LMFDB - Embedded newform 3696.2.a.bo.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3696,2,Mod(1,3696)] 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("3696.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3696, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,4,0,3,0,3,0,3,0,-4,0,-4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	3696.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.18953 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.17313 q^{13} -4.18953 q^{15} +6.85446 q^{17} +0.318669 q^{19} -1.00000 q^{21} +1.87086 q^{23} +12.5522 q^{25} -1.00000 q^{27} -3.17313 q^{29} -9.23353 q^{31} -1.00000 q^{33} +4.18953 q^{35} -7.55220 q^{37} +3.17313 q^{39} +9.36266 q^{41} +10.8873 q^{43} +4.18953 q^{45} +8.06040 q^{47} +1.00000 q^{49} -6.85446 q^{51} +0.508203 q^{53} +4.18953 q^{55} -0.318669 q^{57} +7.04399 q^{59} -2.00000 q^{61} +1.00000 q^{63} -13.2939 q^{65} +2.66492 q^{67} -1.87086 q^{69} +5.01641 q^{71} -4.82687 q^{73} -12.5522 q^{75} +1.00000 q^{77} -5.01641 q^{79} +1.00000 q^{81} -3.52461 q^{83} +28.7170 q^{85} +3.17313 q^{87} -1.74173 q^{89} -3.17313 q^{91} +9.23353 q^{93} +1.33508 q^{95} -12.2499 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 + 8 * q^17 + 8 * q^19 - 3 * q^21 - 10 * q^23 + 15 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 + 4 * q^35 + 4 * q^39 + 14 * q^41 + 14 * q^43 + 4 * q^45 + 3 * q^49 - 8 * q^51 + 4 * q^55 - 8 * q^57 - 6 * q^61 + 3 * q^63 + 14 * q^65 + 4 * q^67 + 10 * q^69 + 12 * q^71 - 20 * q^73 - 15 * q^75 + 3 * q^77 - 12 * q^79 + 3 * q^81 - 6 * q^83 - 6 * q^85 + 4 * q^87 + 26 * q^89 - 4 * q^91 - 2 * q^93 + 8 * q^95 - 4 * q^97 + 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.18953	1.87362	0.936808	−	0.349843i	$-0.113765\pi$
$5$	4.18953	1.87362	0.936808	+	0.349843i	$0.113765\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.17313	−0.880067	−0.440034	−	0.897981i	$-0.645033\pi$
$13$	−3.17313	−0.880067	−0.440034	+	0.897981i	$0.645033\pi$
$14$	0	0
$15$	−4.18953	−1.08173
$16$	0	0
$17$	6.85446	1.66245	0.831225	−	0.555936i	$-0.187640\pi$
$17$	6.85446	1.66245	0.831225	+	0.555936i	$0.187640\pi$
$18$	0	0
$19$	0.318669	0.0731078	0.0365539	−	0.999332i	$-0.488362\pi$
$19$	0.318669	0.0731078	0.0365539	+	0.999332i	$0.488362\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.87086	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$23$	1.87086	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$24$	0	0
$25$	12.5522	2.51044
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.17313	−0.589235	−0.294617	−	0.955615i	$-0.595192\pi$
$29$	−3.17313	−0.589235	−0.294617	+	0.955615i	$0.595192\pi$
$30$	0	0
$31$	−9.23353	−1.65839	−0.829195	−	0.558959i	$-0.811201\pi$
$31$	−9.23353	−1.65839	−0.829195	+	0.558959i	$0.811201\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	4.18953	0.708161
$36$	0	0
$37$	−7.55220	−1.24157	−0.620787	−	0.783980i	$-0.713187\pi$
$37$	−7.55220	−1.24157	−0.620787	+	0.783980i	$0.713187\pi$
$38$	0	0
$39$	3.17313	0.508107
$40$	0	0
$41$	9.36266	1.46220	0.731101	−	0.682269i	$-0.239007\pi$
$41$	9.36266	1.46220	0.731101	+	0.682269i	$0.239007\pi$
$42$	0	0
$43$	10.8873	1.66029	0.830147	−	0.557545i	$-0.188257\pi$
$43$	10.8873	1.66029	0.830147	+	0.557545i	$0.188257\pi$
$44$	0	0
$45$	4.18953	0.624539
$46$	0	0
$47$	8.06040	1.17573	0.587865	−	0.808959i	$-0.299969\pi$
$47$	8.06040	1.17573	0.587865	+	0.808959i	$0.299969\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.85446	−0.959816
$52$	0	0
$53$	0.508203	0.0698071	0.0349036	−	0.999391i	$-0.488888\pi$
$53$	0.508203	0.0698071	0.0349036	+	0.999391i	$0.488888\pi$
$54$	0	0
$55$	4.18953	0.564917
$56$	0	0
$57$	−0.318669	−0.0422088
$58$	0	0
$59$	7.04399	0.917050	0.458525	−	0.888682i	$-0.348378\pi$
$59$	7.04399	0.917050	0.458525	+	0.888682i	$0.348378\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−13.2939	−1.64891
$66$	0	0
$67$	2.66492	0.325572	0.162786	−	0.986661i	$-0.447952\pi$
$67$	2.66492	0.325572	0.162786	+	0.986661i	$0.447952\pi$
$68$	0	0
$69$	−1.87086	−0.225226
$70$	0	0
$71$	5.01641	0.595338	0.297669	−	0.954669i	$-0.403791\pi$
$71$	5.01641	0.595338	0.297669	+	0.954669i	$0.403791\pi$
$72$	0	0
$73$	−4.82687	−0.564943	−0.282471	−	0.959276i	$-0.591154\pi$
$73$	−4.82687	−0.564943	−0.282471	+	0.959276i	$0.591154\pi$
$74$	0	0
$75$	−12.5522	−1.44940
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−5.01641	−0.564390	−0.282195	−	0.959357i	$-0.591062\pi$
$79$	−5.01641	−0.564390	−0.282195	+	0.959357i	$0.591062\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.52461	−0.386876	−0.193438	−	0.981112i	$-0.561964\pi$
