LMFDB - Embedded newform 1617.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.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:	$12.9118100068$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.28734$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.28734 q^{2} -1.00000 q^{3} -0.342766 q^{4} -3.91744 q^{5} -1.28734 q^{6} -3.01593 q^{8} +1.00000 q^{9} -5.04306 q^{10} -1.00000 q^{11} +0.342766 q^{12} +3.04306 q^{13} +3.91744 q^{15} -3.19698 q^{16} -3.97287 q^{17} +1.28734 q^{18} -7.59060 q^{19} +1.34277 q^{20} -1.28734 q^{22} +4.51145 q^{23} +3.01593 q^{24} +10.3463 q^{25} +3.91744 q^{26} -1.00000 q^{27} +3.75572 q^{29} +5.04306 q^{30} +6.74335 q^{31} +1.91627 q^{32} +1.00000 q^{33} -5.11442 q^{34} -0.342766 q^{36} +0.342766 q^{37} -9.77165 q^{38} -3.04306 q^{39} +11.8147 q^{40} +2.79182 q^{41} +11.1222 q^{43} +0.342766 q^{44} -3.91744 q^{45} +5.80775 q^{46} -1.65723 q^{47} +3.19698 q^{48} +13.3192 q^{50} +3.97287 q^{51} -1.04306 q^{52} -12.9403 q^{53} -1.28734 q^{54} +3.91744 q^{55} +7.59060 q^{57} +4.83488 q^{58} +3.66079 q^{59} -1.34277 q^{60} +0.468387 q^{61} +8.68096 q^{62} +8.86084 q^{64} -11.9210 q^{65} +1.28734 q^{66} -2.57823 q^{67} +1.36177 q^{68} -4.51145 q^{69} -5.00355 q^{71} -3.01593 q^{72} +8.73756 q^{73} +0.441256 q^{74} -10.3463 q^{75} +2.60180 q^{76} -3.91744 q^{78} +0.719627 q^{79} +12.5240 q^{80} +1.00000 q^{81} +3.59401 q^{82} +11.5976 q^{83} +15.5635 q^{85} +14.3180 q^{86} -3.75572 q^{87} +3.01593 q^{88} +12.3553 q^{89} -5.04306 q^{90} -1.54637 q^{92} -6.74335 q^{93} -2.13342 q^{94} +29.7357 q^{95} -1.91627 q^{96} +13.1687 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9} - 10 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 12 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{22} + 4 q^{23} - 12 q^{24} + 4 q^{25}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9 - 10 * q^10 - 4 * q^11 - 4 * q^12 + 2 * q^13 + 4 * q^15 + 12 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^22 + 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^27 + 8 * q^29 + 10 * q^30 + 12 * q^31 + 26 * q^32 + 4 * q^33 + 16 * q^34 + 4 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 4 * q^44 - 4 * q^45 - 14 * q^46 - 12 * q^47 - 12 * q^48 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^55 - 4 * q^58 - 12 * q^59 - 2 * q^61 + 26 * q^62 + 56 * q^64 - 4 * q^65 + 2 * q^66 + 28 * q^67 + 48 * q^68 - 4 * q^69 + 12 * q^71 + 12 * q^72 - 6 * q^73 - 16 * q^74 - 4 * q^75 + 18 * q^76 - 4 * q^78 + 2 * q^79 - 16 * q^80 + 4 * q^81 + 12 * q^82 + 12 * q^83 + 18 * q^85 + 36 * q^86 - 8 * q^87 - 12 * q^88 - 8 * q^89 - 10 * q^90 - 16 * q^92 - 12 * q^93 - 20 * q^94 + 34 * q^95 - 26 * q^96 + 44 * q^97 - 4 * 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 $

$2$	1.28734	0.910284	0.455142	−	0.890419i	$-0.349589\pi$
$2$	1.28734	0.910284	0.455142	+	0.890419i	$0.349589\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.342766	−0.171383
$5$	−3.91744	−1.75193	−0.875966	−	0.482373i	$-0.839775\pi$
$5$	−3.91744	−1.75193	−0.875966	+	0.482373i	$0.839775\pi$
$6$	−1.28734	−0.525553
$7$	0	0
$7$	0	0
$8$	−3.01593	−1.06629
$9$	1.00000	0.333333
$10$	−5.04306	−1.59476
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.342766	0.0989482
$13$	3.04306	0.843993	0.421996	−	0.906598i	$-0.361330\pi$
$13$	3.04306	0.843993	0.421996	+	0.906598i	$0.361330\pi$
$14$	0	0
$15$	3.91744	1.01148
$16$	−3.19698	−0.799245
$17$	−3.97287	−0.963562	−0.481781	−	0.876292i	$-0.660010\pi$
$17$	−3.97287	−0.963562	−0.481781	+	0.876292i	$0.660010\pi$
$18$	1.28734	0.303428
$19$	−7.59060	−1.74140	−0.870701	−	0.491812i	$-0.836335\pi$
$19$	−7.59060	−1.74140	−0.870701	+	0.491812i	$0.836335\pi$
$20$	1.34277	0.300252
$21$	0	0
$22$	−1.28734	−0.274461
$23$	4.51145	0.940701	0.470351	−	0.882480i	$-0.344127\pi$
$23$	4.51145	0.940701	0.470351	+	0.882480i	$0.344127\pi$
$24$	3.01593	0.615624
$25$	10.3463	2.06926
$26$	3.91744	0.768273
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.75572	0.697420	0.348710	−	0.937231i	$-0.386620\pi$
$29$	3.75572	0.697420	0.348710	+	0.937231i	$0.386620\pi$
$30$	5.04306	0.920732
$31$	6.74335	1.21114	0.605571	−	0.795791i	$-0.292945\pi$
$31$	6.74335	1.21114	0.605571	+	0.795791i	$0.292945\pi$
$32$	1.91627	0.338752
$33$	1.00000	0.174078
$34$	−5.11442	−0.877115
$35$	0	0
$36$	−0.342766	−0.0571277
$37$	0.342766	0.0563505	0.0281752	−	0.999603i	$-0.491030\pi$
$37$	0.342766	0.0563505	0.0281752	+	0.999603i	$0.491030\pi$
$38$	−9.77165	−1.58517
$39$	−3.04306	−0.487279
$40$	11.8147	1.86807
$41$	2.79182	0.436009	0.218004	−	0.975948i	$-0.430045\pi$
$41$	2.79182	0.436009	0.218004	+	0.975948i	$0.430045\pi$
$42$	0	0
$43$	11.1222	1.69612	0.848061	−	0.529899i	$-0.177770\pi$
$43$	11.1222	1.69612	0.848061	+	0.529899i	$0.177770\pi$
$44$	0.342766	0.0516740
$45$	−3.91744	−0.583977
$46$	5.80775	0.856305
$47$	−1.65723	−0.241732	−0.120866	−	0.992669i	$-0.538567\pi$
$47$	−1.65723	−0.241732	−0.120866	+	0.992669i	$0.538567\pi$
$48$	3.19698	0.461444
$49$	0	0
$50$	13.3192	1.88362
$51$	3.97287	0.556313
$52$	−1.04306	−0.144646
$53$	−12.9403	−1.77749	−0.888745	−	0.458401i	$-0.848422\pi$
$53$	−12.9403	−1.77749	−0.888745	+	0.458401i	$0.848422\pi$
$54$	−1.28734	−0.175184
$55$	3.91744	0.528227
$56$	0	0
$57$	7.59060	1.00540
$58$	4.83488	0.634850
$59$	3.66079	0.476594	0.238297	−	0.971192i	$-0.423411\pi$
$59$	3.66079	0.476594	0.238297	+	0.971192i	$0.423411\pi$
$60$	−1.34277	−0.173350
$61$	0.468387	0.0599708	0.0299854	−	0.999550i	$-0.490454\pi$
$61$	0.468387	0.0599708	0.0299854	+	0.999550i	$0.490454\pi$
