LMFDB - Embedded newform 936.4.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [936,4,Mod(1,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("936.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.2257877654$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.6390848.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 58x^{2} - 152x - 7 $ x^4 - 58x^2 - 152x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.0468916$ of defining polynomial
Character	$\chi$	$=$	936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90622 q^{5} -4.93923 q^{7} -12.7165 q^{11} +13.0000 q^{13} +34.2387 q^{17} +38.4425 q^{19} +88.9725 q^{23} -121.366 q^{25} -255.454 q^{29} -104.901 q^{31} -9.41523 q^{35} +247.186 q^{37} +235.433 q^{41} +522.672 q^{43} +419.488 q^{47} -318.604 q^{49} +603.432 q^{53} -24.2403 q^{55} -338.157 q^{59} -439.250 q^{61} +24.7808 q^{65} -581.045 q^{67} +228.357 q^{71} +573.828 q^{73} +62.8094 q^{77} -57.5271 q^{79} +898.414 q^{83} +65.2664 q^{85} +809.438 q^{89} -64.2099 q^{91} +73.2798 q^{95} -307.495 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 24 q^{7} + 64 q^{11} + 52 q^{13} + 80 q^{17} + 24 q^{19} + 16 q^{23} - 20 q^{25} + 240 q^{29} + 24 q^{31} + 80 q^{35} - 152 q^{37} + 136 q^{41} - 64 q^{43} + 528 q^{47} + 276 q^{49} + 496 q^{53}+ \cdots - 1384 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 24 * q^7 + 64 * q^11 + 52 * q^13 + 80 * q^17 + 24 * q^19 + 16 * q^23 - 20 * q^25 + 240 * q^29 + 24 * q^31 + 80 * q^35 - 152 * q^37 + 136 * q^41 - 64 * q^43 + 528 * q^47 + 276 * q^49 + 496 * q^53 - 352 * q^55 + 832 * q^59 - 104 * q^61 + 104 * q^65 + 1000 * q^67 + 1584 * q^71 - 232 * q^73 + 1584 * q^77 + 464 * q^79 + 1888 * q^83 - 1536 * q^85 + 696 * q^89 + 312 * q^91 + 1808 * q^95 - 1384 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.90622	0.170497	0.0852486	−	0.996360i	$-0.472832\pi$
$5$	1.90622	0.170497	0.0852486	+	0.996360i	$0.472832\pi$
$6$	0	0
$7$	−4.93923	−0.266693	−0.133347	−	0.991069i	$-0.542572\pi$
$7$	−4.93923	−0.266693	−0.133347	+	0.991069i	$0.542572\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−12.7165	−0.348559	−0.174280	−	0.984696i	$-0.555760\pi$
$11$	−12.7165	−0.348559	−0.174280	+	0.984696i	$0.555760\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	34.2387	0.488477	0.244238	−	0.969715i	$-0.421462\pi$
$17$	34.2387	0.488477	0.244238	+	0.969715i	$0.421462\pi$
$18$	0	0
$19$	38.4425	0.464175	0.232087	−	0.972695i	$-0.425444\pi$
$19$	38.4425	0.464175	0.232087	+	0.972695i	$0.425444\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	88.9725	0.806611	0.403305	−	0.915065i	$-0.367861\pi$
$23$	88.9725	0.806611	0.403305	+	0.915065i	$0.367861\pi$
$24$	0	0
$25$	−121.366	−0.970931
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−255.454	−1.63575	−0.817874	−	0.575398i	$-0.804847\pi$
$29$	−255.454	−1.63575	−0.817874	+	0.575398i	$0.804847\pi$
$30$	0	0
$31$	−104.901	−0.607765	−0.303883	−	0.952710i	$-0.598283\pi$
$31$	−104.901	−0.607765	−0.303883	+	0.952710i	$0.598283\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9.41523	−0.0454704
$36$	0	0
$37$	247.186	1.09830	0.549150	−	0.835724i	$-0.314952\pi$
$37$	247.186	1.09830	0.549150	+	0.835724i	$0.314952\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	235.433	0.896790	0.448395	−	0.893835i	$-0.351996\pi$
$41$	235.433	0.896790	0.448395	+	0.893835i	$0.351996\pi$
$42$	0	0
$43$	522.672	1.85364	0.926822	−	0.375502i	$-0.122530\pi$
$43$	522.672	1.85364	0.926822	+	0.375502i	$0.122530\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	419.488	1.30189	0.650943	−	0.759127i	$-0.274374\pi$
$47$	419.488	1.30189	0.650943	+	0.759127i	$0.274374\pi$
$48$	0	0
$49$	−318.604	−0.928875
$50$	0	0
$51$	0	0
$52$	0	0
$53$	603.432	1.56392	0.781960	−	0.623329i	$-0.214220\pi$
$53$	603.432	1.56392	0.781960	+	0.623329i	$0.214220\pi$
$54$	0	0
$55$	−24.2403	−0.0594284
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−338.157	−0.746174	−0.373087	−	0.927796i	$-0.621701\pi$
$59$	−338.157	−0.746174	−0.373087	+	0.927796i	$0.621701\pi$
$60$	0	0
$61$	−439.250	−0.921970	−0.460985	−	0.887408i	$-0.652504\pi$
$61$	−439.250	−0.921970	−0.460985	+	0.887408i	$0.652504\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	24.7808	0.0472874
$66$	0	0
$67$	−581.045	−1.05949	−0.529746	−	0.848156i	$-0.677713\pi$
$67$	−581.045	−1.05949	−0.529746	+	0.848156i	$0.677713\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	228.357	0.381705	0.190852	−	0.981619i	$-0.438875\pi$
$71$	228.357	0.381705	0.190852	+	0.981619i	$0.438875\pi$
$72$	0	0
