LMFDB - Embedded newform 2450.4.a.cd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2450,4,Mod(1,2450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2450.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-6,3,12,0,-6,0,-24,56,0,-26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$144.554679514$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.238585.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 67x - 189 $ x^3 - 67*x - 189
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 350)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.92433$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +6.92433 q^{3} +4.00000 q^{4} -13.8487 q^{6} -8.00000 q^{8} +20.9463 q^{9} +37.3629 q^{11} +27.6973 q^{12} -89.1041 q^{13} +16.0000 q^{16} -70.1578 q^{17} -41.8926 q^{18} +125.401 q^{19} -74.7257 q^{22} +6.26857 q^{23} -55.3946 q^{24} +178.208 q^{26} -41.9179 q^{27} -95.4760 q^{29} +217.763 q^{31} -32.0000 q^{32} +258.713 q^{33} +140.316 q^{34} +83.7851 q^{36} -210.826 q^{37} -250.802 q^{38} -616.986 q^{39} +12.2490 q^{41} +131.746 q^{43} +149.451 q^{44} -12.5371 q^{46} +364.121 q^{47} +110.789 q^{48} -485.795 q^{51} -356.416 q^{52} +498.456 q^{53} +83.8358 q^{54} +868.318 q^{57} +190.952 q^{58} +358.027 q^{59} +154.608 q^{61} -435.526 q^{62} +64.0000 q^{64} -517.425 q^{66} +67.5326 q^{67} -280.631 q^{68} +43.4056 q^{69} -517.772 q^{71} -167.570 q^{72} -674.766 q^{73} +421.651 q^{74} +501.604 q^{76} +1233.97 q^{78} +931.682 q^{79} -855.803 q^{81} -24.4980 q^{82} +1435.72 q^{83} -263.493 q^{86} -661.107 q^{87} -298.903 q^{88} +336.681 q^{89} +25.0743 q^{92} +1507.86 q^{93} -728.242 q^{94} -221.578 q^{96} +179.620 q^{97} +782.613 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 3 q^{3} + 12 q^{4} - 6 q^{6} - 24 q^{8} + 56 q^{9} - 26 q^{11} + 12 q^{12} - 80 q^{13} + 48 q^{16} - 30 q^{17} - 112 q^{18} + 18 q^{19} + 52 q^{22} + 53 q^{23} - 24 q^{24} + 160 q^{26} - 324 q^{27}+ \cdots - 57 q^{99}+O(q^{100}) $ 3 * q - 6 * q^2 + 3 * q^3 + 12 * q^4 - 6 * q^6 - 24 * q^8 + 56 * q^9 - 26 * q^11 + 12 * q^12 - 80 * q^13 + 48 * q^16 - 30 * q^17 - 112 * q^18 + 18 * q^19 + 52 * q^22 + 53 * q^23 - 24 * q^24 + 160 * q^26 - 324 * q^27 - 404 * q^29 + 615 * q^31 - 96 * q^32 + 314 * q^33 + 60 * q^34 + 224 * q^36 - 426 * q^37 - 36 * q^38 - 90 * q^39 + 101 * q^41 + 249 * q^43 - 104 * q^44 - 106 * q^46 - 402 * q^47 + 48 * q^48 - 339 * q^51 - 320 * q^52 - 390 * q^53 + 648 * q^54 + 1667 * q^57 + 808 * q^58 + 91 * q^59 + 647 * q^61 - 1230 * q^62 + 192 * q^64 - 628 * q^66 - 708 * q^67 - 120 * q^68 + 1548 * q^69 + 533 * q^71 - 448 * q^72 - 772 * q^73 + 852 * q^74 + 72 * q^76 + 180 * q^78 + 1421 * q^79 - 1393 * q^81 - 202 * q^82 + 520 * q^83 - 498 * q^86 + 833 * q^87 + 208 * q^88 + 194 * q^89 + 212 * q^92 - 148 * q^93 + 804 * q^94 - 96 * q^96 + 1027 * q^97 - 57 * 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
$3$	6.92433	1.33259	0.666294	−	0.745690i	$-0.267880\pi$
$3$	6.92433	1.33259	0.666294	+	0.745690i	$0.267880\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−13.8487	−0.942281
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	20.9463	0.775788
$10$	0	0
$11$	37.3629	1.02412	0.512060	−	0.858950i	$-0.328882\pi$
$11$	37.3629	1.02412	0.512060	+	0.858950i	$0.328882\pi$
$12$	27.6973	0.666294
$13$	−89.1041	−1.90100	−0.950501	−	0.310722i	$-0.899429\pi$
$13$	−89.1041	−1.90100	−0.950501	+	0.310722i	$0.899429\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−70.1578	−1.00093	−0.500464	−	0.865758i	$-0.666837\pi$
$17$	−70.1578	−1.00093	−0.500464	+	0.865758i	$0.666837\pi$
$18$	−41.8926	−0.548565
$19$	125.401	1.51416	0.757078	−	0.653324i	$-0.226626\pi$
$19$	125.401	1.51416	0.757078	+	0.653324i	$0.226626\pi$
$20$	0	0
$21$	0	0
$22$	−74.7257	−0.724162
$23$	6.26857	0.0568298	0.0284149	−	0.999596i	$-0.490954\pi$
$23$	6.26857	0.0568298	0.0284149	+	0.999596i	$0.490954\pi$
$24$	−55.3946	−0.471141
$25$	0	0
$26$	178.208	1.34421
$27$	−41.9179	−0.298781
$28$	0	0
$29$	−95.4760	−0.611360	−0.305680	−	0.952134i	$-0.598884\pi$
$29$	−95.4760	−0.611360	−0.305680	+	0.952134i	$0.598884\pi$
$30$	0	0
$31$	217.763	1.26166	0.630829	−	0.775922i	$-0.282715\pi$
$31$	217.763	1.26166	0.630829	+	0.775922i	$0.282715\pi$
$32$	−32.0000	−0.176777
$33$	258.713	1.36473
$34$	140.316	0.707762
$35$	0	0
$36$	83.7851	0.387894
$37$	−210.826	−0.936744	−0.468372	−	0.883531i	$-0.655159\pi$
$37$	−210.826	−0.936744	−0.468372	+	0.883531i	$0.655159\pi$
$38$	−250.802	−1.07067
$39$	−616.986	−2.53325
$40$	0	0
$41$	12.2490	0.0466578	0.0233289	−	0.999728i	$-0.492574\pi$
$41$	12.2490	0.0466578	0.0233289	+	0.999728i	$0.492574\pi$
$42$	0	0
$43$	131.746	0.467235	0.233618	−	0.972329i	$-0.424944\pi$
$43$	131.746	0.467235	0.233618	+	0.972329i	$0.424944\pi$
$44$	149.451	0.512060
$45$	0	0
$46$	−12.5371	−0.0401848
$47$	364.121	1.13005	0.565026	−	0.825073i	$-0.308866\pi$
$47$	364.121	1.13005	0.565026	+	0.825073i	$0.308866\pi$
$48$	110.789	0.333147
$49$	0	0
$50$	0	0
$51$	−485.795	−1.33382
$52$	−356.416	−0.950501
$53$	498.456	1.29185	0.645926	−	0.763400i	$-0.276471\pi$
$53$	498.456	1.29185	0.645926	+	0.763400i	$0.276471\pi$
$54$	83.8358	0.211270
$55$	0	0
$56$	0	0
$57$	868.318	2.01775
$58$	190.952	0.432297
