LMFDB - Embedded newform 1665.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1665,2,Mod(1,1665)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1665, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1665.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1665 = 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1665.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,6,-5,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.2950919365$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.552543$ of defining polynomial
Character	$\chi$	$=$	1665.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.180152 q^{2} -1.96755 q^{4} -1.00000 q^{5} +4.41500 q^{7} +0.714762 q^{8} +0.180152 q^{10} -4.27171 q^{11} -2.79978 q^{13} -0.795373 q^{14} +3.80632 q^{16} +4.43948 q^{17} +2.43507 q^{19} +1.96755 q^{20} +0.769559 q^{22} -5.77387 q^{23} +1.00000 q^{25} +0.504387 q^{26} -8.68672 q^{28} -0.409254 q^{29} +7.79180 q^{31} -2.11524 q^{32} -0.799782 q^{34} -4.41500 q^{35} -1.00000 q^{37} -0.438683 q^{38} -0.714762 q^{40} -0.757374 q^{41} +2.19908 q^{43} +8.40479 q^{44} +1.04018 q^{46} +4.26487 q^{47} +12.4922 q^{49} -0.180152 q^{50} +5.50870 q^{52} +0.137540 q^{53} +4.27171 q^{55} +3.15568 q^{56} +0.0737281 q^{58} +3.07119 q^{59} -3.02909 q^{61} -1.40371 q^{62} -7.23158 q^{64} +2.79978 q^{65} -11.4482 q^{67} -8.73487 q^{68} +0.795373 q^{70} +10.7144 q^{71} +8.20680 q^{73} +0.180152 q^{74} -4.79111 q^{76} -18.8596 q^{77} +7.11193 q^{79} -3.80632 q^{80} +0.136443 q^{82} +11.3625 q^{83} -4.43948 q^{85} -0.396170 q^{86} -3.05326 q^{88} +16.2305 q^{89} -12.3610 q^{91} +11.3603 q^{92} -0.768326 q^{94} -2.43507 q^{95} +18.3399 q^{97} -2.25051 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{4} - 5 q^{5} + 7 q^{7} + 6 q^{8} - 7 q^{11} + 2 q^{13} - 4 q^{14} + 8 q^{16} + 8 q^{17} + 14 q^{19} - 6 q^{20} + 2 q^{22} - 2 q^{23} + 5 q^{25} + 20 q^{26} - 14 q^{28} - 2 q^{29} + 8 q^{31} + 22 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) $ 5 * q + 6 * q^4 - 5 * q^5 + 7 * q^7 + 6 * q^8 - 7 * q^11 + 2 * q^13 - 4 * q^14 + 8 * q^16 + 8 * q^17 + 14 * q^19 - 6 * q^20 + 2 * q^22 - 2 * q^23 + 5 * q^25 + 20 * q^26 - 14 * q^28 - 2 * q^29 + 8 * q^31 + 22 * q^32 + 12 * q^34 - 7 * q^35 - 5 * q^37 + 32 * q^38 - 6 * q^40 + 9 * q^41 + 14 * q^43 - 18 * q^44 - 28 * q^46 + 5 * q^47 + 2 * q^49 + 20 * q^52 + 15 * q^53 + 7 * q^55 + 14 * q^56 - 12 * q^59 + 12 * q^61 - 10 * q^62 + 12 * q^64 - 2 * q^65 - 2 * q^67 + 4 * q^68 + 4 * q^70 - 13 * q^71 - 5 * q^73 + 18 * q^76 + 7 * q^77 + 36 * q^79 - 8 * q^80 - 2 * q^82 + 9 * q^83 - 8 * q^85 + 8 * q^86 + 8 * q^88 + 16 * q^89 - 4 * q^91 - 12 * q^92 + 20 * q^94 - 14 * q^95 - 30 * q^98

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 $

$2$	−0.180152	−0.127387	−0.0636934	−	0.997970i	$-0.520288\pi$
$2$	−0.180152	−0.127387	−0.0636934	+	0.997970i	$0.520288\pi$
$3$	0	0
$3$	0	0
$4$	−1.96755	−0.983773
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.41500	1.66871	0.834357	−	0.551224i	$-0.185839\pi$
$7$	4.41500	1.66871	0.834357	+	0.551224i	$0.185839\pi$
$8$	0.714762	0.252707
$9$	0	0
$10$	0.180152	0.0569692
$11$	−4.27171	−1.28797	−0.643985	−	0.765038i	$-0.722720\pi$
$11$	−4.27171	−1.28797	−0.643985	+	0.765038i	$0.722720\pi$
$12$	0	0
$13$	−2.79978	−0.776520	−0.388260	−	0.921550i	$-0.626924\pi$
$13$	−2.79978	−0.776520	−0.388260	+	0.921550i	$0.626924\pi$
$14$	−0.795373	−0.212572
$15$	0	0
$16$	3.80632	0.951581
$17$	4.43948	1.07673	0.538366	−	0.842711i	$-0.319042\pi$
$17$	4.43948	1.07673	0.538366	+	0.842711i	$0.319042\pi$
$18$	0	0
$19$	2.43507	0.558643	0.279321	−	0.960198i	$-0.409890\pi$
$19$	2.43507	0.558643	0.279321	+	0.960198i	$0.409890\pi$
$20$	1.96755	0.439956
$21$	0	0
$22$	0.769559	0.164071
$23$	−5.77387	−1.20393	−0.601967	−	0.798521i	$-0.705616\pi$
$23$	−5.77387	−1.20393	−0.601967	+	0.798521i	$0.705616\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.504387	0.0989184
$27$	0	0
$28$	−8.68672	−1.64164
$29$	−0.409254	−0.0759967	−0.0379983	−	0.999278i	$-0.512098\pi$
$29$	−0.409254	−0.0759967	−0.0379983	+	0.999278i	$0.512098\pi$
$30$	0	0
$31$	7.79180	1.39945	0.699724	−	0.714413i	$-0.253306\pi$
$31$	7.79180	1.39945	0.699724	+	0.714413i	$0.253306\pi$
$32$	−2.11524	−0.373926
$33$	0	0
$34$	−0.799782	−0.137161
$35$	−4.41500	−0.746272
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.438683	−0.0711638
$39$	0	0
$40$	−0.714762	−0.113014
$41$	−0.757374	−0.118282	−0.0591410	−	0.998250i	$-0.518836\pi$
$41$	−0.757374	−0.118282	−0.0591410	+	0.998250i	$0.518836\pi$
$42$	0	0
$43$	2.19908	0.335357	0.167679	−	0.985842i	$-0.446373\pi$
$43$	2.19908	0.335357	0.167679	+	0.985842i	$0.446373\pi$
$44$	8.40479	1.26707
$45$	0	0
$46$	1.04018	0.153366
$47$	4.26487	0.622095	0.311048	−	0.950394i	$-0.399320\pi$
$47$	4.26487	0.622095	0.311048	+	0.950394i	$0.399320\pi$
$48$	0	0
$49$	12.4922	1.78461
$50$	−0.180152	−0.0254774
$51$	0	0
$52$	5.50870	0.763919
$53$	0.137540	0.0188926	0.00944630	−	0.999955i	$-0.496993\pi$
$53$	0.137540	0.0188926	0.00944630	+	0.999955i	$0.496993\pi$
$54$	0	0
$55$	4.27171	0.575998
$56$	3.15568	0.421695
$57$	0	0
$58$	0.0737281	0.00968098
$59$	3.07119	0.399835	0.199918	−	0.979813i	$-0.435933\pi$
