LMFDB - Embedded newform 975.2.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,2,Mod(1,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-5,6,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1821184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 4 $ x^5 - 8x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.329386$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.329386 q^{2} -1.00000 q^{3} -1.89150 q^{4} -0.329386 q^{6} -3.70203 q^{7} -1.28181 q^{8} +1.00000 q^{9} -3.31322 q^{11} +1.89150 q^{12} -1.00000 q^{13} -1.21940 q^{14} +3.36080 q^{16} +4.36080 q^{17} +0.329386 q^{18} +5.21940 q^{19} +3.70203 q^{21} -1.09133 q^{22} -4.92143 q^{23} +1.28181 q^{24} -0.329386 q^{26} -1.00000 q^{27} +7.00241 q^{28} +7.78301 q^{29} +0.0981475 q^{31} +3.67061 q^{32} +3.31322 q^{33} +1.43639 q^{34} -1.89150 q^{36} +2.92143 q^{37} +1.71920 q^{38} +1.00000 q^{39} -0.749608 q^{41} +1.21940 q^{42} +3.78301 q^{43} +6.26698 q^{44} -1.62105 q^{46} -5.67402 q^{47} -3.36080 q^{48} +6.70502 q^{49} -4.36080 q^{51} +1.89150 q^{52} +2.19982 q^{53} -0.329386 q^{54} +4.74529 q^{56} -5.21940 q^{57} +2.56361 q^{58} +0.108987 q^{59} +12.1438 q^{61} +0.0323284 q^{62} -3.70203 q^{63} -5.51255 q^{64} +1.09133 q^{66} -12.4418 q^{67} -8.24848 q^{68} +4.92143 q^{69} +12.0348 q^{71} -1.28181 q^{72} -9.56602 q^{73} +0.962276 q^{74} -9.87251 q^{76} +12.2656 q^{77} +0.329386 q^{78} +14.6093 q^{79} +1.00000 q^{81} -0.246910 q^{82} +16.7001 q^{83} -7.00241 q^{84} +1.24607 q^{86} -7.78301 q^{87} +4.24691 q^{88} -3.59403 q^{89} +3.70203 q^{91} +9.30890 q^{92} -0.0981475 q^{93} -1.86894 q^{94} -3.67061 q^{96} +4.10467 q^{97} +2.20854 q^{98} -3.31322 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9} + 5 q^{11} - 6 q^{12} - 5 q^{13} + 12 q^{14} + 5 q^{17} + 8 q^{19} + 5 q^{21} + 8 q^{22} + 7 q^{23} - 5 q^{27} - 14 q^{28} + 8 q^{29} + 12 q^{31} + 20 q^{32}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9 + 5 * q^11 - 6 * q^12 - 5 * q^13 + 12 * q^14 + 5 * q^17 + 8 * q^19 + 5 * q^21 + 8 * q^22 + 7 * q^23 - 5 * q^27 - 14 * q^28 + 8 * q^29 + 12 * q^31 + 20 * q^32 - 5 * q^33 + 20 * q^34 + 6 * q^36 - 17 * q^37 + 24 * q^38 + 5 * q^39 + 5 * q^41 - 12 * q^42 - 12 * q^43 + 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 - 5 * q^51 - 6 * q^52 + 13 * q^53 - 8 * q^57 + 8 * q^59 + 13 * q^61 + 40 * q^62 - 5 * q^63 - 16 * q^64 - 8 * q^66 - 28 * q^67 + 14 * q^68 - 7 * q^69 + 5 * q^71 + 14 * q^73 + 12 * q^74 + 35 * q^77 + q^79 + 5 * q^81 - 16 * q^82 + 6 * q^83 + 14 * q^84 - 8 * q^87 + 36 * q^88 + 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^93 - 12 * q^94 - 20 * q^96 + 13 * q^97 - 28 * q^98 + 5 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.329386	0.232911	0.116456	−	0.993196i	$-0.462847\pi$
$2$	0.329386	0.232911	0.116456	+	0.993196i	$0.462847\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.89150	−0.945752
$5$	0	0
$5$	0	0
$6$	−0.329386	−0.134471
$7$	−3.70203	−1.39924	−0.699618	−	0.714517i	$-0.746646\pi$
$7$	−3.70203	−1.39924	−0.699618	+	0.714517i	$0.746646\pi$
$8$	−1.28181	−0.453187
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.31322	−0.998974	−0.499487	−	0.866321i	$-0.666478\pi$
$11$	−3.31322	−0.998974	−0.499487	+	0.866321i	$0.666478\pi$
$12$	1.89150	0.546030
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.21940	−0.325897
$15$	0	0
$16$	3.36080	0.840200
$17$	4.36080	1.05765	0.528825	−	0.848731i	$-0.322633\pi$
$17$	4.36080	1.05765	0.528825	+	0.848731i	$0.322633\pi$
$18$	0.329386	0.0776370
$19$	5.21940	1.19741	0.598706	−	0.800969i	$-0.295682\pi$
$19$	5.21940	1.19741	0.598706	+	0.800969i	$0.295682\pi$
$20$	0	0
$21$	3.70203	0.807849
$22$	−1.09133	−0.232672
$23$	−4.92143	−1.02619	−0.513094	−	0.858332i	$-0.671501\pi$
$23$	−4.92143	−1.02619	−0.513094	+	0.858332i	$0.671501\pi$
$24$	1.28181	0.261648
$25$	0	0
$26$	−0.329386	−0.0645979
$27$	−1.00000	−0.192450
$28$	7.00241	1.32333
$29$	7.78301	1.44527	0.722634	−	0.691231i	$-0.242931\pi$
$29$	7.78301	1.44527	0.722634	+	0.691231i	$0.242931\pi$
$30$	0	0
$31$	0.0981475	0.0176278	0.00881390	−	0.999961i	$-0.497194\pi$
$31$	0.0981475	0.0176278	0.00881390	+	0.999961i	$0.497194\pi$
$32$	3.67061	0.648879
$33$	3.31322	0.576758
$34$	1.43639	0.246338
$35$	0	0
$36$	−1.89150	−0.315251
$37$	2.92143	0.480279	0.240140	−	0.970738i	$-0.422807\pi$
$37$	2.92143	0.480279	0.240140	+	0.970738i	$0.422807\pi$
$38$	1.71920	0.278890
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.749608	−0.117069	−0.0585345	−	0.998285i	$-0.518643\pi$
$41$	−0.749608	−0.117069	−0.0585345	+	0.998285i	$0.518643\pi$
$42$	1.21940	0.188157
$43$	3.78301	0.576904	0.288452	−	0.957494i	$-0.406859\pi$
$43$	3.78301	0.576904	0.288452	+	0.957494i	$0.406859\pi$
$44$	6.26698	0.944782
$45$	0	0
$46$	−1.62105	−0.239010
$47$	−5.67402	−0.827641	−0.413821	−	0.910358i	$-0.635806\pi$
$47$	−5.67402	−0.827641	−0.413821	+	0.910358i	$0.635806\pi$
$48$	−3.36080	−0.485090
$49$	6.70502	0.957860
$50$	0	0
$51$	−4.36080	−0.610634
$52$	1.89150	0.262305
$53$	2.19982	0.302169	0.151085	−	0.988521i	$-0.451723\pi$
$53$	2.19982	0.302169	0.151085	+	0.988521i	$0.451723\pi$
$54$	−0.329386	−0.0448237
$55$	0	0
$56$	4.74529	0.634116
$57$	−5.21940	−0.691326
$58$	2.56361	0.336619
$59$	0.108987	0.0141890	0.00709448	−	0.999975i	$-0.497742\pi$
$59$	0.108987	0.0141890	0.00709448	+	0.999975i	$0.497742\pi$
$60$	0	0
$61$	12.1438	1.55486	0.777428	−	0.628972i	$-0.216524\pi$
$61$	12.1438	1.55486	0.777428	+	0.628972i	$0.216524\pi$
