LMFDB - Embedded newform 784.4.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 13 $ x^3 - x^2 - 10*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.90222$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.65024 q^{3} +14.6089 q^{5} +31.5262 q^{9} +O(q^{10})$	$q+7.65024 q^{3} +14.6089 q^{5} +31.5262 q^{9} +17.3418 q^{11} +81.5103 q^{13} +111.762 q^{15} -78.3291 q^{17} +11.1096 q^{19} +133.707 q^{23} +88.4197 q^{25} +34.6265 q^{27} -93.8411 q^{29} -315.431 q^{31} +132.669 q^{33} +179.366 q^{37} +623.574 q^{39} +265.555 q^{41} -6.73466 q^{43} +460.563 q^{45} -172.085 q^{47} -599.236 q^{51} +375.995 q^{53} +253.345 q^{55} +84.9910 q^{57} -113.807 q^{59} -359.527 q^{61} +1190.78 q^{65} -428.597 q^{67} +1022.89 q^{69} -729.780 q^{71} +1157.94 q^{73} +676.432 q^{75} -447.654 q^{79} -586.306 q^{81} +10.3976 q^{83} -1144.30 q^{85} -717.907 q^{87} +766.052 q^{89} -2413.12 q^{93} +162.299 q^{95} +532.824 q^{97} +546.722 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 13 q^{5} + 50 q^{9}+O(q^{10}) $ 3 * q - q^3 + 13 * q^5 + 50 * q^9	$ 3 q - q^{3} + 13 q^{5} + 50 q^{9} + 11 q^{11} + 70 q^{13} + 41 q^{15} + 97 q^{17} + 81 q^{19} - 191 q^{23} + 12 q^{25} - 115 q^{27} - 162 q^{29} - 597 q^{31} + 343 q^{33} + 217 q^{37} + 806 q^{39} + 698 q^{41} + 308 q^{43} + 1118 q^{45} - 139 q^{47} - 1475 q^{51} + 197 q^{53} - 147 q^{55} - 1147 q^{57} - 353 q^{59} + 449 q^{61} + 906 q^{65} + 519 q^{67} + 2557 q^{69} - 224 q^{71} + 1701 q^{73} + 1132 q^{75} - 1143 q^{79} + 251 q^{81} + 1380 q^{83} - 1025 q^{85} - 1458 q^{87} + 1749 q^{89} - 601 q^{93} + 2167 q^{95} + 2602 q^{97} - 1094 q^{99}+O(q^{100}) $ 3 * q - q^3 + 13 * q^5 + 50 * q^9 + 11 * q^11 + 70 * q^13 + 41 * q^15 + 97 * q^17 + 81 * q^19 - 191 * q^23 + 12 * q^25 - 115 * q^27 - 162 * q^29 - 597 * q^31 + 343 * q^33 + 217 * q^37 + 806 * q^39 + 698 * q^41 + 308 * q^43 + 1118 * q^45 - 139 * q^47 - 1475 * q^51 + 197 * q^53 - 147 * q^55 - 1147 * q^57 - 353 * q^59 + 449 * q^61 + 906 * q^65 + 519 * q^67 + 2557 * q^69 - 224 * q^71 + 1701 * q^73 + 1132 * q^75 - 1143 * q^79 + 251 * q^81 + 1380 * q^83 - 1025 * q^85 - 1458 * q^87 + 1749 * q^89 - 601 * q^93 + 2167 * q^95 + 2602 * q^97 - 1094 * 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
$3$	7.65024	1.47229	0.736145	−	0.676824i	$-0.236644\pi$
$3$	7.65024	1.47229	0.736145	+	0.676824i	$0.236644\pi$
$4$	0	0
$5$	14.6089	1.30666	0.653329	−	0.757074i	$-0.273372\pi$
$5$	14.6089	1.30666	0.653329	+	0.757074i	$0.273372\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	31.5262	1.16764
$10$	0	0
$11$	17.3418	0.475342	0.237671	−	0.971346i	$-0.423616\pi$
$11$	17.3418	0.475342	0.237671	+	0.971346i	$0.423616\pi$
$12$	0	0
$13$	81.5103	1.73899	0.869496	−	0.493940i	$-0.164444\pi$
$13$	81.5103	1.73899	0.869496	+	0.493940i	$0.164444\pi$
$14$	0	0
$15$	111.762	1.92378
$16$	0	0
$17$	−78.3291	−1.11751	−0.558753	−	0.829334i	$-0.688720\pi$
$17$	−78.3291	−1.11751	−0.558753	+	0.829334i	$0.688720\pi$
$18$	0	0
$19$	11.1096	0.134143	0.0670714	−	0.997748i	$-0.478634\pi$
$19$	11.1096	0.134143	0.0670714	+	0.997748i	$0.478634\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	133.707	1.21217	0.606085	−	0.795400i	$-0.292739\pi$
$23$	133.707	1.21217	0.606085	+	0.795400i	$0.292739\pi$
$24$	0	0
$25$	88.4197	0.707358
$26$	0	0
$27$	34.6265	0.246810
$28$	0	0
$29$	−93.8411	−0.600892	−0.300446	−	0.953799i	$-0.597135\pi$
$29$	−93.8411	−0.600892	−0.300446	+	0.953799i	$0.597135\pi$
$30$	0	0
$31$	−315.431	−1.82752	−0.913759	−	0.406258i	$-0.866834\pi$
$31$	−315.431	−1.82752	−0.913759	+	0.406258i	$0.866834\pi$
$32$	0	0
$33$	132.669	0.699841
$34$	0	0
$35$	0	0
$36$	0	0
$37$	179.366	0.796962	0.398481	−	0.917177i	$-0.369538\pi$
$37$	179.366	0.796962	0.398481	+	0.917177i	$0.369538\pi$
$38$	0	0
$39$	623.574	2.56030
$40$	0	0
$41$	265.555	1.01153	0.505765	−	0.862671i	$-0.331210\pi$
$41$	265.555	1.01153	0.505765	+	0.862671i	$0.331210\pi$
$42$	0	0
$43$	−6.73466	−0.0238843	−0.0119422	−	0.999929i	$-0.503801\pi$
$43$	−6.73466	−0.0238843	−0.0119422	+	0.999929i	$0.503801\pi$
$44$	0	0
$45$	460.563	1.52570
$46$	0	0
$47$	−172.085	−0.534069	−0.267035	−	0.963687i	$-0.586044\pi$
$47$	−172.085	−0.534069	−0.267035	+	0.963687i	$0.586044\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−599.236	−1.64529
$52$	0	0
$53$	375.995	0.974470	0.487235	−	0.873271i	$-0.338006\pi$
$53$	375.995	0.974470	0.487235	+	0.873271i	$0.338006\pi$
$54$	0	0
$55$	253.345	0.621110
$56$	0	0
$57$	84.9910	0.197497
$58$	0	0
$59$	−113.807	−0.251125	−0.125562	−	0.992086i	$-0.540074\pi$
$59$	−113.807	−0.251125	−0.125562	+	0.992086i	$0.540074\pi$
$60$	0	0
$61$	−359.527	−0.754635	−0.377318	−	0.926084i	$-0.623153\pi$
$61$	−359.527	−0.754635	−0.377318	+	0.926084i	$0.623153\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1190.78	2.27227
$66$	0	0
