LMFDB - Newform orbit 444.2.m.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [444,2,Mod(401,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.401"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 444 = 2^{2} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	444.m (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.54535784974$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
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} - 24x^{10} + 239x^{8} - 1116x^{6} + 2027x^{4} + 708x^{2} + 1369 $ x^12 - 24x^10 + 239x^8 - 1116x^6 + 2027x^4 + 708*x^2 + 1369
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{7} + \beta_1) q^{3} - \beta_{11} q^{5} + ( - \beta_{2} - 1) q^{7} + (2 \beta_{8} + 1) q^{9} + ( - \beta_{9} + \beta_{7}) q^{11} + ( - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{6} + \beta_{4}) q^{15}+ \cdots + ( - \beta_{9} + \beta_{7} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) $ q + (-b7 + b1) * q^3 - b11 * q^5 + (-b2 - 1) * q^7 + (2b8 + 1) q^9 + (-b9 + b7) * q^11 + (-b3 + b2) * q^13 + (-b6 + b4) * q^15 + (-b11 + b8 + b7) * q^17 + (-b5 + b1 + 1) * q^19 + (-b9 + b7 + b3 - b1) * q^21 + (b11 - b10 + b9 + b8 + b7) * q^23 + (-b5 + b4 - b3 - 4b1) q^25 + (b7 + 5b1) q^27 + (b8 - b7 - b6) * q^29 + (-b4 - 3b1 + 3) q^31 + (-b10 - b8 - 2b2 - 2) q^33 + (-b10 + b9) * q^35 + (-b4 + b3 + b1 + 3) * q^37 + (b10 + b9 - b3 - b2) * q^39 + (-b11 - b9 - b7 - b6) * q^41 + (b5 - b3 + b2) * q^43 + (-b11 + 2b5) q^45 + (b11 + b10 - b8 - b6) * q^47 + (-b5 - b4 + 2b2 + 3) q^49 + (-b8 + b7 - b6 + b4 + 2b1 - 2) q^51 + (b10 - b8) * q^53 + (2b3 + 2b2) * q^55 + (b8 - b7 + 2b6 + b4 + b1 - 1) q^57 + (b11 + b10 - b9) * q^59 + (-2b4 - 2b3 - 2b2 + 3b1 - 3) * q^61 + (-2b10 - 2b8 - b2 - 1) * q^63 + (b11 - 2b9 + b6) q^65 + (-b5 + b4 + b3 - 3b1) q^67 + (b10 + b9 - b8 + b7 + b6 - b4 + 2b3 + 2b2 + 2b1 - 2) q^69 + (-b11 - b10 - b8 + b6) * q^71 + (-b5 + b4 - b3 + b1) * q^73 + (-2b11 + b10 - 4b8 + 2b6 + b5 + b4 - b2 + 4) q^75 + (2b11 + 2b9 - 10b7 + 2b6) * q^77 + (b5 - b1 - 1) * q^79 + (4b8 - 7) q^81 + (b11 + 2b10 + 4b8 - b6) * q^83 + (-b3 - 9b1) q^85 + (b11 + b8 + b7 + b5 + 2b1 + 2) q^87 + (b10 + b9 + b8 - b7 + 3b6) q^89 + (2b5 + b3 - b2 - 9b1 - 9) * q^91 + (2b11 - 3b8 - 3b7 - b5 + 3b1 + 3) * q^93 + (b11 + b9 + 9b7 + b6) q^95 + (b3 - b2 - 6b1 - 6) q^97 + (-b9 + b7 + 4b3 - 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7} + 12 q^{9} - 4 q^{13} + 12 q^{19} + 36 q^{31} - 16 q^{33} + 36 q^{37} + 4 q^{39} - 4 q^{43} + 28 q^{49} - 24 q^{51} - 8 q^{55} - 12 q^{57} - 28 q^{61} - 8 q^{63} - 32 q^{69} + 52 q^{75} - 12 q^{79}+ \cdots - 68 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 + 12 * q^9 - 4 * q^13 + 12 * q^19 + 36 * q^31 - 16 * q^33 + 36 * q^37 + 4 * q^39 - 4 * q^43 + 28 * q^49 - 24 * q^51 - 8 * q^55 - 12 * q^57 - 28 * q^61 - 8 * q^63 - 32 * q^69 + 52 * q^75 - 12 * q^79 - 84 * q^81 + 24 * q^87 - 104 * q^91 + 36 * q^93 - 68 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 24x^{10} + 239x^{8} - 1116x^{6} + 2027x^{4} + 708x^{2} + 1369 $ x^12 - 24*x^10 + 239*x^8 - 1116*x^6 + 2027*x^4 + 708*x^2 + 1369 :