$59$	358.027	0.790020	0.395010	−	0.918677i	$-0.370741\pi$
$59$	358.027	0.790020	0.395010	+	0.918677i	$0.370741\pi$
$60$	0	0
$61$	154.608	0.324516	0.162258	−	0.986748i	$-0.448122\pi$
$61$	154.608	0.324516	0.162258	+	0.986748i	$0.448122\pi$
$62$	−435.526	−0.892127
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−517.425	−0.965010
$67$	67.5326	0.123141	0.0615703	−	0.998103i	$-0.480389\pi$
$67$	67.5326	0.123141	0.0615703	+	0.998103i	$0.480389\pi$
$68$	−280.631	−0.500464
$69$	43.4056	0.0757307
$70$	0	0
$71$	−517.772	−0.865469	−0.432734	−	0.901521i	$-0.642451\pi$
$71$	−517.772	−0.865469	−0.432734	+	0.901521i	$0.642451\pi$
$72$	−167.570	−0.274283
$73$	−674.766	−1.08186	−0.540928	−	0.841069i	$-0.681927\pi$
$73$	−674.766	−1.08186	−0.540928	+	0.841069i	$0.681927\pi$
$74$	421.651	0.662378
$75$	0	0
$76$	501.604	0.757078
$77$	0	0
$78$	1233.97	1.79128
$79$	931.682	1.32687	0.663433	−	0.748236i	$-0.269099\pi$
$79$	931.682	1.32687	0.663433	+	0.748236i	$0.269099\pi$
$80$	0	0
$81$	−855.803	−1.17394
$82$	−24.4980	−0.0329920
$83$	1435.72	1.89868	0.949341	−	0.314247i	$-0.101752\pi$
$83$	1435.72	1.89868	0.949341	+	0.314247i	$0.101752\pi$
$84$	0	0
$85$	0	0
$86$	−263.493	−0.330385
$87$	−661.107	−0.814691
$88$	−298.903	−0.362081
$89$	336.681	0.400990	0.200495	−	0.979695i	$-0.435745\pi$
$89$	336.681	0.400990	0.200495	+	0.979695i	$0.435745\pi$
$90$	0	0
$91$	0	0
$92$	25.0743	0.0284149
$93$	1507.86	1.68127
$94$	−728.242	−0.799068
$95$	0	0
$96$	−221.578	−0.235570
$97$	179.620	0.188016	0.0940082	−	0.995571i	$-0.470032\pi$
$97$	179.620	0.188016	0.0940082	+	0.995571i	$0.470032\pi$
$98$	0	0
$99$	782.613	0.794501
$100$	0	0
$101$	894.392	0.881142	0.440571	−	0.897718i	$-0.354776\pi$
$101$	894.392	0.881142	0.440571	+	0.897718i	$0.354776\pi$
$102$	971.591	0.943155
$103$	24.8790	0.0238000	0.0119000	−	0.999929i	$-0.496212\pi$
$103$	24.8790	0.0238000	0.0119000	+	0.999929i	$0.496212\pi$
$104$	712.833	0.672106
$105$	0	0
$106$	−996.912	−0.913478
$107$	−1153.79	−1.04244	−0.521221	−	0.853421i	$-0.674523\pi$
$107$	−1153.79	−1.04244	−0.521221	+	0.853421i	$0.674523\pi$
$108$	−167.672	−0.149391
$109$	1880.97	1.65288	0.826440	−	0.563025i	$-0.190363\pi$
$109$	1880.97	1.65288	0.826440	+	0.563025i	$0.190363\pi$
$110$	0	0
$111$	−1459.83	−1.24829
$112$	0	0
$113$	2073.98	1.72658	0.863289	−	0.504709i	$-0.168400\pi$
$113$	2073.98	1.72658	0.863289	+	0.504709i	$0.168400\pi$
$114$	−1736.64	−1.42676
$115$	0	0
$116$	−381.904	−0.305680
$117$	−1866.40	−1.47478
$118$	−716.054	−0.558628
$119$	0	0
$120$	0	0
$121$	64.9828	0.0488226
$122$	−309.216	−0.229468
$123$	84.8160	0.0621756
$124$	871.053	0.630829
$125$	0	0
$126$	0	0
$127$	−75.3532	−0.0526497	−0.0263249	−	0.999653i	$-0.508380\pi$
$127$	−75.3532	−0.0526497	−0.0263249	+	0.999653i	$0.508380\pi$
$128$	−128.000	−0.0883883
$129$	912.254	0.622632
$130$	0	0
$131$	−2200.18	−1.46741	−0.733705	−	0.679468i	$-0.762211\pi$
$131$	−2200.18	−1.46741	−0.733705	+	0.679468i	$0.762211\pi$
$132$	1034.85	0.682365
$133$	0	0
$134$	−135.065	−0.0870735
$135$	0	0
$136$	561.262	0.353881
$137$	−2055.50	−1.28185	−0.640925	−	0.767604i	$-0.721449\pi$
$137$	−2055.50	−1.28185	−0.640925	+	0.767604i	$0.721449\pi$
$138$	−86.8112	−0.0535497
$139$	453.942	0.276999	0.138499	−	0.990363i	$-0.455772\pi$
$139$	453.942	0.276999	0.138499	+	0.990363i	$0.455772\pi$
$140$	0	0
$141$	2521.29	1.50589
$142$	1035.54	0.611979
$143$	−3329.18	−1.94685
$144$	335.141	0.193947
$145$	0	0
$146$	1349.53	0.764987
$147$	0	0
$148$	−843.303	−0.468372
$149$	234.696	0.129041	0.0645204	−	0.997916i	$-0.479448\pi$
$149$	234.696	0.129041	0.0645204	+	0.997916i	$0.479448\pi$
$150$	0	0
$151$	346.020	0.186481	0.0932407	−	0.995644i	$-0.470277\pi$
$151$	346.020	0.186481	0.0932407	+	0.995644i	$0.470277\pi$
$152$	−1003.21	−0.535335
$153$	−1469.55	−0.776508
$154$	0	0
$155$	0	0
$156$	−2467.94	−1.26663
$157$	3339.12	1.69739	0.848697	−	0.528879i	$-0.177387\pi$
$157$	3339.12	1.69739	0.848697	+	0.528879i	$0.177387\pi$
$158$	−1863.36	−0.938236
$159$	3451.47	1.72151
$160$	0	0
$161$	0	0
$162$	1711.61	0.830101
$163$	2411.64	1.15886	0.579430	−	0.815022i	$-0.303275\pi$
$163$	2411.64	1.15886	0.579430	+	0.815022i	$0.303275\pi$
$164$	48.9959	0.0233289
$165$	0	0
$166$	−2871.44	−1.34257
$167$	1296.58	0.600791	0.300395	−	0.953815i	$-0.402881\pi$
$167$	1296.58	0.600791	0.300395	+	0.953815i	$0.402881\pi$
$168$	0	0
$169$	5742.54	2.61381
$170$	0	0
$171$	2626.69	1.17467
$172$	526.985	0.233618
$173$	2744.51	1.20613	0.603067	−	0.797690i	$-0.293945\pi$
$173$	2744.51	1.20613	0.603067	+	0.797690i	$0.293945\pi$
$174$	1322.21	0.576073
$175$	0	0
$176$	597.806	0.256030
$177$	2479.10	1.05277
$178$	−673.363	−0.283543
$179$	−1405.44	−0.586859	−0.293429	−	0.955981i	$-0.594797\pi$
$179$	−1405.44	−0.586859	−0.293429	+	0.955981i	$0.594797\pi$
$180$	0	0
$181$	637.919	0.261967	0.130984	−	0.991385i	$-0.458186\pi$
$181$	637.919	0.261967	0.130984	+	0.991385i	$0.458186\pi$
$182$	0	0
$183$	1070.55	0.432446
$184$	−50.1485	−0.0200924
$185$	0	0
$186$	−3015.73	−1.18884
$187$	−2621.30	−1.02507
$188$	1456.48	0.565026
$189$	0	0
$190$	0	0
$191$	3655.76	1.38493	0.692465	−	0.721452i	$-0.256525\pi$
$191$	3655.76	1.38493	0.692465	+	0.721452i	$0.256525\pi$
$192$	443.157	0.166573
$193$	−4433.66	−1.65358	−0.826792	−	0.562508i	$-0.809836\pi$
