LMFDB - Embedded newform 375.2.a.f.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [375,2,Mod(1,375)] 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("375.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.99439007580$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.46673$ of defining polynomial
Character	$\chi$	$=$	375.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.52452 q^{2} +1.00000 q^{3} +4.37322 q^{4} +2.52452 q^{6} -4.93346 q^{7} +5.99126 q^{8} +1.00000 q^{9} +0.187020 q^{11} +4.37322 q^{12} +4.16953 q^{13} -12.4546 q^{14} +6.37863 q^{16} -2.85410 q^{17} +2.52452 q^{18} -0.431014 q^{19} -4.93346 q^{21} +0.472136 q^{22} -4.36448 q^{23} +5.99126 q^{24} +10.5261 q^{26} +1.00000 q^{27} -21.5751 q^{28} -1.12048 q^{29} -2.73852 q^{31} +4.12048 q^{32} +0.187020 q^{33} -7.20525 q^{34} +4.37322 q^{36} -5.35165 q^{37} -1.08811 q^{38} +4.16953 q^{39} +7.69740 q^{41} -12.4546 q^{42} -6.58772 q^{43} +0.817879 q^{44} -11.0182 q^{46} -2.43101 q^{47} +6.37863 q^{48} +17.3391 q^{49} -2.85410 q^{51} +18.2343 q^{52} +8.48496 q^{53} +2.52452 q^{54} -29.5576 q^{56} -0.431014 q^{57} -2.82869 q^{58} +10.7573 q^{59} +9.24889 q^{61} -6.91345 q^{62} -4.93346 q^{63} -2.35499 q^{64} +0.472136 q^{66} +9.03156 q^{67} -12.4816 q^{68} -4.36448 q^{69} +1.32336 q^{71} +5.99126 q^{72} +3.04905 q^{73} -13.5104 q^{74} -1.88492 q^{76} -0.922655 q^{77} +10.5261 q^{78} -8.19985 q^{79} +1.00000 q^{81} +19.4323 q^{82} +8.04112 q^{83} -21.5751 q^{84} -16.6309 q^{86} -1.12048 q^{87} +1.12048 q^{88} -3.10300 q^{89} -20.5702 q^{91} -19.0868 q^{92} -2.73852 q^{93} -6.13715 q^{94} +4.12048 q^{96} -17.7622 q^{97} +43.7729 q^{98} +0.187020 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + 4 q^{3} + 7 q^{4} + 3 q^{6} - 4 q^{7} + 9 q^{8} + 4 q^{9} + 6 q^{11} + 7 q^{12} - 8 q^{13} - 2 q^{14} + 5 q^{16} + 2 q^{17} + 3 q^{18} + 8 q^{19} - 4 q^{21} - 16 q^{22} + 8 q^{23} + 9 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) $ 4 * q + 3 * q^2 + 4 * q^3 + 7 * q^4 + 3 * q^6 - 4 * q^7 + 9 * q^8 + 4 * q^9 + 6 * q^11 + 7 * q^12 - 8 * q^13 - 2 * q^14 + 5 * q^16 + 2 * q^17 + 3 * q^18 + 8 * q^19 - 4 * q^21 - 16 * q^22 + 8 * q^23 + 9 * q^24 - 2 * q^26 + 4 * q^27 - 28 * q^28 + 6 * q^29 + 4 * q^31 + 6 * q^32 + 6 * q^33 - 6 * q^34 + 7 * q^36 - 14 * q^37 - 17 * q^38 - 8 * q^39 + 24 * q^41 - 2 * q^42 - 10 * q^43 - 14 * q^44 - 16 * q^46 + 5 * q^48 + 20 * q^49 + 2 * q^51 + 2 * q^52 - 2 * q^53 + 3 * q^54 - 30 * q^56 + 8 * q^57 + 26 * q^58 + 2 * q^59 + 10 * q^61 + 22 * q^62 - 4 * q^63 - 27 * q^64 - 16 * q^66 - 8 * q^67 + 11 * q^68 + 8 * q^69 - 12 * q^71 + 9 * q^72 - 2 * q^73 - 42 * q^74 - 8 * q^76 - 8 * q^77 - 2 * q^78 - 20 * q^79 + 4 * q^81 + 12 * q^82 + 24 * q^83 - 28 * q^84 - 44 * q^86 + 6 * q^87 - 6 * q^88 + 28 * q^89 - 36 * q^91 - 29 * q^92 + 4 * q^93 - 23 * q^94 + 6 * q^96 - 18 * q^97 + 23 * q^98 + 6 * q^99

Coefficient data

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

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

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

$2$	2.52452	1.78511	0.892554	−	0.450940i	$-0.148911\pi$
$2$	2.52452	1.78511	0.892554	+	0.450940i	$0.148911\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.37322	2.18661
$5$	0	0
$5$	0	0
$6$	2.52452	1.03063
$7$	−4.93346	−1.86467	−0.932337	−	0.361591i	$-0.882234\pi$
$7$	−4.93346	−1.86467	−0.932337	+	0.361591i	$0.882234\pi$
$8$	5.99126	2.11823
$9$	1.00000	0.333333
$10$	0	0
$11$	0.187020	0.0563886	0.0281943	−	0.999602i	$-0.491024\pi$
$11$	0.187020	0.0563886	0.0281943	+	0.999602i	$0.491024\pi$
$12$	4.37322	1.26244
$13$	4.16953	1.15642	0.578210	−	0.815888i	$-0.303751\pi$
$13$	4.16953	1.15642	0.578210	+	0.815888i	$0.303751\pi$
$14$	−12.4546	−3.32864
$15$	0	0
$16$	6.37863	1.59466
$17$	−2.85410	−0.692221	−0.346111	−	0.938194i	$-0.612498\pi$
$17$	−2.85410	−0.692221	−0.346111	+	0.938194i	$0.612498\pi$
$18$	2.52452	0.595036
$19$	−0.431014	−0.0988814	−0.0494407	−	0.998777i	$-0.515744\pi$
$19$	−0.431014	−0.0988814	−0.0494407	+	0.998777i	$0.515744\pi$
$20$	0	0
$21$	−4.93346	−1.07657
$22$	0.472136	0.100660
$23$	−4.36448	−0.910057	−0.455028	−	0.890477i	$-0.650371\pi$
$23$	−4.36448	−0.910057	−0.455028	+	0.890477i	$0.650371\pi$
$24$	5.99126	1.22296
$25$	0	0
$26$	10.5261	2.06433
$27$	1.00000	0.192450
$28$	−21.5751	−4.07732
$29$	−1.12048	−0.208069	−0.104034	−	0.994574i	$-0.533175\pi$
$29$	−1.12048	−0.208069	−0.104034	+	0.994574i	$0.533175\pi$
$30$	0	0
$31$	−2.73852	−0.491852	−0.245926	−	0.969289i	$-0.579092\pi$
$31$	−2.73852	−0.491852	−0.245926	+	0.969289i	$0.579092\pi$
$32$	4.12048	0.728405
$33$	0.187020	0.0325560
$34$	−7.20525	−1.23569
$35$	0	0
$36$	4.37322	0.728870
$37$	−5.35165	−0.879806	−0.439903	−	0.898045i	$-0.644987\pi$
$37$	−5.35165	−0.879806	−0.439903	+	0.898045i	$0.644987\pi$
$38$	−1.08811	−0.176514
$39$	4.16953	0.667659
$40$	0	0
$41$	7.69740	1.20213	0.601066	−	0.799200i	$-0.294743\pi$
$41$	7.69740	1.20213	0.601066	+	0.799200i	$0.294743\pi$
$42$	−12.4546	−1.92179
$43$	−6.58772	−1.00462	−0.502309	−	0.864688i	$-0.667516\pi$
$43$	−6.58772	−1.00462	−0.502309	+	0.864688i	$0.667516\pi$
$44$	0.817879	0.123300
$45$	0	0
$46$	−11.0182	−1.62455
$47$	−2.43101	−0.354600	−0.177300	−	0.984157i	$-0.556736\pi$
$47$	−2.43101	−0.354600	−0.177300	+	0.984157i	$0.556736\pi$
$48$	6.37863	0.920675
$49$	17.3391	2.47701
$50$	0	0
$51$	−2.85410	−0.399654
$52$	18.2343	2.52864
$53$	8.48496	1.16550	0.582750	−	0.812652i	$-0.301977\pi$
$53$	8.48496	1.16550	0.582750	+	0.812652i	$0.301977\pi$
