LMFDB - Embedded newform 1716.3.g.a.1429.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1716,3,Mod(1429,1716)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1716, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1716.1429");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1716.g (of order $2$, degree $1$, 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:	$46.7576133642$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1429.13
Character	$\chi$	$=$	1716.1429
Dual form			1716.3.g.a.1429.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +4.63273i q^{5} -1.27920 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +4.63273i q^{5} -1.27920 q^{7} +3.00000 q^{9} +(-8.07659 + 7.46784i) q^{11} +(4.74408 + 12.1035i) q^{13} -8.02413i q^{15} -14.8442i q^{17} +5.37043 q^{19} +2.21563 q^{21} +43.0115 q^{23} +3.53780 q^{25} -5.19615 q^{27} +42.6846i q^{29} +18.1137i q^{31} +(13.9891 - 12.9347i) q^{33} -5.92617i q^{35} +13.1148i q^{37} +(-8.21699 - 20.9638i) q^{39} -34.9706 q^{41} +16.5596i q^{43} +13.8982i q^{45} -13.3599i q^{47} -47.3637 q^{49} +25.7109i q^{51} +54.0314 q^{53} +(-34.5965 - 37.4167i) q^{55} -9.30185 q^{57} +39.4361i q^{59} -13.8555i q^{61} -3.83759 q^{63} +(-56.0721 + 21.9780i) q^{65} -15.1048i q^{67} -74.4981 q^{69} -96.1774i q^{71} -79.3703 q^{73} -6.12765 q^{75} +(10.3315 - 9.55283i) q^{77} -43.3745i q^{79} +9.00000 q^{81} +116.609 q^{83} +68.7692 q^{85} -73.9319i q^{87} +121.738i q^{89} +(-6.06861 - 15.4827i) q^{91} -31.3739i q^{93} +24.8797i q^{95} +104.032i q^{97} +(-24.2298 + 22.4035i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 168 q^{9}+O(q^{10}) $ 56 * q + 168 * q^9	$ 56 q + 168 q^{9} - 72 q^{23} - 336 q^{25} + 576 q^{49} + 96 q^{53} + 32 q^{55} - 48 q^{69} + 96 q^{75} - 496 q^{77} + 504 q^{81} - 504 q^{91}+O(q^{100}) $ 56 * q + 168 * q^9 - 72 * q^23 - 336 * q^25 + 576 * q^49 + 96 * q^53 + 32 * q^55 - 48 * q^69 + 96 * q^75 - 496 * q^77 + 504 * q^81 - 504 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times$.

$n$	$859$	$925$	$937$	$1145$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$			4.63273i			0.926546i	0.886216	+	0.463273i	$0.153325\pi$
$5$			4.63273i			0.926546i	−0.886216	+	0.463273i	$0.846675\pi$
$6$		0			0
$7$	−1.27920			−0.182742			−0.0913711	−	0.995817i	$-0.529125\pi$
$7$	−1.27920			−0.182742			−0.0913711	+	0.995817i	$0.529125\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	−8.07659	+	7.46784i	−0.734236	+	0.678895i
$11$	−8.07659	+	7.46784i	−0.734236	+	0.678895i
$12$		0			0
$13$	4.74408	+	12.1035i	0.364929	+	0.931035i
$13$	4.74408	+	12.1035i	0.364929	+	0.931035i
$14$		0			0
$15$		−	8.02413i		−	0.534942i
$16$		0			0
$17$		−	14.8442i		−	0.873188i	−0.899659	−	0.436594i	$-0.856185\pi$
$17$		−	14.8442i		−	0.873188i	0.899659	−	0.436594i	$-0.143815\pi$
$18$		0			0
$19$	5.37043			0.282654			0.141327	−	0.989963i	$-0.454863\pi$
$19$	5.37043			0.282654			0.141327	+	0.989963i	$0.454863\pi$
$20$		0			0
$21$	2.21563			0.105506
$22$		0			0
$23$	43.0115			1.87007			0.935033	−	0.354561i	$-0.115370\pi$
$23$	43.0115			1.87007			0.935033	+	0.354561i	$0.115370\pi$
$24$		0			0
$25$	3.53780			0.141512
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$			42.6846i			1.47188i	0.677046	+	0.735941i	$0.263260\pi$
$29$			42.6846i			1.47188i	−0.677046	+	0.735941i	$0.736740\pi$
$30$		0			0
$31$			18.1137i			0.584314i	0.956370	+	0.292157i	$0.0943730\pi$
$31$			18.1137i			0.584314i	−0.956370	+	0.292157i	$0.905627\pi$
$32$		0			0
$33$	13.9891	−	12.9347i	0.423911	−	0.391960i
$34$		0			0
$35$		−	5.92617i		−	0.169319i
$36$		0			0
$37$			13.1148i			0.354454i	0.984170	+	0.177227i	$0.0567127\pi$
$37$			13.1148i			0.354454i	−0.984170	+	0.177227i	$0.943287\pi$
$38$		0			0
$39$	−8.21699	−	20.9638i	−0.210692	−	0.537533i
$40$		0			0
$41$	−34.9706			−0.852941			−0.426471	−	0.904501i	$-0.640243\pi$
$41$	−34.9706			−0.852941			−0.426471	+	0.904501i	$0.640243\pi$
$42$		0			0
$43$			16.5596i			0.385108i	0.981286	+	0.192554i	$0.0616770\pi$
$43$			16.5596i			0.385108i	−0.981286	+	0.192554i	$0.938323\pi$
$44$		0			0
$45$			13.8982i			0.308849i
$46$		0			0
$47$		−	13.3599i		−	0.284254i	−0.989848	−	0.142127i	$-0.954606\pi$
$47$		−	13.3599i		−	0.284254i	0.989848	−	0.142127i	$-0.0453941\pi$
$48$		0			0
$49$	−47.3637			−0.966605
$50$		0			0
$51$			25.7109i			0.504135i
$52$		0			0
$53$	54.0314			1.01946			0.509730	−	0.860334i	$-0.329745\pi$
$53$	54.0314			1.01946			0.509730	+	0.860334i	$0.329745\pi$
$54$		0			0
$55$	−34.5965	−	37.4167i	−0.629027	−	0.680303i
$56$		0			0
$57$	−9.30185			−0.163190
$58$		0			0
$59$			39.4361i			0.668408i	0.942501	+	0.334204i	$0.108467\pi$
$59$			39.4361i			0.668408i	−0.942501	+	0.334204i	$0.891533\pi$
$60$		0			0
$61$		−	13.8555i		−	0.227140i	−0.993530	−	0.113570i	$-0.963771\pi$
$61$		−	13.8555i		−	0.227140i	0.993530	−	0.113570i	$-0.0362286\pi$
$62$		0			0
$63$	−3.83759			−0.0609141
$64$		0			0
$65$	−56.0721	+	21.9780i	−0.862647	+	0.338124i
$66$		0			0
$67$		−	15.1048i		−	0.225444i	−0.993627	−	0.112722i	$-0.964043\pi$
$67$		−	15.1048i		−	0.225444i	0.993627	−	0.112722i	$-0.0359570\pi$
$68$		0			0
$69$	−74.4981			−1.07968
$70$		0			0
$71$		−	96.1774i		−	1.35461i	−0.735702	−	0.677306i	$-0.763147\pi$
$71$		−	96.1774i		−	1.35461i	0.735702	−	0.677306i	$-0.236853\pi$
$72$		0			0
$73$	−79.3703			−1.08726			−0.543632	−	0.839324i	$-0.682951\pi$
$73$	−79.3703			−1.08726			−0.543632	+	0.839324i	$0.682951\pi$
