LMFDB - Newform orbit 3525.1.bd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,1,Mod(101,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, base_ring=CyclotomicField(46))

chi = DirichletCharacter(H, H._module([23, 0, 32]))

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

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

chi := DirichletCharacter("3525.101");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3525.bd (of order $46$, degree $22$, 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:	$1.75920416953$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$2$ over $\Q(\zeta_{46})$
Coefficient field:	$\Q(\zeta_{92})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{44} - x^{42} + x^{40} - x^{38} + x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} + \cdots + 1 $ x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Projective image:	$D_{23}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{23} - \cdots)$

$q$-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_{92}^{25} + \zeta_{92}^{3}) q^{2} + \zeta_{92}^{13} q^{3} + ( - \zeta_{92}^{28} + \cdots - \zeta_{92}^{4}) q^{4}+ \cdots + \zeta_{92}^{26} q^{9}+O(q^{10}) $ q + (-z^25 + z^3) * q^2 + z^13 * q^3 + (-z^28 + z^6 - z^4) * q^4 + (-z^38 + z^16) * q^6 + (-z^31 + z^29 + z^9 + z^7) * q^8 + z^26 * q^9	$ q + ( - \zeta_{92}^{25} + \zeta_{92}^{3}) q^{2} + \zeta_{92}^{13} q^{3} + ( - \zeta_{92}^{28} + \cdots - \zeta_{92}^{4}) q^{4}+ \cdots + ( - \zeta_{92}^{39} + \zeta_{92}^{17}) q^{98}+O(q^{100}) $ q + (-z^25 + z^3) * q^2 + z^13 * q^3 + (-z^28 + z^6 - z^4) * q^4 + (-z^38 + z^16) * q^6 + (-z^31 + z^29 + z^9 + z^7) * q^8 + z^26 * q^9 + (-z^41 + z^19 - z^17) * q^12 + (-z^34 + z^32 + z^12 + z^10 + z^8) * q^16 + (-z^45 + z^31) * q^17 + (z^29 + z^5) * q^18 + (-z^40 + z^30) * q^19 + (z^41 - z^23) * q^23 + (-z^44 + z^42 + z^22 - z^20) * q^24 + z^39 * q^27 + (-z^18 - z^2) * q^31 + (-z^37 + z^35 - z^33 + z^15 + z^13 - z^11) * q^32 + (z^34 - z^24 + z^10 + z^2) * q^34 + (z^32 - z^30 + z^8) * q^36 + (-z^43 + z^33 - z^19 + z^9) * q^38 + (z^44 - z^26 + z^20 - z^2) * q^46 - z^21 * q^47 + (z^45 + z^25 - z^23 + z^21 + z) * q^48 + z^14 * q^49 + (z^44 + z^12) * q^51 + (-z^7 + z) * q^53 + (z^42 + z^18) * q^54 + (z^43 + z^7) * q^57 + (-z^38 + z^4) * q^61 + (z^43 + z^27 - z^21 - z^5) * q^62 + (-z^40 + z^38 - z^36 + z^18 + z^16 - z^14 - z^12) * q^64 + (z^37 - z^35 - z^27 + z^13 + z^5 - z^3) * q^68 + (-z^36 - z^8) * q^69 + (z^35 - z^33 + z^11 - z^9) * q^72 + (z^44 + z^36 - z^34 - z^22 + z^12 + 1) * q^76 + (-z^32 - z^16) * q^79 - z^6 * q^81 + (-z^35 - z^27) * q^83 + (-z^45 - z^29 + z^27 + z^23 - z^5 - z) * q^92 + (-z^31 - z^15) * q^93 + (-z^24 - 1) * q^94 + (z^28 - z^26 + z^24 + z^4 - z^2 + 1) * q^96 + (-z^39 + z^17) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 6 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) $ 44 * q + 6 * q^4 - 4 * q^6 + 2 * q^9	$ 44 q + 6 q^{4} - 4 q^{6} + 2 q^{9} - 10 q^{16} + 4 q^{19} + 8 q^{24} - 4 q^{31} + 8 q^{34} - 6 q^{36} - 8 q^{46} + 2 q^{49} - 4 q^{51} + 4 q^{54} - 4 q^{61} + 14 q^{64} + 4 q^{69} + 34 q^{76} + 4 q^{79} - 2 q^{81} - 42 q^{94} + 34 q^{96}+O(q^{100}) $ 44 * q + 6 * q^4 - 4 * q^6 + 2 * q^9 - 10 * q^16 + 4 * q^19 + 8 * q^24 - 4 * q^31 + 8 * q^34 - 6 * q^36 - 8 * q^46 + 2 * q^49 - 4 * q^51 + 4 * q^54 - 4 * q^61 + 14 * q^64 + 4 * q^69 + 34 * q^76 + 4 * q^79 - 2 * q^81 - 42 * q^94 + 34 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$.

