LMFDB - Embedded newform 1840.3.k.d.321.10

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			321.10
Root			$2.98291i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.278523 q^{3} +2.23607i q^{5} +8.51262i q^{7} -8.92243 q^{9} +O(q^{10})$	$q+0.278523 q^{3} +2.23607i q^{5} +8.51262i q^{7} -8.92243 q^{9} -7.57553i q^{11} -2.64076 q^{13} +0.622795i q^{15} -7.56057i q^{17} +24.2676i q^{19} +2.37096i q^{21} +(15.7366 + 16.7738i) q^{23} -5.00000 q^{25} -4.99180 q^{27} -31.8513 q^{29} +56.5071 q^{31} -2.10995i q^{33} -19.0348 q^{35} -39.9378i q^{37} -0.735511 q^{39} -42.5710 q^{41} +20.5721i q^{43} -19.9511i q^{45} -84.3049 q^{47} -23.4647 q^{49} -2.10579i q^{51} -11.9189i q^{53} +16.9394 q^{55} +6.75907i q^{57} -67.6561 q^{59} +35.1621i q^{61} -75.9532i q^{63} -5.90492i q^{65} +44.0660i q^{67} +(4.38299 + 4.67188i) q^{69} -8.86597 q^{71} -87.4150 q^{73} -1.39261 q^{75} +64.4876 q^{77} -154.217i q^{79} +78.9115 q^{81} -141.642i q^{83} +16.9059 q^{85} -8.87131 q^{87} +63.7252i q^{89} -22.4798i q^{91} +15.7385 q^{93} -54.2639 q^{95} -143.322i q^{97} +67.5921i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.278523			0.0928408			0.0464204	−	0.998922i	$-0.485219\pi$
$3$	0.278523			0.0928408			0.0464204	+	0.998922i	$0.485219\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			8.51262i			1.21609i	0.793903	+	0.608044i	$0.208046\pi$
$7$			8.51262i			1.21609i	−0.793903	+	0.608044i	$0.791954\pi$
$8$		0			0
$9$	−8.92243			−0.991381
$10$		0			0
$11$		−	7.57553i		−	0.688684i	−0.938844	−	0.344342i	$-0.888102\pi$
$11$		−	7.57553i		−	0.688684i	0.938844	−	0.344342i	$-0.111898\pi$
$12$		0			0
$13$	−2.64076			−0.203135			−0.101568	−	0.994829i	$-0.532386\pi$
$13$	−2.64076			−0.203135			−0.101568	+	0.994829i	$0.532386\pi$
$14$		0			0
$15$			0.622795i			0.0415197i
$16$		0			0
$17$		−	7.56057i		−	0.444739i	−0.974962	−	0.222370i	$-0.928621\pi$
$17$		−	7.56057i		−	0.444739i	0.974962	−	0.222370i	$-0.0713792\pi$
$18$		0			0
$19$			24.2676i			1.27724i	0.769522	+	0.638620i	$0.220495\pi$
$19$			24.2676i			1.27724i	−0.769522	+	0.638620i	$0.779505\pi$
$20$		0			0
$21$			2.37096i			0.112903i
$22$		0			0
$23$	15.7366	+	16.7738i	0.684199	+	0.729295i
$23$	15.7366	+	16.7738i	0.684199	+	0.729295i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−4.99180			−0.184881
$28$		0			0
$29$	−31.8513			−1.09832			−0.549161	−	0.835717i	$-0.685053\pi$
$29$	−31.8513			−1.09832			−0.549161	+	0.835717i	$0.685053\pi$
$30$		0			0
$31$	56.5071			1.82281			0.911405	−	0.411511i	$-0.134999\pi$
$31$	56.5071			1.82281			0.911405	+	0.411511i	$0.134999\pi$
$32$		0			0
$33$		−	2.10995i		−	0.0639380i
$34$		0			0
$35$	−19.0348			−0.543851
$36$		0			0
$37$		−	39.9378i		−	1.07940i	−0.841858	−	0.539700i	$-0.818538\pi$
$37$		−	39.9378i		−	1.07940i	0.841858	−	0.539700i	$-0.181462\pi$
$38$		0			0
$39$	−0.735511			−0.0188593
$40$		0			0
$41$	−42.5710			−1.03832			−0.519159	−	0.854678i	$-0.673755\pi$
$41$	−42.5710			−1.03832			−0.519159	+	0.854678i	$0.673755\pi$
$42$		0			0
$43$			20.5721i			0.478420i	0.970968	+	0.239210i	$0.0768884\pi$
$43$			20.5721i			0.478420i	−0.970968	+	0.239210i	$0.923112\pi$
$44$		0			0
$45$		−	19.9511i		−	0.443359i
$46$		0			0
$47$	−84.3049			−1.79372			−0.896860	−	0.442314i	$-0.854158\pi$
$47$	−84.3049			−1.79372			−0.896860	+	0.442314i	$0.854158\pi$
$48$		0			0
$49$	−23.4647			−0.478871
$50$		0			0
$51$		−	2.10579i		−	0.0412900i
$52$		0			0
$53$		−	11.9189i		−	0.224885i	−0.993658	−	0.112443i	$-0.964133\pi$
$53$		−	11.9189i		−	0.224885i	0.993658	−	0.112443i	$-0.0358674\pi$
$54$		0			0
$55$	16.9394			0.307989
$56$		0			0
$57$			6.75907i			0.118580i
$58$		0			0
$59$	−67.6561			−1.14671			−0.573357	−	0.819306i	$-0.694359\pi$
$59$	−67.6561			−1.14671			−0.573357	+	0.819306i	$0.694359\pi$
$60$		0			0
$61$			35.1621i			0.576428i	0.957566	+	0.288214i	$0.0930614\pi$
$61$			35.1621i			0.576428i	−0.957566	+	0.288214i	$0.906939\pi$
$62$		0			0
$63$		−	75.9532i		−	1.20561i
$64$		0			0
$65$		−	5.90492i		−	0.0908449i
$66$		0			0
$67$			44.0660i			0.657701i	0.944382	+	0.328850i	$0.106661\pi$
$67$			44.0660i			0.657701i	−0.944382	+	0.328850i	$0.893339\pi$
$68$		0			0
$69$	4.38299	+	4.67188i	0.0635216	+	0.0677084i
$70$		0			0
$71$	−8.86597			−0.124873			−0.0624364	−	0.998049i	$-0.519887\pi$
$71$	−8.86597			−0.124873			−0.0624364	+	0.998049i	$0.519887\pi$
$72$		0			0
$73$	−87.4150			−1.19747			−0.598733	−	0.800949i	$-0.704329\pi$
$73$	−87.4150			−1.19747			−0.598733	+	0.800949i	$0.704329\pi$
$74$		0			0
$75$	−1.39261			−0.0185682
$76$		0			0
$77$	64.4876			0.837501
$78$		0			0
$79$		−	154.217i		−	1.95211i	−0.217517	−	0.976057i	$-0.569796\pi$
$79$		−	154.217i		−	1.95211i	0.217517	−	0.976057i	$-0.430204\pi$
$80$		0			0
$81$	78.9115			0.974216
$82$		0			0
$83$		−	141.642i		−	1.70653i	−0.521476	−	0.853266i	$-0.674618\pi$
$83$		−	141.642i		−	1.70653i	0.521476	−	0.853266i	$-0.325382\pi$
$84$		0			0
$85$	16.9059			0.198893
$86$		0			0
$87$	−8.87131			−0.101969
$88$		0			0
$89$			63.7252i			0.716013i	0.933719	+	0.358007i	$0.116543\pi$
$89$			63.7252i			0.716013i	−0.933719	+	0.358007i	$0.883457\pi$
$90$		0			0
$91$		−	22.4798i		−	0.247030i
$92$		0			0
$93$	15.7385			0.169231