$74$		0			0
$75$	−6.12765			−0.0817020
$76$		0			0
$77$	10.3315	−	9.55283i	0.134176	−	0.124063i
$78$		0			0
$79$		−	43.3745i		−	0.549044i	−0.961581	−	0.274522i	$-0.911480\pi$
$79$		−	43.3745i		−	0.549044i	0.961581	−	0.274522i	$-0.0885196\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	116.609			1.40493			0.702464	−	0.711719i	$-0.252083\pi$
$83$	116.609			1.40493			0.702464	+	0.711719i	$0.252083\pi$
$84$		0			0
$85$	68.7692			0.809049
$86$		0			0
$87$		−	73.9319i		−	0.849792i
$88$		0			0
$89$			121.738i			1.36784i	0.729555	+	0.683922i	$0.239727\pi$
$89$			121.738i			1.36784i	−0.729555	+	0.683922i	$0.760273\pi$
$90$		0			0
$91$	−6.06861	−	15.4827i	−0.0666880	−	0.170139i
$92$		0			0
$93$		−	31.3739i		−	0.337354i
$94$		0			0
$95$			24.8797i			0.261892i
$96$		0			0
$97$			104.032i			1.07249i	0.844061	+	0.536247i	$0.180159\pi$
$97$			104.032i			1.07249i	−0.844061	+	0.536247i	$0.819841\pi$
$98$		0			0
$99$	−24.2298	+	22.4035i	−0.244745	+	0.226298i
$100$		0			0
$101$		−	43.1890i		−	0.427614i	−0.976876	−	0.213807i	$-0.931414\pi$
$101$		−	43.1890i		−	0.427614i	0.976876	−	0.213807i	$-0.0685864\pi$
$102$		0			0
$103$	−43.2843			−0.420236			−0.210118	−	0.977676i	$-0.567385\pi$
$103$	−43.2843			−0.420236			−0.210118	+	0.977676i	$0.567385\pi$
$104$		0			0
$105$			10.2644i			0.0977564i
$106$		0			0
$107$			148.446i			1.38735i	0.720290	+	0.693673i	$0.244009\pi$
$107$			148.446i			1.38735i	−0.720290	+	0.693673i	$0.755991\pi$
$108$		0			0
$109$	−203.977			−1.87135			−0.935673	−	0.352869i	$-0.885206\pi$
$109$	−203.977			−1.87135			−0.935673	+	0.352869i	$0.885206\pi$
$110$		0			0
$111$		−	22.7155i		−	0.204644i
$112$		0			0
$113$	−146.761			−1.29877			−0.649386	−	0.760459i	$-0.724974\pi$
$113$	−146.761			−1.29877			−0.649386	+	0.760459i	$0.724974\pi$
$114$		0			0
$115$			199.261i			1.73270i
$116$		0			0
$117$	14.2322	+	36.3104i	0.121643	+	0.310345i
$118$		0			0
$119$			18.9886i			0.159568i
$120$		0			0
$121$	9.46264	−	120.629i	0.0782036	−	0.996937i
$122$		0			0
$123$	60.5708			0.492446
$124$		0			0
$125$			132.208i			1.05766i
$126$		0			0
$127$		−	140.905i		−	1.10948i	−0.832022	−	0.554742i	$-0.812817\pi$
$127$		−	140.905i		−	1.10948i	0.832022	−	0.554742i	$-0.187183\pi$
$128$		0			0
$129$		−	28.6821i		−	0.222342i
$130$		0			0
$131$		−	162.216i		−	1.23829i	−0.785276	−	0.619146i	$-0.787479\pi$
$131$		−	162.216i		−	1.23829i	0.785276	−	0.619146i	$-0.212521\pi$
$132$		0			0
$133$	−6.86982			−0.0516528
$134$		0			0
$135$		−	24.0724i		−	0.178314i
$136$		0			0
$137$			248.818i			1.81619i	0.418762	+	0.908096i	$0.362464\pi$
$137$			248.818i			1.81619i	−0.418762	+	0.908096i	$0.637536\pi$
$138$		0			0
$139$		−	60.3874i		−	0.434442i	−0.976122	−	0.217221i	$-0.930301\pi$
$139$		−	60.3874i		−	0.434442i	0.976122	−	0.217221i	$-0.0696992\pi$
$140$		0			0
$141$			23.1401i			0.164114i
$142$		0			0
$143$	−128.703	−	62.3266i	−0.900019	−	0.435851i
$144$		0			0
$145$	−197.746			−1.36377
$146$		0			0
$147$	82.0363			0.558070
$148$		0			0
$149$	−94.4519			−0.633905			−0.316953	−	0.948441i	$-0.602660\pi$
$149$	−94.4519			−0.633905			−0.316953	+	0.948441i	$0.602660\pi$
$150$		0			0
$151$	−234.501			−1.55299			−0.776493	−	0.630126i	$-0.783003\pi$
$151$	−234.501			−1.55299			−0.776493	+	0.630126i	$0.783003\pi$
$152$		0			0
$153$		−	44.5326i		−	0.291063i
$154$		0			0
$155$	−83.9161			−0.541394
$156$		0			0
$157$	84.1087			0.535724			0.267862	−	0.963457i	$-0.413683\pi$
$157$	84.1087			0.535724			0.267862	+	0.963457i	$0.413683\pi$
$158$		0			0
$159$	−93.5851			−0.588586
$160$		0			0
$161$	−55.0201			−0.341740
$162$		0			0
$163$		−	85.7012i		−	0.525774i	−0.964827	−	0.262887i	$-0.915325\pi$
$163$		−	85.7012i		−	0.525774i	0.964827	−	0.262887i	$-0.0846747\pi$
$164$		0			0
$165$	59.9229	+	64.8076i	0.363169	+	0.392773i
$166$		0			0
$167$	−208.218			−1.24682			−0.623408	−	0.781897i	$-0.714252\pi$
$167$	−208.218			−1.24682			−0.623408	+	0.781897i	$0.714252\pi$
$168$		0			0
$169$	−123.987	+	114.840i	−0.733653	+	0.679524i
$170$		0			0
$171$	16.1113			0.0942180
$172$		0			0
$173$		−	65.6389i		−	0.379416i	−0.981841	−	0.189708i	$-0.939246\pi$
$173$		−	65.6389i		−	0.379416i	0.981841	−	0.189708i	$-0.0607541\pi$
$174$		0			0
$175$	−4.52554			−0.0258602
$176$		0			0
$177$		−	68.3053i		−	0.385906i
$178$		0			0
$179$	−138.491			−0.773696			−0.386848	−	0.922144i	$-0.626436\pi$
$179$	−138.491			−0.773696			−0.386848	+	0.922144i	$0.626436\pi$
$180$		0			0
$181$	16.8182			0.0929182			0.0464591	−	0.998920i	$-0.485206\pi$
$181$	16.8182			0.0929182			0.0464591	+	0.998920i	$0.485206\pi$
$182$		0			0
$183$			23.9985i			0.131139i
$184$		0			0
$185$	−60.7574			−0.328418
$186$		0			0
$187$	110.854	+	119.890i	0.592803	+	0.641126i
$188$		0			0
$189$	6.64689			0.0351688
$190$		0			0
$191$	185.286			0.970084			0.485042	−	0.874491i	$-0.338804\pi$
$191$	185.286			0.970084			0.485042	+	0.874491i	$0.338804\pi$
$192$		0			0
$193$	18.6039			0.0963934			0.0481967	−	0.998838i	$-0.484653\pi$
$193$	18.6039			0.0963934			0.0481967	+	0.998838i	$0.484653\pi$
$194$		0			0
$195$	97.1197	−	38.0671i	0.498050	−	0.195216i
$196$		0			0
$197$	−19.0118			−0.0965066			−0.0482533	−	0.998835i	$-0.515365\pi$
$197$	−19.0118			−0.0965066			−0.0482533	+	0.998835i	$0.515365\pi$
$198$		0			0
$199$	−346.508			−1.74125			−0.870624	−	0.491950i	$-0.836284\pi$
$199$	−346.508			−1.74125			−0.870624	+	0.491950i	$0.836284\pi$
$200$		0			0
$201$			26.1622i			0.130160i
