LMFDB - Embedded newform 1629.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1629,2,Mod(1,1629)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1629, 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("1629.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1629 = 3^{2} \cdot 181 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1629.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:	$13.0076304893$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 $ x^8 - 3x^7 - 6x^6 + 21x^5 + 5x^4 - 35x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 543)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.57409$ of defining polynomial
Character	$\chi$	$=$	1629.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.57409 q^{2} +0.477759 q^{4} -2.61079 q^{5} +3.43627 q^{7} -2.39614 q^{8} -4.10962 q^{10} -0.961440 q^{11} -2.19899 q^{13} +5.40900 q^{14} -4.72726 q^{16} -1.38921 q^{17} -1.29785 q^{19} -1.24733 q^{20} -1.51339 q^{22} +1.03393 q^{23} +1.81623 q^{25} -3.46140 q^{26} +1.64171 q^{28} -10.3621 q^{29} +1.46354 q^{31} -2.64885 q^{32} -2.18674 q^{34} -8.97139 q^{35} +0.102931 q^{37} -2.04293 q^{38} +6.25583 q^{40} -2.19969 q^{41} -5.89954 q^{43} -0.459336 q^{44} +1.62749 q^{46} -4.32862 q^{47} +4.80796 q^{49} +2.85891 q^{50} -1.05059 q^{52} -9.97667 q^{53} +2.51012 q^{55} -8.23380 q^{56} -16.3109 q^{58} -3.19012 q^{59} -7.75991 q^{61} +2.30375 q^{62} +5.28500 q^{64} +5.74110 q^{65} +4.55325 q^{67} -0.663707 q^{68} -14.1218 q^{70} +13.5850 q^{71} +1.86716 q^{73} +0.162023 q^{74} -0.620058 q^{76} -3.30377 q^{77} +1.27450 q^{79} +12.3419 q^{80} -3.46252 q^{82} -6.39088 q^{83} +3.62693 q^{85} -9.28640 q^{86} +2.30375 q^{88} -0.429627 q^{89} -7.55632 q^{91} +0.493968 q^{92} -6.81364 q^{94} +3.38841 q^{95} +7.74609 q^{97} +7.56817 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 5 q^{4} - 5 q^{5} - 3 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} + 13 q^{13} - 5 q^{14} + 3 q^{16} - 27 q^{17} - 10 q^{19} - 21 q^{20} + 11 q^{22} - 3 q^{23} + 13 q^{25} - 11 q^{26} - 3 q^{28}+ \cdots + 51 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 5 * q^4 - 5 * q^5 - 3 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 + 13 * q^13 - 5 * q^14 + 3 * q^16 - 27 * q^17 - 10 * q^19 - 21 * q^20 + 11 * q^22 - 3 * q^23 + 13 * q^25 - 11 * q^26 - 3 * q^28 - 16 * q^29 - q^31 - 10 * q^32 + 10 * q^34 - 14 * q^35 + 3 * q^37 - 14 * q^38 + 11 * q^40 - 20 * q^41 - 4 * q^43 - 9 * q^44 - 17 * q^46 - 12 * q^47 + 5 * q^49 - 2 * q^50 + 21 * q^52 - 23 * q^53 - 13 * q^55 + 8 * q^56 - 21 * q^58 - 7 * q^59 + q^61 - q^62 - 32 * q^64 - 10 * q^65 - 25 * q^67 + q^68 - 10 * q^70 - 11 * q^71 + 23 * q^73 - 2 * q^74 - 19 * q^76 - 20 * q^77 - 26 * q^79 + 15 * q^80 - 22 * q^82 - 16 * q^83 - 33 * q^85 + 21 * q^86 - q^88 - 14 * q^89 - 35 * q^91 + 18 * q^92 - 52 * q^94 - 10 * q^95 + 28 * q^97 + 51 * q^98

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 $

$2$	1.57409	1.11305	0.556525	−	0.830831i	$-0.312134\pi$
$2$	1.57409	1.11305	0.556525	+	0.830831i	$0.312134\pi$
$3$	0	0
$3$	0	0
$4$	0.477759	0.238880
$5$	−2.61079	−1.16758	−0.583791	−	0.811904i	$-0.698431\pi$
$5$	−2.61079	−1.16758	−0.583791	+	0.811904i	$0.698431\pi$
$6$	0	0
$7$	3.43627	1.29879	0.649394	−	0.760452i	$-0.275022\pi$
$7$	3.43627	1.29879	0.649394	+	0.760452i	$0.275022\pi$
$8$	−2.39614	−0.847165
$9$	0	0
$10$	−4.10962	−1.29958
$11$	−0.961440	−0.289885	−0.144942	−	0.989440i	$-0.546300\pi$
$11$	−0.961440	−0.289885	−0.144942	+	0.989440i	$0.546300\pi$
$12$	0	0
$13$	−2.19899	−0.609889	−0.304945	−	0.952370i	$-0.598638\pi$
$13$	−2.19899	−0.609889	−0.304945	+	0.952370i	$0.598638\pi$
$14$	5.40900	1.44562
$15$	0	0
$16$	−4.72726	−1.18182
$17$	−1.38921	−0.336933	−0.168466	−	0.985707i	$-0.553881\pi$
$17$	−1.38921	−0.336933	−0.168466	+	0.985707i	$0.553881\pi$
$18$	0	0
$19$	−1.29785	−0.297746	−0.148873	−	0.988856i	$-0.547565\pi$
$19$	−1.29785	−0.297746	−0.148873	+	0.988856i	$0.547565\pi$
$20$	−1.24733	−0.278911
$21$	0	0
$22$	−1.51339	−0.322656
$23$	1.03393	0.215589	0.107794	−	0.994173i	$-0.465621\pi$
$23$	1.03393	0.215589	0.107794	+	0.994173i	$0.465621\pi$
$24$	0	0
$25$	1.81623	0.363246
$26$	−3.46140	−0.678837
$27$	0	0
$28$	1.64171	0.310254
$29$	−10.3621	−1.92420	−0.962099	−	0.272701i	$-0.912083\pi$
$29$	−10.3621	−1.92420	−0.962099	+	0.272701i	$0.912083\pi$
$30$	0	0
$31$	1.46354	0.262860	0.131430	−	0.991325i	$-0.458043\pi$
$31$	1.46354	0.262860	0.131430	+	0.991325i	$0.458043\pi$
$32$	−2.64885	−0.468255
$33$	0	0
$34$	−2.18674	−0.375023
$35$	−8.97139	−1.51644
$36$	0	0
$37$	0.102931	0.0169218	0.00846089	−	0.999964i	$-0.497307\pi$
$37$	0.102931	0.0169218	0.00846089	+	0.999964i	$0.497307\pi$
$38$	−2.04293	−0.331407
$39$	0	0
$40$	6.25583	0.989134
$41$	−2.19969	−0.343534	−0.171767	−	0.985138i	$-0.554948\pi$
$41$	−2.19969	−0.343534	−0.171767	+	0.985138i	$0.554948\pi$
$42$	0	0
$43$	−5.89954	−0.899671	−0.449835	−	0.893111i	$-0.648517\pi$
$43$	−5.89954	−0.899671	−0.449835	+	0.893111i	$0.648517\pi$
$44$	−0.459336	−0.0692476
$45$	0	0
$46$	1.62749	0.239961
$47$	−4.32862	−0.631394	−0.315697	−	0.948860i	$-0.602238\pi$
$47$	−4.32862	−0.631394	−0.315697	+	0.948860i	$0.602238\pi$
