LMFDB - Embedded newform 3483.2.a.t.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3483,2,Mod(1,3483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3483, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3483 = 3^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3483.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:	$27.8118950240$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 6 x^{19} - 13 x^{18} + 137 x^{17} - 24 x^{16} - 1247 x^{15} + 1257 x^{14} + 5756 x^{13} + \cdots - 81 $ x^20 - 6x^19 - 13x^18 + 137x^17 - 24x^16 - 1247x^15 + 1257x^14 + 5756x^13 - 8778x^12 - 13888x^11 + 28633x^10 + 15048x^9 - 48400x^8 - 548x^7 + 40321x^6 - 10543x^5 - 13966x^4 + 5450x^3 + 1344x^2 - 441*x - 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 387)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$-2.08982$ of defining polynomial
Character	$\chi$	$=$	3483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08982 q^{2} +2.36736 q^{4} +0.273348 q^{5} -2.42024 q^{7} +0.767710 q^{8} +O(q^{10})$	\(q+2.08982 q^{2} +2.36736 q^{4} +0.273348 q^{5} -2.42024 q^{7} +0.767710 q^{8} +0.571248 q^{10} -1.59894 q^{11} +3.46678 q^{13} -5.05786 q^{14} -3.13034 q^{16} -3.58535 q^{17} +0.324858 q^{19} +0.647112 q^{20} -3.34149 q^{22} -4.35144 q^{23} -4.92528 q^{25} +7.24495 q^{26} -5.72956 q^{28} -5.99369 q^{29} -1.90091 q^{31} -8.07727 q^{32} -7.49274 q^{34} -0.661566 q^{35} +2.23031 q^{37} +0.678896 q^{38} +0.209852 q^{40} -10.4157 q^{41} -1.00000 q^{43} -3.78525 q^{44} -9.09374 q^{46} +8.71476 q^{47} -1.14246 q^{49} -10.2930 q^{50} +8.20710 q^{52} +4.52956 q^{53} -0.437066 q^{55} -1.85804 q^{56} -12.5257 q^{58} +9.90658 q^{59} +11.0170 q^{61} -3.97256 q^{62} -10.6194 q^{64} +0.947636 q^{65} -5.73764 q^{67} -8.48780 q^{68} -1.38256 q^{70} +6.74222 q^{71} +0.832499 q^{73} +4.66095 q^{74} +0.769056 q^{76} +3.86980 q^{77} -8.01597 q^{79} -0.855671 q^{80} -21.7670 q^{82} -6.81066 q^{83} -0.980048 q^{85} -2.08982 q^{86} -1.22752 q^{88} -9.69164 q^{89} -8.39042 q^{91} -10.3014 q^{92} +18.2123 q^{94} +0.0887994 q^{95} +1.69784 q^{97} -2.38754 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8}+O(q^{10}) $ 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8	$ 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8} - 2 q^{10} - 10 q^{11} - q^{13} - 10 q^{14} + 22 q^{16} - 20 q^{17} + 8 q^{19} - 30 q^{20} + 15 q^{22} - 19 q^{23} + 19 q^{25} - 25 q^{26} - 3 q^{28} - 25 q^{29} - 11 q^{31} - 36 q^{32} + 9 q^{34} + 9 q^{37} - 28 q^{38} + 12 q^{40} - 12 q^{41} - 20 q^{43} - 5 q^{44} + 4 q^{46} - 38 q^{47} + 37 q^{49} - 36 q^{50} - 8 q^{52} - 69 q^{53} - 9 q^{55} - 30 q^{56} - 27 q^{58} - 31 q^{59} + 19 q^{61} - 32 q^{62} + 11 q^{64} - 47 q^{65} + 9 q^{67} - 68 q^{68} - 6 q^{70} - 21 q^{71} - 2 q^{73} + 16 q^{74} + 37 q^{76} - 85 q^{77} - 4 q^{79} - 61 q^{80} + q^{82} - 19 q^{83} - 6 q^{85} + 6 q^{86} + 60 q^{88} - 54 q^{89} - 3 q^{91} - 85 q^{92} - 19 q^{94} + 11 q^{95} + 2 q^{97} - 5 q^{98}+O(q^{100}) $ 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8 - 2 * q^10 - 10 * q^11 - q^13 - 10 * q^14 + 22 * q^16 - 20 * q^17 + 8 * q^19 - 30 * q^20 + 15 * q^22 - 19 * q^23 + 19 * q^25 - 25 * q^26 - 3 * q^28 - 25 * q^29 - 11 * q^31 - 36 * q^32 + 9 * q^34 + 9 * q^37 - 28 * q^38 + 12 * q^40 - 12 * q^41 - 20 * q^43 - 5 * q^44 + 4 * q^46 - 38 * q^47 + 37 * q^49 - 36 * q^50 - 8 * q^52 - 69 * q^53 - 9 * q^55 - 30 * q^56 - 27 * q^58 - 31 * q^59 + 19 * q^61 - 32 * q^62 + 11 * q^64 - 47 * q^65 + 9 * q^67 - 68 * q^68 - 6 * q^70 - 21 * q^71 - 2 * q^73 + 16 * q^74 + 37 * q^76 - 85 * q^77 - 4 * q^79 - 61 * q^80 + q^82 - 19 * q^83 - 6 * q^85 + 6 * q^86 + 60 * q^88 - 54 * q^89 - 3 * q^91 - 85 * q^92 - 19 * q^94 + 11 * q^95 + 2 * q^97 - 5 * q^98

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 $

$2$	2.08982	1.47773	0.738864	−	0.673855i	$-0.235363\pi$
$2$	2.08982	1.47773	0.738864	+	0.673855i	$0.235363\pi$
$3$	0	0
$3$	0	0
$4$	2.36736	1.18368
$5$	0.273348	0.122245	0.0611224	−	0.998130i	$-0.480532\pi$
$5$	0.273348	0.122245	0.0611224	+	0.998130i	$0.480532\pi$
$6$	0	0
$7$	−2.42024	−0.914763	−0.457381	−	0.889271i	$-0.651213\pi$
$7$	−2.42024	−0.914763	−0.457381	+	0.889271i	$0.651213\pi$
$8$	0.767710	0.271426
$9$	0	0
$10$	0.571248	0.180645
$11$	−1.59894	−0.482097	−0.241049	−	0.970513i	$-0.577491\pi$
$11$	−1.59894	−0.482097	−0.241049	+	0.970513i	$0.577491\pi$
$12$	0	0
$13$	3.46678	0.961511	0.480756	−	0.876855i	$-0.340362\pi$
$13$	3.46678	0.961511	0.480756	+	0.876855i	$0.340362\pi$
$14$	−5.05786	−1.35177
$15$	0	0
$16$	−3.13034	−0.782584
$17$	−3.58535	−0.869575	−0.434788	−	0.900533i	$-0.643177\pi$
$17$	−3.58535	−0.869575	−0.434788	+	0.900533i	$0.643177\pi$
$18$	0	0
$19$	0.324858	0.0745276	0.0372638	−	0.999305i	$-0.488136\pi$
$19$	0.324858	0.0745276	0.0372638	+	0.999305i	$0.488136\pi$
$20$	0.647112	0.144699
$21$	0	0
$22$	−3.34149	−0.712408
$23$	−4.35144	−0.907339	−0.453669	−	0.891170i	$-0.649885\pi$
$23$	−4.35144	−0.907339	−0.453669	+	0.891170i	$0.649885\pi$
$24$	0	0
$25$	−4.92528	−0.985056
$26$	7.24495	1.42085
$27$	0	0
$28$	−5.72956	−1.08278
$29$	−5.99369	−1.11300	−0.556500	−	0.830847i	$-0.687856\pi$
$29$	−5.99369	−1.11300	−0.556500	+	0.830847i	$0.687856\pi$
$30$	0	0
$31$	−1.90091	−0.341413	−0.170706	−	0.985322i	$-0.554605\pi$
$31$	−1.90091	−0.341413	−0.170706	+	0.985322i	$0.554605\pi$
$32$	−8.07727	−1.42787
$33$	0	0
