LMFDB - Embedded newform 1176.4.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,4,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1176.a (trivial)

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:	yes
Analytic conductor:	$69.3862461668$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.391168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 40x^{2} + 382 $ x^4 - 40*x^2 + 382
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.96955$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -7.81338 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -7.81338 q^{5} +9.00000 q^{9} +42.7627 q^{11} -10.9493 q^{13} -23.4401 q^{15} +80.2889 q^{17} +112.675 q^{19} -70.1884 q^{23} -63.9511 q^{25} +27.0000 q^{27} -147.010 q^{29} -144.529 q^{31} +128.288 q^{33} -0.292803 q^{37} -32.8479 q^{39} -294.097 q^{41} +337.866 q^{43} -70.3204 q^{45} +104.524 q^{47} +240.867 q^{51} -143.050 q^{53} -334.121 q^{55} +338.024 q^{57} +520.440 q^{59} +399.902 q^{61} +85.5509 q^{65} +137.813 q^{67} -210.565 q^{69} +266.418 q^{71} +524.754 q^{73} -191.853 q^{75} +433.874 q^{79} +81.0000 q^{81} +664.921 q^{83} -627.328 q^{85} -441.029 q^{87} +803.579 q^{89} -433.588 q^{93} -880.370 q^{95} +445.813 q^{97} +384.864 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} + 72 q^{17} + 32 q^{19} + 8 q^{23} + 164 q^{25} + 108 q^{27} + 144 q^{29} + 48 q^{31} + 120 q^{33} + 48 q^{37} + 144 q^{39} + 72 q^{41} + 512 q^{43} + 72 q^{45} + 160 q^{47} + 216 q^{51} + 536 q^{53} - 336 q^{55} + 96 q^{57} + 240 q^{59} + 896 q^{61} - 136 q^{65} + 1088 q^{67} + 24 q^{69} + 1288 q^{71} + 1488 q^{73} + 492 q^{75} + 416 q^{79} + 324 q^{81} - 112 q^{83} - 1512 q^{85} + 432 q^{87} + 3160 q^{89} + 144 q^{93} - 240 q^{95} + 2384 q^{97} + 360 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 + 72 * q^17 + 32 * q^19 + 8 * q^23 + 164 * q^25 + 108 * q^27 + 144 * q^29 + 48 * q^31 + 120 * q^33 + 48 * q^37 + 144 * q^39 + 72 * q^41 + 512 * q^43 + 72 * q^45 + 160 * q^47 + 216 * q^51 + 536 * q^53 - 336 * q^55 + 96 * q^57 + 240 * q^59 + 896 * q^61 - 136 * q^65 + 1088 * q^67 + 24 * q^69 + 1288 * q^71 + 1488 * q^73 + 492 * q^75 + 416 * q^79 + 324 * q^81 - 112 * q^83 - 1512 * q^85 + 432 * q^87 + 3160 * q^89 + 144 * q^93 - 240 * q^95 + 2384 * q^97 + 360 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−7.81338	−0.698850	−0.349425	−	0.936964i	$-0.613623\pi$
$5$	−7.81338	−0.698850	−0.349425	+	0.936964i	$0.613623\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	42.7627	1.17213	0.586065	−	0.810264i	$-0.300676\pi$
$11$	42.7627	1.17213	0.586065	+	0.810264i	$0.300676\pi$
$12$	0	0
$13$	−10.9493	−0.233599	−0.116799	−	0.993156i	$-0.537263\pi$
$13$	−10.9493	−0.233599	−0.116799	+	0.993156i	$0.537263\pi$
$14$	0	0
$15$	−23.4401	−0.403481
$16$	0	0
$17$	80.2889	1.14547	0.572733	−	0.819742i	$-0.305883\pi$
$17$	80.2889	1.14547	0.572733	+	0.819742i	$0.305883\pi$
$18$	0	0
$19$	112.675	1.36049	0.680246	−	0.732984i	$-0.261873\pi$
$19$	112.675	1.36049	0.680246	+	0.732984i	$0.261873\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−70.1884	−0.636317	−0.318159	−	0.948037i	$-0.603064\pi$
$23$	−70.1884	−0.636317	−0.318159	+	0.948037i	$0.603064\pi$
$24$	0	0
$25$	−63.9511	−0.511609
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−147.010	−0.941344	−0.470672	−	0.882308i	$-0.655989\pi$
$29$	−147.010	−0.941344	−0.470672	+	0.882308i	$0.655989\pi$
$30$	0	0
$31$	−144.529	−0.837363	−0.418681	−	0.908133i	$-0.637508\pi$
$31$	−144.529	−0.837363	−0.418681	+	0.908133i	$0.637508\pi$
$32$	0	0
$33$	128.288	0.676729
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.292803	−0.00130099	−0.000650494	−	1.00000i	$-0.500207\pi$
$37$	−0.292803	−0.00130099	−0.000650494		1.00000i	$0.500207\pi$
$38$	0	0
$39$	−32.8479	−0.134868
$40$	0	0
$41$	−294.097	−1.12025	−0.560126	−	0.828408i	$-0.689247\pi$
$41$	−294.097	−1.12025	−0.560126	+	0.828408i	$0.689247\pi$
$42$	0	0
$43$	337.866	1.19823	0.599117	−	0.800661i	$-0.295518\pi$
$43$	337.866	1.19823	0.599117	+	0.800661i	$0.295518\pi$
$44$	0	0
$45$	−70.3204	−0.232950
$46$	0	0
$47$	104.524	0.324391	0.162196	−	0.986759i	$-0.448142\pi$
$47$	104.524	0.324391	0.162196	+	0.986759i	$0.448142\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	240.867	0.661335
$52$	0	0
$53$	−143.050	−0.370743	−0.185372	−	0.982668i	$-0.559349\pi$
$53$	−143.050	−0.370743	−0.185372	+	0.982668i	$0.559349\pi$
$54$	0	0
$55$	−334.121	−0.819143
$56$	0	0
$57$	338.024	0.785480
$58$	0	0
$59$	520.440	1.14840	0.574199	−	0.818716i	$-0.305314\pi$
$59$	520.440	1.14840	0.574199	+	0.818716i	$0.305314\pi$
$60$	0	0
$61$	399.902	0.839380	0.419690	−	0.907668i	$-0.362139\pi$
