LMFDB - Newform orbit 2900.1.da.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2900,1,Mod(127,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, base_ring=CyclotomicField(140))

chi = DirichletCharacter(H, H._module([70, 7, 125]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2900.127");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2900.da (of order $140$, degree $48$, 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:	$1.44728853664$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{140})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{38} - x^{36} - x^{34} - x^{32} + x^{28} + x^{26} + x^{24} + x^{22} + x^{20} + \cdots + 1 $ x^48 + x^46 - x^38 - x^36 - x^34 - x^32 + x^28 + x^26 + x^24 + x^22 + x^20 - x^16 - x^14 - x^12 - x^10 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{140}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{140} - \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{140}^{8} q^{2} + \zeta_{140}^{16} q^{4} - \zeta_{140}^{39} q^{5} + \zeta_{140}^{24} q^{8} - \zeta_{140}^{52} q^{9} +O(q^{10}) $ q + z^8 * q^2 + z^16 * q^4 - z^39 * q^5 + z^24 * q^8 - z^52 * q^9	\( q + \zeta_{140}^{8} q^{2} + \zeta_{140}^{16} q^{4} - \zeta_{140}^{39} q^{5} + \zeta_{140}^{24} q^{8} - \zeta_{140}^{52} q^{9} - \zeta_{140}^{47} q^{10} + (\zeta_{140}^{51} - \zeta_{140}^{48}) q^{13} + \zeta_{140}^{32} q^{16} + (\zeta_{140}^{57} - \zeta_{140}^{41}) q^{17} - \zeta_{140}^{60} q^{18} - \zeta_{140}^{55} q^{20} - \zeta_{140}^{8} q^{25} + (\zeta_{140}^{59} - \zeta_{140}^{56}) q^{26} - \zeta_{140}^{53} q^{29} + \zeta_{140}^{40} q^{32} + (\zeta_{140}^{65} - \zeta_{140}^{49}) q^{34} - \zeta_{140}^{68} q^{36} + (\zeta_{140}^{19} + \zeta_{140}^{15}) q^{37} - \zeta_{140}^{63} q^{40} + (\zeta_{140}^{68} - \zeta_{140}^{51}) q^{41} - \zeta_{140}^{21} q^{45} + \zeta_{140}^{45} q^{49} - \zeta_{140}^{16} q^{50} + (\zeta_{140}^{67} - \zeta_{140}^{64}) q^{52} + (\zeta_{140}^{30} - \zeta_{140}^{7}) q^{53} - \zeta_{140}^{61} q^{58} + ( - \zeta_{140}^{27} - \zeta_{140}^{4}) q^{61} + \zeta_{140}^{48} q^{64} + (\zeta_{140}^{20} - \zeta_{140}^{17}) q^{65} + ( - \zeta_{140}^{57} - \zeta_{140}^{3}) q^{68} + \zeta_{140}^{6} q^{72} + (\zeta_{140}^{54} + \zeta_{140}^{42}) q^{73} + (\zeta_{140}^{27} + \zeta_{140}^{23}) q^{74} + \zeta_{140} q^{80} - \zeta_{140}^{34} q^{81} + ( - \zeta_{140}^{59} - \zeta_{140}^{6}) q^{82} + (\zeta_{140}^{26} - \zeta_{140}^{10}) q^{85} + ( - \zeta_{140}^{46} + \zeta_{140}^{5}) q^{89} - \zeta_{140}^{29} q^{90} + ( - \zeta_{140}^{28} + \zeta_{140}^{4}) q^{97} + \zeta_{140}^{53} q^{98} +O(q^{100}) \) q + z^8 * q^2 + z^16 * q^4 - z^39 * q^5 + z^24 * q^8 - z^52 * q^9 - z^47 * q^10 + (z^51 - z^48) * q^13 + z^32 * q^16 + (z^57 - z^41) * q^17 - z^60 * q^18 - z^55 * q^20 - z^8 * q^25 + (z^59 - z^56) * q^26 - z^53 * q^29 + z^40 * q^32 + (z^65 - z^49) * q^34 - z^68 * q^36 + (z^19 + z^15) * q^37 - z^63 * q^40 + (z^68 - z^51) * q^41 - z^21 * q^45 + z^45 * q^49 - z^16 * q^50 + (z^67 - z^64) * q^52 + (z^30 - z^7) * q^53 - z^61 * q^58 + (-z^27 - z^4) * q^61 + z^48 * q^64 + (z^20 - z^17) * q^65 + (-z^57 - z^3) * q^68 + z^6 * q^72 + (z^54 + z^42) * q^73 + (z^27 + z^23) * q^74 + z * q^80 - z^34 * q^81 + (-z^59 - z^6) * q^82 + (z^26 - z^10) * q^85 + (-z^46 + z^5) * q^89 - z^29 * q^90 + (-z^28 + z^4) * q^97 + z^53 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 48 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	$ 48 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{13} + 2 q^{16} + 8 q^{18} - 2 q^{25} + 12 q^{26} - 8 q^{32} - 2 q^{36} + 2 q^{41} - 2 q^{50} - 2 q^{52} + 8 q^{53} - 2 q^{61} + 2 q^{64} - 8 q^{65} - 2 q^{72} + 10 q^{73} + 2 q^{81} + 2 q^{82} - 10 q^{85} + 2 q^{89} + 14 q^{97}+O(q^{100}) $ 48 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^13 + 2 * q^16 + 8 * q^18 - 2 * q^25 + 12 * q^26 - 8 * q^32 - 2 * q^36 + 2 * q^41 - 2 * q^50 - 2 * q^52 + 8 * q^53 - 2 * q^61 + 2 * q^64 - 8 * q^65 - 2 * q^72 + 10 * q^73 + 2 * q^81 + 2 * q^82 - 10 * q^85 + 2 * q^89 + 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$-\zeta_{140}^{65}$	$\zeta_{140}^{21}$	$-1$

