LMFDB - Embedded newform 2400.4.a.ba.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{201}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 50 $ x^2 - x - 50
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.58872$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -26.3549 q^{7} +9.00000 q^{9} -20.0000 q^{11} +40.3549 q^{13} -101.065 q^{17} -29.6451 q^{19} -79.0647 q^{21} +22.3549 q^{23} +27.0000 q^{27} +187.420 q^{29} -293.194 q^{31} -60.0000 q^{33} +1.77447 q^{37} +121.065 q^{39} +229.290 q^{41} +146.580 q^{43} -538.614 q^{47} +351.580 q^{49} -303.194 q^{51} -432.129 q^{53} -88.9353 q^{57} +502.839 q^{59} +937.808 q^{61} -237.194 q^{63} -483.098 q^{67} +67.0647 q^{69} +397.161 q^{71} +56.3881 q^{73} +527.098 q^{77} +327.516 q^{79} +81.0000 q^{81} +473.678 q^{83} +562.259 q^{87} -436.647 q^{89} -1063.55 q^{91} -879.582 q^{93} -1429.87 q^{97} -180.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 4 q^{7} + 18 q^{9} - 40 q^{11} + 24 q^{13} - 32 q^{17} - 116 q^{19} + 12 q^{21} - 12 q^{23} + 54 q^{27} + 148 q^{29} - 76 q^{31} - 120 q^{33} - 280 q^{37} + 72 q^{39} + 572 q^{41} + 520 q^{43}+ \cdots - 360 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 4 * q^7 + 18 * q^9 - 40 * q^11 + 24 * q^13 - 32 * q^17 - 116 * q^19 + 12 * q^21 - 12 * q^23 + 54 * q^27 + 148 * q^29 - 76 * q^31 - 120 * q^33 - 280 * q^37 + 72 * q^39 + 572 * q^41 + 520 * q^43 - 340 * q^47 + 930 * q^49 - 96 * q^51 - 524 * q^53 - 348 * q^57 + 552 * q^59 + 628 * q^61 + 36 * q^63 + 168 * q^67 - 36 * q^69 + 1248 * q^71 - 908 * q^73 - 80 * q^77 + 1052 * q^79 + 162 * q^81 + 40 * q^83 + 444 * q^87 + 828 * q^89 - 1560 * q^91 - 228 * q^93 + 316 * q^97 - 360 * 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
$7$	−26.3549	−1.42303	−0.711515	−	0.702671i	$-0.751991\pi$
$7$	−26.3549	−1.42303	−0.711515	+	0.702671i	$0.751991\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	40.3549	0.860956	0.430478	−	0.902601i	$-0.358345\pi$
$13$	40.3549	0.860956	0.430478	+	0.902601i	$0.358345\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−101.065	−1.44187	−0.720935	−	0.693003i	$-0.756287\pi$
$17$	−101.065	−1.44187	−0.720935	+	0.693003i	$0.756287\pi$
$18$	0	0
$19$	−29.6451	−0.357950	−0.178975	−	0.983854i	$-0.557278\pi$
$19$	−29.6451	−0.357950	−0.178975	+	0.983854i	$0.557278\pi$
$20$	0	0
$21$	−79.0647	−0.821587
$22$	0	0
$23$	22.3549	0.202666	0.101333	−	0.994853i	$-0.467689\pi$
$23$	22.3549	0.202666	0.101333	+	0.994853i	$0.467689\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	187.420	1.20010	0.600051	−	0.799962i	$-0.295147\pi$
$29$	187.420	1.20010	0.600051	+	0.799962i	$0.295147\pi$
$30$	0	0
$31$	−293.194	−1.69868	−0.849342	−	0.527843i	$-0.823001\pi$
$31$	−293.194	−1.69868	−0.849342	+	0.527843i	$0.823001\pi$
$32$	0	0
$33$	−60.0000	−0.316505
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.77447	0.00788435	0.00394217	−	0.999992i	$-0.498745\pi$
$37$	1.77447	0.00788435	0.00394217	+	0.999992i	$0.498745\pi$
$38$	0	0
$39$	121.065	0.497073
$40$	0	0
$41$	229.290	0.873393	0.436697	−	0.899609i	$-0.356148\pi$
$41$	229.290	0.873393	0.436697	+	0.899609i	$0.356148\pi$
$42$	0	0
$43$	146.580	0.519844	0.259922	−	0.965630i	$-0.416303\pi$
$43$	146.580	0.519844	0.259922	+	0.965630i	$0.416303\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−538.614	−1.67159	−0.835796	−	0.549039i	$-0.814994\pi$
$47$	−538.614	−1.67159	−0.835796	+	0.549039i	$0.814994\pi$
$48$	0	0
$49$	351.580	1.02502
$50$	0	0
$51$	−303.194	−0.832464
$52$	0	0
$53$	−432.129	−1.11995	−0.559977	−	0.828508i	$-0.689190\pi$
$53$	−432.129	−1.11995	−0.559977	+	0.828508i	$0.689190\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−88.9353	−0.206663
$58$	0	0
$59$	502.839	1.10956	0.554780	−	0.831997i	$-0.312802\pi$
$59$	502.839	1.10956	0.554780	+	0.831997i	$0.312802\pi$
$60$	0	0
$61$	937.808	1.96843	0.984213	−	0.176989i	$-0.0566357\pi$
$61$	937.808	1.96843	0.984213	+	0.176989i	$0.0566357\pi$
$62$	0	0
$63$	−237.194	−0.474344
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−483.098	−0.880893	−0.440446	−	0.897779i	$-0.645180\pi$
$67$	−483.098	−0.880893	−0.440446	+	0.897779i	$0.645180\pi$
$68$	0	0
$69$	67.0647	0.117009
$70$	0	0
$71$	397.161	0.663864	0.331932	−	0.943303i	$-0.392300\pi$
$71$	397.161	0.663864	0.331932	+	0.943303i	$0.392300\pi$
$72$	0	0
$73$	56.3881	0.0904072	0.0452036	−	0.998978i	$-0.485606\pi$
$73$	56.3881	0.0904072	0.0452036	+	0.998978i	$0.485606\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	527.098	0.780109
$78$	0	0
$79$	327.516	0.466435	0.233218	−	0.972425i	$-0.425075\pi$