$62$	8.68096	1.10248
$63$	0	0
$64$	8.86084	1.10760
$65$	−11.9210	−1.47862
$66$	1.28734	0.158460
$67$	−2.57823	−0.314981	−0.157490	−	0.987521i	$-0.550340\pi$
$67$	−2.57823	−0.314981	−0.157490	+	0.987521i	$0.550340\pi$
$68$	1.36177	0.165138
$69$	−4.51145	−0.543114
$70$	0	0
$71$	−5.00355	−0.593813	−0.296906	−	0.954907i	$-0.595955\pi$
$71$	−5.00355	−0.593813	−0.296906	+	0.954907i	$0.595955\pi$
$72$	−3.01593	−0.355430
$73$	8.73756	1.02265	0.511327	−	0.859386i	$-0.329154\pi$
$73$	8.73756	1.02265	0.511327	+	0.859386i	$0.329154\pi$
$74$	0.441256	0.0512949
$75$	−10.3463	−1.19469
$76$	2.60180	0.298447
$77$	0	0
$78$	−3.91744	−0.443563
$79$	0.719627	0.0809644	0.0404822	−	0.999180i	$-0.487111\pi$
$79$	0.719627	0.0809644	0.0404822	+	0.999180i	$0.487111\pi$
$80$	12.5240	1.40022
$81$	1.00000	0.111111
$82$	3.59401	0.396892
$83$	11.5976	1.27300	0.636499	−	0.771278i	$-0.280382\pi$
$83$	11.5976	1.27300	0.636499	+	0.771278i	$0.280382\pi$
$84$	0	0
$85$	15.5635	1.68810
$86$	14.3180	1.54395
$87$	−3.75572	−0.402656
$88$	3.01593	0.321499
$89$	12.3553	1.30966	0.654829	−	0.755777i	$-0.272741\pi$
$89$	12.3553	1.30966	0.654829	+	0.755777i	$0.272741\pi$
$90$	−5.04306	−0.531585
$91$	0	0
$92$	−1.54637	−0.161220
$93$	−6.74335	−0.699253
$94$	−2.13342	−0.220045
$95$	29.7357	3.05082
$96$	−1.91627	−0.195578
$97$	13.1687	1.33708	0.668538	−	0.743678i	$-0.266920\pi$
$97$	13.1687	1.33708	0.668538	+	0.743678i	$0.266920\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−3.54637	−0.354637
$101$	−7.07019	−0.703510	−0.351755	−	0.936092i	$-0.614415\pi$
$101$	−7.07019	−0.703510	−0.351755	+	0.936092i	$0.614415\pi$
$102$	5.11442	0.506403
$103$	−2.33736	−0.230307	−0.115153	−	0.993348i	$-0.536736\pi$
$103$	−2.33736	−0.230307	−0.115153	+	0.993348i	$0.536736\pi$
$104$	−9.17764	−0.899942
$105$	0	0
$106$	−16.6585	−1.61802
$107$	−10.8532	−1.04922	−0.524609	−	0.851343i	$-0.675788\pi$
$107$	−10.8532	−1.04922	−0.524609	+	0.851343i	$0.675788\pi$
$108$	0.342766	0.0329827
$109$	−2.19240	−0.209994	−0.104997	−	0.994473i	$-0.533483\pi$
$109$	−2.19240	−0.209994	−0.104997	+	0.994473i	$0.533483\pi$
$110$	5.04306	0.480837
$111$	−0.342766	−0.0325340
$112$	0	0
$113$	−11.6415	−1.09514	−0.547568	−	0.836761i	$-0.684446\pi$
$113$	−11.6415	−1.09514	−0.547568	+	0.836761i	$0.684446\pi$
$114$	9.77165	0.915199
$115$	−17.6733	−1.64804
$116$	−1.28734	−0.119526
$117$	3.04306	0.281331
$118$	4.71266	0.433836
$119$	0	0
$120$	−11.8147	−1.07853
$121$	1.00000	0.0909091
$122$	0.602971	0.0545904
$123$	−2.79182	−0.251730
$124$	−2.31139	−0.207569
$125$	−20.9439	−1.87328
$126$	0	0
$127$	5.41837	0.480802	0.240401	−	0.970674i	$-0.422721\pi$
$127$	5.41837	0.480802	0.240401	+	0.970674i	$0.422721\pi$
$128$	7.57434	0.669483
$129$	−11.1222	−0.979256
$130$	−15.3463	−1.34596
$131$	2.06595	0.180503	0.0902514	−	0.995919i	$-0.471233\pi$
$131$	2.06595	0.180503	0.0902514	+	0.995919i	$0.471233\pi$
$132$	−0.342766	−0.0298340
$133$	0	0
$134$	−3.31904	−0.286722
$135$	3.91744	0.337159
$136$	11.9819	1.02744
$137$	−4.18305	−0.357382	−0.178691	−	0.983905i	$-0.557186\pi$
$137$	−4.18305	−0.357382	−0.178691	+	0.983905i	$0.557186\pi$
$138$	−5.80775	−0.494388
$139$	−3.61077	−0.306261	−0.153131	−	0.988206i	$-0.548936\pi$
$139$	−3.61077	−0.306261	−0.153131	+	0.988206i	$0.548936\pi$
$140$	0	0
$141$	1.65723	0.139564
$142$	−6.44126	−0.540538
$143$	−3.04306	−0.254473
$144$	−3.19698	−0.266415
$145$	−14.7128	−1.22183
$146$	11.2482	0.930905
$147$	0	0
$148$	−0.117489	−0.00965752
$149$	13.3192	1.09115	0.545575	−	0.838062i	$-0.316311\pi$
$149$	13.3192	1.09115	0.545575	+	0.838062i	$0.316311\pi$
$150$	−13.3192	−1.08751
$151$	−7.04321	−0.573168	−0.286584	−	0.958055i	$-0.592520\pi$
$151$	−7.04321	−0.573168	−0.286584	+	0.958055i	$0.592520\pi$
$152$	22.8927	1.85684
$153$	−3.97287	−0.321187
$154$	0	0
$155$	−26.4167	−2.12184
$156$	1.04306	0.0835115
$157$	−12.7559	−1.01803	−0.509015	−	0.860758i	$-0.669990\pi$
$157$	−12.7559	−1.01803	−0.509015	+	0.860758i	$0.669990\pi$
$158$	0.926402	0.0737006
$159$	12.9403	1.02623
$160$	−7.50687	−0.593470
$161$	0	0
$162$	1.28734	0.101143
$163$	11.4041	0.893241	0.446621	−	0.894723i	$-0.352627\pi$
$163$	11.4041	0.893241	0.446621	+	0.894723i	$0.352627\pi$
$164$	−0.956942	−0.0747246
$165$	−3.91744	−0.304972
$166$	14.9300	1.15879
$167$	−12.1109	−0.937167	−0.468583	−	0.883419i	$-0.655235\pi$
$167$	−12.1109	−0.937167	−0.468583	+	0.883419i	$0.655235\pi$
$168$	0	0
$169$	−3.73980	−0.287677
$170$	20.0354	1.53665
$171$	−7.59060	−0.580468
$172$	−3.81232	−0.290687
$173$	−3.64247	−0.276932	−0.138466	−	0.990367i	$-0.544217\pi$
$173$	−3.64247	−0.276932	−0.138466	+	0.990367i	$0.544217\pi$
$174$	−4.83488	−0.366531
$175$	0	0
$176$	3.19698	0.240981
$177$	−3.66079	−0.275162
$178$	15.9054	1.19216
$179$	9.92640	0.741934	0.370967	−	0.928646i	$-0.379026\pi$
$179$	9.92640	0.741934	0.370967	+	0.928646i	$0.379026\pi$
$180$	1.34277	0.100084
$181$	8.07219	0.600001	0.300001	−	0.953939i	$-0.403013\pi$
$181$	8.07219	0.600001	0.300001	+	0.953939i	$0.403013\pi$
$182$	0	0
$183$	−0.468387	−0.0346241
$184$	−13.6062	−1.00306
$185$	−1.34277	−0.0987222
$186$	−8.68096	−0.636519
$187$	3.97287	0.290525
$188$	0.568044	0.0414289
$189$	0	0
$190$	38.2798	2.77711
$191$	5.58364	0.404018	0.202009	−	0.979384i	$-0.435253\pi$
$191$	5.58364	0.404018	0.202009	+	0.979384i	$0.435253\pi$
$192$	−8.86084	−0.639476
$193$	11.1338	0.801425	0.400712	−	0.916204i	$-0.368763\pi$
$193$	11.1338	0.801425	0.400712	+	0.916204i	$0.368763\pi$
$194$	16.9525	1.21712
$195$	11.9210	0.853680
$196$	0	0
$197$	−0.644475	−0.0459170	−0.0229585	−	0.999736i	$-0.507309\pi$
