LMFDB - Embedded newform 3364.2.a.p.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3364,2,Mod(1,3364)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3364.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3364, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3364 = 2^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3364.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,5,0,10,0,7,0,11,0,9,0,14,0,10,0,5,0,9,0,-19,0,15,0,16,0, 20,0,0,0,-11,0,22,0,10,0,-16,0,22,0,1,0,15,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(45)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.8616752400$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6456289.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 3x^{3} + 40x^{2} + 6x - 29 $ x^6 - x^5 - 12x^4 + 3x^3 + 40x^2 + 6x - 29
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.08697$ of defining polynomial
Character	$\chi$	$=$	3364.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.08697 q^{3} +3.51360 q^{5} -1.08697 q^{7} +6.52935 q^{9} -1.84938 q^{11} +6.57945 q^{13} +10.8464 q^{15} -0.658856 q^{17} -3.32428 q^{19} -3.35542 q^{21} +3.89074 q^{23} +7.34540 q^{25} +10.8950 q^{27} -3.00066 q^{31} -5.70898 q^{33} -3.81916 q^{35} -6.08844 q^{37} +20.3105 q^{39} -5.19016 q^{41} +1.17695 q^{43} +22.9415 q^{45} -6.21164 q^{47} -5.81851 q^{49} -2.03387 q^{51} -1.40581 q^{53} -6.49799 q^{55} -10.2619 q^{57} -9.55267 q^{59} +8.30080 q^{61} -7.09718 q^{63} +23.1176 q^{65} -12.3977 q^{67} +12.0106 q^{69} +8.12645 q^{71} +8.61593 q^{73} +22.6750 q^{75} +2.01021 q^{77} -5.60250 q^{79} +14.0444 q^{81} -3.72172 q^{83} -2.31496 q^{85} -4.47002 q^{89} -7.15163 q^{91} -9.26292 q^{93} -11.6802 q^{95} -8.00445 q^{97} -12.0753 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} + 10 q^{5} + 7 q^{7} + 11 q^{9} + 9 q^{11} + 14 q^{13} + 10 q^{15} + 5 q^{17} + 9 q^{19} - 19 q^{21} + 15 q^{23} + 16 q^{25} + 20 q^{27} - 11 q^{31} + 22 q^{33} + 10 q^{35} - 16 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 + 10 * q^5 + 7 * q^7 + 11 * q^9 + 9 * q^11 + 14 * q^13 + 10 * q^15 + 5 * q^17 + 9 * q^19 - 19 * q^21 + 15 * q^23 + 16 * q^25 + 20 * q^27 - 11 * q^31 + 22 * q^33 + 10 * q^35 - 16 * q^37 + 22 * q^39 + q^41 + 15 * q^43 + 32 * q^45 - 27 * q^47 - 9 * q^49 - 20 * q^51 + 18 * q^53 + 14 * q^55 + 15 * q^57 + 18 * q^59 - 22 * q^61 - 13 * q^63 + 26 * q^65 + 9 * q^67 - 6 * q^69 - 7 * q^71 + 32 * q^73 + 22 * q^75 - 4 * q^77 - 29 * q^79 - 2 * q^81 + 3 * q^83 + 24 * q^85 - 12 * q^89 + 6 * q^91 + 14 * q^93 - 28 * q^95 + 58 * q^97 + 8 * q^99

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
$3$	3.08697	1.78226	0.891130	−	0.453748i	$-0.149913\pi$
$3$	3.08697	1.78226	0.891130	+	0.453748i	$0.149913\pi$
$4$	0	0
$5$	3.51360	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$5$	3.51360	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$6$	0	0
$7$	−1.08697	−0.410834	−0.205417	−	0.978675i	$-0.565855\pi$
$7$	−1.08697	−0.410834	−0.205417	+	0.978675i	$0.565855\pi$
$8$	0	0
$9$	6.52935	2.17645
$10$	0	0
$11$	−1.84938	−0.557610	−0.278805	−	0.960348i	$-0.589938\pi$
$11$	−1.84938	−0.557610	−0.278805	+	0.960348i	$0.589938\pi$
$12$	0	0
$13$	6.57945	1.82481	0.912406	−	0.409287i	$-0.134222\pi$
$13$	6.57945	1.82481	0.912406	+	0.409287i	$0.134222\pi$
$14$	0	0
$15$	10.8464	2.80052
$16$	0	0
$17$	−0.658856	−0.159796	−0.0798980	−	0.996803i	$-0.525459\pi$
$17$	−0.658856	−0.159796	−0.0798980	+	0.996803i	$0.525459\pi$
$18$	0	0
$19$	−3.32428	−0.762642	−0.381321	−	0.924443i	$-0.624531\pi$
$19$	−3.32428	−0.762642	−0.381321	+	0.924443i	$0.624531\pi$
$20$	0	0
$21$	−3.35542	−0.732213
$22$	0	0
$23$	3.89074	0.811275	0.405638	−	0.914034i	$-0.367050\pi$
$23$	3.89074	0.811275	0.405638	+	0.914034i	$0.367050\pi$
$24$	0	0
$25$	7.34540	1.46908
$26$	0	0
$27$	10.8950	2.09674
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−3.00066	−0.538934	−0.269467	−	0.963010i	$-0.586847\pi$
$31$	−3.00066	−0.538934	−0.269467	+	0.963010i	$0.586847\pi$
$32$	0	0
$33$	−5.70898	−0.993806
$34$	0	0
$35$	−3.81916	−0.645556
$36$	0	0
$37$	−6.08844	−1.00093	−0.500466	−	0.865756i	$-0.666838\pi$
$37$	−6.08844	−1.00093	−0.500466	+	0.865756i	$0.666838\pi$
$38$	0	0
$39$	20.3105	3.25229
$40$	0	0
$41$	−5.19016	−0.810567	−0.405284	−	0.914191i	$-0.632827\pi$
$41$	−5.19016	−0.810567	−0.405284	+	0.914191i	$0.632827\pi$
$42$	0	0
$43$	1.17695	0.179483	0.0897414	−	0.995965i	$-0.471396\pi$
$43$	1.17695	0.179483	0.0897414	+	0.995965i	$0.471396\pi$
$44$	0	0
$45$	22.9415	3.41992
$46$	0	0
$47$	−6.21164	−0.906061	−0.453031	−	0.891495i	$-0.649657\pi$
$47$	−6.21164	−0.906061	−0.453031	+	0.891495i	$0.649657\pi$
$48$	0	0
$49$	−5.81851	−0.831215
$50$	0	0
$51$	−2.03387	−0.284798
$52$	0	0
$53$	−1.40581	−0.193103	−0.0965516	−	0.995328i	$-0.530781\pi$
