LMFDB - Embedded newform 1008.4.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{30}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 30 $ x^2 - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.47723$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.9545 q^{5} -7.00000 q^{7} +O(q^{10})$	$q-12.9545 q^{5} -7.00000 q^{7} -14.8634 q^{11} +37.8178 q^{13} -54.8634 q^{17} -47.8178 q^{19} -16.7723 q^{23} +42.8178 q^{25} -123.818 q^{29} +0.182195 q^{31} +90.6812 q^{35} -216.725 q^{37} -276.590 q^{41} +378.542 q^{43} +67.0021 q^{47} +49.0000 q^{49} -206.820 q^{53} +192.547 q^{55} +492.998 q^{59} +242.182 q^{61} -489.909 q^{65} -186.178 q^{67} +604.954 q^{71} +1074.72 q^{73} +104.043 q^{77} +1180.72 q^{79} +847.271 q^{83} +710.725 q^{85} -822.222 q^{89} -264.725 q^{91} +619.453 q^{95} -437.992 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 14 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - 14 * q^7	$ 2 q - 4 q^{5} - 14 q^{7} + 36 q^{11} - 12 q^{13} - 44 q^{17} - 8 q^{19} + 76 q^{23} - 2 q^{25} - 160 q^{29} + 88 q^{31} + 28 q^{35} + 180 q^{37} - 356 q^{41} + 56 q^{43} + 616 q^{47} + 98 q^{49} - 808 q^{53} + 648 q^{55} + 504 q^{59} + 572 q^{61} - 936 q^{65} + 504 q^{67} + 1188 q^{71} + 572 q^{73} - 252 q^{77} + 784 q^{79} + 1344 q^{83} + 808 q^{85} - 308 q^{89} + 84 q^{91} + 976 q^{95} + 1052 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 14 * q^7 + 36 * q^11 - 12 * q^13 - 44 * q^17 - 8 * q^19 + 76 * q^23 - 2 * q^25 - 160 * q^29 + 88 * q^31 + 28 * q^35 + 180 * q^37 - 356 * q^41 + 56 * q^43 + 616 * q^47 + 98 * q^49 - 808 * q^53 + 648 * q^55 + 504 * q^59 + 572 * q^61 - 936 * q^65 + 504 * q^67 + 1188 * q^71 + 572 * q^73 - 252 * q^77 + 784 * q^79 + 1344 * q^83 + 808 * q^85 - 308 * q^89 + 84 * q^91 + 976 * q^95 + 1052 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−12.9545	−1.15868	−0.579341	−	0.815086i	$-0.696690\pi$
$5$	−12.9545	−1.15868	−0.579341	+	0.815086i	$0.696690\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−14.8634	−0.407406	−0.203703	−	0.979033i	$-0.565298\pi$
$11$	−14.8634	−0.407406	−0.203703	+	0.979033i	$0.565298\pi$
$12$	0	0
$13$	37.8178	0.806829	0.403414	−	0.915017i	$-0.367823\pi$
$13$	37.8178	0.806829	0.403414	+	0.915017i	$0.367823\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−54.8634	−0.782725	−0.391362	−	0.920237i	$-0.627996\pi$
$17$	−54.8634	−0.782725	−0.391362	+	0.920237i	$0.627996\pi$
$18$	0	0
$19$	−47.8178	−0.577377	−0.288688	−	0.957423i	$-0.593219\pi$
$19$	−47.8178	−0.577377	−0.288688	+	0.957423i	$0.593219\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−16.7723	−0.152055	−0.0760273	−	0.997106i	$-0.524224\pi$
$23$	−16.7723	−0.152055	−0.0760273	+	0.997106i	$0.524224\pi$
$24$	0	0
$25$	42.8178	0.342542
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−123.818	−0.792841	−0.396421	−	0.918069i	$-0.629748\pi$
$29$	−123.818	−0.792841	−0.396421	+	0.918069i	$0.629748\pi$
$30$	0	0
$31$	0.182195	0.00105559	0.000527795	−	1.00000i	$-0.499832\pi$
$31$	0.182195	0.00105559	0.000527795		1.00000i	$0.499832\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	90.6812	0.437940
$36$	0	0
$37$	−216.725	−0.962954	−0.481477	−	0.876459i	$-0.659900\pi$
$37$	−216.725	−0.962954	−0.481477	+	0.876459i	$0.659900\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−276.590	−1.05356	−0.526782	−	0.850001i	$-0.676602\pi$
$41$	−276.590	−1.05356	−0.526782	+	0.850001i	$0.676602\pi$
$42$	0	0
$43$	378.542	1.34249	0.671246	−	0.741234i	$-0.265759\pi$
$43$	378.542	1.34249	0.671246	+	0.741234i	$0.265759\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	67.0021	0.207942	0.103971	−	0.994580i	$-0.466845\pi$
$47$	67.0021	0.207942	0.103971	+	0.994580i	$0.466845\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−206.820	−0.536017	−0.268008	−	0.963417i	$-0.586366\pi$
$53$	−206.820	−0.536017	−0.268008	+	0.963417i	$0.586366\pi$
$54$	0	0
$55$	192.547	0.472054
$56$	0	0
$57$	0	0
$58$	0	0
$59$	492.998	1.08785	0.543923	−	0.839135i	$-0.316939\pi$
$59$	492.998	1.08785	0.543923	+	0.839135i	$0.316939\pi$
$60$	0	0
$61$	242.182	0.508332	0.254166	−	0.967161i	$-0.418199\pi$
$61$	242.182	0.508332	0.254166	+	0.967161i	$0.418199\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−489.909	−0.934857
$66$	0	0
$67$	−186.178	−0.339482	−0.169741	−	0.985489i	$-0.554293\pi$
$67$	−186.178	−0.339482	−0.169741	+	0.985489i	$0.554293\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	604.954	1.01120	0.505598	−	0.862769i	$-0.331272\pi$
$71$	604.954	1.01120	0.505598	+	0.862769i	$0.331272\pi$
$72$	0	0
$73$	1074.72	1.72310	0.861551	−	0.507670i	$-0.169493\pi$