$34$	−7.49274	−1.28500
$35$	−0.661566	−0.111825
$36$	0	0
$37$	2.23031	0.366661	0.183330	−	0.983051i	$-0.441312\pi$
$37$	2.23031	0.366661	0.183330	+	0.983051i	$0.441312\pi$
$38$	0.678896	0.110132
$39$	0	0
$40$	0.209852	0.0331805
$41$	−10.4157	−1.62666	−0.813331	−	0.581801i	$-0.802348\pi$
$41$	−10.4157	−1.62666	−0.813331	+	0.581801i	$0.802348\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−3.78525	−0.570648
$45$	0	0
$46$	−9.09374	−1.34080
$47$	8.71476	1.27118	0.635589	−	0.772028i	$-0.280757\pi$
$47$	8.71476	1.27118	0.635589	+	0.772028i	$0.280757\pi$
$48$	0	0
$49$	−1.14246	−0.163209
$50$	−10.2930	−1.45564
$51$	0	0
$52$	8.20710	1.13812
$53$	4.52956	0.622182	0.311091	−	0.950380i	$-0.399306\pi$
$53$	4.52956	0.622182	0.311091	+	0.950380i	$0.399306\pi$
$54$	0	0
$55$	−0.437066	−0.0589339
$56$	−1.85804	−0.248291
$57$	0	0
$58$	−12.5257	−1.64471
$59$	9.90658	1.28973	0.644864	−	0.764298i	$-0.276914\pi$
$59$	9.90658	1.28973	0.644864	+	0.764298i	$0.276914\pi$
$60$	0	0
$61$	11.0170	1.41058	0.705289	−	0.708920i	$-0.250817\pi$
$61$	11.0170	1.41058	0.705289	+	0.708920i	$0.250817\pi$
$62$	−3.97256	−0.504515
$63$	0	0
$64$	−10.6194	−1.32742
$65$	0.947636	0.117540
$66$	0	0
$67$	−5.73764	−0.700964	−0.350482	−	0.936569i	$-0.613982\pi$
$67$	−5.73764	−0.700964	−0.350482	+	0.936569i	$0.613982\pi$
$68$	−8.48780	−1.02930
$69$	0	0
$70$	−1.38256	−0.165247
$71$	6.74222	0.800155	0.400077	−	0.916481i	$-0.368983\pi$
$71$	6.74222	0.800155	0.400077	+	0.916481i	$0.368983\pi$
$72$	0	0
$73$	0.832499	0.0974367	0.0487183	−	0.998813i	$-0.484486\pi$
$73$	0.832499	0.0974367	0.0487183	+	0.998813i	$0.484486\pi$
$74$	4.66095	0.541825
$75$	0	0
$76$	0.769056	0.0882167
$77$	3.86980	0.441005
$78$	0	0
$79$	−8.01597	−0.901867	−0.450934	−	0.892557i	$-0.648909\pi$
$79$	−8.01597	−0.901867	−0.450934	+	0.892557i	$0.648909\pi$
$80$	−0.855671	−0.0956669
$81$	0	0
$82$	−21.7670	−2.40376
$83$	−6.81066	−0.747567	−0.373783	−	0.927516i	$-0.621940\pi$
$83$	−6.81066	−0.747567	−0.373783	+	0.927516i	$0.621940\pi$
$84$	0	0
$85$	−0.980048	−0.106301
$86$	−2.08982	−0.225351
$87$	0	0
$88$	−1.22752	−0.130854
$89$	−9.69164	−1.02731	−0.513656	−	0.857996i	$-0.671709\pi$
$89$	−9.69164	−1.02731	−0.513656	+	0.857996i	$0.671709\pi$
$90$	0	0
$91$	−8.39042	−0.879555
$92$	−10.3014	−1.07400
$93$	0	0
$94$	18.2123	1.87845
$95$	0.0887994	0.00911062
$96$	0	0
$97$	1.69784	0.172389	0.0861946	−	0.996278i	$-0.472529\pi$
$97$	1.69784	0.172389	0.0861946	+	0.996278i	$0.472529\pi$
$98$	−2.38754	−0.241178
$99$	0	0
$100$	−11.6599	−1.16599
$101$	−11.0123	−1.09577	−0.547883	−	0.836555i	$-0.684566\pi$
$101$	−11.0123	−1.09577	−0.547883	+	0.836555i	$0.684566\pi$
$102$	0	0
$103$	4.69031	0.462150	0.231075	−	0.972936i	$-0.425776\pi$
$103$	4.69031	0.462150	0.231075	+	0.972936i	$0.425776\pi$
$104$	2.66148	0.260980
$105$	0	0
$106$	9.46597	0.919416
$107$	−18.1464	−1.75427	−0.877137	−	0.480239i	$-0.840550\pi$
$107$	−18.1464	−1.75427	−0.877137	+	0.480239i	$0.840550\pi$
$108$	0	0
$109$	−2.03854	−0.195257	−0.0976285	−	0.995223i	$-0.531126\pi$
$109$	−2.03854	−0.195257	−0.0976285	+	0.995223i	$0.531126\pi$
$110$	−0.913389	−0.0870883
$111$	0	0
$112$	7.57615	0.715879
$113$	−0.876325	−0.0824377	−0.0412188	−	0.999150i	$-0.513124\pi$
$113$	−0.876325	−0.0824377	−0.0412188	+	0.999150i	$0.513124\pi$
$114$	0	0
$115$	−1.18946	−0.110917
$116$	−14.1892	−1.31743
$117$	0	0
$118$	20.7030	1.90587
$119$	8.67739	0.795455
$120$	0	0
$121$	−8.44340	−0.767582
$122$	23.0235	2.08445
$123$	0	0
$124$	−4.50012	−0.404123
$125$	−2.71305	−0.242663
$126$	0	0
$127$	3.12225	0.277055	0.138527	−	0.990359i	$-0.455763\pi$
$127$	3.12225	0.277055	0.138527	+	0.990359i	$0.455763\pi$
$128$	−6.03807	−0.533695
$129$	0	0
$130$	1.98039	0.173692
$131$	−1.52728	−0.133439	−0.0667196	−	0.997772i	$-0.521253\pi$
$131$	−1.52728	−0.133439	−0.0667196	+	0.997772i	$0.521253\pi$
$132$	0	0
$133$	−0.786234	−0.0681751
$134$	−11.9907	−1.03583
$135$	0	0
$136$	−2.75251	−0.236026
$137$	−4.31689	−0.368817	−0.184408	−	0.982850i	$-0.559037\pi$
$137$	−4.31689	−0.368817	−0.184408	+	0.982850i	$0.559037\pi$
$138$	0	0
$139$	−18.6146	−1.57887	−0.789434	−	0.613835i	$-0.789626\pi$
$139$	−18.6146	−1.57887	−0.789434	+	0.613835i	$0.789626\pi$
$140$	−1.56616	−0.132365
$141$	0	0
$142$	14.0900	1.18241
$143$	−5.54316	−0.463542
$144$	0	0
$145$	−1.63836	−0.136059
$146$	1.73978	0.143985
$147$	0	0
$148$	5.27994	0.434009
$149$	5.11767	0.419256	0.209628	−	0.977781i	$-0.432775\pi$
$149$	5.11767	0.419256	0.209628	+	0.977781i	$0.432775\pi$
$150$	0	0
$151$	12.8966	1.04951	0.524756	−	0.851253i	$-0.324156\pi$
$151$	12.8966	1.04951	0.524756	+	0.851253i	$0.324156\pi$
$152$	0.249397	0.0202288
$153$	0	0
$154$	8.08719	0.651685
$155$	−0.519609	−0.0417360
$156$	0	0
$157$	−11.3197	−0.903411	−0.451705	−	0.892167i	$-0.649184\pi$
$157$	−11.3197	−0.903411	−0.451705	+	0.892167i	$0.649184\pi$
$158$	−16.7520	−1.33271
$159$	0	0
$160$	−2.20790	−0.174550
$161$	10.5315	0.830000
$162$	0	0
$163$	9.46663	0.741484	0.370742	−	0.928736i	$-0.379103\pi$
$163$	9.46663	0.741484	0.370742	+	0.928736i	$0.379103\pi$
$164$	−24.6577	−1.92545
$165$	0	0
$166$	−14.2331	−1.10470
$167$	1.76487	0.136570	0.0682849	−	0.997666i	$-0.478247\pi$
$167$	1.76487	0.136570	0.0682849	+	0.997666i	$0.478247\pi$
$168$	0	0
$169$	−0.981444	−0.0754957
$170$	−2.04813	−0.157084
$171$	0	0
$172$	−2.36736	−0.180509
$173$	21.0721	1.60208	0.801041	−	0.598610i	$-0.204280\pi$
