LMFDB - Embedded newform 1840.3.k.d.321.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not 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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.16
Root			$-0.0431371i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.41949 q^{3} +2.23607i q^{5} +8.24199i q^{7} +20.3709 q^{9} +O(q^{10})$	$q+5.41949 q^{3} +2.23607i q^{5} +8.24199i q^{7} +20.3709 q^{9} +15.8246i q^{11} -14.3219 q^{13} +12.1184i q^{15} +10.1666i q^{17} -36.5359i q^{19} +44.6674i q^{21} +(-22.2445 + 5.84663i) q^{23} -5.00000 q^{25} +61.6245 q^{27} +6.46533 q^{29} +42.8526 q^{31} +85.7611i q^{33} -18.4296 q^{35} +63.6379i q^{37} -77.6175 q^{39} -37.0921 q^{41} +6.00126i q^{43} +45.5507i q^{45} +32.4676 q^{47} -18.9303 q^{49} +55.0977i q^{51} +36.6640i q^{53} -35.3848 q^{55} -198.006i q^{57} -6.65851 q^{59} +55.7093i q^{61} +167.897i q^{63} -32.0248i q^{65} +4.45984i q^{67} +(-120.554 + 31.6858i) q^{69} -118.412 q^{71} +82.2675 q^{73} -27.0975 q^{75} -130.426 q^{77} -133.084i q^{79} +150.636 q^{81} -67.5614i q^{83} -22.7331 q^{85} +35.0388 q^{87} -104.729i q^{89} -118.041i q^{91} +232.240 q^{93} +81.6968 q^{95} +98.6666i q^{97} +322.360i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	5.41949			1.80650			0.903249	−	0.429117i	$-0.141175\pi$
$3$	5.41949			1.80650			0.903249	+	0.429117i	$0.141175\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			8.24199i			1.17743i	0.808342	+	0.588713i	$0.200365\pi$
$7$			8.24199i			1.17743i	−0.808342	+	0.588713i	$0.799635\pi$
$8$		0			0
$9$	20.3709			2.26343
$10$		0			0
$11$			15.8246i			1.43860i	0.694702	+	0.719298i	$0.255536\pi$
$11$			15.8246i			1.43860i	−0.694702	+	0.719298i	$0.744464\pi$
$12$		0			0
$13$	−14.3219			−1.10169			−0.550843	−	0.834609i	$-0.685694\pi$
$13$	−14.3219			−1.10169			−0.550843	+	0.834609i	$0.685694\pi$
$14$		0			0
$15$			12.1184i			0.807890i
$16$		0			0
$17$			10.1666i			0.598034i	0.954248	+	0.299017i	$0.0966587\pi$
$17$			10.1666i			0.598034i	−0.954248	+	0.299017i	$0.903341\pi$
$18$		0			0
$19$		−	36.5359i		−	1.92294i	−0.274904	−	0.961472i	$-0.588646\pi$
$19$		−	36.5359i		−	1.92294i	0.274904	−	0.961472i	$-0.411354\pi$
$20$		0			0
$21$			44.6674i			2.12702i
$22$		0			0
$23$	−22.2445	+	5.84663i	−0.967151	+	0.254201i
$23$	−22.2445	+	5.84663i	−0.967151	+	0.254201i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	61.6245			2.28239
$28$		0			0
$29$	6.46533			0.222942			0.111471	−	0.993768i	$-0.464444\pi$
$29$	6.46533			0.222942			0.111471	+	0.993768i	$0.464444\pi$
$30$		0			0
$31$	42.8526			1.38234			0.691172	−	0.722691i	$-0.257095\pi$
$31$	42.8526			1.38234			0.691172	+	0.722691i	$0.257095\pi$
$32$		0			0
$33$			85.7611i			2.59882i
$34$		0			0
$35$	−18.4296			−0.526561
$36$		0			0
$37$			63.6379i			1.71994i	0.510341	+	0.859972i	$0.329519\pi$
$37$			63.6379i			1.71994i	−0.510341	+	0.859972i	$0.670481\pi$
$38$		0			0
$39$	−77.6175			−1.99019
$40$		0			0
$41$	−37.0921			−0.904685			−0.452342	−	0.891844i	$-0.649412\pi$
$41$	−37.0921			−0.904685			−0.452342	+	0.891844i	$0.649412\pi$
$42$		0			0
$43$			6.00126i			0.139564i	0.997562	+	0.0697821i	$0.0222304\pi$
$43$			6.00126i			0.139564i	−0.997562	+	0.0697821i	$0.977770\pi$
$44$		0			0
$45$			45.5507i			1.01224i
$46$		0			0
$47$	32.4676			0.690800			0.345400	−	0.938455i	$-0.387743\pi$
$47$	32.4676			0.690800			0.345400	+	0.938455i	$0.387743\pi$
$48$		0			0
$49$	−18.9303			−0.386333
$50$		0			0
$51$			55.0977i			1.08035i
$52$		0			0
$53$			36.6640i			0.691774i	0.938276	+	0.345887i	$0.112422\pi$
$53$			36.6640i			0.691774i	−0.938276	+	0.345887i	$0.887578\pi$
$54$		0			0
$55$	−35.3848			−0.643360
$56$		0			0
$57$		−	198.006i		−	3.47379i
$58$		0			0
$59$	−6.65851			−0.112856			−0.0564280	−	0.998407i	$-0.517971\pi$
$59$	−6.65851			−0.112856			−0.0564280	+	0.998407i	$0.517971\pi$
$60$		0			0
$61$			55.7093i			0.913268i	0.889655	+	0.456634i	$0.150945\pi$
$61$			55.7093i			0.913268i	−0.889655	+	0.456634i	$0.849055\pi$
$62$		0			0
$63$			167.897i			2.66503i
$64$		0			0
$65$		−	32.0248i		−	0.492689i
$66$		0			0
$67$			4.45984i			0.0665648i	0.999446	+	0.0332824i	$0.0105961\pi$
$67$			4.45984i			0.0665648i	−0.999446	+	0.0332824i	$0.989404\pi$
$68$		0			0
$69$	−120.554	+	31.6858i	−1.74716	+	0.459214i
$70$		0			0
$71$	−118.412			−1.66777			−0.833886	−	0.551937i	$-0.813889\pi$
$71$	−118.412			−1.66777			−0.833886	+	0.551937i	$0.813889\pi$
$72$		0			0
$73$	82.2675			1.12695			0.563476	−	0.826133i	$-0.309464\pi$
$73$	82.2675			1.12695			0.563476	+	0.826133i	$0.309464\pi$
$74$		0			0
$75$	−27.0975			−0.361300
$76$		0			0
$77$	−130.426			−1.69384
$78$		0			0
$79$		−	133.084i		−	1.68461i	−0.538999	−	0.842307i	$-0.681197\pi$
$79$		−	133.084i		−	1.68461i	0.538999	−	0.842307i	$-0.318803\pi$
$80$		0			0
$81$	150.636			1.85970
$82$		0			0
$83$		−	67.5614i		−	0.813993i	−0.913430	−	0.406996i	$-0.866576\pi$
$83$		−	67.5614i		−	0.813993i	0.913430	−	0.406996i	$-0.133424\pi$
$84$		0			0
$85$	−22.7331			−0.267449
$86$		0			0
$87$	35.0388			0.402745
$88$		0			0
$89$		−	104.729i		−	1.17673i	−0.808595	−	0.588365i	$-0.799772\pi$
$89$		−	104.729i		−	1.17673i	0.808595	−	0.588365i	$-0.200228\pi$
