LMFDB - Embedded newform 2450.4.a.cc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	yes
Analytic conductor:	$144.554679514$
Analytic rank:	$1$
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			1.1
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)

$f(q)$	$=$	$q-2.00000 q^{2} -6.92433 q^{3} +4.00000 q^{4} +13.8487 q^{6} -8.00000 q^{8} +20.9463 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -6.92433 q^{3} +4.00000 q^{4} +13.8487 q^{6} -8.00000 q^{8} +20.9463 q^{9} +37.3629 q^{11} -27.6973 q^{12} +89.1041 q^{13} +16.0000 q^{16} +70.1578 q^{17} -41.8926 q^{18} -125.401 q^{19} -74.7257 q^{22} +6.26857 q^{23} +55.3946 q^{24} -178.208 q^{26} +41.9179 q^{27} -95.4760 q^{29} -217.763 q^{31} -32.0000 q^{32} -258.713 q^{33} -140.316 q^{34} +83.7851 q^{36} -210.826 q^{37} +250.802 q^{38} -616.986 q^{39} -12.2490 q^{41} +131.746 q^{43} +149.451 q^{44} -12.5371 q^{46} -364.121 q^{47} -110.789 q^{48} -485.795 q^{51} +356.416 q^{52} +498.456 q^{53} -83.8358 q^{54} +868.318 q^{57} +190.952 q^{58} -358.027 q^{59} -154.608 q^{61} +435.526 q^{62} +64.0000 q^{64} +517.425 q^{66} +67.5326 q^{67} +280.631 q^{68} -43.4056 q^{69} -517.772 q^{71} -167.570 q^{72} +674.766 q^{73} +421.651 q^{74} -501.604 q^{76} +1233.97 q^{78} +931.682 q^{79} -855.803 q^{81} +24.4980 q^{82} -1435.72 q^{83} -263.493 q^{86} +661.107 q^{87} -298.903 q^{88} -336.681 q^{89} +25.0743 q^{92} +1507.86 q^{93} +728.242 q^{94} +221.578 q^{96} -179.620 q^{97} +782.613 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 3 q^{3} + 12 q^{4} + 6 q^{6} - 24 q^{8} + 56 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 3 * q^3 + 12 * q^4 + 6 * q^6 - 24 * q^8 + 56 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−6.92433	−1.33259	−0.666294	−	0.745690i	$-0.732120\pi$
$3$	−6.92433	−1.33259	−0.666294	+	0.745690i	$0.732120\pi$
$4$	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	$-0.100571\pi$
$13$	89.1041	1.90100	0.950501	+	0.310722i	$0.100571\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	70.1578	1.00093	0.500464	−	0.865758i	$-0.333163\pi$
$17$	70.1578	1.00093	0.500464	+	0.865758i	$0.333163\pi$
$18$	−41.8926	−0.548565
$19$	−125.401	−1.51416	−0.757078	−	0.653324i	$-0.773374\pi$
$19$	−125.401	−1.51416	−0.757078	+	0.653324i	$0.773374\pi$
$20$	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	$-0.717285\pi$
$31$	−217.763	−1.26166	−0.630829	+	0.775922i	$0.717285\pi$
$32$	−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	$-0.507426\pi$
$41$	−12.2490	−0.0466578	−0.0233289	+	0.999728i	$0.507426\pi$
$42$	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	$-0.691134\pi$
$47$	−364.121	−1.13005	−0.565026	+	0.825073i	$0.691134\pi$
$48$	−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	$-0.629259\pi$
$59$	−358.027	−0.790020	−0.395010	+	0.918677i	$0.629259\pi$
$60$	0	0
$61$	−154.608	−0.324516	−0.162258	−	0.986748i	$-0.551878\pi$
$61$	−154.608	−0.324516	−0.162258	+	0.986748i	$0.551878\pi$
$62$	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	$-0.318073\pi$
$73$	674.766	1.08186	0.540928	+	0.841069i	$0.318073\pi$
$74$	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	$-0.898248\pi$
$83$	−1435.72	−1.89868	−0.949341	+	0.314247i	$0.898248\pi$
$84$	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	$-0.564255\pi$
$89$	−336.681	−0.400990	−0.200495	+	0.979695i	$0.564255\pi$
