LMFDB - Newform orbit 575.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(26,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 16]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("575.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	575.k (of order $11$, degree $10$, 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:	$4.59139811622$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$\zeta_{22}^{4}$	$1$

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} $

26.1

−0.415415	−	0.909632i
0.654861	−	0.755750i
−0.841254	+	0.540641i
0.142315	−	0.989821i
0.959493	+	0.281733i
0.959493	−	0.281733i
−0.415415	+	0.909632i
0.142315	+	0.989821i
−0.841254	−	0.540641i
0.654861	+	0.755750i

0.313607

2.18119i

1.04408

−

2.28621i

−2.74024

0.804606i

5.31408

1.56036i

1.99611

−

1.28282i

−0.783524

−

1.71568i

−2.17208

−

2.50672i

101.1

0.226900

−

0.0666238i

0.313607

0.361922i

−1.63546

1.05105i

0.0952700

0.0612263i

1.05773

−

2.31611i

−0.610783

0.704881i

0.394306

−

2.74246i

151.1

1.04408

1.20493i

−0.198939

−

0.127850i

−0.0771283

0.536439i

−0.0536570

−

0.373193i

−0.874908

−

0.256896i

1.95561

−

1.25679i

−1.22301

−

2.67803i

301.1

2.11435

1.35881i

0.226900

1.57812i

1.79329

3.92676i

−1.66463

3.64502i

0.804632

−

0.928595i

−0.828708

5.76379i

0.439490

−

0.129046i

326.1

−0.198939

−

0.435615i

2.11435

−

0.620830i

1.15954

−

1.33818i

−0.691070

−

0.797537i

−0.483568

−

3.36329i

−1.73259

−

0.508735i

1.56130

−

1.00339i

351.1

−0.198939

0.435615i

2.11435

0.620830i

1.15954

1.33818i

−0.691070

0.797537i

−0.483568

3.36329i

−1.73259

0.508735i

1.56130

1.00339i

376.1

0.313607

−

2.18119i

1.04408

2.28621i

−2.74024

−

0.804606i

5.31408

−

1.56036i

1.99611

1.28282i

−0.783524

1.71568i

−2.17208

2.50672i

426.1

2.11435

−

1.35881i

0.226900

−

1.57812i

1.79329

−

3.92676i

−1.66463

−

3.64502i

0.804632

0.928595i

−0.828708

−

5.76379i

0.439490

0.129046i

476.1

1.04408

−

1.20493i

−0.198939

0.127850i

−0.0771283

−

0.536439i

−0.0536570

0.373193i

−0.874908

0.256896i

1.95561

1.25679i

−1.22301

2.67803i

501.1

0.226900

0.0666238i

0.313607

−

0.361922i

−1.63546

−

1.05105i

0.0952700

−

0.0612263i

1.05773

2.31611i

−0.610783

−

0.704881i

0.394306

2.74246i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	575.2.k.b		10
5.b	even	2	1		23.2.c.a	✓	10
5.c	odd	4	2		575.2.p.b		20
15.d	odd	2	1		207.2.i.c		10
20.d	odd	2	1		368.2.m.c		10