$62$	0.0323284	0.00410571
$63$	−3.70203	−0.466412
$64$	−5.51255	−0.689069
$65$	0	0
$66$	1.09133	0.134333
$67$	−12.4418	−1.52001	−0.760003	−	0.649920i	$-0.774802\pi$
$67$	−12.4418	−1.52001	−0.760003	+	0.649920i	$0.774802\pi$
$68$	−8.24848	−1.00027
$69$	4.92143	0.592470
$70$	0	0
$71$	12.0348	1.42827	0.714135	−	0.700008i	$-0.246820\pi$
$71$	12.0348	1.42827	0.714135	+	0.700008i	$0.246820\pi$
$72$	−1.28181	−0.151062
$73$	−9.56602	−1.11962	−0.559809	−	0.828622i	$-0.689125\pi$
$73$	−9.56602	−1.11962	−0.559809	+	0.828622i	$0.689125\pi$
$74$	0.962276	0.111862
$75$	0	0
$76$	−9.87251	−1.13245
$77$	12.2656	1.39780
$78$	0.329386	0.0372956
$79$	14.6093	1.64367	0.821836	−	0.569724i	$-0.192950\pi$
$79$	14.6093	1.64367	0.821836	+	0.569724i	$0.192950\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.246910	−0.0272667
$83$	16.7001	1.83308	0.916538	−	0.399948i	$-0.130972\pi$
$83$	16.7001	1.83308	0.916538	+	0.399948i	$0.130972\pi$
$84$	−7.00241	−0.764025
$85$	0	0
$86$	1.24607	0.134367
$87$	−7.78301	−0.834426
$88$	4.24691	0.452722
$89$	−3.59403	−0.380966	−0.190483	−	0.981690i	$-0.561005\pi$
$89$	−3.59403	−0.380966	−0.190483	+	0.981690i	$0.561005\pi$
$90$	0	0
$91$	3.70203	0.388078
$92$	9.30890	0.970520
$93$	−0.0981475	−0.0101774
$94$	−1.86894	−0.192767
$95$	0	0
$96$	−3.67061	−0.374630
$97$	4.10467	0.416766	0.208383	−	0.978047i	$-0.433180\pi$
$97$	4.10467	0.416766	0.208383	+	0.978047i	$0.433180\pi$
$98$	2.20854	0.223096
$99$	−3.31322	−0.332991
$100$	0	0
$101$	2.77761	0.276383	0.138191	−	0.990406i	$-0.455871\pi$
$101$	2.77761	0.276383	0.138191	+	0.990406i	$0.455871\pi$
$102$	−1.43639	−0.142223
$103$	−16.5046	−1.62625	−0.813124	−	0.582091i	$-0.802235\pi$
$103$	−16.5046	−1.62625	−0.813124	+	0.582091i	$0.802235\pi$
$104$	1.28181	0.125692
$105$	0	0
$106$	0.724591	0.0703785
$107$	1.04326	0.100855	0.0504277	−	0.998728i	$-0.483942\pi$
$107$	1.04326	0.100855	0.0504277	+	0.998728i	$0.483942\pi$
$108$	1.89150	0.182010
$109$	−11.0652	−1.05986	−0.529929	−	0.848042i	$-0.677781\pi$
$109$	−11.0652	−1.05986	−0.529929	+	0.848042i	$0.677781\pi$
$110$	0	0
$111$	−2.92143	−0.277289
$112$	−12.4418	−1.17564
$113$	15.9647	1.50183	0.750915	−	0.660398i	$-0.229613\pi$
$113$	15.9647	1.50183	0.750915	+	0.660398i	$0.229613\pi$
$114$	−1.71920	−0.161017
$115$	0	0
$116$	−14.7216	−1.36687
$117$	−1.00000	−0.0924500
$118$	0.0358989	0.00330476
$119$	−16.1438	−1.47990
$120$	0	0
$121$	−0.0225619	−0.00205108
$122$	4.00000	0.362143
$123$	0.749608	0.0675899
$124$	−0.185646	−0.0166715
$125$	0	0
$126$	−1.21940	−0.108632
$127$	18.9701	1.68332	0.841661	−	0.540006i	$-0.181578\pi$
$127$	18.9701	1.68332	0.841661	+	0.540006i	$0.181578\pi$
$128$	−9.15699	−0.809371
$129$	−3.78301	−0.333075
$130$	0	0
$131$	−0.539929	−0.0471738	−0.0235869	−	0.999722i	$-0.507509\pi$
$131$	−0.539929	−0.0471738	−0.0235869	+	0.999722i	$0.507509\pi$
$132$	−6.26698	−0.545470
$133$	−19.3224	−1.67546
$134$	−4.09815	−0.354026
$135$	0	0
$136$	−5.58970	−0.479313
$137$	12.2376	1.04553	0.522766	−	0.852476i	$-0.324900\pi$
$137$	12.2376	1.04553	0.522766	+	0.852476i	$0.324900\pi$
$138$	1.62105	0.137993
$139$	−9.70502	−0.823169	−0.411584	−	0.911372i	$-0.635024\pi$
$139$	−9.70502	−0.823169	−0.411584	+	0.911372i	$0.635024\pi$
$140$	0	0
$141$	5.67402	0.477839
$142$	3.96410	0.332660
$143$	3.31322	0.277066
$144$	3.36080	0.280067
$145$	0	0
$146$	−3.15091	−0.260771
$147$	−6.70502	−0.553021
$148$	−5.52589	−0.454225
$149$	5.28954	0.433336	0.216668	−	0.976245i	$-0.430481\pi$
$149$	5.28954	0.433336	0.216668	+	0.976245i	$0.430481\pi$
$150$	0	0
$151$	−16.3858	−1.33345	−0.666727	−	0.745302i	$-0.732305\pi$
$151$	−16.3858	−1.33345	−0.666727	+	0.745302i	$0.732305\pi$
$152$	−6.69026	−0.542652
$153$	4.36080	0.352550
$154$	4.04013	0.325563
$155$	0	0
$156$	−1.89150	−0.151442
$157$	−6.44477	−0.514349	−0.257174	−	0.966365i	$-0.582791\pi$
$157$	−6.44477	−0.514349	−0.257174	+	0.966365i	$0.582791\pi$
$158$	4.81209	0.382829
$159$	−2.19982	−0.174457
$160$	0	0
$161$	18.2193	1.43588
$162$	0.329386	0.0258790
$163$	12.7673	1.00001	0.500005	−	0.866023i	$-0.333332\pi$
$163$	12.7673	1.00001	0.500005	+	0.866023i	$0.333332\pi$
$164$	1.41789	0.110718
$165$	0	0
$166$	5.50078	0.426943
$167$	3.98716	0.308535	0.154268	−	0.988029i	$-0.450698\pi$
$167$	3.98716	0.308535	0.154268	+	0.988029i	$0.450698\pi$
$168$	−4.74529	−0.366107
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.21940	0.399137
$172$	−7.15558	−0.545608
$173$	14.8836	1.13158	0.565788	−	0.824551i	$-0.308572\pi$
$173$	14.8836	1.13158	0.565788	+	0.824551i	$0.308572\pi$
$174$	−2.56361	−0.194347
$175$	0	0
$176$	−11.1351	−0.839338
$177$	−0.108987	−0.00819199
$178$	−1.18382	−0.0887312
$179$	4.72160	0.352909	0.176455	−	0.984309i	$-0.443537\pi$
$179$	4.72160	0.352909	0.176455	+	0.984309i	$0.443537\pi$
$180$	0	0
$181$	−11.4614	−0.851916	−0.425958	−	0.904743i	$-0.640063\pi$
$181$	−11.4614	−0.851916	−0.425958	+	0.904743i	$0.640063\pi$
$182$	1.21940	0.0903877
$183$	−12.1438	−0.897696
$184$	6.30832	0.465055
$185$	0	0
$186$	−0.0323284	−0.00237043
$187$	−14.4483	−1.05656
$188$	10.7324	0.782744
$189$	3.70203	0.269283
$190$	0	0
$191$	−16.9994	−1.23003	−0.615017	−	0.788514i	$-0.710851\pi$
$191$	−16.9994	−1.23003	−0.615017	+	0.788514i	$0.710851\pi$
$192$	5.51255	0.397834
$193$	−8.64459	−0.622252	−0.311126	−	0.950369i	$-0.600706\pi$
$193$	−8.64459	−0.622252	−0.311126	+	0.950369i	$0.600706\pi$
$194$	1.35202	0.0970693
$195$	0	0
$196$	−12.6826	−0.905898