$94$		0			0
$95$	−54.2639			−0.571199
$96$		0			0
$97$		−	143.322i		−	1.47755i	−0.673952	−	0.738775i	$-0.735404\pi$
$97$		−	143.322i		−	1.47755i	0.673952	−	0.738775i	$-0.264596\pi$
$98$		0			0
$99$			67.5921i			0.682748i
$100$		0			0
$101$	27.7102			0.274359			0.137179	−	0.990546i	$-0.456196\pi$
$101$	27.7102			0.274359			0.137179	+	0.990546i	$0.456196\pi$
$102$		0			0
$103$		−	133.542i		−	1.29652i	−0.761418	−	0.648261i	$-0.775497\pi$
$103$		−	133.542i		−	1.29652i	0.761418	−	0.648261i	$-0.224503\pi$
$104$		0			0
$105$	−5.30162			−0.0504916
$106$		0			0
$107$			50.3091i			0.470179i	0.971974	+	0.235089i	$0.0755383\pi$
$107$			50.3091i			0.470179i	−0.971974	+	0.235089i	$0.924462\pi$
$108$		0			0
$109$		−	128.819i		−	1.18182i	−0.806737	−	0.590911i	$-0.798768\pi$
$109$		−	128.819i		−	1.18182i	0.806737	−	0.590911i	$-0.201232\pi$
$110$		0			0
$111$		−	11.1236i		−	0.100212i
$112$		0			0
$113$		−	86.3028i		−	0.763742i	−0.924216	−	0.381871i	$-0.875280\pi$
$113$		−	86.3028i		−	0.763742i	0.924216	−	0.381871i	$-0.124720\pi$
$114$		0			0
$115$	−37.5073	+	35.1881i	−0.326151	+	0.305983i
$116$		0			0
$117$	23.5620			0.201384
$118$		0			0
$119$	64.3602			0.540842
$120$		0			0
$121$	63.6114			0.525714
$122$		0			0
$123$	−11.8570			−0.0963983
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	11.0135			0.0867206			0.0433603	−	0.999059i	$-0.486194\pi$
$127$	11.0135			0.0867206			0.0433603	+	0.999059i	$0.486194\pi$
$128$		0			0
$129$			5.72978i			0.0444169i
$130$		0			0
$131$	−3.63941			−0.0277818			−0.0138909	−	0.999904i	$-0.504422\pi$
$131$	−3.63941			−0.0277818			−0.0138909	+	0.999904i	$0.504422\pi$
$132$		0			0
$133$	−206.581			−1.55324
$134$		0			0
$135$		−	11.1620i		−	0.0826815i
$136$		0			0
$137$			9.69785i			0.0707873i	0.999373	+	0.0353936i	$0.0112685\pi$
$137$			9.69785i			0.0707873i	−0.999373	+	0.0353936i	$0.988732\pi$
$138$		0			0
$139$	−8.05485			−0.0579486			−0.0289743	−	0.999580i	$-0.509224\pi$
$139$	−8.05485			−0.0579486			−0.0289743	+	0.999580i	$0.509224\pi$
$140$		0			0
$141$	−23.4808			−0.166531
$142$		0			0
$143$			20.0051i			0.139896i
$144$		0			0
$145$		−	71.2217i		−	0.491184i
$146$		0			0
$147$	−6.53544			−0.0444588
$148$		0			0
$149$			144.062i			0.966859i	0.875383	+	0.483430i	$0.160609\pi$
$149$			144.062i			0.966859i	−0.875383	+	0.483430i	$0.839391\pi$
$150$		0			0
$151$	109.956			0.728188			0.364094	−	0.931362i	$-0.381379\pi$
$151$	109.956			0.728188			0.364094	+	0.931362i	$0.381379\pi$
$152$		0			0
$153$			67.4586i			0.440906i
$154$		0			0
$155$			126.354i			0.815185i
$156$		0			0
$157$		−	24.8208i		−	0.158094i	−0.996871	−	0.0790471i	$-0.974812\pi$
$157$		−	24.8208i		−	0.158094i	0.996871	−	0.0790471i	$-0.0251877\pi$
$158$		0			0
$159$		−	3.31969i		−	0.0208785i
$160$		0			0
$161$	−142.789	+	133.960i	−0.886887	+	0.832047i
$162$		0			0
$163$	−108.964			−0.668489			−0.334244	−	0.942486i	$-0.608481\pi$
$163$	−108.964			−0.668489			−0.334244	+	0.942486i	$0.608481\pi$
$164$		0			0
$165$	4.71800			0.0285940
$166$		0			0
$167$	−72.1383			−0.431966			−0.215983	−	0.976397i	$-0.569296\pi$
$167$	−72.1383			−0.431966			−0.215983	+	0.976397i	$0.569296\pi$
$168$		0			0
$169$	−162.026			−0.958736
$170$		0			0
$171$		−	216.526i		−	1.26623i
$172$		0			0
$173$	−150.077			−0.867497			−0.433748	−	0.901034i	$-0.642809\pi$
$173$	−150.077			−0.867497			−0.433748	+	0.901034i	$0.642809\pi$
$174$		0			0
$175$		−	42.5631i		−	0.243218i
$176$		0			0
$177$	−18.8437			−0.106462
$178$		0			0
$179$	−207.058			−1.15675			−0.578375	−	0.815771i	$-0.696313\pi$
$179$	−207.058			−1.15675			−0.578375	+	0.815771i	$0.696313\pi$
$180$		0			0
$181$		−	331.138i		−	1.82949i	−0.404031	−	0.914746i	$-0.632391\pi$
$181$		−	331.138i		−	1.82949i	0.404031	−	0.914746i	$-0.367609\pi$
$182$		0			0
$183$			9.79343i			0.0535160i
$184$		0			0
$185$	89.3036			0.482722
$186$		0			0
$187$	−57.2753			−0.306285
$188$		0			0
$189$		−	42.4933i		−	0.224832i
$190$		0			0
$191$			172.749i			0.904447i	0.891905	+	0.452223i	$0.149369\pi$
$191$			172.749i			0.904447i	−0.891905	+	0.452223i	$0.850631\pi$
$192$		0			0
$193$	136.450			0.706996			0.353498	−	0.935435i	$-0.384992\pi$
$193$	136.450			0.706996			0.353498	+	0.935435i	$0.384992\pi$
$194$		0			0
$195$		−	1.64465i		−	0.00843412i
$196$		0			0
$197$	−271.090			−1.37609			−0.688046	−	0.725667i	$-0.741531\pi$
$197$	−271.090			−1.37609			−0.688046	+	0.725667i	$0.741531\pi$
$198$		0			0
$199$			257.292i			1.29292i	0.762946	+	0.646462i	$0.223752\pi$
$199$			257.292i			1.29292i	−0.762946	+	0.646462i	$0.776248\pi$
$200$		0			0
$201$			12.2734i			0.0610615i
$202$		0			0
$203$		−	271.138i		−	1.33566i
$204$		0			0
$205$		−	95.1918i		−	0.464350i
$206$		0			0
$207$	−140.408	−	149.663i	−0.678302	−	0.723009i
$208$		0			0
$209$	183.840			0.879615
$210$		0			0
$211$	−54.1944			−0.256846			−0.128423	−	0.991720i	$-0.540991\pi$
$211$	−54.1944			−0.256846			−0.128423	+	0.991720i	$0.540991\pi$
$212$		0			0
$213$	−2.46937			−0.0115933
$214$		0			0
$215$	−46.0005			−0.213956
$216$		0			0
$217$			481.023i			2.21670i
$218$		0			0
$219$	−24.3471			−0.111174
$220$		0			0
$221$			19.9656i			0.0903423i
$222$		0			0
$223$	−104.611			−0.469105			−0.234553	−	0.972103i	$-0.575363\pi$