$\beta_{1}$	$=$	$ ( -4\nu^{10} + 59\nu^{8} - 150\nu^{6} - 1608\nu^{4} + 8248\nu^{2} + 1633 ) / 6246 $ (-4v^10 + 59v^8 - 150v^6 - 1608v^4 + 8248*v^2 + 1633) / 6246
$\beta_{2}$	$=$	$ ( 67\nu^{10} - 34\nu^{8} - 6336\nu^{6} + 38385\nu^{4} + 19037\nu^{2} - 29348 ) / 75993 $ (67v^10 - 34v^8 - 6336v^6 + 38385v^4 + 19037*v^2 - 29348) / 75993
$\beta_{3}$	$=$	$ ( 178\nu^{10} - 3493\nu^{8} + 28536\nu^{6} - 90840\nu^{4} + 111824\nu^{2} - 15587 ) / 151986 $ (178v^10 - 3493v^8 + 28536v^6 - 90840v^4 + 111824*v^2 - 15587) / 151986
$\beta_{4}$	$=$	$ ( -467\nu^{10} + 10445\nu^{8} - 91944\nu^{6} + 386898\nu^{4} - 772393\nu^{2} + 625357 ) / 151986 $ (-467v^10 + 10445v^8 - 91944v^6 + 386898v^4 - 772393*v^2 + 625357) / 151986
$\beta_{5}$	$=$	$ ( 823\nu^{10} - 17431\nu^{8} + 149016\nu^{6} - 568578\nu^{4} + 692069\nu^{2} + 559357 ) / 151986 $ (823v^10 - 17431v^8 + 149016v^6 - 568578v^4 + 692069*v^2 + 559357) / 151986
$\beta_{6}$	$=$	$ ( 2057\nu^{11} - 27131\nu^{9} + 102642\nu^{7} + 637452\nu^{5} - 7305197\nu^{3} + 25820153\nu ) / 5623482 $ (2057v^11 - 27131v^9 + 102642v^7 + 637452v^5 - 7305197v^3 + 25820153v) / 5623482
$\beta_{7}$	$=$	$ ( -23\nu^{11} + 478\nu^{9} - 5016\nu^{7} + 28554\nu^{5} - 83695\nu^{3} + 35738\nu ) / 48618 $ (-23v^11 + 478v^9 - 5016v^7 + 28554v^5 - 83695v^3 + 35738v) / 48618
$\beta_{8}$	$=$	$ ( -4529\nu^{11} + 102110\nu^{9} - 953190\nu^{7} + 3998532\nu^{5} - 5819203\nu^{3} - 7344020\nu ) / 5623482 $ (-4529v^11 + 102110v^9 - 953190v^7 + 3998532v^5 - 5819203v^3 - 7344020v) / 5623482
$\beta_{9}$	$=$	$ ( -4529\nu^{11} + 102110\nu^{9} - 953190\nu^{7} + 3998532\nu^{5} - 5819203\nu^{3} + 3902944\nu ) / 5623482 $ (-4529v^11 + 102110v^9 - 953190v^7 + 3998532v^5 - 5819203v^3 + 3902944v) / 5623482
$\beta_{10}$	$=$	$ ( -9863\nu^{11} + 161528\nu^{9} - 850284\nu^{7} + 407274\nu^{5} + 5171177\nu^{3} + 7074184\nu ) / 5623482 $ (-9863v^11 + 161528v^9 - 850284v^7 + 407274v^5 + 5171177v^3 + 7074184v) / 5623482
$\beta_{11}$	$=$	$ ( -7001\nu^{11} + 177089\nu^{9} - 1803738\nu^{7} + 8634516\nu^{5} - 16131862\nu^{3} - 2926592\nu ) / 2811741 $ (-7001v^11 + 177089v^9 - 1803738v^7 + 8634516v^5 - 16131862v^3 - 2926592v) / 2811741

