LMFDB - Embedded newform 4332.2.a.n.1.3

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			1.3
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.53209 q^{5} -3.22668 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.53209 q^{5} -3.22668 q^{7} +1.00000 q^{9} +3.10607 q^{11} +6.06418 q^{13} -2.53209 q^{15} -5.94356 q^{17} +3.22668 q^{21} -6.71688 q^{23} +1.41147 q^{25} -1.00000 q^{27} -7.12836 q^{29} -7.63816 q^{31} -3.10607 q^{33} -8.17024 q^{35} -2.10607 q^{37} -6.06418 q^{39} +7.04189 q^{41} -9.36959 q^{43} +2.53209 q^{45} -4.65270 q^{47} +3.41147 q^{49} +5.94356 q^{51} +6.35504 q^{53} +7.86484 q^{55} +0.268571 q^{59} +1.58853 q^{61} -3.22668 q^{63} +15.3550 q^{65} +1.30541 q^{67} +6.71688 q^{69} +8.27631 q^{71} +0.921274 q^{73} -1.41147 q^{75} -10.0223 q^{77} +12.0642 q^{79} +1.00000 q^{81} +0.0864665 q^{83} -15.0496 q^{85} +7.12836 q^{87} +7.07192 q^{89} -19.5672 q^{91} +7.63816 q^{93} -3.32770 q^{97} +3.10607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.53209	1.13238	0.566192	−	0.824273i	$-0.308416\pi$
$5$	2.53209	1.13238	0.566192	+	0.824273i	$0.308416\pi$
$6$	0	0
$7$	−3.22668	−1.21957	−0.609786	−	0.792566i	$-0.708744\pi$
$7$	−3.22668	−1.21957	−0.609786	+	0.792566i	$0.708744\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.10607	0.936514	0.468257	−	0.883592i	$-0.344882\pi$
$11$	3.10607	0.936514	0.468257	+	0.883592i	$0.344882\pi$
$12$	0	0
$13$	6.06418	1.68190	0.840950	−	0.541113i	$-0.181997\pi$
$13$	6.06418	1.68190	0.840950	+	0.541113i	$0.181997\pi$
$14$	0	0
$15$	−2.53209	−0.653783
$16$	0	0
$17$	−5.94356	−1.44153	−0.720763	−	0.693182i	$-0.756208\pi$
$17$	−5.94356	−1.44153	−0.720763	+	0.693182i	$0.756208\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.22668	0.704120
$22$	0	0
$23$	−6.71688	−1.40057	−0.700283	−	0.713865i	$-0.746943\pi$
$23$	−6.71688	−1.40057	−0.700283	+	0.713865i	$0.746943\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.12836	−1.32370	−0.661851	−	0.749635i	$-0.730229\pi$
$29$	−7.12836	−1.32370	−0.661851	+	0.749635i	$0.730229\pi$
$30$	0	0
$31$	−7.63816	−1.37185	−0.685927	−	0.727671i	$-0.740603\pi$
$31$	−7.63816	−1.37185	−0.685927	+	0.727671i	$0.740603\pi$
$32$	0	0
$33$	−3.10607	−0.540697
$34$	0	0
$35$	−8.17024	−1.38102
$36$	0	0
$37$	−2.10607	−0.346235	−0.173118	−	0.984901i	$-0.555384\pi$
$37$	−2.10607	−0.346235	−0.173118	+	0.984901i	$0.555384\pi$
$38$	0	0
$39$	−6.06418	−0.971046
$40$	0	0
$41$	7.04189	1.09976	0.549879	−	0.835244i	$-0.314674\pi$
$41$	7.04189	1.09976	0.549879	+	0.835244i	$0.314674\pi$
$42$	0	0
$43$	−9.36959	−1.42885	−0.714424	−	0.699713i	$-0.753311\pi$
$43$	−9.36959	−1.42885	−0.714424	+	0.699713i	$0.753311\pi$
$44$	0	0
$45$	2.53209	0.377462
$46$	0	0
$47$	−4.65270	−0.678667	−0.339333	−	0.940666i	$-0.610201\pi$
$47$	−4.65270	−0.678667	−0.339333	+	0.940666i	$0.610201\pi$
$48$	0	0
$49$	3.41147	0.487353
$50$	0	0
$51$	5.94356	0.832265
$52$	0	0
$53$	6.35504	0.872931	0.436466	−	0.899721i	$-0.356230\pi$
$53$	6.35504	0.872931	0.436466	+	0.899721i	$0.356230\pi$
$54$	0	0
$55$	7.86484	1.06049
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.268571	0.0349649	0.0174825	−	0.999847i	$-0.494435\pi$
$59$	0.268571	0.0349649	0.0174825	+	0.999847i	$0.494435\pi$
$60$	0	0
$61$	1.58853	0.203390	0.101695	−	0.994816i	$-0.467573\pi$
$61$	1.58853	0.203390	0.101695	+	0.994816i	$0.467573\pi$
$62$	0	0
$63$	−3.22668	−0.406524
$64$	0	0
$65$	15.3550	1.90456
$66$	0	0
$67$	1.30541	0.159481	0.0797404	−	0.996816i	$-0.474591\pi$
$67$	1.30541	0.159481	0.0797404	+	0.996816i	$0.474591\pi$
$68$	0	0
$69$	6.71688	0.808617
$70$	0	0
$71$	8.27631	0.982217	0.491109	−	0.871098i	$-0.336592\pi$
$71$	8.27631	0.982217	0.491109	+	0.871098i	$0.336592\pi$
$72$	0	0
$73$	0.921274	0.107827	0.0539135	−	0.998546i	$-0.482830\pi$
$73$	0.921274	0.107827	0.0539135	+	0.998546i	$0.482830\pi$
$74$	0	0
$75$	−1.41147	−0.162983
$76$	0	0
$77$	−10.0223	−1.14215
$78$	0	0
$79$	12.0642	1.35733	0.678663	−	0.734450i	$-0.262560\pi$
$79$	12.0642	1.35733	0.678663	+	0.734450i	$0.262560\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0864665	0.00949093	0.00474546	−	0.999989i	$-0.498489\pi$
$83$	0.0864665	0.00949093	0.00474546	+	0.999989i	$0.498489\pi$
$84$	0	0
$85$	−15.0496	−1.63236
$86$	0	0
$87$	7.12836	0.764240
$88$	0	0
$89$	7.07192	0.749622	0.374811	−	0.927101i	$-0.377708\pi$
$89$	7.07192	0.749622	0.374811	+	0.927101i	$0.377708\pi$
$90$	0	0
$91$	−19.5672	−2.05120
$92$	0	0
$93$	7.63816	0.792040