$73$	573.828	0.920021	0.460010	−	0.887914i	$-0.347846\pi$
$73$	573.828	0.920021	0.460010	+	0.887914i	$0.347846\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	62.8094	0.0929584
$78$	0	0
$79$	−57.5271	−0.0819279	−0.0409639	−	0.999161i	$-0.513043\pi$
$79$	−57.5271	−0.0819279	−0.0409639	+	0.999161i	$0.513043\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	898.414	1.18812	0.594059	−	0.804421i	$-0.297525\pi$
$83$	898.414	1.18812	0.594059	+	0.804421i	$0.297525\pi$
$84$	0	0
$85$	65.2664	0.0832839
$86$	0	0
$87$	0	0
$88$	0	0
$89$	809.438	0.964048	0.482024	−	0.876158i	$-0.339902\pi$
$89$	809.438	0.964048	0.482024	+	0.876158i	$0.339902\pi$
$90$	0	0
$91$	−64.2099	−0.0739674
$92$	0	0
$93$	0	0
$94$	0	0
$95$	73.2798	0.0791405
$96$	0	0
$97$	−307.495	−0.321870	−0.160935	−	0.986965i	$-0.551451\pi$
$97$	−307.495	−0.321870	−0.160935	+	0.986965i	$0.551451\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	253.727	0.249968	0.124984	−	0.992159i	$-0.460112\pi$
$101$	253.727	0.249968	0.124984	+	0.992159i	$0.460112\pi$
$102$	0	0
$103$	1515.76	1.45002	0.725010	−	0.688739i	$-0.241835\pi$
$103$	1515.76	1.45002	0.725010	+	0.688739i	$0.241835\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1769.26	1.59851	0.799257	−	0.600989i	$-0.205226\pi$
$107$	1769.26	1.59851	0.799257	+	0.600989i	$0.205226\pi$
$108$	0	0
$109$	−882.758	−0.775714	−0.387857	−	0.921720i	$-0.626785\pi$
$109$	−882.758	−0.775714	−0.387857	+	0.921720i	$0.626785\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	43.9015	0.0365478	0.0182739	−	0.999833i	$-0.494183\pi$
$113$	43.9015	0.0365478	0.0182739	+	0.999833i	$0.494183\pi$
$114$	0	0
$115$	169.601	0.137525
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−169.113	−0.130273
$120$	0	0
$121$	−1169.29	−0.878506
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−469.628	−0.336038
$126$	0	0
$127$	426.602	0.298069	0.149035	−	0.988832i	$-0.452383\pi$
$127$	426.602	0.298069	0.149035	+	0.988832i	$0.452383\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	979.788	0.653469	0.326735	−	0.945116i	$-0.394052\pi$
$131$	979.788	0.653469	0.326735	+	0.945116i	$0.394052\pi$
$132$	0	0
$133$	−189.876	−0.123792
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2347.08	1.46368	0.731840	−	0.681476i	$-0.238662\pi$
$137$	2347.08	1.46368	0.731840	+	0.681476i	$0.238662\pi$
$138$	0	0
$139$	1127.66	0.688107	0.344053	−	0.938950i	$-0.388200\pi$
$139$	1127.66	0.688107	0.344053	+	0.938950i	$0.388200\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−165.314	−0.0966730
$144$	0	0
$145$	−486.951	−0.278890
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2899.48	1.59419	0.797095	−	0.603853i	$-0.206369\pi$
$149$	2899.48	1.59419	0.797095	+	0.603853i	$0.206369\pi$
$150$	0	0
$151$	−392.022	−0.211274	−0.105637	−	0.994405i	$-0.533688\pi$
$151$	−392.022	−0.211274	−0.105637	+	0.994405i	$0.533688\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−199.963	−0.103622
$156$	0	0
$157$	1798.48	0.914231	0.457116	−	0.889407i	$-0.348883\pi$
$157$	1798.48	0.914231	0.457116	+	0.889407i	$0.348883\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−439.455	−0.215117
$162$	0	0
$163$	756.022	0.363290	0.181645	−	0.983364i	$-0.441858\pi$
$163$	756.022	0.363290	0.181645	+	0.983364i	$0.441858\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	759.697	0.352018	0.176009	−	0.984389i	$-0.443681\pi$
$167$	759.697	0.352018	0.176009	+	0.984389i	$0.443681\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	922.612	0.405462	0.202731	−	0.979234i	$-0.435018\pi$
$173$	922.612	0.405462	0.202731	+	0.979234i	$0.435018\pi$
$174$	0	0
$175$	599.456	0.258941
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1293.49	0.540112	0.270056	−	0.962845i	$-0.412958\pi$
$179$	1293.49	0.540112	0.270056	+	0.962845i	$0.412958\pi$
$180$	0	0
$181$	−2343.02	−0.962183	−0.481091	−	0.876670i	$-0.659759\pi$
$181$	−2343.02	−0.962183	−0.481091	+	0.876670i	$0.659759\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	471.190	0.187257
$186$	0	0
$187$	−435.395	−0.170263
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4005.87	1.51756	0.758781	−	0.651346i	$-0.225795\pi$
$191$	4005.87	1.51756	0.758781	+	0.651346i	$0.225795\pi$
$192$	0	0
$193$	−4925.29	−1.83694	−0.918471	−	0.395488i	$-0.870575\pi$
$193$	−4925.29	−1.83694	−0.918471	+	0.395488i	$0.870575\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	734.419	0.265610	0.132805	−	0.991142i	$-0.457602\pi$
$197$	734.419	0.265610	0.132805	+	0.991142i	$0.457602\pi$
$198$	0	0
$199$	816.002	0.290678	0.145339	−	0.989382i	$-0.453573\pi$