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} $

127.1

−0.722795	−	0.691063i
0.998993	−	0.0448648i
−0.990950	−	0.134233i
0.351375	−	0.936235i
0.919528	−	0.393025i
0.919528	+	0.393025i
−0.990950	+	0.134233i
−0.657939	+	0.753071i
−0.657939	−	0.753071i
−0.351375	−	0.936235i
0.351375	+	0.936235i
0.657939	+	0.753071i
0.657939	−	0.753071i
0.990950	−	0.134233i
−0.919528	−	0.393025i
−0.919528	+	0.393025i
−0.351375	+	0.936235i
0.990950	+	0.134233i
−0.998993	+	0.0448648i
0.722795	+	0.691063i

0.983930

−

0.178557i

0.936235

−

0.351375i

−0.0896393

−

0.995974i

0.858449

−

0.512899i

0.393025

−

0.919528i

−0.266037

−

0.963963i

147.1

0.936235

−

0.351375i

0.753071

−

0.657939i

0.178557

0.983930i

0.473869

−

0.880596i

0.691063

0.722795i

0.512899

0.858449i

163.1

0.473869

0.880596i

−0.550897

0.834573i

0.512899

−

0.858449i

−0.995974

−

0.0896393i

−0.753071

−

0.657939i

0.998993

0.0448648i

363.1

−0.963963

0.266037i

0.858449

−

0.512899i

0.990950

−

0.134233i

−0.691063

0.722795i

−0.983930

0.178557i

−0.919528

0.393025i

367.1

−0.995974

0.0896393i

0.983930

−

0.178557i

0.998993

−

0.0448648i

−0.963963

0.266037i

0.550897

0.834573i

−0.990950

0.134233i

403.1

−0.995974

−

0.0896393i

0.983930

0.178557i

0.998993

0.0448648i

−0.963963

−

0.266037i

0.550897

−

0.834573i

−0.990950

−

0.134233i

427.1

0.473869

−

0.880596i

−0.550897

−

0.834573i

0.512899

0.858449i

−0.995974

0.0896393i

−0.753071

0.657939i

0.998993

−

0.0448648i

467.1

0.858449

−

0.512899i

0.473869

−

0.880596i

−0.266037

−

0.963963i

−0.0448648

−

0.998993i

−0.936235

0.351375i

−0.722795

−

0.691063i

503.1

0.858449

0.512899i

0.473869

0.880596i

−0.266037

0.963963i

−0.0448648

0.998993i

−0.936235

−

0.351375i

−0.722795

0.691063i

723.1

−0.963963

−

0.266037i

0.858449

0.512899i

−0.990950

−

0.134233i

−0.691063

−

0.722795i

−0.983930

−

0.178557i

0.919528

0.393025i

727.1

−0.963963

−

0.266037i

0.858449

0.512899i

0.990950

0.134233i

−0.691063

−

0.722795i

−0.983930

−

0.178557i

−0.919528

−

0.393025i

947.1

0.858449

0.512899i

0.473869

0.880596i

0.266037

−

0.963963i

−0.0448648

0.998993i

−0.936235

−

0.351375i

0.722795

−

0.691063i

983.1

0.858449

−

0.512899i

0.473869

−

0.880596i

0.266037

0.963963i

−0.0448648

−

0.998993i

−0.936235

0.351375i

0.722795

0.691063i

1023.1

0.473869

−

0.880596i

−0.550897

−

0.834573i

−0.512899

−

0.858449i