$223$	−104.611			−0.469105			−0.234553	+	0.972103i	$0.575363\pi$
$224$		0			0
$225$	44.6121			0.198276
$226$		0			0
$227$			200.484i			0.883190i	0.897215	+	0.441595i	$0.145587\pi$
$227$			200.484i			0.883190i	−0.897215	+	0.441595i	$0.854413\pi$
$228$		0			0
$229$		−	22.0718i		−	0.0963834i	−0.998838	−	0.0481917i	$-0.984654\pi$
$229$		−	22.0718i		−	0.0963834i	0.998838	−	0.0481917i	$-0.0153458\pi$
$230$		0			0
$231$	17.9612			0.0777543
$232$		0			0
$233$	−338.632			−1.45335			−0.726677	−	0.686979i	$-0.758936\pi$
$233$	−338.632			−1.45335			−0.726677	+	0.686979i	$0.758936\pi$
$234$		0			0
$235$		−	188.511i		−	0.802176i
$236$		0			0
$237$		−	42.9529i		−	0.181236i
$238$		0			0
$239$	−149.374			−0.624997			−0.312499	−	0.949918i	$-0.601166\pi$
$239$	−149.374			−0.624997			−0.312499	+	0.949918i	$0.601166\pi$
$240$		0			0
$241$		−	133.030i		−	0.551991i	−0.961159	−	0.275995i	$-0.910993\pi$
$241$		−	133.030i		−	0.551991i	0.961159	−	0.275995i	$-0.0890074\pi$
$242$		0			0
$243$	66.9048			0.275328
$244$		0			0
$245$		−	52.4686i		−	0.214158i
$246$		0			0
$247$		−	64.0848i		−	0.259453i
$248$		0			0
$249$		−	39.4505i		−	0.158436i
$250$		0			0
$251$		−	376.920i		−	1.50167i	−0.660489	−	0.750836i	$-0.729651\pi$
$251$		−	376.920i		−	1.50167i	0.660489	−	0.750836i	$-0.270349\pi$
$252$		0			0
$253$	127.070	−	119.213i	0.502254	−	0.471197i
$254$		0			0
$255$	4.70869			0.0184654
$256$		0			0
$257$	226.085			0.879708			0.439854	−	0.898069i	$-0.355030\pi$
$257$	226.085			0.879708			0.439854	+	0.898069i	$0.355030\pi$
$258$		0			0
$259$	339.975			1.31264
$260$		0			0
$261$	284.191			1.08885
$262$		0			0
$263$		−	2.43654i		−	0.00926441i	−0.999989	−	0.00463221i	$-0.998526\pi$
$263$		−	2.43654i		−	0.00926441i	0.999989	−	0.00463221i	$-0.00147448\pi$
$264$		0			0
$265$	26.6515			0.100572
$266$		0			0
$267$			17.7489i			0.0664753i
$268$		0			0
$269$	−311.337			−1.15739			−0.578694	−	0.815545i	$-0.696437\pi$
$269$	−311.337			−1.15739			−0.578694	+	0.815545i	$0.696437\pi$
$270$		0			0
$271$	−260.751			−0.962182			−0.481091	−	0.876671i	$-0.659759\pi$
$271$	−260.751			−0.962182			−0.481091	+	0.876671i	$0.659759\pi$
$272$		0			0
$273$		−	6.26112i		−	0.0229345i
$274$		0			0
$275$			37.8776i			0.137737i
$276$		0			0
$277$	−182.267			−0.658004			−0.329002	−	0.944329i	$-0.606712\pi$
$277$	−182.267			−0.658004			−0.329002	+	0.944329i	$0.606712\pi$
$278$		0			0
$279$	−504.180			−1.80710
$280$		0			0
$281$		−	4.60502i		−	0.0163880i	−0.999966	−	0.00819398i	$-0.997392\pi$
$281$		−	4.60502i		−	0.0163880i	0.999966	−	0.00819398i	$-0.00260825\pi$
$282$		0			0
$283$			332.897i			1.17631i	0.808747	+	0.588157i	$0.200146\pi$
$283$			332.897i			1.17631i	−0.808747	+	0.588157i	$0.799854\pi$
$284$		0			0
$285$	−15.1137			−0.0530306
$286$		0			0
$287$		−	362.391i		−	1.26269i
$288$		0			0
$289$	231.838			0.802207
$290$		0			0
$291$		−	39.9185i		−	0.137177i
$292$		0			0
$293$		−	336.114i		−	1.14715i	−0.819154	−	0.573574i	$-0.805557\pi$
$293$		−	336.114i		−	1.14715i	0.819154	−	0.573574i	$-0.194443\pi$
$294$		0			0
$295$		−	151.284i		−	0.512826i
$296$		0			0
$297$			37.8155i			0.127325i
$298$		0			0
$299$	−41.5565	−	44.2955i	−0.138985	−	0.148146i
$300$		0			0
$301$	−175.122			−0.581801
$302$		0			0
$303$	7.71792			0.0254717
$304$		0			0
$305$	−78.6248			−0.257786
$306$		0			0
$307$	457.934			1.49164			0.745821	−	0.666147i	$-0.232057\pi$
$307$	457.934			1.49164			0.745821	+	0.666147i	$0.232057\pi$
$308$		0			0
$309$		−	37.1944i		−	0.120370i
$310$		0			0
$311$	455.620			1.46502			0.732508	−	0.680758i	$-0.238350\pi$
$311$	455.620			1.46502			0.732508	+	0.680758i	$0.238350\pi$
$312$		0			0
$313$			589.353i			1.88292i	0.337130	+	0.941458i	$0.390544\pi$
$313$			589.353i			1.88292i	−0.337130	+	0.941458i	$0.609456\pi$
$314$		0			0
$315$	169.837			0.539164
$316$		0			0
$317$	−386.251			−1.21846			−0.609229	−	0.792994i	$-0.708521\pi$
$317$	−386.251			−1.21846			−0.609229	+	0.792994i	$0.708521\pi$
$318$		0			0
$319$			241.291i			0.756397i
$320$		0			0
$321$			14.0122i			0.0436518i
$322$		0			0
$323$	183.477			0.568039
$324$		0			0
$325$	13.2038			0.0406271
$326$		0			0
$327$		−	35.8789i		−	0.109721i
$328$		0			0
$329$		−	717.655i		−	2.18132i
$330$		0			0
$331$	−220.712			−0.666803			−0.333402	−	0.942785i	$-0.608196\pi$
$331$	−220.712			−0.666803			−0.333402	+	0.942785i	$0.608196\pi$
$332$		0			0
$333$			356.342i			1.07010i
$334$		0			0
$335$	−98.5345			−0.294133
$336$		0			0
$337$		−	193.563i		−	0.574371i	−0.957875	−	0.287185i	$-0.907280\pi$
$337$		−	193.563i		−	0.574371i	0.957875	−	0.287185i	$-0.0927196\pi$
$338$		0			0
$339$		−	24.0373i		−	0.0709064i
$340$		0			0
$341$		−	428.071i		−	1.25534i
$342$		0			0
$343$			217.373i			0.633739i
$344$		0			0
$345$	−10.4466	+	9.80067i	−0.0302801	+	0.0284077i
$346$		0			0
$347$	262.429			0.756278			0.378139	−	0.925749i	$-0.376564\pi$
$347$	262.429			0.756278			0.378139	+	0.925749i	$0.376564\pi$
$348$		0			0
$349$	229.831			0.658543			0.329271	−	0.944235i	$-0.393197\pi$
$349$	229.831			0.658543			0.329271	+	0.944235i	$0.393197\pi$
$350$		0			0
$351$	13.1821			0.0375560
$352$		0			0
$353$	367.482			1.04103			0.520513	−	0.853854i	$-0.325741\pi$