$59$	3.07119	0.399835	0.199918	+	0.979813i	$0.435933\pi$
$60$	0	0
$61$	−3.02909	−0.387835	−0.193918	−	0.981018i	$-0.562119\pi$
$61$	−3.02909	−0.387835	−0.193918	+	0.981018i	$0.562119\pi$
$62$	−1.40371	−0.178271
$63$	0	0
$64$	−7.23158	−0.903948
$65$	2.79978	0.347270
$66$	0	0
$67$	−11.4482	−1.39862	−0.699310	−	0.714819i	$-0.746509\pi$
$67$	−11.4482	−1.39862	−0.699310	+	0.714819i	$0.746509\pi$
$68$	−8.73487	−1.05926
$69$	0	0
$70$	0.795373	0.0950652
$71$	10.7144	1.27156	0.635781	−	0.771870i	$-0.280678\pi$
$71$	10.7144	1.27156	0.635781	+	0.771870i	$0.280678\pi$
$72$	0	0
$73$	8.20680	0.960534	0.480267	−	0.877122i	$-0.340540\pi$
$73$	8.20680	0.960534	0.480267	+	0.877122i	$0.340540\pi$
$74$	0.180152	0.0209423
$75$	0	0
$76$	−4.79111	−0.549578
$77$	−18.8596	−2.14925
$78$	0	0
$79$	7.11193	0.800155	0.400077	−	0.916481i	$-0.368983\pi$
$79$	7.11193	0.800155	0.400077	+	0.916481i	$0.368983\pi$
$80$	−3.80632	−0.425560
$81$	0	0
$82$	0.136443	0.0150676
$83$	11.3625	1.24719	0.623597	−	0.781746i	$-0.285671\pi$
$83$	11.3625	1.24719	0.623597	+	0.781746i	$0.285671\pi$
$84$	0	0
$85$	−4.43948	−0.481529
$86$	−0.396170	−0.0427201
$87$	0	0
$88$	−3.05326	−0.325479
$89$	16.2305	1.72043	0.860214	−	0.509933i	$-0.170330\pi$
$89$	16.2305	1.72043	0.860214	+	0.509933i	$0.170330\pi$
$90$	0	0
$91$	−12.3610	−1.29579
$92$	11.3603	1.18440
$93$	0	0
$94$	−0.768326	−0.0792468
$95$	−2.43507	−0.249833
$96$	0	0
$97$	18.3399	1.86213	0.931066	−	0.364850i	$-0.118880\pi$
$97$	18.3399	1.86213	0.931066	+	0.364850i	$0.118880\pi$
$98$	−2.25051	−0.227336
$99$	0	0
$100$	−1.96755	−0.196755
$101$	11.1319	1.10767	0.553835	−	0.832627i	$-0.313164\pi$
$101$	11.1319	1.10767	0.553835	+	0.832627i	$0.313164\pi$
$102$	0	0
$103$	6.78105	0.668157	0.334079	−	0.942545i	$-0.391575\pi$
$103$	6.78105	0.668157	0.334079	+	0.942545i	$0.391575\pi$
$104$	−2.00118	−0.196232
$105$	0	0
$106$	−0.0247782	−0.00240667
$107$	−0.536583	−0.0518734	−0.0259367	−	0.999664i	$-0.508257\pi$
$107$	−0.536583	−0.0518734	−0.0259367	+	0.999664i	$0.508257\pi$
$108$	0	0
$109$	9.53465	0.913254	0.456627	−	0.889658i	$-0.349057\pi$
$109$	9.53465	0.913254	0.456627	+	0.889658i	$0.349057\pi$
$110$	−0.769559	−0.0733746
$111$	0	0
$112$	16.8049	1.58792
$113$	−9.16830	−0.862481	−0.431240	−	0.902237i	$-0.641924\pi$
$113$	−9.16830	−0.862481	−0.431240	+	0.902237i	$0.641924\pi$
$114$	0	0
$115$	5.77387	0.538416
$116$	0.805227	0.0747634
$117$	0	0
$118$	−0.553282	−0.0509338
$119$	19.6003	1.79676
$120$	0	0
$121$	7.24754	0.658867
$122$	0.545697	0.0494051
$123$	0	0
$124$	−15.3307	−1.37674
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.32621	0.827567	0.413784	−	0.910375i	$-0.364207\pi$
$127$	9.32621	0.827567	0.413784	+	0.910375i	$0.364207\pi$
$128$	5.53327	0.489077
$129$	0	0
$130$	−0.504387	−0.0442377
$131$	−15.6006	−1.36303	−0.681515	−	0.731804i	$-0.738679\pi$
$131$	−15.6006	−1.36303	−0.681515	+	0.731804i	$0.738679\pi$
$132$	0	0
$133$	10.7508	0.932215
$134$	2.06242	0.178166
$135$	0	0
$136$	3.17317	0.272097
$137$	1.01986	0.0871326	0.0435663	−	0.999051i	$-0.486128\pi$
$137$	1.01986	0.0871326	0.0435663	+	0.999051i	$0.486128\pi$
$138$	0	0
$139$	7.08913	0.601292	0.300646	−	0.953736i	$-0.402798\pi$
$139$	7.08913	0.601292	0.300646	+	0.953736i	$0.402798\pi$
$140$	8.68672	0.734162
$141$	0	0
$142$	−1.93022	−0.161980
$143$	11.9599	1.00013
$144$	0	0
$145$	0.409254	0.0339867
$146$	−1.47847	−0.122359
$147$	0	0
$148$	1.96755	0.161731
$149$	15.5572	1.27449	0.637246	−	0.770660i	$-0.280074\pi$
$149$	15.5572	1.27449	0.637246	+	0.770660i	$0.280074\pi$
$150$	0	0
$151$	−4.00882	−0.326233	−0.163117	−	0.986607i	$-0.552155\pi$
$151$	−4.00882	−0.326233	−0.163117	+	0.986607i	$0.552155\pi$
$152$	1.74049	0.141173
$153$	0	0
$154$	3.39761	0.273787
$155$	−7.79180	−0.625853
$156$	0	0
$157$	3.06585	0.244682	0.122341	−	0.992488i	$-0.460960\pi$
$157$	3.06585	0.244682	0.122341	+	0.992488i	$0.460960\pi$
$158$	−1.28123	−0.101929
$159$	0	0
$160$	2.11524	0.167225
$161$	−25.4916	−2.00902
$162$	0	0
$163$	2.90600	0.227616	0.113808	−	0.993503i	$-0.463695\pi$
$163$	2.90600	0.227616	0.113808	+	0.993503i	$0.463695\pi$
$164$	1.49017	0.116363
$165$	0	0
$166$	−2.04698	−0.158876
$167$	23.5324	1.82099	0.910495	−	0.413519i	$-0.135701\pi$
$167$	23.5324	1.82099	0.910495	+	0.413519i	$0.135701\pi$
$168$	0	0
$169$	−5.16122	−0.397017
$170$	0.799782	0.0613405
$171$	0	0
$172$	−4.32680	−0.329915
$173$	18.6902	1.42099	0.710495	−	0.703703i	$-0.248471\pi$
$173$	18.6902	1.42099	0.710495	+	0.703703i	$0.248471\pi$
$174$	0	0
$175$	4.41500	0.333743
$176$	−16.2595	−1.22561
$177$	0	0
$178$	−2.92396	−0.219160
$179$	1.22059	0.0912310	0.0456155	−	0.998959i	$-0.485475\pi$
$179$	1.22059	0.0912310	0.0456155	+	0.998959i	$0.485475\pi$
$180$	0	0
$181$	−23.8124	−1.76996	−0.884982	−	0.465626i	$-0.845829\pi$
$181$	−23.8124	−1.76996	−0.884982	+	0.465626i	$0.845829\pi$
$182$	2.22687	0.165067
$183$	0	0
$184$	−4.12694	−0.304242
$185$	1.00000	0.0735215
$186$	0	0
$187$	−18.9642	−1.38680
$188$	−8.39132	−0.612000
$189$	0	0
$190$	0.438683	0.0318254
$191$	−14.9462	−1.08147	−0.540736	−	0.841192i	$-0.681854\pi$
$191$	−14.9462	−1.08147	−0.540736	+	0.841192i	$0.681854\pi$
$192$	0	0
$193$	−18.9075	−1.36099	−0.680495	−	0.732753i	$-0.738235\pi$
