LMFDB - Embedded newform 1575.4.a.bf.1.4

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 = [4,-6,0,16,0,0,-28,-93,0,0,-57] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.26729725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 18x^{2} + 19x + 44 $ x^4 - 2x^3 - 18x^2 + 19*x + 44
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			1.4
Root			$-3.56826$ 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)

$f(q)$	$=$	$q+2.56826 q^{2} -1.40404 q^{4} -7.00000 q^{7} -24.1520 q^{8} -66.4181 q^{11} +34.0329 q^{13} -17.9778 q^{14} -50.7963 q^{16} +6.12934 q^{17} -163.424 q^{19} -170.579 q^{22} +35.8799 q^{23} +87.4053 q^{26} +9.82830 q^{28} -27.7256 q^{29} +74.3417 q^{31} +62.7581 q^{32} +15.7417 q^{34} +260.966 q^{37} -419.716 q^{38} +445.485 q^{41} +474.985 q^{43} +93.2538 q^{44} +92.1489 q^{46} -51.0436 q^{47} +49.0000 q^{49} -47.7837 q^{52} -676.667 q^{53} +169.064 q^{56} -71.2066 q^{58} +115.979 q^{59} -390.356 q^{61} +190.929 q^{62} +567.550 q^{64} +713.721 q^{67} -8.60586 q^{68} -810.524 q^{71} -350.196 q^{73} +670.227 q^{74} +229.455 q^{76} +464.927 q^{77} -50.9307 q^{79} +1144.12 q^{82} -84.8904 q^{83} +1219.88 q^{86} +1604.13 q^{88} -1521.82 q^{89} -238.230 q^{91} -50.3769 q^{92} -131.093 q^{94} +1232.08 q^{97} +125.845 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 + 16 * q^4 - 28 * q^7 - 93 * q^8 - 57 * q^11 + 43 * q^13 + 42 * q^14 + 216 * q^16 - 99 * q^17 - 12 * q^19 - 41 * q^22 - 156 * q^23 + 81 * q^26 - 112 * q^28 - 378 * q^29 - 93 * q^31 - 690 * q^32 + 783 * q^34 + 81 * q^37 - 216 * q^38 + 465 * q^41 - 64 * q^43 - 681 * q^44 + 310 * q^46 - 744 * q^47 + 196 * q^49 - 727 * q^52 - 729 * q^53 + 651 * q^56 + 1172 * q^58 - 231 * q^59 - 1353 * q^61 + 165 * q^62 + 3107 * q^64 + 1487 * q^67 - 2577 * q^68 + 1725 * q^71 + 512 * q^73 + 1953 * q^74 - 3046 * q^76 + 399 * q^77 + 1629 * q^79 - 693 * q^82 - 321 * q^83 + 4542 * q^86 + 3482 * q^88 + 978 * q^89 - 301 * q^91 - 852 * q^92 + 2480 * q^94 + 2616 * q^97 - 294 * 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 $

$2$	2.56826	0.908017	0.454008	−	0.890997i	$-0.349994\pi$
$2$	2.56826	0.908017	0.454008	+	0.890997i	$0.349994\pi$
$3$	0	0
$3$	0	0
$4$	−1.40404	−0.175505
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−24.1520	−1.06738
$9$	0	0
$10$	0	0
$11$	−66.4181	−1.82053	−0.910264	−	0.414029i	$-0.864121\pi$
$11$	−66.4181	−1.82053	−0.910264	+	0.414029i	$0.864121\pi$
$12$	0	0
$13$	34.0329	0.726079	0.363040	−	0.931774i	$-0.381739\pi$
$13$	34.0329	0.726079	0.363040	+	0.931774i	$0.381739\pi$
$14$	−17.9778	−0.343198
$15$	0	0
$16$	−50.7963	−0.793692
$17$	6.12934	0.0874461	0.0437231	−	0.999044i	$-0.486078\pi$
$17$	6.12934	0.0874461	0.0437231	+	0.999044i	$0.486078\pi$
$18$	0	0
$19$	−163.424	−1.97327	−0.986634	−	0.162951i	$-0.947899\pi$
$19$	−163.424	−1.97327	−0.986634	+	0.162951i	$0.947899\pi$
$20$	0	0
$21$	0	0
$22$	−170.579	−1.65307
$23$	35.8799	0.325282	0.162641	−	0.986685i	$-0.447999\pi$
$23$	35.8799	0.325282	0.162641	+	0.986685i	$0.447999\pi$
$24$	0	0
$25$	0	0
$26$	87.4053	0.659292
$27$	0	0
$28$	9.82830	0.0663348
$29$	−27.7256	−0.177535	−0.0887676	−	0.996052i	$-0.528293\pi$
$29$	−27.7256	−0.177535	−0.0887676	+	0.996052i	$0.528293\pi$
$30$	0	0
$31$	74.3417	0.430715	0.215358	−	0.976535i	$-0.430908\pi$
$31$	74.3417	0.430715	0.215358	+	0.976535i	$0.430908\pi$
$32$	62.7581	0.346693
$33$	0	0
$34$	15.7417	0.0794026
$35$	0	0
$36$	0	0
$37$	260.966	1.15953	0.579763	−	0.814785i	$-0.303145\pi$
$37$	260.966	1.15953	0.579763	+	0.814785i	$0.303145\pi$
$38$	−419.716	−1.79176
$39$	0	0
$40$	0	0
$41$	445.485	1.69690	0.848452	−	0.529272i	$-0.177535\pi$
$41$	445.485	1.69690	0.848452	+	0.529272i	$0.177535\pi$
$42$	0	0
$43$	474.985	1.68452	0.842261	−	0.539070i	$-0.181224\pi$
$43$	474.985	1.68452	0.842261	+	0.539070i	$0.181224\pi$
$44$	93.2538	0.319512
$45$	0	0
$46$	92.1489	0.295361
$47$	−51.0436	−0.158414	−0.0792072	−	0.996858i	$-0.525239\pi$
$47$	−51.0436	−0.158414	−0.0792072	+	0.996858i	$0.525239\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−47.7837	−0.127431
$53$	−676.667	−1.75372	−0.876862	−	0.480742i	$-0.840367\pi$
$53$	−676.667	−1.75372	−0.876862	+	0.480742i	$0.840367\pi$
$54$	0	0
$55$	0	0
$56$	169.064	0.403431
$57$	0	0
$58$	−71.2066	−0.161205
$59$	115.979	0.255917	0.127959	−	0.991779i	$-0.459158\pi$
$59$	115.979	0.255917	0.127959	+	0.991779i	$0.459158\pi$
$60$	0	0
$61$	−390.356	−0.819344	−0.409672	−	0.912233i	$-0.634357\pi$
$61$	−390.356	−0.819344	−0.409672	+	0.912233i	$0.634357\pi$
$62$	190.929	0.391097
$63$	0	0
$64$	567.550	1.10850
$65$	0	0
$66$	0	0
$67$	713.721	1.30142	0.650708	−	0.759328i	$-0.274472\pi$
$67$	713.721	1.30142	0.650708	+	0.759328i	$0.274472\pi$
$68$	−8.60586	−0.0153473
$69$	0	0
$70$	0	0