$193$	−4433.66	−1.65358	−0.826792	+	0.562508i	$0.809836\pi$
$194$	−359.239	−0.132948
$195$	0	0
$196$	0	0
$197$	5283.75	1.91092	0.955462	−	0.295115i	$-0.0953580\pi$
$197$	5283.75	1.91092	0.955462	+	0.295115i	$0.0953580\pi$
$198$	−1565.23	−0.561797
$199$	4404.91	1.56912	0.784562	−	0.620051i	$-0.212888\pi$
$199$	4404.91	1.56912	0.784562	+	0.620051i	$0.212888\pi$
$200$	0	0
$201$	467.617	0.164095
$202$	−1788.78	−0.623062
$203$	0	0
$204$	−1943.18	−0.666911
$205$	0	0
$206$	−49.7580	−0.0168292
$207$	131.303	0.0440879
$208$	−1425.67	−0.475250
$209$	4685.34	1.55068
$210$	0	0
$211$	−1667.99	−0.544213	−0.272107	−	0.962267i	$-0.587720\pi$
$211$	−1667.99	−0.544213	−0.272107	+	0.962267i	$0.587720\pi$
$212$	1993.82	0.645926
$213$	−3585.23	−1.15331
$214$	2307.59	0.737118
$215$	0	0
$216$	335.343	0.105635
$217$	0	0
$218$	−3761.93	−1.16876
$219$	−4672.30	−1.44167
$220$	0	0
$221$	6251.35	1.90276
$222$	2919.65	0.882677
$223$	−3706.93	−1.11316	−0.556579	−	0.830795i	$-0.687886\pi$
$223$	−3706.93	−1.11316	−0.556579	+	0.830795i	$0.687886\pi$
$224$	0	0
$225$	0	0
$226$	−4147.96	−1.22088
$227$	−4759.43	−1.39160	−0.695802	−	0.718233i	$-0.744951\pi$
$227$	−4759.43	−1.39160	−0.695802	+	0.718233i	$0.744951\pi$
$228$	3473.27	1.00887
$229$	609.382	0.175848	0.0879238	−	0.996127i	$-0.471977\pi$
$229$	609.382	0.175848	0.0879238	+	0.996127i	$0.471977\pi$
$230$	0	0
$231$	0	0
$232$	763.808	0.216149
$233$	3163.96	0.889604	0.444802	−	0.895629i	$-0.353274\pi$
$233$	3163.96	0.889604	0.444802	+	0.895629i	$0.353274\pi$
$234$	3732.80	1.04282
$235$	0	0
$236$	1432.11	0.395010
$237$	6451.27	1.76817
$238$	0	0
$239$	−5180.13	−1.40199	−0.700993	−	0.713168i	$-0.747259\pi$
$239$	−5180.13	−1.40199	−0.700993	+	0.713168i	$0.747259\pi$
$240$	0	0
$241$	−2054.31	−0.549087	−0.274544	−	0.961575i	$-0.588527\pi$
$241$	−2054.31	−0.549087	−0.274544	+	0.961575i	$0.588527\pi$
$242$	−129.966	−0.0345228
$243$	−4794.07	−1.26560
$244$	618.431	0.162258
$245$	0	0
$246$	−169.632	−0.0439648
$247$	−11173.7	−2.87841
$248$	−1742.11	−0.446064
$249$	9941.39	2.53016
$250$	0	0
$251$	−2455.16	−0.617403	−0.308701	−	0.951159i	$-0.599894\pi$
$251$	−2455.16	−0.617403	−0.308701	+	0.951159i	$0.599894\pi$
$252$	0	0
$253$	234.212	0.0582006
$254$	150.706	0.0372290
$255$	0	0
$256$	256.000	0.0625000
$257$	−3690.10	−0.895650	−0.447825	−	0.894121i	$-0.647801\pi$
$257$	−3690.10	−0.895650	−0.447825	+	0.894121i	$0.647801\pi$
$258$	−1824.51	−0.440267
$259$	0	0
$260$	0	0
$261$	−1999.87	−0.474286
$262$	4400.36	1.03762
$263$	5215.13	1.22273	0.611367	−	0.791347i	$-0.290620\pi$
$263$	5215.13	1.22273	0.611367	+	0.791347i	$0.290620\pi$
$264$	−2069.70	−0.482505
$265$	0	0
$266$	0	0
$267$	2331.29	0.534355
$268$	270.130	0.0615703
$269$	3634.39	0.823764	0.411882	−	0.911237i	$-0.364872\pi$
$269$	3634.39	0.823764	0.411882	+	0.911237i	$0.364872\pi$
$270$	0	0
$271$	202.835	0.0454663	0.0227331	−	0.999742i	$-0.492763\pi$
$271$	202.835	0.0454663	0.0227331	+	0.999742i	$0.492763\pi$
$272$	−1122.52	−0.250232
$273$	0	0
$274$	4111.01	0.906405
$275$	0	0
$276$	173.622	0.0378654
$277$	−763.996	−0.165719	−0.0828593	−	0.996561i	$-0.526405\pi$
$277$	−763.996	−0.165719	−0.0828593	+	0.996561i	$0.526405\pi$
$278$	−907.883	−0.195868
$279$	4561.33	0.978780
$280$	0	0
$281$	1451.40	0.308125	0.154063	−	0.988061i	$-0.450764\pi$
$281$	1451.40	0.308125	0.154063	+	0.988061i	$0.450764\pi$
$282$	−5042.58	−1.06483
$283$	7659.78	1.60893	0.804464	−	0.594002i	$-0.202453\pi$
$283$	7659.78	1.60893	0.804464	+	0.594002i	$0.202453\pi$
$284$	−2071.09	−0.432734
$285$	0	0
$286$	6658.37	1.37663
$287$	0	0
$288$	−670.281	−0.137141
$289$	9.11573	0.00185543
$290$	0	0
$291$	1243.74	0.250548
$292$	−2699.07	−0.540928
$293$	−4222.68	−0.841952	−0.420976	−	0.907072i	$-0.638312\pi$
$293$	−4222.68	−0.841952	−0.420976	+	0.907072i	$0.638312\pi$
$294$	0	0
$295$	0	0
$296$	1686.61	0.331189
$297$	−1566.17	−0.305988
$298$	−469.393	−0.0912456
$299$	−558.555	−0.108034
$300$	0	0
$301$	0	0
$302$	−692.040	−0.131862
$303$	6193.06	1.17420
$304$	2006.42	0.378539
$305$	0	0
$306$	2939.09	0.549074
$307$	−1243.11	−0.231101	−0.115551	−	0.993302i	$-0.536863\pi$
$307$	−1243.11	−0.231101	−0.115551	+	0.993302i	$0.536863\pi$
$308$	0	0
$309$	172.270	0.0317156
$310$	0	0
$311$	1368.69	0.249554	0.124777	−	0.992185i	$-0.460179\pi$
$311$	1368.69	0.249554	0.124777	+	0.992185i	$0.460179\pi$
$312$	4935.89	0.895639
$313$	−3210.66	−0.579800	−0.289900	−	0.957057i	$-0.593622\pi$
$313$	−3210.66	−0.579800	−0.289900	+	0.957057i	$0.593622\pi$
$314$	−6678.24	−1.20024
$315$	0	0
$316$	3726.73	0.663433
$317$	−7649.10	−1.35526	−0.677628	−	0.735405i	$-0.736992\pi$
$317$	−7649.10	−1.35526	−0.677628	+	0.735405i	$0.736992\pi$
$318$	−6902.94	−1.21729
$319$	−3567.26	−0.626107
$320$	0	0
$321$	−7989.24	−1.38915
$322$	0	0
$323$	−8797.86	−1.51556
$324$	−3423.21	−0.586970
$325$	0	0
$326$	−4823.28	−0.819437
$327$	13024.4	2.20261
$328$	−97.9919	−0.0164960
$329$	0	0
$330$	0	0
$331$	−7917.29	−1.31472	−0.657362	−	0.753575i	$-0.728328\pi$
$331$	−7917.29	−1.31472	−0.657362	+	0.753575i	$0.728328\pi$
$332$	5742.88	0.949341
$333$	−4416.02	−0.726715
$334$	−2593.15	−0.424823
$335$	0	0
$336$	0	0
$337$	−6878.10	−1.11179	−0.555896	−	0.831252i	$-0.687625\pi$
$337$	−6878.10	−1.11179	−0.555896	+	0.831252i	$0.687625\pi$