$53$	−1.40581	−0.193103	−0.0965516	+	0.995328i	$0.530781\pi$
$54$	0	0
$55$	−6.49799	−0.876189
$56$	0	0
$57$	−10.2619	−1.35923
$58$	0	0
$59$	−9.55267	−1.24365	−0.621826	−	0.783156i	$-0.713609\pi$
$59$	−9.55267	−1.24365	−0.621826	+	0.783156i	$0.713609\pi$
$60$	0	0
$61$	8.30080	1.06281	0.531405	−	0.847118i	$-0.321664\pi$
$61$	8.30080	1.06281	0.531405	+	0.847118i	$0.321664\pi$
$62$	0	0
$63$	−7.09718	−0.894161
$64$	0	0
$65$	23.1176	2.86738
$66$	0	0
$67$	−12.3977	−1.51463	−0.757313	−	0.653052i	$-0.773488\pi$
$67$	−12.3977	−1.51463	−0.757313	+	0.653052i	$0.773488\pi$
$68$	0	0
$69$	12.0106	1.44590
$70$	0	0
$71$	8.12645	0.964433	0.482216	−	0.876052i	$-0.339832\pi$
$71$	8.12645	0.964433	0.482216	+	0.876052i	$0.339832\pi$
$72$	0	0
$73$	8.61593	1.00842	0.504209	−	0.863582i	$-0.331784\pi$
$73$	8.61593	1.00842	0.504209	+	0.863582i	$0.331784\pi$
$74$	0	0
$75$	22.6750	2.61828
$76$	0	0
$77$	2.01021	0.229085
$78$	0	0
$79$	−5.60250	−0.630331	−0.315165	−	0.949037i	$-0.602060\pi$
$79$	−5.60250	−0.630331	−0.315165	+	0.949037i	$0.602060\pi$
$80$	0	0
$81$	14.0444	1.56049
$82$	0	0
$83$	−3.72172	−0.408511	−0.204256	−	0.978918i	$-0.565477\pi$
$83$	−3.72172	−0.408511	−0.204256	+	0.978918i	$0.565477\pi$
$84$	0	0
$85$	−2.31496	−0.251092
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.47002	−0.473821	−0.236910	−	0.971532i	$-0.576135\pi$
$89$	−4.47002	−0.473821	−0.236910	+	0.971532i	$0.576135\pi$
$90$	0	0
$91$	−7.15163	−0.749695
$92$	0	0
$93$	−9.26292	−0.960520
$94$	0	0
$95$	−11.6802	−1.19836
$96$	0	0
$97$	−8.00445	−0.812729	−0.406364	−	0.913711i	$-0.633204\pi$
$97$	−8.00445	−0.812729	−0.406364	+	0.913711i	$0.633204\pi$
$98$	0	0
$99$	−12.0753	−1.21361
$100$	0	0
$101$	2.32228	0.231076	0.115538	−	0.993303i	$-0.463141\pi$
$101$	2.32228	0.231076	0.115538	+	0.993303i	$0.463141\pi$
$102$	0	0
$103$	−7.70034	−0.758737	−0.379368	−	0.925246i	$-0.623859\pi$
$103$	−7.70034	−0.758737	−0.379368	+	0.925246i	$0.623859\pi$
$104$	0	0
$105$	−11.7896	−1.15055
$106$	0	0
$107$	7.43577	0.718843	0.359422	−	0.933175i	$-0.382974\pi$
$107$	7.43577	0.718843	0.359422	+	0.933175i	$0.382974\pi$
$108$	0	0
$109$	5.96077	0.570938	0.285469	−	0.958388i	$-0.407851\pi$
$109$	5.96077	0.570938	0.285469	+	0.958388i	$0.407851\pi$
$110$	0	0
$111$	−18.7948	−1.78392
$112$	0	0
$113$	18.1901	1.71118	0.855589	−	0.517656i	$-0.173195\pi$
$113$	18.1901	1.71118	0.855589	+	0.517656i	$0.173195\pi$
$114$	0	0
$115$	13.6705	1.27478
$116$	0	0
$117$	42.9596	3.97161
$118$	0	0
$119$	0.716154	0.0656497
$120$	0	0
$121$	−7.57978	−0.689071
$122$	0	0
$123$	−16.0219	−1.44464
$124$	0	0
$125$	8.24079	0.737078
$126$	0	0
$127$	4.19960	0.372655	0.186327	−	0.982488i	$-0.440342\pi$
$127$	4.19960	0.372655	0.186327	+	0.982488i	$0.440342\pi$
$128$	0	0
$129$	3.63320	0.319885
$130$	0	0
$131$	11.0211	0.962917	0.481458	−	0.876469i	$-0.340107\pi$
$131$	11.0211	0.962917	0.481458	+	0.876469i	$0.340107\pi$
$132$	0	0
$133$	3.61338	0.313320
$134$	0	0
$135$	38.2807	3.29467
$136$	0	0
$137$	9.32982	0.797100	0.398550	−	0.917147i	$-0.369514\pi$
$137$	9.32982	0.797100	0.398550	+	0.917147i	$0.369514\pi$
$138$	0	0
$139$	17.6662	1.49843	0.749213	−	0.662329i	$-0.230432\pi$
$139$	17.6662	1.49843	0.749213	+	0.662329i	$0.230432\pi$
$140$	0	0
$141$	−19.1751	−1.61484
$142$	0	0
$143$	−12.1679	−1.01753
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−17.9615	−1.48144
$148$	0	0
$149$	16.7796	1.37464	0.687321	−	0.726354i	$-0.258787\pi$
$149$	16.7796	1.37464	0.687321	+	0.726354i	$0.258787\pi$
$150$	0	0
$151$	0.679316	0.0552819	0.0276410	−	0.999618i	$-0.491200\pi$
$151$	0.679316	0.0552819	0.0276410	+	0.999618i	$0.491200\pi$
$152$	0	0
$153$	−4.30190	−0.347788
$154$	0	0
$155$	−10.5431	−0.846843
$156$	0	0
$157$	1.22921	0.0981018	0.0490509	−	0.998796i	$-0.484380\pi$
$157$	1.22921	0.0981018	0.0490509	+	0.998796i	$0.484380\pi$
$158$	0	0
$159$	−4.33970	−0.344160
$160$	0	0
$161$	−4.22910	−0.333300
$162$	0	0
$163$	10.1411	0.794313	0.397156	−	0.917751i	$-0.369997\pi$
$163$	10.1411	0.794313	0.397156	+	0.917751i	$0.369997\pi$
$164$	0	0
$165$	−20.0591	−1.56160
$166$	0	0
$167$	−19.7701	−1.52986	−0.764928	−	0.644116i	$-0.777225\pi$
$167$	−19.7701	−1.52986	−0.764928	+	0.644116i	$0.777225\pi$
$168$	0	0
$169$	30.2892	2.32994
$170$	0	0
$171$	−21.7054	−1.65985
$172$	0	0
$173$	−6.04054	−0.459254	−0.229627	−	0.973279i	$-0.573751\pi$
$173$	−6.04054	−0.459254	−0.229627	+	0.973279i	$0.573751\pi$
$174$	0	0
$175$	−7.98419	−0.603548
$176$	0	0
$177$	−29.4887	−2.21651
$178$	0	0
$179$	−13.4038	−1.00184	−0.500922	−	0.865492i	$-0.667006\pi$
$179$	−13.4038	−1.00184	−0.500922	+	0.865492i	$0.667006\pi$
$180$	0	0
$181$	2.30153	0.171071	0.0855356	−	0.996335i	$-0.472740\pi$
$181$	2.30153	0.171071	0.0855356	+	0.996335i	$0.472740\pi$
$182$	0	0
$183$	25.6243	1.89420
$184$	0	0
$185$	−21.3923	−1.57280
$186$	0	0