$173$	21.0721	1.60208	0.801041	+	0.598610i	$0.204280\pi$
$174$	0	0
$175$	11.9203	0.901093
$176$	5.00521	0.377282
$177$	0	0
$178$	−20.2538	−1.51809
$179$	7.77067	0.580807	0.290403	−	0.956904i	$-0.406210\pi$
$179$	7.77067	0.580807	0.290403	+	0.956904i	$0.406210\pi$
$180$	0	0
$181$	21.4202	1.59215	0.796074	−	0.605199i	$-0.206907\pi$
$181$	21.4202	1.59215	0.796074	+	0.605199i	$0.206907\pi$
$182$	−17.5345	−1.29974
$183$	0	0
$184$	−3.34064	−0.246276
$185$	0.609651	0.0448224
$186$	0	0
$187$	5.73274	0.419220
$188$	20.6309	1.50467
$189$	0	0
$190$	0.185575	0.0134630
$191$	−17.2108	−1.24533	−0.622664	−	0.782489i	$-0.713950\pi$
$191$	−17.2108	−1.24533	−0.622664	+	0.782489i	$0.713950\pi$
$192$	0	0
$193$	−4.95362	−0.356570	−0.178285	−	0.983979i	$-0.557055\pi$
$193$	−4.95362	−0.356570	−0.178285	+	0.983979i	$0.557055\pi$
$194$	3.54817	0.254744
$195$	0	0
$196$	−2.70461	−0.193187
$197$	0.691719	0.0492830	0.0246415	−	0.999696i	$-0.492156\pi$
$197$	0.691719	0.0492830	0.0246415	+	0.999696i	$0.492156\pi$
$198$	0	0
$199$	13.5430	0.960036	0.480018	−	0.877259i	$-0.340630\pi$
$199$	13.5430	0.960036	0.480018	+	0.877259i	$0.340630\pi$
$200$	−3.78119	−0.267370
$201$	0	0
$202$	−23.0138	−1.61924
$203$	14.5061	1.01813
$204$	0	0
$205$	−2.84712	−0.198851
$206$	9.80192	0.682932
$207$	0	0
$208$	−10.8522	−0.752464
$209$	−0.519428	−0.0359296
$210$	0	0
$211$	19.0044	1.30832	0.654159	−	0.756357i	$-0.273023\pi$
$211$	19.0044	1.30832	0.654159	+	0.756357i	$0.273023\pi$
$212$	10.7231	0.736464
$213$	0	0
$214$	−37.9227	−2.59234
$215$	−0.273348	−0.0186422
$216$	0	0
$217$	4.60064	0.312312
$218$	−4.26019	−0.288537
$219$	0	0
$220$	−1.03469	−0.0697588
$221$	−12.4296	−0.836106
$222$	0	0
$223$	4.39519	0.294324	0.147162	−	0.989112i	$-0.452986\pi$
$223$	4.39519	0.294324	0.147162	+	0.989112i	$0.452986\pi$
$224$	19.5489	1.30616
$225$	0	0
$226$	−1.83136	−0.121820
$227$	28.3016	1.87845	0.939223	−	0.343308i	$-0.111547\pi$
$227$	28.3016	1.87845	0.939223	+	0.343308i	$0.111547\pi$
$228$	0	0
$229$	−26.3951	−1.74424	−0.872120	−	0.489292i	$-0.837255\pi$
$229$	−26.3951	−1.74424	−0.872120	+	0.489292i	$0.837255\pi$
$230$	−2.48575	−0.163906
$231$	0	0
$232$	−4.60142	−0.302098
$233$	−28.1007	−1.84094	−0.920470	−	0.390814i	$-0.872194\pi$
$233$	−28.1007	−1.84094	−0.920470	+	0.390814i	$0.872194\pi$
$234$	0	0
$235$	2.38216	0.155395
$236$	23.4524	1.52662
$237$	0	0
$238$	18.1342	1.17547
$239$	−5.58384	−0.361188	−0.180594	−	0.983558i	$-0.557802\pi$
$239$	−5.58384	−0.361188	−0.180594	+	0.983558i	$0.557802\pi$
$240$	0	0
$241$	7.19436	0.463429	0.231715	−	0.972784i	$-0.425566\pi$
$241$	7.19436	0.463429	0.231715	+	0.972784i	$0.425566\pi$
$242$	−17.6452	−1.13428
$243$	0	0
$244$	26.0811	1.66967
$245$	−0.312290	−0.0199514
$246$	0	0
$247$	1.12621	0.0716592
$248$	−1.45934	−0.0926685
$249$	0	0
$250$	−5.66980	−0.358590
$251$	5.34271	0.337229	0.168614	−	0.985682i	$-0.446071\pi$
$251$	5.34271	0.337229	0.168614	+	0.985682i	$0.446071\pi$
$252$	0	0
$253$	6.95768	0.437425
$254$	6.52495	0.409411
$255$	0	0
$256$	8.62025	0.538766
$257$	−16.6949	−1.04140	−0.520698	−	0.853741i	$-0.674328\pi$
$257$	−16.6949	−1.04140	−0.520698	+	0.853741i	$0.674328\pi$
$258$	0	0
$259$	−5.39788	−0.335408
$260$	2.24339	0.139129
$261$	0	0
$262$	−3.19175	−0.197187
$263$	−17.4915	−1.07857	−0.539285	−	0.842123i	$-0.681305\pi$
$263$	−17.4915	−1.07857	−0.539285	+	0.842123i	$0.681305\pi$
$264$	0	0
$265$	1.23814	0.0760586
$266$	−1.64309	−0.100744
$267$	0	0
$268$	−13.5830	−0.829716
$269$	32.3234	1.97079	0.985395	−	0.170286i	$-0.0544690\pi$
$269$	32.3234	1.97079	0.985395	+	0.170286i	$0.0544690\pi$
$270$	0	0
$271$	3.81719	0.231878	0.115939	−	0.993256i	$-0.463012\pi$
$271$	3.81719	0.231878	0.115939	+	0.993256i	$0.463012\pi$
$272$	11.2234	0.680516
$273$	0	0
$274$	−9.02153	−0.545011
$275$	7.87521	0.474893
$276$	0	0
$277$	28.4029	1.70657	0.853283	−	0.521448i	$-0.174608\pi$
$277$	28.4029	1.70657	0.853283	+	0.521448i	$0.174608\pi$
$278$	−38.9012	−2.33314
$279$	0	0
$280$	−0.507891	−0.0303523
$281$	−6.72557	−0.401214	−0.200607	−	0.979672i	$-0.564291\pi$
$281$	−6.72557	−0.401214	−0.200607	+	0.979672i	$0.564291\pi$
$282$	0	0
$283$	30.7959	1.83062	0.915312	−	0.402746i	$-0.131944\pi$
$283$	30.7959	1.83062	0.915312	+	0.402746i	$0.131944\pi$
$284$	15.9612	0.947125
$285$	0	0
$286$	−11.5842	−0.684989
$287$	25.2085	1.48801
$288$	0	0
$289$	−4.14526	−0.243839
$290$	−3.42389	−0.201058
$291$	0	0
$292$	1.97082	0.115334
$293$	−4.47398	−0.261372	−0.130686	−	0.991424i	$-0.541718\pi$
$293$	−4.47398	−0.261372	−0.130686	+	0.991424i	$0.541718\pi$
$294$	0	0
$295$	2.70794	0.157663
$296$	1.71223	0.0995214
$297$	0	0
$298$	10.6950	0.619546
$299$	−15.0855	−0.872416
$300$	0	0
$301$	2.42024	0.139500
$302$	26.9517	1.55089
$303$	0	0
$304$	−1.01692	−0.0583241
$305$	3.01147	0.172436
$306$	0	0
$307$	15.8569	0.904999	0.452500	−	0.891765i	$-0.350532\pi$
$307$	15.8569	0.904999	0.452500	+	0.891765i	$0.350532\pi$
$308$	9.16120	0.522008
$309$	0	0
$310$	−1.08589	−0.0616744
$311$	8.54740	0.484679	0.242339	−	0.970192i	$-0.422085\pi$
$311$	8.54740	0.484679	0.242339	+	0.970192i	$0.422085\pi$
$312$	0	0
$313$	22.8806	1.29329	0.646644	−	0.762792i	$-0.276172\pi$
$313$	22.8806	1.29329	0.646644	+	0.762792i	$0.276172\pi$
$314$	−23.6562	−1.33500
$315$	0	0
$316$	−18.9767	−1.06752
$317$	−27.3972	−1.53878	−0.769391	−	0.638778i	$-0.779440\pi$
$317$	−27.3972	−1.53878	−0.769391	+	0.638778i	$0.779440\pi$
