LMFDB - Embedded newform 864.4.f.b.431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,4,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.9776502450$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Character	$\chi$	$=$	864.431
Dual form			864.4.f.b.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.0227 q^{5} +25.8057i q^{7} +O(q^{10})$	$q-21.0227 q^{5} +25.8057i q^{7} +46.9426i q^{11} +55.4028i q^{13} +64.8085i q^{17} +51.0764 q^{19} -60.4580 q^{23} +316.954 q^{25} -90.5781 q^{29} -8.35979i q^{31} -542.506i q^{35} +228.302i q^{37} +239.750i q^{41} +192.963 q^{43} -38.1976 q^{47} -322.934 q^{49} +6.69418 q^{53} -986.861i q^{55} +5.40778i q^{59} +847.861i q^{61} -1164.72i q^{65} -992.286 q^{67} -494.107 q^{71} -395.627 q^{73} -1211.39 q^{77} +42.1033i q^{79} +328.279i q^{83} -1362.45i q^{85} -1134.87i q^{89} -1429.71 q^{91} -1073.76 q^{95} +991.893 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) $ 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	−21.0227	−1.88033	−0.940164	−	0.340722i	$-0.889328\pi$
$5$	−21.0227	−1.88033	−0.940164	+	0.340722i	$0.889328\pi$
$6$	0	0
$7$	25.8057i	1.39338i	0.717374	+	0.696688i	$0.245344\pi$
$7$	25.8057i	1.39338i	−0.717374	+	0.696688i	$0.754656\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	46.9426i	1.28670i	0.765571	+	0.643352i	$0.222457\pi$
$11$	46.9426i	1.28670i	−0.765571	+	0.643352i	$0.777543\pi$
$12$	0	0
$13$	55.4028i	1.18200i	0.806673	+	0.590999i	$0.201266\pi$
$13$	55.4028i	1.18200i	−0.806673	+	0.590999i	$0.798734\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	64.8085i	0.924609i	0.886721	+	0.462305i	$0.152977\pi$
$17$	64.8085i	0.924609i	−0.886721	+	0.462305i	$0.847023\pi$
$18$	0	0
$19$	51.0764	0.616722	0.308361	−	0.951269i	$-0.400219\pi$
$19$	51.0764	0.616722	0.308361	+	0.951269i	$0.400219\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−60.4580	−0.548102	−0.274051	−	0.961715i	$-0.588364\pi$
$23$	−60.4580	−0.548102	−0.274051	+	0.961715i	$0.588364\pi$
$24$	0	0
$25$	316.954	2.53563
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−90.5781	−0.579998	−0.289999	−	0.957027i	$-0.593655\pi$
$29$	−90.5781	−0.579998	−0.289999	+	0.957027i	$0.593655\pi$
$30$	0	0
$31$	− 8.35979i	− 0.0484343i	−0.999707	−	0.0242171i	$-0.992291\pi$
$31$	− 8.35979i	− 0.0484343i	0.999707	−	0.0242171i	$-0.00770931\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 542.506i	− 2.62001i
$36$	0	0
$37$	228.302i	1.01440i	0.861829	+	0.507198i	$0.169319\pi$
$37$	228.302i	1.01440i	−0.861829	+	0.507198i	$0.830681\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	239.750i	0.913236i	0.889663	+	0.456618i	$0.150939\pi$
$41$	239.750i	0.913236i	−0.889663	+	0.456618i	$0.849061\pi$
$42$	0	0
$43$	192.963	0.684338	0.342169	−	0.939638i	$-0.388838\pi$
$43$	192.963	0.684338	0.342169	+	0.939638i	$0.388838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−38.1976	−0.118547	−0.0592733	−	0.998242i	$-0.518878\pi$
$47$	−38.1976	−0.118547	−0.0592733	+	0.998242i	$0.518878\pi$
$48$	0	0
$49$	−322.934	−0.941499
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.69418	0.0173494	0.00867468	−	0.999962i	$-0.497239\pi$
$53$	6.69418	0.0173494	0.00867468	+	0.999962i	$0.497239\pi$
$54$	0	0
$55$	− 986.861i	− 2.41942i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.40778i	0.0119328i	0.999982	+	0.00596639i	$0.00189917\pi$
$59$	5.40778i	0.0119328i	−0.999982	+	0.00596639i	$0.998101\pi$
$60$	0	0
$61$	847.861i	1.77963i	0.456322	+	0.889815i	$0.349167\pi$
$61$	847.861i	1.77963i	−0.456322	+	0.889815i	$0.650833\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1164.72i	− 2.22254i
$66$	0	0
$67$	−992.286	−1.80936	−0.904679	−	0.426093i	$-0.859890\pi$
$67$	−992.286	−1.80936	−0.904679	+	0.426093i	$0.859890\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−494.107	−0.825911	−0.412956	−	0.910751i	$-0.635504\pi$
$71$	−494.107	−0.825911	−0.412956	+	0.910751i	$0.635504\pi$
$72$	0	0
$73$	−395.627	−0.634310	−0.317155	−	0.948374i	$-0.602727\pi$
$73$	−395.627	−0.634310	−0.317155	+	0.948374i	$0.602727\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1211.39	−1.79286
$78$	0	0
$79$	42.1033i	0.0599620i	0.999550	+	0.0299810i	$0.00954467\pi$
$79$	42.1033i	0.0599620i	−0.999550	+	0.0299810i	$0.990455\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	328.279i	0.434136i	0.976156	+	0.217068i	$0.0696494\pi$
$83$	328.279i	0.434136i	−0.976156	+	0.217068i	$0.930351\pi$
$84$	0	0
$85$	− 1362.45i	− 1.73857i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1134.87i	− 1.35163i	−0.737069	−	0.675817i	$-0.763791\pi$
