LMFDB - Newform orbit 1932.1.bi.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,1,Mod(167,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 11, 18]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1932.167");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1932.bi (of order $22$, degree $10$, 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:	$0.964193604407$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$-1$	$-\zeta_{22}^{7}$	$-1$	$-1$

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} $

167.1

0.959493	+	0.281733i
−0.841254	−	0.540641i
0.142315	−	0.989821i
0.959493	−	0.281733i
−0.415415	+	0.909632i
0.654861	−	0.755750i
0.142315	+	0.989821i
0.654861	+	0.755750i
−0.841254	+	0.540641i
−0.415415	−	0.909632i

−0.415415

0.909632i

−0.959493

−

0.281733i

−0.654861

−

0.755750i

1.10181

−

0.708089i

0.654861

−

0.755750i

−0.142315

0.989821i

0.959493

−

0.281733i

0.841254

0.540641i

0.186393

1.29639i

335.1

0.654861

0.755750i

0.841254

0.540641i

−0.142315

0.989821i

0.118239

−

0.258908i

0.142315

0.989821i

−0.959493

−

0.281733i

−0.841254

0.540641i

0.415415

0.909632i

0.273100

−

0.0801894i

587.1

−0.841254

0.540641i

−0.142315

0.989821i

0.415415

−

0.909632i

0.797176

−

0.234072i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

0.142315

0.989821i

−0.959493

−

0.281733i

−0.544078

0.627899i

671.1

−0.415415

−

0.909632i

−0.959493

0.281733i

−0.654861

0.755750i

1.10181

0.708089i

0.654861

0.755750i

−0.142315

−

0.989821i

0.959493

0.281733i

0.841254

−

0.540641i

0.186393

−

1.29639i

923.1

0.142315

0.989821i

0.415415

−

0.909632i

−0.959493

0.281733i

−1.25667

1.45027i

0.959493

0.281733i

0.841254

−

0.540641i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

−1.61435

−

1.03748i

1007.1

0.959493

0.281733i

−0.654861

0.755750i

0.841254

0.540641i

0.239446

−

1.66538i

−0.841254

0.540641i

0.415415

0.909632i

0.654861

0.755750i

−0.142315

−

0.989821i

0.698939

−

1.53046i

1175.1

−0.841254

−

0.540641i

−0.142315

−

0.989821i

0.415415

0.909632i

0.797176

0.234072i

−0.415415

0.909632i

−0.654861

0.755750i

0.142315

−

0.989821i

−0.959493

0.281733i

−0.544078

−

0.627899i

1343.1

0.959493

−

0.281733i

−0.654861

−

0.755750i

0.841254

−

0.540641i

0.239446

1.66538i

−0.841254

−

0.540641i

0.415415

−

0.909632i

0.654861

−

0.755750i

−0.142315

0.989821i

0.698939

1.53046i

1511.1

0.654861

−

0.755750i

0.841254

−

0.540641i

−0.142315

−

0.989821i

0.118239

0.258908i

0.142315

−

0.989821i

−0.959493

0.281733i

−0.841254

−

0.540641i

0.415415

−

0.909632i

0.273100

0.0801894i

1595.1

0.142315

−

0.989821i

0.415415

0.909632i

−0.959493

−

0.281733i

−1.25667

−

1.45027i

0.959493

−

0.281733i

0.841254

0.540641i

−0.415415

0.909632i

−0.654861

0.755750i

−1.61435

1.03748i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
84.h	odd	2	1	CM by $\Q(\sqrt{-21}) $
23.c	even	11	1	inner
1932.bi	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1932.1.bi.c	yes	10
3.b	odd	2	1		1932.1.bi.a	✓	10
4.b	odd	2	1		1932.1.bi.b	yes	10
7.b	odd	2	1		1932.1.bi.d	yes	10
12.b	even	2	1		1932.1.bi.d	yes	10
21.c	even	2	1		1932.1.bi.b	yes	10
23.c	even	11	1	inner	1932.1.bi.c	yes	10
28.d	even	2	1		1932.1.bi.a	✓	10
69.h	odd	22	1		1932.1.bi.a	✓	10
84.h	odd	2	1	CM	1932.1.bi.c	yes	10
92.g	odd	22	1		1932.1.bi.b	yes	10
161.l	odd	22	1		1932.1.bi.d	yes	10
276.o	even	22	1		1932.1.bi.d	yes	10
483.v	even	22	1		1932.1.bi.b	yes	10
644.t	even	22	1		1932.1.bi.a	✓	10
1932.bi	odd	22	1	inner	1932.1.bi.c	yes	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1932.1.bi.a	✓	10	3.b	odd	2	1