$90$		0			0
$91$		−	118.041i		−	1.29715i
$92$		0			0
$93$	232.240			2.49720
$94$		0			0
$95$	81.6968			0.859966
$96$		0			0
$97$			98.6666i			1.01718i	0.861008	+	0.508591i	$0.169833\pi$
$97$			98.6666i			1.01718i	−0.861008	+	0.508591i	$0.830167\pi$
$98$		0			0
$99$			322.360i			3.25617i
$100$		0			0
$101$	−75.6811			−0.749318			−0.374659	−	0.927163i	$-0.622240\pi$
$101$	−75.6811			−0.749318			−0.374659	+	0.927163i	$0.622240\pi$
$102$		0			0
$103$			86.8499i			0.843203i	0.906781	+	0.421602i	$0.138532\pi$
$103$			86.8499i			0.843203i	−0.906781	+	0.421602i	$0.861468\pi$
$104$		0			0
$105$	−99.8793			−0.951231
$106$		0			0
$107$		−	5.55552i		−	0.0519208i	−0.999663	−	0.0259604i	$-0.991736\pi$
$107$		−	5.55552i		−	0.0519208i	0.999663	−	0.0259604i	$-0.00826438\pi$
$108$		0			0
$109$			71.0003i			0.651379i	0.945477	+	0.325689i	$0.105596\pi$
$109$			71.0003i			0.651379i	−0.945477	+	0.325689i	$0.894404\pi$
$110$		0			0
$111$			344.885i			3.10707i
$112$		0			0
$113$			100.763i			0.891707i	0.895106	+	0.445854i	$0.147100\pi$
$113$			100.763i			0.891707i	−0.895106	+	0.445854i	$0.852900\pi$
$114$		0			0
$115$	−13.0735	−	49.7402i	−0.113682	−	0.432523i
$116$		0			0
$117$	−291.750			−2.49359
$118$		0			0
$119$	−83.7927			−0.704141
$120$		0			0
$121$	−129.417			−1.06956
$122$		0			0
$123$	−201.020			−1.63431
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	43.9602			0.346143			0.173072	−	0.984909i	$-0.444631\pi$
$127$	43.9602			0.346143			0.173072	+	0.984909i	$0.444631\pi$
$128$		0			0
$129$			32.5238i			0.252123i
$130$		0			0
$131$	44.4134			0.339033			0.169517	−	0.985527i	$-0.445779\pi$
$131$	44.4134			0.339033			0.169517	+	0.985527i	$0.445779\pi$
$132$		0			0
$133$	301.129			2.26412
$134$		0			0
$135$			137.797i			1.02072i
$136$		0			0
$137$		−	44.9295i		−	0.327953i	−0.986464	−	0.163976i	$-0.947568\pi$
$137$		−	44.9295i		−	0.327953i	0.986464	−	0.163976i	$-0.0524321\pi$
$138$		0			0
$139$	139.796			1.00573			0.502864	−	0.864365i	$-0.332280\pi$
$139$	139.796			1.00573			0.502864	+	0.864365i	$0.332280\pi$
$140$		0			0
$141$	175.958			1.24793
$142$		0			0
$143$		−	226.638i		−	1.58488i
$144$		0			0
$145$			14.4569i			0.0997029i
$146$		0			0
$147$	−102.593			−0.697910
$148$		0			0
$149$			25.4227i			0.170622i	0.996354	+	0.0853111i	$0.0271884\pi$
$149$			25.4227i			0.170622i	−0.996354	+	0.0853111i	$0.972812\pi$
$150$		0			0
$151$	52.5299			0.347880			0.173940	−	0.984756i	$-0.444350\pi$
$151$	52.5299			0.347880			0.173940	+	0.984756i	$0.444350\pi$
$152$		0			0
$153$			207.102i			1.35361i
$154$		0			0
$155$			95.8214i			0.618203i
$156$		0			0
$157$			74.5874i			0.475079i	0.971378	+	0.237539i	$0.0763409\pi$
$157$			74.5874i			0.475079i	−0.971378	+	0.237539i	$0.923659\pi$
$158$		0			0
$159$			198.700i			1.24969i
$160$		0			0
$161$	−48.1878	−	183.339i	−0.299303	−	1.13875i
$162$		0			0
$163$	18.3238			0.112416			0.0562080	−	0.998419i	$-0.482099\pi$
$163$	18.3238			0.112416			0.0562080	+	0.998419i	$0.482099\pi$
$164$		0			0
$165$	−191.768			−1.16223
$166$		0			0
$167$	−68.9768			−0.413035			−0.206517	−	0.978443i	$-0.566213\pi$
$167$	−68.9768			−0.413035			−0.206517	+	0.978443i	$0.566213\pi$
$168$		0			0
$169$	36.1174			0.213712
$170$		0			0
$171$		−	744.270i		−	4.35245i
$172$		0			0
$173$	118.707			0.686166			0.343083	−	0.939305i	$-0.388529\pi$
$173$	118.707			0.686166			0.343083	+	0.939305i	$0.388529\pi$
$174$		0			0
$175$		−	41.2099i		−	0.235485i
$176$		0			0
$177$	−36.0857			−0.203874
$178$		0			0
$179$	278.892			1.55806			0.779029	−	0.626988i	$-0.215712\pi$
$179$	278.892			1.55806			0.779029	+	0.626988i	$0.215712\pi$
$180$		0			0
$181$		−	66.1123i		−	0.365261i	−0.983182	−	0.182631i	$-0.941539\pi$
$181$		−	66.1123i		−	0.365261i	0.983182	−	0.182631i	$-0.0584613\pi$
$182$		0			0
$183$			301.916i			1.64982i
$184$		0			0
$185$	−142.299			−0.769182
$186$		0			0
$187$	−160.882			−0.860329
$188$		0			0
$189$			507.908i			2.68735i
$190$		0			0
$191$			5.53406i			0.0289741i	0.999895	+	0.0144871i	$0.00461154\pi$
$191$			5.53406i			0.0289741i	−0.999895	+	0.0144871i	$0.995388\pi$
$192$		0			0
$193$	−72.0460			−0.373295			−0.186648	−	0.982427i	$-0.559762\pi$
$193$	−72.0460			−0.373295			−0.186648	+	0.982427i	$0.559762\pi$
$194$		0			0
$195$		−	173.558i		−	0.890041i
$196$		0			0
$197$	191.143			0.970271			0.485136	−	0.874439i	$-0.338770\pi$
$197$	191.143			0.970271			0.485136	+	0.874439i	$0.338770\pi$
$198$		0			0
$199$		−	172.543i		−	0.867051i	−0.901141	−	0.433525i	$-0.857269\pi$
$199$		−	172.543i		−	0.867051i	0.901141	−	0.433525i	$-0.142731\pi$
$200$		0			0
$201$			24.1701i			0.120249i
$202$		0			0
$203$			53.2871i			0.262498i
$204$		0			0
$205$		−	82.9404i		−	0.404587i
$206$		0			0
$207$	−453.140	+	119.101i	−2.18908	+	0.575368i
$208$		0			0
$209$	578.165			2.76634
$210$		0			0
$211$	334.039			1.58312			0.791561	−	0.611090i	$-0.209269\pi$
$211$	334.039			1.58312			0.791561	+	0.611090i	$0.209269\pi$
$212$		0			0
$213$	−641.732			−3.01282
$214$		0			0
$215$	−13.4192			−0.0624150
$216$		0			0
$217$			353.191i			1.62761i
$218$		0			0
$219$	445.848			2.03584
$220$		0			0
$221$		−	145.605i		−	0.658845i
$222$		0			0
