LMFDB - Embedded newform 1575.4.a.bq.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 = [5,4,0,18,0,0,-35,42,0,0,-42] 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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.67516$ 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)

$f(q)$	$=$	$q-1.67516 q^{2} -5.19383 q^{4} -7.00000 q^{7} +22.1018 q^{8} +57.5880 q^{11} +45.5159 q^{13} +11.7261 q^{14} +4.52655 q^{16} +92.0051 q^{17} +125.177 q^{19} -96.4692 q^{22} +158.496 q^{23} -76.2466 q^{26} +36.3568 q^{28} +40.1708 q^{29} +49.5590 q^{31} -184.397 q^{32} -154.123 q^{34} -231.307 q^{37} -209.692 q^{38} -169.556 q^{41} +147.428 q^{43} -299.102 q^{44} -265.507 q^{46} -67.0327 q^{47} +49.0000 q^{49} -236.402 q^{52} +268.647 q^{53} -154.713 q^{56} -67.2926 q^{58} +240.843 q^{59} +90.4579 q^{61} -83.0194 q^{62} +272.683 q^{64} +406.498 q^{67} -477.859 q^{68} -330.782 q^{71} -546.255 q^{73} +387.477 q^{74} -650.149 q^{76} -403.116 q^{77} -25.3087 q^{79} +284.034 q^{82} +376.255 q^{83} -246.965 q^{86} +1272.80 q^{88} -1026.44 q^{89} -318.612 q^{91} -823.203 q^{92} +112.291 q^{94} +942.660 q^{97} -82.0829 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 18 q^{4} - 35 q^{7} + 42 q^{8} - 42 q^{11} - 34 q^{13} - 28 q^{14} + 74 q^{16} + 238 q^{17} - 36 q^{19} - 358 q^{22} + 152 q^{23} + 310 q^{26} - 126 q^{28} + 44 q^{29} + 60 q^{31} + 710 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 18 * q^4 - 35 * q^7 + 42 * q^8 - 42 * q^11 - 34 * q^13 - 28 * q^14 + 74 * q^16 + 238 * q^17 - 36 * q^19 - 358 * q^22 + 152 * q^23 + 310 * q^26 - 126 * q^28 + 44 * q^29 + 60 * q^31 + 710 * q^32 - 482 * q^34 - 312 * q^37 + 280 * q^38 + 426 * q^41 - 304 * q^43 - 712 * q^44 + 88 * q^46 + 370 * q^47 + 245 * q^49 + 1156 * q^52 + 976 * q^53 - 294 * q^56 + 2722 * q^58 + 432 * q^59 - 442 * q^61 - 956 * q^62 + 1362 * q^64 + 804 * q^67 - 420 * q^68 - 440 * q^71 - 564 * q^73 + 1512 * q^74 - 1336 * q^76 + 294 * q^77 + 1790 * q^79 + 3480 * q^82 + 1656 * q^83 - 1216 * q^86 + 1092 * q^88 - 746 * q^89 + 238 * q^91 + 572 * q^92 - 826 * q^94 - 518 * q^97 + 196 * q^98

Coefficient data

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

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

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

$2$	−1.67516	−0.592259	−0.296130	−	0.955148i	$-0.595696\pi$
$2$	−1.67516	−0.592259	−0.296130	+	0.955148i	$0.595696\pi$
$3$	0	0
$3$	0	0
$4$	−5.19383	−0.649229
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	22.1018	0.976771
$9$	0	0
$10$	0	0
$11$	57.5880	1.57849	0.789247	−	0.614076i	$-0.210471\pi$
$11$	57.5880	1.57849	0.789247	+	0.614076i	$0.210471\pi$
$12$	0	0
$13$	45.5159	0.971066	0.485533	−	0.874218i	$-0.338626\pi$
$13$	45.5159	0.971066	0.485533	+	0.874218i	$0.338626\pi$
$14$	11.7261	0.223853
$15$	0	0
$16$	4.52655	0.0707273
$17$	92.0051	1.31262	0.656309	−	0.754492i	$-0.272117\pi$
$17$	92.0051	1.31262	0.656309	+	0.754492i	$0.272117\pi$
$18$	0	0
$19$	125.177	1.51145	0.755726	−	0.654888i	$-0.227284\pi$
$19$	125.177	1.51145	0.755726	+	0.654888i	$0.227284\pi$
$20$	0	0
$21$	0	0
$22$	−96.4692	−0.934878
$23$	158.496	1.43690	0.718451	−	0.695578i	$-0.244851\pi$
$23$	158.496	1.43690	0.718451	+	0.695578i	$0.244851\pi$
$24$	0	0
$25$	0	0
$26$	−76.2466	−0.575123
$27$	0	0
$28$	36.3568	0.245386
$29$	40.1708	0.257225	0.128613	−	0.991695i	$-0.458948\pi$
$29$	40.1708	0.257225	0.128613	+	0.991695i	$0.458948\pi$
$30$	0	0
$31$	49.5590	0.287131	0.143566	−	0.989641i	$-0.454143\pi$
$31$	49.5590	0.287131	0.143566	+	0.989641i	$0.454143\pi$
$32$	−184.397	−1.01866
$33$	0	0
$34$	−154.123	−0.777410
$35$	0	0
$36$	0	0
$37$	−231.307	−1.02775	−0.513874	−	0.857866i	$-0.671790\pi$
$37$	−231.307	−1.02775	−0.513874	+	0.857866i	$0.671790\pi$
$38$	−209.692	−0.895171
$39$	0	0
$40$	0	0
$41$	−169.556	−0.645859	−0.322929	−	0.946423i	$-0.604668\pi$
$41$	−169.556	−0.645859	−0.322929	+	0.946423i	$0.604668\pi$
$42$	0	0
$43$	147.428	0.522849	0.261425	−	0.965224i	$-0.415808\pi$
$43$	147.428	0.522849	0.261425	+	0.965224i	$0.415808\pi$
$44$	−299.102	−1.02480
$45$	0	0
$46$	−265.507	−0.851018
$47$	−67.0327	−0.208037	−0.104018	−	0.994575i	$-0.533170\pi$
$47$	−67.0327	−0.208037	−0.104018	+	0.994575i	$0.533170\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−236.402	−0.630444
$53$	268.647	0.696254	0.348127	−	0.937447i	$-0.386818\pi$
$53$	268.647	0.696254	0.348127	+	0.937447i	$0.386818\pi$
$54$	0	0
$55$	0	0
$56$	−154.713	−0.369185
$57$	0	0
$58$	−67.2926	−0.152344
$59$	240.843	0.531442	0.265721	−	0.964050i	$-0.414390\pi$
$59$	240.843	0.531442	0.265721	+	0.964050i	$0.414390\pi$
$60$	0	0
$61$	90.4579	0.189868	0.0949340	−	0.995484i	$-0.469736\pi$
$61$	90.4579	0.189868	0.0949340	+	0.995484i	$0.469736\pi$
$62$	−83.0194	−0.170056
$63$	0	0
$64$	272.683	0.532583
$65$	0	0
$66$	0	0
$67$	406.498	0.741218	0.370609	−	0.928789i	$-0.379149\pi$
$67$	406.498	0.741218	0.370609	+	0.928789i	$0.379149\pi$
