LMFDB - Embedded newform 4332.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4332,2,Mod(1,4332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4332 = 2^{2} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4332.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:	$34.5911941556$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	4332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.34730 q^{5} +2.41147 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.34730 q^{5} +2.41147 q^{7} +1.00000 q^{9} -5.94356 q^{11} +3.69459 q^{13} -1.34730 q^{15} -0.162504 q^{17} -2.41147 q^{21} -6.57398 q^{23} -3.18479 q^{25} -1.00000 q^{27} -2.38919 q^{29} +2.59627 q^{31} +5.94356 q^{33} +3.24897 q^{35} +6.94356 q^{37} -3.69459 q^{39} +0.361844 q^{41} -11.4534 q^{43} +1.34730 q^{45} -6.87939 q^{47} -1.18479 q^{49} +0.162504 q^{51} -4.02229 q^{53} -8.00774 q^{55} -12.0496 q^{59} +6.18479 q^{61} +2.41147 q^{63} +4.97771 q^{65} +5.75877 q^{67} +6.57398 q^{69} -12.1925 q^{71} -9.17024 q^{73} +3.18479 q^{75} -14.3327 q^{77} +9.69459 q^{79} +1.00000 q^{81} +2.02734 q^{83} -0.218941 q^{85} +2.38919 q^{87} -3.44831 q^{89} +8.90941 q^{91} -2.59627 q^{93} -12.0915 q^{97} -5.94356 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 9 q^{13} - 3 q^{15} - 3 q^{17} + 3 q^{21} - 12 q^{23} - 6 q^{25} - 3 q^{27} - 3 q^{29} - 6 q^{31} + 3 q^{33} - 3 q^{35} + 6 q^{37} - 9 q^{39} + 18 q^{41} - 21 q^{43} + 3 q^{45} - 15 q^{47} + 3 q^{51} - 6 q^{53} - 9 q^{59} + 15 q^{61} - 3 q^{63} + 21 q^{65} + 6 q^{67} + 12 q^{69} - 9 q^{71} - 6 q^{73} + 6 q^{75} - 24 q^{77} + 27 q^{79} + 3 q^{81} - 15 q^{83} - 18 q^{85} + 3 q^{87} - 12 q^{89} - 9 q^{91} + 6 q^{93} - 6 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 9 * q^13 - 3 * q^15 - 3 * q^17 + 3 * q^21 - 12 * q^23 - 6 * q^25 - 3 * q^27 - 3 * q^29 - 6 * q^31 + 3 * q^33 - 3 * q^35 + 6 * q^37 - 9 * q^39 + 18 * q^41 - 21 * q^43 + 3 * q^45 - 15 * q^47 + 3 * q^51 - 6 * q^53 - 9 * q^59 + 15 * q^61 - 3 * q^63 + 21 * q^65 + 6 * q^67 + 12 * q^69 - 9 * q^71 - 6 * q^73 + 6 * q^75 - 24 * q^77 + 27 * q^79 + 3 * q^81 - 15 * q^83 - 18 * q^85 + 3 * q^87 - 12 * q^89 - 9 * q^91 + 6 * q^93 - 6 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.34730	0.602529	0.301265	−	0.953541i	$-0.402591\pi$
$5$	1.34730	0.602529	0.301265	+	0.953541i	$0.402591\pi$
$6$	0	0
$7$	2.41147	0.911452	0.455726	−	0.890120i	$-0.349380\pi$
$7$	2.41147	0.911452	0.455726	+	0.890120i	$0.349380\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.94356	−1.79205	−0.896026	−	0.444002i	$-0.853558\pi$
$11$	−5.94356	−1.79205	−0.896026	+	0.444002i	$0.853558\pi$
$12$	0	0
$13$	3.69459	1.02470	0.512348	−	0.858778i	$-0.328776\pi$
$13$	3.69459	1.02470	0.512348	+	0.858778i	$0.328776\pi$
$14$	0	0
$15$	−1.34730	−0.347870
$16$	0	0
$17$	−0.162504	−0.0394130	−0.0197065	−	0.999806i	$-0.506273\pi$
$17$	−0.162504	−0.0394130	−0.0197065	+	0.999806i	$0.506273\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−2.41147	−0.526227
$22$	0	0
$23$	−6.57398	−1.37077	−0.685385	−	0.728181i	$-0.740366\pi$
$23$	−6.57398	−1.37077	−0.685385	+	0.728181i	$0.740366\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.38919	−0.443661	−0.221830	−	0.975085i	$-0.571203\pi$
$29$	−2.38919	−0.443661	−0.221830	+	0.975085i	$0.571203\pi$
$30$	0	0
$31$	2.59627	0.466303	0.233152	−	0.972440i	$-0.425096\pi$
$31$	2.59627	0.466303	0.233152	+	0.972440i	$0.425096\pi$
$32$	0	0
$33$	5.94356	1.03464
$34$	0	0
$35$	3.24897	0.549176
$36$	0	0
$37$	6.94356	1.14151	0.570757	−	0.821119i	$-0.306650\pi$
$37$	6.94356	1.14151	0.570757	+	0.821119i	$0.306650\pi$
$38$	0	0
$39$	−3.69459	−0.591608
$40$	0	0
$41$	0.361844	0.0565106	0.0282553	−	0.999601i	$-0.491005\pi$
$41$	0.361844	0.0565106	0.0282553	+	0.999601i	$0.491005\pi$
$42$	0	0
$43$	−11.4534	−1.74662	−0.873311	−	0.487164i	$-0.838032\pi$
$43$	−11.4534	−1.74662	−0.873311	+	0.487164i	$0.838032\pi$
$44$	0	0
$45$	1.34730	0.200843
$46$	0	0
$47$	−6.87939	−1.00346	−0.501731	−	0.865024i	$-0.667303\pi$
$47$	−6.87939	−1.00346	−0.501731	+	0.865024i	$0.667303\pi$
$48$	0	0
$49$	−1.18479	−0.169256
$50$	0	0
$51$	0.162504	0.0227551
$52$	0	0
$53$	−4.02229	−0.552504	−0.276252	−	0.961085i	$-0.589092\pi$
$53$	−4.02229	−0.552504	−0.276252	+	0.961085i	$0.589092\pi$
$54$	0	0
$55$	−8.00774	−1.07976
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0496	−1.56873	−0.784364	−	0.620301i	$-0.787011\pi$
$59$	−12.0496	−1.56873	−0.784364	+	0.620301i	$0.787011\pi$
$60$	0	0
$61$	6.18479	0.791882	0.395941	−	0.918276i	$-0.370419\pi$
$61$	6.18479	0.791882	0.395941	+	0.918276i	$0.370419\pi$
$62$	0	0
$63$	2.41147	0.303817
$64$	0	0
$65$	4.97771	0.617409
$66$	0	0
$67$	5.75877	0.703546	0.351773	−	0.936085i	$-0.385579\pi$