$48$	0	0
$49$	4.80796	0.686852
$50$	2.85891	0.404311
$51$	0	0
$52$	−1.05059	−0.145690
$53$	−9.97667	−1.37040	−0.685200	−	0.728355i	$-0.740285\pi$
$53$	−9.97667	−1.37040	−0.685200	+	0.728355i	$0.740285\pi$
$54$	0	0
$55$	2.51012	0.338464
$56$	−8.23380	−1.10029
$57$	0	0
$58$	−16.3109	−2.14173
$59$	−3.19012	−0.415319	−0.207659	−	0.978201i	$-0.566585\pi$
$59$	−3.19012	−0.415319	−0.207659	+	0.978201i	$0.566585\pi$
$60$	0	0
$61$	−7.75991	−0.993555	−0.496778	−	0.867878i	$-0.665484\pi$
$61$	−7.75991	−0.993555	−0.496778	+	0.867878i	$0.665484\pi$
$62$	2.30375	0.292576
$63$	0	0
$64$	5.28500	0.660625
$65$	5.74110	0.712095
$66$	0	0
$67$	4.55325	0.556268	0.278134	−	0.960542i	$-0.410284\pi$
$67$	4.55325	0.556268	0.278134	+	0.960542i	$0.410284\pi$
$68$	−0.663707	−0.0804863
$69$	0	0
$70$	−14.1218	−1.68787
$71$	13.5850	1.61224	0.806121	−	0.591751i	$-0.201563\pi$
$71$	13.5850	1.61224	0.806121	+	0.591751i	$0.201563\pi$
$72$	0	0
$73$	1.86716	0.218535	0.109267	−	0.994012i	$-0.465149\pi$
$73$	1.86716	0.218535	0.109267	+	0.994012i	$0.465149\pi$
$74$	0.162023	0.0188348
$75$	0	0
$76$	−0.620058	−0.0711255
$77$	−3.30377	−0.376499
$78$	0	0
$79$	1.27450	0.143393	0.0716964	−	0.997427i	$-0.477159\pi$
$79$	1.27450	0.143393	0.0716964	+	0.997427i	$0.477159\pi$
$80$	12.3419	1.37987
$81$	0	0
$82$	−3.46252	−0.382371
$83$	−6.39088	−0.701490	−0.350745	−	0.936471i	$-0.614072\pi$
$83$	−6.39088	−0.701490	−0.350745	+	0.936471i	$0.614072\pi$
$84$	0	0
$85$	3.62693	0.393396
$86$	−9.28640	−1.00138
$87$	0	0
$88$	2.30375	0.245580
$89$	−0.429627	−0.0455404	−0.0227702	−	0.999741i	$-0.507249\pi$
$89$	−0.429627	−0.0455404	−0.0227702	+	0.999741i	$0.507249\pi$
$90$	0	0
$91$	−7.55632	−0.792118
$92$	0.493968	0.0514997
$93$	0	0
$94$	−6.81364	−0.702773
$95$	3.38841	0.347643
$96$	0	0
$97$	7.74609	0.786496	0.393248	−	0.919432i	$-0.371351\pi$
$97$	7.74609	0.786496	0.393248	+	0.919432i	$0.371351\pi$
$98$	7.56817	0.764500
$99$	0	0
$100$	0.867720	0.0867720
$101$	−8.32864	−0.828731	−0.414365	−	0.910111i	$-0.635996\pi$
$101$	−8.32864	−0.828731	−0.414365	+	0.910111i	$0.635996\pi$
$102$	0	0
$103$	13.5440	1.33453	0.667265	−	0.744821i	$-0.267465\pi$
$103$	13.5440	1.33453	0.667265	+	0.744821i	$0.267465\pi$
$104$	5.26909	0.516677
$105$	0	0
$106$	−15.7042	−1.52532
$107$	11.6097	1.12235	0.561175	−	0.827697i	$-0.310350\pi$
$107$	11.6097	1.12235	0.561175	+	0.827697i	$0.310350\pi$
$108$	0	0
$109$	−4.66299	−0.446633	−0.223317	−	0.974746i	$-0.571688\pi$
$109$	−4.66299	−0.446633	−0.223317	+	0.974746i	$0.571688\pi$
$110$	3.95115	0.376727
$111$	0	0
$112$	−16.2442	−1.53493
$113$	−10.9955	−1.03437	−0.517187	−	0.855872i	$-0.673021\pi$
$113$	−10.9955	−1.03437	−0.517187	+	0.855872i	$0.673021\pi$
$114$	0	0
$115$	−2.69937	−0.251717
$116$	−4.95060	−0.459651
$117$	0	0
$118$	−5.02154	−0.462270
$119$	−4.77370	−0.437604
$120$	0	0
$121$	−10.0756	−0.915967
$122$	−12.2148	−1.10588
$123$	0	0
$124$	0.699221	0.0627919
$125$	8.31216	0.743462
$126$	0	0
$127$	8.07470	0.716513	0.358257	−	0.933623i	$-0.383371\pi$
$127$	8.07470	0.716513	0.358257	+	0.933623i	$0.383371\pi$
$128$	13.6168	1.20356
$129$	0	0
$130$	9.03700	0.792598
$131$	18.8695	1.64864	0.824320	−	0.566125i	$-0.191558\pi$
$131$	18.8695	1.64864	0.824320	+	0.566125i	$0.191558\pi$
$132$	0	0
$133$	−4.45975	−0.386710
$134$	7.16723	0.619154
$135$	0	0
$136$	3.32875	0.285438
$137$	−17.6058	−1.50417	−0.752085	−	0.659067i	$-0.770952\pi$
$137$	−17.6058	−1.50417	−0.752085	+	0.659067i	$0.770952\pi$
$138$	0	0
$139$	−7.03888	−0.597030	−0.298515	−	0.954405i	$-0.596491\pi$
$139$	−7.03888	−0.597030	−0.298515	+	0.954405i	$0.596491\pi$
$140$	−4.28616	−0.362247
$141$	0	0
$142$	21.3840	1.79451
$143$	2.11419	0.176798
$144$	0	0
$145$	27.0533	2.24666
$146$	2.93908	0.243240
$147$	0	0
$148$	0.0491763	0.00404227
$149$	6.74021	0.552180	0.276090	−	0.961132i	$-0.410961\pi$
$149$	6.74021	0.552180	0.276090	+	0.961132i	$0.410961\pi$
$150$	0	0
$151$	3.82904	0.311603	0.155802	−	0.987788i	$-0.450204\pi$
$151$	3.82904	0.311603	0.155802	+	0.987788i	$0.450204\pi$
$152$	3.10983	0.252240
$153$	0	0
$154$	−5.20043	−0.419062
$155$	−3.82100	−0.306910
$156$	0	0
$157$	−2.71018	−0.216296	−0.108148	−	0.994135i	$-0.534492\pi$
$157$	−2.71018	−0.216296	−0.108148	+	0.994135i	$0.534492\pi$
$158$	2.00618	0.159603
$159$	0	0
$160$	6.91560	0.546726
$161$	3.55285	0.280004
$162$	0	0
$163$	−24.5296	−1.92131	−0.960654	−	0.277749i	$-0.910412\pi$
$163$	−24.5296	−1.92131	−0.960654	+	0.277749i	$0.910412\pi$
$164$	−1.05092	−0.0820633
$165$	0	0
$166$	−10.0598	−0.780793
$167$	7.97488	0.617115	0.308557	−	0.951206i	$-0.400154\pi$
$167$	7.97488	0.617115	0.308557	+	0.951206i	$0.400154\pi$
$168$	0	0
$169$	−8.16445	−0.628035
$170$	5.70912	0.437870
$171$	0	0
$172$	−2.81856	−0.214913
$173$	−23.6725	−1.79979	−0.899895	−	0.436107i	$-0.856357\pi$
$173$	−23.6725	−1.79979	−0.899895	+	0.436107i	$0.856357\pi$
$174$	0	0
$175$	6.24106	0.471779
$176$	4.54498	0.342591
$177$	0	0
$178$	−0.676272	−0.0506887
$179$	14.7917	1.10558	0.552790	−	0.833320i	$-0.313563\pi$
$179$	14.7917	1.10558	0.552790	+	0.833320i	$0.313563\pi$
$180$	0	0
$181$	1.00000	0.0743294
$181$	1.00000	0.0743294
$182$	−11.8943	−0.881666
$183$	0	0
