LMFDB - Newform orbit 9196.2.a.m

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-4,0,4,0,2,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.114134848.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 $ x^6 - 10x^4 - 6x^3 + 20x^2 + 14x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{3} + (\beta_{5} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{3} + \beta_1) q^{15} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 + 2) q^{17}+ \cdots + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \cdots + 7) q^{97}+O(q^{100}) $ q - b1 * q^3 + (b5 - 1) * q^5 + (b5 + b4 + b1) * q^7 + (b2 + b1) * q^9 + (-b3 + b2 + 1) * q^13 + (b5 - b3 + b1) * q^15 + (-b4 - 2b2 - b1 + 2) q^17 - q^19 + (b5 + b4 - b3 - b1 - 4) * q^21 + (b5 - b2 + b1 + 1) * q^23 + (-2b5 - 2b4 - b2 - b1 + 2) * q^25 + (b4 - b3 - b2 - 3) * q^27 + (b5 - 2b3 + b2 + 2b1 - 1) * q^29 + (b5 + 2b4 - b3 - b1 - 2) q^31 + (-3b5 - b4 + 2b3 - b2 - 2b1 + 4) q^35 + (b5 + 3b4 + b2 - 1) q^37 + (b5 + b4 + b2 - 2b1) q^39 + (2b5 + b4 + b3 + b2 - b1 + 1) q^41 + (2b5 - 2b4 + b2 + b1 - 2) * q^43 - b5 * q^45 + (-2b3 + 2b2) * q^47 + (-2b5 + 4b3 - 3b2 - b1 + 5) q^49 + (-3b4 + 2b3 + 3b1 + 4) q^51 + (2b5 + 2b4 - 2b3 + 2b2 + 2b1 - 2) q^53 + b1 * q^57 + (b5 + b4 - b2 + 3) * q^59 + (4b5 + b4 - 2b3 - b1 - 2) * q^61 + (-b5 - 2b4 + 3b2 + 3b1 + 2) q^63 + (b5 - b4 + b3 - b1 - 2) * q^65 + (3b5 + 2b4 - b3 + b1) * q^67 + (b5 - b4 - b2 - 3) * q^69 + (-2b4 - 2b3 + 2b2 + b1) q^71 + (-2b5 + 2b1 + 6) * q^73 + (-2b5 - 3b4 + 3b3 - b2 + b1 + 5) q^75 + (b4 - 2b3 + 2b2 - 3b1) q^79 + (b5 + 2b3 - b2 + 3b1 - 1) * q^81 + (b4 + 2b3 - b2 - 6b1 + 6) * q^83 + (4b5 - b4 - b1) q^85 + (3b5 + b4 - b1 - 6) q^87 + (b5 + b4 + 2b3 + b2 + 2b1 - 5) * q^89 + (-b5 - b4 - 2b3 + 3b2 + b1 - 3) * q^91 + (2b5 + 2b4 + 4b2 + 4b1 + 1) * q^93 + (-b5 + 1) * q^95 + (-b5 - 3b4 + 2b3 - 3b2 + 7) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 2 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 6 q^{31} + 14 q^{35} + 4 q^{37} + 6 q^{39} + 14 q^{41} - 10 q^{43}+ \cdots + 28 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 4 * q^7 + 2 * q^9 + 8 * q^13 + 2 * q^15 + 6 * q^17 - 6 * q^19 - 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 - 2 * q^29 - 6 * q^31 + 14 * q^35 + 4 * q^37 + 6 * q^39 + 14 * q^41 - 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 + 18 * q^51 + 20 * q^59 - 2 * q^61 + 12 * q^63 - 12 * q^65 + 10 * q^67 - 20 * q^69 + 32 * q^73 + 18 * q^75 + 6 * q^79 - 6 * q^81 + 36 * q^83 + 6 * q^85 - 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 + 4 * q^95 + 28 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 $ x^6 - 10*x^4 - 6*x^3 + 20*x^2 + 14*x - 5 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - \nu - 3 $ v^2 - v - 3
$\beta_{3}$	$=$	$ \nu^{5} - \nu^{4} - 8\nu^{3} + 2\nu^{2} + 11\nu - 1 $ v^5 - v^4 - 8v^3 + 2v^2 + 11*v - 1
$\beta_{4}$	$=$	$ \nu^{5} - \nu^{4} - 9\nu^{3} + 3\nu^{2} + 16\nu - 1 $ v^5 - v^4 - 9v^3 + 3v^2 + 16*v - 1
$\beta_{5}$	$=$	$ -2\nu^{5} + 3\nu^{4} + 16\nu^{3} - 12\nu^{2} - 26\nu + 9 $ -2v^5 + 3v^4 + 16v^3 - 12v^2 - 26*v + 9

