LMFDB - Embedded newform 2400.4.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.115636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 69x - 81 $ x^3 - x^2 - 69*x - 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.22200$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +26.7458 q^{7} +9.00000 q^{9} -46.6338 q^{11} -49.6036 q^{13} -32.4098 q^{17} -4.14220 q^{19} -80.2374 q^{21} -213.169 q^{23} -27.0000 q^{27} -272.661 q^{29} +327.756 q^{31} +139.901 q^{33} +399.010 q^{37} +148.811 q^{39} -42.2890 q^{41} -468.747 q^{43} +275.086 q^{47} +372.338 q^{49} +97.2295 q^{51} -158.152 q^{53} +12.4266 q^{57} +566.435 q^{59} +206.493 q^{61} +240.712 q^{63} +351.557 q^{67} +639.507 q^{69} -500.497 q^{71} +987.718 q^{73} -1247.26 q^{77} +172.311 q^{79} +81.0000 q^{81} +238.950 q^{83} +817.982 q^{87} -42.7769 q^{89} -1326.69 q^{91} -983.268 q^{93} +1187.97 q^{97} -419.704 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 3 q^{7} + 27 q^{9} - 44 q^{11} - 13 q^{13} + 36 q^{17} - 71 q^{19} - 9 q^{21} - 160 q^{23} - 81 q^{27} - 184 q^{29} + 59 q^{31} + 132 q^{33} + 350 q^{37} + 39 q^{39} + 166 q^{41} - 341 q^{43}+ \cdots - 396 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 3 * q^7 + 27 * q^9 - 44 * q^11 - 13 * q^13 + 36 * q^17 - 71 * q^19 - 9 * q^21 - 160 * q^23 - 81 * q^27 - 184 * q^29 + 59 * q^31 + 132 * q^33 + 350 * q^37 + 39 * q^39 + 166 * q^41 - 341 * q^43 + 394 * q^47 + 158 * q^49 - 108 * q^51 + 314 * q^53 + 213 * q^57 - 538 * q^59 - 523 * q^61 + 27 * q^63 + 543 * q^67 + 480 * q^69 - 910 * q^71 + 870 * q^73 - 828 * q^77 + 96 * q^79 + 243 * q^81 - 210 * q^83 + 552 * q^87 - 64 * q^89 - 2781 * q^91 - 177 * q^93 + 943 * q^97 - 396 * q^99

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
$7$	26.7458	1.44414	0.722069	−	0.691821i	$-0.243191\pi$
$7$	26.7458	1.44414	0.722069	+	0.691821i	$0.243191\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−46.6338	−1.27824	−0.639119	−	0.769108i	$-0.720701\pi$
$11$	−46.6338	−1.27824	−0.639119	+	0.769108i	$0.720701\pi$
$12$	0	0
$13$	−49.6036	−1.05827	−0.529137	−	0.848536i	$-0.677484\pi$
$13$	−49.6036	−1.05827	−0.529137	+	0.848536i	$0.677484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−32.4098	−0.462384	−0.231192	−	0.972908i	$-0.574263\pi$
$17$	−32.4098	−0.462384	−0.231192	+	0.972908i	$0.574263\pi$
$18$	0	0
$19$	−4.14220	−0.0500150	−0.0250075	−	0.999687i	$-0.507961\pi$
$19$	−4.14220	−0.0500150	−0.0250075	+	0.999687i	$0.507961\pi$
$20$	0	0
$21$	−80.2374	−0.833773
$22$	0	0
$23$	−213.169	−1.93256	−0.966279	−	0.257499i	$-0.917102\pi$
$23$	−213.169	−1.93256	−0.966279	+	0.257499i	$0.917102\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−272.661	−1.74592	−0.872962	−	0.487788i	$-0.837804\pi$
$29$	−272.661	−1.74592	−0.872962	+	0.487788i	$0.837804\pi$
$30$	0	0
$31$	327.756	1.89893	0.949463	−	0.313880i	$-0.101629\pi$
$31$	327.756	1.89893	0.949463	+	0.313880i	$0.101629\pi$
$32$	0	0
$33$	139.901	0.737991
$34$	0	0
$35$	0	0
$36$	0	0
$37$	399.010	1.77289	0.886444	−	0.462836i	$-0.153168\pi$
$37$	399.010	1.77289	0.886444	+	0.462836i	$0.153168\pi$
$38$	0	0
$39$	148.811	0.610995
$40$	0	0
$41$	−42.2890	−0.161084	−0.0805419	−	0.996751i	$-0.525665\pi$
$41$	−42.2890	−0.161084	−0.0805419	+	0.996751i	$0.525665\pi$
$42$	0	0
$43$	−468.747	−1.66240	−0.831201	−	0.555973i	$-0.812346\pi$
$43$	−468.747	−1.66240	−0.831201	+	0.555973i	$0.812346\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	275.086	0.853733	0.426867	−	0.904315i	$-0.359617\pi$
$47$	275.086	0.853733	0.426867	+	0.904315i	$0.359617\pi$
$48$	0	0
$49$	372.338	1.08553
$50$	0	0
$51$	97.2295	0.266958
$52$	0	0
$53$	−158.152	−0.409885	−0.204942	−	0.978774i	$-0.565701\pi$
$53$	−158.152	−0.409885	−0.204942	+	0.978774i	$0.565701\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.4266	0.0288762
$58$	0	0
$59$	566.435	1.24989	0.624945	−	0.780669i	$-0.285121\pi$
$59$	566.435	1.24989	0.624945	+	0.780669i	$0.285121\pi$
$60$	0	0
$61$	206.493	0.433421	0.216711	−	0.976236i	$-0.430467\pi$
$61$	206.493	0.433421	0.216711	+	0.976236i	$0.430467\pi$
$62$	0	0
$63$	240.712	0.481379
$64$	0	0
$65$	0	0
$66$	0	0
$67$	351.557	0.641039	0.320519	−	0.947242i	$-0.396143\pi$
$67$	351.557	0.641039	0.320519	+	0.947242i	$0.396143\pi$
$68$	0	0
$69$	639.507	1.11576
$70$	0	0
$71$	−500.497	−0.836593	−0.418296	−	0.908311i	$-0.637373\pi$
$71$	−500.497	−0.836593	−0.418296	+	0.908311i	$0.637373\pi$
$72$	0	0
$73$	987.718	1.58361	0.791806	−	0.610773i	$-0.209141\pi$
$73$	987.718	1.58361	0.791806	+	0.610773i	$0.209141\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1247.26	−1.84595