$338$	−11485.1	−1.84824
$339$	14360.9	2.30082
$340$	0	0
$341$	8136.25	1.29209
$342$	−5253.37	−0.830614
$343$	0	0
$344$	−1053.97	−0.165193
$345$	0	0
$346$	−5489.02	−0.852866
$347$	1421.66	0.219939	0.109969	−	0.993935i	$-0.464925\pi$
$347$	1421.66	0.219939	0.109969	+	0.993935i	$0.464925\pi$
$348$	−2644.43	−0.407345
$349$	−2837.74	−0.435246	−0.217623	−	0.976033i	$-0.569830\pi$
$349$	−2837.74	−0.435246	−0.217623	+	0.976033i	$0.569830\pi$
$350$	0	0
$351$	3735.05	0.567984
$352$	−1195.61	−0.181041
$353$	−3578.81	−0.539606	−0.269803	−	0.962916i	$-0.586959\pi$
$353$	−3578.81	−0.539606	−0.269803	+	0.962916i	$0.586959\pi$
$354$	−4958.19	−0.744421
$355$	0	0
$356$	1346.73	0.200495
$357$	0	0
$358$	2810.88	0.414972
$359$	−6121.48	−0.899942	−0.449971	−	0.893043i	$-0.648566\pi$
$359$	−6121.48	−0.899942	−0.449971	+	0.893043i	$0.648566\pi$
$360$	0	0
$361$	8866.42	1.29267
$362$	−1275.84	−0.185239
$363$	449.962	0.0650603
$364$	0	0
$365$	0	0
$366$	−2141.11	−0.305786
$367$	12186.8	1.73337	0.866686	−	0.498855i	$-0.166246\pi$
$367$	12186.8	1.73337	0.866686	+	0.498855i	$0.166246\pi$
$368$	100.297	0.0142075
$369$	256.571	0.0361966
$370$	0	0
$371$	0	0
$372$	6031.45	0.840635
$373$	−10446.9	−1.45019	−0.725094	−	0.688650i	$-0.758204\pi$
$373$	−10446.9	−1.45019	−0.725094	+	0.688650i	$0.758204\pi$
$374$	5242.59	0.724834
$375$	0	0
$376$	−2912.97	−0.399534
$377$	8507.30	1.16220
$378$	0	0
$379$	−10274.0	−1.39246	−0.696229	−	0.717820i	$-0.745140\pi$
$379$	−10274.0	−1.39246	−0.696229	+	0.717820i	$0.745140\pi$
$380$	0	0
$381$	−521.770	−0.0701604
$382$	−7311.52	−0.979293
$383$	6487.68	0.865548	0.432774	−	0.901502i	$-0.357535\pi$
$383$	6487.68	0.865548	0.432774	+	0.901502i	$0.357535\pi$
$384$	−886.314	−0.117785
$385$	0	0
$386$	8867.31	1.16926
$387$	2759.59	0.362476
$388$	718.478	0.0940082
$389$	−3706.33	−0.483080	−0.241540	−	0.970391i	$-0.577653\pi$
$389$	−3706.33	−0.483080	−0.241540	+	0.970391i	$0.577653\pi$
$390$	0	0
$391$	−439.789	−0.0568825
$392$	0	0
$393$	−15234.8	−1.95545
$394$	−10567.5	−1.35123
$395$	0	0
$396$	3130.45	0.397250
$397$	9280.26	1.17321	0.586603	−	0.809874i	$-0.300465\pi$
$397$	9280.26	1.17321	0.586603	+	0.809874i	$0.300465\pi$
$398$	−8809.81	−1.10954
$399$	0	0
$400$	0	0
$401$	−2419.08	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$401$	−2419.08	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$402$	−935.235	−0.116033
$403$	−19403.6	−2.39842
$404$	3577.57	0.440571
$405$	0	0
$406$	0	0
$407$	−7877.05	−0.959339
$408$	3886.36	0.471578
$409$	4995.99	0.604000	0.302000	−	0.953308i	$-0.402346\pi$
$409$	4995.99	0.604000	0.302000	+	0.953308i	$0.402346\pi$
$410$	0	0
$411$	−14233.0	−1.70818
$412$	99.5161	0.0119000
$413$	0	0
$414$	−262.606	−0.0311749
$415$	0	0
$416$	2851.33	0.336053
$417$	3143.24	0.369125
$418$	−9370.68	−1.09650
$419$	10746.6	1.25300	0.626500	−	0.779421i	$-0.284487\pi$
$419$	10746.6	1.25300	0.626500	+	0.779421i	$0.284487\pi$
$420$	0	0
$421$	8561.10	0.991075	0.495538	−	0.868587i	$-0.334971\pi$
$421$	8561.10	0.991075	0.495538	+	0.868587i	$0.334971\pi$
$422$	3335.97	0.384817
$423$	7626.98	0.876682
$424$	−3987.65	−0.456739
$425$	0	0
$426$	7170.45	0.815515
$427$	0	0
$428$	−4615.17	−0.521221
$429$	−23052.3	−2.59435
$430$	0	0
$431$	−5235.91	−0.585162	−0.292581	−	0.956241i	$-0.594514\pi$
$431$	−5235.91	−0.585162	−0.292581	+	0.956241i	$0.594514\pi$
$432$	−670.686	−0.0746954
$433$	−3809.88	−0.422844	−0.211422	−	0.977395i	$-0.567809\pi$
$433$	−3809.88	−0.422844	−0.211422	+	0.977395i	$0.567809\pi$
$434$	0	0
$435$	0	0
$436$	7523.86	0.826440
$437$	786.085	0.0860493
$438$	9344.60	1.01941
$439$	12532.8	1.36254	0.681272	−	0.732030i	$-0.261427\pi$
$439$	12532.8	1.36254	0.681272	+	0.732030i	$0.261427\pi$
$440$	0	0
$441$	0	0
$442$	−12502.7	−1.34546
$443$	17794.4	1.90844	0.954218	−	0.299111i	$-0.0966902\pi$
$443$	17794.4	1.90844	0.954218	+	0.299111i	$0.0966902\pi$
$444$	−5839.30	−0.624147
$445$	0	0
$446$	7413.86	0.787122
$447$	1625.11	0.171958
$448$	0	0
$449$	12203.4	1.28266	0.641331	−	0.767265i	$-0.278383\pi$
$449$	12203.4	1.28266	0.641331	+	0.767265i	$0.278383\pi$
$450$	0	0
$451$	457.657	0.0477832
$452$	8295.91	0.863289
$453$	2395.95	0.248503
$454$	9518.85	0.984013
$455$	0	0
$456$	−6946.54	−0.713381
$457$	4749.53	0.486157	0.243079	−	0.970007i	$-0.421843\pi$
$457$	4749.53	0.486157	0.243079	+	0.970007i	$0.421843\pi$
$458$	−1218.76	−0.124343
$459$	2940.87	0.299059
$460$	0	0
$461$	13208.0	1.33440	0.667199	−	0.744880i	$-0.267493\pi$
$461$	13208.0	1.33440	0.667199	+	0.744880i	$0.267493\pi$
$462$	0	0
$463$	11987.9	1.20329	0.601645	−	0.798763i	$-0.294512\pi$
$463$	11987.9	1.20329	0.601645	+	0.798763i	$0.294512\pi$
$464$	−1527.62	−0.152840
$465$	0	0
$466$	−6327.91	−0.629045
$467$	−5181.23	−0.513402	−0.256701	−	0.966491i	$-0.582635\pi$
$467$	−5181.23	−0.513402	−0.256701	+	0.966491i	$0.582635\pi$
$468$	−7465.60	−0.737388
$469$	0	0
$470$	0	0
$471$	23121.2	2.26193
$472$	−2864.22	−0.279314
$473$	4922.42	0.478505
$474$	−12902.5	−1.25028
$475$	0	0
$476$	0	0
$477$	10440.8	1.00220
$478$	10360.3	0.991353
$479$	133.998	0.0127819	0.00639096	−	0.999980i	$-0.497966\pi$
$479$	133.998	0.0127819	0.00639096	+	0.999980i	$0.497966\pi$
$480$	0	0
$481$	18785.4	1.78075
$482$	4108.63	0.388263