$197$	−0.644475	−0.0459170	−0.0229585	+	0.999736i	$0.507309\pi$
$198$	−1.28734	−0.0914870
$199$	25.2355	1.78889	0.894447	−	0.447174i	$-0.147569\pi$
$199$	25.2355	1.78889	0.894447	+	0.447174i	$0.147569\pi$
$200$	−31.2038	−2.20644
$201$	2.57823	0.181854
$202$	−9.10171	−0.640394
$203$	0	0
$204$	−1.36177	−0.0953427
$205$	−10.9368	−0.763857
$206$	−3.00896	−0.209644
$207$	4.51145	0.313567
$208$	−9.72859	−0.674556
$209$	7.59060	0.525053
$210$	0	0
$211$	−17.6357	−1.21409	−0.607044	−	0.794668i	$-0.707645\pi$
$211$	−17.6357	−1.21409	−0.607044	+	0.794668i	$0.707645\pi$
$212$	4.43551	0.304632
$213$	5.00355	0.342838
$214$	−13.9717	−0.955086
$215$	−43.5706	−2.97149
$216$	3.01593	0.205208
$217$	0	0
$218$	−2.82236	−0.191154
$219$	−8.73756	−0.590429
$220$	−1.34277	−0.0905293
$221$	−12.0897	−0.813239
$222$	−0.441256	−0.0296151
$223$	−16.5853	−1.11064	−0.555318	−	0.831638i	$-0.687404\pi$
$223$	−16.5853	−1.11064	−0.555318	+	0.831638i	$0.687404\pi$
$224$	0	0
$225$	10.3463	0.689755
$226$	−14.9865	−0.996884
$227$	2.34818	0.155854	0.0779269	−	0.996959i	$-0.475170\pi$
$227$	2.34818	0.155854	0.0779269	+	0.996959i	$0.475170\pi$
$228$	−2.60180	−0.172309
$229$	17.4504	1.15315	0.576576	−	0.817043i	$-0.304388\pi$
$229$	17.4504	1.15315	0.576576	+	0.817043i	$0.304388\pi$
$230$	−22.7515	−1.50019
$231$	0	0
$232$	−11.3270	−0.743653
$233$	−5.59060	−0.366252	−0.183126	−	0.983089i	$-0.558622\pi$
$233$	−5.59060	−0.366252	−0.183126	+	0.983089i	$0.558622\pi$
$234$	3.91744	0.256091
$235$	6.49211	0.423499
$236$	−1.25480	−0.0816802
$237$	−0.719627	−0.0467448
$238$	0	0
$239$	17.8147	1.15234	0.576169	−	0.817331i	$-0.304547\pi$
$239$	17.8147	1.15234	0.576169	+	0.817331i	$0.304547\pi$
$240$	−12.5240	−0.808418
$241$	−2.98422	−0.192230	−0.0961152	−	0.995370i	$-0.530642\pi$
$241$	−2.98422	−0.192230	−0.0961152	+	0.995370i	$0.530642\pi$
$242$	1.28734	0.0827531
$243$	−1.00000	−0.0641500
$244$	−0.160547	−0.0102780
$245$	0	0
$246$	−3.59401	−0.229146
$247$	−23.0986	−1.46973
$248$	−20.3375	−1.29143
$249$	−11.5976	−0.734966
$250$	−26.9618	−1.70521
$251$	−25.0829	−1.58322	−0.791608	−	0.611029i	$-0.790756\pi$
$251$	−25.0829	−1.58322	−0.791608	+	0.611029i	$0.790756\pi$
$252$	0	0
$253$	−4.51145	−0.283632
$254$	6.97526	0.437666
$255$	−15.5635	−0.974622
$256$	−7.97096	−0.498185
$257$	22.5570	1.40707	0.703535	−	0.710661i	$-0.251604\pi$
$257$	22.5570	1.40707	0.703535	+	0.710661i	$0.251604\pi$
$258$	−14.3180	−0.891401
$259$	0	0
$260$	4.08612	0.253410
$261$	3.75572	0.232473
$262$	2.65957	0.164309
$263$	25.7559	1.58817	0.794087	−	0.607803i	$-0.207949\pi$
$263$	25.7559	1.58817	0.794087	+	0.607803i	$0.207949\pi$
$264$	−3.01593	−0.185617
$265$	50.6929	3.11404
$266$	0	0
$267$	−12.3553	−0.756131
$268$	0.883730	0.0539824
$269$	20.6962	1.26187	0.630935	−	0.775836i	$-0.282672\pi$
$269$	20.6962	1.26187	0.630935	+	0.775836i	$0.282672\pi$
$270$	5.04306	0.306911
$271$	−3.06780	−0.186356	−0.0931779	−	0.995649i	$-0.529703\pi$
$271$	−3.06780	−0.186356	−0.0931779	+	0.995649i	$0.529703\pi$
$272$	12.7012	0.770122
$273$	0	0
$274$	−5.38499	−0.325319
$275$	−10.3463	−0.623907
$276$	1.54637	0.0930807
$277$	−6.68553	−0.401695	−0.200847	−	0.979623i	$-0.564370\pi$
$277$	−6.68553	−0.401695	−0.200847	+	0.979623i	$0.564370\pi$
$278$	−4.64827	−0.278785
$279$	6.74335	0.403714
$280$	0	0
$281$	3.11286	0.185698	0.0928489	−	0.995680i	$-0.470403\pi$
$281$	3.11286	0.185698	0.0928489	+	0.995680i	$0.470403\pi$
$282$	2.13342	0.127043
$283$	22.3599	1.32916	0.664578	−	0.747219i	$-0.268611\pi$
$283$	22.3599	1.32916	0.664578	+	0.747219i	$0.268611\pi$
$284$	1.71505	0.101770
$285$	−29.7357	−1.76139
$286$	−3.91744	−0.231643
$287$	0	0
$288$	1.91627	0.112917
$289$	−1.21631	−0.0715479
$290$	−18.9403	−1.11221
$291$	−13.1687	−0.771962
$292$	−2.99494	−0.175266
$293$	−10.6291	−0.620958	−0.310479	−	0.950580i	$-0.600489\pi$
$293$	−10.6291	−0.620958	−0.310479	+	0.950580i	$0.600489\pi$
$294$	0	0
$295$	−14.3409	−0.834960
$296$	−1.03376	−0.0600860
$297$	1.00000	0.0580259
$298$	17.1463	0.993257
$299$	13.7286	0.793945
$300$	3.54637	0.204750
$301$	0	0
$302$	−9.06697	−0.521746
$303$	7.07019	0.406172
$304$	24.2670	1.39181
$305$	−1.83488	−0.105065
$306$	−5.11442	−0.292372
$307$	18.5229	1.05716	0.528580	−	0.848883i	$-0.322725\pi$
$307$	18.5229	1.05716	0.528580	+	0.848883i	$0.322725\pi$
$308$	0	0
$309$	2.33736	0.132968
$310$	−34.0071	−1.93147
$311$	19.8438	1.12524	0.562620	−	0.826715i	$-0.309793\pi$
$311$	19.8438	1.12524	0.562620	+	0.826715i	$0.309793\pi$
$312$	9.17764	0.519582
$313$	21.1144	1.19346	0.596729	−	0.802443i	$-0.296467\pi$
$313$	21.1144	1.19346	0.596729	+	0.802443i	$0.296467\pi$
$314$	−16.4211	−0.926696
$315$	0	0
$316$	−0.246664	−0.0138759
$317$	−15.8066	−0.887786	−0.443893	−	0.896080i	$-0.646403\pi$
$317$	−15.8066	−0.887786	−0.443893	+	0.896080i	$0.646403\pi$
$318$	16.6585	0.934165
$319$	−3.75572	−0.210280
$320$	−34.7118	−1.94045
$321$	10.8532	0.605766
$322$	0	0
$323$	30.1565	1.67795
$324$	−0.342766	−0.0190426
$325$	31.4845	1.74644
$326$	14.6810	0.813103
$327$	2.19240	0.121240
$328$	−8.41992	−0.464912
$329$	0	0
$330$	−5.04306	−0.277611
$331$	−9.16327	−0.503659	−0.251829	−	0.967772i	$-0.581032\pi$
$331$	−9.16327	−0.503659	−0.251829	+	0.967772i	$0.581032\pi$
$332$	−3.97526	−0.218170
$333$	0.342766	0.0187835
$334$	−15.5907	−0.853088
$335$	10.1000	0.551824
$336$	0	0
$337$	−12.7174	−0.692760	−0.346380	−	0.938094i	$-0.612589\pi$
$337$	−12.7174	−0.692760	−0.346380	+	0.938094i	$0.612589\pi$
$338$	−4.81437	−0.261867
$339$	11.6415	0.632277
$340$	−5.33464	−0.289311
