LMFDB - Embedded newform 1728.3.g.k.703.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.3
Root			$-1.27597 + 0.609843i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.k.703.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76102 q^{5} -12.0363i q^{7} +O(q^{10})$	$q-1.76102 q^{5} -12.0363i q^{7} +13.3544i q^{11} -8.15828 q^{13} -4.15828 q^{17} +8.87027i q^{19} +33.5947i q^{23} -21.8988 q^{25} +36.4649 q^{29} -0.590023i q^{31} +21.1962i q^{35} +69.8156 q^{37} +57.5661 q^{41} +23.5847i q^{43} -24.4804i q^{47} -95.8727 q^{49} -29.9292 q^{53} -23.5174i q^{55} -55.0130i q^{59} +11.1645 q^{61} +14.3669 q^{65} +99.6989i q^{67} -71.2023i q^{71} +36.0508 q^{73} +160.738 q^{77} -108.284i q^{79} -100.556i q^{83} +7.32280 q^{85} +159.616 q^{89} +98.1956i q^{91} -15.6207i q^{95} -24.0242 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 16 * q^13 + 48 * q^17 + 48 * q^25 + 32 * q^29 + 96 * q^37 - 128 * q^41 - 168 * q^53 - 32 * q^61 - 112 * q^65 + 24 * q^73 + 440 * q^77 - 144 * q^85 + 624 * q^89 - 136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	−1.76102	−0.352204	−0.176102	−	0.984372i	$-0.556349\pi$
$5$	−1.76102	−0.352204	−0.176102	+	0.984372i	$0.556349\pi$
$6$	0	0
$7$	− 12.0363i	− 1.71947i	−0.510738	−	0.859736i	$-0.670628\pi$
$7$	− 12.0363i	− 1.71947i	0.510738	−	0.859736i	$-0.329372\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.3544i	1.21404i	0.794687	+	0.607020i	$0.207635\pi$
$11$	13.3544i	1.21404i	−0.794687	+	0.607020i	$0.792365\pi$
$12$	0	0
$13$	−8.15828	−0.627560	−0.313780	−	0.949496i	$-0.601595\pi$
$13$	−8.15828	−0.627560	−0.313780	+	0.949496i	$0.601595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.15828	−0.244605	−0.122302	−	0.992493i	$-0.539028\pi$
$17$	−4.15828	−0.244605	−0.122302	+	0.992493i	$0.539028\pi$
$18$	0	0
$19$	8.87027i	0.466856i	0.972374	+	0.233428i	$0.0749944\pi$
$19$	8.87027i	0.466856i	−0.972374	+	0.233428i	$0.925006\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	33.5947i	1.46064i	0.683107	+	0.730319i	$0.260628\pi$
$23$	33.5947i	1.46064i	−0.683107	+	0.730319i	$0.739372\pi$
$24$	0	0
$25$	−21.8988	−0.875953
$26$	0	0
$27$	0	0
$28$	0	0
$29$	36.4649	1.25741	0.628706	−	0.777643i	$-0.283585\pi$
$29$	36.4649	1.25741	0.628706	+	0.777643i	$0.283585\pi$
$30$	0	0
$31$	− 0.590023i	− 0.0190330i	−0.999955	−	0.00951650i	$-0.996971\pi$
$31$	− 0.590023i	− 0.0190330i	0.999955	−	0.00951650i	$-0.00302924\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	21.1962i	0.605604i
$36$	0	0
$37$	69.8156	1.88691	0.943455	−	0.331502i	$-0.107555\pi$
$37$	69.8156	1.88691	0.943455	+	0.331502i	$0.107555\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	57.5661	1.40405	0.702025	−	0.712152i	$-0.252279\pi$
$41$	57.5661	1.40405	0.702025	+	0.712152i	$0.252279\pi$
$42$	0	0
$43$	23.5847i	0.548482i	0.961661	+	0.274241i	$0.0884267\pi$
$43$	23.5847i	0.548482i	−0.961661	+	0.274241i	$0.911573\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 24.4804i	− 0.520861i	−0.965493	−	0.260430i	$-0.916136\pi$
$47$	− 24.4804i	− 0.520861i	0.965493	−	0.260430i	$-0.0838644\pi$
$48$	0	0
$49$	−95.8727	−1.95659
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−29.9292	−0.564702	−0.282351	−	0.959311i	$-0.591114\pi$
$53$	−29.9292	−0.564702	−0.282351	+	0.959311i	$0.591114\pi$
$54$	0	0
$55$	− 23.5174i	− 0.427589i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 55.0130i	− 0.932424i	−0.884673	−	0.466212i	$-0.845618\pi$
$59$	− 55.0130i	− 0.932424i	0.884673	−	0.466212i	$-0.154382\pi$
$60$	0	0
$61$	11.1645	0.183025	0.0915125	−	0.995804i	$-0.470830\pi$
$61$	11.1645	0.183025	0.0915125	+	0.995804i	$0.470830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.3669	0.221029
$66$	0	0
$67$	99.6989i	1.48804i	0.668155	+	0.744022i	$0.267084\pi$
$67$	99.6989i	1.48804i	−0.668155	+	0.744022i	$0.732916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 71.2023i	− 1.00285i	−0.865201	−	0.501425i	$-0.832809\pi$
$71$	− 71.2023i	− 1.00285i	0.865201	−	0.501425i	$-0.167191\pi$
$72$	0	0
$73$	36.0508	0.493847	0.246924	−	0.969035i	$-0.420580\pi$
$73$	36.0508	0.493847	0.246924	+	0.969035i	$0.420580\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	160.738	2.08751
$78$	0	0
$79$	− 108.284i	− 1.37068i	−0.728223	−	0.685340i	$-0.759654\pi$
$79$	− 108.284i	− 1.37068i	0.728223	−	0.685340i	$-0.240346\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 100.556i	− 1.21152i	−0.795648	−	0.605760i	$-0.792869\pi$
$83$	− 100.556i	− 1.21152i	0.795648	−	0.605760i	$-0.207131\pi$
$84$	0	0
$85$	7.32280	0.0861506
$86$	0	0
