LMFDB - Newform orbit 450.7.g.m

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,7,Mod(307,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.307"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$N$	$=$	$450 = 2 \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$7$
Character orbit:	$[\chi]$	$=$	450.g (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,16,0,0,0,0,202] 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:	$103.524337629$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{129})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 65x^{2} + 1024$ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2\cdot 5^{2}$
Twist minimal:	no (minimal twist has level 10)
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 4 \beta_1 + 4) q^{2} - 32 \beta_1 q^{4} + (11 \beta_{3} - 45 \beta_1 + 56) q^{7} + ( - 128 \beta_1 - 128) q^{8} + (13 \beta_{3} - 13 \beta_{2} + \cdots + 596) q^{11} + (2 \beta_{2} - 199 \beta_1 - 199) q^{13}+ \cdots + (8888 \beta_{2} - 334700 \beta_1 - 334700) q^{98}+O(q^{100})$ q + (-4b1 + 4) q^2 - 32b1 q^4 + (11b3 - 45b1 + 56) * q^7 + (-128b1 - 128) q^8 + (13b3 - 13b2 + 13b1 + 596) q^11 + (2b2 - 199b1 - 199) * q^13 + (44b3 + 44b2 - 404b1) q^14 - 1024 * q^16 + (46b3 - 3069b1 + 3115) * q^17 + (34b3 + 34b2 + 2750b1) q^19 + (104b3 - 2280b1 + 2384) * q^22 + (-81b2 + 8876b1 + 8876) * q^23 + (-8b3 + 8b2 - 8b1 - 1592) q^26 + (352b2 - 1792b1 - 1792) * q^28 + (232b3 + 232b2 - 14320b1) q^29 + (349b3 - 349b2 + 349b1 - 4384) q^31 + (4096b1 - 4096) q^32 + (184b3 + 184b2 - 24736b1) q^34 + (-1356b3 - 17631b1 + 16275) * q^37 + (272b2 + 10864b1 + 10864) * q^38 + (1289b3 - 1289b2 + 1289b1 - 26268) q^41 + (-167b2 + 73468b1 + 73468) * q^43 + (416b3 + 416b2 - 18656b1) q^44 + (324b3 - 324b2 + 324b1 + 71008) q^46 + (-1791b3 - 49755b1 + 47964) * q^47 + (1111b3 + 1111b2 - 82564b1) q^49 + (-64b3 + 6304b1 - 6368) * q^52 + (-4812b2 - 28091b1 - 28091) * q^53 + (-1408b3 + 1408b2 - 1408b1 - 14336) q^56 + (1856b2 - 58208b1 - 58208) * q^58 + (-2586b3 - 2586b2 - 82390b1) q^59 + (-4577b3 + 4577b2 - 4577b1 - 65420) q^61 + (2792b3 + 20328b1 - 17536) * q^62 + 32768b1 q^64 + (-5577b3 - 196189b1 + 190612) * q^67 + (1472b2 - 99680b1 - 99680) * q^68 + (-173b3 + 173b2 - 173b1 + 261360) q^71 + (3392b2 - 232289b1 - 232289) * q^73 + (-5424b3 - 5424b2 - 135624b1) q^74 + (-1088b3 + 1088b2 - 1088b1 + 86912) q^76 + (7726b3 - 256166b1 + 263892) * q^77 + (2704b3 + 2704b2 + 179280b1) q^79 + (10312b3 + 115384b1 - 105072) * q^82 + (-6565b2 - 306232b1 - 306232) * q^83 + (668b3 - 668b2 + 668b1 + 587744) q^86 + (3328b2 - 76288b1 - 76288) * q^88 + (-1920b3 - 1920b2 + 89680b1) q^89 + (-2279b3 + 2279b2 - 2279b1 - 57752) q^91 + (2592b3 - 281440b1 + 284032) * q^92 + (-7164b3 - 7164b2 - 390876b1) q^94 + (8960b3 - 479903b1 + 488863) * q^97 + (8888b2 - 334700b1 - 334700) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 16 q^{2} + 202 q^{7} - 512 q^{8} + 2332 q^{11} - 792 q^{13} - 4096 q^{16} + 12368 q^{17} + 9328 q^{22} + 35342 q^{23} - 6336 q^{26} - 6464 q^{28} - 18932 q^{31} - 16384 q^{32} + 67812 q^{37} + 44000 q^{38}+ \cdots - 1321024 q^{98}+O(q^{100})$ 4 * q + 16 * q^2 + 202 * q^7 - 512 * q^8 + 2332 * q^11 - 792 * q^13 - 4096 * q^16 + 12368 * q^17 + 9328 * q^22 + 35342 * q^23 - 6336 * q^26 - 6464 * q^28 - 18932 * q^31 - 16384 * q^32 + 67812 * q^37 + 44000 * q^38 - 110228 * q^41 + 293538 * q^43 + 282736 * q^46 + 195438 * q^47 - 25344 * q^52 - 121988 * q^53 - 51712 * q^56 - 229120 * q^58 - 243372 * q^61 - 75728 * q^62 + 773602 * q^67 - 395776 * q^68 + 1046132 * q^71 - 922372 * q^73 + 352000 * q^76 + 1040116 * q^77 - 440912 * q^82 - 1238058 * q^83 + 2348304 * q^86 - 298496 * q^88 - 221892 * q^91 + 1130944 * q^92 + 1937532 * q^97 - 1321024 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 65x^{2} + 1024$ x^4 + 65*x^2 + 1024 :

