LMFDB - Embedded newform 2450.4.a.cl.1.2

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:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{29})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 17x^{2} + 18x + 23 $ x^4 - 2x^3 - 17x^2 + 18*x + 23
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.77837$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -1.68657 q^{3} +4.00000 q^{4} +3.37313 q^{6} -8.00000 q^{8} -24.1555 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -1.68657 q^{3} +4.00000 q^{4} +3.37313 q^{6} -8.00000 q^{8} -24.1555 q^{9} +30.0813 q^{11} -6.74626 q^{12} +17.7876 q^{13} +16.0000 q^{16} -32.7468 q^{17} +48.3110 q^{18} -57.2282 q^{19} -60.1626 q^{22} +54.0813 q^{23} +13.4925 q^{24} -35.5752 q^{26} +86.2771 q^{27} -164.021 q^{29} +230.150 q^{31} -32.0000 q^{32} -50.7341 q^{33} +65.4936 q^{34} -96.6220 q^{36} -4.00714 q^{37} +114.456 q^{38} -30.0000 q^{39} -120.711 q^{41} +91.5478 q^{43} +120.325 q^{44} -108.163 q^{46} +62.9900 q^{47} -26.9851 q^{48} +55.2297 q^{51} +71.1505 q^{52} +122.519 q^{53} -172.554 q^{54} +96.5192 q^{57} +328.043 q^{58} -39.3458 q^{59} -661.945 q^{61} -460.299 q^{62} +64.0000 q^{64} +101.468 q^{66} +254.230 q^{67} -130.987 q^{68} -91.2117 q^{69} -163.519 q^{71} +193.244 q^{72} +38.4561 q^{73} +8.01428 q^{74} -228.913 q^{76} +60.0000 q^{78} -253.727 q^{79} +506.686 q^{81} +241.421 q^{82} +1099.67 q^{83} -183.096 q^{86} +276.633 q^{87} -240.651 q^{88} +1071.61 q^{89} +216.325 q^{92} -388.163 q^{93} -125.980 q^{94} +53.9701 q^{96} -500.319 q^{97} -726.629 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} - 32 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 - 32 * q^9	$ 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} - 32 q^{9} - 52 q^{11} + 64 q^{16} + 64 q^{18} + 104 q^{22} + 44 q^{23} + 184 q^{29} - 128 q^{32} - 128 q^{36} + 264 q^{37} - 120 q^{39} - 208 q^{44} - 88 q^{46} + 264 q^{51} + 1244 q^{53} + 1140 q^{57} - 368 q^{58} + 256 q^{64} + 1060 q^{67} - 1408 q^{71} + 256 q^{72} - 528 q^{74} + 240 q^{78} - 2652 q^{79} - 752 q^{81} + 416 q^{88} + 176 q^{92} - 1208 q^{93} - 2368 q^{99}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 - 32 * q^9 - 52 * q^11 + 64 * q^16 + 64 * q^18 + 104 * q^22 + 44 * q^23 + 184 * q^29 - 128 * q^32 - 128 * q^36 + 264 * q^37 - 120 * q^39 - 208 * q^44 - 88 * q^46 + 264 * q^51 + 1244 * q^53 + 1140 * q^57 - 368 * q^58 + 256 * q^64 + 1060 * q^67 - 1408 * q^71 + 256 * q^72 - 528 * q^74 + 240 * q^78 - 2652 * q^79 - 752 * q^81 + 416 * q^88 + 176 * q^92 - 1208 * q^93 - 2368 * 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
$3$	−1.68657	−0.324580	−0.162290	−	0.986743i	$-0.551888\pi$
$3$	−1.68657	−0.324580	−0.162290	+	0.986743i	$0.551888\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.37313	0.229513
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−24.1555	−0.894648
$10$	0	0
$11$	30.0813	0.824533	0.412266	−	0.911063i	$-0.364737\pi$
$11$	30.0813	0.824533	0.412266	+	0.911063i	$0.364737\pi$
$12$	−6.74626	−0.162290
$13$	17.7876	0.379492	0.189746	−	0.981833i	$-0.439234\pi$
$13$	17.7876	0.379492	0.189746	+	0.981833i	$0.439234\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−32.7468	−0.467192	−0.233596	−	0.972334i	$-0.575049\pi$
$17$	−32.7468	−0.467192	−0.233596	+	0.972334i	$0.575049\pi$
$18$	48.3110	0.632612
$19$	−57.2282	−0.691003	−0.345502	−	0.938418i	$-0.612291\pi$
$19$	−57.2282	−0.691003	−0.345502	+	0.938418i	$0.612291\pi$
$20$	0	0
$21$	0	0
$22$	−60.1626	−0.583033
$23$	54.0813	0.490293	0.245146	−	0.969486i	$-0.421164\pi$
$23$	54.0813	0.490293	0.245146	+	0.969486i	$0.421164\pi$
$24$	13.4925	0.114756
$25$	0	0
$26$	−35.5752	−0.268341
$27$	86.2771	0.614964
$28$	0	0
$29$	−164.021	−1.05028	−0.525138	−	0.851017i	$-0.675986\pi$
$29$	−164.021	−1.05028	−0.525138	+	0.851017i	$0.675986\pi$
$30$	0	0
$31$	230.150	1.33342	0.666711	−	0.745316i	$-0.267701\pi$
$31$	230.150	1.33342	0.666711	+	0.745316i	$0.267701\pi$
$32$	−32.0000	−0.176777
$33$	−50.7341	−0.267627
$34$	65.4936	0.330355
$35$	0	0
$36$	−96.6220	−0.447324
$37$	−4.00714	−0.0178046	−0.00890230	−	0.999960i	$-0.502834\pi$
$37$	−4.00714	−0.0178046	−0.00890230	+	0.999960i	$0.502834\pi$
$38$	114.456	0.488613
$39$	−30.0000	−0.123176
$40$	0	0
$41$	−120.711	−0.459800	−0.229900	−	0.973214i	$-0.573840\pi$
$41$	−120.711	−0.459800	−0.229900	+	0.973214i	$0.573840\pi$
$42$	0	0
$43$	91.5478	0.324672	0.162336	−	0.986736i	$-0.448097\pi$
$43$	91.5478	0.324672	0.162336	+	0.986736i	$0.448097\pi$
$44$	120.325	0.412266
$45$	0	0
$46$	−108.163	−0.346689
$47$	62.9900	0.195490	0.0977451	−	0.995211i	$-0.468837\pi$
$47$	62.9900	0.195490	0.0977451	+	0.995211i	$0.468837\pi$
$48$	−26.9851	−0.0811450
$49$	0	0
$50$	0	0
$51$	55.2297	0.151641
$52$	71.1505	0.189746
$53$	122.519	0.317534	0.158767	−	0.987316i	$-0.449248\pi$
$53$	122.519	0.317534	0.158767	+	0.987316i	$0.449248\pi$
$54$	−172.554	−0.434846
$55$	0	0
$56$	0	0
$57$	96.5192	0.224286
$58$	328.043	0.742658
$59$	−39.3458	−0.0868202	−0.0434101	−	0.999057i	$-0.513822\pi$
$59$	−39.3458	−0.0868202	−0.0434101	+	0.999057i	$0.513822\pi$
$60$	0	0
$61$	−661.945	−1.38940	−0.694700	−	0.719300i	$-0.744463\pi$
