LMFDB - Embedded newform 1575.4.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-3,0,-3,0,0,-14,3,0,0,31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.561553 q^{2} -7.68466 q^{4} -7.00000 q^{7} -8.80776 q^{8} +25.8078 q^{11} -15.5464 q^{13} -3.93087 q^{14} +56.5312 q^{16} +95.1619 q^{17} -143.447 q^{19} +14.4924 q^{22} -77.5227 q^{23} -8.73012 q^{26} +53.7926 q^{28} +204.170 q^{29} -40.6998 q^{31} +102.207 q^{32} +53.4384 q^{34} +95.6695 q^{37} -80.5530 q^{38} -24.7301 q^{41} +282.430 q^{43} -198.324 q^{44} -43.5331 q^{46} -257.417 q^{47} +49.0000 q^{49} +119.469 q^{52} -257.978 q^{53} +61.6543 q^{56} +114.652 q^{58} +651.365 q^{59} +451.304 q^{61} -22.8551 q^{62} -394.855 q^{64} +832.071 q^{67} -731.287 q^{68} +174.965 q^{71} -47.4166 q^{73} +53.7235 q^{74} +1102.34 q^{76} -180.654 q^{77} -1.16006 q^{79} -13.8873 q^{82} -1494.03 q^{83} +158.599 q^{86} -227.309 q^{88} -1309.13 q^{89} +108.825 q^{91} +595.736 q^{92} -144.553 q^{94} -1365.33 q^{97} +27.5161 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 3 q^{4} - 14 q^{7} + 3 q^{8} + 31 q^{11} + 39 q^{13} + 21 q^{14} - 23 q^{16} + 79 q^{17} - 56 q^{19} - 4 q^{22} - 254 q^{23} - 203 q^{26} + 21 q^{28} + 62 q^{29} - 135 q^{31} + 291 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 - 3 * q^4 - 14 * q^7 + 3 * q^8 + 31 * q^11 + 39 * q^13 + 21 * q^14 - 23 * q^16 + 79 * q^17 - 56 * q^19 - 4 * q^22 - 254 * q^23 - 203 * q^26 + 21 * q^28 + 62 * q^29 - 135 * q^31 + 291 * q^32 + 111 * q^34 + 113 * q^37 - 392 * q^38 - 235 * q^41 + 804 * q^43 - 174 * q^44 + 585 * q^46 - 152 * q^47 + 98 * q^49 + 375 * q^52 - 149 * q^53 - 21 * q^56 + 621 * q^58 + 441 * q^59 - 223 * q^61 + 313 * q^62 - 431 * q^64 + 1157 * q^67 - 807 * q^68 - 619 * q^71 + 268 * q^73 - 8 * q^74 + 1512 * q^76 - 217 * q^77 - 427 * q^79 + 735 * q^82 - 1211 * q^83 - 1699 * q^86 - 166 * q^88 - 466 * q^89 - 273 * q^91 - 231 * q^92 - 520 * q^94 + 172 * q^97 - 147 * 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 $

$2$	0.561553	0.198539	0.0992695	−	0.995061i	$-0.468349\pi$
$2$	0.561553	0.198539	0.0992695	+	0.995061i	$0.468349\pi$
$3$	0	0
$3$	0	0
$4$	−7.68466	−0.960582
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.80776	−0.389252
$9$	0	0
$10$	0	0
$11$	25.8078	0.707394	0.353697	−	0.935360i	$-0.384924\pi$
$11$	25.8078	0.707394	0.353697	+	0.935360i	$0.384924\pi$
$12$	0	0
$13$	−15.5464	−0.331677	−0.165838	−	0.986153i	$-0.553033\pi$
$13$	−15.5464	−0.331677	−0.165838	+	0.986153i	$0.553033\pi$
$14$	−3.93087	−0.0750407
$15$	0	0
$16$	56.5312	0.883301
$17$	95.1619	1.35766	0.678828	−	0.734297i	$-0.262488\pi$
$17$	95.1619	1.35766	0.678828	+	0.734297i	$0.262488\pi$
$18$	0	0
$19$	−143.447	−1.73205	−0.866026	−	0.499999i	$-0.833334\pi$
$19$	−143.447	−1.73205	−0.866026	+	0.499999i	$0.833334\pi$
$20$	0	0
$21$	0	0
$22$	14.4924	0.140445
$23$	−77.5227	−0.702809	−0.351405	−	0.936224i	$-0.614296\pi$
$23$	−77.5227	−0.702809	−0.351405	+	0.936224i	$0.614296\pi$
$24$	0	0
$25$	0	0
$26$	−8.73012	−0.0658507
$27$	0	0
$28$	53.7926	0.363066
$29$	204.170	1.30736	0.653681	−	0.756770i	$-0.273224\pi$
$29$	204.170	1.30736	0.653681	+	0.756770i	$0.273224\pi$
$30$	0	0
$31$	−40.6998	−0.235803	−0.117902	−	0.993025i	$-0.537617\pi$
$31$	−40.6998	−0.235803	−0.117902	+	0.993025i	$0.537617\pi$
$32$	102.207	0.564621
$33$	0	0
$34$	53.4384	0.269548
$35$	0	0
$36$	0	0
$37$	95.6695	0.425080	0.212540	−	0.977152i	$-0.431826\pi$
$37$	95.6695	0.425080	0.212540	+	0.977152i	$0.431826\pi$
$38$	−80.5530	−0.343880
$39$	0	0
$40$	0	0
$41$	−24.7301	−0.0941999	−0.0471000	−	0.998890i	$-0.514998\pi$
$41$	−24.7301	−0.0941999	−0.0471000	+	0.998890i	$0.514998\pi$
$42$	0	0
$43$	282.430	1.00163	0.500816	−	0.865554i	$-0.333033\pi$
$43$	282.430	1.00163	0.500816	+	0.865554i	$0.333033\pi$
$44$	−198.324	−0.679510
$45$	0	0
$46$	−43.5331	−0.139535
$47$	−257.417	−0.798895	−0.399448	−	0.916756i	$-0.630798\pi$
$47$	−257.417	−0.798895	−0.399448	+	0.916756i	$0.630798\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	119.469	0.318603
$53$	−257.978	−0.668604	−0.334302	−	0.942466i	$-0.608501\pi$
$53$	−257.978	−0.668604	−0.334302	+	0.942466i	$0.608501\pi$
$54$	0	0
$55$	0	0
$56$	61.6543	0.147123
$57$	0	0
$58$	114.652	0.259562
$59$	651.365	1.43730	0.718648	−	0.695374i	$-0.244761\pi$
$59$	651.365	1.43730	0.718648	+	0.695374i	$0.244761\pi$
$60$	0	0
$61$	451.304	0.947271	0.473636	−	0.880721i	$-0.342941\pi$
$61$	451.304	0.947271	0.473636	+	0.880721i	$0.342941\pi$
$62$	−22.8551	−0.0468161
$63$	0	0
$64$	−394.855	−0.771201
$65$	0	0
$66$	0	0
$67$	832.071	1.51722	0.758609	−	0.651546i	$-0.225879\pi$
$67$	832.071	1.51722	0.758609	+	0.651546i	$0.225879\pi$
$68$	−731.287	−1.30414
$69$	0	0
$70$	0	0
$71$	174.965	0.292458	0.146229	−	0.989251i	$-0.453286\pi$