$199$	816.002	0.290678	0.145339	+	0.989382i	$0.453573\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1261.75	0.436243
$204$	0	0
$205$	448.786	0.152900
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−488.853	−0.161793
$210$	0	0
$211$	2579.75	0.841693	0.420846	−	0.907132i	$-0.361733\pi$
$211$	2579.75	0.841693	0.420846	+	0.907132i	$0.361733\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	996.325	0.316041
$216$	0	0
$217$	518.128	0.162087
$218$	0	0
$219$	0	0
$220$	0	0
$221$	445.103	0.135479
$222$	0	0
$223$	5697.69	1.71097	0.855484	−	0.517830i	$-0.173260\pi$
$223$	5697.69	1.71097	0.855484	+	0.517830i	$0.173260\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5870.08	−1.71635	−0.858173	−	0.513360i	$-0.828401\pi$
$227$	−5870.08	−1.71635	−0.858173	+	0.513360i	$0.828401\pi$
$228$	0	0
$229$	−4629.34	−1.33588	−0.667938	−	0.744217i	$-0.732823\pi$
$229$	−4629.34	−1.33588	−0.667938	+	0.744217i	$0.732823\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	310.188	0.0872149	0.0436074	−	0.999049i	$-0.486115\pi$
$233$	310.188	0.0872149	0.0436074	+	0.999049i	$0.486115\pi$
$234$	0	0
$235$	799.635	0.221968
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1613.60	−0.436715	−0.218358	−	0.975869i	$-0.570070\pi$
$239$	−1613.60	−0.436715	−0.218358	+	0.975869i	$0.570070\pi$
$240$	0	0
$241$	1269.18	0.339233	0.169617	−	0.985510i	$-0.445747\pi$
$241$	1269.18	0.339233	0.169617	+	0.985510i	$0.445747\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−607.328	−0.158371
$246$	0	0
$247$	499.753	0.128739
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4184.22	1.05221	0.526107	−	0.850418i	$-0.323651\pi$
$251$	4184.22	1.05221	0.526107	+	0.850418i	$0.323651\pi$
$252$	0	0
$253$	−1131.41	−0.281152
$254$	0	0
$255$	0	0
$256$	0	0
$257$	932.838	0.226416	0.113208	−	0.993571i	$-0.463887\pi$
$257$	932.838	0.226416	0.113208	+	0.993571i	$0.463887\pi$
$258$	0	0
$259$	−1220.91	−0.292909
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3593.74	−0.842584	−0.421292	−	0.906925i	$-0.638423\pi$
$263$	−3593.74	−0.842584	−0.421292	+	0.906925i	$0.638423\pi$
$264$	0	0
$265$	1150.27	0.266644
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3120.38	0.707259	0.353629	−	0.935386i	$-0.384947\pi$
$269$	3120.38	0.707259	0.353629	+	0.935386i	$0.384947\pi$
$270$	0	0
$271$	3665.57	0.821651	0.410825	−	0.911714i	$-0.365241\pi$
$271$	3665.57	0.821651	0.410825	+	0.911714i	$0.365241\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1543.35	0.338427
$276$	0	0
$277$	−2970.99	−0.644439	−0.322219	−	0.946665i	$-0.604429\pi$
$277$	−2970.99	−0.644439	−0.322219	+	0.946665i	$0.604429\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−143.397	−0.0304426	−0.0152213	−	0.999884i	$-0.504845\pi$
$281$	−143.397	−0.0304426	−0.0152213	+	0.999884i	$0.504845\pi$
$282$	0	0
$283$	−4432.53	−0.931047	−0.465524	−	0.885036i	$-0.654134\pi$
$283$	−4432.53	−0.931047	−0.465524	+	0.885036i	$0.654134\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1162.85	−0.239168
$288$	0	0
$289$	−3740.71	−0.761390
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9045.63	−1.80359	−0.901795	−	0.432165i	$-0.857750\pi$
$293$	−9045.63	−1.80359	−0.901795	+	0.432165i	$0.857750\pi$
$294$	0	0
$295$	−644.600	−0.127221
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1156.64	0.223714
$300$	0	0
$301$	−2581.59	−0.494354
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−837.305	−0.157193
$306$	0	0
$307$	8062.78	1.49892	0.749458	−	0.662052i	$-0.230314\pi$
$307$	8062.78	1.49892	0.749458	+	0.662052i	$0.230314\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1097.88	−0.200177	−0.100089	−	0.994979i	$-0.531913\pi$
$311$	−1097.88	−0.200177	−0.100089	+	0.994979i	$0.531913\pi$
$312$	0	0
$313$	−9173.83	−1.65666	−0.828332	−	0.560238i	$-0.810710\pi$
$313$	−9173.83	−1.65666	−0.828332	+	0.560238i	$0.810710\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1824.73	−0.323304	−0.161652	−	0.986848i	$-0.551682\pi$
$317$	−1824.73	−0.323304	−0.161652	+	0.986848i	$0.551682\pi$
$318$	0	0
$319$	3248.47	0.570155
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1316.22	0.226739
$324$	0	0
$325$	−1577.76	−0.269288
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2071.95	−0.347204
$330$	0	0
$331$	4298.62	0.713818	0.356909	−	0.934139i	$-0.383831\pi$
$331$	4298.62	0.713818	0.356909	+	0.934139i	$0.383831\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1107.60	−0.180641
$336$	0	0
$337$	3089.51	0.499396	0.249698	−	0.968324i	$-0.419669\pi$
$337$	3089.51	0.499396	0.249698	+	0.968324i	$0.419669\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1333.96	0.211842