$318$	0	0
$319$	9.58353	0.536575
$320$	−2.90278	−0.162271
$321$	0	0
$322$	22.0090	1.22651
$323$	−1.16473	−0.0648074
$324$	0	0
$325$	−17.0749	−0.947143
$326$	19.7836	1.09571
$327$	0	0
$328$	−7.99625	−0.441519
$329$	−21.0918	−1.16283
$330$	0	0
$331$	−23.0998	−1.26968	−0.634841	−	0.772643i	$-0.718935\pi$
$331$	−23.0998	−1.26968	−0.634841	+	0.772643i	$0.718935\pi$
$332$	−16.1233	−0.884878
$333$	0	0
$334$	3.68826	0.201813
$335$	−1.56837	−0.0856893
$336$	0	0
$337$	22.0062	1.19876	0.599378	−	0.800466i	$-0.295415\pi$
$337$	22.0062	1.19876	0.599378	+	0.800466i	$0.295415\pi$
$338$	−2.05104	−0.111562
$339$	0	0
$340$	−2.32012	−0.125826
$341$	3.03943	0.164594
$342$	0	0
$343$	19.7067	1.06406
$344$	−0.767710	−0.0413921
$345$	0	0
$346$	44.0369	2.36744
$347$	5.40331	0.290065	0.145032	−	0.989427i	$-0.453671\pi$
$347$	5.40331	0.290065	0.145032	+	0.989427i	$0.453671\pi$
$348$	0	0
$349$	4.71207	0.252232	0.126116	−	0.992016i	$-0.459749\pi$
$349$	4.71207	0.252232	0.126116	+	0.992016i	$0.459749\pi$
$350$	24.9114	1.33157
$351$	0	0
$352$	12.9150	0.688373
$353$	−25.0261	−1.33200	−0.666001	−	0.745951i	$-0.731995\pi$
$353$	−25.0261	−1.33200	−0.666001	+	0.745951i	$0.731995\pi$
$354$	0	0
$355$	1.84297	0.0978148
$356$	−22.9436	−1.21601
$357$	0	0
$358$	16.2393	0.858274
$359$	11.4471	0.604156	0.302078	−	0.953283i	$-0.402320\pi$
$359$	11.4471	0.604156	0.302078	+	0.953283i	$0.402320\pi$
$360$	0	0
$361$	−18.8945	−0.994446
$362$	44.7643	2.35276
$363$	0	0
$364$	−19.8631	−1.04111
$365$	0.227562	0.0119111
$366$	0	0
$367$	−16.3275	−0.852291	−0.426146	−	0.904655i	$-0.640129\pi$
$367$	−16.3275	−0.852291	−0.426146	+	0.904655i	$0.640129\pi$
$368$	13.6215	0.710069
$369$	0	0
$370$	1.27406	0.0662353
$371$	−10.9626	−0.569149
$372$	0	0
$373$	16.2983	0.843893	0.421946	−	0.906621i	$-0.361347\pi$
$373$	16.2983	0.843893	0.421946	+	0.906621i	$0.361347\pi$
$374$	11.9804	0.619493
$375$	0	0
$376$	6.69040	0.345031
$377$	−20.7788	−1.07016
$378$	0	0
$379$	34.9308	1.79427	0.897136	−	0.441754i	$-0.145644\pi$
$379$	34.9308	1.79427	0.897136	+	0.441754i	$0.145644\pi$
$380$	0.210220	0.0107840
$381$	0	0
$382$	−35.9675	−1.84026
$383$	−27.5927	−1.40992	−0.704962	−	0.709245i	$-0.749036\pi$
$383$	−27.5927	−1.40992	−0.704962	+	0.709245i	$0.749036\pi$
$384$	0	0
$385$	1.05780	0.0539106
$386$	−10.3522	−0.526913
$387$	0	0
$388$	4.01938	0.204053
$389$	−14.1247	−0.716151	−0.358075	−	0.933693i	$-0.616567\pi$
$389$	−14.1247	−0.716151	−0.358075	+	0.933693i	$0.616567\pi$
$390$	0	0
$391$	15.6014	0.788999
$392$	−0.877079	−0.0442992
$393$	0	0
$394$	1.44557	0.0728268
$395$	−2.19115	−0.110249
$396$	0	0
$397$	29.4385	1.47748	0.738738	−	0.673993i	$-0.235422\pi$
$397$	29.4385	1.47748	0.738738	+	0.673993i	$0.235422\pi$
$398$	28.3024	1.41867
$399$	0	0
$400$	15.4178	0.770889
$401$	−17.0580	−0.851834	−0.425917	−	0.904762i	$-0.640048\pi$
$401$	−17.0580	−0.851834	−0.425917	+	0.904762i	$0.640048\pi$
$402$	0	0
$403$	−6.59002	−0.328272
$404$	−26.0701	−1.29703
$405$	0	0
$406$	30.3153	1.50452
$407$	−3.56612	−0.176766
$408$	0	0
$409$	−37.4226	−1.85043	−0.925215	−	0.379444i	$-0.876115\pi$
$409$	−37.4226	−1.85043	−0.925215	+	0.379444i	$0.876115\pi$
$410$	−5.94996	−0.293848
$411$	0	0
$412$	11.1036	0.547037
$413$	−23.9763	−1.17979
$414$	0	0
$415$	−1.86168	−0.0913862
$416$	−28.0021	−1.37292
$417$	0	0
$418$	−1.08551	−0.0530941
$419$	34.1347	1.66759	0.833794	−	0.552075i	$-0.186164\pi$
$419$	34.1347	1.66759	0.833794	+	0.552075i	$0.186164\pi$
$420$	0	0
$421$	22.5687	1.09993	0.549966	−	0.835187i	$-0.314641\pi$
$421$	22.5687	1.09993	0.549966	+	0.835187i	$0.314641\pi$
$422$	39.7158	1.93334
$423$	0	0
$424$	3.47738	0.168877
$425$	17.6589	0.856580
$426$	0	0
$427$	−26.6637	−1.29034
$428$	−42.9589	−2.07650
$429$	0	0
$430$	−0.571248	−0.0275480
$431$	−2.17813	−0.104917	−0.0524583	−	0.998623i	$-0.516706\pi$
$431$	−2.17813	−0.104917	−0.0524583	+	0.998623i	$0.516706\pi$
$432$	0	0
$433$	−29.5089	−1.41811	−0.709054	−	0.705154i	$-0.750878\pi$
$433$	−29.5089	−1.41811	−0.709054	+	0.705154i	$0.750878\pi$
$434$	9.61452	0.461512
$435$	0	0
$436$	−4.82596	−0.231121
$437$	−1.41360	−0.0676218
$438$	0	0
$439$	10.6539	0.508483	0.254242	−	0.967141i	$-0.418174\pi$
$439$	10.6539	0.508483	0.254242	+	0.967141i	$0.418174\pi$
$440$	−0.335539	−0.0159962
$441$	0	0
$442$	−25.9757	−1.23554
$443$	14.2197	0.675598	0.337799	−	0.941218i	$-0.390318\pi$
$443$	14.2197	0.675598	0.337799	+	0.941218i	$0.390318\pi$
$444$	0	0
$445$	−2.64919	−0.125584
$446$	9.18517	0.434930
$447$	0	0
$448$	25.7014	1.21428
$449$	−11.9133	−0.562221	−0.281111	−	0.959675i	$-0.590703\pi$
$449$	−11.9133	−0.562221	−0.281111	+	0.959675i	$0.590703\pi$
$450$	0	0
$451$	16.6541	0.784210
$452$	−2.07457	−0.0975797
$453$	0	0
$454$	59.1454	2.77583
$455$	−2.29350	−0.107521
$456$	0	0
$457$	15.3608	0.718550	0.359275	−	0.933232i	$-0.383024\pi$
$457$	15.3608	0.718550	0.359275	+	0.933232i	$0.383024\pi$
$458$	−55.1611	−2.57751
$459$	0	0
$460$	−2.81587	−0.131291
$461$	−15.7741	−0.734672	−0.367336	−	0.930088i	$-0.619730\pi$
$461$	−15.7741	−0.734672	−0.367336	+	0.930088i	$0.619730\pi$
$462$	0	0
$463$	−4.72263	−0.219479	−0.109740	−	0.993960i	$-0.535002\pi$
$463$	−4.72263	−0.219479	−0.109740	+	0.993960i	$0.535002\pi$
$464$	18.7623	0.871017
$465$	0	0
$466$	−58.7255	−2.72041
$467$	−24.0572	−1.11324	−0.556618	−	0.830768i	$-0.687901\pi$
$467$	−24.0572	−1.11324	−0.556618	+	0.830768i	$0.687901\pi$