$n$	$1552$	$2026$	$2351$
$\chi(n)$	$1$	$-\zeta_{92}^{22}$	$-1$

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} $

101.1

−0.942261	−	0.334880i
0.942261	+	0.334880i
0.398401	+	0.917211i
−0.398401	−	0.917211i
0.269797	+	0.962917i
−0.269797	−	0.962917i
0.887885	−	0.460065i
−0.887885	+	0.460065i
−0.997669	−	0.0682424i
0.997669	+	0.0682424i
−0.631088	−	0.775711i
0.631088	+	0.775711i
0.519584	+	0.854419i
−0.519584	−	0.854419i
0.136167	+	0.990686i
−0.136167	−	0.990686i
−0.979084	+	0.203456i
0.979084	−	0.203456i
0.269797	−	0.962917i
−0.269797	+	0.962917i

−1.15067

−

0.0787081i

0.269797

0.962917i

0.327165

0.0449678i

−0.234658

−

1.12924i

0.756317

0.157164i

−0.854419

0.519584i

101.2

1.15067

0.0787081i

−0.269797

−

0.962917i

0.327165

0.0449678i

−0.234658

−

1.12924i

−0.756317

−

0.157164i

−0.854419

0.519584i

251.1

−0.211425

0.347674i

−0.816970

0.576680i

0.383889

0.740871i

−0.0277687

−

0.405963i

−0.744708

−

0.0509395i

0.334880

−

0.942261i

251.2

0.211425

−

0.347674i

0.816970

−

0.576680i

0.383889

0.740871i

−0.0277687

−

0.405963i

0.744708

0.0509395i

0.334880

−

0.942261i

401.1

−1.25042

−

1.53697i

−0.398401

−

0.917211i

−0.595279

2.86464i

−0.911560

1.75923i

3.38799

−

1.75551i

−0.682553

0.730836i

401.2

1.25042

1.53697i

0.398401

0.917211i

−0.595279

2.86464i

−0.911560

1.75923i

−3.38799

1.75551i

−0.682553

0.730836i

476.1

−0.680803

−

1.56737i

0.997669

0.0682424i

−1.31059

1.40330i

−0.572255

−

1.61017i

1.48157

0.526549i

0.990686

0.136167i

476.2

0.680803

1.56737i

−0.997669

−

0.0682424i

−1.31059

1.40330i

−0.572255

−

1.61017i

−1.48157

−

0.526549i

0.990686

0.136167i

551.1

−1.11525

0.787230i

−0.631088

−

0.775711i

0.289174

−

0.813657i

1.31448

0.368301i

−0.0502675

−

0.179407i

−0.203456

0.979084i

551.2

1.11525

−

0.787230i

0.631088

0.775711i

0.289174

−

0.813657i

1.31448

0.368301i

0.0502675

0.179407i

−0.203456

0.979084i

776.1

−0.0911989

−

0.663521i

−0.519584

0.854419i

0.530974

−

0.148772i

0.614311

