LMFDB - Embedded newform 1150.4.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.26018$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +3.26018 q^{3} +4.00000 q^{4} -6.52037 q^{6} -27.7921 q^{7} -8.00000 q^{8} -16.3712 q^{9} -10.8182 q^{11} +13.0407 q^{12} -36.9683 q^{13} +55.5842 q^{14} +16.0000 q^{16} -118.996 q^{17} +32.7424 q^{18} -19.3321 q^{19} -90.6073 q^{21} +21.6365 q^{22} -23.0000 q^{23} -26.0815 q^{24} +73.9366 q^{26} -141.398 q^{27} -111.168 q^{28} +234.499 q^{29} +165.319 q^{31} -32.0000 q^{32} -35.2694 q^{33} +237.992 q^{34} -65.4848 q^{36} -202.301 q^{37} +38.6643 q^{38} -120.523 q^{39} -295.846 q^{41} +181.215 q^{42} +65.9221 q^{43} -43.2729 q^{44} +46.0000 q^{46} +110.279 q^{47} +52.1630 q^{48} +429.400 q^{49} -387.949 q^{51} -147.873 q^{52} +688.135 q^{53} +282.796 q^{54} +222.337 q^{56} -63.0263 q^{57} -468.998 q^{58} +10.5847 q^{59} +110.579 q^{61} -330.639 q^{62} +454.990 q^{63} +64.0000 q^{64} +70.5388 q^{66} +643.290 q^{67} -475.984 q^{68} -74.9842 q^{69} +143.216 q^{71} +130.970 q^{72} +158.213 q^{73} +404.601 q^{74} -77.3285 q^{76} +300.661 q^{77} +241.047 q^{78} +1123.34 q^{79} -18.9616 q^{81} +591.692 q^{82} +824.600 q^{83} -362.429 q^{84} -131.844 q^{86} +764.510 q^{87} +86.5458 q^{88} -879.672 q^{89} +1027.43 q^{91} -92.0000 q^{92} +538.971 q^{93} -220.557 q^{94} -104.326 q^{96} +938.437 q^{97} -858.801 q^{98} +177.107 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} - 26 q^{7} - 32 q^{8} + 64 q^{9} + 93 q^{11} - 16 q^{12} - 32 q^{13} + 52 q^{14} + 64 q^{16} - 108 q^{17} - 128 q^{18} + 185 q^{19} + 302 q^{21} - 186 q^{22}+ \cdots + 2013 q^{99}+O(q^{100}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 - 26 * q^7 - 32 * q^8 + 64 * q^9 + 93 * q^11 - 16 * q^12 - 32 * q^13 + 52 * q^14 + 64 * q^16 - 108 * q^17 - 128 * q^18 + 185 * q^19 + 302 * q^21 - 186 * q^22 - 92 * q^23 + 32 * q^24 + 64 * q^26 - 259 * q^27 - 104 * q^28 + 294 * q^29 - 211 * q^31 - 128 * q^32 + 237 * q^33 + 216 * q^34 + 256 * q^36 - 5 * q^37 - 370 * q^38 - 421 * q^39 - 369 * q^41 - 604 * q^42 + 100 * q^43 + 372 * q^44 + 184 * q^46 + 363 * q^47 - 64 * q^48 + 600 * q^49 - 315 * q^51 - 128 * q^52 - 21 * q^53 + 518 * q^54 + 208 * q^56 + 160 * q^57 - 588 * q^58 - 33 * q^59 - 307 * q^61 + 422 * q^62 - 167 * q^63 + 256 * q^64 - 474 * q^66 - 725 * q^67 - 432 * q^68 + 92 * q^69 - 1257 * q^71 - 512 * q^72 - 509 * q^73 + 10 * q^74 + 740 * q^76 + 1962 * q^77 + 842 * q^78 + 1202 * q^79 - 992 * q^81 + 738 * q^82 + 1377 * q^83 + 1208 * q^84 - 200 * q^86 + 2829 * q^87 - 744 * q^88 - 984 * q^89 - 995 * q^91 - 368 * q^92 + 1843 * q^93 - 726 * q^94 + 128 * q^96 - 137 * q^97 - 1200 * q^98 + 2013 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.26018	0.627423	0.313711	−	0.949518i	$-0.398428\pi$
$3$	3.26018	0.627423	0.313711	+	0.949518i	$0.398428\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−6.52037	−0.443655
$7$	−27.7921	−1.50063	−0.750316	−	0.661079i	$-0.770099\pi$
$7$	−27.7921	−1.50063	−0.750316	+	0.661079i	$0.770099\pi$
$8$	−8.00000	−0.353553
$9$	−16.3712	−0.606341
$10$	0	0
$11$	−10.8182	−0.296529	−0.148264	−	0.988948i	$-0.547369\pi$
$11$	−10.8182	−0.296529	−0.148264	+	0.988948i	$0.547369\pi$
$12$	13.0407	0.313711
$13$	−36.9683	−0.788705	−0.394352	−	0.918959i	$-0.629031\pi$
$13$	−36.9683	−0.788705	−0.394352	+	0.918959i	$0.629031\pi$
$14$	55.5842	1.06111
$15$	0	0
$16$	16.0000	0.250000
$17$	−118.996	−1.69769	−0.848847	−	0.528639i	$-0.822703\pi$
$17$	−118.996	−1.69769	−0.848847	+	0.528639i	$0.822703\pi$
$18$	32.7424	0.428748
$19$	−19.3321	−0.233426	−0.116713	−	0.993166i	$-0.537236\pi$
$19$	−19.3321	−0.233426	−0.116713	+	0.993166i	$0.537236\pi$
$20$	0	0
$21$	−90.6073	−0.941531
$22$	21.6365	0.209678
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−26.0815	−0.221827
$25$	0	0
$26$	73.9366	0.557699
$27$	−141.398	−1.00785
$28$	−111.168	−0.750316
$29$	234.499	1.50156	0.750782	−	0.660550i	$-0.229677\pi$
$29$	234.499	1.50156	0.750782	+	0.660550i	$0.229677\pi$
$30$	0	0
$31$	165.319	0.957814	0.478907	−	0.877866i	$-0.341033\pi$
$31$	165.319	0.957814	0.478907	+	0.877866i	$0.341033\pi$
$32$	−32.0000	−0.176777
$33$	−35.2694	−0.186049
$34$	237.992	1.20045
$35$	0	0
$36$	−65.4848	−0.303170
$37$	−202.301	−0.898865	−0.449432	−	0.893314i	$-0.648374\pi$
$37$	−202.301	−0.898865	−0.449432	+	0.893314i	$0.648374\pi$
$38$	38.6643	0.165057
$39$	−120.523	−0.494851
$40$	0	0
$41$	−295.846	−1.12691	−0.563456	−	0.826146i	$-0.690529\pi$
$41$	−295.846	−1.12691	−0.563456	+	0.826146i	$0.690529\pi$
$42$	181.215	0.665763
$43$	65.9221	0.233791	0.116896	−	0.993144i	$-0.462706\pi$
$43$	65.9221	0.233791	0.116896	+	0.993144i	$0.462706\pi$
$44$	−43.2729	−0.148264
$45$	0	0
$46$	46.0000	0.147442
$47$	110.279	0.342251	0.171126	−	0.985249i	$-0.445260\pi$
$47$	110.279	0.342251	0.171126	+	0.985249i	$0.445260\pi$
$48$	52.1630	0.156856
$49$	429.400	1.25190
$50$	0	0
$51$	−387.949	−1.06517
$52$	−147.873	−0.394352
$53$	688.135	1.78345	0.891723	−	0.452582i	$-0.149497\pi$
$53$	688.135	1.78345	0.891723	+	0.452582i	$0.149497\pi$
$54$	282.796	0.712661
$55$	0	0
$56$	222.337	0.530553
$57$	−63.0263	−0.146457
$58$	−468.998	−1.06177
$59$	10.5847	0.0233561	0.0116781	−	0.999932i	$-0.496283\pi$
$59$	10.5847	0.0233561	0.0116781	+	0.999932i	$0.496283\pi$
$60$	0	0
$61$	110.579	0.232101	0.116050	−	0.993243i	$-0.462977\pi$
$61$	110.579	0.232101	0.116050	+	0.993243i	$0.462977\pi$