$68$	−477.859	−0.852190
$69$	0	0
$70$	0	0
$71$	−330.782	−0.552910	−0.276455	−	0.961027i	$-0.589160\pi$
$71$	−330.782	−0.552910	−0.276455	+	0.961027i	$0.589160\pi$
$72$	0	0
$73$	−546.255	−0.875812	−0.437906	−	0.899021i	$-0.644280\pi$
$73$	−546.255	−0.875812	−0.437906	+	0.899021i	$0.644280\pi$
$74$	387.477	0.608693
$75$	0	0
$76$	−650.149	−0.981279
$77$	−403.116	−0.596615
$78$	0	0
$79$	−25.3087	−0.0360436	−0.0180218	−	0.999838i	$-0.505737\pi$
$79$	−25.3087	−0.0360436	−0.0180218	+	0.999838i	$0.505737\pi$
$80$	0	0
$81$	0	0
$82$	284.034	0.382516
$83$	376.255	0.497582	0.248791	−	0.968557i	$-0.419967\pi$
$83$	376.255	0.497582	0.248791	+	0.968557i	$0.419967\pi$
$84$	0	0
$85$	0	0
$86$	−246.965	−0.309662
$87$	0	0
$88$	1272.80	1.54183
$89$	−1026.44	−1.22250	−0.611248	−	0.791439i	$-0.709332\pi$
$89$	−1026.44	−1.22250	−0.611248	+	0.791439i	$0.709332\pi$
$90$	0	0
$91$	−318.612	−0.367028
$92$	−823.203	−0.932878
$93$	0	0
$94$	112.291	0.123212
$95$	0	0
$96$	0	0
$97$	942.660	0.986728	0.493364	−	0.869823i	$-0.335767\pi$
$97$	942.660	0.986728	0.493364	+	0.869823i	$0.335767\pi$
$98$	−82.0829	−0.0846085
$99$	0	0
$100$	0	0
$101$	604.617	0.595659	0.297830	−	0.954619i	$-0.403737\pi$
$101$	604.617	0.595659	0.297830	+	0.954619i	$0.403737\pi$
$102$	0	0
$103$	300.967	0.287914	0.143957	−	0.989584i	$-0.454017\pi$
$103$	300.967	0.287914	0.143957	+	0.989584i	$0.454017\pi$
$104$	1005.98	0.948509
$105$	0	0
$106$	−450.027	−0.412363
$107$	1511.66	1.36577	0.682886	−	0.730525i	$-0.260724\pi$
$107$	1511.66	1.36577	0.682886	+	0.730525i	$0.260724\pi$
$108$	0	0
$109$	−1767.09	−1.55281	−0.776406	−	0.630233i	$-0.782959\pi$
$109$	−1767.09	−1.55281	−0.776406	+	0.630233i	$0.782959\pi$
$110$	0	0
$111$	0	0
$112$	−31.6858	−0.0267324
$113$	1045.27	0.870182	0.435091	−	0.900387i	$-0.356716\pi$
$113$	1045.27	0.870182	0.435091	+	0.900387i	$0.356716\pi$
$114$	0	0
$115$	0	0
$116$	−208.640	−0.166998
$117$	0	0
$118$	−403.451	−0.314751
$119$	−644.036	−0.496123
$120$	0	0
$121$	1985.38	1.49164
$122$	−151.532	−0.112451
$123$	0	0
$124$	−257.401	−0.186414
$125$	0	0
$126$	0	0
$127$	−260.727	−0.182171	−0.0910857	−	0.995843i	$-0.529034\pi$
$127$	−260.727	−0.182171	−0.0910857	+	0.995843i	$0.529034\pi$
$128$	1018.39	0.703233
$129$	0	0
$130$	0	0
$131$	723.522	0.482553	0.241276	−	0.970456i	$-0.422434\pi$
$131$	723.522	0.482553	0.241276	+	0.970456i	$0.422434\pi$
$132$	0	0
$133$	−876.240	−0.571275
$134$	−680.950	−0.438993
$135$	0	0
$136$	2033.48	1.28213
$137$	773.693	0.482490	0.241245	−	0.970464i	$-0.422444\pi$
$137$	773.693	0.482490	0.241245	+	0.970464i	$0.422444\pi$
$138$	0	0
$139$	−2952.97	−1.80192	−0.900961	−	0.433899i	$-0.857137\pi$
$139$	−2952.97	−1.80192	−0.900961	+	0.433899i	$0.857137\pi$
$140$	0	0
$141$	0	0
$142$	554.114	0.327466
$143$	2621.17	1.53282
$144$	0	0
$145$	0	0
$146$	915.066	0.518708
$147$	0	0
$148$	1201.37	0.667244
$149$	−2514.00	−1.38225	−0.691124	−	0.722736i	$-0.742884\pi$
$149$	−2514.00	−1.38225	−0.691124	+	0.722736i	$0.742884\pi$
$150$	0	0
$151$	101.052	0.0544605	0.0272302	−	0.999629i	$-0.491331\pi$
$151$	101.052	0.0544605	0.0272302	+	0.999629i	$0.491331\pi$
$152$	2766.64	1.47634
$153$	0	0
$154$	675.285	0.353351
$155$	0	0
$156$	0	0
$157$	−2338.35	−1.18867	−0.594333	−	0.804219i	$-0.702584\pi$
$157$	−2338.35	−1.18867	−0.594333	+	0.804219i	$0.702584\pi$
$158$	42.3961	0.0213472
$159$	0	0
$160$	0	0
$161$	−1109.47	−0.543098
$162$	0	0
$163$	1325.20	0.636798	0.318399	−	0.947957i	$-0.396855\pi$
$163$	1325.20	0.636798	0.318399	+	0.947957i	$0.396855\pi$
$164$	880.646	0.419310
$165$	0	0
$166$	−630.288	−0.294698
$167$	2086.20	0.966675	0.483338	−	0.875434i	$-0.339424\pi$
$167$	2086.20	0.966675	0.483338	+	0.875434i	$0.339424\pi$
$168$	0	0
$169$	−125.299	−0.0570317
$170$	0	0
$171$	0	0
$172$	−765.715	−0.339449
$173$	1918.19	0.842990	0.421495	−	0.906831i	$-0.361505\pi$
$173$	1918.19	0.842990	0.421495	+	0.906831i	$0.361505\pi$
$174$	0	0
$175$	0	0
$176$	260.675	0.111643
$177$	0	0
$178$	1719.45	0.724035
$179$	−629.046	−0.262665	−0.131333	−	0.991338i	$-0.541926\pi$
$179$	−629.046	−0.262665	−0.131333	+	0.991338i	$0.541926\pi$
$180$	0	0
$181$	−2800.85	−1.15020	−0.575099	−	0.818084i	$-0.695036\pi$
$181$	−2800.85	−1.15020	−0.575099	+	0.818084i	$0.695036\pi$
$182$	533.726	0.217376
$183$	0	0
$184$	3503.05	1.40352
$185$	0	0
$186$	0	0
$187$	5298.39	2.07196
$188$	348.157	0.135063
$189$	0	0
$190$	0	0
$191$	−740.255	−0.280434	−0.140217	−	0.990121i	$-0.544780\pi$
$191$	−740.255	−0.280434	−0.140217	+	0.990121i	$0.544780\pi$
$192$	0	0
$193$	−4082.57	−1.52264	−0.761321	−	0.648375i	$-0.775449\pi$
$193$	−4082.57	−1.52264	−0.761321	+	0.648375i	$0.775449\pi$
$194$	−1579.11	−0.584399
$195$	0	0
$196$	−254.498	−0.0927470
$197$	3414.89	1.23503	0.617515	−	0.786559i	$-0.288140\pi$
$197$	3414.89	1.23503	0.617515	+	0.786559i	$0.288140\pi$
$198$	0	0
$199$	−3392.44	−1.20846	−0.604231	−	0.796809i	$-0.706520\pi$
$199$	−3392.44	−1.20846	−0.604231	+	0.796809i	$0.706520\pi$