$61$	399.902	0.839380	0.419690	+	0.907668i	$0.362139\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	85.5509	0.163251
$66$	0	0
$67$	137.813	0.251292	0.125646	−	0.992075i	$-0.459900\pi$
$67$	137.813	0.251292	0.125646	+	0.992075i	$0.459900\pi$
$68$	0	0
$69$	−210.565	−0.367378
$70$	0	0
$71$	266.418	0.445324	0.222662	−	0.974896i	$-0.428525\pi$
$71$	266.418	0.445324	0.222662	+	0.974896i	$0.428525\pi$
$72$	0	0
$73$	524.754	0.841340	0.420670	−	0.907214i	$-0.361795\pi$
$73$	524.754	0.841340	0.420670	+	0.907214i	$0.361795\pi$
$74$	0	0
$75$	−191.853	−0.295377
$76$	0	0
$77$	0	0
$78$	0	0
$79$	433.874	0.617907	0.308953	−	0.951077i	$-0.400021\pi$
$79$	433.874	0.617907	0.308953	+	0.951077i	$0.400021\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	664.921	0.879332	0.439666	−	0.898161i	$-0.355097\pi$
$83$	664.921	0.879332	0.439666	+	0.898161i	$0.355097\pi$
$84$	0	0
$85$	−627.328	−0.800509
$86$	0	0
$87$	−441.029	−0.543485
$88$	0	0
$89$	803.579	0.957069	0.478535	−	0.878069i	$-0.341168\pi$
$89$	803.579	0.957069	0.478535	+	0.878069i	$0.341168\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−433.588	−0.483452
$94$	0	0
$95$	−880.370	−0.950780
$96$	0	0
$97$	445.813	0.466654	0.233327	−	0.972398i	$-0.425039\pi$
$97$	445.813	0.466654	0.233327	+	0.972398i	$0.425039\pi$
$98$	0	0
$99$	384.864	0.390710
$100$	0	0
$101$	1547.45	1.52453	0.762263	−	0.647267i	$-0.224088\pi$
$101$	1547.45	1.52453	0.762263	+	0.647267i	$0.224088\pi$
$102$	0	0
$103$	679.600	0.650126	0.325063	−	0.945692i	$-0.394614\pi$
$103$	679.600	0.650126	0.325063	+	0.945692i	$0.394614\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1998.86	1.80595	0.902975	−	0.429693i	$-0.141378\pi$
$107$	1998.86	1.80595	0.902975	+	0.429693i	$0.141378\pi$
$108$	0	0
$109$	−1455.60	−1.27909	−0.639547	−	0.768752i	$-0.720878\pi$
$109$	−1455.60	−1.27909	−0.639547	+	0.768752i	$0.720878\pi$
$110$	0	0
$111$	−0.878410	−0.000751126 0
$112$	0	0
$113$	523.808	0.436068	0.218034	−	0.975941i	$-0.430036\pi$
$113$	523.808	0.436068	0.218034	+	0.975941i	$0.430036\pi$
$114$	0	0
$115$	548.409	0.444690
$116$	0	0
$117$	−98.5436	−0.0778663
$118$	0	0
$119$	0	0
$120$	0	0
$121$	497.645	0.373888
$122$	0	0
$123$	−882.292	−0.646777
$124$	0	0
$125$	1476.35	1.05639
$126$	0	0
$127$	2496.08	1.74402	0.872011	−	0.489486i	$-0.162815\pi$
$127$	2496.08	1.74402	0.872011	+	0.489486i	$0.162815\pi$
$128$	0	0
$129$	1013.60	0.691801
$130$	0	0
$131$	407.249	0.271615	0.135807	−	0.990735i	$-0.456637\pi$
$131$	407.249	0.271615	0.135807	+	0.990735i	$0.456637\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−210.961	−0.134494
$136$	0	0
$137$	478.949	0.298682	0.149341	−	0.988786i	$-0.452285\pi$
$137$	478.949	0.298682	0.149341	+	0.988786i	$0.452285\pi$
$138$	0	0
$139$	−123.985	−0.0756569	−0.0378284	−	0.999284i	$-0.512044\pi$
$139$	−123.985	−0.0756569	−0.0378284	+	0.999284i	$0.512044\pi$
$140$	0	0
$141$	313.572	0.187287
$142$	0	0
$143$	−468.221	−0.273808
$144$	0	0
$145$	1148.64	0.657858
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1645.37	0.904658	0.452329	−	0.891851i	$-0.350593\pi$
$149$	1645.37	0.904658	0.452329	+	0.891851i	$0.350593\pi$
$150$	0	0
$151$	−3044.79	−1.64094	−0.820470	−	0.571690i	$-0.806288\pi$
$151$	−3044.79	−1.64094	−0.820470	+	0.571690i	$0.806288\pi$
$152$	0	0
$153$	722.600	0.381822
$154$	0	0
$155$	1129.26	0.585191
$156$	0	0
$157$	−1408.22	−0.715850	−0.357925	−	0.933750i	$-0.616516\pi$
$157$	−1408.22	−0.715850	−0.357925	+	0.933750i	$0.616516\pi$
$158$	0	0
$159$	−429.149	−0.214049
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2083.51	−1.00119	−0.500593	−	0.865683i	$-0.666885\pi$
$163$	−2083.51	−1.00119	−0.500593	+	0.865683i	$0.666885\pi$
$164$	0	0
$165$	−1002.36	−0.472932
$166$	0	0
$167$	−184.235	−0.0853685	−0.0426843	−	0.999089i	$-0.513591\pi$
$167$	−184.235	−0.0853685	−0.0426843	+	0.999089i	$0.513591\pi$
$168$	0	0
$169$	−2077.11	−0.945432
$170$	0	0
$171$	1014.07	0.453497
$172$	0	0
$173$	3332.98	1.46475	0.732375	−	0.680902i	$-0.238412\pi$
$173$	3332.98	1.46475	0.732375	+	0.680902i	$0.238412\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1561.32	0.663028
$178$	0	0
$179$	−3273.98	−1.36709	−0.683544	−	0.729909i	$-0.739562\pi$
$179$	−3273.98	−1.36709	−0.683544	+	0.729909i	$0.739562\pi$
$180$	0	0
$181$	−2708.05	−1.11209	−0.556043	−	0.831154i	$-0.687681\pi$
$181$	−2708.05	−1.11209	−0.556043	+	0.831154i	$0.687681\pi$
$182$	0	0
$183$	1199.71	0.484616
$184$	0	0
$185$	2.28778	0.000909196 0
$186$	0	0
$187$	3433.37	1.34264
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−985.187	−0.373223	−0.186612	−	0.982434i	$-0.559751\pi$
$191$	−985.187	−0.373223	−0.186612	+	0.982434i	$0.559751\pi$
$192$	0	0
$193$	−4627.92	−1.72604	−0.863018	−	0.505172i	$-0.831429\pi$
$193$	−4627.92	−1.72604	−0.863018	+	0.505172i	$0.831429\pi$
