LMFDB - Newform orbit 1860.4.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,4,Mod(1,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1860.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1860.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,-27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$109.743552611$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 1602 x^{7} - 3024 x^{6} + 832872 x^{5} + 2239848 x^{4} - 157824556 x^{3} + \cdots - 36448841472 $ x^9 - x^8 - 1602x^7 - 3024x^6 + 832872x^5 + 2239848x^4 - 157824556x^3 - 62701580x^2 + 10698567888*x - 36448841472
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{8}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 3 q^{3} + 5 q^{5} + (\beta_1 - 4) q^{7} + 9 q^{9} + (\beta_{4} - 5) q^{11} - \beta_{3} q^{13} - 15 q^{15} + ( - \beta_{5} - 5) q^{17} + ( - \beta_{8} + \beta_{4} - 1) q^{19} + ( - 3 \beta_1 + 12) q^{21}+ \cdots + (9 \beta_{4} - 45) q^{99}+O(q^{100}) $ q - 3 * q^3 + 5 * q^5 + (b1 - 4) * q^7 + 9 * q^9 + (b4 - 5) * q^11 - b3 * q^13 - 15 * q^15 + (-b5 - 5) * q^17 + (-b8 + b4 - 1) * q^19 + (-3b1 + 12) q^21 + (b7 + b4 - b2 + 4b1) q^23 + 25 * q^25 - 27 * q^27 + (b8 - b7 + b6 + b5 + 2b4 - b3 + b1 + 31) q^29 - 31 * q^31 + (-3b4 + 15) q^33 + (5b1 - 20) q^35 + (b8 - 3b6 - b5 + b4 - 2b3 + 2b2 + 4b1 + 14) * q^37 + 3b3 q^39 + (-b8 + b6 + b5 + b4 + b3 + 4b2 + 4b1 + 88) * q^41 + (-2b7 + b6 + b5 - 4b4 - b3 - 2b2 + 10b1 - 19) * q^43 + 45 * q^45 + (b8 - 3b7 - 2b5 + b4 + b3 + 3b1 - 73) q^47 + (2b7 - b6 + 2b5 - 3b4 - 2b3 + b2 - 3b1 + 31) q^49 + (3b5 + 15) q^51 + (2b8 + 3b7 - 4b6 - 2b5 + b3 - 4b2 - b1 + 70) q^53 + (5b4 - 25) q^55 + (3b8 - 3b4 + 3) * q^57 + (-b8 + 3b7 + 4b6 + 2b5 - b4 + 4b3 - 7b1 - 5) q^59 + (-2b8 - 4b7 + 5b6 - b5 - b4 + 5b3 - 2b2 + 4b1 + 106) * q^61 + (9b1 - 36) q^63 - 5b3 q^65 + (-3b8 + 5b7 + 2b6 + 3b5 + b3 + 4b2 + 18b1 + 19) * q^67 + (-3b7 - 3b4 + 3b2 - 12b1) * q^69 + (b8 - b7 - 5b6 - b5 + 7b4 + 3b3 - 6b2 + b1 + 136) * q^71 + (-2b8 - 2b7 + 8b6 + 4b5 - 5b4 - 3b3 - 2b2 + 2b1 + 83) * q^73 - 75 * q^75 + (b8 + b6 - 5b5 - 9b4 + 8b3 - 8b2 - 23b1 - 24) q^77 + (3b8 + 7b7 - 7b6 - b5 + 5b4 + 2b3 - 3b2 - 5b1 - 43) q^79 + 81 * q^81 + (-3b8 + 5b7 - 5b6 + b5 + 3b4 - 3b3 + 11b2 - 14b1 - 35) q^83 + (-5b5 - 25) q^85 + (-3b8 + 3b7 - 3b6 - 3b5 - 6b4 + 3b3 - 3b1 - 93) q^87 + (-8b7 - b5 + 3b4 - 3b3 - 2b2 - 16b1 + 104) q^89 + (-b8 - 6b7 - 9b6 - 6b5 - 8b4 - 4b3 + b2 + 3b1 - 125) * q^91 + 93 * q^93 + (-5b8 + 5b4 - 5) * q^95 + (-4b8 + b7 + b6 - 15b4 + b3 - 4b2 + 8b1 + 45) * q^97 + (9b4 - 45) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 27 q^{3} + 45 q^{5} - 35 q^{7} + 81 q^{9} - 45 q^{11} - 4 q^{13} - 135 q^{15} - 42 q^{17} - 11 q^{19} + 105 q^{21} - q^{23} + 225 q^{25} - 243 q^{27} + 280 q^{29} - 279 q^{31} + 135 q^{33} - 175 q^{35}+ \cdots - 405 q^{99}+O(q^{100}) $ 9 * q - 27 * q^3 + 45 * q^5 - 35 * q^7 + 81 * q^9 - 45 * q^11 - 4 * q^13 - 135 * q^15 - 42 * q^17 - 11 * q^19 + 105 * q^21 - q^23 + 225 * q^25 - 243 * q^27 + 280 * q^29 - 279 * q^31 + 135 * q^33 - 175 * q^35 + 126 * q^37 + 12 * q^39 + 800 * q^41 - 161 * q^43 + 405 * q^45 - 630 * q^47 + 254 * q^49 + 126 * q^51 + 623 * q^53 - 225 * q^55 + 33 * q^57 - 52 * q^59 + 