$197$	−19.4540	−1.38604	−0.693022	−	0.720917i	$-0.743721\pi$
$197$	−19.4540	−1.38604	−0.693022	+	0.720917i	$0.743721\pi$
$198$	−1.09133	−0.0775573
$199$	−18.2218	−1.29171	−0.645855	−	0.763460i	$-0.723499\pi$
$199$	−18.2218	−1.29171	−0.645855	+	0.763460i	$0.723499\pi$
$200$	0	0
$201$	12.4418	0.877576
$202$	0.914907	0.0643726
$203$	−28.8129	−2.02227
$204$	8.24848	0.577509
$205$	0	0
$206$	−5.43639	−0.378771
$207$	−4.92143	−0.342063
$208$	−3.36080	−0.233030
$209$	−17.2930	−1.19618
$210$	0	0
$211$	6.41810	0.441840	0.220920	−	0.975292i	$-0.429094\pi$
$211$	6.41810	0.441840	0.220920	+	0.975292i	$0.429094\pi$
$212$	−4.16098	−0.285777
$213$	−12.0348	−0.824612
$214$	0.343634	0.0234903
$215$	0	0
$216$	1.28181	0.0872159
$217$	−0.363345	−0.0246654
$218$	−3.64473	−0.246852
$219$	9.56602	0.646412
$220$	0	0
$221$	−4.36080	−0.293339
$222$	−0.962276	−0.0645838
$223$	2.54292	0.170286	0.0851432	−	0.996369i	$-0.472865\pi$
$223$	2.54292	0.170286	0.0851432	+	0.996369i	$0.472865\pi$
$224$	−13.5887	−0.907935
$225$	0	0
$226$	5.25854	0.349793
$227$	27.9311	1.85386	0.926928	−	0.375240i	$-0.122440\pi$
$227$	27.9311	1.85386	0.926928	+	0.375240i	$0.122440\pi$
$228$	9.87251	0.653823
$229$	2.58730	0.170973	0.0854867	−	0.996339i	$-0.472755\pi$
$229$	2.58730	0.170973	0.0854867	+	0.996339i	$0.472755\pi$
$230$	0	0
$231$	−12.2656	−0.807020
$232$	−9.97632	−0.654977
$233$	23.2651	1.52414	0.762072	−	0.647492i	$-0.224182\pi$
$233$	23.2651	1.52414	0.762072	+	0.647492i	$0.224182\pi$
$234$	−0.329386	−0.0215326
$235$	0	0
$236$	−0.206150	−0.0134192
$237$	−14.6093	−0.948975
$238$	−5.31754	−0.344685
$239$	15.6008	1.00913	0.504567	−	0.863372i	$-0.331652\pi$
$239$	15.6008	1.00913	0.504567	+	0.863372i	$0.331652\pi$
$240$	0	0
$241$	26.8689	1.73078	0.865390	−	0.501098i	$-0.167071\pi$
$241$	26.8689	1.73078	0.865390	+	0.501098i	$0.167071\pi$
$242$	−0.00743158	−0.000477720 0
$243$	−1.00000	−0.0641500
$244$	−22.9701	−1.47051
$245$	0	0
$246$	0.246910	0.0157424
$247$	−5.21940	−0.332102
$248$	−0.125806	−0.00798869
$249$	−16.7001	−1.05833
$250$	0	0
$251$	−2.90426	−0.183315	−0.0916576	−	0.995791i	$-0.529217\pi$
$251$	−2.90426	−0.183315	−0.0916576	+	0.995791i	$0.529217\pi$
$252$	7.00241	0.441110
$253$	16.3058	1.02514
$254$	6.24848	0.392064
$255$	0	0
$256$	8.00892	0.500558
$257$	28.0048	1.74689	0.873446	−	0.486921i	$-0.161880\pi$
$257$	28.0048	1.74689	0.873446	+	0.486921i	$0.161880\pi$
$258$	−1.24607	−0.0775769
$259$	−10.8152	−0.672024
$260$	0	0
$261$	7.78301	0.481756
$262$	−0.177845	−0.0109873
$263$	−2.74828	−0.169466	−0.0847330	−	0.996404i	$-0.527004\pi$
$263$	−2.74828	−0.169466	−0.0847330	+	0.996404i	$0.527004\pi$
$264$	−4.24691	−0.261379
$265$	0	0
$266$	−6.36451	−0.390233
$267$	3.59403	0.219951
$268$	23.5337	1.43755
$269$	−27.1224	−1.65368	−0.826841	−	0.562435i	$-0.809865\pi$
$269$	−27.1224	−1.65368	−0.826841	+	0.562435i	$0.809865\pi$
$270$	0	0
$271$	19.6555	1.19399	0.596994	−	0.802246i	$-0.296362\pi$
$271$	19.6555	1.19399	0.596994	+	0.802246i	$0.296362\pi$
$272$	14.6558	0.888637
$273$	−3.70203	−0.224057
$274$	4.03090	0.243516
$275$	0	0
$276$	−9.30890	−0.560330
$277$	15.1664	0.911259	0.455630	−	0.890170i	$-0.349414\pi$
$277$	15.1664	0.911259	0.455630	+	0.890170i	$0.349414\pi$
$278$	−3.19670	−0.191725
$279$	0.0981475	0.00587593
$280$	0	0
$281$	3.97552	0.237160	0.118580	−	0.992945i	$-0.462166\pi$
$281$	3.97552	0.237160	0.118580	+	0.992945i	$0.462166\pi$
$282$	1.86894	0.111294
$283$	0.768971	0.0457106	0.0228553	−	0.999739i	$-0.492724\pi$
$283$	0.768971	0.0457106	0.0228553	+	0.999739i	$0.492724\pi$
$284$	−22.7639	−1.35079
$285$	0	0
$286$	1.09133	0.0645316
$287$	2.77507	0.163807
$288$	3.67061	0.216293
$289$	2.01658	0.118623
$290$	0	0
$291$	−4.10467	−0.240620
$292$	18.0942	1.05888
$293$	−7.85369	−0.458817	−0.229409	−	0.973330i	$-0.573679\pi$
$293$	−7.85369	−0.458817	−0.229409	+	0.973330i	$0.573679\pi$
$294$	−2.20854	−0.128805
$295$	0	0
$296$	−3.74470	−0.217656
$297$	3.31322	0.192253
$298$	1.74230	0.100929
$299$	4.92143	0.284613
$300$	0	0
$301$	−14.0048	−0.807224
$302$	−5.39724	−0.310576
$303$	−2.77761	−0.159570
$304$	17.5414	1.00607
$305$	0	0
$306$	1.43639	0.0821127
$307$	−0.893913	−0.0510183	−0.0255092	−	0.999675i	$-0.508121\pi$
$307$	−0.893913	−0.0510183	−0.0255092	+	0.999675i	$0.508121\pi$
$308$	−23.2005	−1.32197
$309$	16.5046	0.938915
$310$	0	0
$311$	34.4340	1.95257	0.976286	−	0.216486i	$-0.0694594\pi$
$311$	34.4340	1.95257	0.976286	+	0.216486i	$0.0694594\pi$
$312$	−1.28181	−0.0725680
$313$	20.6008	1.16442	0.582212	−	0.813037i	$-0.302187\pi$
$313$	20.6008	1.16442	0.582212	+	0.813037i	$0.302187\pi$
$314$	−2.12282	−0.119797
$315$	0	0
$316$	−27.6335	−1.55451
$317$	−13.1772	−0.740106	−0.370053	−	0.929011i	$-0.620660\pi$
$317$	−13.1772	−0.740106	−0.370053	+	0.929011i	$0.620660\pi$
$318$	−0.724591	−0.0406330
$319$	−25.7868	−1.44379
$320$	0	0
$321$	−1.04326	−0.0582289
$322$	6.00117	0.334432
$323$	22.7607	1.26644
$324$	−1.89150	−0.105084
$325$	0	0
$326$	4.20536	0.232913
$327$	11.0652	0.611909
$328$	0.960853	0.0530542
$329$	21.0054	1.15806
$330$	0	0
$331$	−19.0536	−1.04728	−0.523640	−	0.851939i	$-0.675426\pi$
$331$	−19.0536	−1.04728	−0.523640	+	0.851939i	$0.675426\pi$
$332$	−31.5883	−1.73364
$333$	2.92143	0.160093
$334$	1.31331	0.0718613
$335$	0	0
$336$	12.4418	0.678755
$337$	−11.6525	−0.634754	−0.317377	−	0.948299i	$-0.602802\pi$
$337$	−11.6525	−0.634754	−0.317377	+	0.948299i	$0.602802\pi$
$338$	0.329386	0.0179162
$339$	−15.9647	−0.867083
$340$	0	0