$194$	0	0
$195$	256.653	0.0942528
$196$	0	0
$197$	−3501.81	−1.26647	−0.633233	−	0.773961i	$-0.718273\pi$
$197$	−3501.81	−1.26647	−0.633233	+	0.773961i	$0.718273\pi$
$198$	0	0
$199$	169.661	0.0604368	0.0302184	−	0.999543i	$-0.490380\pi$
$199$	169.661	0.0604368	0.0302184	+	0.999543i	$0.490380\pi$
$200$	0	0
$201$	413.439	0.145083
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2297.90	0.782888
$206$	0	0
$207$	−631.696	−0.212106
$208$	0	0
$209$	4818.27	1.59467
$210$	0	0
$211$	−1188.76	−0.387857	−0.193929	−	0.981016i	$-0.562123\pi$
$211$	−1188.76	−0.387857	−0.193929	+	0.981016i	$0.562123\pi$
$212$	0	0
$213$	799.255	0.257108
$214$	0	0
$215$	−2639.88	−0.837386
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1574.26	0.485748
$220$	0	0
$221$	−879.106	−0.267580
$222$	0	0
$223$	3395.54	1.01965	0.509826	−	0.860278i	$-0.329710\pi$
$223$	3395.54	1.01965	0.509826	+	0.860278i	$0.329710\pi$
$224$	0	0
$225$	−575.560	−0.170536
$226$	0	0
$227$	1198.05	0.350297	0.175148	−	0.984542i	$-0.443959\pi$
$227$	1198.05	0.350297	0.175148	+	0.984542i	$0.443959\pi$
$228$	0	0
$229$	5820.40	1.67958	0.839788	−	0.542914i	$-0.182679\pi$
$229$	5820.40	1.67958	0.839788	+	0.542914i	$0.182679\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3488.08	0.980737	0.490368	−	0.871515i	$-0.336862\pi$
$233$	3488.08	0.980737	0.490368	+	0.871515i	$0.336862\pi$
$234$	0	0
$235$	−816.685	−0.226701
$236$	0	0
$237$	1301.62	0.356749
$238$	0	0
$239$	4688.08	1.26882	0.634408	−	0.772999i	$-0.281244\pi$
$239$	4688.08	1.26882	0.634408	+	0.772999i	$0.281244\pi$
$240$	0	0
$241$	6419.23	1.71576	0.857881	−	0.513848i	$-0.171780\pi$
$241$	6419.23	1.71576	0.857881	+	0.513848i	$0.171780\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1233.71	−0.317809
$248$	0	0
$249$	1994.76	0.507682
$250$	0	0
$251$	−5928.29	−1.49080	−0.745400	−	0.666618i	$-0.767741\pi$
$251$	−5928.29	−1.49080	−0.745400	+	0.666618i	$0.767741\pi$
$252$	0	0
$253$	−3001.44	−0.745846
$254$	0	0
$255$	−1881.98	−0.462174
$256$	0	0
$257$	−673.329	−0.163428	−0.0817142	−	0.996656i	$-0.526039\pi$
$257$	−673.329	−0.163428	−0.0817142	+	0.996656i	$0.526039\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1323.09	−0.313781
$262$	0	0
$263$	−5270.35	−1.23568	−0.617839	−	0.786304i	$-0.711992\pi$
$263$	−5270.35	−1.23568	−0.617839	+	0.786304i	$0.711992\pi$
$264$	0	0
$265$	1117.70	0.259094
$266$	0	0
$267$	2410.74	0.552564
$268$	0	0
$269$	4064.77	0.921313	0.460656	−	0.887579i	$-0.347614\pi$
$269$	4064.77	0.921313	0.460656	+	0.887579i	$0.347614\pi$
$270$	0	0
$271$	6524.19	1.46242	0.731211	−	0.682151i	$-0.238955\pi$
$271$	6524.19	1.46242	0.731211	+	0.682151i	$0.238955\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2734.72	−0.599672
$276$	0	0
$277$	8345.83	1.81030	0.905149	−	0.425095i	$-0.139759\pi$
$277$	8345.83	1.81030	0.905149	+	0.425095i	$0.139759\pi$
$278$	0	0
$279$	−1300.76	−0.279121
$280$	0	0
$281$	8161.43	1.73263	0.866316	−	0.499496i	$-0.166481\pi$
$281$	8161.43	1.73263	0.866316	+	0.499496i	$0.166481\pi$
$282$	0	0
$283$	−8669.66	−1.82105	−0.910526	−	0.413451i	$-0.864323\pi$
$283$	−8669.66	−1.82105	−0.910526	+	0.413451i	$0.864323\pi$
$284$	0	0
$285$	−2641.11	−0.548933
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1533.31	0.312093
$290$	0	0
$291$	1337.44	0.269423
$292$	0	0
$293$	99.7900	0.0198969	0.00994846	−	0.999951i	$-0.496833\pi$
$293$	99.7900	0.0198969	0.00994846	+	0.999951i	$0.496833\pi$
$294$	0	0
$295$	−4066.39	−0.802558
$296$	0	0
$297$	1154.59	0.225576
$298$	0	0
$299$	768.513	0.148643
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4642.35	0.880186
$304$	0	0
$305$	−3124.58	−0.586601
$306$	0	0
$307$	−8631.55	−1.60465	−0.802327	−	0.596885i	$-0.796405\pi$
$307$	−8631.55	−1.60465	−0.802327	+	0.596885i	$0.796405\pi$
$308$	0	0
$309$	2038.80	0.375351
$310$	0	0
$311$	863.980	0.157530	0.0787650	−	0.996893i	$-0.474902\pi$
$311$	863.980	0.157530	0.0787650	+	0.996893i	$0.474902\pi$
$312$	0	0
$313$	577.824	0.104347	0.0521734	−	0.998638i	$-0.483385\pi$
$313$	577.824	0.104347	0.0521734	+	0.998638i	$0.483385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−196.879	−0.0348828	−0.0174414	−	0.999848i	$-0.505552\pi$
$317$	−196.879	−0.0348828	−0.0174414	+	0.999848i	$0.505552\pi$
$318$	0	0
$319$	−6286.52	−1.10338
$320$	0	0
$321$	5996.57	1.04267
$322$	0	0
$323$	9046.53	1.55840
$324$	0	0
$325$	700.219	0.119511
$326$	0	0
$327$	−4366.80	−0.738485
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4645.40	−0.771402	−0.385701	−	0.922624i	$-0.626040\pi$
$331$	−4645.40	−0.771402	−0.385701	+	0.922624i	$0.626040\pi$
$332$	0	0
$333$	−2.63523	−0.000433663 0
$334$	0	0
$335$	−1076.79	−0.175615
$336$	0	0
