LMFDB - Newform orbit 3236.1.l.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3236,1,Mod(7,3236)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3236, base_ring=CyclotomicField(202))

chi = DirichletCharacter(H, H._module([101, 148]))

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

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

chi := DirichletCharacter("3236.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3236 = 2^{2} \cdot 809 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3236.l (of order $202$, degree $100$, 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:	$1.61497438088$
Analytic rank:	$0$
Dimension:	$100$
Coefficient field:	$\Q(\zeta_{202})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{100} - x^{99} + x^{98} - x^{97} + x^{96} - x^{95} + x^{94} - x^{93} + x^{92} - x^{91} + x^{90} + \cdots + 1 $ x^100 - x^99 + x^98 - x^97 + x^96 - x^95 + x^94 - x^93 + x^92 - x^91 + x^90 - x^89 + x^88 - x^87 + x^86 - x^85 + x^84 - x^83 + x^82 - x^81 + x^80 - x^79 + x^78 - x^77 + x^76 - x^75 + x^74 - x^73 + x^72 - x^71 + x^70 - x^69 + x^68 - x^67 + x^66 - x^65 + x^64 - x^63 + x^62 - x^61 + x^60 - x^59 + x^58 - x^57 + x^56 - x^55 + x^54 - x^53 + x^52 - x^51 + x^50 - x^49 + x^48 - x^47 + x^46 - x^45 + x^44 - x^43 + x^42 - x^41 + x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{101}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{101} - \cdots)$

$q$-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_{202}^{56} q^{2} - \zeta_{202}^{11} q^{4} + (\zeta_{202}^{62} - \zeta_{202}^{35}) q^{5} - \zeta_{202}^{67} q^{8} + \zeta_{202}^{74} q^{9} +O(q^{10}) $ q + z^56 * q^2 - z^11 * q^4 + (z^62 - z^35) * q^5 - z^67 * q^8 + z^74 * q^9	$ q + \zeta_{202}^{56} q^{2} - \zeta_{202}^{11} q^{4} + (\zeta_{202}^{62} - \zeta_{202}^{35}) q^{5} - \zeta_{202}^{67} q^{8} + \zeta_{202}^{74} q^{9} + ( - \zeta_{202}^{91} - \zeta_{202}^{17}) q^{10} + ( - \zeta_{202}^{65} + \zeta_{202}^{22}) q^{13} + \zeta_{202}^{22} q^{16} + ( - \zeta_{202}^{37} - \zeta_{202}^{21}) q^{17} - \zeta_{202}^{29} q^{18} + ( - \zeta_{202}^{73} + \zeta_{202}^{46}) q^{20} + ( - \zeta_{202}^{97} + \cdots - \zeta_{202}^{23}) q^{25} + \cdots + \zeta_{202}^{30} q^{98} +O(q^{100}) $ q + z^56 * q^2 - z^11 * q^4 + (z^62 - z^35) * q^5 - z^67 * q^8 + z^74 * q^9 + (-z^91 - z^17) * q^10 + (-z^65 + z^22) * q^13 + z^22 * q^16 + (-z^37 - z^21) * q^17 - z^29 * q^18 + (-z^73 + z^46) * q^20 + (-z^97 + z^70 - z^23) * q^25 + (z^78 + z^20) * q^26 + (z^98 - z^81) * q^29 + z^78 * q^32 + (-z^93 - z^77) * q^34 - z^85 * q^36 + (-z^93 - z^15) * q^37 + (z^28 - z) * q^40 + (-z^55 + z^6) * q^41 + (-z^35 + z^8) * q^45 - z^75 * q^49 + (-z^79 + z^52 - z^25) * q^50 + (z^76 - z^33) * q^52 + (-z^65 + z^52) * q^53 + (-z^53 + z^36) * q^58 + (z^90 - z^69) * q^61 - z^33 * q^64 + (z^100 + z^84 - z^57 + z^26) * q^65 + (z^48 + z^32) * q^68 + z^40 * q^72 + (z^4 + z^2) * q^73 + (-z^71 + z^48) * q^74 + (z^84 - z^57) * q^80 - z^47 * q^81 + (z^62 + z^10) * q^82 + (-z^99 - z^83 + z^72 + z^56) * q^85 + (-z^69 + z^18) * q^89 + (-z^91 + z^64) * q^90 + (z^14 - z^5) * q^97 + z^30 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 100 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9}+O(q^{10}) $ 100 * q - q^2 - q^4 - 2 * q^5 - q^8 - q^9	$ 100 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9} - 2 q^{10} - 2 q^{13} - q^{16} - 2 q^{17} - q^{18} - 2 q^{20} - 3 q^{25} - 2 q^{26} - 2 q^{29} - q^{32} - 2 q^{34} - q^{36} - 2 q^{37} - 2 q^{40} - 2 q^{41} - 2 q^{45} - q^{49} - 3 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} - 4 q^{65} - 2 q^{68} - q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{80} - q^{81} - 2 q^{82} - 4 q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98}+O(q^{100}) $ 100 * q - q^2 - q^4 - 2 * q^5 - q^8 - q^9 - 2 * q^10 - 2 * q^13 - q^16 - 2 * q^17 - q^18 - 2 * q^20 - 3 * q^25 - 2 * q^26 - 2 * q^29 - q^32 - 2 * q^34 - q^36 - 2 * q^37 - 2 * q^40 - 2 * q^41 - 2 * q^45 - q^49 - 3 * q^50 - 2 * q^52 - 2 * q^53 - 2 * q^58 - 2 * q^61 - q^64 - 4 * q^65 - 2 * q^68 - q^72 - 2 * q^73 - 2 * q^74 - 2 * q^80 - q^81 - 2 * q^82 - 4 * q^85 - 2 * q^89 - 2 * q^90 - 2 * q^97 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3236\mathbb{Z}\right)^\times$.

