LMFDB - Embedded newform 684.5.y.f.145.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.3
Character	$\chi$	$=$	684.145
Dual form			684.5.y.f.217.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-18.6023 - 32.2201i) q^{5} -25.9074 q^{7} +O(q^{10})$	$q+(-18.6023 - 32.2201i) q^{5} -25.9074 q^{7} -189.278 q^{11} +(111.709 + 64.4949i) q^{13} +(-160.296 - 277.640i) q^{17} +(-8.91800 - 360.890i) q^{19} +(-294.572 + 510.213i) q^{23} +(-379.589 + 657.467i) q^{25} +(-788.932 - 455.490i) q^{29} -1657.55i q^{31} +(481.936 + 834.737i) q^{35} +1024.72i q^{37} +(-1389.77 + 802.383i) q^{41} +(905.234 + 1567.91i) q^{43} +(-516.200 + 894.084i) q^{47} -1729.81 q^{49} +(-656.643 - 379.113i) q^{53} +(3521.00 + 6098.56i) q^{55} +(896.448 - 517.564i) q^{59} +(2769.93 - 4797.67i) q^{61} -4799.01i q^{65} +(1014.94 + 585.977i) q^{67} +(6058.10 - 3497.65i) q^{71} +(3097.26 + 5364.62i) q^{73} +4903.70 q^{77} +(4443.93 - 2565.70i) q^{79} +3973.43 q^{83} +(-5963.72 + 10329.5i) q^{85} +(7936.06 + 4581.89i) q^{89} +(-2894.07 - 1670.89i) q^{91} +(-11462.0 + 7000.71i) q^{95} +(-9602.98 + 5544.28i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 304 q^{7}+O(q^{10}) $ 24 * q + 304 * q^7	$ 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) $ 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$5$	−18.6023	−	32.2201i	−0.744091	−	1.28880i	−0.950618	−	0.310362i	$-0.899550\pi$
$5$	−18.6023	−	32.2201i	−0.744091	−	1.28880i	0.206527	−	0.978441i	$-0.433784\pi$
$6$		0			0
$7$	−25.9074			−0.528722			−0.264361	−	0.964424i	$-0.585161\pi$
$7$	−25.9074			−0.528722			−0.264361	+	0.964424i	$0.585161\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−189.278			−1.56428			−0.782141	−	0.623102i	$-0.785872\pi$
$11$	−189.278			−1.56428			−0.782141	+	0.623102i	$0.785872\pi$
$12$		0			0
$13$	111.709	+	64.4949i	0.660997	+	0.381627i	0.792657	−	0.609668i	$-0.208697\pi$
$13$	111.709	+	64.4949i	0.660997	+	0.381627i	−0.131660	+	0.991295i	$0.542031\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−160.296	−	277.640i	−0.554656	−	0.960692i	−0.997930	−	0.0643064i	$-0.979517\pi$
$17$	−160.296	−	277.640i	−0.554656	−	0.960692i	0.443274	−	0.896386i	$-0.353817\pi$
$18$		0			0
$19$	−8.91800	−	360.890i	−0.0247036	−	0.999695i
$19$	−8.91800	−	360.890i	−0.0247036	−	0.999695i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−294.572	+	510.213i	−0.556846	+	0.964486i	0.440911	+	0.897551i	$0.354655\pi$
$23$	−294.572	+	510.213i	−0.556846	+	0.964486i	−0.997757	+	0.0669351i	$0.978678\pi$
$24$		0			0
$25$	−379.589	+	657.467i	−0.607342	+	1.05195i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−788.932	−	455.490i	−0.938088	−	0.541605i	−0.0487275	−	0.998812i	$-0.515517\pi$
$29$	−788.932	−	455.490i	−0.938088	−	0.541605i	−0.889360	+	0.457207i	$0.848850\pi$
$30$		0			0
$31$		−	1657.55i		−	1.72482i	−0.506209	−	0.862411i	$-0.668953\pi$
$31$		−	1657.55i		−	1.72482i	0.506209	−	0.862411i	$-0.331047\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	481.936	+	834.737i	0.393417	+	0.681418i
$36$		0			0
$37$			1024.72i			0.748516i	0.927325	+	0.374258i	$0.122102\pi$
$37$			1024.72i			0.748516i	−0.927325	+	0.374258i	$0.877898\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1389.77	+	802.383i	−0.826751	+	0.477325i	−0.852739	−	0.522337i	$-0.825060\pi$
$41$	−1389.77	+	802.383i	−0.826751	+	0.477325i	0.0259880	+	0.999662i	$0.491727\pi$
$42$		0			0
$43$	905.234	+	1567.91i	0.489580	+	0.847978i	0.999928	−	0.0119902i	$-0.00381670\pi$
$43$	905.234	+	1567.91i	0.489580	+	0.847978i	−0.510348	+	0.859968i	$0.670483\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−516.200	+	894.084i	−0.233680	+	0.404746i	−0.958888	−	0.283784i	$-0.908410\pi$
$47$	−516.200	+	894.084i	−0.233680	+	0.404746i	0.725208	+	0.688530i	$0.241744\pi$
$48$		0			0
$49$	−1729.81			−0.720453
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−656.643	−	379.113i	−0.233764	−	0.134964i	0.378543	−	0.925584i	$-0.376425\pi$
$53$	−656.643	−	379.113i	−0.233764	−	0.134964i	−0.612307	+	0.790620i	$0.709758\pi$
$54$		0			0
$55$	3521.00	+	6098.56i	1.16397	+	2.01605i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	896.448	−	517.564i	0.257526	−	0.148683i	−0.365680	−	0.930741i	$-0.619163\pi$
$59$	896.448	−	517.564i	0.257526	−	0.148683i	0.623205	+	0.782058i	$0.285830\pi$
$60$		0			0
$61$	2769.93	−	4797.67i	0.744406	−	1.28935i	−0.206066	−	0.978538i	$-0.566066\pi$
$61$	2769.93	−	4797.67i	0.744406	−	1.28935i	0.950472	−	0.310810i	$-0.100600\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4799.01i		−	1.13586i
$66$		0			0
$67$	1014.94	+	585.977i	0.226095	+	0.130536i	0.608769	−	0.793347i	$-0.291663\pi$
$67$	1014.94	+	585.977i	0.226095	+	0.130536i	−0.382674	+	0.923883i	$0.624997\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6058.10	−	3497.65i	1.20177	−	0.693840i	0.240818	−	0.970570i	$-0.422584\pi$
$71$	6058.10	−	3497.65i	1.20177	−	0.693840i	0.960948	+	0.276731i	$0.0892510\pi$
$72$		0			0
$73$	3097.26	+	5364.62i	0.581209	+	1.00668i	0.995336	+	0.0964646i	$0.0307535\pi$
$73$	3097.26	+	5364.62i	0.581209	+	1.00668i	−0.414127	+	0.910219i	$0.635913\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4903.70			0.827070
$78$		0			0
$79$	4443.93	−	2565.70i	0.712054	−	0.411105i	−0.0997669	−	0.995011i	$-0.531810\pi$
$79$	4443.93	−	2565.70i	0.712054	−	0.411105i	0.811821	+	0.583906i	$0.198476\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3973.43			0.576779			0.288390	−	0.957513i	$-0.406880\pi$
$83$	3973.43			0.576779			0.288390	+	0.957513i	$0.406880\pi$
$84$		0			0
$85$	−5963.72	+	10329.5i	−0.825429	+	1.42968i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7936.06	+	4581.89i	1.00190	+	0.578448i	0.908810	−	0.417209i	$-0.136992\pi$