0.266833i

−0.413970

−

0.953056i

−0.460065

−

0.887885i

776.2

0.0911989

0.663521i

0.519584

−

0.854419i

0.530974

−

0.148772i

0.614311

0.266833i

0.413970

0.953056i

−0.460065

−

0.887885i

1001.1

−1.88555

−

0.391823i

0.730836

0.682553i

2.48458

1.07920i

−1.11059

−

1.57335i

−2.68860

−

1.89782i

0.0682424

0.997669i

1001.2

1.88555

0.391823i

−0.730836

−

0.682553i

2.48458

1.07920i

−1.11059

−

1.57335i

2.68860

1.89782i

0.0682424

0.997669i

1076.1

−0.128604

0.0457060i

0.979084

−

0.203456i

−0.761261

0.619332i

−0.116615

0.0709153i

0.140510

−

0.231058i

0.917211

−

0.398401i

1076.2

0.128604

−

0.0457060i

−0.979084

0.203456i

−0.761261

0.619332i

−0.116615

0.0709153i

−0.140510

0.231058i

0.917211

−

0.398401i

1226.1

−0.418569

1.49389i

0.887885

0.460065i

−1.20209

−

0.731009i

−1.05893

1.13384i

0.461371

−

0.430891i

0.576680

0.816970i

1226.2

0.418569

−

1.49389i

−0.887885

−

0.460065i

−1.20209

−

0.731009i

−1.05893

1.13384i

−0.461371

0.430891i

0.576680

0.816970i

1301.1

−1.25042

1.53697i

−0.398401

0.917211i

−0.595279

−

2.86464i

−0.911560

−

1.75923i

3.38799

1.75551i

−0.682553

−

0.730836i

1301.2

1.25042

−

1.53697i

0.398401

−

0.917211i

−0.595279

−

2.86464i

−0.911560

−

1.75923i

−3.38799

−

1.75551i

−0.682553

−

0.730836i

See all 44 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
47.c	even	23	1	inner
141.h	odd	46	1	inner
235.i	even	46	1	inner
705.p	odd	46	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3525.1.bd.a		44
3.b	odd	2	1	inner	3525.1.bd.a		44
5.b	even	2	1	inner	3525.1.bd.a		44
5.c	odd	4	1		705.1.p.a	✓	22
5.c	odd	4	1		705.1.p.b	yes	22
15.d	odd	2	1	CM	3525.1.bd.a		44
15.e	even	4	1		705.1.p.a	✓	22
15.e	even	4	1		705.1.p.b	yes	22
47.c	even	23	1	inner	3525.1.bd.a		44
141.h	odd	46	1	inner	3525.1.bd.a		44
235.i	even	46	1	inner	3525.1.bd.a		44
235.k	odd	92	1		705.1.p.a	✓	22
235.k	odd	92	1		705.1.p.b	yes	22
705.p	odd	46	1	inner	3525.1.bd.a		44
705.w	even	92	1		705.1.p.a	✓	22
705.w	even	92	1		705.1.p.b	yes	22

