LMFDB - Newform orbit 310.2.q.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [310,2,Mod(41,310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(310, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 14]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("310.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 310 = 2 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	310.q (of order $15$, degree $8$, 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:	$2.47536246266$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/310\mathbb{Z}\right)^\times$.

$n$	$187$	$251$
$\chi(n)$	$1$	$1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}$

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} $

41.1

−0.978148	−	0.207912i
−0.104528	−	0.994522i
0.913545	−	0.406737i
0.669131	−	0.743145i
0.669131	+	0.743145i
−0.978148	+	0.207912i
0.913545	+	0.406737i
−0.104528	+	0.994522i

−0.809017

−

0.587785i

−0.169131

−

1.60917i

0.309017

0.951057i

−0.500000

−

0.866025i

−0.809017

1.40126i

0.809017

0.171962i

0.309017

−

0.951057i

0.373619

−

0.0794152i

−0.104528

0.994522i

51.1

0.309017

0.951057i

−0.413545

−

0.459289i

−0.809017

0.587785i

−0.500000

−

0.866025i

0.309017

−

0.535233i

−0.309017

−

2.94010i

−0.809017

−

0.587785i

0.273659

−

2.60369i

0.669131

−

0.743145i

71.1

0.309017

−

0.951057i

0.604528

0.128496i

−0.809017

−

0.587785i

−0.500000

−

0.866025i

0.309017

−

0.535233i

−0.309017

0.137583i

−0.809017

0.587785i

−2.39169

−

1.06485i

−0.978148

0.207912i

81.1

−0.809017

−

0.587785i

1.47815

0.658114i

0.309017

0.951057i

−0.500000

0.866025i

−0.809017

−

1.40126i

0.809017

−

0.898504i

0.309017

−

0.951057i

−0.255585

−

0.283856i

0.913545

−

0.406737i

111.1

−0.809017

0.587785i

1.47815

−

0.658114i

0.309017

−

0.951057i

−0.500000

−

0.866025i

−0.809017

1.40126i

0.809017

0.898504i

0.309017

0.951057i

−0.255585

0.283856i

0.913545

0.406737i

121.1

−0.809017

0.587785i

−0.169131

1.60917i

0.309017

−

0.951057i

−0.500000

0.866025i

−0.809017

−

1.40126i

0.809017

−

0.171962i

0.309017

0.951057i

0.373619

0.0794152i

−0.104528

−

0.994522i

131.1

0.309017

0.951057i

0.604528

−

0.128496i