$337$	−7255.28	−1.17276	−0.586380	−	0.810036i	$-0.699447\pi$
$337$	−7255.28	−1.17276	−0.586380	+	0.810036i	$0.699447\pi$
$338$	0	0
$339$	1571.42	0.251764
$340$	0	0
$341$	−6180.46	−0.981498
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1645.23	0.256742
$346$	0	0
$347$	3066.25	0.474366	0.237183	−	0.971465i	$-0.423776\pi$
$347$	3066.25	0.474366	0.237183	+	0.971465i	$0.423776\pi$
$348$	0	0
$349$	884.690	0.135692	0.0678458	−	0.997696i	$-0.478387\pi$
$349$	884.690	0.135692	0.0678458	+	0.997696i	$0.478387\pi$
$350$	0	0
$351$	−295.631	−0.0449561
$352$	0	0
$353$	2463.28	0.371409	0.185704	−	0.982606i	$-0.440543\pi$
$353$	2463.28	0.371409	0.185704	+	0.982606i	$0.440543\pi$
$354$	0	0
$355$	−2081.63	−0.311215
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1999.23	−0.293915	−0.146957	−	0.989143i	$-0.546948\pi$
$359$	−1999.23	−0.293915	−0.146957	+	0.989143i	$0.546948\pi$
$360$	0	0
$361$	5836.59	0.850939
$362$	0	0
$363$	1492.94	0.215865
$364$	0	0
$365$	−4100.10	−0.587971
$366$	0	0
$367$	−6790.82	−0.965880	−0.482940	−	0.875653i	$-0.660431\pi$
$367$	−6790.82	−0.965880	−0.482940	+	0.875653i	$0.660431\pi$
$368$	0	0
$369$	−2646.88	−0.373417
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1222.01	−0.169633	−0.0848164	−	0.996397i	$-0.527030\pi$
$373$	−1222.01	−0.169633	−0.0848164	+	0.996397i	$0.527030\pi$
$374$	0	0
$375$	4429.04	0.609906
$376$	0	0
$377$	1609.65	0.219897
$378$	0	0
$379$	10228.6	1.38630	0.693149	−	0.720794i	$-0.256223\pi$
$379$	10228.6	1.38630	0.693149	+	0.720794i	$0.256223\pi$
$380$	0	0
$381$	7488.23	1.00691
$382$	0	0
$383$	−5452.92	−0.727496	−0.363748	−	0.931497i	$-0.618503\pi$
$383$	−5452.92	−0.727496	−0.363748	+	0.931497i	$0.618503\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3040.79	0.399412
$388$	0	0
$389$	14423.8	1.87999	0.939994	−	0.341190i	$-0.110830\pi$
$389$	14423.8	1.87999	0.939994	+	0.341190i	$0.110830\pi$
$390$	0	0
$391$	−5635.35	−0.728880
$392$	0	0
$393$	1221.75	0.156817
$394$	0	0
$395$	−3390.02	−0.431824
$396$	0	0
$397$	4462.35	0.564129	0.282064	−	0.959395i	$-0.408981\pi$
$397$	4462.35	0.564129	0.282064	+	0.959395i	$0.408981\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12380.6	1.54179	0.770897	−	0.636960i	$-0.219809\pi$
$401$	12380.6	1.54179	0.770897	+	0.636960i	$0.219809\pi$
$402$	0	0
$403$	1582.49	0.195607
$404$	0	0
$405$	−632.884	−0.0776500
$406$	0	0
$407$	−12.5210	−0.00152493
$408$	0	0
$409$	10065.9	1.21694	0.608469	−	0.793578i	$-0.291784\pi$
$409$	10065.9	1.21694	0.608469	+	0.793578i	$0.291784\pi$
$410$	0	0
$411$	1436.85	0.172444
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5195.28	−0.614521
$416$	0	0
$417$	−371.956	−0.0436805
$418$	0	0
$419$	−645.188	−0.0752256	−0.0376128	−	0.999292i	$-0.511975\pi$
$419$	−645.188	−0.0752256	−0.0376128	+	0.999292i	$0.511975\pi$
$420$	0	0
$421$	−16460.0	−1.90549	−0.952745	−	0.303770i	$-0.901755\pi$
$421$	−16460.0	−1.90549	−0.952745	+	0.303770i	$0.901755\pi$
$422$	0	0
$423$	940.716	0.108130
$424$	0	0
$425$	−5134.56	−0.586030
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1404.66	−0.158083
$430$	0	0
$431$	4989.28	0.557599	0.278800	−	0.960349i	$-0.410063\pi$
$431$	4989.28	0.557599	0.278800	+	0.960349i	$0.410063\pi$
$432$	0	0
$433$	−12366.3	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$433$	−12366.3	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$434$	0	0
$435$	3445.92	0.379815
$436$	0	0
$437$	−7908.46	−0.865705
$438$	0	0
$439$	11733.7	1.27567	0.637836	−	0.770172i	$-0.279830\pi$
$439$	11733.7	1.27567	0.637836	+	0.770172i	$0.279830\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4632.30	−0.496811	−0.248406	−	0.968656i	$-0.579907\pi$
$443$	−4632.30	−0.496811	−0.248406	+	0.968656i	$0.579907\pi$
$444$	0	0
$445$	−6278.67	−0.668848
$446$	0	0
$447$	4936.11	0.522304
$448$	0	0
$449$	−4057.84	−0.426506	−0.213253	−	0.976997i	$-0.568406\pi$
$449$	−4057.84	−0.426506	−0.213253	+	0.976997i	$0.568406\pi$
$450$	0	0
$451$	−12576.4	−1.31308
$452$	0	0
$453$	−9134.38	−0.947397
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17692.3	−1.81097	−0.905484	−	0.424381i	$-0.860492\pi$
$457$	−17692.3	−1.81097	−0.905484	+	0.424381i	$0.860492\pi$
$458$	0	0
$459$	2167.80	0.220445
$460$	0	0
$461$	8120.88	0.820449	0.410225	−	0.911984i	$-0.365450\pi$
$461$	8120.88	0.820449	0.410225	+	0.911984i	$0.365450\pi$
$462$	0	0
$463$	−19397.4	−1.94703	−0.973515	−	0.228624i	$-0.926577\pi$
$463$	−19397.4	−1.94703	−0.973515	+	0.228624i	$0.926577\pi$
$464$	0	0
$465$	3387.79	0.337860
$466$	0	0
$467$	−11033.3	−1.09327	−0.546637	−	0.837370i	$-0.684092\pi$
$467$	−11033.3	−1.09327	−0.546637	+	0.837370i	$0.684092\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4224.67	−0.413296
$472$	0	0
$473$	14448.1	1.40449