$71$	−810.524	−1.35481	−0.677405	−	0.735610i	$-0.736896\pi$
$71$	−810.524	−1.35481	−0.677405	+	0.735610i	$0.736896\pi$
$72$	0	0
$73$	−350.196	−0.561470	−0.280735	−	0.959785i	$-0.590578\pi$
$73$	−350.196	−0.561470	−0.280735	+	0.959785i	$0.590578\pi$
$74$	670.227	1.05287
$75$	0	0
$76$	229.455	0.346319
$77$	464.927	0.688095
$78$	0	0
$79$	−50.9307	−0.0725336	−0.0362668	−	0.999342i	$-0.511547\pi$
$79$	−50.9307	−0.0725336	−0.0362668	+	0.999342i	$0.511547\pi$
$80$	0	0
$81$	0	0
$82$	1144.12	1.54082
$83$	−84.8904	−0.112264	−0.0561321	−	0.998423i	$-0.517877\pi$
$83$	−84.8904	−0.112264	−0.0561321	+	0.998423i	$0.517877\pi$
$84$	0	0
$85$	0	0
$86$	1219.88	1.52957
$87$	0	0
$88$	1604.13	1.94319
$89$	−1521.82	−1.81251	−0.906253	−	0.422735i	$-0.861070\pi$
$89$	−1521.82	−1.81251	−0.906253	+	0.422735i	$0.861070\pi$
$90$	0	0
$91$	−238.230	−0.274432
$92$	−50.3769	−0.0570887
$93$	0	0
$94$	−131.093	−0.143843
$95$	0	0
$96$	0	0
$97$	1232.08	1.28968	0.644840	−	0.764317i	$-0.276924\pi$
$97$	1232.08	1.28968	0.644840	+	0.764317i	$0.276924\pi$
$98$	125.845	0.129717
$99$	0	0
$100$	0	0
$101$	1661.64	1.63702	0.818510	−	0.574492i	$-0.194800\pi$
$101$	1661.64	1.63702	0.818510	+	0.574492i	$0.194800\pi$
$102$	0	0
$103$	193.631	0.185233	0.0926167	−	0.995702i	$-0.470477\pi$
$103$	193.631	0.185233	0.0926167	+	0.995702i	$0.470477\pi$
$104$	−821.963	−0.775001
$105$	0	0
$106$	−1737.86	−1.59241
$107$	921.040	0.832153	0.416076	−	0.909330i	$-0.363405\pi$
$107$	921.040	0.832153	0.416076	+	0.909330i	$0.363405\pi$
$108$	0	0
$109$	1304.73	1.14652	0.573260	−	0.819373i	$-0.305678\pi$
$109$	1304.73	1.14652	0.573260	+	0.819373i	$0.305678\pi$
$110$	0	0
$111$	0	0
$112$	355.574	0.299988
$113$	237.585	0.197789	0.0988945	−	0.995098i	$-0.468469\pi$
$113$	237.585	0.197789	0.0988945	+	0.995098i	$0.468469\pi$
$114$	0	0
$115$	0	0
$116$	38.9280	0.0311584
$117$	0	0
$118$	297.863	0.232377
$119$	−42.9054	−0.0330515
$120$	0	0
$121$	3080.36	2.31432
$122$	−1002.54	−0.743978
$123$	0	0
$124$	−104.379	−0.0755928
$125$	0	0
$126$	0	0
$127$	1442.31	1.00775	0.503876	−	0.863776i	$-0.331907\pi$
$127$	1442.31	1.00775	0.503876	+	0.863776i	$0.331907\pi$
$128$	955.550	0.659840
$129$	0	0
$130$	0	0
$131$	−644.035	−0.429539	−0.214770	−	0.976665i	$-0.568900\pi$
$131$	−644.035	−0.429539	−0.214770	+	0.976665i	$0.568900\pi$
$132$	0	0
$133$	1143.97	0.745825
$134$	1833.02	1.18171
$135$	0	0
$136$	−148.036	−0.0933381
$137$	−355.576	−0.221744	−0.110872	−	0.993835i	$-0.535364\pi$
$137$	−355.576	−0.221744	−0.110872	+	0.993835i	$0.535364\pi$
$138$	0	0
$139$	−472.016	−0.288028	−0.144014	−	0.989576i	$-0.546001\pi$
$139$	−472.016	−0.288028	−0.144014	+	0.989576i	$0.546001\pi$
$140$	0	0
$141$	0	0
$142$	−2081.64	−1.23019
$143$	−2260.40	−1.32185
$144$	0	0
$145$	0	0
$146$	−899.393	−0.509824
$147$	0	0
$148$	−366.407	−0.203503
$149$	1091.53	0.600143	0.300072	−	0.953917i	$-0.402989\pi$
$149$	1091.53	0.600143	0.300072	+	0.953917i	$0.402989\pi$
$150$	0	0
$151$	2914.48	1.57071	0.785354	−	0.619047i	$-0.212481\pi$
$151$	2914.48	1.57071	0.785354	+	0.619047i	$0.212481\pi$
$152$	3947.03	2.10622
$153$	0	0
$154$	1194.05	0.624802
$155$	0	0
$156$	0	0
$157$	1476.94	0.750780	0.375390	−	0.926867i	$-0.377509\pi$
$157$	1476.94	0.750780	0.375390	+	0.926867i	$0.377509\pi$
$158$	−130.803	−0.0658617
$159$	0	0
$160$	0	0
$161$	−251.159	−0.122945
$162$	0	0
$163$	−2764.56	−1.32845	−0.664224	−	0.747534i	$-0.731238\pi$
$163$	−2764.56	−1.32845	−0.664224	+	0.747534i	$0.731238\pi$
$164$	−625.480	−0.297816
$165$	0	0
$166$	−218.020	−0.101938
$167$	−255.508	−0.118394	−0.0591970	−	0.998246i	$-0.518854\pi$
$167$	−255.508	−0.118394	−0.0591970	+	0.998246i	$0.518854\pi$
$168$	0	0
$169$	−1038.76	−0.472809
$170$	0	0
$171$	0	0
$172$	−666.899	−0.295643
$173$	−3850.56	−1.69221	−0.846105	−	0.533016i	$-0.821059\pi$
$173$	−3850.56	−1.69221	−0.846105	+	0.533016i	$0.821059\pi$
$174$	0	0
$175$	0	0
$176$	3373.79	1.44494
$177$	0	0
$178$	−3908.44	−1.64579
$179$	−952.804	−0.397854	−0.198927	−	0.980014i	$-0.563746\pi$
$179$	−952.804	−0.397854	−0.198927	+	0.980014i	$0.563746\pi$
$180$	0	0
$181$	1648.83	0.677109	0.338554	−	0.940947i	$-0.390062\pi$
$181$	1648.83	0.677109	0.338554	+	0.940947i	$0.390062\pi$
$182$	−611.837	−0.249189
$183$	0	0
$184$	−866.572	−0.347199
$185$	0	0
$186$	0	0
$187$	−407.099	−0.159198
$188$	71.6675	0.0278026
$189$	0	0
$190$	0	0
$191$	−767.536	−0.290769	−0.145385	−	0.989375i	$-0.546442\pi$
$191$	−767.536	−0.290769	−0.145385	+	0.989375i	$0.546442\pi$
$192$	0	0
$193$	1186.61	0.442561	0.221280	−	0.975210i	$-0.428976\pi$
$193$	1186.61	0.442561	0.221280	+	0.975210i	$0.428976\pi$
$194$	3164.31	1.17105
$195$	0	0
$196$	−68.7981	−0.0250722
$197$	3614.72	1.30730	0.653650	−	0.756797i	$-0.273237\pi$
