LMFDB - Embedded newform 1143.2.a.k.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 $ x^16 - 26x^14 + 269x^12 - 1408x^10 + 3924x^8 - 5655x^6 + 3886x^4 - 1107*x^2 + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.36017$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36017 q^{2} -0.149936 q^{4} +3.81085 q^{5} +4.20131 q^{7} -2.92428 q^{8} +O(q^{10})$	\(q+1.36017 q^{2} -0.149936 q^{4} +3.81085 q^{5} +4.20131 q^{7} -2.92428 q^{8} +5.18341 q^{10} +5.13086 q^{11} -2.63599 q^{13} +5.71450 q^{14} -3.67765 q^{16} -5.08847 q^{17} -3.48160 q^{19} -0.571383 q^{20} +6.97885 q^{22} +6.87803 q^{23} +9.52260 q^{25} -3.58540 q^{26} -0.629927 q^{28} -8.87365 q^{29} +2.71827 q^{31} +0.846330 q^{32} -6.92119 q^{34} +16.0106 q^{35} -6.12586 q^{37} -4.73557 q^{38} -11.1440 q^{40} +2.10756 q^{41} -9.74037 q^{43} -0.769299 q^{44} +9.35530 q^{46} -3.39416 q^{47} +10.6510 q^{49} +12.9524 q^{50} +0.395230 q^{52} -3.88387 q^{53} +19.5530 q^{55} -12.2858 q^{56} -12.0697 q^{58} -0.860571 q^{59} +3.91166 q^{61} +3.69731 q^{62} +8.50645 q^{64} -10.0454 q^{65} +13.1440 q^{67} +0.762944 q^{68} +21.7771 q^{70} -10.5463 q^{71} +15.5787 q^{73} -8.33222 q^{74} +0.522016 q^{76} +21.5564 q^{77} +12.9184 q^{79} -14.0150 q^{80} +2.86665 q^{82} +3.13017 q^{83} -19.3914 q^{85} -13.2486 q^{86} -15.0041 q^{88} +1.06384 q^{89} -11.0746 q^{91} -1.03126 q^{92} -4.61664 q^{94} -13.2679 q^{95} +4.56949 q^{97} +14.4872 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 10 * q^7	$ 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * 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 $

$2$	1.36017	0.961786	0.480893	−	0.876779i	$-0.340313\pi$
$2$	1.36017	0.961786	0.480893	+	0.876779i	$0.340313\pi$
$3$	0	0
$3$	0	0
$4$	−0.149936	−0.0749679
$5$	3.81085	1.70427	0.852133	−	0.523326i	$-0.175309\pi$
$5$	3.81085	1.70427	0.852133	+	0.523326i	$0.175309\pi$
$6$	0	0
$7$	4.20131	1.58795	0.793974	−	0.607952i	$-0.208009\pi$
$7$	4.20131	1.58795	0.793974	+	0.607952i	$0.208009\pi$
$8$	−2.92428	−1.03389
$9$	0	0
$10$	5.18341	1.63914
$11$	5.13086	1.54701	0.773506	−	0.633789i	$-0.218501\pi$
$11$	5.13086	1.54701	0.773506	+	0.633789i	$0.218501\pi$
$12$	0	0
$13$	−2.63599	−0.731093	−0.365547	−	0.930793i	$-0.619118\pi$
$13$	−2.63599	−0.731093	−0.365547	+	0.930793i	$0.619118\pi$
$14$	5.71450	1.52727
$15$	0	0
$16$	−3.67765	−0.919412
$17$	−5.08847	−1.23414	−0.617068	−	0.786910i	$-0.711680\pi$
$17$	−5.08847	−1.23414	−0.617068	+	0.786910i	$0.711680\pi$
$18$	0	0
$19$	−3.48160	−0.798733	−0.399367	−	0.916791i	$-0.630770\pi$
$19$	−3.48160	−0.798733	−0.399367	+	0.916791i	$0.630770\pi$
$20$	−0.571383	−0.127765
$21$	0	0
$22$	6.97885	1.48789
$23$	6.87803	1.43417	0.717084	−	0.696986i	$-0.245476\pi$
$23$	6.87803	1.43417	0.717084	+	0.696986i	$0.245476\pi$
$24$	0	0
$25$	9.52260	1.90452
$26$	−3.58540	−0.703155
$27$	0	0
$28$	−0.629927	−0.119045
$29$	−8.87365	−1.64779	−0.823897	−	0.566739i	$-0.808205\pi$
$29$	−8.87365	−1.64779	−0.823897	+	0.566739i	$0.808205\pi$
$30$	0	0
$31$	2.71827	0.488215	0.244108	−	0.969748i	$-0.421505\pi$
$31$	2.71827	0.488215	0.244108	+	0.969748i	$0.421505\pi$
$32$	0.846330	0.149611
$33$	0	0
$34$	−6.92119	−1.18697
$35$	16.0106	2.70628
$36$	0	0
$37$	−6.12586	−1.00709	−0.503543	−	0.863970i	$-0.667970\pi$
$37$	−6.12586	−1.00709	−0.503543	+	0.863970i	$0.667970\pi$
$38$	−4.73557	−0.768210
$39$	0	0
$40$	−11.1440	−1.76202
$41$	2.10756	0.329146	0.164573	−	0.986365i	$-0.447375\pi$
$41$	2.10756	0.329146	0.164573	+	0.986365i	$0.447375\pi$
$42$	0	0
$43$	−9.74037	−1.48539	−0.742696	−	0.669628i	$-0.766453\pi$
$43$	−9.74037	−1.48539	−0.742696	+	0.669628i	$0.766453\pi$
$44$	−0.769299	−0.115976
$45$	0	0
$46$	9.35530	1.37936
$47$	−3.39416	−0.495089	−0.247545	−	0.968877i	$-0.579624\pi$
$47$	−3.39416	−0.495089	−0.247545	+	0.968877i	$0.579624\pi$
$48$	0	0
$49$	10.6510	1.52158
$50$	12.9524	1.83174
$51$	0	0
$52$	0.395230	0.0548085
$53$	−3.88387	−0.533490	−0.266745	−	0.963767i	$-0.585948\pi$
$53$	−3.88387	−0.533490	−0.266745	+	0.963767i	$0.585948\pi$
$54$	0	0
$55$	19.5530	2.63652
$56$	−12.2858	−1.64176
$57$	0	0
$58$	−12.0697	−1.58483
$59$	−0.860571	−0.112037	−0.0560184	−	0.998430i	$-0.517841\pi$
$59$	−0.860571	−0.112037	−0.0560184	+	0.998430i	$0.517841\pi$
$60$	0	0
$61$	3.91166	0.500837	0.250418	−	0.968138i	$-0.419432\pi$
$61$	3.91166	0.500837	0.250418	+	0.968138i	$0.419432\pi$
$62$	3.69731	0.469559
$63$	0	0
$64$	8.50645	1.06331
$65$	−10.0454	−1.24598
$66$	0	0
$67$	13.1440	1.60580	0.802898	−	0.596117i	$-0.203291\pi$
$67$	13.1440	1.60580	0.802898	+	0.596117i	$0.203291\pi$
$68$	0.762944	0.0925205
$69$	0	0
$70$	21.7771	2.60287
$71$	−10.5463	−1.25162	−0.625809	−	0.779977i	$-0.715231\pi$
$71$	−10.5463	−1.25162	−0.625809	+	0.779977i	$0.715231\pi$
$72$	0	0
$73$	15.5787	1.82335	0.911673	−	0.410916i	$-0.134791\pi$
$73$	15.5787	1.82335	0.911673	+	0.410916i	$0.134791\pi$