$353$	367.482			1.04103			0.520513	+	0.853854i	$0.325741\pi$
$354$		0			0
$355$		−	19.8249i		−	0.0558448i
$356$		0			0
$357$	17.9258			0.0502123
$358$		0			0
$359$			239.971i			0.668442i	0.942495	+	0.334221i	$0.108473\pi$
$359$			239.971i			0.668442i	−0.942495	+	0.334221i	$0.891527\pi$
$360$		0			0
$361$	−227.915			−0.631344
$362$		0			0
$363$	17.7172			0.0488077
$364$		0			0
$365$		−	195.466i		−	0.535523i
$366$		0			0
$367$			389.113i			1.06025i	0.847918	+	0.530127i	$0.177856\pi$
$367$			389.113i			1.06025i	−0.847918	+	0.530127i	$0.822144\pi$
$368$		0			0
$369$	379.837			1.02937
$370$		0			0
$371$	101.461			0.273480
$372$		0			0
$373$			546.309i			1.46463i	0.680964	+	0.732317i	$0.261561\pi$
$373$			546.309i			1.46463i	−0.680964	+	0.732317i	$0.738439\pi$
$374$		0			0
$375$		−	3.11398i		−	0.00830394i
$376$		0			0
$377$	84.1117			0.223108
$378$		0			0
$379$		−	295.837i		−	0.780573i	−0.920693	−	0.390287i	$-0.872376\pi$
$379$		−	295.837i		−	0.780573i	0.920693	−	0.390287i	$-0.127624\pi$
$380$		0			0
$381$	3.06751			0.00805122
$382$		0			0
$383$			566.033i			1.47789i	0.673765	+	0.738946i	$0.264676\pi$
$383$			566.033i			1.47789i	−0.673765	+	0.738946i	$0.735324\pi$
$384$		0			0
$385$			144.199i			0.374542i
$386$		0			0
$387$		−	183.553i		−	0.474296i
$388$		0			0
$389$		−	196.043i		−	0.503966i	−0.967732	−	0.251983i	$-0.918917\pi$
$389$		−	196.043i		−	0.503966i	0.967732	−	0.251983i	$-0.0810826\pi$
$390$		0			0
$391$	126.819	−	118.978i	0.324346	−	0.304290i
$392$		0			0
$393$	−1.01366			−0.00257928
$394$		0			0
$395$	344.840			0.873012
$396$		0			0
$397$	−298.788			−0.752616			−0.376308	−	0.926495i	$-0.622806\pi$
$397$	−298.788			−0.752616			−0.376308	+	0.926495i	$0.622806\pi$
$398$		0			0
$399$	−57.5374			−0.144204
$400$		0			0
$401$			785.114i			1.95789i	0.204121	+	0.978946i	$0.434567\pi$
$401$			785.114i			1.95789i	−0.204121	+	0.978946i	$0.565433\pi$
$402$		0			0
$403$	−149.222			−0.370277
$404$		0			0
$405$			176.451i			0.435683i
$406$		0			0
$407$	−302.550			−0.743365
$408$		0			0
$409$	358.186			0.875761			0.437881	−	0.899033i	$-0.355729\pi$
$409$	358.186			0.875761			0.437881	+	0.899033i	$0.355729\pi$
$410$		0			0
$411$			2.70107i			0.00657195i
$412$		0			0
$413$		−	575.930i		−	1.39450i
$414$		0			0
$415$	316.722			0.763184
$416$		0			0
$417$	−2.24346			−0.00537999
$418$		0			0
$419$			195.271i			0.466042i	0.972472	+	0.233021i	$0.0748610\pi$
$419$			195.271i			0.466042i	−0.972472	+	0.233021i	$0.925139\pi$
$420$		0			0
$421$		−	768.042i		−	1.82433i	−0.409826	−	0.912164i	$-0.634410\pi$
$421$		−	768.042i		−	1.82433i	0.409826	−	0.912164i	$-0.365590\pi$
$422$		0			0
$423$	752.204			1.77826
$424$		0			0
$425$			37.8028i			0.0889479i
$426$		0			0
$427$	−299.321			−0.700987
$428$		0			0
$429$			5.57188i			0.0129881i
$430$		0			0
$431$		−	333.122i		−	0.772905i	−0.922309	−	0.386453i	$-0.873700\pi$
$431$		−	333.122i		−	0.772905i	0.922309	−	0.386453i	$-0.126300\pi$
$432$		0			0
$433$		−	7.49741i		−	0.0173150i	−0.999963	−	0.00865752i	$-0.997244\pi$
$433$		−	7.49741i		−	0.0173150i	0.999963	−	0.00865752i	$-0.00275581\pi$
$434$		0			0
$435$		−	19.8369i		−	0.0456020i
$436$		0			0
$437$	−407.059	+	381.889i	−0.931485	+	0.873887i
$438$		0			0
$439$	303.939			0.692344			0.346172	−	0.938171i	$-0.387481\pi$
$439$	303.939			0.692344			0.346172	+	0.938171i	$0.387481\pi$
$440$		0			0
$441$	209.362			0.474743
$442$		0			0
$443$	−637.145			−1.43825			−0.719125	−	0.694881i	$-0.755457\pi$
$443$	−637.145			−1.43825			−0.719125	+	0.694881i	$0.755457\pi$
$444$		0			0
$445$	−142.494			−0.320211
$446$		0			0
$447$			40.1245i			0.0897640i
$448$		0			0
$449$	88.9331			0.198069			0.0990346	−	0.995084i	$-0.468425\pi$
$449$	88.9331			0.198069			0.0990346	+	0.995084i	$0.468425\pi$
$450$		0			0
$451$			322.498i			0.715073i
$452$		0			0
$453$	30.6253			0.0676056
$454$		0			0
$455$	50.2663			0.110475
$456$		0			0
$457$			377.669i			0.826410i	0.910638	+	0.413205i	$0.135591\pi$
$457$			377.669i			0.826410i	−0.910638	+	0.413205i	$0.864409\pi$
$458$		0			0
$459$			37.7408i			0.0822240i
$460$		0			0
$461$	−585.070			−1.26913			−0.634566	−	0.772869i	$-0.718821\pi$
$461$	−585.070			−1.26913			−0.634566	+	0.772869i	$0.718821\pi$
$462$		0			0
$463$	225.877			0.487854			0.243927	−	0.969794i	$-0.421564\pi$
$463$	225.877			0.487854			0.243927	+	0.969794i	$0.421564\pi$
$464$		0			0
$465$			35.1924i			0.0756825i
$466$		0			0
$467$			702.984i			1.50532i	0.658410	+	0.752659i	$0.271229\pi$
$467$			702.984i			1.50532i	−0.658410	+	0.752659i	$0.728771\pi$
$468$		0			0
$469$	−375.117			−0.799822
$470$		0			0
$471$		−	6.91315i		−	0.0146776i
$472$		0			0
$473$	155.844			0.329480
$474$		0			0
$475$		−	121.338i		−	0.255448i
$476$		0			0
$477$			106.346i			0.222947i
$478$		0			0
$479$			291.706i			0.608989i	0.952514	+	0.304494i	$0.0984875\pi$
$479$			291.706i			0.608989i	−0.952514	+	0.304494i	$0.901512\pi$
$480$		0			0
$481$			105.466i			0.219264i
$482$		0			0
$483$	−39.7699	+	37.3107i	−0.0823394	+	0.0772479i
$484$		0			0
$485$	320.479			0.660781
$486$		0			0
$487$	−205.957			−0.422909			−0.211454	−	0.977388i	$-0.567820\pi$
$487$	−205.957			−0.422909			−0.211454	+	0.977388i	$0.567820\pi$