$341$	−0.325184	−0.0176097
$342$	1.71920	0.0929634
$343$	1.09203	0.0589641
$344$	−4.84909	−0.261445
$345$	0	0
$346$	4.90244	0.263557
$347$	18.4680	0.991415	0.495707	−	0.868490i	$-0.334909\pi$
$347$	18.4680	0.991415	0.495707	+	0.868490i	$0.334909\pi$
$348$	14.7216	0.789161
$349$	18.9309	1.01335	0.506675	−	0.862137i	$-0.330874\pi$
$349$	18.9309	1.01335	0.506675	+	0.862137i	$0.330874\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−12.1616	−0.648213
$353$	−21.9048	−1.16587	−0.582937	−	0.812517i	$-0.698097\pi$
$353$	−21.9048	−1.16587	−0.582937	+	0.812517i	$0.698097\pi$
$354$	−0.0358989	−0.00190801
$355$	0	0
$356$	6.79812	0.360300
$357$	16.1438	0.854421
$358$	1.55523	0.0821964
$359$	23.5630	1.24361	0.621803	−	0.783173i	$-0.286400\pi$
$359$	23.5630	1.24361	0.621803	+	0.783173i	$0.286400\pi$
$360$	0	0
$361$	8.24210	0.433795
$362$	−3.77521	−0.198421
$363$	0.0225619	0.00118419
$364$	−7.00241	−0.367026
$365$	0	0
$366$	−4.00000	−0.209083
$367$	−19.9450	−1.04112	−0.520560	−	0.853825i	$-0.674277\pi$
$367$	−19.9450	−1.04112	−0.520560	+	0.853825i	$0.674277\pi$
$368$	−16.5399	−0.862203
$369$	−0.749608	−0.0390230
$370$	0	0
$371$	−8.14381	−0.422806
$372$	0.185646	0.00962532
$373$	−4.48616	−0.232285	−0.116142	−	0.993233i	$-0.537053\pi$
$373$	−4.48616	−0.232285	−0.116142	+	0.993233i	$0.537053\pi$
$374$	−4.75907	−0.246085
$375$	0	0
$376$	7.27300	0.375076
$377$	−7.78301	−0.400845
$378$	1.21940	0.0627190
$379$	4.65976	0.239356	0.119678	−	0.992813i	$-0.461814\pi$
$379$	4.65976	0.239356	0.119678	+	0.992813i	$0.461814\pi$
$380$	0	0
$381$	−18.9701	−0.971867
$382$	−5.59937	−0.286489
$383$	−4.70011	−0.240165	−0.120082	−	0.992764i	$-0.538316\pi$
$383$	−4.70011	−0.240165	−0.120082	+	0.992764i	$0.538316\pi$
$384$	9.15699	0.467290
$385$	0	0
$386$	−2.84741	−0.144929
$387$	3.78301	0.192301
$388$	−7.76400	−0.394157
$389$	14.6128	0.740899	0.370449	−	0.928853i	$-0.379204\pi$
$389$	14.6128	0.740899	0.370449	+	0.928853i	$0.379204\pi$
$390$	0	0
$391$	−21.4614	−1.08535
$392$	−8.59454	−0.434090
$393$	0.539929	0.0272358
$394$	−6.40789	−0.322825
$395$	0	0
$396$	6.26698	0.314927
$397$	22.2911	1.11876	0.559380	−	0.828911i	$-0.311039\pi$
$397$	22.2911	1.11876	0.559380	+	0.828911i	$0.311039\pi$
$398$	−6.00200	−0.300853
$399$	19.3224	0.967328
$400$	0	0
$401$	3.61021	0.180285	0.0901426	−	0.995929i	$-0.471268\pi$
$401$	3.61021	0.180285	0.0901426	+	0.995929i	$0.471268\pi$
$402$	4.09815	0.204397
$403$	−0.0981475	−0.00488907
$404$	−5.25387	−0.261390
$405$	0	0
$406$	−9.49057	−0.471009
$407$	−9.67933	−0.479787
$408$	5.58970	0.276732
$409$	12.6911	0.627535	0.313767	−	0.949500i	$-0.398409\pi$
$409$	12.6911	0.627535	0.313767	+	0.949500i	$0.398409\pi$
$410$	0	0
$411$	−12.2376	−0.603638
$412$	31.2186	1.53803
$413$	−0.403475	−0.0198537
$414$	−1.62105	−0.0796702
$415$	0	0
$416$	−3.67061	−0.179967
$417$	9.70502	0.475257
$418$	−5.69608	−0.278604
$419$	2.66078	0.129987	0.0649937	−	0.997886i	$-0.479297\pi$
$419$	2.66078	0.129987	0.0649937	+	0.997886i	$0.479297\pi$
$420$	0	0
$421$	−20.0305	−0.976227	−0.488113	−	0.872780i	$-0.662315\pi$
$421$	−20.0305	−0.976227	−0.488113	+	0.872780i	$0.662315\pi$
$422$	2.11403	0.102909
$423$	−5.67402	−0.275880
$424$	−2.81975	−0.136939
$425$	0	0
$426$	−3.96410	−0.192061
$427$	−44.9567	−2.17561
$428$	−1.97333	−0.0953843
$429$	−3.31322	−0.159964
$430$	0	0
$431$	−2.79742	−0.134747	−0.0673736	−	0.997728i	$-0.521462\pi$
$431$	−2.79742	−0.134747	−0.0673736	+	0.997728i	$0.521462\pi$
$432$	−3.36080	−0.161697
$433$	−9.56602	−0.459714	−0.229857	−	0.973224i	$-0.573826\pi$
$433$	−9.56602	−0.459714	−0.229857	+	0.973224i	$0.573826\pi$
$434$	−0.119681	−0.00574485
$435$	0	0
$436$	20.9299	1.00236
$437$	−25.6869	−1.22877
$438$	3.15091	0.150556
$439$	−7.45654	−0.355881	−0.177941	−	0.984041i	$-0.556944\pi$
$439$	−7.45654	−0.355881	−0.177941	+	0.984041i	$0.556944\pi$
$440$	0	0
$441$	6.70502	0.319287
$442$	−1.43639	−0.0683219
$443$	7.92201	0.376386	0.188193	−	0.982132i	$-0.439737\pi$
$443$	7.92201	0.376386	0.188193	+	0.982132i	$0.439737\pi$
$444$	5.52589	0.262247
$445$	0	0
$446$	0.837602	0.0396616
$447$	−5.28954	−0.250187
$448$	20.4076	0.964170
$449$	38.7241	1.82750	0.913752	−	0.406273i	$-0.133172\pi$
$449$	38.7241	1.82750	0.913752	+	0.406273i	$0.133172\pi$
$450$	0	0
$451$	2.48362	0.116949
$452$	−30.1973	−1.42036
$453$	16.3858	0.769871
$454$	9.20012	0.431783
$455$	0	0
$456$	6.69026	0.313300
$457$	−12.8322	−0.600267	−0.300134	−	0.953897i	$-0.597031\pi$
$457$	−12.8322	−0.600267	−0.300134	+	0.953897i	$0.597031\pi$
$458$	0.852220	0.0398216
$459$	−4.36080	−0.203545
$460$	0	0
$461$	−31.1178	−1.44930	−0.724649	−	0.689118i	$-0.757998\pi$
$461$	−31.1178	−1.44930	−0.724649	+	0.689118i	$0.757998\pi$
$462$	−4.04013	−0.187964
$463$	−10.2588	−0.476768	−0.238384	−	0.971171i	$-0.576618\pi$
$463$	−10.2588	−0.476768	−0.238384	+	0.971171i	$0.576618\pi$
$464$	26.1571	1.21431
$465$	0	0
$466$	7.66318	0.354990
$467$	12.3315	0.570632	0.285316	−	0.958434i	$-0.407901\pi$
$467$	12.3315	0.570632	0.285316	+	0.958434i	$0.407901\pi$
$468$	1.89150	0.0874348
$469$	46.0598	2.12685
$470$	0	0
$471$	6.44477	0.296959
$472$	−0.139701	−0.00643025
$473$	−12.5340	−0.576312
$474$	−4.81209	−0.221027
$475$	0	0
$476$	30.5361	1.39962
$477$	2.19982	0.100723
$478$	5.13870	0.235039
$479$	−8.21141	−0.375189	−0.187594	−	0.982247i	$-0.560069\pi$
$479$	−8.21141	−0.375189	−0.187594	+	0.982247i	$0.560069\pi$
$480$	0	0
$481$	−2.92143	−0.133206
$482$	8.85025	0.403118
$483$	−18.2193	−0.829005