23.c	even	11	1	inner	575.2.k.b		10
115.c	odd	2	1		529.2.c.a		10
115.i	odd	22	1		529.2.a.j		5
115.i	odd	22	1		529.2.c.a		10
115.i	odd	22	2		529.2.c.c		10
115.i	odd	22	2		529.2.c.e		10
115.i	odd	22	2		529.2.c.f		10
115.i	odd	22	2		529.2.c.h		10
115.j	even	22	1		23.2.c.a	✓	10
115.j	even	22	1		529.2.a.i		5
115.j	even	22	2		529.2.c.b		10
115.j	even	22	2		529.2.c.d		10
115.j	even	22	2		529.2.c.g		10
115.j	even	22	2		529.2.c.i		10
115.k	odd	44	2		575.2.p.b		20
345.n	even	22	1		4761.2.a.bn		5
345.p	odd	22	1		207.2.i.c		10
345.p	odd	22	1		4761.2.a.bo		5
460.n	odd	22	1		368.2.m.c		10
460.n	odd	22	1		8464.2.a.bs		5
460.o	even	22	1		8464.2.a.bt		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
23.2.c.a	✓	10	5.b	even	2	1
23.2.c.a	✓	10	115.j	even	22	1
207.2.i.c		10	15.d	odd	2	1
207.2.i.c		10	345.p	odd	22	1
368.2.m.c		10	20.d	odd	2	1
368.2.m.c		10	460.n	odd	22	1
529.2.a.i		5	115.j	even	22	1
529.2.a.j		5	115.i	odd	22	1
529.2.c.a		10	115.c	odd	2	1
529.2.c.a		10	115.i	odd	22	1
529.2.c.b		10	115.j	even	22	2
529.2.c.c		10	115.i	odd	22	2
529.2.c.d		10	115.j	even	22	2
529.2.c.e		10	115.i	odd	22	2
529.2.c.f		10	115.i	odd	22	2
529.2.c.g		10	115.j	even	22	2
529.2.c.h		10	115.i	odd	22	2
529.2.c.i		10	115.j	even	22	2
575.2.k.b		10	1.a	even	1	1	trivial
575.2.k.b		10	23.c	even	11	1	inner
575.2.p.b		20	5.c	odd	4	2
575.2.p.b		20	115.k	odd	44	2
4761.2.a.bn		5	345.n	even	22	1
4761.2.a.bo		5	345.p	odd	22	1
8464.2.a.bs		5	460.n	odd	22	1
8464.2.a.bt		5	460.o	even	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 7T_{2}^{9} + 27T_{2}^{8} - 68T_{2}^{7} + 124T_{2}^{6} - 142T_{2}^{5} + 103T_{2}^{4} - 28T_{2}^{3} + 20T_{2}^{2} - 8T_{2} + 1 $ T2^10 - 7*T2^9 + 27*T2^8 - 68*T2^7 + 124*T2^6 - 142*T2^5 + 103*T2^4 - 28*T2^3 + 20*T2^2 - 8*T2 + 1 acting on $S_{2}^{\mathrm{new}}(575, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 7 T^{9} + \cdots + 1 $ T^10 - 7T^9 + 27T^8 - 68T^7 + 124T^6 - 142T^5 + 103T^4 - 28T^3 + 20T^2 - 8*T + 1
$3$	$ T^{10} - 7 T^{9} + \cdots + 1 $ T^10 - 7T^9 + 27T^8 - 68T^7 + 113T^6 - 131T^5 + 103T^4 - 17T^3 - 2T^2 + 3*T + 1
$5$	$ T^{10} $ T^10
$7$	$ T^{10} - 5 T^{9} + \cdots + 529 $ T^10 - 5T^9 + 25T^8 - 81T^7 + 207T^6 - 298T^5 + 170T^4 + 448T^3 - 425T^2 + 46*T + 529
$11$	$ T^{10} - 7 T^{9} + \cdots + 529 $ T^10 - 7T^9 + 16T^8 + 31T^7 - 74T^6 - 813T^5 + 3282T^4 - 6111T^3 + 11317T^2 - 2208*T + 529
$13$	$ T^{10} - 3 T^{9} + \cdots + 1 $ T^10 - 3T^9 + 9T^8 - 71T^7 + 125T^6 - 166T^5 + 1290T^4 + 684T^3 + 159T^2 + 18*T + 1
$17$	$ T^{10} - 10 T^{9} + \cdots + 529 $ T^10 - 10T^9 + 56T^8 - 65T^7 - 274T^6 + 2003T^5 + 10880T^4 + 20197T^3 + 14521T^2 + 1541*T + 529