$67$	5.75877	0.703546	0.351773	+	0.936085i	$0.385579\pi$
$68$	0	0
$69$	6.57398	0.791414
$70$	0	0
$71$	−12.1925	−1.44699	−0.723494	−	0.690331i	$-0.757465\pi$
$71$	−12.1925	−1.44699	−0.723494	+	0.690331i	$0.757465\pi$
$72$	0	0
$73$	−9.17024	−1.07330	−0.536648	−	0.843806i	$-0.680310\pi$
$73$	−9.17024	−1.07330	−0.536648	+	0.843806i	$0.680310\pi$
$74$	0	0
$75$	3.18479	0.367748
$76$	0	0
$77$	−14.3327	−1.63337
$78$	0	0
$79$	9.69459	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$79$	9.69459	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.02734	0.222530	0.111265	−	0.993791i	$-0.464510\pi$
$83$	2.02734	0.222530	0.111265	+	0.993791i	$0.464510\pi$
$84$	0	0
$85$	−0.218941	−0.0237475
$86$	0	0
$87$	2.38919	0.256148
$88$	0	0
$89$	−3.44831	−0.365520	−0.182760	−	0.983158i	$-0.558503\pi$
$89$	−3.44831	−0.365520	−0.182760	+	0.983158i	$0.558503\pi$
$90$	0	0
$91$	8.90941	0.933960
$92$	0	0
$93$	−2.59627	−0.269220
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0915	−1.22771	−0.613854	−	0.789420i	$-0.710382\pi$
$97$	−12.0915	−1.22771	−0.613854	+	0.789420i	$0.710382\pi$
$98$	0	0
$99$	−5.94356	−0.597351
$100$	0	0
$101$	−3.93582	−0.391629	−0.195814	−	0.980641i	$-0.562735\pi$
$101$	−3.93582	−0.391629	−0.195814	+	0.980641i	$0.562735\pi$
$102$	0	0
$103$	12.3131	1.21325	0.606625	−	0.794988i	$-0.292523\pi$
$103$	12.3131	1.21325	0.606625	+	0.794988i	$0.292523\pi$
$104$	0	0
$105$	−3.24897	−0.317067
$106$	0	0
$107$	13.4338	1.29869	0.649345	−	0.760494i	$-0.275043\pi$
$107$	13.4338	1.29869	0.649345	+	0.760494i	$0.275043\pi$
$108$	0	0
$109$	−2.26857	−0.217290	−0.108645	−	0.994081i	$-0.534651\pi$
$109$	−2.26857	−0.217290	−0.108645	+	0.994081i	$0.534651\pi$
$110$	0	0
$111$	−6.94356	−0.659054
$112$	0	0
$113$	12.7101	1.19566	0.597832	−	0.801622i	$-0.296029\pi$
$113$	12.7101	1.19566	0.597832	+	0.801622i	$0.296029\pi$
$114$	0	0
$115$	−8.85710	−0.825929
$116$	0	0
$117$	3.69459	0.341565
$118$	0	0
$119$	−0.391874	−0.0359230
$120$	0	0
$121$	24.3259	2.21145
$122$	0	0
$123$	−0.361844	−0.0326264
$124$	0	0
$125$	−11.0273	−0.986315
$126$	0	0
$127$	−7.73143	−0.686053	−0.343027	−	0.939326i	$-0.611452\pi$
$127$	−7.73143	−0.686053	−0.343027	+	0.939326i	$0.611452\pi$
$128$	0	0
$129$	11.4534	1.00841
$130$	0	0
$131$	12.5672	1.09800	0.548999	−	0.835823i	$-0.315009\pi$
$131$	12.5672	1.09800	0.548999	+	0.835823i	$0.315009\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.34730	−0.115957
$136$	0	0
$137$	15.7101	1.34220	0.671101	−	0.741366i	$-0.265822\pi$
$137$	15.7101	1.34220	0.671101	+	0.741366i	$0.265822\pi$
$138$	0	0
$139$	3.23442	0.274340	0.137170	−	0.990548i	$-0.456199\pi$
$139$	3.23442	0.274340	0.137170	+	0.990548i	$0.456199\pi$
$140$	0	0
$141$	6.87939	0.579349
$142$	0	0
$143$	−21.9590	−1.83631
$144$	0	0
$145$	−3.21894	−0.267318
$146$	0	0
$147$	1.18479	0.0977200
$148$	0	0
$149$	4.46017	0.365391	0.182696	−	0.983170i	$-0.441518\pi$
$149$	4.46017	0.365391	0.182696	+	0.983170i	$0.441518\pi$
$150$	0	0
$151$	−8.94862	−0.728228	−0.364114	−	0.931354i	$-0.618628\pi$
$151$	−8.94862	−0.728228	−0.364114	+	0.931354i	$0.618628\pi$
$152$	0	0
$153$	−0.162504	−0.0131377
$154$	0	0
$155$	3.49794	0.280961
$156$	0	0
$157$	−11.5202	−0.919414	−0.459707	−	0.888071i	$-0.652046\pi$
$157$	−11.5202	−0.919414	−0.459707	+	0.888071i	$0.652046\pi$
$158$	0	0
$159$	4.02229	0.318988
$160$	0	0
$161$	−15.8530	−1.24939
$162$	0	0
$163$	−10.5243	−0.824331	−0.412165	−	0.911109i	$-0.635227\pi$
$163$	−10.5243	−0.824331	−0.412165	+	0.911109i	$0.635227\pi$
$164$	0	0
$165$	8.00774	0.623402
$166$	0	0
$167$	−10.7392	−0.831022	−0.415511	−	0.909588i	$-0.636397\pi$
$167$	−10.7392	−0.831022	−0.415511	+	0.909588i	$0.636397\pi$
$168$	0	0
$169$	0.650015	0.0500012
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.53983	0.573243	0.286621	−	0.958044i	$-0.407468\pi$
$173$	7.53983	0.573243	0.286621	+	0.958044i	$0.407468\pi$
$174$	0	0
$175$	−7.68004	−0.580557
$176$	0	0
$177$	12.0496	0.905706
$178$	0	0
$179$	−26.3628	−1.97045	−0.985223	−	0.171275i	$-0.945211\pi$
$179$	−26.3628	−1.97045	−0.985223	+	0.171275i	$0.945211\pi$
$180$	0	0
$181$	−21.0915	−1.56772	−0.783860	−	0.620937i	$-0.786752\pi$
$181$	−21.0915	−1.56772	−0.783860	+	0.620937i	$0.786752\pi$
$182$	0	0
$183$	−6.18479	−0.457193
$184$	0	0
$185$	9.35504	0.687796
$186$	0	0
$187$	0.965852	0.0706301
$188$	0	0
$189$	−2.41147	−0.175409
$190$	0	0
$191$	−0.255777	−0.0185074	−0.00925370	−	0.999957i	$-0.502946\pi$
$191$	−0.255777	−0.0185074	−0.00925370	+	0.999957i	$0.502946\pi$
$192$	0	0
$193$	−14.3482	−1.03281	−0.516404	−	0.856345i	$-0.672730\pi$
$193$	−14.3482	−1.03281	−0.516404	+	0.856345i	$0.672730\pi$
$194$	0	0
$195$	−4.97771	−0.356461
$196$	0	0
$197$	23.6810	1.68720	0.843600	−	0.536972i	$-0.180432\pi$
$197$	23.6810	1.68720	0.843600	+	0.536972i	$0.180432\pi$