$73$	1074.72	1.72310	0.861551	+	0.507670i	$0.169493\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	104.043	0.153985
$78$	0	0
$79$	1180.72	1.68154	0.840769	−	0.541395i	$-0.182104\pi$
$79$	1180.72	1.68154	0.840769	+	0.541395i	$0.182104\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	847.271	1.12048	0.560242	−	0.828329i	$-0.310708\pi$
$83$	847.271	1.12048	0.560242	+	0.828329i	$0.310708\pi$
$84$	0	0
$85$	710.725	0.906928
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−822.222	−0.979273	−0.489637	−	0.871927i	$-0.662871\pi$
$89$	−822.222	−0.979273	−0.489637	+	0.871927i	$0.662871\pi$
$90$	0	0
$91$	−264.725	−0.304953
$92$	0	0
$93$	0	0
$94$	0	0
$95$	619.453	0.668996
$96$	0	0
$97$	−437.992	−0.458467	−0.229234	−	0.973371i	$-0.573622\pi$
$97$	−437.992	−0.458467	−0.229234	+	0.973371i	$0.573622\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1024.22	1.00904	0.504522	−	0.863399i	$-0.331669\pi$
$101$	1024.22	1.00904	0.504522	+	0.863399i	$0.331669\pi$
$102$	0	0
$103$	−419.810	−0.401603	−0.200801	−	0.979632i	$-0.564355\pi$
$103$	−419.810	−0.401603	−0.200801	+	0.979632i	$0.564355\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	347.132	0.313631	0.156816	−	0.987628i	$-0.449877\pi$
$107$	347.132	0.313631	0.156816	+	0.987628i	$0.449877\pi$
$108$	0	0
$109$	1456.18	1.27960	0.639801	−	0.768541i	$-0.279017\pi$
$109$	1456.18	1.27960	0.639801	+	0.768541i	$0.279017\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−496.729	−0.413525	−0.206762	−	0.978391i	$-0.566293\pi$
$113$	−496.729	−0.413525	−0.206762	+	0.978391i	$0.566293\pi$
$114$	0	0
$115$	217.275	0.176183
$116$	0	0
$117$	0	0
$118$	0	0
$119$	384.043	0.295842
$120$	0	0
$121$	−1110.08	−0.834020
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1064.63	0.761784
$126$	0	0
$127$	904.000	0.631630	0.315815	−	0.948821i	$-0.397722\pi$
$127$	904.000	0.631630	0.315815	+	0.948821i	$0.397722\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−768.174	−0.512333	−0.256167	−	0.966633i	$-0.582460\pi$
$131$	−768.174	−0.512333	−0.256167	+	0.966633i	$0.582460\pi$
$132$	0	0
$133$	334.725	0.218228
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2896.43	−1.80627	−0.903135	−	0.429356i	$-0.858741\pi$
$137$	−2896.43	−1.80627	−0.903135	+	0.429356i	$0.858741\pi$
$138$	0	0
$139$	2481.08	1.51397	0.756986	−	0.653431i	$-0.226671\pi$
$139$	2481.08	1.51397	0.756986	+	0.653431i	$0.226671\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−562.099	−0.328707
$144$	0	0
$145$	1603.99	0.918650
$146$	0	0
$147$	0	0
$148$	0	0
$149$	941.362	0.517580	0.258790	−	0.965934i	$-0.416676\pi$
$149$	941.362	0.517580	0.258790	+	0.965934i	$0.416676\pi$
$150$	0	0
$151$	2265.81	1.22112	0.610558	−	0.791971i	$-0.290945\pi$
$151$	2265.81	1.22112	0.610558	+	0.791971i	$0.290945\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.36024	−0.00122309
$156$	0	0
$157$	1100.73	0.559542	0.279771	−	0.960067i	$-0.409742\pi$
$157$	1100.73	0.559542	0.279771	+	0.960067i	$0.409742\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	117.406	0.0574713
$162$	0	0
$163$	1101.09	0.529106	0.264553	−	0.964371i	$-0.414776\pi$
$163$	1101.09	0.529106	0.264553	+	0.964371i	$0.414776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	600.451	0.278229	0.139115	−	0.990276i	$-0.455574\pi$
$167$	600.451	0.278229	0.139115	+	0.990276i	$0.455574\pi$
$168$	0	0
$169$	−766.814	−0.349028
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−558.847	−0.245597	−0.122799	−	0.992432i	$-0.539187\pi$
$173$	−558.847	−0.245597	−0.122799	+	0.992432i	$0.539187\pi$
$174$	0	0
$175$	−299.725	−0.129469
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4048.20	−1.69037	−0.845186	−	0.534472i	$-0.820511\pi$
$179$	−4048.20	−1.69037	−0.845186	+	0.534472i	$0.820511\pi$
$180$	0	0
$181$	2446.91	1.00485	0.502424	−	0.864621i	$-0.332442\pi$
$181$	2446.91	1.00485	0.502424	+	0.864621i	$0.332442\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2807.55	1.11576
$186$	0	0
$187$	815.453	0.318887
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3681.03	−1.39450	−0.697252	−	0.716826i	$-0.745594\pi$
$191$	−3681.03	−1.39450	−0.697252	+	0.716826i	$0.745594\pi$
$192$	0	0
$193$	1192.36	0.444704	0.222352	−	0.974966i	$-0.428627\pi$
$193$	1192.36	0.444704	0.222352	+	0.974966i	$0.428627\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1208.82	0.437180	0.218590	−	0.975817i	$-0.429854\pi$
$197$	1208.82	0.437180	0.218590	+	0.975817i	$0.429854\pi$
$198$	0	0
$199$	−42.1697	−0.0150218	−0.00751089	−	0.999972i	$-0.502391\pi$