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

$\nu$	$=$	$ ( \beta_{9} - \beta_{8} ) / 2 $ (b9 - b8) / 2
$\nu^{2}$	$=$	$ ( -\beta_{5} - \beta_{4} + 2\beta_{3} + 8 ) / 2 $ (-b5 - b4 + 2*b3 + 8) / 2
$\nu^{3}$	$=$	$ ( 2\beta_{11} + 5\beta_{9} - 13\beta_{8} - 4\beta_{6} ) / 2 $ (2b11 + 5b9 - 13b8 - 4b6) / 2
$\nu^{4}$	$=$	$ -3\beta_{5} - \beta_{4} + 12\beta_{3} - \beta_{2} + 16 $ -3b5 - b4 + 12b3 - b2 + 16
$\nu^{5}$	$=$	$ ( 26\beta_{11} + 5\beta_{10} + 12\beta_{9} - 124\beta_{8} + 9\beta_{7} - 34\beta_{6} ) / 2 $ (26b11 + 5b10 + 12b9 - 124b8 + 9b7 - 34b6) / 2
$\nu^{6}$	$=$	$ ( -21\beta_{5} + 23\beta_{4} + 196\beta_{3} - 12\beta_{2} + 54\beta _1 - 16 ) / 2 $ (-21b5 + 23b4 + 196b3 - 12b2 + 54*b1 - 16) / 2
$\nu^{7}$	$=$	$ ( 228\beta_{11} + 77\beta_{10} - 108\beta_{9} - 874\beta_{8} + 27\beta_{7} - 206\beta_{6} ) / 2 $ (228b11 + 77b10 - 108b9 - 874b8 + 27b7 - 206b6) / 2
$\nu^{8}$	$=$	$ 21\beta_{5} + 201\beta_{4} + 600\beta_{3} + 36\beta_{2} + 360\beta _1 - 923 $ 21b5 + 201b4 + 600b3 + 36b2 + 360*b1 - 923
$\nu^{9}$	$=$	$ ( 1632\beta_{11} + 684\beta_{10} - 1967\beta_{9} - 4477\beta_{8} - 684\beta_{7} - 684\beta_{6} ) / 2 $ (1632b11 + 684b10 - 1967b9 - 4477b8 - 684b7 - 684b6) / 2
$\nu^{10}$	$=$	$ ( 1757\beta_{5} + 3809\beta_{4} + 4826\beta_{3} + 2316\beta_{2} + 5472\beta _1 - 22180 ) / 2 $ (1757b5 + 3809b4 + 4826b3 + 2316b2 + 5472*b1 - 22180) / 2
$\nu^{11}$	$=$	$ ( 9194\beta_{11} + 3630\beta_{10} - 19069\beta_{9} - 10627\beta_{8} - 13158\beta_{7} + 3056\beta_{6} ) / 2 $ (9194b11 + 3630b10 - 19069b9 - 10627b8 - 13158b7 + 3056b6) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/444\mathbb{Z}\right)^\times$.

$n$	$149$	$223$	$409$
$\chi(n)$	$-1$	$1$	$-\beta_{1}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

401.1

0.481488	−	0.707107i
−2.46039	−	0.707107i
2.68601	−	0.707107i
−2.68601	+	0.707107i
2.46039	+	0.707107i
−0.481488	+	0.707107i
0.481488	+	0.707107i
−2.46039	+	0.707107i
2.68601	+	0.707107i
−2.68601	−	0.707107i
2.46039	−	0.707107i
−0.481488	−	0.707107i