$197$	3614.72	1.30730	0.653650	+	0.756797i	$0.273237\pi$
$198$	0	0
$199$	−3077.54	−1.09629	−0.548144	−	0.836384i	$-0.684665\pi$
$199$	−3077.54	−1.09629	−0.548144	+	0.836384i	$0.684665\pi$
$200$	0	0
$201$	0	0
$202$	4267.52	1.48644
$203$	194.079	0.0671020
$204$	0	0
$205$	0	0
$206$	497.295	0.168195
$207$	0	0
$208$	−1728.75	−0.576284
$209$	10854.3	3.59239
$210$	0	0
$211$	145.901	0.0476031	0.0238016	−	0.999717i	$-0.492423\pi$
$211$	145.901	0.0476031	0.0238016	+	0.999717i	$0.492423\pi$
$212$	950.070	0.307788
$213$	0	0
$214$	2365.47	0.755609
$215$	0	0
$216$	0	0
$217$	−520.392	−0.162795
$218$	3350.89	1.04106
$219$	0	0
$220$	0	0
$221$	208.599	0.0634928
$222$	0	0
$223$	−3357.30	−1.00817	−0.504084	−	0.863655i	$-0.668170\pi$
$223$	−3357.30	−1.00817	−0.504084	+	0.863655i	$0.668170\pi$
$224$	−439.306	−0.131037
$225$	0	0
$226$	610.181	0.179596
$227$	−636.529	−0.186114	−0.0930570	−	0.995661i	$-0.529664\pi$
$227$	−636.529	−0.186114	−0.0930570	+	0.995661i	$0.529664\pi$
$228$	0	0
$229$	3652.17	1.05389	0.526947	−	0.849898i	$-0.323337\pi$
$229$	3652.17	1.05389	0.526947	+	0.849898i	$0.323337\pi$
$230$	0	0
$231$	0	0
$232$	669.630	0.189497
$233$	3387.30	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$233$	3387.30	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$234$	0	0
$235$	0	0
$236$	−162.839	−0.0449149
$237$	0	0
$238$	−110.192	−0.0300113
$239$	3382.64	0.915502	0.457751	−	0.889080i	$-0.348655\pi$
$239$	3382.64	0.915502	0.457751	+	0.889080i	$0.348655\pi$
$240$	0	0
$241$	5809.63	1.55283	0.776413	−	0.630225i	$-0.217037\pi$
$241$	5809.63	1.55283	0.776413	+	0.630225i	$0.217037\pi$
$242$	7911.17	2.10144
$243$	0	0
$244$	548.077	0.143799
$245$	0	0
$246$	0	0
$247$	−5561.80	−1.43275
$248$	−1795.50	−0.459736
$249$	0	0
$250$	0	0
$251$	345.006	0.0867593	0.0433796	−	0.999059i	$-0.486187\pi$
$251$	345.006	0.0867593	0.0433796	+	0.999059i	$0.486187\pi$
$252$	0	0
$253$	−2383.07	−0.592184
$254$	3704.23	0.915056
$255$	0	0
$256$	−2086.30	−0.509350
$257$	−3142.39	−0.762712	−0.381356	−	0.924428i	$-0.624543\pi$
$257$	−3142.39	−0.762712	−0.381356	+	0.924428i	$0.624543\pi$
$258$	0	0
$259$	−1826.76	−0.438260
$260$	0	0
$261$	0	0
$262$	−1654.05	−0.390029
$263$	−4949.02	−1.16034	−0.580170	−	0.814495i	$-0.697014\pi$
$263$	−4949.02	−1.16034	−0.580170	+	0.814495i	$0.697014\pi$
$264$	0	0
$265$	0	0
$266$	2938.01	0.677222
$267$	0	0
$268$	−1002.10	−0.228406
$269$	4570.60	1.03596	0.517982	−	0.855391i	$-0.326683\pi$
$269$	4570.60	1.03596	0.517982	+	0.855391i	$0.326683\pi$
$270$	0	0
$271$	−5338.43	−1.19663	−0.598315	−	0.801261i	$-0.704163\pi$
$271$	−5338.43	−1.19663	−0.598315	+	0.801261i	$0.704163\pi$
$272$	−311.348	−0.0694053
$273$	0	0
$274$	−913.213	−0.201347
$275$	0	0
$276$	0	0
$277$	−2225.98	−0.482837	−0.241419	−	0.970421i	$-0.577613\pi$
$277$	−2225.98	−0.482837	−0.241419	+	0.970421i	$0.577613\pi$
$278$	−1212.26	−0.261534
$279$	0	0
$280$	0	0
$281$	−1991.15	−0.422712	−0.211356	−	0.977409i	$-0.567788\pi$
$281$	−1991.15	−0.422712	−0.211356	+	0.977409i	$0.567788\pi$
$282$	0	0
$283$	7350.20	1.54390	0.771951	−	0.635682i	$-0.219281\pi$
$283$	7350.20	1.54390	0.771951	+	0.635682i	$0.219281\pi$
$284$	1138.01	0.237777
$285$	0	0
$286$	−5805.29	−1.20026
$287$	−3118.39	−0.641369
$288$	0	0
$289$	−4875.43	−0.992353
$290$	0	0
$291$	0	0
$292$	491.690	0.0985410
$293$	6250.16	1.24621	0.623103	−	0.782140i	$-0.285872\pi$
$293$	6250.16	1.24621	0.623103	+	0.782140i	$0.285872\pi$
$294$	0	0
$295$	0	0
$296$	−6302.85	−1.23765
$297$	0	0
$298$	2803.32	0.544940
$299$	1221.10	0.236180
$300$	0	0
$301$	−3324.89	−0.636690
$302$	7485.13	1.42623
$303$	0	0
$304$	8301.35	1.56617
$305$	0	0
$306$	0	0
$307$	1259.29	0.234109	0.117054	−	0.993126i	$-0.462655\pi$
$307$	1259.29	0.234109	0.117054	+	0.993126i	$0.462655\pi$
$308$	−652.777	−0.120764
$309$	0	0
$310$	0	0
$311$	6711.92	1.22379	0.611894	−	0.790940i	$-0.290408\pi$
$311$	6711.92	1.22379	0.611894	+	0.790940i	$0.290408\pi$
$312$	0	0
$313$	−2436.27	−0.439955	−0.219978	−	0.975505i	$-0.570598\pi$
$313$	−2436.27	−0.439955	−0.219978	+	0.975505i	$0.570598\pi$
$314$	3793.16	0.681721
$315$	0	0
$316$	71.5089	0.0127300
$317$	1191.81	0.211164	0.105582	−	0.994411i	$-0.466330\pi$
$317$	1191.81	0.211164	0.105582	+	0.994411i	$0.466330\pi$
$318$	0	0
$319$	1841.48	0.323208
$320$	0	0
$321$	0	0
$322$	−645.042	−0.111636
$323$	−1001.68	−0.172555
$324$	0	0
$325$	0	0
$326$	−7100.11	−1.20625
$327$	0	0
$328$	−10759.4	−1.81124
$329$	357.305	0.0598750
$330$	0	0
$331$	5885.63	0.977351	0.488676	−	0.872466i	$-0.337480\pi$
$331$	5885.63	0.977351	0.488676	+	0.872466i	$0.337480\pi$
$332$	119.190	0.0197030
$333$	0	0
$334$	−656.210	−0.107504
$335$	0	0
$336$	0	0
$337$	−1209.73	−0.195544	−0.0977720	−	0.995209i	$-0.531172\pi$
$337$	−1209.73	−0.195544	−0.0977720	+	0.995209i	$0.531172\pi$