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.32770	−0.337876	−0.168938	−	0.985627i	$-0.554034\pi$
$97$	−3.32770	−0.337876	−0.168938	+	0.985627i	$0.554034\pi$
$98$	0	0
$99$	3.10607	0.312171
$100$	0	0
$101$	−10.7588	−1.07054	−0.535269	−	0.844682i	$-0.679790\pi$
$101$	−10.7588	−1.07054	−0.535269	+	0.844682i	$0.679790\pi$
$102$	0	0
$103$	−5.92902	−0.584203	−0.292102	−	0.956387i	$-0.594355\pi$
$103$	−5.92902	−0.584203	−0.292102	+	0.956387i	$0.594355\pi$
$104$	0	0
$105$	8.17024	0.797334
$106$	0	0
$107$	−2.58172	−0.249584	−0.124792	−	0.992183i	$-0.539826\pi$
$107$	−2.58172	−0.249584	−0.124792	+	0.992183i	$0.539826\pi$
$108$	0	0
$109$	−4.78106	−0.457942	−0.228971	−	0.973433i	$-0.573536\pi$
$109$	−4.78106	−0.457942	−0.228971	+	0.973433i	$0.573536\pi$
$110$	0	0
$111$	2.10607	0.199899
$112$	0	0
$113$	−16.6655	−1.56776	−0.783879	−	0.620914i	$-0.786762\pi$
$113$	−16.6655	−1.56776	−0.783879	+	0.620914i	$0.786762\pi$
$114$	0	0
$115$	−17.0077	−1.58598
$116$	0	0
$117$	6.06418	0.560633
$118$	0	0
$119$	19.1780	1.75804
$120$	0	0
$121$	−1.35235	−0.122941
$122$	0	0
$123$	−7.04189	−0.634946
$124$	0	0
$125$	−9.08647	−0.812718
$126$	0	0
$127$	−5.21894	−0.463106	−0.231553	−	0.972822i	$-0.574381\pi$
$127$	−5.21894	−0.463106	−0.231553	+	0.972822i	$0.574381\pi$
$128$	0	0
$129$	9.36959	0.824946
$130$	0	0
$131$	−8.65776	−0.756432	−0.378216	−	0.925717i	$-0.623462\pi$
$131$	−8.65776	−0.756432	−0.378216	+	0.925717i	$0.623462\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.53209	−0.217928
$136$	0	0
$137$	−13.6655	−1.16752	−0.583761	−	0.811925i	$-0.698420\pi$
$137$	−13.6655	−1.16752	−0.583761	+	0.811925i	$0.698420\pi$
$138$	0	0
$139$	−13.6800	−1.16033	−0.580163	−	0.814500i	$-0.697011\pi$
$139$	−13.6800	−1.16033	−0.580163	+	0.814500i	$0.697011\pi$
$140$	0	0
$141$	4.65270	0.391828
$142$	0	0
$143$	18.8357	1.57512
$144$	0	0
$145$	−18.0496	−1.49894
$146$	0	0
$147$	−3.41147	−0.281374
$148$	0	0
$149$	23.7442	1.94520	0.972601	−	0.232480i	$-0.0746839\pi$
$149$	23.7442	1.94520	0.972601	+	0.232480i	$0.0746839\pi$
$150$	0	0
$151$	−8.33544	−0.678328	−0.339164	−	0.940727i	$-0.610144\pi$
$151$	−8.33544	−0.678328	−0.339164	+	0.940727i	$0.610144\pi$
$152$	0	0
$153$	−5.94356	−0.480509
$154$	0	0
$155$	−19.3405	−1.55347
$156$	0	0
$157$	21.6955	1.73149	0.865746	−	0.500484i	$-0.166845\pi$
$157$	21.6955	1.73149	0.865746	+	0.500484i	$0.166845\pi$
$158$	0	0
$159$	−6.35504	−0.503987
$160$	0	0
$161$	21.6732	1.70809
$162$	0	0
$163$	−22.9855	−1.80036	−0.900180	−	0.435519i	$-0.856565\pi$
$163$	−22.9855	−1.80036	−0.900180	+	0.435519i	$0.856565\pi$
$164$	0	0
$165$	−7.86484	−0.612277
$166$	0	0
$167$	7.64590	0.591657	0.295829	−	0.955241i	$-0.404404\pi$
$167$	7.64590	0.591657	0.295829	+	0.955241i	$0.404404\pi$
$168$	0	0
$169$	23.7743	1.82879
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.7442	−0.892897	−0.446448	−	0.894809i	$-0.647311\pi$
$173$	−11.7442	−0.892897	−0.446448	+	0.894809i	$0.647311\pi$
$174$	0	0
$175$	−4.55438	−0.344279
$176$	0	0
$177$	−0.268571	−0.0201870
$178$	0	0
$179$	4.19759	0.313742	0.156871	−	0.987619i	$-0.449859\pi$
$179$	4.19759	0.313742	0.156871	+	0.987619i	$0.449859\pi$
$180$	0	0
$181$	−12.3277	−0.916310	−0.458155	−	0.888872i	$-0.651490\pi$
$181$	−12.3277	−0.916310	−0.458155	+	0.888872i	$0.651490\pi$
$182$	0	0
$183$	−1.58853	−0.117427
$184$	0	0
$185$	−5.33275	−0.392071
$186$	0	0
$187$	−18.4611	−1.35001
$188$	0	0
$189$	3.22668	0.234707
$190$	0	0
$191$	−10.2044	−0.738364	−0.369182	−	0.929357i	$-0.620362\pi$
$191$	−10.2044	−0.738364	−0.369182	+	0.929357i	$0.620362\pi$
$192$	0	0
$193$	21.7074	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$193$	21.7074	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$194$	0	0
$195$	−15.3550	−1.09960
$196$	0	0
$197$	−16.6851	−1.18876	−0.594382	−	0.804183i	$-0.702603\pi$
$197$	−16.6851	−1.18876	−0.594382	+	0.804183i	$0.702603\pi$
$198$	0	0
$199$	−2.46017	−0.174397	−0.0871984	−	0.996191i	$-0.527791\pi$
$199$	−2.46017	−0.174397	−0.0871984	+	0.996191i	$0.527791\pi$
$200$	0	0
$201$	−1.30541	−0.0920763
$202$	0	0
$203$	23.0009	1.61435
$204$	0	0
$205$	17.8307	1.24535
$206$	0	0
$207$	−6.71688	−0.466856
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.23173	0.0847961	0.0423980	−	0.999101i	$-0.486500\pi$
$211$	1.23173	0.0847961	0.0423980	+	0.999101i	$0.486500\pi$
$212$	0	0
$213$	−8.27631	−0.567084
$214$	0	0
$215$	−23.7246	−1.61801
$216$	0	0
$217$	24.6459	1.67307
$218$	0	0
$219$	−0.921274	−0.0622539
$220$	0	0
$221$	−36.0428	−2.42450
$222$	0	0