996 * q^61 - 315 * q^63 - 20 * q^65 + 164 * q^67 + 3 * q^69 + 1235 * q^71 + 735 * q^73 - 675 * q^75 - 197 * q^77 - 413 * q^79 + 729 * q^81 - 364 * q^83 - 210 * q^85 - 840 * q^87 + 941 * q^89 - 1106 * q^91 + 837 * q^93 - 55 * q^95 + 402 * q^97 - 405 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{9} - x^{8} - 1602 x^{7} - 3024 x^{6} + 832872 x^{5} + 2239848 x^{4} - 157824556 x^{3} + \cdots - 36448841472 $ x^9 - x^8 - 1602*x^7 - 3024*x^6 + 832872*x^5 + 2239848*x^4 - 157824556*x^3 - 62701580*x^2 + 10698567888*x - 36448841472 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 14\!\cdots\!87 \nu^{8} + \cdots + 15\!\cdots\!04 ) / 95\!\cdots\!24 $ (-14649774898704887v^8 - 109843825772501375v^7 + 16713235874745291032v^6 + 343426642467522590788v^5 - 4004624362525343854552v^4 - 188939047338816887974340v^3 - 578081333748745273795964v^2 + 18008258069907464631067836v + 15083864800454863041686004) / 959877740205219924626724
$\beta_{3}$	$=$	$ ( - 12\!\cdots\!47 \nu^{8} + \cdots + 42\!\cdots\!24 ) / 47\!\cdots\!62 $ (-12981204298863347v^8 + 1285501769843372575v^7 + 124270863548694812v^6 - 1605627480271369520984v^5 + 8509686496615444737398v^4 + 633111495161294169654334v^3 - 3351894285711217282550192v^2 - 72633050603890099856688294v + 421041180314005235579174124) / 479938870102609962313362
$\beta_{4}$	$=$	$ ( 16\!\cdots\!03 \nu^{8} + \cdots + 50\!\cdots\!80 ) / 19\!\cdots\!48 $ (165438845535184303v^8 - 466050155311658687v^7 - 248903306988483818014v^6 - 140124153576917185280v^5 + 114072507897691706493128v^4 + 265428757158488327166304v^3 - 15743562712130148394694372v^2 - 19207050342007855585071780v + 504882971583739919362877880) / 1919755480410439849253448
$\beta_{5}$	$=$	$ ( - 71\!\cdots\!73 \nu^{8} + \cdots + 50\!\cdots\!20 ) / 63\!\cdots\!16 $ (-71547829443361673v^8 + 1457369815087448493v^7 + 84096411582317136286v^6 - 1390757185989784780208v^5 - 30552890581483505692200v^4 + 416965729335651476551784v^3 + 2983369456186604074508308v^2 - 45566946250995501609731172v + 50433720983435209390078920) / 639918493470146616417816
$\beta_{6}$	$=$	$ ( 98\!\cdots\!53 \nu^{8} + \cdots - 85\!\cdots\!60 ) / 63\!\cdots\!16 $ (98575864066461853v^8 - 4254452513070277297v^7 - 86725917918704646302v^6 + 4870729762117375302776v^5 + 16072166444880267940456v^4 - 1749891518011921279645192v^3 + 2908014780258830894379708v^2 + 188180245545562177096315404v - 856814103345588202976391360) / 639918493470146616417816
$\beta_{7}$	$=$	$ ( 47\!\cdots\!46 \nu^{8} + \cdots + 53\!\cdots\!60 ) / 15\!\cdots\!54 $ (47782542536317146v^8 - 516751045224104815v^7 - 64329102044023718093v^6 + 375186950602253403512v^5 + 27076587800607966794260v^4 - 63017191775259190981834v^3 - 3339420694604807700481250v^2 + 6401731864318668617925360v + 53853897072418274389011360) / 159979623367536654104454
$\beta_{8}$	$=$	$ ( - 22\!\cdots\!37 \nu^{8} + \cdots - 42\!\cdots\!32 ) / 63\!\cdots\!16 $ (-220453325588671537v^8 + 1843273233499252281v^7 + 300235138077432820882v^6 - 1005378818935870576280v^5 - 125886795465445917849312v^4 - 28179564368869803011392v^3 + 15495408803525000308703772v^2 + 12596472097319058187605324v - 422029818391070123851627632) / 639918493470146616417816

