LMFDB - Newform orbit 1224.4.a.q

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(1,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1224.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.a (trivial)

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:	yes
Analytic conductor:	$72.2183378470$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 425x^{5} - 474x^{4} + 46988x^{3} + 195600x^{2} - 182556x - 1029768 $ x^7 - 2x^6 - 425x^5 - 474x^4 + 46988x^3 + 195600x^2 - 182556x - 1029768
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 13 $
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 $1,\beta_1,\ldots,\beta_{6}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{5} + (\beta_{3} + 4) q^{7} + ( - \beta_{4} + \beta_{2} + 9) q^{11} + (\beta_{5} + \beta_{3} + 3) q^{13} + 17 q^{17} + (\beta_{3} - \beta_{2} - \beta_1 + 18) q^{19} + (\beta_{6} + \beta_{5} - 5 \beta_{4} + \cdots - 18) q^{23}+ \cdots + (3 \beta_{6} + 9 \beta_{5} + \cdots + 367) q^{97}+O(q^{100}) $ q - b4 * q^5 + (b3 + 4) * q^7 + (-b4 + b2 + 9) * q^11 + (b5 + b3 + 3) * q^13 + 17 * q^17 + (b3 - b2 - b1 + 18) * q^19 + (b6 + b5 - 5b4 - 2b3 - 18) * q^23 + (-b5 + 3b3 + 2b2 + 26) * q^25 + (-b6 - 2b5 - 9b4 + 4b3 + b2 - 2) q^29 + (-b6 + b5 - 2b4 + 4b3 - b2 + 4b1 + 7) q^31 + (-4b6 - 11b4 + b3 + b1 + 5) * q^35 + (3b6 - 2b5 + 7b4 + b2 + 38) q^37 + (3b6 - 2b5 - 12b4 - b2 - 4b1 - 38) * q^41 + (b6 - b5 - 5b4 - 3b3 - 2b2 - 2b1 + 56) * q^43 + (3b6 + 5b5 + 2b4 - 9b3 - 7b2 + 2b1 - 45) * q^47 + (-8b6 - 2b5 + 4b4 + 6b3 + 4b2 + 173) q^49 + (b6 + 3b5 + 6b4 + 9b3 - b2 + 8b1 - 35) * q^53 + (b6 - 7b5 - 27b4 - 3b3 + 2b2 - 8b1 + 76) q^55 + (5b6 - 5b5 - 5b4 - 12b3 - b2 - 5b1 - 2) q^59 + (2b6 + 5b5 + 25b4 - 15b3 - 10b2 + 97) q^61 + (-7b6 + 4b5 + 2b4 + 2b3 - 7b2 - 4b1 + 30) * q^65 + (7b6 + 9b5 + 3b4 - 12b3 - b2 + 5b1 + 112) q^67 + (-5b6 - 7b5 + 4b4 + 2b3 + 7b2 - 10b1 + 213) * q^71 + (5b6 + b5 + 32b4 - 9b3 + 3b2 + 255) * q^73 + (-3b6 + 5b5 + 19b3 + 7b2 + 8b1 - 75) q^77 + (-7b6 + 7b5 - 36b4 - 10b3 - 5b2 + 4b1 + 147) * q^79 + (-8b5 + 27b4 - 11b3 - 2b2 + 9b1 + 91) q^83 - 17b4 q^85 + (5b6 - 9b5 + 20b4 + 5b3 + 15b2 - 91) q^89 + (-17b6 - 7b5 - 8b4 + 19b3 + 15b2 + 10b1 + 553) * q^91 + (-8b6 + 14b5 - b4 + 12b3 + 7b2 + 14b1 + 43) q^95 + (3b6 + 9b5 + 22b4 - 21b3 - 3b2 + 367) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{5} + 26 q^{7} + 65 q^{11} + 17 q^{13} + 119 q^{17} + 125 q^{19} - 107 q^{23} + 176 q^{25} + 6 q^{29} + 44 q^{31} + 58 q^{35} + 254 q^{37} - 219 q^{41} + 419 q^{43} - 300 q^{47} + 1171 q^{49}+ \cdots + 2536 q^{97}+O(q^{100}) $ 7 * q + 3 * q^5 + 26 * q^7 + 65 * q^11 + 17 * q^13 + 119 * q^17 + 125 * q^19 - 107 * q^23 + 176 * q^25 + 6 * q^29 + 44 * q^31 + 58 * q^35 + 254 * q^37 - 219 * q^41 + 419 * q^43 - 300 * q^47 + 1171 * q^49 - 284 * q^53 + 633 * q^55 + 46 * q^59 + 638 * q^61 + 185 * q^65 + 796 * q^67 + 1472 * q^71 + 1712 * q^73 - 586 * q^77 + 1134 * q^79 + 596 * q^83 + 51 * q^85 - 694 * q^89 + 3822 * q^91 + 229 * q^95 + 2536 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{7} - 2x^{6} - 425x^{5} - 474x^{4} + 46988x^{3} + 195600x^{2} - 182556x - 1029768 $ x^7 - 2*x^6 - 425*x^5 - 474*x^4 + 46988*x^3 + 195600*x^2 - 182556*x - 1029768 :

