LMFDB - Embedded newform 2352.4.a.cn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
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:	no (minimal twist has level 1176)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.96955$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-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	$-0.699324\pi$
$11$	−42.7627	−1.17213	−0.586065	+	0.810264i	$0.699324\pi$
$12$	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	$-0.738127\pi$
$19$	−112.675	−1.36049	−0.680246	+	0.732984i	$0.738127\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	70.1884	0.636317	0.318159	−	0.948037i	$-0.396936\pi$
$23$	70.1884	0.636317	0.318159	+	0.948037i	$0.396936\pi$
$24$	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	$-0.362492\pi$
$31$	144.529	0.837363	0.418681	+	0.908133i	$0.362492\pi$
$32$	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	$-0.704482\pi$
$43$	−337.866	−1.19823	−0.599117	+	0.800661i	$0.704482\pi$
$44$	0	0
$45$	−70.3204	−0.232950
$46$	0	0
$47$	−104.524	−0.324391	−0.162196	−	0.986759i	$-0.551858\pi$
$47$	−104.524	−0.324391	−0.162196	+	0.986759i	$0.551858\pi$
$48$	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	$-0.694686\pi$
$59$	−520.440	−1.14840	−0.574199	+	0.818716i	$0.694686\pi$
$60$	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	$-0.540100\pi$
$67$	−137.813	−0.251292	−0.125646	+	0.992075i	$0.540100\pi$
$68$	0	0
$69$	−210.565	−0.367378
$70$	0	0
$71$	−266.418	−0.445324	−0.222662	−	0.974896i	$-0.571475\pi$
$71$	−266.418	−0.445324	−0.222662	+	0.974896i	$0.571475\pi$
$72$	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	$-0.599979\pi$
$79$	−433.874	−0.617907	−0.308953	+	0.951077i	$0.599979\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−664.921	−0.879332	−0.439666	−	0.898161i	$-0.644903\pi$
$83$	−664.921	−0.879332	−0.439666	+	0.898161i	$0.644903\pi$
$84$	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	$-0.605386\pi$
$103$	−679.600	−0.650126	−0.325063	+	0.945692i	$0.605386\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1998.86	−1.80595	−0.902975	−	0.429693i	$-0.858622\pi$
$107$	−1998.86	−1.80595	−0.902975	+	0.429693i	$0.858622\pi$
$108$	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	$-0.837185\pi$
$127$	−2496.08	−1.74402	−0.872011	+	0.489486i	$0.837185\pi$
$128$	0	0
$129$	1013.60	0.691801
$130$	0	0
$131$	−407.249	−0.271615	−0.135807	−	0.990735i	$-0.543363\pi$
$131$	−407.249	−0.271615	−0.135807	+	0.990735i	$0.543363\pi$
$132$	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	$-0.487956\pi$
$139$	123.985	0.0756569	0.0378284	+	0.999284i	$0.487956\pi$
$140$	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	$-0.193712\pi$
$151$	3044.79	1.64094	0.820470	+	0.571690i	$0.193712\pi$
$152$	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	$-0.333115\pi$
$163$	2083.51	1.00119	0.500593	+	0.865683i	$0.333115\pi$
$164$	0	0
$165$	−1002.36	−0.472932
$166$	0	0
$167$	184.235	0.0853685	0.0426843	−	0.999089i	$-0.486409\pi$
$167$	184.235	0.0853685	0.0426843	+	0.999089i	$0.486409\pi$
$168$	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	$-0.260438\pi$
$179$	3273.98	1.36709	0.683544	+	0.729909i	$0.260438\pi$
$180$	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	$-0.440249\pi$
$191$	985.187	0.373223	0.186612	+	0.982434i	$0.440249\pi$
$192$	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	$-0.509620\pi$
$199$	−169.661	−0.0604368	−0.0302184	+	0.999543i	$0.509620\pi$
$200$	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	$-0.437877\pi$
$211$	1188.76	0.387857	0.193929	+	0.981016i	$0.437877\pi$
$212$	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	$-0.670290\pi$
$223$	−3395.54	−1.01965	−0.509826	+	0.860278i	$0.670290\pi$
$224$	0	0
$225$	−575.560	−0.170536
$226$	0	0
$227$	−1198.05	−0.350297	−0.175148	−	0.984542i	$-0.556041\pi$
$227$	−1198.05	−0.350297	−0.175148	+	0.984542i	$0.556041\pi$
$228$	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	$-0.718756\pi$
$239$	−4688.08	−1.26882	−0.634408	+	0.772999i	$0.718756\pi$
$240$	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	$-0.232259\pi$
$251$	5928.29	1.49080	0.745400	+	0.666618i	$0.232259\pi$
$252$	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	$-0.288008\pi$