$483$	0	0
$484$	259.931	0.0244113
$485$	0	0
$486$	9588.15	0.894912
$487$	−3318.36	−0.308766	−0.154383	−	0.988011i	$-0.549339\pi$
$487$	−3318.36	−0.308766	−0.154383	+	0.988011i	$0.549339\pi$
$488$	−1236.86	−0.114734
$489$	16699.0	1.54428
$490$	0	0
$491$	−2195.90	−0.201832	−0.100916	−	0.994895i	$-0.532177\pi$
$491$	−2195.90	−0.201832	−0.100916	+	0.994895i	$0.532177\pi$
$492$	339.264	0.0310878
$493$	6698.39	0.611927
$494$	22347.5	2.03535
$495$	0	0
$496$	3484.21	0.315415
$497$	0	0
$498$	−19882.8	−1.78909
$499$	11689.6	1.04870	0.524348	−	0.851504i	$-0.324309\pi$
$499$	11689.6	1.04870	0.524348	+	0.851504i	$0.324309\pi$
$500$	0	0
$501$	8977.91	0.800606
$502$	4910.31	0.436570
$503$	9289.98	0.823498	0.411749	−	0.911297i	$-0.364918\pi$
$503$	9289.98	0.823498	0.411749	+	0.911297i	$0.364918\pi$
$504$	0	0
$505$	0	0
$506$	−468.423	−0.0411540
$507$	39763.2	3.48313
$508$	−301.413	−0.0263249
$509$	6690.62	0.582626	0.291313	−	0.956628i	$-0.405908\pi$
$509$	6690.62	0.582626	0.291313	+	0.956628i	$0.405908\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−5256.55	−0.452402
$514$	7380.19	0.633320
$515$	0	0
$516$	3649.02	0.311316
$517$	13604.6	1.15731
$518$	0	0
$519$	19003.9	1.60728
$520$	0	0
$521$	1028.09	0.0864522	0.0432261	−	0.999065i	$-0.486236\pi$
$521$	1028.09	0.0864522	0.0432261	+	0.999065i	$0.486236\pi$
$522$	3999.74	0.335371
$523$	618.266	0.0516919	0.0258460	−	0.999666i	$-0.491772\pi$
$523$	618.266	0.0516919	0.0258460	+	0.999666i	$0.491772\pi$
$524$	−8800.73	−0.733705
$525$	0	0
$526$	−10430.3	−0.864603
$527$	−15277.8	−1.26283
$528$	4139.40	0.341182
$529$	−12127.7	−0.996770
$530$	0	0
$531$	7499.34	0.612888
$532$	0	0
$533$	−1091.43	−0.0886966
$534$	−4662.58	−0.377846
$535$	0	0
$536$	−540.260	−0.0435368
$537$	−9731.74	−0.782040
$538$	−7268.77	−0.582489
$539$	0	0
$540$	0	0
$541$	−835.523	−0.0663991	−0.0331996	−	0.999449i	$-0.510570\pi$
$541$	−835.523	−0.0663991	−0.0331996	+	0.999449i	$0.510570\pi$
$542$	−405.670	−0.0321495
$543$	4417.16	0.349094
$544$	2245.05	0.176941
$545$	0	0
$546$	0	0
$547$	1100.23	0.0860011	0.0430005	−	0.999075i	$-0.486308\pi$
$547$	1100.23	0.0860011	0.0430005	+	0.999075i	$0.486308\pi$
$548$	−8222.01	−0.640925
$549$	3238.46	0.251756
$550$	0	0
$551$	−11972.8	−0.925695
$552$	−347.245	−0.0267749
$553$	0	0
$554$	1527.99	0.117181
$555$	0	0
$556$	1815.77	0.138499
$557$	−5999.65	−0.456397	−0.228199	−	0.973615i	$-0.573284\pi$
$557$	−5999.65	−0.456397	−0.228199	+	0.973615i	$0.573284\pi$
$558$	−9122.66	−0.692102
$559$	−11739.1	−0.888215
$560$	0	0
$561$	−18150.7	−1.36600
$562$	−2902.80	−0.217877
$563$	−24094.2	−1.80364	−0.901819	−	0.432115i	$-0.857768\pi$
$563$	−24094.2	−1.80364	−0.901819	+	0.432115i	$0.857768\pi$
$564$	10085.2	0.752947
$565$	0	0
$566$	−15319.6	−1.13768
$567$	0	0
$568$	4142.18	0.305989
$569$	5509.91	0.405953	0.202977	−	0.979184i	$-0.434938\pi$
$569$	5509.91	0.405953	0.202977	+	0.979184i	$0.434938\pi$
$570$	0	0
$571$	−4438.24	−0.325279	−0.162640	−	0.986686i	$-0.552001\pi$
$571$	−4438.24	−0.325279	−0.162640	+	0.986686i	$0.552001\pi$
$572$	−13316.7	−0.973427
$573$	25313.7	1.84554
$574$	0	0
$575$	0	0
$576$	1340.56	0.0969735
$577$	9201.64	0.663899	0.331949	−	0.943297i	$-0.392294\pi$
$577$	9201.64	0.663899	0.331949	+	0.943297i	$0.392294\pi$
$578$	−18.2315	−0.00131199
$579$	−30700.1	−2.20354
$580$	0	0
$581$	0	0
$582$	−2487.49	−0.177164
$583$	18623.7	1.32301
$584$	5398.13	0.382494
$585$	0	0
$586$	8445.37	0.595350
$587$	−5551.42	−0.390343	−0.195172	−	0.980769i	$-0.562526\pi$
$587$	−5551.42	−0.390343	−0.195172	+	0.980769i	$0.562526\pi$
$588$	0	0
$589$	27307.7	1.91035
$590$	0	0
$591$	36586.4	2.54647
$592$	−3373.21	−0.234186
$593$	5590.69	0.387153	0.193577	−	0.981085i	$-0.437991\pi$
$593$	5590.69	0.387153	0.193577	+	0.981085i	$0.437991\pi$
$594$	3132.34	0.216366
$595$	0	0
$596$	938.786	0.0645204
$597$	30501.0	2.09099
$598$	1117.11	0.0763913
$599$	13675.3	0.932820	0.466410	−	0.884569i	$-0.345547\pi$
$599$	13675.3	0.932820	0.466410	+	0.884569i	$0.345547\pi$
$600$	0	0
$601$	4508.67	0.306011	0.153005	−	0.988225i	$-0.451105\pi$
$601$	4508.67	0.306011	0.153005	+	0.988225i	$0.451105\pi$
$602$	0	0
$603$	1414.56	0.0955310
$604$	1384.08	0.0932407
$605$	0	0
$606$	−12386.1	−0.830284
$607$	−3633.07	−0.242935	−0.121468	−	0.992595i	$-0.538760\pi$
$607$	−3633.07	−0.242935	−0.121468	+	0.992595i	$0.538760\pi$
$608$	−4012.83	−0.267668
$609$	0	0
$610$	0	0
$611$	−32444.6	−2.14823
$612$	−5878.18	−0.388254
$613$	−11909.3	−0.784682	−0.392341	−	0.919820i	$-0.628335\pi$
$613$	−11909.3	−0.784682	−0.392341	+	0.919820i	$0.628335\pi$
$614$	2486.22	0.163413
$615$	0	0
$616$	0	0
$617$	−5510.74	−0.359569	−0.179784	−	0.983706i	$-0.557540\pi$
$617$	−5510.74	−0.359569	−0.179784	+	0.983706i	$0.557540\pi$
$618$	−344.541	−0.0224263
$619$	20935.5	1.35940	0.679699	−	0.733492i	$-0.262111\pi$
$619$	20935.5	1.35940	0.679699	+	0.733492i	$0.262111\pi$
$620$	0	0
$621$	−262.765	−0.0169797
$622$	−2737.38	−0.176461
$623$	0	0
$624$	−9871.77	−0.633313
$625$	0	0
$626$	6421.32	0.409980
$627$	32442.8	2.06641
$628$	13356.5	0.848697
$629$	14791.1	0.937613
$630$	0	0
$631$	−1538.74	−0.0970780	−0.0485390	−	0.998821i	$-0.515457\pi$
$631$	−1538.74	−0.0970780	−0.0485390	+	0.998821i	$0.515457\pi$
$632$	−7453.46	−0.469118