−0.809017

0.587785i

−0.500000

0.866025i

0.309017

0.535233i

−0.309017

−

0.137583i

−0.809017

−

0.587785i

−2.39169

1.06485i

−0.978148

−

0.207912i

231.1

0.309017

−

0.951057i

−0.413545

0.459289i

−0.809017

−

0.587785i

−0.500000

0.866025i

0.309017

0.535233i

−0.309017

2.94010i

−0.809017

0.587785i

0.273659

2.60369i

0.669131

0.743145i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.g	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	310.2.q.b	✓	8
31.g	even	15	1	inner	310.2.q.b	✓	8
31.g	even	15	1		9610.2.a.bw		4
31.h	odd	30	1		9610.2.a.bn		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
310.2.q.b	✓	8	1.a	even	1	1	trivial
310.2.q.b	✓	8	31.g	even	15	1	inner
9610.2.a.bn		4	31.h	odd	30	1
9610.2.a.bw		4	31.g	even	15	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 3T_{3}^{7} + 5T_{3}^{6} - 8T_{3}^{5} + 9T_{3}^{4} - 2T_{3}^{3} - 2T_{3} + 1 $ T3^8 - 3*T3^7 + 5*T3^6 - 8*T3^5 + 9*T3^4 - 2*T3^3 - 2*T3 + 1 acting on $S_{2}^{\mathrm{new}}(310, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$3$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$5$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
$7$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 + 10T^6 - 22T^5 + 24T^4 - 8T^3 - 5T^2 + 2*T + 1
$11$	$ T^{8} - 17 T^{7} + \cdots + 3481 $ T^8 - 17T^7 + 165T^6 - 1117T^5 + 5054T^4 - 13633T^3 + 13965T^2 + 14632*T + 3481
$13$	$ T^{8} - 10 T^{7} + \cdots + 25 $ T^8 - 10T^7 + 430T^5 - 1120T^4 - 11750T^3 + 54825T^2 - 250T + 25
$17$	$ T^{8} - 8 T^{7} + \cdots + 961 $ T^8 - 8T^7 + 2T^5 + 419T^4 + 518T^3 + 3345T^2 + 3193T + 961
$19$	$ T^{8} + 8 T^{7} + \cdots + 151321 $ T^8 + 8T^7 + 45T^6 + 283T^5 + 1124T^4 + 6517T^3 + 43455T^2 + 133427*T + 151321
$23$	$ T^{8} + 2 T^{7} + \cdots + 72361 $ T^8 + 2T^7 + 63T^6 - 196T^5 + 840T^4 + 14T^3 + 27038T^2 + 69402*T + 72361
$29$	$ T^{8} - 11 T^{7} + \cdots + 477481 $ T^8 - 11T^7 + 87T^6 - 583T^5 + 4850T^4 - 14707T^3 + 75927T^2 - 258434*T + 477481
$31$	$ T^{8} + 14 T^{7} + \cdots + 923521 $ T^8 + 14T^7 + 120T^6 + 856T^5 + 5009T^4 + 26536T^3 + 115320T^2 + 417074*T + 923521
$37$	$ T^{8} - 6 T^{7} + \cdots + 961 $ T^8 - 6T^7 + 70T^6 - 114T^5 + 2079T^4 - 5034T^3 + 26335T^2 + 4929*T + 961