$61$	−661.945	−1.38940	−0.694700	+	0.719300i	$0.744463\pi$
$62$	−460.299	−0.942872
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	101.468	0.189241
$67$	254.230	0.463569	0.231784	−	0.972767i	$-0.425544\pi$
$67$	254.230	0.463569	0.231784	+	0.972767i	$0.425544\pi$
$68$	−130.987	−0.233596
$69$	−91.2117	−0.159139
$70$	0	0
$71$	−163.519	−0.273326	−0.136663	−	0.990618i	$-0.543638\pi$
$71$	−163.519	−0.273326	−0.136663	+	0.990618i	$0.543638\pi$
$72$	193.244	0.316306
$73$	38.4561	0.0616568	0.0308284	−	0.999525i	$-0.490185\pi$
$73$	38.4561	0.0616568	0.0308284	+	0.999525i	$0.490185\pi$
$74$	8.01428	0.0125898
$75$	0	0
$76$	−228.913	−0.345502
$77$	0	0
$78$	60.0000	0.0870982
$79$	−253.727	−0.361349	−0.180675	−	0.983543i	$-0.557828\pi$
$79$	−253.727	−0.361349	−0.180675	+	0.983543i	$0.557828\pi$
$80$	0	0
$81$	506.686	0.695043
$82$	241.421	0.325128
$83$	1099.67	1.45427	0.727133	−	0.686496i	$-0.240852\pi$
$83$	1099.67	1.45427	0.727133	+	0.686496i	$0.240852\pi$
$84$	0	0
$85$	0	0
$86$	−183.096	−0.229578
$87$	276.633	0.340899
$88$	−240.651	−0.291516
$89$	1071.61	1.27630	0.638150	−	0.769912i	$-0.279700\pi$
$89$	1071.61	1.27630	0.638150	+	0.769912i	$0.279700\pi$
$90$	0	0
$91$	0	0
$92$	216.325	0.245146
$93$	−388.163	−0.432802
$94$	−125.980	−0.138232
$95$	0	0
$96$	53.9701	0.0573781
$97$	−500.319	−0.523708	−0.261854	−	0.965107i	$-0.584334\pi$
$97$	−500.319	−0.523708	−0.261854	+	0.965107i	$0.584334\pi$
$98$	0	0
$99$	−726.629	−0.737666
$100$	0	0
$101$	177.218	0.174593	0.0872965	−	0.996182i	$-0.472177\pi$
$101$	177.218	0.174593	0.0872965	+	0.996182i	$0.472177\pi$
$102$	−110.459	−0.107227
$103$	−627.965	−0.600731	−0.300365	−	0.953824i	$-0.597109\pi$
$103$	−627.965	−0.600731	−0.300365	+	0.953824i	$0.597109\pi$
$104$	−142.301	−0.134171
$105$	0	0
$106$	−245.038	−0.224531
$107$	2021.37	1.82629	0.913147	−	0.407630i	$-0.133645\pi$
$107$	2021.37	1.82629	0.913147	+	0.407630i	$0.133645\pi$
$108$	345.109	0.307482
$109$	−1879.96	−1.65200	−0.825998	−	0.563672i	$-0.809388\pi$
$109$	−1879.96	−1.65200	−0.825998	+	0.563672i	$0.809388\pi$
$110$	0	0
$111$	6.75831	0.00577901
$112$	0	0
$113$	−1071.38	−0.891920	−0.445960	−	0.895053i	$-0.647138\pi$
$113$	−1071.38	−0.891920	−0.445960	+	0.895053i	$0.647138\pi$
$114$	−193.038	−0.158594
$115$	0	0
$116$	−656.086	−0.525138
$117$	−429.669	−0.339512
$118$	78.6917	0.0613911
$119$	0	0
$120$	0	0
$121$	−426.114	−0.320146
$122$	1323.89	0.982454
$123$	203.586	0.149242
$124$	920.599	0.666711
$125$	0	0
$126$	0	0
$127$	371.768	0.259756	0.129878	−	0.991530i	$-0.458541\pi$
$127$	371.768	0.259756	0.129878	+	0.991530i	$0.458541\pi$
$128$	−128.000	−0.0883883
$129$	−154.401	−0.105382
$130$	0	0
$131$	815.378	0.543816	0.271908	−	0.962323i	$-0.412345\pi$
$131$	815.378	0.543816	0.271908	+	0.962323i	$0.412345\pi$
$132$	−202.937	−0.133813
$133$	0	0
$134$	−508.459	−0.327793
$135$	0	0
$136$	261.975	0.165177
$137$	−219.598	−0.136945	−0.0684727	−	0.997653i	$-0.521813\pi$
$137$	−219.598	−0.136945	−0.0684727	+	0.997653i	$0.521813\pi$
$138$	182.423	0.112528
$139$	−803.999	−0.490607	−0.245303	−	0.969446i	$-0.578888\pi$
$139$	−803.999	−0.490607	−0.245303	+	0.969446i	$0.578888\pi$
$140$	0	0
$141$	−106.237	−0.0634522
$142$	327.038	0.193271
$143$	535.075	0.312904
$144$	−386.488	−0.223662
$145$	0	0
$146$	−76.9122	−0.0435980
$147$	0	0
$148$	−16.0286	−0.00890230
$149$	−1771.52	−0.974017	−0.487008	−	0.873397i	$-0.661912\pi$
$149$	−1771.52	−0.974017	−0.487008	+	0.873397i	$0.661912\pi$
$150$	0	0
$151$	−224.720	−0.121109	−0.0605546	−	0.998165i	$-0.519287\pi$
$151$	−224.720	−0.121109	−0.0605546	+	0.998165i	$0.519287\pi$
$152$	457.826	0.244307
$153$	791.015	0.417973
$154$	0	0
$155$	0	0
$156$	−120.000	−0.0615878
$157$	3211.74	1.63264	0.816320	−	0.577600i	$-0.196010\pi$
$157$	3211.74	1.63264	0.816320	+	0.577600i	$0.196010\pi$
$158$	507.455	0.255512
$159$	−206.637	−0.103065
$160$	0	0
$161$	0	0
$162$	−1013.37	−0.491470
$163$	2935.63	1.41065	0.705327	−	0.708882i	$-0.250800\pi$
$163$	2935.63	1.41065	0.705327	+	0.708882i	$0.250800\pi$
$164$	−482.842	−0.229900
$165$	0	0
$166$	−2199.33	−1.02832
$167$	−3649.80	−1.69120	−0.845599	−	0.533819i	$-0.820756\pi$
$167$	−3649.80	−1.69120	−0.845599	+	0.533819i	$0.820756\pi$
$168$	0	0
$169$	−1880.60	−0.855986
$170$	0	0
$171$	1382.38	0.618205
$172$	366.191	0.162336
$173$	−261.468	−0.114908	−0.0574539	−	0.998348i	$-0.518298\pi$
$173$	−261.468	−0.114908	−0.0574539	+	0.998348i	$0.518298\pi$
$174$	−553.266	−0.241052
$175$	0	0
$176$	481.301	0.206133
$177$	66.3593	0.0281801
$178$	−2143.23	−0.902481
$179$	−2782.56	−1.16189	−0.580945	−	0.813943i	$-0.697317\pi$
$179$	−2782.56	−1.16189	−0.580945	+	0.813943i	$0.697317\pi$
$180$	0	0
$181$	−2071.95	−0.850866	−0.425433	−	0.904990i	$-0.639878\pi$
$181$	−2071.95	−0.850866	−0.425433	+	0.904990i	$0.639878\pi$
$182$	0	0
$183$	1116.41	0.450971
$184$	−432.651	−0.173345
$185$	0	0
$186$	776.325	0.306037
$187$	−985.067	−0.385215
$188$	251.960	0.0977451
$189$	0	0
$190$	0	0
$191$	874.482	0.331284	0.165642	−	0.986186i	$-0.447030\pi$
$191$	874.482	0.331284	0.165642	+	0.986186i	$0.447030\pi$
$192$	−107.940	−0.0405725
$193$	3160.98	1.17892	0.589461	−	0.807796i	$-0.299340\pi$