−0.995974

0.0896393i

−0.753071

0.657939i

−0.998993

0.0448648i

1047.1

−0.995974

−

0.0896393i

0.983930

0.178557i

−0.998993

−

0.0448648i

−0.963963

−

0.266037i

0.550897

−

0.834573i

0.990950

0.134233i

1083.1

−0.995974

0.0896393i

0.983930

−

0.178557i

−0.998993

0.0448648i

−0.963963

0.266037i

0.550897

0.834573i

0.990950

−

0.134233i

1087.1

−0.963963

0.266037i

0.858449

−

0.512899i

−0.990950

0.134233i

−0.691063

0.722795i

−0.983930

0.178557i

0.919528

−

0.393025i

1287.1

0.473869

0.880596i

−0.550897

0.834573i

−0.512899

0.858449i

−0.995974

−

0.0896393i

−0.753071

−

0.657939i

−0.998993

−

0.0448648i

1303.1

0.936235

−

0.351375i

0.753071

−

0.657939i

−0.178557

−

0.983930i

0.473869

−

0.880596i

0.691063

0.722795i

−0.512899

−

0.858449i

1323.1

0.983930

−

0.178557i

0.936235

−

0.351375i

0.0896393

0.995974i

0.858449

−

0.512899i

0.393025

−

0.919528i

0.266037

0.963963i

See all 48 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
725.bn	even	140	1	inner
2900.da	odd	140	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2900.1.da.a	yes	48
4.b	odd	2	1	CM	2900.1.da.a	yes	48
25.f	odd	20	1		2900.1.cr.a	✓	48
29.f	odd	28	1		2900.1.cr.a	✓	48
100.l	even	20	1		2900.1.cr.a	✓	48
116.l	even	28	1		2900.1.cr.a	✓	48
725.bn	even	140	1	inner	2900.1.da.a	yes	48
2900.da	odd	140	1	inner	2900.1.da.a	yes	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2900.1.cr.a	✓	48	25.f	odd	20	1
2900.1.cr.a	✓	48	29.f	odd	28	1
2900.1.cr.a	✓	48	100.l	even	20	1
2900.1.cr.a	✓	48	116.l	even	28	1
2900.1.da.a	yes	48	1.a	even	1	1	trivial
2900.1.da.a	yes	48	4.b	odd	2	1	CM
2900.1.da.a	yes	48	725.bn	even	140	1	inner
2900.1.da.a	yes	48	2900.da	odd	140	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2900, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{24} - T^{23} + T^{19} + \cdots + 1)^{2} $ (T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1)^2
$3$	$ T^{48} $ T^48
$5$	$ T^{48} + T^{46} + \cdots + 1 $ T^48 + T^46 - T^38 - T^36 - T^34 - T^32 + T^28 + T^26 + T^24 + T^22 + T^20 - T^16 - T^14 - T^12 - T^10 + T^2 + 1
$7$	$ T^{48} $ T^48
$11$	$ T^{48} $ T^48
$13$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 2T^46 + 10T^45 + 20T^44 + 4T^43 + 8T^42 + 46T^41 - 119T^40 - 234T^39 + 101T^38 - 40T^37 + 860T^36 + 4544T^35 + 4829T^34 + 11338T^33 + 20826T^32 + 47040T^31 + 83214T^30 + 100128T^29 + 267584T^28 + 444132T^27 + 477550T^26 + 640984T^25 + 1777193T^24 + 1587318T^23 + 549223T^22 + 1746778T^21 + 4337646T^20 + 3357048T^19 - 646851T^18 - 284846T^17 + 5125873T^16 + 3731154T^15 - 1765211T^14 - 3072700T^13 - 3044171T^12 - 122216T^11 + 3800132T^10 + 4621338T^9 + 3692186T^8 + 2325596T^7 + 1099303T^6 + 385266T^5 + 95596T^4 + 14042T^3 + 1185T^2 + 48*T + 1
