LMFDB - Newform orbit 343.2.g.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [343,2,Mod(30,343)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(343, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([32]))

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

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

chi := DirichletCharacter("343.30");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 343 = 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	343.g (of order $21$, degree $12$, 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:	$2.73886878933$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

$q$-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 primitive root of unity $\zeta_{21}$. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/343\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$\zeta_{21}^{2}$

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} $

30.1

−0.733052	+	0.680173i
0.365341	+	0.930874i
0.826239	+	0.563320i
0.955573	+	0.294755i
0.365341	−	0.930874i
0.826239	−	0.563320i
−0.988831	−	0.149042i
0.0747301	−	0.997204i
0.0747301	+	0.997204i
−0.733052	−	0.680173i
0.955573	−	0.294755i
−0.988831	+	0.149042i

1.63402

−

1.11406i

−1.94440

1.80414i

0.698220

−

1.77904i

1.54379

0.476196i

−1.16728

5.11418i

0.0391023

0.171318i

0.301584

−

4.02435i

3.05309

−

0.941754i

67.1

−0.142820

0.0440542i

1.06356

2.70991i

−1.63402

1.11406i

2.17225

0.327414i

−0.271281

−

0.340175i

0.370666

−

0.464800i

−4.01329

3.72379i

−0.324665

0.0489353i

79.1

−1.44973

0.218511i

−0.807782

−

0.550736i

0.142820

−

0.0440542i

−0.0688859

0.919218i

1.29141

0.621909i

2.44440

−

1.17716i

−0.746822

−

1.90287i

−0.100994

−

1.34767i

116.1

−0.0546039

−

0.728639i

1.09839

0.338809i

1.44973

−

0.218511i

0.189617

−

0.175939i

0.186893

−

0.818832i

−0.563561

−

2.46912i

−1.38704

−

0.945669i

−0.138550

−

0.128555i

128.1

−0.142820

−

0.0440542i

1.06356

−

2.70991i

−1.63402

−

1.11406i

2.17225

−

0.327414i

−0.271281

0.340175i

0.370666

0.464800i

−4.01329

−

3.72379i

−0.324665

−

0.0489353i

165.1

−1.44973

−

0.218511i

−0.807782

0.550736i

0.142820

0.0440542i

−0.0688859

−

0.919218i

1.29141

−

0.621909i

2.44440

1.17716i

−0.746822

1.90287i

−0.100994

1.34767i

177.1

1.21135

1.12397i

0.460898

0.0694692i

0.0546039

0.728639i

1.42074

3.61999i

0.480228

0.602187i

1.30778

−

1.63991i

−2.65912

−

0.820229i

−2.34774

5.98195i

214.1

−0.698220

1.77904i

0.129334

−

1.72584i

−1.21135

−

1.12397i

−1.75751

−

1.19825i

2.98003

1.43511i

−0.598393

0.288171i

0.00468671

0.000706408i

3.35886

−

2.29003i

226.1

−0.698220

−

1.77904i

0.129334

1.72584i

−1.21135

1.12397i

−1.75751

1.19825i

2.98003

−

1.43511i

−0.598393

−

0.288171i

0.00468671

0.000706408i

3.35886

2.29003i

263.1

1.63402

1.11406i

−1.94440

−

1.80414i

0.698220

1.77904i

1.54379

−

0.476196i

−1.16728

−

5.11418i

0.0391023

−

0.171318i

0.301584

4.02435i

3.05309

0.941754i

275.1

−0.0546039

0.728639i

1.09839

−

0.338809i

1.44973

0.218511i

0.189617

0.175939i

0.186893

0.818832i

−0.563561

2.46912i

−1.38704

0.945669i

−0.138550

0.128555i

312.1

1.21135

−

1.12397i

0.460898

−

0.0694692i

0.0546039

−

0.728639i

1.42074

−

3.61999i

0.480228

−

0.602187i

1.30778

1.63991i

−2.65912

0.820229i

−2.34774

−

5.98195i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.g	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	343.2.g.c		12
7.b	odd	2	1		343.2.g.b		12
7.c	even	3	1		343.2.e.b		12
7.c	even	3	1		343.2.g.a		12
7.d	odd	6	1		49.2.e.b	✓	12
7.d	odd	6	1		343.2.g.d		12
21.g	even	6	1		441.2.u.b		12
28.f	even	6	1		784.2.u.b		12
49.e	even	7	1		343.2.g.a		12
49.f	odd	14	1		343.2.g.d		12
49.g	even	21	1		343.2.e.b		12
49.g	even	21	1	inner	343.2.g.c		12
49.g	even	21	1		2401.2.a.d		6
49.h	odd	42	1		49.2.e.b	✓	12
49.h	odd	42	1		343.2.g.b		12
49.h	odd	42	1		2401.2.a.c		6
147.o	even	42	1		441.2.u.b		12
196.p	even	42	1		784.2.u.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.2.e.b	✓	12	7.d	odd	6	1
49.2.e.b	✓	12	49.h	odd	42	1
343.2.e.b		12	7.c	even	3	1
343.2.e.b		12	49.g	even	21	1
343.2.g.a		12	7.c	even	3	1
343.2.g.a		12	49.e	even	7	1
343.2.g.b		12	7.b	odd	2	1
343.2.g.b		12	49.h	odd	42	1
343.2.g.c		12	1.a	even	1	1	trivial
343.2.g.c		12	49.g	even	21	1	inner
343.2.g.d		12	7.d	odd	6	1
343.2.g.d		12	49.f	odd	14	1
441.2.u.b		12	21.g	even	6	1
441.2.u.b		12	147.o	even	42	1
784.2.u.b		12	28.f	even	6	1
784.2.u.b		12	196.p	even	42	1
2401.2.a.c		6	49.h	odd	42	1
2401.2.a.d		6	49.g	even	21	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(343, [\chi])$:

$ T_{2}^{12} - T_{2}^{11} + T_{2}^{9} + 6T_{2}^{8} + 21T_{2}^{7} - 20T_{2}^{6} + 69T_{2}^{4} + 29T_{2}^{3} + 49T_{2}^{2} + 13T_{2} + 1 $

