LMFDB - Embedded newform 1200.4.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{181}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 45 $ x^2 - x - 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 600)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.22681$ of defining polynomial
Character	$\chi$	$=$	1200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -29.9072 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -29.9072 q^{7} +9.00000 q^{9} -30.9072 q^{11} -46.8145 q^{13} -26.9072 q^{17} -123.722 q^{19} +89.7217 q^{21} -144.722 q^{23} -27.0000 q^{27} +37.4638 q^{29} +267.351 q^{31} +92.7217 q^{33} -143.443 q^{37} +140.443 q^{39} -310.351 q^{41} -6.64926 q^{43} -485.629 q^{47} +551.443 q^{49} +80.7217 q^{51} +322.351 q^{53} +371.165 q^{57} -217.258 q^{59} -301.742 q^{61} -269.165 q^{63} -568.835 q^{67} +434.165 q^{69} -749.939 q^{71} +52.9275 q^{73} +924.351 q^{77} +1010.14 q^{79} +81.0000 q^{81} +1103.03 q^{83} -112.391 q^{87} +1219.63 q^{89} +1400.09 q^{91} -802.052 q^{93} +1116.03 q^{97} -278.165 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} - 8 q^{11} + 14 q^{13} - 86 q^{19} + 18 q^{21} - 128 q^{23} - 54 q^{27} + 344 q^{29} + 158 q^{31} + 24 q^{33} + 36 q^{37} - 42 q^{39} - 244 q^{41} - 390 q^{43} - 756 q^{47} + 780 q^{49} + 268 q^{53} + 258 q^{57} - 4 q^{59} - 1034 q^{61} - 54 q^{63} - 1622 q^{67} + 384 q^{69} + 276 q^{71} + 644 q^{73} + 1472 q^{77} + 944 q^{79} + 162 q^{81} + 484 q^{83} - 1032 q^{87} + 2224 q^{89} + 2854 q^{91} - 474 q^{93} + 510 q^{97} - 72 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9 - 8 * q^11 + 14 * q^13 - 86 * q^19 + 18 * q^21 - 128 * q^23 - 54 * q^27 + 344 * q^29 + 158 * q^31 + 24 * q^33 + 36 * q^37 - 42 * q^39 - 244 * q^41 - 390 * q^43 - 756 * q^47 + 780 * q^49 + 268 * q^53 + 258 * q^57 - 4 * q^59 - 1034 * q^61 - 54 * q^63 - 1622 * q^67 + 384 * q^69 + 276 * q^71 + 644 * q^73 + 1472 * q^77 + 944 * q^79 + 162 * q^81 + 484 * q^83 - 1032 * q^87 + 2224 * q^89 + 2854 * q^91 - 474 * q^93 + 510 * q^97 - 72 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−29.9072	−1.61484	−0.807420	−	0.589977i	$-0.799137\pi$
$7$	−29.9072	−1.61484	−0.807420	+	0.589977i	$0.799137\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−30.9072	−0.847171	−0.423586	−	0.905856i	$-0.639229\pi$
$11$	−30.9072	−0.847171	−0.423586	+	0.905856i	$0.639229\pi$
$12$	0	0
$13$	−46.8145	−0.998770	−0.499385	−	0.866380i	$-0.666441\pi$
$13$	−46.8145	−0.998770	−0.499385	+	0.866380i	$0.666441\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.9072	−0.383880	−0.191940	−	0.981407i	$-0.561478\pi$
$17$	−26.9072	−0.383880	−0.191940	+	0.981407i	$0.561478\pi$
$18$	0	0
$19$	−123.722	−1.49388	−0.746940	−	0.664892i	$-0.768478\pi$
$19$	−123.722	−1.49388	−0.746940	+	0.664892i	$0.768478\pi$
$20$	0	0
$21$	89.7217	0.932328
$22$	0	0
$23$	−144.722	−1.31202	−0.656012	−	0.754750i	$-0.727758\pi$
$23$	−144.722	−1.31202	−0.656012	+	0.754750i	$0.727758\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	37.4638	0.239891	0.119946	−	0.992780i	$-0.461728\pi$
$29$	37.4638	0.239891	0.119946	+	0.992780i	$0.461728\pi$
$30$	0	0
$31$	267.351	1.54896	0.774478	−	0.632601i	$-0.218013\pi$
$31$	267.351	1.54896	0.774478	+	0.632601i	$0.218013\pi$
$32$	0	0
$33$	92.7217	0.489115
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−143.443	−0.637350	−0.318675	−	0.947864i	$-0.603238\pi$
$37$	−143.443	−0.637350	−0.318675	+	0.947864i	$0.603238\pi$
$38$	0	0
$39$	140.443	0.576640
$40$	0	0
$41$	−310.351	−1.18216	−0.591081	−	0.806612i	$-0.701299\pi$
$41$	−310.351	−1.18216	−0.591081	+	0.806612i	$0.701299\pi$
$42$	0	0
$43$	−6.64926	−0.0235815	−0.0117907	−	0.999930i	$-0.503753\pi$
$43$	−6.64926	−0.0235815	−0.0117907	+	0.999930i	$0.503753\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−485.629	−1.50715	−0.753577	−	0.657359i	$-0.771673\pi$
$47$	−485.629	−1.50715	−0.753577	+	0.657359i	$0.771673\pi$
$48$	0	0
$49$	551.443	1.60771
$50$	0	0
$51$	80.7217	0.221633
$52$	0	0
$53$	322.351	0.835439	0.417720	−	0.908576i	$-0.362829\pi$
$53$	322.351	0.835439	0.417720	+	0.908576i	$0.362829\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	371.165	0.862492
$58$	0	0
$59$	−217.258	−0.479400	−0.239700	−	0.970847i	$-0.577049\pi$
$59$	−217.258	−0.479400	−0.239700	+	0.970847i	$0.577049\pi$
$60$	0	0
$61$	−301.742	−0.633346	−0.316673	−	0.948535i	$-0.602566\pi$
$61$	−301.742	−0.633346	−0.316673	+	0.948535i	$0.602566\pi$
$62$	0	0
$63$	−269.165	−0.538280
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−568.835	−1.03723	−0.518614	−	0.855009i	$-0.673552\pi$
$67$	−568.835	−1.03723	−0.518614	+	0.855009i	$0.673552\pi$
$68$	0	0
$69$	434.165	0.757498
$70$	0	0
$71$	−749.939	−1.25354	−0.626770	−	0.779204i	$-0.715624\pi$