$89$	− 1134.87i	− 1.35163i	0.737069	−	0.675817i	$-0.236209\pi$
$90$	0	0
$91$	−1429.71	−1.64697
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1073.76	−1.15964
$96$	0	0
$97$	991.893	1.03826	0.519131	−	0.854694i	$-0.326256\pi$
$97$	991.893	1.03826	0.519131	+	0.854694i	$0.326256\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	439.547	0.433035	0.216518	−	0.976279i	$-0.430530\pi$
$101$	439.547	0.433035	0.216518	+	0.976279i	$0.430530\pi$
$102$	0	0
$103$	− 863.697i	− 0.826239i	−0.910677	−	0.413119i	$-0.864439\pi$
$103$	− 863.697i	− 0.826239i	0.910677	−	0.413119i	$-0.135561\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	288.856i	0.260979i	0.991450	+	0.130490i	$0.0416549\pi$
$107$	288.856i	0.260979i	−0.991450	+	0.130490i	$0.958345\pi$
$108$	0	0
$109$	584.309i	0.513456i	0.966484	+	0.256728i	$0.0826444\pi$
$109$	584.309i	0.513456i	−0.966484	+	0.256728i	$0.917356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	519.361i	0.432366i	0.976353	+	0.216183i	$0.0693607\pi$
$113$	519.361i	0.432366i	−0.976353	+	0.216183i	$0.930639\pi$
$114$	0	0
$115$	1270.99	1.03061
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1672.43	−1.28833
$120$	0	0
$121$	−872.610	−0.655605
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4035.40	−2.88749
$126$	0	0
$127$	1021.81i	0.713942i	0.934116	+	0.356971i	$0.116190\pi$
$127$	1021.81i	0.713942i	−0.934116	+	0.356971i	$0.883810\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2351.04i	− 1.56802i	−0.620746	−	0.784011i	$-0.713170\pi$
$131$	− 2351.04i	− 1.56802i	0.620746	−	0.784011i	$-0.286830\pi$
$132$	0	0
$133$	1318.06i	0.859327i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	296.307i	0.184782i	0.995723	+	0.0923912i	$0.0294510\pi$
$137$	296.307i	0.184782i	−0.995723	+	0.0923912i	$0.970549\pi$
$138$	0	0
$139$	3225.10	1.96798	0.983991	−	0.178217i	$-0.0570330\pi$
$139$	3225.10	1.96798	0.983991	+	0.178217i	$0.0570330\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2600.75	−1.52088
$144$	0	0
$145$	1904.20	1.09059
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2312.66	1.27155	0.635773	−	0.771876i	$-0.280681\pi$
$149$	2312.66	1.27155	0.635773	+	0.771876i	$0.280681\pi$
$150$	0	0
$151$	− 89.5553i	− 0.0482643i	−0.999709	−	0.0241321i	$-0.992318\pi$
$151$	− 89.5553i	− 0.0482643i	0.999709	−	0.0241321i	$-0.00768225\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	175.745i	0.0910723i
$156$	0	0
$157$	− 1147.61i	− 0.583370i	−0.956514	−	0.291685i	$-0.905784\pi$
$157$	− 1147.61i	− 0.583370i	0.956514	−	0.291685i	$-0.0942159\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1560.16i	− 0.763713i
$162$	0	0
$163$	1392.90	0.669329	0.334664	−	0.942337i	$-0.391377\pi$
$163$	1392.90	0.669329	0.334664	+	0.942337i	$0.391377\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1802.94	0.835422	0.417711	−	0.908580i	$-0.362833\pi$
$167$	1802.94	0.835422	0.417711	+	0.908580i	$0.362833\pi$
$168$	0	0
$169$	−872.466	−0.397117
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2880.31	−1.26582	−0.632908	−	0.774227i	$-0.718139\pi$
$173$	−2880.31	−1.26582	−0.632908	+	0.774227i	$0.718139\pi$
$174$	0	0
$175$	8179.22i	3.53309i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3753.09i	− 1.56715i	−0.621299	−	0.783573i	$-0.713395\pi$
$179$	− 3753.09i	− 1.56715i	0.621299	−	0.783573i	$-0.286605\pi$
$180$	0	0
$181$	1097.17i	0.450565i	0.974294	+	0.225282i	$0.0723304\pi$
$181$	1097.17i	0.450565i	−0.974294	+	0.225282i	$0.927670\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4799.53i	− 1.90740i
$186$	0	0
$187$	−3042.28	−1.18970
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2485.51	0.941597	0.470798	−	0.882241i	$-0.343966\pi$
$191$	2485.51	0.941597	0.470798	+	0.882241i	$0.343966\pi$
$192$	0	0
$193$	−556.275	−0.207469	−0.103735	−	0.994605i	$-0.533079\pi$
$193$	−556.275	−0.207469	−0.103735	+	0.994605i	$0.533079\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	148.033	0.0535376	0.0267688	−	0.999642i	$-0.491478\pi$
$197$	148.033	0.0535376	0.0267688	+	0.999642i	$0.491478\pi$
$198$	0	0
$199$	− 2459.72i	− 0.876207i	−0.898925	−	0.438103i	$-0.855650\pi$
$199$	− 2459.72i	− 0.876207i	0.898925	−	0.438103i	$-0.144350\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2337.43i	− 0.808155i
$204$	0	0
$205$	− 5040.20i	− 1.71718i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2397.66i	0.793539i
$210$	0	0
$211$	1184.61	0.386501	0.193251	−	0.981149i	$-0.438097\pi$
$211$	1184.61	0.386501	0.193251	+	0.981149i	$0.438097\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4056.60	−1.28678