$83$	−3.52461	−0.386876	−0.193438	+	0.981112i	$0.561964\pi$
$84$	0	0
$85$	28.7170	3.11479
$86$	0	0
$87$	3.17313	0.340195
$88$	0	0
$89$	−1.74173	−0.184623	−0.0923115	−	0.995730i	$-0.529426\pi$
$89$	−1.74173	−0.184623	−0.0923115	+	0.995730i	$0.529426\pi$
$90$	0	0
$91$	−3.17313	−0.332634
$92$	0	0
$93$	9.23353	0.957472
$94$	0	0
$95$	1.33508	0.136976
$96$	0	0
$97$	−12.2499	−1.24379	−0.621896	−	0.783100i	$-0.713637\pi$
$97$	−12.2499	−1.24379	−0.621896	+	0.783100i	$0.713637\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	4.88727	0.486302	0.243151	−	0.969988i	$-0.421819\pi$
$101$	4.88727	0.486302	0.243151	+	0.969988i	$0.421819\pi$
$102$	0	0
$103$	0.637339	0.0627988	0.0313994	−	0.999507i	$-0.490004\pi$
$103$	0.637339	0.0627988	0.0313994	+	0.999507i	$0.490004\pi$
$104$	0	0
$105$	−4.18953	−0.408857
$106$	0	0
$107$	−0.956008	−0.0924208	−0.0462104	−	0.998932i	$-0.514714\pi$
$107$	−0.956008	−0.0924208	−0.0462104	+	0.998932i	$0.514714\pi$
$108$	0	0
$109$	−7.61259	−0.729154	−0.364577	−	0.931173i	$-0.618786\pi$
$109$	−7.61259	−0.729154	−0.364577	+	0.931173i	$0.618786\pi$
$110$	0	0
$111$	7.55220	0.716823
$112$	0	0
$113$	−7.70892	−0.725194	−0.362597	−	0.931946i	$-0.618110\pi$
$113$	−7.70892	−0.725194	−0.362597	+	0.931946i	$0.618110\pi$
$114$	0	0
$115$	7.83805	0.730902
$116$	0	0
$117$	−3.17313	−0.293356
$118$	0	0
$119$	6.85446	0.628347
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.36266	−0.844203
$124$	0	0
$125$	31.6402	2.82998
$126$	0	0
$127$	5.49180	0.487318	0.243659	−	0.969861i	$-0.421652\pi$
$127$	5.49180	0.487318	0.243659	+	0.969861i	$0.421652\pi$
$128$	0	0
$129$	−10.8873	−0.958571
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0.318669	0.0276321
$134$	0	0
$135$	−4.18953	−0.360578
$136$	0	0
$137$	15.6126	1.33387	0.666937	−	0.745114i	$-0.267605\pi$
$137$	15.6126	1.33387	0.666937	+	0.745114i	$0.267605\pi$
$138$	0	0
$139$	9.01641	0.764762	0.382381	−	0.924005i	$-0.375104\pi$
$139$	9.01641	0.764762	0.382381	+	0.924005i	$0.375104\pi$
$140$	0	0
$141$	−8.06040	−0.678808
$142$	0	0
$143$	−3.17313	−0.265350
$144$	0	0
$145$	−13.2939	−1.10400
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−5.20594	−0.426487	−0.213244	−	0.976999i	$-0.568403\pi$
$149$	−5.20594	−0.426487	−0.213244	+	0.976999i	$0.568403\pi$
$150$	0	0
$151$	−6.24993	−0.508612	−0.254306	−	0.967124i	$-0.581847\pi$
$151$	−6.24993	−0.508612	−0.254306	+	0.967124i	$0.581847\pi$
$152$	0	0
$153$	6.85446	0.554150
$154$	0	0
$155$	−38.6842	−3.10719
$156$	0	0
$157$	−18.1208	−1.44620	−0.723099	−	0.690745i	$-0.757283\pi$
$157$	−18.1208	−1.44620	−0.723099	+	0.690745i	$0.757283\pi$
$158$	0	0
$159$	−0.508203	−0.0403031
$160$	0	0
$161$	1.87086	0.147445
$162$	0	0
$163$	2.66492	0.208733	0.104366	−	0.994539i	$-0.466719\pi$
$163$	2.66492	0.208733	0.104366	+	0.994539i	$0.466719\pi$
$164$	0	0
$165$	−4.18953	−0.326155
$166$	0	0
$167$	11.1455	0.862468	0.431234	−	0.902240i	$-0.358078\pi$
$167$	11.1455	0.862468	0.431234	+	0.902240i	$0.358078\pi$
$168$	0	0
$169$	−2.93126	−0.225482
$170$	0	0
$171$	0.318669	0.0243693
$172$	0	0
$173$	−24.8461	−1.88902	−0.944508	−	0.328489i	$-0.893461\pi$
$173$	−24.8461	−1.88902	−0.944508	+	0.328489i	$0.893461\pi$
$174$	0	0
$175$	12.5522	0.948857
$176$	0	0
$177$	−7.04399	−0.529459
$178$	0	0
$179$	12.7581	0.953588	0.476794	−	0.879015i	$-0.341799\pi$
$179$	12.7581	0.953588	0.476794	+	0.879015i	$0.341799\pi$
$180$	0	0
$181$	−3.23353	−0.240346	−0.120173	−	0.992753i	$-0.538345\pi$
$181$	−3.23353	−0.240346	−0.120173	+	0.992753i	$0.538345\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−31.6402	−2.32623
$186$	0	0
$187$	6.85446	0.501248
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−20.9753	−1.51772	−0.758858	−	0.651256i	$-0.774242\pi$
$191$	−20.9753	−1.51772	−0.758858	+	0.651256i	$0.774242\pi$
$192$	0	0
$193$	0.249933	0.0179905	0.00899527	−	0.999960i	$-0.497137\pi$
$193$	0.249933	0.0179905	0.00899527	+	0.999960i	$0.497137\pi$
$194$	0	0
$195$	13.2939	0.951998
$196$	0	0
$197$	18.4999	1.31806	0.659030	−	0.752116i	$-0.270967\pi$
$197$	18.4999	1.31806	0.659030	+	0.752116i	$0.270967\pi$
$198$	0	0
$199$	−9.87086	−0.699727	−0.349864	−	0.936801i	$-0.613772\pi$
$199$	−9.87086	−0.699727	−0.349864	+	0.936801i	$0.613772\pi$
$200$	0	0
$201$	−2.66492	−0.187969
$202$	0	0
$203$	−3.17313	−0.222710
$204$	0	0
$205$	39.2252	2.73961
$206$	0	0
$207$	1.87086	0.130034
$208$	0	0
$209$	0.318669	0.0220428
$210$	0	0
$211$	4.63734	0.319248	0.159624	−	0.987178i	$-0.448972\pi$