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{7} - \beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + \beta_{2} + 5\beta _1 + 358 $ 2b7 - b6 + 2b5 - 3b4 - 2b3 + b2 + 5*b1 + 358
$\nu^{3}$	$=$	$ -19\beta_{8} + 13\beta_{7} - 13\beta_{6} + 32\beta_{5} - 58\beta_{4} - 27\beta_{3} + 38\beta_{2} + 576\beta _1 + 1508 $ -19b8 + 13b7 - 13b6 + 32b5 - 58b4 - 27b3 + 38b2 + 576b1 + 1508
$\nu^{4}$	$=$	$ - 177 \beta_{8} + 1596 \beta_{7} - 1121 \beta_{6} + 1271 \beta_{5} - 2933 \beta_{4} - 1950 \beta_{3} + \cdots + 203398 $ -177b8 + 1596b7 - 1121b6 + 1271b5 - 2933b4 - 1950b3 + 1498b2 + 7217b1 + 203398
$\nu^{5}$	$=$	$ - 20148 \beta_{8} + 16973 \beta_{7} - 23257 \beta_{6} + 28266 \beta_{5} - 66069 \beta_{4} + \cdots + 2406059 $ -20148b8 + 16973b7 - 23257b6 + 28266b5 - 66069b4 - 43512b3 + 49194b2 + 400357b1 + 2406059
$\nu^{6}$	$=$	$ - 331120 \beta_{8} + 1201741 \beta_{7} - 1133836 \beta_{6} + 839719 \beta_{5} - 2687149 \beta_{4} + \cdots + 140212240 $ -331120b8 + 1201741b7 - 1133836b6 + 839719b5 - 2687149b4 - 1898720b3 + 1588470b2 + 7614712b1 + 140212240
$\nu^{7}$	$=$	$ - 18801726 \beta_{8} + 18869163 \beta_{7} - 28915796 \beta_{6} + 22456965 \beta_{5} - 66798305 \beta_{4} + \cdots + 2605952876 $ -18801726b8 + 18869163b7 - 28915796b6 + 22456965b5 - 66798305b4 - 51387358b3 + 51192348b2 + 311780530b1 + 2605952876
$\nu^{8}$	$=$	$ - 415674150 \beta_{8} + 944557133 \beta_{7} - 1108375216 \beta_{6} + 613174643 \beta_{5} + \cdots + 108679843804 $ -415674150b8 + 944557133b7 - 1108375216b6 + 613174643b5 - 2445440751b4 - 1840698374b3 + 1577039284b2 + 7365311666b1 + 108679843804

