LMFDB - Newform orbit 3716.1.t.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3716,1,Mod(95,3716)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3716, base_ring=CyclotomicField(232))

chi = DirichletCharacter(H, H._module([116, 169]))

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

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

chi := DirichletCharacter("3716.95");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3716 = 2^{2} \cdot 929 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3716.t (of order $232$, degree $112$, 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:	$1.85452558694$
Analytic rank:	$0$
Dimension:	$112$
Coefficient field:	$\Q(\zeta_{232})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{112} - x^{108} + x^{104} - x^{100} + x^{96} - x^{92} + x^{88} - x^{84} + x^{80} - x^{76} + x^{72} + \cdots + 1 $ x^112 - x^108 + x^104 - x^100 + x^96 - x^92 + x^88 - x^84 + x^80 - x^76 + x^72 - x^68 + x^64 - x^60 + x^56 - x^52 + x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{232}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{232} - \cdots)$

$q$-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_{232}^{5} q^{2} + \zeta_{232}^{10} q^{4} + ( - \zeta_{232}^{27} - \zeta_{232}) q^{5} + \zeta_{232}^{15} q^{8} + \zeta_{232}^{53} q^{9}+O(q^{10}) $ q + z^5 * q^2 + z^10 * q^4 + (-z^27 - z) * q^5 + z^15 * q^8 + z^53 * q^9	$ q + \zeta_{232}^{5} q^{2} + \zeta_{232}^{10} q^{4} + ( - \zeta_{232}^{27} - \zeta_{232}) q^{5} + \zeta_{232}^{15} q^{8} + \zeta_{232}^{53} q^{9} + ( - \zeta_{232}^{32} - \zeta_{232}^{6}) q^{10} + (\zeta_{232}^{47} - \zeta_{232}^{18}) q^{13} + \zeta_{232}^{20} q^{16} + (\zeta_{232}^{88} + \zeta_{232}^{77}) q^{17} + \zeta_{232}^{58} q^{18} + ( - \zeta_{232}^{37} - \zeta_{232}^{11}) q^{20} + (\zeta_{232}^{54} + \cdots + \zeta_{232}^{2}) q^{25}+ \cdots + \zeta_{232}^{90} q^{98}+O(q^{100}) $ q + z^5 * q^2 + z^10 * q^4 + (-z^27 - z) * q^5 + z^15 * q^8 + z^53 * q^9 + (-z^32 - z^6) * q^10 + (z^47 - z^18) * q^13 + z^20 * q^16 + (z^88 + z^77) * q^17 + z^58 * q^18 + (-z^37 - z^11) * q^20 + (z^54 + z^28 + z^2) * q^25 + (z^52 - z^23) * q^26 + (z^61 - z^5) * q^29 + z^25 * q^32 + (z^93 + z^82) * q^34 + z^63 * q^36 + (-z^77 + z^3) * q^37 + (-z^42 - z^16) * q^40 + (-z^98 + z^65) * q^41 + (-z^80 - z^54) * q^45 + z^85 * q^49 + (z^59 + z^33 + z^7) * q^50 + (z^57 - z^28) * q^52 + (-z^40 + z^16) * q^53 + (z^66 - z^10) * q^58 + (-z^96 + z^34) * q^61 + z^30 * q^64 + (-z^74 - z^48 + z^45 + z^19) * q^65 + (z^98 + z^87) * q^68 + z^68 * q^72 + (z^113 - z^102) * q^73 + (-z^82 + z^8) * q^74 + (-z^47 - z^21) * q^80 + z^106 * q^81 + (-z^103 + z^70) * q^82 + (-z^115 - z^104 - z^89 - z^78) * q^85 + (z^110 + z^86) * q^89 + (-z^85 - z^59) * q^90 + (z^84 - z^41) * q^97 + z^90 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 112 q+O(q^{10}) $ 112 * q	$ 112 q + 4 q^{10} + 4 q^{16} - 4 q^{17} + 4 q^{25} + 4 q^{26} + 4 q^{40} + 4 q^{45} - 4 q^{52} + 4 q^{61} + 4 q^{65} + 4 q^{72} - 4 q^{74} + 4 q^{85} + 4 q^{97}+O(q^{100}) $ 112 * q + 4 * q^10 + 4 * q^16 - 4 * q^17 + 4 * q^25 + 4 * q^26 + 4 * q^40 + 4 * q^45 - 4 * q^52 + 4 * q^61 + 4 * q^65 + 4 * q^72 - 4 * q^74 + 4 * q^85 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3716\mathbb{Z}\right)^\times$.

$n$	$1859$	$1861$
$\chi(n)$	$-1$	$\zeta_{232}^{53}$

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} $

95.1

−0.999633	−	0.0270794i
0.895872	+	0.444312i
−0.0270794	+	0.999633i
−0.842582	+	0.538568i
0.135000	−	0.990846i
0.0270794	+	0.999633i
0.779413	−	0.626510i
−0.744352	−	0.667788i
−0.918613	−	0.395159i
−0.583395	−	0.812189i
−0.344846	−	0.938659i
0.293523	−	0.955952i
−0.918613	+	0.395159i
−0.982084	−	0.188445i
0.996701	−	0.0811587i
0.395159	−	0.918613i
−0.870504	+	0.492162i
−0.344846	+	0.938659i
0.955952	−	0.293523i
0.538568	+	0.842582i