$54$	2.52452	0.343544
$55$	0	0
$56$	−29.5576	−3.94981
$57$	−0.431014	−0.0570892
$58$	−2.82869	−0.371425
$59$	10.7573	1.40047	0.700237	−	0.713910i	$-0.253077\pi$
$59$	10.7573	1.40047	0.700237	+	0.713910i	$0.253077\pi$
$60$	0	0
$61$	9.24889	1.18420	0.592100	−	0.805865i	$-0.298299\pi$
$61$	9.24889	1.18420	0.592100	+	0.805865i	$0.298299\pi$
$62$	−6.91345	−0.878009
$63$	−4.93346	−0.621558
$64$	−2.35499	−0.294374
$65$	0	0
$66$	0.472136	0.0581159
$67$	9.03156	1.10338	0.551690	−	0.834049i	$-0.313983\pi$
$67$	9.03156	1.10338	0.551690	+	0.834049i	$0.313983\pi$
$68$	−12.4816	−1.51362
$69$	−4.36448	−0.525421
$70$	0	0
$71$	1.32336	0.157053	0.0785267	−	0.996912i	$-0.474978\pi$
$71$	1.32336	0.157053	0.0785267	+	0.996912i	$0.474978\pi$
$72$	5.99126	0.706076
$73$	3.04905	0.356864	0.178432	−	0.983952i	$-0.442898\pi$
$73$	3.04905	0.356864	0.178432	+	0.983952i	$0.442898\pi$
$74$	−13.5104	−1.57055
$75$	0	0
$76$	−1.88492	−0.216215
$77$	−0.922655	−0.105146
$78$	10.5261	1.19184
$79$	−8.19985	−0.922555	−0.461277	−	0.887256i	$-0.652609\pi$
$79$	−8.19985	−0.922555	−0.461277	+	0.887256i	$0.652609\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	19.4323	2.14593
$83$	8.04112	0.882628	0.441314	−	0.897353i	$-0.354513\pi$
$83$	8.04112	0.882628	0.441314	+	0.897353i	$0.354513\pi$
$84$	−21.5751	−2.35404
$85$	0	0
$86$	−16.6309	−1.79335
$87$	−1.12048	−0.120128
$88$	1.12048	0.119444
$89$	−3.10300	−0.328917	−0.164458	−	0.986384i	$-0.552588\pi$
$89$	−3.10300	−0.328917	−0.164458	+	0.986384i	$0.552588\pi$
$90$	0	0
$91$	−20.5702	−2.15635
$92$	−19.0868	−1.98994
$93$	−2.73852	−0.283971
$94$	−6.13715	−0.632999
$95$	0	0
$96$	4.12048	0.420545
$97$	−17.7622	−1.80347	−0.901737	−	0.432286i	$-0.857707\pi$
$97$	−17.7622	−1.80347	−0.901737	+	0.432286i	$0.857707\pi$
$98$	43.7729	4.42173
$99$	0.187020	0.0187962
$100$	0	0
$101$	11.6368	1.15790	0.578951	−	0.815362i	$-0.303462\pi$
$101$	11.6368	1.15790	0.578951	+	0.815362i	$0.303462\pi$
$102$	−7.20525	−0.713426
$103$	−5.13797	−0.506259	−0.253130	−	0.967432i	$-0.581460\pi$
$103$	−5.13797	−0.506259	−0.253130	+	0.967432i	$0.581460\pi$
$104$	24.9807	2.44956
$105$	0	0
$106$	21.4205	2.08054
$107$	16.3645	1.58201	0.791007	−	0.611807i	$-0.209557\pi$
$107$	16.3645	1.58201	0.791007	+	0.611807i	$0.209557\pi$
$108$	4.37322	0.420813
$109$	1.38686	0.132838	0.0664188	−	0.997792i	$-0.478843\pi$
$109$	1.38686	0.132838	0.0664188	+	0.997792i	$0.478843\pi$
$110$	0	0
$111$	−5.35165	−0.507956
$112$	−31.4687	−2.97351
$113$	−14.9571	−1.40705	−0.703523	−	0.710673i	$-0.748391\pi$
$113$	−14.9571	−1.40705	−0.703523	+	0.710673i	$0.748391\pi$
$114$	−1.08811	−0.101910
$115$	0	0
$116$	−4.90012	−0.454965
$117$	4.16953	0.385473
$118$	27.1569	2.50000
$119$	14.0806	1.29077
$120$	0	0
$121$	−10.9650	−0.996820
$122$	23.3491	2.11392
$123$	7.69740	0.694051
$124$	−11.9761	−1.07549
$125$	0	0
$126$	−12.4546	−1.10955
$127$	−11.3234	−1.00479	−0.502393	−	0.864640i	$-0.667547\pi$
$127$	−11.3234	−1.00479	−0.502393	+	0.864640i	$0.667547\pi$
$128$	−14.1862	−1.25389
$129$	−6.58772	−0.580016
$130$	0	0
$131$	17.1903	1.50192	0.750961	−	0.660347i	$-0.229591\pi$
$131$	17.1903	1.50192	0.750961	+	0.660347i	$0.229591\pi$
$132$	0.817879	0.0711872
$133$	2.12639	0.184382
$134$	22.8004	1.96965
$135$	0	0
$136$	−17.0997	−1.46628
$137$	7.49289	0.640161	0.320080	−	0.947390i	$-0.396290\pi$
$137$	7.49289	0.640161	0.320080	+	0.947390i	$0.396290\pi$
$138$	−11.0182	−0.937934
$139$	−3.89234	−0.330144	−0.165072	−	0.986282i	$-0.552786\pi$
$139$	−3.89234	−0.330144	−0.165072	+	0.986282i	$0.552786\pi$
$140$	0	0
$141$	−2.43101	−0.204728
$142$	3.34084	0.280357
$143$	0.779785	0.0652089
$144$	6.37863	0.531552
$145$	0	0
$146$	7.69740	0.637041
$147$	17.3391	1.43010
$148$	−23.4040	−1.92379
$149$	3.20777	0.262791	0.131395	−	0.991330i	$-0.458054\pi$
$149$	3.20777	0.262791	0.131395	+	0.991330i	$0.458054\pi$
$150$	0	0
$151$	−7.90805	−0.643548	−0.321774	−	0.946817i	$-0.604279\pi$
$151$	−7.90805	−0.643548	−0.321774	+	0.946817i	$0.604279\pi$
$152$	−2.58232	−0.209454
$153$	−2.85410	−0.230740
$154$	−2.32927	−0.187698
$155$	0	0
$156$	18.2343	1.45991
$157$	10.9335	0.872585	0.436293	−	0.899805i	$-0.356291\pi$
$157$	10.9335	0.872585	0.436293	+	0.899805i	$0.356291\pi$
$158$	−20.7007	−1.64686
$159$	8.48496	0.672901
$160$	0	0
$161$	21.5320	1.69696
$162$	2.52452	0.198345
$163$	10.1803	0.797386	0.398693	−	0.917085i	$-0.369464\pi$
$163$	10.1803	0.797386	0.398693	+	0.917085i	$0.369464\pi$
$164$	33.6624	2.62859
$165$	0	0
$166$	20.3000	1.57559
$167$	5.65123	0.437305	0.218653	−	0.975803i	$-0.429834\pi$
$167$	5.65123	0.437305	0.218653	+	0.975803i	$0.429834\pi$
$168$	−29.5576	−2.28042
$169$	4.38499	0.337307
$170$	0	0
$171$	−0.431014	−0.0329605
$172$	−28.8096	−2.19671
$173$	−20.8112	−1.58225	−0.791123	−	0.611657i	$-0.790503\pi$
$173$	−20.8112	−1.58225	−0.791123	+	0.611657i	$0.790503\pi$
$174$	−2.82869	−0.214442
$175$	0	0
$176$	1.19293	0.0899204
$177$	10.7573	0.808565
$178$	−7.83359	−0.587152
$179$	−16.1412	−1.20645	−0.603226	−	0.797570i	$-0.706118\pi$
$179$	−16.1412	−1.20645	−0.603226	+	0.797570i	$0.706118\pi$
$180$	0	0
$181$	8.70028	0.646687	0.323343	−	0.946282i	$-0.395193\pi$
$181$	8.70028	0.646687	0.323343	+	0.946282i	$0.395193\pi$
$182$	−51.9300	−3.84931
$183$	9.24889	0.683698
$184$	−26.1487	−1.92771
$185$	0	0
$186$	−6.91345	−0.506919
$187$	−0.533774	−0.0390334
$188$	−10.6314	−0.775372
$189$	−4.93346	−0.358857
$190$	0	0
$191$	−20.8445	−1.50826	−0.754129	−	0.656726i	$-0.771941\pi$
$191$	−20.8445	−1.50826	−0.754129	+	0.656726i	$0.771941\pi$