$184$	−2.47744	−0.182639
$185$	−0.268732	−0.0197576
$186$	0	0
$187$	1.33564	0.0976717
$188$	−2.06804	−0.150827
$189$	0	0
$190$	5.33366	0.386944
$191$	16.3286	1.18150	0.590749	−	0.806855i	$-0.298832\pi$
$191$	16.3286	1.18150	0.590749	+	0.806855i	$0.298832\pi$
$192$	0	0
$193$	21.4797	1.54614	0.773070	−	0.634321i	$-0.218720\pi$
$193$	21.4797	1.54614	0.773070	+	0.634321i	$0.218720\pi$
$194$	12.1930	0.875409
$195$	0	0
$196$	2.29705	0.164075
$197$	0.649364	0.0462653	0.0231326	−	0.999732i	$-0.492636\pi$
$197$	0.649364	0.0462653	0.0231326	+	0.999732i	$0.492636\pi$
$198$	0	0
$199$	13.6175	0.965321	0.482661	−	0.875807i	$-0.339670\pi$
$199$	13.6175	0.965321	0.482661	+	0.875807i	$0.339670\pi$
$200$	−4.35195	−0.307729
$201$	0	0
$202$	−13.1100	−0.922418
$203$	−35.6071	−2.49913
$204$	0	0
$205$	5.74294	0.401104
$206$	21.3195	1.48540
$207$	0	0
$208$	10.3952	0.720777
$209$	1.24780	0.0863122
$210$	0	0
$211$	6.80702	0.468615	0.234307	−	0.972163i	$-0.424718\pi$
$211$	6.80702	0.468615	0.234307	+	0.972163i	$0.424718\pi$
$212$	−4.76644	−0.327361
$213$	0	0
$214$	18.2747	1.24923
$215$	15.4025	1.05044
$216$	0	0
$217$	5.02913	0.341400
$218$	−7.33996	−0.497125
$219$	0	0
$220$	1.19923	0.0808522
$221$	3.05485	0.205492
$222$	0	0
$223$	−0.115112	−0.00770848	−0.00385424	−	0.999993i	$-0.501227\pi$
$223$	−0.115112	−0.00770848	−0.00385424	+	0.999993i	$0.501227\pi$
$224$	−9.10217	−0.608164
$225$	0	0
$226$	−17.3080	−1.15131
$227$	10.4534	0.693815	0.346908	−	0.937899i	$-0.387232\pi$
$227$	10.4534	0.693815	0.346908	+	0.937899i	$0.387232\pi$
$228$	0	0
$229$	22.5933	1.49301	0.746504	−	0.665381i	$-0.231731\pi$
$229$	22.5933	1.49301	0.746504	+	0.665381i	$0.231731\pi$
$230$	−4.24905	−0.280174
$231$	0	0
$232$	24.8291	1.63011
$233$	8.32806	0.545590	0.272795	−	0.962072i	$-0.412052\pi$
$233$	8.32806	0.545590	0.272795	+	0.962072i	$0.412052\pi$
$234$	0	0
$235$	11.3011	0.737204
$236$	−1.52411	−0.0992112
$237$	0	0
$238$	−7.51423	−0.487075
$239$	−4.65178	−0.300899	−0.150449	−	0.988618i	$-0.548072\pi$
$239$	−4.65178	−0.300899	−0.150449	+	0.988618i	$0.548072\pi$
$240$	0	0
$241$	−15.4970	−0.998251	−0.499125	−	0.866530i	$-0.666345\pi$
$241$	−15.4970	−0.998251	−0.499125	+	0.866530i	$0.666345\pi$
$242$	−15.8600	−1.01952
$243$	0	0
$244$	−3.70737	−0.237340
$245$	−12.5526	−0.801956
$246$	0	0
$247$	2.85395	0.181592
$248$	−3.50686	−0.222686
$249$	0	0
$250$	13.0841	0.827510
$251$	22.2885	1.40684	0.703420	−	0.710775i	$-0.251656\pi$
$251$	22.2885	1.40684	0.703420	+	0.710775i	$0.251656\pi$
$252$	0	0
$253$	−0.994058	−0.0624959
$254$	12.7103	0.797515
$255$	0	0
$256$	10.8640	0.679001
$257$	21.9037	1.36631	0.683157	−	0.730271i	$-0.260606\pi$
$257$	21.9037	1.36631	0.683157	+	0.730271i	$0.260606\pi$
$258$	0	0
$259$	0.353700	0.0219778
$260$	2.74286	0.170105
$261$	0	0
$262$	29.7023	1.83502
$263$	4.62072	0.284926	0.142463	−	0.989800i	$-0.454498\pi$
$263$	4.62072	0.284926	0.142463	+	0.989800i	$0.454498\pi$
$264$	0	0
$265$	26.0470	1.60005
$266$	−7.02005	−0.430427
$267$	0	0
$268$	2.17536	0.132881
$269$	15.4154	0.939895	0.469948	−	0.882694i	$-0.344273\pi$
$269$	15.4154	0.939895	0.469948	+	0.882694i	$0.344273\pi$
$270$	0	0
$271$	−29.3414	−1.78237	−0.891183	−	0.453645i	$-0.850124\pi$
$271$	−29.3414	−1.78237	−0.891183	+	0.453645i	$0.850124\pi$
$272$	6.56716	0.398192
$273$	0	0
$274$	−27.7132	−1.67421
$275$	−1.74619	−0.105299
$276$	0	0
$277$	2.86762	0.172299	0.0861493	−	0.996282i	$-0.472544\pi$
$277$	2.86762	0.172299	0.0861493	+	0.996282i	$0.472544\pi$
$278$	−11.0798	−0.664524
$279$	0	0
$280$	21.4967	1.28468
$281$	−32.5003	−1.93881	−0.969404	−	0.245471i	$-0.921057\pi$
$281$	−32.5003	−1.93881	−0.969404	+	0.245471i	$0.921057\pi$
$282$	0	0
$283$	−20.1377	−1.19706	−0.598532	−	0.801099i	$-0.704249\pi$
$283$	−20.1377	−1.19706	−0.598532	+	0.801099i	$0.704249\pi$
$284$	6.49035	0.385131
$285$	0	0
$286$	3.32793	0.196785
$287$	−7.55875	−0.446179
$288$	0	0
$289$	−15.0701	−0.886476
$290$	42.5844	2.50064
$291$	0	0
$292$	0.892054	0.0522035
$293$	−18.9168	−1.10513	−0.552567	−	0.833468i	$-0.686352\pi$
$293$	−18.9168	−1.10513	−0.552567	+	0.833468i	$0.686352\pi$
$294$	0	0
$295$	8.32875	0.484918
$296$	−0.246638	−0.0143355
$297$	0	0
$298$	10.6097	0.614603
$299$	−2.27359	−0.131485
$300$	0	0
$301$	−20.2724	−1.16848
$302$	6.02726	0.346830
$303$	0	0
$304$	6.13526	0.351881
$305$	20.2595	1.16006
$306$	0	0
$307$	6.24336	0.356327	0.178164	−	0.984001i	$-0.442984\pi$
$307$	6.24336	0.356327	0.178164	+	0.984001i	$0.442984\pi$
$308$	−1.57840	−0.0899380
$309$	0	0
$310$	−6.01460	−0.341607
$311$	−7.45699	−0.422847	−0.211424	−	0.977395i	$-0.567810\pi$
$311$	−7.45699	−0.422847	−0.211424	+	0.977395i	$0.567810\pi$
$312$	0	0
$313$	28.5353	1.61291	0.806457	−	0.591293i	$-0.201382\pi$
$313$	28.5353	1.61291	0.806457	+	0.591293i	$0.201382\pi$
$314$	−4.26606	−0.240748
$315$	0	0
$316$	0.608905	0.0342536
$317$	−9.42774	−0.529515	−0.264757	−	0.964315i	$-0.585292\pi$
$317$	−9.42774	−0.529515	−0.264757	+	0.964315i	$0.585292\pi$
$318$	0	0
$319$	9.96255	0.557796
$320$	−13.7980	−0.771333
$321$	0	0
$322$	5.59251	0.311658
$323$	1.80298	0.100321
$324$	0	0
$325$	−3.99386	−0.221540
$326$	−38.6118	−2.13851
$327$	0	0
$328$	5.27078	0.291030
