LMFDB - Embedded newform 1575.4.a.bp.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.78066700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 $ x^5 - x^4 - 18x^3 + 7x^2 + 30*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.40248$ 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)

$f(q)$	$=$	$q+3.33774 q^{2} +3.14050 q^{4} -7.00000 q^{7} -16.2197 q^{8} +O(q^{10})$	$q+3.33774 q^{2} +3.14050 q^{4} -7.00000 q^{7} -16.2197 q^{8} -18.3258 q^{11} -10.1457 q^{13} -23.3642 q^{14} -79.2613 q^{16} -24.6984 q^{17} +77.4282 q^{19} -61.1666 q^{22} +149.411 q^{23} -33.8636 q^{26} -21.9835 q^{28} +10.2984 q^{29} +124.423 q^{31} -134.796 q^{32} -82.4367 q^{34} -215.905 q^{37} +258.435 q^{38} -495.056 q^{41} +220.606 q^{43} -57.5521 q^{44} +498.696 q^{46} +212.957 q^{47} +49.0000 q^{49} -31.8625 q^{52} +532.455 q^{53} +113.538 q^{56} +34.3732 q^{58} +324.143 q^{59} +653.967 q^{61} +415.292 q^{62} +184.178 q^{64} +819.077 q^{67} -77.5653 q^{68} +466.940 q^{71} -173.066 q^{73} -720.634 q^{74} +243.163 q^{76} +128.280 q^{77} +810.186 q^{79} -1652.37 q^{82} +12.3208 q^{83} +736.326 q^{86} +297.239 q^{88} +33.8297 q^{89} +71.0198 q^{91} +469.227 q^{92} +710.795 q^{94} +810.761 q^{97} +163.549 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 27 q^{4} - 35 q^{7} + 33 q^{8}+O(q^{10}) $ 5 * q + q^2 + 27 * q^4 - 35 * q^7 + 33 * q^8	$ 5 q + q^{2} + 27 q^{4} - 35 q^{7} + 33 q^{8} - 66 q^{11} + 2 q^{13} - 7 q^{14} + 155 q^{16} + 108 q^{17} + 174 q^{19} + 506 q^{22} - 116 q^{23} - 446 q^{26} - 189 q^{28} - 370 q^{29} + 342 q^{31} - 55 q^{32} + 112 q^{34} + 408 q^{37} - 34 q^{38} - 802 q^{41} + 584 q^{43} - 290 q^{44} + 640 q^{46} + 716 q^{47} + 245 q^{49} - 338 q^{52} + 98 q^{53} - 231 q^{56} - 482 q^{58} - 704 q^{59} + 650 q^{61} + 2070 q^{62} + 75 q^{64} - 180 q^{67} + 4520 q^{68} - 1470 q^{71} + 534 q^{73} + 1312 q^{74} + 4370 q^{76} + 462 q^{77} - 820 q^{79} + 1338 q^{82} + 1520 q^{83} - 832 q^{86} + 3258 q^{88} - 286 q^{89} - 14 q^{91} + 1288 q^{92} + 2540 q^{94} - 278 q^{97} + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 27 * q^4 - 35 * q^7 + 33 * q^8 - 66 * q^11 + 2 * q^13 - 7 * q^14 + 155 * q^16 + 108 * q^17 + 174 * q^19 + 506 * q^22 - 116 * q^23 - 446 * q^26 - 189 * q^28 - 370 * q^29 + 342 * q^31 - 55 * q^32 + 112 * q^34 + 408 * q^37 - 34 * q^38 - 802 * q^41 + 584 * q^43 - 290 * q^44 + 640 * q^46 + 716 * q^47 + 245 * q^49 - 338 * q^52 + 98 * q^53 - 231 * q^56 - 482 * q^58 - 704 * q^59 + 650 * q^61 + 2070 * q^62 + 75 * q^64 - 180 * q^67 + 4520 * q^68 - 1470 * q^71 + 534 * q^73 + 1312 * q^74 + 4370 * q^76 + 462 * q^77 - 820 * q^79 + 1338 * q^82 + 1520 * q^83 - 832 * q^86 + 3258 * q^88 - 286 * q^89 - 14 * q^91 + 1288 * q^92 + 2540 * q^94 - 278 * q^97 + 49 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	3.33774	1.18007	0.590035	−	0.807378i	$-0.299114\pi$
$2$	3.33774	1.18007	0.590035	+	0.807378i	$0.299114\pi$
$3$	0	0
$3$	0	0
$4$	3.14050	0.392563
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−16.2197	−0.716818
$9$	0	0
$10$	0	0
$11$	−18.3258	−0.502311	−0.251156	−	0.967947i	$-0.580811\pi$
$11$	−18.3258	−0.502311	−0.251156	+	0.967947i	$0.580811\pi$
$12$	0	0
$13$	−10.1457	−0.216454	−0.108227	−	0.994126i	$-0.534517\pi$
$13$	−10.1457	−0.216454	−0.108227	+	0.994126i	$0.534517\pi$
$14$	−23.3642	−0.446024
$15$	0	0
$16$	−79.2613	−1.23846
$17$	−24.6984	−0.352367	−0.176183	−	0.984357i	$-0.556375\pi$
$17$	−24.6984	−0.352367	−0.176183	+	0.984357i	$0.556375\pi$
$18$	0	0
$19$	77.4282	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$19$	77.4282	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$20$	0	0
$21$	0	0
$22$	−61.1666	−0.592762
$23$	149.411	1.35454	0.677270	−	0.735735i	$-0.263163\pi$
$23$	149.411	1.35454	0.677270	+	0.735735i	$0.263163\pi$
$24$	0	0
$25$	0	0
$26$	−33.8636	−0.255431
$27$	0	0
$28$	−21.9835	−0.148375
$29$	10.2984	0.0659434	0.0329717	−	0.999456i	$-0.489503\pi$
$29$	10.2984	0.0659434	0.0329717	+	0.999456i	$0.489503\pi$
$30$	0	0
$31$	124.423	0.720872	0.360436	−	0.932784i	$-0.382628\pi$
$31$	124.423	0.720872	0.360436	+	0.932784i	$0.382628\pi$
$32$	−134.796	−0.744647
$33$	0	0
$34$	−82.4367	−0.415817
$35$	0	0
$36$	0	0
$37$	−215.905	−0.959312	−0.479656	−	0.877457i	$-0.659239\pi$
$37$	−215.905	−0.959312	−0.479656	+	0.877457i	$0.659239\pi$
$38$	258.435	1.10326
$39$	0	0
$40$	0	0
$41$	−495.056	−1.88573	−0.942863	−	0.333181i	$-0.891878\pi$
$41$	−495.056	−1.88573	−0.942863	+	0.333181i	$0.891878\pi$
$42$	0	0
$43$	220.606	0.782375	0.391188	−	0.920311i	$-0.372064\pi$
$43$	220.606	0.782375	0.391188	+	0.920311i	$0.372064\pi$
$44$	−57.5521	−0.197189
$45$	0	0
$46$	498.696	1.59845
$47$	212.957	0.660915	0.330457	−	0.943821i	$-0.392797\pi$
$47$	212.957	0.660915	0.330457	+	0.943821i	$0.392797\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−31.8625	−0.0849719
$53$	532.455	1.37997	0.689984	−	0.723825i	$-0.257618\pi$
$53$	532.455	1.37997	0.689984	+	0.723825i	$0.257618\pi$