$74$	−8.33222	−0.968601
$75$	0	0
$76$	0.522016	0.0598793
$77$	21.5564	2.45657
$78$	0	0
$79$	12.9184	1.45343	0.726717	−	0.686936i	$-0.241045\pi$
$79$	12.9184	1.45343	0.726717	+	0.686936i	$0.241045\pi$
$80$	−14.0150	−1.56692
$81$	0	0
$82$	2.86665	0.316568
$83$	3.13017	0.343581	0.171790	−	0.985134i	$-0.445045\pi$
$83$	3.13017	0.343581	0.171790	+	0.985134i	$0.445045\pi$
$84$	0	0
$85$	−19.3914	−2.10329
$86$	−13.2486	−1.42863
$87$	0	0
$88$	−15.0041	−1.59944
$89$	1.06384	0.112767	0.0563834	−	0.998409i	$-0.482043\pi$
$89$	1.06384	0.112767	0.0563834	+	0.998409i	$0.482043\pi$
$90$	0	0
$91$	−11.0746	−1.16094
$92$	−1.03126	−0.107517
$93$	0	0
$94$	−4.61664	−0.476170
$95$	−13.2679	−1.36125
$96$	0	0
$97$	4.56949	0.463961	0.231981	−	0.972720i	$-0.425479\pi$
$97$	4.56949	0.463961	0.231981	+	0.972720i	$0.425479\pi$
$98$	14.4872	1.46343
$99$	0	0
$100$	−1.42778	−0.142778
$101$	−6.22093	−0.619005	−0.309503	−	0.950899i	$-0.600163\pi$
$101$	−6.22093	−0.619005	−0.309503	+	0.950899i	$0.600163\pi$
$102$	0	0
$103$	−6.45078	−0.635614	−0.317807	−	0.948155i	$-0.602946\pi$
$103$	−6.45078	−0.635614	−0.317807	+	0.948155i	$0.602946\pi$
$104$	7.70838	0.755869
$105$	0	0
$106$	−5.28272	−0.513103
$107$	−17.2746	−1.67000	−0.834999	−	0.550251i	$-0.814532\pi$
$107$	−17.2746	−1.67000	−0.834999	+	0.550251i	$0.814532\pi$
$108$	0	0
$109$	−4.97212	−0.476242	−0.238121	−	0.971235i	$-0.576532\pi$
$109$	−4.97212	−0.476242	−0.238121	+	0.971235i	$0.576532\pi$
$110$	26.5954	2.53577
$111$	0	0
$112$	−15.4510	−1.45998
$113$	−7.01139	−0.659576	−0.329788	−	0.944055i	$-0.606977\pi$
$113$	−7.01139	−0.659576	−0.329788	+	0.944055i	$0.606977\pi$
$114$	0	0
$115$	26.2112	2.44420
$116$	1.33048	0.123532
$117$	0	0
$118$	−1.17052	−0.107755
$119$	−21.3783	−1.95974
$120$	0	0
$121$	15.3257	1.39325
$122$	5.32052	0.481698
$123$	0	0
$124$	−0.407565	−0.0366005
$125$	17.2350	1.54154
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	9.87756	0.873061
$129$	0	0
$130$	−13.6634	−1.19836
$131$	−13.0678	−1.14174	−0.570869	−	0.821041i	$-0.693394\pi$
$131$	−13.0678	−1.14174	−0.570869	+	0.821041i	$0.693394\pi$
$132$	0	0
$133$	−14.6273	−1.26835
$134$	17.8781	1.54443
$135$	0	0
$136$	14.8801	1.27596
$137$	8.66626	0.740409	0.370204	−	0.928950i	$-0.379288\pi$
$137$	8.66626	0.740409	0.370204	+	0.928950i	$0.379288\pi$
$138$	0	0
$139$	−14.7918	−1.25463	−0.627313	−	0.778767i	$-0.715845\pi$
$139$	−14.7918	−1.25463	−0.627313	+	0.778767i	$0.715845\pi$
$140$	−2.40056	−0.202884
$141$	0	0
$142$	−14.3448	−1.20379
$143$	−13.5249	−1.13101
$144$	0	0
$145$	−33.8162	−2.80828
$146$	21.1897	1.75367
$147$	0	0
$148$	0.918486	0.0754991
$149$	−12.4749	−1.02198	−0.510992	−	0.859586i	$-0.670722\pi$
$149$	−12.4749	−1.02198	−0.510992	+	0.859586i	$0.670722\pi$
$150$	0	0
$151$	2.99628	0.243834	0.121917	−	0.992540i	$-0.461096\pi$
$151$	2.99628	0.243834	0.121917	+	0.992540i	$0.461096\pi$
$152$	10.1812	0.825801
$153$	0	0
$154$	29.3203	2.36270
$155$	10.3589	0.832049
$156$	0	0
$157$	14.6474	1.16899	0.584496	−	0.811396i	$-0.301292\pi$
$157$	14.6474	1.16899	0.584496	+	0.811396i	$0.301292\pi$
$158$	17.5712	1.39789
$159$	0	0
$160$	3.22524	0.254978
$161$	28.8968	2.27738
$162$	0	0
$163$	−17.5686	−1.37608	−0.688039	−	0.725674i	$-0.741528\pi$
$163$	−17.5686	−1.37608	−0.688039	+	0.725674i	$0.741528\pi$
$164$	−0.315999	−0.0246754
$165$	0	0
$166$	4.25756	0.330451
$167$	6.96465	0.538941	0.269471	−	0.963009i	$-0.413151\pi$
$167$	6.96465	0.538941	0.269471	+	0.963009i	$0.413151\pi$
$168$	0	0
$169$	−6.05154	−0.465503
$170$	−26.3756	−2.02292
$171$	0	0
$172$	1.46043	0.111357
$173$	−12.0852	−0.918819	−0.459409	−	0.888225i	$-0.651939\pi$
$173$	−12.0852	−0.918819	−0.459409	+	0.888225i	$0.651939\pi$
$174$	0	0
$175$	40.0075	3.02428
$176$	−18.8695	−1.42234
$177$	0	0
$178$	1.44700	0.108458
$179$	−21.7044	−1.62227	−0.811133	−	0.584862i	$-0.801149\pi$
$179$	−21.7044	−1.62227	−0.811133	+	0.584862i	$0.801149\pi$
$180$	0	0
$181$	−12.2247	−0.908651	−0.454326	−	0.890836i	$-0.650120\pi$
$181$	−12.2247	−0.908651	−0.454326	+	0.890836i	$0.650120\pi$
$182$	−15.0634	−1.11657
$183$	0	0
$184$	−20.1133	−1.48277
$185$	−23.3448	−1.71634
$186$	0	0
$187$	−26.1082	−1.90922
$188$	0.508906	0.0371158
$189$	0	0
$190$	−18.0465	−1.30923
$191$	13.7276	0.993293	0.496646	−	0.867953i	$-0.334565\pi$
$191$	13.7276	0.993293	0.496646	+	0.867953i	$0.334565\pi$
$192$	0	0
$193$	−7.39341	−0.532190	−0.266095	−	0.963947i	$-0.585733\pi$
$193$	−7.39341	−0.532190	−0.266095	+	0.963947i	$0.585733\pi$
$194$	6.21528	0.446231
$195$	0	0
$196$	−1.59697	−0.114069
$197$	4.90823	0.349697	0.174849	−	0.984595i	$-0.444056\pi$
$197$	4.90823	0.349697	0.174849	+	0.984595i	$0.444056\pi$
$198$	0	0
$199$	−4.42470	−0.313659	−0.156829	−	0.987626i	$-0.550127\pi$
$199$	−4.42470	−0.313659	−0.156829	+	0.987626i	$0.550127\pi$
$200$	−27.8468	−1.96906