$187$	1.21848	0.0891039
$188$	0	0
$189$	−11.8425	−0.861413
$190$	0	0
$191$	−26.8511	−1.94287	−0.971437	−	0.237298i	$-0.923738\pi$
$191$	−26.8511	−1.94287	−0.971437	+	0.237298i	$0.923738\pi$
$192$	0	0
$193$	−10.1495	−0.730575	−0.365288	−	0.930895i	$-0.619029\pi$
$193$	−10.1495	−0.730575	−0.365288	+	0.930895i	$0.619029\pi$
$194$	0	0
$195$	71.3631	5.11042
$196$	0	0
$197$	2.47663	0.176453	0.0882264	−	0.996100i	$-0.471880\pi$
$197$	2.47663	0.176453	0.0882264	+	0.996100i	$0.471880\pi$
$198$	0	0
$199$	9.25795	0.656279	0.328139	−	0.944629i	$-0.393578\pi$
$199$	9.25795	0.656279	0.328139	+	0.944629i	$0.393578\pi$
$200$	0	0
$201$	−38.2714	−2.69946
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−18.2362	−1.27367
$206$	0	0
$207$	25.4040	1.76570
$208$	0	0
$209$	6.14787	0.425257
$210$	0	0
$211$	−22.2137	−1.52925	−0.764626	−	0.644474i	$-0.777076\pi$
$211$	−22.2137	−1.52925	−0.764626	+	0.644474i	$0.777076\pi$
$212$	0	0
$213$	25.0861	1.71887
$214$	0	0
$215$	4.13532	0.282027
$216$	0	0
$217$	3.26161	0.221412
$218$	0	0
$219$	26.5971	1.79726
$220$	0	0
$221$	−4.33491	−0.291598
$222$	0	0
$223$	27.0264	1.80982	0.904909	−	0.425605i	$-0.139939\pi$
$223$	27.0264	1.80982	0.904909	+	0.425605i	$0.139939\pi$
$224$	0	0
$225$	47.9607	3.19738
$226$	0	0
$227$	2.17148	0.144126	0.0720632	−	0.997400i	$-0.477042\pi$
$227$	2.17148	0.144126	0.0720632	+	0.997400i	$0.477042\pi$
$228$	0	0
$229$	−14.0851	−0.930768	−0.465384	−	0.885109i	$-0.654084\pi$
$229$	−14.0851	−0.930768	−0.465384	+	0.885109i	$0.654084\pi$
$230$	0	0
$231$	6.20546	0.408289
$232$	0	0
$233$	7.98101	0.522854	0.261427	−	0.965223i	$-0.415807\pi$
$233$	7.98101	0.522854	0.261427	+	0.965223i	$0.415807\pi$
$234$	0	0
$235$	−21.8252	−1.42372
$236$	0	0
$237$	−17.2947	−1.12341
$238$	0	0
$239$	−18.8987	−1.22245	−0.611227	−	0.791455i	$-0.709324\pi$
$239$	−18.8987	−1.22245	−0.611227	+	0.791455i	$0.709324\pi$
$240$	0	0
$241$	−10.6416	−0.685484	−0.342742	−	0.939430i	$-0.611356\pi$
$241$	−10.6416	−0.685484	−0.342742	+	0.939430i	$0.611356\pi$
$242$	0	0
$243$	10.6696	0.684454
$244$	0	0
$245$	−20.4439	−1.30611
$246$	0	0
$247$	−21.8719	−1.39168
$248$	0	0
$249$	−11.4888	−0.728074
$250$	0	0
$251$	−1.26206	−0.0796604	−0.0398302	−	0.999206i	$-0.512682\pi$
$251$	−1.26206	−0.0796604	−0.0398302	+	0.999206i	$0.512682\pi$
$252$	0	0
$253$	−7.19546	−0.452375
$254$	0	0
$255$	−7.14619	−0.447512
$256$	0	0
$257$	−5.78081	−0.360597	−0.180298	−	0.983612i	$-0.557706\pi$
$257$	−5.78081	−0.360597	−0.180298	+	0.983612i	$0.557706\pi$
$258$	0	0
$259$	6.61792	0.411217
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.9520	1.47694	0.738472	−	0.674284i	$-0.235548\pi$
$263$	23.9520	1.47694	0.738472	+	0.674284i	$0.235548\pi$
$264$	0	0
$265$	−4.93947	−0.303429
$266$	0	0
$267$	−13.7988	−0.844472
$268$	0	0
$269$	7.56477	0.461232	0.230616	−	0.973045i	$-0.425926\pi$
$269$	7.56477	0.461232	0.230616	+	0.973045i	$0.425926\pi$
$270$	0	0
$271$	−19.0739	−1.15866	−0.579328	−	0.815095i	$-0.696685\pi$
$271$	−19.0739	−1.15866	−0.579328	+	0.815095i	$0.696685\pi$
$272$	0	0
$273$	−22.0768	−1.33615
$274$	0	0
$275$	−13.5844	−0.819173
$276$	0	0
$277$	−4.14362	−0.248966	−0.124483	−	0.992222i	$-0.539727\pi$
$277$	−4.14362	−0.248966	−0.124483	+	0.992222i	$0.539727\pi$
$278$	0	0
$279$	−19.5923	−1.17296
$280$	0	0
$281$	0.576507	0.0343915	0.0171958	−	0.999852i	$-0.494526\pi$
$281$	0.576507	0.0343915	0.0171958	+	0.999852i	$0.494526\pi$
$282$	0	0
$283$	23.6545	1.40611	0.703056	−	0.711134i	$-0.251818\pi$
$283$	23.6545	1.40611	0.703056	+	0.711134i	$0.251818\pi$
$284$	0	0
$285$	−36.0564	−2.13579
$286$	0	0
$287$	5.64153	0.333009
$288$	0	0
$289$	−16.5659	−0.974465
$290$	0	0
$291$	−24.7095	−1.44849
$292$	0	0
$293$	5.09064	0.297399	0.148699	−	0.988882i	$-0.452491\pi$
$293$	5.09064	0.297399	0.148699	+	0.988882i	$0.452491\pi$
$294$	0	0
$295$	−33.5643	−1.95419
$296$	0	0
$297$	−20.1490	−1.16916
$298$	0	0
$299$	25.5989	1.48042
$300$	0	0
$301$	−1.27930	−0.0737377
$302$	0	0
$303$	7.16881	0.411837
$304$	0	0
$305$	29.1657	1.67002
$306$	0	0
$307$	27.9505	1.59522	0.797609	−	0.603175i	$-0.206098\pi$
$307$	27.9505	1.59522	0.797609	+	0.603175i	$0.206098\pi$
$308$	0	0
$309$	−23.7707	−1.35227
$310$	0	0
$311$	−16.4496	−0.932773	−0.466386	−	0.884581i	$-0.654444\pi$
$311$	−16.4496	−0.932773	−0.466386	+	0.884581i	$0.654444\pi$
$312$	0	0
$313$	2.86030	0.161673	0.0808367	−	0.996727i	$-0.474241\pi$
$313$	2.86030	0.161673	0.0808367	+	0.996727i	$0.474241\pi$
$314$	0	0
$315$	−24.9367	−1.40502
$316$	0	0
$317$	−22.8470	−1.28321	−0.641607	−	0.767034i	$-0.721732\pi$
$317$	−22.8470	−1.28321	−0.641607	+	0.767034i	$0.721732\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	22.9540	1.28117
$322$	0	0
$323$	2.19022	0.121867
$324$	0	0
$325$	48.3287	2.68079
$326$	0	0
$327$	18.4007	1.01756
$328$	0	0
$329$	6.75184	0.372241