$484$	0.0426760	0.00193982
$485$	0	0
$486$	−0.329386	−0.0149412
$487$	23.4860	1.06425	0.532127	−	0.846665i	$-0.321393\pi$
$487$	23.4860	1.06425	0.532127	+	0.846665i	$0.321393\pi$
$488$	−15.5660	−0.704641
$489$	−12.7673	−0.577356
$490$	0	0
$491$	1.21739	0.0549401	0.0274701	−	0.999623i	$-0.491255\pi$
$491$	1.21739	0.0549401	0.0274701	+	0.999623i	$0.491255\pi$
$492$	−1.41789	−0.0639233
$493$	33.9402	1.52859
$494$	−1.71920	−0.0773503
$495$	0	0
$496$	0.329854	0.0148109
$497$	−44.5533	−1.99849
$498$	−5.50078	−0.246496
$499$	−1.60334	−0.0717755	−0.0358877	−	0.999356i	$-0.511426\pi$
$499$	−1.60334	−0.0717755	−0.0358877	+	0.999356i	$0.511426\pi$
$500$	0	0
$501$	−3.98716	−0.178133
$502$	−0.956622	−0.0426961
$503$	27.4355	1.22329	0.611645	−	0.791132i	$-0.290508\pi$
$503$	27.4355	1.22329	0.611645	+	0.791132i	$0.290508\pi$
$504$	4.74529	0.211372
$505$	0	0
$506$	5.37089	0.238765
$507$	−1.00000	−0.0444116
$508$	−35.8820	−1.59201
$509$	11.4124	0.505844	0.252922	−	0.967487i	$-0.418608\pi$
$509$	11.4124	0.505844	0.252922	+	0.967487i	$0.418608\pi$
$510$	0	0
$511$	35.4137	1.56661
$512$	20.9520	0.925956
$513$	−5.21940	−0.230442
$514$	9.22439	0.406870
$515$	0	0
$516$	7.15558	0.315007
$517$	18.7993	0.826792
$518$	−3.56237	−0.156522
$519$	−14.8836	−0.653316
$520$	0	0
$521$	30.6274	1.34181	0.670906	−	0.741542i	$-0.265905\pi$
$521$	30.6274	1.34181	0.670906	+	0.741542i	$0.265905\pi$
$522$	2.56361	0.112206
$523$	24.9956	1.09298	0.546490	−	0.837465i	$-0.315964\pi$
$523$	24.9956	1.09298	0.546490	+	0.837465i	$0.315964\pi$
$524$	1.02128	0.0446148
$525$	0	0
$526$	−0.905243	−0.0394705
$527$	0.428002	0.0186440
$528$	11.1351	0.484592
$529$	1.22042	0.0530620
$530$	0	0
$531$	0.108987	0.00472965
$532$	36.5483	1.58457
$533$	0.749608	0.0324691
$534$	1.18382	0.0512290
$535$	0	0
$536$	15.9480	0.688847
$537$	−4.72160	−0.203752
$538$	−8.93374	−0.385161
$539$	−22.2152	−0.956877
$540$	0	0
$541$	−0.131461	−0.00565193	−0.00282596	−	0.999996i	$-0.500900\pi$
$541$	−0.131461	−0.00565193	−0.00282596	+	0.999996i	$0.500900\pi$
$542$	6.47425	0.278093
$543$	11.4614	0.491854
$544$	16.0068	0.686287
$545$	0	0
$546$	−1.21940	−0.0521853
$547$	21.5955	0.923358	0.461679	−	0.887047i	$-0.347247\pi$
$547$	21.5955	0.923358	0.461679	+	0.887047i	$0.347247\pi$
$548$	−23.1475	−0.988814
$549$	12.1438	0.518285
$550$	0	0
$551$	40.6226	1.73058
$552$	−6.30832	−0.268500
$553$	−54.0840	−2.29988
$554$	4.99559	0.212242
$555$	0	0
$556$	18.3571	0.778514
$557$	−40.1776	−1.70238	−0.851190	−	0.524858i	$-0.824119\pi$
$557$	−40.1776	−1.70238	−0.851190	+	0.524858i	$0.824119\pi$
$558$	0.0323284	0.00136857
$559$	−3.78301	−0.160004
$560$	0	0
$561$	14.4483	0.610008
$562$	1.30948	0.0552371
$563$	3.57240	0.150559	0.0752793	−	0.997162i	$-0.476015\pi$
$563$	3.57240	0.150559	0.0752793	+	0.997162i	$0.476015\pi$
$564$	−10.7324	−0.451917
$565$	0	0
$566$	0.253288	0.0106465
$567$	−3.70203	−0.155471
$568$	−15.4263	−0.647274
$569$	0.744448	0.0312089	0.0156044	−	0.999878i	$-0.495033\pi$
$569$	0.744448	0.0312089	0.0156044	+	0.999878i	$0.495033\pi$
$570$	0	0
$571$	17.4234	0.729145	0.364573	−	0.931175i	$-0.381215\pi$
$571$	17.4234	0.729145	0.364573	+	0.931175i	$0.381215\pi$
$572$	−6.26698	−0.262035
$573$	16.9994	0.710161
$574$	0.914069	0.0381525
$575$	0	0
$576$	−5.51255	−0.229690
$577$	40.6582	1.69262	0.846312	−	0.532687i	$-0.178818\pi$
$577$	40.6582	1.69262	0.846312	+	0.532687i	$0.178818\pi$
$578$	0.664234	0.0276285
$579$	8.64459	0.359257
$580$	0	0
$581$	−61.8243	−2.56490
$582$	−1.35202	−0.0560430
$583$	−7.28850	−0.301859
$584$	12.2618	0.507396
$585$	0	0
$586$	−2.58689	−0.106864
$587$	5.24004	0.216280	0.108140	−	0.994136i	$-0.465511\pi$
$587$	5.24004	0.216280	0.108140	+	0.994136i	$0.465511\pi$
$588$	12.6826	0.523021
$589$	0.512270	0.0211077
$590$	0	0
$591$	19.4540	0.800232
$592$	9.81833	0.403531
$593$	4.71557	0.193645	0.0968227	−	0.995302i	$-0.469132\pi$
$593$	4.71557	0.193645	0.0968227	+	0.995302i	$0.469132\pi$
$594$	1.09133	0.0447778
$595$	0	0
$596$	−10.0052	−0.409828
$597$	18.2218	0.745769
$598$	1.62105	0.0662896
$599$	−17.6268	−0.720213	−0.360107	−	0.932911i	$-0.617260\pi$
$599$	−17.6268	−0.720213	−0.360107	+	0.932911i	$0.617260\pi$
$600$	0	0
$601$	−4.38747	−0.178969	−0.0894844	−	0.995988i	$-0.528522\pi$
$601$	−4.38747	−0.178969	−0.0894844	+	0.995988i	$0.528522\pi$
$602$	−4.61299	−0.188011
$603$	−12.4418	−0.506669
$604$	30.9938	1.26112
$605$	0	0
$606$	−0.914907	−0.0371656
$607$	25.4225	1.03187	0.515934	−	0.856628i	$-0.327445\pi$
$607$	25.4225	1.03187	0.515934	+	0.856628i	$0.327445\pi$
$608$	19.1584	0.776975
$609$	28.8129	1.16756
$610$	0	0
$611$	5.67402	0.229546
$612$	−8.24848	−0.333425
$613$	−15.8464	−0.640029	−0.320015	−	0.947413i	$-0.603688\pi$
$613$	−15.8464	−0.640029	−0.320015	+	0.947413i	$0.603688\pi$
$614$	−0.294442	−0.0118827
$615$	0	0
$616$	−15.7222	−0.633465
$617$	−9.43461	−0.379823	−0.189912	−	0.981801i	$-0.560820\pi$
$617$	−9.43461	−0.379823	−0.189912	+	0.981801i	$0.560820\pi$
$618$	5.43639	0.218684
$619$	−19.0899	−0.767288	−0.383644	−	0.923481i	$-0.625331\pi$
$619$	−19.0899	−0.767288	−0.383644	+	0.923481i	$0.625331\pi$
$620$	0	0
$621$	4.92143	0.197490
$622$	11.3421	0.454775
$623$	13.3052	0.533061
$624$	3.36080	0.134540
$625$	0	0
$626$	6.78560	0.271207
$627$	17.2930	0.690617
$628$	12.1903	0.486447
$629$	12.7398	0.507967
$630$	0	0
$631$	−6.58431	−0.262117	−0.131059	−	0.991375i	$-0.541838\pi$
$631$	−6.58431	−0.262117	−0.131059	+	0.991375i	$0.541838\pi$