$193$	3160.98	1.17892	0.589461	+	0.807796i	$0.299340\pi$
$194$	1000.64	0.370318
$195$	0	0
$196$	0	0
$197$	−2458.21	−0.889037	−0.444518	−	0.895770i	$-0.646625\pi$
$197$	−2458.21	−0.889037	−0.444518	+	0.895770i	$0.646625\pi$
$198$	1453.26	0.521609
$199$	2702.72	0.962767	0.481383	−	0.876510i	$-0.340134\pi$
$199$	2702.72	0.962767	0.481383	+	0.876510i	$0.340134\pi$
$200$	0	0
$201$	−428.775	−0.150465
$202$	−354.437	−0.123456
$203$	0	0
$204$	220.919	0.0758206
$205$	0	0
$206$	1255.93	0.424781
$207$	−1306.36	−0.438639
$208$	284.602	0.0948730
$209$	−1721.50	−0.569755
$210$	0	0
$211$	2.28242	0.000744683 0	0.000372342	−	1.00000i	$-0.499881\pi$
$211$	2.28242	0.000744683 0	0.000372342		1.00000i	$0.499881\pi$
$212$	490.077	0.158767
$213$	275.786	0.0887162
$214$	−4042.75	−1.29138
$215$	0	0
$216$	−690.217	−0.217423
$217$	0	0
$218$	3759.92	1.16814
$219$	−64.8588	−0.0200126
$220$	0	0
$221$	−582.488	−0.177296
$222$	−13.5166	−0.00408638
$223$	489.673	0.147044	0.0735222	−	0.997294i	$-0.476576\pi$
$223$	489.673	0.147044	0.0735222	+	0.997294i	$0.476576\pi$
$224$	0	0
$225$	0	0
$226$	2142.76	0.630682
$227$	4739.95	1.38591	0.692955	−	0.720981i	$-0.256309\pi$
$227$	4739.95	1.38591	0.692955	+	0.720981i	$0.256309\pi$
$228$	386.077	0.112143
$229$	3729.21	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$229$	3729.21	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$230$	0	0
$231$	0	0
$232$	1312.17	0.371329
$233$	−465.970	−0.131016	−0.0655080	−	0.997852i	$-0.520867\pi$
$233$	−465.970	−0.131016	−0.0655080	+	0.997852i	$0.520867\pi$
$234$	859.338	0.240071
$235$	0	0
$236$	−157.383	−0.0434101
$237$	427.928	0.117287
$238$	0	0
$239$	−4282.87	−1.15915	−0.579573	−	0.814921i	$-0.696780\pi$
$239$	−4282.87	−1.15915	−0.579573	+	0.814921i	$0.696780\pi$
$240$	0	0
$241$	6190.50	1.65463	0.827313	−	0.561741i	$-0.189868\pi$
$241$	6190.50	1.65463	0.827313	+	0.561741i	$0.189868\pi$
$242$	852.229	0.226377
$243$	−3184.04	−0.840561
$244$	−2647.78	−0.694700
$245$	0	0
$246$	−407.173	−0.105530
$247$	−1017.95	−0.262230
$248$	−1841.20	−0.471436
$249$	−1854.66	−0.472026
$250$	0	0
$251$	−5811.60	−1.46145	−0.730727	−	0.682669i	$-0.760819\pi$
$251$	−5811.60	−1.46145	−0.730727	+	0.682669i	$0.760819\pi$
$252$	0	0
$253$	1626.84	0.404262
$254$	−743.535	−0.183675
$255$	0	0
$256$	256.000	0.0625000
$257$	−1588.20	−0.385484	−0.192742	−	0.981249i	$-0.561738\pi$
$257$	−1588.20	−0.385484	−0.192742	+	0.981249i	$0.561738\pi$
$258$	308.803	0.0745164
$259$	0	0
$260$	0	0
$261$	3962.02	0.939628
$262$	−1630.76	−0.384536
$263$	5478.27	1.28443	0.642214	−	0.766525i	$-0.278016\pi$
$263$	5478.27	1.28443	0.642214	+	0.766525i	$0.278016\pi$
$264$	405.873	0.0946203
$265$	0	0
$266$	0	0
$267$	−1807.35	−0.414261
$268$	1016.92	0.231784
$269$	1706.64	0.386823	0.193412	−	0.981118i	$-0.438045\pi$
$269$	1706.64	0.386823	0.193412	+	0.981118i	$0.438045\pi$
$270$	0	0
$271$	6783.20	1.52048	0.760240	−	0.649642i	$-0.225081\pi$
$271$	6783.20	1.52048	0.760240	+	0.649642i	$0.225081\pi$
$272$	−523.949	−0.116798
$273$	0	0
$274$	439.196	0.0968350
$275$	0	0
$276$	−364.847	−0.0795696
$277$	−6989.73	−1.51614	−0.758072	−	0.652170i	$-0.773859\pi$
$277$	−6989.73	−1.51614	−0.758072	+	0.652170i	$0.773859\pi$
$278$	1608.00	0.346911
$279$	−5559.38	−1.19294
$280$	0	0
$281$	528.788	0.112259	0.0561297	−	0.998423i	$-0.482124\pi$
$281$	528.788	0.112259	0.0561297	+	0.998423i	$0.482124\pi$
$282$	212.474	0.0448674
$283$	3734.65	0.784459	0.392230	−	0.919867i	$-0.371704\pi$
$283$	3734.65	0.784459	0.392230	+	0.919867i	$0.371704\pi$
$284$	−654.077	−0.136663
$285$	0	0
$286$	−1070.15	−0.221256
$287$	0	0
$288$	772.976	0.158153
$289$	−3840.65	−0.781731
$290$	0	0
$291$	843.821	0.169985
$292$	153.824	0.0308284
$293$	8733.42	1.74134	0.870669	−	0.491869i	$-0.163686\pi$
$293$	8733.42	1.74134	0.870669	+	0.491869i	$0.163686\pi$
$294$	0	0
$295$	0	0
$296$	32.0571	0.00629488
$297$	2595.33	0.507058
$298$	3543.04	0.688734
$299$	961.978	0.186062
$300$	0	0
$301$	0	0
$302$	449.441	0.0856371
$303$	−298.891	−0.0566694
$304$	−915.652	−0.172751
$305$	0	0
$306$	−1582.03	−0.295551
$307$	−9809.54	−1.82365	−0.911824	−	0.410581i	$-0.865326\pi$
$307$	−9809.54	−1.82365	−0.911824	+	0.410581i	$0.865326\pi$
$308$	0	0
$309$	1059.10	0.194985
$310$	0	0
$311$	−4731.52	−0.862701	−0.431350	−	0.902185i	$-0.641963\pi$
$311$	−4731.52	−0.862701	−0.431350	+	0.902185i	$0.641963\pi$
$312$	240.000	0.0435491
$313$	−7397.34	−1.33585	−0.667927	−	0.744227i	$-0.732818\pi$
$313$	−7397.34	−1.33585	−0.667927	+	0.744227i	$0.732818\pi$
$314$	−6423.48	−1.15445
$315$	0	0
$316$	−1014.91	−0.180675
$317$	1000.19	0.177212	0.0886061	−	0.996067i	$-0.471759\pi$
$317$	1000.19	0.177212	0.0886061	+	0.996067i	$0.471759\pi$
$318$	413.274	0.0728781
$319$	−4933.98	−0.865987
$320$	0	0
$321$	−3409.18	−0.592778
$322$	0	0
$323$	1874.04	0.322831
$324$	2026.75	0.347521
$325$	0	0
$326$	−5871.26	−0.997482
$327$	3170.68	0.536205
$328$	965.684	0.162564
$329$	0	0
$330$	0	0
$331$	−329.356	−0.0546920	−0.0273460	−	0.999626i	$-0.508706\pi$
$331$	−329.356	−0.0546920	−0.0273460	+	0.999626i	$0.508706\pi$
$332$	4398.67	0.727133
$333$	96.7945	0.0159288
$334$	7299.60	1.19586
$335$	0	0
$336$	0	0
$337$	−7263.17	−1.17404	−0.587018	−	0.809574i	$-0.699698\pi$