−1.41421

−

1.00000i

−3.01805

−

3.01805i

−0.319072

1.00000

2.82843i

401.2

−1.41421

−

1.00000i

1.09851

1.09851i

−4.47952

1.00000

2.82843i

401.3

−1.41421

−

1.00000i

1.91954

1.91954i

2.79859

1.00000

2.82843i

401.4

1.41421

−

1.00000i

−1.91954

−

1.91954i

2.79859

1.00000

−

2.82843i

401.5

1.41421

−

1.00000i

−1.09851

−

1.09851i

−4.47952

1.00000

−

2.82843i

401.6

1.41421

−

1.00000i

3.01805

3.01805i

−0.319072

1.00000

−

2.82843i

413.1

−1.41421

1.00000i

−3.01805

3.01805i

−0.319072

1.00000

−

2.82843i

413.2

−1.41421

1.00000i

1.09851

−

1.09851i

−4.47952

1.00000

−

2.82843i

413.3

−1.41421

1.00000i

1.91954

−

1.91954i

2.79859

1.00000

−

2.82843i

413.4

1.41421

1.00000i

−1.91954

1.91954i

2.79859

1.00000

2.82843i

413.5

1.41421

1.00000i

−1.09851

1.09851i

−4.47952

1.00000

2.82843i

413.6

1.41421

1.00000i

3.01805

−

3.01805i

−0.319072

1.00000

2.82843i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
37.d	odd	4	1	inner
111.g	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	444.2.m.d	✓	12
3.b	odd	2	1	inner	444.2.m.d	✓	12
37.d	odd	4	1	inner	444.2.m.d	✓	12
111.g	even	4	1	inner	444.2.m.d	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
444.2.m.d	✓	12	1.a	even	1	1	trivial
444.2.m.d	✓	12	3.b	odd	2	1	inner
444.2.m.d	✓	12	37.d	odd	4	1	inner
444.2.m.d	✓	12	111.g	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(444, [\chi])$:

$ T_{5}^{12} + 392T_{5}^{8} + 20272T_{5}^{4} + 104976 $

T5^12 + 392*T5^8 + 20272*T5^4 + 104976

$ T_{7}^{3} + 2T_{7}^{2} - 12T_{7} - 4 $

T7^3 + 2*T7^2 - 12*T7 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{4} - 2 T^{2} + 9)^{3} $ (T^4 - 2*T^2 + 9)^3
$5$	$ T^{12} + 392 T^{8} + \cdots + 104976 $ T^12 + 392T^8 + 20272T^4 + 104976
$7$	$ (T^{3} + 2 T^{2} - 12 T - 4)^{4} $ (T^3 + 2T^2 - 12T - 4)^4
$11$	$ (T^{6} - 56 T^{4} + \cdots - 128)^{2} $ (T^6 - 56T^4 + 640T^2 - 128)^2
$13$	$ (T^{6} + 2 T^{5} + \cdots + 648)^{2} $ (T^6 + 2T^5 + 2T^4 - 16T^3 + 676T^2 + 936*T + 648)^2
$17$	$ T^{12} + 680 T^{8} + \cdots + 2085136 $ T^12 + 680T^8 + 96080T^4 + 2085136
$19$	$ (T^{6} - 6 T^{5} + \cdots + 7688)^{2} $ (T^6 - 6T^5 + 18T^4 + 8T^3 + 484T^2 - 2728*T + 7688)^2
$23$	$ T^{12} + \cdots + 13680577296 $ T^12 + 7432T^8 + 17738352T^4 + 13680577296
$29$	$ T^{12} + 1544 T^{8} + \cdots + 16 $ T^12 + 1544T^8 + 464T^4 + 16
$31$	$ (T^{6} - 18 T^{5} + \cdots + 8712)^{2} $ (T^6 - 18T^5 + 162T^4 - 600T^3 + 676T^2 + 3432*T + 8712)^2
$37$	$ (T^{6} - 18 T^{5} + \cdots + 50653)^{2} $ (T^6 - 18T^5 + 163T^4 - 1068T^3 + 6031T^2 - 24642*T + 50653)^2
$41$	$ (T^{6} - 112 T^{4} + \cdots - 20000)^{2} $ (T^6 - 112T^4 + 3200T^2 - 20000)^2
$43$	$ (T^{6} + 2 T^{5} + \cdots + 16200)^{2} $ (T^6 + 2T^5 + 2T^4 - 296T^3 + 3364T^2 - 10440*T + 16200)^2
$47$	$ (T^{6} + 128 T^{4} + \cdots + 2592)^{2} $ (T^6 + 128T^4 + 1728T^2 + 2592)^2
$53$	$ (T^{6} + 64 T^{4} + \cdots + 3200)^{2} $ (T^6 + 64T^4 + 896T^2 + 3200)^2
$59$	$ T^{12} + \cdots + 688747536 $ T^12 + 12232T^8 + 10521232T^4 + 688747536
$61$	$ (T^{6} + 14 T^{5} + \cdots + 3908808)^{2} $ (T^6 + 14T^5 + 98T^4 - 32T^3 + 40804T^2 + 564792*T + 3908808)^2
$67$	$ (T^{6} + 168 T^{4} + \cdots + 51984)^{2} $ (T^6 + 168T^4 + 6352T^2 + 51984)^2
$71$	$ (T^{6} + 120 T^{4} + \cdots + 43808)^{2} $ (T^6 + 120T^4 + 4320T^2 + 43808)^2
$73$	$ (T^{6} + 132 T^{4} + \cdots + 26896)^{2} $ (T^6 + 132T^4 + 4752T^2 + 26896)^2
$79$	$ (T^{6} + 6 T^{5} + \cdots + 7688)^{2} $ (T^6 + 6T^5 + 18T^4 - 8T^3 + 484T^2 + 2728*T + 7688)^2
$83$	$ (T^{6} + 352 T^{4} + \cdots + 1344800)^{2} $ (T^6 + 352T^4 + 38576T^2 + 1344800)^2
$89$	$ T^{12} + \cdots + 31713911056 $ T^12 + 56360T^8 + 756247280T^4 + 31713911056
$97$	$ (T^{6} + 34 T^{5} + \cdots + 98568)^{2} $ (T^6 + 34T^5 + 578T^4 + 5200T^3 + 27556T^2 + 73704*T + 98568)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	444.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{7} + \beta_1) q^{3} - \beta_{11} q^{5} + ( - \beta_{2} - 1) q^{7} + (2 \beta_{8} + 1) q^{9} + ( - \beta_{9} + \beta_{7}) q^{11} + ( - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{6} + \beta_{4}) q^{15}+ \cdots + ( - \beta_{9} + \beta_{7} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) q + (-b7 + b1) * q^3 - b11 * q^5 + (-b2 - 1) * q^7 + (2b8 + 1) q^9 + (-b9 + b7) * q^11 + (-b3 + b2) * q^13 + (-b6 + b4) * q^15 + (-b11 + b8 + b7) * q^17 + (-b5 + b1 + 1) * q^19 + (-b9 + b7 + b3 - b1) * q^21 + (b11 - b10 + b9 + b8 + b7) * q^23 + (-b5 + b4 - b3 - 4b1) q^25 + (b7 + 5b1) q^27 + (b8 - b7 - b6) * q^29 + (-b4 - 3b1 + 3) q^31 + (-b10 - b8 - 2b2 - 2) q^33 + (-b10 + b9) * q^35 + (-b4 + b3 + b1 + 3) * q^37 + (b10 + b9 - b3 - b2) * q^39 + (-b11 - b9 - b7 - b6) * q^41 + (b5 - b3 + b2) * q^43 + (-b11 + 2b5) q^45 + (b11 + b10 - b8 - b6) * q^47 + (-b5 - b4 + 2b2 + 3) q^49 + (-b8 + b7 - b6 + b4 + 2b1 - 2) q^51 + (b10 - b8) * q^53 + (2b3 + 2b2) * q^55 + (b8 - b7 + 2b6 + b4 + b1 - 1) q^57 + (b11 + b10 - b9) * q^59 + (-2b4 - 2b3 - 2b2 + 3b1 - 3) * q^61 + (-2b10 - 2b8 - b2 - 1) * q^63 + (b11 - 2b9 + b6) q^65 + (-b5 + b4 + b3 - 3b1) q^67 + (b10 + b9 - b8 + b7 + b6 - b4 + 2b3 + 2b2 + 2b1 - 2) q^69 + (-b11 - b10 - b8 + b6) * q^71 + (-b5 + b4 - b3 + b1) * q^73 + (-2b11 + b10 - 4b8 + 2b6 + b5 + b4 - b2 + 4) q^75 + (2b11 + 2b9 - 10b7 + 2b6) * q^77 + (b5 - b1 - 1) * q^79 + (4b8 - 7) q^81 + (b11 + 2b10 + 4b8 - b6) * q^83 + (-b3 - 9b1) q^85 + (b11 + b8 + b7 + b5 + 2b1 + 2) q^87 + (b10 + b9 + b8 - b7 + 3b6) q^89 + (2b5 + b3 - b2 - 9b1 - 9) * q^91 + (2b11 - 3b8 - 3b7 - b5 + 3b1 + 3) * q^93 + (b11 + b9 + 9b7 + b6) q^95 + (b3 - b2 - 6b1 - 6) q^97 + (-b9 + b7 + 4b3 - 4b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7} + 12 q^{9} - 4 q^{13} + 12 q^{19} + 36 q^{31} - 16 q^{33} + 36 q^{37} + 4 q^{39} - 4 q^{43} + 28 q^{49} - 24 q^{51} - 8 q^{55} - 12 q^{57} - 28 q^{61} - 8 q^{63} - 32 q^{69} + 52 q^{75} - 12 q^{79}+ \cdots - 68 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 + 12 * q^9 - 4 * q^13 + 12 * q^19 + 36 * q^31 - 16 * q^33 + 36 * q^37 + 4 * q^39 - 4 * q^43 + 28 * q^49 - 24 * q^51 - 8 * q^55 - 12 * q^57 - 28 * q^61 - 8 * q^63 - 32 * q^69 + 52 * q^75 - 12 * q^79 - 84 * q^81 + 24 * q^87 - 104 * q^91 + 36 * q^93 - 68 * q^97

