LMFDB - Embedded newform 8649.2.a.bu.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22192 q^{2} +2.93691 q^{4} -1.02426 q^{5} +0.246895 q^{7} -2.08174 q^{8} +O(q^{10})$	\(q-2.22192 q^{2} +2.93691 q^{4} -1.02426 q^{5} +0.246895 q^{7} -2.08174 q^{8} +2.27581 q^{10} -4.88554 q^{11} +1.31401 q^{13} -0.548581 q^{14} -1.24838 q^{16} -6.62178 q^{17} -5.60275 q^{19} -3.00815 q^{20} +10.8553 q^{22} +0.476794 q^{23} -3.95090 q^{25} -2.91962 q^{26} +0.725109 q^{28} +6.57245 q^{29} +6.93726 q^{32} +14.7130 q^{34} -0.252884 q^{35} +7.64106 q^{37} +12.4488 q^{38} +2.13223 q^{40} -10.2582 q^{41} -7.93895 q^{43} -14.3484 q^{44} -1.05940 q^{46} +1.81918 q^{47} -6.93904 q^{49} +8.77857 q^{50} +3.85913 q^{52} -12.9101 q^{53} +5.00404 q^{55} -0.513971 q^{56} -14.6034 q^{58} +0.877915 q^{59} -7.43904 q^{61} -12.9173 q^{64} -1.34588 q^{65} -0.438119 q^{67} -19.4476 q^{68} +0.561887 q^{70} -4.43447 q^{71} -1.47426 q^{73} -16.9778 q^{74} -16.4548 q^{76} -1.20622 q^{77} -15.8534 q^{79} +1.27866 q^{80} +22.7929 q^{82} +8.53475 q^{83} +6.78240 q^{85} +17.6397 q^{86} +10.1704 q^{88} -12.7810 q^{89} +0.324423 q^{91} +1.40030 q^{92} -4.04206 q^{94} +5.73865 q^{95} +8.25122 q^{97} +15.4180 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29} + 48 q^{32} + 48 q^{35} + 64 q^{37} + 24 q^{38} + 32 q^{43} - 48 q^{44} + 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} + 64 q^{52} - 80 q^{53} + 48 q^{56} + 32 q^{58} + 32 q^{61} + 56 q^{64} - 16 q^{65} - 16 q^{67} - 80 q^{68} + 8 q^{70} + 32 q^{73} + 56 q^{76} - 96 q^{77} + 32 q^{79} + 72 q^{80} + 8 q^{82} - 48 q^{83} + 96 q^{85} + 32 q^{86} + 96 q^{88} + 16 q^{89} - 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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 $

$2$	−2.22192	−1.57113	−0.785566	−	0.618778i	$-0.787628\pi$
$2$	−2.22192	−1.57113	−0.785566	+	0.618778i	$0.787628\pi$
$3$	0	0
$3$	0	0
$4$	2.93691	1.46846
$5$	−1.02426	−0.458061	−0.229030	−	0.973419i	$-0.573556\pi$
$5$	−1.02426	−0.458061	−0.229030	+	0.973419i	$0.573556\pi$
$6$	0	0
$7$	0.246895	0.0933177	0.0466588	−	0.998911i	$-0.485143\pi$
$7$	0.246895	0.0933177	0.0466588	+	0.998911i	$0.485143\pi$
$8$	−2.08174	−0.736005
$9$	0	0
$10$	2.27581	0.719674
$11$	−4.88554	−1.47305	−0.736523	−	0.676413i	$-0.763533\pi$
$11$	−4.88554	−1.47305	−0.736523	+	0.676413i	$0.763533\pi$
$12$	0	0
$13$	1.31401	0.364441	0.182220	−	0.983258i	$-0.441672\pi$
$13$	1.31401	0.364441	0.182220	+	0.983258i	$0.441672\pi$
$14$	−0.548581	−0.146614
$15$	0	0
$16$	−1.24838	−0.312094
$17$	−6.62178	−1.60602	−0.803009	−	0.595967i	$-0.796769\pi$
$17$	−6.62178	−1.60602	−0.803009	+	0.595967i	$0.796769\pi$
$18$	0	0
$19$	−5.60275	−1.28536	−0.642680	−	0.766135i	$-0.722177\pi$
$19$	−5.60275	−1.28536	−0.642680	+	0.766135i	$0.722177\pi$
$20$	−3.00815	−0.672642
$21$	0	0
$22$	10.8553	2.31435
$23$	0.476794	0.0994184	0.0497092	−	0.998764i	$-0.484171\pi$
$23$	0.476794	0.0994184	0.0497092	+	0.998764i	$0.484171\pi$
$24$	0	0
$25$	−3.95090	−0.790180
$26$	−2.91962	−0.572585
$27$	0	0
$28$	0.725109	0.137033
$29$	6.57245	1.22047	0.610237	−	0.792219i	$-0.291074\pi$
$29$	6.57245	1.22047	0.610237	+	0.792219i	$0.291074\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	6.93726	1.22635
$33$	0	0
$34$	14.7130	2.52327
$35$	−0.252884	−0.0427452
$36$	0	0
$37$	7.64106	1.25618	0.628091	−	0.778140i	$-0.283837\pi$
$37$	7.64106	1.25618	0.628091	+	0.778140i	$0.283837\pi$
$38$	12.4488	2.01947
$39$	0	0
$40$	2.13223	0.337135
$41$	−10.2582	−1.60207	−0.801034	−	0.598619i	$-0.795716\pi$
$41$	−10.2582	−1.60207	−0.801034	+	0.598619i	$0.795716\pi$
$42$	0	0
$43$	−7.93895	−1.21068	−0.605339	−	0.795968i	$-0.706962\pi$
$43$	−7.93895	−1.21068	−0.605339	+	0.795968i	$0.706962\pi$
$44$	−14.3484	−2.16310
$45$	0	0
$46$	−1.05940	−0.156199
$47$	1.81918	0.265354	0.132677	−	0.991159i	$-0.457643\pi$
$47$	1.81918	0.265354	0.132677	+	0.991159i	$0.457643\pi$
$48$	0	0
$49$	−6.93904	−0.991292
$50$	8.77857	1.24148
$51$	0	0
$52$	3.85913	0.535165
$53$	−12.9101	−1.77334	−0.886669	−	0.462405i	$-0.846987\pi$
$53$	−12.9101	−1.77334	−0.886669	+	0.462405i	$0.846987\pi$
$54$	0	0
$55$	5.00404	0.674744
$56$	−0.513971	−0.0686823
$57$	0	0
$58$	−14.6034	−1.91752
$59$	0.877915	0.114295	0.0571474	−	0.998366i	$-0.481800\pi$
$59$	0.877915	0.114295	0.0571474	+	0.998366i	$0.481800\pi$
$60$	0	0
$61$	−7.43904	−0.952471	−0.476236	−	0.879318i	$-0.657999\pi$
$61$	−7.43904	−0.952471	−0.476236	+	0.879318i	$0.657999\pi$
$62$	0	0
$63$	0	0
$64$	−12.9173	−1.61466
$65$	−1.34588	−0.166936
$66$	0	0
$67$	−0.438119	−0.0535248	−0.0267624	−	0.999642i	$-0.508520\pi$
$67$	−0.438119	−0.0535248	−0.0267624	+	0.999642i	$0.508520\pi$
$68$	−19.4476	−2.35837
$69$	0	0
$70$	0.561887	0.0671583
$71$	−4.43447	−0.526275	−0.263137	−	0.964758i	$-0.584757\pi$
$71$	−4.43447	−0.526275	−0.263137	+	0.964758i	$0.584757\pi$
$72$	0	0
$73$	−1.47426	−0.172549	−0.0862745	−	0.996271i	$-0.527496\pi$