$ T_{3}^{12} + 7 T_{3}^{10} + 7 T_{3}^{9} + 42 T_{3}^{8} - 77 T_{3}^{7} + 112 T_{3}^{6} - 196 T_{3}^{5} + \cdots + 49 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 + T^9 + 6T^8 + 21T^7 - 20T^6 + 69T^4 + 29T^3 + 49T^2 + 13*T + 1
$3$	$ T^{12} + 7 T^{10} + \cdots + 49 $ T^12 + 7T^10 + 7T^9 + 42T^8 - 77T^7 + 112T^6 - 196T^5 + 147T^3 + 147T^2 - 196*T + 49
$5$	$ T^{12} - 7 T^{11} + \cdots + 49 $ T^12 - 7T^11 + 28T^10 - 56T^9 + 49T^7 + 455T^6 - 1323T^5 + 1568T^4 - 1470T^3 + 1176T^2 - 343T + 49
$7$	$ T^{12} $ T^12
$11$	$ T^{12} - 4 T^{11} + \cdots + 1849 $ T^12 - 4T^11 + 120T^9 - 655T^8 + 1764T^7 + 4691T^6 - 42014T^5 + 119460T^4 - 109598T^3 + 29491T^2 + 1118T + 1849
$13$	$ T^{12} + 7 T^{11} + \cdots + 49 $ T^12 + 7T^11 + 28T^10 + 77T^9 + 147T^8 + 210T^7 + 406T^6 + 735T^5 + 882T^4 + 637T^3 + 294T^2 + 98*T + 49
$17$	$ T^{12} - 28 T^{10} + \cdots + 3087049 $ T^12 - 28T^10 - 140T^9 + 7T^8 - 2156T^7 - 819T^6 + 125832T^5 + 737156T^4 + 1805356T^3 + 2736405T^2 + 3615906T + 3087049
$19$	$ T^{12} + 7 T^{11} + \cdots + 4439449 $ T^12 + 7T^11 + 70T^10 + 273T^9 + 1960T^8 + 6664T^7 + 36309T^6 + 90895T^5 + 375928T^4 + 808108T^3 + 2561867T^2 + 3303776*T + 4439449
$23$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 - 14T^10 + 43T^9 + 69T^8 - 357T^7 + 736T^6 - 819T^5 + 573T^4 - 202T^3 + 42T^2 - 8T + 1
$29$	$ T^{12} + 11 T^{11} + \cdots + 1681 $ T^12 + 11T^11 + 75T^10 + 508T^9 + 2957T^8 + 1788T^7 + 62377T^6 - 214503T^5 + 822968T^4 - 759692T^3 + 451338T^2 + 51332*T + 1681
$31$	$ T^{12} + 7 T^{11} + \cdots + 66961489 $ T^12 + 7T^11 + 147T^10 + 462T^9 + 11809T^8 + 38661T^7 + 436247T^6 + 429093T^5 + 6957069T^4 + 4730999T^3 + 75535460T^2 - 63753753*T + 66961489
$37$	$ T^{12} + \cdots + 295118041 $ T^12 - 15T^11 + 112T^10 - 1133T^9 + 9855T^8 - 30884T^7 - 65960T^6 + 1001021T^5 - 5294766T^4 + 15024003T^3 + 77099064T^2 - 235592806*T + 295118041
$41$	$ T^{12} + 21 T^{11} + \cdots + 10413529 $ T^12 + 21T^11 + 196T^10 + 1568T^9 + 15120T^8 + 41552T^7 + 99043T^6 + 315364T^5 + 923748T^4 + 1782228T^3 + 7110831T^2 - 9171134*T + 10413529
$43$	$ T^{12} + \cdots + 200307409 $ T^12 - 17T^11 + 82T^10 + 977T^9 - 7683T^8 - 50250T^7 + 854119T^6 - 2984424T^5 + 29767569T^4 - 86491919T^3 + 232868670T^2 + 20988899*T + 200307409