$192$	−2.35499	−0.169957
$193$	−7.13634	−0.513685	−0.256842	−	0.966453i	$-0.582682\pi$
$193$	−7.13634	−0.513685	−0.256842	+	0.966453i	$0.582682\pi$
$194$	−44.8410	−3.21939
$195$	0	0
$196$	75.8276	5.41626
$197$	10.2489	0.730203	0.365102	−	0.930968i	$-0.381034\pi$
$197$	10.2489	0.730203	0.365102	+	0.930968i	$0.381034\pi$
$198$	0.472136	0.0335532
$199$	−20.0885	−1.42404	−0.712019	−	0.702160i	$-0.752219\pi$
$199$	−20.0885	−1.42404	−0.712019	+	0.702160i	$0.752219\pi$
$200$	0	0
$201$	9.03156	0.637037
$202$	29.3773	2.06698
$203$	5.52786	0.387980
$204$	−12.4816	−0.873888
$205$	0	0
$206$	−12.9709	−0.903728
$207$	−4.36448	−0.303352
$208$	26.5959	1.84409
$209$	−0.0806082	−0.00557578
$210$	0	0
$211$	−8.26638	−0.569081	−0.284541	−	0.958664i	$-0.591841\pi$
$211$	−8.26638	−0.569081	−0.284541	+	0.958664i	$0.591841\pi$
$212$	37.1066	2.54849
$213$	1.32336	0.0906749
$214$	41.3125	2.82407
$215$	0	0
$216$	5.99126	0.407653
$217$	13.5104	0.917144
$218$	3.50117	0.237129
$219$	3.04905	0.206036
$220$	0	0
$221$	−11.9003	−0.800499
$222$	−13.5104	−0.906757
$223$	−1.59262	−0.106650	−0.0533248	−	0.998577i	$-0.516982\pi$
$223$	−1.59262	−0.106650	−0.0533248	+	0.998577i	$0.516982\pi$
$224$	−20.3283	−1.35824
$225$	0	0
$226$	−37.7596	−2.51173
$227$	18.6213	1.23594	0.617969	−	0.786202i	$-0.287956\pi$
$227$	18.6213	1.23594	0.617969	+	0.786202i	$0.287956\pi$
$228$	−1.88492	−0.124832
$229$	−5.63965	−0.372679	−0.186339	−	0.982485i	$-0.559662\pi$
$229$	−5.63965	−0.372679	−0.186339	+	0.982485i	$0.559662\pi$
$230$	0	0
$231$	−0.922655	−0.0607063
$232$	−6.71310	−0.440737
$233$	6.57023	0.430430	0.215215	−	0.976567i	$-0.430955\pi$
$233$	6.57023	0.430430	0.215215	+	0.976567i	$0.430955\pi$
$234$	10.5261	0.688112
$235$	0	0
$236$	47.0439	3.06229
$237$	−8.19985	−0.532637
$238$	35.5468	2.30416
$239$	4.32825	0.279972	0.139986	−	0.990154i	$-0.455294\pi$
$239$	4.32825	0.279972	0.139986	+	0.990154i	$0.455294\pi$
$240$	0	0
$241$	6.98717	0.450083	0.225042	−	0.974349i	$-0.427748\pi$
$241$	6.98717	0.450083	0.225042	+	0.974349i	$0.427748\pi$
$242$	−27.6815	−1.77943
$243$	1.00000	0.0641500
$244$	40.4475	2.58938
$245$	0	0
$246$	19.4323	1.23896
$247$	−1.79713	−0.114348
$248$	−16.4072	−1.04186
$249$	8.04112	0.509585
$250$	0	0
$251$	−20.3216	−1.28269	−0.641343	−	0.767254i	$-0.721623\pi$
$251$	−20.3216	−1.28269	−0.641343	+	0.767254i	$0.721623\pi$
$252$	−21.5751	−1.35911
$253$	−0.816244	−0.0513168
$254$	−28.5861	−1.79365
$255$	0	0
$256$	−31.1034	−1.94396
$257$	6.93145	0.432372	0.216186	−	0.976352i	$-0.430638\pi$
$257$	6.93145	0.432372	0.216186	+	0.976352i	$0.430638\pi$
$258$	−16.6309	−1.03539
$259$	26.4022	1.64055
$260$	0	0
$261$	−1.12048	−0.0693562
$262$	43.3973	2.68109
$263$	21.6478	1.33486	0.667431	−	0.744672i	$-0.267394\pi$
$263$	21.6478	1.33486	0.667431	+	0.744672i	$0.267394\pi$
$264$	1.12048	0.0689610
$265$	0	0
$266$	5.36813	0.329141
$267$	−3.10300	−0.189900
$268$	39.4970	2.41266
$269$	−18.2518	−1.11283	−0.556415	−	0.830904i	$-0.687824\pi$
$269$	−18.2518	−1.11283	−0.556415	+	0.830904i	$0.687824\pi$
$270$	0	0
$271$	−4.10766	−0.249522	−0.124761	−	0.992187i	$-0.539816\pi$
$271$	−4.10766	−0.249522	−0.124761	+	0.992187i	$0.539816\pi$
$272$	−18.2052	−1.10386
$273$	−20.5702	−1.24497
$274$	18.9160	1.14276
$275$	0	0
$276$	−19.0868	−1.14889
$277$	−20.8395	−1.25212	−0.626062	−	0.779773i	$-0.715334\pi$
$277$	−20.8395	−1.25212	−0.626062	+	0.779773i	$0.715334\pi$
$278$	−9.82631	−0.589343
$279$	−2.73852	−0.163951
$280$	0	0
$281$	21.3490	1.27357	0.636787	−	0.771039i	$-0.280263\pi$
$281$	21.3490	1.27357	0.636787	+	0.771039i	$0.280263\pi$
$282$	−6.13715	−0.365462
$283$	−2.23016	−0.132569	−0.0662846	−	0.997801i	$-0.521115\pi$
$283$	−2.23016	−0.132569	−0.0662846	+	0.997801i	$0.521115\pi$
$284$	5.78733	0.343415
$285$	0	0
$286$	1.96859	0.116405
$287$	−37.9748	−2.24158
$288$	4.12048	0.242802
$289$	−8.85410	−0.520830
$290$	0	0
$291$	−17.7622	−1.04124
$292$	13.3342	0.780323
$293$	14.0177	0.818924	0.409462	−	0.912327i	$-0.365716\pi$
$293$	14.0177	0.818924	0.409462	+	0.912327i	$0.365716\pi$
$294$	43.7729	2.55289
$295$	0	0
$296$	−32.0631	−1.86363
$297$	0.187020	0.0108520
$298$	8.09810	0.469110
$299$	−18.1978	−1.05241
$300$	0	0
$301$	32.5003	1.87328
$302$	−19.9641	−1.14880
$303$	11.6368	0.668515
$304$	−2.74928	−0.157682
$305$	0	0
$306$	−7.20525	−0.411897
$307$	−17.2294	−0.983333	−0.491667	−	0.870784i	$-0.663612\pi$
$307$	−17.2294	−0.983333	−0.491667	+	0.870784i	$0.663612\pi$
$308$	−4.03498	−0.229914
$309$	−5.13797	−0.292289
$310$	0	0
$311$	1.47703	0.0837550	0.0418775	−	0.999123i	$-0.486666\pi$
$311$	1.47703	0.0837550	0.0418775	+	0.999123i	$0.486666\pi$
$312$	24.9807	1.41426
$313$	30.7937	1.74056	0.870282	−	0.492554i	$-0.163937\pi$
$313$	30.7937	1.74056	0.870282	+	0.492554i	$0.163937\pi$
$314$	27.6018	1.55766
$315$	0	0
$316$	−35.8597	−2.01727
$317$	−7.86693	−0.441851	−0.220925	−	0.975291i	$-0.570908\pi$
$317$	−7.86693	−0.441851	−0.220925	+	0.975291i	$0.570908\pi$
$318$	21.4205	1.20120
$319$	−0.209553	−0.0117327
$320$	0	0
$321$	16.3645	0.913376
$322$	54.3580	3.02925
$323$	1.23016	0.0684478
$324$	4.37322	0.242957
$325$	0	0
$326$	25.7005	1.42342
$327$	1.38686	0.0766938
$328$	46.1171	2.54639
$329$	11.9933	0.661213
$330$	0	0
$331$	25.6794	1.41147	0.705733	−	0.708478i	$-0.250618\pi$
$331$	25.6794	1.41147	0.705733	+	0.708478i	$0.250618\pi$
$332$	35.1656	1.92996
$333$	−5.35165	−0.293269
$334$	14.2667	0.780637
$335$	0	0
$336$	−31.4687	−1.71676
$337$	−13.3059	−0.724817	−0.362408	−	0.932019i	$-0.618045\pi$