$200$	0	0
$201$	0	0
$202$	−1012.83	−0.352785
$203$	−281.196	−0.0972220
$204$	0	0
$205$	0	0
$206$	−504.169	−0.170520
$207$	0	0
$208$	206.030	0.0686809
$209$	7208.70	2.38582
$210$	0	0
$211$	3398.04	1.10867	0.554337	−	0.832292i	$-0.312972\pi$
$211$	3398.04	1.10867	0.554337	+	0.832292i	$0.312972\pi$
$212$	−1395.31	−0.452029
$213$	0	0
$214$	−2532.28	−0.808892
$215$	0	0
$216$	0	0
$217$	−346.913	−0.108525
$218$	2960.16	0.919667
$219$	0	0
$220$	0	0
$221$	4187.70	1.27464
$222$	0	0
$223$	−182.611	−0.0548365	−0.0274183	−	0.999624i	$-0.508729\pi$
$223$	−182.611	−0.0548365	−0.0274183	+	0.999624i	$0.508729\pi$
$224$	1290.78	0.385017
$225$	0	0
$226$	−1750.99	−0.515373
$227$	−3152.33	−0.921707	−0.460854	−	0.887476i	$-0.652457\pi$
$227$	−3152.33	−0.921707	−0.460854	+	0.887476i	$0.652457\pi$
$228$	0	0
$229$	6012.35	1.73497	0.867483	−	0.497466i	$-0.165736\pi$
$229$	6012.35	1.73497	0.867483	+	0.497466i	$0.165736\pi$
$230$	0	0
$231$	0	0
$232$	887.848	0.251250
$233$	−940.660	−0.264484	−0.132242	−	0.991217i	$-0.542218\pi$
$233$	−940.660	−0.264484	−0.132242	+	0.991217i	$0.542218\pi$
$234$	0	0
$235$	0	0
$236$	−1250.90	−0.345027
$237$	0	0
$238$	1078.86	0.293833
$239$	−5158.82	−1.39622	−0.698109	−	0.715991i	$-0.745975\pi$
$239$	−5158.82	−1.39622	−0.698109	+	0.715991i	$0.745975\pi$
$240$	0	0
$241$	−463.836	−0.123976	−0.0619882	−	0.998077i	$-0.519744\pi$
$241$	−463.836	−0.123976	−0.0619882	+	0.998077i	$0.519744\pi$
$242$	−3325.83	−0.883440
$243$	0	0
$244$	−469.823	−0.123268
$245$	0	0
$246$	0	0
$247$	5697.55	1.46772
$248$	1095.34	0.280461
$249$	0	0
$250$	0	0
$251$	−2290.25	−0.575934	−0.287967	−	0.957640i	$-0.592979\pi$
$251$	−2290.25	−0.575934	−0.287967	+	0.957640i	$0.592979\pi$
$252$	0	0
$253$	9127.48	2.26814
$254$	436.760	0.107893
$255$	0	0
$256$	−3887.43	−0.949079
$257$	−802.202	−0.194708	−0.0973541	−	0.995250i	$-0.531038\pi$
$257$	−802.202	−0.194708	−0.0973541	+	0.995250i	$0.531038\pi$
$258$	0	0
$259$	1619.15	0.388452
$260$	0	0
$261$	0	0
$262$	−1212.02	−0.285796
$263$	−286.978	−0.0672845	−0.0336423	−	0.999434i	$-0.510711\pi$
$263$	−286.978	−0.0672845	−0.0336423	+	0.999434i	$0.510711\pi$
$264$	0	0
$265$	0	0
$266$	1467.84	0.338343
$267$	0	0
$268$	−2111.28	−0.481221
$269$	−3561.22	−0.807180	−0.403590	−	0.914940i	$-0.632238\pi$
$269$	−3561.22	−0.807180	−0.403590	+	0.914940i	$0.632238\pi$
$270$	0	0
$271$	1928.81	0.432349	0.216175	−	0.976355i	$-0.430642\pi$
$271$	1928.81	0.432349	0.216175	+	0.976355i	$0.430642\pi$
$272$	416.466	0.0928380
$273$	0	0
$274$	−1296.06	−0.285759
$275$	0	0
$276$	0	0
$277$	−6588.69	−1.42916	−0.714578	−	0.699556i	$-0.753381\pi$
$277$	−6588.69	−1.42916	−0.714578	+	0.699556i	$0.753381\pi$
$278$	4946.70	1.06721
$279$	0	0
$280$	0	0
$281$	−815.552	−0.173138	−0.0865689	−	0.996246i	$-0.527590\pi$
$281$	−815.552	−0.173138	−0.0865689	+	0.996246i	$0.527590\pi$
$282$	0	0
$283$	6513.49	1.36815	0.684076	−	0.729411i	$-0.260206\pi$
$283$	6513.49	1.36815	0.684076	+	0.729411i	$0.260206\pi$
$284$	1718.03	0.358965
$285$	0	0
$286$	−4390.89	−0.907828
$287$	1186.89	0.244112
$288$	0	0
$289$	3551.94	0.722967
$290$	0	0
$291$	0	0
$292$	2837.16	0.568603
$293$	435.520	0.0868373	0.0434186	−	0.999057i	$-0.486175\pi$
$293$	435.520	0.0868373	0.0434186	+	0.999057i	$0.486175\pi$
$294$	0	0
$295$	0	0
$296$	−5112.31	−1.00387
$297$	0	0
$298$	4211.36	0.818650
$299$	7214.10	1.39533
$300$	0	0
$301$	−1031.99	−0.197618
$302$	−169.279	−0.0322547
$303$	0	0
$304$	566.620	0.106901
$305$	0	0
$306$	0	0
$307$	4915.99	0.913910	0.456955	−	0.889490i	$-0.348940\pi$
$307$	4915.99	0.913910	0.456955	+	0.889490i	$0.348940\pi$
$308$	2093.72	0.387340
$309$	0	0
$310$	0	0
$311$	1831.11	0.333868	0.166934	−	0.985968i	$-0.446613\pi$
$311$	1831.11	0.333868	0.166934	+	0.985968i	$0.446613\pi$
$312$	0	0
$313$	2442.96	0.441163	0.220582	−	0.975369i	$-0.429204\pi$
$313$	2442.96	0.441163	0.220582	+	0.975369i	$0.429204\pi$
$314$	3917.11	0.703998
$315$	0	0
$316$	131.449	0.0234006
$317$	1666.19	0.295214	0.147607	−	0.989046i	$-0.452843\pi$
$317$	1666.19	0.295214	0.147607	+	0.989046i	$0.452843\pi$
$318$	0	0
$319$	2313.36	0.406029
$320$	0	0
$321$	0	0
$322$	1858.55	0.321655
$323$	11516.9	1.98396
$324$	0	0
$325$	0	0
$326$	−2219.93	−0.377149
$327$	0	0
$328$	−3747.50	−0.630856
$329$	469.229	0.0786305
$330$	0	0
$331$	−5466.38	−0.907732	−0.453866	−	0.891070i	$-0.649956\pi$
$331$	−5466.38	−0.907732	−0.453866	+	0.891070i	$0.649956\pi$
$332$	−1954.20	−0.323045
$333$	0	0
$334$	−3494.72	−0.572522
$335$	0	0
$336$	0	0
$337$	−10650.5	−1.72157	−0.860784	−	0.508970i	$-0.830027\pi$
$337$	−10650.5	−1.72157	−0.860784	+	0.508970i	$0.830027\pi$
$338$	209.896	0.0337776
$339$	0	0
$340$	0	0
$341$	2854.01	0.453235
$342$	0	0
$343$	−343.000	−0.0539949
$344$	3258.42	0.510704
$345$	0	0
$346$	−3213.28	−0.499269
$347$	4019.13	0.621782	0.310891	−	0.950446i	$-0.399373\pi$
$347$	4019.13	0.621782	0.310891	+	0.950446i	$0.399373\pi$
$348$	0	0