$71$	−749.939	−1.25354	−0.626770	+	0.779204i	$0.715624\pi$
$72$	0	0
$73$	52.9275	0.0848589	0.0424294	−	0.999099i	$-0.486490\pi$
$73$	52.9275	0.0848589	0.0424294	+	0.999099i	$0.486490\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	924.351	1.36805
$78$	0	0
$79$	1010.14	1.43861	0.719305	−	0.694694i	$-0.244460\pi$
$79$	1010.14	1.43861	0.719305	+	0.694694i	$0.244460\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1103.03	1.45872	0.729358	−	0.684132i	$-0.239819\pi$
$83$	1103.03	1.45872	0.729358	+	0.684132i	$0.239819\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−112.391	−0.138501
$88$	0	0
$89$	1219.63	1.45259	0.726294	−	0.687384i	$-0.241241\pi$
$89$	1219.63	1.45259	0.726294	+	0.687384i	$0.241241\pi$
$90$	0	0
$91$	1400.09	1.61285
$92$	0	0
$93$	−802.052	−0.894290
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1116.03	1.16820	0.584102	−	0.811680i	$-0.301447\pi$
$97$	1116.03	1.16820	0.584102	+	0.811680i	$0.301447\pi$
$98$	0	0
$99$	−278.165	−0.282390
$100$	0	0
$101$	−1256.12	−1.23752	−0.618758	−	0.785582i	$-0.712364\pi$
$101$	−1256.12	−1.23752	−0.618758	+	0.785582i	$0.712364\pi$
$102$	0	0
$103$	−1724.99	−1.65018	−0.825090	−	0.565002i	$-0.808875\pi$
$103$	−1724.99	−1.65018	−0.825090	+	0.565002i	$0.808875\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1636.91	−1.47893	−0.739466	−	0.673194i	$-0.764922\pi$
$107$	−1636.91	−1.47893	−0.739466	+	0.673194i	$0.764922\pi$
$108$	0	0
$109$	294.113	0.258449	0.129224	−	0.991615i	$-0.458751\pi$
$109$	294.113	0.258449	0.129224	+	0.991615i	$0.458751\pi$
$110$	0	0
$111$	430.330	0.367974
$112$	0	0
$113$	1209.77	1.00713	0.503566	−	0.863957i	$-0.332021\pi$
$113$	1209.77	1.00713	0.503566	+	0.863957i	$0.332021\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−421.330	−0.332923
$118$	0	0
$119$	804.722	0.619905
$120$	0	0
$121$	−375.742	−0.282301
$122$	0	0
$123$	931.052	0.682522
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−673.072	−0.470280	−0.235140	−	0.971962i	$-0.575555\pi$
$127$	−673.072	−0.470280	−0.235140	+	0.971962i	$0.575555\pi$
$128$	0	0
$129$	19.9478	0.0136148
$130$	0	0
$131$	2224.79	1.48382	0.741910	−	0.670499i	$-0.233920\pi$
$131$	2224.79	1.48382	0.741910	+	0.670499i	$0.233920\pi$
$132$	0	0
$133$	3700.18	2.41238
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1987.28	−1.23931	−0.619653	−	0.784876i	$-0.712727\pi$
$137$	−1987.28	−1.23931	−0.619653	+	0.784876i	$0.712727\pi$
$138$	0	0
$139$	−183.217	−0.111801	−0.0559004	−	0.998436i	$-0.517803\pi$
$139$	−183.217	−0.111801	−0.0559004	+	0.998436i	$0.517803\pi$
$140$	0	0
$141$	1456.89	0.870156
$142$	0	0
$143$	1446.91	0.846129
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1654.33	−0.928210
$148$	0	0
$149$	2179.81	1.19851	0.599253	−	0.800560i	$-0.295464\pi$
$149$	2179.81	1.19851	0.599253	+	0.800560i	$0.295464\pi$
$150$	0	0
$151$	1456.32	0.784858	0.392429	−	0.919782i	$-0.371635\pi$
$151$	1456.32	0.784858	0.392429	+	0.919782i	$0.371635\pi$
$152$	0	0
$153$	−242.165	−0.127960
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2743.00	−1.39436	−0.697182	−	0.716894i	$-0.745563\pi$
$157$	−2743.00	−1.39436	−0.697182	+	0.716894i	$0.745563\pi$
$158$	0	0
$159$	−967.052	−0.482341
$160$	0	0
$161$	4328.23	2.11871
$162$	0	0
$163$	1093.46	0.525436	0.262718	−	0.964873i	$-0.415381\pi$
$163$	1093.46	0.525436	0.262718	+	0.964873i	$0.415381\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1120.79	−0.519335	−0.259668	−	0.965698i	$-0.583613\pi$
$167$	−1120.79	−0.519335	−0.259668	+	0.965698i	$0.583613\pi$
$168$	0	0
$169$	−5.40295	−0.00245924
$170$	0	0
$171$	−1113.50	−0.497960
$172$	0	0
$173$	−1153.63	−0.506989	−0.253494	−	0.967337i	$-0.581580\pi$
$173$	−1153.63	−0.506989	−0.253494	+	0.967337i	$0.581580\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	651.774	0.276782
$178$	0	0
$179$	1385.22	0.578413	0.289207	−	0.957267i	$-0.406608\pi$
$179$	1385.22	0.578413	0.289207	+	0.957267i	$0.406608\pi$
$180$	0	0
$181$	−805.557	−0.330810	−0.165405	−	0.986226i	$-0.552893\pi$
$181$	−805.557	−0.330810	−0.165405	+	0.986226i	$0.552893\pi$
$182$	0	0
$183$	905.226	0.365662
$184$	0	0
$185$	0	0
$186$	0	0
$187$	831.629	0.325212
$188$	0	0
$189$	807.496	0.310776
$190$	0	0
$191$	−3322.50	−1.25868	−0.629339	−	0.777131i	$-0.716674\pi$
$191$	−3322.50	−1.25868	−0.629339	+	0.777131i	$0.716674\pi$
$192$	0	0
$193$	−249.023	−0.0928761	−0.0464381	−	0.998921i	$-0.514787\pi$
$193$	−249.023	−0.0928761	−0.0464381	+	0.998921i	$0.514787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2727.20	0.986320	0.493160	−	0.869939i	$-0.335842\pi$
$197$	2727.20	0.986320	0.493160	+	0.869939i	$0.335842\pi$
$198$	0	0
$199$	3258.42	1.16072	0.580361	−	0.814360i	$-0.302912\pi$