$67$	−428.597	−0.781515	−0.390757	−	0.920494i	$-0.627787\pi$
$67$	−428.597	−0.781515	−0.390757	+	0.920494i	$0.627787\pi$
$68$	0	0
$69$	1022.89	1.78467
$70$	0	0
$71$	−729.780	−1.21984	−0.609922	−	0.792461i	$-0.708799\pi$
$71$	−729.780	−1.21984	−0.609922	+	0.792461i	$0.708799\pi$
$72$	0	0
$73$	1157.94	1.85653	0.928263	−	0.371925i	$-0.121302\pi$
$73$	1157.94	1.85653	0.928263	+	0.371925i	$0.121302\pi$
$74$	0	0
$75$	676.432	1.04144
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−447.654	−0.637532	−0.318766	−	0.947833i	$-0.603268\pi$
$79$	−447.654	−0.637532	−0.318766	+	0.947833i	$0.603268\pi$
$80$	0	0
$81$	−586.306	−0.804261
$82$	0	0
$83$	10.3976	0.0137505	0.00687523	−	0.999976i	$-0.497812\pi$
$83$	10.3976	0.0137505	0.00687523	+	0.999976i	$0.497812\pi$
$84$	0	0
$85$	−1144.30	−1.46020
$86$	0	0
$87$	−717.907	−0.884687
$88$	0	0
$89$	766.052	0.912374	0.456187	−	0.889884i	$-0.349215\pi$
$89$	766.052	0.912374	0.456187	+	0.889884i	$0.349215\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2413.12	−2.69063
$94$	0	0
$95$	162.299	0.175279
$96$	0	0
$97$	532.824	0.557733	0.278866	−	0.960330i	$-0.410041\pi$
$97$	532.824	0.557733	0.278866	+	0.960330i	$0.410041\pi$
$98$	0	0
$99$	546.722	0.555026
$100$	0	0
$101$	−1325.47	−1.30584	−0.652919	−	0.757428i	$-0.726456\pi$
$101$	−1325.47	−1.30584	−0.652919	+	0.757428i	$0.726456\pi$
$102$	0	0
$103$	−297.364	−0.284468	−0.142234	−	0.989833i	$-0.545428\pi$
$103$	−297.364	−0.284468	−0.142234	+	0.989833i	$0.545428\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−827.752	−0.747867	−0.373934	−	0.927455i	$-0.621991\pi$
$107$	−827.752	−0.747867	−0.373934	+	0.927455i	$0.621991\pi$
$108$	0	0
$109$	−89.5118	−0.0786575	−0.0393288	−	0.999226i	$-0.512522\pi$
$109$	−89.5118	−0.0786575	−0.0393288	+	0.999226i	$0.512522\pi$
$110$	0	0
$111$	1372.19	1.17336
$112$	0	0
$113$	267.310	0.222535	0.111267	−	0.993790i	$-0.464509\pi$
$113$	267.310	0.222535	0.111267	+	0.993790i	$0.464509\pi$
$114$	0	0
$115$	1953.32	1.58389
$116$	0	0
$117$	2569.71	2.03051
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1030.26	−0.774050
$122$	0	0
$123$	2031.56	1.48927
$124$	0	0
$125$	−534.397	−0.382384
$126$	0	0
$127$	2133.98	1.49102	0.745511	−	0.666493i	$-0.232206\pi$
$127$	2133.98	1.49102	0.745511	+	0.666493i	$0.232206\pi$
$128$	0	0
$129$	−51.5218	−0.0351646
$130$	0	0
$131$	1439.28	0.959924	0.479962	−	0.877289i	$-0.340650\pi$
$131$	1439.28	0.959924	0.479962	+	0.877289i	$0.340650\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	505.854	0.322496
$136$	0	0
$137$	2281.68	1.42290	0.711448	−	0.702739i	$-0.248040\pi$
$137$	2281.68	1.42290	0.711448	+	0.702739i	$0.248040\pi$
$138$	0	0
$139$	−1494.99	−0.912253	−0.456127	−	0.889915i	$-0.650764\pi$
$139$	−1494.99	−0.912253	−0.456127	+	0.889915i	$0.650764\pi$
$140$	0	0
$141$	−1316.50	−0.786304
$142$	0	0
$143$	1413.54	0.826616
$144$	0	0
$145$	−1370.92	−0.785161
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1281.00	−0.704318	−0.352159	−	0.935940i	$-0.614552\pi$
$149$	−1281.00	−0.704318	−0.352159	+	0.935940i	$0.614552\pi$
$150$	0	0
$151$	−122.383	−0.0659563	−0.0329781	−	0.999456i	$-0.510499\pi$
$151$	−122.383	−0.0659563	−0.0329781	+	0.999456i	$0.510499\pi$
$152$	0	0
$153$	−2469.42	−1.30484
$154$	0	0
$155$	−4608.09	−2.38794
$156$	0	0
$157$	−919.453	−0.467391	−0.233695	−	0.972310i	$-0.575082\pi$
$157$	−919.453	−0.467391	−0.233695	+	0.972310i	$0.575082\pi$
$158$	0	0
$159$	2876.45	1.43470
$160$	0	0
$161$	0	0
$162$	0	0
$163$	283.688	0.136320	0.0681600	−	0.997674i	$-0.478287\pi$
$163$	283.688	0.136320	0.0681600	+	0.997674i	$0.478287\pi$
$164$	0	0
$165$	1938.15	0.914453
$166$	0	0
$167$	276.828	0.128273	0.0641365	−	0.997941i	$-0.479571\pi$
$167$	276.828	0.128273	0.0641365	+	0.997941i	$0.479571\pi$
$168$	0	0
$169$	4446.94	2.02409
$170$	0	0
$171$	350.243	0.156630
$172$	0	0
$173$	1016.80	0.446856	0.223428	−	0.974720i	$-0.428275\pi$
$173$	1016.80	0.446856	0.223428	+	0.974720i	$0.428275\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−870.648	−0.369728
$178$	0	0
$179$	−2774.73	−1.15862	−0.579310	−	0.815107i	$-0.696678\pi$
$179$	−2774.73	−1.15862	−0.579310	+	0.815107i	$0.696678\pi$
$180$	0	0
$181$	3632.00	1.49152	0.745758	−	0.666217i	$-0.232088\pi$
$181$	3632.00	1.49152	0.745758	+	0.666217i	$0.232088\pi$
$182$	0	0
$183$	−2750.47	−1.11104
$184$	0	0
$185$	2620.34	1.04136
$186$	0	0
$187$	−1358.37	−0.531197
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2232.00	−0.845561	−0.422780	−	0.906232i	$-0.638946\pi$
$191$	−2232.00	−0.845561	−0.422780	+	0.906232i	$0.638946\pi$
$192$	0	0
$193$	2800.17	1.04436	0.522178	−	0.852836i	$-0.325119\pi$
$193$	2800.17	1.04436	0.522178	+	0.852836i	$0.325119\pi$
$194$	0	0
$195$	9109.72	3.34544
$196$	0	0
$197$	−3201.39	−1.15781	−0.578907	−	0.815394i	$-0.696520\pi$