$263$	5270.35	1.23568	0.617839	+	0.786304i	$0.288008\pi$
$264$	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	$-0.761045\pi$
$271$	−6524.19	−1.46242	−0.731211	+	0.682151i	$0.761045\pi$
$272$	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	$-0.135677\pi$
$283$	8669.66	1.82105	0.910526	+	0.413451i	$0.135677\pi$
$284$	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	$-0.203595\pi$
$307$	8631.55	1.60465	0.802327	+	0.596885i	$0.203595\pi$
$308$	0	0
$309$	2038.80	0.375351
$310$	0	0
$311$	−863.980	−0.157530	−0.0787650	−	0.996893i	$-0.525098\pi$
$311$	−863.980	−0.157530	−0.0787650	+	0.996893i	$0.525098\pi$
$312$	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	$-0.373960\pi$
$331$	4645.40	0.771402	0.385701	+	0.922624i	$0.373960\pi$
$332$	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	$-0.576224\pi$
$347$	−3066.25	−0.474366	−0.237183	+	0.971465i	$0.576224\pi$
$348$	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	$-0.453052\pi$
$359$	1999.23	0.293915	0.146957	+	0.989143i	$0.453052\pi$
$360$	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	$-0.339569\pi$
$367$	6790.82	0.965880	0.482940	+	0.875653i	$0.339569\pi$
$368$	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	$-0.743777\pi$
$379$	−10228.6	−1.38630	−0.693149	+	0.720794i	$0.743777\pi$
$380$	0	0
$381$	7488.23	1.00691
$382$	0	0
$383$	5452.92	0.727496	0.363748	−	0.931497i	$-0.381497\pi$
$383$	5452.92	0.727496	0.363748	+	0.931497i	$0.381497\pi$
$384$	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	$-0.488025\pi$
$419$	645.188	0.0752256	0.0376128	+	0.999292i	$0.488025\pi$
$420$	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	$-0.589937\pi$
$431$	−4989.28	−0.557599	−0.278800	+	0.960349i	$0.589937\pi$
$432$	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	$-0.720170\pi$
$439$	−11733.7	−1.27567	−0.637836	+	0.770172i	$0.720170\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4632.30	0.496811	0.248406	−	0.968656i	$-0.420093\pi$
$443$	4632.30	0.496811	0.248406	+	0.968656i	$0.420093\pi$
$444$	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	$-0.0734226\pi$
$463$	19397.4	1.94703	0.973515	+	0.228624i	$0.0734226\pi$
$464$	0	0
$465$	3387.79	0.337860
$466$	0	0
$467$	11033.3	1.09327	0.546637	−	0.837370i	$-0.315908\pi$
$467$	11033.3	1.09327	0.546637	+	0.837370i	$0.315908\pi$
$468$	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	$-0.263776\pi$
$479$	14170.5	1.35170	0.675852	+	0.737038i	$0.263776\pi$
$480$	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	$-0.705439\pi$
$487$	−12929.3	−1.20304	−0.601522	+	0.798856i	$0.705439\pi$
$488$	0	0
$489$	−6250.54	−0.578035
$490$	0	0
$491$	−99.6829	−0.00916217	−0.00458109	−	0.999990i	$-0.501458\pi$
$491$	−99.6829	−0.00916217	−0.00458109	+	0.999990i	$0.501458\pi$
$492$	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	$-0.410185\pi$
$499$	6207.28	0.556865	0.278433	+	0.960456i	$0.410185\pi$
$500$	0	0
$501$	−552.705	−0.0492875
$502$	0	0
$503$	−4743.31	−0.420464	−0.210232	−	0.977651i	$-0.567422\pi$
$503$	−4743.31	−0.420464	−0.210232	+	0.977651i	$0.567422\pi$
$504$	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	$-0.604890\pi$
$523$	−7740.65	−0.647180	−0.323590	+	0.946197i	$0.604890\pi$
$524$	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	$-0.458012\pi$
$547$	3365.32	0.263054	0.131527	+	0.991313i	$0.458012\pi$
$548$	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	$-0.689483\pi$
$563$	−14981.4	−1.12148	−0.560740	+	0.827992i	$0.689483\pi$
$564$	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	$-0.596403\pi$
$571$	−8138.91	−0.596503	−0.298251	+	0.954487i	$0.596403\pi$
$572$	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	$-0.311491\pi$
$587$	15877.4	1.11641	0.558203	+	0.829705i	$0.311491\pi$
$588$	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	$-0.369646\pi$
$599$	11674.4	0.796334	0.398167	+	0.917313i	$0.369646\pi$
$600$	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	$-0.268368\pi$
$607$	19894.5	1.33030	0.665150	+	0.746710i	$0.268368\pi$
$608$	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	$-0.477909\pi$
$619$	2135.87	0.138688	0.0693440	+	0.997593i	$0.477909\pi$
$620$	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	$-0.326210\pi$