$202$		0			0
$203$		−	54.6019i		−	0.268975i
$204$		0			0
$205$		−	162.009i		−	0.790290i
$206$		0			0
$207$	129.035			0.623355
$208$		0			0
$209$	−43.3747	+	40.1055i	−0.207535	+	0.191892i
$210$		0			0
$211$			335.806i			1.59150i	0.605628	+	0.795748i	$0.292922\pi$
$211$			335.806i			1.59150i	−0.605628	+	0.795748i	$0.707078\pi$
$212$		0			0
$213$			166.584i			0.782085i
$214$		0			0
$215$	−76.7163			−0.356820
$216$		0			0
$217$		−	23.1710i		−	0.106779i
$218$		0			0
$219$	137.473			0.627732
$220$		0			0
$221$	179.666	−	70.4220i	0.812969	−	0.318652i
$222$		0			0
$223$			442.854i			1.98589i	0.118575	+	0.992945i	$0.462168\pi$
$223$			442.854i			1.98589i	−0.118575	+	0.992945i	$0.537832\pi$
$224$		0			0
$225$	10.6134			0.0471707
$226$		0			0
$227$	−255.197			−1.12422			−0.562108	−	0.827064i	$-0.690009\pi$
$227$	−255.197			−1.12422			−0.562108	+	0.827064i	$0.690009\pi$
$228$		0			0
$229$			258.893i			1.13054i	0.824907	+	0.565268i	$0.191227\pi$
$229$			258.893i			1.13054i	−0.824907	+	0.565268i	$0.808773\pi$
$230$		0			0
$231$	−17.8948	+	16.5460i	−0.0774665	+	0.0716277i
$232$		0			0
$233$		−	113.167i		−	0.485697i	−0.970064	−	0.242848i	$-0.921918\pi$
$233$		−	113.167i		−	0.485697i	0.970064	−	0.242848i	$-0.0780817\pi$
$234$		0			0
$235$	61.8930			0.263375
$236$		0			0
$237$			75.1268i			0.316991i
$238$		0			0
$239$	448.980			1.87858			0.939289	−	0.343126i	$-0.111486\pi$
$239$	448.980			1.87858			0.939289	+	0.343126i	$0.111486\pi$
$240$		0			0
$241$	−419.660			−1.74133			−0.870664	−	0.491879i	$-0.836310\pi$
$241$	−419.660			−1.74133			−0.870664	+	0.491879i	$0.836310\pi$
$242$		0			0
$243$	−15.5885			−0.0641500
$244$		0			0
$245$		−	219.423i		−	0.895605i
$246$		0			0
$247$	25.4777	+	65.0007i	0.103149	+	0.263161i
$248$		0			0
$249$	−201.973			−0.811135
$250$		0			0
$251$	−92.4407			−0.368290			−0.184145	−	0.982899i	$-0.558952\pi$
$251$	−92.4407			−0.368290			−0.184145	+	0.982899i	$0.558952\pi$
$252$		0			0
$253$	−347.386	+	321.203i	−1.37307	+	1.26958i
$254$		0			0
$255$	−119.112			−0.467105
$256$		0			0
$257$	69.7886			0.271551			0.135775	−	0.990740i	$-0.456647\pi$
$257$	69.7886			0.271551			0.135775	+	0.990740i	$0.456647\pi$
$258$		0			0
$259$		−	16.7764i		−	0.0647738i
$260$		0			0
$261$			128.054i			0.490627i
$262$		0			0
$263$		−	199.962i		−	0.760312i	−0.924922	−	0.380156i	$-0.875870\pi$
$263$		−	199.962i		−	0.760312i	0.924922	−	0.380156i	$-0.124130\pi$
$264$		0			0
$265$			250.313i			0.944577i
$266$		0			0
$267$		−	210.857i		−	0.789726i
$268$		0			0
$269$	192.511			0.715655			0.357828	−	0.933788i	$-0.383518\pi$
$269$	192.511			0.715655			0.357828	+	0.933788i	$0.383518\pi$
$270$		0			0
$271$	114.278			0.421689			0.210844	−	0.977520i	$-0.432379\pi$
$271$	114.278			0.421689			0.210844	+	0.977520i	$0.432379\pi$
$272$		0			0
$273$	10.5111	+	26.8168i	0.0385023	+	0.0982301i
$274$		0			0
$275$	−28.5734	+	26.4197i	−0.103903	+	0.0960718i
$276$		0			0
$277$			233.801i			0.844047i	0.906585	+	0.422024i	$0.138680\pi$
$277$			233.801i			0.844047i	−0.906585	+	0.422024i	$0.861320\pi$
$278$		0			0
$279$			54.3412i			0.194771i
$280$		0			0
$281$	439.171			1.56289			0.781444	−	0.623975i	$-0.214483\pi$
$281$	439.171			1.56289			0.781444	+	0.623975i	$0.214483\pi$
$282$		0			0
$283$		−	42.7313i		−	0.150994i	−0.997146	−	0.0754969i	$-0.975946\pi$
$283$		−	42.7313i		−	0.150994i	0.997146	−	0.0754969i	$-0.0240543\pi$
$284$		0			0
$285$		−	43.0930i		−	0.151203i
$286$		0			0
$287$	44.7342			0.155868
$288$		0			0
$289$	68.6498			0.237543
$290$		0			0
$291$		−	180.189i		−	0.619205i
$292$		0			0
$293$	−249.005			−0.849847			−0.424924	−	0.905229i	$-0.639699\pi$
$293$	−249.005			−0.849847			−0.424924	+	0.905229i	$0.639699\pi$
$294$		0			0
$295$	−182.697			−0.619311
$296$		0			0
$297$	41.9672	−	38.8041i	0.141304	−	0.130653i
$298$		0			0
$299$	204.050	+	520.588i	0.682442	+	1.74110i
$300$		0			0
$301$		−	21.1830i		−	0.0703754i
$302$		0			0
$303$			74.8056i			0.246883i
$304$		0			0
$305$	64.1889			0.210456
$306$		0			0
$307$	−68.9824			−0.224698			−0.112349	−	0.993669i	$-0.535838\pi$
$307$	−68.9824			−0.224698			−0.112349	+	0.993669i	$0.535838\pi$
$308$		0			0
$309$	74.9705			0.242623
$310$		0			0
$311$	252.592			0.812194			0.406097	−	0.913830i	$-0.366890\pi$
$311$	252.592			0.812194			0.406097	+	0.913830i	$0.366890\pi$
$312$		0			0
$313$	−17.1856			−0.0549061			−0.0274531	−	0.999623i	$-0.508740\pi$
$313$	−17.1856			−0.0549061			−0.0274531	+	0.999623i	$0.508740\pi$
$314$		0			0
$315$		−	17.7785i		−	0.0564397i
$316$		0			0
$317$			103.663i			0.327012i	0.986542	+	0.163506i	$0.0522804\pi$
$317$			103.663i			0.327012i	−0.986542	+	0.163506i	$0.947720\pi$
$318$		0			0
$319$	−318.762	−	344.746i	−0.999253	−	1.08071i
$320$		0			0
$321$		−	257.116i		−	0.800984i
$322$		0			0
$323$		−	79.7196i		−	0.246810i
$324$		0			0
$325$	16.7836	+	42.8196i	0.0516419	+	0.131753i
$326$		0			0
$327$	353.298			1.08042
$328$		0			0
$329$			17.0900i			0.0519452i
$330$		0			0
$331$		−	102.360i		−	0.309245i	−0.987974	−	0.154622i	$-0.950584\pi$
$331$		−	102.360i		−	0.309245i	0.987974	−	0.154622i	$-0.0494161\pi$
$332$		0			0
$333$			39.3444i			0.118151i
$334$		0			0
$335$	69.9763			0.208884
$336$		0			0
$337$		−	160.170i		−	0.475281i	−0.971353	−	0.237641i	$-0.923626\pi$
$337$		−	160.170i		−	0.475281i	0.971353	−	0.237641i	$-0.0763741\pi$
$338$		0			0
$339$	254.198			0.749846
$340$		0			0
$341$	−135.271	−	146.297i	−0.396688	−	0.429024i