$488$		0			0
$489$	−30.3488			−0.0620630
$490$		0			0
$491$	192.478			0.392012			0.196006	−	0.980603i	$-0.437203\pi$
$491$	192.478			0.392012			0.196006	+	0.980603i	$0.437203\pi$
$492$		0			0
$493$			240.814i			0.488467i
$494$		0			0
$495$	−151.140			−0.305334
$496$		0			0
$497$		−	75.4726i		−	0.151856i
$498$		0			0
$499$	−887.386			−1.77833			−0.889164	−	0.457589i	$-0.848713\pi$
$499$	−887.386			−1.77833			−0.889164	+	0.457589i	$0.848713\pi$
$500$		0			0
$501$	−20.0921			−0.0401041
$502$		0			0
$503$			330.810i			0.657673i	0.944387	+	0.328837i	$0.106657\pi$
$503$			330.810i			0.657673i	−0.944387	+	0.328837i	$0.893343\pi$
$504$		0			0
$505$			61.9619i			0.122697i
$506$		0			0
$507$	−45.1280			−0.0890099
$508$		0			0
$509$	−407.928			−0.801430			−0.400715	−	0.916203i	$-0.631238\pi$
$509$	−407.928			−0.801430			−0.400715	+	0.916203i	$0.631238\pi$
$510$		0			0
$511$		−	744.131i		−	1.45622i
$512$		0			0
$513$		−	121.139i		−	0.236138i
$514$		0			0
$515$	298.609			0.579822
$516$		0			0
$517$			638.654i			1.23531i
$518$		0			0
$519$	−41.7998			−0.0805391
$520$		0			0
$521$		−	458.780i		−	0.880576i	−0.897857	−	0.440288i	$-0.854876\pi$
$521$		−	458.780i		−	0.880576i	0.897857	−	0.440288i	$-0.145124\pi$
$522$		0			0
$523$			404.987i			0.774354i	0.922005	+	0.387177i	$0.126550\pi$
$523$			404.987i			0.774354i	−0.922005	+	0.387177i	$0.873450\pi$
$524$		0			0
$525$		−	11.8548i		−	0.0225805i
$526$		0			0
$527$		−	427.226i		−	0.810675i
$528$		0			0
$529$	−33.7199	+	527.924i	−0.0637427	+	0.997966i
$530$		0			0
$531$	603.656			1.13683
$532$		0			0
$533$	112.420			0.210919
$534$		0			0
$535$	−112.495			−0.210270
$536$		0			0
$537$	−57.6703			−0.107394
$538$		0			0
$539$			177.757i			0.329791i
$540$		0			0
$541$	1047.51			1.93625			0.968123	−	0.250477i	$-0.0805875\pi$
$541$	1047.51			1.93625			0.968123	+	0.250477i	$0.0805875\pi$
$542$		0			0
$543$		−	92.2294i		−	0.169851i
$544$		0			0
$545$	288.047			0.528527
$546$		0			0
$547$	−123.784			−0.226296			−0.113148	−	0.993578i	$-0.536093\pi$
$547$	−123.784			−0.226296			−0.113148	+	0.993578i	$0.536093\pi$
$548$		0			0
$549$		−	313.731i		−	0.571459i
$550$		0			0
$551$		−	772.955i		−	1.40282i
$552$		0			0
$553$	1312.79			2.37394
$554$		0			0
$555$	24.8731			0.0448163
$556$		0			0
$557$			246.292i			0.442176i	0.975254	+	0.221088i	$0.0709608\pi$
$557$			246.292i			0.442176i	−0.975254	+	0.221088i	$0.929039\pi$
$558$		0			0
$559$		−	54.3259i		−	0.0971840i
$560$		0			0
$561$	−15.9525			−0.0284357
$562$		0			0
$563$		−	574.776i		−	1.02092i	−0.859902	−	0.510458i	$-0.829476\pi$
$563$		−	574.776i		−	1.02092i	0.859902	−	0.510458i	$-0.170524\pi$
$564$		0			0
$565$	192.979			0.341556
$566$		0			0
$567$			671.743i			1.18473i
$568$		0			0
$569$			794.332i			1.39601i	0.716091	+	0.698007i	$0.245930\pi$
$569$			794.332i			1.39601i	−0.716091	+	0.698007i	$0.754070\pi$
$570$		0			0
$571$			416.688i			0.729751i	0.931056	+	0.364876i	$0.118889\pi$
$571$			416.688i			0.729751i	−0.931056	+	0.364876i	$0.881111\pi$
$572$		0			0
$573$			48.1146i			0.0839696i
$574$		0			0
$575$	−78.6829	−	83.8689i	−0.136840	−	0.145859i
$576$		0			0
$577$	−282.647			−0.489856			−0.244928	−	0.969541i	$-0.578764\pi$
$577$	−282.647			−0.489856			−0.244928	+	0.969541i	$0.578764\pi$
$578$		0			0
$579$	38.0044			0.0656381
$580$		0			0
$581$	1205.75			2.07529
$582$		0			0
$583$	−90.2920			−0.154875
$584$		0			0
$585$			52.6862i			0.0900618i
$586$		0			0
$587$	168.123			0.286410			0.143205	−	0.989693i	$-0.454259\pi$
$587$	168.123			0.286410			0.143205	+	0.989693i	$0.454259\pi$
$588$		0			0
$589$			1371.29i			2.32817i
$590$		0			0
$591$	−75.5048			−0.127758
$592$		0			0
$593$	−419.364			−0.707191			−0.353595	−	0.935398i	$-0.615041\pi$
$593$	−419.364			−0.707191			−0.353595	+	0.935398i	$0.615041\pi$
$594$		0			0
$595$			143.914i			0.241872i
$596$		0			0
$597$			71.6616i			0.120036i
$598$		0			0
$599$	519.774			0.867736			0.433868	−	0.900977i	$-0.357148\pi$
$599$	519.774			0.867736			0.433868	+	0.900977i	$0.357148\pi$
$600$		0			0
$601$	1089.19			1.81230			0.906151	−	0.422955i	$-0.139007\pi$
$601$	1089.19			1.81230			0.906151	+	0.422955i	$0.139007\pi$
$602$		0			0
$603$		−	393.175i		−	0.652032i
$604$		0			0
$605$			142.239i			0.235107i
$606$		0			0
$607$	−890.673			−1.46734			−0.733668	−	0.679508i	$-0.762193\pi$
$607$	−890.673			−1.46734			−0.733668	+	0.679508i	$0.762193\pi$
$608$		0			0
$609$		−	75.5181i		−	0.124003i
$610$		0			0
$611$	222.629			0.364368
$612$		0			0
$613$		−	629.354i		−	1.02668i	−0.858186	−	0.513339i	$-0.828408\pi$
$613$		−	629.354i		−	1.02668i	0.858186	−	0.513339i	$-0.171592\pi$
$614$		0			0
$615$		−	26.5131i		−	0.0431107i
$616$		0			0
$617$			157.187i			0.254760i	0.991854	+	0.127380i	$0.0406568\pi$
$617$			157.187i			0.254760i	−0.991854	+	0.127380i	$0.959343\pi$
$618$		0			0
$619$		−	462.293i		−	0.746838i	−0.927663	−	0.373419i	$-0.878185\pi$
$619$		−	462.293i		−	0.746838i	0.927663	−	0.373419i	$-0.121815\pi$
$620$		0			0
$621$	−78.5539	−	83.7314i	−0.126496	−	0.134833i
$622$		0			0
$623$	−542.468			−0.870735
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	51.2035			0.0816642
$628$		0			0
$629$	−301.952			−0.480051