$17$	$ T^{48} + 14 T^{46} + \cdots + 1 $ T^48 + 14T^46 + 109T^44 + 633T^42 + 3190T^40 + 13625T^38 + 48824T^36 + 150277T^34 + 415022T^32 + 995662T^30 + 2037255T^28 + 3605607T^26 + 5787436T^24 + 7428851T^22 + 7600811T^20 + 6601886T^18 + 5978410T^16 + 2543877T^14 + 1289991T^12 + 524303T^10 + 133276T^8 + 15376T^6 + 765T^4 - T^2 + 1
$19$	$ T^{48} $ T^48
$23$	$ T^{48} $ T^48
$29$	$ T^{48} + T^{46} + \cdots + 1 $ T^48 + T^46 - T^38 - T^36 - T^34 - T^32 + T^28 + T^26 + T^24 + T^22 + T^20 - T^16 - T^14 - T^12 - T^10 + T^2 + 1
$31$	$ T^{48} $ T^48
$37$	$ T^{48} - T^{46} + \cdots + 1 $ T^48 - T^46 - 5T^44 - 20T^42 - 347T^40 + 371T^38 + 2840T^36 + 3857T^34 + 32923T^32 - 63843T^30 - 348999T^28 + 40612T^26 + 444901T^24 - 107682T^22 + 4603243T^20 + 10905756T^18 + 16703324T^16 + 10819856T^14 + 4065564T^12 + 2611695T^10 + 3203118T^8 + 298369T^6 + 11917T^4 - 150T^2 + 1
$41$	$ T^{48} - 2 T^{47} + \cdots + 1 $ T^48 - 2T^47 + 2T^46 - 10T^45 - T^44 + 24T^43 + 29T^42 + 66T^41 + 14T^40 - 690T^39 + 59T^38 - 632T^37 + 1448T^36 + 7020T^35 - 5265T^34 + 3782T^33 - 4178T^32 - 48440T^31 + 66876T^30 - 83454T^29 + 99675T^28 + 115728T^27 - 131961T^26 - 73970T^25 + 679530T^24 - 2955482T^23 + 5249415T^22 - 7018520T^21 + 8217249T^20 - 3755544T^19 - 1330562T^18 + 543230T^17 - 569250T^16 + 1078224T^15 - 119959T^14 + 58654T^13 + 1018447T^12 + 127984T^11 - 934913T^10 - 1207130T^9 - 363992T^8 + 682570T^7 + 768392T^6 + 334712T^5 + 73427T^4 + 9058T^3 + 737T^2 + 22T + 1
$43$	$ T^{48} $ T^48
$47$	$ T^{48} $ T^48
$53$	$ T^{48} - 8 T^{47} + \cdots + 1 $ T^48 - 8T^47 + 30T^46 - 74T^45 + 154T^44 - 326T^43 + 688T^42 - 1310T^41 + 2241T^40 - 3642T^39 + 5680T^38 - 8578T^37 + 13909T^36 - 22516T^35 + 23807T^34 - 630T^33 - 35084T^32 + 20316T^31 + 157394T^30 - 631014T^29 + 1392270T^28 - 2101986T^27 + 2344502T^26 - 1684576T^25 - 797398T^24 + 5613402T^23 - 11451976T^22 + 14737966T^21 - 11106176T^20 + 2093798T^19 + 4603703T^18 - 5645962T^17 + 5044300T^16 - 7946592T^15 + 15777844T^14 - 19685536T^13 + 16352208T^12 - 9977868T^11 + 3628037T^10 - 83660T^9 - 585670T^8 + 194834T^7 + 108473T^6 - 62154T^5 + 15765T^4 + 2588T^3 + 628T^2 + 38*T + 1
$59$	$ T^{48} $ T^48