$78$	0	0
$79$	172.311	0.245399	0.122700	−	0.992444i	$-0.460845\pi$
$79$	172.311	0.245399	0.122700	+	0.992444i	$0.460845\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	238.950	0.316002	0.158001	−	0.987439i	$-0.449495\pi$
$83$	238.950	0.316002	0.158001	+	0.987439i	$0.449495\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	817.982	1.00801
$88$	0	0
$89$	−42.7769	−0.0509476	−0.0254738	−	0.999675i	$-0.508109\pi$
$89$	−42.7769	−0.0509476	−0.0254738	+	0.999675i	$0.508109\pi$
$90$	0	0
$91$	−1326.69	−1.52829
$92$	0	0
$93$	−983.268	−1.09635
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1187.97	1.24351	0.621753	−	0.783213i	$-0.286421\pi$
$97$	1187.97	1.24351	0.621753	+	0.783213i	$0.286421\pi$
$98$	0	0
$99$	−419.704	−0.426079
$100$	0	0
$101$	−731.979	−0.721135	−0.360568	−	0.932733i	$-0.617417\pi$
$101$	−731.979	−0.721135	−0.360568	+	0.932733i	$0.617417\pi$
$102$	0	0
$103$	−1043.63	−0.998369	−0.499184	−	0.866496i	$-0.666367\pi$
$103$	−1043.63	−0.998369	−0.499184	+	0.866496i	$0.666367\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	323.443	0.292228	0.146114	−	0.989268i	$-0.453323\pi$
$107$	323.443	0.292228	0.146114	+	0.989268i	$0.453323\pi$
$108$	0	0
$109$	−1345.88	−1.18268	−0.591340	−	0.806423i	$-0.701401\pi$
$109$	−1345.88	−1.18268	−0.591340	+	0.806423i	$0.701401\pi$
$110$	0	0
$111$	−1197.03	−1.02358
$112$	0	0
$113$	2226.94	1.85392	0.926959	−	0.375163i	$-0.122413\pi$
$113$	2226.94	1.85392	0.926959	+	0.375163i	$0.122413\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−446.432	−0.352758
$118$	0	0
$119$	−866.827	−0.667747
$120$	0	0
$121$	843.712	0.633894
$122$	0	0
$123$	126.867	0.0930017
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−516.237	−0.360698	−0.180349	−	0.983603i	$-0.557723\pi$
$127$	−516.237	−0.360698	−0.180349	+	0.983603i	$0.557723\pi$
$128$	0	0
$129$	1406.24	0.959788
$130$	0	0
$131$	−241.181	−0.160855	−0.0804277	−	0.996760i	$-0.525629\pi$
$131$	−241.181	−0.160855	−0.0804277	+	0.996760i	$0.525629\pi$
$132$	0	0
$133$	−110.786	−0.0722286
$134$	0	0
$135$	0	0
$136$	0	0
$137$	917.387	0.572100	0.286050	−	0.958215i	$-0.407658\pi$
$137$	917.387	0.572100	0.286050	+	0.958215i	$0.407658\pi$
$138$	0	0
$139$	−1979.43	−1.20786	−0.603932	−	0.797036i	$-0.706400\pi$
$139$	−1979.43	−1.20786	−0.603932	+	0.797036i	$0.706400\pi$
$140$	0	0
$141$	−825.259	−0.492903
$142$	0	0
$143$	2313.21	1.35273
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1117.01	−0.626733
$148$	0	0
$149$	2737.08	1.50490	0.752452	−	0.658647i	$-0.228871\pi$
$149$	2737.08	1.50490	0.752452	+	0.658647i	$0.228871\pi$
$150$	0	0
$151$	1148.59	0.619011	0.309505	−	0.950898i	$-0.399837\pi$
$151$	1148.59	0.619011	0.309505	+	0.950898i	$0.399837\pi$
$152$	0	0
$153$	−291.688	−0.154128
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1814.64	0.922445	0.461222	−	0.887285i	$-0.347411\pi$
$157$	1814.64	0.922445	0.461222	+	0.887285i	$0.347411\pi$
$158$	0	0
$159$	474.457	0.236647
$160$	0	0
$161$	−5701.38	−2.79088
$162$	0	0
$163$	2018.38	0.969887	0.484943	−	0.874546i	$-0.338840\pi$
$163$	2018.38	0.969887	0.484943	+	0.874546i	$0.338840\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2456.67	−1.13834	−0.569171	−	0.822219i	$-0.692736\pi$
$167$	−2456.67	−1.13834	−0.569171	+	0.822219i	$0.692736\pi$
$168$	0	0
$169$	263.518	0.119944
$170$	0	0
$171$	−37.2798	−0.0166717
$172$	0	0
$173$	2988.71	1.31346	0.656728	−	0.754128i	$-0.271940\pi$
$173$	2988.71	1.31346	0.656728	+	0.754128i	$0.271940\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1699.30	−0.721624
$178$	0	0
$179$	−2994.76	−1.25050	−0.625248	−	0.780426i	$-0.715002\pi$
$179$	−2994.76	−1.25050	−0.625248	+	0.780426i	$0.715002\pi$
$180$	0	0
$181$	−118.802	−0.0487873	−0.0243937	−	0.999702i	$-0.507766\pi$
$181$	−118.802	−0.0487873	−0.0243937	+	0.999702i	$0.507766\pi$
$182$	0	0
$183$	−619.479	−0.250236
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1511.39	0.591038
$188$	0	0
$189$	−722.137	−0.277924
$190$	0	0
$191$	−2121.48	−0.803690	−0.401845	−	0.915708i	$-0.631631\pi$
$191$	−2121.48	−0.803690	−0.401845	+	0.915708i	$0.631631\pi$
$192$	0	0
$193$	2915.49	1.08736	0.543682	−	0.839291i	$-0.317030\pi$
$193$	2915.49	1.08736	0.543682	+	0.839291i	$0.317030\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1072.39	−0.387841	−0.193921	−	0.981017i	$-0.562120\pi$
$197$	−1072.39	−0.387841	−0.193921	+	0.981017i	$0.562120\pi$
$198$	0	0
$199$	−664.899	−0.236852	−0.118426	−	0.992963i	$-0.537785\pi$
$199$	−664.899	−0.236852	−0.118426	+	0.992963i	$0.537785\pi$
$200$	0	0