$199$	3258.42	1.16072	0.580361	+	0.814360i	$0.302912\pi$
$200$	0	0
$201$	1706.50	0.598843
$202$	0	0
$203$	−1120.44	−0.387386
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1302.50	−0.437342
$208$	0	0
$209$	3823.90	1.26557
$210$	0	0
$211$	3368.65	1.09909	0.549544	−	0.835465i	$-0.314802\pi$
$211$	3368.65	1.09909	0.549544	+	0.835465i	$0.314802\pi$
$212$	0	0
$213$	2249.82	0.723732
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7995.72	−2.50131
$218$	0	0
$219$	−158.783	−0.0489933
$220$	0	0
$221$	1259.65	0.383408
$222$	0	0
$223$	5251.99	1.57713	0.788564	−	0.614953i	$-0.210825\pi$
$223$	5251.99	1.57713	0.788564	+	0.614953i	$0.210825\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6064.58	1.77322	0.886609	−	0.462520i	$-0.153055\pi$
$227$	6064.58	1.77322	0.886609	+	0.462520i	$0.153055\pi$
$228$	0	0
$229$	595.354	0.171800	0.0858998	−	0.996304i	$-0.472624\pi$
$229$	595.354	0.171800	0.0858998	+	0.996304i	$0.472624\pi$
$230$	0	0
$231$	−2773.05	−0.789842
$232$	0	0
$233$	2241.08	0.630119	0.315060	−	0.949072i	$-0.397976\pi$
$233$	2241.08	0.630119	0.315060	+	0.949072i	$0.397976\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3030.43	−0.830582
$238$	0	0
$239$	4579.17	1.23934	0.619670	−	0.784863i	$-0.287267\pi$
$239$	4579.17	1.23934	0.619670	+	0.784863i	$0.287267\pi$
$240$	0	0
$241$	−5338.97	−1.42702	−0.713512	−	0.700643i	$-0.752897\pi$
$241$	−5338.97	−1.42702	−0.713512	+	0.700643i	$0.752897\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5791.97	1.49204
$248$	0	0
$249$	−3309.10	−0.842190
$250$	0	0
$251$	−2698.93	−0.678704	−0.339352	−	0.940659i	$-0.610208\pi$
$251$	−2698.93	−0.678704	−0.339352	+	0.940659i	$0.610208\pi$
$252$	0	0
$253$	4472.95	1.11151
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7991.64	1.93971	0.969853	−	0.243690i	$-0.0783580\pi$
$257$	7991.64	1.93971	0.969853	+	0.243690i	$0.0783580\pi$
$258$	0	0
$259$	4290.00	1.02922
$260$	0	0
$261$	337.174	0.0799637
$262$	0	0
$263$	7265.82	1.70353	0.851767	−	0.523921i	$-0.175531\pi$
$263$	7265.82	1.70353	0.851767	+	0.523921i	$0.175531\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3658.89	−0.838653
$268$	0	0
$269$	−6046.87	−1.37057	−0.685287	−	0.728273i	$-0.740323\pi$
$269$	−6046.87	−1.37057	−0.685287	+	0.728273i	$0.740323\pi$
$270$	0	0
$271$	5106.39	1.14462	0.572309	−	0.820038i	$-0.306048\pi$
$271$	5106.39	1.14462	0.572309	+	0.820038i	$0.306048\pi$
$272$	0	0
$273$	−4200.28	−0.931181
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8077.70	−1.75214	−0.876069	−	0.482186i	$-0.839843\pi$
$277$	−8077.70	−1.75214	−0.876069	+	0.482186i	$0.839843\pi$
$278$	0	0
$279$	2406.16	0.516318
$280$	0	0
$281$	8509.18	1.80646	0.903230	−	0.429157i	$-0.141189\pi$
$281$	8509.18	1.80646	0.903230	+	0.429157i	$0.141189\pi$
$282$	0	0
$283$	−6873.44	−1.44376	−0.721879	−	0.692019i	$-0.756721\pi$
$283$	−6873.44	−1.44376	−0.721879	+	0.692019i	$0.756721\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9281.74	1.90900
$288$	0	0
$289$	−4189.00	−0.852636
$290$	0	0
$291$	−3348.10	−0.674463
$292$	0	0
$293$	−4981.76	−0.993301	−0.496651	−	0.867950i	$-0.665437\pi$
$293$	−4981.76	−0.993301	−0.496651	+	0.867950i	$0.665437\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	834.496	0.163038
$298$	0	0
$299$	6775.08	1.31041
$300$	0	0
$301$	198.861	0.0380803
$302$	0	0
$303$	3768.37	0.714480
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7039.79	−1.30874	−0.654368	−	0.756176i	$-0.727065\pi$
$307$	−7039.79	−1.30874	−0.654368	+	0.756176i	$0.727065\pi$
$308$	0	0
$309$	5174.97	0.952731
$310$	0	0
$311$	−2221.54	−0.405055	−0.202527	−	0.979277i	$-0.564916\pi$
$311$	−2221.54	−0.405055	−0.202527	+	0.979277i	$0.564916\pi$
$312$	0	0
$313$	−5381.48	−0.971818	−0.485909	−	0.874009i	$-0.661511\pi$
$313$	−5381.48	−0.971818	−0.485909	+	0.874009i	$0.661511\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6282.15	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$317$	−6282.15	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$318$	0	0
$319$	−1157.90	−0.203229
$320$	0	0
$321$	4910.72	0.853862
$322$	0	0
$323$	3329.01	0.573471
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−882.339	−0.149215
$328$	0	0
$329$	14523.8	2.43381
$330$	0	0
$331$	−1727.72	−0.286900	−0.143450	−	0.989658i	$-0.545820\pi$
$331$	−1727.72	−0.286900	−0.143450	+	0.989658i	$0.545820\pi$
$332$	0	0
$333$	−1290.99	−0.212450
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1033.97	0.167134	0.0835670	−	0.996502i	$-0.473369\pi$
$337$	1033.97	0.167134	0.0835670	+	0.996502i	$0.473369\pi$
$338$	0	0
$339$	−3629.32	−0.581468
$340$	0	0
$341$	−8263.08	−1.31223
$342$	0	0
$343$	−6233.97	−0.981349
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9236.90	−1.42900	−0.714500	−	0.699636i	$-0.753346\pi$