$342$		0			0
$343$	123.268			0.359382
$344$		0			0
$345$		−	345.130i		−	1.00038i
$346$		0			0
$347$		−	118.105i		−	0.340359i	−0.985413	−	0.170179i	$-0.945565\pi$
$347$		−	118.105i		−	0.340359i	0.985413	−	0.170179i	$-0.0544347\pi$
$348$		0			0
$349$	−84.3368			−0.241653			−0.120826	−	0.992674i	$-0.538554\pi$
$349$	−84.3368			−0.241653			−0.120826	+	0.992674i	$0.538554\pi$
$350$		0			0
$351$	−24.6510	−	62.8914i	−0.0702307	−	0.179178i
$352$		0			0
$353$		−	268.465i		−	0.760524i	−0.924879	−	0.380262i	$-0.875834\pi$
$353$		−	268.465i		−	0.760524i	0.924879	−	0.380262i	$-0.124166\pi$
$354$		0			0
$355$	445.564			1.25511
$356$		0			0
$357$		−	32.8893i		−	0.0921268i
$358$		0			0
$359$	556.902			1.55126			0.775629	−	0.631189i	$-0.217433\pi$
$359$	556.902			1.55126			0.775629	+	0.631189i	$0.217433\pi$
$360$		0			0
$361$	−332.159			−0.920107
$362$		0			0
$363$	−16.3898	+	208.936i	−0.0451509	+	0.575582i
$364$		0			0
$365$		−	367.701i		−	1.00740i
$366$		0			0
$367$	−63.9806			−0.174334			−0.0871670	−	0.996194i	$-0.527781\pi$
$367$	−63.9806			−0.174334			−0.0871670	+	0.996194i	$0.527781\pi$
$368$		0			0
$369$	−104.912			−0.284314
$370$		0			0
$371$	−69.1167			−0.186298
$372$		0			0
$373$		−	306.415i		−	0.821489i	−0.911750	−	0.410745i	$-0.865269\pi$
$373$		−	306.415i		−	0.821489i	0.911750	−	0.410745i	$-0.134731\pi$
$374$		0			0
$375$		−	228.991i		−	0.610642i
$376$		0			0
$377$	−516.631	+	202.499i	−1.37037	+	0.537133i
$378$		0			0
$379$		−	297.574i		−	0.785156i	−0.919719	−	0.392578i	$-0.871583\pi$
$379$		−	297.574i		−	0.785156i	0.919719	−	0.392578i	$-0.128417\pi$
$380$		0			0
$381$			244.054i			0.640561i
$382$		0			0
$383$		−	681.895i		−	1.78040i	−0.455566	−	0.890202i	$-0.650563\pi$
$383$		−	681.895i		−	1.78040i	0.455566	−	0.890202i	$-0.349437\pi$
$384$		0			0
$385$	44.2557	+	47.8632i	0.114950	+	0.124320i
$386$		0			0
$387$			49.6789i			0.128369i
$388$		0			0
$389$	−669.380			−1.72077			−0.860386	−	0.509643i	$-0.829778\pi$
$389$	−669.380			−1.72077			−0.860386	+	0.509643i	$0.829778\pi$
$390$		0			0
$391$		−	638.471i		−	1.63292i
$392$		0			0
$393$			280.967i			0.714928i
$394$		0			0
$395$	200.942			0.508714
$396$		0			0
$397$			175.239i			0.441408i	0.975341	+	0.220704i	$0.0708355\pi$
$397$			175.239i			0.441408i	−0.975341	+	0.220704i	$0.929164\pi$
$398$		0			0
$399$	11.8989			0.0298218
$400$		0			0
$401$		−	21.4864i		−	0.0535822i	−0.999641	−	0.0267911i	$-0.991471\pi$
$401$		−	21.4864i		−	0.0535822i	0.999641	−	0.0267911i	$-0.00852889\pi$
$402$		0			0
$403$	−219.239	+	85.9330i	−0.544017	+	0.213233i
$404$		0			0
$405$			41.6946i			0.102950i
$406$		0			0
$407$	−97.9393	−	105.923i	−0.240637	−	0.260253i
$408$		0			0
$409$	179.457			0.438771			0.219385	−	0.975638i	$-0.429595\pi$
$409$	179.457			0.438771			0.219385	+	0.975638i	$0.429595\pi$
$410$		0			0
$411$		−	430.966i		−	1.04858i
$412$		0			0
$413$		−	50.4465i		−	0.122146i
$414$		0			0
$415$			540.218i			1.30173i
$416$		0			0
$417$			104.594i			0.250825i
$418$		0			0
$419$	285.423			0.681200			0.340600	−	0.940208i	$-0.389370\pi$
$419$	285.423			0.681200			0.340600	+	0.940208i	$0.389370\pi$
$420$		0			0
$421$			373.211i			0.886486i	0.896401	+	0.443243i	$0.146172\pi$
$421$			373.211i			0.886486i	−0.896401	+	0.443243i	$0.853828\pi$
$422$		0			0
$423$		−	40.0798i		−	0.0947513i
$424$		0			0
$425$		−	52.5158i		−	0.123567i
$426$		0			0
$427$			17.7239i			0.0415080i
$428$		0			0
$429$	222.920	+	107.953i	0.519626	+	0.251638i
$430$		0			0
$431$	184.993			0.429218			0.214609	−	0.976700i	$-0.431152\pi$
$431$	184.993			0.429218			0.214609	+	0.976700i	$0.431152\pi$
$432$		0			0
$433$	−565.339			−1.30563			−0.652816	−	0.757516i	$-0.726413\pi$
$433$	−565.339			−1.30563			−0.652816	+	0.757516i	$0.726413\pi$
$434$		0			0
$435$	342.506			0.787371
$436$		0			0
$437$	230.990			0.528582
$438$		0			0
$439$			253.229i			0.576831i	0.957505	+	0.288415i	$0.0931284\pi$
$439$			253.229i			0.576831i	−0.957505	+	0.288415i	$0.906872\pi$
$440$		0			0
$441$	−142.091			−0.322202
$442$		0			0
$443$	560.693			1.26567			0.632836	−	0.774286i	$-0.281891\pi$
$443$	560.693			1.26567			0.632836	+	0.774286i	$0.281891\pi$
$444$		0			0
$445$	−563.980			−1.26737
$446$		0			0
$447$	163.595			0.365985
$448$		0			0
$449$		−	428.852i		−	0.955128i	−0.878597	−	0.477564i	$-0.841520\pi$
$449$		−	428.852i		−	0.955128i	0.878597	−	0.477564i	$-0.158480\pi$
$450$		0			0
$451$	282.443	−	261.155i	0.626260	−	0.579057i
$452$		0			0
$453$	406.167			0.896616
$454$		0			0
$455$	71.7271	−	28.1142i	0.157642	−	0.0617895i
$456$		0			0
$457$	342.084			0.748542			0.374271	−	0.927319i	$-0.377893\pi$
$457$	342.084			0.748542			0.374271	+	0.927319i	$0.377893\pi$
$458$		0			0
$459$			77.1327i			0.168045i
$460$		0			0
$461$	167.181			0.362649			0.181325	−	0.983423i	$-0.441962\pi$
$461$	167.181			0.362649			0.181325	+	0.983423i	$0.441962\pi$
$462$		0			0
$463$		−	243.541i		−	0.526006i	−0.964795	−	0.263003i	$-0.915287\pi$
$463$		−	243.541i		−	0.526006i	0.964795	−	0.263003i	$-0.0847130\pi$
$464$		0			0
$465$	145.347			0.312574
$466$		0			0
$467$	478.784			1.02523			0.512617	−	0.858617i	$-0.328676\pi$
$467$	478.784			1.02523			0.512617	+	0.858617i	$0.328676\pi$
$468$		0			0
$469$			19.3219i			0.0411981i
$470$		0			0
$471$	−145.681			−0.309301
$472$		0			0
$473$	−123.665	−	133.745i	−0.261448	−	0.282760i
$474$		0			0
$475$	18.9995			0.0399989
$476$		0			0
$477$	162.094			0.339820
