LMFDB - Newform orbit 3420.3.o.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,3,Mod(721,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3420.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3420.o (of order $2$, degree $1$, 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:	$93.1882504112$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 5 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7}+O(q^{10}) $ q - b6 * q^5 + (-b11 + b6 - 1) * q^7	$ q - \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7} + (\beta_{11} + \beta_{8} - \beta_{7} + 2) q^{11} + (\beta_{4} + \beta_1) q^{13} + ( - \beta_{11} + \beta_{8} + \cdots + \beta_{6}) q^{17}+ \cdots + (3 \beta_{9} - 8 \beta_{5} + \cdots - 16 \beta_1) q^{97}+O(q^{100}) $ q - b6 * q^5 + (-b11 + b6 - 1) * q^7 + (b11 + b8 - b7 + 2) * q^11 + (b4 + b1) * q^13 + (-b11 + b8 - 2b7 + b6) q^17 + (b10 + b8 + b2 + b1 + 2) * q^19 + (b11 + 2b10 + 2b8 - 2b7 - 3b6 - 1) * q^23 + 5 * q^25 + (b9 - b4 - 2b3 - 2b2 + b1) * q^29 + (-b5 + b4 - b3 - b1) * q^31 + (-b10 + b8 + b6 - 4) * q^35 + (-b9 - 2b5 + b4 + 2b3 + 2b1) q^37 + (2b9 + b5 + b4 + b3 - 3b1) * q^41 + (2b11 - 2b10 + b8 + 2b7 + 2b6 - 15) * q^43 + (-5b8 - 2b7 + 2b6 + 7) q^47 + (b11 + 2b10 - 5b8 + 2b7 - 5b6 + 1) * q^49 + (2b9 - 2b5 + 4b3 + b1) q^53 + (-b11 + 2b10 + b8 - b7 - 2b6) * q^55 + (-b9 + 5b5 - b3 - 2b2) * q^59 + (b11 - 4b10 - b8 + b7 + 6b6 + 12) * q^61 + (-b5 - b4 - 2b3) q^65 + (2b9 + b4 - 2b3 - 4b2 + b1) q^67 + (-4b9 - b5 - 5b4 - b3 + 11b1) q^71 + (-5b11 - 2b10 + 3b8 - 5b6 - 14) * q^73 + (4b11 + 2b8 - 14b6 - 32) q^77 + (4b9 - 5b5 - b4 - 3b3 + 5b1) * q^79 + (8b11 + 2b10 + 3b8 + 6b7 - 14b6 + 19) q^83 + (b10 + 4b8 - b6 - 1) q^85 + (-10b5 - 4b2 + 8b1) q^89 + (3b9 + 9b5 - 4b4 + 3b3 - 12b1) q^91 + (b11 - 2b9 + 2b8 - 4b7 - b6 - b5 + b4 + b2 + 2b1 - 2) * q^95 + (3b9 - 8b5 - b4 - 4b2 - 16b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7}+O(q^{10}) $ 12 * q - 12 * q^7	$ 12 q - 12 q^{7} + 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 60 q^{25} - 40 q^{35} - 176 q^{43} + 72 q^{47} - 24 q^{49} + 152 q^{61} - 148 q^{73} - 376 q^{77} + 208 q^{83}+O(q^{100}) $ 12 * q - 12 * q^7 + 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 60 * q^25 - 40 * q^35 - 176 * q^43 + 72 * q^47 - 24 * q^49 + 152 * q^61 - 148 * q^73 - 376 * q^77 + 208 * q^83