$630$		0			0
$631$		−	411.630i		−	0.652345i	−0.945310	−	0.326173i	$-0.894241\pi$
$631$		−	411.630i		−	0.652345i	0.945310	−	0.326173i	$-0.105759\pi$
$632$		0			0
$633$	−15.0944			−0.0238458
$634$		0			0
$635$			24.6270i			0.0387827i
$636$		0			0
$637$	61.9645			0.0972756
$638$		0			0
$639$	79.1060			0.123797
$640$		0			0
$641$		−	251.089i		−	0.391715i	−0.980632	−	0.195857i	$-0.937251\pi$
$641$		−	251.089i		−	0.391715i	0.980632	−	0.195857i	$-0.0627490\pi$
$642$		0			0
$643$		−	102.907i		−	0.160042i	−0.996793	−	0.0800211i	$-0.974501\pi$
$643$		−	102.907i		−	0.160042i	0.996793	−	0.0800211i	$-0.0254988\pi$
$644$		0			0
$645$	−12.8122			−0.0198639
$646$		0			0
$647$	437.368			0.675993			0.337997	−	0.941147i	$-0.390251\pi$
$647$	437.368			0.675993			0.337997	+	0.941147i	$0.390251\pi$
$648$		0			0
$649$			512.530i			0.789723i
$650$		0			0
$651$			133.976i			0.205800i
$652$		0			0
$653$	−94.4001			−0.144564			−0.0722819	−	0.997384i	$-0.523028\pi$
$653$	−94.4001			−0.144564			−0.0722819	+	0.997384i	$0.523028\pi$
$654$		0			0
$655$		−	8.13797i		−	0.0124244i
$656$		0			0
$657$	779.954			1.18714
$658$		0			0
$659$			759.727i			1.15285i	0.817151	+	0.576424i	$0.195552\pi$
$659$			759.727i			1.15285i	−0.817151	+	0.576424i	$0.804448\pi$
$660$		0			0
$661$		−	581.754i		−	0.880112i	−0.897970	−	0.440056i	$-0.854959\pi$
$661$		−	581.754i		−	0.880112i	0.897970	−	0.440056i	$-0.145041\pi$
$662$		0			0
$663$			5.56088i			0.00838745i
$664$		0			0
$665$		−	461.928i		−	0.694629i
$666$		0			0
$667$	−501.231	−	534.267i	−0.751471	−	0.801001i
$668$		0			0
$669$	−29.1364			−0.0435521
$670$		0			0
$671$	266.371			0.396977
$672$		0			0
$673$	−831.173			−1.23503			−0.617513	−	0.786560i	$-0.711860\pi$
$673$	−831.173			−1.23503			−0.617513	+	0.786560i	$0.711860\pi$
$674$		0			0
$675$	24.9590			0.0369763
$676$		0			0
$677$		−	106.706i		−	0.157616i	−0.996890	−	0.0788079i	$-0.974889\pi$
$677$		−	106.706i		−	0.157616i	0.996890	−	0.0788079i	$-0.0251114\pi$
$678$		0			0
$679$	1220.05			1.79683
$680$		0			0
$681$			55.8393i			0.0819961i
$682$		0			0
$683$	−380.675			−0.557357			−0.278678	−	0.960385i	$-0.589896\pi$
$683$	−380.675			−0.557357			−0.278678	+	0.960385i	$0.589896\pi$
$684$		0			0
$685$	−21.6851			−0.0316570
$686$		0			0
$687$		−	6.14749i		−	0.00894832i
$688$		0			0
$689$			31.4750i			0.0456821i
$690$		0			0
$691$	318.845			0.461425			0.230713	−	0.973022i	$-0.425894\pi$
$691$	318.845			0.461425			0.230713	+	0.973022i	$0.425894\pi$
$692$		0			0
$693$	−575.385			−0.830282
$694$		0			0
$695$		−	18.0112i		−	0.0259154i
$696$		0			0
$697$			321.861i			0.461781i
$698$		0			0
$699$	−94.3165			−0.134931
$700$		0			0
$701$		−	112.754i		−	0.160848i	−0.996761	−	0.0804239i	$-0.974373\pi$
$701$		−	112.754i		−	0.160848i	0.996761	−	0.0804239i	$-0.0256274\pi$
$702$		0			0
$703$	969.193			1.37865
$704$		0			0
$705$		−	52.5047i		−	0.0744747i
$706$		0			0
$707$			235.887i			0.333644i
$708$		0			0
$709$			465.481i			0.656531i	0.944585	+	0.328266i	$0.106464\pi$
$709$			465.481i			0.656531i	−0.944585	+	0.328266i	$0.893536\pi$
$710$		0			0
$711$			1375.99i			1.93529i
$712$		0			0
$713$	889.229	+	947.838i	1.24717	+	1.32937i
$714$		0			0
$715$	−44.7328			−0.0625634
$716$		0			0
$717$	−41.6041			−0.0580253
$718$		0			0
$719$	347.913			0.483885			0.241943	−	0.970291i	$-0.422215\pi$
$719$	347.913			0.483885			0.241943	+	0.970291i	$0.422215\pi$
$720$		0			0
$721$	1136.79			1.57669
$722$		0			0
$723$		−	37.0518i		−	0.0512473i
$724$		0			0
$725$	159.257			0.219664
$726$		0			0
$727$			88.3077i			0.121469i	0.998154	+	0.0607343i	$0.0193442\pi$
$727$			88.3077i			0.121469i	−0.998154	+	0.0607343i	$0.980656\pi$
$728$		0			0
$729$	−691.569			−0.948654
$730$		0			0
$731$	155.536			0.212772
$732$		0			0
$733$			510.848i			0.696928i	0.937322	+	0.348464i	$0.113297\pi$
$733$			510.848i			0.696928i	−0.937322	+	0.348464i	$0.886703\pi$
$734$		0			0
$735$		−	14.6137i		−	0.0198826i
$736$		0			0
$737$	333.823			0.452948
$738$		0			0
$739$	416.763			0.563955			0.281978	−	0.959421i	$-0.409010\pi$
$739$	416.763			0.563955			0.281978	+	0.959421i	$0.409010\pi$
$740$		0			0
$741$		−	17.8491i		−	0.0240878i
$742$		0			0
$743$			633.469i			0.852583i	0.904586	+	0.426291i	$0.140180\pi$
$743$			633.469i			0.852583i	−0.904586	+	0.426291i	$0.859820\pi$
$744$		0			0
$745$	−322.132			−0.432393
$746$		0			0
$747$			1263.79i			1.69182i
$748$		0			0
$749$	−428.262			−0.571779
$750$		0			0
$751$			1282.52i			1.70775i	0.520475	+	0.853877i	$0.325755\pi$
$751$			1282.52i			1.70775i	−0.520475	+	0.853877i	$0.674245\pi$
$752$		0			0
$753$		−	104.981i		−	0.139416i
$754$		0			0
$755$			245.870i			0.325655i
$756$		0			0
$757$		−	1233.64i		−	1.62964i	−0.579716	−	0.814819i	$-0.696836\pi$
$757$		−	1233.64i		−	1.62964i	0.579716	−	0.814819i	$-0.303164\pi$
$758$		0			0
$759$	35.3919	−	33.2035i	0.0466297	−	0.0437463i
$760$		0			0
$761$	−262.559			−0.345018			−0.172509	−	0.985008i	$-0.555187\pi$
$761$	−262.559			−0.345018			−0.172509	+	0.985008i	$0.555187\pi$
$762$		0			0
$763$	1096.58			1.43720
$764$		0			0
$765$	−150.842			−0.197179
$766$		0			0
$767$	178.663			0.232938
$768$		0			0
$769$			801.510i			1.04228i	0.853473	+	0.521138i	$0.174492\pi$