$342$	0	0
$343$	3267.81	0.514418
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10367.6	−1.60393	−0.801965	−	0.597370i	$-0.796212\pi$
$347$	−10367.6	−1.60393	−0.801965	+	0.597370i	$0.796212\pi$
$348$	0	0
$349$	6824.44	1.04672	0.523358	−	0.852113i	$-0.324679\pi$
$349$	6824.44	1.04672	0.523358	+	0.852113i	$0.324679\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3127.95	−0.471627	−0.235813	−	0.971798i	$-0.575775\pi$
$353$	−3127.95	−0.471627	−0.235813	+	0.971798i	$0.575775\pi$
$354$	0	0
$355$	435.299	0.0650796
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10915.7	−1.60476	−0.802379	−	0.596815i	$-0.796433\pi$
$359$	−10915.7	−1.60476	−0.802379	+	0.596815i	$0.796433\pi$
$360$	0	0
$361$	−5381.17	−0.784542
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1093.84	0.156861
$366$	0	0
$367$	2125.18	0.302272	0.151136	−	0.988513i	$-0.451707\pi$
$367$	2125.18	0.302272	0.151136	+	0.988513i	$0.451707\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2980.49	−0.417087
$372$	0	0
$373$	3758.15	0.521687	0.260844	−	0.965381i	$-0.415999\pi$
$373$	3758.15	0.521687	0.260844	+	0.965381i	$0.415999\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3320.91	−0.453675
$378$	0	0
$379$	−11000.4	−1.49090	−0.745450	−	0.666561i	$-0.767766\pi$
$379$	−11000.4	−1.49090	−0.745450	+	0.666561i	$0.767766\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11409.2	1.52215	0.761074	−	0.648665i	$-0.224672\pi$
$383$	11409.2	1.52215	0.761074	+	0.648665i	$0.224672\pi$
$384$	0	0
$385$	119.728	0.0158491
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9895.04	−1.28971	−0.644857	−	0.764303i	$-0.723083\pi$
$389$	−9895.04	−1.28971	−0.644857	+	0.764303i	$0.723083\pi$
$390$	0	0
$391$	3046.30	0.394011
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−109.659	−0.0139685
$396$	0	0
$397$	−7417.84	−0.937760	−0.468880	−	0.883262i	$-0.655342\pi$
$397$	−7417.84	−0.937760	−0.468880	+	0.883262i	$0.655342\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4839.66	−0.602696	−0.301348	−	0.953514i	$-0.597437\pi$
$401$	−4839.66	−0.602696	−0.301348	+	0.953514i	$0.597437\pi$
$402$	0	0
$403$	−1363.71	−0.168564
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3143.33	−0.382823
$408$	0	0
$409$	−686.094	−0.0829467	−0.0414733	−	0.999140i	$-0.513205\pi$
$409$	−686.094	−0.0829467	−0.0414733	+	0.999140i	$0.513205\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1670.23	0.199000
$414$	0	0
$415$	1712.57	0.202571
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7360.88	0.858240	0.429120	−	0.903247i	$-0.358824\pi$
$419$	7360.88	0.858240	0.429120	+	0.903247i	$0.358824\pi$
$420$	0	0
$421$	−6349.20	−0.735015	−0.367507	−	0.930021i	$-0.619789\pi$
$421$	−6349.20	−0.735015	−0.367507	+	0.930021i	$0.619789\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4155.43	−0.474277
$426$	0	0
$427$	2169.55	0.245883
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4796.00	−0.535998	−0.267999	−	0.963419i	$-0.586362\pi$
$431$	−4796.00	−0.535998	−0.267999	+	0.963419i	$0.586362\pi$
$432$	0	0
$433$	4887.43	0.542436	0.271218	−	0.962518i	$-0.412574\pi$
$433$	4887.43	0.542436	0.271218	+	0.962518i	$0.412574\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3420.33	0.374408
$438$	0	0
$439$	16008.4	1.74041	0.870205	−	0.492690i	$-0.163986\pi$
$439$	16008.4	1.74041	0.870205	+	0.492690i	$0.163986\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6269.14	−0.672361	−0.336181	−	0.941798i	$-0.609135\pi$
$443$	−6269.14	−0.672361	−0.336181	+	0.941798i	$0.609135\pi$
$444$	0	0
$445$	1542.96	0.164367
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4132.48	−0.434352	−0.217176	−	0.976133i	$-0.569684\pi$
$449$	−4132.48	−0.434352	−0.217176	+	0.976133i	$0.569684\pi$
$450$	0	0
$451$	−2993.87	−0.312585
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−122.398	−0.0126112
$456$	0	0
$457$	3335.36	0.341403	0.170702	−	0.985323i	$-0.445397\pi$
$457$	3335.36	0.341403	0.170702	+	0.985323i	$0.445397\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6027.04	−0.608909	−0.304455	−	0.952527i	$-0.598474\pi$
$461$	−6027.04	−0.608909	−0.304455	+	0.952527i	$0.598474\pi$
$462$	0	0
$463$	−6851.24	−0.687698	−0.343849	−	0.939025i	$-0.611731\pi$
$463$	−6851.24	−0.687698	−0.343849	+	0.939025i	$0.611731\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12140.7	−1.20301	−0.601503	−	0.798871i	$-0.705431\pi$
$467$	−12140.7	−1.20301	−0.601503	+	0.798871i	$0.705431\pi$
$468$	0	0
$469$	2869.91	0.282559
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6646.53	−0.646105
$474$	0	0