$73$	−1.47426	−0.172549	−0.0862745	+	0.996271i	$0.527496\pi$
$74$	−16.9778	−1.97363
$75$	0	0
$76$	−16.4548	−1.88749
$77$	−1.20622	−0.137461
$78$	0	0
$79$	−15.8534	−1.78364	−0.891822	−	0.452386i	$-0.850573\pi$
$79$	−15.8534	−1.78364	−0.891822	+	0.452386i	$0.850573\pi$
$80$	1.27866	0.142958
$81$	0	0
$82$	22.7929	2.51706
$83$	8.53475	0.936810	0.468405	−	0.883514i	$-0.344829\pi$
$83$	8.53475	0.936810	0.468405	+	0.883514i	$0.344829\pi$
$84$	0	0
$85$	6.78240	0.735654
$86$	17.6397	1.90213
$87$	0	0
$88$	10.1704	1.08417
$89$	−12.7810	−1.35478	−0.677391	−	0.735623i	$-0.736890\pi$
$89$	−12.7810	−1.35478	−0.677391	+	0.735623i	$0.736890\pi$
$90$	0	0
$91$	0.324423	0.0340088
$92$	1.40030	0.145992
$93$	0	0
$94$	−4.04206	−0.416906
$95$	5.73865	0.588773
$96$	0	0
$97$	8.25122	0.837784	0.418892	−	0.908036i	$-0.362419\pi$
$97$	8.25122	0.837784	0.418892	+	0.908036i	$0.362419\pi$
$98$	15.4180	1.55745
$99$	0	0
$100$	−11.6034	−1.16034
$101$	−0.491992	−0.0489551	−0.0244775	−	0.999700i	$-0.507792\pi$
$101$	−0.491992	−0.0489551	−0.0244775	+	0.999700i	$0.507792\pi$
$102$	0	0
$103$	7.76258	0.764869	0.382435	−	0.923983i	$-0.375086\pi$
$103$	7.76258	0.764869	0.382435	+	0.923983i	$0.375086\pi$
$104$	−2.73542	−0.268230
$105$	0	0
$106$	28.6851	2.78615
$107$	−9.99162	−0.965926	−0.482963	−	0.875641i	$-0.660439\pi$
$107$	−9.99162	−0.965926	−0.482963	+	0.875641i	$0.660439\pi$
$108$	0	0
$109$	10.2578	0.982519	0.491259	−	0.871013i	$-0.336537\pi$
$109$	10.2578	0.982519	0.491259	+	0.871013i	$0.336537\pi$
$110$	−11.1186	−1.06011
$111$	0	0
$112$	−0.308219	−0.0291239
$113$	17.3944	1.63633	0.818165	−	0.574983i	$-0.194991\pi$
$113$	17.3944	1.63633	0.818165	+	0.574983i	$0.194991\pi$
$114$	0	0
$115$	−0.488359	−0.0455397
$116$	19.3027	1.79221
$117$	0	0
$118$	−1.95065	−0.179572
$119$	−1.63489	−0.149870
$120$	0	0
$121$	12.8685	1.16986
$122$	16.5289	1.49646
$123$	0	0
$124$	0	0
$125$	9.16801	0.820012
$126$	0	0
$127$	4.01005	0.355835	0.177917	−	0.984045i	$-0.443064\pi$
$127$	4.01005	0.355835	0.177917	+	0.984045i	$0.443064\pi$
$128$	14.8265	1.31049
$129$	0	0
$130$	2.99044	0.262279
$131$	−0.291671	−0.0254834	−0.0127417	−	0.999919i	$-0.504056\pi$
$131$	−0.291671	−0.0254834	−0.0127417	+	0.999919i	$0.504056\pi$
$132$	0	0
$133$	−1.38329	−0.119947
$134$	0.973464	0.0840944
$135$	0	0
$136$	13.7848	1.18204
$137$	0.382882	0.0327118	0.0163559	−	0.999866i	$-0.494794\pi$
$137$	0.382882	0.0327118	0.0163559	+	0.999866i	$0.494794\pi$
$138$	0	0
$139$	4.82874	0.409568	0.204784	−	0.978807i	$-0.434351\pi$
$139$	4.82874	0.409568	0.204784	+	0.978807i	$0.434351\pi$
$140$	−0.742697	−0.0627694
$141$	0	0
$142$	9.85302	0.826847
$143$	−6.41965	−0.536838
$144$	0	0
$145$	−6.73187	−0.559051
$146$	3.27568	0.271097
$147$	0	0
$148$	22.4411	1.84465
$149$	4.72173	0.386819	0.193410	−	0.981118i	$-0.438045\pi$
$149$	4.72173	0.386819	0.193410	+	0.981118i	$0.438045\pi$
$150$	0	0
$151$	10.7341	0.873531	0.436765	−	0.899575i	$-0.356124\pi$
$151$	10.7341	0.873531	0.436765	+	0.899575i	$0.356124\pi$
$152$	11.6635	0.946031
$153$	0	0
$154$	2.68011	0.215970
$155$	0	0
$156$	0	0
$157$	−23.2962	−1.85924	−0.929618	−	0.368525i	$-0.879863\pi$
$157$	−23.2962	−1.85924	−0.929618	+	0.368525i	$0.879863\pi$
$158$	35.2249	2.80234
$159$	0	0
$160$	−7.10553	−0.561741
$161$	0.117718	0.00927749
$162$	0	0
$163$	−22.9681	−1.79900	−0.899499	−	0.436922i	$-0.856069\pi$
$163$	−22.9681	−1.79900	−0.899499	+	0.436922i	$0.856069\pi$
$164$	−30.1275	−2.35256
$165$	0	0
$166$	−18.9635	−1.47185
$167$	3.08014	0.238348	0.119174	−	0.992873i	$-0.461975\pi$
$167$	3.08014	0.238348	0.119174	+	0.992873i	$0.461975\pi$
$168$	0	0
$169$	−11.2734	−0.867183
$170$	−15.0699	−1.15581
$171$	0	0
$172$	−23.3160	−1.77783
$173$	4.57393	0.347750	0.173875	−	0.984768i	$-0.444371\pi$
$173$	4.57393	0.347750	0.173875	+	0.984768i	$0.444371\pi$
$174$	0	0
$175$	−0.975459	−0.0737378
$176$	6.09899	0.459729
$177$	0	0
$178$	28.3983	2.12854
$179$	−4.89854	−0.366134	−0.183067	−	0.983100i	$-0.558603\pi$
$179$	−4.89854	−0.366134	−0.183067	+	0.983100i	$0.558603\pi$
$180$	0	0
$181$	20.2801	1.50741	0.753705	−	0.657213i	$-0.228265\pi$
$181$	20.2801	1.50741	0.753705	+	0.657213i	$0.228265\pi$
$182$	−0.720841	−0.0534323
$183$	0	0
$184$	−0.992560	−0.0731725
$185$	−7.82640	−0.575408
$186$	0	0
$187$	32.3510	2.36574
$188$	5.34276	0.389661
$189$	0	0
$190$	−12.7508	−0.925040
$191$	17.5005	1.26629	0.633147	−	0.774032i	$-0.281763\pi$
$191$	17.5005	1.26629	0.633147	+	0.774032i	$0.281763\pi$
$192$	0	0
$193$	−6.05965	−0.436183	−0.218091	−	0.975928i	$-0.569983\pi$
$193$	−6.05965	−0.436183	−0.218091	+	0.975928i	$0.569983\pi$
$194$	−18.3335	−1.31627
$195$	0	0
$196$	−20.3793	−1.45567
$197$	−10.5875	−0.754330	−0.377165	−	0.926146i	$-0.623101\pi$
$197$	−10.5875	−0.754330	−0.377165	+	0.926146i	$0.623101\pi$
$198$	0	0
$199$	−26.8385	−1.90253	−0.951265	−	0.308375i	$-0.900215\pi$
$199$	−26.8385	−1.90253	−0.951265	+	0.308375i	$0.900215\pi$