$\beta_{1}$	$=$	$ ( 116393 \nu^{6} + 486006 \nu^{5} - 1013293 \nu^{4} - 244107038 \nu^{3} - 7563919092 \nu^{2} + \cdots + 265998537624 ) / 5010738780 $ (116393v^6 + 486006v^5 - 1013293v^4 - 244107038v^3 - 7563919092v^2 + 13306492716v + 265998537624) / 5010738780
$\beta_{2}$	$=$	$ ( - 162419 \nu^{6} + 1896210 \nu^{5} + 58743901 \nu^{4} - 562367422 \nu^{3} + \cdots + 267208252980 ) / 5010738780 $ (-162419v^6 + 1896210v^5 + 58743901v^4 - 562367422v^3 - 7131771210v^2 + 30156453288v + 267208252980) / 5010738780
$\beta_{3}$	$=$	$ ( - 114489 \nu^{6} + 700084 \nu^{5} + 46505055 \nu^{4} - 189333156 \nu^{3} - 4749925678 \nu^{2} + \cdots + 33287950488 ) / 1670246260 $ (-114489v^6 + 700084v^5 + 46505055v^4 - 189333156v^3 - 4749925678v^2 + 8053670820v + 33287950488) / 1670246260
$\beta_{4}$	$=$	$ ( 240655 \nu^{6} - 1231587 \nu^{5} - 100079438 \nu^{4} + 175008179 \nu^{3} + 10909856379 \nu^{2} + \cdots - 55466746578 ) / 2505369390 $ (240655v^6 - 1231587v^5 - 100079438v^4 + 175008179v^3 + 10909856379v^2 + 20388984156v - 55466746578) / 2505369390
$\beta_{5}$	$=$	$ ( 349808 \nu^{6} + 358560 \nu^{5} - 162492433 \nu^{4} - 378911174 \nu^{3} + 19465832685 \nu^{2} + \cdots - 148552022484 ) / 2505369390 $ (349808v^6 + 358560v^5 - 162492433v^4 - 378911174v^3 + 19465832685v^2 + 59538598686v - 148552022484) / 2505369390
$\beta_{6}$	$=$	$ ( - 1501148 \nu^{6} + 9817563 \nu^{5} + 592913611 \nu^{4} - 1881437179 \nu^{3} + \cdots + 512255618400 ) / 2505369390 $ (-1501148v^6 + 9817563v^5 + 592913611v^4 - 1881437179v^3 - 62749956171v^2 - 25625860092v + 512255618400) / 2505369390