$62$	−330.639	−0.677276
$63$	454.990	0.909894
$64$	64.0000	0.125000
$65$	0	0
$66$	70.5388	0.131557
$67$	643.290	1.17299	0.586496	−	0.809952i	$-0.300507\pi$
$67$	643.290	1.17299	0.586496	+	0.809952i	$0.300507\pi$
$68$	−475.984	−0.848847
$69$	−74.9842	−0.130827
$70$	0	0
$71$	143.216	0.239389	0.119695	−	0.992811i	$-0.461808\pi$
$71$	143.216	0.239389	0.119695	+	0.992811i	$0.461808\pi$
$72$	130.970	0.214374
$73$	158.213	0.253664	0.126832	−	0.991924i	$-0.459519\pi$
$73$	158.213	0.253664	0.126832	+	0.991924i	$0.459519\pi$
$74$	404.601	0.635593
$75$	0	0
$76$	−77.3285	−0.116713
$77$	300.661	0.444981
$78$	241.047	0.349913
$79$	1123.34	1.59982	0.799912	−	0.600118i	$-0.204880\pi$
$79$	1123.34	1.59982	0.799912	+	0.600118i	$0.204880\pi$
$80$	0	0
$81$	−18.9616	−0.0260104
$82$	591.692	0.796847
$83$	824.600	1.09050	0.545251	−	0.838273i	$-0.316434\pi$
$83$	824.600	1.09050	0.545251	+	0.838273i	$0.316434\pi$
$84$	−362.429	−0.470765
$85$	0	0
$86$	−131.844	−0.165316
$87$	764.510	0.942115
$88$	86.5458	0.104839
$89$	−879.672	−1.04770	−0.523849	−	0.851811i	$-0.675504\pi$
$89$	−879.672	−1.04770	−0.523849	+	0.851811i	$0.675504\pi$
$90$	0	0
$91$	1027.43	1.18356
$92$	−92.0000	−0.104257
$93$	538.971	0.600954
$94$	−220.557	−0.242008
$95$	0	0
$96$	−104.326	−0.110914
$97$	938.437	0.982308	0.491154	−	0.871073i	$-0.336575\pi$
$97$	938.437	0.982308	0.491154	+	0.871073i	$0.336575\pi$
$98$	−858.801	−0.885224
$99$	177.107	0.179798
$100$	0	0
$101$	−688.428	−0.678229	−0.339114	−	0.940745i	$-0.610127\pi$
$101$	−688.428	−0.678229	−0.339114	+	0.940745i	$0.610127\pi$
$102$	775.899	0.753190
$103$	−2041.55	−1.95301	−0.976504	−	0.215500i	$-0.930862\pi$
$103$	−2041.55	−1.95301	−0.976504	+	0.215500i	$0.930862\pi$
$104$	295.746	0.278849
$105$	0	0
$106$	−1376.27	−1.26109
$107$	−287.420	−0.259682	−0.129841	−	0.991535i	$-0.541447\pi$
$107$	−287.420	−0.259682	−0.129841	+	0.991535i	$0.541447\pi$
$108$	−565.592	−0.503927
$109$	−1211.29	−1.06441	−0.532204	−	0.846616i	$-0.678636\pi$
$109$	−1211.29	−1.06441	−0.532204	+	0.846616i	$0.678636\pi$
$110$	0	0
$111$	−659.537	−0.563968
$112$	−444.673	−0.375158
$113$	−741.316	−0.617143	−0.308571	−	0.951201i	$-0.599851\pi$
$113$	−741.316	−0.617143	−0.308571	+	0.951201i	$0.599851\pi$
$114$	126.053	0.103561
$115$	0	0
$116$	937.996	0.750782
$117$	605.215	0.478224
$118$	−21.1694	−0.0165153
$119$	3307.15	2.54761
$120$	0	0
$121$	−1213.97	−0.912071
$122$	−221.157	−0.164120
$123$	−964.513	−0.707050
$124$	661.277	0.478907
$125$	0	0
$126$	−909.980	−0.643392
$127$	−1579.99	−1.10395	−0.551976	−	0.833860i	$-0.686126\pi$
$127$	−1579.99	−1.10395	−0.551976	+	0.833860i	$0.686126\pi$
$128$	−128.000	−0.0883883
$129$	214.918	0.146686
$130$	0	0
$131$	−2348.86	−1.56657	−0.783285	−	0.621662i	$-0.786458\pi$
$131$	−2348.86	−1.56657	−0.783285	+	0.621662i	$0.786458\pi$
$132$	−141.078	−0.0930245
$133$	537.280	0.350287
$134$	−1286.58	−0.829430
$135$	0	0
$136$	951.969	0.600225
$137$	617.567	0.385127	0.192563	−	0.981285i	$-0.438320\pi$
$137$	617.567	0.385127	0.192563	+	0.981285i	$0.438320\pi$
$138$	149.968	0.0925084
$139$	1509.75	0.921260	0.460630	−	0.887592i	$-0.347623\pi$
$139$	1509.75	0.921260	0.460630	+	0.887592i	$0.347623\pi$
$140$	0	0
$141$	359.529	0.214736
$142$	−286.432	−0.169274
$143$	399.932	0.233874
$144$	−261.939	−0.151585
$145$	0	0
$146$	−316.427	−0.179368
$147$	1399.92	0.785468
$148$	−809.202	−0.449432
$149$	1025.46	0.563818	0.281909	−	0.959441i	$-0.409032\pi$
$149$	1025.46	0.563818	0.281909	+	0.959441i	$0.409032\pi$
$150$	0	0
$151$	−1516.67	−0.817383	−0.408692	−	0.912672i	$-0.634015\pi$
$151$	−1516.67	−0.817383	−0.408692	+	0.912672i	$0.634015\pi$
$152$	154.657	0.0825286
$153$	1948.11	1.02938
$154$	−601.322	−0.314649
$155$	0	0
$156$	−482.094	−0.247426
$157$	2757.31	1.40164	0.700819	−	0.713339i	$-0.252818\pi$
$157$	2757.31	1.40164	0.700819	+	0.713339i	$0.252818\pi$
$158$	−2246.69	−1.13125
$159$	2243.45	1.11897
$160$	0	0
$161$	639.218	0.312903
$162$	37.9231	0.0183921
$163$	−1263.29	−0.607047	−0.303524	−	0.952824i	$-0.598163\pi$
$163$	−1263.29	−0.607047	−0.303524	+	0.952824i	$0.598163\pi$
$164$	−1183.38	−0.563456
$165$	0	0
$166$	−1649.20	−0.771101
$167$	349.646	0.162014	0.0810072	−	0.996714i	$-0.474186\pi$
$167$	349.646	0.162014	0.0810072	+	0.996714i	$0.474186\pi$
$168$	724.859	0.332881
$169$	−830.344	−0.377945
$170$	0	0
$171$	316.490	0.141536
$172$	263.689	0.116896
$173$	−2313.83	−1.01686	−0.508432	−	0.861102i	$-0.669775\pi$
$173$	−2313.83	−1.01686	−0.508432	+	0.861102i	$0.669775\pi$
$174$	−1529.02	−0.666176
$175$	0	0
$176$	−173.092	−0.0741322
$177$	34.5081	0.0146542
$178$	1759.34	0.740834
$179$	−2347.69	−0.980306	−0.490153	−	0.871636i	$-0.663059\pi$
$179$	−2347.69	−0.980306	−0.490153	+	0.871636i	$0.663059\pi$
$180$	0	0
$181$	4396.31	1.80539	0.902695	−	0.430282i	$-0.141586\pi$
$181$	4396.31	1.80539	0.902695	+	0.430282i	$0.141586\pi$
$182$	−2054.85	−0.836900
$183$	360.507	0.145625
$184$	184.000	0.0737210
$185$	0	0
$186$	−1077.94	−0.424939
$187$	1287.33	0.503415
$188$	441.115	0.171126
$189$	3929.75	1.51242
$190$	0	0
$191$	4153.14	1.57335	0.786677	−	0.617365i	$-0.211800\pi$
$191$	4153.14	1.57335	0.786677	+	0.617365i	$0.211800\pi$
$192$	208.652	0.0784278
$193$	1020.52	0.380616	0.190308	−	0.981724i	$-0.439051\pi$
$193$	1020.52	0.380616	0.190308	+	0.981724i	$0.439051\pi$
$194$	−1876.87	−0.694596
$195$	0	0
$196$	1717.60	0.625948
$197$	398.643	0.144173	0.0720866	−	0.997398i	$-0.477034\pi$