$337$	−13.3059	−0.724817	−0.362408	+	0.932019i	$0.618045\pi$
$338$	11.0700	0.602130
$339$	−14.9571	−0.812358
$340$	0	0
$341$	−0.512157	−0.0277349
$342$	−1.08811	−0.0588380
$343$	−51.0074	−2.75414
$344$	−39.4687	−2.12801
$345$	0	0
$346$	−52.5384	−2.82448
$347$	−12.2630	−0.658310	−0.329155	−	0.944276i	$-0.606764\pi$
$347$	−12.2630	−0.658310	−0.329155	+	0.944276i	$0.606764\pi$
$348$	−4.90012	−0.262674
$349$	−12.5880	−0.673818	−0.336909	−	0.941537i	$-0.609381\pi$
$349$	−12.5880	−0.673818	−0.336909	+	0.941537i	$0.609381\pi$
$350$	0	0
$351$	4.16953	0.222553
$352$	0.770612	0.0410738
$353$	17.9473	0.955238	0.477619	−	0.878567i	$-0.341500\pi$
$353$	17.9473	0.955238	0.477619	+	0.878567i	$0.341500\pi$
$354$	27.1569	1.44338
$355$	0	0
$356$	−13.5701	−0.719213
$357$	14.0806	0.745225
$358$	−40.7489	−2.15365
$359$	−36.8096	−1.94273	−0.971367	−	0.237583i	$-0.923645\pi$
$359$	−36.8096	−1.94273	−0.971367	+	0.237583i	$0.923645\pi$
$360$	0	0
$361$	−18.8142	−0.990222
$362$	21.9641	1.15441
$363$	−10.9650	−0.575514
$364$	−89.9582	−4.71509
$365$	0	0
$366$	23.3491	1.22047
$367$	1.81874	0.0949377	0.0474688	−	0.998873i	$-0.484885\pi$
$367$	1.81874	0.0949377	0.0474688	+	0.998873i	$0.484885\pi$
$368$	−27.8394	−1.45123
$369$	7.69740	0.400710
$370$	0	0
$371$	−41.8602	−2.17328
$372$	−11.9761	−0.620934
$373$	22.7115	1.17596	0.587978	−	0.808877i	$-0.299924\pi$
$373$	22.7115	1.17596	0.587978	+	0.808877i	$0.299924\pi$
$374$	−1.34752	−0.0696788
$375$	0	0
$376$	−14.5648	−0.751124
$377$	−4.67189	−0.240615
$378$	−12.4546	−0.640598
$379$	−23.4164	−1.20282	−0.601410	−	0.798941i	$-0.705394\pi$
$379$	−23.4164	−1.20282	−0.601410	+	0.798941i	$0.705394\pi$
$380$	0	0
$381$	−11.3234	−0.580113
$382$	−52.6225	−2.69240
$383$	−9.35958	−0.478252	−0.239126	−	0.970989i	$-0.576861\pi$
$383$	−9.35958	−0.478252	−0.239126	+	0.970989i	$0.576861\pi$
$384$	−14.1862	−0.723937
$385$	0	0
$386$	−18.0159	−0.916983
$387$	−6.58772	−0.334873
$388$	−77.6778	−3.94349
$389$	−23.4323	−1.18806	−0.594031	−	0.804442i	$-0.702465\pi$
$389$	−23.4323	−1.18806	−0.594031	+	0.804442i	$0.702465\pi$
$390$	0	0
$391$	12.4567	0.629961
$392$	103.883	5.24687
$393$	17.1903	0.867135
$394$	25.8736	1.30349
$395$	0	0
$396$	0.817879	0.0411000
$397$	−22.9915	−1.15391	−0.576956	−	0.816775i	$-0.695760\pi$
$397$	−22.9915	−1.15391	−0.576956	+	0.816775i	$0.695760\pi$
$398$	−50.7140	−2.54206
$399$	2.12639	0.106453
$400$	0	0
$401$	−25.4862	−1.27272	−0.636360	−	0.771392i	$-0.719561\pi$
$401$	−25.4862	−1.27272	−0.636360	+	0.771392i	$0.719561\pi$
$402$	22.8004	1.13718
$403$	−11.4183	−0.568788
$404$	50.8902	2.53188
$405$	0	0
$406$	13.9552	0.692586
$407$	−1.00086	−0.0496110
$408$	−17.0997	−0.846559
$409$	9.14100	0.451993	0.225997	−	0.974128i	$-0.427436\pi$
$409$	9.14100	0.451993	0.225997	+	0.974128i	$0.427436\pi$
$410$	0	0
$411$	7.49289	0.369597
$412$	−22.4695	−1.10699
$413$	−53.0705	−2.61143
$414$	−11.0182	−0.541516
$415$	0	0
$416$	17.1805	0.842343
$417$	−3.89234	−0.190609
$418$	−0.203497	−0.00995338
$419$	4.62581	0.225986	0.112993	−	0.993596i	$-0.463956\pi$
$419$	4.62581	0.225986	0.112993	+	0.993596i	$0.463956\pi$
$420$	0	0
$421$	22.1325	1.07867	0.539337	−	0.842090i	$-0.318675\pi$
$421$	22.1325	1.07867	0.539337	+	0.842090i	$0.318675\pi$
$422$	−20.8687	−1.01587
$423$	−2.43101	−0.118200
$424$	50.8356	2.46879
$425$	0	0
$426$	3.34084	0.161864
$427$	−45.6291	−2.20815
$428$	71.5655	3.45925
$429$	0.779785	0.0376484
$430$	0	0
$431$	−33.1503	−1.59679	−0.798396	−	0.602133i	$-0.794318\pi$
$431$	−33.1503	−1.59679	−0.798396	+	0.602133i	$0.794318\pi$
$432$	6.37863	0.306892
$433$	13.5576	0.651539	0.325769	−	0.945449i	$-0.394377\pi$
$433$	13.5576	0.651539	0.325769	+	0.945449i	$0.394377\pi$
$434$	34.1073	1.63720
$435$	0	0
$436$	6.06507	0.290464
$437$	1.88115	0.0899877
$438$	7.69740	0.367796
$439$	13.1188	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$439$	13.1188	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$440$	0	0
$441$	17.3391	0.825670
$442$	−30.0425	−1.42898
$443$	−0.589737	−0.0280193	−0.0140096	−	0.999902i	$-0.504460\pi$
$443$	−0.589737	−0.0280193	−0.0140096	+	0.999902i	$0.504460\pi$
$444$	−23.4040	−1.11070
$445$	0	0
$446$	−4.02061	−0.190381
$447$	3.20777	0.151722
$448$	11.6183	0.548912
$449$	25.7878	1.21700	0.608501	−	0.793553i	$-0.291771\pi$
$449$	25.7878	1.21700	0.608501	+	0.793553i	$0.291771\pi$
$450$	0	0
$451$	1.43957	0.0677865
$452$	−65.4107	−3.07666
$453$	−7.90805	−0.371553
$454$	47.0099	2.20628
$455$	0	0
$456$	−2.58232	−0.120928
$457$	23.1744	1.08405	0.542027	−	0.840361i	$-0.317657\pi$
$457$	23.1744	1.08405	0.542027	+	0.840361i	$0.317657\pi$
$458$	−14.2374	−0.665272
$459$	−2.85410	−0.133218
$460$	0	0
$461$	7.94605	0.370085	0.185042	−	0.982731i	$-0.440758\pi$
$461$	7.94605	0.370085	0.185042	+	0.982731i	$0.440758\pi$
$462$	−2.32927	−0.108367
$463$	−22.0914	−1.02668	−0.513338	−	0.858187i	$-0.671591\pi$
$463$	−22.0914	−1.02668	−0.513338	+	0.858187i	$0.671591\pi$
$464$	−7.14714	−0.331798
$465$	0	0
$466$	16.5867	0.768364
$467$	25.8124	1.19446	0.597229	−	0.802071i	$-0.296268\pi$
$467$	25.8124	1.19446	0.597229	+	0.802071i	$0.296268\pi$
$468$	18.2343	0.842880
$469$	−44.5569	−2.05745
$470$	0	0
$471$	10.9335	0.503787
$472$	64.4495	2.96653
$473$	−1.23203	−0.0566490
$474$	−20.7007	−0.950815
$475$	0	0
$476$	61.5776	2.82241
$477$	8.48496	0.388500
$478$	10.9268	0.499779
$479$	38.9201	1.77830	0.889152	−	0.457611i	$-0.151295\pi$
$479$	38.9201	1.77830	0.889152	+	0.457611i	$0.151295\pi$
$480$	0	0