$201$	−1054.67	−0.370104
$202$	0	0
$203$	−7292.53	−2.52136
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1918.52	−0.644186
$208$	0	0
$209$	193.167	0.0639311
$210$	0	0
$211$	−2615.36	−0.853311	−0.426656	−	0.904414i	$-0.640308\pi$
$211$	−2615.36	−0.853311	−0.426656	+	0.904414i	$0.640308\pi$
$212$	0	0
$213$	1501.49	0.483007
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8766.09	2.74231
$218$	0	0
$219$	−2963.15	−0.914299
$220$	0	0
$221$	1607.64	0.489330
$222$	0	0
$223$	−3223.81	−0.968082	−0.484041	−	0.875045i	$-0.660831\pi$
$223$	−3223.81	−0.968082	−0.484041	+	0.875045i	$0.660831\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4268.28	1.24800	0.624000	−	0.781424i	$-0.285507\pi$
$227$	4268.28	1.24800	0.624000	+	0.781424i	$0.285507\pi$
$228$	0	0
$229$	5115.74	1.47624	0.738118	−	0.674672i	$-0.235715\pi$
$229$	5115.74	1.47624	0.738118	+	0.674672i	$0.235715\pi$
$230$	0	0
$231$	3741.78	1.06576
$232$	0	0
$233$	−2434.30	−0.684447	−0.342223	−	0.939619i	$-0.611180\pi$
$233$	−2434.30	−0.684447	−0.342223	+	0.939619i	$0.611180\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−516.934	−0.141681
$238$	0	0
$239$	−2824.46	−0.764431	−0.382215	−	0.924073i	$-0.624839\pi$
$239$	−2824.46	−0.764431	−0.382215	+	0.924073i	$0.624839\pi$
$240$	0	0
$241$	626.231	0.167382	0.0836910	−	0.996492i	$-0.473329\pi$
$241$	626.231	0.167382	0.0836910	+	0.996492i	$0.473329\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	205.468	0.0529296
$248$	0	0
$249$	−716.849	−0.182444
$250$	0	0
$251$	1832.85	0.460910	0.230455	−	0.973083i	$-0.425979\pi$
$251$	1832.85	0.460910	0.230455	+	0.973083i	$0.425979\pi$
$252$	0	0
$253$	9940.89	2.47027
$254$	0	0
$255$	0	0
$256$	0	0
$257$	815.323	0.197893	0.0989464	−	0.995093i	$-0.468453\pi$
$257$	815.323	0.197893	0.0989464	+	0.995093i	$0.468453\pi$
$258$	0	0
$259$	10671.8	2.56029
$260$	0	0
$261$	−2453.95	−0.581975
$262$	0	0
$263$	5949.94	1.39501	0.697507	−	0.716578i	$-0.254292\pi$
$263$	5949.94	1.39501	0.697507	+	0.716578i	$0.254292\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	128.331	0.0294146
$268$	0	0
$269$	6545.45	1.48358	0.741790	−	0.670632i	$-0.233977\pi$
$269$	6545.45	1.48358	0.741790	+	0.670632i	$0.233977\pi$
$270$	0	0
$271$	3544.39	0.794488	0.397244	−	0.917713i	$-0.369967\pi$
$271$	3544.39	0.794488	0.397244	+	0.917713i	$0.369967\pi$
$272$	0	0
$273$	3980.07	0.882361
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4238.42	−0.919357	−0.459678	−	0.888086i	$-0.652035\pi$
$277$	−4238.42	−0.919357	−0.459678	+	0.888086i	$0.652035\pi$
$278$	0	0
$279$	2949.80	0.632975
$280$	0	0
$281$	7626.83	1.61914	0.809571	−	0.587022i	$-0.199700\pi$
$281$	7626.83	1.61914	0.809571	+	0.587022i	$0.199700\pi$
$282$	0	0
$283$	4972.15	1.04439	0.522197	−	0.852825i	$-0.325112\pi$
$283$	4972.15	1.04439	0.522197	+	0.852825i	$0.325112\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1131.05	−0.232627
$288$	0	0
$289$	−3862.60	−0.786201
$290$	0	0
$291$	−3563.91	−0.717938
$292$	0	0
$293$	3804.72	0.758614	0.379307	−	0.925271i	$-0.376162\pi$
$293$	3804.72	0.758614	0.379307	+	0.925271i	$0.376162\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1259.11	0.245997
$298$	0	0
$299$	10574.0	2.04518
$300$	0	0
$301$	−12537.0	−2.40074
$302$	0	0
$303$	2195.94	0.416348
$304$	0	0
$305$	0	0
$306$	0	0
$307$	293.524	0.0545677	0.0272839	−	0.999628i	$-0.491314\pi$
$307$	293.524	0.0545677	0.0272839	+	0.999628i	$0.491314\pi$
$308$	0	0
$309$	3130.89	0.576409
$310$	0	0
$311$	7593.46	1.38452	0.692260	−	0.721648i	$-0.256615\pi$
$311$	7593.46	1.38452	0.692260	+	0.721648i	$0.256615\pi$
$312$	0	0
$313$	9630.88	1.73920	0.869600	−	0.493758i	$-0.164377\pi$
$313$	9630.88	1.73920	0.869600	+	0.493758i	$0.164377\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7493.10	1.32762	0.663808	−	0.747903i	$-0.268939\pi$
$317$	7493.10	1.32762	0.663808	+	0.747903i	$0.268939\pi$
$318$	0	0
$319$	12715.2	2.23171
$320$	0	0
$321$	−970.328	−0.168718
$322$	0	0
$323$	134.248	0.0231262
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4037.64	0.682820
$328$	0	0
$329$	7357.41	1.23291
$330$	0	0
$331$	1160.95	0.192785	0.0963923	−	0.995343i	$-0.469270\pi$
$331$	1160.95	0.192785	0.0963923	+	0.995343i	$0.469270\pi$
$332$	0	0
$333$	3591.09	0.590963
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11656.9	−1.88425	−0.942123	−	0.335266i	$-0.891174\pi$
$337$	−11656.9	−1.88425	−0.942123	+	0.335266i	$0.891174\pi$
$338$	0	0
$339$	−6680.81	−1.07036
$340$	0	0
$341$	−15284.5	−2.42728
$342$	0	0
$343$	784.671	0.123523
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3271.21	−0.506074	−0.253037	−	0.967457i	$-0.581429\pi$