$89$	7936.06	+	4581.89i	1.00190	+	0.578448i	0.0930914	+	0.995658i	$0.470325\pi$
$90$		0			0
$91$	−2894.07	−	1670.89i	−0.349484	−	0.201774i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11462.0	+	7000.71i	−1.27003	+	0.775702i
$96$		0			0
$97$	−9602.98	+	5544.28i	−1.02062	+	0.589253i	−0.914282	−	0.405077i	$-0.867245\pi$
$97$	−9602.98	+	5544.28i	−1.02062	+	0.589253i	−0.106334	+	0.994330i	$0.533911\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2172.57	−	3763.01i	0.212977	−	0.368886i	−0.739668	−	0.672972i	$-0.765018\pi$
$101$	2172.57	−	3763.01i	0.212977	−	0.368886i	0.952645	+	0.304086i	$0.0983509\pi$
$102$		0			0
$103$			14532.9i			1.36987i	0.728606	+	0.684933i	$0.240169\pi$
$103$			14532.9i			1.36987i	−0.728606	+	0.684933i	$0.759831\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16840.7i			1.47093i	0.677561	+	0.735467i	$0.263037\pi$
$107$			16840.7i			1.47093i	−0.677561	+	0.735467i	$0.736963\pi$
$108$		0			0
$109$	14906.5	−	8606.29i	1.25465	−	0.724374i	0.282623	−	0.959231i	$-0.408795\pi$
$109$	14906.5	−	8606.29i	1.25465	−	0.724374i	0.972030	+	0.234857i	$0.0754620\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	15278.5i		−	1.19653i	−0.801299	−	0.598265i	$-0.795857\pi$
$113$		−	15278.5i		−	1.19653i	0.801299	−	0.598265i	$-0.204143\pi$
$114$		0			0
$115$	21918.8			1.65738
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4152.84	+	7192.92i	0.293259	+	0.507939i
$120$		0			0
$121$	21185.2			1.44698
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4992.03			0.319490
$126$		0			0
$127$	1191.50	+	687.911i	0.0738729	+	0.0426506i	0.536481	−	0.843912i	$-0.319753\pi$
$127$	1191.50	+	687.911i	0.0738729	+	0.0426506i	−0.462609	+	0.886563i	$0.653086\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2974.72	−	5152.37i	−0.173342	−	0.300237i	0.766244	−	0.642549i	$-0.222123\pi$
$131$	−2974.72	−	5152.37i	−0.173342	−	0.300237i	−0.939586	+	0.342312i	$0.888790\pi$
$132$		0			0
$133$	231.042	+	9349.70i	0.0130613	+	0.528560i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1771.01	+	3067.49i	−0.0943585	+	0.163434i	−0.909341	−	0.416052i	$-0.863413\pi$
$137$	−1771.01	+	3067.49i	−0.0943585	+	0.163434i	0.814982	+	0.579486i	$0.196747\pi$
$138$		0			0
$139$	−15505.8	+	26856.9i	−0.802537	+	1.39003i	0.115405	+	0.993319i	$0.463183\pi$
$139$	−15505.8	+	26856.9i	−0.802537	+	1.39003i	−0.917942	+	0.396716i	$0.870150\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−21144.0	−	12207.5i	−1.03399	−	0.596972i
$144$		0			0
$145$			33892.6i			1.61201i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16399.8	−	28405.3i	−0.738698	−	1.27946i	−0.953082	−	0.302713i	$-0.902108\pi$
$149$	−16399.8	−	28405.3i	−0.738698	−	1.27946i	0.214384	−	0.976749i	$-0.431226\pi$
$150$		0			0
$151$			19290.6i			0.846043i	0.906120	+	0.423021i	$0.139030\pi$
$151$			19290.6i			0.846043i	−0.906120	+	0.423021i	$0.860970\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−53406.5	+	30834.3i	−2.22296	+	1.28342i
$156$		0			0
$157$	13160.1	+	22794.0i	0.533900	+	0.924742i	0.999216	+	0.0395974i	$0.0126075\pi$
$157$	13160.1	+	22794.0i	0.533900	+	0.924742i	−0.465316	+	0.885145i	$0.654059\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7631.57	−	13218.3i	0.294417	−	0.509945i
$162$		0			0
$163$	37908.8			1.42680			0.713402	−	0.700755i	$-0.247153\pi$
$163$	37908.8			1.42680			0.713402	+	0.700755i	$0.247153\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−22176.3	−	12803.5i	−0.795163	−	0.459088i	0.0466139	−	0.998913i	$-0.485157\pi$
$167$	−22176.3	−	12803.5i	−0.795163	−	0.459088i	−0.841777	+	0.539825i	$0.818490\pi$
$168$		0			0
$169$	−5961.30	−	10325.3i	−0.208722	−	0.361517i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−20073.0	+	11589.1i	−0.670687	+	0.387221i	−0.796337	−	0.604854i	$-0.793232\pi$
$173$	−20073.0	+	11589.1i	−0.670687	+	0.387221i	0.125650	+	0.992075i	$0.459898\pi$
$174$		0			0
$175$	9834.15	−	17033.2i	0.321115	−	0.556188i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			34080.4i			1.06365i	0.846854	+	0.531826i	$0.178494\pi$
$179$			34080.4i			1.06365i	−0.846854	+	0.531826i	$0.821506\pi$
$180$		0			0
$181$	−9053.50	−	5227.04i	−0.276350	−	0.159551i	0.355420	−	0.934707i	$-0.384338\pi$
$181$	−9053.50	−	5227.04i	−0.276350	−	0.159551i	−0.631770	+	0.775156i	$0.717671\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	33016.5	−	19062.1i	0.964689	−	0.556964i
$186$		0			0
$187$	30340.4	+	52551.2i	0.867638	+	1.50279i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	28229.8			0.773822			0.386911	−	0.922117i	$-0.373542\pi$
$191$	28229.8			0.773822			0.386911	+	0.922117i	$0.373542\pi$
$192$		0			0
$193$	−12133.7	+	7005.37i	−0.325744	+	0.188069i	−0.653950	−	0.756538i	$-0.726889\pi$
$193$	−12133.7	+	7005.37i	−0.325744	+	0.188069i	0.328206	+	0.944606i	$0.393556\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	60171.8			1.55046			0.775230	−	0.631679i	$-0.217634\pi$
$197$	60171.8			1.55046			0.775230	+	0.631679i	$0.217634\pi$
$198$		0			0
$199$	4076.45	−	7060.62i	0.102938	−	0.178294i	−0.809956	−	0.586491i	$-0.800509\pi$
$199$	4076.45	−	7060.62i	0.102938	−	0.178294i	0.912894	+	0.408197i	$0.133842\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20439.1	+	11800.5i	0.495987	+	0.286358i
$204$		0			0
$205$	51705.7	+	29852.3i	1.23036	+	0.710346i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1687.98	+	68308.5i	0.0386434	+	1.56380i
$210$		0			0
$211$	37338.0	−	21557.1i	0.838660	−	0.484201i	−0.0181482	−	0.999835i	$-0.505777\pi$
$211$	37338.0	−	21557.1i	0.838660	−	0.484201i	0.856809	+	0.515634i	$0.172444\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	33678.8	−	58333.4i	0.728584	−	1.26195i