$223$	441.580			1.98018			0.990089	−	0.140440i	$-0.0448517\pi$
$223$	441.580			1.98018			0.990089	+	0.140440i	$0.0448517\pi$
$224$		0			0
$225$	−101.855			−0.452687
$226$		0			0
$227$		−	54.9222i		−	0.241948i	−0.992656	−	0.120974i	$-0.961398\pi$
$227$		−	54.9222i		−	0.241948i	0.992656	−	0.120974i	$-0.0386018\pi$
$228$		0			0
$229$		−	322.300i		−	1.40743i	−0.710485	−	0.703713i	$-0.751524\pi$
$229$		−	322.300i		−	1.40743i	0.710485	−	0.703713i	$-0.248476\pi$
$230$		0			0
$231$	−706.841			−3.05992
$232$		0			0
$233$	159.208			0.683296			0.341648	−	0.939828i	$-0.389015\pi$
$233$	159.208			0.683296			0.341648	+	0.939828i	$0.389015\pi$
$234$		0			0
$235$			72.5998i			0.308935i
$236$		0			0
$237$		−	721.250i		−	3.04325i
$238$		0			0
$239$	−32.1990			−0.134724			−0.0673620	−	0.997729i	$-0.521458\pi$
$239$	−32.1990			−0.134724			−0.0673620	+	0.997729i	$0.521458\pi$
$240$		0			0
$241$		−	39.8259i		−	0.165253i	−0.996581	−	0.0826263i	$-0.973669\pi$
$241$		−	39.8259i		−	0.165253i	0.996581	−	0.0826263i	$-0.0263308\pi$
$242$		0			0
$243$	261.748			1.07715
$244$		0			0
$245$		−	42.3295i		−	0.172773i
$246$		0			0
$247$			523.265i			2.11848i
$248$		0			0
$249$		−	366.148i		−	1.47048i
$250$		0			0
$251$			232.529i			0.926410i	0.886251	+	0.463205i	$0.153301\pi$
$251$			232.529i			0.926410i	−0.886251	+	0.463205i	$0.846699\pi$
$252$		0			0
$253$	−92.5203	−	352.009i	−0.365693	−	1.39134i
$254$		0			0
$255$	−123.202			−0.483146
$256$		0			0
$257$	325.160			1.26521			0.632606	−	0.774473i	$-0.281985\pi$
$257$	325.160			1.26521			0.632606	+	0.774473i	$0.281985\pi$
$258$		0			0
$259$	−524.503			−2.02511
$260$		0			0
$261$	131.705			0.504615
$262$		0			0
$263$			4.94890i			0.0188171i	0.999956	+	0.00940855i	$0.00299488\pi$
$263$			4.94890i			0.0188171i	−0.999956	+	0.00940855i	$0.997005\pi$
$264$		0			0
$265$	−81.9833			−0.309371
$266$		0			0
$267$		−	567.578i		−	2.12576i
$268$		0			0
$269$	235.047			0.873781			0.436891	−	0.899515i	$-0.356080\pi$
$269$	235.047			0.873781			0.436891	+	0.899515i	$0.356080\pi$
$270$		0			0
$271$	53.5224			0.197500			0.0987498	−	0.995112i	$-0.468516\pi$
$271$	53.5224			0.197500			0.0987498	+	0.995112i	$0.468516\pi$
$272$		0			0
$273$		−	639.723i		−	2.34331i
$274$		0			0
$275$		−	79.1228i		−	0.287719i
$276$		0			0
$277$	−143.576			−0.518326			−0.259163	−	0.965834i	$-0.583447\pi$
$277$	−143.576			−0.518326			−0.259163	+	0.965834i	$0.583447\pi$
$278$		0			0
$279$	872.947			3.12884
$280$		0			0
$281$			187.330i			0.666656i	0.942811	+	0.333328i	$0.108172\pi$
$281$			187.330i			0.666656i	−0.942811	+	0.333328i	$0.891828\pi$
$282$		0			0
$283$		−	486.156i		−	1.71787i	−0.512087	−	0.858934i	$-0.671127\pi$
$283$		−	486.156i		−	1.71787i	0.512087	−	0.858934i	$-0.328873\pi$
$284$		0			0
$285$	442.755			1.55353
$286$		0			0
$287$		−	305.712i		−	1.06520i
$288$		0			0
$289$	185.641			0.642356
$290$		0			0
$291$			534.723i			1.83754i
$292$		0			0
$293$		−	240.733i		−	0.821616i	−0.911722	−	0.410808i	$-0.865247\pi$
$293$		−	240.733i		−	0.821616i	0.911722	−	0.410808i	$-0.134753\pi$
$294$		0			0
$295$		−	14.8889i		−	0.0504708i
$296$		0			0
$297$			975.181i			3.28344i
$298$		0			0
$299$	318.584	−	83.7350i	1.06550	−	0.280050i
$300$		0			0
$301$	−49.4623			−0.164327
$302$		0			0
$303$	−410.153			−1.35364
$304$		0			0
$305$	−124.570			−0.408426
$306$		0			0
$307$	−338.091			−1.10127			−0.550636	−	0.834745i	$-0.685615\pi$
$307$	−338.091			−1.10127			−0.550636	+	0.834745i	$0.685615\pi$
$308$		0			0
$309$			470.683i			1.52324i
$310$		0			0
$311$	131.884			0.424066			0.212033	−	0.977263i	$-0.431992\pi$
$311$	131.884			0.424066			0.212033	+	0.977263i	$0.431992\pi$
$312$		0			0
$313$			160.254i			0.511994i	0.966678	+	0.255997i	$0.0824038\pi$
$313$			160.254i			0.511994i	−0.966678	+	0.255997i	$0.917596\pi$
$314$		0			0
$315$	−375.428			−1.19184
$316$		0			0
$317$	−525.687			−1.65832			−0.829160	−	0.559012i	$-0.811181\pi$
$317$	−525.687			−1.65832			−0.829160	+	0.559012i	$0.811181\pi$
$318$		0			0
$319$			102.311i			0.320724i
$320$		0			0
$321$		−	30.1081i		−	0.0937948i
$322$		0			0
$323$	371.445			1.14998
$324$		0			0
$325$	71.6096			0.220337
$326$		0			0
$327$			384.786i			1.17671i
$328$		0			0
$329$			267.598i			0.813367i
$330$		0			0
$331$	−0.120137			−0.000362953			−0.000181477	−	1.00000i	$-0.500058\pi$
$331$	−0.120137			−0.000362953			−0.000181477		1.00000i	$0.500058\pi$
$332$		0			0
$333$			1296.36i			3.89298i
$334$		0			0
$335$	−9.97251			−0.0297687
$336$		0			0
$337$		−	652.946i		−	1.93752i	−0.247992	−	0.968762i	$-0.579771\pi$
$337$		−	652.946i		−	1.93752i	0.247992	−	0.968762i	$-0.420229\pi$
$338$		0			0
$339$			546.084i			1.61087i
$340$		0			0
$341$			678.124i			1.98863i
$342$		0			0
$343$			247.834i			0.722548i
$344$		0			0
$345$	−70.8515	−	269.566i	−0.205367	−	0.781352i
$346$		0			0
$347$	468.304			1.34958			0.674789	−	0.738010i	$-0.264234\pi$
$347$	468.304			1.34958			0.674789	+	0.738010i	$0.264234\pi$
$348$		0			0
$349$	182.288			0.522315			0.261157	−	0.965296i	$-0.415896\pi$
$349$	182.288			0.522315			0.261157	+	0.965296i	$0.415896\pi$
$350$		0			0
$351$	−882.581			−2.51448
$352$		0			0