$347$	−3271.21	−0.506074	−0.253037	+	0.967457i	$0.581429\pi$
$348$	0	0
$349$	4314.33	0.661722	0.330861	−	0.943680i	$-0.392661\pi$
$349$	4314.33	0.661722	0.330861	+	0.943680i	$0.392661\pi$
$350$	0	0
$351$	1339.30	0.203665
$352$	0	0
$353$	2525.55	0.380798	0.190399	−	0.981707i	$-0.439022\pi$
$353$	2525.55	0.380798	0.190399	+	0.981707i	$0.439022\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2600.48	0.385524
$358$	0	0
$359$	−934.700	−0.137414	−0.0687069	−	0.997637i	$-0.521887\pi$
$359$	−934.700	−0.137414	−0.0687069	+	0.997637i	$0.521887\pi$
$360$	0	0
$361$	−6841.84	−0.997498
$362$	0	0
$363$	−2531.14	−0.365979
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10414.0	1.48122	0.740610	−	0.671936i	$-0.234537\pi$
$367$	10414.0	1.48122	0.740610	+	0.671936i	$0.234537\pi$
$368$	0	0
$369$	−380.601	−0.0536946
$370$	0	0
$371$	−4229.91	−0.591930
$372$	0	0
$373$	−3471.89	−0.481951	−0.240976	−	0.970531i	$-0.577467\pi$
$373$	−3471.89	−0.481951	−0.240976	+	0.970531i	$0.577467\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13525.0	1.84767
$378$	0	0
$379$	−11874.1	−1.60932	−0.804661	−	0.593734i	$-0.797653\pi$
$379$	−11874.1	−1.60932	−0.804661	+	0.593734i	$0.797653\pi$
$380$	0	0
$381$	1548.71	0.208249
$382$	0	0
$383$	5121.40	0.683268	0.341634	−	0.939833i	$-0.389020\pi$
$383$	5121.40	0.683268	0.341634	+	0.939833i	$0.389020\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4218.72	−0.554134
$388$	0	0
$389$	4217.32	0.549683	0.274842	−	0.961490i	$-0.411375\pi$
$389$	4217.32	0.549683	0.274842	+	0.961490i	$0.411375\pi$
$390$	0	0
$391$	6908.77	0.893585
$392$	0	0
$393$	723.542	0.0928699
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10154.9	1.28378	0.641891	−	0.766796i	$-0.278150\pi$
$397$	10154.9	1.28378	0.641891	+	0.766796i	$0.278150\pi$
$398$	0	0
$399$	332.359	0.0417012
$400$	0	0
$401$	−204.637	−0.0254840	−0.0127420	−	0.999919i	$-0.504056\pi$
$401$	−204.637	−0.0254840	−0.0127420	+	0.999919i	$0.504056\pi$
$402$	0	0
$403$	−16257.9	−2.00958
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18607.4	−2.26617
$408$	0	0
$409$	9342.45	1.12947	0.564736	−	0.825271i	$-0.308978\pi$
$409$	9342.45	1.12947	0.564736	+	0.825271i	$0.308978\pi$
$410$	0	0
$411$	−2752.16	−0.330302
$412$	0	0
$413$	15149.7	1.80501
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5938.29	0.697360
$418$	0	0
$419$	−3811.30	−0.444377	−0.222189	−	0.975004i	$-0.571320\pi$
$419$	−3811.30	−0.444377	−0.222189	+	0.975004i	$0.571320\pi$
$420$	0	0
$421$	1274.60	0.147554	0.0737771	−	0.997275i	$-0.476495\pi$
$421$	1274.60	0.147554	0.0737771	+	0.997275i	$0.476495\pi$
$422$	0	0
$423$	2475.78	0.284578
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5522.82	0.625920
$428$	0	0
$429$	−6939.62	−0.780997
$430$	0	0
$431$	5481.09	0.612564	0.306282	−	0.951941i	$-0.400915\pi$
$431$	5481.09	0.612564	0.306282	+	0.951941i	$0.400915\pi$
$432$	0	0
$433$	−5292.42	−0.587384	−0.293692	−	0.955900i	$-0.594884\pi$
$433$	−5292.42	−0.587384	−0.293692	+	0.955900i	$0.594884\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	882.989	0.0966569
$438$	0	0
$439$	3873.74	0.421147	0.210574	−	0.977578i	$-0.432467\pi$
$439$	3873.74	0.421147	0.210574	+	0.977578i	$0.432467\pi$
$440$	0	0
$441$	3351.04	0.361845
$442$	0	0
$443$	2322.09	0.249043	0.124521	−	0.992217i	$-0.460260\pi$
$443$	2322.09	0.249043	0.124521	+	0.992217i	$0.460260\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8211.25	−0.868857
$448$	0	0
$449$	1584.30	0.166521	0.0832603	−	0.996528i	$-0.473467\pi$
$449$	1584.30	0.166521	0.0832603	+	0.996528i	$0.473467\pi$
$450$	0	0
$451$	1972.10	0.205903
$452$	0	0
$453$	−3445.76	−0.357386
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8008.96	−0.819789	−0.409894	−	0.912133i	$-0.634434\pi$
$457$	−8008.96	−0.819789	−0.409894	+	0.912133i	$0.634434\pi$
$458$	0	0
$459$	875.065	0.0889859
$460$	0	0
$461$	−5795.05	−0.585471	−0.292736	−	0.956193i	$-0.594566\pi$
$461$	−5795.05	−0.585471	−0.292736	+	0.956193i	$0.594566\pi$
$462$	0	0
$463$	3813.40	0.382772	0.191386	−	0.981515i	$-0.438702\pi$
$463$	3813.40	0.382772	0.191386	+	0.981515i	$0.438702\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12144.0	1.20334	0.601669	−	0.798746i	$-0.294503\pi$
$467$	12144.0	1.20334	0.601669	+	0.798746i	$0.294503\pi$
$468$	0	0
$469$	9402.69	0.925748
$470$	0	0
$471$	−5443.91	−0.532574
$472$	0	0
$473$	21859.5	2.12495
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1423.37	−0.136628
$478$	0	0
$479$	−15414.5	−1.47037	−0.735184	−	0.677868i	$-0.762904\pi$