$347$	−9236.90	−1.42900	−0.714500	+	0.699636i	$0.753346\pi$
$348$	0	0
$349$	−8658.79	−1.32806	−0.664032	−	0.747704i	$-0.731156\pi$
$349$	−8658.79	−1.32806	−0.664032	+	0.747704i	$0.731156\pi$
$350$	0	0
$351$	1263.99	0.192213
$352$	0	0
$353$	−2572.10	−0.387816	−0.193908	−	0.981020i	$-0.562116\pi$
$353$	−2572.10	−0.387816	−0.193908	+	0.981020i	$0.562116\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2414.17	−0.357902
$358$	0	0
$359$	−1941.12	−0.285371	−0.142685	−	0.989768i	$-0.545574\pi$
$359$	−1941.12	−0.285371	−0.142685	+	0.989768i	$0.545574\pi$
$360$	0	0
$361$	8448.07	1.23168
$362$	0	0
$363$	1127.23	0.162986
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2614.98	0.371937	0.185968	−	0.982556i	$-0.440458\pi$
$367$	2614.98	0.371937	0.185968	+	0.982556i	$0.440458\pi$
$368$	0	0
$369$	−2793.16	−0.394054
$370$	0	0
$371$	−9640.62	−1.34910
$372$	0	0
$373$	2258.81	0.313558	0.156779	−	0.987634i	$-0.449889\pi$
$373$	2258.81	0.313558	0.156779	+	0.987634i	$0.449889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1753.85	−0.239596
$378$	0	0
$379$	3319.24	0.449862	0.224931	−	0.974375i	$-0.427784\pi$
$379$	3319.24	0.449862	0.224931	+	0.974375i	$0.427784\pi$
$380$	0	0
$381$	2019.22	0.271516
$382$	0	0
$383$	−4994.02	−0.666273	−0.333137	−	0.942879i	$-0.608107\pi$
$383$	−4994.02	−0.666273	−0.333137	+	0.942879i	$0.608107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−59.8434	−0.00786049
$388$	0	0
$389$	−7076.79	−0.922385	−0.461192	−	0.887300i	$-0.652578\pi$
$389$	−7076.79	−0.922385	−0.461192	+	0.887300i	$0.652578\pi$
$390$	0	0
$391$	3894.06	0.503661
$392$	0	0
$393$	−6674.36	−0.856684
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9174.72	−1.15986	−0.579932	−	0.814665i	$-0.696921\pi$
$397$	−9174.72	−1.15986	−0.579932	+	0.814665i	$0.696921\pi$
$398$	0	0
$399$	−11100.5	−1.39279
$400$	0	0
$401$	−11194.8	−1.39412	−0.697058	−	0.717015i	$-0.745508\pi$
$401$	−11194.8	−1.39412	−0.697058	+	0.717015i	$0.745508\pi$
$402$	0	0
$403$	−12515.9	−1.54705
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4433.44	0.539945
$408$	0	0
$409$	−11162.6	−1.34952	−0.674762	−	0.738036i	$-0.735754\pi$
$409$	−11162.6	−1.34952	−0.674762	+	0.738036i	$0.735754\pi$
$410$	0	0
$411$	5961.84	0.715514
$412$	0	0
$413$	6497.59	0.774154
$414$	0	0
$415$	0	0
$416$	0	0
$417$	549.652	0.0645482
$418$	0	0
$419$	−9135.55	−1.06516	−0.532579	−	0.846381i	$-0.678777\pi$
$419$	−9135.55	−1.06516	−0.532579	+	0.846381i	$0.678777\pi$
$420$	0	0
$421$	−13668.3	−1.58231	−0.791155	−	0.611615i	$-0.790520\pi$
$421$	−13668.3	−1.58231	−0.791155	+	0.611615i	$0.790520\pi$
$422$	0	0
$423$	−4370.66	−0.502385
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9024.27	1.02275
$428$	0	0
$429$	−4340.72	−0.488513
$430$	0	0
$431$	−5806.70	−0.648953	−0.324477	−	0.945894i	$-0.605188\pi$
$431$	−5806.70	−0.648953	−0.324477	+	0.945894i	$0.605188\pi$
$432$	0	0
$433$	5215.15	0.578809	0.289404	−	0.957207i	$-0.406543\pi$
$433$	5215.15	0.578809	0.289404	+	0.957207i	$0.406543\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17905.2	1.96001
$438$	0	0
$439$	−11398.7	−1.23925	−0.619625	−	0.784898i	$-0.712715\pi$
$439$	−11398.7	−1.23925	−0.619625	+	0.784898i	$0.712715\pi$
$440$	0	0
$441$	4962.99	0.535902
$442$	0	0
$443$	5959.18	0.639118	0.319559	−	0.947566i	$-0.396465\pi$
$443$	5959.18	0.639118	0.319559	+	0.947566i	$0.396465\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6539.44	−0.691958
$448$	0	0
$449$	11864.6	1.24705	0.623526	−	0.781803i	$-0.285700\pi$
$449$	11864.6	1.24705	0.623526	+	0.781803i	$0.285700\pi$
$450$	0	0
$451$	9592.09	1.00149
$452$	0	0
$453$	−4368.96	−0.453138
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13452.6	1.37700	0.688499	−	0.725237i	$-0.258270\pi$
$457$	13452.6	1.37700	0.688499	+	0.725237i	$0.258270\pi$
$458$	0	0
$459$	726.496	0.0738778
$460$	0	0
$461$	−11773.9	−1.18951	−0.594756	−	0.803906i	$-0.702751\pi$
$461$	−11773.9	−1.18951	−0.594756	+	0.803906i	$0.702751\pi$
$462$	0	0
$463$	−3269.36	−0.328164	−0.164082	−	0.986447i	$-0.552466\pi$
$463$	−3269.36	−0.328164	−0.164082	+	0.986447i	$0.552466\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12407.6	1.22945	0.614726	−	0.788741i	$-0.289267\pi$
$467$	12407.6	1.22945	0.614726	+	0.788741i	$0.289267\pi$
$468$	0	0
$469$	17012.3	1.67496
$470$	0	0
$471$	8229.00	0.805037
$472$	0	0
$473$	205.510	0.0199775
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2901.16	0.278480
$478$	0	0
$479$	5046.09	0.481340	0.240670	−	0.970607i	$-0.422633\pi$
$479$	5046.09	0.481340	0.240670	+	0.970607i	$0.422633\pi$
$480$	0	0
$481$	6715.23	0.636566
$482$	0	0
$483$	−12984.7	−1.22324