$329$	−14.8743	−0.820048
$330$	0	0
$331$	−16.6458	−0.914934	−0.457467	−	0.889227i	$-0.651243\pi$
$331$	−16.6458	−0.914934	−0.457467	+	0.889227i	$0.651243\pi$
$332$	−3.05330	−0.167572
$333$	0	0
$334$	12.5532	0.686879
$335$	−11.8876	−0.649489
$336$	0	0
$337$	−34.6785	−1.88906	−0.944530	−	0.328426i	$-0.893482\pi$
$337$	−34.6785	−1.88906	−0.944530	+	0.328426i	$0.893482\pi$
$338$	−12.8516	−0.699034
$339$	0	0
$340$	1.73280	0.0939743
$341$	−1.40711	−0.0761992
$342$	0	0
$343$	−7.53243	−0.406713
$344$	14.1361	0.762169
$345$	0	0
$346$	−37.2627	−2.00325
$347$	12.1433	0.651887	0.325944	−	0.945389i	$-0.394318\pi$
$347$	12.1433	0.651887	0.325944	+	0.945389i	$0.394318\pi$
$348$	0	0
$349$	9.00204	0.481868	0.240934	−	0.970541i	$-0.422546\pi$
$349$	9.00204	0.481868	0.240934	+	0.970541i	$0.422546\pi$
$350$	9.82398	0.525114
$351$	0	0
$352$	2.54671	0.135740
$353$	−9.71810	−0.517242	−0.258621	−	0.965979i	$-0.583268\pi$
$353$	−9.71810	−0.517242	−0.258621	+	0.965979i	$0.583268\pi$
$354$	0	0
$355$	−35.4676	−1.88242
$356$	−0.205258	−0.0108787
$357$	0	0
$358$	23.2834	1.23057
$359$	−27.7228	−1.46315	−0.731577	−	0.681759i	$-0.761215\pi$
$359$	−27.7228	−1.46315	−0.731577	+	0.681759i	$0.761215\pi$
$360$	0	0
$361$	−17.3156	−0.911347
$362$	1.57409	0.0827323
$363$	0	0
$364$	−3.61010	−0.189221
$365$	−4.87477	−0.255157
$366$	0	0
$367$	7.47223	0.390047	0.195024	−	0.980799i	$-0.437522\pi$
$367$	7.47223	0.390047	0.195024	+	0.980799i	$0.437522\pi$
$368$	−4.88765	−0.254786
$369$	0	0
$370$	−0.423008	−0.0219911
$371$	−34.2825	−1.77986
$372$	0	0
$373$	−33.2179	−1.71996	−0.859978	−	0.510331i	$-0.829523\pi$
$373$	−33.2179	−1.71996	−0.859978	+	0.510331i	$0.829523\pi$
$374$	2.10242	0.108713
$375$	0	0
$376$	10.3720	0.534895
$377$	22.7862	1.17355
$378$	0	0
$379$	−4.98384	−0.256003	−0.128001	−	0.991774i	$-0.540856\pi$
$379$	−4.98384	−0.256003	−0.128001	+	0.991774i	$0.540856\pi$
$380$	1.61884	0.0830448
$381$	0	0
$382$	25.7027	1.31507
$383$	18.1335	0.926577	0.463289	−	0.886207i	$-0.346669\pi$
$383$	18.1335	0.926577	0.463289	+	0.886207i	$0.346669\pi$
$384$	0	0
$385$	8.62545	0.439594
$386$	33.8109	1.72093
$387$	0	0
$388$	3.70076	0.187878
$389$	26.2513	1.33099	0.665496	−	0.746402i	$-0.268220\pi$
$389$	26.2513	1.33099	0.665496	+	0.746402i	$0.268220\pi$
$390$	0	0
$391$	−1.43634	−0.0726389
$392$	−11.5206	−0.581877
$393$	0	0
$394$	1.02216	0.0514955
$395$	−3.32746	−0.167423
$396$	0	0
$397$	−0.305813	−0.0153483	−0.00767416	−	0.999971i	$-0.502443\pi$
$397$	−0.305813	−0.0153483	−0.00767416	+	0.999971i	$0.502443\pi$
$398$	21.4352	1.07445
$399$	0	0
$400$	−8.58579	−0.429290
$401$	30.9766	1.54690	0.773448	−	0.633860i	$-0.218531\pi$
$401$	30.9766	1.54690	0.773448	+	0.633860i	$0.218531\pi$
$402$	0	0
$403$	−3.21831	−0.160316
$404$	−3.97908	−0.197967
$405$	0	0
$406$	−56.0487	−2.78165
$407$	−0.0989621	−0.00490537
$408$	0	0
$409$	−4.93720	−0.244129	−0.122065	−	0.992522i	$-0.538951\pi$
$409$	−4.93720	−0.244129	−0.122065	+	0.992522i	$0.538951\pi$
$410$	9.03991	0.446449
$411$	0	0
$412$	6.47077	0.318792
$413$	−10.9621	−0.539411
$414$	0	0
$415$	16.6852	0.819046
$416$	5.82479	0.285584
$417$	0	0
$418$	1.96415	0.0960698
$419$	26.3828	1.28888	0.644442	−	0.764653i	$-0.277090\pi$
$419$	26.3828	1.28888	0.644442	+	0.764653i	$0.277090\pi$
$420$	0	0
$421$	−18.4529	−0.899337	−0.449668	−	0.893196i	$-0.648458\pi$
$421$	−18.4529	−0.899337	−0.449668	+	0.893196i	$0.648458\pi$
$422$	10.7149	0.521591
$423$	0	0
$424$	23.9055	1.16096
$425$	−2.52312	−0.122389
$426$	0	0
$427$	−26.6652	−1.29042
$428$	5.54662	0.268106
$429$	0	0
$430$	24.2448	1.16919
$431$	−41.1292	−1.98112	−0.990562	−	0.137066i	$-0.956233\pi$
$431$	−41.1292	−1.98112	−0.990562	+	0.137066i	$0.956233\pi$
$432$	0	0
$433$	−25.8354	−1.24157	−0.620784	−	0.783982i	$-0.713186\pi$
$433$	−25.8354	−1.24157	−0.620784	+	0.783982i	$0.713186\pi$
$434$	7.91630	0.379995
$435$	0	0
$436$	−2.22778	−0.106692
$437$	−1.34188	−0.0641907
$438$	0	0
$439$	−12.4780	−0.595542	−0.297771	−	0.954637i	$-0.596243\pi$
$439$	−12.4780	−0.595542	−0.297771	+	0.954637i	$0.596243\pi$
$440$	−6.01460	−0.286735
$441$	0	0
$442$	4.80861	0.228722
$443$	−1.99221	−0.0946526	−0.0473263	−	0.998879i	$-0.515070\pi$
$443$	−1.99221	−0.0946526	−0.0473263	+	0.998879i	$0.515070\pi$
$444$	0	0
$445$	1.12167	0.0531721
$446$	−0.181197	−0.00857992
$447$	0	0
$448$	18.1607	0.858012
$449$	−8.59158	−0.405462	−0.202731	−	0.979234i	$-0.564982\pi$
$449$	−8.59158	−0.405462	−0.202731	+	0.979234i	$0.564982\pi$
$450$	0	0
$451$	2.11487	0.0995855
$452$	−5.25322	−0.247091
$453$	0	0
$454$	16.4546	0.772251
$455$	19.7280	0.924861
$456$	0	0
$457$	−18.2645	−0.854378	−0.427189	−	0.904162i	$-0.640496\pi$
$457$	−18.2645	−0.854378	−0.427189	+	0.904162i	$0.640496\pi$
$458$	35.5639	1.66179
$459$	0	0
$460$	−1.28965	−0.0601301
$461$	9.20255	0.428605	0.214303	−	0.976767i	$-0.431252\pi$
$461$	9.20255	0.428605	0.214303	+	0.976767i	$0.431252\pi$
$462$	0	0
$463$	−7.76606	−0.360919	−0.180460	−	0.983582i	$-0.557759\pi$
$463$	−7.76606	−0.360919	−0.180460	+	0.983582i	$0.557759\pi$
$464$	48.9845	2.27405
$465$	0	0
$466$	13.1091	0.607268
$467$	8.59615	0.397782	0.198891	−	0.980022i	$-0.436266\pi$