−0.990846

−

0.135000i

0.963550

0.267528i

1.74399

0.694867i

−0.918613

−

0.395159i

−0.135000

−

0.990846i

−1.63421

−

0.923944i

115.1

−0.667788

0.744352i

−0.108119

−

0.994138i

−1.88672

−

0.309312i

0.812189

0.583395i

0.744352

−

0.667788i

1.49016

−

1.19783i

123.1

−0.135000

0.990846i

−0.963550

−

0.267528i

−0.640708

−

0.255282i

0.395159

−

0.918613i

−0.990846

0.135000i

0.339440

−

0.600380i

131.1

0.955952

0.293523i

0.827689

0.561187i

−0.0960771

−

0.883414i

0.626510

0.779413i

−0.293523

−

0.955952i

0.167457

−

0.872703i

167.1

0.626510

−

0.779413i

−0.214970

−

0.976621i

−0.627162

1.86135i

−0.895872

−

0.444312i

0.779413

−

0.626510i

1.05784

1.65497i

287.1

0.135000

0.990846i

−0.963550

0.267528i

0.640708

−

0.255282i

−0.395159

−

0.918613i

0.990846

0.135000i

0.339440

0.600380i

319.1

−0.970441

0.241338i

0.883512

−

0.468408i

−1.62200

0.0879420i

−0.744352

0.667788i

−0.241338

0.970441i

1.55283

−

0.476792i

327.1

0.870504

0.492162i

0.515554

0.856857i

1.37086

1.44720i

0.0270794

0.999633i

−0.492162

−

0.870504i

0.481084

1.93448i

351.1

0.444312

−

0.895872i

−0.605174

−

0.796093i

0.891534

−

0.604475i

−0.982084

0.188445i

0.895872

−

0.444312i

−0.145413

−

1.06728i

367.1

−0.0270794

0.999633i

−0.998533

−

0.0541389i

1.47927

1.25650i

0.0811587

−

0.996701i

−0.999633

0.0270794i

−1.29610

1.44470i

459.1

−0.982084

0.188445i

0.928977

−

0.370138i

0.426005

1.93536i

−0.842582

0.538568i

0.188445

−

0.982084i

−0.783082

−

1.82041i

479.1

0.996701

−

0.0811587i

0.986827

−

0.161782i

0.688561

1.14440i

0.970441

−

0.241338i

−0.0811587

0.996701i

0.779168

1.08474i

487.1

0.444312

0.895872i

−0.605174

0.796093i

0.891534

0.604475i

−0.982084

−

0.188445i

0.895872

0.444312i

−0.145413

1.06728i

495.1

−0.583395

−

0.812189i

−0.319302

0.947653i

1.37724

−

0.730168i

0.955952

−

0.293523i

0.812189

0.583395i

−1.39651

−

0.692605i

515.1

0.918613

−

0.395159i

0.687699

−

0.725995i

−0.413307

0.893348i

0.344846

−

0.938659i

−0.395159

0.918613i

−0.0266549

0.983962i

543.1

0.895872

0.444312i

0.605174

0.796093i

−1.39479

0.945692i

0.188445

0.982084i

0.444312

0.895872i

−1.66974

0.227497i

587.1

0.842582

0.538568i

0.419889

0.907575i

1.11184

−

1.46260i

−0.135000

0.990846i

0.538568

0.842582i

1.72453

−

0.633561i

591.1

−0.982084

−

0.188445i

0.928977

0.370138i

0.426005

−

1.93536i

−0.842582

−

0.538568i

0.188445

0.982084i

−0.783082

1.82041i

631.1

0.0811587

−

0.996701i

−0.986827

−

0.161782i

−0.767507

1.27561i

−0.241338

0.970441i

−0.996701

0.0811587i

1.20911

0.868502i

639.1

0.293523

−

0.955952i

−0.827689

−

0.561187i

−0.193722

−

1.78124i

−0.779413

0.626510i

−0.955952

0.293523i

−1.75964

−

0.337646i

See next 80 embeddings (of 112 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
929.j	even	232	1	inner
3716.t	odd	232	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3716.1.t.a	✓	112
4.b	odd	2	1	CM	3716.1.t.a	✓	112
929.j	even	232	1	inner	3716.1.t.a	✓	112
3716.t	odd	232	1	inner	3716.1.t.a	✓	112