$474$	0	0
$475$	−7205.67	−0.696039
$476$	0	0
$477$	−1287.45	−0.123581
$478$	0	0
$479$	−14170.5	−1.35170	−0.675852	−	0.737038i	$-0.736224\pi$
$479$	−14170.5	−1.35170	−0.675852	+	0.737038i	$0.736224\pi$
$480$	0	0
$481$	3.20599	0.000303909 0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3483.31	−0.326121
$486$	0	0
$487$	12929.3	1.20304	0.601522	−	0.798856i	$-0.294561\pi$
$487$	12929.3	1.20304	0.601522	+	0.798856i	$0.294561\pi$
$488$	0	0
$489$	−6250.54	−0.578035
$490$	0	0
$491$	99.6829	0.00916217	0.00458109	−	0.999990i	$-0.498542\pi$
$491$	99.6829	0.00916217	0.00458109	+	0.999990i	$0.498542\pi$
$492$	0	0
$493$	−11803.2	−1.07828
$494$	0	0
$495$	−3007.09	−0.273048
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6207.28	−0.556865	−0.278433	−	0.960456i	$-0.589815\pi$
$499$	−6207.28	−0.556865	−0.278433	+	0.960456i	$0.589815\pi$
$500$	0	0
$501$	−552.705	−0.0492875
$502$	0	0
$503$	4743.31	0.420464	0.210232	−	0.977651i	$-0.432578\pi$
$503$	4743.31	0.420464	0.210232	+	0.977651i	$0.432578\pi$
$504$	0	0
$505$	−12090.8	−1.06542
$506$	0	0
$507$	−6231.34	−0.545845
$508$	0	0
$509$	−9836.46	−0.856569	−0.428284	−	0.903644i	$-0.640882\pi$
$509$	−9836.46	−0.856569	−0.428284	+	0.903644i	$0.640882\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3042.22	0.261827
$514$	0	0
$515$	−5309.98	−0.454341
$516$	0	0
$517$	4469.72	0.380229
$518$	0	0
$519$	9998.94	0.845674
$520$	0	0
$521$	−2285.55	−0.192192	−0.0960958	−	0.995372i	$-0.530636\pi$
$521$	−2285.55	−0.192192	−0.0960958	+	0.995372i	$0.530636\pi$
$522$	0	0
$523$	7740.65	0.647180	0.323590	−	0.946197i	$-0.395110\pi$
$523$	7740.65	0.647180	0.323590	+	0.946197i	$0.395110\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11604.1	−0.959171
$528$	0	0
$529$	−7240.59	−0.595100
$530$	0	0
$531$	4683.96	0.382799
$532$	0	0
$533$	3220.16	0.261689
$534$	0	0
$535$	−15617.8	−1.26209
$536$	0	0
$537$	−9821.94	−0.789289
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15682.4	−1.24628	−0.623141	−	0.782110i	$-0.714144\pi$
$541$	−15682.4	−1.24628	−0.623141	+	0.782110i	$0.714144\pi$
$542$	0	0
$543$	−8124.14	−0.642063
$544$	0	0
$545$	11373.2	0.893895
$546$	0	0
$547$	−3365.32	−0.263054	−0.131527	−	0.991313i	$-0.541988\pi$
$547$	−3365.32	−0.263054	−0.131527	+	0.991313i	$0.541988\pi$
$548$	0	0
$549$	3599.12	0.279793
$550$	0	0
$551$	−16564.3	−1.28069
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.86335	0.000524924 0
$556$	0	0
$557$	11968.8	0.910478	0.455239	−	0.890369i	$-0.349554\pi$
$557$	11968.8	0.910478	0.455239	+	0.890369i	$0.349554\pi$
$558$	0	0
$559$	−3699.39	−0.279906
$560$	0	0
$561$	10300.1	0.775171
$562$	0	0
$563$	14981.4	1.12148	0.560740	−	0.827992i	$-0.310517\pi$
$563$	14981.4	1.12148	0.560740	+	0.827992i	$0.310517\pi$
$564$	0	0
$565$	−4092.71	−0.304746
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14947.7	1.10130	0.550652	−	0.834735i	$-0.314379\pi$
$569$	14947.7	1.10130	0.550652	+	0.834735i	$0.314379\pi$
$570$	0	0
$571$	8138.91	0.596503	0.298251	−	0.954487i	$-0.403597\pi$
$571$	8138.91	0.596503	0.298251	+	0.954487i	$0.403597\pi$
$572$	0	0
$573$	−2955.56	−0.215481
$574$	0	0
$575$	4488.63	0.325545
$576$	0	0
$577$	−4092.69	−0.295288	−0.147644	−	0.989041i	$-0.547169\pi$
$577$	−4092.69	−0.295288	−0.147644	+	0.989041i	$0.547169\pi$
$578$	0	0
$579$	−13883.8	−0.996528
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6117.19	−0.434559
$584$	0	0
$585$	769.958	0.0544169
$586$	0	0
$587$	−15877.4	−1.11641	−0.558203	−	0.829705i	$-0.688509\pi$
$587$	−15877.4	−1.11641	−0.558203	+	0.829705i	$0.688509\pi$
$588$	0	0
$589$	−16284.8	−1.13923
$590$	0	0
$591$	−10505.4	−0.731195
$592$	0	0
$593$	21660.8	1.50000	0.750001	−	0.661436i	$-0.230053\pi$
$593$	21660.8	1.50000	0.750001	+	0.661436i	$0.230053\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	508.982	0.0348932
$598$	0	0
$599$	−11674.4	−0.796334	−0.398167	−	0.917313i	$-0.630354\pi$
$599$	−11674.4	−0.796334	−0.398167	+	0.917313i	$0.630354\pi$
$600$	0	0
$601$	5818.05	0.394881	0.197440	−	0.980315i	$-0.436737\pi$
$601$	5818.05	0.394881	0.197440	+	0.980315i	$0.436737\pi$
$602$	0	0
$603$	1240.32	0.0837640
$604$	0	0
$605$	−3888.29	−0.261292
$606$	0	0
$607$	−19894.5	−1.33030	−0.665150	−	0.746710i	$-0.731632\pi$
$607$	−19894.5	−1.33030	−0.665150	+	0.746710i	$0.731632\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1144.46	−0.0757774
$612$	0	0
$613$	23979.8	1.57999	0.789996	−	0.613112i	$-0.210083\pi$
$613$	23979.8	1.57999	0.789996	+	0.613112i	$0.210083\pi$
$614$	0	0
$615$	6893.69	0.452000
$616$	0	0
$617$	−16527.7	−1.07841	−0.539207	−	0.842173i	$-0.681276\pi$
$617$	−16527.7	−1.07841	−0.539207	+	0.842173i	$0.681276\pi$
$618$	0	0
$619$	−2135.87	−0.138688	−0.0693440	−	0.997593i	$-0.522091\pi$