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Newspace parameters

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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	444.2.m.d

Level	$444$
Weight	$2$
Character orbit	444.m
Analytic conductor	$3.545$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.54535784974\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 24x^{10} + 239x^{8} - 1116x^{6} + 2027x^{4} + 708x^{2} + 1369 \) x^12 - 24x^10 + 239x^8 - 1116x^6 + 2027x^4 + 708*x^2 + 1369
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( -4\nu^{10} + 59\nu^{8} - 150\nu^{6} - 1608\nu^{4} + 8248\nu^{2} + 1633 ) / 6246 \) (-4v^10 + 59v^8 - 150v^6 - 1608v^4 + 8248*v^2 + 1633) / 6246
\(\beta_{2}\)	\(=\)	\( ( 67\nu^{10} - 34\nu^{8} - 6336\nu^{6} + 38385\nu^{4} + 19037\nu^{2} - 29348 ) / 75993 \) (67v^10 - 34v^8 - 6336v^6 + 38385v^4 + 19037*v^2 - 29348) / 75993
\(\beta_{3}\)	\(=\)	\( ( 178\nu^{10} - 3493\nu^{8} + 28536\nu^{6} - 90840\nu^{4} + 111824\nu^{2} - 15587 ) / 151986 \) (178v^10 - 3493v^8 + 28536v^6 - 90840v^4 + 111824*v^2 - 15587) / 151986
\(\beta_{4}\)	\(=\)	\( ( -467\nu^{10} + 10445\nu^{8} - 91944\nu^{6} + 386898\nu^{4} - 772393\nu^{2} + 625357 ) / 151986 \) (-467v^10 + 10445v^8 - 91944v^6 + 386898v^4 - 772393*v^2 + 625357) / 151986
\(\beta_{5}\)	\(=\)	\( ( 823\nu^{10} - 17431\nu^{8} + 149016\nu^{6} - 568578\nu^{4} + 692069\nu^{2} + 559357 ) / 151986 \) (823v^10 - 17431v^8 + 149016v^6 - 568578v^4 + 692069*v^2 + 559357) / 151986
\(\beta_{6}\)	\(=\)	\( ( 2057\nu^{11} - 27131\nu^{9} + 102642\nu^{7} + 637452\nu^{5} - 7305197\nu^{3} + 25820153\nu ) / 5623482 \) (2057v^11 - 27131v^9 + 102642v^7 + 637452v^5 - 7305197v^3 + 25820153v) / 5623482
\(\beta_{7}\)	\(=\)	\( ( -23\nu^{11} + 478\nu^{9} - 5016\nu^{7} + 28554\nu^{5} - 83695\nu^{3} + 35738\nu ) / 48618 \) (-23v^11 + 478v^9 - 5016v^7 + 28554v^5 - 83695v^3 + 35738v) / 48618
\(\beta_{8}\)	\(=\)	\( ( -4529\nu^{11} + 102110\nu^{9} - 953190\nu^{7} + 3998532\nu^{5} - 5819203\nu^{3} - 7344020\nu ) / 5623482 \) (-4529v^11 + 102110v^9 - 953190v^7 + 3998532v^5 - 5819203v^3 - 7344020v) / 5623482
\(\beta_{9}\)	\(=\)	\( ( -4529\nu^{11} + 102110\nu^{9} - 953190\nu^{7} + 3998532\nu^{5} - 5819203\nu^{3} + 3902944\nu ) / 5623482 \) (-4529v^11 + 102110v^9 - 953190v^7 + 3998532v^5 - 5819203v^3 + 3902944v) / 5623482
\(\beta_{10}\)	\(=\)	\( ( -9863\nu^{11} + 161528\nu^{9} - 850284\nu^{7} + 407274\nu^{5} + 5171177\nu^{3} + 7074184\nu ) / 5623482 \) (-9863v^11 + 161528v^9 - 850284v^7 + 407274v^5 + 5171177v^3 + 7074184v) / 5623482
\(\beta_{11}\)	\(=\)	\( ( -7001\nu^{11} + 177089\nu^{9} - 1803738\nu^{7} + 8634516\nu^{5} - 16131862\nu^{3} - 2926592\nu ) / 2811741 \) (-7001v^11 + 177089v^9 - 1803738v^7 + 8634516v^5 - 16131862v^3 - 2926592v) / 2811741