$211$	4.63734	0.319248	0.159624	+	0.987178i	$0.448972\pi$
$212$	0	0
$213$	−5.01641	−0.343719
$214$	0	0
$215$	45.6126	3.11075
$216$	0	0
$217$	−9.23353	−0.626813
$218$	0	0
$219$	4.82687	0.326170
$220$	0	0
$221$	−21.7501	−1.46307
$222$	0	0
$223$	18.3463	1.22856	0.614278	−	0.789090i	$-0.289447\pi$
$223$	18.3463	1.22856	0.614278	+	0.789090i	$0.289447\pi$
$224$	0	0
$225$	12.5522	0.836813
$226$	0	0
$227$	−0.379068	−0.0251596	−0.0125798	−	0.999921i	$-0.504004\pi$
$227$	−0.379068	−0.0251596	−0.0125798	+	0.999921i	$0.504004\pi$
$228$	0	0
$229$	24.9424	1.64824	0.824121	−	0.566413i	$-0.191669\pi$
$229$	24.9424	1.64824	0.824121	+	0.566413i	$0.191669\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	23.4506	1.53630	0.768151	−	0.640268i	$-0.221177\pi$
$233$	23.4506	1.53630	0.768151	+	0.640268i	$0.221177\pi$
$234$	0	0
$235$	33.7693	2.20287
$236$	0	0
$237$	5.01641	0.325851
$238$	0	0
$239$	−5.07681	−0.328391	−0.164196	−	0.986428i	$-0.552503\pi$
$239$	−5.07681	−0.328391	−0.164196	+	0.986428i	$0.552503\pi$
$240$	0	0
$241$	19.2939	1.24283	0.621415	−	0.783481i	$-0.286558\pi$
$241$	19.2939	1.24283	0.621415	+	0.783481i	$0.286558\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.18953	0.267660
$246$	0	0
$247$	−1.01118	−0.0643397
$248$	0	0
$249$	3.52461	0.223363
$250$	0	0
$251$	−23.8021	−1.50238	−0.751188	−	0.660088i	$-0.770519\pi$
$251$	−23.8021	−1.50238	−0.751188	+	0.660088i	$0.770519\pi$
$252$	0	0
$253$	1.87086	0.117620
$254$	0	0
$255$	−28.7170	−1.79833
$256$	0	0
$257$	14.9149	0.930363	0.465182	−	0.885215i	$-0.345989\pi$
$257$	14.9149	0.930363	0.465182	+	0.885215i	$0.345989\pi$
$258$	0	0
$259$	−7.55220	−0.469271
$260$	0	0
$261$	−3.17313	−0.196412
$262$	0	0
$263$	−2.92319	−0.180252	−0.0901259	−	0.995930i	$-0.528727\pi$
$263$	−2.92319	−0.180252	−0.0901259	+	0.995930i	$0.528727\pi$
$264$	0	0
$265$	2.12914	0.130792
$266$	0	0
$267$	1.74173	0.106592
$268$	0	0
$269$	11.9672	0.729652	0.364826	−	0.931076i	$-0.381128\pi$
$269$	11.9672	0.729652	0.364826	+	0.931076i	$0.381128\pi$
$270$	0	0
$271$	20.3187	1.23427	0.617136	−	0.786857i	$-0.288293\pi$
$271$	20.3187	1.23427	0.617136	+	0.786857i	$0.288293\pi$
$272$	0	0
$273$	3.17313	0.192046
$274$	0	0
$275$	12.5522	0.756926
$276$	0	0
$277$	18.0552	1.08483	0.542415	−	0.840111i	$-0.317510\pi$
$277$	18.0552	1.08483	0.542415	+	0.840111i	$0.317510\pi$
$278$	0	0
$279$	−9.23353	−0.552797
$280$	0	0
$281$	27.2939	1.62822	0.814110	−	0.580711i	$-0.197226\pi$
$281$	27.2939	1.62822	0.814110	+	0.580711i	$0.197226\pi$
$282$	0	0
$283$	−29.8901	−1.77678	−0.888391	−	0.459087i	$-0.848177\pi$
$283$	−29.8901	−1.77678	−0.888391	+	0.459087i	$0.848177\pi$
$284$	0	0
$285$	−1.33508	−0.0790831
$286$	0	0
$287$	9.36266	0.552660
$288$	0	0
$289$	29.9836	1.76374
$290$	0	0
$291$	12.2499	0.718104
$292$	0	0
$293$	−4.34625	−0.253911	−0.126955	−	0.991908i	$-0.540521\pi$
$293$	−4.34625	−0.253911	−0.126955	+	0.991908i	$0.540521\pi$
$294$	0	0
$295$	29.5110	1.71820
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.93649	−0.343316
$300$	0	0
$301$	10.8873	0.627532
$302$	0	0
$303$	−4.88727	−0.280766
$304$	0	0
$305$	−8.37907	−0.479784
$306$	0	0
$307$	20.7581	1.18473	0.592365	−	0.805670i	$-0.298194\pi$
$307$	20.7581	1.18473	0.592365	+	0.805670i	$0.298194\pi$
$308$	0	0
$309$	−0.637339	−0.0362569
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	8.12914	0.459486	0.229743	−	0.973251i	$-0.426211\pi$
$313$	8.12914	0.459486	0.229743	+	0.973251i	$0.426211\pi$
$314$	0	0
$315$	4.18953	0.236054
$316$	0	0
$317$	−19.9917	−1.12284	−0.561422	−	0.827530i	$-0.689745\pi$
$317$	−19.9917	−1.12284	−0.561422	+	0.827530i	$0.689745\pi$
$318$	0	0
$319$	−3.17313	−0.177661
$320$	0	0
$321$	0.956008	0.0533592
$322$	0	0
$323$	2.18431	0.121538
$324$	0	0
$325$	−39.8297	−2.20935
$326$	0	0
$327$	7.61259	0.420977
$328$	0	0
$329$	8.06040	0.444384
$330$	0	0
$331$	−17.0164	−0.935306	−0.467653	−	0.883912i	$-0.654900\pi$
$331$	−17.0164	−0.935306	−0.467653	+	0.883912i	$0.654900\pi$
$332$	0	0
$333$	−7.55220	−0.413858
$334$	0	0
$335$	11.1648	0.609998
$336$	0	0
$337$	1.52461	0.0830508	0.0415254	−	0.999137i	$-0.486778\pi$
$337$	1.52461	0.0830508	0.0415254	+	0.999137i	$0.486778\pi$
$338$	0	0
$339$	7.70892	0.418691
$340$	0	0
$341$	−9.23353	−0.500023
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.83805	−0.421986
$346$	0	0
$347$	−17.4506	−0.936800	−0.468400	−	0.883517i	$-0.655169\pi$
$347$	−17.4506	−0.936800	−0.468400	+	0.883517i	$0.655169\pi$
$348$	0	0
$349$	6.85969	0.367191	0.183595	−	0.983002i	$-0.441226\pi$
$349$	6.85969	0.367191	0.183595	+	0.983002i	$0.441226\pi$