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

$\nu$	$=$	$ ( 2\beta_{6} - \beta_{5} + 12\beta_{4} - 5\beta_{2} + \beta _1 + 11 ) / 26 $ (2b6 - b5 + 12b4 - 5*b2 + b1 + 11) / 26
$\nu^{2}$	$=$	$ ( \beta_{5} - \beta_{4} + 3\beta_{3} - 5\beta_{2} + 244 ) / 2 $ (b5 - b4 + 3b3 - 5b2 + 244) / 2
$\nu^{3}$	$=$	$ ( 445\beta_{6} - 138\beta_{5} + 2137\beta_{4} - 494\beta_{3} - 1288\beta_{2} + 216\beta _1 + 15207 ) / 26 $ (445b6 - 138b5 + 2137b4 - 494b3 - 1288b2 + 216b1 + 15207) / 26
$\nu^{4}$	$=$	$ ( 38\beta_{6} + 199\beta_{5} - 19\beta_{4} + 779\beta_{3} - 1407\beta_{2} + 198\beta _1 + 52604 ) / 2 $ (38b6 + 199b5 - 19b4 + 779b3 - 1407b2 + 198b1 + 52604) / 2
$\nu^{5}$	$=$	$ ( 107235 \beta_{6} + 2224 \beta_{5} + 438647 \beta_{4} - 130858 \beta_{3} - 345782 \beta_{2} + \cdots + 5241979 ) / 26 $ (107235b6 + 2224b5 + 438647b4 - 130858b3 - 345782b2 + 70134b1 + 5241979) / 26
$\nu^{6}$	$=$	$ ( 20090 \beta_{6} + 52535 \beta_{5} + 33185 \beta_{4} + 164075 \beta_{3} - 389935 \beta_{2} + \cdots + 12416728 ) / 2 $ (20090b6 + 52535b5 + 33185b4 + 164075b3 - 389935b2 + 91350b1 + 12416728) / 2

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

2.21338
15.0295
−4.52275
−10.6413
16.7855
−14.1578
−2.70652

−16.9908

20.1237

1.2

−12.8726

21.2518

1.3

−8.70433

−35.2924

1.4

5.47546

−12.5140

1.5

5.92497

3.18485

1.6

14.4814

35.5447

1.7

15.6859

−6.29875

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$17$	$ -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	1224.4.a.q	yes	7
3.b	odd	2	1		1224.4.a.p	✓	7
4.b	odd	2	1		2448.4.a.bw		7
12.b	even	2	1		2448.4.a.bv		7