$479$	−15414.5	−1.47037	−0.735184	+	0.677868i	$0.762904\pi$
$480$	0	0
$481$	−19792.3	−1.87620
$482$	0	0
$483$	17104.1	1.61131
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20642.1	−1.92070	−0.960350	−	0.278798i	$-0.910064\pi$
$487$	−20642.1	−1.92070	−0.960350	+	0.278798i	$0.910064\pi$
$488$	0	0
$489$	−6055.13	−0.559964
$490$	0	0
$491$	−18144.3	−1.66770	−0.833851	−	0.551990i	$-0.813869\pi$
$491$	−18144.3	−1.66770	−0.833851	+	0.551990i	$0.813869\pi$
$492$	0	0
$493$	8836.88	0.807288
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13386.2	−1.20815
$498$	0	0
$499$	−6942.15	−0.622793	−0.311396	−	0.950280i	$-0.600797\pi$
$499$	−6942.15	−0.622793	−0.311396	+	0.950280i	$0.600797\pi$
$500$	0	0
$501$	7370.02	0.657222
$502$	0	0
$503$	8747.35	0.775398	0.387699	−	0.921786i	$-0.373270\pi$
$503$	8747.35	0.775398	0.387699	+	0.921786i	$0.373270\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−790.553	−0.0692499
$508$	0	0
$509$	−5823.86	−0.507147	−0.253574	−	0.967316i	$-0.581606\pi$
$509$	−5823.86	−0.507147	−0.253574	+	0.967316i	$0.581606\pi$
$510$	0	0
$511$	26417.3	2.28695
$512$	0	0
$513$	111.839	0.00962540
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12828.3	−1.09127
$518$	0	0
$519$	−8966.14	−0.758324
$520$	0	0
$521$	−3169.03	−0.266483	−0.133242	−	0.991084i	$-0.542539\pi$
$521$	−3169.03	−0.266483	−0.133242	+	0.991084i	$0.542539\pi$
$522$	0	0
$523$	−16801.4	−1.40473	−0.702363	−	0.711819i	$-0.747872\pi$
$523$	−16801.4	−1.40473	−0.702363	+	0.711819i	$0.747872\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10622.5	−0.878034
$528$	0	0
$529$	33274.0	2.73478
$530$	0	0
$531$	5097.91	0.416630
$532$	0	0
$533$	2097.69	0.170471
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8984.28	0.721975
$538$	0	0
$539$	−17363.5	−1.38757
$540$	0	0
$541$	8055.67	0.640186	0.320093	−	0.947386i	$-0.396286\pi$
$541$	8055.67	0.640186	0.320093	+	0.947386i	$0.396286\pi$
$542$	0	0
$543$	356.407	0.0281674
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1816.69	−0.142003	−0.0710017	−	0.997476i	$-0.522620\pi$
$547$	−1816.69	−0.142003	−0.0710017	+	0.997476i	$0.522620\pi$
$548$	0	0
$549$	1858.44	0.144474
$550$	0	0
$551$	1129.41	0.0873225
$552$	0	0
$553$	4608.60	0.354390
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19471.0	1.48118	0.740588	−	0.671960i	$-0.234547\pi$
$557$	19471.0	1.48118	0.740588	+	0.671960i	$0.234547\pi$
$558$	0	0
$559$	23251.5	1.75928
$560$	0	0
$561$	−4534.18	−0.341236
$562$	0	0
$563$	312.707	0.0234085	0.0117043	−	0.999932i	$-0.496274\pi$
$563$	312.707	0.0234085	0.0117043	+	0.999932i	$0.496274\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2166.41	0.160460
$568$	0	0
$569$	−19358.8	−1.42630	−0.713148	−	0.701013i	$-0.752731\pi$
$569$	−19358.8	−1.42630	−0.713148	+	0.701013i	$0.752731\pi$
$570$	0	0
$571$	−18362.7	−1.34580	−0.672902	−	0.739732i	$-0.734952\pi$
$571$	−18362.7	−1.34580	−0.672902	+	0.739732i	$0.734952\pi$
$572$	0	0
$573$	6364.44	0.464011
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22141.3	1.59749	0.798746	−	0.601669i	$-0.205497\pi$
$577$	22141.3	1.59749	0.798746	+	0.601669i	$0.205497\pi$
$578$	0	0
$579$	−8746.46	−0.627790
$580$	0	0
$581$	6390.90	0.456350
$582$	0	0
$583$	7375.24	0.523930
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8549.54	0.601154	0.300577	−	0.953758i	$-0.402821\pi$
$587$	8549.54	0.601154	0.300577	+	0.953758i	$0.402821\pi$
$588$	0	0
$589$	−1357.63	−0.0949748
$590$	0	0
$591$	3217.17	0.223920
$592$	0	0
$593$	18064.7	1.25098	0.625489	−	0.780233i	$-0.284900\pi$
$593$	18064.7	1.25098	0.625489	+	0.780233i	$0.284900\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1994.70	0.136746
$598$	0	0
$599$	7052.96	0.481095	0.240548	−	0.970637i	$-0.422673\pi$
$599$	7052.96	0.481095	0.240548	+	0.970637i	$0.422673\pi$
$600$	0	0
$601$	5019.65	0.340692	0.170346	−	0.985384i	$-0.445511\pi$
$601$	5019.65	0.340692	0.170346	+	0.985384i	$0.445511\pi$
$602$	0	0
$603$	3164.02	0.213680
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8722.47	0.583252	0.291626	−	0.956532i	$-0.405804\pi$
$607$	8722.47	0.583252	0.291626	+	0.956532i	$0.405804\pi$
$608$	0	0
$609$	21877.6	1.45571
$610$	0	0
$611$	−13645.3	−0.903484
$612$	0	0
$613$	−11239.7	−0.740565	−0.370282	−	0.928919i	$-0.620739\pi$
$613$	−11239.7	−0.740565	−0.370282	+	0.928919i	$0.620739\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2171.46	−0.141685	−0.0708425	−	0.997488i	$-0.522569\pi$
$617$	−2171.46	−0.141685	−0.0708425	+	0.997488i	$0.522569\pi$
$618$	0	0
$619$	16668.6	1.08234	0.541170	−	0.840913i	$-0.317981\pi$
$619$	16668.6	1.08234	0.541170	+	0.840913i	$0.317981\pi$