$193$	−18.9075	−1.36099	−0.680495	+	0.732753i	$0.738235\pi$
$194$	−3.30397	−0.237211
$195$	0	0
$196$	−24.5791	−1.75565
$197$	−3.72602	−0.265468	−0.132734	−	0.991152i	$-0.542376\pi$
$197$	−3.72602	−0.265468	−0.132734	+	0.991152i	$0.542376\pi$
$198$	0	0
$199$	−18.7237	−1.32729	−0.663645	−	0.748048i	$-0.730991\pi$
$199$	−18.7237	−1.32729	−0.663645	+	0.748048i	$0.730991\pi$
$200$	0.714762	0.0505413
$201$	0	0
$202$	−2.00544	−0.141103
$203$	−1.80686	−0.126817
$204$	0	0
$205$	0.757374	0.0528973
$206$	−1.22162	−0.0851145
$207$	0	0
$208$	−10.6569	−0.738922
$209$	−10.4019	−0.719516
$210$	0	0
$211$	−13.3971	−0.922295	−0.461148	−	0.887323i	$-0.652562\pi$
$211$	−13.3971	−0.922295	−0.461148	+	0.887323i	$0.652562\pi$
$212$	−0.270617	−0.0185860
$213$	0	0
$214$	0.0966666	0.00660800
$215$	−2.19908	−0.149976
$216$	0	0
$217$	34.4008	2.33528
$218$	−1.71769	−0.116337
$219$	0	0
$220$	−8.40479	−0.566651
$221$	−12.4296	−0.836103
$222$	0	0
$223$	−17.4471	−1.16834	−0.584172	−	0.811630i	$-0.698581\pi$
$223$	−17.4471	−1.16834	−0.584172	+	0.811630i	$0.698581\pi$
$224$	−9.33880	−0.623975
$225$	0	0
$226$	1.65169	0.109869
$227$	−20.4988	−1.36056	−0.680278	−	0.732954i	$-0.738141\pi$
$227$	−20.4988	−1.36056	−0.680278	+	0.732954i	$0.738141\pi$
$228$	0	0
$229$	−14.0698	−0.929756	−0.464878	−	0.885375i	$-0.653902\pi$
$229$	−14.0698	−0.929756	−0.464878	+	0.885375i	$0.653902\pi$
$230$	−1.04018	−0.0685872
$231$	0	0
$232$	−0.292520	−0.0192049
$233$	−15.1114	−0.989983	−0.494992	−	0.868898i	$-0.664829\pi$
$233$	−15.1114	−0.989983	−0.494992	+	0.868898i	$0.664829\pi$
$234$	0	0
$235$	−4.26487	−0.278209
$236$	−6.04271	−0.393347
$237$	0	0
$238$	−3.53104	−0.228883
$239$	−0.0747636	−0.00483606	−0.00241803	−	0.999997i	$-0.500770\pi$
$239$	−0.0747636	−0.00483606	−0.00241803	+	0.999997i	$0.500770\pi$
$240$	0	0
$241$	20.4873	1.31971	0.659853	−	0.751395i	$-0.270619\pi$
$241$	20.4873	1.31971	0.659853	+	0.751395i	$0.270619\pi$
$242$	−1.30566	−0.0839311
$243$	0	0
$244$	5.95987	0.381541
$245$	−12.4922	−0.798100
$246$	0	0
$247$	−6.81766	−0.433797
$248$	5.56929	0.353650
$249$	0	0
$250$	0.180152	0.0113938
$251$	6.85355	0.432592	0.216296	−	0.976328i	$-0.430602\pi$
$251$	6.85355	0.432592	0.216296	+	0.976328i	$0.430602\pi$
$252$	0	0
$253$	24.6643	1.55063
$254$	−1.68014	−0.105421
$255$	0	0
$256$	13.4663	0.841646
$257$	10.4751	0.653422	0.326711	−	0.945124i	$-0.394060\pi$
$257$	10.4751	0.653422	0.326711	+	0.945124i	$0.394060\pi$
$258$	0	0
$259$	−4.41500	−0.274335
$260$	−5.50870	−0.341635
$261$	0	0
$262$	2.81048	0.173632
$263$	−5.99787	−0.369844	−0.184922	−	0.982753i	$-0.559203\pi$
$263$	−5.99787	−0.369844	−0.184922	+	0.982753i	$0.559203\pi$
$264$	0	0
$265$	−0.137540	−0.00844903
$266$	−1.93679	−0.118752
$267$	0	0
$268$	22.5248	1.37592
$269$	2.32781	0.141929	0.0709646	−	0.997479i	$-0.477392\pi$
$269$	2.32781	0.141929	0.0709646	+	0.997479i	$0.477392\pi$
$270$	0	0
$271$	−6.13081	−0.372420	−0.186210	−	0.982510i	$-0.559620\pi$
$271$	−6.13081	−0.372420	−0.186210	+	0.982510i	$0.559620\pi$
$272$	16.8981	1.02460
$273$	0	0
$274$	−0.183730	−0.0110996
$275$	−4.27171	−0.257594
$276$	0	0
$277$	−8.30966	−0.499279	−0.249639	−	0.968339i	$-0.580312\pi$
$277$	−8.30966	−0.499279	−0.249639	+	0.968339i	$0.580312\pi$
$278$	−1.27712	−0.0765967
$279$	0	0
$280$	−3.15568	−0.188588
$281$	5.01313	0.299058	0.149529	−	0.988757i	$-0.452224\pi$
$281$	5.01313	0.299058	0.149529	+	0.988757i	$0.452224\pi$
$282$	0	0
$283$	12.5308	0.744876	0.372438	−	0.928057i	$-0.378522\pi$
$283$	12.5308	0.744876	0.372438	+	0.928057i	$0.378522\pi$
$284$	−21.0810	−1.25093
$285$	0	0
$286$	−2.15460	−0.127404
$287$	−3.34381	−0.197379
$288$	0	0
$289$	2.70896	0.159351
$290$	−0.0737281	−0.00432946
$291$	0	0
$292$	−16.1473	−0.944947
$293$	6.13130	0.358194	0.179097	−	0.983831i	$-0.442682\pi$
$293$	6.13130	0.358194	0.179097	+	0.983831i	$0.442682\pi$
$294$	0	0
$295$	−3.07119	−0.178812
$296$	−0.714762	−0.0415447
$297$	0	0
$298$	−2.80266	−0.162354
$299$	16.1656	0.934879
$300$	0	0
$301$	9.70896	0.559615
$302$	0.722198	0.0415578
$303$	0	0
$304$	9.26866	0.531594
$305$	3.02909	0.173445
$306$	0	0
$307$	12.6426	0.721552	0.360776	−	0.932652i	$-0.382512\pi$
$307$	12.6426	0.721552	0.360776	+	0.932652i	$0.382512\pi$
$308$	37.1072	2.11438
$309$	0	0
$310$	1.40371	0.0797254
$311$	−8.05133	−0.456549	−0.228275	−	0.973597i	$-0.573308\pi$
$311$	−8.05133	−0.456549	−0.228275	+	0.973597i	$0.573308\pi$
$312$	0	0
$313$	−2.99549	−0.169315	−0.0846576	−	0.996410i	$-0.526980\pi$
$313$	−2.99549	−0.169315	−0.0846576	+	0.996410i	$0.526980\pi$
$314$	−0.552321	−0.0311693
$315$	0	0
$316$	−13.9930	−0.787170
$317$	−2.24377	−0.126023	−0.0630113	−	0.998013i	$-0.520070\pi$
$317$	−2.24377	−0.126023	−0.0630113	+	0.998013i	$0.520070\pi$
$318$	0	0
$319$	1.74822	0.0978814
$320$	7.23158	0.404258
$321$	0	0
$322$	4.59238	0.255923
$323$	10.8104	0.601508
$324$	0	0
$325$	−2.79978	−0.155304
$326$	−0.523523	−0.0289952
$327$	0	0
$328$	−0.541343	−0.0298906
$329$	18.8294	1.03810
$330$	0	0
$331$	1.73477	0.0953518	0.0476759	−	0.998863i	$-0.484819\pi$
$331$	1.73477	0.0953518	0.0476759	+	0.998863i	$0.484819\pi$
$332$	−22.3562	−1.22696
$333$	0	0
$334$	−4.23941	−0.231970