$353$	−301.039			−0.852802			−0.426401	−	0.904534i	$-0.640219\pi$
$353$	−301.039			−0.852802			−0.426401	+	0.904534i	$0.640219\pi$
$354$		0			0
$355$		−	264.777i		−	0.745850i
$356$		0			0
$357$	−454.114			−1.27203
$358$		0			0
$359$			24.6772i			0.0687388i	0.999409	+	0.0343694i	$0.0109423\pi$
$359$			24.6772i			0.0687388i	−0.999409	+	0.0343694i	$0.989058\pi$
$360$		0			0
$361$	−973.874			−2.69771
$362$		0			0
$363$	−701.372			−1.93215
$364$		0			0
$365$			183.956i			0.503988i
$366$		0			0
$367$		−	427.508i		−	1.16487i	−0.812877	−	0.582435i	$-0.802100\pi$
$367$		−	427.508i		−	1.16487i	0.812877	−	0.582435i	$-0.197900\pi$
$368$		0			0
$369$	−755.599			−2.04769
$370$		0			0
$371$	−302.184			−0.814513
$372$		0			0
$373$			338.652i			0.907913i	0.891024	+	0.453957i	$0.149988\pi$
$373$			338.652i			0.907913i	−0.891024	+	0.453957i	$0.850012\pi$
$374$		0			0
$375$		−	60.5918i		−	0.161578i
$376$		0			0
$377$	−92.5959			−0.245612
$378$		0			0
$379$		−	287.157i		−	0.757671i	−0.925464	−	0.378835i	$-0.876325\pi$
$379$		−	287.157i		−	0.757671i	0.925464	−	0.378835i	$-0.123675\pi$
$380$		0			0
$381$	238.242			0.625307
$382$		0			0
$383$			682.661i			1.78240i	0.453606	+	0.891202i	$0.350137\pi$
$383$			682.661i			1.78240i	−0.453606	+	0.891202i	$0.649863\pi$
$384$		0			0
$385$		−	291.641i		−	0.757509i
$386$		0			0
$387$			122.251i			0.315894i
$388$		0			0
$389$			543.886i			1.39816i	0.715041	+	0.699082i	$0.246408\pi$
$389$			543.886i			1.39816i	−0.715041	+	0.699082i	$0.753592\pi$
$390$		0			0
$391$	−59.4402	−	226.150i	−0.152021	−	0.578389i
$392$		0			0
$393$	240.698			0.612463
$394$		0			0
$395$	297.586			0.753382
$396$		0			0
$397$	99.6609			0.251035			0.125517	−	0.992091i	$-0.459941\pi$
$397$	99.6609			0.251035			0.125517	+	0.992091i	$0.459941\pi$
$398$		0			0
$399$	1631.96			4.09014
$400$		0			0
$401$			404.668i			1.00915i	0.863369	+	0.504573i	$0.168350\pi$
$401$			404.668i			1.00915i	−0.863369	+	0.504573i	$0.831650\pi$
$402$		0			0
$403$	−613.732			−1.52291
$404$		0			0
$405$			336.831i			0.831682i
$406$		0			0
$407$	−1007.04			−2.47430
$408$		0			0
$409$	701.478			1.71510			0.857552	−	0.514397i	$-0.171984\pi$
$409$	701.478			1.71510			0.857552	+	0.514397i	$0.171984\pi$
$410$		0			0
$411$		−	243.495i		−	0.592446i
$412$		0			0
$413$		−	54.8793i		−	0.132880i
$414$		0			0
$415$	151.072			0.364029
$416$		0			0
$417$	757.625			1.81685
$418$		0			0
$419$			685.089i			1.63506i	0.575888	+	0.817529i	$0.304657\pi$
$419$			685.089i			1.63506i	−0.575888	+	0.817529i	$0.695343\pi$
$420$		0			0
$421$		−	205.326i		−	0.487711i	−0.969812	−	0.243856i	$-0.921588\pi$
$421$		−	205.326i		−	0.487711i	0.969812	−	0.243856i	$-0.0784123\pi$
$422$		0			0
$423$	661.395			1.56358
$424$		0			0
$425$		−	50.8329i		−	0.119607i
$426$		0			0
$427$	−459.156			−1.07531
$428$		0			0
$429$		−	1228.26i		−	2.86308i
$430$		0			0
$431$		−	355.926i		−	0.825814i	−0.910773	−	0.412907i	$-0.864513\pi$
$431$		−	355.926i		−	0.825814i	0.910773	−	0.412907i	$-0.135487\pi$
$432$		0			0
$433$		−	17.2870i		−	0.0399238i	−0.999801	−	0.0199619i	$-0.993646\pi$
$433$		−	17.2870i		−	0.0399238i	0.999801	−	0.0199619i	$-0.00635449\pi$
$434$		0			0
$435$			78.3491i			0.180113i
$436$		0			0
$437$	213.612	+	812.723i	0.488815	+	1.85978i
$438$		0			0
$439$	458.708			1.04489			0.522447	−	0.852672i	$-0.325019\pi$
$439$	458.708			1.04489			0.522447	+	0.852672i	$0.325019\pi$
$440$		0			0
$441$	−385.628			−0.874439
$442$		0			0
$443$	196.292			0.443096			0.221548	−	0.975149i	$-0.428889\pi$
$443$	196.292			0.443096			0.221548	+	0.975149i	$0.428889\pi$
$444$		0			0
$445$	234.181			0.526250
$446$		0			0
$447$			137.778i			0.308228i
$448$		0			0
$449$	−344.536			−0.767341			−0.383671	−	0.923470i	$-0.625340\pi$
$449$	−344.536			−0.767341			−0.383671	+	0.923470i	$0.625340\pi$
$450$		0			0
$451$		−	586.966i		−	1.30148i
$452$		0			0
$453$	284.685			0.628445
$454$		0			0
$455$	263.948			0.580105
$456$		0			0
$457$		−	122.214i		−	0.267426i	−0.991020	−	0.133713i	$-0.957310\pi$
$457$		−	122.214i		−	0.267426i	0.991020	−	0.133713i	$-0.0426900\pi$
$458$		0			0
$459$			626.510i			1.36495i
$460$		0			0
$461$	218.326			0.473593			0.236796	−	0.971559i	$-0.423903\pi$
$461$	218.326			0.473593			0.236796	+	0.971559i	$0.423903\pi$
$462$		0			0
$463$	455.380			0.983542			0.491771	−	0.870725i	$-0.336350\pi$
$463$	455.380			0.983542			0.491771	+	0.870725i	$0.336350\pi$
$464$		0			0
$465$			519.304i			1.11678i
$466$		0			0
$467$		−	350.332i		−	0.750176i	−0.926989	−	0.375088i	$-0.877612\pi$
$467$		−	350.332i		−	0.750176i	0.926989	−	0.375088i	$-0.122388\pi$
$468$		0			0
$469$	−36.7580			−0.0783752
$470$		0			0
$471$			404.226i			0.858229i
$472$		0			0
$473$	−94.9673			−0.200777
$474$		0			0
$475$			182.680i			0.384589i
$476$		0			0
$477$			746.879i			1.56578i
$478$		0			0
$479$		−	743.178i		−	1.55152i	−0.631028	−	0.775760i	$-0.717367\pi$
$479$		−	743.178i		−	1.55152i	0.631028	−	0.775760i	$-0.282633\pi$
$480$		0			0
$481$		−	911.417i		−	1.89484i
$482$		0			0
$483$	−261.154	−	993.603i	−0.540691	−	2.05715i
$484$		0			0
$485$	−220.625			−0.454897
$486$		0			0
$487$	366.770			0.753122			0.376561	−	0.926392i	$-0.377107\pi$