$631$	16460.9	1.03851	0.519253	+	0.854620i	$0.326210\pi$
$632$	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	$-0.343351\pi$
$643$	15408.1	0.945002	0.472501	+	0.881330i	$0.343351\pi$
$644$	0	0
$645$	−7919.63	−0.483465
$646$	0	0
$647$	−28067.4	−1.70548	−0.852738	−	0.522338i	$-0.825060\pi$
$647$	−28067.4	−1.70548	−0.852738	+	0.522338i	$0.825060\pi$
$648$	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	$-0.319408\pi$
$659$	18182.5	1.07479	0.537397	+	0.843329i	$0.319408\pi$
$660$	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	$-0.710396\pi$
$683$	−21915.5	−1.22778	−0.613889	+	0.789393i	$0.710396\pi$
$684$	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	$-0.678381\pi$
$691$	−19309.5	−1.06305	−0.531524	+	0.847043i	$0.678381\pi$
$692$	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	$-0.215100\pi$
$719$	30084.9	1.56047	0.780235	+	0.625487i	$0.215100\pi$
$720$	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	$-0.468330\pi$
$727$	3894.17	0.198661	0.0993306	+	0.995054i	$0.468330\pi$
$728$	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	$-0.388911\pi$
$739$	13739.3	0.683907	0.341954	+	0.939717i	$0.388911\pi$
$740$	0	0
$741$	−3701.12	−0.183487
$742$	0	0
$743$	−26261.8	−1.29670	−0.648352	−	0.761341i	$-0.724541\pi$
$743$	−26261.8	−1.29670	−0.648352	+	0.761341i	$0.724541\pi$
$744$	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	$-0.627045\pi$
$751$	−15995.8	−0.777222	−0.388611	+	0.921402i	$0.627045\pi$
$752$	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	$-0.453101\pi$
$787$	6482.32	0.293608	0.146804	+	0.989166i	$0.453101\pi$
$788$	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	$-0.520562\pi$
$811$	−2981.71	−0.129102	−0.0645511	+	0.997914i	$0.520562\pi$
$812$	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	$-0.638601\pi$
$823$	−19917.5	−0.843596	−0.421798	+	0.906690i	$0.638601\pi$
$824$	0	0
$825$	−8204.16	−0.346221
$826$	0	0
$827$	9671.42	0.406660	0.203330	−	0.979110i	$-0.434824\pi$
$827$	9671.42	0.406660	0.203330	+	0.979110i	$0.434824\pi$
$828$	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	$-0.295555\pi$
$839$	29115.1	1.19805	0.599026	+	0.800730i	$0.295555\pi$
$840$	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	$-0.622988\pi$
$859$	−18974.5	−0.753669	−0.376835	+	0.926281i	$0.622988\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21212.2	0.836699	0.418350	−	0.908286i	$-0.362609\pi$
$863$	21212.2	0.836699	0.418350	+	0.908286i	$0.362609\pi$
$864$	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	$-0.306390\pi$
$883$	29986.9	1.14285	0.571427	+	0.820653i	$0.306390\pi$
$884$	0	0
$885$	−12199.2	−0.463357
$886$	0	0
$887$	−32874.6	−1.24444	−0.622222	−	0.782841i	$-0.713770\pi$
$887$	−32874.6	−1.24444	−0.622222	+	0.782841i	$0.713770\pi$
$888$	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	$-0.433204\pi$
$907$	11380.2	0.416620	0.208310	+	0.978063i	$0.433204\pi$
$908$	0	0
$909$	13927.1	0.508176
$910$	0	0
$911$	12842.6	0.467062	0.233531	−	0.972349i	$-0.424972\pi$
$911$	12842.6	0.467062	0.233531	+	0.972349i	$0.424972\pi$
$912$	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	$-0.570177\pi$
$919$	−12184.9	−0.437369	−0.218685	+	0.975796i	$0.570177\pi$
$920$	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	$-0.713343\pi$
$947$	−36204.8	−1.24234	−0.621171	+	0.783675i	$0.713343\pi$
$948$	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	$-0.294599\pi$
$967$	36170.4	1.20286	0.601428	+	0.798927i	$0.294599\pi$
$968$	0	0
$969$	27139.6	0.899741
$970$	0	0
$971$	7565.19	0.250029	0.125015	−	0.992155i	$-0.460102\pi$
$971$	7565.19	0.250029	0.125015	+	0.992155i	$0.460102\pi$
$972$	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	$-0.555185\pi$
$983$	−10632.9	−0.345001	−0.172501	+	0.985009i	$0.555185\pi$
$984$	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	$-0.482589\pi$
$991$	3411.06	0.109340	0.0546699	+	0.998504i	$0.482589\pi$
$992$	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	2352.4.a.cn.1.2		4
4.3	odd	2		1176.4.a.be.1.2	yes	4
7.6	odd	2		2352.4.a.co.1.3		4
28.27	even	2		1176.4.a.z.1.3	✓	4

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.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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