$341$	−6.74335	−0.365173
$342$	−9.77165	−0.528390
$343$	0	0
$344$	−33.5438	−1.80856
$345$	17.6733	0.951499
$346$	−4.68909	−0.252087
$347$	2.54739	0.136751	0.0683756	−	0.997660i	$-0.478218\pi$
$347$	2.54739	0.136751	0.0683756	+	0.997660i	$0.478218\pi$
$348$	1.28734	0.0690084
$349$	25.0183	1.33920	0.669600	−	0.742722i	$-0.266466\pi$
$349$	25.0183	1.33920	0.669600	+	0.742722i	$0.266466\pi$
$350$	0	0
$351$	−3.04306	−0.162426
$352$	−1.91627	−0.102138
$353$	11.4289	0.608298	0.304149	−	0.952624i	$-0.401628\pi$
$353$	11.4289	0.608298	0.304149	+	0.952624i	$0.401628\pi$
$354$	−4.71266	−0.250475
$355$	19.6011	1.04032
$356$	−4.23498	−0.224453
$357$	0	0
$358$	12.7786	0.675371
$359$	29.9622	1.58134	0.790672	−	0.612240i	$-0.209731\pi$
$359$	29.9622	1.58134	0.790672	+	0.612240i	$0.209731\pi$
$360$	11.8147	0.622690
$361$	38.6172	2.03248
$362$	10.3916	0.546171
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−34.2288	−1.79162
$366$	−0.602971	−0.0315178
$367$	14.4626	0.754941	0.377471	−	0.926022i	$-0.376794\pi$
$367$	14.4626	0.754941	0.377471	+	0.926022i	$0.376794\pi$
$368$	−14.4230	−0.751850
$369$	2.79182	0.145336
$370$	−1.72859	−0.0898652
$371$	0	0
$372$	2.31139	0.119840
$373$	30.1452	1.56086	0.780431	−	0.625242i	$-0.215000\pi$
$373$	30.1452	1.56086	0.780431	+	0.625242i	$0.215000\pi$
$374$	5.11442	0.264460
$375$	20.9439	1.08154
$376$	4.99810	0.257757
$377$	11.4289	0.588617
$378$	0	0
$379$	−11.7452	−0.603311	−0.301655	−	0.953417i	$-0.597539\pi$
$379$	−11.7452	−0.603311	−0.301655	+	0.953417i	$0.597539\pi$
$380$	−10.1924	−0.522859
$381$	−5.41837	−0.277591
$382$	7.18801	0.367771
$383$	−27.9371	−1.42752	−0.713759	−	0.700392i	$-0.753009\pi$
$383$	−27.9371	−1.42752	−0.713759	+	0.700392i	$0.753009\pi$
$384$	−7.57434	−0.386526
$385$	0	0
$386$	14.3329	0.729524
$387$	11.1222	0.565374
$388$	−4.51378	−0.229153
$389$	−15.7579	−0.798958	−0.399479	−	0.916742i	$-0.630809\pi$
$389$	−15.7579	−0.798958	−0.399479	+	0.916742i	$0.630809\pi$
$390$	15.3463	0.777091
$391$	−17.9234	−0.906424
$392$	0	0
$393$	−2.06595	−0.104213
$394$	−0.829656	−0.0417975
$395$	−2.81910	−0.141844
$396$	0.342766	0.0172247
$397$	−12.8089	−0.642861	−0.321430	−	0.946933i	$-0.604164\pi$
$397$	−12.8089	−0.642861	−0.321430	+	0.946933i	$0.604164\pi$
$398$	32.4865	1.62840
$399$	0	0
$400$	−33.0770	−1.65385
$401$	−10.9807	−0.548348	−0.274174	−	0.961680i	$-0.588404\pi$
$401$	−10.9807	−0.548348	−0.274174	+	0.961680i	$0.588404\pi$
$402$	3.31904	0.165539
$403$	20.5204	1.02219
$404$	2.42342	0.120570
$405$	−3.91744	−0.194659
$406$	0	0
$407$	−0.342766	−0.0169903
$408$	−11.9819	−0.593192
$409$	−18.0256	−0.891309	−0.445655	−	0.895205i	$-0.647029\pi$
$409$	−18.0256	−0.891309	−0.445655	+	0.895205i	$0.647029\pi$
$410$	−14.0793	−0.695327
$411$	4.18305	0.206335
$412$	0.801168	0.0394707
$413$	0	0
$414$	5.80775	0.285435
$415$	−45.4327	−2.23021
$416$	5.83132	0.285904
$417$	3.61077	0.176820
$418$	9.77165	0.477947
$419$	−11.8542	−0.579116	−0.289558	−	0.957160i	$-0.593508\pi$
$419$	−11.8542	−0.579116	−0.289558	+	0.957160i	$0.593508\pi$
$420$	0	0
$421$	26.4890	1.29100	0.645498	−	0.763762i	$-0.276650\pi$
$421$	26.4890	1.29100	0.645498	+	0.763762i	$0.276650\pi$
$422$	−22.7030	−1.10517
$423$	−1.65723	−0.0805775
$424$	39.0271	1.89532
$425$	−41.1046	−1.99386
$426$	6.44126	0.312080
$427$	0	0
$428$	3.72011	0.179818
$429$	3.04306	0.146920
$430$	−56.0900	−2.70490
$431$	−2.17681	−0.104853	−0.0524266	−	0.998625i	$-0.516696\pi$
$431$	−2.17681	−0.104853	−0.0524266	+	0.998625i	$0.516696\pi$
$432$	3.19698	0.153815
$433$	16.4831	0.792129	0.396065	−	0.918223i	$-0.370376\pi$
$433$	16.4831	0.792129	0.396065	+	0.918223i	$0.370376\pi$
$434$	0	0
$435$	14.7128	0.705425
$436$	0.751482	0.0359895
$437$	−34.2446	−1.63814
$438$	−11.2482	−0.537458
$439$	0.753000	0.0359387	0.0179694	−	0.999839i	$-0.494280\pi$
$439$	0.753000	0.0359387	0.0179694	+	0.999839i	$0.494280\pi$
$440$	−11.8147	−0.563244
$441$	0	0
$442$	−15.5635	−0.740279
$443$	40.0619	1.90340	0.951698	−	0.307035i	$-0.0993370\pi$
$443$	40.0619	1.90340	0.951698	+	0.307035i	$0.0993370\pi$
$444$	0.117489	0.00557577
$445$	−48.4011	−2.29443
$446$	−21.3509	−1.01099
$447$	−13.3192	−0.629976
$448$	0	0
$449$	−7.71924	−0.364294	−0.182147	−	0.983271i	$-0.558305\pi$
$449$	−7.71924	−0.364294	−0.182147	+	0.983271i	$0.558305\pi$
$450$	13.3192	0.627873
$451$	−2.79182	−0.131462
$452$	3.99030	0.187688
$453$	7.04321	0.330919
$454$	3.02289	0.141871
$455$	0	0
$456$	−22.8927	−1.07205
$457$	26.3646	1.23329	0.616643	−	0.787243i	$-0.288492\pi$
$457$	26.3646	1.23329	0.616643	+	0.787243i	$0.288492\pi$
$458$	22.4645	1.04970
$459$	3.97287	0.185438
$460$	6.05782	0.282447
$461$	−12.4685	−0.580718	−0.290359	−	0.956918i	$-0.593775\pi$
$461$	−12.4685	−0.580718	−0.290359	+	0.956918i	$0.593775\pi$
$462$	0	0
$463$	37.7630	1.75499	0.877497	−	0.479582i	$-0.159212\pi$
$463$	37.7630	1.75499	0.877497	+	0.479582i	$0.159212\pi$
$464$	−12.0070	−0.557409
$465$	26.4167	1.22504
$466$	−7.19698	−0.333394
$467$	−22.0208	−1.01900	−0.509502	−	0.860470i	$-0.670170\pi$
$467$	−22.0208	−1.01900	−0.509502	+	0.860470i	$0.670170\pi$
$468$	−1.04306	−0.0482154
$469$	0	0
$470$	8.35753	0.385504
$471$	12.7559	0.587759
$472$	−11.0407	−0.508188
$473$	−11.1222	−0.511400
$474$	−0.926402	−0.0425511
$475$	−78.5348	−3.60342
$476$	0	0
$477$	−12.9403	−0.592497
$478$	22.9335	1.04895
$479$	9.44803	0.431692	0.215846	−	0.976427i	$-0.430749\pi$
$479$	9.44803	0.431692	0.215846	+	0.976427i	$0.430749\pi$
$480$	7.50687	0.342640
$481$	1.04306	0.0475594
$482$	−3.84169	−0.174984
$483$	0	0