$199$	−42.1697	−0.0150218	−0.00751089	+	0.999972i	$0.502391\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	866.725	0.299666
$204$	0	0
$205$	3583.07	1.22074
$206$	0	0
$207$	0	0
$208$	0	0
$209$	710.733	0.235227
$210$	0	0
$211$	−256.720	−0.0837600	−0.0418800	−	0.999123i	$-0.513335\pi$
$211$	−256.720	−0.0837600	−0.0418800	+	0.999123i	$0.513335\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4903.81	−1.55552
$216$	0	0
$217$	−1.27537	−0.000398975 0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2074.81	−0.631525
$222$	0	0
$223$	−549.433	−0.164990	−0.0824949	−	0.996591i	$-0.526289\pi$
$223$	−549.433	−0.164990	−0.0824949	+	0.996591i	$0.526289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5088.82	−1.48791	−0.743957	−	0.668227i	$-0.767053\pi$
$227$	−5088.82	−1.48791	−0.743957	+	0.668227i	$0.767053\pi$
$228$	0	0
$229$	1229.09	0.354675	0.177337	−	0.984150i	$-0.443252\pi$
$229$	1229.09	0.354675	0.177337	+	0.984150i	$0.443252\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3955.73	1.11223	0.556113	−	0.831107i	$-0.312292\pi$
$233$	3955.73	1.11223	0.556113	+	0.831107i	$0.312292\pi$
$234$	0	0
$235$	−867.975	−0.240938
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5321.92	1.44036	0.720181	−	0.693787i	$-0.244059\pi$
$239$	5321.92	1.44036	0.720181	+	0.693787i	$0.244059\pi$
$240$	0	0
$241$	−2452.90	−0.655623	−0.327811	−	0.944743i	$-0.606311\pi$
$241$	−2452.90	−0.655623	−0.327811	+	0.944743i	$0.606311\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−634.768	−0.165526
$246$	0	0
$247$	−1808.36	−0.465844
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3713.69	0.933887	0.466944	−	0.884287i	$-0.345355\pi$
$251$	3713.69	0.933887	0.466944	+	0.884287i	$0.345355\pi$
$252$	0	0
$253$	249.292	0.0619480
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4285.01	1.04004	0.520022	−	0.854153i	$-0.325924\pi$
$257$	4285.01	1.04004	0.520022	+	0.854153i	$0.325924\pi$
$258$	0	0
$259$	1517.07	0.363963
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2651.46	−0.621659	−0.310829	−	0.950466i	$-0.600607\pi$
$263$	−2651.46	−0.621659	−0.310829	+	0.950466i	$0.600607\pi$
$264$	0	0
$265$	2679.24	0.621073
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8365.77	1.89617	0.948086	−	0.318015i	$-0.103016\pi$
$269$	8365.77	1.89617	0.948086	+	0.318015i	$0.103016\pi$
$270$	0	0
$271$	5514.30	1.23605	0.618026	−	0.786158i	$-0.287933\pi$
$271$	5514.30	1.23605	0.618026	+	0.786158i	$0.287933\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−636.416	−0.139554
$276$	0	0
$277$	−2015.24	−0.437127	−0.218563	−	0.975823i	$-0.570137\pi$
$277$	−2015.24	−0.437127	−0.218563	+	0.975823i	$0.570137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1046.23	0.222110	0.111055	−	0.993814i	$-0.464577\pi$
$281$	1046.23	0.222110	0.111055	+	0.993814i	$0.464577\pi$
$282$	0	0
$283$	5029.60	1.05646	0.528231	−	0.849101i	$-0.322856\pi$
$283$	5029.60	1.05646	0.528231	+	0.849101i	$0.322856\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1936.13	0.398210
$288$	0	0
$289$	−1903.01	−0.387342
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1537.82	0.306622	0.153311	−	0.988178i	$-0.451006\pi$
$293$	1537.82	0.306622	0.153311	+	0.988178i	$0.451006\pi$
$294$	0	0
$295$	−6386.52	−1.26047
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−634.290	−0.122682
$300$	0	0
$301$	−2649.80	−0.507414
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3137.34	−0.588995
$306$	0	0
$307$	2161.28	0.401793	0.200897	−	0.979612i	$-0.435614\pi$
$307$	2161.28	0.401793	0.200897	+	0.979612i	$0.435614\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−418.480	−0.0763018	−0.0381509	−	0.999272i	$-0.512147\pi$
$311$	−418.480	−0.0763018	−0.0381509	+	0.999272i	$0.512147\pi$
$312$	0	0
$313$	−9152.88	−1.65288	−0.826440	−	0.563025i	$-0.809638\pi$
$313$	−9152.88	−1.65288	−0.826440	+	0.563025i	$0.809638\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2386.95	−0.422917	−0.211458	−	0.977387i	$-0.567821\pi$
$317$	−2386.95	−0.422917	−0.211458	+	0.977387i	$0.567821\pi$
$318$	0	0
$319$	1840.35	0.323008
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2623.45	0.451927
$324$	0	0
$325$	1619.28	0.276373
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−469.015	−0.0785946
$330$	0	0
$331$	−9493.06	−1.57639	−0.788196	−	0.615424i	$-0.788985\pi$
$331$	−9493.06	−1.57639	−0.788196	+	0.615424i	$0.788985\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2411.83	0.393351
$336$	0	0
$337$	−4665.29	−0.754108	−0.377054	−	0.926191i	$-0.623063\pi$
$337$	−4665.29	−0.754108	−0.377054	+	0.926191i	$0.623063\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.70803	−0.000430054 0