$335$	11.4482	0.625482
$336$	0	0
$337$	−29.9565	−1.63183	−0.815917	−	0.578169i	$-0.803768\pi$
$337$	−29.9565	−1.63183	−0.815917	+	0.578169i	$0.803768\pi$
$338$	0.929806	0.0505748
$339$	0	0
$340$	8.73487	0.473715
$341$	−33.2844	−1.80245
$342$	0	0
$343$	24.2483	1.30928
$344$	1.57182	0.0847470
$345$	0	0
$346$	−3.36708	−0.181015
$347$	14.9978	0.805123	0.402561	−	0.915393i	$-0.368120\pi$
$347$	14.9978	0.805123	0.402561	+	0.915393i	$0.368120\pi$
$348$	0	0
$349$	−7.29913	−0.390714	−0.195357	−	0.980732i	$-0.562586\pi$
$349$	−7.29913	−0.390714	−0.195357	+	0.980732i	$0.562586\pi$
$350$	−0.795373	−0.0425145
$351$	0	0
$352$	9.03571	0.481605
$353$	12.6061	0.670957	0.335479	−	0.942048i	$-0.391102\pi$
$353$	12.6061	0.670957	0.335479	+	0.942048i	$0.391102\pi$
$354$	0	0
$355$	−10.7144	−0.568660
$356$	−31.9342	−1.69251
$357$	0	0
$358$	−0.219892	−0.0116216
$359$	−21.5358	−1.13661	−0.568307	−	0.822817i	$-0.692401\pi$
$359$	−21.5358	−1.13661	−0.568307	+	0.822817i	$0.692401\pi$
$360$	0	0
$361$	−13.0704	−0.687918
$362$	4.28986	0.225470
$363$	0	0
$364$	24.3209	1.27476
$365$	−8.20680	−0.429564
$366$	0	0
$367$	20.1463	1.05163	0.525814	−	0.850600i	$-0.323761\pi$
$367$	20.1463	1.05163	0.525814	+	0.850600i	$0.323761\pi$
$368$	−21.9772	−1.14564
$369$	0	0
$370$	−0.180152	−0.00936567
$371$	0.607241	0.0315264
$372$	0	0
$373$	9.88395	0.511772	0.255886	−	0.966707i	$-0.417633\pi$
$373$	9.88395	0.511772	0.255886	+	0.966707i	$0.417633\pi$
$374$	3.41644	0.176660
$375$	0	0
$376$	3.04837	0.157208
$377$	1.14582	0.0590129
$378$	0	0
$379$	26.5438	1.36346	0.681732	−	0.731602i	$-0.261227\pi$
$379$	26.5438	1.36346	0.681732	+	0.731602i	$0.261227\pi$
$380$	4.79111	0.245779
$381$	0	0
$382$	2.69260	0.137765
$383$	−1.43834	−0.0734959	−0.0367479	−	0.999325i	$-0.511700\pi$
$383$	−1.43834	−0.0734959	−0.0367479	+	0.999325i	$0.511700\pi$
$384$	0	0
$385$	18.8596	0.961176
$386$	3.40623	0.173372
$387$	0	0
$388$	−36.0845	−1.83192
$389$	−28.7600	−1.45819	−0.729094	−	0.684414i	$-0.760058\pi$
$389$	−28.7600	−1.45819	−0.729094	+	0.684414i	$0.760058\pi$
$390$	0	0
$391$	−25.6330	−1.29631
$392$	8.92899	0.450982
$393$	0	0
$394$	0.671250	0.0338171
$395$	−7.11193	−0.357840
$396$	0	0
$397$	33.8486	1.69881	0.849406	−	0.527740i	$-0.176960\pi$
$397$	33.8486	1.69881	0.849406	+	0.527740i	$0.176960\pi$
$398$	3.37312	0.169079
$399$	0	0
$400$	3.80632	0.190316
$401$	20.7788	1.03764	0.518822	−	0.854883i	$-0.326371\pi$
$401$	20.7788	1.03764	0.518822	+	0.854883i	$0.326371\pi$
$402$	0	0
$403$	−21.8153	−1.08670
$404$	−21.9026	−1.08970
$405$	0	0
$406$	0.325510	0.0161548
$407$	4.27171	0.211741
$408$	0	0
$409$	−3.46191	−0.171180	−0.0855902	−	0.996330i	$-0.527278\pi$
$409$	−3.46191	−0.171180	−0.0855902	+	0.996330i	$0.527278\pi$
$410$	−0.136443	−0.00673843
$411$	0	0
$412$	−13.3420	−0.657315
$413$	13.5593	0.667211
$414$	0	0
$415$	−11.3625	−0.557762
$416$	5.92222	0.290361
$417$	0	0
$418$	1.87393	0.0916569
$419$	18.7259	0.914818	0.457409	−	0.889256i	$-0.348777\pi$
$419$	18.7259	0.914818	0.457409	+	0.889256i	$0.348777\pi$
$420$	0	0
$421$	1.91518	0.0933404	0.0466702	−	0.998910i	$-0.485139\pi$
$421$	1.91518	0.0933404	0.0466702	+	0.998910i	$0.485139\pi$
$422$	2.41352	0.117488
$423$	0	0
$424$	0.0983086	0.00477429
$425$	4.43948	0.215346
$426$	0	0
$427$	−13.3734	−0.647186
$428$	1.05575	0.0510317
$429$	0	0
$430$	0.396170	0.0191050
$431$	−6.17915	−0.297639	−0.148820	−	0.988864i	$-0.547547\pi$
$431$	−6.17915	−0.297639	−0.148820	+	0.988864i	$0.547547\pi$
$432$	0	0
$433$	−11.7275	−0.563588	−0.281794	−	0.959475i	$-0.590929\pi$
$433$	−11.7275	−0.563588	−0.281794	+	0.959475i	$0.590929\pi$
$434$	−6.19739	−0.297484
$435$	0	0
$436$	−18.7599	−0.898434
$437$	−14.0598	−0.672570
$438$	0	0
$439$	−18.4724	−0.881640	−0.440820	−	0.897595i	$-0.645312\pi$
$439$	−18.4724	−0.881640	−0.440820	+	0.897595i	$0.645312\pi$
$440$	3.05326	0.145558
$441$	0	0
$442$	2.23922	0.106509
$443$	−3.81261	−0.181142	−0.0905712	−	0.995890i	$-0.528869\pi$
$443$	−3.81261	−0.181142	−0.0905712	+	0.995890i	$0.528869\pi$
$444$	0	0
$445$	−16.2305	−0.769399
$446$	3.14313	0.148832
$447$	0	0
$448$	−31.9275	−1.50843
$449$	4.81563	0.227264	0.113632	−	0.993523i	$-0.463752\pi$
$449$	4.81563	0.227264	0.113632	+	0.993523i	$0.463752\pi$
$450$	0	0
$451$	3.23529	0.152344
$452$	18.0390	0.848485
$453$	0	0
$454$	3.69291	0.173317
$455$	12.3610	0.579495
$456$	0	0
$457$	−27.2012	−1.27242	−0.636210	−	0.771516i	$-0.719499\pi$
$457$	−27.2012	−1.27242	−0.636210	+	0.771516i	$0.719499\pi$
$458$	2.53470	0.118439
$459$	0	0
$460$	−11.3603	−0.529679
$461$	40.8432	1.90226	0.951128	−	0.308796i	$-0.0999261\pi$
$461$	40.8432	1.90226	0.951128	+	0.308796i	$0.0999261\pi$
$462$	0	0
$463$	34.8233	1.61837	0.809187	−	0.587551i	$-0.199908\pi$
$463$	34.8233	1.61837	0.809187	+	0.587551i	$0.199908\pi$
$464$	−1.55776	−0.0723170
$465$	0	0
$466$	2.72236	0.126111
$467$	9.38939	0.434489	0.217245	−	0.976117i	$-0.430293\pi$
$467$	9.38939	0.434489	0.217245	+	0.976117i	$0.430293\pi$
$468$	0	0
$469$	−50.5438	−2.33390
$470$	0.768326	0.0354402
$471$	0	0
$472$	2.19517	0.101041
$473$	−9.39386	−0.431930
$474$	0	0
$475$	2.43507	0.111729
$476$	−38.5645	−1.76760
$477$	0	0
$478$	0.0134688	0.000616050 0