$200$	8.22473	0.581577
$201$	0	0
$202$	1.09317	0.0769149
$203$	1.62271	0.113892
$204$	0	0
$205$	10.5071	0.733845
$206$	−17.2478	−1.20171
$207$	0	0
$208$	−1.64038	−0.113740
$209$	27.3724	1.89339
$210$	0	0
$211$	2.56528	0.176601	0.0883007	−	0.996094i	$-0.471856\pi$
$211$	2.56528	0.176601	0.0883007	+	0.996094i	$0.471856\pi$
$212$	−37.9158	−2.60407
$213$	0	0
$214$	22.2005	1.51760
$215$	8.13151	0.554564
$216$	0	0
$217$	0	0
$218$	−22.7920	−1.54367
$219$	0	0
$220$	14.6964	0.990832
$221$	−8.70109	−0.585299
$222$	0	0
$223$	−6.42752	−0.430419	−0.215209	−	0.976568i	$-0.569043\pi$
$223$	−6.42752	−0.430419	−0.215209	+	0.976568i	$0.569043\pi$
$224$	1.71278	0.114440
$225$	0	0
$226$	−38.6490	−2.57089
$227$	16.4348	1.09082	0.545409	−	0.838170i	$-0.316375\pi$
$227$	16.4348	1.09082	0.545409	+	0.838170i	$0.316375\pi$
$228$	0	0
$229$	−17.3692	−1.14779	−0.573896	−	0.818928i	$-0.694569\pi$
$229$	−17.3692	−1.14779	−0.573896	+	0.818928i	$0.694569\pi$
$230$	1.08509	0.0715489
$231$	0	0
$232$	−13.6821	−0.898274
$233$	17.0566	1.11742	0.558708	−	0.829364i	$-0.311297\pi$
$233$	17.0566	1.11742	0.558708	+	0.829364i	$0.311297\pi$
$234$	0	0
$235$	−1.86330	−0.121548
$236$	2.57836	0.167837
$237$	0	0
$238$	3.63258	0.235465
$239$	−4.66639	−0.301844	−0.150922	−	0.988546i	$-0.548224\pi$
$239$	−4.66639	−0.301844	−0.150922	+	0.988546i	$0.548224\pi$
$240$	0	0
$241$	−6.06181	−0.390476	−0.195238	−	0.980756i	$-0.562548\pi$
$241$	−6.06181	−0.390476	−0.195238	+	0.980756i	$0.562548\pi$
$242$	−28.5927	−1.83801
$243$	0	0
$244$	−21.8478	−1.39866
$245$	7.10735	0.454072
$246$	0	0
$247$	−7.36207	−0.468437
$248$	0	0
$249$	0	0
$250$	−20.3705	−1.28835
$251$	12.6354	0.797536	0.398768	−	0.917052i	$-0.369438\pi$
$251$	12.6354	0.797536	0.398768	+	0.917052i	$0.369438\pi$
$252$	0	0
$253$	−2.32940	−0.146448
$254$	−8.91000	−0.559063
$255$	0	0
$256$	−7.10881	−0.444301
$257$	14.0203	0.874565	0.437283	−	0.899324i	$-0.355941\pi$
$257$	14.0203	0.874565	0.437283	+	0.899324i	$0.355941\pi$
$258$	0	0
$259$	1.88654	0.117224
$260$	−3.95274	−0.245138
$261$	0	0
$262$	0.648069	0.0400379
$263$	17.2115	1.06130	0.530652	−	0.847590i	$-0.321947\pi$
$263$	17.2115	1.06130	0.530652	+	0.847590i	$0.321947\pi$
$264$	0	0
$265$	13.2232	0.812297
$266$	3.07356	0.188452
$267$	0	0
$268$	−1.28672	−0.0785987
$269$	−13.6155	−0.830152	−0.415076	−	0.909787i	$-0.636245\pi$
$269$	−13.6155	−0.830152	−0.415076	+	0.909787i	$0.636245\pi$
$270$	0	0
$271$	9.26824	0.563005	0.281503	−	0.959560i	$-0.409167\pi$
$271$	9.26824	0.563005	0.281503	+	0.959560i	$0.409167\pi$
$272$	8.26648	0.501229
$273$	0	0
$274$	−0.850732	−0.0513946
$275$	19.3023	1.16397
$276$	0	0
$277$	−30.1569	−1.81195	−0.905975	−	0.423332i	$-0.860860\pi$
$277$	−30.1569	−1.81195	−0.905975	+	0.423332i	$0.860860\pi$
$278$	−10.7291	−0.643486
$279$	0	0
$280$	0.526438	0.0314607
$281$	−0.251359	−0.0149948	−0.00749742	−	0.999972i	$-0.502387\pi$
$281$	−0.251359	−0.0149948	−0.00749742	+	0.999972i	$0.502387\pi$
$282$	0	0
$283$	−8.47213	−0.503616	−0.251808	−	0.967777i	$-0.581025\pi$
$283$	−8.47213	−0.503616	−0.251808	+	0.967777i	$0.581025\pi$
$284$	−13.0236	−0.772811
$285$	0	0
$286$	14.2639	0.843443
$287$	−2.53271	−0.149501
$288$	0	0
$289$	26.8480	1.57929
$290$	14.9576	0.878343
$291$	0	0
$292$	−4.32977	−0.253381
$293$	−16.5238	−0.965329	−0.482665	−	0.875805i	$-0.660331\pi$
$293$	−16.5238	−0.965329	−0.482665	+	0.875805i	$0.660331\pi$
$294$	0	0
$295$	−0.899209	−0.0523540
$296$	−15.9067	−0.924556
$297$	0	0
$298$	−10.4913	−0.607744
$299$	0.626512	0.0362321
$300$	0	0
$301$	−1.96009	−0.112978
$302$	−23.8503	−1.37243
$303$	0	0
$304$	6.99435	0.401153
$305$	7.61948	0.436290
$306$	0	0
$307$	3.54132	0.202114	0.101057	−	0.994881i	$-0.467778\pi$
$307$	3.54132	0.202114	0.101057	+	0.994881i	$0.467778\pi$
$308$	−3.54255	−0.201855
$309$	0	0
$310$	0	0
$311$	−25.1123	−1.42399	−0.711993	−	0.702187i	$-0.752207\pi$
$311$	−25.1123	−1.42399	−0.711993	+	0.702187i	$0.752207\pi$
$312$	0	0
$313$	−1.25019	−0.0706647	−0.0353323	−	0.999376i	$-0.511249\pi$
$313$	−1.25019	−0.0706647	−0.0353323	+	0.999376i	$0.511249\pi$
$314$	51.7621	2.92110
$315$	0	0
$316$	−46.5600	−2.61920
$317$	−16.8497	−0.946371	−0.473186	−	0.880963i	$-0.656896\pi$
$317$	−16.8497	−0.946371	−0.473186	+	0.880963i	$0.656896\pi$
$318$	0	0
$319$	−32.1099	−1.79781
$320$	13.2306	0.739612
$321$	0	0
$322$	−0.261560	−0.0145762
$323$	37.1002	2.06431
$324$	0	0
$325$	−5.19152	−0.287974
$326$	51.0331	2.82646
$327$	0	0
$328$	21.3549	1.17913
$329$	0.449146	0.0247622
$330$	0	0
$331$	9.65338	0.530598	0.265299	−	0.964166i	$-0.414529\pi$
$331$	9.65338	0.530598	0.265299	+	0.964166i	$0.414529\pi$
$332$	25.0658	1.37566
$333$	0	0
$334$	−6.84381	−0.374476
$335$	0.448746	0.0245176
$336$	0	0
$337$	7.25309	0.395101	0.197551	−	0.980293i	$-0.436701\pi$
$337$	7.25309	0.395101	0.197551	+	0.980293i	$0.436701\pi$
$338$	25.0485	1.36246
$339$	0	0
$340$	19.9193	1.08028