$71$	174.965	0.292458	0.146229	+	0.989251i	$0.453286\pi$
$72$	0	0
$73$	−47.4166	−0.0760233	−0.0380116	−	0.999277i	$-0.512102\pi$
$73$	−47.4166	−0.0760233	−0.0380116	+	0.999277i	$0.512102\pi$
$74$	53.7235	0.0843950
$75$	0	0
$76$	1102.34	1.66378
$77$	−180.654	−0.267370
$78$	0	0
$79$	−1.16006	−0.00165211	−0.000826057	−	1.00000i	$-0.500263\pi$
$79$	−1.16006	−0.00165211	−0.000826057		1.00000i	$0.500263\pi$
$80$	0	0
$81$	0	0
$82$	−13.8873	−0.0187023
$83$	−1494.03	−1.97580	−0.987898	−	0.155107i	$-0.950428\pi$
$83$	−1494.03	−1.97580	−0.987898	+	0.155107i	$0.950428\pi$
$84$	0	0
$85$	0	0
$86$	158.599	0.198863
$87$	0	0
$88$	−227.309	−0.275354
$89$	−1309.13	−1.55919	−0.779593	−	0.626286i	$-0.784574\pi$
$89$	−1309.13	−1.55919	−0.779593	+	0.626286i	$0.784574\pi$
$90$	0	0
$91$	108.825	0.125362
$92$	595.736	0.675106
$93$	0	0
$94$	−144.553	−0.158612
$95$	0	0
$96$	0	0
$97$	−1365.33	−1.42916	−0.714580	−	0.699553i	$-0.753382\pi$
$97$	−1365.33	−1.42916	−0.714580	+	0.699553i	$0.753382\pi$
$98$	27.5161	0.0283627
$99$	0	0
$100$	0	0
$101$	−1155.99	−1.13887	−0.569434	−	0.822037i	$-0.692838\pi$
$101$	−1155.99	−1.13887	−0.569434	+	0.822037i	$0.692838\pi$
$102$	0	0
$103$	−78.2074	−0.0748156	−0.0374078	−	0.999300i	$-0.511910\pi$
$103$	−78.2074	−0.0748156	−0.0374078	+	0.999300i	$0.511910\pi$
$104$	136.929	0.129106
$105$	0	0
$106$	−144.868	−0.132744
$107$	94.0814	0.0850018	0.0425009	−	0.999096i	$-0.486467\pi$
$107$	94.0814	0.0850018	0.0425009	+	0.999096i	$0.486467\pi$
$108$	0	0
$109$	−2128.25	−1.87018	−0.935088	−	0.354417i	$-0.884679\pi$
$109$	−2128.25	−1.87018	−0.935088	+	0.354417i	$0.884679\pi$
$110$	0	0
$111$	0	0
$112$	−395.719	−0.333856
$113$	596.493	0.496578	0.248289	−	0.968686i	$-0.420132\pi$
$113$	596.493	0.496578	0.248289	+	0.968686i	$0.420132\pi$
$114$	0	0
$115$	0	0
$116$	−1568.98	−1.25583
$117$	0	0
$118$	365.776	0.285359
$119$	−666.133	−0.513146
$120$	0	0
$121$	−664.959	−0.499594
$122$	253.431	0.188070
$123$	0	0
$124$	312.764	0.226508
$125$	0	0
$126$	0	0
$127$	2088.21	1.45904	0.729522	−	0.683957i	$-0.239743\pi$
$127$	2088.21	1.45904	0.729522	+	0.683957i	$0.239743\pi$
$128$	−1039.39	−0.717735
$129$	0	0
$130$	0	0
$131$	−416.142	−0.277546	−0.138773	−	0.990324i	$-0.544316\pi$
$131$	−416.142	−0.277546	−0.138773	+	0.990324i	$0.544316\pi$
$132$	0	0
$133$	1004.13	0.654654
$134$	467.252	0.301227
$135$	0	0
$136$	−838.164	−0.528470
$137$	−619.252	−0.386177	−0.193089	−	0.981181i	$-0.561850\pi$
$137$	−619.252	−0.386177	−0.193089	+	0.981181i	$0.561850\pi$
$138$	0	0
$139$	−266.994	−0.162922	−0.0814610	−	0.996677i	$-0.525959\pi$
$139$	−266.994	−0.162922	−0.0814610	+	0.996677i	$0.525959\pi$
$140$	0	0
$141$	0	0
$142$	98.2520	0.0580643
$143$	−401.218	−0.234626
$144$	0	0
$145$	0	0
$146$	−26.6270	−0.0150936
$147$	0	0
$148$	−735.187	−0.408325
$149$	−2551.80	−1.40303	−0.701515	−	0.712655i	$-0.747493\pi$
$149$	−2551.80	−1.40303	−0.701515	+	0.712655i	$0.747493\pi$
$150$	0	0
$151$	1117.38	0.602195	0.301097	−	0.953593i	$-0.402647\pi$
$151$	1117.38	0.602195	0.301097	+	0.953593i	$0.402647\pi$
$152$	1263.45	0.674204
$153$	0	0
$154$	−101.447	−0.0530833
$155$	0	0
$156$	0	0
$157$	−456.104	−0.231854	−0.115927	−	0.993258i	$-0.536984\pi$
$157$	−456.104	−0.231854	−0.115927	+	0.993258i	$0.536984\pi$
$158$	−0.651435	−0.000328009 0
$159$	0	0
$160$	0	0
$161$	542.659	0.265637
$162$	0	0
$163$	−105.490	−0.0506907	−0.0253453	−	0.999679i	$-0.508069\pi$
$163$	−105.490	−0.0506907	−0.0253453	+	0.999679i	$0.508069\pi$
$164$	190.043	0.0904868
$165$	0	0
$166$	−838.976	−0.392272
$167$	2482.54	1.15033	0.575164	−	0.818038i	$-0.304938\pi$
$167$	2482.54	1.15033	0.575164	+	0.818038i	$0.304938\pi$
$168$	0	0
$169$	−1955.31	−0.889991
$170$	0	0
$171$	0	0
$172$	−2170.38	−0.962150
$173$	−516.708	−0.227079	−0.113539	−	0.993534i	$-0.536219\pi$
$173$	−516.708	−0.227079	−0.113539	+	0.993534i	$0.536219\pi$
$174$	0	0
$175$	0	0
$176$	1458.94	0.624842
$177$	0	0
$178$	−735.146	−0.309559
$179$	−3847.77	−1.60668	−0.803341	−	0.595519i	$-0.796946\pi$
$179$	−3847.77	−1.60668	−0.803341	+	0.595519i	$0.796946\pi$
$180$	0	0
$181$	−3026.41	−1.24282	−0.621411	−	0.783484i	$-0.713440\pi$
$181$	−3026.41	−1.24282	−0.621411	+	0.783484i	$0.713440\pi$
$182$	61.1109	0.0248892
$183$	0	0
$184$	682.802	0.273570
$185$	0	0
$186$	0	0
$187$	2455.92	0.960398
$188$	1978.16	0.767405
$189$	0	0
$190$	0	0
$191$	−1271.09	−0.481532	−0.240766	−	0.970583i	$-0.577399\pi$
$191$	−1271.09	−0.481532	−0.240766	+	0.970583i	$0.577399\pi$
$192$	0	0
$193$	−1896.09	−0.707170	−0.353585	−	0.935402i	$-0.615037\pi$
$193$	−1896.09	−0.707170	−0.353585	+	0.935402i	$0.615037\pi$
$194$	−766.707	−0.283744
$195$	0	0
$196$	−376.548	−0.137226
$197$	−4067.89	−1.47119	−0.735597	−	0.677419i	$-0.763098\pi$
$197$	−4067.89	−1.47119	−0.735597	+	0.677419i	$0.763098\pi$
$198$	0	0