$216$		0			0
$217$			42942.9i			0.911951i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	41353.0i		−	0.846687i
$222$		0			0
$223$	−73783.8	+	42599.1i	−1.48372	+	0.856625i	−0.999829	−	0.0185052i	$-0.994109\pi$
$223$	−73783.8	+	42599.1i	−1.48372	+	0.856625i	−0.483888	+	0.875130i	$0.660776\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			51033.0i			0.990374i	0.868787	+	0.495187i	$0.164900\pi$
$227$			51033.0i			0.990374i	−0.868787	+	0.495187i	$0.835100\pi$
$228$		0			0
$229$	−69442.5			−1.32420			−0.662101	−	0.749414i	$-0.730335\pi$
$229$	−69442.5			−1.32420			−0.662101	+	0.749414i	$0.730335\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6428.13	−	11133.8i	−0.118406	−	0.205085i	0.800730	−	0.599025i	$-0.204445\pi$
$233$	−6428.13	−	11133.8i	−0.118406	−	0.205085i	−0.919136	+	0.393940i	$0.871112\pi$
$234$		0			0
$235$	38410.0			0.695518
$236$		0			0
$237$		0			0
$238$		0			0
$239$	80974.3			1.41759			0.708796	−	0.705414i	$-0.249239\pi$
$239$	80974.3			1.41759			0.708796	+	0.705414i	$0.249239\pi$
$240$		0			0
$241$	20674.8	+	11936.6i	0.355965	+	0.205516i	0.667309	−	0.744781i	$-0.267446\pi$
$241$	20674.8	+	11936.6i	0.355965	+	0.205516i	−0.311344	+	0.950297i	$0.600779\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	32178.4	+	55734.6i	0.536083	+	0.928523i
$246$		0			0
$247$	22279.4	−	40889.6i	0.365181	−	0.670223i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18715.1	−	32415.5i	0.297060	−	0.514523i	−0.678402	−	0.734691i	$-0.737327\pi$
$251$	18715.1	−	32415.5i	0.297060	−	0.514523i	0.975462	+	0.220168i	$0.0706605\pi$
$252$		0			0
$253$	55756.0	−	96572.1i	0.871064	−	1.50873i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−41785.0	−	24124.6i	−0.632636	−	0.365253i	0.149136	−	0.988817i	$-0.452351\pi$
$257$	−41785.0	−	24124.6i	−0.632636	−	0.365253i	−0.781772	+	0.623564i	$0.785684\pi$
$258$		0			0
$259$		−	26547.7i		−	0.395756i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−34898.2	−	60445.5i	−0.504536	−	0.873882i	−0.999986	−	0.00524540i	$-0.998330\pi$
$263$	−34898.2	−	60445.5i	−0.504536	−	0.873882i	0.495450	−	0.868636i	$-0.335003\pi$
$264$		0			0
$265$			28209.5i			0.401701i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−97588.6	+	56342.8i	−1.34864	+	0.778635i	−0.988056	−	0.154093i	$-0.950754\pi$
$269$	−97588.6	+	56342.8i	−1.34864	+	0.778635i	−0.360579	+	0.932729i	$0.617421\pi$
$270$		0			0
$271$	32603.7	+	56471.3i	0.443944	+	0.768934i	0.997978	−	0.0635601i	$-0.0202455\pi$
$271$	32603.7	+	56471.3i	0.443944	+	0.768934i	−0.554034	+	0.832494i	$0.686912\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	71847.9	−	124444.i	0.950055	−	1.64554i
$276$		0			0
$277$	−148562.			−1.93620			−0.968098	−	0.250573i	$-0.919381\pi$
$277$	−148562.			−1.93620			−0.968098	+	0.250573i	$0.919381\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−68337.5	−	39454.7i	−0.865459	−	0.499673i	0.000377502	−	1.00000i	$-0.499880\pi$
$281$	−68337.5	−	39454.7i	−0.865459	−	0.499673i	−0.865837	+	0.500327i	$0.833213\pi$
$282$		0			0
$283$	−27276.6	−	47244.4i	−0.340578	−	0.589899i	0.643962	−	0.765058i	$-0.277290\pi$
$283$	−27276.6	−	47244.4i	−0.340578	−	0.589899i	−0.984540	+	0.175158i	$0.943956\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	36005.2	−	20787.6i	0.437121	−	0.252372i
$288$		0			0
$289$	−9628.86	+	16677.7i	−0.115287	+	0.199682i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			51381.0i			0.598505i	0.954174	+	0.299252i	$0.0967372\pi$
$293$			51381.0i			0.598505i	−0.954174	+	0.299252i	$0.903263\pi$
$294$		0			0
$295$	−33351.9	−	19255.7i	−0.383245	−	0.221267i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−65812.3	+	37996.8i	−0.736147	+	0.425015i
$300$		0			0
$301$	−23452.2	−	40620.4i	−0.258852	−	0.448344i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−206108.			−2.21562
$306$		0			0
$307$	−38744.0	+	22368.9i	−0.411081	+	0.237338i	−0.691254	−	0.722612i	$-0.742942\pi$
$307$	−38744.0	+	22368.9i	−0.411081	+	0.237338i	0.280173	+	0.959950i	$0.409608\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	135143.			1.39724			0.698622	−	0.715491i	$-0.253797\pi$
$311$	135143.			1.39724			0.698622	+	0.715491i	$0.253797\pi$
$312$		0			0
$313$	63850.3	−	110592.i	0.651739	−	1.12885i	−0.330961	−	0.943644i	$-0.607373\pi$
$313$	63850.3	−	110592.i	0.651739	−	1.12885i	0.982701	−	0.185201i	$-0.0592937\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	121666.	+	70243.6i	1.21073	+	0.699018i	0.962919	−	0.269789i	$-0.0869541\pi$
$317$	121666.	+	70243.6i	1.21073	+	0.699018i	0.247815	+	0.968807i	$0.420287\pi$
$318$		0			0
$319$	149328.	+	86214.3i	1.46743	+	0.847223i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−98768.0	+	60325.0i	−0.946697	+	0.578219i
$324$		0			0
$325$	−84806.7	+	48963.1i	−0.802903	+	0.463556i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	13373.4	−	23163.4i	0.123552	−	0.213998i
$330$		0			0
$331$		−	119946.i		−	1.09479i	−0.836874	−	0.547395i	$-0.815619\pi$
$331$		−	119946.i		−	1.09479i	0.836874	−	0.547395i	$-0.184381\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	43602.0i		−	0.388523i
$336$		0			0
$337$	53966.2	−	31157.4i	0.475184	−	0.274348i	−0.243223	−	0.969970i	$-0.578205\pi$
$337$	53966.2	−	31157.4i	0.475184	−	0.274348i	0.718407	+	0.695623i	$0.244871\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			313739.i			2.69811i
$342$		0			0
$343$	107018.			0.909641
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5657.07	−	9798.33i	−0.0469821	−	0.0813754i	0.841578	−	0.540136i	$-0.181627\pi$
$347$	−5657.07	−	9798.33i	−0.0469821	−	0.0813754i	−0.888560	+	0.458760i	$0.848294\pi$
$348$		0			0
$349$	−143009.			−1.17412			−0.587060	−	0.809543i	$-0.699715\pi$