$198$	0	0
$199$	−11.7956	−0.836168	−0.418084	−	0.908408i	$-0.637298\pi$
$199$	−11.7956	−0.836168	−0.418084	+	0.908408i	$0.637298\pi$
$200$	0	0
$201$	−5.75877	−0.406192
$202$	0	0
$203$	−5.76146	−0.404375
$204$	0	0
$205$	0.487511	0.0340493
$206$	0	0
$207$	−6.57398	−0.456923
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−27.8580	−1.91783	−0.958913	−	0.283700i	$-0.908438\pi$
$211$	−27.8580	−1.91783	−0.958913	+	0.283700i	$0.908438\pi$
$212$	0	0
$213$	12.1925	0.835419
$214$	0	0
$215$	−15.4311	−1.05239
$216$	0	0
$217$	6.26083	0.425013
$218$	0	0
$219$	9.17024	0.619668
$220$	0	0
$221$	−0.600385	−0.0403863
$222$	0	0
$223$	24.7374	1.65654	0.828270	−	0.560329i	$-0.189325\pi$
$223$	24.7374	1.65654	0.828270	+	0.560329i	$0.189325\pi$
$224$	0	0
$225$	−3.18479	−0.212320
$226$	0	0
$227$	22.0155	1.46122	0.730609	−	0.682796i	$-0.239236\pi$
$227$	22.0155	1.46122	0.730609	+	0.682796i	$0.239236\pi$
$228$	0	0
$229$	−9.29591	−0.614291	−0.307146	−	0.951663i	$-0.599374\pi$
$229$	−9.29591	−0.614291	−0.307146	+	0.951663i	$0.599374\pi$
$230$	0	0
$231$	14.3327	0.943026
$232$	0	0
$233$	−7.98814	−0.523320	−0.261660	−	0.965160i	$-0.584270\pi$
$233$	−7.98814	−0.523320	−0.261660	+	0.965160i	$0.584270\pi$
$234$	0	0
$235$	−9.26857	−0.604615
$236$	0	0
$237$	−9.69459	−0.629731
$238$	0	0
$239$	−26.4492	−1.71086	−0.855430	−	0.517919i	$-0.826707\pi$
$239$	−26.4492	−1.71086	−0.855430	+	0.517919i	$0.826707\pi$
$240$	0	0
$241$	2.73917	0.176445	0.0882227	−	0.996101i	$-0.471881\pi$
$241$	2.73917	0.176445	0.0882227	+	0.996101i	$0.471881\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.59627	−0.101982
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2.02734	−0.128478
$250$	0	0
$251$	15.2294	0.961269	0.480635	−	0.876921i	$-0.340406\pi$
$251$	15.2294	0.961269	0.480635	+	0.876921i	$0.340406\pi$
$252$	0	0
$253$	39.0729	2.45649
$254$	0	0
$255$	0.218941	0.0137106
$256$	0	0
$257$	−13.0351	−0.813106	−0.406553	−	0.913627i	$-0.633269\pi$
$257$	−13.0351	−0.813106	−0.406553	+	0.913627i	$0.633269\pi$
$258$	0	0
$259$	16.7442	1.04044
$260$	0	0
$261$	−2.38919	−0.147887
$262$	0	0
$263$	−11.9263	−0.735409	−0.367704	−	0.929943i	$-0.619856\pi$
$263$	−11.9263	−0.735409	−0.367704	+	0.929943i	$0.619856\pi$
$264$	0	0
$265$	−5.41921	−0.332900
$266$	0	0
$267$	3.44831	0.211033
$268$	0	0
$269$	−25.8580	−1.57659	−0.788296	−	0.615296i	$-0.789036\pi$
$269$	−25.8580	−1.57659	−0.788296	+	0.615296i	$0.789036\pi$
$270$	0	0
$271$	11.8726	0.721208	0.360604	−	0.932719i	$-0.382571\pi$
$271$	11.8726	0.721208	0.360604	+	0.932719i	$0.382571\pi$
$272$	0	0
$273$	−8.90941	−0.539222
$274$	0	0
$275$	18.9290	1.14146
$276$	0	0
$277$	14.4115	0.865902	0.432951	−	0.901418i	$-0.357472\pi$
$277$	14.4115	0.865902	0.432951	+	0.901418i	$0.357472\pi$
$278$	0	0
$279$	2.59627	0.155434
$280$	0	0
$281$	2.63135	0.156973	0.0784865	−	0.996915i	$-0.474991\pi$
$281$	2.63135	0.156973	0.0784865	+	0.996915i	$0.474991\pi$
$282$	0	0
$283$	6.09152	0.362103	0.181052	−	0.983474i	$-0.442050\pi$
$283$	6.09152	0.362103	0.181052	+	0.983474i	$0.442050\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.872578	0.0515067
$288$	0	0
$289$	−16.9736	−0.998447
$290$	0	0
$291$	12.0915	0.708817
$292$	0	0
$293$	32.7306	1.91214	0.956071	−	0.293134i	$-0.0946983\pi$
$293$	32.7306	1.91214	0.956071	+	0.293134i	$0.0946983\pi$
$294$	0	0
$295$	−16.2344	−0.945205
$296$	0	0
$297$	5.94356	0.344881
$298$	0	0
$299$	−24.2882	−1.40462
$300$	0	0
$301$	−27.6195	−1.59196
$302$	0	0
$303$	3.93582	0.226107
$304$	0	0
$305$	8.33275	0.477132
$306$	0	0
$307$	6.20945	0.354392	0.177196	−	0.984176i	$-0.443297\pi$
$307$	6.20945	0.354392	0.177196	+	0.984176i	$0.443297\pi$
$308$	0	0
$309$	−12.3131	−0.700471
$310$	0	0
$311$	3.10607	0.176129	0.0880644	−	0.996115i	$-0.471932\pi$
$311$	3.10607	0.176129	0.0880644	+	0.996115i	$0.471932\pi$
$312$	0	0
$313$	−16.8726	−0.953695	−0.476847	−	0.878986i	$-0.658221\pi$
$313$	−16.8726	−0.953695	−0.476847	+	0.878986i	$0.658221\pi$
$314$	0	0
$315$	3.24897	0.183059
$316$	0	0
$317$	−14.4260	−0.810246	−0.405123	−	0.914262i	$-0.632771\pi$
$317$	−14.4260	−0.810246	−0.405123	+	0.914262i	$0.632771\pi$
$318$	0	0
$319$	14.2003	0.795063
$320$	0	0
$321$	−13.4338	−0.749800
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−11.7665	−0.652689
$326$	0	0
$327$	2.26857	0.125452
$328$	0	0
$329$	−16.5895	−0.914607
$330$	0	0
$331$	8.99731	0.494537	0.247268	−	0.968947i	$-0.420467\pi$
$331$	8.99731	0.494537	0.247268	+	0.968947i	$0.420467\pi$
$332$	0	0
$333$	6.94356	0.380505
$334$	0	0
$335$	7.75877	0.423907
$336$	0	0
$337$	2.27807	0.124094	0.0620471	−	0.998073i	$-0.480237\pi$
$337$	2.27807	0.124094	0.0620471	+	0.998073i	$0.480237\pi$
$338$	0	0
$339$	−12.7101	−0.690316
$340$	0	0