$n$	$1619$	$1621$
$\chi(n)$	$-1$	$\zeta_{202}^{74}$

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} $

7.1

−0.998066	−	0.0621696i
0.0466404	−	0.998912i
0.667588	−	0.744531i
0.350126	+	0.936702i
−0.490994	+	0.871163i
0.830388	+	0.557186i
−0.982635	−	0.185548i
−0.320826	−	0.947138i
0.517850	+	0.855472i
0.231176	+	0.972912i
0.995649	−	0.0931793i
−0.544204	+	0.838953i
−0.200807	−	0.979631i
0.407684	−	0.913123i
0.794142	+	0.607733i
−0.644110	+	0.764933i
−0.952013	−	0.306057i
−0.847315	+	0.531091i
−0.847315	−	0.531091i
0.407684	+	0.913123i

−0.942034

−

0.335517i

0.774857

0.632136i

−1.32486

0.165694i

−0.517850

−

0.855472i

−0.108652

−

0.994080i

1.30365

0.288422i

51.1

−0.863422

0.504483i

0.490994

−

0.871163i

1.96726

−

0.184109i

0.0155518

0.999879i

0.952013

0.306057i

−1.60570

1.15142i

115.1

−0.995649

−

0.0931793i

0.982635

0.185548i

0.204708

−

1.87291i

−0.961071

−

0.276302i

0.774857

0.632136i

−0.378334

1.84569i

207.1

0.379088

−

0.925360i

−0.712584

−

0.701587i

0.935995

0.813364i

−0.919353

0.393434i

−0.231176

0.972912i

1.10748

−

0.557796i

299.1

−0.893115

−

0.449828i

0.595309

0.803497i

−0.144496

0.238703i

−0.170244

−

0.985402i

−0.961071

−

0.276302i

0.236427

−

0.148191i

319.1

−0.108652

0.994080i

−0.976389

−

0.216018i

0.752316

−

1.83641i

0.320826

−

0.947138i

0.969199

−

0.246279i

1.74380

0.947393i

343.1

−0.517850

−

0.855472i

−0.463664

0.886011i

1.51340

−

0.592674i

0.998066

−

0.0621696i

0.320826

0.947138i

−1.29073

−

0.987758i

407.1

0.847315

0.531091i

0.435884

0.900003i

0.736429

0.563567i

−0.108652

0.994080i

−0.570032

−

0.821622i

0.324682

0.868630i

411.1

0.595309

0.803497i

−0.291215

0.956658i

0.891227

1.70304i

−0.942034

0.335517i

0.847315

0.531091i

−0.837832

1.72993i

435.1

0.878693

−

0.477387i

0.544204

−

0.838953i

1.27284

0.641081i

0.0776838

−

0.996978i

0.0155518

−

0.999879i

1.42448

−

0.0443225i

451.1

0.490994

0.871163i

−0.517850

0.855472i

1.87096

0.353288i

−0.999516

−

0.0310999i

0.812658

−

0.582741i

0.610861

1.80338i

475.1

0.690420

0.723409i

−0.0466404

0.998912i

−0.525187

1.17630i

−0.754823

0.655929i

−0.170244

0.985402i

−1.21355

0.432220i

527.1

0.320826

0.947138i

−0.794142

0.607733i

−1.71210

−

0.732687i

−0.830388

−

0.557186i

0.734059

0.679086i

0.144670

−

1.85666i

555.1

−0.0466404

−

0.998912i

−0.995649

0.0931793i

0.227305

−

0.253503i

0.139515

0.990220i

−0.942034

0.335517i

−0.263829

−

0.215234i

615.1

0.435884

−

0.900003i

−0.620010

−

0.784594i

−0.297924

1.10045i

−0.976389

0.216018i

−0.350126

−

0.936702i

0.860547

0.747801i

623.1

0.0776838

0.996978i

−0.987930

0.154898i

−0.235079

1.36068i

−0.231176

−

0.972912i

−0.0466404

−

0.998912i

−1.37483

−

0.128666i

627.1

0.139515

−

0.990220i

−0.961071

−

0.276302i

0.798020

−

0.572245i

−0.407684

0.913123i

−0.517850

−

0.855472i

−0.455312

−

0.870053i

683.1

0.998066

0.0621696i

0.992270

0.124099i

−0.253872

−

0.524188i

0.982635

0.185548i

−0.830388

0.557186i

−0.220792

−

0.538957i

687.1

0.998066

−

0.0621696i

0.992270

−

0.124099i

−0.253872

0.524188i

0.982635

−

0.185548i

−0.830388

−

0.557186i

−0.220792

0.538957i

723.1

−0.0466404

0.998912i

−0.995649

−

0.0931793i

0.227305

0.253503i

0.139515

−

0.990220i

−0.942034

−

0.335517i

−0.263829

0.215234i

See all 100 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
809.e	even	101	1	inner
3236.l	odd	202	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3236.1.l.a	✓	100
4.b	odd	2	1	CM	3236.1.l.a	✓	100
809.e	even	101	1	inner	3236.1.l.a	✓	100
3236.l	odd	202	1	inner	3236.1.l.a	✓	100