$633$	−11549.7	−0.725211
$634$	15298.2	0.958311
$635$	0	0
$636$	13805.9	0.860753
$637$	0	0
$638$	7134.51	0.442724
$639$	−10845.4	−0.671421
$640$	0	0
$641$	−5922.32	−0.364926	−0.182463	−	0.983213i	$-0.558407\pi$
$641$	−5922.32	−0.364926	−0.182463	+	0.983213i	$0.558407\pi$
$642$	15978.5	0.982274
$643$	22967.5	1.40863	0.704315	−	0.709887i	$-0.251254\pi$
$643$	22967.5	1.40863	0.704315	+	0.709887i	$0.251254\pi$
$644$	0	0
$645$	0	0
$646$	17595.7	1.07166
$647$	−14314.8	−0.869821	−0.434910	−	0.900474i	$-0.643220\pi$
$647$	−14314.8	−0.869821	−0.434910	+	0.900474i	$0.643220\pi$
$648$	6846.42	0.415051
$649$	13376.9	0.809075
$650$	0	0
$651$	0	0
$652$	9646.55	0.579430
$653$	16188.1	0.970119	0.485060	−	0.874481i	$-0.338798\pi$
$653$	16188.1	0.970119	0.485060	+	0.874481i	$0.338798\pi$
$654$	−26048.8	−1.55748
$655$	0	0
$656$	195.984	0.0116645
$657$	−14133.8	−0.839291
$658$	0	0
$659$	5303.87	0.313519	0.156760	−	0.987637i	$-0.449895\pi$
$659$	5303.87	0.313519	0.156760	+	0.987637i	$0.449895\pi$
$660$	0	0
$661$	−22336.6	−1.31436	−0.657180	−	0.753734i	$-0.728251\pi$
$661$	−22336.6	−1.31436	−0.657180	+	0.753734i	$0.728251\pi$
$662$	15834.6	0.929650
$663$	43286.4	2.53560
$664$	−11485.8	−0.671286
$665$	0	0
$666$	8832.03	0.513865
$667$	−598.498	−0.0347435
$668$	5186.30	0.300395
$669$	−25668.0	−1.48338
$670$	0	0
$671$	5776.59	0.332344
$672$	0	0
$673$	22907.3	1.31206	0.656028	−	0.754737i	$-0.272235\pi$
$673$	22907.3	1.31206	0.656028	+	0.754737i	$0.272235\pi$
$674$	13756.2	0.786156
$675$	0	0
$676$	22970.1	1.30690
$677$	32337.4	1.83579	0.917894	−	0.396826i	$-0.129888\pi$
$677$	32337.4	1.83579	0.917894	+	0.396826i	$0.129888\pi$
$678$	−28721.8	−1.62692
$679$	0	0
$680$	0	0
$681$	−32955.8	−1.85443
$682$	−16272.5	−0.913646
$683$	−24675.1	−1.38238	−0.691190	−	0.722673i	$-0.742913\pi$
$683$	−24675.1	−1.38238	−0.691190	+	0.722673i	$0.742913\pi$
$684$	10506.7	0.587333
$685$	0	0
$686$	0	0
$687$	4219.56	0.234332
$688$	2107.94	0.116809
$689$	−44414.5	−2.45581
$690$	0	0
$691$	12860.8	0.708030	0.354015	−	0.935240i	$-0.384816\pi$
$691$	12860.8	0.708030	0.354015	+	0.935240i	$0.384816\pi$
$692$	10978.0	0.603067
$693$	0	0
$694$	−2843.32	−0.155520
$695$	0	0
$696$	5288.86	0.288037
$697$	−859.362	−0.0467011
$698$	5675.49	0.307765
$699$	21908.3	1.18547
$700$	0	0
$701$	−2157.68	−0.116254	−0.0581272	−	0.998309i	$-0.518513\pi$
$701$	−2157.68	−0.116254	−0.0581272	+	0.998309i	$0.518513\pi$
$702$	−7470.11	−0.401625
$703$	−26437.8	−1.41838
$704$	2391.22	0.128015
$705$	0	0
$706$	7157.63	0.381559
$707$	0	0
$708$	9916.39	0.526385
$709$	18549.2	0.982553	0.491277	−	0.871004i	$-0.336530\pi$
$709$	18549.2	0.982553	0.491277	+	0.871004i	$0.336530\pi$
$710$	0	0
$711$	19515.3	1.02937
$712$	−2693.45	−0.141772
$713$	1365.06	0.0716999
$714$	0	0
$715$	0	0
$716$	−5621.77	−0.293429
$717$	−35868.9	−1.86827
$718$	12243.0	0.636355
$719$	−25772.6	−1.33680	−0.668398	−	0.743804i	$-0.733020\pi$
$719$	−25772.6	−1.33680	−0.668398	+	0.743804i	$0.733020\pi$
$720$	0	0
$721$	0	0
$722$	−17732.8	−0.914056
$723$	−14224.7	−0.731706
$724$	2551.67	0.130984
$725$	0	0
$726$	−899.924	−0.0460046
$727$	24295.8	1.23945	0.619727	−	0.784818i	$-0.287243\pi$
$727$	24295.8	1.23945	0.619727	+	0.784818i	$0.287243\pi$
$728$	0	0
$729$	−10089.1	−0.512577
$730$	0	0
$731$	−9243.03	−0.467669
$732$	4282.22	0.216223
$733$	6136.79	0.309232	0.154616	−	0.987975i	$-0.450586\pi$
$733$	6136.79	0.309232	0.154616	+	0.987975i	$0.450586\pi$
$734$	−24373.7	−1.22568
$735$	0	0
$736$	−200.594	−0.0100462
$737$	2523.21	0.126111
$738$	−513.141	−0.0255948
$739$	−23602.0	−1.17485	−0.587424	−	0.809279i	$-0.699858\pi$
$739$	−23602.0	−1.17485	−0.587424	+	0.809279i	$0.699858\pi$
$740$	0	0
$741$	−77370.6	−3.83574
$742$	0	0
$743$	6486.83	0.320294	0.160147	−	0.987093i	$-0.448803\pi$
$743$	6486.83	0.320294	0.160147	+	0.987093i	$0.448803\pi$
$744$	−12062.9	−0.594419
$745$	0	0
$746$	20893.8	1.02544
$747$	30073.0	1.47298
$748$	−10485.2	−0.512535
$749$	0	0
$750$	0	0
$751$	−2592.22	−0.125954	−0.0629770	−	0.998015i	$-0.520059\pi$
$751$	−2592.22	−0.125954	−0.0629770	+	0.998015i	$0.520059\pi$
$752$	5825.93	0.282513
$753$	−17000.3	−0.822743
$754$	−17014.6	−0.821797
$755$	0	0
$756$	0	0
$757$	24578.4	1.18007	0.590037	−	0.807376i	$-0.299113\pi$
$757$	24578.4	1.18007	0.590037	+	0.807376i	$0.299113\pi$
$758$	20548.1	0.984617
$759$	1621.76	0.0775574
$760$	0	0
$761$	−37954.5	−1.80795	−0.903976	−	0.427584i	$-0.859365\pi$
$761$	−37954.5	−1.80795	−0.903976	+	0.427584i	$0.859365\pi$
$762$	1043.54	0.0496109
$763$	0	0
$764$	14623.0	0.692465
$765$	0	0
$766$	−12975.4	−0.612035
$767$	−31901.7	−1.50183
$768$	1772.63	0.0832867
$769$	3017.80	0.141514	0.0707572	−	0.997494i	$-0.477458\pi$
$769$	3017.80	0.141514	0.0707572	+	0.997494i	$0.477458\pi$
$770$	0	0
$771$	−25551.4	−1.19353
$772$	−17734.6	−0.826792
$773$	−102.457	−0.00476728	−0.00238364	−	0.999997i	$-0.500759\pi$
$773$	−102.457	−0.00476728	−0.00238364	+	0.999997i	$0.500759\pi$
$774$	−5519.19	−0.256309
$775$	0	0
$776$	−1436.96	−0.0664739
$777$	0	0
$778$	7412.65	0.341589
$779$	1536.04	0.0706472
$780$	0	0
$781$	−19345.5	−0.886344
$782$	879.578	0.0402220
$783$	4002.15	0.182663
$784$	0	0
$785$	0	0
$786$	30469.6	1.38271
$787$	−12420.0	−0.562546	−0.281273	−	0.959628i	$-0.590757\pi$