$223$	−6.57903	−0.440564	−0.220282	−	0.975436i	$-0.570698\pi$
$223$	−6.57903	−0.440564	−0.220282	+	0.975436i	$0.570698\pi$
$224$	0	0
$225$	1.41147	0.0940983
$226$	0	0
$227$	−9.72967	−0.645781	−0.322891	−	0.946436i	$-0.604655\pi$
$227$	−9.72967	−0.645781	−0.322891	+	0.946436i	$0.604655\pi$
$228$	0	0
$229$	−9.86753	−0.652064	−0.326032	−	0.945359i	$-0.605712\pi$
$229$	−9.86753	−0.652064	−0.326032	+	0.945359i	$0.605712\pi$
$230$	0	0
$231$	10.0223	0.659418
$232$	0	0
$233$	21.8161	1.42922	0.714611	−	0.699522i	$-0.246604\pi$
$233$	21.8161	1.42922	0.714611	+	0.699522i	$0.246604\pi$
$234$	0	0
$235$	−11.7811	−0.768512
$236$	0	0
$237$	−12.0642	−0.783653
$238$	0	0
$239$	21.3114	1.37852	0.689260	−	0.724514i	$-0.257936\pi$
$239$	21.3114	1.37852	0.689260	+	0.724514i	$0.257936\pi$
$240$	0	0
$241$	−15.6459	−1.00784	−0.503920	−	0.863750i	$-0.668110\pi$
$241$	−15.6459	−1.00784	−0.503920	+	0.863750i	$0.668110\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	8.63816	0.551872
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.0864665	−0.00547959
$250$	0	0
$251$	−10.1215	−0.638866	−0.319433	−	0.947609i	$-0.603493\pi$
$251$	−10.1215	−0.638866	−0.319433	+	0.947609i	$0.603493\pi$
$252$	0	0
$253$	−20.8631	−1.31165
$254$	0	0
$255$	15.0496	0.942444
$256$	0	0
$257$	4.77837	0.298067	0.149033	−	0.988832i	$-0.452384\pi$
$257$	4.77837	0.298067	0.149033	+	0.988832i	$0.452384\pi$
$258$	0	0
$259$	6.79561	0.422258
$260$	0	0
$261$	−7.12836	−0.441234
$262$	0	0
$263$	−21.6905	−1.33749	−0.668746	−	0.743491i	$-0.733169\pi$
$263$	−21.6905	−1.33749	−0.668746	+	0.743491i	$0.733169\pi$
$264$	0	0
$265$	16.0915	0.988494
$266$	0	0
$267$	−7.07192	−0.432794
$268$	0	0
$269$	3.23173	0.197042	0.0985212	−	0.995135i	$-0.468589\pi$
$269$	3.23173	0.197042	0.0985212	+	0.995135i	$0.468589\pi$
$270$	0	0
$271$	−11.7219	−0.712057	−0.356028	−	0.934475i	$-0.615869\pi$
$271$	−11.7219	−0.712057	−0.356028	+	0.934475i	$0.615869\pi$
$272$	0	0
$273$	19.5672	1.18426
$274$	0	0
$275$	4.38413	0.264373
$276$	0	0
$277$	8.77332	0.527138	0.263569	−	0.964641i	$-0.415100\pi$
$277$	8.77332	0.527138	0.263569	+	0.964641i	$0.415100\pi$
$278$	0	0
$279$	−7.63816	−0.457284
$280$	0	0
$281$	−25.4165	−1.51622	−0.758111	−	0.652125i	$-0.773878\pi$
$281$	−25.4165	−1.51622	−0.758111	+	0.652125i	$0.773878\pi$
$282$	0	0
$283$	−2.67230	−0.158852	−0.0794260	−	0.996841i	$-0.525309\pi$
$283$	−2.67230	−0.158852	−0.0794260	+	0.996841i	$0.525309\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−22.7219	−1.34123
$288$	0	0
$289$	18.3259	1.07800
$290$	0	0
$291$	3.32770	0.195073
$292$	0	0
$293$	−19.9537	−1.16571	−0.582853	−	0.812578i	$-0.698064\pi$
$293$	−19.9537	−1.16571	−0.582853	+	0.812578i	$0.698064\pi$
$294$	0	0
$295$	0.680045	0.0395937
$296$	0	0
$297$	−3.10607	−0.180232
$298$	0	0
$299$	−40.7324	−2.35561
$300$	0	0
$301$	30.2327	1.74258
$302$	0	0
$303$	10.7588	0.618075
$304$	0	0
$305$	4.02229	0.230316
$306$	0	0
$307$	23.9813	1.36869	0.684343	−	0.729160i	$-0.260089\pi$
$307$	23.9813	1.36869	0.684343	+	0.729160i	$0.260089\pi$
$308$	0	0
$309$	5.92902	0.337290
$310$	0	0
$311$	−0.162504	−0.00921475	−0.00460737	−	0.999989i	$-0.501467\pi$
$311$	−0.162504	−0.00921475	−0.00460737	+	0.999989i	$0.501467\pi$
$312$	0	0
$313$	6.72193	0.379946	0.189973	−	0.981789i	$-0.439160\pi$
$313$	6.72193	0.379946	0.189973	+	0.981789i	$0.439160\pi$
$314$	0	0
$315$	−8.17024	−0.460341
$316$	0	0
$317$	−14.2831	−0.802220	−0.401110	−	0.916030i	$-0.631375\pi$
$317$	−14.2831	−0.802220	−0.401110	+	0.916030i	$0.631375\pi$
$318$	0	0
$319$	−22.1411	−1.23967
$320$	0	0
$321$	2.58172	0.144097
$322$	0	0
$323$	0	0
$324$	0	0
$325$	8.55943	0.474792
$326$	0	0
$327$	4.78106	0.264393
$328$	0	0
$329$	15.0128	0.827682
$330$	0	0
$331$	33.3063	1.83068	0.915341	−	0.402680i	$-0.131921\pi$
$331$	33.3063	1.83068	0.915341	+	0.402680i	$0.131921\pi$
$332$	0	0
$333$	−2.10607	−0.115412
$334$	0	0
$335$	3.30541	0.180594
$336$	0	0
$337$	1.84936	0.100741	0.0503704	−	0.998731i	$-0.483960\pi$
$337$	1.84936	0.100741	0.0503704	+	0.998731i	$0.483960\pi$
$338$	0	0
$339$	16.6655	0.905146
$340$	0	0
$341$	−23.7246	−1.28476
$342$	0	0
$343$	11.5790	0.625209
$344$	0	0
$345$	17.0077	0.915666
$346$	0	0
$347$	−6.89899	−0.370357	−0.185178	−	0.982705i	$-0.559286\pi$
$347$	−6.89899	−0.370357	−0.185178	+	0.982705i	$0.559286\pi$
$348$	0	0
$349$	−11.2216	−0.600680	−0.300340	−	0.953832i	$-0.597100\pi$
$349$	−11.2216	−0.600680	−0.300340	+	0.953832i	$0.597100\pi$
$350$	0	0
$351$	−6.06418	−0.323682
$352$	0	0