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3236.1.l.a	✓	100	1.a	even	1	1	trivial
3236.1.l.a	✓	100	4.b	odd	2	1	CM
3236.1.l.a	✓	100	809.e	even	101	1	inner
3236.1.l.a	✓	100	3236.l	odd	202	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3236, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{100} + T^{99} + \cdots + 1 $ T^100 + T^99 + T^98 + T^97 + T^96 + T^95 + T^94 + T^93 + T^92 + T^91 + T^90 + T^89 + T^88 + T^87 + T^86 + T^85 + T^84 + T^83 + T^82 + T^81 + T^80 + T^79 + T^78 + T^77 + T^76 + T^75 + T^74 + T^73 + T^72 + T^71 + T^70 + T^69 + T^68 + T^67 + T^66 + T^65 + T^64 + T^63 + T^62 + T^61 + T^60 + T^59 + T^58 + T^57 + T^56 + T^55 + T^54 + T^53 + T^52 + T^51 + T^50 + T^49 + T^48 + T^47 + T^46 + T^45 + T^44 + T^43 + T^42 + T^41 + T^40 + T^39 + T^38 + T^37 + T^36 + T^35 + T^34 + T^33 + T^32 + T^31 + T^30 + T^29 + T^28 + T^27 + T^26 + T^25 + T^24 + T^23 + T^22 + T^21 + T^20 + T^19 + T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{100} $ T^100
$5$	$ T^{100} + 2 T^{99} + \cdots + 1 $ T^100 + 2T^99 + 4T^98 + 8T^97 + 16T^96 - 69T^95 - 138T^94 - 276T^93 - 552T^92 - 1104T^91 + 2438T^90 + 4876T^89 + 9752T^88 + 19504T^87 + 39008T^86 - 50860T^85 - 101720T^84 - 203440T^83 - 406880T^82 - 813760T^81 + 778401T^80 + 1556802T^79 + 3113604T^78 + 6227208T^77 + 12454416T^76 - 7039084T^75 - 14078168T^74 - 28156336T^73 - 56312672T^72 - 112638070T^71 + 85717000T^70 + 171434000T^69 + 342868000T^68 + 685736000T^67 + 1356776702T^66 + 459249160T^65 + 918498320T^64 + 1836996640T^63 + 3673993280T^62 + 4562349192T^61 + 21363927672T^60 + 42727855344T^59 + 85455710688T^58 + 170911421376T^57 + 179861368196T^56 + 310051325040T^55 + 620102650080T^54 + 1240205300160T^53 + 2480410600320T^52 + 1268865702810T^51 + 2687041146486T^50 + 5374082292972T^49 + 10748164585944T^48 + 21496329169262T^47 + 5932031919284T^46 + 11537126365888T^45 + 23074252731776T^44 + 46148505463552T^43 + 92297021684917T^42 + 11934762701737T^41 + 24377923454544T^40 + 48755846909088T^39 + 97511693818176T^38 + 195020790533866T^37 + 12292632765278T^36 + 24043994608920T^35 + 48087989217840T^34 + 96175978435680T^33 + 192444012942105T^32 + 5067075698323T^31 + 10508684837334T^30 + 21017369674668T^29 + 42034739349336T^28 + 83411670906232T^27 + 904685094054T^26 + 1653110193852T^25 + 3306220387704T^24 + 6612440775307T^23 + 14264668683219T^22 + 54871599303T^21 + 144890583516T^20 + 289781167032T^19 + 579562155193T^18 + 811632076317T^17 + 1238892089T^16 - 1093340492T^15 - 2186680984T^14 - 4376950498T^13 + 11425447130T^12 - 50382656T^11 + 19728698T^10 + 39457396T^9 + 76631182T^8 + 22808744T^7 + 359792T^6 + 4100T^5 + 8200T^4 - 12385T^3 + 1490T^2 - 50*T + 1
$7$	$ T^{100} $ T^100
$11$	$ T^{100} $ T^100