$330$	0	0
$331$	−26.0822	−1.43361	−0.716803	−	0.697276i	$-0.754395\pi$
$331$	−26.0822	−1.43361	−0.716803	+	0.697276i	$0.754395\pi$
$332$	0	0
$333$	−39.7536	−2.17848
$334$	0	0
$335$	−43.5607	−2.37998
$336$	0	0
$337$	31.6599	1.72462	0.862312	−	0.506377i	$-0.169016\pi$
$337$	31.6599	1.72462	0.862312	+	0.506377i	$0.169016\pi$
$338$	0	0
$339$	56.1521	3.04976
$340$	0	0
$341$	5.54936	0.300515
$342$	0	0
$343$	13.9333	0.752326
$344$	0	0
$345$	42.2004	2.27199
$346$	0	0
$347$	−9.10528	−0.488797	−0.244399	−	0.969675i	$-0.578591\pi$
$347$	−9.10528	−0.488797	−0.244399	+	0.969675i	$0.578591\pi$
$348$	0	0
$349$	−6.07146	−0.324998	−0.162499	−	0.986709i	$-0.551955\pi$
$349$	−6.07146	−0.324998	−0.162499	+	0.986709i	$0.551955\pi$
$350$	0	0
$351$	71.6831	3.82616
$352$	0	0
$353$	0.308919	0.0164421	0.00822106	−	0.999966i	$-0.497383\pi$
$353$	0.308919	0.0164421	0.00822106	+	0.999966i	$0.497383\pi$
$354$	0	0
$355$	28.5531	1.51544
$356$	0	0
$357$	2.21074	0.117005
$358$	0	0
$359$	−18.7602	−0.990123	−0.495062	−	0.868858i	$-0.664854\pi$
$359$	−18.7602	−0.990123	−0.495062	+	0.868858i	$0.664854\pi$
$360$	0	0
$361$	−7.94916	−0.418377
$362$	0	0
$363$	−23.3985	−1.22810
$364$	0	0
$365$	30.2729	1.58456
$366$	0	0
$367$	19.8789	1.03767	0.518835	−	0.854875i	$-0.326366\pi$
$367$	19.8789	1.03767	0.518835	+	0.854875i	$0.326366\pi$
$368$	0	0
$369$	−33.8884	−1.76416
$370$	0	0
$371$	1.52807	0.0793334
$372$	0	0
$373$	13.7685	0.712908	0.356454	−	0.934313i	$-0.383986\pi$
$373$	13.7685	0.712908	0.356454	+	0.934313i	$0.383986\pi$
$374$	0	0
$375$	25.4390	1.31367
$376$	0	0
$377$	0	0
$378$	0	0
$379$	26.2032	1.34597	0.672985	−	0.739656i	$-0.265012\pi$
$379$	26.2032	1.34597	0.672985	+	0.739656i	$0.265012\pi$
$380$	0	0
$381$	12.9640	0.664168
$382$	0	0
$383$	7.17470	0.366610	0.183305	−	0.983056i	$-0.441320\pi$
$383$	7.17470	0.366610	0.183305	+	0.983056i	$0.441320\pi$
$384$	0	0
$385$	7.06309	0.359968
$386$	0	0
$387$	7.68471	0.390636
$388$	0	0
$389$	−14.1936	−0.719642	−0.359821	−	0.933021i	$-0.617162\pi$
$389$	−14.1936	−0.719642	−0.359821	+	0.933021i	$0.617162\pi$
$390$	0	0
$391$	−2.56344	−0.129639
$392$	0	0
$393$	34.0217	1.71617
$394$	0	0
$395$	−19.6850	−0.990458
$396$	0	0
$397$	−28.7763	−1.44424	−0.722121	−	0.691766i	$-0.756833\pi$
$397$	−28.7763	−1.44424	−0.722121	+	0.691766i	$0.756833\pi$
$398$	0	0
$399$	11.1544	0.558417
$400$	0	0
$401$	17.1244	0.855154	0.427577	−	0.903979i	$-0.359367\pi$
$401$	17.1244	0.855154	0.427577	+	0.903979i	$0.359367\pi$
$402$	0	0
$403$	−19.7427	−0.983452
$404$	0	0
$405$	49.3464	2.45204
$406$	0	0
$407$	11.2599	0.558130
$408$	0	0
$409$	2.96364	0.146542	0.0732712	−	0.997312i	$-0.476656\pi$
$409$	2.96364	0.146542	0.0732712	+	0.997312i	$0.476656\pi$
$410$	0	0
$411$	28.8008	1.42064
$412$	0	0
$413$	10.3834	0.510934
$414$	0	0
$415$	−13.0766	−0.641906
$416$	0	0
$417$	54.5349	2.67059
$418$	0	0
$419$	7.46230	0.364557	0.182278	−	0.983247i	$-0.441653\pi$
$419$	7.46230	0.364557	0.182278	+	0.983247i	$0.441653\pi$
$420$	0	0
$421$	4.68009	0.228094	0.114047	−	0.993475i	$-0.463619\pi$
$421$	4.68009	0.228094	0.114047	+	0.993475i	$0.463619\pi$
$422$	0	0
$423$	−40.5580	−1.97200
$424$	0	0
$425$	−4.83956	−0.234753
$426$	0	0
$427$	−9.02268	−0.436638
$428$	0	0
$429$	−37.5620	−1.81351
$430$	0	0
$431$	14.9667	0.720921	0.360461	−	0.932774i	$-0.382620\pi$
$431$	14.9667	0.720921	0.360461	+	0.932774i	$0.382620\pi$
$432$	0	0
$433$	28.0408	1.34755	0.673776	−	0.738935i	$-0.264671\pi$
$433$	28.0408	1.34755	0.673776	+	0.738935i	$0.264671\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.9339	−0.618713
$438$	0	0
$439$	−19.3075	−0.921495	−0.460747	−	0.887531i	$-0.652419\pi$
$439$	−19.3075	−0.921495	−0.460747	+	0.887531i	$0.652419\pi$
$440$	0	0
$441$	−37.9911	−1.80910
$442$	0	0
$443$	30.8972	1.46797	0.733984	−	0.679167i	$-0.237659\pi$
$443$	30.8972	1.46797	0.733984	+	0.679167i	$0.237659\pi$
$444$	0	0
$445$	−15.7059	−0.744529
$446$	0	0
$447$	51.7982	2.44997
$448$	0	0
$449$	−28.8238	−1.36028	−0.680139	−	0.733083i	$-0.738081\pi$
$449$	−28.8238	−1.36028	−0.680139	+	0.733083i	$0.738081\pi$
$450$	0	0
$451$	9.59860	0.451980
$452$	0	0
$453$	2.09702	0.0985268
$454$	0	0
$455$	−25.1280	−1.17802
$456$	0	0
$457$	−35.7451	−1.67209	−0.836044	−	0.548663i	$-0.815137\pi$
$457$	−35.7451	−1.67209	−0.836044	+	0.548663i	$0.815137\pi$
$458$	0	0
$459$	−7.17823	−0.335051
$460$	0	0
$461$	−14.2728	−0.664750	−0.332375	−	0.943147i	$-0.607850\pi$
$461$	−14.2728	−0.664750	−0.332375	+	0.943147i	$0.607850\pi$
$462$	0	0
$463$	−33.7695	−1.56940	−0.784701	−	0.619875i	$-0.787183\pi$
$463$	−33.7695	−1.56940	−0.784701	+	0.619875i	$0.787183\pi$
$464$	0	0
$465$	−32.5462	−1.50929
$466$	0	0
$467$	−31.9927	−1.48044	−0.740222	−	0.672362i	$-0.765280\pi$
$467$	−31.9927	−1.48044	−0.740222	+	0.672362i	$0.765280\pi$
$468$	0	0