$353$	−1.90848	−0.101578	−0.0507891	−	0.998709i	$-0.516174\pi$
$353$	−1.90848	−0.101578	−0.0507891	+	0.998709i	$0.516174\pi$
$354$	0	0
$355$	20.9564	1.11225
$356$	0	0
$357$	−19.1780	−1.01501
$358$	0	0
$359$	9.53033	0.502992	0.251496	−	0.967858i	$-0.419078\pi$
$359$	9.53033	0.502992	0.251496	+	0.967858i	$0.419078\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	1.35235	0.0709799
$364$	0	0
$365$	2.33275	0.122102
$366$	0	0
$367$	17.9222	0.935532	0.467766	−	0.883852i	$-0.345059\pi$
$367$	17.9222	0.935532	0.467766	+	0.883852i	$0.345059\pi$
$368$	0	0
$369$	7.04189	0.366586
$370$	0	0
$371$	−20.5057	−1.06460
$372$	0	0
$373$	30.4688	1.57762	0.788808	−	0.614639i	$-0.210698\pi$
$373$	30.4688	1.57762	0.788808	+	0.614639i	$0.210698\pi$
$374$	0	0
$375$	9.08647	0.469223
$376$	0	0
$377$	−43.2276	−2.22634
$378$	0	0
$379$	−6.57903	−0.337942	−0.168971	−	0.985621i	$-0.554044\pi$
$379$	−6.57903	−0.337942	−0.168971	+	0.985621i	$0.554044\pi$
$380$	0	0
$381$	5.21894	0.267374
$382$	0	0
$383$	−23.0077	−1.17564	−0.587820	−	0.808992i	$-0.700014\pi$
$383$	−23.0077	−1.17564	−0.587820	+	0.808992i	$0.700014\pi$
$384$	0	0
$385$	−25.3773	−1.29335
$386$	0	0
$387$	−9.36959	−0.476283
$388$	0	0
$389$	27.3054	1.38444	0.692220	−	0.721687i	$-0.256633\pi$
$389$	27.3054	1.38444	0.692220	+	0.721687i	$0.256633\pi$
$390$	0	0
$391$	39.9222	2.01895
$392$	0	0
$393$	8.65776	0.436726
$394$	0	0
$395$	30.5476	1.53702
$396$	0	0
$397$	−5.88207	−0.295213	−0.147606	−	0.989046i	$-0.547157\pi$
$397$	−5.88207	−0.295213	−0.147606	+	0.989046i	$0.547157\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.28850	0.214157	0.107079	−	0.994251i	$-0.465850\pi$
$401$	4.28850	0.214157	0.107079	+	0.994251i	$0.465850\pi$
$402$	0	0
$403$	−46.3191	−2.30732
$404$	0	0
$405$	2.53209	0.125821
$406$	0	0
$407$	−6.54158	−0.324254
$408$	0	0
$409$	32.4270	1.60341	0.801705	−	0.597720i	$-0.203927\pi$
$409$	32.4270	1.60341	0.801705	+	0.597720i	$0.203927\pi$
$410$	0	0
$411$	13.6655	0.674069
$412$	0	0
$413$	−0.866592	−0.0426422
$414$	0	0
$415$	0.218941	0.0107474
$416$	0	0
$417$	13.6800	0.669915
$418$	0	0
$419$	9.86577	0.481974	0.240987	−	0.970528i	$-0.422529\pi$
$419$	9.86577	0.481974	0.240987	+	0.970528i	$0.422529\pi$
$420$	0	0
$421$	−22.1070	−1.07743	−0.538715	−	0.842488i	$-0.681090\pi$
$421$	−22.1070	−1.07743	−0.538715	+	0.842488i	$0.681090\pi$
$422$	0	0
$423$	−4.65270	−0.226222
$424$	0	0
$425$	−8.38919	−0.406935
$426$	0	0
$427$	−5.12567	−0.248048
$428$	0	0
$429$	−18.8357	−0.909398
$430$	0	0
$431$	25.3209	1.21966	0.609832	−	0.792531i	$-0.291237\pi$
$431$	25.3209	1.21966	0.609832	+	0.792531i	$0.291237\pi$
$432$	0	0
$433$	−12.8152	−0.615860	−0.307930	−	0.951409i	$-0.599636\pi$
$433$	−12.8152	−0.615860	−0.307930	+	0.951409i	$0.599636\pi$
$434$	0	0
$435$	18.0496	0.865414
$436$	0	0
$437$	0	0
$438$	0	0
$439$	5.22399	0.249328	0.124664	−	0.992199i	$-0.460215\pi$
$439$	5.22399	0.249328	0.124664	+	0.992199i	$0.460215\pi$
$440$	0	0
$441$	3.41147	0.162451
$442$	0	0
$443$	−17.3996	−0.826681	−0.413340	−	0.910577i	$-0.635638\pi$
$443$	−17.3996	−0.826681	−0.413340	+	0.910577i	$0.635638\pi$
$444$	0	0
$445$	17.9067	0.848860
$446$	0	0
$447$	−23.7442	−1.12306
$448$	0	0
$449$	−23.6509	−1.11616	−0.558079	−	0.829788i	$-0.688461\pi$
$449$	−23.6509	−1.11616	−0.558079	+	0.829788i	$0.688461\pi$
$450$	0	0
$451$	21.8726	1.02994
$452$	0	0
$453$	8.33544	0.391633
$454$	0	0
$455$	−49.5458	−2.32274
$456$	0	0
$457$	−24.2695	−1.13528	−0.567640	−	0.823277i	$-0.692143\pi$
$457$	−24.2695	−1.13528	−0.567640	+	0.823277i	$0.692143\pi$
$458$	0	0
$459$	5.94356	0.277422
$460$	0	0
$461$	6.97502	0.324859	0.162430	−	0.986720i	$-0.448067\pi$
$461$	6.97502	0.324859	0.162430	+	0.986720i	$0.448067\pi$
$462$	0	0
$463$	28.0770	1.30485	0.652424	−	0.757854i	$-0.273752\pi$
$463$	28.0770	1.30485	0.652424	+	0.757854i	$0.273752\pi$
$464$	0	0
$465$	19.3405	0.896894
$466$	0	0
$467$	3.82295	0.176905	0.0884525	−	0.996080i	$-0.471808\pi$
$467$	3.82295	0.176905	0.0884525	+	0.996080i	$0.471808\pi$
$468$	0	0
$469$	−4.21213	−0.194498
$470$	0	0
$471$	−21.6955	−0.999677
$472$	0	0
$473$	−29.1026	−1.33814
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.35504	0.290977
$478$	0	0
$479$	−30.9394	−1.41366	−0.706830	−	0.707384i	$-0.749875\pi$
$479$	−30.9394	−1.41366	−0.706830	+	0.707384i	$0.749875\pi$
$480$	0	0
$481$	−12.7716	−0.582333
$482$	0	0
$483$	−21.6732	−0.986166
$484$	0	0
$485$	−8.42602	−0.382606
$486$	0	0
$487$	2.86484	0.129818	0.0649091	−	0.997891i	$-0.479324\pi$