$61$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 2T^46 - 16T^43 + 83T^42 + 146T^41 + 91T^40 - 184T^39 + 101T^38 - 840T^37 + 3525T^36 + 4674T^35 + 2694T^34 - 8212T^33 + 7321T^32 - 17670T^31 + 64744T^30 + 24858T^29 - 6536T^28 - 92048T^27 + 140245T^26 - 261786T^25 + 559973T^24 + 657058T^23 + 168268T^22 + 75598T^21 - 777144T^20 + 44138T^19 + 2969169T^18 - 2845556T^17 - 224182T^16 - 4385636T^15 + 4663779T^14 - 861990T^13 + 2885209T^12 - 126286T^11 + 1889802T^10 + 1154948T^9 + 442706T^8 + 295506T^7 + 142838T^6 + 60616T^5 - 5939T^4 - 4688T^3 + 710T^2 - 32*T + 1
$67$	$ T^{48} $ T^48
$71$	$ T^{48} $ T^48
$73$	$ (T^{24} - 5 T^{23} + \cdots + 1)^{2} $ (T^24 - 5T^23 + 14T^22 - 33T^21 + 72T^20 - 132T^19 + 210T^18 - 324T^17 + 438T^16 - 461T^15 + 519T^14 - 975T^13 + 2504T^12 - 3438T^11 + 1620T^10 + 608T^9 - 2696T^8 + 3687T^7 - 1308T^6 - 1098T^5 + 1526T^4 + 124T^3 + 75T^2 - 18*T + 1)^2
$79$	$ T^{48} $ T^48
$83$	$ T^{48} $ T^48
$89$	$ T^{48} - 2 T^{47} + \cdots + 1 $ T^48 - 2T^47 - 3T^46 - 4T^45 + 33T^44 + 14T^43 - 16T^42 - 300T^41 + 69T^40 + 472T^39 + 1651T^38 - 2172T^37 - 1996T^36 - 4316T^35 + 12149T^34 + 2320T^33 + 18967T^32 - 84622T^31 + 47983T^30 - 16612T^29 + 228931T^28 - 425508T^27 + 640868T^26 - 1012866T^25 + 946609T^24 - 1255706T^23 + 1169466T^22 + 39564T^21 + 1034607T^20 + 309370T^19 - 3153068T^18 - 1104500T^17 - 1824016T^16 + 6450358T^15 + 606946T^14 + 487516T^13 - 479266T^12 - 410992T^11 + 1431497T^10 - 1252532T^9 + 960926T^8 - 859538T^7 + 625377T^6 - 280880T^5 + 70015T^4 - 9170T^3 + 760T^2 - 24*T + 1
$97$	$ (T^{24} - 7 T^{23} + \cdots + 1)^{2} $ (T^24 - 7T^23 + 28T^22 - 79T^21 + 170T^20 - 280T^19 + 340T^18 - 274T^17 + 151T^16 - 330T^15 + 1165T^14 - 2600T^13 + 3524T^12 - 2766T^11 + 726T^10 + 1794T^9 - 2064T^8 + 1220T^7 + 97T^6 - 33T^5 + 138T^4 - 132T^3 + 64T^2 - 13*T + 1)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2900.da (of order \(140\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{140}^{8} q^{2} + \zeta_{140}^{16} q^{4} - \zeta_{140}^{39} q^{5} + \zeta_{140}^{24} q^{8} - \zeta_{140}^{52} q^{9} +O(q^{10}) \) q + z^8 * q^2 + z^16 * q^4 - z^39 * q^5 + z^24 * q^8 - z^52 * q^9	\( q + \zeta_{140}^{8} q^{2} + \zeta_{140}^{16} q^{4} - \zeta_{140}^{39} q^{5} + \zeta_{140}^{24} q^{8} - \zeta_{140}^{52} q^{9} - \zeta_{140}^{47} q^{10} + (\zeta_{140}^{51} - \zeta_{140}^{48}) q^{13} + \zeta_{140}^{32} q^{16} + (\zeta_{140}^{57} - \zeta_{140}^{41}) q^{17} - \zeta_{140}^{60} q^{18} - \zeta_{140}^{55} q^{20} - \zeta_{140}^{8} q^{25} + (\zeta_{140}^{59} - \zeta_{140}^{56}) q^{26} - \zeta_{140}^{53} q^{29