$481$	−22.3139	−1.01743
$482$	17.6393	0.803448
$483$	21.5320	0.979740
$484$	−47.9525	−2.17966
$485$	0	0
$486$	2.52452	0.114515
$487$	27.5335	1.24766	0.623830	−	0.781560i	$-0.285576\pi$
$487$	27.5335	1.24766	0.623830	+	0.781560i	$0.285576\pi$
$488$	55.4125	2.50841
$489$	10.1803	0.460371
$490$	0	0
$491$	10.1531	0.458201	0.229100	−	0.973403i	$-0.426422\pi$
$491$	10.1531	0.458201	0.229100	+	0.973403i	$0.426422\pi$
$492$	33.6624	1.51762
$493$	3.19797	0.144029
$494$	−4.53689	−0.204124
$495$	0	0
$496$	−17.4680	−0.784335
$497$	−6.52873	−0.292854
$498$	20.3000	0.909665
$499$	40.9313	1.83234	0.916168	−	0.400794i	$-0.131266\pi$
$499$	40.9313	1.83234	0.916168	+	0.400794i	$0.131266\pi$
$500$	0	0
$501$	5.65123	0.252478
$502$	−51.3023	−2.28973
$503$	−40.8527	−1.82153	−0.910766	−	0.412923i	$-0.864508\pi$
$503$	−40.8527	−1.82153	−0.910766	+	0.412923i	$0.864508\pi$
$504$	−29.5576	−1.31660
$505$	0	0
$506$	−2.06063	−0.0916060
$507$	4.38499	0.194744
$508$	−49.5195	−2.19707
$509$	−5.04400	−0.223572	−0.111786	−	0.993732i	$-0.535657\pi$
$509$	−5.04400	−0.223572	−0.111786	+	0.993732i	$0.535657\pi$
$510$	0	0
$511$	−15.0424	−0.665435
$512$	−50.1489	−2.21629
$513$	−0.431014	−0.0190297
$514$	17.4986	0.771830
$515$	0	0
$516$	−28.8096	−1.26827
$517$	−0.454648	−0.0199954
$518$	66.6529	2.92856
$519$	−20.8112	−0.913510
$520$	0	0
$521$	22.9426	1.00514	0.502568	−	0.864538i	$-0.332389\pi$
$521$	22.9426	1.00514	0.502568	+	0.864538i	$0.332389\pi$
$522$	−2.82869	−0.123808
$523$	10.7448	0.469838	0.234919	−	0.972015i	$-0.424518\pi$
$523$	10.7448	0.469838	0.234919	+	0.972015i	$0.424518\pi$
$524$	75.1769	3.28412
$525$	0	0
$526$	54.6504	2.38287
$527$	7.81601	0.340471
$528$	1.19293	0.0519156
$529$	−3.95133	−0.171797
$530$	0	0
$531$	10.7573	0.466825
$532$	9.29919	0.403171
$533$	32.0945	1.39017
$534$	−7.83359	−0.338992
$535$	0	0
$536$	54.1104	2.33721
$537$	−16.1412	−0.696546
$538$	−46.0770	−1.98652
$539$	3.24275	0.139675
$540$	0	0
$541$	41.9437	1.80330	0.901651	−	0.432464i	$-0.142356\pi$
$541$	41.9437	1.80330	0.901651	+	0.432464i	$0.142356\pi$
$542$	−10.3699	−0.445425
$543$	8.70028	0.373365
$544$	−11.7603	−0.504218
$545$	0	0
$546$	−51.9300	−2.22240
$547$	−1.09887	−0.0469842	−0.0234921	−	0.999724i	$-0.507478\pi$
$547$	−1.09887	−0.0469842	−0.0234921	+	0.999724i	$0.507478\pi$
$548$	32.7681	1.39978
$549$	9.24889	0.394733
$550$	0	0
$551$	0.482944	0.0205741
$552$	−26.1487	−1.11296
$553$	40.4536	1.72026
$554$	−52.6098	−2.23518
$555$	0	0
$556$	−17.0221	−0.721897
$557$	2.25279	0.0954536	0.0477268	−	0.998860i	$-0.484802\pi$
$557$	2.25279	0.0954536	0.0477268	+	0.998860i	$0.484802\pi$
$558$	−6.91345	−0.292670
$559$	−27.4677	−1.16176
$560$	0	0
$561$	−0.533774	−0.0225359
$562$	53.8961	2.27347
$563$	29.7361	1.25323	0.626614	−	0.779330i	$-0.284440\pi$
$563$	29.7361	1.25323	0.626614	+	0.779330i	$0.284440\pi$
$564$	−10.6314	−0.447661
$565$	0	0
$566$	−5.63009	−0.236650
$567$	−4.93346	−0.207186
$568$	7.92856	0.332675
$569$	27.1667	1.13889	0.569444	−	0.822030i	$-0.307158\pi$
$569$	27.1667	1.13889	0.569444	+	0.822030i	$0.307158\pi$
$570$	0	0
$571$	−1.97482	−0.0826437	−0.0413218	−	0.999146i	$-0.513157\pi$
$571$	−1.97482	−0.0826437	−0.0413218	+	0.999146i	$0.513157\pi$
$572$	3.41017	0.142586
$573$	−20.8445	−0.870793
$574$	−95.8684	−4.00147
$575$	0	0
$576$	−2.35499	−0.0981247
$577$	13.6634	0.568816	0.284408	−	0.958703i	$-0.408203\pi$
$577$	13.6634	0.568816	0.284408	+	0.958703i	$0.408203\pi$
$578$	−22.3524	−0.929737
$579$	−7.13634	−0.296576
$580$	0	0
$581$	−39.6706	−1.64581
$582$	−44.8410	−1.85872
$583$	1.58686	0.0657208
$584$	18.2676	0.755920
$585$	0	0
$586$	35.3881	1.46187
$587$	−36.3511	−1.50037	−0.750186	−	0.661227i	$-0.770036\pi$
$587$	−36.3511	−1.50037	−0.750186	+	0.661227i	$0.770036\pi$
$588$	75.8276	3.12708
$589$	1.18034	0.0486351
$590$	0	0
$591$	10.2489	0.421583
$592$	−34.1362	−1.40299
$593$	−37.7467	−1.55007	−0.775035	−	0.631918i	$-0.782268\pi$
$593$	−37.7467	−1.55007	−0.775035	+	0.631918i	$0.782268\pi$
$594$	0.472136	0.0193720
$595$	0	0
$596$	14.0283	0.574621
$597$	−20.0885	−0.822169
$598$	−45.9409	−1.87866
$599$	−17.3814	−0.710186	−0.355093	−	0.934831i	$-0.615551\pi$
$599$	−17.3814	−0.710186	−0.355093	+	0.934831i	$0.615551\pi$
$600$	0	0
$601$	−22.3120	−0.910126	−0.455063	−	0.890459i	$-0.650383\pi$
$601$	−22.3120	−0.910126	−0.455063	+	0.890459i	$0.650383\pi$
$602$	82.0477	3.34402
$603$	9.03156	0.367794
$604$	−34.5837	−1.40719
$605$	0	0
$606$	29.3773	1.19337
$607$	13.6024	0.552105	0.276053	−	0.961143i	$-0.410974\pi$
$607$	13.6024	0.552105	0.276053	+	0.961143i	$0.410974\pi$
$608$	−1.77599	−0.0720258
$609$	5.52786	0.224000
$610$	0	0
$611$	−10.1362	−0.410066
$612$	−12.4816	−0.504540
$613$	22.2835	0.900021	0.450011	−	0.893023i	$-0.351420\pi$
$613$	22.2835	0.900021	0.450011	+	0.893023i	$0.351420\pi$
$614$	−43.4960	−1.75536
$615$	0	0
$616$	−5.52786	−0.222724
$617$	−0.115820	−0.00466276	−0.00233138	−	0.999997i	$-0.500742\pi$
$617$	−0.115820	−0.00466276	−0.00233138	+	0.999997i	$0.500742\pi$
$618$	−12.9709	−0.521767
$619$	−0.0726817	−0.00292133	−0.00146066	−	0.999999i	$-0.500465\pi$
$619$	−0.0726817	−0.00292133	−0.00146066	+	0.999999i	$0.500465\pi$
$620$	0	0
$621$	−4.36448	−0.175140
$622$	3.72881	0.149512
$623$	15.3085	0.613323
$624$	26.5959	1.06469
$625$	0	0
$626$	77.7395	3.10709
$627$	−0.0806082	−0.00321918
$628$	47.8145	1.90800
$629$	15.2742	0.609021
$630$	0	0
$631$	−29.5462	−1.17622	−0.588108	−	0.808782i	$-0.700127\pi$
$631$	−29.5462	−1.17622	−0.588108	+	0.808782i	$0.700127\pi$