$54$	0	0
$55$	0	0
$56$	113.538	0.270932
$57$	0	0
$58$	34.3732	0.0778177
$59$	324.143	0.715252	0.357626	−	0.933865i	$-0.383586\pi$
$59$	324.143	0.715252	0.357626	+	0.933865i	$0.383586\pi$
$60$	0	0
$61$	653.967	1.37265	0.686327	−	0.727293i	$-0.259222\pi$
$61$	653.967	1.37265	0.686327	+	0.727293i	$0.259222\pi$
$62$	415.292	0.850679
$63$	0	0
$64$	184.178	0.359722
$65$	0	0
$66$	0	0
$67$	819.077	1.49353	0.746763	−	0.665091i	$-0.231607\pi$
$67$	819.077	1.49353	0.746763	+	0.665091i	$0.231607\pi$
$68$	−77.5653	−0.138326
$69$	0	0
$70$	0	0
$71$	466.940	0.780501	0.390250	−	0.920709i	$-0.372388\pi$
$71$	466.940	0.780501	0.390250	+	0.920709i	$0.372388\pi$
$72$	0	0
$73$	−173.066	−0.277478	−0.138739	−	0.990329i	$-0.544305\pi$
$73$	−173.066	−0.277478	−0.138739	+	0.990329i	$0.544305\pi$
$74$	−720.634	−1.13205
$75$	0	0
$76$	243.163	0.367010
$77$	128.280	0.189856
$78$	0	0
$79$	810.186	1.15384	0.576918	−	0.816802i	$-0.304255\pi$
$79$	810.186	1.15384	0.576918	+	0.816802i	$0.304255\pi$
$80$	0	0
$81$	0	0
$82$	−1652.37	−2.22529
$83$	12.3208	0.0162937	0.00814687	−	0.999967i	$-0.497407\pi$
$83$	12.3208	0.0162937	0.00814687	+	0.999967i	$0.497407\pi$
$84$	0	0
$85$	0	0
$86$	736.326	0.923257
$87$	0	0
$88$	297.239	0.360066
$89$	33.8297	0.0402914	0.0201457	−	0.999797i	$-0.493587\pi$
$89$	33.8297	0.0402914	0.0201457	+	0.999797i	$0.493587\pi$
$90$	0	0
$91$	71.0198	0.0818120
$92$	469.227	0.531742
$93$	0	0
$94$	710.795	0.779925
$95$	0	0
$96$	0	0
$97$	810.761	0.848663	0.424331	−	0.905507i	$-0.360509\pi$
$97$	810.761	0.848663	0.424331	+	0.905507i	$0.360509\pi$
$98$	163.549	0.168581
$99$	0	0
$100$	0	0
$101$	−1646.84	−1.62245	−0.811223	−	0.584737i	$-0.801198\pi$
$101$	−1646.84	−1.62245	−0.811223	+	0.584737i	$0.801198\pi$
$102$	0	0
$103$	921.360	0.881401	0.440700	−	0.897654i	$-0.354730\pi$
$103$	921.360	0.881401	0.440700	+	0.897654i	$0.354730\pi$
$104$	164.560	0.155158
$105$	0	0
$106$	1777.19	1.62846
$107$	−1060.84	−0.958464	−0.479232	−	0.877688i	$-0.659085\pi$
$107$	−1060.84	−0.958464	−0.479232	+	0.877688i	$0.659085\pi$
$108$	0	0
$109$	1533.03	1.34713	0.673566	−	0.739127i	$-0.264762\pi$
$109$	1533.03	1.34713	0.673566	+	0.739127i	$0.264762\pi$
$110$	0	0
$111$	0	0
$112$	554.829	0.468093
$113$	−2072.67	−1.72549	−0.862746	−	0.505637i	$-0.831257\pi$
$113$	−2072.67	−1.72549	−0.862746	+	0.505637i	$0.831257\pi$
$114$	0	0
$115$	0	0
$116$	32.3420	0.0258869
$117$	0	0
$118$	1081.91	0.844046
$119$	172.889	0.133182
$120$	0	0
$121$	−995.166	−0.747683
$122$	2182.77	1.61983
$123$	0	0
$124$	390.751	0.282988
$125$	0	0
$126$	0	0
$127$	365.145	0.255129	0.127565	−	0.991830i	$-0.459284\pi$
$127$	365.145	0.255129	0.127565	+	0.991830i	$0.459284\pi$
$128$	1693.10	1.16914
$129$	0	0
$130$	0	0
$131$	−1300.83	−0.867586	−0.433793	−	0.901013i	$-0.642825\pi$
$131$	−1300.83	−0.867586	−0.433793	+	0.901013i	$0.642825\pi$
$132$	0	0
$133$	−541.997	−0.353362
$134$	2733.87	1.76246
$135$	0	0
$136$	400.601	0.252583
$137$	−2578.40	−1.60794	−0.803970	−	0.594670i	$-0.797283\pi$
$137$	−2578.40	−1.60794	−0.803970	+	0.594670i	$0.797283\pi$
$138$	0	0
$139$	3065.49	1.87059	0.935294	−	0.353872i	$-0.115135\pi$
$139$	3065.49	1.87059	0.935294	+	0.353872i	$0.115135\pi$
$140$	0	0
$141$	0	0
$142$	1558.52	0.921044
$143$	185.927	0.108727
$144$	0	0
$145$	0	0
$146$	−577.650	−0.327443
$147$	0	0
$148$	−678.050	−0.376590
$149$	2704.34	1.48690	0.743449	−	0.668793i	$-0.233189\pi$
$149$	2704.34	1.48690	0.743449	+	0.668793i	$0.233189\pi$
$150$	0	0
$151$	388.869	0.209574	0.104787	−	0.994495i	$-0.466584\pi$
$151$	388.869	0.209574	0.104787	+	0.994495i	$0.466584\pi$
$152$	−1255.86	−0.670158
$153$	0	0
$154$	428.166	0.224043
$155$	0	0
$156$	0	0
$157$	2885.78	1.46694	0.733472	−	0.679720i	$-0.237899\pi$
$157$	2885.78	1.46694	0.733472	+	0.679720i	$0.237899\pi$
$158$	2704.19	1.36161
$159$	0	0
$160$	0	0
$161$	−1045.88	−0.511968
$162$	0	0
$163$	−571.954	−0.274840	−0.137420	−	0.990513i	$-0.543881\pi$
$163$	−571.954	−0.274840	−0.137420	+	0.990513i	$0.543881\pi$
$164$	−1554.73	−0.740266
$165$	0	0
$166$	41.1235	0.0192277
$167$	2423.96	1.12319	0.561593	−	0.827414i	$-0.310189\pi$
$167$	2423.96	1.12319	0.561593	+	0.827414i	$0.310189\pi$
$168$	0	0
$169$	−2094.07	−0.953148
$170$	0	0
$171$	0	0
$172$	692.815	0.307131
$173$	626.440	0.275303	0.137651	−	0.990481i	$-0.456045\pi$
$173$	626.440	0.275303	0.137651	+	0.990481i	$0.456045\pi$
$174$	0	0
$175$	0	0
$176$	1452.52	0.622091
$177$	0	0
$178$	112.915	0.0475467
$179$	−2322.11	−0.969623	−0.484812	−	0.874619i	$-0.661112\pi$
$179$	−2322.11	−0.969623	−0.484812	+	0.874619i	$0.661112\pi$
$180$	0	0
$181$	−4664.94	−1.91570	−0.957852	−	0.287263i	$-0.907255\pi$
$181$	−4664.94	−1.91570	−0.957852	+	0.287263i	$0.907255\pi$
$182$	237.045	0.0965438
$183$	0	0
$184$	−2423.41	−0.970958
$185$	0	0
$186$	0	0
$187$	452.617	0.176998
$188$	668.792	0.259451
$189$	0	0
$190$	0	0
$191$	3730.26	1.41315	0.706575	−	0.707638i	$-0.250239\pi$
$191$	3730.26	1.41315	0.706575	+	0.707638i	$0.250239\pi$