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8107.99	−0.754432	−0.377216	−	0.926125i	$-0.623119\pi$
$487$	−8107.99	−0.754432	−0.377216	+	0.926125i	$0.623119\pi$
$488$	0	0
$489$	−3280.37	−0.303360
$490$	0	0
$491$	−13194.7	−1.21276	−0.606381	−	0.795174i	$-0.707379\pi$
$491$	−13194.7	−1.21276	−0.606381	+	0.795174i	$0.707379\pi$
$492$	0	0
$493$	−1008.05	−0.0920895
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22428.6	2.02427
$498$	0	0
$499$	−16840.3	−1.51077	−0.755384	−	0.655282i	$-0.772550\pi$
$499$	−16840.3	−1.51077	−0.755384	+	0.655282i	$0.772550\pi$
$500$	0	0
$501$	3362.36	0.299838
$502$	0	0
$503$	−17.3127	−0.00153466	−0.000767330	−	1.00000i	$-0.500244\pi$
$503$	−17.3127	−0.00153466	−0.000767330		1.00000i	$0.500244\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.2088	0.00141984
$508$	0	0
$509$	−7272.27	−0.633276	−0.316638	−	0.948546i	$-0.602554\pi$
$509$	−7272.27	−0.633276	−0.316638	+	0.948546i	$0.602554\pi$
$510$	0	0
$511$	−1582.92	−0.137033
$512$	0	0
$513$	3340.49	0.287497
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15009.5	1.27682
$518$	0	0
$519$	3460.90	0.292710
$520$	0	0
$521$	4830.58	0.406203	0.203101	−	0.979158i	$-0.434898\pi$
$521$	4830.58	0.406203	0.203101	+	0.979158i	$0.434898\pi$
$522$	0	0
$523$	15788.1	1.32001	0.660006	−	0.751261i	$-0.270554\pi$
$523$	15788.1	1.32001	0.660006	+	0.751261i	$0.270554\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7193.67	−0.594613
$528$	0	0
$529$	8777.38	0.721409
$530$	0	0
$531$	−1955.32	−0.159800
$532$	0	0
$533$	14528.9	1.18071
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4155.65	−0.333947
$538$	0	0
$539$	−17043.6	−1.36200
$540$	0	0
$541$	−19092.5	−1.51729	−0.758644	−	0.651506i	$-0.774138\pi$
$541$	−19092.5	−1.51729	−0.758644	+	0.651506i	$0.774138\pi$
$542$	0	0
$543$	2416.67	0.190993
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6582.19	−0.514504	−0.257252	−	0.966344i	$-0.582817\pi$
$547$	−6582.19	−0.514504	−0.257252	+	0.966344i	$0.582817\pi$
$548$	0	0
$549$	−2715.68	−0.211115
$550$	0	0
$551$	−4635.08	−0.358369
$552$	0	0
$553$	−30210.7	−2.32312
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14331.0	1.09017	0.545086	−	0.838380i	$-0.316497\pi$
$557$	14331.0	1.09017	0.545086	+	0.838380i	$0.316497\pi$
$558$	0	0
$559$	311.282	0.0235525
$560$	0	0
$561$	−2494.89	−0.187762
$562$	0	0
$563$	7759.03	0.580824	0.290412	−	0.956902i	$-0.406208\pi$
$563$	7759.03	0.580824	0.290412	+	0.956902i	$0.406208\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2422.49	−0.179427
$568$	0	0
$569$	−14124.6	−1.04066	−0.520328	−	0.853967i	$-0.674190\pi$
$569$	−14124.6	−1.04066	−0.520328	+	0.853967i	$0.674190\pi$
$570$	0	0
$571$	12036.6	0.882167	0.441083	−	0.897466i	$-0.354594\pi$
$571$	12036.6	0.882167	0.441083	+	0.897466i	$0.354594\pi$
$572$	0	0
$573$	9967.49	0.726698
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14908.7	1.07566	0.537830	−	0.843053i	$-0.319244\pi$
$577$	14908.7	1.07566	0.537830	+	0.843053i	$0.319244\pi$
$578$	0	0
$579$	747.070	0.0536221
$580$	0	0
$581$	−32988.6	−2.35559
$582$	0	0
$583$	−9962.97	−0.707760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14933.8	1.05006	0.525029	−	0.851084i	$-0.324054\pi$
$587$	14933.8	1.05006	0.525029	+	0.851084i	$0.324054\pi$
$588$	0	0
$589$	−33077.1	−2.31395
$590$	0	0
$591$	−8181.60	−0.569452
$592$	0	0
$593$	3451.38	0.239007	0.119503	−	0.992834i	$-0.461870\pi$
$593$	3451.38	0.239007	0.119503	+	0.992834i	$0.461870\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9775.27	−0.670143
$598$	0	0
$599$	733.154	0.0500098	0.0250049	−	0.999687i	$-0.492040\pi$
$599$	733.154	0.0500098	0.0250049	+	0.999687i	$0.492040\pi$
$600$	0	0
$601$	19261.2	1.30729	0.653643	−	0.756803i	$-0.273240\pi$
$601$	19261.2	1.30729	0.653643	+	0.756803i	$0.273240\pi$
$602$	0	0
$603$	−5119.51	−0.345742
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14402.2	−0.963045	−0.481523	−	0.876434i	$-0.659916\pi$
$607$	−14402.2	−0.963045	−0.481523	+	0.876434i	$0.659916\pi$
$608$	0	0
$609$	3361.31	0.223657
$610$	0	0
$611$	22734.5	1.50530
$612$	0	0
$613$	−2394.16	−0.157748	−0.0788738	−	0.996885i	$-0.525132\pi$
$613$	−2394.16	−0.157748	−0.0788738	+	0.996885i	$0.525132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9171.17	0.598407	0.299204	−	0.954189i	$-0.403279\pi$
$617$	9171.17	0.598407	0.299204	+	0.954189i	$0.403279\pi$
$618$	0	0
$619$	−20055.4	−1.30225	−0.651127	−	0.758969i	$-0.725703\pi$
$619$	−20055.4	−1.30225	−0.651127	+	0.758969i	$0.725703\pi$
$620$	0	0
$621$	3907.49	0.252499
$622$	0	0
$623$	−36475.7	−2.34570
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−11471.7	−0.730678
$628$	0	0
$629$	3859.67	0.244666
$630$	0	0
$631$	−3321.72	−0.209565	−0.104783	−	0.994495i	$-0.533415\pi$