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62*x^10 + 1445*x^8 + 15924*x^6 + 83244*x^4 + 170640*x^2 + 55600 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 59\nu^{11} + 5868\nu^{9} + 184075\nu^{7} + 2290126\nu^{5} + 10898236\nu^{3} + 13285720\nu ) / 407880 $ (59v^11 + 5868v^9 + 184075v^7 + 2290126v^5 + 10898236v^3 + 13285720v) / 407880
$\beta_{3}$	$=$	$ ( -239\nu^{11} - 15993\nu^{9} - 373210\nu^{7} - 3571816\nu^{5} - 13082536\nu^{3} - 14157280\nu ) / 407880 $ (-239v^11 - 15993v^9 - 373210v^7 - 3571816v^5 - 13082536v^3 - 14157280v) / 407880
$\beta_{4}$	$=$	$ ( -401\nu^{11} - 20007\nu^{9} - 344590\nu^{7} - 2594164\nu^{5} - 8665084\nu^{3} - 10740520\nu ) / 203940 $ (-401v^11 - 20007v^9 - 344590v^7 - 2594164v^5 - 8665084v^3 - 10740520v) / 203940
$\beta_{5}$	$=$	$ ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1502992\nu^{3} - 417160\nu ) / 37080 $ (-113v^11 - 5661v^9 - 95740v^7 - 650992v^5 - 1502992v^3 - 417160v) / 37080
$\beta_{6}$	$=$	$ ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1502992\nu^{2} - 417160 ) / 37080 $ (-113v^10 - 5661v^8 - 95740v^6 - 650992v^4 - 1502992*v^2 - 417160) / 37080
$\beta_{7}$	$=$	$ ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1465912\nu^{2} - 46360 ) / 37080 $ (-113v^10 - 5661v^8 - 95740v^6 - 650992v^4 - 1465912*v^2 - 46360) / 37080
$\beta_{8}$	$=$	$ ( 301\nu^{10} + 14382\nu^{8} + 228185\nu^{6} + 1462904\nu^{4} + 3463424\nu^{2} + 1849460 ) / 67980 $ (301v^10 + 14382v^8 + 228185v^6 + 1462904v^4 + 3463424*v^2 + 1849460) / 67980
$\beta_{9}$	$=$	$ ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1484452\nu^{3} - 139060\nu ) / 18540 $ (-113v^11 - 5661v^9 - 95740v^7 - 650992v^5 - 1484452v^3 - 139060v) / 18540
$\beta_{10}$	$=$	$ ( -361\nu^{10} - 17757\nu^{8} - 291230\nu^{6} - 1890134\nu^{4} - 4123544\nu^{2} - 1392200 ) / 67980 $ (-361v^10 - 17757v^8 - 291230v^6 - 1890134v^4 - 4123544*v^2 - 1392200) / 67980
$\beta_{11}$	$=$	$ ( 545\nu^{10} + 28107\nu^{8} + 495898\nu^{6} + 3578728\nu^{4} + 9148096\nu^{2} + 3524896 ) / 81576 $ (545v^10 + 28107v^8 + 495898v^6 + 3578728v^4 + 9148096*v^2 + 3524896) / 81576