$90$	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	$-0.529968\pi$
$97$	−179.620	−0.188016	−0.0940082	+	0.995571i	$0.529968\pi$
$98$	0	0
$99$	782.613	0.794501
$100$	0	0
$101$	−894.392	−0.881142	−0.440571	−	0.897718i	$-0.645224\pi$
$101$	−894.392	−0.881142	−0.440571	+	0.897718i	$0.645224\pi$
$102$	971.591	0.943155
$103$	−24.8790	−0.0238000	−0.0119000	−	0.999929i	$-0.503788\pi$
$103$	−24.8790	−0.0238000	−0.0119000	+	0.999929i	$0.503788\pi$
$104$	−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	$-0.237789\pi$
$131$	2200.18	1.46741	0.733705	+	0.679468i	$0.237789\pi$
$132$	−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	$-0.544228\pi$
$139$	−453.942	−0.276999	−0.138499	+	0.990363i	$0.544228\pi$
$140$	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	$-0.822613\pi$
$157$	−3339.12	−1.69739	−0.848697	+	0.528879i	$0.822613\pi$
$158$	−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	$-0.597119\pi$
$167$	−1296.58	−0.600791	−0.300395	+	0.953815i	$0.597119\pi$
$168$	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	$-0.706055\pi$
$173$	−2744.51	−1.20613	−0.603067	+	0.797690i	$0.706055\pi$
$174$	−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	$-0.541814\pi$
$181$	−637.919	−0.261967	−0.130984	+	0.991385i	$0.541814\pi$
$182$	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	$-0.787112\pi$
$199$	−4404.91	−1.56912	−0.784562	+	0.620051i	$0.787112\pi$
$200$	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	$-0.312114\pi$
$223$	3706.93	1.11316	0.556579	+	0.830795i	$0.312114\pi$
$224$	0	0
$225$	0	0
$226$	−4147.96	−1.22088
$227$	4759.43	1.39160	0.695802	−	0.718233i	$-0.255049\pi$
$227$	4759.43	1.39160	0.695802	+	0.718233i	$0.255049\pi$
$228$	3473.27	1.00887
$229$	−609.382	−0.175848	−0.0879238	−	0.996127i	$-0.528023\pi$
$229$	−609.382	−0.175848	−0.0879238	+	0.996127i	$0.528023\pi$
$230$	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	$-0.411473\pi$
$241$	2054.31	0.549087	0.274544	+	0.961575i	$0.411473\pi$
$242$	−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	$-0.400106\pi$
$251$	2455.16	0.617403	0.308701	+	0.951159i	$0.400106\pi$
$252$	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	$-0.352199\pi$
$257$	3690.10	0.895650	0.447825	+	0.894121i	$0.352199\pi$
$258$	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	$-0.635128\pi$
$269$	−3634.39	−0.823764	−0.411882	+	0.911237i	$0.635128\pi$
$270$	0	0
$271$	−202.835	−0.0454663	−0.0227331	−	0.999742i	$-0.507237\pi$
$271$	−202.835	−0.0454663	−0.0227331	+	0.999742i	$0.507237\pi$
$272$	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	$-0.797547\pi$
$283$	−7659.78	−1.60893	−0.804464	+	0.594002i	$0.797547\pi$
$284$	−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	$-0.361688\pi$
$293$	4222.68	0.841952	0.420976	+	0.907072i	$0.361688\pi$
$294$	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	$-0.463137\pi$
$307$	1243.11	0.231101	0.115551	+	0.993302i	$0.463137\pi$
$308$	0	0
$309$	172.270	0.0317156
$310$	0	0
$311$	−1368.69	−0.249554	−0.124777	−	0.992185i	$-0.539821\pi$
$311$	−1368.69	−0.249554	−0.124777	+	0.992185i	$0.539821\pi$
$312$	4935.89	0.895639
$313$	3210.66	0.579800	0.289900	−	0.957057i	$-0.406378\pi$
$313$	3210.66	0.579800	0.289900	+	0.957057i	$0.406378\pi$
$314$	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	$-0.430170\pi$
$349$	2837.74	0.435246	0.217623	+	0.976033i	$0.430170\pi$
$350$	0	0
$351$	3735.05	0.567984