$341$	0	0
$342$	0	0
$343$	−3.44148	−0.185823
$344$	16.5268	0.891065
$345$	0	0
$346$	−10.1629	−0.546360
$347$	−20.2729	−1.08831	−0.544154	−	0.838985i	$-0.683149\pi$
$347$	−20.2729	−1.08831	−0.544154	+	0.838985i	$0.683149\pi$
$348$	0	0
$349$	25.6000	1.37034	0.685169	−	0.728384i	$-0.259728\pi$
$349$	25.6000	1.37034	0.685169	+	0.728384i	$0.259728\pi$
$350$	2.16739	0.115852
$351$	0	0
$352$	−33.8923	−1.80646
$353$	8.02074	0.426901	0.213450	−	0.976954i	$-0.431530\pi$
$353$	8.02074	0.426901	0.213450	+	0.976954i	$0.431530\pi$
$354$	0	0
$355$	4.54203	0.241066
$356$	−37.5366	−1.98944
$357$	0	0
$358$	10.8841	0.575245
$359$	−24.8239	−1.31016	−0.655079	−	0.755560i	$-0.727365\pi$
$359$	−24.8239	−1.31016	−0.655079	+	0.755560i	$0.727365\pi$
$360$	0	0
$361$	12.3908	0.652148
$362$	−45.0607	−2.36834
$363$	0	0
$364$	0.952801	0.0499404
$365$	1.51002	0.0790380
$366$	0	0
$367$	−13.4121	−0.700107	−0.350053	−	0.936730i	$-0.613837\pi$
$367$	−13.4121	−0.700107	−0.350053	+	0.936730i	$0.613837\pi$
$368$	−0.595219	−0.0310279
$369$	0	0
$370$	17.3896	0.904042
$371$	−3.18744	−0.165484
$372$	0	0
$373$	19.5457	1.01204	0.506019	−	0.862522i	$-0.331117\pi$
$373$	19.5457	1.01204	0.506019	+	0.862522i	$0.331117\pi$
$374$	−71.8811	−3.71688
$375$	0	0
$376$	−3.78704	−0.195302
$377$	8.63627	0.444790
$378$	0	0
$379$	32.9292	1.69146	0.845731	−	0.533610i	$-0.179165\pi$
$379$	32.9292	1.69146	0.845731	+	0.533610i	$0.179165\pi$
$380$	16.8539	0.864587
$381$	0	0
$382$	−38.8847	−1.98951
$383$	−8.35280	−0.426808	−0.213404	−	0.976964i	$-0.568455\pi$
$383$	−8.35280	−0.426808	−0.213404	+	0.976964i	$0.568455\pi$
$384$	0	0
$385$	1.23547	0.0629656
$386$	13.4640	0.685301
$387$	0	0
$388$	24.2331	1.23025
$389$	7.06102	0.358008	0.179004	−	0.983848i	$-0.442712\pi$
$389$	7.06102	0.358008	0.179004	+	0.983848i	$0.442712\pi$
$390$	0	0
$391$	−3.15723	−0.159668
$392$	14.4453	0.729596
$393$	0	0
$394$	23.5246	1.18515
$395$	16.2379	0.817018
$396$	0	0
$397$	−29.9950	−1.50540	−0.752702	−	0.658361i	$-0.771250\pi$
$397$	−29.9950	−1.50540	−0.752702	+	0.658361i	$0.771250\pi$
$398$	59.6329	2.98912
$399$	0	0
$400$	4.93222	0.246611
$401$	−24.7567	−1.23629	−0.618145	−	0.786064i	$-0.712115\pi$
$401$	−24.7567	−1.23629	−0.618145	+	0.786064i	$0.712115\pi$
$402$	0	0
$403$	0	0
$404$	−1.44494	−0.0718883
$405$	0	0
$406$	−3.60552	−0.178939
$407$	−37.3307	−1.85041
$408$	0	0
$409$	3.87563	0.191637	0.0958187	−	0.995399i	$-0.469453\pi$
$409$	3.87563	0.191637	0.0958187	+	0.995399i	$0.469453\pi$
$410$	−23.3458	−1.15297
$411$	0	0
$412$	22.7980	1.12318
$413$	0.216753	0.0106657
$414$	0	0
$415$	−8.74176	−0.429116
$416$	9.11564	0.446931
$417$	0	0
$418$	−60.8193	−2.97477
$419$	30.5687	1.49338	0.746689	−	0.665174i	$-0.231642\pi$
$419$	30.5687	1.49338	0.746689	+	0.665174i	$0.231642\pi$
$420$	0	0
$421$	2.10232	0.102461	0.0512303	−	0.998687i	$-0.483686\pi$
$421$	2.10232	0.102461	0.0512303	+	0.998687i	$0.483686\pi$
$422$	−5.69985	−0.277464
$423$	0	0
$424$	26.8754	1.30519
$425$	26.1620	1.26904
$426$	0	0
$427$	−1.83666	−0.0888824
$428$	−29.3445	−1.41842
$429$	0	0
$430$	−18.0675	−0.871294
$431$	−10.1174	−0.487340	−0.243670	−	0.969858i	$-0.578351\pi$
$431$	−10.1174	−0.487340	−0.243670	+	0.969858i	$0.578351\pi$
$432$	0	0
$433$	−16.4597	−0.791003	−0.395502	−	0.918465i	$-0.629429\pi$
$433$	−16.4597	−0.791003	−0.395502	+	0.918465i	$0.629429\pi$
$434$	0	0
$435$	0	0
$436$	30.1262	1.44278
$437$	−2.67136	−0.127788
$438$	0	0
$439$	39.1043	1.86634	0.933172	−	0.359429i	$-0.117029\pi$
$439$	39.1043	1.86634	0.933172	+	0.359429i	$0.117029\pi$
$440$	−10.4171	−0.496615
$441$	0	0
$442$	19.3331	0.919581
$443$	10.0114	0.475657	0.237828	−	0.971307i	$-0.423564\pi$
$443$	10.0114	0.475657	0.237828	+	0.971307i	$0.423564\pi$
$444$	0	0
$445$	13.0910	0.620573
$446$	14.2814	0.676245
$447$	0	0
$448$	−3.18921	−0.150676
$449$	−11.1113	−0.524376	−0.262188	−	0.965017i	$-0.584444\pi$
$449$	−11.1113	−0.524376	−0.262188	+	0.965017i	$0.584444\pi$
$450$	0	0
$451$	50.1170	2.35992
$452$	51.0859	2.40288
$453$	0	0
$454$	−36.5168	−1.71382
$455$	−0.332292	−0.0155781
$456$	0	0
$457$	19.7652	0.924578	0.462289	−	0.886729i	$-0.347028\pi$
$457$	19.7652	0.924578	0.462289	+	0.886729i	$0.347028\pi$
$458$	38.5930	1.80333
$459$	0	0
$460$	−1.43427	−0.0668730
$461$	−24.7521	−1.15282	−0.576410	−	0.817161i	$-0.695547\pi$
$461$	−24.7521	−1.15282	−0.576410	+	0.817161i	$0.695547\pi$
$462$	0	0
$463$	28.7717	1.33714	0.668568	−	0.743651i	$-0.266908\pi$
$463$	28.7717	1.33714	0.668568	+	0.743651i	$0.266908\pi$
$464$	−8.20490	−0.380903
$465$	0	0
$466$	−37.8984	−1.75561
$467$	4.28353	0.198218	0.0991091	−	0.995077i	$-0.468401\pi$
$467$	4.28353	0.198218	0.0991091	+	0.995077i	$0.468401\pi$
$468$	0	0
$469$	−0.108170	−0.00499480
$470$	4.14010	0.190968
$471$	0	0
$472$	−1.82759	−0.0841215
$473$	38.7860	1.78338
$474$	0	0
$475$	22.1359	1.01567
$476$	−4.80152	−0.220077
$477$	0	0
$478$	10.3683	0.474237