$487$	366.770			0.753122			0.376561	+	0.926392i	$0.377107\pi$
$488$		0			0
$489$	99.3057			0.203079
$490$		0			0
$491$	−127.024			−0.258704			−0.129352	−	0.991599i	$-0.541290\pi$
$491$	−127.024			−0.258704			−0.129352	+	0.991599i	$0.541290\pi$
$492$		0			0
$493$			65.7302i			0.133327i
$494$		0			0
$495$	−720.820			−1.45620
$496$		0			0
$497$		−	975.948i		−	1.96368i
$498$		0			0
$499$	−523.599			−1.04930			−0.524648	−	0.851319i	$-0.675803\pi$
$499$	−523.599			−1.04930			−0.524648	+	0.851319i	$0.675803\pi$
$500$		0			0
$501$	−373.819			−0.746147
$502$		0			0
$503$		−	646.170i		−	1.28463i	−0.766440	−	0.642316i	$-0.777974\pi$
$503$		−	646.170i		−	1.28463i	0.766440	−	0.642316i	$-0.222026\pi$
$504$		0			0
$505$		−	169.228i		−	0.335105i
$506$		0			0
$507$	195.738			0.386070
$508$		0			0
$509$	−605.436			−1.18946			−0.594731	−	0.803925i	$-0.702741\pi$
$509$	−605.436			−1.18946			−0.594731	+	0.803925i	$0.702741\pi$
$510$		0			0
$511$			678.047i			1.32690i
$512$		0			0
$513$		−	2251.51i		−	4.38891i
$514$		0			0
$515$	−194.202			−0.377092
$516$		0			0
$517$			513.786i			0.993783i
$518$		0			0
$519$	643.330			1.23956
$520$		0			0
$521$		−	655.952i		−	1.25902i	−0.776990	−	0.629512i	$-0.783255\pi$
$521$		−	655.952i		−	1.25902i	0.776990	−	0.629512i	$-0.216745\pi$
$522$		0			0
$523$			317.109i			0.606328i	0.952938	+	0.303164i	$0.0980430\pi$
$523$			317.109i			0.606328i	−0.952938	+	0.303164i	$0.901957\pi$
$524$		0			0
$525$		−	223.337i		−	0.425404i
$526$		0			0
$527$			435.665i			0.826688i
$528$		0			0
$529$	460.634	−	260.111i	0.870763	−	0.491702i
$530$		0			0
$531$	−135.640			−0.255442
$532$		0			0
$533$	531.230			0.996679
$534$		0			0
$535$	12.4225			0.0232197
$536$		0			0
$537$	1511.46			2.81463
$538$		0			0
$539$		−	299.564i		−	0.555777i
$540$		0			0
$541$	−222.888			−0.411992			−0.205996	−	0.978553i	$-0.566043\pi$
$541$	−222.888			−0.411992			−0.205996	+	0.978553i	$0.566043\pi$
$542$		0			0
$543$		−	358.295i		−	0.659844i
$544$		0			0
$545$	−158.762			−0.291306
$546$		0			0
$547$	552.627			1.01029			0.505144	−	0.863035i	$-0.331439\pi$
$547$	552.627			1.01029			0.505144	+	0.863035i	$0.331439\pi$
$548$		0			0
$549$			1134.85i			2.06712i
$550$		0			0
$551$		−	236.217i		−	0.428706i
$552$		0			0
$553$	1096.88			1.98351
$554$		0			0
$555$	−771.187			−1.38953
$556$		0			0
$557$			110.000i			0.197486i	0.995113	+	0.0987430i	$0.0314822\pi$
$557$			110.000i			0.197486i	−0.995113	+	0.0987430i	$0.968518\pi$
$558$		0			0
$559$		−	85.9496i		−	0.153756i
$560$		0			0
$561$	−871.896			−1.55418
$562$		0			0
$563$		−	360.367i		−	0.640084i	−0.947403	−	0.320042i	$-0.896303\pi$
$563$		−	360.367i		−	0.640084i	0.947403	−	0.320042i	$-0.103697\pi$
$564$		0			0
$565$	−225.313			−0.398784
$566$		0			0
$567$			1241.54i			2.18966i
$568$		0			0
$569$			532.981i			0.936698i	0.883544	+	0.468349i	$0.155151\pi$
$569$			532.981i			0.936698i	−0.883544	+	0.468349i	$0.844849\pi$
$570$		0			0
$571$			582.697i			1.02048i	0.860031	+	0.510242i	$0.170444\pi$
$571$			582.697i			1.02048i	−0.860031	+	0.510242i	$0.829556\pi$
$572$		0			0
$573$			29.9918i			0.0523417i
$574$		0			0
$575$	111.222	−	29.2332i	0.193430	−	0.0508403i
$576$		0			0
$577$	−869.422			−1.50680			−0.753399	−	0.657564i	$-0.771587\pi$
$577$	−869.422			−1.50680			−0.753399	+	0.657564i	$0.771587\pi$
$578$		0			0
$579$	−390.453			−0.674357
$580$		0			0
$581$	556.840			0.958416
$582$		0			0
$583$	−580.192			−0.995183
$584$		0			0
$585$		−	652.374i		−	1.11517i
$586$		0			0
$587$	403.118			0.686742			0.343371	−	0.939200i	$-0.388431\pi$
$587$	403.118			0.686742			0.343371	+	0.939200i	$0.388431\pi$
$588$		0			0
$589$		−	1565.66i		−	2.65817i
$590$		0			0
$591$	1035.90			1.75279
$592$		0			0
$593$	−668.332			−1.12703			−0.563517	−	0.826104i	$-0.690552\pi$
$593$	−668.332			−1.12703			−0.563517	+	0.826104i	$0.690552\pi$
$594$		0			0
$595$		−	187.366i		−	0.314901i
$596$		0			0
$597$		−	935.096i		−	1.56632i
$598$		0			0
$599$	−866.946			−1.44732			−0.723661	−	0.690155i	$-0.757542\pi$
$599$	−866.946			−1.44732			−0.723661	+	0.690155i	$0.757542\pi$
$600$		0			0
$601$	544.425			0.905866			0.452933	−	0.891545i	$-0.350378\pi$
$601$	544.425			0.905866			0.452933	+	0.891545i	$0.350378\pi$
$602$		0			0
$603$			90.8511i			0.150665i
$604$		0			0
$605$		−	289.384i		−	0.478321i
$606$		0			0
$607$	−23.2820			−0.0383559			−0.0191779	−	0.999816i	$-0.506105\pi$
$607$	−23.2820			−0.0383559			−0.0191779	+	0.999816i	$0.506105\pi$
$608$		0			0
$609$			288.789i			0.474202i
$610$		0			0
$611$	−464.999			−0.761045
$612$		0			0
$613$		−	559.419i		−	0.912592i	−0.889828	−	0.456296i	$-0.849176\pi$
$613$		−	559.419i		−	0.912592i	0.889828	−	0.456296i	$-0.150824\pi$
$614$		0			0
$615$		−	449.495i		−	0.730886i
$616$		0			0
$617$		−	1080.07i		−	1.75052i	−0.483649	−	0.875262i	$-0.660689\pi$
$617$		−	1080.07i		−	1.75052i	0.483649	−	0.875262i	$-0.339311\pi$
$618$		0			0
$619$		−	729.225i		−	1.17807i	−0.808107	−	0.589035i	$-0.799508\pi$
$619$		−	729.225i		−	1.17807i	0.808107	−	0.589035i	$-0.200492\pi$
$620$		0			0
$621$	−1370.81	+	360.296i	−2.20742	+	0.580187i
$622$		0			0
$623$	863.175			1.38551
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	3133.36			4.99738