$216$	0	0
$217$	215.730	0.0674872
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3590.57	−1.09289
$222$	0	0
$223$	− 4029.79i	− 1.21011i	−0.796184	−	0.605055i	$-0.793151\pi$
$223$	− 4029.79i	− 1.21011i	0.796184	−	0.605055i	$-0.206849\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2149.19i	0.628401i	0.949357	+	0.314200i	$0.101736\pi$
$227$	2149.19i	0.628401i	−0.949357	+	0.314200i	$0.898264\pi$
$228$	0	0
$229$	− 4733.65i	− 1.36598i	−0.730429	−	0.682988i	$-0.760680\pi$
$229$	− 4733.65i	− 1.36598i	0.730429	−	0.682988i	$-0.239320\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4706.71i	1.32338i	0.749779	+	0.661688i	$0.230160\pi$
$233$	4706.71i	1.32338i	−0.749779	+	0.661688i	$0.769840\pi$
$234$	0	0
$235$	803.016	0.222906
$236$	0	0
$237$	0	0
$238$	0	0
$239$	89.8148	0.0243081	0.0121541	−	0.999926i	$-0.496131\pi$
$239$	89.8148	0.0243081	0.0121541	+	0.999926i	$0.496131\pi$
$240$	0	0
$241$	−6554.77	−1.75199	−0.875996	−	0.482319i	$-0.839795\pi$
$241$	−6554.77	−1.75199	−0.875996	+	0.482319i	$0.839795\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6788.95	1.77033
$246$	0	0
$247$	2829.77i	0.728964i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3092.32i	− 0.777632i	−0.921315	−	0.388816i	$-0.872884\pi$
$251$	− 3092.32i	− 0.777632i	0.921315	−	0.388816i	$-0.127116\pi$
$252$	0	0
$253$	− 2838.06i	− 0.705245i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6481.06i	1.57306i	0.617550	+	0.786532i	$0.288125\pi$
$257$	6481.06i	1.57306i	−0.617550	+	0.786532i	$0.711875\pi$
$258$	0	0
$259$	−5891.50	−1.41344
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1133.90	0.265854	0.132927	−	0.991126i	$-0.457562\pi$
$263$	1133.90	0.265854	0.132927	+	0.991126i	$0.457562\pi$
$264$	0	0
$265$	−140.730	−0.0326225
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5950.58	−1.34875	−0.674375	−	0.738389i	$-0.735587\pi$
$269$	−5950.58	−1.34875	−0.674375	+	0.738389i	$0.735587\pi$
$270$	0	0
$271$	− 1084.06i	− 0.242997i	−0.992592	−	0.121498i	$-0.961230\pi$
$271$	− 1084.06i	− 0.242997i	0.992592	−	0.121498i	$-0.0387699\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14878.7i	3.26261i
$276$	0	0
$277$	− 589.017i	− 0.127764i	−0.997957	−	0.0638819i	$-0.979652\pi$
$277$	− 589.017i	− 0.127764i	0.997957	−	0.0638819i	$-0.0203481\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2488.56i	0.528309i	0.964480	+	0.264155i	$0.0850929\pi$
$281$	2488.56i	0.528309i	−0.964480	+	0.264155i	$0.914907\pi$
$282$	0	0
$283$	−905.542	−0.190208	−0.0951041	−	0.995467i	$-0.530318\pi$
$283$	−905.542	−0.190208	−0.0951041	+	0.995467i	$0.530318\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6186.92	−1.27248
$288$	0	0
$289$	712.864	0.145097
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7084.57	1.41258	0.706289	−	0.707924i	$-0.250368\pi$
$293$	7084.57	1.41258	0.706289	+	0.707924i	$0.250368\pi$
$294$	0	0
$295$	− 113.686i	− 0.0224375i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3349.54i	− 0.647855i
$300$	0	0
$301$	4979.54i	0.953541i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 17824.3i	− 3.34629i
$306$	0	0
$307$	−2630.02	−0.488935	−0.244467	−	0.969658i	$-0.578613\pi$
$307$	−2630.02	−0.488935	−0.244467	+	0.969658i	$0.578613\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3592.70	0.655059	0.327530	−	0.944841i	$-0.393784\pi$
$311$	3592.70	0.655059	0.327530	+	0.944841i	$0.393784\pi$
$312$	0	0
$313$	−1042.95	−0.188343	−0.0941713	−	0.995556i	$-0.530020\pi$
$313$	−1042.95	−0.188343	−0.0941713	+	0.995556i	$0.530020\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4238.10	0.750900	0.375450	−	0.926843i	$-0.377488\pi$
$317$	4238.10	0.750900	0.375450	+	0.926843i	$0.377488\pi$
$318$	0	0
$319$	− 4251.97i	− 0.746285i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3310.18i	0.570227i
$324$	0	0
$325$	17560.1i	2.99711i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 985.715i	− 0.165180i
$330$	0	0
$331$	4481.09	0.744118	0.372059	−	0.928209i	$-0.378652\pi$
$331$	4481.09	0.744118	0.372059	+	0.928209i	$0.378652\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	20860.5	3.40219
$336$	0	0
$337$	9189.85	1.48547	0.742735	−	0.669586i	$-0.233528\pi$
$337$	9189.85	1.48547	0.742735	+	0.669586i	$0.233528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	392.430	0.0623205
$342$	0	0
$343$	517.817i	0.0815146i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7656.58i	1.18452i	0.805749	+	0.592258i	$0.201763\pi$
$347$	7656.58i	1.18452i	−0.805749	+	0.592258i	$0.798237\pi$
$348$	0	0