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3716.1.t.a	✓	112	1.a	even	1	1	trivial
3716.1.t.a	✓	112	4.b	odd	2	1	CM
3716.1.t.a	✓	112	929.j	even	232	1	inner
3716.1.t.a	✓	112	3716.t	odd	232	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3716, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{112} - T^{108} + \cdots + 1 $ T^112 - T^108 + T^104 - T^100 + T^96 - T^92 + T^88 - T^84 + T^80 - T^76 + T^72 - T^68 + T^64 - T^60 + T^56 - T^52 + T^48 - T^44 + T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$3$	$ T^{112} $ T^112
$5$	$ T^{112} - 4 T^{110} + \cdots + 1 $ T^112 - 4T^110 + 12T^108 - 32T^106 + 80T^104 - 192T^102 + 5233T^100 - 20164T^98 + 59724T^96 - 158240T^94 + 394064T^92 - 1082148T^90 + 10645672T^88 - 38254096T^86 + 110433696T^84 - 288718400T^82 + 713140614T^80 + 803966564T^78 - 785111127T^76 - 75395996T^74 + 3442028492T^72 - 13466529984T^70 + 43205053208T^68 - 3575114445464T^66 + 15443153365773T^64 - 47472264189724T^62 + 128116443295804T^60 - 322576585641976T^58 + 792889005616836T^56 - 1273737264027500T^54 + 2015722665506324T^52 - 2967896214178248T^50 + 3808694195156133T^48 - 3366424589314740T^46 + 560310780506468T^44 - 419440048333692T^42 + 542019526553284T^40 - 492318084327836T^38 - 198707588499660T^36 + 3146983560039040T^34 + 398042078334932T^32 - 221220092699856T^30 + 325279928797273T^28 - 410905930116668T^26 + 348329152134522T^24 - 93857235675652T^22 + 5385133969372T^20 - 51794949024T^18 - 36070950777T^16 + 107300576964T^14 + 45421077688T^12 + 4686990728T^10 + 188274654T^8 + 2722476T^6 + 99217T^4 - 552*T^2 + 1
$7$	$ T^{112} $ T^112
$11$	$ T^{112} $ T^112
$13$	$ T^{112} + \cdots + 268435456 $ T^112 - 2T^110 - 4T^109 + 2T^108 + 16T^107 + 16T^106 - 32T^105 - 100T^104 - 32T^103 + 296T^102 + 528T^101 - 264T^100 - 2176T^99 - 2176T^98 + 4352T^97 + 13584T^96 + 4352T^95 - 40224T^94 - 71744T^93 + 35872T^92 + 295680T^91 + 295680T^90 - 591360T^89 - 1845824T^88 - 591360T^87 + 5465728T^86 + 9748736T^85 - 4874368T^84 + 59675136T^83 - 40177664T^82 - 119350272T^81 - 148596480T^80 + 280060928T^79 + 854949376T^78 + 272964608T^77 - 2532949504T^76 - 4525848576T^75 + 2264141824T^74 + 18637565952T^73 + 18641009664T^72 - 37280002048T^71 - 116360566784T^70 - 37279166464T^69 + 344559122432T^68 + 614560604160T^67 - 307280445440T^66 - 2532799365120T^65 - 2532799770624T^64 + 5065599303680T^63 + 15811357892608T^62 + 5065599205376T^61 - 46819513458688T^60 - 83507828588544T^59 + 41753914310656T^58 + 344162512601088T^57 + 344162512650240T^56 - 688325025267712T^55 + 771832853823488T^54 - 688325025267712T^53 + 521309368123392T^52 - 334031314223104T^51 + 167015657177088T^50 - 40524793511936T^49 - 40524793708544T^48 + 81049586761728T^47 - 90882552889344T^46 + 81049588334592T^45 - 61383653851136T^44 + 39331861364736T^43 - 19665939857408T^42 + 4771716005888T^41 + 4771741958144T^40 - 9543395311616T^39 + 10701531774976T^38 - 9543609221120T^37 + 7227033780224T^36 - 4632118034432T^35 + 2317305774080T^34 - 556680609792T^33 - 560206970880T^32 + 1108374192128T^31 - 1287475167232T^30 + 1137440718848T^29 - 738132492288T^28 + 658270846976T^27 - 498547556352T^26 + 319446581248T^25 - 159723290624T^24 + 38755368960T^23 + 38755368960T^22 - 77510737920T^21 + 86914367488T^20 - 77510737920T^19 + 58703478784T^18 - 37614518272T^17 + 18807259136T^16 - 4563402752T^15 - 4563402752T^14 + 9126805504T^13 - 10234101760T^12 + 9126805504T^11 - 6912212992T^10 + 4429185024T^9 - 2214592512T^8 + 536870912T^7 + 536870912T^6 - 1073741824T^5 + 1207959552T^4 - 1073741824T^3 + 805306368T^2 - 536870912T + 268435456
$17$	$ T^{112} + 4 T^{111} + \cdots + 1 $ T^112 + 4T^111 + 10T^110 + 20T^109 + 34T^108 + 48T^107 + 48T^106 - 164T^104 - 560T^103 - 1352T^102 - 2704T^101 - 4616T^100 - 6528T^99 - 6528T^98 + 10556T^97 + 77707T^96 + 234436T^95 + 526782T^94 + 990228T^93 + 1582550T^92 + 2027088T^91 + 1520400T^90 - 1731184T^89 - 11103884T^88 - 32001008T^87 - 71346264T^86 - 134332176T^85 - 215047384T^84 - 275579524T^83 - 173586710T^82 - 617710480T^81 - 2052619813T^80 - 4647404280T^79 - 8397566742T^78 - 12680899580T^77 - 14822575362T^76 - 7500362376T^75 + 25005537832T^74 + 111095823296T^73 + 293993767412T^72 + 624423005952T^71 + 1128101636856T^70 + 1719651934768T^69 + 2019702722585T^68 + 1055741594324T^67 - 565110199948T^66 - 1067602353412T^65 - 648040327199T^64 - 558468193128T^63 - 1485819823070T^62 - 3049426735484T^61 - 4220580121630T^60 - 3412102979640T^59 + 2449001515424T^58 + 19485156923984T^57 + 58039366928092T^56 + 131866738189440T^55 + 252273375854302T^54 + 411079513913944T^53 + 547968031340410T^52 - 62850242119220T^51 - 362850309368136T^50 - 193034148094604T^49 + 64622206656536T^48 + 90989803893652T^47 - 70209998003794T^46 - 182221692561736T^45 - 72911979962644T^44 + 338862635717360T^43 + 1204455648405948T^42 + 2857442244592528T^41 + 5704309590729152T^40 + 9812682217911852T^39 + 14045821046351924T^38 + 14501147529679776T^37 + 5950284166910972T^36 - 4340858320256952T^35 - 6014905579773792T^34 - 980713021856616T^33 + 2948752394614013T^32 + 2301490043000096T^31 - 379595112834604T^30 - 1570885087169708T^29 - 697514288898850T^28 + 513892830084240T^27 + 716306930720436T^26 + 133583761120316T^25 - 312906544773032T^24 - 215926226590736T^23 + 140717234343656T^22 - 19522206416516T^21 + 927280838522T^20 - 63652727328T^19 - 75228576902T^18 - 13096151220T^17 + 36366575822T^16 + 28999171680T^15 - 4076619110T^14 - 18642907404T^13 - 6848379880T^12 + 10159105104T^11 + 15308328020T^10 + 10195106136T^9 + 4082135645T^8 + 1058199972T^7 + 183246888T^6 + 21498628T^5 + 1708749T^4 + 90420T^3 + 3046T^2 + 60T + 1
$19$	$ T^{112} $ T^112
$23$	$ T^{112} $ T^112