$47$	$ T^{12} + \cdots + 3021810841 $ T^12 + 14T^10 - 1162T^9 - 7448T^8 - 23401T^7 + 1689093T^6 + 8599941T^5 - 8763160T^4 + 164353154T^3 + 570459225T^2 - 2376891069T + 3021810841
$53$	$ T^{12} - 18 T^{11} + \cdots + 2985984 $ T^12 - 18T^11 + 252T^10 - 2736T^9 + 20016T^8 - 90720T^7 + 240192T^6 - 362880T^5 + 559872T^4 - 1368576T^3 + 4478976T + 2985984
$59$	$ T^{12} - 7 T^{11} + \cdots + 54125449 $ T^12 - 7T^11 - 14T^10 - 77T^9 + 1197T^8 + 819T^7 + 14812T^6 - 107163T^5 - 312963T^4 + 1899044T^3 + 12288318T^2 + 36667288*T + 54125449
$61$	$ T^{12} + \cdots + 261242569 $ T^12 - 7T^11 - 217T^10 + 2954T^9 - 1036T^8 - 231553T^7 + 2366175T^6 - 12959275T^5 + 47275396T^4 - 120714146T^3 + 221757144T^2 - 273574938*T + 261242569
$67$	$ T^{12} + 24 T^{11} + \cdots + 85849 $ T^12 + 24T^11 + 434T^10 + 3550T^9 + 23754T^8 + 78960T^7 + 307931T^6 + 548898T^5 + 3493096T^4 + 2760990T^3 + 1655598T^2 + 435398*T + 85849
$71$	$ T^{12} + \cdots + 312925003609 $ T^12 + 39T^11 + 1013T^10 + 17644T^9 + 236029T^8 + 2443507T^7 + 21143409T^6 + 163533123T^5 + 1207844693T^4 + 8445075811T^3 + 48568012464T^2 + 180407210691*T + 312925003609
$73$	$ T^{12} + \cdots + 460917961 $ T^12 - 21T^11 + 210T^10 - 1799T^9 + 10598T^8 - 30639T^7 + 116340T^6 - 239708T^5 - 3715523T^4 + 24996811T^3 + 40091849T^2 - 253827987*T + 460917961
$79$	$ T^{12} + \cdots + 467813641 $ T^12 - 8T^11 + 161T^10 - 118T^9 + 9505T^8 + 10388T^7 + 478647T^6 + 1353590T^5 + 10876438T^4 + 12609486T^3 + 79005409T^2 + 41808857*T + 467813641
$83$	$ T^{12} + \cdots + 118178641 $ T^12 - 7T^11 - 14T^10 + 2821T^9 + 13755T^8 - 214004T^7 + 327677T^6 - 249312T^5 + 34397951T^4 + 89724537T^3 + 266644770T^2 - 39189955*T + 118178641
$89$	$ T^{12} + \cdots + 89755965649 $ T^12 + 49T^11 + 1225T^10 + 22463T^9 + 328083T^8 + 3722642T^7 + 33194378T^6 + 248309607T^5 + 1617862694T^4 + 8604200619T^3 + 32896698408T^2 + 78116777599*T + 89755965649
$97$	$ (T^{6} - 14 T^{5} + \cdots + 18571)^{2} $ (T^6 - 14T^5 - 28T^4 + 966T^3 - 1911T^2 - 9898*T + 18571)^2
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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:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	343.g (of order \(21\), degree \(12\), not minimal)