$484$	−0.342766	−0.0155803
$485$	−51.5875	−2.34247
$486$	−1.28734	−0.0583947
$487$	40.2341	1.82318	0.911589	−	0.411102i	$-0.134856\pi$
$487$	40.2341	1.82318	0.911589	+	0.411102i	$0.134856\pi$
$488$	−1.41262	−0.0639463
$489$	−11.4041	−0.515713
$490$	0	0
$491$	20.2964	0.915966	0.457983	−	0.888961i	$-0.348572\pi$
$491$	20.2964	0.915966	0.457983	+	0.888961i	$0.348572\pi$
$492$	0.956942	0.0431423
$493$	−14.9210	−0.672008
$494$	−29.7357	−1.33787
$495$	3.91744	0.176076
$496$	−21.5583	−0.967998
$497$	0	0
$498$	−14.9300	−0.669027
$499$	36.2954	1.62481	0.812403	−	0.583096i	$-0.198159\pi$
$499$	36.2954	1.62481	0.812403	+	0.583096i	$0.198159\pi$
$500$	7.17886	0.321048
$501$	12.1109	0.541073
$502$	−32.2901	−1.44118
$503$	27.6923	1.23474	0.617368	−	0.786674i	$-0.288199\pi$
$503$	27.6923	1.23474	0.617368	+	0.786674i	$0.288199\pi$
$504$	0	0
$505$	27.6970	1.23250
$506$	−5.80775	−0.258186
$507$	3.73980	0.166090
$508$	−1.85723	−0.0824014
$509$	−9.08797	−0.402817	−0.201409	−	0.979507i	$-0.564552\pi$
$509$	−9.08797	−0.402817	−0.201409	+	0.979507i	$0.564552\pi$
$510$	−20.0354	−0.887183
$511$	0	0
$512$	−25.4100	−1.12297
$513$	7.59060	0.335133
$514$	29.0385	1.28083
$515$	9.15645	0.403482
$516$	3.81232	0.167828
$517$	1.65723	0.0728850
$518$	0	0
$519$	3.64247	0.159887
$520$	35.9528	1.57664
$521$	−38.8510	−1.70209	−0.851046	−	0.525092i	$-0.824031\pi$
$521$	−38.8510	−1.70209	−0.851046	+	0.525092i	$0.824031\pi$
$522$	4.83488	0.211617
$523$	21.5001	0.940132	0.470066	−	0.882631i	$-0.344230\pi$
$523$	21.5001	0.940132	0.470066	+	0.882631i	$0.344230\pi$
$524$	−0.708138	−0.0309352
$525$	0	0
$526$	33.1565	1.44569
$527$	−26.7904	−1.16701
$528$	−3.19698	−0.139131
$529$	−2.64686	−0.115081
$530$	65.2588	2.83466
$531$	3.66079	0.158865
$532$	0	0
$533$	8.49566	0.367988
$534$	−15.9054	−0.688294
$535$	42.5167	1.83816
$536$	7.77574	0.335861
$537$	−9.92640	−0.428356
$538$	26.6430	1.14866
$539$	0	0
$540$	−1.34277	−0.0577835
$541$	−29.8940	−1.28524	−0.642622	−	0.766183i	$-0.722153\pi$
$541$	−29.8940	−1.28524	−0.642622	+	0.766183i	$0.722153\pi$
$542$	−3.94929	−0.169637
$543$	−8.07219	−0.346411
$544$	−7.61309	−0.326408
$545$	8.58860	0.367895
$546$	0	0
$547$	−31.9072	−1.36425	−0.682127	−	0.731234i	$-0.738945\pi$
$547$	−31.9072	−1.36425	−0.682127	+	0.731234i	$0.738945\pi$
$548$	1.43381	0.0612493
$549$	0.468387	0.0199903
$550$	−13.3192	−0.567932
$551$	−28.5082	−1.21449
$552$	13.6062	0.579118
$553$	0	0
$554$	−8.60653	−0.365656
$555$	1.34277	0.0569973
$556$	1.23765	0.0524880
$557$	−14.6089	−0.619000	−0.309500	−	0.950900i	$-0.600162\pi$
$557$	−14.6089	−0.619000	−0.309500	+	0.950900i	$0.600162\pi$
$558$	8.68096	0.367494
$559$	33.8455	1.43151
$560$	0	0
$561$	−3.97287	−0.167735
$562$	4.00730	0.169038
$563$	8.48231	0.357487	0.178743	−	0.983896i	$-0.442797\pi$
$563$	8.48231	0.357487	0.178743	+	0.983896i	$0.442797\pi$
$564$	−0.568044	−0.0239190
$565$	45.6047	1.91860
$566$	28.7847	1.20991
$567$	0	0
$568$	15.0904	0.633177
$569$	20.7058	0.868034	0.434017	−	0.900905i	$-0.357096\pi$
$569$	20.7058	0.868034	0.434017	+	0.900905i	$0.357096\pi$
$570$	−38.2798	−1.60337
$571$	1.00882	0.0422177	0.0211088	−	0.999777i	$-0.493280\pi$
$571$	1.00882	0.0422177	0.0211088	+	0.999777i	$0.493280\pi$
$572$	1.04306	0.0436125
$573$	−5.58364	−0.233260
$574$	0	0
$575$	46.6769	1.94656
$576$	8.86084	0.369202
$577$	38.3851	1.59799	0.798996	−	0.601336i	$-0.205365\pi$
$577$	38.3851	1.59799	0.798996	+	0.601336i	$0.205365\pi$
$578$	−1.56580	−0.0651289
$579$	−11.1338	−0.462703
$580$	5.04306	0.209402
$581$	0	0
$582$	−16.9525	−0.702704
$583$	12.9403	0.535934
$584$	−26.3518	−1.09045
$585$	−11.9210	−0.492872
$586$	−13.6832	−0.565248
$587$	−3.95733	−0.163336	−0.0816682	−	0.996660i	$-0.526025\pi$
$587$	−3.95733	−0.163336	−0.0816682	+	0.996660i	$0.526025\pi$
$588$	0	0
$589$	−51.1861	−2.10909
$590$	−18.4616	−0.760051
$591$	0.644475	0.0265102
$592$	−1.09582	−0.0450378
$593$	−8.50263	−0.349161	−0.174581	−	0.984643i	$-0.555857\pi$
$593$	−8.50263	−0.349161	−0.174581	+	0.984643i	$0.555857\pi$
$594$	1.28734	0.0528200
$595$	0	0
$596$	−4.56537	−0.187005
$597$	−25.2355	−1.03282
$598$	17.6733	0.722715
$599$	−6.96152	−0.284440	−0.142220	−	0.989835i	$-0.545424\pi$
$599$	−6.96152	−0.284440	−0.142220	+	0.989835i	$0.545424\pi$
$600$	31.2038	1.27389
$601$	−31.8738	−1.30016	−0.650081	−	0.759865i	$-0.725265\pi$
$601$	−31.8738	−1.30016	−0.650081	+	0.759865i	$0.725265\pi$
$602$	0	0
$603$	−2.57823	−0.104994
$604$	2.41417	0.0982314
$605$	−3.91744	−0.159267
$606$	9.10171	0.369732
$607$	−3.34408	−0.135732	−0.0678660	−	0.997694i	$-0.521619\pi$
$607$	−3.34408	−0.135732	−0.0678660	+	0.997694i	$0.521619\pi$
$608$	−14.5456	−0.589903
$609$	0	0
$610$	−2.36210	−0.0956387
$611$	−5.04306	−0.204020
$612$	1.36177	0.0550461
$613$	−4.22650	−0.170707	−0.0853533	−	0.996351i	$-0.527202\pi$
$613$	−4.22650	−0.170707	−0.0853533	+	0.996351i	$0.527202\pi$
$614$	23.8452	0.962316
$615$	10.9368	0.441013
$616$	0	0
$617$	3.91266	0.157518	0.0787590	−	0.996894i	$-0.474904\pi$
$617$	3.91266	0.157518	0.0787590	+	0.996894i	$0.474904\pi$
$618$	3.00896	0.121038
$619$	−1.73839	−0.0698717	−0.0349359	−	0.999390i	$-0.511123\pi$
$619$	−1.73839	−0.0698717	−0.0349359	+	0.999390i	$0.511123\pi$
$620$	9.05474	0.363647
$621$	−4.51145	−0.181038
$622$	25.5457	1.02429
$623$	0	0
$624$	9.72859	0.389455
$625$	30.3148	1.21259
$626$	27.1813	1.08639
$627$	−7.59060	−0.303139
$628$	4.37228	0.174473
$629$	−1.36177	−0.0542972
$630$	0	0
$631$	−13.3994	−0.533420	−0.266710	−	0.963777i	$-0.585937\pi$
$631$	−13.3994	−0.533420	−0.266710	+	0.963777i	$0.585937\pi$