$192$	0	0
$193$	2585.26	0.964202	0.482101	−	0.876116i	$-0.339874\pi$
$193$	2585.26	0.964202	0.482101	+	0.876116i	$0.339874\pi$
$194$	2706.11	1.00148
$195$	0	0
$196$	153.885	0.0560804
$197$	2405.91	0.870122	0.435061	−	0.900401i	$-0.356727\pi$
$197$	2405.91	0.870122	0.435061	+	0.900401i	$0.356727\pi$
$198$	0	0
$199$	1192.35	0.424740	0.212370	−	0.977189i	$-0.431882\pi$
$199$	1192.35	0.424740	0.212370	+	0.977189i	$0.431882\pi$
$200$	0	0
$201$	0	0
$202$	−5496.74	−1.91460
$203$	−72.0885	−0.0249243
$204$	0	0
$205$	0	0
$206$	3075.26	1.04011
$207$	0	0
$208$	804.160	0.268069
$209$	−1418.93	−0.469615
$210$	0	0
$211$	−764.904	−0.249565	−0.124782	−	0.992184i	$-0.539823\pi$
$211$	−764.904	−0.249565	−0.124782	+	0.992184i	$0.539823\pi$
$212$	1672.18	0.541724
$213$	0	0
$214$	−3540.82	−1.13105
$215$	0	0
$216$	0	0
$217$	−870.961	−0.272464
$218$	5116.84	1.58971
$219$	0	0
$220$	0	0
$221$	250.582	0.0762713
$222$	0	0
$223$	−1585.18	−0.476016	−0.238008	−	0.971263i	$-0.576494\pi$
$223$	−1585.18	−0.476016	−0.238008	+	0.971263i	$0.576494\pi$
$224$	943.569	0.281450
$225$	0	0
$226$	−6918.04	−2.03620
$227$	2363.07	0.690936	0.345468	−	0.938431i	$-0.387720\pi$
$227$	2363.07	0.690936	0.345468	+	0.938431i	$0.387720\pi$
$228$	0	0
$229$	1626.44	0.469337	0.234669	−	0.972075i	$-0.424599\pi$
$229$	1626.44	0.469337	0.234669	+	0.972075i	$0.424599\pi$
$230$	0	0
$231$	0	0
$232$	−167.037	−0.0472694
$233$	1720.34	0.483705	0.241853	−	0.970313i	$-0.422245\pi$
$233$	1720.34	0.483705	0.241853	+	0.970313i	$0.422245\pi$
$234$	0	0
$235$	0	0
$236$	1017.97	0.280781
$237$	0	0
$238$	577.057	0.157164
$239$	−7217.75	−1.95346	−0.976732	−	0.214466i	$-0.931199\pi$
$239$	−7217.75	−1.95346	−0.976732	+	0.214466i	$0.931199\pi$
$240$	0	0
$241$	5094.78	1.36176	0.680879	−	0.732396i	$-0.261598\pi$
$241$	5094.78	1.36176	0.680879	+	0.732396i	$0.261598\pi$
$242$	−3321.61	−0.882318
$243$	0	0
$244$	2053.79	0.538853
$245$	0	0
$246$	0	0
$247$	−785.561	−0.202365
$248$	−2018.11	−0.516734
$249$	0	0
$250$	0	0
$251$	4346.51	1.09303	0.546513	−	0.837451i	$-0.315955\pi$
$251$	4346.51	1.09303	0.546513	+	0.837451i	$0.315955\pi$
$252$	0	0
$253$	−2738.08	−0.680401
$254$	1218.76	0.301070
$255$	0	0
$256$	4177.71	1.01995
$257$	−1428.13	−0.346632	−0.173316	−	0.984866i	$-0.555448\pi$
$257$	−1428.13	−0.346632	−0.173316	+	0.984866i	$0.555448\pi$
$258$	0	0
$259$	1511.33	0.362586
$260$	0	0
$261$	0	0
$262$	−4341.82	−1.02381
$263$	3447.78	0.808362	0.404181	−	0.914679i	$-0.367557\pi$
$263$	3447.78	0.808362	0.404181	+	0.914679i	$0.367557\pi$
$264$	0	0
$265$	0	0
$266$	−1809.05	−0.416991
$267$	0	0
$268$	2572.31	0.586303
$269$	22.3975	0.00507657	0.00253829	−	0.999997i	$-0.499192\pi$
$269$	22.3975	0.00507657	0.00253829	+	0.999997i	$0.499192\pi$
$270$	0	0
$271$	−1557.33	−0.349080	−0.174540	−	0.984650i	$-0.555844\pi$
$271$	−1557.33	−0.349080	−0.174540	+	0.984650i	$0.555844\pi$
$272$	1957.62	0.436391
$273$	0	0
$274$	−8606.03	−1.89748
$275$	0	0
$276$	0	0
$277$	3734.92	0.810142	0.405071	−	0.914285i	$-0.367247\pi$
$277$	3734.92	0.810142	0.405071	+	0.914285i	$0.367247\pi$
$278$	10231.8	2.20742
$279$	0	0
$280$	0	0
$281$	2422.37	0.514258	0.257129	−	0.966377i	$-0.417223\pi$
$281$	2422.37	0.514258	0.257129	+	0.966377i	$0.417223\pi$
$282$	0	0
$283$	−283.232	−0.0594926	−0.0297463	−	0.999557i	$-0.509470\pi$
$283$	−283.232	−0.0594926	−0.0297463	+	0.999557i	$0.509470\pi$
$284$	1466.43	0.306396
$285$	0	0
$286$	620.577	0.128306
$287$	3465.39	0.712737
$288$	0	0
$289$	−4302.99	−0.875838
$290$	0	0
$291$	0	0
$292$	−543.515	−0.108927
$293$	8481.49	1.69111	0.845553	−	0.533892i	$-0.179271\pi$
$293$	8481.49	1.69111	0.845553	+	0.533892i	$0.179271\pi$
$294$	0	0
$295$	0	0
$296$	3501.92	0.687652
$297$	0	0
$298$	9026.37	1.75464
$299$	−1515.88	−0.293196
$300$	0	0
$301$	−1544.24	−0.295710
$302$	1297.94	0.247312
$303$	0	0
$304$	−6137.05	−1.15784
$305$	0	0
$306$	0	0
$307$	2005.50	0.372834	0.186417	−	0.982471i	$-0.440312\pi$
$307$	2005.50	0.372834	0.186417	+	0.982471i	$0.440312\pi$
$308$	402.865	0.0745304
$309$	0	0
$310$	0	0
$311$	−10021.4	−1.82720	−0.913602	−	0.406609i	$-0.866711\pi$
$311$	−10021.4	−1.82720	−0.913602	+	0.406609i	$0.866711\pi$
$312$	0	0
$313$	631.178	0.113982	0.0569908	−	0.998375i	$-0.481849\pi$
$313$	631.178	0.113982	0.0569908	+	0.998375i	$0.481849\pi$
$314$	9631.97	1.73109
$315$	0	0
$316$	2544.39	0.452953
$317$	−1202.71	−0.213095	−0.106547	−	0.994308i	$-0.533980\pi$
$317$	−1202.71	−0.213095	−0.106547	+	0.994308i	$0.533980\pi$
$318$	0	0
$319$	−188.725	−0.0331241
$320$	0	0
$321$	0	0
$322$	−3490.87	−0.604158
$323$	−1912.35	−0.329430
$324$	0	0
$325$	0	0
$326$	−1909.03	−0.324330
$327$	0	0
$328$	8029.68	1.35172
$329$	−1490.70	−0.249802
$330$	0	0
$331$	−9297.55	−1.54393	−0.771963	−	0.635667i	$-0.780725\pi$
$331$	−9297.55	−1.54393	−0.771963	+	0.635667i	$0.780725\pi$
$332$	38.6934	0.00639632
$333$	0	0
$334$	8090.56	1.32544
$335$	0	0
$336$	0	0
$337$	−507.727	−0.0820702	−0.0410351	−	0.999158i	$-0.513066\pi$