$628$		0			0
$629$	−646.980			−1.02858
$630$		0			0
$631$		−	386.633i		−	0.612731i	−0.951914	−	0.306365i	$-0.900887\pi$
$631$		−	386.633i		−	0.612731i	0.951914	−	0.306365i	$-0.0991129\pi$
$632$		0			0
$633$	1810.32			2.85991
$634$		0			0
$635$			98.2979i			0.154800i
$636$		0			0
$637$	271.118			0.425618
$638$		0			0
$639$	−2412.15			−3.77489
$640$		0			0
$641$		−	676.123i		−	1.05479i	−0.849619	−	0.527397i	$-0.823168\pi$
$641$		−	676.123i		−	1.05479i	0.849619	−	0.527397i	$-0.176832\pi$
$642$		0			0
$643$			1063.72i			1.65430i	0.561980	+	0.827151i	$0.310040\pi$
$643$			1063.72i			1.65430i	−0.561980	+	0.827151i	$0.689960\pi$
$644$		0			0
$645$	−72.7254			−0.112753
$646$		0			0
$647$	157.776			0.243857			0.121929	−	0.992539i	$-0.461092\pi$
$647$	157.776			0.243857			0.121929	+	0.992539i	$0.461092\pi$
$648$		0			0
$649$		−	105.368i		−	0.162354i
$650$		0			0
$651$			1914.12i			2.94027i
$652$		0			0
$653$	41.1895			0.0630773			0.0315386	−	0.999503i	$-0.489959\pi$
$653$	41.1895			0.0630773			0.0315386	+	0.999503i	$0.489959\pi$
$654$		0			0
$655$			99.3113i			0.151620i
$656$		0			0
$657$	1675.86			2.55078
$658$		0			0
$659$		−	19.2526i		−	0.0292149i	−0.999893	−	0.0146075i	$-0.995350\pi$
$659$		−	19.2526i		−	0.0292149i	0.999893	−	0.0146075i	$-0.00464986\pi$
$660$		0			0
$661$		−	75.8322i		−	0.114724i	−0.998353	−	0.0573618i	$-0.981731\pi$
$661$		−	75.8322i		−	0.114724i	0.998353	−	0.0573618i	$-0.0182688\pi$
$662$		0			0
$663$		−	789.104i		−	1.19020i
$664$		0			0
$665$			673.344i			1.01255i
$666$		0			0
$667$	−143.818	+	37.8004i	−0.215619	+	0.0566722i
$668$		0			0
$669$	2393.14			3.57719
$670$		0			0
$671$	−881.576			−1.31382
$672$		0			0
$673$	12.6902			0.0188561			0.00942807	−	0.999956i	$-0.496999\pi$
$673$	12.6902			0.0188561			0.00942807	+	0.999956i	$0.496999\pi$
$674$		0			0
$675$	−308.123			−0.456478
$676$		0			0
$677$		−	233.185i		−	0.344439i	−0.985059	−	0.172220i	$-0.944906\pi$
$677$		−	233.185i		−	0.344439i	0.985059	−	0.172220i	$-0.0550939\pi$
$678$		0			0
$679$	−813.209			−1.19766
$680$		0			0
$681$		−	297.650i		−	0.437078i
$682$		0			0
$683$	−367.020			−0.537364			−0.268682	−	0.963229i	$-0.586588\pi$
$683$	−367.020			−0.537364			−0.268682	+	0.963229i	$0.586588\pi$
$684$		0			0
$685$	100.465			0.146665
$686$		0			0
$687$		−	1746.70i		−	2.54251i
$688$		0			0
$689$		−	525.099i		−	0.762118i
$690$		0			0
$691$	1186.21			1.71666			0.858332	−	0.513095i	$-0.171501\pi$
$691$	1186.21			1.71666			0.858332	+	0.513095i	$0.171501\pi$
$692$		0			0
$693$	−2656.89			−3.83390
$694$		0			0
$695$			312.594i			0.449775i
$696$		0			0
$697$		−	377.099i		−	0.541032i
$698$		0			0
$699$	862.826			1.23437
$700$		0			0
$701$			848.508i			1.21043i	0.796064	+	0.605213i	$0.206912\pi$
$701$			848.508i			1.21043i	−0.796064	+	0.605213i	$0.793088\pi$
$702$		0			0
$703$	2325.07			3.30736
$704$		0			0
$705$			393.454i			0.558091i
$706$		0			0
$707$		−	623.763i		−	0.882267i
$708$		0			0
$709$			533.757i			0.752831i	0.926451	+	0.376416i	$0.122844\pi$
$709$			533.757i			0.752831i	−0.926451	+	0.376416i	$0.877156\pi$
$710$		0			0
$711$		−	2711.05i		−	3.81301i
$712$		0			0
$713$	−953.235	+	250.544i	−1.33694	+	0.351394i
$714$		0			0
$715$	506.778			0.708780
$716$		0			0
$717$	−174.502			−0.243379
$718$		0			0
$719$	−1210.11			−1.68304			−0.841520	−	0.540225i	$-0.818339\pi$
$719$	−1210.11			−1.68304			−0.841520	+	0.540225i	$0.818339\pi$
$720$		0			0
$721$	−715.816			−0.992810
$722$		0			0
$723$		−	215.836i		−	0.298528i
$724$		0			0
$725$	−32.3266			−0.0445885
$726$		0			0
$727$			2.43604i			0.00335081i	0.999999	+	0.00167540i	$0.000533298\pi$
$727$			2.43604i			0.00335081i	−0.999999	+	0.00167540i	$0.999467\pi$
$728$		0			0
$729$	62.8190			0.0861715
$730$		0			0
$731$	−61.0123			−0.0834641
$732$		0			0
$733$			495.146i			0.675506i	0.941235	+	0.337753i	$0.109667\pi$
$733$			495.146i			0.675506i	−0.941235	+	0.337753i	$0.890333\pi$
$734$		0			0
$735$		−	229.404i		−	0.312115i
$736$		0			0
$737$	−70.5751			−0.0957599
$738$		0			0
$739$	−1235.77			−1.67222			−0.836112	−	0.548559i	$-0.815177\pi$
$739$	−1235.77			−1.67222			−0.836112	+	0.548559i	$0.815177\pi$
$740$		0			0
$741$			2835.83i			3.82703i
$742$		0			0
$743$		−	1030.67i		−	1.38717i	−0.720373	−	0.693587i	$-0.756029\pi$
$743$		−	1030.67i		−	1.38717i	0.720373	−	0.693587i	$-0.243971\pi$
$744$		0			0
$745$	−56.8469			−0.0763045
$746$		0			0
$747$		−	1376.29i		−	1.84242i
$748$		0			0
$749$	45.7885			0.0611329
$750$		0			0
$751$		−	1112.44i		−	1.48128i	−0.671903	−	0.740639i	$-0.734523\pi$
$751$		−	1112.44i		−	1.48128i	0.671903	−	0.740639i	$-0.265477\pi$
$752$		0			0
$753$			1260.19i			1.67356i
$754$		0			0
$755$			117.460i			0.155577i
$756$		0			0
$757$		−	483.738i		−	0.639019i	−0.947583	−	0.319510i	$-0.896482\pi$
$757$		−	483.738i		−	0.639019i	0.947583	−	0.319510i	$-0.103518\pi$
$758$		0			0
$759$	−501.413	−	1907.71i	−0.660624	−	2.51345i
$760$		0			0
$761$	833.341			1.09506			0.547530	−	0.836786i	$-0.315568\pi$
$761$	833.341			1.09506			0.547530	+	0.836786i	$0.315568\pi$
$762$		0			0
$763$	−585.183			−0.766951
$764$		0			0
$765$	−463.095			−0.605353
$766$		0			0