$632$	−2.17034	−0.0863316
$633$	17.6357	0.700954
$634$	−20.3484	−0.808137
$635$	−21.2261	−0.842332
$636$	−4.43551	−0.175879
$637$	0	0
$638$	−4.83488	−0.191415
$639$	−5.00355	−0.197938
$640$	−29.6720	−1.17289
$641$	10.7820	0.425864	0.212932	−	0.977067i	$-0.431699\pi$
$641$	10.7820	0.425864	0.212932	+	0.977067i	$0.431699\pi$
$642$	13.9717	0.551419
$643$	−30.1180	−1.18774	−0.593868	−	0.804562i	$-0.702400\pi$
$643$	−30.1180	−1.18774	−0.593868	+	0.804562i	$0.702400\pi$
$644$	0	0
$645$	43.5706	1.71559
$646$	38.8215	1.52741
$647$	12.6469	0.497199	0.248600	−	0.968606i	$-0.420030\pi$
$647$	12.6469	0.497199	0.248600	+	0.968606i	$0.420030\pi$
$648$	−3.01593	−0.118477
$649$	−3.66079	−0.143699
$650$	40.5311	1.58976
$651$	0	0
$652$	−3.90896	−0.153087
$653$	1.50229	0.0587892	0.0293946	−	0.999568i	$-0.490642\pi$
$653$	1.50229	0.0587892	0.0293946	+	0.999568i	$0.490642\pi$
$654$	2.82236	0.110363
$655$	−8.09323	−0.316229
$656$	−8.92538	−0.348478
$657$	8.73756	0.340885
$658$	0	0
$659$	−14.4079	−0.561251	−0.280626	−	0.959817i	$-0.590542\pi$
$659$	−14.4079	−0.561251	−0.280626	+	0.959817i	$0.590542\pi$
$660$	1.34277	0.0522671
$661$	16.7854	0.652876	0.326438	−	0.945219i	$-0.394152\pi$
$661$	16.7854	0.652876	0.326438	+	0.945219i	$0.394152\pi$
$662$	−11.7962	−0.458473
$663$	12.0897	0.469524
$664$	−34.9774	−1.35739
$665$	0	0
$666$	0.441256	0.0170983
$667$	16.9437	0.656064
$668$	4.15120	0.160615
$669$	16.5853	0.641226
$670$	13.0021	0.502317
$671$	−0.468387	−0.0180819
$672$	0	0
$673$	−48.1663	−1.85667	−0.928337	−	0.371740i	$-0.878761\pi$
$673$	−48.1663	−1.85667	−0.928337	+	0.371740i	$0.878761\pi$
$674$	−16.3715	−0.630608
$675$	−10.3463	−0.398230
$676$	1.28188	0.0493029
$677$	11.7148	0.450237	0.225118	−	0.974331i	$-0.427723\pi$
$677$	11.7148	0.450237	0.225118	+	0.974331i	$0.427723\pi$
$678$	14.9865	0.575552
$679$	0	0
$680$	−46.9383	−1.80000
$681$	−2.34818	−0.0899823
$682$	−8.68096	−0.332411
$683$	11.4199	0.436971	0.218486	−	0.975840i	$-0.429888\pi$
$683$	11.4199	0.436971	0.218486	+	0.975840i	$0.429888\pi$
$684$	2.60180	0.0994824
$685$	16.3868	0.626109
$686$	0	0
$687$	−17.4504	−0.665773
$688$	−35.5575	−1.35562
$689$	−39.3782	−1.50019
$690$	22.7515	0.866134
$691$	22.9191	0.871885	0.435943	−	0.899974i	$-0.356415\pi$
$691$	22.9191	0.871885	0.435943	+	0.899974i	$0.356415\pi$
$692$	1.24852	0.0474615
$693$	0	0
$694$	3.27935	0.124482
$695$	14.1450	0.536549
$696$	11.3270	0.429348
$697$	−11.0915	−0.420122
$698$	32.2070	1.21905
$699$	5.59060	0.211456
$700$	0	0
$701$	25.3982	0.959277	0.479638	−	0.877466i	$-0.340768\pi$
$701$	25.3982	0.959277	0.479638	+	0.877466i	$0.340768\pi$
$702$	−3.91744	−0.147854
$703$	−2.60180	−0.0981289
$704$	−8.86084	−0.333955
$705$	−6.49211	−0.244507
$706$	14.7128	0.553724
$707$	0	0
$708$	1.25480	0.0471581
$709$	−16.5011	−0.619711	−0.309855	−	0.950784i	$-0.600281\pi$
$709$	−16.5011	−0.619711	−0.309855	+	0.950784i	$0.600281\pi$
$710$	25.2332	0.946986
$711$	0.719627	0.0269881
$712$	−37.2626	−1.39648
$713$	30.4223	1.13932
$714$	0	0
$715$	11.9210	0.445820
$716$	−3.40244	−0.127155
$717$	−17.8147	−0.665302
$718$	38.5714	1.43947
$719$	35.8298	1.33622	0.668112	−	0.744060i	$-0.267103\pi$
$719$	35.8298	1.33622	0.668112	+	0.744060i	$0.267103\pi$
$720$	12.5240	0.466741
$721$	0	0
$722$	49.7133	1.85014
$723$	2.98422	0.110984
$724$	−2.76688	−0.102830
$725$	38.8579	1.44315
$726$	−1.28734	−0.0477775
$727$	18.7401	0.695031	0.347516	−	0.937674i	$-0.387025\pi$
$727$	18.7401	0.695031	0.347516	+	0.937674i	$0.387025\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−44.0640	−1.63088
$731$	−44.1871	−1.63432
$732$	0.160547	0.00593400
$733$	−47.8079	−1.76582	−0.882912	−	0.469538i	$-0.844421\pi$
$733$	−47.8079	−1.76582	−0.882912	+	0.469538i	$0.844421\pi$
$734$	18.6182	0.687211
$735$	0	0
$736$	8.64515	0.318664
$737$	2.57823	0.0949702
$738$	3.59401	0.132297
$739$	−12.1402	−0.446584	−0.223292	−	0.974752i	$-0.571680\pi$
$739$	−12.1402	−0.446584	−0.223292	+	0.974752i	$0.571680\pi$
$740$	0.460255	0.0169193
$741$	23.0986	0.848550
$742$	0	0
$743$	30.9615	1.13587	0.567934	−	0.823074i	$-0.307743\pi$
$743$	30.9615	1.13587	0.567934	+	0.823074i	$0.307743\pi$
$744$	20.3375	0.745607
$745$	−52.1771	−1.91162
$746$	38.8071	1.42083
$747$	11.5976	0.424333
$748$	−1.36177	−0.0497911
$749$	0	0
$750$	26.9618	0.984506
$751$	−32.0437	−1.16929	−0.584646	−	0.811288i	$-0.698767\pi$
$751$	−32.0437	−1.16929	−0.584646	+	0.811288i	$0.698767\pi$
$752$	5.29814	0.193203
$753$	25.0829	0.914070
$754$	14.7128	0.535809
$755$	27.5913	1.00415
$756$	0	0
$757$	32.5209	1.18199	0.590996	−	0.806675i	$-0.298735\pi$
$757$	32.5209	1.18199	0.590996	+	0.806675i	$0.298735\pi$
$758$	−15.1200	−0.549184
$759$	4.51145	0.163755
$760$	−89.6807	−3.25306
$761$	2.46361	0.0893059	0.0446530	−	0.999003i	$-0.485782\pi$
$761$	2.46361	0.0893059	0.0446530	+	0.999003i	$0.485782\pi$
$762$	−6.97526	−0.252687
$763$	0	0
$764$	−1.91388	−0.0692419
$765$	15.5635	0.562698
$766$	−35.9644	−1.29945
$767$	11.1400	0.402242
$768$	7.97096	0.287627
$769$	−49.0232	−1.76782	−0.883911	−	0.467656i	$-0.845099\pi$
$769$	−49.0232	−1.76782	−0.883911	+	0.467656i	$0.845099\pi$
$770$	0	0
$771$	−22.5570	−0.812372
$772$	−3.81628	−0.137351
$773$	−0.416364	−0.0149756	−0.00748779	−	0.999972i	$-0.502383\pi$
$773$	−0.416364	−0.0149756	−0.00748779	+	0.999972i	$0.502383\pi$
$774$	14.3180	0.514651
$775$	69.7689	2.50617
$776$	−39.7158	−1.42571
$777$	0	0
$778$	−20.2857	−0.727278
$779$	−21.1916	−0.759267
$780$	−4.08612	−0.146306
$781$	5.00355	0.179041
$782$	−23.0734	−0.825103
$783$	−3.75572	−0.134219