$87$	0	0
$88$	0	0
$89$	159.616	1.79344	0.896721	−	0.442596i	$-0.145942\pi$
$89$	159.616	1.79344	0.896721	+	0.442596i	$0.145942\pi$
$90$	0	0
$91$	98.1956i	1.07907i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 15.6207i	− 0.164428i
$96$	0	0
$97$	−24.0242	−0.247673	−0.123836	−	0.992303i	$-0.539520\pi$
$97$	−24.0242	−0.247673	−0.123836	+	0.992303i	$0.539520\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	22.5078	0.222850	0.111425	−	0.993773i	$-0.464459\pi$
$101$	22.5078	0.222850	0.111425	+	0.993773i	$0.464459\pi$
$102$	0	0
$103$	− 135.290i	− 1.31350i	−0.754109	−	0.656750i	$-0.771931\pi$
$103$	− 135.290i	− 1.31350i	0.754109	−	0.656750i	$-0.228069\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	29.5388i	0.276063i	0.990428	+	0.138032i	$0.0440776\pi$
$107$	29.5388i	0.276063i	−0.990428	+	0.138032i	$0.955922\pi$
$108$	0	0
$109$	130.246	1.19492	0.597460	−	0.801898i	$-0.296176\pi$
$109$	130.246	1.19492	0.597460	+	0.801898i	$0.296176\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−134.421	−1.18957	−0.594785	−	0.803885i	$-0.702763\pi$
$113$	−134.421	−1.18957	−0.594785	+	0.803885i	$0.702763\pi$
$114$	0	0
$115$	− 59.1608i	− 0.514442i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	50.0503i	0.420591i
$120$	0	0
$121$	−57.3408	−0.473891
$122$	0	0
$123$	0	0
$124$	0	0
$125$	82.5896	0.660717
$126$	0	0
$127$	40.3479i	0.317700i	0.987303	+	0.158850i	$0.0507786\pi$
$127$	40.3479i	0.317700i	−0.987303	+	0.158850i	$0.949221\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	136.966i	1.04555i	0.852472	+	0.522773i	$0.175102\pi$
$131$	136.966i	1.04555i	−0.852472	+	0.522773i	$0.824898\pi$
$132$	0	0
$133$	106.765	0.802747
$134$	0	0
$135$	0	0
$136$	0	0
$137$	245.969	1.79539	0.897697	−	0.440614i	$-0.145239\pi$
$137$	245.969	1.79539	0.897697	+	0.440614i	$0.145239\pi$
$138$	0	0
$139$	− 157.526i	− 1.13328i	−0.823966	−	0.566639i	$-0.808243\pi$
$139$	− 157.526i	− 1.13328i	0.823966	−	0.566639i	$-0.191757\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 108.949i	− 0.761882i
$144$	0	0
$145$	−64.2154	−0.442865
$146$	0	0
$147$	0	0
$148$	0	0
$149$	184.731	1.23981	0.619904	−	0.784678i	$-0.287172\pi$
$149$	184.731	1.23981	0.619904	+	0.784678i	$0.287172\pi$
$150$	0	0
$151$	9.64832i	0.0638961i	0.999490	+	0.0319481i	$0.0101711\pi$
$151$	9.64832i	0.0638961i	−0.999490	+	0.0319481i	$0.989829\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.03904i	0.00670349i
$156$	0	0
$157$	36.3166	0.231316	0.115658	−	0.993289i	$-0.463102\pi$
$157$	36.3166	0.231316	0.115658	+	0.993289i	$0.463102\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	404.356	2.51153
$162$	0	0
$163$	126.095i	0.773588i	0.922166	+	0.386794i	$0.126418\pi$
$163$	126.095i	0.773588i	−0.922166	+	0.386794i	$0.873582\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	302.797i	1.81316i	0.422039	+	0.906578i	$0.361315\pi$
$167$	302.797i	1.81316i	−0.422039	+	0.906578i	$0.638685\pi$
$168$	0	0
$169$	−102.443	−0.606169
$170$	0	0
$171$	0	0
$172$	0	0
$173$	204.895	1.18436	0.592181	−	0.805805i	$-0.298267\pi$
$173$	204.895	1.18436	0.592181	+	0.805805i	$0.298267\pi$
$174$	0	0
$175$	263.581i	1.50618i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	320.815i	1.79226i	0.443789	+	0.896131i	$0.353634\pi$
$179$	320.815i	1.79226i	−0.443789	+	0.896131i	$0.646366\pi$
$180$	0	0
$181$	−285.364	−1.57660	−0.788299	−	0.615292i	$-0.789038\pi$
$181$	−285.364	−1.57660	−0.788299	+	0.615292i	$0.789038\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−122.947	−0.664576
$186$	0	0
$187$	− 55.5314i	− 0.296960i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	132.634i	0.694417i	0.937788	+	0.347208i	$0.112870\pi$
$191$	132.634i	0.694417i	−0.937788	+	0.347208i	$0.887130\pi$
$192$	0	0
$193$	366.794	1.90049	0.950245	−	0.311504i	$-0.100833\pi$
$193$	366.794	1.90049	0.950245	+	0.311504i	$0.100833\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.3110	−0.0726445	−0.0363223	−	0.999340i	$-0.511564\pi$
$197$	−14.3110	−0.0726445	−0.0363223	+	0.999340i	$0.511564\pi$
$198$	0	0
$199$	214.542i	1.07810i	0.842274	+	0.539049i	$0.181216\pi$
$199$	214.542i	1.07810i	−0.842274	+	0.539049i	$0.818784\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 438.903i	− 2.16208i
$204$	0	0
$205$	−101.375	−0.494512
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−118.457	−0.566782
$210$	0	0
$211$	223.579i	1.05962i	0.848118	+	0.529808i	$0.177736\pi$
$211$	223.579i	1.05962i	−0.848118	+	0.529808i	$0.822264\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 41.5331i	− 0.193177i
$216$	0	0