$349$	10544.9	1.61735	0.808674	−	0.588256i	$-0.200185\pi$
$349$	10544.9	1.61735	0.808674	+	0.588256i	$0.200185\pi$
$350$	0	0
$351$	0	0
$352$	−10619.1	−1.60795
$353$	2959.98	0.446300	0.223150	−	0.974784i	$-0.428366\pi$
$353$	2959.98	0.446300	0.223150	+	0.974784i	$0.428366\pi$
$354$	0	0
$355$	0	0
$356$	5331.14	0.793680
$357$	0	0
$358$	1053.75	0.155566
$359$	−2170.17	−0.319045	−0.159523	−	0.987194i	$-0.550995\pi$
$359$	−2170.17	−0.319045	−0.159523	+	0.987194i	$0.550995\pi$
$360$	0	0
$361$	8810.30	1.28449
$362$	4691.88	0.681215
$363$	0	0
$364$	1654.82	0.238285
$365$	0	0
$366$	0	0
$367$	1252.20	0.178105	0.0890523	−	0.996027i	$-0.471616\pi$
$367$	1252.20	0.178105	0.0890523	+	0.996027i	$0.471616\pi$
$368$	717.441	0.101628
$369$	0	0
$370$	0	0
$371$	−1880.53	−0.263159
$372$	0	0
$373$	−4646.02	−0.644938	−0.322469	−	0.946580i	$-0.604513\pi$
$373$	−4646.02	−0.644938	−0.322469	+	0.946580i	$0.604513\pi$
$374$	−8875.66	−1.22714
$375$	0	0
$376$	−1481.54	−0.203204
$377$	1828.41	0.249783
$378$	0	0
$379$	1434.84	0.194466	0.0972331	−	0.995262i	$-0.469001\pi$
$379$	1434.84	0.194466	0.0972331	+	0.995262i	$0.469001\pi$
$380$	0	0
$381$	0	0
$382$	1240.05	0.166090
$383$	−13216.7	−1.76330	−0.881649	−	0.471905i	$-0.843566\pi$
$383$	−13216.7	−1.76330	−0.881649	+	0.471905i	$0.843566\pi$
$384$	0	0
$385$	0	0
$386$	6838.97	0.901799
$387$	0	0
$388$	−4896.02	−0.640612
$389$	7755.01	1.01078	0.505391	−	0.862890i	$-0.331348\pi$
$389$	7755.01	1.01078	0.505391	+	0.862890i	$0.331348\pi$
$390$	0	0
$391$	14582.5	1.88610
$392$	1082.99	0.139539
$393$	0	0
$394$	−5720.49	−0.731457
$395$	0	0
$396$	0	0
$397$	3560.83	0.450158	0.225079	−	0.974341i	$-0.427736\pi$
$397$	3560.83	0.450158	0.225079	+	0.974341i	$0.427736\pi$
$398$	5682.89	0.715723
$399$	0	0
$400$	0	0
$401$	5430.61	0.676288	0.338144	−	0.941094i	$-0.390201\pi$
$401$	5430.61	0.676288	0.338144	+	0.941094i	$0.390201\pi$
$402$	0	0
$403$	2255.73	0.278823
$404$	−3140.28	−0.386719
$405$	0	0
$406$	471.048	0.0575806
$407$	−13320.5	−1.62229
$408$	0	0
$409$	−9698.79	−1.17255	−0.586277	−	0.810111i	$-0.699407\pi$
$409$	−9698.79	−1.17255	−0.586277	+	0.810111i	$0.699407\pi$
$410$	0	0
$411$	0	0
$412$	−1563.17	−0.186922
$413$	−1685.90	−0.200866
$414$	0	0
$415$	0	0
$416$	−8393.01	−0.989186
$417$	0	0
$418$	−12075.7	−1.41302
$419$	−13830.9	−1.61261	−0.806307	−	0.591498i	$-0.798537\pi$
$419$	−13830.9	−1.61261	−0.806307	+	0.591498i	$0.798537\pi$
$420$	0	0
$421$	16703.0	1.93362	0.966810	−	0.255498i	$-0.0822393\pi$
$421$	16703.0	1.93362	0.966810	+	0.255498i	$0.0822393\pi$
$422$	−5692.26	−0.656623
$423$	0	0
$424$	5937.58	0.680081
$425$	0	0
$426$	0	0
$427$	−633.205	−0.0717634
$428$	−7851.31	−0.886699
$429$	0	0
$430$	0	0
$431$	−8174.07	−0.913530	−0.456765	−	0.889588i	$-0.650992\pi$
$431$	−8174.07	−0.913530	−0.456765	+	0.889588i	$0.650992\pi$
$432$	0	0
$433$	−14222.8	−1.57853	−0.789267	−	0.614051i	$-0.789539\pi$
$433$	−14222.8	−1.57853	−0.789267	+	0.614051i	$0.789539\pi$
$434$	581.136	0.0642752
$435$	0	0
$436$	9177.96	1.00813
$437$	19840.1	2.17181
$438$	0	0
$439$	5537.38	0.602016	0.301008	−	0.953622i	$-0.402677\pi$
$439$	5537.38	0.602016	0.301008	+	0.953622i	$0.402677\pi$
$440$	0	0
$441$	0	0
$442$	−7015.07	−0.754916
$443$	3974.09	0.426218	0.213109	−	0.977028i	$-0.431641\pi$
$443$	3974.09	0.426218	0.213109	+	0.977028i	$0.431641\pi$
$444$	0	0
$445$	0	0
$446$	305.903	0.0324774
$447$	0	0
$448$	−1908.78	−0.201298
$449$	15243.1	1.60216	0.801078	−	0.598559i	$-0.204260\pi$
$449$	15243.1	1.60216	0.801078	+	0.598559i	$0.204260\pi$
$450$	0	0
$451$	−9764.40	−1.01948
$452$	−5428.95	−0.564947
$453$	0	0
$454$	5280.67	0.545890
$455$	0	0
$456$	0	0
$457$	10768.9	1.10229	0.551145	−	0.834410i	$-0.314191\pi$
$457$	10768.9	1.10229	0.551145	+	0.834410i	$0.314191\pi$
$458$	−10071.7	−1.02755
$459$	0	0
$460$	0	0
$461$	332.605	0.0336029	0.0168015	−	0.999859i	$-0.494652\pi$
$461$	332.605	0.0336029	0.0168015	+	0.999859i	$0.494652\pi$
$462$	0	0
$463$	8205.35	0.823618	0.411809	−	0.911270i	$-0.364897\pi$
$463$	8205.35	0.823618	0.411809	+	0.911270i	$0.364897\pi$
$464$	181.835	0.0181929
$465$	0	0
$466$	1575.76	0.156643
$467$	−167.628	−0.0166100	−0.00830501	−	0.999966i	$-0.502644\pi$
$467$	−167.628	−0.0166100	−0.00830501	+	0.999966i	$0.502644\pi$
$468$	0	0
$469$	−2845.49	−0.280154
$470$	0	0
$471$	0	0
$472$	5323.06	0.519097
$473$	8490.07	0.825315
$474$	0	0
$475$	0	0
$476$	3345.01	0.322098
$477$	0	0
$478$	8641.86	0.826924
$479$	−6628.58	−0.632292	−0.316146	−	0.948711i	$-0.602389\pi$
$479$	−6628.58	−0.632292	−0.316146	+	0.948711i	$0.602389\pi$
$480$	0	0
$481$	−10528.2	−0.998010
$482$	777.001	0.0734262
$483$	0	0
$484$	−10311.7	−0.968419
$485$	0	0
$486$	0	0
$487$	20641.6	1.92065	0.960327	−	0.278875i	$-0.0899614\pi$
$487$	20641.6	1.92065	0.960327	+	0.278875i	$0.0899614\pi$
$488$	1999.28	0.185458
$489$	0	0
$490$	0	0
$491$	16710.8	1.53594	0.767972	−	0.640484i	$-0.221266\pi$
$491$	16710.8	1.53594	0.767972	+	0.640484i	$0.221266\pi$