$475$	−4665.63	−0.450682
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12894.4	−1.22998	−0.614989	−	0.788535i	$-0.710840\pi$
$479$	−12894.4	−1.22998	−0.614989	+	0.788535i	$0.710840\pi$
$480$	0	0
$481$	3213.42	0.304614
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−586.152	−0.0548779
$486$	0	0
$487$	8547.89	0.795363	0.397682	−	0.917523i	$-0.369815\pi$
$487$	8547.89	0.795363	0.397682	+	0.917523i	$0.369815\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15008.3	−1.37946	−0.689729	−	0.724067i	$-0.742271\pi$
$491$	−15008.3	−1.37946	−0.689729	+	0.724067i	$0.742271\pi$
$492$	0	0
$493$	−8746.43	−0.799025
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1127.91	−0.101798
$498$	0	0
$499$	−1723.40	−0.154609	−0.0773047	−	0.997008i	$-0.524631\pi$
$499$	−1723.40	−0.154609	−0.0773047	+	0.997008i	$0.524631\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13820.8	1.22513	0.612565	−	0.790420i	$-0.290138\pi$
$503$	13820.8	1.22513	0.612565	+	0.790420i	$0.290138\pi$
$504$	0	0
$505$	483.659	0.0426189
$506$	0	0
$507$	0	0
$508$	0	0
$509$	81.7315	0.00711726	0.00355863	−	0.999994i	$-0.498867\pi$
$509$	81.7315	0.00711726	0.00355863	+	0.999994i	$0.498867\pi$
$510$	0	0
$511$	−2834.27	−0.245363
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2889.36	0.247224
$516$	0	0
$517$	−5334.40	−0.453785
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10464.9	0.879993	0.439997	−	0.897999i	$-0.354980\pi$
$521$	10464.9	0.879993	0.439997	+	0.897999i	$0.354980\pi$
$522$	0	0
$523$	−6126.99	−0.512265	−0.256133	−	0.966642i	$-0.582448\pi$
$523$	−6126.99	−0.512265	−0.256133	+	0.966642i	$0.582448\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3591.66	−0.296879
$528$	0	0
$529$	−4250.90	−0.349379
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3060.62	0.248725
$534$	0	0
$535$	3372.60	0.272542
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4051.51	0.323768
$540$	0	0
$541$	981.465	0.0779972	0.0389986	−	0.999239i	$-0.487583\pi$
$541$	981.465	0.0779972	0.0389986	+	0.999239i	$0.487583\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1682.73	−0.132257
$546$	0	0
$547$	−8802.28	−0.688041	−0.344021	−	0.938962i	$-0.611789\pi$
$547$	−8802.28	−0.688041	−0.344021	+	0.938962i	$0.611789\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9820.31	−0.759273
$552$	0	0
$553$	284.139	0.0218496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4371.10	−0.332512	−0.166256	−	0.986083i	$-0.553168\pi$
$557$	−4371.10	−0.332512	−0.166256	+	0.986083i	$0.553168\pi$
$558$	0	0
$559$	6794.73	0.514108
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1506.20	0.112751	0.0563756	−	0.998410i	$-0.482046\pi$
$563$	1506.20	0.112751	0.0563756	+	0.998410i	$0.482046\pi$
$564$	0	0
$565$	83.6858	0.00623130
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12125.1	−0.893339	−0.446669	−	0.894699i	$-0.647390\pi$
$569$	−12125.1	−0.893339	−0.446669	+	0.894699i	$0.647390\pi$
$570$	0	0
$571$	9842.10	0.721329	0.360665	−	0.932696i	$-0.382550\pi$
$571$	9842.10	0.721329	0.360665	+	0.932696i	$0.382550\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10798.3	−0.783163
$576$	0	0
$577$	−5257.04	−0.379296	−0.189648	−	0.981852i	$-0.560735\pi$
$577$	−5257.04	−0.379296	−0.189648	+	0.981852i	$0.560735\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4437.47	−0.316863
$582$	0	0
$583$	−7673.51	−0.545119
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1572.25	−0.110552	−0.0552759	−	0.998471i	$-0.517604\pi$
$587$	−1572.25	−0.110552	−0.0552759	+	0.998471i	$0.517604\pi$
$588$	0	0
$589$	−4032.65	−0.282109
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6737.33	0.466558	0.233279	−	0.972410i	$-0.425054\pi$
$593$	6737.33	0.466558	0.233279	+	0.972410i	$0.425054\pi$
$594$	0	0
$595$	−322.365	−0.0222112
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8480.52	0.578472	0.289236	−	0.957258i	$-0.406599\pi$
$599$	8480.52	0.578472	0.289236	+	0.957258i	$0.406599\pi$
$600$	0	0
$601$	1382.04	0.0938015	0.0469008	−	0.998900i	$-0.485066\pi$
$601$	1382.04	0.0938015	0.0469008	+	0.998900i	$0.485066\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2228.92	−0.149783
$606$	0	0
$607$	−1011.94	−0.0676661	−0.0338330	−	0.999427i	$-0.510771\pi$
$607$	−1011.94	−0.0676661	−0.0338330	+	0.999427i	$0.510771\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5453.35	0.361078
$612$	0	0
$613$	22508.6	1.48306	0.741529	−	0.670921i	$-0.234101\pi$
$613$	22508.6	1.48306	0.741529	+	0.670921i	$0.234101\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10104.1	0.659277	0.329639	−	0.944107i	$-0.393073\pi$