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9039.77	1.39850	0.699251	−	0.714876i	$-0.253517\pi$
$347$	9039.77	1.39850	0.699251	+	0.714876i	$0.253517\pi$
$348$	0	0
$349$	9637.05	1.47811	0.739053	−	0.673647i	$-0.235273\pi$
$349$	9637.05	1.47811	0.739053	+	0.673647i	$0.235273\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9061.84	1.36633	0.683163	−	0.730266i	$-0.260604\pi$
$353$	9061.84	1.36633	0.683163	+	0.730266i	$0.260604\pi$
$354$	0	0
$355$	−7836.85	−1.17165
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12573.7	−1.84851	−0.924254	−	0.381777i	$-0.875312\pi$
$359$	−12573.7	−1.84851	−0.924254	+	0.381777i	$0.875312\pi$
$360$	0	0
$361$	−4572.46	−0.666636
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13922.4	−1.99653
$366$	0	0
$367$	−3306.17	−0.470247	−0.235124	−	0.971966i	$-0.575549\pi$
$367$	−3306.17	−0.470247	−0.235124	+	0.971966i	$0.575549\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1447.74	0.202595
$372$	0	0
$373$	3869.97	0.537210	0.268605	−	0.963250i	$-0.413437\pi$
$373$	3869.97	0.537210	0.268605	+	0.963250i	$0.413437\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4682.52	−0.639687
$378$	0	0
$379$	−2560.75	−0.347064	−0.173532	−	0.984828i	$-0.555518\pi$
$379$	−2560.75	−0.347064	−0.173532	+	0.984828i	$0.555518\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7847.05	1.04691	0.523454	−	0.852054i	$-0.324643\pi$
$383$	7847.05	1.04691	0.523454	+	0.852054i	$0.324643\pi$
$384$	0	0
$385$	−1347.83	−0.178420
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13425.1	−1.74981	−0.874907	−	0.484291i	$-0.839078\pi$
$389$	−13425.1	−1.74981	−0.874907	+	0.484291i	$0.839078\pi$
$390$	0	0
$391$	920.182	0.119017
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15295.6	−1.94837
$396$	0	0
$397$	−2722.51	−0.344178	−0.172089	−	0.985081i	$-0.555052\pi$
$397$	−2722.51	−0.344178	−0.172089	+	0.985081i	$0.555052\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1027.32	0.127935	0.0639673	−	0.997952i	$-0.479625\pi$
$401$	1027.32	0.127935	0.0639673	+	0.997952i	$0.479625\pi$
$402$	0	0
$403$	6.89023	0.000851679 0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3221.25	0.392314
$408$	0	0
$409$	−10399.8	−1.25730	−0.628650	−	0.777688i	$-0.716392\pi$
$409$	−10399.8	−1.25730	−0.628650	+	0.777688i	$0.716392\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3450.99	−0.411167
$414$	0	0
$415$	−10975.9	−1.29828
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7702.85	0.898112	0.449056	−	0.893504i	$-0.351760\pi$
$419$	7702.85	0.898112	0.449056	+	0.893504i	$0.351760\pi$
$420$	0	0
$421$	1334.49	0.154487	0.0772435	−	0.997012i	$-0.475388\pi$
$421$	1334.49	0.154487	0.0772435	+	0.997012i	$0.475388\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2349.13	−0.268116
$426$	0	0
$427$	−1695.28	−0.192131
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14603.6	1.63209	0.816045	−	0.577988i	$-0.196162\pi$
$431$	14603.6	1.63209	0.816045	+	0.577988i	$0.196162\pi$
$432$	0	0
$433$	−7424.10	−0.823971	−0.411986	−	0.911190i	$-0.635165\pi$
$433$	−7424.10	−0.823971	−0.411986	+	0.911190i	$0.635165\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	802.012	0.0877928
$438$	0	0
$439$	−7398.18	−0.804318	−0.402159	−	0.915570i	$-0.631740\pi$
$439$	−7398.18	−0.804318	−0.402159	+	0.915570i	$0.631740\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7436.96	0.797608	0.398804	−	0.917036i	$-0.369425\pi$
$443$	7436.96	0.797608	0.398804	+	0.917036i	$0.369425\pi$
$444$	0	0
$445$	10651.4	1.13467
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6244.93	−0.656384	−0.328192	−	0.944611i	$-0.606439\pi$
$449$	−6244.93	−0.656384	−0.328192	+	0.944611i	$0.606439\pi$
$450$	0	0
$451$	4111.06	0.429229
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3429.36	0.353343
$456$	0	0
$457$	−14927.1	−1.52792	−0.763961	−	0.645262i	$-0.776748\pi$
$457$	−14927.1	−1.52792	−0.763961	+	0.645262i	$0.776748\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14276.9	−1.44238	−0.721192	−	0.692735i	$-0.756406\pi$
$461$	−14276.9	−1.44238	−0.721192	+	0.692735i	$0.756406\pi$
$462$	0	0
$463$	1899.59	0.190673	0.0953365	−	0.995445i	$-0.469607\pi$
$463$	1899.59	0.190673	0.0953365	+	0.995445i	$0.469607\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10826.0	−1.07274	−0.536369	−	0.843984i	$-0.680204\pi$
$467$	−10826.0	−1.07274	−0.536369	+	0.843984i	$0.680204\pi$
$468$	0	0
$469$	1303.25	0.128312
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5626.41	−0.546940
$474$	0	0
$475$	−2047.45	−0.197776
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2205.24	−0.210355	−0.105177	−	0.994453i	$-0.533541\pi$