Display $\beta_i$ in terms of $\nu^j$

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} $

1.1

−22.7141
−20.3712
−18.8062
−14.7458
5.30615
6.28625
11.7195
23.8209
30.5044

−3.00000

5.00000

−26.7141

9.00000

1.2

−3.00000

5.00000

−24.3712

9.00000

1.3

−3.00000

5.00000

−22.8062

9.00000

1.4

−3.00000

5.00000

−18.7458

9.00000

1.5

−3.00000

5.00000

1.30615

9.00000

1.6

−3.00000

5.00000

2.28625

9.00000

1.7

−3.00000

5.00000

7.71954

9.00000

1.8

−3.00000

5.00000

19.8209

9.00000

1.9

−3.00000

5.00000

26.5044

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ -1 $
$31$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1860.4.a.h	✓	9

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1860.4.a.h	✓	9	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{9} + 35 T_{7}^{8} - 1058 T_{7}^{7} - 42952 T_{7}^{6} + 250696 T_{7}^{5} + 14694152 T_{7}^{4} + \cdots - 3370741632 $ T7^9 + 35*T7^8 - 1058*T7^7 - 42952*T7^6 + 250696*T7^5 + 14694152*T7^4 - 6665388*T7^3 - 1254119196*T7^2 + 4196800560*T7 - 3370741632 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1860))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{9} $ T^9
$3$	$ (T + 3)^{9} $ (T + 3)^9
$5$	$ (T - 5)^{9} $ (T - 5)^9
$7$	$ T^{9} + \cdots - 3370741632 $ T^9 + 35T^8 - 1058T^7 - 42952T^6 + 250696T^5 + 14694152T^4 - 6665388T^3 - 1254119196T^2 + 4196800560T - 3370741632
$11$	$ T^{9} + \cdots + 5967992733184 $ T^9 + 45T^8 - 6138T^7 - 190564T^6 + 14256024T^5 + 203154192T^4 - 13241415136T^3 + 2854993728T^2 + 2090547190656T + 5967992733184
$13$	$ T^{9} + \cdots - 15\!\cdots\!72 $ T^9 + 4T^8 - 16052T^7 + 148912T^6 + 87140344T^5 - 1851330536T^4 - 165240075948T^3 + 5303878980216T^2 + 30354781135920T - 1591857056107872
$17$	$ T^{9} + \cdots + 11\!\cdots\!08 $ T^9 + 42T^8 - 36368T^7 - 1136216T^6 + 461810992T^5 + 11143151008T^4 - 2345289304368T^3 - 55465809076640T^2 + 4093561590862848T + 116222097797111808
$19$	$ T^{9} + \cdots - 30\!\cdots\!36 $ T^9 + 11T^8 - 46944T^7 - 631200T^6 + 737814912T^5 + 2844351744T^4 - 4980003481600T^3 + 49687532683264T^2 + 12490062588641280T - 303234041726631936
$23$	$ T^{9} + \cdots - 35\!\cdots\!16 $ T^9 + T^8 - 76846T^7 + 497516T^6 + 1681049640T^5 - 23686483728T^4 - 8804282792272T^3 + 413107402604720T^2 - 4973969138686112*T - 3535743635542016
$29$	$ T^{9} + \cdots - 33\!\cdots\!24 $ T^9 - 280T^8 - 86512T^7 + 24252992T^6 + 2950825748T^5 - 746293586368T^4 - 49968091726196T^3 + 9242054457575064T^2 + 367358475913948416T - 33308191892969293824
$31$	$ (T + 31)^{9} $ (T + 31)^9
$37$	$ T^{9} + \cdots - 30\!\cdots\!76 $ T^9 - 126T^8 - 311356T^7 + 14881968T^6 + 32247601680T^5 + 293515283952T^4 - 1094868253400188T^3 - 10438355384329200T^2 + 11052237372406240528T - 304897482873454814976
$41$	$ T^{9} + \cdots + 18\!\cdots\!68 $ T^9 - 800T^8 - 48516T^7 + 188802976T^6 - 41776107008T^5 - 5042617378944T^4 + 2503513549742592T^3 - 223534456748685312T^2 + 1221028023194836992T + 184955502881006419968
$43$	$ T^{9} + \cdots - 15\!\cdots\!76 $ T^9 + 161T^8 - 478444T^7 - 33552012T^6 + 81620896240T^5 + 307517758768T^4 - 5981251406033856T^3 + 298079720022321344T^2 + 160989949941830689280T - 15235348274276527563776
$47$	$ T^{9} + \cdots + 21\!\cdots\!84 $ T^9 + 630T^8 - 200100T^7 - 164285440T^6 - 4947313584T^5 + 6049714637664T^4 + 162718481230032T^3 - 68878666349525088T^2 + 1644631106513443200T + 21245658322716278784
$53$	$ T^{9} + \cdots - 30\!\cdots\!16 $ T^9 - 623T^8 - 607084T^7 + 430653804T^6 + 60735411160T^5 - 69751891652560T^4 + 3418725894915440T^3 + 2628539435387020944T^2 - 137969050081751664384T - 30394709380417928654016
$59$	$ T^{9} + \cdots + 23\!\cdots\!88 $ T^9 + 52T^8 - 951256T^7 - 88813344T^6 + 305362991692T^5 + 40451828501648T^4 - 34420886927287156T^3 - 5608292657823152712T^2 + 417531273611859328128T + 23327801750748461551488
$61$	$ T^{9} + \cdots + 14\!\cdots\!44 $ T^9 - 996T^8 - 651512T^7 + 820189296T^6 - 6018506048T^5 - 156077549083584T^4 + 28075480486264704T^3 + 7113606078622868736T^2 - 2255152393261663554816T + 147311687593721717305344
$67$	$ T^{9} + \cdots + 23\!\cdots\!96 $ T^9 - 164T^8 - 1681520T^7 + 156113520T^6 + 985976316192T^5 - 32478519213912T^4 - 236646071976084604T^3 - 1852087800043474552T^2 + 18927974964988469223872T + 230925506612096169933696
$71$	$ T^{9} + \cdots - 65\!\cdots\!96 $ T^9 - 1235T^8 - 1159322T^7 + 1523625792T^6 + 494481454980T^5 - 654988809606852T^4 - 84387098396396052T^3 + 114932694107223730548T^2 + 4001358736134209863584T - 6582608754534452430544896
$73$	$ T^{9} + \cdots - 81\!\cdots\!72 $ T^9 - 735T^8 - 1666296T^7 + 1452450428T^6 + 334708181688T^5 - 703595562113664T^4 + 280451138724461140T^3 - 48084435548843083236T^2 + 3523266014344526550888T - 81287573769467385787872
$79$	$ T^{9} + \cdots - 76\!\cdots\!08 $ T^9 + 413T^8 - 2201772T^7 - 445395848T^6 + 1690494239768T^5 + 33288555944496T^4 - 509385359079650928T^3 + 60717411163225975824T^2 + 43827247832975235760320T - 7680241730492172733252608
$83$	$ T^{9} + \cdots - 33\!\cdots\!36 $ T^9 + 364T^8 - 3147852T^7 - 514340192T^6 + 3592009549552T^5 + 8471868838560T^4 - 1683656905726721040T^3 + 162810035517670541184T^2 + 263514146757776255713536T - 33797041483422878906996736
$89$	$ T^{9} + \cdots - 18\!\cdots\!92 $ T^9 - 941T^8 - 1917798T^7 + 1740925332T^6 + 1228061215068T^5 - 1066675795652340T^4 - 313110249472491156T^3 + 250213215290010523740T^2 + 28098312197353441409808T - 18113959234721080526089392
$97$	$ T^{9} + \cdots - 28\!\cdots\!00 $ T^9 - 402T^8 - 2757356T^7 + 777887528T^6 + 2029953863392T^5 - 241258877429632T^4 - 485886745125576960T^3 + 24592488893718935552T^2 + 31198334157592118845440T - 2816151544713601098547200
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)