$79$	327.516	0.466435	0.233218	+	0.972425i	$0.425075\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	473.678	0.626421	0.313211	−	0.949684i	$-0.398595\pi$
$83$	473.678	0.626421	0.313211	+	0.949684i	$0.398595\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	562.259	0.692879
$88$	0	0
$89$	−436.647	−0.520050	−0.260025	−	0.965602i	$-0.583731\pi$
$89$	−436.647	−0.520050	−0.260025	+	0.965602i	$0.583731\pi$
$90$	0	0
$91$	−1063.55	−1.22517
$92$	0	0
$93$	−879.582	−0.980736
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1429.87	−1.49672	−0.748359	−	0.663294i	$-0.769158\pi$
$97$	−1429.87	−1.49672	−0.748359	+	0.663294i	$0.769158\pi$
$98$	0	0
$99$	−180.000	−0.182734
$100$	0	0
$101$	152.063	0.149810	0.0749051	−	0.997191i	$-0.476135\pi$
$101$	152.063	0.149810	0.0749051	+	0.997191i	$0.476135\pi$
$102$	0	0
$103$	160.484	0.153524	0.0767621	−	0.997049i	$-0.475542\pi$
$103$	160.484	0.153524	0.0767621	+	0.997049i	$0.475542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1186.65	1.07213	0.536063	−	0.844178i	$-0.319911\pi$
$107$	1186.65	1.07213	0.536063	+	0.844178i	$0.319911\pi$
$108$	0	0
$109$	1477.10	1.29798	0.648992	−	0.760795i	$-0.275191\pi$
$109$	1477.10	1.29798	0.648992	+	0.760795i	$0.275191\pi$
$110$	0	0
$111$	5.32341	0.00455203
$112$	0	0
$113$	−1385.32	−1.15328	−0.576638	−	0.817000i	$-0.695636\pi$
$113$	−1385.32	−1.15328	−0.576638	+	0.817000i	$0.695636\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	363.194	0.286985
$118$	0	0
$119$	2663.55	2.05182
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	687.871	0.504254
$124$	0	0
$125$	0	0
$126$	0	0
$127$	903.582	0.631338	0.315669	−	0.948869i	$-0.397771\pi$
$127$	903.582	0.631338	0.315669	+	0.948869i	$0.397771\pi$
$128$	0	0
$129$	439.741	0.300132
$130$	0	0
$131$	1100.58	0.734034	0.367017	−	0.930214i	$-0.380379\pi$
$131$	1100.58	0.734034	0.367017	+	0.930214i	$0.380379\pi$
$132$	0	0
$133$	781.294	0.509374
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1839.65	1.14724	0.573619	−	0.819123i	$-0.305539\pi$
$137$	1839.65	1.14724	0.573619	+	0.819123i	$0.305539\pi$
$138$	0	0
$139$	−1858.87	−1.13430	−0.567149	−	0.823615i	$-0.691954\pi$
$139$	−1858.87	−1.13430	−0.567149	+	0.823615i	$0.691954\pi$
$140$	0	0
$141$	−1615.84	−0.965095
$142$	0	0
$143$	−807.098	−0.471978
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1054.74	0.591793
$148$	0	0
$149$	2831.29	1.55670	0.778351	−	0.627829i	$-0.216056\pi$
$149$	2831.29	1.55670	0.778351	+	0.627829i	$0.216056\pi$
$150$	0	0
$151$	2503.71	1.34933	0.674667	−	0.738122i	$-0.264287\pi$
$151$	2503.71	1.34933	0.674667	+	0.738122i	$0.264287\pi$
$152$	0	0
$153$	−909.582	−0.480623
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2152.42	1.09415	0.547076	−	0.837083i	$-0.315741\pi$
$157$	2152.42	1.09415	0.547076	+	0.837083i	$0.315741\pi$
$158$	0	0
$159$	−1296.39	−0.646605
$160$	0	0
$161$	−589.161	−0.288400
$162$	0	0
$163$	2710.84	1.30263	0.651317	−	0.758806i	$-0.274217\pi$
$163$	2710.84	1.30263	0.651317	+	0.758806i	$0.274217\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2175.71	−1.00815	−0.504076	−	0.863659i	$-0.668167\pi$
$167$	−2175.71	−1.00815	−0.504076	+	0.863659i	$0.668167\pi$
$168$	0	0
$169$	−568.483	−0.258754
$170$	0	0
$171$	−266.806	−0.119317
$172$	0	0
$173$	1010.13	0.443922	0.221961	−	0.975056i	$-0.428754\pi$
$173$	1010.13	0.443922	0.221961	+	0.975056i	$0.428754\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1508.52	0.640605
$178$	0	0
$179$	−2266.26	−0.946304	−0.473152	−	0.880981i	$-0.656884\pi$
$179$	−2266.26	−0.946304	−0.473152	+	0.880981i	$0.656884\pi$
$180$	0	0
$181$	4722.01	1.93914	0.969569	−	0.244818i	$-0.0787282\pi$
$181$	4722.01	1.93914	0.969569	+	0.244818i	$0.0787282\pi$
$182$	0	0
$183$	2813.42	1.13647
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2021.29	0.790437
$188$	0	0
$189$	−711.582	−0.273862
$190$	0	0
$191$	333.028	0.126163	0.0630813	−	0.998008i	$-0.479907\pi$
$191$	333.028	0.126163	0.0630813	+	0.998008i	$0.479907\pi$
$192$	0	0
$193$	4503.23	1.67953	0.839767	−	0.542948i	$-0.182692\pi$
$193$	4503.23	1.67953	0.839767	+	0.542948i	$0.182692\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3325.03	1.20253	0.601266	−	0.799049i	$-0.294663\pi$
$197$	3325.03	1.20253	0.601266	+	0.799049i	$0.294663\pi$
$198$	0	0
$199$	3235.32	1.15249	0.576246	−	0.817276i	$-0.304517\pi$
$199$	3235.32	1.15249	0.576246	+	0.817276i	$0.304517\pi$
$200$	0	0
$201$	−1449.29	−0.508584
$202$	0	0
$203$	−4939.42	−1.70778
$204$	0	0
$205$	0	0
$206$	0	0
$207$	201.194	0.0675553
$208$	0	0
$209$	592.902	0.196229
$210$	0	0