$631$	−3321.72	−0.209565	−0.104783	+	0.994495i	$0.533415\pi$
$632$	0	0
$633$	−10105.9	−0.634558
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−25815.5	−1.60573
$638$	0	0
$639$	−6749.45	−0.417847
$640$	0	0
$641$	6189.65	0.381399	0.190699	−	0.981648i	$-0.438924\pi$
$641$	6189.65	0.381399	0.190699	+	0.981648i	$0.438924\pi$
$642$	0	0
$643$	−5362.52	−0.328891	−0.164446	−	0.986386i	$-0.552584\pi$
$643$	−5362.52	−0.328891	−0.164446	+	0.986386i	$0.552584\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12636.0	0.767809	0.383904	−	0.923373i	$-0.374579\pi$
$647$	12636.0	0.767809	0.383904	+	0.923373i	$0.374579\pi$
$648$	0	0
$649$	6714.85	0.406134
$650$	0	0
$651$	23987.2	1.44413
$652$	0	0
$653$	10340.5	0.619687	0.309844	−	0.950788i	$-0.399723\pi$
$653$	10340.5	0.619687	0.309844	+	0.950788i	$0.399723\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	476.348	0.0282863
$658$	0	0
$659$	5232.41	0.309296	0.154648	−	0.987970i	$-0.450576\pi$
$659$	5232.41	0.309296	0.154648	+	0.987970i	$0.450576\pi$
$660$	0	0
$661$	602.256	0.0354388	0.0177194	−	0.999843i	$-0.494359\pi$
$661$	602.256	0.0354388	0.0177194	+	0.999843i	$0.494359\pi$
$662$	0	0
$663$	−3778.95	−0.221361
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5421.82	−0.314743
$668$	0	0
$669$	−15756.0	−0.910555
$670$	0	0
$671$	9326.02	0.536553
$672$	0	0
$673$	1870.31	0.107125	0.0535625	−	0.998565i	$-0.482942\pi$
$673$	1870.31	0.107125	0.0535625	+	0.998565i	$0.482942\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2053.27	0.116563	0.0582817	−	0.998300i	$-0.481438\pi$
$677$	2053.27	0.116563	0.0582817	+	0.998300i	$0.481438\pi$
$678$	0	0
$679$	−33377.4	−1.88646
$680$	0	0
$681$	−18193.7	−1.02377
$682$	0	0
$683$	15169.3	0.849832	0.424916	−	0.905233i	$-0.360304\pi$
$683$	15169.3	0.849832	0.424916	+	0.905233i	$0.360304\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1786.06	−0.0991885
$688$	0	0
$689$	−15090.7	−0.834411
$690$	0	0
$691$	−29469.8	−1.62241	−0.811204	−	0.584764i	$-0.801187\pi$
$691$	−29469.8	−1.62241	−0.811204	+	0.584764i	$0.801187\pi$
$692$	0	0
$693$	8319.16	0.456015
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8350.68	0.453809
$698$	0	0
$699$	−6723.23	−0.363799
$700$	0	0
$701$	−6774.61	−0.365012	−0.182506	−	0.983205i	$-0.558421\pi$
$701$	−6774.61	−0.365012	−0.182506	+	0.983205i	$0.558421\pi$
$702$	0	0
$703$	17747.1	0.952125
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37567.2	1.99839
$708$	0	0
$709$	−22501.2	−1.19189	−0.595944	−	0.803026i	$-0.703222\pi$
$709$	−22501.2	−1.19189	−0.595944	+	0.803026i	$0.703222\pi$
$710$	0	0
$711$	9091.30	0.479537
$712$	0	0
$713$	−38691.5	−2.03227
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−13737.5	−0.715533
$718$	0	0
$719$	10850.6	0.562809	0.281405	−	0.959589i	$-0.409200\pi$
$719$	10850.6	0.562809	0.281405	+	0.959589i	$0.409200\pi$
$720$	0	0
$721$	51589.7	2.66477
$722$	0	0
$723$	16016.9	0.823893
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−27360.6	−1.39580	−0.697902	−	0.716193i	$-0.745883\pi$
$727$	−27360.6	−1.39580	−0.697902	+	0.716193i	$0.745883\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	178.913	0.00905246
$732$	0	0
$733$	930.568	0.0468913	0.0234456	−	0.999725i	$-0.492536\pi$
$733$	930.568	0.0468913	0.0234456	+	0.999725i	$0.492536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17581.1	0.878709
$738$	0	0
$739$	16833.0	0.837903	0.418952	−	0.908009i	$-0.362398\pi$
$739$	16833.0	0.837903	0.418952	+	0.908009i	$0.362398\pi$
$740$	0	0
$741$	−17375.9	−0.861431
$742$	0	0
$743$	−1917.63	−0.0946853	−0.0473426	−	0.998879i	$-0.515075\pi$
$743$	−1917.63	−0.0946853	−0.0473426	+	0.998879i	$0.515075\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9927.29	0.486239
$748$	0	0
$749$	48955.4	2.38824
$750$	0	0
$751$	−30826.2	−1.49782	−0.748910	−	0.662671i	$-0.769423\pi$
$751$	−30826.2	−1.49782	−0.748910	+	0.662671i	$0.769423\pi$
$752$	0	0
$753$	8096.78	0.391850
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21934.2	−1.05312	−0.526559	−	0.850138i	$-0.676518\pi$
$757$	−21934.2	−1.05312	−0.526559	+	0.850138i	$0.676518\pi$
$758$	0	0
$759$	−13418.9	−0.641731
$760$	0	0
$761$	8363.35	0.398385	0.199193	−	0.979960i	$-0.436168\pi$
$761$	8363.35	0.398385	0.199193	+	0.979960i	$0.436168\pi$
$762$	0	0
$763$	−8796.11	−0.417353
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10170.8	0.478810
$768$	0	0
$769$	144.849	0.00679244	0.00339622	−	0.999994i	$-0.498919\pi$
$769$	144.849	0.00679244	0.00339622	+	0.999994i	$0.498919\pi$
$770$	0	0
$771$	−23974.9	−1.11989
$772$	0	0
$773$	3894.51	0.181210	0.0906052	−	0.995887i	$-0.471120\pi$
$773$	3894.51	0.181210	0.0906052	+	0.995887i	$0.471120\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12870.0	−0.594220