$19$	$ T^{10} - 2 T^{9} + \cdots + 541696 $ T^10 - 2T^9 - 40T^8 - 184T^7 + 544T^6 + 8416T^5 + 52864T^4 + 180096T^3 + 450816T^2 + 635904*T + 541696
$23$	$ T^{10} - 12 T^{9} + \cdots + 6436343 $ T^10 - 12T^9 - 10T^8 + 527T^7 - 32T^6 - 14103T^5 - 736T^4 + 278783T^3 - 121670T^2 - 3358092*T + 6436343
$29$	$ T^{10} - 14 T^{9} + \cdots + 4932841 $ T^10 - 14T^9 + 86T^8 + 6T^7 - 2394T^6 + 13100T^5 - 833T^4 - 184633T^3 + 1146117T^2 - 1896734*T + 4932841
$31$	$ T^{10} - 10 T^{9} + \cdots + 17161 $ T^10 - 10T^9 + 56T^8 - 164T^7 + 375T^6 - 505T^5 + 56T^4 + 3862T^3 - 1066T^2 + 2751*T + 17161
$37$	$ T^{10} - 19 T^{9} + \cdots + 529 $ T^10 - 19T^9 + 174T^8 - 1117T^7 + 6956T^6 - 38345T^5 + 148800T^4 - 346183T^3 + 370815T^2 - 4462*T + 529
$41$	$ T^{10} - 7 T^{9} + \cdots + 1849 $ T^10 - 7T^9 + 16T^8 + 9T^7 + 1235T^6 + 4181T^5 + 15173T^4 + 27978T^3 + 23747T^2 + 9331*T + 1849
$43$	$ T^{10} - 11 T^{9} + \cdots + 64009 $ T^10 - 11T^9 + 66T^8 - 341T^7 + 1826T^6 - 6875T^5 + 20449T^4 - 55297T^3 + 106843T^2 - 114103*T + 64009
$47$	$ (T^{5} - 9 T^{4} - 5 T^{3} + \cdots - 1)^{2} $ (T^5 - 9T^4 - 5T^3 + 97T^2 - 106T - 1)^2
$53$	$ T^{10} + \cdots + 517426009 $ T^10 + 29T^9 + 478T^8 + 5579T^7 + 55454T^6 + 448645T^5 + 2974063T^4 + 16104853T^3 + 83193185T^2 + 312339057*T + 517426009
$59$	$ T^{10} + 21 T^{9} + \cdots + 4489 $ T^10 + 21T^9 + 265T^8 + 1649T^7 + 3433T^6 - 22111T^5 - 98174T^4 - 33991T^3 + 1176352T^2 + 20569*T + 4489
$61$	$ T^{10} - 3 T^{9} + \cdots + 529 $ T^10 - 3T^9 + 218T^8 + 281T^7 + 7066T^6 + 46573T^5 + 215097T^4 + 359141T^3 + 236219T^2 - 22103*T + 529
$67$	$ T^{10} + \cdots + 1113757129 $ T^10 + 45T^9 + 1090T^8 + 17513T^7 + 203501T^6 + 1799216T^5 + 12627902T^4 + 70242899T^3 + 291205025T^2 + 798415652*T + 1113757129
$71$	$ T^{10} + 14 T^{9} + \cdots + 529 $ T^10 + 14T^9 + 130T^8 + 566T^7 + 785T^6 - 197T^5 + 1686T^4 + 1560T^3 + 1204T^2 + 851*T + 529
$73$	$ T^{10} + 19 T^{9} + \cdots + 982081 $ T^10 + 19T^9 + 251T^8 + 2415T^7 + 15063T^6 + 73963T^5 + 379118T^4 + 1529101T^3 + 3411556T^2 + 2470563*T + 982081
$79$	$ T^{10} + \cdots + 517426009 $ T^10 + 15T^9 + 115T^8 + 2715T^7 + 26304T^6 + 111816T^5 + 2966242T^4 + 21480574T^3 + 54168013T^2 - 24589507*T + 517426009
$83$	$ T^{10} + 18 T^{9} + \cdots + 529 $ T^10 + 18T^9 + 192T^8 + 321T^7 + 1455T^6 + 2914T^5 + 224T^4 + 17T^3 + 3804T^2 - 2346*T + 529
$89$	$ T^{10} + \cdots + 78310985281 $ T^10 - 25T^9 + 262T^8 - 5615T^7 + 111566T^6 - 1048455T^5 + 11505750T^4 - 150274617T^3 + 1491272689T^2 - 10362512230*T + 78310985281