$479$	41.0615	1.87615	0.938075	−	0.346434i	$-0.112607\pi$
$479$	41.0615	1.87615	0.938075	+	0.346434i	$0.112607\pi$
$480$	0	0
$481$	10.0404	0.457804
$482$	13.4688	0.613489
$483$	0	0
$484$	37.7936	1.71789
$485$	−8.45135	−0.383756
$486$	0	0
$487$	−25.1767	−1.14086	−0.570432	−	0.821345i	$-0.693224\pi$
$487$	−25.1767	−1.14086	−0.570432	+	0.821345i	$0.693224\pi$
$488$	15.4861	0.701024
$489$	0	0
$490$	−15.7919	−0.713407
$491$	−26.7390	−1.20672	−0.603358	−	0.797471i	$-0.706171\pi$
$491$	−26.7390	−1.20672	−0.603358	+	0.797471i	$0.706171\pi$
$492$	0	0
$493$	−43.5213	−1.96010
$494$	16.3579	0.735977
$495$	0	0
$496$	0	0
$497$	−1.09485	−0.0491107
$498$	0	0
$499$	10.1261	0.453308	0.226654	−	0.973975i	$-0.427221\pi$
$499$	10.1261	0.453308	0.226654	+	0.973975i	$0.427221\pi$
$500$	26.9256	1.20415
$501$	0	0
$502$	−28.0747	−1.25303
$503$	5.00755	0.223276	0.111638	−	0.993749i	$-0.464390\pi$
$503$	5.00755	0.223276	0.111638	+	0.993749i	$0.464390\pi$
$504$	0	0
$505$	0.503926	0.0224244
$506$	5.17572	0.230089
$507$	0	0
$508$	11.7772	0.522527
$509$	−11.9995	−0.531867	−0.265933	−	0.963991i	$-0.585680\pi$
$509$	−11.9995	−0.531867	−0.265933	+	0.963991i	$0.585680\pi$
$510$	0	0
$511$	−0.363988	−0.0161019
$512$	−13.8579	−0.612439
$513$	0	0
$514$	−31.1520	−1.37406
$515$	−7.95086	−0.350357
$516$	0	0
$517$	−8.88765	−0.390878
$518$	−4.19174	−0.184174
$519$	0	0
$520$	2.80177	0.122866
$521$	25.1157	1.10034	0.550169	−	0.835054i	$-0.314563\pi$
$521$	25.1157	1.10034	0.550169	+	0.835054i	$0.314563\pi$
$522$	0	0
$523$	29.6809	1.29785	0.648927	−	0.760850i	$-0.275218\pi$
$523$	29.6809	1.29785	0.648927	+	0.760850i	$0.275218\pi$
$524$	−0.856613	−0.0374213
$525$	0	0
$526$	−38.2424	−1.66745
$527$	0	0
$528$	0	0
$529$	−22.7727	−0.990116
$530$	−29.3809	−1.27623
$531$	0	0
$532$	−4.06261	−0.176136
$533$	−13.4794	−0.583859
$534$	0	0
$535$	10.2340	0.442453
$536$	0.912048	0.0393945
$537$	0	0
$538$	30.2525	1.30428
$539$	33.9010	1.46022
$540$	0	0
$541$	5.45842	0.234676	0.117338	−	0.993092i	$-0.462564\pi$
$541$	5.45842	0.234676	0.117338	+	0.993092i	$0.462564\pi$
$542$	−20.5932	−0.884556
$543$	0	0
$544$	−45.9370	−1.96953
$545$	−10.5066	−0.450053
$546$	0	0
$547$	−41.4354	−1.77165	−0.885826	−	0.464018i	$-0.846407\pi$
$547$	−41.4354	−1.77165	−0.885826	+	0.464018i	$0.846407\pi$
$548$	1.12449	0.0480359
$549$	0	0
$550$	−42.8880	−1.82875
$551$	−36.8238	−1.56875
$552$	0	0
$553$	−3.91412	−0.166446
$554$	67.0060	2.84681
$555$	0	0
$556$	14.1816	0.601433
$557$	25.2936	1.07172	0.535861	−	0.844306i	$-0.319987\pi$
$557$	25.2936	1.07172	0.535861	+	0.844306i	$0.319987\pi$
$558$	0	0
$559$	−10.4319	−0.441221
$560$	0.315694	0.0133405
$561$	0	0
$562$	0.558499	0.0235589
$563$	−10.2214	−0.430779	−0.215390	−	0.976528i	$-0.569102\pi$
$563$	−10.2214	−0.430779	−0.215390	+	0.976528i	$0.569102\pi$
$564$	0	0
$565$	−17.8163	−0.749539
$566$	18.8244	0.791247
$567$	0	0
$568$	9.23140	0.387341
$569$	−23.2771	−0.975828	−0.487914	−	0.872892i	$-0.662242\pi$
$569$	−23.2771	−0.975828	−0.487914	+	0.872892i	$0.662242\pi$
$570$	0	0
$571$	11.4224	0.478014	0.239007	−	0.971018i	$-0.423178\pi$
$571$	11.4224	0.478014	0.239007	+	0.971018i	$0.423178\pi$
$572$	−18.8539	−0.788322
$573$	0	0
$574$	5.62747	0.234886
$575$	−1.88377	−0.0785585
$576$	0	0
$577$	−24.7326	−1.02963	−0.514816	−	0.857301i	$-0.672140\pi$
$577$	−24.7326	−1.02963	−0.514816	+	0.857301i	$0.672140\pi$
$578$	−59.6540	−2.48128
$579$	0	0
$580$	−19.7709	−0.820942
$581$	2.10719	0.0874209
$582$	0	0
$583$	63.0728	2.61221
$584$	3.06902	0.126997
$585$	0	0
$586$	36.7144	1.51666
$587$	6.83184	0.281980	0.140990	−	0.990011i	$-0.454971\pi$
$587$	6.83184	0.281980	0.140990	+	0.990011i	$0.454971\pi$
$588$	0	0
$589$	0	0
$590$	1.99797	0.0822550
$591$	0	0
$592$	−9.53892	−0.392047
$593$	−38.6613	−1.58763	−0.793814	−	0.608160i	$-0.791908\pi$
$593$	−38.6613	−1.58763	−0.793814	+	0.608160i	$0.791908\pi$
$594$	0	0
$595$	1.67454	0.0686495
$596$	13.8673	0.568027
$597$	0	0
$598$	−1.39206	−0.0569255
$599$	−17.5339	−0.716414	−0.358207	−	0.933642i	$-0.616612\pi$
$599$	−17.5339	−0.716414	−0.358207	+	0.933642i	$0.616612\pi$
$600$	0	0
$601$	−23.6210	−0.963522	−0.481761	−	0.876303i	$-0.660003\pi$
$601$	−23.6210	−0.963522	−0.481761	+	0.876303i	$0.660003\pi$
$602$	4.35515	0.177503
$603$	0	0
$604$	31.5252	1.28274
$605$	−13.1806	−0.535868
$606$	0	0
$607$	17.4107	0.706679	0.353339	−	0.935495i	$-0.385046\pi$
$607$	17.4107	0.706679	0.353339	+	0.935495i	$0.385046\pi$
$608$	−38.8678	−1.57630
$609$	0	0
$610$	−16.9298	−0.685469
$611$	2.39042	0.0967059
$612$	0	0
$613$	29.9514	1.20972	0.604862	−	0.796330i	$-0.293228\pi$
$613$	29.9514	1.20972	0.604862	+	0.796330i	$0.293228\pi$
$614$	−7.86852	−0.317548
$615$	0	0
$616$	2.51102	0.101172
$617$	10.7265	0.431832	0.215916	−	0.976412i	$-0.430726\pi$
$617$	10.7265	0.431832	0.215916	+	0.976412i	$0.430726\pi$
$618$	0	0
$619$	6.91417	0.277904	0.138952	−	0.990299i	$-0.455627\pi$