$769$			801.510i			1.04228i	−0.853473	+	0.521138i	$0.825508\pi$
$770$		0			0
$771$	62.9698			0.0816729
$772$		0			0
$773$		−	1143.79i		−	1.47967i	−0.672786	−	0.739837i	$-0.734903\pi$
$773$		−	1143.79i		−	1.47967i	0.672786	−	0.739837i	$-0.265097\pi$
$774$		0			0
$775$	−282.536			−0.364562
$776$		0			0
$777$	94.6907			0.121867
$778$		0			0
$779$		−	1033.10i		−	1.32618i
$780$		0			0
$781$			67.1644i			0.0859979i
$782$		0			0
$783$	158.995			0.203059
$784$		0			0
$785$	55.5010			0.0707019
$786$		0			0
$787$			111.467i			0.141635i	0.997489	+	0.0708175i	$0.0225608\pi$
$787$			111.467i			0.141635i	−0.997489	+	0.0708175i	$0.977439\pi$
$788$		0			0
$789$		−	0.678632i		−	0.000860116i
$790$		0			0
$791$	734.663			0.928777
$792$		0			0
$793$		−	92.8546i		−	0.117093i
$794$		0			0
$795$	7.42304			0.00933716
$796$		0			0
$797$		−	231.014i		−	0.289855i	−0.989442	−	0.144927i	$-0.953705\pi$
$797$		−	231.014i		−	0.289855i	0.989442	−	0.144927i	$-0.0462949\pi$
$798$		0			0
$799$			637.393i			0.797738i
$800$		0			0
$801$		−	568.583i		−	0.709842i
$802$		0			0
$803$			662.215i			0.824676i
$804$		0			0
$805$	−299.543	−	319.286i	−0.372103	−	0.396628i
$806$		0			0
$807$	−86.7145			−0.107453
$808$		0			0
$809$	−165.201			−0.204204			−0.102102	−	0.994774i	$-0.532557\pi$
$809$	−165.201			−0.204204			−0.102102	+	0.994774i	$0.532557\pi$
$810$		0			0
$811$	−317.322			−0.391273			−0.195636	−	0.980677i	$-0.562677\pi$
$811$	−317.322			−0.391273			−0.195636	+	0.980677i	$0.562677\pi$
$812$		0			0
$813$	−72.6251			−0.0893298
$814$		0			0
$815$		−	243.650i		−	0.298957i
$816$		0			0
$817$	−499.234			−0.611058
$818$		0			0
$819$			200.574i			0.244901i
$820$		0			0
$821$	−1389.82			−1.69284			−0.846421	−	0.532514i	$-0.821247\pi$
$821$	−1389.82			−1.69284			−0.846421	+	0.532514i	$0.821247\pi$
$822$		0			0
$823$	−948.001			−1.15188			−0.575942	−	0.817490i	$-0.695365\pi$
$823$	−948.001			−1.15188			−0.575942	+	0.817490i	$0.695365\pi$
$824$		0			0
$825$			10.5498i			0.0127876i
$826$		0			0
$827$			1177.54i			1.42387i	0.702245	+	0.711935i	$0.252181\pi$
$827$			1177.54i			1.42387i	−0.702245	+	0.711935i	$0.747819\pi$
$828$		0			0
$829$	−1278.78			−1.54256			−0.771279	−	0.636497i	$-0.780383\pi$
$829$	−1278.78			−1.54256			−0.771279	+	0.636497i	$0.780383\pi$
$830$		0			0
$831$	−50.7655			−0.0610897
$832$		0			0
$833$			177.406i			0.212973i
$834$		0			0
$835$		−	161.306i		−	0.193181i
$836$		0			0
$837$	−282.072			−0.337004
$838$		0			0
$839$		−	1435.77i		−	1.71128i	−0.517570	−	0.855641i	$-0.673163\pi$
$839$		−	1435.77i		−	1.71128i	0.517570	−	0.855641i	$-0.326837\pi$
$840$		0			0
$841$	173.507			0.206311
$842$		0			0
$843$		−	1.28260i		−	0.00152147i
$844$		0			0
$845$		−	362.302i		−	0.428760i
$846$		0			0
$847$			541.500i			0.639315i
$848$		0			0
$849$			92.7193i			0.109210i
$850$		0			0
$851$	669.908	−	628.484i	0.787201	−	0.738524i
$852$		0			0
$853$	812.996			0.953102			0.476551	−	0.879147i	$-0.341887\pi$
$853$	812.996			0.953102			0.476551	+	0.879147i	$0.341887\pi$
$854$		0			0
$855$	484.166			0.566276
$856$		0			0
$857$	−958.841			−1.11883			−0.559417	−	0.828886i	$-0.688975\pi$
$857$	−958.841			−1.11883			−0.559417	+	0.828886i	$0.688975\pi$
$858$		0			0
$859$	383.654			0.446628			0.223314	−	0.974747i	$-0.428312\pi$
$859$	383.654			0.446628			0.223314	+	0.974747i	$0.428312\pi$
$860$		0			0
$861$		−	100.934i		−	0.117229i
$862$		0			0
$863$	−1017.48			−1.17900			−0.589502	−	0.807767i	$-0.700676\pi$
$863$	−1017.48			−1.17900			−0.589502	+	0.807767i	$0.700676\pi$
$864$		0			0
$865$		−	335.582i		−	0.387956i
$866$		0			0
$867$	64.5721			0.0744776
$868$		0			0
$869$	−1168.27			−1.34439
$870$		0			0
$871$		−	116.368i		−	0.133602i
$872$		0			0
$873$			1278.78i			1.46482i
$874$		0			0
$875$	95.1740			0.108770
$876$		0			0
$877$	1443.83			1.64633			0.823163	−	0.567806i	$-0.192207\pi$
$877$	1443.83			1.64633			0.823163	+	0.567806i	$0.192207\pi$
$878$		0			0
$879$		−	93.6154i		−	0.106502i
$880$		0			0
$881$			894.988i			1.01588i	0.861393	+	0.507939i	$0.169592\pi$
$881$			894.988i			1.01588i	−0.861393	+	0.507939i	$0.830408\pi$
$882$		0			0
$883$	870.115			0.985407			0.492704	−	0.870197i	$-0.336009\pi$
$883$	870.115			0.985407			0.492704	+	0.870197i	$0.336009\pi$
$884$		0			0
$885$		−	42.1359i		−	0.0476112i
$886$		0			0
$887$	529.330			0.596764			0.298382	−	0.954446i	$-0.403553\pi$
$887$	529.330			0.596764			0.298382	+	0.954446i	$0.403553\pi$
$888$		0			0
$889$			93.7539i			0.105460i
$890$		0			0
$891$		−	597.796i		−	0.670927i
$892$		0			0
$893$		−	2045.87i		−	2.29101i
$894$		0			0
$895$		−	462.996i		−	0.517314i
$896$		0			0
$897$	−11.5744	−	12.3373i	−0.0129035	−	0.0137540i
$898$		0			0
$899$	−1799.83			−2.00203
$900$		0			0
$901$	−90.1138			−0.100015
$902$		0			0
$903$	−48.7755			−0.0540149
$904$		0			0
$905$	740.447			0.818173
$906$		0			0
$907$			691.482i			0.762384i	0.924496	+	0.381192i	$0.124486\pi$
$907$			691.482i			0.762384i	−0.924496	+	0.381192i	$0.875514\pi$
$908$		0			0
$909$	−247.242			−0.271994
$910$		0			0
$911$		−	1794.49i		−	1.96980i	−0.173115	−	0.984902i	$-0.555383\pi$
$911$		−	1794.49i		−	1.96980i	0.173115	−	0.984902i	$-0.444617\pi$