$211$	−3181.71	−1.03810	−0.519048	−	0.854745i	$-0.673713\pi$
$211$	−3181.71	−1.03810	−0.519048	+	0.854745i	$0.673713\pi$
$212$	0	0
$213$	1191.48	0.383282
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7727.10	2.41728
$218$	0	0
$219$	169.164	0.0521966
$220$	0	0
$221$	−4078.45	−1.24139
$222$	0	0
$223$	1848.03	0.554948	0.277474	−	0.960733i	$-0.410503\pi$
$223$	1848.03	0.554948	0.277474	+	0.960733i	$0.410503\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−592.643	−0.173282	−0.0866412	−	0.996240i	$-0.527613\pi$
$227$	−592.643	−0.173282	−0.0866412	+	0.996240i	$0.527613\pi$
$228$	0	0
$229$	402.060	0.116021	0.0580106	−	0.998316i	$-0.481524\pi$
$229$	402.060	0.116021	0.0580106	+	0.998316i	$0.481524\pi$
$230$	0	0
$231$	1581.29	0.450396
$232$	0	0
$233$	−5961.52	−1.67619	−0.838095	−	0.545524i	$-0.816331\pi$
$233$	−5961.52	−1.67619	−0.838095	+	0.545524i	$0.816331\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	982.547	0.269297
$238$	0	0
$239$	2389.23	0.646637	0.323319	−	0.946290i	$-0.395201\pi$
$239$	2389.23	0.646637	0.323319	+	0.946290i	$0.395201\pi$
$240$	0	0
$241$	608.329	0.162597	0.0812985	−	0.996690i	$-0.474093\pi$
$241$	608.329	0.162597	0.0812985	+	0.996690i	$0.474093\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1196.33	−0.308180
$248$	0	0
$249$	1421.03	0.361664
$250$	0	0
$251$	6852.07	1.72310	0.861551	−	0.507671i	$-0.169493\pi$
$251$	6852.07	1.72310	0.861551	+	0.507671i	$0.169493\pi$
$252$	0	0
$253$	−447.098	−0.111102
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4000.23	0.970924	0.485462	−	0.874258i	$-0.338651\pi$
$257$	4000.23	0.970924	0.485462	+	0.874258i	$0.338651\pi$
$258$	0	0
$259$	−46.7659	−0.0112197
$260$	0	0
$261$	1686.78	0.400034
$262$	0	0
$263$	−3588.03	−0.841246	−0.420623	−	0.907236i	$-0.638188\pi$
$263$	−3588.03	−0.841246	−0.420623	+	0.907236i	$0.638188\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1309.94	−0.300251
$268$	0	0
$269$	−3399.69	−0.770567	−0.385283	−	0.922798i	$-0.625896\pi$
$269$	−3399.69	−0.770567	−0.385283	+	0.922798i	$0.625896\pi$
$270$	0	0
$271$	−7754.49	−1.73820	−0.869099	−	0.494638i	$-0.835300\pi$
$271$	−7754.49	−1.73820	−0.869099	+	0.494638i	$0.835300\pi$
$272$	0	0
$273$	−3190.65	−0.707351
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2247.07	−0.487414	−0.243707	−	0.969849i	$-0.578364\pi$
$277$	−2247.07	−0.487414	−0.243707	+	0.969849i	$0.578364\pi$
$278$	0	0
$279$	−2638.75	−0.566228
$280$	0	0
$281$	−719.101	−0.152662	−0.0763309	−	0.997083i	$-0.524321\pi$
$281$	−719.101	−0.152662	−0.0763309	+	0.997083i	$0.524321\pi$
$282$	0	0
$283$	−5225.16	−1.09754	−0.548770	−	0.835974i	$-0.684904\pi$
$283$	−5225.16	−1.09754	−0.548770	+	0.835974i	$0.684904\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6042.92	−1.24286
$288$	0	0
$289$	5301.07	1.07899
$290$	0	0
$291$	−4289.62	−0.864131
$292$	0	0
$293$	6408.79	1.27783	0.638917	−	0.769276i	$-0.279383\pi$
$293$	6408.79	1.27783	0.638917	+	0.769276i	$0.279383\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−540.000	−0.105502
$298$	0	0
$299$	902.129	0.174487
$300$	0	0
$301$	−3863.11	−0.739754
$302$	0	0
$303$	456.189	0.0864930
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4354.91	−0.809603	−0.404801	−	0.914405i	$-0.632659\pi$
$307$	−4354.91	−0.809603	−0.404801	+	0.914405i	$0.632659\pi$
$308$	0	0
$309$	481.453	0.0886372
$310$	0	0
$311$	−7515.82	−1.37036	−0.685182	−	0.728372i	$-0.740277\pi$
$311$	−7515.82	−1.37036	−0.685182	+	0.728372i	$0.740277\pi$
$312$	0	0
$313$	−8534.34	−1.54118	−0.770590	−	0.637331i	$-0.780038\pi$
$313$	−8534.34	−1.54118	−0.770590	+	0.637331i	$0.780038\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2260.90	0.400583	0.200291	−	0.979736i	$-0.435811\pi$
$317$	2260.90	0.400583	0.200291	+	0.979736i	$0.435811\pi$
$318$	0	0
$319$	−3748.39	−0.657899
$320$	0	0
$321$	3559.94	0.618992
$322$	0	0
$323$	2996.07	0.516118
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4431.29	0.749392
$328$	0	0
$329$	14195.1	2.37873
$330$	0	0
$331$	−1258.62	−0.209003	−0.104502	−	0.994525i	$-0.533325\pi$
$331$	−1258.62	−0.209003	−0.104502	+	0.994525i	$0.533325\pi$
$332$	0	0
$333$	15.9702	0.00262812
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10796.0	1.74509	0.872546	−	0.488532i	$-0.162467\pi$
$337$	10796.0	1.74509	0.872546	+	0.488532i	$0.162467\pi$
$338$	0	0
$339$	−4155.97	−0.665845
$340$	0	0
$341$	5863.88	0.931223
$342$	0	0
$343$	−226.136	−0.0355983
$344$	0	0
$345$	0	0
$346$	0	0
$347$	585.877	0.0906385	0.0453192	−	0.998973i	$-0.485569\pi$
$347$	585.877	0.0906385	0.0453192	+	0.998973i	$0.485569\pi$
$348$	0	0