$469$	13.4759	0.622260
$470$	0	0
$471$	3.79453	0.174843
$472$	0	0
$473$	−2.17663	−0.100081
$474$	0	0
$475$	−24.4182	−1.12038
$476$	0	0
$477$	−9.17905	−0.420280
$478$	0	0
$479$	36.0481	1.64708	0.823539	−	0.567259i	$-0.191996\pi$
$479$	36.0481	1.64708	0.823539	+	0.567259i	$0.191996\pi$
$480$	0	0
$481$	−40.0586	−1.82651
$482$	0	0
$483$	−13.0551	−0.594027
$484$	0	0
$485$	−28.1245	−1.27707
$486$	0	0
$487$	25.8247	1.17023	0.585115	−	0.810951i	$-0.301049\pi$
$487$	25.8247	1.17023	0.585115	+	0.810951i	$0.301049\pi$
$488$	0	0
$489$	31.3052	1.41567
$490$	0	0
$491$	2.05490	0.0927365	0.0463683	−	0.998924i	$-0.485235\pi$
$491$	2.05490	0.0927365	0.0463683	+	0.998924i	$0.485235\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−42.4277	−1.90698
$496$	0	0
$497$	−8.83317	−0.396222
$498$	0	0
$499$	3.22792	0.144501	0.0722507	−	0.997387i	$-0.476982\pi$
$499$	3.22792	0.144501	0.0722507	+	0.997387i	$0.476982\pi$
$500$	0	0
$501$	−61.0296	−2.72660
$502$	0	0
$503$	15.7898	0.704033	0.352017	−	0.935994i	$-0.385496\pi$
$503$	15.7898	0.704033	0.352017	+	0.935994i	$0.385496\pi$
$504$	0	0
$505$	8.15958	0.363096
$506$	0	0
$507$	93.5017	4.15255
$508$	0	0
$509$	−2.86681	−0.127069	−0.0635346	−	0.997980i	$-0.520237\pi$
$509$	−2.86681	−0.127069	−0.0635346	+	0.997980i	$0.520237\pi$
$510$	0	0
$511$	−9.36521	−0.414293
$512$	0	0
$513$	−36.2180	−1.59906
$514$	0	0
$515$	−27.0559	−1.19223
$516$	0	0
$517$	11.4877	0.505229
$518$	0	0
$519$	−18.6469	−0.818509
$520$	0	0
$521$	24.8422	1.08835	0.544177	−	0.838970i	$-0.316842\pi$
$521$	24.8422	1.08835	0.544177	+	0.838970i	$0.316842\pi$
$522$	0	0
$523$	−5.39629	−0.235963	−0.117982	−	0.993016i	$-0.537642\pi$
$523$	−5.39629	−0.235963	−0.117982	+	0.993016i	$0.537642\pi$
$524$	0	0
$525$	−24.6469	−1.07568
$526$	0	0
$527$	1.97700	0.0861195
$528$	0	0
$529$	−7.86215	−0.341833
$530$	0	0
$531$	−62.3727	−2.70675
$532$	0	0
$533$	−34.1484	−1.47913
$534$	0	0
$535$	26.1263	1.12954
$536$	0	0
$537$	−41.3770	−1.78555
$538$	0	0
$539$	10.7606	0.463494
$540$	0	0
$541$	−29.6695	−1.27559	−0.637795	−	0.770206i	$-0.720153\pi$
$541$	−29.6695	−1.27559	−0.637795	+	0.770206i	$0.720153\pi$
$542$	0	0
$543$	7.10473	0.304893
$544$	0	0
$545$	20.9438	0.897133
$546$	0	0
$547$	25.3031	1.08188	0.540941	−	0.841060i	$-0.318068\pi$
$547$	25.3031	1.08188	0.540941	+	0.841060i	$0.318068\pi$
$548$	0	0
$549$	54.1989	2.31315
$550$	0	0
$551$	0	0
$552$	0	0
$553$	6.08973	0.258961
$554$	0	0
$555$	−66.0374	−2.80313
$556$	0	0
$557$	17.7392	0.751636	0.375818	−	0.926694i	$-0.377362\pi$
$557$	17.7392	0.751636	0.375818	+	0.926694i	$0.377362\pi$
$558$	0	0
$559$	7.74367	0.327522
$560$	0	0
$561$	3.76140	0.158806
$562$	0	0
$563$	38.5300	1.62385	0.811923	−	0.583764i	$-0.198421\pi$
$563$	38.5300	1.62385	0.811923	+	0.583764i	$0.198421\pi$
$564$	0	0
$565$	63.9127	2.68883
$566$	0	0
$567$	−15.2658	−0.641102
$568$	0	0
$569$	41.8338	1.75377	0.876883	−	0.480704i	$-0.159619\pi$
$569$	41.8338	1.75377	0.876883	+	0.480704i	$0.159619\pi$
$570$	0	0
$571$	−14.0139	−0.586462	−0.293231	−	0.956042i	$-0.594730\pi$
$571$	−14.0139	−0.586462	−0.293231	+	0.956042i	$0.594730\pi$
$572$	0	0
$573$	−82.8883	−3.46271
$574$	0	0
$575$	28.5790	1.19183
$576$	0	0
$577$	10.5338	0.438529	0.219264	−	0.975665i	$-0.429634\pi$
$577$	10.5338	0.438529	0.219264	+	0.975665i	$0.429634\pi$
$578$	0	0
$579$	−31.3311	−1.30208
$580$	0	0
$581$	4.04538	0.167830
$582$	0	0
$583$	2.59989	0.107676
$584$	0	0
$585$	150.943	6.24072
$586$	0	0
$587$	−11.8349	−0.488481	−0.244240	−	0.969715i	$-0.578539\pi$
$587$	−11.8349	−0.488481	−0.244240	+	0.969715i	$0.578539\pi$
$588$	0	0
$589$	9.97502	0.411014
$590$	0	0
$591$	7.64528	0.314485
$592$	0	0
$593$	0.120568	0.00495114	0.00247557	−	0.999997i	$-0.499212\pi$
$593$	0.120568	0.00495114	0.00247557	+	0.999997i	$0.499212\pi$
$594$	0	0
$595$	2.51628	0.103157
$596$	0	0
$597$	28.5790	1.16966
$598$	0	0
$599$	−11.2858	−0.461125	−0.230562	−	0.973058i	$-0.574057\pi$
$599$	−11.2858	−0.461125	−0.230562	+	0.973058i	$0.574057\pi$
$600$	0	0
$601$	35.1184	1.43251	0.716255	−	0.697839i	$-0.245855\pi$
$601$	35.1184	1.43251	0.716255	+	0.697839i	$0.245855\pi$
$602$	0	0
$603$	−80.9493	−3.29651
$604$	0	0
$605$	−26.6323	−1.08276
$606$	0	0
$607$	10.9968	0.446346	0.223173	−	0.974779i	$-0.428359\pi$
$607$	10.9968	0.446346	0.223173	+	0.974779i	$0.428359\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−40.8692	−1.65339
$612$	0	0
$613$	15.0695	0.608652	0.304326	−	0.952568i	$-0.401569\pi$
$613$	15.0695	0.608652	0.304326	+	0.952568i	$0.401569\pi$
$614$	0	0
$615$	−56.2944	−2.27001
$616$	0	0
$617$	28.5894	1.15096	0.575482	−	0.817814i	$-0.304814\pi$
$617$	28.5894	1.15096	0.575482	+	0.817814i	$0.304814\pi$
$618$	0	0
$619$	−18.3546	−0.737733	−0.368866	−	0.929482i	$-0.620254\pi$
$619$	−18.3546	−0.737733	−0.368866	+	0.929482i	$0.620254\pi$
$620$	0	0