$617$	10104.1	0.659277	0.329639	+	0.944107i	$0.393073\pi$
$618$	0	0
$619$	−4361.32	−0.283193	−0.141596	−	0.989924i	$-0.545223\pi$
$619$	−4361.32	−0.283193	−0.141596	+	0.989924i	$0.545223\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3998.00	−0.257105
$624$	0	0
$625$	14275.6	0.913637
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8463.32	0.536494
$630$	0	0
$631$	10289.5	0.649157	0.324579	−	0.945859i	$-0.394778\pi$
$631$	10289.5	0.649157	0.324579	+	0.945859i	$0.394778\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	813.196	0.0508200
$636$	0	0
$637$	−4141.85	−0.257624
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19292.6	1.18879	0.594394	−	0.804174i	$-0.297392\pi$
$641$	19292.6	1.18879	0.594394	+	0.804174i	$0.297392\pi$
$642$	0	0
$643$	−28099.0	−1.72336	−0.861678	−	0.507456i	$-0.830586\pi$
$643$	−28099.0	−1.72336	−0.861678	+	0.507456i	$0.830586\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31002.3	1.88381	0.941906	−	0.335877i	$-0.109033\pi$
$647$	31002.3	1.88381	0.941906	+	0.335877i	$0.109033\pi$
$648$	0	0
$649$	4300.16	0.260086
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1847.66	0.110726	0.0553632	−	0.998466i	$-0.482368\pi$
$653$	1847.66	0.110726	0.0553632	+	0.998466i	$0.482368\pi$
$654$	0	0
$655$	1867.69	0.111415
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19850.5	1.17339	0.586695	−	0.809808i	$-0.300429\pi$
$659$	19850.5	1.17339	0.586695	+	0.809808i	$0.300429\pi$
$660$	0	0
$661$	−1284.96	−0.0756112	−0.0378056	−	0.999285i	$-0.512037\pi$
$661$	−1284.96	−0.0756112	−0.0378056	+	0.999285i	$0.512037\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−361.946	−0.0211062
$666$	0	0
$667$	−22728.4	−1.31941
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5585.70	0.321361
$672$	0	0
$673$	15502.3	0.887921	0.443960	−	0.896046i	$-0.353573\pi$
$673$	15502.3	0.887921	0.443960	+	0.896046i	$0.353573\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24864.7	−1.41156	−0.705782	−	0.708429i	$-0.749404\pi$
$677$	−24864.7	−1.41156	−0.705782	+	0.708429i	$0.749404\pi$
$678$	0	0
$679$	1518.79	0.0858405
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12324.7	−0.690470	−0.345235	−	0.938516i	$-0.612201\pi$
$683$	−12324.7	−0.690470	−0.345235	+	0.938516i	$0.612201\pi$
$684$	0	0
$685$	4474.04	0.249553
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7844.61	0.433753
$690$	0	0
$691$	417.891	0.0230063	0.0115031	−	0.999934i	$-0.496338\pi$
$691$	417.891	0.0230063	0.0115031	+	0.999934i	$0.496338\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2149.56	0.117320
$696$	0	0
$697$	8060.91	0.438061
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8526.86	−0.459422	−0.229711	−	0.973259i	$-0.573778\pi$
$701$	−8526.86	−0.459422	−0.229711	+	0.973259i	$0.573778\pi$
$702$	0	0
$703$	9502.45	0.509803
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1253.22	−0.0666648
$708$	0	0
$709$	13408.9	0.710273	0.355137	−	0.934814i	$-0.384434\pi$
$709$	13408.9	0.710273	0.355137	+	0.934814i	$0.384434\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9333.27	−0.490230
$714$	0	0
$715$	−315.124	−0.0164825
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21760.4	1.12869	0.564345	−	0.825539i	$-0.309129\pi$
$719$	21760.4	1.12869	0.564345	+	0.825539i	$0.309129\pi$
$720$	0	0
$721$	−7486.67	−0.386710
$722$	0	0
$723$	0	0
$724$	0	0
$725$	31003.6	1.58820
$726$	0	0
$727$	−3416.60	−0.174298	−0.0871490	−	0.996195i	$-0.527776\pi$
$727$	−3416.60	−0.174298	−0.0871490	+	0.996195i	$0.527776\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17895.6	0.905462
$732$	0	0
$733$	25616.7	1.29082	0.645412	−	0.763834i	$-0.276686\pi$
$733$	25616.7	1.29082	0.645412	+	0.763834i	$0.276686\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7388.84	0.369296
$738$	0	0
$739$	−16609.7	−0.826788	−0.413394	−	0.910552i	$-0.635657\pi$
$739$	−16609.7	−0.826788	−0.413394	+	0.910552i	$0.635657\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16364.3	−0.808003	−0.404001	−	0.914758i	$-0.632381\pi$
$743$	−16364.3	−0.808003	−0.404001	+	0.914758i	$0.632381\pi$
$744$	0	0
$745$	5527.03	0.271805
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8738.79	−0.426313
$750$	0	0
$751$	−25436.1	−1.23592	−0.617960	−	0.786210i	$-0.712040\pi$
$751$	−25436.1	−1.23592	−0.617960	+	0.786210i	$0.712040\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−747.279	−0.0360215
$756$	0	0
$757$	−11847.4	−0.568824	−0.284412	−	0.958702i	$-0.591798\pi$
$757$	−11847.4	−0.568824	−0.284412	+	0.958702i	$0.591798\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8819.17	0.420098	0.210049	−	0.977691i	$-0.432638\pi$