$97$	$ T^{10} - 34 T^{9} + \cdots + 2374681 $ T^10 - 34T^9 + 430T^8 - 4423T^7 + 52812T^6 - 183261T^5 + 1071940T^4 - 3733313T^3 + 17156261T^2 - 10511161*T + 2374681
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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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.k (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{22}^{6} - \zeta_{22}^{5} + \cdots + \zeta_{22}) q^{9}+O(q^{10}) \) q + (z^8 - z^7 - z + 1) * q^2 + (-z^9 - z^5 + z^4 + 1) * q^3 + (-2z^9 + 2z^8 - 2z^7 - z^5 + 2z^4 - z^3 + z^2 - 2z + 1) q^4 + (z^6 - 2z^5 + z^3 - 2z + 1) * q^6 + (z^9 - z^8 - z^6 - z^5 - z^4 - z^2 + z) * q^7 + (-z^9 + z^8 + z^7 - z^6 - z^5 + z^4 - 2z) q^8 + (-z^6 - z^5 + z^4 - z^3 + z^2 + z) * q^9	\( q + (\zeta_{22}^{8} - \zeta_{22}^{7} + \cdots + 1) q^{2}+ \cdots + (5 \zeta_{22}^{8} + \cdots + 5 \zeta_{22}^{2}) q^{99}+O(q^{100}) \) q + (z^8 - z^7 - z + 1) * q^2 + (-z^9 - z^5 + z^4 + 1) * q^3 + (-2z^9 + 2z^8 - 2z^7 - z^5 + 2z^4 - z^3 + z^2 - 2z + 1) q^4 + (z^6 - 2z^5 + z^3 - 2z + 1) * q^6 + (z^9 - z^8 - z^6 - z^5 - z^4 - z^2 + z) * q^7 + (-z^9 + z^8 + z^7 - z^6 - z^5 + z^4 - 2z) q^8 + (-z^6 - z^5 + z^4 - z^3 + z^2 + z) * q^9 + (-z^9 + 2z^7 - z^5 - z^3 + z^2 - z + 1) q^11 + (-3z^9 + 2z^8 - 3z^7 + 2z^6 - 3z^5 - z^3 + 4z^2 - 4z + 1) q^12 + (2z^9 - z^8 + 2z^7 - 2z^6 + 2z^5 - z^4 + 2z^3 + z - 1) q^13 + (2z^9 - z^7 + z^6 - z + 1) q^14 + (-z^9 + z^8 + z^7 - 2z^6 - z^5 - 2z^4 + z^3 + z^2 - z) * q^16 + (z^6 + z^5 - z^4 - 3z^3 - z^2 + z + 1) q^17 + (2z^9 - 3z^8 + 3z^7 - 2z^6 + z^3 - 2z^2 + z) q^18 + (2z^9 - 4z^8 + 2z^7 + 2z^5 - 2z^4 + 4z^3 - 4z^2 + 2z - 2) * q^19 + (-z^8 + z^7 - 3z^6 - z^5 - z^4 - z^3 - 3z^2 + z - 1) * q^21 + (-3z^8 + 4z^7 - z^6 + z^5 - 4z^4 + 3z^3 - 1) * q^22 + (-2z^9 + z^8 + 3z^6 - 3z^5 + z^3 + z^2 + z + 2) q^23 + (-z^9 + 2z^7 - 2z^4 + z^2 - 4) * q^24 + (-z^8 + z^7 - 2z^6 + 2z^5 - 2z^4 + 2z^3 - z^2 + z) * q^26 + (3z^8 - z^5 + z^4 - z + 1) q^27 + (-z^9 + 3z^8 - 3z^7 + z^6 + z^4 - 4z^3 - 4z + 1) * q^28 + (-2z^9 + 2z^8 + 2z^6 + 2z^5 + z^4 - 2z^3 + z^2 + 2z + 2) * q^29 + (z^9 + z^8 + z^7 - z^6 - z^5 - z^4 + z^2 - z + 1) * q^31 + (z^9 + 4z^7 - 3z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + 3z - 4) q^32 + (-z^9 - z^8 + z^7 + z^5 - z^4 - z^3 + 2z - 2) q^33 + (4z^9 - 2z^8 - 2z^6 + 4z^5 - z^3 - 4z^2 + 4z + 1) * q^34 + (-z^9 - 2z^7 + 3z^6 - 2z^5 - z^3 - 4z + 4) * q^36 + (-z^9 - 4z^7 + 2z^6 - 3z^5 + 3z^4 - 3z^3 + 3z^2 - 2z + 4) q^37 + (2z^9 - 6z^7 + 6z^6 - 2z^4 + 2z^2 - 2z + 2) * q^38 + (z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z) * q^39 + (-z^9 + z^8 - z^7 + z^6 + z^4 + 2z^3 + 2z^2 + 2z + 1) q^41 + (3z^9 - 2z^8 + 3z^7 - z^5 + 4z^3 - 4z^2 + 1) q^42 + (-z^9 - z^7 + 2z^5 - 2z^4 + z^2 + 1) * q^43 + (-5z^8 + z^7 + z^6 + 3z^5 - 5z^4 + 3z^3 + z^2 + z - 5) * q^44 + (-z^9 - 3z^7 + 6z^6 - 4z^5 - 2z^4 - z^3 + 4z^2 - 2z - 1) * q^46 + (z^9 - 3z^8 + 2z^7 - 3z^6 + 3z^5 - 2z^4 + 3z^3 - z^2) * q^47 + (-2z^8 + z^6 - z^5 - 4z^4 - z^3 + z^2 - 2) * q^48 + (3z^8 - z^7 + 2z^6 - 3z^5 + 3z^4 - 2z^3 + z^2 - 3z) * q^49 + (3z^8 - 3z^7 + 2z^4 - 3z^3 - z^2 - 3z + 2) q^51 + (-2z^9 + 2z^8 + 3z^6 + z^4 - z^3 + z^2 + 3) q^52 + (-2z^9 + 7z^8 - z^7 + 3z^6 - 3z^5 + 3z^4 - z^3 + 7z^2 - 2z) q^53 + (-4z^9 + 5z^8 - z^7 + z^6 - 5z^5 + 4z^4 + 2z^2 - 4z + 2) * q^54 + (-z^9 + 4z^8 + z^6 + 4z^4 - z^3 - z + 1) * q^56 + (4z^9 - 6z^8 + 8z^7 - 6z^6 + 4z^5 + 4z^3 - 8z^2 + 8z - 4) * q^57 + (2z^9 - 3z^8 + z^7 - 3z^6 + 2z^5 - 6z^2 + 6z) * q^58 + (3z^9 - 3z^8 + 3z^7 - 6z^6 + 3z^5 - 3z^4 + 3z^3 + 5z - 5) * q^59 + (3z^9 - 4z^8 - 2z^7 + 2z^6 + 4z^5 - 3z^4 - z^2 + 2z - 1) q^61 + (-2z^9 + 2z^8 + z^7 - z^6 - z^4 + z^3 + 2z^2 - 2z) * q^62 + (z^9 - z^8 - 5z^6 + 2z^5 - 2z^4 + z^3 - 2z^2 + 2z - 5) q^63 + (z^9 - 4z^8 + 4z^7 - z^6 + z^4 - 5z^2 + 1) q^64 + (4z^9 - 8z^8 + 5z^7 - 2z^6 + 4z^5 - 4z^4 + 2z^3 - 5z^2 + 8z - 4) q^66 + (-4z^8 - 3z^7 + z^6 - 2z^5 - 3z^4 - 2z^3 + z^2 - 3z - 4) * q^67 + (7z^9 - 2z^8 - 4z^7 - 5z^6 + 5z^5 + 4z^4 + 2z^3 - 7z^2 + 1) * q^68 + (-2z^9 - 5z^8 + 3z^7 - z^6 + z^5 + z^3 - z^2 + 6z + 1) * q^69 + (-2z^8 - z^7 - 2z^6 - 2z^4 - 2z^2 - z - 2) * q^71 + (-z^9 + 4z^8 - 3z^7 + 4z^6 - 3z^5 + 3z^4 - 4z^3 + 3z^2 - 4z + 1) * q^72 + (-3z^9 + 4z^8 - 3z^7 + z^5 + z^4 + z^3 - z^2 - z - 1) q^73 + (z^9 + 3z^8 - 3z^7 - z^6 + 3z^4 - 5z^3 + 3z^2 - 5z + 3) * q^74 + (2z^9 + 6z^8 - 6z^7 + 4z^5 - 6z^3 + 6z^2 + 2z) q^76 + (3z^8 + z^7 - z^6 - 3z^5 + 2z) q^77 + (-z^9 - z^7 - z^5 + z^4 - z^3 + z^2 + 1) * q^78 + (-6z^9 + 6z^8 - 8z^7 + 2z^6 - 8z^5 + 6z^4 - 6z^3 - 3z + 3) * q^79 + (2z^9 - z^8 + 5z^7 - z^6 + 2z^5 + 7z^3 - 4z^2 + 4z - 7) * q^81 + (-z^9 - 2z^7 + z^6 - 2z^5 - z^3 + z - 1) * q^82 + (-z^9 + 3z^7 - 4z^6 + z^5 - z^2 + 4z - 3) q^83 + (2z^9 + 3z^8 - 3z^7 + 3z^6 - 3z^5 - 2z^4 - 9z) q^84 + (-3z^6 + 5z^5 - 3z^4 - 3z^2 + 5z - 3) q^86 + (-5z^9 + 7z^8 - 7z^7 + 5z^6 + 7z^4 - 4z^3 + 8z^2 - 4z + 7) * q^87 + (3z^9 - 8z^8 + 3z^7 + 5z^5 - 4z^4 - 2z^3 + 2z^2 + 4z - 5) * q^88 + (-5z^9 + 4z^8 - 8z^7 - 3z^6 - z^5 + z^4 + 3z^3 + 8z^2 - 4z + 5) q^89 + (-3z^9 - z^8 - 4z^7 + z^6 - z^5 + 4z^4 + z^3 + 3z^2 + 1) * q^91 + (-5z^9 - 4z^8 - 4z^7 + 3z^6 + 3z^5 - 2z^4 - 4z^3 + 6z^2 + z - 4) * q^92 + (z^9 - z^8 + 2z^7 + z^6 - z^5 - 2z^4 + z^3 - z^2 - 1) * q^93 + (-2z^6 + 4z^5 - 5z^4 + 4z^3 - 2z^2) q^94 + (z^9 + 4z^8 + z^7 + 5z^5 - z^4 - 3z^3 + 3z^2 + z - 5) * q^96 + (5z^9 + 2z^8 - 2z^7 - 5z^6 + 2z^4 + 3z^3 - 4z^2 + 3z + 2) * q^97 + (-4z^9 + 4z^8 + 2z^6 - 6z^5 + 6z^4 - 3z^3 + 6z^2 - 6z + 2) * q^98 + (5z^8 - 4z^7 + 3z^6 - 6z^5 + 3z^4 - 4z^3 + 5z^2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 7 q^{2} + 7 q^{3} - 3 q^{4} + 6 q^{6} + 5 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) 10 * q + 7 * q^2 + 7 * q^3 - 3 * q^4 + 6 * q^6 + 5 * q^7 - 4 * q^8 - 2 * q^9	\( 10 q + 7 q^{2} + 7 q^{3} - 3 q^{4} + 6 q^{6} + 5 q^{7} - 4 q^{8} - 2 q^{9} + 7 q^{11} - 12 q^{12} + 3 q^{13} + 9 q^{14} + q^{16} + 10 q^{17} + 14 q^{18} + 2 q^{19} - 2 q^{21} + 6 q^{22} + 12 q^{23} - 38 q^{24} + 12 q^{26} + 4 q^{27} - 7 q^{28} + 14 q^{29} + 10 q^{31} - 21 q^{32} - 16 q^{33} + 29 q^{34} + 27 q^{36} + 19 q^{37} + 8 q^{38} + q^{39} + 7 q^{41} + 25 q^{42} + 11 q^{43} - 34 q^{44} - 29 q^{46} + 18 q^{47} - 18 q^{48} - 18 q^{49} + 7 q^{51} + 20 q^{52} - 29 q^{53} - 6 q^{54} - 2 q^{56} + 8 q^{57} + 23 q^{58} - 21 q^{59} + 3 q^{61} - 4 q^{62} - 34 q^{63} + 24 q^{64} + 2 q^{66} - 45 q^{67} + 30 q^{68} + 26 q^{69} - 14 q^{71} - 19 q^{72} - 19 q^{73} + 10 q^{74} - 16 q^{76} - 2 q^{77} + 4 q^{78} - 15 q^{79} - 44 q^{81} - 16 q^{82} - 18 q^{83} - 17 q^{84} - 11 q^{86} + 23 q^{87} - 27 q^{88} + 25 q^{89} - 4 q^{91} - 52 q^{92} - 4 q^{93} + 17 q^{94} - 51 q^{96} + 34 q^{97} - 17 q^{98} - 30 q^{99}+O(q^{100}) \) 10 * q + 7 * q^2 + 7 * q^3 - 3 * q^4 + 6 * q^6 + 5 * q^7 - 4 * q^8 - 2 * q^9 + 7 * q^11 - 12 * q^12 + 3 * q^13 + 9 * q^14 + q^16 + 10 * q^17 + 14 * q^18 + 2 * q^19 - 2 * q^21 + 6 * q^22 + 12 * q^23 - 38 * q^24 + 12 * q^26 + 4 * q^27 - 7 * q^28 + 14 * q^29 + 10 * q^31 - 21 * q^32 - 16 * q^33 + 29 * q^34 + 27 * q^36 + 19 * q^37 + 8 * q^38 + q^39 + 7 * q^41 + 25 * q^42 + 11 * q^43 - 34 * q^44 - 29 * q^46 + 18 * q^47 - 18 * q^48 - 18 * q^49 + 7 * q^51 + 20 * q^52 - 29 * q^53 - 6 * q^54 - 2 * q^56 + 8 * q^57 + 23 * q^58 - 21 * q^59 + 3 * q^61 - 4 * q^62 - 34 * q^63 + 24 * q^64 + 2 * q^66 - 45 * q^67 + 30 * q^68 + 26 * q^69 - 14 * q^71 - 19 * q^72 - 19 * q^73 + 10 * q^74 - 16 * q^76 - 2 * q^77 + 4 * q^78 - 15 * q^79 - 44 * q^81 - 16 * q^82 - 18 * q^83 - 17 * q^84 - 11 * q^86 + 23 * q^87 - 27 * q^88 + 25 * q^89 - 4 * q^91 - 52 * q^92 - 4 * q^93 + 17 * q^94 - 51 * q^96 + 34 * q^97 - 17 * q^98 - 30 * q^99