$478$		0			0
$479$	807.939			1.68672			0.843360	−	0.537348i	$-0.180574\pi$
$479$	807.939			1.68672			0.843360	+	0.537348i	$0.180574\pi$
$480$		0			0
$481$	−158.735	+	62.2177i	−0.330009	+	0.129351i
$482$		0			0
$483$	95.2977			0.197304
$484$		0			0
$485$	−481.952			−0.993716
$486$		0			0
$487$		−	151.902i		−	0.311914i	−0.987764	−	0.155957i	$-0.950154\pi$
$487$		−	151.902i		−	0.311914i	0.987764	−	0.155957i	$-0.0498461\pi$
$488$		0			0
$489$			148.439i			0.303556i
$490$		0			0
$491$			727.418i			1.48150i	0.671780	+	0.740751i	$0.265530\pi$
$491$			727.418i			1.48150i	−0.671780	+	0.740751i	$0.734470\pi$
$492$		0			0
$493$	633.618			1.28523
$494$		0			0
$495$	−103.790	−	112.250i	−0.209676	−	0.226768i
$496$		0			0
$497$			123.030i			0.247545i
$498$		0			0
$499$			251.535i			0.504077i	0.967717	+	0.252039i	$0.0811010\pi$
$499$			251.535i			0.504077i	−0.967717	+	0.252039i	$0.918899\pi$
$500$		0			0
$501$	360.645			0.719850
$502$		0			0
$503$			824.504i			1.63917i	0.572956	+	0.819586i	$0.305797\pi$
$503$			824.504i			1.63917i	−0.572956	+	0.819586i	$0.694203\pi$
$504$		0			0
$505$	200.083			0.396204
$506$		0			0
$507$	214.753	−	198.908i	0.423575	−	0.392323i
$508$		0			0
$509$		−	885.337i		−	1.73937i	−0.493611	−	0.869683i	$-0.664323\pi$
$509$		−	885.337i		−	1.73937i	0.493611	−	0.869683i	$-0.335677\pi$
$510$		0			0
$511$	101.530			0.198689
$512$		0			0
$513$	−27.9055			−0.0543968
$514$		0			0
$515$		−	200.524i		−	0.389368i
$516$		0			0
$517$	99.7699	+	107.903i	0.192979	+	0.208709i
$518$		0			0
$519$			113.690i			0.219056i
$520$		0			0
$521$	499.005			0.957783			0.478891	−	0.877874i	$-0.341039\pi$
$521$	499.005			0.957783			0.478891	+	0.877874i	$0.341039\pi$
$522$		0			0
$523$		−	705.580i		−	1.34910i	−0.738228	−	0.674551i	$-0.764337\pi$
$523$		−	705.580i		−	1.34910i	0.738228	−	0.674551i	$-0.235663\pi$
$524$		0			0
$525$	7.83846			0.0149304
$526$		0			0
$527$	268.884			0.510216
$528$		0			0
$529$	1320.99			2.49715
$530$		0			0
$531$			118.308i			0.222803i
$532$		0			0
$533$	−165.903	−	423.265i	−0.311263	−	0.794118i
$534$		0			0
$535$	−687.710			−1.28544
$536$		0			0
$537$	239.874			0.446693
$538$		0			0
$539$	382.537	−	353.704i	0.709716	−	0.656223i
$540$		0			0
$541$	160.613			0.296881			0.148440	−	0.988921i	$-0.452575\pi$
$541$	160.613			0.296881			0.148440	+	0.988921i	$0.452575\pi$
$542$		0			0
$543$	−29.1300			−0.0536464
$544$		0			0
$545$		−	944.969i		−	1.73389i
$546$		0			0
$547$			502.086i			0.917890i	0.888465	+	0.458945i	$0.151772\pi$
$547$			502.086i			0.917890i	−0.888465	+	0.458945i	$0.848228\pi$
$548$		0			0
$549$		−	41.5666i		−	0.0757133i
$550$		0			0
$551$			229.234i			0.416033i
$552$		0			0
$553$			55.4844i			0.100333i
$554$		0			0
$555$	105.235			0.189612
$556$		0			0
$557$	−623.877			−1.12007			−0.560033	−	0.828470i	$-0.689212\pi$
$557$	−623.877			−1.12007			−0.560033	+	0.828470i	$0.689212\pi$
$558$		0			0
$559$	−200.429	+	78.5602i	−0.358549	+	0.140537i
$560$		0			0
$561$	−192.005	−	207.656i	−0.342255	−	0.370154i
$562$		0			0
$563$			482.940i			0.857797i	0.903353	+	0.428898i	$0.141098\pi$
$563$			482.940i			0.857797i	−0.903353	+	0.428898i	$0.858902\pi$
$564$		0			0
$565$		−	679.905i		−	1.20337i
$566$		0			0
$567$	−11.5128			−0.0203047
$568$		0			0
$569$			413.071i			0.725960i	0.931797	+	0.362980i	$0.118241\pi$
$569$			413.071i			0.725960i	−0.931797	+	0.362980i	$0.881759\pi$
$570$		0			0
$571$		−	562.466i		−	0.985054i	−0.870297	−	0.492527i	$-0.836073\pi$
$571$		−	562.466i		−	0.985054i	0.870297	−	0.492527i	$-0.163927\pi$
$572$		0			0
$573$	−320.925			−0.560078
$574$		0			0
$575$	152.166			0.264637
$576$		0			0
$577$			324.878i			0.563046i	0.959554	+	0.281523i	$0.0908396\pi$
$577$			324.878i			0.563046i	−0.959554	+	0.281523i	$0.909160\pi$
$578$		0			0
$579$	−32.2229			−0.0556527
$580$		0			0
$581$	−149.166			−0.256740
$582$		0			0
$583$	−436.389	+	403.498i	−0.748524	+	0.692106i
$584$		0			0
$585$	−168.216	+	65.9341i	−0.287549	+	0.112708i
$586$		0			0
$587$		−	1046.53i		−	1.78284i	−0.453177	−	0.891420i	$-0.649709\pi$
$587$		−	1046.53i		−	1.78284i	0.453177	−	0.891420i	$-0.350291\pi$
$588$		0			0
$589$			97.2785i			0.165159i
$590$		0			0
$591$	32.9294			0.0557181
$592$		0			0
$593$	−746.718			−1.25922			−0.629610	−	0.776911i	$-0.716785\pi$
$593$	−746.718			−1.25922			−0.629610	+	0.776911i	$0.716785\pi$
$594$		0			0
$595$	−87.9692			−0.147847
$596$		0			0
$597$	600.170			1.00531
$598$		0			0
$599$	224.563			0.374896			0.187448	−	0.982274i	$-0.439978\pi$
$599$	224.563			0.374896			0.187448	+	0.982274i	$0.439978\pi$
$600$		0			0
$601$			1019.72i			1.69671i	0.529426	+	0.848356i	$0.322407\pi$
$601$			1019.72i			1.69671i	−0.529426	+	0.848356i	$0.677593\pi$
$602$		0			0
$603$		−	45.3143i		−	0.0751480i
$604$		0			0
$605$	558.844	+	43.8379i	0.923709	+	0.0724593i
$606$		0			0
$607$		−	811.389i		−	1.33672i	−0.743838	−	0.668360i	$-0.766996\pi$
$607$		−	811.389i		−	1.33672i	0.743838	−	0.668360i	$-0.233004\pi$
$608$		0			0
$609$			94.5733i			0.155293i
$610$		0			0
$611$	161.701	−	63.3806i	0.264651	−	0.103733i
$612$		0			0
$613$	−568.697			−0.927728			−0.463864	−	0.885906i	$-0.653537\pi$
$613$	−568.697			−0.927728			−0.463864	+	0.885906i	$0.653537\pi$
$614$		0			0
$615$			280.608i			0.456274i
$616$		0			0
$617$			990.806i			1.60584i	0.596084	+	0.802922i	$0.296722\pi$
$617$			990.806i			1.60584i	−0.596084	+	0.802922i	$0.703278\pi$
$618$		0			0