$467$	8.59615	0.397782	0.198891	+	0.980022i	$0.436266\pi$
$468$	0	0
$469$	15.6462	0.722475
$470$	17.7890	0.820545
$471$	0	0
$472$	7.64400	0.351844
$473$	5.67205	0.260801
$474$	0	0
$475$	−2.35719	−0.108155
$476$	−2.28068	−0.104535
$477$	0	0
$478$	−7.32232	−0.334915
$479$	−21.8628	−0.998935	−0.499468	−	0.866333i	$-0.666471\pi$
$479$	−21.8628	−0.998935	−0.499468	+	0.866333i	$0.666471\pi$
$480$	0	0
$481$	−0.226344	−0.0103204
$482$	−24.3937	−1.11110
$483$	0	0
$484$	−4.81372	−0.218806
$485$	−20.2234	−0.918298
$486$	0	0
$487$	−35.5574	−1.61126	−0.805629	−	0.592420i	$-0.798173\pi$
$487$	−35.5574	−1.61126	−0.805629	+	0.592420i	$0.798173\pi$
$488$	18.5939	0.841705
$489$	0	0
$490$	−19.7589	−0.892616
$491$	23.8887	1.07808	0.539041	−	0.842279i	$-0.318787\pi$
$491$	23.8887	1.07808	0.539041	+	0.842279i	$0.318787\pi$
$492$	0	0
$493$	14.3952	0.648325
$494$	4.49237	0.202121
$495$	0	0
$496$	−6.91855	−0.310652
$497$	46.6817	2.09396
$498$	0	0
$499$	25.1828	1.12734	0.563668	−	0.826002i	$-0.309390\pi$
$499$	25.1828	1.12734	0.563668	+	0.826002i	$0.309390\pi$
$500$	3.97121	0.177598
$501$	0	0
$502$	35.0841	1.56588
$503$	8.57635	0.382400	0.191200	−	0.981551i	$-0.438762\pi$
$503$	8.57635	0.382400	0.191200	+	0.981551i	$0.438762\pi$
$504$	0	0
$505$	21.7443	0.967610
$506$	−1.56474	−0.0695610
$507$	0	0
$508$	3.85776	0.171160
$509$	−2.70721	−0.119995	−0.0599975	−	0.998199i	$-0.519109\pi$
$509$	−2.70721	−0.119995	−0.0599975	+	0.998199i	$0.519109\pi$
$510$	0	0
$511$	6.41608	0.283831
$512$	−10.1326	−0.447802
$513$	0	0
$514$	34.4784	1.52078
$515$	−35.3605	−1.55817
$516$	0	0
$517$	4.16171	0.183032
$518$	0.556755	0.0244624
$519$	0	0
$520$	−13.7565	−0.603262
$521$	26.8480	1.17623	0.588117	−	0.808776i	$-0.299869\pi$
$521$	26.8480	1.17623	0.588117	+	0.808776i	$0.299869\pi$
$522$	0	0
$523$	−26.7930	−1.17157	−0.585787	−	0.810465i	$-0.699215\pi$
$523$	−26.7930	−1.17157	−0.585787	+	0.810465i	$0.699215\pi$
$524$	9.01509	0.393826
$525$	0	0
$526$	7.27343	0.317137
$527$	−2.03317	−0.0885662
$528$	0	0
$529$	−21.9310	−0.953522
$530$	41.0003	1.78094
$531$	0	0
$532$	−2.13069	−0.0923770
$533$	4.83710	0.209518
$534$	0	0
$535$	−30.3104	−1.31043
$536$	−10.9103	−0.471251
$537$	0	0
$538$	24.2653	1.04615
$539$	−4.62257	−0.199108
$540$	0	0
$541$	−3.30744	−0.142198	−0.0710991	−	0.997469i	$-0.522651\pi$
$541$	−3.30744	−0.142198	−0.0710991	+	0.997469i	$0.522651\pi$
$542$	−46.1861	−1.98386
$543$	0	0
$544$	3.67981	0.157770
$545$	12.1741	0.521481
$546$	0	0
$547$	5.26057	0.224926	0.112463	−	0.993656i	$-0.464126\pi$
$547$	5.26057	0.224926	0.112463	+	0.993656i	$0.464126\pi$
$548$	−8.41135	−0.359315
$549$	0	0
$550$	−2.74867	−0.117204
$551$	13.4484	0.572923
$552$	0	0
$553$	4.37954	0.186237
$554$	4.51389	0.191777
$555$	0	0
$556$	−3.36289	−0.142618
$557$	9.12524	0.386649	0.193324	−	0.981135i	$-0.438073\pi$
$557$	9.12524	0.386649	0.193324	+	0.981135i	$0.438073\pi$
$558$	0	0
$559$	12.9730	0.548700
$560$	42.4101	1.79215
$561$	0	0
$562$	−51.1585	−2.15799
$563$	8.22149	0.346495	0.173247	−	0.984878i	$-0.444574\pi$
$563$	8.22149	0.346495	0.173247	+	0.984878i	$0.444574\pi$
$564$	0	0
$565$	28.7071	1.20772
$566$	−31.6986	−1.33239
$567$	0	0
$568$	−32.5516	−1.36583
$569$	−34.5974	−1.45040	−0.725198	−	0.688540i	$-0.758252\pi$
$569$	−34.5974	−1.45040	−0.725198	+	0.688540i	$0.758252\pi$
$570$	0	0
$571$	−26.2483	−1.09846	−0.549229	−	0.835672i	$-0.685078\pi$
$571$	−26.2483	−1.09846	−0.549229	+	0.835672i	$0.685078\pi$
$572$	1.01008	0.0422334
$573$	0	0
$574$	−11.8981	−0.496619
$575$	1.87785	0.0783117
$576$	0	0
$577$	−28.7241	−1.19580	−0.597900	−	0.801570i	$-0.703998\pi$
$577$	−28.7241	−1.19580	−0.597900	+	0.801570i	$0.703998\pi$
$578$	−23.7217	−0.986692
$579$	0	0
$580$	12.9250	0.536680
$581$	−21.9608	−0.911087
$582$	0	0
$583$	9.59197	0.397259
$584$	−4.47399	−0.185135
$585$	0	0
$586$	−29.7768	−1.23007
$587$	−17.2536	−0.712134	−0.356067	−	0.934460i	$-0.615883\pi$
$587$	−17.2536	−0.712134	−0.356067	+	0.934460i	$0.615883\pi$
$588$	0	0
$589$	−1.89945	−0.0782656
$590$	13.1102	0.539738
$591$	0	0
$592$	−0.486583	−0.0199984
$593$	−37.5049	−1.54014	−0.770072	−	0.637957i	$-0.779780\pi$
$593$	−37.5049	−1.54014	−0.770072	+	0.637957i	$0.779780\pi$
$594$	0	0
$595$	12.4631	0.510939
$596$	3.22020	0.131904
$597$	0	0
$598$	−3.57884	−0.146350
$599$	−28.1640	−1.15075	−0.575375	−	0.817890i	$-0.695144\pi$
$599$	−28.1640	−1.15075	−0.575375	+	0.817890i	$0.695144\pi$
$600$	0	0
$601$	28.6925	1.17039	0.585196	−	0.810892i	$-0.301018\pi$
$601$	28.6925	1.17039	0.585196	+	0.810892i	$0.301018\pi$
$602$	−31.9106	−1.30058
$603$	0	0
$604$	1.82936	0.0744356
$605$	26.3054	1.06947
$606$	0	0
$607$	44.1968	1.79389	0.896946	−	0.442141i	$-0.145781\pi$
$607$	44.1968	1.79389	0.896946	+	0.442141i	$0.145781\pi$
$608$	3.43780	0.139421
$609$	0	0
$610$	31.8903	1.29120
$611$	9.51858	0.385081
$612$	0	0
$613$	16.4282	0.663528	0.331764	−	0.943362i	$-0.392356\pi$
$613$	16.4282	0.663528	0.331764	+	0.943362i	$0.392356\pi$
$614$	9.82760	0.396610
$615$	0	0
$616$	7.91630	0.318957
$617$	−29.6980	−1.19560	−0.597798	−	0.801647i	$-0.703958\pi$
$617$	−29.6980	−1.19560	−0.597798	+	0.801647i	$0.703958\pi$
$618$	0	0
$619$	22.7157	0.913021	0.456511	−	0.889718i	$-0.349099\pi$