$338$	−2667.81	−0.429319
$339$	0	0
$340$	0	0
$341$	−4937.63	−0.784129
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−11471.8	−1.79802
$345$	0	0
$346$	−9889.23	−1.53656
$347$	−5065.21	−0.783617	−0.391808	−	0.920047i	$-0.628150\pi$
$347$	−5065.21	−0.783617	−0.391808	+	0.920047i	$0.628150\pi$
$348$	0	0
$349$	5481.35	0.840716	0.420358	−	0.907358i	$-0.361904\pi$
$349$	5481.35	0.840716	0.420358	+	0.907358i	$0.361904\pi$
$350$	0	0
$351$	0	0
$352$	−4168.27	−0.631163
$353$	−602.190	−0.0907970	−0.0453985	−	0.998969i	$-0.514456\pi$
$353$	−602.190	−0.0907970	−0.0453985	+	0.998969i	$0.514456\pi$
$354$	0	0
$355$	0	0
$356$	2136.71	0.318105
$357$	0	0
$358$	−2447.05	−0.361259
$359$	−2004.84	−0.294740	−0.147370	−	0.989081i	$-0.547081\pi$
$359$	−2004.84	−0.294740	−0.147370	+	0.989081i	$0.547081\pi$
$360$	0	0
$361$	19848.5	2.89379
$362$	4234.63	0.614826
$363$	0	0
$364$	334.486	0.0481643
$365$	0	0
$366$	0	0
$367$	7770.37	1.10520	0.552602	−	0.833445i	$-0.313635\pi$
$367$	7770.37	1.10520	0.552602	+	0.833445i	$0.313635\pi$
$368$	−1822.57	−0.258174
$369$	0	0
$370$	0	0
$371$	4736.67	0.662846
$372$	0	0
$373$	−7485.06	−1.03904	−0.519520	−	0.854458i	$-0.673889\pi$
$373$	−7485.06	−1.03904	−0.519520	+	0.854458i	$0.673889\pi$
$374$	−1045.54	−0.144555
$375$	0	0
$376$	1232.81	0.169088
$377$	−943.583	−0.128905
$378$	0	0
$379$	−4053.61	−0.549393	−0.274697	−	0.961531i	$-0.588577\pi$
$379$	−4053.61	−0.549393	−0.274697	+	0.961531i	$0.588577\pi$
$380$	0	0
$381$	0	0
$382$	−1971.23	−0.264023
$383$	−4531.33	−0.604543	−0.302272	−	0.953222i	$-0.597745\pi$
$383$	−4531.33	−0.604543	−0.302272	+	0.953222i	$0.597745\pi$
$384$	0	0
$385$	0	0
$386$	3047.53	0.401853
$387$	0	0
$388$	−1729.90	−0.226346
$389$	1160.19	0.151219	0.0756093	−	0.997138i	$-0.475910\pi$
$389$	1160.19	0.151219	0.0756093	+	0.997138i	$0.475910\pi$
$390$	0	0
$391$	219.920	0.0284446
$392$	−1183.45	−0.152483
$393$	0	0
$394$	9283.53	1.18705
$395$	0	0
$396$	0	0
$397$	−6720.74	−0.849632	−0.424816	−	0.905280i	$-0.639661\pi$
$397$	−6720.74	−0.849632	−0.424816	+	0.905280i	$0.639661\pi$
$398$	−7903.93	−0.995448
$399$	0	0
$400$	0	0
$401$	−2769.06	−0.344839	−0.172419	−	0.985024i	$-0.555158\pi$
$401$	−2769.06	−0.344839	−0.172419	+	0.985024i	$0.555158\pi$
$402$	0	0
$403$	2530.07	0.312733
$404$	−2333.01	−0.287306
$405$	0	0
$406$	498.446	0.0609297
$407$	−17332.8	−2.11095
$408$	0	0
$409$	14438.9	1.74561	0.872806	−	0.488067i	$-0.162298\pi$
$409$	14438.9	1.74561	0.872806	+	0.488067i	$0.162298\pi$
$410$	0	0
$411$	0	0
$412$	−271.866	−0.0325095
$413$	−811.850	−0.0967277
$414$	0	0
$415$	0	0
$416$	2135.84	0.251726
$417$	0	0
$418$	27876.7	3.26195
$419$	−6396.35	−0.745780	−0.372890	−	0.927875i	$-0.621633\pi$
$419$	−6396.35	−0.745780	−0.372890	+	0.927875i	$0.621633\pi$
$420$	0	0
$421$	−8231.07	−0.952868	−0.476434	−	0.879210i	$-0.658071\pi$
$421$	−8231.07	−0.952868	−0.476434	+	0.879210i	$0.658071\pi$
$422$	374.712	0.0432244
$423$	0	0
$424$	16342.9	1.87189
$425$	0	0
$426$	0	0
$427$	2732.49	0.309683
$428$	−1293.18	−0.146047
$429$	0	0
$430$	0	0
$431$	5975.72	0.667843	0.333922	−	0.942601i	$-0.391628\pi$
$431$	5975.72	0.667843	0.333922	+	0.942601i	$0.391628\pi$
$432$	0	0
$433$	−10912.9	−1.21118	−0.605591	−	0.795776i	$-0.707063\pi$
$433$	−10912.9	−1.21118	−0.605591	+	0.795776i	$0.707063\pi$
$434$	−1336.50	−0.147821
$435$	0	0
$436$	−1831.90	−0.201221
$437$	−5863.65	−0.641868
$438$	0	0
$439$	3784.05	0.411396	0.205698	−	0.978616i	$-0.434054\pi$
$439$	3784.05	0.411396	0.205698	+	0.978616i	$0.434054\pi$
$440$	0	0
$441$	0	0
$442$	535.737	0.0576526
$443$	−7869.34	−0.843981	−0.421991	−	0.906600i	$-0.638669\pi$
$443$	−7869.34	−0.843981	−0.421991	+	0.906600i	$0.638669\pi$
$444$	0	0
$445$	0	0
$446$	−8622.42	−0.915434
$447$	0	0
$448$	−3972.85	−0.418972
$449$	15601.8	1.63986	0.819929	−	0.572465i	$-0.194013\pi$
$449$	15601.8	1.63986	0.819929	+	0.572465i	$0.194013\pi$
$450$	0	0
$451$	−29588.3	−3.08926
$452$	−333.580	−0.0347130
$453$	0	0
$454$	−1634.77	−0.168995
$455$	0	0
$456$	0	0
$457$	8840.78	0.904933	0.452466	−	0.891781i	$-0.350544\pi$
$457$	8840.78	0.904933	0.452466	+	0.891781i	$0.350544\pi$
$458$	9379.71	0.956954
$459$	0	0
$460$	0	0
$461$	10523.7	1.06321	0.531605	−	0.846993i	$-0.321589\pi$
$461$	10523.7	1.06321	0.531605	+	0.846993i	$0.321589\pi$
$462$	0	0
$463$	−10781.6	−1.08221	−0.541107	−	0.840954i	$-0.681994\pi$
$463$	−10781.6	−1.08221	−0.541107	+	0.840954i	$0.681994\pi$
$464$	1408.36	0.140908
$465$	0	0
$466$	8699.46	0.864795
$467$	−5592.55	−0.554160	−0.277080	−	0.960847i	$-0.589367\pi$
$467$	−5592.55	−0.554160	−0.277080	+	0.960847i	$0.589367\pi$
$468$	0	0
$469$	−4996.05	−0.491889
$470$	0	0
$471$	0	0
$472$	−2801.12	−0.273161
$473$	−31547.6	−3.06672
$474$	0	0
$475$	0	0