} + \zeta_{140}^{40} q^{32} + (\zeta_{140}^{65} - \zeta_{140}^{49}) q^{34} - \zeta_{140}^{68} q^{36} + (\zeta_{140}^{19} + \zeta_{140}^{15}) q^{37} - \zeta_{140}^{63} q^{40} + (\zeta_{140}^{68} - \zeta_{140}^{51}) q^{41} - \zeta_{140}^{21} q^{45} + \zeta_{140}^{45} q^{49} - \zeta_{140}^{16} q^{50} + (\zeta_{140}^{67} - \zeta_{140}^{64}) q^{52} + (\zeta_{140}^{30} - \zeta_{140}^{7}) q^{53} - \zeta_{140}^{61} q^{58} + ( - \zeta_{140}^{27} - \zeta_{140}^{4}) q^{61} + \zeta_{140}^{48} q^{64} + (\zeta_{140}^{20} - \zeta_{140}^{17}) q^{65} + ( - \zeta_{140}^{57} - \zeta_{140}^{3}) q^{68} + \zeta_{140}^{6} q^{72} + (\zeta_{140}^{54} + \zeta_{140}^{42}) q^{73} + (\zeta_{140}^{27} + \zeta_{140}^{23}) q^{74} + \zeta_{140} q^{80} - \zeta_{140}^{34} q^{81} + ( - \zeta_{140}^{59} - \zeta_{140}^{6}) q^{82} + (\zeta_{140}^{26} - \zeta_{140}^{10}) q^{85} + ( - \zeta_{140}^{46} + \zeta_{140}^{5}) q^{89} - \zeta_{140}^{29} q^{90} + ( - \zeta_{140}^{28} + \zeta_{140}^{4}) q^{97} + \zeta_{140}^{53} q^{98} +O(q^{100}) \) q + z^8 * q^2 + z^16 * q^4 - z^39 * q^5 + z^24 * q^8 - z^52 * q^9 - z^47 * q^10 + (z^51 - z^48) * q^13 + z^32 * q^16 + (z^57 - z^41) * q^17 - z^60 * q^18 - z^55 * q^20 - z^8 * q^25 + (z^59 - z^56) * q^26 - z^53 * q^29 + z^40 * q^32 + (z^65 - z^49) * q^34 - z^68 * q^36 + (z^19 + z^15) * q^37 - z^63 * q^40 + (z^68 - z^51) * q^41 - z^21 * q^45 + z^45 * q^49 - z^16 * q^50 + (z^67 - z^64) * q^52 + (z^30 - z^7) * q^53 - z^61 * q^58 + (-z^27 - z^4) * q^61 + z^48 * q^64 + (z^20 - z^17) * q^65 + (-z^57 - z^3) * q^68 + z^6 * q^72 + (z^54 + z^42) * q^73 + (z^27 + z^23) * q^74 + z * q^80 - z^34 * q^81 + (-z^59 - z^6) * q^82 + (z^26 - z^10) * q^85 + (-z^46 + z^5) * q^89 - z^29 * q^90 + (-z^28 + z^4) * q^97 + z^53 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 48 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	\( 48 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{13} + 2 q^{16} + 8 q^{18} - 2 q^{25} + 12 q^{26} - 8 q^{32} - 2 q^{36} + 2 q^{41} - 2 q^{50} - 2 q^{52} + 8 q^{53} - 2 q^{61} + 2 q^{64} - 8 q^{65} - 2 q^{72} + 10 q^{73} + 2 q^{81} + 2 q^{82} - 10 q^{85} + 2 q^{89} + 14 q^{97}+O(q^{100}) \) 48 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^13 + 2 * q^16 + 8 * q^18 - 2 * q^25 + 12 * q^26 - 8 * q^32 - 2 * q^36 + 2 * q^41 - 2 * q^50 - 2 * q^52 + 8 * q^53 - 2 * q^61 + 2 * q^64 - 8 * q^65 - 2 * q^72 + 10 * q^73 + 2 * q^81 + 2 * q^82 - 10 * q^85 + 2 * q^89 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	2900.1.da.a