$620$	0	0
$621$	5755.56	0.371921
$622$	0	0
$623$	−1144.10	−0.0735754
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−579.500	−0.0369107
$628$	0	0
$629$	−12931.8	−0.819756
$630$	0	0
$631$	−10067.4	−0.635147	−0.317574	−	0.948234i	$-0.602868\pi$
$631$	−10067.4	−0.635147	−0.317574	+	0.948234i	$0.602868\pi$
$632$	0	0
$633$	7846.08	0.492660
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−18469.3	−1.14879
$638$	0	0
$639$	−4504.47	−0.278864
$640$	0	0
$641$	−2111.45	−0.130105	−0.0650524	−	0.997882i	$-0.520721\pi$
$641$	−2111.45	−0.130105	−0.0650524	+	0.997882i	$0.520721\pi$
$642$	0	0
$643$	−26308.5	−1.61354	−0.806771	−	0.590864i	$-0.798787\pi$
$643$	−26308.5	−1.61354	−0.806771	+	0.590864i	$0.798787\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24695.4	1.50058	0.750292	−	0.661107i	$-0.229913\pi$
$647$	24695.4	1.50058	0.750292	+	0.661107i	$0.229913\pi$
$648$	0	0
$649$	−26415.0	−1.59766
$650$	0	0
$651$	−26298.3	−1.58327
$652$	0	0
$653$	−14713.3	−0.881740	−0.440870	−	0.897571i	$-0.645330\pi$
$653$	−14713.3	−0.881740	−0.440870	+	0.897571i	$0.645330\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8889.46	0.527871
$658$	0	0
$659$	−11985.3	−0.708470	−0.354235	−	0.935156i	$-0.615259\pi$
$659$	−11985.3	−0.708470	−0.354235	+	0.935156i	$0.615259\pi$
$660$	0	0
$661$	−897.229	−0.0527960	−0.0263980	−	0.999652i	$-0.508404\pi$
$661$	−897.229	−0.0527960	−0.0263980	+	0.999652i	$0.508404\pi$
$662$	0	0
$663$	−4822.93	−0.282515
$664$	0	0
$665$	0	0
$666$	0	0
$667$	58122.8	3.37410
$668$	0	0
$669$	9671.43	0.558922
$670$	0	0
$671$	−9629.55	−0.554016
$672$	0	0
$673$	17281.2	0.989808	0.494904	−	0.868948i	$-0.335203\pi$
$673$	17281.2	0.989808	0.494904	+	0.868948i	$0.335203\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18929.2	−1.07460	−0.537302	−	0.843390i	$-0.680556\pi$
$677$	−18929.2	−1.07460	−0.537302	+	0.843390i	$0.680556\pi$
$678$	0	0
$679$	31773.2	1.79579
$680$	0	0
$681$	−12804.9	−0.720533
$682$	0	0
$683$	10334.4	0.578967	0.289483	−	0.957183i	$-0.406516\pi$
$683$	10334.4	0.578967	0.289483	+	0.957183i	$0.406516\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15347.2	−0.852305
$688$	0	0
$689$	7844.92	0.433770
$690$	0	0
$691$	−14503.8	−0.798480	−0.399240	−	0.916847i	$-0.630726\pi$
$691$	−14503.8	−0.798480	−0.399240	+	0.916847i	$0.630726\pi$
$692$	0	0
$693$	−11225.3	−0.615317
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1370.58	0.0744826
$698$	0	0
$699$	7302.89	0.395166
$700$	0	0
$701$	11083.4	0.597167	0.298583	−	0.954384i	$-0.403486\pi$
$701$	11083.4	0.597167	0.298583	+	0.954384i	$0.403486\pi$
$702$	0	0
$703$	−1652.78	−0.0886711
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19577.4	−1.04142
$708$	0	0
$709$	−16340.6	−0.865561	−0.432780	−	0.901499i	$-0.642467\pi$
$709$	−16340.6	−0.865561	−0.432780	+	0.901499i	$0.642467\pi$
$710$	0	0
$711$	1550.80	0.0817997
$712$	0	0
$713$	−69867.4	−3.66978
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8473.37	0.441344
$718$	0	0
$719$	18000.4	0.933663	0.466831	−	0.884346i	$-0.345396\pi$
$719$	18000.4	0.933663	0.466831	+	0.884346i	$0.345396\pi$
$720$	0	0
$721$	−27912.7	−1.44178
$722$	0	0
$723$	−1878.69	−0.0966380
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17076.8	0.871176	0.435588	−	0.900146i	$-0.356541\pi$
$727$	17076.8	0.871176	0.435588	+	0.900146i	$0.356541\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	15192.0	0.768669
$732$	0	0
$733$	−13749.8	−0.692852	−0.346426	−	0.938077i	$-0.612605\pi$
$733$	−13749.8	−0.692852	−0.346426	+	0.938077i	$0.612605\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16394.5	−0.819400
$738$	0	0
$739$	28511.3	1.41922	0.709611	−	0.704593i	$-0.248870\pi$
$739$	28511.3	1.41922	0.709611	+	0.704593i	$0.248870\pi$
$740$	0	0
$741$	−616.404	−0.0305589
$742$	0	0
$743$	17563.7	0.867229	0.433614	−	0.901098i	$-0.357238\pi$
$743$	17563.7	0.867229	0.433614	+	0.901098i	$0.357238\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2150.55	0.105334
$748$	0	0
$749$	8650.73	0.422017
$750$	0	0
$751$	31554.5	1.53321	0.766604	−	0.642120i	$-0.221945\pi$
$751$	31554.5	1.53321	0.766604	+	0.642120i	$0.221945\pi$
$752$	0	0
$753$	−5498.54	−0.266106
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23937.9	1.14932	0.574662	−	0.818391i	$-0.305134\pi$
$757$	23937.9	1.14932	0.574662	+	0.818391i	$0.305134\pi$
$758$	0	0
$759$	−29822.7	−1.42621
$760$	0	0
$761$	−12606.7	−0.600518	−0.300259	−	0.953858i	$-0.597073\pi$
$761$	−12606.7	−0.600518	−0.300259	+	0.953858i	$0.597073\pi$
$762$	0	0
$763$	−35996.7	−1.70795