$341$	−15.4311	−0.835640
$342$	0	0
$343$	−19.7374	−1.06572
$344$	0	0
$345$	8.85710	0.476850
$346$	0	0
$347$	7.50299	0.402782	0.201391	−	0.979511i	$-0.435454\pi$
$347$	7.50299	0.402782	0.201391	+	0.979511i	$0.435454\pi$
$348$	0	0
$349$	−29.0351	−1.55421	−0.777106	−	0.629370i	$-0.783313\pi$
$349$	−29.0351	−1.55421	−0.777106	+	0.629370i	$0.783313\pi$
$350$	0	0
$351$	−3.69459	−0.197203
$352$	0	0
$353$	−23.4192	−1.24648	−0.623240	−	0.782031i	$-0.714184\pi$
$353$	−23.4192	−1.24648	−0.623240	+	0.782031i	$0.714184\pi$
$354$	0	0
$355$	−16.4270	−0.871852
$356$	0	0
$357$	0.391874	0.0207402
$358$	0	0
$359$	−35.7178	−1.88511	−0.942557	−	0.334045i	$-0.891586\pi$
$359$	−35.7178	−1.88511	−0.942557	+	0.334045i	$0.891586\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−24.3259	−1.27678
$364$	0	0
$365$	−12.3550	−0.646692
$366$	0	0
$367$	−20.9317	−1.09263	−0.546313	−	0.837581i	$-0.683969\pi$
$367$	−20.9317	−1.09263	−0.546313	+	0.837581i	$0.683969\pi$
$368$	0	0
$369$	0.361844	0.0188369
$370$	0	0
$371$	−9.69965	−0.503580
$372$	0	0
$373$	2.89124	0.149703	0.0748515	−	0.997195i	$-0.476152\pi$
$373$	2.89124	0.149703	0.0748515	+	0.997195i	$0.476152\pi$
$374$	0	0
$375$	11.0273	0.569449
$376$	0	0
$377$	−8.82707	−0.454617
$378$	0	0
$379$	24.7374	1.27068	0.635338	−	0.772234i	$-0.280861\pi$
$379$	24.7374	1.27068	0.635338	+	0.772234i	$0.280861\pi$
$380$	0	0
$381$	7.73143	0.396093
$382$	0	0
$383$	−14.8571	−0.759162	−0.379581	−	0.925158i	$-0.623932\pi$
$383$	−14.8571	−0.759162	−0.379581	+	0.925158i	$0.623932\pi$
$384$	0	0
$385$	−19.3105	−0.984152
$386$	0	0
$387$	−11.4534	−0.582207
$388$	0	0
$389$	31.7588	1.61023	0.805117	−	0.593116i	$-0.202103\pi$
$389$	31.7588	1.61023	0.805117	+	0.593116i	$0.202103\pi$
$390$	0	0
$391$	1.06830	0.0540261
$392$	0	0
$393$	−12.5672	−0.633930
$394$	0	0
$395$	13.0615	0.657195
$396$	0	0
$397$	−17.7716	−0.891929	−0.445965	−	0.895051i	$-0.647139\pi$
$397$	−17.7716	−0.891929	−0.445965	+	0.895051i	$0.647139\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.0333	1.74948	0.874740	−	0.484592i	$-0.161032\pi$
$401$	35.0333	1.74948	0.874740	+	0.484592i	$0.161032\pi$
$402$	0	0
$403$	9.59215	0.477819
$404$	0	0
$405$	1.34730	0.0669477
$406$	0	0
$407$	−41.2695	−2.04565
$408$	0	0
$409$	11.5294	0.570092	0.285046	−	0.958514i	$-0.407991\pi$
$409$	11.5294	0.570092	0.285046	+	0.958514i	$0.407991\pi$
$410$	0	0
$411$	−15.7101	−0.774921
$412$	0	0
$413$	−29.0574	−1.42982
$414$	0	0
$415$	2.73143	0.134081
$416$	0	0
$417$	−3.23442	−0.158390
$418$	0	0
$419$	−34.7692	−1.69859	−0.849293	−	0.527921i	$-0.822972\pi$
$419$	−34.7692	−1.69859	−0.849293	+	0.527921i	$0.822972\pi$
$420$	0	0
$421$	15.7050	0.765416	0.382708	−	0.923869i	$-0.374992\pi$
$421$	15.7050	0.765416	0.382708	+	0.923869i	$0.374992\pi$
$422$	0	0
$423$	−6.87939	−0.334487
$424$	0	0
$425$	0.517541	0.0251044
$426$	0	0
$427$	14.9145	0.721762
$428$	0	0
$429$	21.9590	1.06019
$430$	0	0
$431$	13.4730	0.648970	0.324485	−	0.945891i	$-0.394809\pi$
$431$	13.4730	0.648970	0.324485	+	0.945891i	$0.394809\pi$
$432$	0	0
$433$	−11.7733	−0.565790	−0.282895	−	0.959151i	$-0.591295\pi$
$433$	−11.7733	−0.565790	−0.282895	+	0.959151i	$0.591295\pi$
$434$	0	0
$435$	3.21894	0.154336
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−15.7151	−0.750042	−0.375021	−	0.927016i	$-0.622365\pi$
$439$	−15.7151	−0.750042	−0.375021	+	0.927016i	$0.622365\pi$
$440$	0	0
$441$	−1.18479	−0.0564187
$442$	0	0
$443$	−15.6432	−0.743231	−0.371616	−	0.928387i	$-0.621196\pi$
$443$	−15.6432	−0.743231	−0.371616	+	0.928387i	$0.621196\pi$
$444$	0	0
$445$	−4.64590	−0.220237
$446$	0	0
$447$	−4.46017	−0.210959
$448$	0	0
$449$	18.1857	0.858237	0.429119	−	0.903248i	$-0.358824\pi$
$449$	18.1857	0.858237	0.429119	+	0.903248i	$0.358824\pi$
$450$	0	0
$451$	−2.15064	−0.101270
$452$	0	0
$453$	8.94862	0.420443
$454$	0	0
$455$	12.0036	0.562738
$456$	0	0
$457$	16.8111	0.786390	0.393195	−	0.919455i	$-0.371370\pi$
$457$	16.8111	0.786390	0.393195	+	0.919455i	$0.371370\pi$
$458$	0	0
$459$	0.162504	0.00758503
$460$	0	0
$461$	−12.6364	−0.588536	−0.294268	−	0.955723i	$-0.595076\pi$
$461$	−12.6364	−0.588536	−0.294268	+	0.955723i	$0.595076\pi$
$462$	0	0
$463$	−5.89487	−0.273958	−0.136979	−	0.990574i	$-0.543739\pi$
$463$	−5.89487	−0.273958	−0.136979	+	0.990574i	$0.543739\pi$
$464$	0	0
$465$	−3.49794	−0.162213
$466$	0	0
$467$	−5.36959	−0.248475	−0.124237	−	0.992253i	$-0.539648\pi$
$467$	−5.36959	−0.248475	−0.124237	+	0.992253i	$0.539648\pi$
$468$	0	0
$469$	13.8871	0.641248
$470$	0	0
$471$	11.5202	0.530824
$472$	0	0
$473$	68.0738	3.13004
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.02229	−0.184168
$478$	0	0
$479$	−19.8476	−0.906860	−0.453430	−	0.891292i	$-0.649800\pi$
$479$	−19.8476	−0.906860	−0.453430	+	0.891292i	$0.649800\pi$