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
705.1.p.a	✓	22	5.c	odd	4	1
705.1.p.a	✓	22	15.e	even	4	1
705.1.p.a	✓	22	235.k	odd	92	1
705.1.p.a	✓	22	705.w	even	92	1
705.1.p.b	yes	22	5.c	odd	4	1
705.1.p.b	yes	22	15.e	even	4	1
705.1.p.b	yes	22	235.k	odd	92	1
705.1.p.b	yes	22	705.w	even	92	1
3525.1.bd.a		44	1.a	even	1	1	trivial
3525.1.bd.a		44	3.b	odd	2	1	inner
3525.1.bd.a		44	5.b	even	2	1	inner
3525.1.bd.a		44	15.d	odd	2	1	CM
3525.1.bd.a		44	47.c	even	23	1	inner
3525.1.bd.a		44	141.h	odd	46	1	inner
3525.1.bd.a		44	235.i	even	46	1	inner
3525.1.bd.a		44	705.p	odd	46	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3525, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{44} - 4 T^{42} + \cdots + 1 $ T^44 - 4T^42 + 16T^40 - 64T^38 + 210T^36 - 288T^34 + 1152T^32 - 4608T^30 + 19145T^28 + 9854T^26 + 23788T^24 - 94669T^22 + 374444T^20 - 720146T^18 + 189032T^16 - 17644T^14 + 55534T^12 + 292144T^10 + 141757T^8 + 19633T^6 + 2244T^4 - 75*T^2 + 1
$3$	$ T^{44} - T^{42} + \cdots + 1 $ T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	$ T^{44} $ T^44
$7$	$ T^{44} $ T^44
$11$	$ T^{44} $ T^44
$13$	$ T^{44} $ T^44
$17$	$ T^{44} - 4 T^{42} + \cdots + 1 $ T^44 - 4T^42 - 30T^40 + 120T^38 + 417T^36 - 1668T^34 - 2988T^32 + 11952T^30 + 14936T^28 - 59353T^26 - 14714T^24 + 95932T^22 + 235202T^20 - 358908T^18 + 551144T^16 + 25688T^14 + 565812T^12 - 83239T^10 + 112593T^8 + 3510T^6 + 2911T^4 - 6*T^2 + 1
$19$	$ (T^{22} - 2 T^{21} + \cdots + 1)^{2} $ (T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 41T^16 - 13T^15 + 26T^14 - 52T^13 + 104T^12 - 208T^11 + 554T^10 + 341T^9 + 54T^8 - 85T^7 + 170T^6 - 340T^5 + 519T^4 - 348T^3 + 98T^2 - 12*T + 1)^2
$23$	$ T^{44} + 19 T^{42} + \cdots + 1 $ T^44 + 19T^42 + 177T^40 + 1063T^38 + 4603T^36 + 15237T^34 + 39999T^32 + 85161T^30 + 149670T^28 + 218510T^26 + 270026T^24 + 271974T^22 + 264136T^20 + 91064T^18 + 385636T^16 - 561916T^14 + 858441T^12 - 628661T^10 + 260161T^8 - 51161T^6 + 4521T^4 - 121*T^2 + 1
$29$	$ T^{44} $ T^44
$31$	$ (T^{22} + 2 T^{21} + \cdots + 1)^{2} $ (T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 41T^16 + 13T^15 + 26T^14 + 52T^13 + 104T^12 + 208T^11 + 554T^10 - 341T^9 + 54T^8 + 85T^7 + 170T^6 + 340T^5 + 519T^4 + 348T^3 + 98T^2 + 12*T + 1)^2
$37$	$ T^{44} $ T^44
$41$	$ T^{44} $ T^44
$43$	$ T^{44} $ T^44
$47$	$ T^{44} - T^{42} + \cdots + 1 $ T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$53$	$ T^{44} - 4 T^{42} + \cdots + 1 $ T^44 - 4T^42 + 16T^40 - 64T^38 + 256T^36 - 58T^34 + 232T^32 - 882T^30 - 9030T^28 + 36120T^26 + 32367T^24 - 114173T^22 + 419662T^20 - 140316T^18 + 556319T^16 + 177971T^14 + 178423T^12 - 310663T^10 + 13256T^8 + 19817T^6 + 2635T^4 - 29*T^2 + 1
$59$	$ T^{44} $ T^44
$61$	$ (T^{22} + 2 T^{21} + \cdots + 1)^{2} $ (T^22 + 2T^21 + 4T^20 - 15T^19 - 30T^18 - 60T^17 + 87T^16 + 174T^15 + 348T^14 - 224T^13 - 448T^12 - 896T^11 + 301T^10 + 579T^9 + 1158T^8 + 39T^7 - 198T^6 - 396T^5 + 174T^4 + 26T^3 + 52T^2 + 12*T + 1)^2
$67$	$ T^{44} $ T^44
$71$	$ T^{44} $ T^44
$73$	$ T^{44} $ T^44