$199$	78.1366	0.0278340	0.0139170	−	0.999903i	$-0.495570\pi$
$199$	78.1366	0.0278340	0.0139170	+	0.999903i	$0.495570\pi$
$200$	0	0
$201$	0	0
$202$	−649.152	−0.226110
$203$	−1429.19	−0.494136
$204$	0	0
$205$	0	0
$206$	−43.9176	−0.0148538
$207$	0	0
$208$	−878.857	−0.292970
$209$	−3702.05	−1.22524
$210$	0	0
$211$	−1293.02	−0.421872	−0.210936	−	0.977500i	$-0.567651\pi$
$211$	−1293.02	−0.421872	−0.210936	+	0.977500i	$0.567651\pi$
$212$	1982.47	0.642250
$213$	0	0
$214$	52.8317	0.0168762
$215$	0	0
$216$	0	0
$217$	284.899	0.0891253
$218$	−1195.12	−0.371303
$219$	0	0
$220$	0	0
$221$	−1479.43	−0.450303
$222$	0	0
$223$	−1786.65	−0.536515	−0.268257	−	0.963347i	$-0.586448\pi$
$223$	−1786.65	−0.536515	−0.268257	+	0.963347i	$0.586448\pi$
$224$	−715.452	−0.213407
$225$	0	0
$226$	334.962	0.0985901
$227$	4242.02	1.24032	0.620161	−	0.784475i	$-0.287067\pi$
$227$	4242.02	1.24032	0.620161	+	0.784475i	$0.287067\pi$
$228$	0	0
$229$	3403.36	0.982096	0.491048	−	0.871132i	$-0.336614\pi$
$229$	3403.36	0.982096	0.491048	+	0.871132i	$0.336614\pi$
$230$	0	0
$231$	0	0
$232$	−1798.29	−0.508893
$233$	3904.09	1.09771	0.548853	−	0.835919i	$-0.315065\pi$
$233$	3904.09	1.09771	0.548853	+	0.835919i	$0.315065\pi$
$234$	0	0
$235$	0	0
$236$	−5005.51	−1.38064
$237$	0	0
$238$	−374.069	−0.101879
$239$	−6667.83	−1.80463	−0.902314	−	0.431079i	$-0.858133\pi$
$239$	−6667.83	−1.80463	−0.902314	+	0.431079i	$0.858133\pi$
$240$	0	0
$241$	−6003.99	−1.60478	−0.802388	−	0.596803i	$-0.796437\pi$
$241$	−6003.99	−1.60478	−0.802388	+	0.596803i	$0.796437\pi$
$242$	−373.410	−0.0991888
$243$	0	0
$244$	−3468.12	−0.909932
$245$	0	0
$246$	0	0
$247$	2230.08	0.574481
$248$	358.474	0.0917869
$249$	0	0
$250$	0	0
$251$	−728.467	−0.183189	−0.0915945	−	0.995796i	$-0.529196\pi$
$251$	−728.467	−0.183189	−0.0915945	+	0.995796i	$0.529196\pi$
$252$	0	0
$253$	−2000.69	−0.497163
$254$	1172.64	0.289677
$255$	0	0
$256$	2575.17	0.628703
$257$	1479.86	0.359187	0.179593	−	0.983741i	$-0.442522\pi$
$257$	1479.86	0.359187	0.179593	+	0.983741i	$0.442522\pi$
$258$	0	0
$259$	−669.687	−0.160665
$260$	0	0
$261$	0	0
$262$	−233.686	−0.0551036
$263$	6367.31	1.49287	0.746435	−	0.665458i	$-0.231764\pi$
$263$	6367.31	1.49287	0.746435	+	0.665458i	$0.231764\pi$
$264$	0	0
$265$	0	0
$266$	563.871	0.129974
$267$	0	0
$268$	−6394.18	−1.45741
$269$	−1787.18	−0.405079	−0.202539	−	0.979274i	$-0.564919\pi$
$269$	−1787.18	−0.405079	−0.202539	+	0.979274i	$0.564919\pi$
$270$	0	0
$271$	3907.65	0.875915	0.437957	−	0.898996i	$-0.355702\pi$
$271$	3907.65	0.875915	0.437957	+	0.898996i	$0.355702\pi$
$272$	5379.62	1.19922
$273$	0	0
$274$	−347.743	−0.0766712
$275$	0	0
$276$	0	0
$277$	3978.33	0.862942	0.431471	−	0.902127i	$-0.357995\pi$
$277$	3978.33	0.862942	0.431471	+	0.902127i	$0.357995\pi$
$278$	−149.931	−0.0323464
$279$	0	0
$280$	0	0
$281$	6488.04	1.37738	0.688690	−	0.725055i	$-0.258186\pi$
$281$	6488.04	1.37738	0.688690	+	0.725055i	$0.258186\pi$
$282$	0	0
$283$	−164.336	−0.0345185	−0.0172592	−	0.999851i	$-0.505494\pi$
$283$	−164.336	−0.0345185	−0.0172592	+	0.999851i	$0.505494\pi$
$284$	−1344.55	−0.280930
$285$	0	0
$286$	−225.305	−0.0465824
$287$	173.111	0.0356042
$288$	0	0
$289$	4142.79	0.843231
$290$	0	0
$291$	0	0
$292$	364.381	0.0730266
$293$	−5004.57	−0.997851	−0.498925	−	0.866645i	$-0.666272\pi$
$293$	−5004.57	−0.997851	−0.498925	+	0.866645i	$0.666272\pi$
$294$	0	0
$295$	0	0
$296$	−842.634	−0.165463
$297$	0	0
$298$	−1432.97	−0.278556
$299$	1205.20	0.233105
$300$	0	0
$301$	−1977.01	−0.378581
$302$	627.470	0.119559
$303$	0	0
$304$	−8109.23	−1.52992
$305$	0	0
$306$	0	0
$307$	−190.167	−0.0353531	−0.0176765	−	0.999844i	$-0.505627\pi$
$307$	−190.167	−0.0353531	−0.0176765	+	0.999844i	$0.505627\pi$
$308$	1388.27	0.256831
$309$	0	0
$310$	0	0
$311$	1182.57	0.215619	0.107810	−	0.994172i	$-0.465616\pi$
$311$	1182.57	0.215619	0.107810	+	0.994172i	$0.465616\pi$
$312$	0	0
$313$	−4659.05	−0.841358	−0.420679	−	0.907209i	$-0.638208\pi$
$313$	−4659.05	−0.841358	−0.420679	+	0.907209i	$0.638208\pi$
$314$	−256.127	−0.0460320
$315$	0	0
$316$	8.91467	0.00158699
$317$	3694.55	0.654595	0.327297	−	0.944921i	$-0.393862\pi$
$317$	3694.55	0.654595	0.327297	+	0.944921i	$0.393862\pi$
$318$	0	0
$319$	5269.18	0.924820
$320$	0	0
$321$	0	0
$322$	304.732	0.0527392
$323$	−13650.7	−2.35153
$324$	0	0
$325$	0	0
$326$	−59.2379	−0.0100641
$327$	0	0
$328$	217.817	0.0366675
$329$	1801.92	0.301954
$330$	0	0
$331$	−8632.79	−1.43354	−0.716769	−	0.697310i	$-0.754380\pi$
$331$	−8632.79	−1.43354	−0.716769	+	0.697310i	$0.754380\pi$
$332$	11481.1	1.89791
$333$	0	0
$334$	1394.08	0.228385
$335$	0	0
$336$	0	0
$337$	−8136.61	−1.31522	−0.657610	−	0.753358i	$-0.728433\pi$
$337$	−8136.61	−1.31522	−0.657610	+	0.753358i	$0.728433\pi$
$338$	−1098.01	−0.176698
$339$	0	0
$340$	0	0