$349$	−143009.			−1.17412			−0.587060	+	0.809543i	$0.699715\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−167076.			−1.34080			−0.670400	−	0.742000i	$-0.733878\pi$
$353$	−167076.			−1.34080			−0.670400	+	0.742000i	$0.733878\pi$
$354$		0			0
$355$	−225389.	−	130128.i	−1.78845	−	1.03256i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20058.8	−	34742.9i	−0.155638	−	0.269573i	0.777653	−	0.628694i	$-0.216410\pi$
$359$	−20058.8	−	34742.9i	−0.155638	−	0.269573i	−0.933291	+	0.359120i	$0.883077\pi$
$360$		0			0
$361$	−130162.	+	6436.83i	−0.998779	+	0.0493921i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	115232.	−	199588.i	0.864945	−	1.49813i
$366$		0			0
$367$	10253.8	−	17760.2i	0.0761298	−	0.131861i	−0.825447	−	0.564479i	$-0.809077\pi$
$367$	10253.8	−	17760.2i	0.0761298	−	0.131861i	0.901577	+	0.432619i	$0.142410\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17011.9	+	9821.82i	0.123596	+	0.0713582i
$372$		0			0
$373$			86938.8i			0.624879i	0.949938	+	0.312440i	$0.101146\pi$
$373$			86938.8i			0.624879i	−0.949938	+	0.312440i	$0.898854\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−58753.6	−	101764.i	−0.413382	−	0.715999i
$378$		0			0
$379$		−	145646.i		−	1.01396i	−0.861958	−	0.506980i	$-0.830762\pi$
$379$		−	145646.i		−	1.01396i	0.861958	−	0.506980i	$-0.169238\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−165096.	+	95318.3i	−1.12548	+	0.649799i	−0.942795	−	0.333372i	$-0.891814\pi$
$383$	−165096.	+	95318.3i	−1.12548	+	0.649799i	−0.182689	+	0.983171i	$0.558480\pi$
$384$		0			0
$385$	−91219.9	−	157997.i	−0.615415	−	1.06593i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−76712.0	+	132869.i	−0.506949	+	0.878062i	0.493019	+	0.870019i	$0.335893\pi$
$389$	−76712.0	+	132869.i	−0.506949	+	0.878062i	−0.999968	+	0.00804280i	$0.997440\pi$
$390$		0			0
$391$	188874.			1.23543
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−165334.	−	95455.9i	−1.05967	−	0.611799i
$396$		0			0
$397$	−17906.9	−	31015.7i	−0.113616	−	0.196789i	0.803610	−	0.595157i	$-0.202910\pi$
$397$	−17906.9	−	31015.7i	−0.113616	−	0.196789i	−0.917226	+	0.398368i	$0.869577\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	256007.	−	147806.i	1.59207	−	0.919185i	0.599124	−	0.800657i	$-0.295516\pi$
$401$	256007.	−	147806.i	1.59207	−	0.919185i	0.992951	−	0.118528i	$-0.0378176\pi$
$402$		0			0
$403$	106904.	−	185163.i	0.658239	−	1.14010i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	193957.i		−	1.17089i
$408$		0			0
$409$	−212656.	−	122777.i	−1.27125	−	0.733957i	−0.296028	−	0.955179i	$-0.595662\pi$
$409$	−212656.	−	122777.i	−1.27125	−	0.733957i	−0.975224	+	0.221222i	$0.928996\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−23224.6	+	13408.7i	−0.136160	+	0.0786117i
$414$		0			0
$415$	−73914.9	−	128024.i	−0.429176	−	0.743355i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17960.1			−0.102301			−0.0511505	−	0.998691i	$-0.516289\pi$
$419$	−17960.1			−0.102301			−0.0511505	+	0.998691i	$0.516289\pi$
$420$		0			0
$421$	−24547.6	+	14172.5i	−0.138498	+	0.0799620i	−0.567648	−	0.823271i	$-0.692147\pi$
$421$	−24547.6	+	14172.5i	−0.138498	+	0.0799620i	0.429150	+	0.903233i	$0.358813\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	243386.			1.34746
$426$		0			0
$427$	−71761.7	+	124295.i	−0.393583	+	0.681706i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	117302.	+	67724.3i	0.631467	+	0.364578i	0.781320	−	0.624131i	$-0.214547\pi$
$431$	117302.	+	67724.3i	0.631467	+	0.364578i	−0.149853	+	0.988708i	$0.547880\pi$
$432$		0			0
$433$	−295101.	−	170377.i	−1.57397	−	0.908729i	−0.995676	−	0.0928982i	$-0.970387\pi$
$433$	−295101.	−	170377.i	−1.57397	−	0.908729i	−0.578290	−	0.815831i	$-0.696280\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	186758.	+	101758.i	0.977948	+	0.532850i
$438$		0			0
$439$	−176624.	+	101974.i	−0.916477	+	0.529129i	−0.882510	−	0.470294i	$-0.844148\pi$
$439$	−176624.	+	101974.i	−0.916477	+	0.529129i	−0.0339679	+	0.999423i	$0.510814\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11724.2	−	20307.0i	0.0597417	−	0.103476i	−0.834608	−	0.550845i	$-0.814306\pi$
$443$	11724.2	−	20307.0i	0.0597417	−	0.103476i	0.894349	+	0.447369i	$0.147639\pi$
$444$		0			0
$445$		−	340934.i		−	1.72167i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	239774.i		−	1.18935i	−0.803966	−	0.594675i	$-0.797281\pi$
$449$		−	239774.i		−	1.18935i	0.803966	−	0.594675i	$-0.202719\pi$
$450$		0			0
$451$	263053.	−	151874.i	1.29327	−	0.746671i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			124330.i			0.600554i
$456$		0			0
$457$	−230624.			−1.10426			−0.552130	−	0.833758i	$-0.686185\pi$
$457$	−230624.			−1.10426			−0.552130	+	0.833758i	$0.686185\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	193513.	+	335175.i	0.910561	+	1.57714i	0.813273	+	0.581882i	$0.197684\pi$
$461$	193513.	+	335175.i	0.910561	+	1.57714i	0.0972884	+	0.995256i	$0.468983\pi$
$462$		0			0
$463$	−84329.2			−0.393383			−0.196692	−	0.980465i	$-0.563020\pi$
$463$	−84329.2			−0.393383			−0.196692	+	0.980465i	$0.563020\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	267364.			1.22594			0.612971	−	0.790106i	$-0.289974\pi$
$467$	267364.			1.22594			0.612971	+	0.790106i	$0.289974\pi$
$468$		0			0
$469$	−26294.4	−	15181.1i	−0.119541	−	0.0690173i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−171341.	−	296771.i	−0.765841	−	1.32648i
$474$		0			0
$475$	240659.	+	131127.i	1.06663	+	0.581170i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	23830.8	−	41276.2i	0.103865	−	0.179899i	−0.809409	−	0.587245i	$-0.800212\pi$
$479$	23830.8	−	41276.2i	0.103865	−	0.179899i	0.913274	+	0.407346i	$0.133546\pi$