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{7} - \beta_{6} - 10 $ b7 - b6 - 10
$\nu^{3}$	$=$	$ \beta_{9} - 2\beta_{5} - 15\beta_1 $ b9 - 2b5 - 15b1
$\nu^{4}$	$=$	$ 3\beta_{11} - 4\beta_{10} + \beta_{8} - 22\beta_{7} + 37\beta_{6} + 150 $ 3b11 - 4b10 + b8 - 22b7 + 37b6 + 150
$\nu^{5}$	$=$	$ -29\beta_{9} + 62\beta_{5} - 5\beta_{4} - 6\beta_{3} - 8\beta_{2} + 269\beta_1 $ -29b9 + 62b5 - 5b4 - 6b3 - 8b2 + 269b1
$\nu^{6}$	$=$	$ -93\beta_{11} + 158\beta_{10} - 15\beta_{8} + 470\beta_{7} - 971\beta_{6} - 2674 $ -93b11 + 158b10 - 15b8 + 470b7 - 971*b6 - 2674
$\nu^{7}$	$=$	$ 721\beta_{9} - 1584\beta_{5} + 173\beta_{4} + 236\beta_{3} + 316\beta_{2} - 5403\beta_1 $ 721b9 - 1584b5 + 173b4 + 236b3 + 316b2 - 5403b1
$\nu^{8}$	$=$	$ 2413\beta_{11} - 4526\beta_{10} + 91\beta_{8} - 10310\beta_{7} + 23619\beta_{6} + 53398 $ 2413b11 - 4526b10 + 91b8 - 10310b7 + 23619*b6 + 53398
$\nu^{9}$	$=$	$ -17249\beta_{9} + 38364\beta_{5} - 4617\beta_{4} - 6848\beta_{3} - 9052\beta_{2} + 116231\beta_1 $ -17249b9 + 38364b5 - 4617b4 - 6848b3 - 9052b2 + 116231b1
$\nu^{10}$	$=$	$ -59373\beta_{11} + 115918\beta_{10} + 2389\beta_{8} + 231734\beta_{7} - 560747\beta_{6} - 1144366 $ -59373b11 + 115918b10 + 2389b8 + 231734b7 - 560747*b6 - 1144366
$\nu^{11}$	$=$	$ 407025\beta_{9} - 910788\beta_{5} + 113529\beta_{4} + 177680\beta_{3} + 231836\beta_{2} - 2599023\beta_1 $ 407025b9 - 910788b5 + 113529b4 + 177680b3 + 231836b2 - 2599023b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

721.1

		0.630185i
	−	0.630185i
	−	4.83157i
		4.83157i
	−	3.42504i
		3.42504i
	−	3.14288i
		3.14288i
		3.53755i
	−	3.53755i
		2.03369i
	−	2.03369i