$341$	−1050.37	−0.166806
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−2487.58	−0.389887
$345$	0	0
$346$	−290.159	−0.0450839
$347$	−8646.33	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$347$	−8646.33	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$348$	0	0
$349$	−7455.43	−1.14350	−0.571748	−	0.820429i	$-0.693734\pi$
$349$	−7455.43	−1.14350	−0.571748	+	0.820429i	$0.693734\pi$
$350$	0	0
$351$	0	0
$352$	2637.74	0.399410
$353$	−5450.21	−0.821771	−0.410886	−	0.911687i	$-0.634780\pi$
$353$	−5450.21	−0.821771	−0.410886	+	0.911687i	$0.634780\pi$
$354$	0	0
$355$	0	0
$356$	10060.2	1.49773
$357$	0	0
$358$	−2160.73	−0.318989
$359$	−4775.79	−0.702107	−0.351053	−	0.936355i	$-0.614176\pi$
$359$	−4775.79	−0.702107	−0.351053	+	0.936355i	$0.614176\pi$
$360$	0	0
$361$	13718.0	2.00000
$362$	−1699.49	−0.246749
$363$	0	0
$364$	−836.281	−0.120420
$365$	0	0
$366$	0	0
$367$	9636.30	1.37060	0.685301	−	0.728260i	$-0.259671\pi$
$367$	9636.30	1.37060	0.685301	+	0.728260i	$0.259671\pi$
$368$	−4382.46	−0.620792
$369$	0	0
$370$	0	0
$371$	1805.85	0.252709
$372$	0	0
$373$	12180.4	1.69082	0.845412	−	0.534115i	$-0.179355\pi$
$373$	12180.4	1.69082	0.845412	+	0.534115i	$0.179355\pi$
$374$	1379.13	0.190676
$375$	0	0
$376$	2267.27	0.310971
$377$	−3174.11	−0.433621
$378$	0	0
$379$	1689.39	0.228966	0.114483	−	0.993425i	$-0.463479\pi$
$379$	1689.39	0.228966	0.114483	+	0.993425i	$0.463479\pi$
$380$	0	0
$381$	0	0
$382$	−713.783	−0.0956029
$383$	−1513.13	−0.201872	−0.100936	−	0.994893i	$-0.532184\pi$
$383$	−1513.13	−0.201872	−0.100936	+	0.994893i	$0.532184\pi$
$384$	0	0
$385$	0	0
$386$	−1064.76	−0.140401
$387$	0	0
$388$	10492.1	1.37283
$389$	−12165.5	−1.58564	−0.792822	−	0.609453i	$-0.791389\pi$
$389$	−12165.5	−1.58564	−0.792822	+	0.609453i	$0.791389\pi$
$390$	0	0
$391$	−7377.21	−0.954173
$392$	−431.580	−0.0556074
$393$	0	0
$394$	−2284.34	−0.292089
$395$	0	0
$396$	0	0
$397$	7353.40	0.929613	0.464807	−	0.885412i	$-0.346124\pi$
$397$	7353.40	0.929613	0.464807	+	0.885412i	$0.346124\pi$
$398$	43.8778	0.00552612
$399$	0	0
$400$	0	0
$401$	−7481.62	−0.931707	−0.465853	−	0.884862i	$-0.654253\pi$
$401$	−7481.62	−0.931707	−0.465853	+	0.884862i	$0.654253\pi$
$402$	0	0
$403$	632.735	0.0782104
$404$	8883.42	1.09398
$405$	0	0
$406$	−802.567	−0.0981053
$407$	2469.02	0.300699
$408$	0	0
$409$	9248.94	1.11817	0.559084	−	0.829111i	$-0.311153\pi$
$409$	9248.94	1.11817	0.559084	+	0.829111i	$0.311153\pi$
$410$	0	0
$411$	0	0
$412$	600.997	0.0718665
$413$	−4559.55	−0.543247
$414$	0	0
$415$	0	0
$416$	−1588.96	−0.187272
$417$	0	0
$418$	−2078.89	−0.243258
$419$	3363.39	0.392154	0.196077	−	0.980589i	$-0.437180\pi$
$419$	3363.39	0.392154	0.196077	+	0.980589i	$0.437180\pi$
$420$	0	0
$421$	−3638.86	−0.421252	−0.210626	−	0.977567i	$-0.567550\pi$
$421$	−3638.86	−0.421252	−0.210626	+	0.977567i	$0.567550\pi$
$422$	−726.097	−0.0837579
$423$	0	0
$424$	2272.21	0.260255
$425$	0	0
$426$	0	0
$427$	−3159.13	−0.358035
$428$	−722.983	−0.0816512
$429$	0	0
$430$	0	0
$431$	−11243.2	−1.25653	−0.628264	−	0.778000i	$-0.716234\pi$
$431$	−11243.2	−1.25653	−0.628264	+	0.778000i	$0.716234\pi$
$432$	0	0
$433$	187.332	0.0207912	0.0103956	−	0.999946i	$-0.496691\pi$
$433$	187.332	0.0207912	0.0103956	+	0.999946i	$0.496691\pi$
$434$	159.986	0.0176948
$435$	0	0
$436$	16354.9	1.79646
$437$	11120.4	1.21730
$438$	0	0
$439$	−11479.4	−1.24802	−0.624011	−	0.781415i	$-0.714498\pi$
$439$	−11479.4	−1.24802	−0.624011	+	0.781415i	$0.714498\pi$
$440$	0	0
$441$	0	0
$442$	−830.775	−0.0894026
$443$	9490.56	1.01786	0.508928	−	0.860809i	$-0.330042\pi$
$443$	9490.56	1.01786	0.508928	+	0.860809i	$0.330042\pi$
$444$	0	0
$445$	0	0
$446$	−1003.30	−0.106519
$447$	0	0
$448$	2763.99	0.291487
$449$	5546.58	0.582982	0.291491	−	0.956574i	$-0.405849\pi$
$449$	5546.58	0.582982	0.291491	+	0.956574i	$0.405849\pi$
$450$	0	0
$451$	−638.229	−0.0666364
$452$	−4583.85	−0.477004
$453$	0	0
$454$	2382.12	0.246252
$455$	0	0
$456$	0	0
$457$	1776.94	0.181885	0.0909426	−	0.995856i	$-0.471012\pi$
$457$	1776.94	0.181885	0.0909426	+	0.995856i	$0.471012\pi$
$458$	1911.16	0.194984
$459$	0	0
$460$	0	0
$461$	9059.81	0.915309	0.457655	−	0.889130i	$-0.348690\pi$
$461$	9059.81	0.915309	0.457655	+	0.889130i	$0.348690\pi$
$462$	0	0
$463$	14878.5	1.49344	0.746719	−	0.665139i	$-0.231628\pi$
$463$	14878.5	1.49344	0.746719	+	0.665139i	$0.231628\pi$
$464$	11542.0	1.15479
$465$	0	0
$466$	2192.35	0.217937
$467$	−14275.8	−1.41458	−0.707288	−	0.706926i	$-0.750082\pi$
$467$	−14275.8	−1.41458	−0.707288	+	0.706926i	$0.750082\pi$
$468$	0	0
$469$	−5824.50	−0.573455
$470$	0	0
$471$	0	0
$472$	−5737.07	−0.559470
$473$	7288.89	0.708548
$474$	0	0
$475$	0	0
$476$	5119.01	0.492919
$477$	0	0
$478$	−3744.34	−0.358289
$479$	15377.8	1.46687	0.733433	−	0.679761i	$-0.237917\pi$