$197$	398.643	0.144173	0.0720866	+	0.997398i	$0.477034\pi$
$198$	−354.215	−0.127136
$199$	4131.40	1.47169	0.735847	−	0.677148i	$-0.236784\pi$
$199$	4131.40	1.47169	0.735847	+	0.677148i	$0.236784\pi$
$200$	0	0
$201$	2097.24	0.735961
$202$	1376.86	0.479580
$203$	−6517.22	−2.25329
$204$	−1551.80	−0.532586
$205$	0	0
$206$	4083.10	1.38098
$207$	376.538	0.126431
$208$	−591.493	−0.197176
$209$	209.139	0.0692176
$210$	0	0
$211$	826.315	0.269601	0.134801	−	0.990873i	$-0.456961\pi$
$211$	826.315	0.269601	0.134801	+	0.990873i	$0.456961\pi$
$212$	2752.54	0.891723
$213$	466.911	0.150198
$214$	574.840	0.183623
$215$	0	0
$216$	1131.18	0.356330
$217$	−4594.57	−1.43733
$218$	2422.58	0.752650
$219$	515.805	0.159155
$220$	0	0
$221$	4399.08	1.33898
$222$	1319.07	0.398786
$223$	−1110.09	−0.333351	−0.166676	−	0.986012i	$-0.553303\pi$
$223$	−1110.09	−0.333351	−0.166676	+	0.986012i	$0.553303\pi$
$224$	889.347	0.265277
$225$	0	0
$226$	1482.63	0.436386
$227$	4917.90	1.43794	0.718971	−	0.695040i	$-0.244614\pi$
$227$	4917.90	1.43794	0.718971	+	0.695040i	$0.244614\pi$
$228$	−252.105	−0.0732284
$229$	−1390.62	−0.401288	−0.200644	−	0.979664i	$-0.564304\pi$
$229$	−1390.62	−0.401288	−0.200644	+	0.979664i	$0.564304\pi$
$230$	0	0
$231$	980.211	0.279191
$232$	−1875.99	−0.530883
$233$	3409.59	0.958668	0.479334	−	0.877633i	$-0.340878\pi$
$233$	3409.59	0.958668	0.479334	+	0.877633i	$0.340878\pi$
$234$	−1210.43	−0.338155
$235$	0	0
$236$	42.3388	0.0116781
$237$	3662.31	1.00377
$238$	−6614.30	−1.80143
$239$	2056.35	0.556544	0.278272	−	0.960502i	$-0.410238\pi$
$239$	2056.35	0.556544	0.278272	+	0.960502i	$0.410238\pi$
$240$	0	0
$241$	−1957.03	−0.523085	−0.261543	−	0.965192i	$-0.584231\pi$
$241$	−1957.03	−0.523085	−0.261543	+	0.965192i	$0.584231\pi$
$242$	2427.93	0.644931
$243$	3755.93	0.991535
$244$	442.315	0.116050
$245$	0	0
$246$	1929.03	0.499960
$247$	714.676	0.184104
$248$	−1322.55	−0.338638
$249$	2688.35	0.684205
$250$	0	0
$251$	−1954.13	−0.491410	−0.245705	−	0.969345i	$-0.579019\pi$
$251$	−1954.13	−0.491410	−0.245705	+	0.969345i	$0.579019\pi$
$252$	1819.96	0.454947
$253$	248.819	0.0618306
$254$	3159.99	0.780612
$255$	0	0
$256$	256.000	0.0625000
$257$	−1958.40	−0.475338	−0.237669	−	0.971346i	$-0.576383\pi$
$257$	−1958.40	−0.475338	−0.237669	+	0.971346i	$0.576383\pi$
$258$	−429.837	−0.103723
$259$	5622.35	1.34887
$260$	0	0
$261$	−3839.03	−0.910459
$262$	4697.72	1.10773
$263$	−7205.23	−1.68933	−0.844665	−	0.535295i	$-0.820200\pi$
$263$	−7205.23	−1.68933	−0.844665	+	0.535295i	$0.820200\pi$
$264$	282.155	0.0657783
$265$	0	0
$266$	−1074.56	−0.247690
$267$	−2867.89	−0.657349
$268$	2573.16	0.586496
$269$	−3764.30	−0.853210	−0.426605	−	0.904438i	$-0.640291\pi$
$269$	−3764.30	−0.853210	−0.426605	+	0.904438i	$0.640291\pi$
$270$	0	0
$271$	−5208.88	−1.16759	−0.583795	−	0.811901i	$-0.698433\pi$
$271$	−5208.88	−1.16759	−0.583795	+	0.811901i	$0.698433\pi$
$272$	−1903.94	−0.424423
$273$	3349.60	0.742590
$274$	−1235.13	−0.272326
$275$	0	0
$276$	−299.937	−0.0654133
$277$	−1550.21	−0.336256	−0.168128	−	0.985765i	$-0.553772\pi$
$277$	−1550.21	−0.336256	−0.168128	+	0.985765i	$0.553772\pi$
$278$	−3019.50	−0.651430
$279$	−2706.47	−0.580761
$280$	0	0
$281$	7997.53	1.69784	0.848920	−	0.528522i	$-0.177254\pi$
$281$	7997.53	1.69784	0.848920	+	0.528522i	$0.177254\pi$
$282$	−719.058	−0.151841
$283$	5808.41	1.22005	0.610025	−	0.792382i	$-0.291159\pi$
$283$	5808.41	1.22005	0.610025	+	0.792382i	$0.291159\pi$
$284$	572.864	0.119695
$285$	0	0
$286$	−799.863	−0.165374
$287$	8222.18	1.69108
$288$	523.878	0.107187
$289$	9247.08	1.88216
$290$	0	0
$291$	3059.48	0.616322
$292$	632.854	0.126832
$293$	8854.66	1.76551	0.882756	−	0.469832i	$-0.155686\pi$
$293$	8854.66	1.76551	0.882756	+	0.469832i	$0.155686\pi$
$294$	−2799.85	−0.555410
$295$	0	0
$296$	1618.40	0.317797
$297$	1529.68	0.298858
$298$	−2050.92	−0.398680
$299$	850.271	0.164456
$300$	0	0
$301$	−1832.11	−0.350835
$302$	3033.34	0.577977
$303$	−2244.40	−0.425536
$304$	−309.314	−0.0583565
$305$	0	0
$306$	−3896.22	−0.727882
$307$	1501.03	0.279050	0.139525	−	0.990219i	$-0.455443\pi$
$307$	1501.03	0.279050	0.139525	+	0.990219i	$0.455443\pi$
$308$	1202.64	0.222490
$309$	−6655.83	−1.22536
$310$	0	0
$311$	−7440.48	−1.35663	−0.678313	−	0.734773i	$-0.737289\pi$
$311$	−7440.48	−1.35663	−0.678313	+	0.734773i	$0.737289\pi$
$312$	964.188	0.174956
$313$	2528.63	0.456634	0.228317	−	0.973587i	$-0.426678\pi$
$313$	2528.63	0.456634	0.228317	+	0.973587i	$0.426678\pi$
$314$	−5514.61	−0.991107
$315$	0	0
$316$	4493.38	0.799912
$317$	9636.72	1.70742	0.853710	−	0.520749i	$-0.174347\pi$
$317$	9636.72	1.70742	0.853710	+	0.520749i	$0.174347\pi$
$318$	−4486.89	−0.791234
$319$	−2536.86	−0.445257
$320$	0	0
$321$	−937.042	−0.162930
$322$	−1278.44	−0.221256
$323$	2300.45	0.396286
$324$	−75.8462	−0.0130052
$325$	0	0
$326$	2526.58	0.429247
$327$	−3949.03	−0.667834
$328$	2366.77	0.398424
$329$	−3064.88	−0.513593
$330$	0	0
$331$	2875.82	0.477551	0.238776	−	0.971075i	$-0.423254\pi$
$331$	2875.82	0.477551	0.238776	+	0.971075i	$0.423254\pi$
$332$	3298.40	0.545251
$333$	3311.90	0.545018
$334$	−699.291	−0.114561
$335$	0	0
$336$	−1449.72	−0.235383
$337$	−9911.19	−1.60207	−0.801034	−	0.598619i	$-0.795717\pi$
$337$	−9911.19	−1.60207	−0.801034	+	0.598619i	$0.795717\pi$
$338$	1660.69	0.267247
$339$	−2416.83	−0.387209
$340$	0	0
$341$	−1788.46	−0.284019