$349$	12103.1	1.85634	0.928172	−	0.372151i	$-0.121380\pi$
$349$	12103.1	1.85634	0.928172	+	0.372151i	$0.121380\pi$
$350$	0	0
$351$	1089.58	0.165691
$352$	0	0
$353$	−1689.38	−0.254721	−0.127360	−	0.991857i	$-0.540650\pi$
$353$	−1689.38	−0.254721	−0.127360	+	0.991857i	$0.540650\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7990.65	1.18462
$358$	0	0
$359$	5792.40	0.851563	0.425781	−	0.904826i	$-0.359999\pi$
$359$	5792.40	0.851563	0.425781	+	0.904826i	$0.359999\pi$
$360$	0	0
$361$	−5980.17	−0.871872
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9651.40	1.37275	0.686375	−	0.727248i	$-0.259201\pi$
$367$	9651.40	1.37275	0.686375	+	0.727248i	$0.259201\pi$
$368$	0	0
$369$	2063.61	0.291131
$370$	0	0
$371$	11388.7	1.59373
$372$	0	0
$373$	−9720.23	−1.34931	−0.674657	−	0.738131i	$-0.735709\pi$
$373$	−9720.23	−1.34931	−0.674657	+	0.738131i	$0.735709\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7563.30	1.03324
$378$	0	0
$379$	−2315.32	−0.313800	−0.156900	−	0.987614i	$-0.550150\pi$
$379$	−2315.32	−0.313800	−0.156900	+	0.987614i	$0.550150\pi$
$380$	0	0
$381$	2710.75	0.364503
$382$	0	0
$383$	8955.71	1.19482	0.597409	−	0.801936i	$-0.296197\pi$
$383$	8955.71	1.19482	0.597409	+	0.801936i	$0.296197\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1319.22	0.173281
$388$	0	0
$389$	4201.48	0.547619	0.273809	−	0.961784i	$-0.411716\pi$
$389$	4201.48	0.547619	0.273809	+	0.961784i	$0.411716\pi$
$390$	0	0
$391$	−2259.29	−0.292218
$392$	0	0
$393$	3301.75	0.423795
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1318.23	−0.166649	−0.0833247	−	0.996522i	$-0.526554\pi$
$397$	−1318.23	−0.166649	−0.0833247	+	0.996522i	$0.526554\pi$
$398$	0	0
$399$	2343.88	0.294087
$400$	0	0
$401$	4319.04	0.537862	0.268931	−	0.963159i	$-0.413330\pi$
$401$	4319.04	0.537862	0.268931	+	0.963159i	$0.413330\pi$
$402$	0	0
$403$	−11831.8	−1.46249
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−35.4894	−0.00432222
$408$	0	0
$409$	−2075.17	−0.250882	−0.125441	−	0.992101i	$-0.540035\pi$
$409$	−2075.17	−0.250882	−0.125441	+	0.992101i	$0.540035\pi$
$410$	0	0
$411$	5518.94	0.662358
$412$	0	0
$413$	−13252.3	−1.57894
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5576.62	−0.654888
$418$	0	0
$419$	−13509.1	−1.57509	−0.787545	−	0.616257i	$-0.788648\pi$
$419$	−13509.1	−1.57509	−0.787545	+	0.616257i	$0.788648\pi$
$420$	0	0
$421$	8258.06	0.955993	0.477997	−	0.878362i	$-0.341363\pi$
$421$	8258.06	0.955993	0.477997	+	0.878362i	$0.341363\pi$
$422$	0	0
$423$	−4847.52	−0.557198
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−24715.8	−2.80113
$428$	0	0
$429$	−2421.29	−0.272497
$430$	0	0
$431$	2975.11	0.332497	0.166248	−	0.986084i	$-0.446835\pi$
$431$	2975.11	0.332497	0.166248	+	0.986084i	$0.446835\pi$
$432$	0	0
$433$	−1357.81	−0.150698	−0.0753488	−	0.997157i	$-0.524007\pi$
$433$	−1357.81	−0.150698	−0.0753488	+	0.997157i	$0.524007\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−662.713	−0.0725443
$438$	0	0
$439$	6925.51	0.752931	0.376465	−	0.926431i	$-0.377139\pi$
$439$	6925.51	0.752931	0.376465	+	0.926431i	$0.377139\pi$
$440$	0	0
$441$	3164.22	0.341672
$442$	0	0
$443$	−9300.77	−0.997501	−0.498750	−	0.866746i	$-0.666208\pi$
$443$	−9300.77	−0.997501	−0.498750	+	0.866746i	$0.666208\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8493.88	0.898762
$448$	0	0
$449$	14928.1	1.56904	0.784521	−	0.620103i	$-0.212909\pi$
$449$	14928.1	1.56904	0.784521	+	0.620103i	$0.212909\pi$
$450$	0	0
$451$	−4585.80	−0.478796
$452$	0	0
$453$	7511.14	0.779038
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16323.7	1.67088	0.835438	−	0.549585i	$-0.185214\pi$
$457$	16323.7	1.67088	0.835438	+	0.549585i	$0.185214\pi$
$458$	0	0
$459$	−2728.75	−0.277488
$460$	0	0
$461$	8960.71	0.905297	0.452649	−	0.891689i	$-0.350479\pi$
$461$	8960.71	0.905297	0.452649	+	0.891689i	$0.350479\pi$
$462$	0	0
$463$	12194.3	1.22401	0.612006	−	0.790853i	$-0.290363\pi$
$463$	12194.3	1.22401	0.612006	+	0.790853i	$0.290363\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11673.9	1.15675	0.578376	−	0.815770i	$-0.303687\pi$
$467$	11673.9	1.15675	0.578376	+	0.815770i	$0.303687\pi$
$468$	0	0
$469$	12732.0	1.25354
$470$	0	0
$471$	6457.26	0.631709
$472$	0	0
$473$	−2931.61	−0.284980
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3889.16	−0.373318
$478$	0	0
$479$	−10062.2	−0.959821	−0.479910	−	0.877318i	$-0.659331\pi$
$479$	−10062.2	−0.959821	−0.479910	+	0.877318i	$0.659331\pi$
$480$	0	0
$481$	71.6085	0.00678808
$482$	0	0
$483$	−1767.48	−0.166508
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9345.25	−0.869556	−0.434778	−	0.900538i	$-0.643173\pi$