−2.23607

−3.98530

721.2

−2.23607

−3.98530

721.3

−2.23607

−3.72856

721.4

−2.23607

−3.72856

721.5

−2.23607

9.18599

721.6

−2.23607

9.18599

721.7

2.23607

−12.2192

721.8

2.23607

−12.2192

721.9

2.23607

0.468611

721.10

2.23607

0.468611

721.11

2.23607

4.27843

721.12

2.23607

4.27843

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3420.3.o.a		12
3.b	odd	2	1		380.3.e.a	✓	12
12.b	even	2	1		1520.3.h.b		12
15.d	odd	2	1		1900.3.e.f		12
15.e	even	4	2		1900.3.g.c		24
19.b	odd	2	1	inner	3420.3.o.a		12
57.d	even	2	1		380.3.e.a	✓	12
228.b	odd	2	1		1520.3.h.b		12
285.b	even	2	1		1900.3.e.f		12
285.j	odd	4	2		1900.3.g.c		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.e.a	✓	12	3.b	odd	2	1
380.3.e.a	✓	12	57.d	even	2	1
1520.3.h.b		12	12.b	even	2	1
1520.3.h.b		12	228.b	odd	2	1
1900.3.e.f		12	15.d	odd	2	1
1900.3.e.f		12	285.b	even	2	1
1900.3.g.c		24	15.e	even	4	2
1900.3.g.c		24	285.j	odd	4	2
3420.3.o.a		12	1.a	even	1	1	trivial
3420.3.o.a		12	19.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{6} + 6T_{7}^{5} - 123T_{7}^{4} - 448T_{7}^{3} + 2080T_{7}^{2} + 6272T_{7} - 3344 $ T7^6 + 6*T7^5 - 123*T7^4 - 448*T7^3 + 2080*T7^2 + 6272*T7 - 3344 acting on $S_{3}^{\mathrm{new}}(3420, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ (T^{2} - 5)^{6} $ (T^2 - 5)^6
$7$	$ (T^{6} + 6 T^{5} + \cdots - 3344)^{2} $ (T^6 + 6T^5 - 123T^4 - 448T^3 + 2080T^2 + 6272*T - 3344)^2
$11$	$ (T^{6} - 16 T^{5} + \cdots - 43264)^{2} $ (T^6 - 16T^5 - 164T^4 + 2656T^3 + 5360T^2 - 65728*T - 43264)^2
$13$	$ T^{12} + \cdots + 18127879600 $ T^12 + 1174T^10 + 454605T^8 + 67065412T^6 + 3704449004T^4 + 46566284240*T^2 + 18127879600
$17$	$ (T^{6} - 6 T^{5} + \cdots + 818576)^{2} $ (T^6 - 6T^5 - 703T^4 + 3968T^3 + 111560T^2 - 680352*T + 818576)^2
$19$	$ T^{12} + \cdots + 22\!\cdots\!61 $ T^12 - 24T^11 + 718T^10 - 11448T^9 + 312303T^8 - 3274992T^7 + 69221028T^6 - 1182272112T^5 + 40699639263T^4 - 538581245688T^3 + 12194198263438T^2 - 147145590187224*T + 2213314919066161
$23$	$ (T^{6} + 2 T^{5} + \cdots - 752400)^{2} $ (T^6 + 2T^5 - 1379T^4 - 18400T^3 + 62080T^2 + 748800*T - 752400)^2
$29$	$ T^{12} + \cdots + 24\!\cdots\!00 $ T^12 + 6442T^10 + 14527705T^8 + 13344932784T^6 + 4466410158784T^4 + 577751990604800*T^2 + 24578837040640000
$31$	$ T^{12} + \cdots + 119374584217600 $ T^12 + 3904T^10 + 4308160T^8 + 1469386752T^6 + 189626036224T^4 + 9209673482240*T^2 + 119374584217600
$37$	$ T^{12} + \cdots + 25\!\cdots\!00 $ T^12 + 9204T^10 + 24619340T^8 + 16197016912T^6 + 3535386239344T^4 + 179296520786560*T^2 + 2556834339961600
$41$	$ T^{12} + \cdots + 12\!\cdots\!00 $ T^12 + 7120T^10 + 16359424T^8 + 15782277120T^6 + 7218693013504T^4 + 1540001874575360*T^2 + 122746616519065600
$43$	$ (T^{6} + 88 T^{5} + \cdots - 2451136)^{2} $ (T^6 + 88T^5 + 460T^4 - 110560T^3 - 2018960T^2 + 4996608*T - 2451136)^2
$47$	$ (T^{6} - 36 T^{5} + \cdots - 9167899456)^{2} $ (T^6 - 36T^5 - 7500T^4 + 188128T^3 + 15902576T^2 - 205846592*T - 9167899456)^2
$53$	$ T^{12} + \cdots + 31\!\cdots\!00 $ T^12 + 28390T^10 + 308301741T^8 + 1637397541300T^6 + 4516584793096844T^4 + 6145069971405169040*T^2 + 3193105851428688103600
$59$	$ T^{12} + \cdots + 492787956121600 $ T^12 + 17122T^10 + 81450305T^8 + 123820877984T^6 + 56540303646464T^4 + 7814344462704640*T^2 + 492787956121600