$217$	−7.10170	−0.0327267
$218$	0	0
$219$	0	0
$220$	0	0
$221$	33.9244	0.153504
$222$	0	0
$223$	− 287.564i	− 1.28952i	−0.764384	−	0.644761i	$-0.776957\pi$
$223$	− 287.564i	− 1.28952i	0.764384	−	0.644761i	$-0.223043\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	167.933i	0.739793i	0.929073	+	0.369896i	$0.120607\pi$
$227$	167.933i	0.739793i	−0.929073	+	0.369896i	$0.879393\pi$
$228$	0	0
$229$	−28.4693	−0.124320	−0.0621601	−	0.998066i	$-0.519799\pi$
$229$	−28.4693	−0.124320	−0.0621601	+	0.998066i	$0.519799\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	158.334	0.679545	0.339773	−	0.940508i	$-0.389650\pi$
$233$	158.334	0.679545	0.339773	+	0.940508i	$0.389650\pi$
$234$	0	0
$235$	43.1105i	0.183449i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 29.0236i	− 0.121438i	−0.998155	−	0.0607188i	$-0.980661\pi$
$239$	− 29.0236i	− 0.121438i	0.998155	−	0.0607188i	$-0.0193393\pi$
$240$	0	0
$241$	301.579	1.25137	0.625683	−	0.780078i	$-0.284820\pi$
$241$	301.579	1.25137	0.625683	+	0.780078i	$0.284820\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	168.834	0.689117
$246$	0	0
$247$	− 72.3661i	− 0.292980i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	111.785i	0.445359i	0.974892	+	0.222680i	$0.0714804\pi$
$251$	111.785i	0.445359i	−0.974892	+	0.222680i	$0.928520\pi$
$252$	0	0
$253$	−448.637	−1.77327
$254$	0	0
$255$	0	0
$256$	0	0
$257$	462.857	1.80100	0.900500	−	0.434856i	$-0.143201\pi$
$257$	462.857	1.80100	0.900500	+	0.434856i	$0.143201\pi$
$258$	0	0
$259$	− 840.323i	− 3.24449i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 216.912i	− 0.824759i	−0.911012	−	0.412380i	$-0.864698\pi$
$263$	− 216.912i	− 0.824759i	0.911012	−	0.412380i	$-0.135302\pi$
$264$	0	0
$265$	52.7059	0.198890
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−431.963	−1.60581	−0.802906	−	0.596106i	$-0.796714\pi$
$269$	−431.963	−1.60581	−0.802906	+	0.596106i	$0.796714\pi$
$270$	0	0
$271$	278.164i	1.02644i	0.858258	+	0.513218i	$0.171547\pi$
$271$	278.164i	1.02644i	−0.858258	+	0.513218i	$0.828453\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 292.446i	− 1.06344i
$276$	0	0
$277$	−271.579	−0.980430	−0.490215	−	0.871602i	$-0.663082\pi$
$277$	−271.579	−0.980430	−0.490215	+	0.871602i	$0.663082\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−343.222	−1.22143	−0.610715	−	0.791850i	$-0.709118\pi$
$281$	−343.222	−1.22143	−0.610715	+	0.791850i	$0.709118\pi$
$282$	0	0
$283$	− 242.067i	− 0.855361i	−0.903930	−	0.427680i	$-0.859331\pi$
$283$	− 242.067i	− 0.855361i	0.903930	−	0.427680i	$-0.140669\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 692.883i	− 2.41423i
$288$	0	0
$289$	−271.709	−0.940169
$290$	0	0
$291$	0	0
$292$	0	0
$293$	471.585	1.60950	0.804752	−	0.593611i	$-0.202298\pi$
$293$	471.585	1.60950	0.804752	+	0.593611i	$0.202298\pi$
$294$	0	0
$295$	96.8789i	0.328403i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 274.075i	− 0.916637i
$300$	0	0
$301$	283.873	0.943100
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.6609	−0.0644620
$306$	0	0
$307$	− 115.907i	− 0.377549i	−0.982021	−	0.188774i	$-0.939549\pi$
$307$	− 115.907i	− 0.377549i	0.982021	−	0.188774i	$-0.0604515\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	394.948i	1.26993i	0.772541	+	0.634964i	$0.218985\pi$
$311$	394.948i	1.26993i	−0.772541	+	0.634964i	$0.781015\pi$
$312$	0	0
$313$	298.099	0.952394	0.476197	−	0.879339i	$-0.342015\pi$
$313$	298.099	0.952394	0.476197	+	0.879339i	$0.342015\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.5754	0.0491337	0.0245669	−	0.999698i	$-0.492179\pi$
$317$	15.5754	0.0491337	0.0245669	+	0.999698i	$0.492179\pi$
$318$	0	0
$319$	486.968i	1.52655i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 36.8850i	− 0.114195i
$324$	0	0
$325$	178.657	0.549713
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−294.654	−0.895606
$330$	0	0
$331$	− 132.849i	− 0.401355i	−0.979657	−	0.200678i	$-0.935686\pi$
$331$	− 132.849i	− 0.401355i	0.979657	−	0.200678i	$-0.0643144\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 175.572i	− 0.524094i
$336$	0	0
$337$	64.2930	0.190781	0.0953903	−	0.995440i	$-0.469590\pi$
$337$	64.2930	0.190781	0.0953903	+	0.995440i	$0.469590\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.87942	0.0231068
$342$	0	0
$343$	564.175i	1.64482i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 184.163i	− 0.530728i	−0.964148	−	0.265364i	$-0.914508\pi$
$347$	− 184.163i	− 0.530728i	0.964148	−	0.265364i	$-0.0854922\pi$
$348$	0	0
$349$	107.326	0.307525	0.153763	−	0.988108i	$-0.450861\pi$