$350$	0	0
$351$	3.17313	0.169369
$352$	0	0
$353$	−31.9313	−1.69953	−0.849765	−	0.527162i	$-0.823256\pi$
$353$	−31.9313	−1.69953	−0.849765	+	0.527162i	$0.823256\pi$
$354$	0	0
$355$	21.0164	1.11544
$356$	0	0
$357$	−6.85446	−0.362776
$358$	0	0
$359$	24.4342	1.28959	0.644795	−	0.764356i	$-0.276943\pi$
$359$	24.4342	1.28959	0.644795	+	0.764356i	$0.276943\pi$
$360$	0	0
$361$	−18.8984	−0.994655
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−20.2223	−1.05849
$366$	0	0
$367$	−17.0716	−0.891129	−0.445565	−	0.895250i	$-0.646997\pi$
$367$	−17.0716	−0.891129	−0.445565	+	0.895250i	$0.646997\pi$
$368$	0	0
$369$	9.36266	0.487401
$370$	0	0
$371$	0.508203	0.0263846
$372$	0	0
$373$	3.17836	0.164569	0.0822845	−	0.996609i	$-0.473778\pi$
$373$	3.17836	0.164569	0.0822845	+	0.996609i	$0.473778\pi$
$374$	0	0
$375$	−31.6402	−1.63389
$376$	0	0
$377$	10.0687	0.518566
$378$	0	0
$379$	−3.93960	−0.202364	−0.101182	−	0.994868i	$-0.532262\pi$
$379$	−3.93960	−0.202364	−0.101182	+	0.994868i	$0.532262\pi$
$380$	0	0
$381$	−5.49180	−0.281353
$382$	0	0
$383$	24.7581	1.26508	0.632541	−	0.774527i	$-0.282012\pi$
$383$	24.7581	1.26508	0.632541	+	0.774527i	$0.282012\pi$
$384$	0	0
$385$	4.18953	0.213518
$386$	0	0
$387$	10.8873	0.553431
$388$	0	0
$389$	3.65375	0.185252	0.0926261	−	0.995701i	$-0.470474\pi$
$389$	3.65375	0.185252	0.0926261	+	0.995701i	$0.470474\pi$
$390$	0	0
$391$	12.8238	0.648526
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−21.0164	−1.05745
$396$	0	0
$397$	4.56337	0.229029	0.114515	−	0.993422i	$-0.463469\pi$
$397$	4.56337	0.229029	0.114515	+	0.993422i	$0.463469\pi$
$398$	0	0
$399$	−0.318669	−0.0159534
$400$	0	0
$401$	−7.23353	−0.361225	−0.180613	−	0.983554i	$-0.557808\pi$
$401$	−7.23353	−0.361225	−0.180613	+	0.983554i	$0.557808\pi$
$402$	0	0
$403$	29.2992	1.45949
$404$	0	0
$405$	4.18953	0.208180
$406$	0	0
$407$	−7.55220	−0.374348
$408$	0	0
$409$	5.30749	0.262439	0.131219	−	0.991353i	$-0.458111\pi$
$409$	5.30749	0.262439	0.131219	+	0.991353i	$0.458111\pi$
$410$	0	0
$411$	−15.6126	−0.770112
$412$	0	0
$413$	7.04399	0.346612
$414$	0	0
$415$	−14.7665	−0.724858
$416$	0	0
$417$	−9.01641	−0.441535
$418$	0	0
$419$	11.4231	0.558053	0.279026	−	0.960283i	$-0.409988\pi$
$419$	11.4231	0.558053	0.279026	+	0.960283i	$0.409988\pi$
$420$	0	0
$421$	−27.4147	−1.33611	−0.668056	−	0.744111i	$-0.732873\pi$
$421$	−27.4147	−1.33611	−0.668056	+	0.744111i	$0.732873\pi$
$422$	0	0
$423$	8.06040	0.391910
$424$	0	0
$425$	86.0385	4.17348
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	3.17313	0.153200
$430$	0	0
$431$	2.28586	0.110106	0.0550529	−	0.998483i	$-0.482467\pi$
$431$	2.28586	0.110106	0.0550529	+	0.998483i	$0.482467\pi$
$432$	0	0
$433$	31.5714	1.51723	0.758613	−	0.651541i	$-0.225877\pi$
$433$	31.5714	1.51723	0.758613	+	0.651541i	$0.225877\pi$
$434$	0	0
$435$	13.2939	0.637395
$436$	0	0
$437$	0.596187	0.0285195
$438$	0	0
$439$	18.5275	0.884267	0.442133	−	0.896949i	$-0.354222\pi$
$439$	18.5275	0.884267	0.442133	+	0.896949i	$0.354222\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.4342	−1.35095	−0.675476	−	0.737382i	$-0.736062\pi$
$443$	−28.4342	−1.35095	−0.675476	+	0.737382i	$0.736062\pi$
$444$	0	0
$445$	−7.29703	−0.345913
$446$	0	0
$447$	5.20594	0.246233
$448$	0	0
$449$	−5.68656	−0.268365	−0.134183	−	0.990957i	$-0.542841\pi$
$449$	−5.68656	−0.268365	−0.134183	+	0.990957i	$0.542841\pi$
$450$	0	0
$451$	9.36266	0.440871
$452$	0	0
$453$	6.24993	0.293647
$454$	0	0
$455$	−13.2939	−0.623229
$456$	0	0
$457$	−13.0081	−0.608492	−0.304246	−	0.952594i	$-0.598404\pi$
$457$	−13.0081	−0.608492	−0.304246	+	0.952594i	$0.598404\pi$
$458$	0	0
$459$	−6.85446	−0.319939
$460$	0	0
$461$	−6.37907	−0.297103	−0.148551	−	0.988905i	$-0.547461\pi$
$461$	−6.37907	−0.297103	−0.148551	+	0.988905i	$0.547461\pi$
$462$	0	0
$463$	−34.0932	−1.58445	−0.792223	−	0.610232i	$-0.791076\pi$
$463$	−34.0932	−1.58445	−0.792223	+	0.610232i	$0.791076\pi$
$464$	0	0
$465$	38.6842	1.79394
$466$	0	0
$467$	−18.1484	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$467$	−18.1484	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$468$	0	0
$469$	2.66492	0.123055
$470$	0	0
$471$	18.1208	0.834962
$472$	0	0
$473$	10.8873	0.500597
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0.508203	0.0232690
$478$	0	0
$479$	42.7498	1.95329	0.976644	−	0.214864i	$-0.0689307\pi$
$479$	42.7498	1.95329	0.976644	+	0.214864i	$0.0689307\pi$
$480$	0	0
$481$	23.9641	1.09267
$482$	0	0
$483$	−1.87086	−0.0851273
$484$	0	0
$485$	−51.3215	−2.33039
$486$	0	0