$632$	−49.1274	−1.95418
$633$	−8.26638	−0.328559
$634$	−19.8602	−0.788751
$635$	0	0
$636$	37.1066	1.47137
$637$	72.2958	2.86446
$638$	−0.529020	−0.0209441
$639$	1.32336	0.0523512
$640$	0	0
$641$	4.28099	0.169089	0.0845444	−	0.996420i	$-0.473057\pi$
$641$	4.28099	0.169089	0.0845444	+	0.996420i	$0.473057\pi$
$642$	41.3125	1.63048
$643$	−38.7887	−1.52968	−0.764838	−	0.644223i	$-0.777181\pi$
$643$	−38.7887	−1.52968	−0.764838	+	0.644223i	$0.777181\pi$
$644$	94.1642	3.71059
$645$	0	0
$646$	3.10556	0.122187
$647$	19.5243	0.767580	0.383790	−	0.923421i	$-0.374619\pi$
$647$	19.5243	0.767580	0.383790	+	0.923421i	$0.374619\pi$
$648$	5.99126	0.235359
$649$	2.01182	0.0789708
$650$	0	0
$651$	13.5104	0.529513
$652$	44.5209	1.74357
$653$	16.5733	0.648562	0.324281	−	0.945961i	$-0.394878\pi$
$653$	16.5733	0.648562	0.324281	+	0.945961i	$0.394878\pi$
$654$	3.50117	0.136907
$655$	0	0
$656$	49.0988	1.91699
$657$	3.04905	0.118955
$658$	30.2774	1.18034
$659$	6.79371	0.264645	0.132323	−	0.991207i	$-0.457756\pi$
$659$	6.79371	0.264645	0.132323	+	0.991207i	$0.457756\pi$
$660$	0	0
$661$	13.2090	0.513771	0.256886	−	0.966442i	$-0.417304\pi$
$661$	13.2090	0.513771	0.256886	+	0.966442i	$0.417304\pi$
$662$	64.8282	2.51962
$663$	−11.9003	−0.462168
$664$	48.1764	1.86961
$665$	0	0
$666$	−13.5104	−0.523516
$667$	4.89032	0.189354
$668$	24.7141	0.956216
$669$	−1.59262	−0.0615742
$670$	0	0
$671$	1.72973	0.0667753
$672$	−20.3283	−0.784179
$673$	−47.7353	−1.84006	−0.920031	−	0.391846i	$-0.871837\pi$
$673$	−47.7353	−1.84006	−0.920031	+	0.391846i	$0.871837\pi$
$674$	−33.5910	−1.29388
$675$	0	0
$676$	19.1766	0.737560
$677$	11.6486	0.447691	0.223846	−	0.974625i	$-0.428139\pi$
$677$	11.6486	0.447691	0.223846	+	0.974625i	$0.428139\pi$
$678$	−37.7596	−1.45015
$679$	87.6289	3.36289
$680$	0	0
$681$	18.6213	0.713570
$682$	−1.29295	−0.0495097
$683$	30.1235	1.15264	0.576322	−	0.817223i	$-0.304487\pi$
$683$	30.1235	1.15264	0.576322	+	0.817223i	$0.304487\pi$
$684$	−1.88492	−0.0720717
$685$	0	0
$686$	−128.769	−4.91644
$687$	−5.63965	−0.215166
$688$	−42.0206	−1.60202
$689$	35.3783	1.34781
$690$	0	0
$691$	−34.2787	−1.30402	−0.652011	−	0.758209i	$-0.726074\pi$
$691$	−34.2787	−1.30402	−0.652011	+	0.758209i	$0.726074\pi$
$692$	−91.0120	−3.45976
$693$	−0.922655	−0.0350488
$694$	−30.9582	−1.17516
$695$	0	0
$696$	−6.71310	−0.254460
$697$	−21.9692	−0.832141
$698$	−31.7786	−1.20284
$699$	6.57023	0.248509
$700$	0	0
$701$	−45.2669	−1.70971	−0.854853	−	0.518871i	$-0.826353\pi$
$701$	−45.2669	−1.70971	−0.854853	+	0.518871i	$0.826353\pi$
$702$	10.5261	0.397281
$703$	2.30664	0.0869965
$704$	−0.440430	−0.0165993
$705$	0	0
$706$	45.3084	1.70520
$707$	−57.4096	−2.15911
$708$	47.0439	1.76802
$709$	−27.0295	−1.01512	−0.507558	−	0.861618i	$-0.669452\pi$
$709$	−27.0295	−1.01512	−0.507558	+	0.861618i	$0.669452\pi$
$710$	0	0
$711$	−8.19985	−0.307518
$712$	−18.5908	−0.696721
$713$	11.9522	0.447613
$714$	35.5468	1.33031
$715$	0	0
$716$	−70.5892	−2.63804
$717$	4.32825	0.161642
$718$	−92.9266	−3.46799
$719$	4.57600	0.170656	0.0853279	−	0.996353i	$-0.472806\pi$
$719$	4.57600	0.170656	0.0853279	+	0.996353i	$0.472806\pi$
$720$	0	0
$721$	25.3480	0.944009
$722$	−47.4970	−1.76765
$723$	6.98717	0.259856
$724$	38.0482	1.41405
$725$	0	0
$726$	−27.6815	−1.02736
$727$	44.0914	1.63526	0.817630	−	0.575744i	$-0.195287\pi$
$727$	44.0914	1.63526	0.817630	+	0.575744i	$0.195287\pi$
$728$	−123.242	−4.56763
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.8020	0.695418
$732$	40.4475	1.49498
$733$	−21.3757	−0.789528	−0.394764	−	0.918782i	$-0.629174\pi$
$733$	−21.3757	−0.789528	−0.394764	+	0.918782i	$0.629174\pi$
$734$	4.59146	0.169474
$735$	0	0
$736$	−17.9838	−0.662890
$737$	1.68908	0.0622181
$738$	19.4323	0.715311
$739$	−5.60055	−0.206019	−0.103010	−	0.994680i	$-0.532847\pi$
$739$	−5.60055	−0.206019	−0.103010	+	0.994680i	$0.532847\pi$
$740$	0	0
$741$	−1.79713	−0.0660191
$742$	−105.677	−3.87953
$743$	21.2617	0.780017	0.390008	−	0.920811i	$-0.372472\pi$
$743$	21.2617	0.780017	0.390008	+	0.920811i	$0.372472\pi$
$744$	−16.4072	−0.601516
$745$	0	0
$746$	57.3356	2.09921
$747$	8.04112	0.294209
$748$	−2.33431	−0.0853508
$749$	−80.7336	−2.94994
$750$	0	0
$751$	14.4685	0.527962	0.263981	−	0.964528i	$-0.414964\pi$
$751$	14.4685	0.527962	0.263981	+	0.964528i	$0.414964\pi$
$752$	−15.5065	−0.565465
$753$	−20.3216	−0.740559
$754$	−11.7943	−0.429523
$755$	0	0
$756$	−21.5751	−0.784680
$757$	12.5210	0.455085	0.227542	−	0.973768i	$-0.426931\pi$
$757$	12.5210	0.455085	0.227542	+	0.973768i	$0.426931\pi$
$758$	−59.1153	−2.14716
$759$	−0.816244	−0.0296278
$760$	0	0
$761$	−19.1769	−0.695163	−0.347581	−	0.937650i	$-0.612997\pi$
$761$	−19.1769	−0.695163	−0.347581	+	0.937650i	$0.612997\pi$
$762$	−28.5861	−1.03556
$763$	−6.84205	−0.247699
$764$	−91.1578	−3.29797
$765$	0	0
$766$	−23.6285	−0.853732
$767$	44.8527	1.61954
$768$	−31.1034	−1.12235
$769$	41.8407	1.50882	0.754408	−	0.656406i	$-0.227924\pi$
$769$	41.8407	1.50882	0.754408	+	0.656406i	$0.227924\pi$
$770$	0	0
$771$	6.93145	0.249630
$772$	−31.2088	−1.12323
$773$	−22.9098	−0.824009	−0.412005	−	0.911182i	$-0.635171\pi$
$773$	−22.9098	−0.824009	−0.412005	+	0.911182i	$0.635171\pi$
$774$	−16.6309	−0.597784
$775$	0	0
$776$	−106.418	−3.82017
$777$	26.4022	0.947173
$778$	−59.1553	−2.12082
$779$	−3.31769	−0.118868
$780$	0	0
$781$	0.247494	0.00885602
$782$	31.4472	1.12455
$783$	−1.12048	−0.0400428
$784$	110.599	3.94998
$785$	0	0
$786$	43.3973	1.54793
$787$	−10.7906	−0.384643	−0.192322	−	0.981332i	$-0.561602\pi$