$201$	0	0
$202$	−8.46152	−0.595350
$203$	−37.2810	−2.61661
$204$	0	0
$205$	8.03162	0.560952
$206$	−8.77416	−0.611325
$207$	0	0
$208$	9.69426	0.672176
$209$	−17.8636	−1.23565
$210$	0	0
$211$	15.6596	1.07805	0.539025	−	0.842290i	$-0.318793\pi$
$211$	15.6596	1.07805	0.539025	+	0.842290i	$0.318793\pi$
$212$	0.582331	0.0399946
$213$	0	0
$214$	−23.4964	−1.60618
$215$	−37.1191	−2.53150
$216$	0	0
$217$	11.4203	0.775260
$218$	−6.76293	−0.458043
$219$	0	0
$220$	−2.93169	−0.197654
$221$	13.4132	0.902268
$222$	0	0
$223$	−11.4439	−0.766338	−0.383169	−	0.923678i	$-0.625167\pi$
$223$	−11.4439	−0.766338	−0.383169	+	0.923678i	$0.625167\pi$
$224$	3.55570	0.237575
$225$	0	0
$226$	−9.53669	−0.634371
$227$	2.94281	0.195321	0.0976606	−	0.995220i	$-0.468864\pi$
$227$	2.94281	0.195321	0.0976606	+	0.995220i	$0.468864\pi$
$228$	0	0
$229$	4.17845	0.276119	0.138060	−	0.990424i	$-0.455913\pi$
$229$	4.17845	0.276119	0.138060	+	0.990424i	$0.455913\pi$
$230$	35.6517	2.35080
$231$	0	0
$232$	25.9490	1.70364
$233$	6.10927	0.400231	0.200116	−	0.979772i	$-0.435868\pi$
$233$	6.10927	0.400231	0.200116	+	0.979772i	$0.435868\pi$
$234$	0	0
$235$	−12.9346	−0.843763
$236$	0.129030	0.00839916
$237$	0	0
$238$	−29.0781	−1.88485
$239$	−4.43788	−0.287063	−0.143531	−	0.989646i	$-0.545846\pi$
$239$	−4.43788	−0.287063	−0.143531	+	0.989646i	$0.545846\pi$
$240$	0	0
$241$	4.60706	0.296767	0.148383	−	0.988930i	$-0.452593\pi$
$241$	4.60706	0.296767	0.148383	+	0.988930i	$0.452593\pi$
$242$	20.8456	1.34001
$243$	0	0
$244$	−0.586498	−0.0375467
$245$	40.5896	2.59317
$246$	0	0
$247$	9.17747	0.583948
$248$	−7.94897	−0.504760
$249$	0	0
$250$	23.4425	1.48264
$251$	8.87230	0.560015	0.280007	−	0.959998i	$-0.409663\pi$
$251$	8.87230	0.560015	0.280007	+	0.959998i	$0.409663\pi$
$252$	0	0
$253$	35.2902	2.21868
$254$	1.36017	0.0853447
$255$	0	0
$256$	−3.57773	−0.223608
$257$	20.5081	1.27926	0.639631	−	0.768682i	$-0.279087\pi$
$257$	20.5081	1.27926	0.639631	+	0.768682i	$0.279087\pi$
$258$	0	0
$259$	−25.7367	−1.59920
$260$	1.50616	0.0934082
$261$	0	0
$262$	−17.7744	−1.09811
$263$	7.23530	0.446148	0.223074	−	0.974802i	$-0.428391\pi$
$263$	7.23530	0.446148	0.223074	+	0.974802i	$0.428391\pi$
$264$	0	0
$265$	−14.8009	−0.909209
$266$	−19.8956	−1.21988
$267$	0	0
$268$	−1.97076	−0.120383
$269$	4.48256	0.273306	0.136653	−	0.990619i	$-0.456365\pi$
$269$	4.48256	0.273306	0.136653	+	0.990619i	$0.456365\pi$
$270$	0	0
$271$	−15.3960	−0.935239	−0.467620	−	0.883930i	$-0.654888\pi$
$271$	−15.3960	−0.935239	−0.467620	+	0.883930i	$0.654888\pi$
$272$	18.7136	1.13468
$273$	0	0
$274$	11.7876	0.712115
$275$	48.8592	2.94632
$276$	0	0
$277$	14.7500	0.886242	0.443121	−	0.896462i	$-0.353871\pi$
$277$	14.7500	0.886242	0.443121	+	0.896462i	$0.353871\pi$
$278$	−20.1194	−1.20668
$279$	0	0
$280$	−46.8195	−2.79800
$281$	0.941799	0.0561830	0.0280915	−	0.999605i	$-0.491057\pi$
$281$	0.941799	0.0561830	0.0280915	+	0.999605i	$0.491057\pi$
$282$	0	0
$283$	21.8299	1.29765	0.648827	−	0.760936i	$-0.275260\pi$
$283$	21.8299	1.29765	0.648827	+	0.760936i	$0.275260\pi$
$284$	1.58127	0.0938311
$285$	0	0
$286$	−18.3962	−1.08779
$287$	8.85454	0.522667
$288$	0	0
$289$	8.89253	0.523090
$290$	−45.9958	−2.70096
$291$	0	0
$292$	−2.33580	−0.136692
$293$	−25.7330	−1.50334	−0.751670	−	0.659539i	$-0.770751\pi$
$293$	−25.7330	−1.50334	−0.751670	+	0.659539i	$0.770751\pi$
$294$	0	0
$295$	−3.27951	−0.190940
$296$	17.9137	1.04121
$297$	0	0
$298$	−16.9680	−0.982929
$299$	−18.1304	−1.04851
$300$	0	0
$301$	−40.9224	−2.35873
$302$	4.07545	0.234516
$303$	0	0
$304$	12.8041	0.734365
$305$	14.9068	0.853559
$306$	0	0
$307$	4.98285	0.284386	0.142193	−	0.989839i	$-0.454585\pi$
$307$	4.98285	0.284386	0.142193	+	0.989839i	$0.454585\pi$
$308$	−3.23207	−0.184164
$309$	0	0
$310$	14.0899	0.800253
$311$	24.1803	1.37114	0.685569	−	0.728008i	$-0.259554\pi$
$311$	24.1803	1.37114	0.685569	+	0.728008i	$0.259554\pi$
$312$	0	0
$313$	31.5529	1.78348	0.891738	−	0.452552i	$-0.149486\pi$
$313$	31.5529	1.78348	0.891738	+	0.452552i	$0.149486\pi$
$314$	19.9230	1.12432
$315$	0	0
$316$	−1.93693	−0.108961
$317$	8.71577	0.489526	0.244763	−	0.969583i	$-0.421290\pi$
$317$	8.71577	0.489526	0.244763	+	0.969583i	$0.421290\pi$
$318$	0	0
$319$	−45.5294	−2.54916
$320$	32.4168	1.81216
$321$	0	0
$322$	39.3045	2.19036
$323$	17.7160	0.985745
$324$	0	0
$325$	−25.1015	−1.39238
$326$	−23.8963	−1.32349
$327$	0	0
$328$	−6.16310	−0.340301
$329$	−14.2599	−0.786176
$330$	0	0
$331$	6.75225	0.371137	0.185569	−	0.982631i	$-0.440587\pi$
$331$	6.75225	0.371137	0.185569	+	0.982631i	$0.440587\pi$
$332$	−0.469324	−0.0257575
$333$	0	0
$334$	9.47312	0.518346
$335$	50.0899	2.73670
$336$	0	0
$337$	12.7820	0.696279	0.348139	−	0.937443i	$-0.386814\pi$
$337$	12.7820	0.696279	0.348139	+	0.937443i	$0.386814\pi$
$338$	−8.23112	−0.447714
$339$	0	0
$340$	2.90747	0.157679