$29$	$ T^{112} - 116 T^{100} + \cdots + 707281 $ T^112 - 116T^100 + 31668T^96 + 5394T^88 + 122359497T^84 + 267574242T^80 + 61393T^76 + 20447337284T^72 - 1345814734212T^68 + 529352290975T^64 + 571738461382T^60 + 180074376280044T^56 + 744979583859530T^52 + 122118118585136T^48 - 1648889031297254T^44 + 16238591276874442T^40 - 10422687685646507T^36 + 1939889538197398T^32 + 8358090647880358T^28 + 5407313540797938T^24 + 565283767520893T^20 + 72099157852971T^16 - 7151990815231T^12 + 153328618866T^8 - 138627076T^4 + 707281
$31$	$ T^{112} $ T^112
$37$	$ T^{112} - 4 T^{110} + \cdots + 1 $ T^112 - 4T^110 + 12T^108 - 32T^106 + 80T^104 - 76T^102 - 16T^100 + 252T^98 - 944T^96 + 2768T^94 - 1670T^92 - 4392T^90 + 9490196T^88 - 37943216T^86 + 113817242T^84 - 303348088T^82 + 758123384T^80 - 7052267416T^78 + 25176576128T^76 - 69922947304T^74 + 178987777995T^72 - 436259432964T^70 + 1446485898290T^68 - 4040905861304T^66 + 15895774655406T^64 - 47419453904444T^62 + 126177918971654T^60 - 323108991187656T^58 + 787724290015627T^56 - 1256188875205820T^54 + 1873858455777614T^52 - 2643328848116276T^50 + 3121294661318694T^48 - 1911410354499076T^46 + 1845868492351954T^44 + 262167780615288T^42 - 59847613598536T^40 - 871604861663704T^38 + 4004361672285002T^36 - 3976753866544120T^34 - 110078783664669T^32 - 3323981752868T^30 + 473117773879693T^28 - 291355576506196T^26 + 293153495935269T^24 - 22437904768340T^22 + 21998168409728T^20 + 940835570608T^18 + 572793215572T^16 - 243587399636T^14 - 1283768485T^12 + 3013714740T^10 + 58409957T^8 - 4354916T^6 + 109686T^4 + 144*T^2 + 1
$41$	$ T^{112} - 2 T^{110} + \cdots + 1 $ T^112 - 2T^110 + 4T^109 + 2T^108 - 16T^107 + 16T^106 + 32T^105 - 100T^104 + 32T^103 + 296T^102 - 528T^101 + 4521T^100 + 2176T^99 - 11746T^98 + 14788T^97 + 23154T^96 - 80912T^95 + 36336T^94 + 224864T^93 - 442628T^92 - 142560T^91 + 1712040T^90 - 1935120T^89 + 4784272T^88 + 13233968T^87 - 20733104T^86 - 3460608T^85 + 84833536T^84 - 102479136T^83 - 142043296T^82 + 551213632T^81 - 295497024T^80 - 1465642176T^79 + 3079935168T^78 + 646867020T^77 - 6148127407T^76 - 4728047016T^75 + 8732279900T^74 - 16430149636T^73 - 24080493050T^72 + 77245512904T^71 - 35024172244T^70 - 217952698736T^69 + 427191382204T^68 + 141317682688T^67 - 1656145214864T^66 + 1862052626136T^65 + 4338694192821T^64 + 4615139588240T^63 + 2435084597836T^62 + 4400397685492T^61 + 4913000771646T^60 - 8290736156120T^59 + 2905420003004T^58 + 27432680027840T^57 - 48799786173780T^56 - 26662207731424T^55 + 201519452452914T^54 - 196752373431824T^53 - 319759354347048T^52 - 334912619862952T^51 - 271977918702986T^50 - 216160329109236T^49 - 156175932200121T^48 + 14234223132432T^47 - 8333614630730T^46 - 220851516846876T^45 + 385955986191430T^44 + 379900128905968T^43 - 1638650810074368T^42 + 1226703191605280T^41 + 5129850696467192T^40 + 6582844701268056T^39 + 6296710471905904T^38 + 5072957525951196T^37 + 3478339670301534T^36 + 1875237433295312T^35 + 741729819389908T^34 + 16968762713120T^33 - 939189246201636T^32 - 817493114457232T^31 + 462794112912450T^30 - 2161519211293852T^29 - 1296979905821299T^28 - 745072272789104T^27 - 292299539727284T^26 + 22991024774652T^25 + 203787894744100T^24 + 274961762832224T^23 + 268987389070138T^22 + 219246003762648T^21 + 154296483700420T^20 + 87534023092200T^19 + 30417964386868T^18 + 4462676293824T^17 + 213899976427T^16 + 2652372120T^15 + 68412752T^14 + 73492636T^13 + 71988536T^12 + 55087728T^11 + 22633916T^10 + 30793532T^9 + 31106008T^8 - 81311304T^7 + 39392790T^6 - 8553720T^5 + 1006841T^4 - 68552T^3 + 2700T^2 - 56T + 1
$43$	$ T^{112} $ T^112
$47$	$ T^{112} $ T^112
$53$	$ (T^{28} + 87 T^{21} + \cdots + 29)^{4} $ (T^28 + 87T^21 + 348T^18 + 29T^15 + 1711T^14 - 11194T^11 + 29T^10 + 6873T^8 + 4292T^7 - 1073T^5 + 3712T^4 - 145T^3 + 58T^2 + 174*T + 29)^4
$59$	$ T^{112} $ T^112
$61$	$ (T^{56} - 2 T^{55} + \cdots + 1)^{2} $ (T^56 - 2T^55 + 2T^54 - 4T^52 + 66T^51 - 124T^50 + 116T^49 + 16T^48 - 264T^47 + 1917T^46 - 3306T^45 + 2778T^44 + 1056T^43 - 7668T^42 + 32364T^41 - 49392T^40 + 33592T^39 + 31600T^38 - 130384T^37 + 352138T^36 - 443508T^35 + 352535T^34 + 181946T^33 - 1068962T^32 + 2540154T^31 - 2942384T^30 - 6568268T^29 + 19021304T^28 - 24906070T^27 + 14063780T^26 + 21684580T^25 + 2521372T^24 - 48411904T^23 + 91809716T^22 - 82859744T^21 - 17899944T^20 - 522464T^19 + 36844816T^18 - 70761966T^17 + 71319375T^16 - 1114818T^15 + 3575973T^14 - 4922310T^13 + 10576005T^12 - 9996010T^11 - 1159613T^10 + 1079360T^9 + 160506T^8 - 182294T^7 + 184574T^6 - 12216T^5 + 11926T^4 + 580T^3 + 189T^2 - 30T + 1)^2
$67$	$ T^{112} $ T^112
$71$	$ T^{112} $ T^112
$73$	$ T^{112} - 2 T^{110} + \cdots + 1 $ T^112 - 2T^110 - 4T^109 + 2T^108 + 16T^107 + 16T^106 - 32T^105 - 100T^104 - 32T^103 + 296T^102 + 528T^101 - 264T^100 + 13136T^99 - 2176T^98 - 26272T^97 - 47664T^96 + 34976T^95 + 204768T^94 + 173248T^93 - 454112T^92 - 1235520T^91 - 194304T^90 + 3940992T^89 + 6290271T^88 - 4633728T^87 + 46184570T^86 - 23775612T^85 - 86329945T^84 - 127919600T^83 + 175393198T^82 + 648710204T^81 + 333551894T^80 - 1743154000T^79 - 3612731000T^78 + 854680132T^77 + 13530974212T^76 + 14333764808T^75 - 23255206952T^74 + 39546316320T^73 - 37886593752T^72 - 16612385056T^71 - 35901663872T^70 + 105678514916T^69 + 214026055472T^68 - 34525604232T^67 - 778962842864T^66 - 998410043256T^65 + 1268384714609T^64 + 5181722666432T^63 + 7662703929308T^62 - 13440164104788T^61 + 17846217113897T^60 - 14133932840520T^59 + 5778805949816T^58 - 16236674564972T^57 + 9286377722986T^56 + 37625991011720T^55 + 34816330914286T^54 - 79924147670860T^53 - 238709381321424T^52 - 51956243829056T^51 + 727482691497716T^50 + 210665611111520T^49 - 769721645036360T^48 + 842079104158592T^47 - 758184537368792T^46 + 