$784$	0	0
$785$	49.9703	1.78352
$786$	−2.65957	−0.0948637
$787$	−4.83064	−0.172194	−0.0860968	−	0.996287i	$-0.527439\pi$
$787$	−4.83064	−0.172194	−0.0860968	+	0.996287i	$0.527439\pi$
$788$	0.220905	0.00786940
$789$	−25.7559	−0.916933
$790$	−3.62912	−0.129118
$791$	0	0
$792$	3.01593	0.107166
$793$	1.42533	0.0506149
$794$	−16.4894	−0.585186
$795$	−50.6929	−1.79789
$796$	−8.64987	−0.306587
$797$	−43.5752	−1.54351	−0.771756	−	0.635919i	$-0.780621\pi$
$797$	−43.5752	−1.54351	−0.771756	+	0.635919i	$0.780621\pi$
$798$	0	0
$799$	6.58397	0.232924
$800$	19.8263	0.700967
$801$	12.3553	0.436553
$802$	−14.1358	−0.499153
$803$	−8.73756	−0.308342
$804$	−0.883730	−0.0311667
$805$	0	0
$806$	26.4167	0.930487
$807$	−20.6962	−0.728541
$808$	21.3232	0.750147
$809$	−18.7352	−0.658695	−0.329348	−	0.944209i	$-0.606829\pi$
$809$	−18.7352	−0.658695	−0.329348	+	0.944209i	$0.606829\pi$
$810$	−5.04306	−0.177195
$811$	4.64229	0.163013	0.0815063	−	0.996673i	$-0.474027\pi$
$811$	4.64229	0.163013	0.0815063	+	0.996673i	$0.474027\pi$
$812$	0	0
$813$	3.06780	0.107593
$814$	−0.441256	−0.0154660
$815$	−44.6750	−1.56490
$816$	−12.7012	−0.444630
$817$	−84.4243	−2.95363
$818$	−23.2050	−0.811344
$819$	0	0
$820$	3.74876	0.130912
$821$	17.9116	0.625121	0.312560	−	0.949898i	$-0.398813\pi$
$821$	17.9116	0.625121	0.312560	+	0.949898i	$0.398813\pi$
$822$	5.38499	0.187823
$823$	−27.5943	−0.961877	−0.480938	−	0.876754i	$-0.659704\pi$
$823$	−27.5943	−0.961877	−0.480938	+	0.876754i	$0.659704\pi$
$824$	7.04930	0.245574
$825$	10.3463	0.360213
$826$	0	0
$827$	27.2202	0.946538	0.473269	−	0.880918i	$-0.343074\pi$
$827$	27.2202	0.946538	0.473269	+	0.880918i	$0.343074\pi$
$828$	−1.54637	−0.0537401
$829$	−19.4199	−0.674482	−0.337241	−	0.941418i	$-0.609494\pi$
$829$	−19.4199	−0.674482	−0.337241	+	0.941418i	$0.609494\pi$
$830$	−58.4872	−2.03012
$831$	6.68553	0.231919
$832$	26.9641	0.934810
$833$	0	0
$834$	4.64827	0.160956
$835$	47.4435	1.64185
$836$	−2.60180	−0.0899852
$837$	−6.74335	−0.233084
$838$	−15.2604	−0.527160
$839$	−42.1143	−1.45395	−0.726973	−	0.686666i	$-0.759073\pi$
$839$	−42.1143	−1.45395	−0.726973	+	0.686666i	$0.759073\pi$
$840$	0	0
$841$	−14.8945	−0.513605
$842$	34.1003	1.17517
$843$	−3.11286	−0.107213
$844$	6.04491	0.208074
$845$	14.6504	0.503990
$846$	−2.13342	−0.0733484
$847$	0	0
$848$	41.3699	1.42065
$849$	−22.3599	−0.767388
$850$	−52.9154	−1.81498
$851$	1.54637	0.0530090
$852$	−1.71505	−0.0587567
$853$	53.7179	1.83926	0.919632	−	0.392780i	$-0.128487\pi$
$853$	53.7179	1.83926	0.919632	+	0.392780i	$0.128487\pi$
$854$	0	0
$855$	29.7357	1.01694
$856$	32.7324	1.11877
$857$	6.96848	0.238039	0.119019	−	0.992892i	$-0.462025\pi$
$857$	6.96848	0.238039	0.119019	+	0.992892i	$0.462025\pi$
$858$	3.91744	0.133739
$859$	33.6294	1.14742	0.573710	−	0.819058i	$-0.305503\pi$
$859$	33.6294	1.14742	0.573710	+	0.819058i	$0.305503\pi$
$860$	14.9345	0.509263
$861$	0	0
$862$	−2.80229	−0.0954462
$863$	44.0972	1.50109	0.750544	−	0.660821i	$-0.229792\pi$
$863$	44.0972	1.50109	0.750544	+	0.660821i	$0.229792\pi$
$864$	−1.91627	−0.0651928
$865$	14.2692	0.485166
$866$	21.2193	0.721063
$867$	1.21631	0.0413082
$868$	0	0
$869$	−0.719627	−0.0244117
$870$	18.9403	0.642137
$871$	−7.84569	−0.265841
$872$	6.61212	0.223915
$873$	13.1687	0.445692
$874$	−44.0843	−1.49117
$875$	0	0
$876$	2.99494	0.101190
$877$	10.1450	0.342571	0.171285	−	0.985221i	$-0.445208\pi$
$877$	10.1450	0.342571	0.171285	+	0.985221i	$0.445208\pi$
$878$	0.969364	0.0327145
$879$	10.6291	0.358510
$880$	−12.5240	−0.422183
$881$	−32.4394	−1.09291	−0.546455	−	0.837489i	$-0.684023\pi$
$881$	−32.4394	−1.09291	−0.546455	+	0.837489i	$0.684023\pi$
$882$	0	0
$883$	25.0795	0.843992	0.421996	−	0.906598i	$-0.361330\pi$
$883$	25.0795	0.843992	0.421996	+	0.906598i	$0.361330\pi$
$884$	4.14393	0.139376
$885$	14.3409	0.482064
$886$	51.5731	1.73263
$887$	52.6253	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$887$	52.6253	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$888$	1.03376	0.0346907
$889$	0	0
$890$	−62.3084	−2.08858
$891$	−1.00000	−0.0335013
$892$	5.68490	0.190344
$893$	12.5794	0.420953
$894$	−17.1463	−0.573457
$895$	−38.8861	−1.29982
$896$	0	0
$897$	−13.7286	−0.458384
$898$	−9.93726	−0.331611
$899$	25.3262	0.844675
$900$	−3.54637	−0.118212
$901$	51.4102	1.71272
$902$	−3.59401	−0.119667
$903$	0	0
$904$	35.1098	1.16773
$905$	−31.6223	−1.05116
$906$	9.06697	0.301230
$907$	−36.3628	−1.20741	−0.603704	−	0.797208i	$-0.706309\pi$
$907$	−36.3628	−1.20741	−0.603704	+	0.797208i	$0.706309\pi$
$908$	−0.804876	−0.0267107
$909$	−7.07019	−0.234503
$910$	0	0
$911$	−14.4516	−0.478802	−0.239401	−	0.970921i	$-0.576951\pi$
$911$	−14.4516	−0.478802	−0.239401	+	0.970921i	$0.576951\pi$
$912$	−24.2670	−0.803560
$913$	−11.5976	−0.383823
$914$	33.9401	1.12264
$915$	1.83488	0.0606591
$916$	−5.98140	−0.197631
$917$	0	0
$918$	5.11442	0.168801
$919$	45.6611	1.50622	0.753111	−	0.657894i	$-0.228552\pi$
$919$	45.6611	1.50622	0.753111	+	0.657894i	$0.228552\pi$
$920$	53.3014	1.75730
$921$	−18.5229	−0.610352
$922$	−16.0512	−0.528618
$923$	−15.2261	−0.501174
$924$	0	0
$925$	3.54637	0.116604
$926$	48.6136	1.59754
$927$	−2.33736	−0.0767689
$928$	7.19698	0.236252
$929$	45.3661	1.48841	0.744207	−	0.667949i	$-0.232827\pi$
$929$	45.3661	1.48841	0.744207	+	0.667949i	$0.232827\pi$
$930$	34.0071	1.11514
$931$	0	0
$932$	1.91627	0.0627695
$933$	−19.8438	−0.649658
$934$	−28.3482	−0.927582
$935$	−15.5635	−0.508980
$936$	−9.17764	−0.299981
$937$	−14.7757	−0.482699	−0.241350	−	0.970438i	$-0.577590\pi$