$487$	−9345.25	−0.869556	−0.434778	+	0.900538i	$0.643173\pi$
$488$	0	0
$489$	8132.52	0.752076
$490$	0	0
$491$	15282.5	1.40466	0.702330	−	0.711852i	$-0.252143\pi$
$491$	15282.5	1.40466	0.702330	+	0.711852i	$0.252143\pi$
$492$	0	0
$493$	−18941.5	−1.73039
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10467.1	−0.944698
$498$	0	0
$499$	−7171.02	−0.643324	−0.321662	−	0.946854i	$-0.604242\pi$
$499$	−7171.02	−0.643324	−0.321662	+	0.946854i	$0.604242\pi$
$500$	0	0
$501$	−6527.12	−0.582057
$502$	0	0
$503$	14066.2	1.24688	0.623439	−	0.781872i	$-0.285735\pi$
$503$	14066.2	1.24688	0.623439	+	0.781872i	$0.285735\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1705.45	−0.149392
$508$	0	0
$509$	−968.322	−0.0843224	−0.0421612	−	0.999111i	$-0.513424\pi$
$509$	−968.322	−0.0843224	−0.0421612	+	0.999111i	$0.513424\pi$
$510$	0	0
$511$	−1486.10	−0.128652
$512$	0	0
$513$	−800.418	−0.0688875
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10772.3	0.916371
$518$	0	0
$519$	3030.38	0.256298
$520$	0	0
$521$	8255.03	0.694164	0.347082	−	0.937835i	$-0.387173\pi$
$521$	8255.03	0.694164	0.347082	+	0.937835i	$0.387173\pi$
$522$	0	0
$523$	−8777.73	−0.733888	−0.366944	−	0.930243i	$-0.619596\pi$
$523$	−8777.73	−0.733888	−0.366944	+	0.930243i	$0.619596\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29631.6	2.44928
$528$	0	0
$529$	−11667.3	−0.958926
$530$	0	0
$531$	4525.55	0.369854
$532$	0	0
$533$	9252.98	0.751953
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6798.79	−0.546349
$538$	0	0
$539$	−7031.61	−0.561916
$540$	0	0
$541$	−10814.3	−0.859412	−0.429706	−	0.902969i	$-0.641383\pi$
$541$	−10814.3	−0.859412	−0.429706	+	0.902969i	$0.641383\pi$
$542$	0	0
$543$	14166.0	1.11956
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9580.54	0.748875	0.374437	−	0.927252i	$-0.377836\pi$
$547$	9580.54	0.748875	0.374437	+	0.927252i	$0.377836\pi$
$548$	0	0
$549$	8440.27	0.656142
$550$	0	0
$551$	−5556.07	−0.429577
$552$	0	0
$553$	−8631.64	−0.663752
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17896.0	1.36136	0.680679	−	0.732582i	$-0.261685\pi$
$557$	17896.0	1.36136	0.680679	+	0.732582i	$0.261685\pi$
$558$	0	0
$559$	5915.24	0.447563
$560$	0	0
$561$	6063.88	0.456359
$562$	0	0
$563$	−20954.2	−1.56858	−0.784292	−	0.620392i	$-0.786974\pi$
$563$	−20954.2	−1.56858	−0.784292	+	0.620392i	$0.786974\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2134.75	−0.158115
$568$	0	0
$569$	−6614.54	−0.487339	−0.243670	−	0.969858i	$-0.578351\pi$
$569$	−6614.54	−0.487339	−0.243670	+	0.969858i	$0.578351\pi$
$570$	0	0
$571$	7293.53	0.534544	0.267272	−	0.963621i	$-0.413878\pi$
$571$	7293.53	0.534544	0.267272	+	0.963621i	$0.413878\pi$
$572$	0	0
$573$	999.084	0.0728400
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14724.7	−1.06239	−0.531195	−	0.847250i	$-0.678257\pi$
$577$	−14724.7	−1.06239	−0.531195	+	0.847250i	$0.678257\pi$
$578$	0	0
$579$	13509.7	0.969679
$580$	0	0
$581$	−12483.7	−0.891416
$582$	0	0
$583$	8642.59	0.613961
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3976.96	0.279637	0.139818	−	0.990177i	$-0.455348\pi$
$587$	3976.96	0.279637	0.139818	+	0.990177i	$0.455348\pi$
$588$	0	0
$589$	8691.77	0.608044
$590$	0	0
$591$	9975.09	0.694282
$592$	0	0
$593$	21972.5	1.52159	0.760795	−	0.648993i	$-0.224809\pi$
$593$	21972.5	1.52159	0.760795	+	0.648993i	$0.224809\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9705.97	0.665392
$598$	0	0
$599$	13515.5	0.921916	0.460958	−	0.887422i	$-0.347506\pi$
$599$	13515.5	0.921916	0.460958	+	0.887422i	$0.347506\pi$
$600$	0	0
$601$	−13789.2	−0.935899	−0.467949	−	0.883755i	$-0.655007\pi$
$601$	−13789.2	−0.935899	−0.467949	+	0.883755i	$0.655007\pi$
$602$	0	0
$603$	−4347.88	−0.293631
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29097.6	−1.94569	−0.972845	−	0.231459i	$-0.925650\pi$
$607$	−29097.6	−1.94569	−0.972845	+	0.231459i	$0.925650\pi$
$608$	0	0
$609$	−14818.3	−0.985988
$610$	0	0
$611$	−21735.7	−1.43917
$612$	0	0
$613$	−6066.37	−0.399703	−0.199852	−	0.979826i	$-0.564046\pi$
$613$	−6066.37	−0.399703	−0.199852	+	0.979826i	$0.564046\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17963.1	1.17207	0.586037	−	0.810285i	$-0.300687\pi$
$617$	17963.1	1.17207	0.586037	+	0.810285i	$0.300687\pi$
$618$	0	0
$619$	11980.6	0.777935	0.388968	−	0.921251i	$-0.372832\pi$
$619$	11980.6	0.777935	0.388968	+	0.921251i	$0.372832\pi$
$620$	0	0
$621$	603.582	0.0390031
$622$	0	0
$623$	11507.8	0.740047
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1778.71	0.113293
$628$	0	0
$629$	−179.336	−0.0113682