$487$	2.86484	0.129818	0.0649091	+	0.997891i	$0.479324\pi$
$488$	0	0
$489$	22.9855	1.03944
$490$	0	0
$491$	−41.7502	−1.88416	−0.942080	−	0.335387i	$-0.891133\pi$
$491$	−41.7502	−1.88416	−0.942080	+	0.335387i	$0.891133\pi$
$492$	0	0
$493$	42.3678	1.90815
$494$	0	0
$495$	7.86484	0.353498
$496$	0	0
$497$	−26.7050	−1.19788
$498$	0	0
$499$	9.49020	0.424840	0.212420	−	0.977178i	$-0.431866\pi$
$499$	9.49020	0.424840	0.212420	+	0.977178i	$0.431866\pi$
$500$	0	0
$501$	−7.64590	−0.341593
$502$	0	0
$503$	−12.6408	−0.563627	−0.281814	−	0.959469i	$-0.590936\pi$
$503$	−12.6408	−0.563627	−0.281814	+	0.959469i	$0.590936\pi$
$504$	0	0
$505$	−27.2422	−1.21226
$506$	0	0
$507$	−23.7743	−1.05585
$508$	0	0
$509$	19.4962	0.864153	0.432077	−	0.901837i	$-0.357781\pi$
$509$	19.4962	0.864153	0.432077	+	0.901837i	$0.357781\pi$
$510$	0	0
$511$	−2.97266	−0.131503
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.0128	−0.661543
$516$	0	0
$517$	−14.4516	−0.635581
$518$	0	0
$519$	11.7442	0.515514
$520$	0	0
$521$	−20.2831	−0.888620	−0.444310	−	0.895873i	$-0.646551\pi$
$521$	−20.2831	−0.888620	−0.444310	+	0.895873i	$0.646551\pi$
$522$	0	0
$523$	−15.2668	−0.667571	−0.333786	−	0.942649i	$-0.608326\pi$
$523$	−15.2668	−0.667571	−0.333786	+	0.942649i	$0.608326\pi$
$524$	0	0
$525$	4.55438	0.198769
$526$	0	0
$527$	45.3979	1.97756
$528$	0	0
$529$	22.1165	0.961587
$530$	0	0
$531$	0.268571	0.0116550
$532$	0	0
$533$	42.7033	1.84968
$534$	0	0
$535$	−6.53714	−0.282625
$536$	0	0
$537$	−4.19759	−0.181139
$538$	0	0
$539$	10.5963	0.456414
$540$	0	0
$541$	−20.6159	−0.886345	−0.443173	−	0.896436i	$-0.646147\pi$
$541$	−20.6159	−0.886345	−0.443173	+	0.896436i	$0.646147\pi$
$542$	0	0
$543$	12.3277	0.529032
$544$	0	0
$545$	−12.1061	−0.518567
$546$	0	0
$547$	28.4115	1.21479	0.607393	−	0.794401i	$-0.292215\pi$
$547$	28.4115	1.21479	0.607393	+	0.794401i	$0.292215\pi$
$548$	0	0
$549$	1.58853	0.0677966
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−38.9273	−1.65536
$554$	0	0
$555$	5.33275	0.226363
$556$	0	0
$557$	−37.8188	−1.60244	−0.801218	−	0.598373i	$-0.795814\pi$
$557$	−37.8188	−1.60244	−0.801218	+	0.598373i	$0.795814\pi$
$558$	0	0
$559$	−56.8188	−2.40318
$560$	0	0
$561$	18.4611	0.779428
$562$	0	0
$563$	−9.64084	−0.406313	−0.203157	−	0.979146i	$-0.565120\pi$
$563$	−9.64084	−0.406313	−0.203157	+	0.979146i	$0.565120\pi$
$564$	0	0
$565$	−42.1985	−1.77531
$566$	0	0
$567$	−3.22668	−0.135508
$568$	0	0
$569$	−21.1129	−0.885098	−0.442549	−	0.896744i	$-0.645926\pi$
$569$	−21.1129	−0.885098	−0.442549	+	0.896744i	$0.645926\pi$
$570$	0	0
$571$	−2.11112	−0.0883476	−0.0441738	−	0.999024i	$-0.514066\pi$
$571$	−2.11112	−0.0883476	−0.0441738	+	0.999024i	$0.514066\pi$
$572$	0	0
$573$	10.2044	0.426295
$574$	0	0
$575$	−9.48070	−0.395373
$576$	0	0
$577$	23.1429	0.963452	0.481726	−	0.876322i	$-0.340010\pi$
$577$	23.1429	0.963452	0.481726	+	0.876322i	$0.340010\pi$
$578$	0	0
$579$	−21.7074	−0.902128
$580$	0	0
$581$	−0.279000	−0.0115749
$582$	0	0
$583$	19.7392	0.817513
$584$	0	0
$585$	15.3550	0.634853
$586$	0	0
$587$	−19.1607	−0.790849	−0.395424	−	0.918499i	$-0.629402\pi$
$587$	−19.1607	−0.790849	−0.395424	+	0.918499i	$0.629402\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	16.6851	0.686333
$592$	0	0
$593$	37.2012	1.52767	0.763835	−	0.645411i	$-0.223314\pi$
$593$	37.2012	1.52767	0.763835	+	0.645411i	$0.223314\pi$
$594$	0	0
$595$	48.5604	1.99078
$596$	0	0
$597$	2.46017	0.100688
$598$	0	0
$599$	−30.9050	−1.26274	−0.631371	−	0.775481i	$-0.717508\pi$
$599$	−30.9050	−1.26274	−0.631371	+	0.775481i	$0.717508\pi$
$600$	0	0
$601$	43.5604	1.77686	0.888432	−	0.459008i	$-0.151795\pi$
$601$	43.5604	1.77686	0.888432	+	0.459008i	$0.151795\pi$
$602$	0	0
$603$	1.30541	0.0531603
$604$	0	0
$605$	−3.42427	−0.139216
$606$	0	0
$607$	−20.9659	−0.850978	−0.425489	−	0.904964i	$-0.639898\pi$
$607$	−20.9659	−0.850978	−0.425489	+	0.904964i	$0.639898\pi$
$608$	0	0
$609$	−23.0009	−0.932045
$610$	0	0
$611$	−28.2148	−1.14145
$612$	0	0
$613$	6.30447	0.254635	0.127318	−	0.991862i	$-0.459363\pi$
$613$	6.30447	0.254635	0.127318	+	0.991862i	$0.459363\pi$
$614$	0	0
$615$	−17.8307	−0.719003
$616$	0	0
$617$	−38.4424	−1.54763	−0.773817	−	0.633409i	$-0.781655\pi$
$617$	−38.4424	−1.54763	−0.773817	+	0.633409i	$0.781655\pi$
$618$	0	0
$619$	−18.6382	−0.749131	−0.374565	−	0.927201i	$-0.622208\pi$
$619$	−18.6382	−0.749131	−0.374565	+	0.927201i	$0.622208\pi$
$620$	0	0
$621$	6.71688	0.269539
$622$	0	0
$623$	−22.8188	−0.914217
$624$	0	0