$197$	−3201.39	−1.15781	−0.578907	+	0.815394i	$0.696520\pi$
$198$	0	0
$199$	704.719	0.251036	0.125518	−	0.992091i	$-0.459941\pi$
$199$	704.719	0.251036	0.125518	+	0.992091i	$0.459941\pi$
$200$	0	0
$201$	−3278.87	−1.15062
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3879.47	1.32173
$206$	0	0
$207$	4215.29	1.41538
$208$	0	0
$209$	192.661	0.0637637
$210$	0	0
$211$	−216.098	−0.0705063	−0.0352531	−	0.999378i	$-0.511224\pi$
$211$	−216.098	−0.0705063	−0.0352531	+	0.999378i	$0.511224\pi$
$212$	0	0
$213$	−5582.99	−1.79596
$214$	0	0
$215$	−98.3859	−0.0312087
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8858.51	2.73334
$220$	0	0
$221$	−6384.63	−1.94333
$222$	0	0
$223$	−1251.23	−0.375734	−0.187867	−	0.982194i	$-0.560157\pi$
$223$	−1251.23	−0.375734	−0.187867	+	0.982194i	$0.560157\pi$
$224$	0	0
$225$	2787.54	0.825937
$226$	0	0
$227$	1509.37	0.441324	0.220662	−	0.975350i	$-0.429178\pi$
$227$	1509.37	0.441324	0.220662	+	0.975350i	$0.429178\pi$
$228$	0	0
$229$	−392.301	−0.113205	−0.0566026	−	0.998397i	$-0.518027\pi$
$229$	−392.301	−0.113205	−0.0566026	+	0.998397i	$0.518027\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	501.547	0.141019	0.0705096	−	0.997511i	$-0.477537\pi$
$233$	501.547	0.141019	0.0705096	+	0.997511i	$0.477537\pi$
$234$	0	0
$235$	−2513.98	−0.697846
$236$	0	0
$237$	−3424.66	−0.938632
$238$	0	0
$239$	4654.95	1.25985	0.629924	−	0.776656i	$-0.283086\pi$
$239$	4654.95	1.25985	0.629924	+	0.776656i	$0.283086\pi$
$240$	0	0
$241$	−735.968	−0.196713	−0.0983565	−	0.995151i	$-0.531359\pi$
$241$	−735.968	−0.196713	−0.0983565	+	0.995151i	$0.531359\pi$
$242$	0	0
$243$	−5420.30	−1.43092
$244$	0	0
$245$	0	0
$246$	0	0
$247$	905.546	0.233273
$248$	0	0
$249$	79.5444	0.0202447
$250$	0	0
$251$	1277.98	0.321376	0.160688	−	0.987005i	$-0.448629\pi$
$251$	1277.98	0.321376	0.160688	+	0.987005i	$0.448629\pi$
$252$	0	0
$253$	2318.73	0.576195
$254$	0	0
$255$	−8754.18	−2.14984
$256$	0	0
$257$	−2161.98	−0.524749	−0.262374	−	0.964966i	$-0.584506\pi$
$257$	−2161.98	−0.524749	−0.262374	+	0.964966i	$0.584506\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2958.45	−0.701623
$262$	0	0
$263$	−4650.77	−1.09041	−0.545207	−	0.838301i	$-0.683549\pi$
$263$	−4650.77	−1.09041	−0.545207	+	0.838301i	$0.683549\pi$
$264$	0	0
$265$	5492.87	1.27330
$266$	0	0
$267$	5860.48	1.34328
$268$	0	0
$269$	−1760.04	−0.398927	−0.199464	−	0.979905i	$-0.563920\pi$
$269$	−1760.04	−0.398927	−0.199464	+	0.979905i	$0.563920\pi$
$270$	0	0
$271$	−5723.12	−1.28286	−0.641430	−	0.767182i	$-0.721659\pi$
$271$	−5723.12	−1.28286	−0.641430	+	0.767182i	$0.721659\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1533.36	0.336237
$276$	0	0
$277$	1660.01	0.360074	0.180037	−	0.983660i	$-0.442378\pi$
$277$	1660.01	0.360074	0.180037	+	0.983660i	$0.442378\pi$
$278$	0	0
$279$	−9944.33	−2.13388
$280$	0	0
$281$	−5340.36	−1.13373	−0.566867	−	0.823809i	$-0.691845\pi$
$281$	−5340.36	−1.13373	−0.566867	+	0.823809i	$0.691845\pi$
$282$	0	0
$283$	4637.13	0.974024	0.487012	−	0.873395i	$-0.338087\pi$
$283$	4637.13	0.974024	0.487012	+	0.873395i	$0.338087\pi$
$284$	0	0
$285$	1241.62	0.258061
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1222.45	0.248819
$290$	0	0
$291$	4076.23	0.821144
$292$	0	0
$293$	510.530	0.101793	0.0508967	−	0.998704i	$-0.483792\pi$
$293$	510.530	0.101793	0.0508967	+	0.998704i	$0.483792\pi$
$294$	0	0
$295$	−1662.59	−0.328134
$296$	0	0
$297$	600.486	0.117319
$298$	0	0
$299$	10898.5	2.10796
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10140.2	−1.92257
$304$	0	0
$305$	−5252.30	−0.986051
$306$	0	0
$307$	−4901.40	−0.911198	−0.455599	−	0.890185i	$-0.650575\pi$
$307$	−4901.40	−0.911198	−0.455599	+	0.890185i	$0.650575\pi$
$308$	0	0
$309$	−2274.91	−0.418819
$310$	0	0
$311$	−359.402	−0.0655300	−0.0327650	−	0.999463i	$-0.510431\pi$
$311$	−359.402	−0.0655300	−0.0327650	+	0.999463i	$0.510431\pi$
$312$	0	0
$313$	3866.99	0.698323	0.349161	−	0.937063i	$-0.386466\pi$
$313$	3866.99	0.698323	0.349161	+	0.937063i	$0.386466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5129.68	0.908869	0.454435	−	0.890780i	$-0.349841\pi$
$317$	5129.68	0.908869	0.454435	+	0.890780i	$0.349841\pi$
$318$	0	0
$319$	−1627.38	−0.285629
$320$	0	0
$321$	−6332.50	−1.10108
$322$	0	0
$323$	−870.204	−0.149905
$324$	0	0
$325$	7207.12	1.23009
$326$	0	0
$327$	−684.787	−0.115807
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4067.06	0.675365	0.337682	−	0.941260i	$-0.390357\pi$
$331$	4067.06	0.675365	0.337682	+	0.941260i	$0.390357\pi$
$332$	0	0
$333$	5654.73	0.930562
$334$	0	0
$335$	−6261.33	−1.02117
$336$	0	0
$337$	−7574.84	−1.22441	−0.612207	−	0.790697i	$-0.709718\pi$
$337$	−7574.84	−1.22441	−0.612207	+	0.790697i	$0.709718\pi$
$338$	0	0
$339$	2044.99	0.327636
$340$	0	0
$341$	−5470.15	−0.868695
$342$	0	0
$343$	0	0
$344$	0	0
$345$	14943.4	2.33195