$480$		0			0
$481$	−66089.1	+	114470.i	−0.285654	+	0.494767i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	357274.	+	206272.i	1.51886	+	0.876916i
$486$		0			0
$487$		−	300808.i		−	1.26833i	−0.773199	−	0.634163i	$-0.781345\pi$
$487$		−	300808.i		−	1.26833i	0.773199	−	0.634163i	$-0.218655\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15562.0	−	26954.1i	−0.0645508	−	0.111805i	0.831944	−	0.554860i	$-0.187228\pi$
$491$	−15562.0	−	26954.1i	−0.0645508	−	0.111805i	−0.896495	+	0.443055i	$0.853895\pi$
$492$		0			0
$493$			292052.i			1.20162i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−156949.	+	90614.8i	−0.635399	+	0.366848i
$498$		0			0
$499$	83695.1	+	144964.i	0.336124	+	0.582183i	0.983700	−	0.179817i	$-0.0575507\pi$
$499$	83695.1	+	144964.i	0.336124	+	0.582183i	−0.647576	+	0.762001i	$0.724217\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−193752.	+	335589.i	−0.765793	+	1.32639i	0.174034	+	0.984740i	$0.444320\pi$
$503$	−193752.	+	335589.i	−0.765793	+	1.32639i	−0.939826	+	0.341652i	$0.889013\pi$
$504$		0			0
$505$	−161659.			−0.633896
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−49772.1	−	28735.9i	−0.192110	−	0.110915i	0.400860	−	0.916139i	$-0.368711\pi$
$509$	−49772.1	−	28735.9i	−0.192110	−	0.110915i	−0.592970	+	0.805225i	$0.702045\pi$
$510$		0			0
$511$	−80241.9	−	138983.i	−0.307298	−	0.532255i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	468251.	−	270345.i	1.76549	−	1.01930i
$516$		0			0
$517$	97705.3	−	169231.i	0.365542	−	0.633137i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	253537.i		−	0.934039i	−0.884247	−	0.467020i	$-0.845328\pi$
$521$		−	253537.i		−	0.934039i	0.884247	−	0.467020i	$-0.154672\pi$
$522$		0			0
$523$	303436.	+	175189.i	1.10934	+	0.640477i	0.938658	−	0.344850i	$-0.112070\pi$
$523$	303436.	+	175189.i	1.10934	+	0.640477i	0.170680	+	0.985326i	$0.445403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−460204.	+	265699.i	−1.65702	+	0.956683i
$528$		0			0
$529$	−33624.4	−	58239.1i	−0.120155	−	0.208115i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−206999.			−0.728640
$534$		0			0
$535$	542609.	−	313276.i	1.89574	−	1.09451i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	327415.			1.12699
$540$		0			0
$541$	−90974.1	+	157572.i	−0.310830	+	0.538374i	−0.978542	−	0.206046i	$-0.933940\pi$
$541$	−90974.1	+	157572.i	−0.310830	+	0.538374i	0.667712	+	0.744420i	$0.267274\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−554591.	−	320193.i	−1.86715	−	1.07800i
$546$		0			0
$547$	−65609.1	−	37879.4i	−0.219275	−	0.126598i	0.386340	−	0.922357i	$-0.373739\pi$
$547$	−65609.1	−	37879.4i	−0.219275	−	0.126598i	−0.605615	+	0.795758i	$0.707073\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−157346.	+	288780.i	−0.518266	+	0.951181i
$552$		0			0
$553$	−115131.	+	66470.6i	−0.376478	+	0.217360i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12063.0	+	20893.8i	−0.0388818	+	0.0673453i	−0.884811	−	0.465949i	$-0.845713\pi$
$557$	−12063.0	+	20893.8i	−0.0388818	+	0.0673453i	0.845930	+	0.533295i	$0.179046\pi$
$558$		0			0
$559$			233532.i			0.747348i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			564081.i			1.77961i	0.456343	+	0.889804i	$0.349159\pi$
$563$			564081.i			1.77961i	−0.456343	+	0.889804i	$0.650841\pi$
$564$		0			0
$565$	−492274.	+	284214.i	−1.54209	+	0.890326i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			188598.i			0.582521i	0.956644	+	0.291261i	$0.0940747\pi$
$569$			188598.i			0.582521i	−0.956644	+	0.291261i	$0.905925\pi$
$570$		0			0
$571$	−168254.			−0.516052			−0.258026	−	0.966138i	$-0.583072\pi$
$571$	−168254.			−0.516052			−0.258026	+	0.966138i	$0.583072\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−223632.	−	387343.i	−0.676393	−	1.17155i
$576$		0			0
$577$	346445.			1.04060			0.520298	−	0.853985i	$-0.325821\pi$
$577$	346445.			1.04060			0.520298	+	0.853985i	$0.325821\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−102941.			−0.304956
$582$		0			0
$583$	124288.	+	71757.8i	0.365673	+	0.211121i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−303062.	−	524919.i	−0.879539	−	1.52341i	−0.851847	−	0.523791i	$-0.824517\pi$
$587$	−303062.	−	524919.i	−0.879539	−	1.52341i	−0.0276924	−	0.999616i	$-0.508816\pi$
$588$		0			0
$589$	−598194.	+	14782.1i	−1.72430	+	0.0426093i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	127645.	−	221088.i	0.362991	−	0.628719i	−0.625460	−	0.780256i	$-0.715089\pi$
$593$	127645.	−	221088.i	0.362991	−	0.628719i	0.988452	+	0.151537i	$0.0484222\pi$
$594$		0			0
$595$	154504.	−	267609.i	0.436422	−	0.755905i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	178869.	+	103270.i	0.498520	+	0.287821i	0.728102	−	0.685469i	$-0.240403\pi$
$599$	178869.	+	103270.i	0.498520	+	0.287821i	−0.229582	+	0.973289i	$0.573736\pi$
$600$		0			0
$601$		−	580567.i		−	1.60732i	−0.595086	−	0.803662i	$-0.702882\pi$
$601$		−	580567.i		−	1.60732i	0.595086	−	0.803662i	$-0.297118\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−394093.	−	682589.i	−1.07668	−	1.86487i
$606$		0			0
$607$			19201.3i			0.0521140i	0.999660	+	0.0260570i	$0.00829514\pi$
$607$			19201.3i			0.0521140i	−0.999660	+	0.0260570i	$0.991705\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−115328.	+	66584.6i	−0.308924	+	0.178357i
$612$		0			0
$613$	260010.	+	450351.i	0.691942	+	1.19848i	0.971201	+	0.238262i	$0.0765778\pi$
$613$	260010.	+	450351.i	0.691942	+	1.19848i	−0.279259	+	0.960216i	$0.590089\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−305961.	+	529939.i	−0.803702	+	1.39205i	0.113462	+	0.993542i	$0.463806\pi$
$617$	−305961.	+	529939.i	−0.803702	+	1.39205i	−0.917164	+	0.398510i	$0.869527\pi$