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

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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Hecke characteristic polynomials

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	343.2.g.c

Level	$343$
Weight	$2$
Character orbit	343.g
Analytic conductor	$2.739$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.73886878933\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{21})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 30.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + T^9 + 6T^8 + 21T^7 - 20T^6 + 69T^4 + 29T^3 + 49T^2 + 13*T + 1
$3$	\( T^{12} + 7 T^{10} + \cdots + 49 \) T^12 + 7T^10 + 7T^9 + 42T^8 - 77T^7 + 112T^6 - 196T^5 + 147T^3 + 147T^2 - 196*T + 49
$5$	\( T^{12} - 7 T^{11} + \cdots + 49 \) T^12 - 7T^11 + 28T^10 - 56T^9 + 49T^7 + 455T^6 - 1323T^5 + 1568T^4 - 1470T^3 + 1176T^2 - 343T + 49
$7$	\( T^{12} \) T^12
$11$	\( T^{12} - 4 T^{11} + \cdots + 1849 \) T^12 - 4T^11 + 120T^9 - 655T^8 + 1764T^7 + 4691T^6 - 42014T^5 + 119460T^4 - 109598T^3 + 29491T^2 + 1118T + 1849
$13$	\( T^{12} + 7 T^{11} + \cdots + 49 \) T^12 + 7T^11 + 28T^10 + 77T^9 + 147T^8 + 210T^7 + 406T^6 + 735T^5 + 882T^4 + 637T^3 + 294T^2 + 98*T + 49
$17$	\( T^{12} - 28 T^{10} + \cdots + 3087049 \) T^12 - 28T^10 - 140T^9 + 7T^8 - 2156T^7 - 819T^6 + 125832T^5 + 737156T^4 + 1805356T^3 + 2736405T^2 + 3615906T + 3087049
$19$	\( T^{12} + 7 T^{11} + \cdots + 4439449 \) T^12 + 7T^11 + 70T^10 + 273T^9 + 1960T^8 + 6664T^7 + 36309T^6 + 90895T^5 + 375928T^4 + 808108T^3 + 2561867T^2 + 3303776*T + 4439449
$23$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 - 14T^10 + 43T^9 + 69T^8 - 357T^7 + 736T^6 - 819T^5 + 573T^4 - 202T^3 + 42T^2 - 8T + 1
$29$	\( T^{12} + 11 T^{11} + \cdots + 1681 \) T^12 + 11T^11 + 75T^10 + 508T^9 + 2957T^8 + 1788T^7 + 62377T^6 - 214503T^5 + 822968T^4 - 759692T^3 + 451338T^2 + 51332*T + 1681
$31$	\( T^{12} + 7 T^{11} + \cdots + 66961489 \) T^12 + 7T^11 + 147T^10 + 462T^9 + 11809T^8 + 38661T^7 + 436247T^6 + 429093T^5 + 6957069T^4 + 4730999T^3 + 75535460T^2 - 63753753*T + 66961489
$37$	\( T^{12} + \cdots + 295118041 \) T^12 - 15T^11 + 112T^10 - 1133T^9 + 9855T^8 - 30884T^7 - 65960T^6 + 1001021T^5 - 5294766T^4 + 15024003T^3 + 77099064T^2 - 235592806*T + 295118041
$41$	\( T^{12} + 21 T^{11} + \cdots + 10413529 \) T^12 + 21T^11 + 196T^10 + 1568T^9 + 15120T^8 + 41552T^7 + 99043T^6 + 315364T^5 + 923748T^4 + 1782228T^3 + 7110831T^2 - 9171134*T + 10413529