$787$	−12420.0	−0.562546	−0.281273	+	0.959628i	$0.590757\pi$
$788$	21135.0	0.955462
$789$	36111.3	1.62940
$790$	0	0
$791$	0	0
$792$	−6260.90	−0.280898
$793$	−13776.2	−0.616906
$794$	−18560.5	−0.829582
$795$	0	0
$796$	17619.6	0.784562
$797$	−40547.2	−1.80208	−0.901039	−	0.433739i	$-0.857194\pi$
$797$	−40547.2	−1.80208	−0.901039	+	0.433739i	$0.857194\pi$
$798$	0	0
$799$	−25545.9	−1.13110
$800$	0	0
$801$	7052.22	0.311084
$802$	4838.16	0.213019
$803$	−25211.2	−1.10795
$804$	1870.47	0.0820477
$805$	0	0
$806$	38807.2	1.69594
$807$	25165.7	1.09774
$808$	−7155.14	−0.311531
$809$	−7107.15	−0.308868	−0.154434	−	0.988003i	$-0.549355\pi$
$809$	−7107.15	−0.308868	−0.154434	+	0.988003i	$0.549355\pi$
$810$	0	0
$811$	21314.5	0.922876	0.461438	−	0.887172i	$-0.347334\pi$
$811$	21314.5	0.922876	0.461438	+	0.887172i	$0.347334\pi$
$812$	0	0
$813$	1404.50	0.0605877
$814$	15754.1	0.678355
$815$	0	0
$816$	−7772.73	−0.333456
$817$	16521.1	0.707467
$818$	−9991.98	−0.427092
$819$	0	0
$820$	0	0
$821$	2884.89	0.122635	0.0613175	−	0.998118i	$-0.480470\pi$
$821$	2884.89	0.122635	0.0613175	+	0.998118i	$0.480470\pi$
$822$	28465.9	1.20786
$823$	18429.2	0.780562	0.390281	−	0.920696i	$-0.372378\pi$
$823$	18429.2	0.780562	0.390281	+	0.920696i	$0.372378\pi$
$824$	−199.032	−0.00841458
$825$	0	0
$826$	0	0
$827$	−2056.14	−0.0864559	−0.0432280	−	0.999065i	$-0.513764\pi$
$827$	−2056.14	−0.0864559	−0.0432280	+	0.999065i	$0.513764\pi$
$828$	525.213	0.0220440
$829$	7871.60	0.329785	0.164893	−	0.986312i	$-0.447272\pi$
$829$	7871.60	0.329785	0.164893	+	0.986312i	$0.447272\pi$
$830$	0	0
$831$	−5290.16	−0.220835
$832$	−5702.66	−0.237625
$833$	0	0
$834$	−6286.48	−0.261011
$835$	0	0
$836$	18741.4	0.775339
$837$	−9128.17	−0.376960
$838$	−21493.3	−0.886005
$839$	−34220.8	−1.40815	−0.704073	−	0.710128i	$-0.748637\pi$
$839$	−34220.8	−1.40815	−0.704073	+	0.710128i	$0.748637\pi$
$840$	0	0
$841$	−15273.3	−0.626239
$842$	−17122.2	−0.700796
$843$	10050.0	0.410604
$844$	−6671.95	−0.272107
$845$	0	0
$846$	−15254.0	−0.619908
$847$	0	0
$848$	7975.30	0.322963
$849$	53038.8	2.14404
$850$	0	0
$851$	−1321.58	−0.0532350
$852$	−14340.9	−0.576656
$853$	33509.6	1.34507	0.672536	−	0.740065i	$-0.265205\pi$
$853$	33509.6	1.34507	0.672536	+	0.740065i	$0.265205\pi$
$854$	0	0
$855$	0	0
$856$	9230.34	0.368559
$857$	8614.74	0.343377	0.171688	−	0.985151i	$-0.445078\pi$
$857$	8614.74	0.343377	0.171688	+	0.985151i	$0.445078\pi$
$858$	46104.7	1.83449
$859$	−4363.48	−0.173318	−0.0866589	−	0.996238i	$-0.527619\pi$
$859$	−4363.48	−0.173318	−0.0866589	+	0.996238i	$0.527619\pi$
$860$	0	0
$861$	0	0
$862$	10471.8	0.413772
$863$	−17066.7	−0.673182	−0.336591	−	0.941651i	$-0.609274\pi$
$863$	−17066.7	−0.673182	−0.336591	+	0.941651i	$0.609274\pi$
$864$	1341.37	0.0528176
$865$	0	0
$866$	7619.76	0.298996
$867$	63.1203	0.00247252
$868$	0	0
$869$	34810.3	1.35887
$870$	0	0
$871$	−6017.43	−0.234090
$872$	−15047.7	−0.584381
$873$	3762.36	0.145861
$874$	−1572.17	−0.0608460
$875$	0	0
$876$	−18689.2	−0.720833
$877$	26856.5	1.03407	0.517034	−	0.855965i	$-0.327036\pi$
$877$	26856.5	1.03407	0.517034	+	0.855965i	$0.327036\pi$
$878$	−25065.6	−0.963465
$879$	−29239.2	−1.12197
$880$	0	0
$881$	27044.5	1.03423	0.517113	−	0.855917i	$-0.327007\pi$
$881$	27044.5	1.03423	0.517113	+	0.855917i	$0.327007\pi$
$882$	0	0
$883$	48010.9	1.82978	0.914890	−	0.403702i	$-0.132277\pi$
$883$	48010.9	1.82978	0.914890	+	0.403702i	$0.132277\pi$
$884$	25005.4	0.951382
$885$	0	0
$886$	−35588.8	−1.34947
$887$	3888.02	0.147178	0.0735889	−	0.997289i	$-0.476555\pi$
$887$	3888.02	0.147178	0.0735889	+	0.997289i	$0.476555\pi$
$888$	11678.6	0.441338
$889$	0	0
$890$	0	0
$891$	−31975.2	−1.20226
$892$	−14827.7	−0.556579
$893$	45661.1	1.71108
$894$	−3250.23	−0.121593
$895$	0	0
$896$	0	0
$897$	−3867.62	−0.143964
$898$	−24406.8	−0.906978
$899$	−20791.2	−0.771328
$900$	0	0
$901$	−34970.6	−1.29305
$902$	−915.314	−0.0337878
$903$	0	0
$904$	−16591.8	−0.610438
$905$	0	0
$906$	−4791.91	−0.175718
$907$	9238.85	0.338226	0.169113	−	0.985597i	$-0.445910\pi$
$907$	9238.85	0.338226	0.169113	+	0.985597i	$0.445910\pi$
$908$	−19037.7	−0.695802
$909$	18734.2	0.683580
$910$	0	0
$911$	−54171.6	−1.97013	−0.985064	−	0.172190i	$-0.944916\pi$
$911$	−54171.6	−1.97013	−0.985064	+	0.172190i	$0.944916\pi$
$912$	13893.1	0.504436
$913$	53642.6	1.94448
$914$	−9499.07	−0.343765
$915$	0	0
$916$	2437.53	0.0879238
$917$	0	0
$918$	−5881.73	−0.211466
$919$	−19403.5	−0.696478	−0.348239	−	0.937406i	$-0.613220\pi$
$919$	−19403.5	−0.696478	−0.348239	+	0.937406i	$0.613220\pi$
$920$	0	0
$921$	−8607.71	−0.307963
$922$	−26416.0	−0.943561
$923$	46135.6	1.64526
$924$	0	0
$925$	0	0
$926$	−23975.7	−0.850855
$927$	521.123	0.0184638
$928$	3055.23	0.108074
$929$	39108.2	1.38116	0.690581	−	0.723255i	$-0.257355\pi$
$929$	39108.2	1.38116	0.690581	+	0.723255i	$0.257355\pi$
$930$	0	0
$931$	0	0
$932$	12655.8	0.444802
$933$	9477.24	0.332552
$934$	10362.5	0.363030
$935$	0	0
$936$	14931.2	0.521412
$937$	624.451	0.0217715	0.0108858	−	0.999941i	$-0.496535\pi$
$937$	624.451	0.0217715	0.0108858	+	0.999941i	$0.496535\pi$
$938$	0	0
$939$	−22231.7	−0.772634
$940$	0	0
$941$	−53525.4	−1.85428	−0.927141	−	0.374713i	$-0.877741\pi$
$941$	−53525.4	−1.85428	−0.927141	+	0.374713i	$0.877741\pi$