$621$	42.3896	1.70103
$622$	0	0
$623$	4.85875	0.194662
$624$	0	0
$625$	−7.77214	−0.310886
$626$	0	0
$627$	18.9783	0.757918
$628$	0	0
$629$	4.01140	0.159945
$630$	0	0
$631$	−11.7582	−0.468087	−0.234043	−	0.972226i	$-0.575196\pi$
$631$	−11.7582	−0.468087	−0.234043	+	0.972226i	$0.575196\pi$
$632$	0	0
$633$	−68.5728	−2.72553
$634$	0	0
$635$	14.7557	0.585564
$636$	0	0
$637$	−38.2826	−1.51681
$638$	0	0
$639$	53.0605	2.09904
$640$	0	0
$641$	21.9735	0.867902	0.433951	−	0.900936i	$-0.357119\pi$
$641$	21.9735	0.867902	0.433951	+	0.900936i	$0.357119\pi$
$642$	0	0
$643$	−4.75625	−0.187568	−0.0937841	−	0.995593i	$-0.529896\pi$
$643$	−4.75625	−0.187568	−0.0937841	+	0.995593i	$0.529896\pi$
$644$	0	0
$645$	12.7656	0.502645
$646$	0	0
$647$	19.6242	0.771509	0.385754	−	0.922602i	$-0.373941\pi$
$647$	19.6242	0.771509	0.385754	+	0.922602i	$0.373941\pi$
$648$	0	0
$649$	17.6665	0.693472
$650$	0	0
$651$	10.0685	0.394614
$652$	0	0
$653$	27.1866	1.06389	0.531947	−	0.846778i	$-0.321461\pi$
$653$	27.1866	1.06389	0.531947	+	0.846778i	$0.321461\pi$
$654$	0	0
$655$	38.7237	1.51306
$656$	0	0
$657$	56.2564	2.19477
$658$	0	0
$659$	−46.3597	−1.80592	−0.902959	−	0.429727i	$-0.858610\pi$
$659$	−46.3597	−1.80592	−0.902959	+	0.429727i	$0.858610\pi$
$660$	0	0
$661$	−7.34570	−0.285715	−0.142857	−	0.989743i	$-0.545629\pi$
$661$	−7.34570	−0.285715	−0.142857	+	0.989743i	$0.545629\pi$
$662$	0	0
$663$	−13.3817	−0.519703
$664$	0	0
$665$	12.6960	0.492329
$666$	0	0
$667$	0	0
$668$	0	0
$669$	83.4294	3.22557
$670$	0	0
$671$	−15.3514	−0.592633
$672$	0	0
$673$	−34.4626	−1.32844	−0.664218	−	0.747539i	$-0.731235\pi$
$673$	−34.4626	−1.32844	−0.664218	+	0.747539i	$0.731235\pi$
$674$	0	0
$675$	80.0280	3.08028
$676$	0	0
$677$	−21.9239	−0.842605	−0.421302	−	0.906920i	$-0.638427\pi$
$677$	−21.9239	−0.842605	−0.421302	+	0.906920i	$0.638427\pi$
$678$	0	0
$679$	8.70056	0.333897
$680$	0	0
$681$	6.70329	0.256871
$682$	0	0
$683$	−30.6793	−1.17391	−0.586956	−	0.809619i	$-0.699674\pi$
$683$	−30.6793	−1.17391	−0.586956	+	0.809619i	$0.699674\pi$
$684$	0	0
$685$	32.7813	1.25251
$686$	0	0
$687$	−43.4802	−1.65887
$688$	0	0
$689$	−9.24948	−0.352377
$690$	0	0
$691$	5.81672	0.221279	0.110639	−	0.993861i	$-0.464710\pi$
$691$	5.81672	0.221279	0.110639	+	0.993861i	$0.464710\pi$
$692$	0	0
$693$	13.1254	0.498593
$694$	0	0
$695$	62.0719	2.35452
$696$	0	0
$697$	3.41957	0.129525
$698$	0	0
$699$	24.6371	0.931861
$700$	0	0
$701$	2.40593	0.0908707	0.0454354	−	0.998967i	$-0.485532\pi$
$701$	2.40593	0.0908707	0.0454354	+	0.998967i	$0.485532\pi$
$702$	0	0
$703$	20.2397	0.763354
$704$	0	0
$705$	−67.3737	−2.53744
$706$	0	0
$707$	−2.52424	−0.0949338
$708$	0	0
$709$	13.5346	0.508302	0.254151	−	0.967165i	$-0.418204\pi$
$709$	13.5346	0.508302	0.254151	+	0.967165i	$0.418204\pi$
$710$	0	0
$711$	−36.5807	−1.37188
$712$	0	0
$713$	−11.6748	−0.437223
$714$	0	0
$715$	−42.7532	−1.59888
$716$	0	0
$717$	−58.3396	−2.17873
$718$	0	0
$719$	−1.83647	−0.0684889	−0.0342444	−	0.999413i	$-0.510902\pi$
$719$	−1.83647	−0.0684889	−0.0342444	+	0.999413i	$0.510902\pi$
$720$	0	0
$721$	8.37000	0.311715
$722$	0	0
$723$	−32.8502	−1.22171
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.01846	−0.0377727	−0.0188863	−	0.999822i	$-0.506012\pi$
$727$	−1.01846	−0.0377727	−0.0188863	+	0.999822i	$0.506012\pi$
$728$	0	0
$729$	−9.19654	−0.340613
$730$	0	0
$731$	−0.775439	−0.0286806
$732$	0	0
$733$	−35.3992	−1.30750	−0.653750	−	0.756711i	$-0.726805\pi$
$733$	−35.3992	−1.30750	−0.653750	+	0.756711i	$0.726805\pi$
$734$	0	0
$735$	−63.1097	−2.32783
$736$	0	0
$737$	22.9282	0.844570
$738$	0	0
$739$	3.05344	0.112322	0.0561612	−	0.998422i	$-0.482114\pi$
$739$	3.05344	0.112322	0.0561612	+	0.998422i	$0.482114\pi$
$740$	0	0
$741$	−67.5179	−2.48033
$742$	0	0
$743$	−43.5694	−1.59840	−0.799202	−	0.601062i	$-0.794744\pi$
$743$	−43.5694	−1.59840	−0.799202	+	0.601062i	$0.794744\pi$
$744$	0	0
$745$	58.9570	2.16002
$746$	0	0
$747$	−24.3004	−0.889105
$748$	0	0
$749$	−8.08243	−0.295325
$750$	0	0
$751$	37.2765	1.36024	0.680120	−	0.733101i	$-0.261928\pi$
$751$	37.2765	1.36024	0.680120	+	0.733101i	$0.261928\pi$
$752$	0	0
$753$	−3.89593	−0.141976
$754$	0	0
$755$	2.38684	0.0868662
$756$	0	0
$757$	−36.1528	−1.31399	−0.656997	−	0.753893i	$-0.728174\pi$
$757$	−36.1528	−1.31399	−0.656997	+	0.753893i	$0.728174\pi$
$758$	0	0
$759$	−22.2121	−0.806250
$760$	0	0
$761$	45.3811	1.64506	0.822532	−	0.568719i	$-0.192561\pi$
$761$	45.3811	1.64506	0.822532	+	0.568719i	$0.192561\pi$
$762$	0	0
$763$	−6.47915	−0.234561
$764$	0	0
$765$	−15.1152	−0.546490
$766$	0	0
$767$	−62.8513	−2.26943
$768$	0	0
$769$	7.78374	0.280689	0.140344	−	0.990103i	$-0.455179\pi$
$769$	7.78374	0.280689	0.140344	+	0.990103i	$0.455179\pi$
$770$	0	0
$771$	−17.8451	−0.642677
$772$	0	0
$773$	7.79698	0.280438	0.140219	−	0.990121i	$-0.455219\pi$