$761$	8819.17	0.420098	0.210049	+	0.977691i	$0.432638\pi$
$762$	0	0
$763$	4360.14	0.206878
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4396.04	−0.206951
$768$	0	0
$769$	−21409.4	−1.00396	−0.501979	−	0.864880i	$-0.667395\pi$
$769$	−21409.4	−1.00396	−0.501979	+	0.864880i	$0.667395\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28588.4	1.33021	0.665106	−	0.746749i	$-0.268386\pi$
$773$	28588.4	1.33021	0.665106	+	0.746749i	$0.268386\pi$
$774$	0	0
$775$	12731.4	0.590098
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9050.63	0.416268
$780$	0	0
$781$	−2903.89	−0.133047
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3428.29	0.155874
$786$	0	0
$787$	−34861.5	−1.57901	−0.789503	−	0.613747i	$-0.789662\pi$
$787$	−34861.5	−1.57901	−0.789503	+	0.613747i	$0.789662\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−216.839	−0.00974706
$792$	0	0
$793$	−5710.24	−0.255708
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12351.2	0.548938	0.274469	−	0.961596i	$-0.411498\pi$
$797$	12351.2	0.548938	0.274469	+	0.961596i	$0.411498\pi$
$798$	0	0
$799$	14362.7	0.635941
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7297.06	−0.320682
$804$	0	0
$805$	−837.697	−0.0366769
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43526.2	1.89159	0.945797	−	0.324758i	$-0.105283\pi$
$809$	43526.2	1.89159	0.945797	+	0.324758i	$0.105283\pi$
$810$	0	0
$811$	−26395.4	−1.14287	−0.571434	−	0.820648i	$-0.693613\pi$
$811$	−26395.4	−1.14287	−0.571434	+	0.820648i	$0.693613\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1441.14	0.0619399
$816$	0	0
$817$	20092.8	0.860415
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23973.4	1.01910	0.509548	−	0.860442i	$-0.329813\pi$
$821$	23973.4	1.01910	0.509548	+	0.860442i	$0.329813\pi$
$822$	0	0
$823$	37663.7	1.59523	0.797616	−	0.603166i	$-0.206094\pi$
$823$	37663.7	1.59523	0.797616	+	0.603166i	$0.206094\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27829.5	1.17017	0.585083	−	0.810973i	$-0.301062\pi$
$827$	27829.5	1.17017	0.585083	+	0.810973i	$0.301062\pi$
$828$	0	0
$829$	14780.4	0.619234	0.309617	−	0.950861i	$-0.399799\pi$
$829$	14780.4	0.619234	0.309617	+	0.950861i	$0.399799\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10908.6	−0.453734
$834$	0	0
$835$	1448.15	0.0600182
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34581.6	−1.42299	−0.711495	−	0.702691i	$-0.751982\pi$
$839$	−34581.6	−1.42299	−0.711495	+	0.702691i	$0.751982\pi$
$840$	0	0
$841$	40867.9	1.67567
$842$	0	0
$843$	0	0
$844$	0	0
$845$	322.151	0.0131152
$846$	0	0
$847$	5775.40	0.234292
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21992.7	0.885901
$852$	0	0
$853$	4826.04	0.193717	0.0968583	−	0.995298i	$-0.469121\pi$
$853$	4826.04	0.193717	0.0968583	+	0.995298i	$0.469121\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48299.8	−1.92519	−0.962595	−	0.270943i	$-0.912664\pi$
$857$	−48299.8	−1.92519	−0.962595	+	0.270943i	$0.912664\pi$
$858$	0	0
$859$	−42123.3	−1.67314	−0.836570	−	0.547860i	$-0.815443\pi$
$859$	−42123.3	−1.67314	−0.836570	+	0.547860i	$0.815443\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31958.3	1.26057	0.630286	−	0.776363i	$-0.282938\pi$
$863$	31958.3	1.26057	0.630286	+	0.776363i	$0.282938\pi$
$864$	0	0
$865$	1758.70	0.0691301
$866$	0	0
$867$	0	0
$868$	0	0
$869$	731.540	0.0285567
$870$	0	0
$871$	−7553.59	−0.293850
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2319.60	0.0896191
$876$	0	0
$877$	37396.2	1.43989	0.719943	−	0.694034i	$-0.244168\pi$
$877$	37396.2	1.43989	0.719943	+	0.694034i	$0.244168\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9361.40	−0.357995	−0.178997	−	0.983850i	$-0.557285\pi$
$881$	−9361.40	−0.357995	−0.178997	+	0.983850i	$0.557285\pi$
$882$	0	0
$883$	−16896.3	−0.643949	−0.321974	−	0.946748i	$-0.604346\pi$
$883$	−16896.3	−0.643949	−0.321974	+	0.946748i	$0.604346\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41762.5	1.58089	0.790444	−	0.612535i	$-0.209850\pi$
$887$	41762.5	1.58089	0.790444	+	0.612535i	$0.209850\pi$
$888$	0	0
$889$	−2107.08	−0.0794931
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16126.2	0.604303
$894$	0	0
$895$	2465.68	0.0920877
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26797.3	0.994150
$900$	0	0
$901$	20660.7	0.763938
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4466.30	−0.164049
$906$	0	0
$907$	−31532.2	−1.15437	−0.577183	−	0.816615i	$-0.695848\pi$