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

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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	575.2.k.b

Level	$575$
Weight	$2$
Character orbit	575.k
Analytic conductor	$4.591$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.59139811622\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 26.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - 7 T^{9} + \cdots + 1 \) T^10 - 7T^9 + 27T^8 - 68T^7 + 124T^6 - 142T^5 + 103T^4 - 28T^3 + 20T^2 - 8*T + 1
$3$	\( T^{10} - 7 T^{9} + \cdots + 1 \) T^10 - 7T^9 + 27T^8 - 68T^7 + 113T^6 - 131T^5 + 103T^4 - 17T^3 - 2T^2 + 3*T + 1
$5$	\( T^{10} \) T^10
$7$	\( T^{10} - 5 T^{9} + \cdots + 529 \) T^10 - 5T^9 + 25T^8 - 81T^7 + 207T^6 - 298T^5 + 170T^4 + 448T^3 - 425T^2 + 46*T + 529
$11$	\( T^{10} - 7 T^{9} + \cdots + 529 \) T^10 - 7T^9 + 16T^8 + 31T^7 - 74T^6 - 813T^5 + 3282T^4 - 6111T^3 + 11317T^2 - 2208*T + 529
$13$	\( T^{10} - 3 T^{9} + \cdots + 1 \) T^10 - 3T^9 + 9T^8 - 71T^7 + 125T^6 - 166T^5 + 1290T^4 + 684T^3 + 159T^2 + 18*T + 1
$17$	\( T^{10} - 10 T^{9} + \cdots + 529 \) T^10 - 10T^9 + 56T^8 - 65T^7 - 274T^6 + 2003T^5 + 10880T^4 + 20197T^3 + 14521T^2 + 1541*T + 529
$19$	\( T^{10} - 2 T^{9} + \cdots + 541696 \) T^10 - 2T^9 - 40T^8 - 184T^7 + 544T^6 + 8416T^5 + 52864T^4 + 180096T^3 + 450816T^2 + 635904*T + 541696
$23$	\( T^{10} - 12 T^{9} + \cdots + 6436343 \) T^10 - 12T^9 - 10T^8 + 527T^7 - 32T^6 - 14103T^5 - 736T^4 + 278783T^3 - 121670T^2 - 3358092*T + 6436343
$29$	\( T^{10} - 14 T^{9} + \cdots + 4932841 \) T^10 - 14T^9 + 86T^8 + 6T^7 - 2394T^6 + 13100T^5 - 833T^4 - 184633T^3 + 1146117T^2 - 1896734*T + 4932841
$31$	\( T^{10} - 10 T^{9} + \cdots + 17161 \) T^10 - 10T^9 + 56T^8 - 164T^7 + 375T^6 - 505T^5 + 56T^4 + 3862T^3 - 1066T^2 + 2751*T + 17161
$37$	\( T^{10} - 19 T^{9} + \cdots + 529 \) T^10 - 19T^9 + 174T^8 - 1117T^7 + 6956T^6 - 38345T^5 + 148800T^4 - 346183T^3 + 370815T^2 - 4462*T + 529