$468$	0	0
$469$	13.8864	0.641216
$470$	4.97829	0.229631
$471$	0	0
$472$	7.60538	0.350066
$473$	1.59894	0.0735191
$474$	0	0
$475$	−1.60002	−0.0734139
$476$	20.5425	0.941563
$477$	0	0
$478$	−11.6692	−0.533738
$479$	−24.2562	−1.10830	−0.554148	−	0.832418i	$-0.686956\pi$
$479$	−24.2562	−1.10830	−0.554148	+	0.832418i	$0.686956\pi$
$480$	0	0
$481$	7.73200	0.352549
$482$	15.0349	0.684822
$483$	0	0
$484$	−19.9885	−0.908570
$485$	0.464100	0.0210737
$486$	0	0
$487$	−32.1708	−1.45780	−0.728899	−	0.684621i	$-0.759968\pi$
$487$	−32.1708	−1.45780	−0.728899	+	0.684621i	$0.759968\pi$
$488$	8.45784	0.382868
$489$	0	0
$490$	−0.652630	−0.0294828
$491$	7.19348	0.324637	0.162319	−	0.986738i	$-0.448103\pi$
$491$	7.19348	0.324637	0.162319	+	0.986738i	$0.448103\pi$
$492$	0	0
$493$	21.4895	0.967838
$494$	2.35358	0.105893
$495$	0	0
$496$	5.95048	0.267184
$497$	−16.3178	−0.731952
$498$	0	0
$499$	40.0332	1.79213	0.896066	−	0.443920i	$-0.146413\pi$
$499$	40.0332	1.79213	0.896066	+	0.443920i	$0.146413\pi$
$500$	−6.42277	−0.287235
$501$	0	0
$502$	11.1653	0.498332
$503$	−31.2924	−1.39526	−0.697630	−	0.716458i	$-0.745762\pi$
$503$	−31.2924	−1.39526	−0.697630	+	0.716458i	$0.745762\pi$
$504$	0	0
$505$	−3.01019	−0.133952
$506$	14.5403	0.646396
$507$	0	0
$508$	7.39148	0.327944
$509$	−42.8120	−1.89761	−0.948803	−	0.315867i	$-0.897705\pi$
$509$	−42.8120	−1.89761	−0.948803	+	0.315867i	$0.897705\pi$
$510$	0	0
$511$	−2.01484	−0.0891315
$512$	30.0909	1.32984
$513$	0	0
$514$	−34.8893	−1.53890
$515$	1.28209	0.0564955
$516$	0	0
$517$	−13.9343	−0.612831
$518$	−11.2806	−0.495641
$519$	0	0
$520$	0.727510	0.0319034
$521$	−2.79772	−0.122570	−0.0612851	−	0.998120i	$-0.519520\pi$
$521$	−2.79772	−0.122570	−0.0612851	+	0.998120i	$0.519520\pi$
$522$	0	0
$523$	−7.44564	−0.325575	−0.162788	−	0.986661i	$-0.552049\pi$
$523$	−7.44564	−0.325575	−0.162788	+	0.986661i	$0.552049\pi$
$524$	−3.61562	−0.157949
$525$	0	0
$526$	−36.5540	−1.59383
$527$	6.81542	0.296884
$528$	0	0
$529$	−4.06494	−0.176737
$530$	2.58750	0.112394
$531$	0	0
$532$	−1.86130	−0.0806974
$533$	−36.1090	−1.56405
$534$	0	0
$535$	−4.96027	−0.214451
$536$	−4.40484	−0.190260
$537$	0	0
$538$	67.5501	2.91229
$539$	1.82672	0.0786825
$540$	0	0
$541$	−36.9090	−1.58684	−0.793421	−	0.608673i	$-0.791702\pi$
$541$	−36.9090	−1.58684	−0.793421	+	0.608673i	$0.791702\pi$
$542$	7.97725	0.342652
$543$	0	0
$544$	28.9598	1.24164
$545$	−0.557231	−0.0238692
$546$	0	0
$547$	39.6354	1.69469	0.847343	−	0.531046i	$-0.178201\pi$
$547$	39.6354	1.69469	0.847343	+	0.531046i	$0.178201\pi$
$548$	−10.2196	−0.436560
$549$	0	0
$550$	16.4578	0.701762
$551$	−1.94710	−0.0829493
$552$	0	0
$553$	19.4005	0.824995
$554$	59.3570	2.52184
$555$	0	0
$556$	−44.0674	−1.86887
$557$	24.8972	1.05493	0.527464	−	0.849578i	$-0.323143\pi$
$557$	24.8972	1.05493	0.527464	+	0.849578i	$0.323143\pi$
$558$	0	0
$559$	−3.46678	−0.146629
$560$	2.07092	0.0875125
$561$	0	0
$562$	−14.0552	−0.592885
$563$	13.6122	0.573684	0.286842	−	0.957978i	$-0.407395\pi$
$563$	13.6122	0.573684	0.286842	+	0.957978i	$0.407395\pi$
$564$	0	0
$565$	−0.239541	−0.0100776
$566$	64.3579	2.70516
$567$	0	0
$568$	5.17607	0.217183
$569$	10.6136	0.444945	0.222473	−	0.974939i	$-0.428587\pi$
$569$	10.6136	0.444945	0.222473	+	0.974939i	$0.428587\pi$
$570$	0	0
$571$	34.9349	1.46198	0.730990	−	0.682389i	$-0.239059\pi$
$571$	34.9349	1.46198	0.730990	+	0.682389i	$0.239059\pi$
$572$	−13.1226	−0.548685
$573$	0	0
$574$	52.6813	2.19887
$575$	21.4321	0.893780
$576$	0	0
$577$	−17.0364	−0.709236	−0.354618	−	0.935011i	$-0.615389\pi$
$577$	−17.0364	−0.709236	−0.354618	+	0.935011i	$0.615389\pi$
$578$	−8.66286	−0.360328
$579$	0	0
$580$	−3.87859	−0.161050
$581$	16.4834	0.683846
$582$	0	0
$583$	−7.24247	−0.299952
$584$	0.639118	0.0264469
$585$	0	0
$586$	−9.34981	−0.386237
$587$	−1.64993	−0.0681000	−0.0340500	−	0.999420i	$-0.510841\pi$
$587$	−1.64993	−0.0681000	−0.0340500	+	0.999420i	$0.510841\pi$
$588$	0	0
$589$	−0.617526	−0.0254447
$590$	5.65912	0.232982
$591$	0	0
$592$	−6.98162	−0.286943
$593$	−2.89587	−0.118919	−0.0594596	−	0.998231i	$-0.518938\pi$
$593$	−2.89587	−0.118919	−0.0594596	+	0.998231i	$0.518938\pi$
$594$	0	0
$595$	2.37195	0.0972403
$596$	12.1153	0.496264
$597$	0	0
$598$	−31.5260	−1.28919
$599$	8.40396	0.343377	0.171688	−	0.985151i	$-0.445078\pi$
$599$	8.40396	0.343377	0.171688	+	0.985151i	$0.445078\pi$
$600$	0	0
$601$	30.5408	1.24578	0.622892	−	0.782308i	$-0.285958\pi$
$601$	30.5408	1.24578	0.622892	+	0.782308i	$0.285958\pi$
$602$	5.05786	0.206143
$603$	0	0
$604$	30.5309	1.24229
$605$	−2.30799	−0.0938330
$606$	0	0
$607$	−37.2525	−1.51203	−0.756017	−	0.654552i	$-0.772857\pi$
$607$	−37.2525	−1.51203	−0.756017	+	0.654552i	$0.772857\pi$
$608$	−2.62397	−0.106416
$609$	0	0
$610$	6.29343	0.254813
$611$	30.2121	1.22225
$612$	0	0
$613$	−15.5040	−0.626199	−0.313099	−	0.949720i	$-0.601367\pi$
$613$	−15.5040	−0.626199	−0.313099	+	0.949720i	$0.601367\pi$
$614$	33.1380	1.33734
$615$	0	0
$616$	2.97088	0.119700
$617$	−37.3578	−1.50397	−0.751985	−	0.659180i	$-0.770903\pi$
$617$	−37.3578	−1.50397	−0.751985	+	0.659180i	$0.770903\pi$
$618$	0	0
$619$	−40.8564	−1.64216	−0.821079	−	0.570815i	$-0.806627\pi$
$619$	−40.8564	−1.64216	−0.821079	+	0.570815i	$0.806627\pi$
$620$	−1.23010	−0.0494020
$621$	0	0
$622$	17.8626	0.716223
$623$	23.4560	0.939747
$624$	0	0
$625$	23.8848	0.955392
$626$	47.8164	1.91113