$346$	0	0
$347$	7335.73	1.13488	0.567439	−	0.823416i	$-0.307934\pi$
$347$	7335.73	1.13488	0.567439	+	0.823416i	$0.307934\pi$
$348$	0	0
$349$	−6304.89	−0.967029	−0.483515	−	0.875336i	$-0.660640\pi$
$349$	−6304.89	−0.967029	−0.483515	+	0.875336i	$0.660640\pi$
$350$	0	0
$351$	2822.41	0.429200
$352$	0	0
$353$	4742.76	0.715104	0.357552	−	0.933893i	$-0.383612\pi$
$353$	4742.76	0.715104	0.357552	+	0.933893i	$0.383612\pi$
$354$	0	0
$355$	−10661.3	−1.59392
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10063.1	−1.47942	−0.739708	−	0.672928i	$-0.765037\pi$
$359$	−10063.1	−1.47942	−0.739708	+	0.672928i	$0.765037\pi$
$360$	0	0
$361$	−6735.58	−0.982006
$362$	0	0
$363$	−7881.74	−1.13963
$364$	0	0
$365$	16916.2	2.42585
$366$	0	0
$367$	−12455.7	−1.77161	−0.885805	−	0.464058i	$-0.846393\pi$
$367$	−12455.7	−1.77161	−0.885805	+	0.464058i	$0.846393\pi$
$368$	0	0
$369$	8371.94	1.18110
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6385.33	−0.886380	−0.443190	−	0.896428i	$-0.646153\pi$
$373$	−6385.33	−0.886380	−0.443190	+	0.896428i	$0.646153\pi$
$374$	0	0
$375$	−4088.27	−0.562980
$376$	0	0
$377$	−7649.02	−1.04495
$378$	0	0
$379$	109.028	0.0147767	0.00738835	−	0.999973i	$-0.497648\pi$
$379$	109.028	0.0147767	0.00738835	+	0.999973i	$0.497648\pi$
$380$	0	0
$381$	16325.4	2.19522
$382$	0	0
$383$	−5607.32	−0.748095	−0.374048	−	0.927409i	$-0.622030\pi$
$383$	−5607.32	−0.748095	−0.374048	+	0.927409i	$0.622030\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−212.318	−0.0278882
$388$	0	0
$389$	−14513.8	−1.89172	−0.945862	−	0.324568i	$-0.894781\pi$
$389$	−14513.8	−1.89172	−0.945862	+	0.324568i	$0.894781\pi$
$390$	0	0
$391$	−10473.2	−1.35461
$392$	0	0
$393$	11010.8	1.41329
$394$	0	0
$395$	−6539.73	−0.833037
$396$	0	0
$397$	−10812.7	−1.36694	−0.683471	−	0.729978i	$-0.739530\pi$
$397$	−10812.7	−1.36694	−0.683471	+	0.729978i	$0.739530\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15738.4	−1.95994	−0.979972	−	0.199136i	$-0.936187\pi$
$401$	−15738.4	−1.95994	−0.979972	+	0.199136i	$0.936187\pi$
$402$	0	0
$403$	−25710.9	−3.17804
$404$	0	0
$405$	−8565.29	−1.05090
$406$	0	0
$407$	3110.53	0.378829
$408$	0	0
$409$	−6094.20	−0.736769	−0.368385	−	0.929673i	$-0.620089\pi$
$409$	−6094.20	−0.736769	−0.368385	+	0.929673i	$0.620089\pi$
$410$	0	0
$411$	17455.4	2.09491
$412$	0	0
$413$	0	0
$414$	0	0
$415$	151.898	0.0179672
$416$	0	0
$417$	−11437.0	−1.34310
$418$	0	0
$419$	15324.8	1.78679	0.893395	−	0.449272i	$-0.148317\pi$
$419$	15324.8	1.78679	0.893395	+	0.449272i	$0.148317\pi$
$420$	0	0
$421$	−12916.8	−1.49531	−0.747655	−	0.664087i	$-0.768820\pi$
$421$	−12916.8	−1.49531	−0.747655	+	0.664087i	$0.768820\pi$
$422$	0	0
$423$	−5425.20	−0.623599
$424$	0	0
$425$	−6925.84	−0.790476
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10813.9	1.21702
$430$	0	0
$431$	6287.15	0.702648	0.351324	−	0.936254i	$-0.385732\pi$
$431$	6287.15	0.702648	0.351324	+	0.936254i	$0.385732\pi$
$432$	0	0
$433$	5218.91	0.579226	0.289613	−	0.957144i	$-0.406473\pi$
$433$	5218.91	0.579226	0.289613	+	0.957144i	$0.406473\pi$
$434$	0	0
$435$	−10487.8	−1.15598
$436$	0	0
$437$	1485.43	0.162604
$438$	0	0
$439$	−3011.33	−0.327387	−0.163693	−	0.986511i	$-0.552341\pi$
$439$	−3011.33	−0.327387	−0.163693	+	0.986511i	$0.552341\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2201.75	−0.236136	−0.118068	−	0.993006i	$-0.537670\pi$
$443$	−2201.75	−0.236136	−0.118068	+	0.993006i	$0.537670\pi$
$444$	0	0
$445$	11191.2	1.19216
$446$	0	0
$447$	−9799.94	−1.03696
$448$	0	0
$449$	−10317.0	−1.08439	−0.542193	−	0.840254i	$-0.682406\pi$
$449$	−10317.0	−1.08439	−0.542193	+	0.840254i	$0.682406\pi$
$450$	0	0
$451$	4605.21	0.480823
$452$	0	0
$453$	−936.260	−0.0971067
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5279.44	−0.540398	−0.270199	−	0.962805i	$-0.587089\pi$
$457$	−5279.44	−0.540398	−0.270199	+	0.962805i	$0.587089\pi$
$458$	0	0
$459$	−2712.26	−0.275811
$460$	0	0
$461$	15855.0	1.60182	0.800912	−	0.598782i	$-0.204348\pi$
$461$	15855.0	1.60182	0.800912	+	0.598782i	$0.204348\pi$
$462$	0	0
$463$	8167.07	0.819775	0.409888	−	0.912136i	$-0.365568\pi$
$463$	8167.07	0.819775	0.409888	+	0.912136i	$0.365568\pi$
$464$	0	0
$465$	−35253.0	−3.51574
$466$	0	0
$467$	−16272.3	−1.61240	−0.806202	−	0.591641i	$-0.798480\pi$
$467$	−16272.3	−1.61240	−0.806202	+	0.591641i	$0.798480\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−7034.04	−0.688134
$472$	0	0
$473$	−116.791	−0.0113532
$474$	0	0
$475$	982.306	0.0948870
$476$	0	0
$477$	11853.7	1.13783
$478$	0	0
$479$	8383.38	0.799680	0.399840	−	0.916585i	$-0.369066\pi$
$479$	8383.38	0.799680	0.399840	+	0.916585i	$0.369066\pi$
$480$	0	0
$481$	14620.2	1.38591
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7783.97	0.728766
$486$	0	0
$487$	377.094	0.0350878	0.0175439	−	0.999846i	$-0.494415\pi$