$337$	−507.727	−0.0820702	−0.0410351	+	0.999158i	$0.513066\pi$
$338$	−6989.44	−1.12478
$339$	0	0
$340$	0	0
$341$	−2280.15	−0.362102
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−3578.17	−0.560820
$345$	0	0
$346$	2090.89	0.324876
$347$	6860.59	1.06137	0.530685	−	0.847569i	$-0.321935\pi$
$347$	6860.59	1.06137	0.530685	+	0.847569i	$0.321935\pi$
$348$	0	0
$349$	−9236.45	−1.41666	−0.708332	−	0.705879i	$-0.750552\pi$
$349$	−9236.45	−1.41666	−0.708332	+	0.705879i	$0.750552\pi$
$350$	0	0
$351$	0	0
$352$	2470.23	0.374045
$353$	9419.54	1.42026	0.710129	−	0.704071i	$-0.248636\pi$
$353$	9419.54	1.42026	0.710129	+	0.704071i	$0.248636\pi$
$354$	0	0
$355$	0	0
$356$	106.242	0.0158169
$357$	0	0
$358$	−7750.59	−1.14422
$359$	2890.82	0.424990	0.212495	−	0.977162i	$-0.431841\pi$
$359$	2890.82	0.424990	0.212495	+	0.977162i	$0.431841\pi$
$360$	0	0
$361$	−863.879	−0.125948
$362$	−15570.4	−2.26066
$363$	0	0
$364$	223.038	0.0321164
$365$	0	0
$366$	0	0
$367$	1570.51	0.223379	0.111690	−	0.993743i	$-0.464374\pi$
$367$	1570.51	0.223379	0.111690	+	0.993743i	$0.464374\pi$
$368$	−11842.5	−1.67754
$369$	0	0
$370$	0	0
$371$	−3727.18	−0.521579
$372$	0	0
$373$	8713.38	1.20955	0.604774	−	0.796397i	$-0.293263\pi$
$373$	8713.38	1.20955	0.604774	+	0.796397i	$0.293263\pi$
$374$	1510.72	0.208870
$375$	0	0
$376$	−3454.11	−0.473755
$377$	−104.484	−0.0142737
$378$	0	0
$379$	−5779.65	−0.783326	−0.391663	−	0.920109i	$-0.628100\pi$
$379$	−5779.65	−0.783326	−0.391663	+	0.920109i	$0.628100\pi$
$380$	0	0
$381$	0	0
$382$	12450.6	1.66762
$383$	−12340.2	−1.64636	−0.823181	−	0.567780i	$-0.807803\pi$
$383$	−12340.2	−1.64636	−0.823181	+	0.567780i	$0.807803\pi$
$384$	0	0
$385$	0	0
$386$	8628.92	1.13783
$387$	0	0
$388$	2546.20	0.333153
$389$	−7948.44	−1.03600	−0.517998	−	0.855382i	$-0.673322\pi$
$389$	−7948.44	−1.03600	−0.517998	+	0.855382i	$0.673322\pi$
$390$	0	0
$391$	−3690.22	−0.477295
$392$	−794.767	−0.102403
$393$	0	0
$394$	8030.30	1.02680
$395$	0	0
$396$	0	0
$397$	5363.64	0.678069	0.339034	−	0.940774i	$-0.389900\pi$
$397$	5363.64	0.678069	0.339034	+	0.940774i	$0.389900\pi$
$398$	3979.75	0.501223
$399$	0	0
$400$	0	0
$401$	−6057.13	−0.754310	−0.377155	−	0.926150i	$-0.623098\pi$
$401$	−6057.13	−0.754310	−0.377155	+	0.926150i	$0.623098\pi$
$402$	0	0
$403$	−1262.36	−0.156036
$404$	−5171.92	−0.636912
$405$	0	0
$406$	−240.613	−0.0294123
$407$	3956.62	0.481873
$408$	0	0
$409$	1248.52	0.150942	0.0754709	−	0.997148i	$-0.475954\pi$
$409$	1248.52	0.150942	0.0754709	+	0.997148i	$0.475954\pi$
$410$	0	0
$411$	0	0
$412$	2893.53	0.346005
$413$	−2269.00	−0.270340
$414$	0	0
$415$	0	0
$416$	1367.59	0.161182
$417$	0	0
$418$	−4736.02	−0.554178
$419$	−8324.28	−0.970567	−0.485284	−	0.874357i	$-0.661284\pi$
$419$	−8324.28	−0.970567	−0.485284	+	0.874357i	$0.661284\pi$
$420$	0	0
$421$	1693.40	0.196036	0.0980179	−	0.995185i	$-0.468750\pi$
$421$	1693.40	0.196036	0.0980179	+	0.995185i	$0.468750\pi$
$422$	−2553.05	−0.294504
$423$	0	0
$424$	−8636.27	−0.989185
$425$	0	0
$426$	0	0
$427$	−4577.77	−0.518814
$428$	−3331.58	−0.376257
$429$	0	0
$430$	0	0
$431$	−11754.6	−1.31369	−0.656844	−	0.754026i	$-0.728109\pi$
$431$	−11754.6	−1.31369	−0.656844	+	0.754026i	$0.728109\pi$
$432$	0	0
$433$	14614.3	1.62199	0.810993	−	0.585056i	$-0.198927\pi$
$433$	14614.3	1.62199	0.810993	+	0.585056i	$0.198927\pi$
$434$	−2907.04	−0.321526
$435$	0	0
$436$	4814.48	0.528834
$437$	11568.6	1.26637
$438$	0	0
$439$	1957.52	0.212819	0.106409	−	0.994322i	$-0.466065\pi$
$439$	1957.52	0.212819	0.106409	+	0.994322i	$0.466065\pi$
$440$	0	0
$441$	0	0
$442$	836.377	0.0900054
$443$	−356.774	−0.0382637	−0.0191319	−	0.999817i	$-0.506090\pi$
$443$	−356.774	−0.0382637	−0.0191319	+	0.999817i	$0.506090\pi$
$444$	0	0
$445$	0	0
$446$	−5290.92	−0.561732
$447$	0	0
$448$	−1289.24	−0.135962
$449$	−7395.30	−0.777296	−0.388648	−	0.921386i	$-0.627058\pi$
$449$	−7395.30	−0.777296	−0.388648	+	0.921386i	$0.627058\pi$
$450$	0	0
$451$	9072.28	0.947222
$452$	−6509.23	−0.677364
$453$	0	0
$454$	7887.31	0.815352
$455$	0	0
$456$	0	0
$457$	−12722.2	−1.30223	−0.651115	−	0.758979i	$-0.725698\pi$
$457$	−12722.2	−1.30223	−0.651115	+	0.758979i	$0.725698\pi$
$458$	5428.64	0.553850
$459$	0	0
$460$	0	0
$461$	2206.45	0.222916	0.111458	−	0.993769i	$-0.464448\pi$
$461$	2206.45	0.222916	0.111458	+	0.993769i	$0.464448\pi$
$462$	0	0
$463$	−15912.0	−1.59718	−0.798591	−	0.601875i	$-0.794421\pi$
$463$	−15912.0	−1.59718	−0.798591	+	0.601875i	$0.794421\pi$
$464$	−816.261	−0.0816681
$465$	0	0
$466$	5742.05	0.570806
$467$	7602.65	0.753337	0.376669	−	0.926348i	$-0.377070\pi$
$467$	7602.65	0.753337	0.376669	+	0.926348i	$0.377070\pi$
$468$	0	0
$469$	−5733.54	−0.564499
$470$	0	0
$471$	0	0
$472$	−5257.52	−0.512705
$473$	−4042.78	−0.392996
$474$	0	0
$475$	0	0
$476$	542.957	0.0522824
$477$	0	0
$478$	−24091.0	−2.30522
$479$	−1681.93	−0.160437	−0.0802184	−	0.996777i	$-0.525562\pi$
$479$	−1681.93	−0.160437	−0.0802184	+	0.996777i	$0.525562\pi$