$630$	0	0
$631$	−25246.2	−1.59276	−0.796382	−	0.604794i	$-0.793256\pi$
$631$	−25246.2	−1.59276	−0.796382	+	0.604794i	$0.793256\pi$
$632$	0	0
$633$	−9545.13	−0.599344
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14188.0	0.882494
$638$	0	0
$639$	3574.45	0.221288
$640$	0	0
$641$	3234.97	0.199335	0.0996674	−	0.995021i	$-0.468222\pi$
$641$	3234.97	0.199335	0.0996674	+	0.995021i	$0.468222\pi$
$642$	0	0
$643$	6664.27	0.408730	0.204365	−	0.978895i	$-0.434487\pi$
$643$	6664.27	0.408730	0.204365	+	0.978895i	$0.434487\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5008.94	0.304361	0.152181	−	0.988353i	$-0.451370\pi$
$647$	5008.94	0.304361	0.152181	+	0.988353i	$0.451370\pi$
$648$	0	0
$649$	−10056.8	−0.608264
$650$	0	0
$651$	23181.3	1.39562
$652$	0	0
$653$	4808.71	0.288176	0.144088	−	0.989565i	$-0.453975\pi$
$653$	4808.71	0.288176	0.144088	+	0.989565i	$0.453975\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	507.493	0.0301357
$658$	0	0
$659$	−23465.7	−1.38709	−0.693546	−	0.720413i	$-0.743952\pi$
$659$	−23465.7	−1.38709	−0.693546	+	0.720413i	$0.743952\pi$
$660$	0	0
$661$	5006.86	0.294621	0.147310	−	0.989090i	$-0.452938\pi$
$661$	5006.86	0.294621	0.147310	+	0.989090i	$0.452938\pi$
$662$	0	0
$663$	−12235.4	−0.716715
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4189.74	0.243220
$668$	0	0
$669$	5544.10	0.320400
$670$	0	0
$671$	−18756.2	−1.07910
$672$	0	0
$673$	28544.8	1.63495	0.817475	−	0.575964i	$-0.195373\pi$
$673$	28544.8	1.63495	0.817475	+	0.575964i	$0.195373\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−275.964	−0.0156664	−0.00783321	−	0.999969i	$-0.502493\pi$
$677$	−275.964	−0.0156664	−0.00783321	+	0.999969i	$0.502493\pi$
$678$	0	0
$679$	37684.2	2.12988
$680$	0	0
$681$	−1777.93	−0.100045
$682$	0	0
$683$	32423.7	1.81649	0.908243	−	0.418444i	$-0.137424\pi$
$683$	32423.7	1.81649	0.908243	+	0.418444i	$0.137424\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1206.18	0.0669848
$688$	0	0
$689$	−17438.5	−0.964231
$690$	0	0
$691$	−23410.8	−1.28884	−0.644419	−	0.764672i	$-0.722901\pi$
$691$	−23410.8	−1.28884	−0.644419	+	0.764672i	$0.722901\pi$
$692$	0	0
$693$	4743.88	0.260036
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23173.1	−1.25932
$698$	0	0
$699$	−17884.6	−0.967749
$700$	0	0
$701$	−3314.51	−0.178584	−0.0892920	−	0.996005i	$-0.528460\pi$
$701$	−3314.51	−0.178584	−0.0892920	+	0.996005i	$0.528460\pi$
$702$	0	0
$703$	−52.6043	−0.00282220
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4007.60	−0.213185
$708$	0	0
$709$	−3593.16	−0.190330	−0.0951651	−	0.995462i	$-0.530338\pi$
$709$	−3593.16	−0.190330	−0.0951651	+	0.995462i	$0.530338\pi$
$710$	0	0
$711$	2947.64	0.155478
$712$	0	0
$713$	−6554.32	−0.344265
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7167.68	0.373336
$718$	0	0
$719$	9962.26	0.516731	0.258365	−	0.966047i	$-0.416816\pi$
$719$	9962.26	0.516731	0.258365	+	0.966047i	$0.416816\pi$
$720$	0	0
$721$	−4229.55	−0.218469
$722$	0	0
$723$	1824.99	0.0938754
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35638.3	−1.81809	−0.909045	−	0.416698i	$-0.863187\pi$
$727$	−35638.3	−1.81809	−0.909045	+	0.416698i	$0.863187\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−14814.1	−0.749548
$732$	0	0
$733$	22413.2	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$733$	22413.2	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9661.96	0.482907
$738$	0	0
$739$	1068.64	0.0531941	0.0265971	−	0.999646i	$-0.491533\pi$
$739$	1068.64	0.0531941	0.0265971	+	0.999646i	$0.491533\pi$
$740$	0	0
$741$	−3588.98	−0.177928
$742$	0	0
$743$	9514.38	0.469783	0.234891	−	0.972022i	$-0.424527\pi$
$743$	9514.38	0.469783	0.234891	+	0.972022i	$0.424527\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4263.10	0.208807
$748$	0	0
$749$	−31274.0	−1.52567
$750$	0	0
$751$	−32245.8	−1.56680	−0.783398	−	0.621520i	$-0.786515\pi$
$751$	−32245.8	−1.56680	−0.783398	+	0.621520i	$0.786515\pi$
$752$	0	0
$753$	20556.2	0.994833
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13864.8	0.665688	0.332844	−	0.942982i	$-0.391992\pi$
$757$	13864.8	0.665688	0.332844	+	0.942982i	$0.391992\pi$
$758$	0	0
$759$	−1341.29	−0.0641448
$760$	0	0
$761$	7817.11	0.372365	0.186183	−	0.982515i	$-0.440388\pi$
$761$	7817.11	0.372365	0.186183	+	0.982515i	$0.440388\pi$
$762$	0	0
$763$	−38928.8	−1.84707
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20292.0	0.955283
$768$	0	0
$769$	10091.0	0.473202	0.236601	−	0.971607i	$-0.423967\pi$
$769$	10091.0	0.473202	0.236601	+	0.971607i	$0.423967\pi$
$770$	0	0
$771$	12000.7	0.560563
$772$	0	0