$41$	\( T^{10} - 7 T^{9} + \cdots + 1849 \) T^10 - 7T^9 + 16T^8 + 9T^7 + 1235T^6 + 4181T^5 + 15173T^4 + 27978T^3 + 23747T^2 + 9331*T + 1849
$43$	\( T^{10} - 11 T^{9} + \cdots + 64009 \) T^10 - 11T^9 + 66T^8 - 341T^7 + 1826T^6 - 6875T^5 + 20449T^4 - 55297T^3 + 106843T^2 - 114103*T + 64009
$47$	\( (T^{5} - 9 T^{4} - 5 T^{3} + \cdots - 1)^{2} \) (T^5 - 9T^4 - 5T^3 + 97T^2 - 106T - 1)^2
$53$	\( T^{10} + \cdots + 517426009 \) T^10 + 29T^9 + 478T^8 + 5579T^7 + 55454T^6 + 448645T^5 + 2974063T^4 + 16104853T^3 + 83193185T^2 + 312339057*T + 517426009
$59$	\( T^{10} + 21 T^{9} + \cdots + 4489 \) T^10 + 21T^9 + 265T^8 + 1649T^7 + 3433T^6 - 22111T^5 - 98174T^4 - 33991T^3 + 1176352T^2 + 20569*T + 4489
$61$	\( T^{10} - 3 T^{9} + \cdots + 529 \) T^10 - 3T^9 + 218T^8 + 281T^7 + 7066T^6 + 46573T^5 + 215097T^4 + 359141T^3 + 236219T^2 - 22103*T + 529
$67$	\( T^{10} + \cdots + 1113757129 \) T^10 + 45T^9 + 1090T^8 + 17513T^7 + 203501T^6 + 1799216T^5 + 12627902T^4 + 70242899T^3 + 291205025T^2 + 798415652*T + 1113757129
$71$	\( T^{10} + 14 T^{9} + \cdots + 529 \) T^10 + 14T^9 + 130T^8 + 566T^7 + 785T^6 - 197T^5 + 1686T^4 + 1560T^3 + 1204T^2 + 851*T + 529
$73$	\( T^{10} + 19 T^{9} + \cdots + 982081 \) T^10 + 19T^9 + 251T^8 + 2415T^7 + 15063T^6 + 73963T^5 + 379118T^4 + 1529101T^3 + 3411556T^2 + 2470563*T + 982081
$79$	\( T^{10} + \cdots + 517426009 \) T^10 + 15T^9 + 115T^8 + 2715T^7 + 26304T^6 + 111816T^5 + 2966242T^4 + 21480574T^3 + 54168013T^2 - 24589507*T + 517426009
$83$	\( T^{10} + 18 T^{9} + \cdots + 529 \) T^10 + 18T^9 + 192T^8 + 321T^7 + 1455T^6 + 2914T^5 + 224T^4 + 17T^3 + 3804T^2 - 2346*T + 529
$89$	\( T^{10} + \cdots + 78310985281 \) T^10 - 25T^9 + 262T^8 - 5615T^7 + 111566T^6 - 1048455T^5 + 11505750T^4 - 150274617T^3 + 1491272689T^2 - 10362512230*T + 78310985281
$97$	\( T^{10} - 34 T^{9} + \cdots + 2374681 \) T^10 - 34T^9 + 430T^8 - 4423T^7 + 52812T^6 - 183261T^5 + 1071940T^4 - 3733313T^3 + 17156261T^2 - 10511161*T + 2374681
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\(n\)	\(51\)	\(277\)
\(\chi(n)\)	\(\zeta_{22}^{4}\)	\(1\)