$937$	−14.7757	−0.482699	−0.241350	+	0.970438i	$0.577590\pi$
$938$	0	0
$939$	−21.1144	−0.689043
$940$	−2.22528	−0.0725806
$941$	15.9211	0.519014	0.259507	−	0.965741i	$-0.416440\pi$
$941$	15.9211	0.519014	0.259507	+	0.965741i	$0.416440\pi$
$942$	16.4211	0.535028
$943$	12.5951	0.410154
$944$	−11.7035	−0.380915
$945$	0	0
$946$	−14.3180	−0.465519
$947$	24.3584	0.791540	0.395770	−	0.918350i	$-0.370478\pi$
$947$	24.3584	0.791540	0.395770	+	0.918350i	$0.370478\pi$
$948$	0.246664	0.00801128
$949$	26.5889	0.863112
$950$	−101.101	−3.28014
$951$	15.8066	0.512563
$952$	0	0
$953$	−47.2682	−1.53117	−0.765583	−	0.643337i	$-0.777550\pi$
$953$	−47.2682	−1.53117	−0.765583	+	0.643337i	$0.777550\pi$
$954$	−16.6585	−0.539340
$955$	−21.8735	−0.707811
$956$	−6.10628	−0.197491
$957$	3.75572	0.121405
$958$	12.1628	0.392962
$959$	0	0
$960$	34.7118	1.12032
$961$	14.4728	0.466864
$962$	1.34277	0.0432925
$963$	−10.8532	−0.349739
$964$	1.02289	0.0329451
$965$	−43.6158	−1.40404
$966$	0	0
$967$	37.7062	1.21255	0.606275	−	0.795255i	$-0.292663\pi$
$967$	37.7062	1.21255	0.606275	+	0.795255i	$0.292663\pi$
$968$	−3.01593	−0.0969356
$969$	−30.1565	−0.968765
$970$	−66.4104	−2.13231
$971$	−1.99152	−0.0639109	−0.0319554	−	0.999489i	$-0.510173\pi$
$971$	−1.99152	−0.0639109	−0.0319554	+	0.999489i	$0.510173\pi$
$972$	0.342766	0.0109942
$973$	0	0
$974$	51.7947	1.65961
$975$	−31.4845	−1.00831
$976$	−1.49742	−0.0479313
$977$	−27.2601	−0.872126	−0.436063	−	0.899916i	$-0.643628\pi$
$977$	−27.2601	−0.872126	−0.436063	+	0.899916i	$0.643628\pi$
$978$	−14.6810	−0.469445
$979$	−12.3553	−0.394877
$980$	0	0
$981$	−2.19240	−0.0699980
$982$	26.1283	0.833789
$983$	−10.1422	−0.323487	−0.161744	−	0.986833i	$-0.551712\pi$
$983$	−10.1422	−0.323487	−0.161744	+	0.986833i	$0.551712\pi$
$984$	8.41992	0.268417
$985$	2.52469	0.0804434
$986$	−19.2083	−0.611718
$987$	0	0
$988$	7.91744	0.251887
$989$	50.1773	1.59554
$990$	5.04306	0.160279
$991$	22.1404	0.703312	0.351656	−	0.936129i	$-0.385619\pi$
$991$	22.1404	0.703312	0.351656	+	0.936129i	$0.385619\pi$
$992$	12.9221	0.410276
$993$	9.16327	0.290788
$994$	0	0
$995$	−98.8584	−3.13402
$996$	3.97526	0.125961
$997$	−48.8984	−1.54863	−0.774314	−	0.632802i	$-0.781905\pi$
$997$	−48.8984	−1.54863	−0.774314	+	0.632802i	$0.781905\pi$
$998$	46.7244	1.47904
$999$	−0.342766	−0.0108447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.a.x.1.3		4
3.2	odd	2		4851.2.a.bu.1.2		4
7.3	odd	6		231.2.i.e.100.2	yes	8
7.5	odd	6		231.2.i.e.67.2	✓	8
7.6	odd	2		1617.2.a.z.1.3		4
21.5	even	6		693.2.i.i.298.3		8
21.17	even	6		693.2.i.i.100.3		8
21.20	even	2		4851.2.a.bt.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.i.e.67.2	✓	8	7.5	odd	6
231.2.i.e.100.2	yes	8	7.3	odd	6
693.2.i.i.100.3		8	21.17	even	6
693.2.i.i.298.3		8	21.5	even	6
1617.2.a.x.1.3		4	1.1	even	1	trivial
1617.2.a.z.1.3		4	7.6	odd	2
4851.2.a.bt.1.2		4	21.20	even	2
4851.2.a.bu.1.2		4	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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28734 q^{2} -1.00000 q^{3} -0.342766 q^{4} -3.91744 q^{5} -1.28734 q^{6} -3.01593 q^{8} +1.00000 q^{9} -5.04306 q^{10} -1.00000 q^{11} +0.342766 q^{12} +3.04306 q^{13} +3.91744 q^{15} -3.19698 q^{16} -3.97287 q^{17} +1.28734 q^{18} -7.59060 q^{19} +1.34277 q^{20} -1.28734 q^{22} +4.51145 q^{23} +3.01593 q^{24} +10.3463 q^{25} +3.91744 q^{26} -1.00000 q^{27} +3.75572 q^{29} +5.04306 q^{30} +6.74335 q^{31} +1.91627 q^{32} +1.00000 q^{33} -5.11442 q^{34} -0.342766 q^{36} +0.342766 q^{37} -9.77165 q^{38} -3.04306 q^{39} +11.8147 q^{40} +2.79182 q^{41} +11.1222 q^{43} +0.342766 q^{44} -3.91744 q^{45} +5.80775 q^{46} -1.65723 q^{47} +3.19698 q^{48} +13.3192 q^{50} +3.97287 q^{51} -1.04306 q^{52} -12.9403 q^{53} -1.28734 q^{54} +3.91744 q^{55} +7.59060 q^{57} +4.83488 q^{58} +3.66079 q^{59} -1.34277 q^{60} +0.468387 q^{61} +8.68096 q^{62} +8.86084 q^{64} -11.9210 q^{65} +1.28734 q^{66} -2.57823 q^{67} +1.36177 q^{68} -4.51145 q^{69} -5.00355 q^{71} -3.01593 q^{72} +8.73756 q^{73} +0.441256 q^{74} -10.3463 q^{75} +2.60180 q^{76} -3.91744 q^{78} +0.719627 q^{79} +12.5240 q^{80} +1.00000 q^{81} +3.59401 q^{82} +11.5976 q^{83} +15.5635 q^{85} +14.3180 q^{86} -3.75572 q^{87} +3.01593 q^{88} +12.3553 q^{89} -5.04306 q^{90} -1.54637 q^{92} -6.74335 q^{93} -2.13342 q^{94} +29.7357 q^{95} -1.91627 q^{96} +13.1687 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9} - 10 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 12 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{22} + 4 q^{23} - 12 q^{24} + 4 q^{25}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9 - 10 * q^10 - 4 * q^11 - 4 * q^12 + 2 * q^13 + 4 * q^15 + 12 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^22 + 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^27 + 8 * q^29 + 10 * q^30 + 12 * q^31 + 26 * q^32 + 4 * q^33 + 16 * q^34 + 4 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 4 * q^44 - 4 * q^45 - 14 * q^46 - 12 * q^47 - 12 * q^48 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^55 - 4 * q^58 - 12 * q^59 - 2 * q^61 + 26 * q^62 + 56 * q^64 - 4 * q^65 + 2 * q^66 + 28 * q^67 + 48 * q^68 - 4 * q^69 + 12 * q^71 + 12 * q^72 - 6 * q^73 - 16 * q^74 - 4 * q^75 + 18 * q^76 - 4 * q^78 + 2 * q^79 - 16 * q^80 + 4 * q^81 + 12 * q^82 + 12 * q^83 + 18 * q^85 + 36 * q^86 - 8 * q^87 - 12 * q^88 - 8 * q^89 - 10 * q^90 - 16 * q^92 - 12 * q^93 - 20 * q^94 + 34 * q^95 - 26 * q^96 + 44 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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