$619$		−	40.5138i		−	0.0654503i	−0.999464	−	0.0327252i	$-0.989581\pi$
$619$		−	40.5138i		−	0.0654503i	0.999464	−	0.0327252i	$-0.0104186\pi$
$620$		0			0
$621$	−223.494			−0.359894
$622$		0			0
$623$		−	155.727i		−	0.249963i
$624$		0			0
$625$	−524.039			−0.838462
$626$		0			0
$627$	75.1272	−	69.4648i	0.119820	−	0.110789i
$628$		0			0
$629$	194.679			0.309505
$630$		0			0
$631$			715.037i			1.13318i	0.824000	+	0.566590i	$0.191738\pi$
$631$			715.037i			1.13318i	−0.824000	+	0.566590i	$0.808262\pi$
$632$		0			0
$633$		−	581.632i		−	0.918851i
$634$		0			0
$635$	652.773			1.02799
$636$		0			0
$637$	−224.697	−	573.264i	−0.352742	−	0.899944i
$638$		0			0
$639$		−	288.532i		−	0.451537i
$640$		0			0
$641$	504.456			0.786983			0.393491	−	0.919328i	$-0.371267\pi$
$641$	504.456			0.786983			0.393491	+	0.919328i	$0.371267\pi$
$642$		0			0
$643$			568.613i			0.884313i	0.896938	+	0.442156i	$0.145786\pi$
$643$			568.613i			0.884313i	−0.896938	+	0.442156i	$0.854214\pi$
$644$		0			0
$645$	132.877			0.206010
$646$		0			0
$647$	161.969			0.250338			0.125169	−	0.992135i	$-0.460053\pi$
$647$	161.969			0.250338			0.125169	+	0.992135i	$0.460053\pi$
$648$		0			0
$649$	−294.503	−	318.509i	−0.453779	−	0.490769i
$650$		0			0
$651$			40.1334i			0.0616488i
$652$		0			0
$653$	305.766			0.468249			0.234124	−	0.972207i	$-0.424778\pi$
$653$	305.766			0.468249			0.234124	+	0.972207i	$0.424778\pi$
$654$		0			0
$655$	751.505			1.14734
$656$		0			0
$657$	−238.111			−0.362421
$658$		0			0
$659$			1203.75i			1.82664i	0.407248	+	0.913318i	$0.366489\pi$
$659$			1203.75i			1.82664i	−0.407248	+	0.913318i	$0.633511\pi$
$660$		0			0
$661$			480.501i			0.726931i	0.931608	+	0.363465i	$0.118406\pi$
$661$			480.501i			0.726931i	−0.931608	+	0.363465i	$0.881594\pi$
$662$		0			0
$663$	−311.191	+	121.975i	−0.469368	+	0.183974i
$664$		0			0
$665$		−	31.8260i		−	0.0478587i
$666$		0			0
$667$			1835.93i			2.75252i
$668$		0			0
$669$		−	767.045i		−	1.14655i
$670$		0			0
$671$	103.471	+	111.905i	0.154204	+	0.166774i
$672$		0			0
$673$		−	302.636i		−	0.449682i	−0.974396	−	0.224841i	$-0.927814\pi$
$673$		−	302.636i		−	0.449682i	0.974396	−	0.224841i	$-0.0721862\pi$
$674$		0			0
$675$	−18.3829			−0.0272340
$676$		0			0
$677$		−	583.258i		−	0.861533i	−0.902463	−	0.430767i	$-0.858243\pi$
$677$		−	583.258i		−	0.861533i	0.902463	−	0.430767i	$-0.141757\pi$
$678$		0			0
$679$		−	133.077i		−	0.195990i
$680$		0			0
$681$	442.014			0.649066
$682$		0			0
$683$		−	796.270i		−	1.16584i	−0.812529	−	0.582921i	$-0.801910\pi$
$683$		−	796.270i		−	1.16584i	0.812529	−	0.582921i	$-0.198090\pi$
$684$		0			0
$685$	−1152.71			−1.68279
$686$		0			0
$687$		−	448.415i		−	0.652715i
$688$		0			0
$689$	256.329	+	653.967i	0.372031	+	0.949153i
$690$		0			0
$691$		−	491.533i		−	0.711336i	−0.934612	−	0.355668i	$-0.884253\pi$
$691$		−	491.533i		−	0.711336i	0.934612	−	0.355668i	$-0.115747\pi$
$692$		0			0
$693$	30.9946	−	28.6585i	0.0447253	−	0.0413542i
$694$		0			0
$695$	279.759			0.402530
$696$		0			0
$697$			519.110i			0.744778i
$698$		0			0
$699$			196.011i			0.280417i
$700$		0			0
$701$			939.666i			1.34047i	0.742151	+	0.670233i	$0.233806\pi$
$701$			939.666i			1.34047i	−0.742151	+	0.670233i	$0.766194\pi$
$702$		0			0
$703$			70.4321i			0.100188i
$704$		0			0
$705$	−107.202			−0.152059
$706$		0			0
$707$			55.2472i			0.0781431i
$708$		0			0
$709$		−	978.127i		−	1.37959i	−0.724006	−	0.689793i	$-0.757701\pi$
$709$		−	978.127i		−	1.37959i	0.724006	−	0.689793i	$-0.242299\pi$
$710$		0			0
$711$		−	130.123i		−	0.183015i
$712$		0			0
$713$			779.099i			1.09271i
$714$		0			0
$715$	288.743	−	596.245i	0.403836	−	0.833909i
$716$		0			0
$717$	−777.657			−1.08460
$718$		0			0
$719$	61.5802			0.0856471			0.0428235	−	0.999083i	$-0.486365\pi$
$719$	61.5802			0.0856471			0.0428235	+	0.999083i	$0.486365\pi$
$720$		0			0
$721$	55.3690			0.0767948
$722$		0			0
$723$	726.872			1.00536
$724$		0			0
$725$			151.010i			0.208289i
$726$		0			0
$727$	−866.193			−1.19146			−0.595731	−	0.803184i	$-0.703138\pi$
$727$	−866.193			−1.19146			−0.595731	+	0.803184i	$0.703138\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	245.814			0.336271
$732$		0			0
$733$	702.171			0.957941			0.478971	−	0.877831i	$-0.341010\pi$
$733$	702.171			0.957941			0.478971	+	0.877831i	$0.341010\pi$
$734$		0			0
$735$			380.052i			0.517078i
$736$		0			0
$737$	112.800	+	121.995i	0.153053	+	0.165529i
$738$		0			0
$739$	1012.21			1.36971			0.684854	−	0.728681i	$-0.259866\pi$
$739$	1012.21			1.36971			0.684854	+	0.728681i	$0.259866\pi$
$740$		0			0
$741$	−44.1287	−	112.585i	−0.0595529	−	0.151936i
$742$		0			0
$743$	−722.186			−0.971986			−0.485993	−	0.873963i	$-0.661542\pi$
$743$	−722.186			−0.971986			−0.485993	+	0.873963i	$0.661542\pi$
$744$		0			0
$745$		−	437.570i		−	0.587342i
$746$		0			0
$747$	349.827			0.468309
$748$		0			0
$749$		−	189.891i		−	0.253527i
$750$		0			0
$751$	−1348.19			−1.79519			−0.897595	−	0.440821i	$-0.854687\pi$
$751$	−1348.19			−1.79519			−0.897595	+	0.440821i	$0.854687\pi$
$752$		0			0
$753$	160.112			0.212632
$754$		0			0
$755$		−	1086.38i		−	1.43891i
$756$		0			0
$757$	523.250			0.691215			0.345608	−	0.938379i	$-0.387673\pi$
$757$	523.250			0.691215			0.345608	+	0.938379i	$0.387673\pi$
$758$		0			0
$759$	601.691	−	556.340i	0.792742	−	0.732991i
$760$		0			0
$761$	−1233.13			−1.62041			−0.810205	−	0.586147i	$-0.800644\pi$
$761$	−1233.13			−1.62041			−0.810205	+	0.586147i	$0.800644\pi$