$632$	−18.7263	−0.744891
$633$	−6.41810	−0.255096
$634$	−4.34039	−0.172379
$635$	0	0
$636$	4.16098	0.164993
$637$	−6.70502	−0.265662
$638$	−8.49382	−0.336274
$639$	12.0348	0.476090
$640$	0	0
$641$	20.5606	0.812096	0.406048	−	0.913852i	$-0.366907\pi$
$641$	20.5606	0.812096	0.406048	+	0.913852i	$0.366907\pi$
$642$	−0.343634	−0.0135622
$643$	−14.7084	−0.580043	−0.290022	−	0.957020i	$-0.593662\pi$
$643$	−14.7084	−0.580043	−0.290022	+	0.957020i	$0.593662\pi$
$644$	−34.4618	−1.35799
$645$	0	0
$646$	7.49707	0.294968
$647$	30.1145	1.18392	0.591961	−	0.805967i	$-0.298354\pi$
$647$	30.1145	1.18392	0.591961	+	0.805967i	$0.298354\pi$
$648$	−1.28181	−0.0503541
$649$	−0.361099	−0.0141744
$650$	0	0
$651$	0.363345	0.0142406
$652$	−24.1493	−0.945761
$653$	23.8527	0.933427	0.466713	−	0.884409i	$-0.345438\pi$
$653$	23.8527	0.933427	0.466713	+	0.884409i	$0.345438\pi$
$654$	3.64473	0.142520
$655$	0	0
$656$	−2.51928	−0.0983615
$657$	−9.56602	−0.373206
$658$	6.91888	0.269726
$659$	−26.4789	−1.03147	−0.515736	−	0.856747i	$-0.672482\pi$
$659$	−26.4789	−1.03147	−0.515736	+	0.856747i	$0.672482\pi$
$660$	0	0
$661$	14.7579	0.574016	0.287008	−	0.957928i	$-0.407339\pi$
$661$	14.7579	0.574016	0.287008	+	0.957928i	$0.407339\pi$
$662$	−6.27599	−0.243923
$663$	4.36080	0.169359
$664$	−21.4063	−0.830726
$665$	0	0
$666$	0.962276	0.0372874
$667$	−38.3035	−1.48312
$668$	−7.54172	−0.291798
$669$	−2.54292	−0.0983149
$670$	0	0
$671$	−40.2351	−1.55326
$672$	13.5887	0.524196
$673$	11.2784	0.434750	0.217375	−	0.976088i	$-0.430251\pi$
$673$	11.2784	0.434750	0.217375	+	0.976088i	$0.430251\pi$
$674$	−3.83818	−0.147841
$675$	0	0
$676$	−1.89150	−0.0727502
$677$	−29.1052	−1.11861	−0.559303	−	0.828963i	$-0.688931\pi$
$677$	−29.1052	−1.11861	−0.559303	+	0.828963i	$0.688931\pi$
$678$	−5.25854	−0.201953
$679$	−15.1956	−0.583153
$680$	0	0
$681$	−27.9311	−1.07032
$682$	−0.107111	−0.00410150
$683$	−1.84463	−0.0705827	−0.0352914	−	0.999377i	$-0.511236\pi$
$683$	−1.84463	−0.0705827	−0.0352914	+	0.999377i	$0.511236\pi$
$684$	−9.87251	−0.377485
$685$	0	0
$686$	0.359700	0.0137334
$687$	−2.58730	−0.0987116
$688$	12.7139	0.484715
$689$	−2.19982	−0.0838066
$690$	0	0
$691$	−30.4103	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$691$	−30.4103	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$692$	−28.1523	−1.07019
$693$	12.2656	0.465933
$694$	6.08310	0.230911
$695$	0	0
$696$	9.97632	0.378151
$697$	−3.26889	−0.123818
$698$	6.23558	0.236020
$699$	−23.2651	−0.879965
$700$	0	0
$701$	−48.8119	−1.84360	−0.921801	−	0.387664i	$-0.873282\pi$
$701$	−48.8119	−1.84360	−0.921801	+	0.387664i	$0.873282\pi$
$702$	0.329386	0.0124319
$703$	15.2481	0.575092
$704$	18.2643	0.688362
$705$	0	0
$706$	−7.21513	−0.271545
$707$	−10.2828	−0.386725
$708$	0.206150	0.00774760
$709$	6.74926	0.253474	0.126737	−	0.991936i	$-0.459550\pi$
$709$	6.74926	0.253474	0.126737	+	0.991936i	$0.459550\pi$
$710$	0	0
$711$	14.6093	0.547891
$712$	4.60685	0.172649
$713$	−0.483025	−0.0180894
$714$	5.31754	0.199004
$715$	0	0
$716$	−8.93093	−0.333765
$717$	−15.6008	−0.582624
$718$	7.76131	0.289650
$719$	0.286459	0.0106831	0.00534156	−	0.999986i	$-0.498300\pi$
$719$	0.286459	0.0106831	0.00534156	+	0.999986i	$0.498300\pi$
$720$	0	0
$721$	61.1005	2.27550
$722$	2.71483	0.101036
$723$	−26.8689	−0.999267
$724$	21.6792	0.805701
$725$	0	0
$726$	0.00743158	0.000275812 0
$727$	15.1967	0.563614	0.281807	−	0.959471i	$-0.409066\pi$
$727$	15.1967	0.563614	0.281807	+	0.959471i	$0.409066\pi$
$728$	−4.74529	−0.175872
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.4970	0.610162
$732$	22.9701	0.848998
$733$	8.90456	0.328897	0.164449	−	0.986386i	$-0.447416\pi$
$733$	8.90456	0.328897	0.164449	+	0.986386i	$0.447416\pi$
$734$	−6.56959	−0.242488
$735$	0	0
$736$	−18.0647	−0.665872
$737$	41.2224	1.51845
$738$	−0.246910	−0.00908889
$739$	−31.4422	−1.15662	−0.578310	−	0.815817i	$-0.696287\pi$
$739$	−31.4422	−1.15662	−0.578310	+	0.815817i	$0.696287\pi$
$740$	0	0
$741$	5.21940	0.191739
$742$	−2.68246	−0.0984761
$743$	−41.3314	−1.51630	−0.758150	−	0.652080i	$-0.773897\pi$
$743$	−41.3314	−1.51630	−0.758150	+	0.652080i	$0.773897\pi$
$744$	0.125806	0.00461228
$745$	0	0
$746$	−1.47768	−0.0541016
$747$	16.7001	0.611025
$748$	27.3290	0.999248
$749$	−3.86217	−0.141121
$750$	0	0
$751$	−50.8813	−1.85668	−0.928342	−	0.371726i	$-0.878766\pi$
$751$	−50.8813	−1.85668	−0.928342	+	0.371726i	$0.878766\pi$
$752$	−19.0693	−0.695384
$753$	2.90426	0.105837
$754$	−2.56361	−0.0933613
$755$	0	0
$756$	−7.00241	−0.254675
$757$	−45.8173	−1.66526	−0.832630	−	0.553830i	$-0.813166\pi$
$757$	−45.8173	−1.66526	−0.832630	+	0.553830i	$0.813166\pi$
$758$	1.53486	0.0557486
$759$	−16.3058	−0.591862
$760$	0	0
$761$	−10.9129	−0.395591	−0.197795	−	0.980243i	$-0.563378\pi$
$761$	−10.9129	−0.395591	−0.197795	+	0.980243i	$0.563378\pi$
$762$	−6.24848	−0.226358
$763$	40.9638	1.48299
$764$	32.1545	1.16331
$765$	0	0
$766$	−1.54815	−0.0559370
$767$	−0.108987	−0.00393531
$768$	−8.00892	−0.288997
$769$	28.9338	1.04338	0.521689	−	0.853136i	$-0.325302\pi$
$769$	28.9338	1.04338	0.521689	+	0.853136i	$0.325302\pi$
$770$	0	0
$771$	−28.0048	−1.00857
$772$	16.3513	0.588496
$773$	−6.42314	−0.231024	−0.115512	−	0.993306i	$-0.536851\pi$
$773$	−6.42314	−0.231024	−0.115512	+	0.993306i	$0.536851\pi$
$774$	1.24607	0.0447891
$775$	0	0
$776$	−5.26139	−0.188873
$777$	10.8152	0.387993
$778$	4.81325	0.172563
$779$	−3.91250	−0.140180
$780$	0	0
$781$	−39.8740	−1.42681