$778$	0	0
$779$	38397.1	1.76601
$780$	0	0
$781$	23178.6	1.06196
$782$	0	0
$783$	−1011.52	−0.0461671
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1312.46	−0.0594461	−0.0297231	−	0.999558i	$-0.509463\pi$
$787$	−1312.46	−0.0594461	−0.0297231	+	0.999558i	$0.509463\pi$
$788$	0	0
$789$	−21797.5	−0.983536
$790$	0	0
$791$	−36181.0	−1.62636
$792$	0	0
$793$	14125.9	0.632567
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39493.7	−1.75525	−0.877627	−	0.479344i	$-0.840875\pi$
$797$	−39493.7	−1.75525	−0.877627	+	0.479344i	$0.840875\pi$
$798$	0	0
$799$	13066.9	0.578567
$800$	0	0
$801$	10976.7	0.484196
$802$	0	0
$803$	−1635.84	−0.0718900
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18140.6	0.791301
$808$	0	0
$809$	32500.1	1.41241	0.706207	−	0.708006i	$-0.250405\pi$
$809$	32500.1	1.41241	0.706207	+	0.708006i	$0.250405\pi$
$810$	0	0
$811$	−31191.9	−1.35055	−0.675274	−	0.737567i	$-0.735975\pi$
$811$	−31191.9	−1.35055	−0.675274	+	0.737567i	$0.735975\pi$
$812$	0	0
$813$	−15319.2	−0.660845
$814$	0	0
$815$	0	0
$816$	0	0
$817$	822.658	0.0352279
$818$	0	0
$819$	12600.8	0.537618
$820$	0	0
$821$	−9882.99	−0.420120	−0.210060	−	0.977688i	$-0.567366\pi$
$821$	−9882.99	−0.420120	−0.210060	+	0.977688i	$0.567366\pi$
$822$	0	0
$823$	5785.39	0.245037	0.122519	−	0.992466i	$-0.460903\pi$
$823$	5785.39	0.245037	0.122519	+	0.992466i	$0.460903\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16531.6	0.695114	0.347557	−	0.937659i	$-0.387011\pi$
$827$	16531.6	0.695114	0.347557	+	0.937659i	$0.387011\pi$
$828$	0	0
$829$	24870.4	1.04196	0.520980	−	0.853569i	$-0.325567\pi$
$829$	24870.4	1.04196	0.520980	+	0.853569i	$0.325567\pi$
$830$	0	0
$831$	24233.1	1.01160
$832$	0	0
$833$	−14837.8	−0.617167
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7218.47	−0.298097
$838$	0	0
$839$	9771.47	0.402084	0.201042	−	0.979583i	$-0.435567\pi$
$839$	9771.47	0.402084	0.201042	+	0.979583i	$0.435567\pi$
$840$	0	0
$841$	−22985.5	−0.942452
$842$	0	0
$843$	−25527.5	−1.04296
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11237.4	0.455870
$848$	0	0
$849$	20620.3	0.833554
$850$	0	0
$851$	20759.4	0.836219
$852$	0	0
$853$	−30223.3	−1.21316	−0.606581	−	0.795022i	$-0.707459\pi$
$853$	−30223.3	−1.21316	−0.606581	+	0.795022i	$0.707459\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12389.3	0.493826	0.246913	−	0.969038i	$-0.420584\pi$
$857$	12389.3	0.493826	0.246913	+	0.969038i	$0.420584\pi$
$858$	0	0
$859$	35118.3	1.39490	0.697450	−	0.716633i	$-0.254318\pi$
$859$	35118.3	1.39490	0.697450	+	0.716633i	$0.254318\pi$
$860$	0	0
$861$	−27845.2	−1.10216
$862$	0	0
$863$	−8549.77	−0.337239	−0.168620	−	0.985681i	$-0.553931\pi$
$863$	−8549.77	−0.337239	−0.168620	+	0.985681i	$0.553931\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12567.0	0.492270
$868$	0	0
$869$	−31220.8	−1.21875
$870$	0	0
$871$	26629.7	1.03595
$872$	0	0
$873$	10044.3	0.389402
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25827.7	0.994457	0.497229	−	0.867620i	$-0.334351\pi$
$877$	25827.7	0.994457	0.497229	+	0.867620i	$0.334351\pi$
$878$	0	0
$879$	14945.3	0.573483
$880$	0	0
$881$	22662.3	0.866641	0.433321	−	0.901240i	$-0.357342\pi$
$881$	22662.3	0.866641	0.433321	+	0.901240i	$0.357342\pi$
$882$	0	0
$883$	20406.2	0.777717	0.388859	−	0.921297i	$-0.372869\pi$
$883$	20406.2	0.777717	0.388859	+	0.921297i	$0.372869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21127.3	0.799758	0.399879	−	0.916568i	$-0.369052\pi$
$887$	21127.3	0.799758	0.399879	+	0.916568i	$0.369052\pi$
$888$	0	0
$889$	20129.7	0.759426
$890$	0	0
$891$	−2503.49	−0.0941302
$892$	0	0
$893$	60082.9	2.25151
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−20325.2	−0.756566
$898$	0	0
$899$	10016.0	0.371581
$900$	0	0
$901$	−8673.57	−0.320709
$902$	0	0
$903$	−596.584	−0.0219857
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5421.27	0.198468	0.0992338	−	0.995064i	$-0.468361\pi$
$907$	5421.27	0.198468	0.0992338	+	0.995064i	$0.468361\pi$
$908$	0	0
$909$	−11305.1	−0.412505
$910$	0	0
$911$	47003.1	1.70942	0.854711	−	0.519104i	$-0.173734\pi$
$911$	47003.1	1.70942	0.854711	+	0.519104i	$0.173734\pi$
$912$	0	0
$913$	−34091.7	−1.23578
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−66537.2	−2.39613
$918$	0	0
$919$	6680.57	0.239795	0.119898	−	0.992786i	$-0.461743\pi$
$919$	6680.57	0.239795	0.119898	+	0.992786i	$0.461743\pi$
$920$	0	0
$921$	21119.4	0.755599
$922$	0	0
$923$	35108.0	1.25200
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−15524.9	−0.550060
$928$	0	0
$929$	−39569.9	−1.39747	−0.698733	−	0.715383i	$-0.746252\pi$
$929$	−39569.9	−1.39747	−0.698733	+	0.715383i	$0.746252\pi$
$930$	0	0
$931$	−68225.6	−2.40172
$932$	0	0