$487$	−26.7909	−1.21401	−0.607007	−	0.794697i	$-0.707630\pi$
$487$	−26.7909	−1.21401	−0.607007	+	0.794697i	$0.707630\pi$
$488$	0	0
$489$	−2.66492	−0.120512
$490$	0	0
$491$	−31.6813	−1.42976	−0.714879	−	0.699248i	$-0.753518\pi$
$491$	−31.6813	−1.42976	−0.714879	+	0.699248i	$0.753518\pi$
$492$	0	0
$493$	−21.7501	−0.979574
$494$	0	0
$495$	4.18953	0.188306
$496$	0	0
$497$	5.01641	0.225017
$498$	0	0
$499$	24.1260	1.08003	0.540015	−	0.841656i	$-0.318419\pi$
$499$	24.1260	1.08003	0.540015	+	0.841656i	$0.318419\pi$
$500$	0	0
$501$	−11.1455	−0.497946
$502$	0	0
$503$	−30.8873	−1.37720	−0.688598	−	0.725144i	$-0.741773\pi$
$503$	−30.8873	−1.37720	−0.688598	+	0.725144i	$0.741773\pi$
$504$	0	0
$505$	20.4754	0.911143
$506$	0	0
$507$	2.93126	0.130182
$508$	0	0
$509$	28.2088	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$509$	28.2088	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$510$	0	0
$511$	−4.82687	−0.213528
$512$	0	0
$513$	−0.318669	−0.0140696
$514$	0	0
$515$	2.67015	0.117661
$516$	0	0
$517$	8.06040	0.354496
$518$	0	0
$519$	24.8461	1.09062
$520$	0	0
$521$	−33.7610	−1.47910	−0.739548	−	0.673104i	$-0.764961\pi$
$521$	−33.7610	−1.47910	−0.739548	+	0.673104i	$0.764961\pi$
$522$	0	0
$523$	−12.8185	−0.560515	−0.280258	−	0.959925i	$-0.590420\pi$
$523$	−12.8185	−0.560515	−0.280258	+	0.959925i	$0.590420\pi$
$524$	0	0
$525$	−12.5522	−0.547823
$526$	0	0
$527$	−63.2908	−2.75699
$528$	0	0
$529$	−19.4999	−0.847820
$530$	0	0
$531$	7.04399	0.305683
$532$	0	0
$533$	−29.7089	−1.28684
$534$	0	0
$535$	−4.00523	−0.173161
$536$	0	0
$537$	−12.7581	−0.550554
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−21.8625	−0.939943	−0.469972	−	0.882681i	$-0.655736\pi$
$541$	−21.8625	−0.939943	−0.469972	+	0.882681i	$0.655736\pi$
$542$	0	0
$543$	3.23353	0.138764
$544$	0	0
$545$	−31.8932	−1.36616
$546$	0	0
$547$	−23.4178	−1.00127	−0.500637	−	0.865657i	$-0.666901\pi$
$547$	−23.4178	−1.00127	−0.500637	+	0.865657i	$0.666901\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−1.01118	−0.0430776
$552$	0	0
$553$	−5.01641	−0.213319
$554$	0	0
$555$	31.6402	1.34305
$556$	0	0
$557$	−6.91486	−0.292992	−0.146496	−	0.989211i	$-0.546800\pi$
$557$	−6.91486	−0.292992	−0.146496	+	0.989211i	$0.546800\pi$
$558$	0	0
$559$	−34.5467	−1.46117
$560$	0	0
$561$	−6.85446	−0.289395
$562$	0	0
$563$	−11.8381	−0.498914	−0.249457	−	0.968386i	$-0.580252\pi$
$563$	−11.8381	−0.498914	−0.249457	+	0.968386i	$0.580252\pi$
$564$	0	0
$565$	−32.2968	−1.35874
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	15.2252	0.637154	0.318577	−	0.947897i	$-0.396795\pi$
$571$	15.2252	0.637154	0.318577	+	0.947897i	$0.396795\pi$
$572$	0	0
$573$	20.9753	0.876254
$574$	0	0
$575$	23.4835	0.979328
$576$	0	0
$577$	17.3215	0.721104	0.360552	−	0.932739i	$-0.382588\pi$
$577$	17.3215	0.721104	0.360552	+	0.932739i	$0.382588\pi$
$578$	0	0
$579$	−0.249933	−0.0103868
$580$	0	0
$581$	−3.52461	−0.146225
$582$	0	0
$583$	0.508203	0.0210476
$584$	0	0
$585$	−13.2939	−0.549636
$586$	0	0
$587$	−27.8678	−1.15023	−0.575113	−	0.818074i	$-0.695042\pi$
$587$	−27.8678	−1.15023	−0.575113	+	0.818074i	$0.695042\pi$
$588$	0	0
$589$	−2.94244	−0.121241
$590$	0	0
$591$	−18.4999	−0.760983
$592$	0	0
$593$	−24.3463	−0.999781	−0.499890	−	0.866089i	$-0.666626\pi$
$593$	−24.3463	−0.999781	−0.499890	+	0.866089i	$0.666626\pi$
$594$	0	0
$595$	28.7170	1.17728
$596$	0	0
$597$	9.87086	0.403988
$598$	0	0
$599$	−21.0164	−0.858707	−0.429354	−	0.903136i	$-0.641259\pi$
$599$	−21.0164	−0.858707	−0.429354	+	0.903136i	$0.641259\pi$
$600$	0	0
$601$	40.9477	1.67029	0.835145	−	0.550030i	$-0.185384\pi$
$601$	40.9477	1.67029	0.835145	+	0.550030i	$0.185384\pi$
$602$	0	0
$603$	2.66492	0.108524
$604$	0	0
$605$	4.18953	0.170329
$606$	0	0
$607$	14.0276	0.569362	0.284681	−	0.958622i	$-0.408112\pi$
$607$	14.0276	0.569362	0.284681	+	0.958622i	$0.408112\pi$
$608$	0	0
$609$	3.17313	0.128582
$610$	0	0
$611$	−25.5767	−1.03472
$612$	0	0
$613$	18.5306	0.748442	0.374221	−	0.927339i	$-0.377910\pi$
$613$	18.5306	0.748442	0.374221	+	0.927339i	$0.377910\pi$
$614$	0	0
$615$	−39.2252	−1.58171
$616$	0	0
$617$	2.43424	0.0979987	0.0489994	−	0.998799i	$-0.484397\pi$
$617$	2.43424	0.0979987	0.0489994	+	0.998799i	$0.484397\pi$
$618$	0	0
$619$	30.4259	1.22292	0.611460	−	0.791275i	$-0.290582\pi$
$619$	30.4259	1.22292	0.611460	+	0.791275i	$0.290582\pi$
$620$	0	0
$621$	−1.87086	−0.0750752
$622$	0	0
$623$	−1.74173	−0.0697809
$624$	0	0
$625$	69.7966	2.79187
$626$	0	0
$627$	−0.318669	−0.0127264
$628$	0	0
$629$	−51.7662	−2.06405
$630$	0	0