$342$	−632.980	−0.100081
$343$	−2401.24	−0.378003
$344$	−527.377	−0.0826578
$345$	0	0
$346$	4627.67	0.719031
$347$	−4497.70	−0.695820	−0.347910	−	0.937528i	$-0.613108\pi$
$347$	−4497.70	−0.695820	−0.347910	+	0.937528i	$0.613108\pi$
$348$	3058.04	0.471058
$349$	888.112	0.136216	0.0681082	−	0.997678i	$-0.478304\pi$
$349$	888.112	0.136216	0.0681082	+	0.997678i	$0.478304\pi$
$350$	0	0
$351$	5227.25	0.794900
$352$	346.183	0.0524194
$353$	11722.2	1.76745	0.883727	−	0.468002i	$-0.155026\pi$
$353$	11722.2	1.76745	0.883727	+	0.468002i	$0.155026\pi$
$354$	−69.0161	−0.0103621
$355$	0	0
$356$	−3518.69	−0.523849
$357$	10781.9	1.59843
$358$	4695.39	0.693181
$359$	−1257.49	−0.184869	−0.0924343	−	0.995719i	$-0.529465\pi$
$359$	−1257.49	−0.184869	−0.0924343	+	0.995719i	$0.529465\pi$
$360$	0	0
$361$	−6485.27	−0.945512
$362$	−8792.63	−1.27660
$363$	−3957.75	−0.572254
$364$	4109.71	0.591778
$365$	0	0
$366$	−721.014	−0.102973
$367$	−10122.2	−1.43972	−0.719859	−	0.694120i	$-0.755794\pi$
$367$	−10122.2	−1.43972	−0.719859	+	0.694120i	$0.755794\pi$
$368$	−368.000	−0.0521286
$369$	4843.36	0.683293
$370$	0	0
$371$	−19124.7	−2.67629
$372$	2155.89	0.300477
$373$	−6530.08	−0.906473	−0.453237	−	0.891390i	$-0.649731\pi$
$373$	−6530.08	−0.906473	−0.453237	+	0.891390i	$0.649731\pi$
$374$	−2574.65	−0.355968
$375$	0	0
$376$	−882.230	−0.121004
$377$	−8669.03	−1.18429
$378$	−7859.50	−1.06944
$379$	277.850	0.0376575	0.0188288	−	0.999823i	$-0.494006\pi$
$379$	277.850	0.0376575	0.0188288	+	0.999823i	$0.494006\pi$
$380$	0	0
$381$	−5151.07	−0.692644
$382$	−8306.28	−1.11253
$383$	10679.6	1.42482	0.712408	−	0.701766i	$-0.247605\pi$
$383$	10679.6	1.42482	0.712408	+	0.701766i	$0.247605\pi$
$384$	−417.304	−0.0554569
$385$	0	0
$386$	−2041.05	−0.269136
$387$	−1079.22	−0.141757
$388$	3753.75	0.491154
$389$	1546.05	0.201511	0.100756	−	0.994911i	$-0.467874\pi$
$389$	1546.05	0.201511	0.100756	+	0.994911i	$0.467874\pi$
$390$	0	0
$391$	2736.91	0.353994
$392$	−3435.20	−0.442612
$393$	−7657.71	−0.982902
$394$	−797.285	−0.101946
$395$	0	0
$396$	708.429	0.0898988
$397$	12901.8	1.63103	0.815517	−	0.578733i	$-0.196453\pi$
$397$	12901.8	1.63103	0.815517	+	0.578733i	$0.196453\pi$
$398$	−8262.80	−1.04064
$399$	1751.63	0.219778
$400$	0	0
$401$	−8534.31	−1.06280	−0.531400	−	0.847121i	$-0.678334\pi$
$401$	−8534.31	−1.06280	−0.531400	+	0.847121i	$0.678334\pi$
$402$	−4194.49	−0.520403
$403$	−6111.57	−0.755432
$404$	−2753.71	−0.339114
$405$	0	0
$406$	13034.4	1.59332
$407$	2188.53	0.266539
$408$	3103.59	0.376595
$409$	650.968	0.0787000	0.0393500	−	0.999225i	$-0.487471\pi$
$409$	650.968	0.0787000	0.0393500	+	0.999225i	$0.487471\pi$
$410$	0	0
$411$	2013.38	0.241637
$412$	−8166.20	−0.976504
$413$	−294.171	−0.0350489
$414$	−753.075	−0.0894001
$415$	0	0
$416$	1182.99	0.139425
$417$	4922.06	0.578020
$418$	−418.279	−0.0489442
$419$	13266.2	1.54677	0.773384	−	0.633938i	$-0.218563\pi$
$419$	13266.2	1.54677	0.773384	+	0.633938i	$0.218563\pi$
$420$	0	0
$421$	−8274.69	−0.957918	−0.478959	−	0.877837i	$-0.658986\pi$
$421$	−8274.69	−0.957918	−0.478959	+	0.877837i	$0.658986\pi$
$422$	−1652.63	−0.190637
$423$	−1805.39	−0.207521
$424$	−5505.08	−0.630543
$425$	0	0
$426$	−933.822	−0.106206
$427$	−3073.21	−0.348298
$428$	−1149.68	−0.129841
$429$	1303.85	0.146738
$430$	0	0
$431$	724.084	0.0809232	0.0404616	−	0.999181i	$-0.487117\pi$
$431$	724.084	0.0809232	0.0404616	+	0.999181i	$0.487117\pi$
$432$	−2262.37	−0.251964
$433$	11996.5	1.33145	0.665724	−	0.746198i	$-0.268123\pi$
$433$	11996.5	1.33145	0.665724	+	0.746198i	$0.268123\pi$
$434$	9189.14	1.01634
$435$	0	0
$436$	−4845.16	−0.532204
$437$	444.639	0.0486727
$438$	−1031.61	−0.112539
$439$	−7765.40	−0.844242	−0.422121	−	0.906539i	$-0.638714\pi$
$439$	−7765.40	−0.844242	−0.422121	+	0.906539i	$0.638714\pi$
$440$	0	0
$441$	−7029.80	−0.759075
$442$	−8798.17	−0.946801
$443$	9061.54	0.971844	0.485922	−	0.874002i	$-0.338484\pi$
$443$	9061.54	0.971844	0.485922	+	0.874002i	$0.338484\pi$
$444$	−2638.15	−0.281984
$445$	0	0
$446$	2220.19	0.235715
$447$	3343.19	0.353752
$448$	−1778.69	−0.187579
$449$	−10944.3	−1.15032	−0.575159	−	0.818041i	$-0.695060\pi$
$449$	−10944.3	−1.15032	−0.575159	+	0.818041i	$0.695060\pi$
$450$	0	0
$451$	3200.53	0.334162
$452$	−2965.26	−0.308571
$453$	−4944.63	−0.512845
$454$	−9835.81	−1.01678
$455$	0	0
$456$	504.211	0.0517803
$457$	−10934.9	−1.11928	−0.559641	−	0.828735i	$-0.689061\pi$
$457$	−10934.9	−1.11928	−0.559641	+	0.828735i	$0.689061\pi$
$458$	2781.25	0.283754
$459$	16825.8	1.71103
$460$	0	0
$461$	2652.25	0.267956	0.133978	−	0.990984i	$-0.457225\pi$
$461$	2652.25	0.267956	0.133978	+	0.990984i	$0.457225\pi$
$462$	−1960.42	−0.197418
$463$	−707.699	−0.0710358	−0.0355179	−	0.999369i	$-0.511308\pi$
$463$	−707.699	−0.0710358	−0.0355179	+	0.999369i	$0.511308\pi$
$464$	3751.98	0.375391
$465$	0	0
$466$	−6819.18	−0.677880
$467$	10671.1	1.05738	0.528691	−	0.848814i	$-0.322683\pi$
$467$	10671.1	1.05738	0.528691	+	0.848814i	$0.322683\pi$
$468$	2420.86	0.239112
$469$	−17878.4	−1.76023
$470$	0	0
$471$	8989.33	0.879419
$472$	−84.6776	−0.00825763
$473$	−713.161	−0.0693259
$474$	−7324.62	−0.709770
$475$	0	0
$476$	13228.6	1.27381
$477$	−11265.6	−1.08138
$478$	−4112.69	−0.393536
$479$	1951.17	0.186119	0.0930597	−	0.995661i	$-0.470335\pi$
$479$	1951.17	0.186119	0.0930597	+	0.995661i	$0.470335\pi$
$480$	0	0
$481$	7478.71	0.708939
$482$	3914.06	0.369877
$483$	2083.97	0.196323