$61$	$ (T^{6} - 76 T^{5} + \cdots - 1301500864)^{2} $ (T^6 - 76T^5 - 2952T^4 + 189280T^3 + 3171968T^2 - 116799296*T - 1301500864)^2
$67$	$ T^{12} + \cdots + 39\!\cdots\!00 $ T^12 + 19710T^10 + 112421269T^8 + 153163111300T^6 + 76653074089804T^4 + 13255845003660240*T^2 + 392055999889143600
$71$	$ T^{12} + \cdots + 33\!\cdots\!00 $ T^12 + 52160T^10 + 1014731456T^8 + 8826848235520T^6 + 30856628712812544T^4 + 20557086414940733440*T^2 + 3395808717756144025600
$73$	$ (T^{6} + 74 T^{5} + \cdots + 5637612816)^{2} $ (T^6 + 74T^5 - 7103T^4 - 404672T^3 + 5561000T^2 + 490085088*T + 5637612816)^2
$79$	$ T^{12} + \cdots + 36\!\cdots\!00 $ T^12 + 49000T^10 + 924364304T^8 + 8400762836480T^6 + 37408890436358144T^4 + 72155660704890880000*T^2 + 36124063833702400000000
$83$	$ (T^{6} - 104 T^{5} + \cdots - 2227337984)^{2} $ (T^6 - 104T^5 - 12064T^4 + 941184T^3 + 37448240T^2 - 1670714112*T - 2227337984)^2
$89$	$ T^{12} + \cdots + 10\!\cdots\!00 $ T^12 + 42776T^10 + 630894800T^8 + 4018121756928T^6 + 10690415477484544T^4 + 8824799886414397440*T^2 + 1073529535279823257600
$97$	$ T^{12} + \cdots + 23\!\cdots\!00 $ T^12 + 55476T^10 + 1027748300T^8 + 7602023198288T^6 + 20739756167840624T^4 + 13144169535713183360*T^2 + 2384869939909857913600
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Overview	Random
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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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3420.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7}+O(q^{10}) \) q - b6 * q^5 + (-b11 + b6 - 1) * q^7	\( q - \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7} + (\beta_{11} + \beta_{8} - \beta_{7} + 2) q^{11} + (\beta_{4} + \beta_1) q^{13} + ( - \beta_{11} + \beta_{8} + \cdots + \beta_{6}) q^{17}+ \cdots + (3 \beta_{9} - 8 \beta_{5} + \cdots - 16 \beta_1) q^{97}+O(q^{100}) \) q - b6 * q^5 + (-b11 + b6 - 1) * q^7 + (b11 + b8 - b7 + 2) * q^11 + (b4 + b1) * q^13 + (-b11 + b8 - 2b7 + b6) q^17 + (b10 + b8 + b2 + b1 + 2) * q^19 + (b11 + 2b10 + 2b8 - 2b7 - 3b6 - 1) * q^23 + 5 * q^25 + (b9 - b4 - 2b3 - 2b2 + b1) * q^29 + (-b5 + b4 - b3 - b1) * q^31 + (-b10 + b8 + b6 - 4) * q^35 + (-b9 - 2b5 + b4 + 2b3 + 2b1) q^37 + (2b9 + b5 + b4 + b3 - 3b1) * q^41 + (2b11 - 2b10 + b8 + 2b7 + 2b6 - 15) * q^43 + (-5b8 - 2b7 + 2b6 + 7) q^47 + (b11 + 2b10 - 5b8 + 2b7 - 5b6 + 1) * q^49 + (2b9 - 2b5 + 4b3 + b1) q^53 + (-b11 + 2b10 + b8 - b7 - 2b6) * q^55 + (-b9 + 5b5 - b3 - 2b2) * q^59 + (b11 - 4b10 - b8 + b7 + 6b6 + 12) * q^61 + (-b5 - b4 - 2b3) q^65 + (2b9 + b4 - 2b3 - 4b2 + b1) q^67 + (-4b9 - b5 - 5b4 - b3 + 11b1) q^71 + (-5b11 - 2b10 + 3b8 - 5b6 - 14) * q^73 + (4b11 + 2b8 - 14b6 - 32) q^77 + (4b9 - 5b5 - b4 - 3b3 + 5b1) * q^79 + (8b11 + 2b10 + 3b8 + 6b7 - 14b6 + 19) q^83 + (b10 + 4b8 - b6 - 1) q^85 + (-10b5 - 4b2 + 8b1) q^89 + (3b9 + 9b5 - 4b4 + 3b3 - 12b1) q^91 + (b11 - 2b9 + 2b8 - 4b7 - b6 - b5 + b4 + b2 + 2b1 - 2) * q^95 + (3b9 - 8b5 - b4 - 4b2 - 16b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7}+O(q^{10}) \) 12 * q - 12 * q^7	\( 12 q - 12 q^{7} + 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 60 q^{25} - 40 q^{35} - 176 q^{43} + 72 q^{47} - 24 q^{49} + 152 q^{61} - 148 q^{73} - 376 q^{77} + 208 q^{83}+O(q^{100}) \) 12 * q - 12 * q^7 + 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 60 * q^25 - 40 * q^35 - 176 * q^43 + 72 * q^47 - 24 * q^49 + 152 * q^61 - 148 * q^73 - 376 * q^77 + 208 * q^83