$625$	−30.0651	−1.20260
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.5175	0.499107
$630$	0	0
$631$	15.4570	0.615333	0.307666	−	0.951494i	$-0.400452\pi$
$631$	15.4570	0.615333	0.307666	+	0.951494i	$0.400452\pi$
$632$	0	0
$633$	−1.23173	−0.0489570
$634$	0	0
$635$	−13.2148	−0.524414
$636$	0	0
$637$	20.6878	0.819680
$638$	0	0
$639$	8.27631	0.327406
$640$	0	0
$641$	25.0155	0.988052	0.494026	−	0.869447i	$-0.335525\pi$
$641$	25.0155	0.988052	0.494026	+	0.869447i	$0.335525\pi$
$642$	0	0
$643$	22.8239	0.900086	0.450043	−	0.893007i	$-0.351409\pi$
$643$	22.8239	0.900086	0.450043	+	0.893007i	$0.351409\pi$
$644$	0	0
$645$	23.7246	0.934156
$646$	0	0
$647$	46.1252	1.81337	0.906684	−	0.421811i	$-0.138605\pi$
$647$	46.1252	1.81337	0.906684	+	0.421811i	$0.138605\pi$
$648$	0	0
$649$	0.834198	0.0327452
$650$	0	0
$651$	−24.6459	−0.965949
$652$	0	0
$653$	−14.6774	−0.574369	−0.287185	−	0.957875i	$-0.592719\pi$
$653$	−14.6774	−0.574369	−0.287185	+	0.957875i	$0.592719\pi$
$654$	0	0
$655$	−21.9222	−0.856572
$656$	0	0
$657$	0.921274	0.0359423
$658$	0	0
$659$	21.5175	0.838204	0.419102	−	0.907939i	$-0.362345\pi$
$659$	21.5175	0.838204	0.419102	+	0.907939i	$0.362345\pi$
$660$	0	0
$661$	31.3942	1.22109	0.610547	−	0.791980i	$-0.290950\pi$
$661$	31.3942	1.22109	0.610547	+	0.791980i	$0.290950\pi$
$662$	0	0
$663$	36.0428	1.39979
$664$	0	0
$665$	0	0
$666$	0	0
$667$	47.8803	1.85393
$668$	0	0
$669$	6.57903	0.254360
$670$	0	0
$671$	4.93407	0.190478
$672$	0	0
$673$	−11.6732	−0.449970	−0.224985	−	0.974362i	$-0.572233\pi$
$673$	−11.6732	−0.449970	−0.224985	+	0.974362i	$0.572233\pi$
$674$	0	0
$675$	−1.41147	−0.0543277
$676$	0	0
$677$	−41.3732	−1.59010	−0.795051	−	0.606543i	$-0.792556\pi$
$677$	−41.3732	−1.59010	−0.795051	+	0.606543i	$0.792556\pi$
$678$	0	0
$679$	10.7374	0.412064
$680$	0	0
$681$	9.72967	0.372842
$682$	0	0
$683$	29.5117	1.12923	0.564616	−	0.825354i	$-0.309024\pi$
$683$	29.5117	1.12923	0.564616	+	0.825354i	$0.309024\pi$
$684$	0	0
$685$	−34.6023	−1.32208
$686$	0	0
$687$	9.86753	0.376470
$688$	0	0
$689$	38.5381	1.46818
$690$	0	0
$691$	−4.47834	−0.170364	−0.0851820	−	0.996365i	$-0.527147\pi$
$691$	−4.47834	−0.170364	−0.0851820	+	0.996365i	$0.527147\pi$
$692$	0	0
$693$	−10.0223	−0.380715
$694$	0	0
$695$	−34.6391	−1.31394
$696$	0	0
$697$	−41.8539	−1.58533
$698$	0	0
$699$	−21.8161	−0.825162
$700$	0	0
$701$	5.91353	0.223351	0.111676	−	0.993745i	$-0.464378\pi$
$701$	5.91353	0.223351	0.111676	+	0.993745i	$0.464378\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	11.7811	0.443700
$706$	0	0
$707$	34.7151	1.30560
$708$	0	0
$709$	29.9358	1.12426	0.562132	−	0.827048i	$-0.309981\pi$
$709$	29.9358	1.12426	0.562132	+	0.827048i	$0.309981\pi$
$710$	0	0
$711$	12.0642	0.452442
$712$	0	0
$713$	51.3046	1.92137
$714$	0	0
$715$	47.6938	1.78365
$716$	0	0
$717$	−21.3114	−0.795889
$718$	0	0
$719$	47.7811	1.78193	0.890966	−	0.454069i	$-0.150028\pi$
$719$	47.7811	1.78193	0.890966	+	0.454069i	$0.150028\pi$
$720$	0	0
$721$	19.1310	0.712477
$722$	0	0
$723$	15.6459	0.581877
$724$	0	0
$725$	−10.0615	−0.373674
$726$	0	0
$727$	−41.8949	−1.55379	−0.776897	−	0.629627i	$-0.783208\pi$
$727$	−41.8949	−1.55379	−0.776897	+	0.629627i	$0.783208\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	55.6887	2.05972
$732$	0	0
$733$	21.6135	0.798313	0.399156	−	0.916883i	$-0.369303\pi$
$733$	21.6135	0.798313	0.399156	+	0.916883i	$0.369303\pi$
$734$	0	0
$735$	−8.63816	−0.318623
$736$	0	0
$737$	4.05468	0.149356
$738$	0	0
$739$	32.4243	1.19275	0.596373	−	0.802707i	$-0.296608\pi$
$739$	32.4243	1.19275	0.596373	+	0.802707i	$0.296608\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.8976	0.986776	0.493388	−	0.869809i	$-0.335758\pi$
$743$	26.8976	0.986776	0.493388	+	0.869809i	$0.335758\pi$
$744$	0	0
$745$	60.1225	2.20272
$746$	0	0
$747$	0.0864665	0.00316364
$748$	0	0
$749$	8.33038	0.304386
$750$	0	0
$751$	5.72638	0.208958	0.104479	−	0.994527i	$-0.466682\pi$
$751$	5.72638	0.208958	0.104479	+	0.994527i	$0.466682\pi$
$752$	0	0
$753$	10.1215	0.368850
$754$	0	0
$755$	−21.1061	−0.768128
$756$	0	0
$757$	8.65364	0.314522	0.157261	−	0.987557i	$-0.449734\pi$
$757$	8.65364	0.314522	0.157261	+	0.987557i	$0.449734\pi$
$758$	0	0
$759$	20.8631	0.757282
$760$	0	0
$761$	−37.7279	−1.36764	−0.683818	−	0.729653i	$-0.739682\pi$
$761$	−37.7279	−1.36764	−0.683818	+	0.729653i	$0.739682\pi$
$762$	0	0
$763$	15.4270	0.558493
$764$	0	0
$765$	−15.0496	−0.544121
$766$	0	0
$767$	1.62866	0.0588075
$768$	0	0