Level	$2900$
Weight	$1$
Character orbit	2900.da
Analytic conductor	$1.447$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{140}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.44728853664\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Coefficient field:	\(\Q(\zeta_{140})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{48} + x^{46} - x^{38} - x^{36} - x^{34} - x^{32} + x^{28} + x^{26} + x^{24} + x^{22} + x^{20} + \cdots + 1 \) x^48 + x^46 - x^38 - x^36 - x^34 - x^32 + x^28 + x^26 + x^24 + x^22 + x^20 - x^16 - x^14 - x^12 - x^10 + x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{140}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{140} - \cdots)\)

\(n\)	\(901\)	\(1277\)	\(1451\)
\(\chi(n)\)	\(-\zeta_{140}^{65}\)	\(\zeta_{140}^{21}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 127.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
725.bn	even	140	1	inner
2900.da	odd	140	1	inner

$p$	$F_p(T)$
$2$	\( (T^{24} - T^{23} + T^{19} + \cdots + 1)^{2} \) (T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1)^2
$3$	\( T^{48} \) T^48
$5$	\( T^{48} + T^{46} + \cdots + 1 \) T^48 + T^46 - T^38 - T^36 - T^34 - T^32 + T^28 + T^26 + T^24 + T^22 + T^20 - T^16 - T^14 - T^12 - T^10 + T^2 + 1
$7$	\( T^{48} \) T^48
$11$	\( T^{48} \) T^48
$13$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 2T^46 + 10T^45 + 20T^44 + 4T^43 + 8T^42 + 46T^41 - 119T^40 - 234T^39 + 101T^38 - 40T^37 + 860T^36 + 4544T^35 + 4829T^34 + 11338T^33 + 20826T^32 + 47040T^31 + 83214T^30 + 100128T^29 + 267584T^28 + 444132T^27 + 477550T^26 + 640984T^25 + 1777193T^24 + 1587318T^23 + 549223T^22 + 1746778T^21 + 4337646T^20 + 3357048T^19 - 646851T^18 - 284846T^17 + 5125873T^16 + 3731154T^15 - 1765211T^14 - 3072700T^13 - 3044171T^12 - 122216T^11 + 3800132T^10 + 4621338T^9 + 3692186T^8 + 2325596T^7 + 1099303T^6 + 385266T^5 + 95596T^4 + 14042T^3 + 1185T^2 + 48*T + 1
$17$	\( T^{48} + 14 T^{46} + \cdots + 1 \) T^48 + 14T^46 + 109T^44 + 633T^42 + 3190T^40 + 13625T^38 + 48824T^36 + 150277T^34 + 415022T^32 + 995662T^30 + 2037255T^28 + 3605607T^26 + 5787436T^24 + 7428851T^22 + 7600811T^20 + 6601886T^18 + 5978410T^16 + 2543877T^14 + 1289991T^12 + 524303T^10 + 133276T^8 + 15376T^6 + 765T^4 - T^2 + 1
$19$	\( T^{48} \) T^48
$23$	\( T^{48} \) T^48
$29$	\( T^{48} + T^{46} + \cdots + 1 \) T^48 + T^46 - T^38 - T^36 - T^34 - T^32 + T^28 + T^26 + T^24 + T^22 + T^20 - T^16 - T^14 - T^12 - T^10 + T^2 + 1
$31$	\( T^{48} \) T^48
$37$	\( T^{48} - T^{46} + \cdots + 1 \) T^48 - T^46 - 5T^44 - 20T^42 - 347T^40 + 371T^38 + 2840T^36 + 3857T^34 + 32923T^32 - 63843T^30 - 348999T^28 + 40612T^26 + 444901T^24 - 107682T^22 + 4603243T^20 + 10905756T^18 + 16703324T^16 + 10819856T^14 + 4065564T^12 + 2611695T^10 + 3203118T^8 + 298369T^6 + 11917T^4 - 150T^2 + 1