$492$	0	0
$493$	3695.92	0.337639
$494$	−9544.32	−0.869270
$495$	0	0
$496$	224.331	0.0203080
$497$	2315.47	0.208980
$498$	0	0
$499$	−13728.7	−1.23162	−0.615812	−	0.787893i	$-0.711172\pi$
$499$	−13728.7	−1.23162	−0.615812	+	0.787893i	$0.711172\pi$
$500$	0	0
$501$	0	0
$502$	3836.54	0.341102
$503$	−19523.7	−1.73065	−0.865326	−	0.501209i	$-0.832889\pi$
$503$	−19523.7	−1.73065	−0.865326	+	0.501209i	$0.832889\pi$
$504$	0	0
$505$	0	0
$506$	−15290.0	−1.34333
$507$	0	0
$508$	1354.17	0.118271
$509$	8688.17	0.756574	0.378287	−	0.925688i	$-0.376513\pi$
$509$	8688.17	0.756574	0.378287	+	0.925688i	$0.376513\pi$
$510$	0	0
$511$	3823.78	0.331026
$512$	−1635.04	−0.141131
$513$	0	0
$514$	1343.82	0.115318
$515$	0	0
$516$	0	0
$517$	−3860.28	−0.328385
$518$	−2712.34	−0.230064
$519$	0	0
$520$	0	0
$521$	−6771.36	−0.569402	−0.284701	−	0.958616i	$-0.591894\pi$
$521$	−6771.36	−0.569402	−0.284701	+	0.958616i	$0.591894\pi$
$522$	0	0
$523$	1365.89	0.114200	0.0570998	−	0.998368i	$-0.481815\pi$
$523$	1365.89	0.114200	0.0570998	+	0.998368i	$0.481815\pi$
$524$	−3757.85	−0.313287
$525$	0	0
$526$	480.735	0.0398499
$527$	4559.68	0.376894
$528$	0	0
$529$	12954.0	1.06469
$530$	0	0
$531$	0	0
$532$	4551.04	0.370888
$533$	−7717.51	−0.627171
$534$	0	0
$535$	0	0
$536$	8984.34	0.724001
$537$	0	0
$538$	5965.62	0.478060
$539$	2821.81	0.225499
$540$	0	0
$541$	−23250.1	−1.84769	−0.923844	−	0.382770i	$-0.874970\pi$
$541$	−23250.1	−1.84769	−0.923844	+	0.382770i	$0.874970\pi$
$542$	−3231.06	−0.256063
$543$	0	0
$544$	−16965.5	−1.33711
$545$	0	0
$546$	0	0
$547$	11552.7	0.903033	0.451516	−	0.892263i	$-0.350883\pi$
$547$	11552.7	0.903033	0.451516	+	0.892263i	$0.350883\pi$
$548$	−4018.43	−0.313246
$549$	0	0
$550$	0	0
$551$	5028.47	0.388784
$552$	0	0
$553$	177.161	0.0136232
$554$	11037.1	0.846430
$555$	0	0
$556$	15337.2	1.16986
$557$	−16406.2	−1.24803	−0.624014	−	0.781413i	$-0.714499\pi$
$557$	−16406.2	−1.24803	−0.624014	+	0.781413i	$0.714499\pi$
$558$	0	0
$559$	6710.31	0.507721
$560$	0	0
$561$	0	0
$562$	1366.18	0.102542
$563$	13631.9	1.02045	0.510227	−	0.860040i	$-0.329561\pi$
$563$	13631.9	1.02045	0.510227	+	0.860040i	$0.329561\pi$
$564$	0	0
$565$	0	0
$566$	−10911.2	−0.810300
$567$	0	0
$568$	−7310.88	−0.540067
$569$	3086.83	0.227428	0.113714	−	0.993514i	$-0.463725\pi$
$569$	3086.83	0.227428	0.113714	+	0.993514i	$0.463725\pi$
$570$	0	0
$571$	−3258.06	−0.238784	−0.119392	−	0.992847i	$-0.538095\pi$
$571$	−3258.06	−0.238784	−0.119392	+	0.992847i	$0.538095\pi$
$572$	−13613.9	−0.995152
$573$	0	0
$574$	−1988.24	−0.144577
$575$	0	0
$576$	0	0
$577$	−23758.4	−1.71417	−0.857083	−	0.515178i	$-0.827726\pi$
$577$	−23758.4	−1.71417	−0.857083	+	0.515178i	$0.827726\pi$
$578$	−5950.07	−0.428184
$579$	0	0
$580$	0	0
$581$	−2633.78	−0.188068
$582$	0	0
$583$	15470.8	1.09903
$584$	−12073.2	−0.855468
$585$	0	0
$586$	−729.566	−0.0514302
$587$	596.893	0.0419701	0.0209850	−	0.999780i	$-0.493320\pi$
$587$	596.893	0.0419701	0.0209850	+	0.999780i	$0.493320\pi$
$588$	0	0
$589$	6203.66	0.433985
$590$	0	0
$591$	0	0
$592$	−1047.02	−0.0726898
$593$	19496.3	1.35012	0.675058	−	0.737765i	$-0.264119\pi$
$593$	19496.3	1.35012	0.675058	+	0.737765i	$0.264119\pi$
$594$	0	0
$595$	0	0
$596$	13057.3	0.897396
$597$	0	0
$598$	−12084.8	−0.826395
$599$	−3797.02	−0.259001	−0.129501	−	0.991579i	$-0.541337\pi$
$599$	−3797.02	−0.259001	−0.129501	+	0.991579i	$0.541337\pi$
$600$	0	0
$601$	5789.33	0.392931	0.196466	−	0.980511i	$-0.437054\pi$
$601$	5789.33	0.392931	0.196466	+	0.980511i	$0.437054\pi$
$602$	1728.76	0.117041
$603$	0	0
$604$	−524.850	−0.0353573
$605$	0	0
$606$	0	0
$607$	−18536.4	−1.23949	−0.619745	−	0.784803i	$-0.712764\pi$
$607$	−18536.4	−1.23949	−0.619745	+	0.784803i	$0.712764\pi$
$608$	−23082.3	−1.53966
$609$	0	0
$610$	0	0
$611$	−3051.06	−0.202017
$612$	0	0
$613$	−2163.47	−0.142548	−0.0712738	−	0.997457i	$-0.522706\pi$
$613$	−2163.47	−0.142548	−0.0712738	+	0.997457i	$0.522706\pi$
$614$	−8235.08	−0.541272
$615$	0	0
$616$	−8909.59	−0.582756
$617$	22964.9	1.49843	0.749215	−	0.662327i	$-0.230431\pi$
$617$	22964.9	1.49843	0.749215	+	0.662327i	$0.230431\pi$
$618$	0	0
$619$	−1386.67	−0.0900401	−0.0450200	−	0.998986i	$-0.514335\pi$
$619$	−1386.67	−0.0900401	−0.0450200	+	0.998986i	$0.514335\pi$
$620$	0	0
$621$	0	0
$622$	−3067.41	−0.197736
$623$	7185.06	0.462060
$624$	0	0
$625$	0	0
$626$	−4092.35	−0.261283
$627$	0	0
$628$	12145.0	0.771716
$629$	−21281.4	−1.34904
$630$	0	0
$631$	5969.39	0.376605	0.188303	−	0.982111i	$-0.439701\pi$
$631$	5969.39	0.376605	0.188303	+	0.982111i	$0.439701\pi$
$632$	−559.367	−0.0352064
$633$	0	0
$634$	−2791.14	−0.174843
$635$	0	0
$636$	0	0
$637$	2230.28	0.138724
$638$	−3875.25	−0.240474
$639$	0	0
$640$	0	0
$641$	−30367.1	−1.87118	−0.935592	−	0.353084i	$-0.885133\pi$
$641$	−30367.1	−1.87118	−0.935592	+	0.353084i	$0.885133\pi$
$642$	0	0
$643$	−28592.2	−1.75360	−0.876802	−	0.480851i	$-0.840328\pi$