$337$	−7263.17	−1.17404	−0.587018	+	0.809574i	$0.699698\pi$
$338$	3761.20	0.605273
$339$	1806.95	0.289499
$340$	0	0
$341$	6923.21	1.09945
$342$	−2764.75	−0.437137
$343$	0	0
$344$	−732.382	−0.114789
$345$	0	0
$346$	522.936	0.0812520
$347$	−2707.25	−0.418827	−0.209413	−	0.977827i	$-0.567155\pi$
$347$	−2707.25	−0.418827	−0.209413	+	0.977827i	$0.567155\pi$
$348$	1106.53	0.170449
$349$	−8710.93	−1.33606	−0.668031	−	0.744134i	$-0.732862\pi$
$349$	−8710.93	−1.33606	−0.668031	+	0.744134i	$0.732862\pi$
$350$	0	0
$351$	1534.66	0.233374
$352$	−962.602	−0.145758
$353$	−12973.1	−1.95606	−0.978028	−	0.208475i	$-0.933150\pi$
$353$	−12973.1	−1.95606	−0.978028	+	0.208475i	$0.933150\pi$
$354$	−132.719	−0.0199263
$355$	0	0
$356$	4286.45	0.638150
$357$	0	0
$358$	5565.12	0.821581
$359$	−4314.20	−0.634248	−0.317124	−	0.948384i	$-0.602717\pi$
$359$	−4314.20	−0.634248	−0.317124	+	0.948384i	$0.602717\pi$
$360$	0	0
$361$	−3583.93	−0.522515
$362$	4143.90	0.601653
$363$	718.670	0.103913
$364$	0	0
$365$	0	0
$366$	−2232.83	−0.318885
$367$	−1026.21	−0.145961	−0.0729804	−	0.997333i	$-0.523251\pi$
$367$	−1026.21	−0.145961	−0.0729804	+	0.997333i	$0.523251\pi$
$368$	865.301	0.122573
$369$	2915.82	0.411359
$370$	0	0
$371$	0	0
$372$	−1552.65	−0.216401
$373$	−10249.1	−1.42273	−0.711366	−	0.702821i	$-0.751923\pi$
$373$	−10249.1	−1.42273	−0.711366	+	0.702821i	$0.751923\pi$
$374$	1970.13	0.272388
$375$	0	0
$376$	−503.920	−0.0691162
$377$	−2917.55	−0.398572
$378$	0	0
$379$	−10291.5	−1.39483	−0.697414	−	0.716668i	$-0.745666\pi$
$379$	−10291.5	−1.39483	−0.697414	+	0.716668i	$0.745666\pi$
$380$	0	0
$381$	−627.011	−0.0843116
$382$	−1748.96	−0.234253
$383$	−1974.63	−0.263444	−0.131722	−	0.991287i	$-0.542051\pi$
$383$	−1974.63	−0.263444	−0.131722	+	0.991287i	$0.542051\pi$
$384$	215.880	0.0286891
$385$	0	0
$386$	−6321.95	−0.833624
$387$	−2211.38	−0.290467
$388$	−2001.28	−0.261854
$389$	−14147.9	−1.84402	−0.922011	−	0.387164i	$-0.873455\pi$
$389$	−14147.9	−1.84402	−0.922011	+	0.387164i	$0.873455\pi$
$390$	0	0
$391$	−1770.99	−0.229061
$392$	0	0
$393$	−1375.19	−0.176512
$394$	4916.42	0.628644
$395$	0	0
$396$	−2906.52	−0.368833
$397$	−942.026	−0.119091	−0.0595453	−	0.998226i	$-0.518965\pi$
$397$	−942.026	−0.119091	−0.0595453	+	0.998226i	$0.518965\pi$
$398$	−5405.44	−0.680779
$399$	0	0
$400$	0	0
$401$	−1135.60	−0.141419	−0.0707094	−	0.997497i	$-0.522526\pi$
$401$	−1135.60	−0.141419	−0.0707094	+	0.997497i	$0.522526\pi$
$402$	857.550	0.106395
$403$	4093.81	0.506023
$404$	708.874	0.0872965
$405$	0	0
$406$	0	0
$407$	−120.540	−0.0146805
$408$	−441.837	−0.0536133
$409$	5217.55	0.630785	0.315393	−	0.948961i	$-0.397864\pi$
$409$	5217.55	0.630785	0.315393	+	0.948961i	$0.397864\pi$
$410$	0	0
$411$	370.366	0.0444497
$412$	−2511.86	−0.300365
$413$	0	0
$414$	2612.72	0.310165
$415$	0	0
$416$	−569.204	−0.0670854
$417$	1356.00	0.159241
$418$	3443.00	0.402877
$419$	−11667.2	−1.36033	−0.680166	−	0.733058i	$-0.738092\pi$
$419$	−11667.2	−1.36033	−0.680166	+	0.733058i	$0.738092\pi$
$420$	0	0
$421$	−3456.11	−0.400096	−0.200048	−	0.979786i	$-0.564110\pi$
$421$	−3456.11	−0.400096	−0.200048	+	0.979786i	$0.564110\pi$
$422$	−4.56484	−0.000526571 0
$423$	−1521.55	−0.174895
$424$	−980.154	−0.112265
$425$	0	0
$426$	−551.572	−0.0627318
$427$	0	0
$428$	8085.49	0.913147
$429$	−902.440	−0.101562
$430$	0	0
$431$	−7721.34	−0.862932	−0.431466	−	0.902129i	$-0.642003\pi$
$431$	−7721.34	−0.862932	−0.431466	+	0.902129i	$0.642003\pi$
$432$	1380.43	0.153741
$433$	9897.80	1.09852	0.549259	−	0.835652i	$-0.314910\pi$
$433$	9897.80	1.09852	0.549259	+	0.835652i	$0.314910\pi$
$434$	0	0
$435$	0	0
$436$	−7519.85	−0.825998
$437$	−3094.98	−0.338794
$438$	129.718	0.0141510
$439$	−4327.97	−0.470530	−0.235265	−	0.971931i	$-0.575596\pi$
$439$	−4327.97	−0.470530	−0.235265	+	0.971931i	$0.575596\pi$
$440$	0	0
$441$	0	0
$442$	1164.98	0.125367
$443$	−5400.24	−0.579172	−0.289586	−	0.957152i	$-0.593518\pi$
$443$	−5400.24	−0.579172	−0.289586	+	0.957152i	$0.593518\pi$
$444$	27.0332	0.00288951
$445$	0	0
$446$	−979.345	−0.103976
$447$	2987.78	0.316146
$448$	0	0
$449$	−16211.8	−1.70396	−0.851982	−	0.523571i	$-0.824600\pi$
$449$	−16211.8	−1.70396	−0.851982	+	0.523571i	$0.824600\pi$
$450$	0	0
$451$	−3631.13	−0.379120
$452$	−4285.52	−0.445960
$453$	379.006	0.0393096
$454$	−9479.90	−0.979986
$455$	0	0
$456$	−772.154	−0.0792970
$457$	−12802.1	−1.31041	−0.655207	−	0.755449i	$-0.727419\pi$
$457$	−12802.1	−1.31041	−0.655207	+	0.755449i	$0.727419\pi$
$458$	−7458.43	−0.760937
$459$	−2825.30	−0.287307
$460$	0	0
$461$	2861.43	0.289089	0.144545	−	0.989498i	$-0.453828\pi$
$461$	2861.43	0.289089	0.144545	+	0.989498i	$0.453828\pi$
$462$	0	0
$463$	−7551.04	−0.757941	−0.378970	−	0.925409i	$-0.623722\pi$
$463$	−7551.04	−0.757941	−0.378970	+	0.925409i	$0.623722\pi$
$464$	−2624.34	−0.262569
$465$	0	0
$466$	931.941	0.0926423
$467$	−276.741	−0.0274220	−0.0137110	−	0.999906i	$-0.504364\pi$
$467$	−276.741	−0.0274220	−0.0137110	+	0.999906i	$0.504364\pi$
$468$	−1718.68	−0.169756
$469$	0	0
$470$	0	0
$471$	−5416.81	−0.529922
$472$	314.767	0.0306956
$473$	2753.88	0.267703
$474$	−855.856	−0.0829341
$475$	0	0