$619$	22.7157	0.913021	0.456511	+	0.889718i	$0.349099\pi$
$620$	−1.82552	−0.0733146
$621$	0	0
$622$	−11.7380	−0.470650
$623$	−1.47632	−0.0591473
$624$	0	0
$625$	−30.7825	−1.23130
$626$	44.9172	1.79525
$627$	0	0
$628$	−1.29481	−0.0516686
$629$	−0.142993	−0.00570150
$630$	0	0
$631$	36.5102	1.45345	0.726724	−	0.686930i	$-0.241042\pi$
$631$	36.5102	1.45345	0.726724	+	0.686930i	$0.241042\pi$
$632$	−3.05389	−0.121477
$633$	0	0
$634$	−14.8401	−0.589376
$635$	−21.0813	−0.836588
$636$	0	0
$637$	−10.5727	−0.418904
$638$	15.6820	0.620855
$639$	0	0
$640$	−35.5505	−1.40526
$641$	−3.50020	−0.138250	−0.0691249	−	0.997608i	$-0.522021\pi$
$641$	−3.50020	−0.138250	−0.0691249	+	0.997608i	$0.522021\pi$
$642$	0	0
$643$	46.8651	1.84818	0.924089	−	0.382177i	$-0.124825\pi$
$643$	46.8651	1.84818	0.924089	+	0.382177i	$0.124825\pi$
$644$	1.69741	0.0668872
$645$	0	0
$646$	2.83805	0.111662
$647$	−10.0298	−0.394310	−0.197155	−	0.980372i	$-0.563170\pi$
$647$	−10.0298	−0.394310	−0.197155	+	0.980372i	$0.563170\pi$
$648$	0	0
$649$	3.06711	0.120395
$650$	−6.28670	−0.246585
$651$	0	0
$652$	−11.7192	−0.458961
$653$	6.80902	0.266458	0.133229	−	0.991085i	$-0.457465\pi$
$653$	6.80902	0.266458	0.133229	+	0.991085i	$0.457465\pi$
$654$	0	0
$655$	−49.2644	−1.92492
$656$	10.3985	0.405995
$657$	0	0
$658$	−23.4135	−0.912754
$659$	−39.5734	−1.54156	−0.770780	−	0.637101i	$-0.780133\pi$
$659$	−39.5734	−1.54156	−0.770780	+	0.637101i	$0.780133\pi$
$660$	0	0
$661$	−29.3280	−1.14073	−0.570363	−	0.821393i	$-0.693198\pi$
$661$	−29.3280	−1.14073	−0.570363	+	0.821393i	$0.693198\pi$
$662$	−26.2019	−1.01837
$663$	0	0
$664$	15.3135	0.594277
$665$	11.6435	0.451515
$666$	0	0
$667$	−10.7137	−0.414835
$668$	3.81007	0.147416
$669$	0	0
$670$	−18.7121	−0.722913
$671$	7.46069	0.288017
$672$	0	0
$673$	44.8722	1.72970	0.864848	−	0.502034i	$-0.167415\pi$
$673$	44.8722	1.72970	0.864848	+	0.502034i	$0.167415\pi$
$674$	−54.5871	−2.10262
$675$	0	0
$676$	−3.90064	−0.150025
$677$	−19.9829	−0.768006	−0.384003	−	0.923332i	$-0.625455\pi$
$677$	−19.9829	−0.768006	−0.384003	+	0.923332i	$0.625455\pi$
$678$	0	0
$679$	26.6177	1.02149
$680$	−8.69066	−0.333272
$681$	0	0
$682$	−2.21491	−0.0848135
$683$	−20.2488	−0.774800	−0.387400	−	0.921912i	$-0.626627\pi$
$683$	−20.2488	−0.774800	−0.387400	+	0.921912i	$0.626627\pi$
$684$	0	0
$685$	45.9652	1.75624
$686$	−11.8567	−0.452692
$687$	0	0
$688$	27.8887	1.06325
$689$	21.9386	0.835793
$690$	0	0
$691$	−36.5673	−1.39108	−0.695542	−	0.718485i	$-0.744836\pi$
$691$	−36.5673	−1.39108	−0.695542	+	0.718485i	$0.744836\pi$
$692$	−11.3098	−0.429933
$693$	0	0
$694$	19.1147	0.725583
$695$	18.3770	0.697081
$696$	0	0
$697$	3.05584	0.115748
$698$	14.1700	0.536343
$699$	0	0
$700$	2.98172	0.112698
$701$	18.3854	0.694407	0.347204	−	0.937790i	$-0.387131\pi$
$701$	18.3854	0.694407	0.347204	+	0.937790i	$0.387131\pi$
$702$	0	0
$703$	−0.133589	−0.00503840
$704$	−5.08121	−0.191505
$705$	0	0
$706$	−15.2972	−0.575716
$707$	−28.6195	−1.07635
$708$	0	0
$709$	−14.3088	−0.537380	−0.268690	−	0.963227i	$-0.586591\pi$
$709$	−14.3088	−0.537380	−0.268690	+	0.963227i	$0.586591\pi$
$710$	−55.8291	−2.09523
$711$	0	0
$712$	1.02945	0.0385802
$713$	1.51320	0.0566696
$714$	0	0
$715$	−5.51972	−0.206426
$716$	7.06685	0.264101
$717$	0	0
$718$	−43.6382	−1.62856
$719$	14.5481	0.542552	0.271276	−	0.962502i	$-0.412554\pi$
$719$	14.5481	0.542552	0.271276	+	0.962502i	$0.412554\pi$
$720$	0	0
$721$	46.5409	1.73327
$722$	−27.2563	−1.01437
$723$	0	0
$724$	0.477759	0.0177558
$725$	−18.8200	−0.698957
$726$	0	0
$727$	21.9157	0.812810	0.406405	−	0.913693i	$-0.366782\pi$
$727$	21.9157	0.812810	0.406405	+	0.913693i	$0.366782\pi$
$728$	18.1060	0.671054
$729$	0	0
$730$	−7.67333	−0.284003
$731$	8.19569	0.303129
$732$	0	0
$733$	−37.9899	−1.40319	−0.701595	−	0.712576i	$-0.747528\pi$
$733$	−37.9899	−1.40319	−0.701595	+	0.712576i	$0.747528\pi$
$734$	11.7620	0.434142
$735$	0	0
$736$	−2.73872	−0.100950
$737$	−4.37768	−0.161254
$738$	0	0
$739$	49.5268	1.82187	0.910937	−	0.412545i	$-0.135360\pi$
$739$	49.5268	1.82187	0.910937	+	0.412545i	$0.135360\pi$
$740$	−0.128389	−0.00471967
$741$	0	0
$742$	−53.9638	−1.98107
$743$	−24.6189	−0.903179	−0.451590	−	0.892226i	$-0.649143\pi$
$743$	−24.6189	−0.903179	−0.451590	+	0.892226i	$0.649143\pi$
$744$	0	0
$745$	−17.5973	−0.644714
$746$	−52.2879	−1.91440
$747$	0	0
$748$	0.638114	0.0233318
$749$	39.8940	1.45769
$750$	0	0
$751$	−19.0575	−0.695418	−0.347709	−	0.937603i	$-0.613040\pi$
$751$	−19.0575	−0.695418	−0.347709	+	0.937603i	$0.613040\pi$
$752$	20.4625	0.746192
$753$	0	0
$754$	35.8675	1.30622
$755$	−9.99683	−0.363822
$756$	0	0
$757$	−5.82521	−0.211721	−0.105860	−	0.994381i	$-0.533760\pi$
$757$	−5.82521	−0.211721	−0.105860	+	0.994381i	$0.533760\pi$
$758$	−7.84502	−0.284944
$759$	0	0
$760$	−8.11911	−0.294511
$761$	40.9900	1.48589	0.742943	−	0.669355i	$-0.233429\pi$
$761$	40.9900	1.48589	0.742943	+	0.669355i	$0.233429\pi$
$762$	0	0
$763$	−16.0233	−0.580082
$764$	7.80115	0.282236
$765$	0	0
$766$	28.5437	1.03133
$767$	7.01504	0.253299
$768$	0	0
$769$	1.22318	0.0441088	0.0220544	−	0.999757i	$-0.492979\pi$
$769$	1.22318	0.0441088	0.0220544	+	0.999757i	$0.492979\pi$
