LMFDB - Newform orbit 25.6.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,6,Mod(24,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$25 = 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	25.b (of order $2$ , degree $1$ , not 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:	$4.00959549532$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 121x^{2} + 3600$ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$5^{2}$
Twist minimal:	yes
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + (\beta_{3} - 35) q^{4} + ( - \beta_{3} - 95) q^{6} + (18 \beta_{2} + 4 \beta_1) q^{7} + ( - 69 \beta_{2} + 15 \beta_1) q^{8} + ( - 8 \beta_{3} - 94) q^{9}+ \cdots + ( - 196 \beta_{3} - 110798) q^{99}+O(q^{100})$ q + (b2 - b1) * q^2 + (-b2 - 2b1) q^3 + (b3 - 35) * q^4 + (-b3 - 95) * q^6 + (18b2 + 4b1) * q^7 + (-69b2 + 15b1) * q^8 + (-8b3 - 94) q^9 + (10b3 - 103) q^11 + (-61b2 + 19b1) * q^12 + (52b2 - 32b1) * q^13 + (18b3 - 18) q^14 + (-37b3 + 587) q^16 + (217b2 - 136b1) * q^17 + (434b2 - 2b1) * q^18 + (14b3 + 1583) q^19 + (48b3 + 1458) q^21 + (-763b2 + 223b1) * q^22 + (-150b2 - 12b1) * q^23 + (-93b3 - 1221) q^24 + (52b3 - 2404) q^26 + (619b2 + 110b1) * q^27 + (-630b2 + 362b1) * q^28 + (16b3 + 1952) q^29 + (-220b3 - 438) q^31 + (821b2 - 551b1) * q^32 + (-857b2 - 304b1) * q^33 + (217b3 - 10165) q^34 + (178b3 - 8758) q^36 + (10b2 + 384b1) * q^37 + (659b2 - 1415b1) * q^38 + (8b3 - 2060) q^39 + (80b3 + 13837) q^41 + (-1710b2 - 882b1) * q^42 + (-764b2 + 2128b1) * q^43 + (-443b3 + 18665) q^44 + (-150b3 + 1302) q^46 + (1804b2 + 1544b1) * q^47 + (2965b2 + 713b1) * q^48 + (-160b3 + 5923) q^49 + (26b3 - 8951) q^51 + (-4172b2 + 2004b1) * q^52 + (3074b2 - 752b1) * q^53 + (619b3 - 2107) q^54 + (-54b3 + 27162) q^56 + (-2927b2 - 3880b1) * q^57 + (896b2 - 1760b1) * q^58 + (392b3 - 6176) q^59 + (400b3 - 12398) q^61 + (14082b2 - 2202b1) * q^62 + (-1692b2 - 4392b1) * q^63 + (-363b3 - 21643) q^64 + (-857b3 - 5275) q^66 + (-3213b2 - 1586b1) * q^67 + (-17543b2 + 8417b1) * q^68 + (-336b3 - 9078) q^69 + (200b3 - 43748) q^71 + (-6618b2 + 10830b1) * q^72 + (7585b2 - 1112b1) * q^73 + (10b3 + 20606) q^74 + (1107b3 - 34321) q^76 + (-1854b2 + 4608b1) * q^77 + (-2588b2 + 2156b1) * q^78 + (-1004b3 - 32238) q^79 + (-376b3 + 23329) q^81 + (8557b2 - 12877b1) * q^82 + (-9687b2 + 858b1) * q^83 + (-174b3 + 21258) q^84 + (-764b3 + 124844) q^86 + (-3488b2 - 4720b1) * q^87 + (23487b2 - 16845b1) * q^88 + (2088b3 + 35361) q^89 + (496b3 - 10536) q^91 + (6402b2 - 3486b1) * q^92 + (21558b2 + 12096b1) * q^93 + (1804b3 + 59924) q^94 + (-11b3 - 39115) q^96 + (-18086b2 + 10944b1) * q^97 + (16483b2 - 7843b1) * q^98 + (-196b3 - 110798) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 138 q^{4} - 382 q^{6} - 392 q^{9} - 392 q^{11} - 36 q^{14} + 2274 q^{16} + 6360 q^{19} + 5928 q^{21} - 5070 q^{24} - 9512 q^{26} + 7840 q^{29} - 2192 q^{31} - 40226 q^{34} - 34676 q^{36} - 8224 q^{39}+ \cdots - 443584 q^{99}+O(q^{100})$ 4 * q - 138 * q^4 - 382 * q^6 - 392 * q^9 - 392 * q^11 - 36 * q^14 + 2274 * q^16 + 6360 * q^19 + 5928 * q^21 - 5070 * q^24 - 9512 * q^26 + 7840 * q^29 - 2192 * q^31 - 40226 * q^34 - 34676 * q^36 - 8224 * q^39 + 55508 * q^41 + 73774 * q^44 + 4908 * q^46 + 23372 * q^49 - 35752 * q^51 - 7190 * q^54 + 108540 * q^56 - 23920 * q^59 - 48792 * q^61 - 87298 * q^64 - 22814 * q^66 - 36984 * q^69 - 174592 * q^71 + 82444 * q^74 - 135070 * q^76 - 130960 * q^79 + 92564 * q^81 + 84684 * q^84 + 497848 * q^86 + 145620 * q^89 - 41152 * q^91 + 243304 * q^94 - 156482 * q^96 - 443584 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 121x^{2} + 3600$ x^4 + 121*x^2 + 3600 :

$\beta_{1}$	$=$	$( -\nu^{3} - 81\nu ) / 20$ (-v^3 - 81*v) / 20
$\beta_{2}$	$=$	$( -\nu^{3} - 61\nu ) / 12$ (-v^3 - 61*v) / 12
$\beta_{3}$	$=$	$5\nu^{2} + 303$ 5*v^2 + 303

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

$\nu$	$=$	$( 3\beta_{2} - 5\beta_1 ) / 5$ (3b2 - 5b1) / 5
$\nu^{2}$	$=$	$( \beta_{3} - 303 ) / 5$ (b3 - 303) / 5
$\nu^{3}$	$=$	$( -243\beta_{2} + 305\beta_1 ) / 5$ (-243b2 + 305b1) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$ .