$41$	$ T^{8} + 4 T^{7} + \cdots + 1 $ T^8 + 4T^7 + 90T^6 + 566T^5 + 359T^4 - 4174T^3 + 3495T^2 + 79*T + 1
$43$	$ T^{8} - 14 T^{7} + \cdots + 7921 $ T^8 - 14T^7 + 240T^6 - 2026T^5 + 9269T^4 - 20566T^3 + 11145T^2 + 26611*T + 7921
$47$	$ T^{8} - 6 T^{7} + \cdots + 68121 $ T^8 - 6T^7 + 87T^6 - 828T^5 + 5625T^4 - 8532T^3 + 32967T^2 + 79866*T + 68121
$53$	$ T^{8} + 8 T^{7} + \cdots + 12952801 $ T^8 + 8T^7 - 170T^6 - 2342T^5 + 2859T^4 + 152102T^3 + 1201495T^2 + 3105937*T + 12952801
$59$	$ T^{8} - 8 T^{7} + \cdots + 256 $ T^8 - 8T^7 + 40T^6 - 208T^5 - 1296T^4 + 6208T^3 + 34240T^2 + 3968*T + 256
$61$	$ (T^{4} - 14 T^{3} + \cdots - 599)^{2} $ (T^4 - 14T^3 - 64T^2 + 1121*T - 599)^2
$67$	$ T^{8} - 2 T^{7} + \cdots + 256 $ T^8 - 2T^7 + 40T^6 + 88T^5 + 1264T^4 + 352T^3 + 640T^2 - 128*T + 256
$71$	$ T^{8} - 19 T^{7} + \cdots + 57121 $ T^8 - 19T^7 + 150T^6 - 661T^5 + 2079T^4 - 4321T^3 + 1610T^2 - 5019*T + 57121
$73$	$ T^{8} - 12 T^{7} + \cdots + 17131321 $ T^8 - 12T^7 - 310T^6 + 5563T^5 + 19554T^4 - 606638T^3 + 4672650T^2 - 4250753*T + 17131321
$79$	$ T^{8} + 39 T^{7} + \cdots + 16410601 $ T^8 + 39T^7 + 680T^6 + 8246T^5 + 84159T^4 + 527396T^3 + 701910T^2 - 4014541*T + 16410601
$83$	$ T^{8} - 50 T^{7} + \cdots + 160000 $ T^8 - 50T^7 + 840T^6 - 4000T^5 - 8800T^4 - 292000T^3 + 4176000T^2 + 160000*T + 160000
$89$	$ T^{8} + 12 T^{7} + \cdots + 3345241 $ T^8 + 12T^7 + 143T^6 + 1274T^5 + 33915T^4 - 132886T^3 + 2344203T^2 - 4448128*T + 3345241
$97$	$ T^{8} - 42 T^{7} + \cdots + 128881 $ T^8 - 42T^7 + 803T^6 - 7859T^5 + 48240T^4 - 179939T^3 + 397023T^2 + 92263*T + 128881
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.q (of order \(15\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{15}^{3} q^{2} + (\zeta_{15}^{5} - \zeta_{15}^{4} + 1) q^{3} + \zeta_{15}^{6} q^{4} + \zeta_{15}^{5} q^{5} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} + \cdots - 1) q^{6}+ \cdots + (2 \zeta_{15}^{7} - \zeta_{15}^{6} + \cdots - 2) q^{9}+O(q^{10}) \) q + z^3 * q^2 + (z^5 - z^4 + 1) * q^3 + z^6 * q^4 + z^5 * q^5 + (-z^5 + z^4 + z - 1) * q^6 + (z^7 - z^6 + z^4 - 2z^3 + z^2 + z - 1) q^7 + (z^7 - z^6 - z^3 + z^2 - 1) * q^8 + (2z^7 - z^6 + 2z^4 - 2z^3 + z^2 + z - 2) q^9	\( q + \zeta_{15}^{3} q^{2} + (\zeta_{15}^{5} - \zeta_{15}^{4} + 