$631$	−34.9836	−1.39267	−0.696337	−	0.717715i	$-0.745188\pi$
$631$	−34.9836	−1.39267	−0.696337	+	0.717715i	$0.745188\pi$
$632$	0	0
$633$	−4.63734	−0.184318
$634$	0	0
$635$	23.0081	0.913047
$636$	0	0
$637$	−3.17313	−0.125724
$638$	0	0
$639$	5.01641	0.198446
$640$	0	0
$641$	8.56337	0.338233	0.169116	−	0.985596i	$-0.445909\pi$
$641$	8.56337	0.338233	0.169116	+	0.985596i	$0.445909\pi$
$642$	0	0
$643$	5.11273	0.201626	0.100813	−	0.994905i	$-0.467856\pi$
$643$	5.11273	0.201626	0.100813	+	0.994905i	$0.467856\pi$
$644$	0	0
$645$	−45.6126	−1.79599
$646$	0	0
$647$	−25.7693	−1.01310	−0.506548	−	0.862212i	$-0.669079\pi$
$647$	−25.7693	−1.01310	−0.506548	+	0.862212i	$0.669079\pi$
$648$	0	0
$649$	7.04399	0.276501
$650$	0	0
$651$	9.23353	0.361890
$652$	0	0
$653$	−47.8953	−1.87429	−0.937145	−	0.348941i	$-0.886541\pi$
$653$	−47.8953	−1.87429	−0.937145	+	0.348941i	$0.886541\pi$
$654$	0	0
$655$	−16.7581	−0.654795
$656$	0	0
$657$	−4.82687	−0.188314
$658$	0	0
$659$	−8.95601	−0.348877	−0.174438	−	0.984668i	$-0.555811\pi$
$659$	−8.95601	−0.348877	−0.174438	+	0.984668i	$0.555811\pi$
$660$	0	0
$661$	−10.7993	−0.420044	−0.210022	−	0.977697i	$-0.567353\pi$
$661$	−10.7993	−0.420044	−0.210022	+	0.977697i	$0.567353\pi$
$662$	0	0
$663$	21.7501	0.844703
$664$	0	0
$665$	1.33508	0.0517720
$666$	0	0
$667$	−5.93649	−0.229862
$668$	0	0
$669$	−18.3463	−0.709307
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	32.2499	1.24314	0.621572	−	0.783357i	$-0.286494\pi$
$673$	32.2499	1.24314	0.621572	+	0.783357i	$0.286494\pi$
$674$	0	0
$675$	−12.5522	−0.483134
$676$	0	0
$677$	−27.7089	−1.06494	−0.532470	−	0.846449i	$-0.678736\pi$
$677$	−27.7089	−1.06494	−0.532470	+	0.846449i	$0.678736\pi$
$678$	0	0
$679$	−12.2499	−0.470109
$680$	0	0
$681$	0.379068	0.0145259
$682$	0	0
$683$	−22.3156	−0.853881	−0.426941	−	0.904280i	$-0.640409\pi$
$683$	−22.3156	−0.853881	−0.426941	+	0.904280i	$0.640409\pi$
$684$	0	0
$685$	65.4095	2.49917
$686$	0	0
$687$	−24.9424	−0.951614
$688$	0	0
$689$	−1.61259	−0.0614349
$690$	0	0
$691$	−25.1372	−0.956264	−0.478132	−	0.878288i	$-0.658686\pi$
$691$	−25.1372	−0.956264	−0.478132	+	0.878288i	$0.658686\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	37.7745	1.43287
$696$	0	0
$697$	64.1760	2.43084
$698$	0	0
$699$	−23.4506	−0.886985
$700$	0	0
$701$	−19.2747	−0.727995	−0.363997	−	0.931400i	$-0.618588\pi$
$701$	−19.2747	−0.727995	−0.363997	+	0.931400i	$0.618588\pi$
$702$	0	0
$703$	−2.40665	−0.0907686
$704$	0	0
$705$	−33.7693	−1.27183
$706$	0	0
$707$	4.88727	0.183805
$708$	0	0
$709$	9.76098	0.366581	0.183291	−	0.983059i	$-0.441325\pi$
$709$	9.76098	0.366581	0.183291	+	0.983059i	$0.441325\pi$
$710$	0	0
$711$	−5.01641	−0.188130
$712$	0	0
$713$	−17.2747	−0.646942
$714$	0	0
$715$	−13.2939	−0.497165
$716$	0	0
$717$	5.07681	0.189597
$718$	0	0
$719$	−14.0276	−0.523141	−0.261570	−	0.965184i	$-0.584240\pi$
$719$	−14.0276	−0.523141	−0.261570	+	0.965184i	$0.584240\pi$
$720$	0	0
$721$	0.637339	0.0237357
$722$	0	0
$723$	−19.2939	−0.717549
$724$	0	0
$725$	−39.8297	−1.47924
$726$	0	0
$727$	34.3051	1.27231	0.636153	−	0.771563i	$-0.280525\pi$
$727$	34.3051	1.27231	0.636153	+	0.771563i	$0.280525\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	74.6263	2.76016
$732$	0	0
$733$	−10.7581	−0.397361	−0.198680	−	0.980064i	$-0.563666\pi$
$733$	−10.7581	−0.397361	−0.198680	+	0.980064i	$0.563666\pi$
$734$	0	0
$735$	−4.18953	−0.154533
$736$	0	0
$737$	2.66492	0.0981637
$738$	0	0
$739$	6.38741	0.234965	0.117482	−	0.993075i	$-0.462518\pi$
$739$	6.38741	0.234965	0.117482	+	0.993075i	$0.462518\pi$
$740$	0	0
$741$	1.01118	0.0371466
$742$	0	0
$743$	23.1096	0.847810	0.423905	−	0.905707i	$-0.360659\pi$
$743$	23.1096	0.847810	0.423905	+	0.905707i	$0.360659\pi$
$744$	0	0
$745$	−21.8105	−0.799074
$746$	0	0
$747$	−3.52461	−0.128959
$748$	0	0
$749$	−0.956008	−0.0349318
$750$	0	0
$751$	−31.8678	−1.16287	−0.581435	−	0.813593i	$-0.697509\pi$
$751$	−31.8678	−1.16287	−0.581435	+	0.813593i	$0.697509\pi$
$752$	0	0
$753$	23.8021	0.867398
$754$	0	0
$755$	−26.1843	−0.952944
$756$	0	0
$757$	−48.3103	−1.75587	−0.877934	−	0.478781i	$-0.841079\pi$
$757$	−48.3103	−1.75587	−0.877934	+	0.478781i	$0.841079\pi$
$758$	0	0
$759$	−1.87086	−0.0679081
$760$	0	0
$761$	−17.8625	−0.647516	−0.323758	−	0.946140i	$-0.604946\pi$
$761$	−17.8625	−0.647516	−0.323758	+	0.946140i	$0.604946\pi$
$762$	0	0
$763$	−7.61259	−0.275594
$764$	0	0
$765$	28.7170	1.03826
$766$	0	0
$767$	−22.3515	−0.807065
$768$	0	0
$769$	−24.8820	−0.897269	−0.448635	−	0.893715i	$-0.648090\pi$
$769$	−24.8820	−0.897269	−0.448635	+	0.893715i	$0.648090\pi$