$487$	377.094	0.0350878	0.0175439	+	0.999846i	$0.494415\pi$
$488$	0	0
$489$	2170.28	0.200703
$490$	0	0
$491$	1658.74	0.152460	0.0762299	−	0.997090i	$-0.475712\pi$
$491$	1658.74	0.152460	0.0762299	+	0.997090i	$0.475712\pi$
$492$	0	0
$493$	7350.49	0.671500
$494$	0	0
$495$	7987.00	0.725230
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16054.7	1.44029	0.720146	−	0.693822i	$-0.244075\pi$
$499$	16054.7	1.44029	0.720146	+	0.693822i	$0.244075\pi$
$500$	0	0
$501$	2117.80	0.188855
$502$	0	0
$503$	20948.9	1.85699	0.928494	−	0.371348i	$-0.121104\pi$
$503$	20948.9	1.85699	0.928494	+	0.371348i	$0.121104\pi$
$504$	0	0
$505$	−19363.7	−1.70628
$506$	0	0
$507$	34020.1	2.98005
$508$	0	0
$509$	−2808.07	−0.244530	−0.122265	−	0.992497i	$-0.539016\pi$
$509$	−2808.07	−0.244530	−0.122265	+	0.992497i	$0.539016\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	384.686	0.0331078
$514$	0	0
$515$	−4344.16	−0.371702
$516$	0	0
$517$	−2984.28	−0.253865
$518$	0	0
$519$	7778.79	0.657902
$520$	0	0
$521$	2416.08	0.203168	0.101584	−	0.994827i	$-0.467609\pi$
$521$	2416.08	0.203168	0.101584	+	0.994827i	$0.467609\pi$
$522$	0	0
$523$	−18866.2	−1.57736	−0.788680	−	0.614804i	$-0.789235\pi$
$523$	−18866.2	−1.57736	−0.788680	+	0.614804i	$0.789235\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24707.4	2.04226
$528$	0	0
$529$	5710.68	0.469358
$530$	0	0
$531$	−3587.89	−0.293223
$532$	0	0
$533$	21645.5	1.75904
$534$	0	0
$535$	−12092.5	−0.977208
$536$	0	0
$537$	−21227.3	−1.70582
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8798.61	−0.699227	−0.349614	−	0.936894i	$-0.613687\pi$
$541$	−8798.61	−0.699227	−0.349614	+	0.936894i	$0.613687\pi$
$542$	0	0
$543$	27785.7	2.19594
$544$	0	0
$545$	−1307.67	−0.102779
$546$	0	0
$547$	−19572.3	−1.52989	−0.764947	−	0.644094i	$-0.777235\pi$
$547$	−19572.3	−1.52989	−0.764947	+	0.644094i	$0.777235\pi$
$548$	0	0
$549$	−11334.5	−0.881140
$550$	0	0
$551$	−1042.54	−0.0806053
$552$	0	0
$553$	0	0
$554$	0	0
$555$	20046.2	1.53318
$556$	0	0
$557$	2622.45	0.199491	0.0997457	−	0.995013i	$-0.468197\pi$
$557$	2622.45	0.199491	0.0997457	+	0.995013i	$0.468197\pi$
$558$	0	0
$559$	−548.944	−0.0415347
$560$	0	0
$561$	−10391.9	−0.782076
$562$	0	0
$563$	21790.1	1.63116	0.815579	−	0.578646i	$-0.196419\pi$
$563$	21790.1	1.63116	0.815579	+	0.578646i	$0.196419\pi$
$564$	0	0
$565$	3905.11	0.290777
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6272.55	−0.462143	−0.231071	−	0.972937i	$-0.574223\pi$
$569$	−6272.55	−0.462143	−0.231071	+	0.972937i	$0.574223\pi$
$570$	0	0
$571$	13272.2	0.972718	0.486359	−	0.873759i	$-0.338325\pi$
$571$	13272.2	0.972718	0.486359	+	0.873759i	$0.338325\pi$
$572$	0	0
$573$	−17075.4	−1.24491
$574$	0	0
$575$	11822.4	0.857438
$576$	0	0
$577$	14855.0	1.07179	0.535894	−	0.844285i	$-0.319975\pi$
$577$	14855.0	1.07179	0.535894	+	0.844285i	$0.319975\pi$
$578$	0	0
$579$	21422.0	1.53760
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6520.45	0.463206
$584$	0	0
$585$	37540.6	2.65319
$586$	0	0
$587$	17006.2	1.19578	0.597889	−	0.801579i	$-0.296006\pi$
$587$	17006.2	1.19578	0.597889	+	0.801579i	$0.296006\pi$
$588$	0	0
$589$	−3504.31	−0.245148
$590$	0	0
$591$	−24491.4	−1.70464
$592$	0	0
$593$	20955.1	1.45114	0.725568	−	0.688150i	$-0.241577\pi$
$593$	20955.1	1.45114	0.725568	+	0.688150i	$0.241577\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5391.27	0.369598
$598$	0	0
$599$	−9358.39	−0.638353	−0.319177	−	0.947695i	$-0.603406\pi$
$599$	−9358.39	−0.638353	−0.319177	+	0.947695i	$0.603406\pi$
$600$	0	0
$601$	14036.8	0.952700	0.476350	−	0.879256i	$-0.341960\pi$
$601$	14036.8	0.952700	0.476350	+	0.879256i	$0.341960\pi$
$602$	0	0
$603$	−13512.0	−0.912525
$604$	0	0
$605$	−15051.0	−1.01142
$606$	0	0
$607$	26728.4	1.78727	0.893635	−	0.448794i	$-0.148146\pi$
$607$	26728.4	1.78727	0.893635	+	0.448794i	$0.148146\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14026.7	−0.928742
$612$	0	0
$613$	16053.8	1.05776	0.528880	−	0.848697i	$-0.322612\pi$
$613$	16053.8	1.05776	0.528880	+	0.848697i	$0.322612\pi$
$614$	0	0
$615$	29678.9	1.94596
$616$	0	0
$617$	26329.9	1.71799	0.858996	−	0.511982i	$-0.171088\pi$
$617$	26329.9	1.71799	0.858996	+	0.511982i	$0.171088\pi$
$618$	0	0
$619$	−14165.6	−0.919810	−0.459905	−	0.887968i	$-0.652117\pi$
$619$	−14165.6	−0.919810	−0.459905	+	0.887968i	$0.652117\pi$
$620$	0	0
$621$	4629.82	0.299176
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−18859.4	−1.20700
$626$	0	0
$627$	1473.90	0.0938786
$628$	0	0
$629$	−14049.6	−0.890609
$630$	0	0
$631$	−21047.8	−1.32789	−0.663945	−	0.747781i	$-0.731119\pi$
$631$	−21047.8	−1.32789	−0.663945	+	0.747781i	$0.731119\pi$
$632$	0	0
$633$	−1653.20	−0.103806
$634$	0	0
$635$	31175.0	1.94826
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−23007.2	−1.42434
$640$	0	0