$479$	15377.8	1.46687	0.733433	+	0.679761i	$0.237917\pi$
$480$	0	0
$481$	−1487.32	−0.140989
$482$	−3371.56	−0.318610
$483$	0	0
$484$	5109.99	0.479901
$485$	0	0
$486$	0	0
$487$	19683.3	1.83149	0.915747	−	0.401756i	$-0.131600\pi$
$487$	19683.3	1.83149	0.915747	+	0.401756i	$0.131600\pi$
$488$	−3974.98	−0.368727
$489$	0	0
$490$	0	0
$491$	−19585.1	−1.80013	−0.900065	−	0.435755i	$-0.856481\pi$
$491$	−19585.1	−1.80013	−0.900065	+	0.435755i	$0.856481\pi$
$492$	0	0
$493$	19429.3	1.77495
$494$	1252.31	0.114057
$495$	0	0
$496$	−2300.81	−0.208285
$497$	−1224.75	−0.110539
$498$	0	0
$499$	−6888.47	−0.617977	−0.308988	−	0.951066i	$-0.599990\pi$
$499$	−6888.47	−0.617977	−0.308988	+	0.951066i	$0.599990\pi$
$500$	0	0
$501$	0	0
$502$	−409.073	−0.0363701
$503$	−14878.0	−1.31885	−0.659423	−	0.751772i	$-0.729199\pi$
$503$	−14878.0	−1.31885	−0.659423	+	0.751772i	$0.729199\pi$
$504$	0	0
$505$	0	0
$506$	−1123.49	−0.0987062
$507$	0	0
$508$	−16047.2	−1.40153
$509$	−1935.28	−0.168526	−0.0842629	−	0.996444i	$-0.526854\pi$
$509$	−1935.28	−0.168526	−0.0842629	+	0.996444i	$0.526854\pi$
$510$	0	0
$511$	331.917	0.0287341
$512$	9761.22	0.842557
$513$	0	0
$514$	831.019	0.0713126
$515$	0	0
$516$	0	0
$517$	−6643.35	−0.565134
$518$	−376.064	−0.0318983
$519$	0	0
$520$	0	0
$521$	6892.28	0.579570	0.289785	−	0.957092i	$-0.406416\pi$
$521$	6892.28	0.579570	0.289785	+	0.957092i	$0.406416\pi$
$522$	0	0
$523$	−1074.60	−0.0898447	−0.0449223	−	0.998990i	$-0.514304\pi$
$523$	−1074.60	−0.0898447	−0.0449223	+	0.998990i	$0.514304\pi$
$524$	3197.91	0.266606
$525$	0	0
$526$	3575.58	0.296393
$527$	−3873.07	−0.320140
$528$	0	0
$529$	−6157.23	−0.506060
$530$	0	0
$531$	0	0
$532$	−7716.39	−0.628849
$533$	384.464	0.0312439
$534$	0	0
$535$	0	0
$536$	−7328.69	−0.590580
$537$	0	0
$538$	−1003.60	−0.0804239
$539$	1264.58	0.101056
$540$	0	0
$541$	8660.23	0.688230	0.344115	−	0.938928i	$-0.388179\pi$
$541$	8660.23	0.688230	0.344115	+	0.938928i	$0.388179\pi$
$542$	2194.35	0.173903
$543$	0	0
$544$	9726.25	0.766562
$545$	0	0
$546$	0	0
$547$	−15346.1	−1.19954	−0.599771	−	0.800171i	$-0.704742\pi$
$547$	−15346.1	−1.19954	−0.599771	+	0.800171i	$0.704742\pi$
$548$	4758.74	0.370955
$549$	0	0
$550$	0	0
$551$	−29287.6	−2.26442
$552$	0	0
$553$	8.12042	0.000624440 0
$554$	2234.04	0.171328
$555$	0	0
$556$	2051.76	0.156500
$557$	−15544.9	−1.18251	−0.591257	−	0.806483i	$-0.701368\pi$
$557$	−15544.9	−1.18251	−0.591257	+	0.806483i	$0.701368\pi$
$558$	0	0
$559$	−4390.77	−0.332218
$560$	0	0
$561$	0	0
$562$	3643.38	0.273464
$563$	18511.9	1.38576	0.692881	−	0.721052i	$-0.256341\pi$
$563$	18511.9	1.38576	0.692881	+	0.721052i	$0.256341\pi$
$564$	0	0
$565$	0	0
$566$	−92.2831	−0.00685326
$567$	0	0
$568$	−1541.05	−0.113840
$569$	−2157.36	−0.158948	−0.0794738	−	0.996837i	$-0.525324\pi$
$569$	−2157.36	−0.158948	−0.0794738	+	0.996837i	$0.525324\pi$
$570$	0	0
$571$	16010.3	1.17340	0.586700	−	0.809805i	$-0.300427\pi$
$571$	16010.3	1.17340	0.586700	+	0.809805i	$0.300427\pi$
$572$	3083.22	0.225378
$573$	0	0
$574$	97.2109	0.00706882
$575$	0	0
$576$	0	0
$577$	−2164.01	−0.156134	−0.0780668	−	0.996948i	$-0.524875\pi$
$577$	−2164.01	−0.156134	−0.0780668	+	0.996948i	$0.524875\pi$
$578$	2326.40	0.167414
$579$	0	0
$580$	0	0
$581$	10458.2	0.746780
$582$	0	0
$583$	−6657.84	−0.472967
$584$	417.635	0.0295922
$585$	0	0
$586$	−2810.33	−0.198112
$587$	−27616.6	−1.94184	−0.970918	−	0.239412i	$-0.923045\pi$
$587$	−27616.6	−1.94184	−0.970918	+	0.239412i	$0.923045\pi$
$588$	0	0
$589$	5838.26	0.408424
$590$	0	0
$591$	0	0
$592$	5408.32	0.375474
$593$	10205.4	0.706718	0.353359	−	0.935488i	$-0.385039\pi$
$593$	10205.4	0.706718	0.353359	+	0.935488i	$0.385039\pi$
$594$	0	0
$595$	0	0
$596$	19609.7	1.34773
$597$	0	0
$598$	676.783	0.0462805
$599$	−11090.5	−0.756501	−0.378251	−	0.925703i	$-0.623474\pi$
$599$	−11090.5	−0.756501	−0.378251	+	0.925703i	$0.623474\pi$
$600$	0	0
$601$	−26537.0	−1.80111	−0.900555	−	0.434742i	$-0.856839\pi$
$601$	−26537.0	−1.80111	−0.900555	+	0.434742i	$0.856839\pi$
$602$	−1110.20	−0.0751631
$603$	0	0
$604$	−8586.71	−0.578457
$605$	0	0
$606$	0	0
$607$	−11234.5	−0.751226	−0.375613	−	0.926777i	$-0.622568\pi$
$607$	−11234.5	−0.751226	−0.375613	+	0.926777i	$0.622568\pi$
$608$	−14661.3	−0.977954
$609$	0	0
$610$	0	0
$611$	4001.90	0.264975
$612$	0	0
$613$	−10622.5	−0.699900	−0.349950	−	0.936768i	$-0.613801\pi$
$613$	−10622.5	−0.699900	−0.349950	+	0.936768i	$0.613801\pi$
$614$	−106.789	−0.00701896
$615$	0	0
$616$	1591.16	0.104074
$617$	−20433.3	−1.33325	−0.666623	−	0.745395i	$-0.732261\pi$
$617$	−20433.3	−1.33325	−0.666623	+	0.745395i	$0.732261\pi$
$618$	0	0
$619$	−7963.57	−0.517097	−0.258549	−	0.965998i	$-0.583244\pi$
$619$	−7963.57	−0.517097	−0.258549	+	0.965998i	$0.583244\pi$
$620$	0	0
$621$	0	0
$622$	664.077	0.0428088