$767$	95.3626			0.124332
$768$		0			0
$769$			356.518i			0.463613i	0.972762	+	0.231806i	$0.0744636\pi$
$769$			356.518i			0.463613i	−0.972762	+	0.231806i	$0.925536\pi$
$770$		0			0
$771$	1762.20			2.28560
$772$		0			0
$773$		−	286.633i		−	0.370805i	−0.982663	−	0.185403i	$-0.940641\pi$
$773$		−	286.633i		−	0.370805i	0.982663	−	0.185403i	$-0.0593589\pi$
$774$		0			0
$775$	−214.263			−0.276469
$776$		0			0
$777$	−2842.54			−3.65835
$778$		0			0
$779$			1355.19i			1.73966i
$780$		0			0
$781$		−	1873.81i		−	2.39925i
$782$		0			0
$783$	398.423			0.508841
$784$		0			0
$785$	−166.782			−0.212462
$786$		0			0
$787$			930.798i			1.18272i	0.806409	+	0.591359i	$0.201408\pi$
$787$			930.798i			1.18272i	−0.806409	+	0.591359i	$0.798592\pi$
$788$		0			0
$789$			26.8205i			0.0339931i
$790$		0			0
$791$	−830.487			−1.04992
$792$		0			0
$793$		−	797.865i		−	1.00613i
$794$		0			0
$795$	−444.308			−0.558878
$796$		0			0
$797$			1493.90i			1.87441i	0.348785	+	0.937203i	$0.386594\pi$
$797$			1493.90i			1.87441i	−0.348785	+	0.937203i	$0.613406\pi$
$798$		0			0
$799$			330.084i			0.413122i
$800$		0			0
$801$		−	2133.42i		−	2.66345i
$802$		0			0
$803$			1301.85i			1.62123i
$804$		0			0
$805$	409.958	−	107.751i	0.509264	−	0.133853i
$806$		0			0
$807$	1273.84			1.57848
$808$		0			0
$809$	−850.318			−1.05107			−0.525537	−	0.850771i	$-0.676135\pi$
$809$	−850.318			−1.05107			−0.525537	+	0.850771i	$0.676135\pi$
$810$		0			0
$811$	−1050.33			−1.29510			−0.647552	−	0.762022i	$-0.724207\pi$
$811$	−1050.33			−1.29510			−0.647552	+	0.762022i	$0.724207\pi$
$812$		0			0
$813$	290.064			0.356783
$814$		0			0
$815$			40.9733i			0.0502739i
$816$		0			0
$817$	219.262			0.268374
$818$		0			0
$819$		−	2404.60i		−	2.93602i
$820$		0			0
$821$	−461.343			−0.561929			−0.280964	−	0.959718i	$-0.590654\pi$
$821$	−461.343			−0.561929			−0.280964	+	0.959718i	$0.590654\pi$
$822$		0			0
$823$	−101.176			−0.122936			−0.0614680	−	0.998109i	$-0.519578\pi$
$823$	−101.176			−0.122936			−0.0614680	+	0.998109i	$0.519578\pi$
$824$		0			0
$825$		−	428.805i		−	0.519764i
$826$		0			0
$827$			718.169i			0.868403i	0.900816	+	0.434202i	$0.142969\pi$
$827$			718.169i			0.868403i	−0.900816	+	0.434202i	$0.857031\pi$
$828$		0			0
$829$	634.298			0.765137			0.382568	−	0.923927i	$-0.375040\pi$
$829$	634.298			0.765137			0.382568	+	0.923927i	$0.375040\pi$
$830$		0			0
$831$	−778.111			−0.936356
$832$		0			0
$833$		−	192.456i		−	0.231040i
$834$		0			0
$835$		−	154.237i		−	0.184715i
$836$		0			0
$837$	2640.77			3.15505
$838$		0			0
$839$		−	133.138i		−	0.158687i	−0.996847	−	0.0793434i	$-0.974718\pi$
$839$		−	133.138i		−	0.158687i	0.996847	−	0.0793434i	$-0.0252824\pi$
$840$		0			0
$841$	−799.200			−0.950297
$842$		0			0
$843$			1015.23i			1.20431i
$844$		0			0
$845$			80.7609i			0.0955750i
$846$		0			0
$847$		−	1066.65i		−	1.25933i
$848$		0			0
$849$		−	2634.72i		−	3.10332i
$850$		0			0
$851$	−372.068	−	1415.59i	−0.437212	−	1.66345i
$852$		0			0
$853$	728.808			0.854406			0.427203	−	0.904156i	$-0.359499\pi$
$853$	728.808			0.854406			0.427203	+	0.904156i	$0.359499\pi$
$854$		0			0
$855$	1664.24			1.94648
$856$		0			0
$857$	1137.95			1.32783			0.663914	−	0.747809i	$-0.268894\pi$
$857$	1137.95			1.32783			0.663914	+	0.747809i	$0.268894\pi$
$858$		0			0
$859$	782.016			0.910379			0.455190	−	0.890395i	$-0.349571\pi$
$859$	782.016			0.910379			0.455190	+	0.890395i	$0.349571\pi$
$860$		0			0
$861$		−	1656.81i		−	1.92428i
$862$		0			0
$863$	324.868			0.376440			0.188220	−	0.982127i	$-0.439728\pi$
$863$	324.868			0.376440			0.188220	+	0.982127i	$0.439728\pi$
$864$		0			0
$865$			265.436i			0.306863i
$866$		0			0
$867$	1006.08			1.16041
$868$		0			0
$869$	2106.00			2.42348
$870$		0			0
$871$		−	63.8735i		−	0.0733336i
$872$		0			0
$873$			2009.93i			2.30232i
$874$		0			0
$875$	92.1482			0.105312
$876$		0			0
$877$	−411.966			−0.469745			−0.234872	−	0.972026i	$-0.575467\pi$
$877$	−411.966			−0.469745			−0.234872	+	0.972026i	$0.575467\pi$
$878$		0			0
$879$		−	1304.65i		−	1.48425i
$880$		0			0
$881$			1082.39i			1.22859i	0.789076	+	0.614296i	$0.210560\pi$
$881$			1082.39i			1.22859i	−0.789076	+	0.614296i	$0.789440\pi$
$882$		0			0
$883$	−1343.48			−1.52149			−0.760746	−	0.649049i	$-0.775167\pi$
$883$	−1343.48			−1.52149			−0.760746	+	0.649049i	$0.775167\pi$
$884$		0			0
$885$		−	80.6901i		−	0.0911753i
$886$		0			0
$887$	−377.176			−0.425227			−0.212613	−	0.977136i	$-0.568197\pi$
$887$	−377.176			−0.425227			−0.212613	+	0.977136i	$0.568197\pi$
$888$		0			0
$889$			362.319i			0.407558i
$890$		0			0
$891$			2383.74i			2.67535i
$892$		0			0
$893$		−	1186.23i		−	1.32837i
$894$		0			0
$895$			623.622i			0.696785i
$896$		0			0
$897$	1726.56	−	453.801i	1.92482	−	0.505910i
$898$		0			0
$899$	277.056			0.308183
$900$		0			0
$901$	−372.748			−0.413704
$902$		0			0
$903$	−268.061			−0.296856
$904$		0			0
$905$	147.832			0.163350
$906$		0			0
$907$		−	295.704i		−	0.326024i	−0.986624	−	0.163012i	$-0.947879\pi$
$907$		−	295.704i		−	0.326024i	0.986624	−	0.163012i	$-0.0521209\pi$
$908$		0			0
$909$	−1541.69			−1.69603
$910$		0			0
$911$			419.625i			0.460620i	0.973117	+	0.230310i	$0.0739740\pi$