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	3420.3.o.a

Level	$3420$
Weight	$3$
Character orbit	3420.o
Analytic conductor	$93.188$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(93.1882504112\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 \) x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{9}\cdot 5 \)
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 59\nu^{11} + 5868\nu^{9} + 184075\nu^{7} + 2290126\nu^{5} + 10898236\nu^{3} + 13285720\nu ) / 407880 \) (59v^11 + 5868v^9 + 184075v^7 + 2290126v^5 + 10898236v^3 + 13285720v) / 407880
\(\beta_{3}\)	\(=\)	\( ( -239\nu^{11} - 15993\nu^{9} - 373210\nu^{7} - 3571816\nu^{5} - 13082536\nu^{3} - 14157280\nu ) / 407880 \) (-239v^11 - 15993v^9 - 373210v^7 - 3571816v^5 - 13082536v^3 - 14157280v) / 407880
\(\beta_{4}\)	\(=\)	\( ( -401\nu^{11} - 20007\nu^{9} - 344590\nu^{7} - 2594164\nu^{5} - 8665084\nu^{3} - 10740520\nu ) / 203940 \) (-401v^11 - 20007v^9 - 344590v^7 - 2594164v^5 - 8665084v^3 - 10740520v) / 203940
\(\beta_{5}\)	\(=\)	\( ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1502992\nu^{3} - 417160\nu ) / 37080 \) (-113v^11 - 5661v^9 - 95740v^7 - 650992v^5 - 1502992v^3 - 417160v) / 37080
\(\beta_{6}\)	\(=\)	\( ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1502992\nu^{2} - 417160 ) / 37080 \) (-113v^10 - 5661v^8 - 95740v^6 - 650992v^4 - 1502992*v^2 - 417160) / 37080
\(\beta_{7}\)	\(=\)	\( ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1465912\nu^{2} - 46360 ) / 37080 \) (-113v^10 - 5661v^8 - 95740v^6 - 650992v^4 - 1465912*v^2 - 46360) / 37080
\(\beta_{8}\)	\(=\)	\( ( 301\nu^{10} + 14382\nu^{8} + 228185\nu^{6} + 1462904\nu^{4} + 3463424\nu^{2} + 1849460 ) / 67980 \) (301v^10 + 14382v^8 + 228185v^6 + 1462904v^4 + 3463424*v^2 + 1849460) / 67980
\(\beta_{9}\)	\(=\)	\( ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1484452\nu^{3} - 139060\nu ) / 18540 \) (-113v^11 - 5661v^9 - 95740v^7 - 650992v^5 - 1484452v^3 - 139060v) / 18540
\(\beta_{10}\)	\(=\)	\( ( -361\nu^{10} - 17757\nu^{8} - 291230\nu^{6} - 1890134\nu^{4} - 4123544\nu^{2} - 1392200 ) / 67980 \) (-361v^10 - 17757v^8 - 291230v^6 - 1890134v^4 - 4123544*v^2 - 1392200) / 67980
\(\beta_{11}\)	\(=\)	\( ( 545\nu^{10} + 28107\nu^{8} + 495898\nu^{6} + 3578728\nu^{4} + 9148096\nu^{2} + 3524896 ) / 81576 \) (545v^10 + 28107v^8 + 495898v^6 + 3578728v^4 + 9148096*v^2 + 3524896) / 81576