\(f(q)\)	\(=\)	\( q - 3 q^{3} + 5 q^{5} + (\beta_1 - 4) q^{7} + 9 q^{9} + (\beta_{4} - 5) q^{11} - \beta_{3} q^{13} - 15 q^{15} + ( - \beta_{5} - 5) q^{17} + ( - \beta_{8} + \beta_{4} - 1) q^{19} + ( - 3 \beta_1 + 12) q^{21}+ \cdots + (9 \beta_{4} - 45) q^{99}+O(q^{100}) \) q - 3 * q^3 + 5 * q^5 + (b1 - 4) * q^7 + 9 * q^9 + (b4 - 5) * q^11 - b3 * q^13 - 15 * q^15 + (-b5 - 5) * q^17 + (-b8 + b4 - 1) * q^19 + (-3b1 + 12) q^21 + (b7 + b4 - b2 + 4b1) q^23 + 25 * q^25 - 27 * q^27 + (b8 - b7 + b6 + b5 + 2b4 - b3 + b1 + 31) q^29 - 31 * q^31 + (-3b4 + 15) q^33 + (5b1 - 20) q^35 + (b8 - 3b6 - b5 + b4 - 2b3 + 2b2 + 4b1 + 14) * q^37 + 3b3 q^39 + (-b8 + b6 + b5 + b4 + b3 + 4b2 + 4b1 + 88) * q^41 + (-2b7 + b6 + b5 - 4b4 - b3 - 2b2 + 10b1 - 19) * q^43 + 45 * q^45 + (b8 - 3b7 - 2b5 + b4 + b3 + 3b1 - 73) q^47 + (2b7 - b6 + 2b5 - 3b4 - 2b3 + b2 - 3b1 + 31) q^49 + (3b5 + 15) q^51 + (2b8 + 3b7 - 4b6 - 2b5 + b3 - 4b2 - b1 + 70) q^53 + (5b4 - 25) q^55 + (3b8 - 3b4 + 3) * q^57 + (-b8 + 3b7 + 4b6 + 2b5 - b4 + 4b3 - 7b1 - 5) q^59 + (-2b8 - 4b7 + 5b6 - b5 - b4 + 5b3 - 2b2 + 4b1 + 106) * q^61 + (9b1 - 36) q^63 - 5b3 q^65 + (-3b8 + 5b7 + 2b6 + 3b5 + b3 + 4b2 + 18b1 + 19) * q^67 + (-3b7 - 3b4 + 3b2 - 12b1) * q^69 + (b8 - b7 - 5b6 - b5 + 7b4 + 3b3 - 6b2 + b1 + 136) * q^71 + (-2b8 - 2b7 + 8b6 + 4b5 - 5b4 - 3b3 - 2b2 + 2b1 + 83) * q^73 - 75 * q^75 + (b8 + b6 - 5b5 - 9b4 + 8b3 - 8b2 - 23b1 - 24) q^77 + (3b8 + 7b7 - 7b6 - b5 + 5b4 + 2b3 - 3b2 - 5b1 - 43) q^79 + 81 * q^81 + (-3b8 + 5b7 - 5b6 + b5 + 3b4 - 3b3 + 11b2 - 14b1 - 35) q^83 + (-5b5 - 25) q^85 + (-3b8 + 3b7 - 3b6 - 3b5 - 6b4 + 3b3 - 3b1 - 93) q^87 + (-8b7 - b5 + 3b4 - 3b3 - 2b2 - 16b1 + 104) q^89 + (-b8 - 6b7 - 9b6 - 6b5 - 8b4 - 4b3 + b2 + 3b1 - 125) * q^91 + 93 * q^93 + (-5b8 + 5b4 - 5) * q^95 + (-4b8 + b7 + b6 - 15b4 + b3 - 4b2 + 8b1 + 45) * q^97 + (9b4 - 45) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 27 q^{3} + 45 q^{5} - 35 q^{7} + 81 q^{9} - 45 q^{11} - 4 q^{13} - 135 q^{15} - 42 q^{17} - 11 q^{19} + 105 q^{21} - q^{23} + 225 q^{25} - 243 q^{27} + 280 q^{29} - 279 q^{31} + 135 q^{33} - 175 q^{35}+ \cdots - 405 q^{99}+O(q^{100}) \) 9 * q - 27 * q^3 + 45 * q^5 - 35 * q^7 + 81 * q^9 - 45 * q^11 - 4 * q^13 - 135 * q^15 - 42 * q^17 - 11 * q^19 + 105 * q^21 - q^23 + 225 * q^25 - 243 * q^27 + 280 * q^29 - 279 * q^31 + 135 * q^33 - 175 * q^35 + 126 * q^37 + 12 * q^39 + 800 * q^41 - 161 * q^43 + 405 * q^45 - 630 * q^47 + 254 * q^49 + 126 * q^51 + 623 * q^53 - 225 * q^55 + 33 * q^57 - 52 * q^59 + 996 * q^61 - 315 * q^63 - 20 * q^65 + 164 * q^67 + 3 * q^69 + 1235 * q^71 + 735 * q^73 - 675 * q^75 - 197 * q^77 - 413 * q^79 + 729 * q^81 - 364 * q^83 - 210 * q^85 - 840 * q^87 + 941 * q^89 - 1106 * q^91 + 837 * q^93 - 55 * q^95 + 402 * q^97 - 405 * 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