1932.1.bi.a	✓	10	28.d	even	2	1
1932.1.bi.a	✓	10	69.h	odd	22	1
1932.1.bi.a	✓	10	644.t	even	22	1
1932.1.bi.b	yes	10	4.b	odd	2	1
1932.1.bi.b	yes	10	21.c	even	2	1
1932.1.bi.b	yes	10	92.g	odd	22	1
1932.1.bi.b	yes	10	483.v	even	22	1
1932.1.bi.c	yes	10	1.a	even	1	1	trivial
1932.1.bi.c	yes	10	23.c	even	11	1	inner
1932.1.bi.c	yes	10	84.h	odd	2	1	CM
1932.1.bi.c	yes	10	1932.bi	odd	22	1	inner
1932.1.bi.d	yes	10	7.b	odd	2	1
1932.1.bi.d	yes	10	12.b	even	2	1
1932.1.bi.d	yes	10	161.l	odd	22	1
1932.1.bi.d	yes	10	276.o	even	22	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1932, [\chi])$:

$ T_{5}^{10} - 2T_{5}^{9} + 4T_{5}^{8} - 8T_{5}^{7} + 16T_{5}^{6} - 32T_{5}^{5} + 53T_{5}^{4} - 51T_{5}^{3} + 25T_{5}^{2} - 6T_{5} + 1 $

$ T_{11}^{10} - 2 T_{11}^{9} + 4 T_{11}^{8} - 8 T_{11}^{7} + 16 T_{11}^{6} - 32 T_{11}^{5} + 53 T_{11}^{4} + \cdots + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$7$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$13$	$ T^{10} $ T^10
$17$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$19$	$ T^{10} + 2 T^{9} + \cdots + 1 $ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$23$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$29$	$ T^{10} $ T^10
$31$	$ T^{10} - 9 T^{9} + \cdots + 1 $ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$37$	$ T^{10} - 9 T^{9} + \cdots + 1 $ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$41$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$43$	$ T^{10} $ T^10
$47$	$ T^{10} $ T^10
$53$	$ T^{10} $ T^10
$59$	$ T^{10} $ T^10
$61$	$ T^{10} $ T^10
$67$	$ T^{10} $ T^10
$71$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$73$	$ T^{10} $ T^10
$79$	$ T^{10} $ T^10
$83$	$ T^{10} $ T^10
$89$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$97$	$ T^{10} $ T^10
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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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1932.bi (of order \(22\), degree \(10\), minimal)

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

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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Hecke kernels

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	1932.1.bi.c

Level	$1932$
Weight	$1$
Character orbit	1932.bi
Analytic conductor	$0.964$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{11}$
CM discriminant	-84
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.964193604407\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{11}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

\(n\)	\(829\)	\(925\)	\(967\)	\(1289\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{22}^{7}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 167.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$7$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$13$	\( T^{10} \) T^10
$17$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$19$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$23$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$29$	\( T^{10} \) T^10
$31$	\( T^{10} - 9 T^{9} + \cdots + 1 \) T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$37$	\( T^{10} - 9 T^{9} + \cdots + 1 \) T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$41$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$43$	\( T^{10} \) T^10
$47$	\( T^{10} \) T^10
$53$	\( T^{10} \) T^10
$59$	\( T^{10} \) T^10
$61$	\( T^{10} \) T^10
$67$	\( T^{10} \) T^10
$71$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$73$	\( T^{10} \) T^10
$79$	\( T^{10} \) T^10
$83$	\( T^{10} \) T^10
$89$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$97$	\( T^{10} \) T^10
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