$349$	107.326	0.307525	0.153763	+	0.988108i	$0.450861\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	444.637	1.25960	0.629798	−	0.776759i	$-0.283138\pi$
$353$	444.637	1.25960	0.629798	+	0.776759i	$0.283138\pi$
$354$	0	0
$355$	125.389i	0.353207i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	275.528i	0.767488i	0.923440	+	0.383744i	$0.125365\pi$
$359$	275.528i	0.767488i	−0.923440	+	0.383744i	$0.874635\pi$
$360$	0	0
$361$	282.318	0.782045
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−63.4862	−0.173935
$366$	0	0
$367$	− 594.925i	− 1.62105i	−0.585705	−	0.810525i	$-0.699182\pi$
$367$	− 594.925i	− 1.62105i	0.585705	−	0.810525i	$-0.300818\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	360.237i	0.970990i
$372$	0	0
$373$	−478.902	−1.28392	−0.641960	−	0.766738i	$-0.721878\pi$
$373$	−478.902	−1.28392	−0.641960	+	0.766738i	$0.721878\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−297.491	−0.789101
$378$	0	0
$379$	425.301i	1.12217i	0.827760	+	0.561083i	$0.189615\pi$
$379$	425.301i	1.12217i	−0.827760	+	0.561083i	$0.810385\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 386.549i	− 1.00927i	−0.863334	−	0.504634i	$-0.831628\pi$
$383$	− 386.549i	− 1.00927i	0.863334	−	0.504634i	$-0.168372\pi$
$384$	0	0
$385$	−283.063	−0.735227
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−186.277	−0.478861	−0.239430	−	0.970914i	$-0.576961\pi$
$389$	−186.277	−0.478861	−0.239430	+	0.970914i	$0.576961\pi$
$390$	0	0
$391$	− 139.696i	− 0.357279i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	190.689i	0.482758i
$396$	0	0
$397$	−221.030	−0.556750	−0.278375	−	0.960472i	$-0.589796\pi$
$397$	−221.030	−0.556750	−0.278375	+	0.960472i	$0.589796\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−712.734	−1.77739	−0.888695	−	0.458498i	$-0.848388\pi$
$401$	−712.734	−1.77739	−0.888695	+	0.458498i	$0.848388\pi$
$402$	0	0
$403$	4.81357i	0.0119443i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	932.348i	2.29078i
$408$	0	0
$409$	198.746	0.485933	0.242966	−	0.970035i	$-0.421880\pi$
$409$	198.746	0.485933	0.242966	+	0.970035i	$0.421880\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−662.154	−1.60328
$414$	0	0
$415$	177.081i	0.426701i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	573.831i	1.36953i	0.728766	+	0.684763i	$0.240094\pi$
$419$	573.831i	1.36953i	−0.728766	+	0.684763i	$0.759906\pi$
$420$	0	0
$421$	83.4289	0.198168	0.0990842	−	0.995079i	$-0.468409\pi$
$421$	83.4289	0.198168	0.0990842	+	0.995079i	$0.468409\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	91.0614	0.214262
$426$	0	0
$427$	− 134.380i	− 0.314706i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	328.762i	0.762789i	0.924412	+	0.381394i	$0.124556\pi$
$431$	328.762i	0.762789i	−0.924412	+	0.381394i	$0.875444\pi$
$432$	0	0
$433$	445.063	1.02786	0.513930	−	0.857832i	$-0.328189\pi$
$433$	445.063	1.02786	0.513930	+	0.857832i	$0.328189\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−297.994	−0.681908
$438$	0	0
$439$	− 459.120i	− 1.04583i	−0.852385	−	0.522915i	$-0.824844\pi$
$439$	− 459.120i	− 1.04583i	0.852385	−	0.522915i	$-0.175156\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 463.025i	− 1.04520i	−0.852577	−	0.522602i	$-0.824961\pi$
$443$	− 463.025i	− 1.04520i	0.852577	−	0.522602i	$-0.175039\pi$
$444$	0	0
$445$	−281.087	−0.631657
$446$	0	0
$447$	0	0
$448$	0	0
$449$	271.385	0.604420	0.302210	−	0.953241i	$-0.402276\pi$
$449$	271.385	0.604420	0.302210	+	0.953241i	$0.402276\pi$
$450$	0	0
$451$	768.762i	1.70457i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 172.924i	− 0.380053i
$456$	0	0
$457$	92.0749	0.201477	0.100738	−	0.994913i	$-0.467879\pi$
$457$	92.0749	0.201477	0.100738	+	0.994913i	$0.467879\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−551.486	−1.19628	−0.598141	−	0.801391i	$-0.704094\pi$
$461$	−551.486	−1.19628	−0.598141	+	0.801391i	$0.704094\pi$
$462$	0	0
$463$	− 78.7518i	− 0.170090i	−0.996377	−	0.0850452i	$-0.972897\pi$
$463$	− 78.7518i	− 0.170090i	0.996377	−	0.0850452i	$-0.0271035\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	637.095i	1.36423i	0.731245	+	0.682114i	$0.238939\pi$
$467$	637.095i	1.36423i	−0.731245	+	0.682114i	$0.761061\pi$
$468$	0	0
$469$	1200.01	2.55865
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−314.961	−0.665879
$474$	0	0
$475$	− 194.248i	− 0.408944i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 201.640i	− 0.420961i	−0.977598	−	0.210480i	$-0.932497\pi$
$479$	− 201.640i	− 0.420961i	0.977598	−	0.210480i	$-0.0675028\pi$