Embeddings

Atkin-Lehner signs

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	1860.4.a.h

Level	$1860$
Weight	$4$
Character orbit	1860.a
Self dual	yes
Analytic conductor	$109.744$
Analytic rank	$0$
Dimension	$9$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(109.743552611\)
Analytic rank:	\(0\)
Dimension:	\(9\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{9} - x^{8} - 1602 x^{7} - 3024 x^{6} + 832872 x^{5} + 2239848 x^{4} - 157824556 x^{3} + \cdots - 36448841472 \) x^9 - x^8 - 1602x^7 - 3024x^6 + 832872x^5 + 2239848x^4 - 157824556x^3 - 62701580x^2 + 10698567888*x - 36448841472
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{7}\cdot 3 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 14\!\cdots\!87 \nu^{8} + \cdots + 15\!\cdots\!04 ) / 95\!\cdots\!24 \) (-14649774898704887v^8 - 109843825772501375v^7 + 16713235874745291032v^6 + 343426642467522590788v^5 - 4004624362525343854552v^4 - 188939047338816887974340v^3 - 578081333748745273795964v^2 + 18008258069907464631067836v + 15083864800454863041686004) / 959877740205219924626724
\(\beta_{3}\)	\(=\)	\( ( - 12\!\cdots\!47 \nu^{8} + \cdots + 42\!\cdots\!24 ) / 47\!\cdots\!62 \) (-12981204298863347v^8 + 1285501769843372575v^7 + 124270863548694812v^6 - 1605627480271369520984v^5 + 8509686496615444737398v^4 + 633111495161294169654334v^3 - 3351894285711217282550192v^2 - 72633050603890099856688294v + 421041180314005235579174124) / 479938870102609962313362
\(\beta_{4}\)	\(=\)	\( ( 16\!\cdots\!03 \nu^{8} + \cdots + 50\!\cdots\!80 ) / 19\!\cdots\!48 \) (165438845535184303v^8 - 466050155311658687v^7 - 248903306988483818014v^6 - 140124153576917185280v^5 + 114072507897691706493128v^4 + 265428757158488327166304v^3 - 15743562712130148394694372v^2 - 19207050342007855585071780v + 504882971583739919362877880) / 1919755480410439849253448
\(\beta_{5}\)	\(=\)	\( ( - 71\!\cdots\!73 \nu^{8} + \cdots + 50\!\cdots\!20 ) / 63\!\cdots\!16 \) (-71547829443361673v^8 + 1457369815087448493v^7 + 84096411582317136286v^6 - 1390757185989784780208v^5 - 30552890581483505692200v^4 + 416965729335651476551784v^3 + 2983369456186604074508308v^2 - 45566946250995501609731172v + 50433720983435209390078920) / 639918493470146616417816
\(\beta_{6}\)	\(=\)	\( ( 98\!\cdots\!53 \nu^{8} + \cdots - 85\!\cdots\!60 ) / 63\!\cdots\!16 \) (98575864066461853v^8 - 4254452513070277297v^7 - 86725917918704646302v^6 + 4870729762117375302776v^5 + 16072166444880267940456v^4 - 1749891518011921279645192v^3 + 2908014780258830894379708v^2 + 188180245545562177096315404v - 856814103345588202976391360) / 639918493470146616417816
\(\beta_{7}\)	\(=\)	\( ( 47\!\cdots\!46 \nu^{8} + \cdots + 53\!\cdots\!60 ) / 15\!\cdots\!54 \) (47782542536317146v^8 - 516751045224104815v^7 - 64329102044023718093v^6 + 375186950602253403512v^5 + 27076587800607966794260v^4 - 63017191775259190981834v^3 - 3339420694604807700481250v^2 + 6401731864318668617925360v + 53853897072418274389011360) / 159979623367536654104454
\(\beta_{8}\)	\(=\)	\( ( - 22\!\cdots\!37 \nu^{8} + \cdots - 42\!\cdots\!32 ) / 63\!\cdots\!16 \) (-220453325588671537v^8 + 1843273233499252281v^7 + 300235138077432820882v^6 - 1005378818935870576280v^5 - 125886795465445917849312v^4 - 28179564368869803011392v^3 + 15495408803525000308703772v^2 + 12596472097319058187605324v - 422029818391070123851627632) / 639918493470146616417816