$623$	9163.91	0.589317
$624$	0	0
$625$	0	0
$626$	−2616.30	−0.167042
$627$	0	0
$628$	3505.01	0.222715
$629$	9104.09	0.577113
$630$	0	0
$631$	−14703.1	−0.927608	−0.463804	−	0.885938i	$-0.653516\pi$
$631$	−14703.1	−0.927608	−0.463804	+	0.885938i	$0.653516\pi$
$632$	10.2175	0.000643088 0
$633$	0	0
$634$	2074.68	0.129963
$635$	0	0
$636$	0	0
$637$	−761.773	−0.0473824
$638$	2958.92	0.183613
$639$	0	0
$640$	0	0
$641$	−3353.41	−0.206633	−0.103317	−	0.994649i	$-0.532945\pi$
$641$	−3353.41	−0.206633	−0.103317	+	0.994649i	$0.532945\pi$
$642$	0	0
$643$	31862.0	1.95415	0.977073	−	0.212906i	$-0.0682929\pi$
$643$	31862.0	1.95415	0.977073	+	0.212906i	$0.0682929\pi$
$644$	−4170.15	−0.255166
$645$	0	0
$646$	−7665.58	−0.466870
$647$	8518.46	0.517612	0.258806	−	0.965929i	$-0.416671\pi$
$647$	8518.46	0.517612	0.258806	+	0.965929i	$0.416671\pi$
$648$	0	0
$649$	16810.3	1.01673
$650$	0	0
$651$	0	0
$652$	810.651	0.0486925
$653$	28509.3	1.70851	0.854254	−	0.519856i	$-0.174014\pi$
$653$	28509.3	1.70851	0.854254	+	0.519856i	$0.174014\pi$
$654$	0	0
$655$	0	0
$656$	−1398.02	−0.0832068
$657$	0	0
$658$	1011.87	0.0599496
$659$	27632.4	1.63339	0.816697	−	0.577066i	$-0.195803\pi$
$659$	27632.4	1.63339	0.816697	+	0.577066i	$0.195803\pi$
$660$	0	0
$661$	−27052.8	−1.59188	−0.795941	−	0.605374i	$-0.793023\pi$
$661$	−27052.8	−1.59188	−0.795941	+	0.605374i	$0.793023\pi$
$662$	−4847.77	−0.284613
$663$	0	0
$664$	13159.1	0.769082
$665$	0	0
$666$	0	0
$667$	−15827.9	−0.918826
$668$	−19077.5	−1.10498
$669$	0	0
$670$	0	0
$671$	11647.1	0.670094
$672$	0	0
$673$	−1569.99	−0.0899235	−0.0449618	−	0.998989i	$-0.514317\pi$
$673$	−1569.99	−0.0899235	−0.0449618	+	0.998989i	$0.514317\pi$
$674$	−4569.13	−0.261122
$675$	0	0
$676$	15025.9	0.854909
$677$	−13853.9	−0.786482	−0.393241	−	0.919435i	$-0.628646\pi$
$677$	−13853.9	−0.786482	−0.393241	+	0.919435i	$0.628646\pi$
$678$	0	0
$679$	9557.33	0.540172
$680$	0	0
$681$	0	0
$682$	−589.839	−0.0331174
$683$	28337.1	1.58754	0.793769	−	0.608219i	$-0.208116\pi$
$683$	28337.1	1.58754	0.793769	+	0.608219i	$0.208116\pi$
$684$	0	0
$685$	0	0
$686$	−192.613	−0.0107201
$687$	0	0
$688$	15966.1	0.884742
$689$	4010.63	0.221760
$690$	0	0
$691$	16936.5	0.932410	0.466205	−	0.884677i	$-0.345621\pi$
$691$	16936.5	0.932410	0.466205	+	0.884677i	$0.345621\pi$
$692$	3970.73	0.218128
$693$	0	0
$694$	−4855.37	−0.265572
$695$	0	0
$696$	0	0
$697$	−2353.37	−0.127891
$698$	−4186.62	−0.227028
$699$	0	0
$700$	0	0
$701$	29744.0	1.60259	0.801294	−	0.598271i	$-0.204145\pi$
$701$	29744.0	1.60259	0.801294	+	0.598271i	$0.204145\pi$
$702$	0	0
$703$	−13723.5	−0.736261
$704$	−10190.3	−0.545543
$705$	0	0
$706$	−3060.58	−0.163154
$707$	8091.96	0.430452
$708$	0	0
$709$	34264.7	1.81500	0.907502	−	0.420047i	$-0.137986\pi$
$709$	34264.7	1.81500	0.907502	+	0.420047i	$0.137986\pi$
$710$	0	0
$711$	0	0
$712$	11530.5	0.606916
$713$	3155.16	0.165725
$714$	0	0
$715$	0	0
$716$	29568.8	1.54335
$717$	0	0
$718$	−2681.86	−0.139396
$719$	25469.2	1.32106	0.660529	−	0.750801i	$-0.270332\pi$
$719$	25469.2	1.32106	0.660529	+	0.750801i	$0.270332\pi$
$720$	0	0
$721$	547.452	0.0282776
$722$	7703.40	0.397079
$723$	0	0
$724$	23256.9	1.19383
$725$	0	0
$726$	0	0
$727$	−24294.6	−1.23939	−0.619695	−	0.784843i	$-0.712744\pi$
$727$	−24294.6	−1.23939	−0.619695	+	0.784843i	$0.712744\pi$
$728$	−958.503	−0.0487974
$729$	0	0
$730$	0	0
$731$	26876.6	1.35987
$732$	0	0
$733$	−34870.8	−1.75714	−0.878569	−	0.477615i	$-0.841501\pi$
$733$	−34870.8	−1.75714	−0.878569	+	0.477615i	$0.841501\pi$
$734$	5411.29	0.272118
$735$	0	0
$736$	−7923.40	−0.396821
$737$	21473.9	1.07327
$738$	0	0
$739$	7028.81	0.349877	0.174938	−	0.984579i	$-0.444027\pi$
$739$	7028.81	0.349877	0.174938	+	0.984579i	$0.444027\pi$
$740$	0	0
$741$	0	0
$742$	1014.08	0.0501725
$743$	1368.74	0.0675831	0.0337915	−	0.999429i	$-0.489242\pi$
$743$	1368.74	0.0675831	0.0337915	+	0.999429i	$0.489242\pi$
$744$	0	0
$745$	0	0
$746$	6839.94	0.335694
$747$	0	0
$748$	−18872.9	−0.922541
$749$	−658.570	−0.0321276
$750$	0	0
$751$	37144.0	1.80480	0.902400	−	0.430900i	$-0.141804\pi$
$751$	37144.0	1.80480	0.902400	+	0.430900i	$0.141804\pi$
$752$	−14552.1	−0.705665
$753$	0	0
$754$	−1782.43	−0.0860907
$755$	0	0
$756$	0	0
$757$	−12042.4	−0.578189	−0.289095	−	0.957301i	$-0.593354\pi$
$757$	−12042.4	−0.578189	−0.289095	+	0.957301i	$0.593354\pi$
$758$	948.683	0.0454588
$759$	0	0
$760$	0	0
$761$	−6112.92	−0.291187	−0.145593	−	0.989345i	$-0.546509\pi$
$761$	−6112.92	−0.291187	−0.145593	+	0.989345i	$0.546509\pi$
$762$	0	0
$763$	14897.7	0.706860
$764$	9767.88	0.462552
$765$	0	0
$766$	−849.700	−0.0400795
$767$	−10126.4	−0.476717
$768$	0	0
$769$	−957.146	−0.0448837	−0.0224418	−	0.999748i	$-0.507144\pi$
$769$	−957.146	−0.0448837	−0.0224418	+	0.999748i	$0.507144\pi$
$770$	0	0