$n$	$2$
$\chi(n)$	$-1$

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}

24.1

	−	8.26209i
	−	7.26209i
		7.26209i
		8.26209i

−

10.2621i

−

5.52417i

−73.3104

−56.6896

−

68.9517i

423.931i

212.483

24.2

−

5.26209i

−

25.5242i

4.31044

−134.310

131.048i

−

191.069i

−408.483

24.3

5.26209i

25.5242i

4.31044

−134.310

−

131.048i

191.069i

−408.483

24.4

10.2621i

5.52417i

−73.3104

−56.6896

68.9517i

−

423.931i

212.483

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	25.6.b.b		4
3.b	odd	2	1		225.6.b.i		4
4.b	odd	2	1		400.6.c.n		4
5.b	even	2	1	inner	25.6.b.b		4
5.c	odd	4	1		25.6.a.b	✓	2
5.c	odd	4	1		25.6.a.d	yes	2
15.d	odd	2	1		225.6.b.i		4
15.e	even	4	1		225.6.a.l		2
15.e	even	4	1		225.6.a.s		2
20.d	odd	2	1		400.6.c.n		4
20.e	even	4	1		400.6.a.o		2
20.e	even	4	1		400.6.a.w		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
25.6.a.b	✓	2	5.c	odd	4	1
25.6.a.d	yes	2	5.c	odd	4	1
25.6.b.b		4	1.a	even	1	1	trivial
25.6.b.b		4	5.b	even	2	1	inner
225.6.a.l		2	15.e	even	4	1
225.6.a.s		2	15.e	even	4	1
225.6.b.i		4	3.b	odd	2	1
225.6.b.i		4	15.d	odd	2	1
400.6.a.o		2	20.e	even	4	1
400.6.a.w		2	20.e	even	4	1
400.6.c.n		4	4.b	odd	2	1
400.6.c.n		4	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{4} + 133T_{2}^{2} + 2916$ T2^4 + 133*T2^2 + 2916 acting on $S_{6}^{\mathrm{new}}(25, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 133T^{2} + 2916$ T^4 + 133*T^2 + 2916
$3$	$T^{4} + 682 T^{2} + 19881$ T^4 + 682*T^2 + 19881
$5$	$T^{4}$ T^4
$7$	$T^{4} + 21928 T^{2} + 81649296$ T^4 + 21928*T^2 + 81649296
$11$	$(T^{2} + 196 T - 141021)^{2}$ (T^2 + 196*T - 141021)^2
$13$	$T^{4} + 188192 T^{2} + 858255616$ T^4 + 188192*T^2 + 858255616
$17$	$T^{4} + \cdots + 312882490881$ T^4 + 3338818*T^2 + 312882490881
$19$	$(T^{2} - 3180 T + 2232875)^{2}$ (T^2 - 3180*T + 2232875)^2
$23$	$T^{4} + \cdots + 359668876176$ T^4 + 1234152*T^2 + 359668876176
$29$	$(T^{2} - 3920 T + 3456000)^{2}$ (T^2 - 3920*T + 3456000)^2
$31$	$(T^{2} + 1096 T - 72602196)^{2}$ (T^2 + 1096*T - 72602196)^2
$37$	$T^{4} + \cdots + 61844446287376$ T^4 + 19808648*T^2 + 61844446287376
$41$	$(T^{2} - 27754 T + 182931129)^{2}$ (T^2 - 27754*T + 182931129)^2
$43$	$T^{4} + \cdots + 73\!\cdots\!96$ T^4 + 550170272*T^2 + 73216315824138496
$47$	$T^{4} + \cdots + 495608042209536$ T^4 + 619053088*T^2 + 495608042209536
$53$	$T^{4} + \cdots + 21\!\cdots\!56$ T^4 + 432103432*T^2 + 21876919639178256
$59$	$(T^{2} + 11960 T - 195696000)^{2}$ (T^2 + 11960*T - 195696000)^2
$61$	$(T^{2} + 24396 T - 92208796)^{2}$ (T^2 + 24396*T - 92208796)^2
$67$	$T^{4} + \cdots + 62\!\cdots\!81$ T^4 + 1105507018*T^2 + 62324269199220681
$71$	$(T^{2} + 87296 T + 1844897904)^{2}$ (T^2 + 87296*T + 1844897904)^2
$73$	$T^{4} + \cdots + 13\!\cdots\!01$ T^4 + 2619345602*T^2 + 1347153105574224001
$79$	$(T^{2} + 65480 T - 446416500)^{2}$ (T^2 + 65480*T - 446416500)^2
$83$	$T^{4} + \cdots + 44\!\cdots\!61$ T^4 + 4374235962*T^2 + 4403325447203627961
$89$	$(T^{2} - 72810 T - 5241540375)^{2}$ (T^2 - 72810*T - 5241540375)^2
$97$	$T^{4} + \cdots + 10\!\cdots\!36$ T^4 + 22388071688*T^2 + 10487144169973969936
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