$643$	−28592.2	−1.75360	−0.876802	+	0.480851i	$0.840328\pi$
$644$	5762.42	0.352595
$645$	0	0
$646$	−19292.7	−1.17502
$647$	−14507.9	−0.881555	−0.440778	−	0.897616i	$-0.645297\pi$
$647$	−14507.9	−0.881555	−0.440778	+	0.897616i	$0.645297\pi$
$648$	0	0
$649$	13869.7	0.838877
$650$	0	0
$651$	0	0
$652$	−6882.89	−0.413428
$653$	6999.85	0.419488	0.209744	−	0.977756i	$-0.432737\pi$
$653$	6999.85	0.419488	0.209744	+	0.977756i	$0.432737\pi$
$654$	0	0
$655$	0	0
$656$	−767.504	−0.0456799
$657$	0	0
$658$	−786.035	−0.0465696
$659$	−7308.92	−0.432041	−0.216020	−	0.976389i	$-0.569308\pi$
$659$	−7308.92	−0.432041	−0.216020	+	0.976389i	$0.569308\pi$
$660$	0	0
$661$	−30097.2	−1.77102	−0.885512	−	0.464617i	$-0.846192\pi$
$661$	−30097.2	−1.77102	−0.885512	+	0.464617i	$0.846192\pi$
$662$	9157.07	0.537613
$663$	0	0
$664$	8315.91	0.486024
$665$	0	0
$666$	0	0
$667$	6366.92	0.369608
$668$	−10835.4	−0.627594
$669$	0	0
$670$	0	0
$671$	5209.29	0.299706
$672$	0	0
$673$	5400.26	0.309309	0.154654	−	0.987969i	$-0.450574\pi$
$673$	5400.26	0.309309	0.154654	+	0.987969i	$0.450574\pi$
$674$	17841.3	1.01962
$675$	0	0
$676$	650.781	0.0370267
$677$	6431.09	0.365091	0.182546	−	0.983197i	$-0.441566\pi$
$677$	6431.09	0.365091	0.182546	+	0.983197i	$0.441566\pi$
$678$	0	0
$679$	−6598.62	−0.372948
$680$	0	0
$681$	0	0
$682$	−4780.92	−0.268433
$683$	−20865.8	−1.16897	−0.584486	−	0.811404i	$-0.698704\pi$
$683$	−20865.8	−1.16897	−0.584486	+	0.811404i	$0.698704\pi$
$684$	0	0
$685$	0	0
$686$	574.581	0.0319790
$687$	0	0
$688$	667.339	0.0369797
$689$	12227.7	0.676109
$690$	0	0
$691$	18450.3	1.01575	0.507873	−	0.861432i	$-0.330432\pi$
$691$	18450.3	1.01575	0.507873	+	0.861432i	$0.330432\pi$
$692$	−9962.76	−0.547294
$693$	0	0
$694$	−6732.70	−0.368256
$695$	0	0
$696$	0	0
$697$	−15600.0	−0.847766
$698$	−17664.4	−0.957890
$699$	0	0
$700$	0	0
$701$	−12639.3	−0.680996	−0.340498	−	0.940245i	$-0.610596\pi$
$701$	−12639.3	−0.680996	−0.340498	+	0.940245i	$0.610596\pi$
$702$	0	0
$703$	−28954.4	−1.55339
$704$	15703.3	0.840680
$705$	0	0
$706$	−4958.45	−0.264325
$707$	−4232.32	−0.225138
$708$	0	0
$709$	−23126.8	−1.22503	−0.612514	−	0.790460i	$-0.709842\pi$
$709$	−23126.8	−1.22503	−0.612514	+	0.790460i	$0.709842\pi$
$710$	0	0
$711$	0	0
$712$	−22686.1	−1.19410
$713$	7854.92	0.412579
$714$	0	0
$715$	0	0
$716$	3267.16	0.170530
$717$	0	0
$718$	3635.39	0.188957
$719$	24093.1	1.24968	0.624841	−	0.780752i	$-0.285164\pi$
$719$	24093.1	1.24968	0.624841	+	0.780752i	$0.285164\pi$
$720$	0	0
$721$	−2106.77	−0.108821
$722$	−14758.7	−0.760750
$723$	0	0
$724$	14547.2	0.746741
$725$	0	0
$726$	0	0
$727$	−35983.4	−1.83570	−0.917849	−	0.396931i	$-0.870075\pi$
$727$	−35983.4	−1.83570	−0.917849	+	0.396931i	$0.870075\pi$
$728$	−7041.89	−0.358503
$729$	0	0
$730$	0	0
$731$	13564.1	0.686302
$732$	0	0
$733$	1451.50	0.0731413	0.0365706	−	0.999331i	$-0.488357\pi$
$733$	1451.50	0.0731413	0.0365706	+	0.999331i	$0.488357\pi$
$734$	−2097.64	−0.105484
$735$	0	0
$736$	−29226.3	−1.46371
$737$	23409.4	1.17001
$738$	0	0
$739$	−5891.67	−0.293273	−0.146636	−	0.989190i	$-0.546845\pi$
$739$	−5891.67	−0.293273	−0.146636	+	0.989190i	$0.546845\pi$
$740$	0	0
$741$	0	0
$742$	3150.19	0.155859
$743$	7438.65	0.367292	0.183646	−	0.982992i	$-0.441210\pi$
$743$	7438.65	0.367292	0.183646	+	0.982992i	$0.441210\pi$
$744$	0	0
$745$	0	0
$746$	7782.84	0.381970
$747$	0	0
$748$	−27518.9	−1.34518
$749$	−10581.6	−0.516214
$750$	0	0
$751$	20272.4	0.985018	0.492509	−	0.870307i	$-0.336080\pi$
$751$	20272.4	0.985018	0.492509	+	0.870307i	$0.336080\pi$
$752$	−303.427	−0.0147139
$753$	0	0
$754$	−3062.89	−0.147936
$755$	0	0
$756$	0	0
$757$	10193.8	0.489432	0.244716	−	0.969595i	$-0.421305\pi$
$757$	10193.8	0.489432	0.244716	+	0.969595i	$0.421305\pi$
$758$	−2403.59	−0.115174
$759$	0	0
$760$	0	0
$761$	41117.6	1.95862	0.979311	−	0.202362i	$-0.0648618\pi$
$761$	41117.6	1.95862	0.979311	+	0.202362i	$0.0648618\pi$
$762$	0	0
$763$	12369.6	0.586908
$764$	3844.76	0.182066
$765$	0	0
$766$	22140.2	1.04433
$767$	10962.2	0.516065
$768$	0	0
$769$	11486.6	0.538642	0.269321	−	0.963050i	$-0.413201\pi$
$769$	11486.6	0.538642	0.269321	+	0.963050i	$0.413201\pi$
$770$	0	0
$771$	0	0
$772$	21204.2	0.988544
$773$	−21799.2	−1.01431	−0.507156	−	0.861854i	$-0.669303\pi$
$773$	−21799.2	−1.01431	−0.507156	+	0.861854i	$0.669303\pi$
$774$	0	0
$775$	0	0
$776$	20834.5	0.963807
$777$	0	0
$778$	−12990.9	−0.598645
$779$	−21224.5	−0.976185
$780$	0	0
$781$	−19049.1	−0.872765
$782$	−24428.0	−1.11706
$783$	0	0
$784$	221.801	0.0101039
$785$	0	0
$786$	0	0
$787$	24080.3	1.09068	0.545342	−	0.838214i	$-0.316400\pi$
$787$	24080.3	1.09068	0.545342	+	0.838214i	$0.316400\pi$
$788$	−17736.4	−0.801817
$789$	0	0
$790$	0	0
$791$	−7316.87	−0.328898
$792$	0	0
$793$	4117.28	0.184374
$794$	−5964.96	−0.266610
$795$	0	0
$796$	17619.8	0.784569
$797$	−17194.3	−0.764184	−0.382092	−	0.924124i	$-0.624796\pi$
$797$	−17194.3	−0.764184	−0.382092	+	0.924124i	$0.624796\pi$
$798$	0	0