$480$	0	0
$481$	−569.575	−1.18415
$482$	0	0
$483$	0	0
$484$	0	0
$485$	42.3071	0.0872312
$486$	0	0
$487$	164.086i	0.336932i	0.985707	+	0.168466i	$0.0538814\pi$
$487$	164.086i	0.336932i	−0.985707	+	0.168466i	$0.946119\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.3778i	0.0618692i	0.999521	+	0.0309346i	$0.00984836\pi$
$491$	30.3778i	0.0618692i	−0.999521	+	0.0309346i	$0.990152\pi$
$492$	0	0
$493$	−151.631	−0.307569
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−857.013	−1.72437
$498$	0	0
$499$	37.8381i	0.0758279i	0.999281	+	0.0379140i	$0.0120713\pi$
$499$	37.8381i	0.0758279i	−0.999281	+	0.0379140i	$0.987929\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	252.844i	0.502672i	0.967900	+	0.251336i	$0.0808699\pi$
$503$	252.844i	0.502672i	−0.967900	+	0.251336i	$0.919130\pi$
$504$	0	0
$505$	−39.6366	−0.0784884
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.9887	0.0765986	0.0382993	−	0.999266i	$-0.487806\pi$
$509$	38.9887	0.0765986	0.0382993	+	0.999266i	$0.487806\pi$
$510$	0	0
$511$	− 433.919i	− 0.849157i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	238.249i	0.462619i
$516$	0	0
$517$	326.922	0.632345
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−254.943	−0.489334	−0.244667	−	0.969607i	$-0.578679\pi$
$521$	−254.943	−0.489334	−0.244667	+	0.969607i	$0.578679\pi$
$522$	0	0
$523$	193.808i	0.370571i	0.982685	+	0.185285i	$0.0593209\pi$
$523$	193.808i	0.370571i	−0.982685	+	0.185285i	$0.940679\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.45348i	0.00465556i
$528$	0	0
$529$	−599.601	−1.13346
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−469.640	−0.881126
$534$	0	0
$535$	− 52.0183i	− 0.0972305i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1280.33i	− 2.37537i
$540$	0	0
$541$	519.448	0.960162	0.480081	−	0.877224i	$-0.340607\pi$
$541$	519.448	0.960162	0.480081	+	0.877224i	$0.340607\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−229.366	−0.420855
$546$	0	0
$547$	− 859.658i	− 1.57159i	−0.618490	−	0.785793i	$-0.712255\pi$
$547$	− 859.658i	− 1.57159i	0.618490	−	0.785793i	$-0.287745\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	323.454i	0.587030i
$552$	0	0
$553$	−1303.34	−2.35685
$554$	0	0
$555$	0	0
$556$	0	0
$557$	603.829	1.08407	0.542037	−	0.840354i	$-0.317653\pi$
$557$	603.829	1.08407	0.542037	+	0.840354i	$0.317653\pi$
$558$	0	0
$559$	− 192.411i	− 0.344206i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	102.473i	0.182013i	0.995850	+	0.0910064i	$0.0290084\pi$
$563$	102.473i	0.182013i	−0.995850	+	0.0910064i	$0.970992\pi$
$564$	0	0
$565$	236.718	0.418971
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−237.219	−0.416905	−0.208453	−	0.978032i	$-0.566843\pi$
$569$	−237.219	−0.416905	−0.208453	+	0.978032i	$0.566843\pi$
$570$	0	0
$571$	881.129i	1.54313i	0.636149	+	0.771566i	$0.280526\pi$
$571$	881.129i	1.54313i	−0.636149	+	0.771566i	$0.719474\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 735.683i	− 1.27945i
$576$	0	0
$577$	−197.356	−0.342038	−0.171019	−	0.985268i	$-0.554706\pi$
$577$	−197.356	−0.342038	−0.171019	+	0.985268i	$0.554706\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1210.32	−2.08317
$582$	0	0
$583$	− 399.687i	− 0.685570i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 181.507i	− 0.309210i	−0.987976	−	0.154605i	$-0.950589\pi$
$587$	− 181.507i	− 0.309210i	0.987976	−	0.154605i	$-0.0494106\pi$
$588$	0	0
$589$	5.23366	0.00888568
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−828.164	−1.39657	−0.698283	−	0.715822i	$-0.746052\pi$
$593$	−828.164	−1.39657	−0.698283	+	0.715822i	$0.746052\pi$
$594$	0	0
$595$	− 88.1395i	− 0.148134i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	97.1450i	0.162179i	0.996707	+	0.0810893i	$0.0258399\pi$
$599$	97.1450i	0.162179i	−0.996707	+	0.0810893i	$0.974160\pi$
$600$	0	0
$601$	−729.916	−1.21450	−0.607251	−	0.794510i	$-0.707728\pi$
$601$	−729.916	−1.21450	−0.607251	+	0.794510i	$0.707728\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	100.978	0.166906
$606$	0	0
$607$	− 711.346i	− 1.17191i	−0.810345	−	0.585953i	$-0.800721\pi$
$607$	− 711.346i	− 1.17191i	0.810345	−	0.585953i	$-0.199279\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	199.718i	0.326871i
$612$	0	0
$613$	−317.107	−0.517304	−0.258652	−	0.965971i	$-0.583278\pi$
$613$	−317.107	−0.517304	−0.258652	+	0.965971i	$0.583278\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	682.859	1.10674	0.553371	−	0.832935i	$-0.313341\pi$
$617$	682.859	1.10674	0.553371	+	0.832935i	$0.313341\pi$
$618$	0	0