$769$	36.7912	1.32672	0.663362	−	0.748299i	$-0.269129\pi$
$769$	36.7912	1.32672	0.663362	+	0.748299i	$0.269129\pi$
$770$	0	0
$771$	−4.77837	−0.172089
$772$	0	0
$773$	−19.9685	−0.718218	−0.359109	−	0.933296i	$-0.616919\pi$
$773$	−19.9685	−0.718218	−0.359109	+	0.933296i	$0.616919\pi$
$774$	0	0
$775$	−10.7811	−0.387267
$776$	0	0
$777$	−6.79561	−0.243791
$778$	0	0
$779$	0	0
$780$	0	0
$781$	25.7068	0.919861
$782$	0	0
$783$	7.12836	0.254747
$784$	0	0
$785$	54.9350	1.96071
$786$	0	0
$787$	−8.36783	−0.298281	−0.149140	−	0.988816i	$-0.547651\pi$
$787$	−8.36783	−0.298281	−0.149140	+	0.988816i	$0.547651\pi$
$788$	0	0
$789$	21.6905	0.772201
$790$	0	0
$791$	53.7743	1.91199
$792$	0	0
$793$	9.63310	0.342082
$794$	0	0
$795$	−16.0915	−0.570707
$796$	0	0
$797$	42.1729	1.49384	0.746921	−	0.664913i	$-0.231531\pi$
$797$	42.1729	1.49384	0.746921	+	0.664913i	$0.231531\pi$
$798$	0	0
$799$	27.6536	0.978315
$800$	0	0
$801$	7.07192	0.249874
$802$	0	0
$803$	2.86154	0.100982
$804$	0	0
$805$	54.8786	1.93422
$806$	0	0
$807$	−3.23173	−0.113762
$808$	0	0
$809$	−6.06242	−0.213143	−0.106572	−	0.994305i	$-0.533987\pi$
$809$	−6.06242	−0.213143	−0.106572	+	0.994305i	$0.533987\pi$
$810$	0	0
$811$	−25.6938	−0.902230	−0.451115	−	0.892466i	$-0.648974\pi$
$811$	−25.6938	−0.902230	−0.451115	+	0.892466i	$0.648974\pi$
$812$	0	0
$813$	11.7219	0.411106
$814$	0	0
$815$	−58.2012	−2.03870
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−19.5672	−0.683732
$820$	0	0
$821$	−0.251334	−0.00877163	−0.00438582	−	0.999990i	$-0.501396\pi$
$821$	−0.251334	−0.00877163	−0.00438582	+	0.999990i	$0.501396\pi$
$822$	0	0
$823$	16.3259	0.569087	0.284543	−	0.958663i	$-0.408158\pi$
$823$	16.3259	0.569087	0.284543	+	0.958663i	$0.408158\pi$
$824$	0	0
$825$	−4.38413	−0.152636
$826$	0	0
$827$	−26.4989	−0.921456	−0.460728	−	0.887541i	$-0.652412\pi$
$827$	−26.4989	−0.921456	−0.460728	+	0.887541i	$0.652412\pi$
$828$	0	0
$829$	−41.6296	−1.44586	−0.722928	−	0.690924i	$-0.757204\pi$
$829$	−41.6296	−1.44586	−0.722928	+	0.690924i	$0.757204\pi$
$830$	0	0
$831$	−8.77332	−0.304343
$832$	0	0
$833$	−20.2763	−0.702533
$834$	0	0
$835$	19.3601	0.669984
$836$	0	0
$837$	7.63816	0.264013
$838$	0	0
$839$	−28.4074	−0.980731	−0.490365	−	0.871517i	$-0.663137\pi$
$839$	−28.4074	−0.980731	−0.490365	+	0.871517i	$0.663137\pi$
$840$	0	0
$841$	21.8135	0.752188
$842$	0	0
$843$	25.4165	0.875392
$844$	0	0
$845$	60.1985	2.07089
$846$	0	0
$847$	4.36360	0.149935
$848$	0	0
$849$	2.67230	0.0917132
$850$	0	0
$851$	14.1462	0.484926
$852$	0	0
$853$	−27.1935	−0.931087	−0.465543	−	0.885025i	$-0.654141\pi$
$853$	−27.1935	−0.931087	−0.465543	+	0.885025i	$0.654141\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.9691	−1.43364	−0.716819	−	0.697259i	$-0.754403\pi$
$857$	−41.9691	−1.43364	−0.716819	+	0.697259i	$0.754403\pi$
$858$	0	0
$859$	35.3138	1.20489	0.602445	−	0.798160i	$-0.294193\pi$
$859$	35.3138	1.20489	0.602445	+	0.798160i	$0.294193\pi$
$860$	0	0
$861$	22.7219	0.774361
$862$	0	0
$863$	−12.0104	−0.408840	−0.204420	−	0.978883i	$-0.565531\pi$
$863$	−12.0104	−0.408840	−0.204420	+	0.978883i	$0.565531\pi$
$864$	0	0
$865$	−29.7374	−1.01110
$866$	0	0
$867$	−18.3259	−0.622382
$868$	0	0
$869$	37.4721	1.27116
$870$	0	0
$871$	7.91622	0.268231
$872$	0	0
$873$	−3.32770	−0.112625
$874$	0	0
$875$	29.3191	0.991168
$876$	0	0
$877$	18.0333	0.608942	0.304471	−	0.952522i	$-0.401520\pi$
$877$	18.0333	0.608942	0.304471	+	0.952522i	$0.401520\pi$
$878$	0	0
$879$	19.9537	0.673021
$880$	0	0
$881$	23.8203	0.802525	0.401262	−	0.915963i	$-0.368572\pi$
$881$	23.8203	0.802525	0.401262	+	0.915963i	$0.368572\pi$
$882$	0	0
$883$	11.8334	0.398225	0.199112	−	0.979977i	$-0.436194\pi$
$883$	11.8334	0.398225	0.199112	+	0.979977i	$0.436194\pi$
$884$	0	0
$885$	−0.680045	−0.0228595
$886$	0	0
$887$	36.6996	1.23225	0.616127	−	0.787647i	$-0.288701\pi$
$887$	36.6996	1.23225	0.616127	+	0.787647i	$0.288701\pi$
$888$	0	0
$889$	16.8399	0.564791
$890$	0	0
$891$	3.10607	0.104057
$892$	0	0
$893$	0	0
$894$	0	0
$895$	10.6287	0.355277
$896$	0	0
$897$	40.7324	1.36001
$898$	0	0
$899$	54.4475	1.81593
$900$	0	0
$901$	−37.7716	−1.25835
$902$	0	0
$903$	−30.2327	−1.00608
$904$	0	0
$905$	−31.2148	−1.03762
$906$	0	0
$907$	29.2080	0.969836	0.484918	−	0.874560i	$-0.338849\pi$
$907$	29.2080	0.969836	0.484918	+	0.874560i	$0.338849\pi$
$908$	0	0
$909$	−10.7588	−0.356846
$910$	0	0
$911$	−3.12567	−0.103558	−0.0517790	−	0.998659i	$-0.516489\pi$
$911$	−3.12567	−0.103558	−0.0517790	+	0.998659i	$0.516489\pi$