1) q^{3} + \zeta_{15}^{6} q^{4} + \zeta_{15}^{5} q^{5} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} + \cdots - 1) q^{6}+ \cdots + (3 \zeta_{15}^{7} - 6 \zeta_{15}^{6} + \cdots - 7) q^{99}+O(q^{100}) \) q + z^3 * q^2 + (z^5 - z^4 + 1) * q^3 + z^6 * q^4 + z^5 * q^5 + (-z^5 + z^4 + z - 1) * q^6 + (z^7 - z^6 + z^4 - 2z^3 + z^2 + z - 1) q^7 + (z^7 - z^6 - z^3 + z^2 - 1) * q^8 + (2z^7 - z^6 + 2z^4 - 2z^3 + z^2 + z - 2) q^9 + (z^7 - z^5 + z^4 - z^3 + z - 1) * q^10 + (3z^7 + z^6 + 3z^5 + 3z^2 + z + 3) q^11 + (z^5 - z + 1) * q^12 + (-z^7 + z^6 + z^5 + z^4 + 4z^3 - 3z^2 + 3z + 3) q^13 + (-z^6 + z^4 - z^2) * q^14 + (-z^7 + z^6 + z^3 - z^2) * q^15 + (-z^7 - z^2) * q^16 + (2z^7 - z^5 - 2z^4 - z^3 + 2z) q^17 + (z^7 - z^6 - z^5 + z^4 - z^3 - z^2 - 1) * q^18 + (-7z^7 + 3z^6 + 3z^5 - 5z^4 + 2z^3 - 4z^2 - 2z + 4) q^19 + (-z^6 - z) * q^20 + (z^7 - z^4 + z^2) * q^21 + (4z^7 - z^6 - 3z^5 + 4z^4 - z^3 + z^2 + 3z - 7) * q^22 + (-4z^7 + 4z^6 + 3z^3 - 6z^2 - 2z + 3) q^23 + (z^7 - z^5 + z - 1) * q^24 + (-z^5 - 1) * q^25 + (3z^7 + 3z^6 - 3z^5 + 4z^4 + z^3 + z^2 + z - 1) * q^26 + (4z^7 - 3z^6 + 4z^2 - 4) q^27 + (z^6 - z^5 + z^3 - z^2 + 1) * q^28 + (-z^6 - z^5 + z^4 - 6z^3 + z^2 - z - 1) q^29 + (z^7 - z^3 + z^2) * q^30 + (-4z^6 - 3z^5 + 2z^4 - 2z^3 - 2z^2 + 2z - 5) * q^31 + q^32 + (4z^6 + 3z^5 - 3z^4 + 3z^3 - 3z^2 + 3z + 4) * q^33 + (-3z^7 - z^6 - z^5 + z^4 + z^3 - z - 1) q^34 + (-z^7 + z^6 + z^5 - 2z^4 + z^3 - z + 1) q^35 + (-z^7 - z^5 - z^4 + z^3 - z^2 - z + 1) * q^36 + (z^7 - 2z^6 + 2z^5 + z^4 + 2z^3 + 4z^2 + 1) * q^37 + (z^7 - z^6 + z^4 - 2z^3 + 3z^2 + 3z + 1) q^38 + (-4z^7 + 5z^6 - 2z^5 + z^4 + z^3 - 2z^2 + 4z) q^39 + (-z^7 + z^6 - z^4 + z^3 - z^2 + 1) * q^40 + (-z^7 - 4z^6 + 6z^5 - 3z^4 + 2z^3 + z^2 - 5z + 3) q^41 + (-z^7 - 1) * q^42 + (4z^7 - 6z^6 - 4z^5 + z^4 - 3z^3 + 6z^2 + z - 4) q^43 + (-3z^3 - z^2 - 3z) * q^44 + (-z^7 - 2z^4 - z) q^45 + (4z^7 - z^6 - 2z^5 - 2z^4 - z^3 + 4z^2) * q^46 + (z^7 + 2z^6 - 4z^5 - 4z^4 + 2z^3 + z^2) * q^47 + (-z^7 - z) * q^48 + (-z^7 + z^6 + z^5 - 3z^4 - 4z^2 - z - 1) * q^49 + (-z^7 + z^5 - z^4 - z + 1) * q^50 + (z^7 + 5z^6 - 3z^5 - z^4 - z^3 - 3z^2 + 5z + 1) * q^51 + (4z^7 - 2z^6 + z^5 - 2z^4 - z^3 + 3z^2 - 3z - 3) q^52 + (10z^7 - 6z^6 - z^5 + 6z^4 - 5z^3 + 10z^2 - 4z - 7) * q^53 + (-3z^7 + 3z^6 - z^3 - 3z^2 - 