$484$	−4855.86	−0.456035
$485$	0	0
$486$	−7511.86	−0.701121
$487$	−17071.2	−1.58844	−0.794220	−	0.607630i	$-0.792120\pi$
$487$	−17071.2	−1.58844	−0.794220	+	0.607630i	$0.792120\pi$
$488$	−884.629	−0.0820600
$489$	−4118.57	−0.380875
$490$	0	0
$491$	16979.9	1.56067	0.780337	−	0.625359i	$-0.215047\pi$
$491$	16979.9	1.56067	0.780337	+	0.625359i	$0.215047\pi$
$492$	−3858.05	−0.353525
$493$	−27904.5	−2.54920
$494$	−1429.35	−0.130181
$495$	0	0
$496$	2645.11	0.239453
$497$	−3980.28	−0.359235
$498$	−5376.70	−0.483806
$499$	−2788.83	−0.250191	−0.125095	−	0.992145i	$-0.539924\pi$
$499$	−2788.83	−0.250191	−0.125095	+	0.992145i	$0.539924\pi$
$500$	0	0
$501$	1139.91	0.101651
$502$	3908.27	0.347479
$503$	15525.1	1.37620	0.688102	−	0.725614i	$-0.258444\pi$
$503$	15525.1	1.37620	0.688102	+	0.725614i	$0.258444\pi$
$504$	−3639.92	−0.321696
$505$	0	0
$506$	−497.638	−0.0437208
$507$	−2707.08	−0.237131
$508$	−6319.98	−0.551976
$509$	9039.20	0.787143	0.393571	−	0.919294i	$-0.371239\pi$
$509$	9039.20	0.787143	0.393571	+	0.919294i	$0.371239\pi$
$510$	0	0
$511$	−4397.08	−0.380657
$512$	−512.000	−0.0441942
$513$	2733.53	0.235260
$514$	3916.81	0.336115
$515$	0	0
$516$	859.673	0.0733430
$517$	−1193.02	−0.101487
$518$	−11244.7	−0.953792
$519$	−7543.52	−0.638004
$520$	0	0
$521$	1093.03	0.0919128	0.0459564	−	0.998943i	$-0.485366\pi$
$521$	1093.03	0.0919128	0.0459564	+	0.998943i	$0.485366\pi$
$522$	7678.06	0.643792
$523$	4660.98	0.389695	0.194848	−	0.980834i	$-0.437579\pi$
$523$	4660.98	0.389695	0.194848	+	0.980834i	$0.437579\pi$
$524$	−9395.44	−0.783285
$525$	0	0
$526$	14410.5	1.19454
$527$	−19672.4	−1.62607
$528$	−564.311	−0.0465123
$529$	529.000	0.0434783
$530$	0	0
$531$	−173.284	−0.0141618
$532$	2149.12	0.175143
$533$	10936.9	0.888801
$534$	5735.79	0.464816
$535$	0	0
$536$	−5146.32	−0.414715
$537$	−7653.91	−0.615066
$538$	7528.60	0.603311
$539$	−4645.35	−0.371223
$540$	0	0
$541$	−8072.40	−0.641515	−0.320757	−	0.947161i	$-0.603937\pi$
$541$	−8072.40	−0.641515	−0.320757	+	0.947161i	$0.603937\pi$
$542$	10417.8	0.825611
$543$	14332.8	1.13274
$544$	3807.88	0.300113
$545$	0	0
$546$	−6699.20	−0.525090
$547$	2877.27	0.224905	0.112452	−	0.993657i	$-0.464129\pi$
$547$	2877.27	0.224905	0.112452	+	0.993657i	$0.464129\pi$
$548$	2470.27	0.192563
$549$	−1810.31	−0.140732
$550$	0	0
$551$	−4533.36	−0.350504
$552$	599.874	0.0462542
$553$	−31220.1	−2.40075
$554$	3100.42	0.237769
$555$	0	0
$556$	6038.99	0.460630
$557$	−15928.7	−1.21171	−0.605855	−	0.795575i	$-0.707169\pi$
$557$	−15928.7	−1.21171	−0.605855	+	0.795575i	$0.707169\pi$
$558$	5412.95	0.410660
$559$	−2437.03	−0.184392
$560$	0	0
$561$	4196.92	0.315854
$562$	−15995.1	−1.20055
$563$	−7988.51	−0.598003	−0.299001	−	0.954253i	$-0.596654\pi$
$563$	−7988.51	−0.598003	−0.299001	+	0.954253i	$0.596654\pi$
$564$	1438.12	0.107368
$565$	0	0
$566$	−11616.8	−0.862706
$567$	526.981	0.0390320
$568$	−1145.73	−0.0846368
$569$	−6429.49	−0.473705	−0.236853	−	0.971546i	$-0.576116\pi$
$569$	−6429.49	−0.473705	−0.236853	+	0.971546i	$0.576116\pi$
$570$	0	0
$571$	2886.51	0.211553	0.105776	−	0.994390i	$-0.466267\pi$
$571$	2886.51	0.211553	0.105776	+	0.994390i	$0.466267\pi$
$572$	1599.73	0.116937
$573$	13540.0	0.987158
$574$	−16444.4	−1.19577
$575$	0	0
$576$	−1047.76	−0.0757926
$577$	9966.59	0.719089	0.359545	−	0.933128i	$-0.382932\pi$
$577$	9966.59	0.719089	0.359545	+	0.933128i	$0.382932\pi$
$578$	−18494.2	−1.33089
$579$	3327.10	0.238807
$580$	0	0
$581$	−22917.4	−1.63644
$582$	−6118.95	−0.435806
$583$	−7444.40	−0.528843
$584$	−1265.71	−0.0896838
$585$	0	0
$586$	−17709.3	−1.24841
$587$	3522.91	0.247710	0.123855	−	0.992300i	$-0.460474\pi$
$587$	3522.91	0.247710	0.123855	+	0.992300i	$0.460474\pi$
$588$	5599.70	0.392734
$589$	−3195.97	−0.223579
$590$	0	0
$591$	1299.65	0.0904575
$592$	−3236.81	−0.224716
$593$	28713.0	1.98837	0.994184	−	0.107695i	$-0.0343469\pi$
$593$	28713.0	1.98837	0.994184	+	0.107695i	$0.0343469\pi$
$594$	−3059.35	−0.211325
$595$	0	0
$596$	4101.84	0.281909
$597$	13469.1	0.923374
$598$	−1700.54	−0.116288
$599$	7736.44	0.527717	0.263858	−	0.964561i	$-0.415005\pi$
$599$	7736.44	0.527717	0.263858	+	0.964561i	$0.415005\pi$
$600$	0	0
$601$	−3863.89	−0.262249	−0.131124	−	0.991366i	$-0.541859\pi$
$601$	−3863.89	−0.262249	−0.131124	+	0.991366i	$0.541859\pi$
$602$	3664.23	0.248078
$603$	−10531.4	−0.711232
$604$	−6066.68	−0.408692
$605$	0	0
$606$	4488.80	0.300900
$607$	−24954.6	−1.66866	−0.834330	−	0.551265i	$-0.814145\pi$
$607$	−24954.6	−1.66866	−0.834330	+	0.551265i	$0.814145\pi$
$608$	618.628	0.0412643
$609$	−21247.3	−1.41377
$610$	0	0
$611$	−4076.82	−0.269935
$612$	7792.44	0.514690
$613$	−12712.0	−0.837574	−0.418787	−	0.908085i	$-0.637545\pi$
$613$	−12712.0	−0.837574	−0.418787	+	0.908085i	$0.637545\pi$
$614$	−3002.06	−0.197318
$615$	0	0
$616$	−2405.29	−0.157324
$617$	−11017.6	−0.718884	−0.359442	−	0.933167i	$-0.617033\pi$
$617$	−11017.6	−0.718884	−0.359442	+	0.933167i	$0.617033\pi$
$618$	13311.7	0.866461
$619$	6859.65	0.445416	0.222708	−	0.974885i	$-0.428510\pi$
$619$	6859.65	0.445416	0.222708	+	0.974885i	$0.428510\pi$
$620$	0	0
$621$	3252.16	0.210152
$622$	14881.0	0.959280
$623$	24447.9	1.57221
$624$	−1928.38	−0.123713
$625$	0	0
$626$	−5057.25	−0.322889
$627$	681.833	0.0434287
$628$	11029.2	0.700819
$629$	24073.0	1.52600
$630$	0	0
$631$	−14104.0	−0.889815	−0.444907	−	0.895577i	$-0.646763\pi$