$13$	$ T^{100} + 2 T^{99} + \cdots + 1 $ T^100 + 2T^99 + 4T^98 + 8T^97 + 16T^96 + 32T^95 + 64T^94 + 128T^93 + 256T^92 + 512T^91 + 1024T^90 + 2048T^89 + 4096T^88 + 8192T^87 + 16384T^86 + 32768T^85 + 65536T^84 + 131072T^83 + 262144T^82 + 524288T^81 + 1048576T^80 + 2097051T^79 + 4194102T^78 + 8388204T^77 + 16776408T^76 + 32837332T^75 + 65674664T^74 + 131349328T^73 + 262698656T^72 + 404903302T^71 + 809806604T^70 + 1619613208T^69 + 3239226416T^68 + 2907328162T^67 + 5814656324T^66 + 11629312648T^65 + 23258625296T^64 + 11369865682T^63 + 22739731364T^62 + 45479462728T^61 + 90958925456T^60 + 25657856656T^59 + 51315716342T^58 + 102631432684T^57 + 205262865368T^56 + 35992290048T^55 + 71939322400T^54 + 143878644800T^53 + 287757289600T^52 + 34243657564T^51 + 91388592044T^50 + 182777184088T^49 + 365554368176T^48 + 222710685282T^47 - 1176603889981T^46 - 2353207779962T^45 - 4706415559924T^44 - 9739768592528T^43 + 8994928582079T^42 + 17989857164158T^41 + 35979714328316T^40 + 71810118915766T^39 - 22298418886878T^38 - 44596837800016T^37 - 89193675600032T^36 - 178437022611416T^35 + 22946904963055T^34 + 45893679472490T^33 + 91787358944980T^32 + 183562478660672T^31 - 10623990981110T^30 - 21268161310346T^29 - 42536322620692T^28 - 85074899545628T^27 + 2509481576841T^26 + 4671470919613T^25 + 9342941839226T^24 + 18685572685312T^23 + 310518951384T^22 - 418749229837T^21 - 837498459674T^20 - 1675028867264T^19 + 341897763302T^18 + 25987734164T^17 + 51975497113T^16 + 103948588305T^15 + 45935702054T^14 - 184666637T^13 - 371616884T^12 - 743362644T^11 + 1298912080T^10 + 721674T^9 + 5031878T^8 + 10059110T^7 + 5422922T^6 + 88031T^5 - 2809T^4 - 5719T^3 + 1288T^2 - 50*T + 1
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$19$	$ T^{100} $ T^100
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$31$	$ T^{100} $ T^100
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$59$	$ T^{100} $ T^100
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$67$	$ T^{100} $ T^100
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$79$	$ T^{100} $ T^100
$83$	$ T^{100} $ T^100
$89$	$ T^{100} + 2 T^{99} + \cdots + 1 $ T^100 + 2T^99 + 4T^98 + 8T^97 + 16T^96 + 32T^95 + 64T^94 + 128T^93 + 256T^92 + 512T^91 + 1024T^90 + 533T^89 + 1066T^88 + 2132T^87 + 4163T^86 + 8326T^85 + 16652T^84 + 33304T^83 + 66608T^82 + 133216T^81 + 266432T^80 + 532864T^79 + 1870799T^78 + 3741598T^77 + 7483196T^76 + 6865788T^75 + 13731576T^74 + 27463152T^73 + 54930041T^72 + 109860082T^71 + 219720164T^70 + 439440328T^69 + 878880656T^68 + 1579128672T^67 + 3158257344T^66 + 6316514688T^65 + 935195640T^64 + 1870391280T^63 + 3740782560T^62 + 6663101520T^61 + 13326203040T^60 + 26652404060T^59 + 53304746510T^58 + 106609493020T^57 + 228699754071T^56 + 457399508142T^55 + 914799016284T^54 - 131265832849T^53 - 