$476$	0	0
$477$	−2959.51	−0.284081
$478$	8565.74	0.819639
$479$	19172.8	1.82887	0.914436	−	0.404731i	$-0.132635\pi$
$479$	19172.8	1.82887	0.914436	+	0.404731i	$0.132635\pi$
$480$	0	0
$481$	−71.2775	−0.00675671
$482$	−12381.0	−1.17000
$483$	0	0
$484$	−1704.46	−0.160073
$485$	0	0
$486$	6368.08	0.594367
$487$	−14100.0	−1.31198	−0.655988	−	0.754772i	$-0.727748\pi$
$487$	−14100.0	−1.31198	−0.655988	+	0.754772i	$0.727748\pi$
$488$	5295.56	0.491227
$489$	−4951.14	−0.457870
$490$	0	0
$491$	−837.108	−0.0769413	−0.0384706	−	0.999260i	$-0.512249\pi$
$491$	−837.108	−0.0769413	−0.0384706	+	0.999260i	$0.512249\pi$
$492$	814.345	0.0746210
$493$	5371.18	0.490681
$494$	2035.91	0.185425
$495$	0	0
$496$	3682.39	0.333356
$497$	0	0
$498$	3709.32	0.333773
$499$	−8556.76	−0.767642	−0.383821	−	0.923408i	$-0.625392\pi$
$499$	−8556.76	−0.767642	−0.383821	+	0.923408i	$0.625392\pi$
$500$	0	0
$501$	6155.63	0.548928
$502$	11623.2	1.03340
$503$	17713.8	1.57022	0.785108	−	0.619359i	$-0.212608\pi$
$503$	17713.8	1.57022	0.785108	+	0.619359i	$0.212608\pi$
$504$	0	0
$505$	0	0
$506$	−3253.67	−0.285857
$507$	3171.76	0.277836
$508$	1487.07	0.129878
$509$	−7545.37	−0.657059	−0.328529	−	0.944494i	$-0.606553\pi$
$509$	−7545.37	−0.657059	−0.328529	+	0.944494i	$0.606553\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−4937.49	−0.424942
$514$	3176.41	0.272578
$515$	0	0
$516$	−617.606	−0.0526910
$517$	1894.82	0.161188
$518$	0	0
$519$	440.983	0.0372967
$520$	0	0
$521$	−6.86127	−0.000576963 0	−0.000288481	−	1.00000i	$-0.500092\pi$
$521$	−6.86127	−0.000576963 0	−0.000288481		1.00000i	$0.500092\pi$
$522$	−7924.04	−0.664417
$523$	−19294.0	−1.61313	−0.806564	−	0.591146i	$-0.798676\pi$
$523$	−19294.0	−1.61313	−0.806564	+	0.591146i	$0.798676\pi$
$524$	3261.51	0.271908
$525$	0	0
$526$	−10956.5	−0.908228
$527$	−7536.67	−0.622965
$528$	−811.746	−0.0669067
$529$	−9242.21	−0.759613
$530$	0	0
$531$	950.418	0.0776735
$532$	0	0
$533$	−2147.15	−0.174491
$534$	3614.69	0.292927
$535$	0	0
$536$	−2033.84	−0.163896
$537$	4692.98	0.377126
$538$	−3413.27	−0.273525
$539$	0	0
$540$	0	0
$541$	5694.29	0.452526	0.226263	−	0.974066i	$-0.427349\pi$
$541$	5694.29	0.452526	0.226263	+	0.974066i	$0.427349\pi$
$542$	−13566.4	−1.07514
$543$	3494.48	0.276174
$544$	1047.90	0.0825887
$545$	0	0
$546$	0	0
$547$	−13080.7	−1.02247	−0.511236	−	0.859440i	$-0.670812\pi$
$547$	−13080.7	−1.02247	−0.511236	+	0.859440i	$0.670812\pi$
$548$	−878.391	−0.0684727
$549$	15989.6	1.24302
$550$	0	0
$551$	9386.66	0.725744
$552$	729.694	0.0562642
$553$	0	0
$554$	13979.5	1.07208
$555$	0	0
$556$	−3216.00	−0.245303
$557$	−10009.0	−0.761390	−0.380695	−	0.924701i	$-0.624315\pi$
$557$	−10009.0	−0.761390	−0.380695	+	0.924701i	$0.624315\pi$
$558$	11118.8	0.843539
$559$	1628.42	0.123211
$560$	0	0
$561$	1661.38	0.125033
$562$	−1057.58	−0.0793793
$563$	3655.28	0.273626	0.136813	−	0.990597i	$-0.456314\pi$
$563$	3655.28	0.273626	0.136813	+	0.990597i	$0.456314\pi$
$564$	−424.947	−0.0317261
$565$	0	0
$566$	−7469.30	−0.554697
$567$	0	0
$568$	1308.15	0.0966354
$569$	14426.8	1.06292	0.531460	−	0.847083i	$-0.321644\pi$
$569$	14426.8	1.06292	0.531460	+	0.847083i	$0.321644\pi$
$570$	0	0
$571$	−12753.0	−0.934671	−0.467335	−	0.884080i	$-0.654786\pi$
$571$	−12753.0	−0.934671	−0.467335	+	0.884080i	$0.654786\pi$
$572$	2140.30	0.156452
$573$	−1474.87	−0.107528
$574$	0	0
$575$	0	0
$576$	−1545.95	−0.111831
$577$	−13809.6	−0.996362	−0.498181	−	0.867073i	$-0.665998\pi$
$577$	−13809.6	−0.996362	−0.498181	+	0.867073i	$0.665998\pi$
$578$	7681.29	0.552768
$579$	−5331.20	−0.382655
$580$	0	0
$581$	0	0
$582$	−1687.64	−0.120198
$583$	3685.54	0.261817
$584$	−307.649	−0.0217990
$585$	0	0
$586$	−17466.8	−1.23131
$587$	9424.02	0.662642	0.331321	−	0.943518i	$-0.392506\pi$
$587$	9424.02	0.662642	0.331321	+	0.943518i	$0.392506\pi$
$588$	0	0
$589$	−13171.1	−0.921399
$590$	0	0
$591$	4145.93	0.288563
$592$	−64.1143	−0.00445115
$593$	−25799.4	−1.78660	−0.893301	−	0.449458i	$-0.851617\pi$
$593$	−25799.4	−1.78660	−0.893301	+	0.449458i	$0.851617\pi$
$594$	−5190.66	−0.358544
$595$	0	0
$596$	−7086.08	−0.487008
$597$	−4558.31	−0.312495
$598$	−1923.96	−0.131566
$599$	8220.66	0.560747	0.280373	−	0.959891i	$-0.409542\pi$
$599$	8220.66	0.560747	0.280373	+	0.959891i	$0.409542\pi$
$600$	0	0
$601$	−1024.63	−0.0695433	−0.0347717	−	0.999395i	$-0.511070\pi$
$601$	−1024.63	−0.0695433	−0.0347717	+	0.999395i	$0.511070\pi$
$602$	0	0
$603$	−6141.04	−0.414731
$604$	−898.881	−0.0605546
$605$	0	0
$606$	597.781	0.0400713
$607$	18409.3	1.23099	0.615493	−	0.788142i	$-0.288957\pi$
$607$	18409.3	1.23099	0.615493	+	0.788142i	$0.288957\pi$
$608$	1831.30	0.122153
$609$	0	0
$610$	0	0
$611$	1120.44	0.0741870
$612$	3164.06	0.208986
$613$	−11758.5	−0.774751	−0.387376	−	0.921922i	$-0.626618\pi$
$613$	−11758.5	−0.774751	−0.387376	+	0.921922i	$0.626618\pi$
$614$	19619.1	1.28951
$615$	0	0
$616$	0	0
$617$	15938.7	1.03998	0.519990	−	0.854172i	$-0.325936\pi$
$617$	15938.7	1.03998	0.519990	+	0.854172i	$0.325936\pi$
$618$	−2118.21	−0.137875
$619$	−698.196	−0.0453358	−0.0226679	−	0.999743i	$-0.507216\pi$
$619$	−698.196	−0.0453358	−0.0226679	+	0.999743i	$0.507216\pi$
$620$	0	0