$787$	−10.7906	−0.384643	−0.192322	+	0.981332i	$0.561602\pi$
$788$	44.8207	1.59667
$789$	21.6478	0.770683
$790$	0	0
$791$	73.7903	2.62368
$792$	1.12048	0.0398146
$793$	38.5636	1.36943
$794$	−58.0427	−2.05986
$795$	0	0
$796$	−87.8516	−3.11382
$797$	45.2213	1.60182	0.800910	−	0.598784i	$-0.204349\pi$
$797$	45.2213	1.60182	0.800910	+	0.598784i	$0.204349\pi$
$798$	5.36813	0.190030
$799$	6.93836	0.245462
$800$	0	0
$801$	−3.10300	−0.109639
$802$	−64.3405	−2.27194
$803$	0.570232	0.0201231
$804$	39.4970	1.39295
$805$	0	0
$806$	−28.8259	−1.01535
$807$	−18.2518	−0.642493
$808$	69.7189	2.45270
$809$	24.1895	0.850458	0.425229	−	0.905086i	$-0.360194\pi$
$809$	24.1895	0.850458	0.425229	+	0.905086i	$0.360194\pi$
$810$	0	0
$811$	25.8633	0.908182	0.454091	−	0.890955i	$-0.349964\pi$
$811$	25.8633	0.908182	0.454091	+	0.890955i	$0.349964\pi$
$812$	24.1746	0.848361
$813$	−4.10766	−0.144062
$814$	−2.52671	−0.0885611
$815$	0	0
$816$	−18.2052	−0.637311
$817$	2.83940	0.0993381
$818$	23.0767	0.806857
$819$	−20.5702	−0.718782
$820$	0	0
$821$	34.3622	1.19925	0.599624	−	0.800282i	$-0.295317\pi$
$821$	34.3622	1.19925	0.599624	+	0.800282i	$0.295317\pi$
$822$	18.9160	0.659770
$823$	46.7556	1.62980	0.814899	−	0.579603i	$-0.196792\pi$
$823$	46.7556	1.62980	0.814899	+	0.579603i	$0.196792\pi$
$824$	−30.7829	−1.07237
$825$	0	0
$826$	−133.978	−4.66168
$827$	30.1332	1.04783	0.523916	−	0.851770i	$-0.324470\pi$
$827$	30.1332	1.04783	0.523916	+	0.851770i	$0.324470\pi$
$828$	−19.0868	−0.663313
$829$	−37.5623	−1.30459	−0.652296	−	0.757964i	$-0.726194\pi$
$829$	−37.5623	−1.30459	−0.652296	+	0.757964i	$0.726194\pi$
$830$	0	0
$831$	−20.8395	−0.722914
$832$	−9.81922	−0.340420
$833$	−49.4875	−1.71464
$834$	−9.82631	−0.340257
$835$	0	0
$836$	−0.352517	−0.0121921
$837$	−2.73852	−0.0946570
$838$	11.6780	0.403409
$839$	−15.7230	−0.542820	−0.271410	−	0.962464i	$-0.587490\pi$
$839$	−15.7230	−0.542820	−0.271410	+	0.962464i	$0.587490\pi$
$840$	0	0
$841$	−27.7445	−0.956707
$842$	55.8741	1.92555
$843$	21.3490	0.735299
$844$	−36.1507	−1.24436
$845$	0	0
$846$	−6.13715	−0.211000
$847$	54.0955	1.85874
$848$	54.1224	1.85857
$849$	−2.23016	−0.0765388
$850$	0	0
$851$	23.3572	0.800673
$852$	5.78733	0.198271
$853$	37.6116	1.28780	0.643898	−	0.765111i	$-0.277316\pi$
$853$	37.6116	1.28780	0.643898	+	0.765111i	$0.277316\pi$
$854$	−115.192	−3.94178
$855$	0	0
$856$	98.0438	3.35107
$857$	23.6506	0.807889	0.403945	−	0.914783i	$-0.367639\pi$
$857$	23.6506	0.807889	0.403945	+	0.914783i	$0.367639\pi$
$858$	1.96859	0.0672064
$859$	−2.06312	−0.0703927	−0.0351964	−	0.999380i	$-0.511206\pi$
$859$	−2.06312	−0.0703927	−0.0351964	+	0.999380i	$0.511206\pi$
$860$	0	0
$861$	−37.9748	−1.29418
$862$	−83.6886	−2.85045
$863$	23.8417	0.811583	0.405791	−	0.913966i	$-0.366996\pi$
$863$	23.8417	0.811583	0.405791	+	0.913966i	$0.366996\pi$
$864$	4.12048	0.140182
$865$	0	0
$866$	34.2266	1.16307
$867$	−8.85410	−0.300701
$868$	59.0839	2.00544
$869$	−1.53353	−0.0520216
$870$	0	0
$871$	37.6574	1.27597
$872$	8.30906	0.281380
$873$	−17.7622	−0.601158
$874$	4.74901	0.160638
$875$	0	0
$876$	13.3342	0.450520
$877$	−24.0338	−0.811564	−0.405782	−	0.913970i	$-0.633001\pi$
$877$	−24.0338	−0.811564	−0.405782	+	0.913970i	$0.633001\pi$
$878$	33.1188	1.11771
$879$	14.0177	0.472806
$880$	0	0
$881$	−10.4557	−0.352260	−0.176130	−	0.984367i	$-0.556358\pi$
$881$	−10.4557	−0.352260	−0.176130	+	0.984367i	$0.556358\pi$
$882$	43.7729	1.47391
$883$	−34.6714	−1.16679	−0.583394	−	0.812190i	$-0.698275\pi$
$883$	−34.6714	−1.16679	−0.583394	+	0.812190i	$0.698275\pi$
$884$	−52.0425	−1.75038
$885$	0	0
$886$	−1.48881	−0.0500174
$887$	−3.26739	−0.109708	−0.0548541	−	0.998494i	$-0.517469\pi$
$887$	−3.26739	−0.109708	−0.0548541	+	0.998494i	$0.517469\pi$
$888$	−32.0631	−1.07597
$889$	55.8634	1.87360
$890$	0	0
$891$	0.187020	0.00626540
$892$	−6.96488	−0.233201
$893$	1.04780	0.0350633
$894$	8.09810	0.270841
$895$	0	0
$896$	69.9871	2.33811
$897$	−18.1978	−0.607608
$898$	65.1019	2.17248
$899$	3.06846	0.102339
$900$	0	0
$901$	−24.2169	−0.806783
$902$	3.63422	0.121006
$903$	32.5003	1.08154
$904$	−89.6118	−2.98044
$905$	0	0
$906$	−19.9641	−0.663261
$907$	−23.2742	−0.772806	−0.386403	−	0.922330i	$-0.626283\pi$
$907$	−23.2742	−0.772806	−0.386403	+	0.922330i	$0.626283\pi$
$908$	81.4351	2.70252
$909$	11.6368	0.385967
$910$	0	0
$911$	−19.4852	−0.645573	−0.322787	−	0.946472i	$-0.604620\pi$
$911$	−19.4852	−0.645573	−0.322787	+	0.946472i	$0.604620\pi$
$912$	−2.74928	−0.0910377
$913$	1.50385	0.0497701
$914$	58.5044	1.93515
$915$	0	0
$916$	−24.6634	−0.814903
$917$	−84.8076	−2.80059
$918$	−7.20525	−0.237809
$919$	5.82595	0.192180	0.0960902	−	0.995373i	$-0.469366\pi$
$919$	5.82595	0.192180	0.0960902	+	0.995373i	$0.469366\pi$
$920$	0	0
$921$	−17.2294	−0.567728
$922$	20.0600	0.660641
$923$	5.51777	0.181620
$924$	−4.03498	−0.132741
$925$	0	0
$926$	−55.7703	−1.83273
$927$	−5.13797	−0.168753
$928$	−4.61693	−0.151558
$929$	54.9566	1.80307	0.901533	−	0.432710i	$-0.142443\pi$
$929$	54.9566	1.80307	0.901533	+	0.432710i	$0.142443\pi$
$930$	0	0
$931$	−7.47338	−0.244930
$932$	28.7331	0.941183
$933$	1.47703	0.0483559
$934$	65.1641	2.13224
$935$	0	0
$936$	24.9807	0.816521
$937$	−7.75384	−0.253307	−0.126653	−	0.991947i	$-0.540424\pi$
$937$	−7.75384	−0.253307	−0.126653	+	0.991947i	$0.540424\pi$
$938$	−112.485	−3.67276
$939$	30.7937	1.00491
$940$	0	0
$941$	−11.0480	−0.360156	−0.180078	−	0.983652i	$-0.557635\pi$
$941$	−11.0480	−0.360156	−0.180078	+	0.983652i	$0.557635\pi$