$480$	0	0
$481$	2190.50	0.207647
$482$	17005.0	1.60697
$483$	0	0
$484$	−3125.32	−0.293513
$485$	0	0
$486$	0	0
$487$	14107.0	1.31263	0.656314	−	0.754487i	$-0.272114\pi$
$487$	14107.0	1.31263	0.656314	+	0.754487i	$0.272114\pi$
$488$	−10607.2	−0.983943
$489$	0	0
$490$	0	0
$491$	−9010.20	−0.828156	−0.414078	−	0.910241i	$-0.635896\pi$
$491$	−9010.20	−0.828156	−0.414078	+	0.910241i	$0.635896\pi$
$492$	0	0
$493$	−254.353	−0.0232363
$494$	−2622.00	−0.238804
$495$	0	0
$496$	−9861.92	−0.892769
$497$	−3268.58	−0.295001
$498$	0	0
$499$	2655.07	0.238191	0.119096	−	0.992883i	$-0.462000\pi$
$499$	2655.07	0.238191	0.119096	+	0.992883i	$0.462000\pi$
$500$	0	0
$501$	0	0
$502$	14507.5	1.28985
$503$	16559.4	1.46789	0.733946	−	0.679208i	$-0.237677\pi$
$503$	16559.4	1.46789	0.733946	+	0.679208i	$0.237677\pi$
$504$	0	0
$505$	0	0
$506$	−9138.99	−0.802920
$507$	0	0
$508$	1146.74	0.100154
$509$	12308.6	1.07184	0.535922	−	0.844268i	$-0.319964\pi$
$509$	12308.6	1.07184	0.535922	+	0.844268i	$0.319964\pi$
$510$	0	0
$511$	1211.46	0.104877
$512$	399.294	0.0344658
$513$	0	0
$514$	−4766.74	−0.409050
$515$	0	0
$516$	0	0
$517$	−3902.60	−0.331985
$518$	5044.44	0.427876
$519$	0	0
$520$	0	0
$521$	15045.9	1.26521	0.632605	−	0.774475i	$-0.281986\pi$
$521$	15045.9	1.26521	0.632605	+	0.774475i	$0.281986\pi$
$522$	0	0
$523$	−6372.16	−0.532763	−0.266381	−	0.963868i	$-0.585828\pi$
$523$	−6372.16	−0.532763	−0.266381	+	0.963868i	$0.585828\pi$
$524$	−4085.25	−0.340582
$525$	0	0
$526$	11507.8	0.953923
$527$	−3073.05	−0.254011
$528$	0	0
$529$	10156.8	0.834779
$530$	0	0
$531$	0	0
$532$	−1702.14	−0.138717
$533$	5022.68	0.408173
$534$	0	0
$535$	0	0
$536$	−13285.2	−1.07059
$537$	0	0
$538$	74.7569	0.00599071
$539$	−897.962	−0.0717588
$540$	0	0
$541$	4128.72	0.328110	0.164055	−	0.986451i	$-0.447543\pi$
$541$	4128.72	0.328110	0.164055	+	0.986451i	$0.447543\pi$
$542$	−5197.95	−0.411939
$543$	0	0
$544$	3329.23	0.262389
$545$	0	0
$546$	0	0
$547$	−17987.6	−1.40602	−0.703011	−	0.711179i	$-0.748162\pi$
$547$	−17987.6	−1.40602	−0.703011	+	0.711179i	$0.748162\pi$
$548$	−8097.48	−0.631217
$549$	0	0
$550$	0	0
$551$	797.383	0.0616509
$552$	0	0
$553$	−5671.30	−0.436109
$554$	12466.2	0.956024
$555$	0	0
$556$	9627.19	0.734323
$557$	−13278.6	−1.01011	−0.505057	−	0.863086i	$-0.668529\pi$
$557$	−13278.6	−1.01011	−0.505057	+	0.863086i	$0.668529\pi$
$558$	0	0
$559$	−2238.20	−0.169348
$560$	0	0
$561$	0	0
$562$	8085.24	0.606860
$563$	517.822	0.0387630	0.0193815	−	0.999812i	$-0.493830\pi$
$563$	517.822	0.0387630	0.0193815	+	0.999812i	$0.493830\pi$
$564$	0	0
$565$	0	0
$566$	−945.355	−0.0702054
$567$	0	0
$568$	−7573.64	−0.559477
$569$	−1071.91	−0.0789750	−0.0394875	−	0.999220i	$-0.512573\pi$
$569$	−1071.91	−0.0789750	−0.0394875	+	0.999220i	$0.512573\pi$
$570$	0	0
$571$	−24489.8	−1.79486	−0.897430	−	0.441157i	$-0.854568\pi$
$571$	−24489.8	−1.79486	−0.897430	+	0.441157i	$0.854568\pi$
$572$	583.905	0.0426824
$573$	0	0
$574$	11566.6	0.841079
$575$	0	0
$576$	0	0
$577$	−22765.1	−1.64250	−0.821251	−	0.570568i	$-0.806723\pi$
$577$	−22765.1	−1.64250	−0.821251	+	0.570568i	$0.806723\pi$
$578$	−14362.3	−1.03355
$579$	0	0
$580$	0	0
$581$	−86.2454	−0.00615846
$582$	0	0
$583$	−9757.64	−0.693173
$584$	2807.09	0.198901
$585$	0	0
$586$	28309.0	1.99562
$587$	−10106.8	−0.710651	−0.355325	−	0.934743i	$-0.615630\pi$
$587$	−10106.8	−0.710651	−0.355325	+	0.934743i	$0.615630\pi$
$588$	0	0
$589$	9633.85	0.673949
$590$	0	0
$591$	0	0
$592$	17112.9	1.18807
$593$	−4241.04	−0.293691	−0.146845	−	0.989159i	$-0.546912\pi$
$593$	−4241.04	−0.293691	−0.146845	+	0.989159i	$0.546912\pi$
$594$	0	0
$595$	0	0
$596$	8492.97	0.583701
$597$	0	0
$598$	−5059.61	−0.345992
$599$	16536.6	1.12799	0.563995	−	0.825778i	$-0.309264\pi$
$599$	16536.6	1.12799	0.563995	+	0.825778i	$0.309264\pi$
$600$	0	0
$601$	−6030.83	−0.409322	−0.204661	−	0.978833i	$-0.565609\pi$
$601$	−6030.83	−0.409322	−0.204661	+	0.978833i	$0.565609\pi$
$602$	−5154.28	−0.348958
$603$	0	0
$604$	1221.24	0.0822710
$605$	0	0
$606$	0	0
$607$	−12639.0	−0.845143	−0.422572	−	0.906330i	$-0.638872\pi$
$607$	−12639.0	−0.845143	−0.422572	+	0.906330i	$0.638872\pi$
$608$	−10437.0	−0.696176
$609$	0	0
$610$	0	0
$611$	−2160.59	−0.143058
$612$	0	0
$613$	226.217	0.0149051	0.00745254	−	0.999972i	$-0.497628\pi$
$613$	226.217	0.0149051	0.00745254	+	0.999972i	$0.497628\pi$
$614$	6693.84	0.439970
$615$	0	0
$616$	−2080.67	−0.136092
$617$	−6491.49	−0.423561	−0.211781	−	0.977317i	$-0.567926\pi$
$617$	−6491.49	−0.423561	−0.211781	+	0.977317i	$0.567926\pi$
$618$	0	0
$619$	19224.9	1.24833	0.624163	−	0.781294i	$-0.285440\pi$
$619$	19224.9	1.24833	0.624163	+	0.781294i	$0.285440\pi$
$620$	0	0
$621$	0	0
$622$	−33448.8	−2.15623
$623$	−236.808	−0.0152287
$624$	0	0
$625$	0	0
$626$	2106.71	0.134506
$627$	0	0
$628$	9062.79	0.575868
$629$	5332.50	0.338030
$630$	0	0
$631$	21123.5	1.33267	0.666334	−	0.745653i	$-0.267862\pi$