$349$	6451.66i	0.989541i	0.869024	+	0.494770i	$0.164748\pi$
$349$	6451.66i	0.989541i	−0.869024	+	0.494770i	$0.835252\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8023.93i	1.20983i	0.796289	+	0.604916i	$0.206793\pi$
$353$	8023.93i	1.20983i	−0.796289	+	0.604916i	$0.793207\pi$
$354$	0	0
$355$	10387.5	1.55298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9756.68	1.43437	0.717184	−	0.696884i	$-0.245431\pi$
$359$	9756.68	1.43437	0.717184	+	0.696884i	$0.245431\pi$
$360$	0	0
$361$	−4250.20	−0.619653
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8317.15	1.19271
$366$	0	0
$367$	− 2074.88i	− 0.295116i	−0.989053	−	0.147558i	$-0.952859\pi$
$367$	− 2074.88i	− 0.295116i	0.989053	−	0.147558i	$-0.0471414\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	172.748i	0.0241742i
$372$	0	0
$373$	− 10071.5i	− 1.39808i	−0.715082	−	0.699040i	$-0.753611\pi$
$373$	− 10071.5i	− 1.39808i	0.715082	−	0.699040i	$-0.246389\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5018.28i	− 0.685555i
$378$	0	0
$379$	5470.51	0.741427	0.370714	−	0.928747i	$-0.379113\pi$
$379$	5470.51	0.741427	0.370714	+	0.928747i	$0.379113\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5293.77	−0.706264	−0.353132	−	0.935573i	$-0.614883\pi$
$383$	−5293.77	−0.706264	−0.353132	+	0.935573i	$0.614883\pi$
$384$	0	0
$385$	25466.6	3.37117
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10947.1	1.42684	0.713422	−	0.700735i	$-0.247144\pi$
$389$	10947.1	1.42684	0.713422	+	0.700735i	$0.247144\pi$
$390$	0	0
$391$	− 3918.19i	− 0.506781i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 885.126i	− 0.112748i
$396$	0	0
$397$	− 6624.65i	− 0.837485i	−0.908105	−	0.418743i	$-0.862471\pi$
$397$	− 6624.65i	− 0.837485i	0.908105	−	0.418743i	$-0.137529\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8813.95i	1.09762i	0.835946	+	0.548812i	$0.184920\pi$
$401$	8813.95i	1.09762i	−0.835946	+	0.548812i	$0.815080\pi$
$402$	0	0
$403$	463.155	0.0572492
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10717.1	−1.30523
$408$	0	0
$409$	−9710.22	−1.17393	−0.586967	−	0.809611i	$-0.699678\pi$
$409$	−9710.22	−1.17393	−0.586967	+	0.809611i	$0.699678\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−139.552	−0.0166268
$414$	0	0
$415$	− 6901.32i	− 0.816319i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 15936.0i	− 1.85805i	−0.370012	−	0.929027i	$-0.620647\pi$
$419$	− 15936.0i	− 1.85805i	0.370012	−	0.929027i	$-0.379353\pi$
$420$	0	0
$421$	11978.0i	1.38664i	0.720631	+	0.693319i	$0.243852\pi$
$421$	11978.0i	1.38664i	−0.720631	+	0.693319i	$0.756148\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20541.3i	2.34447i
$426$	0	0
$427$	−21879.6	−2.47969
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7830.77	−0.875162	−0.437581	−	0.899179i	$-0.644165\pi$
$431$	−7830.77	−0.875162	−0.437581	+	0.899179i	$0.644165\pi$
$432$	0	0
$433$	8681.33	0.963506	0.481753	−	0.876307i	$-0.340000\pi$
$433$	8681.33	0.963506	0.481753	+	0.876307i	$0.340000\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3087.97	−0.338027
$438$	0	0
$439$	16031.9i	1.74296i	0.490433	+	0.871479i	$0.336839\pi$
$439$	16031.9i	1.74296i	−0.490433	+	0.871479i	$0.663161\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13242.3i	1.42023i	0.704085	+	0.710116i	$0.251357\pi$
$443$	13242.3i	1.42023i	−0.704085	+	0.710116i	$0.748643\pi$
$444$	0	0
$445$	23857.9i	2.54152i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14212.6i	1.49384i	0.664916	+	0.746918i	$0.268467\pi$
$449$	14212.6i	1.49384i	−0.664916	+	0.746918i	$0.731533\pi$
$450$	0	0
$451$	−11254.5	−1.17506
$452$	0	0
$453$	0	0
$454$	0	0
$455$	30056.3	3.09684
$456$	0	0
$457$	−11455.7	−1.17259	−0.586296	−	0.810097i	$-0.699414\pi$
$457$	−11455.7	−1.17259	−0.586296	+	0.810097i	$0.699414\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11164.3	1.12792	0.563960	−	0.825802i	$-0.309277\pi$
$461$	11164.3	1.12792	0.563960	+	0.825802i	$0.309277\pi$
$462$	0	0
$463$	− 17656.6i	− 1.77229i	−0.463406	−	0.886146i	$-0.653373\pi$
$463$	− 17656.6i	− 1.77229i	0.463406	−	0.886146i	$-0.346627\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6798.24i	− 0.673630i	−0.941571	−	0.336815i	$-0.890650\pi$
$467$	− 6798.24i	− 0.673630i	0.941571	−	0.336815i	$-0.109350\pi$
$468$	0	0
$469$	− 25606.6i	− 2.52112i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9058.18i	0.880540i
$474$	0	0
$475$	16188.9	1.56378
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15852.5	1.51215	0.756073	−	0.654487i	$-0.227115\pi$
$479$	15852.5	1.51215	0.756073	+	0.654487i	$0.227115\pi$