$341$	13.9471	0.755275
$342$	0	0
$343$	15.3392	0.828238
$344$	28.4836	1.53573
$345$	0	0
$346$	−16.4379	−0.883707
$347$	−11.9717	−0.642676	−0.321338	−	0.946965i	$-0.604133\pi$
$347$	−11.9717	−0.642676	−0.321338	+	0.946965i	$0.604133\pi$
$348$	0	0
$349$	24.5092	1.31195	0.655974	−	0.754783i	$-0.272258\pi$
$349$	24.5092	1.31195	0.655974	+	0.754783i	$0.272258\pi$
$350$	54.4170	2.90871
$351$	0	0
$352$	4.34240	0.231451
$353$	−25.6594	−1.36571	−0.682855	−	0.730554i	$-0.739262\pi$
$353$	−25.6594	−1.36571	−0.682855	+	0.730554i	$0.739262\pi$
$354$	0	0
$355$	−40.1905	−2.13309
$356$	−0.159508	−0.00845389
$357$	0	0
$358$	−29.5217	−1.56027
$359$	−19.3783	−1.02275	−0.511373	−	0.859359i	$-0.670863\pi$
$359$	−19.3783	−1.02275	−0.511373	+	0.859359i	$0.670863\pi$
$360$	0	0
$361$	−6.87848	−0.362025
$362$	−16.6276	−0.873928
$363$	0	0
$364$	1.66048	0.0870330
$365$	59.3681	3.10747
$366$	0	0
$367$	−6.68336	−0.348869	−0.174434	−	0.984669i	$-0.555810\pi$
$367$	−6.68336	−0.348869	−0.174434	+	0.984669i	$0.555810\pi$
$368$	−25.2950	−1.31859
$369$	0	0
$370$	−31.7529	−1.65075
$371$	−16.3173	−0.847155
$372$	0	0
$373$	26.1247	1.35269	0.676344	−	0.736586i	$-0.263563\pi$
$373$	26.1247	1.35269	0.676344	+	0.736586i	$0.263563\pi$
$374$	−35.5116	−1.83626
$375$	0	0
$376$	9.92547	0.511867
$377$	23.3909	1.20469
$378$	0	0
$379$	35.3991	1.81833	0.909165	−	0.416436i	$-0.136721\pi$
$379$	35.3991	1.81833	0.909165	+	0.416436i	$0.136721\pi$
$380$	1.98933	0.102050
$381$	0	0
$382$	18.6719	0.955335
$383$	−9.81290	−0.501416	−0.250708	−	0.968063i	$-0.580663\pi$
$383$	−9.81290	−0.501416	−0.250708	+	0.968063i	$0.580663\pi$
$384$	0	0
$385$	82.1481	4.18666
$386$	−10.0563	−0.511852
$387$	0	0
$388$	−0.685130	−0.0347822
$389$	31.6881	1.60665	0.803325	−	0.595540i	$-0.203062\pi$
$389$	31.6881	1.60665	0.803325	+	0.595540i	$0.203062\pi$
$390$	0	0
$391$	−34.9987	−1.76996
$392$	−31.1466	−1.57314
$393$	0	0
$394$	6.67603	0.336334
$395$	49.2302	2.47704
$396$	0	0
$397$	−11.6152	−0.582949	−0.291474	−	0.956579i	$-0.594146\pi$
$397$	−11.6152	−0.582949	−0.291474	+	0.956579i	$0.594146\pi$
$398$	−6.01835	−0.301672
$399$	0	0
$400$	−35.0208	−1.75104
$401$	−6.50746	−0.324967	−0.162483	−	0.986711i	$-0.551950\pi$
$401$	−6.50746	−0.324967	−0.162483	+	0.986711i	$0.551950\pi$
$402$	0	0
$403$	−7.16534	−0.356931
$404$	0.932739	0.0464055
$405$	0	0
$406$	−50.7085	−2.51662
$407$	−31.4309	−1.55797
$408$	0	0
$409$	−19.3726	−0.957915	−0.478957	−	0.877838i	$-0.658985\pi$
$409$	−19.3726	−0.957915	−0.478957	+	0.877838i	$0.658985\pi$
$410$	10.9244	0.539516
$411$	0	0
$412$	0.967203	0.0476507
$413$	−3.61553	−0.177909
$414$	0	0
$415$	11.9286	0.585553
$416$	−2.23092	−0.109380
$417$	0	0
$418$	−24.2975	−1.18843
$419$	−20.4226	−0.997707	−0.498853	−	0.866686i	$-0.666245\pi$
$419$	−20.4226	−0.997707	−0.498853	+	0.866686i	$0.666245\pi$
$420$	0	0
$421$	−1.88195	−0.0917204	−0.0458602	−	0.998948i	$-0.514603\pi$
$421$	−1.88195	−0.0917204	−0.0458602	+	0.998948i	$0.514603\pi$
$422$	21.2997	1.03685
$423$	0	0
$424$	11.3575	0.551570
$425$	−48.4555	−2.35044
$426$	0	0
$427$	16.4341	0.795302
$428$	2.59008	0.125196
$429$	0	0
$430$	−50.4884	−2.43476
$431$	7.73832	0.372742	0.186371	−	0.982479i	$-0.440327\pi$
$431$	7.73832	0.372742	0.186371	+	0.982479i	$0.440327\pi$
$432$	0	0
$433$	35.1363	1.68854	0.844271	−	0.535916i	$-0.180034\pi$
$433$	35.1363	1.68854	0.844271	+	0.535916i	$0.180034\pi$
$434$	15.5336	0.745634
$435$	0	0
$436$	0.745498	0.0357029
$437$	−23.9465	−1.14552
$438$	0	0
$439$	24.1890	1.15448	0.577239	−	0.816575i	$-0.304130\pi$
$439$	24.1890	1.15448	0.577239	+	0.816575i	$0.304130\pi$
$440$	−57.1783	−2.72587
$441$	0	0
$442$	18.2442	0.867789
$443$	1.17625	0.0558852	0.0279426	−	0.999610i	$-0.491104\pi$
$443$	1.17625	0.0558852	0.0279426	+	0.999610i	$0.491104\pi$
$444$	0	0
$445$	4.05414	0.192185
$446$	−15.5656	−0.737053
$447$	0	0
$448$	35.7383	1.68847
$449$	−24.8321	−1.17190	−0.585949	−	0.810348i	$-0.699278\pi$
$449$	−24.8321	−1.17190	−0.585949	+	0.810348i	$0.699278\pi$
$450$	0	0
$451$	10.8136	0.509193
$452$	1.05126	0.0494470
$453$	0	0
$454$	4.00272	0.187857
$455$	−42.2038	−1.97855
$456$	0	0
$457$	−37.1604	−1.73829	−0.869146	−	0.494556i	$-0.835331\pi$
$457$	−37.1604	−1.73829	−0.869146	+	0.494556i	$0.835331\pi$
$458$	5.68340	0.265568
$459$	0	0
$460$	−3.92999	−0.183237
$461$	13.5898	0.632939	0.316470	−	0.948603i	$-0.397502\pi$
$461$	13.5898	0.632939	0.316470	+	0.948603i	$0.397502\pi$
$462$	0	0
$463$	−36.0821	−1.67688	−0.838438	−	0.544996i	$-0.816531\pi$
$463$	−36.0821	−1.67688	−0.838438	+	0.544996i	$0.816531\pi$
$464$	32.6341	1.51500
$465$	0	0
$466$	8.30964	0.384937
$467$	9.39217	0.434618	0.217309	−	0.976103i	$-0.430272\pi$
$467$	9.39217	0.434618	0.217309	+	0.976103i	$0.430272\pi$
$468$	0	0
$469$	55.2221	2.54992
$470$	−17.5933	−0.811520
$471$	0	0
$472$	2.51655	0.115834