$627$	0	0
$628$	−26.7978	−1.06935
$629$	−7.99645	−0.318839
$630$	0	0
$631$	−22.1435	−0.881520	−0.440760	−	0.897625i	$-0.645291\pi$
$631$	−22.1435	−0.881520	−0.440760	+	0.897625i	$0.645291\pi$
$632$	−6.15394	−0.244791
$633$	0	0
$634$	−57.2553	−2.27390
$635$	0.853460	0.0338685
$636$	0	0
$637$	−3.96066	−0.156927
$638$	20.0279	0.792911
$639$	0	0
$640$	−1.65049	−0.0652415
$641$	10.5167	0.415385	0.207693	−	0.978194i	$-0.433405\pi$
$641$	10.5167	0.415385	0.207693	+	0.978194i	$0.433405\pi$
$642$	0	0
$643$	−6.63816	−0.261784	−0.130892	−	0.991397i	$-0.541784\pi$
$643$	−6.63816	−0.261784	−0.130892	+	0.991397i	$0.541784\pi$
$644$	24.9319	0.982453
$645$	0	0
$646$	−2.43408	−0.0957677
$647$	−41.0456	−1.61367	−0.806834	−	0.590779i	$-0.798821\pi$
$647$	−41.0456	−1.61367	−0.806834	+	0.590779i	$0.798821\pi$
$648$	0	0
$649$	−15.8400	−0.621774
$650$	−35.6834	−1.39962
$651$	0	0
$652$	22.4109	0.877678
$653$	−26.0745	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$653$	−26.0745	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$654$	0	0
$655$	−0.417479	−0.0163123
$656$	32.6047	1.27300
$657$	0	0
$658$	−44.0780	−1.71834
$659$	0.966936	0.0376665	0.0188332	−	0.999823i	$-0.494005\pi$
$659$	0.966936	0.0376665	0.0188332	+	0.999823i	$0.494005\pi$
$660$	0	0
$661$	−18.0554	−0.702275	−0.351138	−	0.936324i	$-0.614205\pi$
$661$	−18.0554	−0.702275	−0.351138	+	0.936324i	$0.614205\pi$
$662$	−48.2746	−1.87624
$663$	0	0
$664$	−5.22861	−0.202909
$665$	−0.214915	−0.00833406
$666$	0	0
$667$	26.0812	1.00987
$668$	4.17808	0.161655
$669$	0	0
$670$	−3.27762	−0.126625
$671$	−17.6154	−0.680036
$672$	0	0
$673$	−10.8772	−0.419286	−0.209643	−	0.977778i	$-0.567230\pi$
$673$	−10.8772	−0.419286	−0.209643	+	0.977778i	$0.567230\pi$
$674$	45.9891	1.77143
$675$	0	0
$676$	−2.32343	−0.0893626
$677$	−13.3664	−0.513712	−0.256856	−	0.966450i	$-0.582687\pi$
$677$	−13.3664	−0.513712	−0.256856	+	0.966450i	$0.582687\pi$
$678$	0	0
$679$	−4.10916	−0.157695
$680$	−0.752392	−0.0288529
$681$	0	0
$682$	6.35186	0.243225
$683$	−19.4039	−0.742471	−0.371236	−	0.928539i	$-0.621066\pi$
$683$	−19.4039	−0.742471	−0.371236	+	0.928539i	$0.621066\pi$
$684$	0	0
$685$	−1.18001	−0.0450860
$686$	41.1834	1.57239
$687$	0	0
$688$	3.13034	0.119343
$689$	15.7030	0.598236
$690$	0	0
$691$	−24.8652	−0.945918	−0.472959	−	0.881085i	$-0.656814\pi$
$691$	−24.8652	−0.945918	−0.472959	+	0.881085i	$0.656814\pi$
$692$	49.8851	1.89635
$693$	0	0
$694$	11.2919	0.428637
$695$	−5.08826	−0.193009
$696$	0	0
$697$	37.3440	1.41451
$698$	9.84740	0.372729
$699$	0	0
$700$	28.2197	1.06660
$701$	33.5387	1.26674	0.633369	−	0.773850i	$-0.281672\pi$
$701$	33.5387	1.26674	0.633369	+	0.773850i	$0.281672\pi$
$702$	0	0
$703$	0.724536	0.0273264
$704$	16.9797	0.639946
$705$	0	0
$706$	−52.3000	−1.96834
$707$	26.6524	1.00237
$708$	0	0
$709$	7.22273	0.271255	0.135628	−	0.990760i	$-0.456695\pi$
$709$	7.22273	0.271255	0.135628	+	0.990760i	$0.456695\pi$
$710$	3.85148	0.144544
$711$	0	0
$712$	−7.44037	−0.278840
$713$	8.27169	0.309777
$714$	0	0
$715$	−1.51521	−0.0566656
$716$	18.3959	0.687489
$717$	0	0
$718$	23.9224	0.892777
$719$	−17.6476	−0.658144	−0.329072	−	0.944305i	$-0.606736\pi$
$719$	−17.6476	−0.658144	−0.329072	+	0.944305i	$0.606736\pi$
$720$	0	0
$721$	−11.3517	−0.422758
$722$	−39.4861	−1.46952
$723$	0	0
$724$	50.7092	1.88459
$725$	29.5206	1.09637
$726$	0	0
$727$	7.03178	0.260794	0.130397	−	0.991462i	$-0.458375\pi$
$727$	7.03178	0.260794	0.130397	+	0.991462i	$0.458375\pi$
$728$	−6.44141	−0.238734
$729$	0	0
$730$	0.475564	0.0176014
$731$	3.58535	0.132609
$732$	0	0
$733$	40.1119	1.48157	0.740784	−	0.671744i	$-0.234454\pi$
$733$	40.1119	1.48157	0.740784	+	0.671744i	$0.234454\pi$
$734$	−34.1217	−1.25945
$735$	0	0
$736$	35.1478	1.29556
$737$	9.17412	0.337933
$738$	0	0
$739$	−48.1715	−1.77202	−0.886008	−	0.463670i	$-0.846532\pi$
$739$	−48.1715	−1.77202	−0.886008	+	0.463670i	$0.846532\pi$
$740$	1.44326	0.0530553
$741$	0	0
$742$	−22.9099	−0.841048
$743$	33.3311	1.22280	0.611400	−	0.791322i	$-0.290607\pi$
$743$	33.3311	1.22280	0.611400	+	0.791322i	$0.290607\pi$
$744$	0	0
$745$	1.39890	0.0512519
$746$	34.0605	1.24704
$747$	0	0
$748$	13.5714	0.496221
$749$	43.9185	1.60475
$750$	0	0
$751$	−39.2981	−1.43401	−0.717004	−	0.697069i	$-0.754487\pi$
$751$	−39.2981	−1.43401	−0.717004	+	0.697069i	$0.754487\pi$
$752$	−27.2801	−0.994804
$753$	0	0
$754$	−43.4240	−1.58141
$755$	3.52526	0.128298
$756$	0	0
$757$	23.7890	0.864627	0.432313	−	0.901723i	$-0.357697\pi$
$757$	23.7890	0.864627	0.432313	+	0.901723i	$0.357697\pi$
$758$	72.9991	2.65145
$759$	0	0
$760$	0.0681721	0.00247286
$761$	−24.5736	−0.890791	−0.445395	−	0.895334i	$-0.646937\pi$
$761$	−24.5736	−0.890791	−0.445395	+	0.895334i	$0.646937\pi$
$762$	0	0
$763$	4.93375	0.178614
$764$	−40.7440	−1.47407
$765$	0	0
$766$	−57.6639	−2.08348
$767$	34.3439	1.24009
$768$	0	0
$769$	−24.1816	−0.872010	−0.436005	−	0.899944i	$-0.643607\pi$
$769$	−24.1816	−0.872010	−0.436005	+	0.899944i	$0.643607\pi$
$770$	2.21062	0.0796651
$771$	0	0
$772$	−11.7270	−0.422064
$773$	7.66684	0.275757	0.137878	−	0.990449i	$-0.455972\pi$
$773$	7.66684	0.275757	0.137878	+	0.990449i	$0.455972\pi$
$774$	0	0
$775$	9.36250	0.336311
$776$	1.30344	0.0467909
$777$	0	0
$778$	−29.5181	−1.05828
$779$	−3.38364	−0.121231
$780$	0	0
$781$	−10.7804	−0.385752
$782$	32.6043	1.16593
$783$	0	0
$784$	3.57629	0.127725
$785$	−3.09422	−0.110437