$480$	0	0
$481$	25.6536	1.16971
$482$	0	0
$483$	15.8530	0.721335
$484$	0	0
$485$	−16.2909	−0.739730
$486$	0	0
$487$	−13.0077	−0.589437	−0.294718	−	0.955584i	$-0.595226\pi$
$487$	−13.0077	−0.589437	−0.294718	+	0.955584i	$0.595226\pi$
$488$	0	0
$489$	10.5243	0.475927
$490$	0	0
$491$	29.7478	1.34250	0.671251	−	0.741230i	$-0.265757\pi$
$491$	29.7478	1.34250	0.671251	+	0.741230i	$0.265757\pi$
$492$	0	0
$493$	0.388252	0.0174860
$494$	0	0
$495$	−8.00774	−0.359921
$496$	0	0
$497$	−29.4020	−1.31886
$498$	0	0
$499$	14.9855	0.670841	0.335420	−	0.942069i	$-0.391122\pi$
$499$	14.9855	0.670841	0.335420	+	0.942069i	$0.391122\pi$
$500$	0	0
$501$	10.7392	0.479791
$502$	0	0
$503$	−17.7074	−0.789533	−0.394767	−	0.918781i	$-0.629175\pi$
$503$	−17.7074	−0.789533	−0.394767	+	0.918781i	$0.629175\pi$
$504$	0	0
$505$	−5.30272	−0.235968
$506$	0	0
$507$	−0.650015	−0.0288682
$508$	0	0
$509$	−27.2226	−1.20662	−0.603309	−	0.797507i	$-0.706152\pi$
$509$	−27.2226	−1.20662	−0.603309	+	0.797507i	$0.706152\pi$
$510$	0	0
$511$	−22.1138	−0.978257
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.5895	0.731019
$516$	0	0
$517$	40.8881	1.79825
$518$	0	0
$519$	−7.53983	−0.330962
$520$	0	0
$521$	−20.4260	−0.894880	−0.447440	−	0.894314i	$-0.647664\pi$
$521$	−20.4260	−0.894880	−0.447440	+	0.894314i	$0.647664\pi$
$522$	0	0
$523$	41.1147	1.79782	0.898911	−	0.438131i	$-0.144359\pi$
$523$	41.1147	1.79782	0.898911	+	0.438131i	$0.144359\pi$
$524$	0	0
$525$	7.68004	0.335185
$526$	0	0
$527$	−0.421903	−0.0183784
$528$	0	0
$529$	20.2172	0.879008
$530$	0	0
$531$	−12.0496	−0.522909
$532$	0	0
$533$	1.33687	0.0579061
$534$	0	0
$535$	18.0993	0.782499
$536$	0	0
$537$	26.3628	1.13764
$538$	0	0
$539$	7.04189	0.303316
$540$	0	0
$541$	−6.07098	−0.261012	−0.130506	−	0.991448i	$-0.541660\pi$
$541$	−6.07098	−0.261012	−0.130506	+	0.991448i	$0.541660\pi$
$542$	0	0
$543$	21.0915	0.905124
$544$	0	0
$545$	−3.05644	−0.130923
$546$	0	0
$547$	23.8152	1.01826	0.509132	−	0.860688i	$-0.329966\pi$
$547$	23.8152	1.01826	0.509132	+	0.860688i	$0.329966\pi$
$548$	0	0
$549$	6.18479	0.263961
$550$	0	0
$551$	0	0
$552$	0	0
$553$	23.3783	0.994145
$554$	0	0
$555$	−9.35504	−0.397099
$556$	0	0
$557$	−23.3155	−0.987910	−0.493955	−	0.869487i	$-0.664449\pi$
$557$	−23.3155	−0.987910	−0.493955	+	0.869487i	$0.664449\pi$
$558$	0	0
$559$	−42.3155	−1.78976
$560$	0	0
$561$	−0.965852	−0.0407783
$562$	0	0
$563$	−14.7074	−0.619842	−0.309921	−	0.950762i	$-0.600303\pi$
$563$	−14.7074	−0.619842	−0.309921	+	0.950762i	$0.600303\pi$
$564$	0	0
$565$	17.1242	0.720422
$566$	0	0
$567$	2.41147	0.101272
$568$	0	0
$569$	−32.6750	−1.36981	−0.684903	−	0.728634i	$-0.740156\pi$
$569$	−32.6750	−1.36981	−0.684903	+	0.728634i	$0.740156\pi$
$570$	0	0
$571$	30.3901	1.27179	0.635893	−	0.771777i	$-0.280632\pi$
$571$	30.3901	1.27179	0.635893	+	0.771777i	$0.280632\pi$
$572$	0	0
$573$	0.255777	0.0106853
$574$	0	0
$575$	20.9368	0.873123
$576$	0	0
$577$	30.8648	1.28492	0.642460	−	0.766319i	$-0.277914\pi$
$577$	30.8648	1.28492	0.642460	+	0.766319i	$0.277914\pi$
$578$	0	0
$579$	14.3482	0.596292
$580$	0	0
$581$	4.88888	0.202825
$582$	0	0
$583$	23.9067	0.990115
$584$	0	0
$585$	4.97771	0.205803
$586$	0	0
$587$	28.1712	1.16275	0.581374	−	0.813636i	$-0.302515\pi$
$587$	28.1712	1.16275	0.581374	+	0.813636i	$0.302515\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−23.6810	−0.974105
$592$	0	0
$593$	−6.82058	−0.280088	−0.140044	−	0.990145i	$-0.544724\pi$
$593$	−6.82058	−0.280088	−0.140044	+	0.990145i	$0.544724\pi$
$594$	0	0
$595$	−0.527970	−0.0216447
$596$	0	0
$597$	11.7956	0.482762
$598$	0	0
$599$	35.7110	1.45911	0.729556	−	0.683921i	$-0.239727\pi$
$599$	35.7110	1.45911	0.729556	+	0.683921i	$0.239727\pi$
$600$	0	0
$601$	−5.52797	−0.225491	−0.112745	−	0.993624i	$-0.535964\pi$
$601$	−5.52797	−0.225491	−0.112745	+	0.993624i	$0.535964\pi$
$602$	0	0
$603$	5.75877	0.234515
$604$	0	0
$605$	32.7743	1.33246
$606$	0	0
$607$	−19.4953	−0.791288	−0.395644	−	0.918404i	$-0.629479\pi$
$607$	−19.4953	−0.791288	−0.395644	+	0.918404i	$0.629479\pi$
$608$	0	0
$609$	5.76146	0.233466
$610$	0	0
$611$	−25.4165	−1.02824
$612$	0	0
$613$	39.5202	1.59621	0.798104	−	0.602520i	$-0.205837\pi$
$613$	39.5202	1.59621	0.798104	+	0.602520i	$0.205837\pi$
$614$	0	0
$615$	−0.487511	−0.0196584
$616$	0	0
$617$	−1.24359	−0.0500652	−0.0250326	−	0.999687i	$-0.507969\pi$
$617$	−1.24359	−0.0500652	−0.0250326	+	0.999687i	$0.507969\pi$
$618$	0	0
$619$	−8.40373	−0.337775	−0.168887	−	0.985635i	$-0.554017\pi$
$619$	−8.40373	−0.337775	−0.168887	+	0.985635i	$0.554017\pi$
$620$	0	0
$621$	6.57398	0.263805
$622$	0	0
$623$	−8.31551	−0.333154
$624$	0	0
$625$	1.06687	0.0426746
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.12836	−0.0449905