$476$	60.2410	0.00580072
$477$	0	0
$478$	8687.51	0.831292
$479$	7835.28	0.747397	0.373698	−	0.927550i	$-0.378089\pi$
$479$	7835.28	0.747397	0.373698	+	0.927550i	$0.378089\pi$
$480$	0	0
$481$	8881.42	0.841908
$482$	14920.6	1.40999
$483$	0	0
$484$	−4324.96	−0.406176
$485$	0	0
$486$	0	0
$487$	−5979.24	−0.556356	−0.278178	−	0.960530i	$-0.589731\pi$
$487$	−5979.24	−0.556356	−0.278178	+	0.960530i	$0.589731\pi$
$488$	9427.89	0.874550
$489$	0	0
$490$	0	0
$491$	1792.66	0.164769	0.0823846	−	0.996601i	$-0.473746\pi$
$491$	1792.66	0.164769	0.0823846	+	0.996601i	$0.473746\pi$
$492$	0	0
$493$	−169.940	−0.0155248
$494$	−14284.2	−1.30096
$495$	0	0
$496$	−3776.29	−0.341855
$497$	5673.67	0.512070
$498$	0	0
$499$	−20514.4	−1.84038	−0.920191	−	0.391470i	$-0.871967\pi$
$499$	−20514.4	−1.84038	−0.920191	+	0.391470i	$0.871967\pi$
$500$	0	0
$501$	0	0
$502$	886.065	0.0787789
$503$	16294.5	1.44441	0.722205	−	0.691679i	$-0.243129\pi$
$503$	16294.5	1.44441	0.722205	+	0.691679i	$0.243129\pi$
$504$	0	0
$505$	0	0
$506$	−6120.35	−0.537713
$507$	0	0
$508$	−2025.07	−0.176866
$509$	−12183.8	−1.06097	−0.530487	−	0.847693i	$-0.677991\pi$
$509$	−12183.8	−1.06097	−0.530487	+	0.847693i	$0.677991\pi$
$510$	0	0
$511$	2451.37	0.212216
$512$	−13002.5	−1.12234
$513$	0	0
$514$	−8070.48	−0.692555
$515$	0	0
$516$	0	0
$517$	3390.22	0.288398
$518$	−4691.59	−0.397947
$519$	0	0
$520$	0	0
$521$	11251.3	0.946116	0.473058	−	0.881031i	$-0.343150\pi$
$521$	11251.3	0.946116	0.473058	+	0.881031i	$0.343150\pi$
$522$	0	0
$523$	13493.7	1.12818	0.564090	−	0.825713i	$-0.309227\pi$
$523$	13493.7	1.12818	0.564090	+	0.825713i	$0.309227\pi$
$524$	904.254	0.0753865
$525$	0	0
$526$	−12710.4	−1.05361
$527$	455.666	0.0376644
$528$	0	0
$529$	−10879.6	−0.894192
$530$	0	0
$531$	0	0
$532$	−1606.18	−0.130896
$533$	15161.1	1.23209
$534$	0	0
$535$	0	0
$536$	−17237.8	−1.38910
$537$	0	0
$538$	11738.5	0.940673
$539$	−3254.49	−0.260075
$540$	0	0
$541$	−18810.8	−1.49490	−0.747449	−	0.664319i	$-0.768722\pi$
$541$	−18810.8	−1.49490	−0.747449	+	0.664319i	$0.768722\pi$
$542$	−13710.5	−1.08656
$543$	0	0
$544$	384.666	0.0303169
$545$	0	0
$546$	0	0
$547$	−9822.68	−0.767801	−0.383901	−	0.923374i	$-0.625419\pi$
$547$	−9822.68	−0.767801	−0.383901	+	0.923374i	$0.625419\pi$
$548$	499.245	0.0389173
$549$	0	0
$550$	0	0
$551$	4531.04	0.350324
$552$	0	0
$553$	356.515	0.0274151
$554$	−5716.89	−0.438424
$555$	0	0
$556$	662.731	0.0505504
$557$	23627.9	1.79739	0.898696	−	0.438571i	$-0.144515\pi$
$557$	23627.9	1.79739	0.898696	+	0.438571i	$0.144515\pi$
$558$	0	0
$559$	16165.1	1.22310
$560$	0	0
$561$	0	0
$562$	−5113.79	−0.383830
$563$	−966.441	−0.0723457	−0.0361729	−	0.999346i	$-0.511517\pi$
$563$	−966.441	−0.0723457	−0.0361729	+	0.999346i	$0.511517\pi$
$564$	0	0
$565$	0	0
$566$	18877.2	1.40189
$567$	0	0
$568$	19575.8	1.44610
$569$	−16451.2	−1.21208	−0.606038	−	0.795435i	$-0.707242\pi$
$569$	−16451.2	−1.21208	−0.606038	+	0.795435i	$0.707242\pi$
$570$	0	0
$571$	37.8467	0.00277379	0.00138690	−	0.999999i	$-0.499559\pi$
$571$	37.8467	0.00277379	0.00138690	+	0.999999i	$0.499559\pi$
$572$	3173.70	0.231991
$573$	0	0
$574$	−8008.85	−0.582374
$575$	0	0
$576$	0	0
$577$	23779.1	1.71566	0.857830	−	0.513933i	$-0.171812\pi$
$577$	23779.1	1.71566	0.857830	+	0.513933i	$0.171812\pi$
$578$	−12521.4	−0.901073
$579$	0	0
$580$	0	0
$581$	594.233	0.0424319
$582$	0	0
$583$	44942.9	3.19270
$584$	8457.93	0.599301
$585$	0	0
$586$	16052.0	1.13158
$587$	−12556.0	−0.882863	−0.441432	−	0.897295i	$-0.645529\pi$
$587$	−12556.0	−0.882863	−0.441432	+	0.897295i	$0.645529\pi$
$588$	0	0
$589$	−12149.2	−0.849917
$590$	0	0
$591$	0	0
$592$	−13256.1	−0.920307
$593$	26489.2	1.83437	0.917185	−	0.398461i	$-0.130456\pi$
$593$	26489.2	1.83437	0.917185	+	0.398461i	$0.130456\pi$
$594$	0	0
$595$	0	0
$596$	−1532.55	−0.105328
$597$	0	0
$598$	3136.09	0.214456
$599$	−8910.78	−0.607821	−0.303910	−	0.952701i	$-0.598292\pi$
$599$	−8910.78	−0.607821	−0.303910	+	0.952701i	$0.598292\pi$
$600$	0	0
$601$	2943.50	0.199780	0.0998902	−	0.994998i	$-0.468151\pi$
$601$	2943.50	0.199780	0.0998902	+	0.994998i	$0.468151\pi$
$602$	−8539.18	−0.578125
$603$	0	0
$604$	−4092.05	−0.275668
$605$	0	0
$606$	0	0
$607$	−20445.4	−1.36714	−0.683569	−	0.729886i	$-0.739573\pi$
$607$	−20445.4	−1.36714	−0.683569	+	0.729886i	$0.739573\pi$
$608$	−10256.2	−0.684117
$609$	0	0
$610$	0	0
$611$	−1737.16	−0.115021
$612$	0	0
$613$	−19174.7	−1.26339	−0.631697	−	0.775215i	$-0.717641\pi$
$613$	−19174.7	−1.26339	−0.631697	+	0.775215i	$0.717641\pi$
$614$	3234.18	0.212575
$615$	0	0
$616$	−11228.9	−0.734458
$617$	−21445.6	−1.39930	−0.699650	−	0.714486i	$-0.746661\pi$
$617$	−21445.6	−1.39930	−0.699650	+	0.714486i	$0.746661\pi$
$618$	0	0
$619$	−17031.2	−1.10588	−0.552941	−	0.833220i	$-0.686495\pi$