$480$	0	0
$481$	−12648.6	−1.19901
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20852.3	−1.95227
$486$	0	0
$487$	9101.40i	0.846867i	0.905927	+	0.423433i	$0.139175\pi$
$487$	9101.40i	0.846867i	−0.905927	+	0.423433i	$0.860825\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9196.60i	0.845289i	0.906296	+	0.422645i	$0.138898\pi$
$491$	9196.60i	0.845289i	−0.906296	+	0.422645i	$0.861102\pi$
$492$	0	0
$493$	− 5870.23i	− 0.536271i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12750.8i	− 1.15081i
$498$	0	0
$499$	−13949.8	−1.25146	−0.625729	−	0.780040i	$-0.715198\pi$
$499$	−13949.8	−1.25146	−0.625729	+	0.780040i	$0.715198\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16519.0	−1.46431	−0.732155	−	0.681138i	$-0.761485\pi$
$503$	−16519.0	−1.46431	−0.732155	+	0.681138i	$0.761485\pi$
$504$	0	0
$505$	−9240.47	−0.814249
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17406.4	1.51577	0.757884	−	0.652390i	$-0.226233\pi$
$509$	17406.4	1.51577	0.757884	+	0.652390i	$0.226233\pi$
$510$	0	0
$511$	− 10209.4i	− 0.883832i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18157.2i	1.55360i
$516$	0	0
$517$	− 1793.09i	− 0.152534i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3810.31i	− 0.320408i	−0.987084	−	0.160204i	$-0.948785\pi$
$521$	− 3810.31i	− 0.320408i	0.987084	−	0.160204i	$-0.0512153\pi$
$522$	0	0
$523$	−13252.4	−1.10801	−0.554003	−	0.832515i	$-0.686900\pi$
$523$	−13252.4	−1.10801	−0.554003	+	0.832515i	$0.686900\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	541.785	0.0447828
$528$	0	0
$529$	−8511.84	−0.699584
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−13282.8	−1.07944
$534$	0	0
$535$	− 6072.54i	− 0.490726i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 15159.4i	− 1.21143i
$540$	0	0
$541$	− 9637.46i	− 0.765891i	−0.923771	−	0.382945i	$-0.874910\pi$
$541$	− 9637.46i	− 0.765891i	0.923771	−	0.382945i	$-0.125090\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 12283.8i	− 0.965465i
$546$	0	0
$547$	14827.7	1.15902	0.579512	−	0.814964i	$-0.303243\pi$
$547$	14827.7	1.15902	0.579512	+	0.814964i	$0.303243\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4626.40	−0.357697
$552$	0	0
$553$	−1086.51	−0.0835496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8395.14	0.638624	0.319312	−	0.947650i	$-0.396548\pi$
$557$	8395.14	0.638624	0.319312	+	0.947650i	$0.396548\pi$
$558$	0	0
$559$	10690.7i	0.808886i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4251.95i	− 0.318292i	−0.987255	−	0.159146i	$-0.949126\pi$
$563$	− 4251.95i	− 0.318292i	0.987255	−	0.159146i	$-0.0508741\pi$
$564$	0	0
$565$	− 10918.4i	− 0.812989i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 25967.3i	− 1.91319i	−0.291416	−	0.956596i	$-0.594126\pi$
$569$	− 25967.3i	− 1.91319i	0.291416	−	0.956596i	$-0.405874\pi$
$570$	0	0
$571$	13542.0	0.992495	0.496247	−	0.868181i	$-0.334711\pi$
$571$	13542.0	0.992495	0.496247	+	0.868181i	$0.334711\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−19162.4	−1.38979
$576$	0	0
$577$	−12320.0	−0.888891	−0.444445	−	0.895806i	$-0.646599\pi$
$577$	−12320.0	−0.888891	−0.444445	+	0.895806i	$0.646599\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8471.47	−0.604916
$582$	0	0
$583$	314.242i	0.0223235i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16086.1i	1.13108i	0.824722	+	0.565539i	$0.191332\pi$
$587$	16086.1i	1.13108i	−0.824722	+	0.565539i	$0.808668\pi$
$588$	0	0
$589$	− 426.988i	− 0.0298705i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21688.8i	1.50195i	0.660333	+	0.750973i	$0.270415\pi$
$593$	21688.8i	1.50195i	−0.660333	+	0.750973i	$0.729585\pi$
$594$	0	0
$595$	35159.0	2.42248
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5142.53	−0.350782	−0.175391	−	0.984499i	$-0.556119\pi$
$599$	−5142.53	−0.350782	−0.175391	+	0.984499i	$0.556119\pi$
$600$	0	0
$601$	3458.20	0.234714	0.117357	−	0.993090i	$-0.462558\pi$
$601$	3458.20	0.234714	0.117357	+	0.993090i	$0.462558\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18344.6	1.23275
$606$	0	0
$607$	3691.56i	0.246847i	0.992354	+	0.123423i	$0.0393873\pi$
$607$	3691.56i	0.246847i	−0.992354	+	0.123423i	$0.960613\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2116.25i	− 0.140122i
$612$	0	0
$613$	13078.5i	0.861719i	0.902419	+	0.430860i	$0.141790\pi$
$613$	13078.5i	0.861719i	−0.902419	+	0.430860i	$0.858210\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6273.33i	− 0.409327i	−0.978832	−	0.204664i	$-0.934390\pi$
$617$	− 6273.33i	− 0.409327i	0.978832	−	0.204664i	$-0.0656100\pi$
$618$	0	0