$773$	7.79698	0.280438	0.140219	+	0.990121i	$0.455219\pi$
$774$	0	0
$775$	−22.0410	−0.791736
$776$	0	0
$777$	20.4293	0.732897
$778$	0	0
$779$	17.2536	0.618173
$780$	0	0
$781$	−15.0289	−0.537777
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.31896	0.154150
$786$	0	0
$787$	−33.7649	−1.20359	−0.601794	−	0.798652i	$-0.705547\pi$
$787$	−33.7649	−1.20359	−0.601794	+	0.798652i	$0.705547\pi$
$788$	0	0
$789$	73.9390	2.63230
$790$	0	0
$791$	−19.7720	−0.703010
$792$	0	0
$793$	54.6147	1.93943
$794$	0	0
$795$	−15.2480	−0.540790
$796$	0	0
$797$	−40.4846	−1.43404	−0.717019	−	0.697053i	$-0.754494\pi$
$797$	−40.4846	−1.43404	−0.717019	+	0.697053i	$0.754494\pi$
$798$	0	0
$799$	4.09258	0.144785
$800$	0	0
$801$	−29.1863	−1.03125
$802$	0	0
$803$	−15.9341	−0.562304
$804$	0	0
$805$	−14.8594	−0.523724
$806$	0	0
$807$	23.3522	0.822036
$808$	0	0
$809$	48.1307	1.69219	0.846093	−	0.533036i	$-0.178949\pi$
$809$	48.1307	1.69219	0.846093	+	0.533036i	$0.178949\pi$
$810$	0	0
$811$	26.2797	0.922806	0.461403	−	0.887191i	$-0.347346\pi$
$811$	26.2797	0.922806	0.461403	+	0.887191i	$0.347346\pi$
$812$	0	0
$813$	−58.8804	−2.06503
$814$	0	0
$815$	35.6318	1.24813
$816$	0	0
$817$	−3.91250	−0.136881
$818$	0	0
$819$	−46.6955	−1.63167
$820$	0	0
$821$	−44.2778	−1.54531	−0.772653	−	0.634829i	$-0.781070\pi$
$821$	−44.2778	−1.54531	−0.772653	+	0.634829i	$0.781070\pi$
$822$	0	0
$823$	−9.86111	−0.343737	−0.171868	−	0.985120i	$-0.554980\pi$
$823$	−9.86111	−0.343737	−0.171868	+	0.985120i	$0.554980\pi$
$824$	0	0
$825$	−41.9347	−1.45998
$826$	0	0
$827$	−18.0405	−0.627329	−0.313664	−	0.949534i	$-0.601557\pi$
$827$	−18.0405	−0.627329	−0.313664	+	0.949534i	$0.601557\pi$
$828$	0	0
$829$	−11.5967	−0.402770	−0.201385	−	0.979512i	$-0.564544\pi$
$829$	−11.5967	−0.402770	−0.201385	+	0.979512i	$0.564544\pi$
$830$	0	0
$831$	−12.7912	−0.443722
$832$	0	0
$833$	3.83356	0.132825
$834$	0	0
$835$	−69.4642	−2.40391
$836$	0	0
$837$	−32.6921	−1.13000
$838$	0	0
$839$	−0.831794	−0.0287167	−0.0143584	−	0.999897i	$-0.504571\pi$
$839$	−0.831794	−0.0287167	−0.0143584	+	0.999897i	$0.504571\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	1.77966	0.0612946
$844$	0	0
$845$	106.424	3.66110
$846$	0	0
$847$	8.23896	0.283094
$848$	0	0
$849$	73.0205	2.50606
$850$	0	0
$851$	−23.6885	−0.812032
$852$	0	0
$853$	−3.70379	−0.126815	−0.0634077	−	0.997988i	$-0.520197\pi$
$853$	−3.70379	−0.126815	−0.0634077	+	0.997988i	$0.520197\pi$
$854$	0	0
$855$	−76.2641	−2.60818
$856$	0	0
$857$	−10.1813	−0.347788	−0.173894	−	0.984764i	$-0.555635\pi$
$857$	−10.1813	−0.347788	−0.173894	+	0.984764i	$0.555635\pi$
$858$	0	0
$859$	−21.9534	−0.749039	−0.374519	−	0.927219i	$-0.622192\pi$
$859$	−21.9534	−0.749039	−0.374519	+	0.927219i	$0.622192\pi$
$860$	0	0
$861$	17.4152	0.593508
$862$	0	0
$863$	35.5775	1.21107	0.605537	−	0.795817i	$-0.292959\pi$
$863$	35.5775	1.21107	0.605537	+	0.795817i	$0.292959\pi$
$864$	0	0
$865$	−21.2240	−0.721639
$866$	0	0
$867$	−51.1384	−1.73675
$868$	0	0
$869$	10.3612	0.351479
$870$	0	0
$871$	−81.5704	−2.76391
$872$	0	0
$873$	−52.2639	−1.76886
$874$	0	0
$875$	−8.95745	−0.302817
$876$	0	0
$877$	3.30041	0.111447	0.0557235	−	0.998446i	$-0.482253\pi$
$877$	3.30041	0.111447	0.0557235	+	0.998446i	$0.482253\pi$
$878$	0	0
$879$	15.7146	0.530042
$880$	0	0
$881$	54.4844	1.83563	0.917814	−	0.397011i	$-0.129953\pi$
$881$	54.4844	1.83563	0.917814	+	0.397011i	$0.129953\pi$
$882$	0	0
$883$	−46.5935	−1.56800	−0.783999	−	0.620763i	$-0.786823\pi$
$883$	−46.5935	−1.56800	−0.783999	+	0.620763i	$0.786823\pi$
$884$	0	0
$885$	−103.612	−3.48287
$886$	0	0
$887$	−41.1122	−1.38041	−0.690206	−	0.723613i	$-0.742480\pi$
$887$	−41.1122	−1.38041	−0.690206	+	0.723613i	$0.742480\pi$
$888$	0	0
$889$	−4.56482	−0.153099
$890$	0	0
$891$	−25.9735	−0.870144
$892$	0	0
$893$	20.6492	0.691001
$894$	0	0
$895$	−47.0955	−1.57423
$896$	0	0
$897$	79.0230	2.63850
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.926229	0.0308571
$902$	0	0
$903$	−3.94916	−0.131420
$904$	0	0
$905$	8.08665	0.268809
$906$	0	0
$907$	22.3443	0.741931	0.370966	−	0.928647i	$-0.379027\pi$
$907$	22.3443	0.741931	0.370966	+	0.928647i	$0.379027\pi$
$908$	0	0
$909$	15.1630	0.502925
$910$	0	0
$911$	−32.2237	−1.06762	−0.533810	−	0.845605i	$-0.679240\pi$
$911$	−32.2237	−1.06762	−0.533810	+	0.845605i	$0.679240\pi$
$912$	0	0
$913$	6.88288	0.227790
$914$	0	0
$915$	90.0336	2.97642
$916$	0	0
$917$	−11.9795	−0.395599
$918$	0	0
$919$	−33.8688	−1.11723	−0.558614	−	0.829428i	$-0.688667\pi$
$919$	−33.8688	−1.11723	−0.558614	+	0.829428i	$0.688667\pi$
$920$	0	0
$921$	86.2821	2.84309
$922$	0	0
$923$	53.4676	1.75991
$924$	0	0
$925$	−44.7220	−1.47045
$926$	0	0
$927$	−50.2782	−1.65135
$928$	0	0
$929$	−42.4834	−1.39383	−0.696917	−	0.717152i	$-0.745445\pi$