$621$	4665.98	0.301513
$622$	9463.04	0.610021
$623$	0	0
$624$	−480.000	−0.0307939
$625$	0	0
$626$	14794.7	0.944591
$627$	2903.43	0.184931
$628$	12847.0	0.816320
$629$	131.221	0.00831817
$630$	0	0
$631$	5130.88	0.323704	0.161852	−	0.986815i	$-0.448253\pi$
$631$	5130.88	0.323704	0.161852	+	0.986815i	$0.448253\pi$
$632$	2029.82	0.127756
$633$	−3.84945	−0.000241709 0
$634$	−2000.38	−0.125308
$635$	0	0
$636$	−826.547	−0.0515326
$637$	0	0
$638$	9867.96	0.612345
$639$	3949.89	0.244531
$640$	0	0
$641$	−3352.85	−0.206598	−0.103299	−	0.994650i	$-0.532940\pi$
$641$	−3352.85	−0.206598	−0.103299	+	0.994650i	$0.532940\pi$
$642$	6818.36	0.419157
$643$	15715.1	0.963831	0.481915	−	0.876218i	$-0.339941\pi$
$643$	15715.1	0.963831	0.481915	+	0.876218i	$0.339941\pi$
$644$	0	0
$645$	0	0
$646$	−3748.09	−0.228276
$647$	−5752.20	−0.349524	−0.174762	−	0.984611i	$-0.555916\pi$
$647$	−5752.20	−0.349524	−0.174762	+	0.984611i	$0.555916\pi$
$648$	−4053.49	−0.245735
$649$	−1183.57	−0.0715861
$650$	0	0
$651$	0	0
$652$	11742.5	0.705327
$653$	6433.63	0.385555	0.192778	−	0.981242i	$-0.438250\pi$
$653$	6433.63	0.385555	0.192778	+	0.981242i	$0.438250\pi$
$654$	−6341.36	−0.379154
$655$	0	0
$656$	−1931.37	−0.114950
$657$	−928.927	−0.0551612
$658$	0	0
$659$	15894.1	0.939523	0.469761	−	0.882793i	$-0.344340\pi$
$659$	15894.1	0.939523	0.469761	+	0.882793i	$0.344340\pi$
$660$	0	0
$661$	−22369.7	−1.31631	−0.658153	−	0.752884i	$-0.728662\pi$
$661$	−22369.7	−1.31631	−0.658153	+	0.752884i	$0.728662\pi$
$662$	658.712	0.0386731
$663$	982.404	0.0575466
$664$	−8797.34	−0.514161
$665$	0	0
$666$	−193.589	−0.0112634
$667$	−8870.50	−0.514943
$668$	−14599.2	−0.845599
$669$	−825.865	−0.0477276
$670$	0	0
$671$	−19912.2	−1.14560
$672$	0	0
$673$	−14499.2	−0.830468	−0.415234	−	0.909715i	$-0.636300\pi$
$673$	−14499.2	−0.830468	−0.415234	+	0.909715i	$0.636300\pi$
$674$	14526.3	0.830169
$675$	0	0
$676$	−7522.40	−0.427993
$677$	2304.34	0.130817	0.0654083	−	0.997859i	$-0.479165\pi$
$677$	2304.34	0.130817	0.0654083	+	0.997859i	$0.479165\pi$
$678$	−3613.91	−0.204707
$679$	0	0
$680$	0	0
$681$	−7994.24	−0.449838
$682$	−13846.4	−0.777429
$683$	−3939.03	−0.220678	−0.110339	−	0.993894i	$-0.535194\pi$
$683$	−3939.03	−0.220678	−0.110339	+	0.993894i	$0.535194\pi$
$684$	5529.51	0.309102
$685$	0	0
$686$	0	0
$687$	−6289.56	−0.349289
$688$	1464.76	0.0811681
$689$	2179.33	0.120502
$690$	0	0
$691$	−4148.43	−0.228384	−0.114192	−	0.993459i	$-0.536428\pi$
$691$	−4148.43	−0.228384	−0.114192	+	0.993459i	$0.536428\pi$
$692$	−1045.87	−0.0574539
$693$	0	0
$694$	5414.50	0.296155
$695$	0	0
$696$	−2213.06	−0.120526
$697$	3952.88	0.214815
$698$	17421.9	0.944738
$699$	785.890	0.0425252
$700$	0	0
$701$	19691.9	1.06099	0.530494	−	0.847689i	$-0.322007\pi$
$701$	19691.9	1.06099	0.530494	+	0.847689i	$0.322007\pi$
$702$	−3069.33	−0.165020
$703$	229.322	0.0123030
$704$	1925.20	0.103067
$705$	0	0
$706$	25946.2	1.38314
$707$	0	0
$708$	265.437	0.0140900
$709$	−5887.76	−0.311875	−0.155937	−	0.987767i	$-0.549840\pi$
$709$	−5887.76	−0.311875	−0.155937	+	0.987767i	$0.549840\pi$
$710$	0	0
$711$	6128.91	0.323280
$712$	−8572.90	−0.451240
$713$	12446.8	0.653767
$714$	0	0
$715$	0	0
$716$	−11130.2	−0.580945
$717$	7223.34	0.376235
$718$	8628.41	0.448481
$719$	−17334.7	−0.899133	−0.449566	−	0.893247i	$-0.648422\pi$
$719$	−17334.7	−0.899133	−0.449566	+	0.893247i	$0.648422\pi$
$720$	0	0
$721$	0	0
$722$	7167.85	0.369474
$723$	−10440.7	−0.537058
$724$	−8287.80	−0.425433
$725$	0	0
$726$	−1437.34	−0.0734775
$727$	−13946.4	−0.711478	−0.355739	−	0.934585i	$-0.615771\pi$
$727$	−13946.4	−0.711478	−0.355739	+	0.934585i	$0.615771\pi$
$728$	0	0
$729$	−8310.43	−0.422214
$730$	0	0
$731$	−2997.90	−0.151684
$732$	4465.65	0.225485
$733$	−329.168	−0.0165868	−0.00829339	−	0.999966i	$-0.502640\pi$
$733$	−329.168	−0.0165868	−0.00829339	+	0.999966i	$0.502640\pi$
$734$	2052.41	0.103210
$735$	0	0
$736$	−1730.60	−0.0866723
$737$	7647.56	0.382227
$738$	−5831.64	−0.290875
$739$	6772.01	0.337094	0.168547	−	0.985694i	$-0.446093\pi$
$739$	6772.01	0.337094	0.168547	+	0.985694i	$0.446093\pi$
$740$	0	0
$741$	1716.85	0.0851147
$742$	0	0
$743$	−19959.4	−0.985517	−0.492759	−	0.870166i	$-0.664011\pi$
$743$	−19959.4	−0.985517	−0.492759	+	0.870166i	$0.664011\pi$
$744$	3105.30	0.153019
$745$	0	0
$746$	20498.2	1.00602
$747$	−26563.0	−1.30106
$748$	−3940.27	−0.192608
$749$	0	0
$750$	0	0
$751$	−11138.3	−0.541199	−0.270600	−	0.962692i	$-0.587222\pi$
$751$	−11138.3	−0.541199	−0.270600	+	0.962692i	$0.587222\pi$
$752$	1007.84	0.0488725
$753$	9801.65	0.474359
$754$	5835.10	0.281833
$755$	0	0
$756$	0	0
$757$	−16701.1	−0.801865	−0.400932	−	0.916108i	$-0.631314\pi$
$757$	−16701.1	−0.801865	−0.400932	+	0.916108i	$0.631314\pi$
$758$	20583.0	0.986293
$759$	−2743.77	−0.131215
$760$	0	0
$761$	21451.2	1.02182	0.510909	−	0.859635i	$-0.329309\pi$
$761$	21451.2	1.02182	0.510909	+	0.859635i	$0.329309\pi$
$762$	1254.02	0.0596173
$763$	0	0
$764$	3497.93	0.165642
$765$	0	0
$766$	3949.27	0.186283
$767$	−699.869	−0.0329476
$768$	−431.761	−0.0202862
$769$	32694.4	1.53315	0.766573	−	0.642157i	$-0.221960\pi$
$769$	32694.4	1.53315	0.766573	+	0.642157i	$0.221960\pi$
$770$	0	0