$933$	6664.63	0.233859
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15386.6	−0.536457	−0.268228	−	0.963355i	$-0.586438\pi$
$937$	−15386.6	−0.536457	−0.268228	+	0.963355i	$0.586438\pi$
$938$	0	0
$939$	16144.4	0.561079
$940$	0	0
$941$	11717.6	0.405934	0.202967	−	0.979186i	$-0.434942\pi$
$941$	11717.6	0.405934	0.202967	+	0.979186i	$0.434942\pi$
$942$	0	0
$943$	44914.5	1.55103
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9763.54	−0.335029	−0.167514	−	0.985870i	$-0.553574\pi$
$947$	−9763.54	−0.335029	−0.167514	+	0.985870i	$0.553574\pi$
$948$	0	0
$949$	−2477.78	−0.0847545
$950$	0	0
$951$	18846.5	0.642627
$952$	0	0
$953$	13962.3	0.474590	0.237295	−	0.971438i	$-0.423739\pi$
$953$	13962.3	0.474590	0.237295	+	0.971438i	$0.423739\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3473.71	0.117334
$958$	0	0
$959$	59434.1	2.00128
$960$	0	0
$961$	41685.4	1.39926
$962$	0	0
$963$	−14732.2	−0.492978
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24064.0	0.800255	0.400128	−	0.916459i	$-0.368966\pi$
$967$	24064.0	0.800255	0.400128	+	0.916459i	$0.368966\pi$
$968$	0	0
$969$	−9987.04	−0.331094
$970$	0	0
$971$	−21872.0	−0.722870	−0.361435	−	0.932397i	$-0.617713\pi$
$971$	−21872.0	−0.722870	−0.361435	+	0.932397i	$0.617713\pi$
$972$	0	0
$973$	5479.53	0.180540
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3495.88	0.114476	0.0572381	−	0.998361i	$-0.481771\pi$
$977$	3495.88	0.114476	0.0572381	+	0.998361i	$0.481771\pi$
$978$	0	0
$979$	−37695.4	−1.23059
$980$	0	0
$981$	2647.02	0.0861496
$982$	0	0
$983$	31582.7	1.02475	0.512376	−	0.858761i	$-0.328765\pi$
$983$	31582.7	1.02475	0.512376	+	0.858761i	$0.328765\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−43571.5	−1.40516
$988$	0	0
$989$	962.293	0.0309395
$990$	0	0
$991$	20403.5	0.654025	0.327012	−	0.945020i	$-0.393958\pi$
$991$	20403.5	0.654025	0.327012	+	0.945020i	$0.393958\pi$
$992$	0	0
$993$	5183.15	0.165642
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14760.4	−0.468874	−0.234437	−	0.972131i	$-0.575325\pi$
$997$	−14760.4	−0.468874	−0.234437	+	0.972131i	$0.575325\pi$
$998$	0	0
$999$	3872.97	0.122658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bm.1.1		2
4.3	odd	2		600.4.a.w.1.2	yes	2
5.2	odd	4		1200.4.f.w.49.3		4
5.3	odd	4		1200.4.f.w.49.2		4
5.4	even	2		1200.4.a.bs.1.2		2
12.11	even	2		1800.4.a.bq.1.2		2
20.3	even	4		600.4.f.k.49.3		4
20.7	even	4		600.4.f.k.49.2		4
20.19	odd	2		600.4.a.r.1.1	✓	2
60.23	odd	4		1800.4.f.y.649.1		4
60.47	odd	4		1800.4.f.y.649.4		4
60.59	even	2		1800.4.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.r.1.1	✓	2	20.19	odd	2
600.4.a.w.1.2	yes	2	4.3	odd	2
600.4.f.k.49.2		4	20.7	even	4
600.4.f.k.49.3		4	20.3	even	4
1200.4.a.bm.1.1		2	1.1	even	1	trivial
1200.4.a.bs.1.2		2	5.4	even	2
1200.4.f.w.49.2		4	5.3	odd	4
1200.4.f.w.49.3		4	5.2	odd	4
1800.4.a.bj.1.1		2	60.59	even	2
1800.4.a.bq.1.2		2	12.11	even	2
1800.4.f.y.649.1		4	60.23	odd	4
1800.4.f.y.649.4		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -29.9072 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -29.9072 q^{7} +9.00000 q^{9} -30.9072 q^{11} -46.8145 q^{13} -26.9072 q^{17} -123.722 q^{19} +89.7217 q^{21} -144.722 q^{23} -27.0000 q^{27} +37.4638 q^{29} +267.351 q^{31} +92.7217 q^{33} -143.443 q^{37} +140.443 q^{39} -310.351 q^{41} -6.64926 q^{43} -485.629 q^{47} +551.443 q^{49} +80.7217 q^{51} +322.351 q^{53} +371.165 q^{57} -217.258 q^{59} -301.742 q^{61} -269.165 q^{63} -568.835 q^{67} +434.165 q^{69} -749.939 q^{71} +52.9275 q^{73} +924.351 q^{77} +1010.14 q^{79} +81.0000 q^{81} +1103.03 q^{83} -112.391 q^{87} +1219.63 q^{89} +1400.09 q^{91} -802.052 q^{93} +1116.03 q^{97} -278.165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} - 8 q^{11} + 14 q^{13} - 86 q^{19} + 18 q^{21} - 128 q^{23} - 54 q^{27} + 344 q^{29} + 158 q^{31} + 24 q^{33} + 36 q^{37} - 42 q^{39} - 244 q^{41} - 390 q^{43} - 756 q^{47} + 780 q^{49} + 268 q^{53} + 258 q^{57} - 4 q^{59} - 1034 q^{61} - 54 q^{63} - 1622 q^{67} + 384 q^{69} + 276 q^{71} + 644 q^{73} + 1472 q^{77} + 944 q^{79} + 162 q^{81} + 484 q^{83} - 1032 q^{87} + 2224 q^{89} + 2854 q^{91} - 474 q^{93} + 510 q^{97} - 72 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9 - 8 * q^11 + 14 * q^13 - 86 * q^19 + 18 * q^21 - 128 * q^23 - 54 * q^27 + 344 * q^29 + 158 * q^31 + 24 * q^33 + 36 * q^37 - 42 * q^39 - 244 * q^41 - 390 * q^43 - 756 * q^47 + 780 * q^49 + 268 * q^53 + 258 * q^57 - 4 * q^59 - 1034 * q^61 - 54 * q^63 - 1622 * q^67 + 384 * q^69 + 276 * q^71 + 644 * q^73 + 1472 * q^77 + 944 * q^79 + 162 * q^81 + 484 * q^83 - 1032 * q^87 + 2224 * q^89 + 2854 * q^91 - 474 * q^93 + 510 * q^97 - 72 * q^99

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