$619$	−2135.87	−0.138688	−0.0693440	+	0.997593i	$0.522091\pi$
$620$	0	0
$621$	−1895.09	−0.122459
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3541.37	−0.226648
$626$	0	0
$627$	14454.8	0.920685
$628$	0	0
$629$	−23.5089	−0.00149024
$630$	0	0
$631$	−16460.9	−1.03851	−0.519253	−	0.854620i	$-0.673790\pi$
$631$	−16460.9	−1.03851	−0.519253	+	0.854620i	$0.673790\pi$
$632$	0	0
$633$	−3566.29	−0.223930
$634$	0	0
$635$	−19502.8	−1.21881
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2397.76	0.148441
$640$	0	0
$641$	2506.10	0.154423	0.0772114	−	0.997015i	$-0.475398\pi$
$641$	2506.10	0.154423	0.0772114	+	0.997015i	$0.475398\pi$
$642$	0	0
$643$	−15408.1	−0.945002	−0.472501	−	0.881330i	$-0.656649\pi$
$643$	−15408.1	−0.945002	−0.472501	+	0.881330i	$0.656649\pi$
$644$	0	0
$645$	−7919.63	−0.483465
$646$	0	0
$647$	28067.4	1.70548	0.852738	−	0.522338i	$-0.174940\pi$
$647$	28067.4	1.70548	0.852738	+	0.522338i	$0.174940\pi$
$648$	0	0
$649$	22255.4	1.34607
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3412.13	0.204482	0.102241	−	0.994760i	$-0.467399\pi$
$653$	3412.13	0.204482	0.102241	+	0.994760i	$0.467399\pi$
$654$	0	0
$655$	−3181.99	−0.189818
$656$	0	0
$657$	4722.79	0.280447
$658$	0	0
$659$	−18182.5	−1.07479	−0.537397	−	0.843329i	$-0.680592\pi$
$659$	−18182.5	−1.07479	−0.537397	+	0.843329i	$0.680592\pi$
$660$	0	0
$661$	−23113.9	−1.36010	−0.680049	−	0.733167i	$-0.738042\pi$
$661$	−23113.9	−1.36010	−0.680049	+	0.733167i	$0.738042\pi$
$662$	0	0
$663$	−2637.32	−0.154487
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10318.4	0.598994
$668$	0	0
$669$	10186.6	0.588696
$670$	0	0
$671$	17100.9	0.983862
$672$	0	0
$673$	13586.4	0.778183	0.389092	−	0.921199i	$-0.372789\pi$
$673$	13586.4	0.778183	0.389092	+	0.921199i	$0.372789\pi$
$674$	0	0
$675$	−1726.68	−0.0984591
$676$	0	0
$677$	−24054.6	−1.36557	−0.682786	−	0.730619i	$-0.739232\pi$
$677$	−24054.6	−1.36557	−0.682786	+	0.730619i	$0.739232\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3594.15	0.202244
$682$	0	0
$683$	21915.5	1.22778	0.613889	−	0.789393i	$-0.289604\pi$
$683$	21915.5	1.22778	0.613889	+	0.789393i	$0.289604\pi$
$684$	0	0
$685$	−3742.21	−0.208734
$686$	0	0
$687$	17461.2	0.969704
$688$	0	0
$689$	1566.29	0.0866052
$690$	0	0
$691$	19309.5	1.06305	0.531524	−	0.847043i	$-0.321619\pi$
$691$	19309.5	1.06305	0.531524	+	0.847043i	$0.321619\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	968.745	0.0528728
$696$	0	0
$697$	−23612.8	−1.28321
$698$	0	0
$699$	10464.2	0.566229
$700$	0	0
$701$	10696.2	0.576305	0.288152	−	0.957585i	$-0.406959\pi$
$701$	10696.2	0.576305	0.288152	+	0.957585i	$0.406959\pi$
$702$	0	0
$703$	−32.9915	−0.00176998
$704$	0	0
$705$	−2450.06	−0.130886
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30425.4	−1.61164	−0.805818	−	0.592163i	$-0.798274\pi$
$709$	−30425.4	−1.61164	−0.805818	+	0.592163i	$0.798274\pi$
$710$	0	0
$711$	3904.86	0.205969
$712$	0	0
$713$	10144.3	0.532828
$714$	0	0
$715$	3658.39	0.191351
$716$	0	0
$717$	14064.2	0.732551
$718$	0	0
$719$	−30084.9	−1.56047	−0.780235	−	0.625487i	$-0.784900\pi$
$719$	−30084.9	−1.56047	−0.780235	+	0.625487i	$0.784900\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	19257.7	0.990596
$724$	0	0
$725$	9401.42	0.481600
$726$	0	0
$727$	−3894.17	−0.198661	−0.0993306	−	0.995054i	$-0.531670\pi$
$727$	−3894.17	−0.198661	−0.0993306	+	0.995054i	$0.531670\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	27126.9	1.37254
$732$	0	0
$733$	−10760.8	−0.542235	−0.271117	−	0.962546i	$-0.587393\pi$
$733$	−10760.8	−0.542235	−0.271117	+	0.962546i	$0.587393\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5893.26	0.294547
$738$	0	0
$739$	−13739.3	−0.683907	−0.341954	−	0.939717i	$-0.611089\pi$
$739$	−13739.3	−0.683907	−0.341954	+	0.939717i	$0.611089\pi$
$740$	0	0
$741$	−3701.12	−0.183487
$742$	0	0
$743$	26261.8	1.29670	0.648352	−	0.761341i	$-0.275459\pi$
$743$	26261.8	1.29670	0.648352	+	0.761341i	$0.275459\pi$
$744$	0	0
$745$	−12855.9	−0.632220
$746$	0	0
$747$	5984.29	0.293111
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15995.8	0.777222	0.388611	−	0.921402i	$-0.372955\pi$
$751$	15995.8	0.777222	0.388611	+	0.921402i	$0.372955\pi$
$752$	0	0
$753$	−17784.9	−0.860713
$754$	0	0
$755$	23790.1	1.14677
$756$	0	0
$757$	1837.00	0.0881993	0.0440997	−	0.999027i	$-0.485958\pi$
$757$	1837.00	0.0881993	0.0440997	+	0.999027i	$0.485958\pi$
$758$	0	0
$759$	−9004.33	−0.430615
$760$	0	0
$761$	2992.29	0.142537	0.0712684	−	0.997457i	$-0.477295\pi$
$761$	2992.29	0.142537	0.0712684	+	0.997457i	$0.477295\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5645.95	−0.266836
$766$	0	0
$767$	−5698.44	−0.268264
$768$	0	0