$771$	2678.61	0.125120
$772$	12643.9	0.589461
$773$	14763.3	0.686932	0.343466	−	0.939165i	$-0.388399\pi$
$773$	14763.3	0.686932	0.343466	+	0.939165i	$0.388399\pi$
$774$	4422.76	0.205391
$775$	0	0
$776$	4002.55	0.185159
$777$	0	0
$778$	28295.7	1.30392
$779$	6908.05	0.317724
$780$	0	0
$781$	−4918.87	−0.225366
$782$	3541.98	0.161971
$783$	−14151.3	−0.645883
$784$	0	0
$785$	0	0
$786$	2750.38	0.124813
$787$	17230.4	0.780428	0.390214	−	0.920724i	$-0.372401\pi$
$787$	17230.4	0.780428	0.390214	+	0.920724i	$0.372401\pi$
$788$	−9832.84	−0.444518
$789$	−9239.46	−0.416899
$790$	0	0
$791$	0	0
$792$	5813.03	0.260804
$793$	−11774.4	−0.527266
$794$	1884.05	0.0842097
$795$	0	0
$796$	10810.9	0.481383
$797$	27123.2	1.20546	0.602731	−	0.797944i	$-0.294079\pi$
$797$	27123.2	1.20546	0.602731	+	0.797944i	$0.294079\pi$
$798$	0	0
$799$	−2062.72	−0.0913315
$800$	0	0
$801$	−25885.3	−1.14184
$802$	2271.19	0.0999982
$803$	1156.81	0.0508381
$804$	−1715.10	−0.0752325
$805$	0	0
$806$	−8187.63	−0.357813
$807$	−2878.36	−0.125555
$808$	−1417.75	−0.0617280
$809$	23741.9	1.03180	0.515898	−	0.856650i	$-0.327459\pi$
$809$	23741.9	1.03180	0.515898	+	0.856650i	$0.327459\pi$
$810$	0	0
$811$	−809.829	−0.0350641	−0.0175320	−	0.999846i	$-0.505581\pi$
$811$	−809.829	−0.0350641	−0.0175320	+	0.999846i	$0.505581\pi$
$812$	0	0
$813$	−11440.3	−0.493517
$814$	241.080	0.0103807
$815$	0	0
$816$	883.675	0.0379103
$817$	−5239.12	−0.224350
$818$	−10435.1	−0.446033
$819$	0	0
$820$	0	0
$821$	20867.2	0.887054	0.443527	−	0.896261i	$-0.353727\pi$
$821$	20867.2	0.887054	0.443527	+	0.896261i	$0.353727\pi$
$822$	−740.732	−0.0314307
$823$	−42364.5	−1.79433	−0.897166	−	0.441694i	$-0.854378\pi$
$823$	−42364.5	−1.79433	−0.897166	+	0.441694i	$0.854378\pi$
$824$	5023.72	0.212390
$825$	0	0
$826$	0	0
$827$	−30496.7	−1.28231	−0.641157	−	0.767410i	$-0.721545\pi$
$827$	−30496.7	−1.28231	−0.641157	+	0.767410i	$0.721545\pi$
$828$	−5225.44	−0.219320
$829$	−15121.9	−0.633539	−0.316770	−	0.948503i	$-0.602598\pi$
$829$	−15121.9	−0.633539	−0.316770	+	0.948503i	$0.602598\pi$
$830$	0	0
$831$	11788.6	0.492110
$832$	1138.41	0.0474365
$833$	0	0
$834$	−2712.00	−0.112600
$835$	0	0
$836$	−6886.00	−0.284877
$837$	19856.7	0.820008
$838$	23334.4	0.961900
$839$	39504.1	1.62555	0.812774	−	0.582580i	$-0.197957\pi$
$839$	39504.1	1.62555	0.812774	+	0.582580i	$0.197957\pi$
$840$	0	0
$841$	2514.03	0.103080
$842$	6912.22	0.282911
$843$	−891.837	−0.0364371
$844$	9.12968	0.000372342 0
$845$	0	0
$846$	3043.11	0.123669
$847$	0	0
$848$	1960.31	0.0793835
$849$	−6298.74	−0.254620
$850$	0	0
$851$	−216.712	−0.00872947
$852$	1103.14	0.0443581
$853$	−22526.4	−0.904207	−0.452103	−	0.891966i	$-0.649326\pi$
$853$	−22526.4	−0.904207	−0.452103	+	0.891966i	$0.649326\pi$
$854$	0	0
$855$	0	0
$856$	−16171.0	−0.645692
$857$	2792.33	0.111300	0.0556500	−	0.998450i	$-0.482277\pi$
$857$	2792.33	0.111300	0.0556500	+	0.998450i	$0.482277\pi$
$858$	1804.88	0.0718153
$859$	−6943.38	−0.275792	−0.137896	−	0.990447i	$-0.544034\pi$
$859$	−6943.38	−0.275792	−0.137896	+	0.990447i	$0.544034\pi$
$860$	0	0
$861$	0	0
$862$	15442.7	0.610185
$863$	−18834.7	−0.742920	−0.371460	−	0.928449i	$-0.621143\pi$
$863$	−18834.7	−0.742920	−0.371460	+	0.928449i	$0.621143\pi$
$864$	−2760.87	−0.108711
$865$	0	0
$866$	−19795.6	−0.776769
$867$	6477.50	0.253734
$868$	0	0
$869$	−7632.46	−0.297944
$870$	0	0
$871$	4522.14	0.175921
$872$	15039.7	0.584069
$873$	12085.4	0.468534
$874$	6189.96	0.239563
$875$	0	0
$876$	−259.435	−0.0100063
$877$	28673.0	1.10401	0.552006	−	0.833840i	$-0.313863\pi$
$877$	28673.0	1.10401	0.552006	+	0.833840i	$0.313863\pi$
$878$	8655.94	0.332715
$879$	−14729.5	−0.565203
$880$	0	0
$881$	−16736.0	−0.640012	−0.320006	−	0.947415i	$-0.603685\pi$
$881$	−16736.0	−0.640012	−0.320006	+	0.947415i	$0.603685\pi$
$882$	0	0
$883$	−29530.0	−1.12544	−0.562720	−	0.826648i	$-0.690245\pi$
$883$	−29530.0	−1.12544	−0.562720	+	0.826648i	$0.690245\pi$
$884$	−2329.95	−0.0886479
$885$	0	0
$886$	10800.5	0.409536
$887$	−40767.8	−1.54323	−0.771617	−	0.636087i	$-0.780552\pi$
$887$	−40767.8	−1.54323	−0.771617	+	0.636087i	$0.780552\pi$
$888$	−54.0665	−0.00204319
$889$	0	0
$890$	0	0
$891$	15241.8	0.573086
$892$	1958.69	0.0735222
$893$	−3604.81	−0.135084
$894$	−5975.57	−0.223549
$895$	0	0
$896$	0	0
$897$	−1622.44	−0.0603921
$898$	32423.5	1.20488
$899$	−37749.5	−1.40046
$900$	0	0
$901$	−4012.11	−0.148350
$902$	7262.26	0.268079
$903$	0	0
$904$	8571.04	0.315341
$905$	0	0
$906$	−758.011	−0.0277961
$907$	−14413.6	−0.527670	−0.263835	−	0.964568i	$-0.584987\pi$
$907$	−14413.6	−0.527670	−0.263835	+	0.964568i	$0.584987\pi$
$908$	18959.8	0.692955
$909$	−4280.80	−0.156199
$910$	0	0
$911$	24078.2	0.875681	0.437841	−	0.899053i	$-0.355743\pi$
$911$	24078.2	0.875681	0.437841	+	0.899053i	$0.355743\pi$
$912$	1544.31	0.0560714
$913$	33079.4	1.19909
$914$	25604.3	0.926603
$915$	0	0
$916$	14916.9	0.538064
$917$	0	0
$918$	5650.60	0.203156
$919$	−31043.7	−1.11429	−0.557147	−	0.830414i	$-0.688104\pi$
$919$	−31043.7	−1.11429	−0.557147	+	0.830414i	$0.688104\pi$
$920$	0	0
$921$	16544.4	0.591919
$922$	−5722.86	−0.204417
$923$	−2908.62	−0.103725
$924$	0	0
$925$	0	0
$926$	15102.1	0.535945