$782$	−7.06907	−0.252789
$783$	−7.78301	−0.278142
$784$	22.5342	0.804794
$785$	0	0
$786$	0.177845	0.00634352
$787$	−34.8040	−1.24063	−0.620314	−	0.784354i	$-0.712995\pi$
$787$	−34.8040	−1.24063	−0.620314	+	0.784354i	$0.712995\pi$
$788$	36.7974	1.31085
$789$	2.74828	0.0978412
$790$	0	0
$791$	−59.1017	−2.10142
$792$	4.24691	0.150907
$793$	−12.1438	−0.431239
$794$	7.34239	0.260572
$795$	0	0
$796$	34.4666	1.22164
$797$	−1.30902	−0.0463679	−0.0231839	−	0.999731i	$-0.507380\pi$
$797$	−1.30902	−0.0463679	−0.0231839	+	0.999731i	$0.507380\pi$
$798$	6.36451	0.225301
$799$	−24.7433	−0.875354
$800$	0	0
$801$	−3.59403	−0.126989
$802$	1.18915	0.0419904
$803$	31.6943	1.11847
$804$	−23.5337	−0.829969
$805$	0	0
$806$	−0.0323284	−0.00113872
$807$	27.1224	0.954754
$808$	−3.56037	−0.125253
$809$	17.2262	0.605641	0.302821	−	0.953048i	$-0.402072\pi$
$809$	17.2262	0.605641	0.302821	+	0.953048i	$0.402072\pi$
$810$	0	0
$811$	−6.97289	−0.244851	−0.122426	−	0.992478i	$-0.539067\pi$
$811$	−6.97289	−0.244851	−0.122426	+	0.992478i	$0.539067\pi$
$812$	54.4998	1.91257
$813$	−19.6555	−0.689349
$814$	−3.18823	−0.111748
$815$	0	0
$816$	−14.6558	−0.513055
$817$	19.7450	0.690791
$818$	4.18027	0.146160
$819$	3.70203	0.129359
$820$	0	0
$821$	35.4054	1.23566	0.617828	−	0.786313i	$-0.288013\pi$
$821$	35.4054	1.23566	0.617828	+	0.786313i	$0.288013\pi$
$822$	−4.03090	−0.140594
$823$	−33.3942	−1.16405	−0.582024	−	0.813172i	$-0.697739\pi$
$823$	−33.3942	−1.16405	−0.582024	+	0.813172i	$0.697739\pi$
$824$	21.1557	0.736995
$825$	0	0
$826$	−0.132899	−0.00462414
$827$	17.9643	0.624680	0.312340	−	0.949970i	$-0.398887\pi$
$827$	17.9643	0.624680	0.312340	+	0.949970i	$0.398887\pi$
$828$	9.30890	0.323507
$829$	33.6052	1.16716	0.583578	−	0.812057i	$-0.301652\pi$
$829$	33.6052	1.16716	0.583578	+	0.812057i	$0.301652\pi$
$830$	0	0
$831$	−15.1664	−0.526116
$832$	5.51255	0.191113
$833$	29.2392	1.01308
$834$	3.19670	0.110693
$835$	0	0
$836$	32.7098	1.13129
$837$	−0.0981475	−0.00339247
$838$	0.876422	0.0302755
$839$	−47.1755	−1.62868	−0.814340	−	0.580389i	$-0.802901\pi$
$839$	−47.1755	−1.62868	−0.814340	+	0.580389i	$0.802901\pi$
$840$	0	0
$841$	31.5752	1.08880
$842$	−6.59776	−0.227374
$843$	−3.97552	−0.136924
$844$	−12.1399	−0.417871
$845$	0	0
$846$	−1.86894	−0.0642556
$847$	0.0835249	0.00286995
$848$	7.39317	0.253882
$849$	−0.768971	−0.0263910
$850$	0	0
$851$	−14.3776	−0.492857
$852$	22.7639	0.779879
$853$	2.89307	0.0990569	0.0495284	−	0.998773i	$-0.484228\pi$
$853$	2.89307	0.0990569	0.0495284	+	0.998773i	$0.484228\pi$
$854$	−14.8081	−0.506723
$855$	0	0
$856$	−1.33725	−0.0457064
$857$	−25.7442	−0.879404	−0.439702	−	0.898144i	$-0.644916\pi$
$857$	−25.7442	−0.879404	−0.439702	+	0.898144i	$0.644916\pi$
$858$	−1.09133	−0.0372573
$859$	−30.0417	−1.02501	−0.512505	−	0.858684i	$-0.671282\pi$
$859$	−30.0417	−1.02501	−0.512505	+	0.858684i	$0.671282\pi$
$860$	0	0
$861$	−2.77507	−0.0945741
$862$	−0.921432	−0.0313841
$863$	−23.2852	−0.792636	−0.396318	−	0.918113i	$-0.629712\pi$
$863$	−23.2852	−0.792636	−0.396318	+	0.918113i	$0.629712\pi$
$864$	−3.67061	−0.124877
$865$	0	0
$866$	−3.15091	−0.107072
$867$	−2.01658	−0.0684868
$868$	0.687268	0.0233274
$869$	−48.4038	−1.64199
$870$	0	0
$871$	12.4418	0.421574
$872$	14.1835	0.480314
$873$	4.10467	0.138922
$874$	−8.46089	−0.286194
$875$	0	0
$876$	−18.0942	−0.611345
$877$	23.5946	0.796731	0.398366	−	0.917227i	$-0.369577\pi$
$877$	23.5946	0.796731	0.398366	+	0.917227i	$0.369577\pi$
$878$	−2.45608	−0.0828887
$879$	7.85369	0.264898
$880$	0	0
$881$	54.7266	1.84379	0.921893	−	0.387445i	$-0.126642\pi$
$881$	54.7266	1.84379	0.921893	+	0.387445i	$0.126642\pi$
$882$	2.20854	0.0743654
$883$	−2.10978	−0.0709998	−0.0354999	−	0.999370i	$-0.511302\pi$
$883$	−2.10978	−0.0709998	−0.0354999	+	0.999370i	$0.511302\pi$
$884$	8.24848	0.277426
$885$	0	0
$886$	2.60940	0.0876644
$887$	9.66363	0.324473	0.162236	−	0.986752i	$-0.448129\pi$
$887$	9.66363	0.324473	0.162236	+	0.986752i	$0.448129\pi$
$888$	3.74470	0.125664
$889$	−70.2278	−2.35536
$890$	0	0
$891$	−3.31322	−0.110997
$892$	−4.80994	−0.161049
$893$	−29.6150	−0.991027
$894$	−1.74230	−0.0582712
$895$	0	0
$896$	33.8994	1.13250
$897$	−4.92143	−0.164322
$898$	12.7552	0.425646
$899$	0.763883	0.0254769
$900$	0	0
$901$	9.59299	0.319589
$902$	0.818069	0.0272387
$903$	14.0048	0.466051
$904$	−20.4636	−0.680611
$905$	0	0
$906$	5.39724	0.179311
$907$	12.3350	0.409577	0.204788	−	0.978806i	$-0.434349\pi$
$907$	12.3350	0.409577	0.204788	+	0.978806i	$0.434349\pi$
$908$	−52.8319	−1.75329
$909$	2.77761	0.0921277
$910$	0	0
$911$	−4.24151	−0.140528	−0.0702638	−	0.997528i	$-0.522384\pi$
$911$	−4.24151	−0.140528	−0.0702638	+	0.997528i	$0.522384\pi$
$912$	−17.5414	−0.580852
$913$	−55.3312	−1.83119
$914$	−4.22676	−0.139809
$915$	0	0
$916$	−4.89389	−0.161699
$917$	1.99883	0.0660073
$918$	−1.43639	−0.0474078
$919$	−9.01009	−0.297215	−0.148608	−	0.988896i	$-0.547479\pi$
$919$	−9.01009	−0.297215	−0.148608	+	0.988896i	$0.547479\pi$
$920$	0	0
$921$	0.893913	0.0294554
$922$	−10.2498	−0.337558
$923$	−12.0348	−0.396131
$924$	23.2005	0.763241
$925$	0	0
$926$	−3.37911	−0.111045
$927$	−16.5046	−0.542083
$928$	28.5684	0.937805
$929$	17.1130	0.561458	0.280729	−	0.959787i	$-0.409424\pi$
$929$	17.1130	0.561458	0.280729	+	0.959787i	$0.409424\pi$
$930$	0	0
$931$	34.9961	1.14695
$932$	−44.0060	−1.44146
$933$	−34.4340	−1.12732
$934$	4.06181	0.132906
$935$	0	0
$936$	1.28181	0.0418972
$937$	−15.6965	−0.512782	−0.256391	−	0.966573i	$-0.582533\pi$