$479$	−21.0154	−0.960217	−0.480109	−	0.877209i	$-0.659403\pi$
$479$	−21.0154	−0.960217	−0.480109	+	0.877209i	$0.659403\pi$
$480$	0	0
$481$	2.79978	0.127659
$482$	−3.69084	−0.168113
$483$	0	0
$484$	−14.2599	−0.648176
$485$	−18.3399	−0.832771
$486$	0	0
$487$	−35.7633	−1.62059	−0.810294	−	0.586023i	$-0.800693\pi$
$487$	−35.7633	−1.62059	−0.810294	+	0.586023i	$0.800693\pi$
$488$	−2.16508	−0.0980085
$489$	0	0
$490$	2.25051	0.101668
$491$	−15.0742	−0.680288	−0.340144	−	0.940373i	$-0.610476\pi$
$491$	−15.0742	−0.680288	−0.340144	+	0.940373i	$0.610476\pi$
$492$	0	0
$493$	−1.81688	−0.0818280
$494$	1.22822	0.0552601
$495$	0	0
$496$	29.6581	1.33169
$497$	47.3040	2.12187
$498$	0	0
$499$	32.9424	1.47470	0.737352	−	0.675509i	$-0.236076\pi$
$499$	32.9424	1.47470	0.737352	+	0.675509i	$0.236076\pi$
$500$	1.96755	0.0879913
$501$	0	0
$502$	−1.23468	−0.0551066
$503$	8.61983	0.384339	0.192170	−	0.981362i	$-0.438448\pi$
$503$	8.61983	0.384339	0.192170	+	0.981362i	$0.438448\pi$
$504$	0	0
$505$	−11.1319	−0.495365
$506$	−4.44333	−0.197530
$507$	0	0
$508$	−18.3497	−0.814138
$509$	−37.3100	−1.65374	−0.826869	−	0.562395i	$-0.809880\pi$
$509$	−37.3100	−1.65374	−0.826869	+	0.562395i	$0.809880\pi$
$510$	0	0
$511$	36.2331	1.60286
$512$	−13.4925	−0.596291
$513$	0	0
$514$	−1.88712	−0.0832374
$515$	−6.78105	−0.298809
$516$	0	0
$517$	−18.2183	−0.801240
$518$	0.795373	0.0349467
$519$	0	0
$520$	2.00118	0.0877575
$521$	18.6417	0.816710	0.408355	−	0.912823i	$-0.366103\pi$
$521$	18.6417	0.816710	0.408355	+	0.912823i	$0.366103\pi$
$522$	0	0
$523$	6.56533	0.287082	0.143541	−	0.989644i	$-0.454151\pi$
$523$	6.56533	0.287082	0.143541	+	0.989644i	$0.454151\pi$
$524$	30.6949	1.34091
$525$	0	0
$526$	1.08053	0.0471133
$527$	34.5915	1.50683
$528$	0	0
$529$	10.3376	0.449459
$530$	0.0247782	0.00107630
$531$	0	0
$532$	−21.1527	−0.917088
$533$	2.12048	0.0918483
$534$	0	0
$535$	0.536583	0.0231985
$536$	−8.18274	−0.353441
$537$	0	0
$538$	−0.419361	−0.0180799
$539$	−53.3633	−2.29852
$540$	0	0
$541$	5.25295	0.225842	0.112921	−	0.993604i	$-0.463979\pi$
$541$	5.25295	0.225842	0.112921	+	0.993604i	$0.463979\pi$
$542$	1.10448	0.0474414
$543$	0	0
$544$	−9.39057	−0.402617
$545$	−9.53465	−0.408420
$546$	0	0
$547$	37.4845	1.60272	0.801361	−	0.598181i	$-0.204110\pi$
$547$	37.4845	1.60272	0.801361	+	0.598181i	$0.204110\pi$
$548$	−2.00662	−0.0857187
$549$	0	0
$550$	0.769559	0.0328141
$551$	−0.996563	−0.0424550
$552$	0	0
$553$	31.3992	1.33523
$554$	1.49700	0.0636016
$555$	0	0
$556$	−13.9482	−0.591534
$557$	−12.2789	−0.520275	−0.260138	−	0.965572i	$-0.583768\pi$
$557$	−12.2789	−0.520275	−0.260138	+	0.965572i	$0.583768\pi$
$558$	0	0
$559$	−6.15695	−0.260411
$560$	−16.8049	−0.710138
$561$	0	0
$562$	−0.903127	−0.0380961
$563$	−20.4994	−0.863948	−0.431974	−	0.901886i	$-0.642183\pi$
$563$	−20.4994	−0.863948	−0.431974	+	0.901886i	$0.642183\pi$
$564$	0	0
$565$	9.16830	0.385713
$566$	−2.25744	−0.0948874
$567$	0	0
$568$	7.65823	0.321332
$569$	−1.60123	−0.0671269	−0.0335634	−	0.999437i	$-0.510686\pi$
$569$	−1.60123	−0.0671269	−0.0335634	+	0.999437i	$0.510686\pi$
$570$	0	0
$571$	−41.8238	−1.75027	−0.875136	−	0.483877i	$-0.839228\pi$
$571$	−41.8238	−1.75027	−0.875136	+	0.483877i	$0.839228\pi$
$572$	−23.5316	−0.983905
$573$	0	0
$574$	0.602395	0.0251435
$575$	−5.77387	−0.240787
$576$	0	0
$577$	−14.9405	−0.621981	−0.310991	−	0.950413i	$-0.600661\pi$
$577$	−14.9405	−0.621981	−0.310991	+	0.950413i	$0.600661\pi$
$578$	−0.488025	−0.0202992
$579$	0	0
$580$	−0.805227	−0.0334352
$581$	50.1654	2.08121
$582$	0	0
$583$	−0.587533	−0.0243331
$584$	5.86591	0.242733
$585$	0	0
$586$	−1.10457	−0.0456293
$587$	32.8499	1.35586	0.677930	−	0.735126i	$-0.262877\pi$
$587$	32.8499	1.35586	0.677930	+	0.735126i	$0.262877\pi$
$588$	0	0
$589$	18.9736	0.781792
$590$	0.553282	0.0227783
$591$	0	0
$592$	−3.80632	−0.156439
$593$	−39.7259	−1.63135	−0.815674	−	0.578512i	$-0.803634\pi$
$593$	−39.7259	−1.63135	−0.815674	+	0.578512i	$0.803634\pi$
$594$	0	0
$595$	−19.6003	−0.803534
$596$	−30.6094	−1.25381
$597$	0	0
$598$	−2.91227	−0.119091
$599$	10.4669	0.427668	0.213834	−	0.976870i	$-0.431405\pi$
$599$	10.4669	0.427668	0.213834	+	0.976870i	$0.431405\pi$
$600$	0	0
$601$	−8.56503	−0.349375	−0.174687	−	0.984624i	$-0.555891\pi$
$601$	−8.56503	−0.349375	−0.174687	+	0.984624i	$0.555891\pi$
$602$	−1.74909	−0.0712876
$603$	0	0
$604$	7.88753	0.320939
$605$	−7.24754	−0.294655
$606$	0	0
$607$	8.90888	0.361600	0.180800	−	0.983520i	$-0.442131\pi$
$607$	8.90888	0.361600	0.180800	+	0.983520i	$0.442131\pi$
$608$	−5.15076	−0.208891
$609$	0	0
$610$	−0.545697	−0.0220946
$611$	−11.9407	−0.483069
$612$	0	0
$613$	−10.5868	−0.427598	−0.213799	−	0.976878i	$-0.568584\pi$
$613$	−10.5868	−0.427598	−0.213799	+	0.976878i	$0.568584\pi$
$614$	−2.27760	−0.0919163
$615$	0	0
$616$	−13.4802	−0.543131
$617$	6.83972	0.275357	0.137678	−	0.990477i	$-0.456036\pi$
$617$	6.83972	0.275357	0.137678	+	0.990477i	$0.456036\pi$
$618$	0	0
$619$	−15.5577	−0.625318	−0.312659	−	0.949865i	$-0.601220\pi$
$619$	−15.5577	−0.625318	−0.312659	+	0.949865i	$0.601220\pi$
$620$	15.3307	0.615697
$621$	0	0
$622$	1.45047	0.0581584
$623$	71.6576	2.87090
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.539645	0.0215685