$927$	15168.8	0.537442
$928$	5248.69	0.185664
$929$	−31516.4	−1.11305	−0.556524	−	0.830832i	$-0.687865\pi$
$929$	−31516.4	−1.11305	−0.556524	+	0.830832i	$0.687865\pi$
$930$	0	0
$931$	0	0
$932$	−1863.88	−0.0655080
$933$	7980.02	0.280015
$934$	553.483	0.0193903
$935$	0	0
$936$	3437.35	0.120036
$937$	36900.9	1.28655	0.643277	−	0.765634i	$-0.277574\pi$
$937$	36900.9	1.28655	0.643277	+	0.765634i	$0.277574\pi$
$938$	0	0
$939$	12476.1	0.433591
$940$	0	0
$941$	8631.54	0.299023	0.149511	−	0.988760i	$-0.452230\pi$
$941$	8631.54	0.299023	0.149511	+	0.988760i	$0.452230\pi$
$942$	10833.6	0.374712
$943$	−6528.18	−0.225437
$944$	−629.533	−0.0217050
$945$	0	0
$946$	−5507.76	−0.189295
$947$	−38553.4	−1.32293	−0.661466	−	0.749976i	$-0.730065\pi$
$947$	−38553.4	−1.32293	−0.661466	+	0.749976i	$0.730065\pi$
$948$	1711.71	0.0586433
$949$	684.043	0.0233983
$950$	0	0
$951$	−1686.89	−0.0575195
$952$	0	0
$953$	40691.8	1.38314	0.691572	−	0.722308i	$-0.256919\pi$
$953$	40691.8	1.38314	0.691572	+	0.722308i	$0.256919\pi$
$954$	5919.03	0.200876
$955$	0	0
$956$	−17131.5	−0.579573
$957$	8321.49	0.281082
$958$	−38345.7	−1.29321
$959$	0	0
$960$	0	0
$961$	23177.9	0.778016
$962$	142.555	0.00477771
$963$	−48827.3	−1.63389
$964$	24762.0	0.827313
$965$	0	0
$966$	0	0
$967$	51901.4	1.72599	0.862997	−	0.505210i	$-0.168585\pi$
$967$	51901.4	1.72599	0.862997	+	0.505210i	$0.168585\pi$
$968$	3408.91	0.113189
$969$	−3160.70	−0.104785
$970$	0	0
$971$	−26982.9	−0.891784	−0.445892	−	0.895087i	$-0.647114\pi$
$971$	−26982.9	−0.891784	−0.445892	+	0.895087i	$0.647114\pi$
$972$	−12736.2	−0.420281
$973$	0	0
$974$	28200.0	0.927707
$975$	0	0
$976$	−10591.1	−0.347350
$977$	44232.6	1.44844	0.724220	−	0.689569i	$-0.242200\pi$
$977$	44232.6	1.44844	0.724220	+	0.689569i	$0.242200\pi$
$978$	9902.28	0.323763
$979$	32235.5	1.05235
$980$	0	0
$981$	45411.4	1.47796
$982$	1674.22	0.0544057
$983$	845.302	0.0274272	0.0137136	−	0.999906i	$-0.495635\pi$
$983$	845.302	0.0274272	0.0137136	+	0.999906i	$0.495635\pi$
$984$	−1628.69	−0.0527650
$985$	0	0
$986$	−10742.4	−0.346964
$987$	0	0
$988$	−4071.82	−0.131115
$989$	4951.03	0.159184
$990$	0	0
$991$	46858.1	1.50201	0.751007	−	0.660294i	$-0.229568\pi$
$991$	46858.1	1.50201	0.751007	+	0.660294i	$0.229568\pi$
$992$	−7364.79	−0.235718
$993$	555.481	0.0177519
$994$	0	0
$995$	0	0
$996$	−7418.65	−0.236013
$997$	−16353.3	−0.519474	−0.259737	−	0.965679i	$-0.583636\pi$
$997$	−16353.3	−0.519474	−0.259737	+	0.965679i	$0.583636\pi$
$998$	17113.5	0.542805
$999$	−345.725	−0.0109492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cl.1.2	✓	4
5.4	even	2		2450.4.a.cr.1.3	yes	4
7.6	odd	2	inner	2450.4.a.cl.1.3	yes	4
35.34	odd	2		2450.4.a.cr.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.4.a.cl.1.2	✓	4	1.1	even	1	trivial
2450.4.a.cl.1.3	yes	4	7.6	odd	2	inner
2450.4.a.cr.1.2	yes	4	35.34	odd	2
2450.4.a.cr.1.3	yes	4	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -1.68657 q^{3} +4.00000 q^{4} +3.37313 q^{6} -8.00000 q^{8} -24.1555 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -1.68657 q^{3} +4.00000 q^{4} +3.37313 q^{6} -8.00000 q^{8} -24.1555 q^{9} +30.0813 q^{11} -6.74626 q^{12} +17.7876 q^{13} +16.0000 q^{16} -32.7468 q^{17} +48.3110 q^{18} -57.2282 q^{19} -60.1626 q^{22} +54.0813 q^{23} +13.4925 q^{24} -35.5752 q^{26} +86.2771 q^{27} -164.021 q^{29} +230.150 q^{31} -32.0000 q^{32} -50.7341 q^{33} +65.4936 q^{34} -96.6220 q^{36} -4.00714 q^{37} +114.456 q^{38} -30.0000 q^{39} -120.711 q^{41} +91.5478 q^{43} +120.325 q^{44} -108.163 q^{46} +62.9900 q^{47} -26.9851 q^{48} +55.2297 q^{51} +71.1505 q^{52} +122.519 q^{53} -172.554 q^{54} +96.5192 q^{57} +328.043 q^{58} -39.3458 q^{59} -661.945 q^{61} -460.299 q^{62} +64.0000 q^{64} +101.468 q^{66} +254.230 q^{67} -130.987 q^{68} -91.2117 q^{69} -163.519 q^{71} +193.244 q^{72} +38.4561 q^{73} +8.01428 q^{74} -228.913 q^{76} +60.0000 q^{78} -253.727 q^{79} +506.686 q^{81} +241.421 q^{82} +1099.67 q^{83} -183.096 q^{86} +276.633 q^{87} -240.651 q^{88} +1071.61 q^{89} +216.325 q^{92} -388.163 q^{93} -125.980 q^{94} +53.9701 q^{96} -500.319 q^{97} -726.629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} - 32 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 - 32 * q^9	\( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} - 32 q^{9} - 52 q^{11} + 64 q^{16} + 64 q^{18} + 104 q^{22} + 44 q^{23} + 184 q^{29} - 128 q^{32} - 128 q^{36} + 264 q^{37} - 120 q^{39} - 208 q^{44} - 88 q^{46} + 264 q^{51} + 1244 q^{53} + 1140 q^{57} - 368 q^{58} + 256 q^{64} + 1060 q^{67} - 1408 q^{71} + 256 q^{72} - 528 q^{74} + 240 q^{78} - 2652 q^{79} - 752 q^{81} + 416 q^{88} + 176 q^{92} - 1208 q^{93} - 2368 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 - 32 * q^9 - 52 * q^11 + 64 * q^16 + 64 * q^18 + 104 * q^22 + 44 * q^23 + 184 * q^29 - 128 * q^32 - 128 * q^36 + 264 * q^37 - 120 * q^39 - 208 * q^44 - 88 * q^46 + 264 * q^51 + 1244 * q^53 + 1140 * q^57 - 368 * q^58 + 256 * q^64 + 1060 * q^67 - 1408 * q^71 + 256 * q^72 - 528 * q^74 + 240 * q^78 - 2652 * q^79 - 752 * q^81 + 416 * q^88 + 176 * q^92 - 1208 * q^93 - 2368 * q^99

Introduction

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