$479$	−2205.24	−0.210355	−0.105177	+	0.994453i	$0.533541\pi$
$480$	0	0
$481$	−8196.05	−0.776939
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5673.94	0.531217
$486$	0	0
$487$	8068.67	0.750773	0.375386	−	0.926868i	$-0.377510\pi$
$487$	8068.67	0.750773	0.375386	+	0.926868i	$0.377510\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18729.2	1.72146	0.860730	−	0.509061i	$-0.170007\pi$
$491$	18729.2	1.72146	0.860730	+	0.509061i	$0.170007\pi$
$492$	0	0
$493$	6793.06	0.620576
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4234.68	−0.382196
$498$	0	0
$499$	979.950	0.0879130	0.0439565	−	0.999033i	$-0.486004\pi$
$499$	979.950	0.0879130	0.0439565	+	0.999033i	$0.486004\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9221.76	−0.817452	−0.408726	−	0.912657i	$-0.634027\pi$
$503$	−9221.76	−0.817452	−0.408726	+	0.912657i	$0.634027\pi$
$504$	0	0
$505$	−13268.2	−1.16916
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1919.75	0.167174	0.0835868	−	0.996500i	$-0.473362\pi$
$509$	1919.75	0.167174	0.0835868	+	0.996500i	$0.473362\pi$
$510$	0	0
$511$	−7523.04	−0.651272
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5438.40	0.465329
$516$	0	0
$517$	−995.876	−0.0847167
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12834.3	1.07923	0.539615	−	0.841912i	$-0.318570\pi$
$521$	12834.3	1.07923	0.539615	+	0.841912i	$0.318570\pi$
$522$	0	0
$523$	−4888.75	−0.408738	−0.204369	−	0.978894i	$-0.565514\pi$
$523$	−4888.75	−0.408738	−0.204369	+	0.978894i	$0.565514\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.99585	−0.000826235 0
$528$	0	0
$529$	−11885.7	−0.976879
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10460.0	−0.850045
$534$	0	0
$535$	−4496.91	−0.363399
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−728.304	−0.0582009
$540$	0	0
$541$	−14033.3	−1.11523	−0.557616	−	0.830099i	$-0.688284\pi$
$541$	−14033.3	−1.11523	−0.557616	+	0.830099i	$0.688284\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18864.0	−1.48265
$546$	0	0
$547$	−12558.9	−0.981681	−0.490841	−	0.871249i	$-0.663310\pi$
$547$	−12558.9	−0.981681	−0.490841	+	0.871249i	$0.663310\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5920.70	0.457768
$552$	0	0
$553$	−8265.04	−0.635561
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19674.8	1.49668	0.748338	−	0.663318i	$-0.230852\pi$
$557$	19674.8	1.49668	0.748338	+	0.663318i	$0.230852\pi$
$558$	0	0
$559$	14315.6	1.08316
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25410.4	−1.90217	−0.951083	−	0.308934i	$-0.900028\pi$
$563$	−25410.4	−1.90217	−0.951083	+	0.308934i	$0.900028\pi$
$564$	0	0
$565$	6434.85	0.479144
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18973.4	1.39790	0.698952	−	0.715169i	$-0.253650\pi$
$569$	18973.4	1.39790	0.698952	+	0.715169i	$0.253650\pi$
$570$	0	0
$571$	−13475.2	−0.987597	−0.493798	−	0.869576i	$-0.664392\pi$
$571$	−13475.2	−0.987597	−0.493798	+	0.869576i	$0.664392\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−718.151	−0.0520852
$576$	0	0
$577$	12142.2	0.876061	0.438031	−	0.898960i	$-0.355676\pi$
$577$	12142.2	0.876061	0.438031	+	0.898960i	$0.355676\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5930.90	−0.423503
$582$	0	0
$583$	3074.04	0.218377
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16337.8	1.14878	0.574390	−	0.818582i	$-0.305239\pi$
$587$	16337.8	1.14878	0.574390	+	0.818582i	$0.305239\pi$
$588$	0	0
$589$	−8.71218	−0.000609472 0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14725.2	1.01971	0.509857	−	0.860259i	$-0.329698\pi$
$593$	14725.2	1.01971	0.509857	+	0.860259i	$0.329698\pi$
$594$	0	0
$595$	−4975.07	−0.342787
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23848.5	1.62675	0.813374	−	0.581741i	$-0.197628\pi$
$599$	23848.5	1.62675	0.813374	+	0.581741i	$0.197628\pi$
$600$	0	0
$601$	6780.09	0.460176	0.230088	−	0.973170i	$-0.426099\pi$
$601$	6780.09	0.460176	0.230088	+	0.973170i	$0.426099\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14380.5	0.966363
$606$	0	0
$607$	−18107.9	−1.21083	−0.605416	−	0.795909i	$-0.706993\pi$
$607$	−18107.9	−1.21083	−0.605416	+	0.795909i	$0.706993\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2533.87	0.167773
$612$	0	0
$613$	27618.6	1.81975	0.909874	−	0.414885i	$-0.136178\pi$
$613$	27618.6	1.81975	0.909874	+	0.414885i	$0.136178\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20417.7	−1.33223	−0.666113	−	0.745851i	$-0.732043\pi$
$617$	−20417.7	−1.33223	−0.666113	+	0.745851i	$0.732043\pi$
$618$	0	0
$619$	−16683.1	−1.08328	−0.541641	−	0.840610i	$-0.682197\pi$