$619$	6.91417	0.277904	0.138952	+	0.990299i	$0.455627\pi$
$620$	0	0
$621$	0	0
$622$	55.7973	2.23727
$623$	−3.15557	−0.126425
$624$	0	0
$625$	10.3641	0.414565
$626$	2.77781	0.111024
$627$	0	0
$628$	−68.4187	−2.73020
$629$	−50.5974	−2.01745
$630$	0	0
$631$	−35.8409	−1.42680	−0.713401	−	0.700756i	$-0.752846\pi$
$631$	−35.8409	−1.42680	−0.713401	+	0.700756i	$0.752846\pi$
$632$	33.0026	1.31277
$633$	0	0
$634$	37.4385	1.48687
$635$	−4.10732	−0.162994
$636$	0	0
$637$	−9.11797	−0.361267
$638$	71.3456	2.82460
$639$	0	0
$640$	−15.1862	−0.600286
$641$	−0.559732	−0.0221081	−0.0110540	−	0.999939i	$-0.503519\pi$
$641$	−0.559732	−0.0221081	−0.0110540	+	0.999939i	$0.503519\pi$
$642$	0	0
$643$	−24.2021	−0.954438	−0.477219	−	0.878785i	$-0.658355\pi$
$643$	−24.2021	−0.954438	−0.477219	+	0.878785i	$0.658355\pi$
$644$	0.345728	0.0136236
$645$	0	0
$646$	−82.4335	−3.24330
$647$	21.7594	0.855451	0.427725	−	0.903909i	$-0.359315\pi$
$647$	21.7594	0.855451	0.427725	+	0.903909i	$0.359315\pi$
$648$	0	0
$649$	−4.28909	−0.168361
$650$	11.5351	0.452445
$651$	0	0
$652$	−67.4552	−2.64175
$653$	−6.31383	−0.247079	−0.123540	−	0.992340i	$-0.539425\pi$
$653$	−6.31383	−0.247079	−0.123540	+	0.992340i	$0.539425\pi$
$654$	0	0
$655$	0.298746	0.0116730
$656$	12.8062	0.499996
$657$	0	0
$658$	−0.997965	−0.0389047
$659$	29.7780	1.15999	0.579994	−	0.814621i	$-0.303055\pi$
$659$	29.7780	1.15999	0.579994	+	0.814621i	$0.303055\pi$
$660$	0	0
$661$	8.98626	0.349525	0.174762	−	0.984611i	$-0.444084\pi$
$661$	8.98626	0.349525	0.174762	+	0.984611i	$0.444084\pi$
$662$	−21.4490	−0.833639
$663$	0	0
$664$	−17.7671	−0.689497
$665$	1.41685	0.0549429
$666$	0	0
$667$	3.13370	0.121338
$668$	9.04609	0.350004
$669$	0	0
$670$	−0.997075	−0.0385204
$671$	36.3437	1.40303
$672$	0	0
$673$	7.98073	0.307634	0.153817	−	0.988099i	$-0.450843\pi$
$673$	7.98073	0.307634	0.153817	+	0.988099i	$0.450843\pi$
$674$	−16.1158	−0.620756
$675$	0	0
$676$	−33.1089	−1.27342
$677$	15.1051	0.580534	0.290267	−	0.956946i	$-0.406256\pi$
$677$	15.1051	0.580534	0.290267	+	0.956946i	$0.406256\pi$
$678$	0	0
$679$	2.03719	0.0781801
$680$	−14.1192	−0.541445
$681$	0	0
$682$	0	0
$683$	14.9185	0.570841	0.285420	−	0.958402i	$-0.407867\pi$
$683$	14.9185	0.570841	0.285420	+	0.958402i	$0.407867\pi$
$684$	0	0
$685$	−0.392169	−0.0149840
$686$	7.64669	0.291952
$687$	0	0
$688$	9.91080	0.377846
$689$	−16.9640	−0.646277
$690$	0	0
$691$	38.1161	1.45000	0.725002	−	0.688747i	$-0.241839\pi$
$691$	38.1161	1.45000	0.725002	+	0.688747i	$0.241839\pi$
$692$	13.4332	0.510655
$693$	0	0
$694$	45.0448	1.70988
$695$	−4.94586	−0.187607
$696$	0	0
$697$	67.9278	2.57295
$698$	−56.8811	−2.15298
$699$	0	0
$700$	−2.86484	−0.108281
$701$	9.47697	0.357940	0.178970	−	0.983855i	$-0.442723\pi$
$701$	9.47697	0.357940	0.178970	+	0.983855i	$0.442723\pi$
$702$	0	0
$703$	−42.8109	−1.61465
$704$	63.1078	2.37846
$705$	0	0
$706$	−17.8214	−0.670717
$707$	−0.121471	−0.00456837
$708$	0	0
$709$	30.2319	1.13538	0.567692	−	0.823241i	$-0.307837\pi$
$709$	30.2319	1.13538	0.567692	+	0.823241i	$0.307837\pi$
$710$	−10.0920	−0.378746
$711$	0	0
$712$	26.6067	0.997127
$713$	0	0
$714$	0	0
$715$	6.57536	0.245904
$716$	−14.3866	−0.537652
$717$	0	0
$718$	55.1567	2.05843
$719$	28.1739	1.05071	0.525355	−	0.850883i	$-0.323933\pi$
$719$	28.1739	1.05071	0.525355	+	0.850883i	$0.323933\pi$
$720$	0	0
$721$	1.91654	0.0713758
$722$	−27.5313	−1.02461
$723$	0	0
$724$	59.5609	2.21356
$725$	−25.9671	−0.964394
$726$	0	0
$727$	44.0194	1.63259	0.816294	−	0.577637i	$-0.196025\pi$
$727$	44.0194	1.63259	0.816294	+	0.577637i	$0.196025\pi$
$728$	−0.675363	−0.0250306
$729$	0	0
$730$	−3.35513	−0.124179
$731$	52.5700	1.94437
$732$	0	0
$733$	24.7799	0.915266	0.457633	−	0.889141i	$-0.348697\pi$
$733$	24.7799	0.915266	0.457633	+	0.889141i	$0.348697\pi$
$734$	29.8006	1.09996
$735$	0	0
$736$	3.30765	0.121921
$737$	2.14045	0.0788444
$738$	0	0
$739$	24.3667	0.896345	0.448173	−	0.893947i	$-0.352075\pi$
$739$	24.3667	0.896345	0.448173	+	0.893947i	$0.352075\pi$
$740$	−22.9854	−0.844961
$741$	0	0
$742$	7.08223	0.259997
$743$	19.2865	0.707553	0.353777	−	0.935330i	$-0.384897\pi$
$743$	19.2865	0.707553	0.353777	+	0.935330i	$0.384897\pi$
$744$	0	0
$745$	−4.83626	−0.177187
$746$	−43.4289	−1.59004
$747$	0	0
$748$	95.0119	3.47398
$749$	−2.46688	−0.0901380
$750$	0	0
$751$	−32.7485	−1.19501	−0.597505	−	0.801865i	$-0.703841\pi$
$751$	−32.7485	−1.19501	−0.597505	+	0.801865i	$0.703841\pi$
$752$	−2.27102	−0.0828155
$753$	0	0
$754$	−19.1891	−0.698824
$755$	−10.9945	−0.400130
$756$	0	0
$757$	18.7991	0.683264	0.341632	−	0.939834i	$-0.389020\pi$
$757$	18.7991	0.683264	0.341632	+	0.939834i	$0.389020\pi$
$758$	−73.1660	−2.65751
$759$	0	0
$760$	−11.9464	−0.433340
$761$	40.0426	1.45154	0.725772	−	0.687935i	$-0.241483\pi$
$761$	40.0426	1.45154	0.725772	+	0.687935i	$0.241483\pi$
$762$	0	0
$763$	2.53260	0.0916863
$764$	51.3975	1.85950
$765$	0	0