$619$	−17031.2	−1.10588	−0.552941	+	0.833220i	$0.686495\pi$
$620$	0	0
$621$	0	0
$622$	17237.9	1.11122
$623$	10652.8	0.685063
$624$	0	0
$625$	0	0
$626$	−6256.97	−0.399487
$627$	0	0
$628$	−2073.69	−0.131766
$629$	1599.55	0.101396
$630$	0	0
$631$	25190.1	1.58923	0.794614	−	0.607115i	$-0.207673\pi$
$631$	25190.1	1.58923	0.794614	+	0.607115i	$0.207673\pi$
$632$	1230.08	0.0774208
$633$	0	0
$634$	3060.88	0.191740
$635$	0	0
$636$	0	0
$637$	1667.61	0.103726
$638$	4729.40	0.293478
$639$	0	0
$640$	0	0
$641$	−11755.1	−0.724335	−0.362168	−	0.932113i	$-0.617963\pi$
$641$	−11755.1	−0.724335	−0.362168	+	0.932113i	$0.617963\pi$
$642$	0	0
$643$	−24044.2	−1.47467	−0.737333	−	0.675530i	$-0.763915\pi$
$643$	−24044.2	−1.47467	−0.737333	+	0.675530i	$0.763915\pi$
$644$	352.639	0.0215775
$645$	0	0
$646$	−2572.58	−0.156683
$647$	−5138.50	−0.312234	−0.156117	−	0.987739i	$-0.549898\pi$
$647$	−5138.50	−0.312234	−0.156117	+	0.987739i	$0.549898\pi$
$648$	0	0
$649$	−7703.07	−0.465905
$650$	0	0
$651$	0	0
$652$	3881.56	0.233150
$653$	6665.57	0.399455	0.199727	−	0.979851i	$-0.435994\pi$
$653$	6665.57	0.399455	0.199727	+	0.979851i	$0.435994\pi$
$654$	0	0
$655$	0	0
$656$	−22629.0	−1.34682
$657$	0	0
$658$	917.653	0.0543675
$659$	−8996.31	−0.531785	−0.265893	−	0.964003i	$-0.585667\pi$
$659$	−8996.31	−0.531785	−0.265893	+	0.964003i	$0.585667\pi$
$660$	0	0
$661$	24565.6	1.44553	0.722763	−	0.691096i	$-0.242872\pi$
$661$	24565.6	1.44553	0.722763	+	0.691096i	$0.242872\pi$
$662$	15115.8	0.887451
$663$	0	0
$664$	2050.27	0.119828
$665$	0	0
$666$	0	0
$667$	−994.793	−0.0577489
$668$	358.744	0.0207788
$669$	0	0
$670$	0	0
$671$	25926.7	1.49164
$672$	0	0
$673$	22046.9	1.26277	0.631387	−	0.775468i	$-0.282486\pi$
$673$	22046.9	1.26277	0.631387	+	0.775468i	$0.282486\pi$
$674$	−3106.91	−0.177557
$675$	0	0
$676$	1458.47	0.0829805
$677$	25109.8	1.42548	0.712738	−	0.701430i	$-0.247455\pi$
$677$	25109.8	1.42548	0.712738	+	0.701430i	$0.247455\pi$
$678$	0	0
$679$	−8624.58	−0.487453
$680$	0	0
$681$	0	0
$682$	−12681.1	−0.712002
$683$	20540.1	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$683$	20540.1	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$684$	0	0
$685$	0	0
$686$	−880.913	−0.0490283
$687$	0	0
$688$	−24127.5	−1.33699
$689$	−23029.0	−1.27334
$690$	0	0
$691$	−1635.59	−0.0900446	−0.0450223	−	0.998986i	$-0.514336\pi$
$691$	−1635.59	−0.0900446	−0.0450223	+	0.998986i	$0.514336\pi$
$692$	5406.35	0.296992
$693$	0	0
$694$	−13008.8	−0.711537
$695$	0	0
$696$	0	0
$697$	2730.53	0.148388
$698$	14077.5	0.763385
$699$	0	0
$700$	0	0
$701$	−16895.6	−0.910325	−0.455163	−	0.890408i	$-0.650419\pi$
$701$	−16895.6	−0.910325	−0.455163	+	0.890408i	$0.650419\pi$
$702$	0	0
$703$	−42648.1	−2.28806
$704$	−37695.5	−2.01805
$705$	0	0
$706$	−1546.58	−0.0824452
$707$	−11631.5	−0.618736
$708$	0	0
$709$	−3527.13	−0.186833	−0.0934163	−	0.995627i	$-0.529779\pi$
$709$	−3527.13	−0.186833	−0.0934163	+	0.995627i	$0.529779\pi$
$710$	0	0
$711$	0	0
$712$	36755.1	1.93463
$713$	2667.37	0.140104
$714$	0	0
$715$	0	0
$716$	1337.78	0.0698256
$717$	0	0
$718$	−5148.95	−0.267628
$719$	35693.3	1.85137	0.925687	−	0.378291i	$-0.123488\pi$
$719$	35693.3	1.85137	0.925687	+	0.378291i	$0.123488\pi$
$720$	0	0
$721$	−1355.42	−0.0700116
$722$	50976.1	2.62761
$723$	0	0
$724$	−2315.03	−0.118836
$725$	0	0
$726$	0	0
$727$	18829.9	0.960611	0.480305	−	0.877101i	$-0.340526\pi$
$727$	18829.9	0.960611	0.480305	+	0.877101i	$0.340526\pi$
$728$	5753.74	0.292923
$729$	0	0
$730$	0	0
$731$	2911.34	0.147305
$732$	0	0
$733$	14895.4	0.750577	0.375288	−	0.926908i	$-0.377544\pi$
$733$	14895.4	0.750577	0.375288	+	0.926908i	$0.377544\pi$
$734$	19956.3	1.00354
$735$	0	0
$736$	2251.75	0.112773
$737$	−47404.0	−2.36926
$738$	0	0
$739$	8866.34	0.441345	0.220672	−	0.975348i	$-0.429175\pi$
$739$	8866.34	0.441345	0.220672	+	0.975348i	$0.429175\pi$
$740$	0	0
$741$	0	0
$742$	12165.0	0.601875
$743$	14225.4	0.702394	0.351197	−	0.936302i	$-0.385775\pi$
$743$	14225.4	0.702394	0.351197	+	0.936302i	$0.385775\pi$
$744$	0	0
$745$	0	0
$746$	−19223.6	−0.943465
$747$	0	0
$748$	571.585	0.0279401
$749$	−6447.28	−0.314524
$750$	0	0
$751$	34300.2	1.66662	0.833311	−	0.552804i	$-0.186442\pi$
$751$	34300.2	1.66662	0.833311	+	0.552804i	$0.186442\pi$
$752$	2592.83	0.125732
$753$	0	0
$754$	−2423.37	−0.117047
$755$	0	0
$756$	0	0
$757$	2340.98	0.112397	0.0561984	−	0.998420i	$-0.482102\pi$
$757$	2340.98	0.112397	0.0561984	+	0.998420i	$0.482102\pi$
$758$	−10410.7	−0.498858
$759$	0	0
$760$	0	0
$761$	−23894.9	−1.13823	−0.569113	−	0.822259i	$-0.692713\pi$
$761$	−23894.9	−1.13823	−0.569113	+	0.822259i	$0.692713\pi$
$762$	0	0
$763$	−9133.13	−0.433344
$764$	1077.65	0.0510316
$765$	0	0
$766$	−11637.6	−0.548935