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{7} - \beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + \beta_{2} + 5\beta _1 + 358 \) 2b7 - b6 + 2b5 - 3b4 - 2b3 + b2 + 5*b1 + 358
\(\nu^{3}\)	\(=\)	\( -19\beta_{8} + 13\beta_{7} - 13\beta_{6} + 32\beta_{5} - 58\beta_{4} - 27\beta_{3} + 38\beta_{2} + 576\beta _1 + 1508 \) -19b8 + 13b7 - 13b6 + 32b5 - 58b4 - 27b3 + 38b2 + 576b1 + 1508
\(\nu^{4}\)	\(=\)	\( - 177 \beta_{8} + 1596 \beta_{7} - 1121 \beta_{6} + 1271 \beta_{5} - 2933 \beta_{4} - 1950 \beta_{3} + \cdots + 203398 \) -177b8 + 1596b7 - 1121b6 + 1271b5 - 2933b4 - 1950b3 + 1498b2 + 7217b1 + 203398
\(\nu^{5}\)	\(=\)	\( - 20148 \beta_{8} + 16973 \beta_{7} - 23257 \beta_{6} + 28266 \beta_{5} - 66069 \beta_{4} + \cdots + 2406059 \) -20148b8 + 16973b7 - 23257b6 + 28266b5 - 66069b4 - 43512b3 + 49194b2 + 400357b1 + 2406059
\(\nu^{6}\)	\(=\)	\( - 331120 \beta_{8} + 1201741 \beta_{7} - 1133836 \beta_{6} + 839719 \beta_{5} - 2687149 \beta_{4} + \cdots + 140212240 \) -331120b8 + 1201741b7 - 1133836b6 + 839719b5 - 2687149b4 - 1898720b3 + 1588470b2 + 7614712b1 + 140212240
\(\nu^{7}\)	\(=\)	\( - 18801726 \beta_{8} + 18869163 \beta_{7} - 28915796 \beta_{6} + 22456965 \beta_{5} - 66798305 \beta_{4} + \cdots + 2605952876 \) -18801726b8 + 18869163b7 - 28915796b6 + 22456965b5 - 66798305b4 - 51387358b3 + 51192348b2 + 311780530b1 + 2605952876
\(\nu^{8}\)	\(=\)	\( - 415674150 \beta_{8} + 944557133 \beta_{7} - 1108375216 \beta_{6} + 613174643 \beta_{5} + \cdots + 108679843804 \) -415674150b8 + 944557133b7 - 1108375216b6 + 613174643b5 - 2445440751b4 - 1840698374b3 + 1577039284b2 + 7365311666b1 + 108679843804