$619$	−16683.1	−1.08328	−0.541641	+	0.840610i	$0.682197\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5755.55	0.370130
$624$	0	0
$625$	−19143.9	−1.22521
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11890.2	0.753728
$630$	0	0
$631$	−10531.6	−0.664431	−0.332216	−	0.943203i	$-0.607796\pi$
$631$	−10531.6	−0.664431	−0.332216	+	0.943203i	$0.607796\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11710.8	−0.731858
$636$	0	0
$637$	1853.07	0.115261
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4197.47	0.258643	0.129321	−	0.991603i	$-0.458720\pi$
$641$	4197.47	0.258643	0.129321	+	0.991603i	$0.458720\pi$
$642$	0	0
$643$	29835.0	1.82983	0.914914	−	0.403650i	$-0.132259\pi$
$643$	29835.0	1.82983	0.914914	+	0.403650i	$0.132259\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25333.2	−1.53934	−0.769668	−	0.638444i	$-0.779578\pi$
$647$	−25333.2	−1.53934	−0.769668	+	0.638444i	$0.779578\pi$
$648$	0	0
$649$	−7327.60	−0.443195
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19969.9	1.19676	0.598380	−	0.801213i	$-0.295811\pi$
$653$	19969.9	1.19676	0.598380	+	0.801213i	$0.295811\pi$
$654$	0	0
$655$	9951.27	0.593631
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3176.25	0.187753	0.0938764	−	0.995584i	$-0.470074\pi$
$659$	3176.25	0.187753	0.0938764	+	0.995584i	$0.470074\pi$
$660$	0	0
$661$	13181.5	0.775644	0.387822	−	0.921734i	$-0.373228\pi$
$661$	13181.5	0.775644	0.387822	+	0.921734i	$0.373228\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4336.17	−0.252857
$666$	0	0
$667$	2076.70	0.120555
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3599.64	−0.207098
$672$	0	0
$673$	−23559.2	−1.34939	−0.674696	−	0.738096i	$-0.735725\pi$
$673$	−23559.2	−1.34939	−0.674696	+	0.738096i	$0.735725\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16310.0	0.925914	0.462957	−	0.886381i	$-0.346789\pi$
$677$	16310.0	0.925914	0.462957	+	0.886381i	$0.346789\pi$
$678$	0	0
$679$	3065.94	0.173284
$680$	0	0
$681$	0	0
$682$	0	0
$683$	88.9003	0.00498049	0.00249025	−	0.999997i	$-0.499207\pi$
$683$	88.9003	0.00498049	0.00249025	+	0.999997i	$0.499207\pi$
$684$	0	0
$685$	37521.7	2.09289
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7821.47	−0.432474
$690$	0	0
$691$	9223.87	0.507804	0.253902	−	0.967230i	$-0.418286\pi$
$691$	9223.87	0.507804	0.253902	+	0.967230i	$0.418286\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−32141.0	−1.75421
$696$	0	0
$697$	15174.7	0.824650
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15877.5	0.855471	0.427735	−	0.903904i	$-0.359311\pi$
$701$	15877.5	0.855471	0.427735	+	0.903904i	$0.359311\pi$
$702$	0	0
$703$	10363.3	0.555987
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7169.52	−0.381383
$708$	0	0
$709$	−23507.3	−1.24518	−0.622591	−	0.782547i	$-0.713920\pi$
$709$	−23507.3	−1.24518	−0.622591	+	0.782547i	$0.713920\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.05583	−0.000160507 0
$714$	0	0
$715$	7281.69	0.380867
$716$	0	0
$717$	0	0
$718$	0	0
$719$	57.6892	0.00299227	0.00149614	−	0.999999i	$-0.499524\pi$
$719$	57.6892	0.00299227	0.00149614	+	0.999999i	$0.499524\pi$
$720$	0	0
$721$	2938.67	0.151791
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5301.61	−0.271582
$726$	0	0
$727$	20562.7	1.04901	0.524503	−	0.851408i	$-0.324251\pi$
$727$	20562.7	1.04901	0.524503	+	0.851408i	$0.324251\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20768.1	−1.05080
$732$	0	0
$733$	14965.0	0.754088	0.377044	−	0.926195i	$-0.376941\pi$
$733$	14965.0	0.754088	0.377044	+	0.926195i	$0.376941\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2767.23	0.138307
$738$	0	0
$739$	22596.3	1.12479	0.562395	−	0.826869i	$-0.309880\pi$
$739$	22596.3	1.12479	0.562395	+	0.826869i	$0.309880\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29017.7	−1.43278	−0.716391	−	0.697699i	$-0.754207\pi$
$743$	−29017.7	−1.43278	−0.716391	+	0.697699i	$0.754207\pi$
$744$	0	0
$745$	−12194.8	−0.599710
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2429.93	−0.118542
$750$	0	0
$751$	101.018	0.00490841	0.00245421	−	0.999997i	$-0.499219\pi$
$751$	101.018	0.00490841	0.00245421	+	0.999997i	$0.499219\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−29352.3	−1.41489
$756$	0	0
$757$	120.311	0.00577645	0.00288822	−	0.999996i	$-0.499081\pi$
$757$	120.311	0.00577645	0.00288822	+	0.999996i	$0.499081\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9204.56	−0.438456	−0.219228	−	0.975674i	$-0.570354\pi$
$761$	−9204.56	−0.438456	−0.219228	+	0.975674i	$0.570354\pi$
$762$	0	0
$763$	−10193.2	−0.483644