\(\nu\)	\(=\)	\( ( \beta_{9} - \beta_{8} ) / 2 \) (b9 - b8) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{5} - \beta_{4} + 2\beta_{3} + 8 ) / 2 \) (-b5 - b4 + 2*b3 + 8) / 2
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{11} + 5\beta_{9} - 13\beta_{8} - 4\beta_{6} ) / 2 \) (2b11 + 5b9 - 13b8 - 4b6) / 2
\(\nu^{4}\)	\(=\)	\( -3\beta_{5} - \beta_{4} + 12\beta_{3} - \beta_{2} + 16 \) -3b5 - b4 + 12b3 - b2 + 16
\(\nu^{5}\)	\(=\)	\( ( 26\beta_{11} + 5\beta_{10} + 12\beta_{9} - 124\beta_{8} + 9\beta_{7} - 34\beta_{6} ) / 2 \) (26b11 + 5b10 + 12b9 - 124b8 + 9b7 - 34b6) / 2
\(\nu^{6}\)	\(=\)	\( ( -21\beta_{5} + 23\beta_{4} + 196\beta_{3} - 12\beta_{2} + 54\beta _1 - 16 ) / 2 \) (-21b5 + 23b4 + 196b3 - 12b2 + 54*b1 - 16) / 2
\(\nu^{7}\)	\(=\)	\( ( 228\beta_{11} + 77\beta_{10} - 108\beta_{9} - 874\beta_{8} + 27\beta_{7} - 206\beta_{6} ) / 2 \) (228b11 + 77b10 - 108b9 - 874b8 + 27b7 - 206b6) / 2
\(\nu^{8}\)	\(=\)	\( 21\beta_{5} + 201\beta_{4} + 600\beta_{3} + 36\beta_{2} + 360\beta _1 - 923 \) 21b5 + 201b4 + 600b3 + 36b2 + 360*b1 - 923
\(\nu^{9}\)	\(=\)	\( ( 1632\beta_{11} + 684\beta_{10} - 1967\beta_{9} - 4477\beta_{8} - 684\beta_{7} - 684\beta_{6} ) / 2 \) (1632b11 + 684b10 - 1967b9 - 4477b8 - 684b7 - 684b6) / 2
\(\nu^{10}\)	\(=\)	\( ( 1757\beta_{5} + 3809\beta_{4} + 4826\beta_{3} + 2316\beta_{2} + 5472\beta _1 - 22180 ) / 2 \) (1757b5 + 3809b4 + 4826b3 + 2316b2 + 5472*b1 - 22180) / 2
\(\nu^{11}\)	\(=\)	\( ( 9194\beta_{11} + 3630\beta_{10} - 19069\beta_{9} - 10627\beta_{8} - 13158\beta_{7} + 3056\beta_{6} ) / 2 \) (9194b11 + 3630b10 - 19069b9 - 10627b8 - 13158b7 + 3056b6) / 2