$770$	13.5772	0.489289
$771$	0	0
$772$	10.2621	0.369341
$773$	21.7320	0.781645	0.390823	−	0.920466i	$-0.372191\pi$
$773$	21.7320	0.781645	0.390823	+	0.920466i	$0.372191\pi$
$774$	0	0
$775$	2.65813	0.0954828
$776$	−18.5607	−0.666292
$777$	0	0
$778$	41.3218	1.48146
$779$	2.85487	0.102286
$780$	0	0
$781$	−13.0611	−0.467365
$782$	−2.26093	−0.0808507
$783$	0	0
$784$	−22.7285	−0.811733
$785$	7.07571	0.252543
$786$	0	0
$787$	−17.8059	−0.634712	−0.317356	−	0.948307i	$-0.602795\pi$
$787$	−17.8059	−0.634712	−0.317356	+	0.948307i	$0.602795\pi$
$788$	0.310239	0.0110518
$789$	0	0
$790$	−5.23772	−0.186350
$791$	−37.7837	−1.34343
$792$	0	0
$793$	17.0640	0.605959
$794$	−0.481378	−0.0170835
$795$	0	0
$796$	6.50590	0.230595
$797$	−9.71471	−0.344113	−0.172056	−	0.985087i	$-0.555041\pi$
$797$	−9.71471	−0.344113	−0.172056	+	0.985087i	$0.555041\pi$
$798$	0	0
$799$	6.01336	0.212737
$800$	−4.81092	−0.170092
$801$	0	0
$802$	48.7599	1.72177
$803$	−1.79517	−0.0633500
$804$	0	0
$805$	−9.27576	−0.326927
$806$	−5.06591	−0.178439
$807$	0	0
$808$	19.9566	0.702072
$809$	29.1858	1.02612	0.513060	−	0.858353i	$-0.328512\pi$
$809$	29.1858	1.02612	0.513060	+	0.858353i	$0.328512\pi$
$810$	0	0
$811$	−42.0110	−1.47521	−0.737603	−	0.675235i	$-0.764042\pi$
$811$	−42.0110	−1.47521	−0.737603	+	0.675235i	$0.764042\pi$
$812$	−17.0116	−0.596990
$813$	0	0
$814$	−0.155775	−0.00545992
$815$	64.0417	2.24328
$816$	0	0
$817$	7.65669	0.267874
$818$	−7.77160	−0.271728
$819$	0	0
$820$	2.74374	0.0958156
$821$	2.14389	0.0748222	0.0374111	−	0.999300i	$-0.488089\pi$
$821$	2.14389	0.0748222	0.0374111	+	0.999300i	$0.488089\pi$
$822$	0	0
$823$	30.0484	1.04742	0.523711	−	0.851896i	$-0.324547\pi$
$823$	30.0484	1.04742	0.523711	+	0.851896i	$0.324547\pi$
$824$	−32.4534	−1.13057
$825$	0	0
$826$	−17.2554	−0.600392
$827$	−32.3020	−1.12325	−0.561625	−	0.827392i	$-0.689824\pi$
$827$	−32.3020	−1.12325	−0.561625	+	0.827392i	$0.689824\pi$
$828$	0	0
$829$	30.2603	1.05098	0.525491	−	0.850799i	$-0.323882\pi$
$829$	30.2603	1.05098	0.525491	+	0.850799i	$0.323882\pi$
$830$	26.2641	0.911639
$831$	0	0
$832$	−11.6216	−0.402908
$833$	−6.67927	−0.231423
$834$	0	0
$835$	−20.8207	−0.720531
$836$	0.596148	0.0206182
$837$	0	0
$838$	41.5289	1.43459
$839$	0.881464	0.0304315	0.0152158	−	0.999884i	$-0.495156\pi$
$839$	0.881464	0.0304315	0.0152158	+	0.999884i	$0.495156\pi$
$840$	0	0
$841$	78.3735	2.70254
$842$	−29.0464	−1.00101
$843$	0	0
$844$	3.25212	0.111942
$845$	21.3157	0.733282
$846$	0	0
$847$	−34.6226	−1.18965
$848$	47.1623	1.61956
$849$	0	0
$850$	−3.97162	−0.136225
$851$	0.106423	0.00364814
$852$	0	0
$853$	−3.10967	−0.106473	−0.0532365	−	0.998582i	$-0.516954\pi$
$853$	−3.10967	−0.106473	−0.0532365	+	0.998582i	$0.516954\pi$
$854$	−41.9734	−1.43630
$855$	0	0
$856$	−27.8184	−0.950815
$857$	−31.5844	−1.07890	−0.539451	−	0.842017i	$-0.681368\pi$
$857$	−31.5844	−1.07890	−0.539451	+	0.842017i	$0.681368\pi$
$858$	0	0
$859$	−10.4583	−0.356831	−0.178416	−	0.983955i	$-0.557097\pi$
$859$	−10.4583	−0.356831	−0.178416	+	0.983955i	$0.557097\pi$
$860$	7.35866	0.250928
$861$	0	0
$862$	−64.7411	−2.20509
$863$	−29.2721	−0.996433	−0.498217	−	0.867053i	$-0.666012\pi$
$863$	−29.2721	−0.996433	−0.498217	+	0.867053i	$0.666012\pi$
$864$	0	0
$865$	61.8040	2.10140
$866$	−40.6672	−1.38193
$867$	0	0
$868$	2.40271	0.0815534
$869$	−1.22536	−0.0415674
$870$	0	0
$871$	−10.0125	−0.339262
$872$	11.1732	0.378372
$873$	0	0
$874$	−2.11224	−0.0714475
$875$	28.5628	0.965600
$876$	0	0
$877$	−17.6778	−0.596937	−0.298468	−	0.954419i	$-0.596476\pi$
$877$	−17.6778	−0.596937	−0.298468	+	0.954419i	$0.596476\pi$
$878$	−19.6415	−0.662868
$879$	0	0
$880$	−11.8660	−0.400002
$881$	3.04541	0.102602	0.0513012	−	0.998683i	$-0.483663\pi$
$881$	3.04541	0.102602	0.0513012	+	0.998683i	$0.483663\pi$
$882$	0	0
$883$	−15.7628	−0.530459	−0.265230	−	0.964185i	$-0.585448\pi$
$883$	−15.7628	−0.530459	−0.265230	+	0.964185i	$0.585448\pi$
$884$	1.45948	0.0490878
$885$	0	0
$886$	−3.13591	−0.105353
$887$	21.8406	0.733335	0.366667	−	0.930352i	$-0.380499\pi$
$887$	21.8406	0.733335	0.366667	+	0.930352i	$0.380499\pi$
$888$	0	0
$889$	27.7469	0.930600
$890$	1.76560	0.0591832
$891$	0	0
$892$	−0.0549958	−0.00184140
$893$	5.61788	0.187995
$894$	0	0
$895$	−38.6179	−1.29086
$896$	46.7909	1.56317
$897$	0	0
$898$	−13.5239	−0.451299
$899$	−15.1654	−0.505795
$900$	0	0
$901$	13.8597	0.461733
$902$	3.32900	0.110844
$903$	0	0
$904$	26.3469	0.876286
$905$	−2.61079	−0.0867856
$906$	0	0
$907$	−44.5734	−1.48003	−0.740017	−	0.672588i	$-0.765183\pi$
$907$	−44.5734	−1.48003	−0.740017	+	0.672588i	$0.765183\pi$
$908$	4.99420	0.165738
$909$	0	0
$910$	31.0536	1.02942
$911$	21.3786	0.708305	0.354153	−	0.935188i	$-0.384769\pi$
$911$	21.3786	0.708305	0.354153	+	0.935188i	$0.384769\pi$
$912$	0	0
$913$	6.14444	0.203351
$914$	−28.7500	−0.950965
$915$	0	0
$916$	10.7942	0.356649
$917$	64.8409	2.14123
$918$	0	0
$919$	−45.8883	−1.51371	−0.756857	−	0.653581i	$-0.773266\pi$
$919$	−45.8883	−1.51371	−0.756857	+	0.653581i	$0.773266\pi$
$920$	6.46807	0.213246
$921$	0	0
$922$	14.4856	0.477059
$923$	−29.8732	−0.983289
$924$	0	0
$925$	0.186947	0.00614677
$926$	−12.2245	−0.401721