$352$	−1195.61	−0.181041
$353$	3578.81	0.539606	0.269803	−	0.962916i	$-0.413041\pi$
$353$	3578.81	0.539606	0.269803	+	0.962916i	$0.413041\pi$
$354$	−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	$-0.833754\pi$
$367$	−12186.8	−1.73337	−0.866686	+	0.498855i	$0.833754\pi$
$368$	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	$-0.642465\pi$
$383$	−6487.68	−0.865548	−0.432774	+	0.901502i	$0.642465\pi$
$384$	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	$-0.699535\pi$
$397$	−9280.26	−1.17321	−0.586603	+	0.809874i	$0.699535\pi$
$398$	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	$-0.597654\pi$
$409$	−4995.99	−0.604000	−0.302000	+	0.953308i	$0.597654\pi$
$410$	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	$-0.715513\pi$
$419$	−10746.6	−1.25300	−0.626500	+	0.779421i	$0.715513\pi$
$420$	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	$-0.432191\pi$
$433$	3809.88	0.422844	0.211422	+	0.977395i	$0.432191\pi$
$434$	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	$-0.738573\pi$
$439$	−12532.8	−1.36254	−0.681272	+	0.732030i	$0.738573\pi$
$440$	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	$-0.732507\pi$
$461$	−13208.0	−1.33440	−0.667199	+	0.744880i	$0.732507\pi$
$462$	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	$-0.417365\pi$
$467$	5181.23	0.513402	0.256701	+	0.966491i	$0.417365\pi$
$468$	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	$-0.502034\pi$
$479$	−133.998	−0.0127819	−0.00639096	+	0.999980i	$0.502034\pi$
$480$	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	$-0.635082\pi$
$503$	−9289.98	−0.823498	−0.411749	+	0.911297i	$0.635082\pi$
$504$	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	$-0.594092\pi$
$509$	−6690.62	−0.582626	−0.291313	+	0.956628i	$0.594092\pi$
$510$	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	$-0.513764\pi$
$521$	−1028.09	−0.0864522	−0.0432261	+	0.999065i	$0.513764\pi$
$522$	3999.74	0.335371
$523$	−618.266	−0.0516919	−0.0258460	−	0.999666i	$-0.508228\pi$
$523$	−618.266	−0.0516919	−0.0258460	+	0.999666i	$0.508228\pi$
$524$	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	$-0.142232\pi$
$563$	24094.2	1.80364	0.901819	+	0.432115i	$0.142232\pi$
$564$	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	$-0.607706\pi$
$577$	−9201.64	−0.663899	−0.331949	+	0.943297i	$0.607706\pi$
$578$	−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	$-0.437474\pi$
$587$	5551.42	0.390343	0.195172	+	0.980769i	$0.437474\pi$
$588$	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	$-0.562009\pi$
$593$	−5590.69	−0.387153	−0.193577	+	0.981085i	$0.562009\pi$
$594$	−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	$-0.548895\pi$
$601$	−4508.67	−0.306011	−0.153005	+	0.988225i	$0.548895\pi$
$602$	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	$-0.461240\pi$
$607$	3633.07	0.242935	0.121468	+	0.992595i	$0.461240\pi$
$608$	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	$-0.737889\pi$
$619$	−20935.5	−1.35940	−0.679699	+	0.733492i	$0.737889\pi$
$620$	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	$-0.748746\pi$
$643$	−22967.5	−1.40863	−0.704315	+	0.709887i	$0.748746\pi$
$644$	0	0
$645$	0	0
$646$	17595.7	1.07166
$647$	14314.8	0.869821	0.434910	−	0.900474i	$-0.356780\pi$
$647$	14314.8	0.869821	0.434910	+	0.900474i	$0.356780\pi$
$648$	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	$-0.271749\pi$