$771$	0	0
$772$	14570.8	0.679295
$773$	32867.9	1.52934	0.764668	−	0.644424i	$-0.222903\pi$
$773$	32867.9	1.52934	0.764668	+	0.644424i	$0.222903\pi$
$774$	0	0
$775$	0	0
$776$	12025.5	0.556303
$777$	0	0
$778$	−6831.58	−0.314812
$779$	3547.46	0.163159
$780$	0	0
$781$	4515.45	0.206883
$782$	−4142.69	−0.189440
$783$	0	0
$784$	2770.03	0.126186
$785$	0	0
$786$	0	0
$787$	−15320.0	−0.693900	−0.346950	−	0.937884i	$-0.612783\pi$
$787$	−15320.0	−0.693900	−0.346950	+	0.937884i	$0.612783\pi$
$788$	31260.4	1.41320
$789$	0	0
$790$	0	0
$791$	−4175.45	−0.187689
$792$	0	0
$793$	−7016.15	−0.314188
$794$	4129.32	0.184564
$795$	0	0
$796$	−600.453	−0.0267368
$797$	43629.0	1.93904	0.969522	−	0.245004i	$-0.0787893\pi$
$797$	43629.0	1.93904	0.969522	+	0.245004i	$0.0787893\pi$
$798$	0	0
$799$	−24496.3	−1.08463
$800$	0	0
$801$	0	0
$802$	−4201.33	−0.184980
$803$	−1223.72	−0.0537784
$804$	0	0
$805$	0	0
$806$	355.314	0.0155278
$807$	0	0
$808$	10181.7	0.443307
$809$	−127.735	−0.00555119	−0.00277560	−	0.999996i	$-0.500884\pi$
$809$	−127.735	−0.00555119	−0.00277560	+	0.999996i	$0.500884\pi$
$810$	0	0
$811$	16227.5	0.702618	0.351309	−	0.936260i	$-0.385737\pi$
$811$	16227.5	0.702618	0.351309	+	0.936260i	$0.385737\pi$
$812$	10982.9	0.474659
$813$	0	0
$814$	1386.48	0.0597005
$815$	0	0
$816$	0	0
$817$	−40513.7	−1.73488
$818$	5193.77	0.222000
$819$	0	0
$820$	0	0
$821$	−34249.2	−1.45591	−0.727957	−	0.685623i	$-0.759530\pi$
$821$	−34249.2	−1.45591	−0.727957	+	0.685623i	$0.759530\pi$
$822$	0	0
$823$	6624.51	0.280578	0.140289	−	0.990111i	$-0.455197\pi$
$823$	6624.51	0.280578	0.140289	+	0.990111i	$0.455197\pi$
$824$	688.832	0.0291221
$825$	0	0
$826$	−2560.43	−0.107856
$827$	−33786.8	−1.42065	−0.710327	−	0.703872i	$-0.751453\pi$
$827$	−33786.8	−1.42065	−0.710327	+	0.703872i	$0.751453\pi$
$828$	0	0
$829$	−30283.8	−1.26876	−0.634378	−	0.773023i	$-0.718744\pi$
$829$	−30283.8	−1.26876	−0.634378	+	0.773023i	$0.718744\pi$
$830$	0	0
$831$	0	0
$832$	6138.57	0.255789
$833$	4662.93	0.193951
$834$	0	0
$835$	0	0
$836$	28449.0	1.17695
$837$	0	0
$838$	1888.72	0.0778578
$839$	−16810.9	−0.691750	−0.345875	−	0.938281i	$-0.612418\pi$
$839$	−16810.9	−0.691750	−0.345875	+	0.938281i	$0.612418\pi$
$840$	0	0
$841$	17296.6	0.709195
$842$	−2043.41	−0.0836350
$843$	0	0
$844$	9936.39	0.405242
$845$	0	0
$846$	0	0
$847$	4654.72	0.188829
$848$	−14583.8	−0.590579
$849$	0	0
$850$	0	0
$851$	−7416.56	−0.298750
$852$	0	0
$853$	−14875.8	−0.597112	−0.298556	−	0.954392i	$-0.596505\pi$
$853$	−14875.8	−0.597112	−0.298556	+	0.954392i	$0.596505\pi$
$854$	−1774.02	−0.0710839
$855$	0	0
$856$	−828.647	−0.0330871
$857$	−3987.55	−0.158941	−0.0794703	−	0.996837i	$-0.525323\pi$
$857$	−3987.55	−0.158941	−0.0794703	+	0.996837i	$0.525323\pi$
$858$	0	0
$859$	39344.7	1.56278	0.781388	−	0.624046i	$-0.214512\pi$
$859$	39344.7	1.56278	0.781388	+	0.624046i	$0.214512\pi$
$860$	0	0
$861$	0	0
$862$	−6313.63	−0.249470
$863$	−15627.0	−0.616397	−0.308198	−	0.951322i	$-0.599726\pi$
$863$	−15627.0	−0.616397	−0.308198	+	0.951322i	$0.599726\pi$
$864$	0	0
$865$	0	0
$866$	105.197	0.00412787
$867$	0	0
$868$	−2189.35	−0.0856122
$869$	−29.9386	−0.00116870
$870$	0	0
$871$	−12935.7	−0.503226
$872$	18745.1	0.727969
$873$	0	0
$874$	6244.69	0.241682
$875$	0	0
$876$	0	0
$877$	14519.0	0.559034	0.279517	−	0.960141i	$-0.409826\pi$
$877$	14519.0	0.559034	0.279517	+	0.960141i	$0.409826\pi$
$878$	−6446.29	−0.247781
$879$	0	0
$880$	0	0
$881$	−24177.7	−0.924592	−0.462296	−	0.886726i	$-0.652974\pi$
$881$	−24177.7	−0.924592	−0.462296	+	0.886726i	$0.652974\pi$
$882$	0	0
$883$	−10340.3	−0.394089	−0.197044	−	0.980395i	$-0.563134\pi$
$883$	−10340.3	−0.394089	−0.197044	+	0.980395i	$0.563134\pi$
$884$	11368.9	0.432553
$885$	0	0
$886$	5329.45	0.202084
$887$	−3222.62	−0.121990	−0.0609949	−	0.998138i	$-0.519427\pi$
$887$	−3222.62	−0.121990	−0.0609949	+	0.998138i	$0.519427\pi$
$888$	0	0
$889$	−14617.5	−0.551467
$890$	0	0
$891$	0	0
$892$	13729.8	0.515367
$893$	36925.6	1.38373
$894$	0	0
$895$	0	0
$896$	7275.74	0.271278
$897$	0	0
$898$	3114.70	0.115745
$899$	−8309.70	−0.308280
$900$	0	0
$901$	−24549.7	−0.907735
$902$	−358.399	−0.0132299
$903$	0	0
$904$	−5253.77	−0.193294
$905$	0	0
$906$	0	0
$907$	31692.8	1.16025	0.580123	−	0.814529i	$-0.303005\pi$
$907$	31692.8	1.16025	0.580123	+	0.814529i	$0.303005\pi$
$908$	−32598.5	−1.19143
$909$	0	0
$910$	0	0
$911$	−2403.15	−0.0873985	−0.0436993	−	0.999045i	$-0.513914\pi$
$911$	−2403.15	−0.0873985	−0.0436993	+	0.999045i	$0.513914\pi$
$912$	0	0
$913$	−38557.6	−1.39767
$914$	997.843	0.0361113
$915$	0	0
$916$	−26153.6	−0.943385
$917$	2912.99	0.104902
$918$	0	0
$919$	19622.5	0.704339	0.352170	−	0.935936i	$-0.385444\pi$
$919$	19622.5	0.704339	0.352170	+	0.935936i	$0.385444\pi$
$920$	0	0
$921$	0	0
$922$	5087.56	0.181724