$631$	21123.5	1.33267	0.666334	+	0.745653i	$0.267862\pi$
$632$	−13141.0	−0.827090
$633$	0	0
$634$	−4014.34	−0.251467
$635$	0	0
$636$	0	0
$637$	−497.138	−0.0309220
$638$	−629.916	−0.0390887
$639$	0	0
$640$	0	0
$641$	8079.31	0.497837	0.248918	−	0.968524i	$-0.419925\pi$
$641$	8079.31	0.497837	0.248918	+	0.968524i	$0.419925\pi$
$642$	0	0
$643$	28025.8	1.71887	0.859433	−	0.511249i	$-0.170817\pi$
$643$	28025.8	1.71887	0.859433	+	0.511249i	$0.170817\pi$
$644$	−3284.59	−0.200980
$645$	0	0
$646$	−6382.93	−0.388751
$647$	9585.95	0.582477	0.291239	−	0.956650i	$-0.405933\pi$
$647$	9585.95	0.582477	0.291239	+	0.956650i	$0.405933\pi$
$648$	0	0
$649$	−5940.17	−0.359279
$650$	0	0
$651$	0	0
$652$	−1796.22	−0.107892
$653$	−6356.08	−0.380908	−0.190454	−	0.981696i	$-0.560996\pi$
$653$	−6356.08	−0.380908	−0.190454	+	0.981696i	$0.560996\pi$
$654$	0	0
$655$	0	0
$656$	39238.8	2.33539
$657$	0	0
$658$	−4975.57	−0.294784
$659$	−2394.70	−0.141554	−0.0707772	−	0.997492i	$-0.522548\pi$
$659$	−2394.70	−0.141554	−0.0707772	+	0.997492i	$0.522548\pi$
$660$	0	0
$661$	−23502.0	−1.38294	−0.691470	−	0.722406i	$-0.743036\pi$
$661$	−23502.0	−1.38294	−0.691470	+	0.722406i	$0.743036\pi$
$662$	−31032.8	−1.82194
$663$	0	0
$664$	−199.840	−0.0116796
$665$	0	0
$666$	0	0
$667$	1538.69	0.0893229
$668$	7612.46	0.440921
$669$	0	0
$670$	0	0
$671$	−11984.4	−0.689500
$672$	0	0
$673$	31304.2	1.79300	0.896498	−	0.443047i	$-0.146103\pi$
$673$	31304.2	1.79300	0.896498	+	0.443047i	$0.146103\pi$
$674$	−1694.66	−0.0968486
$675$	0	0
$676$	−6576.42	−0.374170
$677$	16821.3	0.954940	0.477470	−	0.878648i	$-0.341554\pi$
$677$	16821.3	0.954940	0.477470	+	0.878648i	$0.341554\pi$
$678$	0	0
$679$	−5675.32	−0.320764
$680$	0	0
$681$	0	0
$682$	−7610.53	−0.427306
$683$	−106.287	−0.00595453	−0.00297726	−	0.999996i	$-0.500948\pi$
$683$	−106.287	−0.00595453	−0.00297726	+	0.999996i	$0.500948\pi$
$684$	0	0
$685$	0	0
$686$	−1144.84	−0.0637177
$687$	0	0
$688$	−17485.5	−0.968938
$689$	−5402.12	−0.298700
$690$	0	0
$691$	−19383.5	−1.06712	−0.533562	−	0.845761i	$-0.679147\pi$
$691$	−19383.5	−1.06712	−0.533562	+	0.845761i	$0.679147\pi$
$692$	1967.34	0.108074
$693$	0	0
$694$	22898.9	1.25249
$695$	0	0
$696$	0	0
$697$	12227.1	0.664467
$698$	−30828.9	−1.67176
$699$	0	0
$700$	0	0
$701$	13338.2	0.718652	0.359326	−	0.933212i	$-0.383007\pi$
$701$	13338.2	0.718652	0.359326	+	0.933212i	$0.383007\pi$
$702$	0	0
$703$	−16717.1	−0.896868
$704$	−3375.20	−0.180692
$705$	0	0
$706$	31440.0	1.67600
$707$	11527.9	0.613227
$708$	0	0
$709$	15225.3	0.806487	0.403243	−	0.915093i	$-0.367883\pi$
$709$	15225.3	0.806487	0.403243	+	0.915093i	$0.367883\pi$
$710$	0	0
$711$	0	0
$712$	−548.708	−0.0288816
$713$	18590.2	0.976450
$714$	0	0
$715$	0	0
$716$	−7292.59	−0.380638
$717$	0	0
$718$	9648.79	0.501518
$719$	23884.3	1.23885	0.619427	−	0.785054i	$-0.287365\pi$
$719$	23884.3	1.23885	0.619427	+	0.785054i	$0.287365\pi$
$720$	0	0
$721$	−6449.52	−0.333138
$722$	−2883.40	−0.148628
$723$	0	0
$724$	−14650.3	−0.752034
$725$	0	0
$726$	0	0
$727$	27929.5	1.42482	0.712412	−	0.701761i	$-0.247603\pi$
$727$	27929.5	1.42482	0.712412	+	0.701761i	$0.247603\pi$
$728$	−1151.92	−0.0586443
$729$	0	0
$730$	0	0
$731$	−5448.62	−0.275683
$732$	0	0
$733$	17970.8	0.905548	0.452774	−	0.891625i	$-0.350434\pi$
$733$	17970.8	0.905548	0.452774	+	0.891625i	$0.350434\pi$
$734$	5241.97	0.263603
$735$	0	0
$736$	−20140.0	−1.00865
$737$	−15010.2	−0.750215
$738$	0	0
$739$	−1280.61	−0.0637455	−0.0318728	−	0.999492i	$-0.510147\pi$
$739$	−1280.61	−0.0637455	−0.0318728	+	0.999492i	$0.510147\pi$
$740$	0	0
$741$	0	0
$742$	−12440.4	−0.615499
$743$	−28570.1	−1.41068	−0.705340	−	0.708869i	$-0.749206\pi$
$743$	−28570.1	−1.41068	−0.705340	+	0.708869i	$0.749206\pi$
$744$	0	0
$745$	0	0
$746$	29083.0	1.42735
$747$	0	0
$748$	1421.44	0.0694828
$749$	7425.91	0.362265
$750$	0	0
$751$	32836.7	1.59551	0.797754	−	0.602982i	$-0.206021\pi$
$751$	32836.7	1.59551	0.797754	+	0.602982i	$0.206021\pi$
$752$	−16879.2	−0.818514
$753$	0	0
$754$	−348.740	−0.0168440
$755$	0	0
$756$	0	0
$757$	1086.72	0.0521763	0.0260882	−	0.999660i	$-0.491695\pi$
$757$	1086.72	0.0521763	0.0260882	+	0.999660i	$0.491695\pi$
$758$	−19291.0	−0.924379
$759$	0	0
$760$	0	0
$761$	17097.7	0.814443	0.407221	−	0.913329i	$-0.366498\pi$
$761$	17097.7	0.814443	0.407221	+	0.913329i	$0.366498\pi$
$762$	0	0
$763$	−10731.2	−0.509168
$764$	11714.9	0.554751
$765$	0	0
$766$	−41188.5	−1.94282
$767$	−3288.65	−0.154819
$768$	0	0
$769$	−24796.1	−1.16277	−0.581386	−	0.813628i	$-0.697489\pi$
$769$	−24796.1	−1.16277	−0.581386	+	0.813628i	$0.697489\pi$
$770$	0	0
$771$	0	0
$772$	8119.01	0.378510
$773$	−33325.8	−1.55064	−0.775321	−	0.631567i	$-0.782412\pi$
$773$	−33325.8	−1.55064	−0.775321	+	0.631567i	$0.782412\pi$
$774$	0	0
$775$	0	0
$776$	−13150.3	−0.608336
$777$	0	0
$778$	−26529.8	−1.22255
$779$	−38331.3	−1.76298
$780$	0	0
$781$	−8557.02	−0.392054
$782$	−12317.0	−0.563241
$783$	0	0
$784$	−3883.80	−0.176922
$785$	0	0
$786$	0	0