$767$	3947.09	0.185816
$768$	0	0
$769$	22951.1	1.07625	0.538126	−	0.842864i	$-0.319132\pi$
$769$	22951.1	1.07625	0.538126	+	0.842864i	$0.319132\pi$
$770$	0	0
$771$	0	0
$772$	−1666.05	−0.0776718
$773$	13526.7	0.629393	0.314696	−	0.949192i	$-0.398097\pi$
$773$	13526.7	0.629393	0.314696	+	0.949192i	$0.398097\pi$
$774$	0	0
$775$	0	0
$776$	−29757.3	−1.37658
$777$	0	0
$778$	2979.67	0.137309
$779$	−72803.1	−3.34845
$780$	0	0
$781$	53833.5	2.46647
$782$	564.812	0.0258282
$783$	0	0
$784$	−2489.02	−0.113385
$785$	0	0
$786$	0	0
$787$	1611.55	0.0729929	0.0364965	−	0.999334i	$-0.488380\pi$
$787$	1611.55	0.0729929	0.0364965	+	0.999334i	$0.488380\pi$
$788$	−5075.22	−0.229438
$789$	0	0
$790$	0	0
$791$	−1663.10	−0.0747572
$792$	0	0
$793$	−13285.0	−0.594909
$794$	−17260.6	−0.771481
$795$	0	0
$796$	4321.00	0.192404
$797$	−7064.99	−0.313996	−0.156998	−	0.987599i	$-0.550182\pi$
$797$	−7064.99	−0.313996	−0.156998	+	0.987599i	$0.550182\pi$
$798$	0	0
$799$	−312.864	−0.0138527
$800$	0	0
$801$	0	0
$802$	−7111.67	−0.313119
$803$	23259.3	1.02217
$804$	0	0
$805$	0	0
$806$	6497.86	0.283967
$807$	0	0
$808$	−40131.9	−1.74732
$809$	6352.68	0.276079	0.138040	−	0.990427i	$-0.455920\pi$
$809$	6352.68	0.276079	0.138040	+	0.990427i	$0.455920\pi$
$810$	0	0
$811$	−16809.3	−0.727810	−0.363905	−	0.931436i	$-0.618557\pi$
$811$	−16809.3	−0.727810	−0.363905	+	0.931436i	$0.618557\pi$
$812$	−272.496	−0.0117768
$813$	0	0
$814$	−44515.2	−1.91678
$815$	0	0
$816$	0	0
$817$	−77624.0	−3.32401
$818$	37082.7	1.58505
$819$	0	0
$820$	0	0
$821$	41992.9	1.78510	0.892548	−	0.450952i	$-0.148916\pi$
$821$	41992.9	1.78510	0.892548	+	0.450952i	$0.148916\pi$
$822$	0	0
$823$	37046.2	1.56908	0.784538	−	0.620080i	$-0.212900\pi$
$823$	37046.2	1.56908	0.784538	+	0.620080i	$0.212900\pi$
$824$	−4676.58	−0.197714
$825$	0	0
$826$	−2085.04	−0.0878304
$827$	−14371.8	−0.604301	−0.302151	−	0.953260i	$-0.597705\pi$
$827$	−14371.8	−0.604301	−0.302151	+	0.953260i	$0.597705\pi$
$828$	0	0
$829$	−43252.1	−1.81207	−0.906037	−	0.423199i	$-0.860907\pi$
$829$	−43252.1	−1.81207	−0.906037	+	0.423199i	$0.860907\pi$
$830$	0	0
$831$	0	0
$832$	19315.4	0.804855
$833$	300.338	0.0124923
$834$	0	0
$835$	0	0
$836$	−15239.9	−0.630484
$837$	0	0
$838$	−16427.5	−0.677181
$839$	6400.97	0.263392	0.131696	−	0.991290i	$-0.457958\pi$
$839$	6400.97	0.263392	0.131696	+	0.991290i	$0.457958\pi$
$840$	0	0
$841$	−23620.3	−0.968481
$842$	−21139.5	−0.865221
$843$	0	0
$844$	−204.852	−0.00835460
$845$	0	0
$846$	0	0
$847$	−21562.5	−0.874731
$848$	34372.2	1.39192
$849$	0	0
$850$	0	0
$851$	9363.42	0.377173
$852$	0	0
$853$	−17772.0	−0.713368	−0.356684	−	0.934225i	$-0.616093\pi$
$853$	−17772.0	−0.713368	−0.356684	+	0.934225i	$0.616093\pi$
$854$	7017.75	0.281197
$855$	0	0
$856$	−22245.0	−0.888222
$857$	29094.3	1.15968	0.579839	−	0.814731i	$-0.303115\pi$
$857$	29094.3	1.15968	0.579839	+	0.814731i	$0.303115\pi$
$858$	0	0
$859$	−599.045	−0.0237941	−0.0118971	−	0.999929i	$-0.503787\pi$
$859$	−599.045	−0.0237941	−0.0118971	+	0.999929i	$0.503787\pi$
$860$	0	0
$861$	0	0
$862$	15347.2	0.606413
$863$	−19253.4	−0.759435	−0.379717	−	0.925103i	$-0.623979\pi$
$863$	−19253.4	−0.759435	−0.379717	+	0.925103i	$0.623979\pi$
$864$	0	0
$865$	0	0
$866$	−28027.2	−1.09977
$867$	0	0
$868$	730.653	0.0285714
$869$	3382.72	0.132049
$870$	0	0
$871$	24290.0	0.944931
$872$	−31511.9	−1.22377
$873$	0	0
$874$	−15059.4	−0.582827
$875$	0	0
$876$	0	0
$877$	−42907.7	−1.65210	−0.826049	−	0.563598i	$-0.809417\pi$
$877$	−42907.7	−1.65210	−0.826049	+	0.563598i	$0.809417\pi$
$878$	9718.41	0.373554
$879$	0	0
$880$	0	0
$881$	5765.54	0.220484	0.110242	−	0.993905i	$-0.464837\pi$
$881$	5765.54	0.220484	0.110242	+	0.993905i	$0.464837\pi$
$882$	0	0
$883$	35426.8	1.35018	0.675089	−	0.737736i	$-0.264105\pi$
$883$	35426.8	1.35018	0.675089	+	0.737736i	$0.264105\pi$
$884$	−292.883	−0.0111433
$885$	0	0
$886$	−20210.5	−0.766349
$887$	−24413.1	−0.924138	−0.462069	−	0.886844i	$-0.652893\pi$
$887$	−24413.1	−0.924138	−0.462069	+	0.886844i	$0.652893\pi$
$888$	0	0
$889$	−10096.2	−0.380895
$890$	0	0
$891$	0	0
$892$	4713.80	0.176939
$893$	8341.77	0.312594
$894$	0	0
$895$	0	0
$896$	−6688.85	−0.249396
$897$	0	0
$898$	40069.6	1.48902
$899$	−2061.17	−0.0764671
$900$	0	0
$901$	−4147.53	−0.153356
$902$	−75990.3	−2.80510
$903$	0	0
$904$	−5738.17	−0.211116
$905$	0	0
$906$	0	0
$907$	23459.2	0.858818	0.429409	−	0.903110i	$-0.358722\pi$
$907$	23459.2	0.858818	0.429409	+	0.903110i	$0.358722\pi$
$908$	893.714	0.0326640
$909$	0	0
$910$	0	0
$911$	45803.8	1.66580	0.832902	−	0.553421i	$-0.186678\pi$
$911$	45803.8	1.66580	0.832902	+	0.553421i	$0.186678\pi$
$912$	0	0
$913$	5638.25	0.204380
$914$	22705.4	0.821694
$915$	0	0
$916$	−5127.80	−0.184964
$917$	4508.25	0.162351
$918$	0	0