$907$	−31532.2	−1.15437	−0.577183	+	0.816615i	$0.695848\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15530.9	0.564832	0.282416	−	0.959292i	$-0.408864\pi$
$911$	15530.9	0.564832	0.282416	+	0.959292i	$0.408864\pi$
$912$	0	0
$913$	−11424.6	−0.414130
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4839.39	−0.174276
$918$	0	0
$919$	−32364.2	−1.16169	−0.580846	−	0.814013i	$-0.697278\pi$
$919$	−32364.2	−1.16169	−0.580846	+	0.814013i	$0.697278\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2968.65	0.105866
$924$	0	0
$925$	−30000.0	−1.06637
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27523.7	0.972038	0.486019	−	0.873948i	$-0.338449\pi$
$929$	27523.7	0.972038	0.486019	+	0.873948i	$0.338449\pi$
$930$	0	0
$931$	−12248.0	−0.431160
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−829.957	−0.0290294
$936$	0	0
$937$	31981.3	1.11503	0.557515	−	0.830167i	$-0.311755\pi$
$937$	31981.3	1.11503	0.557515	+	0.830167i	$0.311755\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−23448.2	−0.812318	−0.406159	−	0.913803i	$-0.633132\pi$
$941$	−23448.2	−0.812318	−0.406159	+	0.913803i	$0.633132\pi$
$942$	0	0
$943$	20947.0	0.723360
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−54392.8	−1.86645	−0.933225	−	0.359292i	$-0.883018\pi$
$947$	−54392.8	−1.86645	−0.933225	+	0.359292i	$0.883018\pi$
$948$	0	0
$949$	7459.77	0.255168
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54219.9	1.84297	0.921487	−	0.388408i	$-0.126975\pi$
$953$	54219.9	1.84297	0.921487	+	0.388408i	$0.126975\pi$
$954$	0	0
$955$	7636.05	0.258740
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11592.7	−0.390353
$960$	0	0
$961$	−18786.9	−0.630622
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9388.66	−0.313193
$966$	0	0
$967$	23586.8	0.784386	0.392193	−	0.919883i	$-0.371716\pi$
$967$	23586.8	0.784386	0.392193	+	0.919883i	$0.371716\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1673.11	−0.0552962	−0.0276481	−	0.999618i	$-0.508802\pi$
$971$	−1673.11	−0.0552962	−0.0276481	+	0.999618i	$0.508802\pi$
$972$	0	0
$973$	−5569.77	−0.183513
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7217.95	−0.236359	−0.118179	−	0.992992i	$-0.537706\pi$
$977$	−7217.95	−0.236359	−0.118179	+	0.992992i	$0.537706\pi$
$978$	0	0
$979$	−10293.2	−0.336028
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13878.1	−0.450298	−0.225149	−	0.974324i	$-0.572287\pi$
$983$	−13878.1	−0.450298	−0.225149	+	0.974324i	$0.572287\pi$
$984$	0	0
$985$	1399.96	0.0452858
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46503.4	1.49517
$990$	0	0
$991$	13033.0	0.417766	0.208883	−	0.977941i	$-0.433017\pi$
$991$	13033.0	0.417766	0.208883	+	0.977941i	$0.433017\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1555.48	0.0495597
$996$	0	0
$997$	−18062.5	−0.573767	−0.286884	−	0.957965i	$-0.592619\pi$
$997$	−18062.5	−0.573767	−0.286884	+	0.957965i	$0.592619\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.4.a.o.1.3	yes	4
3.2	odd	2		936.4.a.n.1.2	✓	4
4.3	odd	2		1872.4.a.bq.1.3		4
12.11	even	2		1872.4.a.bn.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.4.a.n.1.2	✓	4	3.2	odd	2
936.4.a.o.1.3	yes	4	1.1	even	1	trivial
1872.4.a.bn.1.2		4	12.11	even	2
1872.4.a.bq.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	936.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90622 q^{5} -4.93923 q^{7} -12.7165 q^{11} +13.0000 q^{13} +34.2387 q^{17} +38.4425 q^{19} +88.9725 q^{23} -121.366 q^{25} -255.454 q^{29} -104.901 q^{31} -9.41523 q^{35} +247.186 q^{37} +235.433 q^{41} +522.672 q^{43} +419.488 q^{47} -318.604 q^{49} +603.432 q^{53} -24.2403 q^{55} -338.157 q^{59} -439.250 q^{61} +24.7808 q^{65} -581.045 q^{67} +228.357 q^{71} +573.828 q^{73} +62.8094 q^{77} -57.5271 q^{79} +898.414 q^{83} +65.2664 q^{85} +809.438 q^{89} -64.2099 q^{91} +73.2798 q^{95} -307.495 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 24 q^{7} + 64 q^{11} + 52 q^{13} + 80 q^{17} + 24 q^{19} + 16 q^{23} - 20 q^{25} + 240 q^{29} + 24 q^{31} + 80 q^{35} - 152 q^{37} + 136 q^{41} - 64 q^{43} + 528 q^{47} + 276 q^{49} + 496 q^{53}+ \cdots - 1384 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 24 * q^7 + 64 * q^11 + 52 * q^13 + 80 * q^17 + 24 * q^19 + 16 * q^23 - 20 * q^25 + 240 * q^29 + 24 * q^31 + 80 * q^35 - 152 * q^37 + 136 * q^41 - 64 * q^43 + 528 * q^47 + 276 * q^49 + 496 * q^53 - 352 * q^55 + 832 * q^59 - 104 * q^61 + 104 * q^65 + 1000 * q^67 + 1584 * q^71 - 232 * q^73 + 1584 * q^77 + 464 * q^79 + 1888 * q^83 - 1536 * q^85 + 696 * q^89 + 312 * q^91 + 1808 * q^95 - 1384 * q^97

Introduction

L-functions

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