$641$	1003.91	0.0618595	0.0309297	−	0.999522i	$-0.490153\pi$
$641$	1003.91	0.0618595	0.0309297	+	0.999522i	$0.490153\pi$
$642$	0	0
$643$	9759.07	0.598538	0.299269	−	0.954169i	$-0.403257\pi$
$643$	9759.07	0.598538	0.299269	+	0.954169i	$0.403257\pi$
$644$	0	0
$645$	−752.676	−0.0459482
$646$	0	0
$647$	2212.99	0.134469	0.0672346	−	0.997737i	$-0.478582\pi$
$647$	2212.99	0.134469	0.0672346	+	0.997737i	$0.478582\pi$
$648$	0	0
$649$	−1973.62	−0.119370
$650$	0	0
$651$	0	0
$652$	0	0
$653$	527.882	0.0316350	0.0158175	−	0.999875i	$-0.494965\pi$
$653$	527.882	0.0316350	0.0158175	+	0.999875i	$0.494965\pi$
$654$	0	0
$655$	21026.2	1.25429
$656$	0	0
$657$	36505.4	2.16775
$658$	0	0
$659$	9476.80	0.560188	0.280094	−	0.959973i	$-0.409634\pi$
$659$	9476.80	0.560188	0.280094	+	0.959973i	$0.409634\pi$
$660$	0	0
$661$	−32877.5	−1.93463	−0.967313	−	0.253586i	$-0.918390\pi$
$661$	−32877.5	−1.93463	−0.967313	+	0.253586i	$0.918390\pi$
$662$	0	0
$663$	−48844.0	−2.86115
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12547.3	−0.728384
$668$	0	0
$669$	−9572.22	−0.553189
$670$	0	0
$671$	−6234.86	−0.358710
$672$	0	0
$673$	17749.9	1.01665	0.508327	−	0.861164i	$-0.330264\pi$
$673$	17749.9	1.01665	0.508327	+	0.861164i	$0.330264\pi$
$674$	0	0
$675$	3061.66	0.174583
$676$	0	0
$677$	10445.3	0.592977	0.296489	−	0.955036i	$-0.404184\pi$
$677$	10445.3	0.592977	0.296489	+	0.955036i	$0.404184\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11547.1	0.649756
$682$	0	0
$683$	−3368.81	−0.188732	−0.0943659	−	0.995538i	$-0.530082\pi$
$683$	−3368.81	−0.188732	−0.0943659	+	0.995538i	$0.530082\pi$
$684$	0	0
$685$	33332.7	1.85924
$686$	0	0
$687$	−3001.20	−0.166671
$688$	0	0
$689$	30647.5	1.69460
$690$	0	0
$691$	8727.02	0.480451	0.240225	−	0.970717i	$-0.422779\pi$
$691$	8727.02	0.480451	0.240225	+	0.970717i	$0.422779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21840.1	−1.19200
$696$	0	0
$697$	−20800.7	−1.13039
$698$	0	0
$699$	3836.96	0.207621
$700$	0	0
$701$	−227.187	−0.0122407	−0.00612036	−	0.999981i	$-0.501948\pi$
$701$	−227.187	−0.0122407	−0.00612036	+	0.999981i	$0.501948\pi$
$702$	0	0
$703$	1992.68	0.106907
$704$	0	0
$705$	−19232.5	−1.02743
$706$	0	0
$707$	0	0
$708$	0	0
$709$	18440.8	0.976813	0.488406	−	0.872616i	$-0.337578\pi$
$709$	18440.8	0.976813	0.488406	+	0.872616i	$0.337578\pi$
$710$	0	0
$711$	−14112.8	−0.744406
$712$	0	0
$713$	−42175.4	−2.21526
$714$	0	0
$715$	20650.2	1.08010
$716$	0	0
$717$	35611.5	1.85486
$718$	0	0
$719$	20602.1	1.06861	0.534304	−	0.845293i	$-0.320574\pi$
$719$	20602.1	1.06861	0.534304	+	0.845293i	$0.320574\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5630.33	−0.289619
$724$	0	0
$725$	−8297.41	−0.425045
$726$	0	0
$727$	−32847.4	−1.67571	−0.837856	−	0.545892i	$-0.816191\pi$
$727$	−32847.4	−1.67571	−0.837856	+	0.545892i	$0.816191\pi$
$728$	0	0
$729$	−25636.3	−1.30246
$730$	0	0
$731$	527.520	0.0266909
$732$	0	0
$733$	29778.6	1.50054	0.750270	−	0.661131i	$-0.229923\pi$
$733$	29778.6	1.50054	0.750270	+	0.661131i	$0.229923\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7432.66	−0.371487
$738$	0	0
$739$	−30356.0	−1.51105	−0.755523	−	0.655122i	$-0.772617\pi$
$739$	−30356.0	−1.51105	−0.755523	+	0.655122i	$0.772617\pi$
$740$	0	0
$741$	6927.65	0.343446
$742$	0	0
$743$	−16361.0	−0.807843	−0.403921	−	0.914794i	$-0.632353\pi$
$743$	−16361.0	−0.807843	−0.403921	+	0.914794i	$0.632353\pi$
$744$	0	0
$745$	−18713.9	−0.920303
$746$	0	0
$747$	327.798	0.0160555
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4051.35	0.196852	0.0984261	−	0.995144i	$-0.468619\pi$
$751$	4051.35	0.196852	0.0984261	+	0.995144i	$0.468619\pi$
$752$	0	0
$753$	9776.84	0.473158
$754$	0	0
$755$	−1787.88	−0.0861823
$756$	0	0
$757$	−34778.4	−1.66981	−0.834903	−	0.550397i	$-0.814476\pi$
$757$	−34778.4	−1.66981	−0.834903	+	0.550397i	$0.814476\pi$
$758$	0	0
$759$	17738.9	0.848327
$760$	0	0
$761$	12538.2	0.597254	0.298627	−	0.954370i	$-0.403471\pi$
$761$	12538.2	0.597254	0.298627	+	0.954370i	$0.403471\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−36075.5	−1.70498
$766$	0	0
$767$	−9276.42	−0.436704
$768$	0	0
$769$	7227.06	0.338900	0.169450	−	0.985539i	$-0.445801\pi$
$769$	7227.06	0.338900	0.169450	+	0.985539i	$0.445801\pi$
$770$	0	0
$771$	−16539.7	−0.772582
$772$	0	0
$773$	30894.3	1.43750	0.718752	−	0.695266i	$-0.244714\pi$
$773$	30894.3	1.43750	0.718752	+	0.695266i	$0.244714\pi$
$774$	0	0
$775$	−27890.3	−1.29271
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2950.21	0.135690
$780$	0	0
$781$	−12655.7	−0.579843
$782$	0	0
$783$	−3249.39	−0.148306
$784$	0	0
$785$	−13432.2	−0.610720
$786$	0	0
$787$	18446.3	0.835503	0.417751	−	0.908561i	$-0.362818\pi$
$787$	18446.3	0.835503	0.417751	+	0.908561i	$0.362818\pi$
$788$	0	0
$789$	−35579.5	−1.60541
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−29305.2	−1.31230