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{7} - \beta_{6} - 10 \) b7 - b6 - 10
\(\nu^{3}\)	\(=\)	\( \beta_{9} - 2\beta_{5} - 15\beta_1 \) b9 - 2b5 - 15b1
\(\nu^{4}\)	\(=\)	\( 3\beta_{11} - 4\beta_{10} + \beta_{8} - 22\beta_{7} + 37\beta_{6} + 150 \) 3b11 - 4b10 + b8 - 22b7 + 37b6 + 150
\(\nu^{5}\)	\(=\)	\( -29\beta_{9} + 62\beta_{5} - 5\beta_{4} - 6\beta_{3} - 8\beta_{2} + 269\beta_1 \) -29b9 + 62b5 - 5b4 - 6b3 - 8b2 + 269b1
\(\nu^{6}\)	\(=\)	\( -93\beta_{11} + 158\beta_{10} - 15\beta_{8} + 470\beta_{7} - 971\beta_{6} - 2674 \) -93b11 + 158b10 - 15b8 + 470b7 - 971*b6 - 2674
\(\nu^{7}\)	\(=\)	\( 721\beta_{9} - 1584\beta_{5} + 173\beta_{4} + 236\beta_{3} + 316\beta_{2} - 5403\beta_1 \) 721b9 - 1584b5 + 173b4 + 236b3 + 316b2 - 5403b1
\(\nu^{8}\)	\(=\)	\( 2413\beta_{11} - 4526\beta_{10} + 91\beta_{8} - 10310\beta_{7} + 23619\beta_{6} + 53398 \) 2413b11 - 4526b10 + 91b8 - 10310b7 + 23619*b6 + 53398
\(\nu^{9}\)	\(=\)	\( -17249\beta_{9} + 38364\beta_{5} - 4617\beta_{4} - 6848\beta_{3} - 9052\beta_{2} + 116231\beta_1 \) -17249b9 + 38364b5 - 4617b4 - 6848b3 - 9052b2 + 116231b1
\(\nu^{10}\)	\(=\)	\( -59373\beta_{11} + 115918\beta_{10} + 2389\beta_{8} + 231734\beta_{7} - 560747\beta_{6} - 1144366 \) -59373b11 + 115918b10 + 2389b8 + 231734b7 - 560747*b6 - 1144366
\(\nu^{11}\)	\(=\)	\( 407025\beta_{9} - 910788\beta_{5} + 113529\beta_{4} + 177680\beta_{3} + 231836\beta_{2} - 2599023\beta_1 \) 407025b9 - 910788b5 + 113529b4 + 177680b3 + 231836b2 - 2599023b1