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28097.2	−1.32273
$768$	0	0
$769$	7666.35	0.359500	0.179750	−	0.983712i	$-0.442471\pi$
$769$	7666.35	0.359500	0.179750	+	0.983712i	$0.442471\pi$
$770$	0	0
$771$	−2445.97	−0.114253
$772$	0	0
$773$	30999.7	1.44241	0.721203	−	0.692724i	$-0.243589\pi$
$773$	30999.7	1.44241	0.721203	+	0.692724i	$0.243589\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−32015.5	−1.47819
$778$	0	0
$779$	175.169	0.00805661
$780$	0	0
$781$	23340.1	1.06936
$782$	0	0
$783$	7361.84	0.336003
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15999.9	0.724696	0.362348	−	0.932043i	$-0.381975\pi$
$787$	15999.9	0.724696	0.362348	+	0.932043i	$0.381975\pi$
$788$	0	0
$789$	−17849.8	−0.805412
$790$	0	0
$791$	59561.2	2.67731
$792$	0	0
$793$	−10242.8	−0.458679
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14714.8	0.653982	0.326991	−	0.945027i	$-0.393965\pi$
$797$	14714.8	0.653982	0.326991	+	0.945027i	$0.393965\pi$
$798$	0	0
$799$	−8915.50	−0.394753
$800$	0	0
$801$	−384.992	−0.0169825
$802$	0	0
$803$	−46061.1	−2.02423
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19636.4	−0.856546
$808$	0	0
$809$	17761.2	0.771881	0.385940	−	0.922524i	$-0.373877\pi$
$809$	17761.2	0.771881	0.385940	+	0.922524i	$0.373877\pi$
$810$	0	0
$811$	19307.1	0.835960	0.417980	−	0.908456i	$-0.362738\pi$
$811$	19307.1	0.835960	0.417980	+	0.908456i	$0.362738\pi$
$812$	0	0
$813$	−10633.2	−0.458698
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1941.64	0.0831451
$818$	0	0
$819$	−11940.2	−0.509431
$820$	0	0
$821$	6344.12	0.269685	0.134842	−	0.990867i	$-0.456947\pi$
$821$	6344.12	0.269685	0.134842	+	0.990867i	$0.456947\pi$
$822$	0	0
$823$	−678.688	−0.0287456	−0.0143728	−	0.999897i	$-0.504575\pi$
$823$	−678.688	−0.0287456	−0.0143728	+	0.999897i	$0.504575\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9890.28	−0.415863	−0.207932	−	0.978143i	$-0.566673\pi$
$827$	−9890.28	−0.415863	−0.207932	+	0.978143i	$0.566673\pi$
$828$	0	0
$829$	−17731.8	−0.742884	−0.371442	−	0.928456i	$-0.621137\pi$
$829$	−17731.8	−0.742884	−0.371442	+	0.928456i	$0.621137\pi$
$830$	0	0
$831$	12715.2	0.530791
$832$	0	0
$833$	−12067.4	−0.501934
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8849.41	−0.365448
$838$	0	0
$839$	−21286.1	−0.875897	−0.437949	−	0.899000i	$-0.644295\pi$
$839$	−21286.1	−0.875897	−0.437949	+	0.899000i	$0.644295\pi$
$840$	0	0
$841$	49954.8	2.04825
$842$	0	0
$843$	−22880.5	−0.934812
$844$	0	0
$845$	0	0
$846$	0	0
$847$	22565.8	0.915430
$848$	0	0
$849$	−14916.5	−0.602982
$850$	0	0
$851$	−85056.6	−3.42621
$852$	0	0
$853$	14452.2	0.580108	0.290054	−	0.957010i	$-0.406327\pi$
$853$	14452.2	0.580108	0.290054	+	0.957010i	$0.406327\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27547.1	−1.09801	−0.549003	−	0.835820i	$-0.684992\pi$
$857$	−27547.1	−1.09801	−0.549003	+	0.835820i	$0.684992\pi$
$858$	0	0
$859$	40442.1	1.60636	0.803182	−	0.595734i	$-0.203139\pi$
$859$	40442.1	1.60636	0.803182	+	0.595734i	$0.203139\pi$
$860$	0	0
$861$	3393.16	0.134307
$862$	0	0
$863$	−30084.4	−1.18666	−0.593328	−	0.804961i	$-0.702186\pi$
$863$	−30084.4	−1.18666	−0.593328	+	0.804961i	$0.702186\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11587.8	0.453913
$868$	0	0
$869$	−8035.53	−0.313679
$870$	0	0
$871$	−17438.5	−0.678395
$872$	0	0
$873$	10691.7	0.414502
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33411.0	1.28644	0.643221	−	0.765681i	$-0.277598\pi$
$877$	33411.0	1.28644	0.643221	+	0.765681i	$0.277598\pi$
$878$	0	0
$879$	−11414.2	−0.437986
$880$	0	0
$881$	28291.3	1.08190	0.540952	−	0.841054i	$-0.318064\pi$
$881$	28291.3	1.08190	0.540952	+	0.841054i	$0.318064\pi$
$882$	0	0
$883$	5099.71	0.194359	0.0971794	−	0.995267i	$-0.469018\pi$
$883$	5099.71	0.194359	0.0971794	+	0.995267i	$0.469018\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17410.2	−0.659051	−0.329525	−	0.944147i	$-0.606889\pi$
$887$	−17410.2	−0.659051	−0.329525	+	0.944147i	$0.606889\pi$
$888$	0	0
$889$	−13807.2	−0.520898
$890$	0	0
$891$	−3777.34	−0.142026
$892$	0	0
$893$	−1139.46	−0.0426995
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−31721.9	−1.18078
$898$	0	0
$899$	−89366.1	−3.31538
$900$	0	0
$901$	5125.69	0.189524
$902$	0	0
$903$	37611.1	1.38607
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6704.04	−0.245429	−0.122714	−	0.992442i	$-0.539160\pi$
$907$	−6704.04	−0.245429	−0.122714	+	0.992442i	$0.539160\pi$
$908$	0	0
$909$	−6587.82	−0.240378
$910$	0	0
$911$	−9442.28	−0.343399	−0.171700	−	0.985149i	$-0.554926\pi$
$911$	−9442.28	−0.343399	−0.171700	+	0.985149i	$0.554926\pi$