$794$	0	0
$795$	42021.8	1.87467
$796$	0	0
$797$	22292.7	0.990773	0.495387	−	0.868673i	$-0.335026\pi$
$797$	22292.7	0.990773	0.495387	+	0.868673i	$0.335026\pi$
$798$	0	0
$799$	13479.3	0.596825
$800$	0	0
$801$	24150.7	1.06532
$802$	0	0
$803$	20080.8	0.882484
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13464.7	−0.587336
$808$	0	0
$809$	−5126.24	−0.222780	−0.111390	−	0.993777i	$-0.535530\pi$
$809$	−5126.24	−0.222780	−0.111390	+	0.993777i	$0.535530\pi$
$810$	0	0
$811$	16954.8	0.734109	0.367054	−	0.930200i	$-0.380366\pi$
$811$	16954.8	0.734109	0.367054	+	0.930200i	$0.380366\pi$
$812$	0	0
$813$	−43783.3	−1.88874
$814$	0	0
$815$	4144.37	0.178124
$816$	0	0
$817$	−74.8193	−0.00320391
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16594.5	−0.705422	−0.352711	−	0.935732i	$-0.614740\pi$
$821$	−16594.5	−0.705422	−0.352711	+	0.935732i	$0.614740\pi$
$822$	0	0
$823$	16972.9	0.718880	0.359440	−	0.933168i	$-0.382968\pi$
$823$	16972.9	0.718880	0.359440	+	0.933168i	$0.382968\pi$
$824$	0	0
$825$	11730.6	0.495038
$826$	0	0
$827$	34042.7	1.43142	0.715709	−	0.698399i	$-0.246104\pi$
$827$	34042.7	1.43142	0.715709	+	0.698399i	$0.246104\pi$
$828$	0	0
$829$	−47.2172	−0.00197819	−0.000989096	−	1.00000i	$-0.500315\pi$
$829$	−47.2172	−0.00197819	−0.000989096		1.00000i	$0.500315\pi$
$830$	0	0
$831$	12699.5	0.530133
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4044.15	0.167609
$836$	0	0
$837$	−10922.3	−0.451049
$838$	0	0
$839$	−5261.84	−0.216519	−0.108259	−	0.994123i	$-0.534528\pi$
$839$	−5261.84	−0.216519	−0.108259	+	0.994123i	$0.534528\pi$
$840$	0	0
$841$	−15582.8	−0.638929
$842$	0	0
$843$	−40855.1	−1.66919
$844$	0	0
$845$	64964.8	2.64480
$846$	0	0
$847$	0	0
$848$	0	0
$849$	35475.2	1.43404
$850$	0	0
$851$	23982.6	0.966054
$852$	0	0
$853$	26959.2	1.08214	0.541069	−	0.840978i	$-0.318020\pi$
$853$	26959.2	1.08214	0.541069	+	0.840978i	$0.318020\pi$
$854$	0	0
$855$	5116.66	0.204662
$856$	0	0
$857$	15617.2	0.622490	0.311245	−	0.950330i	$-0.399254\pi$
$857$	15617.2	0.622490	0.311245	+	0.950330i	$0.399254\pi$
$858$	0	0
$859$	−48281.0	−1.91773	−0.958863	−	0.283869i	$-0.908382\pi$
$859$	−48281.0	−1.91773	−0.958863	+	0.283869i	$0.908382\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35327.2	−1.39346	−0.696728	−	0.717336i	$-0.745361\pi$
$863$	−35327.2	−1.39346	−0.696728	+	0.717336i	$0.745361\pi$
$864$	0	0
$865$	14854.4	0.583888
$866$	0	0
$867$	9352.01	0.366333
$868$	0	0
$869$	−7763.15	−0.303046
$870$	0	0
$871$	−34935.1	−1.35905
$872$	0	0
$873$	16797.9	0.651229
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3072.39	0.118298	0.0591490	−	0.998249i	$-0.481161\pi$
$877$	3072.39	0.118298	0.0591490	+	0.998249i	$0.481161\pi$
$878$	0	0
$879$	3905.68	0.149869
$880$	0	0
$881$	−23851.6	−0.912124	−0.456062	−	0.889948i	$-0.650740\pi$
$881$	−23851.6	−0.912124	−0.456062	+	0.889948i	$0.650740\pi$
$882$	0	0
$883$	−42506.1	−1.61998	−0.809991	−	0.586443i	$-0.800528\pi$
$883$	−42506.1	−1.61998	−0.809991	+	0.586443i	$0.800528\pi$
$884$	0	0
$885$	−12719.2	−0.483109
$886$	0	0
$887$	27616.4	1.04540	0.522698	−	0.852518i	$-0.324925\pi$
$887$	27616.4	1.04540	0.522698	+	0.852518i	$0.324925\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−10167.6	−0.382299
$892$	0	0
$893$	−1911.80	−0.0716415
$894$	0	0
$895$	−40535.7	−1.51392
$896$	0	0
$897$	83376.5	3.10352
$898$	0	0
$899$	29600.4	1.09814
$900$	0	0
$901$	−29451.4	−1.08898
$902$	0	0
$903$	0	0
$904$	0	0
$905$	53059.5	1.94890
$906$	0	0
$907$	−5959.86	−0.218185	−0.109093	−	0.994032i	$-0.534794\pi$
$907$	−5959.86	−0.218185	−0.109093	+	0.994032i	$0.534794\pi$
$908$	0	0
$909$	−41787.1	−1.52474
$910$	0	0
$911$	−10070.8	−0.366257	−0.183128	−	0.983089i	$-0.558622\pi$
$911$	−10070.8	−0.366257	−0.183128	+	0.983089i	$0.558622\pi$
$912$	0	0
$913$	180.314	0.00653617
$914$	0	0
$915$	−40181.3	−1.45175
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−32070.7	−1.15116	−0.575580	−	0.817746i	$-0.695224\pi$
$919$	−32070.7	−1.15116	−0.575580	+	0.817746i	$0.695224\pi$
$920$	0	0
$921$	−37496.9	−1.34155
$922$	0	0
$923$	−59484.6	−2.12130
$924$	0	0
$925$	15859.5	0.563737
$926$	0	0
$927$	−9374.76	−0.332155
$928$	0	0
$929$	−22081.6	−0.779844	−0.389922	−	0.920848i	$-0.627498\pi$
$929$	−22081.6	−0.779844	−0.389922	+	0.920848i	$0.627498\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2749.52	−0.0964792
$934$	0	0
$935$	−19844.3	−0.694093
$936$	0	0
$937$	−31573.1	−1.10080	−0.550399	−	0.834902i	$-0.685524\pi$
$937$	−31573.1	−1.10080	−0.550399	+	0.834902i	$0.685524\pi$
$938$	0	0
$939$	29583.4	1.02813
$940$	0	0
$941$	−32516.2	−1.12646	−0.563230	−	0.826300i	$-0.690442\pi$
$941$	−32516.2	−1.12646	−0.563230	+	0.826300i	$0.690442\pi$
$942$	0	0
$943$	35506.7	1.22615
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37818.3	1.29771	0.648854	−	0.760913i	$-0.275249\pi$