$786$	0	0
$787$	−14.0010	−0.499081	−0.249540	−	0.968364i	$-0.580280\pi$
$787$	−14.0010	−0.499081	−0.249540	+	0.968364i	$0.580280\pi$
$788$	1.63755	0.0583352
$789$	0	0
$790$	−4.57911	−0.162917
$791$	2.12091	0.0754109
$792$	0	0
$793$	38.1934	1.35629
$794$	61.5212	2.18331
$795$	0	0
$796$	32.0611	1.13637
$797$	−7.12692	−0.252449	−0.126224	−	0.992002i	$-0.540286\pi$
$797$	−7.12692	−0.252449	−0.126224	+	0.992002i	$0.540286\pi$
$798$	0	0
$799$	−31.2455	−1.10538
$800$	39.7828	1.40653
$801$	0	0
$802$	−35.6481	−1.25878
$803$	−1.33111	−0.0469740
$804$	0	0
$805$	2.87877	0.101463
$806$	−13.7720	−0.485097
$807$	0	0
$808$	−8.45426	−0.297420
$809$	28.5298	1.00306	0.501528	−	0.865141i	$-0.332771\pi$
$809$	28.5298	1.00306	0.501528	+	0.865141i	$0.332771\pi$
$810$	0	0
$811$	21.0869	0.740463	0.370231	−	0.928940i	$-0.379278\pi$
$811$	21.0869	0.740463	0.370231	+	0.928940i	$0.379278\pi$
$812$	34.3412	1.20514
$813$	0	0
$814$	−7.45257	−0.261212
$815$	2.58768	0.0906426
$816$	0	0
$817$	−0.324858	−0.0113654
$818$	−78.2066	−2.73443
$819$	0	0
$820$	−6.74014	−0.235376
$821$	6.76105	0.235962	0.117981	−	0.993016i	$-0.462358\pi$
$821$	6.76105	0.235962	0.117981	+	0.993016i	$0.462358\pi$
$822$	0	0
$823$	51.4798	1.79447	0.897237	−	0.441550i	$-0.145571\pi$
$823$	51.4798	1.79447	0.897237	+	0.441550i	$0.145571\pi$
$824$	3.60080	0.125440
$825$	0	0
$826$	−50.1061	−1.74342
$827$	−32.7541	−1.13897	−0.569486	−	0.822001i	$-0.692858\pi$
$827$	−32.7541	−1.13897	−0.569486	+	0.822001i	$0.692858\pi$
$828$	0	0
$829$	−31.7417	−1.10243	−0.551217	−	0.834362i	$-0.685836\pi$
$829$	−31.7417	−1.10243	−0.551217	+	0.834362i	$0.685836\pi$
$830$	−3.89058	−0.135044
$831$	0	0
$832$	−36.8150	−1.27633
$833$	4.09613	0.141922
$834$	0	0
$835$	0.482423	0.0166950
$836$	−1.22967	−0.0425291
$837$	0	0
$838$	71.3354	2.46424
$839$	29.1488	1.00633	0.503164	−	0.864191i	$-0.332169\pi$
$839$	29.1488	1.00633	0.503164	+	0.864191i	$0.332169\pi$
$840$	0	0
$841$	6.92434	0.238770
$842$	47.1646	1.62540
$843$	0	0
$844$	44.9902	1.54863
$845$	−0.268276	−0.00922896
$846$	0	0
$847$	20.4350	0.702156
$848$	−14.1790	−0.486910
$849$	0	0
$850$	36.9039	1.26579
$851$	−9.70507	−0.332686
$852$	0	0
$853$	−54.1664	−1.85462	−0.927312	−	0.374290i	$-0.877887\pi$
$853$	−54.1664	−1.85462	−0.927312	+	0.374290i	$0.877887\pi$
$854$	−55.7223	−1.90678
$855$	0	0
$856$	−13.9311	−0.476156
$857$	−1.72499	−0.0589246	−0.0294623	−	0.999566i	$-0.509380\pi$
$857$	−1.72499	−0.0589246	−0.0294623	+	0.999566i	$0.509380\pi$
$858$	0	0
$859$	15.2815	0.521397	0.260699	−	0.965420i	$-0.416047\pi$
$859$	15.2815	0.521397	0.260699	+	0.965420i	$0.416047\pi$
$860$	−0.647112	−0.0220663
$861$	0	0
$862$	−4.55190	−0.155038
$863$	31.2885	1.06507	0.532537	−	0.846407i	$-0.321239\pi$
$863$	31.2885	1.06507	0.532537	+	0.846407i	$0.321239\pi$
$864$	0	0
$865$	5.76001	0.195846
$866$	−61.6684	−2.09558
$867$	0	0
$868$	10.8914	0.369677
$869$	12.8170	0.434788
$870$	0	0
$871$	−19.8911	−0.673985
$872$	−1.56501	−0.0529979
$873$	0	0
$874$	−2.95418	−0.0999266
$875$	6.56623	0.221979
$876$	0	0
$877$	10.5431	0.356014	0.178007	−	0.984029i	$-0.443035\pi$
$877$	10.5431	0.356014	0.178007	+	0.984029i	$0.443035\pi$
$878$	22.2648	0.751400
$879$	0	0
$880$	1.36816	0.0461207
$881$	−42.4065	−1.42871	−0.714356	−	0.699783i	$-0.753280\pi$
$881$	−42.4065	−1.42871	−0.714356	+	0.699783i	$0.753280\pi$
$882$	0	0
$883$	−40.6960	−1.36953	−0.684765	−	0.728764i	$-0.740095\pi$
$883$	−40.6960	−1.36953	−0.684765	+	0.728764i	$0.740095\pi$
$884$	−29.4253	−0.989681
$885$	0	0
$886$	29.7166	0.998350
$887$	−24.5321	−0.823706	−0.411853	−	0.911250i	$-0.635118\pi$
$887$	−24.5321	−0.823706	−0.411853	+	0.911250i	$0.635118\pi$
$888$	0	0
$889$	−7.55658	−0.253439
$890$	−5.53633	−0.185578
$891$	0	0
$892$	10.4050	0.348385
$893$	2.83106	0.0947379
$894$	0	0
$895$	2.12410	0.0710007
$896$	14.6136	0.488205
$897$	0	0
$898$	−24.8966	−0.830810
$899$	11.3934	0.379993
$900$	0	0
$901$	−16.2400	−0.541034
$902$	34.8040	1.15885
$903$	0	0
$904$	−0.672763	−0.0223758
$905$	5.85515	0.194632
$906$	0	0
$907$	−30.1628	−1.00154	−0.500770	−	0.865581i	$-0.666950\pi$
$907$	−30.1628	−1.00154	−0.500770	+	0.865581i	$0.666950\pi$
$908$	67.0001	2.22348
$909$	0	0
$910$	−4.79301	−0.158887
$911$	−7.46680	−0.247386	−0.123693	−	0.992321i	$-0.539474\pi$
$911$	−7.46680	−0.247386	−0.123693	+	0.992321i	$0.539474\pi$
$912$	0	0
$913$	10.8898	0.360400
$914$	32.1014	1.06182
$915$	0	0
$916$	−62.4867	−2.06462
$917$	3.69638	0.122065
$918$	0	0
$919$	28.7562	0.948579	0.474289	−	0.880369i	$-0.342705\pi$
$919$	28.7562	0.948579	0.474289	+	0.880369i	$0.342705\pi$
$920$	−0.913158	−0.0301059
$921$	0	0
$922$	−32.9650	−1.08564
$923$	23.3738	0.769358
$924$	0	0
$925$	−10.9849	−0.361182
$926$	−9.86945	−0.324330
$927$	0	0
$928$	48.4126	1.58922
$929$	35.4718	1.16379	0.581896	−	0.813263i	$-0.302311\pi$
$929$	35.4718	1.16379	0.581896	+	0.813263i	$0.302311\pi$
$930$	0	0
$931$	−0.371138	−0.0121636
$932$	−66.5244	−2.17908
$933$	0	0
$934$	−50.2754	−1.64506
$935$	1.56703	0.0512475
$936$	0	0
$937$	−37.7639	−1.23369	−0.616847	−	0.787083i	$-0.711590\pi$
$937$	−37.7639	−1.23369	−0.616847	+	0.787083i	$0.711590\pi$
$938$	29.0202	0.947543
$939$	0	0
$940$	5.63942	0.183938
$941$	18.4224	0.600553	0.300277	−	0.953852i	$-0.402921\pi$
$941$	18.4224	0.600553	0.300277	+	0.953852i	$0.402921\pi$
$942$	0	0
$943$	45.3234	1.47593
$944$	−31.0109	−1.00932
$945$	0	0