$912$	0	0
$913$	0.268571	0.00888839
$914$	0	0
$915$	−4.02229	−0.132973
$916$	0	0
$917$	27.9358	0.922522
$918$	0	0
$919$	−8.78644	−0.289838	−0.144919	−	0.989444i	$-0.546292\pi$
$919$	−8.78644	−0.289838	−0.144919	+	0.989444i	$0.546292\pi$
$920$	0	0
$921$	−23.9813	−0.790212
$922$	0	0
$923$	50.1890	1.65199
$924$	0	0
$925$	−2.97266	−0.0977404
$926$	0	0
$927$	−5.92902	−0.194734
$928$	0	0
$929$	25.1548	0.825301	0.412651	−	0.910889i	$-0.364603\pi$
$929$	25.1548	0.825301	0.412651	+	0.910889i	$0.364603\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0.162504	0.00532014
$934$	0	0
$935$	−46.7452	−1.52873
$936$	0	0
$937$	−26.4730	−0.864834	−0.432417	−	0.901674i	$-0.642339\pi$
$937$	−26.4730	−0.864834	−0.432417	+	0.901674i	$0.642339\pi$
$938$	0	0
$939$	−6.72193	−0.219362
$940$	0	0
$941$	−5.26950	−0.171781	−0.0858905	−	0.996305i	$-0.527374\pi$
$941$	−5.26950	−0.171781	−0.0858905	+	0.996305i	$0.527374\pi$
$942$	0	0
$943$	−47.2995	−1.54028
$944$	0	0
$945$	8.17024	0.265778
$946$	0	0
$947$	24.3783	0.792187	0.396093	−	0.918210i	$-0.370366\pi$
$947$	24.3783	0.792187	0.396093	+	0.918210i	$0.370366\pi$
$948$	0	0
$949$	5.58677	0.181354
$950$	0	0
$951$	14.2831	0.463162
$952$	0	0
$953$	−49.6195	−1.60733	−0.803666	−	0.595080i	$-0.797120\pi$
$953$	−49.6195	−1.60733	−0.803666	+	0.595080i	$0.797120\pi$
$954$	0	0
$955$	−25.8384	−0.836112
$956$	0	0
$957$	22.1411	0.715722
$958$	0	0
$959$	44.0942	1.42388
$960$	0	0
$961$	27.3414	0.881981
$962$	0	0
$963$	−2.58172	−0.0831947
$964$	0	0
$965$	54.9650	1.76939
$966$	0	0
$967$	−25.8922	−0.832636	−0.416318	−	0.909219i	$-0.636680\pi$
$967$	−25.8922	−0.832636	−0.416318	+	0.909219i	$0.636680\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.6260	−0.405187	−0.202593	−	0.979263i	$-0.564937\pi$
$971$	−12.6260	−0.405187	−0.202593	+	0.979263i	$0.564937\pi$
$972$	0	0
$973$	44.1411	1.41510
$974$	0	0
$975$	−8.55943	−0.274121
$976$	0	0
$977$	−3.38144	−0.108182	−0.0540910	−	0.998536i	$-0.517226\pi$
$977$	−3.38144	−0.108182	−0.0540910	+	0.998536i	$0.517226\pi$
$978$	0	0
$979$	21.9659	0.702032
$980$	0	0
$981$	−4.78106	−0.152647
$982$	0	0
$983$	0.222563	0.00709865	0.00354933	−	0.999994i	$-0.498870\pi$
$983$	0.222563	0.00709865	0.00354933	+	0.999994i	$0.498870\pi$
$984$	0	0
$985$	−42.2481	−1.34614
$986$	0	0
$987$	−15.0128	−0.477862
$988$	0	0
$989$	62.9344	2.00120
$990$	0	0
$991$	4.82893	0.153396	0.0766981	−	0.997054i	$-0.475562\pi$
$991$	4.82893	0.153396	0.0766981	+	0.997054i	$0.475562\pi$
$992$	0	0
$993$	−33.3063	−1.05694
$994$	0	0
$995$	−6.22937	−0.197484
$996$	0	0
$997$	−12.6418	−0.400369	−0.200185	−	0.979758i	$-0.564154\pi$
$997$	−12.6418	−0.400369	−0.200185	+	0.979758i	$0.564154\pi$
$998$	0	0
$999$	2.10607	0.0666330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4332.2.a.n.1.3		3
19.2	odd	18		228.2.q.a.61.1	✓	6
19.10	odd	18		228.2.q.a.157.1	yes	6
19.18	odd	2		4332.2.a.o.1.3		3
57.2	even	18		684.2.bo.a.289.1		6
57.29	even	18		684.2.bo.a.613.1		6
76.59	even	18		912.2.bo.e.289.1		6
76.67	even	18		912.2.bo.e.385.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.q.a.61.1	✓	6	19.2	odd	18
228.2.q.a.157.1	yes	6	19.10	odd	18
684.2.bo.a.289.1		6	57.2	even	18
684.2.bo.a.613.1		6	57.29	even	18
912.2.bo.e.289.1		6	76.59	even	18
912.2.bo.e.385.1		6	76.67	even	18
4332.2.a.n.1.3		3	1.1	even	1	trivial
4332.2.a.o.1.3		3	19.18	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:	\( N \)	\(=\)	\( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4332.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.53209 q^{5} -3.22668 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.53209 q^{5} -3.22668 q^{7} +1.00000 q^{9} +3.10607 q^{11} +6.06418 q^{13} -2.53209 q^{15} -5.94356 q^{17} +3.22668 q^{21} -6.71688 q^{23} +1.41147 q^{25} -1.00000 q^{27} -7.12836 q^{29} -7.63816 q^{31} -3.10607 q^{33} -8.17024 q^{35} -2.10607 q^{37} -6.06418 q^{39} +7.04189 q^{41} -9.36959 q^{43} +2.53209 q^{45} -4.65270 q^{47} +3.41147 q^{49} +5.94356 q^{51} +6.35504 q^{53} +7.86484 q^{55} +0.268571 q^{59} +1.58853 q^{61} -3.22668 q^{63} +15.3550 q^{65} +1.30541 q^{67} +6.71688 q^{69} +8.27631 q^{71} +0.921274 q^{73} -1.41147 q^{75} -10.0223 q^{77} +12.0642 q^{79} +1.00000 q^{81} +0.0864665 q^{83} -15.0496 q^{85} +7.12836 q^{87} +7.07192 q^{89} -19.5672 q^{91} +7.63816 q^{93} -3.32770 q^{97} +3.10607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 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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