$762$		0			0
$763$	260.926			0.341974
$764$		0			0
$765$	206.308			0.269683
$766$		0			0
$767$	−477.313	+	187.088i	−0.622312	+	0.243922i
$768$		0			0
$769$	−901.606			−1.17244			−0.586220	−	0.810152i	$-0.699384\pi$
$769$	−901.606			−1.17244			−0.586220	+	0.810152i	$0.699384\pi$
$770$		0			0
$771$	−120.877			−0.156780
$772$		0			0
$773$			957.704i			1.23894i	0.785019	+	0.619472i	$0.212653\pi$
$773$			957.704i			1.23894i	−0.785019	+	0.619472i	$0.787347\pi$
$774$		0			0
$775$			64.0828i			0.0826874i
$776$		0			0
$777$			29.0576i			0.0373972i
$778$		0			0
$779$	−187.807			−0.241087
$780$		0			0
$781$	718.238	+	776.786i	0.919639	+	0.994604i
$782$		0			0
$783$		−	221.796i		−	0.283264i
$784$		0			0
$785$			389.653i			0.496373i
$786$		0			0
$787$	−394.595			−0.501391			−0.250696	−	0.968066i	$-0.580659\pi$
$787$	−394.595			−0.501391			−0.250696	+	0.968066i	$0.580659\pi$
$788$		0			0
$789$			346.344i			0.438966i
$790$		0			0
$791$	187.736			0.237340
$792$		0			0
$793$	167.700	−	65.7317i	0.211475	−	0.0828899i
$794$		0			0
$795$		−	433.555i		−	0.545352i
$796$		0			0
$797$	−46.5727			−0.0584350			−0.0292175	−	0.999573i	$-0.509302\pi$
$797$	−46.5727			−0.0584350			−0.0292175	+	0.999573i	$0.509302\pi$
$798$		0			0
$799$	−198.318			−0.248207
$800$		0			0
$801$			365.215i			0.455948i
$802$		0			0
$803$	641.041	−	592.725i	0.798308	−	0.738138i
$804$		0			0
$805$		−	254.894i		−	0.316638i
$806$		0			0
$807$	−333.439			−0.413184
$808$		0			0
$809$		−	1253.77i		−	1.54978i	−0.632095	−	0.774891i	$-0.717805\pi$
$809$		−	1253.77i		−	1.54978i	0.632095	−	0.774891i	$-0.282195\pi$
$810$		0			0
$811$	−105.160			−0.129667			−0.0648333	−	0.997896i	$-0.520652\pi$
$811$	−105.160			−0.129667			−0.0648333	+	0.997896i	$0.520652\pi$
$812$		0			0
$813$	−197.935			−0.243462
$814$		0			0
$815$	397.031			0.487154
$816$		0			0
$817$			88.9322i			0.108852i
$818$		0			0
$819$	−18.2058	−	46.4481i	−0.0222293	−	0.0567132i
$820$		0			0
$821$	418.169			0.509341			0.254671	−	0.967028i	$-0.418033\pi$
$821$	418.169			0.509341			0.254671	+	0.967028i	$0.418033\pi$
$822$		0			0
$823$	−170.244			−0.206858			−0.103429	−	0.994637i	$-0.532981\pi$
$823$	−170.244			−0.206858			−0.103429	+	0.994637i	$0.532981\pi$
$824$		0			0
$825$	49.4905	−	45.7603i	0.0599885	−	0.0554671i
$826$		0			0
$827$	−86.8711			−0.105044			−0.0525218	−	0.998620i	$-0.516726\pi$
$827$	−86.8711			−0.105044			−0.0525218	+	0.998620i	$0.516726\pi$
$828$		0			0
$829$	1326.18			1.59973			0.799865	−	0.600180i	$-0.204905\pi$
$829$	1326.18			1.59973			0.799865	+	0.600180i	$0.204905\pi$
$830$		0			0
$831$		−	404.955i		−	0.487311i
$832$		0			0
$833$			703.075i			0.844028i
$834$		0			0
$835$		−	964.619i		−	1.15523i
$836$		0			0
$837$		−	94.1217i		−	0.112451i
$838$		0			0
$839$		−	1464.44i		−	1.74546i	−0.488207	−	0.872728i	$-0.662349\pi$
$839$		−	1464.44i		−	1.74546i	0.488207	−	0.872728i	$-0.337651\pi$
$840$		0			0
$841$	−980.974			−1.16644
$842$		0			0
$843$	−760.667			−0.902334
$844$		0			0
$845$	−532.021	−	574.400i	−0.629610	−	0.679764i
$846$		0			0
$847$	−12.1046	+	154.309i	−0.0142911	+	0.182183i
$848$		0			0
$849$			74.0127i			0.0871763i
$850$		0			0
$851$			564.088i			0.662853i
$852$		0			0
$853$	557.179			0.653199			0.326600	−	0.945163i	$-0.394097\pi$
$853$	557.179			0.653199			0.326600	+	0.945163i	$0.394097\pi$
$854$		0			0
$855$			74.6392i			0.0872973i
$856$		0			0
$857$			289.853i			0.338218i	0.985597	+	0.169109i	$0.0540890\pi$
$857$			289.853i			0.338218i	−0.985597	+	0.169109i	$0.945911\pi$
$858$		0			0
$859$	−724.117			−0.842977			−0.421489	−	0.906834i	$-0.638492\pi$
$859$	−724.117			−0.842977			−0.421489	+	0.906834i	$0.638492\pi$
$860$		0			0
$861$	−77.4820			−0.0899907
$862$		0			0
$863$			1073.56i			1.24399i	0.783022	+	0.621994i	$0.213677\pi$
$863$			1073.56i			1.24399i	−0.783022	+	0.621994i	$0.786323\pi$
$864$		0			0
$865$	304.087			0.351546
$866$		0			0
$867$	−118.905			−0.137145
$868$		0			0
$869$	323.914	+	350.318i	0.372743	+	0.403127i
$870$		0			0
$871$	182.820	−	71.6581i	0.209896	−	0.0822711i
$872$		0			0
$873$			312.096i			0.357498i
$874$		0			0
$875$		−	169.120i		−	0.193280i
$876$		0			0
$877$	−689.735			−0.786471			−0.393235	−	0.919438i	$-0.628644\pi$
$877$	−689.735			−0.786471			−0.393235	+	0.919438i	$0.628644\pi$
$878$		0			0
$879$	431.290			0.490659
$880$		0			0
$881$	1240.81			1.40842			0.704208	−	0.709994i	$-0.251302\pi$
$881$	1240.81			1.40842			0.704208	+	0.709994i	$0.251302\pi$
$882$		0			0
$883$	713.902			0.808496			0.404248	−	0.914649i	$-0.367533\pi$
$883$	713.902			0.808496			0.404248	+	0.914649i	$0.367533\pi$
$884$		0			0
$885$	316.440			0.357559
$886$		0			0
$887$		−	279.934i		−	0.315597i	−0.987471	−	0.157798i	$-0.949560\pi$
$887$		−	279.934i		−	0.315597i	0.987471	−	0.157798i	$-0.0504396\pi$
$888$		0			0
$889$			180.244i			0.202750i
$890$		0			0
$891$	−72.6893	+	67.2106i	−0.0815817	+	0.0754328i
$892$		0			0
$893$		−	71.7486i		−	0.0803455i
$894$		0			0
$895$		−	641.594i		−	0.716865i
$896$		0			0
$897$	−353.425	−	901.685i	−0.394008	−	1.00522i
$898$		0			0
$899$	−773.177			−0.860041
$900$		0			0
$901$		−	802.052i		−	0.890180i
$902$		0			0
$903$			36.6900i			0.0406313i
$904$		0			0
$905$			77.9142i			0.0860930i
$906$		0			0
$907$	−368.133			−0.405880			−0.202940	−	0.979191i	$-0.565050\pi$
$907$	−368.133			−0.405880			−0.202940	+	0.979191i	$0.565050\pi$
$908$		0			0
$909$		−	129.567i		−	0.142538i
$910$		0			0
