\( ( 164\beta_{7} - 164\beta_{4} - 3213\beta_{3} + 3213\beta_{2} ) / 432 \) (164b7 - 164b4 - 3213b3 + 3213b2) / 432
\(\nu^{4}\)	\(=\)	\( ( 5\beta_{6} + 45333\beta_1 ) / 27 \) (5b6 + 45333b1) / 27
\(\nu^{5}\)	\(=\)	\( ( 7036\beta_{7} - 215973\beta_{3} ) / 432 \) (7036b7 - 215973b3) / 432
\(\nu^{6}\)	\(=\)	\( ( 4879\beta_{5} - 31743360 ) / 432 \) (4879*b5 - 31743360) / 432
\(\nu^{7}\)	\(=\)	\( ( 313436\beta_{4} - 12390867\beta_{2} ) / 432 \) (313436b4 - 12390867b2) / 432

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 53.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - 128 T^{2} + 16384)^{2} \) (T^4 - 128*T^2 + 16384)^2
$3$	\( T^{8} \) T^8
$5$	\( T^{8} + \cdots + 26\!\cdots\!00 \) T^8 - 1034352T^6 + 906441741504T^4 - 169056888921676800*T^2 + 26713391443966978560000
$7$	\( (T^{4} + \cdots + 217315212808225)^{2} \) (T^4 + 82T^3 + 14748339T^2 - 1208812430*T + 217315212808225)^2
$11$	\( T^{8} + \cdots + 22\!\cdots\!00 \) T^8 - 775057392T^6 + 450622958804201664T^4 - 116329140642262928618880000*T^2 + 22527308908272318846595329600000000
$13$	\( (T^{4} + \cdots + 36\!\cdots\!25)^{2} \) (T^4 + 49822T^3 + 1876417059T^2 + 30182896246750*T + 367011359863890625)^2
$17$	\( (T^{4} + \cdots + 59\!\cdots\!16)^{2} \) (T^4 + 20561526000*T^2 + 59172911543388465216)^2
$19$	\( (T^{2} - 256018 T + 9250548817)^{4} \) (T^2 - 256018*T + 9250548817)^4
$23$	\( T^{8} + \cdots + 56\!\cdots\!76 \) T^8 - 140957633520T^6 + 19793944027052151263424T^4 - 10587387127284982884438697835520*T^2 + 5641575268659699534967649627100065304576
$29$	\( T^{8} + \cdots + 12\!\cdots\!76 \) T^8 - 1957816428480T^6 + 3481680380409402230836224T^4 - 687907752802392797121774896614932480*T^2 + 123457213695893595067597042097860319642020478976
$31$	\( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \) (T^4 + 297772T^3 + 680058129324T^2 - 176099372759222480*T + 349742091104846401315600)^2
$37$	\( (T^{2} + \cdots - 3696029027375)^{4} \) (T^2 + 505490*T - 3696029027375)^4
$41$	\( T^{8} + \cdots + 77\!\cdots\!00 \) T^8 - 10775894113728T^6 + 88213225718974066944297984T^4 - 300719301927465627679308705838817280000*T^2 + 778782131772054713813100316017683483722137600000000
$43$	\( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \) (T^4 - 4047092T^3 + 17381621374572T^2 + 4057888500613141936*T + 1005342552935903751099664)^2
$47$	\( T^{8} + \cdots + 63\!\cdots\!00 \) T^8 - 27212771060208T^6 + 660806678767363006800003264T^4 - 2169626070241432563555234958808688000000*T^2 + 6356590660197553898015315289897455852721000000000000
$53$	\( (T^{4} + \cdots + 23\!\cdots\!44)^{2} \) (T^4 + 17084039126976*T^2 + 23941073785562085552620544)^2
$59$	\( T^{8} + \cdots + 32\!\cdots\!36 \) T^8 - 531992852563824T^6 + 226444705946937819985135856832T^4 - 30095734328916041637239638422749820167494656*T^2 + 3200356022568406757220892511401581691858334377260318068736
$61$	\( (T^{4} + \cdots + 19\!\cdots\!25)^{2} \) (T^4 + 2392318T^3 + 50013133894659T^2 - 105955640971448848130*T + 1961599536497024452235956225)^2
$67$	\( (T^{4} + \cdots + 80\!\cdots\!81)^{2} \) (T^4 + 28578850T^3 + 1099674187657059T^2 - 8085628849113311477150*T + 80045718358499619967289724481)^2
$71$	\( (T^{4} + \cdots + 33\!\cdots\!84)^{2} \) (T^4 + 39962350169856*T^2 + 339440623474026614344925184)^2
$73$	\( (T^{2} + \cdots - 531549935014847)^{4} \) (T^2 - 6618046*T - 531549935014847)^4
$79$	\( (T^{4} + \cdots + 22\!\cdots\!89)^{2} \) (T^4 + 24791842T^3 + 764458017454947T^2 - 3714377922338705622686*T + 22446807785718387672822132289)^2
$83$	\( T^{8} + \cdots + 50\!\cdots\!76 \) T^8 - 9500192811334080T^6 + 67690373876871514989204969114624T^4 - 214355601426660362685560131740257680125175726080*T^2 + 509102036474735937739388820632539993160361358963815738875314176
$89$	\( (T^{4} + \cdots + 25\!\cdots\!44)^{2} \) (T^4 + 10392887324026224*T^2 + 25004295098998028639850457134144)^2
$97$	\( (T^{4} + \cdots + 56\!\cdots\!25)^{2} \) (T^4 + 204172462T^3 + 34201733900062899T^2 + 1528161528663421344027790*T + 56020140391375360213659679317025)^2
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\(n\)	\(83\)
\(\chi(n)\)	\(-\beta_{1}\)