$618$		0			0
$619$	561440.			1.46528			0.732642	−	0.680614i	$-0.238287\pi$
$619$	561440.			1.46528			0.732642	+	0.680614i	$0.238287\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−205602.	−	118705.i	−0.529727	−	0.305838i
$624$		0			0
$625$	144380.	+	250073.i	0.369613	+	0.640188i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	284503.	−	164258.i	0.719093	−	0.415169i
$630$		0			0
$631$	−330147.	+	571831.i	−0.829179	+	1.43618i	0.0695045	+	0.997582i	$0.477858\pi$
$631$	−330147.	+	571831.i	−0.829179	+	1.43618i	−0.898683	+	0.438598i	$0.855475\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	51186.8i		−	0.126944i
$636$		0			0
$637$	−193234.	−	111564.i	−0.476218	−	0.274944i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−45344.7	+	26179.8i	−0.110360	+	0.0637162i	−0.554164	−	0.832408i	$-0.686962\pi$
$641$	−45344.7	+	26179.8i	−0.110360	+	0.0637162i	0.443804	+	0.896124i	$0.353629\pi$
$642$		0			0
$643$	128987.	+	223412.i	0.311978	+	0.540362i	0.978791	−	0.204864i	$-0.0656751\pi$
$643$	128987.	+	223412.i	0.311978	+	0.540362i	−0.666812	+	0.745226i	$0.732342\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	295878.			0.706813			0.353407	−	0.935470i	$-0.385023\pi$
$647$	295878.			0.706813			0.353407	+	0.935470i	$0.385023\pi$
$648$		0			0
$649$	−169678.	+	97963.6i	−0.402843	+	0.232582i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	605903.			1.42094			0.710472	−	0.703726i	$-0.248482\pi$
$653$	605903.			1.42094			0.710472	+	0.703726i	$0.248482\pi$
$654$		0			0
$655$	−110673.	+	191691.i	−0.257964	+	0.446807i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−512302.	−	295778.i	−1.17965	−	0.681074i	−0.223720	−	0.974653i	$-0.571820\pi$
$659$	−512302.	−	295778.i	−1.17965	−	0.681074i	−0.955935	+	0.293579i	$0.905154\pi$
$660$		0			0
$661$	313351.	+	180913.i	0.717179	+	0.414063i	0.813713	−	0.581266i	$-0.197442\pi$
$661$	313351.	+	180913.i	0.717179	+	0.414063i	−0.0965345	+	0.995330i	$0.530776\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	296950.	−	181370.i	0.671491	−	0.410130i
$666$		0			0
$667$	464794.	−	268349.i	1.04474	−	0.603182i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−524288.	+	908093.i	−1.16446	+	2.01690i
$672$		0			0
$673$			172441.i			0.380724i	0.981714	+	0.190362i	$0.0609661\pi$
$673$			172441.i			0.380724i	−0.981714	+	0.190362i	$0.939034\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			708240.i			1.54526i	0.634854	+	0.772632i	$0.281060\pi$
$677$			708240.i			1.54526i	−0.634854	+	0.772632i	$0.718940\pi$
$678$		0			0
$679$	248788.	−	143638.i	0.539622	−	0.311551i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			766639.i			1.64342i	0.569904	+	0.821712i	$0.306981\pi$
$683$			766639.i			1.64342i	−0.569904	+	0.821712i	$0.693019\pi$
$684$		0			0
$685$	131780.			0.280845
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−48901.8	−	84700.3i	−0.103012	−	0.178421i
$690$		0			0
$691$	41957.7			0.0878729			0.0439365	−	0.999034i	$-0.486010\pi$
$691$	41957.7			0.0878729			0.0439365	+	0.999034i	$0.486010\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.15377e6			2.38864
$696$		0			0
$697$	445547.	+	257237.i	0.917125	+	0.529502i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−397795.	−	689002.i	−0.809513	−	1.40212i	−0.913202	−	0.407508i	$-0.866398\pi$
$701$	−397795.	−	689002.i	−0.809513	−	1.40212i	0.103689	−	0.994610i	$-0.466935\pi$
$702$		0			0
$703$	369810.	−	9138.44i	0.748287	−	0.0184910i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−56285.6	+	97489.6i	−0.112605	+	0.195038i
$708$		0			0
$709$	−104386.	+	180802.i	−0.207659	+	0.359676i	−0.950977	−	0.309263i	$-0.899918\pi$
$709$	−104386.	+	180802.i	−0.207659	+	0.359676i	0.743318	+	0.668939i	$0.233251\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	845706.	+	488268.i	1.66357	+	0.960461i
$714$		0			0
$715$			908347.i			1.77681i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	121432.	+	210327.i	0.234897	+	0.406853i	0.959243	−	0.282584i	$-0.0911915\pi$
$719$	121432.	+	210327.i	0.234897	+	0.406853i	−0.724346	+	0.689437i	$0.757858\pi$
$720$		0			0
$721$		−	376509.i		−	0.724278i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	598940.	−	345798.i	1.13948	−	0.657880i
$726$		0			0
$727$	−58207.6	−	100819.i	−0.110131	−	0.190753i	0.805692	−	0.592335i	$-0.201794\pi$
$727$	−58207.6	−	100819.i	−0.110131	−	0.190753i	−0.915823	+	0.401582i	$0.868461\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	290210.	−	502658.i	0.543097	−	0.940672i
$732$		0			0
$733$	696231.			1.29582			0.647911	−	0.761716i	$-0.275643\pi$
$733$	696231.			1.29582			0.647911	+	0.761716i	$0.275643\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−192106.	−	110913.i	−0.353677	−	0.204195i
$738$		0			0
$739$	24355.4	+	42184.8i	0.0445971	+	0.0772444i	0.887462	−	0.460880i	$-0.152466\pi$
$739$	24355.4	+	42184.8i	0.0445971	+	0.0772444i	−0.842865	+	0.538125i	$0.819133\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	82349.9	−	47544.7i	0.149171	−	0.0861241i	−0.423557	−	0.905870i	$-0.639218\pi$
$743$	82349.9	−	47544.7i	0.149171	−	0.0861241i	0.572728	+	0.819746i	$0.305885\pi$
$744$		0			0
$745$	−610148.	+	1.05681e6i	−1.09932	+	1.90407i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	436298.i		−	0.777714i
$750$		0			0
$751$	120915.	+	69810.4i	0.214388	+	0.123777i	0.603349	−	0.797477i	$-0.293833\pi$
$751$	120915.	+	69810.4i	0.214388	+	0.123777i	−0.388961	+	0.921254i	$0.627166\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	621545.	−	358849.i	1.09038	−	0.629533i
$756$		0			0
$757$	−43704.0	−	75697.5i	−0.0762657	−	0.132096i	0.825370	−	0.564592i	$-0.190966\pi$