$627$	0	0
$628$	−6.03221	−0.240711
$629$	−4.43948	−0.177014
$630$	0	0
$631$	12.0975	0.481596	0.240798	−	0.970575i	$-0.422591\pi$
$631$	12.0975	0.481596	0.240798	+	0.970575i	$0.422591\pi$
$632$	5.08334	0.202204
$633$	0	0
$634$	0.404220	0.0160536
$635$	−9.32621	−0.370099
$636$	0	0
$637$	−34.9756	−1.38578
$638$	−0.314946	−0.0124688
$639$	0	0
$640$	−5.53327	−0.218722
$641$	39.4179	1.55691	0.778457	−	0.627698i	$-0.216003\pi$
$641$	39.4179	1.55691	0.778457	+	0.627698i	$0.216003\pi$
$642$	0	0
$643$	−2.80810	−0.110741	−0.0553704	−	0.998466i	$-0.517634\pi$
$643$	−2.80810	−0.110741	−0.0553704	+	0.998466i	$0.517634\pi$
$644$	50.1560	1.97642
$645$	0	0
$646$	−1.94752	−0.0766243
$647$	47.0232	1.84867	0.924337	−	0.381577i	$-0.124619\pi$
$647$	47.0232	1.84867	0.924337	+	0.381577i	$0.124619\pi$
$648$	0	0
$649$	−13.1193	−0.514976
$650$	0.504387	0.0197837
$651$	0	0
$652$	−5.71769	−0.223922
$653$	−21.7426	−0.850852	−0.425426	−	0.904993i	$-0.639876\pi$
$653$	−21.7426	−0.850852	−0.425426	+	0.904993i	$0.639876\pi$
$654$	0	0
$655$	15.6006	0.609566
$656$	−2.88281	−0.112555
$657$	0	0
$658$	−3.39216	−0.132240
$659$	26.3436	1.02620	0.513100	−	0.858329i	$-0.328497\pi$
$659$	26.3436	1.02620	0.513100	+	0.858329i	$0.328497\pi$
$660$	0	0
$661$	20.5259	0.798364	0.399182	−	0.916872i	$-0.369294\pi$
$661$	20.5259	0.798364	0.399182	+	0.916872i	$0.369294\pi$
$662$	−0.312523	−0.0121466
$663$	0	0
$664$	8.12147	0.315174
$665$	−10.7508	−0.416899
$666$	0	0
$667$	2.36298	0.0914950
$668$	−46.3010	−1.79144
$669$	0	0
$670$	−2.06242	−0.0796782
$671$	12.9394	0.499520
$672$	0	0
$673$	−5.66357	−0.218315	−0.109157	−	0.994024i	$-0.534815\pi$
$673$	−5.66357	−0.218315	−0.109157	+	0.994024i	$0.534815\pi$
$674$	5.39673	0.207874
$675$	0	0
$676$	10.1549	0.390574
$677$	−6.93823	−0.266658	−0.133329	−	0.991072i	$-0.542567\pi$
$677$	−6.93823	−0.266658	−0.133329	+	0.991072i	$0.542567\pi$
$678$	0	0
$679$	80.9706	3.10737
$680$	−3.17317	−0.121686
$681$	0	0
$682$	5.99625	0.229608
$683$	−19.4849	−0.745570	−0.372785	−	0.927918i	$-0.621597\pi$
$683$	−19.4849	−0.745570	−0.372785	+	0.927918i	$0.621597\pi$
$684$	0	0
$685$	−1.01986	−0.0389669
$686$	−4.36838	−0.166786
$687$	0	0
$688$	8.37043	0.319119
$689$	−0.385083	−0.0146705
$690$	0	0
$691$	−19.9927	−0.760557	−0.380278	−	0.924872i	$-0.624172\pi$
$691$	−19.9927	−0.760557	−0.380278	+	0.924872i	$0.624172\pi$
$692$	−36.7738	−1.39793
$693$	0	0
$694$	−2.70188	−0.102562
$695$	−7.08913	−0.268906
$696$	0	0
$697$	−3.36235	−0.127358
$698$	1.31496	0.0497718
$699$	0	0
$700$	−8.68672	−0.328327
$701$	25.5726	0.965863	0.482932	−	0.875658i	$-0.339572\pi$
$701$	25.5726	0.965863	0.482932	+	0.875658i	$0.339572\pi$
$702$	0	0
$703$	−2.43507	−0.0918403
$704$	30.8913	1.16426
$705$	0	0
$706$	−2.27103	−0.0854712
$707$	49.1476	1.84838
$708$	0	0
$709$	−36.7156	−1.37888	−0.689441	−	0.724342i	$-0.742144\pi$
$709$	−36.7156	−1.37888	−0.689441	+	0.724342i	$0.742144\pi$
$710$	1.93022	0.0724398
$711$	0	0
$712$	11.6009	0.434764
$713$	−44.9888	−1.68485
$714$	0	0
$715$	−11.9599	−0.447274
$716$	−2.40156	−0.0897505
$717$	0	0
$718$	3.87971	0.144790
$719$	−4.11485	−0.153458	−0.0767290	−	0.997052i	$-0.524448\pi$
$719$	−4.11485	−0.153458	−0.0767290	+	0.997052i	$0.524448\pi$
$720$	0	0
$721$	29.9384	1.11496
$722$	2.35467	0.0876317
$723$	0	0
$724$	46.8520	1.74124
$725$	−0.409254	−0.0151993
$726$	0	0
$727$	27.2507	1.01067	0.505336	−	0.862923i	$-0.331369\pi$
$727$	27.2507	1.01067	0.505336	+	0.862923i	$0.331369\pi$
$728$	−8.83521	−0.327455
$729$	0	0
$730$	1.47847	0.0547208
$731$	9.76278	0.361090
$732$	0	0
$733$	27.7634	1.02546	0.512732	−	0.858549i	$-0.328634\pi$
$733$	27.7634	1.02546	0.512732	+	0.858549i	$0.328634\pi$
$734$	−3.62940	−0.133964
$735$	0	0
$736$	12.2131	0.450182
$737$	48.9034	1.80138
$738$	0	0
$739$	−41.6263	−1.53125	−0.765624	−	0.643288i	$-0.777570\pi$
$739$	−41.6263	−1.53125	−0.765624	+	0.643288i	$0.777570\pi$
$740$	−1.96755	−0.0723284
$741$	0	0
$742$	−0.109396	−0.00401605
$743$	48.9645	1.79633	0.898166	−	0.439657i	$-0.144900\pi$
$743$	48.9645	1.79633	0.898166	+	0.439657i	$0.144900\pi$
$744$	0	0
$745$	−15.5572	−0.569970
$746$	−1.78062	−0.0651930
$747$	0	0
$748$	37.3129	1.36429
$749$	−2.36902	−0.0865619
$750$	0	0
$751$	7.34688	0.268091	0.134046	−	0.990975i	$-0.457203\pi$
$751$	7.34688	0.268091	0.134046	+	0.990975i	$0.457203\pi$
$752$	16.2335	0.591974
$753$	0	0
$754$	−0.206423	−0.00751747
$755$	4.00882	0.145896
$756$	0	0
$757$	−4.47852	−0.162775	−0.0813873	−	0.996683i	$-0.525935\pi$
$757$	−4.47852	−0.162775	−0.0813873	+	0.996683i	$0.525935\pi$
$758$	−4.78193	−0.173687
$759$	0	0
$760$	−1.74049	−0.0631344
$761$	−33.7349	−1.22289	−0.611445	−	0.791287i	$-0.709411\pi$
$761$	−33.7349	−1.22289	−0.611445	+	0.791287i	$0.709411\pi$
$762$	0	0
$763$	42.0955	1.52396
$764$	29.4074	1.06392
$765$	0	0
$766$	0.259121	0.00936241
$767$	−8.59867	−0.310480
$768$	0	0
$769$	25.1523	0.907014	0.453507	−	0.891253i	$-0.350173\pi$
$769$	25.1523	0.907014	0.453507	+	0.891253i	$0.350173\pi$
$770$	−3.39761	−0.122441
$771$	0	0
$772$	37.2013	1.33890
$773$	−13.1660	−0.473549	−0.236774	−	0.971565i	$-0.576090\pi$
$773$	−13.1660	−0.473549	−0.236774	+	0.971565i	$0.576090\pi$
$774$	0	0