$799$	−6167.35	−0.273073
$800$	0	0
$801$	0	0
$802$	−9097.15	−0.400538
$803$	−31457.7	−1.38246
$804$	0	0
$805$	0	0
$806$	−3778.71	−0.165136
$807$	0	0
$808$	13363.1	0.581823
$809$	−33349.1	−1.44931	−0.724656	−	0.689111i	$-0.758001\pi$
$809$	−33349.1	−1.44931	−0.724656	+	0.689111i	$0.758001\pi$
$810$	0	0
$811$	4577.87	0.198213	0.0991066	−	0.995077i	$-0.468402\pi$
$811$	4577.87	0.198213	0.0991066	+	0.995077i	$0.468402\pi$
$812$	1460.48	0.0631194
$813$	0	0
$814$	22314.0	0.960819
$815$	0	0
$816$	0	0
$817$	18454.6	0.790262
$818$	16247.0	0.694455
$819$	0	0
$820$	0	0
$821$	5832.52	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$821$	5832.52	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$822$	0	0
$823$	−37974.3	−1.60838	−0.804192	−	0.594369i	$-0.797402\pi$
$823$	−37974.3	−1.60838	−0.804192	+	0.594369i	$0.797402\pi$
$824$	6651.92	0.281226
$825$	0	0
$826$	2824.15	0.118965
$827$	15796.2	0.664193	0.332096	−	0.943245i	$-0.392244\pi$
$827$	15796.2	0.664193	0.332096	+	0.943245i	$0.392244\pi$
$828$	0	0
$829$	−12714.1	−0.532666	−0.266333	−	0.963881i	$-0.585812\pi$
$829$	−12714.1	−0.532666	−0.266333	+	0.963881i	$0.585812\pi$
$830$	0	0
$831$	0	0
$832$	12411.4	0.517173
$833$	4508.25	0.187517
$834$	0	0
$835$	0	0
$836$	−37440.8	−1.54894
$837$	0	0
$838$	23169.0	0.955085
$839$	42924.2	1.76628	0.883140	−	0.469109i	$-0.155425\pi$
$839$	42924.2	1.76628	0.883140	+	0.469109i	$0.155425\pi$
$840$	0	0
$841$	−22775.3	−0.933835
$842$	−27980.2	−1.14520
$843$	0	0
$844$	−17648.8	−0.719784
$845$	0	0
$846$	0	0
$847$	−13897.6	−0.563788
$848$	1216.04	0.0492442
$849$	0	0
$850$	0	0
$851$	−36661.3	−1.47677
$852$	0	0
$853$	36172.5	1.45196	0.725980	−	0.687716i	$-0.241387\pi$
$853$	36172.5	1.45196	0.725980	+	0.687716i	$0.241387\pi$
$854$	1060.72	0.0425025
$855$	0	0
$856$	33410.4	1.33405
$857$	−32039.9	−1.27709	−0.638544	−	0.769586i	$-0.720463\pi$
$857$	−32039.9	−1.27709	−0.638544	+	0.769586i	$0.720463\pi$
$858$	0	0
$859$	−6798.07	−0.270020	−0.135010	−	0.990844i	$-0.543107\pi$
$859$	−6798.07	−0.270020	−0.135010	+	0.990844i	$0.543107\pi$
$860$	0	0
$861$	0	0
$862$	13692.9	0.541046
$863$	−30179.3	−1.19040	−0.595200	−	0.803577i	$-0.702927\pi$
$863$	−30179.3	−1.19040	−0.595200	+	0.803577i	$0.702927\pi$
$864$	0	0
$865$	0	0
$866$	23825.5	0.934901
$867$	0	0
$868$	1801.81	0.0704578
$869$	−1457.47	−0.0568946
$870$	0	0
$871$	18502.1	0.719772
$872$	−39055.9	−1.51674
$873$	0	0
$874$	−33235.4	−1.28627
$875$	0	0
$876$	0	0
$877$	1700.51	0.0654758	0.0327379	−	0.999464i	$-0.489577\pi$
$877$	1700.51	0.0654758	0.0327379	+	0.999464i	$0.489577\pi$
$878$	−9276.02	−0.356549
$879$	0	0
$880$	0	0
$881$	−1678.46	−0.0641869	−0.0320935	−	0.999485i	$-0.510217\pi$
$881$	−1678.46	−0.0641869	−0.0320935	+	0.999485i	$0.510217\pi$
$882$	0	0
$883$	−10285.8	−0.392009	−0.196005	−	0.980603i	$-0.562797\pi$
$883$	−10285.8	−0.392009	−0.196005	+	0.980603i	$0.562797\pi$
$884$	−21750.2	−0.827532
$885$	0	0
$886$	−6657.24	−0.252432
$887$	−12167.5	−0.460593	−0.230296	−	0.973121i	$-0.573970\pi$
$887$	−12167.5	−0.460593	−0.230296	+	0.973121i	$0.573970\pi$
$888$	0	0
$889$	1825.09	0.0688543
$890$	0	0
$891$	0	0
$892$	948.452	0.0356015
$893$	−8390.96	−0.314438
$894$	0	0
$895$	0	0
$896$	−7128.73	−0.265797
$897$	0	0
$898$	−25534.7	−0.948892
$899$	1990.83	0.0738574
$900$	0	0
$901$	24716.9	0.913916
$902$	16356.9	0.603799
$903$	0	0
$904$	23102.3	0.849968
$905$	0	0
$906$	0	0
$907$	50766.0	1.85850	0.929250	−	0.369452i	$-0.120455\pi$
$907$	50766.0	1.85850	0.929250	+	0.369452i	$0.120455\pi$
$908$	16372.7	0.598399
$909$	0	0
$910$	0	0
$911$	−18451.1	−0.671033	−0.335517	−	0.942034i	$-0.608911\pi$
$911$	−18451.1	−0.671033	−0.335517	+	0.942034i	$0.608911\pi$
$912$	0	0
$913$	21667.8	0.785431
$914$	−18039.6	−0.652841
$915$	0	0
$916$	−31227.1	−1.12639
$917$	−5064.65	−0.182388
$918$	0	0
$919$	393.861	0.0141374	0.00706870	−	0.999975i	$-0.497750\pi$
$919$	393.861	0.0141374	0.00706870	+	0.999975i	$0.497750\pi$
$920$	0	0
$921$	0	0
$922$	−557.167	−0.0199016
$923$	−15055.9	−0.536912
$924$	0	0
$925$	0	0
$926$	−13745.3	−0.487795
$927$	0	0
$928$	−7407.39	−0.262025
$929$	−27452.3	−0.969517	−0.484759	−	0.874648i	$-0.661093\pi$
$929$	−27452.3	−0.969517	−0.484759	+	0.874648i	$0.661093\pi$
$930$	0	0
$931$	6133.68	0.215922
$932$	4885.63	0.171710
$933$	0	0
$934$	280.803	0.00983744
$935$	0	0
$936$	0	0
$937$	−21608.3	−0.753375	−0.376688	−	0.926340i	$-0.622937\pi$
$937$	−21608.3	−0.753375	−0.376688	+	0.926340i	$0.622937\pi$
$938$	4766.65	0.165924
$939$	0	0
$940$	0	0
$941$	16708.5	0.578832	0.289416	−	0.957203i	$-0.406539\pi$
$941$	16708.5	0.578832	0.289416	+	0.957203i	$0.406539\pi$
$942$	0	0
$943$	−26874.0	−0.928036
$944$	1090.19	0.0375874
$945$	0	0
$946$	−14222.2	−0.488800
$947$	13972.8	0.479466	0.239733	−	0.970839i	$-0.422940\pi$
$947$	13972.8	0.479466	0.239733	+	0.970839i	$0.422940\pi$
$948$	0	0
$949$	−24863.3	−0.850471
$950$	0	0
$951$	0	0
$952$	−14234.4	−0.484599