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1224.4.a.p	✓	7	3.b	odd	2	1
1224.4.a.q	yes	7	1.a	even	1	1	trivial
2448.4.a.bv		7	12.b	even	2	1
2448.4.a.bw		7	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{7} - 3T_{5}^{6} - 521T_{5}^{5} + 1715T_{5}^{4} + 79488T_{5}^{3} - 274832T_{5}^{2} - 3265940T_{5} + 14029500 $ T5^7 - 3*T5^6 - 521*T5^5 + 1715*T5^4 + 79488*T5^3 - 274832*T5^2 - 3265940*T5 + 14029500 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1224))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{7} $ T^7
$3$	$ T^{7} $ T^7
$5$	$ T^{7} - 3 T^{6} + \cdots + 14029500 $ T^7 - 3T^6 - 521T^5 + 1715T^4 + 79488T^3 - 274832T^2 - 3265940T + 14029500
$7$	$ T^{7} - 26 T^{6} + \cdots + 134678560 $ T^7 - 26T^6 - 1448T^5 + 38000T^4 + 267988T^3 - 7199944T^2 - 23145936T + 134678560
$11$	$ T^{7} + \cdots - 46024334784 $ T^7 - 65T^6 - 3857T^5 + 319345T^4 + 189648T^3 - 369500784T^2 + 7812117696T - 46024334784
$13$	$ T^{7} + \cdots - 648750174768 $ T^7 - 17T^6 - 11813T^5 + 68117T^4 + 41572288T^3 + 186487576T^2 - 43385160368T - 648750174768
$17$	$ (T - 17)^{7} $ (T - 17)^7
$19$	$ T^{7} + \cdots + 1455850962000 $ T^7 - 125T^6 - 15449T^5 + 1218933T^4 + 78151272T^3 - 3066703464T^2 - 123514932240T + 1455850962000
$23$	$ T^{7} + \cdots - 17364541460172 $ T^7 + 107T^6 - 31229T^5 - 3568175T^4 + 166532968T^3 + 23521041576T^2 + 180244667388T - 17364541460172
$29$	$ T^{7} + \cdots + 9310477266144 $ T^7 - 6T^6 - 94352T^5 - 4273984T^4 + 1940421300T^3 + 166363106696T^2 + 2377067416368T + 9310477266144
$31$	$ T^{7} + \cdots - 761122651480320 $ T^7 - 44T^6 - 159332T^5 + 6442832T^4 + 6328513348T^3 - 315785468144T^2 - 40019187066944T - 761122651480320
$37$	$ T^{7} + \cdots + 16\!\cdots\!80 $ T^7 - 254T^6 - 202000T^5 + 50563616T^4 + 9660104052T^3 - 1981505776920T^2 - 165922908024912T + 16139091685546080
$41$	$ T^{7} + \cdots - 12\!\cdots\!36 $ T^7 + 219T^6 - 297261T^5 - 35450255T^4 + 24655433568T^3 + 761371016040T^2 - 246980655178800T - 12073877057162736
$43$	$ T^{7} + \cdots + 37575792956160 $ T^7 - 419T^6 - 17681T^5 + 28792259T^4 - 3983509648T^3 + 102177014048T^2 + 6243031317504T + 37575792956160
$47$	$ T^{7} + \cdots + 12\!\cdots\!12 $ T^7 + 300T^6 - 385792T^5 - 151501888T^4 + 18506155328T^3 + 15791836564224T^2 + 2471700868893696T + 123560825626509312
$53$	$ T^{7} + \cdots + 13\!\cdots\!12 $ T^7 + 284T^6 - 703368T^5 - 269723200T^4 + 103868270992T^3 + 55510404592704T^2 + 5481147965817600T + 131586060208269312
$59$	$ T^{7} + \cdots - 47\!\cdots\!84 $ T^7 - 46T^6 - 776220T^5 + 41739272T^4 + 131482465648T^3 - 2495498037792T^2 - 1182272845450304T - 4724757057453184
$61$	$ T^{7} + \cdots + 15\!\cdots\!44 $ T^7 - 638T^6 - 792496T^5 + 609247968T^4 + 102185823988T^3 - 143786329545112T^2 + 21004812176880944T + 1529693864383515744
$67$	$ T^{7} + \cdots + 16\!\cdots\!48 $ T^7 - 796T^6 - 1261896T^5 + 961980512T^4 + 432701230096T^3 - 316483667289024T^2 - 20459525821076736T + 16500180538076802048
$71$	$ T^{7} + \cdots - 20\!\cdots\!40 $ T^7 - 1472T^6 - 610132T^5 + 1671145312T^4 - 539628889084T^3 - 95798112848576T^2 + 37948104662961664T - 2045770659999959040