$769$	−27602.4	−1.29437	−0.647183	−	0.762335i	$-0.724053\pi$
$769$	−27602.4	−1.29437	−0.647183	+	0.762335i	$0.724053\pi$
$770$	0	0
$771$	−2019.99	−0.0943554
$772$	0	0
$773$	−29608.1	−1.37766	−0.688830	−	0.724923i	$-0.741875\pi$
$773$	−29608.1	−1.37766	−0.688830	+	0.724923i	$0.741875\pi$
$774$	0	0
$775$	9242.81	0.428402
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−33137.3	−1.52409
$780$	0	0
$781$	11392.8	0.521978
$782$	0	0
$783$	−3969.26	−0.181162
$784$	0	0
$785$	11003.0	0.500272
$786$	0	0
$787$	−6482.32	−0.293608	−0.146804	−	0.989166i	$-0.546899\pi$
$787$	−6482.32	−0.293608	−0.146804	+	0.989166i	$0.546899\pi$
$788$	0	0
$789$	−15811.0	−0.713420
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4378.64	−0.196078
$794$	0	0
$795$	3353.11	0.149588
$796$	0	0
$797$	−27622.2	−1.22764	−0.613821	−	0.789446i	$-0.710368\pi$
$797$	−27622.2	−1.22764	−0.613821	+	0.789446i	$0.710368\pi$
$798$	0	0
$799$	8392.12	0.371579
$800$	0	0
$801$	7232.21	0.319023
$802$	0	0
$803$	22439.9	0.986160
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12194.3	0.531920
$808$	0	0
$809$	−12740.7	−0.553695	−0.276848	−	0.960914i	$-0.589290\pi$
$809$	−12740.7	−0.553695	−0.276848	+	0.960914i	$0.589290\pi$
$810$	0	0
$811$	2981.71	0.129102	0.0645511	−	0.997914i	$-0.479438\pi$
$811$	2981.71	0.129102	0.0645511	+	0.997914i	$0.479438\pi$
$812$	0	0
$813$	19572.6	0.844330
$814$	0	0
$815$	16279.3	0.699679
$816$	0	0
$817$	38069.0	1.63019
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39713.6	1.68820	0.844100	−	0.536185i	$-0.180135\pi$
$821$	39713.6	1.68820	0.844100	+	0.536185i	$0.180135\pi$
$822$	0	0
$823$	19917.5	0.843596	0.421798	−	0.906690i	$-0.361399\pi$
$823$	19917.5	0.843596	0.421798	+	0.906690i	$0.361399\pi$
$824$	0	0
$825$	−8204.16	−0.346221
$826$	0	0
$827$	−9671.42	−0.406660	−0.203330	−	0.979110i	$-0.565176\pi$
$827$	−9671.42	−0.406660	−0.203330	+	0.979110i	$0.565176\pi$
$828$	0	0
$829$	43001.4	1.80157	0.900784	−	0.434267i	$-0.142993\pi$
$829$	43001.4	1.80157	0.900784	+	0.434267i	$0.142993\pi$
$830$	0	0
$831$	25037.5	1.04518
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1439.50	0.0596598
$836$	0	0
$837$	−3902.29	−0.161151
$838$	0	0
$839$	−29115.1	−1.19805	−0.599026	−	0.800730i	$-0.704445\pi$
$839$	−29115.1	−1.19805	−0.599026	+	0.800730i	$0.704445\pi$
$840$	0	0
$841$	−2777.20	−0.113871
$842$	0	0
$843$	24484.3	1.00034
$844$	0	0
$845$	16229.3	0.660715
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−26009.0	−1.05139
$850$	0	0
$851$	20.5514	0.000827841 0
$852$	0	0
$853$	−19860.6	−0.797201	−0.398601	−	0.917125i	$-0.630504\pi$
$853$	−19860.6	−0.797201	−0.398601	+	0.917125i	$0.630504\pi$
$854$	0	0
$855$	−7923.33	−0.316927
$856$	0	0
$857$	−26027.5	−1.03744	−0.518719	−	0.854945i	$-0.673591\pi$
$857$	−26027.5	−1.03744	−0.518719	+	0.854945i	$0.673591\pi$
$858$	0	0
$859$	18974.5	0.753669	0.376835	−	0.926281i	$-0.377012\pi$
$859$	18974.5	0.753669	0.376835	+	0.926281i	$0.377012\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21212.2	−0.836699	−0.418350	−	0.908286i	$-0.637391\pi$
$863$	−21212.2	−0.836699	−0.418350	+	0.908286i	$0.637391\pi$
$864$	0	0
$865$	−26041.8	−1.02364
$866$	0	0
$867$	4599.93	0.180187
$868$	0	0
$869$	18553.6	0.724267
$870$	0	0
$871$	−1508.96	−0.0587015
$872$	0	0
$873$	4012.32	0.155551
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13227.6	0.509310	0.254655	−	0.967032i	$-0.418038\pi$
$877$	13227.6	0.509310	0.254655	+	0.967032i	$0.418038\pi$
$878$	0	0
$879$	299.370	0.0114875
$880$	0	0
$881$	−7026.27	−0.268696	−0.134348	−	0.990934i	$-0.542894\pi$
$881$	−7026.27	−0.268696	−0.134348	+	0.990934i	$0.542894\pi$
$882$	0	0
$883$	−29986.9	−1.14285	−0.571427	−	0.820653i	$-0.693610\pi$
$883$	−29986.9	−1.14285	−0.571427	+	0.820653i	$0.693610\pi$
$884$	0	0
$885$	−12199.2	−0.463357
$886$	0	0
$887$	32874.6	1.24444	0.622222	−	0.782841i	$-0.286230\pi$
$887$	32874.6	1.24444	0.622222	+	0.782841i	$0.286230\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3463.78	0.130237
$892$	0	0
$893$	11777.2	0.441332
$894$	0	0
$895$	25580.9	0.955390
$896$	0	0
$897$	2305.54	0.0858191
$898$	0	0
$899$	21247.2	0.788247
$900$	0	0
$901$	−11485.3	−0.424674
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21159.0	0.777181
$906$	0	0
$907$	−11380.2	−0.416620	−0.208310	−	0.978063i	$-0.566796\pi$
$907$	−11380.2	−0.416620	−0.208310	+	0.978063i	$0.566796\pi$
$908$	0	0
$909$	13927.1	0.508176
$910$	0	0
$911$	−12842.6	−0.467062	−0.233531	−	0.972349i	$-0.575028\pi$
$911$	−12842.6	−0.467062	−0.233531	+	0.972349i	$0.575028\pi$
$912$	0	0
$913$	28433.8	1.03069
$914$	0	0
$915$	−9373.75	−0.338674
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12184.9	0.437369	0.218685	−	0.975796i	$-0.429823\pi$