262531665698T^52 - 525063331396T^51 + 213458128415T^50 + 426916256830T^49 + 853830525980T^48 + 1697942204851T^47 + 3395884409702T^46 + 6383031778206T^45 + 12766063993136T^44 + 25532127986272T^43 + 8549015104654T^42 + 17098030209308T^41 + 34196060418616T^40 + 746157000648T^39 + 1492314001296T^38 + 2984440522453T^37 + 1599624673937T^36 + 3199249347874T^35 + 8422006836717T^34 + 16829913388452T^33 + 33659826776904T^32 - 15426811917148T^31 - 30853624891968T^30 - 61707249784037T^29 + 2283415671984T^28 + 4566831343968T^27 + 9132149232125T^26 + 351994722351T^25 + 703989444702T^24 + 598689921950T^23 + 1566275163406T^22 + 3132550326812T^21 + 417771028298T^20 + 834325086184T^19 + 1668650211253T^18 + 41328372674T^17 + 82657188637T^16 + 164865265826T^15 + 128812090T^14 + 257624180T^13 + 5001475147T^12 + 430072021T^11 + 860144042T^10 + 46590723T^9 - 14478191T^8 - 28999711T^7 + 332320T^6 + 125906T^5 + 130915T^4 + 947T^3 + 682T^2 - 50*T + 1
$97$	$ T^{100} + 2 T^{99} + \cdots + 1 $ T^100 + 2T^99 + 4T^98 + 8T^97 + 16T^96 + 32T^95 + 64T^94 + 128T^93 + 256T^92 + 512T^91 + 1024T^90 + 2048T^89 + 4096T^88 + 2637T^87 + 5274T^86 + 10548T^85 + 21096T^84 + 28052T^83 + 56104T^82 + 112208T^81 + 224416T^80 + 448832T^79 + 897664T^78 + 1795328T^77 + 3590656T^76 + 7181312T^75 + 23722799T^74 + 47445598T^73 + 94891196T^72 + 189782392T^71 + 54920989T^70 + 109841978T^69 + 219683956T^68 + 439367912T^67 + 924771018T^66 + 1849523149T^65 + 3699046298T^64 + 7398092596T^63 + 14796162770T^62 + 24607367116T^61 + 49214734232T^60 + 98429468464T^59 + 196858936928T^58 + 30000785061T^57 + 60001570122T^56 + 120003140244T^55 + 240006280488T^54 + 34774873844T^53 + 69499737134T^52 + 138999474268T^51 + 277998948536T^50 + 533172821222T^49 + 1653894937699T^48 + 3307789875398T^47 + 6615579750796T^46 + 13231103899577T^45 - 3973749645863T^44 - 7947499294554T^43 - 15894998589108T^42 - 31789997181145T^41 + 1289449529208T^40 + 2570483325730T^39 + 5140966651460T^38 + 10281933302920T^37 + 6931445272695T^36 + 6898822970179T^35 + 13797645940358T^34 + 27595291880716T^33 + 55627889097434T^32 + 7624764253282T^31 + 15249533393853T^30 + 30499066787706T^29 + 60995612081223T^28 + 1429662875220T^27 + 2803600048517T^26 + 5607200097034T^25 + 11214402450105T^24 - 823802323387T^23 + 327510332379T^22 + 655020664657T^21 + 1310041329213T^20 + 1419156876550T^19 - 22745738297T^18 - 45548439786T^17 - 91096879572T^16 - 204893337343T^15 + 1410052640T^14 + 1206803596T^13 + 2413607192T^12 + 4638175714T^11 + 325943427T^10 - 3456999T^9 - 6928441T^8 - 14611049T^7 + 4024072T^6 - 28422T^5 + 4968T^4 + 8522T^3 + 1389T^2 + 51*T + 1