$929$	−42.4834	−1.39383	−0.696917	+	0.717152i	$0.745445\pi$
$930$	0	0
$931$	19.3424	0.633920
$932$	0	0
$933$	−50.7794	−1.66244
$934$	0	0
$935$	4.28124	0.140012
$936$	0	0
$937$	18.3549	0.599629	0.299814	−	0.953998i	$-0.403075\pi$
$937$	18.3549	0.599629	0.299814	+	0.953998i	$0.403075\pi$
$938$	0	0
$939$	8.82963	0.288144
$940$	0	0
$941$	55.0896	1.79587	0.897934	−	0.440130i	$-0.145067\pi$
$941$	55.0896	1.79587	0.897934	+	0.440130i	$0.145067\pi$
$942$	0	0
$943$	−20.1936	−0.657593
$944$	0	0
$945$	−41.6097	−1.35356
$946$	0	0
$947$	48.2394	1.56757	0.783785	−	0.621032i	$-0.213286\pi$
$947$	48.2394	1.56757	0.783785	+	0.621032i	$0.213286\pi$
$948$	0	0
$949$	56.6881	1.84017
$950$	0	0
$951$	−70.5278	−2.28702
$952$	0	0
$953$	31.6040	1.02375	0.511877	−	0.859059i	$-0.328950\pi$
$953$	31.6040	1.02375	0.511877	+	0.859059i	$0.328950\pi$
$954$	0	0
$955$	−94.3439	−3.05290
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.1412	−0.327476
$960$	0	0
$961$	−21.9961	−0.709551
$962$	0	0
$963$	48.5508	1.56453
$964$	0	0
$965$	−35.6612	−1.14798
$966$	0	0
$967$	−7.76191	−0.249606	−0.124803	−	0.992182i	$-0.539830\pi$
$967$	−7.76191	−0.249606	−0.124803	+	0.992182i	$0.539830\pi$
$968$	0	0
$969$	6.76114	0.217199
$970$	0	0
$971$	−1.30184	−0.0417779	−0.0208890	−	0.999782i	$-0.506650\pi$
$971$	−1.30184	−0.0417779	−0.0208890	+	0.999782i	$0.506650\pi$
$972$	0	0
$973$	−19.2025	−0.615605
$974$	0	0
$975$	149.189	4.77787
$976$	0	0
$977$	25.6371	0.820203	0.410101	−	0.912040i	$-0.365493\pi$
$977$	25.6371	0.820203	0.410101	+	0.912040i	$0.365493\pi$
$978$	0	0
$979$	8.26677	0.264207
$980$	0	0
$981$	38.9200	1.24262
$982$	0	0
$983$	10.7333	0.342338	0.171169	−	0.985242i	$-0.445246\pi$
$983$	10.7333	0.342338	0.171169	+	0.985242i	$0.445246\pi$
$984$	0	0
$985$	8.70190	0.277266
$986$	0	0
$987$	20.8427	0.663430
$988$	0	0
$989$	4.57920	0.145610
$990$	0	0
$991$	22.9452	0.728877	0.364438	−	0.931227i	$-0.381261\pi$
$991$	22.9452	0.728877	0.364438	+	0.931227i	$0.381261\pi$
$992$	0	0
$993$	−80.5148	−2.55506
$994$	0	0
$995$	32.5287	1.03123
$996$	0	0
$997$	−42.8898	−1.35833	−0.679167	−	0.733984i	$-0.737659\pi$
$997$	−42.8898	−1.35833	−0.679167	+	0.733984i	$0.737659\pi$
$998$	0	0
$999$	−66.3335	−2.09870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3364.2.a.p.1.6		6
29.4	even	14		116.2.g.b.45.1	✓	12
29.12	odd	4		3364.2.c.j.1681.12		12
29.17	odd	4		3364.2.c.j.1681.1		12
29.22	even	14		116.2.g.b.49.1	yes	12
29.28	even	2		3364.2.a.m.1.1		6
87.62	odd	14		1044.2.u.c.973.2		12
87.80	odd	14		1044.2.u.c.397.2		12
116.51	odd	14		464.2.u.g.49.2		12
116.91	odd	14		464.2.u.g.161.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.g.b.45.1	✓	12	29.4	even	14
116.2.g.b.49.1	yes	12	29.22	even	14
464.2.u.g.49.2		12	116.51	odd	14
464.2.u.g.161.2		12	116.91	odd	14
1044.2.u.c.397.2		12	87.80	odd	14
1044.2.u.c.973.2		12	87.62	odd	14
3364.2.a.m.1.1		6	29.28	even	2
3364.2.a.p.1.6		6	1.1	even	1	trivial
3364.2.c.j.1681.1		12	29.17	odd	4
3364.2.c.j.1681.12		12	29.12	odd	4

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:	\( N \)	\(=\)	\( 3364 = 2^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3364.a (trivial)

\(f(q)\)	\(=\)	\(q+3.08697 q^{3} +3.51360 q^{5} -1.08697 q^{7} +6.52935 q^{9} -1.84938 q^{11} +6.57945 q^{13} +10.8464 q^{15} -0.658856 q^{17} -3.32428 q^{19} -3.35542 q^{21} +3.89074 q^{23} +7.34540 q^{25} +10.8950 q^{27} -3.00066 q^{31} -5.70898 q^{33} -3.81916 q^{35} -6.08844 q^{37} +20.3105 q^{39} -5.19016 q^{41} +1.17695 q^{43} +22.9415 q^{45} -6.21164 q^{47} -5.81851 q^{49} -2.03387 q^{51} -1.40581 q^{53} -6.49799 q^{55} -10.2619 q^{57} -9.55267 q^{59} +8.30080 q^{61} -7.09718 q^{63} +23.1176 q^{65} -12.3977 q^{67} +12.0106 q^{69} +8.12645 q^{71} +8.61593 q^{73} +22.6750 q^{75} +2.01021 q^{77} -5.60250 q^{79} +14.0444 q^{81} -3.72172 q^{83} -2.31496 q^{85} -4.47002 q^{89} -7.15163 q^{91} -9.26292 q^{93} -11.6802 q^{95} -8.00445 q^{97} -12.0753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} + 10 q^{5} + 7 q^{7} + 11 q^{9} + 9 q^{11} + 14 q^{13} + 10 q^{15} + 5 q^{17} + 9 q^{19} - 19 q^{21} + 15 q^{23} + 16 q^{25} + 20 q^{27} - 11 q^{31} + 22 q^{33} + 10 q^{35} - 16 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 + 10 * q^5 + 7 * q^7 + 11 * q^9 + 9 * q^11 + 14 * q^13 + 10 * q^15 + 5 * q^17 + 9 * q^19 - 19 * q^21 + 15 * q^23 + 16 * q^25 + 20 * q^27 - 11 * q^31 + 22 * q^33 + 10 * q^35 - 16 * q^37 + 22 * q^39 + q^41 + 15 * q^43 + 32 * q^45 - 27 * q^47 - 9 * q^49 - 20 * q^51 + 18 * q^53 + 14 * q^55 + 15 * q^57 + 18 * q^59 - 22 * q^61 - 13 * q^63 + 26 * q^65 + 9 * q^67 - 6 * q^69 - 7 * q^71 + 32 * q^73 + 22 * q^75 - 4 * q^77 - 29 * q^79 - 2 * q^81 + 3 * q^83 + 24 * q^85 - 12 * q^89 + 6 * q^91 + 14 * q^93 - 28 * q^95 + 58 * q^97 + 8 * q^99

Introduction

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