$773$	−11257.8	−0.523825	−0.261912	−	0.965092i	$-0.584353\pi$
$773$	−11257.8	−0.523825	−0.261912	+	0.965092i	$0.584353\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−140.298	−0.00647768
$778$	0	0
$779$	−6797.33	−0.312631
$780$	0	0
$781$	−7943.22	−0.363932
$782$	0	0
$783$	5060.33	0.230960
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−37864.2	−1.71501	−0.857505	−	0.514475i	$-0.827987\pi$
$787$	−37864.2	−1.71501	−0.857505	+	0.514475i	$0.827987\pi$
$788$	0	0
$789$	−10764.1	−0.485693
$790$	0	0
$791$	36510.1	1.64115
$792$	0	0
$793$	37845.1	1.69473
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33482.9	−1.48811	−0.744056	−	0.668117i	$-0.767100\pi$
$797$	−33482.9	−1.48811	−0.744056	+	0.668117i	$0.767100\pi$
$798$	0	0
$799$	54434.8	2.41022
$800$	0	0
$801$	−3929.82	−0.173350
$802$	0	0
$803$	−1127.76	−0.0495615
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10199.1	−0.444887
$808$	0	0
$809$	40425.0	1.75682	0.878410	−	0.477909i	$-0.158605\pi$
$809$	40425.0	1.75682	0.878410	+	0.477909i	$0.158605\pi$
$810$	0	0
$811$	−2720.26	−0.117782	−0.0588910	−	0.998264i	$-0.518756\pi$
$811$	−2720.26	−0.117782	−0.0588910	+	0.998264i	$0.518756\pi$
$812$	0	0
$813$	−23263.5	−1.00355
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4345.39	−0.186078
$818$	0	0
$819$	−9571.94	−0.408389
$820$	0	0
$821$	−38982.6	−1.65713	−0.828563	−	0.559896i	$-0.810841\pi$
$821$	−38982.6	−1.65713	−0.828563	+	0.559896i	$0.810841\pi$
$822$	0	0
$823$	−15194.2	−0.643545	−0.321772	−	0.946817i	$-0.604279\pi$
$823$	−15194.2	−0.643545	−0.321772	+	0.946817i	$0.604279\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5073.07	−0.213311	−0.106655	−	0.994296i	$-0.534014\pi$
$827$	−5073.07	−0.213311	−0.106655	+	0.994296i	$0.534014\pi$
$828$	0	0
$829$	−29064.4	−1.21767	−0.608834	−	0.793298i	$-0.708362\pi$
$829$	−29064.4	−1.21767	−0.608834	+	0.793298i	$0.708362\pi$
$830$	0	0
$831$	−6741.22	−0.281409
$832$	0	0
$833$	−35532.4	−1.47794
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7916.24	−0.326912
$838$	0	0
$839$	30858.6	1.26979	0.634897	−	0.772597i	$-0.281042\pi$
$839$	30858.6	1.26979	0.634897	+	0.772597i	$0.281042\pi$
$840$	0	0
$841$	10737.1	0.440243
$842$	0	0
$843$	−2157.30	−0.0881394
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24536.4	0.995373
$848$	0	0
$849$	−15675.5	−0.633665
$850$	0	0
$851$	39.6681	0.00159789
$852$	0	0
$853$	−15449.4	−0.620138	−0.310069	−	0.950714i	$-0.600352\pi$
$853$	−15449.4	−0.620138	−0.310069	+	0.950714i	$0.600352\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30036.3	−1.19722	−0.598612	−	0.801039i	$-0.704281\pi$
$857$	−30036.3	−1.19722	−0.598612	+	0.801039i	$0.704281\pi$
$858$	0	0
$859$	−23949.6	−0.951278	−0.475639	−	0.879640i	$-0.657783\pi$
$859$	−23949.6	−0.951278	−0.475639	+	0.879640i	$0.657783\pi$
$860$	0	0
$861$	−18128.8	−0.717568
$862$	0	0
$863$	13084.4	0.516105	0.258052	−	0.966131i	$-0.416919\pi$
$863$	13084.4	0.516105	0.258052	+	0.966131i	$0.416919\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15903.2	0.622954
$868$	0	0
$869$	−6550.31	−0.255701
$870$	0	0
$871$	−19495.4	−0.758410
$872$	0	0
$873$	−12868.9	−0.498906
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33296.4	−1.28203	−0.641015	−	0.767528i	$-0.721486\pi$
$877$	−33296.4	−1.28203	−0.641015	+	0.767528i	$0.721486\pi$
$878$	0	0
$879$	19226.4	0.737758
$880$	0	0
$881$	−27460.6	−1.05014	−0.525068	−	0.851060i	$-0.675960\pi$
$881$	−27460.6	−1.05014	−0.525068	+	0.851060i	$0.675960\pi$
$882$	0	0
$883$	29831.5	1.13693	0.568466	−	0.822707i	$-0.307537\pi$
$883$	29831.5	1.13693	0.568466	+	0.822707i	$0.307537\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3402.93	−0.128815	−0.0644077	−	0.997924i	$-0.520516\pi$
$887$	−3402.93	−0.128815	−0.0644077	+	0.997924i	$0.520516\pi$
$888$	0	0
$889$	−23813.8	−0.898413
$890$	0	0
$891$	−1620.00	−0.0609114
$892$	0	0
$893$	15967.3	0.598347
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2706.39	0.100740
$898$	0	0
$899$	−54950.3	−2.03859
$900$	0	0
$901$	43673.0	1.61483
$902$	0	0
$903$	−11589.3	−0.427097
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−10442.8	−0.382300	−0.191150	−	0.981561i	$-0.561222\pi$
$907$	−10442.8	−0.382300	−0.191150	+	0.981561i	$0.561222\pi$
$908$	0	0
$909$	1368.57	0.0499367
$910$	0	0
$911$	43858.4	1.59505	0.797527	−	0.603283i	$-0.206141\pi$
$911$	43858.4	1.59505	0.797527	+	0.603283i	$0.206141\pi$
$912$	0	0
$913$	−9473.57	−0.343406
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−29005.8	−1.04455
$918$	0	0
$919$	−19120.3	−0.686313	−0.343156	−	0.939278i	$-0.611496\pi$