$73$	$ T^{7} + \cdots + 86\!\cdots\!00 $ T^7 - 1712T^6 + 121112T^5 + 1142619552T^4 - 772351681008T^3 + 196758138724480T^2 - 18139005596569600T + 86403507110528000
$79$	$ T^{7} + \cdots + 81\!\cdots\!20 $ T^7 - 1134T^6 - 1798776T^5 + 2359218416T^4 + 574553190228T^3 - 1279289930074200T^2 + 192277041318439024T + 81463614916681309920
$83$	$ T^{7} + \cdots - 35\!\cdots\!36 $ T^7 - 596T^6 - 2555912T^5 + 2104147680T^4 + 1287906948368T^3 - 1646387185071424T^2 + 407507130683919104T - 3575041666920308736
$89$	$ T^{7} + \cdots - 38\!\cdots\!52 $ T^7 + 694T^6 - 2312556T^5 - 1871681160T^4 + 1198163181872T^3 + 1183559315671584T^2 + 83464006183961024T - 38205191937076743552
$97$	$ T^{7} + \cdots - 68\!\cdots\!00 $ T^7 - 2536T^6 + 1246312T^5 + 923364384T^4 - 711263050096T^3 + 5115556195072T^2 + 41003571126821888T - 681102240139545600
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{5} + (\beta_{3} + 4) q^{7} + ( - \beta_{4} + \beta_{2} + 9) q^{11} + (\beta_{5} + \beta_{3} + 3) q^{13} + 17 q^{17} + (\beta_{3} - \beta_{2} - \beta_1 + 18) q^{19} + (\beta_{6} + \beta_{5} - 5 \beta_{4} + \cdots - 18) q^{23}+ \cdots + (3 \beta_{6} + 9 \beta_{5} + \cdots + 367) q^{97}+O(q^{100}) \) q - b4 * q^5 + (b3 + 4) * q^7 + (-b4 + b2 + 9) * q^11 + (b5 + b3 + 3) * q^13 + 17 * q^17 + (b3 - b2 - b1 + 18) * q^19 + (b6 + b5 - 5b4 - 2b3 - 18) * q^23 + (-b5 + 3b3 + 2b2 + 26) * q^25 + (-b6 - 2b5 - 9b4 + 4b3 + b2 - 2) q^29 + (-b6 + b5 - 2b4 + 4b3 - b2 + 4b1 + 7) q^31 + (-4b6 - 11b4 + b3 + b1 + 5) * q^35 + (3b6 - 2b5 + 7b4 + b2 + 38) q^37 + (3b6 - 2b5 - 12b4 - b2 - 4b1 - 38) * q^41 + (b6 - b5 - 5b4 - 3b3 - 2b2 - 2b1 + 56) * q^43 + (3b6 + 5b5 + 2b4 - 9b3 - 7b2 + 2b1 - 45) * q^47 + (-8b6 - 2b5 + 4b4 + 6b3 + 4b2 + 173) q^49 + (b6 + 3b5 + 6b4 + 9b3 - b2 + 8b1 - 35) * q^53 + (b6 - 7b5 - 27b4 - 3b3 + 2b2 - 8b1 + 76) q^55 + (5b6 - 5b5 - 5b4 - 12b3 - b2 - 5b1 - 2) q^59 + (2b6 + 5b5 + 25b4 - 15b3 - 10b2 + 97) q^61 + (-7b6 + 4b5 + 2b4 + 2b3 - 7b2 - 4b1 + 30) * q^65 + (7b6 + 9b5 + 3b4 - 12b3 - b2 + 5b1 + 112) q^67 + (-5b6 - 7b5 + 4b4 + 2b3 + 7b2 - 10b1 + 213) * q^71 + (5b6 + b5 + 32b4 - 9b3 + 3b2 + 255) * q^73 + (-3b6 + 5b5 + 19b3 + 7b2 + 8b1 - 75) q^77 + (-7b6 + 7b5 - 36b4 - 10b3 - 5b2 + 4b1 + 147) * q^79 + (-8b5 + 27b4 - 11b3 - 2b2 + 9b1 + 91) q^83 - 17b4 q^85 + (5b6 - 9b5 + 20b4 + 5b3 + 15b2 - 91) q^89 + (-17b6 - 7b5 - 8b4 + 19b3 + 15b2 + 10b1 + 553) * q^91 + (-8b6 + 14b5 - b4 + 12b3 + 7b2 + 14b1 + 43) q^95 + (3b6 + 9b5 + 22b4 - 21b3 - 3b2 + 367) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{5} + 26 q^{7} + 65 q^{11} + 17 q^{13} + 119 q^{17} + 125 q^{19} - 107 q^{23} + 176 q^{25} + 6 q^{29} + 44 q^{31} + 58 q^{35} + 254 q^{37} - 219 q^{41} + 419 q^{43} - 300 q^{47} + 1171 q^{49}+ \cdots + 2536 q^{97}+O(q^{100}) \) 7 * q + 3 * q^5 + 26 * q^7 + 65 * q^11 + 17 * q^13 + 119 * q^17 + 125 * q^19 - 107 * q^23 + 176 * q^25 + 6 * q^29 + 44 * q^31 + 58 * q^35 + 254 * q^37 - 219 * q^41 + 419 * q^43 - 300 * q^47 + 1171 * q^49 - 284 * q^53 + 633 * q^55 + 46 * q^59 + 638 * q^61 + 185 * q^65 + 796 * q^67 + 1472 * q^71 + 1712 * q^73 - 586 * q^77 + 1134 * q^79 + 596 * q^83 + 51 * q^85 - 694 * q^89 + 3822 * q^91 + 229 * q^95 + 2536 * q^97

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