$619$	−18878.7	−1.22585	−0.612924	−	0.790142i	$-0.710007\pi$
$619$	−18878.7	−1.22585	−0.612924	+	0.790142i	$0.710007\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	29286.0	1.88334
$624$	0	0
$625$	45215.7	2.89380
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14795.9	−0.937921
$630$	0	0
$631$	14835.8i	0.935982i	0.883733	+	0.467991i	$0.155022\pi$
$631$	14835.8i	0.935982i	−0.883733	+	0.467991i	$0.844978\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 21481.1i	− 1.34244i
$636$	0	0
$637$	− 17891.4i	− 1.11285i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3702.09i	− 0.228118i	−0.993474	−	0.114059i	$-0.963615\pi$
$641$	− 3702.09i	− 0.228118i	0.993474	−	0.114059i	$-0.0363853\pi$
$642$	0	0
$643$	−12974.2	−0.795727	−0.397863	−	0.917445i	$-0.630248\pi$
$643$	−12974.2	−0.795727	−0.397863	+	0.917445i	$0.630248\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18049.8	1.09677	0.548386	−	0.836225i	$-0.315242\pi$
$647$	18049.8	1.09677	0.548386	+	0.836225i	$0.315242\pi$
$648$	0	0
$649$	−253.856	−0.0153539
$650$	0	0
$651$	0	0
$652$	0	0
$653$	220.468	0.0132122	0.00660610	−	0.999978i	$-0.497897\pi$
$653$	220.468	0.0132122	0.00660610	+	0.999978i	$0.497897\pi$
$654$	0	0
$655$	49425.1i	2.94840i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 30349.1i	− 1.79398i	−0.442052	−	0.896989i	$-0.645749\pi$
$659$	− 30349.1i	− 1.79398i	0.442052	−	0.896989i	$-0.354251\pi$
$660$	0	0
$661$	− 4775.98i	− 0.281035i	−0.990078	−	0.140517i	$-0.955123\pi$
$661$	− 4775.98i	− 0.281035i	0.990078	−	0.140517i	$-0.0448766\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 27709.2i	− 1.61582i
$666$	0	0
$667$	5476.17	0.317898
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39800.8	−2.28986
$672$	0	0
$673$	10172.8	0.582665	0.291332	−	0.956622i	$-0.405901\pi$
$673$	10172.8	0.582665	0.291332	+	0.956622i	$0.405901\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13505.1	−0.766680	−0.383340	−	0.923607i	$-0.625226\pi$
$677$	−13505.1	−0.766680	−0.383340	+	0.923607i	$0.625226\pi$
$678$	0	0
$679$	25596.5i	1.44669i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5512.09i	0.308805i	0.988008	+	0.154403i	$0.0493453\pi$
$683$	5512.09i	0.308805i	−0.988008	+	0.154403i	$0.950655\pi$
$684$	0	0
$685$	− 6229.17i	− 0.347452i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	370.876i	0.0205069i
$690$	0	0
$691$	−16835.7	−0.926862	−0.463431	−	0.886133i	$-0.653382\pi$
$691$	−16835.7	−0.926862	−0.463431	+	0.886133i	$0.653382\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−67800.4	−3.70045
$696$	0	0
$697$	−15537.8	−0.844387
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4576.61	−0.246585	−0.123293	−	0.992370i	$-0.539345\pi$
$701$	−4576.61	−0.246585	−0.123293	+	0.992370i	$0.539345\pi$
$702$	0	0
$703$	11660.9i	0.625601i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11342.8i	0.603381i
$708$	0	0
$709$	− 754.010i	− 0.0399399i	−0.999801	−	0.0199700i	$-0.993643\pi$
$709$	− 754.010i	− 0.0399399i	0.999801	−	0.0199700i	$-0.00635706\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	505.416i	0.0265469i
$714$	0	0
$715$	54674.8	2.85975
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−768.569	−0.0398648	−0.0199324	−	0.999801i	$-0.506345\pi$
$719$	−768.569	−0.0398648	−0.0199324	+	0.999801i	$0.506345\pi$
$720$	0	0
$721$	22288.3	1.15126
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−28709.1	−1.47066
$726$	0	0
$727$	− 37696.9i	− 1.92311i	−0.274614	−	0.961555i	$-0.588550\pi$
$727$	− 37696.9i	− 1.92311i	0.274614	−	0.961555i	$-0.411450\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12505.6i	0.632746i
$732$	0	0
$733$	23625.7i	1.19050i	0.803540	+	0.595250i	$0.202947\pi$
$733$	23625.7i	1.19050i	−0.803540	+	0.595250i	$0.797053\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 46580.5i	− 2.32811i
$738$	0	0
$739$	16064.4	0.799645	0.399822	−	0.916593i	$-0.369072\pi$
$739$	16064.4	0.799645	0.399822	+	0.916593i	$0.369072\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31366.8	1.54877	0.774385	−	0.632715i	$-0.218059\pi$
$743$	31366.8	1.54877	0.774385	+	0.632715i	$0.218059\pi$
$744$	0	0
$745$	−48618.3	−2.39092
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7454.13	−0.363642
$750$	0	0
$751$	− 37857.4i	− 1.83946i	−0.392547	−	0.919732i	$-0.628406\pi$
$751$	− 37857.4i	− 1.83946i	0.392547	−	0.919732i	$-0.371594\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1882.70i	0.0907527i
$756$	0	0
$757$	23676.8i	1.13679i	0.822757	+	0.568393i	$0.192435\pi$