566078941679684T^45 - 312504051824064T^44 + 217491551631800T^43 - 122938588330298T^42 - 317124806653992T^41 + 1043626891478T^40 + 691020863365576T^39 + 1856770726400474T^38 - 751966620989080T^37 + 5693280692095615T^36 - 7305191390354888T^35 + 7744658083588638T^34 - 6658806745694524T^33 + 4893647265516850T^32 - 3050636062255728T^31 + 1387984014872792T^30 - 155702349583980T^29 - 360696202027409T^28 + 853383403835304T^27 - 1431766227082478T^26 + 633056352230812T^25 + 171391242878558T^24 - 81672770680392T^23 - 11475440515408T^22 + 420741692896T^21 + 6859477995268T^20 - 8208855836272T^19 + 7548961256044T^18 - 5895821923372T^17 + 4034182186214T^16 - 2023076173776T^15 + 1165959753936T^14 - 298896790196T^13 + 159557737095T^12 - 19893088728T^11 + 10274597162T^10 - 651190532T^9 + 330057670T^8 - 9827112T^7 + 7285962T^6 - 818152T^5 + 16810T^4 + 18440T^3 + 380T^2 - 60T + 1
$79$	$ T^{112} $ T^112
$83$	$ T^{112} $ T^112
$89$	$ (T^{56} - 4 T^{54} + \cdots + 1)^{2} $ (T^56 - 4T^54 + 16T^52 - 64T^50 + 256T^48 - 1024T^46 + 4096T^44 - 14876T^42 + 59504T^40 - 97018T^38 + 388072T^36 - 240908T^34 + 963574T^32 - 369221T^30 + 1833526T^28 - 3398224T^26 - 7638990T^24 + 32850208T^22 + 12584255T^20 - 49545697T^18 - 3859052T^16 + 17888216T^14 + 2465228T^12 - 1958441T^10 + 461355T^8 + 38739T^6 + 7183T^4 - 22*T^2 + 1)^2
$97$	$ T^{112} - 4 T^{111} + \cdots + 1 $ T^112 - 4T^111 + 10T^110 - 20T^109 + 34T^108 - 48T^107 + 48T^106 - 164T^104 + 560T^103 - 1352T^102 + 4444T^101 - 11576T^100 + 23928T^99 - 41328T^98 + 59160T^97 - 61232T^96 + 7424T^95 + 183712T^94 - 652784T^93 + 1629733T^92 - 3351924T^91 + 6993396T^90 - 13075404T^89 + 20489470T^88 - 24775192T^87 + 14478772T^86 + 34928992T^85 - 168466772T^84 + 455928436T^83 - 981586624T^82 + 1315616824T^81 - 859727752T^80 - 1111267472T^79 + 6304142352T^78 - 17741287216T^77 + 39304820144T^76 - 73765591744T^75 + 117590310272T^74 - 149504396768T^73 + 109100268602T^72 + 137795248600T^71 - 311249371338T^70 + 280833712880T^69 + 18836458695T^68 - 790941123908T^67 + 2549909637228T^66 - 6091005066004T^65 + 12190654018890T^64 - 20834342369812T^63 + 29457646355468T^62 - 27626793344928T^61 - 7282643311628T^60 + 27049534834728T^59 - 12910398800368T^58 - 23885722651140T^57 + 79372430691112T^56 - 176632865305320T^55 + 351660565279910T^54 - 616563346749888T^53 + 904210448678780T^52 - 970838244724792T^51 + 848478283539394T^50 + 47401232975832T^49 - 532050859304696T^48 + 391559394655800T^47 - 219877544921036T^46 + 563554783016464T^45 - 1437089015209692T^44 + 2463419402285648T^43 - 3045607560108224T^42 + 1894537420301892T^41 + 3715996100718568T^40 - 6310690592116316T^39 + 1459851203162736T^38 + 2847455719122812T^37 - 2338159928439536T^36 - 808491501845440T^35 + 2953400295485876T^34 - 3304940361674692T^33 + 3412209493625495T^32 - 4015813982706336T^31 + 4590886400094656T^30 - 1307618955293204T^29 - 947755056509003T^28 + 1304816322829284T^27 - 189608938621030T^26 - 664204045869160T^25 + 521396180584201T^24 + 48432216799924T^23 - 286038058690660T^22 + 112323729018596T^21 + 30412207786303T^20 - 14793253192276T^19 - 519032641778T^18 + 384047995668T^17 - 73398935506T^16 - 43249324616T^15 + 115277594434T^14 - 106596785960T^13 + 70175821902T^12 - 17498190204T^11 + 9580733074T^10 - 843057144T^9 + 437071966T^8 - 16496888T^7 + 9338296T^6 - 551812T^5 + 42989T^4 + 15952T^3 + 436T^2 - 60T + 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:	\( N \)	\(=\)	\( 3716 = 2^{2} \cdot 929 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3716.t (of order \(232\), degree \(112\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{232}^{5} q^{2} + \zeta_{232}^{10} q^{4} + ( - \zeta_{232}^{27} - \zeta_{232}) q^{5} + \zeta_{232}^{15} q^{8} + \zeta_{232}^{53} q^{9}+O(q^{10}) \) q + z^5 * q^2 + z^10 * q^4 + (-z^27 - z) * q^5 + z^15 * q^8 + z^53 * q^9	\( q + \zeta_{232}^{5} q^{2} + \zeta_{232}^{10} q^{4} + ( - \zeta_{232}^{27} - \zeta_{232}) q^{5} + \zeta_{232}^{15} q^{8} + \zeta_{232}^{53} q^{9} + ( - \zeta_{232}^{32} - \zeta_{232}^{6}) q^{10} + (\zeta_{232}^{47} - \zeta_{232}^{18}) q^{13} + \zeta_{232}^{20} q^{16} + (\zeta_{232}^{88} + \zeta_{232}^{77}) q^{17} + \zeta_{232}^{58} q^{18} + ( - \zeta_{232}^{37} - \zeta_{232}^{11}) q^{20} + (\zeta_{232}^{54} + \cdots + \zeta_{232}^{2}) q^{25}+ \cdots + \zeta_{232}^{90} q^{98}+O(q^{100}) \) q + z^5 * q^2 + z^10 * q^4 + (-z^27 - z) * q^5 + z^15 * q^8 + z^53 * q^9 + (-z^32 - z^6) * q^10 + (z^47 - z^18) * q^13 + z^20 * q^16 + (z^88 + z^77) * q^17 + z^58 * q^18 + (-z^37 - z^11) * q^20 + (z^54 + z^28 + z^2) * q^25 + (z^52 - z^23) * q^26 + (z^61 - z^5) * q^29 + z^25 * q^32 + (z^93 + z^82) * q^34 + z^63 * q^36 + (-z^77 + z^3) * q^37 + (-z^42 - z^16) * q^40 + (-z^98 + z^65) * q^41 + (-z^80 - z^54) * q^45 + z^85 * q^49 + (z^59 + z^33 + z^7) * q^50 + (z^57 - z^28) * q^52 + (-z^40 + z^16) * q^53 + (z^66 - z^10) * q^58 + (-z^96 + z^34) * q^61 + z^30 * q^64 + (-z^74 - z^48 + z^45 + z^19) * q^65 + (z^98 + z^87) * q^68 + z^68 * q^72 + (z^113 - z^102) * q^73 + (-z^82 + z^8) * q^74 + (-z^47 - z^21) * q^80 + z^106 * q^81 + (-z^103 + z^70) * q^82 + (-z^115 - z^104 - z^89 - z^78) * q^85 + (z^110 + z^86) * q^89 + (-z^85 - z^59) * q^90 + (z^84 - z^41) * q^97 + z^90 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q+O(q^{10}) \) 112 * q	\( 112 q + 4 q^{10} + 4 q^{16} - 4 q^{17} + 4 q^{25} + 4 q^{26} + 4 q^{40} + 4 q^{45} - 4 q^{52} + 4 q^{61} + 4 q^{65} + 4 q^{72} - 4 q^{74} + 4 q^{85} + 4 q^{97}+O(q^{100}) \) 112 * q + 4 * q^10 + 4 * q^16 - 4 * q^17 + 4 * q^25 + 4 * q^26 + 4 * q^40 + 4 * q^45 - 4 * q^52 + 4 * q^61 + 4 * q^65 + 4 * q^72 - 4 * q^74 + 4 * q^85 + 4 * 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	3716.1.t.a