$661$	22336.6	1.31436	0.657180	+	0.753734i	$0.271749\pi$
$662$	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	$-0.870112\pi$
$677$	−32337.4	−1.83579	−0.917894	+	0.396826i	$0.870112\pi$
$678$	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	$-0.615184\pi$
$691$	−12860.8	−0.708030	−0.354015	+	0.935240i	$0.615184\pi$
$692$	−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	$-0.266980\pi$
$719$	25772.6	1.33680	0.668398	+	0.743804i	$0.266980\pi$
$720$	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	$-0.712757\pi$
$727$	−24295.8	−1.23945	−0.619727	+	0.784818i	$0.712757\pi$
$728$	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	$-0.549414\pi$
$733$	−6136.79	−0.309232	−0.154616	+	0.987975i	$0.549414\pi$
$734$	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	$-0.140635\pi$
$761$	37954.5	1.80795	0.903976	+	0.427584i	$0.140635\pi$
$762$	−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	$-0.522542\pi$
$769$	−3017.80	−0.141514	−0.0707572	+	0.997494i	$0.522542\pi$
$770$	0	0
$771$	−25551.4	−1.19353
$772$	−17734.6	−0.826792
$773$	102.457	0.00476728	0.00238364	−	0.999997i	$-0.499241\pi$
$773$	102.457	0.00476728	0.00238364	+	0.999997i	$0.499241\pi$
$774$	−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	$-0.409243\pi$
$787$	12420.0	0.562546	0.281273	+	0.959628i	$0.409243\pi$
$788$	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	$-0.142806\pi$
$797$	40547.2	1.80208	0.901039	+	0.433739i	$0.142806\pi$
$798$	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	$-0.652666\pi$
$811$	−21314.5	−0.922876	−0.461438	+	0.887172i	$0.652666\pi$
$812$	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	$-0.552728\pi$
$829$	−7871.60	−0.329785	−0.164893	+	0.986312i	$0.552728\pi$
$830$	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	$-0.251363\pi$
$839$	34220.8	1.40815	0.704073	+	0.710128i	$0.251363\pi$
$840$	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	$-0.734795\pi$
$853$	−33509.6	−1.34507	−0.672536	+	0.740065i	$0.734795\pi$
$854$	0	0
$855$	0	0
$856$	9230.34	0.368559
$857$	−8614.74	−0.343377	−0.171688	−	0.985151i	$-0.554922\pi$
$857$	−8614.74	−0.343377	−0.171688	+	0.985151i	$0.554922\pi$
$858$	46104.7	1.83449
$859$	4363.48	0.173318	0.0866589	−	0.996238i	$-0.472381\pi$
$859$	4363.48	0.173318	0.0866589	+	0.996238i	$0.472381\pi$
$860$	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	$-0.672993\pi$
$881$	−27044.5	−1.03423	−0.517113	+	0.855917i	$0.672993\pi$
$882$	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	$-0.523445\pi$
$887$	−3888.02	−0.147178	−0.0735889	+	0.997289i	$0.523445\pi$
$888$	−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	$-0.742645\pi$
$929$	−39108.2	−1.38116	−0.690581	+	0.723255i	$0.742645\pi$
$930$	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	$-0.503465\pi$
$937$	−624.451	−0.0217715	−0.0108858	+	0.999941i	$0.503465\pi$
$938$	0	0
$939$	−22231.7	−0.772634
$940$	0	0
$941$	53525.4	1.85428	0.927141	−	0.374713i	$-0.122259\pi$
$941$	53525.4	1.85428	0.927141	+	0.374713i	$0.122259\pi$
$942$	−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	$-0.356624\pi$
$971$	26345.1	0.870704	0.435352	+	0.900260i	$0.356624\pi$
$972$	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	$-0.655361\pi$
$983$	−28904.8	−0.937864	−0.468932	+	0.883234i	$0.655361\pi$
$984$	−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	$-0.301879\pi$