$757$	23676.8i	1.13679i	−0.822757	+	0.568393i	$0.807565\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34417.1i	1.63944i	0.572761	+	0.819722i	$0.305872\pi$
$761$	34417.1i	1.63944i	−0.572761	+	0.819722i	$0.694128\pi$
$762$	0	0
$763$	−15078.5	−0.715437
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−299.606	−0.0141045
$768$	0	0
$769$	25686.4	1.20452	0.602259	−	0.798301i	$-0.294267\pi$
$769$	25686.4	1.20452	0.602259	+	0.798301i	$0.294267\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18976.1	−0.882954	−0.441477	−	0.897272i	$-0.645545\pi$
$773$	−18976.1	−0.882954	−0.441477	+	0.897272i	$0.645545\pi$
$774$	0	0
$775$	− 2649.67i	− 0.122812i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12245.6i	0.563213i
$780$	0	0
$781$	− 23194.7i	− 1.06270i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24125.8i	1.09693i
$786$	0	0
$787$	−11337.5	−0.513518	−0.256759	−	0.966475i	$-0.582655\pi$
$787$	−11337.5	−0.513518	−0.256759	+	0.966475i	$0.582655\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13402.5	−0.602448
$792$	0	0
$793$	−46973.8	−2.10352
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15288.3	−0.679472	−0.339736	−	0.940521i	$-0.610338\pi$
$797$	−15288.3	−0.679472	−0.339736	+	0.940521i	$0.610338\pi$
$798$	0	0
$799$	− 2475.53i	− 0.109609i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 18571.8i	− 0.816168i
$804$	0	0
$805$	32798.8i	1.43603i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17214.2i	0.748108i	0.927407	+	0.374054i	$0.122033\pi$
$809$	17214.2i	0.748108i	−0.927407	+	0.374054i	$0.877967\pi$
$810$	0	0
$811$	−21003.7	−0.909421	−0.454711	−	0.890639i	$-0.650257\pi$
$811$	−21003.7	−0.909421	−0.454711	+	0.890639i	$0.650257\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−29282.6	−1.25856
$816$	0	0
$817$	9855.84	0.422047
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35917.2	−1.52682	−0.763409	−	0.645915i	$-0.776476\pi$
$821$	−35917.2	−1.52682	−0.763409	+	0.645915i	$0.776476\pi$
$822$	0	0
$823$	− 27671.0i	− 1.17199i	−0.810313	−	0.585997i	$-0.800703\pi$
$823$	− 27671.0i	− 1.17199i	0.810313	−	0.585997i	$-0.199297\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11321.8i	− 0.476055i	−0.971258	−	0.238028i	$-0.923499\pi$
$827$	− 11321.8i	− 0.476055i	0.971258	−	0.238028i	$-0.0765009\pi$
$828$	0	0
$829$	− 35680.4i	− 1.49485i	−0.664345	−	0.747426i	$-0.731289\pi$
$829$	− 35680.4i	− 1.49485i	0.664345	−	0.747426i	$-0.268711\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 20928.9i	− 0.870518i
$834$	0	0
$835$	−37902.6	−1.57087
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4841.63	0.199227	0.0996136	−	0.995026i	$-0.468239\pi$
$839$	4841.63	0.199227	0.0996136	+	0.995026i	$0.468239\pi$
$840$	0	0
$841$	−16184.6	−0.663603
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18341.6	0.746711
$846$	0	0
$847$	− 22518.3i	− 0.913504i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 13802.7i	− 0.555993i
$852$	0	0
$853$	− 11167.6i	− 0.448266i	−0.974559	−	0.224133i	$-0.928045\pi$
$853$	− 11167.6i	− 0.448266i	0.974559	−	0.224133i	$-0.0719550\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 934.475i	− 0.0372474i	−0.999827	−	0.0186237i	$-0.994072\pi$
$857$	− 934.475i	− 0.0372474i	0.999827	−	0.0186237i	$-0.00592846\pi$
$858$	0	0
$859$	−34803.8	−1.38241	−0.691205	−	0.722658i	$-0.742920\pi$
$859$	−34803.8	−1.38241	−0.691205	+	0.722658i	$0.742920\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4119.93	−0.162507	−0.0812537	−	0.996693i	$-0.525892\pi$
$863$	−4119.93	−0.162507	−0.0812537	+	0.996693i	$0.525892\pi$
$864$	0	0
$865$	60551.9	2.38015
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1976.44	−0.0771532
$870$	0	0
$871$	− 54975.4i	− 2.13866i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 104136.i	− 4.02337i
$876$	0	0
$877$	− 18557.8i	− 0.714539i	−0.934001	−	0.357270i	$-0.883708\pi$
$877$	− 18557.8i	− 0.714539i	0.934001	−	0.357270i	$-0.116292\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1170.36i	− 0.0447563i	−0.999750	−	0.0223782i	$-0.992876\pi$
$881$	− 1170.36i	− 0.0447563i	0.999750	−	0.0223782i	$-0.00712379\pi$
$882$	0	0
$883$	1080.22	0.0411689	0.0205845	−	0.999788i	$-0.493447\pi$
$883$	1080.22	0.0411689	0.0205845	+	0.999788i	$0.493447\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34016.2	−1.28766	−0.643830	−	0.765169i	$-0.722655\pi$
$887$	−34016.2	−1.28766	−0.643830	+	0.765169i	$0.722655\pi$
$888$	0	0
$889$	−26368.4	−0.994790
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1950.99	−0.0731103
$894$	0	0
$895$	78900.2i	2.94675i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	757.214i	0.0280918i