Level	$3716$
Weight	$1$
Character orbit	3716.t
Analytic conductor	$1.855$
Analytic rank	$0$
Dimension	$112$
Projective image	$D_{232}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.85452558694\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Coefficient field:	\(\Q(\zeta_{232})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{112} - x^{108} + x^{104} - x^{100} + x^{96} - x^{92} + x^{88} - x^{84} + x^{80} - x^{76} + x^{72} + \cdots + 1 \) x^112 - x^108 + x^104 - x^100 + x^96 - x^92 + x^88 - x^84 + x^80 - x^76 + x^72 - x^68 + x^64 - x^60 + x^56 - x^52 + x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{232}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{232} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 95.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
929.j	even	232	1	inner
3716.t	odd	232	1	inner

$p$	$F_p(T)$
$2$	\( T^{112} - T^{108} + \cdots + 1 \) T^112 - T^108 + T^104 - T^100 + T^96 - T^92 + T^88 - T^84 + T^80 - T^76 + T^72 - T^68 + T^64 - T^60 + T^56 - T^52 + T^48 - T^44 + T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$3$	\( T^{112} \) T^112
$5$	\( T^{112} - 4 T^{110} + \cdots + 1 \) T^112 - 4T^110 + 12T^108 - 32T^106 + 80T^104 - 192T^102 + 5233T^100 - 20164T^98 + 59724T^96 - 158240T^94 + 394064T^92 - 1082148T^90 + 10645672T^88 - 38254096T^86 + 110433696T^84 - 288718400T^82 + 713140614T^80 + 803966564T^78 - 785111127T^76 - 75395996T^74 + 3442028492T^72 - 13466529984T^70 + 43205053208T^68 - 3575114445464T^66 + 15443153365773T^64 - 47472264189724T^62 + 128116443295804T^60 - 322576585641976T^58 + 792889005616836T^56 - 1273737264027500T^54 + 2015722665506324T^52 - 2967896214178248T^50 + 3808694195156133T^48 - 3366424589314740T^46 + 560310780506468T^44 - 419440048333692T^42 + 542019526553284T^40 - 492318084327836T^38 - 198707588499660T^36 + 3146983560039040T^34 + 398042078334932T^32 - 221220092699856T^30 + 325279928797273T^28 - 410905930116668T^26 + 348329152134522T^24 - 93857235675652T^22 + 5385133969372T^20 - 51794949024T^18 - 36070950777T^16 + 107300576964T^14 + 45421077688T^12 + 4686990728T^10 + 188274654T^8 + 2722476T^6 + 99217T^4 - 552*T^2 + 1
$7$	\( T^{112} \) T^112
$11$	\( T^{112} \) T^112
$13$	\( T^{112} + \cdots + 268435456 \) T^112 - 2T^110 - 4T^109 + 2T^108 + 16T^107 + 16T^106 - 32T^105 - 100T^104 - 32T^103 + 296T^102 + 528T^101 - 264T^100 - 2176T^99 - 2176T^98 + 4352T^97 + 13584T^96 + 4352T^95 - 40224T^94 - 71744T^93 + 35872T^92 + 295680T^91 + 295680T^90 - 591360T^89 - 1845824T^88 - 591360T^87 + 5465728T^86 + 9748736T^85 - 4874368T^84 + 59675136T^83 - 40177664T^82 - 119350272T^81 - 148596480T^80 + 280060928T^79 + 854949376T^78 + 272964608T^77 - 2532949504T^76 - 4525848576T^75 + 2264141824T^74 + 18637565952T^73 + 18641009664T^72 - 37280002048T^71 - 116360566784T^70 - 37279166464T^69 + 344559122432T^68 + 614560604160T^67 - 307280445440T^66 - 2532799365120T^65 - 2532799770624T^64 + 5065599303680T^63 + 15811357892608T^62 + 5065599205376T^61 - 46819513458688T^60 - 83507828588544T^59 + 41753914310656T^58 + 344162512601088T^57 + 344162512650240T^56 - 688325025267712T^55 + 771832853823488T^54 - 688325025267712T^53 + 521309368123392T^52 - 334031314223104T^51 + 167015657177088T^50 - 40524793511936T^49 - 40524793708544T^48 + 81049586761728T^47 - 90882552889344T^46 + 81049588334592T^45 - 61383653851136T^44 + 39331861364736T^43 - 19665939857408T^42 + 4771716005888T^41 + 4771741958144T^40 - 9543395311616T^39 + 10701531774976T^38 - 9543609221120T^37 + 7227033780224T^36 - 4632118034432T^35 + 2317305774080T^34 - 556680609792T^33 - 560206970880T^32 + 1108374192128T^31 - 1287475167232T^30 + 1137440718848T^29 - 738132492288T^28 + 658270846976T^27 - 498547556352T^26 + 319446581248T^25 - 159723290624T^24 + 38755368960T^23 + 38755368960T^22 - 77510737920T^21 + 86914367488T^20 - 77510737920T^19 + 58703478784T^18 - 37614518272T^17 + 18807259136T^16 - 4563402752T^15 - 4563402752T^14 + 9126805504T^13 - 10234101760T^12 + 9126805504T^11 - 6912212992T^10 + 4429185024T^9 - 2214592512T^8 + 536870912T^7 + 536870912T^6 - 1073741824T^5 + 1207959552T^4 - 1073741824T^3 + 805306368T^2 - 536870912T + 268435456
$17$	\( T^{112} + 4 T^{111} + \cdots + 1 \) T^112 + 4T^111 + 10T^110 + 20T^109 + 34T^108 + 48T^107 + 48T^106 - 164T^104 - 560T^103 - 1352T^102 - 2704T^101 - 4616T^100 - 6528T^99 - 6528T^98 + 10556T^97 + 77707T^96 + 234436T^95 + 526782T^94 + 990228T^93 + 1582550T^92 + 2027088T^91 + 1520400T^90 - 1731184T^89 - 11103884T^88 - 32001008T^87 - 71346264T^86 - 134332176T^85 - 215047384T^84 - 275579524T^83 - 173586710T^82 - 617710480T^81 - 2052619813T^80 - 4647404280T^79 - 8397566742T^78 - 12680899580T^77 - 14822575362T^76 - 7500362376T^75 + 25005537832T^74 + 111095823296T^73 + 293993767412T^72 + 624423005952T^71 + 1128101636856T^70 + 1719651934768T^69 + 2019702722585T^68 + 1055741594324T^67 - 565110199948T^66 - 1067602353412T^65 - 648040327199T^64 - 558468193128T^63 - 1485819823070T^62 - 3049426735484T^61 - 4220580121630T^60 - 3412102979640T^59 + 2449001515424T^58 + 19485156923984T^57 + 58039366928092T^56 + 131866738189440T^55 + 252273375854302T^54 + 411079513913944T^53 + 547968031340410T^52 - 62850242119220T^51 - 362850309368136T^50 - 193034148094604T^49 + 64622206656536T^48 + 90989803893652T^47 - 70209998003794T^46 - 182221692561736T^45 - 72911979962644T^44 + 338862635717360T^43 + 1204455648405948T^42 + 2857442244592528T^41 + 5704309590729152T^40 + 9812682217911852T^39 + 14045821046351924T^38 + 14501147529679776T^37 + 5950284166910972T^36 - 4340858320256952T^35 - 6014905579773792T^34 - 980713021856616T^33 + 2948752394614013T^32 + 2301490043000096T^31 - 379595112834604T^30 - 1570885087169708T^29 - 697514288898850T^28 + 513892830084240T^27 + 716306930720436T^26 + 133583761120316T^25 - 312906544773032T^24 - 215926226590736T^23 + 140717234343656T^22 - 19522206416516T^21 + 927280838522T^20 - 63652727328T^19 - 75228576902T^18 - 13096151220T^17 + 36366575822T^16 + 28999171680T^15 - 4076619110T^14 - 18642907404T^13 - 6848379880T^12 + 10159105104T^11 + 15308328020T^10 + 10195106136T^9 + 4082135645T^8 + 1058199972T^7 + 183246888T^6 + 21498628T^5 + 1708749T^4 + 90420T^3 + 3046T^2 + 60T + 1
$19$	\( T^{112} \) T^112
$23$	\( T^{112} \) T^112
$29$	\( T^{112} - 116 T^{100} + \cdots + 707281 \) T^112 - 116T^100 + 31668T^96 + 5394T^88 + 122359497T^84 + 267574242T^80 + 61393T^76 + 20447337284T^72 - 1345814734212T^68 + 529352290975T^64 + 571738461382T^60 + 180074376280044T^56 + 744979583859530T^52 + 122118118585136T^48 - 1648889031297254T^44 + 16238591276874442T^40 - 10422687685646507T^36 + 1939889538197398T^32 + 8358090647880358T^28 + 5407313540797938T^24 + 565283767520893T^20 + 72099157852971T^16 - 7151990815231T^12 + 153328618866T^8 - 138627076T^4 + 707281
$31$	\( T^{112} \) T^112
$37$	\( T^{112} - 4 T^{110} + \cdots + 1 \) T^112 - 4T^110 + 12T^108 - 32T^106 + 80T^104 - 76T^102 - 16T^100 + 252T^98 - 944T^96 + 2768T^94 - 1670T^92 - 4392T^90 + 9490196T^88 - 37943216T^86 + 113817242T^84 - 303348088T^82 + 758123384T^80 - 7052267416T^78 + 25176576128T^76 - 69922947304T^74 + 178987777995T^72 - 436259432964T^70 + 1446485898290T^68 - 4040905861304T^66 + 15895774655406T^64 - 47419453904444T^62 + 126177918971654T^60 - 323108991187656T^58 + 787724290015627T^56 - 1256188875205820T^54 + 1873858455777614T^52 - 2643328848116276T^50 + 3121294661318694T^48 - 1911410354499076T^46 + 1845868492351954T^44 + 262167780615288T^42 - 59847613598536T^40 - 871604861663704T^38 + 4004361672285002T^36 - 3976753866544120T^34 - 110078783664669T^32 - 3323981752868T^30 + 473117773879693T^28 - 291355576506196T^26 + 293153495935269T^24 - 22437904768340T^22 + 21998168409728T^20 + 940835570608T^18 + 572793215572T^16 - 243587399636T^14 - 1283768485T^12 + 3013714740T^10 + 58409957T^8 - 4354916T^6 + 109686T^4 + 144*T^2 + 1
$41$	\( T^{112} - 2 T^{110} + \cdots + 1 \) T^112 - 2T^110 + 4T^109 + 2T^108 - 16T^107 + 16T^106 + 32T^105 - 100T^104 + 32T^103 + 296T^102 - 528T^101 + 4521T^100 + 2176T^99 - 11746T^98 + 14788T^97 + 23154T^96 - 80912T^95 + 36336T^94 + 224864T^93 - 442628T^92 - 142560T^91 + 1712040T^90 - 1935120T^89 + 4784272T^88 + 13233968T^87 - 20733104T^86 - 3460608T^85 + 84833536T^84 - 102479136T^83 - 142043296T^82 + 551213632T^81 - 295497024T^80 - 1465642176T^79 + 3079935168T^78 + 646867020T^77 - 6148127407T^76 - 4728047016T^75 + 8732279900T^74 - 16430149636T^73 - 24080493050T^72 + 77245512904T^71 - 35024172244T^70 - 217952698736T^69 + 427191382204T^68 + 141317682688T^67 - 1656145214864T^66 + 1862052626136T^65 + 4338694192821T^64 + 4615139588240T^63 + 2435084597836T^62 + 4400397685492T^61 + 4913000771646T^60 - 8290736156120T^59 + 2905420003004T^58 + 27432680027840T^57 - 48799786173780T^56 - 26662207731424T^55 + 201519452452914T^54 - 196752373431824T^53 - 319759354347048T^52 - 334912619862952T^51 - 271977918702986T^50 - 216160329109236T^49 - 156175932200121T^48 + 14234223132432T^47 - 8333614630730T^46 - 220851516846876T^45 + 385955986191430T^44 + 379900128905968T^43 - 1638650810074368T^42 + 1226703191605280T^41 + 5129850696467192T^40 + 6582844701268056T^39 + 6296710471905904T^38 + 5072957525951196T^37 + 3478339670301534T^36 + 1875237433295312T^35 + 741729819389908T^34 + 16968762713120T^33 - 939189246201636T^32 - 817493114457232T^31 + 462794112912450T^30 - 2161519211293852T^29 - 1296979905821299T^28 - 745072272789104T^27 - 292299539727284T^26 + 22991024774652T^25 + 203787894744100T^24 + 274961762832224T^23 + 268987389070138T^22 + 219246003762648T^21 + 154296483700420T^20 + 87534023092200T^19 + 30417964386868T^18 + 4462676293824T^17 + 213899976427T^16 + 2652372120T^15 + 68412752T^14 + 73492636T^13 + 71988536T^12 + 55087728T^11 + 22633916T^10 + 30793532T^9 + 31106008T^8 - 81311304T^7 + 39392790T^6 - 8553720T^5 + 1006841T^4 - 68552T^3 + 2700T^2 - 56T + 1
$43$	\( T^{112} \) T^112
$47$	\( T^{112} \) T^112
$53$	\( (T^{28} + 87 T^{21} + \cdots + 29)^{4} \) (T^28 + 87T^21 + 348T^18 + 29T^15 + 1711T^14 - 11194T^11 + 29T^10 + 6873T^8 + 4292T^7 - 1073T^5 + 3712T^4 - 145T^3 + 58T^2 + 174*T + 29)^4
$59$	\( T^{112} \) T^112