$631$	−14104.0	−0.889815	−0.444907	+	0.895577i	$0.646763\pi$
$632$	−8986.75	−0.565623
$633$	2693.94	0.169154
$634$	−19273.4	−1.20733
$635$	0	0
$636$	8973.79	0.559487
$637$	−15874.2	−0.987376
$638$	5073.73	0.314844
$639$	−2344.62	−0.145151
$640$	0	0
$641$	−144.931	−0.00893047	−0.00446524	−	0.999990i	$-0.501421\pi$
$641$	−144.931	−0.00893047	−0.00446524	+	0.999990i	$0.501421\pi$
$642$	1874.08	0.115209
$643$	−30891.4	−1.89461	−0.947307	−	0.320328i	$-0.896207\pi$
$643$	−30891.4	−1.89461	−0.947307	+	0.320328i	$0.896207\pi$
$644$	2556.87	0.156452
$645$	0	0
$646$	−4600.90	−0.280217
$647$	−22784.1	−1.38444	−0.692222	−	0.721684i	$-0.743368\pi$
$647$	−22784.1	−1.38444	−0.692222	+	0.721684i	$0.743368\pi$
$648$	151.692	0.00919605
$649$	−114.508	−0.00692576
$650$	0	0
$651$	−14979.1	−0.901811
$652$	−5053.17	−0.303524
$653$	32125.7	1.92523	0.962617	−	0.270868i	$-0.0873106\pi$
$653$	32125.7	1.92523	0.962617	+	0.270868i	$0.0873106\pi$
$654$	7898.05	0.472230
$655$	0	0
$656$	−4733.54	−0.281728
$657$	−2590.14	−0.153807
$658$	6129.75	0.363165
$659$	15521.7	0.917508	0.458754	−	0.888563i	$-0.348296\pi$
$659$	15521.7	0.917508	0.458754	+	0.888563i	$0.348296\pi$
$660$	0	0
$661$	19428.4	1.14323	0.571617	−	0.820521i	$-0.306316\pi$
$661$	19428.4	1.14323	0.571617	+	0.820521i	$0.306316\pi$
$662$	−5751.64	−0.337680
$663$	14341.8	0.840106
$664$	−6596.80	−0.385550
$665$	0	0
$666$	−6623.80	−0.385386
$667$	−5393.48	−0.313098
$668$	1398.58	0.0810072
$669$	−3619.11	−0.209152
$670$	0	0
$671$	−1196.27	−0.0688246
$672$	2899.43	0.166441
$673$	−18992.7	−1.08784	−0.543919	−	0.839138i	$-0.683060\pi$
$673$	−18992.7	−1.08784	−0.543919	+	0.839138i	$0.683060\pi$
$674$	19822.4	1.13283
$675$	0	0
$676$	−3321.38	−0.188972
$677$	−19104.3	−1.08455	−0.542273	−	0.840202i	$-0.682436\pi$
$677$	−19104.3	−1.08455	−0.542273	+	0.840202i	$0.682436\pi$
$678$	4833.65	0.273798
$679$	−26081.1	−1.47408
$680$	0	0
$681$	16033.3	0.902197
$682$	3576.92	0.200832
$683$	24737.2	1.38586	0.692931	−	0.721004i	$-0.256319\pi$
$683$	24737.2	1.38586	0.692931	+	0.721004i	$0.256319\pi$
$684$	1265.96	0.0707679
$685$	0	0
$686$	4802.49	0.267288
$687$	−4533.69	−0.251778
$688$	1054.75	0.0584479
$689$	−25439.2	−1.40661
$690$	0	0
$691$	12957.9	0.713373	0.356687	−	0.934224i	$-0.383906\pi$
$691$	12957.9	0.713373	0.356687	+	0.934224i	$0.383906\pi$
$692$	−9255.33	−0.508432
$693$	−4922.18	−0.269810
$694$	8995.41	0.492019
$695$	0	0
$696$	−6116.08	−0.333088
$697$	35204.5	1.91315
$698$	−1776.22	−0.0963195
$699$	11115.9	0.601490
$700$	0	0
$701$	20730.7	1.11696	0.558478	−	0.829519i	$-0.311385\pi$
$701$	20730.7	1.11696	0.558478	+	0.829519i	$0.311385\pi$
$702$	−10454.5	−0.562079
$703$	3910.90	0.209819
$704$	−692.367	−0.0370661
$705$	0	0
$706$	−23444.5	−1.24978
$707$	19132.8	1.01777
$708$	138.032	0.00732708
$709$	−22191.2	−1.17547	−0.587735	−	0.809053i	$-0.699980\pi$
$709$	−22191.2	−1.17547	−0.587735	+	0.809053i	$0.699980\pi$
$710$	0	0
$711$	−18390.5	−0.970038
$712$	7037.38	0.370417
$713$	−3802.34	−0.199718
$714$	−21563.8	−1.13026
$715$	0	0
$716$	−9390.77	−0.490153
$717$	6704.07	0.349188
$718$	2514.98	0.130722
$719$	−773.962	−0.0401445	−0.0200723	−	0.999799i	$-0.506390\pi$
$719$	−773.962	−0.0401445	−0.0200723	+	0.999799i	$0.506390\pi$
$720$	0	0
$721$	56738.9	2.93075
$722$	12970.5	0.668578
$723$	−6380.29	−0.328196
$724$	17585.3	0.902695
$725$	0	0
$726$	7915.51	0.404645
$727$	32484.7	1.65721	0.828604	−	0.559835i	$-0.189136\pi$
$727$	32484.7	1.65721	0.828604	+	0.559835i	$0.189136\pi$
$728$	−8219.41	−0.418450
$729$	12757.0	0.648122
$730$	0	0
$731$	−7844.48	−0.396906
$732$	1442.03	0.0728127
$733$	13701.5	0.690416	0.345208	−	0.938526i	$-0.387808\pi$
$733$	13701.5	0.690416	0.345208	+	0.938526i	$0.387808\pi$
$734$	20244.5	1.01803
$735$	0	0
$736$	736.000	0.0368605
$737$	−6959.26	−0.347826
$738$	−9686.71	−0.483161
$739$	−31474.2	−1.56671	−0.783354	−	0.621576i	$-0.786493\pi$
$739$	−31474.2	−1.56671	−0.783354	+	0.621576i	$0.786493\pi$
$740$	0	0
$741$	2329.98	0.115511
$742$	38249.4	1.89243
$743$	−28664.2	−1.41533	−0.707664	−	0.706549i	$-0.750251\pi$
$743$	−28664.2	−1.41533	−0.707664	+	0.706549i	$0.750251\pi$
$744$	−4311.77	−0.212469
$745$	0	0
$746$	13060.2	0.640974
$747$	−13499.7	−0.661215
$748$	5149.31	0.251708
$749$	7988.00	0.389687
$750$	0	0
$751$	−25904.0	−1.25866	−0.629328	−	0.777140i	$-0.716670\pi$
$751$	−25904.0	−1.25866	−0.629328	+	0.777140i	$0.716670\pi$
$752$	1764.46	0.0855628
$753$	−6370.84	−0.308322
$754$	17338.1	0.837420
$755$	0	0
$756$	15719.0	0.756209
$757$	−2569.70	−0.123378	−0.0616892	−	0.998095i	$-0.519649\pi$
$757$	−2569.70	−0.123378	−0.0616892	+	0.998095i	$0.519649\pi$
$758$	−555.700	−0.0266279
$759$	811.197	0.0387939
$760$	0	0
$761$	−23219.4	−1.10605	−0.553024	−	0.833165i	$-0.686526\pi$
$761$	−23219.4	−1.10605	−0.553024	+	0.833165i	$0.686526\pi$
$762$	10302.1	0.489774
$763$	33664.3	1.59728
$764$	16612.6	0.786677
$765$	0	0
$766$	−21359.3	−1.00750
$767$	−391.298	−0.0184211
$768$	834.607	0.0392139
$769$	34771.1	1.63053	0.815265	−	0.579088i	$-0.196591\pi$
$769$	34771.1	1.63053	0.815265	+	0.579088i	$0.196591\pi$
$770$	0	0
$771$	−6384.76	−0.298238
$772$	4082.10	0.190308
$773$	36377.2	1.69262	0.846312	−	0.532687i	$-0.178818\pi$
$773$	36377.2	1.69262	0.846312	+	0.532687i	$0.178818\pi$
$774$	2158.45	0.100238
$775$	0	0
$776$	−7507.50	−0.347298
$777$	18329.9	0.846309
$778$	−3092.10	−0.142490
$779$	5719.34	0.263051
$780$	0	0
$781$	−1549.34	−0.0709858