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.9
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{9} \) T^9
$3$	\( (T + 3)^{9} \) (T + 3)^9
$5$	\( (T - 5)^{9} \) (T - 5)^9
$7$	\( T^{9} + \cdots - 3370741632 \) T^9 + 35T^8 - 1058T^7 - 42952T^6 + 250696T^5 + 14694152T^4 - 6665388T^3 - 1254119196T^2 + 4196800560T - 3370741632
$11$	\( T^{9} + \cdots + 5967992733184 \) T^9 + 45T^8 - 6138T^7 - 190564T^6 + 14256024T^5 + 203154192T^4 - 13241415136T^3 + 2854993728T^2 + 2090547190656T + 5967992733184
$13$	\( T^{9} + \cdots - 15\!\cdots\!72 \) T^9 + 4T^8 - 16052T^7 + 148912T^6 + 87140344T^5 - 1851330536T^4 - 165240075948T^3 + 5303878980216T^2 + 30354781135920T - 1591857056107872
$17$	\( T^{9} + \cdots + 11\!\cdots\!08 \) T^9 + 42T^8 - 36368T^7 - 1136216T^6 + 461810992T^5 + 11143151008T^4 - 2345289304368T^3 - 55465809076640T^2 + 4093561590862848T + 116222097797111808
$19$	\( T^{9} + \cdots - 30\!\cdots\!36 \) T^9 + 11T^8 - 46944T^7 - 631200T^6 + 737814912T^5 + 2844351744T^4 - 4980003481600T^3 + 49687532683264T^2 + 12490062588641280T - 303234041726631936
$23$	\( T^{9} + \cdots - 35\!\cdots\!16 \) T^9 + T^8 - 76846T^7 + 497516T^6 + 1681049640T^5 - 23686483728T^4 - 8804282792272T^3 + 413107402604720T^2 - 4973969138686112*T - 3535743635542016
$29$	\( T^{9} + \cdots - 33\!\cdots\!24 \) T^9 - 280T^8 - 86512T^7 + 24252992T^6 + 2950825748T^5 - 746293586368T^4 - 49968091726196T^3 + 9242054457575064T^2 + 367358475913948416T - 33308191892969293824
$31$	\( (T + 31)^{9} \) (T + 31)^9
$37$	\( T^{9} + \cdots - 30\!\cdots\!76 \) T^9 - 126T^8 - 311356T^7 + 14881968T^6 + 32247601680T^5 + 293515283952T^4 - 1094868253400188T^3 - 10438355384329200T^2 + 11052237372406240528T - 304897482873454814976
$41$	\( T^{9} + \cdots + 18\!\cdots\!68 \) T^9 - 800T^8 - 48516T^7 + 188802976T^6 - 41776107008T^5 - 5042617378944T^4 + 2503513549742592T^3 - 223534456748685312T^2 + 1221028023194836992T + 184955502881006419968
$43$	\( T^{9} + \cdots - 15\!\cdots\!76 \) T^9 + 161T^8 - 478444T^7 - 33552012T^6 + 81620896240T^5 + 307517758768T^4 - 5981251406033856T^3 + 298079720022321344T^2 + 160989949941830689280T - 15235348274276527563776
$47$	\( T^{9} + \cdots + 21\!\cdots\!84 \) T^9 + 630T^8 - 200100T^7 - 164285440T^6 - 4947313584T^5 + 6049714637664T^4 + 162718481230032T^3 - 68878666349525088T^2 + 1644631106513443200T + 21245658322716278784
$53$	\( T^{9} + \cdots - 30\!\cdots\!16 \) T^9 - 623T^8 - 607084T^7 + 430653804T^6 + 60735411160T^5 - 69751891652560T^4 + 3418725894915440T^3 + 2628539435387020944T^2 - 137969050081751664384T - 30394709380417928654016
$59$	\( T^{9} + \cdots + 23\!\cdots\!88 \) T^9 + 52T^8 - 951256T^7 - 88813344T^6 + 305362991692T^5 + 40451828501648T^4 - 34420886927287156T^3 - 5608292657823152712T^2 + 417531273611859328128T + 23327801750748461551488
$61$	\( T^{9} + \cdots + 14\!\cdots\!44 \) T^9 - 996T^8 - 651512T^7 + 820189296T^6 - 6018506048T^5 - 156077549083584T^4 + 28075480486264704T^3 + 7113606078622868736T^2 - 2255152393261663554816T + 147311687593721717305344
$67$	\( T^{9} + \cdots + 23\!\cdots\!96 \) T^9 - 164T^8 - 1681520T^7 + 156113520T^6 + 985976316192T^5 - 32478519213912T^4 - 236646071976084604T^3 - 1852087800043474552T^2 + 18927974964988469223872T + 230925506612096169933696
$71$	\( T^{9} + \cdots - 65\!\cdots\!96 \) T^9 - 1235T^8 - 1159322T^7 + 1523625792T^6 + 494481454980T^5 - 654988809606852T^4 - 84387098396396052T^3 + 114932694107223730548T^2 + 4001358736134209863584T - 6582608754534452430544896
$73$	\( T^{9} + \cdots - 81\!\cdots\!72 \) T^9 - 735T^8 - 1666296T^7 + 1452450428T^6 + 334708181688T^5 - 703595562113664T^4 + 280451138724461140T^3 - 48084435548843083236T^2 + 3523266014344526550888T - 81287573769467385787872
$79$	\( T^{9} + \cdots - 76\!\cdots\!08 \) T^9 + 413T^8 - 2201772T^7 - 445395848T^6 + 1690494239768T^5 + 33288555944496T^4 - 509385359079650928T^3 + 60717411163225975824T^2 + 43827247832975235760320T - 7680241730492172733252608
$83$	\( T^{9} + \cdots - 33\!\cdots\!36 \) T^9 + 364T^8 - 3147852T^7 - 514340192T^6 + 3592009549552T^5 + 8471868838560T^4 - 1683656905726721040T^3 + 162810035517670541184T^2 + 263514146757776255713536T - 33797041483422878906996736
$89$	\( T^{9} + \cdots - 18\!\cdots\!92 \) T^9 - 941T^8 - 1917798T^7 + 1740925332T^6 + 1228061215068T^5 - 1066675795652340T^4 - 313110249472491156T^3 + 250213215290010523740T^2 + 28098312197353441409808T - 18113959234721080526089392
$97$	\( T^{9} + \cdots - 28\!\cdots\!00 \) T^9 - 402T^8 - 2757356T^7 + 777887528T^6 + 2029953863392T^5 - 241258877429632T^4 - 485886745125576960T^3 + 24592488893718935552T^2 + 31198334157592118845440T - 2816151544713601098547200
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\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( +1 \)
\(5\)	\( -1 \)
\(31\)	\( +1 \)