$923$	−2720.07	−0.0970014
$924$	0	0
$925$	0	0
$926$	8355.06	0.296506
$927$	0	0
$928$	20867.7	0.738165
$929$	12930.5	0.456660	0.228330	−	0.973584i	$-0.426673\pi$
$929$	12930.5	0.456660	0.228330	+	0.973584i	$0.426673\pi$
$930$	0	0
$931$	−7028.90	−0.247436
$932$	−30001.6	−1.05444
$933$	0	0
$934$	−8016.64	−0.280848
$935$	0	0
$936$	0	0
$937$	18717.1	0.652573	0.326287	−	0.945271i	$-0.394203\pi$
$937$	18717.1	0.652573	0.326287	+	0.945271i	$0.394203\pi$
$938$	−3270.76	−0.113853
$939$	0	0
$940$	0	0
$941$	10152.9	0.351727	0.175863	−	0.984415i	$-0.443728\pi$
$941$	10152.9	0.351727	0.175863	+	0.984415i	$0.443728\pi$
$942$	0	0
$943$	1917.15	0.0662045
$944$	36822.4	1.26956
$945$	0	0
$946$	4093.09	0.140674
$947$	−15836.2	−0.543408	−0.271704	−	0.962381i	$-0.587587\pi$
$947$	−15836.2	−0.543408	−0.271704	+	0.962381i	$0.587587\pi$
$948$	0	0
$949$	737.158	0.0252151
$950$	0	0
$951$	0	0
$952$	5867.15	0.199743
$953$	−4847.86	−0.164782	−0.0823911	−	0.996600i	$-0.526256\pi$
$953$	−4847.86	−0.164782	−0.0823911	+	0.996600i	$0.526256\pi$
$954$	0	0
$955$	0	0
$956$	51240.0	1.73349
$957$	0	0
$958$	8635.44	0.291230
$959$	4334.76	0.145961
$960$	0	0
$961$	−28134.5	−0.944397
$962$	−835.207	−0.0279918
$963$	0	0
$964$	46138.6	1.54152
$965$	0	0
$966$	0	0
$967$	17153.8	0.570454	0.285227	−	0.958460i	$-0.407931\pi$
$967$	17153.8	0.570454	0.285227	+	0.958460i	$0.407931\pi$
$968$	5856.80	0.194468
$969$	0	0
$970$	0	0
$971$	−50352.2	−1.66414	−0.832070	−	0.554670i	$-0.812844\pi$
$971$	−50352.2	−1.66414	−0.832070	+	0.554670i	$0.812844\pi$
$972$	0	0
$973$	1868.96	0.0615788
$974$	11053.2	0.363623
$975$	0	0
$976$	25512.8	0.836725
$977$	−1510.03	−0.0494474	−0.0247237	−	0.999694i	$-0.507871\pi$
$977$	−1510.03	−0.0494474	−0.0247237	+	0.999694i	$0.507871\pi$
$978$	0	0
$979$	−33785.7	−1.10296
$980$	0	0
$981$	0	0
$982$	−10998.1	−0.357396
$983$	5310.89	0.172321	0.0861603	−	0.996281i	$-0.472540\pi$
$983$	5310.89	0.172321	0.0861603	+	0.996281i	$0.472540\pi$
$984$	0	0
$985$	0	0
$986$	10910.6	0.352396
$987$	0	0
$988$	−17137.4	−0.551836
$989$	−21894.7	−0.703956
$990$	0	0
$991$	35845.4	1.14901	0.574505	−	0.818501i	$-0.305195\pi$
$991$	35845.4	1.14901	0.574505	+	0.818501i	$0.305195\pi$
$992$	−4159.82	−0.133140
$993$	0	0
$994$	−687.764	−0.0219462
$995$	0	0
$996$	0	0
$997$	−17857.5	−0.567253	−0.283627	−	0.958935i	$-0.591538\pi$
$997$	−17857.5	−0.567253	−0.283627	+	0.958935i	$0.591538\pi$
$998$	−3868.24	−0.122692
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.o.1.2		2
3.2	odd	2		525.4.a.m.1.1	yes	2
5.4	even	2		1575.4.a.x.1.1		2
15.2	even	4		525.4.d.m.274.2		4
15.8	even	4		525.4.d.m.274.3		4
15.14	odd	2		525.4.a.j.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.j.1.2	✓	2	15.14	odd	2
525.4.a.m.1.1	yes	2	3.2	odd	2
525.4.d.m.274.2		4	15.2	even	4
525.4.d.m.274.3		4	15.8	even	4
1575.4.a.o.1.2		2	1.1	even	1	trivial
1575.4.a.x.1.1		2	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+0.561553 q^{2} -7.68466 q^{4} -7.00000 q^{7} -8.80776 q^{8} +25.8078 q^{11} -15.5464 q^{13} -3.93087 q^{14} +56.5312 q^{16} +95.1619 q^{17} -143.447 q^{19} +14.4924 q^{22} -77.5227 q^{23} -8.73012 q^{26} +53.7926 q^{28} +204.170 q^{29} -40.6998 q^{31} +102.207 q^{32} +53.4384 q^{34} +95.6695 q^{37} -80.5530 q^{38} -24.7301 q^{41} +282.430 q^{43} -198.324 q^{44} -43.5331 q^{46} -257.417 q^{47} +49.0000 q^{49} +119.469 q^{52} -257.978 q^{53} +61.6543 q^{56} +114.652 q^{58} +651.365 q^{59} +451.304 q^{61} -22.8551 q^{62} -394.855 q^{64} +832.071 q^{67} -731.287 q^{68} +174.965 q^{71} -47.4166 q^{73} +53.7235 q^{74} +1102.34 q^{76} -180.654 q^{77} -1.16006 q^{79} -13.8873 q^{82} -1494.03 q^{83} +158.599 q^{86} -227.309 q^{88} -1309.13 q^{89} +108.825 q^{91} +595.736 q^{92} -144.553 q^{94} -1365.33 q^{97} +27.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{4} - 14 q^{7} + 3 q^{8} + 31 q^{11} + 39 q^{13} + 21 q^{14} - 23 q^{16} + 79 q^{17} - 56 q^{19} - 4 q^{22} - 254 q^{23} - 203 q^{26} + 21 q^{28} + 62 q^{29} - 135 q^{31} + 291 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^4 - 14 * q^7 + 3 * q^8 + 31 * q^11 + 39 * q^13 + 21 * q^14 - 23 * q^16 + 79 * q^17 - 56 * q^19 - 4 * q^22 - 254 * q^23 - 203 * q^26 + 21 * q^28 + 62 * q^29 - 135 * q^31 + 291 * q^32 + 111 * q^34 + 113 * q^37 - 392 * q^38 - 235 * q^41 + 804 * q^43 - 174 * q^44 + 585 * q^46 - 152 * q^47 + 98 * q^49 + 375 * q^52 - 149 * q^53 - 21 * q^56 + 621 * q^58 + 441 * q^59 - 223 * q^61 + 313 * q^62 - 431 * q^64 + 1157 * q^67 - 807 * q^68 - 619 * q^71 + 268 * q^73 - 8 * q^74 + 1512 * q^76 - 217 * q^77 - 427 * q^79 + 735 * q^82 - 1211 * q^83 - 1699 * q^86 - 166 * q^88 - 466 * q^89 - 273 * q^91 - 231 * q^92 - 520 * q^94 + 172 * q^97 - 147 * q^98

Introduction

L-functions

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