$770$	0	0
$771$	−14.9149	−0.537145
$772$	0	0
$773$	−34.9700	−1.25778	−0.628892	−	0.777492i	$-0.716491\pi$
$773$	−34.9700	−1.25778	−0.628892	+	0.777492i	$0.716491\pi$
$774$	0	0
$775$	−115.901	−4.16329
$776$	0	0
$777$	7.55220	0.270933
$778$	0	0
$779$	2.98359	0.106898
$780$	0	0
$781$	5.01641	0.179501
$782$	0	0
$783$	3.17313	0.113398
$784$	0	0
$785$	−75.9177	−2.70962
$786$	0	0
$787$	15.1648	0.540566	0.270283	−	0.962781i	$-0.412883\pi$
$787$	15.1648	0.540566	0.270283	+	0.962781i	$0.412883\pi$
$788$	0	0
$789$	2.92319	0.104068
$790$	0	0
$791$	−7.70892	−0.274097
$792$	0	0
$793$	6.34625	0.225362
$794$	0	0
$795$	−2.12914	−0.0755126
$796$	0	0
$797$	8.24470	0.292042	0.146021	−	0.989281i	$-0.453353\pi$
$797$	8.24470	0.292042	0.146021	+	0.989281i	$0.453353\pi$
$798$	0	0
$799$	55.2497	1.95459
$800$	0	0
$801$	−1.74173	−0.0615410
$802$	0	0
$803$	−4.82687	−0.170337
$804$	0	0
$805$	7.83805	0.276255
$806$	0	0
$807$	−11.9672	−0.421265
$808$	0	0
$809$	29.8433	1.04923	0.524617	−	0.851338i	$-0.324209\pi$
$809$	29.8433	1.04923	0.524617	+	0.851338i	$0.324209\pi$
$810$	0	0
$811$	16.6321	0.584032	0.292016	−	0.956413i	$-0.405674\pi$
$811$	16.6321	0.584032	0.292016	+	0.956413i	$0.405674\pi$
$812$	0	0
$813$	−20.3187	−0.712607
$814$	0	0
$815$	11.1648	0.391086
$816$	0	0
$817$	3.46944	0.121380
$818$	0	0
$819$	−3.17313	−0.110878
$820$	0	0
$821$	30.3327	1.05862	0.529309	−	0.848429i	$-0.322451\pi$
$821$	30.3327	1.05862	0.529309	+	0.848429i	$0.322451\pi$
$822$	0	0
$823$	18.4067	0.641616	0.320808	−	0.947144i	$-0.396046\pi$
$823$	18.4067	0.641616	0.320808	+	0.947144i	$0.396046\pi$
$824$	0	0
$825$	−12.5522	−0.437011
$826$	0	0
$827$	−45.4559	−1.58066	−0.790328	−	0.612684i	$-0.790090\pi$
$827$	−45.4559	−1.58066	−0.790328	+	0.612684i	$0.790090\pi$
$828$	0	0
$829$	−20.3463	−0.706655	−0.353327	−	0.935500i	$-0.614950\pi$
$829$	−20.3463	−0.706655	−0.353327	+	0.935500i	$0.614950\pi$
$830$	0	0
$831$	−18.0552	−0.626327
$832$	0	0
$833$	6.85446	0.237493
$834$	0	0
$835$	46.6946	1.61593
$836$	0	0
$837$	9.23353	0.319157
$838$	0	0
$839$	−16.1812	−0.558637	−0.279318	−	0.960199i	$-0.590109\pi$
$839$	−16.1812	−0.558637	−0.279318	+	0.960199i	$0.590109\pi$
$840$	0	0
$841$	−18.9313	−0.652802
$842$	0	0
$843$	−27.2939	−0.940053
$844$	0	0
$845$	−12.2806	−0.422466
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	29.8901	1.02583
$850$	0	0
$851$	−14.1291	−0.484341
$852$	0	0
$853$	−20.9836	−0.718465	−0.359232	−	0.933248i	$-0.616961\pi$
$853$	−20.9836	−0.718465	−0.359232	+	0.933248i	$0.616961\pi$
$854$	0	0
$855$	1.33508	0.0456586
$856$	0	0
$857$	4.79095	0.163656	0.0818279	−	0.996646i	$-0.473924\pi$
$857$	4.79095	0.163656	0.0818279	+	0.996646i	$0.473924\pi$
$858$	0	0
$859$	9.96719	0.340076	0.170038	−	0.985438i	$-0.445611\pi$
$859$	9.96719	0.340076	0.170038	+	0.985438i	$0.445611\pi$
$860$	0	0
$861$	−9.36266	−0.319079
$862$	0	0
$863$	−25.5470	−0.869629	−0.434814	−	0.900520i	$-0.643186\pi$
$863$	−25.5470	−0.869629	−0.434814	+	0.900520i	$0.643186\pi$
$864$	0	0
$865$	−104.094	−3.53929
$866$	0	0
$867$	−29.9836	−1.01830
$868$	0	0
$869$	−5.01641	−0.170170
$870$	0	0
$871$	−8.45614	−0.286525
$872$	0	0
$873$	−12.2499	−0.414597
$874$	0	0
$875$	31.6402	1.06963
$876$	0	0
$877$	−50.9893	−1.72179	−0.860893	−	0.508787i	$-0.830094\pi$
$877$	−50.9893	−1.72179	−0.860893	+	0.508787i	$0.830094\pi$
$878$	0	0
$879$	4.34625	0.146596
$880$	0	0
$881$	32.1895	1.08449	0.542246	−	0.840219i	$-0.317574\pi$
$881$	32.1895	1.08449	0.542246	+	0.840219i	$0.317574\pi$
$882$	0	0
$883$	−36.6154	−1.23221	−0.616104	−	0.787665i	$-0.711290\pi$
$883$	−36.6154	−1.23221	−0.616104	+	0.787665i	$0.711290\pi$
$884$	0	0
$885$	−29.5110	−0.992003
$886$	0	0
$887$	48.2004	1.61841	0.809206	−	0.587525i	$-0.199897\pi$
$887$	48.2004	1.61841	0.809206	+	0.587525i	$0.199897\pi$
$888$	0	0
$889$	5.49180	0.184189
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	2.56860	0.0859550
$894$	0	0
$895$	53.4506	1.78666
$896$	0	0
$897$	5.93649	0.198214
$898$	0	0
$899$	29.2992	0.977181
$900$	0	0
$901$	3.48346	0.116051
$902$	0	0
$903$	−10.8873	−0.362306
$904$	0	0
$905$	−13.5470	−0.450316
$906$	0	0
$907$	−22.9013	−0.760425	−0.380212	−	0.924899i	$-0.624149\pi$
$907$	−22.9013	−0.760425	−0.380212	+	0.924899i	$0.624149\pi$
$908$	0	0
$909$	4.88727	0.162101
$910$	0	0
$911$	36.0552	1.19456	0.597281	−	0.802032i	$-0.296248\pi$
$911$	36.0552	1.19456	0.597281	+	0.802032i	$0.296248\pi$
$912$	0	0
$913$	−3.52461	−0.116648
$914$	0	0
$915$	8.37907	0.277003