$911$	521.392			0.572329			0.286165	−	0.958180i	$-0.407620\pi$
$911$	521.392			0.572329			0.286165	+	0.958180i	$0.407620\pi$
$912$		0			0
$913$	−941.803	+	870.818i	−1.03155	+	0.953798i
$914$		0			0
$915$	−111.179			−0.121507
$916$		0			0
$917$			207.506i			0.226288i
$918$		0			0
$919$		−	73.6419i		−	0.0801326i	−0.999197	−	0.0400663i	$-0.987243\pi$
$919$		−	73.6419i		−	0.0801326i	0.999197	−	0.0400663i	$-0.0127569\pi$
$920$		0			0
$921$	119.481			0.129730
$922$		0			0
$923$	1164.08	−	456.273i	1.26119	−	0.494337i
$924$		0			0
$925$			46.3976i			0.0501595i
$926$		0			0
$927$	−129.853			−0.140079
$928$		0			0
$929$		−	135.750i		−	0.146125i	−0.997327	−	0.0730627i	$-0.976723\pi$
$929$		−	135.750i		−	0.146125i	0.997327	−	0.0730627i	$-0.0232773\pi$
$930$		0			0
$931$	−254.363			−0.273215
$932$		0			0
$933$	−437.503			−0.468920
$934$		0			0
$935$	−555.420	+	513.557i	−0.594033	+	0.549259i
$936$		0			0
$937$		−	279.870i		−	0.298687i	−0.988785	−	0.149343i	$-0.952284\pi$
$937$		−	279.870i		−	0.298687i	0.988785	−	0.149343i	$-0.0477160\pi$
$938$		0			0
$939$	29.7664			0.0317001
$940$		0			0
$941$	1312.78			1.39509			0.697544	−	0.716542i	$-0.254276\pi$
$941$	1312.78			1.39509			0.697544	+	0.716542i	$0.254276\pi$
$942$		0			0
$943$	−1504.14			−1.59506
$944$		0			0
$945$			30.7933i			0.0325855i
$946$		0			0
$947$		−	924.043i		−	0.975758i	−0.872911	−	0.487879i	$-0.837771\pi$
$947$		−	924.043i		−	0.975758i	0.872911	−	0.487879i	$-0.162229\pi$
$948$		0			0
$949$	−376.539	−	960.655i	−0.396774	−	1.01228i
$950$		0			0
$951$		−	179.549i		−	0.188801i
$952$		0			0
$953$			122.246i			0.128275i	0.997941	+	0.0641375i	$0.0204296\pi$
$953$			122.246i			0.128275i	−0.997941	+	0.0641375i	$0.979570\pi$
$954$		0			0
$955$			858.381i			0.898828i
$956$		0			0
$957$	552.112	+	597.117i	0.576919	+	0.623947i
$958$		0			0
$959$		−	318.287i		−	0.331895i
$960$		0			0
$961$	632.893			0.658577
$962$		0			0
$963$			445.338i			0.462448i
$964$		0			0
$965$			86.1870i			0.0893129i
$966$		0			0
$967$	961.764			0.994585			0.497293	−	0.867583i	$-0.334327\pi$
$967$	961.764			0.994585			0.497293	+	0.867583i	$0.334327\pi$
$968$		0			0
$969$			138.078i			0.142496i
$970$		0			0
$971$	−851.727			−0.877165			−0.438583	−	0.898691i	$-0.644519\pi$
$971$	−851.727			−0.877165			−0.438583	+	0.898691i	$0.644519\pi$
$972$		0			0
$973$			77.2473i			0.0793908i
$974$		0			0
$975$	−29.0701	−	74.1657i	−0.0298154	−	0.0760674i
$976$		0			0
$977$			823.958i			0.843355i	0.906746	+	0.421678i	$0.138559\pi$
$977$			823.958i			0.843355i	−0.906746	+	0.421678i	$0.861441\pi$
$978$		0			0
$979$	−909.122	−	983.230i	−0.928623	−	1.00432i
$980$		0			0
$981$	−611.930			−0.623782
$982$		0			0
$983$		−	73.0474i		−	0.0743107i	−0.999310	−	0.0371553i	$-0.988170\pi$
$983$		−	73.0474i		−	0.0743107i	0.999310	−	0.0371553i	$-0.0118296\pi$
$984$		0			0
$985$		−	88.0766i		−	0.0894179i
$986$		0			0
$987$		−	29.6007i		−	0.0299906i
$988$		0			0
$989$			712.255i			0.720177i
$990$		0			0
$991$	−145.246			−0.146565			−0.0732826	−	0.997311i	$-0.523348\pi$
$991$	−145.246			−0.146565			−0.0732826	+	0.997311i	$0.523348\pi$
$992$		0			0
$993$			177.293i			0.178543i
$994$		0			0
$995$		−	1605.28i		−	1.61335i
$996$		0			0
$997$			458.808i			0.460189i	0.973168	+	0.230094i	$0.0739035\pi$
$997$			458.808i			0.460189i	−0.973168	+	0.230094i	$0.926097\pi$
$998$		0			0
$999$		−	68.1465i		−	0.0682148i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1716.3.g.a.1429.13	✓	56
11.10	odd	2	inner	1716.3.g.a.1429.16	yes	56
13.12	even	2	inner	1716.3.g.a.1429.14	yes	56
143.142	odd	2	inner	1716.3.g.a.1429.15	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.3.g.a.1429.13	✓	56	1.1	even	1	trivial
1716.3.g.a.1429.14	yes	56	13.12	even	2	inner
1716.3.g.a.1429.15	yes	56	143.142	odd	2	inner
1716.3.g.a.1429.16	yes	56	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1716.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +4.63273i q^{5} -1.27920 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +4.63273i q^{5} -1.27920 q^{7} +3.00000 q^{9} +(-8.07659 + 7.46784i) q^{11} +(4.74408 + 12.1035i) q^{13} -8.02413i q^{15} -14.8442i q^{17} +5.37043 q^{19} +2.21563 q^{21} +43.0115 q^{23} +3.53780 q^{25} -5.19615 q^{27} +42.6846i q^{29} +18.1137i q^{31} +(13.9891 - 12.9347i) q^{33} -5.92617i q^{35} +13.1148i q^{37} +(-8.21699 - 20.9638i) q^{39} -34.9706 q^{41} +16.5596i q^{43} +13.8982i q^{45} -13.3599i q^{47} -47.3637 q^{49} +25.7109i q^{51} +54.0314 q^{53} +(-34.5965 - 37.4167i) q^{55} -9.30185 q^{57} +39.4361i q^{59} -13.8555i q^{61} -3.83759 q^{63} +(-56.0721 + 21.9780i) q^{65} -15.1048i q^{67} -74.4981 q^{69} -96.1774i q^{71} -79.3703 q^{73} -6.12765 q^{75} +(10.3315 - 9.55283i) q^{77} -43.3745i q^{79} +9.00000 q^{81} +116.609 q^{83} +68.7692 q^{85} -73.9319i q^{87} +121.738i q^{89} +(-6.06861 - 15.4827i) q^{91} -31.3739i q^{93} +24.8797i q^{95} +104.032i q^{97} +(-24.2298 + 22.4035i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 168 q^{9}+O(q^{10}) \) 56 * q + 168 * q^9	\( 56 q + 168 q^{9} - 72 q^{23} - 336 q^{25} + 576 q^{49} + 96 q^{53} + 32 q^{55} - 48 q^{69} + 96 q^{75} - 496 q^{77} + 504 q^{81} - 504 q^{91}+O(q^{100}) \) 56 * q + 168 * q^9 - 72 * q^23 - 336 * q^25 + 576 * q^49 + 96 * q^53 + 32 * q^55 - 48 * q^69 + 96 * q^75 - 496 * q^77 + 504 * q^81 - 504 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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