$900$	0	0
$901$	433.839i	0.0160414i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 23065.5i	− 0.847209i
$906$	0	0
$907$	34414.6	1.25989	0.629943	−	0.776641i	$-0.283078\pi$
$907$	34414.6	1.25989	0.629943	+	0.776641i	$0.283078\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21678.0	0.788390	0.394195	−	0.919027i	$-0.371023\pi$
$911$	21678.0	0.788390	0.394195	+	0.919027i	$0.371023\pi$
$912$	0	0
$913$	−15410.3	−0.558605
$914$	0	0
$915$	0	0
$916$	0	0
$917$	60670.1	2.18485
$918$	0	0
$919$	− 34780.5i	− 1.24842i	−0.781255	−	0.624212i	$-0.785420\pi$
$919$	− 34780.5i	− 1.24842i	0.781255	−	0.624212i	$-0.214580\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 27374.9i	− 0.976225i
$924$	0	0
$925$	72361.4i	2.57214i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10475.6i	0.369961i	0.982742	+	0.184980i	$0.0592222\pi$
$929$	10475.6i	0.369961i	−0.982742	+	0.184980i	$0.940778\pi$
$930$	0	0
$931$	−16494.3	−0.580643
$932$	0	0
$933$	0	0
$934$	0	0
$935$	63956.9	2.23702
$936$	0	0
$937$	−3735.50	−0.130238	−0.0651192	−	0.997877i	$-0.520743\pi$
$937$	−3735.50	−0.130238	−0.0651192	+	0.997877i	$0.520743\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12748.2	−0.441636	−0.220818	−	0.975315i	$-0.570873\pi$
$941$	−12748.2	−0.441636	−0.220818	+	0.975315i	$0.570873\pi$
$942$	0	0
$943$	− 14494.8i	− 0.500547i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39828.7i	1.36669i	0.730095	+	0.683346i	$0.239476\pi$
$947$	39828.7i	1.36669i	−0.730095	+	0.683346i	$0.760524\pi$
$948$	0	0
$949$	− 21918.8i	− 0.749752i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32132.0i	1.09219i	0.837723	+	0.546095i	$0.183886\pi$
$953$	32132.0i	1.09219i	−0.837723	+	0.546095i	$0.816114\pi$
$954$	0	0
$955$	−52252.1	−1.77051
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7646.40	−0.257471
$960$	0	0
$961$	29721.1	0.997654
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11694.4	0.390110
$966$	0	0
$967$	− 16136.2i	− 0.536614i	−0.963333	−	0.268307i	$-0.913536\pi$
$967$	− 16136.2i	− 0.536614i	0.963333	−	0.268307i	$-0.0864642\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5400.54i	0.178488i	0.996010	+	0.0892438i	$0.0284450\pi$
$971$	5400.54i	0.178488i	−0.996010	+	0.0892438i	$0.971555\pi$
$972$	0	0
$973$	83226.0i	2.74214i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54220.0i	− 1.77549i	−0.460337	−	0.887744i	$-0.652271\pi$
$977$	− 54220.0i	− 1.77549i	0.460337	−	0.887744i	$-0.347729\pi$
$978$	0	0
$979$	53273.5	1.73915
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32973.3	−1.06987	−0.534936	−	0.844892i	$-0.679664\pi$
$983$	−32973.3	−1.06987	−0.534936	+	0.844892i	$0.679664\pi$
$984$	0	0
$985$	−3112.05	−0.100668
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11666.1	−0.375087
$990$	0	0
$991$	18477.3i	0.592282i	0.955144	+	0.296141i	$0.0956998\pi$
$991$	18477.3i	0.592282i	−0.955144	+	0.296141i	$0.904300\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	51710.0i	1.64756i
$996$	0	0
$997$	31842.0i	1.01148i	0.862686	+	0.505741i	$0.168781\pi$
$997$	31842.0i	1.01148i	−0.862686	+	0.505741i	$0.831219\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.f.b.431.2		24
3.2	odd	2	inner	864.4.f.b.431.24		24
4.3	odd	2		216.4.f.b.107.6	yes	24
8.3	odd	2	inner	864.4.f.b.431.23		24
8.5	even	2		216.4.f.b.107.20	yes	24
12.11	even	2		216.4.f.b.107.19	yes	24
24.5	odd	2		216.4.f.b.107.5	✓	24
24.11	even	2	inner	864.4.f.b.431.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.f.b.107.5	✓	24	24.5	odd	2
216.4.f.b.107.6	yes	24	4.3	odd	2
216.4.f.b.107.19	yes	24	12.11	even	2
216.4.f.b.107.20	yes	24	8.5	even	2
864.4.f.b.431.1		24	24.11	even	2	inner
864.4.f.b.431.2		24	1.1	even	1	trivial
864.4.f.b.431.23		24	8.3	odd	2	inner
864.4.f.b.431.24		24	3.2	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.0227 q^{5} +25.8057i q^{7} +O(q^{10})\)	\(q-21.0227 q^{5} +25.8057i q^{7} +46.9426i q^{11} +55.4028i q^{13} +64.8085i q^{17} +51.0764 q^{19} -60.4580 q^{23} +316.954 q^{25} -90.5781 q^{29} -8.35979i q^{31} -542.506i q^{35} +228.302i q^{37} +239.750i q^{41} +192.963 q^{43} -38.1976 q^{47} -322.934 q^{49} +6.69418 q^{53} -986.861i q^{55} +5.40778i q^{59} +847.861i q^{61} -1164.72i q^{65} -992.286 q^{67} -494.107 q^{71} -395.627 q^{73} -1211.39 q^{77} +42.1033i q^{79} +328.279i q^{83} -1362.45i q^{85} -1134.87i q^{89} -1429.71 q^{91} -1073.76 q^{95} +991.893 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) \) 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

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