$79$	$ (T^{22} - 2 T^{21} + \cdots + 1)^{2} $ (T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 64T^16 - 128T^15 + 256T^14 - 512T^13 + 1024T^12 - 2048T^11 + 4073T^10 - 7640T^9 + 11991T^8 - 14115T^7 + 11785T^6 - 6826T^5 + 2704T^4 - 716T^3 + 121T^2 - 12*T + 1)^2
$83$	$ T^{44} - 4 T^{42} + \cdots + 1 $ T^44 - 4T^42 + 16T^40 - 64T^38 + 256T^36 - 1024T^34 + 4096T^32 - 16384T^30 + 65490T^28 - 254140T^26 + 838264T^24 - 2000978T^22 + 3219636T^20 - 3480859T^18 + 2576386T^16 - 1331059T^14 + 486163T^12 - 108332T^10 + 63442T^8 + 18575T^6 + 3486T^4 + 109*T^2 + 1
$89$	$ T^{44} $ T^44
$97$	$ T^{44} $ T^44
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3525.bd (of order \(46\), degree \(22\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{92}^{25} + \zeta_{92}^{3}) q^{2} + \zeta_{92}^{13} q^{3} + ( - \zeta_{92}^{28} + \cdots - \zeta_{92}^{4}) q^{4}+ \cdots + \zeta_{92}^{26} q^{9}+O(q^{10}) \) q + (-z^25 + z^3) * q^2 + z^13 * q^3 + (-z^28 + z^6 - z^4) * q^4 + (-z^38 + z^16) * q^6 + (-z^31 + z^29 + z^9 + z^7) * q^8 + z^26 * q^9	\( q + ( - \zeta_{92}^{25} + \zeta_{92}^{3}) q^{2} + \zeta_{92}^{13} q^{3} + ( - \zeta_{92}^{28} + \cdots - \zeta_{92}^{4}) q^{4}+ \cdots + ( - \zeta_{92}^{39} + \zeta_{92}^{17}) q^{98}+O(q^{100}) \) q + (-z^25 + z^3) * q^2 + z^13 * q^3 + (-z^28 + z^6 - z^4) * q^4 + (-z^38 + z^16) * q^6 + (-z^31 + z^29 + z^9 + z^7) * q^8 + z^26 * q^9 + (-z^41 + z^19 - z^17) * q^12 + (-z^34 + z^32 + z^12 + z^10 + z^8) * q^16 + (-z^45 + z^31) * q^17 + (z^29 + z^5) * q^18 + (-z^40 + z^30) * q^19 + (z^41 - z^23) * q^23 + (-z^44 + z^42 + z^22 - z^20) * q^24 + z^39 * q^27 + (-z^18 - z^2) * q^31 + (-z^37 + z^35 - z^33 + z^15 + z^13 - z^11) * q^32 + (z^34 - z^24 + z^10 + z^2) * q^34 + (z^32 - z^30 + z^8) * q^36 + (-z^43 + z^33 - z^19 + z^9) * q^38 + (z^44 - z^26 + z^20 - z^2) * q^46 - z^21 * q^47 + (z^45 + z^25 - z^23 + z^21 + z) * q^48 + z^14 * q^49 + (z^44 + z^12) * q^51 + (-z^7 + z) * q^53 + (z^42 + z^18) * q^54 + (z^43 + z^7) * q^57 + (-z^38 + z^4) * q^61 + (z^43 + z^27 - z^21 - z^5) * q^62 + (-z^40 + z^38 - z^36 + z^18 + z^16 - z^14 - z^12) * q^64 + (z^37 - z^35 - z^27 + z^13 + z^5 - z^3) * q^68 + (-z^36 - z^8) * q^69 + (z^35 - z^33 + z^11 - z^9) * q^72 + (z^44 + z^36 - z^34 - z^22 + z^12 + 1) * q^76 + (-z^32 - z^16) * q^79 - z^6 * q^81 + (-z^35 - z^27) * q^83 + (-z^45 - z^29 + z^27 + z^23 - z^5 - z) * q^92 + (-z^31 - z^15) * q^93 + (-z^24 - 1) * q^94 + (z^28 - z^26 + z^24 + z^4 - z^2 + 1) * q^96 + (-z^39 + z^17) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 6 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) 44 * q + 6 * q^4 - 4 * q^6 + 2 * q^9	\( 44 q + 6 q^{4} - 4 q^{6} + 2 q^{9} - 10 q^{16} + 4 q^{19} + 8 q^{24} - 4 q^{31} + 8 q^{34} - 6 q^{36} - 8 q^{46} + 2 q^{49} - 4 q^{51} + 4 q^{54} - 4 q^{61} + 14 q^{64} + 4 q^{69} + 34 q^{76} + 4 q^{79} - 2 q^{81} - 42 q^{94} + 34 q^{96}+O(q^{100}) \) 44 * q + 6 * q^4 - 4 * q^6 + 2 * q^9 - 10 * q^16 + 4 * q^19 + 8 * q^24 - 4 * q^31 + 8 * q^34 - 6 * q^36 - 8 * q^46 + 2 * q^49 - 4 * q^51 + 4 * q^54 - 4 * q^61 + 14 * q^64 + 4 * q^69 + 34 * q^76 + 4 * q^79 - 2 * q^81 - 42 * q^94 + 34 * q^96