$919$	−19120.3	−0.686313	−0.343156	+	0.939278i	$0.611496\pi$
$920$	0	0
$921$	−13064.7	−0.467424
$922$	0	0
$923$	16027.4	0.571558
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1444.36	0.0511747
$928$	0	0
$929$	−39572.8	−1.39757	−0.698784	−	0.715333i	$-0.746275\pi$
$929$	−39572.8	−1.39757	−0.698784	+	0.715333i	$0.746275\pi$
$930$	0	0
$931$	−10422.6	−0.366905
$932$	0	0
$933$	−22547.5	−0.791180
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16251.0	−0.566593	−0.283296	−	0.959032i	$-0.591428\pi$
$937$	−16251.0	−0.566593	−0.283296	+	0.959032i	$0.591428\pi$
$938$	0	0
$939$	−25603.0	−0.889801
$940$	0	0
$941$	13456.6	0.466176	0.233088	−	0.972456i	$-0.425117\pi$
$941$	13456.6	0.466176	0.233088	+	0.972456i	$0.425117\pi$
$942$	0	0
$943$	5125.76	0.177007
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25628.9	0.879437	0.439718	−	0.898136i	$-0.355078\pi$
$947$	25628.9	0.879437	0.439718	+	0.898136i	$0.355078\pi$
$948$	0	0
$949$	2275.54	0.0778367
$950$	0	0
$951$	6782.70	0.231277
$952$	0	0
$953$	15570.2	0.529244	0.264622	−	0.964352i	$-0.414753\pi$
$953$	15570.2	0.529244	0.264622	+	0.964352i	$0.414753\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−11245.2	−0.379838
$958$	0	0
$959$	−48483.7	−1.63255
$960$	0	0
$961$	56171.7	1.88553
$962$	0	0
$963$	10679.8	0.357375
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20978.9	−0.697659	−0.348830	−	0.937186i	$-0.613421\pi$
$967$	−20978.9	−0.697659	−0.348830	+	0.937186i	$0.613421\pi$
$968$	0	0
$969$	8988.22	0.297981
$970$	0	0
$971$	53556.2	1.77003	0.885016	−	0.465561i	$-0.154148\pi$
$971$	53556.2	1.77003	0.885016	+	0.465561i	$0.154148\pi$
$972$	0	0
$973$	48990.4	1.61414
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10356.5	0.339135	0.169568	−	0.985519i	$-0.445763\pi$
$977$	10356.5	0.339135	0.169568	+	0.985519i	$0.445763\pi$
$978$	0	0
$979$	8732.94	0.285093
$980$	0	0
$981$	13293.9	0.432662
$982$	0	0
$983$	41887.6	1.35911	0.679556	−	0.733624i	$-0.262172\pi$
$983$	41887.6	1.35911	0.679556	+	0.733624i	$0.262172\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	42585.3	1.37336
$988$	0	0
$989$	3276.79	0.105355
$990$	0	0
$991$	−31978.8	−1.02507	−0.512533	−	0.858668i	$-0.671293\pi$
$991$	−31978.8	−1.02507	−0.512533	+	0.858668i	$0.671293\pi$
$992$	0	0
$993$	−3775.86	−0.120668
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−56184.5	−1.78474	−0.892368	−	0.451309i	$-0.850957\pi$
$997$	−56184.5	−1.78474	−0.892368	+	0.451309i	$0.850957\pi$
$998$	0	0
$999$	47.9107	0.00151734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.ba.1.1		2
4.3	odd	2		2400.4.a.z.1.2		2
5.4	even	2		480.4.a.n.1.2	✓	2
15.14	odd	2		1440.4.a.ba.1.2		2
20.19	odd	2		480.4.a.p.1.1	yes	2
40.19	odd	2		960.4.a.bl.1.1		2
40.29	even	2		960.4.a.bp.1.2		2
60.59	even	2		1440.4.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.n.1.2	✓	2	5.4	even	2
480.4.a.p.1.1	yes	2	20.19	odd	2
960.4.a.bl.1.1		2	40.19	odd	2
960.4.a.bp.1.2		2	40.29	even	2
1440.4.a.ba.1.2		2	15.14	odd	2
1440.4.a.bf.1.1		2	60.59	even	2
2400.4.a.z.1.2		2	4.3	odd	2
2400.4.a.ba.1.1		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -26.3549 q^{7} +9.00000 q^{9} -20.0000 q^{11} +40.3549 q^{13} -101.065 q^{17} -29.6451 q^{19} -79.0647 q^{21} +22.3549 q^{23} +27.0000 q^{27} +187.420 q^{29} -293.194 q^{31} -60.0000 q^{33} +1.77447 q^{37} +121.065 q^{39} +229.290 q^{41} +146.580 q^{43} -538.614 q^{47} +351.580 q^{49} -303.194 q^{51} -432.129 q^{53} -88.9353 q^{57} +502.839 q^{59} +937.808 q^{61} -237.194 q^{63} -483.098 q^{67} +67.0647 q^{69} +397.161 q^{71} +56.3881 q^{73} +527.098 q^{77} +327.516 q^{79} +81.0000 q^{81} +473.678 q^{83} +562.259 q^{87} -436.647 q^{89} -1063.55 q^{91} -879.582 q^{93} -1429.87 q^{97} -180.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 4 q^{7} + 18 q^{9} - 40 q^{11} + 24 q^{13} - 32 q^{17} - 116 q^{19} + 12 q^{21} - 12 q^{23} + 54 q^{27} + 148 q^{29} - 76 q^{31} - 120 q^{33} - 280 q^{37} + 72 q^{39} + 572 q^{41} + 520 q^{43}+ \cdots - 360 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 4 * q^7 + 18 * q^9 - 40 * q^11 + 24 * q^13 - 32 * q^17 - 116 * q^19 + 12 * q^21 - 12 * q^23 + 54 * q^27 + 148 * q^29 - 76 * q^31 - 120 * q^33 - 280 * q^37 + 72 * q^39 + 572 * q^41 + 520 * q^43 - 340 * q^47 + 930 * q^49 - 96 * q^51 - 524 * q^53 - 348 * q^57 + 552 * q^59 + 628 * q^61 + 36 * q^63 + 168 * q^67 - 36 * q^69 + 1248 * q^71 - 908 * q^73 - 80 * q^77 + 1052 * q^79 + 162 * q^81 + 40 * q^83 + 444 * q^87 + 828 * q^89 - 1560 * q^91 - 228 * q^93 + 316 * q^97 - 360 * q^99

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