$782$	−5473.82	−0.250311
$783$	−33157.7	−1.51336
$784$	6870.40	0.312974
$785$	0	0
$786$	15315.4	0.695017
$787$	1814.43	0.0821824	0.0410912	−	0.999155i	$-0.486917\pi$
$787$	1814.43	0.0821824	0.0410912	+	0.999155i	$0.486917\pi$
$788$	1594.57	0.0720866
$789$	−23490.4	−1.05992
$790$	0	0
$791$	20602.7	0.926104
$792$	−1416.86	−0.0635680
$793$	−4087.91	−0.183059
$794$	−25803.5	−1.15332
$795$	0	0
$796$	16525.6	0.735847
$797$	7958.23	0.353695	0.176848	−	0.984238i	$-0.443410\pi$
$797$	7958.23	0.353695	0.176848	+	0.984238i	$0.443410\pi$
$798$	−3503.27	−0.155406
$799$	−13122.7	−0.581038
$800$	0	0
$801$	14401.3	0.635261
$802$	17068.6	0.751513
$803$	−1711.59	−0.0752188
$804$	8388.98	0.367981
$805$	0	0
$806$	12223.1	0.534171
$807$	−12272.3	−0.535323
$808$	5507.42	0.239790
$809$	20506.9	0.891203	0.445601	−	0.895232i	$-0.352990\pi$
$809$	20506.9	0.891203	0.445601	+	0.895232i	$0.352990\pi$
$810$	0	0
$811$	1962.00	0.0849506	0.0424753	−	0.999098i	$-0.486476\pi$
$811$	1962.00	0.0849506	0.0424753	+	0.999098i	$0.486476\pi$
$812$	−26068.9	−1.12665
$813$	−16981.9	−0.732573
$814$	−4377.07	−0.188472
$815$	0	0
$816$	−6207.19	−0.266293
$817$	−1274.42	−0.0545730
$818$	−1301.94	−0.0556493
$819$	−16820.2	−0.717638
$820$	0	0
$821$	25168.1	1.06988	0.534941	−	0.844889i	$-0.320334\pi$
$821$	25168.1	1.06988	0.534941	+	0.844889i	$0.320334\pi$
$822$	−4026.77	−0.170863
$823$	−3337.33	−0.141351	−0.0706756	−	0.997499i	$-0.522516\pi$
$823$	−3337.33	−0.141351	−0.0706756	+	0.997499i	$0.522516\pi$
$824$	16332.4	0.690492
$825$	0	0
$826$	588.342	0.0247833
$827$	13087.4	0.550294	0.275147	−	0.961402i	$-0.411273\pi$
$827$	13087.4	0.550294	0.275147	+	0.961402i	$0.411273\pi$
$828$	1506.15	0.0632154
$829$	−38486.8	−1.61243	−0.806213	−	0.591626i	$-0.798486\pi$
$829$	−38486.8	−1.61243	−0.806213	+	0.591626i	$0.798486\pi$
$830$	0	0
$831$	−5053.97	−0.210975
$832$	−2365.97	−0.0985881
$833$	−51097.0	−2.12534
$834$	−9844.12	−0.408722
$835$	0	0
$836$	836.558	0.0346088
$837$	−23375.8	−0.965337
$838$	−26532.4	−1.09373
$839$	8192.40	0.337107	0.168554	−	0.985692i	$-0.446090\pi$
$839$	8192.40	0.337107	0.168554	+	0.985692i	$0.446090\pi$
$840$	0	0
$841$	30600.7	1.25469
$842$	16549.4	0.677350
$843$	26073.4	1.06526
$844$	3305.26	0.134801
$845$	0	0
$846$	3610.79	0.146739
$847$	33738.7	1.36868
$848$	11010.2	0.445861
$849$	18936.5	0.765487
$850$	0	0
$851$	4652.91	0.187426
$852$	1867.64	0.0750991
$853$	6774.83	0.271941	0.135971	−	0.990713i	$-0.456585\pi$
$853$	6774.83	0.271941	0.135971	+	0.990713i	$0.456585\pi$
$854$	6146.42	0.246284
$855$	0	0
$856$	2299.36	0.0918113
$857$	−19272.3	−0.768177	−0.384089	−	0.923296i	$-0.625484\pi$
$857$	−19272.3	−0.768177	−0.384089	+	0.923296i	$0.625484\pi$
$858$	−2607.70	−0.103759
$859$	−5991.11	−0.237968	−0.118984	−	0.992896i	$-0.537964\pi$
$859$	−5991.11	−0.237968	−0.118984	+	0.992896i	$0.537964\pi$
$860$	0	0
$861$	26805.8	1.06102
$862$	−1448.17	−0.0572214
$863$	−20711.4	−0.816945	−0.408472	−	0.912771i	$-0.633938\pi$
$863$	−20711.4	−0.816945	−0.408472	+	0.912771i	$0.633938\pi$
$864$	4524.74	0.178165
$865$	0	0
$866$	−23993.1	−0.941476
$867$	30147.2	1.18091
$868$	−18378.3	−0.718663
$869$	−12152.6	−0.474394
$870$	0	0
$871$	−23781.4	−0.925144
$872$	9690.31	0.376325
$873$	−15363.3	−0.595613
$874$	−889.278	−0.0344168
$875$	0	0
$876$	2063.22	0.0795774
$877$	45757.4	1.76182	0.880912	−	0.473281i	$-0.156930\pi$
$877$	45757.4	1.76182	0.880912	+	0.473281i	$0.156930\pi$
$878$	15530.8	0.596969
$879$	28867.8	1.10772
$880$	0	0
$881$	37019.6	1.41569	0.707845	−	0.706367i	$-0.249667\pi$
$881$	37019.6	1.41569	0.707845	+	0.706367i	$0.249667\pi$
$882$	14059.6	0.536747
$883$	9311.25	0.354868	0.177434	−	0.984133i	$-0.443220\pi$
$883$	9311.25	0.354868	0.177434	+	0.984133i	$0.443220\pi$
$884$	17596.3	0.669490
$885$	0	0
$886$	−18123.1	−0.687197
$887$	−33041.8	−1.25077	−0.625387	−	0.780315i	$-0.715059\pi$
$887$	−33041.8	−1.25077	−0.625387	+	0.780315i	$0.715059\pi$
$888$	5276.30	0.199393
$889$	43911.4	1.65662
$890$	0	0
$891$	205.130	0.00771283
$892$	−4440.37	−0.166676
$893$	−2131.92	−0.0798903
$894$	−6686.37	−0.250141
$895$	0	0
$896$	3557.39	0.132638
$897$	2772.04	0.103184
$898$	21888.6	0.813398
$899$	38767.2	1.43822
$900$	0	0
$901$	−81885.4	−3.02774
$902$	−6401.06	−0.236288
$903$	−5973.03	−0.220122
$904$	5930.53	0.218193
$905$	0	0
$906$	9889.25	0.362636
$907$	41359.1	1.51412	0.757060	−	0.653346i	$-0.226635\pi$
$907$	41359.1	1.51412	0.757060	+	0.653346i	$0.226635\pi$
$908$	19671.6	0.718971
$909$	11270.4	0.411238
$910$	0	0
$911$	−16960.3	−0.616818	−0.308409	−	0.951254i	$-0.599797\pi$
$911$	−16960.3	−0.616818	−0.308409	+	0.951254i	$0.599797\pi$
$912$	−1008.42	−0.0366142
$913$	−8920.71	−0.323365
$914$	21869.8	0.791452
$915$	0	0
$916$	−5562.50	−0.200644
$917$	65279.7	2.35085
$918$	−33651.7	−1.20988
$919$	3085.38	0.110748	0.0553739	−	0.998466i	$-0.482365\pi$
$919$	3085.38	0.110748	0.0553739	+	0.998466i	$0.482365\pi$
$920$	0	0
$921$	4893.63	0.175082
$922$	−5304.51	−0.189474
$923$	−5294.46	−0.188807
$924$	3920.84	0.139596
$925$	0	0
$926$	1415.40	0.0502299
$927$	33422.6	1.18419
$928$	−7503.97	−0.265442
$929$	−42292.0	−1.49360	−0.746801	−	0.665048i	$-0.768411\pi$
$929$	−42292.0	−1.49360	−0.746801	+	0.665048i	$0.768411\pi$
$930$	0	0
$931$	−8301.22	−0.292225
$932$	13638.4	0.479334
$933$	−24257.3	−0.851178
$934$	−21342.1	−0.747682
$935$	0	0
$936$	−4841.72	−0.169078