$942$	−46242.3	−1.59942
$943$	76.7836	0.00265156
$944$	5728.43	0.197505
$945$	0	0
$946$	−9844.83	−0.338354
$947$	2846.06	0.0976606	0.0488303	−	0.998807i	$-0.484451\pi$
$947$	2846.06	0.0976606	0.0488303	+	0.998807i	$0.484451\pi$
$948$	25805.1	0.884083
$949$	60124.4	2.05661
$950$	0	0
$951$	−52964.9	−1.80600
$952$	0	0
$953$	−29720.3	−1.01021	−0.505107	−	0.863057i	$-0.668547\pi$
$953$	−29720.3	−1.01021	−0.505107	+	0.863057i	$0.668547\pi$
$954$	−20881.6	−0.708665
$955$	0	0
$956$	−20720.5	−0.700993
$957$	−24700.8	−0.834342
$958$	−267.997	−0.00903819
$959$	0	0
$960$	0	0
$961$	17629.8	0.591783
$962$	−37570.9	−1.25918
$963$	−24167.7	−0.808715
$964$	−8217.26	−0.274544
$965$	0	0
$966$	0	0
$967$	−7604.19	−0.252879	−0.126440	−	0.991974i	$-0.540355\pi$
$967$	−7604.19	−0.252879	−0.126440	+	0.991974i	$0.540355\pi$
$968$	−519.863	−0.0172614
$969$	−60919.3	−2.01962
$970$	0	0
$971$	−26345.1	−0.870704	−0.435352	−	0.900260i	$-0.643376\pi$
$971$	−26345.1	−0.870704	−0.435352	+	0.900260i	$0.643376\pi$
$972$	−19176.3	−0.632798
$973$	0	0
$974$	6636.72	0.218331
$975$	0	0
$976$	2473.72	0.0811291
$977$	−13491.0	−0.441775	−0.220888	−	0.975299i	$-0.570895\pi$
$977$	−13491.0	−0.441775	−0.220888	+	0.975299i	$0.570895\pi$
$978$	−33397.9	−1.09197
$979$	12579.4	0.410663
$980$	0	0
$981$	39399.3	1.28228
$982$	4391.80	0.142717
$983$	28904.8	0.937864	0.468932	−	0.883234i	$-0.344639\pi$
$983$	28904.8	0.937864	0.468932	+	0.883234i	$0.344639\pi$
$984$	−678.528	−0.0219824
$985$	0	0
$986$	−13396.8	−0.432698
$987$	0	0
$988$	−44695.0	−1.43921
$989$	825.860	0.0265529
$990$	0	0
$991$	−9201.76	−0.294958	−0.147479	−	0.989065i	$-0.547116\pi$
$991$	−9201.76	−0.294958	−0.147479	+	0.989065i	$0.547116\pi$
$992$	−6968.42	−0.223032
$993$	−54821.9	−1.75198
$994$	0	0
$995$	0	0
$996$	39765.5	1.26508
$997$	−36706.4	−1.16600	−0.583000	−	0.812472i	$-0.698121\pi$
$997$	−36706.4	−1.16600	−0.583000	+	0.812472i	$0.698121\pi$
$998$	−23379.2	−0.741540
$999$	8837.37	0.279882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cd.1.3		3
5.4	even	2		2450.4.a.ch.1.1		3
7.3	odd	6		350.4.e.j.51.3	yes	6
7.5	odd	6		350.4.e.j.151.3	yes	6
7.6	odd	2		2450.4.a.cc.1.1		3
35.3	even	12		350.4.j.h.149.1		12
35.12	even	12		350.4.j.h.249.1		12
35.17	even	12		350.4.j.h.149.6		12
35.19	odd	6		350.4.e.i.151.1	yes	6
35.24	odd	6		350.4.e.i.51.1	✓	6
35.33	even	12		350.4.j.h.249.6		12
35.34	odd	2		2450.4.a.ci.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.e.i.51.1	✓	6	35.24	odd	6
350.4.e.i.151.1	yes	6	35.19	odd	6
350.4.e.j.51.3	yes	6	7.3	odd	6
350.4.e.j.151.3	yes	6	7.5	odd	6
350.4.j.h.149.1		12	35.3	even	12
350.4.j.h.149.6		12	35.17	even	12
350.4.j.h.249.1		12	35.12	even	12
350.4.j.h.249.6		12	35.33	even	12
2450.4.a.cc.1.1		3	7.6	odd	2
2450.4.a.cd.1.3		3	1.1	even	1	trivial
2450.4.a.ch.1.1		3	5.4	even	2
2450.4.a.ci.1.3		3	35.34	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +6.92433 q^{3} +4.00000 q^{4} -13.8487 q^{6} -8.00000 q^{8} +20.9463 q^{9} +37.3629 q^{11} +27.6973 q^{12} -89.1041 q^{13} +16.0000 q^{16} -70.1578 q^{17} -41.8926 q^{18} +125.401 q^{19} -74.7257 q^{22} +6.26857 q^{23} -55.3946 q^{24} +178.208 q^{26} -41.9179 q^{27} -95.4760 q^{29} +217.763 q^{31} -32.0000 q^{32} +258.713 q^{33} +140.316 q^{34} +83.7851 q^{36} -210.826 q^{37} -250.802 q^{38} -616.986 q^{39} +12.2490 q^{41} +131.746 q^{43} +149.451 q^{44} -12.5371 q^{46} +364.121 q^{47} +110.789 q^{48} -485.795 q^{51} -356.416 q^{52} +498.456 q^{53} +83.8358 q^{54} +868.318 q^{57} +190.952 q^{58} +358.027 q^{59} +154.608 q^{61} -435.526 q^{62} +64.0000 q^{64} -517.425 q^{66} +67.5326 q^{67} -280.631 q^{68} +43.4056 q^{69} -517.772 q^{71} -167.570 q^{72} -674.766 q^{73} +421.651 q^{74} +501.604 q^{76} +1233.97 q^{78} +931.682 q^{79} -855.803 q^{81} -24.4980 q^{82} +1435.72 q^{83} -263.493 q^{86} -661.107 q^{87} -298.903 q^{88} +336.681 q^{89} +25.0743 q^{92} +1507.86 q^{93} -728.242 q^{94} -221.578 q^{96} +179.620 q^{97} +782.613 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 3 q^{3} + 12 q^{4} - 6 q^{6} - 24 q^{8} + 56 q^{9} - 26 q^{11} + 12 q^{12} - 80 q^{13} + 48 q^{16} - 30 q^{17} - 112 q^{18} + 18 q^{19} + 52 q^{22} + 53 q^{23} - 24 q^{24} + 160 q^{26} - 324 q^{27}+ \cdots - 57 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 3 * q^3 + 12 * q^4 - 6 * q^6 - 24 * q^8 + 56 * q^9 - 26 * q^11 + 12 * q^12 - 80 * q^13 + 48 * q^16 - 30 * q^17 - 112 * q^18 + 18 * q^19 + 52 * q^22 + 53 * q^23 - 24 * q^24 + 160 * q^26 - 324 * q^27 - 404 * q^29 + 615 * q^31 - 96 * q^32 + 314 * q^33 + 60 * q^34 + 224 * q^36 - 426 * q^37 - 36 * q^38 - 90 * q^39 + 101 * q^41 + 249 * q^43 - 104 * q^44 - 106 * q^46 - 402 * q^47 + 48 * q^48 - 339 * q^51 - 320 * q^52 - 390 * q^53 + 648 * q^54 + 1667 * q^57 + 808 * q^58 + 91 * q^59 + 647 * q^61 - 1230 * q^62 + 192 * q^64 - 628 * q^66 - 708 * q^67 - 120 * q^68 + 1548 * q^69 + 533 * q^71 - 448 * q^72 - 772 * q^73 + 852 * q^74 + 72 * q^76 + 180 * q^78 + 1421 * q^79 - 1393 * q^81 - 202 * q^82 + 520 * q^83 - 498 * q^86 + 833 * q^87 + 208 * q^88 + 194 * q^89 + 212 * q^92 - 148 * q^93 + 804 * q^94 - 96 * q^96 + 1027 * q^97 - 57 * q^99

Introduction

L-functions

Modular forms

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