$927$	0	0
$928$	27.4477	0.901015
$929$	−9.13738	−0.299788	−0.149894	−	0.988702i	$-0.547893\pi$
$929$	−9.13738	−0.299788	−0.149894	+	0.988702i	$0.547893\pi$
$930$	0	0
$931$	−6.24000	−0.204508
$932$	3.97881	0.130330
$933$	0	0
$934$	13.5311	0.442752
$935$	−3.48708	−0.114040
$936$	0	0
$937$	34.6066	1.13055	0.565275	−	0.824903i	$-0.308770\pi$
$937$	34.6066	1.13055	0.565275	+	0.824903i	$0.308770\pi$
$938$	24.6286	0.804151
$939$	0	0
$940$	5.39921	0.176103
$941$	−15.0445	−0.490437	−0.245219	−	0.969468i	$-0.578860\pi$
$941$	−15.0445	−0.490437	−0.245219	+	0.969468i	$0.578860\pi$
$942$	0	0
$943$	−2.27432	−0.0740621
$944$	15.0806	0.490830
$945$	0	0
$946$	8.92831	0.290284
$947$	−20.2433	−0.657820	−0.328910	−	0.944361i	$-0.606681\pi$
$947$	−20.2433	−0.657820	−0.328910	+	0.944361i	$0.606681\pi$
$948$	0	0
$949$	−4.10587	−0.133282
$950$	−3.71042	−0.120382
$951$	0	0
$952$	11.4385	0.370723
$953$	40.3903	1.30837	0.654185	−	0.756334i	$-0.273012\pi$
$953$	40.3903	1.30837	0.654185	+	0.756334i	$0.273012\pi$
$954$	0	0
$955$	−42.6307	−1.37950
$956$	−2.22243	−0.0718785
$957$	0	0
$958$	−34.4140	−1.11186
$959$	−60.4985	−1.95360
$960$	0	0
$961$	−28.8580	−0.930905
$962$	−0.356286	−0.0114871
$963$	0	0
$964$	−7.40384	−0.238462
$965$	−56.0789	−1.80524
$966$	0	0
$967$	−21.2417	−0.683087	−0.341543	−	0.939866i	$-0.610950\pi$
$967$	−21.2417	−0.683087	−0.341543	+	0.939866i	$0.610950\pi$
$968$	24.1427	0.775975
$969$	0	0
$970$	−31.8335	−1.02211
$971$	−24.9595	−0.800990	−0.400495	−	0.916299i	$-0.631162\pi$
$971$	−24.9595	−0.800990	−0.400495	+	0.916299i	$0.631162\pi$
$972$	0	0
$973$	−24.1875	−0.775416
$974$	−55.9705	−1.79341
$975$	0	0
$976$	36.6832	1.17420
$977$	25.7236	0.822972	0.411486	−	0.911416i	$-0.365010\pi$
$977$	25.7236	0.822972	0.411486	+	0.911416i	$0.365010\pi$
$978$	0	0
$979$	0.413061	0.0132015
$980$	−5.99711	−0.191571
$981$	0	0
$982$	37.6030	1.19996
$983$	48.1254	1.53496	0.767480	−	0.641073i	$-0.221510\pi$
$983$	48.1254	1.53496	0.767480	+	0.641073i	$0.221510\pi$
$984$	0	0
$985$	−1.69535	−0.0540185
$986$	22.6593	0.721618
$987$	0	0
$988$	1.36350	0.0433787
$989$	−6.09969	−0.193959
$990$	0	0
$991$	12.6416	0.401572	0.200786	−	0.979635i	$-0.435650\pi$
$991$	12.6416	0.401572	0.200786	+	0.979635i	$0.435650\pi$
$992$	−3.87671	−0.123086
$993$	0	0
$994$	73.4812	2.33068
$995$	−35.5525	−1.12709
$996$	0	0
$997$	34.3583	1.08814	0.544069	−	0.839041i	$-0.316883\pi$
$997$	34.3583	1.08814	0.544069	+	0.839041i	$0.316883\pi$
$998$	39.6399	1.25478
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1629.2.a.e.1.7		8
3.2	odd	2		543.2.a.d.1.2	✓	8
12.11	even	2		8688.2.a.bf.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.2.a.d.1.2	✓	8	3.2	odd	2
1629.2.a.e.1.7		8	1.1	even	1	trivial
8688.2.a.bf.1.6		8	12.11	even	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:	\( N \)	\(=\)	\( 1629 = 3^{2} \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1629.a (trivial)

\(f(q)\)	\(=\)	\(q+1.57409 q^{2} +0.477759 q^{4} -2.61079 q^{5} +3.43627 q^{7} -2.39614 q^{8} -4.10962 q^{10} -0.961440 q^{11} -2.19899 q^{13} +5.40900 q^{14} -4.72726 q^{16} -1.38921 q^{17} -1.29785 q^{19} -1.24733 q^{20} -1.51339 q^{22} +1.03393 q^{23} +1.81623 q^{25} -3.46140 q^{26} +1.64171 q^{28} -10.3621 q^{29} +1.46354 q^{31} -2.64885 q^{32} -2.18674 q^{34} -8.97139 q^{35} +0.102931 q^{37} -2.04293 q^{38} +6.25583 q^{40} -2.19969 q^{41} -5.89954 q^{43} -0.459336 q^{44} +1.62749 q^{46} -4.32862 q^{47} +4.80796 q^{49} +2.85891 q^{50} -1.05059 q^{52} -9.97667 q^{53} +2.51012 q^{55} -8.23380 q^{56} -16.3109 q^{58} -3.19012 q^{59} -7.75991 q^{61} +2.30375 q^{62} +5.28500 q^{64} +5.74110 q^{65} +4.55325 q^{67} -0.663707 q^{68} -14.1218 q^{70} +13.5850 q^{71} +1.86716 q^{73} +0.162023 q^{74} -0.620058 q^{76} -3.30377 q^{77} +1.27450 q^{79} +12.3419 q^{80} -3.46252 q^{82} -6.39088 q^{83} +3.62693 q^{85} -9.28640 q^{86} +2.30375 q^{88} -0.429627 q^{89} -7.55632 q^{91} +0.493968 q^{92} -6.81364 q^{94} +3.38841 q^{95} +7.74609 q^{97} +7.56817 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 5 q^{4} - 5 q^{5} - 3 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} + 13 q^{13} - 5 q^{14} + 3 q^{16} - 27 q^{17} - 10 q^{19} - 21 q^{20} + 11 q^{22} - 3 q^{23} + 13 q^{25} - 11 q^{26} - 3 q^{28}+ \cdots + 51 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 5 * q^4 - 5 * q^5 - 3 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 + 13 * q^13 - 5 * q^14 + 3 * q^16 - 27 * q^17 - 10 * q^19 - 21 * q^20 + 11 * q^22 - 3 * q^23 + 13 * q^25 - 11 * q^26 - 3 * q^28 - 16 * q^29 - q^31 - 10 * q^32 + 10 * q^34 - 14 * q^35 + 3 * q^37 - 14 * q^38 + 11 * q^40 - 20 * q^41 - 4 * q^43 - 9 * q^44 - 17 * q^46 - 12 * q^47 + 5 * q^49 - 2 * q^50 + 21 * q^52 - 23 * q^53 - 13 * q^55 + 8 * q^56 - 21 * q^58 - 7 * q^59 + q^61 - q^62 - 32 * q^64 - 10 * q^65 - 25 * q^67 + q^68 - 10 * q^70 - 11 * q^71 + 23 * q^73 - 2 * q^74 - 19 * q^76 - 20 * q^77 - 26 * q^79 + 15 * q^80 - 22 * q^82 - 16 * q^83 - 33 * q^85 + 21 * q^86 - q^88 - 14 * q^89 - 35 * q^91 + 18 * q^92 - 52 * q^94 - 10 * q^95 + 28 * q^97 + 51 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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