$630$	0	0
$631$	−9.28075	−0.369461	−0.184731	−	0.982789i	$-0.559141\pi$
$631$	−9.28075	−0.369461	−0.184731	+	0.982789i	$0.559141\pi$
$632$	0	0
$633$	27.8580	1.10726
$634$	0	0
$635$	−10.4165	−0.413367
$636$	0	0
$637$	−4.37733	−0.173436
$638$	0	0
$639$	−12.1925	−0.482329
$640$	0	0
$641$	8.71419	0.344190	0.172095	−	0.985080i	$-0.444946\pi$
$641$	8.71419	0.344190	0.172095	+	0.985080i	$0.444946\pi$
$642$	0	0
$643$	−15.1310	−0.596710	−0.298355	−	0.954455i	$-0.596438\pi$
$643$	−15.1310	−0.596710	−0.298355	+	0.954455i	$0.596438\pi$
$644$	0	0
$645$	15.4311	0.607598
$646$	0	0
$647$	7.31282	0.287497	0.143748	−	0.989614i	$-0.454084\pi$
$647$	7.31282	0.287497	0.143748	+	0.989614i	$0.454084\pi$
$648$	0	0
$649$	71.6177	2.81124
$650$	0	0
$651$	−6.26083	−0.245381
$652$	0	0
$653$	17.5381	0.686318	0.343159	−	0.939277i	$-0.388503\pi$
$653$	17.5381	0.686318	0.343159	+	0.939277i	$0.388503\pi$
$654$	0	0
$655$	16.9317	0.661576
$656$	0	0
$657$	−9.17024	−0.357765
$658$	0	0
$659$	7.87164	0.306636	0.153318	−	0.988177i	$-0.451004\pi$
$659$	7.87164	0.306636	0.153318	+	0.988177i	$0.451004\pi$
$660$	0	0
$661$	−0.964097	−0.0374990	−0.0187495	−	0.999824i	$-0.505969\pi$
$661$	−0.964097	−0.0374990	−0.0187495	+	0.999824i	$0.505969\pi$
$662$	0	0
$663$	0.600385	0.0233170
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.7065	0.608156
$668$	0	0
$669$	−24.7374	−0.956404
$670$	0	0
$671$	−36.7597	−1.41909
$672$	0	0
$673$	25.8530	0.996559	0.498280	−	0.867016i	$-0.333965\pi$
$673$	25.8530	0.996559	0.498280	+	0.867016i	$0.333965\pi$
$674$	0	0
$675$	3.18479	0.122583
$676$	0	0
$677$	−29.9956	−1.15282	−0.576411	−	0.817160i	$-0.695547\pi$
$677$	−29.9956	−1.15282	−0.576411	+	0.817160i	$0.695547\pi$
$678$	0	0
$679$	−29.1584	−1.11900
$680$	0	0
$681$	−22.0155	−0.843635
$682$	0	0
$683$	−33.5084	−1.28216	−0.641081	−	0.767473i	$-0.721514\pi$
$683$	−33.5084	−1.28216	−0.641081	+	0.767473i	$0.721514\pi$
$684$	0	0
$685$	21.1661	0.808716
$686$	0	0
$687$	9.29591	0.354661
$688$	0	0
$689$	−14.8607	−0.566148
$690$	0	0
$691$	−12.8135	−0.487447	−0.243723	−	0.969845i	$-0.578369\pi$
$691$	−12.8135	−0.487447	−0.243723	+	0.969845i	$0.578369\pi$
$692$	0	0
$693$	−14.3327	−0.544456
$694$	0	0
$695$	4.35773	0.165298
$696$	0	0
$697$	−0.0588011	−0.00222725
$698$	0	0
$699$	7.98814	0.302139
$700$	0	0
$701$	3.97266	0.150045	0.0750226	−	0.997182i	$-0.476097\pi$
$701$	3.97266	0.150045	0.0750226	+	0.997182i	$0.476097\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	9.26857	0.349075
$706$	0	0
$707$	−9.49113	−0.356951
$708$	0	0
$709$	32.3054	1.21326	0.606628	−	0.794986i	$-0.292522\pi$
$709$	32.3054	1.21326	0.606628	+	0.794986i	$0.292522\pi$
$710$	0	0
$711$	9.69459	0.363576
$712$	0	0
$713$	−17.0678	−0.639194
$714$	0	0
$715$	−29.5853	−1.10643
$716$	0	0
$717$	26.4492	0.987765
$718$	0	0
$719$	45.2686	1.68823	0.844116	−	0.536160i	$-0.180126\pi$
$719$	45.2686	1.68823	0.844116	+	0.536160i	$0.180126\pi$
$720$	0	0
$721$	29.6928	1.10582
$722$	0	0
$723$	−2.73917	−0.101871
$724$	0	0
$725$	7.60906	0.282593
$726$	0	0
$727$	−22.1821	−0.822689	−0.411344	−	0.911480i	$-0.634941\pi$
$727$	−22.1821	−0.822689	−0.411344	+	0.911480i	$0.634941\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.86122	0.0688395
$732$	0	0
$733$	45.8212	1.69244	0.846222	−	0.532830i	$-0.178872\pi$
$733$	45.8212	1.69244	0.846222	+	0.532830i	$0.178872\pi$
$734$	0	0
$735$	1.59627	0.0588792
$736$	0	0
$737$	−34.2276	−1.26079
$738$	0	0
$739$	−3.77425	−0.138838	−0.0694191	−	0.997588i	$-0.522115\pi$
$739$	−3.77425	−0.138838	−0.0694191	+	0.997588i	$0.522115\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.4858	0.824922	0.412461	−	0.910975i	$-0.364669\pi$
$743$	22.4858	0.824922	0.412461	+	0.910975i	$0.364669\pi$
$744$	0	0
$745$	6.00917	0.220159
$746$	0	0
$747$	2.02734	0.0741765
$748$	0	0
$749$	32.3952	1.18369
$750$	0	0
$751$	41.4962	1.51422	0.757109	−	0.653289i	$-0.226611\pi$
$751$	41.4962	1.51422	0.757109	+	0.653289i	$0.226611\pi$
$752$	0	0
$753$	−15.2294	−0.554989
$754$	0	0
$755$	−12.0564	−0.438779
$756$	0	0
$757$	−17.8821	−0.649935	−0.324968	−	0.945725i	$-0.605353\pi$
$757$	−17.8821	−0.649935	−0.324968	+	0.945725i	$0.605353\pi$
$758$	0	0
$759$	−39.0729	−1.41825
$760$	0	0
$761$	38.0806	1.38042	0.690210	−	0.723609i	$-0.257518\pi$
$761$	38.0806	1.38042	0.690210	+	0.723609i	$0.257518\pi$
$762$	0	0
$763$	−5.47060	−0.198049
$764$	0	0
$765$	−0.218941	−0.00791582
$766$	0	0
$767$	−44.5185	−1.60747
$768$	0	0
$769$	−12.6245	−0.455253	−0.227626	−	0.973749i	$-0.573096\pi$
$769$	−12.6245	−0.455253	−0.227626	+	0.973749i	$0.573096\pi$
$770$	0	0
$771$	13.0351	0.469447
$772$	0	0
$773$	−33.7989	−1.21566	−0.607831	−	0.794066i	$-0.707960\pi$
$773$	−33.7989	−1.21566	−0.607831	+	0.794066i	$0.707960\pi$