1) q^54 + (-3z^2 - z - 3) q^55 + (-z^4 + z^3 + z^2 - z) * q^56 + (-7z^7 + 3z^6 + 2z^5 - 2z^4 + 3z^3 - 5z^2 - 5z + 5) q^57 + (-z^7 - 5z^6 + 2z^5 - 2z^4 + z^3 - z^2 - z + 2) q^58 + (-6z^7 + 2z^6 - 2z^5 + 2z^3 - 2z^2 + 2) q^59 + (-z^6 - 1) * q^60 + (-8z^7 - 3z^6 + 5z^5 - 2z^4 + 3z^3 + 2z^2 - 8z + 8) q^61 + (-5z^7 + 2z^6 + z^5 - z^4 + 2z^3 - 4z^2 - 3z + 7) q^62 + (-2z^7 + z^6 + 2z^5 - 3z^4 + 2z^2 - z) * q^63 + z^3 * q^64 + (3z^7 + z^6 - 2z^5 + 4z^4 - 5z^3 + 2z^2 + 3z - 6) * q^65 + (4z^7 - z^6 - 6z^5 + 6z^4 - 3z^3 + 4z^2 + 3z - 7) * q^66 + (2z^7 + 2z^6 - 2z^5 + 4z^4 - 2z^3 + 2z^2 + 2z - 2) q^67 + (-z^7 + 2z^6 + 4z^5 - 2z^4 + z^3 - z^2 - z + 5) q^68 + (-6z^7 + 6z^5 - 2z^2 - 7z + 4) * q^69 + (-z^3 + z^2 + z - 1) * q^70 + (-3z^7 - 3z^6 + 2z^5 - z^4 - z^3 + z^2 - 2z + 3) * q^71 + (-2z^7 + z^6 + z^5 - 2z^4 + 2z^3 - z + 2) q^72 + (z^7 - 8z^6 + 5z^4 + 5z^3 - 8z + 1) * q^73 + (z^7 + 4z^6 + z^5 + 2z^4 + z^3 - 2z^2 + 2z - 1) * q^74 + (z^7 - z^6 - z^5 + z^4 - z^3 + z^2 - 1) * q^75 + (-z^6 + 2z^5 + 3z^4 + 2z^3 - z^2) q^76 + (3z^7 - 2z^6 - z^5 - z^4 - 2z^3 + 3z^2) * q^77 + (4z^7 - 4z^6 + 4z^5 + 2z^4 - 3z^3 + 5z^2 - 2z + 1) q^78 + (-10z^7 - z^6 + 2z^5 - 8z^4 + 2z^3 - z^2 - 10z) q^79 + z^2 * q^80 + (-2z^7 - 4z^5 - 2z^4 + 4z - 4) * q^81 + (-z^7 + 6z^6 - 4z^5 + z^4 + z^3 - 4z^2 + 6z - 1) * q^82 + (-4z^7 + 6z^6 - 8z^4 + 6z^3 + 2z^2 + 4z + 10) * q^83 + (z^5 - z^3 + 1) * q^84 + (-5z^7 + 4z^6 + 2z^5 - z^4 + 3z^3 - 4z^2 - z + 4) q^85 + (-9z^7 + 3z^6 + 6z^5 - 3z^4 + 6z^3 - 6z^2 - 4z + 6) q^86 + (2z^7 - 4z^6 + 6z^5 - 5z^4 - z^3 + 3z^2 - 7z + 4) * q^87 + (-3z^6 - z^5 - 3z^4) * q^88 + (6z^7 + 7z^6 - 2z^5 + 8z^4 - 4z^3 + 4z - 4) * q^89 + (-2z^7 + z^5 - z^4 + 1) q^90 + (-4z^6 + 4z^5 + z^4 - 7z^3 + z^2 + 4z - 4) * q^91 + (-5z^7 + 2z^5 - 2z^4 + 3z^3 - z^2 - 2z - 1) q^92 + (z^7 - z^6 - 7z^5 + 3z^4 - 3z^3 + 3z^2 + 2z - 7) q^93 + (-6z^7 + 4z^5 - 4z^4 + 2z^3 + 2z^2 - 4z + 1) * q^94 + (-z^5 + 2z^4 + 3z^3 + 2z^2 - z) q^95 + (z^5 - z^4 + 1) * q^96 + (-2z^7 - 3z^6 - 3z^5 + 6z^4 - 3z^3 - 5z^2 + 3z + 2) q^97 + (-z^7 - z^6 - 4z^5 - 3z^3 + z^2 + z - 1) * q^98 + (3z^7 - 6z^6 - 4z^5 - z^4 - 6z^3 - 4z - 7) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 3 q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 3 * q^3 - 