$473$	−49.9765	−2.29792
$474$	0	0
$475$	−33.1539	−1.52120
$476$	3.20537	0.146918
$477$	0	0
$478$	−6.03628	−0.276093
$479$	16.8408	0.769478	0.384739	−	0.923025i	$-0.374292\pi$
$479$	16.8408	0.769478	0.384739	+	0.923025i	$0.374292\pi$
$480$	0	0
$481$	16.1477	0.736273
$482$	6.26639	0.285426
$483$	0	0
$484$	−2.29787	−0.104449
$485$	17.4137	0.790713
$486$	0	0
$487$	−1.74688	−0.0791585	−0.0395792	−	0.999216i	$-0.512602\pi$
$487$	−1.74688	−0.0791585	−0.0395792	+	0.999216i	$0.512602\pi$
$488$	−11.4388	−0.517809
$489$	0	0
$490$	55.2087	2.49408
$491$	−1.83853	−0.0829717	−0.0414858	−	0.999139i	$-0.513209\pi$
$491$	−1.83853	−0.0829717	−0.0414858	+	0.999139i	$0.513209\pi$
$492$	0	0
$493$	45.1533	2.03360
$494$	12.4829	0.561633
$495$	0	0
$496$	−9.99683	−0.448871
$497$	−44.3084	−1.98750
$498$	0	0
$499$	−28.9253	−1.29488	−0.647438	−	0.762118i	$-0.724159\pi$
$499$	−28.9253	−1.29488	−0.647438	+	0.762118i	$0.724159\pi$
$500$	−2.58414	−0.115566
$501$	0	0
$502$	12.0678	0.538614
$503$	−28.3575	−1.26440	−0.632199	−	0.774806i	$-0.717847\pi$
$503$	−28.3575	−1.26440	−0.632199	+	0.774806i	$0.717847\pi$
$504$	0	0
$505$	−23.7070	−1.05495
$506$	48.0007	2.13389
$507$	0	0
$508$	−0.149936	−0.00665232
$509$	21.7242	0.962906	0.481453	−	0.876472i	$-0.340109\pi$
$509$	21.7242	0.962906	0.481453	+	0.876472i	$0.340109\pi$
$510$	0	0
$511$	65.4509	2.89538
$512$	−24.6214	−1.08812
$513$	0	0
$514$	27.8945	1.23038
$515$	−24.5830	−1.08326
$516$	0	0
$517$	−17.4150	−0.765909
$518$	−35.0063	−1.53809
$519$	0	0
$520$	29.3755	1.28820
$521$	35.3198	1.54739	0.773694	−	0.633560i	$-0.218407\pi$
$521$	35.3198	1.54739	0.773694	+	0.633560i	$0.218407\pi$
$522$	0	0
$523$	−19.0682	−0.833793	−0.416897	−	0.908954i	$-0.636882\pi$
$523$	−19.0682	−0.833793	−0.416897	+	0.908954i	$0.636882\pi$
$524$	1.95933	0.0855936
$525$	0	0
$526$	9.84124	0.429099
$527$	−13.8318	−0.602524
$528$	0	0
$529$	24.3073	1.05684
$530$	−20.1317	−0.874464
$531$	0	0
$532$	2.19315	0.0950852
$533$	−5.55552	−0.240636
$534$	0	0
$535$	−65.8310	−2.84612
$536$	−38.4367	−1.66021
$537$	0	0
$538$	6.09705	0.262862
$539$	54.6490	2.35390
$540$	0	0
$541$	37.3314	1.60500	0.802501	−	0.596651i	$-0.203502\pi$
$541$	37.3314	1.60500	0.802501	+	0.596651i	$0.203502\pi$
$542$	−20.9412	−0.899500
$543$	0	0
$544$	−4.30653	−0.184641
$545$	−18.9480	−0.811644
$546$	0	0
$547$	−3.81943	−0.163307	−0.0816535	−	0.996661i	$-0.526020\pi$
$547$	−3.81943	−0.163307	−0.0816535	+	0.996661i	$0.526020\pi$
$548$	−1.29938	−0.0555069
$549$	0	0
$550$	66.4568	2.83373
$551$	30.8945	1.31615
$552$	0	0
$553$	54.2743	2.30798
$554$	20.0625	0.852375
$555$	0	0
$556$	2.21782	0.0940567
$557$	6.50616	0.275675	0.137837	−	0.990455i	$-0.455985\pi$
$557$	6.50616	0.275675	0.137837	+	0.990455i	$0.455985\pi$
$558$	0	0
$559$	25.6756	1.08596
$560$	−58.8813	−2.48819
$561$	0	0
$562$	1.28101	0.0540360
$563$	−30.3002	−1.27700	−0.638500	−	0.769621i	$-0.720445\pi$
$563$	−30.3002	−1.27700	−0.638500	+	0.769621i	$0.720445\pi$
$564$	0	0
$565$	−26.7194	−1.12409
$566$	29.6924	1.24806
$567$	0	0
$568$	30.8404	1.29403
$569$	−28.3544	−1.18868	−0.594340	−	0.804214i	$-0.702587\pi$
$569$	−28.3544	−1.18868	−0.594340	+	0.804214i	$0.702587\pi$
$570$	0	0
$571$	−11.4841	−0.480595	−0.240297	−	0.970699i	$-0.577245\pi$
$571$	−11.4841	−0.480595	−0.240297	+	0.970699i	$0.577245\pi$
$572$	2.02787	0.0847894
$573$	0	0
$574$	12.0437	0.502694
$575$	65.4968	2.73140
$576$	0	0
$577$	33.8969	1.41115	0.705573	−	0.708637i	$-0.250690\pi$
$577$	33.8969	1.41115	0.705573	+	0.708637i	$0.250690\pi$
$578$	12.0954	0.503100
$579$	0	0
$580$	5.07025	0.210531
$581$	13.1508	0.545588
$582$	0	0
$583$	−19.9276	−0.825316
$584$	−45.5564	−1.88514
$585$	0	0
$586$	−35.0013	−1.44589
$587$	31.6046	1.30446	0.652231	−	0.758020i	$-0.273833\pi$
$587$	31.6046	1.30446	0.652231	+	0.758020i	$0.273833\pi$
$588$	0	0
$589$	−9.46391	−0.389954
$590$	−4.46069	−0.183644
$591$	0	0
$592$	22.5288	0.925926
$593$	2.45836	0.100953	0.0504765	−	0.998725i	$-0.483926\pi$
$593$	2.45836	0.100953	0.0504765	+	0.998725i	$0.483926\pi$
$594$	0	0
$595$	−81.4694	−3.33992
$596$	1.87043	0.0766159
$597$	0	0
$598$	−24.6605	−1.00844
$599$	32.4195	1.32462	0.662312	−	0.749228i	$-0.269575\pi$
$599$	32.4195	1.32462	0.662312	+	0.749228i	$0.269575\pi$
$600$	0	0
$601$	−30.9869	−1.26398	−0.631992	−	0.774975i	$-0.717762\pi$
$601$	−30.9869	−1.26398	−0.631992	+	0.774975i	$0.717762\pi$
$602$	−55.6614	−2.26859
$603$	0	0
$604$	−0.449249	−0.0182797
$605$	58.4041	2.37446
$606$	0	0
$607$	31.8833	1.29410	0.647052	−	0.762446i	$-0.276002\pi$
$607$	31.8833	1.29410	0.647052	+	0.762446i	$0.276002\pi$
$608$	−2.94658	−0.119500
$609$	0	0
$610$	20.2757	0.820941
$611$	8.94699	0.361956
$612$	0	0
$613$	−10.3436	−0.417774	−0.208887	−	0.977940i	$-0.566984\pi$
$613$	−10.3436	−0.417774	−0.208887	+	0.977940i	$0.566984\pi$
$614$	6.77753	0.273519
$615$	0	0
$616$	−63.0368	−2.53983