$787$	30195.2	1.36765	0.683826	−	0.729645i	$-0.260315\pi$
$787$	30195.2	1.36765	0.683826	+	0.729645i	$0.260315\pi$
$788$	7555.77	0.341578
$789$	0	0
$790$	0	0
$791$	14508.7	0.652175
$792$	0	0
$793$	−6634.94	−0.297117
$794$	17902.4	0.800168
$795$	0	0
$796$	3744.57	0.166737
$797$	24226.2	1.07671	0.538353	−	0.842720i	$-0.319047\pi$
$797$	24226.2	1.07671	0.538353	+	0.842720i	$0.319047\pi$
$798$	0	0
$799$	−5259.69	−0.232884
$800$	0	0
$801$	0	0
$802$	−20217.1	−0.890138
$803$	3171.57	0.139380
$804$	0	0
$805$	0	0
$806$	−4213.42	−0.184133
$807$	0	0
$808$	26711.4	1.16300
$809$	16039.7	0.697065	0.348533	−	0.937297i	$-0.386680\pi$
$809$	16039.7	0.697065	0.348533	+	0.937297i	$0.386680\pi$
$810$	0	0
$811$	33767.0	1.46205	0.731024	−	0.682352i	$-0.239043\pi$
$811$	33767.0	1.46205	0.731024	+	0.682352i	$0.239043\pi$
$812$	−226.394	−0.00978434
$813$	0	0
$814$	13206.2	0.568644
$815$	0	0
$816$	0	0
$817$	17081.1	0.731448
$818$	4167.22	0.178122
$819$	0	0
$820$	0	0
$821$	−631.743	−0.0268550	−0.0134275	−	0.999910i	$-0.504274\pi$
$821$	−631.743	−0.0268550	−0.0134275	+	0.999910i	$0.504274\pi$
$822$	0	0
$823$	30565.3	1.29458	0.647290	−	0.762244i	$-0.275902\pi$
$823$	30565.3	1.29458	0.647290	+	0.762244i	$0.275902\pi$
$824$	−14944.2	−0.631804
$825$	0	0
$826$	−7573.34	−0.319020
$827$	24472.5	1.02901	0.514505	−	0.857487i	$-0.327976\pi$
$827$	24472.5	1.02901	0.514505	+	0.857487i	$0.327976\pi$
$828$	0	0
$829$	−9613.32	−0.402756	−0.201378	−	0.979514i	$-0.564542\pi$
$829$	−9613.32	−0.402756	−0.201378	+	0.979514i	$0.564542\pi$
$830$	0	0
$831$	0	0
$832$	−1868.61	−0.0778634
$833$	−1210.22	−0.0503381
$834$	0	0
$835$	0	0
$836$	−4456.15	−0.184353
$837$	0	0
$838$	−27784.3	−1.14534
$839$	23320.5	0.959609	0.479805	−	0.877375i	$-0.340707\pi$
$839$	23320.5	0.959609	0.479805	+	0.877375i	$0.340707\pi$
$840$	0	0
$841$	−24282.9	−0.995651
$842$	5652.11	0.231336
$843$	0	0
$844$	−2402.18	−0.0979699
$845$	0	0
$846$	0	0
$847$	6966.17	0.282598
$848$	−42203.0	−1.70903
$849$	0	0
$850$	0	0
$851$	−32258.6	−1.29943
$852$	0	0
$853$	−2992.19	−0.120106	−0.0600531	−	0.998195i	$-0.519127\pi$
$853$	−2992.19	−0.120106	−0.0600531	+	0.998195i	$0.519127\pi$
$854$	−15279.4	−0.612237
$855$	0	0
$856$	17206.6	0.687044
$857$	20796.2	0.828919	0.414459	−	0.910068i	$-0.363971\pi$
$857$	20796.2	0.828919	0.414459	+	0.910068i	$0.363971\pi$
$858$	0	0
$859$	−5302.17	−0.210603	−0.105301	−	0.994440i	$-0.533581\pi$
$859$	−5302.17	−0.210603	−0.105301	+	0.994440i	$0.533581\pi$
$860$	0	0
$861$	0	0
$862$	−39233.8	−1.55024
$863$	−27066.7	−1.06762	−0.533812	−	0.845603i	$-0.679241\pi$
$863$	−27066.7	−1.06762	−0.533812	+	0.845603i	$0.679241\pi$
$864$	0	0
$865$	0	0
$866$	48778.8	1.91406
$867$	0	0
$868$	−2735.26	−0.106959
$869$	−14847.3	−0.579585
$870$	0	0
$871$	−8310.09	−0.323280
$872$	−24865.3	−0.965648
$873$	0	0
$874$	38613.1	1.49440
$875$	0	0
$876$	0	0
$877$	−26825.0	−1.03286	−0.516429	−	0.856330i	$-0.672739\pi$
$877$	−26825.0	−1.03286	−0.516429	+	0.856330i	$0.672739\pi$
$878$	6533.69	0.251141
$879$	0	0
$880$	0	0
$881$	47701.8	1.82420	0.912098	−	0.409973i	$-0.134462\pi$
$881$	47701.8	1.82420	0.912098	+	0.409973i	$0.134462\pi$
$882$	0	0
$883$	−28447.5	−1.08418	−0.542092	−	0.840319i	$-0.682368\pi$
$883$	−28447.5	−1.08418	−0.542092	+	0.840319i	$0.682368\pi$
$884$	786.953	0.0299413
$885$	0	0
$886$	−1190.82	−0.0451538
$887$	−35046.9	−1.32667	−0.663337	−	0.748321i	$-0.730860\pi$
$887$	−35046.9	−1.32667	−0.663337	+	0.748321i	$0.730860\pi$
$888$	0	0
$889$	−2556.02	−0.0964297
$890$	0	0
$891$	0	0
$892$	−4978.27	−0.186866
$893$	16488.9	0.617894
$894$	0	0
$895$	0	0
$896$	−11851.7	−0.441895
$897$	0	0
$898$	−24683.6	−0.917262
$899$	1281.35	0.0475367
$900$	0	0
$901$	−13150.8	−0.486255
$902$	30280.9	1.11779
$903$	0	0
$904$	33618.2	1.23686
$905$	0	0
$906$	0	0
$907$	19761.2	0.723439	0.361720	−	0.932287i	$-0.382190\pi$
$907$	19761.2	0.723439	0.361720	+	0.932287i	$0.382190\pi$
$908$	7421.23	0.271236
$909$	0	0
$910$	0	0
$911$	−23563.4	−0.856960	−0.428480	−	0.903551i	$-0.640951\pi$
$911$	−23563.4	−0.856960	−0.428480	+	0.903551i	$0.640951\pi$
$912$	0	0
$913$	−225.788	−0.00818453
$914$	−42463.4	−1.53672
$915$	0	0
$916$	5107.84	0.184244
$917$	9105.78	0.327917
$918$	0	0
$919$	−37601.1	−1.34967	−0.674834	−	0.737970i	$-0.735785\pi$
$919$	−37601.1	−1.34967	−0.674834	+	0.737970i	$0.735785\pi$
$920$	0	0
$921$	0	0
$922$	7364.54	0.263057
$923$	−4737.42	−0.168943
$924$	0	0
$925$	0	0
$926$	−53110.2	−1.88478
$927$	0	0
$928$	−1388.17	−0.0491046
$929$	17616.6	0.622154	0.311077	−	0.950385i	$-0.399310\pi$
$929$	17616.6	0.622154	0.311077	+	0.950385i	$0.399310\pi$
$930$	0	0
$931$	3793.98	0.133558
$932$	5402.74	0.189885
$933$	0	0
$934$	25375.6	0.888990
$935$	0	0
$936$	0	0
$937$	36717.2	1.28015	0.640074	−	0.768313i	$-0.278904\pi$
$937$	36717.2	1.28015	0.640074	+	0.768313i	$0.278904\pi$
$938$	−19137.1	−0.666148
$939$	0	0
$940$	0	0
$941$	−21861.7	−0.757354	−0.378677	−	0.925529i	$-0.623621\pi$