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18644.1	0.877705
$768$	0	0
$769$	13642.8	0.639754	0.319877	−	0.947459i	$-0.396358\pi$
$769$	13642.8	0.639754	0.319877	+	0.947459i	$0.396358\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33355.7	1.55203	0.776016	−	0.630713i	$-0.217237\pi$
$773$	33355.7	1.55203	0.776016	+	0.630713i	$0.217237\pi$
$774$	0	0
$775$	7.80121	0.000361584 0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13225.9	0.608303
$780$	0	0
$781$	−8991.65	−0.411967
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−14259.4	−0.648331
$786$	0	0
$787$	26733.5	1.21086	0.605430	−	0.795899i	$-0.293001\pi$
$787$	26733.5	1.21086	0.605430	+	0.795899i	$0.293001\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3477.10	0.156298
$792$	0	0
$793$	9158.80	0.410137
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15227.7	−0.676780	−0.338390	−	0.941006i	$-0.609882\pi$
$797$	−15227.7	−0.676780	−0.338390	+	0.941006i	$0.609882\pi$
$798$	0	0
$799$	−3675.96	−0.162761
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15974.0	−0.702003
$804$	0	0
$805$	−1520.93	−0.0665909
$806$	0	0
$807$	0	0
$808$	0	0
$809$	968.083	0.0420717	0.0210358	−	0.999779i	$-0.493304\pi$
$809$	968.083	0.0420717	0.0210358	+	0.999779i	$0.493304\pi$
$810$	0	0
$811$	10960.8	0.474583	0.237291	−	0.971439i	$-0.423740\pi$
$811$	10960.8	0.474583	0.237291	+	0.971439i	$0.423740\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14264.1	−0.613065
$816$	0	0
$817$	−18101.1	−0.775124
$818$	0	0
$819$	0	0
$820$	0	0
$821$	562.050	0.0238924	0.0119462	−	0.999929i	$-0.496197\pi$
$821$	562.050	0.0238924	0.0119462	+	0.999929i	$0.496197\pi$
$822$	0	0
$823$	−15160.8	−0.642130	−0.321065	−	0.947057i	$-0.604041\pi$
$823$	−15160.8	−0.642130	−0.321065	+	0.947057i	$0.604041\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22065.5	0.927801	0.463900	−	0.885887i	$-0.346450\pi$
$827$	22065.5	0.927801	0.463900	+	0.885887i	$0.346450\pi$
$828$	0	0
$829$	29564.0	1.23860	0.619300	−	0.785155i	$-0.287417\pi$
$829$	29564.0	1.23860	0.619300	+	0.785155i	$0.287417\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2688.30	−0.111818
$834$	0	0
$835$	−7778.52	−0.322379
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3961.78	−0.163022	−0.0815111	−	0.996672i	$-0.525975\pi$
$839$	−3961.78	−0.163022	−0.0815111	+	0.996672i	$0.525975\pi$
$840$	0	0
$841$	−9058.15	−0.371403
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9933.65	0.404412
$846$	0	0
$847$	7770.57	0.315230
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3634.96	0.146422
$852$	0	0
$853$	36920.0	1.48196	0.740982	−	0.671524i	$-0.234360\pi$
$853$	36920.0	1.48196	0.740982	+	0.671524i	$0.234360\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19243.2	0.767018	0.383509	−	0.923537i	$-0.374716\pi$
$857$	19243.2	0.767018	0.383509	+	0.923537i	$0.374716\pi$
$858$	0	0
$859$	−10510.3	−0.417470	−0.208735	−	0.977972i	$-0.566935\pi$
$859$	−10510.3	−0.417470	−0.208735	+	0.977972i	$0.566935\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45152.7	−1.78101	−0.890507	−	0.454969i	$-0.849650\pi$
$863$	−45152.7	−1.78101	−0.890507	+	0.454969i	$0.849650\pi$
$864$	0	0
$865$	7239.55	0.284569
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−17549.5	−0.685069
$870$	0	0
$871$	−7040.84	−0.273903
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7452.38	−0.287927
$876$	0	0
$877$	47931.1	1.84551	0.922757	−	0.385381i	$-0.125930\pi$
$877$	47931.1	1.84551	0.922757	+	0.385381i	$0.125930\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	644.698	0.0246543	0.0123271	−	0.999924i	$-0.496076\pi$
$881$	644.698	0.0246543	0.0123271	+	0.999924i	$0.496076\pi$
$882$	0	0
$883$	−23995.7	−0.914519	−0.457259	−	0.889333i	$-0.651169\pi$
$883$	−23995.7	−0.914519	−0.457259	+	0.889333i	$0.651169\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42649.3	1.61446	0.807228	−	0.590239i	$-0.200967\pi$
$887$	42649.3	1.61446	0.807228	+	0.590239i	$0.200967\pi$
$888$	0	0
$889$	−6328.00	−0.238734
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3203.89	−0.120061
$894$	0	0
$895$	52442.2	1.95860
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−22.5590	−0.000836914 0
$900$	0	0
$901$	11346.8	0.419554
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31698.4	−1.16430
$906$	0	0
$907$	42527.2	1.55688	0.778442	−	0.627717i	$-0.216010\pi$
$907$	42527.2	1.55688	0.778442	+	0.627717i	$0.216010\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16969.8	0.617162	0.308581	−	0.951198i	$-0.400146\pi$
$911$	16969.8	0.617162	0.308581	+	0.951198i	$0.400146\pi$
$912$	0	0
$913$	−12593.3	−0.456492