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	3525.1.bd.a

Level	$3525$
Weight	$1$
Character orbit	3525.bd
Analytic conductor	$1.759$
Analytic rank	$0$
Dimension	$44$
Projective image	$D_{23}$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.75920416953\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(2\) over \(\Q(\zeta_{46})\)
Coefficient field:	\(\Q(\zeta_{92})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{44} - x^{42} + x^{40} - x^{38} + x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} + \cdots + 1 \) x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 705)
Projective image:	\(D_{23}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{23} - \cdots)\)

\(n\)	\(1552\)	\(2026\)	\(2351\)
\(\chi(n)\)	\(1\)	\(-\zeta_{92}^{22}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 101.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{44} - 4 T^{42} + \cdots + 1 \) T^44 - 4T^42 + 16T^40 - 64T^38 + 210T^36 - 288T^34 + 1152T^32 - 4608T^30 + 19145T^28 + 9854T^26 + 23788T^24 - 94669T^22 + 374444T^20 - 720146T^18 + 189032T^16 - 17644T^14 + 55534T^12 + 292144T^10 + 141757T^8 + 19633T^6 + 2244T^4 - 75*T^2 + 1
$3$	\( T^{44} - T^{42} + \cdots + 1 \) T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	\( T^{44} \) T^44
$7$	\( T^{44} \) T^44
$11$	\( T^{44} \) T^44
$13$	\( T^{44} \) T^44
$17$	\( T^{44} - 4 T^{42} + \cdots + 1 \) T^44 - 4T^42 - 30T^40 + 120T^38 + 417T^36 - 1668T^34 - 2988T^32 + 11952T^30 + 14936T^28 - 59353T^26 - 14714T^24 + 95932T^22 + 235202T^20 - 358908T^18 + 551144T^16 + 25688T^14 + 565812T^12 - 83239T^10 + 112593T^8 + 3510T^6 + 2911T^4 - 6*T^2 + 1
$19$	\( (T^{22} - 2 T^{21} + \cdots + 1)^{2} \) (T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 41T^16 - 13T^15 + 26T^14 - 52T^13 + 104T^12 - 208T^11 + 554T^10 + 341T^9 + 54T^8 - 85T^7 + 170T^6 - 340T^5 + 519T^4 - 348T^3 + 98T^2 - 12*T + 1)^2
$23$	\( T^{44} + 19 T^{42} + \cdots + 1 \) T^44 + 19T^42 + 177T^40 + 1063T^38 + 4603T^36 + 15237T^34 + 39999T^32 + 85161T^30 + 149670T^28 + 218510T^26 + 270026T^24 + 271974T^22 + 264136T^20 + 91064T^18 + 385636T^16 - 561916T^14 + 858441T^12 - 628661T^10 + 260161T^8 - 51161T^6 + 4521T^4 - 121*T^2 + 1
$29$	\( T^{44} \) T^44
$31$	\( (T^{22} + 2 T^{21} + \cdots + 1)^{2} \) (T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 41T^16 + 13T^15 + 26T^14 + 52T^13 + 104T^12 + 208T^11 + 554T^10 - 341T^9 + 54T^8 + 85T^7 + 170T^6 + 340T^5 + 519T^4 + 348T^3 + 98T^2 + 12*T + 1)^2
$37$	\( T^{44} \) T^44
$41$	\( T^{44} \) T^44
$43$	\( T^{44} \) T^44
$47$	\( T^{44} - T^{42} + \cdots + 1 \) T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$53$	\( T^{44} - 4 T^{42} + \cdots + 1 \) T^44 - 4T^42 + 16T^40 - 64T^38 + 256T^36 - 58T^34 + 232T^32 - 882T^30 - 9030T^28 + 36120T^26 + 32367T^24 - 114173T^22 + 419662T^20 - 140316T^18 + 556319T^16 + 177971T^14 + 178423T^12 - 310663T^10 + 13256T^8 + 19817T^6 + 2635T^4 - 29*T^2 + 1
$59$	\( T^{44} \) T^44
$61$	\( (T^{22} + 2 T^{21} + \cdots + 1)^{2} \) (T^22 + 2T^21 + 4T^20 - 15T^19 - 30T^18 - 60T^17 + 87T^16 + 174T^15 + 348T^14 - 224T^13 - 448T^12 - 896T^11 + 301T^10 + 579T^9 + 1158T^8 + 39T^7 - 198T^6 - 396T^5 + 174T^4 + 26T^3 + 52T^2 + 12*T + 1)^2
$67$	\( T^{44} \) T^44
$71$	\( T^{44} \) T^44
$73$	\( T^{44} \) T^44
$79$	\( (T^{22} - 2 T^{21} + \cdots + 1)^{2} \) (T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 64T^16 - 128T^15 + 256T^14 - 512T^13 + 1024T^12 - 2048T^11 + 4073T^10 - 7640T^9 + 11991T^8 - 14115T^7 + 11785T^6 - 6826T^5 + 2704T^4 - 716T^3 + 121T^2 - 12*T + 1)^2
$83$	\( T^{44} - 4 T^{42} + \cdots + 1 \) T^44 - 4T^42 + 16T^40 - 64T^38 + 256T^36 - 1024T^34 + 4096T^32 - 16384T^30 + 65490T^28 - 254140T^26 + 838264T^24 - 2000978T^22 + 3219636T^20 - 3480859T^18 + 2576386T^16 - 1331059T^14 + 486163T^12 - 108332T^10 + 63442T^8 + 18575T^6 + 3486T^4 + 109*T^2 + 1
$89$	\( T^{44} \) T^44
$97$	\( T^{44} \) T^44
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