\(n\)	\(149\)	\(223\)	\(409\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\beta_{1}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 401.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{4} - 2 T^{2} + 9)^{3} \) (T^4 - 2*T^2 + 9)^3
$5$	\( T^{12} + 392 T^{8} + \cdots + 104976 \) T^12 + 392T^8 + 20272T^4 + 104976
$7$	\( (T^{3} + 2 T^{2} - 12 T - 4)^{4} \) (T^3 + 2T^2 - 12T - 4)^4
$11$	\( (T^{6} - 56 T^{4} + \cdots - 128)^{2} \) (T^6 - 56T^4 + 640T^2 - 128)^2
$13$	\( (T^{6} + 2 T^{5} + \cdots + 648)^{2} \) (T^6 + 2T^5 + 2T^4 - 16T^3 + 676T^2 + 936*T + 648)^2
$17$	\( T^{12} + 680 T^{8} + \cdots + 2085136 \) T^12 + 680T^8 + 96080T^4 + 2085136
$19$	\( (T^{6} - 6 T^{5} + \cdots + 7688)^{2} \) (T^6 - 6T^5 + 18T^4 + 8T^3 + 484T^2 - 2728*T + 7688)^2
$23$	\( T^{12} + \cdots + 13680577296 \) T^12 + 7432T^8 + 17738352T^4 + 13680577296
$29$	\( T^{12} + 1544 T^{8} + \cdots + 16 \) T^12 + 1544T^8 + 464T^4 + 16
$31$	\( (T^{6} - 18 T^{5} + \cdots + 8712)^{2} \) (T^6 - 18T^5 + 162T^4 - 600T^3 + 676T^2 + 3432*T + 8712)^2
$37$	\( (T^{6} - 18 T^{5} + \cdots + 50653)^{2} \) (T^6 - 18T^5 + 163T^4 - 1068T^3 + 6031T^2 - 24642*T + 50653)^2
$41$	\( (T^{6} - 112 T^{4} + \cdots - 20000)^{2} \) (T^6 - 112T^4 + 3200T^2 - 20000)^2
$43$	\( (T^{6} + 2 T^{5} + \cdots + 16200)^{2} \) (T^6 + 2T^5 + 2T^4 - 296T^3 + 3364T^2 - 10440*T + 16200)^2
$47$	\( (T^{6} + 128 T^{4} + \cdots + 2592)^{2} \) (T^6 + 128T^4 + 1728T^2 + 2592)^2
$53$	\( (T^{6} + 64 T^{4} + \cdots + 3200)^{2} \) (T^6 + 64T^4 + 896T^2 + 3200)^2
$59$	\( T^{12} + \cdots + 688747536 \) T^12 + 12232T^8 + 10521232T^4 + 688747536
$61$	\( (T^{6} + 14 T^{5} + \cdots + 3908808)^{2} \) (T^6 + 14T^5 + 98T^4 - 32T^3 + 40804T^2 + 564792*T + 3908808)^2
$67$	\( (T^{6} + 168 T^{4} + \cdots + 51984)^{2} \) (T^6 + 168T^4 + 6352T^2 + 51984)^2
$71$	\( (T^{6} + 120 T^{4} + \cdots + 43808)^{2} \) (T^6 + 120T^4 + 4320T^2 + 43808)^2
$73$	\( (T^{6} + 132 T^{4} + \cdots + 26896)^{2} \) (T^6 + 132T^4 + 4752T^2 + 26896)^2
$79$	\( (T^{6} + 6 T^{5} + \cdots + 7688)^{2} \) (T^6 + 6T^5 + 18T^4 - 8T^3 + 484T^2 + 2728*T + 7688)^2
$83$	\( (T^{6} + 352 T^{4} + \cdots + 1344800)^{2} \) (T^6 + 352T^4 + 38576T^2 + 1344800)^2
$89$	\( T^{12} + \cdots + 31713911056 \) T^12 + 56360T^8 + 756247280T^4 + 31713911056
$97$	\( (T^{6} + 34 T^{5} + \cdots + 98568)^{2} \) (T^6 + 34T^5 + 578T^4 + 5200T^3 + 27556T^2 + 73704*T + 98568)^2
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