$919$	12184.9	0.437369	0.218685	+	0.975796i	$0.429823\pi$
$920$	0	0
$921$	−25894.7	−0.926447
$922$	0	0
$923$	−2917.09	−0.104027
$924$	0	0
$925$	18.7251	0.000665597 0
$926$	0	0
$927$	6116.40	0.216709
$928$	0	0
$929$	32603.6	1.15144	0.575721	−	0.817646i	$-0.304721\pi$
$929$	32603.6	1.15144	0.575721	+	0.817646i	$0.304721\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2591.94	0.0909499
$934$	0	0
$935$	−26826.2	−0.938301
$936$	0	0
$937$	15788.1	0.550453	0.275227	−	0.961379i	$-0.411247\pi$
$937$	15788.1	0.550453	0.275227	+	0.961379i	$0.411247\pi$
$938$	0	0
$939$	1733.47	0.0602446
$940$	0	0
$941$	−45207.2	−1.56611	−0.783057	−	0.621950i	$-0.786341\pi$
$941$	−45207.2	−1.56611	−0.783057	+	0.621950i	$0.786341\pi$
$942$	0	0
$943$	20642.2	0.712835
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36204.8	1.24234	0.621171	−	0.783675i	$-0.286657\pi$
$947$	36204.8	1.24234	0.621171	+	0.783675i	$0.286657\pi$
$948$	0	0
$949$	−5745.68	−0.196536
$950$	0	0
$951$	−590.637	−0.0201396
$952$	0	0
$953$	41627.8	1.41496	0.707481	−	0.706733i	$-0.249832\pi$
$953$	41627.8	1.41496	0.707481	+	0.706733i	$0.249832\pi$
$954$	0	0
$955$	7697.64	0.260827
$956$	0	0
$957$	−18859.6	−0.637035
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8902.25	−0.298824
$962$	0	0
$963$	17989.7	0.601983
$964$	0	0
$965$	36159.7	1.20624
$966$	0	0
$967$	−36170.4	−1.20286	−0.601428	−	0.798927i	$-0.705401\pi$
$967$	−36170.4	−1.20286	−0.601428	+	0.798927i	$0.705401\pi$
$968$	0	0
$969$	27139.6	0.899741
$970$	0	0
$971$	−7565.19	−0.250029	−0.125015	−	0.992155i	$-0.539898\pi$
$971$	−7565.19	−0.250029	−0.125015	+	0.992155i	$0.539898\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2100.66	0.0689998
$976$	0	0
$977$	−47949.1	−1.57014	−0.785070	−	0.619407i	$-0.787373\pi$
$977$	−47949.1	−1.57014	−0.785070	+	0.619407i	$0.787373\pi$
$978$	0	0
$979$	34363.2	1.12181
$980$	0	0
$981$	−13100.4	−0.426365
$982$	0	0
$983$	10632.9	0.345001	0.172501	−	0.985009i	$-0.444815\pi$
$983$	10632.9	0.345001	0.172501	+	0.985009i	$0.444815\pi$
$984$	0	0
$985$	27361.0	0.885070
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23714.3	−0.762457
$990$	0	0
$991$	−3411.06	−0.109340	−0.0546699	−	0.998504i	$-0.517411\pi$
$991$	−3411.06	−0.109340	−0.0546699	+	0.998504i	$0.517411\pi$
$992$	0	0
$993$	−13936.2	−0.445369
$994$	0	0
$995$	−1325.62	−0.0422362
$996$	0	0
$997$	17965.3	0.570678	0.285339	−	0.958427i	$-0.407894\pi$
$997$	17965.3	0.570678	0.285339	+	0.958427i	$0.407894\pi$
$998$	0	0
$999$	−7.90569	−0.000250375 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.be.1.2	yes	4
4.3	odd	2		2352.4.a.cn.1.2		4
7.6	odd	2		1176.4.a.z.1.3	✓	4
28.27	even	2		2352.4.a.co.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.z.1.3	✓	4	7.6	odd	2
1176.4.a.be.1.2	yes	4	1.1	even	1	trivial
2352.4.a.cn.1.2		4	4.3	odd	2
2352.4.a.co.1.3		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -7.81338 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -7.81338 q^{5} +9.00000 q^{9} +42.7627 q^{11} -10.9493 q^{13} -23.4401 q^{15} +80.2889 q^{17} +112.675 q^{19} -70.1884 q^{23} -63.9511 q^{25} +27.0000 q^{27} -147.010 q^{29} -144.529 q^{31} +128.288 q^{33} -0.292803 q^{37} -32.8479 q^{39} -294.097 q^{41} +337.866 q^{43} -70.3204 q^{45} +104.524 q^{47} +240.867 q^{51} -143.050 q^{53} -334.121 q^{55} +338.024 q^{57} +520.440 q^{59} +399.902 q^{61} +85.5509 q^{65} +137.813 q^{67} -210.565 q^{69} +266.418 q^{71} +524.754 q^{73} -191.853 q^{75} +433.874 q^{79} +81.0000 q^{81} +664.921 q^{83} -627.328 q^{85} -441.029 q^{87} +803.579 q^{89} -433.588 q^{93} -880.370 q^{95} +445.813 q^{97} +384.864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} + 72 q^{17} + 32 q^{19} + 8 q^{23} + 164 q^{25} + 108 q^{27} + 144 q^{29} + 48 q^{31} + 120 q^{33} + 48 q^{37} + 144 q^{39} + 72 q^{41} + 512 q^{43} + 72 q^{45} + 160 q^{47} + 216 q^{51} + 536 q^{53} - 336 q^{55} + 96 q^{57} + 240 q^{59} + 896 q^{61} - 136 q^{65} + 1088 q^{67} + 24 q^{69} + 1288 q^{71} + 1488 q^{73} + 492 q^{75} + 416 q^{79} + 324 q^{81} - 112 q^{83} - 1512 q^{85} + 432 q^{87} + 3160 q^{89} + 144 q^{93} - 240 q^{95} + 2384 q^{97} + 360 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 + 72 * q^17 + 32 * q^19 + 8 * q^23 + 164 * q^25 + 108 * q^27 + 144 * q^29 + 48 * q^31 + 120 * q^33 + 48 * q^37 + 144 * q^39 + 72 * q^41 + 512 * q^43 + 72 * q^45 + 160 * q^47 + 216 * q^51 + 536 * q^53 - 336 * q^55 + 96 * q^57 + 240 * q^59 + 896 * q^61 - 136 * q^65 + 1088 * q^67 + 24 * q^69 + 1288 * q^71 + 1488 * q^73 + 492 * q^75 + 416 * q^79 + 324 * q^81 - 112 * q^83 - 1512 * q^85 + 432 * q^87 + 3160 * q^89 + 144 * q^93 - 240 * q^95 + 2384 * q^97 + 360 * q^99

Introduction

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