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 - 4 * q^9	\( 8 q - 2 q^{2} + 3 q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} - 4 q^{9} + q^{10} + 17 q^{11} + 3 q^{12} + 10 q^{13} + 2 q^{14} - 6 q^{15} - 2 q^{16} + 8 q^{17} + q^{18} - 8 q^{19} + q^{20} + q^{21} - 28 q^{22} - 2 q^{23} - 2 q^{24} - 4 q^{25} + 5 q^{26} - 18 q^{27} + 7 q^{28} + 11 q^{29} + 4 q^{30} - 14 q^{31} + 8 q^{32} + 3 q^{33} - 7 q^{34} - 4 q^{35} + 6 q^{36} + 6 q^{37} + 22 q^{38} - 5 q^{39} + q^{40} - 4 q^{41} - 9 q^{42} + 14 q^{43} + 2 q^{44} - 4 q^{45} + 18 q^{46} + 6 q^{47} - 2 q^{48} - 23 q^{49} + q^{50} + 14 q^{51} - 20 q^{52} - 8 q^{53} - 18 q^{54} - 28 q^{55} - 3 q^{56} + q^{57} + 11 q^{58} + 8 q^{59} - 6 q^{60} + 28 q^{61} + 31 q^{62} - 14 q^{63} - 2 q^{64} - 20 q^{65} - 7 q^{66} + 2 q^{67} + 13 q^{68} - 7 q^{69} - 4 q^{70} + 19 q^{71} + q^{72} + 12 q^{73} - 19 q^{74} + 3 q^{75} - 8 q^{76} + 17 q^{77} + 15 q^{78} - 39 q^{79} + q^{80} - 16 q^{81} - 4 q^{82} + 50 q^{83} + 6 q^{84} - q^{85} - 16 q^{86} + 11 q^{87} + 7 q^{88} - 12 q^{89} + q^{90} - 20 q^{91} - 32 q^{92} - 11 q^{93} - 24 q^{94} + q^{95} + 3 q^{96} + 42 q^{97} + 17 q^{98} - 18 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 3 * q^3 - 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 - 4 * q^9 + q^10 + 17 * q^11 + 3 * q^12 + 10 * q^13 + 2 * q^14 - 6 * q^15 - 2 * q^16 + 8 * q^17 + q^18 - 8 * q^19 + q^20 + q^21 - 28 * q^22 - 2 * q^23 - 2 * q^24 - 4 * q^25 + 5 * q^26 - 18 * q^27 + 7 * q^28 + 11 * q^29 + 4 * q^30 - 14 * q^31 + 8 * q^32 + 3 * q^33 - 7 * q^34 - 4 * q^35 + 6 * q^36 + 6 * q^37 + 22 * q^38 - 5 * q^39 + q^40 - 4 * q^41 - 9 * q^42 + 14 * q^43 + 2 * q^44 - 4 * q^45 + 18 * q^46 + 6 * q^47 - 2 * q^48 - 23 * q^49 + q^50 + 14 * q^51 - 20 * q^52 - 8 * q^53 - 18 * q^54 - 28 * q^55 - 3 * q^56 + q^57 + 11 * q^58 + 8 * q^59 - 6 * q^60 + 28 * q^61 + 31 * q^62 - 14 * q^63 - 2 * q^64 - 20 * q^65 - 7 * q^66 + 2 * q^67 + 13 * q^68 - 7 * q^69 - 4 * q^70 + 19 * q^71 + q^72 + 12 * q^73 - 19 * q^74 + 3 * q^75 - 8 * q^76 + 17 * q^77 + 15 * q^78 - 39 * q^79 + q^80 - 16 * q^81 - 4 * q^82 + 50 * q^83 + 6 * q^84 - q^85 - 16 * q^86 + 11 * q^87 + 7 * q^88 - 12 * q^89 + q^90 - 20 * q^91 - 32 * q^92 - 11 * q^93 - 24 * q^94 + q^95 + 3 * q^96 + 42 * q^97 + 17 * q^98 - 18 * q^99