$61$	\( (T^{56} - 2 T^{55} + \cdots + 1)^{2} \) (T^56 - 2T^55 + 2T^54 - 4T^52 + 66T^51 - 124T^50 + 116T^49 + 16T^48 - 264T^47 + 1917T^46 - 3306T^45 + 2778T^44 + 1056T^43 - 7668T^42 + 32364T^41 - 49392T^40 + 33592T^39 + 31600T^38 - 130384T^37 + 352138T^36 - 443508T^35 + 352535T^34 + 181946T^33 - 1068962T^32 + 2540154T^31 - 2942384T^30 - 6568268T^29 + 19021304T^28 - 24906070T^27 + 14063780T^26 + 21684580T^25 + 2521372T^24 - 48411904T^23 + 91809716T^22 - 82859744T^21 - 17899944T^20 - 522464T^19 + 36844816T^18 - 70761966T^17 + 71319375T^16 - 1114818T^15 + 3575973T^14 - 4922310T^13 + 10576005T^12 - 9996010T^11 - 1159613T^10 + 1079360T^9 + 160506T^8 - 182294T^7 + 184574T^6 - 12216T^5 + 11926T^4 + 580T^3 + 189T^2 - 30T + 1)^2
$67$	\( T^{112} \) T^112
$71$	\( T^{112} \) T^112
$73$	\( T^{112} - 2 T^{110} + \cdots + 1 \) T^112 - 2T^110 - 4T^109 + 2T^108 + 16T^107 + 16T^106 - 32T^105 - 100T^104 - 32T^103 + 296T^102 + 528T^101 - 264T^100 + 13136T^99 - 2176T^98 - 26272T^97 - 47664T^96 + 34976T^95 + 204768T^94 + 173248T^93 - 454112T^92 - 1235520T^91 - 194304T^90 + 3940992T^89 + 6290271T^88 - 4633728T^87 + 46184570T^86 - 23775612T^85 - 86329945T^84 - 127919600T^83 + 175393198T^82 + 648710204T^81 + 333551894T^80 - 1743154000T^79 - 3612731000T^78 + 854680132T^77 + 13530974212T^76 + 14333764808T^75 - 23255206952T^74 + 39546316320T^73 - 37886593752T^72 - 16612385056T^71 - 35901663872T^70 + 105678514916T^69 + 214026055472T^68 - 34525604232T^67 - 778962842864T^66 - 998410043256T^65 + 1268384714609T^64 + 5181722666432T^63 + 7662703929308T^62 - 13440164104788T^61 + 17846217113897T^60 - 14133932840520T^59 + 5778805949816T^58 - 16236674564972T^57 + 9286377722986T^56 + 37625991011720T^55 + 34816330914286T^54 - 79924147670860T^53 - 238709381321424T^52 - 51956243829056T^51 + 727482691497716T^50 + 210665611111520T^49 - 769721645036360T^48 + 842079104158592T^47 - 758184537368792T^46 + 566078941679684T^45 - 312504051824064T^44 + 217491551631800T^43 - 122938588330298T^42 - 317124806653992T^41 + 1043626891478T^40 + 691020863365576T^39 + 1856770726400474T^38 - 751966620989080T^37 + 5693280692095615T^36 - 7305191390354888T^35 + 7744658083588638T^34 - 6658806745694524T^33 + 4893647265516850T^32 - 3050636062255728T^31 + 1387984014872792T^30 - 155702349583980T^29 - 360696202027409T^28 + 853383403835304T^27 - 1431766227082478T^26 + 633056352230812T^25 + 171391242878558T^24 - 81672770680392T^23 - 11475440515408T^22 + 420741692896T^21 + 6859477995268T^20 - 8208855836272T^19 + 7548961256044T^18 - 5895821923372T^17 + 4034182186214T^16 - 2023076173776T^15 + 1165959753936T^14 - 298896790196T^13 + 159557737095T^12 - 19893088728T^11 + 10274597162T^10 - 651190532T^9 + 330057670T^8 - 9827112T^7 + 7285962T^6 - 818152T^5 + 16810T^4 + 18440T^3 + 380T^2 - 60T + 1
$79$	\( T^{112} \) T^112
$83$	\( T^{112} \) T^112
$89$	\( (T^{56} - 4 T^{54} + \cdots + 1)^{2} \) (T^56 - 4T^54 + 16T^52 - 64T^50 + 256T^48 - 1024T^46 + 4096T^44 - 14876T^42 + 59504T^40 - 97018T^38 + 388072T^36 - 240908T^34 + 963574T^32 - 369221T^30 + 1833526T^28 - 3398224T^26 - 7638990T^24 + 32850208T^22 + 12584255T^20 - 49545697T^18 - 3859052T^16 + 17888216T^14 + 2465228T^12 - 1958441T^10 + 461355T^8 + 38739T^6 + 7183T^4 - 22*T^2 + 1)^2
$97$	\( T^{112} - 4 T^{111} + \cdots + 1 \) T^112 - 4T^111 + 10T^110 - 20T^109 + 34T^108 - 48T^107 + 48T^106 - 164T^104 + 560T^103 - 1352T^102 + 4444T^101 - 11576T^100 + 23928T^99 - 41328T^98 + 59160T^97 - 61232T^96 + 7424T^95 + 183712T^94 - 652784T^93 + 1629733T^92 - 3351924T^91 + 6993396T^90 - 13075404T^89 + 20489470T^88 - 24775192T^87 + 14478772T^86 + 34928992T^85 - 168466772T^84 + 455928436T^83 - 981586624T^82 + 1315616824T^81 - 859727752T^80 - 1111267472T^79 + 6304142352T^78 - 17741287216T^77 + 39304820144T^76 - 73765591744T^75 + 117590310272T^74 - 149504396768T^73 + 109100268602T^72 + 137795248600T^71 - 311249371338T^70 + 280833712880T^69 + 18836458695T^68 - 790941123908T^67 + 2549909637228T^66 - 6091005066004T^65 + 12190654018890T^64 - 20834342369812T^63 + 29457646355468T^62 - 27626793344928T^61 - 7282643311628T^60 + 27049534834728T^59 - 12910398800368T^58 - 23885722651140T^57 + 79372430691112T^56 - 176632865305320T^55 + 351660565279910T^54 - 616563346749888T^53 + 904210448678780T^52 - 970838244724792T^51 + 848478283539394T^50 + 47401232975832T^49 - 532050859304696T^48 + 391559394655800T^47 - 219877544921036T^46 + 563554783016464T^45 - 1437089015209692T^44 + 2463419402285648T^43 - 3045607560108224T^42 + 1894537420301892T^41 + 3715996100718568T^40 - 6310690592116316T^39 + 1459851203162736T^38 + 2847455719122812T^37 - 2338159928439536T^36 - 808491501845440T^35 + 2953400295485876T^34 - 3304940361674692T^33 + 3412209493625495T^32 - 4015813982706336T^31 + 4590886400094656T^30 - 1307618955293204T^29 - 947755056509003T^28 + 1304816322829284T^27 - 189608938621030T^26 - 664204045869160T^25 + 521396180584201T^24 + 48432216799924T^23 - 286038058690660T^22 + 112323729018596T^21 + 30412207786303T^20 - 14793253192276T^19 - 519032641778T^18 + 384047995668T^17 - 73398935506T^16 - 43249324616T^15 + 115277594434T^14 - 106596785960T^13 + 70175821902T^12 - 17498190204T^11 + 9580733074T^10 - 843057144T^9 + 437071966T^8 - 16496888T^7 + 9338296T^6 - 551812T^5 + 42989T^4 + 15952T^3 + 436T^2 - 60T + 1
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\(n\)	\(1859\)	\(1861\)
\(\chi(n)\)	\(-1\)	\(\zeta_{232}^{53}\)