$997$	36706.4	1.16600	0.583000	+	0.812472i	$0.301879\pi$
$998$	−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.cc.1.1		3
5.4	even	2		2450.4.a.ci.1.3		3
7.2	even	3		350.4.e.j.151.3	yes	6
7.4	even	3		350.4.e.j.51.3	yes	6
7.6	odd	2		2450.4.a.cd.1.3		3
35.2	odd	12		350.4.j.h.249.1		12
35.4	even	6		350.4.e.i.51.1	✓	6
35.9	even	6		350.4.e.i.151.1	yes	6
35.18	odd	12		350.4.j.h.149.1		12
35.23	odd	12		350.4.j.h.249.6		12
35.32	odd	12		350.4.j.h.149.6		12
35.34	odd	2		2450.4.a.ch.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.e.i.51.1	✓	6	35.4	even	6
350.4.e.i.151.1	yes	6	35.9	even	6
350.4.e.j.51.3	yes	6	7.4	even	3
350.4.e.j.151.3	yes	6	7.2	even	3
350.4.j.h.149.1		12	35.18	odd	12
350.4.j.h.149.6		12	35.32	odd	12
350.4.j.h.249.1		12	35.2	odd	12
350.4.j.h.249.6		12	35.23	odd	12
2450.4.a.cc.1.1		3	1.1	even	1	trivial
2450.4.a.cd.1.3		3	7.6	odd	2
2450.4.a.ch.1.1		3	35.34	odd	2
2450.4.a.ci.1.3		3	5.4	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -6.92433 q^{3} +4.00000 q^{4} +13.8487 q^{6} -8.00000 q^{8} +20.9463 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -6.92433 q^{3} +4.00000 q^{4} +13.8487 q^{6} -8.00000 q^{8} +20.9463 q^{9} +37.3629 q^{11} -27.6973 q^{12} +89.1041 q^{13} +16.0000 q^{16} +70.1578 q^{17} -41.8926 q^{18} -125.401 q^{19} -74.7257 q^{22} +6.26857 q^{23} +55.3946 q^{24} -178.208 q^{26} +41.9179 q^{27} -95.4760 q^{29} -217.763 q^{31} -32.0000 q^{32} -258.713 q^{33} -140.316 q^{34} +83.7851 q^{36} -210.826 q^{37} +250.802 q^{38} -616.986 q^{39} -12.2490 q^{41} +131.746 q^{43} +149.451 q^{44} -12.5371 q^{46} -364.121 q^{47} -110.789 q^{48} -485.795 q^{51} +356.416 q^{52} +498.456 q^{53} -83.8358 q^{54} +868.318 q^{57} +190.952 q^{58} -358.027 q^{59} -154.608 q^{61} +435.526 q^{62} +64.0000 q^{64} +517.425 q^{66} +67.5326 q^{67} +280.631 q^{68} -43.4056 q^{69} -517.772 q^{71} -167.570 q^{72} +674.766 q^{73} +421.651 q^{74} -501.604 q^{76} +1233.97 q^{78} +931.682 q^{79} -855.803 q^{81} +24.4980 q^{82} -1435.72 q^{83} -263.493 q^{86} +661.107 q^{87} -298.903 q^{88} -336.681 q^{89} +25.0743 q^{92} +1507.86 q^{93} +728.242 q^{94} +221.578 q^{96} -179.620 q^{97} +782.613 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 3 q^{3} + 12 q^{4} + 6 q^{6} - 24 q^{8} + 56 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 3 * q^3 + 12 * q^4 + 6 * q^6 - 24 * q^8 + 56 * q^9	\( 3 q - 6 q^{2} - 3 q^{3} + 12 q^{4} + 6 q^{6} - 24 q^{8} + 56 q^{9} - 26 q^{11} - 12 q^{12} + 80 q^{13} + 48 q^{16} + 30 q^{17} - 112 q^{18} - 18 q^{19} + 52 q^{22} + 53 q^{23} + 24 q^{24} - 160 q^{26} + 324 q^{27} - 404 q^{29} - 615 q^{31} - 96 q^{32} - 314 q^{33} - 60 q^{34} + 224 q^{36} - 426 q^{37} + 36 q^{38} - 90 q^{39} - 101 q^{41} + 249 q^{43} - 104 q^{44} - 106 q^{46} + 402 q^{47} - 48 q^{48} - 339 q^{51} + 320 q^{52} - 390 q^{53} - 648 q^{54} + 1667 q^{57} + 808 q^{58} - 91 q^{59} - 647 q^{61} + 1230 q^{62} + 192 q^{64} + 628 q^{66} - 708 q^{67} + 120 q^{68} - 1548 q^{69} + 533 q^{71} - 448 q^{72} + 772 q^{73} + 852 q^{74} - 72 q^{76} + 180 q^{78} + 1421 q^{79} - 1393 q^{81} + 202 q^{82} - 520 q^{83} - 498 q^{86} - 833 q^{87} + 208 q^{88} - 194 q^{89} + 212 q^{92} - 148 q^{93} - 804 q^{94} + 96 q^{96} - 1027 q^{97} - 57 q^{99}+O(q^{100}) \) 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

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