$766$	18.5592	0.670572
$767$	1.15359	0.0416537
$768$	0	0
$769$	−18.3446	−0.661522	−0.330761	−	0.943715i	$-0.607305\pi$
$769$	−18.3446	−0.661522	−0.330761	+	0.943715i	$0.607305\pi$
$770$	−2.74512	−0.0989272
$771$	0	0
$772$	−17.7966	−0.640515
$773$	−19.0710	−0.685935	−0.342968	−	0.939347i	$-0.611432\pi$
$773$	−19.0710	−0.685935	−0.342968	+	0.939347i	$0.611432\pi$
$774$	0	0
$775$	0	0
$776$	−17.1769	−0.616613
$777$	0	0
$778$	−15.6890	−0.562478
$779$	57.4743	2.05923
$780$	0	0
$781$	21.6648	0.775227
$782$	7.01509	0.250859
$783$	0	0
$784$	8.66254	0.309377
$785$	23.8612	0.851643
$786$	0	0
$787$	−0.519440	−0.0185160	−0.00925802	−	0.999957i	$-0.502947\pi$
$787$	−0.519440	−0.0185160	−0.00925802	+	0.999957i	$0.502947\pi$
$788$	−31.0946	−1.10770
$789$	0	0
$790$	−36.0793	−1.28364
$791$	4.29460	0.152699
$792$	0	0
$793$	−9.77497	−0.347120
$794$	66.6463	2.36519
$795$	0	0
$796$	−78.8222	−2.79378
$797$	−25.5899	−0.906442	−0.453221	−	0.891398i	$-0.649725\pi$
$797$	−25.5899	−0.906442	−0.453221	+	0.891398i	$0.649725\pi$
$798$	0	0
$799$	−12.0462	−0.426163
$800$	−27.4084	−0.969035
$801$	0	0
$802$	55.0073	1.94237
$803$	7.20255	0.254172
$804$	0	0
$805$	−0.120574	−0.00424966
$806$	0	0
$807$	0	0
$808$	1.02420	0.0360312
$809$	0.476507	0.0167531	0.00837655	−	0.999965i	$-0.497334\pi$
$809$	0.476507	0.0167531	0.00837655	+	0.999965i	$0.497334\pi$
$810$	0	0
$811$	−13.8014	−0.484634	−0.242317	−	0.970197i	$-0.577907\pi$
$811$	−13.8014	−0.484634	−0.242317	+	0.970197i	$0.577907\pi$
$812$	4.76574	0.167245
$813$	0	0
$814$	82.9456	2.90724
$815$	23.5252	0.824051
$816$	0	0
$817$	44.4799	1.55616
$818$	−8.61132	−0.301088
$819$	0	0
$820$	30.8583	1.07762
$821$	−4.57783	−0.159767	−0.0798836	−	0.996804i	$-0.525455\pi$
$821$	−4.57783	−0.159767	−0.0798836	+	0.996804i	$0.525455\pi$
$822$	0	0
$823$	41.4156	1.44366	0.721829	−	0.692071i	$-0.243302\pi$
$823$	41.4156	1.44366	0.721829	+	0.692071i	$0.243302\pi$
$824$	−16.1596	−0.562948
$825$	0	0
$826$	−0.481607	−0.0167572
$827$	−42.8847	−1.49125	−0.745623	−	0.666368i	$-0.767848\pi$
$827$	−42.8847	−1.49125	−0.745623	+	0.666368i	$0.767848\pi$
$828$	0	0
$829$	−24.6524	−0.856213	−0.428107	−	0.903728i	$-0.640819\pi$
$829$	−24.6524	−0.856213	−0.428107	+	0.903728i	$0.640819\pi$
$830$	19.4235	0.674198
$831$	0	0
$832$	−16.9734	−0.588447
$833$	45.9488	1.59203
$834$	0	0
$835$	−3.15485	−0.109178
$836$	80.3904	2.78036
$837$	0	0
$838$	−67.9210	−2.34629
$839$	32.2000	1.11167	0.555834	−	0.831293i	$-0.312399\pi$
$839$	32.2000	1.11167	0.555834	+	0.831293i	$0.312399\pi$
$840$	0	0
$841$	14.1971	0.489555
$842$	−4.67117	−0.160979
$843$	0	0
$844$	7.53401	0.259331
$845$	11.5468	0.397223
$846$	0	0
$847$	3.17717	0.109169
$848$	16.1167	0.553449
$849$	0	0
$850$	−58.1298	−1.99383
$851$	3.64321	0.124888
$852$	0	0
$853$	43.8610	1.50177	0.750886	−	0.660431i	$-0.229627\pi$
$853$	43.8610	1.50177	0.750886	+	0.660431i	$0.229627\pi$
$854$	4.08091	0.139646
$855$	0	0
$856$	20.7999	0.710927
$857$	18.0501	0.616581	0.308290	−	0.951292i	$-0.400243\pi$
$857$	18.0501	0.616581	0.308290	+	0.951292i	$0.400243\pi$
$858$	0	0
$859$	−42.0106	−1.43338	−0.716692	−	0.697390i	$-0.754345\pi$
$859$	−42.0106	−1.43338	−0.716692	+	0.697390i	$0.754345\pi$
$860$	23.8815	0.814353
$861$	0	0
$862$	22.4801	0.765676
$863$	−16.3390	−0.556186	−0.278093	−	0.960554i	$-0.589702\pi$
$863$	−16.3390	−0.556186	−0.278093	+	0.960554i	$0.589702\pi$
$864$	0	0
$865$	−4.68487	−0.159290
$866$	36.5721	1.24277
$867$	0	0
$868$	0	0
$869$	77.4523	2.62739
$870$	0	0
$871$	−0.575693	−0.0195066
$872$	−21.3540	−0.723139
$873$	0	0
$874$	5.93553	0.200772
$875$	2.26354	0.0765216
$876$	0	0
$877$	33.9713	1.14713	0.573564	−	0.819161i	$-0.305560\pi$
$877$	33.9713	1.14713	0.573564	+	0.819161i	$0.305560\pi$
$878$	−86.8864	−2.93227
$879$	0	0
$880$	−6.24693	−0.210584
$881$	−47.4247	−1.59778	−0.798889	−	0.601478i	$-0.794579\pi$
$881$	−47.4247	−1.59778	−0.798889	+	0.601478i	$0.794579\pi$
$882$	0	0
$883$	32.4675	1.09262	0.546310	−	0.837583i	$-0.316032\pi$
$883$	32.4675	1.09262	0.546310	+	0.837583i	$0.316032\pi$
$884$	−25.5543	−0.859485
$885$	0	0
$886$	−22.2445	−0.747320
$887$	−21.9399	−0.736670	−0.368335	−	0.929693i	$-0.620072\pi$
$887$	−21.9399	−0.736670	−0.368335	+	0.929693i	$0.620072\pi$
$888$	0	0
$889$	0.990063	0.0332057
$890$	−29.0871	−0.975002
$891$	0	0
$892$	−18.8771	−0.632051
$893$	−10.1924	−0.341075
$894$	0	0
$895$	5.01736	0.167712
$896$	3.66060	0.122292
$897$	0	0
$898$	24.6884	0.823864
$899$	0	0
$900$	0	0
$901$	85.4878	2.84801
$902$	−111.356	−3.70774
$903$	0	0
$904$	−36.2106	−1.20435
$905$	−20.7720	−0.690485
$906$	0	0
$907$	8.76232	0.290948	0.145474	−	0.989362i	$-0.453529\pi$
$907$	8.76232	0.290948	0.145474	+	0.989362i	$0.453529\pi$
$908$	48.2676	1.60182
$909$	0	0
$910$	0.738325	0.0244752
$911$	−23.6888	−0.784845	−0.392422	−	0.919785i	$-0.628363\pi$
$911$	−23.6888	−0.784845	−0.392422	+	0.919785i	$0.628363\pi$
$912$	0	0
$913$	−41.6968	−1.37996