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5377.22	0.193644
$918$	0	0
$919$	−41155.7	−1.47726	−0.738630	−	0.674112i	$-0.764527\pi$
$919$	−41155.7	−1.47726	−0.738630	+	0.674112i	$0.764527\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22878.0	0.815861
$924$	0	0
$925$	−9279.67	−0.329853
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19854.4	0.701186	0.350593	−	0.936528i	$-0.385980\pi$
$929$	19854.4	0.701186	0.350593	+	0.936528i	$0.385980\pi$
$930$	0	0
$931$	−2343.07	−0.0824824
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10563.8	−0.369488
$936$	0	0
$937$	45750.0	1.59508	0.797539	−	0.603267i	$-0.206135\pi$
$937$	45750.0	1.59508	0.797539	+	0.603267i	$0.206135\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26656.7	−0.923470	−0.461735	−	0.887018i	$-0.652773\pi$
$941$	−26656.7	−0.923470	−0.461735	+	0.887018i	$0.652773\pi$
$942$	0	0
$943$	4639.04	0.160199
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47537.6	1.63122	0.815609	−	0.578604i	$-0.196402\pi$
$947$	47537.6	1.63122	0.815609	+	0.578604i	$0.196402\pi$
$948$	0	0
$949$	40643.6	1.39025
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38971.9	1.32468	0.662342	−	0.749201i	$-0.269563\pi$
$953$	38971.9	1.32468	0.662342	+	0.749201i	$0.269563\pi$
$954$	0	0
$955$	47685.8	1.61579
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20275.0	0.682706
$960$	0	0
$961$	−29791.0	−0.999999
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15446.4	−0.515271
$966$	0	0
$967$	−5600.03	−0.186231	−0.0931153	−	0.995655i	$-0.529683\pi$
$967$	−5600.03	−0.186231	−0.0931153	+	0.995655i	$0.529683\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7822.72	−0.258541	−0.129270	−	0.991609i	$-0.541264\pi$
$971$	−7822.72	−0.258541	−0.129270	+	0.991609i	$0.541264\pi$
$972$	0	0
$973$	−17367.5	−0.572228
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9072.74	−0.297096	−0.148548	−	0.988905i	$-0.547460\pi$
$977$	−9072.74	−0.297096	−0.148548	+	0.988905i	$0.547460\pi$
$978$	0	0
$979$	12221.0	0.398962
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57868.1	1.87762	0.938812	−	0.344430i	$-0.111928\pi$
$983$	57868.1	1.87762	0.938812	+	0.344430i	$0.111928\pi$
$984$	0	0
$985$	−15659.5	−0.506553
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6349.01	−0.204132
$990$	0	0
$991$	47196.7	1.51287	0.756434	−	0.654070i	$-0.226940\pi$
$991$	47196.7	1.51287	0.756434	+	0.654070i	$0.226940\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	546.286	0.0174055
$996$	0	0
$997$	20863.4	0.662740	0.331370	−	0.943501i	$-0.392489\pi$
$997$	20863.4	0.662740	0.331370	+	0.943501i	$0.392489\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.a.bb.1.1		2
3.2	odd	2		1008.4.a.bf.1.2		2
4.3	odd	2		504.4.a.l.1.1	✓	2
12.11	even	2		504.4.a.m.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.a.l.1.1	✓	2	4.3	odd	2
504.4.a.m.1.2	yes	2	12.11	even	2
1008.4.a.bb.1.1		2	1.1	even	1	trivial
1008.4.a.bf.1.2		2	3.2	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q-12.9545 q^{5} -7.00000 q^{7} +O(q^{10})\)	\(q-12.9545 q^{5} -7.00000 q^{7} -14.8634 q^{11} +37.8178 q^{13} -54.8634 q^{17} -47.8178 q^{19} -16.7723 q^{23} +42.8178 q^{25} -123.818 q^{29} +0.182195 q^{31} +90.6812 q^{35} -216.725 q^{37} -276.590 q^{41} +378.542 q^{43} +67.0021 q^{47} +49.0000 q^{49} -206.820 q^{53} +192.547 q^{55} +492.998 q^{59} +242.182 q^{61} -489.909 q^{65} -186.178 q^{67} +604.954 q^{71} +1074.72 q^{73} +104.043 q^{77} +1180.72 q^{79} +847.271 q^{83} +710.725 q^{85} -822.222 q^{89} -264.725 q^{91} +619.453 q^{95} -437.992 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 14 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - 14 * q^7	\( 2 q - 4 q^{5} - 14 q^{7} + 36 q^{11} - 12 q^{13} - 44 q^{17} - 8 q^{19} + 76 q^{23} - 2 q^{25} - 160 q^{29} + 88 q^{31} + 28 q^{35} + 180 q^{37} - 356 q^{41} + 56 q^{43} + 616 q^{47} + 98 q^{49} - 808 q^{53} + 648 q^{55} + 504 q^{59} + 572 q^{61} - 936 q^{65} + 504 q^{67} + 1188 q^{71} + 572 q^{73} - 252 q^{77} + 784 q^{79} + 1344 q^{83} + 808 q^{85} - 308 q^{89} + 84 q^{91} + 976 q^{95} + 1052 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 14 * q^7 + 36 * q^11 - 12 * q^13 - 44 * q^17 - 8 * q^19 + 76 * q^23 - 2 * q^25 - 160 * q^29 + 88 * q^31 + 28 * q^35 + 180 * q^37 - 356 * q^41 + 56 * q^43 + 616 * q^47 + 98 * q^49 - 808 * q^53 + 648 * q^55 + 504 * q^59 + 572 * q^61 - 936 * q^65 + 504 * q^67 + 1188 * q^71 + 572 * q^73 - 252 * q^77 + 784 * q^79 + 1344 * q^83 + 808 * q^85 - 308 * q^89 + 84 * q^91 + 976 * q^95 + 1052 * q^97

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