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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:	\( N \)	\(=\)	\( 3236 = 2^{2} \cdot 809 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3236.l (of order \(202\), degree \(100\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{202}^{56} q^{2} - \zeta_{202}^{11} q^{4} + (\zeta_{202}^{62} - \zeta_{202}^{35}) q^{5} - \zeta_{202}^{67} q^{8} + \zeta_{202}^{74} q^{9} +O(q^{10}) \) q + z^56 * q^2 - z^11 * q^4 + (z^62 - z^35) * q^5 - z^67 * q^8 + z^74 * q^9	\( q + \zeta_{202}^{56} q^{2} - \zeta_{202}^{11} q^{4} + (\zeta_{202}^{62} - \zeta_{202}^{35}) q^{5} - \zeta_{202}^{67} q^{8} + \zeta_{202}^{74} q^{9} + ( - \zeta_{202}^{91} - \zeta_{202}^{17}) q^{10} + ( - \zeta_{202}^{65} + \zeta_{202}^{22}) q^{13} + \zeta_{202}^{22} q^{16} + ( - \zeta_{202}^{37} - \zeta_{202}^{21}) q^{17} - \zeta_{202}^{29} q^{18} + ( - \zeta_{202}^{73} + \zeta_{202}^{46}) q^{20} + ( - \zeta_{202}^{97} + \cdots - \zeta_{202}^{23}) q^{25} + \cdots + \zeta_{202}^{30} q^{98} +O(q^{100}) \) q + z^56 * q^2 - z^11 * q^4 + (z^62 - z^35) * q^5 - z^67 * q^8 + z^74 * q^9 + (-z^91 - z^17) * q^10 + (-z^65 + z^22) * q^13 + z^22 * q^16 + (-z^37 - z^21) * q^17 - z^29 * q^18 + (-z^73 + z^46) * q^20 + (-z^97 + z^70 - z^23) * q^25 + (z^78 + z^20) * q^26 + (z^98 - z^81) * q^29 + z^78 * q^32 + (-z^93 - z^77) * q^34 - z^85 * q^36 + (-z^93 - z^15) * q^37 + (z^28 - z) * q^40 + (-z^55 + z^6) * q^41 + (-z^35 + z^8) * q^45 - z^75 * q^49 + (-z^79 + z^52 - z^25) * q^50 + (z^76 - z^33) * q^52 + (-z^65 + z^52) * q^53 + (-z^53 + z^36) * q^58 + (z^90 - z^69) * q^61 - z^33 * q^64 + (z^100 + z^84 - z^57 + z^26) * q^65 + (z^48 + z^32) * q^68 + z^40 * q^72 + (z^4 + z^2) * q^73 + (-z^71 + z^48) * q^74 + (z^84 - z^57) * q^80 - z^47 * q^81 + (z^62 + z^10) * q^82 + (-z^99 - z^83 + z^72 + z^56) * q^85 + (-z^69 + z^18) * q^89 + (-z^91 + z^64) * q^90 + (z^14 - z^5) * q^97 + z^30 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9}+O(q^{10}) \) 100 * q - q^2 - q^4 - 2 * q^5 - q^8 - q^9	\( 100 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9} - 2 q^{10} - 2 q^{13} - q^{16} - 2 q^{17} - q^{18} - 2 q^{20} - 3 q^{25} - 2 q^{26} - 2 q^{29} - q^{32} - 2 q^{34} - q^{36} - 2 q^{37} - 2 q^{40} - 2 q^{41} - 2 q^{45} - q^{49} - 3 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} - 4 q^{65} - 2 q^{68} - q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{80} - q^{81} - 2 q^{82} - 4 q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98}+O(q^{100}) \) 100 * q - q^2 - q^4 - 2 * q^5 - q^8 - q^9 - 2 * q^10 - 2 * q^13 - q^16 - 2 * q^17 - q^18 - 2 * q^20 - 3 * q^25 - 2 * q^26 - 2 * q^29 - q^32 - 2 * q^34 - q^36 - 2 * q^37 - 2 * q^40 - 2 * q^41 - 2 * q^45 - q^49 - 3 * q^50 - 2 * q^52 - 2 * q^53 - 2 * q^58 - 2 * q^61 - q^64 - 4 * q^65 - 2 * q^68 - q^72 - 2 * q^73 - 2 * q^74 - 2 * q^80 - q^81 - 2 * q^82 - 4 * q^85 - 2 * q^89 - 2 * q^90 - 2 * q^97 - q^98