$619$	− 87.7695i	− 0.141792i	−0.997484	−	0.0708962i	$-0.977414\pi$
$619$	− 87.7695i	− 0.141792i	0.997484	−	0.0708962i	$-0.0225859\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1921.19i	− 3.08378i
$624$	0	0
$625$	402.029	0.643246
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−290.313	−0.461547
$630$	0	0
$631$	− 150.666i	− 0.238773i	−0.992848	−	0.119386i	$-0.961907\pi$
$631$	− 150.666i	− 0.238773i	0.992848	−	0.119386i	$-0.0380927\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 71.0534i	− 0.111895i
$636$	0	0
$637$	782.156	1.22788
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−447.618	−0.698312	−0.349156	−	0.937065i	$-0.613532\pi$
$641$	−447.618	−0.698312	−0.349156	+	0.937065i	$0.613532\pi$
$642$	0	0
$643$	400.436i	0.622762i	0.950285	+	0.311381i	$0.100791\pi$
$643$	400.436i	0.622762i	−0.950285	+	0.311381i	$0.899209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	253.680i	0.392086i	0.980595	+	0.196043i	$0.0628092\pi$
$647$	253.680i	0.392086i	−0.980595	+	0.196043i	$0.937191\pi$
$648$	0	0
$649$	734.668	1.13200
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−719.542	−1.10190	−0.550951	−	0.834537i	$-0.685735\pi$
$653$	−719.542	−1.10190	−0.550951	+	0.834537i	$0.685735\pi$
$654$	0	0
$655$	− 241.200i	− 0.368245i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	475.052i	0.720868i	0.932785	+	0.360434i	$0.117371\pi$
$659$	475.052i	0.720868i	−0.932785	+	0.360434i	$0.882629\pi$
$660$	0	0
$661$	−132.074	−0.199809	−0.0999046	−	0.994997i	$-0.531854\pi$
$661$	−132.074	−0.199809	−0.0999046	+	0.994997i	$0.531854\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−188.016	−0.282730
$666$	0	0
$667$	1225.03i	1.83662i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	149.096i	0.222199i
$672$	0	0
$673$	919.525	1.36631	0.683154	−	0.730275i	$-0.260608\pi$
$673$	919.525	1.36631	0.683154	+	0.730275i	$0.260608\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−200.276	−0.295829	−0.147914	−	0.989000i	$-0.547256\pi$
$677$	−200.276	−0.295829	−0.147914	+	0.989000i	$0.547256\pi$
$678$	0	0
$679$	289.163i	0.425866i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1049.18i	− 1.53613i	−0.640373	−	0.768064i	$-0.721220\pi$
$683$	− 1049.18i	− 1.53613i	0.640373	−	0.768064i	$-0.278780\pi$
$684$	0	0
$685$	−433.156	−0.632344
$686$	0	0
$687$	0	0
$688$	0	0
$689$	244.171	0.354384
$690$	0	0
$691$	1039.19i	1.50389i	0.659225	+	0.751946i	$0.270884\pi$
$691$	1039.19i	1.50389i	−0.659225	+	0.751946i	$0.729116\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	277.405i	0.399145i
$696$	0	0
$697$	−239.376	−0.343437
$698$	0	0
$699$	0	0
$700$	0	0
$701$	903.635	1.28907	0.644533	−	0.764576i	$-0.277052\pi$
$701$	903.635	1.28907	0.644533	+	0.764576i	$0.277052\pi$
$702$	0	0
$703$	619.284i	0.880915i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 270.911i	− 0.383184i
$708$	0	0
$709$	−990.096	−1.39647	−0.698234	−	0.715869i	$-0.746031\pi$
$709$	−990.096	−1.39647	−0.698234	+	0.715869i	$0.746031\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.8216	0.0278003
$714$	0	0
$715$	191.861i	0.268338i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	775.482i	1.07856i	0.842128	+	0.539278i	$0.181303\pi$
$719$	775.482i	1.07856i	−0.842128	+	0.539278i	$0.818697\pi$
$720$	0	0
$721$	−1628.40	−2.25853
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−798.539	−1.10143
$726$	0	0
$727$	− 862.588i	− 1.18650i	−0.805017	−	0.593252i	$-0.797844\pi$
$727$	− 862.588i	− 1.18650i	0.805017	−	0.593252i	$-0.202156\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 98.0719i	− 0.134161i
$732$	0	0
$733$	1013.12	1.38215	0.691075	−	0.722783i	$-0.257137\pi$
$733$	1013.12	1.38215	0.691075	+	0.722783i	$0.257137\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1331.42	−1.80654
$738$	0	0
$739$	1205.31i	1.63100i	0.578755	+	0.815502i	$0.303539\pi$
$739$	1205.31i	1.63100i	−0.578755	+	0.815502i	$0.696461\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 762.230i	− 1.02588i	−0.858424	−	0.512941i	$-0.828556\pi$
$743$	− 762.230i	− 1.02588i	0.858424	−	0.512941i	$-0.171444\pi$
$744$	0	0
$745$	−325.315	−0.436664
$746$	0	0
$747$	0	0
$748$	0	0
$749$	355.538	0.474684
$750$	0	0
$751$	122.219i	0.162742i	0.996684	+	0.0813709i	$0.0259298\pi$
$751$	122.219i	0.162742i	−0.996684	+	0.0813709i	$0.974070\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 16.9909i	− 0.0225044i
$756$	0	0
$757$	279.583	0.369330	0.184665	−	0.982802i	$-0.440880\pi$
$757$	279.583	0.369330	0.184665	+	0.982802i	$0.440880\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	250.171	0.328740	0.164370	−	0.986399i	$-0.447441\pi$