\(n\)	\(1711\)	\(1901\)	\(2737\)	\(3061\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 721.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( (T^{2} - 5)^{6} \) (T^2 - 5)^6
$7$	\( (T^{6} + 6 T^{5} + \cdots - 3344)^{2} \) (T^6 + 6T^5 - 123T^4 - 448T^3 + 2080T^2 + 6272*T - 3344)^2
$11$	\( (T^{6} - 16 T^{5} + \cdots - 43264)^{2} \) (T^6 - 16T^5 - 164T^4 + 2656T^3 + 5360T^2 - 65728*T - 43264)^2
$13$	\( T^{12} + \cdots + 18127879600 \) T^12 + 1174T^10 + 454605T^8 + 67065412T^6 + 3704449004T^4 + 46566284240*T^2 + 18127879600
$17$	\( (T^{6} - 6 T^{5} + \cdots + 818576)^{2} \) (T^6 - 6T^5 - 703T^4 + 3968T^3 + 111560T^2 - 680352*T + 818576)^2
$19$	\( T^{12} + \cdots + 22\!\cdots\!61 \) T^12 - 24T^11 + 718T^10 - 11448T^9 + 312303T^8 - 3274992T^7 + 69221028T^6 - 1182272112T^5 + 40699639263T^4 - 538581245688T^3 + 12194198263438T^2 - 147145590187224*T + 2213314919066161
$23$	\( (T^{6} + 2 T^{5} + \cdots - 752400)^{2} \) (T^6 + 2T^5 - 1379T^4 - 18400T^3 + 62080T^2 + 748800*T - 752400)^2
$29$	\( T^{12} + \cdots + 24\!\cdots\!00 \) T^12 + 6442T^10 + 14527705T^8 + 13344932784T^6 + 4466410158784T^4 + 577751990604800*T^2 + 24578837040640000
$31$	\( T^{12} + \cdots + 119374584217600 \) T^12 + 3904T^10 + 4308160T^8 + 1469386752T^6 + 189626036224T^4 + 9209673482240*T^2 + 119374584217600
$37$	\( T^{12} + \cdots + 25\!\cdots\!00 \) T^12 + 9204T^10 + 24619340T^8 + 16197016912T^6 + 3535386239344T^4 + 179296520786560*T^2 + 2556834339961600
$41$	\( T^{12} + \cdots + 12\!\cdots\!00 \) T^12 + 7120T^10 + 16359424T^8 + 15782277120T^6 + 7218693013504T^4 + 1540001874575360*T^2 + 122746616519065600
$43$	\( (T^{6} + 88 T^{5} + \cdots - 2451136)^{2} \) (T^6 + 88T^5 + 460T^4 - 110560T^3 - 2018960T^2 + 4996608*T - 2451136)^2
$47$	\( (T^{6} - 36 T^{5} + \cdots - 9167899456)^{2} \) (T^6 - 36T^5 - 7500T^4 + 188128T^3 + 15902576T^2 - 205846592*T - 9167899456)^2
$53$	\( T^{12} + \cdots + 31\!\cdots\!00 \) T^12 + 28390T^10 + 308301741T^8 + 1637397541300T^6 + 4516584793096844T^4 + 6145069971405169040*T^2 + 3193105851428688103600
$59$	\( T^{12} + \cdots + 492787956121600 \) T^12 + 17122T^10 + 81450305T^8 + 123820877984T^6 + 56540303646464T^4 + 7814344462704640*T^2 + 492787956121600
$61$	\( (T^{6} - 76 T^{5} + \cdots - 1301500864)^{2} \) (T^6 - 76T^5 - 2952T^4 + 189280T^3 + 3171968T^2 - 116799296*T - 1301500864)^2
$67$	\( T^{12} + \cdots + 39\!\cdots\!00 \) T^12 + 19710T^10 + 112421269T^8 + 153163111300T^6 + 76653074089804T^4 + 13255845003660240*T^2 + 392055999889143600
$71$	\( T^{12} + \cdots + 33\!\cdots\!00 \) T^12 + 52160T^10 + 1014731456T^8 + 8826848235520T^6 + 30856628712812544T^4 + 20557086414940733440*T^2 + 3395808717756144025600
$73$	\( (T^{6} + 74 T^{5} + \cdots + 5637612816)^{2} \) (T^6 + 74T^5 - 7103T^4 - 404672T^3 + 5561000T^2 + 490085088*T + 5637612816)^2
$79$	\( T^{12} + \cdots + 36\!\cdots\!00 \) T^12 + 49000T^10 + 924364304T^8 + 8400762836480T^6 + 37408890436358144T^4 + 72155660704890880000*T^2 + 36124063833702400000000
$83$	\( (T^{6} - 104 T^{5} + \cdots - 2227337984)^{2} \) (T^6 - 104T^5 - 12064T^4 + 941184T^3 + 37448240T^2 - 1670714112*T - 2227337984)^2
$89$	\( T^{12} + \cdots + 10\!\cdots\!00 \) T^12 + 42776T^10 + 630894800T^8 + 4018121756928T^6 + 10690415477484544T^4 + 8824799886414397440*T^2 + 1073529535279823257600
$97$	\( T^{12} + \cdots + 23\!\cdots\!00 \) T^12 + 55476T^10 + 1027748300T^8 + 7602023198288T^6 + 20739756167840624T^4 + 13144169535713183360*T^2 + 2384869939909857913600
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