$942$	27.6018	0.899315
$943$	−33.5951	−1.09401
$944$	68.6165	2.23328
$945$	0	0
$946$	−3.11030	−0.101125
$947$	2.38966	0.0776534	0.0388267	−	0.999246i	$-0.487638\pi$
$947$	2.38966	0.0776534	0.0388267	+	0.999246i	$0.487638\pi$
$948$	−35.8597	−1.16467
$949$	12.7131	0.412685
$950$	0	0
$951$	−7.86693	−0.255103
$952$	84.3605	2.73414
$953$	−22.4858	−0.728387	−0.364194	−	0.931323i	$-0.618655\pi$
$953$	−22.4858	−0.728387	−0.364194	+	0.931323i	$0.618655\pi$
$954$	21.4205	0.693514
$955$	0	0
$956$	18.9284	0.612189
$957$	−0.209553	−0.00677387
$958$	98.2547	3.17447
$959$	−36.9659	−1.19369
$960$	0	0
$961$	−23.5005	−0.758081
$962$	−56.3319	−1.81621
$963$	16.3645	0.527338
$964$	30.5565	0.984157
$965$	0	0
$966$	54.3580	1.74894
$967$	43.7138	1.40574	0.702871	−	0.711318i	$-0.251901\pi$
$967$	43.7138	1.40574	0.702871	+	0.711318i	$0.251901\pi$
$968$	−65.6943	−2.11149
$969$	1.23016	0.0395184
$970$	0	0
$971$	−18.2818	−0.586692	−0.293346	−	0.956006i	$-0.594769\pi$
$971$	−18.2818	−0.586692	−0.293346	+	0.956006i	$0.594769\pi$
$972$	4.37322	0.140271
$973$	19.2027	0.615611
$974$	69.5089	2.22721
$975$	0	0
$976$	58.9952	1.88839
$977$	−42.4081	−1.35675	−0.678377	−	0.734714i	$-0.737317\pi$
$977$	−42.4081	−1.35675	−0.678377	+	0.734714i	$0.737317\pi$
$978$	25.7005	0.821812
$979$	−0.580321	−0.0185472
$980$	0	0
$981$	1.38686	0.0442792
$982$	25.6316	0.817938
$983$	26.6561	0.850198	0.425099	−	0.905147i	$-0.360239\pi$
$983$	26.6561	0.850198	0.425099	+	0.905147i	$0.360239\pi$
$984$	46.1171	1.47016
$985$	0	0
$986$	8.07336	0.257108
$987$	11.9933	0.381752
$988$	−7.85924	−0.250036
$989$	28.7520	0.914259
$990$	0	0
$991$	22.2630	0.707206	0.353603	−	0.935396i	$-0.384956\pi$
$991$	22.2630	0.707206	0.353603	+	0.935396i	$0.384956\pi$
$992$	−11.2840	−0.358268
$993$	25.6794	0.814910
$994$	−16.4819	−0.522775
$995$	0	0
$996$	35.1656	1.11426
$997$	−22.6781	−0.718223	−0.359112	−	0.933295i	$-0.616920\pi$
$997$	−22.6781	−0.718223	−0.359112	+	0.933295i	$0.616920\pi$
$998$	103.332	3.27092
$999$	−5.35165	−0.169319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	375.2.a.f.1.4	yes	4
3.2	odd	2		1125.2.a.h.1.1		4
4.3	odd	2		6000.2.a.bg.1.4		4
5.2	odd	4		375.2.b.c.124.8		8
5.3	odd	4		375.2.b.c.124.1		8
5.4	even	2		375.2.a.e.1.1	✓	4
15.2	even	4		1125.2.b.g.874.1		8
15.8	even	4		1125.2.b.g.874.8		8
15.14	odd	2		1125.2.a.l.1.4		4
20.3	even	4		6000.2.f.o.1249.1		8
20.7	even	4		6000.2.f.o.1249.8		8
20.19	odd	2		6000.2.a.bh.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.e.1.1	✓	4	5.4	even	2
375.2.a.f.1.4	yes	4	1.1	even	1	trivial
375.2.b.c.124.1		8	5.3	odd	4
375.2.b.c.124.8		8	5.2	odd	4
1125.2.a.h.1.1		4	3.2	odd	2
1125.2.a.l.1.4		4	15.14	odd	2
1125.2.b.g.874.1		8	15.2	even	4
1125.2.b.g.874.8		8	15.8	even	4
6000.2.a.bg.1.4		4	4.3	odd	2
6000.2.a.bh.1.1		4	20.19	odd	2
6000.2.f.o.1249.1		8	20.3	even	4
6000.2.f.o.1249.8		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52452 q^{2} +1.00000 q^{3} +4.37322 q^{4} +2.52452 q^{6} -4.93346 q^{7} +5.99126 q^{8} +1.00000 q^{9} +0.187020 q^{11} +4.37322 q^{12} +4.16953 q^{13} -12.4546 q^{14} +6.37863 q^{16} -2.85410 q^{17} +2.52452 q^{18} -0.431014 q^{19} -4.93346 q^{21} +0.472136 q^{22} -4.36448 q^{23} +5.99126 q^{24} +10.5261 q^{26} +1.00000 q^{27} -21.5751 q^{28} -1.12048 q^{29} -2.73852 q^{31} +4.12048 q^{32} +0.187020 q^{33} -7.20525 q^{34} +4.37322 q^{36} -5.35165 q^{37} -1.08811 q^{38} +4.16953 q^{39} +7.69740 q^{41} -12.4546 q^{42} -6.58772 q^{43} +0.817879 q^{44} -11.0182 q^{46} -2.43101 q^{47} +6.37863 q^{48} +17.3391 q^{49} -2.85410 q^{51} +18.2343 q^{52} +8.48496 q^{53} +2.52452 q^{54} -29.5576 q^{56} -0.431014 q^{57} -2.82869 q^{58} +10.7573 q^{59} +9.24889 q^{61} -6.91345 q^{62} -4.93346 q^{63} -2.35499 q^{64} +0.472136 q^{66} +9.03156 q^{67} -12.4816 q^{68} -4.36448 q^{69} +1.32336 q^{71} +5.99126 q^{72} +3.04905 q^{73} -13.5104 q^{74} -1.88492 q^{76} -0.922655 q^{77} +10.5261 q^{78} -8.19985 q^{79} +1.00000 q^{81} +19.4323 q^{82} +8.04112 q^{83} -21.5751 q^{84} -16.6309 q^{86} -1.12048 q^{87} +1.12048 q^{88} -3.10300 q^{89} -20.5702 q^{91} -19.0868 q^{92} -2.73852 q^{93} -6.13715 q^{94} +4.12048 q^{96} -17.7622 q^{97} +43.7729 q^{98} +0.187020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 4 q^{3} + 7 q^{4} + 3 q^{6} - 4 q^{7} + 9 q^{8} + 4 q^{9} + 6 q^{11} + 7 q^{12} - 8 q^{13} - 2 q^{14} + 5 q^{16} + 2 q^{17} + 3 q^{18} + 8 q^{19} - 4 q^{21} - 16 q^{22} + 8 q^{23} + 9 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) \) 4 * q + 3 * q^2 + 4 * q^3 + 7 * q^4 + 3 * q^6 - 4 * q^7 + 9 * q^8 + 4 * q^9 + 6 * q^11 + 7 * q^12 - 8 * q^13 - 2 * q^14 + 5 * q^16 + 2 * q^17 + 3 * q^18 + 8 * q^19 - 4 * q^21 - 16 * q^22 + 8 * q^23 + 9 * q^24 - 2 * q^26 + 4 * q^27 - 28 * q^28 + 6 * q^29 + 4 * q^31 + 6 * q^32 + 6 * q^33 - 6 * q^34 + 7 * q^36 - 14 * q^37 - 17 * q^38 - 8 * q^39 + 24 * q^41 - 2 * q^42 - 10 * q^43 - 14 * q^44 - 16 * q^46 + 5 * q^48 + 20 * q^49 + 2 * q^51 + 2 * q^52 - 2 * q^53 + 3 * q^54 - 30 * q^56 + 8 * q^57 + 26 * q^58 + 2 * q^59 + 10 * q^61 + 22 * q^62 - 4 * q^63 - 27 * q^64 - 16 * q^66 - 8 * q^67 + 11 * q^68 + 8 * q^69 - 12 * q^71 + 9 * q^72 - 2 * q^73 - 42 * q^74 - 8 * q^76 - 8 * q^77 - 2 * q^78 - 20 * q^79 + 4 * q^81 + 12 * q^82 + 24 * q^83 - 28 * q^84 - 44 * q^86 + 6 * q^87 - 6 * q^88 + 28 * q^89 - 36 * q^91 - 29 * q^92 + 4 * q^93 - 23 * q^94 + 6 * q^96 - 18 * q^97 + 23 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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