$947$	37818.3	1.29771	0.648854	+	0.760913i	$0.275249\pi$
$948$	0	0
$949$	94383.9	3.22848
$950$	0	0
$951$	39243.3	1.33812
$952$	0	0
$953$	25258.5	0.858554	0.429277	−	0.903173i	$-0.358768\pi$
$953$	25258.5	0.858554	0.429277	+	0.903173i	$0.358768\pi$
$954$	0	0
$955$	−32607.1	−1.10486
$956$	0	0
$957$	−12449.8	−0.420529
$958$	0	0
$959$	0	0
$960$	0	0
$961$	69705.6	2.33982
$962$	0	0
$963$	−26095.9	−0.873237
$964$	0	0
$965$	40907.4	1.36462
$966$	0	0
$967$	57660.1	1.91750	0.958751	−	0.284246i	$-0.0917433\pi$
$967$	57660.1	1.91750	0.958751	+	0.284246i	$0.0917433\pi$
$968$	0	0
$969$	−6657.27	−0.220704
$970$	0	0
$971$	34536.6	1.14143	0.570717	−	0.821147i	$-0.306665\pi$
$971$	34536.6	1.14143	0.570717	+	0.821147i	$0.306665\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	55136.2	1.81105
$976$	0	0
$977$	5953.28	0.194946	0.0974731	−	0.995238i	$-0.468924\pi$
$977$	5953.28	0.194946	0.0974731	+	0.995238i	$0.468924\pi$
$978$	0	0
$979$	13284.7	0.433690
$980$	0	0
$981$	−2821.97	−0.0918434
$982$	0	0
$983$	−25434.5	−0.825263	−0.412632	−	0.910898i	$-0.635390\pi$
$983$	−25434.5	−0.825263	−0.412632	+	0.910898i	$0.635390\pi$
$984$	0	0
$985$	−46768.7	−1.51287
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−900.474	−0.0289519
$990$	0	0
$991$	−643.073	−0.0206134	−0.0103067	−	0.999947i	$-0.503281\pi$
$991$	−643.073	−0.0206134	−0.0103067	+	0.999947i	$0.503281\pi$
$992$	0	0
$993$	31114.0	0.994332
$994$	0	0
$995$	10295.2	0.328019
$996$	0	0
$997$	9254.11	0.293963	0.146981	−	0.989139i	$-0.453044\pi$
$997$	9254.11	0.293963	0.146981	+	0.989139i	$0.453044\pi$
$998$	0	0
$999$	6210.81	0.196698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bc.1.3		3
4.3	odd	2		392.4.a.k.1.1		3
7.2	even	3		112.4.i.f.81.1		6
7.4	even	3		112.4.i.f.65.1		6
7.6	odd	2		784.4.a.bd.1.1		3
28.3	even	6		392.4.i.n.177.1		6
28.11	odd	6		56.4.i.a.9.3	✓	6
28.19	even	6		392.4.i.n.361.1		6
28.23	odd	6		56.4.i.a.25.3	yes	6
28.27	even	2		392.4.a.j.1.3		3
56.11	odd	6		448.4.i.l.65.1		6
56.37	even	6		448.4.i.k.193.3		6
56.51	odd	6		448.4.i.l.193.1		6
56.53	even	6		448.4.i.k.65.3		6
84.11	even	6		504.4.s.i.289.3		6
84.23	even	6		504.4.s.i.361.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.i.a.9.3	✓	6	28.11	odd	6
56.4.i.a.25.3	yes	6	28.23	odd	6
112.4.i.f.65.1		6	7.4	even	3
112.4.i.f.81.1		6	7.2	even	3
392.4.a.j.1.3		3	28.27	even	2
392.4.a.k.1.1		3	4.3	odd	2
392.4.i.n.177.1		6	28.3	even	6
392.4.i.n.361.1		6	28.19	even	6
448.4.i.k.65.3		6	56.53	even	6
448.4.i.k.193.3		6	56.37	even	6
448.4.i.l.65.1		6	56.11	odd	6
448.4.i.l.193.1		6	56.51	odd	6
504.4.s.i.289.3		6	84.11	even	6
504.4.s.i.361.3		6	84.23	even	6
784.4.a.bc.1.3		3	1.1	even	1	trivial
784.4.a.bd.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+7.65024 q^{3} +14.6089 q^{5} +31.5262 q^{9} +O(q^{10})\)	\(q+7.65024 q^{3} +14.6089 q^{5} +31.5262 q^{9} +17.3418 q^{11} +81.5103 q^{13} +111.762 q^{15} -78.3291 q^{17} +11.1096 q^{19} +133.707 q^{23} +88.4197 q^{25} +34.6265 q^{27} -93.8411 q^{29} -315.431 q^{31} +132.669 q^{33} +179.366 q^{37} +623.574 q^{39} +265.555 q^{41} -6.73466 q^{43} +460.563 q^{45} -172.085 q^{47} -599.236 q^{51} +375.995 q^{53} +253.345 q^{55} +84.9910 q^{57} -113.807 q^{59} -359.527 q^{61} +1190.78 q^{65} -428.597 q^{67} +1022.89 q^{69} -729.780 q^{71} +1157.94 q^{73} +676.432 q^{75} -447.654 q^{79} -586.306 q^{81} +10.3976 q^{83} -1144.30 q^{85} -717.907 q^{87} +766.052 q^{89} -2413.12 q^{93} +162.299 q^{95} +532.824 q^{97} +546.722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 13 q^{5} + 50 q^{9}+O(q^{10}) \) 3 * q - q^3 + 13 * q^5 + 50 * q^9	\( 3 q - q^{3} + 13 q^{5} + 50 q^{9} + 11 q^{11} + 70 q^{13} + 41 q^{15} + 97 q^{17} + 81 q^{19} - 191 q^{23} + 12 q^{25} - 115 q^{27} - 162 q^{29} - 597 q^{31} + 343 q^{33} + 217 q^{37} + 806 q^{39} + 698 q^{41} + 308 q^{43} + 1118 q^{45} - 139 q^{47} - 1475 q^{51} + 197 q^{53} - 147 q^{55} - 1147 q^{57} - 353 q^{59} + 449 q^{61} + 906 q^{65} + 519 q^{67} + 2557 q^{69} - 224 q^{71} + 1701 q^{73} + 1132 q^{75} - 1143 q^{79} + 251 q^{81} + 1380 q^{83} - 1025 q^{85} - 1458 q^{87} + 1749 q^{89} - 601 q^{93} + 2167 q^{95} + 2602 q^{97} - 1094 q^{99}+O(q^{100}) \) 3 * q - q^3 + 13 * q^5 + 50 * q^9 + 11 * q^11 + 70 * q^13 + 41 * q^15 + 97 * q^17 + 81 * q^19 - 191 * q^23 + 12 * q^25 - 115 * q^27 - 162 * q^29 - 597 * q^31 + 343 * q^33 + 217 * q^37 + 806 * q^39 + 698 * q^41 + 308 * q^43 + 1118 * q^45 - 139 * q^47 - 1475 * q^51 + 197 * q^53 - 147 * q^55 - 1147 * q^57 - 353 * q^59 + 449 * q^61 + 906 * q^65 + 519 * q^67 + 2557 * q^69 - 224 * q^71 + 1701 * q^73 + 1132 * q^75 - 1143 * q^79 + 251 * q^81 + 1380 * q^83 - 1025 * q^85 - 1458 * q^87 + 1749 * q^89 - 601 * q^93 + 2167 * q^95 + 2602 * q^97 - 1094 * q^99

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