$43$	\( T^{12} + \cdots + 200307409 \) T^12 - 17T^11 + 82T^10 + 977T^9 - 7683T^8 - 50250T^7 + 854119T^6 - 2984424T^5 + 29767569T^4 - 86491919T^3 + 232868670T^2 + 20988899*T + 200307409
$47$	\( T^{12} + \cdots + 3021810841 \) T^12 + 14T^10 - 1162T^9 - 7448T^8 - 23401T^7 + 1689093T^6 + 8599941T^5 - 8763160T^4 + 164353154T^3 + 570459225T^2 - 2376891069T + 3021810841
$53$	\( T^{12} - 18 T^{11} + \cdots + 2985984 \) T^12 - 18T^11 + 252T^10 - 2736T^9 + 20016T^8 - 90720T^7 + 240192T^6 - 362880T^5 + 559872T^4 - 1368576T^3 + 4478976T + 2985984
$59$	\( T^{12} - 7 T^{11} + \cdots + 54125449 \) T^12 - 7T^11 - 14T^10 - 77T^9 + 1197T^8 + 819T^7 + 14812T^6 - 107163T^5 - 312963T^4 + 1899044T^3 + 12288318T^2 + 36667288*T + 54125449
$61$	\( T^{12} + \cdots + 261242569 \) T^12 - 7T^11 - 217T^10 + 2954T^9 - 1036T^8 - 231553T^7 + 2366175T^6 - 12959275T^5 + 47275396T^4 - 120714146T^3 + 221757144T^2 - 273574938*T + 261242569
$67$	\( T^{12} + 24 T^{11} + \cdots + 85849 \) T^12 + 24T^11 + 434T^10 + 3550T^9 + 23754T^8 + 78960T^7 + 307931T^6 + 548898T^5 + 3493096T^4 + 2760990T^3 + 1655598T^2 + 435398*T + 85849
$71$	\( T^{12} + \cdots + 312925003609 \) T^12 + 39T^11 + 1013T^10 + 17644T^9 + 236029T^8 + 2443507T^7 + 21143409T^6 + 163533123T^5 + 1207844693T^4 + 8445075811T^3 + 48568012464T^2 + 180407210691*T + 312925003609
$73$	\( T^{12} + \cdots + 460917961 \) T^12 - 21T^11 + 210T^10 - 1799T^9 + 10598T^8 - 30639T^7 + 116340T^6 - 239708T^5 - 3715523T^4 + 24996811T^3 + 40091849T^2 - 253827987*T + 460917961
$79$	\( T^{12} + \cdots + 467813641 \) T^12 - 8T^11 + 161T^10 - 118T^9 + 9505T^8 + 10388T^7 + 478647T^6 + 1353590T^5 + 10876438T^4 + 12609486T^3 + 79005409T^2 + 41808857*T + 467813641
$83$	\( T^{12} + \cdots + 118178641 \) T^12 - 7T^11 - 14T^10 + 2821T^9 + 13755T^8 - 214004T^7 + 327677T^6 - 249312T^5 + 34397951T^4 + 89724537T^3 + 266644770T^2 - 39189955*T + 118178641
$89$	\( T^{12} + \cdots + 89755965649 \) T^12 + 49T^11 + 1225T^10 + 22463T^9 + 328083T^8 + 3722642T^7 + 33194378T^6 + 248309607T^5 + 1617862694T^4 + 8604200619T^3 + 32896698408T^2 + 78116777599*T + 89755965649
$97$	\( (T^{6} - 14 T^{5} + \cdots + 18571)^{2} \) (T^6 - 14T^5 - 28T^4 + 966T^3 - 1911T^2 - 9898*T + 18571)^2
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\(n\)	\(3\)
\(\chi(n)\)	\(\zeta_{21}^{2}\)