$774$	0	0
$775$	−8.26857	−0.297016
$776$	0	0
$777$	−16.7442	−0.600696
$778$	0	0
$779$	0	0
$780$	0	0
$781$	72.4671	2.59308
$782$	0	0
$783$	2.38919	0.0853825
$784$	0	0
$785$	−15.5212	−0.553974
$786$	0	0
$787$	33.6117	1.19813	0.599065	−	0.800701i	$-0.295539\pi$
$787$	33.6117	1.19813	0.599065	+	0.800701i	$0.295539\pi$
$788$	0	0
$789$	11.9263	0.424588
$790$	0	0
$791$	30.6500	1.08979
$792$	0	0
$793$	22.8503	0.811438
$794$	0	0
$795$	5.41921	0.192200
$796$	0	0
$797$	46.0547	1.63134	0.815670	−	0.578517i	$-0.196368\pi$
$797$	46.0547	1.63134	0.815670	+	0.578517i	$0.196368\pi$
$798$	0	0
$799$	1.11793	0.0395494
$800$	0	0
$801$	−3.44831	−0.121840
$802$	0	0
$803$	54.5039	1.92340
$804$	0	0
$805$	−21.3587	−0.752794
$806$	0	0
$807$	25.8580	0.910246
$808$	0	0
$809$	40.3705	1.41935	0.709676	−	0.704528i	$-0.248841\pi$
$809$	40.3705	1.41935	0.709676	+	0.704528i	$0.248841\pi$
$810$	0	0
$811$	51.5853	1.81141	0.905703	−	0.423912i	$-0.139344\pi$
$811$	51.5853	1.81141	0.905703	+	0.423912i	$0.139344\pi$
$812$	0	0
$813$	−11.8726	−0.416389
$814$	0	0
$815$	−14.1794	−0.496683
$816$	0	0
$817$	0	0
$818$	0	0
$819$	8.90941	0.311320
$820$	0	0
$821$	39.8289	1.39004	0.695020	−	0.718991i	$-0.255396\pi$
$821$	39.8289	1.39004	0.695020	+	0.718991i	$0.255396\pi$
$822$	0	0
$823$	−18.9736	−0.661378	−0.330689	−	0.943740i	$-0.607281\pi$
$823$	−18.9736	−0.661378	−0.330689	+	0.943740i	$0.607281\pi$
$824$	0	0
$825$	−18.9290	−0.659024
$826$	0	0
$827$	4.91891	0.171047	0.0855236	−	0.996336i	$-0.472744\pi$
$827$	4.91891	0.171047	0.0855236	+	0.996336i	$0.472744\pi$
$828$	0	0
$829$	33.2799	1.15586	0.577930	−	0.816086i	$-0.303861\pi$
$829$	33.2799	1.15586	0.577930	+	0.816086i	$0.303861\pi$
$830$	0	0
$831$	−14.4115	−0.499928
$832$	0	0
$833$	0.192533	0.00667088
$834$	0	0
$835$	−14.4688	−0.500715
$836$	0	0
$837$	−2.59627	−0.0897401
$838$	0	0
$839$	−18.5003	−0.638701	−0.319351	−	0.947637i	$-0.603465\pi$
$839$	−18.5003	−0.638701	−0.319351	+	0.947637i	$0.603465\pi$
$840$	0	0
$841$	−23.2918	−0.803165
$842$	0	0
$843$	−2.63135	−0.0906285
$844$	0	0
$845$	0.875763	0.0301272
$846$	0	0
$847$	58.6614	2.01563
$848$	0	0
$849$	−6.09152	−0.209060
$850$	0	0
$851$	−45.6468	−1.56475
$852$	0	0
$853$	8.67768	0.297118	0.148559	−	0.988904i	$-0.452536\pi$
$853$	8.67768	0.297118	0.148559	+	0.988904i	$0.452536\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.0164	0.922863	0.461432	−	0.887176i	$-0.347336\pi$
$857$	27.0164	0.922863	0.461432	+	0.887176i	$0.347336\pi$
$858$	0	0
$859$	−51.1995	−1.74690	−0.873451	−	0.486911i	$-0.838123\pi$
$859$	−51.1995	−1.74690	−0.873451	+	0.486911i	$0.838123\pi$
$860$	0	0
$861$	−0.872578	−0.0297374
$862$	0	0
$863$	−19.1607	−0.652239	−0.326120	−	0.945328i	$-0.605741\pi$
$863$	−19.1607	−0.652239	−0.326120	+	0.945328i	$0.605741\pi$
$864$	0	0
$865$	10.1584	0.345395
$866$	0	0
$867$	16.9736	0.576453
$868$	0	0
$869$	−57.6204	−1.95464
$870$	0	0
$871$	21.2763	0.720920
$872$	0	0
$873$	−12.0915	−0.409236
$874$	0	0
$875$	−26.5921	−0.898979
$876$	0	0
$877$	−53.3218	−1.80055	−0.900275	−	0.435322i	$-0.856635\pi$
$877$	−53.3218	−1.80055	−0.900275	+	0.435322i	$0.856635\pi$
$878$	0	0
$879$	−32.7306	−1.10398
$880$	0	0
$881$	−0.673238	−0.0226820	−0.0113410	−	0.999936i	$-0.503610\pi$
$881$	−0.673238	−0.0226820	−0.0113410	+	0.999936i	$0.503610\pi$
$882$	0	0
$883$	9.79116	0.329499	0.164749	−	0.986335i	$-0.447318\pi$
$883$	9.79116	0.329499	0.164749	+	0.986335i	$0.447318\pi$
$884$	0	0
$885$	16.2344	0.545714
$886$	0	0
$887$	8.79467	0.295296	0.147648	−	0.989040i	$-0.452830\pi$
$887$	8.79467	0.295296	0.147648	+	0.989040i	$0.452830\pi$
$888$	0	0
$889$	−18.6441	−0.625304
$890$	0	0
$891$	−5.94356	−0.199117
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−35.5185	−1.18725
$896$	0	0
$897$	24.2882	0.810958
$898$	0	0
$899$	−6.20296	−0.206880
$900$	0	0
$901$	0.653637	0.0217758
$902$	0	0
$903$	27.6195	0.919119
$904$	0	0
$905$	−28.4165	−0.944597
$906$	0	0
$907$	5.79797	0.192518	0.0962592	−	0.995356i	$-0.469312\pi$
$907$	5.79797	0.192518	0.0962592	+	0.995356i	$0.469312\pi$
$908$	0	0
$909$	−3.93582	−0.130543
$910$	0	0
$911$	16.9145	0.560401	0.280201	−	0.959941i	$-0.409599\pi$
$911$	16.9145	0.560401	0.280201	+	0.959941i	$0.409599\pi$
$912$	0	0
$913$	−12.0496	−0.398785
$914$	0	0
$915$	−8.33275	−0.275472
$916$	0	0
$917$	30.3054	1.00077
$918$	0	0
$919$	−36.8759	−1.21642	−0.608211	−	0.793775i	$-0.708113\pi$
$919$	−36.8759	−1.21642	−0.608211	+	0.793775i	$0.708113\pi$
$920$	0	0
$921$	−6.20945	−0.204608
$922$	0	0
$923$	−45.0464	−1.48272
$924$	0	0
$925$	−22.1138	−0.727098
$926$	0	0
$927$	12.3131	0.404417
$928$	0	0
$929$	30.0368	0.985477	0.492738	−	0.870178i	$-0.335996\pi$
$929$	30.0368	0.985477	0.492738	+	0.870178i	$0.335996\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.10607	−0.101688