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	28.3379	0.934782	0.467391	−	0.884051i	$-0.345194\pi$
$919$	28.3379	0.934782	0.467391	+	0.884051i	$0.345194\pi$
$920$	0	0
$921$	−20.7581	−0.684004
$922$	0	0
$923$	−15.9177	−0.523937
$924$	0	0
$925$	−94.7966	−3.11689
$926$	0	0
$927$	0.637339	0.0209329
$928$	0	0
$929$	5.08514	0.166838	0.0834191	−	0.996515i	$-0.473416\pi$
$929$	5.08514	0.166838	0.0834191	+	0.996515i	$0.473416\pi$
$930$	0	0
$931$	0.318669	0.0104440
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	28.7170	0.939146
$936$	0	0
$937$	32.2088	1.05222	0.526108	−	0.850418i	$-0.323651\pi$
$937$	32.2088	1.05222	0.526108	+	0.850418i	$0.323651\pi$
$938$	0	0
$939$	−8.12914	−0.265284
$940$	0	0
$941$	32.7805	1.06861	0.534307	−	0.845291i	$-0.320573\pi$
$941$	32.7805	1.06861	0.534307	+	0.845291i	$0.320573\pi$
$942$	0	0
$943$	17.5163	0.570408
$944$	0	0
$945$	−4.18953	−0.136286
$946$	0	0
$947$	27.3627	0.889167	0.444584	−	0.895737i	$-0.353352\pi$
$947$	27.3627	0.889167	0.444584	+	0.895737i	$0.353352\pi$
$948$	0	0
$949$	15.3163	0.497188
$950$	0	0
$951$	19.9917	0.648274
$952$	0	0
$953$	−24.6894	−0.799768	−0.399884	−	0.916566i	$-0.630950\pi$
$953$	−24.6894	−0.799768	−0.399884	+	0.916566i	$0.630950\pi$
$954$	0	0
$955$	−87.8765	−2.84362
$956$	0	0
$957$	3.17313	0.102573
$958$	0	0
$959$	15.6126	0.504157
$960$	0	0
$961$	54.2580	1.75026
$962$	0	0
$963$	−0.956008	−0.0308069
$964$	0	0
$965$	1.04710	0.0337074
$966$	0	0
$967$	−21.3298	−0.685922	−0.342961	−	0.939350i	$-0.611430\pi$
$967$	−21.3298	−0.685922	−0.342961	+	0.939350i	$0.611430\pi$
$968$	0	0
$969$	−2.18431	−0.0701700
$970$	0	0
$971$	−28.4946	−0.914436	−0.457218	−	0.889355i	$-0.651154\pi$
$971$	−28.4946	−0.914436	−0.457218	+	0.889355i	$0.651154\pi$
$972$	0	0
$973$	9.01641	0.289053
$974$	0	0
$975$	39.8297	1.27557
$976$	0	0
$977$	−19.8157	−0.633960	−0.316980	−	0.948432i	$-0.602669\pi$
$977$	−19.8157	−0.633960	−0.316980	+	0.948432i	$0.602669\pi$
$978$	0	0
$979$	−1.74173	−0.0556659
$980$	0	0
$981$	−7.61259	−0.243051
$982$	0	0
$983$	53.7745	1.71514	0.857571	−	0.514366i	$-0.171973\pi$
$983$	53.7745	1.71514	0.857571	+	0.514366i	$0.171973\pi$
$984$	0	0
$985$	77.5058	2.46954
$986$	0	0
$987$	−8.06040	−0.256565
$988$	0	0
$989$	20.3686	0.647684
$990$	0	0
$991$	−49.7693	−1.58097	−0.790487	−	0.612479i	$-0.790173\pi$
$991$	−49.7693	−1.58097	−0.790487	+	0.612479i	$0.790173\pi$
$992$	0	0
$993$	17.0164	0.539999
$994$	0	0
$995$	−41.3543	−1.31102
$996$	0	0
$997$	0.659696	0.0208928	0.0104464	−	0.999945i	$-0.496675\pi$
$997$	0.659696	0.0208928	0.0104464	+	0.999945i	$0.496675\pi$
$998$	0	0
$999$	7.55220	0.238941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bo.1.3		3
4.3	odd	2		231.2.a.e.1.1	✓	3
12.11	even	2		693.2.a.l.1.3		3
20.19	odd	2		5775.2.a.bp.1.3		3
28.27	even	2		1617.2.a.t.1.1		3
44.43	even	2		2541.2.a.bg.1.3		3
84.83	odd	2		4851.2.a.bi.1.3		3
132.131	odd	2		7623.2.a.cd.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.1	✓	3	4.3	odd	2
693.2.a.l.1.3		3	12.11	even	2
1617.2.a.t.1.1		3	28.27	even	2
2541.2.a.bg.1.3		3	44.43	even	2
3696.2.a.bo.1.3		3	1.1	even	1	trivial
4851.2.a.bi.1.3		3	84.83	odd	2
5775.2.a.bp.1.3		3	20.19	odd	2
7623.2.a.cd.1.1		3	132.131	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.18953 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.17313 q^{13} -4.18953 q^{15} +6.85446 q^{17} +0.318669 q^{19} -1.00000 q^{21} +1.87086 q^{23} +12.5522 q^{25} -1.00000 q^{27} -3.17313 q^{29} -9.23353 q^{31} -1.00000 q^{33} +4.18953 q^{35} -7.55220 q^{37} +3.17313 q^{39} +9.36266 q^{41} +10.8873 q^{43} +4.18953 q^{45} +8.06040 q^{47} +1.00000 q^{49} -6.85446 q^{51} +0.508203 q^{53} +4.18953 q^{55} -0.318669 q^{57} +7.04399 q^{59} -2.00000 q^{61} +1.00000 q^{63} -13.2939 q^{65} +2.66492 q^{67} -1.87086 q^{69} +5.01641 q^{71} -4.82687 q^{73} -12.5522 q^{75} +1.00000 q^{77} -5.01641 q^{79} +1.00000 q^{81} -3.52461 q^{83} +28.7170 q^{85} +3.17313 q^{87} -1.74173 q^{89} -3.17313 q^{91} +9.23353 q^{93} +1.33508 q^{95} -12.2499 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 + 8 * q^17 + 8 * q^19 - 3 * q^21 - 10 * q^23 + 15 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 + 4 * q^35 + 4 * q^39 + 14 * q^41 + 14 * q^43 + 4 * q^45 + 3 * q^49 - 8 * q^51 + 4 * q^55 - 8 * q^57 - 6 * q^61 + 3 * q^63 + 14 * q^65 + 4 * q^67 + 10 * q^69 + 12 * q^71 - 20 * q^73 - 15 * q^75 + 3 * q^77 - 12 * q^79 + 3 * q^81 - 6 * q^83 - 6 * q^85 + 4 * q^87 + 26 * q^89 - 4 * q^91 - 2 * q^93 + 8 * q^95 - 4 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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