Introduction

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Newspace parameters

Newform invariants

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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	310.2.q.b

Level	$310$
Weight	$2$
Character orbit	310.q
Analytic conductor	$2.475$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.47536246266\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(1\)	\(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 41.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$3$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$5$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
$7$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 + 10T^6 - 22T^5 + 24T^4 - 8T^3 - 5T^2 + 2*T + 1
$11$	\( T^{8} - 17 T^{7} + \cdots + 3481 \) T^8 - 17T^7 + 165T^6 - 1117T^5 + 5054T^4 - 13633T^3 + 13965T^2 + 14632*T + 3481
$13$	\( T^{8} - 10 T^{7} + \cdots + 25 \) T^8 - 10T^7 + 430T^5 - 1120T^4 - 11750T^3 + 54825T^2 - 250T + 25
$17$	\( T^{8} - 8 T^{7} + \cdots + 961 \) T^8 - 8T^7 + 2T^5 + 419T^4 + 518T^3 + 3345T^2 + 3193T + 961
$19$	\( T^{8} + 8 T^{7} + \cdots + 151321 \) T^8 + 8T^7 + 45T^6 + 283T^5 + 1124T^4 + 6517T^3 + 43455T^2 + 133427*T + 151321
$23$	\( T^{8} + 2 T^{7} + \cdots + 72361 \) T^8 + 2T^7 + 63T^6 - 196T^5 + 840T^4 + 14T^3 + 27038T^2 + 69402*T + 72361
$29$	\( T^{8} - 11 T^{7} + \cdots + 477481 \) T^8 - 11T^7 + 87T^6 - 583T^5 + 4850T^4 - 14707T^3 + 75927T^2 - 258434*T + 477481
$31$	\( T^{8} + 14 T^{7} + \cdots + 923521 \) T^8 + 14T^7 + 120T^6 + 856T^5 + 5009T^4 + 26536T^3 + 115320T^2 + 417074*T + 923521
$37$	\( T^{8} - 6 T^{7} + \cdots + 961 \) T^8 - 6T^7 + 70T^6 - 114T^5 + 2079T^4 - 5034T^3 + 26335T^2 + 4929*T + 961
$41$	\( T^{8} + 4 T^{7} + \cdots + 1 \) T^8 + 4T^7 + 90T^6 + 566T^5 + 359T^4 - 4174T^3 + 3495T^2 + 79*T + 1
$43$	\( T^{8} - 14 T^{7} + \cdots + 7921 \) T^8 - 14T^7 + 240T^6 - 2026T^5 + 9269T^4 - 20566T^3 + 11145T^2 + 26611*T + 7921
$47$	\( T^{8} - 6 T^{7} + \cdots + 68121 \) T^8 - 6T^7 + 87T^6 - 828T^5 + 5625T^4 - 8532T^3 + 32967T^2 + 79866*T + 68121
$53$	\( T^{8} + 8 T^{7} + \cdots + 12952801 \) T^8 + 8T^7 - 170T^6 - 2342T^5 + 2859T^4 + 152102T^3 + 1201495T^2 + 3105937*T + 12952801
$59$	\( T^{8} - 8 T^{7} + \cdots + 256 \) T^8 - 8T^7 + 40T^6 - 208T^5 - 1296T^4 + 6208T^3 + 34240T^2 + 3968*T + 256
$61$	\( (T^{4} - 14 T^{3} + \cdots - 599)^{2} \) (T^4 - 14T^3 - 64T^2 + 1121*T - 599)^2
$67$	\( T^{8} - 2 T^{7} + \cdots + 256 \) T^8 - 2T^7 + 40T^6 + 88T^5 + 1264T^4 + 352T^3 + 640T^2 - 128*T + 256
$71$	\( T^{8} - 19 T^{7} + \cdots + 57121 \) T^8 - 19T^7 + 150T^6 - 661T^5 + 2079T^4 - 4321T^3 + 1610T^2 - 5019*T + 57121
$73$	\( T^{8} - 12 T^{7} + \cdots + 17131321 \) T^8 - 12T^7 - 310T^6 + 5563T^5 + 19554T^4 - 606638T^3 + 4672650T^2 - 4250753*T + 17131321
$79$	\( T^{8} + 39 T^{7} + \cdots + 16410601 \) T^8 + 39T^7 + 680T^6 + 8246T^5 + 84159T^4 + 527396T^3 + 701910T^2 - 4014541*T + 16410601
$83$	\( T^{8} - 50 T^{7} + \cdots + 160000 \) T^8 - 50T^7 + 840T^6 - 4000T^5 - 8800T^4 - 292000T^3 + 4176000T^2 + 160000*T + 160000
$89$	\( T^{8} + 12 T^{7} + \cdots + 3345241 \) T^8 + 12T^7 + 143T^6 + 1274T^5 + 33915T^4 - 132886T^3 + 2344203T^2 - 4448128*T + 3345241
$97$	\( T^{8} - 42 T^{7} + \cdots + 128881 \) T^8 - 42T^7 + 803T^6 - 7859T^5 + 48240T^4 - 179939T^3 + 397023T^2 + 92263*T + 128881
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