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	3236.1.l.a

Level	$3236$
Weight	$1$
Character orbit	3236.l
Analytic conductor	$1.615$
Analytic rank	$0$
Dimension	$100$
Projective image	$D_{101}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.61497438088\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Coefficient field:	\(\Q(\zeta_{202})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{100} - x^{99} + x^{98} - x^{97} + x^{96} - x^{95} + x^{94} - x^{93} + x^{92} - x^{91} + x^{90} + \cdots + 1 \) x^100 - x^99 + x^98 - x^97 + x^96 - x^95 + x^94 - x^93 + x^92 - x^91 + x^90 - x^89 + x^88 - x^87 + x^86 - x^85 + x^84 - x^83 + x^82 - x^81 + x^80 - x^79 + x^78 - x^77 + x^76 - x^75 + x^74 - x^73 + x^72 - x^71 + x^70 - x^69 + x^68 - x^67 + x^66 - x^65 + x^64 - x^63 + x^62 - x^61 + x^60 - x^59 + x^58 - x^57 + x^56 - x^55 + x^54 - x^53 + x^52 - x^51 + x^50 - x^49 + x^48 - x^47 + x^46 - x^45 + x^44 - x^43 + x^42 - x^41 + x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{101}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{101} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
809.e	even	101	1	inner
3236.l	odd	202	1	inner

\(n\)	\(1619\)	\(1621\)
\(\chi(n)\)	\(-1\)	\(\zeta_{202}^{74}\)