$934$	0	0
$935$	1.30129	0.0425567
$936$	0	0
$937$	−4.20615	−0.137409	−0.0687044	−	0.997637i	$-0.521887\pi$
$937$	−4.20615	−0.137409	−0.0687044	+	0.997637i	$0.521887\pi$
$938$	0	0
$939$	16.8726	0.550616
$940$	0	0
$941$	35.8111	1.16741	0.583704	−	0.811966i	$-0.301603\pi$
$941$	35.8111	1.16741	0.583704	+	0.811966i	$0.301603\pi$
$942$	0	0
$943$	−2.37876	−0.0774630
$944$	0	0
$945$	−3.24897	−0.105689
$946$	0	0
$947$	−10.4510	−0.339612	−0.169806	−	0.985478i	$-0.554314\pi$
$947$	−10.4510	−0.339612	−0.169806	+	0.985478i	$0.554314\pi$
$948$	0	0
$949$	−33.8803	−1.09980
$950$	0	0
$951$	14.4260	0.467796
$952$	0	0
$953$	−21.6132	−0.700120	−0.350060	−	0.936727i	$-0.613839\pi$
$953$	−21.6132	−0.700120	−0.350060	+	0.936727i	$0.613839\pi$
$954$	0	0
$955$	−0.344608	−0.0111513
$956$	0	0
$957$	−14.2003	−0.459030
$958$	0	0
$959$	37.8844	1.22335
$960$	0	0
$961$	−24.2594	−0.782561
$962$	0	0
$963$	13.4338	0.432897
$964$	0	0
$965$	−19.3313	−0.622297
$966$	0	0
$967$	9.12155	0.293329	0.146665	−	0.989186i	$-0.453146\pi$
$967$	9.12155	0.293329	0.146665	+	0.989186i	$0.453146\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	48.8221	1.56678	0.783388	−	0.621533i	$-0.213490\pi$
$971$	48.8221	1.56678	0.783388	+	0.621533i	$0.213490\pi$
$972$	0	0
$973$	7.79973	0.250048
$974$	0	0
$975$	11.7665	0.376830
$976$	0	0
$977$	−2.62536	−0.0839928	−0.0419964	−	0.999118i	$-0.513372\pi$
$977$	−2.62536	−0.0839928	−0.0419964	+	0.999118i	$0.513372\pi$
$978$	0	0
$979$	20.4953	0.655031
$980$	0	0
$981$	−2.26857	−0.0724299
$982$	0	0
$983$	−10.7264	−0.342118	−0.171059	−	0.985261i	$-0.554719\pi$
$983$	−10.7264	−0.342118	−0.171059	+	0.985261i	$0.554719\pi$
$984$	0	0
$985$	31.9053	1.01659
$986$	0	0
$987$	16.5895	0.528048
$988$	0	0
$989$	75.2942	2.39421
$990$	0	0
$991$	−56.5776	−1.79725	−0.898623	−	0.438721i	$-0.855432\pi$
$991$	−56.5776	−1.79725	−0.898623	+	0.438721i	$0.855432\pi$
$992$	0	0
$993$	−8.99731	−0.285521
$994$	0	0
$995$	−15.8922	−0.503816
$996$	0	0
$997$	11.0541	0.350086	0.175043	−	0.984561i	$-0.443994\pi$
$997$	11.0541	0.350086	0.175043	+	0.984561i	$0.443994\pi$
$998$	0	0
$999$	−6.94356	−0.219685

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4332.2.a.n.1.2		3
19.14	odd	18		228.2.q.a.25.1	✓	6
19.15	odd	18		228.2.q.a.73.1	yes	6
19.18	odd	2		4332.2.a.o.1.2		3
57.14	even	18		684.2.bo.a.253.1		6
57.53	even	18		684.2.bo.a.73.1		6
76.15	even	18		912.2.bo.e.529.1		6
76.71	even	18		912.2.bo.e.481.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.q.a.25.1	✓	6	19.14	odd	18
228.2.q.a.73.1	yes	6	19.15	odd	18
684.2.bo.a.73.1		6	57.53	even	18
684.2.bo.a.253.1		6	57.14	even	18
912.2.bo.e.481.1		6	76.71	even	18
912.2.bo.e.529.1		6	76.15	even	18
4332.2.a.n.1.2		3	1.1	even	1	trivial
4332.2.a.o.1.2		3	19.18	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4332.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.34730 q^{5} +2.41147 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.34730 q^{5} +2.41147 q^{7} +1.00000 q^{9} -5.94356 q^{11} +3.69459 q^{13} -1.34730 q^{15} -0.162504 q^{17} -2.41147 q^{21} -6.57398 q^{23} -3.18479 q^{25} -1.00000 q^{27} -2.38919 q^{29} +2.59627 q^{31} +5.94356 q^{33} +3.24897 q^{35} +6.94356 q^{37} -3.69459 q^{39} +0.361844 q^{41} -11.4534 q^{43} +1.34730 q^{45} -6.87939 q^{47} -1.18479 q^{49} +0.162504 q^{51} -4.02229 q^{53} -8.00774 q^{55} -12.0496 q^{59} +6.18479 q^{61} +2.41147 q^{63} +4.97771 q^{65} +5.75877 q^{67} +6.57398 q^{69} -12.1925 q^{71} -9.17024 q^{73} +3.18479 q^{75} -14.3327 q^{77} +9.69459 q^{79} +1.00000 q^{81} +2.02734 q^{83} -0.218941 q^{85} +2.38919 q^{87} -3.44831 q^{89} +8.90941 q^{91} -2.59627 q^{93} -12.0915 q^{97} -5.94356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 9 q^{13} - 3 q^{15} - 3 q^{17} + 3 q^{21} - 12 q^{23} - 6 q^{25} - 3 q^{27} - 3 q^{29} - 6 q^{31} + 3 q^{33} - 3 q^{35} + 6 q^{37} - 9 q^{39} + 18 q^{41} - 21 q^{43} + 3 q^{45} - 15 q^{47} + 3 q^{51} - 6 q^{53} - 9 q^{59} + 15 q^{61} - 3 q^{63} + 21 q^{65} + 6 q^{67} + 12 q^{69} - 9 q^{71} - 6 q^{73} + 6 q^{75} - 24 q^{77} + 27 q^{79} + 3 q^{81} - 15 q^{83} - 18 q^{85} + 3 q^{87} - 12 q^{89} - 9 q^{91} + 6 q^{93} - 6 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 9 * q^13 - 3 * q^15 - 3 * q^17 + 3 * q^21 - 12 * q^23 - 6 * q^25 - 3 * q^27 - 3 * q^29 - 6 * q^31 + 3 * q^33 - 3 * q^35 + 6 * q^37 - 9 * q^39 + 18 * q^41 - 21 * q^43 + 3 * q^45 - 15 * q^47 + 3 * q^51 - 6 * q^53 - 9 * q^59 + 15 * q^61 - 3 * q^63 + 21 * q^65 + 6 * q^67 + 12 * q^69 - 9 * q^71 - 6 * q^73 + 6 * q^75 - 24 * q^77 + 27 * q^79 + 3 * q^81 - 15 * q^83 - 18 * q^85 + 3 * q^87 - 12 * q^89 - 9 * q^91 + 6 * q^93 - 6 * q^97 - 3 * q^99

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