$914$	−43.9167	−1.45263
$915$	0	0
$916$	−51.0119	−1.68548
$917$	−0.0720123	−0.00237806
$918$	0	0
$919$	27.4194	0.904481	0.452241	−	0.891896i	$-0.350625\pi$
$919$	27.4194	0.904481	0.452241	+	0.891896i	$0.350625\pi$
$920$	1.01663	0.0335175
$921$	0	0
$922$	54.9970	1.81123
$923$	−5.82694	−0.191796
$924$	0	0
$925$	−30.1891	−0.992610
$926$	−63.9284	−2.10082
$927$	0	0
$928$	45.5948	1.49672
$929$	−24.5453	−0.805304	−0.402652	−	0.915353i	$-0.631912\pi$
$929$	−24.5453	−0.805304	−0.402652	+	0.915353i	$0.631912\pi$
$930$	0	0
$931$	38.8777	1.27417
$932$	50.0938	1.64088
$933$	0	0
$934$	−9.51764	−0.311427
$935$	−33.1356	−1.08365
$936$	0	0
$937$	−6.50507	−0.212511	−0.106256	−	0.994339i	$-0.533886\pi$
$937$	−6.50507	−0.212511	−0.106256	+	0.994339i	$0.533886\pi$
$938$	0.240344	0.00784750
$939$	0	0
$940$	−5.47235	−0.178488
$941$	−30.8349	−1.00519	−0.502595	−	0.864522i	$-0.667621\pi$
$941$	−30.8349	−1.00519	−0.502595	+	0.864522i	$0.667621\pi$
$942$	0	0
$943$	−4.89107	−0.159275
$944$	−1.09597	−0.0356707
$945$	0	0
$946$	−86.1793	−2.80193
$947$	−28.8158	−0.936389	−0.468194	−	0.883625i	$-0.655095\pi$
$947$	−28.8158	−0.936389	−0.468194	+	0.883625i	$0.655095\pi$
$948$	0	0
$949$	−1.93719	−0.0628839
$950$	−49.1841	−1.59574
$951$	0	0
$952$	3.40340	0.110305
$953$	−24.6753	−0.799312	−0.399656	−	0.916665i	$-0.630871\pi$
$953$	−24.6753	−0.799312	−0.399656	+	0.916665i	$0.630871\pi$
$954$	0	0
$955$	−17.9250	−0.580039
$956$	−13.7048	−0.443244
$957$	0	0
$958$	−91.2353	−2.94768
$959$	0.0945318	0.00305259
$960$	0	0
$961$	0	0
$962$	−22.3090	−0.719271
$963$	0	0
$964$	−17.8030	−0.573396
$965$	6.20662	0.199798
$966$	0	0
$967$	−31.0647	−0.998973	−0.499486	−	0.866322i	$-0.666478\pi$
$967$	−31.0647	−0.998973	−0.499486	+	0.866322i	$0.666478\pi$
$968$	−26.7888	−0.861024
$969$	0	0
$970$	18.7782	0.602932
$971$	43.4539	1.39450	0.697250	−	0.716828i	$-0.254407\pi$
$971$	43.4539	1.39450	0.697250	+	0.716828i	$0.254407\pi$
$972$	0	0
$973$	1.19219	0.0382199
$974$	55.9405	1.79245
$975$	0	0
$976$	9.28673	0.297261
$977$	30.7060	0.982372	0.491186	−	0.871055i	$-0.336563\pi$
$977$	30.7060	0.982372	0.491186	+	0.871055i	$0.336563\pi$
$978$	0	0
$979$	62.4420	1.99566
$980$	20.8737	0.666785
$981$	0	0
$982$	59.4119	1.89591
$983$	−56.0862	−1.78887	−0.894436	−	0.447196i	$-0.852423\pi$
$983$	−56.0862	−1.78887	−0.894436	+	0.447196i	$0.852423\pi$
$984$	0	0
$985$	10.8443	0.345529
$986$	96.7007	3.07958
$987$	0	0
$988$	−21.6217	−0.687879
$989$	−3.78524	−0.120364
$990$	0	0
$991$	16.8077	0.533915	0.266957	−	0.963708i	$-0.413982\pi$
$991$	16.8077	0.533915	0.266957	+	0.963708i	$0.413982\pi$
$992$	0	0
$993$	0	0
$994$	2.43266	0.0771594
$995$	27.4895	0.871475
$996$	0	0
$997$	39.4217	1.24850	0.624248	−	0.781226i	$-0.285405\pi$
$997$	39.4217	1.24850	0.624248	+	0.781226i	$0.285405\pi$
$998$	−22.4994	−0.712207
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bu.1.4		24
3.2	odd	2		2883.2.a.v.1.21	yes	24
31.30	odd	2		8649.2.a.bv.1.4		24
93.92	even	2		2883.2.a.u.1.21	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.21	✓	24	93.92	even	2
2883.2.a.v.1.21	yes	24	3.2	odd	2
8649.2.a.bu.1.4		24	1.1	even	1	trivial
8649.2.a.bv.1.4		24	31.30	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22192 q^{2} +2.93691 q^{4} -1.02426 q^{5} +0.246895 q^{7} -2.08174 q^{8} +O(q^{10})\)	\(q-2.22192 q^{2} +2.93691 q^{4} -1.02426 q^{5} +0.246895 q^{7} -2.08174 q^{8} +2.27581 q^{10} -4.88554 q^{11} +1.31401 q^{13} -0.548581 q^{14} -1.24838 q^{16} -6.62178 q^{17} -5.60275 q^{19} -3.00815 q^{20} +10.8553 q^{22} +0.476794 q^{23} -3.95090 q^{25} -2.91962 q^{26} +0.725109 q^{28} +6.57245 q^{29} +6.93726 q^{32} +14.7130 q^{34} -0.252884 q^{35} +7.64106 q^{37} +12.4488 q^{38} +2.13223 q^{40} -10.2582 q^{41} -7.93895 q^{43} -14.3484 q^{44} -1.05940 q^{46} +1.81918 q^{47} -6.93904 q^{49} +8.77857 q^{50} +3.85913 q^{52} -12.9101 q^{53} +5.00404 q^{55} -0.513971 q^{56} -14.6034 q^{58} +0.877915 q^{59} -7.43904 q^{61} -12.9173 q^{64} -1.34588 q^{65} -0.438119 q^{67} -19.4476 q^{68} +0.561887 q^{70} -4.43447 q^{71} -1.47426 q^{73} -16.9778 q^{74} -16.4548 q^{76} -1.20622 q^{77} -15.8534 q^{79} +1.27866 q^{80} +22.7929 q^{82} +8.53475 q^{83} +6.78240 q^{85} +17.6397 q^{86} +10.1704 q^{88} -12.7810 q^{89} +0.324423 q^{91} +1.40030 q^{92} -4.04206 q^{94} +5.73865 q^{95} +8.25122 q^{97} +15.4180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29} + 48 q^{32} + 48 q^{35} + 64 q^{37} + 24 q^{38} + 32 q^{43} - 48 q^{44} + 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} + 64 q^{52} - 80 q^{53} + 48 q^{56} + 32 q^{58} + 32 q^{61} + 56 q^{64} - 16 q^{65} - 16 q^{67} - 80 q^{68} + 8 q^{70} + 32 q^{73} + 56 q^{76} - 96 q^{77} + 32 q^{79} + 72 q^{80} + 8 q^{82} - 48 q^{83} + 96 q^{85} + 32 q^{86} + 96 q^{88} + 16 q^{89} - 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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