$937$	−15.6965	−0.512782	−0.256391	+	0.966573i	$0.582533\pi$
$938$	15.1715	0.495366
$939$	−20.6008	−0.672280
$940$	0	0
$941$	2.53477	0.0826311	0.0413156	−	0.999146i	$-0.486845\pi$
$941$	2.53477	0.0826311	0.0413156	+	0.999146i	$0.486845\pi$
$942$	2.12282	0.0691651
$943$	3.68914	0.120135
$944$	0.366285	0.0119216
$945$	0	0
$946$	−4.12851	−0.134229
$947$	4.66480	0.151586	0.0757928	−	0.997124i	$-0.475851\pi$
$947$	4.66480	0.151586	0.0757928	+	0.997124i	$0.475851\pi$
$948$	27.6335	0.897495
$949$	9.56602	0.310526
$950$	0	0
$951$	13.1772	0.427300
$952$	20.6932	0.670672
$953$	−41.4489	−1.34266	−0.671330	−	0.741158i	$-0.734277\pi$
$953$	−41.4489	−1.34266	−0.671330	+	0.741158i	$0.734277\pi$
$954$	0.724591	0.0234595
$955$	0	0
$956$	−29.5091	−0.954392
$957$	25.7868	0.833570
$958$	−2.70472	−0.0873856
$959$	−45.3041	−1.46295
$960$	0	0
$961$	−30.9904	−0.999689
$962$	−0.962276	−0.0310250
$963$	1.04326	0.0336185
$964$	−50.8227	−1.63689
$965$	0	0
$966$	−6.00117	−0.193084
$967$	−40.1592	−1.29143	−0.645716	−	0.763578i	$-0.723441\pi$
$967$	−40.1592	−1.29143	−0.645716	+	0.763578i	$0.723441\pi$
$968$	0.0289200	0.000929525 0
$969$	−22.7607	−0.731181
$970$	0	0
$971$	−3.79988	−0.121944	−0.0609719	−	0.998139i	$-0.519420\pi$
$971$	−3.79988	−0.121944	−0.0609719	+	0.998139i	$0.519420\pi$
$972$	1.89150	0.0606700
$973$	35.9283	1.15181
$974$	7.73597	0.247876
$975$	0	0
$976$	40.8129	1.30639
$977$	−53.7828	−1.72066	−0.860332	−	0.509734i	$-0.829744\pi$
$977$	−53.7828	−1.72066	−0.860332	+	0.509734i	$0.829744\pi$
$978$	−4.20536	−0.134472
$979$	11.9078	0.380575
$980$	0	0
$981$	−11.0652	−0.353286
$982$	0.400992	0.0127962
$983$	25.6007	0.816536	0.408268	−	0.912862i	$-0.366133\pi$
$983$	25.6007	0.816536	0.408268	+	0.912862i	$0.366133\pi$
$984$	−0.960853	−0.0306309
$985$	0	0
$986$	11.1794	0.356025
$987$	−21.0054	−0.668609
$988$	9.87251	0.314086
$989$	−18.6178	−0.592012
$990$	0	0
$991$	−51.0382	−1.62128	−0.810640	−	0.585544i	$-0.800881\pi$
$991$	−51.0382	−1.62128	−0.810640	+	0.585544i	$0.800881\pi$
$992$	0.360261	0.0114383
$993$	19.0536	0.604648
$994$	−14.6752	−0.465470
$995$	0	0
$996$	31.5883	1.00092
$997$	−6.08651	−0.192762	−0.0963809	−	0.995345i	$-0.530727\pi$
$997$	−6.08651	−0.192762	−0.0963809	+	0.995345i	$0.530727\pi$
$998$	−0.528118	−0.0167173
$999$	−2.92143	−0.0924298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.r.1.3		5
3.2	odd	2		2925.2.a.bl.1.3		5
5.2	odd	4		195.2.c.b.79.6	yes	10
5.3	odd	4		195.2.c.b.79.5	✓	10
5.4	even	2		975.2.a.s.1.3		5
15.2	even	4		585.2.c.c.469.5		10
15.8	even	4		585.2.c.c.469.6		10
15.14	odd	2		2925.2.a.bm.1.3		5
20.3	even	4		3120.2.l.p.1249.10		10
20.7	even	4		3120.2.l.p.1249.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.5	✓	10	5.3	odd	4
195.2.c.b.79.6	yes	10	5.2	odd	4
585.2.c.c.469.5		10	15.2	even	4
585.2.c.c.469.6		10	15.8	even	4
975.2.a.r.1.3		5	1.1	even	1	trivial
975.2.a.s.1.3		5	5.4	even	2
2925.2.a.bl.1.3		5	3.2	odd	2
2925.2.a.bm.1.3		5	15.14	odd	2
3120.2.l.p.1249.5		10	20.7	even	4
3120.2.l.p.1249.10		10	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+0.329386 q^{2} -1.00000 q^{3} -1.89150 q^{4} -0.329386 q^{6} -3.70203 q^{7} -1.28181 q^{8} +1.00000 q^{9} -3.31322 q^{11} +1.89150 q^{12} -1.00000 q^{13} -1.21940 q^{14} +3.36080 q^{16} +4.36080 q^{17} +0.329386 q^{18} +5.21940 q^{19} +3.70203 q^{21} -1.09133 q^{22} -4.92143 q^{23} +1.28181 q^{24} -0.329386 q^{26} -1.00000 q^{27} +7.00241 q^{28} +7.78301 q^{29} +0.0981475 q^{31} +3.67061 q^{32} +3.31322 q^{33} +1.43639 q^{34} -1.89150 q^{36} +2.92143 q^{37} +1.71920 q^{38} +1.00000 q^{39} -0.749608 q^{41} +1.21940 q^{42} +3.78301 q^{43} +6.26698 q^{44} -1.62105 q^{46} -5.67402 q^{47} -3.36080 q^{48} +6.70502 q^{49} -4.36080 q^{51} +1.89150 q^{52} +2.19982 q^{53} -0.329386 q^{54} +4.74529 q^{56} -5.21940 q^{57} +2.56361 q^{58} +0.108987 q^{59} +12.1438 q^{61} +0.0323284 q^{62} -3.70203 q^{63} -5.51255 q^{64} +1.09133 q^{66} -12.4418 q^{67} -8.24848 q^{68} +4.92143 q^{69} +12.0348 q^{71} -1.28181 q^{72} -9.56602 q^{73} +0.962276 q^{74} -9.87251 q^{76} +12.2656 q^{77} +0.329386 q^{78} +14.6093 q^{79} +1.00000 q^{81} -0.246910 q^{82} +16.7001 q^{83} -7.00241 q^{84} +1.24607 q^{86} -7.78301 q^{87} +4.24691 q^{88} -3.59403 q^{89} +3.70203 q^{91} +9.30890 q^{92} -0.0981475 q^{93} -1.86894 q^{94} -3.67061 q^{96} +4.10467 q^{97} +2.20854 q^{98} -3.31322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9} + 5 q^{11} - 6 q^{12} - 5 q^{13} + 12 q^{14} + 5 q^{17} + 8 q^{19} + 5 q^{21} + 8 q^{22} + 7 q^{23} - 5 q^{27} - 14 q^{28} + 8 q^{29} + 12 q^{31} + 20 q^{32}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9 + 5 * q^11 - 6 * q^12 - 5 * q^13 + 12 * q^14 + 5 * q^17 + 8 * q^19 + 5 * q^21 + 8 * q^22 + 7 * q^23 - 5 * q^27 - 14 * q^28 + 8 * q^29 + 12 * q^31 + 20 * q^32 - 5 * q^33 + 20 * q^34 + 6 * q^36 - 17 * q^37 + 24 * q^38 + 5 * q^39 + 5 * q^41 - 12 * q^42 - 12 * q^43 + 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 - 5 * q^51 - 6 * q^52 + 13 * q^53 - 8 * q^57 + 8 * q^59 + 13 * q^61 + 40 * q^62 - 5 * q^63 - 16 * q^64 - 8 * q^66 - 28 * q^67 + 14 * q^68 - 7 * q^69 + 5 * q^71 + 14 * q^73 + 12 * q^74 + 35 * q^77 + q^79 + 5 * q^81 - 16 * q^82 + 6 * q^83 + 14 * q^84 - 8 * q^87 + 36 * q^88 + 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^93 - 12 * q^94 - 20 * q^96 + 13 * q^97 - 28 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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