$775$	7.79180	0.279890
$776$	13.1087	0.470573
$777$	0	0
$778$	5.18117	0.185754
$779$	−1.84426	−0.0660774
$780$	0	0
$781$	−45.7687	−1.63773
$782$	4.61784	0.165134
$783$	0	0
$784$	47.5495	1.69820
$785$	−3.06585	−0.109425
$786$	0	0
$787$	19.5552	0.697066	0.348533	−	0.937296i	$-0.386680\pi$
$787$	19.5552	0.697066	0.348533	+	0.937296i	$0.386680\pi$
$788$	7.33111	0.261160
$789$	0	0
$790$	1.28123	0.0455841
$791$	−40.4781	−1.43923
$792$	0	0
$793$	8.48079	0.301162
$794$	−6.09790	−0.216406
$795$	0	0
$796$	36.8398	1.30575
$797$	−51.4656	−1.82300	−0.911502	−	0.411295i	$-0.865077\pi$
$797$	−51.4656	−1.82300	−0.911502	+	0.411295i	$0.865077\pi$
$798$	0	0
$799$	18.9338	0.669829
$800$	−2.11524	−0.0747851
$801$	0	0
$802$	−3.74335	−0.132182
$803$	−35.0571	−1.23714
$804$	0	0
$805$	25.4916	0.898463
$806$	3.93008	0.138431
$807$	0	0
$808$	7.95669	0.279915
$809$	−42.6242	−1.49859	−0.749293	−	0.662238i	$-0.769607\pi$
$809$	−42.6242	−1.49859	−0.749293	+	0.662238i	$0.769607\pi$
$810$	0	0
$811$	−26.7451	−0.939147	−0.469574	−	0.882893i	$-0.655592\pi$
$811$	−26.7451	−0.939147	−0.469574	+	0.882893i	$0.655592\pi$
$812$	3.55508	0.124759
$813$	0	0
$814$	−0.769559	−0.0269730
$815$	−2.90600	−0.101793
$816$	0	0
$817$	5.35492	0.187345
$818$	0.623671	0.0218061
$819$	0	0
$820$	−1.49017	−0.0520389
$821$	−43.8050	−1.52880	−0.764402	−	0.644740i	$-0.776966\pi$
$821$	−43.8050	−1.52880	−0.764402	+	0.644740i	$0.776966\pi$
$822$	0	0
$823$	−5.72621	−0.199603	−0.0998016	−	0.995007i	$-0.531821\pi$
$823$	−5.72621	−0.199603	−0.0998016	+	0.995007i	$0.531821\pi$
$824$	4.84684	0.168848
$825$	0	0
$826$	−2.44274	−0.0849939
$827$	35.4306	1.23204	0.616021	−	0.787730i	$-0.288744\pi$
$827$	35.4306	1.23204	0.616021	+	0.787730i	$0.288744\pi$
$828$	0	0
$829$	−20.7040	−0.719081	−0.359541	−	0.933129i	$-0.617067\pi$
$829$	−20.7040	−0.719081	−0.359541	+	0.933129i	$0.617067\pi$
$830$	2.04698	0.0710516
$831$	0	0
$832$	20.2469	0.701933
$833$	55.4590	1.92154
$834$	0	0
$835$	−23.5324	−0.814372
$836$	20.4662	0.707840
$837$	0	0
$838$	−3.37351	−0.116536
$839$	−10.9086	−0.376607	−0.188303	−	0.982111i	$-0.560299\pi$
$839$	−10.9086	−0.376607	−0.188303	+	0.982111i	$0.560299\pi$
$840$	0	0
$841$	−28.8325	−0.994225
$842$	−0.345025	−0.0118903
$843$	0	0
$844$	26.3594	0.907329
$845$	5.16122	0.177551
$846$	0	0
$847$	31.9979	1.09946
$848$	0.523523	0.0179778
$849$	0	0
$850$	−0.799782	−0.0274323
$851$	5.77387	0.197926
$852$	0	0
$853$	−38.8833	−1.33134	−0.665670	−	0.746247i	$-0.731854\pi$
$853$	−38.8833	−1.33134	−0.665670	+	0.746247i	$0.731854\pi$
$854$	2.40925	0.0824430
$855$	0	0
$856$	−0.383529	−0.0131088
$857$	−40.6975	−1.39020	−0.695100	−	0.718913i	$-0.744640\pi$
$857$	−40.6975	−1.39020	−0.695100	+	0.718913i	$0.744640\pi$
$858$	0	0
$859$	28.1532	0.960574	0.480287	−	0.877111i	$-0.340533\pi$
$859$	28.1532	0.960574	0.480287	+	0.877111i	$0.340533\pi$
$860$	4.32680	0.147543
$861$	0	0
$862$	1.11319	0.0379154
$863$	−18.5236	−0.630552	−0.315276	−	0.949000i	$-0.602097\pi$
$863$	−18.5236	−0.630552	−0.315276	+	0.949000i	$0.602097\pi$
$864$	0	0
$865$	−18.6902	−0.635486
$866$	2.11274	0.0717937
$867$	0	0
$868$	−67.6852	−2.29738
$869$	−30.3801	−1.03058
$870$	0	0
$871$	32.0525	1.08606
$872$	6.81501	0.230785
$873$	0	0
$874$	2.53290	0.0856766
$875$	−4.41500	−0.149254
$876$	0	0
$877$	14.0422	0.474172	0.237086	−	0.971489i	$-0.423808\pi$
$877$	14.0422	0.474172	0.237086	+	0.971489i	$0.423808\pi$
$878$	3.32785	0.112309
$879$	0	0
$880$	16.2595	0.548109
$881$	−7.68268	−0.258836	−0.129418	−	0.991590i	$-0.541311\pi$
$881$	−7.68268	−0.258836	−0.129418	+	0.991590i	$0.541311\pi$
$882$	0	0
$883$	35.0479	1.17945	0.589727	−	0.807602i	$-0.299235\pi$
$883$	35.0479	1.17945	0.589727	+	0.807602i	$0.299235\pi$
$884$	24.4557	0.822535
$885$	0	0
$886$	0.686850	0.0230752
$887$	9.30606	0.312467	0.156233	−	0.987720i	$-0.450065\pi$
$887$	9.30606	0.312467	0.156233	+	0.987720i	$0.450065\pi$
$888$	0	0
$889$	41.1752	1.38097
$890$	2.92396	0.0980113
$891$	0	0
$892$	34.3280	1.14938
$893$	10.3852	0.347529
$894$	0	0
$895$	−1.22059	−0.0407997
$896$	24.4294	0.816129
$897$	0	0
$898$	−0.867547	−0.0289504
$899$	−3.18883	−0.106353
$900$	0	0
$901$	0.610607	0.0203423
$902$	−0.582844	−0.0194066
$903$	0	0
$904$	−6.55315	−0.217955
$905$	23.8124	0.791552
$906$	0	0
$907$	−4.56415	−0.151550	−0.0757751	−	0.997125i	$-0.524143\pi$
$907$	−4.56415	−0.151550	−0.0757751	+	0.997125i	$0.524143\pi$
$908$	40.3324	1.33848
$909$	0	0
$910$	−2.22687	−0.0738200
$911$	0.294602	0.00976060	0.00488030	−	0.999988i	$-0.498447\pi$
$911$	0.294602	0.00976060	0.00488030	+	0.999988i	$0.498447\pi$
$912$	0	0
$913$	−48.5373	−1.60635
$914$	4.90036	0.162090
$915$	0	0
$916$	27.6829	0.914668
$917$	−68.8767	−2.27451
$918$	0	0
$919$	33.9110	1.11862	0.559311	−	0.828958i	$-0.311066\pi$
$919$	33.9110	1.11862	0.559311	+	0.828958i	$0.311066\pi$
$920$	4.12694	0.136061
$921$	0	0
$922$	−7.35799	−0.242323
$923$	−29.9979	−0.987393
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−6.27349	−0.206160
$927$	0	0
$928$	0.865673	0.0284171
$929$	−22.4955	−0.738055	−0.369028	−	0.929418i	$-0.620309\pi$
$929$	−22.4955	−0.738055	−0.369028	+	0.929418i	$0.620309\pi$
$930$	0	0
$931$	30.4195	0.996958
$932$	29.7324	0.973918