$617$	45.6478	1.83771	0.918856	−	0.394593i	$-0.129114\pi$
$617$	45.6478	1.83771	0.918856	+	0.394593i	$0.129114\pi$
$618$	0	0
$619$	26.0010	1.04507	0.522535	−	0.852618i	$-0.324986\pi$
$619$	26.0010	1.04507	0.522535	+	0.852618i	$0.324986\pi$
$620$	−1.55317	−0.0623769
$621$	0	0
$622$	32.8893	1.31874
$623$	4.46953	0.179068
$624$	0	0
$625$	18.0670	0.722679
$626$	42.9173	1.71532
$627$	0	0
$628$	−2.19617	−0.0876369
$629$	31.1713	1.24288
$630$	0	0
$631$	23.2177	0.924282	0.462141	−	0.886807i	$-0.347081\pi$
$631$	23.2177	0.924282	0.462141	+	0.886807i	$0.347081\pi$
$632$	−37.7770	−1.50269
$633$	0	0
$634$	11.8549	0.470820
$635$	3.81085	0.151229
$636$	0	0
$637$	−28.0761	−1.11241
$638$	−61.9278	−2.45175
$639$	0	0
$640$	37.6419	1.48793
$641$	−3.46335	−0.136794	−0.0683971	−	0.997658i	$-0.521788\pi$
$641$	−3.46335	−0.136794	−0.0683971	+	0.997658i	$0.521788\pi$
$642$	0	0
$643$	−20.3858	−0.803937	−0.401968	−	0.915654i	$-0.631674\pi$
$643$	−20.3858	−0.803937	−0.401968	+	0.915654i	$0.631674\pi$
$644$	−4.33266	−0.170731
$645$	0	0
$646$	24.0968	0.948075
$647$	−15.1587	−0.595949	−0.297974	−	0.954574i	$-0.596311\pi$
$647$	−15.1587	−0.595949	−0.297974	+	0.954574i	$0.596311\pi$
$648$	0	0
$649$	−4.41547	−0.173322
$650$	−34.1424	−1.33917
$651$	0	0
$652$	2.63416	0.103162
$653$	18.5147	0.724536	0.362268	−	0.932074i	$-0.382003\pi$
$653$	18.5147	0.724536	0.362268	+	0.932074i	$0.382003\pi$
$654$	0	0
$655$	−49.7994	−1.94582
$656$	−7.75088	−0.302621
$657$	0	0
$658$	−19.3959	−0.756133
$659$	−20.0515	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$659$	−20.0515	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$660$	0	0
$661$	8.67512	0.337423	0.168711	−	0.985665i	$-0.446039\pi$
$661$	8.67512	0.337423	0.168711	+	0.985665i	$0.446039\pi$
$662$	9.18422	0.356955
$663$	0	0
$664$	−9.15349	−0.355224
$665$	−55.7424	−2.16160
$666$	0	0
$667$	−61.0332	−2.36322
$668$	−1.04425	−0.0404033
$669$	0	0
$670$	68.1308	2.63212
$671$	20.0702	0.774801
$672$	0	0
$673$	−31.9367	−1.23107	−0.615535	−	0.788109i	$-0.711060\pi$
$673$	−31.9367	−1.23107	−0.615535	+	0.788109i	$0.711060\pi$
$674$	17.3857	0.669671
$675$	0	0
$676$	0.907341	0.0348977
$677$	−36.6170	−1.40731	−0.703653	−	0.710543i	$-0.748449\pi$
$677$	−36.6170	−1.40731	−0.703653	+	0.710543i	$0.748449\pi$
$678$	0	0
$679$	19.1979	0.736746
$680$	56.7059	2.17457
$681$	0	0
$682$	18.9704	0.726413
$683$	−5.47798	−0.209609	−0.104805	−	0.994493i	$-0.533422\pi$
$683$	−5.47798	−0.209609	−0.104805	+	0.994493i	$0.533422\pi$
$684$	0	0
$685$	33.0258	1.26185
$686$	20.8639	0.796587
$687$	0	0
$688$	35.8217	1.36569
$689$	10.2379	0.390031
$690$	0	0
$691$	9.38430	0.356996	0.178498	−	0.983940i	$-0.442876\pi$
$691$	9.38430	0.356996	0.178498	+	0.983940i	$0.442876\pi$
$692$	1.81200	0.0688819
$693$	0	0
$694$	−16.2836	−0.618116
$695$	−56.3695	−2.13822
$696$	0	0
$697$	−10.7243	−0.406211
$698$	33.3367	1.26181
$699$	0	0
$700$	−5.99855	−0.226724
$701$	−11.5491	−0.436203	−0.218101	−	0.975926i	$-0.569986\pi$
$701$	−11.5491	−0.436203	−0.218101	+	0.975926i	$0.569986\pi$
$702$	0	0
$703$	21.3278	0.804393
$704$	43.6454	1.64495
$705$	0	0
$706$	−34.9011	−1.31352
$707$	−26.1361	−0.982948
$708$	0	0
$709$	−20.4046	−0.766311	−0.383155	−	0.923684i	$-0.625163\pi$
$709$	−20.4046	−0.766311	−0.383155	+	0.923684i	$0.625163\pi$
$710$	−54.6659	−2.05157
$711$	0	0
$712$	−3.11097	−0.116588
$713$	18.6963	0.700183
$714$	0	0
$715$	−51.5415	−1.92754
$716$	3.25427	0.121618
$717$	0	0
$718$	−26.3577	−0.983662
$719$	41.2162	1.53710	0.768552	−	0.639787i	$-0.220977\pi$
$719$	41.2162	1.53710	0.768552	+	0.639787i	$0.220977\pi$
$720$	0	0
$721$	−27.1018	−1.00932
$722$	−9.35591	−0.348191
$723$	0	0
$724$	1.83291	0.0681197
$725$	−84.5002	−3.13826
$726$	0	0
$727$	0.381582	0.0141521	0.00707605	−	0.999975i	$-0.497748\pi$
$727$	0.381582	0.0141521	0.00707605	+	0.999975i	$0.497748\pi$
$728$	32.3853	1.20028
$729$	0	0
$730$	80.7507	2.98872
$731$	49.5636	1.83318
$732$	0	0
$733$	30.7194	1.13465	0.567324	−	0.823495i	$-0.307979\pi$
$733$	30.7194	1.13465	0.567324	+	0.823495i	$0.307979\pi$
$734$	−9.09051	−0.335537
$735$	0	0
$736$	5.82109	0.214568
$737$	67.4400	2.48419
$738$	0	0
$739$	18.8373	0.692940	0.346470	−	0.938061i	$-0.387380\pi$
$739$	18.8373	0.692940	0.346470	+	0.938061i	$0.387380\pi$
$740$	3.50021	0.128670
$741$	0	0
$742$	−22.1944	−0.814781
$743$	17.3527	0.636609	0.318305	−	0.947989i	$-0.396887\pi$
$743$	17.3527	0.636609	0.318305	+	0.947989i	$0.396887\pi$
$744$	0	0
$745$	−47.5400	−1.74173
$746$	35.5341	1.30100
$747$	0	0
$748$	3.91456	0.143130
$749$	−72.5760	−2.65187
$750$	0	0
$751$	−46.3700	−1.69206	−0.846032	−	0.533132i	$-0.821015\pi$
$751$	−46.3700	−1.69206	−0.846032	+	0.533132i	$0.821015\pi$
$752$	12.4825	0.455191
$753$	0	0
$754$	31.8156	1.15866
$755$	11.4184	0.415557
$756$	0	0
$757$	−11.0648	−0.402156	−0.201078	−	0.979575i	$-0.564444\pi$