$761$	250.171	0.328740	0.164370	+	0.986399i	$0.447441\pi$
$762$	0	0
$763$	− 1567.69i	− 2.05463i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	448.812i	0.585152i
$768$	0	0
$769$	993.027	1.29132	0.645662	−	0.763624i	$-0.276582\pi$
$769$	993.027	1.29132	0.645662	+	0.763624i	$0.276582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−631.978	−0.817565	−0.408782	−	0.912632i	$-0.634047\pi$
$773$	−631.978	−0.817565	−0.408782	+	0.912632i	$0.634047\pi$
$774$	0	0
$775$	12.9208i	0.0166720i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	510.627i	0.655490i
$780$	0	0
$781$	950.867	1.21750
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−63.9541	−0.0814702
$786$	0	0
$787$	− 881.763i	− 1.12041i	−0.828354	−	0.560205i	$-0.810722\pi$
$787$	− 881.763i	− 1.12041i	0.828354	−	0.560205i	$-0.189278\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1617.94i	2.04543i
$792$	0	0
$793$	−91.0833	−0.114859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−442.306	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$797$	−442.306	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$798$	0	0
$799$	101.796i	0.127405i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	481.439i	0.599550i
$804$	0	0
$805$	−712.077	−0.884568
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−665.367	−0.822456	−0.411228	−	0.911533i	$-0.634900\pi$
$809$	−665.367	−0.822456	−0.411228	+	0.911533i	$0.634900\pi$
$810$	0	0
$811$	− 1193.47i	− 1.47160i	−0.677198	−	0.735801i	$-0.736806\pi$
$811$	− 1193.47i	− 1.47160i	0.677198	−	0.735801i	$-0.263194\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 222.055i	− 0.272461i
$816$	0	0
$817$	−209.203	−0.256062
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−359.355	−0.437704	−0.218852	−	0.975758i	$-0.570231\pi$
$821$	−359.355	−0.437704	−0.218852	+	0.975758i	$0.570231\pi$
$822$	0	0
$823$	1159.12i	1.40841i	0.709999	+	0.704203i	$0.248696\pi$
$823$	1159.12i	1.40841i	−0.709999	+	0.704203i	$0.751304\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	160.002i	0.193473i	0.995310	+	0.0967364i	$0.0308404\pi$
$827$	160.002i	0.193473i	−0.995310	+	0.0967364i	$0.969160\pi$
$828$	0	0
$829$	−992.384	−1.19709	−0.598543	−	0.801091i	$-0.704253\pi$
$829$	−992.384	−1.19709	−0.598543	+	0.801091i	$0.704253\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	398.665	0.478590
$834$	0	0
$835$	− 533.231i	− 0.638600i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 84.8821i	− 0.101171i	−0.998720	−	0.0505853i	$-0.983891\pi$
$839$	− 84.8821i	− 0.101171i	0.998720	−	0.0505853i	$-0.0161087\pi$
$840$	0	0
$841$	488.691	0.581083
$842$	0	0
$843$	0	0
$844$	0	0
$845$	180.403	0.213495
$846$	0	0
$847$	690.172i	0.814843i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2345.43i	2.75609i
$852$	0	0
$853$	626.933	0.734974	0.367487	−	0.930029i	$-0.380218\pi$
$853$	626.933	0.734974	0.367487	+	0.930029i	$0.380218\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1560.76	1.82119	0.910594	−	0.413302i	$-0.135625\pi$
$857$	1560.76	1.82119	0.910594	+	0.413302i	$0.135625\pi$
$858$	0	0
$859$	172.848i	0.201220i	0.994926	+	0.100610i	$0.0320794\pi$
$859$	172.848i	0.201220i	−0.994926	+	0.100610i	$0.967921\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 66.7620i	− 0.0773604i	−0.999252	−	0.0386802i	$-0.987685\pi$
$863$	− 66.7620i	− 0.0773604i	0.999252	−	0.0386802i	$-0.0123154\pi$
$864$	0	0
$865$	−360.823	−0.417136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1446.07	1.66406
$870$	0	0
$871$	− 813.372i	− 0.933836i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 994.074i	− 1.13609i
$876$	0	0
$877$	903.899	1.03067	0.515336	−	0.856988i	$-0.327667\pi$
$877$	903.899	1.03067	0.515336	+	0.856988i	$0.327667\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	514.281	0.583747	0.291873	−	0.956457i	$-0.405721\pi$
$881$	514.281	0.583747	0.291873	+	0.956457i	$0.405721\pi$
$882$	0	0
$883$	− 1668.73i	− 1.88984i	−0.327299	−	0.944921i	$-0.606139\pi$
$883$	− 1668.73i	− 1.88984i	0.327299	−	0.944921i	$-0.393861\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1155.43i	1.30262i	0.758810	+	0.651312i	$0.225781\pi$
$887$	1155.43i	1.30262i	−0.758810	+	0.651312i	$0.774219\pi$
$888$	0	0
$889$	485.640	0.546277
$890$	0	0
$891$	0	0
$892$	0	0
$893$	217.148	0.243167
$894$	0	0
$895$	− 564.961i	− 0.631241i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 21.5151i	− 0.0239323i
$900$	0	0
$901$	124.454	0.138129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	502.531	0.555283