$912$	0	0
$913$	−11143.1	−0.403925
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6450.57	−0.232297
$918$	0	0
$919$	37528.3	1.34706	0.673529	−	0.739161i	$-0.264778\pi$
$919$	37528.3	1.34706	0.673529	+	0.739161i	$0.264778\pi$
$920$	0	0
$921$	−880.572	−0.0315047
$922$	0	0
$923$	24826.5	0.885344
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−9392.68	−0.332790
$928$	0	0
$929$	−32419.4	−1.14494	−0.572468	−	0.819927i	$-0.694014\pi$
$929$	−32419.4	−1.14494	−0.572468	+	0.819927i	$0.694014\pi$
$930$	0	0
$931$	−1542.30	−0.0542930
$932$	0	0
$933$	−22780.4	−0.799353
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39352.8	−1.37204	−0.686020	−	0.727583i	$-0.740644\pi$
$937$	−39352.8	−1.37204	−0.686020	+	0.727583i	$0.740644\pi$
$938$	0	0
$939$	−28892.6	−1.00413
$940$	0	0
$941$	−48896.5	−1.69392	−0.846961	−	0.531655i	$-0.821570\pi$
$941$	−48896.5	−1.69392	−0.846961	+	0.531655i	$0.821570\pi$
$942$	0	0
$943$	9014.71	0.311304
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36128.3	1.23972	0.619858	−	0.784714i	$-0.287190\pi$
$947$	36128.3	1.23972	0.619858	+	0.784714i	$0.287190\pi$
$948$	0	0
$949$	−48994.4	−1.67590
$950$	0	0
$951$	−22479.3	−0.766500
$952$	0	0
$953$	−17959.1	−0.610444	−0.305222	−	0.952281i	$-0.598731\pi$
$953$	−17959.1	−0.610444	−0.305222	+	0.952281i	$0.598731\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−38145.6	−1.28848
$958$	0	0
$959$	24536.2	0.826191
$960$	0	0
$961$	77632.9	2.60592
$962$	0	0
$963$	2910.98	0.0974092
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−22352.5	−0.743339	−0.371669	−	0.928365i	$-0.621215\pi$
$967$	−22352.5	−0.743339	−0.371669	+	0.928365i	$0.621215\pi$
$968$	0	0
$969$	−402.744	−0.0133519
$970$	0	0
$971$	−28099.6	−0.928690	−0.464345	−	0.885654i	$-0.653710\pi$
$971$	−28099.6	−0.928690	−0.464345	+	0.885654i	$0.653710\pi$
$972$	0	0
$973$	−52941.4	−1.74432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23533.0	0.770612	0.385306	−	0.922789i	$-0.374096\pi$
$977$	23533.0	0.770612	0.385306	+	0.922789i	$0.374096\pi$
$978$	0	0
$979$	1994.85	0.0651232
$980$	0	0
$981$	−12112.9	−0.394226
$982$	0	0
$983$	59414.3	1.92779	0.963897	−	0.266276i	$-0.0857933\pi$
$983$	59414.3	1.92779	0.963897	+	0.266276i	$0.0857933\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−22072.2	−0.711820
$988$	0	0
$989$	99922.4	3.21269
$990$	0	0
$991$	32548.1	1.04332	0.521658	−	0.853155i	$-0.325314\pi$
$991$	32548.1	1.04332	0.521658	+	0.853155i	$0.325314\pi$
$992$	0	0
$993$	−3482.86	−0.111304
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4644.79	0.147545	0.0737723	−	0.997275i	$-0.476496\pi$
$997$	4644.79	0.147545	0.0737723	+	0.997275i	$0.476496\pi$
$998$	0	0
$999$	−10773.3	−0.341192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bi.1.3	✓	3
4.3	odd	2		2400.4.a.bt.1.1	yes	3
5.4	even	2		2400.4.a.bs.1.1	yes	3
20.19	odd	2		2400.4.a.bj.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bi.1.3	✓	3	1.1	even	1	trivial
2400.4.a.bj.1.3	yes	3	20.19	odd	2
2400.4.a.bs.1.1	yes	3	5.4	even	2
2400.4.a.bt.1.1	yes	3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +26.7458 q^{7} +9.00000 q^{9} -46.6338 q^{11} -49.6036 q^{13} -32.4098 q^{17} -4.14220 q^{19} -80.2374 q^{21} -213.169 q^{23} -27.0000 q^{27} -272.661 q^{29} +327.756 q^{31} +139.901 q^{33} +399.010 q^{37} +148.811 q^{39} -42.2890 q^{41} -468.747 q^{43} +275.086 q^{47} +372.338 q^{49} +97.2295 q^{51} -158.152 q^{53} +12.4266 q^{57} +566.435 q^{59} +206.493 q^{61} +240.712 q^{63} +351.557 q^{67} +639.507 q^{69} -500.497 q^{71} +987.718 q^{73} -1247.26 q^{77} +172.311 q^{79} +81.0000 q^{81} +238.950 q^{83} +817.982 q^{87} -42.7769 q^{89} -1326.69 q^{91} -983.268 q^{93} +1187.97 q^{97} -419.704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 3 q^{7} + 27 q^{9} - 44 q^{11} - 13 q^{13} + 36 q^{17} - 71 q^{19} - 9 q^{21} - 160 q^{23} - 81 q^{27} - 184 q^{29} + 59 q^{31} + 132 q^{33} + 350 q^{37} + 39 q^{39} + 166 q^{41} - 341 q^{43}+ \cdots - 396 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 3 * q^7 + 27 * q^9 - 44 * q^11 - 13 * q^13 + 36 * q^17 - 71 * q^19 - 9 * q^21 - 160 * q^23 - 81 * q^27 - 184 * q^29 + 59 * q^31 + 132 * q^33 + 350 * q^37 + 39 * q^39 + 166 * q^41 - 341 * q^43 + 394 * q^47 + 158 * q^49 - 108 * q^51 + 314 * q^53 + 213 * q^57 - 538 * q^59 - 523 * q^61 + 27 * q^63 + 543 * q^67 + 480 * q^69 - 910 * q^71 + 870 * q^73 - 828 * q^77 + 96 * q^79 + 243 * q^81 - 210 * q^83 + 552 * q^87 - 64 * q^89 - 2781 * q^91 - 177 * q^93 + 943 * q^97 - 396 * q^99

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