$757$	−43704.0	−	75697.5i	−0.0762657	−	0.132096i	−0.901636	+	0.432496i	$0.857633\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−510619.			−0.881714			−0.440857	−	0.897577i	$-0.645325\pi$
$761$	−510619.			−0.881714			−0.440857	+	0.897577i	$0.645325\pi$
$762$		0			0
$763$	−386189.	+	222966.i	−0.663362	+	0.382992i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	133521.			0.226965
$768$		0			0
$769$	12429.1	−	21527.9i	0.0210178	−	0.0364039i	−0.855325	−	0.518092i	$-0.826643\pi$
$769$	12429.1	−	21527.9i	0.0210178	−	0.0364039i	0.876343	+	0.481688i	$0.159976\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7645.09	−	4413.89i	−0.0127945	−	0.00738691i	0.493589	−	0.869695i	$-0.335685\pi$
$773$	−7645.09	−	4413.89i	−0.0127945	−	0.00738691i	−0.506384	+	0.862308i	$0.669018\pi$
$774$		0			0
$775$	1.08979e6	+	629189.i	1.81442	+	1.04756i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	301966.	+	494398.i	0.497603	+	0.814707i
$780$		0			0
$781$	−1.14667e6	+	662028.i	−1.87990	+	1.08536i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	489616.	−	848039.i	0.794540	−	1.37618i
$786$		0			0
$787$			209129.i			0.337649i	0.985646	+	0.168824i	$0.0539971\pi$
$787$			209129.i			0.337649i	−0.985646	+	0.168824i	$0.946003\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			395825.i			0.632631i
$792$		0			0
$793$	618850.	−	357293.i	0.984100	−	0.568170i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			975546.i			1.53579i	0.640578	+	0.767894i	$0.278695\pi$
$797$			975546.i			1.53579i	−0.640578	+	0.767894i	$0.721305\pi$
$798$		0			0
$799$	330978.			0.518449
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−586244.	−	1.01540e6i	−0.909175	−	1.57474i
$804$		0			0
$805$	−567858.			−0.876291
$806$		0			0
$807$		0			0
$808$		0			0
$809$	219203.			0.334926			0.167463	−	0.985878i	$-0.446443\pi$
$809$	219203.			0.334926			0.167463	+	0.985878i	$0.446443\pi$
$810$		0			0
$811$	804012.	+	464197.i	1.22242	+	0.705765i	0.965434	−	0.260649i	$-0.0839366\pi$
$811$	804012.	+	464197.i	1.22242	+	0.705765i	0.256988	+	0.966415i	$0.417270\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−705189.	−	1.22142e6i	−1.06167	−	1.83887i
$816$		0			0
$817$	557770.	−	340672.i	0.835625	−	0.510379i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−213229.	+	369324.i	−0.316344	+	0.547925i	−0.979722	−	0.200360i	$-0.935789\pi$
$821$	−213229.	+	369324.i	−0.316344	+	0.547925i	0.663378	+	0.748284i	$0.269122\pi$
$822$		0			0
$823$	250617.	−	434082.i	0.370008	−	0.640873i	−0.619558	−	0.784951i	$-0.712688\pi$
$823$	250617.	−	434082.i	0.370008	−	0.640873i	0.989566	+	0.144078i	$0.0460216\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−367431.	−	212137.i	−0.537236	−	0.310173i	0.206722	−	0.978400i	$-0.433720\pi$
$827$	−367431.	−	212137.i	−0.537236	−	0.310173i	−0.743958	+	0.668226i	$0.767054\pi$
$828$		0			0
$829$			653309.i			0.950626i	0.879817	+	0.475313i	$0.157665\pi$
$829$			653309.i			0.950626i	−0.879817	+	0.475313i	$0.842335\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	277281.	+	480264.i	0.399604	+	0.692134i
$834$		0			0
$835$			952696.i			1.36641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−382493.	+	220833.i	−0.543376	+	0.313718i	−0.746446	−	0.665446i	$-0.768241\pi$
$839$	−382493.	+	220833.i	−0.543376	+	0.313718i	0.203070	+	0.979164i	$0.434908\pi$
$840$		0			0
$841$	61302.0	+	106178.i	0.0866727	+	0.150122i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−221788.	+	384147.i	−0.310616	+	0.538003i
$846$		0			0
$847$	−548853.			−0.765048
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−522824.	−	301853.i	−0.721933	−	0.416808i
$852$		0			0
$853$	−407216.	−	705318.i	−0.559663	−	0.969364i	−0.997524	−	0.0703220i	$-0.977597\pi$
$853$	−407216.	−	705318.i	−0.559663	−	0.969364i	0.437862	−	0.899042i	$-0.355736\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	193851.	−	111920.i	0.263940	−	0.152386i	−0.362190	−	0.932104i	$-0.617971\pi$
$857$	193851.	−	111920.i	0.263940	−	0.152386i	0.626131	+	0.779718i	$0.284638\pi$
$858$		0			0
$859$	393015.	−	680722.i	0.532627	−	0.922537i	−0.466647	−	0.884443i	$-0.654538\pi$
$859$	393015.	−	680722.i	0.532627	−	0.922537i	0.999274	−	0.0380933i	$-0.0121284\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			21594.6i			0.0289950i	0.999895	+	0.0144975i	$0.00461487\pi$
$863$			21594.6i			0.0289950i	−0.999895	+	0.0144975i	$0.995385\pi$
$864$		0			0
$865$	746806.	+	431169.i	0.998104	+	0.576255i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−841139.	+	485632.i	−1.11385	+	0.643084i
$870$		0			0
$871$	75585.0	+	130917.i	0.0996322	+	0.172568i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−129330.			−0.168921
$876$		0			0
$877$	−71775.7	+	41439.7i	−0.0933208	+	0.0538788i	−0.545934	−	0.837828i	$-0.683825\pi$
$877$	−71775.7	+	41439.7i	−0.0933208	+	0.0538788i	0.452613	+	0.891707i	$0.350492\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.09279e6			−1.40794			−0.703970	−	0.710230i	$-0.748591\pi$
$881$	−1.09279e6			−1.40794			−0.703970	+	0.710230i	$0.748591\pi$
$882$		0			0
$883$	169370.	−	293358.i	0.217228	−	0.376250i	−0.736732	−	0.676185i	$-0.763632\pi$
$883$	169370.	−	293358.i	0.217228	−	0.376250i	0.953959	+	0.299936i	$0.0969652\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	117292.	+	67718.4i	0.149080	+	0.0860715i	0.572685	−	0.819776i	$-0.305902\pi$
$887$	117292.	+	67718.4i	0.149080	+	0.0860715i	−0.423604	+	0.905847i	$0.639235\pi$
$888$		0			0
$889$	−30868.5	−	17822.0i	−0.0390582	−	0.0225503i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	327269.	+	178318.i	0.410395	+	0.223610i
$894$		0			0
$895$	1.09807e6	−	633974.i	1.37084	−	0.791453i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−755000.	+	1.30770e6i	−0.934173	+	1.61803i