$937$	−15746.4	−0.549000	−0.274500	−	0.961587i	$-0.588512\pi$
$937$	−15746.4	−0.549000	−0.274500	+	0.961587i	$0.588512\pi$
$938$	35756.8	1.24467
$939$	8243.79	0.286502
$940$	0	0
$941$	52033.4	1.80259	0.901296	−	0.433204i	$-0.142617\pi$
$941$	52033.4	1.80259	0.901296	+	0.433204i	$0.142617\pi$
$942$	−17978.7	−0.621843
$943$	6804.46	0.234977
$944$	169.355	0.00583903
$945$	0	0
$946$	1426.32	0.0490208
$947$	17293.2	0.593405	0.296702	−	0.954970i	$-0.404113\pi$
$947$	17293.2	0.593405	0.296702	+	0.954970i	$0.404113\pi$
$948$	14649.2	0.501883
$949$	−5848.88	−0.200066
$950$	0	0
$951$	31417.5	1.07127
$952$	−26457.2	−0.900717
$953$	−8989.54	−0.305561	−0.152781	−	0.988260i	$-0.548823\pi$
$953$	−8989.54	−0.305561	−0.152781	+	0.988260i	$0.548823\pi$
$954$	22531.2	0.764648
$955$	0	0
$956$	8225.39	0.278272
$957$	−8270.64	−0.279365
$958$	−3902.34	−0.131606
$959$	−17163.5	−0.577933
$960$	0	0
$961$	−2460.53	−0.0825931
$962$	−14957.4	−0.501296
$963$	4705.41	0.157456
$964$	−7828.13	−0.261543
$965$	0	0
$966$	−4167.94	−0.138821
$967$	−23285.3	−0.774358	−0.387179	−	0.922005i	$-0.626550\pi$
$967$	−23285.3	−0.774358	−0.387179	+	0.922005i	$0.626550\pi$
$968$	9711.73	0.322466
$969$	7499.89	0.248639
$970$	0	0
$971$	44316.5	1.46466	0.732330	−	0.680950i	$-0.238433\pi$
$971$	44316.5	1.46466	0.732330	+	0.680950i	$0.238433\pi$
$972$	15023.7	0.495768
$973$	−41959.1	−1.38247
$974$	34142.4	1.12320
$975$	0	0
$976$	1769.26	0.0580252
$977$	−761.665	−0.0249415	−0.0124707	−	0.999922i	$-0.503970\pi$
$977$	−761.665	−0.0249415	−0.0124707	+	0.999922i	$0.503970\pi$
$978$	8237.13	0.269319
$979$	9516.49	0.310673
$980$	0	0
$981$	19830.3	0.645394
$982$	−33959.8	−1.10356
$983$	17423.3	0.565328	0.282664	−	0.959219i	$-0.408782\pi$
$983$	17423.3	0.565328	0.282664	+	0.959219i	$0.408782\pi$
$984$	7716.10	0.249980
$985$	0	0
$986$	55808.9	1.80255
$987$	−9992.06	−0.322240
$988$	2858.70	0.0920521
$989$	−1516.21	−0.0487489
$990$	0	0
$991$	−33873.1	−1.08579	−0.542894	−	0.839801i	$-0.682671\pi$
$991$	−33873.1	−1.08579	−0.542894	+	0.839801i	$0.682671\pi$
$992$	−5290.22	−0.169319
$993$	9375.71	0.299626
$994$	7960.55	0.254017
$995$	0	0
$996$	10753.4	0.342103
$997$	31163.3	0.989921	0.494960	−	0.868915i	$-0.335183\pi$
$997$	31163.3	0.989921	0.494960	+	0.868915i	$0.335183\pi$
$998$	5577.66	0.176912
$999$	28604.9	0.905925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.o.1.3		4
5.2	odd	4		1150.4.b.m.599.2		8
5.3	odd	4		1150.4.b.m.599.7		8
5.4	even	2		230.4.a.i.1.2	✓	4
15.14	odd	2		2070.4.a.bi.1.4		4
20.19	odd	2		1840.4.a.l.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.2	✓	4	5.4	even	2
1150.4.a.o.1.3		4	1.1	even	1	trivial
1150.4.b.m.599.2		8	5.2	odd	4
1150.4.b.m.599.7		8	5.3	odd	4
1840.4.a.l.1.3		4	20.19	odd	2
2070.4.a.bi.1.4		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.26018 q^{3} +4.00000 q^{4} -6.52037 q^{6} -27.7921 q^{7} -8.00000 q^{8} -16.3712 q^{9} -10.8182 q^{11} +13.0407 q^{12} -36.9683 q^{13} +55.5842 q^{14} +16.0000 q^{16} -118.996 q^{17} +32.7424 q^{18} -19.3321 q^{19} -90.6073 q^{21} +21.6365 q^{22} -23.0000 q^{23} -26.0815 q^{24} +73.9366 q^{26} -141.398 q^{27} -111.168 q^{28} +234.499 q^{29} +165.319 q^{31} -32.0000 q^{32} -35.2694 q^{33} +237.992 q^{34} -65.4848 q^{36} -202.301 q^{37} +38.6643 q^{38} -120.523 q^{39} -295.846 q^{41} +181.215 q^{42} +65.9221 q^{43} -43.2729 q^{44} +46.0000 q^{46} +110.279 q^{47} +52.1630 q^{48} +429.400 q^{49} -387.949 q^{51} -147.873 q^{52} +688.135 q^{53} +282.796 q^{54} +222.337 q^{56} -63.0263 q^{57} -468.998 q^{58} +10.5847 q^{59} +110.579 q^{61} -330.639 q^{62} +454.990 q^{63} +64.0000 q^{64} +70.5388 q^{66} +643.290 q^{67} -475.984 q^{68} -74.9842 q^{69} +143.216 q^{71} +130.970 q^{72} +158.213 q^{73} +404.601 q^{74} -77.3285 q^{76} +300.661 q^{77} +241.047 q^{78} +1123.34 q^{79} -18.9616 q^{81} +591.692 q^{82} +824.600 q^{83} -362.429 q^{84} -131.844 q^{86} +764.510 q^{87} +86.5458 q^{88} -879.672 q^{89} +1027.43 q^{91} -92.0000 q^{92} +538.971 q^{93} -220.557 q^{94} -104.326 q^{96} +938.437 q^{97} -858.801 q^{98} +177.107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} - 26 q^{7} - 32 q^{8} + 64 q^{9} + 93 q^{11} - 16 q^{12} - 32 q^{13} + 52 q^{14} + 64 q^{16} - 108 q^{17} - 128 q^{18} + 185 q^{19} + 302 q^{21} - 186 q^{22}+ \cdots + 2013 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 - 26 * q^7 - 32 * q^8 + 64 * q^9 + 93 * q^11 - 16 * q^12 - 32 * q^13 + 52 * q^14 + 64 * q^16 - 108 * q^17 - 128 * q^18 + 185 * q^19 + 302 * q^21 - 186 * q^22 - 92 * q^23 + 32 * q^24 + 64 * q^26 - 259 * q^27 - 104 * q^28 + 294 * q^29 - 211 * q^31 - 128 * q^32 + 237 * q^33 + 216 * q^34 + 256 * q^36 - 5 * q^37 - 370 * q^38 - 421 * q^39 - 369 * q^41 - 604 * q^42 + 100 * q^43 + 372 * q^44 + 184 * q^46 + 363 * q^47 - 64 * q^48 + 600 * q^49 - 315 * q^51 - 128 * q^52 - 21 * q^53 + 518 * q^54 + 208 * q^56 + 160 * q^57 - 588 * q^58 - 33 * q^59 - 307 * q^61 + 422 * q^62 - 167 * q^63 + 256 * q^64 - 474 * q^66 - 725 * q^67 - 432 * q^68 + 92 * q^69 - 1257 * q^71 - 512 * q^72 - 509 * q^73 + 10 * q^74 + 740 * q^76 + 1962 * q^77 + 842 * q^78 + 1202 * q^79 - 992 * q^81 + 738 * q^82 + 1377 * q^83 + 1208 * q^84 - 200 * q^86 + 2829 * q^87 - 744 * q^88 - 984 * q^89 - 995 * q^91 - 368 * q^92 + 1843 * q^93 - 726 * q^94 + 128 * q^96 - 137 * q^97 - 1200 * q^98 + 2013 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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