$919$	33360.4	1.19745	0.598725	−	0.800955i	$-0.295674\pi$
$919$	33360.4	1.19745	0.598725	+	0.800955i	$0.295674\pi$
$920$	0	0
$921$	0	0
$922$	27027.7	0.965412
$923$	−27584.5	−0.983700
$924$	0	0
$925$	0	0
$926$	−27690.0	−0.982668
$927$	0	0
$928$	−1740.01	−0.0615501
$929$	−11194.3	−0.395341	−0.197670	−	0.980269i	$-0.563338\pi$
$929$	−11194.3	−0.395341	−0.197670	+	0.980269i	$0.563338\pi$
$930$	0	0
$931$	−8007.79	−0.281895
$932$	−4755.91	−0.167151
$933$	0	0
$934$	−14363.1	−0.503186
$935$	0	0
$936$	0	0
$937$	−4383.20	−0.152821	−0.0764103	−	0.997076i	$-0.524346\pi$
$937$	−4383.20	−0.152821	−0.0764103	+	0.997076i	$0.524346\pi$
$938$	−12831.1	−0.446644
$939$	0	0
$940$	0	0
$941$	27642.7	0.957626	0.478813	−	0.877917i	$-0.341067\pi$
$941$	27642.7	0.957626	0.478813	+	0.877917i	$0.341067\pi$
$942$	0	0
$943$	15984.0	0.551972
$944$	−5891.29	−0.203120
$945$	0	0
$946$	−81022.3	−2.78463
$947$	25657.8	0.880428	0.440214	−	0.897893i	$-0.354902\pi$
$947$	25657.8	0.880428	0.440214	+	0.897893i	$0.354902\pi$
$948$	0	0
$949$	−11918.2	−0.407672
$950$	0	0
$951$	0	0
$952$	1036.25	0.0352785
$953$	48210.6	1.63871	0.819356	−	0.573285i	$-0.194331\pi$
$953$	48210.6	1.63871	0.819356	+	0.573285i	$0.194331\pi$
$954$	0	0
$955$	0	0
$956$	−4749.38	−0.160676
$957$	0	0
$958$	20123.0	0.678649
$959$	2489.04	0.0838114
$960$	0	0
$961$	−24264.3	−0.814484
$962$	22809.8	0.764467
$963$	0	0
$964$	−8156.97	−0.272529
$965$	0	0
$966$	0	0
$967$	17711.5	0.589002	0.294501	−	0.955651i	$-0.404847\pi$
$967$	17711.5	0.589002	0.294501	+	0.955651i	$0.404847\pi$
$968$	−74396.9	−2.47026
$969$	0	0
$970$	0	0
$971$	−35777.8	−1.18245	−0.591227	−	0.806505i	$-0.701356\pi$
$971$	−35777.8	−1.18245	−0.591227	+	0.806505i	$0.701356\pi$
$972$	0	0
$973$	3304.11	0.108864
$974$	−15356.2	−0.505181
$975$	0	0
$976$	19828.7	0.650307
$977$	30436.2	0.996662	0.498331	−	0.866987i	$-0.333946\pi$
$977$	30436.2	0.996662	0.498331	+	0.866987i	$0.333946\pi$
$978$	0	0
$979$	101077.	3.29972
$980$	0	0
$981$	0	0
$982$	4604.02	0.149613
$983$	−57841.2	−1.87675	−0.938376	−	0.345617i	$-0.887670\pi$
$983$	−57841.2	−1.87675	−0.938376	+	0.345617i	$0.887670\pi$
$984$	0	0
$985$	0	0
$986$	−436.450	−0.0140967
$987$	0	0
$988$	7809.01	0.251455
$989$	17042.4	0.547944
$990$	0	0
$991$	1521.84	0.0487819	0.0243909	−	0.999702i	$-0.492235\pi$
$991$	1521.84	0.0487819	0.0243909	+	0.999702i	$0.492235\pi$
$992$	4665.54	0.149326
$993$	0	0
$994$	14571.5	0.464968
$995$	0	0
$996$	0	0
$997$	−46899.5	−1.48979	−0.744896	−	0.667181i	$-0.767501\pi$
$997$	−46899.5	−1.48979	−0.744896	+	0.667181i	$0.767501\pi$
$998$	−52686.3	−1.67110
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bf.1.4		4
3.2	odd	2		525.4.a.v.1.1	yes	4
5.4	even	2		1575.4.a.bm.1.1		4
15.2	even	4		525.4.d.o.274.3		8
15.8	even	4		525.4.d.o.274.6		8
15.14	odd	2		525.4.a.s.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.s.1.4	✓	4	15.14	odd	2
525.4.a.v.1.1	yes	4	3.2	odd	2
525.4.d.o.274.3		8	15.2	even	4
525.4.d.o.274.6		8	15.8	even	4
1575.4.a.bf.1.4		4	1.1	even	1	trivial
1575.4.a.bm.1.1		4	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)

\(f(q)\)	\(=\)	\(q+2.56826 q^{2} -1.40404 q^{4} -7.00000 q^{7} -24.1520 q^{8} -66.4181 q^{11} +34.0329 q^{13} -17.9778 q^{14} -50.7963 q^{16} +6.12934 q^{17} -163.424 q^{19} -170.579 q^{22} +35.8799 q^{23} +87.4053 q^{26} +9.82830 q^{28} -27.7256 q^{29} +74.3417 q^{31} +62.7581 q^{32} +15.7417 q^{34} +260.966 q^{37} -419.716 q^{38} +445.485 q^{41} +474.985 q^{43} +93.2538 q^{44} +92.1489 q^{46} -51.0436 q^{47} +49.0000 q^{49} -47.7837 q^{52} -676.667 q^{53} +169.064 q^{56} -71.2066 q^{58} +115.979 q^{59} -390.356 q^{61} +190.929 q^{62} +567.550 q^{64} +713.721 q^{67} -8.60586 q^{68} -810.524 q^{71} -350.196 q^{73} +670.227 q^{74} +229.455 q^{76} +464.927 q^{77} -50.9307 q^{79} +1144.12 q^{82} -84.8904 q^{83} +1219.88 q^{86} +1604.13 q^{88} -1521.82 q^{89} -238.230 q^{91} -50.3769 q^{92} -131.093 q^{94} +1232.08 q^{97} +125.845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 + 16 * q^4 - 28 * q^7 - 93 * q^8 - 57 * q^11 + 43 * q^13 + 42 * q^14 + 216 * q^16 - 99 * q^17 - 12 * q^19 - 41 * q^22 - 156 * q^23 + 81 * q^26 - 112 * q^28 - 378 * q^29 - 93 * q^31 - 690 * q^32 + 783 * q^34 + 81 * q^37 - 216 * q^38 + 465 * q^41 - 64 * q^43 - 681 * q^44 + 310 * q^46 - 744 * q^47 + 196 * q^49 - 727 * q^52 - 729 * q^53 + 651 * q^56 + 1172 * q^58 - 231 * q^59 - 1353 * q^61 + 165 * q^62 + 3107 * q^64 + 1487 * q^67 - 2577 * q^68 + 1725 * q^71 + 512 * q^73 + 1953 * q^74 - 3046 * q^76 + 399 * q^77 + 1629 * q^79 - 693 * q^82 - 321 * q^83 + 4542 * q^86 + 3482 * q^88 + 978 * q^89 - 301 * q^91 - 852 * q^92 + 2480 * q^94 + 2616 * q^97 - 294 * q^98

Introduction

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