$900$		0			0
$901$			243081.i			0.299434i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			388939.i			0.474881i
$906$		0			0
$907$	−4012.56	+	2316.65i	−0.00487761	+	0.00281609i	−0.502437	−	0.864614i	$-0.667563\pi$
$907$	−4012.56	+	2316.65i	−0.00487761	+	0.00281609i	0.497559	+	0.867430i	$0.334230\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	228474.i		−	0.275296i	−0.990481	−	0.137648i	$-0.956046\pi$
$911$		−	228474.i		−	0.275296i	0.990481	−	0.137648i	$-0.0439543\pi$
$912$		0			0
$913$	−752084.			−0.902245
$914$		0			0
$915$		0			0
$916$		0			0
$917$	77067.1	+	133484.i	0.0916496	+	0.158742i
$918$		0			0
$919$	252976.			0.299535			0.149768	−	0.988721i	$-0.452147\pi$
$919$	252976.			0.299535			0.149768	+	0.988721i	$0.452147\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	902322.			1.05915
$924$		0			0
$925$	−673719.	−	388972.i	−0.787399	−	0.454605i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	308399.	+	534163.i	0.357340	+	0.618931i	0.987516	−	0.157522i	$-0.0503505\pi$
$929$	308399.	+	534163.i	0.357340	+	0.618931i	−0.630176	+	0.776453i	$0.717017\pi$
$930$		0			0
$931$	15426.4	+	624270.i	0.0177978	+	0.720234i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.12880e6	−	1.95514e6i	1.29120	−	2.23643i
$936$		0			0
$937$	−698080.	+	1.20911e6i	−0.795108	+	1.37717i	0.127663	+	0.991818i	$0.459253\pi$
$937$	−698080.	+	1.20911e6i	−0.795108	+	1.37717i	−0.922771	+	0.385350i	$0.874081\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	119887.	+	69216.8i	0.135392	+	0.0781686i	0.566166	−	0.824291i	$-0.308426\pi$
$941$	119887.	+	69216.8i	0.135392	+	0.0781686i	−0.430774	+	0.902460i	$0.641759\pi$
$942$		0			0
$943$		−	945437.i		−	1.06319i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−171569.	−	297167.i	−0.191311	−	0.331360i	0.754374	−	0.656445i	$-0.227941\pi$
$947$	−171569.	−	297167.i	−0.191311	−	0.331360i	−0.945685	+	0.325085i	$0.894607\pi$
$948$		0			0
$949$			799031.i			0.887220i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.06953e6	+	617492.i	−1.17762	+	0.679901i	−0.955464	−	0.295109i	$-0.904644\pi$
$953$	−1.06953e6	+	617492.i	−1.17762	+	0.679901i	−0.222160	+	0.975010i	$0.571311\pi$
$954$		0			0
$955$	−525139.	−	909567.i	−0.575794	−	0.997305i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	45882.3	−	79470.5i	0.0498894	−	0.0864109i
$960$		0			0
$961$	−1.82396e6			−1.97501
$962$		0			0
$963$		0			0
$964$		0			0
$965$	451427.	+	260632.i	0.484767	+	0.279880i
$966$		0			0
$967$	847953.	+	1.46870e6i	0.906815	+	1.57065i	0.818462	+	0.574561i	$0.194827\pi$
$967$	847953.	+	1.46870e6i	0.906815	+	1.57065i	0.0883534	+	0.996089i	$0.471840\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−699151.	+	403655.i	−0.741536	+	0.428126i	−0.822628	−	0.568581i	$-0.807493\pi$
$971$	−699151.	+	403655.i	−0.741536	+	0.428126i	0.0810915	+	0.996707i	$0.474159\pi$
$972$		0			0
$973$	401715.	−	695790.i	0.424318	−	0.734941i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			90445.0i			0.0947536i	0.998877	+	0.0473768i	$0.0150861\pi$
$977$			90445.0i			0.0947536i	−0.998877	+	0.0473768i	$0.984914\pi$
$978$		0			0
$979$	−1.50212e6	−	867251.i	−1.56726	−	0.904856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	947240.	−	546889.i	0.980286	−	0.565968i	0.0779295	−	0.996959i	$-0.475169\pi$
$983$	947240.	−	546889.i	0.980286	−	0.565968i	0.902356	+	0.430990i	$0.141836\pi$
$984$		0			0
$985$	−1.11933e6	−	1.93874e6i	−1.15368	−	1.99824i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.06662e6			−1.09048
$990$		0			0
$991$	718383.	−	414759.i	0.731491	−	0.422326i	−0.0874766	−	0.996167i	$-0.527880\pi$
$991$	718383.	−	414759.i	0.731491	−	0.422326i	0.818967	+	0.573840i	$0.194547\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−303325.			−0.306381
$996$		0			0
$997$	−619962.	+	1.07381e6i	−0.623699	+	1.08028i	0.365092	+	0.930972i	$0.381038\pi$
$997$	−619962.	+	1.07381e6i	−0.623699	+	1.08028i	−0.988791	+	0.149307i	$0.952296\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.y.f.145.3	✓	24
3.2	odd	2	inner	684.5.y.f.145.10	yes	24
19.8	odd	6	inner	684.5.y.f.217.3	yes	24
57.8	even	6	inner	684.5.y.f.217.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.5.y.f.145.3	✓	24	1.1	even	1	trivial
684.5.y.f.145.10	yes	24	3.2	odd	2	inner
684.5.y.f.217.3	yes	24	19.8	odd	6	inner
684.5.y.f.217.10	yes	24	57.8	even	6	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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-18.6023 - 32.2201i) q^{5} -25.9074 q^{7} +O(q^{10})\)	\(q+(-18.6023 - 32.2201i) q^{5} -25.9074 q^{7} -189.278 q^{11} +(111.709 + 64.4949i) q^{13} +(-160.296 - 277.640i) q^{17} +(-8.91800 - 360.890i) q^{19} +(-294.572 + 510.213i) q^{23} +(-379.589 + 657.467i) q^{25} +(-788.932 - 455.490i) q^{29} -1657.55i q^{31} +(481.936 + 834.737i) q^{35} +1024.72i q^{37} +(-1389.77 + 802.383i) q^{41} +(905.234 + 1567.91i) q^{43} +(-516.200 + 894.084i) q^{47} -1729.81 q^{49} +(-656.643 - 379.113i) q^{53} +(3521.00 + 6098.56i) q^{55} +(896.448 - 517.564i) q^{59} +(2769.93 - 4797.67i) q^{61} -4799.01i q^{65} +(1014.94 + 585.977i) q^{67} +(6058.10 - 3497.65i) q^{71} +(3097.26 + 5364.62i) q^{73} +4903.70 q^{77} +(4443.93 - 2565.70i) q^{79} +3973.43 q^{83} +(-5963.72 + 10329.5i) q^{85} +(7936.06 + 4581.89i) q^{89} +(-2894.07 - 1670.89i) q^{91} +(-11462.0 + 7000.71i) q^{95} +(-9602.98 + 5544.28i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 304 q^{7}+O(q^{10}) \) 24 * q + 304 * q^7	\( 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) \) 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

Introduction

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