$911$			419.625i			0.460620i	−0.973117	+	0.230310i	$0.926026\pi$
$912$		0			0
$913$	1069.13			1.17101
$914$		0			0
$915$	−675.105			−0.737820
$916$		0			0
$917$			366.054i			0.399187i
$918$		0			0
$919$		−	1654.94i		−	1.80080i	−0.435062	−	0.900401i	$-0.643274\pi$
$919$		−	1654.94i		−	1.80080i	0.435062	−	0.900401i	$-0.356726\pi$
$920$		0			0
$921$	−1832.28			−1.98945
$922$		0			0
$923$	1695.88			1.83736
$924$		0			0
$925$		−	318.190i		−	0.343989i
$926$		0			0
$927$			1769.21i			1.90853i
$928$		0			0
$929$	−1291.79			−1.39052			−0.695260	−	0.718758i	$-0.744711\pi$
$929$	−1291.79			−1.39052			−0.695260	+	0.718758i	$0.744711\pi$
$930$		0			0
$931$			691.637i			0.742897i
$932$		0			0
$933$	714.747			0.766074
$934$		0			0
$935$		−	359.742i		−	0.384751i
$936$		0			0
$937$			212.428i			0.226711i	0.993554	+	0.113355i	$0.0361599\pi$
$937$			212.428i			0.226711i	−0.993554	+	0.113355i	$0.963840\pi$
$938$		0			0
$939$			868.496i			0.924916i
$940$		0			0
$941$		−	980.930i		−	1.04243i	−0.853424	−	0.521217i	$-0.825478\pi$
$941$		−	980.930i		−	1.04243i	0.853424	−	0.521217i	$-0.174522\pi$
$942$		0			0
$943$	825.094	−	216.864i	0.874967	−	0.229972i
$944$		0			0
$945$	−1135.72			−1.20182
$946$		0			0
$947$	953.755			1.00713			0.503567	−	0.863956i	$-0.332021\pi$
$947$	953.755			1.00713			0.503567	+	0.863956i	$0.332021\pi$
$948$		0			0
$949$	−1178.23			−1.24155
$950$		0			0
$951$	−2848.96			−2.99575
$952$		0			0
$953$			248.522i			0.260779i	0.991463	+	0.130389i	$0.0416227\pi$
$953$			248.522i			0.260779i	−0.991463	+	0.130389i	$0.958377\pi$
$954$		0			0
$955$	−12.3745			−0.0129576
$956$		0			0
$957$			554.473i			0.579387i
$958$		0			0
$959$	370.308			0.386140
$960$		0			0
$961$	875.349			0.910873
$962$		0			0
$963$		−	113.171i		−	0.117519i
$964$		0			0
$965$		−	161.100i		−	0.166943i
$966$		0			0
$967$	1112.21			1.15016			0.575081	−	0.818096i	$-0.304970\pi$
$967$	1112.21			1.15016			0.575081	+	0.818096i	$0.304970\pi$
$968$		0			0
$969$	2013.04			2.07745
$970$		0			0
$971$		−	548.535i		−	0.564918i	−0.959279	−	0.282459i	$-0.908850\pi$
$971$		−	548.535i		−	0.564918i	0.959279	−	0.282459i	$-0.0911500\pi$
$972$		0			0
$973$			1152.20i			1.18417i
$974$		0			0
$975$	388.088			0.398039
$976$		0			0
$977$			262.933i			0.269122i	0.990905	+	0.134561i	$0.0429625\pi$
$977$			262.933i			0.269122i	−0.990905	+	0.134561i	$0.957038\pi$
$978$		0			0
$979$	1657.29			1.69284
$980$		0			0
$981$			1446.34i			1.47435i
$982$		0			0
$983$			1516.23i			1.54245i	0.636562	+	0.771226i	$0.280356\pi$
$983$			1516.23i			1.54245i	−0.636562	+	0.771226i	$0.719644\pi$
$984$		0			0
$985$			427.410i			0.433918i
$986$		0			0
$987$			1450.24i			1.46934i
$988$		0			0
$989$	−35.0872	−	133.495i	−0.0354774	−	0.134980i
$990$		0			0
$991$	−705.633			−0.712042			−0.356021	−	0.934478i	$-0.615867\pi$
$991$	−705.633			−0.712042			−0.356021	+	0.934478i	$0.615867\pi$
$992$		0			0
$993$	−0.651084			−0.000655674
$994$		0			0
$995$	385.818			0.387757
$996$		0			0
$997$	480.892			0.482339			0.241169	−	0.970483i	$-0.422469\pi$
$997$	480.892			0.482339			0.241169	+	0.970483i	$0.422469\pi$
$998$		0			0
$999$			3921.66i			3.92558i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.16		16
4.3	odd	2		230.3.d.a.91.2	yes	16
12.11	even	2		2070.3.c.a.91.10		16
20.3	even	4		1150.3.c.c.1149.30		32
20.7	even	4		1150.3.c.c.1149.3		32
20.19	odd	2		1150.3.d.b.551.16		16
23.22	odd	2	inner	1840.3.k.d.321.15		16
92.91	even	2		230.3.d.a.91.1	✓	16
276.275	odd	2		2070.3.c.a.91.15		16
460.183	odd	4		1150.3.c.c.1149.4		32
460.367	odd	4		1150.3.c.c.1149.29		32
460.459	even	2		1150.3.d.b.551.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.1	✓	16	92.91	even	2
230.3.d.a.91.2	yes	16	4.3	odd	2
1150.3.c.c.1149.3		32	20.7	even	4
1150.3.c.c.1149.4		32	460.183	odd	4
1150.3.c.c.1149.29		32	460.367	odd	4
1150.3.c.c.1149.30		32	20.3	even	4
1150.3.d.b.551.15		16	460.459	even	2
1150.3.d.b.551.16		16	20.19	odd	2
1840.3.k.d.321.15		16	23.22	odd	2	inner
1840.3.k.d.321.16		16	1.1	even	1	trivial
2070.3.c.a.91.10		16	12.11	even	2
2070.3.c.a.91.15		16	276.275	odd	2

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.41949 q^{3} +2.23607i q^{5} +8.24199i q^{7} +20.3709 q^{9} +O(q^{10})\)	\(q+5.41949 q^{3} +2.23607i q^{5} +8.24199i q^{7} +20.3709 q^{9} +15.8246i q^{11} -14.3219 q^{13} +12.1184i q^{15} +10.1666i q^{17} -36.5359i q^{19} +44.6674i q^{21} +(-22.2445 + 5.84663i) q^{23} -5.00000 q^{25} +61.6245 q^{27} +6.46533 q^{29} +42.8526 q^{31} +85.7611i q^{33} -18.4296 q^{35} +63.6379i q^{37} -77.6175 q^{39} -37.0921 q^{41} +6.00126i q^{43} +45.5507i q^{45} +32.4676 q^{47} -18.9303 q^{49} +55.0977i q^{51} +36.6640i q^{53} -35.3848 q^{55} -198.006i q^{57} -6.65851 q^{59} +55.7093i q^{61} +167.897i q^{63} -32.0248i q^{65} +4.45984i q^{67} +(-120.554 + 31.6858i) q^{69} -118.412 q^{71} +82.2675 q^{73} -27.0975 q^{75} -130.426 q^{77} -133.084i q^{79} +150.636 q^{81} -67.5614i q^{83} -22.7331 q^{85} +35.0388 q^{87} -104.729i q^{89} -118.041i q^{91} +232.240 q^{93} +81.6968 q^{95} +98.6666i q^{97} +322.360i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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