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + \beta _1 + 3 $ b2 + b1 + 3
$\nu^{3}$	$=$	$ -\beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 3 $ -b4 + b3 + b2 + 6*b1 + 3
$\nu^{4}$	$=$	$ \beta_{5} + 2\beta_{3} + 8\beta_{2} + 12\beta _1 + 17 $ b5 + 2b3 + 8b2 + 12*b1 + 17
$\nu^{5}$	$=$	$ \beta_{5} - 8\beta_{4} + 11\beta_{3} + 14\beta_{2} + 47\beta _1 + 36 $ b5 - 8b4 + 11b3 + 14b2 + 47b1 + 36

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

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} $

1.1

3.06597
1.52628
0.267039
−1.28435
−1.50521
−2.06973

−3.06597

−0.135132

3.35676

6.40017

1.2

−1.52628

−3.03462

0.757059

−0.670469

1.3

−0.267039

0.518487

5.09697

−2.92869

1.4

1.28435

2.85345

−1.18021

−1.35045

1.5

1.50521

−3.76432

−4.72308

−0.734348

1.6

2.06973

−0.437868

0.692506

1.28379

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$11$	$ +1 $
$19$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9196.2.a.m	yes	6
11.b	odd	2	1		9196.2.a.l	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9196.2.a.l	✓	6	11.b	odd	2	1
9196.2.a.m	yes	6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9196))$:

$ T_{3}^{6} - 10T_{3}^{4} + 6T_{3}^{3} + 20T_{3}^{2} - 14T_{3} - 5 $

T3^6 - 10*T3^4 + 6*T3^3 + 20*T3^2 - 14*T3 - 5

$ T_{5}^{6} + 4T_{5}^{5} - 8T_{5}^{4} - 34T_{5}^{3} + 8T_{5} + 1 $

T5^6 + 4*T5^5 - 8*T5^4 - 34*T5^3 + 8*T5 + 1

$ T_{7}^{6} - 4T_{7}^{5} - 23T_{7}^{4} + 92T_{7}^{3} + 3T_{7}^{2} - 110T_{7} + 50 $

T7^6 - 4*T7^5 - 23*T7^4 + 92*T7^3 + 3*T7^2 - 110*T7 + 50

$ T_{13}^{6} - 8T_{13}^{5} + T_{13}^{4} + 122T_{13}^{3} - 303T_{13}^{2} + 230T_{13} - 50 $

T13^6 - 8*T13^5 + T13^4 + 122*T13^3 - 303*T13^2 + 230*T13 - 50

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 10 T^{4} + \cdots - 5 $ T^6 - 10T^4 + 6T^3 + 20T^2 - 14T - 5
$5$	$ T^{6} + 4 T^{5} + \cdots + 1 $ T^6 + 4T^5 - 8T^4 - 34T^3 + 8T + 1
$7$	$ T^{6} - 4 T^{5} + \cdots + 50 $ T^6 - 4T^5 - 23T^4 + 92T^3 + 3T^2 - 110*T + 50
$11$	$ T^{6} $ T^6
$13$	$ T^{6} - 8 T^{5} + \cdots - 50 $ T^6 - 8T^5 + T^4 + 122T^3 - 303T^2 + 230T - 50
$17$	$ T^{6} - 6 T^{5} + \cdots + 10400 $ T^6 - 6T^5 - 66T^4 + 448T^3 + 552T^2 - 7200*T + 10400
$19$	$ (T + 1)^{6} $ (T + 1)^6
$23$	$ T^{6} - 6 T^{5} + \cdots + 1456 $ T^6 - 6T^5 - 27T^4 + 180T^3 + 80T^2 - 1360*T + 1456
$29$	$ T^{6} + 2 T^{5} + \cdots - 50 $ T^6 + 2T^5 - 93T^4 - 76T^3 + 1637T^2 - 2330*T - 50
$31$	$ T^{6} + 6 T^{5} + \cdots - 7561 $ T^6 + 6T^5 - 68T^4 - 262T^3 + 1382T^2 + 2312*T - 7561
$37$	$ T^{6} - 4 T^{5} + \cdots - 5392 $ T^6 - 4T^5 - 111T^4 + 280T^3 + 2764T^2 - 672*T - 5392
$41$	$ T^{6} - 14 T^{5} + \cdots + 22400 $ T^6 - 14T^5 - 49T^4 + 1572T^3 - 7349T^2 + 4720*T + 22400
$43$	$ T^{6} + 10 T^{5} + \cdots + 200000 $ T^6 + 10T^5 - 161T^4 - 2002T^3 + 1945T^2 + 78000*T + 200000
$47$	$ T^{6} - 4 T^{5} + \cdots - 448 $ T^6 - 4T^5 - 96T^4 + 528T^3 + 64T^2 - 1280*T - 448
$53$	$ T^{6} - 136 T^{4} + \cdots - 10240 $ T^6 - 136T^4 + 480T^3 + 1728T^2 - 4608T - 10240
$59$	$ T^{6} - 20 T^{5} + \cdots + 16 $ T^6 - 20T^5 + 121T^4 - 244T^3 - 24T^2 + 368*T + 16
$61$	$ T^{6} + 2 T^{5} + \cdots - 4000 $ T^6 + 2T^5 - 258T^4 - 8T^3 + 12760T^2 - 32000*T - 4000
$67$	$ T^{6} - 10 T^{5} + \cdots - 6349 $ T^6 - 10T^5 - 92T^4 + 1494T^3 - 6618T^2 + 11584*T - 6349
$71$	$ T^{6} - 194 T^{4} + \cdots + 6871 $ T^6 - 194T^4 + 1278T^3 - 1516T^2 - 4666T + 6871
$73$	$ T^{6} - 32 T^{5} + \cdots - 12800 $ T^6 - 32T^5 + 320T^4 - 640T^3 - 5632T^2 + 21760*T - 12800
$79$	$ T^{6} - 6 T^{5} + \cdots + 20000 $ T^6 - 6T^5 - 206T^4 + 344T^3 + 5600T^2 - 20800*T + 20000
$83$	$ T^{6} - 36 T^{5} + \cdots + 2849050 $ T^6 - 36T^5 + 149T^4 + 6648T^3 - 54157T^2 - 313010*T + 2849050
$89$	$ T^{6} + 24 T^{5} + \cdots + 57328 $ T^6 + 24T^5 + 17T^4 - 3096T^3 - 22336T^2 - 30560*T + 57328
$97$	$ T^{6} - 28 T^{5} + \cdots - 111808 $ T^6 - 28T^5 + 129T^4 + 2764T^3 - 33024T^2 + 116384*T - 111808
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + (\beta_{5} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{3} + \beta_1) q^{15} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 + 2) q^{17}+ \cdots + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \cdots + 7) q^{97}+O(q^{100}) \) q - b1 * q^3 + (b5 - 1) * q^5 + (b5 + b4 + b1) * q^7 + (b2 + b1) * q^9 + (-b3 + b2 + 1) * q^13 + (b5 - b3 + b1) * q^15 + (-b4 - 2b2 - b1 + 2) q^17 - q^19 + (b5 + b4 - b3 - b1 - 4) * q^21 + (b5 - b2 + b1 + 1) * q^23 + (-2b5 - 2b4 - b2 - b1 + 2) * q^25 + (b4 - b3 - b2 - 3) * q^27 + (b5 - 2b3 + b2 + 2b1 - 1) * q^29 + (b5 + 2b4 - b3 - b1 - 2) q^31 + (-3b5 - b4 + 2b3 - b2 - 2b1 + 4) q^35 + (b5 + 3b4 + b2 - 1) q^37 + (b5 + b4 + b2 - 2b1) q^39 + (2b5 + b4 + b3 + b2 - b1 + 1) q^41 + (2b5 - 2b4 + b2 + b1 - 2) * q^43 - b5 * q^45 + (-2b3 + 2b2) * q^47 + (-2b5 + 4b3 - 3b2 - b1 + 5) q^49 + (-3b4 + 2b3 + 3b1 + 4) q^51 + (2b5 + 2b4 - 2b3 + 2b2 + 2b1 - 2) q^53 + b1 * q^57 + (b5 + b4 - b2 + 3) * q^59 + (4b5 + b4 - 2b3 - b1 - 2) * q^61 + (-b5 - 2b4 + 3b2 + 3b1 + 2) q^63 + (b5 - b4 + b3 - b1 - 2) * q^65 + (3b5 + 2b4 - b3 + b1) * q^67 + (b5 - b4 - b2 - 3) * q^69 + (-2b4 - 2b3 + 2b2 + b1) q^71 + (-2b5 + 2b1 + 6) * q^73 + (-2b5 - 3b4 + 3b3 - b2 + b1 + 5) q^75 + (b4 - 2b3 + 2b2 - 3b1) q^79 + (b5 + 2b3 - b2 + 3b1 - 1) * q^81 + (b4 + 2b3 - b2 - 6b1 + 6) * q^83 + (4b5 - b4 - b1) q^85 + (3b5 + b4 - b1 - 6) q^87 + (b5 + b4 + 2b3 + b2 + 2b1 - 5) * q^89 + (-b5 - b4 - 2b3 + 3b2 + b1 - 3) * q^91 + (2b5 + 2b4 + 4b2 + 4b1 + 1) * q^93 + (-b5 + 1) * q^95 + (-b5 - 3b4 + 2b3 - 3b2 + 7) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 2 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 6 q^{31} + 14 q^{35} + 4 q^{37} + 6 q^{39} + 14 q^{41} - 10 q^{43}+ \cdots + 28 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 4 * q^7 + 2 * q^9 + 8 * q^13 + 2 * q^15 + 6 * q^17 - 6 * q^19 - 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 - 2 * q^29 - 6 * q^31 + 14 * q^35 + 4 * q^37 + 6 * q^39 + 14 * q^41 - 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 + 18 * q^51 + 20 * q^59 - 2 * q^61 + 12 * q^63 - 12 * q^65 + 10 * q^67 - 20 * q^69 + 32 * q^73 + 18 * q^75 + 6 * q^79 - 6 * q^81 + 36 * q^83 + 6 * q^85 - 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 + 4 * q^95 + 28 * q^97

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