$41$	\( T^{48} - 2 T^{47} + \cdots + 1 \) T^48 - 2T^47 + 2T^46 - 10T^45 - T^44 + 24T^43 + 29T^42 + 66T^41 + 14T^40 - 690T^39 + 59T^38 - 632T^37 + 1448T^36 + 7020T^35 - 5265T^34 + 3782T^33 - 4178T^32 - 48440T^31 + 66876T^30 - 83454T^29 + 99675T^28 + 115728T^27 - 131961T^26 - 73970T^25 + 679530T^24 - 2955482T^23 + 5249415T^22 - 7018520T^21 + 8217249T^20 - 3755544T^19 - 1330562T^18 + 543230T^17 - 569250T^16 + 1078224T^15 - 119959T^14 + 58654T^13 + 1018447T^12 + 127984T^11 - 934913T^10 - 1207130T^9 - 363992T^8 + 682570T^7 + 768392T^6 + 334712T^5 + 73427T^4 + 9058T^3 + 737T^2 + 22T + 1
$43$	\( T^{48} \) T^48
$47$	\( T^{48} \) T^48
$53$	\( T^{48} - 8 T^{47} + \cdots + 1 \) T^48 - 8T^47 + 30T^46 - 74T^45 + 154T^44 - 326T^43 + 688T^42 - 1310T^41 + 2241T^40 - 3642T^39 + 5680T^38 - 8578T^37 + 13909T^36 - 22516T^35 + 23807T^34 - 630T^33 - 35084T^32 + 20316T^31 + 157394T^30 - 631014T^29 + 1392270T^28 - 2101986T^27 + 2344502T^26 - 1684576T^25 - 797398T^24 + 5613402T^23 - 11451976T^22 + 14737966T^21 - 11106176T^20 + 2093798T^19 + 4603703T^18 - 5645962T^17 + 5044300T^16 - 7946592T^15 + 15777844T^14 - 19685536T^13 + 16352208T^12 - 9977868T^11 + 3628037T^10 - 83660T^9 - 585670T^8 + 194834T^7 + 108473T^6 - 62154T^5 + 15765T^4 + 2588T^3 + 628T^2 + 38*T + 1
$59$	\( T^{48} \) T^48
$61$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 2T^46 - 16T^43 + 83T^42 + 146T^41 + 91T^40 - 184T^39 + 101T^38 - 840T^37 + 3525T^36 + 4674T^35 + 2694T^34 - 8212T^33 + 7321T^32 - 17670T^31 + 64744T^30 + 24858T^29 - 6536T^28 - 92048T^27 + 140245T^26 - 261786T^25 + 559973T^24 + 657058T^23 + 168268T^22 + 75598T^21 - 777144T^20 + 44138T^19 + 2969169T^18 - 2845556T^17 - 224182T^16 - 4385636T^15 + 4663779T^14 - 861990T^13 + 2885209T^12 - 126286T^11 + 1889802T^10 + 1154948T^9 + 442706T^8 + 295506T^7 + 142838T^6 + 60616T^5 - 5939T^4 - 4688T^3 + 710T^2 - 32*T + 1
$67$	\( T^{48} \) T^48
$71$	\( T^{48} \) T^48
$73$	\( (T^{24} - 5 T^{23} + \cdots + 1)^{2} \) (T^24 - 5T^23 + 14T^22 - 33T^21 + 72T^20 - 132T^19 + 210T^18 - 324T^17 + 438T^16 - 461T^15 + 519T^14 - 975T^13 + 2504T^12 - 3438T^11 + 1620T^10 + 608T^9 - 2696T^8 + 3687T^7 - 1308T^6 - 1098T^5 + 1526T^4 + 124T^3 + 75T^2 - 18*T + 1)^2
$79$	\( T^{48} \) T^48
$83$	\( T^{48} \) T^48
$89$	\( T^{48} - 2 T^{47} + \cdots + 1 \) T^48 - 2T^47 - 3T^46 - 4T^45 + 33T^44 + 14T^43 - 16T^42 - 300T^41 + 69T^40 + 472T^39 + 1651T^38 - 2172T^37 - 1996T^36 - 4316T^35 + 12149T^34 + 2320T^33 + 18967T^32 - 84622T^31 + 47983T^30 - 16612T^29 + 228931T^28 - 425508T^27 + 640868T^26 - 1012866T^25 + 946609T^24 - 1255706T^23 + 1169466T^22 + 39564T^21 + 1034607T^20 + 309370T^19 - 3153068T^18 - 1104500T^17 - 1824016T^16 + 6450358T^15 + 606946T^14 + 487516T^13 - 479266T^12 - 410992T^11 + 1431497T^10 - 1252532T^9 + 960926T^8 - 859538T^7 + 625377T^6 - 280880T^5 + 70015T^4 - 9170T^3 + 760T^2 - 24*T + 1
$97$	\( (T^{24} - 7 T^{23} + \cdots + 1)^{2} \) (T^24 - 7T^23 + 28T^22 - 79T^21 + 170T^20 - 280T^19 + 340T^18 - 274T^17 + 151T^16 - 330T^15 + 1165T^14 - 2600T^13 + 3524T^12 - 2766T^11 + 726T^10 + 1794T^9 - 2064T^8 + 1220T^7 + 97T^6 - 33T^5 + 138T^4 - 132T^3 + 64T^2 - 13*T + 1)^2
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