$\beta_{1}$	$=$	$( \nu^{3} + 33\nu ) / 32$ (v^3 + 33*v) / 32
$\beta_{2}$	$=$	$( 3\nu^{3} + 160\nu^{2} + 259\nu + 5216 ) / 32$ (3v^3 + 160v^2 + 259*v + 5216) / 32
$\beta_{3}$	$=$	$( \nu^{3} - 80\nu^{2} + 113\nu - 2608 ) / 16$ (v^3 - 80v^2 + 113v - 2608) / 16

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

$\nu$	$=$	$( \beta_{3} + \beta_{2} - 5\beta_1 ) / 10$ (b3 + b2 - 5*b1) / 10
$\nu^{2}$	$=$	$( -\beta_{3} + \beta_{2} - \beta _1 - 326 ) / 10$ (-b3 + b2 - b1 - 326) / 10
$\nu^{3}$	$=$	$( -33\beta_{3} - 33\beta_{2} + 485\beta_1 ) / 10$ (-33b3 - 33b2 + 485*b1) / 10

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$ .

$n$	$101$	$127$
$\chi(n)$	$1$	$-\beta_{1}$

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}

307.1

	−	5.17891i
		6.17891i
		5.17891i
	−	6.17891i

4.00000

4.00000i

32.0000i

−261.840

−

261.840i

−128.000

128.000i

307.2

4.00000

4.00000i

32.0000i

362.840

362.840i

−128.000

128.000i

343.1

4.00000

−

4.00000i

−

32.0000i

−261.840

261.840i

−128.000

−

128.000i

343.2

4.00000

−

4.00000i

−

32.0000i

362.840

−

362.840i

−128.000

−

128.000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	450.7.g.m		4
3.b	odd	2	1		50.7.c.d		4
5.b	even	2	1		90.7.g.b		4
5.c	odd	4	1		90.7.g.b		4
5.c	odd	4	1	inner	450.7.g.m		4
15.d	odd	2	1		10.7.c.b	✓	4
15.e	even	4	1		10.7.c.b	✓	4
15.e	even	4	1		50.7.c.d		4
60.h	even	2	1		80.7.p.c		4
60.l	odd	4	1		80.7.p.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.7.c.b	✓	4	15.d	odd	2	1
10.7.c.b	✓	4	15.e	even	4	1
50.7.c.d		4	3.b	odd	2	1
50.7.c.d		4	15.e	even	4	1
80.7.p.c		4	60.h	even	2	1
80.7.p.c		4	60.l	odd	4	1
90.7.g.b		4	5.b	even	2	1
90.7.g.b		4	5.c	odd	4	1
450.7.g.m		4	1.a	even	1	1	trivial
450.7.g.m		4	5.c	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{7}^{\mathrm{new}}(450, [\chi])$ :