$941$	−21861.7	−0.757354	−0.378677	+	0.925529i	$0.623621\pi$
$942$	0	0
$943$	−73967.0	−2.55429
$944$	−25692.0	−0.885809
$945$	0	0
$946$	−13493.7	−0.463762
$947$	−17752.1	−0.609149	−0.304575	−	0.952488i	$-0.598514\pi$
$947$	−17752.1	−0.609149	−0.304575	+	0.952488i	$0.598514\pi$
$948$	0	0
$949$	1755.87	0.0600612
$950$	0	0
$951$	0	0
$952$	−2804.21	−0.0954673
$953$	26764.9	0.909758	0.454879	−	0.890553i	$-0.349683\pi$
$953$	26764.9	0.909758	0.454879	+	0.890553i	$0.349683\pi$
$954$	0	0
$955$	0	0
$956$	−22667.4	−0.766857
$957$	0	0
$958$	−5613.83	−0.189326
$959$	18048.8	0.607744
$960$	0	0
$961$	−14309.9	−0.480344
$962$	7311.32	0.245038
$963$	0	0
$964$	16000.2	0.534575
$965$	0	0
$966$	0	0
$967$	31173.9	1.03669	0.518347	−	0.855170i	$-0.326547\pi$
$967$	31173.9	1.03669	0.518347	+	0.855170i	$0.326547\pi$
$968$	16141.3	0.535953
$969$	0	0
$970$	0	0
$971$	35414.7	1.17046	0.585228	−	0.810869i	$-0.301005\pi$
$971$	35414.7	1.17046	0.585228	+	0.810869i	$0.301005\pi$
$972$	0	0
$973$	−21458.5	−0.707016
$974$	47085.6	1.54899
$975$	0	0
$976$	−51834.3	−1.69997
$977$	20952.6	0.686112	0.343056	−	0.939315i	$-0.388538\pi$
$977$	20952.6	0.686112	0.343056	+	0.939315i	$0.388538\pi$
$978$	0	0
$979$	−619.955	−0.0202388
$980$	0	0
$981$	0	0
$982$	−30073.7	−0.977281
$983$	20858.6	0.676793	0.338396	−	0.941004i	$-0.390116\pi$
$983$	20858.6	0.676793	0.338396	+	0.941004i	$0.390116\pi$
$984$	0	0
$985$	0	0
$986$	−848.963	−0.0274204
$987$	0	0
$988$	−2467.06	−0.0794409
$989$	32961.1	1.05976
$990$	0	0
$991$	−13565.0	−0.434820	−0.217410	−	0.976080i	$-0.569761\pi$
$991$	−13565.0	−0.434820	−0.217410	+	0.976080i	$0.569761\pi$
$992$	−16771.7	−0.536795
$993$	0	0
$994$	−10909.7	−0.348122
$995$	0	0
$996$	0	0
$997$	−12520.7	−0.397727	−0.198864	−	0.980027i	$-0.563725\pi$
$997$	−12520.7	−0.397727	−0.198864	+	0.980027i	$0.563725\pi$
$998$	8861.94	0.281082
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bp.1.4		5
3.2	odd	2		525.4.a.w.1.2		5
5.2	odd	4		315.4.d.b.64.8		10
5.3	odd	4		315.4.d.b.64.3		10
5.4	even	2		1575.4.a.bo.1.2		5
15.2	even	4		105.4.d.b.64.3	✓	10
15.8	even	4		105.4.d.b.64.8	yes	10
15.14	odd	2		525.4.a.x.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.b.64.3	✓	10	15.2	even	4
105.4.d.b.64.8	yes	10	15.8	even	4
315.4.d.b.64.3		10	5.3	odd	4
315.4.d.b.64.8		10	5.2	odd	4
525.4.a.w.1.2		5	3.2	odd	2
525.4.a.x.1.4		5	15.14	odd	2
1575.4.a.bo.1.2		5	5.4	even	2
1575.4.a.bp.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.33774 q^{2} +3.14050 q^{4} -7.00000 q^{7} -16.2197 q^{8} +O(q^{10})\)	\(q+3.33774 q^{2} +3.14050 q^{4} -7.00000 q^{7} -16.2197 q^{8} -18.3258 q^{11} -10.1457 q^{13} -23.3642 q^{14} -79.2613 q^{16} -24.6984 q^{17} +77.4282 q^{19} -61.1666 q^{22} +149.411 q^{23} -33.8636 q^{26} -21.9835 q^{28} +10.2984 q^{29} +124.423 q^{31} -134.796 q^{32} -82.4367 q^{34} -215.905 q^{37} +258.435 q^{38} -495.056 q^{41} +220.606 q^{43} -57.5521 q^{44} +498.696 q^{46} +212.957 q^{47} +49.0000 q^{49} -31.8625 q^{52} +532.455 q^{53} +113.538 q^{56} +34.3732 q^{58} +324.143 q^{59} +653.967 q^{61} +415.292 q^{62} +184.178 q^{64} +819.077 q^{67} -77.5653 q^{68} +466.940 q^{71} -173.066 q^{73} -720.634 q^{74} +243.163 q^{76} +128.280 q^{77} +810.186 q^{79} -1652.37 q^{82} +12.3208 q^{83} +736.326 q^{86} +297.239 q^{88} +33.8297 q^{89} +71.0198 q^{91} +469.227 q^{92} +710.795 q^{94} +810.761 q^{97} +163.549 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 27 q^{4} - 35 q^{7} + 33 q^{8}+O(q^{10}) \) 5 * q + q^2 + 27 * q^4 - 35 * q^7 + 33 * q^8	\( 5 q + q^{2} + 27 q^{4} - 35 q^{7} + 33 q^{8} - 66 q^{11} + 2 q^{13} - 7 q^{14} + 155 q^{16} + 108 q^{17} + 174 q^{19} + 506 q^{22} - 116 q^{23} - 446 q^{26} - 189 q^{28} - 370 q^{29} + 342 q^{31} - 55 q^{32} + 112 q^{34} + 408 q^{37} - 34 q^{38} - 802 q^{41} + 584 q^{43} - 290 q^{44} + 640 q^{46} + 716 q^{47} + 245 q^{49} - 338 q^{52} + 98 q^{53} - 231 q^{56} - 482 q^{58} - 704 q^{59} + 650 q^{61} + 2070 q^{62} + 75 q^{64} - 180 q^{67} + 4520 q^{68} - 1470 q^{71} + 534 q^{73} + 1312 q^{74} + 4370 q^{76} + 462 q^{77} - 820 q^{79} + 1338 q^{82} + 1520 q^{83} - 832 q^{86} + 3258 q^{88} - 286 q^{89} - 14 q^{91} + 1288 q^{92} + 2540 q^{94} - 278 q^{97} + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 27 * q^4 - 35 * q^7 + 33 * q^8 - 66 * q^11 + 2 * q^13 - 7 * q^14 + 155 * q^16 + 108 * q^17 + 174 * q^19 + 506 * q^22 - 116 * q^23 - 446 * q^26 - 189 * q^28 - 370 * q^29 + 342 * q^31 - 55 * q^32 + 112 * q^34 + 408 * q^37 - 34 * q^38 - 802 * q^41 + 584 * q^43 - 290 * q^44 + 640 * q^46 + 716 * q^47 + 245 * q^49 - 338 * q^52 + 98 * q^53 - 231 * q^56 - 482 * q^58 - 704 * q^59 + 650 * q^61 + 2070 * q^62 + 75 * q^64 - 180 * q^67 + 4520 * q^68 - 1470 * q^71 + 534 * q^73 + 1312 * q^74 + 4370 * q^76 + 462 * q^77 - 820 * q^79 + 1338 * q^82 + 1520 * q^83 - 832 * q^86 + 3258 * q^88 - 286 * q^89 - 14 * q^91 + 1288 * q^92 + 2540 * q^94 - 278 * q^97 + 49 * q^98

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