$933$	0	0
$934$	−1.69152	−0.0553482
$935$	18.9642	0.620195
$936$	0	0
$937$	25.4994	0.833028	0.416514	−	0.909129i	$-0.363252\pi$
$937$	25.4994	0.833028	0.416514	+	0.909129i	$0.363252\pi$
$938$	9.10558	0.297308
$939$	0	0
$940$	8.39132	0.273695
$941$	−52.3734	−1.70732	−0.853662	−	0.520827i	$-0.825624\pi$
$941$	−52.3734	−1.70732	−0.853662	+	0.520827i	$0.825624\pi$
$942$	0	0
$943$	4.37298	0.142404
$944$	11.6900	0.380476
$945$	0	0
$946$	1.69232	0.0550222
$947$	50.7001	1.64753	0.823766	−	0.566931i	$-0.191869\pi$
$947$	50.7001	1.64753	0.823766	+	0.566931i	$0.191869\pi$
$948$	0	0
$949$	−22.9773	−0.745874
$950$	−0.438683	−0.0142328
$951$	0	0
$952$	14.0096	0.454052
$953$	−18.5912	−0.602228	−0.301114	−	0.953588i	$-0.597359\pi$
$953$	−18.5912	−0.602228	−0.301114	+	0.953588i	$0.597359\pi$
$954$	0	0
$955$	14.9462	0.483649
$956$	0.147101	0.00475758
$957$	0	0
$958$	3.78597	0.122319
$959$	4.50269	0.145399
$960$	0	0
$961$	29.7122	0.958457
$962$	−0.504387	−0.0162621
$963$	0	0
$964$	−40.3098	−1.29829
$965$	18.9075	0.608653
$966$	0	0
$967$	−20.5566	−0.661056	−0.330528	−	0.943796i	$-0.607227\pi$
$967$	−20.5566	−0.661056	−0.330528	+	0.943796i	$0.607227\pi$
$968$	5.18027	0.166500
$969$	0	0
$970$	3.30397	0.106084
$971$	−40.7627	−1.30814	−0.654069	−	0.756435i	$-0.726939\pi$
$971$	−40.7627	−1.30814	−0.654069	+	0.756435i	$0.726939\pi$
$972$	0	0
$973$	31.2985	1.00338
$974$	6.44284	0.206442
$975$	0	0
$976$	−11.5297	−0.369056
$977$	−5.60203	−0.179225	−0.0896124	−	0.995977i	$-0.528563\pi$
$977$	−5.60203	−0.179225	−0.0896124	+	0.995977i	$0.528563\pi$
$978$	0	0
$979$	−69.3320	−2.21586
$980$	24.5791	0.785149
$981$	0	0
$982$	2.71565	0.0866598
$983$	−16.0493	−0.511894	−0.255947	−	0.966691i	$-0.582387\pi$
$983$	−16.0493	−0.511894	−0.255947	+	0.966691i	$0.582387\pi$
$984$	0	0
$985$	3.72602	0.118721
$986$	0.327314	0.0104238
$987$	0	0
$988$	13.4141	0.426758
$989$	−12.6972	−0.403748
$990$	0	0
$991$	−36.8868	−1.17175	−0.585874	−	0.810402i	$-0.699249\pi$
$991$	−36.8868	−1.17175	−0.585874	+	0.810402i	$0.699249\pi$
$992$	−16.4816	−0.523290
$993$	0	0
$994$	−8.52192	−0.270299
$995$	18.7237	0.593582
$996$	0	0
$997$	−0.491984	−0.0155813	−0.00779064	−	0.999970i	$-0.502480\pi$
$997$	−0.491984	−0.0155813	−0.00779064	+	0.999970i	$0.502480\pi$
$998$	−5.93465	−0.187858
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1665.2.a.q.1.3		5
3.2	odd	2		185.2.a.d.1.3	✓	5
5.4	even	2		8325.2.a.cc.1.3		5
12.11	even	2		2960.2.a.ba.1.5		5
15.2	even	4		925.2.b.g.149.6		10
15.8	even	4		925.2.b.g.149.5		10
15.14	odd	2		925.2.a.h.1.3		5
21.20	even	2		9065.2.a.j.1.3		5
111.110	odd	2		6845.2.a.g.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.d.1.3	✓	5	3.2	odd	2
925.2.a.h.1.3		5	15.14	odd	2
925.2.b.g.149.5		10	15.8	even	4
925.2.b.g.149.6		10	15.2	even	4
1665.2.a.q.1.3		5	1.1	even	1	trivial
2960.2.a.ba.1.5		5	12.11	even	2
6845.2.a.g.1.3		5	111.110	odd	2
8325.2.a.cc.1.3		5	5.4	even	2
9065.2.a.j.1.3		5	21.20	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1665.a (trivial)

\(f(q)\)	\(=\)	\(q-0.180152 q^{2} -1.96755 q^{4} -1.00000 q^{5} +4.41500 q^{7} +0.714762 q^{8} +0.180152 q^{10} -4.27171 q^{11} -2.79978 q^{13} -0.795373 q^{14} +3.80632 q^{16} +4.43948 q^{17} +2.43507 q^{19} +1.96755 q^{20} +0.769559 q^{22} -5.77387 q^{23} +1.00000 q^{25} +0.504387 q^{26} -8.68672 q^{28} -0.409254 q^{29} +7.79180 q^{31} -2.11524 q^{32} -0.799782 q^{34} -4.41500 q^{35} -1.00000 q^{37} -0.438683 q^{38} -0.714762 q^{40} -0.757374 q^{41} +2.19908 q^{43} +8.40479 q^{44} +1.04018 q^{46} +4.26487 q^{47} +12.4922 q^{49} -0.180152 q^{50} +5.50870 q^{52} +0.137540 q^{53} +4.27171 q^{55} +3.15568 q^{56} +0.0737281 q^{58} +3.07119 q^{59} -3.02909 q^{61} -1.40371 q^{62} -7.23158 q^{64} +2.79978 q^{65} -11.4482 q^{67} -8.73487 q^{68} +0.795373 q^{70} +10.7144 q^{71} +8.20680 q^{73} +0.180152 q^{74} -4.79111 q^{76} -18.8596 q^{77} +7.11193 q^{79} -3.80632 q^{80} +0.136443 q^{82} +11.3625 q^{83} -4.43948 q^{85} -0.396170 q^{86} -3.05326 q^{88} +16.2305 q^{89} -12.3610 q^{91} +11.3603 q^{92} -0.768326 q^{94} -2.43507 q^{95} +18.3399 q^{97} -2.25051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{4} - 5 q^{5} + 7 q^{7} + 6 q^{8} - 7 q^{11} + 2 q^{13} - 4 q^{14} + 8 q^{16} + 8 q^{17} + 14 q^{19} - 6 q^{20} + 2 q^{22} - 2 q^{23} + 5 q^{25} + 20 q^{26} - 14 q^{28} - 2 q^{29} + 8 q^{31} + 22 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) 5 * q + 6 * q^4 - 5 * q^5 + 7 * q^7 + 6 * q^8 - 7 * q^11 + 2 * q^13 - 4 * q^14 + 8 * q^16 + 8 * q^17 + 14 * q^19 - 6 * q^20 + 2 * q^22 - 2 * q^23 + 5 * q^25 + 20 * q^26 - 14 * q^28 - 2 * q^29 + 8 * q^31 + 22 * q^32 + 12 * q^34 - 7 * q^35 - 5 * q^37 + 32 * q^38 - 6 * q^40 + 9 * q^41 + 14 * q^43 - 18 * q^44 - 28 * q^46 + 5 * q^47 + 2 * q^49 + 20 * q^52 + 15 * q^53 + 7 * q^55 + 14 * q^56 - 12 * q^59 + 12 * q^61 - 10 * q^62 + 12 * q^64 - 2 * q^65 - 2 * q^67 + 4 * q^68 + 4 * q^70 - 13 * q^71 - 5 * q^73 + 18 * q^76 + 7 * q^77 + 36 * q^79 - 8 * q^80 - 2 * q^82 + 9 * q^83 - 8 * q^85 + 8 * q^86 + 8 * q^88 + 16 * q^89 - 4 * q^91 - 12 * q^92 + 20 * q^94 - 14 * q^95 - 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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