$757$	−11.0648	−0.402156	−0.201078	+	0.979575i	$0.564444\pi$
$758$	48.1488	1.74884
$759$	0	0
$760$	38.7989	1.40738
$761$	−27.8839	−1.01079	−0.505395	−	0.862888i	$-0.668653\pi$
$761$	−27.8839	−1.01079	−0.505395	+	0.862888i	$0.668653\pi$
$762$	0	0
$763$	−20.8894	−0.756248
$764$	−2.05825	−0.0744650
$765$	0	0
$766$	−13.3472	−0.482254
$767$	2.26846	0.0819093
$768$	0	0
$769$	−3.33132	−0.120130	−0.0600652	−	0.998194i	$-0.519131\pi$
$769$	−3.33132	−0.120130	−0.0600652	+	0.998194i	$0.519131\pi$
$770$	111.735	4.02667
$771$	0	0
$772$	1.10854	0.0398971
$773$	11.8837	0.427426	0.213713	−	0.976896i	$-0.431444\pi$
$773$	11.8837	0.427426	0.213713	+	0.976896i	$0.431444\pi$
$774$	0	0
$775$	25.8850	0.929816
$776$	−13.3625	−0.479684
$777$	0	0
$778$	43.1012	1.54525
$779$	−7.33769	−0.262900
$780$	0	0
$781$	−54.1117	−1.93627
$782$	−47.6041	−1.70232
$783$	0	0
$784$	−39.1708	−1.39896
$785$	55.8192	1.99227
$786$	0	0
$787$	39.4099	1.40481	0.702406	−	0.711776i	$-0.252109\pi$
$787$	39.4099	1.40481	0.702406	+	0.711776i	$0.252109\pi$
$788$	−0.735919	−0.0262160
$789$	0	0
$790$	66.9614	2.38238
$791$	−29.4571	−1.04737
$792$	0	0
$793$	−10.3111	−0.366158
$794$	−15.7986	−0.560672
$795$	0	0
$796$	0.663421	0.0235143
$797$	−55.4958	−1.96576	−0.982881	−	0.184241i	$-0.941017\pi$
$797$	−55.4958	−1.96576	−0.982881	+	0.184241i	$0.941017\pi$
$798$	0	0
$799$	17.2711	0.611007
$800$	8.05927	0.284938
$801$	0	0
$802$	−8.85125	−0.312549
$803$	79.9320	2.82074
$804$	0	0
$805$	110.121	3.88127
$806$	−9.74608	−0.343291
$807$	0	0
$808$	18.1917	0.639983
$809$	−31.3721	−1.10298	−0.551491	−	0.834181i	$-0.685941\pi$
$809$	−31.3721	−1.10298	−0.551491	+	0.834181i	$0.685941\pi$
$810$	0	0
$811$	22.7537	0.798991	0.399496	−	0.916735i	$-0.369185\pi$
$811$	22.7537	0.798991	0.399496	+	0.916735i	$0.369185\pi$
$812$	5.58975	0.196162
$813$	0	0
$814$	−42.7514	−1.49844
$815$	−66.9513	−2.34520
$816$	0	0
$817$	33.9121	1.18643
$818$	−26.3501	−0.921309
$819$	0	0
$820$	−1.20423	−0.0420534
$821$	34.8957	1.21787	0.608934	−	0.793221i	$-0.291597\pi$
$821$	34.8957	1.21787	0.608934	+	0.793221i	$0.291597\pi$
$822$	0	0
$823$	16.3189	0.568840	0.284420	−	0.958700i	$-0.408199\pi$
$823$	16.3189	0.568840	0.284420	+	0.958700i	$0.408199\pi$
$824$	18.8639	0.657155
$825$	0	0
$826$	−4.91774	−0.171110
$827$	39.3623	1.36876	0.684380	−	0.729126i	$-0.260073\pi$
$827$	39.3623	1.36876	0.684380	+	0.729126i	$0.260073\pi$
$828$	0	0
$829$	−42.0024	−1.45880	−0.729402	−	0.684086i	$-0.760201\pi$
$829$	−42.0024	−1.45880	−0.729402	+	0.684086i	$0.760201\pi$
$830$	16.2249	0.563176
$831$	0	0
$832$	−22.4230	−0.777376
$833$	−54.1975	−1.87783
$834$	0	0
$835$	26.5413	0.918499
$836$	2.67839	0.0926341
$837$	0	0
$838$	−27.7782	−0.959581
$839$	8.89866	0.307216	0.153608	−	0.988132i	$-0.450911\pi$
$839$	8.89866	0.307216	0.153608	+	0.988132i	$0.450911\pi$
$840$	0	0
$841$	49.7416	1.71523
$842$	−2.55977	−0.0882154
$843$	0	0
$844$	−2.34793	−0.0808191
$845$	−23.0615	−0.793340
$846$	0	0
$847$	64.3882	2.21240
$848$	14.2835	0.490497
$849$	0	0
$850$	−65.9077	−2.26062
$851$	−42.1339	−1.44433
$852$	0	0
$853$	43.2679	1.48146	0.740732	−	0.671801i	$-0.234479\pi$
$853$	43.2679	1.48146	0.740732	+	0.671801i	$0.234479\pi$
$854$	22.3532	0.764911
$855$	0	0
$856$	50.5158	1.72659
$857$	−36.6021	−1.25031	−0.625153	−	0.780502i	$-0.714963\pi$
$857$	−36.6021	−1.25031	−0.625153	+	0.780502i	$0.714963\pi$
$858$	0	0
$859$	9.50827	0.324418	0.162209	−	0.986756i	$-0.448138\pi$
$859$	9.50827	0.324418	0.162209	+	0.986756i	$0.448138\pi$
$860$	5.56548	0.189781
$861$	0	0
$862$	10.5254	0.358498
$863$	46.1979	1.57260	0.786298	−	0.617848i	$-0.211995\pi$
$863$	46.1979	1.57260	0.786298	+	0.617848i	$0.211995\pi$
$864$	0	0
$865$	−46.0548	−1.56591
$866$	47.7914	1.62402
$867$	0	0
$868$	−1.71231	−0.0581196
$869$	66.2826	2.24848
$870$	0	0
$871$	−34.6475	−1.17399
$872$	14.5399	0.492382
$873$	0	0
$874$	−32.5714	−1.10174
$875$	72.4096	2.44789
$876$	0	0
$877$	4.81661	0.162645	0.0813227	−	0.996688i	$-0.474086\pi$
$877$	4.81661	0.162645	0.0813227	+	0.996688i	$0.474086\pi$
$878$	32.9012	1.11036
$879$	0	0
$880$	−71.9089	−2.42405
$881$	−26.8459	−0.904462	−0.452231	−	0.891901i	$-0.649372\pi$
$881$	−26.8459	−0.904462	−0.452231	+	0.891901i	$0.649372\pi$
$882$	0	0
$883$	9.05018	0.304563	0.152281	−	0.988337i	$-0.451338\pi$
$883$	9.05018	0.304563	0.152281	+	0.988337i	$0.451338\pi$
$884$	−2.01111	−0.0676411
$885$	0	0
$886$	1.59990	0.0537496
$887$	21.0242	0.705925	0.352962	−	0.935637i	$-0.385174\pi$
$887$	21.0242	0.705925	0.352962	+	0.935637i	$0.385174\pi$
$888$	0	0
$889$	4.20131	0.140908
$890$	5.51432	0.184840
$891$	0	0
$892$	1.71585	0.0574508
$893$	11.8171	0.395444
$894$	0	0
$895$	−82.7124	−2.76477
$896$	41.4987	1.38638
$897$	0	0
$898$	−33.7759	−1.12712
$899$	−24.1209	−0.804479
$900$	0	0
$901$	19.7629	0.658399
$902$	14.7084	0.489735