$906$	0	0
$907$	229.965i	0.253545i	0.991932	+	0.126772i	$0.0404618\pi$
$907$	229.965i	0.253545i	−0.991932	+	0.126772i	$0.959538\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 20.1180i	− 0.0220835i	−0.999939	−	0.0110417i	$-0.996485\pi$
$911$	− 20.1180i	− 0.0220835i	0.999939	−	0.0110417i	$-0.00351476\pi$
$912$	0	0
$913$	1342.87	1.47083
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1648.57	1.79779
$918$	0	0
$919$	− 580.742i	− 0.631929i	−0.948771	−	0.315964i	$-0.897672\pi$
$919$	− 580.742i	− 0.631929i	0.948771	−	0.315964i	$-0.102328\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	580.889i	0.629348i
$924$	0	0
$925$	−1528.88	−1.65284
$926$	0	0
$927$	0	0
$928$	0	0
$929$	375.347	0.404033	0.202017	−	0.979382i	$-0.435250\pi$
$929$	375.347	0.404033	0.202017	+	0.979382i	$0.435250\pi$
$930$	0	0
$931$	− 850.417i	− 0.913445i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	97.7918i	0.104590i
$936$	0	0
$937$	339.330	0.362145	0.181073	−	0.983470i	$-0.442043\pi$
$937$	339.330	0.362145	0.181073	+	0.983470i	$0.442043\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	459.867	0.488700	0.244350	−	0.969687i	$-0.421425\pi$
$941$	459.867	0.488700	0.244350	+	0.969687i	$0.421425\pi$
$942$	0	0
$943$	1933.91i	2.05081i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	424.359i	0.448109i	0.974577	+	0.224055i	$0.0719294\pi$
$947$	424.359i	0.448109i	−0.974577	+	0.224055i	$0.928071\pi$
$948$	0	0
$949$	−294.113	−0.309919
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−140.791	−0.147734	−0.0738670	−	0.997268i	$-0.523534\pi$
$953$	−140.791	−0.147734	−0.0738670	+	0.997268i	$0.523534\pi$
$954$	0	0
$955$	− 233.570i	− 0.244576i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2960.56i	− 3.08713i
$960$	0	0
$961$	960.652	0.999638
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−645.932	−0.669359
$966$	0	0
$967$	683.919i	0.707258i	0.935386	+	0.353629i	$0.115053\pi$
$967$	683.919i	0.707258i	−0.935386	+	0.353629i	$0.884947\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 983.197i	− 1.01256i	−0.862369	−	0.506281i	$-0.831020\pi$
$971$	− 983.197i	− 1.01256i	0.862369	−	0.506281i	$-0.168980\pi$
$972$	0	0
$973$	−1896.03	−1.94864
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1008.88	1.03263	0.516315	−	0.856399i	$-0.327303\pi$
$977$	1008.88	1.03263	0.516315	+	0.856399i	$0.327303\pi$
$978$	0	0
$979$	2131.59i	2.17731i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1681.75i	1.71084i	0.517936	+	0.855419i	$0.326700\pi$
$983$	1681.75i	1.71084i	−0.517936	+	0.855419i	$0.673300\pi$
$984$	0	0
$985$	25.2019	0.0255857
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−792.321	−0.801134
$990$	0	0
$991$	81.5243i	0.0822647i	0.999154	+	0.0411323i	$0.0130965\pi$
$991$	81.5243i	0.0822647i	−0.999154	+	0.0411323i	$0.986903\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 377.812i	− 0.379710i
$996$	0	0
$997$	−549.116	−0.550768	−0.275384	−	0.961334i	$-0.588805\pi$
$997$	−549.116	−0.550768	−0.275384	+	0.961334i	$0.588805\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.k.703.3		8
3.2	odd	2		1728.3.g.n.703.5		8
4.3	odd	2	inner	1728.3.g.k.703.4		8
8.3	odd	2		864.3.g.c.703.6	yes	8
8.5	even	2		864.3.g.c.703.5	yes	8
12.11	even	2		1728.3.g.n.703.6		8
24.5	odd	2		864.3.g.a.703.3	✓	8
24.11	even	2		864.3.g.a.703.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.a.703.3	✓	8	24.5	odd	2
864.3.g.a.703.4	yes	8	24.11	even	2
864.3.g.c.703.5	yes	8	8.5	even	2
864.3.g.c.703.6	yes	8	8.3	odd	2
1728.3.g.k.703.3		8	1.1	even	1	trivial
1728.3.g.k.703.4		8	4.3	odd	2	inner
1728.3.g.n.703.5		8	3.2	odd	2
1728.3.g.n.703.6		8	12.11	even	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.76102 q^{5} -12.0363i q^{7} +O(q^{10})\)	\(q-1.76102 q^{5} -12.0363i q^{7} +13.3544i q^{11} -8.15828 q^{13} -4.15828 q^{17} +8.87027i q^{19} +33.5947i q^{23} -21.8988 q^{25} +36.4649 q^{29} -0.590023i q^{31} +21.1962i q^{35} +69.8156 q^{37} +57.5661 q^{41} +23.5847i q^{43} -24.4804i q^{47} -95.8727 q^{49} -29.9292 q^{53} -23.5174i q^{55} -55.0130i q^{59} +11.1645 q^{61} +14.3669 q^{65} +99.6989i q^{67} -71.2023i q^{71} +36.0508 q^{73} +160.738 q^{77} -108.284i q^{79} -100.556i q^{83} +7.32280 q^{85} +159.616 q^{89} +98.1956i q^{91} -15.6207i q^{95} -24.0242 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 16 * q^13 + 48 * q^17 + 48 * q^25 + 32 * q^29 + 96 * q^37 - 128 * q^41 - 168 * q^53 - 32 * q^61 - 112 * q^65 + 24 * q^73 + 440 * q^77 - 144 * q^85 + 624 * q^89 - 136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more