T_{7}^{4} - 202T_{7}^{3} + 20402T_{7}^{2} + 38382424T_{7} + 36104560144

T7^4 - 202*T7^3 + 20402*T7^2 + 38382424*T7 + 36104560144

T_{11}^{2} - 1166T_{11} - 205136

T11^2 - 1166*T11 - 205136

T_{17}^{4} - 12368T_{17}^{3} + 76483712T_{17}^{2} - 194287403104T_{17} + 246768848018884

T17^4 - 12368*T17^3 + 76483712*T17^2 - 194287403104*T17 + 246768848018884

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} - 8 T + 32)^{2}$ (T^2 - 8*T + 32)^2
$3$	$T^{4}$ T^4
$5$	$T^{4}$ T^4
$7$	$T^{4} + \cdots + 36104560144$ T^4 - 202T^3 + 20402T^2 + 38382424*T + 36104560144
$11$	$(T^{2} - 1166 T - 205136)^{2}$ (T^2 - 1166*T - 205136)^2
$13$	$T^{4} + \cdots + 5177953764$ T^4 + 792T^3 + 313632T^2 + 56990736*T + 5177953764
$17$	$T^{4} + \cdots + 246768848018884$ T^4 - 12368T^3 + 76483712T^2 - 194287403104*T + 246768848018884
$19$	$T^{4} + \cdots + 14702623360000$ T^4 + 22581200*T^2 + 14702623360000
$23$	$T^{4} + \cdots + 21\!\cdots\!64$ T^4 - 35342T^3 + 624528482T^2 - 5144116737736*T + 21185532585090064
$29$	$T^{4} + \cdots + 990990400000000$ T^4 + 757289600*T^2 + 990990400000000
$31$	$(T^{2} + 9466 T - 370406936)^{2}$ (T^2 + 9466*T - 370406936)^2
$37$	$T^{4} + \cdots + 57\!\cdots\!24$ T^4 - 67812T^3 + 2299233672T^2 + 162081081140184*T + 5712833189486037924
$41$	$(T^{2} + 55114 T - 4599016976)^{2}$ (T^2 + 55114*T - 4599016976)^2
$43$	$T^{4} + \cdots + 11\!\cdots\!24$ T^4 - 293538T^3 + 43082278722T^2 - 3148370781807384*T + 115038466787003374224
$47$	$T^{4} + \cdots + 15\!\cdots\!24$ T^4 - 195438T^3 + 19098005922T^2 + 77761678989816*T + 158311782497393424
$53$	$T^{4} + \cdots + 12\!\cdots\!24$ T^4 + 121988T^3 + 7440536072T^2 - 4327872963905816*T + 1258678421182104345124
$59$	$T^{4} + \cdots + 21\!\cdots\!00$ T^4 + 56709928400*T^2 + 218411155987600000000
$61$	$(T^{2} + 121686 T - 63858425376)^{2}$ (T^2 + 121686*T - 63858425376)^2
$67$	$T^{4} + \cdots + 60\!\cdots\!44$ T^4 - 773602T^3 + 299230027202T^2 - 19072409846464376*T + 607821382019845628944
$71$	$(T^{2} - 523066 T + 68302989064)^{2}$ (T^2 - 523066*T + 68302989064)^2
$73$	$T^{4} + \cdots + 77\!\cdots\!04$ T^4 + 922372T^3 + 425385053192T^2 + 80978155587752456*T + 7707677589031902489604
$79$	$T^{4} + \cdots + 73\!\cdots\!00$ T^4 + 111442560000*T^2 + 73296830256906240000
$83$	$T^{4} + \cdots + 14\!\cdots\!64$ T^4 + 1238058T^3 + 766393805682T^2 + 151168059326868264*T + 14908642261478206427664
$89$	$T^{4} + \cdots + 14\!\cdots\!00$ T^4 + 39862284800*T^2 + 14792774438133760000
$97$	$T^{4} + \cdots + 11\!\cdots\!84$ T^4 - 1937532T^3 + 1877015125512T^2 - 658372795010319096*T + 115463837056577975098884
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Elliptic curves over $\Q$
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

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	450.7.g.m

Level	$450$
Weight	$7$
Character orbit	450.g
Analytic conductor	$103.524$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$