$903$	0	0
$904$	20.5033	0.681929
$905$	−46.5864	−1.54858
$906$	0	0
$907$	−9.49947	−0.315425	−0.157712	−	0.987485i	$-0.550412\pi$
$907$	−9.49947	−0.315425	−0.157712	+	0.987485i	$0.550412\pi$
$908$	−0.441232	−0.0146428
$909$	0	0
$910$	−57.4044	−1.90294
$911$	7.22549	0.239391	0.119696	−	0.992811i	$-0.461808\pi$
$911$	7.22549	0.239391	0.119696	+	0.992811i	$0.461808\pi$
$912$	0	0
$913$	16.0605	0.531523
$914$	−50.5445	−1.67186
$915$	0	0
$916$	−0.626498	−0.0207001
$917$	−54.9019	−1.81302
$918$	0	0
$919$	25.6819	0.847169	0.423584	−	0.905857i	$-0.360772\pi$
$919$	25.6819	0.847169	0.423584	+	0.905857i	$0.360772\pi$
$920$	−76.6488	−2.52704
$921$	0	0
$922$	18.4844	0.608752
$923$	27.8000	0.915049
$924$	0	0
$925$	−58.3342	−1.91802
$926$	−49.0778	−1.61280
$927$	0	0
$928$	−7.51004	−0.246529
$929$	−29.8044	−0.977850	−0.488925	−	0.872326i	$-0.662611\pi$
$929$	−29.8044	−0.977850	−0.488925	+	0.872326i	$0.662611\pi$
$930$	0	0
$931$	−37.0826	−1.21533
$932$	−0.915997	−0.0300045
$933$	0	0
$934$	12.7750	0.418009
$935$	−99.4946	−3.25382
$936$	0	0
$937$	45.1050	1.47352	0.736759	−	0.676156i	$-0.236355\pi$
$937$	45.1050	1.47352	0.736759	+	0.676156i	$0.236355\pi$
$938$	75.1115	2.45248
$939$	0	0
$940$	1.93937	0.0632551
$941$	−46.2269	−1.50695	−0.753477	−	0.657474i	$-0.771625\pi$
$941$	−46.2269	−1.50695	−0.753477	+	0.657474i	$0.771625\pi$
$942$	0	0
$943$	14.4959	0.472051
$944$	3.16488	0.103008
$945$	0	0
$946$	−67.9766	−2.21011
$947$	30.4501	0.989494	0.494747	−	0.869037i	$-0.335261\pi$
$947$	30.4501	0.989494	0.494747	+	0.869037i	$0.335261\pi$
$948$	0	0
$949$	−41.0653	−1.33304
$950$	−45.0949	−1.46307
$951$	0	0
$952$	62.5160	2.02616
$953$	−4.84074	−0.156807	−0.0784035	−	0.996922i	$-0.524982\pi$
$953$	−4.84074	−0.156807	−0.0784035	+	0.996922i	$0.524982\pi$
$954$	0	0
$955$	52.3138	1.69283
$956$	0.665397	0.0215205
$957$	0	0
$958$	22.9064	0.740073
$959$	36.4097	1.17573
$960$	0	0
$961$	−23.6110	−0.761646
$962$	21.9637	0.708137
$963$	0	0
$964$	−0.690763	−0.0222480
$965$	−28.1752	−0.906992
$966$	0	0
$967$	−40.2445	−1.29417	−0.647087	−	0.762416i	$-0.724013\pi$
$967$	−40.2445	−1.29417	−0.647087	+	0.762416i	$0.724013\pi$
$968$	−44.8167	−1.44046
$969$	0	0
$970$	23.6855	0.760497
$971$	−11.2966	−0.362525	−0.181262	−	0.983435i	$-0.558018\pi$
$971$	−11.2966	−0.362525	−0.181262	+	0.983435i	$0.558018\pi$
$972$	0	0
$973$	−62.1451	−1.99228
$974$	−2.37605	−0.0761335
$975$	0	0
$976$	−14.3857	−0.460475
$977$	37.4195	1.19716	0.598578	−	0.801064i	$-0.295733\pi$
$977$	37.4195	1.19716	0.598578	+	0.801064i	$0.295733\pi$
$978$	0	0
$979$	5.45841	0.174452
$980$	−6.08583	−0.194405
$981$	0	0
$982$	−2.50071	−0.0798010
$983$	−7.98240	−0.254599	−0.127299	−	0.991864i	$-0.540631\pi$
$983$	−7.98240	−0.254599	−0.127299	+	0.991864i	$0.540631\pi$
$984$	0	0
$985$	18.7046	0.595977
$986$	61.4162	1.95589
$987$	0	0
$988$	−1.37603	−0.0437774
$989$	−66.9946	−2.13030
$990$	0	0
$991$	18.3554	0.583080	0.291540	−	0.956559i	$-0.405832\pi$
$991$	18.3554	0.583080	0.291540	+	0.956559i	$0.405832\pi$
$992$	2.30055	0.0730426
$993$	0	0
$994$	−60.2670	−1.91155
$995$	−16.8619	−0.534558
$996$	0	0
$997$	−38.2142	−1.21026	−0.605128	−	0.796128i	$-0.706878\pi$
$997$	−38.2142	−1.21026	−0.605128	+	0.796128i	$0.706878\pi$
$998$	−39.3434	−1.24539
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.k.1.12	yes	16
3.2	odd	2	inner	1143.2.a.k.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.k.1.5	✓	16	3.2	odd	2	inner
1143.2.a.k.1.12	yes	16	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36017 q^{2} -0.149936 q^{4} +3.81085 q^{5} +4.20131 q^{7} -2.92428 q^{8} +O(q^{10})\)	\(q+1.36017 q^{2} -0.149936 q^{4} +3.81085 q^{5} +4.20131 q^{7} -2.92428 q^{8} +5.18341 q^{10} +5.13086 q^{11} -2.63599 q^{13} +5.71450 q^{14} -3.67765 q^{16} -5.08847 q^{17} -3.48160 q^{19} -0.571383 q^{20} +6.97885 q^{22} +6.87803 q^{23} +9.52260 q^{25} -3.58540 q^{26} -0.629927 q^{28} -8.87365 q^{29} +2.71827 q^{31} +0.846330 q^{32} -6.92119 q^{34} +16.0106 q^{35} -6.12586 q^{37} -4.73557 q^{38} -11.1440 q^{40} +2.10756 q^{41} -9.74037 q^{43} -0.769299 q^{44} +9.35530 q^{46} -3.39416 q^{47} +10.6510 q^{49} +12.9524 q^{50} +0.395230 q^{52} -3.88387 q^{53} +19.5530 q^{55} -12.2858 q^{56} -12.0697 q^{58} -0.860571 q^{59} +3.91166 q^{61} +3.69731 q^{62} +8.50645 q^{64} -10.0454 q^{65} +13.1440 q^{67} +0.762944 q^{68} +21.7771 q^{70} -10.5463 q^{71} +15.5787 q^{73} -8.33222 q^{74} +0.522016 q^{76} +21.5564 q^{77} +12.9184 q^{79} -14.0150 q^{80} +2.86665 q^{82} +3.13017 q^{83} -19.3914 q^{85} -13.2486 q^{86} -15.0041 q^{88} +1.06384 q^{89} -11.0746 q^{91} -1.03126 q^{92} -4.61664 q^{94} -13.2679 q^{95} +4.56949 q^{97} +14.4872 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 10 * q^7	\( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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