$946$	3.34149	0.108641
$947$	−16.5267	−0.537047	−0.268523	−	0.963273i	$-0.586536\pi$
$947$	−16.5267	−0.537047	−0.268523	+	0.963273i	$0.586536\pi$
$948$	0	0
$949$	2.88609	0.0936865
$950$	−3.34376	−0.108486
$951$	0	0
$952$	6.66172	0.215907
$953$	−25.4326	−0.823842	−0.411921	−	0.911219i	$-0.635142\pi$
$953$	−25.4326	−0.823842	−0.411921	+	0.911219i	$0.635142\pi$
$954$	0	0
$955$	−4.70453	−0.152235
$956$	−13.2189	−0.427531
$957$	0	0
$958$	−50.6912	−1.63776
$959$	10.4479	0.337380
$960$	0	0
$961$	−27.3866	−0.883437
$962$	16.1585	0.520971
$963$	0	0
$964$	17.0316	0.548551
$965$	−1.35406	−0.0435888
$966$	0	0
$967$	−14.1414	−0.454757	−0.227379	−	0.973806i	$-0.573016\pi$
$967$	−14.1414	−0.454757	−0.227379	+	0.973806i	$0.573016\pi$
$968$	−6.48208	−0.208342
$969$	0	0
$970$	0.969886	0.0311412
$971$	−27.1607	−0.871627	−0.435814	−	0.900037i	$-0.643539\pi$
$971$	−27.1607	−0.871627	−0.435814	+	0.900037i	$0.643539\pi$
$972$	0	0
$973$	45.0517	1.44429
$974$	−67.2313	−2.15423
$975$	0	0
$976$	−34.4868	−1.10390
$977$	−7.27440	−0.232729	−0.116364	−	0.993207i	$-0.537124\pi$
$977$	−7.27440	−0.232729	−0.116364	+	0.993207i	$0.537124\pi$
$978$	0	0
$979$	15.4963	0.495264
$980$	−0.739301	−0.0236161
$981$	0	0
$982$	15.0331	0.479725
$983$	39.7774	1.26870	0.634351	−	0.773045i	$-0.281267\pi$
$983$	39.7774	1.26870	0.634351	+	0.773045i	$0.281267\pi$
$984$	0	0
$985$	0.189080	0.00602459
$986$	44.9092	1.43020
$987$	0	0
$988$	2.66615	0.0848214
$989$	4.35144	0.138368
$990$	0	0
$991$	−7.74113	−0.245905	−0.122953	−	0.992413i	$-0.539236\pi$
$991$	−7.74113	−0.245905	−0.122953	+	0.992413i	$0.539236\pi$
$992$	15.3541	0.487494
$993$	0	0
$994$	−34.1012	−1.08162
$995$	3.70194	0.117359
$996$	0	0
$997$	−4.37867	−0.138674	−0.0693369	−	0.997593i	$-0.522088\pi$
$997$	−4.37867	−0.138674	−0.0693369	+	0.997593i	$0.522088\pi$
$998$	83.6623	2.64828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3483.2.a.t.1.19		20
3.2	odd	2		3483.2.a.u.1.2		20
9.2	odd	6		1161.2.f.d.388.19		40
9.4	even	3		387.2.f.d.259.2	yes	40
9.5	odd	6		1161.2.f.d.775.19		40
9.7	even	3		387.2.f.d.130.2	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.f.d.130.2	✓	40	9.7	even	3
387.2.f.d.259.2	yes	40	9.4	even	3
1161.2.f.d.388.19		40	9.2	odd	6
1161.2.f.d.775.19		40	9.5	odd	6
3483.2.a.t.1.19		20	1.1	even	1	trivial
3483.2.a.u.1.2		20	3.2	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:	\( N \)	\(=\)	\( 3483 = 3^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3483.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08982 q^{2} +2.36736 q^{4} +0.273348 q^{5} -2.42024 q^{7} +0.767710 q^{8} +O(q^{10})\)	\(q+2.08982 q^{2} +2.36736 q^{4} +0.273348 q^{5} -2.42024 q^{7} +0.767710 q^{8} +0.571248 q^{10} -1.59894 q^{11} +3.46678 q^{13} -5.05786 q^{14} -3.13034 q^{16} -3.58535 q^{17} +0.324858 q^{19} +0.647112 q^{20} -3.34149 q^{22} -4.35144 q^{23} -4.92528 q^{25} +7.24495 q^{26} -5.72956 q^{28} -5.99369 q^{29} -1.90091 q^{31} -8.07727 q^{32} -7.49274 q^{34} -0.661566 q^{35} +2.23031 q^{37} +0.678896 q^{38} +0.209852 q^{40} -10.4157 q^{41} -1.00000 q^{43} -3.78525 q^{44} -9.09374 q^{46} +8.71476 q^{47} -1.14246 q^{49} -10.2930 q^{50} +8.20710 q^{52} +4.52956 q^{53} -0.437066 q^{55} -1.85804 q^{56} -12.5257 q^{58} +9.90658 q^{59} +11.0170 q^{61} -3.97256 q^{62} -10.6194 q^{64} +0.947636 q^{65} -5.73764 q^{67} -8.48780 q^{68} -1.38256 q^{70} +6.74222 q^{71} +0.832499 q^{73} +4.66095 q^{74} +0.769056 q^{76} +3.86980 q^{77} -8.01597 q^{79} -0.855671 q^{80} -21.7670 q^{82} -6.81066 q^{83} -0.980048 q^{85} -2.08982 q^{86} -1.22752 q^{88} -9.69164 q^{89} -8.39042 q^{91} -10.3014 q^{92} +18.2123 q^{94} +0.0887994 q^{95} +1.69784 q^{97} -2.38754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8}+O(q^{10}) \) 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8	\( 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8} - 2 q^{10} - 10 q^{11} - q^{13} - 10 q^{14} + 22 q^{16} - 20 q^{17} + 8 q^{19} - 30 q^{20} + 15 q^{22} - 19 q^{23} + 19 q^{25} - 25 q^{26} - 3 q^{28} - 25 q^{29} - 11 q^{31} - 36 q^{32} + 9 q^{34} + 9 q^{37} - 28 q^{38} + 12 q^{40} - 12 q^{41} - 20 q^{43} - 5 q^{44} + 4 q^{46} - 38 q^{47} + 37 q^{49} - 36 q^{50} - 8 q^{52} - 69 q^{53} - 9 q^{55} - 30 q^{56} - 27 q^{58} - 31 q^{59} + 19 q^{61} - 32 q^{62} + 11 q^{64} - 47 q^{65} + 9 q^{67} - 68 q^{68} - 6 q^{70} - 21 q^{71} - 2 q^{73} + 16 q^{74} + 37 q^{76} - 85 q^{77} - 4 q^{79} - 61 q^{80} + q^{82} - 19 q^{83} - 6 q^{85} + 6 q^{86} + 60 q^{88} - 54 q^{89} - 3 q^{91} - 85 q^{92} - 19 q^{94} + 11 q^{95} + 2 q^{97} - 5 q^{98}+O(q^{100}) \) 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8 - 2 * q^10 - 10 * q^11 - q^13 - 10 * q^14 + 22 * q^16 - 20 * q^17 + 8 * q^19 - 30 * q^20 + 15 * q^22 - 19 * q^23 + 19 * q^25 - 25 * q^26 - 3 * q^28 - 25 * q^29 - 11 * q^31 - 36 * q^32 + 9 * q^34 + 9 * q^37 - 28 * q^38 + 12 * q^40 - 12 * q^41 - 20 * q^43 - 5 * q^44 + 4 * q^46 - 38 * q^47 + 37 * q^49 - 36 * q^50 - 8 * q^52 - 69 * q^53 - 9 * q^55 - 30 * q^56 - 27 * q^58 - 31 * q^59 + 19 * q^61 - 32 * q^62 + 11 * q^64 - 47 * q^65 + 9 * q^67 - 68 * q^68 - 6 * q^70 - 21 * q^71 - 2 * q^73 + 16 * q^74 + 37 * q^76 - 85 * q^77 - 4 * q^79 - 61 * q^80 + q^82 - 19 * q^83 - 6 * q^85 + 6 * q^86 + 60 * q^88 - 54 * q^89 - 3 * q^91 - 85 * q^92 - 19 * q^94 + 11 * q^95 + 2 * q^97 - 5 * q^98

Introduction

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