$912$		0			0
$913$	−1073.01			−1.17526
$914$		0			0
$915$	−21.8988			−0.0239331
$916$		0			0
$917$		−	30.9809i		−	0.0337851i
$918$		0			0
$919$			428.705i			0.466491i	0.972418	+	0.233245i	$0.0749345\pi$
$919$			428.705i			0.466491i	−0.972418	+	0.233245i	$0.925065\pi$
$920$		0			0
$921$	127.545			0.138485
$922$		0			0
$923$	23.4129			0.0253661
$924$		0			0
$925$			199.689i			0.215880i
$926$		0			0
$927$			1191.52i			1.28535i
$928$		0			0
$929$	1577.01			1.69754			0.848768	−	0.528766i	$-0.177345\pi$
$929$	1577.01			1.69754			0.848768	+	0.528766i	$0.177345\pi$
$930$		0			0
$931$		−	569.431i		−	0.611633i
$932$		0			0
$933$	126.900			0.136013
$934$		0			0
$935$		−	128.071i		−	0.136975i
$936$		0			0
$937$		−	141.317i		−	0.150818i	−0.997153	−	0.0754091i	$-0.975974\pi$
$937$		−	141.317i		−	0.150818i	0.997153	−	0.0754091i	$-0.0240263\pi$
$938$		0			0
$939$			164.148i			0.174812i
$940$		0			0
$941$		−	1506.21i		−	1.60065i	−0.599566	−	0.800325i	$-0.704660\pi$
$941$		−	1506.21i		−	1.60065i	0.599566	−	0.800325i	$-0.295340\pi$
$942$		0			0
$943$	−669.923	−	714.078i	−0.710417	−	0.757240i
$944$		0			0
$945$	95.0179			0.100548
$946$		0			0
$947$	−1689.84			−1.78441			−0.892205	−	0.451632i	$-0.850842\pi$
$947$	−1689.84			−1.78441			−0.892205	+	0.451632i	$0.850842\pi$
$948$		0			0
$949$	230.842			0.243248
$950$		0			0
$951$	−107.580			−0.113123
$952$		0			0
$953$		−	990.482i		−	1.03933i	−0.854370	−	0.519665i	$-0.826057\pi$
$953$		−	990.482i		−	1.03933i	0.854370	−	0.519665i	$-0.173943\pi$
$954$		0			0
$955$	−386.279			−0.404481
$956$		0			0
$957$			67.2049i			0.0702245i
$958$		0			0
$959$	−82.5541			−0.0860836
$960$		0			0
$961$	2232.05			2.32264
$962$		0			0
$963$		−	448.879i		−	0.466126i
$964$		0			0
$965$			305.112i			0.316178i
$966$		0			0
$967$	−1528.09			−1.58024			−0.790118	−	0.612954i	$-0.789981\pi$
$967$	−1528.09			−1.58024			−0.790118	+	0.612954i	$0.789981\pi$
$968$		0			0
$969$	51.1024			0.0527372
$970$		0			0
$971$		−	616.239i		−	0.634643i	−0.948318	−	0.317322i	$-0.897217\pi$
$971$		−	616.239i		−	0.634643i	0.948318	−	0.317322i	$-0.102783\pi$
$972$		0			0
$973$		−	68.5679i		−	0.0704706i
$974$		0			0
$975$	3.67755			0.00377185
$976$		0			0
$977$			1393.79i			1.42660i	0.700859	+	0.713299i	$0.252800\pi$
$977$			1393.79i			1.42660i	−0.700859	+	0.713299i	$0.747200\pi$
$978$		0			0
$979$	482.752			0.493107
$980$		0			0
$981$			1149.37i			1.17164i
$982$		0			0
$983$		−	1033.22i		−	1.05109i	−0.850767	−	0.525543i	$-0.823862\pi$
$983$		−	1033.22i		−	1.05109i	0.850767	−	0.525543i	$-0.176138\pi$
$984$		0			0
$985$		−	606.176i		−	0.615407i
$986$		0			0
$987$		−	199.883i		−	0.202516i
$988$		0			0
$989$	−345.071	+	323.734i	−0.348909	+	0.327335i
$990$		0			0
$991$	1122.80			1.13300			0.566501	−	0.824061i	$-0.308297\pi$
$991$	1122.80			1.13300			0.566501	+	0.824061i	$0.308297\pi$
$992$		0			0
$993$	−61.4732			−0.0619066
$994$		0			0
$995$	−575.323			−0.578214
$996$		0			0
$997$	−718.650			−0.720812			−0.360406	−	0.932795i	$-0.617362\pi$
$997$	−718.650			−0.720812			−0.360406	+	0.932795i	$0.617362\pi$
$998$		0			0
$999$			199.361i			0.199561i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.10		16
4.3	odd	2		230.3.d.a.91.4	yes	16
12.11	even	2		2070.3.c.a.91.9		16
20.3	even	4		1150.3.c.c.1149.26		32
20.7	even	4		1150.3.c.c.1149.7		32
20.19	odd	2		1150.3.d.b.551.14		16
23.22	odd	2	inner	1840.3.k.d.321.9		16
92.91	even	2		230.3.d.a.91.3	✓	16
276.275	odd	2		2070.3.c.a.91.16		16
460.183	odd	4		1150.3.c.c.1149.8		32
460.367	odd	4		1150.3.c.c.1149.25		32
460.459	even	2		1150.3.d.b.551.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.3	✓	16	92.91	even	2
230.3.d.a.91.4	yes	16	4.3	odd	2
1150.3.c.c.1149.7		32	20.7	even	4
1150.3.c.c.1149.8		32	460.183	odd	4
1150.3.c.c.1149.25		32	460.367	odd	4
1150.3.c.c.1149.26		32	20.3	even	4
1150.3.d.b.551.13		16	460.459	even	2
1150.3.d.b.551.14		16	20.19	odd	2
1840.3.k.d.321.9		16	23.22	odd	2	inner
1840.3.k.d.321.10		16	1.1	even	1	trivial
2070.3.c.a.91.9		16	12.11	even	2
2070.3.c.a.91.16		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)

\(f(q)\)	\(=\)	\(q+0.278523 q^{3} +2.23607i q^{5} +8.51262i q^{7} -8.92243 q^{9} +O(q^{10})\)	\(q+0.278523 q^{3} +2.23607i q^{5} +8.51262i q^{7} -8.92243 q^{9} -7.57553i q^{11} -2.64076 q^{13} +0.622795i q^{15} -7.56057i q^{17} +24.2676i q^{19} +2.37096i q^{21} +(15.7366 + 16.7738i) q^{23} -5.00000 q^{25} -4.99180 q^{27} -31.8513 q^{29} +56.5071 q^{31} -2.10995i q^{33} -19.0348 q^{35} -39.9378i q^{37} -0.735511 q^{39} -42.5710 q^{41} +20.5721i q^{43} -19.9511i q^{45} -84.3049 q^{47} -23.4647 q^{49} -2.10579i q^{51} -11.9189i q^{53} +16.9394 q^{55} +6.75907i q^{57} -67.6561 q^{59} +35.1621i q^{61} -75.9532i q^{63} -5.90492i q^{65} +44.0660i q^{67} +(4.38299 + 4.67188i) q^{69} -8.86597 q^{71} -87.4150 q^{73} -1.39261 q^{75} +64.4876 q^{77} -154.217i q^{79} +78.9115 q^{81} -141.642i q^{83} +16.9059 q^{85} -8.87131 q^{87} +63.7252i q^{89} -22.4798i q^{91} +15.7385 q^{93} -54.2639 q^{95} -143.322i q^{97} +67.5921i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 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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