$953$	−11155.9	−0.379197	−0.189598	−	0.981862i	$-0.560719\pi$
$953$	−11155.9	−0.379197	−0.189598	+	0.981862i	$0.560719\pi$
$954$	0	0
$955$	0	0
$956$	26794.0	0.906466
$957$	0	0
$958$	11104.0	0.374481
$959$	−5415.85	−0.182364
$960$	0	0
$961$	−27334.9	−0.917556
$962$	17636.4	0.591081
$963$	0	0
$964$	2409.09	0.0804891
$965$	0	0
$966$	0	0
$967$	1212.55	0.0403238	0.0201619	−	0.999797i	$-0.493582\pi$
$967$	1212.55	0.0403238	0.0201619	+	0.999797i	$0.493582\pi$
$968$	43880.4	1.45699
$969$	0	0
$970$	0	0
$971$	48832.2	1.61390	0.806952	−	0.590617i	$-0.201115\pi$
$971$	48832.2	1.61390	0.806952	+	0.590617i	$0.201115\pi$
$972$	0	0
$973$	20670.8	0.681063
$974$	−34578.0	−1.13753
$975$	0	0
$976$	409.462	0.0134289
$977$	5939.19	0.194485	0.0972424	−	0.995261i	$-0.468998\pi$
$977$	5939.19	0.194485	0.0972424	+	0.995261i	$0.468998\pi$
$978$	0	0
$979$	−59110.5	−1.92970
$980$	0	0
$981$	0	0
$982$	−27993.3	−0.909677
$983$	−21863.5	−0.709398	−0.354699	−	0.934980i	$-0.615417\pi$
$983$	−21863.5	−0.709398	−0.354699	+	0.934980i	$0.615417\pi$
$984$	0	0
$985$	0	0
$986$	−6191.26	−0.199970
$987$	0	0
$988$	−29592.1	−0.952886
$989$	23366.7	0.751283
$990$	0	0
$991$	−44618.5	−1.43023	−0.715113	−	0.699009i	$-0.753625\pi$
$991$	−44618.5	−1.43023	−0.715113	+	0.699009i	$0.753625\pi$
$992$	−9138.55	−0.292489
$993$	0	0
$994$	−3878.80	−0.123771
$995$	0	0
$996$	0	0
$997$	37923.9	1.20468	0.602338	−	0.798241i	$-0.294236\pi$
$997$	37923.9	1.20468	0.602338	+	0.798241i	$0.294236\pi$
$998$	22997.8	0.729441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bq.1.2		5
3.2	odd	2		175.4.a.i.1.4		5
5.2	odd	4		315.4.d.c.64.4		10
5.3	odd	4		315.4.d.c.64.7		10
5.4	even	2		1575.4.a.bn.1.4		5
15.2	even	4		35.4.b.a.29.7	yes	10
15.8	even	4		35.4.b.a.29.4	✓	10
15.14	odd	2		175.4.a.j.1.2		5
21.20	even	2		1225.4.a.be.1.4		5
60.23	odd	4		560.4.g.f.449.4		10
60.47	odd	4		560.4.g.f.449.7		10
105.2	even	12		245.4.j.e.214.7		20
105.17	odd	12		245.4.j.f.79.4		20
105.23	even	12		245.4.j.e.214.4		20
105.32	even	12		245.4.j.e.79.4		20
105.38	odd	12		245.4.j.f.79.7		20
105.47	odd	12		245.4.j.f.214.7		20
105.53	even	12		245.4.j.e.79.7		20
105.62	odd	4		245.4.b.d.99.7		10
105.68	odd	12		245.4.j.f.214.4		20
105.83	odd	4		245.4.b.d.99.4		10
105.104	even	2		1225.4.a.bh.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.4	✓	10	15.8	even	4
35.4.b.a.29.7	yes	10	15.2	even	4
175.4.a.i.1.4		5	3.2	odd	2
175.4.a.j.1.2		5	15.14	odd	2
245.4.b.d.99.4		10	105.83	odd	4
245.4.b.d.99.7		10	105.62	odd	4
245.4.j.e.79.4		20	105.32	even	12
245.4.j.e.79.7		20	105.53	even	12
245.4.j.e.214.4		20	105.23	even	12
245.4.j.e.214.7		20	105.2	even	12
245.4.j.f.79.4		20	105.17	odd	12
245.4.j.f.79.7		20	105.38	odd	12
245.4.j.f.214.4		20	105.68	odd	12
245.4.j.f.214.7		20	105.47	odd	12
315.4.d.c.64.4		10	5.2	odd	4
315.4.d.c.64.7		10	5.3	odd	4
560.4.g.f.449.4		10	60.23	odd	4
560.4.g.f.449.7		10	60.47	odd	4
1225.4.a.be.1.4		5	21.20	even	2
1225.4.a.bh.1.2		5	105.104	even	2
1575.4.a.bn.1.4		5	5.4	even	2
1575.4.a.bq.1.2		5	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.67516 q^{2} -5.19383 q^{4} -7.00000 q^{7} +22.1018 q^{8} +57.5880 q^{11} +45.5159 q^{13} +11.7261 q^{14} +4.52655 q^{16} +92.0051 q^{17} +125.177 q^{19} -96.4692 q^{22} +158.496 q^{23} -76.2466 q^{26} +36.3568 q^{28} +40.1708 q^{29} +49.5590 q^{31} -184.397 q^{32} -154.123 q^{34} -231.307 q^{37} -209.692 q^{38} -169.556 q^{41} +147.428 q^{43} -299.102 q^{44} -265.507 q^{46} -67.0327 q^{47} +49.0000 q^{49} -236.402 q^{52} +268.647 q^{53} -154.713 q^{56} -67.2926 q^{58} +240.843 q^{59} +90.4579 q^{61} -83.0194 q^{62} +272.683 q^{64} +406.498 q^{67} -477.859 q^{68} -330.782 q^{71} -546.255 q^{73} +387.477 q^{74} -650.149 q^{76} -403.116 q^{77} -25.3087 q^{79} +284.034 q^{82} +376.255 q^{83} -246.965 q^{86} +1272.80 q^{88} -1026.44 q^{89} -318.612 q^{91} -823.203 q^{92} +112.291 q^{94} +942.660 q^{97} -82.0829 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 18 q^{4} - 35 q^{7} + 42 q^{8} - 42 q^{11} - 34 q^{13} - 28 q^{14} + 74 q^{16} + 238 q^{17} - 36 q^{19} - 358 q^{22} + 152 q^{23} + 310 q^{26} - 126 q^{28} + 44 q^{29} + 60 q^{31} + 710 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 18 * q^4 - 35 * q^7 + 42 * q^8 - 42 * q^11 - 34 * q^13 - 28 * q^14 + 74 * q^16 + 238 * q^17 - 36 * q^19 - 358 * q^22 + 152 * q^23 + 310 * q^26 - 126 * q^28 + 44 * q^29 + 60 * q^31 + 710 * q^32 - 482 * q^34 - 312 * q^37 + 280 * q^38 + 426 * q^41 - 304 * q^43 - 712 * q^44 + 88 * q^46 + 370 * q^47 + 245 * q^49 + 1156 * q^52 + 976 * q^53 - 294 * q^56 + 2722 * q^58 + 432 * q^59 - 442 * q^61 - 956 * q^62 + 1362 * q^64 + 804 * q^67 - 420 * q^68 - 440 * q^71 - 564 * q^73 + 1512 * q^74 - 1336 * q^76 + 294 * q^77 + 1790 * q^79 + 3480 * q^82 + 1656 * q^83 - 1216 * q^86 + 1092 * q^88 - 746 * q^89 + 238 * q^91 + 572 * q^92 - 826 * q^94 - 518 * q^97 + 196 * q^98

Introduction

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